Properties

Label 195.2.b.b.181.2
Level $195$
Weight $2$
Character 195.181
Analytic conductor $1.557$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [195,2,Mod(181,195)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(195, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("195.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 195 = 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 195.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: \(1.55708283941\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 181.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 195.181
Dual form 195.2.b.b.181.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} +3.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000i q^{5} +1.00000i q^{6} -2.00000i q^{7} +3.00000i q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +(-3.00000 - 2.00000i) q^{13} +2.00000 q^{14} +1.00000i q^{15} -1.00000 q^{16} +2.00000 q^{17} +1.00000i q^{18} +2.00000i q^{19} +1.00000i q^{20} -2.00000i q^{21} -8.00000 q^{23} +3.00000i q^{24} -1.00000 q^{25} +(2.00000 - 3.00000i) q^{26} +1.00000 q^{27} -2.00000i q^{28} +2.00000 q^{29} -1.00000 q^{30} -2.00000i q^{31} +5.00000i q^{32} +2.00000i q^{34} +2.00000 q^{35} +1.00000 q^{36} -8.00000i q^{37} -2.00000 q^{38} +(-3.00000 - 2.00000i) q^{39} -3.00000 q^{40} -2.00000i q^{41} +2.00000 q^{42} -4.00000 q^{43} +1.00000i q^{45} -8.00000i q^{46} -4.00000i q^{47} -1.00000 q^{48} +3.00000 q^{49} -1.00000i q^{50} +2.00000 q^{51} +(-3.00000 - 2.00000i) q^{52} -6.00000 q^{53} +1.00000i q^{54} +6.00000 q^{56} +2.00000i q^{57} +2.00000i q^{58} -12.0000i q^{59} +1.00000i q^{60} +10.0000 q^{61} +2.00000 q^{62} -2.00000i q^{63} -7.00000 q^{64} +(2.00000 - 3.00000i) q^{65} -6.00000i q^{67} +2.00000 q^{68} -8.00000 q^{69} +2.00000i q^{70} +8.00000i q^{71} +3.00000i q^{72} +16.0000i q^{73} +8.00000 q^{74} -1.00000 q^{75} +2.00000i q^{76} +(2.00000 - 3.00000i) q^{78} -8.00000 q^{79} -1.00000i q^{80} +1.00000 q^{81} +2.00000 q^{82} +12.0000i q^{83} -2.00000i q^{84} +2.00000i q^{85} -4.00000i q^{86} +2.00000 q^{87} +6.00000i q^{89} -1.00000 q^{90} +(-4.00000 + 6.00000i) q^{91} -8.00000 q^{92} -2.00000i q^{93} +4.00000 q^{94} -2.00000 q^{95} +5.00000i q^{96} +16.0000i q^{97} +3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} - 6 q^{13} + 4 q^{14} - 2 q^{16} + 4 q^{17} - 16 q^{23} - 2 q^{25} + 4 q^{26} + 2 q^{27} + 4 q^{29} - 2 q^{30} + 4 q^{35} + 2 q^{36} - 4 q^{38} - 6 q^{39} - 6 q^{40} + 4 q^{42} - 8 q^{43} - 2 q^{48} + 6 q^{49} + 4 q^{51} - 6 q^{52} - 12 q^{53} + 12 q^{56} + 20 q^{61} + 4 q^{62} - 14 q^{64} + 4 q^{65} + 4 q^{68} - 16 q^{69} + 16 q^{74} - 2 q^{75} + 4 q^{78} - 16 q^{79} + 2 q^{81} + 4 q^{82} + 4 q^{87} - 2 q^{90} - 8 q^{91} - 16 q^{92} + 8 q^{94} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/195\mathbb{Z}\right)^\times\).

\(n\) \(106\) \(131\) \(157\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000i 0.447214i
\(6\) 1.00000i 0.408248i
\(7\) 2.00000i 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 3.00000i 1.06066i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.00000 0.288675
\(13\) −3.00000 2.00000i −0.832050 0.554700i
\(14\) 2.00000 0.534522
\(15\) 1.00000i 0.258199i
\(16\) −1.00000 −0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 1.00000i 0.223607i
\(21\) 2.00000i 0.436436i
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 3.00000i 0.612372i
\(25\) −1.00000 −0.200000
\(26\) 2.00000 3.00000i 0.392232 0.588348i
\(27\) 1.00000 0.192450
\(28\) 2.00000i 0.377964i
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) −1.00000 −0.182574
\(31\) 2.00000i 0.359211i −0.983739 0.179605i \(-0.942518\pi\)
0.983739 0.179605i \(-0.0574821\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 0 0
\(34\) 2.00000i 0.342997i
\(35\) 2.00000 0.338062
\(36\) 1.00000 0.166667
\(37\) 8.00000i 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) −2.00000 −0.324443
\(39\) −3.00000 2.00000i −0.480384 0.320256i
\(40\) −3.00000 −0.474342
\(41\) 2.00000i 0.312348i −0.987730 0.156174i \(-0.950084\pi\)
0.987730 0.156174i \(-0.0499160\pi\)
\(42\) 2.00000 0.308607
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) 1.00000i 0.149071i
\(46\) 8.00000i 1.17954i
\(47\) 4.00000i 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) −1.00000 −0.144338
\(49\) 3.00000 0.428571
\(50\) 1.00000i 0.141421i
\(51\) 2.00000 0.280056
\(52\) −3.00000 2.00000i −0.416025 0.277350i
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.00000i 0.136083i
\(55\) 0 0
\(56\) 6.00000 0.801784
\(57\) 2.00000i 0.264906i
\(58\) 2.00000i 0.262613i
\(59\) 12.0000i 1.56227i −0.624364 0.781133i \(-0.714642\pi\)
0.624364 0.781133i \(-0.285358\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 2.00000 0.254000
\(63\) 2.00000i 0.251976i
\(64\) −7.00000 −0.875000
