Atlas › Modular forms › Weight 2 › Level 701 › 701.2.a

Newspace 701.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(701))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
701.2.a.a	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 4q^{6} - q^{7} + O(q^{8})$	−	$+1$	701a1 rank 0
701.2.a.b	21	degree 21	$21q - 3q^{2} - 14q^{3} + 7q^{4} - 7q^{5} - 7q^{6} - 9q^{7} + O(q^{8})$	+	$-1$
701.2.a.c	36	degree 36	$36q - q^{2} + 14q^{3} + 49q^{4} + q^{5} + 3q^{6} + 8q^{7} + O(q^{8})$	−	$+1$

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 $T$
701.2.a.a	$x - 4$
701.2.a.b	$x^{21} + 17x^{20} + 91x^{19} - 1593x^{17} - 3869x^{16} + 7626x^{15} + 37382x^{14} - 883x^{13} - 153625x^{12} - 93442x^{11} + 328433x^{10} + 290780x^{9} - 385183x^{8} - 376011x^{7} + 250361x^{6} + 226782x^{5} - 87079x^{4} - 58567x^{3} + 14627x^{2} + 4799x - 798$
701.2.a.c	$x^{36} - 13x^{35} - 56x^{34} + 1365x^{33} - 800x^{32} - 59933x^{31} + 143346x^{30} + 1450278x^{29} - 5178655x^{28} - 21281758x^{27} + 100781212x^{26} + 193839778x^{25} - 1239509292x^{24} - 1040133571x^{23} + 10258758276x^{22} + 2344652103x^{21} - 58949677654x^{20} + 7665549249x^{19} + 239177929956x^{18} - 79819505234x^{17} - 690955231086x^{16} + 297485829245x^{15} + 1424099151387x^{14} - 635102799522x^{13} - 2085403211329x^{12} + 823884763097x^{11} + 2142931310486x^{10} - 622123656482x^{9} - 1503226788483x^{8} + 221456669403x^{7} + 678178898782x^{6} + 9196195342x^{5} - 172068088339x^{4} - 29754574558x^{3} + 16691610612x^{2} + 5629059399x + 419410201$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(701))$	58
Old forms	0
Cusp forms $S_{2}(\Gamma_0(701))$	58
Eisenstein series $E_{2}(\Gamma_0(701))$	1
Modular forms $M_{2}(\Gamma_0(701))$	59

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.238 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, 701]: [0, 59] (certified at ell = 2147479601)
T = [(2, 1), (3, 1)]; h of degree 58 = 1 x 58 (phi(ord eps) x dim S_k^new); 6 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(701, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 701:2 1000

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