Atlas › Modular forms › Weight 2 › Level 307 › 307.2.a

Newspace 307.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(307))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
307.2.a.a	1	$\mathbb{Q}$	$q - 2q^{4} + 4q^{5} + O(q^{8})$	−	$+1$	307a1 rank 0
307.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} - q^{4} + 2q^{6} + 3q^{7} + O(q^{8})$	−	$+1$	307b1 rank 0
307.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + 2q^{5} + 3q^{7} + O(q^{8})$	−	$+1$	307c1 rank 0
307.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{3} + 2q^{4} + 4q^{6} - 3q^{7} + O(q^{8})$	−	$+1$	307d1 rank 0
307.2.a.e	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} - 3q^{3} + 3q^{4} + 6q^{5} - 5q^{6} + 5q^{7} + O(q^{8})$	−	$+1$
307.2.a.f	9	degree 9	$9q + 3q^{2} + q^{3} + 13q^{4} + 5q^{5} - 6q^{6} - 5q^{7} + O(q^{8})$	−	$+1$
307.2.a.g	10	degree 10	$10q - 7q^{2} - 4q^{3} + 9q^{4} - 15q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+	$-1$

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 $T$
307.2.a.a	$x$
307.2.a.b	$x - 7$
307.2.a.c	$x - 2$
307.2.a.d	$x - 8$
307.2.a.e	$x^{2} + 10x + 12$
307.2.a.f	$x^{9} - 6x^{8} - 173x^{7} + 840x^{6} + 10807x^{5} - 34968x^{4} - 299362x^{3} + 304682x^{2} + 3415922x + 4051801$
307.2.a.g	$x^{10} + 19x^{9} + 34x^{8} - 1102x^{7} - 5887x^{6} + 2605x^{5} + 55115x^{4} + 40175x^{3} - 66063x^{2} - 26982x + 17348$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(307))$	25
Old forms	0
Cusp forms $S_{2}(\Gamma_0(307))$	25
Eisenstein series $E_{2}(\Gamma_0(307))$	1
Modular forms $M_{2}(\Gamma_0(307))$	26

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, 307]: [0, 26] (certified at ell = 2147478013)
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(307, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 307:2 1000

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