\(65\) 2.00000 3.00000i 0.248069 0.372104i
\(66\) 0 0
\(67\) 6.00000i 0.733017i −0.930415 0.366508i \(-0.880553\pi\)
0.930415 0.366508i \(-0.119447\pi\)
\(68\) 2.00000 0.242536
\(69\) −8.00000 −0.963087
\(70\) 2.00000i 0.239046i
\(71\) 8.00000i 0.949425i 0.880141 + 0.474713i \(0.157448\pi\)
−0.880141 + 0.474713i \(0.842552\pi\)
\(72\) 3.00000i 0.353553i
\(73\) 16.0000i 1.87266i 0.351123 + 0.936329i \(0.385800\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 8.00000 0.929981
\(75\) −1.00000 −0.115470
\(76\) 2.00000i 0.229416i
\(77\) 0 0
\(78\) 2.00000 3.00000i 0.226455 0.339683i
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.00000i 0.111803i
\(81\) 1.00000 0.111111
\(82\) 2.00000 0.220863
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 2.00000i 0.218218i
\(85\) 2.00000i 0.216930i
\(86\) 4.00000i 0.431331i
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 6.00000i 0.635999i 0.948091 + 0.317999i \(0.103011\pi\)
−0.948091 + 0.317999i \(0.896989\pi\)
\(90\) −1.00000 −0.105409
\(91\) −4.00000 + 6.00000i −0.419314 + 0.628971i
\(92\) −8.00000 −0.834058
\(93\) 2.00000i 0.207390i
\(94\) 4.00000 0.412568
\(95\) −2.00000 −0.205196
\(96\) 5.00000i 0.510310i
\(97\) 16.0000i 1.62455i 0.583272 + 0.812277i \(0.301772\pi\)
−0.583272 + 0.812277i \(0.698228\pi\)
\(98\) 3.00000i 0.303046i
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 2.00000i 0.198030i
\(103\) 16.0000 1.57653 0.788263 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 6.00000 9.00000i 0.588348 0.882523i
\(105\) 2.00000 0.195180
\(106\) 6.00000i 0.582772i
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 1.00000 0.0962250
\(109\) 16.0000i 1.53252i 0.642529 + 0.766261i \(0.277885\pi\)
−0.642529 + 0.766261i \(0.722115\pi\)
\(110\) 0 0
\(111\) 8.00000i 0.759326i
\(112\) 2.00000i 0.188982i
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) −2.00000 −0.187317
\(115\) 8.00000i 0.746004i
\(116\) 2.00000 0.185695
\(117\) −3.00000 2.00000i −0.277350 0.184900i
\(118\) 12.0000 1.10469
\(119\) 4.00000i 0.366679i
\(120\) −3.00000 −0.273861
\(121\) 11.0000 1.00000
\(122\) 10.0000i 0.905357i
\(123\) 2.00000i 0.180334i
\(124\) 2.00000i 0.179605i
\(125\) 1.00000i 0.0894427i
\(126\) 2.00000 0.178174
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 3.00000i 0.265165i
\(129\) −4.00000 −0.352180
\(130\) 3.00000 + 2.00000i 0.263117 + 0.175412i
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 6.00000 0.518321
\(135\) 1.00000i 0.0860663i
\(136\) 6.00000i 0.514496i
\(137\) 6.00000i 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 8.00000i 0.681005i
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 2.00000 0.169031
\(141\) 4.00000i 0.336861i
\(142\) −8.00000 −0.671345
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 2.00000i 0.166091i
\(146\) −16.0000 −1.32417
\(147\) 3.00000 0.247436
\(148\) 8.00000i 0.657596i
\(149\) 14.0000i 1.14692i −0.819232 0.573462i \(-0.805600\pi\)
0.819232 0.573462i \(-0.194400\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) 14.0000i 1.13930i 0.821886 + 0.569652i \(0.192922\pi\)
−0.821886 + 0.569652i \(0.807078\pi\)
\(152\) −6.00000 −0.486664
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) −3.00000 2.00000i −0.240192 0.160128i
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 8.00000i 0.636446i
\(159\) −6.00000 −0.475831
\(160\) −5.00000 −0.395285
\(161\) 16.0000i 1.26098i
\(162\) 1.00000i 0.0785674i
\(163\) 18.0000i 1.40987i −0.709273 0.704934i \(-0.750976\pi\)
0.709273 0.704934i \(-0.249024\pi\)
\(164\) 2.00000i 0.156174i
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 20.0000i 1.54765i −0.633402 0.773823i \(-0.718342\pi\)
0.633402 0.773823i \(-0.281658\pi\)
\(168\) 6.00000 0.462910
\(169\) 5.00000 + 12.0000i 0.384615 + 0.923077i
\(170\) −2.00000 −0.153393
\(171\) 2.00000i 0.152944i
\(172\) −4.00000 −0.304997
\(173\) 10.0000 0.760286 0.380143 0.924928i \(-0.375875\pi\)
0.380143 + 0.924928i \(0.375875\pi\)
\(174\) 2.00000i 0.151620i
\(175\) 2.00000i 0.151186i
\(176\) 0 0
\(177\) 12.0000i 0.901975i
\(178\) −6.00000 −0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) −6.00000 4.00000i −0.444750 0.296500i
\(183\) 10.0000 0.739221
\(184\) 24.0000i 1.76930i
\(185\) 8.00000 0.588172
\(186\) 2.00000 0.146647
\(187\) 0 0
\(188\) 4.00000i 0.291730i
\(189\) 2.00000i 0.145479i
\(190\) 2.00000i 0.145095i
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) −7.00000 −0.505181
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) −16.0000 −1.14873
\(195\) 2.00000 3.00000i 0.143223 0.214834i
\(196\) 3.00000 0.214286
\(197\) 22.0000i 1.56744i 0.621117 + 0.783718i \(0.286679\pi\)
−0.621117 + 0.783718i \(0.713321\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 3.00000i 0.212132i
\(201\) 6.00000i 0.423207i
\(202\) 2.00000i 0.140720i
