Atlas › Modular forms › Weight 2 › Level 705 › 705.2.a

Newspace 705.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(705))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
705.2.a.a	1	$\mathbb{Q}$	$q - q^{2} - q^{3} - q^{4} + q^{5} + q^{6} - 5q^{7} + O(q^{8})$	+ − −	$-1$	705b1 rank 1
705.2.a.b	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + q^{7} + O(q^{8})$	− + −	$-1$	705e1 rank 1
705.2.a.c	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} - q^{5} + 2q^{7} + O(q^{8})$	+ + +	$-1$	705a1 rank 1
705.2.a.d	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{5} + 2q^{7} + O(q^{8})$	− + +	$+1$	705c1 rank 0
705.2.a.e	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - 3q^{7} + O(q^{8})$	− + −	$-1$	705d1 rank 1
705.2.a.f	1	$\mathbb{Q}$	$q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} + O(q^{8})$	− − −	$+1$	705f1 rank 0
705.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} - 2q^{6} - 6q^{7} + O(q^{8})$	− − +	$-1$
705.2.a.h	2	$\mathbb{Q}(\sqrt{17})$	$2q - q^{2} - 2q^{3} + 5q^{4} + 2q^{5} + q^{6} - 3q^{7} + O(q^{8})$	+ − −	$-1$
705.2.a.i	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{7} + O(q^{8})$	+ + +	$-1$
705.2.a.j	4	degree 4	$4q + 2q^{2} - 4q^{3} + 4q^{4} - 4q^{5} - 2q^{6} + O(q^{8})$	+ + −	$+1$
705.2.a.k	4	degree 4	$4q + 4q^{2} - 4q^{3} + 8q^{4} + 4q^{5} - 4q^{6} + 8q^{7} + O(q^{8})$	+ − +	$+1$
705.2.a.l	5	degree 5	$5q + 5q^{3} + 10q^{4} + 5q^{5} + 10q^{7} + O(q^{8})$	− − −	$+1$
705.2.a.m	6	degree 6	$6q + 2q^{2} + 6q^{3} + 14q^{4} - 6q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	− + +	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{7} + 3T_{11} + 4T_{13} + 5T_{17} + 6T_{19} + 7T_{23} + 8T_{29}$ 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 $T$
705.2.a.a	$x + 93$
705.2.a.b	$x - 15$
705.2.a.c	$x + 129$
705.2.a.d	$x - 63$
705.2.a.e	$x + 121$
705.2.a.f	$x - 35$
705.2.a.g	$x^{2} + 210x + 8977$
705.2.a.h	$x^{2} + 52x + 523$
705.2.a.i	$x^{2} + 78x - 507$
705.2.a.j	$x^{4} - 220x^{3} + 9822x^{2} + 112372x - 1986963$
705.2.a.k	$x^{4} - 132x^{3} + 6234x^{2} - 124780x + 886361$
705.2.a.l	$x^{5} - 9x^{4} - 4034x^{3} + 104106x^{2} + 370125x - 17264749$
705.2.a.m	$x^{6} - 112x^{5} - 8735x^{4} + 917984x^{3} + 36513159x^{2} - 2070741728x - 75792058457$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(705))$	31
Old forms	62
Cusp forms $S_{2}(\Gamma_0(705))$	93
Eisenstein series $E_{2}(\Gamma_0(705))$	7
Modular forms $M_{2}(\Gamma_0(705))$	100

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(705)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(47))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(141))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(235))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.622 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, 3, 5, 15, 47, 141, 235, 705]: [0, 1, 1, 4, 5, 18, 26, 100] (certified at ell = 2147483137)
T = [(2, 1), (7, 2), (11, 3), (13, 4), (17, 5), (19, 6), (23, 7), (29, 8)]; h of degree 31 = 1 x 31 (phi(ord eps) x dim S_k^new); 9 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(705, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 705:2 1000

← 704.2.a · 706.2.a →