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Properties

Label 200.2.d.e
Level $200$
Weight $2$
Character orbit 200.d
Analytic conductor $1.597$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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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: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 40)
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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} + 1) q^{4} + (\beta_{3} - 1) q^{6} + (\beta_{2} - 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} + 1) q^{4} + (\beta_{3} - 1) q^{6} + (\beta_{2} - 2 \beta_1) q^{7} + 2 \beta_{2} q^{8} + q^{9} - 2 \beta_{3} q^{11} + (2 \beta_{2} - 2 \beta_1) q^{12} + ( - \beta_{3} - 3) q^{14} + (2 \beta_{3} - 2) q^{16} + ( - 2 \beta_{2} + 4 \beta_1) q^{17} + \beta_1 q^{18} - 2 \beta_{3} q^{19} - 2 \beta_{3} q^{21} + ( - 4 \beta_{2} + 2 \beta_1) q^{22} + (\beta_{2} - 2 \beta_1) q^{23} - 4 q^{24} + 4 \beta_{2} q^{27} + ( - 2 \beta_{2} - 2 \beta_1) q^{28} + 4 q^{31} + (4 \beta_{2} - 4 \beta_1) q^{32} + ( - 2 \beta_{2} + 4 \beta_1) q^{33} + (2 \beta_{3} + 6) q^{34} + (\beta_{3} + 1) q^{36} - 6 \beta_{2} q^{37} + ( - 4 \beta_{2} + 2 \beta_1) q^{38} + ( - 4 \beta_{2} + 2 \beta_1) q^{42} - 3 \beta_{2} q^{43} + ( - 2 \beta_{3} + 6) q^{44} + ( - \beta_{3} - 3) q^{46} + (3 \beta_{2} - 6 \beta_1) q^{47} - 4 \beta_1 q^{48} - q^{49} + 4 \beta_{3} q^{51} - 4 \beta_{2} q^{53} + (4 \beta_{3} - 4) q^{54} - 4 \beta_{3} q^{56} + ( - 2 \beta_{2} + 4 \beta_1) q^{57} + 6 \beta_{3} q^{59} + 2 \beta_{3} q^{61} + 4 \beta_1 q^{62} + (\beta_{2} - 2 \beta_1) q^{63} - 8 q^{64} + (2 \beta_{3} + 6) q^{66} - 3 \beta_{2} q^{67} + (4 \beta_{2} + 4 \beta_1) q^{68} - 2 \beta_{3} q^{69} - 12 q^{71} + 2 \beta_{2} q^{72} + (2 \beta_{2} - 4 \beta_1) q^{73} + ( - 6 \beta_{3} + 6) q^{74} + ( - 2 \beta_{3} + 6) q^{76} + 6 \beta_{2} q^{77} + 4 q^{79} - 5 q^{81} + 7 \beta_{2} q^{83} + ( - 2 \beta_{3} + 6) q^{84} + ( - 3 \beta_{3} + 3) q^{86} + ( - 4 \beta_{2} + 8 \beta_1) q^{88} - 6 q^{89} + ( - 2 \beta_{2} - 2 \beta_1) q^{92} + 4 \beta_{2} q^{93} + ( - 3 \beta_{3} - 9) q^{94} + ( - 4 \beta_{3} - 4) q^{96} + (2 \beta_{2} - 4 \beta_1) q^{97} - \beta_1 q^{98} - 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 4 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 4 q^{6} + 4 q^{9} - 12 q^{14} - 8 q^{16} - 16 q^{24} + 16 q^{31} + 24 q^{34} + 4 q^{36} + 24 q^{44} - 12 q^{46} - 4 q^{49} - 16 q^{54} - 32 q^{64} + 24 q^{66} - 48 q^{71} + 24 q^{74} + 24 q^{76} + 16 q^{79} - 20 q^{81} + 24 q^{84} + 12 q^{86} - 24 q^{89} - 36 q^{94} - 16 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display

