Atlas › Modular forms › Weight 2 › Level 827 › 827.2.a

Newspace 827.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(827))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
827.2.a.a	1	$\mathbb{Q}$	$q - 3q^{3} - 2q^{4} + O(q^{8})$	+	$-1$	827a1 rank 1
827.2.a.b	3	degree 3	$3q + 3q^{3} + 2q^{4} - 4q^{7} + O(q^{8})$	+	$-1$
827.2.a.c	24	degree 24	$24q - 5q^{2} - 11q^{3} + 19q^{4} - 19q^{5} - 9q^{6} - 16q^{7} + O(q^{8})$	+	$-1$
827.2.a.d	41	degree 41	$41q + 3q^{2} + 11q^{3} + 47q^{4} + 21q^{5} + 3q^{6} + 20q^{7} + O(q^{8})$	−	$+1$

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 $T$
827.2.a.a	$x + 3$
827.2.a.b	$x^{3} - 3x^{2} - x + 5$
827.2.a.c	$x^{24} + 16x^{23} + 58x^{22} - 323x^{21} - 2478x^{20} - 184x^{19} + 31183x^{18} + 40400x^{17} - 197960x^{16} - 373240x^{15} + 751215x^{14} + 1662132x^{13} - 1863481x^{12} - 4227628x^{11} + 3164882x^{10} + 6289960x^{9} - 3612046x^{8} - 5249853x^{7} + 2496680x^{6} + 2199234x^{5} - 845287x^{4} - 371983x^{3} + 88878x^{2} + 20374x + 889$
827.2.a.d	$x^{41} - 14x^{40} - 55x^{39} + 1585x^{38} - 1520x^{37} - 77111x^{36} + 223611x^{35} + 2099489x^{34} - 9078291x^{33} - 34637276x^{32} + 207233332x^{31} + 340895154x^{30} - 3065444496x^{29} - 1575636864x^{28} + 31130978322x^{27} - 5059693789x^{26} - 223247976930x^{25} + 133698946494x^{24} + 1145287003477x^{23} - 1021574016762x^{22} - 4213316571429x^{21} + 4604442594456x^{20} + 11030241166298x^{19} - 13610094704867x^{18} - 20127972946967x^{17} + 27062856301568x^{16} + 24541892129813x^{15} - 36031777739909x^{14} - 18294386337542x^{13} + 31272657946153x^{12} + 6463117653645x^{11} - 16704930200660x^{10} + 495753674719x^{9} + 4862675877976x^{8} - 1062563496283x^{7} - 564785340118x^{6} + 225856979606x^{5} + 182736257x^{4} - 9442907082x^{3} + 908414577x^{2} + 77806101x - 10387783$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(827))$	69
Old forms	0
Cusp forms $S_{2}(\Gamma_0(827))$	69
Eisenstein series $E_{2}(\Gamma_0(827))$	1
Modular forms $M_{2}(\Gamma_0(827))$	70

Decomposition of the old subspace

The old subspace is 0.

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.362 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, 827]: [0, 70] (certified at ell = 2147481883)
T = [(2, 1), (3, 1)]; h of degree 69 = 1 x 69 (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(827, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 827:2 1000

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