Properties

Label 845.2.a.n.1.6
Level $845$
Weight $2$
Character 845.1
Self dual yes
Analytic conductor $6.747$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [845,2,Mod(1,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.a (trivial)

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: yes
Analytic conductor: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 17x^{7} - 9x^{6} + 59x^{5} + 32x^{4} - 44x^{3} - 23x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.07331\) of defining polynomial
Character \(\chi\) \(=\) 845.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.271374 q^{2} -0.319618 q^{3} -1.92636 q^{4} -1.00000 q^{5} -0.0867358 q^{6} +3.38151 q^{7} -1.06551 q^{8} -2.89784 q^{9} +O(q^{10})\) \(q+0.271374 q^{2} -0.319618 q^{3} -1.92636 q^{4} -1.00000 q^{5} -0.0867358 q^{6} +3.38151 q^{7} -1.06551 q^{8} -2.89784 q^{9} -0.271374 q^{10} +1.75204 q^{11} +0.615697 q^{12} +0.917654 q^{14} +0.319618 q^{15} +3.56356 q^{16} +1.95419 q^{17} -0.786399 q^{18} -7.13261 q^{19} +1.92636 q^{20} -1.08079 q^{21} +0.475459 q^{22} +7.61420 q^{23} +0.340556 q^{24} +1.00000 q^{25} +1.88505 q^{27} -6.51400 q^{28} +3.98380 q^{29} +0.0867358 q^{30} +4.86923 q^{31} +3.09808 q^{32} -0.559984 q^{33} +0.530316 q^{34} -3.38151 q^{35} +5.58228 q^{36} +10.5010 q^{37} -1.93560 q^{38} +1.06551 q^{40} +0.911110 q^{41} -0.293298 q^{42} +4.58553 q^{43} -3.37506 q^{44} +2.89784 q^{45} +2.06629 q^{46} -8.58491 q^{47} -1.13898 q^{48} +4.43464 q^{49} +0.271374 q^{50} -0.624593 q^{51} +11.9646 q^{53} +0.511554 q^{54} -1.75204 q^{55} -3.60304 q^{56} +2.27971 q^{57} +1.08110 q^{58} +3.82297 q^{59} -0.615697 q^{60} +7.98476 q^{61} +1.32138 q^{62} -9.79910 q^{63} -6.28638 q^{64} -0.151965 q^{66} +0.472749 q^{67} -3.76446 q^{68} -2.43363 q^{69} -0.917654 q^{70} -5.50312 q^{71} +3.08768 q^{72} +2.93857 q^{73} +2.84969 q^{74} -0.319618 q^{75} +13.7400 q^{76} +5.92456 q^{77} +4.09938 q^{79} -3.56356 q^{80} +8.09104 q^{81} +0.247252 q^{82} -11.7733 q^{83} +2.08199 q^{84} -1.95419 q^{85} +1.24439 q^{86} -1.27329 q^{87} -1.86682 q^{88} -3.85992 q^{89} +0.786399 q^{90} -14.6677 q^{92} -1.55629 q^{93} -2.32972 q^{94} +7.13261 q^{95} -0.990200 q^{96} -6.95775 q^{97} +1.20345 q^{98} -5.07715 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{2} + 7 q^{3} + 17 q^{4} - 9 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 3 q^{2} + 7 q^{3} + 17 q^{4} - 9 q^{5} + 2 q^{6} - 7 q^{7} - 12 q^{8} + 16 q^{9} + 3 q^{10} + 9 q^{11} + 12 q^{12} - 2 q^{14} - 7 q^{15} + 37 q^{16} - q^{17} + 10 q^{18} + 4 q^{19} - 17 q^{20} + q^{21} + 12 q^{22} + 14 q^{23} + 35 q^{24} + 9 q^{25} + 22 q^{27} - 18 q^{28} + 12 q^{29} - 2 q^{30} + 7 q^{31} - 22 q^{32} + 8 q^{33} + 30 q^{34} + 7 q^{35} + 3 q^{36} + 5 q^{37} - 47 q^{38} + 12 q^{40} + 10 q^{41} - 11 q^{42} + 39 q^{43} + 25 q^{44} - 16 q^{45} - 6 q^{46} - 36 q^{47} - 3 q^{48} + 16 q^{49} - 3 q^{50} + 43 q^{51} - 8 q^{53} + 2 q^{54} - 9 q^{55} - 29 q^{56} + 32 q^{57} - 21 q^{58} + 21 q^{59} - 12 q^{60} - 3 q^{61} - 10 q^{62} - 35 q^{63} + 34 q^{64} - 49 q^{66} - q^{67} - 20 q^{68} - 13 q^{69} + 2 q^{70} + q^{71} + 3 q^{72} - 15 q^{74} + 7 q^{75} + 5 q^{76} - 4 q^{77} + 39 q^{79} - 37 q^{80} + 29 q^{81} - 4 q^{82} - 7 q^{83} - 12 q^{84} + q^{85} + 24 q^{86} + 16 q^{87} + 42 q^{88} + 19 q^{89} - 10 q^{90} - 27 q^{92} - 31 q^{93} + 16 q^{94} - 4 q^{95} + 7 q^{96} + 34 q^{97} - 48 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.271374 0.191890 0.0959451 0.995387i \(-0.469413\pi\)
0.0959451 + 0.995387i \(0.469413\pi\)
\(3\) −0.319618 −0.184531 −0.0922656 0.995734i \(-0.529411\pi\)
−0.0922656 + 0.995734i \(0.529411\pi\)
\(4\) −1.92636 −0.963178
\(5\) −1.00000 −0.447214
\(6\) −0.0867358 −0.0354098
\(7\) 3.38151 1.27809 0.639046 0.769168i \(-0.279329\pi\)
0.639046 + 0.769168i \(0.279329\pi\)
\(8\) −1.06551 −0.376715
\(9\) −2.89784 −0.965948
\(10\) −0.271374 −0.0858159
\(11\) 1.75204 0.528261 0.264131 0.964487i \(-0.414915\pi\)
0.264131 + 0.964487i \(0.414915\pi\)
\(12\) 0.615697 0.177736
\(13\) 0 0
\(14\) 0.917654 0.245253
\(15\) 0.319618 0.0825249
\(16\) 3.56356 0.890890
\(17\) 1.95419 0.473960 0.236980 0.971514i \(-0.423842\pi\)
0.236980 + 0.971514i \(0.423842\pi\)
\(18\) −0.786399 −0.185356
\(19\) −7.13261 −1.63633 −0.818167 0.574981i \(-0.805010\pi\)
−0.818167 + 0.574981i \(0.805010\pi\)
\(20\) 1.92636 0.430746
\(21\) −1.08079 −0.235848
\(22\) 0.475459 0.101368
\(23\) 7.61420 1.58767 0.793835 0.608133i \(-0.208081\pi\)
0.793835 + 0.608133i \(0.208081\pi\)
\(24\) 0.340556 0.0695157
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.88505 0.362779
\(28\) −6.51400 −1.23103
\(29\) 3.98380 0.739773 0.369886 0.929077i \(-0.379397\pi\)
0.369886 + 0.929077i \(0.379397\pi\)
\(30\) 0.0867358 0.0158357
\(31\) 4.86923 0.874540 0.437270 0.899330i \(-0.355945\pi\)
0.437270 + 0.899330i \(0.355945\pi\)
\(32\) 3.09808 0.547668
\(33\) −0.559984 −0.0974807
\(34\) 0.530316 0.0909484
\(35\) −3.38151 −0.571580
\(36\) 5.58228 0.930380
\(37\) 10.5010 1.72635 0.863176 0.504903i \(-0.168472\pi\)
0.863176 + 0.504903i \(0.168472\pi\)
\(38\) −1.93560 −0.313997
\(39\) 0 0
\(40\) 1.06551 0.168472
\(41\) 0.911110 0.142292 0.0711458 0.997466i \(-0.477334\pi\)
0.0711458 + 0.997466i \(0.477334\pi\)
\(42\) −0.293298 −0.0452569
