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Properties

Label 9.8.c.a.4.3
Level $9$
Weight $8$
Character 9.4
Analytic conductor $2.811$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,8,Mod(4,9)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9.4"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 9.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.81146522936\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 375 x^{10} - 1820 x^{9} + 50808 x^{8} - 192378 x^{7} + 3002887 x^{6} + \cdots + 754412211 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{15} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 4.3
Root \(0.500000 - 2.70685i\) of defining polynomial
Character \(\chi\) \(=\) 9.4
Dual form 9.8.c.a.7.3

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.09420 - 5.35931i) q^{2} +(36.5784 + 29.1379i) q^{3} +(44.8519 - 77.6857i) q^{4} +(167.952 - 290.901i) q^{5} +(42.9781 - 286.194i) q^{6} +(442.025 + 765.610i) q^{7} -1347.24 q^{8} +(488.965 + 2131.64i) q^{9} -2078.70 q^{10} +(-2106.95 - 3649.35i) q^{11} +(3904.21 - 1534.73i) q^{12} +(-6257.63 + 10838.5i) q^{13} +(2735.43 - 4737.90i) q^{14} +(14619.6 - 5746.93i) q^{15} +(-1572.42 - 2723.51i) q^{16} -742.627 q^{17} +(9911.16 - 9216.23i) q^{18} +9111.12 q^{19} +(-15065.9 - 26094.9i) q^{20} +(-6139.68 + 40884.5i) q^{21} +(-13038.7 + 22583.6i) q^{22} +(-22651.2 + 39233.1i) q^{23} +(-49279.8 - 39255.7i) q^{24} +(-17352.9 - 30056.2i) q^{25} +77449.4 q^{26} +(-44225.9 + 92219.4i) q^{27} +79302.6 q^{28} +(17291.8 + 29950.3i) q^{29} +(-76035.7 - 60569.0i) q^{30} +(138773. - 240361. i) q^{31} +(-95953.9 + 166197. i) q^{32} +(29265.4 - 194880. i) q^{33} +(2297.84 + 3979.97i) q^{34} +296955. q^{35} +(187529. + 57622.4i) q^{36} -209817. q^{37} +(-28191.6 - 48829.3i) q^{38} +(-544707. + 214122. i) q^{39} +(-226271. + 391912. i) q^{40} +(-53466.0 + 92605.8i) q^{41} +(238110. - 93600.3i) q^{42} +(-8512.90 - 14744.8i) q^{43} -378003. q^{44} +(702217. + 215772. i) q^{45} +280350. q^{46} +(-675738. - 1.17041e6i) q^{47} +(21840.7 - 145439. i) q^{48} +(20999.2 - 36371.6i) q^{49} +(-107387. + 186000. i) q^{50} +(-27164.1 - 21638.6i) q^{51} +(561333. + 972257. i) q^{52} +1.83419e6 q^{53} +(631076. - 48324.8i) q^{54} -1.41546e6 q^{55} +(-595513. - 1.03146e6i) q^{56} +(333270. + 265479. i) q^{57} +(107009. - 185345. i) q^{58} +(-435574. + 754437. i) q^{59} +(209264. - 1.39350e6i) q^{60} +(-487289. - 844009. i) q^{61} -1.71756e6 q^{62} +(-1.41587e6 + 1.31659e6i) q^{63} +785063. q^{64} +(2.10196e6 + 3.64070e6i) q^{65} +(-1.13497e6 + 446154. i) q^{66} +(143227. - 248076. i) q^{67} +(-33308.2 + 57691.5i) q^{68} +(-1.97172e6 + 775076. i) q^{69} +(-918838. - 