Properties

Label 196.3.c.g
Level $196$
Weight $3$
Character orbit 196.c
Analytic conductor $5.341$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [196,3,Mod(99,196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(196, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.99");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 196.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.34061318146\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1539727.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{3} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{3} q^{3} + \beta_{4} q^{4} + ( - \beta_{5} - \beta_1 + 1) q^{5} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{6} + ( - 2 \beta_{3} + \beta_{2} - 2) q^{8} + ( - \beta_{4} + \beta_{2} - 2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - \beta_{3} q^{3} + \beta_{4} q^{4} + ( - \beta_{5} - \beta_1 + 1) q^{5} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 1) q^{6} + ( - 2 \beta_{3} + \beta_{2} - 2) q^{8} + ( - \beta_{4} + \beta_{2} - 2 \beta_1 - 1) q^{9} + ( - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 5) q^{10} + ( - 2 \beta_{5} - 3 \beta_{4} - 2 \beta_{3} - \beta_{2} + 4 \beta_1) q^{11} + (2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{12} + (\beta_{5} - 2 \beta_{4} + 2 \beta_{2} - 3 \beta_1 - 1) q^{13} + (2 \beta_{5} + 3 \beta_{4} + \beta_{2} - 4 \beta_1) q^{15} + (4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2) q^{16} + (2 \beta_{5} + 2 \beta_1) q^{17} + (2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 6) q^{18} + ( - 4 \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 8 \beta_1) q^{19} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 6) q^{20} + ( - 2 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 10) q^{22} + ( - 2 \beta_{5} + \beta_{4} + 3 \beta_{2} + 4 \beta_1) q^{23} + (4 \beta_{3} + 2 \beta_{2} - 20) q^{24} + ( - 4 \beta_{5} - \beta_{4} + \beta_{2} - 6 \beta_1 - 3) q^{25} + (5 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} + \beta_1 + 7) q^{26} + (4 \beta_{4} - 2 \beta_{3} + 4 \beta_{2}) q^{27} + (6 \beta_{5} + 6 \beta_1 - 8) q^{29} + (4 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} - 12) q^{30} + (4 \beta_{5} + 2 \beta_{4} + 6 \beta_{3} - 2 \beta_{2} - 8 \beta_1) q^{31} + (4 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 5 \beta_{2} - 18) q^{32} + ( - 8 \beta_{5} - 4 \beta_{4} + 4 \beta_{2} - 16 \beta_1 - 8) q^{33} + (2 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 10) q^{34} + ( - 4 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} - 4 \beta_1 - 20) q^{36} + ( - 6 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} - 10 \beta_1 + 8) q^{37} + ( - \beta_{5} - 9 \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 33) q^{38} + ( - 2 \beta_{5} + 5 \beta_{4} + 12 \beta_{3} + 7 \beta_{2} + 4 \beta_1) q^{39} + (8 \beta_{5} + 8 \beta_{4} - 8 \beta_1 - 8) q^{40} + ( - 6 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 4) q^{41} + (2 \beta_{5} - 5 \beta_{4} + 2 \beta_{3} - 7 \beta_{2} - 4 \beta_1) q^{43} + ( - 4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - 8 \beta_{2} - 4 \beta_1 + 