Atlas › Modular forms › Weight 2 › Level 642 › 642.2.a

Newspace 642.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(642))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
642.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + 2q^{5} + q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$	642a1 rank 0
642.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 3q^{5} - q^{6} + 2q^{7} + O(q^{8})$	+ − +	$+1$	642b1 rank 0
642.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$	642c1 rank 1
642.2.a.d	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - q^{5} - 2q^{6} - 7q^{7} + O(q^{8})$	+ − −	$-1$
642.2.a.e	2	$\mathbb{Q}(\sqrt{10})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 4q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$
642.2.a.f	3	degree 3	$3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$
642.2.a.g	4	degree 4	$4q + 4q^{2} - 4q^{3} + 4q^{4} + 5q^{5} - 4q^{6} + 3q^{7} + O(q^{8})$	− + +	$+1$
642.2.a.h	5	degree 5	$5q + 5q^{2} + 5q^{3} + 5q^{4} + 5q^{6} + 5q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7}$ 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 $T$
642.2.a.a	$x - 8$
642.2.a.b	$x - 3$
642.2.a.c	$x + 7$
642.2.a.d	$x^{2} + 22x + 101$
642.2.a.e	$x^{2} - 16x - 26$
642.2.a.f	$x^{3} + 9x^{2} - 13x - 5$
642.2.a.g	$x^{4} - 14x^{3} - 89x^{2} + 286x + 796$
642.2.a.h	$x^{5} - 15x^{4} - 23x^{3} + 741x^{2} - 414x - 6306$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(642))$	19
Old forms	86
Cusp forms $S_{2}(\Gamma_0(642))$	105
Eisenstein series $E_{2}(\Gamma_0(642))$	7
Modular forms $M_{2}(\Gamma_0(642))$	112

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(642)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(107))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(214))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(321))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.473 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, 107, 214, 321, 642]: [0, 1, 1, 3, 10, 29, 38, 112] (certified at ell = 2147482621)
T = [(5, 1), (7, 3)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 3 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(642, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 642:2 1000

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