1.59147e6i) q^{70} +967923. q^{71} +(-658751. - 2.87182e6i) q^{72} +4.50531e6 q^{73} +(649216. + 1.12448e6i) q^{74} +(241031. - 1.60504e6i) q^{75} +(408651. - 707804. i) q^{76} +(1.86265e6 - 3.22621e6i) q^{77} +(2.83298e6 + 2.25671e6i) q^{78} +(-1.22765e6 - 2.12634e6i) q^{79} -1.05636e6 q^{80} +(-4.30480e6 + 2.08459e6i) q^{81} +661738. q^{82} +(-695940. - 1.20540e6i) q^{83} +(2.90077e6 + 2.31071e6i) q^{84} +(-124725. + 216031. i) q^{85} +(-52681.2 + 91246.5i) q^{86} +(-240182. + 1.59938e6i) q^{87} +(2.83856e6 + 4.91654e6i) q^{88} -7.88308e6 q^{89} +(-1.01641e6 - 4.43104e6i) q^{90} -1.10641e7 q^{91} +(2.03190e6 + 3.51936e6i) q^{92} +(1.20797e7 - 4.74850e6i) q^{93} +(-4.18174e6 + 7.24298e6i) q^{94} +(1.53023e6 - 2.65043e6i) q^{95} +(-8.35248e6 + 3.28333e6i) q^{96} +(3.43723e6 + 5.95346e6i) q^{97} -259902. q^{98} +(6.74887e6 - 6.27566e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 9 q^{2} + 24 q^{3} - 321 q^{4} - 180 q^{5} - 1233 q^{6} - 84 q^{7} + 5922 q^{8} + 990 q^{9} + 252 q^{10} - 8460 q^{11} + 8052 q^{12} - 1848 q^{13} - 16272 q^{14} - 1188 q^{15} - 12417 q^{16} + 30564 q^{17}+ \cdots - 49382676 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.09420 − 5.35931i −0.273491 − 0.473701i 0.696262 − 0.717788i \(-0.254845\pi\)
−0.969753 + 0.244087i \(0.921512\pi\)
\(3\) 36.5784 + 29.1379i 0.782169 + 0.623066i
\(4\) 44.8519 − 77.6857i 0.350405 − 0.606920i
\(5\) 167.952 − 290.901i 0.600882 − 1.04076i −0.391806 − 0.920048i \(-0.628150\pi\)
0.992688 − 0.120710i \(-0.0385171\pi\)
\(6\) 42.9781 − 286.194i 0.0812303 − 0.540917i
\(7\) 442.025 + 765.610i 0.487084 + 0.843654i 0.999890 − 0.0148506i \(-0.00472726\pi\)
−0.512806 + 0.858505i \(0.671394\pi\)
\(8\) −1347.24 −0.930313
\(9\) 488.965 + 2131.64i 0.223578 + 0.974686i
\(10\) −2078.70 −0.657343
\(11\) −2106.95 − 3649.35i −0.477288 − 0.826687i 0.522373 − 0.852717i \(-0.325047\pi\)
−0.999661 + 0.0260303i \(0.991713\pi\)
\(12\) 3904.21 − 1534.73i 0.652227 − 0.256388i
\(13\) −6257.63 + 10838.5i −0.789966 + 1.36826i 0.136021 + 0.990706i \(0.456568\pi\)
−0.925987 + 0.377555i \(0.876765\pi\)
\(14\) 2735.43 − 4737.90i 0.266426 − 0.461464i
\(15\) 14619.6 − 5746.93i 1.11845 − 0.439660i
\(16\) −1572.42 − 2723.51i −0.0959728 − 0.166230i
\(17\) −742.627 −0.0366606 −0.0183303 − 0.999832i \(-0.505835\pi\)
−0.0183303 + 0.999832i \(0.505835\pi\)
\(18\) 9911.16 − 9216.23i 0.400563 − 0.372477i
\(19\) 9111.12 0.304743 0.152372 − 0.988323i \(-0.451309\pi\)
0.152372 + 0.988323i \(0.451309\pi\)
\(20\) −15065.9 − 26094.9i −0.421104 − 0.729374i
\(21\) −6139.68 + 40884.5i −0.144670 + 0.963366i
\(22\) −13038.7 + 22583.6i −0.261068 + 0.452183i
\(23\) −22651.2 + 39233.1i −0.388190 + 0.672365i −0.992206 − 0.124607i \(-0.960233\pi\)
