Properties

Label 1875.2.b.f.1249.7
Level $1875$
Weight $2$
Character 1875.1249
Analytic conductor $14.972$
Analytic rank $0$
Dimension $12$
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] = [1875,2,Mod(1249,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1875.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.b (of order \(2\), degree \(1\), not 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: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 22x^{10} + 179x^{8} + 641x^{6} + 869x^{4} + 67x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.7
Root \(0.141689i\) of defining polynomial
Character \(\chi\) \(=\) 1875.1249
Dual form 1875.2.b.f.1249.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.141689i q^{2} -1.00000i q^{3} +1.97992 q^{4} +0.141689 q^{6} +0.858311i q^{7} +0.563913i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+0.141689i q^{2} -1.00000i q^{3} +1.97992 q^{4} +0.141689 q^{6} +0.858311i q^{7} +0.563913i q^{8} -1.00000 q^{9} -3.67993 q^{11} -1.97992i q^{12} +4.58134i q^{13} -0.121614 q^{14} +3.87995 q^{16} +5.30702i q^{17} -0.141689i q^{18} -6.36870 q^{19} +0.858311 q^{21} -0.521407i q^{22} +3.42379i q^{23} +0.563913 q^{24} -0.649128 q^{26} +1.00000i q^{27} +1.69939i q^{28} -3.73405 q^{29} +1.25290 q^{31} +1.67757i q^{32} +3.67993i q^{33} -0.751949 q^{34} -1.97992 q^{36} +7.45067i q^{37} -0.902378i q^{38} +4.58134 q^{39} -2.53168 q^{41} +0.121614i q^{42} -3.37972i q^{43} -7.28598 q^{44} -0.485114 q^{46} +8.49937i q^{47} -3.87995i q^{48} +6.26330 q^{49} +5.30702 q^{51} +9.07071i q^{52} -2.34827i q^{53} -0.141689 q^{54} -0.484013 q^{56} +6.36870i q^{57} -0.529076i q^{58} -13.1264 q^{59} +10.3476 q^{61} +0.177523i q^{62} -0.858311i q^{63} +7.52220 q^{64} -0.521407 q^{66} -3.34649i q^{67} +10.5075i q^{68} +3.42379 q^{69} -4.32289 q^{71} -0.563913i q^{72} -9.08007i q^{73} -1.05568 q^{74} -12.6095 q^{76} -3.15852i q^{77} +0.649128i q^{78} +3.24730 q^{79} +1.00000 q^{81} -0.358712i q^{82} -7.39269i q^{83} +1.69939 q^{84} +0.478870 q^{86} +3.73405i q^{87} -2.07516i q^{88} +15.4975 q^{89} -3.93221 q^{91} +6.77883i q^{92} -1.25290i q^{93} -1.20427 q^{94} +1.67757 q^{96} +10.0386i q^{97} +0.887444i q^{98} +3.67993 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 20 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 20 q^{4} - 12 q^{9} + 6 q^{11} + 44 q^{14} + 36 q^{16} - 22 q^{19} + 12 q^{21} - 6 q^{24} - 56 q^{26} + 6 q^{29} - 22 q^{31} - 30 q^{34} + 20 q^{36} - 12 q^{39} - 2 q^{41} - 18 q^{44} + 38 q^{46} + 28 q^{49} + 26 q^{51} - 70 q^{56} - 18 q^{59} + 22 q^{61} + 46 q^{64} + 32 q^{66} - 26 q^{69} - 16 q^{71} + 44 q^{74} - 52 q^{76} + 10 q^{79} + 12 q^{81} - 14 q^{84} - 74 q^{86} + 8 q^{89} + 68 q^{91} - 82 q^{94} + 32 q^{96} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\) \(626\) \(1252\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.141689i 0.100190i 0.998744 + 0.0500948i \(0.0159523\pi\)
−0.998744 + 0.0500948i \(0.984048\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 1.97992 0.989962
\(5\) 0 0
\(6\) 0.141689 0.0578445
\(7\) 0.858311i 0.324411i 0.986757 + 0.162205i \(0.0518607\pi\)
−0.986757 + 0.162205i \(0.948139\pi\)
\(8\) 0.563913i 0.199374i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.67993 −1.10954 −0.554770 0.832004i \(-0.687194\pi\)
−0.554770 + 0.832004i \(0.687194\pi\)
\(12\) − 1.97992i − 0.571555i
\(13\) 4.58134i 1.27064i 0.772251 + 0.635318i \(0.219131\pi\)
−0.772251 + 0.635318i \(0.780869\pi\)
\(14\) −0.121614 −0.0325026
\(15\) 0 0
\(16\) 3.87995 0.969987
\(17\) 5.30702i 1.28714i 0.765387 + 0.643571i \(0.222548\pi\)
−0.765387 + 0.643571i \(0.777452\pi\)
\(18\) − 0.141689i − 0.0333965i
\(19\) −6.36870 −1.46108 −0.730540 0.682870i \(-0.760732\pi\)
−0.730540 + 0.682870i \(0.760732\pi\)
\(20\) 0 0
\(21\) 0.858311 0.187299
\(22\) − 0.521407i − 0.111164i
\(23\) 3.42379i 0.713909i 0.934122 + 0.356954i \(0.116185\pi\)
−0.934122 + 0.356954i \(0.883815\pi\)
\(24\) 0.563913 0.115108
\(25\) 0 0
\(26\) −0.649128 −0.127304
\(27\) 1.00000i 0.192450i
\(28\) 1.69939i 0.321154i
\(29\) −3.73405 −0.693396 −0.346698 0.937977i \(-0.612697\pi\)
−0.346698 + 0.937977i \(0.612697\pi\)
\(30\) 0 0
\(31\) 1.25290 0.225028 0.112514 0.993650i \(-0.464110\pi\)
0.112514 + 0.993650i \(0.464110\pi\)
\(32\) 1.67757i 0.296556i
\(33\) 3.67993i 0.640593i
\(34\) −0.751949 −0.128958
\(35\) 0 0
\(36\) −1.97992 −0.329987
\(37\) 7.45067i 1.22488i 0.790516 + 0.612441i \(0.209812\pi\)
−0.790516 + 0.612441i \(0.790188\pi\)
\(38\) − 0.902378i − 0.146385i
\(39\) 4.58134 0.733602
\(40\) 0 0
\(41\) −2.53168 −0.395381 −0.197691 0.980264i \(-0.563344\pi\)
−0.197691 + 0.980264i \(0.563344\pi\)
\(42\) 0.121614i 0.0187654i
\(43\) − 3.37972i − 0.515402i −0.966225 0.257701i \(-0.917035\pi\)
0.966225 0.257701i \(-0.0829650\pi\)
\(44\) −7.28598 −1.09840
\(45\) 0 0
\(46\) −0.485114 −0.0715262
\(47\) 8.49937i 1.23976i 0.784696 + 0.619880i \(0.212819\pi\)
−0.784696 + 0.619880i \(0.787181\pi\)
\(48\) − 3.87995i − 0.560022i
\(49\) 6.26330 0.894758
\(50\) 0 0
\(51\) 5.30702 0.743132
\(52\) 9.07071i 1.25788i
\(53\) − 2.34827i − 0.322560i −0.986909 0.161280i \(-0.948438\pi\)
