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Properties

Label 333.3.bn.a
Level $333$
Weight $3$
Character orbit 333.bn
Analytic conductor $9.074$
Analytic rank $0$
Dimension $444$
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] = [333,3,Mod(86,333)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(333, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([15, 14]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("333.86");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 333 = 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 333.bn (of order \(18\), degree \(6\), 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: \(9.07359280320\)
Analytic rank: \(0\)
Dimension: \(444\)
Relative dimension: \(74\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 444 q - 9 q^{2} - 12 q^{3} - 3 q^{4} - 9 q^{5} - 12 q^{6} - 6 q^{7} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 444 q - 9 q^{2} - 12 q^{3} - 3 q^{4} - 9 q^{5} - 12 q^{6} - 6 q^{7} + 24 q^{9} - 6 q^{10} - 9 q^{11} + 12 q^{13} - 9 q^{14} + 63 q^{15} - 27 q^{16} + 15 q^{18} - 12 q^{19} - 162 q^{20} - 24 q^{21} - 27 q^{22} - 171 q^{23} + 330 q^{24} - 3 q^{25} + 93 q^{27} + 24 q^{28} - 9 q^{29} - 468 q^{30} - 96 q^{31} - 153 q^{32} - 192 q^{33} - 3 q^{34} - 324 q^{35} - 12 q^{36} + 21 q^{37} - 18 q^{38} + 69 q^{39} + 168 q^{40} - 9 q^{41} - 45 q^{42} - 6 q^{43} - 144 q^{44} + 249 q^{45} + 105 q^{48} - 42 q^{49} - 9 q^{50} - 123 q^{51} - 147 q^{52} + 540 q^{53} - 708 q^{54} + 63 q^{55} - 387 q^{56} - 225 q^{57} - 27 q^{58} - 144 q^{59} - 384 q^{60} - 3 q^{61} + 972 q^{62} - 279 q^{63} + 1434 q^{64} - 9 q^{65} + 111 q^{66} - 447 q^{67} - 369 q^{68} + 63 q^{69} - 300 q^{70} + 567 q^{71} - 321 q^{72} - 24 q^{73} + 423 q^{74} - 96 q^{75} - 27 q^{76} + 855 q^{77} - 111 q^{78} + 48 q^{79} + 480 q^{81} - 6 q^{82} + 666 q^{83} - 888 q^{84} - 6 q^{85} - 9 q^{86} + 615 q^{87} - 774 q^{88} + 1524 q^{90} + 219 q^{91} + 504 q^{92} - 666 q^{93} + 45 q^{94} - 9 q^{95} + 1830 q^{96} - 6 q^{97} - 441 q^{98} - 663 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
86.1 −1.31848 + 3.62250i −1.95451 + 2.27593i −8.31995 − 6.98127i 4.99548 + 0.880837i −5.66758 − 10.0810i −5.43254 − 4.55844i 22.9053 − 13.2244i −1.35974 − 8.89669i −9.77729 + 16.9348i
86.2 −1.31473 + 3.61218i 0.597099 − 2.93998i −8.25519 − 6.92693i −2.93326 − 0.517213i 9.83472 + 6.02210i 6.90707 + 5.79572i 22.5587 − 13.0242i −8.28695 − 3.51092i 5.72471 − 9.91549i
86.3 −1.26967 + 3.48838i 2.90227 + 0.759493i −7.49256 − 6.28700i −3.05473 − 0.538632i −6.33431 + 9.15991i 1.11526 + 0.935816i 18.5849 − 10.7300i 7.84634 + 4.40851i 5.75744 − 9.97218i
86.4 −1.25706 + 3.45374i 1.50356 + 2.59602i −7.28394 − 6.11195i 6.69956 + 1.18131i −10.8560 + 1.92955i 3.81763 + 3.20337i 17.5335 − 10.1230i −4.47862 + 7.80653i −12.5017 + 21.6536i
86.5 −1.25142 + 3.43825i −2.53951 − 1.59715i −7.19136 − 6.03427i −4.75110 − 0.837746i 8.66941 − 6.73277i −8.33595 − 6.99470i 17.0719 − 9.85648i 3.89822 + 8.11196i 8.82601 − 15.2871i
86.6 −1.20891 + 3.32146i 1.92792 − 2.29851i −6.50643 − 5.45954i 6.57435 + 1.15923i 5.30371 + 9.18219i −8.73811 − 7.33215i 13.7551 − 7.94148i −1.56626 − 8.86266i −11.7982 + 20.4350i
