Properties

Label 1148.2.i.d.165.8
Level $1148$
Weight $2$
Character 1148.165
Analytic conductor $9.167$
Analytic rank $0$
Dimension $16$
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] = [1148,2,Mod(165,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.165");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.i (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: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 14 x^{14} - 8 x^{13} + 136 x^{12} - 87 x^{11} + 706 x^{10} - 568 x^{9} + 2685 x^{8} - 2100 x^{7} + 5529 x^{6} - 4919 x^{5} + 8145 x^{4} - 5182 x^{3} + 2775 x^{2} + \cdots + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 165.8
Root \(-1.43353 + 2.48295i\) of defining polynomial
Character \(\chi\) \(=\) 1148.165
Dual form 1148.2.i.d.821.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.43353 - 2.48295i) q^{3} +(-0.601031 - 1.04102i) q^{5} +(-2.46485 + 0.961519i) q^{7} +(-2.61002 - 4.52069i) q^{9} +O(q^{10})\) \(q+(1.43353 - 2.48295i) q^{3} +(-0.601031 - 1.04102i) q^{5} +(-2.46485 + 0.961519i) q^{7} +(-2.61002 - 4.52069i) q^{9} +(0.423540 - 0.733593i) q^{11} -1.77566 q^{13} -3.44639 q^{15} +(-0.665195 + 1.15215i) q^{17} +(-2.56015 - 4.43430i) q^{19} +(-1.14604 + 7.49846i) q^{21} +(0.408156 + 0.706947i) q^{23} +(1.77752 - 3.07876i) q^{25} -6.36501 q^{27} -8.14823 q^{29} +(-2.94452 + 5.10006i) q^{31} +(-1.21432 - 2.10326i) q^{33} +(2.48241 + 1.98805i) q^{35} +(-0.829482 - 1.43670i) q^{37} +(-2.54546 + 4.40887i) q^{39} +1.00000 q^{41} -3.82843 q^{43} +(-3.13741 + 5.43415i) q^{45} +(-2.33648 - 4.04690i) q^{47} +(5.15096 - 4.74000i) q^{49} +(1.90716 + 3.30329i) q^{51} +(1.82475 - 3.16055i) q^{53} -1.01824 q^{55} -14.6802 q^{57} +(0.425709 - 0.737349i) q^{59} +(4.56564 + 7.90792i) q^{61} +(10.7800 + 8.63324i) q^{63} +(1.06723 + 1.84849i) q^{65} +(4.19546 - 7.26675i) q^{67} +2.34042 q^{69} +1.66604 q^{71} +(-1.54331 + 2.67310i) q^{73} +(-5.09627 - 8.82700i) q^{75} +(-0.338599 + 2.21544i) q^{77} +(-4.06100 - 7.03386i) q^{79} +(-1.29437 + 2.24192i) q^{81} +3.90385 q^{83} +1.59921 q^{85} +(-11.6807 + 20.2316i) q^{87} +(4.76473 + 8.25275i) q^{89} +(4.37673 - 1.70733i) q^{91} +(8.44213 + 14.6222i) q^{93} +(-3.07746 + 5.33031i) q^{95} +13.3723 q^{97} -4.42180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{9} + 8 q^{11} - 14 q^{13} - 2 q^{15} + q^{17} - 4 q^{19} + 13 q^{21} + 3 q^{23} + 4 q^{25} - 24 q^{27} - 8 q^{29} - 4 q^{31} - 23 q^{33} + 12 q^{35} + 31 q^{37} - 5 q^{39} + 16 q^{41} - 16 q^{43} - q^{45} - 24 q^{47} + 16 q^{49} + 23 q^{51} + q^{53} + 4 q^{55} - 30 q^{57} - 4 q^{59} + 4 q^{61} + 23 q^{63} + 24 q^{65} - 42 q^{69} + 16 q^{71} - 11 q^{73} + 15 q^{75} + 25 q^{77} - 14 q^{79} + 28 q^{81} - 84 q^{83} - 40 q^{85} - 25 q^{87} + 11 q^{89} + 7 q^{91} + 27 q^{93} + 15 q^{95} - 32 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) 1.43353 2.48295i 0.827650 1.43353i −0.0722278 0.997388i \(-0.523011\pi\)
0.899877 0.436143i \(-0.143656\pi\)
\(4\) 0 0
\(5\) −0.601031 1.04102i −0.268789 0.465557i 0.699760 0.714378i \(-0.253290\pi\)
−0.968549 + 0.248821i \(0.919957\pi\)
\(6\) 0 0
\(7\) −2.46485 + 0.961519i −0.931625 + 0.363420i
\(8\) 0 0
\(9\) −2.61002 4.52069i −0.870008 1.50690i
\(10\) 0 0
\(11\) 0.423540 0.733593i 0.127702 0.221187i −0.795084 0.606500i \(-0.792573\pi\)
0.922786 + 0.385313i \(0.125907\pi\)
\(12\) 0 0
\(13\) −1.77566 −0.492479 −0.246239 0.969209i \(-0.579195\pi\)
−0.246239 + 0.969209i \(0.579195\pi\)
\(14\) 0 0
\(15\) −3.44639 −0.889853
\(16\) 0 0
\(17\) −0.665195 + 1.15215i −0.161334 + 0.279438i −0.935347 0.353731i \(-0.884913\pi\)
0.774014 + 0.633169i \(0.218246\pi\)
\(18\) 0 0
\(19\) −2.56015 4.43430i −0.587338 1.01730i −0.994580 0.103979i \(-0.966843\pi\)
0.407242 0.913320i \(-0.366491\pi\)
\(20\) 0 0
\(21\) −1.14604 + 7.49846i −0.250086 + 1.63630i
\(22\) 0 0
\(23\) 0.408156 + 0.706947i 0.0851064 + 0.147409i 0.905437 0.424482i \(-0.139544\pi\)
−0.820330 + 0.571890i \(0.806210\pi\)
\(24\) 0 0
\(25\) 1.77752 3.07876i 0.355505 0.615752i
\(26\) 0 0
\(27\) −6.36501 −1.22495
\(28\) 0 0
\(29\) −8.14823 −1.51309 −0.756544 0.653942i \(-0.773114\pi\)
−0.756544 + 0.653942i \(0.773114\pi\)
\(30\) 0 0
\(31\) −2.94452 + 5.10006i −0.528852 + 0.915998i 0.470582 + 0.882356i \(0.344044\pi\)
−0.999434 + 0.0336419i \(0.989289\pi\)
\(32\) 0 0
\(33\) −1.21432 2.10326i −0.211385 0.366130i
\(34\) 0 0
\(35\) 2.48241 + 1.98805i 0.419604 + 0.336041i
\(36\) 0 0
\(37\) −0.829482 1.43670i −0.136366 0.236193i 0.789752 0.613426i \(-0.210209\pi\)
−0.926118 + 0.377233i \(0.876876\pi\)
\(38\) 0 0
\(39\) −2.54546 + 4.40887i −0.407600 + 0.705983i
\(40\) 0 0
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −3.82843 −0.583831 −0.291915 0.956444i \(-0.594293\pi\)
−0.291915 + 0.956444i \(0.594293\pi\)
\(44\) 0 0
\(45\) −3.13741 + 5.43415i −0.467698 + 0.810076i
\(46\) 0 0
\(47\) −2.33648 4.04690i −0.340810 0.590301i 0.643773 0.765216i \(-0.277368\pi\)
−0.984583 + 0.174916i \(0.944035\pi\)
\(48\) 0 0
\(49\) 5.15096 4.74000i 0.735852 0.677142i
\(50\) 0 0
\(51\) 1.90716 + 3.30329i 0.267055 + 0.462553i
\(52\) 0 0
\(53\) 1.82475 3.16055i 0.250648 0.434136i −0.713056 0.701107i \(-0.752690\pi\)
0.963704 + 0.266971i \(0.0860229\pi\)
\(54\) 0 0
\(55\) −1.01824 −0.137300
\(56\) 0 0
\(57\) −14.6802 −1.94444
\(58\) 0 0
