Atlas › Modular forms › Weight 2 › Level 941 › 941.2.a

Newspace 941.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(941))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
941.2.a.a	28	degree 28	$28q - 3q^{2} - 10q^{3} + 9q^{4} - 8q^{5} - 9q^{6} - 21q^{7} + O(q^{8})$	+	$-1$
941.2.a.b	50	degree 50	$50q + 3q^{2} + 12q^{3} + 71q^{4} + 6q^{5} + 9q^{6} + 23q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ generates each orbit's coefficient field; on the newspace its characteristic polynomial factors over $\mathbb{Q}$ as one irreducible factor per orbit:

Orbit	Characteristic polynomial of $T$
941.2.a.a	$x^{28} + 13x^{27} + 13x^{26} - 479x^{25} - 1599x^{24} + 6974x^{23} + 35583x^{22} - 48635x^{21} - 402471x^{20} + 123423x^{19} + 2777536x^{18} + 515305x^{17} - 12562191x^{16} - 5317000x^{15} + 38327743x^{14} + 19772983x^{13} - 79102266x^{12} - 40500406x^{11} + 108122766x^{10} + 47416378x^{9} - 92936657x^{8} - 29277715x^{7} + 45240156x^{6} + 7227330x^{5} - 9969926x^{4} + 62850x^{3} + 567256x^{2} - 80295x + 3175$
941.2.a.b	$x^{50} - 15x^{49} - 91x^{48} + 2405x^{47} - 648x^{46} - 167841x^{45} + 455383x^{44} + 6675619x^{43} - 29492201x^{42} - 164852752x^{41} + 1035683780x^{40} + 2511487606x^{39} - 23756846245x^{38} - 19211512124x^{37} + 381736056987x^{36} - 71900988134x^{35} - 4444366015934x^{34} + 3959705404792x^{33} + 38102079756843x^{32} - 55324718354443x^{31} - 241492632348839x^{30} + 475701405838497x^{29} + 1121821352750146x^{28} - 2854006694126434x^{27} - 3710861821113324x^{26} + 12439812687957743x^{25} + 8068256527828257x^{24} - 39913627504088182x^{23} - 8255170517171357x^{22} + 94188012164303786x^{21} - 10900528053585801x^{20} - 161555158212848586x^{19} + 59824480554610783x^{18} + 196553145601890652x^{17} - 116977479777776983x^{16} - 161971525190447815x^{15} + 135589686323037493x^{14} + 81676807630669627x^{13} - 99107559001094796x^{12} - 17433094906603102x^{11} + 44360186740658804x^{10} - 4318703780150171x^{9} - 10802755024686874x^{8} + 3333412867817183x^{7} + 943277630009320x^{6} - 579656150528744x^{5} + 55395231575072x^{4} + 10180239804112x^{3} - 1225610491194x^{2} - 83141163651x + 1764870211$

Orbit

Characteristic polynomial of

T

941.2.a.a

x^{28} + 13x^{27} + 13x^{26} - 479x^{25} - 1599x^{24} + 6974x^{23} + 35583x^{22} - 48635x^{21} - 402471x^{20} + 123423x^{19} + 2777536x^{18} + 515305x^{17} - 12562191x^{16} - 5317000x^{15} + 38327743x^{14} + 19772983x^{13} - 79102266x^{12} - 40500406x^{11} + 108122766x^{10} + 47416378x^{9} - 92936657x^{8} - 29277715x^{7} + 45240156x^{6} + 7227330x^{5} - 9969926x^{4} + 62850x^{3} + 567256x^{2} - 80295x + 3175

941.2.a.b

x^{50} - 15x^{49} - 91x^{48} + 2405x^{47} - 648x^{46} - 167841x^{45} + 455383x^{44} + 6675619x^{43} - 29492201x^{42} - 164852752x^{41} + 1035683780x^{40} + 2511487606x^{39} - 23756846245x^{38} - 19211512124x^{37} + 381736056987x^{36} - 71900988134x^{35} - 4444366015934x^{34} + 3959705404792x^{33} + 38102079756843x^{32} - 55324718354443x^{31} - 241492632348839x^{30} + 475701405838497x^{29} + 1121821352750146x^{28} - 2854006694126434x^{27} - 3710861821113324x^{26} + 12439812687957743x^{25} + 8068256527828257x^{24} - 39913627504088182x^{23} - 8255170517171357x^{22} + 94188012164303786x^{21} - 10900528053585801x^{20} - 161555158212848586x^{19} + 59824480554610783x^{18} + 196553145601890652x^{17} - 116977479777776983x^{16} - 161971525190447815x^{15} + 135589686323037493x^{14} + 81676807630669627x^{13} - 99107559001094796x^{12} - 17433094906603102x^{11} + 44360186740658804x^{10} - 4318703780150171x^{9} - 10802755024686874x^{8} + 3333412867817183x^{7} + 943277630009320x^{6} - 579656150528744x^{5} + 55395231575072x^{4} + 10180239804112x^{3} - 1225610491194x^{2} - 83141163651x + 1764870211

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(941))$	78
Old forms	0
Cusp forms $S_{2}(\Gamma_0(941))$	78
Eisenstein series $E_{2}(\Gamma_0(941))$	1
Modular forms $M_{2}(\Gamma_0(941))$	79

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.3 s (one thread) and proven: the dimensions are certified, and the characteristic polynomial of $T$ over $\mathbb{Z}$ is reconstructed by CRT past a rigorous coefficient bound.

The engine's checks

dim M^+ at levels [1, 941]: [0, 79] (certified at ell = 2147473241)
T = [(2, 1), (3, 1)]; h of degree 78 = 1 x 78 (phi(ord eps) x dim S_k^new); 8 primes, 0 rejected

Open in the notebook · JSON · stored data

Recompute it yourself

In Sage mode at sagebrush.space (or npx sagebrush --sage):

newform_orbits(941, 2)

With the Rust engine (source), in engine/:

cargo run --release -p sagebrush-web --example atlas -- 941:2 1000

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