\(43\) 4.58553 0.699286 0.349643 0.936883i \(-0.386303\pi\)
0.349643 + 0.936883i \(0.386303\pi\)
\(44\) −3.37506 −0.508810
\(45\) 2.89784 0.431985
\(46\) 2.06629 0.304658
\(47\) −8.58491 −1.25224 −0.626119 0.779728i \(-0.715357\pi\)
−0.626119 + 0.779728i \(0.715357\pi\)
\(48\) −1.13898 −0.164397
\(49\) 4.43464 0.633520
\(50\) 0.271374 0.0383781
\(51\) −0.624593 −0.0874605
\(52\) 0 0
\(53\) 11.9646 1.64347 0.821734 0.569871i \(-0.193007\pi\)
0.821734 + 0.569871i \(0.193007\pi\)
\(54\) 0.511554 0.0696137
\(55\) −1.75204 −0.236246
\(56\) −3.60304 −0.481476
\(57\) 2.27971 0.301955
\(58\) 1.08110 0.141955
\(59\) 3.82297 0.497708 0.248854 0.968541i \(-0.419946\pi\)
0.248854 + 0.968541i \(0.419946\pi\)
\(60\) −0.615697 −0.0794862
\(61\) 7.98476 1.02234 0.511172 0.859479i \(-0.329212\pi\)
0.511172 + 0.859479i \(0.329212\pi\)
\(62\) 1.32138 0.167816
\(63\) −9.79910 −1.23457
\(64\) −6.28638 −0.785798
\(65\) 0 0
\(66\) −0.151965 −0.0187056
\(67\) 0.472749 0.0577554 0.0288777 0.999583i \(-0.490807\pi\)
0.0288777 + 0.999583i \(0.490807\pi\)
\(68\) −3.76446 −0.456508
\(69\) −2.43363 −0.292975
\(70\) −0.917654 −0.109681
\(71\) −5.50312 −0.653101 −0.326550 0.945180i \(-0.605886\pi\)
−0.326550 + 0.945180i \(0.605886\pi\)
\(72\) 3.08768 0.363887
\(73\) 2.93857 0.343934 0.171967 0.985103i \(-0.444988\pi\)
0.171967 + 0.985103i \(0.444988\pi\)
\(74\) 2.84969 0.331270
\(75\) −0.319618 −0.0369063
\(76\) 13.7400 1.57608
\(77\) 5.92456 0.675167
\(78\) 0 0
\(79\) 4.09938 0.461216 0.230608 0.973047i \(-0.425928\pi\)
0.230608 + 0.973047i \(0.425928\pi\)
\(80\) −3.56356 −0.398418
\(81\) 8.09104 0.899004
\(82\) 0.247252 0.0273044
\(83\) −11.7733 −1.29229 −0.646144 0.763215i \(-0.723619\pi\)
−0.646144 + 0.763215i \(0.723619\pi\)
\(84\) 2.08199 0.227164
\(85\) −1.95419 −0.211961
\(86\) 1.24439 0.134186
\(87\) −1.27329 −0.136511
\(88\) −1.86682 −0.199004
\(89\) −3.85992 −0.409151 −0.204576 0.978851i \(-0.565581\pi\)
−0.204576 + 0.978851i \(0.565581\pi\)
\(90\) 0.786399 0.0828937
\(91\) 0 0
\(92\) −14.6677 −1.52921
\(93\) −1.55629 −0.161380
\(94\) −2.32972 −0.240292
\(95\) 7.13261 0.731791
\(96\) −0.990200 −0.101062
\(97\) −6.95775 −0.706453 −0.353226 0.935538i \(-0.614915\pi\)
−0.353226 + 0.935538i \(0.614915\pi\)
\(98\) 1.20345 0.121566
\(99\) −5.07715 −0.510273
\(100\) −1.92636 −0.192636
\(101\) −17.8923 −1.78035 −0.890175 0.455619i \(-0.849418\pi\)
−0.890175 + 0.455619i \(0.849418\pi\)
\(102\) −0.169498 −0.0167828
\(103\) 10.2561 1.01056 0.505281 0.862955i \(-0.331389\pi\)
0.505281 + 0.862955i \(0.331389\pi\)
\(104\) 0 0
\(105\) 1.08079 0.105474
\(106\) 3.24689 0.315366
\(107\) 12.8986 1.24696 0.623478 0.781841i \(-0.285719\pi\)
0.623478 + 0.781841i \(0.285719\pi\)
\(108\) −3.63129 −0.349421
\(109\) 10.4476 1.00070 0.500348 0.865825i \(-0.333206\pi\)
0.500348 + 0.865825i \(0.333206\pi\)
\(110\) −0.475459 −0.0453332
\(111\) −3.35630 −0.318566
\(112\) 12.0502 1.13864
\(113\) −18.9766 −1.78516 −0.892582 0.450884i \(-0.851109\pi\)
−0.892582 + 0.450884i \(0.851109\pi\)
\(114\) 0.618653 0.0579422
\(115\) −7.61420 −0.710028
\(116\) −7.67421 −0.712533
\(117\) 0 0
\(118\) 1.03745 0.0955054
\(119\) 6.60812 0.605765
\(120\) −0.340556 −0.0310883
\(121\) −7.93034 −0.720940
\(122\) 2.16685 0.196178
\(123\) −0.291207 −0.0262572
\(124\) −9.37987 −0.842337
\(125\) −1.00000 −0.0894427
\(126\) −2.65922 −0.236902
\(127\) −9.88489 −0.877142 −0.438571 0.898697i \(-0.644515\pi\)
−0.438571 + 0.898697i \(0.644515\pi\)
\(128\) −7.90212 −0.698455
\(129\) −1.46561 −0.129040
\(130\) 0 0
\(131\) −11.4705 −1.00218 −0.501089 0.865396i \(-0.667067\pi\)
−0.501089 + 0.865396i \(0.667067\pi\)
\(132\) 1.07873 0.0938913
\(133\) −24.1190 −2.09139
\(134\) 0.128292 0.0110827
\(135\) −1.88505 −0.162240
\(136\) −2.08221 −0.178548
\(137\) −13.0826 −1.11772 −0.558861 0.829261i \(-0.688761\pi\)
−0.558861 + 0.829261i \(0.688761\pi\)
\(138\) −0.660424 −0.0562190
\(139\) 11.9762 1.01581 0.507904 0.861414i \(-0.330421\pi\)
0.507904 + 0.861414i \(0.330421\pi\)
\(140\) 6.51400 0.550534
\(141\) 2.74389 0.231077
\(142\) −1.49340 −0.125324
\(143\) 0 0
\(144\) −10.3266 −0.860554
\(145\) −3.98380 −0.330836
\(146\) 0.797452 0.0659976
\(147\) −1.41739 −0.116904
\(148\) −20.2287 −1.66278
\(149\) 16.3052 1.33577 0.667887 0.744262i \(-0.267199\pi\)
0.667887 + 0.744262i \(0.267199\pi\)
\(150\) −0.0867358 −0.00708195
\(151\) 8.75415 0.712402 0.356201 0.934409i \(-0.384072\pi\)
0.356201 + 0.934409i \(0.384072\pi\)
\(152\) 7.59987 0.616431
\(153\) −5.66293 −0.457821
\(154\) 1.60777 0.129558
\(155\) −4.86923 −0.391106
\(156\) 0 0
\(157\) 6.48821 0.517816 0.258908 0.965902i \(-0.416637\pi\)
0.258908 + 0.965902i \(0.416637\pi\)
\(158\) 1.11246 0.0885029
\(159\) −3.82411 −0.303271
\(160\) −3.09808 −0.244925
\(161\) 25.7475 2.02919
\(162\) 2.19570 0.172510
\(163\) −3.84933 −0.301503 −0.150751 0.988572i \(-0.548169\pi\)
−0.150751 + 0.988572i \(0.548169\pi\)
\(164\) −1.75512 −0.137052
\(165\) 0.559984 0.0435947
\(166\) −3.19497 −0.247977
\(167\) −6.85894 −0.530760 −0.265380 0.964144i \(-0.585497\pi\)
−0.265380 + 0.964144i \(0.585497\pi\)
\(168\) 1.15159 0.0888474
\(169\) 0 0
\(170\) −0.530316 −0.0406733
\(171\) 20.6692 1.58061
\(172\) −8.83336 −0.673537
\(173\) 16.0204 1.21801 0.609005 0.793166i \(-0.291569\pi\)
0.609005 + 0.793166i \(0.291569\pi\)
