Atlas › Modular forms › Weight 2 › Level 774 › 774.2.a

Newspace 774.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(774))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
774.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 3q^{5} - q^{7} + O(q^{8})$	+ +	774c1 rank 0
774.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	+ −	774e1 rank 1
774.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} + q^{7} + O(q^{8})$	+ −	774d1 rank 1
774.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	+ +	774b1 rank 0
774.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 3q^{5} - q^{7} + O(q^{8})$	+ −	774a1 rank 0
774.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 3q^{5} - q^{7} + O(q^{8})$	− −	774f1 rank 1
774.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - q^{5} - 5q^{7} + O(q^{8})$	− +	774g1 rank 1
774.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	− −	774h1 rank 0
774.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 3q^{5} - 3q^{7} + O(q^{8})$	− −	774i1 rank 0
774.2.a.j	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} + 2q^{4} - 3q^{5} + 3q^{7} + O(q^{8})$	+ +
774.2.a.k	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{4} + 3q^{5} + O(q^{8})$	+ +
774.2.a.l	2	$\mathbb{Q}(\sqrt{21})$	$2q + 2q^{2} + 2q^{4} - 3q^{5} + 4q^{7} + O(q^{8})$	− −
774.2.a.m	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{2} + 2q^{4} + 3q^{5} + 3q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{47}$ 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 $T$
774.2.a.a	$x - 346$
774.2.a.b	$x + 300$
774.2.a.c	$x - 24$
774.2.a.d	$x + 114$
774.2.a.e	$x - 74$
774.2.a.f	$x + 130$
774.2.a.g	$x + 186$
774.2.a.h	$x - 8$
774.2.a.i	$x + 16$
774.2.a.j	$x^{2} + 384x + 3952$
774.2.a.k	$x^{2} - 207x + 10711$
774.2.a.l	$x^{2} - 205x + 3319$
774.2.a.m	$x^{2} - 80x - 5200$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(774))$	17
Old forms	108
Cusp forms $S_{2}(\Gamma_0(774))$	125
Eisenstein series $E_{2}(\Gamma_0(774))$	15
Modular forms $M_{2}(\Gamma_0(774))$	140

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(774)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(86))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(129))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(258))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(387))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.274 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, 3, 6, 9, 18, 43, 86, 129, 258, 387, 774]: [0, 1, 1, 3, 2, 5, 4, 13, 16, 48, 46, 136] (certified at ell = 2147483647)
T = [(5, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (47, 12)]; h of degree 17 = 1 x 17 (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(774, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 774:2 1000

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