Properties

Label 196.2.d.c
Level $196$
Weight $2$
Character orbit 196.d
Analytic conductor $1.565$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 196.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.56506787962\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1212153856.10
Defining polynomial: \(x^{8} - 4 x^{6} + 10 x^{4} - 16 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{5} ) q^{2} + ( -2 \beta_{1} - \beta_{7} ) q^{3} + ( \beta_{3} - \beta_{5} + \beta_{6} ) q^{4} + ( \beta_{2} + \beta_{7} ) q^{5} + ( 2 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{7} ) q^{6} + ( 1 - \beta_{5} - 2 \beta_{6} ) q^{8} + ( 4 + \beta_{3} - \beta_{5} ) q^{9} +O(q^{10})\) \( q + ( -1 + \beta_{5} ) q^{2} + ( -2 \beta_{1} - \beta_{7} ) q^{3} + ( \beta_{3} - \beta_{5} + \beta_{6} ) q^{4} + ( \beta_{2} + \beta_{7} ) q^{5} + ( 2 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{7} ) q^{6} + ( 1 - \beta_{5} - 2 \beta_{6} ) q^{8} + ( 4 + \beta_{3} - \beta_{5} ) q^{9} + ( \beta_{1} - \beta_{4} + \beta_{7} ) q^{10} + ( 1 - \beta_{3} - \beta_{5} - 2 \beta_{6} ) q^{11} + ( -\beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{7} ) q^{12} + ( -\beta_{2} + \beta_{7} ) q^{13} + ( 2 \beta_{3} + 2 \beta_{5} ) q^{15} + ( -2 - 3 \beta_{3} + \beta_{5} + \beta_{6} ) q^{16} + ( -2 \beta_{2} + \beta_{7} ) q^{17} + ( -4 - \beta_{3} + 3 \beta_{5} - \beta_{6} ) q^{18} + ( -\beta_{2} + 2 \beta_{4} ) q^{19} + ( -\beta_{1} - \beta_{4} - 3 \beta_{7} ) q^{20} + ( -1 - 3 \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{22} + ( 2 - 4 \beta_{6} ) q^{23} + ( 3 \beta_{2} - \beta_{4} - 4 \beta_{7} ) q^{24} + ( 3 + 2 \beta_{3} - 2 \beta_{5} ) q^{25} + ( -\beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{7} ) q^{26} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{7} ) q^{27} + ( -2 + 2 \beta_{3} - 2 \beta_{5} ) q^{29} + ( -4 + 2 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{30} + ( 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{7} ) q^{31} + ( 5 + 2 \beta_{3} + \beta_{5} ) q^{32} + ( \beta_{2} - 5 \beta_{7} ) q^{33} + ( -2 \beta_{1} + 3 \beta_{2} - \beta_{4} - 2 \beta_{7} ) q^{34} + ( 1 + 2 \beta_{3} - 3 \beta_{5} + 4 \beta_{6} ) q^{36} -4 q^{37} + ( -\beta_{1} + \beta_{2} + 3 \beta_{7} ) q^{38} + ( -2 + 4 \beta_{6} ) q^{39} + ( \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - \beta_{7} ) q^{40} -3 \beta_{2} q^{41} + ( -1 - 3 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} ) q^{43} + ( 3 + 3 \beta_{3} + 2 \beta_{5} + \beta_{6} ) q^{44} + ( 3 \beta_{2} + \beta_{7} ) q^{45} + ( -6 - 4 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} ) q^{46} + ( 4 \beta_{1} + 2 \beta_{7} ) q^{47} + ( 3 \beta_{1} - 7 \beta_{2} + 4 \beta_{4} + \beta_{7} ) q^{48} + ( -3 - 2 \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{50} + ( -3 - \beta_{3} - \beta_{5} + 6 \beta_{6} ) q^{51} + ( 3 \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{7} ) q^{52} + ( -4 - 6 \beta_{3} + 6 \beta_{5} ) q^{53} + ( 3 \beta_{1} + \beta_{4} - \beta_{7} ) q^{54} + ( -2 \beta_{2} + 4 \beta_{4} ) q^{55} + ( -1 - 5 \beta_{3} + 5 \beta_{5} ) q^{57} + ( 2 - 2 \beta_{3} - 4 \beta_{5} - 2 \beta_{6} ) q^{58} + ( -4 \beta_{1} + 3 \beta_{2} - 6 \beta_{4} - 2 \beta_{7} ) q^{59} + ( 6 - 6 \beta_{5} - 4 \beta_{6} ) q^{60} + ( -5 \beta_{2} + 5 \beta_{7} ) q^{61} + ( -6 \beta_{1} - 2 \beta_{4} + 2 \beta_{7} ) q^{62} + ( -8 + \beta_{3} + 3 \beta_{5} + \beta_{6} ) q^{64} + ( 2 + 2 \beta_{3} - 2 \beta_{5} ) q^{65} + ( \beta_{1} - 6 \beta_{2} + 5 \beta_{4} + \beta_{7} ) q^{66} + ( 2 + 2 \beta_{3} + 2 \beta_{5} - 4 \beta_{6} ) q^{67} + ( 5 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} + 3 \beta_{7} ) q^{68} + ( 8 \beta_{2} - 6 \beta_{7} ) q^{69} + ( 4 + \beta_{3} - \beta_{5} - 7 \beta_{6} ) q^{72} + ( \beta_{2} - 8 \beta_{7} ) q^{73} + ( 4 - 4 \beta_{5} ) q^{74} + ( -6 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} - 3 \beta_{7} ) q^{75} + ( 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} + 2 \beta_{7} ) q^{76} + ( 6 + 4 \beta_{3} - 2 \beta_{5} - 4 \beta_{6} ) q^{78} + ( -2 + 2 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} ) q^{79} + ( -3 \beta_{1} + \beta_{4} + 3 \beta_{7} ) q^{80} + ( -4 + 3 \beta_{3} - 3 \beta_{5} ) q^{81} + ( -3 \beta_{1} + 3 \beta_{2} - 3 \beta_{7} ) q^{82} + ( 4 \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{7} ) q^{83} + ( 4 + 2 \beta_{3} - 2 \beta_{5} ) q^{85} + ( 9 - \beta_{3} + 2 \beta_{5} - 5 \beta_{6} ) q^{86} + ( 4 \beta_{1} + 2 \beta_{2} - 4 \beta_{4} + 2 \beta_{7} ) q^{87} + ( -7 + 3 \beta_{3} + \beta_{6} ) q^{88} + ( 3 \beta_{2} + 2 \beta_{7} ) q^{89} + ( 3 \beta_{1} - 2 \beta_{2} - \beta_{4} + 3 \beta_{7} ) q^{90} + ( 12 + 6 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{92} + ( -16 - 12 \beta_{3} + 12 \beta_{5} ) q^{93} + ( -4 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 4 \beta_{7} ) q^{94} + ( -2 + 4 \beta_{3} + 4 \beta_{5} + 4 \beta_{6} ) q^{95} + ( -10 \beta_{1} + 5 \beta_{2} - \beta_{4} - 2 \beta_{7} ) q^{96} + 5 \beta_{7} q^{97} + ( 3 - \beta_{3} - \beta_{5} - 6 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{2} - 4q^{4} - 4q^{8} + 24q^{9} + O(q^{10}) \) \( 8q - 4q^{2} - 4q^{4} - 4q^{8} + 24q^{9} + 4q^{16} - 20q^{18} + 16q^{22} + 8q^{25} - 32q^{29} - 40q^{30} + 36q^{32} + 4q^{36} - 32q^{37} + 24q^{44} - 8q^{46} - 20q^{50} + 16q^{53} + 32q^{57} + 8q^{60} - 52q^{64} - 4q^{72} + 16q^{74} + 8q^{78} - 56q^{81} + 16q^{85} + 64q^{86} - 64q^{88} + 56q^{92} - 32q^{93} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 10 x^{4} - 16 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{7} + 2 \nu^{3} \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} + 2 \nu^{4} - 2 \nu^{2} \)\()/4\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} - 2 \nu^{3} \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{4} + 2 \nu^{2} - 2 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{6} - 2 \nu^{4} + 6 \nu^{2} - 4 \)\()/4\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} - 10 \nu^{3} + 8 \nu \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{3} + 1\)
\(\nu^{3}\)\(=\)\(-\beta_{7} + \beta_{4} + \beta_{1}\)
\(\nu^{4}\)\(=\)\(2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3}\)
\(\nu^{5}\)\(=\)\(2 \beta_{4} + 2 \beta_{2}\)
\(\nu^{6}\)\(=\)\(2 \beta_{6} - 4 \beta_{5} - 2 \beta_{3} - 2\)
\(\nu^{7}\)\(=\)\(2 \beta_{7} - 2 \beta_{4} + 8 \beta_{2} - 2 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/196\mathbb{Z}\right)^\times\).

