Properties

Label 1815.2.a.r.1.4
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $4$
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] = [1815,2,Mod(1,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1815.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.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: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5725.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 8x^{2} + 6x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.86205\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.48008 q^{2} -1.00000 q^{3} +4.15081 q^{4} -1.00000 q^{5} -2.48008 q^{6} +4.39482 q^{7} +5.33418 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.48008 q^{2} -1.00000 q^{3} +4.15081 q^{4} -1.00000 q^{5} -2.48008 q^{6} +4.39482 q^{7} +5.33418 q^{8} +1.00000 q^{9} -2.48008 q^{10} -4.15081 q^{12} +2.13795 q^{13} +10.8995 q^{14} +1.00000 q^{15} +4.92760 q^{16} -3.96017 q^{17} +2.48008 q^{18} +2.15081 q^{19} -4.15081 q^{20} -4.39482 q^{21} +5.16367 q^{23} -5.33418 q^{24} +1.00000 q^{25} +5.30230 q^{26} -1.00000 q^{27} +18.2421 q^{28} -7.93936 q^{29} +2.48008 q^{30} +6.19132 q^{31} +1.55248 q^{32} -9.82154 q^{34} -4.39482 q^{35} +4.15081 q^{36} -8.57025 q^{37} +5.33418 q^{38} -2.13795 q^{39} -5.33418 q^{40} +7.63089 q^{41} -10.8995 q^{42} +2.22321 q^{43} -1.00000 q^{45} +12.8063 q^{46} +6.88285 q^{47} -4.92760 q^{48} +12.3145 q^{49} +2.48008 q^{50} +3.96017 q^{51} +8.87423 q^{52} -5.69644 q^{53} -2.48008 q^{54} +23.4428 q^{56} -2.15081 q^{57} -19.6903 q^{58} +3.78170 q^{59} +4.15081 q^{60} -4.83439 q^{61} +15.3550 q^{62} +4.39482 q^{63} -6.00491 q^{64} -2.13795 q^{65} +9.57025 q^{67} -16.4379 q^{68} -5.16367 q^{69} -10.8995 q^{70} -2.11030 q^{71} +5.33418 q^{72} +14.6506 q^{73} -21.2549 q^{74} -1.00000 q^{75} +8.92760 q^{76} -5.30230 q^{78} -3.26111 q^{79} -4.92760 q^{80} +1.00000 q^{81} +18.9252 q^{82} -10.6635 q^{83} -18.2421 q^{84} +3.96017 q^{85} +5.51374 q^{86} +7.93936 q^{87} -7.64375 q^{89} -2.48008 q^{90} +9.39592 q^{91} +21.4334 q^{92} -6.19132 q^{93} +17.0700 q^{94} -2.15081 q^{95} -1.55248 q^{96} -5.67073 q^{97} +30.5409 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 4 q^{3} + 9 q^{4} - 4 q^{5} + q^{6} + 8 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 4 q^{3} + 9 q^{4} - 4 q^{5} + q^{6} + 8 q^{7} - 3 q^{8} + 4 q^{9} + q^{10} - 9 q^{12} + 15 q^{13} + 7 q^{14} + 4 q^{15} + 7 q^{16} + 6 q^{17} - q^{18} + q^{19} - 9 q^{20} - 8 q^{21} - q^{23} + 3 q^{24} + 4 q^{25} - 18 q^{26} - 4 q^{27} + 31 q^{28} - 17 q^{29} - q^{30} + 15 q^{31} + 8 q^{32} - 35 q^{34} - 8 q^{35} + 9 q^{36} - q^{37} - 3 q^{38} - 15 q^{39} + 3 q^{40} + 12 q^{41} - 7 q^{42} + 14 q^{43} - 4 q^{45} + 9 q^{46} + 14 q^{47} - 7 q^{48} + 20 q^{49} - q^{50} - 6 q^{51} + 39 q^{52} + 2 q^{53} + q^{54} - 12 q^{56} - q^{57} - 11 q^{58} - 11 q^{59} + 9 q^{60} - q^{61} + 30 q^{62} + 8 q^{63} - 3 q^{64} - 15 q^{65} + 5 q^{67} + 19 q^{68} + q^{69} - 7 q^{70} - 3 q^{71} - 3 q^{72} + 45 q^{73} - 29 q^{74} - 4 q^{75} + 23 q^{76} + 18 q^{78} - 7 q^{80} + 4 q^{81} + 11 q^{82} - 15 q^{83} - 31 q^{84} - 6 q^{85} - 10 q^{86} + 17 q^{87} + 2 q^{89} + q^{90} + 16 q^{91} + 34 q^{92} - 15 q^{93} + 29 q^{94} - q^{95} - 8 q^{96} - 26 q^{97} + q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.48008 1.75368 0.876842 0.480779i \(-0.159646\pi\)
0.876842 + 0.480779i \(0.159646\pi\)
\(3\) −1.00000 −0.577350
\(4\) 4.15081 2.07540
\(5\) −1.00000 −0.447214
\(6\) −2.48008 −1.01249
\(7\) 4.39482 1.66109 0.830544 0.556954i \(-0.188030\pi\)
0.830544 + 0.556954i \(0.188030\pi\)
\(8\) 5.33418 1.88592
\(9\) 1.00000 0.333333
\(10\) −2.48008 −0.784271
\(11\) 0 0
\(12\) −4.15081 −1.19824
\(13\) 2.13795 0.592961 0.296481 0.955039i \(-0.404187\pi\)
0.296481 + 0.955039i \(0.404187\pi\)
\(14\) 10.8995 2.91302
\(15\) 1.00000 0.258199
\(16\) 4.92760 1.23190
\(17\) −3.96017 −0.960481 −0.480241 0.877137i \(-0.659451\pi\)
−0.480241 + 0.877137i \(0.659451\pi\)
\(18\) 2.48008 0.584561
\(19\) 2.15081 0.493429 0.246715 0.969088i \(-0.420649\pi\)
0.246715 + 0.969088i \(0.420649\pi\)
\(20\) −4.15081 −0.928149
\(21\) −4.39482 −0.959029
\(22\) 0 0
\(23\) 5.16367 1.07670 0.538350 0.842722i \(-0.319048\pi\)
0.538350 + 0.842722i \(0.319048\pi\)
\(24\) −5.33418 −1.08884
\(25\) 1.00000 0.200000
\(26\) 5.30230 1.03987
\(27\) −1.00000 −0.192450
\(28\) 18.2421 3.44743
\(29\) −7.93936 −1.47430 −0.737151 0.675728i \(-0.763829\pi\)
−0.737151 + 0.675728i \(0.763829\pi\)
\(30\) 2.48008 0.452799
\(31\) 6.19132 1.11199 0.555997 0.831184i \(-0.312337\pi\)
0.555997 + 0.831184i \(0.312337\pi\)
\(32\) 1.55248 0.274443
\(33\) 0 0
\(34\) −9.82154 −1.68438
\(35\) −4.39482 −0.742861
\(36\) 4.15081 0.691802
\(37\) −8.57025 −1.40894 −0.704470 0.709733i \(-0.748815\pi\)
−0.704470 + 0.709733i \(0.748815\pi\)
\(38\) 5.33418 0.865319
\(39\) −2.13795 −0.342346
\(40\) −5.33418 −0.843409
\(41\) 7.63089 1.19175 0.595873 0.803079i \(-0.296806\pi\)
0.595873 + 0.803079i \(0.296806\pi\)
\(42\) −10.8995 −1.68183
\(43\) 2.22321 0.339036 0.169518 0.985527i \(-0.445779\pi\)
0.169518 + 0.985527i \(0.445779\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 12.8063 1.88819
