Properties

Label 3420.2.t.v.1261.1
Level $3420$
Weight $2$
Character 3420.1261
Analytic conductor $27.309$
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] = [3420,2,Mod(1261,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3420.1261");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.t (of order \(3\), degree \(2\), 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: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1783323.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1261.1
Root \(0.356769 - 0.617942i\) of defining polynomial
Character \(\chi\) \(=\) 3420.1261
Dual form 3420.2.t.v.3241.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{5} -2.20440 q^{7} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{5} -2.20440 q^{7} +1.20440 q^{11} +(0.500000 - 0.866025i) q^{13} +(-1.07031 - 1.85383i) q^{17} +(4.30660 - 0.673184i) q^{19} +(-4.63409 + 8.02649i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(4.16599 - 7.21570i) q^{29} +8.26819 q^{31} +(1.10220 + 1.90907i) q^{35} -2.20440 q^{37} +(-3.30660 - 5.72720i) q^{41} +(-1.17251 - 2.03084i) q^{43} +(-4.16599 + 7.21570i) q^{47} -2.14061 q^{49} +(0.134095 - 0.232259i) q^{53} +(-0.602201 - 1.04304i) q^{55} +(-4.16599 - 7.21570i) q^{59} +(1.70440 - 2.95211i) q^{61} -1.00000 q^{65} +(1.06379 - 1.84253i) q^{67} +(-5.23630 - 9.06953i) q^{71} +(2.42969 + 4.20835i) q^{73} -2.65498 q^{77} +(-8.20440 - 14.2104i) q^{79} +5.73181 q^{83} +(-1.07031 + 1.85383i) q^{85} +(-5.40880 + 9.36832i) q^{89} +(-1.10220 + 1.90907i) q^{91} +(-2.73630 - 3.39304i) q^{95} +(-7.87039 - 13.6319i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{5} + 4 q^{7} - 10 q^{11} + 3 q^{13} - 3 q^{17} - 14 q^{23} - 3 q^{25} + 6 q^{29} + 22 q^{31} - 2 q^{35} + 4 q^{37} + 6 q^{41} + 5 q^{43} - 6 q^{47} - 6 q^{49} - 13 q^{53} + 5 q^{55} - 6 q^{59} - 7 q^{61} - 6 q^{65} - 4 q^{67} - 9 q^{71} + 18 q^{73} - 40 q^{77} - 32 q^{79} + 62 q^{83} - 3 q^{85} + 2 q^{89} + 2 q^{91} + 6 q^{95} - 11 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) −0.500000 0.866025i −0.223607 0.387298i
\(6\) 0 0
\(7\) −2.20440 −0.833185 −0.416593 0.909093i \(-0.636776\pi\)
−0.416593 + 0.909093i \(0.636776\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.20440 0.363141 0.181570 0.983378i \(-0.441882\pi\)
0.181570 + 0.983378i \(0.441882\pi\)
\(12\) 0 0
\(13\) 0.500000 0.866025i 0.138675 0.240192i −0.788320 0.615265i \(-0.789049\pi\)
0.926995 + 0.375073i \(0.122382\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.07031 1.85383i −0.259588 0.449619i 0.706544 0.707669i \(-0.250253\pi\)
−0.966131 + 0.258050i \(0.916920\pi\)
\(18\) 0 0
\(19\) 4.30660 0.673184i 0.988002 0.154439i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.63409 + 8.02649i −0.966276 + 1.67364i −0.260127 + 0.965574i \(0.583765\pi\)
−0.706148 + 0.708064i \(0.749569\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.16599 7.21570i 0.773605 1.33992i −0.161971 0.986796i \(-0.551785\pi\)
0.935575 0.353127i \(-0.114882\pi\)
\(30\) 0 0
\(31\) 8.26819 1.48501 0.742505 0.669840i \(-0.233637\pi\)
0.742505 + 0.669840i \(0.233637\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.10220 + 1.90907i 0.186306 + 0.322691i
\(36\) 0 0
\(37\) −2.20440 −0.362401 −0.181201 0.983446i \(-0.557998\pi\)
−0.181201 + 0.983446i \(0.557998\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.30660 5.72720i −0.516405 0.894439i −0.999819 0.0190471i \(-0.993937\pi\)
0.483414 0.875392i \(-0.339397\pi\)
\(42\) 0 0
\(43\) −1.17251 2.03084i −0.178806 0.309701i 0.762666 0.646793i \(-0.223890\pi\)
−0.941472 + 0.337092i \(0.890557\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.16599 + 7.21570i −0.607672 + 1.05252i 0.383951 + 0.923353i \(0.374563\pi\)
−0.991623 + 0.129165i \(0.958770\pi\)
\(48\) 0 0
\(49\) −2.14061 −0.305802
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.134095 0.232259i 0.0184193 0.0319032i −0.856669 0.515867i \(-0.827470\pi\)
0.875088 + 0.483964i \(0.160803\pi\)
\(54\) 0 0
\(55\) −0.602201 1.04304i −0.0812007 0.140644i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.16599 7.21570i −0.542366 0.939405i −0.998768 0.0496310i \(-0.984195\pi\)
0.456402 0.889774i \(-0.349138\pi\)
\(60\) 0 0
\(61\) 1.70440 2.95211i 0.218226 0.377979i −0.736040 0.676939i \(-0.763306\pi\)
0.954266 + 0.298960i \(0.0966396\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 1.06379 1.84253i 0.129962 0.225101i −0.793699 0.608310i \(-0.791848\pi\)
0.923662 + 0.383209i \(0.125181\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.23630 9.06953i −0.621434 1.07636i −0.989219 0.146444i \(-0.953217\pi\)
0.367785 0.929911i \(-0.380116\pi\)
\(72\) 0 0
\(73\) 2.42969 + 4.20835i 0.284374 + 0.492550i 0.972457 0.233082i \(-0.0748809\pi\)
