Properties

Label 1520.2.q.n.961.1
Level $1520$
Weight $2$
Character 1520.961
Analytic conductor $12.137$
Analytic rank $0$
Dimension $8$
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] = [1520,2,Mod(881,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, 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("1520.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.q (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: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1500534351369.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 13x^{6} - 18x^{5} + 147x^{4} - 156x^{3} + 369x^{2} + 180x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.1
Root \(-1.72425 + 2.98649i\) of defining polynomial
Character \(\chi\) \(=\) 1520.961
Dual form 1520.2.q.n.881.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.72425 + 2.98649i) q^{3} +(-0.500000 + 0.866025i) q^{5} +4.44850 q^{7} +(-4.44607 - 7.70083i) q^{9} +O(q^{10})\) \(q+(-1.72425 + 2.98649i) q^{3} +(-0.500000 + 0.866025i) q^{5} +4.44850 q^{7} +(-4.44607 - 7.70083i) q^{9} +0.918574 q^{11} +(1.76496 + 3.05701i) q^{13} +(-1.72425 - 2.98649i) q^{15} +(1.94607 - 3.37070i) q^{17} +(4.22425 + 1.07504i) q^{19} +(-7.67032 + 13.2854i) q^{21} +(3.44607 + 5.96878i) q^{23} +(-0.500000 - 0.866025i) q^{25} +20.3191 q^{27} +(0.0407130 + 0.0705170i) q^{29} +3.00485 q^{31} +(-1.58385 + 2.74331i) q^{33} +(-2.22425 + 3.85251i) q^{35} +6.44850 q^{37} -12.1729 q^{39} +(0.224250 - 0.388412i) q^{41} +(-1.90536 + 3.30018i) q^{43} +8.89215 q^{45} +(-2.40779 - 4.17041i) q^{47} +12.7891 q^{49} +(6.71104 + 11.6239i) q^{51} +(2.76739 + 4.79326i) q^{53} +(-0.459287 + 0.795508i) q^{55} +(-10.4943 + 10.7620i) q^{57} +(-1.04071 + 1.80257i) q^{59} +(-5.20861 - 9.02158i) q^{61} +(-19.7784 - 34.2571i) q^{63} -3.52993 q^{65} +(-4.44365 - 7.69663i) q^{67} -23.7676 q^{69} +(-1.46171 + 2.53176i) q^{71} +(8.07326 - 13.9833i) q^{73} +3.44850 q^{75} +4.08628 q^{77} +(-6.44850 + 11.1691i) q^{79} +(-21.6969 + 37.5802i) q^{81} -5.16770 q^{83} +(1.94607 + 3.37070i) q^{85} -0.280798 q^{87} +(1.26011 + 2.18258i) q^{89} +(7.85144 + 13.5991i) q^{91} +(-5.18111 + 8.97395i) q^{93} +(-3.04314 + 3.12079i) q^{95} +(-3.27575 + 5.67377i) q^{97} +(-4.08405 - 7.07378i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - 4 q^{5} + 6 q^{7} - 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} - 4 q^{5} + 6 q^{7} - 13 q^{9} + 8 q^{11} - q^{13} + q^{15} - 7 q^{17} + 19 q^{19} - 24 q^{21} + 5 q^{23} - 4 q^{25} + 28 q^{27} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 22 q^{37} - 38 q^{39} - 13 q^{41} + 7 q^{43} + 26 q^{45} + 10 q^{47} - 2 q^{49} + 16 q^{51} - 4 q^{55} - 25 q^{57} - 8 q^{59} - 11 q^{61} - 24 q^{63} + 2 q^{65} - 20 q^{67} - 26 q^{69} - 5 q^{71} + 12 q^{73} - 2 q^{75} + 18 q^{77} - 22 q^{79} - 40 q^{81} - 26 q^{83} - 7 q^{85} + 12 q^{87} + 9 q^{89} + 18 q^{91} - 34 q^{93} - 17 q^{95} - 41 q^{97} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) −1.72425 + 2.98649i −0.995496 + 1.72425i −0.415646 + 0.909526i \(0.636445\pi\)
−0.579850 + 0.814723i \(0.696889\pi\)
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) 4.44850 1.68137 0.840687 0.541521i \(-0.182151\pi\)
0.840687 + 0.541521i \(0.182151\pi\)
\(8\) 0 0
\(9\) −4.44607 7.70083i −1.48202 2.56694i
\(10\) 0 0
\(11\) 0.918574 0.276960 0.138480 0.990365i \(-0.455778\pi\)
0.138480 + 0.990365i \(0.455778\pi\)
\(12\) 0 0
\(13\) 1.76496 + 3.05701i 0.489513 + 0.847861i 0.999927 0.0120678i \(-0.00384138\pi\)
−0.510415 + 0.859928i \(0.670508\pi\)
\(14\) 0 0
\(15\) −1.72425 2.98649i −0.445199 0.771108i
\(16\) 0 0
\(17\) 1.94607 3.37070i 0.471992 0.817515i −0.527494 0.849559i \(-0.676868\pi\)
0.999486 + 0.0320439i \(0.0102016\pi\)
\(18\) 0 0
\(19\) 4.22425 + 1.07504i 0.969109 + 0.246631i
\(20\) 0 0
\(21\) −7.67032 + 13.2854i −1.67380 + 2.89911i
\(22\) 0 0
\(23\) 3.44607 + 5.96878i 0.718556 + 1.24458i 0.961572 + 0.274554i \(0.0885300\pi\)
−0.243016 + 0.970022i \(0.578137\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 20.3191 3.91041
\(28\) 0 0
\(29\) 0.0407130 + 0.0705170i 0.00756022 + 0.0130947i 0.869781 0.493438i \(-0.164260\pi\)
−0.862221 + 0.506533i \(0.830927\pi\)
\(30\) 0 0
\(31\) 3.00485 0.539687 0.269843 0.962904i \(-0.413028\pi\)
0.269843 + 0.962904i \(0.413028\pi\)
\(32\) 0 0
\(33\) −1.58385 + 2.74331i −0.275713 + 0.477549i
\(34\) 0 0
\(35\) −2.22425 + 3.85251i −0.375967 + 0.651194i
\(36\) 0 0
\(37\) 6.44850 1.06013 0.530063 0.847958i \(-0.322168\pi\)
0.530063 + 0.847958i \(0.322168\pi\)
\(38\) 0 0
\(39\) −12.1729 −1.94923
\(40\) 0 0
\(41\) 0.224250 0.388412i 0.0350219 0.0606598i −0.847983 0.530023i \(-0.822183\pi\)
0.883005 + 0.469363i \(0.155517\pi\)
\(42\) 0 0
\(43\) −1.90536 + 3.30018i −0.290565 + 0.503273i −0.973943 0.226791i \(-0.927177\pi\)
0.683379 + 0.730064i \(0.260510\pi\)
\(44\) 0 0
\(45\) 8.89215 1.32556
\(46\) 0 0
\(47\) −2.40779 4.17041i −0.351212 0.608317i 0.635250 0.772306i \(-0.280897\pi\)
−0.986462 + 0.163990i \(0.947564\pi\)
\(48\) 0 0
\(49\) 12.7891 1.82702
\(50\) 0 0
\(51\) 6.71104 + 11.6239i 0.939733 + 1.62767i
\(52\) 0 0
\(53\) 2.76739 + 4.79326i 0.380130 + 0.658404i 0.991081 0.133264i \(-0.0425459\pi\)
−0.610951 + 0.791669i \(0.709213\pi\)
\(54\) 0 0
