Atlas › Modular forms › Weight 2 › Level 999 › 999.2.a

Newspace 999.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(999))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
999.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{4} - q^{5} - q^{7} + O(q^{8})$	+	999b1 rank 1
999.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - q^{4} + q^{5} - q^{7} + O(q^{8})$	+	999a1 rank 1
999.2.a.c	2	$\mathbb{Q}(\sqrt{3})$	$2q + 2q^{4} - 2q^{7} + O(q^{8})$	−
999.2.a.d	2	$\mathbb{Q}(\sqrt{7})$	$2q + 10q^{4} - 2q^{7} + O(q^{8})$	+
999.2.a.e	6	degree 6	$6q - 3q^{2} + 5q^{4} - 4q^{5} - q^{7} + O(q^{8})$	+
999.2.a.f	6	degree 6	$6q - 3q^{2} + 5q^{4} - 4q^{5} - q^{7} + O(q^{8})$	−
999.2.a.g	6	degree 6	$6q + 3q^{2} + 5q^{4} + 4q^{5} - q^{7} + O(q^{8})$	+
999.2.a.h	6	degree 6	$6q + 3q^{2} + 5q^{4} + 4q^{5} - q^{7} + O(q^{8})$	−
999.2.a.i	8	degree 8	$8q + 8q^{4} + 8q^{7} + O(q^{8})$	+
999.2.a.j	10	degree 10	$10q + 14q^{4} + 6q^{7} + O(q^{8})$	−

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
999.2.a.a	$x + 4$
999.2.a.b	$x - 4$
999.2.a.c	$x^{2} - 12$
999.2.a.d	$x^{2} - 112$
999.2.a.e	$x^{6} + 15x^{5} - 70x^{4} - 1859x^{3} - 6193x^{2} + 6962x + 23467$
999.2.a.f	$x^{6} + 15x^{5} - 22x^{4} - 569x^{3} - 501x^{2} + 2202x + 1497$
999.2.a.g	$x^{6} - 15x^{5} - 70x^{4} + 1859x^{3} - 6193x^{2} - 6962x + 23467$
999.2.a.h	$x^{6} - 15x^{5} - 22x^{4} + 569x^{3} - 501x^{2} - 2202x + 1497$
999.2.a.i	$x^{8} - 384x^{6} + 48352x^{4} - 2010400x^{2} + 3430000$
999.2.a.j	$x^{10} - 332x^{8} + 38884x^{6} - 1835968x^{4} + 27920976x^{2} - 110316288$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(999))$	48
Old forms	61
Cusp forms $S_{2}(\Gamma_0(999))$	109
Eisenstein series $E_{2}(\Gamma_0(999))$	11
Modular forms $M_{2}(\Gamma_0(999))$	120

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(999)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(27))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(111))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(333))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.522 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, 9, 27, 37, 111, 333, 999]: [0, 1, 2, 4, 3, 14, 40, 116] (certified at ell = 2147483629)
T = [(2, 1), (5, 3)]; h of degree 48 = 1 x 48 (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(999, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 999:2 1000

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