Properties

Label 585.2.c.b.469.6
Level $585$
Weight $2$
Character 585.469
Analytic conductor $4.671$
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] = [585,2,Mod(469,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, 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("585.469");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.c (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: \(4.67124851824\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 469.6
Root \(0.403032 + 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 585.469
Dual form 585.2.c.b.469.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.67513i q^{2} -5.15633 q^{4} +(-1.67513 - 1.48119i) q^{5} -0.806063i q^{7} -8.44358i q^{8} +O(q^{10})\) \(q+2.67513i q^{2} -5.15633 q^{4} +(-1.67513 - 1.48119i) q^{5} -0.806063i q^{7} -8.44358i q^{8} +(3.96239 - 4.48119i) q^{10} +3.67513 q^{11} -1.00000i q^{13} +2.15633 q^{14} +12.2750 q^{16} -1.35026i q^{17} +1.67513 q^{19} +(8.63752 + 7.63752i) q^{20} +9.83146i q^{22} -6.48119i q^{23} +(0.612127 + 4.96239i) q^{25} +2.67513 q^{26} +4.15633i q^{28} +2.41819 q^{29} -5.28726 q^{31} +15.9502i q^{32} +3.61213 q^{34} +(-1.19394 + 1.35026i) q^{35} -3.76845i q^{37} +4.48119i q^{38} +(-12.5066 + 14.1441i) q^{40} +8.31265 q^{41} -6.79384i q^{43} -18.9502 q^{44} +17.3380 q^{46} +3.19394i q^{47} +6.35026 q^{49} +(-13.2750 + 1.63752i) q^{50} +5.15633i q^{52} -5.73813i q^{53} +(-6.15633 - 5.44358i) q^{55} -6.80606 q^{56} +6.46898i q^{58} +5.98778 q^{59} -1.76845 q^{61} -14.1441i q^{62} -18.1187 q^{64} +(-1.48119 + 1.67513i) q^{65} -9.89446i q^{67} +6.96239i q^{68} +(-3.61213 - 3.19394i) q^{70} -8.56230 q^{71} -11.7685i q^{73} +10.0811 q^{74} -8.63752 q^{76} -2.96239i q^{77} +2.26187 q^{79} +(-20.5623 - 18.1817i) q^{80} +22.2374i q^{82} -3.84367i q^{83} +(-2.00000 + 2.26187i) q^{85} +18.1744 q^{86} -31.0313i q^{88} +2.77575 q^{89} -0.806063 q^{91} +33.4191i q^{92} -8.54420 q^{94} +(-2.80606 - 2.48119i) q^{95} +1.87399i q^{97} +16.9878i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} + 2 q^{10} + 12 q^{11} - 8 q^{14} + 10 q^{16} + 20 q^{20} + 2 q^{25} + 6 q^{26} + 12 q^{29} - 20 q^{31} + 20 q^{34} - 8 q^{35} - 34 q^{40} + 8 q^{41} - 40 q^{44} + 32 q^{46} + 18 q^{49} - 16 q^{50} - 16 q^{55} - 40 q^{56} - 16 q^{59} + 12 q^{61} - 66 q^{64} + 2 q^{65} - 20 q^{70} + 24 q^{71} - 4 q^{74} - 20 q^{76} + 32 q^{79} - 48 q^{80} - 12 q^{85} + 32 q^{86} + 20 q^{89} - 4 q^{91} - 32 q^{94} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.67513i 1.89160i 0.324745 + 0.945802i \(0.394721\pi\)
−0.324745 + 0.945802i \(0.605279\pi\)
\(3\) 0 0
\(4\) −5.15633 −2.57816
\(5\) −1.67513 1.48119i −0.749141 0.662410i
\(6\) 0 0
\(7\) 0.806063i 0.304663i −0.988329 0.152332i \(-0.951322\pi\)
0.988329 0.152332i \(-0.0486782\pi\)
\(8\) 8.44358i 2.98526i
\(9\) 0 0
\(10\) 3.96239 4.48119i 1.25302 1.41708i
\(11\) 3.67513 1.10809 0.554047 0.832486i \(-0.313083\pi\)
0.554047 + 0.832486i \(0.313083\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 2.15633 0.576302
\(15\) 0 0
\(16\) 12.2750 3.06876
\(17\) 1.35026i 0.327487i −0.986503 0.163743i \(-0.947643\pi\)
0.986503 0.163743i \(-0.0523569\pi\)
\(18\) 0 0
\(19\) 1.67513 0.384301 0.192151 0.981365i \(-0.438454\pi\)
0.192151 + 0.981365i \(0.438454\pi\)
\(20\) 8.63752 + 7.63752i 1.93141 + 1.70780i
\(21\) 0 0
\(22\) 9.83146i 2.09607i
\(23\) 6.48119i 1.35142i −0.737166 0.675711i \(-0.763837\pi\)
0.737166 0.675711i \(-0.236163\pi\)
\(24\) 0 0
\(25\) 0.612127 + 4.96239i 0.122425 + 0.992478i
\(26\) 2.67513 0.524636
\(27\) 0 0
\(28\) 4.15633i 0.785472i
\(29\) 2.41819 0.449047 0.224523 0.974469i \(-0.427917\pi\)
0.224523 + 0.974469i \(0.427917\pi\)
\(30\) 0 0
\(31\) −5.28726 −0.949620 −0.474810 0.880088i \(-0.657483\pi\)
−0.474810 + 0.880088i \(0.657483\pi\)
\(32\) 15.9502i 2.81962i
\(33\) 0 0
\(34\) 3.61213 0.619475
\(35\) −1.19394 + 1.35026i −0.201812 + 0.228236i
\(36\) 0 0
\(37\) 3.76845i 0.619530i −0.950813 0.309765i \(-0.899750\pi\)
0.950813 0.309765i \(-0.100250\pi\)
\(38\) 4.48119i 0.726946i
\(39\) 0 0
\(40\) −12.5066 + 14.1441i −1.97747 + 2.23638i
\(41\) 8.31265 1.29822 0.649109 0.760695i \(-0.275142\pi\)
0.649109 + 0.760695i \(0.275142\pi\)
\(42\) 0 0
\(43\) 6.79384i 1.03605i −0.855365 0.518026i \(-0.826667\pi\)
0.855365 0.518026i \(-0.173333\pi\)
\(44\) −18.9502 −2.85685
\(45\) 0 0
\(46\) 17.3380 2.55635
\(47\) 3.19394i 0.465884i 0.972491 + 0.232942i \(0.0748352\pi\)
−0.972491 + 0.232942i \(0.925165\pi\)
\(48\) 0 0
\(49\) 6.35026 0.907180
\(50\) −13.2750 + 1.63752i −1.87737 + 0.231580i
\(51\) 0 0
\(52\) 5.15633i 0.715054i
\(53\) 5.73813i 0.788193i −0.919069 0.394097i \(-0.871057\pi\)
0.919069 0.394097i \(-0.128943\pi\)
\(54\) 0 0
\(55\) −6.15633 5.44358i −0.830119 0.734013i
\(56\) −6.80606 −0.909498
\(57\) 0 0
\(58\) 6.46898i 0.849418i
\(59\) 5.98778 0.779543 0.389771 0.920912i \(-0.372554\pi\)
0.389771 + 0.920912i \(0.372554\pi\)
\(60\) 0 0
\(61\) −1.76845 −0.226427 −0.113214 0.993571i \(-0.536114\pi\)
−0.113214 + 0.993571i \(0.536114\pi\)
\(62\) 14.1441i 1.79630i
\(63\) 0 0
\(64\) −18.1187 −2.26484
\(65\) −1.48119 + 1.67513i −0.183720 + 0.207774i
\(66\) 0 0
\(67\) 9.89446i 1.20880i −0.796681 0.604400i \(-0.793413\pi\)