\(174\) −0.345538 −0.0261952
\(175\) 3.38151 0.255618
\(176\) 6.24352 0.470623
\(177\) −1.22189 −0.0918428
\(178\) −1.04748 −0.0785121
\(179\) −8.08309 −0.604158 −0.302079 0.953283i \(-0.597681\pi\)
−0.302079 + 0.953283i \(0.597681\pi\)
\(180\) −5.58228 −0.416079
\(181\) 11.0898 0.824298 0.412149 0.911117i \(-0.364778\pi\)
0.412149 + 0.911117i \(0.364778\pi\)
\(182\) 0 0
\(183\) −2.55207 −0.188654
\(184\) −8.11301 −0.598099
\(185\) −10.5010 −0.772048
\(186\) −0.422337 −0.0309672
\(187\) 3.42382 0.250375
\(188\) 16.5376 1.20613
\(189\) 6.37434 0.463665
\(190\) 1.93560 0.140424
\(191\) 21.7532 1.57401 0.787003 0.616949i \(-0.211631\pi\)
0.787003 + 0.616949i \(0.211631\pi\)
\(192\) 2.00924 0.145004
\(193\) −22.5285 −1.62164 −0.810820 0.585296i \(-0.800978\pi\)
−0.810820 + 0.585296i \(0.800978\pi\)
\(194\) −1.88815 −0.135561
\(195\) 0 0
\(196\) −8.54270 −0.610193
\(197\) 17.2749 1.23078 0.615392 0.788221i \(-0.288998\pi\)
0.615392 + 0.788221i \(0.288998\pi\)
\(198\) −1.37781 −0.0979164
\(199\) 17.4901 1.23984 0.619919 0.784666i \(-0.287166\pi\)
0.619919 + 0.784666i \(0.287166\pi\)
\(200\) −1.06551 −0.0753430
\(201\) −0.151099 −0.0106577
\(202\) −4.85550 −0.341632
\(203\) 13.4713 0.945498
\(204\) 1.20319 0.0842400
\(205\) −0.911110 −0.0636347
\(206\) 2.78323 0.193917
\(207\) −22.0648 −1.53361
\(208\) 0 0
\(209\) −12.4967 −0.864412
\(210\) 0.293298 0.0202395
\(211\) 6.33563 0.436163 0.218081 0.975931i \(-0.430020\pi\)
0.218081 + 0.975931i \(0.430020\pi\)
\(212\) −23.0481 −1.58295
\(213\) 1.75889 0.120517
\(214\) 3.50034 0.239279
\(215\) −4.58553 −0.312730
\(216\) −2.00855 −0.136664
\(217\) 16.4654 1.11774
\(218\) 2.83520 0.192024
\(219\) −0.939220 −0.0634666
\(220\) 3.37506 0.227547
\(221\) 0 0
\(222\) −0.910812 −0.0611297
\(223\) −23.4948 −1.57333 −0.786664 0.617381i \(-0.788194\pi\)
−0.786664 + 0.617381i \(0.788194\pi\)
\(224\) 10.4762 0.699970
\(225\) −2.89784 −0.193190
\(226\) −5.14974 −0.342556
\(227\) −9.59397 −0.636774 −0.318387 0.947961i \(-0.603141\pi\)
−0.318387 + 0.947961i \(0.603141\pi\)
\(228\) −4.39153 −0.290836
\(229\) 7.23000 0.477772 0.238886 0.971048i \(-0.423218\pi\)
0.238886 + 0.971048i \(0.423218\pi\)
\(230\) −2.06629 −0.136247
\(231\) −1.89359 −0.124589
\(232\) −4.24478 −0.278683
\(233\) −0.429924 −0.0281652 −0.0140826 0.999901i \(-0.504483\pi\)
−0.0140826 + 0.999901i \(0.504483\pi\)
\(234\) 0 0
\(235\) 8.58491 0.560017
\(236\) −7.36440 −0.479382
\(237\) −1.31023 −0.0851088
\(238\) 1.79327 0.116240
\(239\) 12.7450 0.824405 0.412203 0.911092i \(-0.364760\pi\)
0.412203 + 0.911092i \(0.364760\pi\)
\(240\) 1.13898 0.0735206
\(241\) −19.1338 −1.23252 −0.616258 0.787545i \(-0.711352\pi\)
−0.616258 + 0.787545i \(0.711352\pi\)
\(242\) −2.15209 −0.138341
\(243\) −8.24120 −0.528673
\(244\) −15.3815 −0.984699
\(245\) −4.43464 −0.283319
\(246\) −0.0790259 −0.00503851
\(247\) 0 0
\(248\) −5.18821 −0.329452
\(249\) 3.76295 0.238468
\(250\) −0.271374 −0.0171632
\(251\) 0.411815 0.0259935 0.0129968 0.999916i \(-0.495863\pi\)
0.0129968 + 0.999916i \(0.495863\pi\)
\(252\) 18.8766 1.18911
\(253\) 13.3404 0.838705
\(254\) −2.68250 −0.168315
\(255\) 0.624593 0.0391135
\(256\) 10.4283 0.651771
\(257\) −2.47556 −0.154421 −0.0772105 0.997015i \(-0.524601\pi\)
−0.0772105 + 0.997015i \(0.524601\pi\)
\(258\) −0.397729 −0.0247616
\(259\) 35.5093 2.20644
\(260\) 0 0
\(261\) −11.5444 −0.714582
\(262\) −3.11278 −0.192308
\(263\) −11.3591 −0.700430 −0.350215 0.936669i \(-0.613892\pi\)
−0.350215 + 0.936669i \(0.613892\pi\)
\(264\) 0.596669 0.0367224
\(265\) −11.9646 −0.734981
\(266\) −6.54527 −0.401317
\(267\) 1.23370 0.0755012
\(268\) −0.910682 −0.0556288
\(269\) −29.6888 −1.81016 −0.905078 0.425245i \(-0.860188\pi\)
−0.905078 + 0.425245i \(0.860188\pi\)
\(270\) −0.511554 −0.0311322
\(271\) 14.7761 0.897583 0.448792 0.893636i \(-0.351855\pi\)
0.448792 + 0.893636i \(0.351855\pi\)
\(272\) 6.96387 0.422247
\(273\) 0 0
\(274\) −3.55028 −0.214480
\(275\) 1.75204 0.105652
\(276\) 4.68804 0.282187
\(277\) −1.94652 −0.116955 −0.0584774 0.998289i \(-0.518625\pi\)
−0.0584774 + 0.998289i \(0.518625\pi\)
\(278\) 3.25002 0.194924
\(279\) −14.1103 −0.844760
\(280\) 3.60304 0.215323
\(281\) −18.5256 −1.10515 −0.552573 0.833464i \(-0.686354\pi\)
−0.552573 + 0.833464i \(0.686354\pi\)
\(282\) 0.744619 0.0443414
\(283\) 13.0952 0.778427 0.389214 0.921148i \(-0.372747\pi\)
0.389214 + 0.921148i \(0.372747\pi\)
\(284\) 10.6010 0.629052
\(285\) −2.27971 −0.135038
\(286\) 0 0
\(287\) 3.08093 0.181862
\(288\) −8.97775 −0.529019
\(289\) −13.1811 −0.775362
\(290\) −1.08110 −0.0634843
\(291\) 2.22382 0.130363
\(292\) −5.66074 −0.331270
\(293\) 1.20753 0.0705445 0.0352722 0.999378i \(-0.488770\pi\)
0.0352722 + 0.999378i \(0.488770\pi\)
\(294\) −0.384642 −0.0224328
\(295\) −3.82297 −0.222582
\(296\) −11.1889 −0.650342
\(297\) 3.30270 0.191642
\(298\) 4.42481 0.256322
\(299\) 0 0
\(300\) 0.615697 0.0355473
\(301\) 15.5060 0.893752
\(302\) 2.37565 0.136703
\(303\) 5.71869 0.328530
\(304\) −25.4175 −1.45779
\(305\) −7.98476 −0.457206
\(306\) −1.53677 −0.0878514
\(307\) 21.9275 1.25147 0.625733 0.780037i \(-0.284800\pi\)
0.625733 + 0.780037i \(0.284800\pi\)
\(308\) −11.4128 −0.650306
\(309\) −3.27802 −0.186480
\(310\) −1.32138 −0.0750494
\(311\) −12.5966 −0.714288 −0.357144 0.934049i \(-0.616249\pi\)
