Properties

Label 360.2.bi.b.49.12
Level $360$
Weight $2$
Character 360.49
Analytic conductor $2.875$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,2,Mod(49,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.bi (of order \(6\), 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: \(2.87461447277\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 49.12
Character \(\chi\) \(=\) 360.49
Dual form 360.2.bi.b.169.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.19594 + 1.25289i) q^{3} +(1.61222 + 1.54943i) q^{5} +(1.98012 - 1.14322i) q^{7} +(-0.139458 + 2.99676i) q^{9} +O(q^{10})\) \(q+(1.19594 + 1.25289i) q^{3} +(1.61222 + 1.54943i) q^{5} +(1.98012 - 1.14322i) q^{7} +(-0.139458 + 2.99676i) q^{9} +(0.864628 + 1.49758i) q^{11} +(-5.20118 - 3.00291i) q^{13} +(-0.0131467 + 3.87296i) q^{15} -4.65465i q^{17} -0.888934 q^{19} +(3.80044 + 1.11364i) q^{21} +(5.17546 + 2.98806i) q^{23} +(0.198516 + 4.99606i) q^{25} +(-3.92138 + 3.40922i) q^{27} +(-2.34652 - 4.06430i) q^{29} +(-2.98190 + 5.16479i) q^{31} +(-0.842257 + 2.87430i) q^{33} +(4.96374 + 1.22493i) q^{35} -8.68683i q^{37} +(-2.45800 - 10.1078i) q^{39} +(-3.15468 + 5.46406i) q^{41} +(1.46119 - 0.843618i) q^{43} +(-4.86811 + 4.61536i) q^{45} +(10.8177 - 6.24559i) q^{47} +(-0.886081 + 1.53474i) q^{49} +(5.83175 - 5.56667i) q^{51} +2.66110i q^{53} +(-0.926427 + 3.75411i) q^{55} +(-1.06311 - 1.11374i) q^{57} +(1.62355 - 2.81207i) q^{59} +(-6.08732 - 10.5436i) q^{61} +(3.14982 + 6.09337i) q^{63} +(-3.73266 - 12.9002i) q^{65} +(-10.8344 - 6.25524i) q^{67} +(2.44584 + 10.0578i) q^{69} +7.25413 q^{71} -1.69305i q^{73} +(-6.02209 + 6.22370i) q^{75} +(3.42413 + 1.97693i) q^{77} +(-1.99175 - 3.44982i) q^{79} +(-8.96110 - 0.835841i) q^{81} +(-3.63578 + 2.09912i) q^{83} +(7.21206 - 7.50432i) q^{85} +(2.28581 - 7.80059i) q^{87} +2.60891 q^{89} -13.7320 q^{91} +(-10.0371 + 2.44080i) q^{93} +(-1.43316 - 1.37734i) q^{95} +(-3.46355 + 1.99968i) q^{97} +(-4.60846 + 2.38223i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q - 2 q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 32 q - 2 q^{5} + 4 q^{9} + 16 q^{11} - 10 q^{15} + 8 q^{19} - 4 q^{21} - 6 q^{25} + 20 q^{29} - 12 q^{31} + 4 q^{35} - 28 q^{39} - 8 q^{41} + 38 q^{45} + 36 q^{49} - 84 q^{51} + 20 q^{55} - 20 q^{61} + 10 q^{65} - 4 q^{69} + 16 q^{71} - 10 q^{75} + 4 q^{79} - 52 q^{81} + 36 q^{85} - 96 q^{89} - 8 q^{91} - 32 q^{95} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(217\) \(271\) \(281\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 1.19594 + 1.25289i 0.690476 + 0.723355i
\(4\) 0 0
\(5\) 1.61222 + 1.54943i 0.721007 + 0.692927i
\(6\) 0 0
\(7\) 1.98012 1.14322i 0.748415 0.432098i −0.0767057 0.997054i \(-0.524440\pi\)
0.825121 + 0.564956i \(0.191107\pi\)
\(8\) 0 0
\(9\) −0.139458 + 2.99676i −0.0464858 + 0.998919i
\(10\) 0 0
\(11\) 0.864628 + 1.49758i 0.260695 + 0.451537i 0.966427 0.256942i \(-0.0827149\pi\)
−0.705732 + 0.708479i \(0.749382\pi\)
\(12\) 0 0
\(13\) −5.20118 3.00291i −1.44255 0.832856i −0.444530 0.895764i \(-0.646629\pi\)
−0.998019 + 0.0629081i \(0.979962\pi\)
\(14\) 0 0
\(15\) −0.0131467 + 3.87296i −0.00339446 + 0.999994i
\(16\) 0 0
\(17\) 4.65465i 1.12892i −0.825461 0.564459i \(-0.809085\pi\)
0.825461 0.564459i \(-0.190915\pi\)
\(18\) 0 0
\(19\) −0.888934 −0.203936 −0.101968 0.994788i \(-0.532514\pi\)
−0.101968 + 0.994788i \(0.532514\pi\)
\(20\) 0 0
\(21\) 3.80044 + 1.11364i 0.829323 + 0.243017i
\(22\) 0 0
\(23\) 5.17546 + 2.98806i 1.07916 + 0.623053i 0.930669 0.365861i \(-0.119226\pi\)
0.148489 + 0.988914i \(0.452559\pi\)
\(24\) 0 0
\(25\) 0.198516 + 4.99606i 0.0397032 + 0.999212i
\(26\) 0 0
\(27\) −3.92138 + 3.40922i −0.754671 + 0.656104i
\(28\) 0 0
\(29\) −2.34652 4.06430i −0.435738 0.754721i 0.561617 0.827397i \(-0.310179\pi\)
−0.997356 + 0.0726761i \(0.976846\pi\)
\(30\) 0 0
\(31\) −2.98190 + 5.16479i −0.535564 + 0.927625i 0.463572 + 0.886059i \(0.346568\pi\)
−0.999136 + 0.0415650i \(0.986766\pi\)
\(32\) 0 0
\(33\) −0.842257 + 2.87430i −0.146618 + 0.500351i
\(34\) 0 0
\(35\) 4.96374 + 1.22493i 0.839025 + 0.207052i
\(36\) 0 0
\(37\) 8.68683i 1.42811i −0.700091 0.714053i \(-0.746857\pi\)
0.700091 0.714053i \(-0.253143\pi\)
\(38\) 0 0
\(39\) −2.45800 10.1078i −0.393595 1.61854i
\(40\) 0 0
\(41\) −3.15468 + 5.46406i −0.492678 + 0.853343i −0.999964 0.00843414i \(-0.997315\pi\)
0.507286 + 0.861778i \(0.330649\pi\)
\(42\) 0 0
\(43\) 1.46119 0.843618i 0.222829 0.128651i −0.384430 0.923154i \(-0.625602\pi\)
0.607260 + 0.794503i \(0.292269\pi\)
\(44\) 0 0
\(45\) −4.86811 + 4.61536i −0.725695 + 0.688017i
\(46\) 0 0
\(47\) 10.8177 6.24559i 1.57792 0.911014i 0.582773 0.812635i \(-0.301968\pi\)
0.995149 0.0983785i \(-0.0313656\pi\)
\(48\) 0 0
\(49\) −0.886081 + 1.53474i −0.126583 + 0.219248i
\(50\) 0 0
\(51\) 5.83175 5.56667i 0.816608 0.779490i
\(52\) 0 0
\(53\) 2.66110i 0.365530i 0.983157 + 0.182765i \(0.0585047\pi\)
−0.983157 + 0.182765i \(0.941495\pi\)
\(54\) 0 0
\(55\) −0.926427 + 3.75411i −0.124919 + 0.506204i
\(56\) 0 0
\(57\) −1.06311 1.11374i −0.140813 0.147518i
\(58\) 0 0
\(59\) 1.62355 2.81207i 0.211368 0.366100i −0.740775 0.671753i \(-0.765541\pi\)
0.952143 + 0.305653i \(0.0988748\pi\)
\(60\) 0 0
\(61\) −6.08732 10.5436i −0.779402 1.34996i −0.932287 0.361720i \(-0.882190\pi\)
0.152885 0.988244i \(-0.451144\pi\)
\(62\) 0 0
\(63\) 3.14982 + 6.09337i 0.396840 + 0.767693i
\(64\) 0 0
\(65\) −3.73266 12.9002i −0.462980 1.60008i
\(66\) 0 0
\(67\) −10.8344 6.25524i −1.32363 0.764199i −0.339325 0.940669i \(-0.610199\pi\)
