Properties

Label 196.2.f.a
Level $196$
Weight $2$
Character orbit 196.f
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.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.56506787962\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

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

Character values

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

\(n\) \(99\) \(101\)
\(\chi(n)\) \(-1\) \(\zeta_{12}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−1.36603 + 0.366025i −0.866025 1.50000i 1.73205 1.00000i 1.50000 + 0.866025i 1.73205 + 1.73205i 0 −2.00000 + 2.00000i 0 −2.36603 0.633975i
19.2 0.366025 + 1.36603i 0.866025 + 1.50000i −1.73205 + 1.00000i 1.50000 + 0.866025i −1.73205 + 1.73205i 0 −2.00000 2.00000i 0 −0.633975 + 2.36603i
31.1 −1.36603 0.366025i −0.866025 + 1.50000i 1.73205 + 1.00000i 1.50000 0.866025i 1.73205 1.73205i 0 −2.00000 2.00000i 0 −2.36603 + 0.633975i
31.2 0.366025 1.36603i 0.866025 1.50000i −1.73205 1.00000i 1.50000 0.866025i −1.73205 1.73205i 0 −2.00000 + 2.00000i 0 −0.633975 2.36603i
\(n\): 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.d odd 6 1 inner
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.2.f.a 4
4.b odd 2 1 inner 196.2.f.a 4
7.b odd 2 1 28.2.f.a 4
7.c even 3 1 28.2.f.a 4
7.c even 3 1 196.2.d.b 4
7.d odd 6 1 196.2.d.b 4
7.d odd 6 1 inner 196.2.f.a 4
21.c even 2 1 252.2.bf.e 4
21.g even 6 1 1764.2.b.a 4
21.h odd 6 1 252.2.bf.e 4
21.h odd 6 1 1764.2.b.a 4
28.d even 2 1 28.2.f.a 4
28.f even 6 1 196.2.d.b 4
28.f even 6 1 inner 196.2.f.a 4
28.g odd 6 1 28.2.f.a 4
28.g odd 6 1 196.2.d.b 4
35.c odd 2 1 700.2.p.a 4
35.f even 4 1 700.2.t.a 4
35.f even 4 1 700.2.t.b 4
35.j even 6 1 700.2.p.a 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 1 448.2.p.d 4
56.h odd 2 1 448.2.p.d 4
56.j odd 6 1 3136.2.f.e 4
56.k odd 6 1 448.2.p.d 4
56.k odd 6 1 3136.2.f.e 4
56.m even 6 1 3136.2.f.e 4
56.p even 6 1 448.2.p.d 4
56.p even 6 1 3136.2.f.e 4
84.h odd 2 1 252.2.bf.e 4
84.j odd 6 1 1764.2.b.a 4
84.n even 6 1 252.2.bf.e 4
84.n even 6 1 1764.2.b.a 4
140.c even 2 1 700.2.p.a 4
140.j odd 4 1 700.2.t.a 4
140.j odd 4 1 700.2.t.b 4
140.p odd 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
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 7.b odd 2 1
28.2.f.a 4 7.c even 3 1
28.2.f.a 4 28.d even 2 1
28.2.f.a 4 28.g odd 6 1
196.2.d.b 4 7.c even 3 1
196.2.d.b 4 7.d odd 6 1
196.2.d.b 4 28.f even 6 1
196.2.d.b 4 28.g odd 6 1
196.2.f.a 4 1.a even 1 1 trivial
196.2.f.a 4 4.b odd 2 1 inner
196.2.f.a 4 7.d odd 6 1 inner
196.2.f.a 4 28.f even 6 1 inner
252.2.bf.e 4 21.c even 2 1
252.2.bf.e 4 21.h odd 6 1
252.2.bf.e 4 84.h odd 2 1
252.2.bf.e 4 84.n even 6 1
448.2.p.d 4 56.e even 2 1
448.2.p.d 4 56.h odd 2 1
448.2.p.d 4 56.k odd 6 1
448.2.p.d 4 56.p even 6 1
700.2.p.a 4 35.c odd 2 1
700.2.p.a 4 35.j even 6 1
700.2.p.a 4 140.c even 2 1
700.2.p.a 4 140.p odd 6 1
700.2.t.a 4 35.f even 4 1
700.2.t.a 4 35.l odd 12 1
700.2.t.a 4 140.j odd 4 1
700.2.t.a 4 140.w even 12 1
700.2.t.b 4 35.f even 4 1
700.2.t.b 4 35.l odd 12 1
700.2.t.b 4 140.j odd 4 1
700.2.t.b 4 140.w even 12 1
1764.2.b.a 4 21.g even 6 1
1764.2.b.a 4 21.h odd 6 1
1764.2.b.a 4 84.j odd 6 1
1764.2.b.a 4 84.n even 6 1
3136.2.f.e 4 56.j odd 6 1
3136.2.f.e 4 56.k odd 6 1
3136.2.f.e 4 56.m even 6 1
3136.2.f.e 4 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(196, [\chi])\):

\( T_{3}^{4} + 3 T_{3}^{2} + 9 \)
\( T_{5}^{2} - 3 T_{5} + 3 \)