Properties

Label 432.6.i.b.145.1
Level $432$
Weight $6$
Character 432.145
Analytic conductor $69.286$
Analytic rank $0$
Dimension $6$
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] = [432,6,Mod(145,432)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(432, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("432.145");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 432.i (of order \(3\), degree \(2\), 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: \(69.2858101592\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.47347183152.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{9} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 145.1
Root \(0.500000 + 5.23712i\) of defining polynomial
Character \(\chi\) \(=\) 432.145
Dual form 432.6.i.b.289.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-33.0434 + 57.2329i) q^{5} +(57.0952 + 98.8918i) q^{7} +O(q^{10})\) \(q+(-33.0434 + 57.2329i) q^{5} +(57.0952 + 98.8918i) q^{7} +(192.812 + 333.961i) q^{11} +(-516.287 + 894.236i) q^{13} -959.020 q^{17} +464.576 q^{19} +(-1151.70 + 1994.81i) q^{23} +(-621.235 - 1076.01i) q^{25} +(3549.13 + 6147.28i) q^{29} +(3881.53 - 6723.00i) q^{31} -7546.48 q^{35} +9317.57 q^{37} +(-6661.64 + 11538.3i) q^{41} +(-1056.29 - 1829.55i) q^{43} +(-1247.40 - 2160.56i) q^{47} +(1883.78 - 3262.80i) q^{49} +10044.2 q^{53} -25484.7 q^{55} +(-2720.33 + 4711.75i) q^{59} +(-17094.4 - 29608.4i) q^{61} +(-34119.8 - 59097.2i) q^{65} +(26792.6 - 46406.2i) q^{67} -970.010 q^{71} -72400.3 q^{73} +(-22017.3 + 38135.1i) q^{77} +(16098.9 + 27884.0i) q^{79} +(18046.6 + 31257.6i) q^{83} +(31689.3 - 54887.5i) q^{85} -42622.2 q^{89} -117910. q^{91} +(-15351.2 + 26589.0i) q^{95} +(21926.9 + 37978.6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 54 q^{5} + 132 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 54 q^{5} + 132 q^{7} - 315 q^{11} - 744 q^{13} - 2898 q^{17} - 2262 q^{19} - 3168 q^{23} - 2883 q^{25} + 5148 q^{29} + 8610 q^{31} + 2700 q^{35} + 39936 q^{37} - 5049 q^{41} + 31389 q^{43} + 12924 q^{47} - 52857 q^{49} + 96048 q^{53} - 126252 q^{55} + 62955 q^{59} - 75966 q^{61} - 108702 q^{65} + 32991 q^{67} - 129672 q^{71} - 8466 q^{73} - 88740 q^{77} - 89202 q^{79} + 32634 q^{83} + 71388 q^{85} - 66132 q^{89} + 301836 q^{91} - 82944 q^{95} + 46245 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(271\) \(325\) \(353\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

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



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
\(3\) 0 0
\(4\) 0 0
\(5\) −33.0434 + 57.2329i −0.591099 + 1.02381i 0.402986 + 0.915206i \(0.367972\pi\)
−0.994085 + 0.108607i \(0.965361\pi\)
\(6\) 0 0
\(7\) 57.0952 + 98.8918i 0.440407 + 0.762808i 0.997720 0.0674952i \(-0.0215007\pi\)
−0.557312 + 0.830303i \(0.688167\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 192.812 + 333.961i 0.480455 + 0.832173i 0.999749 0.0224231i \(-0.00713811\pi\)
−0.519293 + 0.854596i \(0.673805\pi\)
\(12\) 0 0
\(13\) −516.287 + 894.236i −0.847292 + 1.46755i 0.0363242 + 0.999340i \(0.488435\pi\)
−0.883616 + 0.468212i \(0.844898\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −959.020 −0.804832 −0.402416 0.915457i \(-0.631829\pi\)
−0.402416 + 0.915457i \(0.631829\pi\)
\(18\) 0 0
\(19\) 464.576 0.295239 0.147619 0.989044i \(-0.452839\pi\)
0.147619 + 0.989044i \(0.452839\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1151.70 + 1994.81i −0.453963 + 0.786288i −0.998628 0.0523662i \(-0.983324\pi\)
0.544664 + 0.838654i \(0.316657\pi\)
\(24\) 0 0
\(25\) −621.235 1076.01i −0.198795 0.344323i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3549.13 + 6147.28i 0.783659 + 1.35734i 0.929797 + 0.368073i \(0.119982\pi\)
−0.146138 + 0.989264i \(0.546684\pi\)
\(30\) 0 0
\(31\) 3881.53 6723.00i 0.725435 1.25649i −0.233360 0.972391i \(-0.574972\pi\)
0.958795 0.284100i \(-0.0916947\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7546.48 −1.04130
\(36\) 0 0
\(37\) 9317.57 1.11892 0.559459 0.828858i \(-0.311009\pi\)
0.559459 + 0.828858i \(0.311009\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6661.64 + 11538.3i −0.618901 + 1.07197i 0.370785 + 0.928719i \(0.379089\pi\)
−0.989687 + 0.143250i \(0.954245\pi\)
\(42\) 0 0
\(43\) −1056.29 1829.55i −0.0871190 0.150895i 0.819173 0.573546i \(-0.194433\pi\)
−0.906292 + 0.422652i \(0.861099\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1247.40 2160.56i −0.0823686 0.142667i 0.821898 0.569634i \(-0.192915\pi\)
−0.904267 + 0.426968i \(0.859582\pi\)
\(48\) 0 0
\(49\) 1883.78 3262.80i 0.112083 0.194133i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10044.2 0.491163 0.245582 0.969376i \(-0.421021\pi\)
0.245582 + 0.969376i \(0.421021\pi\)
\(54\) 0 0
\(55\) −25484.7 −1.13599
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2720.33 + 4711.75i −0.101740 + 0.176219i −0.912402 0.409296i \(-0.865774\pi\)
