Properties

Label 960.2.b.c.671.12
Level $960$
Weight $2$
Character 960.671
Analytic conductor $7.666$
Analytic rank $0$
Dimension $12$
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] = [960,2,Mod(671,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.671");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.b (of order \(2\), degree \(1\), 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: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 671.12
Root \(2.17840 + 0.583700i\) of defining polynomial
Character \(\chi\) \(=\) 960.671
Dual form 960.2.b.c.671.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.59470 + 0.675970i) q^{3} -1.00000 q^{5} +0.648061i q^{7} +(2.08613 + 2.15594i) q^{9} +O(q^{10})\) \(q+(1.59470 + 0.675970i) q^{3} -1.00000 q^{5} +0.648061i q^{7} +(2.08613 + 2.15594i) q^{9} +4.17226i q^{11} -0.847771i q^{13} +(-1.59470 - 0.675970i) q^{15} +2.91469i q^{17} -0.847771 q^{19} +(-0.438069 + 1.03346i) q^{21} +2.34163 q^{23} +1.00000 q^{25} +(1.86940 + 4.84823i) q^{27} -6.87614 q^{29} +2.17226i q^{31} +(-2.82032 + 6.65350i) q^{33} -0.648061i q^{35} +5.53103i q^{37} +(0.573067 - 1.35194i) q^{39} -4.31187i q^{41} +9.56819 q^{43} +(-2.08613 - 2.15594i) q^{45} +10.9654 q^{47} +6.58002 q^{49} +(-1.97024 + 4.64806i) q^{51} -11.6406 q^{53} -4.17226i q^{55} +(-1.35194 - 0.573067i) q^{57} +6.87614i q^{59} +10.6907i q^{61} +(-1.39718 + 1.35194i) q^{63} +0.847771i q^{65} -5.43435 q^{67} +(3.73419 + 1.58287i) q^{69} +13.3070 q^{71} -13.0484 q^{73} +(1.59470 + 0.675970i) q^{75} -2.70388 q^{77} -17.2207i q^{79} +(-0.296122 + 8.99513i) q^{81} -10.6481i q^{83} -2.91469i q^{85} +(-10.9654 - 4.64806i) q^{87} -17.9903i q^{89} +0.549407 q^{91} +(-1.46838 + 3.46410i) q^{93} +0.847771 q^{95} +4.70388 q^{97} +(-8.99513 + 8.70388i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{5} - 4 q^{9} + 32 q^{21} + 12 q^{25} - 8 q^{29} + 16 q^{33} + 4 q^{45} - 12 q^{49} - 40 q^{53} - 8 q^{57} + 24 q^{69} - 24 q^{73} - 16 q^{77} - 20 q^{81} + 24 q^{93} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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
\(3\) 1.59470 + 0.675970i 0.920700 + 0.390271i
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.648061i 0.244944i 0.992472 + 0.122472i \(0.0390822\pi\)
−0.992472 + 0.122472i \(0.960918\pi\)
\(8\) 0 0
\(9\) 2.08613 + 2.15594i 0.695377 + 0.718645i
\(10\) 0 0
\(11\) 4.17226i 1.25798i 0.777412 + 0.628992i \(0.216532\pi\)
−0.777412 + 0.628992i \(0.783468\pi\)
\(12\) 0 0
\(13\) 0.847771i 0.235129i −0.993065 0.117565i \(-0.962491\pi\)
0.993065 0.117565i \(-0.0375087\pi\)
\(14\) 0 0
\(15\) −1.59470 0.675970i −0.411750 0.174535i
\(16\) 0 0
\(17\) 2.91469i 0.706917i 0.935450 + 0.353459i \(0.114994\pi\)
−0.935450 + 0.353459i \(0.885006\pi\)
\(18\) 0 0
\(19\) −0.847771 −0.194492 −0.0972460 0.995260i \(-0.531003\pi\)
−0.0972460 + 0.995260i \(0.531003\pi\)
\(20\) 0 0
\(21\) −0.438069 + 1.03346i −0.0955946 + 0.225520i
\(22\) 0 0
\(23\) 2.34163 0.488263 0.244132 0.969742i \(-0.421497\pi\)
0.244132 + 0.969742i \(0.421497\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.86940 + 4.84823i 0.359767 + 0.933042i
\(28\) 0 0
\(29\) −6.87614 −1.27687 −0.638433 0.769677i \(-0.720417\pi\)
−0.638433 + 0.769677i \(0.720417\pi\)
\(30\) 0 0
\(31\) 2.17226i 0.390149i 0.980788 + 0.195075i \(0.0624949\pi\)
−0.980788 + 0.195075i \(0.937505\pi\)
\(32\) 0 0
\(33\) −2.82032 + 6.65350i −0.490955 + 1.15823i
\(34\) 0 0
\(35\) 0.648061i 0.109542i
\(36\) 0 0
\(37\) 5.53103i 0.909295i 0.890671 + 0.454647i \(0.150235\pi\)
−0.890671 + 0.454647i \(0.849765\pi\)
\(38\) 0 0
\(39\) 0.573067 1.35194i 0.0917642 0.216484i
\(40\) 0 0
\(41\) 4.31187i 0.673401i −0.941612 0.336701i \(-0.890689\pi\)
0.941612 0.336701i \(-0.109311\pi\)
\(42\) 0 0
\(43\) 9.56819 1.45914 0.729568 0.683908i \(-0.239721\pi\)
0.729568 + 0.683908i \(0.239721\pi\)
\(44\) 0 0
\(45\) −2.08613 2.15594i −0.310982 0.321388i
\(46\) 0 0
\(47\) 10.9654 1.59946 0.799732 0.600357i \(-0.204975\pi\)
0.799732 + 0.600357i \(0.204975\pi\)
\(48\) 0 0
\(49\) 6.58002 0.940002
\(50\) 0 0
\(51\) −1.97024 + 4.64806i −0.275889 + 0.650859i
\(52\) 0 0
\(53\) −11.6406 −1.59897 −0.799483 0.600689i \(-0.794893\pi\)
−0.799483 + 0.600689i \(0.794893\pi\)
\(54\) 0 0
\(55\) 4.17226i 0.562587i
\(56\) 0 0
\(57\) −1.35194 0.573067i −0.179069 0.0759046i
\(58\) 0 0
\(59\) 6.87614i 0.895197i 0.894235 + 0.447599i \(0.147721\pi\)
−0.894235 + 0.447599i \(0.852279\pi\)
\(60\) 0 0
\(61\) 10.6907i 1.36880i 0.729107 + 0.684400i \(0.239936\pi\)
−0.729107 + 0.684400i \(0.760064\pi\)
\(62\) 0 0
\(63\) −1.39718 + 1.35194i −0.176028 + 0.170328i
\(64\) 0 0
\(65\) 0.847771i 0.105153i
\(66\) 0 0
\(67\) −5.43435 −0.663911 −0.331956 0.943295i \(-0.607708\pi\)