\(55\) −0.459287 + 0.795508i −0.0619302 + 0.107266i
\(56\) 0 0
\(57\) −10.4943 + 10.7620i −1.39000 + 1.42547i
\(58\) 0 0
\(59\) −1.04071 + 1.80257i −0.135489 + 0.234674i −0.925784 0.378052i \(-0.876594\pi\)
0.790295 + 0.612727i \(0.209927\pi\)
\(60\) 0 0
\(61\) −5.20861 9.02158i −0.666894 1.15510i −0.978768 0.204971i \(-0.934290\pi\)
0.311873 0.950124i \(-0.399043\pi\)
\(62\) 0 0
\(63\) −19.7784 34.2571i −2.49184 4.31599i
\(64\) 0 0
\(65\) −3.52993 −0.437833
\(66\) 0 0
\(67\) −4.44365 7.69663i −0.542878 0.940293i −0.998737 0.0502406i \(-0.984001\pi\)
0.455859 0.890052i \(-0.349332\pi\)
\(68\) 0 0
\(69\) −23.7676 −2.86128
\(70\) 0 0
\(71\) −1.46171 + 2.53176i −0.173473 + 0.300465i −0.939632 0.342187i \(-0.888832\pi\)
0.766159 + 0.642652i \(0.222166\pi\)
\(72\) 0 0
\(73\) 8.07326 13.9833i 0.944904 1.63662i 0.188960 0.981985i \(-0.439488\pi\)
0.755944 0.654636i \(-0.227178\pi\)
\(74\) 0 0
\(75\) 3.44850 0.398198
\(76\) 0 0
\(77\) 4.08628 0.465674
\(78\) 0 0
\(79\) −6.44850 + 11.1691i −0.725513 + 1.25663i 0.233250 + 0.972417i \(0.425064\pi\)
−0.958763 + 0.284208i \(0.908269\pi\)
\(80\) 0 0
\(81\) −21.6969 + 37.5802i −2.41077 + 4.17558i
\(82\) 0 0
\(83\) −5.16770 −0.567229 −0.283614 0.958938i \(-0.591534\pi\)
−0.283614 + 0.958938i \(0.591534\pi\)
\(84\) 0 0
\(85\) 1.94607 + 3.37070i 0.211081 + 0.365604i
\(86\) 0 0
\(87\) −0.280798 −0.0301047
\(88\) 0 0
\(89\) 1.26011 + 2.18258i 0.133572 + 0.231353i 0.925051 0.379843i \(-0.124022\pi\)
−0.791479 + 0.611196i \(0.790689\pi\)
\(90\) 0 0
\(91\) 7.85144 + 13.5991i 0.823054 + 1.42557i
\(92\) 0 0
\(93\) −5.18111 + 8.97395i −0.537256 + 0.930555i
\(94\) 0 0
\(95\) −3.04314 + 3.12079i −0.312219 + 0.320186i
\(96\) 0 0
\(97\) −3.27575 + 5.67377i −0.332602 + 0.576084i −0.983021 0.183492i \(-0.941260\pi\)
0.650419 + 0.759575i \(0.274593\pi\)
\(98\) 0 0
\(99\) −4.08405 7.07378i −0.410462 0.710942i
\(100\) 0 0
\(101\) −4.47600 7.75266i −0.445379 0.771418i 0.552700 0.833380i \(-0.313598\pi\)
−0.998079 + 0.0619619i \(0.980264\pi\)
\(102\) 0 0
\(103\) 4.89215 0.482038 0.241019 0.970520i \(-0.422518\pi\)
0.241019 + 0.970520i \(0.422518\pi\)
\(104\) 0 0
\(105\) −7.67032 13.2854i −0.748547 1.29652i
\(106\) 0 0
\(107\) −9.58493 −0.926610 −0.463305 0.886199i \(-0.653337\pi\)
−0.463305 + 0.886199i \(0.653337\pi\)
\(108\) 0 0
\(109\) 4.54314 7.86895i 0.435154 0.753708i −0.562154 0.827032i \(-0.690027\pi\)
0.997308 + 0.0733239i \(0.0233607\pi\)
\(110\) 0 0
\(111\) −11.1188 + 19.2584i −1.05535 + 1.82792i
\(112\) 0 0
\(113\) −8.33580 −0.784166 −0.392083 0.919930i \(-0.628245\pi\)
−0.392083 + 0.919930i \(0.628245\pi\)
\(114\) 0 0
\(115\) −6.89215 −0.642696
\(116\) 0 0
\(117\) 15.6943 27.1833i 1.45094 2.51310i
\(118\) 0 0
\(119\) 8.65711 14.9946i 0.793596 1.37455i
\(120\) 0 0
\(121\) −10.1562 −0.923293
\(122\) 0 0
\(123\) 0.773325 + 1.33944i 0.0697284 + 0.120773i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 2.26496 + 3.92303i 0.200983 + 0.348113i 0.948845 0.315741i \(-0.102253\pi\)
−0.747862 + 0.663854i \(0.768920\pi\)
\(128\) 0 0
\(129\) −6.57064 11.3807i −0.578512 1.00201i
\(130\) 0 0
\(131\) −6.67275 + 11.5575i −0.583001 + 1.00979i 0.412121 + 0.911129i \(0.364788\pi\)
−0.995121 + 0.0986577i \(0.968545\pi\)
\(132\) 0 0
\(133\) 18.7916 + 4.78232i 1.62944 + 0.414680i
\(134\) 0 0
\(135\) −10.1595 + 17.5968i −0.874394 + 1.51449i
\(136\) 0 0
\(137\) −1.37544 2.38233i −0.117511 0.203536i 0.801269 0.598304i \(-0.204158\pi\)
−0.918781 + 0.394768i \(0.870825\pi\)
\(138\) 0 0
\(139\) 5.76011 + 9.97681i 0.488566 + 0.846222i 0.999914 0.0131524i \(-0.00418668\pi\)
−0.511347 + 0.859374i \(0.670853\pi\)
\(140\) 0 0
\(141\) 16.6065 1.39852
\(142\) 0 0
\(143\) 1.62125 + 2.80809i 0.135576 + 0.234824i
\(144\) 0 0
\(145\) −0.0814260 −0.00676206
\(146\) 0 0
\(147\) −22.0517 + 38.1946i −1.81879 + 3.15024i
\(148\) 0 0
\(149\) 0.900314 1.55939i 0.0737566 0.127750i −0.826788 0.562513i \(-0.809835\pi\)
0.900545 + 0.434763i \(0.143168\pi\)
\(150\) 0 0
\(151\) 21.3671 1.73883 0.869414 0.494084i \(-0.164497\pi\)
0.869414 + 0.494084i \(0.164497\pi\)
\(152\) 0 0
\(153\) −34.6096 −2.79802
\(154\) 0 0
\(155\) −1.50242 + 2.60228i −0.120678 + 0.209020i
\(156\) 0 0
\(157\) −6.38379 + 11.0570i −0.509482 + 0.882448i 0.490458 + 0.871465i \(0.336829\pi\)
−0.999940 + 0.0109833i \(0.996504\pi\)
\(158\) 0 0
\(159\) −19.0867 −1.51367
\(160\) 0 0
\(161\) 15.3299 + 26.5521i 1.20816 + 2.09260i
\(162\) 0 0
\(163\) 3.71960 0.291341 0.145671 0.989333i \(-0.453466\pi\)
0.145671 + 0.989333i \(0.453466\pi\)
\(164\) 0 0
\(165\) −1.58385 2.74331i −0.123303 0.213566i
\(166\) 0 0
\(167\) −12.0398 20.8536i −0.931669 1.61370i −0.780468 0.625196i \(-0.785019\pi\)
−0.151202 0.988503i \(-0.548314\pi\)
\(168\) 0 0
\(169\) 0.269813 0.467329i 0.0207548 0.0359484i
\(170\) 0 0
\(171\) −10.5026 37.3099i −0.803156 2.85316i
\(172\) 0 0
\(173\) −6.21940 + 10.7723i −0.472852 + 0.819004i −0.999517 0.0310688i \(-0.990109\pi\)
0.526665 + 0.850073i \(0.323442\pi\)
\(174\) 0 0
\(175\) −2.22425 3.85251i −0.168137 0.291223i
\(176\) 0 0
\(177\) −3.58890 6.21615i −0.269758 0.467235i
\(178\) 0 0
\(179\) 4.72930 0.353484 0.176742 0.984257i \(-0.443444\pi\)
0.176742 + 0.984257i \(0.443444\pi\)
\(180\) 0 0
\(181\) 2.20861 + 3.82543i 0.164165 + 0.284342i 0.936358 0.351046i \(-0.114174\pi\)