−0.984306 + 0.176470i \(0.943532\pi\)
\(68\) 0 0
\(69\) 2.44584 + 10.0578i 0.294445 + 1.21082i
\(70\) 0 0
\(71\) 7.25413 0.860906 0.430453 0.902613i \(-0.358354\pi\)
0.430453 + 0.902613i \(0.358354\pi\)
\(72\) 0 0
\(73\) 1.69305i 0.198156i −0.995080 0.0990781i \(-0.968411\pi\)
0.995080 0.0990781i \(-0.0315894\pi\)
\(74\) 0 0
\(75\) −6.02209 + 6.22370i −0.695371 + 0.718651i
\(76\) 0 0
\(77\) 3.42413 + 1.97693i 0.390216 + 0.225292i
\(78\) 0 0
\(79\) −1.99175 3.44982i −0.224090 0.388135i 0.731956 0.681352i \(-0.238608\pi\)
−0.956046 + 0.293217i \(0.905274\pi\)
\(80\) 0 0
\(81\) −8.96110 0.835841i −0.995678 0.0928712i
\(82\) 0 0
\(83\) −3.63578 + 2.09912i −0.399079 + 0.230408i −0.686087 0.727520i \(-0.740673\pi\)
0.287007 + 0.957928i \(0.407340\pi\)
\(84\) 0 0
\(85\) 7.21206 7.50432i 0.782258 0.813958i
\(86\) 0 0
\(87\) 2.28581 7.80059i 0.245065 0.836311i
\(88\) 0 0
\(89\) 2.60891 0.276544 0.138272 0.990394i \(-0.455845\pi\)
0.138272 + 0.990394i \(0.455845\pi\)
\(90\) 0 0
\(91\) −13.7320 −1.43950
\(92\) 0 0
\(93\) −10.0371 + 2.44080i −1.04080 + 0.253099i
\(94\) 0 0
\(95\) −1.43316 1.37734i −0.147039 0.141313i
\(96\) 0 0
\(97\) −3.46355 + 1.99968i −0.351670 + 0.203037i −0.665420 0.746469i \(-0.731748\pi\)
0.313751 + 0.949505i \(0.398414\pi\)
\(98\) 0 0
\(99\) −4.60846 + 2.38223i −0.463168 + 0.239423i
\(100\) 0 0
\(101\) −3.10733 5.38206i −0.309191 0.535535i 0.668995 0.743267i \(-0.266725\pi\)
−0.978186 + 0.207733i \(0.933392\pi\)
\(102\) 0 0
\(103\) 0.00503185 + 0.00290514i 0.000495803 + 0.000286252i 0.500248 0.865882i \(-0.333242\pi\)
−0.499752 + 0.866168i \(0.666576\pi\)
\(104\) 0 0
\(105\) 4.40163 + 7.68396i 0.429555 + 0.749878i
\(106\) 0 0
\(107\) 4.54091i 0.438986i −0.975614 0.219493i \(-0.929560\pi\)
0.975614 0.219493i \(-0.0704403\pi\)
\(108\) 0 0
\(109\) 15.5326 1.48776 0.743878 0.668315i \(-0.232984\pi\)
0.743878 + 0.668315i \(0.232984\pi\)
\(110\) 0 0
\(111\) 10.8836 10.3889i 1.03303 0.986073i
\(112\) 0 0
\(113\) 1.45560 + 0.840393i 0.136932 + 0.0790575i 0.566901 0.823786i \(-0.308142\pi\)
−0.429969 + 0.902844i \(0.641476\pi\)
\(114\) 0 0
\(115\) 3.71420 + 12.8364i 0.346351 + 1.19700i
\(116\) 0 0
\(117\) 9.72432 15.1679i 0.899014 1.40227i
\(118\) 0 0
\(119\) −5.32130 9.21676i −0.487803 0.844899i
\(120\) 0 0
\(121\) 4.00484 6.93658i 0.364076 0.630598i
\(122\) 0 0
\(123\) −10.6187 + 2.58223i −0.957453 + 0.232832i
\(124\) 0 0
\(125\) −7.42100 + 8.36234i −0.663755 + 0.747950i
\(126\) 0 0
\(127\) 10.5663i 0.937612i −0.883301 0.468806i \(-0.844684\pi\)
0.883301 0.468806i \(-0.155316\pi\)
\(128\) 0 0
\(129\) 2.80445 + 0.821791i 0.246918 + 0.0723547i
\(130\) 0 0
\(131\) −10.0803 + 17.4596i −0.880718 + 1.52545i −0.0301741 + 0.999545i \(0.509606\pi\)
−0.850544 + 0.525904i \(0.823727\pi\)
\(132\) 0 0
\(133\) −1.76020 + 1.01625i −0.152628 + 0.0881201i
\(134\) 0 0
\(135\) −11.6045 0.579511i −0.998755 0.0498764i
\(136\) 0 0
\(137\) −13.3242 + 7.69273i −1.13836 + 0.657234i −0.946025 0.324095i \(-0.894940\pi\)
−0.192338 + 0.981329i \(0.561607\pi\)
\(138\) 0 0
\(139\) −3.49575 + 6.05482i −0.296506 + 0.513564i −0.975334 0.220734i \(-0.929155\pi\)
0.678828 + 0.734297i \(0.262488\pi\)
\(140\) 0 0
\(141\) 20.7623 + 6.08400i 1.74850 + 0.512365i
\(142\) 0 0
\(143\) 10.3856i 0.868486i
\(144\) 0 0
\(145\) 2.51424 10.1883i 0.208796 0.846095i
\(146\) 0 0
\(147\) −2.98255 + 0.725293i −0.245997 + 0.0598211i
\(148\) 0 0
\(149\) −7.84354 + 13.5854i −0.642568 + 1.11296i 0.342290 + 0.939594i \(0.388798\pi\)
−0.984858 + 0.173366i \(0.944536\pi\)
\(150\) 0 0
\(151\) −2.79420 4.83970i −0.227389 0.393849i 0.729644 0.683827i \(-0.239686\pi\)
−0.957034 + 0.289977i \(0.906352\pi\)
\(152\) 0 0
\(153\) 13.9488 + 0.649125i 1.12770 + 0.0524787i
\(154\) 0 0
\(155\) −12.8100 + 3.70655i −1.02892 + 0.297717i
\(156\) 0 0
\(157\) 18.9780 + 10.9569i 1.51461 + 0.874458i 0.999853 + 0.0171197i \(0.00544964\pi\)
0.514753 + 0.857339i \(0.327884\pi\)
\(158\) 0 0
\(159\) −3.33406 + 3.18251i −0.264408 + 0.252390i
\(160\) 0 0
\(161\) 13.6641 1.07688
\(162\) 0 0
\(163\) 22.8885i 1.79277i 0.443280 + 0.896383i \(0.353815\pi\)
−0.443280 + 0.896383i \(0.646185\pi\)
\(164\) 0 0
\(165\) −5.81143 + 3.32898i −0.452419 + 0.259161i
\(166\) 0 0
\(167\) −3.64205 2.10274i −0.281830 0.162715i 0.352421 0.935841i \(-0.385358\pi\)
−0.634252 + 0.773127i \(0.718692\pi\)
\(168\) 0 0
\(169\) 11.5349 + 19.9790i 0.887298 + 1.53685i
\(170\) 0 0
\(171\) 0.123969 2.66392i 0.00948011 0.203715i
\(172\) 0 0
\(173\) 7.10273 4.10077i 0.540011 0.311775i −0.205072 0.978747i \(-0.565743\pi\)
0.745083 + 0.666971i \(0.232410\pi\)
\(174\) 0 0
\(175\) 6.10470 + 9.66585i 0.461472 + 0.730669i
\(176\) 0 0
\(177\) 5.46488 1.32894i 0.410765 0.0998892i
\(178\) 0 0
\(179\) 22.4354 1.67690 0.838449 0.544980i \(-0.183463\pi\)
0.838449 + 0.544980i \(0.183463\pi\)
\(180\) 0 0
\(181\) −20.3832 −1.51507 −0.757537 0.652792i \(-0.773597\pi\)
−0.757537 + 0.652792i \(0.773597\pi\)
\(182\) 0 0
\(183\) 5.92982 20.2362i 0.438345 1.49590i
\(184\) 0 0
\(185\) 13.4597 14.0051i 0.989574 1.02968i
\(186\) 0 0
\(187\) 6.97070 4.02454i 0.509748 0.294303i
\(188\) 0 0
\(189\) −3.86732 + 11.2337i −0.281306 + 0.817130i
\(190\) 0 0
\(191\) 13.0270 + 22.5634i 0.942598 + 1.63263i 0.760491 + 0.649348i \(0.224958\pi\)
0.182107 + 0.983279i \(0.441708\pi\)
\(192\) 0 0
\(193\) 10.9966 + 6.34891i 0.791555 + 0.457004i 0.840510 0.541797i \(-0.182256\pi\)
−0.0489549 + 0.998801i \(0.515589\pi\)
\(194\) 0 0
\(195\) 11.6985 20.1045i 0.837748 1.43971i
\(196\) 0 0
\(197\) 6.79988i 0.484471i −0.970217 0.242236i \(-0.922119\pi\)
