Atlas › Modular forms › Weight 2 › Level 815 › 815.2.a

Newspace 815.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(815))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
815.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - 2q^{4} + q^{5} + 2q^{7} + O(q^{8})$	− −	$-1$	815a1 rank 1
815.2.a.b	5	degree 5	$5q - 2q^{2} + 2q^{4} + 5q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	− −	$-1$
815.2.a.c	6	degree 6	$6q - 5q^{2} + 8q^{3} + 7q^{4} + 6q^{5} + q^{6} + O(q^{8})$	− +	$+1$
815.2.a.d	7	degree 7	$7q - 2q^{2} - 2q^{3} + 2q^{4} - 7q^{5} + 5q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
815.2.a.e	16	degree 16	$16q + 7q^{2} - 6q^{3} + 21q^{4} + 16q^{5} + 4q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$
815.2.a.f	20	degree 20	$20q + 5q^{2} + 6q^{3} + 27q^{4} - 20q^{5} + q^{6} + 11q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
815.2.a.a	$x + 2$
815.2.a.b	$x^{5} + 2x^{4} - 6x^{3} - 3x^{2} + 6x + 1$
815.2.a.c	$x^{6} - 3x^{5} - 21x^{4} + 57x^{3} + 49x^{2} - 40x + 4$
815.2.a.d	$x^{7} + 4x^{6} - 14x^{5} - 45x^{4} + 68x^{3} + 103x^{2} - 60x - 20$
815.2.a.e	$x^{16} - x^{15} - 67x^{14} + 72x^{13} + 1665x^{12} - 1693x^{11} - 20106x^{10} + 17880x^{9} + 126883x^{8} - 91398x^{7} - 413878x^{6} + 211350x^{5} + 639873x^{4} - 157591x^{3} - 394535x^{2} - 6300x + 35095$
815.2.a.f	$x^{20} - 11x^{19} - 20x^{18} + 557x^{17} - 575x^{16} - 10671x^{15} + 19592x^{14} + 101916x^{13} - 201949x^{12} - 559503x^{11} + 942588x^{10} + 1846111x^{9} - 2072493x^{8} - 3363255x^{7} + 2162424x^{6} + 3066091x^{5} - 1066497x^{4} - 1263704x^{3} + 215190x^{2} + 179029x - 5066$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(815))$	55
Old forms	26
Cusp forms $S_{2}(\Gamma_0(815))$	81
Eisenstein series $E_{2}(\Gamma_0(815))$	3
Modular forms $M_{2}(\Gamma_0(815))$	84

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(815)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(163))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.339 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, 5, 163, 815]: [0, 1, 14, 84] (certified at ell = 2147482693)
T = [(2, 1), (3, 1)]; h of degree 55 = 1 x 55 (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(815, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 815:2 1000

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