Atlas › Modular forms › Weight 2 › Level 430 › 430.2.a

Newspace 430.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(430))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
430.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - q^{5} + q^{7} + O(q^{8})$	+ + +	$-1$	430a1 rank 1
430.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + q^{5} - 3q^{7} + O(q^{8})$	+ − −	$-1$	430b1 rank 1
430.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - q^{5} - 2q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	430c1 rank 1
430.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} + q^{5} - 2q^{6} - 5q^{7} + O(q^{8})$	− − +	$-1$	430d1 rank 1
430.2.a.e	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$
430.2.a.f	2	$\mathbb{Q}(\sqrt{6})$	$2q + 2q^{2} + 2q^{4} - 2q^{5} + 2q^{7} + O(q^{8})$	− + +	$+1$
430.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} + 2q^{7} + O(q^{8})$	− − −	$+1$
430.2.a.h	3	degree 3	$3q - 3q^{2} - 2q^{3} + 3q^{4} - 3q^{5} + 2q^{6} - 6q^{7} + O(q^{8})$	+ + −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7}$ 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 $T$
430.2.a.a	$x - 3$
430.2.a.b	$x + 9$
430.2.a.c	$x + 5$
430.2.a.d	$x + 17$
430.2.a.e	$x^{2} - 8x - 59$
430.2.a.f	$x^{2} - 6x + 3$
430.2.a.g	$x^{2} - 6x + 7$
430.2.a.h	$x^{3} + 20x^{2} + 33x - 668$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(430))$	13
Old forms	50
Cusp forms $S_{2}(\Gamma_0(430))$	63
Eisenstein series $E_{2}(\Gamma_0(430))$	7
Modular forms $M_{2}(\Gamma_0(430))$	70

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(430)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(86))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(215))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.325 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, 5, 10, 43, 86, 215, 430]: [0, 1, 1, 3, 4, 13, 24, 70] (certified at ell = 2147483353)
T = [(3, 1), (7, 3)]; h of degree 13 = 1 x 13 (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(430, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 430:2 1000

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