0.970217 0.242236i \(-0.0778807\pi\)
\(198\) 0 0
\(199\) 4.13197 0.292907 0.146454 0.989218i \(-0.453214\pi\)
0.146454 + 0.989218i \(0.453214\pi\)
\(200\) 0 0
\(201\) −5.12016 21.0552i −0.361148 1.48512i
\(202\) 0 0
\(203\) −9.29280 5.36520i −0.652227 0.376563i
\(204\) 0 0
\(205\) −13.5522 + 3.92132i −0.946530 + 0.273877i
\(206\) 0 0
\(207\) −9.67623 + 15.0929i −0.672545 + 1.04903i
\(208\) 0 0
\(209\) −0.768597 1.33125i −0.0531650 0.0920845i
\(210\) 0 0
\(211\) −1.22846 + 2.12776i −0.0845710 + 0.146481i −0.905208 0.424968i \(-0.860285\pi\)
0.820637 + 0.571449i \(0.193619\pi\)
\(212\) 0 0
\(213\) 8.67549 + 9.08861i 0.594435 + 0.622741i
\(214\) 0 0
\(215\) 3.66289 + 0.903916i 0.249807 + 0.0616465i
\(216\) 0 0
\(217\) 13.6359i 0.925664i
\(218\) 0 0
\(219\) 2.12120 2.02478i 0.143337 0.136822i
\(220\) 0 0
\(221\) −13.9775 + 24.2097i −0.940226 + 1.62852i
\(222\) 0 0
\(223\) 14.3020 8.25727i 0.957733 0.552947i 0.0622584 0.998060i \(-0.480170\pi\)
0.895475 + 0.445113i \(0.146836\pi\)
\(224\) 0 0
\(225\) −14.9997 0.101833i −0.999977 0.00678888i
\(226\) 0 0
\(227\) −2.88936 + 1.66817i −0.191773 + 0.110720i −0.592813 0.805341i \(-0.701982\pi\)
0.401039 + 0.916061i \(0.368649\pi\)
\(228\) 0 0
\(229\) 4.22081 7.31066i 0.278919 0.483102i −0.692197 0.721708i \(-0.743357\pi\)
0.971116 + 0.238606i \(0.0766905\pi\)
\(230\) 0 0
\(231\) 1.61819 + 6.65434i 0.106469 + 0.437823i
\(232\) 0 0
\(233\) 5.85320i 0.383456i 0.981448 + 0.191728i \(0.0614091\pi\)
−0.981448 + 0.191728i \(0.938591\pi\)
\(234\) 0 0
\(235\) 27.1176 + 6.69200i 1.76896 + 0.436538i
\(236\) 0 0
\(237\) 1.94022 6.62122i 0.126031 0.430095i
\(238\) 0 0
\(239\) −1.54025 + 2.66779i −0.0996304 + 0.172565i −0.911532 0.411230i \(-0.865099\pi\)
0.811901 + 0.583795i \(0.198433\pi\)
\(240\) 0 0
\(241\) 1.14795 + 1.98831i 0.0739459 + 0.128078i 0.900627 0.434592i \(-0.143107\pi\)
−0.826681 + 0.562670i \(0.809774\pi\)
\(242\) 0 0
\(243\) −9.66972 12.2269i −0.620313 0.784354i
\(244\) 0 0
\(245\) −3.80653 + 1.10141i −0.243190 + 0.0703667i
\(246\) 0 0
\(247\) 4.62351 + 2.66939i 0.294187 + 0.169849i
\(248\) 0 0
\(249\) −6.97814 2.04481i −0.442222 0.129585i
\(250\) 0 0
\(251\) −2.66918 −0.168477 −0.0842387 0.996446i \(-0.526846\pi\)
−0.0842387 + 0.996446i \(0.526846\pi\)
\(252\) 0 0
\(253\) 10.3342i 0.649707i
\(254\) 0 0
\(255\) 18.0273 + 0.0611931i 1.12891 + 0.00383206i
\(256\) 0 0
\(257\) −18.9713 10.9531i −1.18340 0.683236i −0.226600 0.973988i \(-0.572761\pi\)
−0.956798 + 0.290752i \(0.906094\pi\)
\(258\) 0 0
\(259\) −9.93099 17.2010i −0.617082 1.06882i
\(260\) 0 0
\(261\) 12.5070 6.46516i 0.774161 0.400184i
\(262\) 0 0
\(263\) −21.3919 + 12.3506i −1.31908 + 0.761570i −0.983580 0.180470i \(-0.942238\pi\)
−0.335498 + 0.942041i \(0.608905\pi\)
\(264\) 0 0
\(265\) −4.12319 + 4.29028i −0.253286 + 0.263550i
\(266\) 0 0
\(267\) 3.12010 + 3.26867i 0.190947 + 0.200039i
\(268\) 0 0
\(269\) −3.77073 −0.229905 −0.114953 0.993371i \(-0.536672\pi\)
−0.114953 + 0.993371i \(0.536672\pi\)
\(270\) 0 0
\(271\) −4.44729 −0.270154 −0.135077 0.990835i \(-0.543128\pi\)
−0.135077 + 0.990835i \(0.543128\pi\)
\(272\) 0 0
\(273\) −16.4226 17.2046i −0.993941 1.04127i
\(274\) 0 0
\(275\) −7.31035 + 4.61702i −0.440831 + 0.278417i
\(276\) 0 0
\(277\) −7.73881 + 4.46801i −0.464980 + 0.268456i −0.714136 0.700007i \(-0.753180\pi\)
0.249156 + 0.968463i \(0.419847\pi\)
\(278\) 0 0
\(279\) −15.0618 9.65629i −0.901726 0.578107i
\(280\) 0 0
\(281\) 4.53764 + 7.85942i 0.270693 + 0.468854i 0.969039 0.246906i \(-0.0794139\pi\)
−0.698347 + 0.715760i \(0.746081\pi\)
\(282\) 0 0
\(283\) −18.4847 10.6722i −1.09880 0.634394i −0.162896 0.986643i \(-0.552084\pi\)
−0.935906 + 0.352249i \(0.885417\pi\)
\(284\) 0 0
\(285\) 0.0116865 3.44281i 0.000692251 0.203934i
\(286\) 0 0
\(287\) 14.4260i 0.851540i
\(288\) 0 0
\(289\) −4.66573 −0.274455
\(290\) 0 0
\(291\) −6.64757 1.94794i −0.389687 0.114190i
\(292\) 0 0
\(293\) −5.36286 3.09625i −0.313301 0.180885i 0.335101 0.942182i \(-0.391229\pi\)
−0.648403 + 0.761297i \(0.724563\pi\)
\(294\) 0 0
\(295\) 6.97463 2.01810i 0.406079 0.117498i
\(296\) 0 0
\(297\) −8.49611 2.92488i −0.492994 0.169719i
\(298\) 0 0
\(299\) −17.9457 31.0829i −1.03783 1.79757i
\(300\) 0 0
\(301\) 1.92889 3.34093i 0.111179 0.192568i
\(302\) 0 0
\(303\) 3.02693 10.3298i 0.173893 0.593429i
\(304\) 0 0
\(305\) 6.52241 26.4304i 0.373472 1.51340i
\(306\) 0 0
\(307\) 14.8211i 0.845887i 0.906156 + 0.422944i \(0.139003\pi\)
−0.906156 + 0.422944i \(0.860997\pi\)
\(308\) 0 0
\(309\) 0.00237797 + 0.00977871i 0.000135278 + 0.000556291i
\(310\) 0 0
\(311\) 1.96694 3.40683i 0.111535 0.193184i −0.804854 0.593472i \(-0.797757\pi\)
0.916389 + 0.400288i \(0.131090\pi\)
\(312\) 0 0
\(313\) 15.0551 8.69208i 0.850966 0.491305i −0.0100110 0.999950i \(-0.503187\pi\)
0.860977 + 0.508645i \(0.169853\pi\)
\(314\) 0 0
\(315\) −4.36306 + 14.7043i −0.245831 + 0.828493i
\(316\) 0 0
\(317\) −22.0187 + 12.7125i −1.23669 + 0.714006i −0.968417 0.249336i \(-0.919788\pi\)
−0.268277 + 0.963342i \(0.586454\pi\)
\(318\) 0 0
\(319\) 4.05774 7.02821i 0.227190 0.393504i
\(320\) 0 0
\(321\) 5.68925 5.43065i 0.317543 0.303109i
\(322\) 0 0
\(323\) 4.13767i 0.230226i
\(324\) 0 0
\(325\) 13.9702 26.5815i 0.774925 1.47448i
\(326\) 0 0
\(327\) 18.5761 + 19.4606i 1.02726 + 1.07618i
\(328\) 0 0
\(329\) 14.2802 24.7341i 0.787294 1.36363i
\(330\) 0 0
\(331\) −7.50445 12.9981i −0.412482 0.714439i 0.582679 0.812703i \(-0.302005\pi\)
−0.995160 + 0.0982634i \(0.968671\pi\)
\(332\) 0 0
\(333\) 26.0323 + 1.21144i 1.42656 + 0.0663867i
