Atlas › Modular forms › Weight 4 › Level 161 › 161.4.a

Newspace 161.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(161))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
161.4.a.a	5	degree 5	$5q - 4q^{2} - 11q^{3} - 4q^{5} + 35q^{7} + O(q^{8})$	− +	$-1$
161.4.a.b	8	degree 8	$8q - 3q^{3} + 24q^{4} - 24q^{5} - 41q^{6} - 56q^{7} + O(q^{8})$	+ −	$-1$
161.4.a.c	9	degree 9	$9q + 9q^{3} + 48q^{4} - 4q^{5} + 46q^{6} - 63q^{7} + O(q^{8})$	+ +	$+1$
161.4.a.d	12	degree 12	$12q + 4q^{2} + q^{3} + 72q^{4} + 16q^{5} - 15q^{6} + 84q^{7} + O(q^{8})$	− −	$+1$

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 $T$
161.4.a.a	$x^{5} + 15x^{4} + 18x^{3} - 312x^{2} + 73x + 453$
161.4.a.b	$x^{8} + 3x^{7} - 154x^{6} + 7x^{5} + 4258x^{4} - 5641x^{3} - 4596x^{2} + 343x + 147$
161.4.a.c	$x^{9} - 9x^{8} - 248x^{7} + 1842x^{6} + 21679x^{5} - 120041x^{4} - 799840x^{3} + 2960374x^{2} + 10668992x - 21456414$
161.4.a.d	$x^{12} - 5x^{11} - 320x^{10} + 1493x^{9} + 32188x^{8} - 118513x^{7} - 1432230x^{6} + 3344891x^{5} + 29772725x^{4} - 30228170x^{3} - 252581562x^{2} + 83941312x + 730973278$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(161))$	34
Old forms	12
Cusp forms $S_{4}(\Gamma_0(161))$	46
Eisenstein series $E_{4}(\Gamma_0(161))$	4
Modular forms $M_{4}(\Gamma_0(161))$	50

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(161)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(7))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.119 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, 23, 161]: [1, 3, 7, 50] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 34 = 1 x 34 (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(161, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 161:4 1000

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