Atlas › Modular forms › Weight 2 › Level 947 › 947.2.a › 947.2.a.b

Newform orbit 947.2.a.b

Trace form

The 44 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$44q + 5q^{2} + 8q^{3} + 47q^{4} + 17q^{5} + 25q^{7} + 12q^{8} + 46q^{9} + 24q^{10} + 3q^{11} + 8q^{12} + 72q^{13} - 3q^{14} - 2q^{15} + 45q^{16} + 16q^{17} + 5q^{18} + 26q^{19} + 17q^{20} + 25q^{21} + 8q^{22} + 5q^{23} - 16q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 44. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{44} - 13x^{43} - 72x^{42} + 1591x^{41} - 11x^{40} - 84262x^{39} + 173142x^{38} + 2542193x^{37} - 8545869x^{36} - 48127444x^{35} + 221046249x^{34} + 587062161x^{33} - 3669984532x^{32} - 4409040707x^{31} + 42068169741x^{30} + 15001975094x^{29} - 344718106408x^{28} + 62704450538x^{27} + 2055276296520x^{26} - 1116829942411x^{25} - 8984984232243x^{24} + 7108728396272x^{23} + 28815453432938x^{22} - 27611869445350x^{21} - 67404277866877x^{20} + 71504403689829x^{19} + 113621357279878x^{18} - 125430387648463x^{17} - 135412470920748x^{16} + 146702748815661x^{15} + 110941471241648x^{14} - 109691676225083x^{13} - 59830040514226x^{12} + 48695793831709x^{11} + 19625751075384x^{10} - 11483615907741x^{9} - 3318717954016x^{8} + 1320251232422x^{7} + 244078868143x^{6} - 74350870868x^{5} - 6684581655x^{4} + 1910409948x^{3} + 25803437x^{2} - 15185807x + 446357$

Atkin–Lehner signs

$p$	$w_p$
947	$-1$

For $p \parallel N$ every form in the orbit has $a_p = -p^{k/2-1} w_p$ . The Fricke sign is $w_N = -1$ , so the sign of the functional equation is $(-1)^{k/2} w_N = +1$ : each L-function in the orbit has even analytic rank.

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
$\operatorname{tr} a_p$	5	8	17	25	3	72	16	26	5	16	36	94	19	31	3	10	−4	99	41	−13	101	17	6	21	117

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.293 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, 947]: [0, 80] (certified at ell = 2147481491)
T = [(2, 1), (3, 1)]; h of degree 79 = 1 x 79 (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(947, 2)[1]   # 947.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 947:2 1000