\(59\) 0.425709 0.737349i 0.0554225 0.0959946i −0.836983 0.547229i \(-0.815683\pi\)
0.892406 + 0.451234i \(0.149016\pi\)
\(60\) 0 0
\(61\) 4.56564 + 7.90792i 0.584570 + 1.01250i 0.994929 + 0.100581i \(0.0320701\pi\)
−0.410359 + 0.911924i \(0.634597\pi\)
\(62\) 0 0
\(63\) 10.7800 + 8.63324i 1.35816 + 1.08769i
\(64\) 0 0
\(65\) 1.06723 + 1.84849i 0.132373 + 0.229277i
\(66\) 0 0
\(67\) 4.19546 7.26675i 0.512557 0.887775i −0.487337 0.873214i \(-0.662032\pi\)
0.999894 0.0145610i \(-0.00463506\pi\)
\(68\) 0 0
\(69\) 2.34042 0.281753
\(70\) 0 0
\(71\) 1.66604 0.197723 0.0988615 0.995101i \(-0.468480\pi\)
0.0988615 + 0.995101i \(0.468480\pi\)
\(72\) 0 0
\(73\) −1.54331 + 2.67310i −0.180631 + 0.312863i −0.942096 0.335344i \(-0.891147\pi\)
0.761464 + 0.648207i \(0.224481\pi\)
\(74\) 0 0
\(75\) −5.09627 8.82700i −0.588466 1.01925i
\(76\) 0 0
\(77\) −0.338599 + 2.21544i −0.0385870 + 0.252472i
\(78\) 0 0
\(79\) −4.06100 7.03386i −0.456898 0.791371i 0.541897 0.840445i \(-0.317706\pi\)
−0.998795 + 0.0490737i \(0.984373\pi\)
\(80\) 0 0
\(81\) −1.29437 + 2.24192i −0.143819 + 0.249102i
\(82\) 0 0
\(83\) 3.90385 0.428503 0.214251 0.976779i \(-0.431269\pi\)
0.214251 + 0.976779i \(0.431269\pi\)
\(84\) 0 0
\(85\) 1.59921 0.173459
\(86\) 0 0
\(87\) −11.6807 + 20.2316i −1.25231 + 2.16906i
\(88\) 0 0
\(89\) 4.76473 + 8.25275i 0.505060 + 0.874790i 0.999983 + 0.00585279i \(0.00186301\pi\)
−0.494923 + 0.868937i \(0.664804\pi\)
\(90\) 0 0
\(91\) 4.37673 1.70733i 0.458806 0.178977i
\(92\) 0 0
\(93\) 8.44213 + 14.6222i 0.875408 + 1.51625i
\(94\) 0 0
\(95\) −3.07746 + 5.33031i −0.315740 + 0.546878i
\(96\) 0 0
\(97\) 13.3723 1.35776 0.678878 0.734251i \(-0.262466\pi\)
0.678878 + 0.734251i \(0.262466\pi\)
\(98\) 0 0
\(99\) −4.42180 −0.444407
\(100\) 0 0
\(101\) −1.37820 + 2.38712i −0.137136 + 0.237527i −0.926411 0.376513i \(-0.877123\pi\)
0.789275 + 0.614040i \(0.210456\pi\)
\(102\) 0 0
\(103\) 0.730755 + 1.26570i 0.0720034 + 0.124714i 0.899779 0.436345i \(-0.143727\pi\)
−0.827776 + 0.561059i \(0.810394\pi\)
\(104\) 0 0
\(105\) 8.49483 3.31377i 0.829010 0.323390i
\(106\) 0 0
\(107\) −6.31141 10.9317i −0.610147 1.05681i −0.991215 0.132258i \(-0.957777\pi\)
0.381069 0.924547i \(-0.375556\pi\)
\(108\) 0 0
\(109\) 10.2890 17.8210i 0.985504 1.70694i 0.345831 0.938297i \(-0.387597\pi\)
0.639674 0.768647i \(-0.279069\pi\)
\(110\) 0 0
\(111\) −4.75635 −0.451453
\(112\) 0 0
\(113\) 4.11456 0.387065 0.193532 0.981094i \(-0.438006\pi\)
0.193532 + 0.981094i \(0.438006\pi\)
\(114\) 0 0
\(115\) 0.490629 0.849795i 0.0457514 0.0792438i
\(116\) 0 0
\(117\) 4.63451 + 8.02720i 0.428460 + 0.742115i
\(118\) 0 0
\(119\) 0.531791 3.47948i 0.0487492 0.318963i
\(120\) 0 0
\(121\) 5.14123 + 8.90487i 0.467384 + 0.809533i
\(122\) 0 0
\(123\) 1.43353 2.48295i 0.129257 0.223880i
\(124\) 0 0
\(125\) −10.2837 −0.919802
\(126\) 0 0
\(127\) 1.96838 0.174665 0.0873327 0.996179i \(-0.472166\pi\)
0.0873327 + 0.996179i \(0.472166\pi\)
\(128\) 0 0
\(129\) −5.48818 + 9.50580i −0.483207 + 0.836939i
\(130\) 0 0
\(131\) −6.58282 11.4018i −0.575143 0.996178i −0.996026 0.0890625i \(-0.971613\pi\)
0.420883 0.907115i \(-0.361720\pi\)
\(132\) 0 0
\(133\) 10.5740 + 8.46826i 0.916886 + 0.734291i
\(134\) 0 0
\(135\) 3.82557 + 6.62608i 0.329253 + 0.570282i
\(136\) 0 0
\(137\) −2.07245 + 3.58958i −0.177061 + 0.306679i −0.940873 0.338761i \(-0.889992\pi\)
0.763812 + 0.645439i \(0.223326\pi\)
\(138\) 0 0
\(139\) −12.0961 −1.02598 −0.512990 0.858394i \(-0.671462\pi\)
−0.512990 + 0.858394i \(0.671462\pi\)
\(140\) 0 0
\(141\) −13.3977 −1.12829
\(142\) 0 0
\(143\) −0.752062 + 1.30261i −0.0628906 + 0.108930i
\(144\) 0 0
\(145\) 4.89734 + 8.48244i 0.406702 + 0.704429i
\(146\) 0 0
\(147\) −4.38510 19.5845i −0.361677 1.61530i
\(148\) 0 0
\(149\) −3.55140 6.15120i −0.290942 0.503926i 0.683091 0.730333i \(-0.260635\pi\)
−0.974033 + 0.226407i \(0.927302\pi\)
\(150\) 0 0
\(151\) 4.85040 8.40113i 0.394720 0.683674i −0.598346 0.801238i \(-0.704175\pi\)
0.993065 + 0.117564i \(0.0375084\pi\)
\(152\) 0 0
\(153\) 6.94470 0.561446
\(154\) 0 0
\(155\) 7.07900 0.568599
\(156\) 0 0
\(157\) 1.46100 2.53052i 0.116600 0.201958i −0.801818 0.597568i \(-0.796134\pi\)
0.918418 + 0.395611i \(0.129467\pi\)
\(158\) 0 0
\(159\) −5.23166 9.06151i −0.414898 0.718624i
\(160\) 0 0
\(161\) −1.68579 1.35007i −0.132859 0.106400i
\(162\) 0 0
\(163\) −9.78609 16.9500i −0.766506 1.32763i −0.939447 0.342695i \(-0.888660\pi\)
0.172941 0.984932i \(-0.444673\pi\)
\(164\) 0 0
\(165\) −1.45968 + 2.52825i −0.113636 + 0.196824i
\(166\) 0 0
\(167\) −14.8666 −1.15042 −0.575208 0.818007i \(-0.695079\pi\)
−0.575208 + 0.818007i \(0.695079\pi\)
\(168\) 0 0
\(169\) −9.84704 −0.757465
\(170\) 0 0
\(171\) −13.3641 + 23.1473i −1.02198 + 1.77012i
\(172\) 0 0
\(173\) −8.21200 14.2236i −0.624347 1.08140i −0.988667 0.150126i \(-0.952032\pi\)
0.364320 0.931274i \(-0.381301\pi\)
\(174\) 0 0
\(175\) −1.42104 + 9.29780i −0.107421 + 0.702848i
\(176\) 0 0
\(177\) −1.22053 2.11402i −0.0917409 0.158900i
\(178\) 0 0
\(179\) 12.8807 22.3101i 0.962750 1.66753i 0.247207 0.968963i \(-0.420487\pi\)
0.715543 0.698569i \(-0.246180\pi\)
\(180\) 0 0
\(181\) 2.11304 0.157061 0.0785305 0.996912i \(-0.474977\pi\)
0.0785305 + 0.996912i \(0.474977\pi\)
\(182\) 0 0
\(183\) 26.1799 1.93528
\(184\) 0 0
