Properties

Label 6042.2.a.o
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: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

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

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
3.37228
−2.37228
1.00000 −1.00000 1.00000 −4.37228 −1.00000 1.00000 1.00000 1.00000 −4.37228
1.2 1.00000 −1.00000 1.00000 1.37228 −1.00000 1.00000 1.00000 1.00000 1.37228
\(n\): 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.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.o 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6042))\):

\( T_{5}^{2} + 3 T_{5} - 6 \)
\( T_{7} - 1 \)
\( T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4} \)
$7$ \( ( 1 - T + 7 T^{2} )^{2} \)
$11$ \( ( 1 + 4 T + 11 T^{2} )^{2} \)
$13$ \( 1 + 3 T + 20 T^{2} + 39 T^{3} + 169 T^{4} \)
$17$ \( 1 - 7 T + 38 T^{2} - 119 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - T )^{2} \)
$23$ \( 1 - 3 T + 40 T^{2} - 69 T^{3} + 529 T^{4} \)
$29$ \( 1 + 8 T + 41 T^{2} + 232 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 7 T + 31 T^{2} )^{2} \)
$37$ \( 1 - 9 T + 86 T^{2} - 333 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 15 T + 134 T^{2} - 645 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 3 T + 22 T^{2} + 141 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 + T )^{2} \)
$59$ \( 1 - 16 T + 149 T^{2} - 944 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 5 T + 120 T^{2} + 305 T^{3} + 3721 T^{4} \)
$67$ \( 1 + T + 126 T^{2} + 67 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 19 T + 224 T^{2} + 1349 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 - 4 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 25 T + 314 T^{2} - 2075 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 21 T + 280 T^{2} + 1869 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 14 T + 97 T^{2} )^{2} \)
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