\(47\) 6.88285 1.00397 0.501984 0.864877i \(-0.332604\pi\)
0.501984 + 0.864877i \(0.332604\pi\)
\(48\) −4.92760 −0.711238
\(49\) 12.3145 1.75921
\(50\) 2.48008 0.350737
\(51\) 3.96017 0.554534
\(52\) 8.87423 1.23063
\(53\) −5.69644 −0.782467 −0.391233 0.920292i \(-0.627951\pi\)
−0.391233 + 0.920292i \(0.627951\pi\)
\(54\) −2.48008 −0.337496
\(55\) 0 0
\(56\) 23.4428 3.13268
\(57\) −2.15081 −0.284882
\(58\) −19.6903 −2.58546
\(59\) 3.78170 0.492336 0.246168 0.969227i \(-0.420829\pi\)
0.246168 + 0.969227i \(0.420829\pi\)
\(60\) 4.15081 0.535867
\(61\) −4.83439 −0.618981 −0.309490 0.950903i \(-0.600158\pi\)
−0.309490 + 0.950903i \(0.600158\pi\)
\(62\) 15.3550 1.95009
\(63\) 4.39482 0.553696
\(64\) −6.00491 −0.750614
\(65\) −2.13795 −0.265180
\(66\) 0 0
\(67\) 9.57025 1.16919 0.584596 0.811324i \(-0.301253\pi\)
0.584596 + 0.811324i \(0.301253\pi\)
\(68\) −16.4379 −1.99339
\(69\) −5.16367 −0.621632
\(70\) −10.8995 −1.30274
\(71\) −2.11030 −0.250446 −0.125223 0.992129i \(-0.539965\pi\)
−0.125223 + 0.992129i \(0.539965\pi\)
\(72\) 5.33418 0.628640
\(73\) 14.6506 1.71472 0.857362 0.514715i \(-0.172102\pi\)
0.857362 + 0.514715i \(0.172102\pi\)
\(74\) −21.2549 −2.47084
\(75\) −1.00000 −0.115470
\(76\) 8.92760 1.02407
\(77\) 0 0
\(78\) −5.30230 −0.600367
\(79\) −3.26111 −0.366903 −0.183452 0.983029i \(-0.558727\pi\)
−0.183452 + 0.983029i \(0.558727\pi\)
\(80\) −4.92760 −0.550922
\(81\) 1.00000 0.111111
\(82\) 18.9252 2.08994
\(83\) −10.6635 −1.17047 −0.585233 0.810865i \(-0.698997\pi\)
−0.585233 + 0.810865i \(0.698997\pi\)
\(84\) −18.2421 −1.99037
\(85\) 3.96017 0.429540
\(86\) 5.51374 0.594562
\(87\) 7.93936 0.851189
\(88\) 0 0
\(89\) −7.64375 −0.810236 −0.405118 0.914264i \(-0.632770\pi\)
−0.405118 + 0.914264i \(0.632770\pi\)
\(90\) −2.48008 −0.261424
\(91\) 9.39592 0.984960
\(92\) 21.4334 2.23459
\(93\) −6.19132 −0.642010
\(94\) 17.0700 1.76064
\(95\) −2.15081 −0.220668
\(96\) −1.55248 −0.158450
\(97\) −5.67073 −0.575775 −0.287888 0.957664i \(-0.592953\pi\)
−0.287888 + 0.957664i \(0.592953\pi\)
\(98\) 30.5409 3.08510
\(99\) 0 0
\(100\) 4.15081 0.415081
\(101\) 7.03366 0.699876 0.349938 0.936773i \(-0.386203\pi\)
0.349938 + 0.936773i \(0.386203\pi\)
\(102\) 9.82154 0.972477
\(103\) 0.622945 0.0613806 0.0306903 0.999529i \(-0.490229\pi\)
0.0306903 + 0.999529i \(0.490229\pi\)
\(104\) 11.4042 1.11828
\(105\) 4.39482 0.428891
\(106\) −14.1276 −1.37220
\(107\) −17.8847 −1.72898 −0.864491 0.502648i \(-0.832359\pi\)
−0.864491 + 0.502648i \(0.832359\pi\)
\(108\) −4.15081 −0.399412
\(109\) −13.1041 −1.25515 −0.627574 0.778557i \(-0.715952\pi\)
−0.627574 + 0.778557i \(0.715952\pi\)
\(110\) 0 0
\(111\) 8.57025 0.813452
\(112\) 21.6559 2.04629
\(113\) −8.38755 −0.789035 −0.394517 0.918888i \(-0.629088\pi\)
−0.394517 + 0.918888i \(0.629088\pi\)
\(114\) −5.33418 −0.499592
\(115\) −5.16367 −0.481514
\(116\) −32.9548 −3.05977
\(117\) 2.13795 0.197654
\(118\) 9.37893 0.863401
\(119\) −17.4042 −1.59544
\(120\) 5.33418 0.486942
\(121\) 0 0
\(122\) −11.9897 −1.08550
\(123\) −7.63089 −0.688054
\(124\) 25.6990 2.30784
\(125\) −1.00000 −0.0894427
\(126\) 10.8995 0.971007
\(127\) −10.9828 −0.974570 −0.487285 0.873243i \(-0.662013\pi\)
−0.487285 + 0.873243i \(0.662013\pi\)
\(128\) −17.9976 −1.59078
\(129\) −2.22321 −0.195743
\(130\) −5.30230 −0.465042
\(131\) 7.33115 0.640525 0.320263 0.947329i \(-0.396229\pi\)
0.320263 + 0.947329i \(0.396229\pi\)
\(132\) 0 0
\(133\) 9.45243 0.819629
\(134\) 23.7350 2.05039
\(135\) 1.00000 0.0860663
\(136\) −21.1243 −1.81139
\(137\) −16.6774 −1.42485 −0.712424 0.701750i \(-0.752403\pi\)
−0.712424 + 0.701750i \(0.752403\pi\)
\(138\) −12.8063 −1.09015
\(139\) 11.0386 0.936280 0.468140 0.883654i \(-0.344924\pi\)
0.468140 + 0.883654i \(0.344924\pi\)
\(140\) −18.2421 −1.54174
\(141\) −6.88285 −0.579641
\(142\) −5.23371 −0.439203
\(143\) 0 0
\(144\) 4.92760 0.410633
\(145\) 7.93936 0.659328
\(146\) 36.3347 3.00708
\(147\) −12.3145 −1.01568
\(148\) −35.5735 −2.92412
\(149\) −4.61118 −0.377763 −0.188881 0.982000i \(-0.560486\pi\)
−0.188881 + 0.982000i \(0.560486\pi\)
\(150\) −2.48008 −0.202498
\(151\) 9.00914 0.733154 0.366577 0.930388i \(-0.380530\pi\)
0.366577 + 0.930388i \(0.380530\pi\)
\(152\) 11.4728 0.930568
\(153\) −3.96017 −0.320160
\(154\) 0 0
\(155\) −6.19132 −0.497299
\(156\) −8.87423 −0.710507
\(157\) 2.98902 0.238550 0.119275 0.992861i \(-0.461943\pi\)
0.119275 + 0.992861i \(0.461943\pi\)
\(158\) −8.08781 −0.643432
\(159\) 5.69644 0.451757
\(160\) −1.55248 −0.122735
\(161\) 22.6934 1.78849
\(162\) 2.48008 0.194854
\(163\) −2.89571 −0.226810 −0.113405 0.993549i \(-0.536176\pi\)
−0.113405 + 0.993549i \(0.536176\pi\)
\(164\) 31.6744 2.47335
\(165\) 0 0
\(166\) −26.4463 −2.05263
\(167\) 3.80512 0.294449 0.147225 0.989103i \(-0.452966\pi\)
0.147225 + 0.989103i \(0.452966\pi\)
\(168\) −23.4428 −1.80865
\(169\) −8.42916 −0.648397
\(170\) 9.82154 0.753277
\(171\) 2.15081 0.164476
\(172\) 9.22812 0.703638
\(173\) 1.11291 0.0846132 0.0423066 0.999105i \(-0.486529\pi\)
0.0423066 + 0.999105i \(0.486529\pi\)
\(174\) 19.6903 1.49272
\(175\) 4.39482 0.332217
\(176\) 0 0
\(177\) −3.78170 −0.284250
\(178\) −18.9571 −1.42090