\(185\) −0.997089 + 1.72701i −0.0733074 + 0.126972i
\(186\) 0 0
\(187\) 0.563474 + 0.975965i 0.0412053 + 0.0713696i
\(188\) 0 0
\(189\) 15.6888 6.12007i 1.14119 0.445170i
\(190\) 0 0
\(191\) −9.26542 16.0482i −0.670422 1.16121i −0.977785 0.209613i \(-0.932780\pi\)
0.307362 0.951593i \(-0.400554\pi\)
\(192\) 0 0
\(193\) −1.43037 + 2.47747i −0.102960 + 0.178332i −0.912903 0.408177i \(-0.866165\pi\)
0.809943 + 0.586509i \(0.199498\pi\)
\(194\) 0 0
\(195\) 6.11960 0.438234
\(196\) 0 0
\(197\) 8.05021 0.573553 0.286777 0.957997i \(-0.407416\pi\)
0.286777 + 0.957997i \(0.407416\pi\)
\(198\) 0 0
\(199\) 0.395395 0.684845i 0.0280288 0.0485474i −0.851671 0.524077i \(-0.824410\pi\)
0.879700 + 0.475530i \(0.157744\pi\)
\(200\) 0 0
\(201\) −12.0286 20.8342i −0.848435 1.46953i
\(202\) 0 0
\(203\) 20.0842 7.83468i 1.40963 0.549886i
\(204\) 0 0
\(205\) −0.601031 1.04102i −0.0419778 0.0727078i
\(206\) 0 0
\(207\) 2.13059 3.69030i 0.148086 0.256493i
\(208\) 0 0
\(209\) −4.33730 −0.300017
\(210\) 0 0
\(211\) 17.9976 1.23900 0.619502 0.784995i \(-0.287334\pi\)
0.619502 + 0.784995i \(0.287334\pi\)
\(212\) 0 0
\(213\) 2.38833 4.13670i 0.163645 0.283442i
\(214\) 0 0
\(215\) 2.30101 + 3.98546i 0.156927 + 0.271806i
\(216\) 0 0
\(217\) 2.35400 15.4021i 0.159800 1.04556i
\(218\) 0 0
\(219\) 4.42478 + 7.66394i 0.298999 + 0.517881i
\(220\) 0 0
\(221\) 1.18116 2.04583i 0.0794533 0.137617i
\(222\) 0 0
\(223\) 20.5976 1.37932 0.689659 0.724134i \(-0.257760\pi\)
0.689659 + 0.724134i \(0.257760\pi\)
\(224\) 0 0
\(225\) −18.5575 −1.23717
\(226\) 0 0
\(227\) 0.545207 0.944326i 0.0361867 0.0626771i −0.847365 0.531011i \(-0.821812\pi\)
0.883552 + 0.468334i \(0.155146\pi\)
\(228\) 0 0
\(229\) 6.78465 + 11.7514i 0.448342 + 0.776552i 0.998278 0.0586551i \(-0.0186812\pi\)
−0.549936 + 0.835207i \(0.685348\pi\)
\(230\) 0 0
\(231\) 5.01542 + 4.01662i 0.329991 + 0.264274i
\(232\) 0 0
\(233\) −8.74924 15.1541i −0.573182 0.992780i −0.996237 0.0866761i \(-0.972375\pi\)
0.423055 0.906104i \(-0.360958\pi\)
\(234\) 0 0
\(235\) −2.80859 + 4.86462i −0.183212 + 0.317333i
\(236\) 0 0
\(237\) −23.2863 −1.51261
\(238\) 0 0
\(239\) 5.17257 0.334586 0.167293 0.985907i \(-0.446497\pi\)
0.167293 + 0.985907i \(0.446497\pi\)
\(240\) 0 0
\(241\) −11.7092 + 20.2810i −0.754257 + 1.30641i 0.191486 + 0.981495i \(0.438669\pi\)
−0.945743 + 0.324916i \(0.894664\pi\)
\(242\) 0 0
\(243\) −5.83647 10.1091i −0.374410 0.648497i
\(244\) 0 0
\(245\) −8.03031 2.51335i −0.513037 0.160572i
\(246\) 0 0
\(247\) 4.54594 + 7.87380i 0.289251 + 0.500998i
\(248\) 0 0
\(249\) 5.59629 9.69305i 0.354650 0.614272i
\(250\) 0 0
\(251\) −4.39194 −0.277217 −0.138608 0.990347i \(-0.544263\pi\)
−0.138608 + 0.990347i \(0.544263\pi\)
\(252\) 0 0
\(253\) 0.691482 0.0434731
\(254\) 0 0
\(255\) 2.29252 3.97076i 0.143563 0.248659i
\(256\) 0 0
\(257\) 9.66509 + 16.7404i 0.602892 + 1.04424i 0.992381 + 0.123208i \(0.0393183\pi\)
−0.389489 + 0.921031i \(0.627348\pi\)
\(258\) 0 0
\(259\) 3.42597 + 2.74370i 0.212879 + 0.170485i
\(260\) 0 0
\(261\) 21.2671 + 36.8356i 1.31640 + 2.28007i
\(262\) 0 0
\(263\) −2.14257 + 3.71103i −0.132116 + 0.228832i −0.924492 0.381201i \(-0.875511\pi\)
0.792376 + 0.610033i \(0.208844\pi\)
\(264\) 0 0
\(265\) −4.38692 −0.269486
\(266\) 0 0
\(267\) 27.3215 1.67205
\(268\) 0 0
\(269\) 5.09292 8.82120i 0.310521 0.537838i −0.667954 0.744202i \(-0.732830\pi\)
0.978475 + 0.206364i \(0.0661632\pi\)
\(270\) 0 0
\(271\) 14.7973 + 25.6297i 0.898873 + 1.55689i 0.828937 + 0.559342i \(0.188946\pi\)
0.0699357 + 0.997552i \(0.477721\pi\)
\(272\) 0 0
\(273\) 2.03497 13.3147i 0.123162 0.805842i
\(274\) 0 0
\(275\) −1.50570 2.60796i −0.0907974 0.157266i
\(276\) 0 0
\(277\) −3.00530 + 5.20532i −0.180571 + 0.312758i −0.942075 0.335402i \(-0.891128\pi\)
0.761504 + 0.648160i \(0.224461\pi\)
\(278\) 0 0
\(279\) 30.7411 1.84042
\(280\) 0 0
\(281\) −24.4365 −1.45776 −0.728881 0.684641i \(-0.759959\pi\)
−0.728881 + 0.684641i \(0.759959\pi\)
\(282\) 0 0
\(283\) 0.195832 0.339192i 0.0116410 0.0201628i −0.860146 0.510048i \(-0.829628\pi\)
0.871787 + 0.489885i \(0.162961\pi\)
\(284\) 0 0
\(285\) 8.82326 + 15.2823i 0.522645 + 0.905247i
\(286\) 0 0
\(287\) −2.46485 + 0.961519i −0.145495 + 0.0567566i
\(288\) 0 0
\(289\) 7.61503 + 13.1896i 0.447943 + 0.775860i
\(290\) 0 0
\(291\) 19.1697 33.2029i 1.12375 1.94639i
\(292\) 0 0
\(293\) 1.12533 0.0657427 0.0328713 0.999460i \(-0.489535\pi\)
0.0328713 + 0.999460i \(0.489535\pi\)
\(294\) 0 0
\(295\) −1.02346 −0.0595879
\(296\) 0 0
\(297\) −2.69584 + 4.66932i −0.156428 + 0.270942i
\(298\) 0 0
\(299\) −0.724745 1.25530i −0.0419131 0.0725956i
\(300\) 0 0
\(301\) 9.43651 3.68111i 0.543911 0.212176i
\(302\) 0 0
\(303\) 3.95139 + 6.84401i 0.227002 + 0.393178i
\(304\) 0 0
\(305\) 5.48818 9.50581i 0.314252 0.544301i
\(306\) 0 0
\(307\) −10.8412 −0.618740 −0.309370 0.950942i \(-0.600118\pi\)
−0.309370 + 0.950942i \(0.600118\pi\)
\(308\) 0 0
\(309\) 4.19024 0.238374
\(310\) 0 0
\(311\) 2.82635 4.89538i 0.160268 0.277592i −0.774697 0.632333i \(-0.782098\pi\)
0.934965 + 0.354741i \(0.115431\pi\)
\(312\) 0 0
\(313\) 2.70635 + 4.68753i 0.152972 + 0.264955i 0.932319 0.361638i \(-0.117782\pi\)
−0.779347 + 0.626593i \(0.784449\pi\)
\(314\) 0 0
\(315\) 2.50820 16.4110i 0.141321 0.924658i
\(316\) 0 0
