Atlas › Modular forms › Weight 2 › Level 450 › 450.2.a

Newspace 450.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(450))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
450.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{7} + O(q^{8})$	+	450f1 rank 1
450.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{7} + O(q^{8})$	+	450c1 rank 1
450.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{7} + O(q^{8})$	+	450d1 rank 0
450.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 4q^{7} + O(q^{8})$	+	450g1 rank 0
450.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 2q^{7} + O(q^{8})$	−	450e1 rank 0
450.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{7} + O(q^{8})$	−	450a1 rank 0
450.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 2q^{7} + O(q^{8})$	−	450b1 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 $T$
450.2.a.a	$x + 36$
450.2.a.b	$x + 104$
450.2.a.c	$x - 81$
450.2.a.d	$x - 108$
450.2.a.e	$x + 48$
450.2.a.f	$x + 32$
450.2.a.g	$x - 13$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(450))$	7
Old forms	60
Cusp forms $S_{2}(\Gamma_0(450))$	67
Eisenstein series $E_{2}(\Gamma_0(450))$	47
Modular forms $M_{2}(\Gamma_0(450))$	114

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(450)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 8}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(30))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(50))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(75))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(90))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(150))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(225))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.452 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, 9, 10, 15, 18, 25, 30, 45, 50, 75, 90, 150, 225, 450]: [0, 1, 1, 1, 3, 2, 3, 4, 5, 3, 10, 8, 9, 12, 22, 34, 32, 94] (certified at ell = 2147482921)
T = [(7, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8)]; h of degree 7 = 1 x 7 (phi(ord eps) x dim S_k^new); 2 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(450, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 450:2 1000

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