Properties

Label 102.2.a.c
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

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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

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

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
1.00000 1.00000 1.00000 −2.00000 1.00000 0 1.00000 1.00000 −2.00000
\(n\): 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 102.2.a.c 1
3.b odd 2 1 306.2.a.b 1
4.b odd 2 1 816.2.a.b 1
5.b even 2 1 2550.2.a.c 1
5.c odd 4 2 2550.2.d.m 2
7.b odd 2 1 4998.2.a.be 1
8.b even 2 1 3264.2.a.m 1
8.d odd 2 1 3264.2.a.bc 1
12.b even 2 1 2448.2.a.p 1
15.d odd 2 1 7650.2.a.ca 1
17.b even 2 1 1734.2.a.j 1
17.c even 4 2 1734.2.b.b 2
17.d even 8 4 1734.2.f.e 4
24.f even 2 1 9792.2.a.l 1
24.h odd 2 1 9792.2.a.k 1
51.c odd 2 1 5202.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.a.c 1 1.a even 1 1 trivial
306.2.a.b 1 3.b odd 2 1
816.2.a.b 1 4.b odd 2 1
1734.2.a.j 1 17.b even 2 1
1734.2.b.b 2 17.c even 4 2
1734.2.f.e 4 17.d even 8 4
2448.2.a.p 1 12.b even 2 1
2550.2.a.c 1 5.b even 2 1
2550.2.d.m 2 5.c odd 4 2
3264.2.a.m 1 8.b even 2 1
3264.2.a.bc 1 8.d odd 2 1
4998.2.a.be 1 7.b odd 2 1
5202.2.a.c 1 51.c odd 2 1
7650.2.a.ca 1 15.d odd 2 1
9792.2.a.k 1 24.h odd 2 1
9792.2.a.l 1 24.f even 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 \( T_{5} + 2 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(102))\).

Hecke characteristic polynomials

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