\(317\) 12.3641 + 21.4153i 0.694439 + 1.20280i 0.970369 + 0.241626i \(0.0776808\pi\)
−0.275930 + 0.961178i \(0.588986\pi\)
\(318\) 0 0
\(319\) −3.45110 + 5.97748i −0.193225 + 0.334675i
\(320\) 0 0
\(321\) −36.1904 −2.01995
\(322\) 0 0
\(323\) 6.81199 0.379029
\(324\) 0 0
\(325\) −3.15627 + 5.46682i −0.175078 + 0.303245i
\(326\) 0 0
\(327\) −29.4991 51.0940i −1.63130 2.82550i
\(328\) 0 0
\(329\) 9.65023 + 7.72843i 0.532035 + 0.426082i
\(330\) 0 0
\(331\) 8.94191 + 15.4879i 0.491492 + 0.851289i 0.999952 0.00979641i \(-0.00311834\pi\)
−0.508460 + 0.861086i \(0.669785\pi\)
\(332\) 0 0
\(333\) −4.32993 + 7.49966i −0.237279 + 0.410979i
\(334\) 0 0
\(335\) −10.0864 −0.551080
\(336\) 0 0
\(337\) −22.1922 −1.20888 −0.604442 0.796649i \(-0.706604\pi\)
−0.604442 + 0.796649i \(0.706604\pi\)
\(338\) 0 0
\(339\) 5.89834 10.2162i 0.320354 0.554870i
\(340\) 0 0
\(341\) 2.49425 + 4.32016i 0.135071 + 0.233950i
\(342\) 0 0
\(343\) −8.13876 + 16.6361i −0.439452 + 0.898266i
\(344\) 0 0
\(345\) −1.40666 2.43641i −0.0757323 0.131172i
\(346\) 0 0
\(347\) 13.6538 23.6491i 0.732975 1.26955i −0.222632 0.974903i \(-0.571465\pi\)
0.955606 0.294647i \(-0.0952020\pi\)
\(348\) 0 0
\(349\) 23.0020 1.23127 0.615633 0.788033i \(-0.288900\pi\)
0.615633 + 0.788033i \(0.288900\pi\)
\(350\) 0 0
\(351\) 11.3021 0.603260
\(352\) 0 0
\(353\) 11.6241 20.1336i 0.618689 1.07160i −0.371036 0.928618i \(-0.620997\pi\)
0.989725 0.142982i \(-0.0456692\pi\)
\(354\) 0 0
\(355\) −1.00134 1.73438i −0.0531458 0.0920513i
\(356\) 0 0
\(357\) −7.87703 6.30835i −0.416897 0.333873i
\(358\) 0 0
\(359\) 10.2417 + 17.7392i 0.540538 + 0.936240i 0.998873 + 0.0474602i \(0.0151127\pi\)
−0.458335 + 0.888780i \(0.651554\pi\)
\(360\) 0 0
\(361\) −3.60870 + 6.25045i −0.189932 + 0.328971i
\(362\) 0 0
\(363\) 29.4804 1.54732
\(364\) 0 0
\(365\) 3.71032 0.194207
\(366\) 0 0
\(367\) −0.202540 + 0.350809i −0.0105725 + 0.0183121i −0.871263 0.490816i \(-0.836699\pi\)
0.860691 + 0.509128i \(0.170032\pi\)
\(368\) 0 0
\(369\) −2.61002 4.52069i −0.135872 0.235338i
\(370\) 0 0
\(371\) −1.45879 + 9.54482i −0.0757368 + 0.495542i
\(372\) 0 0
\(373\) 1.35692 + 2.35026i 0.0702589 + 0.121692i 0.899015 0.437918i \(-0.144284\pi\)
−0.828756 + 0.559610i \(0.810951\pi\)
\(374\) 0 0
\(375\) −14.7420 + 25.5339i −0.761274 + 1.31856i
\(376\) 0 0
\(377\) 14.4685 0.745164
\(378\) 0 0
\(379\) 10.2270 0.525327 0.262663 0.964888i \(-0.415399\pi\)
0.262663 + 0.964888i \(0.415399\pi\)
\(380\) 0 0
\(381\) 2.82173 4.88738i 0.144562 0.250388i
\(382\) 0 0
\(383\) −16.5025 28.5832i −0.843241 1.46054i −0.887140 0.461500i \(-0.847311\pi\)
0.0438996 0.999036i \(-0.486022\pi\)
\(384\) 0 0
\(385\) 2.50982 0.979059i 0.127912 0.0498975i
\(386\) 0 0
\(387\) 9.99230 + 17.3072i 0.507937 + 0.879773i
\(388\) 0 0
\(389\) −13.6827 + 23.6992i −0.693742 + 1.20160i 0.276861 + 0.960910i \(0.410706\pi\)
−0.970603 + 0.240686i \(0.922628\pi\)
\(390\) 0 0
\(391\) −1.08601 −0.0549221
\(392\) 0 0
\(393\) −37.7467 −1.90407
\(394\) 0 0
\(395\) −4.88158 + 8.45514i −0.245619 + 0.425424i
\(396\) 0 0
\(397\) 7.34821 + 12.7275i 0.368796 + 0.638774i 0.989378 0.145368i \(-0.0464367\pi\)
−0.620582 + 0.784142i \(0.713103\pi\)
\(398\) 0 0
\(399\) 36.1845 14.1153i 1.81149 0.706648i
\(400\) 0 0
\(401\) 4.70877 + 8.15582i 0.235145 + 0.407282i 0.959315 0.282339i \(-0.0911103\pi\)
−0.724170 + 0.689621i \(0.757777\pi\)
\(402\) 0 0
\(403\) 5.22846 9.05596i 0.260448 0.451109i
\(404\) 0 0
\(405\) 3.11183 0.154628
\(406\) 0 0
\(407\) −1.40527 −0.0696569
\(408\) 0 0
\(409\) 14.7879 25.6133i 0.731213 1.26650i −0.225152 0.974324i \(-0.572288\pi\)
0.956365 0.292174i \(-0.0943788\pi\)
\(410\) 0 0
\(411\) 5.94183 + 10.2916i 0.293089 + 0.507645i
\(412\) 0 0
\(413\) −0.340333 + 2.22678i −0.0167467 + 0.109573i
\(414\) 0 0
\(415\) −2.34633 4.06397i −0.115177 0.199492i
\(416\) 0 0
\(417\) −17.3402 + 30.0341i −0.849152 + 1.47078i
\(418\) 0 0
\(419\) 13.4890 0.658979 0.329489 0.944159i \(-0.393123\pi\)
0.329489 + 0.944159i \(0.393123\pi\)
\(420\) 0 0
\(421\) 13.6752 0.666489 0.333244 0.942841i \(-0.391857\pi\)
0.333244 + 0.942841i \(0.391857\pi\)
\(422\) 0 0
\(423\) −12.1965 + 21.1250i −0.593015 + 1.02713i
\(424\) 0 0
\(425\) 2.36480 + 4.09595i 0.114710 + 0.198683i
\(426\) 0 0
\(427\) −18.8572 15.1019i −0.912565 0.730831i
\(428\) 0 0
\(429\) 2.15621 + 3.73466i 0.104103 + 0.180311i
\(430\) 0 0
\(431\) 8.86833 15.3604i 0.427173 0.739885i −0.569448 0.822027i \(-0.692843\pi\)
0.996621 + 0.0821427i \(0.0261763\pi\)
\(432\) 0 0
\(433\) 12.5658 0.603874 0.301937 0.953328i \(-0.402367\pi\)
0.301937 + 0.953328i \(0.402367\pi\)
\(434\) 0 0
\(435\) 28.0820 1.34643
\(436\) 0 0
\(437\) 2.08988 3.61978i 0.0999725 0.173157i
\(438\) 0 0
\(439\) −0.276998 0.479774i −0.0132204 0.0228984i 0.859340 0.511405i \(-0.170875\pi\)
−0.872560 + 0.488507i \(0.837542\pi\)
\(440\) 0 0
\(441\) −34.8722 10.9144i −1.66058 0.519734i
\(442\) 0 0
\(443\) 17.7952 + 30.8222i 0.845477 + 1.46441i 0.885206 + 0.465199i \(0.154017\pi\)
−0.0397291 + 0.999210i \(0.512650\pi\)
\(444\) 0 0
\(445\) 5.72750 9.92032i 0.271510 0.470268i
\(446\) 0 0
\(447\) −20.3642 −0.963191
\(448\) 0 0
\(449\) 9.72773 0.459080 0.229540 0.973299i \(-0.426278\pi\)
0.229540 + 0.973299i \(0.426278\pi\)
\(450\) 0 0
\(451\) 0.423540 0.733593i 0.0199437 0.0345435i
\(452\) 0 0
