Properties

Label 1875.2.b.h.1249.5
Level $1875$
Weight $2$
Character 1875.1249
Analytic conductor $14.972$
Analytic rank $0$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.5
Root \(2.35083i\) of defining polynomial
Character \(\chi\) \(=\) 1875.1249
Dual form 1875.2.b.h.1249.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.35083i q^{2} +1.00000i q^{3} +0.175259 q^{4} +1.35083 q^{6} +1.59580i q^{7} -2.93840i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.35083i q^{2} +1.00000i q^{3} +0.175259 q^{4} +1.35083 q^{6} +1.59580i q^{7} -2.93840i q^{8} -1.00000 q^{9} +3.33277 q^{11} +0.175259i q^{12} -7.05132i q^{13} +2.15565 q^{14} -3.61877 q^{16} +4.09625i q^{17} +1.35083i q^{18} -0.567535 q^{19} -1.59580 q^{21} -4.50200i q^{22} -6.30400i q^{23} +2.93840 q^{24} -9.52513 q^{26} -1.00000i q^{27} +0.279678i q^{28} +2.78357 q^{29} -0.995824 q^{31} -0.988473i q^{32} +3.33277i q^{33} +5.53333 q^{34} -0.175259 q^{36} -3.55334i q^{37} +0.766643i q^{38} +7.05132 q^{39} +1.16293 q^{41} +2.15565i q^{42} -0.117022i q^{43} +0.584098 q^{44} -8.51563 q^{46} +7.64173i q^{47} -3.61877i q^{48} +4.45343 q^{49} -4.09625 q^{51} -1.23581i q^{52} -0.523635i q^{53} -1.35083 q^{54} +4.68910 q^{56} -0.567535i q^{57} -3.76013i q^{58} -0.983998 q^{59} +10.6137 q^{61} +1.34519i q^{62} -1.59580i q^{63} -8.57279 q^{64} +4.50200 q^{66} -15.2159i q^{67} +0.717905i q^{68} +6.30400 q^{69} -10.6639 q^{71} +2.93840i q^{72} -5.55832i q^{73} -4.79996 q^{74} -0.0994657 q^{76} +5.31842i q^{77} -9.52513i q^{78} +14.5969 q^{79} +1.00000 q^{81} -1.57091i q^{82} -5.02398i q^{83} -0.279678 q^{84} -0.158076 q^{86} +2.78357i q^{87} -9.79302i q^{88} -2.82350 q^{89} +11.2525 q^{91} -1.10483i q^{92} -0.995824i q^{93} +10.3227 q^{94} +0.988473 q^{96} -1.70592i q^{97} -6.01583i q^{98} -3.33277 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.35083i − 0.955181i −0.878583 0.477590i \(-0.841510\pi\)
0.878583 0.477590i \(-0.158490\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 0.175259 0.0876296
\(5\) 0 0
\(6\) 1.35083 0.551474
\(7\) 1.59580i 0.603155i 0.953442 + 0.301577i \(0.0975131\pi\)
−0.953442 + 0.301577i \(0.902487\pi\)
\(8\) − 2.93840i − 1.03888i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.33277 1.00487 0.502434 0.864616i \(-0.332438\pi\)
0.502434 + 0.864616i \(0.332438\pi\)
\(12\) 0.175259i 0.0505930i
\(13\) − 7.05132i − 1.95568i −0.209345 0.977842i \(-0.567133\pi\)
0.209345 0.977842i \(-0.432867\pi\)
