Atlas › Modular forms › Weight 2 › Level 988 › 988.2.a

Newspace 988.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(988))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
988.2.a.a	1	$\mathbb{Q}$	$q - 2q^{3} + 2q^{7} + O(q^{8})$	− −	988d1 rank 0
988.2.a.b	1	$\mathbb{Q}$	$q - 3q^{5} + 3q^{7} + O(q^{8})$	+ −	988c1 rank 1
988.2.a.c	1	$\mathbb{Q}$	$q + 2q^{5} - 2q^{7} + O(q^{8})$	+ −	988b1 rank 1
988.2.a.d	1	$\mathbb{Q}$	$q + 2q^{3} + 4q^{5} + 2q^{7} + O(q^{8})$	+ +	988a1 rank 0
988.2.a.e	3	degree 3	$3q - 3q^{3} - 3q^{5} - 3q^{7} + O(q^{8})$	+ −
988.2.a.f	3	degree 3	$3q - q^{3} - q^{5} + 7q^{7} + O(q^{8})$	+ +
988.2.a.g	4	degree 4	$4q - 5q^{3} - 3q^{5} - q^{7} + O(q^{8})$	− +
988.2.a.h	4	degree 4	$4q + 5q^{3} + 2q^{5} - 2q^{7} + O(q^{8})$	− −

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 2T_{5} + 3T_{7} + 4T_{11} + 5T_{17} + 6T_{23} + 7T_{29} + 8T_{31}$ 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 $T$
988.2.a.a	$x - 41$
988.2.a.b	$x + 60$
988.2.a.c	$x + 85$
988.2.a.d	$x + 83$
988.2.a.e	$x^{3} + 129x^{2} + 4410x + 15929$
988.2.a.f	$x^{3} - 105x^{2} + 3504x - 36377$
988.2.a.g	$x^{4} + 152x^{3} - 1101x^{2} - 542715x - 11067359$
988.2.a.h	$x^{4} - 139x^{3} + 5102x^{2} - 36513x - 481204$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(988))$	18
Old forms	117
Cusp forms $S_{2}(\Gamma_0(988))$	135
Eisenstein series $E_{2}(\Gamma_0(988))$	11
Modular forms $M_{2}(\Gamma_0(988))$	146

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(988)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 6}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(26))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(38))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(52))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(76))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(247))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(494))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.659 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, 4, 13, 19, 26, 38, 52, 76, 247, 494, 988]: [0, 1, 2, 1, 2, 5, 7, 10, 13, 24, 74, 146] (certified at ell = 2147483629)
T = [(3, 1), (5, 2), (7, 3), (11, 4), (17, 5), (23, 6), (29, 7), (31, 8)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 5 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(988, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 988:2 1000

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