Properties

Label 432.3.bc.b.113.6
Level $432$
Weight $3$
Character 432.113
Analytic conductor $11.771$
Analytic rank $0$
Dimension $36$
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] = [432,3,Mod(65,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 13]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.65");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 432.bc (of order \(18\), degree \(6\), not 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: \(11.7711474204\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 113.6
Character \(\chi\) \(=\) 432.113
Dual form 432.3.bc.b.65.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.92720 + 0.656882i) q^{3} +(0.740753 + 2.03520i) q^{5} +(-1.08248 - 6.13906i) q^{7} +(8.13701 + 3.84565i) q^{9} +O(q^{10})\) \(q+(2.92720 + 0.656882i) q^{3} +(0.740753 + 2.03520i) q^{5} +(-1.08248 - 6.13906i) q^{7} +(8.13701 + 3.84565i) q^{9} +(5.10694 - 14.0312i) q^{11} +(9.95988 + 8.35733i) q^{13} +(0.831446 + 6.44403i) q^{15} +(-3.36308 - 1.94167i) q^{17} +(-6.39866 - 11.0828i) q^{19} +(0.863993 - 18.6813i) q^{21} +(35.3663 + 6.23604i) q^{23} +(15.5578 - 13.0545i) q^{25} +(21.2925 + 16.6020i) q^{27} +(7.13197 + 8.49955i) q^{29} +(-6.25720 + 35.4864i) q^{31} +(24.1659 - 37.7175i) q^{33} +(11.6924 - 6.75059i) q^{35} +(-16.3431 + 28.3071i) q^{37} +(23.6648 + 31.0060i) q^{39} +(-14.5164 + 17.3000i) q^{41} +(-58.2594 - 21.2047i) q^{43} +(-1.79915 + 19.4091i) q^{45} +(3.36255 - 0.592908i) q^{47} +(9.52866 - 3.46815i) q^{49} +(-8.56895 - 7.89281i) q^{51} -79.3144i q^{53} +32.3393 q^{55} +(-11.4501 - 36.6448i) q^{57} +(-7.36318 - 20.2302i) q^{59} +(13.7025 + 77.7109i) q^{61} +(14.8005 - 54.1165i) q^{63} +(-9.63104 + 26.4611i) q^{65} +(84.5379 + 70.9357i) q^{67} +(99.4280 + 41.4856i) q^{69} +(-71.2183 - 41.1179i) q^{71} +(-3.09644 - 5.36320i) q^{73} +(54.1160 - 27.9936i) q^{75} +(-91.6666 - 16.1633i) q^{77} +(-49.2237 + 41.3036i) q^{79} +(51.4220 + 62.5842i) q^{81} +(-97.6080 - 116.325i) q^{83} +(1.46049 - 8.28284i) q^{85} +(15.2935 + 29.5648i) q^{87} +(97.5489 - 56.3199i) q^{89} +(40.5248 - 70.1909i) q^{91} +(-41.6264 + 99.7655i) q^{93} +(17.8159 - 21.2322i) q^{95} +(-52.0684 - 18.9513i) q^{97} +(95.5143 - 94.5326i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q - 9 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 36 q - 9 q^{5} + 6 q^{9} - 36 q^{11} - 45 q^{15} + 42 q^{21} + 18 q^{23} - 9 q^{25} - 18 q^{29} - 45 q^{31} - 153 q^{33} + 243 q^{35} + 123 q^{39} - 198 q^{41} - 90 q^{43} - 333 q^{45} + 243 q^{47} + 72 q^{49} + 99 q^{51} + 243 q^{57} - 252 q^{59} - 144 q^{61} - 381 q^{63} + 747 q^{65} - 108 q^{67} + 585 q^{69} - 324 q^{71} - 63 q^{73} - 597 q^{75} + 495 q^{77} - 36 q^{79} - 54 q^{81} + 27 q^{83} - 180 q^{85} + 441 q^{87} - 567 q^{89} - 99 q^{91} - 699 q^{93} + 1044 q^{95} - 216 q^{97} + 945 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{5}{18}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.92720 + 0.656882i 0.975734 + 0.218961i
\(4\) 0 0
\(5\) 0.740753 + 2.03520i 0.148151 + 0.407040i 0.991464 0.130382i \(-0.0416204\pi\)
−0.843313 + 0.537422i \(0.819398\pi\)
\(6\) 0 0
\(7\) −1.08248 6.13906i −0.154640 0.877008i −0.959114 0.283020i \(-0.908664\pi\)
0.804474 0.593988i \(-0.202447\pi\)
\(8\) 0 0
\(9\) 8.13701 + 3.84565i 0.904113 + 0.427294i
\(10\) 0 0
\(11\) 5.10694 14.0312i 0.464267 1.27556i −0.457979 0.888963i \(-0.651427\pi\)
0.922247 0.386601i \(-0.126351\pi\)
\(12\) 0 0
\(13\) 9.95988 + 8.35733i 0.766144 + 0.642871i 0.939718 0.341949i \(-0.111087\pi\)
−0.173574 + 0.984821i \(0.555532\pi\)
\(14\) 0 0
\(15\) 0.831446 + 6.44403i 0.0554297 + 0.429602i
\(16\) 0 0
\(17\) −3.36308 1.94167i −0.197828 0.114216i 0.397814 0.917466i \(-0.369769\pi\)
−0.595642 + 0.803250i \(0.703102\pi\)
\(18\) 0 0
\(19\) −6.39866 11.0828i −0.336772 0.583306i 0.647052 0.762446i \(-0.276002\pi\)
−0.983824 + 0.179140i \(0.942668\pi\)
\(20\) 0 0
\(21\) 0.863993 18.6813i 0.0411425 0.889587i
\(22\) 0 0
\(23\) 35.3663 + 6.23604i 1.53767 + 0.271132i 0.877348 0.479854i \(-0.159310\pi\)
0.660318 + 0.750986i \(0.270422\pi\)
\(24\) 0 0
\(25\) 15.5578 13.0545i 0.622311 0.522181i
\(26\) 0 0
\(27\) 21.2925 + 16.6020i 0.788613 + 0.614890i
\(28\) 0 0
\(29\) 7.13197 + 8.49955i 0.245930 + 0.293088i 0.874862 0.484372i \(-0.160952\pi\)
−0.628932 + 0.777460i \(0.716508\pi\)
\(30\) 0 0
\(31\) −6.25720 + 35.4864i −0.201845 + 1.14472i 0.700482 + 0.713670i \(0.252968\pi\)
−0.902327 + 0.431052i \(0.858143\pi\)
\(32\) 0 0
\(33\) 24.1659 37.7175i 0.732300 1.14295i
\(34\) 0 0
\(35\) 11.6924 6.75059i 0.334068 0.192874i
\(36\) 0 0
\(37\) −16.3431 + 28.3071i −0.441705 + 0.765056i −0.997816 0.0660523i \(-0.978960\pi\)
0.556111 + 0.831108i \(0.312293\pi\)
\(38\) 0 0
\(39\) 23.6648 + 31.0060i 0.606789 + 0.795027i
\(40\) 0 0
\(41\) −14.5164 + 17.3000i −0.354059 + 0.421951i −0.913449 0.406954i \(-0.866591\pi\)
0.559390 + 0.828905i \(0.311036\pi\)
\(42\) 0 0
\(43\) −58.2594 21.2047i −1.35487 0.493133i −0.440406 0.897799i \(-0.645166\pi\)
−0.914465 + 0.404666i \(0.867388\pi\)
\(44\) 0 0
\(45\) −1.79915 + 19.4091i −0.0399812 + 0.431314i
\(46\) 0 0
\(47\) 3.36255 0.592908i 0.0715435 0.0126151i −0.137762 0.990465i \(-0.543991\pi\)
0.209305 + 0.977850i \(0.432880\pi\)
\(48\) 0 0
\(49\) 9.52866 3.46815i 0.194462 0.0707785i
\(50\) 0 0
\(51\) −8.56895 7.89281i −0.168019 0.154761i
\(52\) 0 0
\(53\) 79.3144i 1.49650i −0.663418 0.748249i \(-0.730895\pi\)
0.663418 0.748249i \(-0.269105\pi\)
\(54\) 0 0
\(55\) 32.3393 0.587987
\(56\) 0 0
\(57\) −11.4501 36.6448i −0.200879 0.642891i
\(58\) 0 0
\(59\) −7.36318 20.2302i −0.124800 0.342884i 0.861521 0.507722i \(-0.169512\pi\)