−0.331956 + 0.943295i \(0.607708\pi\)
\(68\) 0 0
\(69\) 3.73419 + 1.58287i 0.449544 + 0.190555i
\(70\) 0 0
\(71\) 13.3070 1.57925 0.789625 0.613590i \(-0.210275\pi\)
0.789625 + 0.613590i \(0.210275\pi\)
\(72\) 0 0
\(73\) −13.0484 −1.52720 −0.763600 0.645690i \(-0.776570\pi\)
−0.763600 + 0.645690i \(0.776570\pi\)
\(74\) 0 0
\(75\) 1.59470 + 0.675970i 0.184140 + 0.0780542i
\(76\) 0 0
\(77\) −2.70388 −0.308136
\(78\) 0 0
\(79\) 17.2207i 1.93748i −0.248087 0.968738i \(-0.579802\pi\)
0.248087 0.968738i \(-0.420198\pi\)
\(80\) 0 0
\(81\) −0.296122 + 8.99513i −0.0329024 + 0.999459i
\(82\) 0 0
\(83\) 10.6481i 1.16878i −0.811474 0.584388i \(-0.801335\pi\)
0.811474 0.584388i \(-0.198665\pi\)
\(84\) 0 0
\(85\) 2.91469i 0.316143i
\(86\) 0 0
\(87\) −10.9654 4.64806i −1.17561 0.498324i
\(88\) 0 0
\(89\) 17.9903i 1.90696i −0.301451 0.953482i \(-0.597471\pi\)
0.301451 0.953482i \(-0.402529\pi\)
\(90\) 0 0
\(91\) 0.549407 0.0575935
\(92\) 0 0
\(93\) −1.46838 + 3.46410i −0.152264 + 0.359211i
\(94\) 0 0
\(95\) 0.847771 0.0869794
\(96\) 0 0
\(97\) 4.70388 0.477606 0.238803 0.971068i \(-0.423245\pi\)
0.238803 + 0.971068i \(0.423245\pi\)
\(98\) 0 0
\(99\) −8.99513 + 8.70388i −0.904044 + 0.874773i
\(100\) 0 0
\(101\) 1.46838 0.146109 0.0730547 0.997328i \(-0.476725\pi\)
0.0730547 + 0.997328i \(0.476725\pi\)
\(102\) 0 0
\(103\) 7.69646i 0.758355i 0.925324 + 0.379177i \(0.123793\pi\)
−0.925324 + 0.379177i \(0.876207\pi\)
\(104\) 0 0
\(105\) 0.438069 1.03346i 0.0427512 0.100856i
\(106\) 0 0
\(107\) 1.35194i 0.130697i −0.997863 0.0653484i \(-0.979184\pi\)
0.997863 0.0653484i \(-0.0208159\pi\)
\(108\) 0 0
\(109\) 17.6189i 1.68758i −0.536672 0.843791i \(-0.680319\pi\)
0.536672 0.843791i \(-0.319681\pi\)
\(110\) 0 0
\(111\) −3.73881 + 8.82032i −0.354872 + 0.837188i
\(112\) 0 0
\(113\) 7.59795i 0.714755i 0.933960 + 0.357377i \(0.116329\pi\)
−0.933960 + 0.357377i \(0.883671\pi\)
\(114\) 0 0
\(115\) −2.34163 −0.218358
\(116\) 0 0
\(117\) 1.82774 1.76856i 0.168975 0.163503i
\(118\) 0 0
\(119\) −1.88890 −0.173155
\(120\) 0 0
\(121\) −6.40776 −0.582523
\(122\) 0 0
\(123\) 2.91469 6.87614i 0.262809 0.620001i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 11.6965i 1.03789i 0.854807 + 0.518946i \(0.173676\pi\)
−0.854807 + 0.518946i \(0.826324\pi\)
\(128\) 0 0
\(129\) 15.2584 + 6.46781i 1.34343 + 0.569459i
\(130\) 0 0
\(131\) 9.58002i 0.837010i −0.908214 0.418505i \(-0.862554\pi\)
0.908214 0.418505i \(-0.137446\pi\)
\(132\) 0 0
\(133\) 0.549407i 0.0476396i
\(134\) 0 0
\(135\) −1.86940 4.84823i −0.160893 0.417269i
\(136\) 0 0
\(137\) 16.2217i 1.38591i −0.720979 0.692956i \(-0.756308\pi\)
0.720979 0.692956i \(-0.243692\pi\)
\(138\) 0 0
\(139\) −4.98162 −0.422535 −0.211268 0.977428i \(-0.567759\pi\)
−0.211268 + 0.977428i \(0.567759\pi\)
\(140\) 0 0
\(141\) 17.4865 + 7.41226i 1.47263 + 0.624225i
\(142\) 0 0
\(143\) 3.53712 0.295789
\(144\) 0 0
\(145\) 6.87614 0.571032
\(146\) 0 0
\(147\) 10.4931 + 4.44789i 0.865460 + 0.366856i
\(148\) 0 0
\(149\) 2.34452 0.192071 0.0960353 0.995378i \(-0.469384\pi\)
0.0960353 + 0.995378i \(0.469384\pi\)
\(150\) 0 0
\(151\) 11.5800i 0.942368i 0.882035 + 0.471184i \(0.156173\pi\)
−0.882035 + 0.471184i \(0.843827\pi\)
\(152\) 0 0
\(153\) −6.28390 + 6.08043i −0.508023 + 0.491574i
\(154\) 0 0
\(155\) 2.17226i 0.174480i
\(156\) 0 0
\(157\) 1.39718i 0.111507i 0.998445 + 0.0557535i \(0.0177561\pi\)
−0.998445 + 0.0557535i \(0.982244\pi\)
\(158\) 0 0
\(159\) −18.5633 7.86872i −1.47217 0.624030i
\(160\) 0 0
\(161\) 1.51752i 0.119597i
\(162\) 0 0
\(163\) 11.8131 0.925277 0.462638 0.886547i \(-0.346903\pi\)
0.462638 + 0.886547i \(0.346903\pi\)
\(164\) 0 0
\(165\) 2.82032 6.65350i 0.219562 0.517974i
\(166\) 0 0
\(167\) −6.28212 −0.486125 −0.243062 0.970011i \(-0.578152\pi\)
−0.243062 + 0.970011i \(0.578152\pi\)
\(168\) 0 0
\(169\) 12.2813 0.944714
\(170\) 0 0
\(171\) −1.76856 1.82774i −0.135245 0.139771i
\(172\) 0 0
\(173\) 16.0968 1.22382 0.611908 0.790929i \(-0.290402\pi\)
0.611908 + 0.790929i \(0.290402\pi\)
\(174\) 0 0
\(175\) 0.648061i 0.0489888i
\(176\) 0 0
\(177\) −4.64806 + 10.9654i −0.349370 + 0.824208i
\(178\) 0 0
\(179\) 13.4684i 1.00667i −0.864090 0.503337i \(-0.832105\pi\)
0.864090 0.503337i \(-0.167895\pi\)
\(180\) 0 0
\(181\) 11.0621i 0.822236i −0.911582 0.411118i \(-0.865138\pi\)
0.911582 0.411118i \(-0.134862\pi\)
\(182\) 0 0
\(183\) −7.22657 + 17.0484i −0.534203 + 1.26025i
\(184\) 0 0
\(185\) 5.53103i 0.406649i
\(186\) 0 0
\(187\) −12.1609 −0.889291
\(188\) 0 0
\(189\) −3.14195 + 1.21149i −0.228543 + 0.0881227i
\(190\) 0 0
\(191\) −1.14613 −0.0829314 −0.0414657 0.999140i \(-0.513203\pi\)
−0.0414657 + 0.999140i \(0.513203\pi\)
\(192\) 0 0
\(193\) −17.3929 −1.25197 −0.625985 0.779835i \(-0.715303\pi\)