0.986909 0.161280i \(-0.0515622\pi\)
\(54\) −0.141689 −0.0192815
\(55\) 0 0
\(56\) −0.484013 −0.0646789
\(57\) 6.36870i 0.843555i
\(58\) − 0.529076i − 0.0694710i
\(59\) −13.1264 −1.70891 −0.854455 0.519525i \(-0.826109\pi\)
−0.854455 + 0.519525i \(0.826109\pi\)
\(60\) 0 0
\(61\) 10.3476 1.32488 0.662439 0.749116i \(-0.269521\pi\)
0.662439 + 0.749116i \(0.269521\pi\)
\(62\) 0.177523i 0.0225454i
\(63\) − 0.858311i − 0.108137i
\(64\) 7.52220 0.940275
\(65\) 0 0
\(66\) −0.521407 −0.0641808
\(67\) − 3.34649i − 0.408838i −0.978883 0.204419i \(-0.934469\pi\)
0.978883 0.204419i \(-0.0655305\pi\)
\(68\) 10.5075i 1.27422i
\(69\) 3.42379 0.412175
\(70\) 0 0
\(71\) −4.32289 −0.513032 −0.256516 0.966540i \(-0.582575\pi\)
−0.256516 + 0.966540i \(0.582575\pi\)
\(72\) − 0.563913i − 0.0664578i
\(73\) − 9.08007i − 1.06274i −0.847139 0.531371i \(-0.821677\pi\)
0.847139 0.531371i \(-0.178323\pi\)
\(74\) −1.05568 −0.122721
\(75\) 0 0
\(76\) −12.6095 −1.44641
\(77\) − 3.15852i − 0.359947i
\(78\) 0.649128i 0.0734993i
\(79\) 3.24730 0.365350 0.182675 0.983173i \(-0.441524\pi\)
0.182675 + 0.983173i \(0.441524\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 0.358712i − 0.0396131i
\(83\) − 7.39269i − 0.811453i −0.913994 0.405727i \(-0.867019\pi\)
0.913994 0.405727i \(-0.132981\pi\)
\(84\) 1.69939 0.185419
\(85\) 0 0
\(86\) 0.478870 0.0516379
\(87\) 3.73405i 0.400332i
\(88\) − 2.07516i − 0.221213i
\(89\) 15.4975 1.64273 0.821367 0.570400i \(-0.193212\pi\)
0.821367 + 0.570400i \(0.193212\pi\)
\(90\) 0 0
\(91\) −3.93221 −0.412208
\(92\) 6.77883i 0.706742i
\(93\) − 1.25290i − 0.129920i
\(94\) −1.20427 −0.124211
\(95\) 0 0
\(96\) 1.67757 0.171217
\(97\) 10.0386i 1.01926i 0.860393 + 0.509631i \(0.170218\pi\)
−0.860393 + 0.509631i \(0.829782\pi\)
\(98\) 0.887444i 0.0896454i
\(99\) 3.67993 0.369847
\(100\) 0 0
\(101\) −0.714616 −0.0711070 −0.0355535 0.999368i \(-0.511319\pi\)
−0.0355535 + 0.999368i \(0.511319\pi\)
\(102\) 0.751949i 0.0744541i
\(103\) 0.780764i 0.0769310i 0.999260 + 0.0384655i \(0.0122470\pi\)
−0.999260 + 0.0384655i \(0.987753\pi\)
\(104\) −2.58348 −0.253331
\(105\) 0 0
\(106\) 0.332725 0.0323171
\(107\) − 11.9601i − 1.15623i −0.815957 0.578113i \(-0.803789\pi\)
0.815957 0.578113i \(-0.196211\pi\)
\(108\) 1.97992i 0.190518i
\(109\) −2.43384 −0.233119 −0.116560 0.993184i \(-0.537187\pi\)
−0.116560 + 0.993184i \(0.537187\pi\)
\(110\) 0 0
\(111\) 7.45067 0.707186
\(112\) 3.33020i 0.314674i
\(113\) − 16.0354i − 1.50848i −0.656599 0.754240i \(-0.728006\pi\)
0.656599 0.754240i \(-0.271994\pi\)
\(114\) −0.902378 −0.0845154
\(115\) 0 0
\(116\) −7.39313 −0.686435
\(117\) − 4.58134i − 0.423545i
\(118\) − 1.85987i − 0.171215i
\(119\) −4.55507 −0.417563
\(120\) 0 0
\(121\) 2.54188 0.231080
\(122\) 1.46615i 0.132739i
\(123\) 2.53168i 0.228274i
\(124\) 2.48065 0.222769
\(125\) 0 0
\(126\) 0.121614 0.0108342
\(127\) 3.83848i 0.340610i 0.985391 + 0.170305i \(0.0544753\pi\)
−0.985391 + 0.170305i \(0.945525\pi\)
\(128\) 4.42097i 0.390762i
\(129\) −3.37972 −0.297568
\(130\) 0 0
\(131\) −12.4495 −1.08771 −0.543857 0.839178i \(-0.683037\pi\)
−0.543857 + 0.839178i \(0.683037\pi\)
\(132\) 7.28598i 0.634163i
\(133\) − 5.46632i − 0.473990i
\(134\) 0.474162 0.0409613
\(135\) 0 0
\(136\) −2.99270 −0.256622
\(137\) 11.8589i 1.01317i 0.862189 + 0.506587i \(0.169093\pi\)
−0.862189 + 0.506587i \(0.830907\pi\)
\(138\) 0.485114i 0.0412957i
\(139\) 4.66089 0.395332 0.197666 0.980269i \(-0.436664\pi\)
0.197666 + 0.980269i \(0.436664\pi\)
\(140\) 0 0
\(141\) 8.49937 0.715776
\(142\) − 0.612508i − 0.0514005i
\(143\) − 16.8590i − 1.40982i
\(144\) −3.87995 −0.323329
\(145\) 0 0
\(146\) 1.28655 0.106476
\(147\) − 6.26330i − 0.516589i
\(148\) 14.7518i 1.21259i
\(149\) 11.5480 0.946053 0.473026 0.881048i \(-0.343162\pi\)
0.473026 + 0.881048i \(0.343162\pi\)
\(150\) 0 0
\(151\) 24.4694 1.99129 0.995646 0.0932103i \(-0.0297129\pi\)
0.995646 + 0.0932103i \(0.0297129\pi\)
\(152\) − 3.59140i − 0.291301i
\(153\) − 5.30702i − 0.429047i
\(154\) 0.447529 0.0360629
\(155\) 0 0
\(156\) 9.07071 0.726238
\(157\) 22.3660i 1.78500i 0.451045 + 0.892501i \(0.351052\pi\)
−0.451045 + 0.892501i \(0.648948\pi\)
\(158\) 0.460109i 0.0366043i
\(159\) −2.34827 −0.186230
\(160\) 0 0
\(161\) −2.93867 −0.231600
\(162\) 0.141689i 0.0111322i
\(163\) − 0.813450i − 0.0637143i −0.999492 0.0318572i \(-0.989858\pi\)
0.999492 0.0318572i \(-0.0101422\pi\)
\(164\) −5.01253 −0.391413
\(165\) 0 0
\(166\) 1.04747 0.0812992
\(167\) 25.1660i 1.94740i 0.227833 + 0.973700i \(0.426836\pi\)
−0.227833 + 0.973700i \(0.573164\pi\)
\(168\) 0.484013i 0.0373424i
\(169\) −7.98869 −0.614515
\(170\) 0 0
\(171\) 6.36870 0.487027
\(172\) − 6.69158i − 0.510229i
\(173\) 0.410945i 0.0312436i 0.999878 + 0.0156218i \(0.00497277\pi\)
−0.999878 + 0.0156218i \(0.995027\pi\)
\(174\) −0.529076 −0.0401091
\(175\) 0 0
\(176\) −14.2779 −1.07624
\(177\) 13.1264i 0.986640i
\(178\) 2.19584i 0.164585i
\(179\) 14.8502 1.10996 0.554978 0.831865i \(-0.312727\pi\)
0.554978 + 0.831865i \(0.312727\pi\)
\(180\) 0 0
\(181\) 0.739223 0.0549460 0.0274730 0.999623i \(-0.491254\pi\)
0.0274730 + 0.999623i \(0.491254\pi\)
\(182\) − 0.557153i − 0.0412990i