\(179\) −4.81918 −0.360202 −0.180101 0.983648i \(-0.557643\pi\)
−0.180101 + 0.983648i \(0.557643\pi\)
\(180\) −4.15081 −0.309383
\(181\) −21.4623 −1.59528 −0.797638 0.603136i \(-0.793918\pi\)
−0.797638 + 0.603136i \(0.793918\pi\)
\(182\) 23.3027 1.72731
\(183\) 4.83439 0.357369
\(184\) 27.5440 2.03057
\(185\) 8.57025 0.630097
\(186\) −15.3550 −1.12588
\(187\) 0 0
\(188\) 28.5694 2.08364
\(189\) −4.39482 −0.319676
\(190\) −5.33418 −0.386982
\(191\) −9.36184 −0.677399 −0.338699 0.940895i \(-0.609987\pi\)
−0.338699 + 0.940895i \(0.609987\pi\)
\(192\) 6.00491 0.433367
\(193\) 18.4088 1.32509 0.662546 0.749021i \(-0.269476\pi\)
0.662546 + 0.749021i \(0.269476\pi\)
\(194\) −14.0639 −1.00973
\(195\) 2.13795 0.153102
\(196\) 51.1150 3.65107
\(197\) 4.67182 0.332854 0.166427 0.986054i \(-0.446777\pi\)
0.166427 + 0.986054i \(0.446777\pi\)
\(198\) 0 0
\(199\) −16.2018 −1.14852 −0.574258 0.818674i \(-0.694710\pi\)
−0.574258 + 0.818674i \(0.694710\pi\)
\(200\) 5.33418 0.377184
\(201\) −9.57025 −0.675034
\(202\) 17.4441 1.22736
\(203\) −34.8921 −2.44894
\(204\) 16.4379 1.15088
\(205\) −7.63089 −0.532965
\(206\) 1.54496 0.107642
\(207\) 5.16367 0.358900
\(208\) 10.5350 0.730469
\(209\) 0 0
\(210\) 10.8995 0.752139
\(211\) 6.40465 0.440914 0.220457 0.975397i \(-0.429245\pi\)
0.220457 + 0.975397i \(0.429245\pi\)
\(212\) −23.6448 −1.62393
\(213\) 2.11030 0.144595
\(214\) −44.3556 −3.03209
\(215\) −2.22321 −0.151622
\(216\) −5.33418 −0.362945
\(217\) 27.2098 1.84712
\(218\) −32.4993 −2.20113
\(219\) −14.6506 −0.989996
\(220\) 0 0
\(221\) −8.46664 −0.569528
\(222\) 21.2549 1.42654
\(223\) −14.3520 −0.961078 −0.480539 0.876973i \(-0.659559\pi\)
−0.480539 + 0.876973i \(0.659559\pi\)
\(224\) 6.82289 0.455874
\(225\) 1.00000 0.0666667
\(226\) −20.8018 −1.38372
\(227\) −18.3113 −1.21537 −0.607683 0.794180i \(-0.707901\pi\)
−0.607683 + 0.794180i \(0.707901\pi\)
\(228\) −8.92760 −0.591245
\(229\) −18.9191 −1.25021 −0.625104 0.780542i \(-0.714943\pi\)
−0.625104 + 0.780542i \(0.714943\pi\)
\(230\) −12.8063 −0.844424
\(231\) 0 0
\(232\) −42.3500 −2.78041
\(233\) 6.35813 0.416535 0.208267 0.978072i \(-0.433218\pi\)
0.208267 + 0.978072i \(0.433218\pi\)
\(234\) 5.30230 0.346622
\(235\) −6.88285 −0.448988
\(236\) 15.6971 1.02180
\(237\) 3.26111 0.211832
\(238\) −43.1639 −2.79790
\(239\) 13.6150 0.880681 0.440341 0.897831i \(-0.354858\pi\)
0.440341 + 0.897831i \(0.354858\pi\)
\(240\) 4.92760 0.318075
\(241\) −10.1564 −0.654231 −0.327116 0.944984i \(-0.606077\pi\)
−0.327116 + 0.944984i \(0.606077\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) −20.0666 −1.28464
\(245\) −12.3145 −0.786743
\(246\) −18.9252 −1.20663
\(247\) 4.59833 0.292584
\(248\) 33.0257 2.09713
\(249\) 10.6635 0.675769
\(250\) −2.48008 −0.156854
\(251\) 15.7579 0.994627 0.497314 0.867571i \(-0.334320\pi\)
0.497314 + 0.867571i \(0.334320\pi\)
\(252\) 18.2421 1.14914
\(253\) 0 0
\(254\) −27.2384 −1.70909
\(255\) −3.96017 −0.247995
\(256\) −32.6258 −2.03911
\(257\) −1.36335 −0.0850437 −0.0425218 0.999096i \(-0.513539\pi\)
−0.0425218 + 0.999096i \(0.513539\pi\)
\(258\) −5.51374 −0.343271
\(259\) −37.6647 −2.34037
\(260\) −8.87423 −0.550356
\(261\) −7.93936 −0.491434
\(262\) 18.1819 1.12328
\(263\) 8.12002 0.500702 0.250351 0.968155i \(-0.419454\pi\)
0.250351 + 0.968155i \(0.419454\pi\)
\(264\) 0 0
\(265\) 5.69644 0.349930
\(266\) 23.4428 1.43737
\(267\) 7.64375 0.467790
\(268\) 39.7243 2.42655
\(269\) −11.7112 −0.714047 −0.357023 0.934095i \(-0.616208\pi\)
−0.357023 + 0.934095i \(0.616208\pi\)
\(270\) 2.48008 0.150933
\(271\) −2.18651 −0.132821 −0.0664106 0.997792i \(-0.521155\pi\)
−0.0664106 + 0.997792i \(0.521155\pi\)
\(272\) −19.5141 −1.18322
\(273\) −9.39592 −0.568667
\(274\) −41.3614 −2.49873
\(275\) 0 0
\(276\) −21.4334 −1.29014
\(277\) −10.6733 −0.641295 −0.320648 0.947199i \(-0.603901\pi\)
−0.320648 + 0.947199i \(0.603901\pi\)
\(278\) 27.3766 1.64194
\(279\) 6.19132 0.370665
\(280\) −23.4428 −1.40098
\(281\) −5.43272 −0.324089 −0.162044 0.986783i \(-0.551809\pi\)
−0.162044 + 0.986783i \(0.551809\pi\)
\(282\) −17.0700 −1.01651
\(283\) −18.2401 −1.08426 −0.542132 0.840293i \(-0.682383\pi\)
−0.542132 + 0.840293i \(0.682383\pi\)
\(284\) −8.75944 −0.519777
\(285\) 2.15081 0.127403
\(286\) 0 0
\(287\) 33.5364 1.97959
\(288\) 1.55248 0.0914810
\(289\) −1.31709 −0.0774761
\(290\) 19.6903 1.15625
\(291\) 5.67073 0.332424
\(292\) 60.8118 3.55874
\(293\) 15.8594 0.926518 0.463259 0.886223i \(-0.346680\pi\)
0.463259 + 0.886223i \(0.346680\pi\)
\(294\) −30.5409 −1.78118
\(295\) −3.78170 −0.220179
\(296\) −45.7153 −2.65715
\(297\) 0 0
\(298\) −11.4361 −0.662476
\(299\) 11.0397 0.638441
\(300\) −4.15081 −0.239647
\(301\) 9.77062 0.563169
\(302\) 22.3434 1.28572
\(303\) −7.03366 −0.404073
\(304\) 10.5983 0.607856
\(305\) 4.83439 0.276817
\(306\) −9.82154 −0.561460
\(307\) 18.1026 1.03317 0.516585 0.856236i \(-0.327203\pi\)
0.516585 + 0.856236i \(0.327203\pi\)
\(308\) 0 0
\(309\) −0.622945 −0.0354381
\(310\) −15.3550 −0.872105
\(311\) −18.3391 −1.03991 −0.519957 0.854192i \(-0.674052\pi\)
−0.519957 + 0.854192i \(0.674052\pi\)
\(312\) −11.4042 −0.645637
\(313\) 30.2483 1.70974 0.854869 0.518844i \(-0.173638\pi\)
