Properties

Label 6042.2.a.r
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 0
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2456129013\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.788.1
Defining polynomial: \(x^{3} - x^{2} - 7 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 7 x - 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 4\)

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
−0.476452
−1.87740
3.35386
1.00000 −1.00000 1.00000 −3.29654 −1.00000 2.00000 1.00000 1.00000 −3.29654
1.2 1.00000 −1.00000 1.00000 1.40205 −1.00000 2.00000 1.00000 1.00000 1.40205
1.3 1.00000 −1.00000 1.00000 3.89449 −1.00000 2.00000 1.00000 1.00000 3.89449
\(n\): 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 6042.2.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.r 3 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(19\) \(-1\)
\(53\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{3} - 2 T_{5}^{2} - 12 T_{5} + 18 \)
\( T_{7} - 2 \)
\( T_{11}^{3} - 2 T_{11}^{2} - 6 T_{11} + 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 1 - 2 T + 3 T^{2} - 2 T^{3} + 15 T^{4} - 50 T^{5} + 125 T^{6} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{3} \)
$11$ \( 1 - 2 T + 27 T^{2} - 34 T^{3} + 297 T^{4} - 242 T^{5} + 1331 T^{6} \)
$13$ \( 1 - 4 T + 35 T^{2} - 92 T^{3} + 455 T^{4} - 676 T^{5} + 2197 T^{6} \)
$17$ \( 1 + 2 T + 15 T^{2} + 28 T^{3} + 255 T^{4} + 578 T^{5} + 4913 T^{6} \)
$19$ \( ( 1 - T )^{3} \)
$23$ \( 1 + 2 T + 61 T^{2} + 88 T^{3} + 1403 T^{4} + 1058 T^{5} + 12167 T^{6} \)
$29$ \( 1 - 16 T + 163 T^{2} - 1028 T^{3} + 4727 T^{4} - 13456 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 2 T + 49 T^{2} - 160 T^{3} + 1519 T^{4} - 1922 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 4 T + 71 T^{2} - 172 T^{3} + 2627 T^{4} - 5476 T^{5} + 50653 T^{6} \)
$41$ \( ( 1 + 4 T + 41 T^{2} )^{3} \)
$43$ \( 1 + 6 T + 61 T^{2} + 92 T^{3} + 2623 T^{4} + 11094 T^{5} + 79507 T^{6} \)
$47$ \( 1 - 22 T + 289 T^{2} - 2374 T^{3} + 13583 T^{4} - 48598 T^{5} + 103823 T^{6} \)
$53$ \( ( 1 - T )^{3} \)
$59$ \( 1 - 8 T + 169 T^{2} - 928 T^{3} + 9971 T^{4} - 27848 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 6 T + 67 T^{2} + 132 T^{3} + 4087 T^{4} + 22326 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 73 T^{2} - 352 T^{3} + 4891 T^{4} + 300763 T^{6} \)
$71$ \( 1 - 2 T + 169 T^{2} - 320 T^{3} + 11999 T^{4} - 10082 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 8 T + 187 T^{2} - 1120 T^{3} + 13651 T^{4} - 42632 T^{5} + 389017 T^{6} \)
$79$ \( 1 - 14 T + 169 T^{2} - 1128 T^{3} + 13351 T^{4} - 87374 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 4 T + 217 T^{2} + 632 T^{3} + 18011 T^{4} + 27556 T^{5} + 571787 T^{6} \)
$89$ \( 1 - 30 T + 487 T^{2} - 5268 T^{3} + 43343 T^{4} - 237630 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 2 T + 239 T^{2} - 428 T^{3} + 23183 T^{4} - 18818 T^{5} + 912673 T^{6} \)
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