Properties

Label 8034.2.a.m
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{65}) \)
Defining polynomial: \(x^{2} - x - 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{65})\). We also show the integral \(q\)-expansion of the trace form.

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

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
4.53113
−3.53113
−1.00000 −1.00000 1.00000 0 1.00000 1.00000 −1.00000 1.00000 0
1.2 −1.00000 −1.00000 1.00000 0 1.00000 1.00000 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(13\) \(-1\)
\(103\) \(-1\)

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 8034.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.m 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5} \)
\( T_{7} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( T^{2} \)
$7$ \( ( -1 + T )^{2} \)
$11$ \( ( -3 + T )^{2} \)
$13$ \( ( -1 + T )^{2} \)
$17$ \( -14 - 3 T + T^{2} \)
$19$ \( T^{2} \)
$23$ \( -10 + 5 T + T^{2} \)
$29$ \( T^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( -4 + 7 T + T^{2} \)
$41$ \( -16 + T + T^{2} \)
$43$ \( -16 + T + T^{2} \)
$47$ \( 4 + 9 T + T^{2} \)
$53$ \( -14 - 3 T + T^{2} \)
$59$ \( -64 + 2 T + T^{2} \)
$61$ \( 26 + 13 T + T^{2} \)
$67$ \( 56 + 17 T + T^{2} \)
$71$ \( -16 + 14 T + T^{2} \)
$73$ \( -126 + 9 T + T^{2} \)
$79$ \( -64 + 2 T + T^{2} \)
$83$ \( -64 - 2 T + T^{2} \)
$89$ \( -40 + 10 T + T^{2} \)
$97$ \( -256 + 4 T + T^{2} \)
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