\(203\) 4.00000i 0.280745i
\(204\) 2.00000 0.140028
\(205\) 2.00000 0.139686
\(206\) 16.0000i 1.11477i
\(207\) −8.00000 −0.556038
\(208\) 3.00000 + 2.00000i 0.208013 + 0.138675i
\(209\) 0 0
\(210\) 2.00000i 0.138013i
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) −6.00000 −0.412082
\(213\) 8.00000i 0.548151i
\(214\) 8.00000i 0.546869i
\(215\) 4.00000i 0.272798i
\(216\) 3.00000i 0.204124i
\(217\) −4.00000 −0.271538
\(218\) −16.0000 −1.08366
\(219\) 16.0000i 1.08118i
\(220\) 0 0
\(221\) −6.00000 4.00000i −0.403604 0.269069i
\(222\) 8.00000 0.536925
\(223\) 22.0000i 1.47323i −0.676313 0.736614i \(-0.736423\pi\)
0.676313 0.736614i \(-0.263577\pi\)
\(224\) 10.0000 0.668153
\(225\) −1.00000 −0.0666667
\(226\) 14.0000i 0.931266i
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 2.00000i 0.132453i
\(229\) 8.00000i 0.528655i 0.964433 + 0.264327i \(0.0851500\pi\)
−0.964433 + 0.264327i \(0.914850\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) 14.0000 0.917170 0.458585 0.888650i \(-0.348356\pi\)
0.458585 + 0.888650i \(0.348356\pi\)
\(234\) 2.00000 3.00000i 0.130744 0.196116i
\(235\) 4.00000 0.260931
\(236\) 12.0000i 0.781133i
\(237\) −8.00000 −0.519656
\(238\) 4.00000 0.259281
\(239\) 8.00000i 0.517477i 0.965947 + 0.258738i \(0.0833068\pi\)
−0.965947 + 0.258738i \(0.916693\pi\)
\(240\) 1.00000i 0.0645497i
\(241\) 12.0000i 0.772988i 0.922292 + 0.386494i \(0.126314\pi\)
−0.922292 + 0.386494i \(0.873686\pi\)
\(242\) 11.0000i 0.707107i
\(243\) 1.00000 0.0641500
\(244\) 10.0000 0.640184
\(245\) 3.00000i 0.191663i
\(246\) 2.00000 0.127515
\(247\) 4.00000 6.00000i 0.254514 0.381771i
\(248\) 6.00000 0.381000
\(249\) 12.0000i 0.760469i
\(250\) 1.00000 0.0632456
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 2.00000i 0.125988i
\(253\) 0 0
\(254\) 0 0
\(255\) 2.00000i 0.125245i
\(256\) −17.0000 −1.06250
\(257\) 30.0000 1.87135 0.935674 0.352865i \(-0.114792\pi\)
0.935674 + 0.352865i \(0.114792\pi\)
\(258\) 4.00000i 0.249029i
\(259\) −16.0000 −0.994192
\(260\) 2.00000 3.00000i 0.124035 0.186052i
\(261\) 2.00000 0.123797
\(262\) 12.0000i 0.741362i
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 6.00000i 0.368577i
\(266\) 4.00000i 0.245256i
\(267\) 6.00000i 0.367194i
\(268\) 6.00000i 0.366508i
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) −1.00000 −0.0608581
\(271\) 14.0000i 0.850439i −0.905090 0.425220i \(-0.860197\pi\)
0.905090 0.425220i \(-0.139803\pi\)
\(272\) −2.00000 −0.121268
\(273\) −4.00000 + 6.00000i −0.242091 + 0.363137i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 4.00000i 0.239904i
\(279\) 2.00000i 0.119737i
\(280\) 6.00000i 0.358569i
\(281\) 6.00000i 0.357930i −0.983855 0.178965i \(-0.942725\pi\)
0.983855 0.178965i \(-0.0572749\pi\)
\(282\) 4.00000 0.238197
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 8.00000i 0.474713i
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 5.00000i 0.294628i
\(289\) −13.0000 −0.764706
\(290\) −2.00000 −0.117444
\(291\) 16.0000i 0.937937i
\(292\) 16.0000i 0.936329i
\(293\) 18.0000i 1.05157i −0.850617 0.525786i \(-0.823771\pi\)
0.850617 0.525786i \(-0.176229\pi\)
\(294\) 3.00000i 0.174964i
\(295\) 12.0000 0.698667
\(296\) 24.0000 1.39497
\(297\) 0 0
\(298\) 14.0000 0.810998
\(299\) 24.0000 + 16.0000i 1.38796 + 0.925304i
\(300\) −1.00000 −0.0577350
\(301\) 8.00000i 0.461112i
\(302\) −14.0000 −0.805609
\(303\) −2.00000 −0.114897
\(304\) 2.00000i 0.114708i
\(305\) 10.0000i 0.572598i
\(306\) 2.00000i 0.114332i
\(307\) 26.0000i 1.48390i 0.670456 + 0.741949i \(0.266098\pi\)
−0.670456 + 0.741949i \(0.733902\pi\)
\(308\) 0 0
\(309\) 16.0000 0.910208
\(310\) 2.00000i 0.113592i
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 6.00000 9.00000i 0.339683 0.509525i
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 14.0000i 0.790066i
\(315\) 2.00000 0.112687
\(316\) −8.00000 −0.450035
\(317\) 26.0000i 1.46031i −0.683284 0.730153i \(-0.739449\pi\)
0.683284 0.730153i \(-0.260551\pi\)
\(318\) 6.00000i 0.336463i
\(319\) 0 0
\(320\) 7.00000i 0.391312i
\(321\) 8.00000 0.446516
\(322\) −16.0000 −0.891645
\(323\) 4.00000i 0.222566i
\(324\) 1.00000 0.0555556
\(325\) 3.00000 + 2.00000i 0.166410 + 0.110940i
\(326\) 18.0000 0.996928
\(327\) 16.0000i 0.884802i
\(328\) 6.00000 0.331295
\(329\) −8.00000 −0.441054
\(330\) 0 0
\(331\) 26.0000i 1.42909i −0.699590 0.714545i \(-0.746634\pi\)
0.699590 0.714545i \(-0.253366\pi\)
\(332\) 12.0000i 0.658586i
\(333\) 8.00000i 0.438397i
\(334\) 20.0000 1.09435
\(335\) 6.00000 0.327815
\(336\) 2.00000i 0.109109i
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) −12.0000 + 5.00000i −0.652714 + 0.271964i
\(339\) −14.0000 −0.760376
\(340\) 2.00000i 0.108465i
\(341\) 0 0
\(342\) −2.00000 −0.108148
\(343\) 20.0000i 1.07990i
\(344\) 12.0000i 0.646997i
\(345\) 8.00000i 0.430706i