−0.986321 + 0.164838i \(0.947290\pi\)
\(60\) 0 0
\(61\) 13.7025 + 77.7109i 0.224632 + 1.27395i 0.863388 + 0.504540i \(0.168338\pi\)
−0.638757 + 0.769409i \(0.720551\pi\)
\(62\) 0 0
\(63\) 14.8005 54.1165i 0.234929 0.858991i
\(64\) 0 0
\(65\) −9.63104 + 26.4611i −0.148170 + 0.407093i
\(66\) 0 0
\(67\) 84.5379 + 70.9357i 1.26176 + 1.05874i 0.995492 + 0.0948429i \(0.0302349\pi\)
0.266267 + 0.963899i \(0.414210\pi\)
\(68\) 0 0
\(69\) 99.4280 + 41.4856i 1.44099 + 0.601241i
\(70\) 0 0
\(71\) −71.2183 41.1179i −1.00307 0.579125i −0.0939180 0.995580i \(-0.529939\pi\)
−0.909157 + 0.416455i \(0.863272\pi\)
\(72\) 0 0
\(73\) −3.09644 5.36320i −0.0424170 0.0734685i 0.844037 0.536284i \(-0.180172\pi\)
−0.886454 + 0.462816i \(0.846839\pi\)
\(74\) 0 0
\(75\) 54.1160 27.9936i 0.721547 0.373248i
\(76\) 0 0
\(77\) −91.6666 16.1633i −1.19048 0.209913i
\(78\) 0 0
\(79\) −49.2237 + 41.3036i −0.623085 + 0.522830i −0.898772 0.438417i \(-0.855539\pi\)
0.275687 + 0.961248i \(0.411095\pi\)
\(80\) 0 0
\(81\) 51.4220 + 62.5842i 0.634839 + 0.772644i
\(82\) 0 0
\(83\) −97.6080 116.325i −1.17600 1.40150i −0.897470 0.441075i \(-0.854597\pi\)
−0.278530 0.960428i \(-0.589847\pi\)
\(84\) 0 0
\(85\) 1.46049 8.28284i 0.0171822 0.0974451i
\(86\) 0 0
\(87\) 15.2935 + 29.5648i 0.175788 + 0.339825i
\(88\) 0 0
\(89\) 97.5489 56.3199i 1.09606 0.632808i 0.160873 0.986975i \(-0.448569\pi\)
0.935182 + 0.354167i \(0.115236\pi\)
\(90\) 0 0
\(91\) 40.5248 70.1909i 0.445327 0.771329i
\(92\) 0 0
\(93\) −41.6264 + 99.7655i −0.447596 + 1.07275i
\(94\) 0 0
\(95\) 17.8159 21.2322i 0.187536 0.223497i
\(96\) 0 0
\(97\) −52.0684 18.9513i −0.536787 0.195375i 0.0593790 0.998236i \(-0.481088\pi\)
−0.596166 + 0.802861i \(0.703310\pi\)
\(98\) 0 0
\(99\) 95.5143 94.5326i 0.964791 0.954875i
\(100\) 0 0
\(101\) 77.5927 13.6817i 0.768245 0.135462i 0.224228 0.974537i \(-0.428014\pi\)
0.544017 + 0.839074i \(0.316903\pi\)
\(102\) 0 0
\(103\) −49.2174 + 17.9137i −0.477839 + 0.173919i −0.569700 0.821853i \(-0.692941\pi\)
0.0918613 + 0.995772i \(0.470718\pi\)
\(104\) 0 0
\(105\) 38.6603 12.0798i 0.368193 0.115046i
\(106\) 0 0
\(107\) 189.920i 1.77495i 0.460854 + 0.887476i \(0.347543\pi\)
−0.460854 + 0.887476i \(0.652457\pi\)
\(108\) 0 0
\(109\) −161.394 −1.48068 −0.740338 0.672234i \(-0.765335\pi\)
−0.740338 + 0.672234i \(0.765335\pi\)
\(110\) 0 0
\(111\) −66.4339 + 72.1250i −0.598504 + 0.649775i
\(112\) 0 0
\(113\) −71.3701 196.088i −0.631594 1.73529i −0.676649 0.736306i \(-0.736568\pi\)
0.0450545 0.998985i \(-0.485654\pi\)
\(114\) 0 0
\(115\) 13.5061 + 76.5969i 0.117444 + 0.666060i
\(116\) 0 0
\(117\) 48.9043 + 106.306i 0.417985 + 0.908597i
\(118\) 0 0
\(119\) −8.27958 + 22.7480i −0.0695763 + 0.191159i
\(120\) 0 0
\(121\) −78.1025 65.5358i −0.645475 0.541618i
\(122\) 0 0
\(123\) −53.8565 + 41.1050i −0.437858 + 0.334187i
\(124\) 0 0
\(125\) 84.9844 + 49.0657i 0.679875 + 0.392526i
\(126\) 0 0
\(127\) 28.4088 + 49.2055i 0.223691 + 0.387445i 0.955926 0.293608i \(-0.0948559\pi\)
−0.732235 + 0.681052i \(0.761523\pi\)
\(128\) 0 0
\(129\) −156.608 100.340i −1.21402 0.777829i
\(130\) 0 0
\(131\) −136.396 24.0503i −1.04119 0.183590i −0.373194 0.927753i \(-0.621737\pi\)
−0.667998 + 0.744163i \(0.732849\pi\)
\(132\) 0 0
\(133\) −61.1116 + 51.2787i −0.459486 + 0.385554i
\(134\) 0 0
\(135\) −18.0160 + 55.6326i −0.133452 + 0.412093i
\(136\) 0 0
\(137\) −34.3399 40.9247i −0.250656 0.298720i 0.626014 0.779811i \(-0.284685\pi\)
−0.876671 + 0.481091i \(0.840241\pi\)
\(138\) 0 0
\(139\) −34.7093 + 196.846i −0.249707 + 1.41616i 0.559595 + 0.828767i \(0.310957\pi\)
−0.809302 + 0.587393i \(0.800154\pi\)
\(140\) 0 0
\(141\) 10.2323 + 0.473235i 0.0725696 + 0.00335627i
\(142\) 0 0
\(143\) 168.128 97.0687i 1.17572 0.678802i
\(144\) 0 0
\(145\) −12.0153 + 20.8111i −0.0828639 + 0.143525i
\(146\) 0 0
\(147\) 30.1705 3.89277i 0.205241 0.0264814i
\(148\) 0 0
\(149\) −43.0873 + 51.3494i −0.289176 + 0.344627i −0.891001 0.454001i \(-0.849996\pi\)
0.601825 + 0.798628i \(0.294441\pi\)
\(150\) 0 0
\(151\) −209.460 76.2371i −1.38715 0.504881i −0.462812 0.886456i \(-0.653160\pi\)
−0.924338 + 0.381575i \(0.875382\pi\)
\(152\) 0 0
\(153\) −19.8984 28.7326i −0.130055 0.187795i
\(154\) 0 0
\(155\) −76.8569 + 13.5520i −0.495851 + 0.0874320i
\(156\) 0 0
\(157\) −113.832 + 41.4314i −0.725043 + 0.263894i −0.678065 0.735002i \(-0.737181\pi\)
−0.0469779 + 0.998896i \(0.514959\pi\)
\(158\) 0 0
\(159\) 52.1002 232.169i 0.327674 1.46018i
\(160\) 0 0
\(161\) 223.866i 1.39047i
\(162\) 0 0
\(163\) −70.9962 −0.435560 −0.217780 0.975998i \(-0.569881\pi\)
−0.217780 + 0.975998i \(0.569881\pi\)
\(164\) 0 0
\(165\) 94.6637 + 21.2431i 0.573719 + 0.128746i
\(166\) 0 0
\(167\) −57.0403 156.717i −0.341559 0.938425i −0.984943 0.172882i \(-0.944692\pi\)
0.643384 0.765544i \(-0.277530\pi\)
\(168\) 0 0
\(169\) 0.00765212 + 0.0433973i 4.52788e−5 + 0.000256789i
\(170\) 0 0
\(171\) −9.44541 114.788i −0.0552363 0.671275i
\(172\) 0 0
\(173\) −78.3577 + 215.286i −0.452935 + 1.24443i 0.477714 + 0.878515i \(0.341465\pi\)
−0.930649 + 0.365913i \(0.880757\pi\)
\(174\) 0 0
\(175\) −96.9835 81.3789i −0.554192 0.465022i
\(176\) 0 0
\(177\) −8.26468 64.0545i −0.0466931 0.361890i
\(178\) 0 0
\(179\) −129.909 75.0033i −0.725751 0.419013i 0.0911147 0.995840i \(-0.470957\pi\)
−0.816866 + 0.576828i \(0.804290\pi\)
\(180\) 0 0
\(181\) 74.7777 + 129.519i 0.413137 + 0.715574i 0.995231 0.0975476i \(-0.0310998\pi\)
−0.582094 + 0.813121i \(0.697767\pi\)
\(182\) 0 0
\(183\) −10.9368 + 236.476i −0.0597639 + 1.29222i
\(184\) 0 0
\(185\) −69.7168 12.2929i −0.376847 0.0664483i
\(186\) 0 0
\(187\) −44.4191 + 37.2720i −0.237535 + 0.199316i