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

Character values

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

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

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
−1.22474 − 0.707107i
−1.22474 + 0.707107i
1.22474 − 0.707107i
1.22474 + 0.707107i
−1.22474 − 0.707107i − 1.41421i 1.00000 + 1.73205i 0 −1.00000 + 1.73205i 2.44949 − 2.82843i 1.00000 0
101.2 −1.22474 + 0.707107i 1.41421i 1.00000 − 1.73205i 0 −1.00000 − 1.73205i 2.44949 2.82843i 1.00000 0
101.3 1.22474 − 0.707107i − 1.41421i 1.00000 − 1.73205i 0 −1.00000 − 1.73205i −2.44949 − 2.82843i 1.00000 0
101.4 1.22474 + 0.707107i 1.41421i 1.00000 + 1.73205i 0 −1.00000 + 1.73205i −2.44949 2.82843i 1.00000 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
5.b even 2 1 inner
8.b even 2 1 inner
40.f 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.e 4
3.b odd 2 1 1800.2.k.m 4
4.b odd 2 1 800.2.d.f 4
5.b even 2 1 inner 200.2.d.e 4
5.c odd 4 2 40.2.f.a ✓ 4
8.b even 2 1 inner 200.2.d.e 4
8.d odd 2 1 800.2.d.f 4
12.b even 2 1 7200.2.k.l 4
15.d odd 2 1 1800.2.k.m 4
15.e even 4 2 360.2.d.b 4
16.e even 4 2 6400.2.a.co 4
16.f odd 4 2 6400.2.a.cm 4
20.d odd 2 1 800.2.d.f 4
20.e even 4 2 160.2.f.a 4
24.f even 2 1 7200.2.k.l 4
24.h odd 2 1 1800.2.k.m 4
40.e odd 2 1 800.2.d.f 4
40.f even 2 1 inner 200.2.d.e 4
40.i odd 4 2 40.2.f.a ✓ 4
40.k even 4 2 160.2.f.a 4
60.h even 2 1 7200.2.k.l 4
60.l odd 4 2 1440.2.d.c 4
80.i odd 4 2 1280.2.c.i 4
80.j even 4 2 1280.2.c.k 4
80.k odd 4 2 6400.2.a.cm 4
80.q even 4 2 6400.2.a.co 4
80.s even 4 2 1280.2.c.k 4
80.t odd 4 2 1280.2.c.i 4
120.i odd 2 1 1800.2.k.m 4
120.m even 2 1 7200.2.k.l 4
120.q odd 4 2 1440.2.d.c 4
120.w even 4 2 360.2.d.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.f.a ✓ 4 5.c odd 4 2
40.2.f.a ✓ 4 40.i odd 4 2
160.2.f.a 4 20.e even 4 2
160.2.f.a 4 40.k even 4 2
200.2.d.e 4 1.a even 1 1 trivial
200.2.d.e 4 5.b even 2 1 inner
200.2.d.e 4 8.b even 2 1 inner
200.2.d.e 4 40.f even 2 1 inner
360.2.d.b 4 15.e even 4 2
360.2.d.b 4 120.w even 4 2
800.2.d.f 4 4.b odd 2 1
800.2.d.f 4 8.d odd 2 1
800.2.d.f 4 20.d odd 2 1
800.2.d.f 4 40.e odd 2 1
1280.2.c.i 4 80.i odd 4 2
1280.2.c.i 4 80.t odd 4 2
1280.2.c.k 4 80.j even 4 2
1280.2.c.k 4 80.s even 4 2
1440.2.d.c 4 60.l odd 4 2
1440.2.d.c 4 120.q odd 4 2
1800.2.k.m 4 3.b odd 2 1
1800.2.k.m 4 15.d odd 2 1
1800.2.k.m 4 24.h odd 2 1
1800.2.k.m 4 120.i odd 2 1
6400.2.a.cm 4 16.f odd 4 2
6400.2.a.cm 4 80.k odd 4 2
6400.2.a.co 4 16.e even 4 2
6400.2.a.co 4 80.q even 4 2
7200.2.k.l 4 12.b even 2 1
7200.2.k.l 4 24.f even 2 1
7200.2.k.l 4 60.h even 2 1
7200.2.k.l 4 120.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 \(S_{2}^{\mathrm{new}}(200, [\chi])\):

\( T_{3}^{2} + 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T - 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 54)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$71$ \( (T + 12)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 98)^{2} \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
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