−0.772194 + 0.635387i \(0.780840\pi\)
\(182\) 0 0
\(183\) 35.9238 2.65556
\(184\) 0 0
\(185\) −3.22425 + 5.58456i −0.237052 + 0.410585i
\(186\) 0 0
\(187\) 1.78761 3.09624i 0.130723 0.226419i
\(188\) 0 0
\(189\) 90.3894 6.57486
\(190\) 0 0
\(191\) −19.0696 −1.37982 −0.689912 0.723893i \(-0.742351\pi\)
−0.689912 + 0.723893i \(0.742351\pi\)
\(192\) 0 0
\(193\) 3.42100 5.92534i 0.246249 0.426516i −0.716233 0.697861i \(-0.754135\pi\)
0.962482 + 0.271346i \(0.0874686\pi\)
\(194\) 0 0
\(195\) 6.08647 10.5421i 0.435861 0.754934i
\(196\) 0 0
\(197\) −12.4053 −0.883845 −0.441922 0.897053i \(-0.645703\pi\)
−0.441922 + 0.897053i \(0.645703\pi\)
\(198\) 0 0
\(199\) 5.01321 + 8.68314i 0.355377 + 0.615531i 0.987182 0.159596i \(-0.0510191\pi\)
−0.631805 + 0.775127i \(0.717686\pi\)
\(200\) 0 0
\(201\) 30.6478 2.16173
\(202\) 0 0
\(203\) 0.181112 + 0.313695i 0.0127116 + 0.0220171i
\(204\) 0 0
\(205\) 0.224250 + 0.388412i 0.0156623 + 0.0271279i
\(206\) 0 0
\(207\) 30.6430 53.0752i 2.12984 3.68898i
\(208\) 0 0
\(209\) 3.88029 + 0.987505i 0.268405 + 0.0683071i
\(210\) 0 0
\(211\) 3.37281 5.84188i 0.232194 0.402172i −0.726259 0.687421i \(-0.758743\pi\)
0.958454 + 0.285249i \(0.0920762\pi\)
\(212\) 0 0
\(213\) −5.04071 8.73077i −0.345384 0.598223i
\(214\) 0 0
\(215\) −1.90536 3.30018i −0.129945 0.225071i
\(216\) 0 0
\(217\) 13.3671 0.907416
\(218\) 0 0
\(219\) 27.8406 + 48.2214i 1.88130 + 3.25850i
\(220\) 0 0
\(221\) 13.7390 0.924185
\(222\) 0 0
\(223\) −1.07064 + 1.85440i −0.0716953 + 0.124180i −0.899644 0.436623i \(-0.856174\pi\)
0.827949 + 0.560803i \(0.189508\pi\)
\(224\) 0 0
\(225\) −4.44607 + 7.70083i −0.296405 + 0.513388i
\(226\) 0 0
\(227\) −2.94015 −0.195145 −0.0975723 0.995228i \(-0.531108\pi\)
−0.0975723 + 0.995228i \(0.531108\pi\)
\(228\) 0 0
\(229\) −8.50835 −0.562248 −0.281124 0.959672i \(-0.590707\pi\)
−0.281124 + 0.959672i \(0.590707\pi\)
\(230\) 0 0
\(231\) −7.04576 + 12.2036i −0.463577 + 0.802939i
\(232\) 0 0
\(233\) 13.6390 23.6235i 0.893524 1.54763i 0.0579029 0.998322i \(-0.481559\pi\)
0.835621 0.549306i \(-0.185108\pi\)
\(234\) 0 0
\(235\) 4.81557 0.314133
\(236\) 0 0
\(237\) −22.2376 38.5167i −1.44449 2.50193i
\(238\) 0 0
\(239\) 4.99515 0.323109 0.161555 0.986864i \(-0.448349\pi\)
0.161555 + 0.986864i \(0.448349\pi\)
\(240\) 0 0
\(241\) −4.97358 8.61448i −0.320376 0.554908i 0.660189 0.751099i \(-0.270476\pi\)
−0.980566 + 0.196191i \(0.937143\pi\)
\(242\) 0 0
\(243\) −44.3433 76.8048i −2.84462 4.92703i
\(244\) 0 0
\(245\) −6.39457 + 11.0757i −0.408534 + 0.707602i
\(246\) 0 0
\(247\) 4.16924 + 14.8110i 0.265282 + 0.942399i
\(248\) 0 0
\(249\) 8.91041 15.4333i 0.564674 0.978044i
\(250\) 0 0
\(251\) −11.6978 20.2612i −0.738360 1.27888i −0.953233 0.302235i \(-0.902267\pi\)
0.214873 0.976642i \(-0.431066\pi\)
\(252\) 0 0
\(253\) 3.16547 + 5.48276i 0.199012 + 0.344698i
\(254\) 0 0
\(255\) −13.4221 −0.840523
\(256\) 0 0
\(257\) 12.6261 + 21.8690i 0.787594 + 1.36415i 0.927437 + 0.373980i \(0.122007\pi\)
−0.139842 + 0.990174i \(0.544660\pi\)
\(258\) 0 0
\(259\) 28.6861 1.78247
\(260\) 0 0
\(261\) 0.362026 0.627048i 0.0224089 0.0388133i
\(262\) 0 0
\(263\) 10.9128 18.9016i 0.672914 1.16552i −0.304160 0.952621i \(-0.598376\pi\)
0.977074 0.212900i \(-0.0682910\pi\)
\(264\) 0 0
\(265\) −5.53478 −0.339999
\(266\) 0 0
\(267\) −8.69100 −0.531880
\(268\) 0 0
\(269\) 4.68354 8.11212i 0.285560 0.494605i −0.687185 0.726483i \(-0.741154\pi\)
0.972745 + 0.231878i \(0.0744870\pi\)
\(270\) 0 0
\(271\) −2.70619 + 4.68725i −0.164389 + 0.284730i −0.936438 0.350833i \(-0.885899\pi\)
0.772049 + 0.635563i \(0.219232\pi\)
\(272\) 0 0
\(273\) −54.1513 −3.27739
\(274\) 0 0
\(275\) −0.459287 0.795508i −0.0276960 0.0479710i
\(276\) 0 0
\(277\) 9.45859 0.568312 0.284156 0.958778i \(-0.408287\pi\)
0.284156 + 0.958778i \(0.408287\pi\)
\(278\) 0 0
\(279\) −13.3598 23.1398i −0.799830 1.38535i
\(280\) 0 0
\(281\) 8.56821 + 14.8406i 0.511137 + 0.885315i 0.999917 + 0.0129077i \(0.00410877\pi\)
−0.488780 + 0.872407i \(0.662558\pi\)
\(282\) 0 0
\(283\) −11.5576 + 20.0184i −0.687029 + 1.18997i 0.285765 + 0.958300i \(0.407752\pi\)
−0.972794 + 0.231670i \(0.925581\pi\)
\(284\) 0 0
\(285\) −4.07306 14.4693i −0.241267 0.857088i
\(286\) 0 0
\(287\) 0.997575 1.72785i 0.0588850 0.101992i
\(288\) 0 0
\(289\) 0.925587 + 1.60316i 0.0544463 + 0.0943038i
\(290\) 0 0
\(291\) −11.2964 19.5660i −0.662208 1.14698i
\(292\) 0 0
\(293\) 9.05500 0.528999 0.264499 0.964386i \(-0.414793\pi\)
0.264499 + 0.964386i \(0.414793\pi\)
\(294\) 0 0
\(295\) −1.04071 1.80257i −0.0605927 0.104950i
\(296\) 0 0
\(297\) 18.6646 1.08303
\(298\) 0 0
\(299\) −12.1644 + 21.0693i −0.703485 + 1.21847i
\(300\) 0 0
\(301\) −8.47600 + 14.6809i −0.488549 + 0.846191i
\(302\) 0 0
\(303\) 30.8710 1.77349
\(304\) 0 0
\(305\) 10.4172 0.596489
\(306\) 0 0
\(307\) −10.4271 + 18.0603i −0.595107 + 1.03076i 0.398424 + 0.917201i \(0.369557\pi\)
−0.993532 + 0.113555i \(0.963776\pi\)
\(308\) 0 0
\(309\) −8.43529 + 14.6103i −0.479867 + 0.831154i
\(310\) 0 0
\(311\) −3.02157 −0.171338 −0.0856689 0.996324i \(-0.527303\pi\)
−0.0856689 + 0.996324i \(0.527303\pi\)
\(312\) 0 0
\(313\) 2.71104 + 4.69565i 0.153237 + 0.265414i 0.932416 0.361388i \(-0.117697\pi\)
