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Properties

Label 200.3.e.b
Level $200$
Weight $3$
Character orbit 200.e
Analytic conductor $5.450$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [200,3,Mod(99,200)]
 
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([1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.99");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 200.e (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: \(5.44960528721\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
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)
 

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.

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} - 4 q^{4} + ( - 2 \beta_{3} - 2) q^{6} - 8 \beta_1 q^{8} + ( - 2 \beta_{3} - 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} - 4 q^{4} + ( - 2 \beta_{3} - 2) q^{6} - 8 \beta_1 q^{8} + ( - 2 \beta_{3} - 16) q^{9} + (3 \beta_{3} - 7) q^{11} + ( - 4 \beta_{2} - 4 \beta_1) q^{12} + 16 q^{16} + (6 \beta_{2} + \beta_1) q^{17} + ( - 4 \beta_{2} - 32 \beta_1) q^{18} + (3 \beta_{3} - 17) q^{19} + (6 \beta_{2} - 14 \beta_1) q^{22} + (8 \beta_{3} + 8) q^{24} + ( - 9 \beta_{2} - 55 \beta_1) q^{27} + 32 \beta_1 q^{32} + ( - 4 \beta_{2} + 65 \beta_1) q^{33} + ( - 12 \beta_{3} - 2) q^{34} + (8 \beta_{3} + 64) q^{36} + (6 \beta_{2} - 34 \beta_1) q^{38} + ( - 12 \beta_{3} + 23) q^{41} + 14 \beta_1 q^{43} + ( - 12 \beta_{3} + 28) q^{44} + (16 \beta_{2} + 16 \beta_1) q^{48} - 49 q^{49} + ( - 7 \beta_{3} - 145) q^{51} + (18 \beta_{3} + 110) q^{54} + ( - 14 \beta_{2} + 55 \beta_1) q^{57} + 82 q^{59} - 64 q^{64} + (8 \beta_{3} - 130) q^{66} + (21 \beta_{2} + 31 \beta_1) q^{67} + ( - 24 \beta_{2} - 4 \beta_1) q^{68} + (16 \beta_{2} + 128 \beta_1) q^{72} + (6 \beta_{2} + 71 \beta_1) q^{73} + ( - 12 \beta_{3} + 68) q^{76} + (46 \beta_{3} + 127) q^{81} + ( - 24 \beta_{2} + 46 \beta_1) q^{82} + ( - 9 \beta_{2} - 79 \beta_1) q^{83} - 28 q^{86} + ( - 24 \beta_{2} + 56 \beta_1) q^{88} + (18 \beta_{3} + 73) q^{89} + ( - 32 \beta_{3} - 32) q^{96} + 94 \beta_1 q^{97} - 98 \beta_1 q^{98} + ( - 34 \beta_{3} - 32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{4} - 8 q^{6} - 64 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{4} - 8 q^{6} - 64 q^{9} - 28 q^{11} + 64 q^{16} - 68 q^{19} + 32 q^{24} - 8 q^{34} + 256 q^{36} + 92 q^{41} + 112 q^{44} - 196 q^{49} - 580 q^{51} + 440 q^{54} + 328 q^{59} - 256 q^{64} - 520 q^{66} + 272 q^{76} + 508 q^{81} - 112 q^{86} + 292 q^{89} - 128 q^{96} - 128 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

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

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 3\beta_{2} ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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
1.22474 − 1.22474i
−1.22474 + 1.22474i
−1.22474 − 1.22474i
1.22474 + 1.22474i
− 2.00000i − 5.89898i −4.00000 0 −11.7980 0 8.00000i −25.7980 0
99.2 − 2.00000i 3.89898i −4.00000 0 7.79796 0 8.00000i −6.20204 0
99.3 2.00000i − 3.89898i −4.00000 0 7.79796 0 − 8.00000i −6.20204 0
99.4 2.00000i 5.89898i −4.00000 0 −11.7980 0 − 8.00000i −25.7980 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.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
5.b even 2 1 inner
40.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 200.3.e.b 4
4.b odd 2 1 800.3.e.b 4
5.b even 2 1 inner 200.3.e.b 4
5.c odd 4 1 200.3.g.b ✓ 2
5.c odd 4 1 200.3.g.d yes 2
8.b even 2 1 800.3.e.b 4
8.d odd 2 1 CM 200.3.e.b 4
20.d odd 2 1 800.3.e.b 4
20.e even 4 1 800.3.g.b 2
20.e even 4 1 800.3.g.d 2
40.e odd 2 1 inner 200.3.e.b 4
40.f even 2 1 800.3.e.b 4
40.i odd 4 1 800.3.g.b 2
40.i odd 4 1 800.3.g.d 2
40.k even 4 1 200.3.g.b ✓ 2
40.k even 4 1 200.3.g.d yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.3.e.b 4 1.a even 1 1 trivial
200.3.e.b 4 5.b even 2 1 inner
200.3.e.b 4 8.d odd 2 1 CM
200.3.e.b 4 40.e odd 2 1 inner
200.3.g.b ✓ 2 5.c odd 4 1
200.3.g.b ✓ 2 40.k even 4 1
200.3.g.d yes 2 5.c odd 4 1
200.3.g.d yes 2 40.k even 4 1
800.3.e.b 4 4.b odd 2 1
800.3.e.b 4 8.b even 2 1
800.3.e.b 4 20.d odd 2 1
800.3.e.b 4 40.f even 2 1
800.3.g.b 2 20.e even 4 1
800.3.g.b 2 40.i odd 4 1
800.3.g.d 2 20.e even 4 1
800.3.g.d 2 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 50T_{3}^{2} + 529 \) acting on \(S_{3}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 50T^{2} + 529 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 14 T - 167)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 1730 T^{2} + 744769 \) Copy content Toggle raw display
$19$ \( (T^{2} + 34 T + 73)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 46 T - 2927)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T - 82)^{4} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 23090 T^{2} + 92602129 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 11810 T^{2} + 17447329 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 16370 T^{2} + 18464209 \) Copy content Toggle raw display
$89$ \( (T^{2} - 146 T - 2447)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8836)^{2} \) Copy content Toggle raw display
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