Atlas › Modular forms › Weight 2 › Level 409 › 409.2.a

Newspace 409.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(409))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
409.2.a.a	13	degree 13	$13q - 6q^{2} - 3q^{3} + 6q^{4} - 10q^{5} - 8q^{6} - 9q^{7} + O(q^{8})$	+	$-1$
409.2.a.b	20	degree 20	$20q + 5q^{2} + q^{3} + 23q^{4} + 8q^{5} - 4q^{6} + 5q^{7} + O(q^{8})$	−	$+1$

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 $T$
409.2.a.a	$x^{13} + 9x^{12} + 13x^{11} - 80x^{10} - 197x^{9} + 225x^{8} + 707x^{7} - 299x^{6} - 944x^{5} + 363x^{4} + 437x^{3} - 257x^{2} + 32x - 1$
409.2.a.b	$x^{20} - 6x^{19} - 53x^{18} + 371x^{17} + 884x^{16} - 8541x^{15} - 3292x^{14} + 93301x^{13} - 45060x^{12} - 515110x^{11} + 444762x^{10} + 1447059x^{9} - 1304306x^{8} - 2181912x^{7} + 1332857x^{6} + 1858280x^{5} - 294748x^{4} - 603955x^{3} - 24502x^{2} + 61633x + 6929$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(409))$	33
Old forms	0
Cusp forms $S_{2}(\Gamma_0(409))$	33
Eisenstein series $E_{2}(\Gamma_0(409))$	1
Modular forms $M_{2}(\Gamma_0(409))$	34

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.186 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, 409]: [0, 34] (certified at ell = 2147477809)
T = [(2, 1), (3, 1)]; h of degree 33 = 1 x 33 (phi(ord eps) x dim S_k^new); 4 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(409, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 409:2 1000

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