Atlas › Modular forms › Weight 2 › Level 675 › 675.2.a

Newspace 675.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(675))$

Newforms

Label	Dim	Coefficient field	Trace form	CM	Elliptic curve
675.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} + 3q^{7} + O(q^{8})$		675h1 rank 0
675.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + O(q^{8})$		675b1 rank 1
675.2.a.c	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + O(q^{8})$		675d1 rank 0
675.2.a.d	1	$\mathbb{Q}$	$q - 2q^{4} - 4q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	675c1 rank 0
675.2.a.e	1	$\mathbb{Q}$	$q - 2q^{4} + q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	675a1 rank 1
675.2.a.f	1	$\mathbb{Q}$	$q - 2q^{4} + 4q^{7} + O(q^{8})$	$\mathbb{Q}(\sqrt{-3})$	675e1 rank 0
675.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + O(q^{8})$		675i1 rank 1
675.2.a.h	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + O(q^{8})$		675f1 rank 0
675.2.a.i	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + 3q^{7} + O(q^{8})$		675g1 rank 0
675.2.a.j	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} + 3q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-15})$
675.2.a.k	2	$\mathbb{Q}(\sqrt{13})$	$2q - q^{2} + 3q^{4} - 2q^{7} + O(q^{8})$
675.2.a.l	2	$\mathbb{Q}(\sqrt{2})$	$2q - 6q^{7} + O(q^{8})$
675.2.a.m	2	$\mathbb{Q}(\sqrt{2})$	$2q + 6q^{7} + O(q^{8})$
675.2.a.n	2	$\mathbb{Q}(\sqrt{7})$	$2q + 10q^{4} - 6q^{7} + O(q^{8})$
675.2.a.o	2	$\mathbb{Q}(\sqrt{7})$	$2q + 10q^{4} + 6q^{7} + O(q^{8})$
675.2.a.p	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{2} + 3q^{4} - 2q^{7} + O(q^{8})$
675.2.a.q	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} + 3q^{4} + O(q^{8})$	$\mathbb{Q}(\sqrt{-15})$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29} + 9T_{31} + 10T_{37} + 11T_{41} + 12T_{43}$ 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 $T$
675.2.a.a	$x - 34$
675.2.a.b	$x + 313$
675.2.a.c	$x - 397$
675.2.a.d	$x + 7$
675.2.a.e	$x + 302$
675.2.a.f	$x - 301$
675.2.a.g	$x + 13$
675.2.a.h	$x - 97$
675.2.a.i	$x + 166$
675.2.a.j	$x^{2} + 201x + 6319$
675.2.a.k	$x^{2} + 159x + 6161$
675.2.a.l	$x^{2} + 312x + 13384$
675.2.a.m	$x^{2} - 408x + 40648$
675.2.a.n	$x^{2} + 42x - 11326$
675.2.a.o	$x^{2} + 42x - 3262$
675.2.a.p	$x^{2} - 95x - 16025$
675.2.a.q	$x^{2} - 9x - 431$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(675))$	25
Old forms	48
Cusp forms $S_{2}(\Gamma_0(675))$	73
Eisenstein series $E_{2}(\Gamma_0(675))$	35
Modular forms $M_{2}(\Gamma_0(675))$	108

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(675)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(75))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(135))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(225))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.887 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, 3, 5, 9, 15, 25, 27, 45, 75, 135, 225, 675]: [0, 1, 1, 2, 4, 3, 4, 8, 12, 20, 32, 92] (certified at ell = 2147482801)
T = [(2, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8), (31, 9), (37, 10), (41, 11), (43, 12)]; h of degree 25 = 1 x 25 (phi(ord eps) x dim S_k^new); 8 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(675, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 675:2 1000

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