\(14\) 2.15565 0.576122
\(15\) 0 0
\(16\) −3.61877 −0.904691
\(17\) 4.09625i 0.993486i 0.867898 + 0.496743i \(0.165471\pi\)
−0.867898 + 0.496743i \(0.834529\pi\)
\(18\) 1.35083i 0.318394i
\(19\) −0.567535 −0.130201 −0.0651007 0.997879i \(-0.520737\pi\)
−0.0651007 + 0.997879i \(0.520737\pi\)
\(20\) 0 0
\(21\) −1.59580 −0.348231
\(22\) − 4.50200i − 0.959830i
\(23\) − 6.30400i − 1.31448i −0.753683 0.657238i \(-0.771725\pi\)
0.753683 0.657238i \(-0.228275\pi\)
\(24\) 2.93840 0.599799
\(25\) 0 0
\(26\) −9.52513 −1.86803
\(27\) − 1.00000i − 0.192450i
\(28\) 0.279678i 0.0528542i
\(29\) 2.78357 0.516897 0.258448 0.966025i \(-0.416789\pi\)
0.258448 + 0.966025i \(0.416789\pi\)
\(30\) 0 0
\(31\) −0.995824 −0.178855 −0.0894276 0.995993i \(-0.528504\pi\)
−0.0894276 + 0.995993i \(0.528504\pi\)
\(32\) − 0.988473i − 0.174739i
\(33\) 3.33277i 0.580161i
\(34\) 5.53333 0.948959
\(35\) 0 0
\(36\) −0.175259 −0.0292099
\(37\) − 3.55334i − 0.584165i −0.956393 0.292083i \(-0.905652\pi\)
0.956393 0.292083i \(-0.0943483\pi\)
\(38\) 0.766643i 0.124366i
\(39\) 7.05132 1.12911
\(40\) 0 0
\(41\) 1.16293 0.181618 0.0908092 0.995868i \(-0.471055\pi\)
0.0908092 + 0.995868i \(0.471055\pi\)
\(42\) 2.15565i 0.332624i
\(43\) − 0.117022i − 0.0178456i −0.999960 0.00892281i \(-0.997160\pi\)
0.999960 0.00892281i \(-0.00284026\pi\)
\(44\) 0.584098 0.0880561
\(45\) 0 0
\(46\) −8.51563 −1.25556
\(47\) 7.64173i 1.11466i 0.830291 + 0.557331i \(0.188174\pi\)
−0.830291 + 0.557331i \(0.811826\pi\)
\(48\) − 3.61877i − 0.522324i
\(49\) 4.45343 0.636205
\(50\) 0 0
\(51\) −4.09625 −0.573589
\(52\) − 1.23581i − 0.171376i
\(53\) − 0.523635i − 0.0719268i −0.999353 0.0359634i \(-0.988550\pi\)
0.999353 0.0359634i \(-0.0114500\pi\)
\(54\) −1.35083 −0.183825
\(55\) 0 0
\(56\) 4.68910 0.626607
\(57\) − 0.567535i − 0.0751718i
\(58\) − 3.76013i − 0.493730i
\(59\) −0.983998 −0.128106 −0.0640528 0.997947i \(-0.520403\pi\)
−0.0640528 + 0.997947i \(0.520403\pi\)
\(60\) 0 0
\(61\) 10.6137 1.35895 0.679473 0.733701i \(-0.262208\pi\)
0.679473 + 0.733701i \(0.262208\pi\)
\(62\) 1.34519i 0.170839i
\(63\) − 1.59580i − 0.201052i
\(64\) −8.57279 −1.07160
\(65\) 0 0
\(66\) 4.50200 0.554158
\(67\) − 15.2159i − 1.85892i −0.368920 0.929461i \(-0.620272\pi\)
0.368920 0.929461i \(-0.379728\pi\)
\(68\) 0.717905i 0.0870588i
\(69\) 6.30400 0.758913
\(70\) 0 0
\(71\) −10.6639 −1.26558 −0.632788 0.774325i \(-0.718090\pi\)
−0.632788 + 0.774325i \(0.718090\pi\)
\(72\) 2.93840i 0.346294i
