Atlas › Modular forms › Weight 2 › Level 852 › 852.2.a

Newspace 852.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(852))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs
852.2.a.a	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} - q^{5} - 2q^{7} + O(q^{8})$	+ −
852.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} - 3q^{5} - 4q^{7} + O(q^{8})$	− +
852.2.a.c	4	degree 4	$4q - 4q^{3} - q^{5} + 4q^{7} + O(q^{8})$	+ +
852.2.a.d	4	degree 4	$4q + 4q^{3} + 5q^{5} + 2q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + T_{7}$ 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 $T$
852.2.a.a	$x^{2} + 3x + 1$
852.2.a.b	$x^{2} + 7x + 11$
852.2.a.c	$x^{4} - 3x^{3} - 37x^{2} + 70x - 30$
852.2.a.d	$x^{4} - 7x^{3} - 3x^{2} + 54x - 54$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(852))$	12
Old forms	127
Cusp forms $S_{2}(\Gamma_0(852))$	139
Eisenstein series $E_{2}(\Gamma_0(852))$	11
Modular forms $M_{2}(\Gamma_0(852))$	150

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(852)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(71))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(142))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(213))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(284))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(426))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.395 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, 4, 6, 12, 71, 142, 213, 284, 426, 852]: [0, 1, 1, 2, 3, 5, 7, 20, 26, 39, 76, 150] (certified at ell = 2147483171)
T = [(5, 1), (7, 1)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 2 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(852, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 852:2 1000

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