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Properties

Label 342.3.be.a
Level $342$
Weight $3$
Character orbit 342.be
Analytic conductor $9.319$
Analytic rank $0$
Dimension $240$
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] = [342,3,Mod(23,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([15, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.23");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 342.be (of order \(18\), degree \(6\), 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: \(9.31882504112\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(40\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 6 q^{3} - 12 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 6 q^{3} - 12 q^{6} + 6 q^{9} - 30 q^{13} + 6 q^{15} - 162 q^{17} + 48 q^{18} + 42 q^{19} - 36 q^{22} - 48 q^{24} + 30 q^{27} - 12 q^{28} - 162 q^{29} - 12 q^{33} - 12 q^{36} - 108 q^{38} + 60 q^{39} - 54 q^{41} + 96 q^{43} - 342 q^{45} + 24 q^{48} + 1680 q^{49} + 432 q^{50} + 66 q^{51} + 60 q^{52} - 144 q^{54} - 444 q^{57} - 648 q^{59} - 12 q^{60} - 84 q^{61} - 24 q^{63} + 960 q^{64} - 672 q^{66} - 84 q^{67} + 234 q^{69} - 48 q^{72} + 258 q^{73} + 204 q^{78} - 102 q^{79} + 426 q^{81} - 144 q^{82} - 360 q^{84} - 408 q^{87} - 648 q^{89} - 1032 q^{90} - 192 q^{91} - 432 q^{92} + 384 q^{93} - 90 q^{97} + 648 q^{98} + 96 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −1.39273 + 0.245576i −2.98605 − 0.288979i 1.87939 − 0.684040i 7.34011 − 1.29426i 4.22972 − 0.330831i 6.56446 −2.44949 + 1.41421i 8.83298 + 1.72581i −9.90494 + 3.60510i
23.2 −1.39273 + 0.245576i −2.96750 + 0.440375i 1.87939 − 0.684040i −0.493879 + 0.0870841i 4.02478 − 1.34207i −1.83524 −2.44949 + 1.41421i 8.61214 − 2.61363i 0.666453 − 0.242569i
23.3 −1.39273 + 0.245576i −2.77642 − 1.13643i 1.87939 − 0.684040i −7.46255 + 1.31585i 4.14588 + 0.900920i 8.12903 −2.44949 + 1.41421i 6.41704 + 6.31043i 10.0702 − 3.66524i
23.4 −1.39273 + 0.245576i −2.46060 + 1.71623i 1.87939 − 0.684040i −2.58457 + 0.455729i 3.00548 − 2.99451i −12.6623 −2.44949 + 1.41421i 3.10911 − 8.44591i 3.48768 − 1.26941i
23.5 −1.39273 + 0.245576i −1.76119 − 2.42862i 1.87939 − 0.684040i 5.09080 − 0.897645i 3.04927 + 2.94991i −6.10361 −2.44949 + 1.41421i −2.79643 + 8.55453i −6.86966 + 2.50035i
23.6 −1.39273 + 0.245576i −1.69820 + 2.47308i 1.87939 − 0.684040i −1.02170 + 0.180153i 1.75780 − 3.86136i 9.76145 −2.44949 + 1.41421i −3.23223 − 8.39956i 1.37870 − 0.501807i
23.7 −1.39273 + 0.245576i −1.61189 − 2.53018i 1.87939 − 0.684040i −0.673995 + 0.118844i 2.86627 + 3.12802i −5.79312 −2.44949 + 1.41421i −3.80365 + 8.15673i 0.909507 − 0.331034i
