Atlas › Modular forms › Weight 2 › Level 150 › 150.2.a

Newspace 150.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(150))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
150.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{6} + 2q^{7} + O(q^{8})$	+ +	150b1 rank 0
150.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} + 4q^{7} + O(q^{8})$	− +	150c1 rank 0
150.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{6} - 2q^{7} + O(q^{8})$	− −	150a1 rank 0

Hecke characteristic polynomials

The eigenvalue of $T = T_{7} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31}$ 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 $T$
150.2.a.a	$x + 56$
150.2.a.b	$x + 24$
150.2.a.c	$x + 64$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(150))$	3
Old forms	16
Cusp forms $S_{2}(\Gamma_0(150))$	19
Eisenstein series $E_{2}(\Gamma_0(150))$	23
Modular forms $M_{2}(\Gamma_0(150))$	42

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(150)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(75))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.138 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]: [0, 1, 1, 1, 3, 3, 4, 3, 10, 9, 12, 34] (certified at ell = 2147482921)
T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; h of degree 3 = 1 x 3 (phi(ord eps) x dim S_k^new); 1 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, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 150:2 1000

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