Atlas › Modular forms › Weight 2 › Level 325 › 325.2.a › 325.2.a.a

Newform orbit 325.2.a.a

$q$ -expansion

$q - 2q^{2} - q^{3} + 2q^{4} + 2q^{6} - 2q^{7} - 2q^{9} + 2q^{11} - 2q^{12} + q^{13} + 4q^{14} - 4q^{16} - 2q^{17} + 4q^{18} + 2q^{21} - 4q^{22} + 9q^{23} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate $\mathbb{Q}$ . The eigenvalue of $T = T_{2} + 3T_{3}$ is a root of

$x + 5$

Atkin–Lehner signs

$p$	$w_p$
5	— ( $5^2 \mid 325$ )
13	$-1$

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 325.a (Cremona's 325e). Its optimal curve 325e1 is

$y^2 + y = x^3 - x^2 - 98x + 378$

of rank 0 with torsion of order 5. 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
$\operatorname{tr} a_p$	−2	−1	0	−2	2	1	−2	0	9	5	2	8	12	−1	8	−11	0	−13	−2	12	−6	15	4	−10	8

$\operatorname{tr} a_n$ 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 325, against the semicircle $\tfrac{2}{\pi}\sqrt{1 - x^2}$ :

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.375 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, 5, 13, 25, 65, 325]: [0, 1, 1, 3, 8, 36] (certified at ell = 2147482921)
T = [(2, 1), (3, 3)]; h of degree 19 = 1 x 19 (phi(ord eps) x dim S_k^new); 3 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(325, 2)[0]   # 325.2.a.a

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

cargo run --release -p sagebrush-web --example atlas -- 325:2 1000