Atlas › Modular forms › Weight 2 › Level 791 › 791.2.a

Newspace 791.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(791))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
791.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + 2q^{5} + q^{7} + O(q^{8})$	− −	$-1$	791c1 rank 1
791.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} - q^{4} - 4q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+ −	$+1$	791b1 rank 0
791.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} - q^{4} - 2q^{6} - q^{7} + O(q^{8})$	+ −	$+1$	791a1 rank 0
791.2.a.d	4	degree 4	$4q - q^{3} - 4q^{5} - 4q^{6} + 4q^{7} + O(q^{8})$	− −	$-1$
791.2.a.e	8	degree 8	$8q - q^{3} + 4q^{4} - 4q^{5} - 4q^{6} - 8q^{7} + O(q^{8})$	+ +	$-1$
791.2.a.f	18	degree 18	$18q - 2q^{2} + 9q^{3} + 26q^{4} + 12q^{5} + 14q^{6} - 18q^{7} + O(q^{8})$	+ −	$+1$
791.2.a.g	24	degree 24	$24q + 4q^{2} + q^{3} + 36q^{4} + 4q^{5} - 2q^{6} + 24q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
791.2.a.a	$x + 25$
791.2.a.b	$x - 63$
791.2.a.c	$x - 67$
791.2.a.d	$x^{4} + 115x^{3} + 3782x^{2} + 30252x - 186347$
791.2.a.e	$x^{8} + 227x^{7} + 18544x^{6} + 611082x^{5} + 2764412x^{4} - 272227029x^{3} - 4400129944x^{2} + 15838657232x + 484919542787$
791.2.a.f	$x^{18} - 341x^{17} + 27625x^{16} + 3031323x^{15} - 579332950x^{14} + 11963583945x^{13} + 2924921156924x^{12} - 185830560248457x^{11} - 2687688378995646x^{10} + 553581362812294739x^{9} - 11249604020098197800x^{8} - 450348321872986721019x^{7} + 21356050344542396829970x^{6} - 166087803168872775588425x^{5} - 5549655927551330651323304x^{4} + 129363953098817091274064513x^{3} - 992255898942148508372782559x^{2} + 2358503616042503709826174138x + 2087568437849369020875416155$
791.2.a.g	$x^{24} - 175x^{23} - 25674x^{22} + 5445670x^{21} + 251808924x^{20} - 70636834607x^{19} - 1159664606634x^{18} + 498379136553540x^{17} + 2398544049849903x^{16} - 2091589580304252854x^{15} - 1756794161961520844x^{14} + 5363204960593568651100x^{13} + 4092117637778718092740x^{12} - 8310942286232253588933722x^{11} - 22599875118512434578711128x^{10} + 7401897397780104326951316796x^{9} + 39929216656840860355361133815x^{8} - 3380190549953127616529673081235x^{7} - 24960265592288210055345848948234x^{6} + 637884323502465015031047092108942x^{5} + 3847266408950287706754192177560320x^{4} - 59860245023943809659867846716173103x^{3} - 142201845673299615906033274206728542x^{2} + 2448958630194483238362485853591205856x - 4623763302763408325486066852876059847$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(791))$	57
Old forms	18
Cusp forms $S_{2}(\Gamma_0(791))$	75
Eisenstein series $E_{2}(\Gamma_0(791))$	3
Modular forms $M_{2}(\Gamma_0(791))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(791)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(113))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.367 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, 7, 113, 791]: [0, 1, 10, 78] (certified at ell = 2147480161)
T = [(2, 1), (3, 3), (13, 5), (17, 7), (19, 9)]; h of degree 57 = 1 x 57 (phi(ord eps) x dim S_k^new); 15 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(791, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 791:2 1000

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