Properties

Label 200.8.c.c
Level $200$
Weight $8$
Character orbit 200.c
Analytic conductor $62.477$
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] = [200,8,Mod(49,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.49");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 200.c (of order \(2\), degree \(1\), 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: \(62.4770050968\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 8)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 22 \beta q^{3} + 612 \beta q^{7} + 251 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 22 \beta q^{3} + 612 \beta q^{7} + 251 q^{9} - 3164 q^{11} + 3059 \beta q^{13} + 8135 \beta q^{17} + 5476 q^{19} - 53856 q^{21} + 788 \beta q^{23} + 53636 \beta q^{27} - 122838 q^{29} + 251360 q^{31} - 69608 \beta q^{33} + 26169 \beta q^{37} - 269192 q^{39} - 319398 q^{41} + 355394 \beta q^{43} - 142056 \beta q^{47} - 674633 q^{49} - 715880 q^{51} + 148031 \beta q^{53} + 120472 \beta q^{57} + 897548 q^{59} - 884810 q^{61} + 153612 \beta q^{63} - 2329846 \beta q^{67} - 69344 q^{69} - 2710792 q^{71} - 2835427 \beta q^{73} - 1936368 \beta q^{77} + 5124176 q^{79} - 4171031 q^{81} - 781778 \beta q^{83} - 2702436 \beta q^{87} - 11605674 q^{89} - 7488432 q^{91} + 5529920 \beta q^{93} - 5465809 \beta q^{97} - 794164 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 502 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 502 q^{9} - 6328 q^{11} + 10952 q^{19} - 107712 q^{21} - 245676 q^{29} + 502720 q^{31} - 538384 q^{39} - 638796 q^{41} - 1349266 q^{49} - 1431760 q^{51} + 1795096 q^{59} - 1769620 q^{61} - 138688 q^{69} - 5421584 q^{71} + 10248352 q^{79} - 8342062 q^{81} - 23211348 q^{89} - 14976864 q^{91} - 1588328 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/200\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 44.0000i 0 0 0 1224.00i 0 251.000 0
49.2 0 44.0000i 0 0 0 1224.00i 0 251.000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.8.c.c 2
4.b odd 2 1 400.8.c.f 2
5.b even 2 1 inner 200.8.c.c 2
5.c odd 4 1 8.8.a.b 1
5.c odd 4 1 200.8.a.b 1
15.e even 4 1 72.8.a.a 1
20.d odd 2 1 400.8.c.f 2
20.e even 4 1 16.8.a.a 1
20.e even 4 1 400.8.a.p 1
35.f even 4 1 392.8.a.b 1
40.i odd 4 1 64.8.a.b 1
40.k even 4 1 64.8.a.f 1
60.l odd 4 1 144.8.a.a 1
80.i odd 4 1 256.8.b.g 2
80.j even 4 1 256.8.b.a 2
80.s even 4 1 256.8.b.a 2
80.t odd 4 1 256.8.b.g 2
120.q odd 4 1 576.8.a.z 1
120.w even 4 1 576.8.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.b 1 5.c odd 4 1
16.8.a.a 1 20.e even 4 1
64.8.a.b 1 40.i odd 4 1
64.8.a.f 1 40.k even 4 1
72.8.a.a 1 15.e even 4 1
144.8.a.a 1 60.l odd 4 1
200.8.a.b 1 5.c odd 4 1
200.8.c.c 2 1.a even 1 1 trivial
200.8.c.c 2 5.b even 2 1 inner
256.8.b.a 2 80.j even 4 1
256.8.b.a 2 80.s even 4 1
256.8.b.g 2 80.i odd 4 1
256.8.b.g 2 80.t odd 4 1
392.8.a.b 1 35.f even 4 1
400.8.a.p 1 20.e even 4 1
400.8.c.f 2 4.b odd 2 1
400.8.c.f 2 20.d odd 2 1
576.8.a.y 1 120.w even 4 1
576.8.a.z 1 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 1936 \) acting on \(S_{8}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1936 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1498176 \) Copy content Toggle raw display
$11$ \( (T + 3164)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 37429924 \) Copy content Toggle raw display
$17$ \( T^{2} + 264712900 \) Copy content Toggle raw display
$19$ \( (T - 5476)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 2483776 \) Copy content Toggle raw display
$29$ \( (T + 122838)^{2} \) Copy content Toggle raw display
$31$ \( (T - 251360)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 2739266244 \) Copy content Toggle raw display
$41$ \( (T + 319398)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 505219580944 \) Copy content Toggle raw display
$47$ \( T^{2} + 80719628544 \) Copy content Toggle raw display
$53$ \( T^{2} + 87652707844 \) Copy content Toggle raw display
$59$ \( (T - 897548)^{2} \) Copy content Toggle raw display
$61$ \( (T + 884810)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 21712729534864 \) Copy content Toggle raw display
$71$ \( (T + 2710792)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 32158585089316 \) Copy content Toggle raw display
$79$ \( (T - 5124176)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 2444707365136 \) Copy content Toggle raw display
$89$ \( (T + 11605674)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 119500272097924 \) Copy content Toggle raw display
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