\(73\) − 5.55832i − 0.650552i −0.945619 0.325276i \(-0.894543\pi\)
0.945619 0.325276i \(-0.105457\pi\)
\(74\) −4.79996 −0.557984
\(75\) 0 0
\(76\) −0.0994657 −0.0114095
\(77\) 5.31842i 0.606090i
\(78\) − 9.52513i − 1.07851i
\(79\) 14.5969 1.64227 0.821137 0.570731i \(-0.193340\pi\)
0.821137 + 0.570731i \(0.193340\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 1.57091i − 0.173478i
\(83\) − 5.02398i − 0.551453i −0.961236 0.275727i \(-0.911082\pi\)
0.961236 0.275727i \(-0.0889184\pi\)
\(84\) −0.279678 −0.0305154
\(85\) 0 0
\(86\) −0.158076 −0.0170458
\(87\) 2.78357i 0.298430i
\(88\) − 9.79302i − 1.04394i
\(89\) −2.82350 −0.299291 −0.149645 0.988740i \(-0.547813\pi\)
−0.149645 + 0.988740i \(0.547813\pi\)
\(90\) 0 0
\(91\) 11.2525 1.17958
\(92\) − 1.10483i − 0.115187i
\(93\) − 0.995824i − 0.103262i
\(94\) 10.3227 1.06470
\(95\) 0 0
\(96\) 0.988473 0.100886
\(97\) − 1.70592i − 0.173210i −0.996243 0.0866049i \(-0.972398\pi\)
0.996243 0.0866049i \(-0.0276018\pi\)
\(98\) − 6.01583i − 0.607690i
\(99\) −3.33277 −0.334956
\(100\) 0 0
\(101\) 13.1747 1.31093 0.655464 0.755226i \(-0.272473\pi\)
0.655464 + 0.755226i \(0.272473\pi\)
\(102\) 5.53333i 0.547882i
\(103\) − 10.8720i − 1.07125i −0.844456 0.535624i \(-0.820076\pi\)
0.844456 0.535624i \(-0.179924\pi\)
\(104\) −20.7196 −2.03173
\(105\) 0 0
\(106\) −0.707341 −0.0687031
\(107\) 9.37236i 0.906060i 0.891495 + 0.453030i \(0.149657\pi\)
−0.891495 + 0.453030i \(0.850343\pi\)
\(108\) − 0.175259i − 0.0168643i
\(109\) 15.5899 1.49324 0.746622 0.665249i \(-0.231674\pi\)
0.746622 + 0.665249i \(0.231674\pi\)
\(110\) 0 0
\(111\) 3.55334 0.337268
\(112\) − 5.77482i − 0.545669i
\(113\) 10.0481i 0.945243i 0.881265 + 0.472622i \(0.156692\pi\)
−0.881265 + 0.472622i \(0.843308\pi\)
\(114\) −0.766643 −0.0718027
\(115\) 0 0
\(116\) 0.487847 0.0452954
\(117\) 7.05132i 0.651895i
\(118\) 1.32921i 0.122364i
\(119\) −6.53678 −0.599226
\(120\) 0 0
\(121\) 0.107347 0.00975880
\(122\) − 14.3373i − 1.29804i
\(123\) 1.16293i 0.104857i
\(124\) −0.174527 −0.0156730
\(125\) 0 0
\(126\) −2.15565 −0.192041
\(127\) 0.976784i 0.0866756i 0.999060 + 0.0433378i \(0.0137992\pi\)
−0.999060 + 0.0433378i \(0.986201\pi\)
\(128\) 9.60343i 0.848832i
\(129\) 0.117022 0.0103032
\(130\) 0 0
\(131\) 10.0616 0.879086 0.439543 0.898221i \(-0.355140\pi\)
0.439543 + 0.898221i \(0.355140\pi\)
\(132\) 0.584098i 0.0508392i
\(133\) − 0.905670i − 0.0785316i
\(134\) −20.5541 −1.77561
\(135\) 0 0
