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Properties

Label 165.4.w.a
Level $165$
Weight $4$
Character orbit 165.w
Analytic conductor $9.735$
Analytic rank $0$
Dimension $288$
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] = [165,4,Mod(7,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 5, 14]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.7");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.w (of order \(20\), degree \(8\), 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: \(288\)
Relative dimension: \(36\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q + 32 q^{5} - 40 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q + 32 q^{5} - 40 q^{7} - 56 q^{11} + 48 q^{12} - 252 q^{15} + 1264 q^{16} - 640 q^{17} + 1140 q^{20} + 356 q^{22} - 224 q^{23} - 232 q^{25} + 240 q^{26} + 720 q^{28} - 2040 q^{30} - 432 q^{31} - 228 q^{33} + 2592 q^{36} - 1104 q^{37} - 2072 q^{38} + 4440 q^{41} + 1224 q^{42} + 9600 q^{46} - 2280 q^{47} - 576 q^{48} - 2720 q^{50} - 1680 q^{51} - 8520 q^{52} - 1872 q^{53} - 1780 q^{55} - 13600 q^{56} - 1920 q^{57} + 2152 q^{58} - 1152 q^{60} - 480 q^{61} + 160 q^{62} + 360 q^{63} + 1128 q^{66} + 5504 q^{67} + 10240 q^{68} + 3528 q^{70} - 224 q^{71} - 7380 q^{73} + 720 q^{75} + 3312 q^{77} - 288 q^{78} + 4520 q^{80} + 5832 q^{81} - 5616 q^{82} + 13400 q^{83} + 320 q^{85} - 13840 q^{86} + 7188 q^{88} + 9928 q^{91} + 12340 q^{92} + 4104 q^{93} + 2760 q^{95} + 9744 q^{97}+O(q^{100}) \) 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 −5.44513 + 0.862425i 1.36197 + 2.67302i 21.2973 − 6.91990i 8.73683 + 6.97622i −9.72140 − 13.3804i −20.8292 − 10.6130i −70.7016 + 36.0242i −5.29007 + 7.28115i −53.5897 − 30.4516i
7.2 −5.25561 + 0.832407i −1.36197 − 2.67302i 19.3201 − 6.27748i 1.73793 + 11.0444i 9.38303 + 12.9146i 25.3389 + 12.9108i −58.3843 + 29.7483i −5.29007 + 7.28115i −18.3274 − 56.5986i
7.3 −5.16234 + 0.817634i −1.36197 − 2.67302i 18.3728 − 5.96968i −9.96941 − 5.06072i 9.21651 + 12.6854i −12.2125 − 6.22258i −52.7094 + 26.8568i −5.29007 + 7.28115i 55.6033 + 17.9738i
7.4 −4.67076 + 0.739776i 1.36197 + 2.67302i 13.6603 − 4.43851i −9.96728 + 5.06492i −8.33888 − 11.4775i 7.62618 + 3.88573i −26.8122 + 13.6615i −5.29007 + 7.28115i 42.8079 − 31.0306i
7.5 −4.36734 + 0.691718i 1.36197 + 2.67302i 10.9867 − 3.56980i 9.87026 − 5.25148i −7.79716 − 10.7319i 27.8060 + 14.1679i −13.9947 + 7.13066i −5.29007 + 7.28115i −39.4742 + 29.7624i
7.6 −4.17582 + 0.661385i −1.36197 − 2.67302i 9.39161 − 3.05152i 11.0215 − 1.87788i 7.45525 + 10.2613i 7.52484 + 3.83410i −7.06295 + 3.59875i −5.29007 + 7.28115i −44.7818 + 15.1312i
7.7 −3.57801 + 0.566702i −1.36197 − 2.67302i 4.87258 − 1.58320i −0.220318 + 11.1782i 6.38796 + 8.79227i −16.4570 − 8.38526i 9.28522 − 4.73106i −5.29007 + 7.28115i −5.54639 − 40.1205i