0.810662 + 0.585515i \(0.199108\pi\)
\(60\) 0 0
\(61\) −17094.4 29608.4i −0.588206 1.01880i −0.994467 0.105046i \(-0.966501\pi\)
0.406261 0.913757i \(-0.366832\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −34119.8 59097.2i −1.00167 1.73494i
\(66\) 0 0
\(67\) 26792.6 46406.2i 0.729169 1.26296i −0.228066 0.973646i \(-0.573240\pi\)
0.957235 0.289311i \(-0.0934263\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −970.010 −0.0228365 −0.0114183 0.999935i \(-0.503635\pi\)
−0.0114183 + 0.999935i \(0.503635\pi\)
\(72\) 0 0
\(73\) −72400.3 −1.59013 −0.795066 0.606523i \(-0.792564\pi\)
−0.795066 + 0.606523i \(0.792564\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −22017.3 + 38135.1i −0.423192 + 0.732990i
\(78\) 0 0
\(79\) 16098.9 + 27884.0i 0.290220 + 0.502676i 0.973862 0.227142i \(-0.0729382\pi\)
−0.683642 + 0.729818i \(0.739605\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 18046.6 + 31257.6i 0.287541 + 0.498036i 0.973222 0.229866i \(-0.0738289\pi\)
−0.685681 + 0.727902i \(0.740496\pi\)
\(84\) 0 0
\(85\) 31689.3 54887.5i 0.475735 0.823998i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −42622.2 −0.570375 −0.285188 0.958472i \(-0.592056\pi\)
−0.285188 + 0.958472i \(0.592056\pi\)
\(90\) 0 0
\(91\) −117910. −1.49261
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −15351.2 + 26589.0i −0.174515 + 0.302269i
\(96\) 0 0
\(97\) 21926.9 + 37978.6i 0.236618 + 0.409835i 0.959742 0.280884i \(-0.0906275\pi\)
−0.723123 + 0.690719i \(0.757294\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 12056.7 + 20882.8i 0.117604 + 0.203697i 0.918818 0.394682i \(-0.129145\pi\)
−0.801213 + 0.598379i \(0.795812\pi\)
\(102\) 0 0
\(103\) −68610.4 + 118837.i −0.637231 + 1.10372i 0.348807 + 0.937195i \(0.386587\pi\)
−0.986038 + 0.166522i \(0.946746\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −167979. −1.41839 −0.709195 0.705012i \(-0.750941\pi\)
−0.709195 + 0.705012i \(0.750941\pi\)
\(108\) 0 0
\(109\) −21224.0 −0.171104 −0.0855521 0.996334i \(-0.527265\pi\)
−0.0855521 + 0.996334i \(0.527265\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −14420.8 + 24977.5i −0.106241 + 0.184015i −0.914245 0.405163i \(-0.867215\pi\)
0.808003 + 0.589178i \(0.200548\pi\)
\(114\) 0 0
\(115\) −76112.4 131831.i −0.536674 0.929547i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −54755.4 94839.2i −0.354454 0.613932i
\(120\) 0 0
\(121\) 6172.35 10690.8i 0.0383255 0.0663816i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −124410. −0.712167
\(126\) 0 0
\(127\) 215062. 1.18319 0.591594 0.806236i \(-0.298499\pi\)
0.591594 + 0.806236i \(0.298499\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 94654.7 163947.i 0.481908 0.834688i −0.517877 0.855455i \(-0.673277\pi\)
0.999784 + 0.0207668i \(0.00661075\pi\)
\(132\) 0 0
\(133\) 26525.1 + 45942.8i 0.130025 + 0.225210i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1809.78 3134.63i −0.00823805 0.0142687i 0.861877 0.507117i \(-0.169289\pi\)
−0.870115 + 0.492849i \(0.835956\pi\)
\(138\) 0 0
\(139\) 60440.1 104685.i 0.265331 0.459567i −0.702319 0.711862i \(-0.747852\pi\)
0.967650 + 0.252295i \(0.0811854\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −398186. −1.62834
\(144\) 0 0
\(145\) −469102. −1.85288
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 37408.2 64792.9i 0.138039 0.239090i −0.788715 0.614758i \(-0.789254\pi\)
0.926754 + 0.375668i \(0.122587\pi\)
\(150\) 0 0
\(151\) −179976. 311728.i −0.642352 1.11259i −0.984906 0.173088i \(-0.944625\pi\)
0.342554 0.939498i \(-0.388708\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 256518. + 444302.i 0.857607 + 1.48542i
\(156\) 0 0
\(157\) −23983.3 + 41540.3i −0.0776533 + 0.134499i −0.902237 0.431241i \(-0.858076\pi\)
0.824584 + 0.565740i \(0.191409\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −263027. −0.799715
\(162\) 0 0
\(163\) −361063. −1.06442 −0.532212 0.846611i \(-0.678639\pi\)
−0.532212 + 0.846611i \(0.678639\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −58233.2 + 100863.i −0.161577 + 0.279860i −0.935434 0.353500i \(-0.884991\pi\)
0.773857 + 0.633360i \(0.218325\pi\)
\(168\) 0 0
\(169\) −347458. 601816.i −0.935807 1.62086i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 51762.9 + 89656.0i 0.131493 + 0.227753i 0.924252 0.381782i \(-0.124689\pi\)
−0.792759 + 0.609535i \(0.791356\pi\)
\(174\) 0 0
\(175\) 70939.0 122870.i 0.175102 0.303285i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −78941.2 −0.184150 −0.0920748 0.995752i \(-0.529350\pi\)
−0.0920748 + 0.995752i \(0.529350\pi\)
\(180\) 0 0
\(181\) 586108. 1.32978 0.664892 0.746940i \(-0.268478\pi\)
0.664892 + 0.746940i \(0.268478\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −307884. + 533271.i −0.661391 + 1.14556i
\(186\) 0 0
