Atlas › Modular forms › Weight 2 › Level 794 › 794.2.a

Newspace 794.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(794))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
794.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} - 4q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$	794a1 rank 2
794.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} - q^{7} + O(q^{8})$	− −	$-1$	794d1 rank 1
794.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + O(q^{8})$	− −	$-1$	794c1 rank 1
794.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$	794b1 rank 1
794.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
794.2.a.f	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 2q^{7} + O(q^{8})$	+ +	$-1$
794.2.a.g	12	degree 12	$12q - 12q^{2} + 7q^{3} + 12q^{4} + 3q^{5} - 7q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$
794.2.a.h	14	degree 14	$14q + 14q^{2} + q^{3} + 14q^{4} + 7q^{5} + q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{17} + 7T_{19} + 8T_{23}$ 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 $T$
794.2.a.a	$x + 177$
794.2.a.b	$x + 59$
794.2.a.c	$x + 55$
794.2.a.d	$x + 29$
794.2.a.e	$x^{2} + 50x - 625$
794.2.a.f	$x^{2} + 46x + 527$
794.2.a.g	$x^{12} - 372x^{11} + 45863x^{10} - 990356x^{9} - 229042651x^{8} + 16883873327x^{7} - 102640865666x^{6} - 22279977144575x^{5} + 468470586676603x^{4} + 5874589987407947x^{3} - 123657782597534379x^{2} - 73401935288021843x + 624528591824845429$
794.2.a.h	$x^{14} - 304x^{13} + 18246x^{12} + 2889966x^{11} - 381868768x^{10} + 3291591549x^{9} + 1395826007465x^{8} - 57029832757814x^{7} - 698095759532095x^{6} + 63320410142608730x^{5} - 179988530847046130x^{4} - 19964170861983579080x^{3} + 41245320493348547910x^{2} + 1087301063449428030905x + 1023143530380871056011$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(794))$	34
Old forms	64
Cusp forms $S_{2}(\Gamma_0(794))$	98
Eisenstein series $E_{2}(\Gamma_0(794))$	3
Modular forms $M_{2}(\Gamma_0(794))$	101

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(794)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(397))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.56 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, 2, 397, 794]: [0, 1, 33, 101] (certified at ell = 2147483053)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (17, 6), (19, 7), (23, 8)]; h of degree 34 = 1 x 34 (phi(ord eps) x dim S_k^new); 10 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(794, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 794:2 1000

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