Properties

Label 8030.2.a.k
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: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.30278
−1.30278
−1.00000 −2.30278 1.00000 −1.00000 2.30278 2.60555 −1.00000 2.30278 1.00000
1.2 −1.00000 1.30278 1.00000 −1.00000 −1.30278 −4.60555 −1.00000 −1.30278 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.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.k 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\( T_{3}^{2} + T_{3} - 3 \)
\( T_{7}^{2} + 2 T_{7} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( 1 + T + 3 T^{2} + 3 T^{3} + 9 T^{4} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( 1 + 2 T + 2 T^{2} + 14 T^{3} + 49 T^{4} \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( 1 - 3 T + 25 T^{2} - 39 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 17 T^{2} )^{2} \)
$19$ \( 1 + T + 35 T^{2} + 19 T^{3} + 361 T^{4} \)
$23$ \( ( 1 - 2 T + 23 T^{2} )^{2} \)
$29$ \( 1 - 2 T + 46 T^{2} - 58 T^{3} + 841 T^{4} \)
$31$ \( 1 + 6 T + 58 T^{2} + 186 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 2 T + 37 T^{2} )^{2} \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 5 T + 89 T^{2} - 215 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 3 T + 15 T^{2} + 141 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 4 T + 58 T^{2} - 212 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 11 T + 145 T^{2} + 649 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 19 T + 183 T^{2} + 1159 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 3 T + 107 T^{2} + 201 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 5 T - 11 T^{2} - 355 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 - T )^{2} \)
$79$ \( 1 + 10 T + 170 T^{2} + 790 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 11 T + 115 T^{2} - 913 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 15 T + 205 T^{2} + 1335 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 12 T + 178 T^{2} + 1164 T^{3} + 9409 T^{4} \)
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