Atlas › Modular forms › Weight 2 › Level 770 › 770.2.a

Newspace 770.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(770))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
770.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} - q^{5} + 2q^{6} + q^{7} + O(q^{8})$	+ + − +	$+1$	770b1 rank 0
770.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + q^{5} + 2q^{6} - q^{7} + O(q^{8})$	+ − + −	$-1$	770d1 rank 1
770.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - q^{7} + O(q^{8})$	+ − + +	$+1$	770c1 rank 0
770.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} + q^{7} + O(q^{8})$	+ − − +	$-1$	770e1 rank 1
770.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - q^{5} - 2q^{6} + q^{7} + O(q^{8})$	+ + − +	$+1$	770a1 rank 0
770.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} + q^{7} + O(q^{8})$	− + − +	$-1$	770f1 rank 1
770.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} + q^{5} - 2q^{6} + q^{7} + O(q^{8})$	− − − +	$+1$	770g1 rank 0
770.2.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	+ − − −	$+1$
770.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	− + + +	$+1$
770.2.a.j	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− + − −	$+1$
770.2.a.k	2	$\mathbb{Q}(\sqrt{33})$	$2q + 2q^{2} + 4q^{3} + 2q^{4} + 2q^{5} + 4q^{6} + 2q^{7} + O(q^{8})$	− − − +	$+1$
770.2.a.l	3	degree 3	$3q - 3q^{2} + 3q^{4} - 3q^{5} - 3q^{7} + O(q^{8})$	+ + + −	$+1$
770.2.a.m	3	degree 3	$3q + 3q^{2} + 2q^{3} + 3q^{4} + 3q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	− − + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{13} + 3T_{19} + 4T_{23} + 5T_{29} + 6T_{17} + 7T_{31} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53}$ 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 $T$
770.2.a.a	$x - 172$
770.2.a.b	$x + 256$
770.2.a.c	$x - 304$
770.2.a.d	$x + 112$
770.2.a.e	$x + 4$
770.2.a.f	$x + 260$
770.2.a.g	$x - 244$
770.2.a.h	$x^{2} - 20x - 5708$
770.2.a.i	$x^{2} + 80x - 57568$
770.2.a.j	$x^{2} + 124x + 3652$
770.2.a.k	$x^{2} + 264x + 15312$
770.2.a.l	$x^{3} - 120x^{2} + 1508x + 48160$
770.2.a.m	$x^{3} - 248x^{2} - 23132x + 6425504$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(770))$	21
Old forms	116
Cusp forms $S_{2}(\Gamma_0(770))$	137
Eisenstein series $E_{2}(\Gamma_0(770))$	15
Modular forms $M_{2}(\Gamma_0(770))$	152

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.872 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, 5, 7, 10, 11, 14, 22, 35, 55, 70, 77, 110, 154, 385, 770]: [0, 1, 1, 1, 3, 2, 4, 5, 6, 8, 16, 10, 22, 28, 52, 152] (certified at ell = 2147482921)
T = [(3, 1), (13, 2), (19, 3), (23, 4), (29, 5), (17, 6), (31, 7), (37, 8), (41, 9), (43, 10), (47, 11), (53, 12)]; h of degree 21 = 1 x 21 (phi(ord eps) x dim S_k^new); 7 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(770, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 770:2 1000

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