\(187\) −184911. 320275.i −0.386686 0.669760i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −386026. 668617.i −0.765656 1.32615i −0.939899 0.341452i \(-0.889081\pi\)
0.174243 0.984703i \(-0.444252\pi\)
\(192\) 0 0
\(193\) 8719.67 15102.9i 0.0168503 0.0291855i −0.857477 0.514522i \(-0.827969\pi\)
0.874328 + 0.485336i \(0.161303\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −340729. −0.625524 −0.312762 0.949832i \(-0.601254\pi\)
−0.312762 + 0.949832i \(0.601254\pi\)
\(198\) 0 0
\(199\) −14239.7 −0.0254899 −0.0127450 0.999919i \(-0.504057\pi\)
−0.0127450 + 0.999919i \(0.504057\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −405277. + 701960.i −0.690258 + 1.19556i
\(204\) 0 0
\(205\) −440246. 762529.i −0.731663 1.26728i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 89576.0 + 155150.i 0.141849 + 0.245690i
\(210\) 0 0
\(211\) −24049.2 + 41654.4i −0.0371873 + 0.0644102i −0.884020 0.467449i \(-0.845173\pi\)
0.846833 + 0.531859i \(0.178506\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 139614. 0.205984
\(216\) 0 0
\(217\) 886466. 1.27795
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 495130. 857590.i 0.681928 1.18113i
\(222\) 0 0
\(223\) 391988. + 678944.i 0.527851 + 0.914264i 0.999473 + 0.0324633i \(0.0103352\pi\)
−0.471622 + 0.881801i \(0.656331\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 277898. + 481334.i 0.357949 + 0.619986i 0.987618 0.156878i \(-0.0501428\pi\)
−0.629669 + 0.776863i \(0.716810\pi\)
\(228\) 0 0
\(229\) −57346.2 + 99326.6i −0.0722630 + 0.125163i −0.899893 0.436111i \(-0.856355\pi\)
0.827630 + 0.561274i \(0.189689\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 111177. 0.134161 0.0670805 0.997748i \(-0.478632\pi\)
0.0670805 + 0.997748i \(0.478632\pi\)
\(234\) 0 0
\(235\) 164874. 0.194752
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 369523. 640033.i 0.418453 0.724782i −0.577331 0.816510i \(-0.695906\pi\)
0.995784 + 0.0917279i \(0.0292390\pi\)
\(240\) 0 0
\(241\) 645180. + 1.11748e6i 0.715547 + 1.23936i 0.962748 + 0.270400i \(0.0871559\pi\)
−0.247201 + 0.968964i \(0.579511\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 124493. + 215628.i 0.132504 + 0.229504i
\(246\) 0 0
\(247\) −239855. + 415441.i −0.250153 + 0.433278i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.29383e6 1.29626 0.648129 0.761530i \(-0.275552\pi\)
0.648129 + 0.761530i \(0.275552\pi\)
\(252\) 0 0
\(253\) −888250. −0.872437
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 312181. 540713.i 0.294831 0.510663i −0.680114 0.733106i \(-0.738070\pi\)
0.974946 + 0.222443i \(0.0714032\pi\)
\(258\) 0 0
\(259\) 531988. + 921431.i 0.492780 + 0.853519i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 582495. + 1.00891e6i 0.519281 + 0.899421i 0.999749 + 0.0224089i \(0.00713358\pi\)
−0.480468 + 0.877012i \(0.659533\pi\)
\(264\) 0 0
\(265\) −331895. + 574859.i −0.290326 + 0.502859i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.58951e6 −1.33932 −0.669659 0.742669i \(-0.733560\pi\)
−0.669659 + 0.742669i \(0.733560\pi\)
\(270\) 0 0
\(271\) 977878. 0.808837 0.404419 0.914574i \(-0.367474\pi\)
0.404419 + 0.914574i \(0.367474\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 239563. 414936.i 0.191024 0.330864i
\(276\) 0 0
\(277\) 645243. + 1.11759e6i 0.505271 + 0.875154i 0.999981 + 0.00609677i \(0.00194067\pi\)
−0.494711 + 0.869058i \(0.664726\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 833662. + 1.44395e6i 0.629831 + 1.09090i 0.987585 + 0.157084i \(0.0502094\pi\)
−0.357754 + 0.933816i \(0.616457\pi\)
\(282\) 0 0
\(283\) 564568. 977861.i 0.419035 0.725790i −0.576808 0.816880i \(-0.695702\pi\)
0.995843 + 0.0910901i \(0.0290351\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.52139e6 −1.09027
\(288\) 0 0
\(289\) −500138. −0.352245
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.02815e6 1.78081e6i 0.699663 1.21185i −0.268921 0.963162i \(-0.586667\pi\)
0.968583 0.248689i \(-0.0799997\pi\)
\(294\) 0 0
\(295\) −179778. 311385.i −0.120277 0.208325i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.18922e6 2.05979e6i −0.769279 1.33243i
\(300\) 0 0
\(301\) 120618. 208917.i 0.0767357 0.132910i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.25943e6 1.39075
\(306\) 0 0
\(307\) −1.52149e6 −0.921346 −0.460673 0.887570i \(-0.652392\pi\)
−0.460673 + 0.887570i \(0.652392\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.35420e6 2.34554e6i 0.793928 1.37512i −0.129589 0.991568i \(-0.541366\pi\)
0.923517 0.383556i \(-0.125301\pi\)
\(312\) 0 0
\(313\) 325583. + 563926.i 0.187845 + 0.325358i 0.944532 0.328420i \(-0.106516\pi\)
−0.756686 + 0.653778i \(0.773183\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −58055.3 100555.i −0.0324484 0.0562024i 0.849345 0.527838i \(-0.176997\pi\)
−0.881794 + 0.471636i \(0.843664\pi\)
\(318\) 0 0
