Properties

Label 315.2.d.e.64.2
Level $315$
Weight $2$
Character 315.64
Analytic conductor $2.515$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,2,Mod(64,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("315.64");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.d (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: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 64.2
Root \(1.45161 + 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 315.64
Dual form 315.2.d.e.64.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.90321i q^{2} -1.62222 q^{4} +(-0.311108 - 2.21432i) q^{5} +1.00000i q^{7} -0.719004i q^{8} +O(q^{10})\) \(q-1.90321i q^{2} -1.62222 q^{4} +(-0.311108 - 2.21432i) q^{5} +1.00000i q^{7} -0.719004i q^{8} +(-4.21432 + 0.592104i) q^{10} -2.00000 q^{11} -6.42864i q^{13} +1.90321 q^{14} -4.61285 q^{16} +4.42864i q^{17} +2.42864 q^{19} +(0.504684 + 3.59210i) q^{20} +3.80642i q^{22} -1.37778i q^{23} +(-4.80642 + 1.37778i) q^{25} -12.2351 q^{26} -1.62222i q^{28} +0.755569 q^{29} +5.18421 q^{31} +7.34122i q^{32} +8.42864 q^{34} +(2.21432 - 0.311108i) q^{35} -7.61285i q^{37} -4.62222i q^{38} +(-1.59210 + 0.223688i) q^{40} +8.23506 q^{41} +10.1017i q^{43} +3.24443 q^{44} -2.62222 q^{46} +2.75557i q^{47} -1.00000 q^{49} +(2.62222 + 9.14764i) q^{50} +10.4286i q^{52} -9.18421i q^{53} +(0.622216 + 4.42864i) q^{55} +0.719004 q^{56} -1.43801i q^{58} +14.1017 q^{59} +6.85728 q^{61} -9.86665i q^{62} +4.74620 q^{64} +(-14.2351 + 2.00000i) q^{65} -2.75557i q^{67} -7.18421i q^{68} +(-0.592104 - 4.21432i) q^{70} -2.00000 q^{71} -1.57136i q^{73} -14.4889 q^{74} -3.93978 q^{76} -2.00000i q^{77} +4.85728 q^{79} +(1.43509 + 10.2143i) q^{80} -15.6731i q^{82} -11.6128i q^{83} +(9.80642 - 1.37778i) q^{85} +19.2257 q^{86} +1.43801i q^{88} +4.62222 q^{89} +6.42864 q^{91} +2.23506i q^{92} +5.24443 q^{94} +(-0.755569 - 5.37778i) q^{95} +11.9398i q^{97} +1.90321i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} - 2 q^{5} - 12 q^{10} - 12 q^{11} - 2 q^{14} + 26 q^{16} - 12 q^{19} + 30 q^{20} - 2 q^{25} - 20 q^{26} + 4 q^{29} + 4 q^{31} + 24 q^{34} + 4 q^{40} - 4 q^{41} + 20 q^{44} - 16 q^{46} - 6 q^{49} + 16 q^{50} + 4 q^{55} + 18 q^{56} + 32 q^{59} - 12 q^{61} - 26 q^{64} - 32 q^{65} + 10 q^{70} - 12 q^{71} - 88 q^{74} + 4 q^{76} - 24 q^{79} - 46 q^{80} + 32 q^{85} + 8 q^{86} + 28 q^{89} + 12 q^{91} + 32 q^{94} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.90321i 1.34577i −0.739745 0.672887i \(-0.765054\pi\)
0.739745 0.672887i \(-0.234946\pi\)
\(3\) 0 0
\(4\) −1.62222 −0.811108
\(5\) −0.311108 2.21432i −0.139132 0.990274i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0.719004i 0.254206i
\(9\) 0 0
\(10\) −4.21432 + 0.592104i −1.33268 + 0.187240i
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 6.42864i 1.78298i −0.453037 0.891492i \(-0.649659\pi\)
0.453037 0.891492i \(-0.350341\pi\)
\(14\) 1.90321 0.508655
\(15\) 0 0
\(16\) −4.61285 −1.15321
\(17\) 4.42864i 1.07410i 0.843550 + 0.537051i \(0.180462\pi\)
−0.843550 + 0.537051i \(0.819538\pi\)
\(18\) 0 0
\(19\) 2.42864 0.557168 0.278584 0.960412i \(-0.410135\pi\)
0.278584 + 0.960412i \(0.410135\pi\)
\(20\) 0.504684 + 3.59210i 0.112851 + 0.803219i
\(21\) 0 0
\(22\) 3.80642i 0.811532i
\(23\) 1.37778i 0.287288i −0.989629 0.143644i \(-0.954118\pi\)
0.989629 0.143644i \(-0.0458820\pi\)
\(24\) 0 0
\(25\) −4.80642 + 1.37778i −0.961285 + 0.275557i
\(26\) −12.2351 −2.39949
\(27\) 0 0
\(28\) 1.62222i 0.306570i
\(29\) 0.755569 0.140306 0.0701528 0.997536i \(-0.477651\pi\)
0.0701528 + 0.997536i \(0.477651\pi\)
\(30\) 0 0
\(31\) 5.18421 0.931111 0.465556 0.885019i \(-0.345855\pi\)
0.465556 + 0.885019i \(0.345855\pi\)
\(32\) 7.34122i 1.29776i
\(33\) 0 0
\(34\) 8.42864 1.44550
\(35\) 2.21432 0.311108i 0.374288 0.0525868i
\(36\) 0 0
\(37\) 7.61285i 1.25154i −0.780006 0.625772i \(-0.784784\pi\)
0.780006 0.625772i \(-0.215216\pi\)
\(38\) 4.62222i 0.749822i
\(39\) 0 0
\(40\) −1.59210 + 0.223688i −0.251734 + 0.0353681i
\(41\) 8.23506 1.28610 0.643050 0.765824i \(-0.277669\pi\)
0.643050 + 0.765824i \(0.277669\pi\)
\(42\) 0 0
\(43\) 10.1017i 1.54050i 0.637744 + 0.770248i \(0.279868\pi\)
−0.637744 + 0.770248i \(0.720132\pi\)
\(44\) 3.24443 0.489116
\(45\) 0 0
\(46\) −2.62222 −0.386625
\(47\) 2.75557i 0.401941i 0.979597 + 0.200971i \(0.0644095\pi\)
−0.979597 + 0.200971i \(0.935590\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 2.62222 + 9.14764i 0.370837 + 1.29367i
\(51\) 0 0
\(52\) 10.4286i 1.44619i
\(53\) 9.18421i 1.26155i −0.775967 0.630774i \(-0.782737\pi\)
0.775967 0.630774i \(-0.217263\pi\)
\(54\) 0 0
\(55\) 0.622216 + 4.42864i 0.0838995 + 0.597158i
\(56\) 0.719004 0.0960809
\(57\) 0 0
\(58\) 1.43801i 0.188820i
\(59\) 14.1017 1.83589 0.917943 0.396712i \(-0.129849\pi\)
0.917943 + 0.396712i \(0.129849\pi\)
\(60\) 0 0
\(61\) 6.85728 0.877985 0.438992 0.898491i \(-0.355336\pi\)
0.438992 + 0.898491i \(0.355336\pi\)
\(62\) 9.86665i 1.25307i
\(63\) 0 0
\(64\) 4.74620 0.593275
\(65\) −14.2351 + 2.00000i −1.76564 + 0.248069i
\(66\) 0 0
\(67\) 2.75557i 0.336646i −0.985732 0.168323i \(-0.946165\pi\)
0.985732 0.168323i \(-0.0538352\pi\)