\(183\) − 10.3476i − 0.764919i
\(184\) −1.93072 −0.142334
\(185\) 0 0
\(186\) 0.177523 0.0130166
\(187\) − 19.5295i − 1.42814i
\(188\) 16.8281i 1.22732i
\(189\) −0.858311 −0.0624329
\(190\) 0 0
\(191\) −2.72506 −0.197179 −0.0985893 0.995128i \(-0.531433\pi\)
−0.0985893 + 0.995128i \(0.531433\pi\)
\(192\) − 7.52220i − 0.542868i
\(193\) 14.2040i 1.02243i 0.859453 + 0.511215i \(0.170804\pi\)
−0.859453 + 0.511215i \(0.829196\pi\)
\(194\) −1.42236 −0.102120
\(195\) 0 0
\(196\) 12.4009 0.885776
\(197\) 5.54591i 0.395130i 0.980290 + 0.197565i \(0.0633033\pi\)
−0.980290 + 0.197565i \(0.936697\pi\)
\(198\) 0.521407i 0.0370548i
\(199\) −5.96371 −0.422756 −0.211378 0.977404i \(-0.567795\pi\)
−0.211378 + 0.977404i \(0.567795\pi\)
\(200\) 0 0
\(201\) −3.34649 −0.236043
\(202\) − 0.101254i − 0.00712418i
\(203\) − 3.20497i − 0.224945i
\(204\) 10.5075 0.735672
\(205\) 0 0
\(206\) −0.110626 −0.00770768
\(207\) − 3.42379i − 0.237970i
\(208\) 17.7754i 1.23250i
\(209\) 23.4364 1.62113
\(210\) 0 0
\(211\) −13.7183 −0.944407 −0.472204 0.881490i \(-0.656541\pi\)
−0.472204 + 0.881490i \(0.656541\pi\)
\(212\) − 4.64940i − 0.319322i
\(213\) 4.32289i 0.296199i
\(214\) 1.69462 0.115842
\(215\) 0 0
\(216\) −0.563913 −0.0383694
\(217\) 1.07538i 0.0730014i
\(218\) − 0.344849i − 0.0233561i
\(219\) −9.08007 −0.613574
\(220\) 0 0
\(221\) −24.3133 −1.63549
\(222\) 1.05568i 0.0708527i
\(223\) 3.24170i 0.217081i 0.994092 + 0.108540i \(0.0346176\pi\)
−0.994092 + 0.108540i \(0.965382\pi\)
\(224\) −1.43988 −0.0962060
\(225\) 0 0
\(226\) 2.27204 0.151134
\(227\) − 13.1799i − 0.874779i −0.899272 0.437390i \(-0.855903\pi\)
0.899272 0.437390i \(-0.144097\pi\)
\(228\) 12.6095i 0.835087i
\(229\) −5.83810 −0.385793 −0.192896 0.981219i \(-0.561788\pi\)
−0.192896 + 0.981219i \(0.561788\pi\)
\(230\) 0 0
\(231\) −3.15852 −0.207815
\(232\) − 2.10568i − 0.138245i
\(233\) 9.33453i 0.611526i 0.952108 + 0.305763i \(0.0989115\pi\)
−0.952108 + 0.305763i \(0.901089\pi\)
\(234\) 0.649128 0.0424348
\(235\) 0 0
\(236\) −25.9893 −1.69176
\(237\) − 3.24730i − 0.210935i
\(238\) − 0.645406i − 0.0418354i
\(239\) −17.1502 −1.10935 −0.554676 0.832066i \(-0.687158\pi\)
−0.554676 + 0.832066i \(0.687158\pi\)
\(240\) 0 0
\(241\) 7.22073 0.465128 0.232564 0.972581i \(-0.425289\pi\)
0.232564 + 0.972581i \(0.425289\pi\)
\(242\) 0.360157i 0.0231518i
\(243\) − 1.00000i − 0.0641500i
\(244\) 20.4875 1.31158
\(245\) 0 0
\(246\) −0.358712 −0.0228706
\(247\) − 29.1772i − 1.85650i
\(248\) 0.706527i 0.0448645i
\(249\) −7.39269 −0.468493
\(250\) 0 0
\(251\) −5.75708 −0.363383 −0.181692 0.983356i \(-0.558157\pi\)
−0.181692 + 0.983356i \(0.558157\pi\)
\(252\) − 1.69939i − 0.107051i
\(253\) − 12.5993i − 0.792110i
\(254\) −0.543872 −0.0341256
\(255\) 0 0
\(256\) 14.4180 0.901125
\(257\) − 26.9602i − 1.68173i −0.541242 0.840867i \(-0.682046\pi\)
0.541242 0.840867i \(-0.317954\pi\)
\(258\) − 0.478870i − 0.0298132i
\(259\) −6.39499 −0.397365
\(260\) 0 0
\(261\) 3.73405 0.231132
\(262\) − 1.76396i − 0.108978i
\(263\) 20.6673i 1.27440i 0.770699 + 0.637200i \(0.219907\pi\)
−0.770699 + 0.637200i \(0.780093\pi\)
\(264\) −2.07516 −0.127717
\(265\) 0 0
\(266\) 0.774520 0.0474889
\(267\) − 15.4975i − 0.948433i
\(268\) − 6.62579i − 0.404734i
\(269\) 13.7435 0.837958 0.418979 0.907996i \(-0.362388\pi\)
0.418979 + 0.907996i \(0.362388\pi\)
\(270\) 0 0
\(271\) 3.34392 0.203128 0.101564 0.994829i \(-0.467615\pi\)
0.101564 + 0.994829i \(0.467615\pi\)
\(272\) 20.5910i 1.24851i
\(273\) 3.93221i 0.237988i
\(274\) −1.68028 −0.101510
\(275\) 0 0
\(276\) 6.77883 0.408038
\(277\) − 8.90947i − 0.535318i −0.963514 0.267659i \(-0.913750\pi\)
0.963514 0.267659i \(-0.0862500\pi\)
\(278\) 0.660400i 0.0396081i
\(279\) −1.25290 −0.0750092
\(280\) 0 0
\(281\) −22.4913 −1.34172 −0.670859 0.741585i \(-0.734074\pi\)
−0.670859 + 0.741585i \(0.734074\pi\)
\(282\) 1.20427i 0.0717133i
\(283\) 1.30694i 0.0776894i 0.999245 + 0.0388447i \(0.0123678\pi\)
−0.999245 + 0.0388447i \(0.987632\pi\)
\(284\) −8.55899 −0.507883
\(285\) 0 0
\(286\) 2.38874 0.141249
\(287\) − 2.17296i − 0.128266i
\(288\) − 1.67757i − 0.0988520i
\(289\) −11.1645 −0.656733
\(290\) 0 0
\(291\) 10.0386 0.588472
\(292\) − 17.9779i − 1.05207i
\(293\) 1.97058i 0.115123i 0.998342 + 0.0575613i \(0.0183325\pi\)
−0.998342 + 0.0575613i \(0.981668\pi\)
\(294\) 0.887444 0.0517568
\(295\) 0 0
\(296\) −4.20153 −0.244209
\(297\) − 3.67993i − 0.213531i
\(298\) 1.63624i 0.0947847i
\(299\) −15.6855 −0.907118
\(300\) 0 0
\(301\) 2.90085 0.167202
\(302\) 3.46706i 0.199507i
\(303\) 0.714616i 0.0410536i
\(304\) −24.7102 −1.41723
\(305\) 0 0
\(306\) 0.751949 0.0429861
\(307\) 15.2544i 0.870617i 0.900281 + 0.435308i \(0.143361\pi\)
−0.900281 + 0.435308i \(0.856639\pi\)
\(308\) − 6.25363i − 0.356334i
\(309\) 0.780764 0.0444161
\(310\) 0 0
\(311\) 17.7452 1.00624 0.503120 0.864217i \(-0.332185\pi\)
0.503120 + 0.864217i \(0.332185\pi\)
\(312\) 2.58348i 0.146261i
\(313\) − 27.3869i − 1.54800i −0.633187 0.773999i \(-0.718253\pi\)
0.633187 0.773999i \(-0.281747\pi\)
\(314\) −3.16903 −0.178839
\(315\) 0 0
\(316\) 6.42941 0.361683
\(317\) − 8.64563i − 0.485587i −0.970078 0.242793i \(-0.921936\pi\)