\(319\) −1.36863e6 + 2.37054e6i −0.753026 + 1.30428i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −445538. −0.237618
\(324\) 0 0
\(325\) 1.28294e6 0.673750
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 142441. 246715.i 0.0725514 0.125663i
\(330\) 0 0
\(331\) 158828. + 275099.i 0.0796816 + 0.138013i 0.903113 0.429404i \(-0.141276\pi\)
−0.823431 + 0.567416i \(0.807943\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.77064e6 + 3.06684e6i 0.862021 + 1.49306i
\(336\) 0 0
\(337\) −215275. + 372867.i −0.103257 + 0.178846i −0.913025 0.407904i \(-0.866260\pi\)
0.809768 + 0.586750i \(0.199593\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.99363e6 1.39416
\(342\) 0 0
\(343\) 2.34942e6 1.07826
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.07020e6 3.58569e6i 0.922971 1.59863i 0.128178 0.991751i \(-0.459087\pi\)
0.794793 0.606881i \(-0.207580\pi\)
\(348\) 0 0
\(349\) 1.73048e6 + 2.99727e6i 0.760505 + 1.31723i 0.942590 + 0.333951i \(0.108382\pi\)
−0.182085 + 0.983283i \(0.558285\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.32229e6 + 2.29027e6i 0.564792 + 0.978249i 0.997069 + 0.0765076i \(0.0243770\pi\)
−0.432277 + 0.901741i \(0.642290\pi\)
\(354\) 0 0
\(355\) 32052.5 55516.5i 0.0134987 0.0233804i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.13123e6 0.872761 0.436380 0.899762i \(-0.356260\pi\)
0.436380 + 0.899762i \(0.356260\pi\)
\(360\) 0 0
\(361\) −2.26027e6 −0.912834
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.39235e6 4.14368e6i 0.939925 1.62800i
\(366\) 0 0
\(367\) −640035. 1.10857e6i −0.248049 0.429634i 0.714935 0.699191i \(-0.246456\pi\)
−0.962985 + 0.269557i \(0.913123\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 573476. + 993289.i 0.216312 + 0.374663i
\(372\) 0 0
\(373\) 1.94733e6 3.37288e6i 0.724716 1.25524i −0.234375 0.972146i \(-0.575304\pi\)
0.959091 0.283099i \(-0.0913623\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.32949e6 −2.65595
\(378\) 0 0
\(379\) 2.83335e6 1.01322 0.506609 0.862176i \(-0.330899\pi\)
0.506609 + 0.862176i \(0.330899\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.18249e6 + 2.04814e6i −0.411909 + 0.713448i −0.995099 0.0988883i \(-0.968471\pi\)
0.583189 + 0.812336i \(0.301805\pi\)
\(384\) 0 0
\(385\) −1.45505e6 2.52023e6i −0.500296 0.866539i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −871692. 1.50981e6i −0.292071 0.505882i 0.682228 0.731139i \(-0.261011\pi\)
−0.974299 + 0.225257i \(0.927678\pi\)
\(390\) 0 0
\(391\) 1.10451e6 1.91306e6i 0.365364 0.632830i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.12785e6 −0.686195
\(396\) 0 0
\(397\) 4.81109e6 1.53203 0.766015 0.642823i \(-0.222237\pi\)
0.766015 + 0.642823i \(0.222237\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.27703e6 + 3.94394e6i −0.707145 + 1.22481i 0.258767 + 0.965940i \(0.416684\pi\)
−0.965912 + 0.258871i \(0.916649\pi\)
\(402\) 0 0
\(403\) 4.00797e6 + 6.94200e6i 1.22931 + 2.12923i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.79654e6 + 3.11170e6i 0.537590 + 0.931133i
\(408\) 0 0
\(409\) −956414. + 1.65656e6i −0.282708 + 0.489664i −0.972051 0.234771i \(-0.924566\pi\)
0.689343 + 0.724435i \(0.257899\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −621271. −0.179228
\(414\) 0 0
\(415\) −2.38528e6 −0.679860
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.20697e6 + 3.82259e6i −0.614132 + 1.06371i 0.376404 + 0.926456i \(0.377161\pi\)
−0.990536 + 0.137252i \(0.956173\pi\)
\(420\) 0 0
\(421\) 872253. + 1.51079e6i 0.239848 + 0.415430i 0.960671 0.277690i \(-0.0895689\pi\)
−0.720822 + 0.693120i \(0.756236\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 595777. + 1.03192e6i 0.159997 + 0.277122i
\(426\) 0 0
\(427\) 1.95202e6 3.38099e6i 0.518100 0.897376i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.21740e6 −0.574976 −0.287488 0.957784i \(-0.592820\pi\)
−0.287488 + 0.957784i \(0.592820\pi\)
\(432\) 0 0
\(433\) −3.63511e6 −0.931746 −0.465873 0.884851i \(-0.654260\pi\)
−0.465873 + 0.884851i \(0.654260\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −535054. + 926741.i −0.134028 + 0.232143i
\(438\) 0 0
\(439\) −1.54413e6 2.67452e6i −0.382405 0.662345i 0.609001 0.793170i \(-0.291571\pi\)
−0.991405 + 0.130825i \(0.958237\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.50048e6 + 6.06302e6i 0.847459 + 1.46784i 0.883468 + 0.468491i \(0.155202\pi\)
−0.0360092 + 0.999351i \(0.511465\pi\)
\(444\) 0 0
\(445\) 1.40838e6 2.43939e6i 0.337148 0.583958i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5.00196e6 1.17091 0.585456 0.810704i \(-0.300916\pi\)
0.585456 + 0.810704i \(0.300916\pi\)
\(450\) 0 0
\(451\) −5.13778e6 −1.18942
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.89615e6 6.74833e6i 0.882282 1.52816i
\(456\) 0 0
\(457\) −1.78581e6 3.09311e6i −0.399985 0.692795i 0.593738 0.804658i \(-0.297651\pi\)
