Atlas › Modular forms › Weight 8 › Level 34 › 34.8.a

Newspace 34.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(34))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
34.8.a.a	1	$\mathbb{Q}$	$q - 8q^{2} - 68q^{3} + 64q^{4} + 270q^{5} + 544q^{6} + 680q^{7} + O(q^{8})$	+ −	$-1$
34.8.a.b	1	$\mathbb{Q}$	$q - 8q^{2} + 84q^{3} + 64q^{4} - 450q^{5} - 672q^{6} - 1408q^{7} + O(q^{8})$	+ −	$-1$
34.8.a.c	1	$\mathbb{Q}$	$q + 8q^{2} - 42q^{3} + 64q^{4} + 72q^{5} - 336q^{6} - 580q^{7} + O(q^{8})$	− +	$-1$
34.8.a.d	2	$\mathbb{Q}(\sqrt{577})$	$2q - 16q^{2} + 16q^{3} + 128q^{4} + 296q^{5} - 128q^{6} - 1136q^{7} + O(q^{8})$	+ +	$+1$
34.8.a.e	3	degree 3	$3q + 24q^{2} + 12q^{3} + 192q^{4} + 346q^{5} + 96q^{6} + 1200q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
34.8.a.a	$x - 202$
34.8.a.b	$x + 366$
34.8.a.c	$x - 30$
34.8.a.d	$x^{2} - 312x - 88756$
34.8.a.e	$x^{3} - 358x^{2} - 93748x + 9645976$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(34))$	8
Old forms	22
Cusp forms $S_{8}(\Gamma_0(34))$	30
Eisenstein series $E_{8}(\Gamma_0(34))$	4
Modular forms $M_{8}(\Gamma_0(34))$	34

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(34)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.277 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, 17, 34]: [1, 3, 12, 34] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 8 = 1 x 8 (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(34, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 34:8 1000

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