Atlas › Modular forms › Weight 8 › Level 46 › 46.8.a

Newspace 46.8.a: $S_{8}^{\mathrm{new}}(\Gamma_0(46))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
46.8.a.a	2	$\mathbb{Q}(\sqrt{85})$	$2q - 16q^{2} - 28q^{3} + 128q^{4} - 110q^{5} + 224q^{6} + 74q^{7} + O(q^{8})$	+ −	$-1$
46.8.a.b	3	degree 3	$3q - 24q^{2} - 28q^{3} + 192q^{4} + 390q^{5} + 224q^{6} - 470q^{7} + O(q^{8})$	+ +	$+1$
46.8.a.c	3	degree 3	$3q + 24q^{2} - 12q^{3} + 192q^{4} - 570q^{5} - 96q^{6} - 1382q^{7} + O(q^{8})$	− +	$-1$
46.8.a.d	4	degree 4	$4q + 32q^{2} + 96q^{3} + 256q^{4} + 180q^{5} + 768q^{6} + 534q^{7} + O(q^{8})$	− −	$+1$

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 $T$
46.8.a.a	$x^{2} + 138x - 3739$
46.8.a.b	$x^{3} - 362x^{2} - 252599x + 76043764$
46.8.a.c	$x^{3} + 582x^{2} + 53961x - 9376060$
46.8.a.d	$x^{4} - 276x^{3} - 168383x^{2} + 24068794x + 8344577640$

Dimensions

Newforms $S_{8}^{\mathrm{new}}(\Gamma_0(46))$	12
Old forms	28
Cusp forms $S_{8}(\Gamma_0(46))$	40
Eisenstein series $E_{8}(\Gamma_0(46))$	4
Modular forms $M_{8}(\Gamma_0(46))$	44

Decomposition of the old subspace

$S_{8}^{\mathrm{old}}(\Gamma_0(46)) =$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))^{\oplus 2}$ $\oplus$ $S_{8}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.285 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, 23, 46]: [1, 3, 15, 44] (certified at ell = 2147483647)
T = [(3, 1), (5, 1)]; h of degree 12 = 1 x 12 (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(46, 8)

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

cargo run --release -p sagebrush-web --example atlas -- 46:8 1000

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