−0.993724 + 0.111863i \(0.964318\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.75061e6 + 4.76420e6i 0.602805 + 1.04409i 0.992394 + 0.123100i \(0.0392837\pi\)
−0.389589 + 0.920989i \(0.627383\pi\)
\(462\) 0 0
\(463\) −1.42708e6 + 2.47177e6i −0.309382 + 0.535866i −0.978227 0.207536i \(-0.933456\pi\)
0.668845 + 0.743402i \(0.266789\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.16552e6 −1.52039 −0.760196 0.649694i \(-0.774897\pi\)
−0.760196 + 0.649694i \(0.774897\pi\)
\(468\) 0 0
\(469\) 6.11892e6 1.28452
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 407332. 705520.i 0.0837136 0.144996i
\(474\) 0 0
\(475\) −288611. 499889.i −0.0586920 0.101658i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1.64185e6 2.84377e6i −0.326961 0.566312i 0.654947 0.755675i \(-0.272691\pi\)
−0.981907 + 0.189363i \(0.939358\pi\)
\(480\) 0 0
\(481\) −4.81054e6 + 8.33210e6i −0.948050 + 1.64207i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −2.89816e6 −0.559459
\(486\) 0 0
\(487\) −2.18097e6 −0.416704 −0.208352 0.978054i \(-0.566810\pi\)
−0.208352 + 0.978054i \(0.566810\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.96974e6 8.60785e6i 0.930315 1.61135i 0.147534 0.989057i \(-0.452866\pi\)
0.782782 0.622297i \(-0.213800\pi\)
\(492\) 0 0
\(493\) −3.40369e6 5.89536e6i −0.630714 1.09243i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −55382.9 95926.0i −0.0100574 0.0174199i
\(498\) 0 0
\(499\) −2.26386e6 + 3.92112e6i −0.407004 + 0.704951i −0.994552 0.104238i \(-0.966760\pi\)
0.587549 + 0.809189i \(0.300093\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4.40282e6 −0.775909 −0.387955 0.921679i \(-0.626818\pi\)
−0.387955 + 0.921679i \(0.626818\pi\)
\(504\) 0 0
\(505\) −1.59357e6 −0.278063
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.91063e6 5.04135e6i 0.497957 0.862487i −0.502040 0.864844i \(-0.667417\pi\)
0.999997 + 0.00235703i \(0.000750266\pi\)
\(510\) 0 0
\(511\) −4.13371e6 7.15979e6i −0.700306 1.21296i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.53424e6 7.85354e6i −0.753333 1.30481i
\(516\) 0 0
\(517\) 481028. 833166.i 0.0791488 0.137090i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.09649e7 −1.76975 −0.884875 0.465829i \(-0.845756\pi\)
−0.884875 + 0.465829i \(0.845756\pi\)
\(522\) 0 0
\(523\) 3.83705e6 0.613399 0.306700 0.951806i \(-0.400775\pi\)
0.306700 + 0.951806i \(0.400775\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.72246e6 + 6.44750e6i −0.583854 + 1.01126i
\(528\) 0 0
\(529\) 565332. + 979183.i 0.0878343 + 0.152133i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.87864e6 1.19141e7i −1.04878 1.81654i
\(534\) 0 0
\(535\) 5.55060e6 9.61392e6i 0.838408 1.45217i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.45286e6 0.215403
\(540\) 0 0
\(541\) −4.88767e6 −0.717974 −0.358987 0.933343i \(-0.616878\pi\)
−0.358987 + 0.933343i \(0.616878\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 701313. 1.21471e6i 0.101139 0.175179i
\(546\) 0 0
\(547\) −3.46672e6 6.00454e6i −0.495394 0.858047i 0.504592 0.863358i \(-0.331643\pi\)
−0.999986 + 0.00531061i \(0.998310\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.64884e6 + 2.85588e6i 0.231366 + 0.400739i
\(552\) 0 0
\(553\) −1.83834e6 + 3.18409e6i −0.255630 + 0.442764i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.83771e6 1.34356 0.671778 0.740752i \(-0.265531\pi\)
0.671778 + 0.740752i \(0.265531\pi\)
\(558\) 0 0
\(559\) 2.18140e6 0.295261
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.36609e6 + 2.36613e6i −0.181638 + 0.314607i −0.942439 0.334379i \(-0.891473\pi\)
0.760800 + 0.648986i \(0.224807\pi\)
\(564\) 0 0
\(565\) −953024. 1.65069e6i −0.125598 0.217542i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 4.71998e6 + 8.17525e6i 0.611167 + 1.05857i 0.991044 + 0.133535i \(0.0426329\pi\)
−0.379877 + 0.925037i \(0.624034\pi\)
\(570\) 0 0
\(571\) −6.33948e6 + 1.09803e7i −0.813698 + 1.40937i 0.0965611 + 0.995327i \(0.469216\pi\)
−0.910259 + 0.414039i \(0.864118\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.86191e6 0.360983
\(576\) 0 0
\(577\) −1.51657e7 −1.89637 −0.948187 0.317712i \(-0.897086\pi\)
−0.948187 + 0.317712i \(0.897086\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.06075e6 + 3.56932e6i −0.253270 + 0.438677i
\(582\) 0 0
\(583\) 1.93665e6 + 3.35437e6i 0.235982 + 0.408733i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −911702. 1.57911e6i −0.109209 0.189155i 0.806241 0.591587i \(-0.201498\pi\)
−0.915450 + 0.402432i \(0.868165\pi\)
\(588\) 0 0
\(589\) 1.80327e6 3.12335e6i 0.214176 0.370965i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.27957e7 −1.49427 −0.747133 0.664674i \(-0.768570\pi\)
−0.747133 + 0.664674i \(0.768570\pi\)
\(594\) 0 0
\(595\) 7.23723e6 0.838069
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −4.38212e6 + 7.59006e6i −0.499020 + 0.864327i −0.999999 0.00113166i \(-0.999640\pi\)
