Atlas › Modular forms › Weight 2 › Level 448 › 448.2.a

Newspace 448.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(448))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
448.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} - q^{7} + O(q^{8})$	+	448f1 rank 0
448.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} + q^{7} + O(q^{8})$	−	448g1 rank 1
448.2.a.c	1	$\mathbb{Q}$	$q - 2q^{3} + 4q^{5} + q^{7} + O(q^{8})$	−	448h1 rank 0
448.2.a.d	1	$\mathbb{Q}$	$q - 2q^{5} - q^{7} + O(q^{8})$	+	448a1 rank 1
448.2.a.e	1	$\mathbb{Q}$	$q - 2q^{5} + q^{7} + O(q^{8})$	−	448b1 rank 1
448.2.a.f	1	$\mathbb{Q}$	$q + 2q^{3} - q^{7} + O(q^{8})$	+	448d1 rank 0
448.2.a.g	1	$\mathbb{Q}$	$q + 2q^{3} + q^{7} + O(q^{8})$	−	448c1 rank 0
448.2.a.h	1	$\mathbb{Q}$	$q + 2q^{3} + 4q^{5} - q^{7} + O(q^{8})$	+	448e1 rank 0
448.2.a.i	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} - 2q^{5} - 2q^{7} + O(q^{8})$	+
448.2.a.j	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = 8661T_{3} + 7879T_{5} + 8882T_{11} + 1822T_{13} + 7435T_{17} + 9474T_{19} + 6076T_{23} + 2656T_{29}$ 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 $T$
448.2.a.a	$x - 69460$
448.2.a.b	$x + 171196$
448.2.a.c	$x - 61568$
448.2.a.d	$x + 120212$
448.2.a.e	$x + 39684$
448.2.a.f	$x - 145408$
448.2.a.g	$x - 66208$
448.2.a.h	$x + 38900$
448.2.a.i	$x^{2} + 109200x + 2882058480$
448.2.a.j	$x^{2} - 55820x - 4568135920$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(448))$	12
Old forms	41
Cusp forms $S_{2}(\Gamma_0(448))$	53
Eisenstein series $E_{2}(\Gamma_0(448))$	23
Modular forms $M_{2}(\Gamma_0(448))$	76

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(448)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(32))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(56))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(64))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(112))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(224))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.704 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, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 224, 448]: [0, 1, 2, 1, 3, 4, 4, 7, 6, 12, 10, 20, 36, 68] (certified at ell = 2147483137)
T = [(3, 8661), (5, 7879), (11, 8882), (13, 1822), (17, 7435), (19, 9474), (23, 6076), (29, 2656)]; h of degree 12 = 1 x 12 (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(448, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 448:2 1000

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