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Properties

Label 196.4.e.g
Level $196$
Weight $4$
Character orbit 196.e
Analytic conductor $11.564$
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] = [196,4,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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{37})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 10x^{2} + 9x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 7) q^{5} + (10 \beta_1 - 10) q^{9} + ( - 7 \beta_{2} - 16 \beta_1) q^{11} + ( - 4 \beta_{3} + 14) q^{13} + ( - 7 \beta_{3} - 74) q^{15} + (4 \beta_{2} - 77 \beta_1) q^{17}+ \cdots + ( - 70 \beta_{3} + 160) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{5} - 20 q^{9} - 32 q^{11} + 56 q^{13} - 296 q^{15} - 154 q^{17} - 224 q^{19} - 68 q^{23} - 144 q^{25} - 472 q^{29} - 196 q^{31} - 518 q^{33} - 346 q^{37} - 296 q^{39} + 840 q^{41} - 688 q^{43}+ \cdots + 640 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 10x^{2} + 9x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 10\nu^{2} - 10\nu + 81 ) / 90 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 10\nu^{2} + 190\nu - 81 ) / 90 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 14 ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 19\beta _1 - 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 14 \) 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
1.77069 − 3.06693i
−1.27069 + 2.20090i
1.77069 + 3.06693i
−1.27069 − 2.20090i
0 −3.04138 + 5.26783i 0 2.58276 + 4.47348i 0 0 0 −5.00000 − 8.66025i 0
165.2 0 3.04138 − 5.26783i 0 −9.58276 − 16.5978i 0 0 0 −5.00000 − 8.66025i 0
177.1 0 −3.04138 − 5.26783i 0 2.58276 − 4.47348i 0 0 0 −5.00000 + 8.66025i 0
177.2 0 3.04138 + 5.26783i 0 −9.58276 + 16.5978i 0 0 0 −5.00000 + 8.66025i 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.g 4
3.b odd 2 1 1764.4.k.ba 4
7.b odd 2 1 28.4.e.a ✓ 4
7.c even 3 1 196.4.a.g 2
7.c even 3 1 inner 196.4.e.g 4
7.d odd 6 1 28.4.e.a ✓ 4
7.d odd 6 1 196.4.a.e 2
21.c even 2 1 252.4.k.c 4
21.g even 6 1 252.4.k.c 4
21.g even 6 1 1764.4.a.z 2
21.h odd 6 1 1764.4.a.n 2
21.h odd 6 1 1764.4.k.ba 4
28.d even 2 1 112.4.i.d 4
28.f even 6 1 112.4.i.d 4
28.f even 6 1 784.4.a.u 2
28.g odd 6 1 784.4.a.ba 2
35.c odd 2 1 700.4.i.g 4
35.f even 4 2 700.4.r.d 8
35.i odd 6 1 700.4.i.g 4
35.k even 12 2 700.4.r.d 8
56.e even 2 1 448.4.i.g 4
56.h odd 2 1 448.4.i.h 4
56.j odd 6 1 448.4.i.h 4
56.m even 6 1 448.4.i.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.e.a ✓ 4 7.b odd 2 1
28.4.e.a ✓ 4 7.d odd 6 1
112.4.i.d 4 28.d even 2 1
112.4.i.d 4 28.f even 6 1
196.4.a.e 2 7.d odd 6 1
196.4.a.g 2 7.c even 3 1
196.4.e.g 4 1.a even 1 1 trivial
196.4.e.g 4 7.c even 3 1 inner
252.4.k.c 4 21.c even 2 1
252.4.k.c 4 21.g even 6 1
448.4.i.g 4 56.e even 2 1
448.4.i.g 4 56.m even 6 1
448.4.i.h 4 56.h odd 2 1
448.4.i.h 4 56.j odd 6 1
700.4.i.g 4 35.c odd 2 1
700.4.i.g 4 35.i odd 6 1
700.4.r.d 8 35.f even 4 2
700.4.r.d 8 35.k even 12 2
784.4.a.u 2 28.f even 6 1
784.4.a.ba 2 28.g odd 6 1
1764.4.a.n 2 21.h odd 6 1
1764.4.a.z 2 21.g even 6 1
1764.4.k.ba 4 3.b odd 2 1
1764.4.k.ba 4 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 \(S_{4}^{\mathrm{new}}(196, [\chi])\):

\( T_{3}^{4} + 37T_{3}^{2} + 1369 \) Copy content Toggle raw display
\( T_{5}^{4} + 14T_{5}^{3} + 295T_{5}^{2} - 1386T_{5} + 9801 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 37T^{2} + 1369 \) Copy content Toggle raw display
$5$ \( T^{4} + 14 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 32 T^{3} + \cdots + 2424249 \) Copy content Toggle raw display
$13$ \( (T^{2} - 28 T - 396)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 154 T^{3} + \cdots + 28483569 \) Copy content Toggle raw display
$19$ \( T^{4} + 224 T^{3} + \cdots + 149108521 \) Copy content Toggle raw display
$23$ \( T^{4} + 68 T^{3} + \cdots + 431649 \) Copy content Toggle raw display
$29$ \( (T^{2} + 236 T - 15084)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 196 T^{3} + \cdots + 60699681 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 1248844921 \) Copy content Toggle raw display
$41$ \( (T^{2} - 420 T + 38772)^{2} \) Copy content Toggle raw display
$43$ \( (T + 172)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} - 84 T^{3} + \cdots + 43046721 \) Copy content Toggle raw display
$53$ \( T^{4} - 438 T^{3} + \cdots + 299532249 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 2486917161 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 40084444521 \) Copy content Toggle raw display
$67$ \( T^{4} + 336 T^{3} + \cdots + 697540921 \) Copy content Toggle raw display
$71$ \( (T^{2} - 896 T + 84672)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 47737443121 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 427476669489 \) Copy content Toggle raw display
$83$ \( (T^{2} + 392 T - 248112)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 12452551281 \) Copy content Toggle raw display
$97$ \( (T^{2} - 420 T - 169612)^{2} \) Copy content Toggle raw display
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