0.604016 + 0.796972i \(0.293566\pi\)
\(24\) −49279.8 − 39255.7i −0.727662 − 0.579646i
\(25\) −17352.9 − 30056.2i −0.222118 − 0.384719i
\(26\) 77449.4 0.864195
\(27\) −44225.9 + 92219.4i −0.432418 + 0.901673i
\(28\) 79302.6 0.682707
\(29\) 17291.8 + 29950.3i 0.131658 + 0.228039i 0.924316 − 0.381628i \(-0.124637\pi\)
−0.792658 + 0.609667i \(0.791303\pi\)
\(30\) −76035.7 − 60569.0i −0.514154 − 0.409568i
\(31\) 138773. − 240361.i 0.836639 − 1.44910i −0.0560492 − 0.998428i \(-0.517850\pi\)
0.892689 − 0.450674i \(-0.148816\pi\)
\(32\) −95953.9 + 166197.i −0.517652 + 0.896600i
\(33\) 29265.4 − 194880.i 0.141760 − 0.943991i
\(34\) 2297.84 + 3979.97i 0.0100263 + 0.0173661i
\(35\) 296955. 1.17072
\(36\) 187529. + 57622.4i 0.669899 + 0.205841i
\(37\) −209817. −0.680981 −0.340491 − 0.940248i \(-0.610593\pi\)
−0.340491 + 0.940248i \(0.610593\pi\)
\(38\) −28191.6 − 48829.3i −0.0833446 − 0.144357i
\(39\) −544707. + 214122.i −1.47040 + 0.578011i
\(40\) −226271. + 391912.i −0.559008 + 0.968231i
\(41\) −53466.0 + 92605.8i −0.121153 + 0.209843i −0.920223 − 0.391395i \(-0.871993\pi\)
0.799070 + 0.601239i \(0.205326\pi\)
\(42\) 238110. − 93600.3i 0.495913 − 0.194942i
\(43\) −8512.90 − 14744.8i −0.0163282 − 0.0282812i 0.857746 − 0.514074i \(-0.171864\pi\)
−0.874074 + 0.485793i \(0.838531\pi\)
\(44\) −378003. −0.668976
\(45\) 702217. + 215772.i 1.14876 + 0.352981i
\(46\) 280350. 0.424666
\(47\) −675738. − 1.17041e6i −0.949371 − 1.64436i −0.746754 − 0.665100i \(-0.768389\pi\)
−0.202616 − 0.979258i \(-0.564944\pi\)
\(48\) 21840.7 − 145439.i 0.0285051 − 0.189817i
\(49\) 20999.2 − 36371.6i 0.0254986 − 0.0441648i
\(50\) −107387. + 186000.i −0.121494 + 0.210435i
\(51\) −27164.1 − 21638.6i −0.0286748 − 0.0228420i
\(52\) 561333. + 972257.i 0.553616 + 0.958892i
\(53\) 1.83419e6 1.69230 0.846152 − 0.532942i \(-0.178914\pi\)
0.846152 + 0.532942i \(0.178914\pi\)
\(54\) 631076. − 48324.8i 0.545386 − 0.0417630i
\(55\) −1.41546e6 −1.14717
\(56\) −595513. − 1.03146e6i −0.453141 − 0.784862i
\(57\) 333270. + 265479.i 0.238361 + 0.189875i
\(58\) 107009. − 185345.i 0.0720147 − 0.124733i
\(59\) −435574. + 754437.i −0.276109 + 0.478235i −0.970414 − 0.241446i \(-0.922378\pi\)
0.694305 + 0.719680i \(0.255712\pi\)
\(60\) 209264. − 1.39350e6i 0.125073 − 0.832869i
\(61\) −487289. − 844009.i −0.274873 − 0.476094i 0.695230 − 0.718787i \(-0.255302\pi\)
−0.970103 + 0.242693i \(0.921969\pi\)
\(62\) −1.71756e6 −0.915254
\(63\) −1.41587e6 + 1.31659e6i −0.713397 + 0.663376i
\(64\) 785063. 0.374347
\(65\) 2.10196e6 + 3.64070e6i 0.949352 + 1.64433i
\(66\) −1.13497e6 + 446154.i −0.485939 + 0.191021i
