Atlas › Modular forms › Weight 2 › Level 899 › 899.2.a

Newspace 899.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(899))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
899.2.a.a	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} - q^{4} + q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	+ +	$-1$	899a1 rank 1
899.2.a.b	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 5q^{7} + O(q^{8})$	− +	$+1$	899b1 rank 0
899.2.a.c	10	degree 10	$10q - 3q^{3} + 4q^{4} - 7q^{5} - 2q^{7} + O(q^{8})$	− −	$-1$
899.2.a.d	11	degree 11	$11q - 4q^{2} + q^{3} + 4q^{4} - 6q^{5} + 2q^{6} - 8q^{7} + O(q^{8})$	+ +	$-1$
899.2.a.e	23	degree 23	$23q + 2q^{2} + 5q^{3} + 32q^{4} + 3q^{5} + 4q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$
899.2.a.f	25	degree 25	$25q - 2q^{3} + 36q^{4} + 9q^{5} + 2q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$

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 $T$
899.2.a.a	$x + 1$
899.2.a.b	$x - 3$
899.2.a.c	$x^{10} + 3x^{9} - 21x^{8} - 64x^{7} + 105x^{6} + 322x^{5} - 223x^{4} - 575x^{3} + 238x^{2} + 335x - 113$
899.2.a.d	$x^{11} + 3x^{10} - 28x^{9} - 50x^{8} + 320x^{7} + 173x^{6} - 1569x^{5} + 511x^{4} + 2231x^{3} - 1005x^{2} - 1051x + 293$
899.2.a.e	$x^{23} - 7x^{22} - 66x^{21} + 519x^{20} + 1611x^{19} - 15295x^{18} - 18004x^{17} + 235873x^{16} + 80277x^{15} - 2087070x^{14} + 101023x^{13} + 10904937x^{12} - 2092410x^{11} - 33510686x^{10} + 5789342x^{9} + 58616424x^{8} - 2102136x^{7} - 54131013x^{6} - 8576251x^{5} + 21126139x^{4} + 7332479x^{3} - 1377034x^{2} - 710444x - 40433$
899.2.a.f	$x^{25} + 2x^{24} - 104x^{23} - 182x^{22} + 4474x^{21} + 7029x^{20} - 103454x^{19} - 150060x^{18} + 1401750x^{17} + 1906090x^{16} - 11377594x^{15} - 14360794x^{14} + 54423008x^{13} + 60351329x^{12} - 147150478x^{11} - 123107762x^{10} + 215097972x^{9} + 90124661x^{8} - 157629042x^{7} - 5492989x^{6} + 37750569x^{5} - 911334x^{4} - 3733805x^{3} - 64775x^{2} + 131725x + 10575$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(899))$	71
Old forms	8
Cusp forms $S_{2}(\Gamma_0(899))$	79
Eisenstein series $E_{2}(\Gamma_0(899))$	3
Modular forms $M_{2}(\Gamma_0(899))$	82

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(899)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(29))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(31))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.271 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, 29, 31, 899]: [0, 3, 3, 82] (certified at ell = 2147482681)
T = [(2, 1), (3, 1)]; h of degree 71 = 1 x 71 (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(899, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 899:2 1000

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