Atlas › Modular forms › Weight 2 › Level 830 › 830.2.a

Newspace 830.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(830))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
830.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 5q^{7} + O(q^{8})$	+ − +	$+1$	830a1 rank 0
830.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 3q^{3} + q^{4} + q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	− − +	$-1$	830c1 rank 1
830.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− − +	$-1$	830b1 rank 1
830.2.a.d	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} - 3q^{3} + 2q^{4} - 2q^{5} + 3q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$
830.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ + +	$-1$
830.2.a.f	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} + 3q^{5} + q^{6} - 5q^{7} + O(q^{8})$	+ − −	$-1$
830.2.a.g	3	degree 3	$3q - 3q^{2} + 3q^{4} + 3q^{5} + 3q^{7} + O(q^{8})$	+ − +	$+1$
830.2.a.h	3	degree 3	$3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	− + −	$-1$
830.2.a.i	4	degree 4	$4q - 4q^{2} + 3q^{3} + 4q^{4} - 4q^{5} - 3q^{6} + 3q^{7} + O(q^{8})$	+ + −	$+1$
830.2.a.j	4	degree 4	$4q + 4q^{2} + 3q^{3} + 4q^{4} - 4q^{5} + 3q^{6} + 8q^{7} + O(q^{8})$	− + +	$+1$
830.2.a.k	5	degree 5	$5q + 5q^{2} + 2q^{3} + 5q^{4} + 5q^{5} + 2q^{6} + 5q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
830.2.a.a	$x - 89$
830.2.a.b	$x + 39$
830.2.a.c	$x + 75$
830.2.a.d	$x^{2} - 62x + 909$
830.2.a.e	$x^{2} + 118x + 3193$
830.2.a.f	$x^{3} + 187x^{2} + 10419x + 154361$
830.2.a.g	$x^{3} - 63x^{2} - 2865x + 911$
830.2.a.h	$x^{3} + 107x^{2} - 125x - 142775$
830.2.a.i	$x^{4} - 45x^{3} - 3989x^{2} + 26849x + 981832$
830.2.a.j	$x^{4} - 138x^{3} + 2564x^{2} + 214610x - 4852909$
830.2.a.k	$x^{5} - 93x^{4} - 546x^{3} + 154034x^{2} + 78609x - 37997477$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(830))$	29
Old forms	94
Cusp forms $S_{2}(\Gamma_0(830))$	123
Eisenstein series $E_{2}(\Gamma_0(830))$	7
Modular forms $M_{2}(\Gamma_0(830))$	130

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(830)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(83))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(166))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(415))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.792 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, 83, 166, 415, 830]: [0, 1, 1, 3, 8, 23, 44, 130] (certified at ell = 2147483077)
T = [(3, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 29 = 1 x 29 (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(830, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 830:2 1000

← 829.2.a · 831.2.a →