0.796681 0.604400i \(-0.206587\pi\)
\(68\) 6.96239i 0.844314i
\(69\) 0 0
\(70\) −3.61213 3.19394i −0.431732 0.381748i
\(71\) −8.56230 −1.01616 −0.508079 0.861311i \(-0.669644\pi\)
−0.508079 + 0.861311i \(0.669644\pi\)
\(72\) 0 0
\(73\) 11.7685i 1.37739i −0.725050 0.688697i \(-0.758183\pi\)
0.725050 0.688697i \(-0.241817\pi\)
\(74\) 10.0811 1.17190
\(75\) 0 0
\(76\) −8.63752 −0.990791
\(77\) 2.96239i 0.337596i
\(78\) 0 0
\(79\) 2.26187 0.254480 0.127240 0.991872i \(-0.459388\pi\)
0.127240 + 0.991872i \(0.459388\pi\)
\(80\) −20.5623 18.1817i −2.29893 2.03278i
\(81\) 0 0
\(82\) 22.2374i 2.45571i
\(83\) 3.84367i 0.421898i −0.977497 0.210949i \(-0.932345\pi\)
0.977497 0.210949i \(-0.0676554\pi\)
\(84\) 0 0
\(85\) −2.00000 + 2.26187i −0.216930 + 0.245334i
\(86\) 18.1744 1.95980
\(87\) 0 0
\(88\) 31.0313i 3.30794i
\(89\) 2.77575 0.294229 0.147114 0.989120i \(-0.453001\pi\)
0.147114 + 0.989120i \(0.453001\pi\)
\(90\) 0 0
\(91\) −0.806063 −0.0844984
\(92\) 33.4191i 3.48419i
\(93\) 0 0
\(94\) −8.54420 −0.881267
\(95\) −2.80606 2.48119i −0.287896 0.254565i
\(96\) 0 0
\(97\) 1.87399i 0.190275i 0.995464 + 0.0951375i \(0.0303291\pi\)
−0.995464 + 0.0951375i \(0.969671\pi\)
\(98\) 16.9878i 1.71603i
\(99\) 0 0
\(100\) −3.15633 25.5877i −0.315633 2.55877i
\(101\) −10.4993 −1.04472 −0.522359 0.852725i \(-0.674948\pi\)
−0.522359 + 0.852725i \(0.674948\pi\)
\(102\) 0 0
\(103\) 15.3684i 1.51429i 0.653247 + 0.757145i \(0.273406\pi\)
−0.653247 + 0.757145i \(0.726594\pi\)
\(104\) −8.44358 −0.827961
\(105\) 0 0
\(106\) 15.3503 1.49095
\(107\) 11.1309i 1.07607i 0.842923 + 0.538034i \(0.180833\pi\)
−0.842923 + 0.538034i \(0.819167\pi\)
\(108\) 0 0
\(109\) −9.58769 −0.918334 −0.459167 0.888350i \(-0.651852\pi\)
−0.459167 + 0.888350i \(0.651852\pi\)
\(110\) 14.5623 16.4690i 1.38846 1.57026i
\(111\) 0 0
\(112\) 9.89446i 0.934939i
\(113\) 0.574515i 0.0540459i −0.999635 0.0270229i \(-0.991397\pi\)
0.999635 0.0270229i \(-0.00860271\pi\)
\(114\) 0 0
\(115\) −9.59991 + 10.8568i −0.895196 + 1.01241i
\(116\) −12.4690 −1.15772
\(117\) 0 0
\(118\) 16.0181i 1.47459i
\(119\) −1.08840 −0.0997732
\(120\) 0 0
\(121\) 2.50659 0.227872
\(122\) 4.73084i 0.428310i
\(123\) 0 0
\(124\) 27.2628 2.44827
\(125\) 6.32487 9.21933i 0.565713 0.824602i
\(126\) 0 0
\(127\) 4.29455i 0.381080i 0.981679 + 0.190540i \(0.0610239\pi\)
−0.981679 + 0.190540i \(0.938976\pi\)
\(128\) 16.5696i 1.46456i
\(129\) 0 0
\(130\) −4.48119 3.96239i −0.393027 0.347524i
\(131\) 0.836381 0.0730749 0.0365375 0.999332i \(-0.488367\pi\)
0.0365375 + 0.999332i \(0.488367\pi\)
\(132\) 0 0
\(133\) 1.35026i 0.117083i
\(134\) 26.4690 2.28657
\(135\) 0 0
\(136\) −11.4010 −0.977632
\(137\) 14.9380i 1.27624i −0.769939 0.638118i \(-0.779713\pi\)
0.769939 0.638118i \(-0.220287\pi\)
\(138\) 0 0
\(139\) 8.43866 0.715758 0.357879 0.933768i \(-0.383500\pi\)
0.357879 + 0.933768i \(0.383500\pi\)
\(140\) 6.15633 6.96239i 0.520304 0.588429i
\(141\) 0 0
\(142\) 22.9053i 1.92217i
\(143\) 3.67513i 0.307330i
\(144\) 0 0
\(145\) −4.05079 3.58181i −0.336399 0.297453i
\(146\) 31.4821 2.60548
\(147\) 0 0
\(148\) 19.4314i 1.59725i
\(149\) −11.3503 −0.929850 −0.464925 0.885350i \(-0.653919\pi\)
−0.464925 + 0.885350i \(0.653919\pi\)
\(150\) 0 0
\(151\) 13.9878 1.13831 0.569155 0.822230i \(-0.307271\pi\)
0.569155 + 0.822230i \(0.307271\pi\)
\(152\) 14.1441i 1.14724i
\(153\) 0 0
\(154\) 7.92478 0.638597
\(155\) 8.85685 + 7.83146i 0.711399 + 0.629038i
\(156\) 0 0
\(157\) 2.77575i 0.221529i −0.993847 0.110764i \(-0.964670\pi\)
0.993847 0.110764i \(-0.0353299\pi\)
\(158\) 6.05079i 0.481375i
\(159\) 0 0
\(160\) 23.6253 26.7186i 1.86774 2.11229i
\(161\) −5.22425 −0.411729
\(162\) 0 0
\(163\) 2.23155i 0.174788i 0.996174 + 0.0873942i \(0.0278540\pi\)
−0.996174 + 0.0873942i \(0.972146\pi\)
\(164\) −42.8627 −3.34702
\(165\) 0 0
\(166\) 10.2823 0.798064
\(167\) 15.6932i 1.21438i −0.794557 0.607189i \(-0.792297\pi\)
0.794557 0.607189i \(-0.207703\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) −6.05079 5.35026i −0.464074 0.410346i
\(171\) 0 0
\(172\) 35.0313i 2.67111i
\(173\) 25.5877i 1.94540i 0.232075 + 0.972698i \(0.425449\pi\)
−0.232075 + 0.972698i \(0.574551\pi\)
\(174\) 0 0
\(175\) 4.00000 0.493413i 0.302372 0.0372985i
\(176\) 45.1124 3.40047
\(177\) 0 0
\(178\) 7.42548i 0.556564i
\(179\) 12.1260 0.906340 0.453170 0.891424i \(-0.350293\pi\)
0.453170 + 0.891424i \(0.350293\pi\)
\(180\) 0 0
\(181\) −2.73084 −0.202982 −0.101491 0.994836i \(-0.532361\pi\)
−0.101491 + 0.994836i \(0.532361\pi\)
\(182\) 2.15633i 0.159837i
\(183\) 0 0
\(184\) −54.7245 −4.03434
\(185\) −5.58181 + 6.31265i −0.410383 + 0.464115i
\(186\) 0 0
\(187\) 4.96239i 0.362886i
\(188\) 16.4690i 1.20112i
\(189\) 0 0
\(190\) 6.63752 7.50659i 0.481536 0.544585i
\(191\) −20.6253 −1.49239 −0.746197 0.665725i \(-0.768122\pi\)
−0.746197 + 0.665725i \(0.768122\pi\)
\(192\) 0 0
\(193\) 21.7889i 1.56840i −0.620508 0.784200i \(-0.713073\pi\)
0.620508 0.784200i \(-0.286927\pi\)
\(194\) −5.01317 −0.359925
\(195\) 0 0
\(196\) −32.7440 −2.33886
\(197\) 2.00000i 0.142494i 0.997459 + 0.0712470i \(0.0226979\pi\)
−0.997459 + 0.0712470i \(0.977302\pi\)
\(198\) 0 0
\(199\) 16.7513 1.18747 0.593734 0.804661i \(-0.297653\pi\)