\(188\) 0 0
\(189\) 78.8721 148.688i 0.417313 0.786707i
\(190\) 0 0
\(191\) 48.7526 + 58.1011i 0.255249 + 0.304194i 0.878418 0.477893i \(-0.158599\pi\)
−0.623169 + 0.782087i \(0.714155\pi\)
\(192\) 0 0
\(193\) −23.6459 + 134.103i −0.122518 + 0.694833i 0.860234 + 0.509900i \(0.170318\pi\)
−0.982751 + 0.184932i \(0.940793\pi\)
\(194\) 0 0
\(195\) −45.5738 + 71.1304i −0.233712 + 0.364771i
\(196\) 0 0
\(197\) −285.910 + 165.070i −1.45132 + 0.837919i −0.998556 0.0537122i \(-0.982895\pi\)
−0.452762 + 0.891631i \(0.649561\pi\)
\(198\) 0 0
\(199\) 29.8782 51.7505i 0.150141 0.260053i −0.781138 0.624359i \(-0.785360\pi\)
0.931279 + 0.364306i \(0.118694\pi\)
\(200\) 0 0
\(201\) 200.863 + 263.175i 0.999319 + 1.30933i
\(202\) 0 0
\(203\) 44.4590 52.9842i 0.219010 0.261006i
\(204\) 0 0
\(205\) −45.9621 16.7288i −0.224205 0.0816040i
\(206\) 0 0
\(207\) 263.795 + 186.749i 1.27437 + 0.902170i
\(208\) 0 0
\(209\) −188.183 + 33.1817i −0.900396 + 0.158764i
\(210\) 0 0
\(211\) −10.6114 + 3.86224i −0.0502911 + 0.0183045i −0.367043 0.930204i \(-0.619630\pi\)
0.316752 + 0.948508i \(0.397408\pi\)
\(212\) 0 0
\(213\) −181.461 167.142i −0.851928 0.784706i
\(214\) 0 0
\(215\) 134.277i 0.624545i
\(216\) 0 0
\(217\) 224.626 1.03514
\(218\) 0 0
\(219\) −5.54093 17.7332i −0.0253010 0.0809733i
\(220\) 0 0
\(221\) −17.2686 47.4452i −0.0781386 0.214684i
\(222\) 0 0
\(223\) −19.2101 108.946i −0.0861441 0.488547i −0.997104 0.0760521i \(-0.975768\pi\)
0.910960 0.412495i \(-0.135343\pi\)
\(224\) 0 0
\(225\) 176.797 46.3951i 0.785764 0.206200i
\(226\) 0 0
\(227\) −112.933 + 310.280i −0.497500 + 1.36687i 0.396182 + 0.918172i \(0.370335\pi\)
−0.893683 + 0.448699i \(0.851887\pi\)
\(228\) 0 0
\(229\) 346.931 + 291.110i 1.51498 + 1.27122i 0.853273 + 0.521464i \(0.174614\pi\)
0.661711 + 0.749759i \(0.269831\pi\)
\(230\) 0 0
\(231\) −257.709 107.527i −1.11562 0.465486i
\(232\) 0 0
\(233\) 312.892 + 180.648i 1.34289 + 0.775315i 0.987230 0.159302i \(-0.0509243\pi\)
0.355655 + 0.934617i \(0.384258\pi\)
\(234\) 0 0
\(235\) 3.69750 + 6.40426i 0.0157340 + 0.0272522i
\(236\) 0 0
\(237\) −171.219 + 88.5698i −0.722444 + 0.373712i
\(238\) 0 0
\(239\) 125.627 + 22.1514i 0.525636 + 0.0926838i 0.430168 0.902749i \(-0.358454\pi\)
0.0954677 + 0.995433i \(0.469565\pi\)
\(240\) 0 0
\(241\) −100.014 + 83.9215i −0.414995 + 0.348222i −0.826255 0.563296i \(-0.809533\pi\)
0.411260 + 0.911518i \(0.365089\pi\)
\(242\) 0 0
\(243\) 109.412 + 216.975i 0.450255 + 0.892900i
\(244\) 0 0
\(245\) 14.1168 + 16.8237i 0.0576194 + 0.0686681i
\(246\) 0 0
\(247\) 28.8928 163.859i 0.116975 0.663397i
\(248\) 0 0
\(249\) −209.307 404.623i −0.840589 1.62499i
\(250\) 0 0
\(251\) 376.247 217.226i 1.49899 0.865444i 0.498994 0.866606i \(-0.333703\pi\)
0.999999 + 0.00116146i \(0.000369704\pi\)
\(252\) 0 0
\(253\) 268.113 464.385i 1.05973 1.83551i
\(254\) 0 0
\(255\) 9.71598 23.2862i 0.0381019 0.0913183i
\(256\) 0 0
\(257\) 104.428 124.452i 0.406334 0.484250i −0.523606 0.851960i \(-0.675414\pi\)
0.929941 + 0.367710i \(0.119858\pi\)
\(258\) 0 0
\(259\) 191.470 + 69.6893i 0.739266 + 0.269071i
\(260\) 0 0
\(261\) 25.3466 + 96.5880i 0.0971136 + 0.370069i
\(262\) 0 0
\(263\) 297.767 52.5043i 1.13219 0.199636i 0.424005 0.905660i \(-0.360624\pi\)
0.708188 + 0.706024i \(0.249513\pi\)
\(264\) 0 0
\(265\) 161.421 58.7524i 0.609135 0.221707i
\(266\) 0 0
\(267\) 322.541 100.782i 1.20802 0.377459i
\(268\) 0 0
\(269\) 252.041i 0.936956i −0.883475 0.468478i \(-0.844803\pi\)
0.883475 0.468478i \(-0.155197\pi\)
\(270\) 0 0
\(271\) −107.217 −0.395635 −0.197818 0.980239i \(-0.563385\pi\)
−0.197818 + 0.980239i \(0.563385\pi\)
\(272\) 0 0
\(273\) 164.731 178.843i 0.603411 0.655103i
\(274\) 0 0
\(275\) −103.718 284.963i −0.377157 1.03623i
\(276\) 0 0
\(277\) −46.8272 265.570i −0.169051 0.958738i −0.944788 0.327681i \(-0.893733\pi\)
0.775737 0.631056i \(-0.217378\pi\)
\(278\) 0 0
\(279\) −187.383 + 264.690i −0.671624 + 0.948710i
\(280\) 0 0
\(281\) −5.79982 + 15.9349i −0.0206399 + 0.0567077i −0.949585 0.313511i \(-0.898495\pi\)
0.928945 + 0.370218i \(0.120717\pi\)
\(282\) 0 0
\(283\) 250.525 + 210.215i 0.885247 + 0.742810i 0.967251 0.253822i \(-0.0816876\pi\)
−0.0820041 + 0.996632i \(0.526132\pi\)
\(284\) 0 0
\(285\) 66.0978 50.4479i 0.231922 0.177010i
\(286\) 0 0
\(287\) 121.919 + 70.3903i 0.424807 + 0.245262i
\(288\) 0 0
\(289\) −136.960 237.221i −0.473909 0.820835i
\(290\) 0 0
\(291\) −139.966 89.6771i −0.480982 0.308169i
\(292\) 0 0
\(293\) 264.414 + 46.6234i 0.902437 + 0.159124i 0.605568 0.795794i \(-0.292946\pi\)
0.296870 + 0.954918i \(0.404057\pi\)
\(294\) 0 0
\(295\) 35.7182 29.9711i 0.121078 0.101597i
\(296\) 0 0
\(297\) 341.686 213.974i 1.15046 0.720452i
\(298\) 0 0
\(299\) 300.128 + 357.678i 1.00377 + 1.19625i
\(300\) 0 0
\(301\) −67.1121 + 380.612i −0.222964 + 1.26449i
\(302\) 0 0
\(303\) 236.117 + 10.9202i 0.779263 + 0.0360401i
\(304\) 0 0
\(305\) −148.007 + 85.4519i −0.485269 + 0.280170i
\(306\) 0 0
\(307\) 51.1411 88.5790i 0.166583 0.288531i −0.770633 0.637279i \(-0.780060\pi\)
0.937216 + 0.348748i \(0.113393\pi\)
\(308\) 0 0
\(309\) −155.836 + 20.1069i −0.504325 + 0.0650709i
\(310\) 0 0
\(311\) −300.925 + 358.629i −0.967605 + 1.15315i 0.0205661 + 0.999788i \(0.493453\pi\)
−0.988171 + 0.153358i \(0.950991\pi\)
\(312\) 0 0
\(313\) −173.348 63.0934i −0.553827 0.201576i 0.0499192 0.998753i \(-0.484104\pi\)
−0.603746 + 0.797177i \(0.706326\pi\)
\(314\) 0 0
\(315\) 121.101 9.96492i 0.384449 0.0316347i
\(316\) 0 0
\(317\) 96.3583 16.9906i 0.303969 0.0535980i −0.0195831 0.999808i \(-0.506234\pi\)
0.323552 + 0.946210i \(0.395123\pi\)
\(318\) 0 0
