Atlas › Modular forms › Weight 2 › Level 645 › 645.2.a

Newspace 645.2.a: $S_{2}^{\mathrm{new}}(\Gamma_0(645))$

Newforms

Label	Dim	Coefficient field	Trace form	AL signs	Sign	Elliptic curve
645.2.a.a	1	$\mathbb{Q}$	$q - 2q^{2} - q^{3} + 2q^{4} + q^{5} + 2q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$	645d1 rank 0
645.2.a.b	1	$\mathbb{Q}$	$q - 2q^{2} + q^{3} + 2q^{4} + q^{5} - 2q^{6} - 4q^{7} + O(q^{8})$	− − +	$-1$	645f1 rank 1
645.2.a.c	1	$\mathbb{Q}$	$q + q^{3} - 2q^{4} + q^{5} - 2q^{7} + O(q^{8})$	− − +	$-1$	645e1 rank 1
645.2.a.d	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} + O(q^{8})$	+ + −	$+1$	645a1 rank 0
645.2.a.e	1	$\mathbb{Q}$	$q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + 4q^{7} + O(q^{8})$	+ − +	$+1$	645b1 rank 0
645.2.a.f	1	$\mathbb{Q}$	$q + 2q^{2} - q^{3} + 2q^{4} + q^{5} - 2q^{6} + O(q^{8})$	+ − +	$+1$	645c1 rank 0
645.2.a.g	2	$\mathbb{Q}(\sqrt{2})$	$2q - 2q^{3} - 2q^{5} + O(q^{8})$	+ + +	$-1$
645.2.a.h	3	degree 3	$3q - 2q^{2} + 3q^{3} + 2q^{4} - 3q^{5} - 2q^{6} - 2q^{7} + O(q^{8})$	− + −	$-1$
645.2.a.i	3	degree 3	$3q - 3q^{3} + 2q^{4} + 3q^{5} - 8q^{7} + O(q^{8})$	+ − −	$-1$
645.2.a.j	3	degree 3	$3q + q^{2} + 3q^{3} + 3q^{4} - 3q^{5} + q^{6} - 2q^{7} + O(q^{8})$	− + +	$+1$
645.2.a.k	5	degree 5	$5q - 4q^{2} - 5q^{3} + 12q^{4} - 5q^{5} + 4q^{6} + O(q^{8})$	+ + −	$+1$
645.2.a.l	5	degree 5	$5q + 2q^{2} + 5q^{3} + 4q^{4} + 5q^{5} + 2q^{6} + 2q^{7} + O(q^{8})$	− − −	$+1$

Hecke characteristic polynomials

The eigenvalue of $T = T_{2} + 2T_{7} + 3T_{13} + 4T_{17} + 5T_{19} + 6T_{11} + 7T_{23} + 8T_{29}$ 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 $T$
645.2.a.a	$x + 28$
645.2.a.b	$x + 104$
645.2.a.c	$x + 58$
645.2.a.d	$x - 79$
645.2.a.e	$x - 113$
645.2.a.f	$x - 32$
645.2.a.g	$x^{2} + 40x + 398$
645.2.a.h	$x^{3} + 186x^{2} + 8756x + 25562$
645.2.a.i	$x^{3} + 152x^{2} - 2882x - 612146$
645.2.a.j	$x^{3} - 181x^{2} + 7444x + 49474$
645.2.a.k	$x^{5} + 54x^{4} - 3127x^{3} - 83278x^{2} + 528692x + 14122$
645.2.a.l	$x^{5} - 120x^{4} - 797x^{3} + 415752x^{2} - 8473238x - 29639794$

Dimensions

Newforms $S_{2}^{\mathrm{new}}(\Gamma_0(645))$	27
Old forms	58
Cusp forms $S_{2}(\Gamma_0(645))$	85
Eisenstein series $E_{2}(\Gamma_0(645))$	7
Modular forms $M_{2}(\Gamma_0(645))$	92

Decomposition of the old subspace

$S_{2}^{\mathrm{old}}(\Gamma_0(645)) =$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(43))^{\oplus 4}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(129))^{\oplus 2}$ $\oplus$ $S_{2}^{\mathrm{new}}(\Gamma_0(215))^{\oplus 2}$

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.659 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, 43, 129, 215, 645]: [0, 1, 1, 4, 4, 16, 24, 92] (certified at ell = 2147483353)
T = [(2, 1), (7, 2), (13, 3), (17, 4), (19, 5), (11, 6), (23, 7), (29, 8)]; h of degree 27 = 1 x 27 (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(645, 2)

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

cargo run --release -p sagebrush-web --example atlas -- 645:2 1000

← 644.2.a · 646.2.a →