Atlas › Modular forms › Weight 4 › Level 46 › 46.4.a

Newspace 46.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(46))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
46.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 4q^{4} - 10q^{5} + 2q^{6} - 12q^{7} + O(q^{8})$	+ −	$-1$
46.4.a.b	1	$\mathbb{Q}$	$q + 2q^{2} - 9q^{3} + 4q^{4} - 20q^{5} - 18q^{6} + 2q^{7} + O(q^{8})$	− +	$-1$
46.4.a.c	2	$\mathbb{Q}(\sqrt{41})$	$2q - 4q^{2} - q^{3} + 8q^{4} + 10q^{5} + 2q^{6} + 6q^{7} + O(q^{8})$	+ +	$+1$
46.4.a.d	2	$\mathbb{Q}(\sqrt{73})$	$2q + 4q^{2} + 3q^{3} + 8q^{4} + 10q^{5} + 6q^{6} + 12q^{7} + O(q^{8})$	− −	$+1$

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 $T$
46.4.a.a	$x + 31$
46.4.a.b	$x + 69$
46.4.a.c	$x^{2} - 29x + 118$
46.4.a.d	$x^{2} - 33x - 184$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(46))$	6
Old forms	10
Cusp forms $S_{4}(\Gamma_0(46))$	16
Eisenstein series $E_{4}(\Gamma_0(46))$	4
Modular forms $M_{4}(\Gamma_0(46))$	20

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(46)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.102 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, 2, 7, 20] (certified at ell = 2147483647)
T = [(3, 1), (5, 3)]; h of degree 6 = 1 x 6 (phi(ord eps) x dim S_k^new); 2 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, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 46:4 1000

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