Properties

Label 196.2.d.b
Level $196$
Weight $2$
Character orbit 196.d
Analytic conductor $1.565$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \zeta_{12}^{3} ) q^{2} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{3} q^{4} + ( 1 - 2 \zeta_{12}^{2} ) q^{5} + ( -1 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{6} + ( -2 + 2 \zeta_{12}^{3} ) q^{8} +O(q^{10})\) \( q + ( 1 + \zeta_{12}^{3} ) q^{2} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{3} + 2 \zeta_{12}^{3} q^{4} + ( 1 - 2 \zeta_{12}^{2} ) q^{5} + ( -1 + 2 \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{6} + ( -2 + 2 \zeta_{12}^{3} ) q^{8} + ( 1 + 2 \zeta_{12} - 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{10} + \zeta_{12}^{3} q^{11} + ( -2 + 4 \zeta_{12}^{2} ) q^{12} + ( 2 - 4 \zeta_{12}^{2} ) q^{13} -3 \zeta_{12}^{3} q^{15} -4 q^{16} + ( -1 + 2 \zeta_{12}^{2} ) q^{17} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{19} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{20} + ( -1 + \zeta_{12}^{3} ) q^{22} -\zeta_{12}^{3} q^{23} + ( -2 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{24} + 2 q^{25} + ( 2 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{26} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{27} + 4 q^{29} + ( 3 - 3 \zeta_{12}^{3} ) q^{30} + ( -2 \zeta_{12} + \zeta_{12}^{3} ) q^{31} + ( -4 - 4 \zeta_{12}^{3} ) q^{32} + ( -1 + 2 \zeta_{12}^{2} ) q^{33} + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{34} + 3 q^{37} + ( 3 - 6 \zeta_{12} - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{38} -6 \zeta_{12}^{3} q^{39} + ( -2 + 4 \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{40} + ( -2 + 4 \zeta_{12}^{2} ) q^{41} -2 \zeta_{12}^{3} q^{43} -2 q^{44} + ( 1 - \zeta_{12}^{3} ) q^{46} + ( 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{47} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{48} + ( 2 + 2 \zeta_{12}^{3} ) q^{50} + 3 \zeta_{12}^{3} q^{51} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{52} - q^{53} + ( 3 - 6 \zeta_{12} - 6 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{54} + ( 2 \zeta_{12} - \zeta_{12}^{3} ) q^{55} -9 q^{57} + ( 4 + 4 \zeta_{12}^{3} ) q^{58} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{59} + 6 q^{60} + ( 3 - 6 \zeta_{12}^{2} ) q^{61} + ( 1 - 2 \zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{62} -8 \zeta_{12}^{3} q^{64} -6 q^{65} + ( -1 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{66} + 3 \zeta_{12}^{3} q^{67} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{68} + ( 1 - 2 \zeta_{12}^{2} ) q^{69} + 14 \zeta_{12}^{3} q^{71} + ( 5 - 10 \zeta_{12}^{2} ) q^{73} + ( 3 + 3 \zeta_{12}^{3} ) q^{74} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{75} + ( 6 - 12 \zeta_{12}^{2} ) q^{76} + ( 6 - 6 \zeta_{12}^{3} ) q^{78} -9 \zeta_{12}^{3} q^{79} + ( -4 + 8 \zeta_{12}^{2} ) q^{80} -9 q^{81} + ( -2 - 4 \zeta_{12} + 4 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{82} + ( 16 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{83} + 3 q^{85} + ( 2 - 2 \zeta_{12}^{3} ) q^{86} + ( 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{87} + ( -2 - 2 \zeta_{12}^{3} ) q^{88} + ( -9 + 18 \zeta_{12}^{2} ) q^{89} + 2 q^{92} -3 q^{93} + ( -5 + 10 \zeta_{12} + 10 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{94} + 9 \zeta_{12}^{3} q^{95} + ( 4 - 8 \zeta_{12} - 8 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{96} + ( -10 + 20 \zeta_{12}^{2} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{2} - 8q^{8} + O(q^{10}) \) \( 4q + 4q^{2} - 8q^{8} - 16q^{16} - 4q^{22} + 8q^{25} + 16q^{29} + 12q^{30} - 16q^{32} + 12q^{37} - 8q^{44} + 4q^{46} + 8q^{50} - 4q^{53} - 36q^{57} + 16q^{58} + 24q^{60} - 24q^{65} + 12q^{74} + 24q^{78} - 36q^{81} + 12q^{85} + 8q^{86} - 8q^{88} + 8q^{92} - 12q^{93} + O(q^{100}) \)

