Properties

Label 195.2.a.a
Level $195$
Weight $2$
Character orbit 195.a
Self dual yes
Analytic conductor $1.557$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.55708283941\)
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

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

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 1.00000 −1.00000 0 3.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(-1\)

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 195.2.a.a 1
3.b odd 2 1 585.2.a.g 1
4.b odd 2 1 3120.2.a.k 1
5.b even 2 1 975.2.a.i 1
5.c odd 4 2 975.2.c.e 2
7.b odd 2 1 9555.2.a.b 1
12.b even 2 1 9360.2.a.o 1
13.b even 2 1 2535.2.a.k 1
15.d odd 2 1 2925.2.a.d 1
15.e even 4 2 2925.2.c.f 2
39.d odd 2 1 7605.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.a 1 1.a even 1 1 trivial
585.2.a.g 1 3.b odd 2 1
975.2.a.i 1 5.b even 2 1
975.2.c.e 2 5.c odd 4 2
2535.2.a.k 1 13.b even 2 1
2925.2.a.d 1 15.d odd 2 1
2925.2.c.f 2 15.e even 4 2
3120.2.a.k 1 4.b odd 2 1
7605.2.a.h 1 39.d odd 2 1
9360.2.a.o 1 12.b even 2 1
9555.2.a.b 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(195))\):

\( T_{2} + 1 \)
\( T_{7} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -1 + T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( -4 + T \)
$13$ \( -1 + T \)
$17$ \( -2 + T \)
$19$ \( 4 + T \)
$23$ \( -8 + T \)
$29$ \( 2 + T \)
$31$ \( 8 + T \)
$37$ \( -6 + T \)
$41$ \( 6 + T \)
$43$ \( 4 + T \)
$47$ \( 8 + T \)
$53$ \( -6 + T \)
$59$ \( 12 + T \)
$61$ \( 2 + T \)
$67$ \( 4 + T \)
$71$ \( T \)
$73$ \( 6 + T \)
$79$ \( -16 + T \)
$83$ \( 4 + T \)
$89$ \( -10 + T \)
$97$ \( -18 + T \)
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