−0.625985 + 0.779835i \(0.715303\pi\)
\(194\) 0 0
\(195\) −0.573067 + 1.35194i −0.0410382 + 0.0968144i
\(196\) 0 0
\(197\) 10.4562 0.744970 0.372485 0.928038i \(-0.378506\pi\)
0.372485 + 0.928038i \(0.378506\pi\)
\(198\) 0 0
\(199\) 14.2839i 1.01256i −0.862370 0.506279i \(-0.831021\pi\)
0.862370 0.506279i \(-0.168979\pi\)
\(200\) 0 0
\(201\) −8.66615 3.67345i −0.611263 0.259105i
\(202\) 0 0
\(203\) 4.45616i 0.312761i
\(204\) 0 0
\(205\) 4.31187i 0.301154i
\(206\) 0 0
\(207\) 4.88494 + 5.04840i 0.339527 + 0.350888i
\(208\) 0 0
\(209\) 3.53712i 0.244668i
\(210\) 0 0
\(211\) 8.92211 0.614223 0.307112 0.951673i \(-0.400637\pi\)
0.307112 + 0.951673i \(0.400637\pi\)
\(212\) 0 0
\(213\) 21.2207 + 8.99513i 1.45402 + 0.616336i
\(214\) 0 0
\(215\) −9.56819 −0.652545
\(216\) 0 0
\(217\) −1.40776 −0.0955648
\(218\) 0 0
\(219\) −20.8083 8.82032i −1.40609 0.596022i
\(220\) 0 0
\(221\) 2.47099 0.166217
\(222\) 0 0
\(223\) 17.1042i 1.14538i 0.819771 + 0.572692i \(0.194101\pi\)
−0.819771 + 0.572692i \(0.805899\pi\)
\(224\) 0 0
\(225\) 2.08613 + 2.15594i 0.139075 + 0.143729i
\(226\) 0 0
\(227\) 13.3519i 0.886199i −0.896472 0.443100i \(-0.853879\pi\)
0.896472 0.443100i \(-0.146121\pi\)
\(228\) 0 0
\(229\) 17.9903i 1.18883i −0.804159 0.594415i \(-0.797384\pi\)
0.804159 0.594415i \(-0.202616\pi\)
\(230\) 0 0
\(231\) −4.31187 1.82774i −0.283700 0.120256i
\(232\) 0 0
\(233\) 20.1622i 1.32087i −0.750884 0.660434i \(-0.770372\pi\)
0.750884 0.660434i \(-0.229628\pi\)
\(234\) 0 0
\(235\) −10.9654 −0.715302
\(236\) 0 0
\(237\) 11.6406 27.4618i 0.756141 1.78383i
\(238\) 0 0
\(239\) −12.1609 −0.786621 −0.393310 0.919406i \(-0.628670\pi\)
−0.393310 + 0.919406i \(0.628670\pi\)
\(240\) 0 0
\(241\) −8.51678 −0.548614 −0.274307 0.961642i \(-0.588448\pi\)
−0.274307 + 0.961642i \(0.588448\pi\)
\(242\) 0 0
\(243\) −6.55266 + 14.1444i −0.420353 + 0.907361i
\(244\) 0 0
\(245\) −6.58002 −0.420382
\(246\) 0 0
\(247\) 0.718715i 0.0457308i
\(248\) 0 0
\(249\) 7.19777 16.9805i 0.456140 1.07609i
\(250\) 0 0
\(251\) 12.5168i 0.790052i −0.918670 0.395026i \(-0.870736\pi\)
0.918670 0.395026i \(-0.129264\pi\)
\(252\) 0 0
\(253\) 9.76988i 0.614227i
\(254\) 0 0
\(255\) 1.97024 4.64806i 0.123382 0.291073i
\(256\) 0 0
\(257\) 1.76856i 0.110320i 0.998478 + 0.0551599i \(0.0175668\pi\)
−0.998478 + 0.0551599i \(0.982433\pi\)
\(258\) 0 0
\(259\) −3.58444 −0.222726
\(260\) 0 0
\(261\) −14.3445 14.8245i −0.887904 0.917615i
\(262\) 0 0
\(263\) 30.1018 1.85615 0.928077 0.372388i \(-0.121461\pi\)
0.928077 + 0.372388i \(0.121461\pi\)
\(264\) 0 0
\(265\) 11.6406 0.715079
\(266\) 0 0
\(267\) 12.1609 28.6890i 0.744233 1.75574i
\(268\) 0 0
\(269\) −3.06324 −0.186769 −0.0933844 0.995630i \(-0.529769\pi\)
−0.0933844 + 0.995630i \(0.529769\pi\)
\(270\) 0 0
\(271\) 8.76450i 0.532406i −0.963917 0.266203i \(-0.914231\pi\)
0.963917 0.266203i \(-0.0857691\pi\)
\(272\) 0 0
\(273\) 0.876139 + 0.371382i 0.0530263 + 0.0224771i
\(274\) 0 0
\(275\) 4.17226i 0.251597i
\(276\) 0 0
\(277\) 28.0112i 1.68303i 0.540235 + 0.841514i \(0.318335\pi\)
−0.540235 + 0.841514i \(0.681665\pi\)
\(278\) 0 0
\(279\) −4.68325 + 4.53162i −0.280379 + 0.271301i
\(280\) 0 0
\(281\) 10.1413i 0.604977i −0.953153 0.302488i \(-0.902183\pi\)
0.953153 0.302488i \(-0.0978174\pi\)
\(282\) 0 0
\(283\) −4.88494 −0.290380 −0.145190 0.989404i \(-0.546379\pi\)
−0.145190 + 0.989404i \(0.546379\pi\)
\(284\) 0 0
\(285\) 1.35194 + 0.573067i 0.0800820 + 0.0339456i
\(286\) 0 0
\(287\) 2.79436 0.164946
\(288\) 0 0
\(289\) 8.50456 0.500268
\(290\) 0 0
\(291\) 7.50127 + 3.17968i 0.439732 + 0.186396i
\(292\) 0 0
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) 6.87614i 0.400344i
\(296\) 0 0
\(297\) −20.2281 + 7.79963i −1.17375 + 0.452581i
\(298\) 0 0
\(299\) 1.98516i 0.114805i
\(300\) 0 0
\(301\) 6.20077i 0.357407i
\(302\) 0 0
\(303\) 2.34163 + 0.992582i 0.134523 + 0.0570223i
\(304\) 0 0
\(305\) 10.6907i 0.612146i
\(306\) 0 0
\(307\) −8.42206 −0.480672 −0.240336 0.970690i \(-0.577258\pi\)
−0.240336 + 0.970690i \(0.577258\pi\)
\(308\) 0 0
\(309\) −5.20257 + 12.2735i −0.295964 + 0.698217i
\(310\) 0 0
\(311\) −21.9307 −1.24358 −0.621789 0.783185i \(-0.713594\pi\)
−0.621789 + 0.783185i \(0.713594\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) 1.39718 1.35194i 0.0787220 0.0761731i
\(316\) 0 0
\(317\) −17.0484 −0.957533 −0.478767 0.877942i \(-0.658916\pi\)
−0.478767 + 0.877942i \(0.658916\pi\)
\(318\) 0 0
\(319\) 28.6890i 1.60628i
\(320\) 0 0
\(321\) 0.913870 2.15594i 0.0510072 0.120333i
\(322\) 0 0
\(323\) 2.47099i 0.137490i
\(324\) 0 0
\(325\) 0.847771i 0.0470259i
\(326\) 0 0
\(327\) 11.9098 28.0968i 0.658615 1.55376i
\(328\) 0 0
\(329\) 7.10623i 0.391779i
\(330\) 0 0
\(331\) 2.73667 0.150421 0.0752105 0.997168i \(-0.476037\pi\)
