Atlas › Modular forms › Weight 4 › Level 150 › 150.4.a

Newspace 150.4.a: $S_{4}^{\mathrm{new}}(\Gamma_0(150))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
150.4.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 4q^{4} + 6q^{6} - q^{7} + O(q^{8})$	+ +
150.4.a.b	1	$\mathbb{Q}$	$q - 2q^{2} - 3q^{3} + 4q^{4} + 6q^{6} + 4q^{7} + O(q^{8})$	+ +
150.4.a.c	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{6} - 23q^{7} + O(q^{8})$	+ −
150.4.a.d	1	$\mathbb{Q}$	$q - 2q^{2} + 3q^{3} + 4q^{4} - 6q^{6} + 2q^{7} + O(q^{8})$	+ −
150.4.a.e	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} - 6q^{6} - 32q^{7} + O(q^{8})$	− +
150.4.a.f	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} - 6q^{6} - 2q^{7} + O(q^{8})$	− +
150.4.a.g	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 4q^{4} - 6q^{6} + 23q^{7} + O(q^{8})$	− +
150.4.a.h	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 4q^{4} + 6q^{6} + q^{7} + O(q^{8})$	− −
150.4.a.i	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 4q^{4} + 6q^{6} + 16q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 3T_{11}$ 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 $T$
150.4.a.a	$x - 125$
150.4.a.b	$x + 140$
150.4.a.c	$x + 113$
150.4.a.d	$x - 212$
150.4.a.e	$x + 212$
150.4.a.f	$x - 208$
150.4.a.g	$x + 67$
150.4.a.h	$x - 127$
150.4.a.i	$x - 52$

Dimensions

Newforms $S_{4}^{\mathrm{new}}(\Gamma_0(150))$	9
Old forms	69
Cusp forms $S_{4}(\Gamma_0(150))$	78
Eisenstein series $E_{4}(\Gamma_0(150))$	24
Modular forms $M_{4}(\Gamma_0(150))$	102

Decomposition of the old subspace

$S_{4}^{\mathrm{old}}(\Gamma_0(150)) =$ $S_{4}^{\mathrm{new}}(\Gamma_0(5))^{\oplus 8}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(6))^{\oplus 3}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(10))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(25))^{\oplus 4}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 2}$ $\oplus$ $S_{4}^{\mathrm{new}}(\Gamma_0(75))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.174 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, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150]: [1, 2, 2, 3, 5, 7, 8, 9, 22, 25, 32, 94] (certified at ell = 2147482921)
T = [(7, 1), (11, 3)]; h of degree 9 = 1 x 9 (phi(ord eps) x dim S_k^new); 3 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(150, 4)

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

cargo run --release -p sagebrush-web --example atlas -- 150:4 1000

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