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Properties

Label 201.4.m.b
Level $201$
Weight $4$
Character orbit 201.m
Analytic conductor $11.859$
Analytic rank $0$
Dimension $360$
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] = [201,4,Mod(4,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 201.m (of order \(33\), degree \(20\), 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: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(360\)
Relative dimension: \(18\) over \(\Q(\zeta_{33})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 360 q - 2 q^{2} + 108 q^{3} + 90 q^{4} + 4 q^{5} - 27 q^{6} - 22 q^{7} + 183 q^{8} - 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 360 q - 2 q^{2} + 108 q^{3} + 90 q^{4} + 4 q^{5} - 27 q^{6} - 22 q^{7} + 183 q^{8} - 324 q^{9} + 19 q^{10} + 192 q^{11} - 105 q^{12} + 46 q^{13} - 14 q^{14} + 54 q^{15} + 346 q^{16} + 228 q^{17} - 18 q^{18} + 88 q^{19} + 180 q^{20} + 66 q^{21} - 247 q^{22} + 676 q^{23} + 144 q^{24} - 1120 q^{25} + 333 q^{26} + 972 q^{27} - 3894 q^{28} - 2210 q^{29} + 702 q^{30} + 96 q^{31} - 1970 q^{32} - 708 q^{33} - 26 q^{34} + 1272 q^{35} + 315 q^{36} - 3616 q^{37} - 20 q^{38} - 138 q^{39} - 5357 q^{40} + 288 q^{41} - 618 q^{42} - 1680 q^{43} + 2951 q^{44} + 36 q^{45} - 1125 q^{46} - 40 q^{47} - 1038 q^{48} - 4954 q^{49} - 3590 q^{50} - 90 q^{51} + 9120 q^{52} + 3896 q^{53} + 54 q^{54} - 1076 q^{55} + 3557 q^{56} + 660 q^{57} + 3232 q^{58} + 116 q^{59} + 3618 q^{60} - 672 q^{61} + 3882 q^{62} - 1584 q^{63} - 11581 q^{64} - 15380 q^{65} - 5166 q^{66} - 288 q^{67} + 13898 q^{68} - 1632 q^{69} - 5058 q^{70} - 15102 q^{71} - 2016 q^{72} - 146 q^{73} - 3531 q^{74} + 3360 q^{75} + 10770 q^{76} + 3594 q^{77} + 1905 q^{78} + 2852 q^{79} + 27997 q^{80} - 2916 q^{81} + 8824 q^{82} + 5278 q^{83} + 6765 q^{84} - 5290 q^{85} + 5973 q^{86} + 228 q^{87} + 13872 q^{88} + 1462 q^{89} + 171 q^{90} + 74 q^{91} - 3144 q^{92} + 3144 q^{93} + 3262 q^{94} + 1600 q^{95} - 855 q^{96} + 6626 q^{97} + 15491 q^{98} + 144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −5.53116 − 0.528162i −2.52376 − 1.62192i 22.4593 + 4.32868i 0.468128 + 3.25590i 13.1027 + 10.3041i −14.8464 + 20.8489i −79.2899 − 23.2816i 3.73874 + 8.18669i −0.869647 − 18.2561i
4.2 −4.34497 − 0.414894i −2.52376 − 1.62192i 10.8512 + 2.09139i 0.231938 + 1.61316i 10.2927 + 8.09430i 0.778258 − 1.09291i −12.7769 − 3.75164i 3.73874 + 8.18669i −0.338470 − 7.10537i
4.3 −4.30544 − 0.411120i −2.52376 − 1.62192i 10.5124 + 2.02609i 2.02353 + 14.0740i 10.1991 + 8.02066i 12.0391 − 16.9066i −11.2287 − 3.29705i 3.73874 + 8.18669i −2.92611 − 61.4266i
4.4 −4.10740 − 0.392210i −2.52376 − 1.62192i 8.86152 + 1.70792i −1.30857 − 9.10131i 9.72997 + 7.65174i 6.79116 − 9.53685i −4.05636 − 1.19105i 3.73874 + 8.18669i 1.80521 + 37.8960i
4.5 −3.31558 − 0.316600i −2.52376 − 1.62192i 3.03741 + 0.585414i −2.57817 − 17.9316i 7.85423 + 6.17664i −18.7844 + 26.3790i 15.6805 + 4.60422i 3.73874 + 8.18669i 2.87100 + 60.2698i
4.6 −2.29557 − 0.219201i −2.52376 − 1.62192i −2.63381 − 0.507626i 2.82242 + 19.6304i 5.43795 + 4.27645i −16.2624 + 22.8374i 23.6357 + 6.94007i 3.73874 + 8.18669i −2.17608 − 45.6816i
