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Properties

Label 9.8.c.a.7.3
Level $9$
Weight $8$
Character 9.7
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 7.3
Root \(0.500000 + 2.70685i\) of defining polynomial
Character \(\chi\) \(=\) 9.7
Dual form 9.8.c.a.4.3

$q$-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 \(\left(\mathbb{Z}/9\mathbb{Z}\right)^\times\).

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

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 −0.969753 − 0.244087i \(-0.921512\pi\)
0.696262 + 0.717788i \(0.254845\pi\)
\(3\) 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 0.992688 + 0.120710i \(0.0385171\pi\)
−0.391806 + 0.920048i \(0.628150\pi\)
\(6\) 42.9781 + 286.194i 0.0812303 + 0.540917i
\(7\) 442.025 − 765.610i 0.487084 − 0.843654i −0.512806 − 0.858505i \(-0.671394\pi\)
0.999890 + 0.0148506i \(0.00472726\pi\)
\(8\) −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 −0.999661 − 0.0260303i \(-0.991713\pi\)
0.522373 + 0.852717i \(0.325047\pi\)
\(12\) 3904.21 + 1534.73i 0.652227 + 0.256388i
\(13\) −6257.63 − 10838.5i −0.789966 − 1.36826i −0.925987 − 0.377555i \(-0.876765\pi\)
0.136021 − 0.990706i \(-0.456568\pi\)
\(14\) 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 0.604016 − 0.796972i \(-0.293566\pi\)
−0.992206 + 0.124607i \(0.960233\pi\)
\(24\) −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 −0.792658 − 0.609667i \(-0.791303\pi\)
0.924316 + 0.381628i \(0.124637\pi\)
\(30\) −76035.7 + 60569.0i −0.514154 + 0.409568i
\(31\) 138773. + 240361.i 0.836639 + 1.44910i 0.892689 + 0.450674i \(0.148816\pi\)
−0.0560492 + 0.998428i \(0.517850\pi\)
\(32\) −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 0.799070 − 0.601239i \(-0.205326\pi\)
−0.920223 + 0.391395i \(0.871993\pi\)
\(42\) 238110. + 93600.3i 0.495913 + 0.194942i
\(43\) −8512.90 + 14744.8i −0.0163282 + 0.0282812i −0.874074 − 0.485793i \(-0.838531\pi\)
0.857746 + 0.514074i \(0.171864\pi\)
\(44\) −378003. −0.668976
\(45\) 702217. − 215772.i 1.14876 − 0.352981i
\(46\) 280350. 0.424666
\(47\) −675738. + 1.17041e6i −0.949371 + 1.64436i −0.202616 + 0.979258i \(0.564944\pi\)
−0.746754 + 0.665100i \(0.768389\pi\)
\(48\) 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 0.694305 − 0.719680i \(-0.255712\pi\)
−0.970414 + 0.241446i \(0.922378\pi\)
\(60\) 209264. + 1.39350e6i 0.125073 + 0.832869i
\(61\) −487289. + 844009.i −0.274873 + 0.476094i −0.970103 − 0.242693i \(-0.921969\pi\)
0.695230 + 0.718787i \(0.255302\pi\)
\(62\) −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 0.893648 − 0.448769i \(-0.148137\pi\)
−0.835469 + 0.549537i \(0.814804\pi\)
\(68\) −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 −0.971419 − 0.237369i \(-0.923715\pi\)
0.691278 + 0.722589i \(0.257048\pi\)
\(80\) −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 −0.925061 − 0.379819i \(-0.875986\pi\)
0.791463 + 0.611217i \(0.209320\pi\)
\(84\) 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 −0.609012 − 0.793161i \(-0.708434\pi\)
0.991404 + 0.130840i \(0.0417673\pi\)
\(98\) −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.7.3 yes 12
3.2 odd 2 27.8.c.a.19.4 12
4.3 odd 2 144.8.i.c.97.2 12
9.2 odd 6 81.8.a.c.1.3 6
9.4 even 3 inner 9.8.c.a.4.3 ✓ 12
9.5 odd 6 27.8.c.a.10.4 12
9.7 even 3 81.8.a.e.1.4 6
12.11 even 2 432.8.i.c.289.1 12
36.23 even 6 432.8.i.c.145.1 12
36.31 odd 6 144.8.i.c.49.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.3 ✓ 12 9.4 even 3 inner
9.8.c.a.7.3 yes 12 1.1 even 1 trivial
27.8.c.a.10.4 12 9.5 odd 6
27.8.c.a.19.4 12 3.2 odd 2
81.8.a.c.1.3 6 9.2 odd 6
81.8.a.e.1.4 6 9.7 even 3
144.8.i.c.49.2 12 36.31 odd 6
144.8.i.c.97.2 12 4.3 odd 2
432.8.i.c.145.1 12 36.23 even 6
432.8.i.c.289.1 12 12.11 even 2