Atlas › Modular forms › Weight 2 › Level 672 › 672.2.a

Newspace 672.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(672))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
672.2.a.a	1	$\mathbb{Q}$	$q - q^{3} - 4q^{5} - q^{7} + O(q^{8})$	+ +	672d1 rank 0
672.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{5} + q^{7} + O(q^{8})$	+ −	672e1 rank 1
672.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - q^{7} + O(q^{8})$	+ +	672a1 rank 1
672.2.a.d	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} - q^{7} + O(q^{8})$	+ +	672c1 rank 0
672.2.a.e	1	$\mathbb{Q}$	$q + q^{3} - 4q^{5} + q^{7} + O(q^{8})$	− −	672b1 rank 1
672.2.a.f	1	$\mathbb{Q}$	$q + q^{3} - 2q^{5} - q^{7} + O(q^{8})$	− +	672f1 rank 1
672.2.a.g	1	$\mathbb{Q}$	$q + q^{3} + q^{7} + O(q^{8})$	− −	672g1 rank 0
672.2.a.h	1	$\mathbb{Q}$	$q + q^{3} + 2q^{5} + q^{7} + O(q^{8})$	− −	672h1 rank 0
672.2.a.i	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{7} + O(q^{8})$	+ −
672.2.a.j	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{3} - 2q^{7} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{5} + 2T_{11} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{23} + 7T_{29} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47} + 13T_{53} + 14T_{59} + 15T_{61} + 16T_{67}$ 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 $T$
672.2.a.a	$x - 94$
672.2.a.b	$x - 24$
672.2.a.c	$x + 398$
672.2.a.d	$x - 572$
672.2.a.e	$x + 74$
672.2.a.f	$x + 288$
672.2.a.g	$x + 134$
672.2.a.h	$x + 12$
672.2.a.i	$x^{2} - 292x - 872$
672.2.a.j	$x^{2} + 124x - 7688$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(672))$	12
Old forms	101
Cusp forms $S_{2}(\Gamma_0(672))$	113
Eisenstein series $E_{2}(\Gamma_0(672))$	31
Modular forms $M_{2}(\Gamma_0(672))$	144

Decomposition of the old subspace

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.544 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, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, 336, 672]: [0, 1, 1, 2, 3, 1, 3, 5, 4, 4, 4, 8, 7, 6, 12, 12, 12, 22, 20, 20, 40, 36, 72, 136] (certified at ell = 2147483497)
T = [(5, 1), (11, 2), (13, 3), (17, 4), (19, 5), (23, 6), (29, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12), (53, 13), (59, 14), (61, 15), (67, 16)]; h of degree 12 = 1 x 12 (phi(ord eps) x dim S_k^new); 5 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(672, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 672:2 1000

← 671.2.a · 673.2.a →