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Properties

Label 368.3.k.b
Level $368$
Weight $3$
Character orbit 368.k
Analytic conductor $10.027$
Analytic rank $0$
Dimension $176$
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] = [368,3,Mod(45,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.45");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 368.k (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: \(10.0272737285\)
Analytic rank: \(0\)
Dimension: \(176\)
Relative dimension: \(88\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 176 q - 4 q^{2} - 4 q^{3} - 16 q^{4} - 64 q^{6} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 176 q - 4 q^{2} - 4 q^{3} - 16 q^{4} - 64 q^{6} - 4 q^{8} - 28 q^{12} - 4 q^{13} - 80 q^{16} + 72 q^{18} + 68 q^{24} + 96 q^{26} + 212 q^{27} + 28 q^{29} - 8 q^{31} + 136 q^{32} - 104 q^{35} - 264 q^{36} - 32 q^{46} - 8 q^{47} - 520 q^{48} + 1728 q^{49} + 88 q^{50} - 56 q^{52} - 148 q^{54} + 540 q^{58} + 296 q^{59} - 812 q^{62} - 352 q^{64} - 80 q^{69} + 124 q^{70} + 712 q^{72} + 132 q^{75} + 24 q^{77} - 320 q^{78} - 440 q^{81} - 844 q^{82} + 416 q^{85} - 492 q^{92} - 1084 q^{93} - 596 q^{94} - 8 q^{95} + 664 q^{96} - 240 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
45.1 −1.99091 − 0.190503i −3.82125 − 3.82125i 3.92742 + 0.758546i −5.18968 − 5.18968i 6.87979 + 8.33570i 6.04746 −7.67462 − 2.25838i 20.2039i 9.34352 + 11.3208i
45.2 −1.99091 − 0.190503i −3.82125 − 3.82125i 3.92742 + 0.758546i 5.18968 + 5.18968i 6.87979 + 8.33570i −6.04746 −7.67462 − 2.25838i 20.2039i −9.34352 − 11.3208i
45.3 −1.98630 + 0.233722i −0.369206 − 0.369206i 3.89075 − 0.928481i −1.12325 − 1.12325i 0.819645 + 0.647062i 9.16844 −7.51117 + 2.75359i − 8.72737i 2.49363 + 1.96858i
45.4 −1.98630 + 0.233722i −0.369206 − 0.369206i 3.89075 − 0.928481i 1.12325 + 1.12325i 0.819645 + 0.647062i −9.16844 −7.51117 + 2.75359i − 8.72737i −2.49363 − 1.96858i
45.5 −1.94739 − 0.455721i 2.84929 + 2.84929i 3.58464 + 1.77493i −4.50863 − 4.50863i −4.25019 − 6.84716i 6.89373 −6.17180 − 5.09007i 7.23694i 6.72537 + 10.8347i
45.6 −1.94739 − 0.455721i 2.84929 + 2.84929i 3.58464 + 1.77493i 4.50863 + 4.50863i −4.25019 − 6.84716i −6.89373 −6.17180 − 5.09007i 7.23694i −6.72537 − 10.8347i
45.7 −1.92151 − 0.554793i 0.957217 + 0.957217i 3.38441 + 2.13208i −3.52863 − 3.52863i −1.30825 − 2.37036i −0.532746 −5.32032 − 5.97447i − 7.16747i 4.82265 + 8.73797i
45.8 −1.92151 − 0.554793i 0.957217 + 0.957217i 3.38441 + 2.13208i 3.52863 + 3.52863i −1.30825 − 2.37036i 0.532746 −5.32032 − 5.97447i − 7.16747i −4.82265 − 8.73797i
45.9 −1.89910 + 0.627220i 3.04556 + 3.04556i 3.21319 − 2.38231i −2.97493 − 2.97493i −7.69407 − 3.87360i −6.61236 −4.60795 + 6.53964i 9.55085i 7.51564 + 3.78377i
45.10 −1.89910 + 0.627220i 3.04556 + 3.04556i 3.21319 − 2.38231i 2.97493 + 2.97493i −7.69407 − 3.87360i 6.61236 −4.60795 + 6.53964i 9.55085i −7.51564 − 3.78377i
45.11 −1.88889 − 0.657339i −0.875057 − 0.875057i 3.13581 + 2.48328i −6.27246 − 6.27246i 1.07768 + 2.22809i −13.5040 −4.29085 − 6.75194i − 7.46855i 7.72485 + 15.9711i
45.12 −1.88889 − 0.657339i −0.875057 − 0.875057i 3.13581 + 2.48328i 6.27246 + 6.27246i 1.07768 + 2.22809i 13.5040 −4.29085 − 6.75194i − 7.46855i −7.72485 − 15.9711i
45.13 −1.68730 + 1.07378i −1.06385 − 1.06385i 1.69399 − 3.62359i −5.50618 − 5.50618i 2.93739 + 0.652699i 6.92562 1.03267 + 7.93307i − 6.73643i 15.2030 + 3.37817i
45.14 −1.68730 + 1.07378i −1.06385 − 1.06385i 1.69399 − 3.62359i 5.50618 + 5.50618i 2.93739 + 0.652699i −6.92562 1.03267 + 7.93307i − 6.73643i −15.2030 − 3.37817i
45.15 −1.66928 + 1.10159i −2.86022 − 2.86022i 1.57301 − 3.67772i −4.07644 − 4.07644i 7.92530 + 1.62373i −8.10417 1.42553 + 7.87197i 7.36172i 11.2953 + 2.31417i
45.16 −1.66928 + 1.10159i −2.86022 − 2.86022i 1.57301 − 3.67772i 4.07644 + 4.07644i 7.92530 + 1.62373i 8.10417 1.42553 + 7.87197i 7.36172i −11.2953 − 2.31417i
45.17 −1.65101 − 1.12879i −2.32244 − 2.32244i 1.45165 + 3.72729i −0.541772 − 0.541772i 1.21281 + 6.45593i 0.211665 1.81065 − 7.79240i 1.78749i 0.282921 + 1.50602i
45.18 −1.65101 − 1.12879i −2.32244 − 2.32244i 1.45165 + 3.72729i 0.541772 + 0.541772i 1.21281 + 6.45593i −0.211665 1.81065 − 7.79240i 1.78749i −0.282921 − 1.50602i
45.19 −1.52002 + 1.29982i 0.968686 + 0.968686i 0.620917 − 3.95151i −3.86184 − 3.86184i −2.73154 − 0.213300i −4.96433 4.19246 + 6.81346i − 7.12330i 10.8898 + 0.850361i
45.20 −1.52002 + 1.29982i 0.968686 + 0.968686i 0.620917 − 3.95151i 3.86184 + 3.86184i −2.73154 − 0.213300i 4.96433 4.19246 + 6.81346i − 7.12330i −10.8898 − 0.850361i
See next 80 embeddings (of 176 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 45.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner
23.b odd 2 1 inner
368.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.3.k.b ✓ 176
16.e even 4 1 inner 368.3.k.b ✓ 176
23.b odd 2 1 inner 368.3.k.b ✓ 176
368.k odd 4 1 inner 368.3.k.b ✓ 176
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
368.3.k.b ✓ 176 1.a even 1 1 trivial
368.3.k.b ✓ 176 16.e even 4 1 inner
368.3.k.b ✓ 176 23.b odd 2 1 inner
368.3.k.b ✓ 176 368.k odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{88} + 2 T_{3}^{87} + 2 T_{3}^{86} - 46 T_{3}^{85} + 4983 T_{3}^{84} + 9104 T_{3}^{83} + \cdots + 13\!\cdots\!64 \) acting on \(S_{3}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display