Atlas › Modular forms › Weight 2 › Level 389 › 389.2.a

Newspace 389.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(389))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
389.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - 2q^{3} + 2q^{4} - 3q^{5} + 4q^{6} - 5q^{7} + O(q^{8})$	−	$+1$	389a1 rank 2
389.2.a.b	2	$\mathbb{Q}(\sqrt{2})$	$2q - 4q^{3} - 2q^{5} + 4q^{6} - 2q^{7} + O(q^{8})$	+	$-1$
389.2.a.c	3	degree 3	$3q + 2q^{4} - 5q^{5} - 8q^{6} - 3q^{7} + O(q^{8})$	+	$-1$
389.2.a.d	6	degree 6	$6q - 3q^{2} - 5q^{3} + q^{4} + 3q^{5} - q^{6} - 4q^{7} + O(q^{8})$	+	$-1$
389.2.a.e	20	degree 20	$20q + 3q^{2} + 11q^{3} + 27q^{4} + q^{5} + q^{6} + 12q^{7} + O(q^{8})$	−	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
389.2.a.a	$x + 8$
389.2.a.b	$x^{2} + 12x + 4$
389.2.a.c	$x^{3} - 16x + 16$
389.2.a.d	$x^{6} + 18x^{5} + 82x^{4} - 248x^{3} - 3034x^{2} - 8381x - 7489$
389.2.a.e	$x^{20} - 36x^{19} + 238x^{18} + 5938x^{17} - 89638x^{16} - 98459x^{15} + 8381042x^{14} - 31945454x^{13} - 300466488x^{12} + 2266802048x^{11} + 2323820867x^{10} - 58089218087x^{9} + 92671844985x^{8} + 561570921194x^{7} - 1869681316392x^{6} - 683117287589x^{5} + 8599815908447x^{4} - 6905582251608x^{3} - 6096196967352x^{2} + 5699822108448x + 21372607552$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(389))$	32
Old forms	0
Cusp forms $S_{2}(\Gamma_0(389))$	32
Eisenstein series $E_{2}(\Gamma_0(389))$	1
Modular forms $M_{2}(\Gamma_0(389))$	33

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.187 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, 389]: [0, 33] (certified at ell = 2147479897)
T = [(2, 1), (3, 3)]; h of degree 32 = 1 x 32 (phi(ord eps) x dim S_k^new); 5 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(389, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 389:2 1000

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