Properties

Label 6034.2.a.h
Level 6034
Weight 2
Character orbit 6034.a
Self dual yes
Analytic conductor 48.182
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6034 = 2 \cdot 7 \cdot 431 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1817325796\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

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

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
1.61803
−0.618034
1.00000 −1.00000 1.00000 −2.23607 −1.00000 −1.00000 1.00000 −2.00000 −2.23607
1.2 1.00000 −1.00000 1.00000 2.23607 −1.00000 −1.00000 1.00000 −2.00000 2.23607
\(n\): 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 6034.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6034.2.a.h 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(431\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6034))\):

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( ( 1 + T + 3 T^{2} )^{2} \)
$5$ \( 1 + 5 T^{2} + 25 T^{4} \)
$7$ \( ( 1 + T )^{2} \)
$11$ \( 1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4} \)
$13$ \( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} \)
$17$ \( 1 + 6 T + 38 T^{2} + 102 T^{3} + 289 T^{4} \)
$19$ \( ( 1 - 3 T + 19 T^{2} )^{2} \)
$23$ \( 1 + 8 T + 57 T^{2} + 184 T^{3} + 529 T^{4} \)
$29$ \( 1 + 2 T + 39 T^{2} + 58 T^{3} + 841 T^{4} \)
$31$ \( 1 + 6 T + 26 T^{2} + 186 T^{3} + 961 T^{4} \)
$37$ \( 1 - 4 T - 2 T^{2} - 148 T^{3} + 1369 T^{4} \)
$41$ \( 1 + 62 T^{2} + 1681 T^{4} \)
$43$ \( 1 + 4 T + 10 T^{2} + 172 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 4 T + 78 T^{2} - 188 T^{3} + 2209 T^{4} \)
$53$ \( ( 1 - 3 T + 53 T^{2} )^{2} \)
$59$ \( ( 1 + 3 T + 59 T^{2} )^{2} \)
$61$ \( 1 + 16 T + 166 T^{2} + 976 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 6 T + 98 T^{2} - 402 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 6 T + 106 T^{2} + 426 T^{3} + 5041 T^{4} \)
$73$ \( 1 + 10 T + 166 T^{2} + 730 T^{3} + 5329 T^{4} \)
$79$ \( ( 1 + 12 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 22 T + 282 T^{2} + 1826 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 4 T + 162 T^{2} + 356 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 4 T + 153 T^{2} + 388 T^{3} + 9409 T^{4} \)
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