−0.357144 + 0.934049i \(0.616249\pi\)
\(312\) 0 0
\(313\) 26.8268 1.51634 0.758169 0.652058i \(-0.226094\pi\)
0.758169 + 0.652058i \(0.226094\pi\)
\(314\) 1.76073 0.0993638
\(315\) 9.79910 0.552117
\(316\) −7.89687 −0.444234
\(317\) −17.1279 −0.961999 −0.481000 0.876721i \(-0.659726\pi\)
−0.481000 + 0.876721i \(0.659726\pi\)
\(318\) −1.03776 −0.0581948
\(319\) 6.97979 0.390793
\(320\) 6.28638 0.351420
\(321\) −4.12262 −0.230102
\(322\) 6.98720 0.389382
\(323\) −13.9385 −0.775557
\(324\) −15.5862 −0.865901
\(325\) 0 0
\(326\) −1.04461 −0.0578555
\(327\) −3.33923 −0.184660
\(328\) −0.970798 −0.0536033
\(329\) −29.0300 −1.60047
\(330\) 0.151965 0.00836540
\(331\) 1.43062 0.0786338 0.0393169 0.999227i \(-0.487482\pi\)
0.0393169 + 0.999227i \(0.487482\pi\)
\(332\) 22.6796 1.24470
\(333\) −30.4302 −1.66757
\(334\) −1.86134 −0.101848
\(335\) −0.472749 −0.0258290
\(336\) −3.85147 −0.210115
\(337\) −7.48872 −0.407937 −0.203968 0.978977i \(-0.565384\pi\)
−0.203968 + 0.978977i \(0.565384\pi\)
\(338\) 0 0
\(339\) 6.06524 0.329419
\(340\) 3.76446 0.204157
\(341\) 8.53111 0.461985
\(342\) 5.60908 0.303304
\(343\) −8.67480 −0.468395
\(344\) −4.88593 −0.263431
\(345\) 2.43363 0.131022
\(346\) 4.34752 0.233724
\(347\) 6.96813 0.374069 0.187034 0.982353i \(-0.440112\pi\)
0.187034 + 0.982353i \(0.440112\pi\)
\(348\) 2.45281 0.131485
\(349\) 21.9738 1.17623 0.588116 0.808777i \(-0.299870\pi\)
0.588116 + 0.808777i \(0.299870\pi\)
\(350\) 0.917654 0.0490507
\(351\) 0 0
\(352\) 5.42797 0.289312
\(353\) 13.0184 0.692899 0.346450 0.938069i \(-0.387387\pi\)
0.346450 + 0.938069i \(0.387387\pi\)
\(354\) −0.331589 −0.0176237
\(355\) 5.50312 0.292075
\(356\) 7.43559 0.394085
\(357\) −2.11207 −0.111783
\(358\) −2.19354 −0.115932
\(359\) 9.99211 0.527363 0.263682 0.964610i \(-0.415063\pi\)
0.263682 + 0.964610i \(0.415063\pi\)
\(360\) −3.08768 −0.162735
\(361\) 31.8742 1.67759
\(362\) 3.00948 0.158175
\(363\) 2.53468 0.133036
\(364\) 0 0
\(365\) −2.93857 −0.153812
\(366\) −0.692564 −0.0362009
\(367\) 25.2882 1.32004 0.660018 0.751250i \(-0.270549\pi\)
0.660018 + 0.751250i \(0.270549\pi\)
\(368\) 27.1337 1.41444
\(369\) −2.64026 −0.137446
\(370\) −2.84969 −0.148149
\(371\) 40.4586 2.10050
\(372\) 2.99797 0.155438
\(373\) 14.2703 0.738887 0.369444 0.929253i \(-0.379548\pi\)
0.369444 + 0.929253i \(0.379548\pi\)
\(374\) 0.929136 0.0480445
\(375\) 0.319618 0.0165050
\(376\) 9.14730 0.471736
\(377\) 0 0
\(378\) 1.72983 0.0889728
\(379\) 7.01341 0.360255 0.180127 0.983643i \(-0.442349\pi\)
0.180127 + 0.983643i \(0.442349\pi\)
\(380\) −13.7400 −0.704845
\(381\) 3.15938 0.161860
\(382\) 5.90325 0.302036
\(383\) 4.69336 0.239820 0.119910 0.992785i \(-0.461739\pi\)
0.119910 + 0.992785i \(0.461739\pi\)
\(384\) 2.52565 0.128887
\(385\) −5.92456 −0.301944
\(386\) −6.11365 −0.311177
\(387\) −13.2881 −0.675474
\(388\) 13.4031 0.680440
\(389\) −9.34682 −0.473902 −0.236951 0.971522i \(-0.576148\pi\)
−0.236951 + 0.971522i \(0.576148\pi\)
\(390\) 0 0
\(391\) 14.8796 0.752493
\(392\) −4.72515 −0.238656
\(393\) 3.66616 0.184933
\(394\) 4.68795 0.236176
\(395\) −4.09938 −0.206262
\(396\) 9.78041 0.491484
\(397\) −21.2133 −1.06467 −0.532334 0.846535i \(-0.678685\pi\)
−0.532334 + 0.846535i \(0.678685\pi\)
\(398\) 4.74634 0.237913
\(399\) 7.70887 0.385926
\(400\) 3.56356 0.178178
\(401\) 18.2172 0.909725 0.454863 0.890562i \(-0.349688\pi\)
0.454863 + 0.890562i \(0.349688\pi\)
\(402\) −0.0410042 −0.00204511
\(403\) 0 0
\(404\) 34.4669 1.71479
\(405\) −8.09104 −0.402047
\(406\) 3.65575 0.181432
\(407\) 18.3982 0.911965
\(408\) 0.665510 0.0329477
\(409\) −4.36136 −0.215655 −0.107828 0.994170i \(-0.534389\pi\)
−0.107828 + 0.994170i \(0.534389\pi\)
\(410\) −0.247252 −0.0122109
\(411\) 4.18143 0.206255
\(412\) −19.7569 −0.973351
\(413\) 12.9274 0.636117
\(414\) −5.98780 −0.294284
\(415\) 11.7733 0.577929
\(416\) 0 0
\(417\) −3.82780 −0.187448
\(418\) −3.39127 −0.165872
\(419\) 17.0555 0.833214 0.416607 0.909087i \(-0.363219\pi\)
0.416607 + 0.909087i \(0.363219\pi\)
\(420\) −2.08199 −0.101591
\(421\) 1.27996 0.0623816 0.0311908 0.999513i \(-0.490070\pi\)
0.0311908 + 0.999513i \(0.490070\pi\)
\(422\) 1.71932 0.0836954
\(423\) 24.8777 1.20960
\(424\) −12.7484 −0.619119
\(425\) 1.95419 0.0947921
\(426\) 0.477318 0.0231261
\(427\) 27.0006 1.30665
\(428\) −24.8473 −1.20104
\(429\) 0 0
\(430\) −1.24439 −0.0600099
\(431\) −32.8733 −1.58345 −0.791725 0.610878i \(-0.790817\pi\)
−0.791725 + 0.610878i \(0.790817\pi\)
\(432\) 6.71751 0.323196
\(433\) −28.2622 −1.35819 −0.679097 0.734048i \(-0.737628\pi\)
−0.679097 + 0.734048i \(0.737628\pi\)
\(434\) 4.46827 0.214484
\(435\) 1.27329 0.0610497
\(436\) −20.1257 −0.963848
\(437\) −54.3091 −2.59796
\(438\) −0.254880 −0.0121786
\(439\) 26.2621 1.25342 0.626711 0.779252i \(-0.284401\pi\)
0.626711 + 0.779252i \(0.284401\pi\)
\(440\) 1.86682 0.0889972
\(441\) −12.8509 −0.611948
\(442\) 0 0
\(443\) −14.0477 −0.667427 −0.333714 0.942674i \(-0.608302\pi\)
−0.333714 + 0.942674i \(0.608302\pi\)
\(444\) 6.46543 0.306836
\(445\) 3.85992 0.182978
\(446\) −6.37588 −0.301906
\(447\) −5.21143 −0.246492
\(448\) −21.2575 −1.00432
\(449\) 4.45904 0.210435 0.105218 0.994449i \(-0.466446\pi\)
0.105218 + 0.994449i \(0.466446\pi\)
\(450\) −0.786399 −0.0370712