\(453\) −13.9064 24.0866i −0.653379 1.13169i
\(454\) 0 0
\(455\) −4.40791 3.53009i −0.206646 0.165493i
\(456\) 0 0
\(457\) 18.4013 + 31.8721i 0.860778 + 1.49091i 0.871179 + 0.490966i \(0.163356\pi\)
−0.0104002 + 0.999946i \(0.503311\pi\)
\(458\) 0 0
\(459\) 4.23398 7.33346i 0.197625 0.342297i
\(460\) 0 0
\(461\) −33.6797 −1.56862 −0.784310 0.620369i \(-0.786983\pi\)
−0.784310 + 0.620369i \(0.786983\pi\)
\(462\) 0 0
\(463\) 3.14990 0.146388 0.0731942 0.997318i \(-0.476681\pi\)
0.0731942 + 0.997318i \(0.476681\pi\)
\(464\) 0 0
\(465\) 10.1480 17.5768i 0.470601 0.815104i
\(466\) 0 0
\(467\) −16.3931 28.3937i −0.758584 1.31391i −0.943573 0.331165i \(-0.892558\pi\)
0.184989 0.982741i \(-0.440775\pi\)
\(468\) 0 0
\(469\) −3.35406 + 21.9455i −0.154876 + 1.01335i
\(470\) 0 0
\(471\) −4.18877 7.25517i −0.193008 0.334301i
\(472\) 0 0
\(473\) −1.62149 + 2.80851i −0.0745564 + 0.129135i
\(474\) 0 0
\(475\) −18.2029 −0.835205
\(476\) 0 0
\(477\) −19.0505 −0.872264
\(478\) 0 0
\(479\) 7.91416 13.7077i 0.361607 0.626322i −0.626618 0.779326i \(-0.715562\pi\)
0.988226 + 0.153004i \(0.0488948\pi\)
\(480\) 0 0
\(481\) 1.47287 + 2.55109i 0.0671573 + 0.116320i
\(482\) 0 0
\(483\) −5.76878 + 2.25036i −0.262488 + 0.102395i
\(484\) 0 0
\(485\) −8.03720 13.9208i −0.364950 0.632113i
\(486\) 0 0
\(487\) −5.85215 + 10.1362i −0.265186 + 0.459316i −0.967612 0.252440i \(-0.918767\pi\)
0.702426 + 0.711757i \(0.252100\pi\)
\(488\) 0 0
\(489\) −56.1146 −2.53759
\(490\) 0 0
\(491\) −25.0759 −1.13166 −0.565830 0.824522i \(-0.691444\pi\)
−0.565830 + 0.824522i \(0.691444\pi\)
\(492\) 0 0
\(493\) 5.42017 9.38800i 0.244112 0.422814i
\(494\) 0 0
\(495\) 2.65764 + 4.60316i 0.119452 + 0.206897i
\(496\) 0 0
\(497\) −4.10655 + 1.60193i −0.184204 + 0.0718565i
\(498\) 0 0
\(499\) 16.0710 + 27.8357i 0.719435 + 1.24610i 0.961224 + 0.275769i \(0.0889324\pi\)
−0.241789 + 0.970329i \(0.577734\pi\)
\(500\) 0 0
\(501\) −21.3118 + 36.9131i −0.952141 + 1.64916i
\(502\) 0 0
\(503\) −20.8480 −0.929567 −0.464784 0.885424i \(-0.653868\pi\)
−0.464784 + 0.885424i \(0.653868\pi\)
\(504\) 0 0
\(505\) 3.31337 0.147443
\(506\) 0 0
\(507\) −14.1160 + 24.4497i −0.626915 + 1.08585i
\(508\) 0 0
\(509\) 14.3549 + 24.8634i 0.636270 + 1.10205i 0.986244 + 0.165293i \(0.0528571\pi\)
−0.349974 + 0.936759i \(0.613810\pi\)
\(510\) 0 0
\(511\) 1.23380 8.07271i 0.0545802 0.357116i
\(512\) 0 0
\(513\) 16.2954 + 28.2244i 0.719458 + 1.24614i
\(514\) 0 0
\(515\) 0.878413 1.52146i 0.0387075 0.0670434i
\(516\) 0 0
\(517\) −3.95837 −0.174089
\(518\) 0 0
\(519\) −47.0886 −2.06696
\(520\) 0 0
\(521\) −0.325114 + 0.563115i −0.0142435 + 0.0246705i −0.873059 0.487614i \(-0.837867\pi\)
0.858816 + 0.512285i \(0.171201\pi\)
\(522\) 0 0
\(523\) −3.33127 5.76993i −0.145666 0.252301i 0.783955 0.620818i \(-0.213199\pi\)
−0.929621 + 0.368516i \(0.879866\pi\)
\(524\) 0 0
\(525\) 21.0489 + 16.8571i 0.918647 + 0.735702i
\(526\) 0 0
\(527\) −3.91736 6.78507i −0.170643 0.295562i
\(528\) 0 0
\(529\) 11.1668 19.3415i 0.485514 0.840935i
\(530\) 0 0
\(531\) −4.44444 −0.192872
\(532\) 0 0
\(533\) −1.77566 −0.0769122
\(534\) 0 0
\(535\) −7.58670 + 13.1406i −0.328002 + 0.568116i
\(536\) 0 0
\(537\) −36.9298 63.9643i −1.59364 2.76026i
\(538\) 0 0
\(539\) −1.29559 5.78629i −0.0558049 0.249233i
\(540\) 0 0
\(541\) 16.9160 + 29.2993i 0.727274 + 1.25968i 0.958031 + 0.286665i \(0.0925465\pi\)
−0.230757 + 0.973012i \(0.574120\pi\)
\(542\) 0 0
\(543\) 3.02911 5.24657i 0.129991 0.225152i
\(544\) 0 0
\(545\) −24.7360 −1.05957
\(546\) 0 0
\(547\) −31.0612 −1.32808 −0.664040 0.747697i \(-0.731160\pi\)
−0.664040 + 0.747697i \(0.731160\pi\)
\(548\) 0 0
\(549\) 23.8328 41.2797i 1.01716 1.76177i
\(550\) 0 0
\(551\) 20.8607 + 36.1317i 0.888694 + 1.53926i
\(552\) 0 0
\(553\) 16.7729 + 13.4327i 0.713258 + 0.571216i
\(554\) 0 0
\(555\) 2.85872 + 4.95144i 0.121346 + 0.210177i
\(556\) 0 0
\(557\) 13.1148 22.7155i 0.555692 0.962488i −0.442157 0.896938i \(-0.645787\pi\)
0.997849 0.0655498i \(-0.0208801\pi\)
\(558\) 0 0
\(559\) 6.79798 0.287524
\(560\) 0 0
\(561\) 3.23103 0.136414
\(562\) 0 0
\(563\) −7.59923 + 13.1623i −0.320269 + 0.554723i −0.980544 0.196302i \(-0.937107\pi\)
0.660274 + 0.751025i \(0.270440\pi\)
\(564\) 0 0
\(565\) −2.47298 4.28332i −0.104039 0.180201i
\(566\) 0 0
\(567\) 1.03478 6.77054i 0.0434569 0.284336i
\(568\) 0 0
\(569\) −14.0806 24.3882i −0.590288 1.02241i −0.994193 0.107607i \(-0.965681\pi\)
0.403906 0.914801i \(-0.367652\pi\)
\(570\) 0 0
\(571\) 14.9883 25.9604i 0.627239 1.08641i −0.360864 0.932619i \(-0.617518\pi\)
0.988103 0.153792i \(-0.0491486\pi\)
\(572\) 0 0
\(573\) −53.1291 −2.21950
\(574\) 0 0
\(575\) 2.90203 0.121023
\(576\) 0 0
\(577\) −11.6161 + 20.1197i −0.483585 + 0.837595i −0.999822 0.0188513i \(-0.993999\pi\)
0.516237 + 0.856446i \(0.327332\pi\)
\(578\) 0 0
\(579\) 4.10095 + 7.10305i 0.170430 + 0.295193i
\(580\) 0 0
\(581\) −9.62239 + 3.75362i −0.399204 + 0.155726i
\(582\) 0 0
\(583\) −1.54571 2.67724i −0.0640166 0.110880i
\(584\) 0 0
\(585\) 5.57096 9.64919i 0.230331 0.398945i
\(586\) 0 0
\(587\) −29.8264 −1.23107 −0.615533 0.788111i \(-0.711059\pi\)
−0.615533 + 0.788111i \(0.711059\pi\)
\(588\) 0 0
\(589\) 30.1536 1.24246
\(590\) 0 0
\(591\) 11.5402 19.9882i 0.474701 0.822207i
\(592\) 0 0
\(593\) 1.00043 + 1.73279i 0.0410825 + 0.0711570i 0.885836 0.463999i \(-0.153586\pi\)
