Atlas › Modular forms › Weight 4 › Level 203 › 203.4.a

Newspace 203.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(203))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
203.4.a.a	1	$\mathbb{Q}$	$q + q^{2} - 4q^{3} - 7q^{4} + 14q^{5} - 4q^{6} + 7q^{7} + O(q^{8})$	− −	$+1$
203.4.a.b	1	$\mathbb{Q}$	$q + 4q^{2} + 8q^{3} + 8q^{4} + 2q^{5} + 32q^{6} + 7q^{7} + O(q^{8})$	− −	$+1$
203.4.a.c	7	degree 7	$7q - 2q^{2} - 2q^{3} + 10q^{4} - 24q^{5} - 39q^{6} + 49q^{7} + O(q^{8})$	− +	$-1$
203.4.a.d	10	degree 10	$10q - 4q^{2} - 8q^{3} + 46q^{4} - 24q^{5} - 29q^{6} - 70q^{7} + O(q^{8})$	+ −	$-1$
203.4.a.e	11	degree 11	$11q + 2q^{2} + 4q^{3} + 50q^{4} + 16q^{5} + 7q^{6} - 77q^{7} + O(q^{8})$	+ +	$+1$
203.4.a.f	12	degree 12	$12q + 3q^{2} + 6q^{3} + 85q^{4} - 31q^{6} + 84q^{7} + O(q^{8})$	− −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
203.4.a.a	$x + 11$
203.4.a.b	$x - 28$
203.4.a.c	$x^{7} + 8x^{6} - 658x^{5} - 1240x^{4} + 107857x^{3} - 309332x^{2} - 379384x + 639724$
203.4.a.d	$x^{10} + 28x^{9} - 1231x^{8} - 28236x^{7} + 555833x^{6} + 8295138x^{5} - 106911303x^{4} - 844985954x^{3} + 8116982132x^{2} + 28366531368x - 199698087232$
203.4.a.e	$x^{11} - 14x^{10} - 2062x^{9} + 26050x^{8} + 1532584x^{7} - 16854222x^{6} - 505537818x^{5} + 4665386354x^{4} + 73172841815x^{3} - 547863096324x^{2} - 3555745884952x + 21747374959852$
203.4.a.f	$x^{12} - 21x^{11} - 1712x^{10} + 35992x^{9} + 1031576x^{8} - 21651554x^{7} - 265003614x^{6} + 5582036764x^{5} + 27750793687x^{4} - 607333894825x^{3} - 882520114042x^{2} + 23037537147660x - 11773978663944$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(203))$	42
Old forms	16
Cusp forms $S_{4}(\Gamma_0(203))$	58
Eisenstein series $E_{4}(\Gamma_0(203))$	4
Modular forms $M_{4}(\Gamma_0(203))$	62

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(203)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.149 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, 7, 29, 203]: [1, 3, 9, 62] (certified at ell = 2147483353)
T = [(2, 1), (3, 3)]; h of degree 42 = 1 x 42 (phi(ord eps) x dim S_k^new); 8 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(203, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 203:4 1000

← 202.4.a · 204.4.a →