−0.688083 + 0.725632i \(0.741548\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.65498 −0.302564
\(78\) 0 0
\(79\) −8.20440 14.2104i −0.923067 1.59880i −0.794640 0.607080i \(-0.792341\pi\)
−0.128427 0.991719i \(-0.540993\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.73181 0.629148 0.314574 0.949233i \(-0.398138\pi\)
0.314574 + 0.949233i \(0.398138\pi\)
\(84\) 0 0
\(85\) −1.07031 + 1.85383i −0.116091 + 0.201076i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.40880 + 9.36832i −0.573332 + 0.993040i 0.422889 + 0.906182i \(0.361016\pi\)
−0.996221 + 0.0868585i \(0.972317\pi\)
\(90\) 0 0
\(91\) −1.10220 + 1.90907i −0.115542 + 0.200125i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.73630 3.39304i −0.280738 0.348118i
\(96\) 0 0
\(97\) −7.87039 13.6319i −0.799117 1.38411i −0.920192 0.391468i \(-0.871967\pi\)
0.121075 0.992643i \(-0.461366\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.27471 + 9.13606i −0.524853 + 0.909072i 0.474728 + 0.880133i \(0.342546\pi\)
−0.999581 + 0.0289397i \(0.990787\pi\)
\(102\) 0 0
\(103\) −0.731811 −0.0721075 −0.0360537 0.999350i \(-0.511479\pi\)
−0.0360537 + 0.999350i \(0.511479\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.3958 1.29502 0.647509 0.762058i \(-0.275811\pi\)
0.647509 + 0.762058i \(0.275811\pi\)
\(108\) 0 0
\(109\) −2.96811 5.14091i −0.284293 0.492410i 0.688144 0.725574i \(-0.258426\pi\)
−0.972437 + 0.233164i \(0.925092\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −15.6132 −1.46877 −0.734383 0.678735i \(-0.762529\pi\)
−0.734383 + 0.678735i \(0.762529\pi\)
\(114\) 0 0
\(115\) 9.26819 0.864263
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.35939 + 4.08658i 0.216285 + 0.374616i
\(120\) 0 0
\(121\) −9.54942 −0.868129
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 7.67699 13.2969i 0.681223 1.17991i −0.293385 0.955994i \(-0.594782\pi\)
0.974608 0.223918i \(-0.0718849\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.89780 6.75119i −0.340552 0.589854i 0.643983 0.765040i \(-0.277281\pi\)
−0.984535 + 0.175186i \(0.943947\pi\)
\(132\) 0 0
\(133\) −9.49348 + 1.48397i −0.823189 + 0.128676i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.23630 14.2657i 0.703674 1.21880i −0.263494 0.964661i \(-0.584875\pi\)
0.967168 0.254138i \(-0.0817919\pi\)
\(138\) 0 0
\(139\) 9.76819 16.9190i 0.828527 1.43505i −0.0706667 0.997500i \(-0.522513\pi\)
0.899194 0.437551i \(-0.144154\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.602201 1.04304i 0.0503586 0.0872236i
\(144\) 0 0
\(145\) −8.33198 −0.691933
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.83850 4.91642i −0.232539 0.402769i 0.726016 0.687678i \(-0.241370\pi\)
−0.958555 + 0.284909i \(0.908037\pi\)
\(150\) 0 0
\(151\) −14.2264 −1.15773 −0.578864 0.815424i \(-0.696504\pi\)
−0.578864 + 0.815424i \(0.696504\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.13409 7.16046i −0.332058 0.575142i
\(156\) 0 0
\(157\) 0.327492 + 0.567233i 0.0261367 + 0.0452701i 0.878798 0.477194i \(-0.158346\pi\)
−0.852661 + 0.522464i \(0.825013\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 10.2154 17.6936i 0.805087 1.39445i
\(162\) 0 0
\(163\) −12.6770 −0.992939 −0.496469 0.868054i \(-0.665370\pi\)
−0.496469 + 0.868054i \(0.665370\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.89780 + 6.75119i −0.301621 + 0.522422i −0.976503 0.215503i \(-0.930861\pi\)
0.674882 + 0.737925i \(0.264194\pi\)
\(168\) 0 0
\(169\) 6.00000 + 10.3923i 0.461538 + 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.37039 9.30179i −0.408303 0.707202i 0.586397 0.810024i \(-0.300546\pi\)
−0.994700 + 0.102822i \(0.967213\pi\)
\(174\) 0 0
\(175\) 1.10220 1.90907i 0.0833185 0.144312i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.54942 0.115809 0.0579044 0.998322i \(-0.481558\pi\)
0.0579044 + 0.998322i \(0.481558\pi\)
\(180\) 0 0
\(181\) 4.11320 7.12428i 0.305732 0.529544i −0.671692 0.740831i \(-0.734432\pi\)
0.977424 + 0.211287i \(0.0677655\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.10220 + 1.90907i 0.0810354 + 0.140357i
\(186\) 0 0
\(187\) −1.28908 2.23275i −0.0942668 0.163275i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.2812 −1.17807 −0.589034 0.808108i \(-0.700492\pi\)
−0.589034 + 0.808108i \(0.700492\pi\)
\(192\) 0 0
\(193\) −8.60669 14.9072i −0.619523 1.07304i −0.989573 0.144033i \(-0.953993\pi\)
0.370050 0.929012i \(-0.379341\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.2044 0.727034 0.363517 0.931588i \(-0.381576\pi\)
0.363517 + 0.931588i \(0.381576\pi\)
\(198\) 0 0
\(199\) −5.03189 + 8.71550i −0.356701 + 0.617825i −0.987408 0.158197i \(-0.949432\pi\)
0.630706 + 0.776022i \(0.282765\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −9.18351 + 15.9063i −0.644556 + 1.11640i
