Atlas › Modular forms › Weight 2 › Level 606 › 606.2.a

Newspace 606.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(606))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
606.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} - 3q^{7} + O(q^{8})$	+ − −	$-1$	606b1 rank 1
606.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} + 2q^{5} - q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$	606a1 rank 0
606.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	606c1 rank 0
606.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} + 3q^{5} - q^{6} + 2q^{7} + O(q^{8})$	− + +	$+1$	606d1 rank 0
606.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 4q^{5} + q^{6} - 5q^{7} + O(q^{8})$	− − +	$-1$	606e1 rank 1
606.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} + O(q^{8})$	− − −	$+1$	606f1 rank 0
606.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+ + +	$-1$
606.2.a.h	2	$\mathbb{Q}(\sqrt{17})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + q^{5} + 2q^{6} - 5q^{7} + O(q^{8})$	+ + −	$+1$
606.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
606.2.a.j	2	$\mathbb{Q}(\sqrt{6})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 6q^{7} + O(q^{8})$	− − −	$+1$
606.2.a.k	3	degree 3	$3q - 3q^{2} + 3q^{3} + 3q^{4} - q^{5} - 3q^{6} + O(q^{8})$	+ − +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 3T_{7} + 5T_{11} + 7T_{13}$ 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 $T$
606.2.a.a	$x + 61$
606.2.a.b	$x - 20$
606.2.a.c	$x - 21$
606.2.a.d	$x + 9$
606.2.a.e	$x + 43$
606.2.a.f	$x - 33$
606.2.a.g	$x^{2} - 30x - 657$
606.2.a.h	$x^{2} + 8x - 1361$
606.2.a.i	$x^{2} + 78x + 1279$
606.2.a.j	$x^{2} + 2x - 725$
606.2.a.k	$x^{3} - 45x^{2} + 481x - 1369$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(606))$	17
Old forms	82
Cusp forms $S_{2}(\Gamma_0(606))$	99
Eisenstein series $E_{2}(\Gamma_0(606))$	7
Modular forms $M_{2}(\Gamma_0(606))$	106

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(606)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(101))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(202))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(303))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.416 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, 101, 202, 303, 606]: [0, 1, 1, 3, 9, 27, 36, 106] (certified at ell = 2147482801)
T = [(5, 1), (7, 3), (11, 5), (13, 7)]; h of degree 17 = 1 x 17 (phi(ord eps) x dim S_k^new); 4 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(606, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 606:2 1000

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