Atlas › Modular forms › Weight 2 › Level 921 › 921.2.a

Newspace 921.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(921))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
921.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} + O(q^{8})$	+ −	$+1$	921a1 rank 0
921.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{7} + O(q^{8})$	− −	$-1$	921b1 rank 1
921.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} + 4q^{7} + O(q^{8})$	− +	$+1$
921.2.a.d	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$
921.2.a.e	7	degree 7	$7q - 2q^{2} + 7q^{3} + 2q^{4} - 7q^{5} - 2q^{6} - 13q^{7} + O(q^{8})$	− −	$-1$
921.2.a.f	11	degree 11	$11q + 4q^{2} - 11q^{3} + 10q^{4} + 7q^{5} - 4q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$
921.2.a.g	13	degree 13	$13q - 2q^{2} - 13q^{3} + 12q^{4} - 9q^{5} + 2q^{6} - 8q^{7} + O(q^{8})$	+ +	$-1$
921.2.a.h	14	degree 14	$14q - 3q^{2} + 14q^{3} + 21q^{4} + 3q^{5} - 3q^{6} + 6q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
921.2.a.a	$x + 2$
921.2.a.b	$x$
921.2.a.c	$x^{2} - 18$
921.2.a.d	$x^{2} - 2x - 1$
921.2.a.e	$x^{7} + 9x^{6} + 22x^{5} - x^{4} - 65x^{3} - 80x^{2} - 37x - 6$
921.2.a.f	$x^{11} - 11x^{10} + 11x^{9} + 226x^{8} - 666x^{7} - 898x^{6} + 4972x^{5} - 1329x^{4} - 11302x^{3} + 8166x^{2} + 7187x - 6008$
921.2.a.g	$x^{13} + 11x^{12} + 7x^{11} - 252x^{10} - 569x^{9} + 1721x^{8} + 5530x^{7} - 2698x^{6} - 17795x^{5} - 5136x^{4} + 18358x^{3} + 8584x^{2} - 6160x - 2560$
921.2.a.h	$x^{14} - 74x^{12} + 55x^{11} + 1960x^{10} - 2624x^{9} - 21838x^{8} + 39381x^{7} + 93119x^{6} - 214438x^{5} - 87331x^{4} + 398971x^{3} - 183780x^{2} - 59912x + 37952$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(921))$	51
Old forms	50
Cusp forms $S_{2}(\Gamma_0(921))$	101
Eisenstein series $E_{2}(\Gamma_0(921))$	3
Modular forms $M_{2}(\Gamma_0(921))$	104

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(921)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(307))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.275 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, 307, 921]: [0, 1, 26, 104] (certified at ell = 2147478013)
T = [(2, 1), (5, 1)]; h of degree 51 = 1 x 51 (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(921, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 921:2 1000

← 920.2.a · 922.2.a →