\(204\) 0 0
\(205\) −3.30660 + 5.72720i −0.230943 + 0.400005i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.18688 0.810784i 0.358784 0.0560831i
\(210\) 0 0
\(211\) 6.12758 + 10.6133i 0.421840 + 0.730648i 0.996119 0.0880113i \(-0.0280512\pi\)
−0.574280 + 0.818659i \(0.694718\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.17251 + 2.03084i −0.0799644 + 0.138502i
\(216\) 0 0
\(217\) −18.2264 −1.23729
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.14061 −0.143993
\(222\) 0 0
\(223\) 10.0748 + 17.4501i 0.674658 + 1.16854i 0.976569 + 0.215206i \(0.0690422\pi\)
−0.301911 + 0.953336i \(0.597624\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.25515 0.547914 0.273957 0.961742i \(-0.411667\pi\)
0.273957 + 0.961742i \(0.411667\pi\)
\(228\) 0 0
\(229\) −26.6860 −1.76346 −0.881729 0.471756i \(-0.843620\pi\)
−0.881729 + 0.471756i \(0.843620\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.00448 + 12.1321i 0.458879 + 0.794802i 0.998902 0.0468485i \(-0.0149178\pi\)
−0.540023 + 0.841650i \(0.681584\pi\)
\(234\) 0 0
\(235\) 8.33198 0.543518
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.2682 0.793563 0.396782 0.917913i \(-0.370127\pi\)
0.396782 + 0.917913i \(0.370127\pi\)
\(240\) 0 0
\(241\) −4.93621 + 8.54977i −0.317969 + 0.550739i −0.980064 0.198681i \(-0.936334\pi\)
0.662095 + 0.749420i \(0.269668\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.07031 + 1.85383i 0.0683794 + 0.118437i
\(246\) 0 0
\(247\) 1.57031 4.06622i 0.0999162 0.258727i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 14.4198 24.9758i 0.910170 1.57646i 0.0963474 0.995348i \(-0.469284\pi\)
0.813823 0.581113i \(-0.197383\pi\)
\(252\) 0 0
\(253\) −5.58131 + 9.66711i −0.350894 + 0.607766i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.39780 + 4.15311i −0.149571 + 0.259064i −0.931069 0.364844i \(-0.881122\pi\)
0.781498 + 0.623907i \(0.214456\pi\)
\(258\) 0 0
\(259\) 4.85939 0.301948
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 10.2044 + 17.6745i 0.629230 + 1.08986i 0.987706 + 0.156320i \(0.0499631\pi\)
−0.358476 + 0.933539i \(0.616704\pi\)
\(264\) 0 0
\(265\) −0.268189 −0.0164747
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.563788 + 0.976509i 0.0343747 + 0.0595388i 0.882701 0.469935i \(-0.155723\pi\)
−0.848326 + 0.529474i \(0.822389\pi\)
\(270\) 0 0
\(271\) −9.83850 17.0408i −0.597646 1.03515i −0.993168 0.116697i \(-0.962769\pi\)
0.395522 0.918457i \(-0.370564\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.602201 + 1.04304i −0.0363141 + 0.0628978i
\(276\) 0 0
\(277\) 7.35398 0.441858 0.220929 0.975290i \(-0.429091\pi\)
0.220929 + 0.975290i \(0.429091\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.73630 11.6676i 0.401854 0.696031i −0.592096 0.805867i \(-0.701699\pi\)
0.993950 + 0.109836i \(0.0350327\pi\)
\(282\) 0 0
\(283\) −10.6595 18.4627i −0.633640 1.09750i −0.986802 0.161934i \(-0.948227\pi\)
0.353162 0.935562i \(-0.385106\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.28908 + 12.6251i 0.430261 + 0.745233i
\(288\) 0 0
\(289\) 6.20889 10.7541i 0.365229 0.632594i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8.08580 −0.472377 −0.236189 0.971707i \(-0.575898\pi\)
−0.236189 + 0.971707i \(0.575898\pi\)
\(294\) 0 0
\(295\) −4.16599 + 7.21570i −0.242553 + 0.420115i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.63409 + 8.02649i 0.267997 + 0.464184i
\(300\) 0 0
\(301\) 2.58468 + 4.47679i 0.148978 + 0.258038i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.40880 −0.195187
\(306\) 0 0
\(307\) −14.2922 24.7549i −0.815701 1.41284i −0.908824 0.417180i \(-0.863018\pi\)
0.0931229 0.995655i \(-0.470315\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 10.7408 0.609054 0.304527 0.952504i \(-0.401502\pi\)
0.304527 + 0.952504i \(0.401502\pi\)
\(312\) 0 0
\(313\) −5.71092 + 9.89160i −0.322800 + 0.559107i −0.981065 0.193680i \(-0.937958\pi\)
0.658264 + 0.752787i \(0.271291\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.62309 2.81128i 0.0911619 0.157897i −0.816838 0.576867i \(-0.804275\pi\)
0.908000 + 0.418970i \(0.137609\pi\)
\(318\) 0 0
\(319\) 5.01752 8.69060i 0.280927 0.486580i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.85735 7.26318i −0.325912 0.404134i
\(324\) 0 0
\(325\) 0.500000 + 0.866025i 0.0277350 + 0.0480384i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.18351 15.9063i 0.506303 0.876943i
\(330\) 0 0
\(331\) 35.1208 1.93042 0.965208 0.261483i \(-0.0842116\pi\)
0.965208 + 0.261483i \(0.0842116\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.12758 −0.116242
\(336\) 0 0
\(337\) −7.93621 13.7459i −0.432313 0.748788i 0.564759 0.825256i \(-0.308969\pi\)
