Atlas › Modular forms › Weight 2 › Level 995 › 995.2.a

Newspace 995.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(995))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
995.2.a.a	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + q^{5} + 2q^{7} + O(q^{8})$	− −	$-1$	995b1 rank 1
995.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} - q^{4} + q^{5} - 2q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$	995a1 rank 0
995.2.a.c	3	degree 3	$3q - 6q^{2} - q^{3} + 6q^{4} + 3q^{5} + 2q^{6} + O(q^{8})$	− −	$-1$
995.2.a.d	9	degree 9	$9q - q^{2} - 5q^{3} + 7q^{4} + 9q^{5} - 3q^{6} - 15q^{7} + O(q^{8})$	− −	$-1$
995.2.a.e	13	degree 13	$13q + q^{2} - q^{3} + 7q^{4} - 13q^{5} - 11q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
995.2.a.f	19	degree 19	$19q + 5q^{2} + 7q^{3} + 25q^{4} + 19q^{5} + 3q^{6} + 7q^{7} + O(q^{8})$	− +	$+1$
995.2.a.g	21	degree 21	$21q - 3q^{2} + q^{3} + 27q^{4} - 21q^{5} + 3q^{6} + 11q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
995.2.a.a	$x - 1$
995.2.a.b	$x + 1$
995.2.a.c	$x^{3} + 7x^{2} + 13x + 5$
995.2.a.d	$x^{9} + 6x^{8} - 10x^{7} - 73x^{6} + 28x^{5} + 286x^{4} - 21x^{3} - 400x^{2} + 3x + 153$
995.2.a.e	$x^{13} - 28x^{11} + 13x^{10} + 281x^{9} - 268x^{8} - 1127x^{7} + 1711x^{6} + 1053x^{5} - 3225x^{4} + 1605x^{3} + 364x^{2} - 475x + 97$
995.2.a.f	$x^{19} - 12x^{18} + x^{17} + 503x^{16} - 1520x^{15} - 6305x^{14} + 33353x^{13} + 14028x^{12} - 279357x^{11} + 238371x^{10} + 967936x^{9} - 1613216x^{8} - 867930x^{7} + 3045679x^{6} - 1125219x^{5} - 1000302x^{4} + 756078x^{3} - 129429x^{2} - 4959x + 1611$
995.2.a.g	$x^{21} + 2x^{20} - 82x^{19} - 149x^{18} + 2605x^{17} + 4484x^{16} - 40805x^{15} - 71145x^{14} + 330596x^{13} + 637823x^{12} - 1301177x^{11} - 3066945x^{10} + 1696858x^{9} + 6688693x^{8} + 1630233x^{7} - 3951835x^{6} - 1606073x^{5} + 841455x^{4} + 365927x^{3} - 57630x^{2} - 19971x + 1247$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(995))$	67
Old forms	32
Cusp forms $S_{2}(\Gamma_0(995))$	99
Eisenstein series $E_{2}(\Gamma_0(995))$	3
Modular forms $M_{2}(\Gamma_0(995))$	102

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(995)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(199))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.376 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, 199, 995]: [0, 1, 17, 102] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 67 = 1 x 67 (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(995, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 995:2 1000

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