Properties

Label 6002.2.a.a
Level 6002
Weight 2
Character orbit 6002.a
Self dual Yes
Analytic conductor 47.926
Analytic rank 1
Dimension 47
CM No

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Newspace parameters

Level: \( N \) = \( 6002 = 2 \cdot 3001 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6002.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9262112932\)
Analytic rank: \(1\)
Dimension: \(47\)
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.

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

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 1.00000 −3.27359 1.00000 0.938843 −3.27359 3.32454 1.00000 7.71641 0.938843
1.2 1.00000 −3.25356 1.00000 −1.47632 −3.25356 0.915630 1.00000 7.58563 −1.47632
1.3 1.00000 −2.97652 1.00000 2.57540 −2.97652 −1.79126 1.00000 5.85967 2.57540
1.4 1.00000 −2.82438 1.00000 −1.67949 −2.82438 4.44281 1.00000 4.97711 −1.67949
1.5 1.00000 −2.77545 1.00000 1.33751 −2.77545 −0.156144 1.00000 4.70313 1.33751
1.6 1.00000 −2.69765 1.00000 −1.67926 −2.69765 −4.07202 1.00000 4.27732 −1.67926
1.7 1.00000 −2.53917 1.00000 0.927459 −2.53917 −2.39755 1.00000 3.44739 0.927459
1.8 1.00000 −2.32207 1.00000 −3.20826 −2.32207 3.53388 1.00000 2.39199 −3.20826
1.9 1.00000 −2.18587 1.00000 0.199820 −2.18587 0.0442522 1.00000 1.77803 0.199820
1.10 1.00000 −2.03423 1.00000 −3.81435 −2.03423 −2.54300 1.00000 1.13810 −3.81435
1.11 1.00000 −1.93355 1.00000 2.56764 −1.93355 0.413652 1.00000 0.738635 2.56764
1.12 1.00000 −1.68417 1.00000 −2.87245 −1.68417 −3.82153 1.00000 −0.163575 −2.87245
1.13 1.00000 −1.63738 1.00000 3.71883 −1.63738 2.15943 1.00000 −0.318981 3.71883
1.14 1.00000 −1.60528 1.00000 1.61064 −1.60528 −2.90280 1.00000 −0.423077 1.61064
1.15 1.00000 −1.49159 1.00000 1.25928 −1.49159 −0.926965 1.00000 −0.775168 1.25928
1.16 1.00000 −1.48674 1.00000 1.68773 −1.48674 3.31084 1.00000 −0.789607 1.68773
1.17 1.00000 −1.41473 1.00000 −1.78933 −1.41473 1.98770 1.00000 −0.998550 −1.78933
1.18 1.00000 −1.29467 1.00000 −3.90714 −1.29467 −0.238571 1.00000 −1.32383 −3.90714
1.19 1.00000 −0.985702 1.00000 −1.57345 −0.985702 −1.72023 1.00000 −2.02839 −1.57345
1.20 1.00000 −0.879988 1.00000 −1.86923 −0.879988 3.94056 1.00000 −2.22562 −1.86923
See all 47 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.47
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3001\) \(-1\)