0.854869 + 0.518844i \(0.173638\pi\)
\(314\) 7.41301 0.418340
\(315\) −4.39482 −0.247620
\(316\) −13.5362 −0.761472
\(317\) 30.5985 1.71858 0.859292 0.511485i \(-0.170904\pi\)
0.859292 + 0.511485i \(0.170904\pi\)
\(318\) 14.1276 0.792239
\(319\) 0 0
\(320\) 6.00491 0.335685
\(321\) 17.8847 0.998228
\(322\) 56.2815 3.13645
\(323\) −8.51756 −0.473930
\(324\) 4.15081 0.230601
\(325\) 2.13795 0.118592
\(326\) −7.18160 −0.397752
\(327\) 13.1041 0.724660
\(328\) 40.7046 2.24753
\(329\) 30.2489 1.66768
\(330\) 0 0
\(331\) −1.09362 −0.0601110 −0.0300555 0.999548i \(-0.509568\pi\)
−0.0300555 + 0.999548i \(0.509568\pi\)
\(332\) −44.2620 −2.42919
\(333\) −8.57025 −0.469647
\(334\) 9.43702 0.516371
\(335\) −9.57025 −0.522879
\(336\) −21.6559 −1.18143
\(337\) −16.2674 −0.886140 −0.443070 0.896487i \(-0.646111\pi\)
−0.443070 + 0.896487i \(0.646111\pi\)
\(338\) −20.9050 −1.13708
\(339\) 8.38755 0.455549
\(340\) 16.4379 0.891470
\(341\) 0 0
\(342\) 5.33418 0.288440
\(343\) 23.3562 1.26112
\(344\) 11.8590 0.639395
\(345\) 5.16367 0.278002
\(346\) 2.76012 0.148385
\(347\) −7.88599 −0.423342 −0.211671 0.977341i \(-0.567891\pi\)
−0.211671 + 0.977341i \(0.567891\pi\)
\(348\) 32.9548 1.76656
\(349\) 12.4119 0.664395 0.332197 0.943210i \(-0.392210\pi\)
0.332197 + 0.943210i \(0.392210\pi\)
\(350\) 10.8995 0.582604
\(351\) −2.13795 −0.114115
\(352\) 0 0
\(353\) −13.7984 −0.734413 −0.367207 0.930139i \(-0.619686\pi\)
−0.367207 + 0.930139i \(0.619686\pi\)
\(354\) −9.37893 −0.498485
\(355\) 2.11030 0.112003
\(356\) −31.7277 −1.68157
\(357\) 17.4042 0.921129
\(358\) −11.9520 −0.631681
\(359\) 21.6248 1.14131 0.570657 0.821188i \(-0.306689\pi\)
0.570657 + 0.821188i \(0.306689\pi\)
\(360\) −5.33418 −0.281136
\(361\) −14.3740 −0.756527
\(362\) −53.2282 −2.79761
\(363\) 0 0
\(364\) 39.0007 2.04419
\(365\) −14.6506 −0.766847
\(366\) 11.9897 0.626711
\(367\) 23.1283 1.20729 0.603644 0.797254i \(-0.293715\pi\)
0.603644 + 0.797254i \(0.293715\pi\)
\(368\) 25.4445 1.32639
\(369\) 7.63089 0.397248
\(370\) 21.2549 1.10499
\(371\) −25.0349 −1.29975
\(372\) −25.6990 −1.33243
\(373\) 23.7304 1.22871 0.614356 0.789029i \(-0.289416\pi\)
0.614356 + 0.789029i \(0.289416\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 36.7144 1.89340
\(377\) −16.9740 −0.874204
\(378\) −10.8995 −0.560611
\(379\) −33.0484 −1.69758 −0.848791 0.528728i \(-0.822669\pi\)
−0.848791 + 0.528728i \(0.822669\pi\)
\(380\) −8.92760 −0.457976
\(381\) 10.9828 0.562668
\(382\) −23.2181 −1.18794
\(383\) −7.97370 −0.407437 −0.203719 0.979029i \(-0.565303\pi\)
−0.203719 + 0.979029i \(0.565303\pi\)
\(384\) 17.9976 0.918438
\(385\) 0 0
\(386\) 45.6553 2.32379
\(387\) 2.22321 0.113012
\(388\) −23.5381 −1.19497
\(389\) 4.28766 0.217393 0.108697 0.994075i \(-0.465332\pi\)
0.108697 + 0.994075i \(0.465332\pi\)
\(390\) 5.30230 0.268492
\(391\) −20.4490 −1.03415
\(392\) 65.6877 3.31773
\(393\) −7.33115 −0.369808
\(394\) 11.5865 0.583720
\(395\) 3.26111 0.164084
\(396\) 0 0
\(397\) −2.21087 −0.110961 −0.0554803 0.998460i \(-0.517669\pi\)
−0.0554803 + 0.998460i \(0.517669\pi\)
\(398\) −40.1819 −2.01413
\(399\) −9.45243 −0.473213
\(400\) 4.92760 0.246380
\(401\) 12.1478 0.606631 0.303315 0.952890i \(-0.401906\pi\)
0.303315 + 0.952890i \(0.401906\pi\)
\(402\) −23.7350 −1.18379
\(403\) 13.2367 0.659369
\(404\) 29.1954 1.45252
\(405\) −1.00000 −0.0496904
\(406\) −86.5353 −4.29467
\(407\) 0 0
\(408\) 21.1243 1.04581
\(409\) −33.8603 −1.67428 −0.837142 0.546985i \(-0.815775\pi\)
−0.837142 + 0.546985i \(0.815775\pi\)
\(410\) −18.9252 −0.934651
\(411\) 16.6774 0.822636
\(412\) 2.58573 0.127390
\(413\) 16.6199 0.817812
\(414\) 12.8063 0.629396
\(415\) 10.6635 0.523449
\(416\) 3.31913 0.162734
\(417\) −11.0386 −0.540561
\(418\) 0 0
\(419\) 18.1583 0.887093 0.443546 0.896251i \(-0.353720\pi\)
0.443546 + 0.896251i \(0.353720\pi\)
\(420\) 18.2421 0.890122
\(421\) 39.1558 1.90834 0.954169 0.299269i \(-0.0967428\pi\)
0.954169 + 0.299269i \(0.0967428\pi\)
\(422\) 15.8841 0.773224
\(423\) 6.88285 0.334656
\(424\) −30.3859 −1.47567
\(425\) −3.96017 −0.192096
\(426\) 5.23371 0.253574
\(427\) −21.2463 −1.02818
\(428\) −74.2361 −3.58834
\(429\) 0 0
\(430\) −5.51374 −0.265896
\(431\) −22.8319 −1.09978 −0.549888 0.835239i \(-0.685330\pi\)
−0.549888 + 0.835239i \(0.685330\pi\)
\(432\) −4.92760 −0.237079
\(433\) 5.89190 0.283146 0.141573 0.989928i \(-0.454784\pi\)
0.141573 + 0.989928i \(0.454784\pi\)
\(434\) 67.4825 3.23926
\(435\) −7.93936 −0.380663
\(436\) −54.3927 −2.60494
\(437\) 11.1061 0.531275
\(438\) −36.3347 −1.73614
\(439\) −5.46570 −0.260864 −0.130432 0.991457i \(-0.541636\pi\)
−0.130432 + 0.991457i \(0.541636\pi\)
\(440\) 0 0
\(441\) 12.3145 0.586404
\(442\) −20.9980 −0.998771
\(443\) 25.2148 1.19799 0.598995 0.800753i \(-0.295567\pi\)
0.598995 + 0.800753i \(0.295567\pi\)
\(444\) 35.5735 1.68824
\(445\) 7.64375 0.362348
\(446\) −35.5940 −1.68543
\(447\) 4.61118 0.218102
\(448\) −26.3905 −1.24684
\(449\) 14.8612 0.701344 0.350672 0.936498i \(-0.385953\pi\)
0.350672 + 0.936498i \(0.385953\pi\)
\(450\) 2.48008 0.116912
\(451\) 0 0
\(452\) −34.8151 −1.63757
\(453\) −9.00914 −0.423287
\(454\) −45.4136 −2.13137
