Atlas › Modular forms › Weight 2 › Level 423 › 423.2.a

Newspace 423.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(423))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
423.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - 3q^{5} + q^{7} + O(q^{8})$	+	423g1 rank 1
423.2.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} + q^{5} - 3q^{7} + O(q^{8})$	−	423f1 rank 1
423.2.a.c	1	$\mathbb{Q}$	$q - 2q^{4} + q^{5} - 3q^{7} + O(q^{8})$	−	423a1 rank 1
423.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{4} - 2q^{5} + O(q^{8})$	−	423c1 rank 1
423.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + 4q^{7} + O(q^{8})$	+	423b1 rank 0
423.2.a.f	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + 3q^{5} - 3q^{7} + O(q^{8})$	+	423e1 rank 0
423.2.a.g	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{4} + 3q^{5} + q^{7} + O(q^{8})$	−	423d1 rank 0
423.2.a.h	2	$\mathbb{Q}(\sqrt{17})$	$2q + q^{2} + 5q^{4} - q^{5} + q^{7} + O(q^{8})$	+
423.2.a.i	3	degree 3	$3q - 2q^{2} + 4q^{4} - q^{5} - 3q^{7} + O(q^{8})$	+
423.2.a.j	3	degree 3	$3q + 2q^{2} + 4q^{4} + q^{5} - 3q^{7} + O(q^{8})$	−
423.2.a.k	4	degree 4	$4q - q^{2} + 3q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{7} + 7T_{11}$ 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 $T$
423.2.a.a	$x - 15$
423.2.a.b	$x + 21$
423.2.a.c	$x - 9$
423.2.a.d	$x + 33$
423.2.a.e	$x - 21$
423.2.a.f	$x - 31$
423.2.a.g	$x + 5$
423.2.a.h	$x^{2} + 46x + 257$
423.2.a.i	$x^{3} + 83x^{2} + 2267x + 20393$
423.2.a.j	$x^{3} - 53x^{2} + 7x + 4157$
423.2.a.k	$x^{4} - 67x^{3} + 281x^{2} + 32585x - 111093$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(423))$	19
Old forms	26
Cusp forms $S_{2}(\Gamma_0(423))$	45
Eisenstein series $E_{2}(\Gamma_0(423))$	7
Modular forms $M_{2}(\Gamma_0(423))$	52

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(423)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(141))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.313 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, 9, 47, 141, 423]: [0, 1, 2, 5, 18, 50] (certified at ell = 2147483137)
T = [(2, 1), (5, 3), (7, 5), (11, 7)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 5 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(423, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 423:2 1000

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