\(319\) 155.682 56.6634i 0.488030 0.177628i
\(320\) 0 0
\(321\) −124.755 + 555.933i −0.388644 + 1.73188i
\(322\) 0 0
\(323\) 49.6965i 0.153859i
\(324\) 0 0
\(325\) 264.055 0.812476
\(326\) 0 0
\(327\) −472.432 106.017i −1.44475 0.324210i
\(328\) 0 0
\(329\) −7.27979 20.0011i −0.0221270 0.0607935i
\(330\) 0 0
\(331\) 18.3892 + 104.290i 0.0555564 + 0.315076i 0.999904 0.0138738i \(-0.00441632\pi\)
−0.944347 + 0.328950i \(0.893305\pi\)
\(332\) 0 0
\(333\) −241.843 + 167.485i −0.726255 + 0.502958i
\(334\) 0 0
\(335\) −81.7468 + 224.597i −0.244020 + 0.670440i
\(336\) 0 0
\(337\) −3.17508 2.66420i −0.00942159 0.00790565i 0.638065 0.769983i \(-0.279735\pi\)
−0.647486 + 0.762077i \(0.724180\pi\)
\(338\) 0 0
\(339\) −80.1083 620.870i −0.236308 1.83148i
\(340\) 0 0
\(341\) 465.961 + 269.023i 1.36646 + 0.788924i
\(342\) 0 0
\(343\) −184.333 319.274i −0.537414 0.930828i
\(344\) 0 0
\(345\) −10.7800 + 233.087i −0.0312464 + 0.675613i
\(346\) 0 0
\(347\) −387.103 68.2568i −1.11557 0.196705i −0.414676 0.909969i \(-0.636105\pi\)
−0.700896 + 0.713264i \(0.747216\pi\)
\(348\) 0 0
\(349\) −246.286 + 206.658i −0.705690 + 0.592144i −0.923386 0.383872i \(-0.874590\pi\)
0.217696 + 0.976017i \(0.430146\pi\)
\(350\) 0 0
\(351\) 73.3223 + 343.303i 0.208896 + 0.978071i
\(352\) 0 0
\(353\) 112.600 + 134.192i 0.318982 + 0.380147i 0.901580 0.432613i \(-0.142408\pi\)
−0.582598 + 0.812760i \(0.697964\pi\)
\(354\) 0 0
\(355\) 30.9281 175.402i 0.0871213 0.494089i
\(356\) 0 0
\(357\) −39.1787 + 61.1491i −0.109744 + 0.171286i
\(358\) 0 0
\(359\) 455.354 262.899i 1.26840 0.732309i 0.293711 0.955894i \(-0.405110\pi\)
0.974684 + 0.223586i \(0.0717762\pi\)
\(360\) 0 0
\(361\) 98.6142 170.805i 0.273170 0.473143i
\(362\) 0 0
\(363\) −185.573 243.141i −0.511219 0.669809i
\(364\) 0 0
\(365\) 8.62149 10.2747i 0.0236205 0.0281498i
\(366\) 0 0
\(367\) 304.854 + 110.958i 0.830664 + 0.302337i 0.722131 0.691756i \(-0.243163\pi\)
0.108533 + 0.994093i \(0.465385\pi\)
\(368\) 0 0
\(369\) −184.650 + 84.9453i −0.500407 + 0.230204i
\(370\) 0 0
\(371\) −486.916 + 85.8564i −1.31244 + 0.231419i
\(372\) 0 0
\(373\) 227.629 82.8503i 0.610267 0.222119i −0.0183533 0.999832i \(-0.505842\pi\)
0.628620 + 0.777713i \(0.283620\pi\)
\(374\) 0 0
\(375\) 216.536 + 199.450i 0.577429 + 0.531866i
\(376\) 0 0
\(377\) 144.259i 0.382649i
\(378\) 0 0
\(379\) 605.266 1.59701 0.798504 0.601989i \(-0.205625\pi\)
0.798504 + 0.601989i \(0.205625\pi\)
\(380\) 0 0
\(381\) 50.8361 + 162.696i 0.133428 + 0.427023i
\(382\) 0 0
\(383\) −92.4520 254.010i −0.241389 0.663211i −0.999933 0.0115889i \(-0.996311\pi\)
0.758544 0.651622i \(-0.225911\pi\)
\(384\) 0 0
\(385\) −35.0067 198.533i −0.0909265 0.515670i
\(386\) 0 0
\(387\) −392.512 396.588i −1.01424 1.02478i
\(388\) 0 0
\(389\) 56.4976 155.226i 0.145238 0.399038i −0.845648 0.533741i \(-0.820786\pi\)
0.990886 + 0.134703i \(0.0430079\pi\)
\(390\) 0 0
\(391\) −106.831 89.6421i −0.273226 0.229264i
\(392\) 0 0
\(393\) −383.461 159.996i −0.975727 0.407115i
\(394\) 0 0
\(395\) −120.524 69.5844i −0.305123 0.176163i
\(396\) 0 0
\(397\) −192.854 334.032i −0.485778 0.841392i 0.514089 0.857737i \(-0.328130\pi\)
−0.999866 + 0.0163455i \(0.994797\pi\)
\(398\) 0 0
\(399\) −212.570 + 109.960i −0.532757 + 0.275589i
\(400\) 0 0
\(401\) 238.103 + 41.9840i 0.593773 + 0.104698i 0.462456 0.886642i \(-0.346968\pi\)
0.131317 + 0.991340i \(0.458079\pi\)
\(402\) 0 0
\(403\) −358.892 + 301.146i −0.890551 + 0.747261i
\(404\) 0 0
\(405\) −89.2805 + 151.013i −0.220446 + 0.372873i
\(406\) 0 0
\(407\) 313.719 + 373.876i 0.770808 + 0.918614i
\(408\) 0 0
\(409\) 26.9195 152.668i 0.0658180 0.373272i −0.934052 0.357137i \(-0.883753\pi\)
0.999870 0.0161348i \(-0.00513609\pi\)
\(410\) 0 0
\(411\) −73.6371 142.352i −0.179166 0.346355i
\(412\) 0 0
\(413\) −116.224 + 67.1018i −0.281413 + 0.162474i
\(414\) 0 0
\(415\) 164.441 284.820i 0.396243 0.686313i
\(416\) 0 0
\(417\) −230.906 + 553.409i −0.553731 + 1.32712i
\(418\) 0 0
\(419\) −308.155 + 367.245i −0.735453 + 0.876479i −0.996034 0.0889728i \(-0.971642\pi\)
0.260581 + 0.965452i \(0.416086\pi\)
\(420\) 0 0
\(421\) −647.463 235.657i −1.53792 0.559756i −0.572372 0.819994i \(-0.693977\pi\)
−0.965545 + 0.260238i \(0.916199\pi\)
\(422\) 0 0
\(423\) 29.6412 + 8.10667i 0.0700737 + 0.0191647i
\(424\) 0 0
\(425\) −77.6696 + 13.6953i −0.182752 + 0.0322241i
\(426\) 0 0
\(427\) 462.239 168.241i 1.08253 0.394008i
\(428\) 0 0
\(429\) 555.907 173.699i 1.29582 0.404894i
\(430\) 0 0
\(431\) 506.298i 1.17471i 0.809331 + 0.587353i \(0.199830\pi\)
−0.809331 + 0.587353i \(0.800170\pi\)
\(432\) 0 0
\(433\) 425.824 0.983427 0.491714 0.870757i \(-0.336371\pi\)
0.491714 + 0.870757i \(0.336371\pi\)
\(434\) 0 0
\(435\) −48.8415 + 53.0256i −0.112279 + 0.121898i
\(436\) 0 0
\(437\) −157.184 431.861i −0.359690 0.988239i
\(438\) 0 0
\(439\) −79.7889 452.505i −0.181751 1.03076i −0.930059 0.367411i \(-0.880244\pi\)
0.748307 0.663353i \(-0.230867\pi\)
\(440\) 0 0
\(441\) 90.8721 + 8.42351i 0.206059 + 0.0191009i
\(442\) 0 0
\(443\) −18.6549 + 51.2541i −0.0421105 + 0.115698i −0.958966 0.283523i \(-0.908497\pi\)
0.916855 + 0.399220i \(0.130719\pi\)
\(444\) 0 0
\(445\) 186.882 + 156.813i 0.419959 + 0.352388i
\(446\) 0 0
\(447\) −159.856 + 122.007i −0.357619 + 0.272946i
\(448\) 0 0
\(449\) 182.250 + 105.222i 0.405902 + 0.234348i 0.689027 0.724735i \(-0.258038\pi\)
−0.283125 + 0.959083i \(0.591371\pi\)
\(450\) 0 0
\(451\) 168.605 + 292.033i 0.373848 + 0.647523i
\(452\) 0 0
\(453\) −563.052 360.751i −1.24294 0.796361i
\(454\) 0 0
\(455\) 172.871 + 30.4819i 0.379937 + 0.0669932i
\(456\) 0 0
\(457\) −150.910 + 126.628i −0.330218 + 0.277086i −0.792789 0.609497i \(-0.791372\pi\)
