Properties

Label 8030.2.a.q
Level 8030
Weight 2
Character orbit 8030.a
Self dual yes
Analytic conductor 64.120
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8030 = 2 \cdot 5 \cdot 11 \cdot 73 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8030.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1198728231\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.56155
2.56155
1.00000 −1.56155 1.00000 −1.00000 −1.56155 −0.438447 1.00000 −0.561553 −1.00000
1.2 1.00000 2.56155 1.00000 −1.00000 2.56155 −4.56155 1.00000 3.56155 −1.00000
\(n\): 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 8030.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.q 2 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(11\) \(-1\)
\(73\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( 1 - T + 2 T^{2} - 3 T^{3} + 9 T^{4} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( 1 + 5 T + 16 T^{2} + 35 T^{3} + 49 T^{4} \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( ( 1 + 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 17 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - T + 19 T^{2} )^{2} \)
$23$ \( 1 + T + 8 T^{2} + 23 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 7 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 4 T + 49 T^{2} - 124 T^{3} + 961 T^{4} \)
$37$ \( 1 + 57 T^{2} + 1369 T^{4} \)
$41$ \( 1 - 3 T + 46 T^{2} - 123 T^{3} + 1681 T^{4} \)
$43$ \( 1 - 5 T + 88 T^{2} - 215 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 14 T + 126 T^{2} - 658 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 3 T + 2 T^{2} + 159 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 10 T + 126 T^{2} + 590 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 15 T + 174 T^{2} + 915 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 + 12 T + 67 T^{2} )^{2} \)
$71$ \( 1 - T - 66 T^{2} - 71 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 + T )^{2} \)
$79$ \( ( 1 + 2 T + 79 T^{2} )^{2} \)
$83$ \( ( 1 - 4 T + 83 T^{2} )^{2} \)
$89$ \( ( 1 - T + 89 T^{2} )^{2} \)
$97$ \( 1 + 18 T + 258 T^{2} + 1746 T^{3} + 9409 T^{4} \)
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