Properties

Label 165.4.m.d
Level $165$
Weight $4$
Character orbit 165.m
Analytic conductor $9.735$
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] = [165,4,Mod(16,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.16");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.m (of order \(5\), 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: \(9.73531515095\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} - 18 q^{3} - 16 q^{4} + 30 q^{5} + 6 q^{6} + 23 q^{7} + 139 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 2 q^{2} - 18 q^{3} - 16 q^{4} + 30 q^{5} + 6 q^{6} + 23 q^{7} + 139 q^{8} - 54 q^{9} + 90 q^{10} + 28 q^{11} + 342 q^{12} + 99 q^{13} - 451 q^{14} + 90 q^{15} - 120 q^{16} + 229 q^{17} + 63 q^{18} + 48 q^{19} + 205 q^{20} - 126 q^{21} + 415 q^{22} - 68 q^{23} - 48 q^{24} - 150 q^{25} + 901 q^{26} - 162 q^{27} - 727 q^{28} - 487 q^{29} - 105 q^{30} - 30 q^{31} - 3156 q^{32} + 99 q^{33} - 20 q^{34} + 10 q^{35} - 144 q^{36} + 1018 q^{37} + 2203 q^{38} + 297 q^{39} - 695 q^{40} + 299 q^{41} + 567 q^{42} + 222 q^{43} + 555 q^{44} - 1080 q^{45} - 1163 q^{46} + 451 q^{47} - 315 q^{48} + 985 q^{49} + 50 q^{50} + 252 q^{51} - 3209 q^{52} - 1896 q^{53} - 486 q^{54} - 115 q^{55} + 3086 q^{56} + 9 q^{57} - 1191 q^{58} + 1504 q^{59} + 615 q^{60} - 1238 q^{61} + 754 q^{62} + 207 q^{63} + 881 q^{64} + 780 q^{65} - 2010 q^{66} - 714 q^{67} + 3636 q^{68} - 1059 q^{69} + 2255 q^{70} + 1110 q^{71} - 144 q^{72} + 1966 q^{73} - 3867 q^{74} - 450 q^{75} - 756 q^{76} + 540 q^{77} + 168 q^{78} + 905 q^{79} + 525 q^{80} - 486 q^{81} + 4221 q^{82} - 1026 q^{83} + 3309 q^{84} - 420 q^{85} - 1341 q^{86} - 186 q^{87} - 690 q^{88} + 3126 q^{89} - 90 q^{90} - 1907 q^{91} + 5783 q^{92} - 90 q^{93} - 4054 q^{94} - 240 q^{95} + 1752 q^{96} + 1978 q^{97} - 13172 q^{98} - 918 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.73360 5.33547i −2.42705 1.76336i −18.9898 + 13.7969i −1.54508 + 4.75528i −5.20080 + 16.0064i 24.0593 17.4801i 70.2246 + 51.0211i 2.78115 + 8.55951i 28.0502
16.2 −1.08489 3.33894i −2.42705 1.76336i −3.49938 + 2.54245i −1.54508 + 4.75528i −3.25466 + 10.0168i −11.4322 + 8.30596i −10.4366 7.58266i 2.78115 + 8.55951i 17.5538
16.3 −0.914984 2.81603i −2.42705 1.76336i −0.620704 + 0.450968i −1.54508 + 4.75528i −2.74495 + 8.44810i 20.5786 14.9512i −17.3258 12.5879i 2.78115 + 8.55951i 14.8048
16.4 0.237147 + 0.729862i −2.42705 1.76336i 5.99568 4.35611i −1.54508 + 4.75528i 0.711440 2.18959i 5.09134 3.69907i 9.56808 + 6.95162i 2.78115 + 8.55951i −3.83711
16.5 0.609711 + 1.87650i −2.42705 1.76336i 3.32264 2.41404i −1.54508 + 4.75528i 1.82913 5.62949i −13.8429 + 10.0574i 19.3257 + 14.0410i 2.78115 + 8.55951i −9.86533
16.6 1.15055 + 3.54102i −2.42705 1.76336i −4.74290 + 3.44592i −1.54508 + 4.75528i 3.45164 10.6230i −11.4369 + 8.30940i 6.43837 + 4.67775i 2.78115 + 8.55951i −18.6162
