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Properties

Label 230.3.f.b
Level $230$
Weight $3$
Character orbit 230.f
Analytic conductor $6.267$
Analytic rank $0$
Dimension $24$
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] = [230,3,Mod(47,230)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(230, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("230.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 230.f (of order \(4\), degree \(2\), 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: \(6.26704608029\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{2} + 4 q^{5} + 8 q^{7} - 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{2} + 4 q^{5} + 8 q^{7} - 48 q^{8} + 16 q^{10} - 8 q^{11} - 24 q^{13} - 24 q^{15} - 96 q^{16} - 12 q^{17} + 88 q^{18} + 24 q^{20} - 24 q^{21} - 8 q^{22} - 48 q^{25} - 48 q^{26} + 60 q^{27} - 16 q^{28} + 12 q^{30} + 12 q^{31} - 96 q^{32} + 92 q^{33} + 48 q^{35} + 176 q^{36} - 100 q^{37} + 56 q^{38} + 16 q^{40} + 116 q^{41} - 24 q^{42} - 120 q^{43} - 204 q^{45} + 56 q^{47} - 104 q^{50} + 176 q^{51} - 48 q^{52} - 192 q^{53} + 180 q^{55} - 32 q^{56} + 28 q^{58} + 72 q^{60} - 152 q^{61} + 12 q^{62} + 364 q^{63} + 40 q^{65} + 184 q^{66} + 72 q^{67} + 24 q^{68} - 100 q^{70} - 28 q^{71} + 176 q^{72} - 364 q^{73} + 276 q^{75} + 112 q^{76} - 92 q^{77} - 32 q^{78} - 16 q^{80} - 440 q^{81} + 116 q^{82} + 360 q^{83} + 232 q^{85} - 240 q^{86} + 176 q^{87} + 16 q^{88} - 84 q^{90} - 432 q^{91} + 192 q^{93} + 144 q^{95} - 432 q^{97} - 484 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 1.00000 + 1.00000i −3.78907 + 3.78907i 2.00000i 0.818077 − 4.93262i −7.57813 −3.70466 − 3.70466i −2.00000 + 2.00000i − 19.7140i 5.75070 − 4.11454i
47.2 1.00000 + 1.00000i −3.54993 + 3.54993i 2.00000i 3.53474 + 3.53633i −7.09985 9.46554 + 9.46554i −2.00000 + 2.00000i − 16.2040i −0.00158807 + 7.07107i
47.3 1.00000 + 1.00000i −2.98637 + 2.98637i 2.00000i −3.91488 + 3.11026i −5.97274 −3.94695 − 3.94695i −2.00000 + 2.00000i − 8.83680i −7.02514 − 0.804611i
47.4 1.00000 + 1.00000i −1.36040 + 1.36040i 2.00000i −3.17264 − 3.86450i −2.72079 7.21891 + 7.21891i −2.00000 + 2.00000i 5.29864i 0.691863 − 7.03714i
47.5 1.00000 + 1.00000i −1.03594 + 1.03594i 2.00000i 3.53271 − 3.53835i −2.07189 1.37334 + 1.37334i −2.00000 + 2.00000i 6.85364i 7.07107 − 0.00564363i
47.6 1.00000 + 1.00000i −0.646303 + 0.646303i 2.00000i 1.67140 + 4.71237i −1.29261 −2.86792 − 2.86792i −2.00000 + 2.00000i 8.16459i −3.04097 + 6.38377i
47.7 1.00000 + 1.00000i 0.561611 − 0.561611i 2.00000i −4.87730 − 1.10088i 1.12322 −8.38816 − 8.38816i −2.00000 + 2.00000i 8.36919i −3.77642 − 5.97818i
47.8 1.00000 + 1.00000i 1.74954 − 1.74954i 2.00000i −3.63965 + 3.42826i 3.49907 4.68530 + 4.68530i −2.00000 + 2.00000i 2.87825i −7.06791 − 0.211383i
47.9 1.00000 + 1.00000i 1.83776 − 1.83776i 2.00000i 4.99936 + 0.0803026i 3.67552 5.10553 + 5.10553i −2.00000 + 2.00000i 2.24526i 4.91905 + 5.07966i
47.10 1.00000 + 1.00000i 2.20008 − 2.20008i 2.00000i 0.183624 − 4.99663i 4.40016 −8.35722 − 8.35722i −2.00000 + 2.00000i − 0.680698i 5.18025 − 4.81300i
47.11 1.00000 + 1.00000i 2.98334 − 2.98334i 2.00000i 4.43275 + 2.31317i 5.96668 −2.75384 − 2.75384i −2.00000 + 2.00000i − 8.80061i 2.11958 + 6.74591i
47.12 1.00000 + 1.00000i 4.03568 − 4.03568i 2.00000i −1.56820 − 4.74771i 8.07136 6.17013 + 6.17013i −2.00000 + 2.00000i − 23.5735i 3.17952 − 6.31591i
93.1 1.00000 − 1.00000i −3.78907 − 3.78907i − 2.00000i 0.818077 + 4.93262i −7.57813 −3.70466 + 3.70466i −2.00000 − 2.00000i 19.7140i 5.75070 + 4.11454i
93.2 1.00000 − 1.00000i −3.54993 − 3.54993i − 2.00000i 3.53474 − 3.53633i −7.09985 9.46554 − 9.46554i −2.00000 − 2.00000i 16.2040i −0.00158807 − 7.07107i
93.3 1.00000 − 1.00000i −2.98637 − 2.98637i − 2.00000i −3.91488 − 3.11026i −5.97274 −3.94695 + 3.94695i −2.00000 − 2.00000i 8.83680i −7.02514 + 0.804611i
93.4 1.00000 − 1.00000i −1.36040 − 1.36040i − 2.00000i −3.17264 + 3.86450i −2.72079 7.21891 − 7.21891i −2.00000 − 2.00000i − 5.29864i 0.691863 + 7.03714i
93.5 1.00000 − 1.00000i −1.03594 − 1.03594i − 2.00000i 3.53271 + 3.53835i −2.07189 1.37334 − 1.37334i −2.00000 − 2.00000i − 6.85364i 7.07107 + 0.00564363i
93.6 1.00000 − 1.00000i −0.646303 − 0.646303i − 2.00000i 1.67140 − 4.71237i −1.29261 −2.86792 + 2.86792i −2.00000 − 2.00000i − 8.16459i −3.04097 − 6.38377i
93.7 1.00000 − 1.00000i 0.561611 + 0.561611i − 2.00000i −4.87730 + 1.10088i 1.12322 −8.38816 + 8.38816i −2.00000 − 2.00000i − 8.36919i −3.77642 + 5.97818i
93.8 1.00000 − 1.00000i 1.74954 + 1.74954i − 2.00000i −3.63965 − 3.42826i 3.49907 4.68530 − 4.68530i −2.00000 − 2.00000i − 2.87825i −7.06791 + 0.211383i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.3.f.b ✓ 24
5.c odd 4 1 inner 230.3.f.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.3.f.b ✓ 24 1.a even 1 1 trivial
230.3.f.b ✓ 24 5.c odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 20 T_{3}^{21} + 1676 T_{3}^{20} - 308 T_{3}^{19} + 200 T_{3}^{18} - 32292 T_{3}^{17} + \cdots + 12544000000 \) acting on \(S_{3}^{\mathrm{new}}(230, [\chi])\). Copy content Toggle raw display