Atlas › Modular forms › Weight 2 › Level 892 › 892.2.a

Newspace 892.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(892))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
892.2.a.a	1	$\mathbb{Q}$	$q - q^{3} + 2q^{5} - 2q^{7} + O(q^{8})$	−	892c1 rank 1
892.2.a.b	1	$\mathbb{Q}$	$q + q^{3} - 4q^{7} + O(q^{8})$	−	892b1 rank 1
892.2.a.c	1	$\mathbb{Q}$	$q + 3q^{3} + 4q^{7} + O(q^{8})$	+	892a1 rank 0
892.2.a.d	7	degree 7	$7q - 4q^{3} - 7q^{5} - q^{7} + O(q^{8})$	−
892.2.a.e	8	degree 8	$8q - q^{3} + 5q^{5} + 5q^{7} + O(q^{8})$	+

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + 3T_{5}$ 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 $T$
892.2.a.a	$x - 5$
892.2.a.b	$x - 1$
892.2.a.c	$x - 3$
892.2.a.d	$x^{7} + 25x^{6} + 121x^{5} - 1397x^{4} - 14443x^{3} - 35412x^{2} - 16508x + 3768$
892.2.a.e	$x^{8} - 14x^{7} - 162x^{6} + 2710x^{5} + 4242x^{4} - 142337x^{3} + 130758x^{2} + 2245652x - 4388504$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(892))$	18
Old forms	92
Cusp forms $S_{2}(\Gamma_0(892))$	110
Eisenstein series $E_{2}(\Gamma_0(892))$	5
Modular forms $M_{2}(\Gamma_0(892))$	115

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(892)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(223))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(446))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.264 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, 2, 4, 223, 446, 892]: [0, 1, 2, 19, 58, 115] (certified at ell = 2147482591)
T = [(3, 1), (5, 3)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 3 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(892, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 892:2 1000

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