0.593734 + 0.804661i \(0.297653\pi\)
\(200\) 41.9003 5.16854i 2.96280 0.365471i
\(201\) 0 0
\(202\) 28.0870i 1.97619i
\(203\) 1.94921i 0.136808i
\(204\) 0 0
\(205\) −13.9248 12.3127i −0.972549 0.859953i
\(206\) −41.1124 −2.86443
\(207\) 0 0
\(208\) 12.2750i 0.851121i
\(209\) 6.15633 0.425842
\(210\) 0 0
\(211\) −4.90175 −0.337451 −0.168725 0.985663i \(-0.553965\pi\)
−0.168725 + 0.985663i \(0.553965\pi\)
\(212\) 29.5877i 2.03209i
\(213\) 0 0
\(214\) −29.7767 −2.03549
\(215\) −10.0630 + 11.3806i −0.686291 + 0.776149i
\(216\) 0 0
\(217\) 4.26187i 0.289314i
\(218\) 25.6483i 1.73712i
\(219\) 0 0
\(220\) 31.7440 + 28.0689i 2.14018 + 1.89240i
\(221\) −1.35026 −0.0908284
\(222\) 0 0
\(223\) 24.9076i 1.66794i 0.551811 + 0.833969i \(0.313937\pi\)
−0.551811 + 0.833969i \(0.686063\pi\)
\(224\) 12.8568 0.859034
\(225\) 0 0
\(226\) 1.53690 0.102233
\(227\) 9.95509i 0.660743i 0.943851 + 0.330371i \(0.107174\pi\)
−0.943851 + 0.330371i \(0.892826\pi\)
\(228\) 0 0
\(229\) −5.35026 −0.353555 −0.176778 0.984251i \(-0.556567\pi\)
−0.176778 + 0.984251i \(0.556567\pi\)
\(230\) −29.0435 25.6810i −1.91507 1.69336i
\(231\) 0 0
\(232\) 20.4182i 1.34052i
\(233\) 10.7612i 0.704987i 0.935814 + 0.352493i \(0.114666\pi\)
−0.935814 + 0.352493i \(0.885334\pi\)
\(234\) 0 0
\(235\) 4.73084 5.35026i 0.308606 0.349013i
\(236\) −30.8749 −2.00979
\(237\) 0 0
\(238\) 2.91160i 0.188731i
\(239\) −11.8618 −0.767274 −0.383637 0.923484i \(-0.625329\pi\)
−0.383637 + 0.923484i \(0.625329\pi\)
\(240\) 0 0
\(241\) −28.6253 −1.84392 −0.921959 0.387288i \(-0.873412\pi\)
−0.921959 + 0.387288i \(0.873412\pi\)
\(242\) 6.70545i 0.431043i
\(243\) 0 0
\(244\) 9.11871 0.583766
\(245\) −10.6375 9.40597i −0.679606 0.600925i
\(246\) 0 0
\(247\) 1.67513i 0.106586i
\(248\) 44.6434i 2.83486i
\(249\) 0 0
\(250\) 24.6629 + 16.9199i 1.55982 + 1.07011i
\(251\) 19.3865 1.22366 0.611831 0.790988i \(-0.290433\pi\)
0.611831 + 0.790988i \(0.290433\pi\)
\(252\) 0 0
\(253\) 23.8192i 1.49750i
\(254\) −11.4885 −0.720852
\(255\) 0 0
\(256\) 8.08840 0.505525
\(257\) 22.8627i 1.42614i 0.701094 + 0.713069i \(0.252695\pi\)
−0.701094 + 0.713069i \(0.747305\pi\)
\(258\) 0 0
\(259\) −3.03761 −0.188748
\(260\) 7.63752 8.63752i 0.473659 0.535676i
\(261\) 0 0
\(262\) 2.23743i 0.138229i
\(263\) 21.8822i 1.34932i −0.738130 0.674658i \(-0.764291\pi\)
0.738130 0.674658i \(-0.235709\pi\)
\(264\) 0 0
\(265\) −8.49929 + 9.61213i −0.522107 + 0.590468i
\(266\) 3.61213 0.221474
\(267\) 0 0
\(268\) 51.0191i 3.11648i
\(269\) 22.7513 1.38717 0.693586 0.720374i \(-0.256030\pi\)
0.693586 + 0.720374i \(0.256030\pi\)
\(270\) 0 0
\(271\) 0.123638 0.00751049 0.00375525 0.999993i \(-0.498805\pi\)
0.00375525 + 0.999993i \(0.498805\pi\)
\(272\) 16.5745i 1.00498i
\(273\) 0 0
\(274\) 39.9610 2.41413
\(275\) 2.24965 + 18.2374i 0.135659 + 1.09976i
\(276\) 0 0
\(277\) 15.3503i 0.922308i 0.887320 + 0.461154i \(0.152564\pi\)
−0.887320 + 0.461154i \(0.847436\pi\)
\(278\) 22.5745i 1.35393i
\(279\) 0 0
\(280\) 11.4010 + 10.0811i 0.681343 + 0.602461i
\(281\) −13.9248 −0.830683 −0.415341 0.909666i \(-0.636338\pi\)
−0.415341 + 0.909666i \(0.636338\pi\)
\(282\) 0 0
\(283\) 20.3815i 1.21156i 0.795634 + 0.605778i \(0.207138\pi\)
−0.795634 + 0.605778i \(0.792862\pi\)
\(284\) 44.1500 2.61982
\(285\) 0 0
\(286\) 9.83146 0.581346
\(287\) 6.70052i 0.395519i
\(288\) 0 0
\(289\) 15.1768 0.892753
\(290\) 9.58181 10.8364i 0.562663 0.636334i
\(291\) 0 0
\(292\) 60.6820i 3.55114i
\(293\) 5.38058i 0.314337i −0.987572 0.157168i \(-0.949763\pi\)
0.987572 0.157168i \(-0.0502365\pi\)
\(294\) 0 0
\(295\) −10.0303 8.86907i −0.583988 0.516377i
\(296\) −31.8192 −1.84946
\(297\) 0 0
\(298\) 30.3634i 1.75891i
\(299\) −6.48119 −0.374817
\(300\) 0 0
\(301\) −5.47627 −0.315647
\(302\) 37.4191i 2.15323i
\(303\) 0 0
\(304\) 20.5623 1.17933
\(305\) 2.96239 + 2.61942i 0.169626 + 0.149988i
\(306\) 0 0
\(307\) 19.1695i 1.09406i 0.837113 + 0.547031i \(0.184242\pi\)
−0.837113 + 0.547031i \(0.815758\pi\)
\(308\) 15.2750i 0.870376i
\(309\) 0 0
\(310\) −20.9502 + 23.6932i −1.18989 + 1.34568i
\(311\) 25.2506 1.43183 0.715915 0.698187i \(-0.246010\pi\)
0.715915 + 0.698187i \(0.246010\pi\)
\(312\) 0 0
\(313\) 2.81194i 0.158940i 0.996837 + 0.0794702i \(0.0253229\pi\)
−0.996837 + 0.0794702i \(0.974677\pi\)
\(314\) 7.42548 0.419044
\(315\) 0 0
\(316\) −11.6629 −0.656090
\(317\) 23.7685i 1.33497i −0.744624 0.667485i \(-0.767371\pi\)
0.744624 0.667485i \(-0.232629\pi\)
\(318\) 0 0
\(319\) 8.88717 0.497586
\(320\) 30.3512 + 26.8373i 1.69668 + 1.50025i
\(321\) 0 0
\(322\) 13.9756i 0.778828i
\(323\) 2.26187i 0.125854i
\(324\) 0 0
\(325\) 4.96239 0.612127i 0.275264 0.0339547i
\(326\) −5.96968 −0.330630
\(327\) 0 0
\(328\) 70.1886i 3.87551i
\(329\) 2.57452 0.141938
\(330\) 0 0
\(331\) −11.8011 −0.648649 −0.324325 0.945946i \(-0.605137\pi\)
−0.324325 + 0.945946i \(0.605137\pi\)
\(332\) 19.8192i 1.08772i
\(333\) 0 0
\(334\) 41.9814 2.29712
\(335\) −14.6556 + 16.5745i −0.800722 + 0.905563i
\(336\) 0 0
\(337\) 16.1114i 0.877645i −0.898574 0.438822i \(-0.855396\pi\)
0.898574 0.438822i \(-0.144604\pi\)
\(338\) 2.67513i 0.145508i
\(339\) 0 0
\(340\) 10.3127 11.6629i 0.559282 0.632510i
\(341\) −19.4314 −1.05227
