Atlas › Modular forms › Weight 2 › Level 939 › 939.2.a

Newspace 939.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(939))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
939.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} - q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$	939c1 rank 1
939.2.a.b	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} - q^{5} + 4q^{7} + O(q^{8})$	+ +	$-1$	939a1 rank 1
939.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} + q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$	939b1 rank 1
939.2.a.d	6	degree 6	$6q - q^{2} + 6q^{3} + q^{4} - 6q^{5} - q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
939.2.a.e	10	degree 10	$10q - 3q^{2} - 10q^{3} + 9q^{4} - 8q^{5} + 3q^{6} + 6q^{7} + O(q^{8})$	+ +	$-1$
939.2.a.f	15	degree 15	$15q + 3q^{2} - 15q^{3} + 15q^{4} + 13q^{5} - 3q^{6} - 10q^{7} + O(q^{8})$	+ −	$+1$
939.2.a.g	19	degree 19	$19q + q^{2} + 19q^{3} + 27q^{4} + 9q^{5} + q^{6} + 18q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
939.2.a.a	$x + 3$
939.2.a.b	$x + 1$
939.2.a.c	$x - 1$
939.2.a.d	$x^{6} + 7x^{5} + 9x^{4} - 34x^{3} - 103x^{2} - 90x - 25$
939.2.a.e	$x^{10} + 11x^{9} + 17x^{8} - 144x^{7} - 340x^{6} + 573x^{5} + 1344x^{4} - 846x^{3} - 1621x^{2} + 426x + 575$
939.2.a.f	$x^{15} - 16x^{14} + 56x^{13} + 303x^{12} - 1953x^{11} - 1485x^{10} + 22241x^{9} - 1317x^{8} - 120033x^{7} + 10754x^{6} + 308680x^{5} + 43921x^{4} - 306979x^{3} - 82517x^{2} + 98019x + 30677$
939.2.a.g	$x^{19} - 10x^{18} - 54x^{17} + 839x^{16} - 226x^{15} - 24231x^{14} + 56275x^{13} + 278244x^{12} - 1084982x^{11} - 1054183x^{10} + 8348837x^{9} - 1754104x^{8} - 31565290x^{7} + 22970355x^{6} + 60838989x^{5} - 59445832x^{4} - 54860511x^{3} + 56460085x^{2} + 16485649x - 11476187$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(939))$	53
Old forms	50
Cusp forms $S_{2}(\Gamma_0(939))$	103
Eisenstein series $E_{2}(\Gamma_0(939))$	3
Modular forms $M_{2}(\Gamma_0(939))$	106

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(939)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(313))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.375 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, 313, 939]: [0, 1, 26, 106] (certified at ell = 2147482273)
T = [(2, 1), (5, 1)]; h of degree 53 = 1 x 53 (phi(ord eps) x dim S_k^new); 6 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(939, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 939:2 1000

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