Atlas › Modular forms › Weight 2 › Level 213 › 213.2.a

Newspace 213.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(213))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
213.2.a.a	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} + 2q^{5} + q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$	213a1 rank 0
213.2.a.b	2	$\mathbb{Q}(\sqrt{5})$	$2q - 3q^{2} + 2q^{3} + 3q^{4} - 5q^{5} - 3q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
213.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} - 2q^{3} - q^{4} + q^{5} + q^{6} - 6q^{7} + O(q^{8})$	+ +	$-1$
213.2.a.d	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{2} + 2q^{3} + 3q^{4} - q^{5} + q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$
213.2.a.e	4	degree 4	$4q + 3q^{2} - 4q^{3} + 5q^{4} - 3q^{5} - 3q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
213.2.a.a	$x - 7$
213.2.a.b	$x^{2} + 18x + 76$
213.2.a.c	$x^{2} - 2x - 4$
213.2.a.d	$x^{2} + 2x - 12$
213.2.a.e	$x^{4} + 6x^{3} - 89x^{2} + 106x + 76$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(213))$	11
Old forms	12
Cusp forms $S_{2}(\Gamma_0(213))$	23
Eisenstein series $E_{2}(\Gamma_0(213))$	3
Modular forms $M_{2}(\Gamma_0(213))$	26

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(213)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(71))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.307 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, 71, 213]: [0, 1, 7, 26] (certified at ell = 2147483171)
T = [(2, 1), (5, 3)]; h of degree 11 = 1 x 11 (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(213, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 213:2 1000

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