\(346\) 10.0000i 0.537603i
\(347\) −20.0000 −1.07366 −0.536828 0.843692i \(-0.680378\pi\)
−0.536828 + 0.843692i \(0.680378\pi\)
\(348\) 2.00000 0.107211
\(349\) 20.0000i 1.07058i −0.844670 0.535288i \(-0.820203\pi\)
0.844670 0.535288i \(-0.179797\pi\)
\(350\) −2.00000 −0.106904
\(351\) −3.00000 2.00000i −0.160128 0.106752i
\(352\) 0 0
\(353\) 30.0000i 1.59674i 0.602168 + 0.798369i \(0.294304\pi\)
−0.602168 + 0.798369i \(0.705696\pi\)
\(354\) 12.0000 0.637793
\(355\) −8.00000 −0.424596
\(356\) 6.00000i 0.317999i
\(357\) 4.00000i 0.211702i
\(358\) 12.0000i 0.634220i
\(359\) 4.00000i 0.211112i 0.994413 + 0.105556i \(0.0336622\pi\)
−0.994413 + 0.105556i \(0.966338\pi\)
\(360\) −3.00000 −0.158114
\(361\) 15.0000 0.789474
\(362\) 6.00000i 0.315353i
\(363\) 11.0000 0.577350
\(364\) −4.00000 + 6.00000i −0.209657 + 0.314485i
\(365\) −16.0000 −0.837478
\(366\) 10.0000i 0.522708i
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 8.00000 0.417029
\(369\) 2.00000i 0.104116i
\(370\) 8.00000i 0.415900i
\(371\) 12.0000i 0.623009i
\(372\) 2.00000i 0.103695i
\(373\) −38.0000 −1.96757 −0.983783 0.179364i \(-0.942596\pi\)
−0.983783 + 0.179364i \(0.942596\pi\)
\(374\) 0 0
\(375\) 1.00000i 0.0516398i
\(376\) 12.0000 0.618853
\(377\) −6.00000 4.00000i −0.309016 0.206010i
\(378\) 2.00000 0.102869
\(379\) 30.0000i 1.54100i −0.637442 0.770498i \(-0.720007\pi\)
0.637442 0.770498i \(-0.279993\pi\)
\(380\) −2.00000 −0.102598
\(381\) 0 0
\(382\) 20.0000i 1.02329i
\(383\) 4.00000i 0.204390i 0.994764 + 0.102195i \(0.0325866\pi\)
−0.994764 + 0.102195i \(0.967413\pi\)
\(384\) 3.00000i 0.153093i
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) −4.00000 −0.203331
\(388\) 16.0000i 0.812277i
\(389\) −38.0000 −1.92668 −0.963338 0.268290i \(-0.913542\pi\)
−0.963338 + 0.268290i \(0.913542\pi\)
\(390\) 3.00000 + 2.00000i 0.151911 + 0.101274i
\(391\) −16.0000 −0.809155
\(392\) 9.00000i 0.454569i
\(393\) −12.0000 −0.605320
\(394\) −22.0000 −1.10834
\(395\) 8.00000i 0.402524i
\(396\) 0 0
\(397\) 8.00000i 0.401508i 0.979642 + 0.200754i \(0.0643393\pi\)
−0.979642 + 0.200754i \(0.935661\pi\)
\(398\) 24.0000i 1.20301i
\(399\) 4.00000 0.200250
\(400\) 1.00000 0.0500000
\(401\) 10.0000i 0.499376i −0.968326 0.249688i \(-0.919672\pi\)
0.968326 0.249688i \(-0.0803281\pi\)
\(402\) 6.00000 0.299253
\(403\) −4.00000 + 6.00000i −0.199254 + 0.298881i
\(404\) −2.00000 −0.0995037
\(405\) 1.00000i 0.0496904i
\(406\) 4.00000 0.198517
\(407\) 0 0
\(408\) 6.00000i 0.297044i
\(409\) 12.0000i 0.593362i −0.954977 0.296681i \(-0.904120\pi\)
0.954977 0.296681i \(-0.0958798\pi\)
\(410\) 2.00000i 0.0987730i
\(411\) 6.00000i 0.295958i
\(412\) 16.0000 0.788263
\(413\) −24.0000 −1.18096
\(414\) 8.00000i 0.393179i
\(415\) −12.0000 −0.589057
\(416\) 10.0000 15.0000i 0.490290 0.735436i
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 2.00000 0.0975900
\(421\) 40.0000i 1.94948i −0.223341 0.974740i \(-0.571696\pi\)
0.223341 0.974740i \(-0.428304\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 4.00000i 0.194487i
\(424\) 18.0000i 0.874157i
\(425\) −2.00000 −0.0970143
\(426\) −8.00000 −0.387601
\(427\) 20.0000i 0.967868i
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 12.0000i 0.578020i 0.957326 + 0.289010i \(0.0933260\pi\)
−0.957326 + 0.289010i \(0.906674\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 4.00000i 0.192006i
\(435\) 2.00000i 0.0958927i
\(436\) 16.0000i 0.766261i
\(437\) 16.0000i 0.765384i
\(438\) −16.0000 −0.764510
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 3.00000 0.142857
\(442\) 4.00000 6.00000i 0.190261 0.285391i
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 8.00000i 0.379663i
\(445\) −6.00000 −0.284427
\(446\) 22.0000 1.04173
\(447\) 14.0000i 0.662177i
\(448\) 14.0000i 0.661438i
\(449\) 6.00000i 0.283158i 0.989927 + 0.141579i \(0.0452178\pi\)
−0.989927 + 0.141579i \(0.954782\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 0 0
\(452\) −14.0000 −0.658505
\(453\) 14.0000i 0.657777i
\(454\) 0 0
\(455\) −6.00000 4.00000i −0.281284 0.187523i
\(456\) −6.00000 −0.280976
\(457\) 8.00000i 0.374224i 0.982339 + 0.187112i \(0.0599128\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) −8.00000 −0.373815
\(459\) 2.00000 0.0933520
\(460\) 8.00000i 0.373002i
\(461\) 34.0000i 1.58354i −0.610821 0.791769i \(-0.709160\pi\)
0.610821 0.791769i \(-0.290840\pi\)
\(462\) 0 0
\(463\) 14.0000i 0.650635i 0.945605 + 0.325318i \(0.105471\pi\)
−0.945605 + 0.325318i \(0.894529\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 2.00000 0.0927478
\(466\) 14.0000i 0.648537i
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) −3.00000 2.00000i −0.138675 0.0924500i
\(469\) −12.0000 −0.554109
\(470\) 4.00000i 0.184506i
\(471\) 14.0000 0.645086
\(472\) 36.0000 1.65703
\(473\) 0 0
\(474\) 8.00000i 0.367452i
