Properties

Label 6046.2.a.c
Level 6046
Weight 2
Character orbit 6046.a
Self dual yes
Analytic conductor 48.278
Analytic rank 2
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6046 = 2 \cdot 3023 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6046.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.2775530621\)
Analytic rank: \(2\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
1.61803
−0.618034
1.00000 −2.61803 1.00000 −3.00000 −2.61803 −1.38197 1.00000 3.85410 −3.00000
1.2 1.00000 −0.381966 1.00000 −3.00000 −0.381966 −3.61803 1.00000 −2.85410 −3.00000
\(n\): 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 6046.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6046.2.a.c 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3023\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6046))\):

\( T_{3}^{2} + 3 T_{3} + 1 \)
\( T_{11}^{2} + 3 T_{11} + 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( 1 + 3 T + 7 T^{2} + 9 T^{3} + 9 T^{4} \)
$5$ \( ( 1 + 3 T + 5 T^{2} )^{2} \)
$7$ \( 1 + 5 T + 19 T^{2} + 35 T^{3} + 49 T^{4} \)
$11$ \( 1 + 3 T + 23 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( 1 + 8 T + 37 T^{2} + 104 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 5 T + 17 T^{2} )^{2} \)
$19$ \( 1 + 11 T + 67 T^{2} + 209 T^{3} + 361 T^{4} \)
$23$ \( 1 + 8 T + 57 T^{2} + 184 T^{3} + 529 T^{4} \)
$29$ \( 1 - 2 T + 54 T^{2} - 58 T^{3} + 841 T^{4} \)
$31$ \( 1 + 42 T^{2} + 961 T^{4} \)
$37$ \( 1 + 6 T + 63 T^{2} + 222 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 13 T + 93 T^{2} + 533 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 3 T + 77 T^{2} - 129 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 6 T + 83 T^{2} + 282 T^{3} + 2209 T^{4} \)
$53$ \( 1 - T + 105 T^{2} - 53 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 8 T + 89 T^{2} + 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 7 T + 103 T^{2} + 427 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 + 8 T + 67 T^{2} )^{2} \)
$71$ \( 1 - 6 T + 106 T^{2} - 426 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 66 T^{2} + 5329 T^{4} \)
$79$ \( 1 - 10 T + 163 T^{2} - 790 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 8 T + 102 T^{2} + 664 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 158 T^{2} + 7921 T^{4} \)
$97$ \( 1 + 4 T + 178 T^{2} + 388 T^{3} + 9409 T^{4} \)
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