\(68\) 7.18421i 0.871213i
\(69\) 0 0
\(70\) −0.592104 4.21432i −0.0707700 0.503708i
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) 1.57136i 0.183914i −0.995763 0.0919569i \(-0.970688\pi\)
0.995763 0.0919569i \(-0.0293122\pi\)
\(74\) −14.4889 −1.68430
\(75\) 0 0
\(76\) −3.93978 −0.451923
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) 4.85728 0.546487 0.273243 0.961945i \(-0.411904\pi\)
0.273243 + 0.961945i \(0.411904\pi\)
\(80\) 1.43509 + 10.2143i 0.160448 + 1.14200i
\(81\) 0 0
\(82\) 15.6731i 1.73080i
\(83\) 11.6128i 1.27468i −0.770585 0.637338i \(-0.780036\pi\)
0.770585 0.637338i \(-0.219964\pi\)
\(84\) 0 0
\(85\) 9.80642 1.37778i 1.06366 0.149442i
\(86\) 19.2257 2.07316
\(87\) 0 0
\(88\) 1.43801i 0.153292i
\(89\) 4.62222 0.489954 0.244977 0.969529i \(-0.421220\pi\)
0.244977 + 0.969529i \(0.421220\pi\)
\(90\) 0 0
\(91\) 6.42864 0.673905
\(92\) 2.23506i 0.233021i
\(93\) 0 0
\(94\) 5.24443 0.540922
\(95\) −0.755569 5.37778i −0.0775197 0.551749i
\(96\) 0 0
\(97\) 11.9398i 1.21230i 0.795350 + 0.606150i \(0.207287\pi\)
−0.795350 + 0.606150i \(0.792713\pi\)
\(98\) 1.90321i 0.192253i
\(99\) 0 0
\(100\) 7.79706 2.23506i 0.779706 0.223506i
\(101\) −1.47949 −0.147215 −0.0736076 0.997287i \(-0.523451\pi\)
−0.0736076 + 0.997287i \(0.523451\pi\)
\(102\) 0 0
\(103\) 8.85728i 0.872734i −0.899769 0.436367i \(-0.856265\pi\)
0.899769 0.436367i \(-0.143735\pi\)
\(104\) −4.62222 −0.453246
\(105\) 0 0
\(106\) −17.4795 −1.69776
\(107\) 1.76494i 0.170623i 0.996354 + 0.0853114i \(0.0271885\pi\)
−0.996354 + 0.0853114i \(0.972811\pi\)
\(108\) 0 0
\(109\) −5.61285 −0.537613 −0.268807 0.963194i \(-0.586629\pi\)
−0.268807 + 0.963194i \(0.586629\pi\)
\(110\) 8.42864 1.18421i 0.803639 0.112910i
\(111\) 0 0
\(112\) 4.61285i 0.435873i
\(113\) 11.2859i 1.06169i 0.847469 + 0.530845i \(0.178125\pi\)
−0.847469 + 0.530845i \(0.821875\pi\)
\(114\) 0 0
\(115\) −3.05086 + 0.428639i −0.284494 + 0.0399708i
\(116\) −1.22570 −0.113803
\(117\) 0 0
\(118\) 26.8385i 2.47069i
\(119\) −4.42864 −0.405973
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 13.0509i 1.18157i
\(123\) 0 0
\(124\) −8.40990 −0.755232
\(125\) 4.54617 + 10.2143i 0.406622 + 0.913597i
\(126\) 0 0
\(127\) 12.8573i 1.14090i 0.821333 + 0.570450i \(0.193231\pi\)
−0.821333 + 0.570450i \(0.806769\pi\)
\(128\) 5.64941i 0.499342i
\(129\) 0 0
\(130\) 3.80642 + 27.0923i 0.333845 + 2.37616i
\(131\) 2.10171 0.183627 0.0918136 0.995776i \(-0.470734\pi\)
0.0918136 + 0.995776i \(0.470734\pi\)
\(132\) 0 0
\(133\) 2.42864i 0.210590i
\(134\) −5.24443 −0.453050
\(135\) 0 0
\(136\) 3.18421 0.273044
\(137\) 15.9398i 1.36183i 0.732364 + 0.680914i \(0.238417\pi\)
−0.732364 + 0.680914i \(0.761583\pi\)
\(138\) 0 0
\(139\) −11.6731 −0.990097 −0.495048 0.868865i \(-0.664850\pi\)
−0.495048 + 0.868865i \(0.664850\pi\)
\(140\) −3.59210 + 0.504684i −0.303588 + 0.0426536i
\(141\) 0 0
\(142\) 3.80642i 0.319428i
\(143\) 12.8573i 1.07518i
\(144\) 0 0
\(145\) −0.235063 1.67307i −0.0195209 0.138941i
\(146\) −2.99063 −0.247506
\(147\) 0 0
\(148\) 12.3497i 1.01514i
\(149\) −21.2257 −1.73888 −0.869438 0.494041i \(-0.835519\pi\)
−0.869438 + 0.494041i \(0.835519\pi\)
\(150\) 0 0
\(151\) 16.8573 1.37183 0.685913 0.727684i \(-0.259403\pi\)
0.685913 + 0.727684i \(0.259403\pi\)
\(152\) 1.74620i 0.141636i
\(153\) 0 0
\(154\) −3.80642 −0.306730
\(155\) −1.61285 11.4795i −0.129547 0.922055i
\(156\) 0 0
\(157\) 10.4286i 0.832296i 0.909297 + 0.416148i \(0.136620\pi\)
−0.909297 + 0.416148i \(0.863380\pi\)
\(158\) 9.24443i 0.735447i
\(159\) 0 0
\(160\) 16.2558 2.28391i 1.28513 0.180559i
\(161\) 1.37778 0.108585
\(162\) 0 0
\(163\) 20.8573i 1.63367i 0.576873 + 0.816834i \(0.304273\pi\)
−0.576873 + 0.816834i \(0.695727\pi\)
\(164\) −13.3590 −1.04317
\(165\) 0 0
\(166\) −22.1017 −1.71543
\(167\) 15.3461i 1.18752i 0.804642 + 0.593760i \(0.202357\pi\)
−0.804642 + 0.593760i \(0.797643\pi\)
\(168\) 0 0
\(169\) −28.3274 −2.17903
\(170\) −2.62222 18.6637i −0.201115 1.43144i
\(171\) 0 0
\(172\) 16.3872i 1.24951i
\(173\) 2.06022i 0.156636i −0.996928 0.0783179i \(-0.975045\pi\)
0.996928 0.0783179i \(-0.0249549\pi\)
\(174\) 0 0
\(175\) −1.37778 4.80642i −0.104151 0.363331i
\(176\) 9.22570 0.695413
\(177\) 0 0
\(178\) 8.79706i 0.659367i
\(179\) 10.0000 0.747435 0.373718 0.927543i \(-0.378083\pi\)
0.373718 + 0.927543i \(0.378083\pi\)
\(180\) 0 0
\(181\) −12.1017 −0.899513 −0.449757 0.893151i \(-0.648489\pi\)
−0.449757 + 0.893151i \(0.648489\pi\)
\(182\) 12.2351i 0.906923i
\(183\) 0 0
\(184\) −0.990632 −0.0730304
\(185\) −16.8573 + 2.36842i −1.23937 + 0.174129i
\(186\) 0 0
\(187\) 8.85728i 0.647708i
\(188\) 4.47013i 0.326017i
\(189\) 0 0
\(190\) −10.2351 + 1.43801i −0.742530 + 0.104324i
\(191\) −0.488863 −0.0353729 −0.0176864 0.999844i \(-0.505630\pi\)
−0.0176864 + 0.999844i \(0.505630\pi\)
\(192\) 0 0
\(193\) 22.9590i 1.65262i −0.563212 0.826312i \(-0.690435\pi\)
0.563212 0.826312i \(-0.309565\pi\)
\(194\) 22.7239 1.63148
\(195\) 0 0
\(196\) 1.62222 0.115873
\(197\) 1.18421i 0.0843713i −0.999110 0.0421857i \(-0.986568\pi\)
0.999110 0.0421857i \(-0.0134321\pi\)
\(198\) 0 0
