Atlas › Modular forms › Weight 2 › Level 871 › 871.2.a

Newspace 871.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(871))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
871.2.a.a	1	$\mathbb{Q}$	$q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 4q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$	871a1 rank 0
871.2.a.b	3	degree 3	$3q - 6q^{4} + O(q^{8})$	− −	$-1$
871.2.a.c	8	degree 8	$8q - 4q^{2} - 4q^{3} + 12q^{4} - 8q^{5} - q^{6} + O(q^{8})$	− −	$-1$
871.2.a.d	12	degree 12	$12q - 4q^{2} + 2q^{3} + 8q^{4} - 8q^{5} - 3q^{6} - 4q^{7} + O(q^{8})$	+ +	$-1$
871.2.a.e	20	degree 20	$20q + 5q^{2} + 23q^{4} + 10q^{5} + 3q^{6} + O(q^{8})$	+ −	$+1$
871.2.a.f	23	degree 23	$23q + 4q^{2} + 4q^{3} + 26q^{4} + 6q^{5} - 3q^{6} - 2q^{7} + O(q^{8})$	− +	$+1$

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 $T$
871.2.a.a	$x - 4$
871.2.a.b	$x^{3} - 6x + 2$
871.2.a.c	$x^{8} + 8x^{7} + 10x^{6} - 56x^{5} - 133x^{4} + 9x^{3} + 142x^{2} + 69x + 9$
871.2.a.d	$x^{12} + 2x^{11} - 26x^{10} - 20x^{9} + 235x^{8} - 27x^{7} - 766x^{6} + 411x^{5} + 719x^{4} - 324x^{3} - 266x^{2} + 24x + 18$
871.2.a.e	$x^{20} - 5x^{19} - 71x^{18} + 361x^{17} + 1900x^{16} - 9839x^{15} - 24739x^{14} + 130056x^{13} + 170877x^{12} - 900727x^{11} - 626997x^{10} + 3308844x^{9} + 1068970x^{8} - 6350364x^{7} - 527544x^{6} + 5928735x^{5} - 223248x^{4} - 2346028x^{3} + 144695x^{2} + 233658x + 22834$
871.2.a.f	$x^{23} - 8x^{22} - 60x^{21} + 610x^{20} + 919x^{19} - 17331x^{18} + 6158x^{17} + 235093x^{16} - 267843x^{15} - 1699634x^{14} + 2502496x^{13} + 7118905x^{12} - 10914842x^{11} - 18487636x^{10} + 25407834x^{9} + 30528661x^{8} - 32465613x^{7} - 31246998x^{6} + 22332731x^{5} + 18672541x^{4} - 7667302x^{3} - 5854232x^{2} + 1021734x + 733790$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(871))$	67
Old forms	10
Cusp forms $S_{2}(\Gamma_0(871))$	77
Eisenstein series $E_{2}(\Gamma_0(871))$	3
Modular forms $M_{2}(\Gamma_0(871))$	80

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(871)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(67))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.248 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, 13, 67, 871]: [0, 1, 6, 80] (certified at ell = 2147483053)
T = [(2, 1), (3, 1)]; h of degree 67 = 1 x 67 (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(871, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 871:2 1000

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