Atlas › Modular forms › Weight 2 › Level 347 › 347.2.a

Newspace 347.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(347))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
347.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} - 2q^{6} - 2q^{7} + O(q^{8})$	+	$-1$	347a1 rank 1
347.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - q^{3} - 2q^{4} - 2q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+	$-1$
347.2.a.c	7	degree 7	$7q - 2q^{2} - 7q^{3} + 4q^{4} - 8q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	+	$-1$
347.2.a.d	19	degree 19	$19q + 7q^{3} + 22q^{4} + 10q^{5} - 6q^{6} + 9q^{7} + O(q^{8})$	−	$+1$

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 $T$
347.2.a.a	$x + 1$
347.2.a.b	$x^{2} - x - 1$
347.2.a.c	$x^{7} + 9x^{6} + 19x^{5} - 13x^{4} - 35x^{3} + 8x^{2} + 11x - 1$
347.2.a.d	$x^{19} - 7x^{18} - 39x^{17} + 358x^{16} + 349x^{15} - 6773x^{14} + 2782x^{13} + 62168x^{12} - 63594x^{11} - 300197x^{10} + 394450x^{9} + 780463x^{8} - 1077353x^{7} - 1146282x^{6} + 1343363x^{5} + 1032701x^{4} - 688395x^{3} - 496010x^{2} + 67126x + 56047$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(347))$	29
Old forms	0
Cusp forms $S_{2}(\Gamma_0(347))$	29
Eisenstein series $E_{2}(\Gamma_0(347))$	1
Modular forms $M_{2}(\Gamma_0(347))$	30

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.26 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, 347]: [0, 30] (certified at ell = 2147481179)
T = [(2, 1), (3, 1)]; h of degree 29 = 1 x 29 (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(347, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 347:2 1000

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