Atlas › Modular forms › Weight 2 › Level 669 › 669.2.a

Newspace 669.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(669))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
669.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + 3q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$	669a1 rank 1
669.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} - 6q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	+ +	$-1$
669.2.a.c	2	$\mathbb{Q}(\sqrt{5})$	$2q - q^{2} + 2q^{3} - q^{4} + 2q^{5} - q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
669.2.a.d	2	$\mathbb{Q}(\sqrt{13})$	$2q + q^{2} - 2q^{3} + 3q^{4} - 6q^{5} - q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
669.2.a.e	3	degree 3	$3q - 2q^{2} - 3q^{3} + 4q^{4} + 2q^{6} + q^{7} + O(q^{8})$	+ +	$-1$
669.2.a.f	3	degree 3	$3q - q^{2} - 3q^{3} + 3q^{4} + 2q^{5} + q^{6} - q^{7} + O(q^{8})$	+ +	$-1$
669.2.a.g	3	degree 3	$3q - q^{2} + 3q^{3} + q^{4} - 5q^{5} - q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
669.2.a.h	7	degree 7	$7q + 2q^{2} - 7q^{3} + 6q^{4} + 3q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ −	$+1$
669.2.a.i	14	degree 14	$14q + 14q^{3} + 16q^{4} + q^{5} + 14q^{7} + O(q^{8})$	− +	$+1$

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 $T$
669.2.a.a	$x - 10$
669.2.a.b	$x^{2} + 20x + 68$
669.2.a.c	$x^{2} - 5x + 5$
669.2.a.d	$x^{2} + 17x + 69$
669.2.a.e	$x^{3} + 2x^{2} - 3x - 3$
669.2.a.f	$x^{3} - 5x^{2} - 100x + 375$
669.2.a.g	$x^{3} + 16x^{2} + 72x + 80$
669.2.a.h	$x^{7} - 11x^{6} - 89x^{5} + 1044x^{4} + 480x^{3} - 12032x^{2} - 1792x + 12160$
669.2.a.i	$x^{14} - 3x^{13} - 457x^{12} + 2116x^{11} + 76096x^{10} - 457448x^{9} - 5394108x^{8} + 39345031x^{7} + 143795513x^{6} - 1304401083x^{5} - 997911510x^{4} + 14564377200x^{3} + 4672037680x^{2} - 52339106256x - 46238449376$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(669))$	37
Old forms	36
Cusp forms $S_{2}(\Gamma_0(669))$	73
Eisenstein series $E_{2}(\Gamma_0(669))$	3
Modular forms $M_{2}(\Gamma_0(669))$	76

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(669)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(223))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.335 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, 223, 669]: [0, 1, 19, 76] (certified at ell = 2147482591)
T = [(2, 1), (5, 3)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 6 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(669, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 669:2 1000

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