0.500980 + 0.865459i \(0.332973\pi\)
\(600\) 0 0
\(601\) −834793. 1.44590e6i −0.0942741 0.163288i 0.815031 0.579417i \(-0.196720\pi\)
−0.909305 + 0.416129i \(0.863386\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 407911. + 706523.i 0.0453082 + 0.0784762i
\(606\) 0 0
\(607\) −5.08535e6 + 8.80809e6i −0.560208 + 0.970308i 0.437270 + 0.899330i \(0.355945\pi\)
−0.997478 + 0.0709782i \(0.977388\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.57607e6 0.279161
\(612\) 0 0
\(613\) 1.37829e7 1.48146 0.740730 0.671803i \(-0.234480\pi\)
0.740730 + 0.671803i \(0.234480\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.05409e6 + 1.04860e7i −0.640230 + 1.10891i 0.345152 + 0.938547i \(0.387827\pi\)
−0.985381 + 0.170363i \(0.945506\pi\)
\(618\) 0 0
\(619\) −8.48665e6 1.46993e7i −0.890245 1.54195i −0.839581 0.543234i \(-0.817200\pi\)
−0.0506641 0.998716i \(-0.516134\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.43352e6 4.21498e6i −0.251197 0.435087i
\(624\) 0 0
\(625\) 6.05231e6 1.04829e7i 0.619756 1.07345i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −8.93574e6 −0.900541
\(630\) 0 0
\(631\) −7.90092e6 −0.789958 −0.394979 0.918690i \(-0.629248\pi\)
−0.394979 + 0.918690i \(0.629248\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −7.10637e6 + 1.23086e7i −0.699381 + 1.21136i
\(636\) 0 0
\(637\) 1.94514e6 + 3.36908e6i 0.189934 + 0.328975i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5.33807e6 + 9.24581e6i 0.513144 + 0.888792i 0.999884 + 0.0152446i \(0.00485269\pi\)
−0.486740 + 0.873547i \(0.661814\pi\)
\(642\) 0 0
\(643\) 4.68885e6 8.12132e6i 0.447238 0.774639i −0.550967 0.834527i \(-0.685741\pi\)
0.998205 + 0.0598882i \(0.0190744\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.44013e6 −0.886579 −0.443289 0.896379i \(-0.646189\pi\)
−0.443289 + 0.896379i \(0.646189\pi\)
\(648\) 0 0
\(649\) −2.09805e6 −0.195526
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.44455e6 1.28943e7i 0.683212 1.18336i −0.290783 0.956789i \(-0.593916\pi\)
0.973995 0.226569i \(-0.0727509\pi\)
\(654\) 0 0
\(655\) 6.25543e6 + 1.08347e7i 0.569710 + 0.986766i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.82391e6 + 6.62320e6i 0.343000 + 0.594093i 0.984988 0.172621i \(-0.0552237\pi\)
−0.641989 + 0.766714i \(0.721890\pi\)
\(660\) 0 0
\(661\) 2.50269e6 4.33479e6i 0.222794 0.385891i −0.732861 0.680378i \(-0.761816\pi\)
0.955655 + 0.294487i \(0.0951489\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.50592e6 −0.307431
\(666\) 0 0
\(667\) −1.63502e7 −1.42301
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6.59202e6 1.14177e7i 0.565214 0.978979i
\(672\) 0 0
\(673\) −1.07220e6 1.85710e6i −0.0912511 0.158051i 0.816787 0.576940i \(-0.195753\pi\)
−0.908038 + 0.418888i \(0.862420\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −4.75463e6 8.23526e6i −0.398699 0.690567i 0.594867 0.803824i \(-0.297205\pi\)
−0.993566 + 0.113257i \(0.963871\pi\)
\(678\) 0 0
\(679\) −2.50385e6 + 4.33679e6i −0.208417 + 0.360989i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −5.47266e6 −0.448897 −0.224449 0.974486i \(-0.572058\pi\)
−0.224449 + 0.974486i \(0.572058\pi\)
\(684\) 0 0
\(685\) 239205. 0.0194780
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.18570e6 + 8.98189e6i −0.416159 + 0.720808i
\(690\) 0 0
\(691\) 3.44084e6 + 5.95970e6i 0.274138 + 0.474821i 0.969917 0.243435i \(-0.0782742\pi\)
−0.695779 + 0.718255i \(0.744941\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.99430e6 + 6.91832e6i 0.313674 + 0.543299i
\(696\) 0 0
\(697\) 6.38864e6 1.10655e7i 0.498112 0.862755i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −5.19615e6 −0.399380 −0.199690 0.979859i \(-0.563994\pi\)
−0.199690 + 0.979859i \(0.563994\pi\)
\(702\) 0 0
\(703\) 4.32872e6 0.330348
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.37676e6 + 2.38461e6i −0.103588 + 0.179419i
\(708\) 0 0
\(709\) −287410. 497809.i −0.0214727 0.0371918i 0.855089 0.518481i \(-0.173502\pi\)
−0.876562 + 0.481289i \(0.840169\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.94074e6 + 1.54858e7i 0.658642 + 1.14080i
\(714\) 0 0
\(715\) 1.31574e7 2.27893e7i 0.962511 1.66712i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −400906. −0.0289215 −0.0144607 0.999895i \(-0.504603\pi\)
−0.0144607 + 0.999895i \(0.504603\pi\)
\(720\) 0 0
\(721\) −1.56693e7 −1.12256
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.40969e6 7.63781e6i 0.311575 0.539664i
\(726\) 0 0
\(727\) 5.27154e6 + 9.13058e6i 0.369915 + 0.640711i 0.989552 0.144177i \(-0.0460535\pi\)
−0.619637 + 0.784889i \(0.712720\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.01301e6 + 1.75458e6i 0.0701162 + 0.121445i
\(732\) 0 0
\(733\) −6.96868e6 + 1.20701e7i −0.479061 + 0.829758i −0.999712 0.0240119i \(-0.992356\pi\)
0.520651 + 0.853770i \(0.325689\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.06638e7 1.40133
