Atlas › Modular forms › Weight 2 › Level 807 › 807.2.a

Newspace 807.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(807))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
807.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + 3q^{5} + 2q^{7} + O(q^{8})$	− +	$+1$	807a1 rank 0
807.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} - 4q^{4} + 2q^{7} + O(q^{8})$	+ +	$-1$
807.2.a.c	6	degree 6	$6q - q^{2} - 6q^{3} + 9q^{4} - 2q^{5} + q^{6} - 9q^{7} + O(q^{8})$	+ +	$-1$
807.2.a.d	8	degree 8	$8q - 5q^{2} + 8q^{3} + 5q^{4} - 6q^{5} - 5q^{6} - 5q^{7} + O(q^{8})$	− −	$-1$
807.2.a.e	13	degree 13	$13q + 4q^{2} + 13q^{3} + 14q^{4} + 5q^{5} + 4q^{6} - q^{7} + O(q^{8})$	− +	$+1$
807.2.a.f	15	degree 15	$15q + q^{2} - 15q^{3} + 19q^{4} + 2q^{5} - q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
807.2.a.a	$x - 3$
807.2.a.b	$x^{2} - 5$
807.2.a.c	$x^{6} + 3x^{5} - 7x^{4} - 22x^{3} - 4x^{2} + 15x + 5$
807.2.a.d	$x^{8} + 11x^{7} + 34x^{6} - 19x^{5} - 247x^{4} - 243x^{3} + 313x^{2} + 401x - 73$
807.2.a.e	$x^{13} - 9x^{12} - 17x^{11} + 328x^{10} - 344x^{9} - 3285x^{8} + 6547x^{7} + 6587x^{6} - 17507x^{5} - 4807x^{4} + 15592x^{3} + 1452x^{2} - 3825x + 381$
807.2.a.f	$x^{15} - 3x^{14} - 88x^{13} + 243x^{12} + 2881x^{11} - 7361x^{10} - 45687x^{9} + 108430x^{8} + 374108x^{7} - 828740x^{6} - 1544057x^{5} + 3204501x^{4} + 2862737x^{3} - 5519751x^{2} - 1811633x + 3138827$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(807))$	45
Old forms	44
Cusp forms $S_{2}(\Gamma_0(807))$	89
Eisenstein series $E_{2}(\Gamma_0(807))$	3
Modular forms $M_{2}(\Gamma_0(807))$	92

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(807)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(269))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.345 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, 269, 807]: [0, 1, 23, 92] (certified at ell = 2147482661)
T = [(2, 1), (5, 1)]; h of degree 45 = 1 x 45 (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(807, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 807:2 1000

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