0.0752105 + 0.997168i \(0.476037\pi\)
\(332\) 0 0
\(333\) −11.9245 + 11.5384i −0.653461 + 0.632303i
\(334\) 0 0
\(335\) 5.43435 0.296910
\(336\) 0 0
\(337\) 5.65548 0.308074 0.154037 0.988065i \(-0.450773\pi\)
0.154037 + 0.988065i \(0.450773\pi\)
\(338\) 0 0
\(339\) −5.13598 + 12.1164i −0.278948 + 0.658075i
\(340\) 0 0
\(341\) −9.06324 −0.490802
\(342\) 0 0
\(343\) 8.80068i 0.475192i
\(344\) 0 0
\(345\) −3.73419 1.58287i −0.201042 0.0852188i
\(346\) 0 0
\(347\) 16.0558i 0.861921i 0.902371 + 0.430961i \(0.141825\pi\)
−0.902371 + 0.430961i \(0.858175\pi\)
\(348\) 0 0
\(349\) 25.5152i 1.36580i −0.730513 0.682898i \(-0.760719\pi\)
0.730513 0.682898i \(-0.239281\pi\)
\(350\) 0 0
\(351\) 4.11019 1.58482i 0.219386 0.0845917i
\(352\) 0 0
\(353\) 16.2217i 0.863394i 0.902019 + 0.431697i \(0.142085\pi\)
−0.902019 + 0.431697i \(0.857915\pi\)
\(354\) 0 0
\(355\) −13.3070 −0.706262
\(356\) 0 0
\(357\) −3.01223 1.27684i −0.159424 0.0675775i
\(358\) 0 0
\(359\) 23.0769 1.21795 0.608976 0.793189i \(-0.291581\pi\)
0.608976 + 0.793189i \(0.291581\pi\)
\(360\) 0 0
\(361\) −18.2813 −0.962173
\(362\) 0 0
\(363\) −10.2184 4.33145i −0.536329 0.227342i
\(364\) 0 0
\(365\) 13.0484 0.682984
\(366\) 0 0
\(367\) 19.9293i 1.04030i 0.854074 + 0.520152i \(0.174125\pi\)
−0.854074 + 0.520152i \(0.825875\pi\)
\(368\) 0 0
\(369\) 9.29612 8.99513i 0.483937 0.468268i
\(370\) 0 0
\(371\) 7.54384i 0.391657i
\(372\) 0 0
\(373\) 19.9842i 1.03474i 0.855762 + 0.517370i \(0.173089\pi\)
−0.855762 + 0.517370i \(0.826911\pi\)
\(374\) 0 0
\(375\) −1.59470 0.675970i −0.0823499 0.0349069i
\(376\) 0 0
\(377\) 5.82939i 0.300229i
\(378\) 0 0
\(379\) 18.8380 0.967644 0.483822 0.875166i \(-0.339248\pi\)
0.483822 + 0.875166i \(0.339248\pi\)
\(380\) 0 0
\(381\) −7.90645 + 18.6523i −0.405060 + 0.955588i
\(382\) 0 0
\(383\) −9.81924 −0.501740 −0.250870 0.968021i \(-0.580717\pi\)
−0.250870 + 0.968021i \(0.580717\pi\)
\(384\) 0 0
\(385\) 2.70388 0.137802
\(386\) 0 0
\(387\) 19.9605 + 20.6284i 1.01465 + 1.04860i
\(388\) 0 0
\(389\) −4.09680 −0.207716 −0.103858 0.994592i \(-0.533119\pi\)
−0.103858 + 0.994592i \(0.533119\pi\)
\(390\) 0 0
\(391\) 6.82513i 0.345162i
\(392\) 0 0
\(393\) 6.47580 15.2772i 0.326661 0.770635i
\(394\) 0 0
\(395\) 17.2207i 0.866465i
\(396\) 0 0
\(397\) 5.57835i 0.279969i −0.990154 0.139985i \(-0.955295\pi\)
0.990154 0.139985i \(-0.0447053\pi\)
\(398\) 0 0
\(399\) 0.371382 0.876139i 0.0185924 0.0438618i
\(400\) 0 0
\(401\) 29.4084i 1.46858i 0.678834 + 0.734292i \(0.262485\pi\)
−0.678834 + 0.734292i \(0.737515\pi\)
\(402\) 0 0
\(403\) 1.84158 0.0917356
\(404\) 0 0
\(405\) 0.296122 8.99513i 0.0147144 0.446971i
\(406\) 0 0
\(407\) −23.0769 −1.14388
\(408\) 0 0
\(409\) −0.172260 −0.00851773 −0.00425886 0.999991i \(-0.501356\pi\)
−0.00425886 + 0.999991i \(0.501356\pi\)
\(410\) 0 0
\(411\) 10.9654 25.8687i 0.540882 1.27601i
\(412\) 0 0
\(413\) −4.45616 −0.219273
\(414\) 0 0
\(415\) 10.6481i 0.522693i
\(416\) 0 0
\(417\) −7.94418 3.36742i −0.389028 0.164903i
\(418\) 0 0
\(419\) 20.6284i 1.00776i 0.863772 + 0.503882i \(0.168095\pi\)
−0.863772 + 0.503882i \(0.831905\pi\)
\(420\) 0 0
\(421\) 26.8393i 1.30807i −0.756464 0.654035i \(-0.773075\pi\)
0.756464 0.654035i \(-0.226925\pi\)
\(422\) 0 0
\(423\) 22.8752 + 23.6406i 1.11223 + 1.14945i
\(424\) 0 0
\(425\) 2.91469i 0.141383i
\(426\) 0 0
\(427\) −6.92820 −0.335279
\(428\) 0 0
\(429\) 5.64064 + 2.39099i 0.272333 + 0.115438i
\(430\) 0 0
\(431\) 24.9658 1.20256 0.601280 0.799039i \(-0.294658\pi\)
0.601280 + 0.799039i \(0.294658\pi\)
\(432\) 0 0
\(433\) 18.5774 0.892773 0.446387 0.894840i \(-0.352711\pi\)
0.446387 + 0.894840i \(0.352711\pi\)
\(434\) 0 0
\(435\) 10.9654 + 4.64806i 0.525749 + 0.222857i
\(436\) 0 0
\(437\) −1.98516 −0.0949632
\(438\) 0 0
\(439\) 1.71610i 0.0819052i −0.999161 0.0409526i \(-0.986961\pi\)
0.999161 0.0409526i \(-0.0130393\pi\)
\(440\) 0 0
\(441\) 13.7268 + 14.1861i 0.653656 + 0.675528i
\(442\) 0 0
\(443\) 32.7449i 1.55576i 0.628416 + 0.777878i \(0.283704\pi\)
−0.628416 + 0.777878i \(0.716296\pi\)
\(444\) 0 0
\(445\) 17.9903i 0.852820i
\(446\) 0 0
\(447\) 3.73881 + 1.58482i 0.176839 + 0.0749596i
\(448\) 0 0
\(449\) 4.31187i 0.203490i 0.994811 + 0.101745i \(0.0324426\pi\)
−0.994811 + 0.101745i \(0.967557\pi\)
\(450\) 0 0
\(451\) 17.9903 0.847128
\(452\) 0 0
\(453\) −7.82774 + 18.4666i −0.367779 + 0.867639i
\(454\) 0 0
\(455\) −0.549407 −0.0257566
\(456\) 0 0
\(457\) 13.6555 0.638776 0.319388 0.947624i \(-0.396523\pi\)
0.319388 + 0.947624i \(0.396523\pi\)
\(458\) 0 0
\(459\) −14.1311 + 5.44874i −0.659584 + 0.254325i
\(460\) 0 0
\(461\) −10.9974 −0.512199 −0.256100 0.966650i \(-0.582438\pi\)
−0.256100 + 0.966650i \(0.582438\pi\)
\(462\) 0 0