\(342\) 0 0
\(343\) 10.7612i 0.581048i
\(344\) −57.3644 −3.09288
\(345\) 0 0
\(346\) −68.4504 −3.67992
\(347\) 27.4944i 1.47598i 0.674814 + 0.737988i \(0.264224\pi\)
−0.674814 + 0.737988i \(0.735776\pi\)
\(348\) 0 0
\(349\) 17.6023 0.942228 0.471114 0.882072i \(-0.343852\pi\)
0.471114 + 0.882072i \(0.343852\pi\)
\(350\) 1.31994 + 10.7005i 0.0705540 + 0.571967i
\(351\) 0 0
\(352\) 58.6190i 3.12440i
\(353\) 15.7685i 0.839270i −0.907693 0.419635i \(-0.862158\pi\)
0.907693 0.419635i \(-0.137842\pi\)
\(354\) 0 0
\(355\) 14.3430 + 12.6824i 0.761246 + 0.673113i
\(356\) −14.3127 −0.758569
\(357\) 0 0
\(358\) 32.4387i 1.71444i
\(359\) −14.8242 −0.782389 −0.391195 0.920308i \(-0.627938\pi\)
−0.391195 + 0.920308i \(0.627938\pi\)
\(360\) 0 0
\(361\) −16.1939 −0.852312
\(362\) 7.30536i 0.383961i
\(363\) 0 0
\(364\) 4.15633 0.217851
\(365\) −17.4314 + 19.7137i −0.912399 + 1.03186i
\(366\) 0 0
\(367\) 27.0313i 1.41102i 0.708700 + 0.705510i \(0.249282\pi\)
−0.708700 + 0.705510i \(0.750718\pi\)
\(368\) 79.5569i 4.14719i
\(369\) 0 0
\(370\) −16.8872 14.9321i −0.877922 0.776281i
\(371\) −4.62530 −0.240134
\(372\) 0 0
\(373\) 12.9525i 0.670657i 0.942101 + 0.335329i \(0.108847\pi\)
−0.942101 + 0.335329i \(0.891153\pi\)
\(374\) 13.2750 0.686436
\(375\) 0 0
\(376\) 26.9683 1.39078
\(377\) 2.41819i 0.124543i
\(378\) 0 0
\(379\) 30.2858 1.55568 0.777840 0.628463i \(-0.216316\pi\)
0.777840 + 0.628463i \(0.216316\pi\)
\(380\) 14.4690 + 12.7938i 0.742243 + 0.656310i
\(381\) 0 0
\(382\) 55.1754i 2.82302i
\(383\) 21.0943i 1.07787i 0.842348 + 0.538934i \(0.181173\pi\)
−0.842348 + 0.538934i \(0.818827\pi\)
\(384\) 0 0
\(385\) −4.38787 + 4.96239i −0.223627 + 0.252907i
\(386\) 58.2882 2.96679
\(387\) 0 0
\(388\) 9.66291i 0.490560i
\(389\) −6.77575 −0.343544 −0.171772 0.985137i \(-0.554949\pi\)
−0.171772 + 0.985137i \(0.554949\pi\)
\(390\) 0 0
\(391\) −8.75131 −0.442573
\(392\) 53.6190i 2.70817i
\(393\) 0 0
\(394\) −5.35026 −0.269542
\(395\) −3.78892 3.35026i −0.190641 0.168570i
\(396\) 0 0
\(397\) 10.4690i 0.525423i −0.964874 0.262711i \(-0.915383\pi\)
0.964874 0.262711i \(-0.0846167\pi\)
\(398\) 44.8119i 2.24622i
\(399\) 0 0
\(400\) 7.51388 + 60.9135i 0.375694 + 3.04568i
\(401\) −5.01317 −0.250346 −0.125173 0.992135i \(-0.539949\pi\)
−0.125173 + 0.992135i \(0.539949\pi\)
\(402\) 0 0
\(403\) 5.28726i 0.263377i
\(404\) 54.1378 2.69345
\(405\) 0 0
\(406\) 5.21440 0.258787
\(407\) 13.8496i 0.686497i
\(408\) 0 0
\(409\) 14.3879 0.711435 0.355717 0.934594i \(-0.384237\pi\)
0.355717 + 0.934594i \(0.384237\pi\)
\(410\) 32.9380 37.2506i 1.62669 1.83968i
\(411\) 0 0
\(412\) 79.2443i 3.90408i
\(413\) 4.82653i 0.237498i
\(414\) 0 0
\(415\) −5.69323 + 6.43866i −0.279470 + 0.316061i
\(416\) 15.9502 0.782021
\(417\) 0 0
\(418\) 16.4690i 0.805524i
\(419\) 17.4617 0.853059 0.426529 0.904474i \(-0.359736\pi\)
0.426529 + 0.904474i \(0.359736\pi\)
\(420\) 0 0
\(421\) 2.88717 0.140712 0.0703559 0.997522i \(-0.477586\pi\)
0.0703559 + 0.997522i \(0.477586\pi\)
\(422\) 13.1128i 0.638323i
\(423\) 0 0
\(424\) −48.4504 −2.35296
\(425\) 6.70052 0.826531i 0.325023 0.0400927i
\(426\) 0 0
\(427\) 1.42548i 0.0689840i
\(428\) 57.3947i 2.77428i
\(429\) 0 0
\(430\) −30.4445 26.9199i −1.46817 1.29819i
\(431\) −0.889535 −0.0428474 −0.0214237 0.999770i \(-0.506820\pi\)
−0.0214237 + 0.999770i \(0.506820\pi\)
\(432\) 0 0
\(433\) 25.2506i 1.21347i −0.794906 0.606733i \(-0.792480\pi\)
0.794906 0.606733i \(-0.207520\pi\)
\(434\) −11.4010 −0.547268
\(435\) 0 0
\(436\) 49.4372 2.36761
\(437\) 10.8568i 0.519354i
\(438\) 0 0
\(439\) −28.8119 −1.37512 −0.687560 0.726128i \(-0.741318\pi\)
−0.687560 + 0.726128i \(0.741318\pi\)
\(440\) −45.9633 + 51.9814i −2.19122 + 2.47812i
\(441\) 0 0
\(442\) 3.61213i 0.171811i
\(443\) 36.9805i 1.75700i −0.477746 0.878498i \(-0.658546\pi\)
0.477746 0.878498i \(-0.341454\pi\)
\(444\) 0 0
\(445\) −4.64974 4.11142i −0.220419 0.194900i
\(446\) −66.6312 −3.15508
\(447\) 0 0
\(448\) 14.6048i 0.690013i
\(449\) −12.6859 −0.598686 −0.299343 0.954146i \(-0.596768\pi\)
−0.299343 + 0.954146i \(0.596768\pi\)
\(450\) 0 0
\(451\) 30.5501 1.43855
\(452\) 2.96239i 0.139339i
\(453\) 0 0
\(454\) −26.6312 −1.24986
\(455\) 1.35026 + 1.19394i 0.0633012 + 0.0559726i
\(456\) 0 0
\(457\) 25.0494i 1.17176i 0.810398 + 0.585880i \(0.199251\pi\)
−0.810398 + 0.585880i \(0.800749\pi\)
\(458\) 14.3127i 0.668786i
\(459\) 0 0
\(460\) 49.5002 55.9814i 2.30796 2.61015i
\(461\) 36.8872 1.71801 0.859003 0.511970i \(-0.171084\pi\)
0.859003 + 0.511970i \(0.171084\pi\)
\(462\) 0 0
\(463\) 39.0191i 1.81337i −0.421809 0.906685i \(-0.638605\pi\)
0.421809 0.906685i \(-0.361395\pi\)
\(464\) 29.6834 1.37802
\(465\) 0 0
\(466\) −28.7875 −1.33356
\(467\) 32.7694i 1.51639i 0.652029 + 0.758194i \(0.273918\pi\)
−0.652029 + 0.758194i \(0.726082\pi\)
\(468\) 0 0
\(469\) −7.97556 −0.368277
\(470\) 14.3127 + 12.6556i 0.660193 + 0.583760i
\(471\) 0 0
\(472\) 50.5583i 2.32714i
\(473\) 24.9683i 1.14804i
\(474\) 0 0
\(475\) 1.02539 + 8.31265i 0.0470482 + 0.381411i
\(476\) 5.61213 0.257231
\(477\) 0 0
\(478\) 31.7318i 1.45138i
\(479\) 16.8749 0.771036 0.385518 0.922700i \(-0.374023\pi\)
0.385518 + 0.922700i \(0.374023\pi\)
\(480\) 0 0
\(481\) −3.76845 −0.171827
