Properties

Label 8018.2.a.c
Level $8018$
Weight $2$
Character orbit 8018.a
Self dual yes
Analytic conductor $64.024$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8018 = 2 \cdot 19 \cdot 211 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8018.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0240523407\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Defining polynomial: \(x^{2} - x - 1\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−0.618034
1.61803
−1.00000 0.381966 1.00000 −2.23607 −0.381966 2.00000 −1.00000 −2.85410 2.23607
1.2 −1.00000 2.61803 1.00000 2.23607 −2.61803 2.00000 −1.00000 3.85410 −2.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(19\) \(-1\)
\(211\) \(1\)

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 8018.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8018.2.a.c 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8018))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( 1 - 3 T + 7 T^{2} - 9 T^{3} + 9 T^{4} \)
$5$ \( 1 + 5 T^{2} + 25 T^{4} \)
$7$ \( ( 1 - 2 T + 7 T^{2} )^{2} \)
$11$ \( 1 + 9 T + 41 T^{2} + 99 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )^{2} \)
$17$ \( 1 + 29 T^{2} + 289 T^{4} \)
$19$ \( ( 1 - T )^{2} \)
$23$ \( 1 + 12 T + 77 T^{2} + 276 T^{3} + 529 T^{4} \)
$29$ \( 1 + 3 T + 49 T^{2} + 87 T^{3} + 841 T^{4} \)
$31$ \( 1 - T - 39 T^{2} - 31 T^{3} + 961 T^{4} \)
$37$ \( 1 - 2 T + 30 T^{2} - 74 T^{3} + 1369 T^{4} \)
$41$ \( 1 - 6 T + 71 T^{2} - 246 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 7 T + 87 T^{2} + 301 T^{3} + 1849 T^{4} \)
$47$ \( 1 - 12 T + 125 T^{2} - 564 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 12 T + 122 T^{2} - 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 15 T + 163 T^{2} - 885 T^{3} + 3481 T^{4} \)
$61$ \( ( 1 - 3 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 16 T + 153 T^{2} - 1072 T^{3} + 4489 T^{4} \)
$71$ \( 1 - 9 T + 161 T^{2} - 639 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 4 T + 105 T^{2} - 292 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 10 T + 138 T^{2} - 790 T^{3} + 6241 T^{4} \)
$83$ \( 1 - 3 T + 137 T^{2} - 249 T^{3} + 6889 T^{4} \)
$89$ \( 1 - 12 T + 209 T^{2} - 1068 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 13 T + 97 T^{2} )^{2} \)
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