\(199\) −8.79706 −0.623607 −0.311803 0.950147i \(-0.600933\pi\)
−0.311803 + 0.950147i \(0.600933\pi\)
\(200\) 0.990632 + 3.45584i 0.0700483 + 0.244365i
\(201\) 0 0
\(202\) 2.81579i 0.198118i
\(203\) 0.755569i 0.0530305i
\(204\) 0 0
\(205\) −2.56199 18.2351i −0.178937 1.27359i
\(206\) −16.8573 −1.17450
\(207\) 0 0
\(208\) 29.6543i 2.05616i
\(209\) −4.85728 −0.335985
\(210\) 0 0
\(211\) 23.2257 1.59892 0.799461 0.600717i \(-0.205118\pi\)
0.799461 + 0.600717i \(0.205118\pi\)
\(212\) 14.8988i 1.02325i
\(213\) 0 0
\(214\) 3.35905 0.229620
\(215\) 22.3684 3.14272i 1.52551 0.214332i
\(216\) 0 0
\(217\) 5.18421i 0.351927i
\(218\) 10.6824i 0.723506i
\(219\) 0 0
\(220\) −1.00937 7.18421i −0.0680516 0.484359i
\(221\) 28.4701 1.91511
\(222\) 0 0
\(223\) 15.2257i 1.01959i −0.860297 0.509794i \(-0.829722\pi\)
0.860297 0.509794i \(-0.170278\pi\)
\(224\) −7.34122 −0.490506
\(225\) 0 0
\(226\) 21.4795 1.42879
\(227\) 14.3684i 0.953665i −0.878994 0.476833i \(-0.841785\pi\)
0.878994 0.476833i \(-0.158215\pi\)
\(228\) 0 0
\(229\) 5.61285 0.370907 0.185454 0.982653i \(-0.440625\pi\)
0.185454 + 0.982653i \(0.440625\pi\)
\(230\) 0.815792 + 5.80642i 0.0537917 + 0.382864i
\(231\) 0 0
\(232\) 0.543257i 0.0356666i
\(233\) 23.2859i 1.52551i −0.646687 0.762756i \(-0.723846\pi\)
0.646687 0.762756i \(-0.276154\pi\)
\(234\) 0 0
\(235\) 6.10171 0.857279i 0.398032 0.0559227i
\(236\) −22.8760 −1.48910
\(237\) 0 0
\(238\) 8.42864i 0.546348i
\(239\) −8.48886 −0.549099 −0.274549 0.961573i \(-0.588529\pi\)
−0.274549 + 0.961573i \(0.588529\pi\)
\(240\) 0 0
\(241\) −7.24443 −0.466655 −0.233327 0.972398i \(-0.574961\pi\)
−0.233327 + 0.972398i \(0.574961\pi\)
\(242\) 13.3225i 0.856402i
\(243\) 0 0
\(244\) −11.1240 −0.712140
\(245\) 0.311108 + 2.21432i 0.0198759 + 0.141468i
\(246\) 0 0
\(247\) 15.6128i 0.993422i
\(248\) 3.72746i 0.236694i
\(249\) 0 0
\(250\) 19.4400 8.65233i 1.22949 0.547221i
\(251\) −27.6128 −1.74291 −0.871454 0.490478i \(-0.836822\pi\)
−0.871454 + 0.490478i \(0.836822\pi\)
\(252\) 0 0
\(253\) 2.75557i 0.173241i
\(254\) 24.4701 1.53539
\(255\) 0 0
\(256\) 20.2444 1.26528
\(257\) 0.428639i 0.0267378i −0.999911 0.0133689i \(-0.995744\pi\)
0.999911 0.0133689i \(-0.00425558\pi\)
\(258\) 0 0
\(259\) 7.61285 0.473039
\(260\) 23.0923 3.24443i 1.43213 0.201211i
\(261\) 0 0
\(262\) 4.00000i 0.247121i
\(263\) 9.37778i 0.578259i 0.957290 + 0.289129i \(0.0933658\pi\)
−0.957290 + 0.289129i \(0.906634\pi\)
\(264\) 0 0
\(265\) −20.3368 + 2.85728i −1.24928 + 0.175521i
\(266\) 4.62222 0.283406
\(267\) 0 0
\(268\) 4.47013i 0.273056i
\(269\) 1.74620 0.106468 0.0532339 0.998582i \(-0.483047\pi\)
0.0532339 + 0.998582i \(0.483047\pi\)
\(270\) 0 0
\(271\) −2.69535 −0.163731 −0.0818653 0.996643i \(-0.526088\pi\)
−0.0818653 + 0.996643i \(0.526088\pi\)
\(272\) 20.4286i 1.23867i
\(273\) 0 0
\(274\) 30.3368 1.83271
\(275\) 9.61285 2.75557i 0.579677 0.166167i
\(276\) 0 0
\(277\) 5.12399i 0.307870i 0.988081 + 0.153935i \(0.0491947\pi\)
−0.988081 + 0.153935i \(0.950805\pi\)
\(278\) 22.2163i 1.33245i
\(279\) 0 0
\(280\) −0.223688 1.59210i −0.0133679 0.0951464i
\(281\) −23.9813 −1.43060 −0.715301 0.698816i \(-0.753710\pi\)
−0.715301 + 0.698816i \(0.753710\pi\)
\(282\) 0 0
\(283\) 2.36842i 0.140788i 0.997519 + 0.0703939i \(0.0224256\pi\)
−0.997519 + 0.0703939i \(0.977574\pi\)
\(284\) 3.24443 0.192522
\(285\) 0 0
\(286\) 24.4701 1.44695
\(287\) 8.23506i 0.486100i
\(288\) 0 0
\(289\) −2.61285 −0.153697
\(290\) −3.18421 + 0.447375i −0.186983 + 0.0262708i
\(291\) 0 0
\(292\) 2.54909i 0.149174i
\(293\) 8.42864i 0.492406i −0.969218 0.246203i \(-0.920817\pi\)
0.969218 0.246203i \(-0.0791831\pi\)
\(294\) 0 0
\(295\) −4.38715 31.2257i −0.255430 1.81803i
\(296\) −5.47367 −0.318150
\(297\) 0 0
\(298\) 40.3970i 2.34014i
\(299\) −8.85728 −0.512230
\(300\) 0 0
\(301\) −10.1017 −0.582253
\(302\) 32.0830i 1.84617i
\(303\) 0 0
\(304\) −11.2029 −0.642533
\(305\) −2.13335 15.1842i −0.122155 0.869445i
\(306\) 0 0
\(307\) 22.5718i 1.28824i 0.764923 + 0.644121i \(0.222777\pi\)
−0.764923 + 0.644121i \(0.777223\pi\)
\(308\) 3.24443i 0.184869i
\(309\) 0 0
\(310\) −21.8479 + 3.06959i −1.24088 + 0.174341i
\(311\) 24.0830 1.36562 0.682810 0.730596i \(-0.260758\pi\)
0.682810 + 0.730596i \(0.260758\pi\)
\(312\) 0 0
\(313\) 9.65433i 0.545695i 0.962057 + 0.272848i \(0.0879655\pi\)
−0.962057 + 0.272848i \(0.912035\pi\)
\(314\) 19.8479 1.12008
\(315\) 0 0
\(316\) −7.87955 −0.443260
\(317\) 6.04149i 0.339324i −0.985502 0.169662i \(-0.945732\pi\)
0.985502 0.169662i \(-0.0542676\pi\)
\(318\) 0 0
\(319\) −1.51114 −0.0846075
\(320\) −1.47658 10.5096i −0.0825433 0.587505i
\(321\) 0 0
\(322\) 2.62222i 0.146130i
\(323\) 10.7556i 0.598456i
\(324\) 0 0
\(325\) 8.85728 + 30.8988i 0.491313 + 1.71396i
\(326\) 39.6958 2.19855
\(327\) 0 0
\(328\) 5.92104i 0.326935i
\(329\) −2.75557 −0.151919
\(330\) 0 0
\(331\) 13.5111 0.742639 0.371320 0.928505i \(-0.378905\pi\)
0.371320 + 0.928505i \(0.378905\pi\)
\(332\) 18.8385i 1.03390i
\(333\) 0 0
\(334\) 29.2070 1.59813
\(335\) −6.10171 + 0.857279i −0.333372 + 0.0468382i
\(336\) 0 0
\(337\) 10.4889i 0.571365i −0.958324 0.285682i \(-0.907780\pi\)
