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Properties

Label 200.3.e.d
Level $200$
Weight $3$
Character orbit 200.e
Analytic conductor $5.450$
Analytic rank $0$
Dimension $16$
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] = [200,3,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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 2 x^{14} - x^{12} + 4 x^{11} - 6 x^{10} + 14 x^{9} - 15 x^{8} + 28 x^{7} + \cdots + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{27}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + \beta_{2} q^{3} + (\beta_{11} + 1) q^{4} + ( - \beta_{3} - 2) q^{6} + (\beta_{5} - \beta_{4}) q^{7} + (\beta_{7} + \beta_{6} + \cdots - 2 \beta_{4}) q^{8}+ \cdots + ( - \beta_{14} - \beta_{11} - \beta_{10} + \cdots - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + \beta_{2} q^{3} + (\beta_{11} + 1) q^{4} + ( - \beta_{3} - 2) q^{6} + (\beta_{5} - \beta_{4}) q^{7} + (\beta_{7} + \beta_{6} + \cdots - 2 \beta_{4}) q^{8}+ \cdots + ( - 3 \beta_{15} + 5 \beta_{14} + \cdots + 44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} - 24 q^{6} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{4} - 24 q^{6} - 48 q^{9} - 64 q^{11} + 40 q^{14} + 16 q^{16} - 64 q^{19} + 16 q^{24} + 120 q^{26} - 24 q^{34} - 288 q^{36} + 96 q^{41} + 176 q^{44} - 280 q^{46} + 176 q^{49} + 128 q^{51} - 112 q^{54} + 560 q^{56} - 448 q^{59} + 256 q^{64} + 448 q^{66} - 120 q^{74} - 384 q^{76} - 112 q^{81} + 80 q^{84} - 888 q^{86} + 96 q^{89} + 200 q^{94} + 896 q^{96} + 704 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} - x^{12} + 4 x^{11} - 6 x^{10} + 14 x^{9} - 15 x^{8} + 28 x^{7} + \cdots + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2 \nu^{15} + 7 \nu^{14} + 8 \nu^{13} + 18 \nu^{12} + 26 \nu^{11} + 29 \nu^{10} + 54 \nu^{9} + \cdots + 832 ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 9 \nu^{15} + 21 \nu^{14} - 42 \nu^{13} + 50 \nu^{12} - 67 \nu^{11} + 73 \nu^{10} - 60 \nu^{9} + \cdots - 320 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2 \nu^{15} + 3 \nu^{14} - 24 \nu^{13} + 34 \nu^{12} - 70 \nu^{11} + 49 \nu^{10} - 114 \nu^{9} + \cdots - 2496 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 23 \nu^{15} - 2 \nu^{14} + 6 \nu^{13} - 64 \nu^{12} + 47 \nu^{11} - 124 \nu^{10} + 118 \nu^{9} + \cdots + 4224 ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5 \nu^{15} + 48 \nu^{14} + 54 \nu^{13} + 60 \nu^{12} + 219 \nu^{11} + 138 \nu^{10} + 458 \nu^{9} + \cdots + 9216 ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 37 \nu^{15} - 20 \nu^{14} - 10 \nu^{13} + 68 \nu^{12} - 117 \nu^{11} + 142 \nu^{10} - 318 \nu^{9} + \cdots - 9472 ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12 \nu^{15} + 13 \nu^{14} - 16 \nu^{13} + 6 \nu^{12} - 8 \nu^{11} - 5 \nu^{10} + 26 \nu^{9} + \cdots + 1600 ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11 \nu^{15} + 8 \nu^{14} + 6 \nu^{13} - 20 \nu^{12} + 43 \nu^{11} - 46 \nu^{10} + 106 \nu^{9} + \cdots + 3072 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 53 \nu^{15} - 36 \nu^{14} - 10 \nu^{13} + 100 \nu^{12} - 133 \nu^{11} + 190 \nu^{10} - 350 \nu^{9} + \cdots - 11520 ) / 256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27 \nu^{15} - 6 \nu^{14} + 2 \nu^{13} + 64 \nu^{12} - 51 \nu^{11} + 140 \nu^{10} - 142 \nu^{9} + \cdots - 5248 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 37 \nu^{15} - 24 \nu^{14} + 70 \nu^{13} - 4 \nu^{12} + 91 \nu^{11} + 50 \nu^{10} + 42 \nu^{9} + \cdots - 2176 ) / 128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 15 \nu^{15} + 49 \nu^{14} + 46 \nu^{13} + 106 \nu^{12} + 165 \nu^{11} + 221 \nu^{10} + 292 \nu^{9} + \cdots + 5312 ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 35 \nu^{15} - 14 \nu^{14} + 30 \nu^{13} - 168 \nu^{12} + 139 \nu^{11} - 296 \nu^{10} + 262 \nu^{9} + \cdots + 8576 ) / 128 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 27 \nu^{15} + 25 \nu^{14} - 22 \nu^{13} - 6 \nu^{12} + 15 \nu^{11} - 47 \nu^{10} + 120 \nu^{9} + \cdots + 4736 ) / 64 