Properties

Label 6018.2.a.r
Level 6018
Weight 2
Character orbit 6018.a
Self dual yes
Analytic conductor 48.054
Analytic rank 1
Dimension 6
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6018.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.0539719364\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.18461324.1
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} -\beta_{1} q^{5} + q^{6} + ( -2 + \beta_{1} + \beta_{5} ) q^{7} + q^{8} + q^{9} +O(q^{10})\) \( q + q^{2} + q^{3} + q^{4} -\beta_{1} q^{5} + q^{6} + ( -2 + \beta_{1} + \beta_{5} ) q^{7} + q^{8} + q^{9} -\beta_{1} q^{10} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{11} + q^{12} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{13} + ( -2 + \beta_{1} + \beta_{5} ) q^{14} -\beta_{1} q^{15} + q^{16} - q^{17} + q^{18} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{5} ) q^{19} -\beta_{1} q^{20} + ( -2 + \beta_{1} + \beta_{5} ) q^{21} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{22} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{23} + q^{24} + ( -3 + \beta_{1} + \beta_{2} ) q^{25} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{26} + q^{27} + ( -2 + \beta_{1} + \beta_{5} ) q^{28} + ( -3 + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{29} -\beta_{1} q^{30} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{31} + q^{32} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{33} - q^{34} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{35} + q^{36} + ( -4 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{37} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{5} ) q^{38} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{39} -\beta_{1} q^{40} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{41} + ( -2 + \beta_{1} + \beta_{5} ) q^{42} + ( -1 + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{43} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{44} -\beta_{1} q^{45} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{46} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{47} + q^{48} + ( 2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{49} + ( -3 + \beta_{1} + \beta_{2} ) q^{50} - q^{51} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{52} + ( -5 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{53} + q^{54} + ( -3 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{55} + ( -2 + \beta_{1} + \beta_{5} ) q^{56} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{5} ) q^{57} + ( -3 + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{58} - q^{59} -\beta_{1} q^{60} + ( \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{61} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{62} + ( -2 + \beta_{1} + \beta_{5} ) q^{63} + q^{64} + ( 1 + 2 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{65} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{66} + ( -3 - \beta_{2} + 3 \beta_{4} + 2 \beta_{5} ) q^{67} - q^{68} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{69} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{70} + ( -\beta_{2} - 6 \beta_{4} - 2 \beta_{5} ) q^{71} + q^{72} + ( 1 - \beta_{1} - 4 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} ) q^{73} + ( -4 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{74} + ( -3 + \beta_{1} + \beta_{2} ) q^{75} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{5} ) q^{76} + ( 3 - 5 \beta_{1} + 2 \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{77} + ( -1 - \beta_{3} + \beta_{4} - \beta_{5} ) q^{78} + ( -2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} ) q^{79} -\beta_{1} q^{80} + q^{81} + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{82} + ( 3 - 3 \beta_{1} + \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{83} + ( -2 + \beta_{1} + \beta_{5} ) q^{84} + \beta_{1} q^{85} + ( -1 + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} ) q^{86} + ( -3 + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{87} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{88} + ( -3 + 2 \beta_{1} - \beta_{4} + \beta_{5} ) q^{89} -\beta_{1} q^{90} + ( -3 - \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{91} + ( -1 + \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{5} ) q^{92} + ( -2 + \beta_{1} + 2 \beta_{2} ) q^{93} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{94} + ( 5 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{95} + q^{96} + ( \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{97} + ( 2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{98} + ( -1 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 6q^{3} + 6q^{4} - 3q^{5} + 6q^{6} - 7q^{7} + 6q^{8} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{2} + 6q^{3} + 6q^{4} - 3q^{5} + 6q^{6} - 7q^{7} + 6q^{8} + 6q^{9} - 3q^{10} - 8q^{11} + 6q^{12} - 6q^{13} - 7q^{14} - 3q^{15} + 6q^{16} - 6q^{17} + 6q^{18} - 14q^{19} - 3q^{20} - 7q^{21} - 8q^{22} - 8q^{23} + 6q^{24} - 13q^{25} - 6q^{26} + 6q^{27} - 7q^{28} - 21q^{29} - 3q^{30} - 5q^{31} + 6q^{32} - 8q^{33} - 6q^{34} - 12q^{35} + 6q^{36} - 13q^{37} - 14q^{38} - 6q^{39} - 3q^{40} - 18q^{41} - 7q^{42} - 4q^{43} - 8q^{44} - 3q^{45} - 8q^{46} - 4q^{47} + 6q^{48} + 5q^{49} - 13q^{50} - 6q^{51} - 6q^{52} - 25q^{53} + 6q^{54} - 8q^{55} - 7q^{56} - 14q^{57} - 21q^{58} - 6q^{59} - 3q^{60} - 5q^{62} - 7q^{63} + 6q^{64} + 6q^{65} - 8q^{66} - 13q^{67} - 6q^{68} - 8q^{69} - 12q^{70} - 12q^{71} + 6q^{72} - 4q^{73} - 13q^{74} - 13q^{75} - 14q^{76} + 4q^{77} - 6q^{78} - 12q^{79} - 3q^{80} + 6q^{81} - 18q^{82} + 9q^{83} - 7q^{84} + 3q^{85} - 4q^{86} - 21q^{87} - 8q^{88} - 11q^{89} - 3q^{90} - 31q^{91} - 8q^{92} - 5q^{93} - 4q^{94} + 35q^{95} + 6q^{96} + 16q^{97} + 5q^{98} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 3 x^{5} - 4 x^{4} + 12 x^{3} + 3 x^{2} - 6 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{5} - 3 \nu^{4} - 3 \nu^{3} + 10 \nu^{2} - \nu - 2 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - 4 \nu^{4} - \nu^{3} + 15 \nu^{2} - 7 \nu - 5 \)
\(\beta_{5}\)\(=\)\( -2 \nu^{5} + 7 \nu^{4} + 5 \nu^{3} - 27 \nu^{2} + 5 \nu + 10 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(\beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{5} + \beta_{4} + 3 \beta_{3} + 9 \beta_{2} + 9 \beta_{1} + 9\)
\(\nu^{5}\)\(=\)\(9 \beta_{5} + 6 \beta_{4} + 13 \beta_{3} + 23 \beta_{2} + 33 \beta_{1} + 12\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.94685
1.91714
0.908132
−0.380739
−0.558656
−1.83273
1.00000 1.00000 1.00000 −2.94685 1.00000 2.59293 1.00000 1.00000 −2.94685
1.2 1.00000 1.00000 1.00000 −1.91714 1.00000 −1.73720 1.00000 1.00000 −1.91714
1.3 1.00000 1.00000 1.00000 −0.908132 1.00000 −1.54786 1.00000 1.00000 −0.908132
1.4 1.00000 1.00000 1.00000 0.380739 1.00000 1.68872 1.00000 1.00000 0.380739
1.5 1.00000 1.00000 1.00000 0.558656 1.00000 −3.85965 1.00000 1.00000 0.558656
1.6 1.00000 1.00000 1.00000 1.83273 1.00000 −4.13695 1.00000 1.00000 1.83273
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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 6018.2.a.r 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.r 6 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(17\) \(1\)
\(59\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{6} + 3 T_{5}^{5} - 4 T_{5}^{4} - 12 T_{5}^{3} + 3 T_{5}^{2} + 6 T_{5} - 2 \)
\( T_{7}^{6} + 7 T_{7}^{5} + T_{7}^{4} - 69 T_{7}^{3} - 77 T_{7}^{2} + 140 T_{7} + 188 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{6} \)
$3$ \( ( 1 - T )^{6} \)
$5$ \( 1 + 3 T + 26 T^{2} + 63 T^{3} + 298 T^{4} + 576 T^{5} + 1928 T^{6} + 2880 T^{7} + 7450 T^{8} + 7875 T^{9} + 16250 T^{10} + 9375 T^{11} + 15625 T^{12} \)
$7$ \( 1 + 7 T + 43 T^{2} + 176 T^{3} + 686 T^{4} + 2121 T^{5} + 6264 T^{6} + 14847 T^{7} + 33614 T^{8} + 60368 T^{9} + 103243 T^{10} + 117649 T^{11} + 117649 T^{12} \)
