Atlas › Modular forms › Weight 2 › Level 690 › 690.2.a

Newspace 690.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(690))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
690.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ + + +	$-1$	690a1 rank 1
690.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 4q^{7} + O(q^{8})$	+ + + −	$+1$	690b1 rank 0
690.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ + − +	$+1$	690c1 rank 0
690.2.a.d	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 4q^{7} + O(q^{8})$	+ + − +	$+1$	690d1 rank 0
690.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + O(q^{8})$	+ − + −	$-1$	690e1 rank 1
690.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + O(q^{8})$	+ − − −	$+1$	690f1 rank 0
690.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + + −	$-1$	690h1 rank 1
690.2.a.h	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + O(q^{8})$	− + + +	$+1$	690g1 rank 0
690.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + O(q^{8})$	− − + −	$+1$	690i1 rank 0
690.2.a.j	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + O(q^{8})$	− − − +	$+1$	690j1 rank 0
690.2.a.k	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + O(q^{8})$	− − − +	$+1$	690k1 rank 0
690.2.a.l	2	$\mathbb{Q}(\sqrt{17})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− + − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{29} + 7T_{31} + 8T_{37} + 9T_{41} + 10T_{43} + 11T_{47} + 12T_{53} + 13T_{59} + 14T_{61} + 15T_{67} + 16T_{71}$ 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 $T$
690.2.a.a	$x + 382$
690.2.a.b	$x - 300$
690.2.a.c	$x - 786$
690.2.a.d	$x - 408$
690.2.a.e	$x - 128$
690.2.a.f	$x - 480$
690.2.a.g	$x + 30$
690.2.a.h	$x + 64$
690.2.a.i	$x + 292$
690.2.a.j	$x$
690.2.a.k	$x + 408$
690.2.a.l	$x^{2} + 310x + 3200$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(690))$	13
Old forms	124
Cusp forms $S_{2}(\Gamma_0(690))$	137
Eisenstein series $E_{2}(\Gamma_0(690))$	15
Modular forms $M_{2}(\Gamma_0(690))$	152

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.177 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, 5, 6, 10, 15, 23, 30, 46, 69, 115, 138, 230, 345, 690]: [0, 1, 1, 1, 3, 3, 4, 3, 10, 8, 10, 14, 28, 40, 52, 152] (certified at ell = 2147483053)
T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (29, 6), (31, 7), (37, 8), (41, 9), (43, 10), (47, 11), (53, 12), (59, 13), (61, 14), (67, 15), (71, 16)]; h of degree 13 = 1 x 13 (phi(ord eps) x dim S_k^new); 5 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(690, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 690:2 1000

← 689.2.a · 691.2.a →