\(334\) 0 0
\(335\) −7.77536 26.8720i −0.424814 1.46817i
\(336\) 0 0
\(337\) 17.8677 + 10.3159i 0.973317 + 0.561945i 0.900246 0.435382i \(-0.143387\pi\)
0.0730711 + 0.997327i \(0.476720\pi\)
\(338\) 0 0
\(339\) 0.687895 + 2.82877i 0.0373613 + 0.153637i
\(340\) 0 0
\(341\) −10.3129 −0.558476
\(342\) 0 0
\(343\) 20.0571i 1.08298i
\(344\) 0 0
\(345\) −11.6407 + 20.0051i −0.626712 + 1.07704i
\(346\) 0 0
\(347\) 30.2610 + 17.4712i 1.62450 + 0.937904i 0.985696 + 0.168532i \(0.0539025\pi\)
0.638801 + 0.769372i \(0.279431\pi\)
\(348\) 0 0
\(349\) −7.96471 13.7953i −0.426341 0.738445i 0.570203 0.821504i \(-0.306864\pi\)
−0.996545 + 0.0830588i \(0.973531\pi\)
\(350\) 0 0
\(351\) 30.6334 5.95641i 1.63509 0.317930i
\(352\) 0 0
\(353\) 1.48879 0.859556i 0.0792405 0.0457495i −0.459856 0.887993i \(-0.652099\pi\)
0.539097 + 0.842244i \(0.318766\pi\)
\(354\) 0 0
\(355\) 11.6953 + 11.2398i 0.620720 + 0.596546i
\(356\) 0 0
\(357\) 5.18362 17.6897i 0.274346 0.936237i
\(358\) 0 0
\(359\) −34.6658 −1.82959 −0.914797 0.403915i \(-0.867649\pi\)
−0.914797 + 0.403915i \(0.867649\pi\)
\(360\) 0 0
\(361\) −18.2098 −0.958410
\(362\) 0 0
\(363\) 13.4803 3.27812i 0.707533 0.172057i
\(364\) 0 0
\(365\) 2.62326 2.72957i 0.137308 0.142872i
\(366\) 0 0
\(367\) −22.4914 + 12.9854i −1.17404 + 0.677834i −0.954629 0.297798i \(-0.903748\pi\)
−0.219414 + 0.975632i \(0.570414\pi\)
\(368\) 0 0
\(369\) −15.9345 10.2158i −0.829518 0.531814i
\(370\) 0 0
\(371\) 3.04223 + 5.26929i 0.157945 + 0.273568i
\(372\) 0 0
\(373\) −25.0807 14.4803i −1.29863 0.749763i −0.318461 0.947936i \(-0.603166\pi\)
−0.980167 + 0.198172i \(0.936499\pi\)
\(374\) 0 0
\(375\) −19.3521 + 0.703164i −0.999341 + 0.0363112i
\(376\) 0 0
\(377\) 28.1855i 1.45163i
\(378\) 0 0
\(379\) 22.8816 1.17535 0.587674 0.809098i \(-0.300044\pi\)
0.587674 + 0.809098i \(0.300044\pi\)
\(380\) 0 0
\(381\) 13.2385 12.6367i 0.678226 0.647398i
\(382\) 0 0
\(383\) −0.0595976 0.0344087i −0.00304529 0.00175820i 0.498477 0.866903i \(-0.333893\pi\)
−0.501522 + 0.865145i \(0.667226\pi\)
\(384\) 0 0
\(385\) 2.45735 + 8.49271i 0.125238 + 0.432828i
\(386\) 0 0
\(387\) 2.32434 + 4.49648i 0.118153 + 0.228569i
\(388\) 0 0
\(389\) 9.31848 + 16.1401i 0.472466 + 0.818334i 0.999504 0.0315074i \(-0.0100308\pi\)
−0.527038 + 0.849842i \(0.676697\pi\)
\(390\) 0 0
\(391\) 13.9083 24.0900i 0.703375 1.21828i
\(392\) 0 0
\(393\) −33.9303 + 8.25111i −1.71156 + 0.416213i
\(394\) 0 0
\(395\) 2.13411 8.64796i 0.107379 0.435126i
\(396\) 0 0
\(397\) 18.8002i 0.943554i 0.881718 + 0.471777i \(0.156387\pi\)
−0.881718 + 0.471777i \(0.843613\pi\)
\(398\) 0 0
\(399\) −3.37834 0.989956i −0.169128 0.0495598i
\(400\) 0 0
\(401\) 1.19535 2.07041i 0.0596930 0.103391i −0.834635 0.550804i \(-0.814321\pi\)
0.894328 + 0.447413i \(0.147654\pi\)
\(402\) 0 0
\(403\) 31.0188 17.9087i 1.54516 0.892096i
\(404\) 0 0
\(405\) −13.1522 15.2322i −0.653538 0.756893i
\(406\) 0 0
\(407\) 13.0092 7.51088i 0.644843 0.372300i
\(408\) 0 0
\(409\) −9.65718 + 16.7267i −0.477517 + 0.827083i −0.999668 0.0257696i \(-0.991796\pi\)
0.522151 + 0.852853i \(0.325130\pi\)
\(410\) 0 0
\(411\) −25.5731 7.49369i −1.26143 0.369636i
\(412\) 0 0
\(413\) 7.42432i 0.365327i
\(414\) 0 0
\(415\) −9.11414 2.24916i −0.447395 0.110407i
\(416\) 0 0
\(417\) −11.7667 + 2.86141i −0.576219 + 0.140124i
\(418\) 0 0
\(419\) −3.52444 + 6.10450i −0.172180 + 0.298224i −0.939182 0.343421i \(-0.888414\pi\)
0.767002 + 0.641645i \(0.221748\pi\)
\(420\) 0 0
\(421\) −7.07655 12.2569i −0.344890 0.597367i 0.640444 0.768005i \(-0.278750\pi\)
−0.985334 + 0.170638i \(0.945417\pi\)
\(422\) 0 0
\(423\) 17.2079 + 33.2890i 0.836678 + 1.61857i
\(424\) 0 0
\(425\) 23.2549 0.924022i 1.12803 0.0448217i
\(426\) 0 0
\(427\) −24.1073 13.9183i −1.16663 0.673556i
\(428\) 0 0
\(429\) 13.0120 12.4205i 0.628224 0.599669i
\(430\) 0 0
\(431\) 0.388711 0.0187236 0.00936178 0.999956i \(-0.497020\pi\)
0.00936178 + 0.999956i \(0.497020\pi\)
\(432\) 0 0
\(433\) 29.2319i 1.40479i −0.711786 0.702397i \(-0.752113\pi\)
0.711786 0.702397i \(-0.247887\pi\)
\(434\) 0 0
\(435\) 15.7717 9.03456i 0.756196 0.433174i
\(436\) 0 0
\(437\) −4.60065 2.65619i −0.220079 0.127063i
\(438\) 0 0
\(439\) 19.5275 + 33.8226i 0.931995 + 1.61426i 0.779907 + 0.625895i \(0.215266\pi\)
0.152088 + 0.988367i \(0.451400\pi\)
\(440\) 0 0
\(441\) −4.47567 2.86940i −0.213127 0.136638i
\(442\) 0 0
\(443\) 17.3321 10.0067i 0.823472 0.475432i −0.0281404 0.999604i \(-0.508959\pi\)
0.851612 + 0.524172i \(0.175625\pi\)
\(444\) 0 0
\(445\) 4.20614 + 4.04233i 0.199390 + 0.191625i
\(446\) 0 0
\(447\) −26.4014 + 6.42025i −1.24874 + 0.303667i
\(448\) 0 0
\(449\) 23.4031 1.10446 0.552230 0.833692i \(-0.313777\pi\)
0.552230 + 0.833692i \(0.313777\pi\)
\(450\) 0 0
\(451\) −10.9105 −0.513755
\(452\) 0 0
\(453\) 2.72191 9.28881i 0.127886 0.436427i
\(454\) 0 0
\(455\) −22.1390 21.2768i −1.03789 0.997470i
\(456\) 0 0
\(457\) −16.3691 + 9.45069i −0.765713 + 0.442084i −0.831343 0.555760i \(-0.812428\pi\)
0.0656303 + 0.997844i \(0.479094\pi\)
\(458\) 0 0
\(459\) 15.8687 + 18.2527i 0.740687 + 0.851961i
\(460\) 0 0
\(461\) 9.29895 + 16.1063i 0.433095 + 0.750143i 0.997138 0.0756026i \(-0.0240880\pi\)
−0.564043 + 0.825746i \(0.690755\pi\)
\(462\) 0 0
\(463\) −21.7115 12.5352i −1.00902 0.582558i −0.0981153 0.995175i \(-0.531281\pi\)
−0.910905 + 0.412617i \(0.864615\pi\)
\(464\) 0 0
\(465\) −19.9638 11.6167i −0.925801 0.538710i
\(466\) 0 0
\(467\) 4.90450i 0.226953i −0.993541 0.113476i \(-0.963801\pi\)
0.993541 0.113476i \(-0.0361987\pi\)
\(468\) 0 0
