Atlas › Modular forms › Weight 2 › Level 869 › 869.2.a

Newspace 869.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(869))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
869.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} + q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$	869b1 rank 1
869.2.a.b	1	$\mathbb{Q}$	$q - q^{2} - 2q^{3} - q^{4} + 2q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	− +	$+1$	869c1 rank 0
869.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + q^{7} + O(q^{8})$	− −	$-1$	869d1 rank 1
869.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$	869a1 rank 1
869.2.a.e	11	degree 11	$11q - 3q^{2} - 3q^{3} + 7q^{4} - 8q^{5} - 11q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
869.2.a.f	11	degree 11	$11q + q^{2} - 9q^{3} + 7q^{4} - 8q^{5} - 3q^{6} - 3q^{7} + O(q^{8})$	+ +	$-1$
869.2.a.g	18	degree 18	$18q + 4q^{2} + 8q^{3} + 24q^{4} - 3q^{5} + 2q^{6} + 10q^{7} + O(q^{8})$	− +	$+1$
869.2.a.h	21	degree 21	$21q + 7q^{3} + 32q^{4} + 6q^{5} + 4q^{6} + 10q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
869.2.a.a	$x - 1$
869.2.a.b	$x + 7$
869.2.a.c	$x + 2$
869.2.a.d	$x - 4$
869.2.a.e	$x^{11} + 12x^{10} - 103x^{9} - 1391x^{8} + 2487x^{7} + 44499x^{6} - 3083x^{5} - 386728x^{4} - 69767x^{3} + 703677x^{2} + 78584x - 336559$
869.2.a.f	$x^{11} + 26x^{10} + 139x^{9} - 1545x^{8} - 16015x^{7} + 9303x^{6} + 451503x^{5} + 553868x^{4} - 4137139x^{3} - 3801413x^{2} + 15060526x - 3353129$
869.2.a.g	$x^{18} - 28x^{17} + 32x^{16} + 5743x^{15} - 45827x^{14} - 305966x^{13} + 4643000x^{12} - 1448520x^{11} - 177211612x^{10} + 512216945x^{9} + 2794384365x^{8} - 13916499743x^{7} - 12094454138x^{6} + 144153191840x^{5} - 98582750833x^{4} - 523413433970x^{3} + 771016321344x^{2} + 11735199073x - 183397332980$
869.2.a.h	$x^{21} - 21x^{20} - 247x^{19} + 7614x^{18} + 8946x^{17} - 1067704x^{16} + 2144338x^{15} + 77775781x^{14} - 260515540x^{13} - 3297543415x^{12} + 12570628426x^{11} + 86726515131x^{10} - 309241587595x^{9} - 1481669123910x^{8} + 3896096140668x^{7} + 16474383476931x^{6} - 21034782958423x^{5} - 105593824977146x^{4} + 854981931625x^{3} + 277236063945801x^{2} + 266270431842605x + 59634834333366$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(869))$	65
Old forms	14
Cusp forms $S_{2}(\Gamma_0(869))$	79
Eisenstein series $E_{2}(\Gamma_0(869))$	3
Modular forms $M_{2}(\Gamma_0(869))$	82

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(869)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(11))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(79))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.276 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, 11, 79, 869]: [0, 2, 7, 82] (certified at ell = 2147481571)
T = [(2, 1), (3, 3)]; h of degree 65 = 1 x 65 (phi(ord eps) x dim S_k^new); 9 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(869, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 869:2 1000

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