Atlas › Modular forms › Weight 2 › Level 911 › 911.2.a › 911.2.a.c

Newform orbit 911.2.a.c

Trace form

The 53 newforms in this orbit are conjugate over $\mathbb{Q}$ ; the sum of their $q$ -expansions is

$53q + 2q^{2} + 6q^{3} + 66q^{4} + 10q^{5} + 17q^{6} + 14q^{7} + 6q^{8} + 79q^{9} + 31q^{10} + 6q^{11} + 9q^{12} + 33q^{13} + 3q^{14} - 8q^{15} + 84q^{16} + 28q^{17} - 4q^{18} + 42q^{19} + 10q^{20} + 10q^{21} - 4q^{22} - 2q^{23} + 39q^{24} + O(q^{25})$

Coefficient field

The coefficients $a_n$ generate a number field of degree 53. It is generated by the eigenvalue $\alpha$ of $T = T_{2} + T_{3}$ , whose minimal polynomial is

$x^{53} - 8x^{52} - 190x^{51} + 1646x^{50} + 16001x^{49} - 153631x^{48} - 785649x^{47} + 8645993x^{46} + 24752198x^{45} - 328943611x^{44} - 511718334x^{43} + 8991321710x^{42} + 6490683487x^{41} - 183174562967x^{40} - 29373009111x^{39} + 2847428420015x^{38} - 652958181159x^{37} - 34292006238830x^{36} + 17243535850666x^{35} + 322968591001636x^{34} - 224103954202472x^{33} - 2390277296968724x^{32} + 1970619629531648x^{31} + 13913182372815715x^{30} - 12662201359347269x^{29} - 63504653413188194x^{28} + 61155645028061658x^{27} + 225707032570010839x^{26} - 224265121263079304x^{25} - 617438605477097764x^{24} + 624618659829086036x^{23} + 1277583378902561936x^{22} - 1312089849777179121x^{21} - 1949476123407312982x^{20} + 2049275375925845884x^{19} + 2112496229319094176x^{18} - 2324488308358379506x^{17} - 1529974228132608945x^{16} + 1845872229727455001x^{15} + 659103143185166977x^{14} - 968976677029574406x^{13} - 118335133776544227x^{12} + 306750865693429779x^{11} - 15319602267798499x^{10} - 50328244492228539x^{9} + 8135142817585598x^{8} + 3436707949688009x^{7} - 830003386881494x^{6} - 56947690521566x^{5} + 26967770716802x^{4} - 1500239509621x^{3} - 123843165530x^{2} + 14159727911x - 350223237$

Atkin–Lehner signs

$p$	$w_p$
911	$-1$

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
$\operatorname{tr} a_p$	2	6	10	14	6	33	28	42	−2	21	50	13	68	26	0	−9	3	136	−5	−17	131	50	−24	46	114

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

Provenance

Stored in the atlas: computed by Sagebrush's modular symbols engine in 0.292 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, 911]: [0, 77] (certified at ell = 2147480791)
T = [(2, 1), (3, 1)]; h of degree 76 = 1 x 76 (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(911, 2)[2]   # 911.2.a.c

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

cargo run --release -p sagebrush-web --example atlas -- 911:2 1000