Atlas › Modular forms › Weight 2 › Level 341 › 341.2.a

Newspace 341.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(341))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
341.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q + q^{2} - 2q^{3} - q^{4} - 3q^{5} - q^{6} + q^{7} + O(q^{8})$	− −	$-1$
341.2.a.b	4	degree 4	$4q - 2q^{2} - 2q^{3} - 2q^{4} - q^{5} + q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
341.2.a.c	8	degree 8	$8q + q^{2} + 4q^{3} + 13q^{4} - 5q^{5} + 9q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$
341.2.a.d	11	degree 11	$11q + q^{2} + 4q^{3} + 19q^{4} + 3q^{5} - 5q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
341.2.a.a	$x^{2} + 5x + 5$
341.2.a.b	$x^{4} + 8x^{3} - 37x^{2} - 122x + 331$
341.2.a.c	$x^{8} - 13x^{7} - 83x^{6} + 1382x^{5} + 1227x^{4} - 42529x^{3} + 29686x^{2} + 289193x - 413445$
341.2.a.d	$x^{11} - 13x^{10} - 173x^{9} + 2819x^{8} + 4206x^{7} - 177570x^{6} + 358358x^{5} + 3031316x^{4} - 9365668x^{3} - 8491980x^{2} + 26721433x - 7025263$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(341))$	25
Old forms	6
Cusp forms $S_{2}(\Gamma_0(341))$	31
Eisenstein series $E_{2}(\Gamma_0(341))$	3
Modular forms $M_{2}(\Gamma_0(341))$	34

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(341)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.185 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, 31, 341]: [0, 2, 3, 34] (certified at ell = 2147482951)
T = [(2, 1), (3, 3)]; h of degree 25 = 1 x 25 (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(341, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 341:2 1000

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