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 7 \nu^{15} - 14 \nu^{14} + 10 \nu^{13} - 16 \nu^{12} + \nu^{11} - 20 \nu^{10} - 38 \nu^{9} + \cdots - 1376 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} - \beta_{11} + \beta_{10} + 2\beta_{8} + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} - \beta_{14} + \beta_{13} + \beta_{11} + 2\beta_{9} + 2\beta_{7} - 2\beta_{5} - \beta_{3} + \beta _1 - 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{13} + \beta_{10} + \beta_{3} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{15} + \beta_{13} + 2 \beta_{12} - 2 \beta_{11} - \beta_{10} + 2 \beta_{9} - 4 \beta_{7} + \cdots - 8 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{12} - 3\beta_{8} + 5\beta_{7} + 2\beta_{6} + 3\beta_{5} + \beta_{4} - 4\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2 \beta_{15} + \beta_{14} - 2 \beta_{12} + 13 \beta_{11} - \beta_{10} + 4 \beta_{9} + 6 \beta_{8} + \cdots - 26 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2\beta_{15} + 8\beta_{14} - \beta_{13} + 2\beta_{11} + 9\beta_{10} - 3\beta_{3} - 5\beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9 \beta_{15} - \beta_{14} - 3 \beta_{13} - 6 \beta_{12} - 11 \beta_{11} + 2 \beta_{9} - 2 \beta_{8} + \cdots - 78 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( \beta_{12} - 12\beta_{9} - 18\beta_{8} - 16\beta_{7} - 3\beta_{6} + 5\beta_{5} + 26\beta_{4} + 15\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 23 \beta_{15} + 4 \beta_{14} - 9 \beta_{13} + 18 \beta_{12} - 26 \beta_{11} - 11 \beta_{10} - 6 \beta_{9} + \cdots - 108 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 5\beta_{15} + 4\beta_{14} - 8\beta_{13} - \beta_{11} - 13\beta_{10} + \beta_{3} + 10\beta _1 + 29 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 26 \beta_{15} - 43 \beta_{14} - 26 \beta_{13} + 4 \beta_{12} + 41 \beta_{11} - 23 \beta_{10} + \cdots + 158 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( -12\beta_{12} - 52\beta_{9} - 3\beta_{8} + 35\beta_{7} + 29\beta_{6} + 12\beta_{5} - 123\beta_{4} - 39\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 45 \beta_{15} - 13 \beta_{14} + 29 \beta_{13} - 8 \beta_{12} + 61 \beta_{11} - 192 \beta_{10} + \cdots + 342 ) / 8 \) 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
−1.06626 − 0.929025i
−1.06626 + 0.929025i
−0.595955 + 1.28251i
−0.595955 − 1.28251i
−1.41421 + 0.00487727i
−1.41421 − 0.00487727i
0.579920 + 1.28984i
0.579920 − 1.28984i
1.28984 + 0.579920i
1.28984 − 0.579920i
−0.00487727 + 1.41421i
−0.00487727 − 1.41421i
1.28251 − 0.595955i
1.28251 + 0.595955i
0.929025 + 1.06626i
0.929025 − 1.06626i
−1.99529 − 0.137237i 5.01222i 3.96233 + 0.547652i 0 0.687859 − 10.0008i −8.92613 −7.83083 − 1.63650i −16.1223 0
99.2 −1.99529 + 0.137237i − 5.01222i 3.96233 − 0.547652i 0 0.687859 + 10.0008i −8.92613 −7.83083 + 1.63650i −16.1223 0
99.3 −1.87847 − 0.686557i 2.08343i 3.05728 + 2.57935i 0 1.43039 − 3.91366i 1.18656 −3.97213 − 6.94422i 4.65931 0
99.4 −1.87847 + 0.686557i − 2.08343i 3.05728 − 2.57935i 0 1.43039 + 3.91366i 1.18656 −3.97213 + 6.94422i 4.65931 0
99.5 −1.40933 − 1.41908i − 4.13348i −0.0275898 + 3.99990i 0 −5.86575 + 5.82543i −2.28473 5.71508 − 5.59803i −8.08569 0
99.6 −1.40933 + 1.41908i 4.13348i −0.0275898 − 3.99990i 0 −5.86575 − 5.82543i −2.28473 5.71508 + 5.59803i −8.08569 0
99.7 −0.709922 − 1.86976i − 1.20470i −2.99202 + 2.65477i 0 −2.25250 + 0.855241i 12.3974 7.08789 + 3.70969i 7.54870 0
99.8 −0.709922 + 1.86976i 1.20470i −2.99202 − 2.65477i 0 −2.25250 − 0.855241i 12.3974 7.08789 − 3.70969i 7.54870 0
99.9 0.709922 − 1.86976i − 1.20470i −2.99202 − 2.65477i 0 −2.25250 − 0.855241i −12.3974 −7.08789 + 3.70969i 7.54870 0
99.10 0.709922 + 1.86976i 1.20470i −2.99202 + 2.65477i 0 −2.25250 + 0.855241i −12.3974 −7.08789 − 3.70969i 7.54870 0
99.11 1.40933 − 1.41908i − 4.13348i −0.0275898 − 3.99990i 0 −5.86575 − 5.82543i 2.28473 −5.71508 − 5.59803i −8.08569 0
99.12 1.40933 + 1.41908i 4.13348i −0.0275898 + 3.99990i 0 −5.86575 + 5.82543i 2.28473 −5.71508 + 5.59803i −8.08569 0