\(738\) 0 0
\(739\) 9.88034e6 0.665519 0.332760 0.943012i \(-0.392020\pi\)
0.332760 + 0.943012i \(0.392020\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.66695e6 8.08339e6i 0.310142 0.537182i −0.668251 0.743936i \(-0.732957\pi\)
0.978393 + 0.206754i \(0.0662899\pi\)
\(744\) 0 0
\(745\) 2.47219e6 + 4.28195e6i 0.163189 + 0.282652i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.59080e6 1.66117e7i −0.624669 1.08196i
\(750\) 0 0
\(751\) 1.43761e7 2.49001e7i 0.930124 1.61102i 0.147018 0.989134i \(-0.453032\pi\)
0.783106 0.621888i \(-0.213634\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.37881e7 1.51877
\(756\) 0 0
\(757\) −4.94587e6 −0.313691 −0.156846 0.987623i \(-0.550133\pi\)
−0.156846 + 0.987623i \(0.550133\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.56829e6 7.91251e6i 0.285951 0.495282i −0.686888 0.726763i \(-0.741024\pi\)
0.972839 + 0.231481i \(0.0743571\pi\)
\(762\) 0 0
\(763\) −1.21179e6 2.09888e6i −0.0753555 0.130520i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.80894e6 4.86523e6i −0.172407 0.298617i
\(768\) 0 0
\(769\) −8.26221e6 + 1.43106e7i −0.503826 + 0.872652i 0.496164 + 0.868229i \(0.334741\pi\)
−0.999990 + 0.00442349i \(0.998592\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.01763e7 −0.612550 −0.306275 0.951943i \(-0.599083\pi\)
−0.306275 + 0.951943i \(0.599083\pi\)
\(774\) 0 0
\(775\) −9.64536e6 −0.576852
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.09484e6 + 5.36042e6i −0.182724 + 0.316486i
\(780\) 0 0
\(781\) −187030. 323945.i −0.0109719 0.0190040i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1.58498e6 2.74527e6i −0.0918015 0.159005i
\(786\) 0 0
\(787\) −9.38173e6 + 1.62496e7i −0.539941 + 0.935205i 0.458966 + 0.888454i \(0.348220\pi\)
−0.998907 + 0.0467507i \(0.985113\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.29343e6 −0.187158
\(792\) 0 0
\(793\) 3.53025e7 1.99353
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.23982e6 + 2.14744e6i −0.0691376 + 0.119750i −0.898522 0.438929i \(-0.855358\pi\)
0.829384 + 0.558678i \(0.188691\pi\)
\(798\) 0 0
\(799\) 1.19628e6 + 2.07202e6i 0.0662929 + 0.114823i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.39597e7 2.41788e7i −0.763987 1.32326i
\(804\) 0 0
\(805\) 8.69130e6 1.50538e7i 0.472711 0.818759i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.50880e7 1.34770 0.673852 0.738867i \(-0.264639\pi\)
0.673852 + 0.738867i \(0.264639\pi\)
\(810\) 0 0
\(811\) 4.28465e6 0.228751 0.114376 0.993438i \(-0.463513\pi\)
0.114376 + 0.993438i \(0.463513\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.19308e7 2.06647e7i 0.629179 1.08977i
\(816\) 0 0
\(817\) −490728. 849967.i −0.0257209 0.0445499i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.01563e7 + 1.75913e7i 0.525871 + 0.910836i 0.999546 + 0.0301356i \(0.00959392\pi\)
−0.473675 + 0.880700i \(0.657073\pi\)
\(822\) 0 0
\(823\) −6.58313e6 + 1.14023e7i −0.338792 + 0.586804i −0.984206 0.177029i \(-0.943351\pi\)
0.645414 + 0.763833i \(0.276685\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.82859e7 1.43816 0.719079 0.694928i \(-0.244564\pi\)
0.719079 + 0.694928i \(0.244564\pi\)
\(828\) 0 0
\(829\) 1.36608e7 0.690381 0.345190 0.938533i \(-0.387814\pi\)
0.345190 + 0.938533i \(0.387814\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.80658e6 + 3.12909e6i −0.0902080 + 0.156245i
\(834\) 0 0
\(835\) −3.84845e6 6.66571e6i −0.191016 0.330849i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 75641.6 + 131015.i 0.00370984 + 0.00642564i 0.867874 0.496784i \(-0.165486\pi\)
−0.864165 + 0.503209i \(0.832152\pi\)
\(840\) 0 0
\(841\) −1.49371e7 + 2.58718e7i −0.728244 + 1.26135i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.59249e7 2.21262
\(846\) 0 0
\(847\) 1.40965e6 0.0675152
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.07311e7 + 1.85868e7i −0.507948 + 0.879792i
\(852\) 0 0
\(853\) 1.38831e7 + 2.40462e7i 0.653301 + 1.13155i 0.982317 + 0.187226i \(0.0599498\pi\)
−0.329015 + 0.944325i \(0.606717\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.76136e7 3.05077e7i −0.819212 1.41892i −0.906263 0.422713i \(-0.861078\pi\)
0.0870512 0.996204i \(-0.472256\pi\)
\(858\) 0 0
\(859\) −2.02341e7 + 3.50464e7i −0.935622 + 1.62054i −0.162100 + 0.986774i \(0.551827\pi\)
−0.773522 + 0.633770i \(0.781507\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.96865e7 −0.899789 −0.449895 0.893082i \(-0.648538\pi\)
−0.449895 + 0.893082i \(0.648538\pi\)
\(864\) 0 0
\(865\) −6.84170e6 −0.310902
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6.20812e6 + 1.07528e7i −0.278876 + 0.483027i
\(870\) 0 0
\(871\) 2.76654e7 + 4.79178e7i 1.23564 + 2.14019i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −7.10324e6 1.23032e7i −0.313644 0.543247i
\(876\) 0 0
\(877\) −1.31221e7 + 2.27282e7i −0.576109 + 0.997850i 0.419811 + 0.907611i \(0.362096\pi\)
