Atlas › Modular forms › Weight 2 › Level 831 › 831.2.a

Newspace 831.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(831))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
831.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$	831a1 rank 1
831.2.a.b	3	degree 3	$3q + 3q^{3} - 3q^{5} - 3q^{7} + O(q^{8})$	− −	$-1$
831.2.a.c	6	degree 6	$6q - q^{2} - 6q^{3} + q^{4} - 3q^{5} + q^{6} + 3q^{7} + O(q^{8})$	+ +	$-1$
831.2.a.d	17	degree 17	$17q + 4q^{2} - 17q^{3} + 20q^{4} + 9q^{5} - 4q^{6} - 5q^{7} + O(q^{8})$	+ −	$+1$
831.2.a.e	20	degree 20	$20q + q^{2} + 20q^{3} + 29q^{4} + 3q^{5} + q^{6} + 11q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{5}$ 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 $T$
831.2.a.a	$x + 1$
831.2.a.b	$x^{3} + 9x^{2} + 15x - 17$
831.2.a.c	$x^{6} + 10x^{5} - 39x^{4} - 378x^{3} - 19x^{2} + 1280x + 953$
831.2.a.d	$x^{17} - 31x^{16} - 79x^{15} + 10769x^{14} - 57519x^{13} - 1303771x^{12} + 11569500x^{11} + 62285178x^{10} - 811381342x^{9} - 794475550x^{8} + 23798762769x^{7} - 8960806659x^{6} - 290409176787x^{5} + 53592720177x^{4} + 1486760130450x^{3} + 986997713456x^{2} - 1110086276833x - 859513119073$
831.2.a.e	$x^{20} - 10x^{19} - 673x^{18} + 7314x^{17} + 176634x^{16} - 2150838x^{15} - 22587333x^{14} + 325659710x^{13} + 1384156985x^{12} - 27179094718x^{11} - 25647808995x^{10} + 1249578364666x^{9} - 1114675498941x^{8} - 30189911724282x^{7} + 57505846032545x^{6} + 337794428499706x^{5} - 816263818980278x^{4} - 1200178044812408x^{3} + 3468285481367016x^{2} - 1907388992535700x + 92566751152991$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(831))$	47
Old forms	44
Cusp forms $S_{2}(\Gamma_0(831))$	91
Eisenstein series $E_{2}(\Gamma_0(831))$	3
Modular forms $M_{2}(\Gamma_0(831))$	94

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(831)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(277))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.27 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, 3, 277, 831]: [0, 1, 23, 94] (certified at ell = 2147483137)
T = [(2, 1), (5, 3)]; h of degree 47 = 1 x 47 (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(831, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 831:2 1000

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