Atlas › Modular forms › Weight 2 › Level 301 › 301.2.a

Newspace 301.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(301))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
301.2.a.a	4	degree 4	$4q - 4q^{2} - 3q^{3} + 4q^{4} - 4q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	− −	$-1$
301.2.a.b	5	degree 5	$5q - 3q^{3} + 2q^{4} - 6q^{5} - 10q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
301.2.a.c	5	degree 5	$5q + q^{2} + 5q^{3} + 3q^{4} + 4q^{5} - 5q^{7} + O(q^{8})$	+ −	$+1$
301.2.a.d	7	degree 7	$7q + 4q^{2} + q^{3} + 8q^{4} + 7q^{7} + O(q^{8})$	− +	$+1$

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 $T$
301.2.a.a	$x^{4} + 7x^{3} + 10x^{2} - 2x - 7$
301.2.a.b	$x^{5} + 3x^{4} - 2x^{3} - 8x^{2} + x + 4$
301.2.a.c	$x^{5} - 6x^{4} + x^{3} + 36x^{2} - 53x + 17$
301.2.a.d	$x^{7} - 5x^{6} - 13x^{5} + 83x^{4} + 11x^{3} - 334x^{2} + 201x + 4$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(301))$	21
Old forms	6
Cusp forms $S_{2}(\Gamma_0(301))$	27
Eisenstein series $E_{2}(\Gamma_0(301))$	3
Modular forms $M_{2}(\Gamma_0(301))$	30

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(301)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.181 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, 7, 43, 301]: [0, 1, 4, 30] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 21 = 1 x 21 (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(301, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 301:2 1000

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