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
Inner twists 1

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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: \(149\)
Coefficient ring index: multiple of None
Twist minimal: yes
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.

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

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 admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4001.2.a.a 149
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4001.2.a.a 149 1.a even 1 1 trivial

Atkin-Lehner signs

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