Properties

Label 196.6.e.g
Level $196$
Weight $6$
Character orbit 196.e
Analytic conductor $31.435$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [196,6,Mod(165,196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(196, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.165");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 196.e (of order \(3\), degree \(2\), not 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: \(31.4352286833\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 12 \zeta_{6} + 12) q^{3} - 54 \zeta_{6} q^{5} + 99 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 12 \zeta_{6} + 12) q^{3} - 54 \zeta_{6} q^{5} + 99 \zeta_{6} q^{9} + (540 \zeta_{6} - 540) q^{11} - 418 q^{13} - 648 q^{15} + (594 \zeta_{6} - 594) q^{17} - 836 \zeta_{6} q^{19} + 4104 \zeta_{6} q^{23} + ( - 209 \zeta_{6} + 209) q^{25} + 4104 q^{27} - 594 q^{29} + (4256 \zeta_{6} - 4256) q^{31} + 6480 \zeta_{6} q^{33} + 298 \zeta_{6} q^{37} + (5016 \zeta_{6} - 5016) q^{39} + 17226 q^{41} - 12100 q^{43} + ( - 5346 \zeta_{6} + 5346) q^{45} + 1296 \zeta_{6} q^{47} + 7128 \zeta_{6} q^{51} + (19494 \zeta_{6} - 19494) q^{53} + 29160 q^{55} - 10032 q^{57} + ( - 7668 \zeta_{6} + 7668) q^{59} + 34738 \zeta_{6} q^{61} + 22572 \zeta_{6} q^{65} + (21812 \zeta_{6} - 21812) q^{67} + 49248 q^{69} - 46872 q^{71} + (67562 \zeta_{6} - 67562) q^{73} - 2508 \zeta_{6} q^{75} + 76912 \zeta_{6} q^{79} + ( - 25191 \zeta_{6} + 25191) q^{81} + 67716 q^{83} + 32076 q^{85} + (7128 \zeta_{6} - 7128) q^{87} - 29754 \zeta_{6} q^{89} + 51072 \zeta_{6} q^{93} + (45144 \zeta_{6} - 45144) q^{95} - 122398 q^{97} - 53460 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 12 q^{3} - 54 q^{5} + 99 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 12 q^{3} - 54 q^{5} + 99 q^{9} - 540 q^{11} - 836 q^{13} - 1296 q^{15} - 594 q^{17} - 836 q^{19} + 4104 q^{23} + 209 q^{25} + 8208 q^{27} - 1188 q^{29} - 4256 q^{31} + 6480 q^{33} + 298 q^{37} - 5016 q^{39} + 34452 q^{41} - 24200 q^{43} + 5346 q^{45} + 1296 q^{47} + 7128 q^{51} - 19494 q^{53} + 58320 q^{55} - 20064 q^{57} + 7668 q^{59} + 34738 q^{61} + 22572 q^{65} - 21812 q^{67} + 98496 q^{69} - 93744 q^{71} - 67562 q^{73} - 2508 q^{75} + 76912 q^{79} + 25191 q^{81} + 135432 q^{83} + 64152 q^{85} - 7128 q^{87} - 29754 q^{89} + 51072 q^{93} - 45144 q^{95} - 244796 q^{97} - 106920 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(99\) \(101\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
0.500000 + 0.866025i
0.500000 0.866025i
0 6.00000 10.3923i 0 −27.0000 46.7654i 0 0 0 49.5000 + 85.7365i 0
177.1 0 6.00000 + 10.3923i 0 −27.0000 + 46.7654i 0 0 0 49.5000 85.7365i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.6.e.g 2
7.b odd 2 1 196.6.e.d 2
7.c even 3 1 4.6.a.a 1
7.c even 3 1 inner 196.6.e.g 2
7.d odd 6 1 196.6.a.e 1
7.d odd 6 1 196.6.e.d 2
21.h odd 6 1 36.6.a.a 1
28.f even 6 1 784.6.a.d 1
28.g odd 6 1 16.6.a.b 1
35.j even 6 1 100.6.a.b 1