−0.997072 + 0.0764677i \(0.975636\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 9.95822 0.539268
\(342\) 0 0
\(343\) 20.1496 1.08798
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −16.3221 28.2707i −0.876216 1.51765i −0.855462 0.517866i \(-0.826727\pi\)
−0.0207537 0.999785i \(-0.506607\pi\)
\(348\) 0 0
\(349\) −21.7538 −1.16446 −0.582228 0.813026i \(-0.697819\pi\)
−0.582228 + 0.813026i \(0.697819\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −8.67699 −0.461830 −0.230915 0.972974i \(-0.574172\pi\)
−0.230915 + 0.972974i \(0.574172\pi\)
\(354\) 0 0
\(355\) −5.23630 + 9.06953i −0.277914 + 0.481361i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.81109 + 10.0651i 0.306697 + 0.531216i 0.977638 0.210296i \(-0.0674427\pi\)
−0.670940 + 0.741511i \(0.734109\pi\)
\(360\) 0 0
\(361\) 18.0936 5.79827i 0.952297 0.305172i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.42969 4.20835i 0.127176 0.220275i
\(366\) 0 0
\(367\) −15.3046 + 26.5083i −0.798892 + 1.38372i 0.121446 + 0.992598i \(0.461247\pi\)
−0.920338 + 0.391123i \(0.872087\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −0.295598 + 0.511992i −0.0153467 + 0.0265813i
\(372\) 0 0
\(373\) 4.70980 0.243864 0.121932 0.992538i \(-0.461091\pi\)
0.121932 + 0.992538i \(0.461091\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.16599 7.21570i −0.214559 0.371628i
\(378\) 0 0
\(379\) 10.0638 0.516942 0.258471 0.966019i \(-0.416781\pi\)
0.258471 + 0.966019i \(0.416781\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 13.7219 + 23.7671i 0.701158 + 1.21444i 0.968060 + 0.250717i \(0.0806664\pi\)
−0.266903 + 0.963723i \(0.586000\pi\)
\(384\) 0 0
\(385\) 1.32749 + 2.29928i 0.0676553 + 0.117182i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0.257185 0.445458i 0.0130398 0.0225856i −0.859432 0.511250i \(-0.829183\pi\)
0.872472 + 0.488665i \(0.162516\pi\)
\(390\) 0 0
\(391\) 19.8396 1.00333
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.20440 + 14.2104i −0.412808 + 0.715005i
\(396\) 0 0
\(397\) −1.55278 2.68950i −0.0779320 0.134982i 0.824426 0.565970i \(-0.191498\pi\)
−0.902358 + 0.430988i \(0.858165\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.51752 11.2887i −0.325470 0.563730i 0.656138 0.754641i \(-0.272189\pi\)
−0.981607 + 0.190911i \(0.938856\pi\)
\(402\) 0 0
\(403\) 4.13409 7.16046i 0.205934 0.356688i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.65498 −0.131603
\(408\) 0 0
\(409\) −13.7747 + 23.8585i −0.681115 + 1.17973i 0.293525 + 0.955951i \(0.405172\pi\)
−0.974641 + 0.223775i \(0.928162\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9.18351 + 15.9063i 0.451891 + 0.782698i
\(414\) 0 0
\(415\) −2.86591 4.96389i −0.140682 0.243668i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −16.5494 −0.808492 −0.404246 0.914650i \(-0.632466\pi\)
−0.404246 + 0.914650i \(0.632466\pi\)
\(420\) 0 0
\(421\) 7.93418 + 13.7424i 0.386688 + 0.669764i 0.992002 0.126223i \(-0.0402856\pi\)
−0.605314 + 0.795987i \(0.706952\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.14061 0.103835
\(426\) 0 0
\(427\) −3.75719 + 6.50764i −0.181823 + 0.314927i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 11.4472 19.8272i 0.551393 0.955041i −0.446781 0.894643i \(-0.647430\pi\)
0.998174 0.0603975i \(-0.0192368\pi\)
\(432\) 0 0
\(433\) 10.6660 18.4740i 0.512575 0.887805i −0.487319 0.873224i \(-0.662025\pi\)
0.999894 0.0145814i \(-0.00464157\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −14.5539 + 37.6865i −0.696208 + 1.80279i
\(438\) 0 0
\(439\) 5.08783 + 8.81238i 0.242829 + 0.420592i 0.961519 0.274739i \(-0.0885913\pi\)
−0.718690 + 0.695331i \(0.755258\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −3.30660 + 5.72720i −0.157101 + 0.272108i −0.933822 0.357737i \(-0.883548\pi\)
0.776721 + 0.629845i \(0.216882\pi\)
\(444\) 0 0
\(445\) 10.8176 0.512804
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −10.9870 −0.518507 −0.259253 0.965809i \(-0.583476\pi\)
−0.259253 + 0.965809i \(0.583476\pi\)
\(450\) 0 0
\(451\) −3.98248 6.89785i −0.187528 0.324807i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.20440 0.103344
\(456\) 0 0
\(457\) 12.5494 0.587037 0.293518 0.955953i \(-0.405174\pi\)
0.293518 + 0.955953i \(0.405174\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −11.6386 20.1586i −0.542063 0.938880i −0.998785 0.0492710i \(-0.984310\pi\)
0.456723 0.889609i \(-0.349023\pi\)
\(462\) 0 0
\(463\) −23.3958 −1.08729 −0.543647 0.839314i \(-0.682957\pi\)
−0.543647 + 0.839314i \(0.682957\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.6352 0.584688 0.292344 0.956313i \(-0.405565\pi\)
0.292344 + 0.956313i \(0.405565\pi\)
\(468\) 0 0