−0.844753 + 0.535156i \(0.820253\pi\)
\(594\) 0 0
\(595\) −3.94182 + 1.53767i −0.161599 + 0.0630384i
\(596\) 0 0
\(597\) −1.13362 1.96349i −0.0463961 0.0803604i
\(598\) 0 0
\(599\) −17.6729 + 30.6103i −0.722094 + 1.25070i 0.238065 + 0.971249i \(0.423487\pi\)
−0.960159 + 0.279454i \(0.909847\pi\)
\(600\) 0 0
\(601\) −11.2751 −0.459920 −0.229960 0.973200i \(-0.573860\pi\)
−0.229960 + 0.973200i \(0.573860\pi\)
\(602\) 0 0
\(603\) −43.8010 −1.78371
\(604\) 0 0
\(605\) 6.18008 10.7042i 0.251256 0.435188i
\(606\) 0 0
\(607\) −5.60312 9.70489i −0.227424 0.393909i 0.729620 0.683853i \(-0.239697\pi\)
−0.957044 + 0.289943i \(0.906364\pi\)
\(608\) 0 0
\(609\) 9.33817 61.0992i 0.378402 2.47586i
\(610\) 0 0
\(611\) 4.14878 + 7.18590i 0.167842 + 0.290711i
\(612\) 0 0
\(613\) 3.89025 6.73812i 0.157126 0.272150i −0.776705 0.629864i \(-0.783110\pi\)
0.933831 + 0.357714i \(0.116444\pi\)
\(614\) 0 0
\(615\) −3.44639 −0.138972
\(616\) 0 0
\(617\) −40.9869 −1.65007 −0.825035 0.565082i \(-0.808845\pi\)
−0.825035 + 0.565082i \(0.808845\pi\)
\(618\) 0 0
\(619\) 7.65595 13.2605i 0.307719 0.532985i −0.670144 0.742231i \(-0.733768\pi\)
0.977863 + 0.209246i \(0.0671011\pi\)
\(620\) 0 0
\(621\) −2.59792 4.49973i −0.104251 0.180568i
\(622\) 0 0
\(623\) −19.6795 15.7604i −0.788443 0.631428i
\(624\) 0 0
\(625\) −2.70679 4.68830i −0.108272 0.187532i
\(626\) 0 0
\(627\) −6.21765 + 10.7693i −0.248309 + 0.430084i
\(628\) 0 0
\(629\) 2.20707 0.0880016
\(630\) 0 0
\(631\) −20.4649 −0.814696 −0.407348 0.913273i \(-0.633546\pi\)
−0.407348 + 0.913273i \(0.633546\pi\)
\(632\) 0 0
\(633\) 25.8001 44.6871i 1.02546 1.77615i
\(634\) 0 0
\(635\) −1.18306 2.04911i −0.0469482 0.0813166i
\(636\) 0 0
\(637\) −9.14635 + 8.41661i −0.362391 + 0.333478i
\(638\) 0 0
\(639\) −4.34841 7.53167i −0.172021 0.297948i
\(640\) 0 0
\(641\) −0.358245 + 0.620499i −0.0141498 + 0.0245082i −0.873014 0.487696i \(-0.837838\pi\)
0.858864 + 0.512204i \(0.171171\pi\)
\(642\) 0 0
\(643\) −43.2936 −1.70733 −0.853667 0.520819i \(-0.825627\pi\)
−0.853667 + 0.520819i \(0.825627\pi\)
\(644\) 0 0
\(645\) 13.1943 0.519524
\(646\) 0 0
\(647\) −0.371451 + 0.643372i −0.0146032 + 0.0252936i −0.873235 0.487300i \(-0.837982\pi\)
0.858631 + 0.512593i \(0.171315\pi\)
\(648\) 0 0
\(649\) −0.360609 0.624593i −0.0141551 0.0245174i
\(650\) 0 0
\(651\) −34.8681 27.9242i −1.36659 1.09444i
\(652\) 0 0
\(653\) −25.0663 43.4161i −0.980919 1.69900i −0.658828 0.752294i \(-0.728948\pi\)
−0.322091 0.946709i \(-0.604386\pi\)
\(654\) 0 0
\(655\) −7.91296 + 13.7056i −0.309185 + 0.535524i
\(656\) 0 0
\(657\) 16.1123 0.628602
\(658\) 0 0
\(659\) 14.7410 0.574226 0.287113 0.957897i \(-0.407304\pi\)
0.287113 + 0.957897i \(0.407304\pi\)
\(660\) 0 0
\(661\) −7.79181 + 13.4958i −0.303066 + 0.524926i −0.976829 0.214022i \(-0.931344\pi\)
0.673763 + 0.738948i \(0.264677\pi\)
\(662\) 0 0
\(663\) −3.38646 5.86551i −0.131519 0.227798i
\(664\) 0 0
\(665\) 2.46027 16.0974i 0.0954053 0.624232i
\(666\) 0 0
\(667\) −3.32575 5.76037i −0.128774 0.223042i
\(668\) 0 0
\(669\) 29.5273 51.1428i 1.14159 1.97730i
\(670\) 0 0
\(671\) 7.73492 0.298603
\(672\) 0 0
\(673\) 8.12856 0.313333 0.156666 0.987652i \(-0.449925\pi\)
0.156666 + 0.987652i \(0.449925\pi\)
\(674\) 0 0
\(675\) −11.3140 + 19.5963i −0.435474 + 0.754263i
\(676\) 0 0
\(677\) 22.7593 + 39.4202i 0.874709 + 1.51504i 0.857072 + 0.515196i \(0.172281\pi\)
0.0176367 + 0.999844i \(0.494386\pi\)
\(678\) 0 0
\(679\) −32.9608 + 12.8578i −1.26492 + 0.493436i
\(680\) 0 0
\(681\) −1.56314 2.70744i −0.0598997 0.103749i
\(682\) 0 0
\(683\) 9.76614 16.9155i 0.373691 0.647252i −0.616439 0.787403i \(-0.711425\pi\)
0.990130 + 0.140151i \(0.0447587\pi\)
\(684\) 0 0
\(685\) 4.98242 0.190368
\(686\) 0 0
\(687\) 38.9040 1.48428
\(688\) 0 0
\(689\) −3.24012 + 5.61206i −0.123439 + 0.213802i
\(690\) 0 0
\(691\) 11.5181 + 19.9499i 0.438168 + 0.758930i 0.997548 0.0699814i \(-0.0222940\pi\)
−0.559380 + 0.828911i \(0.688961\pi\)
\(692\) 0 0
\(693\) 10.8991 4.25164i 0.414021 0.161506i
\(694\) 0 0
\(695\) 7.27015 + 12.5923i 0.275773 + 0.477652i
\(696\) 0 0
\(697\) −0.665195 + 1.15215i −0.0251961 + 0.0436409i
\(698\) 0 0
\(699\) −50.1692 −1.89758
\(700\) 0 0
\(701\) 13.2873 0.501854 0.250927 0.968006i \(-0.419265\pi\)
0.250927 + 0.968006i \(0.419265\pi\)
\(702\) 0 0
\(703\) −4.24719 + 7.35635i −0.160186 + 0.277450i
\(704\) 0 0
\(705\) 8.05241 + 13.9472i 0.303271 + 0.525281i
\(706\) 0 0
\(707\) 1.10180 7.20905i 0.0414376 0.271124i
\(708\) 0 0
\(709\) 12.5437 + 21.7263i 0.471089 + 0.815949i 0.999453 0.0330682i \(-0.0105279\pi\)
−0.528364 + 0.849018i \(0.677195\pi\)
\(710\) 0 0
\(711\) −21.1986 + 36.7171i −0.795010 + 1.37700i
\(712\) 0 0
\(713\) −4.80730 −0.180035
\(714\) 0 0
\(715\) 1.80805 0.0676172
\(716\) 0 0
\(717\) 7.41505 12.8432i 0.276920 0.479639i
\(718\) 0 0
\(719\) 7.22565 + 12.5152i 0.269471 + 0.466738i 0.968725 0.248135i \(-0.0798177\pi\)
−0.699254 + 0.714873i \(0.746484\pi\)
\(720\) 0 0
\(721\) −3.01820 2.41714i −0.112404 0.0900189i
\(722\) 0 0
\(723\) 33.5710 + 58.1468i 1.24852 + 2.16250i
\(724\) 0 0
\(725\) −14.4837 + 25.0865i −0.537910 + 0.931687i
\(726\) 0 0
\(727\) −13.3781 −0.496167 −0.248084 0.968739i \(-0.579801\pi\)
−0.248084 + 0.968739i \(0.579801\pi\)
\(728\) 0 0