\(463\) 7.24030i 0.336485i 0.985746 + 0.168243i \(0.0538092\pi\)
−0.985746 + 0.168243i \(0.946191\pi\)
\(464\) 0 0
\(465\) 1.46838 3.46410i 0.0680946 0.160644i
\(466\) 0 0
\(467\) 10.6481i 0.492733i −0.969177 0.246367i \(-0.920763\pi\)
0.969177 0.246367i \(-0.0792368\pi\)
\(468\) 0 0
\(469\) 3.52179i 0.162621i
\(470\) 0 0
\(471\) −0.944450 + 2.22808i −0.0435179 + 0.102664i
\(472\) 0 0
\(473\) 39.9210i 1.83557i
\(474\) 0 0
\(475\) −0.847771 −0.0388984
\(476\) 0 0
\(477\) −24.2839 25.0965i −1.11188 1.14909i
\(478\) 0 0
\(479\) −23.0769 −1.05441 −0.527205 0.849738i \(-0.676760\pi\)
−0.527205 + 0.849738i \(0.676760\pi\)
\(480\) 0 0
\(481\) 4.68904 0.213802
\(482\) 0 0
\(483\) −1.02580 + 2.41998i −0.0466753 + 0.110113i
\(484\) 0 0
\(485\) −4.70388 −0.213592
\(486\) 0 0
\(487\) 10.4003i 0.471284i −0.971840 0.235642i \(-0.924281\pi\)
0.971840 0.235642i \(-0.0757193\pi\)
\(488\) 0 0
\(489\) 18.8384 + 7.98533i 0.851902 + 0.361109i
\(490\) 0 0
\(491\) 36.5168i 1.64798i 0.566605 + 0.823990i \(0.308257\pi\)
−0.566605 + 0.823990i \(0.691743\pi\)
\(492\) 0 0
\(493\) 20.0418i 0.902639i
\(494\) 0 0
\(495\) 8.99513 8.70388i 0.404301 0.391210i
\(496\) 0 0
\(497\) 8.62374i 0.386828i
\(498\) 0 0
\(499\) 32.6944 1.46360 0.731802 0.681517i \(-0.238680\pi\)
0.731802 + 0.681517i \(0.238680\pi\)
\(500\) 0 0
\(501\) −10.0181 4.24652i −0.447575 0.189721i
\(502\) 0 0
\(503\) −22.6242 −1.00876 −0.504381 0.863482i \(-0.668279\pi\)
−0.504381 + 0.863482i \(0.668279\pi\)
\(504\) 0 0
\(505\) −1.46838 −0.0653421
\(506\) 0 0
\(507\) 19.5850 + 8.30178i 0.869798 + 0.368695i
\(508\) 0 0
\(509\) 5.12386 0.227111 0.113556 0.993532i \(-0.463776\pi\)
0.113556 + 0.993532i \(0.463776\pi\)
\(510\) 0 0
\(511\) 8.45616i 0.374078i
\(512\) 0 0
\(513\) −1.58482 4.11019i −0.0699717 0.181469i
\(514\) 0 0
\(515\) 7.69646i 0.339147i
\(516\) 0 0
\(517\) 45.7504i 2.01210i
\(518\) 0 0
\(519\) 25.6695 + 10.8809i 1.12677 + 0.477621i
\(520\) 0 0
\(521\) 28.9063i 1.26641i 0.773986 + 0.633203i \(0.218260\pi\)
−0.773986 + 0.633203i \(0.781740\pi\)
\(522\) 0 0
\(523\) 14.6075 0.638741 0.319371 0.947630i \(-0.396529\pi\)
0.319371 + 0.947630i \(0.396529\pi\)
\(524\) 0 0
\(525\) −0.438069 + 1.03346i −0.0191189 + 0.0451040i
\(526\) 0 0
\(527\) −6.33148 −0.275803
\(528\) 0 0
\(529\) −17.5168 −0.761599
\(530\) 0 0
\(531\) −14.8245 + 14.3445i −0.643329 + 0.622499i
\(532\) 0 0
\(533\) −3.65548 −0.158336
\(534\) 0 0
\(535\) 1.35194i 0.0584494i
\(536\) 0 0
\(537\) 9.10422 21.4780i 0.392876 0.926845i
\(538\) 0 0
\(539\) 27.4535i 1.18251i
\(540\) 0 0
\(541\) 31.3446i 1.34761i 0.738910 + 0.673804i \(0.235341\pi\)
−0.738910 + 0.673804i \(0.764659\pi\)
\(542\) 0 0
\(543\) 7.47761 17.6406i 0.320895 0.757032i
\(544\) 0 0
\(545\) 17.6189i 0.754710i
\(546\) 0 0
\(547\) −17.5952 −0.752317 −0.376158 0.926555i \(-0.622755\pi\)
−0.376158 + 0.926555i \(0.622755\pi\)
\(548\) 0 0
\(549\) −23.0484 + 22.3021i −0.983682 + 0.951832i
\(550\) 0 0
\(551\) 5.82939 0.248340
\(552\) 0 0
\(553\) 11.1600 0.474573
\(554\) 0 0
\(555\) 3.73881 8.82032i 0.158703 0.374402i
\(556\) 0 0
\(557\) −18.5678 −0.786743 −0.393371 0.919380i \(-0.628691\pi\)
−0.393371 + 0.919380i \(0.628691\pi\)
\(558\) 0 0
\(559\) 8.11164i 0.343086i
\(560\) 0 0
\(561\) −19.3929 8.22038i −0.818770 0.347065i
\(562\) 0 0
\(563\) 13.1191i 0.552902i 0.961028 + 0.276451i \(0.0891584\pi\)
−0.961028 + 0.276451i \(0.910842\pi\)
\(564\) 0 0
\(565\) 7.59795i 0.319648i
\(566\) 0 0
\(567\) −5.82939 0.191905i −0.244811 0.00805924i
\(568\) 0 0
\(569\) 10.8840i 0.456282i 0.973628 + 0.228141i \(0.0732647\pi\)
−0.973628 + 0.228141i \(0.926735\pi\)
\(570\) 0 0
\(571\) 12.4592 0.521402 0.260701 0.965420i \(-0.416046\pi\)
0.260701 + 0.965420i \(0.416046\pi\)
\(572\) 0 0
\(573\) −1.82774 0.774752i −0.0763549 0.0323657i
\(574\) 0 0
\(575\) 2.34163 0.0976526
\(576\) 0 0
\(577\) 21.2813 0.885951 0.442976 0.896534i \(-0.353923\pi\)
0.442976 + 0.896534i \(0.353923\pi\)
\(578\) 0 0
\(579\) −27.7365 11.7571i −1.15269 0.488608i
\(580\) 0 0
\(581\) 6.90059 0.286285
\(582\) 0 0
\(583\) 48.5678i 2.01147i
\(584\) 0 0
\(585\) −1.82774 + 1.76856i −0.0755677 + 0.0731210i
\(586\) 0 0
\(587\) 30.9926i 1.27920i 0.768708 + 0.639600i \(0.220900\pi\)
−0.768708 + 0.639600i \(0.779100\pi\)
\(588\) 0 0
\(589\) 1.84158i 0.0758809i
\(590\) 0 0
\(591\) 16.6744 + 7.06804i 0.685894 + 0.290740i
\(592\) 0 0
\(593\) 39.2986i 1.61380i −0.590689 0.806899i \(-0.701144\pi\)
0.590689 0.806899i \(-0.298856\pi\)
\(594\) 0 0
\(595\) 1.88890 0.0774373
\(596\) 0 0
\(597\) 9.65548 22.7785i 0.395172 0.932263i
\(598\) 0 0
\(599\) −29.4084 −1.20159 −0.600796 0.799402i \(-0.705150\pi\)
−0.600796 + 0.799402i \(0.705150\pi\)
\(600\) 0 0
\(601\) 19.3323 0.788581 0.394290 0.918986i \(-0.370990\pi\)
