Properties

Label 6041.2.a.f
Level $6041$
Weight $2$
Character orbit 6041.a
Self dual yes
Analytic conductor $48.238$
Analytic rank $0$
Dimension $132$
CM no
Inner twists $1$

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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: \(0\)
Dimension: \(132\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 132q + 8q^{2} + 10q^{3} + 174q^{4} + 11q^{5} + 16q^{6} + 132q^{7} + 30q^{8} + 178q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 132q + 8q^{2} + 10q^{3} + 174q^{4} + 11q^{5} + 16q^{6} + 132q^{7} + 30q^{8} + 178q^{9} + 22q^{10} + 32q^{11} + 30q^{12} + 16q^{13} + 8q^{14} + 59q^{15} + 254q^{16} + 11q^{17} + 33q^{18} + 40q^{19} + 4q^{20} + 10q^{21} + 66q^{22} + 62q^{23} + 36q^{24} + 235q^{25} + 25q^{26} + 37q^{27} + 174q^{28} + 28q^{29} + 45q^{30} + 121q^{31} + 53q^{32} - 13q^{33} + 44q^{34} + 11q^{35} + 274q^{36} + 61q^{37} - 28q^{38} + 114q^{39} + 32q^{40} - q^{41} + 16q^{42} + 105q^{43} + 54q^{44} + 29q^{45} + 104q^{46} + 33q^{47} + 16q^{48} + 132q^{49} - 14q^{50} + 53q^{51} - 11q^{52} + 48q^{53} + 11q^{54} + 118q^{55} + 30q^{56} + 93q^{57} + 87q^{58} + 12q^{59} + 41q^{60} + 54q^{61} - 28q^{62} + 178q^{63} + 376q^{64} + 22q^{65} + 6q^{66} + 123q^{67} - 47q^{68} + 58q^{69} + 22q^{70} + 108q^{71} + 97q^{72} + q^{73} - 10q^{74} + 23q^{75} + 71q^{76} + 32q^{77} + 5q^{78} + 204q^{79} - 10q^{80} + 296q^{81} + 80q^{82} - 10q^{83} + 30q^{84} + 94q^{85} + 48q^{86} + 4q^{87} + 155q^{88} + q^{89} - 66q^{90} + 16q^{91} + 49q^{92} + 90q^{93} + 79q^{94} + 100q^{95} + q^{96} + 18q^{97} + 8q^{98} + 96q^{99} + O(q^{100}) \)

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 −2.81389 0.225620 5.91799 −1.24518 −0.634869 1.00000 −11.0248 −2.94910 3.50380
1.2 −2.78355 3.33951 5.74813 2.68755 −9.29568 1.00000 −10.4331 8.15233 −7.48091
1.3 −2.77966 −1.97929 5.72652 −3.34292 5.50177 1.00000 −10.3585 0.917605 9.29218
1.4 −2.75252 −2.26942 5.57639 4.03697 6.24664 1.00000 −9.84409 2.15028 −11.1119
1.5 −2.71866 −0.898623 5.39111 3.58669 2.44305 1.00000 −9.21926 −2.19248 −9.75100
1.6 −2.65744 2.58260 5.06196 1.09780 −6.86308 1.00000 −8.13696 3.66980 −2.91732
1.7 −2.62362 −1.27143 4.88340 1.41870 3.33576 1.00000 −7.56495 −1.38346 −3.72213
1.8 −2.60157 0.915852 4.76817 −4.25711 −2.38265 1.00000 −7.20159 −2.16122 11.0752
1.9 −2.58774 −2.55272 4.69637 −0.926438 6.60577 1.00000 −6.97750 3.51640 2.39738
1.10 −2.58389 1.40748 4.67649 −2.16822 −3.63678 1.00000 −6.91577 −1.01899 5.60245
1.11 −2.57785 −0.937633 4.64529 −3.87896 2.41707 1.00000 −6.81916 −2.12084 9.99936
1.12 −2.56875 −3.19690 4.59849 −1.91049 8.21205 1.00000 −6.67489 7.22017 4.90758
1.13 −2.49223 0.173328 4.21120 2.29751 −0.431973 1.00000 −5.51083 −2.96996 −5.72592
1.14 −2.47641 1.36965 4.13259 −1.36117 −3.39181 1.00000 −5.28115 −1.12406 3.37081
1.15 −2.46575 3.35396 4.07990 −4.25898 −8.27001 1.00000 −5.12851 8.24904 10.5016
1.16 −2.32322 2.41590 3.39733 0.249961 −5.61265 1.00000 −3.24629 2.83657 −0.580712
1.17 −2.26422 −1.83534 3.12671 1.84374 4.15563 1.00000 −2.55113 0.368489 −4.17465
1.18 −2.25429 −3.03067 3.08183 3.72305 6.83200 1.00000 −2.43875 6.18494 −8.39284
1.19 −2.23793 1.90424 3.00834 4.23440 −4.26157 1.00000 −2.25659 0.626144 −9.47631
1.20 −2.22730 1.19588 2.96085 2.18062 −2.66358 1.00000 −2.14009 −1.56986 −4.85688
See next 80 embeddings (of 132 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.132
Significant digits:
Format:

