Atlas › Modular forms › Weight 2 › Level 802 › 802.2.a

Newspace 802.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(802))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
802.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{5} - 2q^{6} + O(q^{8})$	− +	$+1$	802b1 rank 0
802.2.a.b	1	$\mathbb{Q}$	$q + q^{2} + q^{4} + 4q^{5} - 2q^{7} + O(q^{8})$	− +	$+1$	802a1 rank 0
802.2.a.c	5	degree 5	$5q + 5q^{2} - 6q^{3} + 5q^{4} - 9q^{5} - 6q^{6} - q^{7} + O(q^{8})$	− −	$-1$
802.2.a.d	7	degree 7	$7q - 7q^{2} + 6q^{3} + 7q^{4} + 5q^{5} - 6q^{6} + q^{7} + O(q^{8})$	+ −	$+1$
802.2.a.e	9	degree 9	$9q - 9q^{2} - 6q^{3} + 9q^{4} - 5q^{5} + 6q^{6} - 7q^{7} + O(q^{8})$	+ +	$-1$
802.2.a.f	10	degree 10	$10q + 10q^{2} + 10q^{3} + 10q^{4} + 9q^{5} + 10q^{6} + q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
802.2.a.a	$x + 4$
802.2.a.b	$x - 4$
802.2.a.c	$x^{5} + 15x^{4} + 82x^{3} + 195x^{2} + 187x + 59$
802.2.a.d	$x^{7} - 11x^{6} + 18x^{5} + 137x^{4} - 399x^{3} - 143x^{2} + 1012x - 568$
802.2.a.e	$x^{9} + 11x^{8} + 22x^{7} - 79x^{6} - 177x^{5} + 175x^{4} + 358x^{3} - 122x^{2} - 192x - 28$
802.2.a.f	$x^{10} - 19x^{9} + 132x^{8} - 375x^{7} + 25x^{6} + 2471x^{5} - 6164x^{4} + 6480x^{3} - 2908x^{2} + 228x + 128$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(802))$	33
Old forms	66
Cusp forms $S_{2}(\Gamma_0(802))$	99
Eisenstein series $E_{2}(\Gamma_0(802))$	3
Modular forms $M_{2}(\Gamma_0(802))$	102

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(802)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(401))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.247 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, 401, 802]: [0, 1, 34, 102] (certified at ell = 2147482801)
T = [(3, 1), (5, 1)]; h of degree 33 = 1 x 33 (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(802, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 802:2 1000

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