Properties

Label 196.6.e.f
Level $196$
Weight $6$
Character orbit 196.e
Analytic conductor $31.435$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{3} + 96 \zeta_{6} q^{5} + 239 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{3} + 96 \zeta_{6} q^{5} + 239 \zeta_{6} q^{9} + ( - 720 \zeta_{6} + 720) q^{11} + 572 q^{13} + 192 q^{15} + (1254 \zeta_{6} - 1254) q^{17} + 94 \zeta_{6} q^{19} - 96 \zeta_{6} q^{23} + (6091 \zeta_{6} - 6091) q^{25} + 964 q^{27} - 4374 q^{29} + ( - 6244 \zeta_{6} + 6244) q^{31} - 1440 \zeta_{6} q^{33} + 10798 \zeta_{6} q^{37} + ( - 1144 \zeta_{6} + 1144) q^{39} + 12006 q^{41} - 9160 q^{43} + (22944 \zeta_{6} - 22944) q^{45} + 25836 \zeta_{6} q^{47} + 2508 \zeta_{6} q^{51} + (1014 \zeta_{6} - 1014) q^{53} + 69120 q^{55} + 188 q^{57} + (1242 \zeta_{6} - 1242) q^{59} - 7592 \zeta_{6} q^{61} + 54912 \zeta_{6} q^{65} + (41132 \zeta_{6} - 41132) q^{67} - 192 q^{69} - 37632 q^{71} + ( - 13438 \zeta_{6} + 13438) q^{73} + 12182 \zeta_{6} q^{75} - 6248 \zeta_{6} q^{79} + (56149 \zeta_{6} - 56149) q^{81} - 25254 q^{83} - 120384 q^{85} + (8748 \zeta_{6} - 8748) q^{87} + 45126 \zeta_{6} q^{89} - 12488 \zeta_{6} q^{93} + (9024 \zeta_{6} - 9024) q^{95} + 107222 q^{97} + 172080 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 96 q^{5} + 239 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 96 q^{5} + 239 q^{9} + 720 q^{11} + 1144 q^{13} + 384 q^{15} - 1254 q^{17} + 94 q^{19} - 96 q^{23} - 6091 q^{25} + 1928 q^{27} - 8748 q^{29} + 6244 q^{31} - 1440 q^{33} + 10798 q^{37} + 1144 q^{39} + 24012 q^{41} - 18320 q^{43} - 22944 q^{45} + 25836 q^{47} + 2508 q^{51} - 1014 q^{53} + 138240 q^{55} + 376 q^{57} - 1242 q^{59} - 7592 q^{61} + 54912 q^{65} - 41132 q^{67} - 384 q^{69} - 75264 q^{71} + 13438 q^{73} + 12182 q^{75} - 6248 q^{79} - 56149 q^{81} - 50508 q^{83} - 240768 q^{85} - 8748 q^{87} + 45126 q^{89} - 12488 q^{93} - 9024 q^{95} + 214444 q^{97} + 344160 q^{99}+O(q^{100}) \) 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 1.00000 1.73205i 0 48.0000 + 83.1384i 0 0 0 119.500 + 206.980i 0
177.1 0 1.00000 + 1.73205i 0 48.0000 83.1384i 0 0 0 119.500 206.980i 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.f 2
7.b odd 2 1 196.6.e.e 2
7.c even 3 1 28.6.a.a 1
7.c even 3 1 inner 196.6.e.f 2
7.d odd 6 1 196.6.a.d 1
7.d odd 6 1 196.6.e.e 2
21.h odd 6 1 252.6.a.d 1
28.f even 6 1 784.6.a.f 1
28.g odd 6 1 112.6.a.e 1
35.j even 6 1 700.6.a.d 1
35.l odd 12 2 700.6.e.d 2
56.k odd 6 1 448.6.a.h 1
56.p even 6 1 448.6.a.i 1
84.n even 6 1 1008.6.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.a.a 1 7.c even 3 1
112.6.a.e 1 28.g odd 6 1
196.6.a.d 1 7.d odd 6 1
196.6.e.e 2 7.b odd 2 1
196.6.e.e 2 7.d odd 6 1
196.6.e.f 2 1.a even 1 1 trivial
196.6.e.f 2 7.c even 3 1 inner
252.6.a.d 1 21.h odd 6 1
448.6.a.h 1 56.k odd 6 1
448.6.a.i 1 56.p even 6 1
700.6.a.d 1 35.j even 6 1
700.6.e.d 2 35.l odd 12 2
784.6.a.f 1 28.f even 6 1
1008.6.a.bb 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} + 4 \) 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
$3$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} - 96T + 9216 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 720T + 518400 \) Copy content Toggle raw display
$13$ \( (T - 572)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 1254 T + 1572516 \) Copy content Toggle raw display
$19$ \( T^{2} - 94T + 8836 \) Copy content Toggle raw display
$23$ \( T^{2} + 96T + 9216 \) Copy content Toggle raw display
$29$ \( (T + 4374)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 6244 T + 38987536 \) Copy content Toggle raw display
$37$ \( T^{2} - 10798 T + 116596804 \) Copy content Toggle raw display
$41$ \( (T - 12006)^{2} \) Copy content Toggle raw display
$43$ \( (T + 9160)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 25836 T + 667498896 \) Copy content Toggle raw display
$53$ \( T^{2} + 1014 T + 1028196 \) Copy content Toggle raw display
$59$ \( T^{2} + 1242 T + 1542564 \) Copy content Toggle raw display
$61$ \( T^{2} + 7592 T + 57638464 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 1691841424 \) Copy content Toggle raw display
$71$ \( (T + 37632)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 13438 T + 180579844 \) Copy content Toggle raw display
$79$ \( T^{2} + 6248 T + 39037504 \) Copy content Toggle raw display
$83$ \( (T + 25254)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 2036355876 \) Copy content Toggle raw display
$97$ \( (T - 107222)^{2} \) Copy content Toggle raw display
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