−0.779179 + 0.626802i \(0.784364\pi\)
\(314\) 0 0
\(315\) 39.5567 2.22877
\(316\) 0 0
\(317\) 7.70025 + 13.3372i 0.432489 + 0.749093i 0.997087 0.0762730i \(-0.0243021\pi\)
−0.564598 + 0.825366i \(0.690969\pi\)
\(318\) 0 0
\(319\) 0.0373979 + 0.0647751i 0.00209388 + 0.00362671i
\(320\) 0 0
\(321\) 16.5268 28.6253i 0.922436 1.59771i
\(322\) 0 0
\(323\) 11.8443 12.1466i 0.659037 0.675853i
\(324\) 0 0
\(325\) 1.76496 3.05701i 0.0979025 0.169572i
\(326\) 0 0
\(327\) 15.6670 + 27.1361i 0.866388 + 1.50063i
\(328\) 0 0
\(329\) −10.7110 18.5521i −0.590519 1.02281i
\(330\) 0 0
\(331\) 9.09113 0.499693 0.249847 0.968285i \(-0.419620\pi\)
0.249847 + 0.968285i \(0.419620\pi\)
\(332\) 0 0
\(333\) −28.6705 49.6588i −1.57113 2.72128i
\(334\) 0 0
\(335\) 8.88730 0.485565
\(336\) 0 0
\(337\) −12.1778 + 21.0926i −0.663367 + 1.14899i 0.316358 + 0.948640i \(0.397540\pi\)
−0.979725 + 0.200345i \(0.935794\pi\)
\(338\) 0 0
\(339\) 14.3730 24.8948i 0.780634 1.35210i
\(340\) 0 0
\(341\) 2.76018 0.149472
\(342\) 0 0
\(343\) 25.7530 1.39053
\(344\) 0 0
\(345\) 11.8838 20.5833i 0.639802 1.10817i
\(346\) 0 0
\(347\) 1.09949 1.90437i 0.0590236 0.102232i −0.835004 0.550244i \(-0.814535\pi\)
0.894027 + 0.448012i \(0.147868\pi\)
\(348\) 0 0
\(349\) −29.6962 −1.58960 −0.794802 0.606869i \(-0.792425\pi\)
−0.794802 + 0.606869i \(0.792425\pi\)
\(350\) 0 0
\(351\) 35.8624 + 62.1155i 1.91419 + 3.31548i
\(352\) 0 0
\(353\) −7.38340 −0.392979 −0.196489 0.980506i \(-0.562954\pi\)
−0.196489 + 0.980506i \(0.562954\pi\)
\(354\) 0 0
\(355\) −1.46171 2.53176i −0.0775796 0.134372i
\(356\) 0 0
\(357\) 29.8540 + 51.7087i 1.58004 + 2.73672i
\(358\) 0 0
\(359\) −7.29157 + 12.6294i −0.384835 + 0.666553i −0.991746 0.128216i \(-0.959075\pi\)
0.606912 + 0.794769i \(0.292408\pi\)
\(360\) 0 0
\(361\) 16.6886 + 9.08249i 0.878346 + 0.478026i
\(362\) 0 0
\(363\) 17.5119 30.3314i 0.919134 1.59199i
\(364\) 0 0
\(365\) 8.07326 + 13.9833i 0.422574 + 0.731919i
\(366\) 0 0
\(367\) 2.60211 + 4.50699i 0.135829 + 0.235263i 0.925914 0.377735i \(-0.123297\pi\)
−0.790085 + 0.612998i \(0.789964\pi\)
\(368\) 0 0
\(369\) −3.98812 −0.207613
\(370\) 0 0
\(371\) 12.3107 + 21.3228i 0.639141 + 1.10702i
\(372\) 0 0
\(373\) 22.2781 1.15352 0.576758 0.816915i \(-0.304318\pi\)
0.576758 + 0.816915i \(0.304318\pi\)
\(374\) 0 0
\(375\) −1.72425 + 2.98649i −0.0890399 + 0.154222i
\(376\) 0 0
\(377\) −0.143714 + 0.248920i −0.00740164 + 0.0128200i
\(378\) 0 0
\(379\) −29.5950 −1.52019 −0.760097 0.649810i \(-0.774849\pi\)
−0.760097 + 0.649810i \(0.774849\pi\)
\(380\) 0 0
\(381\) −15.6214 −0.800311
\(382\) 0 0
\(383\) 8.19101 14.1872i 0.418541 0.724934i −0.577252 0.816566i \(-0.695875\pi\)
0.995793 + 0.0916318i \(0.0292083\pi\)
\(384\) 0 0
\(385\) −2.04314 + 3.53882i −0.104128 + 0.180355i
\(386\) 0 0
\(387\) 33.8855 1.72250
\(388\) 0 0
\(389\) 12.5350 + 21.7112i 0.635548 + 1.10080i 0.986399 + 0.164370i \(0.0525593\pi\)
−0.350850 + 0.936432i \(0.614107\pi\)
\(390\) 0 0
\(391\) 26.8253 1.35661
\(392\) 0 0
\(393\) −23.0110 39.8562i −1.16075 2.01048i
\(394\) 0 0
\(395\) −6.44850 11.1691i −0.324459 0.561980i
\(396\) 0 0
\(397\) 12.2793 21.2683i 0.616278 1.06742i −0.373881 0.927477i \(-0.621973\pi\)
0.990159 0.139948i \(-0.0446935\pi\)
\(398\) 0 0
\(399\) −46.6837 + 47.8749i −2.33711 + 2.39674i
\(400\) 0 0
\(401\) −9.27244 + 16.0603i −0.463043 + 0.802015i −0.999111 0.0421606i \(-0.986576\pi\)
0.536068 + 0.844175i \(0.319909\pi\)
\(402\) 0 0
\(403\) 5.30345 + 9.18584i 0.264184 + 0.457579i
\(404\) 0 0
\(405\) −21.6969 37.5802i −1.07813 1.86737i
\(406\) 0 0
\(407\) 5.92342 0.293613
\(408\) 0 0
\(409\) 3.28322 + 5.68671i 0.162345 + 0.281190i 0.935709 0.352772i \(-0.114761\pi\)
−0.773364 + 0.633962i \(0.781428\pi\)
\(410\) 0 0
\(411\) 9.48638 0.467929
\(412\) 0 0
\(413\) −4.62961 + 8.01872i −0.227808 + 0.394576i
\(414\) 0 0
\(415\) 2.58385 4.47536i 0.126836 0.219687i
\(416\) 0 0
\(417\) −39.7275 −1.94546
\(418\) 0 0
\(419\) −21.6448 −1.05742 −0.528708 0.848804i \(-0.677323\pi\)
−0.528708 + 0.848804i \(0.677323\pi\)
\(420\) 0 0
\(421\) 0.194324 0.336579i 0.00947078 0.0164039i −0.861251 0.508180i \(-0.830319\pi\)
0.870722 + 0.491776i \(0.163652\pi\)
\(422\) 0 0
\(423\) −21.4104 + 37.0839i −1.04101 + 1.80308i
\(424\) 0 0
\(425\) −3.89215 −0.188797
\(426\) 0 0
\(427\) −23.1705 40.1325i −1.12130 1.94215i
\(428\) 0 0
\(429\) −11.1818 −0.539860
\(430\) 0 0
\(431\) −1.42936 2.47573i −0.0688499 0.119252i 0.829545 0.558439i \(-0.188600\pi\)
−0.898395 + 0.439188i \(0.855266\pi\)
\(432\) 0 0
\(433\) −15.2326 26.3836i −0.732032 1.26792i −0.956013 0.293323i \(-0.905239\pi\)
0.223981 0.974593i \(-0.428095\pi\)
\(434\) 0 0
\(435\) 0.140399 0.243178i 0.00673161 0.0116595i
\(436\) 0 0
\(437\) 8.14040 + 28.9183i 0.389408 + 1.38335i
\(438\) 0 0
\(439\) −5.52527 + 9.57005i −0.263707 + 0.456754i −0.967224 0.253925i \(-0.918279\pi\)
0.703517 + 0.710678i \(0.251612\pi\)
\(440\) 0 0
\(441\) −56.8615 98.4870i −2.70769 4.68986i
\(442\) 0 0
\(443\) 19.7484 + 34.2053i 0.938276 + 1.62514i 0.768684 + 0.639628i \(0.220912\pi\)
0.169592 + 0.985514i \(0.445755\pi\)
\(444\) 0 0
\(445\) −2.52023 −0.119470
\(446\) 0 0
\(447\) 3.10473 + 5.37756i 0.146849 + 0.254350i
\(448\) 0 0
\(449\) −36.3604 −1.71596 −0.857978 0.513687i \(-0.828279\pi\)