0.394290 + 0.918986i \(0.370990\pi\)
\(602\) 0 0
\(603\) −11.3368 11.7161i −0.461668 0.477117i
\(604\) 0 0
\(605\) 6.40776 0.260512
\(606\) 0 0
\(607\) 4.87133i 0.197721i 0.995101 + 0.0988606i \(0.0315198\pi\)
−0.995101 + 0.0988606i \(0.968480\pi\)
\(608\) 0 0
\(609\) 3.01223 7.10623i 0.122062 0.287959i
\(610\) 0 0
\(611\) 9.29612i 0.376081i
\(612\) 0 0
\(613\) 12.5065i 0.505135i −0.967579 0.252567i \(-0.918725\pi\)
0.967579 0.252567i \(-0.0812749\pi\)
\(614\) 0 0
\(615\) −2.91469 + 6.87614i −0.117532 + 0.277273i
\(616\) 0 0
\(617\) 44.4839i 1.79085i 0.445207 + 0.895427i \(0.353130\pi\)
−0.445207 + 0.895427i \(0.646870\pi\)
\(618\) 0 0
\(619\) 14.1074 0.567026 0.283513 0.958968i \(-0.408500\pi\)
0.283513 + 0.958968i \(0.408500\pi\)
\(620\) 0 0
\(621\) 4.37744 + 11.3527i 0.175661 + 0.455570i
\(622\) 0 0
\(623\) 11.6588 0.467099
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 2.39099 5.64064i 0.0954868 0.225266i
\(628\) 0 0
\(629\) −16.1213 −0.642796
\(630\) 0 0
\(631\) 8.64325i 0.344082i 0.985090 + 0.172041i \(0.0550362\pi\)
−0.985090 + 0.172041i \(0.944964\pi\)
\(632\) 0 0
\(633\) 14.2281 + 6.03107i 0.565515 + 0.239714i
\(634\) 0 0
\(635\) 11.6965i 0.464160i
\(636\) 0 0
\(637\) 5.57835i 0.221022i
\(638\) 0 0
\(639\) 27.7601 + 28.6890i 1.09817 + 1.13492i
\(640\) 0 0
\(641\) 36.0125i 1.42241i −0.702986 0.711204i \(-0.748150\pi\)
0.702986 0.711204i \(-0.251850\pi\)
\(642\) 0 0
\(643\) 16.5437 0.652421 0.326210 0.945297i \(-0.394228\pi\)
0.326210 + 0.945297i \(0.394228\pi\)
\(644\) 0 0
\(645\) −15.2584 6.46781i −0.600799 0.254670i
\(646\) 0 0
\(647\) 37.0773 1.45766 0.728829 0.684696i \(-0.240065\pi\)
0.728829 + 0.684696i \(0.240065\pi\)
\(648\) 0 0
\(649\) −28.6890 −1.12614
\(650\) 0 0
\(651\) −2.24495 0.951601i −0.0879865 0.0372962i
\(652\) 0 0
\(653\) −13.3929 −0.524105 −0.262053 0.965054i \(-0.584399\pi\)
−0.262053 + 0.965054i \(0.584399\pi\)
\(654\) 0 0
\(655\) 9.58002i 0.374322i
\(656\) 0 0
\(657\) −27.2207 28.1315i −1.06198 1.09751i
\(658\) 0 0
\(659\) 29.6917i 1.15662i −0.815816 0.578311i \(-0.803712\pi\)
0.815816 0.578311i \(-0.196288\pi\)
\(660\) 0 0
\(661\) 44.8296i 1.74367i −0.489800 0.871835i \(-0.662930\pi\)
0.489800 0.871835i \(-0.337070\pi\)
\(662\) 0 0
\(663\) 3.94049 + 1.67032i 0.153036 + 0.0648697i
\(664\) 0 0
\(665\) 0.549407i 0.0213051i
\(666\) 0 0
\(667\) −16.1014 −0.623447
\(668\) 0 0
\(669\) −11.5619 + 27.2761i −0.447010 + 1.05455i
\(670\) 0 0
\(671\) −44.6043 −1.72193
\(672\) 0 0
\(673\) −33.3929 −1.28720 −0.643601 0.765361i \(-0.722560\pi\)
−0.643601 + 0.765361i \(0.722560\pi\)
\(674\) 0 0
\(675\) 1.86940 + 4.84823i 0.0719533 + 0.186608i
\(676\) 0 0
\(677\) 36.2084 1.39160 0.695802 0.718234i \(-0.255049\pi\)
0.695802 + 0.718234i \(0.255049\pi\)
\(678\) 0 0
\(679\) 3.04840i 0.116987i
\(680\) 0 0
\(681\) 9.02551 21.2923i 0.345858 0.815924i
\(682\) 0 0
\(683\) 8.51197i 0.325702i 0.986651 + 0.162851i \(0.0520689\pi\)
−0.986651 + 0.162851i \(0.947931\pi\)
\(684\) 0 0
\(685\) 16.2217i 0.619799i
\(686\) 0 0
\(687\) 12.1609 28.6890i 0.463966 1.09456i
\(688\) 0 0
\(689\) 9.86860i 0.375964i
\(690\) 0 0
\(691\) −48.2937 −1.83718 −0.918589 0.395213i \(-0.870671\pi\)
−0.918589 + 0.395213i \(0.870671\pi\)
\(692\) 0 0
\(693\) −5.64064 5.82939i −0.214270 0.221440i
\(694\) 0 0
\(695\) 4.98162 0.188964
\(696\) 0 0
\(697\) 12.5678 0.476039
\(698\) 0 0
\(699\) 13.6290 32.1526i 0.515497 1.21612i
\(700\) 0 0
\(701\) 40.0968 1.51444 0.757218 0.653163i \(-0.226558\pi\)
0.757218 + 0.653163i \(0.226558\pi\)
\(702\) 0 0
\(703\) 4.68904i 0.176851i
\(704\) 0 0
\(705\) −17.4865 7.41226i −0.658579 0.279162i
\(706\) 0 0
\(707\) 0.951601i 0.0357886i
\(708\) 0 0
\(709\) 16.1487i 0.606476i 0.952915 + 0.303238i \(0.0980678\pi\)
−0.952915 + 0.303238i \(0.901932\pi\)
\(710\) 0 0
\(711\) 37.1266 35.9245i 1.39236 1.34728i
\(712\) 0 0
\(713\) 5.08662i 0.190496i
\(714\) 0 0
\(715\) −3.53712 −0.132281
\(716\) 0 0
\(717\) −19.3929 8.22038i −0.724242 0.306996i
\(718\) 0 0
\(719\) 6.33148 0.236124 0.118062 0.993006i \(-0.462332\pi\)
0.118062 + 0.993006i \(0.462332\pi\)
\(720\) 0 0
\(721\) −4.98777 −0.185754
\(722\) 0 0
\(723\) −13.5817 5.75709i −0.505109 0.214108i
\(724\) 0 0
\(725\) −6.87614 −0.255373
\(726\) 0 0
\(727\) 35.6965i 1.32391i 0.749544 + 0.661954i \(0.230273\pi\)
−0.749544 + 0.661954i \(0.769727\pi\)
\(728\) 0 0
\(729\) −20.0107 + 18.1266i −0.741136 + 0.671355i
\(730\) 0 0
\(731\) 27.8884i 1.03149i
\(732\) 0 0
\(733\) 31.0462i 1.14672i −0.819304 0.573359i \(-0.805640\pi\)
0.819304 0.573359i \(-0.194360\pi\)
\(734\) 0 0
\(735\) −10.4931 4.44789i −0.387046 0.164063i
\(736\) 0 0
\(737\) 22.6735i 0.835189i
\(738\) 0 0
\(739\) −11.1671 −0.410787 −0.205393 0.978679i \(-0.565847\pi\)