\(482\) 76.5764i 3.48796i
\(483\) 0 0
\(484\) −12.9248 −0.587490
\(485\) 2.77575 3.13918i 0.126040 0.142543i
\(486\) 0 0
\(487\) 9.24472i 0.418918i 0.977817 + 0.209459i \(0.0671703\pi\)
−0.977817 + 0.209459i \(0.932830\pi\)
\(488\) 14.9321i 0.675943i
\(489\) 0 0
\(490\) 25.1622 28.4568i 1.13671 1.28555i
\(491\) 25.7499 1.16208 0.581038 0.813876i \(-0.302647\pi\)
0.581038 + 0.813876i \(0.302647\pi\)
\(492\) 0 0
\(493\) 3.26519i 0.147057i
\(494\) 4.48119 0.201618
\(495\) 0 0
\(496\) −64.9013 −2.91415
\(497\) 6.90175i 0.309586i
\(498\) 0 0
\(499\) 27.7015 1.24009 0.620044 0.784567i \(-0.287115\pi\)
0.620044 + 0.784567i \(0.287115\pi\)
\(500\) −32.6131 + 47.5379i −1.45850 + 2.12596i
\(501\) 0 0
\(502\) 51.8613i 2.31468i
\(503\) 2.35519i 0.105013i −0.998621 0.0525063i \(-0.983279\pi\)
0.998621 0.0525063i \(-0.0167210\pi\)
\(504\) 0 0
\(505\) 17.5877 + 15.5515i 0.782642 + 0.692032i
\(506\) 63.7196 2.83268
\(507\) 0 0
\(508\) 22.1441i 0.982486i
\(509\) 21.5125 0.953523 0.476762 0.879033i \(-0.341810\pi\)
0.476762 + 0.879033i \(0.341810\pi\)
\(510\) 0 0
\(511\) −9.48612 −0.419641
\(512\) 11.5017i 0.508306i
\(513\) 0 0
\(514\) −61.1608 −2.69769
\(515\) 22.7635 25.7440i 1.00308 1.13442i
\(516\) 0 0
\(517\) 11.7381i 0.516243i
\(518\) 8.12601i 0.357036i
\(519\) 0 0
\(520\) 14.1441 + 12.5066i 0.620260 + 0.548450i
\(521\) −37.7440 −1.65360 −0.826798 0.562499i \(-0.809840\pi\)
−0.826798 + 0.562499i \(0.809840\pi\)
\(522\) 0 0
\(523\) 23.7416i 1.03815i 0.854729 + 0.519075i \(0.173723\pi\)
−0.854729 + 0.519075i \(0.826277\pi\)
\(524\) −4.31265 −0.188399
\(525\) 0 0
\(526\) 58.5379 2.55237
\(527\) 7.13918i 0.310988i
\(528\) 0 0
\(529\) −19.0059 −0.826343
\(530\) −25.7137 22.7367i −1.11693 0.987620i
\(531\) 0 0
\(532\) 6.96239i 0.301858i
\(533\) 8.31265i 0.360061i
\(534\) 0 0
\(535\) 16.4871 18.6458i 0.712798 0.806127i
\(536\) −83.5447 −3.60858
\(537\) 0 0
\(538\) 60.8627i 2.62398i
\(539\) 23.3380 1.00524
\(540\) 0 0
\(541\) 13.0376 0.560531 0.280265 0.959923i \(-0.409578\pi\)
0.280265 + 0.959923i \(0.409578\pi\)
\(542\) 0.330749i 0.0142069i
\(543\) 0 0
\(544\) 21.5369 0.923387
\(545\) 16.0606 + 14.2012i 0.687962 + 0.608314i
\(546\) 0 0
\(547\) 8.43041i 0.360458i −0.983625 0.180229i \(-0.942316\pi\)
0.983625 0.180229i \(-0.0576839\pi\)
\(548\) 77.0249i 3.29034i
\(549\) 0 0
\(550\) −48.7875 + 6.01810i −2.08031 + 0.256613i
\(551\) 4.05079 0.172569
\(552\) 0 0
\(553\) 1.82321i 0.0775306i
\(554\) −41.0640 −1.74464
\(555\) 0 0
\(556\) −43.5125 −1.84534
\(557\) 13.6932i 0.580201i 0.956996 + 0.290100i \(0.0936887\pi\)
−0.956996 + 0.290100i \(0.906311\pi\)
\(558\) 0 0
\(559\) −6.79384 −0.287349
\(560\) −14.6556 + 16.5745i −0.619313 + 0.700401i
\(561\) 0 0
\(562\) 37.2506i 1.57132i
\(563\) 8.86907i 0.373787i 0.982380 + 0.186893i \(0.0598419\pi\)
−0.982380 + 0.186893i \(0.940158\pi\)
\(564\) 0 0
\(565\) −0.850969 + 0.962389i −0.0358005 + 0.0404880i
\(566\) −54.5233 −2.29178
\(567\) 0 0
\(568\) 72.2965i 3.03349i
\(569\) −32.7816 −1.37428 −0.687139 0.726526i \(-0.741134\pi\)
−0.687139 + 0.726526i \(0.741134\pi\)
\(570\) 0 0
\(571\) 40.2882 1.68601 0.843005 0.537906i \(-0.180785\pi\)
0.843005 + 0.537906i \(0.180785\pi\)
\(572\) 18.9502i 0.792346i
\(573\) 0 0
\(574\) 17.9248 0.748166
\(575\) 32.1622 3.96731i 1.34126 0.165448i
\(576\) 0 0
\(577\) 28.8568i 1.20133i −0.799502 0.600663i \(-0.794903\pi\)
0.799502 0.600663i \(-0.205097\pi\)
\(578\) 40.5999i 1.68873i
\(579\) 0 0
\(580\) 20.8872 + 18.4690i 0.867292 + 0.766882i
\(581\) −3.09825 −0.128537
\(582\) 0 0
\(583\) 21.0884i 0.873392i
\(584\) −99.3679 −4.11187
\(585\) 0 0
\(586\) 14.3938 0.594600
\(587\) 41.6786i 1.72026i 0.510074 + 0.860131i \(0.329618\pi\)
−0.510074 + 0.860131i \(0.670382\pi\)
\(588\) 0 0
\(589\) −8.85685 −0.364940
\(590\) 23.7259 26.8324i 0.976781 1.10467i
\(591\) 0 0
\(592\) 46.2579i 1.90119i
\(593\) 22.4993i 0.923935i −0.886897 0.461968i \(-0.847144\pi\)
0.886897 0.461968i \(-0.152856\pi\)
\(594\) 0 0
\(595\) 1.82321 + 1.61213i 0.0747442 + 0.0660908i
\(596\) 58.5256 2.39730
\(597\) 0 0
\(598\) 17.3380i 0.709005i
\(599\) 4.15045 0.169583 0.0847913 0.996399i \(-0.472978\pi\)
0.0847913 + 0.996399i \(0.472978\pi\)
\(600\) 0 0
\(601\) 27.9248 1.13908 0.569538 0.821965i \(-0.307122\pi\)
0.569538 + 0.821965i \(0.307122\pi\)
\(602\) 14.6497i 0.597079i
\(603\) 0 0
\(604\) −72.1255 −2.93475
\(605\) −4.19886 3.71274i −0.170708 0.150944i
\(606\) 0 0
\(607\) 8.19489i 0.332620i −0.986073 0.166310i \(-0.946815\pi\)
0.986073 0.166310i \(-0.0531853\pi\)
\(608\) 26.7186i 1.08358i
\(609\) 0 0
\(610\) −7.00729 + 7.92478i −0.283717 + 0.320865i
\(611\) 3.19394 0.129213
\(612\) 0 0
\(613\) 33.1392i 1.33848i 0.743047 + 0.669239i \(0.233380\pi\)
−0.743047 + 0.669239i \(0.766620\pi\)
\(614\) −51.2809 −2.06953
\(615\) 0 0
\(616\) −25.0132 −1.00781
\(617\) 29.0132i 1.16803i −0.811744 0.584013i \(-0.801482\pi\)
0.811744 0.584013i \(-0.198518\pi\)
\(618\) 0 0
\(619\) −12.2134 −0.490900 −0.245450 0.969409i \(-0.578936\pi\)
−0.245450 + 0.969409i \(0.578936\pi\)
\(620\) −45.6688 40.3815i −1.83410 1.62176i
\(621\) 0 0
\(622\) 67.5487i 2.70845i
\(623\) 2.23743i 0.0896406i
\(624\) 0 0
\(625\) −24.2506 + 6.07522i −0.970024 + 0.243009i