\(451\) 1.59631 0.0751671
\(452\) 36.5556 1.71943
\(453\) −2.79798 −0.131461
\(454\) −2.60355 −0.122191
\(455\) 0 0
\(456\) −2.42905 −0.113751
\(457\) −16.4149 −0.767855 −0.383927 0.923363i \(-0.625429\pi\)
−0.383927 + 0.923363i \(0.625429\pi\)
\(458\) 1.96203 0.0916798
\(459\) 3.68375 0.171943
\(460\) 14.6677 0.683883
\(461\) −20.5455 −0.956899 −0.478449 0.878115i \(-0.658801\pi\)
−0.478449 + 0.878115i \(0.658801\pi\)
\(462\) −0.513872 −0.0239075
\(463\) −1.58359 −0.0735958 −0.0367979 0.999323i \(-0.511716\pi\)
−0.0367979 + 0.999323i \(0.511716\pi\)
\(464\) 14.1965 0.659056
\(465\) 1.55629 0.0721713
\(466\) −0.116670 −0.00540463
\(467\) 5.14277 0.237979 0.118990 0.992896i \(-0.462034\pi\)
0.118990 + 0.992896i \(0.462034\pi\)
\(468\) 0 0
\(469\) 1.59861 0.0738168
\(470\) 2.32972 0.107462
\(471\) −2.07375 −0.0955532
\(472\) −4.07341 −0.187494
\(473\) 8.03405 0.369406
\(474\) −0.355563 −0.0163316
\(475\) −7.13261 −0.327267
\(476\) −12.7296 −0.583460
\(477\) −34.6716 −1.58751
\(478\) 3.45866 0.158195
\(479\) −22.2914 −1.01852 −0.509260 0.860613i \(-0.670081\pi\)
−0.509260 + 0.860613i \(0.670081\pi\)
\(480\) 0.990200 0.0451962
\(481\) 0 0
\(482\) −5.19241 −0.236508
\(483\) −8.22936 −0.374449
\(484\) 15.2767 0.694394
\(485\) 6.95775 0.315935
\(486\) −2.23645 −0.101447
\(487\) 23.4185 1.06119 0.530597 0.847624i \(-0.321968\pi\)
0.530597 + 0.847624i \(0.321968\pi\)
\(488\) −8.50784 −0.385132
\(489\) 1.23031 0.0556367
\(490\) −1.20345 −0.0543661
\(491\) 3.39791 0.153345 0.0766727 0.997056i \(-0.475570\pi\)
0.0766727 + 0.997056i \(0.475570\pi\)
\(492\) 0.560968 0.0252904
\(493\) 7.78509 0.350623
\(494\) 0 0
\(495\) 5.07715 0.228201
\(496\) 17.3518 0.779119
\(497\) −18.6089 −0.834723
\(498\) 1.02117 0.0457596
\(499\) 4.87712 0.218330 0.109165 0.994024i \(-0.465182\pi\)
0.109165 + 0.994024i \(0.465182\pi\)
\(500\) 1.92636 0.0861493
\(501\) 2.19224 0.0979419
\(502\) 0.111756 0.00498791
\(503\) −13.9853 −0.623572 −0.311786 0.950152i \(-0.600927\pi\)
−0.311786 + 0.950152i \(0.600927\pi\)
\(504\) 10.4410 0.465081
\(505\) 17.8923 0.796196
\(506\) 3.62024 0.160939
\(507\) 0 0
\(508\) 19.0418 0.844844
\(509\) −37.0157 −1.64069 −0.820347 0.571866i \(-0.806220\pi\)
−0.820347 + 0.571866i \(0.806220\pi\)
\(510\) 0.169498 0.00750550
\(511\) 9.93683 0.439579
\(512\) 18.6342 0.823524
\(513\) −13.4454 −0.593627
\(514\) −0.671801 −0.0296319
\(515\) −10.2561 −0.451937
\(516\) 2.82330 0.124289
\(517\) −15.0411 −0.661508
\(518\) 9.63628 0.423394
\(519\) −5.12041 −0.224761
\(520\) 0 0
\(521\) 21.5328 0.943370 0.471685 0.881767i \(-0.343646\pi\)
0.471685 + 0.881767i \(0.343646\pi\)
\(522\) −3.13286 −0.137121
\(523\) 36.0055 1.57441 0.787204 0.616692i \(-0.211528\pi\)
0.787204 + 0.616692i \(0.211528\pi\)
\(524\) 22.0962 0.965277
\(525\) −1.08079 −0.0471696
\(526\) −3.08255 −0.134406
\(527\) 9.51539 0.414497
\(528\) −1.99554 −0.0868446
\(529\) 34.9760 1.52070
\(530\) −3.24689 −0.141036
\(531\) −11.0784 −0.480761
\(532\) 46.4619 2.01438
\(533\) 0 0
\(534\) 0.334794 0.0144879
\(535\) −12.8986 −0.557656
\(536\) −0.503718 −0.0217573
\(537\) 2.58350 0.111486
\(538\) −8.05676 −0.347351
\(539\) 7.76969 0.334664
\(540\) 3.63129 0.156266
\(541\) −0.123280 −0.00530020 −0.00265010 0.999996i \(-0.500844\pi\)
−0.00265010 + 0.999996i \(0.500844\pi\)
\(542\) 4.00984 0.172238
\(543\) −3.54449 −0.152109
\(544\) 6.05423 0.259573
\(545\) −10.4476 −0.447525
\(546\) 0 0
\(547\) 2.52166 0.107818 0.0539092 0.998546i \(-0.482832\pi\)
0.0539092 + 0.998546i \(0.482832\pi\)
\(548\) 25.2018 1.07657
\(549\) −23.1386 −0.987531
\(550\) 0.475459 0.0202736
\(551\) −28.4149 −1.21052
\(552\) 2.59306 0.110368
\(553\) 13.8621 0.589477
\(554\) −0.528233 −0.0224425
\(555\) 3.35630 0.142467
\(556\) −23.0704 −0.978403
\(557\) 0.974960 0.0413104 0.0206552 0.999787i \(-0.493425\pi\)
0.0206552 + 0.999787i \(0.493425\pi\)
\(558\) −3.82916 −0.162101
\(559\) 0 0
\(560\) −12.0502 −0.509215
\(561\) −1.09431 −0.0462020
\(562\) −5.02737 −0.212067
\(563\) −31.5376 −1.32915 −0.664576 0.747220i \(-0.731388\pi\)
−0.664576 + 0.747220i \(0.731388\pi\)
\(564\) −5.28570 −0.222568
\(565\) 18.9766 0.798350
\(566\) 3.55369 0.149373
\(567\) 27.3600 1.14901
\(568\) 5.86363 0.246033
\(569\) 24.8762 1.04287 0.521433 0.853292i \(-0.325398\pi\)
0.521433 + 0.853292i \(0.325398\pi\)
\(570\) −0.618653 −0.0259125
\(571\) −7.92958 −0.331843 −0.165921 0.986139i \(-0.553060\pi\)
−0.165921 + 0.986139i \(0.553060\pi\)
\(572\) 0 0
\(573\) −6.95271 −0.290453
\(574\) 0.836085 0.0348975
\(575\) 7.61420 0.317534
\(576\) 18.2170 0.759040
\(577\) −19.0769 −0.794180 −0.397090 0.917780i \(-0.629980\pi\)
−0.397090 + 0.917780i \(0.629980\pi\)
\(578\) −3.57702 −0.148784
\(579\) 7.20051 0.299243
\(580\) 7.67421 0.318654
\(581\) −39.8116 −1.65166
\(582\) 0.603487 0.0250153
\(583\) 20.9626 0.868181
\(584\) −3.13108 −0.129565
\(585\) 0 0
\(586\) 0.327691 0.0135368
\(587\) 19.1196 0.789149 0.394574 0.918864i \(-0.370892\pi\)
0.394574 + 0.918864i \(0.370892\pi\)
\(588\) 2.73040 0.112600
\(589\) −34.7303 −1.43104
\(590\) −1.03745 −0.0427113
\(591\) −5.52136 −0.227118
\(592\) 37.4209 1.53799
\(593\) 36.9354 1.51676 0.758379 0.651814i \(-0.225992\pi\)
0.758379 + 0.651814i \(0.225992\pi\)
\(594\) 0.896266 0.0367742
\(595\) −6.60812 −0.270906
\(596\) −31.4096 −1.28659
