Properties

Label 102.2.a.b
Level 102
Weight 2
Character orbit 102.a
Self dual yes
Analytic conductor 0.814
Analytic rank 0
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 102.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(0.814474100617\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{6} + 2q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{6} + 2q^{7} - q^{8} + q^{9} + q^{12} + 2q^{13} - 2q^{14} + q^{16} - q^{17} - q^{18} - 4q^{19} + 2q^{21} - 6q^{23} - q^{24} - 5q^{25} - 2q^{26} + q^{27} + 2q^{28} - 10q^{31} - q^{32} + q^{34} + q^{36} + 8q^{37} + 4q^{38} + 2q^{39} + 6q^{41} - 2q^{42} - 4q^{43} + 6q^{46} + 12q^{47} + q^{48} - 3q^{49} + 5q^{50} - q^{51} + 2q^{52} + 6q^{53} - q^{54} - 2q^{56} - 4q^{57} - 12q^{59} + 8q^{61} + 10q^{62} + 2q^{63} + q^{64} - 4q^{67} - q^{68} - 6q^{69} + 6q^{71} - q^{72} + 2q^{73} - 8q^{74} - 5q^{75} - 4q^{76} - 2q^{78} - 10q^{79} + q^{81} - 6q^{82} + 12q^{83} + 2q^{84} + 4q^{86} - 18q^{89} + 4q^{91} - 6q^{92} - 10q^{93} - 12q^{94} - q^{96} + 14q^{97} + 3q^{98} + O(q^{100}) \)

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
−1.00000 1.00000 1.00000 0 −1.00000 2.00000 −1.00000 1.00000 0
\(n\): 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 102.2.a.b 1
3.b odd 2 1 306.2.a.c 1
4.b odd 2 1 816.2.a.d 1
5.b even 2 1 2550.2.a.u 1
5.c odd 4 2 2550.2.d.g 2
7.b odd 2 1 4998.2.a.d 1
8.b even 2 1 3264.2.a.i 1
8.d odd 2 1 3264.2.a.w 1
12.b even 2 1 2448.2.a.i 1
15.d odd 2 1 7650.2.a.j 1
17.b even 2 1 1734.2.a.b 1
17.c even 4 2 1734.2.b.f 2
17.d even 8 4 1734.2.f.b 4
24.f even 2 1 9792.2.a.ba 1
24.h odd 2 1 9792.2.a.bg 1
51.c odd 2 1 5202.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.b 1 1.a even 1 1 trivial
306.2.a.c 1 3.b odd 2 1
816.2.a.d 1 4.b odd 2 1
1734.2.a.b 1 17.b even 2 1
1734.2.b.f 2 17.c even 4 2
1734.2.f.b 4 17.d even 8 4
2448.2.a.i 1 12.b even 2 1
2550.2.a.u 1 5.b even 2 1
2550.2.d.g 2 5.c odd 4 2
3264.2.a.i 1 8.b even 2 1
3264.2.a.w 1 8.d odd 2 1
4998.2.a.d 1 7.b odd 2 1
5202.2.a.j 1 51.c odd 2 1
7650.2.a.j 1 15.d odd 2 1
9792.2.a.ba 1 24.f even 2 1
9792.2.a.bg 1 24.h odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(17\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(102))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 - T \)
$5$ \( 1 + 5 T^{2} \)
$7$ \( 1 - 2 T + 7 T^{2} \)
$11$ \( 1 + 11 T^{2} \)
$13$ \( 1 - 2 T + 13 T^{2} \)
$17$ \( 1 + T \)
$19$ \( 1 + 4 T + 19 T^{2} \)
$23$ \( 1 + 6 T + 23 T^{2} \)
$29$ \( 1 + 29 T^{2} \)
$31$ \( 1 + 10 T + 31 T^{2} \)
$37$ \( 1 - 8 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 + 4 T + 43 T^{2} \)
$47$ \( 1 - 12 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 12 T + 59 T^{2} \)
$61$ \( 1 - 8 T + 61 T^{2} \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 - 6 T + 71 T^{2} \)
$73$ \( 1 - 2 T + 73 T^{2} \)
$79$ \( 1 + 10 T + 79 T^{2} \)
$83$ \( 1 - 12 T + 83 T^{2} \)
$89$ \( 1 + 18 T + 89 T^{2} \)
$97$ \( 1 - 14 T + 97 T^{2} \)
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