\(469\) −2.34502 + 4.06169i −0.108283 + 0.187551i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.41217 2.44595i −0.0649316 0.112465i
\(474\) 0 0
\(475\) −1.57031 + 4.06622i −0.0720506 + 0.186571i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2.01752 3.49445i 0.0921830 0.159666i −0.816246 0.577704i \(-0.803949\pi\)
0.908429 + 0.418038i \(0.137282\pi\)
\(480\) 0 0
\(481\) −1.10220 + 1.90907i −0.0502560 + 0.0870460i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.87039 + 13.6319i −0.357376 + 0.618993i
\(486\) 0 0
\(487\) 6.54942 0.296782 0.148391 0.988929i \(-0.452591\pi\)
0.148391 + 0.988929i \(0.452591\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.04290 + 1.80635i 0.0470653 + 0.0815195i 0.888598 0.458686i \(-0.151680\pi\)
−0.841533 + 0.540206i \(0.818346\pi\)
\(492\) 0 0
\(493\) −17.8355 −0.803273
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 11.5429 + 19.9929i 0.517770 + 0.896803i
\(498\) 0 0
\(499\) 12.0539 + 20.8780i 0.539607 + 0.934626i 0.998925 + 0.0463546i \(0.0147604\pi\)
−0.459318 + 0.888272i \(0.651906\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4.13858 + 7.16823i −0.184530 + 0.319616i −0.943418 0.331606i \(-0.892410\pi\)
0.758888 + 0.651221i \(0.225743\pi\)
\(504\) 0 0
\(505\) 10.5494 0.469443
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.86591 + 4.96389i −0.127029 + 0.220021i −0.922524 0.385939i \(-0.873877\pi\)
0.795495 + 0.605960i \(0.207211\pi\)
\(510\) 0 0
\(511\) −5.35602 9.27690i −0.236936 0.410386i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.365905 + 0.633767i 0.0161237 + 0.0279271i
\(516\) 0 0
\(517\) −5.01752 + 8.69060i −0.220670 + 0.382212i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −14.8086 −0.648778 −0.324389 0.945924i \(-0.605159\pi\)
−0.324389 + 0.945924i \(0.605159\pi\)
\(522\) 0 0
\(523\) −7.79356 + 13.4988i −0.340789 + 0.590263i −0.984579 0.174938i \(-0.944027\pi\)
0.643791 + 0.765202i \(0.277361\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.84950 15.3278i −0.385490 0.667689i
\(528\) 0 0
\(529\) −31.4497 54.4724i −1.36738 2.36837i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.61320 −0.286450
\(534\) 0 0
\(535\) −6.69788 11.6011i −0.289575 0.501558i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.57816 −0.111049
\(540\) 0 0
\(541\) −14.0539 + 24.3421i −0.604224 + 1.04655i 0.387949 + 0.921681i \(0.373184\pi\)
−0.992174 + 0.124867i \(0.960150\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.96811 + 5.14091i −0.127140 + 0.220212i
\(546\) 0 0
\(547\) 21.6835 37.5569i 0.927120 1.60582i 0.139004 0.990292i \(-0.455610\pi\)
0.788116 0.615527i \(-0.211057\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 13.0838 33.8796i 0.557387 1.44332i
\(552\) 0 0
\(553\) 18.0858 + 31.3255i 0.769086 + 1.33210i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.65814 14.9963i 0.366857 0.635415i −0.622215 0.782846i \(-0.713767\pi\)
0.989072 + 0.147431i \(0.0471005\pi\)
\(558\) 0 0
\(559\) −2.34502 −0.0991836
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.35398 −0.352078 −0.176039 0.984383i \(-0.556329\pi\)
−0.176039 + 0.984383i \(0.556329\pi\)
\(564\) 0 0
\(565\) 7.80660 + 13.5214i 0.328426 + 0.568851i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.6640 0.447056 0.223528 0.974697i \(-0.428243\pi\)
0.223528 + 0.974697i \(0.428243\pi\)
\(570\) 0 0
\(571\) −2.56512 −0.107347 −0.0536735 0.998559i \(-0.517093\pi\)
−0.0536735 + 0.998559i \(0.517093\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.63409 8.02649i −0.193255 0.334728i
\(576\) 0 0
\(577\) 20.4218 0.850172 0.425086 0.905153i \(-0.360244\pi\)
0.425086 + 0.905153i \(0.360244\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −12.6352 −0.524197
\(582\) 0 0
\(583\) 0.161504 0.279733i 0.00668880 0.0115853i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.40432 7.62850i −0.181786 0.314862i 0.760703 0.649100i \(-0.224854\pi\)
−0.942489 + 0.334238i \(0.891521\pi\)
\(588\) 0 0
\(589\) 35.6078 5.56601i 1.46719 0.229343i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 16.4726 28.5314i 0.676448 1.17164i −0.299595 0.954066i \(-0.596852\pi\)
0.976043 0.217576i \(-0.0698151\pi\)
\(594\) 0 0
\(595\) 2.35939 4.08658i 0.0967254 0.167533i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −9.56379 + 16.5650i −0.390766 + 0.676826i −0.992551 0.121832i \(-0.961123\pi\)
0.601785 + 0.798658i \(0.294456\pi\)
\(600\) 0 0
\(601\) 19.3100 0.787670 0.393835 0.919181i \(-0.371148\pi\)
0.393835 + 0.919181i \(0.371148\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.77471 + 8.27004i 0.194120 + 0.336225i
\(606\) 0 0
\(607\) 36.8773 1.49680 0.748402 0.663245i \(-0.230821\pi\)
0.748402 + 0.663245i \(0.230821\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.16599 + 7.21570i 0.168538 + 0.291916i
