Atlas › Modular forms › Weight 4 › Level 237 › 237.4.a

Newspace 237.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(237))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
237.4.a.a	6	degree 6	$6q - 6q^{2} + 18q^{3} + 4q^{4} - 8q^{5} - 18q^{6} - 37q^{7} + O(q^{8})$	− +	$-1$
237.4.a.b	9	degree 9	$9q + 4q^{2} - 27q^{3} + 40q^{4} + 8q^{5} - 12q^{6} + 5q^{7} + O(q^{8})$	+ +	$+1$
237.4.a.c	11	degree 11	$11q - 6q^{2} - 33q^{3} + 44q^{4} - 12q^{5} + 18q^{6} - 23q^{7} + O(q^{8})$	+ −	$-1$
237.4.a.d	14	degree 14	$14q + 4q^{2} + 42q^{3} + 80q^{4} + 12q^{5} + 12q^{6} + 47q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
237.4.a.a	$x^{6} + 14x^{5} - 104x^{4} - 1958x^{3} - 1300x^{2} + 52386x + 129443$
237.4.a.b	$x^{9} - 12x^{8} - 622x^{7} + 5020x^{6} + 112550x^{5} - 280182x^{4} - 5251333x^{3} - 11969298x^{2} + 3374048x + 21072992$
237.4.a.c	$x^{11} + 18x^{10} - 629x^{9} - 8880x^{8} + 143787x^{7} + 1360797x^{6} - 13001237x^{5} - 89716956x^{4} + 417376709x^{3} + 2418378732x^{2} - 1420065157x - 9393958125$
237.4.a.d	$x^{14} - 16x^{13} - 1251x^{12} + 20120x^{11} + 581867x^{10} - 9667599x^{9} - 122064467x^{8} + 2227308818x^{7} + 10656232273x^{6} - 253739295188x^{5} - 151043773545x^{4} + 13438039702717x^{3} - 23094999063722x^{2} - 262419627089376x + 804673493253504$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(237))$	40
Old forms	38
Cusp forms $S_{4}(\Gamma_0(237))$	78
Eisenstein series $E_{4}(\Gamma_0(237))$	4
Modular forms $M_{4}(\Gamma_0(237))$	82

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(237)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(79))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.153 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, 3, 79, 237]: [1, 2, 21, 82] (certified at ell = 2147483053)
T = [(2, 1), (5, 1)]; h of degree 40 = 1 x 40 (phi(ord eps) x dim S_k^new); 7 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(237, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 237:4 1000

← 236.4.a · 238.4.a →