Atlas › Modular forms › Weight 2 › Level 898 › 898.2.a

Newspace 898.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(898))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
898.2.a.a	1	$\mathbb{Q}$	$q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} + q^{7} + O(q^{8})$	+ +	$-1$	898a1 rank 1
898.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + 2q^{3} + q^{4} - 2q^{5} - 2q^{6} + O(q^{8})$	+ −	$+1$	898b1 rank 0
898.2.a.c	1	$\mathbb{Q}$	$q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + 3q^{7} + O(q^{8})$	− −	$-1$	898d1 rank 1
898.2.a.d	1	$\mathbb{Q}$	$q + q^{2} + 2q^{3} + q^{4} + 2q^{5} + 2q^{6} + O(q^{8})$	− +	$+1$	898c1 rank 0
898.2.a.e	2	$\mathbb{Q}(\sqrt{13})$	$2q - 2q^{2} - 3q^{3} + 2q^{4} - 3q^{5} + 3q^{6} - 2q^{7} + O(q^{8})$	+ +	$-1$
898.2.a.f	3	degree 3	$3q - 3q^{2} + 2q^{3} + 3q^{4} + 6q^{5} - 2q^{6} + 6q^{7} + O(q^{8})$	+ −	$+1$
898.2.a.g	4	degree 4	$4q - 4q^{2} + 2q^{3} + 4q^{4} + 3q^{5} - 2q^{6} - 3q^{7} + O(q^{8})$	+ −	$+1$
898.2.a.h	5	degree 5	$5q + 5q^{2} - 5q^{3} + 5q^{4} - 9q^{5} - 5q^{6} - 6q^{7} + O(q^{8})$	− −	$-1$
898.2.a.i	7	degree 7	$7q - 7q^{2} - 4q^{3} + 7q^{4} - 2q^{5} + 4q^{6} - 8q^{7} + O(q^{8})$	+ +	$-1$
898.2.a.j	12	degree 12	$12q + 12q^{2} + 6q^{3} + 12q^{4} + 11q^{5} + 6q^{6} + q^{7} + O(q^{8})$	− +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{3} + T_{5}$ 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 $T$
898.2.a.a	$x + 1$
898.2.a.b	$x$
898.2.a.c	$x + 3$
898.2.a.d	$x - 4$
898.2.a.e	$x^{2} + 6x - 4$
898.2.a.f	$x^{3} - 8x^{2} + 17x - 8$
898.2.a.g	$x^{4} - 5x^{3} - 25x^{2} + 75x + 225$
898.2.a.h	$x^{5} + 14x^{4} + 66x^{3} + 136x^{2} + 122x + 37$
898.2.a.i	$x^{7} + 6x^{6} - 8x^{5} - 66x^{4} + 22x^{3} + 99x^{2} - 26x - 4$
898.2.a.j	$x^{12} - 17x^{11} + 82x^{10} + 120x^{9} - 2298x^{8} + 6637x^{7} + 357x^{6} - 37486x^{5} + 74422x^{4} - 48244x^{3} - 11996x^{2} + 25504x - 7040$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(898))$	37
Old forms	74
Cusp forms $S_{2}(\Gamma_0(898))$	111
Eisenstein series $E_{2}(\Gamma_0(898))$	3
Modular forms $M_{2}(\Gamma_0(898))$	114

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(898)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(449))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.377 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, 449, 898]: [0, 1, 38, 114] (certified at ell = 2147479937)
T = [(3, 1), (5, 1)]; h of degree 37 = 1 x 37 (phi(ord eps) x dim S_k^new); 4 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(898, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 898:2 1000

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