Properties

Label 196.6.e.a
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.

\(f(q)\) \(=\) \( q + (26 \zeta_{6} - 26) q^{3} - 16 \zeta_{6} q^{5} - 433 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (26 \zeta_{6} - 26) q^{3} - 16 \zeta_{6} q^{5} - 433 \zeta_{6} q^{9} + (8 \zeta_{6} - 8) q^{11} + 684 q^{13} + 416 q^{15} + ( - 2218 \zeta_{6} + 2218) q^{17} + 2698 \zeta_{6} q^{19} - 3344 \zeta_{6} q^{23} + ( - 2869 \zeta_{6} + 2869) q^{25} + 4940 q^{27} - 3254 q^{29} + (4788 \zeta_{6} - 4788) q^{31} - 208 \zeta_{6} q^{33} + 11470 \zeta_{6} q^{37} + (17784 \zeta_{6} - 17784) q^{39} + 13350 q^{41} - 928 q^{43} + (6928 \zeta_{6} - 6928) q^{45} - 1212 \zeta_{6} q^{47} + 57668 \zeta_{6} q^{51} + (13110 \zeta_{6} - 13110) q^{53} + 128 q^{55} - 70148 q^{57} + (34702 \zeta_{6} - 34702) q^{59} + 1032 \zeta_{6} q^{61} - 10944 \zeta_{6} q^{65} + (10108 \zeta_{6} - 10108) q^{67} + 86944 q^{69} + 62720 q^{71} + ( - 18926 \zeta_{6} + 18926) q^{73} + 74594 \zeta_{6} q^{75} - 11400 \zeta_{6} q^{79} + (23221 \zeta_{6} - 23221) q^{81} + 88958 q^{83} - 35488 q^{85} + ( - 84604 \zeta_{6} + 84604) q^{87} - 19722 \zeta_{6} q^{89} - 124488 \zeta_{6} q^{93} + ( - 43168 \zeta_{6} + 43168) q^{95} + 17062 q^{97} + 3464 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 26 q^{3} - 16 q^{5} - 433 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 26 q^{3} - 16 q^{5} - 433 q^{9} - 8 q^{11} + 1368 q^{13} + 832 q^{15} + 2218 q^{17} + 2698 q^{19} - 3344 q^{23} + 2869 q^{25} + 9880 q^{27} - 6508 q^{29} - 4788 q^{31} - 208 q^{33} + 11470 q^{37} - 17784 q^{39} + 26700 q^{41} - 1856 q^{43} - 6928 q^{45} - 1212 q^{47} + 57668 q^{51} - 13110 q^{53} + 256 q^{55} - 140296 q^{57} - 34702 q^{59} + 1032 q^{61} - 10944 q^{65} - 10108 q^{67} + 173888 q^{69} + 125440 q^{71} + 18926 q^{73} + 74594 q^{75} - 11400 q^{79} - 23221 q^{81} + 177916 q^{83} - 70976 q^{85} + 84604 q^{87} - 19722 q^{89} - 124488 q^{93} + 43168 q^{95} + 34124 q^{97} + 6928 q^{99}+O(q^{100}) \) 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 −13.0000 + 22.5167i 0 −8.00000 13.8564i 0 0 0 −216.500 374.989i 0
177.1 0 −13.0000 22.5167i 0 −8.00000 + 13.8564i 0 0 0 −216.500 + 374.989i 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.a 2
7.b odd 2 1 196.6.e.i 2
7.c even 3 1 28.6.a.b 1
7.c even 3 1 inner 196.6.e.a 2
7.d odd 6 1 196.6.a.a 1
7.d odd 6 1 196.6.e.i 2
21.h odd 6 1 252.6.a.a 1
28.f even 6 1 784.6.a.m 1
28.g odd 6 1 112.6.a.b 1
35.j even 6 1 700.6.a.b 1
35.l odd 12 2 700.6.e.b 2
56.k odd 6 1 448.6.a.o 1
56.p even 6 1 448.6.a.b 1
84.n even 6 1 1008.6.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.a.b 1 7.c even 3 1
112.6.a.b 1 28.g odd 6 1
196.6.a.a 1 7.d odd 6 1
196.6.e.a 2 1.a even 1 1 trivial
196.6.e.a 2 7.c even 3 1 inner
196.6.e.i 2 7.b odd 2 1
196.6.e.i 2 7.d odd 6 1
252.6.a.a 1 21.h odd 6 1
448.6.a.b 1 56.p even 6 1
448.6.a.o 1 56.k odd 6 1
700.6.a.b 1 35.j even 6 1
700.6.e.b 2 35.l odd 12 2
784.6.a.m 1 28.f even 6 1
1008.6.a.l 1 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 26T_{3} + 676 \) 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
$3$ \( T^{2} + 26T + 676 \) Copy content Toggle raw display
$5$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$13$ \( (T - 684)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2218 T + 4919524 \) Copy content Toggle raw display
$19$ \( T^{2} - 2698 T + 7279204 \) Copy content Toggle raw display
$23$ \( T^{2} + 3344 T + 11182336 \) Copy content Toggle raw display
$29$ \( (T + 3254)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 4788 T + 22924944 \) Copy content Toggle raw display
$37$ \( T^{2} - 11470 T + 131560900 \) Copy content Toggle raw display
$41$ \( (T - 13350)^{2} \) Copy content Toggle raw display
$43$ \( (T + 928)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 1212 T + 1468944 \) Copy content Toggle raw display
$53$ \( T^{2} + 13110 T + 171872100 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 1204228804 \) Copy content Toggle raw display
$61$ \( T^{2} - 1032 T + 1065024 \) Copy content Toggle raw display
$67$ \( T^{2} + 10108 T + 102171664 \) Copy content Toggle raw display
$71$ \( (T - 62720)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 18926 T + 358193476 \) Copy content Toggle raw display
$79$ \( T^{2} + 11400 T + 129960000 \) Copy content Toggle raw display
$83$ \( (T - 88958)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 19722 T + 388957284 \) Copy content Toggle raw display
$97$ \( (T - 17062)^{2} \) Copy content Toggle raw display
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