\(597\) −5.59013 −0.228789
\(598\) 0 0
\(599\) −5.19953 −0.212447 −0.106224 0.994342i \(-0.533876\pi\)
−0.106224 + 0.994342i \(0.533876\pi\)
\(600\) 0.340556 0.0139031
\(601\) −28.3207 −1.15523 −0.577613 0.816311i \(-0.696016\pi\)
−0.577613 + 0.816311i \(0.696016\pi\)
\(602\) 4.20793 0.171502
\(603\) −1.36995 −0.0557887
\(604\) −16.8636 −0.686170
\(605\) 7.93034 0.322414
\(606\) 1.55190 0.0630417
\(607\) 20.1615 0.818332 0.409166 0.912460i \(-0.365820\pi\)
0.409166 + 0.912460i \(0.365820\pi\)
\(608\) −22.0974 −0.896168
\(609\) −4.30565 −0.174474
\(610\) −2.16685 −0.0877333
\(611\) 0 0
\(612\) 10.9088 0.440963
\(613\) 18.8190 0.760094 0.380047 0.924967i \(-0.375908\pi\)
0.380047 + 0.924967i \(0.375908\pi\)
\(614\) 5.95054 0.240144
\(615\) 0.291207 0.0117426
\(616\) −6.31268 −0.254345
\(617\) −2.82492 −0.113727 −0.0568634 0.998382i \(-0.518110\pi\)
−0.0568634 + 0.998382i \(0.518110\pi\)
\(618\) −0.889570 −0.0357837
\(619\) 15.6686 0.629773 0.314886 0.949129i \(-0.398034\pi\)
0.314886 + 0.949129i \(0.398034\pi\)
\(620\) 9.37987 0.376705
\(621\) 14.3532 0.575973
\(622\) −3.41839 −0.137065
\(623\) −13.0524 −0.522933
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 7.28008 0.290971
\(627\) 3.99415 0.159511
\(628\) −12.4986 −0.498749
\(629\) 20.5209 0.818223
\(630\) 2.65922 0.105946
\(631\) 10.1149 0.402667 0.201333 0.979523i \(-0.435473\pi\)
0.201333 + 0.979523i \(0.435473\pi\)
\(632\) −4.36793 −0.173747
\(633\) −2.02498 −0.0804857
\(634\) −4.64807 −0.184598
\(635\) 9.88489 0.392270
\(636\) 7.36659 0.292104
\(637\) 0 0
\(638\) 1.89413 0.0749894
\(639\) 15.9472 0.630861
\(640\) 7.90212 0.312359
\(641\) −17.6173 −0.695843 −0.347922 0.937524i \(-0.613112\pi\)
−0.347922 + 0.937524i \(0.613112\pi\)
\(642\) −1.11877 −0.0441544
\(643\) −19.0312 −0.750519 −0.375260 0.926920i \(-0.622446\pi\)
−0.375260 + 0.926920i \(0.622446\pi\)
\(644\) −49.5989 −1.95447
\(645\) 1.46561 0.0577085
\(646\) −3.78254 −0.148822
\(647\) −26.6972 −1.04957 −0.524787 0.851233i \(-0.675855\pi\)
−0.524787 + 0.851233i \(0.675855\pi\)
\(648\) −8.62108 −0.338668
\(649\) 6.69802 0.262920
\(650\) 0 0
\(651\) −5.26262 −0.206258
\(652\) 7.41518 0.290401
\(653\) −3.16076 −0.123690 −0.0618451 0.998086i \(-0.519698\pi\)
−0.0618451 + 0.998086i \(0.519698\pi\)
\(654\) −0.906178 −0.0354344
\(655\) 11.4705 0.448188
\(656\) 3.24680 0.126766
\(657\) −8.51553 −0.332222
\(658\) −7.87798 −0.307116
\(659\) −18.0060 −0.701413 −0.350706 0.936486i \(-0.614058\pi\)
−0.350706 + 0.936486i \(0.614058\pi\)
\(660\) −1.07873 −0.0419895
\(661\) −3.51786 −0.136829 −0.0684145 0.997657i \(-0.521794\pi\)
−0.0684145 + 0.997657i \(0.521794\pi\)
\(662\) 0.388232 0.0150891
\(663\) 0 0
\(664\) 12.5446 0.486824
\(665\) 24.1190 0.935296
\(666\) −8.25797 −0.319990
\(667\) 30.3334 1.17452
\(668\) 13.2128 0.511217
\(669\) 7.50935 0.290328
\(670\) −0.128292 −0.00495634
\(671\) 13.9896 0.540064
\(672\) −3.34838 −0.129166
\(673\) −26.0032 −1.00235 −0.501175 0.865346i \(-0.667099\pi\)
−0.501175 + 0.865346i \(0.667099\pi\)
\(674\) −2.03224 −0.0782791
\(675\) 1.88505 0.0725558
\(676\) 0 0
\(677\) −28.5157 −1.09595 −0.547975 0.836495i \(-0.684601\pi\)
−0.547975 + 0.836495i \(0.684601\pi\)
\(678\) 1.64595 0.0632122
\(679\) −23.5277 −0.902912
\(680\) 2.08221 0.0798490
\(681\) 3.06640 0.117505
\(682\) 2.31512 0.0886505
\(683\) −28.9493 −1.10771 −0.553857 0.832612i \(-0.686845\pi\)
−0.553857 + 0.832612i \(0.686845\pi\)
\(684\) −39.8163 −1.52241
\(685\) 13.0826 0.499861
\(686\) −2.35411 −0.0898805
\(687\) −2.31084 −0.0881639
\(688\) 16.3408 0.622987
\(689\) 0 0
\(690\) 0.660424 0.0251419
\(691\) −21.6566 −0.823854 −0.411927 0.911217i \(-0.635144\pi\)
−0.411927 + 0.911217i \(0.635144\pi\)
\(692\) −30.8611 −1.17316
\(693\) −17.1685 −0.652176
\(694\) 1.89097 0.0717801
\(695\) −11.9762 −0.454283
\(696\) 1.35671 0.0514258
\(697\) 1.78048 0.0674405
\(698\) 5.96312 0.225707
\(699\) 0.137411 0.00519737
\(700\) −6.51400 −0.246206
\(701\) 25.0376 0.945656 0.472828 0.881155i \(-0.343233\pi\)
0.472828 + 0.881155i \(0.343233\pi\)
\(702\) 0 0
\(703\) −74.8995 −2.82489
\(704\) −11.0140 −0.415107
\(705\) −2.74389 −0.103341
\(706\) 3.53285 0.132961
\(707\) −60.5030 −2.27545
\(708\) 2.35379 0.0884610
\(709\) −25.6247 −0.962355 −0.481178 0.876623i \(-0.659791\pi\)
−0.481178 + 0.876623i \(0.659791\pi\)
\(710\) 1.49340 0.0560464
\(711\) −11.8794 −0.445511
\(712\) 4.11279 0.154133
\(713\) 37.0753 1.38848
\(714\) −0.573160 −0.0214500
\(715\) 0 0
\(716\) 15.5709 0.581912
\(717\) −4.07352 −0.152129
\(718\) 2.71160 0.101196
\(719\) 14.8064 0.552186 0.276093 0.961131i \(-0.410960\pi\)
0.276093 + 0.961131i \(0.410960\pi\)
\(720\) 10.3266 0.384851
\(721\) 34.6811 1.29159
\(722\) 8.64982 0.321913
\(723\) 6.11549 0.227438
\(724\) −21.3629 −0.793945
\(725\) 3.98380 0.147955
\(726\) 0.687845 0.0255283
\(727\) −30.1688 −1.11890 −0.559449 0.828865i \(-0.688987\pi\)
−0.559449 + 0.828865i \(0.688987\pi\)
\(728\) 0 0
\(729\) −21.6391 −0.801447
\(730\) −0.797452 −0.0295150
\(731\) 8.96098 0.331434
\(732\) 4.91619 0.181708
\(733\) 49.8997 1.84309 0.921544 0.388275i \(-0.126929\pi\)
0.921544 + 0.388275i \(0.126929\pi\)
\(734\) 6.86257 0.253302
\(735\) 1.41739 0.0522812
\(736\) 23.5894 0.869516
\(737\) 0.828276 0.0305100
\(738\) −0.716496 −0.0263746
