Atlas › Modular forms › Weight 2 › Level 855 › 855.2.a

Newspace 855.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(855))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
855.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - q^{5} + 4q^{7} + O(q^{8})$	+ +	855a1 rank 0
855.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{4} + q^{5} - 2q^{7} + O(q^{8})$	− +	855b1 rank 1
855.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + q^{5} - 2q^{7} + O(q^{8})$	− −	855c1 rank 0
855.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 2q^{5} + O(q^{8})$	− +
855.2.a.e	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	− −
855.2.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	+ −
855.2.a.g	2	$\mathbb{Q}(\sqrt{7})$	$2q + 10q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	+ +
855.2.a.h	3	degree 3	$3q - q^{2} + q^{4} - 3q^{5} + O(q^{8})$	+ +
855.2.a.i	3	degree 3	$3q - q^{2} + q^{4} - 3q^{5} + O(q^{8})$	+ +
855.2.a.j	3	degree 3	$3q - q^{2} + q^{4} + 3q^{5} - 4q^{7} + O(q^{8})$	− −
855.2.a.k	3	degree 3	$3q + q^{2} + q^{4} - 3q^{5} - 4q^{7} + O(q^{8})$	+ −
855.2.a.l	3	degree 3	$3q + q^{2} + q^{4} + 3q^{5} + O(q^{8})$	− +
855.2.a.m	4	degree 4	$4q + 2q^{2} + 8q^{4} + 4q^{5} + 4q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{23} + 7T_{29} + 8T_{31}$ 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 $T$
855.2.a.a	$x - 31$
855.2.a.b	$x + 29$
855.2.a.c	$x - 65$
855.2.a.d	$x^{2} + 190x + 9007$
855.2.a.e	$x^{2} - 50x - 4177$
855.2.a.f	$x^{2} + 40x + 397$
855.2.a.g	$x^{2} - 28x - 3507$
855.2.a.h	$x^{3} + 219x^{2} + 9491x + 114335$
855.2.a.i	$x^{3} - 171x^{2} + 9629x - 178825$
855.2.a.j	$x^{3} + 107x^{2} - 9717x - 729389$
855.2.a.k	$x^{3} - 171x^{2} + 6539x + 58981$
855.2.a.l	$x^{3} - 107x^{2} + 1955x + 7225$
855.2.a.m	$x^{4} - 38x^{3} - 1706x^{2} + 91432x - 979675$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(855))$	30
Old forms	83
Cusp forms $S_{2}(\Gamma_0(855))$	113
Eisenstein series $E_{2}(\Gamma_0(855))$	15
Modular forms $M_{2}(\Gamma_0(855))$	128

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(855)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(45))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(57))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(95))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(171))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(285))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.739 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, 3, 5, 9, 15, 19, 45, 57, 95, 171, 285, 855]: [0, 1, 1, 2, 4, 2, 8, 8, 12, 22, 44, 124] (certified at ell = 2147483629)
T = [(2, 1), (7, 2), (11, 3), (13, 4), (17, 5), (23, 6), (29, 7), (31, 8)]; h of degree 30 = 1 x 30 (phi(ord eps) x dim S_k^new); 9 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(855, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 855:2 1000

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