Properties

Label 196.3.b.b
Level $196$
Weight $3$
Character orbit 196.b
Analytic conductor $5.341$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 196.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.34061318146\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
Defining polynomial: \(x^{4} + 4 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 7^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( \beta_{1} + 2 \beta_{2} ) q^{5} + ( -1 - \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( \beta_{1} + 2 \beta_{2} ) q^{5} + ( -1 - \beta_{3} ) q^{9} + ( -6 + \beta_{3} ) q^{11} + ( 3 \beta_{1} - 4 \beta_{2} ) q^{13} + ( -18 - 2 \beta_{3} ) q^{15} + 3 \beta_{2} q^{17} + ( 5 \beta_{1} - \beta_{2} ) q^{19} + 26 q^{23} + ( -27 - 2 \beta_{3} ) q^{25} + ( -\beta_{1} - 2 \beta_{2} ) q^{27} + ( 8 - 2 \beta_{3} ) q^{29} + ( -2 \beta_{1} + 8 \beta_{2} ) q^{31} + ( \beta_{1} + 4 \beta_{2} ) q^{33} -32 q^{37} + ( 46 + 4 \beta_{3} ) q^{39} + ( 3 \beta_{1} + 5 \beta_{2} ) q^{41} + ( 10 + 7 \beta_{3} ) q^{43} + ( 7 \beta_{1} - 20 \beta_{2} ) q^{45} + ( -12 \beta_{1} + 2 \beta_{2} ) q^{47} + ( -30 - 3 \beta_{3} ) q^{51} + ( 6 - 2 \beta_{3} ) q^{53} + ( -14 \beta_{1} + 6 \beta_{2} ) q^{55} + ( 20 + \beta_{3} ) q^{57} + ( -11 \beta_{1} - 3 \beta_{2} ) q^{59} + ( -9 \beta_{1} - 8 \beta_{2} ) q^{61} + ( 24 + 14 \beta_{3} ) q^{65} + ( 28 - 6 \beta_{3} ) q^{67} + 26 \beta_{2} q^{69} + ( 56 + 4 \beta_{3} ) q^{71} + ( 11 \beta_{1} - 3 \beta_{2} ) q^{73} + ( -2 \beta_{1} - 47 \beta_{2} ) q^{75} + ( 60 + 2 \beta_{3} ) q^{79} + ( 9 - 7 \beta_{3} ) q^{81} + ( -13 \beta_{1} - 33 \beta_{2} ) q^{83} + ( -54 - 6 \beta_{3} ) q^{85} + ( -2 \beta_{1} - 12 \beta_{2} ) q^{87} + ( -11 \beta_{1} + 25 \beta_{2} ) q^{89} + ( -84 - 8 \beta_{3} ) q^{93} + ( -62 + 12 \beta_{3} ) q^{95} + ( 22 \beta_{1} + 33 \beta_{2} ) q^{97} + ( -92 + 5 \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{9} - 24q^{11} - 72q^{15} + 104q^{23} - 108q^{25} + 32q^{29} - 128q^{37} + 184q^{39} + 40q^{43} - 120q^{51} + 24q^{53} + 80q^{57} + 96q^{65} + 112q^{67} + 224q^{71} + 240q^{79} + 36q^{81} - 216q^{85} - 336q^{93} - 248q^{95} - 368q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 7 \nu \)
\(\beta_{3}\)\(=\)\( 7 \nu^{2} + 14 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - 2 \beta_{1}\)\()/7\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 14\)\()/7\)
\(\nu^{3}\)\(=\)\(\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1
0.765367i
1.84776i
1.84776i
0.765367i
0 4.46088i 0 8.47343i 0 0 0 −10.8995 0
97.2 0 0.317025i 0 5.67459i 0 0 0 8.89949 0
97.3 0 0.317025i 0 5.67459i 0 0 0 8.89949 0
97.4 0 4.46088i 0 8.47343i 0 0 0 −10.8995 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.3.b.b 4
3.b odd 2 1 1764.3.d.e 4
4.b odd 2 1 784.3.c.d 4
7.b odd 2 1 inner 196.3.b.b 4
7.c even 3 2 196.3.h.c 8
7.d odd 6 2 196.3.h.c 8
21.c even 2 1 1764.3.d.e 4
21.g even 6 2 1764.3.z.k 8
21.h odd 6 2 1764.3.z.k 8
28.d even 2 1 784.3.c.d 4
28.f even 6 2 784.3.s.g 8
28.g odd 6 2 784.3.s.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.3.b.b 4 1.a even 1 1 trivial
196.3.b.b 4 7.b odd 2 1 inner
196.3.h.c 8 7.c even 3 2
196.3.h.c 8 7.d odd 6 2
784.3.c.d 4 4.b odd 2 1
784.3.c.d 4 28.d even 2 1
784.3.s.g 8 28.f even 6 2
784.3.s.g 8 28.g odd 6 2
1764.3.d.e 4 3.b odd 2 1
1764.3.d.e 4 21.c even 2 1
1764.3.z.k 8 21.g even 6 2
1764.3.z.k 8 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 20 T_{3}^{2} + 2 \) acting on \(S_{3}^{\mathrm{new}}(196, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 2 + 20 T^{2} + T^{4} \)
$5$ \( 2312 + 104 T^{2} + T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( -62 + 12 T + T^{2} )^{2} \)
$13$ \( 150152 + 776 T^{2} + T^{4} \)
$17$ \( 162 + 180 T^{2} + T^{4} \)
$19$ \( 45602 + 1060 T^{2} + T^{4} \)
$23$ \( ( -26 + T )^{4} \)
$29$ \( ( -328 - 16 T + T^{2} )^{2} \)
$31$ \( 307328 + 1568 T^{2} + T^{4} \)
$37$ \( ( 32 + T )^{4} \)
$41$ \( 132098 + 740 T^{2} + T^{4} \)
$43$ \( ( -4702 - 20 T + T^{2} )^{2} \)
$47$ \( 1191968 + 6032 T^{2} + T^{4} \)
$53$ \( ( -356 - 12 T + T^{2} )^{2} \)
$59$ \( 334562 + 4756 T^{2} + T^{4} \)
$61$ \( 2947592 + 3944 T^{2} + T^{4} \)
$67$ \( ( -2744 - 56 T + T^{2} )^{2} \)
$71$ \( ( 1568 - 112 T + T^{2} )^{2} \)
$73$ \( 1659842 + 5284 T^{2} + T^{4} \)
$79$ \( ( 3208 - 120 T + T^{2} )^{2} \)
$83$ \( 102330818 + 25108 T^{2} + T^{4} \)
$89$ \( 81077378 + 19540 T^{2} + T^{4} \)
$97$ \( 310652738 + 35332 T^{2} + T^{4} \)
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