−0.857978 + 0.513687i \(0.828279\pi\)
\(450\) 0 0
\(451\) 0.205990 0.356785i 0.00969969 0.0168004i
\(452\) 0 0
\(453\) −36.8422 + 63.8125i −1.73100 + 2.99817i
\(454\) 0 0
\(455\) −15.7029 −0.736162
\(456\) 0 0
\(457\) 13.0145 0.608795 0.304397 0.952545i \(-0.401545\pi\)
0.304397 + 0.952545i \(0.401545\pi\)
\(458\) 0 0
\(459\) 39.5424 68.4895i 1.84568 3.19682i
\(460\) 0 0
\(461\) 2.56579 4.44408i 0.119501 0.206981i −0.800069 0.599908i \(-0.795204\pi\)
0.919570 + 0.392926i \(0.128537\pi\)
\(462\) 0 0
\(463\) 3.78915 0.176097 0.0880483 0.996116i \(-0.471937\pi\)
0.0880483 + 0.996116i \(0.471937\pi\)
\(464\) 0 0
\(465\) −5.18111 8.97395i −0.240268 0.416157i
\(466\) 0 0
\(467\) 29.4339 1.36204 0.681020 0.732265i \(-0.261537\pi\)
0.681020 + 0.732265i \(0.261537\pi\)
\(468\) 0 0
\(469\) −19.7676 34.2384i −0.912782 1.58098i
\(470\) 0 0
\(471\) −22.0145 38.1302i −1.01437 1.75695i
\(472\) 0 0
\(473\) −1.75022 + 3.03146i −0.0804750 + 0.139387i
\(474\) 0 0
\(475\) −1.18111 4.19583i −0.0541931 0.192518i
\(476\) 0 0
\(477\) 24.6080 42.6223i 1.12672 1.95154i
\(478\) 0 0
\(479\) −9.78303 16.9447i −0.446998 0.774223i 0.551191 0.834379i \(-0.314174\pi\)
−0.998189 + 0.0601560i \(0.980840\pi\)
\(480\) 0 0
\(481\) 11.3814 + 19.7131i 0.518945 + 0.898840i
\(482\) 0 0
\(483\) −105.730 −4.81088
\(484\) 0 0
\(485\) −3.27575 5.67377i −0.148744 0.257632i
\(486\) 0 0
\(487\) 4.62808 0.209718 0.104859 0.994487i \(-0.466561\pi\)
0.104859 + 0.994487i \(0.466561\pi\)
\(488\) 0 0
\(489\) −6.41351 + 11.1085i −0.290029 + 0.502345i
\(490\) 0 0
\(491\) −4.05743 + 7.02767i −0.183109 + 0.317154i −0.942938 0.332969i \(-0.891949\pi\)
0.759829 + 0.650123i \(0.225283\pi\)
\(492\) 0 0
\(493\) 0.316922 0.0142735
\(494\) 0 0
\(495\) 8.16810 0.367129
\(496\) 0 0
\(497\) −6.50242 + 11.2625i −0.291674 + 0.505194i
\(498\) 0 0
\(499\) 22.0625 38.2133i 0.987652 1.71066i 0.358151 0.933664i \(-0.383407\pi\)
0.629501 0.777000i \(-0.283259\pi\)
\(500\) 0 0
\(501\) 83.0386 3.70989
\(502\) 0 0
\(503\) −18.3083 31.7109i −0.816326 1.41392i −0.908372 0.418164i \(-0.862674\pi\)
0.0920456 0.995755i \(-0.470659\pi\)
\(504\) 0 0
\(505\) 8.95200 0.398359
\(506\) 0 0
\(507\) 0.930449 + 1.61158i 0.0413227 + 0.0715730i
\(508\) 0 0
\(509\) −4.31315 7.47059i −0.191177 0.331128i 0.754464 0.656342i \(-0.227897\pi\)
−0.945641 + 0.325214i \(0.894564\pi\)
\(510\) 0 0
\(511\) 35.9139 62.2047i 1.58874 2.75177i
\(512\) 0 0
\(513\) 85.8328 + 21.8438i 3.78961 + 0.964429i
\(514\) 0 0
\(515\) −2.44607 + 4.23673i −0.107787 + 0.186692i
\(516\) 0 0
\(517\) −2.21173 3.83083i −0.0972718 0.168480i
\(518\) 0 0
\(519\) −21.4476 37.1483i −0.941445 1.63063i
\(520\) 0 0
\(521\) −13.2878 −0.582149 −0.291075 0.956700i \(-0.594013\pi\)
−0.291075 + 0.956700i \(0.594013\pi\)
\(522\) 0 0
\(523\) −13.5037 23.3891i −0.590475 1.02273i −0.994168 0.107839i \(-0.965607\pi\)
0.403693 0.914895i \(-0.367726\pi\)
\(524\) 0 0
\(525\) 15.3406 0.669521
\(526\) 0 0
\(527\) 5.84766 10.1284i 0.254728 0.441202i
\(528\) 0 0
\(529\) −12.2509 + 21.2191i −0.532646 + 0.922570i
\(530\) 0 0
\(531\) 18.5084 0.803194
\(532\) 0 0
\(533\) 1.58317 0.0685747
\(534\) 0 0
\(535\) 4.79246 8.30079i 0.207196 0.358874i
\(536\) 0 0
\(537\) −8.15449 + 14.1240i −0.351892 + 0.609495i
\(538\) 0 0
\(539\) 11.7478 0.506013
\(540\) 0 0
\(541\) −19.3587 33.5303i −0.832296 1.44158i −0.896213 0.443624i \(-0.853693\pi\)
0.0639171 0.997955i \(-0.479641\pi\)
\(542\) 0 0
\(543\) −15.2328 −0.653702
\(544\) 0 0
\(545\) 4.54314 + 7.86895i 0.194607 + 0.337069i
\(546\) 0 0
\(547\) 0.574151 + 0.994459i 0.0245489 + 0.0425200i 0.878039 0.478589i \(-0.158852\pi\)
−0.853490 + 0.521109i \(0.825518\pi\)
\(548\) 0 0
\(549\) −46.3158 + 80.2212i −1.97671 + 3.42376i
\(550\) 0 0
\(551\) 0.0961732 + 0.341650i 0.00409712 + 0.0145548i
\(552\) 0 0
\(553\) −28.6861 + 49.6859i −1.21986 + 2.11286i
\(554\) 0 0
\(555\) −11.1188 19.2584i −0.471968 0.817472i
\(556\) 0 0
\(557\) −18.4441 31.9461i −0.781501 1.35360i −0.931067 0.364848i \(-0.881121\pi\)
0.149565 0.988752i \(-0.452213\pi\)
\(558\) 0 0
\(559\) −13.4516 −0.568941
\(560\) 0 0
\(561\) 6.16458 + 10.6774i 0.260269 + 0.450799i
\(562\) 0 0
\(563\) 9.52993 0.401638 0.200819 0.979628i \(-0.435640\pi\)
0.200819 + 0.979628i \(0.435640\pi\)
\(564\) 0 0
\(565\) 4.16790 7.21901i 0.175345 0.303706i
\(566\) 0 0
\(567\) −96.5188 + 167.175i −4.05341 + 7.02071i
\(568\) 0 0
\(569\) −13.9454 −0.584620 −0.292310 0.956324i \(-0.594424\pi\)
−0.292310 + 0.956324i \(0.594424\pi\)
\(570\) 0 0
\(571\) −30.6197 −1.28139 −0.640697 0.767794i \(-0.721354\pi\)
−0.640697 + 0.767794i \(0.721354\pi\)
\(572\) 0 0
\(573\) 32.8807 56.9510i 1.37361 2.37916i
\(574\) 0 0
\(575\) 3.44607 5.96878i 0.143711 0.248915i
\(576\) 0 0
\(577\) 42.4106 1.76558 0.882788 0.469771i \(-0.155664\pi\)
0.882788 + 0.469771i \(0.155664\pi\)
\(578\) 0 0
\(579\) 11.7973 + 20.4335i 0.490280 + 0.849189i
\(580\) 0 0
\(581\) −22.9885 −0.953725
\(582\) 0 0
\(583\) 2.54205 + 4.40296i 0.105281 + 0.182352i
\(584\) 0 0
\(585\) 15.6943 + 27.1833i 0.648880 + 1.12389i
\(586\) 0 0
\(587\) 1.25310 2.17043i 0.0517210 0.0895833i −0.839006 0.544122i \(-0.816863\pi\)
0.890727 + 0.454539i \(0.150196\pi\)
\(588\) 0 0
\(589\) 12.6932 + 3.23034i 0.523016 + 0.133104i
\(590\) 0 0
