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Properties

Label 196.4.f.a
Level $196$
Weight $4$
Character orbit 196.f
Analytic conductor $11.564$
Analytic rank $0$
Dimension $4$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [196,4,Mod(19,196)]
 
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([3, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.19");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 196.f (of order \(6\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) 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: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 2 \beta_{2}) q^{2} + (7 \beta_{2} + 5 \beta_1 - 7) q^{4} + (17 \beta_{3} - 17 \beta_{2} + \cdots + 11) q^{8}+ \cdots + 27 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - 2 \beta_{2}) q^{2} + (7 \beta_{2} + 5 \beta_1 - 7) q^{4} + (17 \beta_{3} - 17 \beta_{2} + \cdots + 11) q^{8}+ \cdots + (540 \beta_{3} - 540 \beta_{2} + \cdots + 270) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5 q^{2} - 9 q^{4} + 10 q^{8} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 5 q^{2} - 9 q^{4} + 10 q^{8} + 54 q^{9} + 47 q^{16} + 135 q^{18} + 140 q^{22} - 250 q^{25} + 664 q^{29} + 275 q^{32} - 486 q^{36} + 900 q^{37} - 350 q^{44} + 574 q^{46} + 1250 q^{50} - 1180 q^{53} - 830 q^{58} - 1998 q^{64} + 135 q^{72} + 2250 q^{74} - 1458 q^{81} - 1414 q^{86} - 1190 q^{88} - 5740 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

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

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

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

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

Character values

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

\(n\) \(99\) \(101\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
1.39564 − 0.228425i
−0.895644 + 1.09445i
1.39564 + 0.228425i
−0.895644 − 1.09445i
−2.39564 + 1.50363i 0 3.47822 − 7.20430i 0 0 0 2.50000 + 22.4889i 13.5000 − 23.3827i 0
19.2 −0.104356 + 2.82650i 0 −7.97822 − 0.589925i 0 0 0 2.50000 − 22.4889i 13.5000 − 23.3827i 0
31.1 −2.39564 − 1.50363i 0 3.47822 + 7.20430i 0 0 0 2.50000 − 22.4889i 13.5000 + 23.3827i 0
31.2 −0.104356 − 2.82650i 0 −7.97822 + 0.589925i 0 0 0 2.50000 + 22.4889i 13.5000 + 23.3827i 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.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
4.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.4.f.a 4
4.b odd 2 1 inner 196.4.f.a 4
7.b odd 2 1 CM 196.4.f.a 4
7.c even 3 1 28.4.d.a ✓ 2
7.c even 3 1 inner 196.4.f.a 4
7.d odd 6 1 28.4.d.a ✓ 2
7.d odd 6 1 inner 196.4.f.a 4
21.g even 6 1 252.4.b.a 2
21.h odd 6 1 252.4.b.a 2
28.d even 2 1 inner 196.4.f.a 4
28.f even 6 1 28.4.d.a ✓ 2
28.f even 6 1 inner 196.4.f.a 4
28.g odd 6 1 28.4.d.a ✓ 2
28.g odd 6 1 inner 196.4.f.a 4
56.j odd 6 1 448.4.f.a 2
56.k odd 6 1 448.4.f.a 2
56.m even 6 1 448.4.f.a 2
56.p even 6 1 448.4.f.a 2
84.j odd 6 1 252.4.b.a 2
84.n even 6 1 252.4.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.d.a ✓ 2 7.c even 3 1
28.4.d.a ✓ 2 7.d odd 6 1
28.4.d.a ✓ 2 28.f even 6 1
28.4.d.a ✓ 2 28.g odd 6 1
196.4.f.a 4 1.a even 1 1 trivial
196.4.f.a 4 4.b odd 2 1 inner
196.4.f.a 4 7.b odd 2 1 CM
196.4.f.a 4 7.c even 3 1 inner
196.4.f.a 4 7.d odd 6 1 inner
196.4.f.a 4 28.d even 2 1 inner
196.4.f.a 4 28.f even 6 1 inner
196.4.f.a 4 28.g odd 6 1 inner
252.4.b.a 2 21.g even 6 1
252.4.b.a 2 21.h odd 6 1
252.4.b.a 2 84.j odd 6 1
252.4.b.a 2 84.n even 6 1
448.4.f.a 2 56.j odd 6 1
448.4.f.a 2 56.k odd 6 1
448.4.f.a 2 56.m even 6 1
448.4.f.a 2 56.p even 6 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 5 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 700 T^{2} + 490000 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 2215396624 \) Copy content Toggle raw display
$29$ \( (T - 166)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 450 T + 202500)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 285628)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 590 T + 348100)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 429617324304 \) Copy content Toggle raw display
$71$ \( (T^{2} + 958300)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 3214890000 \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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