Properties

Label 8030.2.a.n
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.

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

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 −3.30278 1.00000 1.00000 −3.30278 0 1.00000 7.90833 1.00000
1.2 1.00000 0.302776 1.00000 1.00000 0.302776 0 1.00000 −2.90833 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.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.n 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

\( T_{3}^{2} + 3 T_{3} - 1 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( 1 + 3 T + 5 T^{2} + 9 T^{3} + 9 T^{4} \)
$5$ \( ( 1 - T )^{2} \)
$7$ \( ( 1 + 7 T^{2} )^{2} \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( 1 - 7 T + 35 T^{2} - 91 T^{3} + 169 T^{4} \)
$17$ \( 1 + 6 T + 30 T^{2} + 102 T^{3} + 289 T^{4} \)
$19$ \( 1 + 7 T + 47 T^{2} + 133 T^{3} + 361 T^{4} \)
$23$ \( 1 + 2 T + 34 T^{2} + 46 T^{3} + 529 T^{4} \)
$29$ \( 1 - 2 T + 46 T^{2} - 58 T^{3} + 841 T^{4} \)
$31$ \( ( 1 + 6 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 + 6 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 10 T + 94 T^{2} + 410 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 5 T + 89 T^{2} + 215 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 7 T + 77 T^{2} + 329 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 18 T + 174 T^{2} - 954 T^{3} + 2809 T^{4} \)
$59$ \( 1 + T + 115 T^{2} + 59 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 7 T + 105 T^{2} + 427 T^{3} + 3721 T^{4} \)
$67$ \( 1 - 7 T + 117 T^{2} - 469 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 11 T + 91 T^{2} + 781 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 - T )^{2} \)
$79$ \( ( 1 + 2 T + 79 T^{2} )^{2} \)
$83$ \( 1 + 7 T + 175 T^{2} + 581 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 15 T + 205 T^{2} + 1335 T^{3} + 7921 T^{4} \)
$97$ \( 1 + 22 T + 302 T^{2} + 2134 T^{3} + 9409 T^{4} \)
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