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Properties

Label 200.2.k.g
Level $200$
Weight $2$
Character orbit 200.k
Analytic conductor $1.597$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [200,2,Mod(43,200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(200, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.k (of order \(4\), degree \(2\), 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: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.3317760000.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{4}) q^{3} + \beta_{2} q^{4} + (\beta_{7} + 2) q^{6} + ( - 2 \beta_{5} - 2 \beta_1) q^{7} + (\beta_{6} + 2 \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} + \beta_{4}) q^{3} + \beta_{2} q^{4} + (\beta_{7} + 2) q^{6} + ( - 2 \beta_{5} - 2 \beta_1) q^{7} + (\beta_{6} + 2 \beta_{4}) q^{8} - q^{11} + (2 \beta_{5} + \beta_1) q^{12} + (2 \beta_{6} - 2 \beta_{4}) q^{13} + ( - 4 \beta_{3} - 2 \beta_{2}) q^{14} + (\beta_{7} + 4) q^{16} + (3 \beta_{5} - 3 \beta_1) q^{17} - 3 \beta_{3} q^{19} + ( - 4 \beta_{7} - 2) q^{21} - \beta_1 q^{22} + (4 \beta_{3} + \beta_{2}) q^{24} + (2 \beta_{7} - 4) q^{26} + ( - 3 \beta_{5} + 3 \beta_1) q^{27} + ( - 6 \beta_{6} - 4 \beta_{4}) q^{28} + ( - 2 \beta_{3} + 4 \beta_{2}) q^{29} + (2 \beta_{5} + 3 \beta_1) q^{32} + ( - \beta_{6} - \beta_{4}) q^{33} + (6 \beta_{3} - 3 \beta_{2}) q^{34} + (2 \beta_{5} + 2 \beta_1) q^{37} - 3 \beta_{6} q^{38} + (2 \beta_{3} - 4 \beta_{2}) q^{39} - q^{41} + ( - 8 \beta_{5} + 2 \beta_1) q^{42} + ( - 2 \beta_{6} - 2 \beta_{4}) q^{43} - \beta_{2} q^{44} + ( - 2 \beta_{5} - 2 \beta_1) q^{47} + (5 \beta_{6} + 2 \beta_{4}) q^{48} + 13 \beta_{3} q^{49} - 9 q^{51} + (4 \beta_{5} - 6 \beta_1) q^{52} + ( - 4 \beta_{6} + 4 \beta_{4}) q^{53} + ( - 6 \beta_{3} + 3 \beta_{2}) q^{54} + ( - 6 \beta_{7} - 8) q^{56} + ( - 3 \beta_{5} + 3 \beta_1) q^{57} + (2 \beta_{6} + 8 \beta_{4}) q^{58} - 4 \beta_{3} q^{59} + (4 \beta_{7} + 2) q^{61} + (4 \beta_{3} + 3 \beta_{2}) q^{64} + ( - \beta_{7} - 2) q^{66} + (3 \beta_{5} - 3 \beta_1) q^{67} + (3 \beta_{6} - 6 \beta_{4}) q^{68} + (4 \beta_{7} + 2) q^{71} + (\beta_{6} + \beta_{4}) q^{73} + (4 \beta_{3} + 2 \beta_{2}) q^{74} - 3 \beta_{7} q^{76} + (2 \beta_{5} + 2 \beta_1) q^{77} + ( - 2 \beta_{6} - 8 \beta_{4}) q^{78} + (2 \beta_{3} - 4 \beta_{2}) q^{79} + 9 q^{81} - \beta_1 q^{82} + (\beta_{6} + \beta_{4}) q^{83} + ( - 16 \beta_{3} + 2 \beta_{2}) q^{84} + ( - 2 \beta_{7} - 4) q^{86} + (6 \beta_{5} + 6 \beta_1) q^{87} + ( - \beta_{6} - 2 \beta_{4}) q^{88} - 13 \beta_{3} q^{89} + 20 q^{91} + ( - 4 \beta_{3} - 2 \beta_{2}) q^{94} + (5 \beta_{7} + 4) q^{96} + (4 \beta_{5} - 4 \beta_1) q^{97} + 13 \beta_{6} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{6} - 8 q^{11} + 28 q^{16} - 40 q^{26} - 8 q^{41} - 72 q^{51} - 40 q^{56} - 12 q^{66} + 12 q^{76} + 72 q^{81} - 24 q^{86} + 160 q^{91} + 12 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 3\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 7\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 3\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 3\nu^{3} ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{4} - 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} + 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7\beta_{6} + 6\beta_{4} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(151\) \(177\)
