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Properties

Label 164.4.o.b
Level $164$
Weight $4$
Character orbit 164.o
Analytic conductor $9.676$
Analytic rank $0$
Dimension $960$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [164,4,Mod(7,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 39]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 164.o (of order \(40\), degree \(16\), 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: \(9.67631324094\)
Analytic rank: \(0\)
Dimension: \(960\)
Relative dimension: \(60\) over \(\Q(\zeta_{40})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 960 q - 24 q^{2} - 20 q^{4} - 32 q^{5} + 44 q^{6} + 48 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 960 q - 24 q^{2} - 20 q^{4} - 32 q^{5} + 44 q^{6} + 48 q^{8} + 8 q^{9} - 12 q^{10} + 296 q^{12} - 216 q^{13} + 116 q^{14} - 268 q^{16} - 328 q^{17} - 12 q^{18} - 620 q^{20} - 32 q^{21} + 488 q^{22} - 376 q^{24} - 40 q^{25} + 424 q^{26} + 240 q^{28} - 8 q^{29} - 940 q^{30} - 3124 q^{32} + 1128 q^{33} - 2684 q^{34} + 748 q^{36} - 24 q^{37} + 40 q^{38} + 688 q^{41} - 592 q^{42} + 240 q^{44} - 40 q^{45} + 856 q^{46} - 164 q^{48} - 240 q^{49} + 6324 q^{50} + 3236 q^{52} - 1640 q^{53} - 4712 q^{54} + 524 q^{56} - 24 q^{57} + 268 q^{58} + 2680 q^{60} - 3408 q^{61} - 476 q^{62} - 20 q^{64} + 9312 q^{65} + 12700 q^{66} + 9092 q^{68} + 184 q^{69} + 4792 q^{70} + 1600 q^{72} - 32 q^{73} - 4104 q^{74} - 13296 q^{76} - 32 q^{77} - 18316 q^{78} - 9648 q^{80} - 12760 q^{82} - 16888 q^{84} + 7688 q^{85} - 12220 q^{86} - 14256 q^{88} - 1560 q^{89} + 2072 q^{90} + 3204 q^{92} + 184 q^{93} + 14300 q^{94} + 33288 q^{96} - 5760 q^{97} + 15424 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −2.82831 − 0.0261826i 2.20725 + 0.914273i 7.99863 + 0.148105i 1.58435 + 10.0032i −6.21884 − 2.64364i 2.14917 − 27.3078i −22.6187 − 0.628310i −15.0558 − 15.0558i −4.21913 − 28.3337i
7.2 −2.81046 − 0.318301i −4.11359 − 1.70390i 7.79737 + 1.78914i −1.57850 − 9.96623i 11.0187 + 6.09811i −1.64453 + 20.8957i −21.3447 − 7.51022i −5.07356 − 5.07356i 1.26404 + 28.5121i
7.3 −2.79760 + 0.416423i 7.97398 + 3.30293i 7.65318 − 2.32998i −2.03548 − 12.8515i −23.6835 − 5.91975i −1.55305 + 19.7334i −20.4403 + 9.70531i 33.5832 + 33.5832i 11.0461 + 35.1058i
7.4 −2.77117 + 0.566254i −9.45316 − 3.91563i 7.35871 − 3.13836i −0.204169 − 1.28907i 28.4135 + 5.49797i 0.346791 − 4.40640i −18.6151 + 12.8638i 54.9382 + 54.9382i 1.29573 + 3.45661i
7.5 −2.75263 + 0.650386i 2.69636 + 1.11687i 7.15400 − 3.58055i 1.98639 + 12.5416i −8.14848 − 1.32066i −1.10255 + 14.0092i −17.3636 + 14.5088i −13.0689 − 13.0689i −13.6247 − 33.2305i
7.6 −2.74788 − 0.670173i 5.02669 + 2.08212i 7.10174 + 3.68312i −0.772955 − 4.88025i −12.4174 − 9.09018i 0.776421 − 9.86537i −17.0464 − 14.8802i 1.84048 + 1.84048i −1.14662 + 13.9284i
7.7 −2.72509 + 0.757553i −1.70615 − 0.706712i 6.85223 − 4.12880i −3.02887 − 19.1236i 5.18480 + 0.633353i 2.19277 − 27.8618i −15.5452 + 16.4423i −16.6804 − 16.6804i 22.7410 + 49.8189i
