Atlas › Modular forms › Weight 2 › Level 703 › 703.2.a

Newspace 703.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(703))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
703.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} + 2q^{4} - 3q^{5} + 3q^{7} + O(q^{8})$	− −	$-1$	703b1 rank 1
703.2.a.b	1	$\mathbb{Q}$	$q + 3q^{3} - 2q^{4} - 2q^{5} - q^{7} + O(q^{8})$	− +	$+1$	703a1 rank 0
703.2.a.c	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} - 4q^{5} - 4q^{6} + 2q^{7} + O(q^{8})$	+ +	$-1$
703.2.a.d	4	degree 4	$4q - 4q^{3} + 4q^{4} - 4q^{5} + 8q^{6} + O(q^{8})$	− −	$-1$
703.2.a.e	4	degree 4	$4q + 3q^{2} + 8q^{3} + 5q^{4} + 3q^{5} + 6q^{6} + 8q^{7} + O(q^{8})$	− +	$+1$
703.2.a.f	5	degree 5	$5q - 3q^{2} - 2q^{3} + 5q^{4} + q^{5} - 5q^{6} - 7q^{7} + O(q^{8})$	− −	$-1$
703.2.a.g	9	degree 9	$9q - 3q^{2} - 2q^{3} + 9q^{4} - 7q^{5} - 5q^{6} - 7q^{7} + O(q^{8})$	+ +	$-1$
703.2.a.h	11	degree 11	$11q + 5q^{2} - 3q^{3} + 15q^{4} + 7q^{5} - 9q^{6} - 7q^{7} + O(q^{8})$	− +	$+1$
703.2.a.i	18	degree 18	$18q + 3q^{2} + 2q^{3} + 23q^{4} + 9q^{5} - 3q^{6} + 7q^{7} + O(q^{8})$	+ −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{3}$ 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 $T$
703.2.a.a	$x + 2$
703.2.a.b	$x - 9$
703.2.a.c	$x^{2} - 6x + 1$
703.2.a.d	$x^{4} + 12x^{3} - 30x^{2} - 524x - 127$
703.2.a.e	$x^{4} - 27x^{3} + 268x^{2} - 1155x + 1814$
703.2.a.f	$x^{5} + 9x^{4} - 15x^{3} - 101x^{2} + 80x + 159$
703.2.a.g	$x^{9} + 9x^{8} - 66x^{7} - 640x^{6} + 933x^{5} + 10242x^{4} - 9306x^{3} - 49049x^{2} + 30944x + 62944$
703.2.a.h	$x^{11} + 4x^{10} - 186x^{9} - 1101x^{8} + 8240x^{7} + 66618x^{6} - 20987x^{5} - 941865x^{4} - 1284380x^{3} + 2950189x^{2} + 7102168x + 3697047$
703.2.a.i	$x^{18} - 9x^{17} - 367x^{16} + 3213x^{15} + 53717x^{14} - 446060x^{13} - 4155595x^{12} + 31429778x^{11} + 188234979x^{10} - 1212261984x^{9} - 5194910775x^{8} + 25435760575x^{7} + 86956339588x^{6} - 266405047917x^{5} - 815935875729x^{4} + 1051855386476x^{3} + 3328195169729x^{2} + 198142312108x - 1589648048920$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(703))$	55
Old forms	6
Cusp forms $S_{2}(\Gamma_0(703))$	61
Eisenstein series $E_{2}(\Gamma_0(703))$	3
Modular forms $M_{2}(\Gamma_0(703))$	64

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(703)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(37))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.245 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, 19, 37, 703]: [0, 2, 3, 64] (certified at ell = 2147483629)
T = [(2, 1), (3, 3)]; h of degree 55 = 1 x 55 (phi(ord eps) x dim S_k^new); 8 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(703, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 703:2 1000

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