Atkin-Lehner signs

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

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 6041.2.a.f 132
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6041.2.a.f 132 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(14\!\cdots\!53\)\( T_{2}^{117} + \)\(11\!\cdots\!11\)\( T_{2}^{116} - \)\(29\!\cdots\!17\)\( T_{2}^{115} + \)\(95\!\cdots\!54\)\( T_{2}^{114} + \)\(54\!\cdots\!62\)\( T_{2}^{113} - \)\(35\!\cdots\!93\)\( T_{2}^{112} - \)\(86\!\cdots\!60\)\( T_{2}^{111} + \)\(81\!\cdots\!42\)\( T_{2}^{110} + \)\(12\!\cdots\!01\)\( T_{2}^{109} - \)\(14\!\cdots\!57\)\( T_{2}^{108} - \)\(15\!\cdots\!17\)\( T_{2}^{107} + \)\(21\!\cdots\!70\)\( T_{2}^{106} + \)\(16\!\cdots\!47\)\( T_{2}^{105} - \)\(27\!\cdots\!89\)\( T_{2}^{104} - \)\(16\!\cdots\!99\)\( T_{2}^{103} + \)\(30\!\cdots\!04\)\( T_{2}^{102} + \)\(14\!\cdots\!74\)\( T_{2}^{101} - \)\(30\!\cdots\!42\)\( T_{2}^{100} - \)\(12\!\cdots\!82\)\( T_{2}^{99} + \)\(26\!\cdots\!50\)\( T_{2}^{98} + \)\(88\!\cdots\!30\)\( T_{2}^{97} - \)\(21\!\cdots\!94\)\( T_{2}^{96} - \)\(59\!\cdots\!16\)\( T_{2}^{95} + \)\(15\!\cdots\!38\)\( T_{2}^{94} + \)\(36\!\cdots\!47\)\( T_{2}^{93} - \)\(10\!\cdots\!94\)\( T_{2}^{92} - \)\(20\!\cdots\!79\)\( T_{2}^{91} + \)\(60\!\cdots\!64\)\( T_{2}^{90} + \)\(10\!\cdots\!64\)\( T_{2}^{89} - \)\(32\!\cdots\!62\)\( T_{2}^{88} - \)\(46\!\cdots\!94\)\( T_{2}^{87} + \)\(16\!\cdots\!32\)\( T_{2}^{86} + \)\(19\!\cdots\!19\)\( T_{2}^{85} - \)\(74\!\cdots\!07\)\( T_{2}^{84} - \)\(76\!\cdots\!06\)\( T_{2}^{83} + \)\(31\!\cdots\!88\)\( T_{2}^{82} + \)\(26\!\cdots\!95\)\( T_{2}^{81} - \)\(12\!\cdots\!68\)\( T_{2}^{80} - \)\(84\!\cdots\!27\)\( T_{2}^{79} + \)\(42\!\cdots\!56\)\( T_{2}^{78} + \)\(24\!\cdots\!26\)\( T_{2}^{77} - \)\(13\!\cdots\!10\)\( T_{2}^{76} - \)\(63\!\cdots\!97\)\( T_{2}^{75} + \)\(41\!\cdots\!42\)\( T_{2}^{74} + \)\(14\!\cdots\!40\)\( T_{2}^{73} - \)\(11\!\cdots\!09\)\( T_{2}^{72} - \)\(29\!\cdots\!24\)\( T_{2}^{71} + \)\(28\!\cdots\!83\)\( T_{2}^{70} + \)\(51\!\cdots\!79\)\( T_{2}^{69} - \)\(67\!\cdots\!61\)\( T_{2}^{68} - \)\(70\!\cdots\!16\)\( T_{2}^{67} + \)\(14\!\cdots\!40\)\( T_{2}^{66} + \)\(54\!\cdots\!52\)\( T_{2}^{65} - \)\(28\!\cdots\!76\)\( T_{2}^{64} + \)\(61\!\cdots\!37\)\( T_{2}^{63} + \)\(50\!\cdots\!31\)\( T_{2}^{62} - \)\(38\!