Atlas › Modular forms › Weight 2 › Level 698 › 698.2.a

Newspace 698.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(698))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign
698.2.a.a	3	degree 3	$3q + 3q^{2} - 2q^{3} + 3q^{4} - 4q^{5} - 2q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
698.2.a.b	6	degree 6	$6q - 6q^{2} - 4q^{3} + 6q^{4} + 3q^{5} + 4q^{6} - 8q^{7} + O(q^{8})$	+ +	$-1$
698.2.a.c	9	degree 9	$9q - 9q^{2} + 3q^{3} + 9q^{4} - 2q^{5} - 3q^{6} + 10q^{7} + O(q^{8})$	+ −	$+1$
698.2.a.d	12	degree 12	$12q + 12q^{2} + q^{3} + 12q^{4} + 5q^{5} + q^{6} + 9q^{7} + O(q^{8})$	− +	$+1$

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 $T$
698.2.a.a	$x^{3} + 14x^{2} + 56x + 56$
698.2.a.b	$x^{6} - 5x^{5} - 78x^{4} + 147x^{3} + 1248x^{2} - 1280x - 4096$
698.2.a.c	$x^{9} + 3x^{8} - 303x^{7} - 536x^{6} + 30696x^{5} + 22519x^{4} - 1190501x^{3} - 299770x^{2} + 13639604x + 11921656$
698.2.a.d	$x^{12} - 16x^{11} - 304x^{10} + 6312x^{9} + 3851x^{8} - 613196x^{7} + 2878012x^{6} + 10416513x^{5} - 103601405x^{4} + 140131587x^{3} + 634570800x^{2} - 2029316532x + 1576255080$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(698))$	30
Old forms	56
Cusp forms $S_{2}(\Gamma_0(698))$	86
Eisenstein series $E_{2}(\Gamma_0(698))$	3
Modular forms $M_{2}(\Gamma_0(698))$	89

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(698)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(349))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.237 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, 349, 698]: [0, 1, 29, 89] (certified at ell = 2147481901)
T = [(3, 1), (5, 3)]; h of degree 30 = 1 x 30 (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(698, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 698:2 1000

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