Atlas › Modular forms › Weight 2 › Level 842 › 842.2.a

Newspace 842.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(842))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
842.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 2q^{5} + 2q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$	842a1 rank 1
842.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{5} - 2q^{6} + q^{7} + O(q^{8})$	− −	$-1$	842b1 rank 1
842.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
842.2.a.d	4	degree 4	$4q - 4q^{2} - 3q^{3} + 4q^{4} + 4q^{5} + 3q^{6} + q^{7} + O(q^{8})$	+ +	$-1$
842.2.a.e	4	degree 4	$4q + 4q^{2} - 3q^{3} + 4q^{4} - 4q^{5} - 3q^{6} - 9q^{7} + O(q^{8})$	− −	$-1$
842.2.a.f	11	degree 11	$11q - 11q^{2} + 8q^{3} + 11q^{4} - 2q^{5} - 8q^{6} + 8q^{7} + O(q^{8})$	+ −	$+1$
842.2.a.g	13	degree 13	$13q + 13q^{2} + 6q^{3} + 13q^{4} + 7q^{5} + 6q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$

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 $T$
842.2.a.a	$x$
842.2.a.b	$x + 4$
842.2.a.c	$x^{2} + 5x - 5$
842.2.a.d	$x^{4} - x^{3} - 8x^{2} + 14x - 5$
842.2.a.e	$x^{4} + 7x^{3} + 14x^{2} + 4x - 7$
842.2.a.f	$x^{11} - 6x^{10} - 54x^{9} + 323x^{8} + 967x^{7} - 5750x^{6} - 6451x^{5} + 38441x^{4} + 17498x^{3} - 86699x^{2} - 18485x + 3056$
842.2.a.g	$x^{13} - 13x^{12} + 9x^{11} + 549x^{10} - 2400x^{9} - 1888x^{8} + 30520x^{7} - 46672x^{6} - 57286x^{5} + 165356x^{4} + 14500x^{3} - 201020x^{2} + 20859x + 89892$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(842))$	36
Old forms	68
Cusp forms $S_{2}(\Gamma_0(842))$	104
Eisenstein series $E_{2}(\Gamma_0(842))$	3
Modular forms $M_{2}(\Gamma_0(842))$	107

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(842)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(421))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.26 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, 421, 842]: [0, 1, 35, 107] (certified at ell = 2147482681)
T = [(3, 1), (5, 1)]; h of degree 36 = 1 x 36 (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(842, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 842:2 1000

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