Properties

Label 196.6.a.e
Level $196$
Weight $6$
Character orbit 196.a
Self dual yes
Analytic conductor $31.435$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 196.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.4352286833\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 12 q^{3} - 54 q^{5} - 99 q^{9} + O(q^{10}) \) \( q + 12 q^{3} - 54 q^{5} - 99 q^{9} + 540 q^{11} + 418 q^{13} - 648 q^{15} - 594 q^{17} - 836 q^{19} - 4104 q^{23} - 209 q^{25} - 4104 q^{27} - 594 q^{29} - 4256 q^{31} + 6480 q^{33} - 298 q^{37} + 5016 q^{39} - 17226 q^{41} - 12100 q^{43} + 5346 q^{45} + 1296 q^{47} - 7128 q^{51} + 19494 q^{53} - 29160 q^{55} - 10032 q^{57} + 7668 q^{59} + 34738 q^{61} - 22572 q^{65} + 21812 q^{67} - 49248 q^{69} - 46872 q^{71} - 67562 q^{73} - 2508 q^{75} - 76912 q^{79} - 25191 q^{81} - 67716 q^{83} + 32076 q^{85} - 7128 q^{87} - 29754 q^{89} - 51072 q^{93} + 45144 q^{95} + 122398 q^{97} - 53460 q^{99} + O(q^{100}) \)

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 12.0000 0 −54.0000 0 0 0 −99.0000 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 196.6.a.e 1
4.b odd 2 1 784.6.a.d 1
7.b odd 2 1 4.6.a.a 1
7.c even 3 2 196.6.e.d 2
7.d odd 6 2 196.6.e.g 2
21.c even 2 1 36.6.a.a 1
28.d even 2 1 16.6.a.b 1
35.c odd 2 1 100.6.a.b 1
35.f even 4 2 100.6.c.b 2
56.e even 2 1 64.6.a.b 1
56.h odd 2 1 64.6.a.f 1
63.l odd 6 2 324.6.e.a 2
63.o even 6 2 324.6.e.d 2
77.b even 2 1 484.6.a.a 1
84.h odd 2 1 144.6.a.c 1
91.b odd 2 1 676.6.a.a 1
91.i even 4 2 676.6.d.a 2
105.g even 2 1 900.6.a.h 1
105.k odd 4 2 900.6.d.a 2
112.j even 4 2 256.6.b.c 2
112.l odd 4 2 256.6.b.g 2
140.c even 2 1 400.6.a.d 1
140.j odd 4 2 400.6.c.f 2
168.e odd 2 1 576.6.a.bd 1
168.i even 2 1 576.6.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 7.b odd 2 1
16.6.a.b 1 28.d even 2 1
36.6.a.a 1 21.c even 2 1
64.6.a.b 1 56.e even 2 1
64.6.a.f 1 56.h odd 2 1
100.6.a.b 1 35.c odd 2 1
100.6.c.b 2 35.f even 4 2
144.6.a.c 1 84.h odd 2 1
196.6.a.e 1 1.a even 1 1 trivial
196.6.e.d 2 7.c even 3 2
196.6.e.g 2 7.d odd 6 2
256.6.b.c 2 112.j even 4 2
256.6.b.g 2 112.l odd 4 2
324.6.e.a 2 63.l odd 6 2
324.6.e.d 2 63.o even 6 2
400.6.a.d 1 140.c even 2 1
400.6.c.f 2 140.j odd 4 2
484.6.a.a 1 77.b even 2 1
576.6.a.bc 1 168.i even 2 1
576.6.a.bd 1 168.e odd 2 1
676.6.a.a 1 91.b odd 2 1
676.6.d.a 2 91.i even 4 2
784.6.a.d 1 4.b odd 2 1
900.6.a.h 1 105.g even 2 1
900.6.d.a 2 105.k odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 12 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(196))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -12 + T \)
$5$ \( 54 + T \)
$7$ \( T \)
$11$ \( -540 + T \)
$13$ \( -418 + T \)
$17$ \( 594 + T \)
$19$ \( 836 + T \)
$23$ \( 4104 + T \)
$29$ \( 594 + T \)
$31$ \( 4256 + T \)
$37$ \( 298 + T \)
$41$ \( 17226 + T \)
$43$ \( 12100 + T \)
$47$ \( -1296 + T \)
$53$ \( -19494 + T \)
$59$ \( -7668 + T \)
$61$ \( -34738 + T \)
$67$ \( -21812 + T \)
$71$ \( 46872 + T \)
$73$ \( 67562 + T \)
$79$ \( 76912 + T \)
$83$ \( 67716 + T \)
$89$ \( 29754 + T \)
$97$ \( -122398 + T \)
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