\(729\) −41.2333 −1.52716
\(730\) 0 0
\(731\) 2.54666 4.41094i 0.0941915 0.163144i
\(732\) 0 0
\(733\) −16.9828 29.4151i −0.627275 1.08647i −0.988096 0.153837i \(-0.950837\pi\)
0.360821 0.932635i \(-0.382496\pi\)
\(734\) 0 0
\(735\) −17.7522 + 16.3359i −0.654800 + 0.602557i
\(736\) 0 0
\(737\) −3.55389 6.15552i −0.130909 0.226741i
\(738\) 0 0
\(739\) −3.80161 + 6.58459i −0.139845 + 0.242218i −0.927438 0.373978i \(-0.877994\pi\)
0.787593 + 0.616196i \(0.211327\pi\)
\(740\) 0 0
\(741\) 26.0670 0.957595
\(742\) 0 0
\(743\) 8.92565 0.327450 0.163725 0.986506i \(-0.447649\pi\)
0.163725 + 0.986506i \(0.447649\pi\)
\(744\) 0 0
\(745\) −4.26900 + 7.39413i −0.156404 + 0.270900i
\(746\) 0 0
\(747\) −10.1891 17.6481i −0.372801 0.645710i
\(748\) 0 0
\(749\) 26.0677 + 20.8764i 0.952492 + 0.762807i
\(750\) 0 0
\(751\) −5.54662 9.60703i −0.202399 0.350566i 0.746902 0.664934i \(-0.231540\pi\)
−0.949301 + 0.314369i \(0.898207\pi\)
\(752\) 0 0
\(753\) −6.29598 + 10.9050i −0.229438 + 0.397399i
\(754\) 0 0
\(755\) −11.6610 −0.424386
\(756\) 0 0
\(757\) 3.86787 0.140580 0.0702900 0.997527i \(-0.477608\pi\)
0.0702900 + 0.997527i \(0.477608\pi\)
\(758\) 0 0
\(759\) 0.991261 1.71691i 0.0359805 0.0623200i
\(760\) 0 0
\(761\) 24.9211 + 43.1646i 0.903390 + 1.56472i 0.823064 + 0.567949i \(0.192263\pi\)
0.0803258 + 0.996769i \(0.474404\pi\)
\(762\) 0 0
\(763\) −8.22552 + 53.8191i −0.297784 + 1.94838i
\(764\) 0 0
\(765\) −4.17398 7.22955i −0.150911 0.261385i
\(766\) 0 0
\(767\) −0.755912 + 1.30928i −0.0272944 + 0.0472753i
\(768\) 0 0
\(769\) 48.6248 1.75345 0.876727 0.480988i \(-0.159722\pi\)
0.876727 + 0.480988i \(0.159722\pi\)
\(770\) 0 0
\(771\) 55.4208 1.99593
\(772\) 0 0
\(773\) 11.8943 20.6015i 0.427808 0.740984i −0.568870 0.822427i \(-0.692619\pi\)
0.996678 + 0.0814427i \(0.0259528\pi\)
\(774\) 0 0
\(775\) 10.4679 + 18.1310i 0.376018 + 0.651283i
\(776\) 0 0
\(777\) 11.7237 4.57332i 0.420585 0.164067i
\(778\) 0 0
\(779\) −2.56015 4.43430i −0.0917268 0.158875i
\(780\) 0 0
\(781\) 0.705636 1.22220i 0.0252496 0.0437337i
\(782\) 0 0
\(783\) 51.8636 1.85345
\(784\) 0 0
\(785\) −3.51242 −0.125364
\(786\) 0 0
\(787\) −7.10531 + 12.3068i −0.253277 + 0.438689i −0.964426 0.264352i \(-0.914842\pi\)
0.711149 + 0.703041i \(0.248175\pi\)
\(788\) 0 0
\(789\) 6.14287 + 10.6398i 0.218692 + 0.378786i
\(790\) 0 0
\(791\) −10.1418 + 3.95622i −0.360599 + 0.140667i
\(792\) 0 0
\(793\) −8.10701 14.0417i −0.287888 0.498637i
\(794\) 0 0
\(795\) −6.28879 + 10.8925i −0.223040 + 0.386317i
\(796\) 0 0
\(797\) −52.4563 −1.85810 −0.929049 0.369956i \(-0.879373\pi\)
−0.929049 + 0.369956i \(0.879373\pi\)
\(798\) 0 0
\(799\) 6.21686 0.219937
\(800\) 0 0
\(801\) 24.8721 43.0797i 0.878812 1.52215i
\(802\) 0 0
\(803\) 1.30731 + 2.26433i 0.0461340 + 0.0799064i
\(804\) 0 0
\(805\) −0.392234 + 2.56636i −0.0138244 + 0.0904525i
\(806\) 0 0
\(807\) −14.6017 25.2909i −0.514005 0.890282i
\(808\) 0 0
\(809\) −18.1383 + 31.4164i −0.637707 + 1.10454i 0.348228 + 0.937410i \(0.386784\pi\)
−0.985935 + 0.167131i \(0.946550\pi\)
\(810\) 0 0
\(811\) −25.3129 −0.888857 −0.444428 0.895814i \(-0.646593\pi\)
−0.444428 + 0.895814i \(0.646593\pi\)
\(812\) 0 0
\(813\) 84.8496 2.97581
\(814\) 0 0
\(815\) −11.7635 + 20.3750i −0.412057 + 0.713704i
\(816\) 0 0
\(817\) 9.80135 + 16.9764i 0.342906 + 0.593930i
\(818\) 0 0
\(819\) −19.1417 15.3297i −0.668864 0.535662i
\(820\) 0 0
\(821\) 23.2657 + 40.2974i 0.811979 + 1.40639i 0.911477 + 0.411351i \(0.134943\pi\)
−0.0994979 + 0.995038i \(0.531724\pi\)
\(822\) 0 0
\(823\) 13.4627 23.3181i 0.469280 0.812817i −0.530103 0.847933i \(-0.677847\pi\)
0.999383 + 0.0351160i \(0.0111801\pi\)
\(824\) 0 0
\(825\) −8.63389 −0.300594
\(826\) 0 0
\(827\) −2.99334 −0.104089 −0.0520444 0.998645i \(-0.516574\pi\)
−0.0520444 + 0.998645i \(0.516574\pi\)
\(828\) 0 0
\(829\) −12.3597 + 21.4076i −0.429270 + 0.743518i −0.996809 0.0798290i \(-0.974563\pi\)
0.567538 + 0.823347i \(0.307896\pi\)
\(830\) 0 0
\(831\) 8.61637 + 14.9240i 0.298899 + 0.517707i
\(832\) 0 0
\(833\) 2.03480 + 9.08772i 0.0705016 + 0.314871i
\(834\) 0 0
\(835\) 8.93532 + 15.4764i 0.309219 + 0.535584i
\(836\) 0 0
\(837\) 18.7419 32.4619i 0.647815 1.12205i
\(838\) 0 0
\(839\) −6.82342 −0.235571 −0.117785 0.993039i \(-0.537579\pi\)
−0.117785 + 0.993039i \(0.537579\pi\)
\(840\) 0 0
\(841\) 37.3937 1.28944
\(842\) 0 0
\(843\) −35.0305 + 60.6747i −1.20652 + 2.08975i
\(844\) 0 0
\(845\) 5.91838 + 10.2509i 0.203598 + 0.352643i
\(846\) 0 0
\(847\) −21.2345 17.0058i −0.729628 0.584325i
\(848\) 0 0
\(849\) −0.561464 0.972483i −0.0192694 0.0333755i
\(850\) 0 0
\(851\) 0.677116 1.17280i 0.0232112 0.0402030i
\(852\) 0 0
\(853\) −7.11780 −0.243709 −0.121854 0.992548i \(-0.538884\pi\)
−0.121854 + 0.992548i \(0.538884\pi\)
\(854\) 0 0
\(855\) 32.1289 1.09879
\(856\) 0 0
\(857\) 14.6216 25.3253i 0.499464 0.865097i −0.500536 0.865716i \(-0.666864\pi\)
1.00000 0.000619066i \(0.000197055\pi\)
\(858\) 0 0
\(859\) 26.3171 + 45.5825i 0.897927 + 1.55525i 0.830140 + 0.557555i \(0.188260\pi\)
0.0677863 + 0.997700i \(0.478406\pi\)
\(860\) 0 0
\(861\) −1.14604 + 7.49846i −0.0390568 + 0.255547i
\(862\) 0 0
\(863\) −10.6440 18.4360i −0.362326 0.627567i 0.626017 0.779809i \(-0.284684\pi\)
−0.988343 + 0.152242i \(0.951351\pi\)
\(864\) 0 0
