Atlas › Modular forms › Weight 2 › Level 443 › 443.2.a

Newspace 443.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(443))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
443.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} - q^{4} + 2q^{6} + q^{7} + O(q^{8})$	+	$-1$	443b1 rank 1
443.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - 2q^{5} + 2q^{7} + O(q^{8})$	+	$-1$	443a1 rank 1
443.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} - q^{4} + 4q^{5} - 2q^{6} - q^{7} + O(q^{8})$	−	$+1$	443c1 rank 0
443.2.a.d	12	degree 12	$12q - 3q^{2} - 7q^{3} + 11q^{4} - 7q^{5} - 8q^{6} - 16q^{7} + O(q^{8})$	+	$-1$
443.2.a.e	22	degree 22	$22q + q^{2} + 8q^{3} + 27q^{4} + 3q^{5} - 4q^{6} + 12q^{7} + O(q^{8})$	−	$+1$

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 $T$
443.2.a.a	$x + 3$
443.2.a.b	$x - 1$
443.2.a.c	$x + 1$
443.2.a.d	$x^{12} + 10x^{11} + 18x^{10} - 85x^{9} - 265x^{8} + 49x^{7} + 659x^{6} + 340x^{5} - 373x^{4} - 345x^{3} - 67x^{2} + 3x + 1$
443.2.a.e	$x^{22} - 9x^{21} - 36x^{20} + 515x^{19} + 57x^{18} - 11644x^{17} + 13973x^{16} + 133029x^{15} - 253020x^{14} - 813220x^{13} + 1971022x^{12} + 2606520x^{11} - 7798367x^{10} - 4143410x^{9} + 16128887x^{8} + 3328032x^{7} - 17767694x^{6} - 1460634x^{5} + 9842103x^{4} + 324352x^{3} - 2216577x^{2} + 78971x + 123703$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(443))$	37
Old forms	0
Cusp forms $S_{2}(\Gamma_0(443))$	37
Eisenstein series $E_{2}(\Gamma_0(443))$	1
Modular forms $M_{2}(\Gamma_0(443))$	38

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.287 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, 443]: [0, 38] (certified at ell = 2147481311)
T = [(2, 1), (3, 1)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 4 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(443, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 443:2 1000

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