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Properties

Label 196.8.e.d
Level $196$
Weight $8$
Character orbit 196.e
Analytic conductor $61.227$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [196,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.165");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
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: \(61.2274649949\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{3529})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 883x^{2} + 882x + 777924 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 7 \beta_1) q^{3} + ( - 3 \beta_{3} - 3 \beta_{2} + \cdots - 21) q^{5} + (14 \beta_{3} + 14 \beta_{2} + \cdots - 1391) q^{9} + ( - 42 \beta_{2} - 3714 \beta_1) q^{11} + (9 \beta_{3} + 5915) q^{13}+ \cdots + ( - 110418 \beta_{3} + 7241226) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 14 q^{3} - 42 q^{5} - 2782 q^{9} - 7428 q^{11} + 23660 q^{13} + 41760 q^{15} - 15792 q^{17} - 26614 q^{19} - 32640 q^{23} + 91846 q^{25} - 175336 q^{27} - 316032 q^{29} + 180740 q^{31} + 348432 q^{33}+ \cdots + 28964904 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 883x^{2} + 882x + 777924 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 883\nu^{2} - 883\nu + 777924 ) / 778806 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 883\nu^{2} + 1558495\nu - 777924 ) / 778806 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + 2647 ) / 883 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 1765\beta _1 - 1765 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 883\beta_{3} - 2647 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(99\) \(101\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
165.1
−14.6013 + 25.2903i
15.1013 − 26.1563i
−14.6013 − 25.2903i
15.1013 + 26.1563i
0 −26.2027 + 45.3844i 0 −99.6081 − 172.526i 0 0 0 −279.662 − 484.389i 0
165.2 0 33.2027 − 57.5088i 0 78.6081 + 136.153i 0 0 0 −1111.34 − 1924.89i 0
177.1 0 −26.2027 − 45.3844i 0 −99.6081 + 172.526i 0 0 0 −279.662 + 484.389i 0
177.2 0 33.2027 + 57.5088i 0 78.6081 − 136.153i 0 0 0 −1111.34 + 1924.89i 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 196.8.e.d 4
7.b odd 2 1 196.8.e.a 4
7.c even 3 1 28.8.a.a ✓ 2
7.c even 3 1 inner 196.8.e.d 4
7.d odd 6 1 196.8.a.b 2
7.d odd 6 1 196.8.e.a 4
21.h odd 6 1 252.8.a.e 2
28.g odd 6 1 112.8.a.i 2
56.k odd 6 1 448.8.a.n 2
56.p even 6 1 448.8.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.8.a.a ✓ 2 7.c even 3 1
112.8.a.i 2 28.g odd 6 1
196.8.a.b 2 7.d odd 6 1
196.8.e.a 4 7.b odd 2 1
196.8.e.a 4 7.d odd 6 1
196.8.e.d 4 1.a even 1 1 trivial
196.8.e.d 4 7.c even 3 1 inner
252.8.a.e 2 21.h odd 6 1
448.8.a.n 2 56.k odd 6 1
448.8.a.p 2 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 14T_{3}^{3} + 3676T_{3}^{2} + 48720T_{3} + 12110400 \) acting on \(S_{8}^{\mathrm{new}}(196, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 14 T^{3} + \cdots + 12110400 \) Copy content Toggle raw display
$5$ \( T^{4} + 42 T^{3} + \cdots + 980942400 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 57284311449600 \) Copy content Toggle raw display
$13$ \( (T^{2} - 11830 T + 34701376)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{2} + 158016 T + 5489020188)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 27\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{2} + 321720 T - 208069107156)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 1023868 T + 127557024352)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 43\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots + 4028431841280)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 27\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( (T^{2} + \cdots + 14563855983960)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 2955690377596)^{2} \) Copy content Toggle raw display
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