Atlas › Modular forms › Weight 2 › Level 848 › 848.2.a

Newspace 848.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(848))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
848.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + q^{5} + 2q^{7} + O(q^{8})$	−	848g1 rank 1
848.2.a.b	1	$\mathbb{Q}$	$q - 2q^{3} + 2q^{5} + O(q^{8})$	+	848d1 rank 0
848.2.a.c	1	$\mathbb{Q}$	$q - q^{3} + 4q^{7} + O(q^{8})$	+	848b1 rank 0
848.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - 4q^{5} + O(q^{8})$	+	848a1 rank 0
848.2.a.e	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} + 2q^{7} + O(q^{8})$	−	848f1 rank 1
848.2.a.f	1	$\mathbb{Q}$	$q + 2q^{3} + 3q^{5} - 2q^{7} + O(q^{8})$	+	848c1 rank 0
848.2.a.g	1	$\mathbb{Q}$	$q + 3q^{3} + 4q^{7} + O(q^{8})$	+	848e1 rank 0
848.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} - 4q^{5} - 4q^{7} + O(q^{8})$	+
848.2.a.i	3	degree 3	$3q - 3q^{3} - 2q^{5} - 4q^{7} + O(q^{8})$	−
848.2.a.j	3	degree 3	$3q - 2q^{3} + 3q^{5} - 6q^{7} + O(q^{8})$	+
848.2.a.k	3	degree 3	$3q + q^{3} - 2q^{5} + 8q^{7} + O(q^{8})$	−
848.2.a.l	3	degree 3	$3q + 3q^{3} - 6q^{7} + O(q^{8})$	+
848.2.a.m	5	degree 5	$5q - q^{3} + 5q^{5} + 6q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
848.2.a.a	$x + 25$
848.2.a.b	$x - 28$
848.2.a.c	$x - 7$
848.2.a.d	$x - 61$
848.2.a.e	$x + 91$
848.2.a.f	$x - 25$
848.2.a.g	$x + 45$
848.2.a.h	$x^{2} + 86x + 791$
848.2.a.i	$x^{3} + 13x^{2} - 261x - 535$
848.2.a.j	$x^{3} + 87x^{2} - 1959x - 3961$
848.2.a.k	$x^{3} + 37x^{2} - 3527x - 74761$
848.2.a.l	$x^{3} - 105x^{2} - 1071x + 133133$
848.2.a.m	$x^{5} - 124x^{4} + 2642x^{3} + 65024x^{2} + 190707x - 641366$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(848))$	26
Old forms	77
Cusp forms $S_{2}(\Gamma_0(848))$	103
Eisenstein series $E_{2}(\Gamma_0(848))$	11
Modular forms $M_{2}(\Gamma_0(848))$	114

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(848)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(53))^{\oplus 5}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(106))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(212))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(424))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.522 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, 4, 8, 16, 53, 106, 212, 424, 848]: [0, 1, 2, 3, 4, 5, 15, 30, 58, 112] (certified at ell = 2147483053)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 26 = 1 x 26 (phi(ord eps) x dim S_k^new); 7 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(848, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 848:2 1000

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