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Properties

Label 200.2.d.c
Level $200$
Weight $2$
Character orbit 200.d
Analytic conductor $1.597$
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] = [200,2,Mod(101,200)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 200.d (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: no
Analytic conductor: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(\beta = \frac{1}{2}(1 + \sqrt{-7})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + (2 \beta - 1) q^{3} + (\beta - 2) q^{4} + (\beta - 4) q^{6} + 4 q^{7} + ( - \beta - 2) q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + (2 \beta - 1) q^{3} + (\beta - 2) q^{4} + (\beta - 4) q^{6} + 4 q^{7} + ( - \beta - 2) q^{8} - 4 q^{9} + ( - 2 \beta + 1) q^{11} + ( - 3 \beta - 2) q^{12} + 4 \beta q^{14} + ( - 3 \beta + 2) q^{16} - 3 q^{17} - 4 \beta q^{18} + ( - 2 \beta + 1) q^{19} + (8 \beta - 4) q^{21} + ( - \beta + 4) q^{22} + 4 q^{23} + ( - 5 \beta + 6) q^{24} + ( - 2 \beta + 1) q^{27} + (4 \beta - 8) q^{28} + 4 q^{31} + ( - \beta + 6) q^{32} + 7 q^{33} - 3 \beta q^{34} + ( - 4 \beta + 8) q^{36} + (8 \beta - 4) q^{37} + ( - \beta + 4) q^{38} - 5 q^{41} + (4 \beta - 16) q^{42} + (4 \beta - 2) q^{43} + (3 \beta + 2) q^{44} + 4 \beta q^{46} - 8 q^{47} + (\beta + 10) q^{48} + 9 q^{49} + ( - 6 \beta + 3) q^{51} + ( - 8 \beta + 4) q^{53} + ( - \beta + 4) q^{54} + ( - 4 \beta - 8) q^{56} + 7 q^{57} + ( - 4 \beta + 2) q^{59} + ( - 8 \beta + 4) q^{61} + 4 \beta q^{62} - 16 q^{63} + (5 \beta + 2) q^{64} + 7 \beta q^{66} + ( - 6 \beta + 3) q^{67} + ( - 3 \beta + 6) q^{68} + (8 \beta - 4) q^{69} + 8 q^{71} + (4 \beta + 8) q^{72} - 7 q^{73} + (4 \beta - 16) q^{74} + (3 \beta + 2) q^{76} + ( - 8 \beta + 4) q^{77} + 4 q^{79} - 5 q^{81} - 5 \beta q^{82} + ( - 6 \beta + 3) q^{83} + ( - 12 \beta - 8) q^{84} + (2 \beta - 8) q^{86} + (5 \beta - 6) q^{88} - q^{89} + (4 \beta - 8) q^{92} + (8 \beta - 4) q^{93} - 8 \beta q^{94} + (11 \beta - 2) q^{96} - 2 q^{97} + 9 \beta q^{98} + (8 \beta - 4) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 3 q^{4} - 7 q^{6} + 8 q^{7} - 5 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 3 q^{4} - 7 q^{6} + 8 q^{7} - 5 q^{8} - 8 q^{9} - 7 q^{12} + 4 q^{14} + q^{16} - 6 q^{17} - 4 q^{18} + 7 q^{22} + 8 q^{23} + 7 q^{24} - 12 q^{28} + 8 q^{31} + 11 q^{32} + 14 q^{33} - 3 q^{34} + 12 q^{36} + 7 q^{38} - 10 q^{41} - 28 q^{42} + 7 q^{44} + 4 q^{46} - 16 q^{47} + 21 q^{48} + 18 q^{49} + 7 q^{54} - 20 q^{56} + 14 q^{57} + 4 q^{62} - 32 q^{63} + 9 q^{64} + 7 q^{66} + 9 q^{68} + 16 q^{71} + 20 q^{72} - 14 q^{73} - 28 q^{74} + 7 q^{76} + 8 q^{79} - 10 q^{81} - 5 q^{82} - 28 q^{84} - 14 q^{86} - 7 q^{88} - 2 q^{89} - 12 q^{92} - 8 q^{94} + 7 q^{96} - 4 q^{97} + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.500000 − 1.32288i
0.500000 + 1.32288i
