Atlas › Modular forms › Weight 2 › Level 979 › 979.2.a

Newspace 979.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(979))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
979.2.a.a	1	$\mathbb{Q}$	$q + 2q^{3} - 2q^{4} + 3q^{5} + O(q^{8})$	+ −	$+1$	979a1 rank 0
979.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} - q^{4} - 2q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	979b1 rank 1
979.2.a.c	11	degree 11	$11q - 2q^{2} - 5q^{3} + 4q^{4} - 10q^{5} - 10q^{6} + q^{7} + O(q^{8})$	− −	$-1$
979.2.a.d	17	degree 17	$17q - 4q^{2} - q^{3} + 14q^{4} - 14q^{5} - 6q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
979.2.a.e	18	degree 18	$18q + 4q^{2} - q^{3} + 18q^{4} + 11q^{5} + 10q^{6} + 3q^{7} + O(q^{8})$	+ −	$+1$
979.2.a.f	25	degree 25	$25q + 4q^{2} + 3q^{3} + 34q^{4} + 18q^{5} - 4q^{6} - q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{5} + 7T_{7}$ 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 $T$
979.2.a.a	$x - 21$
979.2.a.b	$x + 17$
979.2.a.c	$x^{11} + 60x^{10} - 533x^{9} - 63839x^{8} + 30105x^{7} + 19577771x^{6} + 15225415x^{5} - 1957633640x^{4} - 2096951120x^{3} + 53448702771x^{2} + 463369961x - 4814151707$
979.2.a.d	$x^{17} + 112x^{16} + 3444x^{15} - 35759x^{14} - 3288975x^{13} - 19908032x^{12} + 1086422346x^{11} + 11163033040x^{10} - 183873168441x^{9} - 2051191025195x^{8} + 18304874495285x^{7} + 172348894245979x^{6} - 1062946265549308x^{5} - 6234288449629378x^{4} + 28584877669322783x^{3} + 61270800485537220x^{2} - 88524445861712275x - 95808647635531819$
979.2.a.e	$x^{18} - 77x^{17} - 3502x^{16} + 384027x^{15} + 2103612x^{14} - 738718801x^{13} + 5626739810x^{12} + 697234937024x^{11} - 10007674775261x^{10} - 334130490862850x^{9} + 6448000183200078x^{8} + 72467654071175696x^{7} - 1881207461876597119x^{6} - 3527278427567334568x^{5} + 226847101713296352301x^{4} - 591894352049510353527x^{3} - 8257209133352728803133x^{2} + 49447806279951105927828x - 76063429295826416917139$
979.2.a.f	$x^{25} - 96x^{24} - 3650x^{23} + 612853x^{22} - 2682063x^{21} - 1518174652x^{20} + 31369043824x^{19} + 1821595660710x^{18} - 61028065922216x^{17} - 1001301034966767x^{16} + 56179648921397855x^{15} + 64872139126608092x^{14} - 27043021839487685071x^{13} + 183346173173820720068x^{12} + 6484030403964784645333x^{11} - 77509090625587413468778x^{10} - 666298234383381832141318x^{9} + 11151151992525459471878902x^{8} + 34677183605420208839310899x^{7} - 754671862983684640529038385x^{6} - 1236160138629896194450729554x^{5} + 24722288192343297019292096576x^{4} + 34452802742977666287814160875x^{3} - 336454860327398427183540087260x^{2} - 460433485897048970178233814371x + 936958585176353020813869299473$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(979))$	73
Old forms	16
Cusp forms $S_{2}(\Gamma_0(979))$	89
Eisenstein series $E_{2}(\Gamma_0(979))$	3
Modular forms $M_{2}(\Gamma_0(979))$	92

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(979)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(89))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.366 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, 11, 89, 979]: [0, 2, 8, 92] (certified at ell = 2147482921)
T = [(2, 1), (3, 3), (5, 5), (7, 7)]; h of degree 73 = 1 x 73 (phi(ord eps) x dim S_k^new); 16 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(979, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 979:2 1000

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