0.958324 0.285682i \(-0.0922202\pi\)
\(338\) 53.9131i 2.93248i
\(339\) 0 0
\(340\) −15.9081 + 2.23506i −0.862740 + 0.121213i
\(341\) −10.3684 −0.561481
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 7.26317 0.391604
\(345\) 0 0
\(346\) −3.92104 −0.210796
\(347\) 16.7239i 0.897787i −0.893585 0.448894i \(-0.851818\pi\)
0.893585 0.448894i \(-0.148182\pi\)
\(348\) 0 0
\(349\) 16.3684 0.876181 0.438091 0.898931i \(-0.355655\pi\)
0.438091 + 0.898931i \(0.355655\pi\)
\(350\) −9.14764 + 2.62222i −0.488962 + 0.140163i
\(351\) 0 0
\(352\) 14.6824i 0.782577i
\(353\) 0.549086i 0.0292249i −0.999893 0.0146124i \(-0.995349\pi\)
0.999893 0.0146124i \(-0.00465145\pi\)
\(354\) 0 0
\(355\) 0.622216 + 4.42864i 0.0330238 + 0.235048i
\(356\) −7.49823 −0.397405
\(357\) 0 0
\(358\) 19.0321i 1.00588i
\(359\) −0.285442 −0.0150651 −0.00753253 0.999972i \(-0.502398\pi\)
−0.00753253 + 0.999972i \(0.502398\pi\)
\(360\) 0 0
\(361\) −13.1017 −0.689564
\(362\) 23.0321i 1.21054i
\(363\) 0 0
\(364\) −10.4286 −0.546609
\(365\) −3.47949 + 0.488863i −0.182125 + 0.0255882i
\(366\) 0 0
\(367\) 1.71456i 0.0894992i −0.998998 0.0447496i \(-0.985751\pi\)
0.998998 0.0447496i \(-0.0142490\pi\)
\(368\) 6.35551i 0.331304i
\(369\) 0 0
\(370\) 4.50760 + 32.0830i 0.234339 + 1.66791i
\(371\) 9.18421 0.476820
\(372\) 0 0
\(373\) 16.0000i 0.828449i 0.910175 + 0.414224i \(0.135947\pi\)
−0.910175 + 0.414224i \(0.864053\pi\)
\(374\) −16.8573 −0.871669
\(375\) 0 0
\(376\) 1.98126 0.102176
\(377\) 4.85728i 0.250163i
\(378\) 0 0
\(379\) 4.85728 0.249502 0.124751 0.992188i \(-0.460187\pi\)
0.124751 + 0.992188i \(0.460187\pi\)
\(380\) 1.22570 + 8.72393i 0.0628768 + 0.447528i
\(381\) 0 0
\(382\) 0.930409i 0.0476039i
\(383\) 8.38715i 0.428563i 0.976772 + 0.214282i \(0.0687411\pi\)
−0.976772 + 0.214282i \(0.931259\pi\)
\(384\) 0 0
\(385\) −4.42864 + 0.622216i −0.225704 + 0.0317110i
\(386\) −43.6958 −2.22406
\(387\) 0 0
\(388\) 19.3689i 0.983307i
\(389\) 8.95899 0.454239 0.227119 0.973867i \(-0.427069\pi\)
0.227119 + 0.973867i \(0.427069\pi\)
\(390\) 0 0
\(391\) 6.10171 0.308577
\(392\) 0.719004i 0.0363152i
\(393\) 0 0
\(394\) −2.25380 −0.113545
\(395\) −1.51114 10.7556i −0.0760336 0.541171i
\(396\) 0 0
\(397\) 2.54909i 0.127935i 0.997952 + 0.0639675i \(0.0203754\pi\)
−0.997952 + 0.0639675i \(0.979625\pi\)
\(398\) 16.7427i 0.839234i
\(399\) 0 0
\(400\) 22.1713 6.35551i 1.10857 0.317775i
\(401\) −0.958989 −0.0478896 −0.0239448 0.999713i \(-0.507623\pi\)
−0.0239448 + 0.999713i \(0.507623\pi\)
\(402\) 0 0
\(403\) 33.3274i 1.66016i
\(404\) 2.40006 0.119407
\(405\) 0 0
\(406\) 1.43801 0.0713671
\(407\) 15.2257i 0.754710i
\(408\) 0 0
\(409\) 31.9813 1.58137 0.790686 0.612222i \(-0.209724\pi\)
0.790686 + 0.612222i \(0.209724\pi\)
\(410\) −34.7052 + 4.87601i −1.71397 + 0.240809i
\(411\) 0 0
\(412\) 14.3684i 0.707881i
\(413\) 14.1017i 0.693900i
\(414\) 0 0
\(415\) −25.7146 + 3.61285i −1.26228 + 0.177348i
\(416\) 47.1941 2.31388
\(417\) 0 0
\(418\) 9.24443i 0.452160i
\(419\) −0.470127 −0.0229672 −0.0114836 0.999934i \(-0.503655\pi\)
−0.0114836 + 0.999934i \(0.503655\pi\)
\(420\) 0 0
\(421\) −33.6128 −1.63819 −0.819095 0.573658i \(-0.805524\pi\)
−0.819095 + 0.573658i \(0.805524\pi\)
\(422\) 44.2034i 2.15179i
\(423\) 0 0
\(424\) −6.60348 −0.320693
\(425\) −6.10171 21.2859i −0.295976 1.03252i
\(426\) 0 0
\(427\) 6.85728i 0.331847i
\(428\) 2.86311i 0.138394i
\(429\) 0 0
\(430\) −5.98126 42.5718i −0.288442 2.05300i
\(431\) −11.7146 −0.564270 −0.282135 0.959375i \(-0.591043\pi\)
−0.282135 + 0.959375i \(0.591043\pi\)
\(432\) 0 0
\(433\) 0.0602231i 0.00289414i −0.999999 0.00144707i \(-0.999539\pi\)
0.999999 0.00144707i \(-0.000460616\pi\)
\(434\) 9.86665 0.473614
\(435\) 0 0
\(436\) 9.10525 0.436062
\(437\) 3.34614i 0.160068i
\(438\) 0 0
\(439\) 22.4286 1.07046 0.535230 0.844706i \(-0.320225\pi\)
0.535230 + 0.844706i \(0.320225\pi\)
\(440\) 3.18421 0.447375i 0.151801 0.0213278i
\(441\) 0 0
\(442\) 54.1847i 2.57730i
\(443\) 23.9496i 1.13788i 0.822379 + 0.568940i \(0.192647\pi\)
−0.822379 + 0.568940i \(0.807353\pi\)
\(444\) 0 0
\(445\) −1.43801 10.2351i −0.0681681 0.485189i
\(446\) −28.9777 −1.37214
\(447\) 0 0
\(448\) 4.74620i 0.224237i
\(449\) 29.4291 1.38885 0.694423 0.719567i \(-0.255660\pi\)
0.694423 + 0.719567i \(0.255660\pi\)
\(450\) 0 0
\(451\) −16.4701 −0.775548
\(452\) 18.3082i 0.861145i
\(453\) 0 0
\(454\) −27.3461 −1.28342
\(455\) −2.00000 14.2351i −0.0937614 0.667350i
\(456\) 0 0
\(457\) 3.14272i 0.147010i 0.997295 + 0.0735051i \(0.0234185\pi\)
−0.997295 + 0.0735051i \(0.976581\pi\)
\(458\) 10.6824i 0.499158i
\(459\) 0 0
\(460\) 4.94914 0.695346i 0.230755 0.0324207i
\(461\) 3.37778 0.157319 0.0786596 0.996902i \(-0.474936\pi\)
0.0786596 + 0.996902i \(0.474936\pi\)
\(462\) 0 0
\(463\) 20.8573i 0.969320i 0.874703 + 0.484660i \(0.161057\pi\)
−0.874703 + 0.484660i \(0.838943\pi\)
\(464\) −3.48532 −0.161802
\(465\) 0 0
\(466\) −44.3180 −2.05299
\(467\) 14.3684i 0.664891i −0.943122 0.332446i \(-0.892126\pi\)
0.943122 0.332446i \(-0.107874\pi\)
\(468\) 0 0
\(469\) 2.75557 0.127240
\(470\) −1.63158 11.6128i −0.0752593 0.535661i
\(471\) 0 0