\(591\) 21.3899 37.0484i 0.879864 1.52397i
\(592\) 0 0
\(593\) −0.556350 0.963627i −0.0228466 0.0395714i 0.854376 0.519655i \(-0.173940\pi\)
−0.877223 + 0.480084i \(0.840606\pi\)
\(594\) 0 0
\(595\) 8.65711 + 14.9946i 0.354907 + 0.614717i
\(596\) 0 0
\(597\) −34.5761 −1.41511
\(598\) 0 0
\(599\) 15.4727 + 26.7995i 0.632197 + 1.09500i 0.987102 + 0.160095i \(0.0511799\pi\)
−0.354905 + 0.934902i \(0.615487\pi\)
\(600\) 0 0
\(601\) −6.75612 −0.275588 −0.137794 0.990461i \(-0.544001\pi\)
−0.137794 + 0.990461i \(0.544001\pi\)
\(602\) 0 0
\(603\) −39.5136 + 68.4396i −1.60912 + 2.78707i
\(604\) 0 0
\(605\) 5.07811 8.79555i 0.206455 0.357590i
\(606\) 0 0
\(607\) 24.6958 1.00237 0.501187 0.865339i \(-0.332897\pi\)
0.501187 + 0.865339i \(0.332897\pi\)
\(608\) 0 0
\(609\) −1.24913 −0.0506172
\(610\) 0 0
\(611\) 8.49931 14.7212i 0.343845 0.595557i
\(612\) 0 0
\(613\) 8.39107 14.5338i 0.338912 0.587013i −0.645316 0.763916i \(-0.723274\pi\)
0.984228 + 0.176902i \(0.0566077\pi\)
\(614\) 0 0
\(615\) −1.54665 −0.0623670
\(616\) 0 0
\(617\) 2.21239 + 3.83197i 0.0890673 + 0.154269i 0.907117 0.420878i \(-0.138278\pi\)
−0.818050 + 0.575147i \(0.804945\pi\)
\(618\) 0 0
\(619\) −30.0787 −1.20897 −0.604483 0.796618i \(-0.706620\pi\)
−0.604483 + 0.796618i \(0.706620\pi\)
\(620\) 0 0
\(621\) 70.0210 + 121.280i 2.80985 + 4.86680i
\(622\) 0 0
\(623\) 5.60561 + 9.70920i 0.224584 + 0.388991i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) −9.63975 + 9.88572i −0.384975 + 0.394798i
\(628\) 0 0
\(629\) 12.5493 21.7360i 0.500372 0.866669i
\(630\) 0 0
\(631\) 7.77905 + 13.4737i 0.309679 + 0.536380i 0.978292 0.207230i \(-0.0664450\pi\)
−0.668613 + 0.743611i \(0.733112\pi\)
\(632\) 0 0
\(633\) 11.6311 + 20.1457i 0.462297 + 0.800721i
\(634\) 0 0
\(635\) −4.52993 −0.179765
\(636\) 0 0
\(637\) 22.5724 + 39.0965i 0.894350 + 1.54906i
\(638\) 0 0
\(639\) 25.9955 1.02837
\(640\) 0 0
\(641\) 6.30345 10.9179i 0.248971 0.431231i −0.714269 0.699871i \(-0.753241\pi\)
0.963241 + 0.268640i \(0.0865742\pi\)
\(642\) 0 0
\(643\) −1.98086 + 3.43095i −0.0781176 + 0.135304i −0.902438 0.430820i \(-0.858224\pi\)
0.824320 + 0.566124i \(0.191558\pi\)
\(644\) 0 0
\(645\) 13.1413 0.517437
\(646\) 0 0
\(647\) 5.09776 0.200413 0.100207 0.994967i \(-0.468050\pi\)
0.100207 + 0.994967i \(0.468050\pi\)
\(648\) 0 0
\(649\) −0.955972 + 1.65579i −0.0375252 + 0.0649955i
\(650\) 0 0
\(651\) −23.0482 + 39.9206i −0.903329 + 1.56461i
\(652\) 0 0
\(653\) 18.1413 0.709923 0.354962 0.934881i \(-0.384494\pi\)
0.354962 + 0.934881i \(0.384494\pi\)
\(654\) 0 0
\(655\) −6.67275 11.5575i −0.260726 0.451590i
\(656\) 0 0
\(657\) −143.577 −5.60148
\(658\) 0 0
\(659\) −4.61378 7.99130i −0.179727 0.311297i 0.762060 0.647507i \(-0.224188\pi\)
−0.941787 + 0.336210i \(0.890855\pi\)
\(660\) 0 0
\(661\) −19.7317 34.1763i −0.767475 1.32930i −0.938928 0.344113i \(-0.888180\pi\)
0.171454 0.985192i \(-0.445154\pi\)
\(662\) 0 0
\(663\) −23.6895 + 41.0314i −0.920022 + 1.59353i
\(664\) 0 0
\(665\) −13.5374 + 13.8828i −0.524958 + 0.538353i
\(666\) 0 0
\(667\) −0.280600 + 0.486014i −0.0108649 + 0.0188185i
\(668\) 0 0
\(669\) −3.69210 6.39490i −0.142745 0.247241i
\(670\) 0 0
\(671\) −4.78450 8.28699i −0.184703 0.319916i
\(672\) 0 0
\(673\) 29.9863 1.15589 0.577944 0.816076i \(-0.303855\pi\)
0.577944 + 0.816076i \(0.303855\pi\)
\(674\) 0 0
\(675\) −10.1595 17.5968i −0.391041 0.677302i
\(676\) 0 0
\(677\) 16.0343 0.616250 0.308125 0.951346i \(-0.400299\pi\)
0.308125 + 0.951346i \(0.400299\pi\)
\(678\) 0 0
\(679\) −14.5722 + 25.2397i −0.559229 + 0.968613i
\(680\) 0 0
\(681\) 5.06955 8.78072i 0.194266 0.336478i
\(682\) 0 0
\(683\) −41.0268 −1.56985 −0.784923 0.619593i \(-0.787298\pi\)
−0.784923 + 0.619593i \(0.787298\pi\)
\(684\) 0 0
\(685\) 2.75087 0.105105
\(686\) 0 0
\(687\) 14.6705 25.4101i 0.559715 0.969455i
\(688\) 0 0
\(689\) −9.76867 + 16.9198i −0.372157 + 0.644594i
\(690\) 0 0
\(691\) 12.0602 0.458793 0.229397 0.973333i \(-0.426325\pi\)
0.229397 + 0.973333i \(0.426325\pi\)
\(692\) 0 0
\(693\) −18.1679 31.4677i −0.690141 1.19536i
\(694\) 0 0
\(695\) −11.5202 −0.436987
\(696\) 0 0
\(697\) −0.872814 1.51176i −0.0330602 0.0572619i
\(698\) 0 0
\(699\) 47.0343 + 81.4657i 1.77900 + 3.08132i
\(700\) 0 0
\(701\) 11.4219 19.7833i 0.431398 0.747204i −0.565596 0.824683i \(-0.691354\pi\)
0.996994 + 0.0774788i \(0.0246870\pi\)
\(702\) 0 0
\(703\) 27.2401 + 6.93240i 1.02738 + 0.261461i
\(704\) 0 0
\(705\) −8.30325 + 14.3817i −0.312719 + 0.541644i
\(706\) 0 0
\(707\) −19.9115 34.4877i −0.748848 1.29704i
\(708\) 0 0
\(709\) −0.770010 1.33370i −0.0289183 0.0500880i 0.851204 0.524835i \(-0.175873\pi\)
−0.880122 + 0.474747i \(0.842540\pi\)
\(710\) 0 0
\(711\) 114.682 4.30091
\(712\) 0 0
\(713\) 10.3549 + 17.9353i 0.387795 + 0.671681i
\(714\) 0 0
\(715\) −3.24250 −0.121263
\(716\) 0 0
\(717\) −8.61289 + 14.9180i −0.321654 + 0.557121i
\(718\) 0 0
\(719\) 14.8133 25.6574i 0.552444 0.956861i −0.445653 0.895206i \(-0.647029\pi\)
0.998097 0.0616557i \(-0.0196381\pi\)
\(720\) 0 0
\(721\) 21.7627 0.810486
\(722\) 0 0
\(723\) 34.3027 1.27573
\(724\) 0 0
\(725\) 0.0407130 0.0705170i 0.00151204 0.00261894i
\(726\) 0 0
\(727\) 8.74251 15.1425i 0.324242 0.561603i −0.657117 0.753789i \(-0.728224\pi\)
0.981359 + 0.192186i \(0.0615575\pi\)
