Atlas › Modular forms › Weight 10 › Level 26 › 26.10.a

Newspace 26.10.a: $S_{10}^{\mathrm{new}}(\Gamma_0(26))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
26.10.a.a	1	$\mathbb{Q}$	$q - 16q^{2} - 273q^{3} + 256q^{4} + 1015q^{5} + 4368q^{6} + 3955q^{7} + O(q^{8})$	+ +	$-1$
26.10.a.b	1	$\mathbb{Q}$	$q - 16q^{2} + 192q^{3} + 256q^{4} - 1310q^{5} - 3072q^{6} - 5810q^{7} + O(q^{8})$	+ +	$-1$
26.10.a.c	1	$\mathbb{Q}$	$q + 16q^{2} + 75q^{3} + 256q^{4} - 1979q^{5} + 1200q^{6} - 10115q^{7} + O(q^{8})$	− −	$-1$
26.10.a.d	3	degree 3	$3q - 48q^{2} + 768q^{4} + 248q^{5} - 2956q^{7} + O(q^{8})$	+ −	$+1$
26.10.a.e	3	degree 3	$3q + 48q^{2} + 156q^{3} + 768q^{4} - 1272q^{5} + 2496q^{6} + 17058q^{7} + O(q^{8})$	− +	$+1$

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 $T$
26.10.a.a	$x - 742$
26.10.a.b	$x + 1118$
26.10.a.c	$x + 1904$
26.10.a.d	$x^{3} - 248x^{2} - 5347316x + 4069822848$
26.10.a.e	$x^{3} + 1116x^{2} - 4420020x - 2206425872$

Dimensions

Newforms $S_{10}^{\mathrm{new}}(\Gamma_0(26))$	9
Old forms	20
Cusp forms $S_{10}(\Gamma_0(26))$	29
Eisenstein series $E_{10}(\Gamma_0(26))$	4
Modular forms $M_{10}(\Gamma_0(26))$	33

Decomposition of the old subspace

$S_{10}^{\mathrm{old}}(\Gamma_0(26)) =$ $S_{10}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 2}$ $\oplus$ $S_{10}^{\mathrm{new}}(\Gamma_0(13))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.315 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, 13, 26]: [1, 3, 11, 33] (certified at ell = 2147483629)
T = [(3, 1), (5, 1)]; h of degree 9 = 1 x 9 (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(26, 10)

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

cargo run --release -p sagebrush-web --example atlas -- 26:10 1000

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