\(472\) 10.1392i 0.466694i
\(473\) 20.2034i 0.928954i
\(474\) 0 0
\(475\) −11.6731 + 3.34614i −0.535597 + 0.153532i
\(476\) 7.18421 0.329288
\(477\) 0 0
\(478\) 16.1561i 0.738963i
\(479\) 6.36842 0.290980 0.145490 0.989360i \(-0.453524\pi\)
0.145490 + 0.989360i \(0.453524\pi\)
\(480\) 0 0
\(481\) −48.9403 −2.23148
\(482\) 13.7877i 0.628012i
\(483\) 0 0
\(484\) 11.3555 0.516160
\(485\) 26.4385 3.71456i 1.20051 0.168669i
\(486\) 0 0
\(487\) 17.3274i 0.785180i −0.919714 0.392590i \(-0.871579\pi\)
0.919714 0.392590i \(-0.128421\pi\)
\(488\) 4.93041i 0.223189i
\(489\) 0 0
\(490\) 4.21432 0.592104i 0.190384 0.0267485i
\(491\) −2.00000 −0.0902587 −0.0451294 0.998981i \(-0.514370\pi\)
−0.0451294 + 0.998981i \(0.514370\pi\)
\(492\) 0 0
\(493\) 3.34614i 0.150703i
\(494\) −29.7146 −1.33692
\(495\) 0 0
\(496\) −23.9140 −1.07377
\(497\) 2.00000i 0.0897123i
\(498\) 0 0
\(499\) −23.3461 −1.04512 −0.522558 0.852603i \(-0.675022\pi\)
−0.522558 + 0.852603i \(0.675022\pi\)
\(500\) −7.37487 16.5698i −0.329814 0.741025i
\(501\) 0 0
\(502\) 52.5531i 2.34556i
\(503\) 0.387152i 0.0172623i −0.999963 0.00863113i \(-0.997253\pi\)
0.999963 0.00863113i \(-0.00274741\pi\)
\(504\) 0 0
\(505\) 0.460282 + 3.27607i 0.0204823 + 0.145783i
\(506\) 5.24443 0.233143
\(507\) 0 0
\(508\) 20.8573i 0.925392i
\(509\) −29.9496 −1.32749 −0.663747 0.747957i \(-0.731035\pi\)
−0.663747 + 0.747957i \(0.731035\pi\)
\(510\) 0 0
\(511\) 1.57136 0.0695129
\(512\) 27.2306i 1.20343i
\(513\) 0 0
\(514\) −0.815792 −0.0359830
\(515\) −19.6128 + 2.75557i −0.864245 + 0.121425i
\(516\) 0 0
\(517\) 5.51114i 0.242380i
\(518\) 14.4889i 0.636604i
\(519\) 0 0
\(520\) 1.43801 + 10.2351i 0.0630608 + 0.448837i
\(521\) 18.5205 0.811398 0.405699 0.914007i \(-0.367028\pi\)
0.405699 + 0.914007i \(0.367028\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) −3.40943 −0.148942
\(525\) 0 0
\(526\) 17.8479 0.778206
\(527\) 22.9590i 1.00011i
\(528\) 0 0
\(529\) 21.1017 0.917466
\(530\) 5.43801 + 38.7052i 0.236212 + 1.68125i
\(531\) 0 0
\(532\) 3.93978i 0.170811i
\(533\) 52.9403i 2.29310i
\(534\) 0 0
\(535\) 3.90813 0.549086i 0.168963 0.0237390i
\(536\) −1.98126 −0.0855776
\(537\) 0 0
\(538\) 3.32339i 0.143282i
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 14.5906 0.627298 0.313649 0.949539i \(-0.398449\pi\)
0.313649 + 0.949539i \(0.398449\pi\)
\(542\) 5.12981i 0.220344i
\(543\) 0 0
\(544\) −32.5116 −1.39392
\(545\) 1.74620 + 12.4286i 0.0747990 + 0.532384i
\(546\) 0 0
\(547\) 18.7556i 0.801930i 0.916093 + 0.400965i \(0.131325\pi\)
−0.916093 + 0.400965i \(0.868675\pi\)
\(548\) 25.8578i 1.10459i
\(549\) 0 0
\(550\) −5.24443 18.2953i −0.223623 0.780114i
\(551\) 1.83500 0.0781738
\(552\) 0 0
\(553\) 4.85728i 0.206553i
\(554\) 9.75203 0.414324
\(555\) 0 0
\(556\) 18.9362 0.803075
\(557\) 31.8765i 1.35065i 0.737520 + 0.675325i \(0.235997\pi\)
−0.737520 + 0.675325i \(0.764003\pi\)
\(558\) 0 0
\(559\) 64.9403 2.74668
\(560\) −10.2143 + 1.43509i −0.431634 + 0.0606437i
\(561\) 0 0
\(562\) 45.6414i 1.92527i
\(563\) 2.01874i 0.0850796i 0.999095 + 0.0425398i \(0.0135449\pi\)
−0.999095 + 0.0425398i \(0.986455\pi\)
\(564\) 0 0
\(565\) 24.9906 3.51114i 1.05136 0.147715i
\(566\) 4.50760 0.189468
\(567\) 0 0
\(568\) 1.43801i 0.0603375i
\(569\) −28.9590 −1.21402 −0.607012 0.794693i \(-0.707632\pi\)
−0.607012 + 0.794693i \(0.707632\pi\)
\(570\) 0 0
\(571\) 8.97773 0.375706 0.187853 0.982197i \(-0.439847\pi\)
0.187853 + 0.982197i \(0.439847\pi\)
\(572\) 20.8573i 0.872087i
\(573\) 0 0
\(574\) 15.6731 0.654181
\(575\) 1.89829 + 6.62222i 0.0791642 + 0.276165i
\(576\) 0 0
\(577\) 28.6766i 1.19382i −0.802307 0.596911i \(-0.796394\pi\)
0.802307 0.596911i \(-0.203606\pi\)
\(578\) 4.97280i 0.206841i
\(579\) 0 0
\(580\) 0.381323 + 2.71408i 0.0158336 + 0.112696i
\(581\) 11.6128 0.481782
\(582\) 0 0
\(583\) 18.3684i 0.760742i
\(584\) −1.12981 −0.0467520
\(585\) 0 0
\(586\) −16.0415 −0.662668
\(587\) 45.2070i 1.86589i 0.360018 + 0.932945i \(0.382771\pi\)
−0.360018 + 0.932945i \(0.617229\pi\)
\(588\) 0 0
\(589\) 12.5906 0.518786
\(590\) −59.4291 + 8.34968i −2.44666 + 0.343751i
\(591\) 0 0
\(592\) 35.1169i 1.44330i
\(593\) 18.2636i 0.749998i 0.927025 + 0.374999i \(0.122357\pi\)
−0.927025 + 0.374999i \(0.877643\pi\)
\(594\) 0 0
\(595\) 1.37778 + 9.80642i 0.0564837 + 0.402024i
\(596\) 34.4327 1.41042
\(597\) 0 0
\(598\) 16.8573i 0.689345i
\(599\) −22.7368 −0.929002 −0.464501 0.885573i \(-0.653766\pi\)
−0.464501 + 0.885573i \(0.653766\pi\)
\(600\) 0 0
\(601\) 0.488863 0.0199411 0.00997056 0.999950i \(-0.496826\pi\)
0.00997056 + 0.999950i \(0.496826\pi\)
\(602\) 19.2257i 0.783581i
\(603\) 0 0
\(604\) −27.3461 −1.11270
\(605\) 2.17775 + 15.5002i 0.0885383 + 0.630174i
\(606\) 0 0
\(607\) 20.2034i 0.820032i −0.912078 0.410016i \(-0.865523\pi\)
0.912078 0.410016i \(-0.134477\pi\)
\(608\) 17.8292i 0.723069i
\(609\) 0 0
\(610\) −28.8988 + 4.06022i −1.17008 + 0.164394i
\(611\) 17.7146 0.716654
\(612\) 0 0
\(613\) 10.3684i 0.418776i −0.977833 0.209388i \(-0.932853\pi\)
0.977833 0.209388i \(-0.0671472\pi\)
\(614\) 42.9590 1.73368
\(615\) 0 0
\(616\) −1.43801 −0.0579390
\(617\) 39.2859i 1.58159i 0.612080 + 0.790796i \(0.290333\pi\)