\(739\) −31.5796 −1.16167 −0.580836 0.814020i \(-0.697274\pi\)
−0.580836 + 0.814020i \(0.697274\pi\)
\(740\) 20.2287 0.743620
\(741\) 0 0
\(742\) 10.9794 0.403066
\(743\) 1.60448 0.0588625 0.0294312 0.999567i \(-0.490630\pi\)
0.0294312 + 0.999567i \(0.490630\pi\)
\(744\) 1.65824 0.0607942
\(745\) −16.3052 −0.597377
\(746\) 3.87258 0.141785
\(747\) 34.1172 1.24828
\(748\) −6.59551 −0.241156
\(749\) 43.6168 1.59372
\(750\) 0.0867358 0.00316714
\(751\) 10.7873 0.393634 0.196817 0.980440i \(-0.436940\pi\)
0.196817 + 0.980440i \(0.436940\pi\)
\(752\) −30.5928 −1.11561
\(753\) −0.131623 −0.00479662
\(754\) 0 0
\(755\) −8.75415 −0.318596
\(756\) −12.2792 −0.446592
\(757\) −6.40389 −0.232753 −0.116377 0.993205i \(-0.537128\pi\)
−0.116377 + 0.993205i \(0.537128\pi\)
\(758\) 1.90326 0.0691294
\(759\) −4.26383 −0.154767
\(760\) −7.59987 −0.275676
\(761\) −50.2655 −1.82212 −0.911062 0.412269i \(-0.864736\pi\)
−0.911062 + 0.412269i \(0.864736\pi\)
\(762\) 0.857374 0.0310594
\(763\) 35.3286 1.27898
\(764\) −41.9044 −1.51605
\(765\) 5.66293 0.204744
\(766\) 1.27366 0.0460191
\(767\) 0 0
\(768\) −3.33308 −0.120272
\(769\) −4.10457 −0.148015 −0.0740074 0.997258i \(-0.523579\pi\)
−0.0740074 + 0.997258i \(0.523579\pi\)
\(770\) −1.60777 −0.0579401
\(771\) 0.791232 0.0284955
\(772\) 43.3980 1.56193
\(773\) 1.17851 0.0423882 0.0211941 0.999775i \(-0.493253\pi\)
0.0211941 + 0.999775i \(0.493253\pi\)
\(774\) −3.60605 −0.129617
\(775\) 4.86923 0.174908
\(776\) 7.41356 0.266131
\(777\) −11.3494 −0.407157
\(778\) −2.53648 −0.0909373
\(779\) −6.49860 −0.232836
\(780\) 0 0
\(781\) −9.64172 −0.345008
\(782\) 4.03793 0.144396
\(783\) 7.50968 0.268374
\(784\) 15.8031 0.564397
\(785\) −6.48821 −0.231574
\(786\) 0.994900 0.0354869
\(787\) −33.4916 −1.19385 −0.596923 0.802298i \(-0.703610\pi\)
−0.596923 + 0.802298i \(0.703610\pi\)
\(788\) −33.2776 −1.18546
\(789\) 3.63056 0.129251
\(790\) −1.11246 −0.0395797
\(791\) −64.1695 −2.28161
\(792\) 5.40976 0.192227
\(793\) 0 0
\(794\) −5.75675 −0.204299
\(795\) 3.82411 0.135627
\(796\) −33.6921 −1.19418
\(797\) −24.9394 −0.883400 −0.441700 0.897163i \(-0.645624\pi\)
−0.441700 + 0.897163i \(0.645624\pi\)
\(798\) 2.09198 0.0740555
\(799\) −16.7765 −0.593511
\(800\) 3.09808 0.109534
\(801\) 11.1855 0.395219
\(802\) 4.94368 0.174567
\(803\) 5.14851 0.181687
\(804\) 0.291070 0.0102652
\(805\) −25.7475 −0.907481
\(806\) 0 0
\(807\) 9.48906 0.334031
\(808\) 19.0644 0.670684
\(809\) 12.8221 0.450802 0.225401 0.974266i \(-0.427631\pi\)
0.225401 + 0.974266i \(0.427631\pi\)
\(810\) −2.19570 −0.0771489
\(811\) −29.8424 −1.04791 −0.523954 0.851746i \(-0.675544\pi\)
−0.523954 + 0.851746i \(0.675544\pi\)
\(812\) −25.9505 −0.910683
\(813\) −4.72270 −0.165632
\(814\) 4.99279 0.174997
\(815\) 3.84933 0.134836
\(816\) −2.22577 −0.0779177
\(817\) −32.7068 −1.14427
\(818\) −1.18356 −0.0413822
\(819\) 0 0
\(820\) 1.75512 0.0612916
\(821\) −19.6918 −0.687249 −0.343624 0.939107i \(-0.611655\pi\)
−0.343624 + 0.939107i \(0.611655\pi\)
\(822\) 1.13473 0.0395783
\(823\) 41.9069 1.46078 0.730390 0.683030i \(-0.239338\pi\)
0.730390 + 0.683030i \(0.239338\pi\)
\(824\) −10.9280 −0.380693
\(825\) −0.559984 −0.0194961
\(826\) 3.50817 0.122065
\(827\) −21.7734 −0.757134 −0.378567 0.925574i \(-0.623583\pi\)
−0.378567 + 0.925574i \(0.623583\pi\)
\(828\) 42.5046 1.47714
\(829\) −5.87683 −0.204111 −0.102055 0.994779i \(-0.532542\pi\)
−0.102055 + 0.994779i \(0.532542\pi\)
\(830\) 3.19497 0.110899
\(831\) 0.622141 0.0215818
\(832\) 0 0
\(833\) 8.66612 0.300263
\(834\) −1.03876 −0.0359695
\(835\) 6.85894 0.237363
\(836\) 24.0730 0.832583
\(837\) 9.17876 0.317265
\(838\) 4.62841 0.159886
\(839\) 54.1116 1.86814 0.934069 0.357092i \(-0.116232\pi\)
0.934069 + 0.357092i \(0.116232\pi\)
\(840\) −1.15159 −0.0397338
\(841\) −13.1294 −0.452736
\(842\) 0.347348 0.0119704
\(843\) 5.92112 0.203934
\(844\) −12.2047 −0.420103
\(845\) 0 0
\(846\) 6.75116 0.232110
\(847\) −26.8166 −0.921428
\(848\) 42.6367 1.46415
\(849\) −4.18545 −0.143644
\(850\) 0.530316 0.0181897
\(851\) 79.9567 2.74088
\(852\) −3.38826 −0.116080
\(853\) −10.8001 −0.369787 −0.184893 0.982759i \(-0.559194\pi\)
−0.184893 + 0.982759i \(0.559194\pi\)
\(854\) 7.32725 0.250733
\(855\) −20.6692 −0.706872
\(856\) −13.7436 −0.469747
\(857\) −12.0558 −0.411817 −0.205908 0.978571i \(-0.566015\pi\)
−0.205908 + 0.978571i \(0.566015\pi\)
\(858\) 0 0
\(859\) −47.6819 −1.62688 −0.813442 0.581646i \(-0.802409\pi\)
−0.813442 + 0.581646i \(0.802409\pi\)
\(860\) 8.83336 0.301215
\(861\) −0.984720 −0.0335592
\(862\) −8.92095 −0.303849
\(863\) 46.2292 1.57366 0.786831 0.617169i \(-0.211721\pi\)
0.786831 + 0.617169i \(0.211721\pi\)
\(864\) 5.84005 0.198682
\(865\) −16.0204 −0.544711
\(866\) −7.66962 −0.260624
\(867\) 4.21293 0.143078
\(868\) −31.7182 −1.07658
\(869\) 7.18230 0.243643
\(870\) 0.345538 0.0117148
\(871\) 0 0
\(872\) −11.1320 −0.376977
\(873\) 20.1625 0.682397
\(874\) −14.7381 −0.498523
\(875\) −3.38151 −0.114316
\(876\) 1.80927 0.0611296
\(877\) 25.2281 0.851893 0.425947 0.904748i \(-0.359941\pi\)
0.425947 + 0.904748i \(0.359941\pi\)
\(878\) 7.12684 0.240519
\(879\) −0.385947 −0.0130177
\(880\) −6.24352 −0.210469
\(881\) −12.5086 −0.421425 −0.210712 0.977548i \(-0.567578\pi\)
−0.210712 + 0.977548i \(0.567578\pi\)
