Atlas › Modular forms › Weight 2 › Level 822 › 822.2.a

Newspace 822.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(822))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
822.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ + +	$-1$	822a1 rank 1
822.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ − −	$-1$	822d1 rank 1
822.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$	822b1 rank 0
822.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 3q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$	822c1 rank 0
822.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + O(q^{8})$	− − −	$+1$	822e1 rank 0
822.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 3q^{5} + q^{6} - 3q^{7} + O(q^{8})$	− − −	$+1$	822f1 rank 0
822.2.a.g	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	+ + −	$+1$
822.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− + −	$-1$
822.2.a.i	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} - 4q^{5} + 3q^{6} - q^{7} + O(q^{8})$	+ + +	$-1$
822.2.a.j	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 5q^{7} + O(q^{8})$	+ − +	$+1$
822.2.a.k	3	degree 3	$3q + 3q^{2} - 3q^{3} + 3q^{4} - q^{5} - 3q^{6} + 2q^{7} + O(q^{8})$	− + +	$+1$
822.2.a.l	4	degree 4	$4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 9q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
822.2.a.a	$x - 5$
822.2.a.b	$x + 10$
822.2.a.c	$x - 12$
822.2.a.d	$x - 9$
822.2.a.e	$x - 2$
822.2.a.f	$x + 6$
822.2.a.g	$x^{2} + 7x - 19$
822.2.a.h	$x^{2} + 14x - 1$
822.2.a.i	$x^{3} + 7x^{2} - 56x - 16$
822.2.a.j	$x^{3} + 18x^{2} + 10x - 669$
822.2.a.k	$x^{3} - 5x^{2} - 91x + 184$
822.2.a.l	$x^{4} - 23x^{3} + 70x^{2} + 1319x - 7041$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(822))$	23
Old forms	112
Cusp forms $S_{2}(\Gamma_0(822))$	135
Eisenstein series $E_{2}(\Gamma_0(822))$	7
Modular forms $M_{2}(\Gamma_0(822))$	142

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(822)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(137))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(274))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(411))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.353 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, 6, 137, 274, 411, 822]: [0, 1, 1, 3, 12, 36, 48, 142] (certified at ell = 2147483249)
T = [(5, 1), (7, 3)]; h of degree 23 = 1 x 23 (phi(ord eps) x dim S_k^new); 4 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(822, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 822:2 1000

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