Properties

Label 6042.2.a.p
Level 6042
Weight 2
Character orbit 6042.a
Self dual yes
Analytic conductor 48.246
Analytic rank 1
Dimension 2
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.56155
2.56155
1.00000 1.00000 1.00000 −1.56155 1.00000 1.56155 1.00000 1.00000 −1.56155
1.2 1.00000 1.00000 1.00000 2.56155 1.00000 −2.56155 1.00000 1.00000 2.56155
\(n\): 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.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.p 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{2} - T_{5} - 4 \)
\( T_{7}^{2} + T_{7} - 4 \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( ( 1 - T )^{2} \)
$5$ \( 1 - T + 6 T^{2} - 5 T^{3} + 25 T^{4} \)
$7$ \( 1 + T + 10 T^{2} + 7 T^{3} + 49 T^{4} \)
$11$ \( ( 1 + 4 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 + 4 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 + 2 T + 17 T^{2} )^{2} \)
$19$ \( ( 1 - T )^{2} \)
$23$ \( 1 + 5 T + 48 T^{2} + 115 T^{3} + 529 T^{4} \)
$29$ \( 1 - T + 20 T^{2} - 29 T^{3} + 841 T^{4} \)
$31$ \( 1 + 5 T + 64 T^{2} + 155 T^{3} + 961 T^{4} \)
$37$ \( 1 - 6 T + 66 T^{2} - 222 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 2 T + 66 T^{2} + 82 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 15 T + 138 T^{2} + 645 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 26 T^{2} + 2209 T^{4} \)
$53$ \( ( 1 - T )^{2} \)
$59$ \( 1 - 5 T + 18 T^{2} - 295 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 18 T + 186 T^{2} + 1098 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 11 T + 126 T^{2} - 737 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 4 T + 78 T^{2} - 284 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 + 10 T + 73 T^{2} )^{2} \)
$79$ \( 1 + 18 T + 222 T^{2} + 1422 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 - 4 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 21 T + 284 T^{2} + 1869 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 126 T^{2} + 9409 T^{4} \)
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