−0.612080 + 0.790796i \(0.709667\pi\)
\(618\) 0 0
\(619\) −42.8988 −1.72425 −0.862123 0.506698i \(-0.830866\pi\)
−0.862123 + 0.506698i \(0.830866\pi\)
\(620\) 2.61639 + 18.6222i 0.105077 + 0.747886i
\(621\) 0 0
\(622\) 45.8350i 1.83782i
\(623\) 4.62222i 0.185185i
\(624\) 0 0
\(625\) 21.2034 13.2444i 0.848137 0.529777i
\(626\) 18.3742 0.734383
\(627\) 0 0
\(628\) 16.9175i 0.675082i
\(629\) 33.7146 1.34429
\(630\) 0 0
\(631\) 15.3461 0.610920 0.305460 0.952205i \(-0.401190\pi\)
0.305460 + 0.952205i \(0.401190\pi\)
\(632\) 3.49240i 0.138920i
\(633\) 0 0
\(634\) −11.4982 −0.456653
\(635\) 28.4701 4.00000i 1.12980 0.158735i
\(636\) 0 0
\(637\) 6.42864i 0.254712i
\(638\) 2.87601i 0.113863i
\(639\) 0 0
\(640\) 12.5096 1.75758i 0.494486 0.0694743i
\(641\) −30.6735 −1.21153 −0.605766 0.795643i \(-0.707133\pi\)
−0.605766 + 0.795643i \(0.707133\pi\)
\(642\) 0 0
\(643\) 49.0607i 1.93477i 0.253320 + 0.967383i \(0.418477\pi\)
−0.253320 + 0.967383i \(0.581523\pi\)
\(644\) −2.23506 −0.0880738
\(645\) 0 0
\(646\) 20.4701 0.805386
\(647\) 15.3461i 0.603319i 0.953416 + 0.301660i \(0.0975406\pi\)
−0.953416 + 0.301660i \(0.902459\pi\)
\(648\) 0 0
\(649\) −28.2034 −1.10708
\(650\) 58.8069 16.8573i 2.30660 0.661197i
\(651\) 0 0
\(652\) 33.8350i 1.32508i
\(653\) 19.4697i 0.761906i −0.924594 0.380953i \(-0.875596\pi\)
0.924594 0.380953i \(-0.124404\pi\)
\(654\) 0 0
\(655\) −0.653858 4.65386i −0.0255484 0.181841i
\(656\) −37.9871 −1.48315
\(657\) 0 0
\(658\) 5.24443i 0.204449i
\(659\) 30.9403 1.20526 0.602631 0.798020i \(-0.294119\pi\)
0.602631 + 0.798020i \(0.294119\pi\)
\(660\) 0 0
\(661\) 47.7975 1.85911 0.929554 0.368685i \(-0.120192\pi\)
0.929554 + 0.368685i \(0.120192\pi\)
\(662\) 25.7146i 0.999425i
\(663\) 0 0
\(664\) −8.34968 −0.324030
\(665\) 5.37778 0.755569i 0.208542 0.0292997i
\(666\) 0 0
\(667\) 1.04101i 0.0403081i
\(668\) 24.8948i 0.963207i
\(669\) 0 0
\(670\) 1.63158 + 11.6128i 0.0630336 + 0.448643i
\(671\) −13.7146 −0.529445
\(672\) 0 0
\(673\) 27.8163i 1.07224i −0.844142 0.536119i \(-0.819890\pi\)
0.844142 0.536119i \(-0.180110\pi\)
\(674\) −19.9625 −0.768928
\(675\) 0 0
\(676\) 45.9532 1.76743
\(677\) 19.0005i 0.730248i 0.930959 + 0.365124i \(0.118973\pi\)
−0.930959 + 0.365124i \(0.881027\pi\)
\(678\) 0 0
\(679\) −11.9398 −0.458207
\(680\) −0.990632 7.05086i −0.0379890 0.270388i
\(681\) 0 0
\(682\) 19.7333i 0.755627i
\(683\) 4.52051i 0.172972i 0.996253 + 0.0864862i \(0.0275638\pi\)
−0.996253 + 0.0864862i \(0.972436\pi\)
\(684\) 0 0
\(685\) 35.2958 4.95899i 1.34858 0.189473i
\(686\) −1.90321 −0.0726650
\(687\) 0 0
\(688\) 46.5977i 1.77652i
\(689\) −59.0420 −2.24932
\(690\) 0 0
\(691\) −1.18421 −0.0450494 −0.0225247 0.999746i \(-0.507170\pi\)
−0.0225247 + 0.999746i \(0.507170\pi\)
\(692\) 3.34213i 0.127049i
\(693\) 0 0
\(694\) −31.8292 −1.20822
\(695\) 3.63158 + 25.8479i 0.137754 + 0.980467i
\(696\) 0 0
\(697\) 36.4701i 1.38140i
\(698\) 31.1526i 1.17914i
\(699\) 0 0
\(700\) 2.23506 + 7.79706i 0.0844775 + 0.294701i
\(701\) 26.6735 1.00745 0.503723 0.863865i \(-0.331963\pi\)
0.503723 + 0.863865i \(0.331963\pi\)
\(702\) 0 0
\(703\) 18.4889i 0.697321i
\(704\) −9.49240 −0.357758
\(705\) 0 0
\(706\) −1.04503 −0.0393301
\(707\) 1.47949i 0.0556421i
\(708\) 0 0
\(709\) 18.2034 0.683644 0.341822 0.939765i \(-0.388956\pi\)
0.341822 + 0.939765i \(0.388956\pi\)
\(710\) 8.42864 1.18421i 0.316321 0.0444425i
\(711\) 0 0
\(712\) 3.32339i 0.124549i
\(713\) 7.14272i 0.267497i
\(714\) 0 0
\(715\) 28.4701 4.00000i 1.06472 0.149592i
\(716\) −16.2222 −0.606250
\(717\) 0 0
\(718\) 0.543257i 0.0202742i
\(719\) 4.85728 0.181146 0.0905730 0.995890i \(-0.471130\pi\)
0.0905730 + 0.995890i \(0.471130\pi\)
\(720\) 0 0
\(721\) 8.85728 0.329862
\(722\) 24.9353i 0.927997i
\(723\) 0 0
\(724\) 19.6316 0.729602
\(725\) −3.63158 + 1.04101i −0.134874 + 0.0386622i
\(726\) 0 0
\(727\) 21.0607i 0.781098i 0.920582 + 0.390549i \(0.127715\pi\)
−0.920582 + 0.390549i \(0.872285\pi\)
\(728\) 4.62222i 0.171311i
\(729\) 0 0
\(730\) 0.930409 + 6.62222i 0.0344360 + 0.245099i
\(731\) −44.7368 −1.65465
\(732\) 0 0
\(733\) 9.45091i 0.349077i −0.984650 0.174539i \(-0.944157\pi\)
0.984650 0.174539i \(-0.0558434\pi\)
\(734\) −3.26317 −0.120446
\(735\) 0 0
\(736\) 10.1146 0.372830
\(737\) 5.51114i 0.203005i
\(738\) 0 0
\(739\) 8.20342 0.301768 0.150884 0.988551i \(-0.451788\pi\)
0.150884 + 0.988551i \(0.451788\pi\)
\(740\) 27.3461 3.84208i 1.00526 0.141238i
\(741\) 0 0
\(742\) 17.4795i 0.641692i
\(743\) 8.33677i 0.305847i 0.988238 + 0.152923i \(0.0488687\pi\)
−0.988238 + 0.152923i \(0.951131\pi\)
\(744\) 0 0
\(745\) 6.60348 + 47.0005i 0.241933 + 1.72196i
\(746\) 30.4514 1.11490
\(747\) 0 0
\(748\) 14.3684i 0.525361i
\(749\) −1.76494 −0.0644894
\(750\) 0 0
\(751\) −25.9180 −0.945760 −0.472880 0.881127i \(-0.656786\pi\)
−0.472880 + 0.881127i \(0.656786\pi\)
\(752\) 12.7110i 0.463523i
\(753\) 0 0
\(754\) −9.24443 −0.336662
\(755\) −5.24443 37.3274i −0.190864 1.35848i
\(756\) 0 0
\(757\) 8.94025i 0.324939i 0.986714 + 0.162470i \(0.0519459\pi\)
−0.986714 + 0.162470i \(0.948054\pi\)
\(758\) 9.24443i 0.335773i
\(759\) 0 0