\(728\) 0 0
\(729\) 175.654 6.50570
\(730\) 0 0
\(731\) 7.41595 + 12.8448i 0.274289 + 0.475082i
\(732\) 0 0
\(733\) −14.7557 −0.545015 −0.272508 0.962154i \(-0.587853\pi\)
−0.272508 + 0.962154i \(0.587853\pi\)
\(734\) 0 0
\(735\) −22.0517 38.1946i −0.813389 1.40883i
\(736\) 0 0
\(737\) −4.08182 7.06992i −0.150356 0.260424i
\(738\) 0 0
\(739\) −2.79866 + 4.84743i −0.102950 + 0.178315i −0.912899 0.408186i \(-0.866162\pi\)
0.809949 + 0.586501i \(0.199495\pi\)
\(740\) 0 0
\(741\) −51.4216 13.0864i −1.88902 0.480742i
\(742\) 0 0
\(743\) −11.7984 + 20.4354i −0.432841 + 0.749703i −0.997117 0.0758835i \(-0.975822\pi\)
0.564275 + 0.825587i \(0.309156\pi\)
\(744\) 0 0
\(745\) 0.900314 + 1.55939i 0.0329850 + 0.0571316i
\(746\) 0 0
\(747\) 22.9760 + 39.7956i 0.840647 + 1.45604i
\(748\) 0 0
\(749\) −42.6385 −1.55798
\(750\) 0 0
\(751\) 14.9064 + 25.8187i 0.543944 + 0.942138i 0.998673 + 0.0515081i \(0.0164028\pi\)
−0.454729 + 0.890630i \(0.650264\pi\)
\(752\) 0 0
\(753\) 80.6799 2.94014
\(754\) 0 0
\(755\) −10.6835 + 18.5044i −0.388814 + 0.673445i
\(756\) 0 0
\(757\) 13.5048 23.3911i 0.490842 0.850163i −0.509102 0.860706i \(-0.670023\pi\)
0.999944 + 0.0105425i \(0.00335585\pi\)
\(758\) 0 0
\(759\) −21.8323 −0.792461
\(760\) 0 0
\(761\) −33.9286 −1.22991 −0.614956 0.788561i \(-0.710826\pi\)
−0.614956 + 0.788561i \(0.710826\pi\)
\(762\) 0 0
\(763\) 20.2101 35.0050i 0.731656 1.26727i
\(764\) 0 0
\(765\) 17.3048 29.9728i 0.625656 1.08367i
\(766\) 0 0
\(767\) −7.34728 −0.265295
\(768\) 0 0
\(769\) −17.1023 29.6220i −0.616725 1.06820i −0.990079 0.140510i \(-0.955126\pi\)
0.373355 0.927689i \(-0.378207\pi\)
\(770\) 0 0
\(771\) −87.0822 −3.13619
\(772\) 0 0
\(773\) −2.94716 5.10463i −0.106002 0.183601i 0.808145 0.588983i \(-0.200472\pi\)
−0.914147 + 0.405382i \(0.867138\pi\)
\(774\) 0 0
\(775\) −1.50242 2.60228i −0.0539687 0.0934765i
\(776\) 0 0
\(777\) −49.4621 + 85.6708i −1.77444 + 3.07342i
\(778\) 0 0
\(779\) 1.36485 1.39967i 0.0489007 0.0501484i
\(780\) 0 0
\(781\) −1.34269 + 2.32561i −0.0480452 + 0.0832168i
\(782\) 0 0
\(783\) 0.827251 + 1.43284i 0.0295635 + 0.0512055i
\(784\) 0 0
\(785\) −6.38379 11.0570i −0.227847 0.394643i
\(786\) 0 0
\(787\) −31.7055 −1.13018 −0.565090 0.825029i \(-0.691159\pi\)
−0.565090 + 0.825029i \(0.691159\pi\)
\(788\) 0 0
\(789\) 37.6329 + 65.1821i 1.33977 + 2.32054i
\(790\) 0 0
\(791\) −37.0818 −1.31848
\(792\) 0 0
\(793\) 18.3860 31.8455i 0.652907 1.13087i
\(794\) 0 0
\(795\) 9.54334 16.5295i 0.338467 0.586242i
\(796\) 0 0
\(797\) 20.1941 0.715313 0.357656 0.933853i \(-0.383576\pi\)
0.357656 + 0.933853i \(0.383576\pi\)
\(798\) 0 0
\(799\) −18.7429 −0.663077
\(800\) 0 0
\(801\) 11.2051 19.4078i 0.395913 0.685742i
\(802\) 0 0
\(803\) 7.41589 12.8447i 0.261701 0.453279i
\(804\) 0 0
\(805\) −30.6597 −1.08061
\(806\) 0 0
\(807\) 16.1512 + 27.9747i 0.568548 + 0.984754i
\(808\) 0 0
\(809\) 8.24774 0.289975 0.144988 0.989433i \(-0.453686\pi\)
0.144988 + 0.989433i \(0.453686\pi\)
\(810\) 0 0
\(811\) 12.7689 + 22.1164i 0.448378 + 0.776613i 0.998281 0.0586158i \(-0.0186687\pi\)
−0.549903 + 0.835228i \(0.685335\pi\)
\(812\) 0 0
\(813\) −9.33229 16.1640i −0.327298 0.566896i
\(814\) 0 0
\(815\) −1.85980 + 3.22127i −0.0651459 + 0.112836i
\(816\) 0 0
\(817\) −11.5966 + 11.8925i −0.405712 + 0.416064i
\(818\) 0 0
\(819\) 69.8161 120.925i 2.43957 4.22547i
\(820\) 0 0
\(821\) 13.9128 + 24.0977i 0.485561 + 0.841016i 0.999862 0.0165934i \(-0.00528209\pi\)
−0.514301 + 0.857609i \(0.671949\pi\)
\(822\) 0 0
\(823\) 11.3297 + 19.6236i 0.394927 + 0.684034i 0.993092 0.117339i \(-0.0374365\pi\)
−0.598165 + 0.801373i \(0.704103\pi\)
\(824\) 0 0
\(825\) 3.16770 0.110285
\(826\) 0 0
\(827\) −8.45355 14.6420i −0.293959 0.509151i 0.680783 0.732485i \(-0.261639\pi\)
−0.974742 + 0.223333i \(0.928306\pi\)
\(828\) 0 0
\(829\) −43.0299 −1.49449 −0.747245 0.664549i \(-0.768624\pi\)
−0.747245 + 0.664549i \(0.768624\pi\)
\(830\) 0 0
\(831\) −16.3090 + 28.2480i −0.565752 + 0.979912i
\(832\) 0 0
\(833\) 24.8886 43.1084i 0.862340 1.49362i
\(834\) 0 0
\(835\) 24.0796 0.833311
\(836\) 0 0
\(837\) 61.0558 2.11040
\(838\) 0 0
\(839\) 19.6534 34.0407i 0.678511 1.17522i −0.296919 0.954903i \(-0.595959\pi\)
0.975429 0.220312i \(-0.0707077\pi\)
\(840\) 0 0
\(841\) 14.4967 25.1090i 0.499886 0.865827i
\(842\) 0 0
\(843\) −59.0950 −2.03534
\(844\) 0 0
\(845\) 0.269813 + 0.467329i 0.00928184 + 0.0160766i
\(846\) 0 0
\(847\) −45.1799 −1.55240
\(848\) 0 0
\(849\) −39.8565 69.0334i −1.36787 2.36922i
\(850\) 0 0
\(851\) 22.2220 + 38.4897i 0.761761 + 1.31941i
\(852\) 0 0
\(853\) −1.02974 + 1.78356i −0.0352576 + 0.0610680i −0.883116 0.469155i \(-0.844559\pi\)
0.847858 + 0.530223i \(0.177892\pi\)
\(854\) 0 0
\(855\) 37.5627 + 9.55943i 1.28462 + 0.326926i
\(856\) 0 0
\(857\) 0.760310 1.31690i 0.0259717 0.0449843i −0.852747 0.522323i \(-0.825065\pi\)
0.878719 + 0.477339i \(0.158399\pi\)
\(858\) 0 0
\(859\) 12.8895 + 22.3253i 0.439785 + 0.761730i 0.997673 0.0681866i \(-0.0217213\pi\)
−0.557888 + 0.829917i \(0.688388\pi\)
\(860\) 0 0
\(861\) 3.44014 + 5.95849i 0.117240 + 0.203065i
\(862\) 0 0
\(863\) −7.30020 −0.248502 −0.124251 0.992251i \(-0.539653\pi\)
−0.124251 + 0.992251i \(0.539653\pi\)
\(864\) 0 0
\(865\) −6.21940 10.7723i −0.211466 0.366270i