−0.205393 + 0.978679i \(0.565847\pi\)
\(740\) 0 0
\(741\) −0.485830 + 1.14613i −0.0178474 + 0.0421043i
\(742\) 0 0
\(743\) −36.4332 −1.33661 −0.668303 0.743889i \(-0.732979\pi\)
−0.668303 + 0.743889i \(0.732979\pi\)
\(744\) 0 0
\(745\) −2.34452 −0.0858966
\(746\) 0 0
\(747\) 22.9565 22.2132i 0.839936 0.812740i
\(748\) 0 0
\(749\) 0.876139 0.0320134
\(750\) 0 0
\(751\) 25.4832i 0.929896i −0.885338 0.464948i \(-0.846073\pi\)
0.885338 0.464948i \(-0.153927\pi\)
\(752\) 0 0
\(753\) 8.46096 19.9605i 0.308335 0.727401i
\(754\) 0 0
\(755\) 11.5800i 0.421440i
\(756\) 0 0
\(757\) 44.1599i 1.60502i 0.596641 + 0.802509i \(0.296502\pi\)
−0.596641 + 0.802509i \(0.703498\pi\)
\(758\) 0 0
\(759\) −6.60414 + 15.5800i −0.239715 + 0.565519i
\(760\) 0 0
\(761\) 6.33148i 0.229516i 0.993393 + 0.114758i \(0.0366092\pi\)
−0.993393 + 0.114758i \(0.963391\pi\)
\(762\) 0 0
\(763\) 11.4181 0.413363
\(764\) 0 0
\(765\) 6.28390 6.08043i 0.227195 0.219839i
\(766\) 0 0
\(767\) 5.82939 0.210487
\(768\) 0 0
\(769\) −42.9123 −1.54746 −0.773729 0.633517i \(-0.781611\pi\)
−0.773729 + 0.633517i \(0.781611\pi\)
\(770\) 0 0
\(771\) −1.19549 + 2.82032i −0.0430546 + 0.101571i
\(772\) 0 0
\(773\) −6.23289 −0.224181 −0.112091 0.993698i \(-0.535755\pi\)
−0.112091 + 0.993698i \(0.535755\pi\)
\(774\) 0 0
\(775\) 2.17226i 0.0780299i
\(776\) 0 0
\(777\) −5.71610 2.42297i −0.205064 0.0869237i
\(778\) 0 0
\(779\) 3.65548i 0.130971i
\(780\) 0 0
\(781\) 55.5203i 1.98667i
\(782\) 0 0
\(783\) −12.8543 33.3371i −0.459374 1.19137i
\(784\) 0 0
\(785\) 1.39718i 0.0498674i
\(786\) 0 0
\(787\) −29.8034 −1.06238 −0.531188 0.847254i \(-0.678254\pi\)
−0.531188 + 0.847254i \(0.678254\pi\)
\(788\) 0 0
\(789\) 48.0032 + 20.3479i 1.70896 + 0.724404i
\(790\) 0 0
\(791\) −4.92393 −0.175075
\(792\) 0 0
\(793\) 9.06324 0.321845
\(794\) 0 0
\(795\) 18.5633 + 7.86872i 0.658373 + 0.279075i
\(796\) 0 0
\(797\) 51.6110 1.82815 0.914077 0.405540i \(-0.132917\pi\)
0.914077 + 0.405540i \(0.132917\pi\)
\(798\) 0 0
\(799\) 31.9607i 1.13069i
\(800\) 0 0
\(801\) 38.7858 37.5300i 1.37043 1.32606i
\(802\) 0 0
\(803\) 54.4413i 1.92119i
\(804\) 0 0
\(805\) 1.51752i 0.0534854i
\(806\) 0 0
\(807\) −4.88494 2.07065i −0.171958 0.0728905i
\(808\) 0 0
\(809\) 15.1959i 0.534259i 0.963661 + 0.267130i \(0.0860752\pi\)
−0.963661 + 0.267130i \(0.913925\pi\)
\(810\) 0 0
\(811\) −7.96933 −0.279841 −0.139921 0.990163i \(-0.544685\pi\)
−0.139921 + 0.990163i \(0.544685\pi\)
\(812\) 0 0
\(813\) 5.92454 13.9767i 0.207783 0.490186i
\(814\) 0 0
\(815\) −11.8131 −0.413796
\(816\) 0 0
\(817\) −8.11164 −0.283790
\(818\) 0 0
\(819\) 1.14613 + 1.18449i 0.0400492 + 0.0413893i
\(820\) 0 0
\(821\) −30.5678 −1.06682 −0.533412 0.845856i \(-0.679090\pi\)
−0.533412 + 0.845856i \(0.679090\pi\)
\(822\) 0 0
\(823\) 28.2739i 0.985565i −0.870153 0.492783i \(-0.835980\pi\)
0.870153 0.492783i \(-0.164020\pi\)
\(824\) 0 0
\(825\) −2.82032 + 6.65350i −0.0981910 + 0.231645i
\(826\) 0 0
\(827\) 29.0894i 1.01154i −0.862669 0.505768i \(-0.831209\pi\)
0.862669 0.505768i \(-0.168791\pi\)
\(828\) 0 0
\(829\) 30.7325i 1.06738i −0.845679 0.533692i \(-0.820804\pi\)
0.845679 0.533692i \(-0.179196\pi\)
\(830\) 0 0
\(831\) −18.9347 + 44.6694i −0.656838 + 1.54956i
\(832\) 0 0
\(833\) 19.1787i 0.664504i
\(834\) 0 0
\(835\) 6.28212 0.217402
\(836\) 0 0
\(837\) −10.5316 + 4.06083i −0.364026 + 0.140363i
\(838\) 0 0
\(839\) 24.5624 0.847989 0.423994 0.905665i \(-0.360628\pi\)
0.423994 + 0.905665i \(0.360628\pi\)
\(840\) 0 0
\(841\) 18.2813 0.630389
\(842\) 0 0
\(843\) 6.85518 16.1723i 0.236105 0.557002i
\(844\) 0 0
\(845\) −12.2813 −0.422489
\(846\) 0 0
\(847\) 4.15262i 0.142686i
\(848\) 0 0
\(849\) −7.79001 3.30207i −0.267352 0.113327i
\(850\) 0 0
\(851\) 12.9516i 0.443975i
\(852\) 0 0
\(853\) 55.1746i 1.88914i −0.328308 0.944571i \(-0.606478\pi\)
0.328308 0.944571i \(-0.393522\pi\)
\(854\) 0 0
\(855\) 1.76856 + 1.82774i 0.0604835 + 0.0625074i
\(856\) 0 0
\(857\) 4.80359i 0.164088i 0.996629 + 0.0820438i \(0.0261447\pi\)
−0.996629 + 0.0820438i \(0.973855\pi\)
\(858\) 0 0
\(859\) 30.8529 1.05269 0.526343 0.850272i \(-0.323563\pi\)
0.526343 + 0.850272i \(0.323563\pi\)
\(860\) 0 0
\(861\) 4.45616 + 1.88890i 0.151865 + 0.0643735i
\(862\) 0 0
\(863\) −31.9907 −1.08897 −0.544487 0.838769i \(-0.683276\pi\)
−0.544487 + 0.838769i \(0.683276\pi\)
\(864\) 0 0
\(865\) −16.0968 −0.547308
\(866\) 0 0
\(867\) 13.5622 + 5.74882i 0.460597 + 0.195240i
\(868\) 0 0
\(869\) 71.8491 2.43731
\(870\) 0 0
\(871\) 4.60708i 0.156105i
\(872\) 0 0
\(873\) 9.81290 + 10.1413i 0.332116 + 0.343230i
\(874\) 0 0
\(875\) 0.648061i 0.0219085i
\(876\) 0 0
\(877\) 24.1180i 0.814407i −0.913337 0.407204i \(-0.866504\pi\)
0.913337 0.407204i \(-0.133496\pi\)
\(878\) 0 0