0.462571 + 0.886582i \(0.346927\pi\)
\(458\) 0 0
\(459\) −39.3727 97.1771i −0.0857793 0.211715i
\(460\) 0 0
\(461\) 128.532 + 153.179i 0.278812 + 0.332275i 0.887218 0.461351i \(-0.152635\pi\)
−0.608406 + 0.793626i \(0.708191\pi\)
\(462\) 0 0
\(463\) −67.1681 + 380.929i −0.145071 + 0.822741i 0.822238 + 0.569143i \(0.192725\pi\)
−0.967310 + 0.253598i \(0.918386\pi\)
\(464\) 0 0
\(465\) −233.878 10.8166i −0.502963 0.0232615i
\(466\) 0 0
\(467\) −300.701 + 173.610i −0.643900 + 0.371756i −0.786115 0.618080i \(-0.787911\pi\)
0.142215 + 0.989836i \(0.454577\pi\)
\(468\) 0 0
\(469\) 343.968 595.770i 0.733407 1.27030i
\(470\) 0 0
\(471\) −360.424 + 46.5040i −0.765231 + 0.0987346i
\(472\) 0 0
\(473\) −595.055 + 709.159i −1.25804 + 1.49928i
\(474\) 0 0
\(475\) −244.230 88.8924i −0.514168 0.187142i
\(476\) 0 0
\(477\) 305.015 645.382i 0.639445 1.35300i
\(478\) 0 0
\(479\) 239.163 42.1710i 0.499297 0.0880396i 0.0816727 0.996659i \(-0.473974\pi\)
0.417625 + 0.908620i \(0.362863\pi\)
\(480\) 0 0
\(481\) −399.347 + 145.350i −0.830242 + 0.302183i
\(482\) 0 0
\(483\) 147.054 655.302i 0.304459 1.35673i
\(484\) 0 0
\(485\) 120.008i 0.247439i
\(486\) 0 0
\(487\) −294.321 −0.604355 −0.302177 0.953252i \(-0.597713\pi\)
−0.302177 + 0.953252i \(0.597713\pi\)
\(488\) 0 0
\(489\) −207.820 46.6361i −0.424990 0.0953704i
\(490\) 0 0
\(491\) 81.7251 + 224.538i 0.166446 + 0.457307i 0.994672 0.103087i \(-0.0328719\pi\)
−0.828226 + 0.560394i \(0.810650\pi\)
\(492\) 0 0
\(493\) −7.48201 42.4326i −0.0151765 0.0860702i
\(494\) 0 0
\(495\) 263.145 + 124.366i 0.531607 + 0.251244i
\(496\) 0 0
\(497\) −175.333 + 481.723i −0.352782 + 0.969261i
\(498\) 0 0
\(499\) 184.698 + 154.980i 0.370136 + 0.310581i 0.808815 0.588063i \(-0.200109\pi\)
−0.438679 + 0.898644i \(0.644554\pi\)
\(500\) 0 0
\(501\) −64.0240 496.211i −0.127792 0.990441i
\(502\) 0 0
\(503\) 138.798 + 80.1352i 0.275941 + 0.159314i 0.631584 0.775307i \(-0.282405\pi\)
−0.355644 + 0.934622i \(0.615738\pi\)
\(504\) 0 0
\(505\) 85.3220 + 147.782i 0.168954 + 0.292638i
\(506\) 0 0
\(507\) −0.00610761 + 0.132059i −1.20466e−5 + 0.000260472i
\(508\) 0 0
\(509\) 78.7326 + 13.8827i 0.154681 + 0.0272744i 0.250452 0.968129i \(-0.419421\pi\)
−0.0957713 + 0.995403i \(0.530532\pi\)
\(510\) 0 0
\(511\) −29.5731 + 24.8148i −0.0578731 + 0.0485613i
\(512\) 0 0
\(513\) 47.7535 342.212i 0.0930867 0.667080i
\(514\) 0 0
\(515\) −72.9158 86.8977i −0.141584 0.168733i
\(516\) 0 0
\(517\) 8.85312 50.2085i 0.0171240 0.0971151i
\(518\) 0 0
\(519\) −370.786 + 578.714i −0.714424 + 1.11506i
\(520\) 0 0
\(521\) 599.564 346.158i 1.15079 0.664411i 0.201713 0.979445i \(-0.435349\pi\)
0.949081 + 0.315033i \(0.102016\pi\)
\(522\) 0 0
\(523\) 33.5131 58.0464i 0.0640786 0.110987i −0.832206 0.554466i \(-0.812923\pi\)
0.896285 + 0.443479i \(0.146256\pi\)
\(524\) 0 0
\(525\) −230.434 301.919i −0.438922 0.575084i
\(526\) 0 0
\(527\) 89.9464 107.194i 0.170676 0.203404i
\(528\) 0 0
\(529\) 714.791 + 260.163i 1.35121 + 0.491801i
\(530\) 0 0
\(531\) 17.8838 192.929i 0.0336795 0.363332i
\(532\) 0 0
\(533\) −289.164 + 50.9873i −0.542521 + 0.0956611i
\(534\) 0 0
\(535\) −386.525 + 140.684i −0.722477 + 0.262960i
\(536\) 0 0
\(537\) −331.003 304.885i −0.616393 0.567756i
\(538\) 0 0
\(539\) 151.410i 0.280909i
\(540\) 0 0
\(541\) −584.288 −1.08002 −0.540008 0.841660i \(-0.681579\pi\)
−0.540008 + 0.841660i \(0.681579\pi\)
\(542\) 0 0
\(543\) 133.811 + 428.248i 0.246429 + 0.788670i
\(544\) 0 0
\(545\) −119.553 328.469i −0.219363 0.602695i
\(546\) 0 0
\(547\) −13.0135 73.8033i −0.0237907 0.134924i 0.970599 0.240702i \(-0.0773777\pi\)
−0.994390 + 0.105778i \(0.966267\pi\)
\(548\) 0 0
\(549\) −187.351 + 685.030i −0.341259 + 1.24778i
\(550\) 0 0
\(551\) 48.5638 133.428i 0.0881377 0.242156i
\(552\) 0 0
\(553\) 306.849 + 257.477i 0.554881 + 0.465600i
\(554\) 0 0
\(555\) −196.000 81.7796i −0.353153 0.147351i
\(556\) 0 0
\(557\) −460.828 266.059i −0.827340 0.477665i 0.0256012 0.999672i \(-0.491850\pi\)
−0.852941 + 0.522007i \(0.825183\pi\)
\(558\) 0 0
\(559\) −403.042 698.090i −0.721006 1.24882i
\(560\) 0 0
\(561\) −154.507 + 79.9246i −0.275413 + 0.142468i
\(562\) 0 0
\(563\) 457.883 + 80.7371i 0.813291 + 0.143405i 0.564799 0.825229i \(-0.308954\pi\)
0.248492 + 0.968634i \(0.420065\pi\)
\(564\) 0 0
\(565\) 346.211 290.505i 0.612762 0.514168i
\(566\) 0 0
\(567\) 328.545 383.429i 0.579444 0.676241i
\(568\) 0 0
\(569\) 324.973 + 387.288i 0.571130 + 0.680647i 0.971863 0.235549i \(-0.0756887\pi\)
−0.400732 + 0.916195i \(0.631244\pi\)
\(570\) 0 0
\(571\) −4.63998 + 26.3146i −0.00812605 + 0.0460851i −0.988601 0.150558i \(-0.951893\pi\)
0.980475 + 0.196643i \(0.0630041\pi\)
\(572\) 0 0
\(573\) 104.543 + 202.098i 0.182449 + 0.352702i
\(574\) 0 0
\(575\) 631.630 364.672i 1.09849 0.634212i
\(576\) 0 0
\(577\) −568.853 + 985.283i −0.985881 + 1.70760i −0.347926 + 0.937522i \(0.613114\pi\)
−0.637955 + 0.770074i \(0.720219\pi\)
\(578\) 0 0
\(579\) −157.306 + 377.013i −0.271686 + 0.651145i
\(580\) 0 0
\(581\) −608.465 + 725.141i −1.04727 + 1.24809i
\(582\) 0 0
\(583\) −1112.88 405.054i −1.90888 0.694775i
\(584\) 0 0
\(585\) −180.128 + 178.276i −0.307911 + 0.304746i
\(586\) 0 0
\(587\) −413.543 + 72.9187i −0.704502 + 0.124223i −0.514411 0.857544i \(-0.671989\pi\)
−0.190091 + 0.981767i \(0.560878\pi\)
\(588\) 0 0
\(589\) 433.327 157.718i 0.735699 0.267772i
\(590\) 0 0
\(591\) −945.347 + 295.384i −1.59957 + 0.499805i
\(592\) 0 0
\(593\) 608.635i 1.02637i 0.858279 + 0.513183i \(0.171534\pi\)
−0.858279 + 0.513183i \(0.828466\pi\)
\(594\) 0 0
\(595\) −52.4298 −0.0881173
\(596\) 0 0
\(597\) 121.453 131.858i 0.203439 0.220867i