7.8 −3.51033 + 0.555981i 1.36197 + 2.67302i 4.40484 − 1.43122i −9.82516 − 5.33538i −6.26712 − 8.62595i −17.3324 − 8.83131i 10.6670 − 5.43511i −5.29007 + 7.28115i 37.4559 + 13.2663i
7.9 −3.47948 + 0.551095i 1.36197 + 2.67302i 4.19460 − 1.36291i 5.38082 − 9.80035i −6.21204 − 8.55013i −10.5883 − 5.39501i 11.2671 − 5.74089i −5.29007 + 7.28115i −13.3215 + 37.0654i
7.10 −3.38245 + 0.535727i −1.36197 − 2.67302i 3.54550 − 1.15200i −5.14737 − 9.92495i 6.03881 + 8.31170i 14.1418 + 7.20561i 13.0355 − 6.64192i −5.29007 + 7.28115i 22.7278 + 30.8130i
7.11 −2.47669 + 0.392269i 1.36197 + 2.67302i −1.62833 + 0.529076i 0.780290 + 11.1531i −4.42173 − 6.08598i 20.6972 + 10.5457i 21.6994 − 11.0564i −5.29007 + 7.28115i −6.30755 − 27.3166i
7.12 −2.07190 + 0.328156i −1.36197 − 2.67302i −3.42338 + 1.11232i −11.1670 + 0.545159i 3.69903 + 5.09128i −20.9600 − 10.6796i 21.6806 − 11.0468i −5.29007 + 7.28115i 22.9581 − 4.79405i
7.13 −2.02304 + 0.320417i −1.36197 − 2.67302i −3.61845 + 1.17571i 7.04080 + 8.68488i 3.61180 + 4.97121i 4.18773 + 2.13375i 21.5436 − 10.9770i −5.29007 + 7.28115i −17.0266 − 15.3138i
7.14 −1.01863 + 0.161335i 1.36197 + 2.67302i −6.59687 + 2.14345i 11.1507 − 0.813732i −1.81860 − 2.50309i −27.5385 − 14.0316i 13.7253 − 6.99341i −5.29007 + 7.28115i −11.2272 + 2.62789i
7.15 −0.939558 + 0.148811i −1.36197 − 2.67302i −6.74783 + 2.19250i 2.41879 − 10.9156i 1.67743 + 2.30878i −24.2230 − 12.3422i 12.7944 − 6.51908i −5.29007 + 7.28115i −0.648231 + 10.6157i
7.16 −0.869919 + 0.137782i 1.36197 + 2.67302i −6.87068 + 2.23242i −10.1398 − 4.70998i −1.55310 − 2.13766i 10.0336 + 5.11235i 11.9475 − 6.08754i −5.29007 + 7.28115i 9.46978 + 2.70022i
7.17 −0.626725 + 0.0992634i 1.36197 + 2.67302i −7.22552 + 2.34771i −1.03890 − 11.1320i −1.11891 − 1.54005i 20.7390 + 10.5670i 8.81838 − 4.49319i −5.29007 + 7.28115i 1.75610 + 6.87355i
7.18 −0.623925 + 0.0988201i −1.36197 − 2.67302i −7.22893 + 2.34882i 9.11856 − 6.46930i 1.11392 + 1.53317i 19.1840 + 9.77474i 8.78101 − 4.47415i −5.29007 + 7.28115i −5.05000 + 4.93746i
7.19 −0.524292 + 0.0830397i 1.36197 + 2.67302i −7.34047 + 2.38506i −4.51540 + 10.2280i −0.936038 − 1.28835i −7.54341 − 3.84356i 7.43426 − 3.78795i −5.29007 + 7.28115i 1.51806 − 5.73740i
7.20 −0.0494138 + 0.00782638i −1.36197 − 2.67302i −7.60607 + 2.47136i −8.47625 + 7.29062i 0.0882203 + 0.121425i 9.39386 + 4.78641i 0.713118 − 0.363352i −5.29007 + 7.28115i 0.361785 − 0.426596i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.d odd 10 1 inner
55.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 165.4.w.a ✓ 288
5.c odd 4 1 inner 165.4.w.a ✓ 288
11.d odd 10 1 inner 165.4.w.a ✓ 288
55.l even 20 1 inner 165.4.w.a ✓ 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.w.a ✓ 288 1.a even 1 1 trivial
165.4.w.a ✓ 288 5.c odd 4 1 inner
165.4.w.a ✓ 288 11.d odd 10 1 inner
165.4.w.a ✓ 288 55.l even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(165, [\chi])\).