\(455\) −9.39592 −0.440488
\(456\) −11.4728 −0.537264
\(457\) 38.6831 1.80952 0.904760 0.425923i \(-0.140050\pi\)
0.904760 + 0.425923i \(0.140050\pi\)
\(458\) −46.9209 −2.19247
\(459\) 3.96017 0.184845
\(460\) −21.4334 −0.999337
\(461\) 4.32624 0.201493 0.100746 0.994912i \(-0.467877\pi\)
0.100746 + 0.994912i \(0.467877\pi\)
\(462\) 0 0
\(463\) 0.723001 0.0336007 0.0168003 0.999859i \(-0.494652\pi\)
0.0168003 + 0.999859i \(0.494652\pi\)
\(464\) −39.1220 −1.81619
\(465\) 6.19132 0.287116
\(466\) 15.7687 0.730470
\(467\) −19.8965 −0.920700 −0.460350 0.887738i \(-0.652276\pi\)
−0.460350 + 0.887738i \(0.652276\pi\)
\(468\) 8.87423 0.410211
\(469\) 42.0596 1.94213
\(470\) −17.0700 −0.787382
\(471\) −2.98902 −0.137727
\(472\) 20.1723 0.928505
\(473\) 0 0
\(474\) 8.08781 0.371486
\(475\) 2.15081 0.0986859
\(476\) −72.2416 −3.31119
\(477\) −5.69644 −0.260822
\(478\) 33.7663 1.54444
\(479\) 18.7395 0.856230 0.428115 0.903724i \(-0.359178\pi\)
0.428115 + 0.903724i \(0.359178\pi\)
\(480\) 1.55248 0.0708608
\(481\) −18.3228 −0.835447
\(482\) −25.1887 −1.14731
\(483\) −22.6934 −1.03259
\(484\) 0 0
\(485\) 5.67073 0.257494
\(486\) −2.48008 −0.112499
\(487\) −0.524245 −0.0237558 −0.0118779 0.999929i \(-0.503781\pi\)
−0.0118779 + 0.999929i \(0.503781\pi\)
\(488\) −25.7875 −1.16735
\(489\) 2.89571 0.130949
\(490\) −30.5409 −1.37970
\(491\) −20.4709 −0.923838 −0.461919 0.886922i \(-0.652839\pi\)
−0.461919 + 0.886922i \(0.652839\pi\)
\(492\) −31.6744 −1.42799
\(493\) 31.4412 1.41604
\(494\) 11.4042 0.513100
\(495\) 0 0
\(496\) 30.5084 1.36987
\(497\) −9.27438 −0.416013
\(498\) 26.4463 1.18509
\(499\) −4.56237 −0.204240 −0.102120 0.994772i \(-0.532563\pi\)
−0.102120 + 0.994772i \(0.532563\pi\)
\(500\) −4.15081 −0.185630
\(501\) −3.80512 −0.170000
\(502\) 39.0808 1.74426
\(503\) 23.4683 1.04640 0.523199 0.852211i \(-0.324738\pi\)
0.523199 + 0.852211i \(0.324738\pi\)
\(504\) 23.4428 1.04423
\(505\) −7.03366 −0.312994
\(506\) 0 0
\(507\) 8.42916 0.374352
\(508\) −45.5877 −2.02263
\(509\) 4.74119 0.210149 0.105075 0.994464i \(-0.466492\pi\)
0.105075 + 0.994464i \(0.466492\pi\)
\(510\) −9.82154 −0.434905
\(511\) 64.3868 2.84830
\(512\) −44.9194 −1.98518
\(513\) −2.15081 −0.0949605
\(514\) −3.38123 −0.149140
\(515\) −0.622945 −0.0274502
\(516\) −9.22812 −0.406245
\(517\) 0 0
\(518\) −93.4117 −4.10427
\(519\) −1.11291 −0.0488515
\(520\) −11.4042 −0.500108
\(521\) −33.8092 −1.48121 −0.740604 0.671942i \(-0.765460\pi\)
−0.740604 + 0.671942i \(0.765460\pi\)
\(522\) −19.6903 −0.861820
\(523\) 28.1112 1.22922 0.614609 0.788832i \(-0.289314\pi\)
0.614609 + 0.788832i \(0.289314\pi\)
\(524\) 30.4302 1.32935
\(525\) −4.39482 −0.191806
\(526\) 20.1383 0.878072
\(527\) −24.5187 −1.06805
\(528\) 0 0
\(529\) 3.66346 0.159281
\(530\) 14.1276 0.613666
\(531\) 3.78170 0.164112
\(532\) 39.2352 1.70106
\(533\) 16.3145 0.706658
\(534\) 18.9571 0.820355
\(535\) 17.8847 0.773224
\(536\) 51.0495 2.20500
\(537\) 4.81918 0.207963
\(538\) −29.0448 −1.25221
\(539\) 0 0
\(540\) 4.15081 0.178622
\(541\) −14.1474 −0.608242 −0.304121 0.952633i \(-0.598363\pi\)
−0.304121 + 0.952633i \(0.598363\pi\)
\(542\) −5.42273 −0.232926
\(543\) 21.4623 0.921033
\(544\) −6.14809 −0.263597
\(545\) 13.1041 0.561319
\(546\) −23.3027 −0.997262
\(547\) 31.6553 1.35348 0.676742 0.736220i \(-0.263391\pi\)
0.676742 + 0.736220i \(0.263391\pi\)
\(548\) −69.2248 −2.95713
\(549\) −4.83439 −0.206327
\(550\) 0 0
\(551\) −17.0761 −0.727464
\(552\) −27.5440 −1.17235
\(553\) −14.3320 −0.609458
\(554\) −26.4706 −1.12463
\(555\) −8.57025 −0.363787
\(556\) 45.8190 1.94316
\(557\) 27.1124 1.14879 0.574395 0.818578i \(-0.305237\pi\)
0.574395 + 0.818578i \(0.305237\pi\)
\(558\) 15.3550 0.650029
\(559\) 4.75312 0.201035
\(560\) −21.6559 −0.915130
\(561\) 0 0
\(562\) −13.4736 −0.568349
\(563\) 24.0556 1.01382 0.506911 0.861999i \(-0.330787\pi\)
0.506911 + 0.861999i \(0.330787\pi\)
\(564\) −28.5694 −1.20299
\(565\) 8.38755 0.352867
\(566\) −45.2370 −1.90145
\(567\) 4.39482 0.184565
\(568\) −11.2567 −0.472321
\(569\) −27.8406 −1.16714 −0.583569 0.812063i \(-0.698344\pi\)
−0.583569 + 0.812063i \(0.698344\pi\)
\(570\) 5.33418 0.223424
\(571\) 33.5792 1.40525 0.702624 0.711562i \(-0.252012\pi\)
0.702624 + 0.711562i \(0.252012\pi\)
\(572\) 0 0
\(573\) 9.36184 0.391096
\(574\) 83.1731 3.47158
\(575\) 5.16367 0.215340
\(576\) −6.00491 −0.250205
\(577\) −19.1012 −0.795191 −0.397596 0.917561i \(-0.630155\pi\)
−0.397596 + 0.917561i \(0.630155\pi\)
\(578\) −3.26650 −0.135868
\(579\) −18.4088 −0.765043
\(580\) 32.9548 1.36837
\(581\) −46.8640 −1.94425
\(582\) 14.0639 0.582966
\(583\) 0 0
\(584\) 78.1490 3.23383
\(585\) −2.13795 −0.0883934
\(586\) 39.3327 1.62482
\(587\) −21.3648 −0.881820 −0.440910 0.897551i \(-0.645344\pi\)
−0.440910 + 0.897551i \(0.645344\pi\)
\(588\) −51.1150 −2.10795
\(589\) 13.3164 0.548691
\(590\) −9.37893 −0.386124
\(591\) −4.67182 −0.192173
\(592\) −42.2308 −1.73567
\(593\) 9.16002 0.376157 0.188078 0.982154i \(-0.439774\pi\)
0.188078 + 0.982154i \(0.439774\pi\)
\(594\) 0 0
\(595\) 17.4042 0.713504
\(596\) −19.1401 −0.784011
\(597\) 16.2018 0.663096
\(598\) 27.3793 1.11962