\(475\) 2.00000i 0.0917663i
\(476\) 4.00000i 0.183340i
\(477\) −6.00000 −0.274721
\(478\) −8.00000 −0.365911
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −5.00000 −0.228218
\(481\) −16.0000 + 24.0000i −0.729537 + 1.09431i
\(482\) −12.0000 −0.546585
\(483\) 16.0000i 0.728025i
\(484\) 11.0000 0.500000
\(485\) −16.0000 −0.726523
\(486\) 1.00000i 0.0453609i
\(487\) 18.0000i 0.815658i 0.913058 + 0.407829i \(0.133714\pi\)
−0.913058 + 0.407829i \(0.866286\pi\)
\(488\) 30.0000i 1.35804i
\(489\) 18.0000i 0.813988i
\(490\) −3.00000 −0.135526
\(491\) −40.0000 −1.80517 −0.902587 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) 2.00000i 0.0901670i
\(493\) 4.00000 0.180151
\(494\) 6.00000 + 4.00000i 0.269953 + 0.179969i
\(495\) 0 0
\(496\) 2.00000i 0.0898027i
\(497\) 16.0000 0.717698
\(498\) −12.0000 −0.537733
\(499\) 10.0000i 0.447661i −0.974628 0.223831i \(-0.928144\pi\)
0.974628 0.223831i \(-0.0718563\pi\)
\(500\) 1.00000i 0.0447214i
\(501\) 20.0000i 0.893534i
\(502\) 0 0
\(503\) −4.00000 −0.178351 −0.0891756 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 6.00000 0.267261
\(505\) 2.00000i 0.0889988i
\(506\) 0 0
\(507\) 5.00000 + 12.0000i 0.222058 + 0.532939i
\(508\) 0 0
\(509\) 6.00000i 0.265945i 0.991120 + 0.132973i \(0.0424523\pi\)
−0.991120 + 0.132973i \(0.957548\pi\)
\(510\) −2.00000 −0.0885615
\(511\) 32.0000 1.41560
\(512\) 11.0000i 0.486136i
\(513\) 2.00000i 0.0883022i
\(514\) 30.0000i 1.32324i
\(515\) 16.0000i 0.705044i
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 16.0000i 0.703000i
\(519\) 10.0000 0.438951
\(520\) 9.00000 + 6.00000i 0.394676 + 0.263117i
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) −12.0000 −0.524222
\(525\) 2.00000i 0.0872872i
\(526\) 12.0000i 0.523225i
\(527\) 4.00000i 0.174243i
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 6.00000 0.260623
\(531\) 12.0000i 0.520756i
\(532\) 4.00000 0.173422
\(533\) −4.00000 + 6.00000i −0.173259 + 0.259889i
\(534\) −6.00000 −0.259645
\(535\) 8.00000i 0.345870i
\(536\) 18.0000 0.777482
\(537\) 12.0000 0.517838
\(538\) 10.0000i 0.431131i
\(539\) 0 0
\(540\) 1.00000i 0.0430331i
\(541\) 8.00000i 0.343947i −0.985102 0.171973i \(-0.944986\pi\)
0.985102 0.171973i \(-0.0550143\pi\)
\(542\) 14.0000 0.601351
\(543\) 6.00000 0.257485
\(544\) 10.0000i 0.428746i
\(545\) −16.0000 −0.685365
\(546\) −6.00000 4.00000i −0.256776 0.171184i
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 4.00000i 0.170406i
\(552\) 24.0000i 1.02151i
\(553\) 16.0000i 0.680389i
\(554\) 2.00000i 0.0849719i
\(555\) 8.00000 0.339581
\(556\) 4.00000 0.169638
\(557\) 6.00000i 0.254228i 0.991888 + 0.127114i \(0.0405714\pi\)
−0.991888 + 0.127114i \(0.959429\pi\)
\(558\) 2.00000 0.0846668
\(559\) 12.0000 + 8.00000i 0.507546 + 0.338364i
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 6.00000 0.253095
\(563\) 24.0000 1.01148 0.505740 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 4.00000i 0.168430i
\(565\) 14.0000i 0.588984i
\(566\) 4.00000i 0.168133i
\(567\) 2.00000i 0.0839921i
\(568\) −24.0000 −1.00702
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 2.00000i 0.0837708i
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) −20.0000 −0.835512
\(574\) 4.00000i 0.166957i
\(575\) 8.00000 0.333623
\(576\) −7.00000 −0.291667
\(577\) 4.00000i 0.166522i 0.996528 + 0.0832611i \(0.0265335\pi\)
−0.996528 + 0.0832611i \(0.973466\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 4.00000i 0.166234i
\(580\) 2.00000i 0.0830455i
\(581\) 24.0000 0.995688
\(582\) −16.0000 −0.663221
\(583\) 0 0
\(584\) −48.0000 −1.98625
\(585\) 2.00000 3.00000i 0.0826898 0.124035i
\(586\) 18.0000 0.743573
\(587\) 28.0000i 1.15568i 0.816149 + 0.577842i \(0.196105\pi\)
−0.816149 + 0.577842i \(0.803895\pi\)
\(588\) 3.00000 0.123718
\(589\) 4.00000 0.164817
\(590\) 12.0000i 0.494032i
\(591\) 22.0000i 0.904959i
\(592\) 8.00000i 0.328798i
\(593\) 10.0000i 0.410651i −0.978694 0.205325i \(-0.934175\pi\)
0.978694 0.205325i \(-0.0658253\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 14.0000i 0.573462i
\(597\) −24.0000 −0.982255
\(598\) −16.0000 + 24.0000i −0.654289 + 0.981433i
\(599\) 16.0000 0.653742 0.326871 0.945069i \(-0.394006\pi\)
0.326871 + 0.945069i \(0.394006\pi\)
\(600\) 3.00000i 0.122474i
\(601\) 38.0000 1.55005 0.775026 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) −8.00000 −0.326056
\(603\) 6.00000i 0.244339i
\(604\) 14.0000i 0.569652i
\(605\) 11.0000i 0.447214i
\(606\) 2.00000i 0.0812444i
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −10.0000 −0.405554
\(609\) 4.00000i 0.162088i
\(610\) −10.0000 −0.404888
\(611\) −8.00000 + 12.0000i −0.323645 + 0.485468i
\(612\) 2.00000 0.0808452
\(613\) 16.0000i 0.646234i 0.946359 + 0.323117i \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) −26.0000 −1.04927
