Atlas › Modular forms › Weight 10 › Level 34 › 34.10.a

Newspace 34.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(34))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
34.10.a.a	2	$\mathbb{Q}(\sqrt{43})$	$2q + 32q^{2} + 64q^{3} + 512q^{4} - 2788q^{5} + 1024q^{6} - 728q^{7} + O(q^{8})$	− −	$-1$
34.10.a.b	3	degree 3	$3q - 48q^{2} + 84q^{3} + 768q^{4} - 1304q^{5} - 1344q^{6} + 690q^{7} + O(q^{8})$	+ +	$-1$
34.10.a.c	3	degree 3	$3q - 48q^{2} + 84q^{3} + 768q^{4} + 1314q^{5} - 1344q^{6} + 6912q^{7} + O(q^{8})$	+ −	$+1$
34.10.a.d	4	degree 4	$4q + 64q^{2} + 226q^{3} + 1024q^{4} - 1656q^{5} + 3616q^{6} + 2654q^{7} + O(q^{8})$	− +	$+1$

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 $T$
34.10.a.a	$x^{2} + 2724x + 1193876$
34.10.a.b	$x^{3} + 1220x^{2} - 159790x - 84290348$
34.10.a.c	$x^{3} - 1398x^{2} - 2315220x + 2180455704$
34.10.a.d	$x^{4} + 1430x^{3} - 6730662x^{2} - 8501880936x + 2537598862440$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(34))$	12
Old forms	26
Cusp forms $S_{10}(\Gamma_0(34))$	38
Eisenstein series $E_{10}(\Gamma_0(34))$	4
Modular forms $M_{10}(\Gamma_0(34))$	42

Decomposition of the old subspace

$S_{10}^{\mathrm{old}}(\Gamma_0(34)) =$ $S_{10}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 2}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.329 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, 14, 42] (certified at ell = 2147483489)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (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(34, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 34:10 1000

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