Atlas › Modular forms › Weight 2 › Level 503 › 503.2.a

Newspace 503.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(503))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
503.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - 4q^{5} - q^{6} - 3q^{7} + O(q^{8})$	−	$+1$	503c1 rank 0
503.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} - 2q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+	$-1$	503a1 rank 1
503.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + 3q^{3} - q^{4} - 2q^{5} + 3q^{6} + 3q^{7} + O(q^{8})$	−	$+1$	503b1 rank 0
503.2.a.d	3	degree 3	$3q + q^{3} + 4q^{4} + q^{6} + q^{7} + O(q^{8})$	−	$+1$
503.2.a.e	10	degree 10	$10q - 4q^{2} - 8q^{3} + 4q^{4} - q^{5} - 2q^{6} - 5q^{7} + O(q^{8})$	+	$-1$
503.2.a.f	26	degree 26	$26q + 4q^{2} + 4q^{3} + 36q^{4} + 9q^{5} - 4q^{6} + 11q^{7} + O(q^{8})$	−	$+1$

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 $T$
503.2.a.a	$x$
503.2.a.b	$x - 2$
503.2.a.c	$x - 4$
503.2.a.d	$x^{3} - x^{2} - 10x - 9$
503.2.a.e	$x^{10} + 12x^{9} + 48x^{8} + 59x^{7} - 44x^{6} - 91x^{5} + 28x^{4} + 41x^{3} - 16x^{2} - 2x + 1$
503.2.a.f	$x^{26} - 8x^{25} - 68x^{24} + 656x^{23} + 1624x^{22} - 22208x^{21} - 12260x^{20} + 410522x^{19} - 147661x^{18} - 4604083x^{17} + 4062371x^{16} + 32724437x^{15} - 40208191x^{14} - 148387452x^{13} + 222867363x^{12} + 416913685x^{11} - 743947956x^{10} - 665018660x^{9} + 1479426047x^{8} + 460903078x^{7} - 1634532440x^{6} + 54910880x^{5} + 871942090x^{4} - 163245078x^{3} - 195350329x^{2} + 36711663x + 14483700$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(503))$	42
Old forms	0
Cusp forms $S_{2}(\Gamma_0(503))$	42
Eisenstein series $E_{2}(\Gamma_0(503))$	1
Modular forms $M_{2}(\Gamma_0(503))$	43

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.198 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, 503]: [0, 43] (certified at ell = 2147476183)
T = [(2, 1), (3, 1)]; h of degree 42 = 1 x 42 (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(503, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 503:2 1000

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