4.7 −1.73979 − 0.166129i −2.52376 − 1.62192i −4.85617 − 0.935951i −1.62576 − 11.3074i 4.12135 + 3.24107i 11.0364 − 15.4984i 21.7085 + 6.37418i 3.73874 + 8.18669i 0.949981 + 19.9426i
4.8 −1.65101 − 0.157652i −2.52376 − 1.62192i −5.15445 − 0.993439i 0.854289 + 5.94171i 3.91105 + 3.07569i 2.57806 − 3.62038i 21.0841 + 6.19086i 3.73874 + 8.18669i −0.473715 − 9.94449i
4.9 0.315936 + 0.0301682i −2.52376 − 1.62192i −7.75652 − 1.49495i 0.375986 + 2.61504i −0.748416 − 0.588561i −0.540924 + 0.759621i −4.84160 − 1.42162i 3.73874 + 8.18669i 0.0398963 + 0.837527i
4.10 0.598865 + 0.0571847i −2.52376 − 1.62192i −7.50006 − 1.44552i −2.98516 − 20.7622i −1.41864 − 1.11563i 11.0160 − 15.4698i −9.02663 − 2.65046i 3.73874 + 8.18669i −0.600425 − 12.6045i
4.11 0.793373 + 0.0757579i −2.52376 − 1.62192i −7.23173 − 1.39380i −1.23463 − 8.58703i −1.87941 − 1.47798i −17.8196 + 25.0241i −11.7495 − 3.44995i 3.73874 + 8.18669i −0.328985 − 6.90625i
4.12 1.55324 + 0.148316i −2.52376 − 1.62192i −5.46489 − 1.05327i 1.80350 + 12.5436i −3.67944 − 2.89354i 18.8939 − 26.5327i −20.3088 − 5.96321i 3.73874 + 8.18669i 0.940843 + 19.7507i
4.13 1.84819 + 0.176481i −2.52376 − 1.62192i −4.47077 − 0.861670i 1.71935 + 11.9584i −4.37815 − 3.44302i 1.67021 − 2.34548i −22.3619 − 6.56604i 3.73874 + 8.18669i 1.06727 + 22.4048i
4.14 3.24171 + 0.309546i −2.52376 − 1.62192i 2.55743 + 0.492904i −1.28041 − 8.90545i −7.67924 − 6.03902i −6.81145 + 9.56535i −16.8585 − 4.95010i 3.73874 + 8.18669i −1.39407 − 29.2652i
4.15 4.06355 + 0.388022i −2.52376 − 1.62192i 8.50643 + 1.63948i 1.57648 + 10.9647i −9.62608 − 7.57003i −13.4370 + 18.8697i 2.59669 + 0.762458i 3.73874 + 8.18669i 2.15158 + 45.1672i
4.16 4.21003 + 0.402009i −2.52376 − 1.62192i 9.70733 + 1.87093i −0.265215 − 1.84461i −9.97308 − 7.84292i 17.7472 − 24.9224i 7.65306 + 2.24714i 3.73874 + 8.18669i −0.375013 − 7.87250i
4.17 5.11914 + 0.488818i −2.52376 − 1.62192i 18.1112 + 3.49064i 1.77055 + 12.3144i −12.1266 − 9.53650i −2.95569 + 4.15068i 51.5344 + 15.1319i 3.73874 + 8.18669i 3.04415 + 63.9046i
4.18 5.26356 + 0.502609i −2.52376 − 1.62192i 19.5970 + 3.77701i −2.69847 − 18.7683i −12.4688 − 9.80555i −5.28570 + 7.42273i 60.6650 + 17.8129i 3.73874 + 8.18669i −4.77045 − 100.144i
10.1 −4.01070 − 3.82419i 0.426945 + 2.96946i 1.08059 + 22.6844i −5.58319 + 12.2255i 9.64345 − 13.5423i −8.14672 + 33.5812i 53.3833 − 61.6077i −8.63544 + 2.53559i 69.1451 − 27.6815i
10.2 −3.39327 − 3.23548i 0.426945 + 2.96946i 0.665321 + 13.9668i −1.46096 + 3.19905i 8.15890 − 11.4576i 3.19725 − 13.1792i 18.3688 − 21.1988i −8.63544 + 2.53559i 15.3079 − 6.12835i
See next 80 embeddings (of 360 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.g even 33 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.m.b ✓ 360
67.g even 33 1 inner 201.4.m.b ✓ 360
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.m.b ✓ 360 1.a even 1 1 trivial
201.4.m.b ✓ 360 67.g even 33 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{360} + 2 T_{2}^{359} - 115 T_{2}^{358} - 315 T_{2}^{357} + 5244 T_{2}^{356} + \cdots + 78\!\cdots\!24 \) acting on \(S_{4}^{\mathrm{new}}(201, [\chi])\). Copy content Toggle raw display