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) 10.0000i 0.402585i −0.979531 0.201292i \(-0.935486\pi\)
0.979531 0.201292i \(-0.0645141\pi\)
\(618\) 16.0000i 0.643614i
\(619\) 38.0000i 1.52735i 0.645601 + 0.763674i \(0.276607\pi\)
−0.645601 + 0.763674i \(0.723393\pi\)
\(620\) 2.00000 0.0803219
\(621\) −8.00000 −0.321029
\(622\) 4.00000i 0.160385i
\(623\) 12.0000 0.480770
\(624\) 3.00000 + 2.00000i 0.120096 + 0.0800641i
\(625\) 1.00000 0.0400000
\(626\) 10.0000i 0.399680i
\(627\) 0 0
\(628\) 14.0000 0.558661
\(629\) 16.0000i 0.637962i
\(630\) 2.00000i 0.0796819i
\(631\) 38.0000i 1.51276i 0.654135 + 0.756378i \(0.273033\pi\)
−0.654135 + 0.756378i \(0.726967\pi\)
\(632\) 24.0000i 0.954669i
\(633\) −12.0000 −0.476957
\(634\) 26.0000 1.03259
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) −9.00000 6.00000i −0.356593 0.237729i
\(638\) 0 0
\(639\) 8.00000i 0.316475i
\(640\) −3.00000 −0.118585
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 8.00000i 0.315735i
\(643\) 14.0000i 0.552106i −0.961142 0.276053i \(-0.910973\pi\)
0.961142 0.276053i \(-0.0890266\pi\)
\(644\) 16.0000i 0.630488i
\(645\) 4.00000i 0.157500i
\(646\) −4.00000 −0.157378
\(647\) 28.0000 1.10079 0.550397 0.834903i \(-0.314476\pi\)
0.550397 + 0.834903i \(0.314476\pi\)
\(648\) 3.00000i 0.117851i
\(649\) 0 0
\(650\) −2.00000 + 3.00000i −0.0784465 + 0.117670i
\(651\) −4.00000 −0.156772
\(652\) 18.0000i 0.704934i
\(653\) −2.00000 −0.0782660 −0.0391330 0.999234i \(-0.512460\pi\)
−0.0391330 + 0.999234i \(0.512460\pi\)
\(654\) −16.0000 −0.625650
\(655\) 12.0000i 0.468879i
\(656\) 2.00000i 0.0780869i
\(657\) 16.0000i 0.624219i
\(658\) 8.00000i 0.311872i
\(659\) 48.0000 1.86981 0.934907 0.354892i \(-0.115482\pi\)
0.934907 + 0.354892i \(0.115482\pi\)
\(660\) 0 0
\(661\) 4.00000i 0.155582i −0.996970 0.0777910i \(-0.975213\pi\)
0.996970 0.0777910i \(-0.0247867\pi\)
\(662\) 26.0000 1.01052
\(663\) −6.00000 4.00000i −0.233021 0.155347i
\(664\) −36.0000 −1.39707
\(665\) 4.00000i 0.155113i
\(666\) 8.00000 0.309994
\(667\) −16.0000 −0.619522
\(668\) 20.0000i 0.773823i
\(669\) 22.0000i 0.850569i
\(670\) 6.00000i 0.231800i
\(671\) 0 0
\(672\) 10.0000 0.385758
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 6.00000i 0.231111i
\(675\) −1.00000 −0.0384900
\(676\) 5.00000 + 12.0000i 0.192308 + 0.461538i
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 14.0000i 0.537667i
\(679\) 32.0000 1.22805
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) 0 0
\(683\) 16.0000i 0.612223i −0.951996 0.306111i \(-0.900972\pi\)
0.951996 0.306111i \(-0.0990280\pi\)
\(684\) 2.00000i 0.0764719i
\(685\) 6.00000 0.229248
\(686\) 20.0000 0.763604
\(687\) 8.00000i 0.305219i
\(688\) 4.00000 0.152499
\(689\) 18.0000 + 12.0000i 0.685745 + 0.457164i
\(690\) 8.00000 0.304555
\(691\) 10.0000i 0.380418i −0.981744 0.190209i \(-0.939083\pi\)
0.981744 0.190209i \(-0.0609166\pi\)
\(692\) 10.0000 0.380143
\(693\) 0 0
\(694\) 20.0000i 0.759190i
\(695\) 4.00000i 0.151729i
\(696\) 6.00000i 0.227429i
\(697\) 4.00000i 0.151511i
\(698\) 20.0000 0.757011
\(699\) 14.0000 0.529529
\(700\) 2.00000i 0.0755929i
\(701\) 18.0000 0.679851 0.339925 0.940452i \(-0.389598\pi\)
0.339925 + 0.940452i \(0.389598\pi\)
\(702\) 2.00000 3.00000i 0.0754851 0.113228i
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) 4.00000 0.150649
\(706\) −30.0000 −1.12906
\(707\) 4.00000i 0.150435i
\(708\) 12.0000i 0.450988i
\(709\) 32.0000i 1.20179i −0.799330 0.600893i \(-0.794812\pi\)
0.799330 0.600893i \(-0.205188\pi\)
\(710\) 8.00000i 0.300235i
\(711\) −8.00000 −0.300023
\(712\) −18.0000 −0.674579
\(713\) 16.0000i 0.599205i
\(714\) 4.00000 0.149696
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 8.00000i 0.298765i
\(718\) −4.00000 −0.149279
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 1.00000i 0.0372678i
\(721\) 32.0000i 1.19174i
\(722\) 15.0000i 0.558242i
\(723\) 12.0000i 0.446285i
\(724\) 6.00000 0.222988
\(725\) −2.00000 −0.0742781
\(726\) 11.0000i 0.408248i
\(727\) −40.0000 −1.48352 −0.741759 0.670667i \(-0.766008\pi\)
−0.741759 + 0.670667i \(0.766008\pi\)
\(728\) −18.0000 12.0000i −0.667124 0.444750i
\(729\) 1.00000 0.0370370
\(730\) 16.0000i 0.592187i
\(731\) −8.00000 −0.295891
\(732\) 10.0000 0.369611
\(733\) 12.0000i 0.443230i 0.975134 + 0.221615i \(0.0711328\pi\)
−0.975134 + 0.221615i \(0.928867\pi\)
\(734\) 0 0
\(735\) 3.00000i 0.110657i
\(736\) 40.0000i 1.47442i
\(737\) 0 0
\(738\) 2.00000 0.0736210
\(739\) 6.00000i 0.220714i 0.993892 + 0.110357i \(0.0351994\pi\)
−0.993892 + 0.110357i \(0.964801\pi\)
\(740\) 8.00000 0.294086
\(741\) 4.00000 6.00000i 0.146944 0.220416i
\(742\) −12.0000 −0.440534
\(743\) 8.00000i 0.293492i −0.989174 0.146746i \(-0.953120\pi\)
0.989174 0.146746i \(-0.0468799\pi\)
\(744\) 6.00000 0.219971
\(745\) 14.0000 0.512920
