Properties

Label 8034.2.a.n
Level $8034$
Weight $2$
Character orbit 8034.a
Self dual yes
Analytic conductor $64.152$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.72329.1
Defining polynomial: \(x^{4} - x^{3} - 9 x^{2} + 2 x + 15\)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 9 x^{2} + 2 x + 15\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 5 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 5 \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 2 \beta_{2} + 7 \beta_{1} + 5\)

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
1.46917
−1.52193
−2.11664
3.16940
1.00000 1.00000 1.00000 −3.31071 1.00000 3.84154 1.00000 1.00000 −3.31071
1.2 1.00000 1.00000 1.00000 −0.161792 1.00000 3.68372 1.00000 1.00000 −0.161792
1.3 1.00000 1.00000 1.00000 2.59680 1.00000 1.51984 1.00000 1.00000 2.59680
1.4 1.00000 1.00000 1.00000 2.87570 1.00000 −4.04510 1.00000 1.00000 2.87570
\(n\): 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.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.n 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\( T_{5}^{4} - 2 T_{5}^{3} - 11 T_{5}^{2} + 23 T_{5} + 4 \)
\( T_{7}^{4} - 5 T_{7}^{3} - 11 T_{7}^{2} + 82 T_{7} - 87 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( 4 + 23 T - 11 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( -87 + 82 T - 11 T^{2} - 5 T^{3} + T^{4} \)
$11$ \( -72 + 64 T + 10 T^{2} - 9 T^{3} + T^{4} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( ( -2 + T )^{4} \)
$19$ \( 128 - 120 T - 24 T^{2} + 6 T^{3} + T^{4} \)
$23$ \( 191 + 78 T - 24 T^{2} - 6 T^{3} + T^{4} \)
$29$ \( 120 - 13 T - 27 T^{2} + T^{4} \)
$31$ \( 128 + 120 T - 24 T^{2} - 6 T^{3} + T^{4} \)
$37$ \( -72 + 116 T + 88 T^{2} + 17 T^{3} + T^{4} \)
$41$ \( 9 + 56 T + 46 T^{2} - 16 T^{3} + T^{4} \)
$43$ \( 568 - 544 T + 178 T^{2} - 23 T^{3} + T^{4} \)
$47$ \( 1585 - 44 T - 85 T^{2} + T^{3} + T^{4} \)
$53$ \( -4848 + 1696 T - 36 T^{2} - 18 T^{3} + T^{4} \)
$59$ \( 120 - 223 T + 113 T^{2} - 20 T^{3} + T^{4} \)
$61$ \( -232 - 208 T - 22 T^{2} + 9 T^{3} + T^{4} \)
$67$ \( 268 + 161 T - 29 T^{2} - 8 T^{3} + T^{4} \)
$71$ \( -20880 - 3961 T - 77 T^{2} + 21 T^{3} + T^{4} \)
$73$ \( -642 + 113 T + 115 T^{2} - 22 T^{3} + T^{4} \)
$79$ \( -1152 - 264 T + 88 T^{2} + 22 T^{3} + T^{4} \)
$83$ \( 120 + 241 T + 109 T^{2} + 18 T^{3} + T^{4} \)
$89$ \( -1312 - 696 T - 32 T^{2} + 14 T^{3} + T^{4} \)
$97$ \( -41472 + 7544 T - 244 T^{2} - 20 T^{3} + T^{4} \)
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