\(612\) 0 0
\(613\) 14.2603 + 24.6996i 0.575970 + 0.997609i 0.995936 + 0.0900688i \(0.0287087\pi\)
−0.419966 + 0.907540i \(0.637958\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.64713 + 9.78112i −0.227345 + 0.393773i −0.957020 0.290021i \(-0.906338\pi\)
0.729675 + 0.683794i \(0.239671\pi\)
\(618\) 0 0
\(619\) 27.2044 1.09344 0.546719 0.837316i \(-0.315877\pi\)
0.546719 + 0.837316i \(0.315877\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 11.9232 20.6515i 0.477692 0.827387i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.35939 + 4.08658i 0.0940749 + 0.162942i
\(630\) 0 0
\(631\) −4.19136 + 7.25965i −0.166856 + 0.289002i −0.937313 0.348490i \(-0.886695\pi\)
0.770457 + 0.637492i \(0.220028\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −15.3540 −0.609304
\(636\) 0 0
\(637\) −1.07031 + 1.85383i −0.0424071 + 0.0734513i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.55594 + 16.5514i 0.377437 + 0.653740i 0.990689 0.136148i \(-0.0434722\pi\)
−0.613252 + 0.789888i \(0.710139\pi\)
\(642\) 0 0
\(643\) 24.1112 + 41.7618i 0.950852 + 1.64692i 0.743588 + 0.668638i \(0.233123\pi\)
0.207264 + 0.978285i \(0.433544\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30.0817 1.18263 0.591317 0.806439i \(-0.298608\pi\)
0.591317 + 0.806439i \(0.298608\pi\)
\(648\) 0 0
\(649\) −5.01752 8.69060i −0.196955 0.341136i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −41.9124 −1.64016 −0.820079 0.572250i \(-0.806071\pi\)
−0.820079 + 0.572250i \(0.806071\pi\)
\(654\) 0 0
\(655\) −3.89780 + 6.75119i −0.152300 + 0.263791i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −20.0936 + 34.8032i −0.782737 + 1.35574i 0.147604 + 0.989047i \(0.452844\pi\)
−0.930341 + 0.366694i \(0.880489\pi\)
\(660\) 0 0
\(661\) 0.757185 1.31148i 0.0294511 0.0510108i −0.850924 0.525289i \(-0.823957\pi\)
0.880375 + 0.474278i \(0.157291\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.03189 + 7.47961i 0.233907 + 0.290047i
\(666\) 0 0
\(667\) 38.6112 + 66.8765i 1.49503 + 2.58947i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.05278 3.55553i 0.0792468 0.137260i
\(672\) 0 0
\(673\) −7.76686 −0.299390 −0.149695 0.988732i \(-0.547829\pi\)
−0.149695 + 0.988732i \(0.547829\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.19136 −0.237953 −0.118977 0.992897i \(-0.537961\pi\)
−0.118977 + 0.992897i \(0.537961\pi\)
\(678\) 0 0
\(679\) 17.3495 + 30.0502i 0.665813 + 1.15322i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2.92317 −0.111852 −0.0559261 0.998435i \(-0.517811\pi\)
−0.0559261 + 0.998435i \(0.517811\pi\)
\(684\) 0 0
\(685\) −16.4726 −0.629385
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.134095 0.232259i −0.00510860 0.00884835i
\(690\) 0 0
\(691\) −2.35805 −0.0897046 −0.0448523 0.998994i \(-0.514282\pi\)
−0.0448523 + 0.998994i \(0.514282\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −19.5364 −0.741057
\(696\) 0 0
\(697\) −7.07816 + 12.2597i −0.268104 + 0.464370i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 12.2792 + 21.2682i 0.463779 + 0.803288i 0.999145 0.0413314i \(-0.0131599\pi\)
−0.535367 + 0.844620i \(0.679827\pi\)
\(702\) 0 0
\(703\) −9.49348 + 1.48397i −0.358053 + 0.0559689i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 11.6276 20.1396i 0.437300 0.757426i
\(708\) 0 0
\(709\) −3.63858 + 6.30220i −0.136650 + 0.236684i −0.926226 0.376968i \(-0.876967\pi\)
0.789577 + 0.613652i \(0.210300\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −38.3156 + 66.3645i −1.43493 + 2.48537i
\(714\) 0 0
\(715\) −1.20440 −0.0450421
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16.4452 + 28.4839i 0.613302 + 1.06227i 0.990680 + 0.136211i \(0.0434924\pi\)
−0.377378 + 0.926059i \(0.623174\pi\)
\(720\) 0 0
\(721\) 1.61320 0.0600789
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.16599 + 7.21570i 0.154721 + 0.267985i
\(726\) 0 0
\(727\) −13.9551 24.1709i −0.517565 0.896449i −0.999792 0.0204023i \(-0.993505\pi\)
0.482227 0.876046i \(-0.339828\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.50989 + 4.34725i −0.0928315 + 0.160789i
\(732\) 0 0
\(733\) −4.89710 −0.180878 −0.0904392 0.995902i \(-0.528827\pi\)
−0.0904392 + 0.995902i \(0.528827\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.28123 2.21915i 0.0471946 0.0817435i
\(738\) 0 0
\(739\) 11.5936 + 20.0808i 0.426479 + 0.738684i 0.996557 0.0829069i \(-0.0264204\pi\)
−0.570078 + 0.821591i \(0.693087\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 19.9901 + 34.6239i 0.733366 + 1.27023i 0.955436 + 0.295197i \(0.0953852\pi\)
−0.222070 + 0.975031i \(0.571281\pi\)
\(744\) 0 0
\(745\) −2.83850 + 4.91642i −0.103994 + 0.180124i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −29.5296 −1.07899
