Atlas › Modular forms › Weight 2 › Level 750 › 750.2.a

Newspace 750.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(750))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
750.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 6q^{7} + O(q^{8})$	+ +
750.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - q^{7} + O(q^{8})$	+ +
750.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 2q^{7} + O(q^{8})$	+ −
750.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 7q^{7} + O(q^{8})$	+ −
750.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} - 7q^{7} + O(q^{8})$	− +
750.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} - 2q^{7} + O(q^{8})$	− +
750.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + q^{7} + O(q^{8})$	− −
750.2.a.h	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 6q^{7} + O(q^{8})$	− −

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 $T$
750.2.a.a	$x^{2} + 9x - 131$
750.2.a.b	$x^{2} + 4x - 1$
750.2.a.c	$x^{2} - 17x + 41$
750.2.a.d	$x^{2} + 8x - 109$
750.2.a.e	$x^{2} + 22x + 41$
750.2.a.f	$x^{2} - 13x + 41$
750.2.a.g	$x^{2} + 2x - 19$
750.2.a.h	$x^{2} - 3x - 59$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(750))$	16
Old forms	115
Cusp forms $S_{2}(\Gamma_0(750))$	131
Eisenstein series $E_{2}(\Gamma_0(750))$	39
Modular forms $M_{2}(\Gamma_0(750))$	170

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(750)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(75))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(125))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(150))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(250))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(375))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.375 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, 125, 150, 250, 375, 750]: [0, 1, 1, 1, 3, 3, 4, 3, 10, 9, 12, 13, 34, 39, 52, 154] (certified at ell = 2147482801)
T = [(7, 1), (11, 3)]; h of degree 16 = 1 x 16 (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(750, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 750:2 1000

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