Atlas › Modular forms › Weight 2 › Level 879 › 879.2.a

Newspace 879.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(879))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
879.2.a.a	7	degree 7	$7q - 5q^{2} + 7q^{3} + 5q^{4} - 10q^{5} - 5q^{6} - 3q^{7} + O(q^{8})$	− −	$-1$
879.2.a.b	7	degree 7	$7q + q^{2} - 7q^{3} + q^{4} - 6q^{5} - q^{6} - 5q^{7} + O(q^{8})$	+ +	$-1$
879.2.a.c	17	degree 17	$17q + 6q^{2} + 17q^{3} + 20q^{4} + 12q^{5} + 6q^{6} - q^{7} + O(q^{8})$	− +	$+1$
879.2.a.d	18	degree 18	$18q - 3q^{2} - 18q^{3} + 23q^{4} + 6q^{5} + 3q^{6} + 5q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + T_{5}$ 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 $T$
879.2.a.a	$x^{7} + 15x^{6} + 80x^{5} + 168x^{4} + 64x^{3} - 144x^{2} - 56x + 41$
879.2.a.b	$x^{7} + 5x^{6} - 6x^{5} - 36x^{4} - 32x^{3} - 4x^{2} + 4x + 1$
879.2.a.c	$x^{17} - 18x^{16} + 77x^{15} + 426x^{14} - 4321x^{13} + 5743x^{12} + 48969x^{11} - 172234x^{10} - 47395x^{9} + 986562x^{8} - 1035247x^{7} - 1548354x^{6} + 3110486x^{5} - 238555x^{4} - 2036184x^{3} + 680726x^{2} + 345214x - 94139$
879.2.a.d	$x^{18} - 3x^{17} - 107x^{16} + 297x^{15} + 4563x^{14} - 11694x^{13} - 99662x^{12} + 237981x^{11} + 1182893x^{10} - 2701703x^{9} - 7371793x^{8} + 16872495x^{7} + 20595152x^{6} - 50900971x^{5} - 14530501x^{4} + 49051062x^{3} + 6967696x^{2} - 15916765x - 3550737$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(879))$	49
Old forms	48
Cusp forms $S_{2}(\Gamma_0(879))$	97
Eisenstein series $E_{2}(\Gamma_0(879))$	3
Modular forms $M_{2}(\Gamma_0(879))$	100

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(879)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(293))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.351 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, 293, 879]: [0, 1, 25, 100] (certified at ell = 2147478961)
T = [(2, 1), (5, 1)]; h of degree 49 = 1 x 49 (phi(ord eps) x dim S_k^new); 5 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(879, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 879:2 1000

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