\(136\) 12.0364 1.03212
\(137\) 4.87244i 0.416281i 0.978099 + 0.208140i \(0.0667411\pi\)
−0.978099 + 0.208140i \(0.933259\pi\)
\(138\) − 8.51563i − 0.724899i
\(139\) −0.185784 −0.0157580 −0.00787898 0.999969i \(-0.502508\pi\)
−0.00787898 + 0.999969i \(0.502508\pi\)
\(140\) 0 0
\(141\) −7.64173 −0.643550
\(142\) 14.4052i 1.20885i
\(143\) − 23.5004i − 1.96520i
\(144\) 3.61877 0.301564
\(145\) 0 0
\(146\) −7.50834 −0.621395
\(147\) 4.45343i 0.367313i
\(148\) − 0.622755i − 0.0511902i
\(149\) −3.88889 −0.318590 −0.159295 0.987231i \(-0.550922\pi\)
−0.159295 + 0.987231i \(0.550922\pi\)
\(150\) 0 0
\(151\) −22.1146 −1.79966 −0.899829 0.436242i \(-0.856309\pi\)
−0.899829 + 0.436242i \(0.856309\pi\)
\(152\) 1.66765i 0.135264i
\(153\) − 4.09625i − 0.331162i
\(154\) 7.18428 0.578926
\(155\) 0 0
\(156\) 1.23581 0.0989438
\(157\) − 13.6058i − 1.08586i −0.839777 0.542931i \(-0.817314\pi\)
0.839777 0.542931i \(-0.182686\pi\)
\(158\) − 19.7179i − 1.56867i
\(159\) 0.523635 0.0415269
\(160\) 0 0
\(161\) 10.0599 0.792832
\(162\) − 1.35083i − 0.106131i
\(163\) 8.62895i 0.675871i 0.941169 + 0.337936i \(0.109729\pi\)
−0.941169 + 0.337936i \(0.890271\pi\)
\(164\) 0.203813 0.0159151
\(165\) 0 0
\(166\) −6.78654 −0.526737
\(167\) − 6.59891i − 0.510639i −0.966857 0.255319i \(-0.917819\pi\)
0.966857 0.255319i \(-0.0821806\pi\)
\(168\) 4.68910i 0.361772i
\(169\) −36.7211 −2.82470
\(170\) 0 0
\(171\) 0.567535 0.0434005
\(172\) − 0.0205091i − 0.00156381i
\(173\) 13.6595i 1.03851i 0.854618 + 0.519257i \(0.173791\pi\)
−0.854618 + 0.519257i \(0.826209\pi\)
\(174\) 3.76013 0.285055
\(175\) 0 0
\(176\) −12.0605 −0.909095
\(177\) − 0.983998i − 0.0739618i
\(178\) 3.81407i 0.285877i
\(179\) −9.82880 −0.734639 −0.367320 0.930095i \(-0.619724\pi\)
−0.367320 + 0.930095i \(0.619724\pi\)
\(180\) 0 0
\(181\) 17.8687 1.32817 0.664085 0.747657i \(-0.268821\pi\)
0.664085 + 0.747657i \(0.268821\pi\)
\(182\) − 15.2002i − 1.12671i
\(183\) 10.6137i 0.784588i
\(184\) −18.5237 −1.36559
\(185\) 0 0
\(186\) −1.34519 −0.0986340
\(187\) 13.6518i 0.998322i
\(188\) 1.33928i 0.0976773i
\(189\) 1.59580 0.116077
\(190\) 0 0
\(191\) 0.325813 0.0235750 0.0117875 0.999931i \(-0.496248\pi\)
0.0117875 + 0.999931i \(0.496248\pi\)
\(192\) − 8.57279i − 0.618688i
\(193\) − 2.90187i − 0.208881i −0.994531 0.104441i \(-0.966695\pi\)
0.994531 0.104441i \(-0.0333052\pi\)
\(194\) −2.30441 −0.165447
\(195\) 0 0
\(196\) 0.780505 0.0557503