23.8 −1.39273 + 0.245576i −1.08554 + 2.79671i 1.87939 − 0.684040i 1.73287 − 0.305552i 0.825061 − 4.16164i −1.74204 −2.44949 + 1.41421i −6.64320 − 6.07190i −2.33838 + 0.851102i
23.9 −1.39273 + 0.245576i −0.0278605 + 2.99987i 1.87939 − 0.684040i −9.40864 + 1.65900i −0.697893 − 4.18485i −3.27152 −2.44949 + 1.41421i −8.99845 − 0.167156i 12.6963 − 4.62107i
23.10 −1.39273 + 0.245576i 0.0746888 − 2.99907i 1.87939 − 0.684040i −5.20697 + 0.918129i 0.632477 + 4.19523i 3.05046 −2.44949 + 1.41421i −8.98884 − 0.447994i 7.02642 − 2.55741i
23.11 −1.39273 + 0.245576i 0.439909 − 2.96757i 1.87939 − 0.684040i −2.63684 + 0.464947i 0.116089 + 4.24105i 7.14836 −2.44949 + 1.41421i −8.61296 − 2.61092i 3.55823 − 1.29509i
23.12 −1.39273 + 0.245576i 0.873051 + 2.87015i 1.87939 − 0.684040i 3.19136 − 0.562724i −1.92076 − 3.78294i −3.01106 −2.44949 + 1.41421i −7.47556 + 5.01158i −4.30651 + 1.56744i
23.13 −1.39273 + 0.245576i 1.02416 + 2.81977i 1.87939 − 0.684040i 5.65031 − 0.996302i −2.11884 − 3.67566i 6.84259 −2.44949 + 1.41421i −6.90219 + 5.77579i −7.62468 + 2.77516i
23.14 −1.39273 + 0.245576i 1.86146 − 2.35265i 1.87939 − 0.684040i 4.21040 − 0.742407i −2.01476 + 3.73373i −4.83186 −2.44949 + 1.41421i −2.06992 − 8.75874i −5.68162 + 2.06794i
23.15 −1.39273 + 0.245576i 2.26478 − 1.96743i 1.87939 − 0.684040i 3.59752 − 0.634340i −2.67106 + 3.29627i −0.775306 −2.44949 + 1.41421i 1.25842 − 8.91159i −4.85459 + 1.76693i
23.16 −1.39273 + 0.245576i 2.65579 − 1.39527i 1.87939 − 0.684040i −6.56503 + 1.15759i −3.35615 + 2.59543i −4.54258 −2.44949 + 1.41421i 5.10644 − 7.41109i 8.85903 − 3.22442i
23.17 −1.39273 + 0.245576i 2.73017 + 1.24345i 1.87939 − 0.684040i 7.87372 − 1.38835i −4.10775 − 1.06132i −10.5753 −2.44949 + 1.41421i 5.90768 + 6.78965i −10.6250 + 3.86719i
23.18 −1.39273 + 0.245576i 2.74867 + 1.20200i 1.87939 − 0.684040i −3.93820 + 0.694410i −4.12334 − 0.999050i 9.05498 −2.44949 + 1.41421i 6.11040 + 6.60780i 5.31431 − 1.93425i
23.19 −1.39273 + 0.245576i 2.88880 + 0.809210i 1.87939 − 0.684040i −4.14275 + 0.730479i −4.22204 − 0.417590i −9.32874 −2.44949 + 1.41421i 7.69036 + 4.67530i 5.59034 − 2.03472i
23.20 −1.39273 + 0.245576i 2.96189 − 0.476644i 1.87939 − 0.684040i 5.44802 − 0.960634i −4.00806 + 1.39120i 12.3892 −2.44949 + 1.41421i 8.54562 − 2.82354i −7.35171 + 2.67580i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.bf odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 342.3.be.a yes 240
9.d odd 6 1 342.3.y.a ✓ 240
19.e even 9 1 342.3.y.a ✓ 240
171.bf odd 18 1 inner 342.3.be.a yes 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
342.3.y.a ✓ 240 9.d odd 6 1
342.3.y.a ✓ 240 19.e even 9 1
342.3.be.a yes 240 1.a even 1 1 trivial
342.3.be.a yes 240 171.bf odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(342, [\chi])\).