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Properties

Label 165.4.k.a
Level $165$
Weight $4$
Character orbit 165.k
Analytic conductor $9.735$
Analytic rank $0$
Dimension $60$
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(23,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.23");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.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: \(9.73531515095\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 6 q^{3} + 8 q^{6} + 36 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 6 q^{3} + 8 q^{6} + 36 q^{8} + 20 q^{9} - 160 q^{12} - 24 q^{13} + 96 q^{14} + 202 q^{15} - 1188 q^{16} + 216 q^{17} + 432 q^{18} - 576 q^{20} + 376 q^{21} - 216 q^{23} - 364 q^{24} + 144 q^{25} - 930 q^{27} - 432 q^{28} + 144 q^{29} - 132 q^{30} - 408 q^{31} + 288 q^{32} + 110 q^{33} - 216 q^{35} + 336 q^{36} + 864 q^{37} - 864 q^{38} - 448 q^{39} + 204 q^{40} - 460 q^{42} + 2772 q^{44} - 924 q^{45} - 600 q^{46} - 180 q^{47} + 5276 q^{48} + 96 q^{50} - 16 q^{51} + 228 q^{52} - 3624 q^{53} - 3184 q^{54} + 396 q^{55} - 156 q^{57} - 588 q^{58} + 3816 q^{59} - 196 q^{60} - 120 q^{61} - 288 q^{62} + 2288 q^{63} + 1176 q^{65} - 108 q^{67} - 7152 q^{68} - 5672 q^{69} + 3900 q^{70} + 1908 q^{72} + 1800 q^{73} + 3336 q^{74} + 1942 q^{75} + 1344 q^{76} + 10188 q^{78} + 1728 q^{80} + 1132 q^{81} - 972 q^{82} - 2640 q^{83} - 12072 q^{84} - 3864 q^{85} + 3596 q^{87} - 396 q^{88} + 6816 q^{89} + 2000 q^{90} + 2448 q^{91} - 3456 q^{92} + 2492 q^{93} + 2448 q^{95} - 2588 q^{96} - 2784 q^{97} - 9588 q^{98} - 308 q^{99}+O(q^{100}) \) 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 −3.84806 − 3.84806i −0.102591 + 5.19514i 21.6151i 10.9719 + 2.14897i 20.3860 − 19.5964i 1.08722 − 1.08722i 52.3919 − 52.3919i −26.9790 − 1.06595i −33.9511 − 50.4898i
23.2 −3.71906 − 3.71906i −5.18346 + 0.362980i 19.6628i −11.1632 + 0.618938i 20.6275 + 17.9277i 22.1514 − 22.1514i 43.3748 − 43.3748i 26.7365 − 3.76299i 43.8185 + 39.2147i
23.3 −3.65593 − 3.65593i −3.84206 − 3.49837i 18.7316i 3.79149 + 10.5178i 1.25650 + 26.8361i −21.7958 + 21.7958i 39.2338 − 39.2338i 2.52281 + 26.8819i 24.5910 − 52.3138i
23.4 −3.34648 − 3.34648i 3.87247 − 3.46467i 14.3979i −11.0198 − 1.88764i −24.5536 − 1.36468i −15.7391 + 15.7391i 21.4103 − 21.4103i 2.99206 − 26.8337i 30.5607 + 43.1946i
23.5 −3.07824 − 3.07824i 5.03512 + 1.28359i 10.9512i 0.199123 − 11.1786i −11.5481 − 19.4505i 12.4286 − 12.4286i 9.08442 − 9.08442i 23.7048 + 12.9260i −35.0233 + 33.7974i
23.6 −2.54428 − 2.54428i −1.88116 + 4.84368i 4.94674i −8.63011 + 7.10783i 17.1099 − 7.53749i −13.1466 + 13.1466i −7.76835 + 7.76835i −19.9225 − 18.2235i 40.0417 + 3.87311i
23.7 −2.42487 − 2.42487i 2.59987 − 4.49896i 3.75995i 9.26056 + 6.26435i −17.2137 + 4.60504i 5.31387 − 5.31387i −10.2816 + 10.2816i −13.4813 − 23.3935i −7.26540 − 37.6458i