28) q^{44} + ( - \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 3 \beta_1 - 3) q^{45} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 20) q^{46} + ( - 8 \beta_{5} - 8 \beta_{4} - 6 \beta_{3} + 16 \beta_1) q^{47} + ( - 4 \beta_{4} - 4 \beta_{3} - 6 \beta_{2} + 24 \beta_1 + 12) q^{48} + ( - 2 \beta_{5} + 6 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 26) q^{50} + ( - 4 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 8 \beta_1) q^{51} + ( - 10 \beta_{5} - 6 \beta_{4} - 6 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 46) q^{52} + ( - 4 \beta_{5} - 2 \beta_{4} + 2 \beta_{2} - 8 \beta_1 + 18) q^{53} + (10 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 6) q^{54} + ( - 4 \beta_{5} - 10 \beta_{4} + 12 \beta_{3} - 6 \beta_{2} + 8 \beta_1) q^{55} + ( - 8 \beta_{5} + 5 \beta_{4} - 5 \beta_{2} + 2 \beta_1 + 26) q^{57} + (6 \beta_{5} - 6 \beta_{4} + 6 \beta_{3} - 6 \beta_{2} + 8 \beta_1 - 30) q^{58} + (4 \beta_{4} - 11 \beta_{3} + 4 \beta_{2}) q^{59} + (8 \beta_{5} + 4 \beta_{4} + 4 \beta_{2} + 8 \beta_1 - 32) q^{60} + (3 \beta_{5} - 4 \beta_{4} + 4 \beta_{2} - 5 \beta_1 + 29) q^{61} + ( - 6 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} - 6 \beta_{2} + 6 \beta_1 - 26) q^{62} + ( - 4 \beta_{5} + 2 \beta_{4} + 10 \beta_{3} + \beta_{2} + 16 \beta_1 - 38) q^{64} + (5 \beta_{4} - 5 \beta_{2} + 10 \beta_1 - 26) q^{65} + (16 \beta_{4} + 8 \beta_1 + 64) q^{66} + (2 \beta_{5} + 3 \beta_{4} - 24 \beta_{3} + \beta_{2} - 4 \beta_1) q^{67} + ( - 4 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 12 \beta_1 - 12) q^{68} + (6 \beta_{4} - 6 \beta_{2} + 12 \beta_1 + 4) q^{69} + ( - 4 \beta_{5} - 2 \beta_{4} - 12 \beta_{3} + 2 \beta_{2} + 8 \beta_1) q^{71} + (8 \beta_{4} + 6 \beta_{3} + 5 \beta_{2} + 16 \beta_1 + 22) q^{72} + (4 \beta_{5} + 4 \beta_{4} - 4 \beta_{2} + 12 \beta_1 + 18) q^{73} + ( - 2 \beta_{5} + 10 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 8 \beta_1 + 42) q^{74} + (8 \beta_{5} + 16 \beta_{4} + 13 \beta_{3} + 8 \beta_{2} - 16 \beta_1) q^{75} + ( - 2 \beta_{5} + 18 \beta_{3} - 6 \beta_{2} - 34 \beta_1 + 18) q^{76} + ( - 16 \beta_{4} - 24 \beta_{3} - 12 \beta_{2} + 12 \beta_1 + 40) q^{78} + ( - 4 \beta_{4} + 12 \beta_{3} - 4 \beta_{2}) q^{79} + (8 \beta_{5} + 8 \beta_{4} - 8 \beta_{3} + 8 \beta_1 - 56) q^{80} + (8 \beta_{5} - 3 \beta_{4} + 3 \beta_{2} + 2 \beta_1 - 37) q^{81} + ( - 10 \beta_{5} + 2 \beta_{4} - 10 \beta_{3} + 10 \beta_{2} - 4 \beta_1 + 18) q^{82} + ( - 4 \beta_{5} - 14 \beta_{4} - 23 \beta_{3} - 10 \beta_{2} + 8 \beta_1) q^{83} + (6 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 10 \beta_1 - 42) q^{85} + ( - 14 \beta_{5} + 2 \beta_{4} + 10 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 26) q^{86} + ( - 12 \beta_{5} - 18 \beta_{4} + 2 \beta_{3} - 6 \beta_{2} + 24 \beta_1) q^{87} + ( - 16 \beta_{5} + 8 \beta_{4} + 8 \beta_{3} + 12 \beta_{2} - 32 \beta_1 - 8) q^{88} + (4 \beta_{5} + 10 \beta_{4} - 10 \beta_{2} + 24 \beta_1 - 66) q^{89} + ( - 5 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + 5 \beta_{2} + 3 \beta_1 - 7) q^{90} + (8 \beta_{5} + 4 \beta_{4} + 16 \beta_{3} - 4 \beta_{2} - 24 \beta_1 - 