\(879\) 28.7046 + 12.1675i 0.968182 + 0.410398i
\(880\) 0 0
\(881\) 23.0449i 0.776402i −0.921575 0.388201i \(-0.873097\pi\)
0.921575 0.388201i \(-0.126903\pi\)
\(882\) 0 0
\(883\) −35.8261 −1.20565 −0.602823 0.797875i \(-0.705957\pi\)
−0.602823 + 0.797875i \(0.705957\pi\)
\(884\) 0 0
\(885\) 4.64806 10.9654i 0.156243 0.368597i
\(886\) 0 0
\(887\) −48.8348 −1.63971 −0.819856 0.572570i \(-0.805946\pi\)
−0.819856 + 0.572570i \(0.805946\pi\)
\(888\) 0 0
\(889\) −7.58002 −0.254226
\(890\) 0 0
\(891\) −37.5300 1.23550i −1.25730 0.0413907i
\(892\) 0 0
\(893\) −9.29612 −0.311083
\(894\) 0 0
\(895\) 13.4684i 0.450198i
\(896\) 0 0
\(897\) 1.34191 3.16574i 0.0448051 0.105701i
\(898\) 0 0
\(899\) 14.9368i 0.498169i
\(900\) 0 0
\(901\) 33.9289i 1.13034i
\(902\) 0 0
\(903\) −4.19153 + 9.88836i −0.139485 + 0.329064i
\(904\) 0 0
\(905\) 11.0621i 0.367715i
\(906\) 0 0
\(907\) −32.0483 −1.06415 −0.532074 0.846698i \(-0.678587\pi\)
−0.532074 + 0.846698i \(0.678587\pi\)
\(908\) 0 0
\(909\) 3.06324 + 3.16574i 0.101601 + 0.105001i
\(910\) 0 0
\(911\) −24.9658 −0.827153 −0.413577 0.910469i \(-0.635721\pi\)
−0.413577 + 0.910469i \(0.635721\pi\)
\(912\) 0 0
\(913\) 44.4265 1.47030
\(914\) 0 0
\(915\) 7.22657 17.0484i 0.238903 0.563603i
\(916\) 0 0
\(917\) 6.20843 0.205021
\(918\) 0 0
\(919\) 37.5652i 1.23916i 0.784933 + 0.619580i \(0.212697\pi\)
−0.784933 + 0.619580i \(0.787303\pi\)
\(920\) 0 0
\(921\) −13.4307 5.69306i −0.442555 0.187593i
\(922\) 0 0
\(923\) 11.2813i 0.371328i
\(924\) 0 0
\(925\) 5.53103i 0.181859i
\(926\) 0 0
\(927\) −16.5931 + 16.0558i −0.544988 + 0.527342i
\(928\) 0 0
\(929\) 33.2181i 1.08985i −0.838484 0.544926i \(-0.816558\pi\)
0.838484 0.544926i \(-0.183442\pi\)
\(930\) 0 0
\(931\) −5.57835 −0.182823
\(932\) 0 0
\(933\) −34.9729 14.8245i −1.14496 0.485333i
\(934\) 0 0
\(935\) 12.1609 0.397703
\(936\) 0 0
\(937\) 33.0484 1.07964 0.539822 0.841779i \(-0.318492\pi\)
0.539822 + 0.841779i \(0.318492\pi\)
\(938\) 0 0
\(939\) −22.3258 9.46357i −0.728575 0.308832i
\(940\) 0 0
\(941\) 47.5652 1.55058 0.775290 0.631605i \(-0.217604\pi\)
0.775290 + 0.631605i \(0.217604\pi\)
\(942\) 0 0
\(943\) 10.0968i 0.328797i
\(944\) 0 0
\(945\) 3.14195 1.21149i 0.102208 0.0394097i
\(946\) 0 0
\(947\) 0.551262i 0.0179136i −0.999960 0.00895681i \(-0.997149\pi\)
0.999960 0.00895681i \(-0.00285108\pi\)
\(948\) 0 0
\(949\) 11.0621i 0.359089i
\(950\) 0 0
\(951\) −27.1871 11.5242i −0.881601 0.373698i
\(952\) 0 0
\(953\) 11.1351i 0.360700i −0.983602 0.180350i \(-0.942277\pi\)
0.983602 0.180350i \(-0.0577231\pi\)
\(954\) 0 0
\(955\) 1.14613 0.0370880
\(956\) 0 0
\(957\) 19.3929 45.7504i 0.626884 1.47890i
\(958\) 0 0
\(959\) 10.5126 0.339471
\(960\) 0 0
\(961\) 26.2813 0.847783
\(962\) 0 0
\(963\) 2.91469 2.82032i 0.0939247 0.0908836i
\(964\) 0 0
\(965\) 17.3929 0.559898
\(966\) 0 0
\(967\) 37.5700i 1.20817i 0.796920 + 0.604085i \(0.206461\pi\)
−0.796920 + 0.604085i \(0.793539\pi\)
\(968\) 0 0
\(969\) 1.67032 3.94049i 0.0536583 0.126587i
\(970\) 0 0
\(971\) 38.2691i 1.22811i −0.789262 0.614056i \(-0.789537\pi\)
0.789262 0.614056i \(-0.210463\pi\)
\(972\) 0 0
\(973\) 3.22839i 0.103497i
\(974\) 0 0
\(975\) 0.573067 1.35194i 0.0183528 0.0432967i
\(976\) 0 0
\(977\) 0.120339i 0.00384999i −0.999998 0.00192500i \(-0.999387\pi\)
0.999998 0.00192500i \(-0.000612746\pi\)
\(978\) 0 0
\(979\) 75.0600 2.39893
\(980\) 0 0
\(981\) 37.9852 36.7553i 1.21277 1.17351i
\(982\) 0 0
\(983\) −30.8445 −0.983788 −0.491894 0.870655i \(-0.663695\pi\)
−0.491894 + 0.870655i \(0.663695\pi\)
\(984\) 0 0
\(985\) −10.4562 −0.333161
\(986\) 0 0
\(987\) −4.80359 + 11.3323i −0.152900 + 0.360711i
\(988\) 0 0
\(989\) 22.4051 0.712442
\(990\) 0 0
\(991\) 9.82774i 0.312188i −0.987742 0.156094i \(-0.950110\pi\)
0.987742 0.156094i \(-0.0498903\pi\)
\(992\) 0 0
\(993\) 4.36417 + 1.84991i 0.138493 + 0.0587050i
\(994\) 0 0
\(995\) 14.2839i 0.452830i
\(996\) 0 0
\(997\) 35.7295i 1.13156i 0.824555 + 0.565782i \(0.191426\pi\)
−0.824555 + 0.565782i \(0.808574\pi\)
\(998\) 0 0
\(999\) −26.8157 + 10.3397i −0.848411 + 0.327134i
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 960.2.b.c.671.12 yes 12
3.2 odd 2 960.2.b.d.671.11 yes 12
4.3 odd 2 inner 960.2.b.c.671.1 12
8.3 odd 2 960.2.b.d.671.12 yes 12
8.5 even 2 960.2.b.d.671.1 yes 12
12.11 even 2 960.2.b.d.671.2 yes 12
24.5 odd 2 inner 960.2.b.c.671.2 yes 12
24.11 even 2 inner 960.2.b.c.671.11 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
960.2.b.c.671.1 12 4.3 odd 2 inner
960.2.b.c.671.2 yes 12 24.5 odd 2 inner
960.2.b.c.671.11 yes 12 24.11 even 2 inner
960.2.b.c.671.12 yes 12 1.1 even 1 trivial
960.2.b.d.671.1 yes 12 8.5 even 2
960.2.b.d.671.2 yes 12 12.11 even 2
960.2.b.d.671.11 yes 12 3.2 odd 2
960.2.b.d.671.12 yes 12 8.3 odd 2