\(599\) −3.39567 −0.138743 −0.0693716 0.997591i \(-0.522099\pi\)
−0.0693716 + 0.997591i \(0.522099\pi\)
\(600\) −5.33418 −0.217767
\(601\) 14.4385 0.588958 0.294479 0.955658i \(-0.404854\pi\)
0.294479 + 0.955658i \(0.404854\pi\)
\(602\) 24.2319 0.987620
\(603\) 9.57025 0.389731
\(604\) 37.3952 1.52159
\(605\) 0 0
\(606\) −17.4441 −0.708617
\(607\) −1.37689 −0.0558862 −0.0279431 0.999610i \(-0.508896\pi\)
−0.0279431 + 0.999610i \(0.508896\pi\)
\(608\) 3.33910 0.135418
\(609\) 34.8921 1.41390
\(610\) 11.9897 0.485449
\(611\) 14.7152 0.595313
\(612\) −16.4379 −0.664462
\(613\) −1.19074 −0.0480934 −0.0240467 0.999711i \(-0.507655\pi\)
−0.0240467 + 0.999711i \(0.507655\pi\)
\(614\) 44.8960 1.81185
\(615\) 7.63089 0.307707
\(616\) 0 0
\(617\) −10.9076 −0.439122 −0.219561 0.975599i \(-0.570463\pi\)
−0.219561 + 0.975599i \(0.570463\pi\)
\(618\) −1.54496 −0.0621472
\(619\) −28.7339 −1.15491 −0.577457 0.816421i \(-0.695955\pi\)
−0.577457 + 0.816421i \(0.695955\pi\)
\(620\) −25.6990 −1.03210
\(621\) −5.16367 −0.207211
\(622\) −45.4825 −1.82368
\(623\) −33.5929 −1.34587
\(624\) −10.5350 −0.421736
\(625\) 1.00000 0.0400000
\(626\) 75.0184 2.99834
\(627\) 0 0
\(628\) 12.4068 0.495087
\(629\) 33.9396 1.35326
\(630\) −10.8995 −0.434247
\(631\) 33.7376 1.34307 0.671536 0.740972i \(-0.265635\pi\)
0.671536 + 0.740972i \(0.265635\pi\)
\(632\) −17.3953 −0.691950
\(633\) −6.40465 −0.254562
\(634\) 75.8869 3.01385
\(635\) 10.9828 0.435841
\(636\) 23.6448 0.937579
\(637\) 26.3278 1.04314
\(638\) 0 0
\(639\) −2.11030 −0.0834821
\(640\) 17.9976 0.711419
\(641\) −17.5210 −0.692036 −0.346018 0.938228i \(-0.612466\pi\)
−0.346018 + 0.938228i \(0.612466\pi\)
\(642\) 44.3556 1.75058
\(643\) −0.699059 −0.0275682 −0.0137841 0.999905i \(-0.504388\pi\)
−0.0137841 + 0.999905i \(0.504388\pi\)
\(644\) 94.1960 3.71184
\(645\) 2.22321 0.0875388
\(646\) −21.1243 −0.831123
\(647\) 45.4242 1.78581 0.892905 0.450246i \(-0.148664\pi\)
0.892905 + 0.450246i \(0.148664\pi\)
\(648\) 5.33418 0.209547
\(649\) 0 0
\(650\) 5.30230 0.207973
\(651\) −27.2098 −1.06643
\(652\) −12.0195 −0.470722
\(653\) 37.1061 1.45207 0.726036 0.687656i \(-0.241360\pi\)
0.726036 + 0.687656i \(0.241360\pi\)
\(654\) 32.4993 1.27082
\(655\) −7.33115 −0.286452
\(656\) 37.6020 1.46811
\(657\) 14.6506 0.571574
\(658\) 75.0198 2.92458
\(659\) −16.0716 −0.626061 −0.313031 0.949743i \(-0.601344\pi\)
−0.313031 + 0.949743i \(0.601344\pi\)
\(660\) 0 0
\(661\) 42.1478 1.63936 0.819680 0.572822i \(-0.194151\pi\)
0.819680 + 0.572822i \(0.194151\pi\)
\(662\) −2.71228 −0.105416
\(663\) 8.46664 0.328817
\(664\) −56.8809 −2.20741
\(665\) −9.45243 −0.366549
\(666\) −21.2549 −0.823612
\(667\) −40.9962 −1.58738
\(668\) 15.7943 0.611101
\(669\) 14.3520 0.554879
\(670\) −23.7350 −0.916964
\(671\) 0 0
\(672\) −6.82289 −0.263199
\(673\) 38.9658 1.50202 0.751010 0.660291i \(-0.229567\pi\)
0.751010 + 0.660291i \(0.229567\pi\)
\(674\) −40.3444 −1.55401
\(675\) −1.00000 −0.0384900
\(676\) −34.9879 −1.34569
\(677\) 1.42510 0.0547709 0.0273854 0.999625i \(-0.491282\pi\)
0.0273854 + 0.999625i \(0.491282\pi\)
\(678\) 20.8018 0.798889
\(679\) −24.9218 −0.956413
\(680\) 21.1243 0.810078
\(681\) 18.3113 0.701692
\(682\) 0 0
\(683\) −21.1490 −0.809245 −0.404623 0.914484i \(-0.632597\pi\)
−0.404623 + 0.914484i \(0.632597\pi\)
\(684\) 8.92760 0.341355
\(685\) 16.6774 0.637211
\(686\) 57.9253 2.21160
\(687\) 18.9191 0.721807
\(688\) 10.9551 0.417659
\(689\) −12.1787 −0.463972
\(690\) 12.8063 0.487528
\(691\) 27.8197 1.05831 0.529155 0.848525i \(-0.322509\pi\)
0.529155 + 0.848525i \(0.322509\pi\)
\(692\) 4.61949 0.175607
\(693\) 0 0
\(694\) −19.5579 −0.742408
\(695\) −11.0386 −0.418717
\(696\) 42.3500 1.60527
\(697\) −30.2196 −1.14465
\(698\) 30.7826 1.16514
\(699\) −6.35813 −0.240486
\(700\) 18.2421 0.689486
\(701\) −1.45437 −0.0549307 −0.0274653 0.999623i \(-0.508744\pi\)
−0.0274653 + 0.999623i \(0.508744\pi\)
\(702\) −5.30230 −0.200122
\(703\) −18.4330 −0.695213
\(704\) 0 0
\(705\) 6.88285 0.259223
\(706\) −34.2211 −1.28793
\(707\) 30.9117 1.16255
\(708\) −15.6971 −0.589934
\(709\) −12.3696 −0.464551 −0.232275 0.972650i \(-0.574617\pi\)
−0.232275 + 0.972650i \(0.574617\pi\)
\(710\) 5.23371 0.196418
\(711\) −3.26111 −0.122301
\(712\) −40.7732 −1.52804
\(713\) 31.9699 1.19728
\(714\) 43.1639 1.61537
\(715\) 0 0
\(716\) −20.0035 −0.747566
\(717\) −13.6150 −0.508461
\(718\) 53.6313 2.00150
\(719\) 30.4067 1.13398 0.566989 0.823725i \(-0.308108\pi\)
0.566989 + 0.823725i \(0.308108\pi\)
\(720\) −4.92760 −0.183641
\(721\) 2.73773 0.101959
\(722\) −35.6488 −1.32671
\(723\) 10.1564 0.377721
\(724\) −89.0857 −3.31084
\(725\) −7.93936 −0.294860
\(726\) 0 0
\(727\) 36.7184 1.36181 0.680906 0.732371i \(-0.261586\pi\)
0.680906 + 0.732371i \(0.261586\pi\)
\(728\) 50.1196 1.85755
\(729\) 1.00000 0.0370370
\(730\) −36.3347 −1.34481
\(731\) −8.80428 −0.325638
\(732\) 20.0666 0.741685
\(733\) −42.8001 −1.58086 −0.790429 0.612554i \(-0.790142\pi\)
−0.790429 + 0.612554i \(0.790142\pi\)
\(734\) 57.3602 2.11720
\(735\) 12.3145 0.454226
\(736\) 8.01651 0.295492
\(737\) 0 0
\(738\) 18.9252 0.696648
\(739\) 32.2616 1.18676 0.593381 0.804921i \(-0.297793\pi\)
0.593381 + 0.804921i \(0.297793\pi\)