\(626\) −7.52232 −0.300652
\(627\) 0 0
\(628\) 14.3127i 0.571137i
\(629\) −5.08840 −0.202888
\(630\) 0 0
\(631\) −1.22188 −0.0486424 −0.0243212 0.999704i \(-0.507742\pi\)
−0.0243212 + 0.999704i \(0.507742\pi\)
\(632\) 19.0982i 0.759687i
\(633\) 0 0
\(634\) 63.5837 2.52523
\(635\) 6.36107 7.19394i 0.252431 0.285483i
\(636\) 0 0
\(637\) 6.35026i 0.251607i
\(638\) 23.7743i 0.941235i
\(639\) 0 0
\(640\) −24.5428 + 27.7562i −0.970139 + 1.09716i
\(641\) 22.1016 0.872960 0.436480 0.899714i \(-0.356225\pi\)
0.436480 + 0.899714i \(0.356225\pi\)
\(642\) 0 0
\(643\) 11.6688i 0.460172i 0.973170 + 0.230086i \(0.0739008\pi\)
−0.973170 + 0.230086i \(0.926099\pi\)
\(644\) 26.9380 1.06150
\(645\) 0 0
\(646\) 6.05079 0.238065
\(647\) 11.9575i 0.470096i 0.971984 + 0.235048i \(0.0755248\pi\)
−0.971984 + 0.235048i \(0.924475\pi\)
\(648\) 0 0
\(649\) 22.0059 0.863806
\(650\) 1.63752 + 13.2750i 0.0642288 + 0.520690i
\(651\) 0 0
\(652\) 11.5066i 0.450633i
\(653\) 10.9986i 0.430408i 0.976569 + 0.215204i \(0.0690416\pi\)
−0.976569 + 0.215204i \(0.930958\pi\)
\(654\) 0 0
\(655\) −1.40105 1.23884i −0.0547434 0.0484056i
\(656\) 102.038 3.98392
\(657\) 0 0
\(658\) 6.88717i 0.268490i
\(659\) −2.63989 −0.102835 −0.0514177 0.998677i \(-0.516374\pi\)
−0.0514177 + 0.998677i \(0.516374\pi\)
\(660\) 0 0
\(661\) 18.3028 0.711896 0.355948 0.934506i \(-0.384158\pi\)
0.355948 + 0.934506i \(0.384158\pi\)
\(662\) 31.5696i 1.22699i
\(663\) 0 0
\(664\) −32.4544 −1.25947
\(665\) −2.00000 + 2.26187i −0.0775567 + 0.0877114i
\(666\) 0 0
\(667\) 15.6728i 0.606852i
\(668\) 80.9194i 3.13087i
\(669\) 0 0
\(670\) −44.3390 39.2057i −1.71296 1.51465i
\(671\) −6.49929 −0.250902
\(672\) 0 0
\(673\) 6.71037i 0.258666i 0.991601 + 0.129333i \(0.0412836\pi\)
−0.991601 + 0.129333i \(0.958716\pi\)
\(674\) 43.1002 1.66016
\(675\) 0 0
\(676\) 5.15633 0.198320
\(677\) 1.57593i 0.0605679i −0.999541 0.0302840i \(-0.990359\pi\)
0.999541 0.0302840i \(-0.00964116\pi\)
\(678\) 0 0
\(679\) 1.51056 0.0579698
\(680\) 19.0982 + 16.8872i 0.732384 + 0.647593i
\(681\) 0 0
\(682\) 51.9814i 1.99047i
\(683\) 15.1939i 0.581380i −0.956817 0.290690i \(-0.906115\pi\)
0.956817 0.290690i \(-0.0938848\pi\)
\(684\) 0 0
\(685\) −22.1260 + 25.0230i −0.845391 + 0.956081i
\(686\) 28.7875 1.09911
\(687\) 0 0
\(688\) 83.3947i 3.17939i
\(689\) −5.73813 −0.218606
\(690\) 0 0
\(691\) −18.7127 −0.711866 −0.355933 0.934511i \(-0.615837\pi\)
−0.355933 + 0.934511i \(0.615837\pi\)
\(692\) 131.938i 5.01555i
\(693\) 0 0
\(694\) −73.5510 −2.79196
\(695\) −14.1359 12.4993i −0.536204 0.474125i
\(696\) 0 0
\(697\) 11.2243i 0.425149i
\(698\) 47.0884i 1.78232i
\(699\) 0 0
\(700\) −20.6253 + 2.54420i −0.779563 + 0.0961617i
\(701\) −24.3028 −0.917904 −0.458952 0.888461i \(-0.651775\pi\)
−0.458952 + 0.888461i \(0.651775\pi\)
\(702\) 0 0
\(703\) 6.31265i 0.238086i
\(704\) −66.5886 −2.50965
\(705\) 0 0
\(706\) 42.1827 1.58757
\(707\) 8.46310i 0.318287i
\(708\) 0 0
\(709\) 9.66291 0.362898 0.181449 0.983400i \(-0.441921\pi\)
0.181449 + 0.983400i \(0.441921\pi\)
\(710\) −33.9271 + 38.3693i −1.27326 + 1.43997i
\(711\) 0 0
\(712\) 23.4372i 0.878348i
\(713\) 34.2677i 1.28334i
\(714\) 0 0
\(715\) −5.44358 + 6.15633i −0.203578 + 0.230234i
\(716\) −62.5256 −2.33669
\(717\) 0 0
\(718\) 39.6566i 1.47997i
\(719\) 28.4142 1.05967 0.529836 0.848100i \(-0.322254\pi\)
0.529836 + 0.848100i \(0.322254\pi\)
\(720\) 0 0
\(721\) 12.3879 0.461349
\(722\) 43.3209i 1.61224i
\(723\) 0 0
\(724\) 14.0811 0.523320
\(725\) 1.48024 + 12.0000i 0.0549747 + 0.445669i
\(726\) 0 0
\(727\) 34.8545i 1.29268i 0.763049 + 0.646341i \(0.223701\pi\)
−0.763049 + 0.646341i \(0.776299\pi\)
\(728\) 6.80606i 0.252249i
\(729\) 0 0
\(730\) −52.7367 46.6312i −1.95187 1.72590i
\(731\) −9.17347 −0.339293
\(732\) 0 0
\(733\) 6.25202i 0.230923i 0.993312 + 0.115462i \(0.0368348\pi\)
−0.993312 + 0.115462i \(0.963165\pi\)
\(734\) −72.3122 −2.66909
\(735\) 0 0
\(736\) 103.376 3.81050
\(737\) 36.3634i 1.33946i
\(738\) 0 0
\(739\) 32.0846 1.18025 0.590126 0.807311i \(-0.299078\pi\)
0.590126 + 0.807311i \(0.299078\pi\)
\(740\) 28.7816 32.5501i 1.05803 1.19656i
\(741\) 0 0
\(742\) 12.3733i 0.454238i
\(743\) 30.5442i 1.12056i 0.828304 + 0.560279i \(0.189306\pi\)
−0.828304 + 0.560279i \(0.810694\pi\)
\(744\) 0 0
\(745\) 19.0132 + 16.8119i 0.696589 + 0.615942i
\(746\) −34.6497 −1.26862
\(747\) 0 0
\(748\) 25.5877i 0.935579i
\(749\) 8.97224 0.327838
\(750\) 0 0
\(751\) 28.1622 1.02765 0.513827 0.857894i \(-0.328227\pi\)
0.513827 + 0.857894i \(0.328227\pi\)
\(752\) 39.2057i 1.42968i
\(753\) 0 0
\(754\) 6.46898 0.235586
\(755\) −23.4314 20.7186i −0.852755 0.754028i
\(756\) 0 0
\(757\) 35.4109i 1.28703i −0.765433 0.643515i \(-0.777475\pi\)
0.765433 0.643515i \(-0.222525\pi\)
\(758\) 81.0186i 2.94273i
\(759\) 0 0
\(760\) −20.9502 + 23.6932i −0.759943 + 0.859444i
\(761\) 19.2388 0.697407 0.348704 0.937233i \(-0.386622\pi\)
0.348704 + 0.937233i \(0.386622\pi\)
\(762\) 0 0
\(763\) 7.72829i 0.279783i
\(764\) 106.351 3.84764
\(765\) 0 0
\(766\) −56.4299 −2.03890
\(767\) 5.98778i 0.216206i
\(768\) 0 0
\(769\) −48.9643 −1.76570 −0.882849 0.469657i \(-0.844378\pi\)
−0.882849 + 0.469657i \(0.844378\pi\)
\(770\) −13.2750 11.7381i −0.478399 0.423013i