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
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
1.00000 1.00000i −1.73205 2.00000i 1.73205i −1.73205 + 1.73205i 0 −2.00000 2.00000i 0 −1.73205 1.73205i
195.2 1.00000 1.00000i 1.73205 2.00000i 1.73205i 1.73205 1.73205i 0 −2.00000 2.00000i 0 1.73205 + 1.73205i
195.3 1.00000 + 1.00000i −1.73205 2.00000i 1.73205i −1.73205 1.73205i 0 −2.00000 + 2.00000i 0 −1.73205 + 1.73205i
195.4 1.00000 + 1.00000i 1.73205 2.00000i 1.73205i 1.73205 + 1.73205i 0 −2.00000 + 2.00000i 0 1.73205 1.73205i
\(n\): e.g. 2-40 or 990-1000
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.b 4
3.b odd 2 1 1764.2.b.a 4
4.b odd 2 1 inner 196.2.d.b 4
7.b odd 2 1 inner 196.2.d.b 4
7.c even 3 1 28.2.f.a 4
7.c even 3 1 196.2.f.a 4
7.d odd 6 1 28.2.f.a 4
7.d odd 6 1 196.2.f.a 4
8.b even 2 1 3136.2.f.e 4
8.d odd 2 1 3136.2.f.e 4
12.b even 2 1 1764.2.b.a 4
21.c even 2 1 1764.2.b.a 4
21.g even 6 1 252.2.bf.e 4
21.h odd 6 1 252.2.bf.e 4
28.d even 2 1 inner 196.2.d.b 4
28.f even 6 1 28.2.f.a 4
28.f even 6 1 196.2.f.a 4
28.g odd 6 1 28.2.f.a 4
28.g odd 6 1 196.2.f.a 4
35.i odd 6 1 700.2.p.a 4
35.j even 6 1 700.2.p.a 4
35.k even 12 1 700.2.t.a 4
35.k even 12 1 700.2.t.b 4
35.l odd 12 1 700.2.t.a 4
35.l odd 12 1 700.2.t.b 4
56.e even 2 1 3136.2.f.e 4
56.h odd 2 1 3136.2.f.e 4
56.j odd 6 1 448.2.p.d 4
56.k odd 6 1 448.2.p.d 4
56.m even 6 1 448.2.p.d 4
56.p even 6 1 448.2.p.d 4
84.h odd 2 1 1764.2.b.a 4
84.j odd 6 1 252.2.bf.e 4
84.n even 6 1 252.2.bf.e 4
140.p odd 6 1 700.2.p.a 4
140.s even 6 1 700.2.p.a 4
140.w even 12 1 700.2.t.a 4
140.w even 12 1 700.2.t.b 4
140.x odd 12 1 700.2.t.a 4
140.x odd 12 1 700.2.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 7.c even 3 1
28.2.f.a 4 7.d odd 6 1
28.2.f.a 4 28.f even 6 1
28.2.f.a 4 28.g odd 6 1
196.2.d.b 4 1.a even 1 1 trivial
196.2.d.b 4 4.b odd 2 1 inner
196.2.d.b 4 7.b odd 2 1 inner
196.2.d.b 4 28.d even 2 1 inner
196.2.f.a 4 7.c even 3 1
196.2.f.a 4 7.d odd 6 1
196.2.f.a 4 28.f even 6 1
196.2.f.a 4 28.g odd 6 1
252.2.bf.e 4 21.g even 6 1
252.2.bf.e 4 21.h odd 6 1
252.2.bf.e 4 84.j odd 6 1
252.2.bf.e 4 84.n even 6 1
448.2.p.d 4 56.j odd 6 1
448.2.p.d 4 56.k odd 6 1
448.2.p.d 4 56.m even 6 1
448.2.p.d 4 56.p even 6 1
700.2.p.a 4 35.i odd 6 1
700.2.p.a 4 35.j even 6 1
700.2.p.a 4 140.p odd 6 1
700.2.p.a 4 140.s even 6 1
700.2.t.a 4 35.k even 12 1
700.2.t.a 4 35.l odd 12 1
700.2.t.a 4 140.w even 12 1
700.2.t.a 4 140.x odd 12 1
700.2.t.b 4 35.k even 12 1
700.2.t.b 4 35.l odd 12 1
700.2.t.b 4 140.w even 12 1
700.2.t.b 4 140.x odd 12 1
1764.2.b.a 4 3.b odd 2 1
1764.2.b.a 4 12.b even 2 1
1764.2.b.a 4 21.c even 2 1
1764.2.b.a 4 84.h odd 2 1
3136.2.f.e 4 8.b even 2 1
3136.2.f.e 4 8.d odd 2 1
3136.2.f.e 4 56.e even 2 1
3136.2.f.e 4 56.h odd 2 1

Hecke kernels

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