Properties

Label 196.4.a.d
Level $196$
Weight $4$
Character orbit 196.a
Self dual yes
Analytic conductor $11.564$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 196.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.5643743611\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 10q^{3} + 8q^{5} + 73q^{9} + O(q^{10}) \) \( q + 10q^{3} + 8q^{5} + 73q^{9} - 40q^{11} + 12q^{13} + 80q^{15} + 58q^{17} - 26q^{19} - 64q^{23} - 61q^{25} + 460q^{27} - 62q^{29} - 252q^{31} - 400q^{33} + 26q^{37} + 120q^{39} - 6q^{41} + 416q^{43} + 584q^{45} + 396q^{47} + 580q^{51} - 450q^{53} - 320q^{55} - 260q^{57} - 274q^{59} + 576q^{61} + 96q^{65} - 476q^{67} - 640q^{69} - 448q^{71} + 158q^{73} - 610q^{75} - 936q^{79} + 2629q^{81} - 530q^{83} + 464q^{85} - 620q^{87} + 390q^{89} - 2520q^{93} - 208q^{95} - 214q^{97} - 2920q^{99} + O(q^{100}) \)

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
0 10.0000 0 8.00000 0 0 0 73.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(-1\)

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 196.4.a.d 1
3.b odd 2 1 1764.4.a.c 1
4.b odd 2 1 784.4.a.a 1
7.b odd 2 1 28.4.a.a 1
7.c even 3 2 196.4.e.a 2
7.d odd 6 2 196.4.e.f 2
21.c even 2 1 252.4.a.d 1
21.g even 6 2 1764.4.k.d 2
21.h odd 6 2 1764.4.k.m 2
28.d even 2 1 112.4.a.g 1
35.c odd 2 1 700.4.a.n 1
35.f even 4 2 700.4.e.a 2
56.e even 2 1 448.4.a.a 1
56.h odd 2 1 448.4.a.p 1
84.h odd 2 1 1008.4.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 7.b odd 2 1
112.4.a.g 1 28.d even 2 1
196.4.a.d 1 1.a even 1 1 trivial
196.4.e.a 2 7.c even 3 2
196.4.e.f 2 7.d odd 6 2
252.4.a.d 1 21.c even 2 1
448.4.a.a 1 56.e even 2 1
448.4.a.p 1 56.h odd 2 1
700.4.a.n 1 35.c odd 2 1
700.4.e.a 2 35.f even 4 2
784.4.a.a 1 4.b odd 2 1
1008.4.a.o 1 84.h odd 2 1
1764.4.a.c 1 3.b odd 2 1
1764.4.k.d 2 21.g even 6 2
1764.4.k.m 2 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(196))\):

\( T_{3} - 10 \)
\( T_{5} - 8 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -10 + T \)
$5$ \( -8 + T \)
$7$ \( T \)
$11$ \( 40 + T \)
$13$ \( -12 + T \)
$17$ \( -58 + T \)
$19$ \( 26 + T \)
$23$ \( 64 + T \)
$29$ \( 62 + T \)
$31$ \( 252 + T \)
$37$ \( -26 + T \)
$41$ \( 6 + T \)
$43$ \( -416 + T \)
$47$ \( -396 + T \)
$53$ \( 450 + T \)
$59$ \( 274 + T \)
$61$ \( -576 + T \)
$67$ \( 476 + T \)
$71$ \( 448 + T \)
$73$ \( -158 + T \)
$79$ \( 936 + T \)
$83$ \( 530 + T \)
$89$ \( -390 + T \)
$97$ \( 214 + T \)
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