Atlas › Modular forms › Weight 2 › Level 735 › 735.2.a

Newspace 735.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(735))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Elliptic curve
735.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} - q^{5} + 2q^{6} + O(q^{8})$	+ +	735b1 rank 0
735.2.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} + q^{5} - 2q^{6} + O(q^{8})$	− −	735f1 rank 1
735.2.a.c	1	$\mathbb{Q}$	$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + O(q^{8})$	− +	735e1 rank 1
735.2.a.d	1	$\mathbb{Q}$	$q - q^{3} - 2q^{4} + q^{5} + O(q^{8})$	+ −	735c1 rank 1
735.2.a.e	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} - q^{5} + O(q^{8})$	− +	735d1 rank 0
735.2.a.f	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} + O(q^{8})$	+ +	735a1 rank 0
735.2.a.g	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} - 2q^{3} + 4q^{4} + 2q^{5} + 2q^{6} + O(q^{8})$	+ −
735.2.a.h	2	$\mathbb{Q}(\sqrt{3})$	$2q - 2q^{2} + 2q^{3} + 4q^{4} - 2q^{5} - 2q^{6} + O(q^{8})$	− +
735.2.a.i	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 2q^{5} + O(q^{8})$	+ +
735.2.a.j	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{3} + 2q^{5} + O(q^{8})$	− −
735.2.a.k	2	$\mathbb{Q}(\sqrt{5})$	$2q + 2q^{3} + 6q^{4} + 2q^{5} + O(q^{8})$	− −
735.2.a.l	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{5} - 2q^{6} + O(q^{8})$	+ +
735.2.a.m	2	$\mathbb{Q}(\sqrt{2})$	$2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + O(q^{8})$	− −
735.2.a.n	4	degree 4	$4q + 4q^{2} - 4q^{3} + 8q^{4} + 4q^{5} - 4q^{6} + O(q^{8})$	+ −
735.2.a.o	4	degree 4	$4q + 4q^{2} + 4q^{3} + 8q^{4} - 4q^{5} + 4q^{6} + O(q^{8})$	− +

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 3T_{11} + 5T_{13} + 7T_{17} + 9T_{19}$ 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 $T$
735.2.a.a	$x - 14$
735.2.a.b	$x + 54$
735.2.a.c	$x + 53$
735.2.a.d	$x + 82$
735.2.a.e	$x - 82$
735.2.a.f	$x - 89$
735.2.a.g	$x^{2} - 120x + 3408$
735.2.a.h	$x^{2} + 136x + 4576$
735.2.a.i	$x^{2} + 64x - 3584$
735.2.a.j	$x^{2} - 88x - 4336$
735.2.a.k	$x^{2} - 4x - 841$
735.2.a.l	$x^{2} - 14x - 7889$
735.2.a.m	$x^{2} + 10x - 4777$
735.2.a.n	$x^{4} - 44x^{3} - 520x^{2} - 1516x - 1279$
735.2.a.o	$x^{4} - 12x^{3} - 1040x^{2} + 628x + 174697$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(735))$	28
Old forms	69
Cusp forms $S_{2}(\Gamma_0(735))$	97
Eisenstein series $E_{2}(\Gamma_0(735))$	31
Modular forms $M_{2}(\Gamma_0(735))$	128

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(735)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 3}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(21))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(35))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(49))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(105))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(147))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(245))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.693 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, 7, 15, 21, 35, 49, 105, 147, 245, 735]: [0, 1, 1, 1, 4, 4, 6, 5, 20, 20, 30, 116] (certified at ell = 2147483353)
T = [(2, 1), (11, 3), (13, 5), (17, 7), (19, 9)]; h of degree 28 = 1 x 28 (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(735, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 735:2 1000

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