\(197\) 18.1220i 1.29114i 0.763702 + 0.645568i \(0.223380\pi\)
−0.763702 + 0.645568i \(0.776620\pi\)
\(198\) 4.50200i 0.319943i
\(199\) −1.53256 −0.108640 −0.0543201 0.998524i \(-0.517299\pi\)
−0.0543201 + 0.998524i \(0.517299\pi\)
\(200\) 0 0
\(201\) 15.2159 1.07325
\(202\) − 17.7967i − 1.25217i
\(203\) 4.44202i 0.311769i
\(204\) −0.717905 −0.0502634
\(205\) 0 0
\(206\) −14.6862 −1.02324
\(207\) 6.30400i 0.438158i
\(208\) 25.5171i 1.76929i
\(209\) −1.89146 −0.130835
\(210\) 0 0
\(211\) −11.3698 −0.782727 −0.391363 0.920236i \(-0.627996\pi\)
−0.391363 + 0.920236i \(0.627996\pi\)
\(212\) − 0.0917718i − 0.00630291i
\(213\) − 10.6639i − 0.730681i
\(214\) 12.6605 0.865451
\(215\) 0 0
\(216\) −2.93840 −0.199933
\(217\) − 1.58913i − 0.107877i
\(218\) − 21.0593i − 1.42632i
\(219\) 5.55832 0.375596
\(220\) 0 0
\(221\) 28.8839 1.94294
\(222\) − 4.79996i − 0.322152i
\(223\) 16.9507i 1.13510i 0.823339 + 0.567550i \(0.192109\pi\)
−0.823339 + 0.567550i \(0.807891\pi\)
\(224\) 1.57740 0.105395
\(225\) 0 0
\(226\) 13.5732 0.902878
\(227\) − 14.1117i − 0.936627i −0.883562 0.468314i \(-0.844862\pi\)
0.883562 0.468314i \(-0.155138\pi\)
\(228\) − 0.0994657i − 0.00658728i
\(229\) −0.0619945 −0.00409671 −0.00204835 0.999998i \(-0.500652\pi\)
−0.00204835 + 0.999998i \(0.500652\pi\)
\(230\) 0 0
\(231\) −5.31842 −0.349926
\(232\) − 8.17927i − 0.536995i
\(233\) − 26.0191i − 1.70457i −0.523081 0.852283i \(-0.675218\pi\)
0.523081 0.852283i \(-0.324782\pi\)
\(234\) 9.52513 0.622677
\(235\) 0 0
\(236\) −0.172455 −0.0112258
\(237\) 14.5969i 0.948168i
\(238\) 8.83008i 0.572369i
\(239\) −19.4970 −1.26116 −0.630579 0.776125i \(-0.717183\pi\)
−0.630579 + 0.776125i \(0.717183\pi\)
\(240\) 0 0
\(241\) −4.09860 −0.264014 −0.132007 0.991249i \(-0.542142\pi\)
−0.132007 + 0.991249i \(0.542142\pi\)
\(242\) − 0.145007i − 0.00932141i
\(243\) 1.00000i 0.0641500i
\(244\) 1.86015 0.119084
\(245\) 0 0
\(246\) 1.57091 0.100158
\(247\) 4.00187i 0.254633i
\(248\) 2.92613i 0.185810i
\(249\) 5.02398 0.318382
\(250\) 0 0
\(251\) −1.02933 −0.0649704 −0.0324852 0.999472i \(-0.510342\pi\)
−0.0324852 + 0.999472i \(0.510342\pi\)
\(252\) − 0.279678i − 0.0176181i
\(253\) − 21.0098i − 1.32087i
\(254\) 1.31947 0.0827909
\(255\) 0 0
\(256\) −4.17298 −0.260811
\(257\) 18.5597i 1.15772i 0.815426 + 0.578862i \(0.196503\pi\)
−0.815426 + 0.578862i \(0.803497\pi\)
\(258\) − 0.158076i − 0.00984140i
\(259\) 5.67041 0.352342
\(260\) 0 0
\(261\) −2.78357 −0.172299
\(262\) − 13.5915i − 0.839686i