\(n\) \(99\) \(101\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
195.1
1.36145 + 0.382683i
−1.36145 0.382683i
1.36145 0.382683i
−1.36145 + 0.382683i
1.07072 0.923880i
−1.07072 + 0.923880i
1.07072 + 0.923880i
−1.07072 0.923880i
−1.20711 0.736813i −2.72291 0.914214 + 1.77882i 1.08239i 3.28684 + 2.00627i 0 0.207107 2.82083i 4.41421 0.797521 1.30656i
195.2 −1.20711 0.736813i 2.72291 0.914214 + 1.77882i 1.08239i −3.28684 2.00627i 0 0.207107 2.82083i 4.41421 −0.797521 + 1.30656i
195.3 −1.20711 + 0.736813i −2.72291 0.914214 1.77882i 1.08239i 3.28684 2.00627i 0 0.207107 + 2.82083i 4.41421 0.797521 + 1.30656i
195.4 −1.20711 + 0.736813i 2.72291 0.914214 1.77882i 1.08239i −3.28684 + 2.00627i 0 0.207107 + 2.82083i 4.41421 −0.797521 1.30656i
195.5 0.207107 1.39897i −2.14144 −1.91421 0.579471i 2.61313i −0.443508 + 2.99581i 0 −1.20711 + 2.55791i 1.58579 3.65568 + 0.541196i
195.6 0.207107 1.39897i 2.14144 −1.91421 0.579471i 2.61313i 0.443508 2.99581i 0 −1.20711 + 2.55791i 1.58579 −3.65568 0.541196i
195.7 0.207107 + 1.39897i −2.14144 −1.91421 + 0.579471i 2.61313i −0.443508 2.99581i 0 −1.20711 2.55791i 1.58579 3.65568 0.541196i
195.8 0.207107 + 1.39897i 2.14144 −1.91421 + 0.579471i 2.61313i 0.443508 + 2.99581i 0 −1.20711 2.55791i 1.58579 −3.65568 + 0.541196i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 195.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.2.d.c 8
3.b odd 2 1 1764.2.b.k 8
4.b odd 2 1 inner 196.2.d.c 8
7.b odd 2 1 inner 196.2.d.c 8
7.c even 3 2 196.2.f.d 16
7.d odd 6 2 196.2.f.d 16
8.b even 2 1 3136.2.f.i 8
8.d odd 2 1 3136.2.f.i 8
12.b even 2 1 1764.2.b.k 8
21.c even 2 1 1764.2.b.k 8
28.d even 2 1 inner 196.2.d.c 8
28.f even 6 2 196.2.f.d 16
28.g odd 6 2 196.2.f.d 16
56.e even 2 1 3136.2.f.i 8
56.h odd 2 1 3136.2.f.i 8
84.h odd 2 1 1764.2.b.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.d.c 8 1.a even 1 1 trivial
196.2.d.c 8 4.b odd 2 1 inner
196.2.d.c 8 7.b odd 2 1 inner
196.2.d.c 8 28.d even 2 1 inner
196.2.f.d 16 7.c even 3 2
196.2.f.d 16 7.d odd 6 2
196.2.f.d 16 28.f even 6 2
196.2.f.d 16 28.g odd 6 2
1764.2.b.k 8 3.b odd 2 1
1764.2.b.k 8 12.b even 2 1
1764.2.b.k 8 21.c even 2 1
1764.2.b.k 8 84.h odd 2 1
3136.2.f.i 8 8.b even 2 1
3136.2.f.i 8 8.d odd 2 1
3136.2.f.i 8 56.e even 2 1
3136.2.f.i 8 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 12 T_{3}^{2} + 34 \) acting on \(S_{2}^{\mathrm{new}}(196, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 4 T + 3 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$3$ \( ( 34 - 12 T^{2} + T^{4} )^{2} \)
$5$ \( ( 8 + 8 T^{2} + T^{4} )^{2} \)
$7$ \( T^{8} \)
$11$ \( ( 68 + 20 T^{2} + T^{4} )^{2} \)
$13$ \( ( 8 + 8 T^{2} + T^{4} )^{2} \)
$17$ \( ( 2 + 20 T^{2} + T^{4} )^{2} \)
$19$ \( ( 34 - 28 T^{2} + T^{4} )^{2} \)
$23$ \( ( 272 + 56 T^{2} + T^{4} )^{2} \)
$29$ \( ( 8 + 8 T + T^{2} )^{4} \)
$31$ \( ( 2176 - 96 T^{2} + T^{4} )^{2} \)
$37$ \( ( 4 + T )^{8} \)
$41$ \( ( 162 + 36 T^{2} + T^{4} )^{2} \)
$43$ \( ( 3332 + 116 T^{2} + T^{4} )^{2} \)
$47$ \( ( 544 - 48 T^{2} + T^{4} )^{2} \)
$53$ \( ( -68 - 4 T + T^{2} )^{4} \)
$59$ \( ( 9826 - 204 T^{2} + T^{4} )^{2} \)
$61$ \( ( 5000 + 200 T^{2} + T^{4} )^{2} \)
$67$ \( ( 1088 + 112 T^{2} + T^{4} )^{2} \)
$71$ \( T^{8} \)
$73$ \( ( 12482 + 260 T^{2} + T^{4} )^{2} \)
$79$ \( ( 1088 + 80 T^{2} + T^{4} )^{2} \)
$83$ \( ( 1666 - 108 T^{2} + T^{4} )^{2} \)
$89$ \( ( 578 + 52 T^{2} + T^{4} )^{2} \)
$97$ \( ( 1250 + 100 T^{2} + T^{4} )^{2} \)
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