Atlas › Modular forms › Weight 2 › Level 798 › 798.2.a

Newspace 798.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(798))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
798.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} + O(q^{8})$	+ + + +	$-1$	798a1 rank 1
798.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + O(q^{8})$	+ − + −	$-1$	798c1 rank 1
798.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − − +	$-1$	798d1 rank 1
798.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} + O(q^{8})$	+ − − −	$+1$	798e1 rank 0
798.2.a.e	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} - q^{7} + O(q^{8})$	+ − + +	$+1$	798b1 rank 0
798.2.a.f	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 4q^{5} - q^{6} + q^{7} + O(q^{8})$	+ − − −	$+1$	798f1 rank 0
798.2.a.g	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + O(q^{8})$	− + + −	$-1$	798g1 rank 1
798.2.a.h	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 4q^{5} + q^{6} - q^{7} + O(q^{8})$	− − + +	$-1$	798h1 rank 1
798.2.a.i	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} - q^{7} + O(q^{8})$	− − + −	$+1$	798i1 rank 0
798.2.a.j	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+ + − +	$+1$
798.2.a.k	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{6} - 2q^{7} + O(q^{8})$	+ + + −	$+1$
798.2.a.l	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 2q^{7} + O(q^{8})$	− + − −	$+1$
798.2.a.m	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− − − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{23}$ 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 $T$
798.2.a.a	$x - 4$
798.2.a.b	$x + 26$
798.2.a.c	$x + 74$
798.2.a.d	$x - 52$
798.2.a.e	$x - 98$
798.2.a.f	$x$
798.2.a.g	$x + 90$
798.2.a.h	$x + 52$
798.2.a.i	$x - 34$
798.2.a.j	$x^{2} - 6x - 236$
798.2.a.k	$x^{2} - 8x - 1436$
798.2.a.l	$x^{2} - 48x + 376$
798.2.a.m	$x^{2} + 42x - 9684$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(798))$	17
Old forms	136
Cusp forms $S_{2}(\Gamma_0(798))$	153
Eisenstein series $E_{2}(\Gamma_0(798))$	15
Modular forms $M_{2}(\Gamma_0(798))$	168

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.877 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, 7, 14, 19, 21, 38, 42, 57, 114, 133, 266, 399, 798]: [0, 1, 1, 3, 1, 4, 2, 4, 7, 12, 8, 24, 14, 44, 56, 168] (certified at ell = 2147483647)
T = [(5, 1), (11, 3), (13, 5), (17, 7), (23, 9)]; h of degree 17 = 1 x 17 (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(798, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 798:2 1000

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