Properties

Label 4001.2.a.b
Level 4001
Weight 2
Character orbit 4001.a
Self dual Yes
Analytic conductor 31.948
Analytic rank 0
Dimension 184
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: \(0\)
Dimension: \(184\)
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.

\(\operatorname{Tr}(f)(q) = \) \(184q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 28q^{3} \) \(\mathstrut +\mathstrut 217q^{4} \) \(\mathstrut +\mathstrut 15q^{5} \) \(\mathstrut +\mathstrut 31q^{6} \) \(\mathstrut +\mathstrut 49q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 210q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(184q \) \(\mathstrut +\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 28q^{3} \) \(\mathstrut +\mathstrut 217q^{4} \) \(\mathstrut +\mathstrut 15q^{5} \) \(\mathstrut +\mathstrut 31q^{6} \) \(\mathstrut +\mathstrut 49q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 210q^{9} \) \(\mathstrut +\mathstrut 46q^{10} \) \(\mathstrut +\mathstrut 25q^{11} \) \(\mathstrut +\mathstrut 61q^{12} \) \(\mathstrut +\mathstrut 52q^{13} \) \(\mathstrut +\mathstrut 28q^{14} \) \(\mathstrut +\mathstrut 59q^{15} \) \(\mathstrut +\mathstrut 279q^{16} \) \(\mathstrut +\mathstrut 16q^{17} \) \(\mathstrut -\mathstrut 2q^{18} \) \(\mathstrut +\mathstrut 86q^{19} \) \(\mathstrut +\mathstrut 26q^{20} \) \(\mathstrut +\mathstrut 22q^{21} \) \(\mathstrut +\mathstrut 54q^{22} \) \(\mathstrut +\mathstrut 55q^{23} \) \(\mathstrut +\mathstrut 72q^{24} \) \(\mathstrut +\mathstrut 241q^{25} \) \(\mathstrut +\mathstrut 32q^{26} \) \(\mathstrut +\mathstrut 97q^{27} \) \(\mathstrut +\mathstrut 75q^{28} \) \(\mathstrut +\mathstrut 27q^{29} \) \(\mathstrut -\mathstrut 10q^{30} \) \(\mathstrut +\mathstrut 276q^{31} \) \(\mathstrut +\mathstrut 20q^{33} \) \(\mathstrut +\mathstrut 122q^{34} \) \(\mathstrut +\mathstrut 30q^{35} \) \(\mathstrut +\mathstrut 278q^{36} \) \(\mathstrut +\mathstrut 42q^{37} \) \(\mathstrut +\mathstrut 14q^{38} \) \(\mathstrut +\mathstrut 113q^{39} \) \(\mathstrut +\mathstrut 115q^{40} \) \(\mathstrut +\mathstrut 39q^{41} \) \(\mathstrut +\mathstrut 15q^{42} \) \(\mathstrut +\mathstrut 65q^{43} \) \(\mathstrut +\mathstrut 32q^{44} \) \(\mathstrut +\mathstrut 54q^{45} \) \(\mathstrut +\mathstrut 65q^{46} \) \(\mathstrut +\mathstrut 82q^{47} \) \(\mathstrut +\mathstrut 117q^{48} \) \(\mathstrut +\mathstrut 297q^{49} \) \(\mathstrut +\mathstrut 4q^{50} \) \(\mathstrut +\mathstrut 45q^{51} \) \(\mathstrut +\mathstrut 136q^{52} \) \(\mathstrut +\mathstrut 21q^{53} \) \(\mathstrut +\mathstrut 93q^{54} \) \(\mathstrut +\mathstrut 252q^{55} \) \(\mathstrut +\mathstrut 74q^{56} \) \(\mathstrut +\mathstrut 14q^{57} \) \(\mathstrut +\mathstrut 54q^{58} \) \(\mathstrut +\mathstrut 95q^{59} \) \(\mathstrut +\mathstrut 58q^{60} \) \(\mathstrut +\mathstrut 131q^{61} \) \(\mathstrut +\mathstrut 14q^{62} \) \(\mathstrut +\mathstrut 88q^{63} \) \(\mathstrut +\mathstrut 368q^{64} \) \(\mathstrut -\mathstrut 9q^{65} \) \(\mathstrut +\mathstrut 52q^{66} \) \(\mathstrut +\mathstrut 90q^{67} \) \(\mathstrut +\mathstrut 27q^{68} \) \(\mathstrut +\mathstrut 101q^{69} \) \(\mathstrut +\mathstrut 18q^{70} \) \(\mathstrut +\mathstrut 117q^{71} \) \(\mathstrut -\mathstrut 15q^{72} \) \(\mathstrut +\mathstrut 72q^{73} \) \(\mathstrut +\mathstrut 7q^{74} \) \(\mathstrut +\mathstrut 150q^{75} \) \(\mathstrut +\mathstrut 148q^{76} \) \(\mathstrut +\mathstrut 7q^{77} \) \(\mathstrut +\mathstrut 22q^{78} \) \(\mathstrut +\mathstrut 287q^{79} \) \(\mathstrut +\mathstrut 43q^{80} \) \(\mathstrut +\mathstrut 244q^{81} \) \(\mathstrut +\mathstrut 86q^{82} \) \(\mathstrut +\mathstrut 25q^{83} \) \(\mathstrut +\mathstrut 14q^{84} \) \(\mathstrut +\mathstrut 41q^{85} \) \(\mathstrut +\mathstrut 25q^{86} \) \(\mathstrut +\mathstrut 82q^{87} \) \(\mathstrut +\mathstrut 115q^{88} \) \(\mathstrut +\mathstrut 48q^{89} \) \(\mathstrut +\mathstrut 78q^{90} \) \(\mathstrut +\mathstrut 272q^{91} \) \(\mathstrut +\mathstrut 69q^{92} \) \(\mathstrut +\mathstrut 44q^{93} \) \(\mathstrut +\mathstrut 161q^{94} \) \(\mathstrut +\mathstrut 37q^{95} \) \(\mathstrut +\mathstrut 129q^{96} \) \(\mathstrut +\mathstrut 106q^{97} \) \(\mathstrut -\mathstrut 46q^{98} \) \(\mathstrut +\mathstrut 53q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.80114 −3.24231 5.84639 −0.362239 9.08218 0.683027 −10.7743 7.51260 1.01468
1.2 −2.79357 0.695080 5.80404 −1.43429 −1.94176 3.16464 −10.6269 −2.51686 4.00680
1.3 −2.77961 −0.970175 5.72623 0.973772 2.69671 0.392591 −10.3575 −2.05876 −2.70671
1.4 −2.76351 −0.782711 5.63697 4.06144 2.16303 1.66325 −10.0508 −2.38736 −11.2238
1.5 −2.73134 2.60660 5.46019 −2.78013 −7.11951 −4.77010 −9.45095 3.79438 7.59346
1.6 −2.71583 2.95270 5.37572 2.81232 −8.01903 −1.71576 −9.16786 5.71846 −7.63778
1.7 −2.69288 −1.22667 5.25159 2.75099 3.30328 −4.02180 −8.75615 −1.49527 −7.40807
1.8 −2.68105 −2.59933 5.18802 −3.48579 6.96892 −5.16960 −8.54725 3.75650 9.34557
1.9 −2.66682 0.260886 5.11190 −1.12952 −0.695735 −1.67133 −8.29887 −2.93194 3.01221
1.10 −2.65266 3.05290 5.03661 −1.94746 −8.09830 2.48037 −8.05510 6.32018 5.16596
1.11 −2.64594 −1.99415 5.00101 −3.20075 5.27640 3.40947 −7.94051 0.976617 8.46900
1.12 −2.63285 1.55445 4.93191 −4.32053 −4.09264 −0.850828 −7.71930 −0.583679 11.3753
1.13 −2.61467 2.84813 4.83651 0.542328 −7.44692 4.00529 −7.41654 5.11183 −1.41801
1.14 −2.61059 −0.259768 4.81518 −4.09316 0.678147 2.08780 −7.34928 −2.93252 10.6856
1.15 −2.55462 2.40673 4.52611 3.51168 −6.14829 3.52515 −6.45325 2.79234 −8.97103
1.16 −2.43211 0.645473 3.91518 −1.15456 −1.56986 1.52754 −4.65795 −2.58336 2.80801
1.17 −2.42514 −1.95794 3.88128 1.35530 4.74827 4.91337 −4.56237 0.833523 −3.28679
1.18 −2.40164 −1.91938 3.76787 −2.41584 4.60966 −1.95333 −4.24578 0.684021 5.80197
1.19 −2.38954 1.02374 3.70989 −0.691971 −2.44626 1.59189 −4.08584 −1.95196 1.65349
1.20 −2.38517 0.919938 3.68905 3.44498 −2.19421 −3.87561 −4.02868 −2.15371 −8.21687
See next 80 embeddings (of 184 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.184
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

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