Properties

Label 6018.2.a.o
Level 6018
Weight 2
Character orbit 6018.a
Self dual yes
Analytic conductor 48.054
Analytic rank 1
Dimension 4
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: \(4\)
Coefficient field: 4.4.725.1
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 3 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 1 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 2 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 1\)
\(\nu^{3}\)\(=\)\(\beta_{3} + \beta_{2} + 3 \beta_{1}\)

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
−1.35567
0.737640
−0.477260
2.09529
1.00000 1.00000 1.00000 −2.97371 1.00000 4.16724 1.00000 1.00000 −2.97371
1.2 1.00000 1.00000 1.00000 −0.880394 1.00000 −1.31313 1.00000 1.00000 −0.880394
1.3 1.00000 1.00000 1.00000 0.140774 1.00000 −1.43574 1.00000 1.00000 0.140774
1.4 1.00000 1.00000 1.00000 2.71333 1.00000 −2.41837 1.00000 1.00000 2.71333
\(n\): e.g. 2-40 or 990-1000
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.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.o 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

\( T_{5}^{4} + T_{5}^{3} - 8 T_{5}^{2} - 6 T_{5} + 1 \)
\( T_{7}^{4} + T_{7}^{3} - 13 T_{7}^{2} - 31 T_{7} - 19 \)