\(740\) 35.5735 1.30771
\(741\) −4.59833 −0.168924
\(742\) −62.0885 −2.27934
\(743\) −40.5472 −1.48753 −0.743767 0.668439i \(-0.766963\pi\)
−0.743767 + 0.668439i \(0.766963\pi\)
\(744\) −33.0257 −1.21078
\(745\) 4.61118 0.168941
\(746\) 58.8533 2.15477
\(747\) −10.6635 −0.390156
\(748\) 0 0
\(749\) −78.6002 −2.87199
\(750\) 2.48008 0.0905598
\(751\) −33.5276 −1.22344 −0.611720 0.791074i \(-0.709522\pi\)
−0.611720 + 0.791074i \(0.709522\pi\)
\(752\) 33.9159 1.23679
\(753\) −15.7579 −0.574248
\(754\) −42.0968 −1.53308
\(755\) −9.00914 −0.327876
\(756\) −18.2421 −0.663458
\(757\) 10.8866 0.395679 0.197839 0.980234i \(-0.436608\pi\)
0.197839 + 0.980234i \(0.436608\pi\)
\(758\) −81.9627 −2.97702
\(759\) 0 0
\(760\) −11.4728 −0.416163
\(761\) 21.7351 0.787896 0.393948 0.919133i \(-0.371109\pi\)
0.393948 + 0.919133i \(0.371109\pi\)
\(762\) 27.2384 0.986742
\(763\) −57.5903 −2.08491
\(764\) −38.8592 −1.40588
\(765\) 3.96017 0.143180
\(766\) −19.7754 −0.714516
\(767\) 8.08509 0.291936
\(768\) 32.6258 1.17728
\(769\) 38.6384 1.39333 0.696667 0.717394i \(-0.254665\pi\)
0.696667 + 0.717394i \(0.254665\pi\)
\(770\) 0 0
\(771\) 1.36335 0.0491000
\(772\) 76.4113 2.75010
\(773\) −7.50909 −0.270083 −0.135042 0.990840i \(-0.543117\pi\)
−0.135042 + 0.990840i \(0.543117\pi\)
\(774\) 5.51374 0.198187
\(775\) 6.19132 0.222399
\(776\) −30.2487 −1.08587
\(777\) 37.6647 1.35122
\(778\) 10.6338 0.381239
\(779\) 16.4126 0.588042
\(780\) 8.87423 0.317748
\(781\) 0 0
\(782\) −50.7151 −1.81357
\(783\) 7.93936 0.283730
\(784\) 60.6808 2.16717
\(785\) −2.98902 −0.106683
\(786\) −18.1819 −0.648525
\(787\) 37.4105 1.33354 0.666770 0.745264i \(-0.267676\pi\)
0.666770 + 0.745264i \(0.267676\pi\)
\(788\) 19.3918 0.690806
\(789\) −8.12002 −0.289080
\(790\) 8.08781 0.287751
\(791\) −36.8618 −1.31066
\(792\) 0 0
\(793\) −10.3357 −0.367031
\(794\) −5.48315 −0.194590
\(795\) −5.69644 −0.202032
\(796\) −67.2507 −2.38364
\(797\) 10.7695 0.381476 0.190738 0.981641i \(-0.438912\pi\)
0.190738 + 0.981641i \(0.438912\pi\)
\(798\) −23.4428 −0.829866
\(799\) −27.2572 −0.964292
\(800\) 1.55248 0.0548886
\(801\) −7.64375 −0.270079
\(802\) 30.1275 1.06384
\(803\) 0 0
\(804\) −39.7243 −1.40097
\(805\) −22.6934 −0.799838
\(806\) 32.8282 1.15632
\(807\) 11.7112 0.412255
\(808\) 37.5189 1.31991
\(809\) −20.1705 −0.709155 −0.354578 0.935027i \(-0.615375\pi\)
−0.354578 + 0.935027i \(0.615375\pi\)
\(810\) −2.48008 −0.0871412
\(811\) −4.33664 −0.152280 −0.0761399 0.997097i \(-0.524260\pi\)
−0.0761399 + 0.997097i \(0.524260\pi\)
\(812\) −144.830 −5.08255
\(813\) 2.18651 0.0766843
\(814\) 0 0
\(815\) 2.89571 0.101432
\(816\) 19.5141 0.683130
\(817\) 4.78170 0.167291
\(818\) −83.9764 −2.93616
\(819\) 9.39592 0.328320
\(820\) −31.6744 −1.10612
\(821\) −44.7129 −1.56049 −0.780246 0.625472i \(-0.784906\pi\)
−0.780246 + 0.625472i \(0.784906\pi\)
\(822\) 41.3614 1.44264
\(823\) −23.8721 −0.832130 −0.416065 0.909335i \(-0.636591\pi\)
−0.416065 + 0.909335i \(0.636591\pi\)
\(824\) 3.32290 0.115759
\(825\) 0 0
\(826\) 41.2187 1.43418
\(827\) −3.43278 −0.119370 −0.0596848 0.998217i \(-0.519010\pi\)
−0.0596848 + 0.998217i \(0.519010\pi\)
\(828\) 21.4334 0.744862
\(829\) 23.8429 0.828097 0.414048 0.910255i \(-0.364114\pi\)
0.414048 + 0.910255i \(0.364114\pi\)
\(830\) 26.4463 0.917963
\(831\) 10.6733 0.370252
\(832\) −12.8382 −0.445085
\(833\) −48.7674 −1.68969
\(834\) −27.3766 −0.947973
\(835\) −3.80512 −0.131682
\(836\) 0 0
\(837\) −6.19132 −0.214003
\(838\) 45.0342 1.55568
\(839\) −16.6444 −0.574629 −0.287315 0.957836i \(-0.592763\pi\)
−0.287315 + 0.957836i \(0.592763\pi\)
\(840\) 23.4428 0.808853
\(841\) 34.0334 1.17357
\(842\) 97.1096 3.34662
\(843\) 5.43272 0.187113
\(844\) 26.5845 0.915075
\(845\) 8.42916 0.289972
\(846\) 17.0700 0.586880
\(847\) 0 0
\(848\) −28.0698 −0.963920
\(849\) 18.2401 0.626000
\(850\) −9.82154 −0.336876
\(851\) −44.2539 −1.51701
\(852\) 8.75944 0.300093
\(853\) −4.76800 −0.163253 −0.0816266 0.996663i \(-0.526011\pi\)
−0.0816266 + 0.996663i \(0.526011\pi\)
\(854\) −52.6926 −1.80310
\(855\) −2.15081 −0.0735561
\(856\) −95.4004 −3.26072
\(857\) 27.8195 0.950296 0.475148 0.879906i \(-0.342395\pi\)
0.475148 + 0.879906i \(0.342395\pi\)
\(858\) 0 0
\(859\) −25.1802 −0.859139 −0.429569 0.903034i \(-0.641335\pi\)
−0.429569 + 0.903034i \(0.641335\pi\)
\(860\) −9.22812 −0.314676
\(861\) −33.5364 −1.14292
\(862\) −56.6251 −1.92866
\(863\) 21.9684 0.747812 0.373906 0.927467i \(-0.378018\pi\)
0.373906 + 0.927467i \(0.378018\pi\)
\(864\) −1.55248 −0.0528166
\(865\) −1.11291 −0.0378402
\(866\) 14.6124 0.496549
\(867\) 1.31709 0.0447308
\(868\) 112.943 3.83352
\(869\) 0 0
\(870\) −19.6903 −0.667563
\(871\) 20.4607 0.693286
\(872\) −69.8998 −2.36711
\(873\) −5.67073 −0.191925
\(874\) 27.5440 0.931688
\(875\) −4.39482 −0.148572
\(876\) −60.8118 −2.05464
\(877\) −33.0169 −1.11490 −0.557451 0.830210i \(-0.688221\pi\)
−0.557451 + 0.830210i \(0.688221\pi\)
\(878\) −13.5554 −0.457473
\(879\) −15.8594 −0.534925
\(880\) 0 0
\(881\) 2.26357 0.0762615 0.0381307 0.999273i \(-0.487860\pi\)
0.0381307 + 0.999273i \(0.487860\pi\)
\(882\) 30.5409 1.02837
\(883\) −28.9534 −0.974360 −0.487180 0.873302i \(-0.661974\pi\)