\(263\) 12.3938i 0.764231i 0.924114 + 0.382116i \(0.124804\pi\)
−0.924114 + 0.382116i \(0.875196\pi\)
\(264\) 9.79302 0.602719
\(265\) 0 0
\(266\) −1.22341 −0.0750119
\(267\) − 2.82350i − 0.172796i
\(268\) − 2.66673i − 0.162897i
\(269\) 5.30032 0.323166 0.161583 0.986859i \(-0.448340\pi\)
0.161583 + 0.986859i \(0.448340\pi\)
\(270\) 0 0
\(271\) 0.797428 0.0484403 0.0242201 0.999707i \(-0.492290\pi\)
0.0242201 + 0.999707i \(0.492290\pi\)
\(272\) − 14.8234i − 0.898798i
\(273\) 11.2525i 0.681031i
\(274\) 6.58184 0.397624
\(275\) 0 0
\(276\) 1.10483 0.0665032
\(277\) 4.74425i 0.285054i 0.989791 + 0.142527i \(0.0455228\pi\)
−0.989791 + 0.142527i \(0.954477\pi\)
\(278\) 0.250962i 0.0150517i
\(279\) 0.995824 0.0596184
\(280\) 0 0
\(281\) −18.7398 −1.11792 −0.558961 0.829194i \(-0.688800\pi\)
−0.558961 + 0.829194i \(0.688800\pi\)
\(282\) 10.3227i 0.614707i
\(283\) 11.7612i 0.699130i 0.936912 + 0.349565i \(0.113671\pi\)
−0.936912 + 0.349565i \(0.886329\pi\)
\(284\) −1.86895 −0.110902
\(285\) 0 0
\(286\) −31.7451 −1.87712
\(287\) 1.85579i 0.109544i
\(288\) 0.988473i 0.0582463i
\(289\) 0.220750 0.0129853
\(290\) 0 0
\(291\) 1.70592 0.100003
\(292\) − 0.974146i − 0.0570076i
\(293\) 22.2819i 1.30172i 0.759198 + 0.650860i \(0.225592\pi\)
−0.759198 + 0.650860i \(0.774408\pi\)
\(294\) 6.01583 0.350850
\(295\) 0 0
\(296\) −10.4411 −0.606879
\(297\) − 3.33277i − 0.193387i
\(298\) 5.25322i 0.304311i
\(299\) −44.4515 −2.57070
\(300\) 0 0
\(301\) 0.186743 0.0107637
\(302\) 29.8730i 1.71900i
\(303\) 13.1747i 0.756865i
\(304\) 2.05378 0.117792
\(305\) 0 0
\(306\) −5.53333 −0.316320
\(307\) 15.3063i 0.873574i 0.899565 + 0.436787i \(0.143884\pi\)
−0.899565 + 0.436787i \(0.856116\pi\)
\(308\) 0.932102i 0.0531115i
\(309\) 10.8720 0.618486
\(310\) 0 0
\(311\) −12.8545 −0.728913 −0.364456 0.931220i \(-0.618745\pi\)
−0.364456 + 0.931220i \(0.618745\pi\)
\(312\) − 20.7196i − 1.17302i
\(313\) − 10.5072i − 0.593901i −0.954893 0.296951i \(-0.904030\pi\)
0.954893 0.296951i \(-0.0959697\pi\)
\(314\) −18.3791 −1.03720
\(315\) 0 0
\(316\) 2.55823 0.143912
\(317\) 19.4806i 1.09414i 0.837086 + 0.547071i \(0.184257\pi\)
−0.837086 + 0.547071i \(0.815743\pi\)
\(318\) − 0.707341i − 0.0396657i
\(319\) 9.27701 0.519413
\(320\) 0 0
\(321\) −9.37236 −0.523114
\(322\) − 13.5892i − 0.757298i
\(323\) − 2.32476i − 0.129353i
\(324\) 0.175259 0.00973662
\(325\) 0 0
\(326\) 11.6562 0.645579
\(327\) 15.5899i 0.862125i
\(328\) − 3.41714i − 0.188680i