\(598\) 0 0
\(599\) −222.904 612.422i −0.372126 1.02241i −0.974538 0.224223i \(-0.928016\pi\)
0.602412 0.798185i \(-0.294206\pi\)
\(600\) 0 0
\(601\) 29.5147 + 167.386i 0.0491092 + 0.278512i 0.999467 0.0326472i \(-0.0103938\pi\)
−0.950358 + 0.311160i \(0.899283\pi\)
\(602\) 0 0
\(603\) 415.092 + 902.308i 0.688378 + 1.49636i
\(604\) 0 0
\(605\) 75.5239 207.500i 0.124833 0.342976i
\(606\) 0 0
\(607\) 275.061 + 230.804i 0.453149 + 0.380237i 0.840603 0.541652i \(-0.182201\pi\)
−0.387454 + 0.921889i \(0.626645\pi\)
\(608\) 0 0
\(609\) 164.945 125.891i 0.270845 0.206718i
\(610\) 0 0
\(611\) 38.4457 + 22.1966i 0.0629225 + 0.0363283i
\(612\) 0 0
\(613\) 112.374 + 194.637i 0.183317 + 0.317515i 0.943008 0.332769i \(-0.107983\pi\)
−0.759691 + 0.650284i \(0.774650\pi\)
\(614\) 0 0
\(615\) −123.551 79.1603i −0.200897 0.128716i
\(616\) 0 0
\(617\) −369.095 65.0815i −0.598210 0.105481i −0.133660 0.991027i \(-0.542673\pi\)
−0.464550 + 0.885547i \(0.653784\pi\)
\(618\) 0 0
\(619\) 511.061 428.831i 0.825623 0.692780i −0.128658 0.991689i \(-0.541067\pi\)
0.954282 + 0.298909i \(0.0966226\pi\)
\(620\) 0 0
\(621\) 649.508 + 719.934i 1.04591 + 1.15931i
\(622\) 0 0
\(623\) −451.346 537.893i −0.724472 0.863392i
\(624\) 0 0
\(625\) 51.2603 290.712i 0.0820165 0.465139i
\(626\) 0 0
\(627\) −572.645 26.4843i −0.913310 0.0422397i
\(628\) 0 0
\(629\) 109.926 63.4659i 0.174763 0.100900i
\(630\) 0 0
\(631\) 288.004 498.837i 0.456425 0.790551i −0.542344 0.840156i \(-0.682463\pi\)
0.998769 + 0.0496058i \(0.0157965\pi\)
\(632\) 0 0
\(633\) −33.5988 + 4.33511i −0.0530786 + 0.00684851i
\(634\) 0 0
\(635\) −79.0992 + 94.2667i −0.124566 + 0.148452i
\(636\) 0 0
\(637\) 123.889 + 45.0918i 0.194488 + 0.0707877i
\(638\) 0 0
\(639\) −421.379 608.457i −0.659435 0.952203i
\(640\) 0 0
\(641\) −732.193 + 129.105i −1.14227 + 0.201412i −0.712597 0.701574i \(-0.752481\pi\)
−0.429670 + 0.902986i \(0.641370\pi\)
\(642\) 0 0
\(643\) −717.060 + 260.989i −1.11518 + 0.405892i −0.832891 0.553437i \(-0.813316\pi\)
−0.282289 + 0.959330i \(0.591094\pi\)
\(644\) 0 0
\(645\) 88.2042 393.056i 0.136751 0.609389i
\(646\) 0 0
\(647\) 128.575i 0.198725i 0.995051 + 0.0993624i \(0.0316803\pi\)
−0.995051 + 0.0993624i \(0.968320\pi\)
\(648\) 0 0
\(649\) −321.457 −0.495311
\(650\) 0 0
\(651\) 657.526 + 147.553i 1.01002 + 0.226656i
\(652\) 0 0
\(653\) −61.1625 168.042i −0.0936638 0.257339i 0.884010 0.467468i \(-0.154834\pi\)
−0.977674 + 0.210129i \(0.932612\pi\)
\(654\) 0 0
\(655\) −52.0886 295.409i −0.0795245 0.451006i
\(656\) 0 0
\(657\) −4.57083 55.5482i −0.00695712 0.0845483i
\(658\) 0 0
\(659\) 69.7138 191.537i 0.105787 0.290648i −0.875494 0.483229i \(-0.839464\pi\)
0.981281 + 0.192581i \(0.0616859\pi\)
\(660\) 0 0
\(661\) −588.511 493.819i −0.890334 0.747079i 0.0779433 0.996958i \(-0.475165\pi\)
−0.968277 + 0.249879i \(0.919609\pi\)
\(662\) 0 0
\(663\) −19.3829 150.225i −0.0292351 0.226584i
\(664\) 0 0
\(665\) −149.631 86.3896i −0.225009 0.129909i
\(666\) 0 0
\(667\) 199.228 + 345.073i 0.298693 + 0.517351i
\(668\) 0 0
\(669\) 15.3327 331.526i 0.0229189 0.495554i
\(670\) 0 0
\(671\) 1160.36 + 204.602i 1.72929 + 0.304921i
\(672\) 0 0
\(673\) 897.798 753.342i 1.33402 1.11938i 0.350905 0.936411i \(-0.385874\pi\)
0.983119 0.182967i \(-0.0585703\pi\)
\(674\) 0 0
\(675\) 547.997 19.6731i 0.811847 0.0291454i
\(676\) 0 0
\(677\) 192.459 + 229.364i 0.284283 + 0.338795i 0.889221 0.457477i \(-0.151247\pi\)
−0.604939 + 0.796272i \(0.706802\pi\)
\(678\) 0 0
\(679\) −59.9803 + 340.165i −0.0883363 + 0.500980i
\(680\) 0 0
\(681\) −534.393 + 834.068i −0.784719 + 1.22477i
\(682\) 0 0
\(683\) −812.585 + 469.146i −1.18973 + 0.686890i −0.958245 0.285949i \(-0.907691\pi\)
−0.231484 + 0.972839i \(0.574358\pi\)
\(684\) 0 0
\(685\) 57.8526 100.204i 0.0844564 0.146283i
\(686\) 0 0
\(687\) 824.313 + 1080.03i 1.19987 + 1.57210i
\(688\) 0 0
\(689\) 662.856 789.962i 0.962056 1.14653i
\(690\) 0 0
\(691\) 31.6500 + 11.5197i 0.0458032 + 0.0166710i 0.364820 0.931078i \(-0.381130\pi\)
−0.319017 + 0.947749i \(0.603353\pi\)
\(692\) 0 0
\(693\) −683.734 484.038i −0.986629 0.698468i
\(694\) 0 0
\(695\) −426.333 + 75.1739i −0.613428 + 0.108164i
\(696\) 0 0
\(697\) 82.4108 29.9951i 0.118236 0.0430345i
\(698\) 0 0
\(699\) 797.234 + 734.328i 1.14054 + 1.05054i
\(700\) 0 0
\(701\) 1259.81i 1.79716i −0.438814 0.898578i \(-0.644601\pi\)
0.438814 0.898578i \(-0.355399\pi\)
\(702\) 0 0
\(703\) 418.296 0.595015
\(704\) 0 0
\(705\) 6.61649 + 21.1754i 0.00938509 + 0.0300360i
\(706\) 0 0
\(707\) −167.985 461.536i −0.237603 0.652809i
\(708\) 0 0
\(709\) 51.2300 + 290.540i 0.0722567 + 0.409788i 0.999386 + 0.0350457i \(0.0111577\pi\)
−0.927129 + 0.374742i \(0.877731\pi\)
\(710\) 0 0
\(711\) −559.373 + 146.791i −0.786742 + 0.206457i
\(712\) 0 0
\(713\) −442.589 + 1216.00i −0.620741 + 1.70547i
\(714\) 0 0
\(715\) 322.096 + 270.270i 0.450483 + 0.378000i
\(716\) 0 0
\(717\) 353.185 + 147.364i 0.492587 + 0.205528i
\(718\) 0 0
\(719\) 719.610 + 415.467i 1.00085 + 0.577840i 0.908499 0.417886i \(-0.137229\pi\)
0.0923493 + 0.995727i \(0.470562\pi\)
\(720\) 0 0
\(721\) 163.250 + 282.757i 0.226422 + 0.392174i
\(722\) 0 0
\(723\) −347.887 + 179.958i −0.481172 + 0.248905i
\(724\) 0 0
\(725\) 221.915 + 39.1297i 0.306090 + 0.0539719i
\(726\) 0 0
\(727\) −171.551 + 143.949i −0.235972 + 0.198004i −0.753104 0.657902i \(-0.771444\pi\)
0.517132 + 0.855906i \(0.327000\pi\)
\(728\) 0 0
\(729\) 177.744 + 706.999i 0.243820 + 0.969821i
\(730\) 0 0
\(731\) 154.758 + 184.434i 0.211708 + 0.252303i
\(732\) 0 0
\(733\) −47.7555 + 270.835i −0.0651508 + 0.369488i 0.934749 + 0.355309i \(0.115625\pi\)
