Atlas › Modular forms › Weight 2 › Level 805 › 805.2.a

Newspace 805.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(805))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
805.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - q^{5} - q^{7} + O(q^{8})$	+ + −	$+1$	805c1 rank 0
805.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - q^{5} - q^{7} + O(q^{8})$	+ + −	$+1$	805b1 rank 0
805.2.a.c	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} - q^{5} - 2q^{6} - q^{7} + O(q^{8})$	+ + +	$-1$	805a1 rank 1
805.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 2q^{4} - q^{5} + 6q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	805d1 rank 0
805.2.a.e	2	$\mathbb{Q}(\sqrt{5})$	$2q + 3q^{2} + 3q^{4} - 2q^{5} - 5q^{6} - 2q^{7} + O(q^{8})$	+ + −	$+1$
805.2.a.f	3	degree 3	$3q - 2q^{2} - 3q^{5} - 3q^{7} + O(q^{8})$	+ + +	$-1$
805.2.a.g	4	degree 4	$4q - 2q^{2} - 5q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	− − +	$-1$
805.2.a.h	4	degree 4	$4q - q^{2} + 11q^{4} - 4q^{5} - 4q^{7} + O(q^{8})$	+ + −	$+1$
805.2.a.i	4	degree 4	$4q + q^{2} + 4q^{3} + 3q^{4} - 4q^{5} + 4q^{7} + O(q^{8})$	+ − +	$+1$
805.2.a.j	4	degree 4	$4q + 3q^{2} + 6q^{3} + 3q^{4} + 4q^{5} + 4q^{6} - 4q^{7} + O(q^{8})$	− + +	$+1$
805.2.a.k	5	degree 5	$5q - 3q^{2} - 6q^{3} + 3q^{4} + 5q^{5} + 7q^{6} - 5q^{7} + O(q^{8})$	− + −	$-1$
805.2.a.l	5	degree 5	$5q - q^{2} - 4q^{3} + 3q^{4} - 5q^{5} - 5q^{6} + 5q^{7} + O(q^{8})$	+ − −	$-1$
805.2.a.m	8	degree 8	$8q + q^{2} + 7q^{3} + 7q^{4} + 8q^{5} + q^{6} + 8q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
805.2.a.a	$x - 19$
805.2.a.b	$x + 17$
805.2.a.c	$x + 50$
805.2.a.d	$x - 82$
805.2.a.e	$x^{2} - 123x + 3571$
805.2.a.f	$x^{3} - 43x^{2} - 1318x - 3037$
805.2.a.g	$x^{4} + 167x^{3} + 7636x^{2} + 58537x + 116108$
805.2.a.h	$x^{4} + 176x^{3} + 8721x^{2} + 30407x - 4577687$
805.2.a.i	$x^{4} - 164x^{3} + 9551x^{2} - 233075x + 2006801$
805.2.a.j	$x^{4} - 66x^{3} - 585x^{2} + 90845x - 1261861$
805.2.a.k	$x^{5} + 116x^{4} + 4308x^{3} + 44832x^{2} - 440629x - 7597009$
805.2.a.l	$x^{5} + 146x^{4} + 712x^{3} - 458398x^{2} - 7999969x + 222451805$
805.2.a.m	$x^{8} - 144x^{7} + 1441x^{6} + 486796x^{5} - 12381705x^{4} - 409882615x^{3} + 10803582913x^{2} + 8227739133x - 65142174112$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(805))$	43
Old forms	50
Cusp forms $S_{2}(\Gamma_0(805))$	93
Eisenstein series $E_{2}(\Gamma_0(805))$	7
Modular forms $M_{2}(\Gamma_0(805))$	100

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(805)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(115))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(161))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.402 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, 5, 7, 23, 35, 115, 161, 805]: [0, 1, 1, 3, 6, 14, 18, 100] (certified at ell = 2147483053)
T = [(2, 1), (3, 3), (11, 5), (13, 7), (17, 9)]; h of degree 43 = 1 x 43 (phi(ord eps) x dim S_k^new); 11 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(805, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 805:2 1000

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