Atlas › Modular forms › Weight 2 › Level 421 › 421.2.a

Newspace 421.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(421))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	CM	Elliptic curve
421.2.a.a	15	degree 15	$15q - 6q^{2} - 9q^{3} + 10q^{4} - 11q^{5} - 10q^{6} - 5q^{7} + O(q^{8})$	+	$-1$
421.2.a.b	19	degree 19	$19q + 4q^{2} + 7q^{3} + 18q^{4} + 7q^{5} + 8q^{6} + 3q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
421.2.a.a	$x^{15} + 15x^{14} + 76x^{13} + 93x^{12} - 408x^{11} - 1203x^{10} + 144x^{9} + 3259x^{8} + 1796x^{7} - 2812x^{6} - 2259x^{5} + 371x^{4} + 593x^{3} + 126x^{2} + 2x - 1$
421.2.a.b	$x^{19} - 11x^{18} - 14x^{17} + 535x^{16} - 1031x^{15} - 8646x^{14} + 28978x^{13} + 56322x^{12} - 297854x^{11} - 80329x^{10} + 1448541x^{9} - 665130x^{8} - 3284627x^{7} + 2675780x^{6} + 2960543x^{5} - 2515146x^{4} - 1425513x^{3} + 705866x^{2} + 371316x + 18823$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(421))$	34
Old forms	0
Cusp forms $S_{2}(\Gamma_0(421))$	34
Eisenstein series $E_{2}(\Gamma_0(421))$	1
Modular forms $M_{2}(\Gamma_0(421))$	35

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.187 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, 421]: [0, 35] (certified at ell = 2147482681)
T = [(2, 1), (3, 1)]; h of degree 34 = 1 x 34 (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(421, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 421:2 1000

← 420.2.a · 422.2.a →