$11$ \( 1 + 8 T + 64 T^{2} + 285 T^{3} + 1390 T^{4} + 4713 T^{5} + 18422 T^{6} + 51843 T^{7} + 168190 T^{8} + 379335 T^{9} + 937024 T^{10} + 1288408 T^{11} + 1771561 T^{12} \)
$13$ \( 1 + 6 T + 51 T^{2} + 202 T^{3} + 1168 T^{4} + 3626 T^{5} + 17130 T^{6} + 47138 T^{7} + 197392 T^{8} + 443794 T^{9} + 1456611 T^{10} + 2227758 T^{11} + 4826809 T^{12} \)
$17$ \( ( 1 + T )^{6} \)
$19$ \( 1 + 14 T + 142 T^{2} + 1032 T^{3} + 6430 T^{4} + 33630 T^{5} + 158058 T^{6} + 638970 T^{7} + 2321230 T^{8} + 7078488 T^{9} + 18505582 T^{10} + 34665386 T^{11} + 47045881 T^{12} \)
$23$ \( 1 + 8 T + 58 T^{2} + 239 T^{3} + 1660 T^{4} + 5627 T^{5} + 33542 T^{6} + 129421 T^{7} + 878140 T^{8} + 2907913 T^{9} + 16230778 T^{10} + 51490744 T^{11} + 148035889 T^{12} \)
$29$ \( 1 + 21 T + 308 T^{2} + 3233 T^{3} + 27644 T^{4} + 192116 T^{5} + 1134652 T^{6} + 5571364 T^{7} + 23248604 T^{8} + 78849637 T^{9} + 217842548 T^{10} + 430734129 T^{11} + 594823321 T^{12} \)
$31$ \( 1 + 5 T + 120 T^{2} + 335 T^{3} + 6220 T^{4} + 10152 T^{5} + 216224 T^{6} + 314712 T^{7} + 5977420 T^{8} + 9979985 T^{9} + 110822520 T^{10} + 143145755 T^{11} + 887503681 T^{12} \)
$37$ \( 1 + 13 T + 174 T^{2} + 1079 T^{3} + 7478 T^{4} + 24414 T^{5} + 184720 T^{6} + 903318 T^{7} + 10237382 T^{8} + 54654587 T^{9} + 326104014 T^{10} + 901471441 T^{11} + 2565726409 T^{12} \)
$41$ \( 1 + 18 T + 312 T^{2} + 3403 T^{3} + 34028 T^{4} + 263831 T^{5} + 1877422 T^{6} + 10817071 T^{7} + 57201068 T^{8} + 234538163 T^{9} + 881637432 T^{10} + 2085411618 T^{11} + 4750104241 T^{12} \)
$43$ \( 1 + 4 T + 82 T^{2} - 119 T^{3} + 3276 T^{4} - 7875 T^{5} + 196154 T^{6} - 338625 T^{7} + 6057324 T^{8} - 9461333 T^{9} + 280341682 T^{10} + 588033772 T^{11} + 6321363049 T^{12} \)
$47$ \( 1 + 4 T + 138 T^{2} + 619 T^{3} + 10402 T^{4} + 47159 T^{5} + 541662 T^{6} + 2216473 T^{7} + 22978018 T^{8} + 64266437 T^{9} + 673395978 T^{10} + 917380028 T^{11} + 10779215329 T^{12} \)
$53$ \( 1 + 25 T + 485 T^{2} + 6416 T^{3} + 71862 T^{4} + 651131 T^{5} + 5175170 T^{6} + 34509943 T^{7} + 201860358 T^{8} + 955194832 T^{9} + 3826883285 T^{10} + 10454887325 T^{11} + 22164361129 T^{12} \)
$59$ \( ( 1 + T )^{6} \)
$61$ \( 1 + 308 T^{2} + 129 T^{3} + 42246 T^{4} + 21563 T^{5} + 3317184 T^{6} + 1315343 T^{7} + 157197366 T^{8} + 29280549 T^{9} + 4264519028 T^{10} + 51520374361 T^{12} \)
$67$ \( 1 + 13 T + 353 T^{2} + 3480 T^{3} + 54244 T^{4} + 421823 T^{5} + 4719308 T^{6} + 28262141 T^{7} + 243501316 T^{8} + 1046655240 T^{9} + 7113345713 T^{10} + 17551626391 T^{11} + 90458382169 T^{12} \)
$71$ \( 1 + 12 T + 208 T^{2} + 1643 T^{3} + 14798 T^{4} + 127067 T^{5} + 921380 T^{6} + 9021757 T^{7} + 74596718 T^{8} + 588047773 T^{9} + 5285629648 T^{10} + 21650752212 T^{11} + 128100283921 T^{12} \)
$73$ \( 1 + 4 T + 84 T^{2} - 1103 T^{3} + 3268 T^{4} - 32085 T^{5} + 1185158 T^{6} - 2342205 T^{7} + 17415172 T^{8} - 429085751 T^{9} + 2385452244 T^{10} + 8292286372 T^{11} + 151334226289 T^{12} \)
$79$ \( 1 + 12 T + 412 T^{2} + 3560 T^{3} + 70624 T^{4} + 471276 T^{5} + 7011018 T^{6} + 37230804 T^{7} + 440764384 T^{8} + 1755218840 T^{9} + 16047433372 T^{10} + 36924676788 T^{11} + 243087455521 T^{12} \)
$83$ \( 1 - 9 T + 316 T^{2} - 3677 T^{3} + 48348 T^{4} - 599848 T^{5} + 4785622 T^{6} - 49787384 T^{7} + 333069372 T^{8} - 2102460799 T^{9} + 14996829436 T^{10} - 35451365787 T^{11} + 326940373369 T^{12} \)
$89$ \( 1 + 11 T + 526 T^{2} + 4607 T^{3} + 115612 T^{4} + 794778 T^{5} + 13655926 T^{6} + 70735242 T^{7} + 915762652 T^{8} + 3247792183 T^{9} + 33002418766 T^{10} + 61424653939 T^{11} + 496981290961 T^{12} \)
$97$ \( 1 - 16 T + 291 T^{2} - 4292 T^{3} + 61432 T^{4} - 686420 T^{5} + 6799904 T^{6} - 66582740 T^{7} + 578013688 T^{8} - 3917192516 T^{9} + 25762020771 T^{10} - 137397444112 T^{11} + 832972004929 T^{12} \)
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