Atlas › Modular forms › Weight 2 › Level 777 › 777.2.a

Newspace 777.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(777))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
777.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + 2q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	777b1 rank 0
777.2.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} + q^{5} - 2q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	777f1 rank 1
777.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} - 2q^{5} + q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	777d1 rank 1
777.2.a.d	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} + 3q^{5} + q^{7} + O(q^{8})$	+ − +	$+1$	777c1 rank 0
777.2.a.e	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{5} + q^{7} + O(q^{8})$	− − +	$-1$	777g1 rank 1
777.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - 2q^{5} - q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	777a1 rank 0
777.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} - 2q^{5} + q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	777e1 rank 1
777.2.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{4} + 2q^{7} + O(q^{8})$	+ − +	$+1$
777.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} - 2q^{7} + O(q^{8})$	+ + −	$+1$
777.2.a.j	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 2q^{7} + O(q^{8})$	− + +	$+1$
777.2.a.k	5	degree 5	$5q - 2q^{2} - 5q^{3} + 4q^{4} - 3q^{5} + 2q^{6} - 5q^{7} + O(q^{8})$	+ + +	$-1$
777.2.a.l	5	degree 5	$5q - 2q^{2} - 5q^{3} + 8q^{4} - 5q^{5} + 2q^{6} + 5q^{7} + O(q^{8})$	+ − −	$-1$
777.2.a.m	5	degree 5	$5q + 5q^{3} + 12q^{4} + 5q^{5} - 5q^{7} + O(q^{8})$	− + +	$+1$
777.2.a.n	7	degree 7	$7q + 7q^{3} + 10q^{4} + 3q^{5} + 7q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{5} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{41} + 11T_{43} + 12T_{47}$ 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 $T$
777.2.a.a	$x - 156$
777.2.a.b	$x + 68$
777.2.a.c	$x + 247$
777.2.a.d	$x - 278$
777.2.a.e	$x + 270$
777.2.a.f	$x - 159$
777.2.a.g	$x + 89$
777.2.a.h	$x^{2} - 340x + 28033$
777.2.a.i	$x^{2} - 182x - 25001$
777.2.a.j	$x^{2} - 198x - 10601$
777.2.a.k	$x^{5} + 470x^{4} + 48483x^{3} - 2252734x^{2} - 290758320x - 277176166$
777.2.a.l	$x^{5} + 458x^{4} + 48683x^{3} - 5508672x^{2} - 1290523594x - 61314404642$
777.2.a.m	$x^{5} - 28x^{4} - 111171x^{3} - 944118x^{2} + 1140434120x - 41647165522$
777.2.a.n	$x^{7} - 480x^{6} + 23298x^{5} + 18070988x^{4} - 2663725613x^{3} + 37727831238x^{2} + 7473120686674x - 104522809354254$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(777))$	35
Old forms	62
Cusp forms $S_{2}(\Gamma_0(777))$	97
Eisenstein series $E_{2}(\Gamma_0(777))$	7
Modular forms $M_{2}(\Gamma_0(777))$	104

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(777)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(111))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(259))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.007 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, 3, 7, 21, 37, 111, 259, 777]: [0, 1, 1, 4, 3, 14, 26, 104] (certified at ell = 2147483629)
T = [(2, 1), (5, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (41, 10), (43, 11), (47, 12)]; h of degree 35 = 1 x 35 (phi(ord eps) x dim S_k^new); 12 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(777, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 777:2 1000

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