0.970078 0.242793i \(-0.0780637\pi\)
\(318\) − 0.332725i − 0.0186583i
\(319\) 13.7410 0.769350
\(320\) 0 0
\(321\) −11.9601 −0.667547
\(322\) − 0.416379i − 0.0232039i
\(323\) − 33.7988i − 1.88062i
\(324\) 1.97992 0.109996
\(325\) 0 0
\(326\) 0.115257 0.00638351
\(327\) 2.43384i 0.134592i
\(328\) − 1.42765i − 0.0788286i
\(329\) −7.29510 −0.402192
\(330\) 0 0
\(331\) 4.96260 0.272769 0.136385 0.990656i \(-0.456452\pi\)
0.136385 + 0.990656i \(0.456452\pi\)
\(332\) − 14.6370i − 0.803308i
\(333\) − 7.45067i − 0.408294i
\(334\) −3.56575 −0.195109
\(335\) 0 0
\(336\) 3.33020 0.181677
\(337\) 7.30529i 0.397945i 0.980005 + 0.198972i \(0.0637604\pi\)
−0.980005 + 0.198972i \(0.936240\pi\)
\(338\) − 1.13191i − 0.0615680i
\(339\) −16.0354 −0.870921
\(340\) 0 0
\(341\) −4.61058 −0.249677
\(342\) 0.902378i 0.0487950i
\(343\) 11.3840i 0.614680i
\(344\) 1.90587 0.102758
\(345\) 0 0
\(346\) −0.0582266 −0.00313028
\(347\) − 15.9374i − 0.855565i −0.903882 0.427783i \(-0.859295\pi\)
0.903882 0.427783i \(-0.140705\pi\)
\(348\) 7.39313i 0.396314i
\(349\) −16.5844 −0.887743 −0.443871 0.896091i \(-0.646395\pi\)
−0.443871 + 0.896091i \(0.646395\pi\)
\(350\) 0 0
\(351\) −4.58134 −0.244534
\(352\) − 6.17336i − 0.329041i
\(353\) 12.9691i 0.690277i 0.938552 + 0.345138i \(0.112168\pi\)
−0.938552 + 0.345138i \(0.887832\pi\)
\(354\) −1.85987 −0.0988511
\(355\) 0 0
\(356\) 30.6839 1.62624
\(357\) 4.55507i 0.241080i
\(358\) 2.10412i 0.111206i
\(359\) 13.4920 0.712082 0.356041 0.934470i \(-0.384126\pi\)
0.356041 + 0.934470i \(0.384126\pi\)
\(360\) 0 0
\(361\) 21.5603 1.13475
\(362\) 0.104740i 0.00550502i
\(363\) − 2.54188i − 0.133414i
\(364\) −7.78549 −0.408070
\(365\) 0 0
\(366\) 1.46615 0.0766369
\(367\) − 4.57463i − 0.238794i −0.992847 0.119397i \(-0.961904\pi\)
0.992847 0.119397i \(-0.0380961\pi\)
\(368\) 13.2841i 0.692482i
\(369\) 2.53168 0.131794
\(370\) 0 0
\(371\) 2.01555 0.104642
\(372\) − 2.48065i − 0.128616i
\(373\) − 20.7463i − 1.07420i −0.843518 0.537101i \(-0.819519\pi\)
0.843518 0.537101i \(-0.180481\pi\)
\(374\) 2.76712 0.143084
\(375\) 0 0
\(376\) −4.79291 −0.247175
\(377\) − 17.1070i − 0.881053i
\(378\) − 0.121614i − 0.00625513i
\(379\) 5.78163 0.296982 0.148491 0.988914i \(-0.452558\pi\)
0.148491 + 0.988914i \(0.452558\pi\)
\(380\) 0 0
\(381\) 3.83848 0.196651
\(382\) − 0.386113i − 0.0197553i
\(383\) 25.8124i 1.31895i 0.751726 + 0.659476i \(0.229222\pi\)
−0.751726 + 0.659476i \(0.770778\pi\)
\(384\) 4.42097 0.225606
\(385\) 0 0
\(386\) −2.01256 −0.102437
\(387\) 3.37972i 0.171801i
\(388\) 19.8756i 1.00903i
\(389\) 15.7046 0.796254 0.398127 0.917330i \(-0.369660\pi\)
0.398127 + 0.917330i \(0.369660\pi\)
\(390\) 0 0
\(391\) −18.1701 −0.918901
\(392\) 3.53196i 0.178391i
\(393\) 12.4495i 0.627992i
\(394\) −0.785797 −0.0395879
\(395\) 0 0
\(396\) 7.28598 0.366134
\(397\) 19.6040i 0.983895i 0.870625 + 0.491948i \(0.163715\pi\)
−0.870625 + 0.491948i \(0.836285\pi\)
\(398\) − 0.844995i − 0.0423558i
\(399\) −5.46632 −0.273658
\(400\) 0 0
\(401\) −14.4239 −0.720297 −0.360148 0.932895i \(-0.617274\pi\)
−0.360148 + 0.932895i \(0.617274\pi\)
\(402\) − 0.474162i − 0.0236490i
\(403\) 5.73996i 0.285928i
\(404\) −1.41489 −0.0703932
\(405\) 0 0
\(406\) 0.454111 0.0225372
\(407\) − 27.4179i − 1.35906i
\(408\) 2.99270i 0.148161i
\(409\) 27.8742 1.37829 0.689146 0.724622i \(-0.257986\pi\)
0.689146 + 0.724622i \(0.257986\pi\)
\(410\) 0 0
\(411\) 11.8589 0.584957
\(412\) 1.54585i 0.0761588i
\(413\) − 11.2665i − 0.554389i
\(414\) 0.485114 0.0238421
\(415\) 0 0
\(416\) −7.68554 −0.376815
\(417\) − 4.66089i − 0.228245i
\(418\) 3.32069i 0.162420i
\(419\) 36.2881 1.77279 0.886396 0.462928i \(-0.153201\pi\)
0.886396 + 0.462928i \(0.153201\pi\)
\(420\) 0 0
\(421\) −3.37600 −0.164536 −0.0822681 0.996610i \(-0.526216\pi\)
−0.0822681 + 0.996610i \(0.526216\pi\)
\(422\) − 1.94374i − 0.0946198i
\(423\) − 8.49937i − 0.413254i
\(424\) 1.32422 0.0643099
\(425\) 0 0
\(426\) −0.612508 −0.0296761
\(427\) 8.88148i 0.429805i
\(428\) − 23.6801i − 1.14462i
\(429\) −16.8590 −0.813961
\(430\) 0 0
\(431\) −25.6625 −1.23612 −0.618059 0.786131i \(-0.712081\pi\)
−0.618059 + 0.786131i \(0.712081\pi\)
\(432\) 3.87995i 0.186674i
\(433\) − 39.8070i − 1.91300i −0.291730 0.956501i \(-0.594231\pi\)
0.291730 0.956501i \(-0.405769\pi\)
\(434\) −0.152370 −0.00731398
\(435\) 0 0
\(436\) −4.81882 −0.230779
\(437\) − 21.8051i − 1.04308i
\(438\) − 1.28655i − 0.0614738i
\(439\) −33.9180 −1.61882 −0.809408 0.587247i \(-0.800212\pi\)
−0.809408 + 0.587247i \(0.800212\pi\)
\(440\) 0 0
\(441\) −6.26330 −0.298253
\(442\) − 3.44494i − 0.163859i
\(443\) − 28.9300i − 1.37451i −0.726418 0.687253i \(-0.758816\pi\)
0.726418 0.687253i \(-0.241184\pi\)
\(444\) 14.7518 0.700088
\(445\) 0 0
\(446\) −0.459315 −0.0217492
\(447\) − 11.5480i − 0.546204i
\(448\) 6.45638i 0.305035i
\(449\) 10.2089 0.481788 0.240894 0.970551i \(-0.422559\pi\)
0.240894 + 0.970551i \(0.422559\pi\)
\(450\) 0 0
\(451\) 9.31639 0.438692
\(452\) − 31.7488i − 1.49334i
\(453\) − 24.4694i − 1.14967i
\(454\) 1.86745 0.0876438
\(455\) 0 0
\(456\) −3.59140 −0.168182
\(457\) − 32.4952i − 1.52006i −0.649888 0.760030i \(-0.725184\pi\)