\(750\) 0 0
\(751\) 10.2936 17.8290i 0.375617 0.650589i −0.614802 0.788682i \(-0.710764\pi\)
0.990419 + 0.138093i \(0.0440973\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 7.11320 + 12.3204i 0.258876 + 0.448386i
\(756\) 0 0
\(757\) 17.3176 + 29.9950i 0.629419 + 1.09019i 0.987668 + 0.156560i \(0.0500404\pi\)
−0.358249 + 0.933626i \(0.616626\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −24.3890 −0.884102 −0.442051 0.896990i \(-0.645749\pi\)
−0.442051 + 0.896990i \(0.645749\pi\)
\(762\) 0 0
\(763\) 6.54290 + 11.3326i 0.236869 + 0.410269i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.33198 −0.300850
\(768\) 0 0
\(769\) −3.27471 + 5.67196i −0.118089 + 0.204536i −0.919010 0.394233i \(-0.871010\pi\)
0.800921 + 0.598770i \(0.204343\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −18.7363 + 32.4522i −0.673898 + 1.16723i 0.302892 + 0.953025i \(0.402048\pi\)
−0.976790 + 0.214200i \(0.931286\pi\)
\(774\) 0 0
\(775\) −4.13409 + 7.16046i −0.148501 + 0.257211i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −18.0957 22.4388i −0.648345 0.803955i
\(780\) 0 0
\(781\) −6.30660 10.9234i −0.225668 0.390868i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.327492 0.567233i 0.0116887 0.0202454i
\(786\) 0 0
\(787\) −12.8306 −0.457363 −0.228682 0.973501i \(-0.573441\pi\)
−0.228682 + 0.973501i \(0.573441\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 34.4178 1.22376
\(792\) 0 0
\(793\) −1.70440 2.95211i −0.0605251 0.104833i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 28.3032 1.00255 0.501276 0.865287i \(-0.332864\pi\)
0.501276 + 0.865287i \(0.332864\pi\)
\(798\) 0 0
\(799\) 17.8355 0.630976
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.92633 + 5.06855i 0.103268 + 0.178865i
\(804\) 0 0
\(805\) −20.4308 −0.720091
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 55.4946 1.95109 0.975543 0.219808i \(-0.0705432\pi\)
0.975543 + 0.219808i \(0.0705432\pi\)
\(810\) 0 0
\(811\) 11.8540 20.5317i 0.416250 0.720966i −0.579309 0.815108i \(-0.696678\pi\)
0.995559 + 0.0941424i \(0.0300109\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6.33850 + 10.9786i 0.222028 + 0.384563i
\(816\) 0 0
\(817\) −6.41665 7.95672i −0.224490 0.278370i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 11.3625 19.6805i 0.396555 0.686854i −0.596743 0.802432i \(-0.703539\pi\)
0.993298 + 0.115578i \(0.0368722\pi\)
\(822\) 0 0
\(823\) 8.00000 13.8564i 0.278862 0.483004i −0.692240 0.721668i \(-0.743376\pi\)
0.971102 + 0.238664i \(0.0767093\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.94722 17.2291i 0.345899 0.599114i −0.639618 0.768693i \(-0.720907\pi\)
0.985517 + 0.169579i \(0.0542408\pi\)
\(828\) 0 0
\(829\) 4.37109 0.151814 0.0759071 0.997115i \(-0.475815\pi\)
0.0759071 + 0.997115i \(0.475815\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.29111 + 3.96833i 0.0793824 + 0.137494i
\(834\) 0 0
\(835\) 7.79560 0.269778
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.22396 + 2.11996i 0.0422558 + 0.0731891i 0.886380 0.462959i \(-0.153212\pi\)
−0.844124 + 0.536148i \(0.819879\pi\)
\(840\) 0 0
\(841\) −20.2109 35.0063i −0.696928 1.20712i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6.00000 10.3923i 0.206406 0.357506i
\(846\) 0 0
\(847\) 21.0507 0.723312
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 10.2154 17.6936i 0.350180 0.606529i
\(852\) 0 0
\(853\) 8.96607 + 15.5297i 0.306992 + 0.531727i 0.977703 0.209993i \(-0.0673440\pi\)
−0.670711 + 0.741719i \(0.734011\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −29.0474 50.3115i −0.992240 1.71861i −0.603801 0.797135i \(-0.706348\pi\)
−0.388438 0.921475i \(-0.626985\pi\)
\(858\) 0 0
\(859\) −10.0409 + 17.3913i −0.342590 + 0.593383i −0.984913 0.173051i \(-0.944637\pi\)
0.642323 + 0.766434i \(0.277971\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 28.6262 0.974449 0.487224 0.873277i \(-0.338009\pi\)
0.487224 + 0.873277i \(0.338009\pi\)
\(864\) 0 0
\(865\) −5.37039 + 9.30179i −0.182599 + 0.316270i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −9.88139 17.1151i −0.335203 0.580589i
\(870\) 0 0
\(871\) −1.06379 1.84253i −0.0360451 0.0624319i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.20440 −0.0745224
\(876\) 0 0
\(877\) −25.5200 44.2019i −0.861748 1.49259i −0.870240 0.492628i \(-0.836036\pi\)
0.00849182 0.999964i \(-0.497297\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 14.2992 0.481751 0.240876 0.970556i \(-0.422565\pi\)
0.240876 + 0.970556i \(0.422565\pi\)
\(882\) 0 0
\(883\) −14.6999 + 25.4610i −0.494692 + 0.856831i −0.999981 0.00611882i \(-0.998052\pi\)
0.505290 + 0.862950i \(0.331386\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.46811 11.2031i 0.217178 0.376163i −0.736766 0.676147i \(-0.763648\pi\)