\(\chi(n)\) \(-1\) \(-1\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
−1.40294 − 0.178197i
−0.178197 − 1.40294i
0.178197 + 1.40294i
1.40294 + 0.178197i
−1.40294 + 0.178197i
−0.178197 + 1.40294i
0.178197 − 1.40294i
1.40294 − 0.178197i
−1.40294 − 0.178197i −1.22474 − 1.22474i 1.93649 + 0.500000i 0 1.50000 + 1.93649i 3.16228 + 3.16228i −2.62769 − 1.04655i 0 0
43.2 −0.178197 − 1.40294i 1.22474 + 1.22474i −1.93649 + 0.500000i 0 1.50000 − 1.93649i 3.16228 + 3.16228i 1.04655 + 2.62769i 0 0
43.3 0.178197 + 1.40294i −1.22474 − 1.22474i −1.93649 + 0.500000i 0 1.50000 − 1.93649i −3.16228 − 3.16228i −1.04655 − 2.62769i 0 0
43.4 1.40294 + 0.178197i 1.22474 + 1.22474i 1.93649 + 0.500000i 0 1.50000 + 1.93649i −3.16228 − 3.16228i 2.62769 + 1.04655i 0 0
107.1 −1.40294 + 0.178197i −1.22474 + 1.22474i 1.93649 − 0.500000i 0 1.50000 − 1.93649i 3.16228 − 3.16228i −2.62769 + 1.04655i 0 0
107.2 −0.178197 + 1.40294i 1.22474 − 1.22474i −1.93649 − 0.500000i 0 1.50000 + 1.93649i 3.16228 − 3.16228i 1.04655 − 2.62769i 0 0
107.3 0.178197 − 1.40294i −1.22474 + 1.22474i −1.93649 − 0.500000i 0 1.50000 + 1.93649i −3.16228 + 3.16228i −1.04655 + 2.62769i 0 0
107.4 1.40294 − 0.178197i 1.22474 − 1.22474i 1.93649 − 0.500000i 0 1.50000 − 1.93649i −3.16228 + 3.16228i 2.62769 − 1.04655i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
8.d odd 2 1 inner
40.e odd 2 1 inner
40.k even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.2.k.g ✓ 8
4.b odd 2 1 800.2.o.h 8
5.b even 2 1 inner 200.2.k.g ✓ 8
5.c odd 4 2 inner 200.2.k.g ✓ 8
8.b even 2 1 800.2.o.h 8
8.d odd 2 1 inner 200.2.k.g ✓ 8
20.d odd 2 1 800.2.o.h 8
20.e even 4 2 800.2.o.h 8
40.e odd 2 1 inner 200.2.k.g ✓ 8
40.f even 2 1 800.2.o.h 8
40.i odd 4 2 800.2.o.h 8
40.k even 4 2 inner 200.2.k.g ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.k.g ✓ 8 1.a even 1 1 trivial
200.2.k.g ✓ 8 5.b even 2 1 inner
200.2.k.g ✓ 8 5.c odd 4 2 inner
200.2.k.g ✓ 8 8.d odd 2 1 inner
200.2.k.g ✓ 8 40.e odd 2 1 inner
200.2.k.g ✓ 8 40.k even 4 2 inner
800.2.o.h 8 4.b odd 2 1
800.2.o.h 8 8.b even 2 1
800.2.o.h 8 20.d odd 2 1
800.2.o.h 8 20.e even 4 2
800.2.o.h 8 40.f even 2 1
800.2.o.h 8 40.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(200, [\chi])\):

\( T_{3}^{4} + 9 \) Copy content Toggle raw display
\( T_{7}^{4} + 400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 7T^{4} + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
$11$ \( (T + 1)^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 729)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{2} - 60)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
$41$ \( (T + 1)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 6400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 729)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 60)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 169)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
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