16) q^{92} + (8 \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 12 \beta_1 + 44) q^{93} + ( - 2 \beta_{5} - 10 \beta_{4} + 14 \beta_{3} + 6 \beta_{2} - 6 \beta_1 + 50) q^{94} + ( - 2 \beta_{5} - 19 \beta_{4} + 16 \beta_{3} - 17 \beta_{2} + 4 \beta_1) q^{95} + ( - 8 \beta_{5} - 20 \beta_{4} + 12 \beta_{3} + 6 \beta_{2} - 16 \beta_1 - 20) q^{96} + (6 \beta_{5} - 6 \beta_{4} + 6 \beta_{2} - 6 \beta_1 - 40) q^{97} + ( - 2 \beta_{5} + 13 \beta_{4} + 22 \beta_{3} + 15 \beta_{2} + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{2} + q^{4} + 4 q^{5} - 6 q^{6} - 13 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{2} + q^{4} + 4 q^{5} - 6 q^{6} - 13 q^{8} - 10 q^{9} + 28 q^{10} - 6 q^{12} - 12 q^{13} + 17 q^{16} + 4 q^{17} + 43 q^{18} + 32 q^{20} + 52 q^{22} - 122 q^{24} - 30 q^{25} + 56 q^{26} - 36 q^{29} - 64 q^{30} - 101 q^{32} - 80 q^{33} - 58 q^{34} - 131 q^{36} + 28 q^{37} + 190 q^{38} - 40 q^{40} + 20 q^{41} + 164 q^{44} - 12 q^{45} + 120 q^{46} + 98 q^{48} + 161 q^{50} - 292 q^{52} + 92 q^{53} + 44 q^{54} + 160 q^{57} - 166 q^{58} - 176 q^{60} + 164 q^{61} - 148 q^{62} - 215 q^{64} - 136 q^{65} + 408 q^{66} - 62 q^{68} + 48 q^{69} + 151 q^{72} + 132 q^{73} + 250 q^{74} + 78 q^{76} + 248 q^{78} - 312 q^{80} - 218 q^{81} + 86 q^{82} - 232 q^{85} - 164 q^{86} - 100 q^{88} - 348 q^{89} - 52 q^{90} - 104 q^{92} + 288 q^{93} + 276 q^{94} - 170 q^{96} - 252 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{3} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} - 2\nu^{3} + \nu^{2} + 4\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} + \nu^{3} - \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} - 3\nu^{2} - 4\nu + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - \nu^{4} - \nu^{3} + 2\nu - 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + \nu^{4} + \nu^{3} - 2\nu^{2} + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{3} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - 2\beta_{4} - \beta_{3} + \beta _1 + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{5} - 2\beta_{4} + \beta_{3} + 2\beta_{2} + 3\beta _1 - 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{5} + 2\beta_{4} + \beta_{3} + 4\beta_{2} - \beta _1 + 9 ) / 4 \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0.841985 1.13625i
0.841985 + 1.13625i
1.35935 + 0.390070i
1.35935 0.390070i
−1.20134 0.746179i
−1.20134 + 0.746179i
−1.92411 0.545716i 4.54500i 3.40439 + 2.10003i −1.36794 −2.48028 + 8.74507i 0 −5.40439 5.89853i −11.6570 2.63206 + 0.746506i
99.2 −1.92411 + 0.545716i 4.54500i 3.40439 2.10003i −1.36794 −2.48028 8.74507i 0 −5.40439 + 5.89853i −11.6570 2.63206 0.746506i
99.3 −0.163664 1.99329i 1.56028i −3.94643 + 0.652459i −3.43742 3.11009 0.255361i 0 1.94643 + 7.75960i 6.56553 0.562581 + 6.85178i
99.4 −0.163664 + 1.99329i 1.56028i −3.94643 0.652459i −3.43742 3.11009 + 0.255361i 0 1.94643 7.75960i 6.56553 0.562581 6.85178i
99.5 1.58777 1.21613i 2.98472i 1.04204 3.86188i 6.80536 −3.62981 4.73905i 0 −3.04204 7.39905i 0.0914622 10.8054 8.27622i
99.6 1.58777 + 1.21613i 2.98472i 1.04204 + 3.86188i 6.80536 −3.62981 + 4.73905i 0 −3.04204 + 7.39905i 0.0914622 10.8054 + 8.27622i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.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.c.g 6
