Properties

Label 6018.2.a.w
Level 6018
Weight 2
Character orbit 6018.a
Self dual yes
Analytic conductor 48.054
Analytic rank 0
Dimension 9
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: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - x^{8} - 22 x^{7} + 20 x^{6} + 129 x^{5} - 106 x^{4} - 126 x^{3} + 48 x^{2} + 24 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{8} - 4 \nu^{7} + 74 \nu^{6} + 91 \nu^{5} - 543 \nu^{4} - 533 \nu^{3} + 1171 \nu^{2} + 504 \nu - 300 \)\()/26\)
\(\beta_{3}\)\(=\)\((\)\( -45 \nu^{8} + 5 \nu^{7} + 1032 \nu^{6} - 26 \nu^{5} - 6637 \nu^{4} - 234 \nu^{3} + 9908 \nu^{2} + 1840 \nu - 1900 \)\()/52\)
\(\beta_{4}\)\(=\)\((\)\( -21 \nu^{8} - 28 \nu^{7} + 518 \nu^{6} + 637 \nu^{5} - 3827 \nu^{4} - 3653 \nu^{3} + 8405 \nu^{2} + 2774 \nu - 1866 \)\()/26\)
\(\beta_{5}\)\(=\)\((\)\( 41 \nu^{8} - 84 \nu^{7} - 838 \nu^{6} + 1729 \nu^{5} + 3989 \nu^{4} - 9139 \nu^{3} + 1633 \nu^{2} + 3434 \nu - 1152 \)\()/26\)
\(\beta_{6}\)\(=\)\((\)\( -137 \nu^{8} + 125 \nu^{7} + 2998 \nu^{6} - 2444 \nu^{5} - 17309 \nu^{4} + 12350 \nu^{3} + 15182 \nu^{2} - 1944 \nu - 1740 \)\()/52\)
\(\beta_{7}\)\(=\)\((\)\( 11 \nu^{8} - 9 \nu^{7} - 242 \nu^{6} + 174 \nu^{5} + 1415 \nu^{4} - 868 \nu^{3} - 1344 \nu^{2} + 68 \nu + 160 \)\()/4\)
\(\beta_{8}\)\(=\)\((\)\( -134 \nu^{8} + 129 \nu^{7} + 2924 \nu^{6} - 2535 \nu^{5} - 16766 \nu^{4} + 12857 \nu^{3} + 14011 \nu^{2} - 2162 \nu - 1414 \)\()/26\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(-\beta_{8} + 2 \beta_{6} - \beta_{2} + 11 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-3 \beta_{8} + 8 \beta_{7} + 14 \beta_{6} - \beta_{4} + 12 \beta_{2} + 12 \beta_{1} + 52\)
\(\nu^{5}\)\(=\)\(-17 \beta_{8} + 3 \beta_{7} + 29 \beta_{6} - 7 \beta_{5} + 4 \beta_{4} + 8 \beta_{3} - 15 \beta_{2} + 125 \beta_{1} + 22\)
\(\nu^{6}\)\(=\)\(-61 \beta_{8} + 58 \beta_{7} + 186 \beta_{6} + 9 \beta_{5} - 34 \beta_{4} + 10 \beta_{3} + 146 \beta_{2} + 150 \beta_{1} + 595\)
\(\nu^{7}\)\(=\)\(-248 \beta_{8} + 66 \beta_{7} + 398 \beta_{6} - 139 \beta_{5} + 68 \beta_{4} + 184 \beta_{3} - 193 \beta_{2} + 1469 \beta_{1} + 408\)
\(\nu^{8}\)\(=\)\(-969 \beta_{8} + 376 \beta_{7} + 2438 \beta_{6} + 195 \beta_{5} - 627 \beta_{4} + 244 \beta_{3} + 1791 \beta_{2} + 1965 \beta_{1} + 7062\)

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
−3.44871
−2.94492
−0.637772
−0.621481
0.354931
0.374254
1.44979
2.83873
3.63517
−1.00000 1.00000 1.00000 −3.44871 −1.00000 −0.188516 −1.00000 1.00000 3.44871
1.2 −1.00000 1.00000 1.00000 −2.94492 −1.00000 −3.69752 −1.00000 1.00000 2.94492
1.3 −1.00000 1.00000 1.00000 −0.637772 −1.00000 −2.55112 −1.00000 1.00000 0.637772
1.4 −1.00000 1.00000 1.00000 −0.621481 −1.00000 2.55275 −1.00000 1.00000 0.621481
1.5 −1.00000 1.00000 1.00000 0.354931 −1.00000 1.61788 −1.00000 1.00000 −0.354931
1.6 −1.00000 1.00000 1.00000 0.374254 −1.00000 −3.42020 −1.00000 1.00000 −0.374254
1.7 −1.00000 1.00000 1.00000 1.44979 −1.00000 4.37806 −1.00000 1.00000 −1.44979
1.8 −1.00000 1.00000 1.00000 2.83873 −1.00000 −0.0272276 −1.00000 1.00000 −2.83873
1.9 −1.00000 1.00000 1.00000 3.63517 −1.00000 1.33590 −1.00000 1.00000 −3.63517
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.w 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.w 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\(T_{5}^{9} - \cdots\)
\( T_{7}^{9} - 31 T_{7}^{7} + 284 T_{7}^{5} - 93 T_{7}^{4} - 836 T_{7}^{3} + 605 T_{7}^{2} + 164 T_{7} + 4 \)