Atlas › Modular forms › Weight 2 › Level 986 › 986.2.a

Newspace 986.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(986))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
986.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 2q^{6} + 5q^{7} + O(q^{8})$	+ − +	$+1$	986a1 rank 0
986.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - 2q^{7} + O(q^{8})$	+ − −	$-1$	986b1 rank 1
986.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ − −	$-1$	986c1 rank 1
986.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} + q^{7} + O(q^{8})$	− + −	$-1$	986e1 rank 1
986.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 2q^{5} - q^{7} + O(q^{8})$	− − +	$-1$	986f1 rank 1
986.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} + q^{4} - 3q^{5} + q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$	986d1 rank 1
986.2.a.g	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	+ − −	$-1$
986.2.a.h	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{4} - 2q^{7} + O(q^{8})$	+ + +	$-1$
986.2.a.i	6	degree 6	$6q - 6q^{2} + 5q^{3} + 6q^{4} + q^{5} - 5q^{6} - 3q^{7} + O(q^{8})$	+ − +	$+1$
986.2.a.j	7	degree 7	$7q - 7q^{2} + 4q^{3} + 7q^{4} - 4q^{6} + 2q^{7} + O(q^{8})$	+ + −	$+1$
986.2.a.k	7	degree 7	$7q + 7q^{2} + 3q^{3} + 7q^{4} + 5q^{5} + 3q^{6} + 9q^{7} + O(q^{8})$	− + +	$+1$
986.2.a.l	9	degree 9	$9q + 9q^{2} + 9q^{4} + q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{13} + 6T_{19} + 7T_{23} + 8T_{31} + 9T_{37} + 10T_{41} + 11T_{43} + 12T_{47}$ 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 $T$
986.2.a.a	$x - 34$
986.2.a.b	$x + 147$
986.2.a.c	$x + 414$
986.2.a.d	$x + 254$
986.2.a.e	$x + 174$
986.2.a.f	$x + 47$
986.2.a.g	$x^{2} + 12x - 3432$
986.2.a.h	$x^{2} + 64x + 824$
986.2.a.i	$x^{6} - 251x^{5} - 57363x^{4} + 13848693x^{3} + 862997382x^{2} - 153871246467x - 7388325992758$
986.2.a.j	$x^{7} - 526x^{6} + 69374x^{5} + 2636474x^{4} - 1082542997x^{3} + 62502410473x^{2} - 369536091445x - 35288609103026$
986.2.a.k	$x^{7} - 567x^{6} + 27591x^{5} + 24622543x^{4} - 2598123538x^{3} - 203065643385x^{2} + 34988269315552x - 1141621776419072$
986.2.a.l	$x^{9} - 34x^{8} - 69240x^{7} - 1919742x^{6} + 1090632009x^{5} + 36944551571x^{4} - 5614845303285x^{3} - 150108531924244x^{2} + 5816030905801728x + 910095759341568$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(986))$	39
Old forms	92
Cusp forms $S_{2}(\Gamma_0(986))$	131
Eisenstein series $E_{2}(\Gamma_0(986))$	7
Modular forms $M_{2}(\Gamma_0(986))$	138

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(986)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(17))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(34))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(58))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(493))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 1.466 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, 17, 29, 34, 58, 493, 986]: [0, 1, 2, 3, 6, 9, 46, 138] (certified at ell = 2147480161)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (13, 5), (19, 6), (23, 7), (31, 8), (37, 9), (41, 10), (43, 11), (47, 12)]; h of degree 39 = 1 x 39 (phi(ord eps) x dim S_k^new); 13 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(986, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 986:2 1000

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