\(469\) −28.6045 −1.32083
\(470\) 0 0
\(471\) 8.96868 + 36.8811i 0.413255 + 1.69939i
\(472\) 0 0
\(473\) 2.52677 + 1.45883i 0.116181 + 0.0670771i
\(474\) 0 0
\(475\) −0.176468 4.44117i −0.00809690 0.203775i
\(476\) 0 0
\(477\) −7.97466 0.371110i −0.365135 0.0169920i
\(478\) 0 0
\(479\) 10.5302 + 18.2388i 0.481136 + 0.833353i 0.999766 0.0216466i \(-0.00689087\pi\)
−0.518629 + 0.854999i \(0.673558\pi\)
\(480\) 0 0
\(481\) −26.0857 + 45.1818i −1.18941 + 2.06011i
\(482\) 0 0
\(483\) 16.3414 + 17.1195i 0.743559 + 0.778966i
\(484\) 0 0
\(485\) −8.68237 2.14261i −0.394246 0.0972907i
\(486\) 0 0
\(487\) 31.2503i 1.41609i −0.706170 0.708043i \(-0.749578\pi\)
0.706170 0.708043i \(-0.250422\pi\)
\(488\) 0 0
\(489\) −28.6767 + 27.3733i −1.29681 + 1.23786i
\(490\) 0 0
\(491\) 7.95579 13.7798i 0.359040 0.621876i −0.628761 0.777599i \(-0.716437\pi\)
0.987801 + 0.155723i \(0.0497708\pi\)
\(492\) 0 0
\(493\) −18.9179 + 10.9222i −0.852018 + 0.491913i
\(494\) 0 0
\(495\) −11.1210 3.29982i −0.499850 0.148316i
\(496\) 0 0
\(497\) 14.3640 8.29308i 0.644315 0.371996i
\(498\) 0 0
\(499\) 12.0304 20.8373i 0.538556 0.932806i −0.460426 0.887698i \(-0.652303\pi\)
0.998982 0.0451079i \(-0.0143632\pi\)
\(500\) 0 0
\(501\) −1.72118 7.07783i −0.0768964 0.316214i
\(502\) 0 0
\(503\) 5.52657i 0.246417i 0.992381 + 0.123209i \(0.0393185\pi\)
−0.992381 + 0.123209i \(0.960682\pi\)
\(504\) 0 0
\(505\) 3.32943 13.4917i 0.148158 0.600371i
\(506\) 0 0
\(507\) −11.2364 + 38.3456i −0.499027 + 1.70299i
\(508\) 0 0
\(509\) 4.57604 7.92593i 0.202829 0.351311i −0.746610 0.665262i \(-0.768320\pi\)
0.949439 + 0.313952i \(0.101653\pi\)
\(510\) 0 0
\(511\) −1.93553 3.35244i −0.0856228 0.148303i
\(512\) 0 0
\(513\) 3.48585 3.03057i 0.153904 0.133803i
\(514\) 0 0
\(515\) 0.00361114 + 0.0124802i 0.000159126 + 0.000549945i
\(516\) 0 0
\(517\) 18.7065 + 10.8002i 0.822713 + 0.474994i
\(518\) 0 0
\(519\) 13.6322 + 3.99466i 0.598389 + 0.175346i
\(520\) 0 0
\(521\) 28.6259 1.25412 0.627061 0.778970i \(-0.284258\pi\)
0.627061 + 0.778970i \(0.284258\pi\)
\(522\) 0 0
\(523\) 8.60074i 0.376084i 0.982161 + 0.188042i \(0.0602141\pi\)
−0.982161 + 0.188042i \(0.939786\pi\)
\(524\) 0 0
\(525\) −4.80938 + 19.2083i −0.209899 + 0.838318i
\(526\) 0 0
\(527\) 24.0403 + 13.8797i 1.04721 + 0.604608i
\(528\) 0 0
\(529\) 6.35695 + 11.0106i 0.276389 + 0.478720i
\(530\) 0 0
\(531\) 8.20067 + 5.25755i 0.355879 + 0.228158i
\(532\) 0 0
\(533\) 32.8161 18.9464i 1.42142 0.820660i
\(534\) 0 0
\(535\) 7.03583 7.32095i 0.304186 0.316512i
\(536\) 0 0
\(537\) 26.8313 + 28.1090i 1.15786 + 1.21299i
\(538\) 0 0
\(539\) −3.06452 −0.131998
\(540\) 0 0
\(541\) 14.2900 0.614377 0.307188 0.951649i \(-0.400612\pi\)
0.307188 + 0.951649i \(0.400612\pi\)
\(542\) 0 0
\(543\) −24.3771 25.5379i −1.04612 1.09594i
\(544\) 0 0
\(545\) 25.0420 + 24.0668i 1.07268 + 1.03091i
\(546\) 0 0
\(547\) 14.4638 8.35071i 0.618429 0.357050i −0.157828 0.987467i \(-0.550449\pi\)
0.776257 + 0.630416i \(0.217116\pi\)
\(548\) 0 0
\(549\) 32.4454 16.7718i 1.38474 0.715805i
\(550\) 0 0
\(551\) 2.08591 + 3.61289i 0.0888626 + 0.153914i
\(552\) 0 0
\(553\) −7.88783 4.55404i −0.335425 0.193657i
\(554\) 0 0
\(555\) 33.6438 + 0.114203i 1.42810 + 0.00484765i
\(556\) 0 0
\(557\) 5.13418i 0.217542i 0.994067 + 0.108771i \(0.0346915\pi\)
−0.994067 + 0.108771i \(0.965308\pi\)
\(558\) 0 0
\(559\) −10.1332 −0.428590
\(560\) 0 0
\(561\) 13.3788 + 3.92041i 0.564855 + 0.165520i
\(562\) 0 0
\(563\) −1.22820 0.709103i −0.0517625 0.0298851i 0.473895 0.880581i \(-0.342847\pi\)
−0.525658 + 0.850696i \(0.676181\pi\)
\(564\) 0 0
\(565\) 1.04462 + 3.61026i 0.0439476 + 0.151885i
\(566\) 0 0
\(567\) −18.6996 + 8.58948i −0.785310 + 0.360724i
\(568\) 0 0
\(569\) 4.08734 + 7.07948i 0.171350 + 0.296787i 0.938892 0.344211i \(-0.111854\pi\)
−0.767542 + 0.640999i \(0.778520\pi\)
\(570\) 0 0
\(571\) −13.0685 + 22.6352i −0.546898 + 0.947255i 0.451587 + 0.892227i \(0.350858\pi\)
−0.998485 + 0.0550277i \(0.982475\pi\)
\(572\) 0 0
\(573\) −12.6899 + 43.3057i −0.530128 + 1.80912i
\(574\) 0 0
\(575\) −13.9011 + 26.4501i −0.579715 + 1.10305i
\(576\) 0 0
\(577\) 15.8687i 0.660622i −0.943872 0.330311i \(-0.892846\pi\)
0.943872 0.330311i \(-0.107154\pi\)
\(578\) 0 0
\(579\) 5.19683 + 21.3705i 0.215973 + 0.888126i
\(580\) 0 0
\(581\) −4.79953 + 8.31303i −0.199118 + 0.344882i
\(582\) 0 0
\(583\) −3.98520 + 2.30086i −0.165050 + 0.0952918i
\(584\) 0 0
\(585\) 39.1794 9.38684i 1.61987 0.388098i
\(586\) 0 0
\(587\) 15.5489 8.97715i 0.641771 0.370527i −0.143526 0.989647i \(-0.545844\pi\)
0.785296 + 0.619120i \(0.212511\pi\)
\(588\) 0 0
\(589\) 2.65071 4.59116i 0.109221 0.189176i
\(590\) 0 0
\(591\) 8.51949 8.13225i 0.350445 0.334516i
\(592\) 0 0
\(593\) 0.875924i 0.0359699i −0.999838 0.0179849i \(-0.994275\pi\)
0.999838 0.0179849i \(-0.00572510\pi\)
\(594\) 0 0
\(595\) 5.70164 23.1045i 0.233744 0.947190i
\(596\) 0 0
\(597\) 4.94158 + 5.17689i 0.202246 + 0.211876i
\(598\) 0 0
\(599\) 8.16182 14.1367i 0.333483 0.577609i −0.649709 0.760183i \(-0.725109\pi\)
0.983192 + 0.182573i \(0.0584427\pi\)
\(600\) 0 0
\(601\) 7.79862 + 13.5076i 0.318112 + 0.550987i 0.980094 0.198533i \(-0.0636178\pi\)
−0.661982 + 0.749520i \(0.730284\pi\)
\(602\) 0 0
\(603\) 20.2564 31.5957i 0.824903 1.28668i
\(604\) 0 0
\(605\) 17.2045 4.97808i 0.699460 0.202388i
\(606\) 0 0
\(607\) −11.2066 6.47016i −0.454864 0.262616i 0.255018 0.966936i \(-0.417918\pi\)
−0.709882 + 0.704321i \(0.751252\pi\)
\(608\) 0 0
\(609\) −4.39163 18.0593i −0.177958 0.731799i
\(610\) 0 0
\(611\) −75.0197 −3.03497