\(746\) 38.0000i 1.39128i
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) 16.0000i 0.584627i
\(750\) 1.00000 0.0365148
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 4.00000i 0.145865i
\(753\) 0 0
\(754\) 4.00000 6.00000i 0.145671 0.218507i
\(755\) −14.0000 −0.509512
\(756\) 2.00000i 0.0727393i
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 30.0000 1.08965
\(759\) 0 0
\(760\) 6.00000i 0.217643i
\(761\) 38.0000i 1.37750i −0.724999 0.688749i \(-0.758160\pi\)
0.724999 0.688749i \(-0.241840\pi\)
\(762\) 0 0
\(763\) 32.0000 1.15848
\(764\) −20.0000 −0.723575
\(765\) 2.00000i 0.0723102i
\(766\) −4.00000 −0.144526
\(767\) −24.0000 + 36.0000i −0.866590 + 1.29988i
\(768\) −17.0000 −0.613435
\(769\) 40.0000i 1.44244i 0.692708 + 0.721218i \(0.256418\pi\)
−0.692708 + 0.721218i \(0.743582\pi\)
\(770\) 0 0
\(771\) 30.0000 1.08042
\(772\) 4.00000i 0.143963i
\(773\) 54.0000i 1.94225i −0.238581 0.971123i \(-0.576682\pi\)
0.238581 0.971123i \(-0.423318\pi\)
\(774\) 4.00000i 0.143777i
\(775\) 2.00000i 0.0718421i
\(776\) −48.0000 −1.72310
\(777\) −16.0000 −0.573997
\(778\) 38.0000i 1.36237i
\(779\) 4.00000 0.143315
\(780\) 2.00000 3.00000i 0.0716115 0.107417i
\(781\) 0 0
\(782\) 16.0000i 0.572159i
\(783\) 2.00000 0.0714742
\(784\) −3.00000 −0.107143
\(785\) 14.0000i 0.499681i
\(786\) 12.0000i 0.428026i
\(787\) 38.0000i 1.35455i −0.735728 0.677277i \(-0.763160\pi\)
0.735728 0.677277i \(-0.236840\pi\)
\(788\) 22.0000i 0.783718i
\(789\) 12.0000 0.427211
\(790\) 8.00000 0.284627
\(791\) 28.0000i 0.995565i
\(792\) 0 0
\(793\) −30.0000 20.0000i −1.06533 0.710221i
\(794\) −8.00000 −0.283909
\(795\) 6.00000i 0.212798i
\(796\) −24.0000 −0.850657
\(797\) −50.0000 −1.77109 −0.885545 0.464553i \(-0.846215\pi\)
−0.885545 + 0.464553i \(0.846215\pi\)
\(798\) 4.00000i 0.141598i
\(799\) 8.00000i 0.283020i
\(800\) 5.00000i 0.176777i
\(801\) 6.00000i 0.212000i
\(802\) 10.0000 0.353112
\(803\) 0 0
\(804\) 6.00000i 0.211604i
\(805\) −16.0000 −0.563926
\(806\) −6.00000 4.00000i −0.211341 0.140894i
\(807\) 10.0000 0.352017
\(808\) 6.00000i 0.211079i
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 14.0000i 0.491606i −0.969320 0.245803i \(-0.920948\pi\)
0.969320 0.245803i \(-0.0790517\pi\)
\(812\) 4.00000i 0.140372i
\(813\) 14.0000i 0.491001i
\(814\) 0 0
\(815\) 18.0000 0.630512
\(816\) −2.00000 −0.0700140
\(817\) 8.00000i 0.279885i
\(818\) 12.0000 0.419570
\(819\) −4.00000 + 6.00000i −0.139771 + 0.209657i
\(820\) 2.00000 0.0698430
\(821\) 18.0000i 0.628204i 0.949389 + 0.314102i \(0.101703\pi\)
−0.949389 + 0.314102i \(0.898297\pi\)
\(822\) 6.00000 0.209274
\(823\) 24.0000 0.836587 0.418294 0.908312i \(-0.362628\pi\)
0.418294 + 0.908312i \(0.362628\pi\)
\(824\) 48.0000i 1.67216i
\(825\) 0 0
\(826\) 24.0000i 0.835067i
\(827\) 8.00000i 0.278187i 0.990279 + 0.139094i \(0.0444189\pi\)
−0.990279 + 0.139094i \(0.955581\pi\)
\(828\) −8.00000 −0.278019
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 12.0000i 0.416526i
\(831\) −2.00000 −0.0693792
\(832\) 21.0000 + 14.0000i 0.728044 + 0.485363i
\(833\) 6.00000 0.207888
\(834\) 4.00000i 0.138509i
\(835\) 20.0000 0.692129
\(836\) 0 0
\(837\) 2.00000i 0.0691301i
\(838\) 12.0000i 0.414533i
\(839\) 24.0000i 0.828572i 0.910147 + 0.414286i \(0.135969\pi\)
−0.910147 + 0.414286i \(0.864031\pi\)
\(840\) 6.00000i 0.207020i
\(841\) −25.0000 −0.862069
\(842\) 40.0000 1.37849
\(843\) 6.00000i 0.206651i
\(844\) −12.0000 −0.413057
\(845\) −12.0000 + 5.00000i −0.412813 + 0.172005i
\(846\) 4.00000 0.137523
\(847\) 22.0000i 0.755929i
\(848\) 6.00000 0.206041
\(849\) −4.00000 −0.137280
\(850\) 2.00000i 0.0685994i
\(851\) 64.0000i 2.19389i
\(852\) 8.00000i 0.274075i
\(853\) 24.0000i 0.821744i 0.911693 + 0.410872i \(0.134776\pi\)
−0.911693 + 0.410872i \(0.865224\pi\)
\(854\) 20.0000 0.684386
\(855\) −2.00000 −0.0683986
\(856\) 24.0000i 0.820303i
\(857\) 38.0000 1.29806 0.649028 0.760765i \(-0.275176\pi\)
0.649028 + 0.760765i \(0.275176\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 4.00000i 0.136399i
\(861\) −4.00000 −0.136320
\(862\) −12.0000 −0.408722
\(863\) 36.0000i 1.22545i −0.790295 0.612727i \(-0.790072\pi\)
0.790295 0.612727i \(-0.209928\pi\)
\(864\) 5.00000i 0.170103i
\(865\) 10.0000i 0.340010i
\(866\) 14.0000i 0.475739i
\(867\) −13.0000 −0.441503
\(868\) −4.00000 −0.135769
\(869\) 0 0
\(870\) −2.00000 −0.0678064
\(871\) −12.0000 + 18.0000i −0.406604 + 0.609907i
\(872\) −48.0000 −1.62549
\(873\) 16.0000i 0.541518i
\(874\) 16.0000 0.541208
\(875\) −2.00000 −0.0676123
\(876\) 16.0000i 0.540590i
\(877\) 16.0000i 0.540282i −0.962821 0.270141i \(-0.912930\pi\)
0.962821 0.270141i \(-0.0870703\pi\)
\(878\) 16.0000i 0.539974i
\(879\) 18.0000i 0.607125i
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 3.00000i 0.101015i