0.649888 0.760030i \(-0.274816\pi\)
\(458\) − 0.827198i − 0.0386524i
\(459\) −5.30702 −0.247711
\(460\) 0 0
\(461\) −0.700568 −0.0326287 −0.0163144 0.999867i \(-0.505193\pi\)
−0.0163144 + 0.999867i \(0.505193\pi\)
\(462\) − 0.447529i − 0.0208209i
\(463\) 2.17126i 0.100907i 0.998726 + 0.0504535i \(0.0160667\pi\)
−0.998726 + 0.0504535i \(0.983933\pi\)
\(464\) −14.4879 −0.672585
\(465\) 0 0
\(466\) −1.32261 −0.0612685
\(467\) 3.87710i 0.179411i 0.995968 + 0.0897053i \(0.0285925\pi\)
−0.995968 + 0.0897053i \(0.971407\pi\)
\(468\) − 9.07071i − 0.419294i
\(469\) 2.87232 0.132632
\(470\) 0 0
\(471\) 22.3660 1.03057
\(472\) − 7.40215i − 0.340711i
\(473\) 12.4371i 0.571859i
\(474\) 0.460109 0.0211335
\(475\) 0 0
\(476\) −9.01870 −0.413371
\(477\) 2.34827i 0.107520i
\(478\) − 2.43000i − 0.111146i
\(479\) 21.9994 1.00518 0.502588 0.864526i \(-0.332381\pi\)
0.502588 + 0.864526i \(0.332381\pi\)
\(480\) 0 0
\(481\) −34.1341 −1.55638
\(482\) 1.02310i 0.0466010i
\(483\) 2.93867i 0.133714i
\(484\) 5.03272 0.228760
\(485\) 0 0
\(486\) 0.141689 0.00642717
\(487\) − 28.3997i − 1.28692i −0.765482 0.643458i \(-0.777499\pi\)
0.765482 0.643458i \(-0.222501\pi\)
\(488\) 5.83517i 0.264146i
\(489\) −0.813450 −0.0367855
\(490\) 0 0
\(491\) −14.0468 −0.633925 −0.316963 0.948438i \(-0.602663\pi\)
−0.316963 + 0.948438i \(0.602663\pi\)
\(492\) 5.01253i 0.225982i
\(493\) − 19.8167i − 0.892498i
\(494\) 4.13410 0.186002
\(495\) 0 0
\(496\) 4.86119 0.218274
\(497\) − 3.71038i − 0.166433i
\(498\) − 1.04747i − 0.0469381i
\(499\) −13.0842 −0.585731 −0.292866 0.956154i \(-0.594609\pi\)
−0.292866 + 0.956154i \(0.594609\pi\)
\(500\) 0 0
\(501\) 25.1660 1.12433
\(502\) − 0.815717i − 0.0364072i
\(503\) 12.3044i 0.548625i 0.961641 + 0.274312i \(0.0884502\pi\)
−0.961641 + 0.274312i \(0.911550\pi\)
\(504\) 0.484013 0.0215596
\(505\) 0 0
\(506\) 1.78519 0.0793612
\(507\) 7.98869i 0.354790i
\(508\) 7.59990i 0.337191i
\(509\) −0.0783897 −0.00347456 −0.00173728 0.999998i \(-0.500553\pi\)
−0.00173728 + 0.999998i \(0.500553\pi\)
\(510\) 0 0
\(511\) 7.79352 0.344765
\(512\) 10.8848i 0.481045i
\(513\) − 6.36870i − 0.281185i
\(514\) 3.81998 0.168492
\(515\) 0 0
\(516\) −6.69158 −0.294581
\(517\) − 31.2771i − 1.37556i
\(518\) − 0.906103i − 0.0398119i
\(519\) 0.410945 0.0180385
\(520\) 0 0
\(521\) 18.1194 0.793824 0.396912 0.917857i \(-0.370082\pi\)
0.396912 + 0.917857i \(0.370082\pi\)
\(522\) 0.529076i 0.0231570i
\(523\) 11.6303i 0.508558i 0.967131 + 0.254279i \(0.0818382\pi\)
−0.967131 + 0.254279i \(0.918162\pi\)
\(524\) −24.6490 −1.07680
\(525\) 0 0
\(526\) −2.92834 −0.127682
\(527\) 6.64917i 0.289642i
\(528\) 14.2779i 0.621367i
\(529\) 11.2777 0.490335
\(530\) 0 0
\(531\) 13.1264 0.569637
\(532\) − 10.8229i − 0.469232i
\(533\) − 11.5985i − 0.502386i
\(534\) 2.19584 0.0950231
\(535\) 0 0
\(536\) 1.88713 0.0815115
\(537\) − 14.8502i − 0.640833i
\(538\) 1.94732i 0.0839547i
\(539\) −23.0485 −0.992770
\(540\) 0 0
\(541\) 29.2716 1.25848 0.629242 0.777209i \(-0.283365\pi\)
0.629242 + 0.777209i \(0.283365\pi\)
\(542\) 0.473798i 0.0203514i
\(543\) − 0.739223i − 0.0317231i
\(544\) −8.90292 −0.381710
\(545\) 0 0
\(546\) −0.557153 −0.0238440
\(547\) − 12.0675i − 0.515971i −0.966149 0.257985i \(-0.916941\pi\)
0.966149 0.257985i \(-0.0830587\pi\)
\(548\) 23.4797i 1.00300i
\(549\) −10.3476 −0.441626
\(550\) 0 0
\(551\) 23.7810 1.01311
\(552\) 1.93072i 0.0821768i
\(553\) 2.78719i 0.118524i
\(554\) 1.26238 0.0536333
\(555\) 0 0
\(556\) 9.22821 0.391363
\(557\) 41.4154i 1.75483i 0.479737 + 0.877413i \(0.340732\pi\)
−0.479737 + 0.877413i \(0.659268\pi\)
\(558\) − 0.177523i − 0.00751514i
\(559\) 15.4836 0.654888
\(560\) 0 0
\(561\) −19.5295 −0.824534
\(562\) − 3.18678i − 0.134426i
\(563\) − 32.1467i − 1.35482i −0.735604 0.677411i \(-0.763102\pi\)
0.735604 0.677411i \(-0.236898\pi\)
\(564\) 16.8281 0.708591
\(565\) 0 0
\(566\) −0.185179 −0.00778367
\(567\) 0.858311i 0.0360457i
\(568\) − 2.43773i − 0.102285i
\(569\) −21.1080 −0.884893 −0.442446 0.896795i \(-0.645889\pi\)
−0.442446 + 0.896795i \(0.645889\pi\)
\(570\) 0 0
\(571\) −24.7493 −1.03573 −0.517863 0.855464i \(-0.673272\pi\)
−0.517863 + 0.855464i \(0.673272\pi\)
\(572\) − 33.3796i − 1.39567i
\(573\) 2.72506i 0.113841i
\(574\) 0.307886 0.0128509
\(575\) 0 0
\(576\) −7.52220 −0.313425
\(577\) 18.7710i 0.781448i 0.920508 + 0.390724i \(0.127775\pi\)
−0.920508 + 0.390724i \(0.872225\pi\)
\(578\) − 1.58189i − 0.0657979i
\(579\) 14.2040 0.590300
\(580\) 0 0
\(581\) 6.34522 0.263244
\(582\) 1.42236i 0.0589587i
\(583\) 8.64147i 0.357893i
\(584\) 5.12037 0.211883
\(585\) 0 0
\(586\) −0.279211 −0.0115341
\(587\) − 14.7572i − 0.609095i −0.952497 0.304547i \(-0.901495\pi\)
0.952497 0.304547i \(-0.0985052\pi\)
\(588\) − 12.4009i − 0.511403i
\(589\) −7.97935 −0.328783
\(590\) 0 0
\(591\) 5.54591 0.228128
\(592\) 28.9082i 1.18812i
\(593\) 8.01859i 0.329284i 0.986353 + 0.164642i \(0.0526469\pi\)
−0.986353 + 0.164642i \(0.947353\pi\)
\(594\) 0.521407 0.0213936
\(595\) 0 0
\(596\) 22.8643 0.936556
\(597\) 5.96371i 0.244078i
\(598\) − 2.22247i − 0.0908838i
\(599\) 1.28951 0.0526878 0.0263439 0.999653i \(-0.491614\pi\)
0.0263439 + 0.999653i \(0.491614\pi\)
