Atlas › Modular forms › Weight 2 › Level 941 › 941.2.a › 941.2.a.b

Newform orbit 941.2.a.b

Trace form

The 50 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$50q + 3q^{2} + 12q^{3} + 71q^{4} + 6q^{5} + 9q^{6} + 23q^{7} + 9q^{8} + 68q^{9} + 5q^{10} + 37q^{11} + 11q^{12} + 6q^{13} + 3q^{14} + 16q^{15} + 109q^{16} + 3q^{17} - 2q^{18} + 62q^{19} + 3q^{20} + 5q^{21} + 10q^{22} + 13q^{23} + 14q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 50. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{50} - 15x^{49} - 91x^{48} + 2405x^{47} - 648x^{46} - 167841x^{45} + 455383x^{44} + 6675619x^{43} - 29492201x^{42} - 164852752x^{41} + 1035683780x^{40} + 2511487606x^{39} - 23756846245x^{38} - 19211512124x^{37} + 381736056987x^{36} - 71900988134x^{35} - 4444366015934x^{34} + 3959705404792x^{33} + 38102079756843x^{32} - 55324718354443x^{31} - 241492632348839x^{30} + 475701405838497x^{29} + 1121821352750146x^{28} - 2854006694126434x^{27} - 3710861821113324x^{26} + 12439812687957743x^{25} + 8068256527828257x^{24} - 39913627504088182x^{23} - 8255170517171357x^{22} + 94188012164303786x^{21} - 10900528053585801x^{20} - 161555158212848586x^{19} + 59824480554610783x^{18} + 196553145601890652x^{17} - 116977479777776983x^{16} - 161971525190447815x^{15} + 135589686323037493x^{14} + 81676807630669627x^{13} - 99107559001094796x^{12} - 17433094906603102x^{11} + 44360186740658804x^{10} - 4318703780150171x^{9} - 10802755024686874x^{8} + 3333412867817183x^{7} + 943277630009320x^{6} - 579656150528744x^{5} + 55395231575072x^{4} + 10180239804112x^{3} - 1225610491194x^{2} - 83141163651x + 1764870211$

Atkin–Lehner signs

$p$	$w_p$
941	$-1$

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
$\operatorname{tr} a_p$	3	12	6	23	37	6	3	62	13	7	60	31	3	54	−8	4	59	22	117	52	10	44	−6	−19	−4

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.3 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, 941]: [0, 79] (certified at ell = 2147473241)
T = [(2, 1), (3, 1)]; h of degree 78 = 1 x 78 (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(941, 2)[1]   # 941.2.a.b

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

cargo run --release -p sagebrush-web --example atlas -- 941:2 1000