35.l odd 12 2 100.6.c.b 2
56.k odd 6 1 64.6.a.b 1
56.p even 6 1 64.6.a.f 1
63.g even 3 1 324.6.e.a 2
63.h even 3 1 324.6.e.a 2
63.j odd 6 1 324.6.e.d 2
63.n odd 6 1 324.6.e.d 2
77.h odd 6 1 484.6.a.a 1
84.n even 6 1 144.6.a.c 1
91.r even 6 1 676.6.a.a 1
91.z odd 12 2 676.6.d.a 2
105.o odd 6 1 900.6.a.h 1
105.x even 12 2 900.6.d.a 2
112.u odd 12 2 256.6.b.c 2
112.w even 12 2 256.6.b.g 2
140.p odd 6 1 400.6.a.d 1
140.w even 12 2 400.6.c.f 2
168.s odd 6 1 576.6.a.bc 1
168.v even 6 1 576.6.a.bd 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.6.a.a 1 7.c even 3 1
16.6.a.b 1 28.g odd 6 1
36.6.a.a 1 21.h odd 6 1
64.6.a.b 1 56.k odd 6 1
64.6.a.f 1 56.p even 6 1
100.6.a.b 1 35.j even 6 1
100.6.c.b 2 35.l odd 12 2
144.6.a.c 1 84.n even 6 1
196.6.a.e 1 7.d odd 6 1
196.6.e.d 2 7.b odd 2 1
196.6.e.d 2 7.d odd 6 1
196.6.e.g 2 1.a even 1 1 trivial
196.6.e.g 2 7.c even 3 1 inner
256.6.b.c 2 112.u odd 12 2
256.6.b.g 2 112.w even 12 2
324.6.e.a 2 63.g even 3 1
324.6.e.a 2 63.h even 3 1
324.6.e.d 2 63.j odd 6 1
324.6.e.d 2 63.n odd 6 1
400.6.a.d 1 140.p odd 6 1
400.6.c.f 2 140.w even 12 2
484.6.a.a 1 77.h odd 6 1
576.6.a.bc 1 168.s odd 6 1
576.6.a.bd 1 168.v even 6 1
676.6.a.a 1 91.r even 6 1
676.6.d.a 2 91.z odd 12 2
784.6.a.d 1 28.f even 6 1
900.6.a.h 1 105.o odd 6 1
900.6.d.a 2 105.x even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 12T_{3} + 144 \) acting on \(S_{6}^{\mathrm{new}}(196, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$5$ \( T^{2} + 54T + 2916 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 540T + 291600 \) Copy content Toggle raw display
$13$ \( (T + 418)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 594T + 352836 \) Copy content Toggle raw display
$19$ \( T^{2} + 836T + 698896 \) Copy content Toggle raw display
$23$ \( T^{2} - 4104 T + 16842816 \) Copy content Toggle raw display
$29$ \( (T + 594)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4256 T + 18113536 \) Copy content Toggle raw display
$37$ \( T^{2} - 298T + 88804 \) Copy content Toggle raw display
$41$ \( (T - 17226)^{2} \) Copy content Toggle raw display
$43$ \( (T + 12100)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 1296 T + 1679616 \) Copy content Toggle raw display
$53$ \( T^{2} + 19494 T + 380016036 \) Copy content Toggle raw display
$59$ \( T^{2} - 7668 T + 58798224 \) Copy content Toggle raw display
$61$ \( T^{2} - 34738 T + 1206728644 \) Copy content Toggle raw display
$67$ \( T^{2} + 21812 T + 475763344 \) Copy content Toggle raw display
$71$ \( (T + 46872)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 67562 T + 4564623844 \) Copy content Toggle raw display
$79$ \( T^{2} - 76912 T + 5915455744 \) Copy content Toggle raw display
$83$ \( (T - 67716)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 29754 T + 885300516 \) Copy content Toggle raw display
$97$ \( (T + 122398)^{2} \) Copy content Toggle raw display
show more
show less