Properties

Label 196.5.c.a
Level $196$
Weight $5$
Character orbit 196.c
Self dual yes
Analytic conductor $20.261$
Analytic rank $0$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [196,5,Mod(99,196)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(196, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("196.99");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 196.c (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(20.2605127644\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 16 q^{4} + 14 q^{5} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 16 q^{4} + 14 q^{5} - 64 q^{8} + 81 q^{9} - 56 q^{10} + 238 q^{13} + 256 q^{16} - 322 q^{17} - 324 q^{18} + 224 q^{20} - 429 q^{25} - 952 q^{26} + 82 q^{29} - 1024 q^{32} + 1288 q^{34} + 1296 q^{36} + 2162 q^{37} - 896 q^{40} + 3038 q^{41} + 1134 q^{45} + 1716 q^{50} + 3808 q^{52} + 2482 q^{53} - 328 q^{58} + 6958 q^{61} + 4096 q^{64} + 3332 q^{65} - 5152 q^{68} - 5184 q^{72} - 1442 q^{73} - 8648 q^{74} + 3584 q^{80} + 6561 q^{81} - 12152 q^{82} - 4508 q^{85} + 9758 q^{89} - 4536 q^{90} + 1918 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(99\) \(101\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0
−4.00000 0 16.0000 14.0000 0 0 −64.0000 81.0000 −56.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 196.5.c.a 1
4.b odd 2 1 CM 196.5.c.a 1
7.b odd 2 1 4.5.b.a 1
21.c even 2 1 36.5.d.a 1
28.d even 2 1 4.5.b.a 1
35.c odd 2 1 100.5.b.a 1
35.f even 4 2 100.5.d.a 2
56.e even 2 1 64.5.c.a 1
56.h odd 2 1 64.5.c.a 1
84.h odd 2 1 36.5.d.a 1
112.j even 4 2 256.5.d.c 2
112.l odd 4 2 256.5.d.c 2
140.c even 2 1 100.5.b.a 1
140.j odd 4 2 100.5.d.a 2
168.e odd 2 1 576.5.g.b 1
168.i even 2 1 576.5.g.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.5.b.a 1 7.b odd 2 1
4.5.b.a 1 28.d even 2 1
36.5.d.a 1 21.c even 2 1
36.5.d.a 1 84.h odd 2 1
64.5.c.a 1 56.e even 2 1
64.5.c.a 1 56.h odd 2 1
100.5.b.a 1 35.c odd 2 1
100.5.b.a 1 140.c even 2 1
100.5.d.a 2 35.f even 4 2
100.5.d.a 2 140.j odd 4 2
196.5.c.a 1 1.a even 1 1 trivial
196.5.c.a 1 4.b odd 2 1 CM
256.5.d.c 2 112.j even 4 2
256.5.d.c 2 112.l odd 4 2
576.5.g.b 1 168.e odd 2 1
576.5.g.b 1 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(196, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 14 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 238 \) Copy content Toggle raw display
$17$ \( T + 322 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 82 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 2162 \) Copy content Toggle raw display
$41$ \( T - 3038 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 2482 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 6958 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 1442 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T - 9758 \) Copy content Toggle raw display
$97$ \( T - 1918 \) Copy content Toggle raw display
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