31.1 −1.73360 + 5.33547i −2.42705 + 1.76336i −18.9898 13.7969i −1.54508 4.75528i −5.20080 16.0064i 24.0593 + 17.4801i 70.2246 51.0211i 2.78115 8.55951i 28.0502
31.2 −1.08489 + 3.33894i −2.42705 + 1.76336i −3.49938 2.54245i −1.54508 4.75528i −3.25466 10.0168i −11.4322 8.30596i −10.4366 + 7.58266i 2.78115 8.55951i 17.5538
31.3 −0.914984 + 2.81603i −2.42705 + 1.76336i −0.620704 0.450968i −1.54508 4.75528i −2.74495 8.44810i 20.5786 + 14.9512i −17.3258 + 12.5879i 2.78115 8.55951i 14.8048
31.4 0.237147 0.729862i −2.42705 + 1.76336i 5.99568 + 4.35611i −1.54508 4.75528i 0.711440 + 2.18959i 5.09134 + 3.69907i 9.56808 6.95162i 2.78115 8.55951i −3.83711
31.5 0.609711 1.87650i −2.42705 + 1.76336i 3.32264 + 2.41404i −1.54508 4.75528i 1.82913 + 5.62949i −13.8429 10.0574i 19.3257 14.0410i 2.78115 8.55951i −9.86533
31.6 1.15055 3.54102i −2.42705 + 1.76336i −4.74290 3.44592i −1.54508 4.75528i 3.45164 + 10.6230i −11.4369 8.30940i 6.43837 4.67775i 2.78115 8.55951i −18.6162
91.1 −3.76750 + 2.73725i 0.927051 + 2.85317i 4.22938 13.0167i 4.04508 + 2.93893i −11.3025 8.21174i 1.04715 3.22280i 8.18331 + 25.1856i −7.28115 + 5.29007i −23.2844
91.2 −2.79760 + 2.03258i 0.927051 + 2.85317i 1.22307 3.76423i 4.04508 + 2.93893i −8.39281 6.09773i −0.169994 + 0.523189i −4.31930 13.2934i −7.28115 + 5.29007i −17.2901
91.3 0.0713397 0.0518313i 0.927051 + 2.85317i −2.46973 + 7.60106i 4.04508 + 2.93893i 0.214019 + 0.155494i −6.20592 + 19.0999i 0.435778 + 1.34119i −7.28115 + 5.29007i 0.440904
91.4 1.82807 1.32817i 0.927051 + 2.85317i −0.894328 + 2.75246i 4.04508 + 2.93893i 5.48422 + 3.98452i 7.37247 22.6901i 7.60694 + 23.4117i −7.28115 + 5.29007i 11.2981
91.5 3.00512 2.18335i 0.927051 + 2.85317i 1.79161 5.51401i 4.04508 + 2.93893i 9.01537 + 6.55005i −6.55262 + 20.1669i 2.52784 + 7.77988i −7.28115 + 5.29007i 18.5727
91.6 4.39663 3.19434i 0.927051 + 2.85317i 6.65444 20.4802i 4.04508 + 2.93893i 13.1899 + 9.58302i 2.99170 9.20750i −22.7289 69.9523i −7.28115 + 5.29007i 27.1727
136.1 −3.76750 2.73725i 0.927051 2.85317i 4.22938 + 13.0167i 4.04508 2.93893i −11.3025 + 8.21174i 1.04715 + 3.22280i 8.18331 25.1856i −7.28115 5.29007i −23.2844
136.2 −2.79760 2.03258i 0.927051 2.85317i 1.22307 + 3.76423i 4.04508 2.93893i −8.39281 + 6.09773i −0.169994 0.523189i −4.31930 + 13.2934i −7.28115 5.29007i −17.2901
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.4.m.d 24
11.c even 5 1 inner 165.4.m.d 24
11.c even 5 1 1815.4.a.bg 12
11.d odd 10 1 1815.4.a.bo 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.m.d 24 1.a even 1 1 trivial
165.4.m.d 24 11.c even 5 1 inner
1815.4.a.bg 12 11.c even 5 1
1815.4.a.bo 12 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} - 2 T_{2}^{23} + 34 T_{2}^{22} - 149 T_{2}^{21} + 1212 T_{2}^{20} - 1490 T_{2}^{19} + 31819 T_{2}^{18} - 68287 T_{2}^{17} + 658346 T_{2}^{16} - 1541489 T_{2}^{15} + 10159597 T_{2}^{14} - 22123599 T_{2}^{13} + \cdots + 453519616 \) acting on \(S_{4}^{\mathrm{new}}(165, [\chi])\). Copy content Toggle raw display