\(612\) 0 0
\(613\) 1.16228i 0.0469439i 0.999724 + 0.0234719i \(0.00747204\pi\)
−0.999724 + 0.0234719i \(0.992528\pi\)
\(614\) 0 0
\(615\) −21.1206 12.2898i −0.851666 0.495572i
\(616\) 0 0
\(617\) −15.9879 9.23060i −0.643647 0.371610i 0.142371 0.989813i \(-0.454527\pi\)
−0.786018 + 0.618203i \(0.787861\pi\)
\(618\) 0 0
\(619\) 18.0563 + 31.2744i 0.725742 + 1.25702i 0.958668 + 0.284528i \(0.0918369\pi\)
−0.232925 + 0.972495i \(0.574830\pi\)
\(620\) 0 0
\(621\) −30.4819 + 5.92696i −1.22320 + 0.237841i
\(622\) 0 0
\(623\) 5.16595 2.98256i 0.206970 0.119494i
\(624\) 0 0
\(625\) −24.9212 + 1.98360i −0.996847 + 0.0793439i
\(626\) 0 0
\(627\) 0.748711 2.55506i 0.0299006 0.102039i
\(628\) 0 0
\(629\) −40.4341 −1.61221
\(630\) 0 0
\(631\) 40.6445 1.61803 0.809017 0.587786i \(-0.200000\pi\)
0.809017 + 0.587786i \(0.200000\pi\)
\(632\) 0 0
\(633\) −4.13502 + 1.00555i −0.164352 + 0.0399669i
\(634\) 0 0
\(635\) 16.3718 17.0353i 0.649697 0.676025i
\(636\) 0 0
\(637\) 9.21734 5.32164i 0.365204 0.210851i
\(638\) 0 0
\(639\) −1.01164 + 21.7388i −0.0400200 + 0.859976i
\(640\) 0 0
\(641\) −12.2380 21.1968i −0.483371 0.837223i 0.516447 0.856319i \(-0.327254\pi\)
−0.999818 + 0.0190966i \(0.993921\pi\)
\(642\) 0 0
\(643\) −7.99487 4.61584i −0.315287 0.182031i 0.334003 0.942572i \(-0.391600\pi\)
−0.649290 + 0.760541i \(0.724934\pi\)
\(644\) 0 0
\(645\) 3.24809 + 5.67022i 0.127893 + 0.223265i
\(646\) 0 0
\(647\) 20.0509i 0.788282i −0.919050 0.394141i \(-0.871042\pi\)
0.919050 0.394141i \(-0.128958\pi\)
\(648\) 0 0
\(649\) 5.61506 0.220411
\(650\) 0 0
\(651\) −17.0842 + 16.3077i −0.669584 + 0.639149i
\(652\) 0 0
\(653\) 27.0382 + 15.6105i 1.05809 + 0.610887i 0.924903 0.380204i \(-0.124146\pi\)
0.133185 + 0.991091i \(0.457480\pi\)
\(654\) 0 0
\(655\) −43.3041 + 12.5300i −1.69203 + 0.489586i
\(656\) 0 0
\(657\) 5.07365 + 0.236108i 0.197942 + 0.00921145i
\(658\) 0 0
\(659\) 21.6530 + 37.5041i 0.843480 + 1.46095i 0.886934 + 0.461895i \(0.152830\pi\)
−0.0434542 + 0.999055i \(0.513836\pi\)
\(660\) 0 0
\(661\) 17.2404 29.8612i 0.670573 1.16147i −0.307168 0.951655i \(-0.599381\pi\)
0.977742 0.209812i \(-0.0672852\pi\)
\(662\) 0 0
\(663\) −47.0482 + 11.4411i −1.82720 + 0.444336i
\(664\) 0 0
\(665\) −4.41244 1.08889i −0.171107 0.0422252i
\(666\) 0 0
\(667\) 28.0462i 1.08595i
\(668\) 0 0
\(669\) 27.4498 + 8.04362i 1.06127 + 0.310984i
\(670\) 0 0
\(671\) 10.5265 18.2325i 0.406372 0.703858i
\(672\) 0 0
\(673\) 26.2234 15.1401i 1.01084 0.583606i 0.0993997 0.995048i \(-0.468308\pi\)
0.911436 + 0.411441i \(0.134974\pi\)
\(674\) 0 0
\(675\) −17.8111 18.9147i −0.685549 0.728026i
\(676\) 0 0
\(677\) −8.03094 + 4.63666i −0.308654 + 0.178201i −0.646324 0.763063i \(-0.723695\pi\)
0.337670 + 0.941265i \(0.390361\pi\)
\(678\) 0 0
\(679\) −4.57216 + 7.91921i −0.175463 + 0.303912i
\(680\) 0 0
\(681\) −5.54553 1.62501i −0.212505 0.0622705i
\(682\) 0 0
\(683\) 18.0856i 0.692028i 0.938229 + 0.346014i \(0.112465\pi\)
−0.938229 + 0.346014i \(0.887535\pi\)
\(684\) 0 0
\(685\) −33.4009 8.24256i −1.27618 0.314932i
\(686\) 0 0
\(687\) 14.2073 3.45490i 0.542041 0.131813i
\(688\) 0 0
\(689\) 7.99102 13.8409i 0.304434 0.527295i
\(690\) 0 0
\(691\) −20.7805 35.9928i −0.790527 1.36923i −0.925641 0.378402i \(-0.876474\pi\)
0.135114 0.990830i \(-0.456860\pi\)
\(692\) 0 0
\(693\) −6.40189 + 9.98560i −0.243188 + 0.379322i
\(694\) 0 0
\(695\) −15.0175 + 4.34528i −0.569645 + 0.164826i
\(696\) 0 0
\(697\) 25.4333 + 14.6839i 0.963354 + 0.556193i
\(698\) 0 0
\(699\) −7.33340 + 7.00007i −0.277375 + 0.264767i
\(700\) 0 0
\(701\) 23.4059 0.884030 0.442015 0.897008i \(-0.354264\pi\)
0.442015 + 0.897008i \(0.354264\pi\)
\(702\) 0 0
\(703\) 7.72203i 0.291242i
\(704\) 0 0
\(705\) 24.0467 + 41.9786i 0.905652 + 1.58101i
\(706\) 0 0
\(707\) −12.3058 7.10475i −0.462807 0.267201i
\(708\) 0 0
\(709\) −4.93075 8.54030i −0.185178 0.320738i 0.758458 0.651721i \(-0.225953\pi\)
−0.943637 + 0.330984i \(0.892620\pi\)
\(710\) 0 0
\(711\) 10.6160 5.48770i 0.398132 0.205805i
\(712\) 0 0
\(713\) −30.8654 + 17.8201i −1.15592 + 0.667369i
\(714\) 0 0
\(715\) 16.0918 16.7439i 0.601798 0.626185i
\(716\) 0 0
\(717\) −5.18449 + 1.26075i −0.193618 + 0.0470837i
\(718\) 0 0
\(719\) 7.32159 0.273049 0.136525 0.990637i \(-0.456407\pi\)
0.136525 + 0.990637i \(0.456407\pi\)
\(720\) 0 0
\(721\) 0.0132849 0.000494755
\(722\) 0 0
\(723\) −1.11825 + 3.81615i −0.0415881 + 0.141924i
\(724\) 0 0
\(725\) 19.8396 12.5302i 0.736826 0.465360i
\(726\) 0 0
\(727\) −31.1998 + 18.0132i −1.15714 + 0.668073i −0.950616 0.310369i \(-0.899547\pi\)
−0.206521 + 0.978442i \(0.566214\pi\)
\(728\) 0 0
\(729\) 3.75450 26.7377i 0.139056 0.990285i
\(730\) 0 0
\(731\) −3.92674 6.80132i −0.145236 0.251556i
\(732\) 0 0
\(733\) −24.8324 14.3370i −0.917207 0.529550i −0.0344639 0.999406i \(-0.510972\pi\)
−0.882743 + 0.469856i \(0.844306\pi\)
\(734\) 0 0
\(735\) −5.93233 3.45193i −0.218817 0.127327i
\(736\) 0 0
\(737\) 21.6338i 0.796892i
\(738\) 0 0
\(739\) −34.2481 −1.25984 −0.629918 0.776662i \(-0.716912\pi\)
−0.629918 + 0.776662i \(0.716912\pi\)
\(740\) 0 0
\(741\) 2.18500 + 8.98517i 0.0802679 + 0.330078i
\(742\) 0 0
\(743\) 0.562189 + 0.324580i 0.0206247 + 0.0119077i 0.510277 0.860010i \(-0.329543\pi\)
−0.489652 + 0.871918i \(0.662876\pi\)
\(744\) 0 0
\(745\) −33.6952 + 9.74965i −1.23450 + 0.357200i
\(746\) 0 0
\(747\) −5.78352 11.1883i −0.211608 0.409359i
\(748\) 0 0
\(749\) −5.19127 8.99155i −0.189685 0.328544i
\(750\) 0 0
\(751\) −6.93481 + 12.0114i −0.253055 + 0.438304i −0.964365 0.264574i \(-0.914769\pi\)