23.8 −2.29370 − 2.29370i −2.17404 − 4.71949i 2.52213i −7.54936 − 8.24664i −5.83850 + 15.8117i −1.24165 + 1.24165i −12.5646 + 12.5646i −17.5471 + 20.5207i −1.59934 + 36.2313i
23.9 −2.14832 − 2.14832i 2.94234 + 4.28283i 1.23057i 8.94968 − 6.70099i 2.87980 − 15.5220i −22.9362 + 22.9362i −14.5429 + 14.5429i −9.68525 + 25.2031i −33.6227 − 4.83090i
23.10 −2.09883 − 2.09883i 3.40209 + 3.92757i 0.810178i 1.55928 + 11.0711i 1.10290 − 15.3837i 19.5040 − 19.5040i −15.0902 + 15.0902i −3.85163 + 26.7239i 19.9636 − 26.5090i
23.11 −1.22725 − 1.22725i −4.15648 + 3.11828i − 4.98770i −5.15890 − 9.91896i 8.92797 + 1.27414i 8.44060 − 8.44060i −15.9392 + 15.9392i 7.55266 − 25.9221i −5.84179 + 18.5044i
23.12 −0.996188 − 0.996188i −4.56236 + 2.48693i − 6.01522i 11.1056 + 1.29023i 7.02242 + 2.06752i −14.2549 + 14.2549i −13.9618 + 13.9618i 14.6303 − 22.6926i −9.77799 − 12.3486i
23.13 −0.544935 − 0.544935i 3.31640 − 4.00018i − 7.40609i 8.80657 − 6.88799i −3.98706 + 0.372615i 11.5215 − 11.5215i −8.39532 + 8.39532i −5.00293 − 26.5324i −8.55251 − 1.04550i
23.14 −0.297503 − 0.297503i −0.519840 − 5.17008i − 7.82298i −6.44325 + 9.13698i −1.38346 + 1.69277i −1.14479 + 1.14479i −4.70739 + 4.70739i −26.4595 + 5.37523i 4.63517 − 0.801393i
23.15 −0.293559 − 0.293559i −4.78108 − 2.03502i − 7.82765i 4.05612 + 10.4186i 0.806131 + 2.00093i 21.2667 − 21.2667i −4.64635 + 4.64635i 18.7174 + 19.4592i 1.86778 − 4.24920i
23.16 0.237835 + 0.237835i 1.46376 + 4.98572i − 7.88687i −11.1425 + 0.918650i −0.837644 + 1.53391i 9.16412 − 9.16412i 3.77845 − 3.77845i −22.7148 + 14.5958i −2.86857 − 2.43159i
23.17 0.255953 + 0.255953i 5.18326 − 0.365760i − 7.86898i −3.35143 − 10.6662i 1.42029 + 1.23305i −7.57961 + 7.57961i 4.06171 − 4.06171i 26.7324 − 3.79166i 1.87224 − 3.58786i
23.18 0.919027 + 0.919027i −4.66471 − 2.28921i − 6.31078i −9.57507 − 5.77217i −2.18315 − 6.39084i −18.9648 + 18.9648i 13.1520 − 13.1520i 16.5190 + 21.3570i −3.49497 − 14.1045i
23.19 1.32917 + 1.32917i 3.58033 + 3.76580i − 4.46660i 9.69092 + 5.57550i −0.246512 + 9.76428i −7.67291 + 7.67291i 16.5703 − 16.5703i −1.36243 + 26.9656i 5.47011 + 20.2917i
23.20 1.62914 + 1.62914i −1.32893 − 5.02334i − 2.69180i 10.3779 − 4.15919i 6.01871 − 10.3487i −5.14632 + 5.14632i 17.4184 − 17.4184i −23.4679 + 13.3514i 23.6830 + 10.1312i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.4.k.a ✓ 60
3.b odd 2 1 165.4.k.b yes 60
5.c odd 4 1 165.4.k.b yes 60
15.e even 4 1 inner 165.4.k.a ✓ 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.k.a ✓ 60 1.a even 1 1 trivial
165.4.k.a ✓ 60 15.e even 4 1 inner
165.4.k.b yes 60 3.b odd 2 1
165.4.k.b yes 60 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{60} - 12 T_{2}^{57} + 3729 T_{2}^{56} - 288 T_{2}^{55} + 72 T_{2}^{54} - 36900 T_{2}^{53} + \cdots + 24\!\cdots\!36 \) acting on \(S_{4}^{\mathrm{new}}(165, [\chi])\). Copy content Toggle raw display