\(882\) −3.48740 −0.117427
\(883\) −11.2289 −0.377881 −0.188941 0.981989i \(-0.560505\pi\)
−0.188941 + 0.981989i \(0.560505\pi\)
\(884\) 0 0
\(885\) 1.22189 0.0410733
\(886\) −3.81218 −0.128073
\(887\) −54.5383 −1.83122 −0.915609 0.402069i \(-0.868291\pi\)
−0.915609 + 0.402069i \(0.868291\pi\)
\(888\) 3.57617 0.120009
\(889\) −33.4259 −1.12107
\(890\) 1.04748 0.0351117
\(891\) 14.1759 0.474909
\(892\) 45.2594 1.51540
\(893\) 61.2328 2.04908
\(894\) −1.41425 −0.0472995
\(895\) 8.08309 0.270188
\(896\) −26.7211 −0.892690
\(897\) 0 0
\(898\) 1.21007 0.0403805
\(899\) 19.3980 0.646960
\(900\) 5.58228 0.186076
\(901\) 23.3811 0.778939
\(902\) 0.433196 0.0144238
\(903\) −4.95600 −0.164925
\(904\) 20.2197 0.672498
\(905\) −11.0898 −0.368637
\(906\) −0.759298 −0.0252260
\(907\) 28.9083 0.959883 0.479942 0.877300i \(-0.340658\pi\)
0.479942 + 0.877300i \(0.340658\pi\)
\(908\) 18.4814 0.613327
\(909\) 51.8491 1.71973
\(910\) 0 0
\(911\) −7.04371 −0.233369 −0.116684 0.993169i \(-0.537227\pi\)
−0.116684 + 0.993169i \(0.537227\pi\)
\(912\) 8.12388 0.269009
\(913\) −20.6274 −0.682666
\(914\) −4.45456 −0.147344
\(915\) 2.55207 0.0843688
\(916\) −13.9276 −0.460180
\(917\) −38.7875 −1.28088
\(918\) 0.999674 0.0329942
\(919\) −17.3638 −0.572780 −0.286390 0.958113i \(-0.592455\pi\)
−0.286390 + 0.958113i \(0.592455\pi\)
\(920\) 8.11301 0.267478
\(921\) −7.00840 −0.230935
\(922\) −5.57551 −0.183620
\(923\) 0 0
\(924\) 3.64774 0.120002
\(925\) 10.5010 0.345271
\(926\) −0.429745 −0.0141223
\(927\) −29.7205 −0.976150
\(928\) 12.3421 0.405150
\(929\) 30.9381 1.01504 0.507522 0.861639i \(-0.330561\pi\)
0.507522 + 0.861639i \(0.330561\pi\)
\(930\) 0.422337 0.0138490
\(931\) −31.6306 −1.03665
\(932\) 0.828186 0.0271281
\(933\) 4.02610 0.131808
\(934\) 1.39561 0.0456659
\(935\) −3.42382 −0.111971
\(936\) 0 0
\(937\) 30.5009 0.996422 0.498211 0.867056i \(-0.333990\pi\)
0.498211 + 0.867056i \(0.333990\pi\)
\(938\) 0.433820 0.0141647
\(939\) −8.57431 −0.279812
\(940\) −16.5376 −0.539397
\(941\) −32.0423 −1.04455 −0.522275 0.852777i \(-0.674917\pi\)
−0.522275 + 0.852777i \(0.674917\pi\)
\(942\) −0.562761 −0.0183357
\(943\) 6.93738 0.225912
\(944\) 13.6234 0.443404
\(945\) −6.37434 −0.207357
\(946\) 2.18023 0.0708854
\(947\) −8.63395 −0.280566 −0.140283 0.990111i \(-0.544801\pi\)
−0.140283 + 0.990111i \(0.544801\pi\)
\(948\) 2.52398 0.0819750
\(949\) 0 0
\(950\) −1.93560 −0.0627993
\(951\) 5.47438 0.177519
\(952\) −7.04102 −0.228201
\(953\) −32.4260 −1.05038 −0.525191 0.850984i \(-0.676006\pi\)
−0.525191 + 0.850984i \(0.676006\pi\)
\(954\) −9.40897 −0.304627
\(955\) −21.7532 −0.703917
\(956\) −24.5514 −0.794049
\(957\) −2.23086 −0.0721136
\(958\) −6.04930 −0.195444
\(959\) −44.2390 −1.42855
\(960\) −2.00924 −0.0648479
\(961\) −7.29060 −0.235181
\(962\) 0 0
\(963\) −37.3782 −1.20449
\(964\) 36.8585 1.18713
\(965\) 22.5285 0.725219
\(966\) −2.23323 −0.0718531
\(967\) −27.0744 −0.870656 −0.435328 0.900272i \(-0.643368\pi\)
−0.435328 + 0.900272i \(0.643368\pi\)
\(968\) 8.44986 0.271589
\(969\) 4.45498 0.143115
\(970\) 1.88815 0.0606249
\(971\) 31.8039 1.02063 0.510317 0.859986i \(-0.329528\pi\)
0.510317 + 0.859986i \(0.329528\pi\)
\(972\) 15.8755 0.509207
\(973\) 40.4977 1.29830
\(974\) 6.35517 0.203633
\(975\) 0 0
\(976\) 28.4542 0.910796
\(977\) −31.3672 −1.00353 −0.501763 0.865005i \(-0.667315\pi\)
−0.501763 + 0.865005i \(0.667315\pi\)
\(978\) 0.333875 0.0106761
\(979\) −6.76276 −0.216139
\(980\) 8.54270 0.272886
\(981\) −30.2754 −0.966620
\(982\) 0.922103 0.0294255
\(983\) 35.5044 1.13241 0.566207 0.824263i \(-0.308410\pi\)
0.566207 + 0.824263i \(0.308410\pi\)
\(984\) 0.310284 0.00989149
\(985\) −17.2749 −0.550424
\(986\) 2.11267 0.0672811
\(987\) 9.27849 0.295338
\(988\) 0 0
\(989\) 34.9151 1.11024
\(990\) 1.37781 0.0437896
\(991\) −41.0638 −1.30443 −0.652217 0.758032i \(-0.726161\pi\)
−0.652217 + 0.758032i \(0.726161\pi\)
\(992\) 15.0853 0.478957
\(993\) −0.457250 −0.0145104
\(994\) −5.04997 −0.160175
\(995\) −17.4901 −0.554472
\(996\) −7.24879 −0.229687
\(997\) −44.7904 −1.41853 −0.709264 0.704943i \(-0.750972\pi\)
−0.709264 + 0.704943i \(0.750972\pi\)
\(998\) 1.32352 0.0418954
\(999\) 19.7949 0.626284
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 845.2.a.n.1.6 9
3.2 odd 2 7605.2.a.cs.1.4 9
5.4 even 2 4225.2.a.bt.1.4 9
13.2 odd 12 845.2.m.j.316.11 36
13.3 even 3 845.2.e.p.191.4 18
13.4 even 6 845.2.e.o.146.6 18
13.5 odd 4 845.2.c.h.506.8 18
13.6 odd 12 845.2.m.j.361.8 36
13.7 odd 12 845.2.m.j.361.11 36
13.8 odd 4 845.2.c.h.506.11 18
13.9 even 3 845.2.e.p.146.4 18
13.10 even 6 845.2.e.o.191.6 18
13.11 odd 12 845.2.m.j.316.8 36
13.12 even 2 845.2.a.o.1.4 yes 9
39.38 odd 2 7605.2.a.cp.1.6 9
65.64 even 2 4225.2.a.bs.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
845.2.a.n.1.6 9 1.1 even 1 trivial
845.2.a.o.1.4 yes 9 13.12 even 2
845.2.c.h.506.8 18 13.5 odd 4
845.2.c.h.506.11 18 13.8 odd 4
845.2.e.o.146.6 18 13.4 even 6
845.2.e.o.191.6 18 13.10 even 6
845.2.e.p.146.4 18 13.9 even 3
845.2.e.p.191.4 18 13.3 even 3
845.2.m.j.316.8 36 13.11 odd 12
845.2.m.j.316.11 36 13.2 odd 12
845.2.m.j.361.8 36 13.6 odd 12
845.2.m.j.361.11 36 13.7 odd 12
4225.2.a.bs.1.6 9 65.64 even 2
4225.2.a.bt.1.4 9 5.4 even 2
7605.2.a.cp.1.6 9 39.38 odd 2
7605.2.a.cs.1.4 9 3.2 odd 2