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Properties

Label 32.4.g.a
Level $32$
Weight $4$
Character orbit 32.g
Analytic conductor $1.888$
Analytic rank $0$
Dimension $44$
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] = [32,4,Mod(5,32)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(32, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("32.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 32.g (of order \(8\), degree \(4\), 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: \(1.88806112018\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{2} - 4 q^{3} - 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} - 4 q^{9} + 116 q^{10} - 4 q^{11} - 52 q^{12} - 4 q^{13} - 212 q^{14} - 304 q^{16} - 184 q^{18} - 4 q^{19} + 76 q^{20} - 4 q^{21} + 192 q^{22} + 324 q^{23} - 48 q^{24} - 4 q^{25} + 16 q^{26} - 268 q^{27} + 376 q^{28} - 4 q^{29} + 1188 q^{30} - 752 q^{31} + 616 q^{32} - 8 q^{33} + 528 q^{34} - 460 q^{35} + 1456 q^{36} - 4 q^{37} + 980 q^{38} + 596 q^{39} - 536 q^{40} - 4 q^{41} - 2264 q^{42} + 804 q^{43} - 2044 q^{44} + 104 q^{45} - 1444 q^{46} - 2448 q^{48} - 3564 q^{50} - 1384 q^{51} - 2524 q^{52} + 748 q^{53} - 1088 q^{54} - 292 q^{55} + 1192 q^{56} - 4 q^{57} + 3200 q^{58} + 1372 q^{59} + 5752 q^{60} - 1828 q^{61} + 3384 q^{62} + 2512 q^{63} + 4952 q^{64} - 8 q^{65} + 5996 q^{66} + 2036 q^{67} + 2768 q^{68} - 1060 q^{69} + 1400 q^{70} + 220 q^{71} - 1708 q^{72} - 4 q^{73} - 3476 q^{74} - 1712 q^{75} - 5124 q^{76} + 1900 q^{77} - 11916 q^{78} - 10312 q^{80} - 6404 q^{82} + 2436 q^{83} - 6560 q^{84} + 496 q^{85} - 928 q^{86} - 1292 q^{87} + 1248 q^{88} - 4 q^{89} + 7400 q^{90} - 3604 q^{91} + 10152 q^{92} - 112 q^{93} + 12840 q^{94} - 6088 q^{95} + 17792 q^{96} - 8 q^{97} + 11224 q^{98} - 5424 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −2.82381 − 0.161516i −0.477813 − 1.15354i 7.94783 + 0.912182i −16.3468 − 6.77105i 1.16294 + 3.33456i −18.0222 − 18.0222i −22.2958 − 3.85953i 17.9895 − 17.9895i 45.0666 + 21.7604i
5.2 −2.51802 + 1.28824i −0.998206 − 2.40988i 4.68088 − 6.48763i 17.4005 + 7.20752i 5.61801 + 4.78221i 4.37099 + 4.37099i −3.42892 + 22.3661i 14.2808 − 14.2808i −53.0999 + 4.26731i
5.3 −2.10117 − 1.89344i 1.94138 + 4.68690i 0.829801 + 7.95685i 4.93338 + 2.04347i 4.79518 − 13.5238i 14.0755 + 14.0755i 13.3222 − 18.2898i 0.893826 − 0.893826i −6.49667 − 13.6347i
5.4 −1.75970 + 2.21438i 3.54796 + 8.56554i −1.80692 − 7.79327i −7.55322 − 3.12865i −25.2107 − 7.21625i 7.16166 + 7.16166i 20.4369 + 9.71261i −41.6886 + 41.6886i 20.2194 − 11.2202i
5.5 −0.719102 + 2.73549i −3.21198 − 7.75440i −6.96578 − 3.93419i −13.6472 − 5.65283i 23.5218 − 3.21012i 9.07689 + 9.07689i 15.7710 − 16.2257i −30.7220 + 30.7220i 25.2770 − 33.2666i
5.6 −0.582457 − 2.76780i −1.90169 − 4.59109i −7.32149 + 3.22425i 0.188811 + 0.0782080i −11.5996 + 7.93763i −11.4103 − 11.4103i 13.1886 + 18.3865i 1.63019 − 1.63019i 0.106490 − 0.568145i
5.7 1.17521 + 2.57272i 0.729459 + 1.76107i −5.23776 + 6.04698i 4.29822 + 1.78038i −3.67347 + 3.94632i 1.47807 + 1.47807i −21.7126 − 6.36880i 16.5226 − 16.5226i 0.470898 + 13.1504i
