Atlas › Modular forms › Weight 2 › Level 994 › 994.2.a

Newspace 994.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(994))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
994.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} + q^{4} + 2q^{5} + 2q^{6} - q^{7} + O(q^{8})$	+ + −	$+1$	994b1 rank 0
994.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} + O(q^{8})$	+ + +	$-1$	994a1 rank 1
994.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - q^{6} + q^{7} + O(q^{8})$	+ − −	$-1$	994d1 rank 1
994.2.a.d	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - 2q^{5} - 2q^{6} + q^{7} + O(q^{8})$	+ − +	$+1$	994c1 rank 0
994.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} - 2q^{6} + q^{7} + O(q^{8})$	− − +	$-1$	994g1 rank 1
994.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} + q^{4} + 4q^{5} - 2q^{6} - q^{7} + O(q^{8})$	− + +	$+1$	994e1 rank 0
994.2.a.g	1	$\mathbb{Q}$	$q + q^{2} + q^{4} - 2q^{5} + q^{7} + O(q^{8})$	− − +	$-1$	994f1 rank 1
994.2.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ + −	$+1$
994.2.a.i	3	degree 3	$3q - 3q^{2} - 2q^{3} + 3q^{4} + 4q^{5} + 2q^{6} + 3q^{7} + O(q^{8})$	+ − +	$+1$
994.2.a.j	3	degree 3	$3q - 3q^{2} - q^{3} + 3q^{4} - 2q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+ − −	$-1$
994.2.a.k	3	degree 3	$3q + 3q^{2} - 2q^{3} + 3q^{4} - 4q^{5} - 2q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$
994.2.a.l	3	degree 3	$3q + 3q^{2} + 3q^{4} - 3q^{7} + O(q^{8})$	− + +	$+1$
994.2.a.m	3	degree 3	$3q + 3q^{2} + 6q^{3} + 3q^{4} + 2q^{5} + 6q^{6} - 3q^{7} + O(q^{8})$	− + +	$+1$
994.2.a.n	4	degree 4	$4q - 4q^{2} - q^{3} + 4q^{4} - 4q^{5} + q^{6} - 4q^{7} + O(q^{8})$	+ + +	$-1$
994.2.a.o	7	degree 7	$7q + 7q^{2} + 7q^{4} + 2q^{5} + 7q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5} + 5T_{11} + 7T_{13} + 9T_{17}$ 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 $T$
994.2.a.a	$x + 12$
994.2.a.b	$x + 7$
994.2.a.c	$x + 75$
994.2.a.d	$x - 16$
994.2.a.e	$x + 72$
994.2.a.f	$x - 44$
994.2.a.g	$x + 14$
994.2.a.h	$x^{2} - 42x + 366$
994.2.a.i	$x^{3} - 16x^{2} - 1962x + 26406$
994.2.a.j	$x^{3} + 29x^{2} - 2232x + 18286$
994.2.a.k	$x^{3} + 116x^{2} + 3880x + 38278$
994.2.a.l	$x^{3} - 54x^{2} - 2727x + 146784$
994.2.a.m	$x^{3} - 172x^{2} + 4552x + 111232$
994.2.a.n	$x^{4} + 79x^{3} - 3902x^{2} - 222084x + 6962314$
994.2.a.o	$x^{7} - 78x^{6} - 2219x^{5} + 219430x^{4} + 122664x^{3} - 92914482x^{2} + 334745408x + 972135392$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(994))$	35
Old forms	106
Cusp forms $S_{2}(\Gamma_0(994))$	141
Eisenstein series $E_{2}(\Gamma_0(994))$	7
Modular forms $M_{2}(\Gamma_0(994))$	148

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(994)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(14))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(71))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(142))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(497))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.571 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, 7, 14, 71, 142, 497, 994]: [0, 1, 1, 4, 7, 20, 50, 148] (certified at ell = 2147482681)
T = [(3, 1), (5, 3), (11, 5), (13, 7), (17, 9)]; h of degree 35 = 1 x 35 (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(994, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 994:2 1000

← 993.2.a · 995.2.a →