\(67\) 143227. − 248076.i 0.0581785 − 0.100768i −0.835469 − 0.549537i \(-0.814804\pi\)
0.893648 + 0.448769i \(0.148137\pi\)
\(68\) −33308.2 + 57691.5i −0.0128461 + 0.0222500i
\(69\) −1.97172e6 + 775076.i −0.722558 + 0.284035i
\(70\) −918838. − 1.59147e6i −0.320181 − 0.554570i
\(71\) 967923. 0.320950 0.160475 − 0.987040i \(-0.448697\pi\)
0.160475 + 0.987040i \(0.448697\pi\)
\(72\) −658751. − 2.87182e6i −0.207997 − 0.906763i
\(73\) 4.50531e6 1.35548 0.677742 − 0.735299i \(-0.262958\pi\)
0.677742 + 0.735299i \(0.262958\pi\)
\(74\) 649216. + 1.12448e6i 0.186242 + 0.322581i
\(75\) 241031. − 1.60504e6i 0.0659717 − 0.439309i
\(76\) 408651. − 707804.i 0.106784 − 0.184955i
\(77\) 1.86265e6 − 3.22621e6i 0.464958 − 0.805331i
\(78\) 2.83298e6 + 2.25671e6i 0.675947 + 0.538450i
\(79\) −1.22765e6 − 2.12634e6i −0.280142 − 0.485220i 0.691278 − 0.722589i \(-0.257048\pi\)
−0.971419 + 0.237369i \(0.923715\pi\)
\(80\) −1.05636e6 −0.230673
\(81\) −4.30480e6 + 2.08459e6i −0.900026 + 0.435836i
\(82\) 661738. 0.132537
\(83\) −695940. − 1.20540e6i −0.133598 − 0.231398i 0.791463 − 0.611217i \(-0.209320\pi\)
−0.925061 + 0.379819i \(0.875986\pi\)
\(84\) 2.90077e6 + 2.31071e6i 0.533992 + 0.425371i
\(85\) −124725. + 216031.i −0.0220287 + 0.0381548i
\(86\) −52681.2 + 91246.5i −0.00893123 + 0.0154693i
\(87\) −240182. + 1.59938e6i −0.0391041 + 0.260397i
\(88\) 2.83856e6 + 4.91654e6i 0.444027 + 0.769077i
\(89\) −7.88308e6 −1.18531 −0.592653 − 0.805458i \(-0.701920\pi\)
−0.592653 + 0.805458i \(0.701920\pi\)
\(90\) −1.01641e6 − 4.43104e6i −0.146967 − 0.640703i
\(91\) −1.10641e7 −1.53912
\(92\) 2.03190e6 + 3.51936e6i 0.272048 + 0.471201i
\(93\) 1.20797e7 − 4.74850e6i 1.55728 − 0.612162i
\(94\) −4.18174e6 + 7.24298e6i −0.519289 + 0.899435i
\(95\) 1.53023e6 − 2.65043e6i 0.183115 − 0.317164i
\(96\) −8.35248e6 + 3.28333e6i −0.963532 + 0.378761i
\(97\) 3.43723e6 + 5.95346e6i 0.382391 + 0.662321i 0.991404 − 0.130840i \(-0.0417673\pi\)
−0.609012 + 0.793161i \(0.708434\pi\)
\(98\) −259902. −0.0278945
\(99\) 6.74887e6 − 6.27566e6i 0.699049 − 0.650034i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9.8.c.a.4.3 ✓ 12
3.2 odd 2 27.8.c.a.10.4 12
4.3 odd 2 144.8.i.c.49.2 12
9.2 odd 6 27.8.c.a.19.4 12
9.4 even 3 81.8.a.e.1.4 6
9.5 odd 6 81.8.a.c.1.3 6
9.7 even 3 inner 9.8.c.a.7.3 yes 12
12.11 even 2 432.8.i.c.145.1 12
36.7 odd 6 144.8.i.c.97.2 12
36.11 even 6 432.8.i.c.289.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.3 ✓ 12 1.1 even 1 trivial
9.8.c.a.7.3 yes 12 9.7 even 3 inner
27.8.c.a.10.4 12 3.2 odd 2
27.8.c.a.19.4 12 9.2 odd 6
81.8.a.c.1.3 6 9.5 odd 6
81.8.a.e.1.4 6 9.4 even 3
144.8.i.c.49.2 12 4.3 odd 2
144.8.i.c.97.2 12 36.7 odd 6
432.8.i.c.145.1 12 12.11 even 2
432.8.i.c.289.1 12 36.11 even 6