−0.487180 + 0.873302i \(0.661974\pi\)
\(884\) −35.1434 −1.18200
\(885\) 3.78170 0.127120
\(886\) 62.5347 2.10090
\(887\) −36.6047 −1.22907 −0.614533 0.788891i \(-0.710655\pi\)
−0.614533 + 0.788891i \(0.710655\pi\)
\(888\) 45.7153 1.53411
\(889\) −48.2677 −1.61885
\(890\) 18.9571 0.635444
\(891\) 0 0
\(892\) −59.5722 −1.99463
\(893\) 14.8037 0.495387
\(894\) 11.4361 0.382481
\(895\) 4.81918 0.161087
\(896\) −79.0965 −2.64243
\(897\) −11.0397 −0.368604
\(898\) 36.8570 1.22993
\(899\) −49.1551 −1.63942
\(900\) 4.15081 0.138360
\(901\) 22.5589 0.751544
\(902\) 0 0
\(903\) −9.77062 −0.325146
\(904\) −44.7408 −1.48806
\(905\) 21.4623 0.713429
\(906\) −22.3434 −0.742311
\(907\) 23.2651 0.772506 0.386253 0.922393i \(-0.373769\pi\)
0.386253 + 0.922393i \(0.373769\pi\)
\(908\) −76.0069 −2.52238
\(909\) 7.03366 0.233292
\(910\) −23.3027 −0.772476
\(911\) 32.5851 1.07959 0.539796 0.841796i \(-0.318501\pi\)
0.539796 + 0.841796i \(0.318501\pi\)
\(912\) −10.5983 −0.350946
\(913\) 0 0
\(914\) 95.9373 3.17332
\(915\) −4.83439 −0.159820
\(916\) −78.5294 −2.59469
\(917\) 32.2191 1.06397
\(918\) 9.82154 0.324159
\(919\) −4.19869 −0.138502 −0.0692511 0.997599i \(-0.522061\pi\)
−0.0692511 + 0.997599i \(0.522061\pi\)
\(920\) −27.5440 −0.908097
\(921\) −18.1026 −0.596501
\(922\) 10.7294 0.353355
\(923\) −4.51171 −0.148505
\(924\) 0 0
\(925\) −8.57025 −0.281788
\(926\) 1.79310 0.0589250
\(927\) 0.622945 0.0204602
\(928\) −12.3257 −0.404612
\(929\) −35.9774 −1.18038 −0.590189 0.807265i \(-0.700947\pi\)
−0.590189 + 0.807265i \(0.700947\pi\)
\(930\) 15.3550 0.503510
\(931\) 26.4861 0.868047
\(932\) 26.3914 0.864478
\(933\) 18.3391 0.600395
\(934\) −49.3449 −1.61462
\(935\) 0 0
\(936\) 11.4042 0.372759
\(937\) −20.6219 −0.673689 −0.336845 0.941560i \(-0.609360\pi\)
−0.336845 + 0.941560i \(0.609360\pi\)
\(938\) 104.311 3.40588
\(939\) −30.2483 −0.987117
\(940\) −28.5694 −0.931831
\(941\) 37.4439 1.22064 0.610318 0.792156i \(-0.291042\pi\)
0.610318 + 0.792156i \(0.291042\pi\)
\(942\) −7.41301 −0.241529
\(943\) 39.4034 1.28315
\(944\) 18.6347 0.606508
\(945\) 4.39482 0.142964
\(946\) 0 0
\(947\) 46.3928 1.50756 0.753781 0.657126i \(-0.228228\pi\)
0.753781 + 0.657126i \(0.228228\pi\)
\(948\) 13.5362 0.439636
\(949\) 31.3223 1.01676
\(950\) 5.33418 0.173064
\(951\) −30.5985 −0.992225
\(952\) −92.8374 −3.00888
\(953\) 44.0854 1.42806 0.714032 0.700113i \(-0.246867\pi\)
0.714032 + 0.700113i \(0.246867\pi\)
\(954\) −14.1276 −0.457399
\(955\) 9.36184 0.302942
\(956\) 56.5133 1.82777
\(957\) 0 0
\(958\) 46.4755 1.50156
\(959\) −73.2943 −2.36680
\(960\) −6.00491 −0.193808
\(961\) 7.33247 0.236531
\(962\) −45.4420 −1.46511
\(963\) −17.8847 −0.576327
\(964\) −42.1573 −1.35779
\(965\) −18.4088 −0.592600
\(966\) −56.2815 −1.81083
\(967\) −18.7661 −0.603476 −0.301738 0.953391i \(-0.597567\pi\)
−0.301738 + 0.953391i \(0.597567\pi\)
\(968\) 0 0
\(969\) 8.51756 0.273623
\(970\) 14.0639 0.451564
\(971\) −48.7690 −1.56507 −0.782536 0.622605i \(-0.786074\pi\)
−0.782536 + 0.622605i \(0.786074\pi\)
\(972\) −4.15081 −0.133137
\(973\) 48.5126 1.55524
\(974\) −1.30017 −0.0416601
\(975\) −2.13795 −0.0684692
\(976\) −23.8220 −0.762522
\(977\) 18.6971 0.598174 0.299087 0.954226i \(-0.403318\pi\)
0.299087 + 0.954226i \(0.403318\pi\)
\(978\) 7.18160 0.229642
\(979\) 0 0
\(980\) −51.1150 −1.63281
\(981\) −13.1041 −0.418382
\(982\) −50.7695 −1.62012
\(983\) −39.6527 −1.26473 −0.632363 0.774672i \(-0.717915\pi\)
−0.632363 + 0.774672i \(0.717915\pi\)
\(984\) −40.7046 −1.29761
\(985\) −4.67182 −0.148857
\(986\) 77.9767 2.48328
\(987\) −30.2489 −0.962834
\(988\) 19.0868 0.607231
\(989\) 11.4799 0.365040
\(990\) 0 0
\(991\) −15.4204 −0.489846 −0.244923 0.969542i \(-0.578763\pi\)
−0.244923 + 0.969542i \(0.578763\pi\)
\(992\) 9.61192 0.305179
\(993\) 1.09362 0.0347051
\(994\) −23.0012 −0.729555
\(995\) 16.2018 0.513632
\(996\) 44.2620 1.40249
\(997\) −15.3415 −0.485869 −0.242934 0.970043i \(-0.578110\pi\)
−0.242934 + 0.970043i \(0.578110\pi\)
\(998\) −11.3151 −0.358172
\(999\) 8.57025 0.271151
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 1815.2.a.r.1.4 4
3.2 odd 2 5445.2.a.br.1.1 4
5.4 even 2 9075.2.a.dg.1.1 4
11.2 odd 10 165.2.m.b.136.2 yes 8
11.6 odd 10 165.2.m.b.91.2 8
11.10 odd 2 1815.2.a.v.1.1 4
33.2 even 10 495.2.n.b.136.1 8
33.17 even 10 495.2.n.b.91.1 8
33.32 even 2 5445.2.a.bk.1.4 4
55.2 even 20 825.2.bx.g.499.1 16
55.13 even 20 825.2.bx.g.499.4 16
55.17 even 20 825.2.bx.g.124.4 16
55.24 odd 10 825.2.n.i.301.1 8
55.28 even 20 825.2.bx.g.124.1 16
55.39 odd 10 825.2.n.i.751.1 8
55.54 odd 2 9075.2.a.cq.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.b.91.2 8 11.6 odd 10
165.2.m.b.136.2 yes 8 11.2 odd 10
495.2.n.b.91.1 8 33.17 even 10
495.2.n.b.136.1 8 33.2 even 10
825.2.n.i.301.1 8 55.24 odd 10
825.2.n.i.751.1 8 55.39 odd 10
825.2.bx.g.124.1 16 55.28 even 20
825.2.bx.g.124.4 16 55.17 even 20
825.2.bx.g.499.1 16 55.2 even 20
825.2.bx.g.499.4 16 55.13 even 20
1815.2.a.r.1.4 4 1.1 even 1 trivial
1815.2.a.v.1.1 4 11.10 odd 2
5445.2.a.bk.1.4 4 33.32 even 2
5445.2.a.br.1.1 4 3.2 odd 2
9075.2.a.cq.1.4 4 55.54 odd 2
9075.2.a.dg.1.1 4 5.4 even 2