\(771\) 0 0
\(772\) 112.351i 4.04359i
\(773\) 46.1681i 1.66055i −0.557354 0.830275i \(-0.688183\pi\)
0.557354 0.830275i \(-0.311817\pi\)
\(774\) 0 0
\(775\) −3.23647 26.2374i −0.116258 0.942476i
\(776\) 15.8232 0.568020
\(777\) 0 0
\(778\) 18.1260i 0.649849i
\(779\) 13.9248 0.498907
\(780\) 0 0
\(781\) −31.4676 −1.12600
\(782\) 23.4109i 0.837172i
\(783\) 0 0
\(784\) 77.9497 2.78392
\(785\) −4.11142 + 4.64974i −0.146743 + 0.165956i
\(786\) 0 0
\(787\) 22.6458i 0.807234i 0.914928 + 0.403617i \(0.132247\pi\)
−0.914928 + 0.403617i \(0.867753\pi\)
\(788\) 10.3127i 0.367373i
\(789\) 0 0
\(790\) 8.96239 10.1359i 0.318867 0.360618i
\(791\) −0.463096 −0.0164658
\(792\) 0 0
\(793\) 1.76845i 0.0627996i
\(794\) 28.0059 0.993891
\(795\) 0 0
\(796\) −86.3752 −3.06149
\(797\) 8.23743i 0.291785i 0.989300 + 0.145892i \(0.0466053\pi\)
−0.989300 + 0.145892i \(0.953395\pi\)
\(798\) 0 0
\(799\) 4.31265 0.152571
\(800\) −79.1509 + 9.76353i −2.79841 + 0.345193i
\(801\) 0 0
\(802\) 13.4109i 0.473555i
\(803\) 43.2506i 1.52628i
\(804\) 0 0
\(805\) 8.75131 + 7.73813i 0.308443 + 0.272733i
\(806\) −14.1441 −0.498205
\(807\) 0 0
\(808\) 88.6516i 3.11875i
\(809\) 44.1319 1.55159 0.775797 0.630982i \(-0.217348\pi\)
0.775797 + 0.630982i \(0.217348\pi\)
\(810\) 0 0
\(811\) 22.6883 0.796694 0.398347 0.917235i \(-0.369584\pi\)
0.398347 + 0.917235i \(0.369584\pi\)
\(812\) 10.0508i 0.352713i
\(813\) 0 0
\(814\) 37.0494 1.29858
\(815\) 3.30536 3.73813i 0.115782 0.130941i
\(816\) 0 0
\(817\) 11.3806i 0.398156i
\(818\) 38.4894i 1.34575i
\(819\) 0 0
\(820\) 71.8007 + 63.4880i 2.50739 + 2.21710i
\(821\) 50.2736 1.75456 0.877281 0.479978i \(-0.159355\pi\)
0.877281 + 0.479978i \(0.159355\pi\)
\(822\) 0 0
\(823\) 5.13093i 0.178853i 0.995993 + 0.0894265i \(0.0285034\pi\)
−0.995993 + 0.0894265i \(0.971497\pi\)
\(824\) 129.764 4.52054
\(825\) 0 0
\(826\) 12.9116 0.449252
\(827\) 18.6946i 0.650076i 0.945701 + 0.325038i \(0.105377\pi\)
−0.945701 + 0.325038i \(0.894623\pi\)
\(828\) 0 0
\(829\) 3.44121 0.119518 0.0597591 0.998213i \(-0.480967\pi\)
0.0597591 + 0.998213i \(0.480967\pi\)
\(830\) −17.2243 15.2301i −0.597863 0.528646i
\(831\) 0 0
\(832\) 18.1187i 0.628153i
\(833\) 8.57452i 0.297089i
\(834\) 0 0
\(835\) −23.2447 + 26.2882i −0.804417 + 0.909741i
\(836\) −31.7440 −1.09789
\(837\) 0 0
\(838\) 46.7123i 1.61365i
\(839\) −52.6248 −1.81681 −0.908406 0.418090i \(-0.862700\pi\)
−0.908406 + 0.418090i \(0.862700\pi\)
\(840\) 0 0
\(841\) −23.1524 −0.798357
\(842\) 7.72355i 0.266171i
\(843\) 0 0
\(844\) 25.2750 0.870003
\(845\) 1.67513 + 1.48119i 0.0576263 + 0.0509546i
\(846\) 0 0
\(847\) 2.02047i 0.0694241i
\(848\) 70.4358i 2.41878i
\(849\) 0 0
\(850\) 2.21108 + 17.9248i 0.0758394 + 0.614815i
\(851\) −24.4241 −0.837246
\(852\) 0 0
\(853\) 6.31853i 0.216342i −0.994132 0.108171i \(-0.965501\pi\)
0.994132 0.108171i \(-0.0344995\pi\)
\(854\) −3.81336 −0.130490
\(855\) 0 0
\(856\) 93.9850 3.21234
\(857\) 0.775746i 0.0264990i −0.999912 0.0132495i \(-0.995782\pi\)
0.999912 0.0132495i \(-0.00421757\pi\)
\(858\) 0 0
\(859\) −3.24869 −0.110844 −0.0554220 0.998463i \(-0.517650\pi\)
−0.0554220 + 0.998463i \(0.517650\pi\)
\(860\) 51.8881 58.6820i 1.76937 2.00104i
\(861\) 0 0
\(862\) 2.37962i 0.0810503i
\(863\) 19.9208i 0.678112i 0.940766 + 0.339056i \(0.110108\pi\)
−0.940766 + 0.339056i \(0.889892\pi\)
\(864\) 0 0
\(865\) 37.9003 42.8627i 1.28865 1.45738i
\(866\) 67.5487 2.29540
\(867\) 0 0
\(868\) 21.9756i 0.745899i
\(869\) 8.31265 0.281987
\(870\) 0 0
\(871\) −9.89446 −0.335261
\(872\) 80.9544i 2.74146i
\(873\) 0 0
\(874\) 29.0435 0.982411
\(875\) −7.43136 5.09825i −0.251226 0.172352i
\(876\) 0 0
\(877\) 22.1378i 0.747539i −0.927522 0.373770i \(-0.878065\pi\)
0.927522 0.373770i \(-0.121935\pi\)
\(878\) 77.0757i 2.60118i
\(879\) 0 0
\(880\) −75.5691 66.8202i −2.54743 2.25251i
\(881\) −2.23155 −0.0751828 −0.0375914 0.999293i \(-0.511969\pi\)
−0.0375914 + 0.999293i \(0.511969\pi\)
\(882\) 0 0
\(883\) 4.30440i 0.144855i −0.997374 0.0724273i \(-0.976925\pi\)
0.997374 0.0724273i \(-0.0230745\pi\)
\(884\) 6.96239 0.234170
\(885\) 0 0
\(886\) 98.9276 3.32354
\(887\) 15.9330i 0.534979i 0.963561 + 0.267489i \(0.0861940\pi\)
−0.963561 + 0.267489i \(0.913806\pi\)
\(888\) 0 0
\(889\) 3.46168 0.116101
\(890\) 10.9986 12.4387i 0.368673 0.416945i
\(891\) 0 0
\(892\) 128.432i 4.30022i
\(893\) 5.35026i 0.179040i
\(894\) 0 0
\(895\) −20.3127 17.9610i −0.678977 0.600369i
\(896\) −13.3561 −0.446197
\(897\) 0 0
\(898\) 33.9365i 1.13248i
\(899\) −12.7856 −0.426423
\(900\) 0 0
\(901\) −7.74798 −0.258123
\(902\) 81.7255i 2.72116i
\(903\) 0 0
\(904\) −4.85097 −0.161341
\(905\) 4.57452 + 4.04491i 0.152062 + 0.134457i
\(906\) 0 0
\(907\) 51.9086i 1.72360i 0.507251 + 0.861798i \(0.330662\pi\)
−0.507251 + 0.861798i \(0.669338\pi\)
\(908\) 51.3317i 1.70350i
\(909\) 0 0
\(910\) −3.19394 + 3.61213i −0.105878 + 0.119741i
\(911\) 9.67750 0.320630 0.160315 0.987066i \(-0.448749\pi\)
0.160315 + 0.987066i \(0.448749\pi\)
\(912\) 0 0
\(913\) 14.1260i 0.467503i
\(914\) −67.0103 −2.21651
\(915\) 0 0
\(916\) 27.5877 0.911523
\(917\) 0.674176i 0.0222632i
\(918\) 0 0
\(919\) −13.5515 −0.447022 −0.223511 0.974701i \(-0.571752\pi\)