0.500000 − 1.32288i − 2.64575i −1.50000 − 1.32288i 0 −3.50000 − 1.32288i 4.00000 −2.50000 + 1.32288i −4.00000 0
101.2 0.500000 + 1.32288i 2.64575i −1.50000 + 1.32288i 0 −3.50000 + 1.32288i 4.00000 −2.50000 − 1.32288i −4.00000 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
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.2.d.c yes 2
3.b odd 2 1 1800.2.k.d 2
4.b odd 2 1 800.2.d.a 2
5.b even 2 1 200.2.d.b ✓ 2
5.c odd 4 2 200.2.f.d 4
8.b even 2 1 inner 200.2.d.c yes 2
8.d odd 2 1 800.2.d.a 2
12.b even 2 1 7200.2.k.b 2
15.d odd 2 1 1800.2.k.f 2
15.e even 4 2 1800.2.d.m 4
16.e even 4 2 6400.2.a.bg 2
16.f odd 4 2 6400.2.a.cb 2
20.d odd 2 1 800.2.d.d 2
20.e even 4 2 800.2.f.d 4
24.f even 2 1 7200.2.k.b 2
24.h odd 2 1 1800.2.k.d 2
40.e odd 2 1 800.2.d.d 2
40.f even 2 1 200.2.d.b ✓ 2
40.i odd 4 2 200.2.f.d 4
40.k even 4 2 800.2.f.d 4
60.h even 2 1 7200.2.k.i 2
60.l odd 4 2 7200.2.d.m 4
80.k odd 4 2 6400.2.a.bh 2
80.q even 4 2 6400.2.a.cc 2
120.i odd 2 1 1800.2.k.f 2
120.m even 2 1 7200.2.k.i 2
120.q odd 4 2 7200.2.d.m 4
120.w even 4 2 1800.2.d.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.d.b ✓ 2 5.b even 2 1
200.2.d.b ✓ 2 40.f even 2 1
200.2.d.c yes 2 1.a even 1 1 trivial
200.2.d.c yes 2 8.b even 2 1 inner
200.2.f.d 4 5.c odd 4 2
200.2.f.d 4 40.i odd 4 2
800.2.d.a 2 4.b odd 2 1
800.2.d.a 2 8.d odd 2 1
800.2.d.d 2 20.d odd 2 1
800.2.d.d 2 40.e odd 2 1
800.2.f.d 4 20.e even 4 2
800.2.f.d 4 40.k even 4 2
1800.2.d.m 4 15.e even 4 2
1800.2.d.m 4 120.w even 4 2
1800.2.k.d 2 3.b odd 2 1
1800.2.k.d 2 24.h odd 2 1
1800.2.k.f 2 15.d odd 2 1
1800.2.k.f 2 120.i odd 2 1
6400.2.a.bg 2 16.e even 4 2
6400.2.a.bh 2 80.k odd 4 2
6400.2.a.cb 2 16.f odd 4 2
6400.2.a.cc 2 80.q even 4 2
7200.2.d.m 4 60.l odd 4 2
7200.2.d.m 4 120.q odd 4 2
7200.2.k.b 2 12.b even 2 1
7200.2.k.b 2 24.f even 2 1
7200.2.k.i 2 60.h even 2 1
7200.2.k.i 2 120.m even 2 1

Hecke kernels

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

\( T_{3}^{2} + 7 \) Copy content Toggle raw display
\( T_{7} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} + 7 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 7 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 7 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 112 \) Copy content Toggle raw display
$41$ \( (T + 5)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 28 \) Copy content Toggle raw display
$47$ \( (T + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 112 \) Copy content Toggle raw display
$59$ \( T^{2} + 28 \) Copy content Toggle raw display
$61$ \( T^{2} + 112 \) Copy content Toggle raw display
$67$ \( T^{2} + 63 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( (T + 7)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 63 \) Copy content Toggle raw display
$89$ \( (T + 1)^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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