Properties

Label 6042.2.a.q
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: 3.3.257.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 3\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

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

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
2.19869
−1.91223
0.713538
−1.00000 −1.00000 1.00000 −2.83424 1.00000 −4.03293 −1.00000 1.00000 2.83424
1.2 −1.00000 −1.00000 1.00000 −1.65662 1.00000 1.25561 −1.00000 1.00000 1.65662
1.3 −1.00000 −1.00000 1.00000 1.49086 1.00000 1.77733 −1.00000 1.00000 −1.49086
\(n\): 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.q 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.q 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{3} + 3 T_{5}^{2} - 2 T_{5} - 7 \)
\( T_{7}^{3} + T_{7}^{2} - 10 T_{7} + 9 \)
\( T_{11}^{3} - 5 T_{11}^{2} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( 1 + 3 T + 13 T^{2} + 23 T^{3} + 65 T^{4} + 75 T^{5} + 125 T^{6} \)
$7$ \( 1 + T + 11 T^{2} + 23 T^{3} + 77 T^{4} + 49 T^{5} + 343 T^{6} \)
$11$ \( 1 - 5 T + 33 T^{2} - 101 T^{3} + 363 T^{4} - 605 T^{5} + 1331 T^{6} \)
$13$ \( 1 - 3 T + 37 T^{2} - 71 T^{3} + 481 T^{4} - 507 T^{5} + 2197 T^{6} \)
$17$ \( 1 - 7 T + 57 T^{2} - 239 T^{3} + 969 T^{4} - 2023 T^{5} + 4913 T^{6} \)
$19$ \( ( 1 - T )^{3} \)
$23$ \( 1 - 11 T + 87 T^{2} - 439 T^{3} + 2001 T^{4} - 5819 T^{5} + 12167 T^{6} \)
$29$ \( 1 - 2 T + 10 T^{2} + 11 T^{3} + 290 T^{4} - 1682 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 4 T + 65 T^{2} - 208 T^{3} + 2015 T^{4} - 3844 T^{5} + 29791 T^{6} \)
$37$ \( 1 - 11 T + 143 T^{2} - 833 T^{3} + 5291 T^{4} - 15059 T^{5} + 50653 T^{6} \)
$41$ \( 1 - 14 T + 162 T^{2} - 1103 T^{3} + 6642 T^{4} - 23534 T^{5} + 68921 T^{6} \)
$43$ \( 1 + 11 T + 147 T^{2} + 879 T^{3} + 6321 T^{4} + 20339 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 11 T + 164 T^{2} + 1007 T^{3} + 7708 T^{4} + 24299 T^{5} + 103823 T^{6} \)
$53$ \( ( 1 + T )^{3} \)
$59$ \( 1 - 6 T + 34 T^{2} + 225 T^{3} + 2006 T^{4} - 20886 T^{5} + 205379 T^{6} \)
$61$ \( 1 + 10 T + 138 T^{2} + 1071 T^{3} + 8418 T^{4} + 37210 T^{5} + 226981 T^{6} \)
$67$ \( 1 - 19 T + 297 T^{2} - 2609 T^{3} + 19899 T^{4} - 85291 T^{5} + 300763 T^{6} \)
$71$ \( 1 - 16 T + 281 T^{2} - 2344 T^{3} + 19951 T^{4} - 80656 T^{5} + 357911 T^{6} \)
$73$ \( 1 - 9 T + 205 T^{2} - 1319 T^{3} + 14965 T^{4} - 47961 T^{5} + 389017 T^{6} \)
$79$ \( 1 + 21 T + 183 T^{2} + 1325 T^{3} + 14457 T^{4} + 131061 T^{5} + 493039 T^{6} \)
$83$ \( 1 - 15 T + 193 T^{2} - 1491 T^{3} + 16019 T^{4} - 103335 T^{5} + 571787 T^{6} \)
$89$ \( 1 + 4 T + 220 T^{2} + 659 T^{3} + 19580 T^{4} + 31684 T^{5} + 704969 T^{6} \)
$97$ \( 1 + 11 T + 251 T^{2} + 1613 T^{3} + 24347 T^{4} + 103499 T^{5} + 912673 T^{6} \)
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