0.711311 + 0.702878i \(0.248102\pi\)
\(752\) 0 0
\(753\) −3.19218 3.34419i −0.116330 0.121869i
\(754\) 0 0
\(755\) 2.99392 12.1321i 0.108960 0.441532i
\(756\) 0 0
\(757\) 9.30142i 0.338066i −0.985610 0.169033i \(-0.945936\pi\)
0.985610 0.169033i \(-0.0540644\pi\)
\(758\) 0 0
\(759\) −12.9476 + 12.3591i −0.469969 + 0.448607i
\(760\) 0 0
\(761\) 13.4133 23.2326i 0.486233 0.842181i −0.513642 0.858005i \(-0.671704\pi\)
0.999875 + 0.0158243i \(0.00503723\pi\)
\(762\) 0 0
\(763\) 30.7565 17.7573i 1.11346 0.642856i
\(764\) 0 0
\(765\) 21.4828 + 22.6593i 0.776714 + 0.819250i
\(766\) 0 0
\(767\) −16.8888 + 9.75073i −0.609818 + 0.352078i
\(768\) 0 0
\(769\) 19.1864 33.2317i 0.691878 1.19837i −0.279344 0.960191i \(-0.590117\pi\)
0.971222 0.238176i \(-0.0765495\pi\)
\(770\) 0 0
\(771\) −8.96555 36.8682i −0.322886 1.32778i
\(772\) 0 0
\(773\) 41.6235i 1.49709i −0.663082 0.748546i \(-0.730752\pi\)
0.663082 0.748546i \(-0.269248\pi\)
\(774\) 0 0
\(775\) −26.3956 13.8724i −0.948157 0.498312i
\(776\) 0 0
\(777\) 9.67404 33.0138i 0.347054 1.18436i
\(778\) 0 0
\(779\) 2.80430 4.85719i 0.100475 0.174027i
\(780\) 0 0
\(781\) 6.27212 + 10.8636i 0.224434 + 0.388731i
\(782\) 0 0
\(783\) 23.0577 + 7.93787i 0.824014 + 0.283676i
\(784\) 0 0
\(785\) 13.6196 + 47.0701i 0.486106 + 1.68000i
\(786\) 0 0
\(787\) −18.1836 10.4983i −0.648177 0.374225i 0.139581 0.990211i \(-0.455425\pi\)
−0.787757 + 0.615986i \(0.788758\pi\)
\(788\) 0 0
\(789\) −41.0573 12.0310i −1.46168 0.428317i
\(790\) 0 0
\(791\) 3.84303 0.136642
\(792\) 0 0
\(793\) 73.1186i 2.59652i
\(794\) 0 0
\(795\) −10.3063 0.0349846i −0.365528 0.00124078i
\(796\) 0 0
\(797\) 1.31920 + 0.761643i 0.0467286 + 0.0269788i 0.523182 0.852221i \(-0.324745\pi\)
−0.476454 + 0.879200i \(0.658078\pi\)
\(798\) 0 0
\(799\) −29.0710 50.3525i −1.02846 1.78134i
\(800\) 0 0
\(801\) −0.363832 + 7.81826i −0.0128554 + 0.276245i
\(802\) 0 0
\(803\) 2.53547 1.46386i 0.0894749 0.0516583i
\(804\) 0 0
\(805\) 22.0295 + 21.1715i 0.776437 + 0.746199i
\(806\) 0 0
\(807\) −4.50956 4.72430i −0.158744 0.166303i
\(808\) 0 0
\(809\) −35.7106 −1.25552 −0.627758 0.778408i \(-0.716027\pi\)
−0.627758 + 0.778408i \(0.716027\pi\)
\(810\) 0 0
\(811\) 30.3963 1.06736 0.533679 0.845687i \(-0.320809\pi\)
0.533679 + 0.845687i \(0.320809\pi\)
\(812\) 0 0
\(813\) −5.31869 5.57196i −0.186535 0.195417i
\(814\) 0 0
\(815\) −35.4642 + 36.9013i −1.24226 + 1.29260i
\(816\) 0 0
\(817\) −1.29890 + 0.749921i −0.0454428 + 0.0262364i
\(818\) 0 0
\(819\) 1.91503 41.1514i 0.0669164 1.43794i
\(820\) 0 0
\(821\) −3.00587 5.20632i −0.104906 0.181702i 0.808794 0.588092i \(-0.200121\pi\)
−0.913700 + 0.406390i \(0.866787\pi\)
\(822\) 0 0
\(823\) 29.2509 + 16.8880i 1.01962 + 0.588680i 0.913995 0.405726i \(-0.132981\pi\)
0.105629 + 0.994406i \(0.466314\pi\)
\(824\) 0 0
\(825\) −14.5274 3.63737i −0.505777 0.126637i
\(826\) 0 0
\(827\) 3.36732i 0.117093i 0.998285 + 0.0585466i \(0.0186466\pi\)
−0.998285 + 0.0585466i \(0.981353\pi\)
\(828\) 0 0
\(829\) −6.57350 −0.228307 −0.114154 0.993463i \(-0.536416\pi\)
−0.114154 + 0.993463i \(0.536416\pi\)
\(830\) 0 0
\(831\) −14.8531 4.35240i −0.515247 0.150983i
\(832\) 0 0
\(833\) 7.14366 + 4.12439i 0.247513 + 0.142902i
\(834\) 0 0
\(835\) −2.61374 9.03319i −0.0904522 0.312607i
\(836\) 0 0
\(837\) −5.91474 30.4191i −0.204443 1.05144i
\(838\) 0 0
\(839\) −2.40071 4.15814i −0.0828816 0.143555i 0.821605 0.570057i \(-0.193079\pi\)
−0.904487 + 0.426502i \(0.859746\pi\)
\(840\) 0 0
\(841\) 3.48766 6.04080i 0.120264 0.208303i
\(842\) 0 0
\(843\) −4.42023 + 15.0845i −0.152241 + 0.519539i
\(844\) 0 0
\(845\) −12.3593 + 50.0831i −0.425174 + 1.72291i
\(846\) 0 0
\(847\) 18.3137i 0.629266i
\(848\) 0 0
\(849\) −8.73559 35.9225i −0.299805 1.23286i
\(850\) 0 0
\(851\) 25.9567 44.9584i 0.889786 1.54115i
\(852\) 0 0
\(853\) 13.0511 7.53504i 0.446860 0.257995i −0.259643 0.965705i \(-0.583605\pi\)
0.706503 + 0.707710i \(0.250272\pi\)
\(854\) 0 0
\(855\) 4.32743 4.10275i 0.147995 0.140311i
\(856\) 0 0
\(857\) 27.8833 16.0984i 0.952476 0.549912i 0.0586268 0.998280i \(-0.481328\pi\)
0.893849 + 0.448368i \(0.147994\pi\)
\(858\) 0 0
\(859\) −14.4187 + 24.9740i −0.491961 + 0.852102i −0.999957 0.00925765i \(-0.997053\pi\)
0.507996 + 0.861359i \(0.330386\pi\)
\(860\) 0 0
\(861\) −18.0742 + 17.2526i −0.615966 + 0.587968i
\(862\) 0 0
\(863\) 19.7667i 0.672866i −0.941707 0.336433i \(-0.890779\pi\)
0.941707 0.336433i \(-0.109221\pi\)
\(864\) 0 0
\(865\) 17.8050 + 4.39387i 0.605390 + 0.149396i
\(866\) 0 0
\(867\) −5.57993 5.84563i −0.189504 0.198528i
\(868\) 0 0
\(869\) 3.44425 5.96562i 0.116838 0.202370i
\(870\) 0 0
\(871\) 37.5678 + 65.0693i 1.27294 + 2.20479i
\(872\) 0 0
\(873\) −5.50954 10.6583i −0.186470 0.360728i
\(874\) 0 0
\(875\) −5.13446 + 25.0423i −0.173577 + 0.846584i
\(876\) 0 0
\(877\) −17.7188 10.2300i −0.598322 0.345441i 0.170059 0.985434i \(-0.445604\pi\)
−0.768381 + 0.639993i \(0.778937\pi\)
\(878\) 0 0
\(879\) −2.53440 10.4220i −0.0854832 0.351525i
\(880\) 0 0
\(881\) −20.4830 −0.690090 −0.345045 0.938586i \(-0.612136\pi\)
−0.345045 + 0.938586i \(0.612136\pi\)
\(882\) 0 0
\(883\) 45.9847i 1.54751i 0.633486 + 0.773754i \(0.281624\pi\)
−0.633486 + 0.773754i \(0.718376\pi\)
\(884\) 0 0
\(885\) 10.8697 + 6.32491i 0.365381 + 0.212610i
\(886\) 0 0
\(887\) −8.92804 5.15461i −0.299774 0.173075i 0.342567 0.939493i \(-0.388704\pi\)
−0.642341 + 0.766419i \(0.722037\pi\)
\(888\) 0 0
\(889\) −12.0797 20.9226i −0.405140 0.701723i
\(890\) 0 0
\(891\) −6.49628 14.1427i −0.217634 0.473797i
\(892\) 0 0