0.953944 + 0.299985i \(0.0969815\pi\)
\(888\) 0 0
\(889\) −16.9232 + 29.3118i −0.567585 + 0.983086i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −13.0838 + 33.8796i −0.437831 + 1.13374i
\(894\) 0 0
\(895\) −0.774708 1.34183i −0.0258956 0.0448526i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 34.4452 59.6608i 1.14881 1.98980i
\(900\) 0 0
\(901\) −0.574090 −0.0191257
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.22641 −0.273455
\(906\) 0 0
\(907\) 5.00000 + 8.66025i 0.166022 + 0.287559i 0.937018 0.349281i \(-0.113574\pi\)
−0.770996 + 0.636841i \(0.780241\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.10964 0.103027 0.0515134 0.998672i \(-0.483596\pi\)
0.0515134 + 0.998672i \(0.483596\pi\)
\(912\) 0 0
\(913\) 6.90340 0.228469
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.59231 + 14.8823i 0.283743 + 0.491458i
\(918\) 0 0
\(919\) 23.4987 0.775150 0.387575 0.921838i \(-0.373313\pi\)
0.387575 + 0.921838i \(0.373313\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −10.4726 −0.344710
\(924\) 0 0
\(925\) 1.10220 1.90907i 0.0362401 0.0627698i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −22.3221 38.6630i −0.732364 1.26849i −0.955870 0.293789i \(-0.905084\pi\)
0.223506 0.974703i \(-0.428250\pi\)
\(930\) 0 0
\(931\) −9.21877 + 1.44103i −0.302133 + 0.0472277i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.28908 + 2.23275i −0.0421574 + 0.0730188i
\(936\) 0 0
\(937\) 11.7363 20.3279i 0.383408 0.664082i −0.608139 0.793831i \(-0.708084\pi\)
0.991547 + 0.129748i \(0.0414170\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 20.0858 34.7896i 0.654778 1.13411i −0.327171 0.944965i \(-0.606095\pi\)
0.981949 0.189144i \(-0.0605713\pi\)
\(942\) 0 0
\(943\) 61.2924 1.99596
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.68148 + 16.7688i 0.314606 + 0.544913i 0.979354 0.202155i \(-0.0647944\pi\)
−0.664748 + 0.747068i \(0.731461\pi\)
\(948\) 0 0
\(949\) 4.85939 0.157742
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 2.41981 + 4.19123i 0.0783852 + 0.135767i 0.902553 0.430578i \(-0.141690\pi\)
−0.824168 + 0.566345i \(0.808357\pi\)
\(954\) 0 0
\(955\) 8.14061 + 14.1000i 0.263424 + 0.456264i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −18.1561 + 31.4473i −0.586291 + 1.01549i
\(960\) 0 0
\(961\) 37.3630 1.20526
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −8.60669 + 14.9072i −0.277059 + 0.479880i
\(966\) 0 0
\(967\) −13.5364 23.4457i −0.435301 0.753963i 0.562020 0.827124i \(-0.310025\pi\)
−0.997320 + 0.0731612i \(0.976691\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.61769 + 2.80192i 0.0519141 + 0.0899179i 0.890815 0.454367i \(-0.150134\pi\)
−0.838901 + 0.544285i \(0.816801\pi\)
\(972\) 0 0
\(973\) −21.5330 + 37.2963i −0.690317 + 1.19566i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.93214 0.0618147 0.0309074 0.999522i \(-0.490160\pi\)
0.0309074 + 0.999522i \(0.490160\pi\)
\(978\) 0 0
\(979\) −6.51437 + 11.2832i −0.208200 + 0.360613i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −7.18555 12.4457i −0.229183 0.396957i 0.728383 0.685170i \(-0.240272\pi\)
−0.957566 + 0.288213i \(0.906939\pi\)
\(984\) 0 0
\(985\) −5.10220 8.83727i −0.162570 0.281579i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 21.7340 0.691102
\(990\) 0 0
\(991\) −17.7956 30.8229i −0.565296 0.979121i −0.997022 0.0771164i \(-0.975429\pi\)
0.431726 0.902005i \(-0.357905\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 10.0638 0.319044
\(996\) 0 0
\(997\) −23.7902 + 41.2058i −0.753443 + 1.30500i 0.192702 + 0.981257i \(0.438275\pi\)
−0.946145 + 0.323744i \(0.895058\pi\)
\(998\) 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 3420.2.t.v.1261.1 6
3.2 odd 2 380.2.i.b.121.3 6
12.11 even 2 1520.2.q.i.881.1 6
15.2 even 4 1900.2.s.c.349.6 12
15.8 even 4 1900.2.s.c.349.1 12
15.14 odd 2 1900.2.i.c.501.1 6
19.11 even 3 inner 3420.2.t.v.3241.1 6
57.11 odd 6 380.2.i.b.201.3 yes 6
57.26 odd 6 7220.2.a.n.1.1 3
57.50 even 6 7220.2.a.o.1.3 3
228.11 even 6 1520.2.q.i.961.1 6
285.68 even 12 1900.2.s.c.49.6 12
285.182 even 12 1900.2.s.c.49.1 12
285.239 odd 6 1900.2.i.c.201.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.i.b.121.3 6 3.2 odd 2
380.2.i.b.201.3 yes 6 57.11 odd 6
1520.2.q.i.881.1 6 12.11 even 2
1520.2.q.i.961.1 6 228.11 even 6
1900.2.i.c.201.1 6 285.239 odd 6
1900.2.i.c.501.1 6 15.14 odd 2
1900.2.s.c.49.1 12 285.182 even 12
1900.2.s.c.49.6 12 285.68 even 12
1900.2.s.c.349.1 12 15.8 even 4
1900.2.s.c.349.6 12 15.2 even 4
3420.2.t.v.1261.1 6 1.1 even 1 trivial
3420.2.t.v.3241.1 6 19.11 even 3 inner
7220.2.a.n.1.1 3 57.26 odd 6
7220.2.a.o.1.3 3 57.50 even 6