Atlas › Modular forms › Weight 2 › Level 846 › 846.2.a

Newspace 846.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(846))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
846.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{4} - 2q^{5} + O(q^{8})$	+ −	846c1 rank 1
846.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + O(q^{8})$	+ −	846b1 rank 1
846.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{4} + 4q^{5} - 4q^{7} + O(q^{8})$	+ +	846a1 rank 0
846.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{4} - 4q^{5} - 4q^{7} + O(q^{8})$	− +
846.2.a.e	2	$\mathbb{Q}(\sqrt{6})$	$2q + 2q^{2} + 2q^{4} + 4q^{7} + O(q^{8})$	− −
846.2.a.f	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{2} + 2q^{4} + 2q^{5} + O(q^{8})$	− −
846.2.a.g	3	degree 3	$3q - 3q^{2} + 3q^{4} - 2q^{5} + O(q^{8})$	+ +
846.2.a.h	4	degree 4	$4q - 4q^{2} + 4q^{4} - 2q^{5} + 4q^{7} + O(q^{8})$	+ −
846.2.a.i	4	degree 4	$4q + 4q^{2} + 4q^{4} + 2q^{5} + 4q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
846.2.a.a	$x + 6$
846.2.a.b	$x + 24$
846.2.a.c	$x - 78$
846.2.a.d	$x^{2} + 108x + 2338$
846.2.a.e	$x^{2} - 68x - 578$
846.2.a.f	$x^{2} + 6x - 858$
846.2.a.g	$x^{3} + 24x^{2} - 5860x + 78836$
846.2.a.h	$x^{4} - 186x^{3} + 10956x^{2} - 209440x + 730110$
846.2.a.i	$x^{4} - 78x^{3} - 7284x^{2} + 140760x + 4308950$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(846))$	20
Old forms	117
Cusp forms $S_{2}(\Gamma_0(846))$	137
Eisenstein series $E_{2}(\Gamma_0(846))$	15
Modular forms $M_{2}(\Gamma_0(846))$	152

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(846)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(94))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(141))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(282))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(423))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.654 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, 9, 18, 47, 94, 141, 282, 423, 846]: [0, 1, 1, 3, 2, 5, 5, 14, 18, 52, 50, 148] (certified at ell = 2147483137)
T = [(5, 1), (7, 3), (13, 5), (17, 7), (19, 9)]; h of degree 20 = 1 x 20 (phi(ord eps) x dim S_k^new); 6 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(846, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 846:2 1000

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