Atlas › Modular forms › Weight 2 › Level 349 › 349.2.a

Newspace 349.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(349))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
349.2.a.a	11	degree 11	$11q - 5q^{2} - 6q^{3} + 5q^{4} - 9q^{5} - 5q^{6} - 3q^{7} + O(q^{8})$	+	$-1$
349.2.a.b	17	degree 17	$17q + 5q^{2} + 6q^{3} + 19q^{4} + 5q^{5} - 3q^{6} + q^{7} + O(q^{8})$	−	$+1$

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 $T$
349.2.a.a	$x^{11} + 11x^{10} + 34x^{9} - 14x^{8} - 218x^{7} - 265x^{6} + 52x^{5} + 240x^{4} + 118x^{3} + 8x^{2} - 6x - 1$
349.2.a.b	$x^{17} - 11x^{16} - x^{15} + 399x^{14} - 1051x^{13} - 4198x^{12} + 18687x^{11} + 11308x^{10} - 127618x^{9} + 50877x^{8} + 413606x^{7} - 340358x^{6} - 671456x^{5} + 673829x^{4} + 518852x^{3} - 511909x^{2} - 148020x + 94951$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(349))$	28
Old forms	0
Cusp forms $S_{2}(\Gamma_0(349))$	28
Eisenstein series $E_{2}(\Gamma_0(349))$	1
Modular forms $M_{2}(\Gamma_0(349))$	29

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.176 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, 349]: [0, 29] (certified at ell = 2147481901)
T = [(2, 1), (3, 1)]; h of degree 28 = 1 x 28 (phi(ord eps) x dim S_k^new); 3 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(349, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 349:2 1000

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