99.13 1.87847 − 0.686557i 2.08343i 3.05728 − 2.57935i 0 1.43039 + 3.91366i −1.18656 3.97213 − 6.94422i 4.65931 0
99.14 1.87847 + 0.686557i − 2.08343i 3.05728 + 2.57935i 0 1.43039 − 3.91366i −1.18656 3.97213 + 6.94422i 4.65931 0
99.15 1.99529 − 0.137237i 5.01222i 3.96233 − 0.547652i 0 0.687859 + 10.0008i 8.92613 7.83083 − 1.63650i −16.1223 0
99.16 1.99529 + 0.137237i − 5.01222i 3.96233 + 0.547652i 0 0.687859 − 10.0008i 8.92613 7.83083 + 1.63650i −16.1223 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.d odd 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.d 16
4.b odd 2 1 800.3.e.d 16
5.b even 2 1 inner 200.3.e.d 16
5.c odd 4 1 40.3.g.a ✓ 8
5.c odd 4 1 200.3.g.g 8
8.b even 2 1 800.3.e.d 16
8.d odd 2 1 inner 200.3.e.d 16
15.e even 4 1 360.3.g.a 8
20.d odd 2 1 800.3.e.d 16
20.e even 4 1 160.3.g.a 8
20.e even 4 1 800.3.g.g 8
40.e odd 2 1 inner 200.3.e.d 16
40.f even 2 1 800.3.e.d 16
40.i odd 4 1 160.3.g.a 8
40.i odd 4 1 800.3.g.g 8
40.k even 4 1 40.3.g.a ✓ 8
40.k even 4 1 200.3.g.g 8
60.l odd 4 1 1440.3.g.a 8
80.i odd 4 1 1280.3.b.i 16
80.j even 4 1 1280.3.b.i 16
80.s even 4 1 1280.3.b.i 16
80.t odd 4 1 1280.3.b.i 16
120.q odd 4 1 360.3.g.a 8
120.w even 4 1 1440.3.g.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.3.g.a ✓ 8 5.c odd 4 1
40.3.g.a ✓ 8 40.k even 4 1
160.3.g.a 8 20.e even 4 1
160.3.g.a 8 40.i odd 4 1
200.3.e.d 16 1.a even 1 1 trivial
200.3.e.d 16 5.b even 2 1 inner
200.3.e.d 16 8.d odd 2 1 inner
200.3.e.d 16 40.e odd 2 1 inner
200.3.g.g 8 5.c odd 4 1
200.3.g.g 8 40.k even 4 1
360.3.g.a 8 15.e even 4 1
360.3.g.a 8 120.q odd 4 1
800.3.e.d 16 4.b odd 2 1
800.3.e.d 16 8.b even 2 1
800.3.e.d 16 20.d odd 2 1
800.3.e.d 16 40.f even 2 1
800.3.g.g 8 20.e even 4 1
800.3.g.g 8 40.i odd 4 1
1280.3.b.i 16 80.i odd 4 1
1280.3.b.i 16 80.j even 4 1
1280.3.b.i 16 80.s even 4 1
1280.3.b.i 16 80.t odd 4 1
1440.3.g.a 8 60.l odd 4 1
1440.3.g.a 8 120.w even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 48T_{3}^{6} + 680T_{3}^{4} + 2752T_{3}^{2} + 2704 \) acting on \(S_{3}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 8 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( (T^{8} + 48 T^{6} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 240 T^{6} + \cdots + 90000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 16 T^{3} + \cdots - 7792)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} - 880 T^{6} + \cdots + 138297600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1328 T^{6} + \cdots + 5269598464)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 16 T^{3} + \cdots - 2672)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} - 2320 T^{6} + \cdots + 47610000)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 4480 T^{6} + \cdots + 184968806400)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 2240 T^{6} + \cdots + 23040000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 6480 T^{6} + \cdots + 134307590400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 24 T^{3} + \cdots - 1472)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 31337648784016)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 1101366291600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 3957553209600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 112 T^{3} + \cdots - 4307824)^{4} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 39126526214400)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 2928 T^{6} + \cdots + 485232784)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 2536502169600)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 149450820600064)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 976317515366400)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 22288 T^{6} + \cdots + 520862437264)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 24 T^{3} + \cdots - 3224432)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + 11504 T^{6} + \cdots + 7053312256)^{2} \) Copy content Toggle raw display
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