Atlas › Modular forms › Weight 2 › Level 755 › 755.2.a

Newspace 755.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(755))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
755.2.a.a	1	$\mathbb{Q}$	$q - 2q^{4} - q^{5} + 3q^{7} + O(q^{8})$	+ +	$-1$	755a1 rank 1
755.2.a.b	1	$\mathbb{Q}$	$q + q^{2} - 2q^{3} - q^{4} - q^{5} - 2q^{6} + 2q^{7} + O(q^{8})$	+ −	$+1$	755c1 rank 0
755.2.a.c	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} + O(q^{8})$	+ +	$-1$	755b1 rank 1
755.2.a.d	1	$\mathbb{Q}$	$q + 2q^{2} - 3q^{3} + 2q^{4} + q^{5} - 6q^{6} - q^{7} + O(q^{8})$	− +	$+1$	755f1 rank 0
755.2.a.e	1	$\mathbb{Q}$	$q + 2q^{2} + q^{3} + 2q^{4} + q^{5} + 2q^{6} + 3q^{7} + O(q^{8})$	− +	$+1$	755d1 rank 0
755.2.a.f	1	$\mathbb{Q}$	$q + 2q^{2} + 3q^{3} + 2q^{4} + q^{5} + 6q^{6} - q^{7} + O(q^{8})$	− +	$+1$	755e1 rank 0
755.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} - 2q^{3} - 2q^{4} + 2q^{5} + 2q^{6} - 4q^{7} + O(q^{8})$	− −	$-1$
755.2.a.h	5	degree 5	$5q - q^{2} - 2q^{3} + 3q^{4} + 5q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− −	$-1$
755.2.a.i	5	degree 5	$5q - 3q^{3} + 4q^{4} - 5q^{5} - 3q^{6} - 7q^{7} + O(q^{8})$	+ +	$-1$
755.2.a.j	15	degree 15	$15q - 2q^{2} + 5q^{3} + 18q^{4} + 15q^{5} - 4q^{6} + 11q^{7} + O(q^{8})$	− +	$+1$
755.2.a.k	18	degree 18	$18q - 2q^{2} + 2q^{3} + 32q^{4} - 18q^{5} + 10q^{6} + 4q^{7} + O(q^{8})$	+ −	$+1$

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 $T$
755.2.a.a	$x$
755.2.a.b	$x + 5$
755.2.a.c	$x - 4$
755.2.a.d	$x + 7$
755.2.a.e	$x - 5$
755.2.a.f	$x - 11$
755.2.a.g	$x^{2} + 8x - 2$
755.2.a.h	$x^{5} + 7x^{4} - 30x^{3} - 206x^{2} + 45x + 202$
755.2.a.i	$x^{5} + 9x^{4} - 34x^{3} - 344x^{2} + 155x + 2000$
755.2.a.j	$x^{15} - 13x^{14} - 175x^{13} + 2703x^{12} + 9447x^{11} - 205392x^{10} - 152495x^{9} + 7473756x^{8} - 1184684x^{7} - 139849744x^{6} + 23791312x^{5} + 1313783744x^{4} + 523747008x^{3} - 4834572288x^{2} - 6878601216x - 2675624960$
755.2.a.k	$x^{18} - 4x^{17} - 443x^{16} + 1766x^{15} + 77928x^{14} - 292435x^{13} - 7104675x^{12} + 23260438x^{11} + 368493989x^{10} - 923171351x^{9} - 11298093764x^{8} + 16313850778x^{7} + 205503871384x^{6} - 50432127932x^{5} - 2033497525136x^{4} - 1831294924424x^{3} + 7781252419472x^{2} + 16023366691424x + 8217727555008$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(755))$	51
Old forms	24
Cusp forms $S_{2}(\Gamma_0(755))$	75
Eisenstein series $E_{2}(\Gamma_0(755))$	3
Modular forms $M_{2}(\Gamma_0(755))$	78

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(755)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(151))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.253 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, 5, 151, 755]: [0, 1, 13, 78] (certified at ell = 2147482801)
T = [(2, 1), (3, 3)]; h of degree 51 = 1 x 51 (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(755, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 755:2 1000

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