\(866\) 0 0
\(867\) −6.38377 −0.216804
\(868\) 0 0
\(869\) −5.92342 + 10.2597i −0.200938 + 0.348036i
\(870\) 0 0
\(871\) 15.6858 27.1685i 0.531491 0.920570i
\(872\) 0 0
\(873\) 58.2569 1.97170
\(874\) 0 0
\(875\) 4.44850 0.150387
\(876\) 0 0
\(877\) 8.95315 15.5073i 0.302326 0.523645i −0.674336 0.738425i \(-0.735570\pi\)
0.976662 + 0.214780i \(0.0689034\pi\)
\(878\) 0 0
\(879\) −15.6131 + 27.0427i −0.526616 + 0.912126i
\(880\) 0 0
\(881\) −3.53693 −0.119162 −0.0595811 0.998223i \(-0.518976\pi\)
−0.0595811 + 0.998223i \(0.518976\pi\)
\(882\) 0 0
\(883\) 25.3323 + 43.8768i 0.852499 + 1.47657i 0.878946 + 0.476922i \(0.158248\pi\)
−0.0264464 + 0.999650i \(0.508419\pi\)
\(884\) 0 0
\(885\) 7.17780 0.241279
\(886\) 0 0
\(887\) −17.7354 30.7186i −0.595497 1.03143i −0.993477 0.114037i \(-0.963622\pi\)
0.397979 0.917394i \(-0.369712\pi\)
\(888\) 0 0
\(889\) 10.0757 + 17.4516i 0.337928 + 0.585308i
\(890\) 0 0
\(891\) −19.9302 + 34.5202i −0.667688 + 1.15647i
\(892\) 0 0
\(893\) −5.68773 20.2053i −0.190333 0.676145i
\(894\) 0 0
\(895\) −2.36465 + 4.09569i −0.0790415 + 0.136904i
\(896\) 0 0
\(897\) −41.9489 72.6576i −1.40063 2.42597i
\(898\) 0 0
\(899\) 0.122336 + 0.211893i 0.00408015 + 0.00706703i
\(900\) 0 0
\(901\) 21.5422 0.717674
\(902\) 0 0
\(903\) −29.2295 50.6269i −0.972696 1.68476i
\(904\) 0 0
\(905\) −4.41722 −0.146834
\(906\) 0 0
\(907\) 4.79400 8.30345i 0.159182 0.275711i −0.775392 0.631480i \(-0.782448\pi\)
0.934574 + 0.355769i \(0.115781\pi\)
\(908\) 0 0
\(909\) −39.8013 + 68.9378i −1.32012 + 2.28652i
\(910\) 0 0
\(911\) −30.7728 −1.01955 −0.509774 0.860308i \(-0.670271\pi\)
−0.509774 + 0.860308i \(0.670271\pi\)
\(912\) 0 0
\(913\) −4.74692 −0.157100
\(914\) 0 0
\(915\) −17.9619 + 31.1109i −0.593802 + 1.02850i
\(916\) 0 0
\(917\) −29.6837 + 51.4137i −0.980243 + 1.69783i
\(918\) 0 0
\(919\) −26.0316 −0.858705 −0.429352 0.903137i \(-0.641258\pi\)
−0.429352 + 0.903137i \(0.641258\pi\)
\(920\) 0 0
\(921\) −35.9579 62.2810i −1.18485 2.05223i
\(922\) 0 0
\(923\) −10.3195 −0.339669
\(924\) 0 0
\(925\) −3.22425 5.58456i −0.106013 0.183619i
\(926\) 0 0
\(927\) −21.7509 37.6736i −0.714392 1.23736i
\(928\) 0 0
\(929\) 27.6623 47.9126i 0.907572 1.57196i 0.0901452 0.995929i \(-0.471267\pi\)
0.817427 0.576032i \(-0.195400\pi\)
\(930\) 0 0
\(931\) 54.0246 + 13.7489i 1.77058 + 0.450601i
\(932\) 0 0
\(933\) 5.20995 9.02390i 0.170566 0.295429i
\(934\) 0 0
\(935\) 1.78761 + 3.09624i 0.0584612 + 0.101258i
\(936\) 0 0
\(937\) 14.6703 + 25.4097i 0.479259 + 0.830100i 0.999717 0.0237867i \(-0.00757227\pi\)
−0.520458 + 0.853887i \(0.674239\pi\)
\(938\) 0 0
\(939\) −18.6980 −0.610187
\(940\) 0 0
\(941\) 14.8393 + 25.7024i 0.483747 + 0.837875i 0.999826 0.0186664i \(-0.00594203\pi\)
−0.516078 + 0.856541i \(0.672609\pi\)
\(942\) 0 0
\(943\) 3.09113 0.100661
\(944\) 0 0
\(945\) −45.1947 + 78.2795i −1.47018 + 2.54643i
\(946\) 0 0
\(947\) −10.0158 + 17.3479i −0.325471 + 0.563731i −0.981607 0.190910i \(-0.938856\pi\)
0.656137 + 0.754642i \(0.272189\pi\)
\(948\) 0 0
\(949\) 56.9960 1.85017
\(950\) 0 0
\(951\) −53.1086 −1.72216
\(952\) 0 0
\(953\) −14.2691 + 24.7148i −0.462222 + 0.800592i −0.999071 0.0430860i \(-0.986281\pi\)
0.536849 + 0.843678i \(0.319614\pi\)
\(954\) 0 0
\(955\) 9.53478 16.5147i 0.308538 0.534404i
\(956\) 0 0
\(957\) −0.257933 −0.00833780
\(958\) 0 0
\(959\) −6.11863 10.5978i −0.197581 0.342220i
\(960\) 0 0
\(961\) −21.9709 −0.708738
\(962\) 0 0
\(963\) 42.6153 + 73.8119i 1.37326 + 2.37855i
\(964\) 0 0
\(965\) 3.42100 + 5.92534i 0.110126 + 0.190744i
\(966\) 0 0
\(967\) 9.69278 16.7884i 0.311699 0.539878i −0.667032 0.745029i \(-0.732435\pi\)
0.978730 + 0.205151i \(0.0657687\pi\)
\(968\) 0 0
\(969\) 15.8530 + 56.3167i 0.509271 + 1.80915i
\(970\) 0 0
\(971\) 9.36310 16.2174i 0.300476 0.520440i −0.675768 0.737115i \(-0.736188\pi\)
0.976244 + 0.216675i \(0.0695211\pi\)
\(972\) 0 0
\(973\) 25.6239 + 44.3818i 0.821463 + 1.42282i
\(974\) 0 0
\(975\) 6.08647 + 10.5421i 0.194923 + 0.337617i
\(976\) 0 0
\(977\) 40.7847 1.30482 0.652409 0.757867i \(-0.273759\pi\)
0.652409 + 0.757867i \(0.273759\pi\)
\(978\) 0 0
\(979\) 1.15751 + 2.00486i 0.0369941 + 0.0640756i
\(980\) 0 0
\(981\) −80.7965 −2.57963
\(982\) 0 0
\(983\) 24.0145 41.5943i 0.765943 1.32665i −0.173804 0.984780i \(-0.555606\pi\)
0.939747 0.341872i \(-0.111061\pi\)
\(984\) 0 0
\(985\) 6.20267 10.7433i 0.197634 0.342312i
\(986\) 0 0
\(987\) 73.8740 2.35144
\(988\) 0 0
\(989\) −26.2641 −0.835149
\(990\) 0 0
\(991\) 17.3455 30.0433i 0.550998 0.954356i −0.447205 0.894432i \(-0.647581\pi\)
0.998203 0.0599249i \(-0.0190861\pi\)
\(992\) 0 0
\(993\) −15.6754 + 27.1505i −0.497443 + 0.861596i
\(994\) 0 0
\(995\) −10.0264 −0.317859
\(996\) 0 0
\(997\) 13.5286 + 23.4322i 0.428454 + 0.742105i 0.996736 0.0807294i \(-0.0257250\pi\)
−0.568282 + 0.822834i \(0.692392\pi\)
\(998\) 0 0
\(999\) 131.028 4.14553
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 1520.2.q.n.961.1 8
4.3 odd 2 760.2.q.d.201.4 yes 8
19.7 even 3 inner 1520.2.q.n.881.1 8
76.7 odd 6 760.2.q.d.121.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.q.d.121.4 8 76.7 odd 6
760.2.q.d.201.4 yes 8 4.3 odd 2
1520.2.q.n.881.1 8 19.7 even 3 inner
1520.2.q.n.961.1 8 1.1 even 1 trivial