\(600\) 0 0
\(601\) −16.8813 −0.688603 −0.344302 0.938859i \(-0.611884\pi\)
−0.344302 + 0.938859i \(0.611884\pi\)
\(602\) 0.411020i 0.0167519i
\(603\) 3.34649i 0.136279i
\(604\) 48.4476 1.97130
\(605\) 0 0
\(606\) −0.101254 −0.00411315
\(607\) 0.499318i 0.0202667i 0.999949 + 0.0101334i \(0.00322560\pi\)
−0.999949 + 0.0101334i \(0.996774\pi\)
\(608\) − 10.6840i − 0.433292i
\(609\) −3.20497 −0.129872
\(610\) 0 0
\(611\) −38.9385 −1.57528
\(612\) − 10.5075i − 0.424740i
\(613\) 27.4241i 1.10765i 0.832633 + 0.553825i \(0.186832\pi\)
−0.832633 + 0.553825i \(0.813168\pi\)
\(614\) −2.16139 −0.0872267
\(615\) 0 0
\(616\) 1.78113 0.0717639
\(617\) 28.3205i 1.14014i 0.821596 + 0.570070i \(0.193084\pi\)
−0.821596 + 0.570070i \(0.806916\pi\)
\(618\) 0.110626i 0.00445003i
\(619\) 0.371804 0.0149440 0.00747202 0.999972i \(-0.497622\pi\)
0.00747202 + 0.999972i \(0.497622\pi\)
\(620\) 0 0
\(621\) −3.42379 −0.137392
\(622\) 2.51431i 0.100815i
\(623\) 13.3017i 0.532921i
\(624\) 17.7754 0.711584
\(625\) 0 0
\(626\) 3.88043 0.155093
\(627\) − 23.4364i − 0.935958i
\(628\) 44.2830i 1.76709i
\(629\) −39.5409 −1.57660
\(630\) 0 0
\(631\) 16.4507 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(632\) 1.83120i 0.0728411i
\(633\) 13.7183i 0.545254i
\(634\) 1.22499 0.0486508
\(635\) 0 0
\(636\) −4.64940 −0.184361
\(637\) 28.6943i 1.13691i
\(638\) 1.94696i 0.0770809i
\(639\) 4.32289 0.171011
\(640\) 0 0
\(641\) −2.02604 −0.0800239 −0.0400119 0.999199i \(-0.512740\pi\)
−0.0400119 + 0.999199i \(0.512740\pi\)
\(642\) − 1.69462i − 0.0668813i
\(643\) − 33.2034i − 1.30941i −0.755883 0.654706i \(-0.772792\pi\)
0.755883 0.654706i \(-0.227208\pi\)
\(644\) −5.81834 −0.229275
\(645\) 0 0
\(646\) 4.78894 0.188418
\(647\) 40.5797i 1.59535i 0.603085 + 0.797677i \(0.293938\pi\)
−0.603085 + 0.797677i \(0.706062\pi\)
\(648\) 0.563913i 0.0221526i
\(649\) 48.3042 1.89611
\(650\) 0 0
\(651\) 1.07538 0.0421474
\(652\) − 1.61057i − 0.0630748i
\(653\) − 9.29712i − 0.363824i −0.983315 0.181912i \(-0.941771\pi\)
0.983315 0.181912i \(-0.0582286\pi\)
\(654\) −0.344849 −0.0134847
\(655\) 0 0
\(656\) −9.82277 −0.383515
\(657\) 9.08007i 0.354247i
\(658\) − 1.03364i − 0.0402954i
\(659\) 3.47563 0.135391 0.0676957 0.997706i \(-0.478435\pi\)
0.0676957 + 0.997706i \(0.478435\pi\)
\(660\) 0 0
\(661\) 4.07303 0.158422 0.0792112 0.996858i \(-0.474760\pi\)
0.0792112 + 0.996858i \(0.474760\pi\)
\(662\) 0.703148i 0.0273286i
\(663\) 24.3133i 0.944249i
\(664\) 4.16884 0.161782
\(665\) 0 0
\(666\) 1.05568 0.0409068
\(667\) − 12.7846i − 0.495021i
\(668\) 49.8267i 1.92785i
\(669\) 3.24170 0.125332
\(670\) 0 0
\(671\) −38.0785 −1.47001
\(672\) 1.43988i 0.0555446i
\(673\) 21.3843i 0.824305i 0.911115 + 0.412153i \(0.135223\pi\)
−0.911115 + 0.412153i \(0.864777\pi\)
\(674\) −1.03508 −0.0398699
\(675\) 0 0
\(676\) −15.8170 −0.608346
\(677\) 20.3257i 0.781181i 0.920565 + 0.390590i \(0.127729\pi\)
−0.920565 + 0.390590i \(0.872271\pi\)
\(678\) − 2.27204i − 0.0872573i
\(679\) −8.61621 −0.330660
\(680\) 0 0
\(681\) −13.1799 −0.505054
\(682\) − 0.653271i − 0.0250150i
\(683\) − 18.9591i − 0.725449i −0.931896 0.362725i \(-0.881847\pi\)
0.931896 0.362725i \(-0.118153\pi\)
\(684\) 12.6095 0.482138
\(685\) 0 0
\(686\) −1.61300 −0.0615845
\(687\) 5.83810i 0.222737i
\(688\) − 13.1131i − 0.499933i
\(689\) 10.7582 0.409856
\(690\) 0 0
\(691\) 14.8195 0.563762 0.281881 0.959449i \(-0.409042\pi\)
0.281881 + 0.959449i \(0.409042\pi\)
\(692\) 0.813641i 0.0309300i
\(693\) 3.15852i 0.119982i
\(694\) 2.25816 0.0857187
\(695\) 0 0
\(696\) −2.10568 −0.0798156
\(697\) − 13.4357i − 0.508912i
\(698\) − 2.34984i − 0.0889426i
\(699\) 9.33453 0.353064
\(700\) 0 0
\(701\) 31.3996 1.18595 0.592973 0.805222i \(-0.297954\pi\)
0.592973 + 0.805222i \(0.297954\pi\)
\(702\) − 0.649128i − 0.0244998i
\(703\) − 47.4511i − 1.78965i
\(704\) −27.6812 −1.04327
\(705\) 0 0
\(706\) −1.83759 −0.0691586
\(707\) − 0.613363i − 0.0230679i
\(708\) 25.9893i 0.976736i
\(709\) −25.3324 −0.951379 −0.475690 0.879613i \(-0.657801\pi\)
−0.475690 + 0.879613i \(0.657801\pi\)
\(710\) 0 0
\(711\) −3.24730 −0.121783
\(712\) 8.73926i 0.327518i
\(713\) 4.28966i 0.160649i
\(714\) −0.645406 −0.0241537
\(715\) 0 0
\(716\) 29.4023 1.09881
\(717\) 17.1502i 0.640485i
\(718\) 1.91168i 0.0713432i
\(719\) 4.12711 0.153915 0.0769575 0.997034i \(-0.475479\pi\)
0.0769575 + 0.997034i \(0.475479\pi\)
\(720\) 0 0
\(721\) −0.670138 −0.0249572
\(722\) 3.05487i 0.113691i
\(723\) − 7.22073i − 0.268542i
\(724\) 1.46361 0.0543945
\(725\) 0 0
\(726\) 0.360157 0.0133667
\(727\) − 23.0039i − 0.853168i −0.904448 0.426584i \(-0.859717\pi\)
0.904448 0.426584i \(-0.140283\pi\)
\(728\) − 2.21743i − 0.0821834i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 17.9362 0.663396
\(732\) − 20.4875i − 0.757240i
\(733\) − 9.35692i − 0.345606i −0.984956 0.172803i \(-0.944718\pi\)
0.984956 0.172803i \(-0.0552823\pi\)
\(734\) 0.648176 0.0239246
\(735\) 0 0
\(736\) −5.74366 −0.211714
\(737\) 12.3148i 0.453623i
\(738\) 0.358712i 0.0132044i
\(739\) 47.1754 1.73538 0.867688 0.497109i \(-0.165605\pi\)
0.867688 + 0.497109i \(0.165605\pi\)
\(740\) 0 0
\(741\) −29.1772 −1.07185
\(742\) 0.285582i 0.0104840i
