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Properties

Label 165.4.k.b
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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 10 q^{3} + 8 q^{6} - 36 q^{8} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q + 10 q^{3} + 8 q^{6} - 36 q^{8} - 20 q^{9} + 96 q^{12} - 24 q^{13} - 96 q^{14} - 378 q^{15} - 1188 q^{16} - 216 q^{17} + 12 q^{18} + 576 q^{20} + 376 q^{21} + 216 q^{23} + 364 q^{24} + 144 q^{25} - 182 q^{27} - 432 q^{28} - 144 q^{29} - 692 q^{30} - 408 q^{31} - 288 q^{32} + 66 q^{33} + 216 q^{35} + 336 q^{36} + 864 q^{37} + 864 q^{38} + 448 q^{39} + 204 q^{40} - 348 q^{42} - 2772 q^{44} - 868 q^{45} - 600 q^{46} + 180 q^{47} - 1428 q^{48} - 96 q^{50} - 16 q^{51} + 228 q^{52} + 3624 q^{53} + 3184 q^{54} + 396 q^{55} - 884 q^{57} - 588 q^{58} - 3816 q^{59} + 956 q^{60} - 120 q^{61} + 288 q^{62} - 4936 q^{63} - 1176 q^{65} - 108 q^{67} + 7152 q^{68} + 5672 q^{69} + 3900 q^{70} - 1452 q^{72} + 1800 q^{73} - 3336 q^{74} + 4794 q^{75} + 1344 q^{76} - 4948 q^{78} - 1728 q^{80} + 1132 q^{81} - 972 q^{82} + 2640 q^{83} + 12072 q^{84} - 3864 q^{85} - 1564 q^{87} - 396 q^{88} - 6816 q^{89} + 4944 q^{90} + 2448 q^{91} + 3456 q^{92} - 6928 q^{93} - 2448 q^{95} - 2588 q^{96} - 2784 q^{97} + 9588 q^{98} + 308 q^{99}+O(q^{100}) \) 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 −3.92061 − 3.92061i 0.959234 − 5.10685i 22.7423i 5.55520 − 9.70256i −23.7827 + 16.2612i 10.4367 − 10.4367i 57.7989 − 57.7989i −25.1597 − 9.79732i −59.8197 + 16.2602i
23.2 −3.57956 − 3.57956i 4.20024 + 3.05908i 17.6265i −7.78453 + 8.02503i −4.08484 − 25.9852i −2.31019 + 2.31019i 34.4588 − 34.4588i 8.28402 + 25.6978i 56.5913 − 0.860855i
23.3 −3.34243 − 3.34243i −5.06177 + 1.17410i 14.3437i 0.137356 − 11.1795i 20.8430 + 12.9942i −23.9613 + 23.9613i 21.2033 − 21.2033i 24.2430 − 11.8861i −37.8258 + 36.9076i
23.4 −3.20901 − 3.20901i −0.511035 + 5.17096i 12.5954i −5.99544 − 9.43688i 18.2336 − 14.9537i 3.33254 − 3.33254i 14.7468 − 14.7468i −26.4777 − 5.28508i −11.0436 + 49.5224i
23.5 −3.16443 − 3.16443i 5.14633 − 0.717832i 12.0273i 10.8258 + 2.79320i −18.5568 − 14.0137i −7.69021 + 7.69021i 12.7441 − 12.7441i 25.9694 − 7.38840i −25.4187 − 43.0964i
23.6 −2.99137 − 2.99137i −0.408678 − 5.18006i 9.89661i −4.50498 + 10.2326i −14.2730 + 16.7180i 9.44394 − 9.44394i 5.67346 − 5.67346i −26.6660 + 4.23395i 44.0854 − 17.1333i
23.7 −2.82630 − 2.82630i −4.44678 + 2.68815i 7.97591i 6.24850 + 9.27126i 20.1654 + 4.97041i 7.60335 − 7.60335i −0.0680788 + 0.0680788i 12.5477 − 23.9072i 8.54320 − 43.8635i
23.8 −1.79097 − 1.79097i 4.48076 − 2.63112i − 1.58489i −11.1426 − 0.918066i −12.7371 − 3.31264i 23.8991 − 23.8991i −17.1662 + 17.1662i 13.1544 − 23.5788i 18.3118 + 21.6002i
23.9 −1.68243 − 1.68243i −0.458584 + 5.17588i − 2.33888i 9.10251 − 6.49186i 9.47956 − 7.93650i 15.2701 − 15.2701i −17.3944 + 17.3944i −26.5794 − 4.74715i −26.2364 − 4.39223i
23.10 −1.63898 − 1.63898i 1.91285 − 4.83125i − 2.62749i 2.29362 − 10.9425i −11.0534 + 4.78320i −17.2793 + 17.2793i −17.4182 + 17.4182i −19.6820 − 18.4829i −21.6938 + 14.1754i
23.11 −1.62914 − 1.62914i −5.02334 − 1.32893i − 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
23.12 −1.32917 − 1.32917i 3.76580 + 3.58033i − 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
23.13 −0.919027 − 0.919027i −2.28921 − 4.66471i − 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
23.14 −0.255953 − 0.255953i −0.365760 + 5.18326i − 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
23.15 −0.237835 − 0.237835i 4.98572 + 1.46376i − 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
23.16 0.293559 + 0.293559i −2.03502 − 4.78108i − 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
23.17 0.297503 + 0.297503i −5.17008 − 0.519840i − 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
23.18 0.544935 + 0.544935i −4.00018 + 3.31640i − 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
23.19 0.996188 + 0.996188i 2.48693 − 4.56236i − 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
23.20 1.22725 + 1.22725i 3.11828 − 4.15648i − 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
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.30
Significant digits:
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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.b yes 60
3.b odd 2 1 165.4.k.a ✓ 60
5.c odd 4 1 165.4.k.a ✓ 60
15.e even 4 1 inner 165.4.k.b yes 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.k.a ✓ 60 3.b odd 2 1
165.4.k.a ✓ 60 5.c odd 4 1
165.4.k.b yes 60 1.a even 1 1 trivial
165.4.k.b yes 60 15.e even 4 1 inner

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