\(760\) −3.86665 + 0.543257i −0.140258 + 0.0197060i
\(761\) 0.825636 0.0299293 0.0149646 0.999888i \(-0.495236\pi\)
0.0149646 + 0.999888i \(0.495236\pi\)
\(762\) 0 0
\(763\) 5.61285i 0.203199i
\(764\) 0.793040 0.0286912
\(765\) 0 0
\(766\) 15.9625 0.576750
\(767\) 90.6548i 3.27336i
\(768\) 0 0
\(769\) −21.2257 −0.765418 −0.382709 0.923869i \(-0.625009\pi\)
−0.382709 + 0.923869i \(0.625009\pi\)
\(770\) 1.18421 + 8.42864i 0.0426759 + 0.303747i
\(771\) 0 0
\(772\) 37.2444i 1.34046i
\(773\) 29.4893i 1.06066i 0.847792 + 0.530329i \(0.177932\pi\)
−0.847792 + 0.530329i \(0.822068\pi\)
\(774\) 0 0
\(775\) −24.9175 + 7.14272i −0.895063 + 0.256574i
\(776\) 8.58474 0.308174
\(777\) 0 0
\(778\) 17.0509i 0.611303i
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 11.6128i 0.415275i
\(783\) 0 0
\(784\) 4.61285 0.164745
\(785\) 23.0923 3.24443i 0.824201 0.115799i
\(786\) 0 0
\(787\) 34.4514i 1.22806i −0.789283 0.614030i \(-0.789547\pi\)
0.789283 0.614030i \(-0.210453\pi\)
\(788\) 1.92104i 0.0684343i
\(789\) 0 0
\(790\) −20.4701 + 2.87601i −0.728294 + 0.102324i
\(791\) −11.2859 −0.401281
\(792\) 0 0
\(793\) 44.0830i 1.56543i
\(794\) 4.85145 0.172172
\(795\) 0 0
\(796\) 14.2707 0.505812
\(797\) 18.9175i 0.670092i −0.942202 0.335046i \(-0.891248\pi\)
0.942202 0.335046i \(-0.108752\pi\)
\(798\) 0 0
\(799\) −12.2034 −0.431726
\(800\) −10.1146 35.2850i −0.357606 1.24751i
\(801\) 0 0
\(802\) 1.82516i 0.0644486i
\(803\) 3.14272i 0.110904i
\(804\) 0 0
\(805\) −0.428639 3.05086i −0.0151076 0.107529i
\(806\) −63.4291 −2.23420
\(807\) 0 0
\(808\) 1.06376i 0.0374230i
\(809\) 21.2257 0.746256 0.373128 0.927780i \(-0.378285\pi\)
0.373128 + 0.927780i \(0.378285\pi\)
\(810\) 0 0
\(811\) −21.5081 −0.755251 −0.377625 0.925958i \(-0.623259\pi\)
−0.377625 + 0.925958i \(0.623259\pi\)
\(812\) 1.22570i 0.0430135i
\(813\) 0 0
\(814\) 28.9777 1.01567
\(815\) 46.1847 6.48886i 1.61778 0.227295i
\(816\) 0 0
\(817\) 24.5334i 0.858315i
\(818\) 60.8671i 2.12817i
\(819\) 0 0
\(820\) 4.15610 + 29.5812i 0.145137 + 1.03302i
\(821\) 46.2034 1.61251 0.806255 0.591568i \(-0.201491\pi\)
0.806255 + 0.591568i \(0.201491\pi\)
\(822\) 0 0
\(823\) 17.8350i 0.621689i 0.950461 + 0.310845i \(0.100612\pi\)
−0.950461 + 0.310845i \(0.899388\pi\)
\(824\) −6.36842 −0.221854
\(825\) 0 0
\(826\) 26.8385 0.933832
\(827\) 35.2128i 1.22447i −0.790676 0.612234i \(-0.790271\pi\)
0.790676 0.612234i \(-0.209729\pi\)
\(828\) 0 0
\(829\) −14.3872 −0.499686 −0.249843 0.968286i \(-0.580379\pi\)
−0.249843 + 0.968286i \(0.580379\pi\)
\(830\) 6.87601 + 48.9403i 0.238670 + 1.69874i
\(831\) 0 0
\(832\) 30.5116i 1.05780i
\(833\) 4.42864i 0.153443i
\(834\) 0 0
\(835\) 33.9813 4.77430i 1.17597 0.165222i
\(836\) 7.87955 0.272520
\(837\) 0 0
\(838\) 0.894751i 0.0309087i
\(839\) 1.51114 0.0521703 0.0260851 0.999660i \(-0.491696\pi\)
0.0260851 + 0.999660i \(0.491696\pi\)
\(840\) 0 0
\(841\) −28.4291 −0.980314
\(842\) 63.9724i 2.20463i
\(843\) 0 0
\(844\) −37.6771 −1.29690
\(845\) 8.81288 + 62.7259i 0.303172 + 2.15784i
\(846\) 0 0
\(847\) 7.00000i 0.240523i
\(848\) 42.3654i 1.45483i
\(849\) 0 0
\(850\) −40.5116 + 11.6128i −1.38954 + 0.398317i
\(851\) −10.4889 −0.359554
\(852\) 0 0
\(853\) 15.4064i 0.527504i 0.964591 + 0.263752i \(0.0849600\pi\)
−0.964591 + 0.263752i \(0.915040\pi\)
\(854\) 13.0509 0.446591
\(855\) 0 0
\(856\) 1.26900 0.0433734
\(857\) 19.8578i 0.678328i −0.940727 0.339164i \(-0.889856\pi\)
0.940727 0.339164i \(-0.110144\pi\)
\(858\) 0 0
\(859\) −2.42864 −0.0828641 −0.0414321 0.999141i \(-0.513192\pi\)
−0.0414321 + 0.999141i \(0.513192\pi\)
\(860\) −36.2864 + 5.09817i −1.23736 + 0.173846i
\(861\) 0 0
\(862\) 22.2953i 0.759380i
\(863\) 39.2958i 1.33764i −0.743423 0.668822i \(-0.766799\pi\)
0.743423 0.668822i \(-0.233201\pi\)
\(864\) 0 0
\(865\) −4.56199 + 0.640951i −0.155112 + 0.0217930i
\(866\) −0.114617 −0.00389485
\(867\) 0 0
\(868\) 8.40990i 0.285451i
\(869\) −9.71456 −0.329544
\(870\) 0 0
\(871\) −17.7146 −0.600235
\(872\) 4.03566i 0.136665i
\(873\) 0 0
\(874\) −6.36842 −0.215415
\(875\) −10.2143 + 4.54617i −0.345307 + 0.153689i
\(876\) 0 0
\(877\) 56.2864i 1.90066i −0.311249 0.950328i \(-0.600747\pi\)
0.311249 0.950328i \(-0.399253\pi\)
\(878\) 42.6865i 1.44060i
\(879\) 0 0
\(880\) −2.87019 20.4286i −0.0967539 0.688649i
\(881\) 2.33677 0.0787279 0.0393640 0.999225i \(-0.487467\pi\)
0.0393640 + 0.999225i \(0.487467\pi\)
\(882\) 0 0
\(883\) 33.7146i 1.13459i −0.823516 0.567293i \(-0.807991\pi\)
0.823516 0.567293i \(-0.192009\pi\)
\(884\) −46.1847 −1.55336
\(885\) 0 0
\(886\) 45.5812 1.53133
\(887\) 47.8992i 1.60830i −0.594427 0.804150i \(-0.702621\pi\)
0.594427 0.804150i \(-0.297379\pi\)
\(888\) 0 0
\(889\) −12.8573 −0.431219
\(890\) −19.4795 + 2.73683i −0.652954 + 0.0917388i
\(891\) 0 0
\(892\) 24.6994i 0.826996i
\(893\) 6.69228i 0.223949i
\(894\) 0 0
\(895\) −3.11108 22.1432i −0.103992 0.740165i
\(896\) −5.64941 −0.188734
\(897\) 0 0
\(898\) 56.0098i 1.86907i
\(899\) 3.91703 0.130640
\(900\) 0 0
\(901\) 40.6735 1.35503
\(902\) 31.3461i 1.04371i
\(903\) 0 0
\(904\) 8.11462 0.269888
\(905\) 3.76494 + 26.7971i 0.125151 + 0.890764i
\(906\) 0 0