−0.999899 + 0.0141788i \(0.995487\pi\)
\(734\) 0 0
\(735\) 30.2714 + 58.5194i 0.0411856 + 0.0796182i
\(736\) 0 0
\(737\) 1427.04 823.904i 1.93629 1.11792i
\(738\) 0 0
\(739\) −235.936 + 408.653i −0.319264 + 0.552981i −0.980335 0.197343i \(-0.936769\pi\)
0.661071 + 0.750323i \(0.270102\pi\)
\(740\) 0 0
\(741\) 192.211 460.670i 0.259394 0.621686i
\(742\) 0 0
\(743\) 245.380 292.432i 0.330255 0.393583i −0.575208 0.818007i \(-0.695079\pi\)
0.905464 + 0.424424i \(0.139523\pi\)
\(744\) 0 0
\(745\) −136.423 49.6541i −0.183119 0.0666497i
\(746\) 0 0
\(747\) −346.894 1321.90i −0.464383 1.76961i
\(748\) 0 0
\(749\) 1165.93 205.585i 1.55665 0.274479i
\(750\) 0 0
\(751\) 1053.31 383.374i 1.40255 0.510485i 0.473613 0.880733i \(-0.342949\pi\)
0.928933 + 0.370248i \(0.120727\pi\)
\(752\) 0 0
\(753\) 1244.04 388.716i 1.65212 0.516223i
\(754\) 0 0
\(755\) 482.765i 0.639424i
\(756\) 0 0
\(757\) 157.155 0.207602 0.103801 0.994598i \(-0.466899\pi\)
0.103801 + 0.994598i \(0.466899\pi\)
\(758\) 0 0
\(759\) 1089.87 1183.23i 1.43592 1.55893i
\(760\) 0 0
\(761\) −440.078 1209.11i −0.578290 1.58884i −0.791062 0.611736i \(-0.790472\pi\)
0.212773 0.977102i \(-0.431751\pi\)
\(762\) 0 0
\(763\) 174.706 + 990.806i 0.228972 + 1.29857i
\(764\) 0 0
\(765\) 43.7369 61.7810i 0.0571724 0.0807595i
\(766\) 0 0
\(767\) 95.7338 263.026i 0.124816 0.342929i
\(768\) 0 0
\(769\) −687.236 576.660i −0.893675 0.749882i 0.0752688 0.997163i \(-0.476019\pi\)
−0.968944 + 0.247281i \(0.920463\pi\)
\(770\) 0 0
\(771\) 387.432 295.700i 0.502506 0.383528i
\(772\) 0 0
\(773\) −211.789 122.276i −0.273983 0.158184i 0.356713 0.934214i \(-0.383897\pi\)
−0.630696 + 0.776030i \(0.717231\pi\)
\(774\) 0 0
\(775\) 365.910 + 633.774i 0.472141 + 0.817773i
\(776\) 0 0
\(777\) 514.693 + 329.768i 0.662411 + 0.424411i
\(778\) 0 0
\(779\) 284.618 + 50.1859i 0.365364 + 0.0644235i
\(780\) 0 0
\(781\) −940.641 + 789.292i −1.20441 + 1.01062i
\(782\) 0 0
\(783\) 10.7479 + 299.382i 0.0137265 + 0.382353i
\(784\) 0 0
\(785\) −168.642 200.980i −0.214831 0.256026i
\(786\) 0 0
\(787\) −75.9828 + 430.920i −0.0965474 + 0.547547i 0.897715 + 0.440577i \(0.145226\pi\)
−0.994262 + 0.106970i \(0.965885\pi\)
\(788\) 0 0
\(789\) 906.112 + 41.9068i 1.14843 + 0.0531138i
\(790\) 0 0
\(791\) −1126.54 + 650.407i −1.42419 + 0.822259i
\(792\) 0 0
\(793\) −512.980 + 888.507i −0.646885 + 1.12044i
\(794\) 0 0
\(795\) 511.104 65.9456i 0.642899 0.0829505i
\(796\) 0 0
\(797\) −275.128 + 327.885i −0.345204 + 0.411399i −0.910513 0.413481i \(-0.864313\pi\)
0.565308 + 0.824880i \(0.308757\pi\)
\(798\) 0 0
\(799\) −12.4597 4.53497i −0.0155942 0.00567581i
\(800\) 0 0
\(801\) 1010.34 83.1368i 1.26135 0.103791i
\(802\) 0 0
\(803\) −91.0655 + 16.0573i −0.113407 + 0.0199966i
\(804\) 0 0
\(805\) 455.613 165.830i 0.565979 0.205999i
\(806\) 0 0
\(807\) 165.561 737.775i 0.205156 0.914220i
\(808\) 0 0
\(809\) 1161.28i 1.43545i 0.696326 + 0.717725i \(0.254817\pi\)
−0.696326 + 0.717725i \(0.745183\pi\)
\(810\) 0 0
\(811\) 611.140 0.753563 0.376782 0.926302i \(-0.377031\pi\)
0.376782 + 0.926302i \(0.377031\pi\)
\(812\) 0 0
\(813\) −313.846 70.4289i −0.386034 0.0866284i
\(814\) 0 0
\(815\) −52.5907 144.492i −0.0645284 0.177290i
\(816\) 0 0
\(817\) 137.775 + 781.360i 0.168635 + 0.956377i
\(818\) 0 0
\(819\) 599.680 415.301i 0.732210 0.507082i
\(820\) 0 0
\(821\) 531.207 1459.48i 0.647024 1.77769i 0.0185843 0.999827i \(-0.494084\pi\)
0.628440 0.777858i \(-0.283694\pi\)
\(822\) 0 0
\(823\) −463.298 388.753i −0.562938 0.472361i 0.316356 0.948641i \(-0.397541\pi\)
−0.879294 + 0.476279i \(0.841985\pi\)
\(824\) 0 0
\(825\) −116.417 902.275i −0.141111 1.09367i
\(826\) 0 0
\(827\) −79.8064 46.0762i −0.0965010 0.0557149i 0.450973 0.892538i \(-0.351077\pi\)
−0.547474 + 0.836823i \(0.684410\pi\)
\(828\) 0 0
\(829\) −418.250 724.429i −0.504523 0.873859i −0.999986 0.00523058i \(-0.998335\pi\)
0.495463 0.868629i \(-0.334998\pi\)
\(830\) 0 0
\(831\) 37.3756 808.138i 0.0449766 0.972488i
\(832\) 0 0
\(833\) −38.7796 6.83789i −0.0465541 0.00820875i
\(834\) 0 0
\(835\) 276.698 232.177i 0.331375 0.278056i
\(836\) 0 0
\(837\) −722.378 + 651.713i −0.863056 + 0.778629i
\(838\) 0 0
\(839\) −278.941 332.429i −0.332469 0.396221i 0.573750 0.819031i \(-0.305488\pi\)
−0.906218 + 0.422810i \(0.861044\pi\)
\(840\) 0 0
\(841\) 124.661 706.986i 0.148229 0.840649i
\(842\) 0 0
\(843\) −27.4445 + 42.8348i −0.0325558 + 0.0508123i
\(844\) 0 0
\(845\) −0.0826539 + 0.0477203i −9.78153e−5 + 5.64737e-5i
\(846\) 0 0
\(847\) −317.784 + 550.417i −0.375187 + 0.649843i
\(848\) 0 0
\(849\) 595.250 + 779.908i 0.701119 + 0.918619i
\(850\) 0 0
\(851\) −754.519 + 899.200i −0.886626 + 1.05664i
\(852\) 0 0
\(853\) 375.612 + 136.712i 0.440342 + 0.160271i 0.552671 0.833400i \(-0.313609\pi\)
−0.112329 + 0.993671i \(0.535831\pi\)
\(854\) 0 0
\(855\) 226.620 104.253i 0.265053 0.121933i
\(856\) 0 0
\(857\) 314.017 55.3697i 0.366414 0.0646087i 0.0125903 0.999921i \(-0.495992\pi\)
0.353824 + 0.935312i \(0.384881\pi\)
\(858\) 0 0
\(859\) −259.079 + 94.2970i −0.301605 + 0.109775i −0.488390 0.872626i \(-0.662416\pi\)
0.186785 + 0.982401i \(0.440193\pi\)
\(860\) 0 0
\(861\) 310.645 + 286.133i 0.360795 + 0.332326i
\(862\) 0 0
\(863\) 764.036i 0.885325i −0.896688 0.442663i \(-0.854034\pi\)
0.896688 0.442663i \(-0.145966\pi\)
\(864\) 0 0
\(865\) −496.194 −0.573635
\(866\) 0 0
\(867\) −245.083 784.361i −0.282679 0.904684i
\(868\) 0 0
\(869\) 328.157 + 901.603i 0.377626 + 1.03752i
\(870\) 0 0
\(871\) 249.154 + 1413.02i 0.286055 + 1.62230i
\(872\) 0 0
\(873\) −350.801 354.444i −0.401834 0.406007i
\(874\) 0 0
\(875\) 209.223 574.837i 0.239113 0.656956i
\(876\) 0 0
