Atlas › Modular forms › Weight 2 › Level 829 › 829.2.a

Newspace 829.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(829))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
829.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 2q^{4} - 3q^{5} + 4q^{7} + O(q^{8})$	+	$-1$	829a1 rank 1
829.2.a.b	28	degree 28	$28q - 9q^{2} - 6q^{3} + 21q^{4} - 8q^{5} - 9q^{6} - 10q^{7} + O(q^{8})$	+	$-1$
829.2.a.c	39	degree 39	$39q + 9q^{2} + 7q^{3} + 45q^{4} + 7q^{5} + q^{6} + 6q^{7} + O(q^{8})$	−	$+1$

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 $T$
829.2.a.a	$x + 3$
829.2.a.b	$x^{28} + 15x^{27} + 38x^{26} - 442x^{25} - 2311x^{24} + 4117x^{23} + 41433x^{22} - 154x^{21} - 382819x^{20} - 261487x^{19} + 2102918x^{18} + 2076438x^{17} - 7280608x^{16} - 7989367x^{15} + 16175913x^{14} + 17693502x^{13} - 22774081x^{12} - 23463460x^{11} + 19524035x^{10} + 18274999x^{9} - 9445605x^{8} - 7692213x^{7} + 2282442x^{6} + 1460787x^{5} - 262310x^{4} - 113424x^{3} + 13469x^{2} + 3014x - 243$
829.2.a.c	$x^{39} - 16x^{38} - 20x^{37} + 1600x^{36} - 5170x^{35} - 61654x^{34} + 360419x^{33} + 1065653x^{32} - 10964792x^{31} - 3380249x^{30} + 191289760x^{29} - 199332954x^{28} - 2075548331x^{27} + 4148486911x^{26} + 14262056547x^{25} - 42185681477x^{24} - 59644020508x^{23} + 265098099625x^{22} + 122748765596x^{21} - 1097781663839x^{20} + 84807718110x^{19} + 3060150156757x^{18} - 1278937709651x^{17} - 5749345263883x^{16} + 3732662562311x^{15} + 7172365277621x^{14} - 5865355550830x^{13} - 5741257547544x^{12} + 5466969005195x^{11} + 2771606512484x^{10} - 2979227613278x^{9} - 729409974124x^{8} + 879443876850x^{7} + 93584701421x^{6} - 123021792480x^{5} - 6702528485x^{4} + 7082573422x^{3} + 156110780x^{2} - 125441967x + 3471453$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(829))$	68
Old forms	0
Cusp forms $S_{2}(\Gamma_0(829))$	68
Eisenstein series $E_{2}(\Gamma_0(829))$	1
Modular forms $M_{2}(\Gamma_0(829))$	69

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.261 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, 829]: [0, 69] (certified at ell = 2147479273)
T = [(2, 1), (3, 1)]; h of degree 68 = 1 x 68 (phi(ord eps) x dim S_k^new); 7 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(829, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 829:2 1000

← 828.2.a · 830.2.a →