\(907\) 23.7591i 0.788908i 0.918916 + 0.394454i \(0.129066\pi\)
−0.918916 + 0.394454i \(0.870934\pi\)
\(908\) 23.3087i 0.773525i
\(909\) 0 0
\(910\) −27.0923 + 3.80642i −0.898102 + 0.126182i
\(911\) −22.9403 −0.760045 −0.380022 0.924977i \(-0.624084\pi\)
−0.380022 + 0.924977i \(0.624084\pi\)
\(912\) 0 0
\(913\) 23.2257i 0.768658i
\(914\) 5.98126 0.197843
\(915\) 0 0
\(916\) −9.10525 −0.300846
\(917\) 2.10171i 0.0694046i
\(918\) 0 0
\(919\) −16.9777 −0.560043 −0.280022 0.959994i \(-0.590342\pi\)
−0.280022 + 0.959994i \(0.590342\pi\)
\(920\) 0.308193 + 2.19358i 0.0101608 + 0.0723201i
\(921\) 0 0
\(922\) 6.42864i 0.211716i
\(923\) 12.8573i 0.423202i
\(924\) 0 0
\(925\) 10.4889 + 36.5906i 0.344872 + 1.20309i
\(926\) 39.6958 1.30449
\(927\) 0 0
\(928\) 5.54680i 0.182082i
\(929\) 39.3403 1.29071 0.645357 0.763881i \(-0.276709\pi\)
0.645357 + 0.763881i \(0.276709\pi\)
\(930\) 0 0
\(931\) −2.42864 −0.0795954
\(932\) 37.7748i 1.23735i
\(933\) 0 0
\(934\) −27.3461 −0.894793
\(935\) −19.6128 + 2.75557i −0.641409 + 0.0901167i
\(936\) 0 0
\(937\) 17.7748i 0.580677i −0.956924 0.290338i \(-0.906232\pi\)
0.956924 0.290338i \(-0.0937679\pi\)
\(938\) 5.24443i 0.171237i
\(939\) 0 0
\(940\) −9.89829 + 1.39069i −0.322847 + 0.0453594i
\(941\) −35.5812 −1.15991 −0.579957 0.814647i \(-0.696931\pi\)
−0.579957 + 0.814647i \(0.696931\pi\)
\(942\) 0 0
\(943\) 11.3461i 0.369481i
\(944\) −65.0490 −2.11717
\(945\) 0 0
\(946\) −38.4514 −1.25016
\(947\) 30.5018i 0.991174i −0.868558 0.495587i \(-0.834953\pi\)
0.868558 0.495587i \(-0.165047\pi\)
\(948\) 0 0
\(949\) −10.1017 −0.327915
\(950\) 6.36842 + 22.2163i 0.206619 + 0.720793i
\(951\) 0 0
\(952\) 3.18421i 0.103201i
\(953\) 51.1655i 1.65741i −0.559684 0.828706i \(-0.689077\pi\)
0.559684 0.828706i \(-0.310923\pi\)
\(954\) 0 0
\(955\) 0.152089 + 1.08250i 0.00492148 + 0.0350288i
\(956\) 13.7708 0.445378
\(957\) 0 0
\(958\) 12.1204i 0.391594i
\(959\) −15.9398 −0.514722
\(960\) 0 0
\(961\) −4.12399 −0.133032
\(962\) 93.1437i 3.00307i
\(963\) 0 0
\(964\) 11.7520 0.378507
\(965\) −50.8385 + 7.14272i −1.63655 + 0.229932i
\(966\) 0 0
\(967\) 47.8992i 1.54034i 0.637841 + 0.770168i \(0.279828\pi\)
−0.637841 + 0.770168i \(0.720172\pi\)
\(968\) 5.03303i 0.161768i
\(969\) 0 0
\(970\) −7.06959 50.3180i −0.226991 1.61561i
\(971\) −40.6735 −1.30528 −0.652638 0.757670i \(-0.726338\pi\)
−0.652638 + 0.757670i \(0.726338\pi\)
\(972\) 0 0
\(973\) 11.6731i 0.374221i
\(974\) −32.9777 −1.05667
\(975\) 0 0
\(976\) −31.6316 −1.01250
\(977\) 27.4893i 0.879462i 0.898130 + 0.439731i \(0.144926\pi\)
−0.898130 + 0.439731i \(0.855074\pi\)
\(978\) 0 0
\(979\) −9.24443 −0.295453
\(980\) −0.504684 3.59210i −0.0161215 0.114746i
\(981\) 0 0
\(982\) 3.80642i 0.121468i
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) 0 0
\(985\) −2.62222 + 0.368416i −0.0835507 + 0.0117387i
\(986\) 6.36842 0.202812
\(987\) 0 0
\(988\) 25.3274i 0.805772i
\(989\) 13.9180 0.442566
\(990\) 0 0
\(991\) −34.6923 −1.10204 −0.551018 0.834493i \(-0.685761\pi\)
−0.551018 + 0.834493i \(0.685761\pi\)
\(992\) 38.0584i 1.20836i
\(993\) 0 0
\(994\) −3.80642 −0.120732
\(995\) 2.73683 + 19.4795i 0.0867634 + 0.617541i
\(996\) 0 0
\(997\) 28.6766i 0.908197i −0.890951 0.454099i \(-0.849961\pi\)
0.890951 0.454099i \(-0.150039\pi\)
\(998\) 44.4327i 1.40649i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 315.2.d.e.64.2 6
3.2 odd 2 105.2.d.b.64.5 yes 6
4.3 odd 2 5040.2.t.v.1009.3 6
5.2 odd 4 1575.2.a.w.1.3 3
5.3 odd 4 1575.2.a.x.1.1 3
5.4 even 2 inner 315.2.d.e.64.5 6
7.6 odd 2 2205.2.d.l.1324.2 6
12.11 even 2 1680.2.t.k.1009.5 6
15.2 even 4 525.2.a.k.1.1 3
15.8 even 4 525.2.a.j.1.3 3
15.14 odd 2 105.2.d.b.64.2 6
20.19 odd 2 5040.2.t.v.1009.4 6
21.2 odd 6 735.2.q.e.214.5 12
21.5 even 6 735.2.q.f.214.5 12
21.11 odd 6 735.2.q.e.79.2 12
21.17 even 6 735.2.q.f.79.2 12
21.20 even 2 735.2.d.b.589.5 6
35.34 odd 2 2205.2.d.l.1324.5 6
60.23 odd 4 8400.2.a.dg.1.3 3
60.47 odd 4 8400.2.a.dj.1.1 3
60.59 even 2 1680.2.t.k.1009.2 6
105.44 odd 6 735.2.q.e.214.2 12
105.59 even 6 735.2.q.f.79.5 12
105.62 odd 4 3675.2.a.bj.1.1 3
105.74 odd 6 735.2.q.e.79.5 12
105.83 odd 4 3675.2.a.bi.1.3 3
105.89 even 6 735.2.q.f.214.2 12
105.104 even 2 735.2.d.b.589.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.2 6 15.14 odd 2
105.2.d.b.64.5 yes 6 3.2 odd 2
315.2.d.e.64.2 6 1.1 even 1 trivial
315.2.d.e.64.5 6 5.4 even 2 inner
525.2.a.j.1.3 3 15.8 even 4
525.2.a.k.1.1 3 15.2 even 4
735.2.d.b.589.2 6 105.104 even 2
735.2.d.b.589.5 6 21.20 even 2
735.2.q.e.79.2 12 21.11 odd 6
735.2.q.e.79.5 12 105.74 odd 6
735.2.q.e.214.2 12 105.44 odd 6
735.2.q.e.214.5 12 21.2 odd 6
735.2.q.f.79.2 12 21.17 even 6
735.2.q.f.79.5 12 105.59 even 6
735.2.q.f.214.2 12 105.89 even 6
735.2.q.f.214.5 12 21.5 even 6
1575.2.a.w.1.3 3 5.2 odd 4
1575.2.a.x.1.1 3 5.3 odd 4
1680.2.t.k.1009.2 6 60.59 even 2
1680.2.t.k.1009.5 6 12.11 even 2
2205.2.d.l.1324.2 6 7.6 odd 2
2205.2.d.l.1324.5 6 35.34 odd 2
3675.2.a.bi.1.3 3 105.83 odd 4
3675.2.a.bj.1.1 3 105.62 odd 4
5040.2.t.v.1009.3 6 4.3 odd 2
5040.2.t.v.1009.4 6 20.19 odd 2
8400.2.a.dg.1.3 3 60.23 odd 4
8400.2.a.dj.1.1 3 60.47 odd 4