\(743\) 2.39450i 0.0878455i 0.999035 + 0.0439228i \(0.0139855\pi\)
−0.999035 + 0.0439228i \(0.986014\pi\)
\(744\) 0.706527 0.0259025
\(745\) 0 0
\(746\) 2.93953 0.107624
\(747\) 7.39269i 0.270484i
\(748\) − 38.6668i − 1.41380i
\(749\) 10.2655 0.375092
\(750\) 0 0
\(751\) −7.21632 −0.263327 −0.131664 0.991294i \(-0.542032\pi\)
−0.131664 + 0.991294i \(0.542032\pi\)
\(752\) 32.9771i 1.20255i
\(753\) 5.75708i 0.209800i
\(754\) 2.42388 0.0882724
\(755\) 0 0
\(756\) −1.69939 −0.0618062
\(757\) − 13.3742i − 0.486094i −0.970015 0.243047i \(-0.921853\pi\)
0.970015 0.243047i \(-0.0781469\pi\)
\(758\) 0.819196i 0.0297545i
\(759\) −12.5993 −0.457325
\(760\) 0 0
\(761\) 21.7541 0.788584 0.394292 0.918985i \(-0.370990\pi\)
0.394292 + 0.918985i \(0.370990\pi\)
\(762\) 0.543872i 0.0197024i
\(763\) − 2.08899i − 0.0756265i
\(764\) −5.39542 −0.195199
\(765\) 0 0
\(766\) −3.65735 −0.132145
\(767\) − 60.1365i − 2.17140i
\(768\) − 14.4180i − 0.520265i
\(769\) −14.2003 −0.512077 −0.256038 0.966667i \(-0.582417\pi\)
−0.256038 + 0.966667i \(0.582417\pi\)
\(770\) 0 0
\(771\) −26.9602 −0.970949
\(772\) 28.1229i 1.01217i
\(773\) 22.8365i 0.821373i 0.911777 + 0.410687i \(0.134711\pi\)
−0.911777 + 0.410687i \(0.865289\pi\)
\(774\) −0.478870 −0.0172126
\(775\) 0 0
\(776\) −5.66089 −0.203214
\(777\) 6.39499i 0.229419i
\(778\) 2.22517i 0.0797763i
\(779\) 16.1235 0.577684
\(780\) 0 0
\(781\) 15.9079 0.569230
\(782\) − 2.57451i − 0.0920644i
\(783\) − 3.73405i − 0.133444i
\(784\) 24.3013 0.867903
\(785\) 0 0
\(786\) −1.76396 −0.0629183
\(787\) 38.4761i 1.37153i 0.727825 + 0.685763i \(0.240531\pi\)
−0.727825 + 0.685763i \(0.759469\pi\)
\(788\) 10.9805i 0.391163i
\(789\) 20.6673 0.735775
\(790\) 0 0
\(791\) 13.7633 0.489367
\(792\) 2.07516i 0.0737376i
\(793\) 47.4060i 1.68344i
\(794\) −2.77768 −0.0985760
\(795\) 0 0
\(796\) −11.8077 −0.418512
\(797\) 1.86739i 0.0661464i 0.999453 + 0.0330732i \(0.0105294\pi\)
−0.999453 + 0.0330732i \(0.989471\pi\)
\(798\) − 0.774520i − 0.0274177i
\(799\) −45.1063 −1.59575
\(800\) 0 0
\(801\) −15.4975 −0.547578
\(802\) − 2.04372i − 0.0721662i
\(803\) 33.4140i 1.17916i
\(804\) −6.62579 −0.233673
\(805\) 0 0
\(806\) −0.813293 −0.0286470
\(807\) − 13.7435i − 0.483796i
\(808\) − 0.402982i − 0.0141768i
\(809\) −35.1514 −1.23586 −0.617929 0.786234i \(-0.712028\pi\)
−0.617929 + 0.786234i \(0.712028\pi\)
\(810\) 0 0
\(811\) −6.59158 −0.231462 −0.115731 0.993281i \(-0.536921\pi\)
−0.115731 + 0.993281i \(0.536921\pi\)
\(812\) − 6.34561i − 0.222687i
\(813\) − 3.34392i − 0.117276i
\(814\) 3.88483 0.136163
\(815\) 0 0
\(816\) 20.5910 0.720828
\(817\) 21.5244i 0.753044i
\(818\) 3.94949i 0.138091i
\(819\) 3.93221 0.137403
\(820\) 0 0
\(821\) 5.25743 0.183486 0.0917428 0.995783i \(-0.470756\pi\)
0.0917428 + 0.995783i \(0.470756\pi\)
\(822\) 1.68028i 0.0586066i
\(823\) − 22.1919i − 0.773560i −0.922172 0.386780i \(-0.873587\pi\)
0.922172 0.386780i \(-0.126413\pi\)
\(824\) −0.440283 −0.0153380
\(825\) 0 0
\(826\) 1.59635 0.0555440
\(827\) 6.56256i 0.228203i 0.993469 + 0.114101i \(0.0363988\pi\)
−0.993469 + 0.114101i \(0.963601\pi\)
\(828\) − 6.77883i − 0.235581i
\(829\) −25.0574 −0.870279 −0.435140 0.900363i \(-0.643301\pi\)
−0.435140 + 0.900363i \(0.643301\pi\)
\(830\) 0 0
\(831\) −8.90947 −0.309066
\(832\) 34.4618i 1.19475i
\(833\) 33.2395i 1.15168i
\(834\) 0.660400 0.0228678
\(835\) 0 0
\(836\) 46.4022 1.60485
\(837\) 1.25290i 0.0433066i
\(838\) 5.14165i 0.177615i
\(839\) 42.1485 1.45513 0.727564 0.686040i \(-0.240652\pi\)
0.727564 + 0.686040i \(0.240652\pi\)
\(840\) 0 0
\(841\) −15.0569 −0.519203
\(842\) − 0.478344i − 0.0164848i
\(843\) 22.4913i 0.774641i
\(844\) −27.1612 −0.934927
\(845\) 0 0
\(846\) 1.20427 0.0414037
\(847\) 2.18172i 0.0749648i
\(848\) − 9.11117i − 0.312879i
\(849\) 1.30694 0.0448540
\(850\) 0 0
\(851\) −25.5095 −0.874454
\(852\) 8.55899i 0.293226i
\(853\) 43.2473i 1.48076i 0.672189 + 0.740379i \(0.265354\pi\)
−0.672189 + 0.740379i \(0.734646\pi\)
\(854\) −1.25841 −0.0430620
\(855\) 0 0
\(856\) 6.74446 0.230521
\(857\) 39.2430i 1.34052i 0.742128 + 0.670258i \(0.233817\pi\)
−0.742128 + 0.670258i \(0.766183\pi\)
\(858\) − 2.38874i − 0.0815504i
\(859\) −52.6092 −1.79500 −0.897501 0.441013i \(-0.854619\pi\)
−0.897501 + 0.441013i \(0.854619\pi\)
\(860\) 0 0
\(861\) −2.17296 −0.0740544
\(862\) − 3.63611i − 0.123846i
\(863\) − 42.8744i − 1.45946i −0.683735 0.729731i \(-0.739645\pi\)
0.683735 0.729731i \(-0.260355\pi\)
\(864\) −1.67757 −0.0570722
\(865\) 0 0
\(866\) 5.64023 0.191663
\(867\) 11.1645i 0.379165i
\(868\) 2.12917i 0.0722686i
\(869\) −11.9498 −0.405371
\(870\) 0 0
\(871\) 15.3314 0.519484
\(872\) − 1.37247i − 0.0464778i
\(873\) − 10.0386i − 0.339754i
\(874\) 3.08955 0.104506
\(875\) 0 0
\(876\) −17.9779 −0.607415
\(877\) 0.390314i 0.0131800i 0.999978 + 0.00658999i \(0.00209767\pi\)
−0.999978 + 0.00658999i \(0.997902\pi\)
\(878\) − 4.80582i − 0.162188i
\(879\) 1.97058 0.0664660
\(880\) 0 0
\(881\) 44.7002 1.50599 0.752994 0.658028i \(-0.228609\pi\)
0.752994 + 0.658028i \(0.228609\pi\)
\(882\) − 0.887444i − 0.0298818i
\(883\) 32.7567i 1.10235i 0.834390 + 0.551175i \(0.185820\pi\)
