Properties

Label 4001.2.a.a
Level 4001
Weight 2
Character orbit 4001.a
Self dual Yes
Analytic conductor 31.948
Analytic rank 1
Dimension 149
CM No

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Newspace parameters

Level: \( N \) = \( 4001 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4001.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(31.9481458487\)
Analytic rank: \(1\)
Dimension: \(149\)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \(149q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 28q^{3} \) \(\mathstrut +\mathstrut 116q^{4} \) \(\mathstrut -\mathstrut 19q^{5} \) \(\mathstrut -\mathstrut 31q^{6} \) \(\mathstrut -\mathstrut 47q^{7} \) \(\mathstrut -\mathstrut 15q^{8} \) \(\mathstrut +\mathstrut 115q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(149q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 28q^{3} \) \(\mathstrut +\mathstrut 116q^{4} \) \(\mathstrut -\mathstrut 19q^{5} \) \(\mathstrut -\mathstrut 31q^{6} \) \(\mathstrut -\mathstrut 47q^{7} \) \(\mathstrut -\mathstrut 15q^{8} \) \(\mathstrut +\mathstrut 115q^{9} \) \(\mathstrut -\mathstrut 48q^{10} \) \(\mathstrut -\mathstrut 31q^{11} \) \(\mathstrut -\mathstrut 61q^{12} \) \(\mathstrut -\mathstrut 54q^{13} \) \(\mathstrut -\mathstrut 44q^{14} \) \(\mathstrut -\mathstrut 65q^{15} \) \(\mathstrut +\mathstrut 58q^{16} \) \(\mathstrut -\mathstrut 26q^{17} \) \(\mathstrut -\mathstrut 23q^{18} \) \(\mathstrut -\mathstrut 86q^{19} \) \(\mathstrut -\mathstrut 52q^{20} \) \(\mathstrut -\mathstrut 30q^{21} \) \(\mathstrut -\mathstrut 56q^{22} \) \(\mathstrut -\mathstrut 63q^{23} \) \(\mathstrut -\mathstrut 90q^{24} \) \(\mathstrut +\mathstrut 92q^{25} \) \(\mathstrut -\mathstrut 38q^{26} \) \(\mathstrut -\mathstrut 103q^{27} \) \(\mathstrut -\mathstrut 77q^{28} \) \(\mathstrut -\mathstrut 51q^{29} \) \(\mathstrut -\mathstrut 22q^{30} \) \(\mathstrut -\mathstrut 256q^{31} \) \(\mathstrut -\mathstrut 21q^{32} \) \(\mathstrut -\mathstrut 36q^{33} \) \(\mathstrut -\mathstrut 124q^{34} \) \(\mathstrut -\mathstrut 50q^{35} \) \(\mathstrut +\mathstrut 45q^{36} \) \(\mathstrut -\mathstrut 42q^{37} \) \(\mathstrut -\mathstrut 14q^{38} \) \(\mathstrut -\mathstrut 119q^{39} \) \(\mathstrut -\mathstrut 131q^{40} \) \(\mathstrut -\mathstrut 55q^{41} \) \(\mathstrut -\mathstrut 5q^{42} \) \(\mathstrut -\mathstrut 55q^{43} \) \(\mathstrut -\mathstrut 54q^{44} \) \(\mathstrut -\mathstrut 68q^{45} \) \(\mathstrut -\mathstrut 59q^{46} \) \(\mathstrut -\mathstrut 82q^{47} \) \(\mathstrut -\mathstrut 89q^{48} \) \(\mathstrut +\mathstrut 30q^{49} \) \(\mathstrut +\mathstrut 13q^{50} \) \(\mathstrut -\mathstrut 83q^{51} \) \(\mathstrut -\mathstrut 126q^{52} \) \(\mathstrut -\mathstrut 23q^{53} \) \(\mathstrut -\mathstrut 83q^{54} \) \(\mathstrut -\mathstrut 244q^{55} \) \(\mathstrut -\mathstrut 94q^{56} \) \(\mathstrut -\mathstrut 14q^{57} \) \(\mathstrut -\mathstrut 60q^{58} \) \(\mathstrut -\mathstrut 93q^{59} \) \(\mathstrut -\mathstrut 70q^{60} \) \(\mathstrut -\mathstrut 139q^{61} \) \(\mathstrut -\mathstrut 10q^{62} \) \(\mathstrut -\mathstrut 120q^{63} \) \(\mathstrut -\mathstrut 45q^{64} \) \(\mathstrut -\mathstrut 37q^{65} \) \(\mathstrut -\mathstrut 28q^{66} \) \(\mathstrut -\mathstrut 110q^{67} \) \(\mathstrut -\mathstrut 27q^{68} \) \(\mathstrut -\mathstrut 79q^{69} \) \(\mathstrut -\mathstrut 78q^{70} \) \(\mathstrut -\mathstrut 123q^{71} \) \(\mathstrut -\mathstrut 74q^{73} \) \(\mathstrut -\mathstrut 25q^{74} \) \(\mathstrut -\mathstrut 146q^{75} \) \(\mathstrut -\mathstrut 192q^{76} \) \(\mathstrut -\mathstrut q^{77} \) \(\mathstrut +\mathstrut 26q^{78} \) \(\mathstrut -\mathstrut 273q^{79} \) \(\mathstrut -\mathstrut 51q^{80} \) \(\mathstrut +\mathstrut 41q^{81} \) \(\mathstrut -\mathstrut 120q^{82} \) \(\mathstrut -\mathstrut 27q^{83} \) \(\mathstrut -\mathstrut 14q^{84} \) \(\mathstrut -\mathstrut 71q^{85} \) \(\mathstrut +\mathstrut 3q^{86} \) \(\mathstrut -\mathstrut 98q^{87} \) \(\mathstrut -\mathstrut 143q^{88} \) \(\mathstrut -\mathstrut 70q^{89} \) \(\mathstrut -\mathstrut 24q^{90} \) \(\mathstrut -\mathstrut 256q^{91} \) \(\mathstrut -\mathstrut 47q^{92} \) \(\mathstrut +\mathstrut 16q^{93} \) \(\mathstrut -\mathstrut 151q^{94} \) \(\mathstrut -\mathstrut 83q^{95} \) \(\mathstrut -\mathstrut 137q^{96} \) \(\mathstrut -\mathstrut 108q^{97} \) \(\mathstrut +\mathstrut 17q^{98} \) \(\mathstrut -\mathstrut 131q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.71370 0.987518 5.36416 1.22965 −2.67983 4.88451 −9.12930 −2.02481 −3.33689
1.2 −2.68009 1.97360 5.18289 2.92338 −5.28944 −1.04102 −8.53045 0.895108 −7.83492
1.3 −2.67250 0.814974 5.14227 −1.39982 −2.17802 −2.73506 −8.39771 −2.33582 3.74102
1.4 −2.66308 −1.99633 5.09200 2.13614 5.31640 2.50209 −8.23426 0.985342 −5.68872
1.5 −2.62536 1.08746 4.89252 0.733437 −2.85498 −4.12609 −7.59390 −1.81743 −1.92554
1.6 −2.59431 −3.00898 4.73047 −0.189685 7.80624 1.12756 −7.08370 6.05397 0.492103
1.7 −2.59150 −1.41020 4.71587 −1.43230 3.65455 −1.66447 −7.03818 −1.01132 3.71180
1.8 −2.58646 −1.10071 4.68977 −0.828720 2.84694 −1.43853 −6.95698 −1.78844 2.14345
1.9 −2.47526 2.78845 4.12689 −0.919151 −6.90213 −1.88527 −5.26460 4.77546 2.27513
1.10 −2.46316 2.96188 4.06716 −3.16778 −7.29560 1.16261 −5.09175 5.77276 7.80275
1.11 −2.45256 0.529200 4.01507 2.50676 −1.29790 −0.602620 −4.94208 −2.71995 −6.14798
1.12 −2.44683 −2.82076 3.98699 4.21697 6.90192 −1.62820 −4.86182 4.95666 −10.3182
1.13 −2.41976 −1.68750 3.85525 −0.342140 4.08335 2.33028 −4.48927 −0.152339 0.827898
1.14 −2.31463 1.63767 3.35751 −2.89581 −3.79059 −1.05145 −3.14214 −0.318053 6.70274
1.15 −2.27915 2.08452 3.19451 −0.864719 −4.75093 2.39591 −2.72247 1.34523 1.97082
1.16 −2.27484 −0.745234 3.17488 3.93220 1.69529 2.00217 −2.67267 −2.44463 −8.94513
1.17 −2.25752 −0.289421 3.09639 −1.87801 0.653373 3.41261 −2.47511 −2.91624 4.23965
1.18 −2.24881 −2.62223 3.05717 −2.07631 5.89691 −2.22773 −2.37738 3.87609 4.66924
1.19 −2.21148 1.63586 2.89063 1.70947 −3.61768 0.528860 −1.96961 −0.323947 −3.78045
1.20 −2.16569 −0.380840 2.69021 −3.90103 0.824780 −2.04947 −1.49478 −2.85496 8.44841
See next 80 embeddings (of 149 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.149
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Atkin-Lehner signs

\( p \) Sign
\(4001\) \(1\)