Properties

Label 196.4.e.a
Level $196$
Weight $4$
Character orbit 196.e
Analytic conductor $11.564$
Analytic rank $0$
Dimension $2$
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] = [196,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.165");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(11.5643743611\)
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.

\(f(q)\) \(=\) \( q + (10 \zeta_{6} - 10) q^{3} - 8 \zeta_{6} q^{5} - 73 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (10 \zeta_{6} - 10) q^{3} - 8 \zeta_{6} q^{5} - 73 \zeta_{6} q^{9} + ( - 40 \zeta_{6} + 40) q^{11} + 12 q^{13} + 80 q^{15} + (58 \zeta_{6} - 58) q^{17} + 26 \zeta_{6} q^{19} + 64 \zeta_{6} q^{23} + ( - 61 \zeta_{6} + 61) q^{25} + 460 q^{27} - 62 q^{29} + ( - 252 \zeta_{6} + 252) q^{31} + 400 \zeta_{6} q^{33} - 26 \zeta_{6} q^{37} + (120 \zeta_{6} - 120) q^{39} - 6 q^{41} + 416 q^{43} + (584 \zeta_{6} - 584) q^{45} - 396 \zeta_{6} q^{47} - 580 \zeta_{6} q^{51} + ( - 450 \zeta_{6} + 450) q^{53} - 320 q^{55} - 260 q^{57} + ( - 274 \zeta_{6} + 274) q^{59} - 576 \zeta_{6} q^{61} - 96 \zeta_{6} q^{65} + ( - 476 \zeta_{6} + 476) q^{67} - 640 q^{69} - 448 q^{71} + (158 \zeta_{6} - 158) q^{73} + 610 \zeta_{6} q^{75} + 936 \zeta_{6} q^{79} + (2629 \zeta_{6} - 2629) q^{81} - 530 q^{83} + 464 q^{85} + ( - 620 \zeta_{6} + 620) q^{87} - 390 \zeta_{6} q^{89} + 2520 \zeta_{6} q^{93} + ( - 208 \zeta_{6} + 208) q^{95} - 214 q^{97} - 2920 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{3} - 8 q^{5} - 73 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{3} - 8 q^{5} - 73 q^{9} + 40 q^{11} + 24 q^{13} + 160 q^{15} - 58 q^{17} + 26 q^{19} + 64 q^{23} + 61 q^{25} + 920 q^{27} - 124 q^{29} + 252 q^{31} + 400 q^{33} - 26 q^{37} - 120 q^{39} - 12 q^{41} + 832 q^{43} - 584 q^{45} - 396 q^{47} - 580 q^{51} + 450 q^{53} - 640 q^{55} - 520 q^{57} + 274 q^{59} - 576 q^{61} - 96 q^{65} + 476 q^{67} - 1280 q^{69} - 896 q^{71} - 158 q^{73} + 610 q^{75} + 936 q^{79} - 2629 q^{81} - 1060 q^{83} + 928 q^{85} + 620 q^{87} - 390 q^{89} + 2520 q^{93} + 208 q^{95} - 428 q^{97} - 5840 q^{99}+O(q^{100}) \) 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 −5.00000 + 8.66025i 0 −4.00000 6.92820i 0 0 0 −36.5000 63.2199i 0
177.1 0 −5.00000 8.66025i 0 −4.00000 + 6.92820i 0 0 0 −36.5000 + 63.2199i 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 196.4.e.a 2
3.b odd 2 1 1764.4.k.m 2
7.b odd 2 1 196.4.e.f 2
7.c even 3 1 196.4.a.d 1
7.c even 3 1 inner 196.4.e.a 2
7.d odd 6 1 28.4.a.a 1
7.d odd 6 1 196.4.e.f 2
21.c even 2 1 1764.4.k.d 2
21.g even 6 1 252.4.a.d 1
21.g even 6 1 1764.4.k.d 2
21.h odd 6 1 1764.4.a.c 1
21.h odd 6 1 1764.4.k.m 2
28.f even 6 1 112.4.a.g 1
28.g odd 6 1 784.4.a.a 1
35.i odd 6 1 700.4.a.n 1
35.k even 12 2 700.4.e.a 2
56.j odd 6 1 448.4.a.p 1
56.m even 6 1 448.4.a.a 1
84.j odd 6 1 1008.4.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 7.d odd 6 1
112.4.a.g 1 28.f even 6 1
196.4.a.d 1 7.c even 3 1
196.4.e.a 2 1.a even 1 1 trivial
196.4.e.a 2 7.c even 3 1 inner
196.4.e.f 2 7.b odd 2 1
196.4.e.f 2 7.d odd 6 1
252.4.a.d 1 21.g even 6 1
448.4.a.a 1 56.m even 6 1
448.4.a.p 1 56.j odd 6 1
700.4.a.n 1 35.i odd 6 1
700.4.e.a 2 35.k even 12 2
784.4.a.a 1 28.g odd 6 1
1008.4.a.o 1 84.j odd 6 1
1764.4.a.c 1 21.h odd 6 1
1764.4.k.d 2 21.c even 2 1
1764.4.k.d 2 21.g even 6 1
1764.4.k.m 2 3.b odd 2 1
1764.4.k.m 2 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(196, [\chi])\):

\( T_{3}^{2} + 10T_{3} + 100 \) Copy content Toggle raw display
\( T_{5}^{2} + 8T_{5} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$5$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 40T + 1600 \) Copy content Toggle raw display
$13$ \( (T - 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 58T + 3364 \) Copy content Toggle raw display
$19$ \( T^{2} - 26T + 676 \) Copy content Toggle raw display
$23$ \( T^{2} - 64T + 4096 \) Copy content Toggle raw display
$29$ \( (T + 62)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 252T + 63504 \) Copy content Toggle raw display
$37$ \( T^{2} + 26T + 676 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( (T - 416)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 396T + 156816 \) Copy content Toggle raw display
$53$ \( T^{2} - 450T + 202500 \) Copy content Toggle raw display
$59$ \( T^{2} - 274T + 75076 \) Copy content Toggle raw display
$61$ \( T^{2} + 576T + 331776 \) Copy content Toggle raw display
$67$ \( T^{2} - 476T + 226576 \) Copy content Toggle raw display
$71$ \( (T + 448)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 158T + 24964 \) Copy content Toggle raw display
$79$ \( T^{2} - 936T + 876096 \) Copy content Toggle raw display
$83$ \( (T + 530)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 390T + 152100 \) Copy content Toggle raw display
$97$ \( (T + 214)^{2} \) Copy content Toggle raw display
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