Atlas › Modular forms › Weight 4 › Level 202 › 202.4.a

Newspace 202.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(202))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
202.4.a.a	1	$\mathbb{Q}$	$q + 2q^{2} - 8q^{3} + 4q^{4} + 18q^{5} - 16q^{6} - 13q^{7} + O(q^{8})$	− −	$+1$
202.4.a.b	1	$\mathbb{Q}$	$q + 2q^{2} - 2q^{3} + 4q^{4} + 3q^{5} - 4q^{6} - 30q^{7} + O(q^{8})$	− +	$-1$
202.4.a.c	3	degree 3	$3q + 6q^{2} - 5q^{3} + 12q^{4} - 27q^{5} - 10q^{6} - 7q^{7} + O(q^{8})$	− +	$-1$
202.4.a.d	5	degree 5	$5q - 10q^{2} + 7q^{3} + 20q^{4} + 6q^{5} - 14q^{6} + 14q^{7} + O(q^{8})$	+ +	$+1$
202.4.a.e	7	degree 7	$7q - 14q^{2} - 2q^{3} + 28q^{4} - 19q^{5} + 4q^{6} - 14q^{7} + O(q^{8})$	+ −	$-1$
202.4.a.f	8	degree 8	$8q + 16q^{2} + 10q^{3} + 32q^{4} + 3q^{5} + 20q^{6} + 46q^{7} + O(q^{8})$	− −	$+1$

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 $T$
202.4.a.a	$x - 10$
202.4.a.b	$x - 1$
202.4.a.c	$x^{3} + 32x^{2} + 321x + 1031$
202.4.a.d	$x^{5} - 13x^{4} - 321x^{3} + 4362x^{2} + 14849x - 225206$
202.4.a.e	$x^{7} + 21x^{6} - 348x^{5} - 6960x^{4} + 14173x^{3} + 343493x^{2} - 369675x - 72123$
202.4.a.f	$x^{8} - 13x^{7} - 1138x^{6} + 17660x^{5} + 292897x^{4} - 5896917x^{3} + 17933007x^{2} + 39217743x - 5098998$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(202))$	25
Old forms	50
Cusp forms $S_{4}(\Gamma_0(202))$	75
Eisenstein series $E_{4}(\Gamma_0(202))$	4
Modular forms $M_{4}(\Gamma_0(202))$	79

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(202)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(101))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.138 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, 101, 202]: [1, 2, 27, 79] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 25 = 1 x 25 (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(202, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 202:4 1000

← 201.4.a · 203.4.a →