Properties

Label 6001.2.a.c
Level 6001
Weight 2
Character orbit 6001.a
Self dual yes
Analytic conductor 47.918
Analytic rank 0
Dimension 121
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6001 = 17 \cdot 353 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9182262530\)
Analytic rank: \(0\)
Dimension: \(121\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 121q + 9q^{2} + 13q^{3} + 127q^{4} + 21q^{5} + 19q^{6} - 13q^{7} + 24q^{8} + 134q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 121q + 9q^{2} + 13q^{3} + 127q^{4} + 21q^{5} + 19q^{6} - 13q^{7} + 24q^{8} + 134q^{9} - q^{10} + 40q^{11} + 41q^{12} + 14q^{13} + 32q^{14} + 49q^{15} + 135q^{16} - 121q^{17} + 28q^{18} + 34q^{19} + 64q^{20} + 34q^{21} - 18q^{22} + 37q^{23} + 54q^{24} + 128q^{25} + 91q^{26} + 55q^{27} - 28q^{28} + 45q^{29} + 30q^{30} + 67q^{31} + 47q^{32} + 40q^{33} - 9q^{34} + 59q^{35} + 138q^{36} - 16q^{37} + 30q^{38} + 37q^{39} + 14q^{40} + 89q^{41} + 33q^{42} + 16q^{43} + 90q^{44} + 83q^{45} - 9q^{46} + 135q^{47} + 96q^{48} + 128q^{49} + 71q^{50} - 13q^{51} + 47q^{52} + 52q^{53} + 90q^{54} + 93q^{55} + 69q^{56} - 4q^{57} + 5q^{58} + 170q^{59} + 78q^{60} - 2q^{61} + 46q^{62} - 10q^{63} + 182q^{64} + 50q^{65} + 68q^{66} + 46q^{67} - 127q^{68} + 97q^{69} + 46q^{70} + 191q^{71} + 57q^{72} - 12q^{73} + 68q^{74} + 86q^{75} + 108q^{76} + 62q^{77} - 10q^{78} + 130q^{80} + 149q^{81} + 14q^{82} + 83q^{83} + 126q^{84} - 21q^{85} + 132q^{86} + 50q^{87} - 42q^{88} + 144q^{89} + 9q^{90} + 13q^{91} + 50q^{92} + 43q^{93} + 41q^{94} + 82q^{95} + 110q^{96} - 3q^{97} + 36q^{98} + 89q^{99} + O(q^{100}) \)

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 −2.80099 1.23587 5.84553 −1.08657 −3.46165 −3.96716 −10.7713 −1.47263 3.04347
1.2 −2.75847 −2.11283 5.60915 3.70679 5.82818 −2.88565 −9.95572 1.46406 −10.2251
1.3 −2.69106 2.90048 5.24179 1.92081 −7.80536 −1.20204 −8.72383 5.41280 −5.16901
1.4 −2.63511 −2.54509 4.94380 −1.75130 6.70660 −0.500367 −7.75725 3.47750 4.61488
1.5 −2.61678 −3.27387 4.84754 0.197154 8.56700 −1.64878 −7.45139 7.71823 −0.515910
1.6 −2.60140 −0.594038 4.76726 2.91382 1.54533 0.0965456 −7.19873 −2.64712 −7.57999
1.7 −2.58940 0.995654 4.70501 0.868437 −2.57815 −0.888525 −7.00437 −2.00867 −2.24873
1.8 −2.54984 3.38084 4.50169 −1.07763 −8.62059 1.13594 −6.37892 8.43005 2.74778
1.9 −2.53200 0.341990 4.41103 2.95840 −0.865919 4.74675 −6.10474 −2.88304 −7.49068
1.10 −2.48167 1.61831 4.15867 −3.04222 −4.01610 0.428830 −5.35710 −0.381079 7.54977
1.11 −2.44940 0.192595 3.99956 1.46027 −0.471742 −3.96634 −4.89773 −2.96291 −3.57679
1.12 −2.29935 −2.65633 3.28702 −0.0208883 6.10783 4.05842 −2.95931 4.05607 0.0480296
1.13 −2.25146 1.98627 3.06909 4.11708 −4.47201 2.76168 −2.40701 0.945268 −9.26946
1.14 −2.24876 −0.983386 3.05693 −2.62112 2.21140 0.682123 −2.37679 −2.03295 5.89428
1.15 −2.22966 1.54651 2.97139 −1.67307 −3.44820 −3.79558 −2.16587 −0.608295 3.73037
1.16 −2.18200 −1.07317 2.76112 −3.39433 2.34166 −4.74202 −1.66076 −1.84830 7.40643
1.17 −2.13704 −0.352862 2.56694 0.777890 0.754080 0.156355 −1.21157 −2.87549 −1.66238
1.18 −2.08608 1.06648 2.35172 −3.22907 −2.22475 0.560788 −0.733710 −1.86263 6.73610
1.19 −2.05839 1.11600 2.23697 1.51993 −2.29716 0.698036 −0.487784 −1.75454 −3.12862
1.20 −2.05373 −1.71490 2.21782 −3.91336 3.52195 1.58894 −0.447336 −0.0591109 8.03700
See next 80 embeddings (of 121 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.121
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 6001.2.a.c 121
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6001.2.a.c 121 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(17\) \(1\)
\(353\) \(-1\)