Properties

Label 8030.2.a.p
Level 8030
Weight 2
Character orbit 8030.a
Self dual yes
Analytic conductor 64.120
Analytic rank 0
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: \(0\)
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.

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

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 4.30278 1.00000 2.30278 −1.00000
1.2 1.00000 1.30278 1.00000 −1.00000 1.30278 0.697224 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.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.p 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8030))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( 1 + T + 3 T^{2} + 3 T^{3} + 9 T^{4} \)
$5$ \( ( 1 + T )^{2} \)
$7$ \( 1 - 5 T + 17 T^{2} - 35 T^{3} + 49 T^{4} \)
$11$ \( ( 1 - T )^{2} \)
$13$ \( 1 - 9 T + 43 T^{2} - 117 T^{3} + 169 T^{4} \)
$17$ \( 1 + 11 T + 61 T^{2} + 187 T^{3} + 289 T^{4} \)
$19$ \( 1 - 2 T + 26 T^{2} - 38 T^{3} + 361 T^{4} \)
$23$ \( 1 + 5 T + 23 T^{2} + 115 T^{3} + 529 T^{4} \)
$29$ \( 1 - 3 T + 31 T^{2} - 87 T^{3} + 841 T^{4} \)
$31$ \( 1 + 10 T^{2} + 961 T^{4} \)
$37$ \( 1 + 5 T + 51 T^{2} + 185 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 12 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 8 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 - 2 T + 47 T^{2} )^{2} \)
$53$ \( 1 - 10 T + 118 T^{2} - 530 T^{3} + 2809 T^{4} \)
$59$ \( ( 1 + 2 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 2 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 7 T + 143 T^{2} + 469 T^{3} + 4489 T^{4} \)
$71$ \( 1 + 5 T + 145 T^{2} + 355 T^{3} + 5041 T^{4} \)
$73$ \( ( 1 - T )^{2} \)
$79$ \( 1 - 8 T + 122 T^{2} - 632 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 11 T + 115 T^{2} - 913 T^{3} + 6889 T^{4} \)
$89$ \( 1 + 17 T + 169 T^{2} + 1513 T^{3} + 7921 T^{4} \)
$97$ \( 1 - 11 T + 221 T^{2} - 1067 T^{3} + 9409 T^{4} \)
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