−0.995920 + 0.0902384i \(0.971237\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.29423e7 0.561786 0.280893 0.959739i \(-0.409369\pi\)
0.280893 + 0.959739i \(0.409369\pi\)
\(882\) 0 0
\(883\) −3.39097e6 −0.146360 −0.0731800 0.997319i \(-0.523315\pi\)
−0.0731800 + 0.997319i \(0.523315\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −3.83796e6 + 6.64753e6i −0.163791 + 0.283695i −0.936225 0.351400i \(-0.885706\pi\)
0.772434 + 0.635095i \(0.219039\pi\)
\(888\) 0 0
\(889\) 1.22790e7 + 2.12678e7i 0.521085 + 0.902545i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −579513. 1.00375e6i −0.0243184 0.0421207i
\(894\) 0 0
\(895\) 2.60849e6 4.51803e6i 0.108851 0.188535i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.51042e7 2.27398
\(900\) 0 0
\(901\) −9.63260e6 −0.395304
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.93670e7 + 3.35446e7i −0.786033 + 1.36145i
\(906\) 0 0
\(907\) 5.44435e6 + 9.42990e6i 0.219750 + 0.380618i 0.954731 0.297469i \(-0.0961426\pi\)
−0.734982 + 0.678087i \(0.762809\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 90239.2 + 156299.i 0.00360246 + 0.00623964i 0.867821 0.496877i \(-0.165520\pi\)
−0.864219 + 0.503117i \(0.832187\pi\)
\(912\) 0 0
\(913\) −6.95920e6 + 1.20537e7i −0.276301 + 0.478568i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.16173e7 0.848942
\(918\) 0 0
\(919\) 2.24831e7 0.878148 0.439074 0.898451i \(-0.355307\pi\)
0.439074 + 0.898451i \(0.355307\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 500804. 867418.i 0.0193492 0.0335138i
\(924\) 0 0
\(925\) −5.78840e6 1.00258e7i −0.222436 0.385270i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2.06620e7 3.57877e7i −0.785477 1.36049i −0.928714 0.370797i \(-0.879084\pi\)
0.143237 0.989688i \(-0.454249\pi\)
\(930\) 0 0
\(931\) 875159. 1.51582e6i 0.0330912 0.0573157i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2.44403e7 0.914278
\(936\) 0 0
\(937\) 2.56470e7 0.954305 0.477153 0.878820i \(-0.341669\pi\)
0.477153 + 0.878820i \(0.341669\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 8.99880e6 1.55864e7i 0.331292 0.573814i −0.651474 0.758671i \(-0.725849\pi\)
0.982765 + 0.184857i \(0.0591823\pi\)
\(942\) 0 0
\(943\) −1.53445e7 2.65774e7i −0.561917 0.973269i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.25398e7 + 3.90400e7i 0.816723 + 1.41461i 0.908084 + 0.418788i \(0.137545\pi\)
−0.0913614 + 0.995818i \(0.529122\pi\)
\(948\) 0 0
\(949\) 3.73793e7 6.47429e7i 1.34731 2.33360i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2.21374e7 −0.789577 −0.394788 0.918772i \(-0.629182\pi\)
−0.394788 + 0.918772i \(0.629182\pi\)
\(954\) 0 0
\(955\) 5.10225e7 1.81031
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 206660. 357945.i 0.00725619 0.0125681i
\(960\) 0 0
\(961\) −1.58180e7 2.73975e7i −0.552512 0.956979i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 576255. + 998103.i 0.0199203 + 0.0345030i
\(966\) 0 0
\(967\) −9.81614e6 + 1.70020e7i −0.337578 + 0.584703i −0.983977 0.178297i \(-0.942941\pi\)
0.646398 + 0.763000i \(0.276274\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −3.46539e7 −1.17952 −0.589758 0.807580i \(-0.700777\pi\)
−0.589758 + 0.807580i \(0.700777\pi\)
\(972\) 0 0
\(973\) 1.38034e7 0.467415
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.34006e7 2.32106e7i 0.449148 0.777947i −0.549183 0.835702i \(-0.685061\pi\)
0.998331 + 0.0577553i \(0.0183943\pi\)
\(978\) 0 0
\(979\) −8.21808e6 1.42341e7i −0.274040 0.474651i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.19040e7 + 2.06183e7i 0.392924 + 0.680565i 0.992834 0.119503i \(-0.0381301\pi\)
−0.599910 + 0.800068i \(0.704797\pi\)
\(984\) 0 0
\(985\) 1.12589e7 1.95009e7i 0.369746 0.640419i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.86614e6 0.158195
\(990\) 0 0
\(991\) 2.37156e7 0.767096 0.383548 0.923521i \(-0.374702\pi\)
0.383548 + 0.923521i \(0.374702\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 470529. 814980.i 0.0150671 0.0260969i
\(996\) 0 0
\(997\) 2.04170e7 + 3.53633e7i 0.650511 + 1.12672i 0.982999 + 0.183610i \(0.0587785\pi\)
−0.332488 + 0.943107i \(0.607888\pi\)
\(998\) 0 0
\(999\) 0 0
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 432.6.i.b.145.1 6
3.2 odd 2 144.6.i.b.49.1 6
4.3 odd 2 54.6.c.b.37.1 6
9.2 odd 6 144.6.i.b.97.1 6
9.7 even 3 inner 432.6.i.b.289.1 6
12.11 even 2 18.6.c.b.13.3 yes 6
36.7 odd 6 54.6.c.b.19.1 6
36.11 even 6 18.6.c.b.7.3 6
36.23 even 6 162.6.a.j.1.1 3
36.31 odd 6 162.6.a.i.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.6.c.b.7.3 6 36.11 even 6
18.6.c.b.13.3 yes 6 12.11 even 2
54.6.c.b.19.1 6 36.7 odd 6
54.6.c.b.37.1 6 4.3 odd 2
144.6.i.b.49.1 6 3.2 odd 2
144.6.i.b.97.1 6 9.2 odd 6
162.6.a.i.1.3 3 36.31 odd 6
162.6.a.j.1.1 3 36.23 even 6
432.6.i.b.145.1 6 1.1 even 1 trivial
432.6.i.b.289.1 6 9.7 even 3 inner