7.8 −2.61321 − 1.08218i −2.75655 − 1.14180i 5.65777 + 5.65594i −0.104572 − 0.660240i 5.96781 + 5.96685i 0.120842 − 1.53544i −8.66421 − 20.9029i −12.7970 − 12.7970i −0.441231 + 1.83851i
7.9 −2.60667 − 1.09784i −6.11856 − 2.53439i 5.58949 + 5.72343i 3.15837 + 19.9412i 13.1667 + 13.3235i −0.491353 + 6.24324i −8.28655 − 21.0555i 11.9218 + 11.9218i 13.6594 − 55.4474i
7.10 −2.41995 − 1.46419i 8.37655 + 3.46968i 3.71232 + 7.08652i 2.18824 + 13.8160i −15.1906 − 20.6613i −0.643432 + 8.17557i 1.39237 − 22.5845i 39.0361 + 39.0361i 14.9338 − 36.6380i
7.11 −2.39010 + 1.51242i −3.33971 − 1.38335i 3.42516 − 7.22968i −0.179380 − 1.13256i 10.0745 − 1.74470i −1.78240 + 22.6476i 2.74784 + 22.4600i −9.85188 − 9.85188i 2.14164 + 2.43563i
7.12 −2.33834 + 1.59128i −4.63804 − 1.92114i 2.93566 − 7.44190i 2.51545 + 15.8819i 13.9024 − 2.88815i 1.23040 − 15.6337i 4.97759 + 22.0731i −1.27122 − 1.27122i −31.1546 − 33.1345i
7.13 −2.17344 + 1.81001i 8.34877 + 3.45817i 1.44772 − 7.86792i 1.05704 + 6.67386i −24.4049 + 7.59522i 2.44206 − 31.0293i 11.0945 + 19.7209i 38.6511 + 38.6511i −14.3772 − 12.5920i
7.14 −2.16503 − 1.82007i 2.75819 + 1.14248i 1.37470 + 7.88100i −2.71222 − 17.1243i −3.89217 − 7.49361i −0.748433 + 9.50975i 11.3677 − 19.5646i −12.7895 − 12.7895i −25.2953 + 42.0110i
7.15 −2.09121 − 1.90443i −6.33828 − 2.62540i 0.746293 + 7.96511i −0.748299 − 4.72458i 8.25476 + 17.5611i 2.87641 − 36.5483i 13.6083 − 18.0780i 14.1892 + 14.1892i −7.43277 + 11.3051i
7.16 −2.07973 + 1.91695i 4.44102 + 1.83953i 0.650578 − 7.97350i −1.30459 − 8.23688i −12.7624 + 4.68749i 0.0837532 − 1.06419i 13.9318 + 17.8299i −2.75313 − 2.75313i 18.5029 + 14.6297i
7.17 −1.86814 − 2.12369i 3.34557 + 1.38578i −1.02011 + 7.93469i 0.677438 + 4.27717i −3.30702 − 9.69379i 0.392387 − 4.98575i 18.7565 − 12.6567i −9.81943 − 9.81943i 7.81784 − 9.42902i
7.18 −1.58526 + 2.34242i −4.86593 − 2.01553i −2.97388 − 7.42671i 0.136385 + 0.861102i 12.4350 − 8.20290i 1.41846 − 18.0233i 22.1109 + 4.80719i 0.522982 + 0.522982i −2.23327 − 1.04560i
7.19 −1.57060 − 2.35228i −7.99141 − 3.31015i −3.06644 + 7.38897i −1.55639 − 9.82669i 4.76490 + 23.9970i −1.83801 + 23.3541i 22.1971 − 4.39198i 33.8137 + 33.8137i −20.6707 + 19.0949i
7.20 −1.56722 − 2.35453i −0.376671 − 0.156022i −3.08761 + 7.38015i 2.05617 + 12.9821i 0.222969 + 1.13140i −2.24931 + 28.5802i 22.2158 − 4.29647i −18.9743 − 18.9743i 27.3444 − 25.1872i
See next 80 embeddings (of 960 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
41.h odd 40 1 inner
164.o even 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.4.o.b ✓ 960
4.b odd 2 1 inner 164.4.o.b ✓ 960
41.h odd 40 1 inner 164.4.o.b ✓ 960
164.o even 40 1 inner 164.4.o.b ✓ 960
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.4.o.b ✓ 960 1.a even 1 1 trivial
164.4.o.b ✓ 960 4.b odd 2 1 inner
164.4.o.b ✓ 960 41.h odd 40 1 inner
164.4.o.b ✓ 960 164.o even 40 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{960} - 4 T_{3}^{958} + 8 T_{3}^{956} + 50400 T_{3}^{954} + 1576527422 T_{3}^{952} + \cdots + 12\!\cdots\!76 \) acting on \(S_{4}^{\mathrm{new}}(164, [\chi])\). Copy content Toggle raw display