Atlas › Modular forms › Weight 2 › Level 912 › 912.2.a › 912.2.a.d

Newform orbit 912.2.a.d

$q$ -expansion

$q - q^{3} + q^{5} - 3q^{7} + q^{9} + 3q^{11} - 6q^{13} - q^{15} + 3q^{17} + q^{19} + 3q^{21} - 4q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = 2555T_{5} + 3163T_{7} + 2311T_{13} + 3911T_{17} + 4960T_{23} + 3573T_{29} + 9540T_{11} + 2170T_{31}$ is a root of

$x + 40357$

Atkin–Lehner signs

$p$	$w_p$
2	— ( $2^2 \mid 912$ )
3	$+1$
19	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ .

Elliptic curve

By modularity this newform is the one attached to the isogeny class 912.d (Cremona's 912f). Its optimal curve 912f1 is

$y^2 = x^3 - x^2 + 315x + 2349$

of rank 1 with torsion of order 1. Its $a_p$ , computed by Sagebrush's point-counting engine, equal the newform's for every prime $p < 1000$ .

Traces of Hecke eigenvalues

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
$\operatorname{tr} a_p$	0	−1	1	−3	3	−6	3	1	−4	−10	−2	8	−8	1	−3	−6	0	7	−8	−12	−11	0	−4	10	−2

$\operatorname{tr} a_n$ for every $n \le 1000$ is in the JSON.

Sato–Tate

The normalized $a_p / 2p^{1/2}$ for the primes $p < 1000$ not dividing 912, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 2.263 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 label, dimension and $\operatorname{tr} a_p$ for every $p < 1000$ equal LMFDB's.

The engine's checks

dim M^+ at levels [1, 2, 3, 4, 6, 8, 12, 16, 19, 24, 38, 48, 57, 76, 114, 152, 228, 304, 456, 912]: [0, 1, 1, 2, 3, 3, 5, 4, 2, 8, 7, 12, 8, 13, 24, 24, 46, 44, 88, 168] (certified at ell = 2147483629)
T = [(5, 2555), (7, 3163), (13, 2311), (17, 3911), (23, 4960), (29, 3573), (11, 9540), (31, 2170)]; h of degree 18 = 1 x 18 (phi(ord eps) x dim S_k^new); 11 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(912, 2)[3]   # 912.2.a.d

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

cargo run --release -p sagebrush-web --example atlas -- 912:2 1000