\(883\) 28.0000 0.942275 0.471138 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) −6.00000 4.00000i −0.201802 0.134535i
\(885\) 12.0000 0.403376
\(886\) 24.0000i 0.806296i
\(887\) −16.0000 −0.537227 −0.268614 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 24.0000 0.805387
\(889\) 0 0
\(890\) 6.00000i 0.201120i
\(891\) 0 0
\(892\) 22.0000i 0.736614i
\(893\) 8.00000 0.267710
\(894\) 14.0000 0.468230
\(895\) 12.0000i 0.401116i
\(896\) 6.00000 0.200446
\(897\) 24.0000 + 16.0000i 0.801337 + 0.534224i
\(898\) −6.00000 −0.200223
\(899\) 4.00000i 0.133407i
\(900\) −1.00000 −0.0333333
\(901\) −12.0000 −0.399778
\(902\) 0 0
\(903\) 8.00000i 0.266223i
\(904\) 42.0000i 1.39690i
\(905\) 6.00000i 0.199447i
\(906\) −14.0000 −0.465119
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 0 0
\(909\) −2.00000 −0.0663358
\(910\) 4.00000 6.00000i 0.132599 0.198898i
\(911\) 20.0000 0.662630 0.331315 0.943520i \(-0.392508\pi\)
0.331315 + 0.943520i \(0.392508\pi\)
\(912\) 2.00000i 0.0662266i
\(913\) 0 0
\(914\) −8.00000 −0.264616
\(915\) 10.0000i 0.330590i
\(916\) 8.00000i 0.264327i
\(917\) 24.0000i 0.792550i
\(918\) 2.00000i 0.0660098i
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 24.0000 0.791257
\(921\) 26.0000i 0.856729i
\(922\) 34.0000 1.11973
\(923\) 16.0000 24.0000i 0.526646 0.789970i
\(924\) 0 0
\(925\) 8.00000i 0.263038i
\(926\) −14.0000 −0.460069
\(927\) 16.0000 0.525509
\(928\) 10.0000i 0.328266i
\(929\) 22.0000i 0.721797i 0.932605 + 0.360898i \(0.117530\pi\)
−0.932605 + 0.360898i \(0.882470\pi\)
\(930\) 2.00000i 0.0655826i
\(931\) 6.00000i 0.196642i
\(932\) 14.0000 0.458585
\(933\) 4.00000 0.130954
\(934\) 36.0000i 1.17796i
\(935\) 0 0
\(936\) 6.00000 9.00000i 0.196116 0.294174i
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 12.0000i 0.391814i
\(939\) −10.0000 −0.326338
\(940\) 4.00000 0.130466
\(941\) 50.0000i 1.62995i 0.579494 + 0.814977i \(0.303250\pi\)
−0.579494 + 0.814977i \(0.696750\pi\)
\(942\) 14.0000i 0.456145i
\(943\) 16.0000i 0.521032i
\(944\) 12.0000i 0.390567i
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) −8.00000 −0.259828
\(949\) 32.0000 48.0000i 1.03876 1.55815i
\(950\) 2.00000 0.0648886
\(951\) 26.0000i 0.843108i
\(952\) 12.0000 0.388922
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 6.00000i 0.194257i
\(955\) 20.0000i 0.647185i
\(956\) 8.00000i 0.258738i
\(957\) 0 0
\(958\) 0 0
\(959\) −12.0000 −0.387500
\(960\) 7.00000i 0.225924i
\(961\) 27.0000 0.870968
\(962\) −24.0000 16.0000i −0.773791 0.515861i
\(963\) 8.00000 0.257796
\(964\) 12.0000i 0.386494i
\(965\) −4.00000 −0.128765
\(966\) −16.0000 −0.514792
\(967\) 58.0000i 1.86515i 0.360971 + 0.932577i \(0.382445\pi\)
−0.360971 + 0.932577i \(0.617555\pi\)
\(968\) 33.0000i 1.06066i
\(969\) 4.00000i 0.128499i
\(970\) 16.0000i 0.513729i
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 1.00000 0.0320750
\(973\) 8.00000i 0.256468i
\(974\) −18.0000 −0.576757
\(975\) 3.00000 + 2.00000i 0.0960769 + 0.0640513i
\(976\) −10.0000 −0.320092
\(977\) 38.0000i 1.21573i −0.794041 0.607864i \(-0.792027\pi\)
0.794041 0.607864i \(-0.207973\pi\)
\(978\) 18.0000 0.575577
\(979\) 0 0
\(980\) 3.00000i 0.0958315i
\(981\) 16.0000i 0.510841i
\(982\) 40.0000i 1.27645i
\(983\) 32.0000i 1.02064i 0.859984 + 0.510321i \(0.170473\pi\)
−0.859984 + 0.510321i \(0.829527\pi\)
\(984\) 6.00000 0.191273
\(985\) −22.0000 −0.700978
\(986\) 4.00000i 0.127386i
\(987\) −8.00000 −0.254643
\(988\) 4.00000 6.00000i 0.127257 0.190885i
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 10.0000 0.317500
\(993\) 26.0000i 0.825085i
\(994\) 16.0000i 0.507489i
\(995\) 24.0000i 0.760851i
\(996\) 12.0000i 0.380235i
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 10.0000 0.316544
\(999\) 8.00000i 0.253109i
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 195.2.b.b.181.2 yes 2
3.2 odd 2 585.2.b.a.181.1 2
4.3 odd 2 3120.2.g.a.961.2 2
5.2 odd 4 975.2.h.a.649.1 2
5.3 odd 4 975.2.h.d.649.2 2
5.4 even 2 975.2.b.b.376.1 2
13.5 odd 4 2535.2.a.l.1.1 1
13.8 odd 4 2535.2.a.e.1.1 1
13.12 even 2 inner 195.2.b.b.181.1 2
39.5 even 4 7605.2.a.d.1.1 1
39.8 even 4 7605.2.a.p.1.1 1
39.38 odd 2 585.2.b.a.181.2 2
52.51 odd 2 3120.2.g.a.961.1 2
65.12 odd 4 975.2.h.d.649.1 2
65.38 odd 4 975.2.h.a.649.2 2
65.64 even 2 975.2.b.b.376.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.b.b.181.1 2 13.12 even 2 inner
195.2.b.b.181.2 yes 2 1.1 even 1 trivial
585.2.b.a.181.1 2 3.2 odd 2
585.2.b.a.181.2 2 39.38 odd 2
975.2.b.b.376.1 2 5.4 even 2
975.2.b.b.376.2 2 65.64 even 2
975.2.h.a.649.1 2 5.2 odd 4
975.2.h.a.649.2 2 65.38 odd 4
975.2.h.d.649.1 2 65.12 odd 4
975.2.h.d.649.2 2 5.3 odd 4
2535.2.a.e.1.1 1 13.8 odd 4
2535.2.a.l.1.1 1 13.5 odd 4
3120.2.g.a.961.1 2 52.51 odd 2
3120.2.g.a.961.2 2 4.3 odd 2
7605.2.a.d.1.1 1 39.5 even 4
7605.2.a.p.1.1 1 39.8 even 4