\(893\) −9.61621 + 5.55192i −0.321794 + 0.185788i
\(894\) 0 0
\(895\) 36.1708 + 34.7621i 1.20906 + 1.16197i
\(896\) 0 0
\(897\) 17.4814 59.6572i 0.583686 1.99189i
\(898\) 0 0
\(899\) 27.9884 0.933464
\(900\) 0 0
\(901\) 12.3865 0.412653
\(902\) 0 0
\(903\) 6.49265 1.57887i 0.216062 0.0525416i
\(904\) 0 0
\(905\) −32.8623 31.5825i −1.09238 1.04984i
\(906\) 0 0
\(907\) 25.4011 14.6653i 0.843428 0.486953i −0.0149999 0.999887i \(-0.504775\pi\)
0.858428 + 0.512934i \(0.171441\pi\)
\(908\) 0 0
\(909\) 16.5621 8.56135i 0.549329 0.283962i
\(910\) 0 0
\(911\) 9.03057 + 15.6414i 0.299196 + 0.518223i 0.975952 0.217984i \(-0.0699482\pi\)
−0.676756 + 0.736207i \(0.736615\pi\)
\(912\) 0 0
\(913\) −6.28720 3.62992i −0.208076 0.120133i
\(914\) 0 0
\(915\) 40.9148 23.4374i 1.35260 0.774815i
\(916\) 0 0
\(917\) 46.0960i 1.52223i
\(918\) 0 0
\(919\) 6.69380 0.220808 0.110404 0.993887i \(-0.464786\pi\)
0.110404 + 0.993887i \(0.464786\pi\)
\(920\) 0 0
\(921\) −18.5692 + 17.7252i −0.611877 + 0.584065i
\(922\) 0 0
\(923\) −37.7300 21.7834i −1.24190 0.717011i
\(924\) 0 0
\(925\) 43.3999 1.72448i 1.42698 0.0567004i
\(926\) 0 0
\(927\) −0.00940772 + 0.0146741i −0.000308990 + 0.000481960i
\(928\) 0 0
\(929\) −9.17475 15.8911i −0.301014 0.521371i 0.675352 0.737495i \(-0.263992\pi\)
−0.976366 + 0.216124i \(0.930658\pi\)
\(930\) 0 0
\(931\) 0.787668 1.36428i 0.0258148 0.0447125i
\(932\) 0 0
\(933\) 6.62072 1.61002i 0.216753 0.0527096i
\(934\) 0 0
\(935\) 17.4741 + 4.31219i 0.571463 + 0.141024i
\(936\) 0 0
\(937\) 5.15287i 0.168337i 0.996452 + 0.0841684i \(0.0268234\pi\)
−0.996452 + 0.0841684i \(0.973177\pi\)
\(938\) 0 0
\(939\) 28.8952 + 8.46718i 0.942959 + 0.276316i
\(940\) 0 0
\(941\) −12.2567 + 21.2293i −0.399557 + 0.692054i −0.993671 0.112327i \(-0.964170\pi\)
0.594114 + 0.804381i \(0.297503\pi\)
\(942\) 0 0
\(943\) −32.6539 + 18.8527i −1.06336 + 0.613929i
\(944\) 0 0
\(945\) −23.6408 + 12.1190i −0.769035 + 0.394232i
\(946\) 0 0
\(947\) 45.6887 26.3784i 1.48468 0.857183i 0.484836 0.874605i \(-0.338879\pi\)
0.999848 + 0.0174217i \(0.00554579\pi\)
\(948\) 0 0
\(949\) −5.08406 + 8.80585i −0.165036 + 0.285850i
\(950\) 0 0
\(951\) −42.2604 12.3836i −1.37039 0.401565i
\(952\) 0 0
\(953\) 43.2040i 1.39952i 0.714380 + 0.699758i \(0.246709\pi\)
−0.714380 + 0.699758i \(0.753291\pi\)
\(954\) 0 0
\(955\) −13.9581 + 56.5615i −0.451672 + 1.83029i
\(956\) 0 0
\(957\) 13.6584 3.32142i 0.441512 0.107366i
\(958\) 0 0
\(959\) −17.5890 + 30.4651i −0.567979 + 0.983768i
\(960\) 0 0
\(961\) −2.28340 3.95497i −0.0736582 0.127580i
\(962\) 0 0
\(963\) 13.6080 + 0.633264i 0.438512 + 0.0204066i
\(964\) 0 0
\(965\) 7.89180 + 27.2744i 0.254046 + 0.877993i
\(966\) 0 0
\(967\) 21.2701 + 12.2803i 0.683999 + 0.394907i 0.801360 0.598182i \(-0.204110\pi\)
−0.117361 + 0.993089i \(0.537443\pi\)
\(968\) 0 0
\(969\) −5.18404 + 4.94841i −0.166535 + 0.158966i
\(970\) 0 0
\(971\) −28.9959 −0.930524 −0.465262 0.885173i \(-0.654040\pi\)
−0.465262 + 0.885173i \(0.654040\pi\)
\(972\) 0 0
\(973\) 15.9857i 0.512478i
\(974\) 0 0
\(975\) 50.0112 14.2869i 1.60164 0.457546i
\(976\) 0 0
\(977\) −33.8566 19.5471i −1.08317 0.625368i −0.151420 0.988469i \(-0.548385\pi\)
−0.931750 + 0.363101i \(0.881718\pi\)
\(978\) 0 0
\(979\) 2.25573 + 3.90705i 0.0720936 + 0.124870i
\(980\) 0 0
\(981\) −2.16614 + 46.5475i −0.0691596 + 1.48615i
\(982\) 0 0
\(983\) −3.60163 + 2.07940i −0.114874 + 0.0663227i −0.556336 0.830957i \(-0.687793\pi\)
0.441462 + 0.897280i \(0.354460\pi\)
\(984\) 0 0
\(985\) 10.5360 10.9629i 0.335704 0.349307i
\(986\) 0 0
\(987\) 48.0673 11.6889i 1.53000 0.372063i
\(988\) 0 0
\(989\) 10.0831 0.320624
\(990\) 0 0
\(991\) −9.79485 −0.311144 −0.155572 0.987825i \(-0.549722\pi\)
−0.155572 + 0.987825i \(0.549722\pi\)
\(992\) 0 0
\(993\) 7.31028 24.9471i 0.231985 0.791674i
\(994\) 0 0
\(995\) 6.66165 + 6.40221i 0.211188 + 0.202964i
\(996\) 0 0
\(997\) 4.98551 2.87839i 0.157893 0.0911594i −0.418972 0.907999i \(-0.637609\pi\)
0.576865 + 0.816840i \(0.304276\pi\)
\(998\) 0 0
\(999\) 29.6153 + 34.0644i 0.936986 + 1.07775i
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 360.2.bi.b.49.12 yes 32
3.2 odd 2 1080.2.bi.b.1009.2 32
4.3 odd 2 720.2.by.f.49.5 32
5.4 even 2 inner 360.2.bi.b.49.5 32
9.2 odd 6 1080.2.bi.b.289.14 32
9.4 even 3 3240.2.f.k.649.7 16
9.5 odd 6 3240.2.f.i.649.10 16
9.7 even 3 inner 360.2.bi.b.169.5 yes 32
12.11 even 2 2160.2.by.f.1009.2 32
15.14 odd 2 1080.2.bi.b.1009.14 32
20.19 odd 2 720.2.by.f.49.12 32
36.7 odd 6 720.2.by.f.529.12 32
36.11 even 6 2160.2.by.f.289.14 32
45.4 even 6 3240.2.f.k.649.8 16
45.14 odd 6 3240.2.f.i.649.9 16
45.29 odd 6 1080.2.bi.b.289.2 32
45.34 even 6 inner 360.2.bi.b.169.12 yes 32
60.59 even 2 2160.2.by.f.1009.14 32
180.79 odd 6 720.2.by.f.529.5 32
180.119 even 6 2160.2.by.f.289.2 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.bi.b.49.5 32 5.4 even 2 inner
360.2.bi.b.49.12 yes 32 1.1 even 1 trivial
360.2.bi.b.169.5 yes 32 9.7 even 3 inner
360.2.bi.b.169.12 yes 32 45.34 even 6 inner
720.2.by.f.49.5 32 4.3 odd 2
720.2.by.f.49.12 32 20.19 odd 2
720.2.by.f.529.5 32 180.79 odd 6
720.2.by.f.529.12 32 36.7 odd 6
1080.2.bi.b.289.2 32 45.29 odd 6
1080.2.bi.b.289.14 32 9.2 odd 6
1080.2.bi.b.1009.2 32 3.2 odd 2
1080.2.bi.b.1009.14 32 15.14 odd 2
2160.2.by.f.289.2 32 180.119 even 6
2160.2.by.f.289.14 32 36.11 even 6
2160.2.by.f.1009.2 32 12.11 even 2
2160.2.by.f.1009.14 32 60.59 even 2
3240.2.f.i.649.9 16 45.14 odd 6
3240.2.f.i.649.10 16 9.5 odd 6
3240.2.f.k.649.7 16 9.4 even 3
3240.2.f.k.649.8 16 45.4 even 6