Atlas › Modular forms › Weight 2 › Level 803 › 803.2.a

Newspace 803.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(803))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
803.2.a.a	2	$\mathbb{Q}(\sqrt{21})$	$2q - 2q^{2} + q^{3} - 2q^{4} - 4q^{5} - q^{6} - q^{7} + O(q^{8})$	+ +	$-1$
803.2.a.b	5	degree 5	$5q + 2q^{3} - 10q^{4} + q^{5} - q^{7} + O(q^{8})$	+ −	$+1$
803.2.a.c	9	degree 9	$9q + 3q^{2} - 5q^{3} + 5q^{4} - q^{5} - 3q^{6} - q^{7} + O(q^{8})$	+ +	$-1$
803.2.a.d	10	degree 10	$10q - 5q^{2} - 3q^{3} + 5q^{4} - 2q^{5} - 8q^{7} + O(q^{8})$	− −	$-1$
803.2.a.e	16	degree 16	$16q + 4q^{3} + 38q^{4} + 4q^{5} + 3q^{7} + O(q^{8})$	+ −	$+1$
803.2.a.f	19	degree 19	$19q + 7q^{2} + 3q^{3} + 23q^{4} + 2q^{5} + 16q^{7} + O(q^{8})$	− +	$+1$

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 $T$
803.2.a.a	$x^{2} + x - 5$
803.2.a.b	$x^{5} - 2x^{4} - 9x^{3} + 8x^{2} + 25x + 7$
803.2.a.c	$x^{9} + 2x^{8} - 19x^{7} - 27x^{6} + 105x^{5} + 46x^{4} - 194x^{3} + 107x^{2} - 20x + 1$
803.2.a.d	$x^{10} + 8x^{9} + 2x^{8} - 86x^{7} - 71x^{6} + 261x^{5} + 278x^{4} - 223x^{3} - 316x^{2} - 40x + 25$
803.2.a.e	$x^{16} - 4x^{15} - 64x^{14} + 224x^{13} + 1555x^{12} - 4313x^{11} - 18751x^{10} + 33455x^{9} + 118952x^{8} - 79806x^{7} - 346660x^{6} - 78915x^{5} + 207802x^{4} + 54997x^{3} - 41132x^{2} - 6600x + 2839$
803.2.a.f	$x^{19} - 10x^{18} - 20x^{17} + 490x^{16} - 770x^{15} - 7558x^{14} + 23934x^{13} + 35422x^{12} - 207958x^{11} + 38679x^{10} + 714386x^{9} - 566200x^{8} - 1047599x^{7} + 1125222x^{6} + 634467x^{5} - 706188x^{4} - 154569x^{3} + 72773x^{2} + 2069x + 3$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(803))$	61
Old forms	12
Cusp forms $S_{2}(\Gamma_0(803))$	73
Eisenstein series $E_{2}(\Gamma_0(803))$	3
Modular forms $M_{2}(\Gamma_0(803))$	76

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(803)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(73))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.24 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, 11, 73, 803]: [0, 2, 6, 76] (certified at ell = 2147482801)
T = [(2, 1), (3, 1)]; h of degree 61 = 1 x 61 (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(803, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 803:2 1000

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