\(865\) −9.87133 + 17.0977i −0.335635 + 0.581338i
\(866\) 0 0
\(867\) 43.6655 1.48296
\(868\) 0 0
\(869\) −6.87999 −0.233388
\(870\) 0 0
\(871\) −7.44970 + 12.9033i −0.252423 + 0.437210i
\(872\) 0 0
\(873\) −34.9021 60.4523i −1.18126 2.04600i
\(874\) 0 0
\(875\) 25.3478 9.88797i 0.856911 0.334274i
\(876\) 0 0
\(877\) −20.7075 35.8664i −0.699242 1.21112i −0.968729 0.248119i \(-0.920187\pi\)
0.269487 0.963004i \(-0.413146\pi\)
\(878\) 0 0
\(879\) 1.61320 2.79415i 0.0544119 0.0942442i
\(880\) 0 0
\(881\) 28.0441 0.944831 0.472415 0.881376i \(-0.343382\pi\)
0.472415 + 0.881376i \(0.343382\pi\)
\(882\) 0 0
\(883\) 42.7112 1.43735 0.718674 0.695347i \(-0.244750\pi\)
0.718674 + 0.695347i \(0.244750\pi\)
\(884\) 0 0
\(885\) −1.46716 + 2.54119i −0.0493179 + 0.0854212i
\(886\) 0 0
\(887\) 13.7012 + 23.7311i 0.460040 + 0.796813i 0.998962 0.0455427i \(-0.0145017\pi\)
−0.538922 + 0.842355i \(0.681168\pi\)
\(888\) 0 0
\(889\) −4.85176 + 1.89263i −0.162723 + 0.0634768i
\(890\) 0 0
\(891\) 1.09644 + 1.89908i 0.0367320 + 0.0636216i
\(892\) 0 0
\(893\) −11.9635 + 20.7213i −0.400342 + 0.693412i
\(894\) 0 0
\(895\) −30.9668 −1.03511
\(896\) 0 0
\(897\) −4.15578 −0.138757
\(898\) 0 0
\(899\) 23.9926 41.5565i 0.800200 1.38599i
\(900\) 0 0
\(901\) 2.42763 + 4.20477i 0.0808760 + 0.140081i
\(902\) 0 0
\(903\) 4.38753 28.7074i 0.146008 0.955321i
\(904\) 0 0
\(905\) −1.27000 2.19971i −0.0422163 0.0731208i
\(906\) 0 0
\(907\) 18.9462 32.8158i 0.629099 1.08963i −0.358634 0.933478i \(-0.616757\pi\)
0.987733 0.156153i \(-0.0499092\pi\)
\(908\) 0 0
\(909\) 14.3886 0.477238
\(910\) 0 0
\(911\) 32.0315 1.06125 0.530626 0.847606i \(-0.321957\pi\)
0.530626 + 0.847606i \(0.321957\pi\)
\(912\) 0 0
\(913\) 1.65344 2.86383i 0.0547207 0.0947791i
\(914\) 0 0
\(915\) −15.7350 27.2537i −0.520182 0.900981i
\(916\) 0 0
\(917\) 27.1887 + 21.7742i 0.897849 + 0.719046i
\(918\) 0 0
\(919\) 5.55296 + 9.61802i 0.183175 + 0.317269i 0.942960 0.332906i \(-0.108029\pi\)
−0.759785 + 0.650175i \(0.774696\pi\)
\(920\) 0 0
\(921\) −15.5412 + 26.9181i −0.512100 + 0.886983i
\(922\) 0 0
\(923\) −2.95832 −0.0973744
\(924\) 0 0
\(925\) −5.89769 −0.193915
\(926\) 0 0
\(927\) 3.81458 6.60704i 0.125287 0.217004i
\(928\) 0 0
\(929\) 5.47383 + 9.48096i 0.179591 + 0.311060i 0.941740 0.336341i \(-0.109189\pi\)
−0.762150 + 0.647401i \(0.775856\pi\)
\(930\) 0 0
\(931\) −34.2058 10.7059i −1.12105 0.350870i
\(932\) 0 0
\(933\) −8.10333 14.0354i −0.265291 0.459497i
\(934\) 0 0
\(935\) 0.677331 1.17317i 0.0221511 0.0383668i
\(936\) 0 0
\(937\) 31.2474 1.02081 0.510404 0.859935i \(-0.329496\pi\)
0.510404 + 0.859935i \(0.329496\pi\)
\(938\) 0 0
\(939\) 15.5185 0.506429
\(940\) 0 0
\(941\) −30.0361 + 52.0240i −0.979147 + 1.69593i −0.313639 + 0.949542i \(0.601548\pi\)
−0.665508 + 0.746391i \(0.731785\pi\)
\(942\) 0 0
\(943\) 0.408156 + 0.706947i 0.0132914 + 0.0230214i
\(944\) 0 0
\(945\) −15.8006 12.6539i −0.513992 0.411632i
\(946\) 0 0
\(947\) 14.2131 + 24.6178i 0.461864 + 0.799971i 0.999054 0.0434899i \(-0.0138476\pi\)
−0.537190 + 0.843461i \(0.680514\pi\)
\(948\) 0 0
\(949\) 2.74040 4.74651i 0.0889571 0.154078i
\(950\) 0 0
\(951\) 70.8975 2.29901
\(952\) 0 0
\(953\) 8.26316 0.267670 0.133835 0.991004i \(-0.457271\pi\)
0.133835 + 0.991004i \(0.457271\pi\)
\(954\) 0 0
\(955\) −11.1376 + 19.2909i −0.360405 + 0.624239i
\(956\) 0 0
\(957\) 9.89452 + 17.1378i 0.319845 + 0.553987i
\(958\) 0 0
\(959\) 1.65682 10.8405i 0.0535014 0.350057i
\(960\) 0 0
\(961\) −1.84042 3.18769i −0.0593683 0.102829i
\(962\) 0 0
\(963\) −32.9458 + 57.0639i −1.06166 + 1.83886i
\(964\) 0 0
\(965\) 3.43878 0.110698
\(966\) 0 0
\(967\) −2.49199 −0.0801369 −0.0400684 0.999197i \(-0.512758\pi\)
−0.0400684 + 0.999197i \(0.512758\pi\)
\(968\) 0 0
\(969\) 9.76520 16.9138i 0.313703 0.543350i
\(970\) 0 0
\(971\) −29.2701 50.6974i −0.939323 1.62696i −0.766737 0.641962i \(-0.778121\pi\)
−0.172587 0.984994i \(-0.555212\pi\)
\(972\) 0 0
\(973\) 29.8151 11.6307i 0.955830 0.372862i
\(974\) 0 0
\(975\) 9.04923 + 15.6737i 0.289807 + 0.501961i
\(976\) 0 0
\(977\) 18.2617 31.6302i 0.584244 1.01194i −0.410726 0.911759i \(-0.634725\pi\)
0.994969 0.100181i \(-0.0319421\pi\)
\(978\) 0 0
\(979\) 8.07221 0.257989
\(980\) 0 0
\(981\) −107.418 −3.42959
\(982\) 0 0
\(983\) −26.3570 + 45.6516i −0.840657 + 1.45606i 0.0486839 + 0.998814i \(0.484497\pi\)
−0.889340 + 0.457246i \(0.848836\pi\)
\(984\) 0 0
\(985\) −4.83842 8.38040i −0.154165 0.267022i
\(986\) 0 0
\(987\) 33.0232 12.8821i 1.05114 0.410042i
\(988\) 0 0
\(989\) −1.56260 2.70650i −0.0496877 0.0860617i
\(990\) 0 0
\(991\) 7.00757 12.1375i 0.222603 0.385559i −0.732995 0.680234i \(-0.761878\pi\)
0.955598 + 0.294675i \(0.0952114\pi\)
\(992\) 0 0
\(993\) 51.2741 1.62713
\(994\) 0 0
\(995\) −0.950580 −0.0301354
\(996\) 0 0
\(997\) 29.1037 50.4092i 0.921725 1.59647i 0.124979 0.992159i \(-0.460114\pi\)
0.796746 0.604315i \(-0.206553\pi\)
\(998\) 0 0
\(999\) 5.27966 + 9.14464i 0.167041 + 0.289324i
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 1148.2.i.d.165.8 16
7.2 even 3 inner 1148.2.i.d.821.8 yes 16
7.3 odd 6 8036.2.a.n.1.8 8
7.4 even 3 8036.2.a.m.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1148.2.i.d.165.8 16 1.1 even 1 trivial
1148.2.i.d.821.8 yes 16 7.2 even 3 inner
8036.2.a.m.1.1 8 7.4 even 3
8036.2.a.n.1.8 8 7.3 odd 6