\cdots\!42\)\( T_{2}^{61} - \)\(82\!\cdots\!50\)\( T_{2}^{60} + \)\(10\!\cdots\!71\)\( T_{2}^{59} + \)\(12\!\cdots\!76\)\( T_{2}^{58} - \)\(20\!\cdots\!43\)\( T_{2}^{57} - \)\(17\!\cdots\!50\)\( T_{2}^{56} + \)\(34\!\cdots\!96\)\( T_{2}^{55} + \)\(21\!\cdots\!64\)\( T_{2}^{54} - \)\(50\!\cdots\!89\)\( T_{2}^{53} - \)\(24\!\cdots\!25\)\( T_{2}^{52} + \)\(65\!\cdots\!21\)\( T_{2}^{51} + \)\(24\!\cdots\!20\)\( T_{2}^{50} - \)\(73\!\cdots\!47\)\( T_{2}^{49} - \)\(23\!\cdots\!18\)\( T_{2}^{48} + \)\(73\!\cdots\!59\)\( T_{2}^{47} + \)\(19\!\cdots\!71\)\( T_{2}^{46} - \)\(65\!\cdots\!70\)\( T_{2}^{45} - \)\(14\!\cdots\!43\)\( T_{2}^{44} + \)\(52\!\cdots\!99\)\( T_{2}^{43} + \)\(99\!\cdots\!50\)\( T_{2}^{42} - \)\(36\!\cdots\!34\)\( T_{2}^{41} - \)\(59\!\cdots\!08\)\( T_{2}^{40} + \)\(22\!\cdots\!22\)\( T_{2}^{39} + \)\(31\!\cdots\!48\)\( T_{2}^{38} - \)\(12\!\cdots\!38\)\( T_{2}^{37} - \)\(14\!\cdots\!32\)\( T_{2}^{36} + \)\(57\!\cdots\!30\)\( T_{2}^{35} + \)\(60\!\cdots\!18\)\( T_{2}^{34} - \)\(23\!\cdots\!96\)\( T_{2}^{33} - \)\(21\!\cdots\!77\)\( T_{2}^{32} + \)\(84\!\cdots\!76\)\( T_{2}^{31} + \)\(65\!\cdots\!16\)\( T_{2}^{30} - \)\(25\!\cdots\!28\)\( T_{2}^{29} - \)\(17\!\cdots\!47\)\( T_{2}^{28} + \)\(65\!\cdots\!27\)\( T_{2}^{27} + \)\(36\!\cdots\!09\)\( T_{2}^{26} - \)\(13\!\cdots\!81\)\( T_{2}^{25} - \)\(66\!\cdots\!81\)\( T_{2}^{24} + \)\(24\!\cdots\!89\)\( T_{2}^{23} + \)\(97\!\cdots\!37\)\( T_{2}^{22} - \)\(34\!\cdots\!39\)\( T_{2}^{21} - \)\(11\!\cdots\!03\)\( T_{2}^{20} + \)\(38\!\cdots\!02\)\( T_{2}^{19} + \)\(10\!\cdots\!38\)\( T_{2}^{18} - \)\(32\!\cdots\!76\)\( T_{2}^{17} - \)\(70\!\cdots\!69\)\( T_{2}^{16} + \)\(20\!\cdots\!02\)\( T_{2}^{15} + \)\(35\!\cdots\!41\)\( T_{2}^{14} - \)\(94\!\cdots\!01\)\( T_{2}^{13} - \)\(12\!\cdots\!44\)\( T_{2}^{12} + \)\(28\!\cdots\!95\)\( T_{2}^{11} + \)\(30\!\cdots\!08\)\( T_{2}^{10} - \)\(52\!\cdots\!53\)\( T_{2}^{9} - \)\(45\!\cdots\!38\)\( T_{2}^{8} + \)\(53\!\cdots\!53\)\( T_{2}^{7} + \)\(35\!\cdots\!59\)\( T_{2}^{6} - \)\(22\!\cdots\!03\)\( T_{2}^{5} - \)\(75\!\cdots\!90\)\( T_{2}^{4} + \)\(46\!\cdots\!60\)\( T_{2}^{3} - 314366227737 T_{2}^{2} - 6361432074 T_{2} + 57534003 \)">\(T_{2}^{132} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6041))\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database