Atlas › Modular forms › Weight 2 › Level 987 › 987.2.a

Newspace 987.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(987))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
987.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	987b1 rank 0
987.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} - q^{7} + O(q^{8})$	− + −	$-1$	987e1 rank 1
987.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + 4q^{5} - q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	987a1 rank 0
987.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} - 2q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	987c1 rank 0
987.2.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + 4q^{5} + 2q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	987d1 rank 0
987.2.a.f	2	$\mathbb{Q}(\sqrt{5})$	$2q - 2q^{3} + 6q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	+ + +	$-1$
987.2.a.g	3	degree 3	$3q - q^{2} - 3q^{3} - q^{4} - q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+ − −	$-1$
987.2.a.h	3	degree 3	$3q - q^{2} + 3q^{3} + 3q^{4} - q^{5} - q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$
987.2.a.i	3	degree 3	$3q + q^{2} - 3q^{3} - q^{4} - 3q^{5} - q^{6} - 3q^{7} + O(q^{8})$	+ + +	$-1$
987.2.a.j	4	degree 4	$4q - 2q^{2} + 4q^{3} + 2q^{4} - 7q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	− − +	$-1$
987.2.a.k	4	degree 4	$4q - q^{2} - 4q^{3} + 9q^{4} + q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+ + −	$+1$
987.2.a.l	5	degree 5	$5q + 5q^{3} + 4q^{4} + 5q^{5} - 5q^{7} + O(q^{8})$	− + +	$+1$
987.2.a.m	9	degree 9	$9q + 3q^{2} - 9q^{3} + 15q^{4} + 3q^{5} - 3q^{6} + 9q^{7} + O(q^{8})$	+ − +	$+1$
987.2.a.n	9	degree 9	$9q + 3q^{2} + 9q^{3} + 11q^{4} + 7q^{5} + 3q^{6} + 9q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
987.2.a.a	$x - 37$
987.2.a.b	$x - 65$
987.2.a.c	$x - 77$
987.2.a.d	$x - 86$
987.2.a.e	$x + 22$
987.2.a.f	$x^{2} + 78x + 1501$
987.2.a.g	$x^{3} + 6x^{2} - 387x + 2241$
987.2.a.h	$x^{3} + 138x^{2} + 6257x + 93475$
987.2.a.i	$x^{3} + 70x^{2} - 1519x - 22589$
987.2.a.j	$x^{4} + 233x^{3} + 16823x^{2} + 442492x + 2937307$
987.2.a.k	$x^{4} + 20x^{3} - 4375x^{2} + 25929x + 2076046$
987.2.a.l	$x^{5} - 193x^{4} + 13501x^{3} - 431036x^{2} + 6265909x - 33117538$
987.2.a.m	$x^{9} - 110x^{8} - 3627x^{7} + 335193x^{6} + 6147237x^{5} - 251565038x^{4} - 4149721973x^{3} + 45179805305x^{2} + 966643318818x + 3689014690370$
987.2.a.n	$x^{9} - 210x^{8} + 9765x^{7} + 767401x^{6} - 90004227x^{5} + 3009418922x^{4} - 30946939269x^{3} - 90411419091x^{2} + 781519717298x - 330431752142$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(987))$	47
Old forms	78
Cusp forms $S_{2}(\Gamma_0(987))$	125
Eisenstein series $E_{2}(\Gamma_0(987))$	7
Modular forms $M_{2}(\Gamma_0(987))$	132

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(987)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(141))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(329))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.658 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, 7, 21, 47, 141, 329, 987]: [0, 1, 1, 4, 5, 18, 34, 132] (certified at ell = 2147483137)
T = [(2, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 47 = 1 x 47 (phi(ord eps) x dim S_k^new); 12 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(987, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 987:2 1000

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