86.7 −1.20666 + 3.31527i −2.98068 + 0.339954i −6.47082 − 5.42966i −0.827028 − 0.145827i 2.46962 − 10.2920i 6.18608 + 5.19074i 13.5874 − 7.84470i 8.76886 − 2.02659i 1.48140 − 2.56586i
86.8 −1.19088 + 3.27192i −0.689400 + 2.91971i −6.22307 − 5.22178i −7.61686 − 1.34306i −8.73207 − 5.73269i 2.92709 + 2.45612i 12.4345 − 7.17905i −8.04946 − 4.02570i 13.4651 − 23.3223i
86.9 −1.11395 + 3.06055i 2.73400 − 1.23501i −5.06191 − 4.24744i −7.00508 − 1.23518i 0.734288 + 9.74328i −4.04970 − 3.39810i 7.35575 − 4.24685i 5.94948 − 6.75305i 11.5836 − 20.0635i
86.10 −1.08426 + 2.97899i 2.95994 + 0.488639i −4.63456 − 3.88886i 3.82049 + 0.673655i −4.66500 + 8.28780i 0.241452 + 0.202602i 5.62813 − 3.24940i 8.52246 + 2.89268i −6.14922 + 10.6508i
86.11 −1.03321 + 2.83873i −0.580253 − 2.94335i −3.92669 − 3.29488i 3.21504 + 0.566898i 8.95491 + 1.39393i −0.318514 − 0.267265i 2.94565 − 1.70067i −8.32661 + 3.41577i −4.93109 + 8.54090i
86.12 −1.01780 + 2.79638i −2.98255 + 0.323129i −3.71967 − 3.12117i 4.99582 + 0.880898i 2.13205 − 8.66923i 5.22765 + 4.38652i 2.20524 − 1.27320i 8.79118 − 1.92750i −7.54808 + 13.0737i
86.13 −0.961211 + 2.64090i 1.31409 + 2.69688i −2.98627 − 2.50578i −1.65563 − 0.291932i −8.38532 + 0.878108i −8.10155 − 6.79801i −0.247518 + 0.142905i −5.54635 + 7.08788i 2.36237 − 4.09175i
86.14 −0.955950 + 2.62645i −2.22228 − 2.01531i −2.92023 − 2.45037i 1.37887 + 0.243132i 7.41750 − 3.91019i −0.260707 − 0.218759i −0.454839 + 0.262601i 0.877089 + 8.95716i −1.95671 + 3.38912i
86.15 −0.924319 + 2.53955i −0.934435 + 2.85076i −2.53075 − 2.12355i 4.08474 + 0.720250i −6.37592 − 5.00805i −0.00592698 − 0.00497333i −1.62976 + 0.940944i −7.25366 − 5.32770i −5.60471 + 9.70765i
86.16 −0.922568 + 2.53473i 2.25362 − 1.98020i −2.50957 − 2.10578i 6.91929 + 1.22006i 2.94016 + 7.53920i 8.86570 + 7.43920i −1.69126 + 0.976447i 1.15761 − 8.92524i −9.47604 + 16.4130i
86.17 −0.797793 + 2.19192i −1.93625 − 2.29149i −1.10386 − 0.926245i −9.63279 − 1.69852i 6.56749 − 2.41597i 8.97432 + 7.53035i −5.16943 + 2.98457i −1.50186 + 8.87381i 11.4080 − 19.7592i
86.18 −0.795273 + 2.18500i 2.55083 + 1.57901i −1.07757 − 0.904186i −5.76712 − 1.01690i −5.47873 + 4.31781i 7.21417 + 6.05340i −5.22221 + 3.01504i 4.01348 + 8.05556i 6.80835 − 11.7924i
86.19 −0.758723 + 2.08457i −2.88520 + 0.821957i −0.705607 − 0.592075i −6.74119 − 1.18865i 0.475637 − 6.63805i −5.55578 − 4.66185i −5.91503 + 3.41504i 7.64877 − 4.74302i 7.59253 − 13.1506i
86.20 −0.757552 + 2.08136i −2.99153 + 0.225279i −0.693979 − 0.582318i 9.06413 + 1.59825i 1.79735 − 6.39710i −8.59736 − 7.21404i −5.93502 + 3.42658i 8.89850 − 1.34786i −10.1931 + 17.6549i
See next 80 embeddings (of 444 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 86.74
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
333.bn odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 333.3.bn.a yes 444
9.d odd 6 1 333.3.bk.a ✓ 444
37.f even 9 1 333.3.bk.a ✓ 444
333.bn odd 18 1 inner 333.3.bn.a yes 444
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
333.3.bk.a ✓ 444 9.d odd 6 1
333.3.bk.a ✓ 444 37.f even 9 1
333.3.bn.a yes 444 1.a even 1 1 trivial
333.3.bn.a yes 444 333.bn odd 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(333, [\chi])\).