−0.223511 + 0.974701i \(0.571752\pi\)
\(920\) 91.6707 + 81.0576i 3.02229 + 2.67239i
\(921\) 0 0
\(922\) 98.6780i 3.24979i
\(923\) 8.56230i 0.281831i
\(924\) 0 0
\(925\) 18.7005 2.30677i 0.614869 0.0758462i
\(926\) 104.381 3.43017
\(927\) 0 0
\(928\) 38.5705i 1.26614i
\(929\) −9.44992 −0.310042 −0.155021 0.987911i \(-0.549545\pi\)
−0.155021 + 0.987911i \(0.549545\pi\)
\(930\) 0 0
\(931\) 10.6375 0.348631
\(932\) 55.4880i 1.81757i
\(933\) 0 0
\(934\) −87.6625 −2.86840
\(935\) −7.35026 + 8.31265i −0.240379 + 0.271853i
\(936\) 0 0
\(937\) 16.0409i 0.524035i −0.965063 0.262017i \(-0.915612\pi\)
0.965063 0.262017i \(-0.0843877\pi\)
\(938\) 21.3357i 0.696634i
\(939\) 0 0
\(940\) −24.3938 + 27.5877i −0.795636 + 0.899811i
\(941\) 21.6747 0.706574 0.353287 0.935515i \(-0.385064\pi\)
0.353287 + 0.935515i \(0.385064\pi\)
\(942\) 0 0
\(943\) 53.8759i 1.75444i
\(944\) 73.5002 2.39223
\(945\) 0 0
\(946\) 66.7934 2.17164
\(947\) 4.63118i 0.150493i −0.997165 0.0752466i \(-0.976026\pi\)
0.997165 0.0752466i \(-0.0239744\pi\)
\(948\) 0 0
\(949\) −11.7685 −0.382020
\(950\) −22.2374 + 2.74306i −0.721477 + 0.0889966i
\(951\) 0 0
\(952\) 9.18997i 0.297849i
\(953\) 26.2981i 0.851878i −0.904752 0.425939i \(-0.859944\pi\)
0.904752 0.425939i \(-0.140056\pi\)
\(954\) 0 0
\(955\) 34.5501 + 30.5501i 1.11801 + 0.988577i
\(956\) 61.1632 1.97816
\(957\) 0 0
\(958\) 45.1427i 1.45849i
\(959\) −12.0409 −0.388822
\(960\) 0 0
\(961\) −3.04491 −0.0982228
\(962\) 10.0811i 0.325028i
\(963\) 0 0
\(964\) 147.601 4.75392
\(965\) −32.2736 + 36.4993i −1.03892 + 1.17495i
\(966\) 0 0
\(967\) 11.9405i 0.383981i 0.981397 + 0.191990i \(0.0614942\pi\)
−0.981397 + 0.191990i \(0.938506\pi\)
\(968\) 21.1646i 0.680255i
\(969\) 0 0
\(970\) 8.39772 + 7.42548i 0.269635 + 0.238418i
\(971\) −30.1524 −0.967635 −0.483818 0.875169i \(-0.660750\pi\)
−0.483818 + 0.875169i \(0.660750\pi\)
\(972\) 0 0
\(973\) 6.80209i 0.218065i
\(974\) −24.7308 −0.792427
\(975\) 0 0
\(976\) −21.7078 −0.694850
\(977\) 26.9321i 0.861633i 0.902439 + 0.430817i \(0.141774\pi\)
−0.902439 + 0.430817i \(0.858226\pi\)
\(978\) 0 0
\(979\) 10.2012 0.326033
\(980\) 54.8505 + 48.5002i 1.75214 + 1.54928i
\(981\) 0 0
\(982\) 68.8843i 2.19819i
\(983\) 20.5902i 0.656727i 0.944551 + 0.328363i \(0.106497\pi\)
−0.944551 + 0.328363i \(0.893503\pi\)
\(984\) 0 0
\(985\) 2.96239 3.35026i 0.0943895 0.106748i
\(986\) 8.73481 0.278173
\(987\) 0 0
\(988\) 8.63752i 0.274796i
\(989\) −44.0322 −1.40014
\(990\) 0 0
\(991\) −48.1378 −1.52915 −0.764573 0.644537i \(-0.777050\pi\)
−0.764573 + 0.644537i \(0.777050\pi\)
\(992\) 84.3327i 2.67756i
\(993\) 0 0
\(994\) −18.4631 −0.585614
\(995\) −28.0606 24.8119i −0.889582 0.786591i
\(996\) 0 0
\(997\) 33.4255i 1.05860i −0.848436 0.529298i \(-0.822455\pi\)
0.848436 0.529298i \(-0.177545\pi\)
\(998\) 74.1051i 2.34576i
\(999\) 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 585.2.c.b.469.6 6
3.2 odd 2 65.2.b.a.14.1 6
5.2 odd 4 2925.2.a.bf.1.1 3
5.3 odd 4 2925.2.a.bj.1.3 3
5.4 even 2 inner 585.2.c.b.469.1 6
12.11 even 2 1040.2.d.c.209.4 6
15.2 even 4 325.2.a.k.1.3 3
15.8 even 4 325.2.a.j.1.1 3
15.14 odd 2 65.2.b.a.14.6 yes 6
39.2 even 12 845.2.l.e.654.6 12
39.5 even 4 845.2.d.a.844.1 6
39.8 even 4 845.2.d.b.844.5 6
39.11 even 12 845.2.l.d.654.2 12
39.17 odd 6 845.2.n.g.484.6 12
39.20 even 12 845.2.l.d.699.1 12
39.23 odd 6 845.2.n.g.529.1 12
39.29 odd 6 845.2.n.f.529.6 12
39.32 even 12 845.2.l.e.699.5 12
39.35 odd 6 845.2.n.f.484.1 12
39.38 odd 2 845.2.b.c.339.6 6
60.23 odd 4 5200.2.a.cj.1.1 3
60.47 odd 4 5200.2.a.cb.1.3 3
60.59 even 2 1040.2.d.c.209.3 6
195.29 odd 6 845.2.n.f.529.1 12
195.38 even 4 4225.2.a.bh.1.3 3
195.44 even 4 845.2.d.b.844.6 6
195.59 even 12 845.2.l.e.699.6 12
195.74 odd 6 845.2.n.f.484.6 12
195.77 even 4 4225.2.a.ba.1.1 3
195.89 even 12 845.2.l.e.654.5 12
195.119 even 12 845.2.l.d.654.1 12
195.134 odd 6 845.2.n.g.484.1 12
195.149 even 12 845.2.l.d.699.2 12
195.164 even 4 845.2.d.a.844.2 6
195.179 odd 6 845.2.n.g.529.6 12
195.194 odd 2 845.2.b.c.339.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.b.a.14.1 6 3.2 odd 2
65.2.b.a.14.6 yes 6 15.14 odd 2
325.2.a.j.1.1 3 15.8 even 4
325.2.a.k.1.3 3 15.2 even 4
585.2.c.b.469.1 6 5.4 even 2 inner
585.2.c.b.469.6 6 1.1 even 1 trivial
845.2.b.c.339.1 6 195.194 odd 2
845.2.b.c.339.6 6 39.38 odd 2
845.2.d.a.844.1 6 39.5 even 4
845.2.d.a.844.2 6 195.164 even 4
845.2.d.b.844.5 6 39.8 even 4
845.2.d.b.844.6 6 195.44 even 4
845.2.l.d.654.1 12 195.119 even 12
845.2.l.d.654.2 12 39.11 even 12
845.2.l.d.699.1 12 39.20 even 12
845.2.l.d.699.2 12 195.149 even 12
845.2.l.e.654.5 12 195.89 even 12
845.2.l.e.654.6 12 39.2 even 12
845.2.l.e.699.5 12 39.32 even 12
845.2.l.e.699.6 12 195.59 even 12
845.2.n.f.484.1 12 39.35 odd 6
845.2.n.f.484.6 12 195.74 odd 6
845.2.n.f.529.1 12 195.29 odd 6
845.2.n.f.529.6 12 39.29 odd 6
845.2.n.g.484.1 12 195.134 odd 6
845.2.n.g.484.6 12 39.17 odd 6
845.2.n.g.529.1 12 39.23 odd 6
845.2.n.g.529.6 12 195.179 odd 6
1040.2.d.c.209.3 6 60.59 even 2
1040.2.d.c.209.4 6 12.11 even 2
2925.2.a.bf.1.1 3 5.2 odd 4
2925.2.a.bj.1.3 3 5.3 odd 4
4225.2.a.ba.1.1 3 195.77 even 4
4225.2.a.bh.1.3 3 195.38 even 4
5200.2.a.cb.1.3 3 60.47 odd 4
5200.2.a.cj.1.1 3 60.23 odd 4