\(877\) 611.740 + 513.311i 0.697537 + 0.585303i 0.921072 0.389393i \(-0.127315\pi\)
−0.223535 + 0.974696i \(0.571760\pi\)
\(878\) 0 0
\(879\) 743.367 + 310.165i 0.845697 + 0.352861i
\(880\) 0 0
\(881\) −659.646 380.847i −0.748747 0.432290i 0.0764937 0.997070i \(-0.475627\pi\)
−0.825241 + 0.564781i \(0.808961\pi\)
\(882\) 0 0
\(883\) −463.143 802.187i −0.524511 0.908479i −0.999593 0.0285375i \(-0.990915\pi\)
0.475082 0.879941i \(-0.342418\pi\)
\(884\) 0 0
\(885\) 124.242 64.2688i 0.140386 0.0726201i
\(886\) 0 0
\(887\) −169.111 29.8188i −0.190655 0.0336176i 0.0775053 0.996992i \(-0.475305\pi\)
−0.268160 + 0.963374i \(0.586416\pi\)
\(888\) 0 0
\(889\) 271.323 227.667i 0.305201 0.256094i
\(890\) 0 0
\(891\) 1140.74 401.898i 1.28029 0.451065i
\(892\) 0 0
\(893\) −28.0869 33.4726i −0.0314523 0.0374834i
\(894\) 0 0
\(895\) 56.4159 319.951i 0.0630346 0.357487i
\(896\) 0 0
\(897\) 643.582 + 1244.14i 0.717482 + 1.38701i
\(898\) 0 0
\(899\) −346.244 + 199.904i −0.385144 + 0.222363i
\(900\) 0 0
\(901\) −154.003 + 266.740i −0.170924 + 0.296049i
\(902\) 0 0
\(903\) −446.468 + 1070.04i −0.494427 + 1.18499i
\(904\) 0 0
\(905\) −208.205 + 248.129i −0.230061 + 0.274176i
\(906\) 0 0
\(907\) 479.601 + 174.560i 0.528777 + 0.192459i 0.592592 0.805503i \(-0.298105\pi\)
−0.0638151 + 0.997962i \(0.520327\pi\)
\(908\) 0 0
\(909\) 683.988 + 187.066i 0.752462 + 0.205793i
\(910\) 0 0
\(911\) 653.228 115.182i 0.717045 0.126434i 0.196791 0.980446i \(-0.436948\pi\)
0.520255 + 0.854011i \(0.325837\pi\)
\(912\) 0 0
\(913\) −2130.65 + 775.495i −2.33369 + 0.849392i
\(914\) 0 0
\(915\) −489.378 + 152.912i −0.534840 + 0.167117i
\(916\) 0 0
\(917\) 863.378i 0.941525i
\(918\) 0 0
\(919\) −63.4363 −0.0690275 −0.0345138 0.999404i \(-0.510988\pi\)
−0.0345138 + 0.999404i \(0.510988\pi\)
\(920\) 0 0
\(921\) 207.886 225.695i 0.225718 0.245054i
\(922\) 0 0
\(923\) −365.690 1004.72i −0.396197 1.08854i
\(924\) 0 0
\(925\) 115.273 + 653.746i 0.124620 + 0.706753i
\(926\) 0 0
\(927\) −469.372 43.5091i −0.506334 0.0469353i
\(928\) 0 0
\(929\) 444.979 1222.57i 0.478987 1.31601i −0.431368 0.902176i \(-0.641969\pi\)
0.910354 0.413829i \(-0.135809\pi\)
\(930\) 0 0
\(931\) −99.4075 83.4128i −0.106775 0.0895948i
\(932\) 0 0
\(933\) −1116.44 + 852.106i −1.19662 + 0.913297i
\(934\) 0 0
\(935\) −108.760 62.7924i −0.116320 0.0671576i
\(936\) 0 0
\(937\) 88.1210 + 152.630i 0.0940459 + 0.162892i 0.909210 0.416338i \(-0.136687\pi\)
−0.815164 + 0.579230i \(0.803353\pi\)
\(938\) 0 0
\(939\) −465.979 298.556i −0.496250 0.317951i
\(940\) 0 0
\(941\) 1167.87 + 205.926i 1.24109 + 0.218838i 0.755383 0.655284i \(-0.227451\pi\)
0.485708 + 0.874121i \(0.338562\pi\)
\(942\) 0 0
\(943\) −621.276 + 521.312i −0.658829 + 0.552823i
\(944\) 0 0
\(945\) 361.034 + 50.3799i 0.382046 + 0.0533121i
\(946\) 0 0
\(947\) −169.717 202.261i −0.179216 0.213581i 0.668957 0.743302i \(-0.266741\pi\)
−0.848172 + 0.529721i \(0.822297\pi\)
\(948\) 0 0
\(949\) 13.9818 79.2948i 0.0147332 0.0835561i
\(950\) 0 0
\(951\) 293.221 + 13.5612i 0.308329 + 0.0142599i
\(952\) 0 0
\(953\) −117.730 + 67.9713i −0.123536 + 0.0713235i −0.560495 0.828158i \(-0.689389\pi\)
0.436959 + 0.899482i \(0.356056\pi\)
\(954\) 0 0
\(955\) −82.1337 + 142.260i −0.0860039 + 0.148963i
\(956\) 0 0
\(957\) 492.932 63.6010i 0.515081 0.0664587i
\(958\) 0 0
\(959\) −214.067 + 255.115i −0.223219 + 0.266022i
\(960\) 0 0
\(961\) −317.085 115.410i −0.329954 0.120093i
\(962\) 0 0
\(963\) −730.365 + 1545.38i −0.758427 + 1.60476i
\(964\) 0 0
\(965\) −290.442 + 51.2127i −0.300976 + 0.0530702i
\(966\) 0 0
\(967\) −434.941 + 158.306i −0.449784 + 0.163708i −0.556973 0.830531i \(-0.688037\pi\)
0.107189 + 0.994239i \(0.465815\pi\)
\(968\) 0 0
\(969\) −32.6447 + 145.472i −0.0336890 + 0.150125i
\(970\) 0 0
\(971\) 241.390i 0.248599i −0.992245 0.124299i \(-0.960332\pi\)
0.992245 0.124299i \(-0.0396683\pi\)
\(972\) 0 0
\(973\) 1246.02 1.28060
\(974\) 0 0
\(975\) 772.941 + 173.453i 0.792760 + 0.177900i
\(976\) 0 0
\(977\) −543.473 1493.18i −0.556267 1.52833i −0.825009 0.565119i \(-0.808830\pi\)
0.268742 0.963212i \(-0.413392\pi\)
\(978\) 0 0
\(979\) −292.059 1656.35i −0.298324 1.69188i
\(980\) 0 0
\(981\) −1313.26 620.664i −1.33870 0.632685i
\(982\) 0 0
\(983\) 11.4309 31.4063i 0.0116286 0.0319494i −0.933743 0.357945i \(-0.883477\pi\)
0.945371 + 0.325996i \(0.105700\pi\)
\(984\) 0 0
\(985\) −547.739 459.608i −0.556080 0.466607i
\(986\) 0 0
\(987\) −8.17108 63.3291i −0.00827871 0.0641632i
\(988\) 0 0
\(989\) −1928.19 1113.24i −1.94963 1.12562i
\(990\) 0 0
\(991\) −90.3973 156.573i −0.0912182 0.157995i 0.816806 0.576913i \(-0.195743\pi\)
−0.908024 + 0.418918i \(0.862409\pi\)
\(992\) 0 0
\(993\) −14.6775 + 317.358i −0.0147810 + 0.319595i
\(994\) 0 0
\(995\) 127.455 + 22.4737i 0.128095 + 0.0225867i
\(996\) 0 0
\(997\) −371.647 + 311.849i −0.372766 + 0.312787i −0.809854 0.586631i \(-0.800454\pi\)
0.437089 + 0.899418i \(0.356010\pi\)
\(998\) 0 0
\(999\) −817.941 + 331.401i −0.818760 + 0.331732i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 432.3.bc.b.113.6 36
4.3 odd 2 108.3.k.a.5.1 36
12.11 even 2 324.3.k.a.125.3 36
27.11 odd 18 inner 432.3.bc.b.65.6 36
108.11 even 18 108.3.k.a.65.1 yes 36
108.23 even 18 2916.3.c.b.1457.22 36
108.31 odd 18 2916.3.c.b.1457.15 36
108.43 odd 18 324.3.k.a.197.3 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.3.k.a.5.1 36 4.3 odd 2
108.3.k.a.65.1 yes 36 108.11 even 18
324.3.k.a.125.3 36 12.11 even 2
324.3.k.a.197.3 36 108.43 odd 18
432.3.bc.b.65.6 36 27.11 odd 18 inner
432.3.bc.b.113.6 36 1.1 even 1 trivial
2916.3.c.b.1457.15 36 108.31 odd 18
2916.3.c.b.1457.22 36 108.23 even 18