Properties

Label 6041.2.a.c
Level 6041
Weight 2
Character orbit 6041.a
Self dual Yes
Analytic conductor 48.238
Analytic rank 1
Dimension 83
CM No

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Newspace parameters

Level: \( N \) = \( 6041 = 7 \cdot 863 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6041.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2376278611\)
Analytic rank: \(1\)
Dimension: \(83\)
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.

\(\operatorname{Tr}(f)(q) = \) \(83q \) \(\mathstrut -\mathstrut 8q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 48q^{4} \) \(\mathstrut -\mathstrut 11q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut +\mathstrut 83q^{7} \) \(\mathstrut -\mathstrut 18q^{8} \) \(\mathstrut +\mathstrut 39q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(83q \) \(\mathstrut -\mathstrut 8q^{2} \) \(\mathstrut -\mathstrut 12q^{3} \) \(\mathstrut +\mathstrut 48q^{4} \) \(\mathstrut -\mathstrut 11q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut +\mathstrut 83q^{7} \) \(\mathstrut -\mathstrut 18q^{8} \) \(\mathstrut +\mathstrut 39q^{9} \) \(\mathstrut -\mathstrut 20q^{10} \) \(\mathstrut -\mathstrut 26q^{11} \) \(\mathstrut -\mathstrut 14q^{12} \) \(\mathstrut -\mathstrut 22q^{13} \) \(\mathstrut -\mathstrut 8q^{14} \) \(\mathstrut -\mathstrut 37q^{15} \) \(\mathstrut -\mathstrut 10q^{16} \) \(\mathstrut -\mathstrut 9q^{17} \) \(\mathstrut -\mathstrut 27q^{18} \) \(\mathstrut -\mathstrut 42q^{19} \) \(\mathstrut -\mathstrut 22q^{20} \) \(\mathstrut -\mathstrut 12q^{21} \) \(\mathstrut -\mathstrut 44q^{22} \) \(\mathstrut -\mathstrut 46q^{23} \) \(\mathstrut -\mathstrut 24q^{24} \) \(\mathstrut -\mathstrut 20q^{25} \) \(\mathstrut -\mathstrut 9q^{26} \) \(\mathstrut -\mathstrut 39q^{27} \) \(\mathstrut +\mathstrut 48q^{28} \) \(\mathstrut -\mathstrut 36q^{29} \) \(\mathstrut -\mathstrut 11q^{30} \) \(\mathstrut -\mathstrut 107q^{31} \) \(\mathstrut -\mathstrut 19q^{32} \) \(\mathstrut -\mathstrut 25q^{33} \) \(\mathstrut -\mathstrut 24q^{34} \) \(\mathstrut -\mathstrut 11q^{35} \) \(\mathstrut -\mathstrut 32q^{36} \) \(\mathstrut -\mathstrut 75q^{37} \) \(\mathstrut -\mathstrut 16q^{38} \) \(\mathstrut -\mathstrut 78q^{39} \) \(\mathstrut -\mathstrut 34q^{40} \) \(\mathstrut -\mathstrut 17q^{41} \) \(\mathstrut -\mathstrut 8q^{42} \) \(\mathstrut -\mathstrut 87q^{43} \) \(\mathstrut -\mathstrut 32q^{44} \) \(\mathstrut -\mathstrut 17q^{45} \) \(\mathstrut -\mathstrut 56q^{46} \) \(\mathstrut -\mathstrut 39q^{47} \) \(\mathstrut -\mathstrut 16q^{48} \) \(\mathstrut +\mathstrut 83q^{49} \) \(\mathstrut -\mathstrut 26q^{50} \) \(\mathstrut -\mathstrut 71q^{51} \) \(\mathstrut -\mathstrut 53q^{52} \) \(\mathstrut -\mathstrut 28q^{53} \) \(\mathstrut -\mathstrut 25q^{54} \) \(\mathstrut -\mathstrut 94q^{55} \) \(\mathstrut -\mathstrut 18q^{56} \) \(\mathstrut -\mathstrut 79q^{57} \) \(\mathstrut -\mathstrut 69q^{58} \) \(\mathstrut -\mathstrut 26q^{59} \) \(\mathstrut -\mathstrut 43q^{60} \) \(\mathstrut -\mathstrut 56q^{61} \) \(\mathstrut -\mathstrut 6q^{62} \) \(\mathstrut +\mathstrut 39q^{63} \) \(\mathstrut -\mathstrut 108q^{64} \) \(\mathstrut -\mathstrut 