5.8 1.49948 − 2.39824i 2.92731 + 7.06715i −3.50313 − 7.19223i 13.5234 + 5.60159i 21.3382 + 3.57665i −23.0737 − 23.0737i −22.5016 − 2.38325i −22.2836 + 22.2836i 33.7121 − 24.0330i
5.9 2.01560 − 1.98428i −1.20185 − 2.90153i 0.125267 − 7.99902i −3.98512 − 1.65069i −8.17991 − 3.46351i 22.4050 + 22.4050i −15.6198 − 16.3714i 12.1174 − 12.1174i −11.3078 + 4.58047i
5.10 2.70658 + 0.821250i −3.28810 − 7.93817i 6.65110 + 4.44555i 11.2895 + 4.67626i −2.38026 − 24.1856i −11.8490 − 11.8490i 14.3508 + 17.4944i −33.1111 + 33.1111i 26.7154 + 21.9281i
5.11 2.81450 + 0.280314i 1.64064 + 3.96085i 7.84285 + 1.57789i −11.8087 − 4.89132i 3.50730 + 11.6077i −5.11236 − 5.11236i 21.6314 + 6.63943i 6.09524 − 6.09524i −31.8645 − 17.0768i
13.1 −2.82381 + 0.161516i −0.477813 + 1.15354i 7.94783 − 0.912182i −16.3468 + 6.77105i 1.16294 − 3.33456i −18.0222 + 18.0222i −22.2958 + 3.85953i 17.9895 + 17.9895i 45.0666 − 21.7604i
13.2 −2.51802 − 1.28824i −0.998206 + 2.40988i 4.68088 + 6.48763i 17.4005 − 7.20752i 5.61801 − 4.78221i 4.37099 − 4.37099i −3.42892 − 22.3661i 14.2808 + 14.2808i −53.0999 − 4.26731i
13.3 −2.10117 + 1.89344i 1.94138 − 4.68690i 0.829801 − 7.95685i 4.93338 − 2.04347i 4.79518 + 13.5238i 14.0755 − 14.0755i 13.3222 + 18.2898i 0.893826 + 0.893826i −6.49667 + 13.6347i
13.4 −1.75970 − 2.21438i 3.54796 − 8.56554i −1.80692 + 7.79327i −7.55322 + 3.12865i −25.2107 + 7.21625i 7.16166 − 7.16166i 20.4369 − 9.71261i −41.6886 − 41.6886i 20.2194 + 11.2202i
13.5 −0.719102 − 2.73549i −3.21198 + 7.75440i −6.96578 + 3.93419i −13.6472 + 5.65283i 23.5218 + 3.21012i 9.07689 − 9.07689i 15.7710 + 16.2257i −30.7220 − 30.7220i 25.2770 + 33.2666i
13.6 −0.582457 + 2.76780i −1.90169 + 4.59109i −7.32149 − 3.22425i 0.188811 − 0.0782080i −11.5996 − 7.93763i −11.4103 + 11.4103i 13.1886 − 18.3865i 1.63019 + 1.63019i 0.106490 + 0.568145i
13.7 1.17521 − 2.57272i 0.729459 − 1.76107i −5.23776 − 6.04698i 4.29822 − 1.78038i −3.67347 − 3.94632i 1.47807 − 1.47807i −21.7126 + 6.36880i 16.5226 + 16.5226i 0.470898 − 13.1504i
13.8 1.49948 + 2.39824i 2.92731 − 7.06715i −3.50313 + 7.19223i 13.5234 − 5.60159i 21.3382 − 3.57665i −23.0737 + 23.0737i −22.5016 + 2.38325i −22.2836 − 22.2836i 33.7121 + 24.0330i
13.9 2.01560 + 1.98428i −1.20185 + 2.90153i 0.125267 + 7.99902i −3.98512 + 1.65069i −8.17991 + 3.46351i 22.4050 − 22.4050i −15.6198 + 16.3714i 12.1174 + 12.1174i −11.3078 − 4.58047i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.11
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.4.g.a ✓ 44
4.b odd 2 1 128.4.g.a 44
8.b even 2 1 256.4.g.b 44
8.d odd 2 1 256.4.g.a 44
32.g even 8 1 inner 32.4.g.a ✓ 44
32.g even 8 1 256.4.g.b 44
32.h odd 8 1 128.4.g.a 44
32.h odd 8 1 256.4.g.a 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.g.a ✓ 44 1.a even 1 1 trivial
32.4.g.a ✓ 44 32.g even 8 1 inner
128.4.g.a 44 4.b odd 2 1
128.4.g.a 44 32.h odd 8 1
256.4.g.a 44 8.d odd 2 1
256.4.g.a 44 32.h odd 8 1
256.4.g.b 44 8.b even 2 1
256.4.g.b 44 32.g even 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(32, [\chi])\).