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Properties

Label 200.6.f.d
Level $200$
Weight $6$
Character orbit 200.f
Analytic conductor $32.077$
Analytic rank $0$
Dimension $40$
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] = [200,6,Mod(149,200)]
 
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, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("200.149");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 200 = 2^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 200.f (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: \(32.0767639626\)
Analytic rank: \(0\)
Dimension: \(40\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{4} + 66 q^{6} + 3240 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 2 q^{4} + 66 q^{6} + 3240 q^{9} + 848 q^{14} - 110 q^{16} - 18918 q^{24} + 18344 q^{26} + 14320 q^{31} + 19182 q^{34} + 29656 q^{36} - 44904 q^{39} - 11608 q^{41} + 23186 q^{44} - 75224 q^{46} - 125304 q^{49} - 177894 q^{54} - 73816 q^{56} - 230354 q^{64} + 262878 q^{66} - 15448 q^{71} - 4224 q^{74} + 111902 q^{76} + 15560 q^{79} + 193968 q^{81} + 195112 q^{84} - 131972 q^{86} + 6320 q^{89} + 117080 q^{94} + 115582 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1 −5.59127 − 0.858917i 2.58812 30.5245 + 9.60487i 0 −14.4709 − 2.22298i − 27.4015i −162.421 − 79.9214i −236.302 0
149.2 −5.59127 + 0.858917i 2.58812 30.5245 − 9.60487i 0 −14.4709 + 2.22298i 27.4015i −162.421 + 79.9214i −236.302 0
149.3 −5.36768 − 1.78549i 29.6034 25.6240 + 19.1679i 0 −158.902 − 52.8566i − 90.3364i −103.318 − 148.639i 633.360 0
149.4 −5.36768 + 1.78549i 29.6034 25.6240 − 19.1679i 0 −158.902 + 52.8566i 90.3364i −103.318 + 148.639i 633.360 0
149.5 −4.95437 − 2.73024i −18.7068 17.0916 + 27.0532i 0 92.6806 + 51.0741i − 207.924i −10.8164 − 180.696i 106.946 0
149.6 −4.95437 + 2.73024i −18.7068 17.0916 − 27.0532i 0 92.6806 − 51.0741i 207.924i −10.8164 + 180.696i 106.946 0
149.7 −4.94424 − 2.74854i −23.5470 16.8910 + 27.1789i 0 116.422 + 64.7200i − 39.5751i −8.81071 − 180.805i 311.462 0
149.8 −4.94424 + 2.74854i −23.5470 16.8910 − 27.1789i 0 116.422 − 64.7200i 39.5751i −8.81071 + 180.805i 311.462 0
149.9 −4.83252 − 2.94053i −0.457817 14.7065 + 28.4204i 0 2.21241 + 1.34622i 195.837i 12.5013 − 180.587i −242.790 0
149.10 −4.83252 + 2.94053i −0.457817 14.7065 − 28.4204i 0 2.21241 − 1.34622i − 195.837i 12.5013 + 180.587i −242.790 0
149.11 −3.87282 − 4.12326i 18.2848 −2.00256 + 31.9373i 0 −70.8136 − 75.3929i − 2.25573i 139.441 − 115.430i 91.3327 0
149.12 −3.87282 + 4.12326i 18.2848 −2.00256 − 31.9373i 0 −70.8136 + 75.3929i 2.25573i 139.441 + 115.430i 91.3327 0
149.13 −2.17296 − 5.22286i −8.95730 −22.5565 + 22.6981i 0 19.4639 + 46.7827i 179.198i 167.563 + 68.4872i −162.767 0
149.14 −2.17296 + 5.22286i −8.95730 −22.5565 − 22.6981i 0 19.4639 − 46.7827i − 179.198i 167.563 − 68.4872i −162.767 0
149.15 −2.08722 − 5.25771i 10.4354 −23.2871 + 21.9480i 0 −21.7810 − 54.8666i − 66.0164i 164.001 + 76.6266i −134.101 0
149.16 −2.08722 + 5.25771i 10.4354 −23.2871 − 21.9480i 0 −21.7810 + 54.8666i 66.0164i 164.001 − 76.6266i −134.101 0
149.17 −1.74979 − 5.37943i −25.2673 −25.8765 + 18.8258i 0 44.2124 + 135.923i 185.199i 146.550 + 106.259i 395.434 0
149.18 −1.74979 + 5.37943i −25.2673 −25.8765 − 18.8258i 0 44.2124 − 135.923i − 185.199i 146.550 − 106.259i 395.434 0
149.19 −0.438664 − 5.63982i −17.0419 −31.6151 + 4.94797i 0 7.47566 + 96.1131i − 191.207i 41.7741 + 176.133i 47.4256 0
149.20 −0.438664 + 5.63982i −17.0419 −31.6151 − 4.94797i 0 7.47566 − 96.1131i 191.207i 41.7741 − 176.133i 47.4256 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.40
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 200.6.f.d 40
4.b odd 2 1 800.6.f.d 40
5.b even 2 1 inner 200.6.f.d 40
5.c odd 4 1 200.6.d.c ✓ 20
5.c odd 4 1 200.6.d.d yes 20
8.b even 2 1 inner 200.6.f.d 40
8.d odd 2 1 800.6.f.d 40
20.d odd 2 1 800.6.f.d 40
20.e even 4 1 800.6.d.b 20
20.e even 4 1 800.6.d.d 20
40.e odd 2 1 800.6.f.d 40
40.f even 2 1 inner 200.6.f.d 40
40.i odd 4 1 200.6.d.c ✓ 20
40.i odd 4 1 200.6.d.d yes 20
40.k even 4 1 800.6.d.b 20
40.k even 4 1 800.6.d.d 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.6.d.c ✓ 20 5.c odd 4 1
200.6.d.c ✓ 20 40.i odd 4 1
200.6.d.d yes 20 5.c odd 4 1
200.6.d.d yes 20 40.i odd 4 1
200.6.f.d 40 1.a even 1 1 trivial
200.6.f.d 40 5.b even 2 1 inner
200.6.f.d 40 8.b even 2 1 inner
200.6.f.d 40 40.f even 2 1 inner
800.6.d.b 20 20.e even 4 1
800.6.d.b 20 40.k even 4 1
800.6.d.d 20 20.e even 4 1
800.6.d.d 20 40.k even 4 1
800.6.f.d 40 4.b odd 2 1
800.6.f.d 40 8.d odd 2 1
800.6.f.d 40 20.d odd 2 1
800.6.f.d 40 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{20} - 3240 T_{3}^{18} + 4338819 T_{3}^{16} - 3123786848 T_{3}^{14} + 1315141752042 T_{3}^{12} + \cdots + 12\!\cdots\!83 \) acting on \(S_{6}^{\mathrm{new}}(200, [\chi])\). Copy content Toggle raw display