−0.834390 + 0.551175i \(0.814180\pi\)
\(884\) −48.1384 −1.61907
\(885\) 0 0
\(886\) 4.09908 0.137711
\(887\) 15.7130i 0.527591i 0.964579 + 0.263796i \(0.0849745\pi\)
−0.964579 + 0.263796i \(0.915026\pi\)
\(888\) 4.20153i 0.140994i
\(889\) −3.29461 −0.110498
\(890\) 0 0
\(891\) −3.67993 −0.123282
\(892\) 6.41833i 0.214902i
\(893\) − 54.1299i − 1.81139i
\(894\) 1.63624 0.0547239
\(895\) 0 0
\(896\) −3.79456 −0.126767
\(897\) 15.6855i 0.523725i
\(898\) 1.44649i 0.0482701i
\(899\) −4.67839 −0.156033
\(900\) 0 0
\(901\) 12.4623 0.415180
\(902\) 1.32003i 0.0439523i
\(903\) − 2.90085i − 0.0965342i
\(904\) 9.04256 0.300751
\(905\) 0 0
\(906\) 3.46706 0.115185
\(907\) 1.50466i 0.0499613i 0.999688 + 0.0249806i \(0.00795241\pi\)
−0.999688 + 0.0249806i \(0.992048\pi\)
\(908\) − 26.0952i − 0.865998i
\(909\) 0.714616 0.0237023
\(910\) 0 0
\(911\) 11.4509 0.379386 0.189693 0.981843i \(-0.439251\pi\)
0.189693 + 0.981843i \(0.439251\pi\)
\(912\) 24.7102i 0.818237i
\(913\) 27.2046i 0.900340i
\(914\) 4.60422 0.152294
\(915\) 0 0
\(916\) −11.5590 −0.381920
\(917\) − 10.6855i − 0.352866i
\(918\) − 0.751949i − 0.0248180i
\(919\) 21.8001 0.719119 0.359559 0.933122i \(-0.382927\pi\)
0.359559 + 0.933122i \(0.382927\pi\)
\(920\) 0 0
\(921\) 15.2544 0.502651
\(922\) − 0.0992632i − 0.00326906i
\(923\) − 19.8046i − 0.651877i
\(924\) −6.25363 −0.205729
\(925\) 0 0
\(926\) −0.307645 −0.0101098
\(927\) − 0.780764i − 0.0256437i
\(928\) − 6.26415i − 0.205631i
\(929\) 30.7507 1.00890 0.504449 0.863442i \(-0.331696\pi\)
0.504449 + 0.863442i \(0.331696\pi\)
\(930\) 0 0
\(931\) −39.8891 −1.30731
\(932\) 18.4817i 0.605387i
\(933\) − 17.7452i − 0.580953i
\(934\) −0.549344 −0.0179751
\(935\) 0 0
\(936\) 2.58348 0.0844437
\(937\) 3.81060i 0.124487i 0.998061 + 0.0622435i \(0.0198255\pi\)
−0.998061 + 0.0622435i \(0.980174\pi\)
\(938\) 0.406978i 0.0132883i
\(939\) −27.3869 −0.893737
\(940\) 0 0
\(941\) −20.7043 −0.674939 −0.337470 0.941336i \(-0.609571\pi\)
−0.337470 + 0.941336i \(0.609571\pi\)
\(942\) 3.16903i 0.103253i
\(943\) − 8.66792i − 0.282266i
\(944\) −50.9297 −1.65762
\(945\) 0 0
\(946\) −1.76221 −0.0572944
\(947\) − 15.6833i − 0.509639i −0.966989 0.254819i \(-0.917984\pi\)
0.966989 0.254819i \(-0.0820160\pi\)
\(948\) − 6.42941i − 0.208818i
\(949\) 41.5989 1.35036
\(950\) 0 0
\(951\) −8.64563 −0.280354
\(952\) − 2.56867i − 0.0832509i
\(953\) − 7.75207i − 0.251114i −0.992086 0.125557i \(-0.959928\pi\)
0.992086 0.125557i \(-0.0400718\pi\)
\(954\) −0.332725 −0.0107724
\(955\) 0 0
\(956\) −33.9560 −1.09822
\(957\) − 13.7410i − 0.444185i
\(958\) 3.11708i 0.100708i
\(959\) −10.1786 −0.328685
\(960\) 0 0
\(961\) −29.4302 −0.949363
\(962\) − 4.83644i − 0.155933i
\(963\) 11.9601i 0.385409i
\(964\) 14.2965 0.460459
\(965\) 0 0
\(966\) −0.416379 −0.0133968
\(967\) 48.4273i 1.55732i 0.627447 + 0.778659i \(0.284100\pi\)
−0.627447 + 0.778659i \(0.715900\pi\)
\(968\) 1.43340i 0.0460712i
\(969\) −33.7988 −1.08577
\(970\) 0 0
\(971\) 25.3350 0.813039 0.406520 0.913642i \(-0.366742\pi\)
0.406520 + 0.913642i \(0.366742\pi\)
\(972\) − 1.97992i − 0.0635061i
\(973\) 4.00049i 0.128250i
\(974\) 4.02394 0.128936
\(975\) 0 0
\(976\) 40.1483 1.28511
\(977\) 1.16885i 0.0373948i 0.999825 + 0.0186974i \(0.00595192\pi\)
−0.999825 + 0.0186974i \(0.994048\pi\)
\(978\) − 0.115257i − 0.00368552i
\(979\) −57.0298 −1.82268
\(980\) 0 0
\(981\) 2.43384 0.0777065
\(982\) − 1.99029i − 0.0635127i
\(983\) − 23.9943i − 0.765299i −0.923894 0.382649i \(-0.875012\pi\)
0.923894 0.382649i \(-0.124988\pi\)
\(984\) −1.42765 −0.0455117
\(985\) 0 0
\(986\) 2.80782 0.0894190
\(987\) 7.29510i 0.232206i
\(988\) − 57.7686i − 1.83786i
\(989\) 11.5714 0.367950
\(990\) 0 0
\(991\) −36.6726 −1.16494 −0.582472 0.812851i \(-0.697914\pi\)
−0.582472 + 0.812851i \(0.697914\pi\)
\(992\) 2.10183i 0.0667333i
\(993\) − 4.96260i − 0.157483i
\(994\) 0.525722 0.0166749
\(995\) 0 0
\(996\) −14.6370 −0.463790
\(997\) 57.0203i 1.80585i 0.429798 + 0.902925i \(0.358585\pi\)
−0.429798 + 0.902925i \(0.641415\pi\)
\(998\) − 1.85390i − 0.0586842i
\(999\) −7.45067 −0.235729
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 1875.2.b.f.1249.7 12
5.2 odd 4 1875.2.a.j.1.3 6
5.3 odd 4 1875.2.a.k.1.4 6
5.4 even 2 inner 1875.2.b.f.1249.6 12
15.2 even 4 5625.2.a.p.1.4 6
15.8 even 4 5625.2.a.q.1.3 6
25.3 odd 20 375.2.g.c.76.2 12
25.4 even 10 375.2.i.d.49.3 24
25.6 even 5 375.2.i.d.199.3 24
25.8 odd 20 375.2.g.c.301.2 12
25.17 odd 20 75.2.g.c.61.2 yes 12
25.19 even 10 375.2.i.d.199.4 24
25.21 even 5 375.2.i.d.49.4 24
25.22 odd 20 75.2.g.c.16.2 12
75.17 even 20 225.2.h.d.136.2 12
75.47 even 20 225.2.h.d.91.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.g.c.16.2 12 25.22 odd 20
75.2.g.c.61.2 yes 12 25.17 odd 20
225.2.h.d.91.2 12 75.47 even 20
225.2.h.d.136.2 12 75.17 even 20
375.2.g.c.76.2 12 25.3 odd 20
375.2.g.c.301.2 12 25.8 odd 20
375.2.i.d.49.3 24 25.4 even 10
375.2.i.d.49.4 24 25.21 even 5
375.2.i.d.199.3 24 25.6 even 5
375.2.i.d.199.4 24 25.19 even 10
1875.2.a.j.1.3 6 5.2 odd 4
1875.2.a.k.1.4 6 5.3 odd 4
1875.2.b.f.1249.6 12 5.4 even 2 inner
1875.2.b.f.1249.7 12 1.1 even 1 trivial
5625.2.a.p.1.4 6 15.2 even 4
5625.2.a.q.1.3 6 15.8 even 4