Atlas › Modular forms › Weight 2 › Level 989 › 989.2.a

Newspace 989.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(989))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
989.2.a.a	1	$\mathbb{Q}$	$q + q^{2} + 3q^{3} - q^{4} + 2q^{5} + 3q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$	989a1 rank 0
989.2.a.b	4	degree 4	$4q + 2q^{2} - 2q^{3} - 2q^{4} - 7q^{5} - q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$
989.2.a.c	10	degree 10	$10q - q^{2} - 5q^{3} + 11q^{4} - q^{5} - 2q^{7} + O(q^{8})$	− −	$-1$
989.2.a.d	16	degree 16	$16q - 2q^{2} - 11q^{3} + 8q^{4} - 6q^{6} - 8q^{7} + O(q^{8})$	+ +	$-1$
989.2.a.e	22	degree 22	$22q + q^{2} + 11q^{3} + 29q^{4} + 10q^{5} + q^{6} + 8q^{7} + O(q^{8})$	− +	$+1$
989.2.a.f	24	degree 24	$24q + 8q^{3} + 36q^{4} - 6q^{5} + 5q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
989.2.a.a	$x - 4$
989.2.a.b	$x^{4} - 9x^{2} - 10x - 1$
989.2.a.c	$x^{10} + 6x^{9} - 15x^{8} - 112x^{7} + 84x^{6} + 666x^{5} - 266x^{4} - 1276x^{3} + 191x^{2} + 644x - 71$
989.2.a.d	$x^{16} + 13x^{15} + 41x^{14} - 118x^{13} - 786x^{12} - 116x^{11} + 4963x^{10} + 4273x^{9} - 15128x^{8} - 16737x^{7} + 24057x^{6} + 26196x^{5} - 18910x^{4} - 14776x^{3} + 5655x^{2} - 99x - 61$
989.2.a.e	$x^{22} - 12x^{21} - 9x^{20} + 628x^{19} - 1610x^{18} - 11128x^{17} + 51124x^{16} + 61536x^{15} - 613838x^{14} + 294382x^{13} + 3366633x^{12} - 4439168x^{11} - 8332873x^{10} + 16942608x^{9} + 7821960x^{8} - 28262018x^{7} + 1864943x^{6} + 20340658x^{5} - 6928780x^{4} - 4128574x^{3} + 2502469x^{2} - 412176x + 21024$
989.2.a.f	$x^{24} - 8x^{23} - 67x^{22} + 635x^{21} + 1492x^{20} - 19315x^{19} - 11986x^{18} + 300142x^{17} - 20480x^{16} - 2622040x^{15} + 914225x^{14} + 13203652x^{13} - 5814579x^{12} - 37607009x^{11} + 16791472x^{10} + 57636021x^{9} - 22235579x^{8} - 43526668x^{7} + 9733288x^{6} + 15276009x^{5} - 502487x^{4} - 1572304x^{3} + 38996x^{2} + 40041x - 2372$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(989))$	77
Old forms	10
Cusp forms $S_{2}(\Gamma_0(989))$	87
Eisenstein series $E_{2}(\Gamma_0(989))$	3
Modular forms $M_{2}(\Gamma_0(989))$	90

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(989)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(23))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.284 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, 23, 43, 989]: [0, 3, 4, 90] (certified at ell = 2147483647)
T = [(2, 1), (3, 1)]; h of degree 77 = 1 x 77 (phi(ord eps) x dim S_k^new); 8 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(989, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 989:2 1000

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