Atlas › Modular forms › Weight 2 › Level 973 › 973.2.a

Newspace 973.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(973))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
973.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + q^{7} + O(q^{8})$	− −	$-1$	973b1 rank 1
973.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + 4q^{5} + 2q^{6} + q^{7} + O(q^{8})$	− +	$+1$	973a1 rank 0
973.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} + 4q^{6} + 2q^{7} + O(q^{8})$	− +	$+1$
973.2.a.d	3	degree 3	$3q + q^{2} + 2q^{3} + q^{4} - 2q^{5} - 4q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$
973.2.a.e	4	degree 4	$4q + q^{2} + 9q^{4} - 4q^{5} - 6q^{6} - 4q^{7} + O(q^{8})$	+ −	$+1$
973.2.a.f	12	degree 12	$12q + 4q^{2} + 6q^{3} + 14q^{4} + 11q^{5} + 7q^{6} + 12q^{7} + O(q^{8})$	− +	$+1$
973.2.a.g	15	degree 15	$15q - 7q^{2} - 12q^{3} + 17q^{4} - 19q^{5} + q^{6} + 15q^{7} + O(q^{8})$	− −	$-1$
973.2.a.h	15	degree 15	$15q + 7q^{2} + 7q^{3} + 15q^{4} - q^{5} + 9q^{6} - 15q^{7} + O(q^{8})$	+ −	$+1$
973.2.a.i	16	degree 16	$16q - 9q^{2} - 7q^{3} + 15q^{4} + 5q^{5} - q^{6} - 16q^{7} + O(q^{8})$	+ +	$-1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
973.2.a.a	$x - 3$
973.2.a.b	$x - 5$
973.2.a.c	$x^{2} - 6x - 23$
973.2.a.d	$x^{3} - 7x^{2} - 21x + 25$
973.2.a.e	$x^{4} - x^{3} - 134x^{2} + 112x + 3999$
973.2.a.f	$x^{12} - 22x^{11} - 32x^{10} + 3601x^{9} - 16074x^{8} - 164292x^{7} + 1323628x^{6} + 388594x^{5} - 26755995x^{4} + 77385915x^{3} - 64554803x^{2} - 12324201x + 25111279$
973.2.a.g	$x^{15} + 43x^{14} + 619x^{13} + 1486x^{12} - 46951x^{11} - 416305x^{10} + 63935x^{9} + 14843174x^{8} + 52348747x^{7} - 101926834x^{6} - 916268921x^{5} - 1402130768x^{4} + 886192332x^{3} + 2770627633x^{2} + 420505622x - 334131157$
973.2.a.h	$x^{15} - 28x^{14} + 113x^{13} + 2917x^{12} - 21138x^{11} - 127116x^{10} + 1091983x^{9} + 3268088x^{8} - 25396112x^{7} - 56685489x^{6} + 251890354x^{5} + 535177186x^{4} - 540002487x^{3} - 736784243x^{2} + 442946622x + 25418565$
973.2.a.i	$x^{16} + 30x^{15} + 164x^{14} - 2971x^{13} - 31409x^{12} + 74952x^{11} + 1616124x^{10} + 153607x^{9} - 39051503x^{8} - 24734705x^{7} + 492853475x^{6} + 212562575x^{5} - 3003374010x^{4} + 27359165x^{3} + 5910988649x^{2} - 816857081x - 74824127$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(973))$	69
Old forms	22
Cusp forms $S_{2}(\Gamma_0(973))$	91
Eisenstein series $E_{2}(\Gamma_0(973))$	3
Modular forms $M_{2}(\Gamma_0(973))$	94

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(973)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(139))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.39 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, 7, 139, 973]: [0, 1, 12, 94] (certified at ell = 2147483137)
T = [(2, 1), (3, 3)]; h of degree 69 = 1 x 69 (phi(ord eps) x dim S_k^new); 9 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(973, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 973:2 1000

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