4.b odd 2 1 inner 196.3.c.g 6
7.b odd 2 1 28.3.c.a 6
7.c even 3 2 196.3.g.j 12
7.d odd 6 2 196.3.g.k 12
21.c even 2 1 252.3.g.a 6
28.d even 2 1 28.3.c.a 6
28.f even 6 2 196.3.g.k 12
28.g odd 6 2 196.3.g.j 12
56.e even 2 1 448.3.d.d 6
56.h odd 2 1 448.3.d.d 6
84.h odd 2 1 252.3.g.a 6
112.j even 4 2 1792.3.g.g 12
112.l odd 4 2 1792.3.g.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.3.c.a 6 7.b odd 2 1
28.3.c.a 6 28.d even 2 1
196.3.c.g 6 1.a even 1 1 trivial
196.3.c.g 6 4.b odd 2 1 inner
196.3.g.j 12 7.c even 3 2
196.3.g.j 12 28.g odd 6 2
196.3.g.k 12 7.d odd 6 2
196.3.g.k 12 28.f even 6 2
252.3.g.a 6 21.c even 2 1
252.3.g.a 6 84.h odd 2 1
448.3.d.d 6 56.e even 2 1
448.3.d.d 6 56.h odd 2 1
1792.3.g.g 12 112.j even 4 2
1792.3.g.g 12 112.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(196, [\chi])\):

\( T_{3}^{6} + 32T_{3}^{4} + 256T_{3}^{2} + 448 \) Copy content Toggle raw display
\( T_{5}^{3} - 2T_{5}^{2} - 28T_{5} - 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + T^{5} + 4 T^{3} + 16 T + 64 \) Copy content Toggle raw display
$3$ \( T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 448 \) Copy content Toggle raw display
$5$ \( (T^{3} - 2 T^{2} - 28 T - 32)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 512 T^{4} + 79872 T^{2} + \cdots + 3469312 \) Copy content Toggle raw display
$13$ \( (T^{3} + 6 T^{2} - 364 T - 1712)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} - 2 T^{2} - 116 T + 488)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 1600 T^{4} + \cdots + 81683392 \) Copy content Toggle raw display
$23$ \( T^{6} + 928 T^{4} + 157952 T^{2} + \cdots + 114688 \) Copy content Toggle raw display
$29$ \( (T^{3} + 18 T^{2} - 948 T + 4952)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 2048 T^{4} + \cdots + 1404928 \) Copy content Toggle raw display
$37$ \( (T^{3} - 14 T^{2} - 1300 T - 4328)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 10 T^{2} - 1396 T + 15368)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 4096 T^{4} + \cdots + 1477439488 \) Copy content Toggle raw display
$47$ \( T^{6} + 5440 T^{4} + \cdots + 4610486272 \) Copy content Toggle raw display
$53$ \( (T^{3} - 46 T^{2} - 84 T + 8536)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 5216 T^{4} + \cdots + 3560148928 \) Copy content Toggle raw display
$61$ \( (T^{3} - 82 T^{2} + 580 T + 7552)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 20064 T^{4} + \cdots + 92073361408 \) Copy content Toggle raw display
$71$ \( T^{6} + 4928 T^{4} + \cdots + 1438646272 \) Copy content Toggle raw display
$73$ \( (T^{3} - 66 T^{2} - 340 T + 3688)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 5952 T^{4} + \cdots + 5754585088 \) Copy content Toggle raw display
$83$ \( T^{6} + 25600 T^{4} + \cdots + 95209408 \) Copy content Toggle raw display
$89$ \( (T^{3} + 174 T^{2} + 1196 T - 620888)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 126 T^{2} + 1068 T - 196184)^{2} \) Copy content Toggle raw display
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