Atlas › Modular forms › Weight 2 › Level 734 › 734.2.a

Newspace 734.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(734))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
734.2.a.a	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{6} + O(q^{8})$	− +	$+1$	734a1 rank 0
734.2.a.b	2	$\mathbb{Q}(\sqrt{21})$	$2q - 2q^{2} + q^{3} + 2q^{4} - 3q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$
734.2.a.c	3	degree 3	$3q - 3q^{2} + q^{3} + 3q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
734.2.a.d	3	degree 3	$3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
734.2.a.e	11	degree 11	$11q - 11q^{2} - 2q^{3} + 11q^{4} + 6q^{5} + 2q^{6} + 9q^{7} + O(q^{8})$	+ −	$+1$
734.2.a.f	11	degree 11	$11q + 11q^{2} + 3q^{3} + 11q^{4} + 5q^{5} + 3q^{6} + 10q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
734.2.a.a	$x - 2$
734.2.a.b	$x^{2} + 8x - 5$
734.2.a.c	$x^{3} + 2x^{2} - 8x - 8$
734.2.a.d	$x^{3} + 12x^{2} + 36x + 8$
734.2.a.e	$x^{11} - 16x^{10} - 349x^{9} + 6576x^{8} + 29921x^{7} - 914246x^{6} + 1056819x^{5} + 48153014x^{4} - 200986140x^{3} - 635068848x^{2} + 5236156992x - 8412464992$
734.2.a.f	$x^{11} - 18x^{10} - 194x^{9} + 5176x^{8} - 5006x^{7} - 418766x^{6} + 2248346x^{5} + 5624844x^{4} - 71787431x^{3} + 165205720x^{2} - 12399028x - 217741864$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(734))$	31
Old forms	60
Cusp forms $S_{2}(\Gamma_0(734))$	91
Eisenstein series $E_{2}(\Gamma_0(734))$	3
Modular forms $M_{2}(\Gamma_0(734))$	94

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(734)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(367))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.246 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, 2, 367, 734]: [0, 1, 31, 94] (certified at ell = 2147479381)
T = [(3, 1), (5, 3)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 5 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(734, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 734:2 1000

← 733.2.a · 735.2.a →