26q^{65} \) \(\mathstrut +\mathstrut 10q^{66} \) \(\mathstrut -\mathstrut 123q^{67} \) \(\mathstrut -\mathstrut 11q^{68} \) \(\mathstrut +\mathstrut 2q^{69} \) \(\mathstrut -\mathstrut 20q^{70} \) \(\mathstrut -\mathstrut 96q^{71} \) \(\mathstrut -\mathstrut 11q^{72} \) \(\mathstrut -\mathstrut 53q^{73} \) \(\mathstrut -\mathstrut 26q^{74} \) \(\mathstrut -\mathstrut 27q^{75} \) \(\mathstrut -\mathstrut 65q^{76} \) \(\mathstrut -\mathstrut 26q^{77} \) \(\mathstrut -\mathstrut 43q^{78} \) \(\mathstrut -\mathstrut 160q^{79} \) \(\mathstrut +\mathstrut 12q^{80} \) \(\mathstrut -\mathstrut 53q^{81} \) \(\mathstrut -\mathstrut 20q^{82} \) \(\mathstrut -\mathstrut 2q^{83} \) \(\mathstrut -\mathstrut 14q^{84} \) \(\mathstrut -\mathstrut 110q^{85} \) \(\mathstrut +\mathstrut 24q^{86} \) \(\mathstrut -\mathstrut 52q^{87} \) \(\mathstrut -\mathstrut 79q^{88} \) \(\mathstrut -\mathstrut 5q^{89} \) \(\mathstrut -\mathstrut 4q^{90} \) \(\mathstrut -\mathstrut 22q^{91} \) \(\mathstrut -\mathstrut 51q^{92} \) \(\mathstrut -\mathstrut 30q^{93} \) \(\mathstrut -\mathstrut 9q^{94} \) \(\mathstrut -\mathstrut 76q^{95} \) \(\mathstrut -\mathstrut 3q^{96} \) \(\mathstrut -\mathstrut 44q^{97} \) \(\mathstrut -\mathstrut 8q^{98} \) \(\mathstrut -\mathstrut 82q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 −2.73002 −0.0142784 5.45300 −0.484021 0.0389802 1.00000 −9.42677 −2.99980 1.32139
1.2 −2.58676 −0.792673 4.69133 0.0189688 2.05046 1.00000 −6.96183 −2.37167 −0.0490676
1.3 −2.50552 2.64911 4.27764 −1.70463 −6.63740 1.00000 −5.70669 4.01777 4.27099
1.4 −2.50484 1.86042 4.27422 2.20949 −4.66004 1.00000 −5.69655 0.461144 −5.53442
1.5 −2.49510 −2.54767 4.22552 0.711578 6.35668 1.00000 −5.55291 3.49060 −1.77546
1.6 −2.36693 2.60578 3.60235 0.153258 −6.16771 1.00000 −3.79266 3.79011 −0.362751
1.7 −2.28673 0.00653898 3.22911 3.50344 −0.0149528 1.00000 −2.81064 −2.99996 −8.01141
1.8 −2.27758 −1.23058 3.18737 1.22240 2.80274 1.00000 −2.70432 −1.48568 −2.78412
1.9 −2.25188 −3.15195 3.07097 1.48598 7.09783 1.00000 −2.41170 6.93481 −3.34625
1.10 −2.19298 0.436268 2.80917 2.07310 −0.956728 1.00000 −1.77450 −2.80967 −4.54628
1.11 −2.16622 −1.36958 2.69252 −2.62137 2.96681 1.00000 −1.50014 −1.12425 5.67847
1.12 −2.15056 −0.709425 2.62493 −2.42733 1.52566 1.00000 −1.34395 −2.49672 5.22013
1.13 −2.14754 −2.08287 2.61192 −2.63815 4.47305 1.00000 −1.31412 1.33837 5.66553
1.14 −2.14224 0.933242 2.58920 −2.33430 −1.99923 1.00000 −1.26222 −2.12906 5.00064
1.15 −1.96601 2.65332 1.86520 0.0208201 −5.21646 1.00000 0.265017 4.04013 −0.0409325
1.16 −1.90567 −1.35998 1.63156 3.50685 2.59167 1.00000 0.702124 −1.15045 −6.68289
1.17 −1.79680 0.970719 1.22848 −1.22425 −1.74419 1.00000 1.38627 −2.05770 2.19973
1.18 −1.78752 2.42627 1.19524 −3.07648 −4.33701 1.00000 1.43852 2.88678 5.49928
1.19 −1.77588 −2.45572 1.15374 3.79745 4.36107 1.00000 1.50285 3.03058 −6.74380
1.20 −1.71609 −2.62448 0.944964 −3.91107 4.50383 1.00000 1.81054 3.88787 6.71175
See all 83 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.83
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(863\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{83} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\).