Atlas › Modular forms › Weight 2 › Level 985 › 985.2.a

Newspace 985.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(985))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
985.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 2q^{3} + 2q^{4} + q^{5} + 4q^{6} + q^{7} + O(q^{8})$	− −	$-1$	985b1 rank 1
985.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + q^{5} + 3q^{7} + O(q^{8})$	− +	$+1$	985a1 rank 0
985.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{4} + 2q^{5} + 6q^{7} + O(q^{8})$	− +	$+1$
985.2.a.d	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} + 5q^{4} - 2q^{5} - 3q^{7} + O(q^{8})$	+ +	$-1$
985.2.a.e	10	degree 10	$10q - 2q^{2} - 3q^{3} + 2q^{4} + 10q^{5} - 9q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
985.2.a.f	14	degree 14	$14q - 6q^{2} - 7q^{3} + 10q^{4} - 14q^{5} + 5q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
985.2.a.g	17	degree 17	$17q + 6q^{2} + 5q^{3} + 18q^{4} - 17q^{5} + 3q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$
985.2.a.h	18	degree 18	$18q + 7q^{2} + 3q^{3} + 33q^{4} + 18q^{5} + q^{6} - 8q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{3}$ 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 $T$
985.2.a.a	$x + 4$
985.2.a.b	$x - 1$
985.2.a.c	$x^{2} + 2x - 7$
985.2.a.d	$x^{2} + x - 4$
985.2.a.e	$x^{10} + 5x^{9} - 2x^{8} - 41x^{7} - 35x^{6} + 87x^{5} + 122x^{4} - 12x^{3} - 62x^{2} - 17x + 1$
985.2.a.f	$x^{14} + 13x^{13} + 33x^{12} - 200x^{11} - 918x^{10} + 809x^{9} + 7379x^{8} + 712x^{7} - 25165x^{6} - 8872x^{5} + 36013x^{4} + 11429x^{3} - 20147x^{2} - 4035x + 3551$
985.2.a.g	$x^{17} - 11x^{16} - 5x^{15} + 404x^{14} - 710x^{13} - 5321x^{12} + 13196x^{11} + 31561x^{10} - 85179x^{9} - 95919x^{8} + 220428x^{7} + 166922x^{6} - 199619x^{5} - 105905x^{4} + 62145x^{3} + 20165x^{2} - 5393x + 104$
985.2.a.h	$x^{18} - 10x^{17} - 23x^{16} + 491x^{15} - 516x^{14} - 7848x^{13} + 15664x^{12} + 54572x^{11} - 122745x^{10} - 207194x^{9} + 394264x^{8} + 479275x^{7} - 479907x^{6} - 564346x^{5} + 118728x^{4} + 201417x^{3} + 25036x^{2} - 6035x + 8$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(985))$	65
Old forms	32
Cusp forms $S_{2}(\Gamma_0(985))$	97
Eisenstein series $E_{2}(\Gamma_0(985))$	3
Modular forms $M_{2}(\Gamma_0(985))$	100

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(985)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(197))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.287 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, 5, 197, 985]: [0, 1, 17, 100] (certified at ell = 2147479489)
T = [(2, 1), (3, 1)]; h of degree 65 = 1 x 65 (phi(ord eps) x dim S_k^new); 7 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(985, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 985:2 1000

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