\(329\) −12.1947 −0.672313
\(330\) 0 0
\(331\) −14.4925 −0.796579 −0.398289 0.917260i \(-0.630396\pi\)
−0.398289 + 0.917260i \(0.630396\pi\)
\(332\) − 0.880498i − 0.0483236i
\(333\) 3.55334i 0.194722i
\(334\) −8.91400 −0.487753
\(335\) 0 0
\(336\) 5.77482 0.315042
\(337\) − 9.33225i − 0.508360i −0.967157 0.254180i \(-0.918194\pi\)
0.967157 0.254180i \(-0.0818056\pi\)
\(338\) 49.6039i 2.69810i
\(339\) −10.0481 −0.545737
\(340\) 0 0
\(341\) −3.31885 −0.179726
\(342\) − 0.766643i − 0.0414553i
\(343\) 18.2774i 0.986884i
\(344\) −0.343857 −0.0185395
\(345\) 0 0
\(346\) 18.4517 0.991968
\(347\) 1.05341i 0.0565499i 0.999600 + 0.0282750i \(0.00900140\pi\)
−0.999600 + 0.0282750i \(0.990999\pi\)
\(348\) 0.487847i 0.0261513i
\(349\) 13.0715 0.699700 0.349850 0.936806i \(-0.386233\pi\)
0.349850 + 0.936806i \(0.386233\pi\)
\(350\) 0 0
\(351\) −7.05132 −0.376371
\(352\) − 3.29435i − 0.175590i
\(353\) 33.9473i 1.80683i 0.428767 + 0.903415i \(0.358948\pi\)
−0.428767 + 0.903415i \(0.641052\pi\)
\(354\) −1.32921 −0.0706469
\(355\) 0 0
\(356\) −0.494845 −0.0262267
\(357\) − 6.53678i − 0.345963i
\(358\) 13.2770i 0.701713i
\(359\) 6.09450 0.321656 0.160828 0.986982i \(-0.448584\pi\)
0.160828 + 0.986982i \(0.448584\pi\)
\(360\) 0 0
\(361\) −18.6779 −0.983048
\(362\) − 24.1376i − 1.26864i
\(363\) 0.107347i 0.00563424i
\(364\) 1.97210 0.103366
\(365\) 0 0
\(366\) 14.3373 0.749423
\(367\) 21.8636i 1.14127i 0.821203 + 0.570636i \(0.193303\pi\)
−0.821203 + 0.570636i \(0.806697\pi\)
\(368\) 22.8127i 1.18919i
\(369\) −1.16293 −0.0605395
\(370\) 0 0
\(371\) 0.835615 0.0433830
\(372\) − 0.174527i − 0.00904882i
\(373\) − 24.4559i − 1.26628i −0.774037 0.633140i \(-0.781766\pi\)
0.774037 0.633140i \(-0.218234\pi\)
\(374\) 18.4413 0.953578
\(375\) 0 0
\(376\) 22.4545 1.15800
\(377\) − 19.6279i − 1.01089i
\(378\) − 2.15565i − 0.110875i
\(379\) 6.27821 0.322490 0.161245 0.986914i \(-0.448449\pi\)
0.161245 + 0.986914i \(0.448449\pi\)
\(380\) 0 0
\(381\) −0.976784 −0.0500422
\(382\) − 0.440118i − 0.0225184i
\(383\) 24.6876i 1.26148i 0.775996 + 0.630738i \(0.217248\pi\)
−0.775996 + 0.630738i \(0.782752\pi\)
\(384\) −9.60343 −0.490073
\(385\) 0 0
\(386\) −3.91993 −0.199519
\(387\) 0.117022i 0.00594854i
\(388\) − 0.298978i − 0.0151783i
\(389\) 12.9236 0.655251 0.327626 0.944808i \(-0.393752\pi\)
0.327626 + 0.944808i \(0.393752\pi\)
\(390\) 0 0
\(391\) 25.8228 1.30591
\(392\) − 13.0860i − 0.660942i
\(393\) 10.0616i 0.507541i
\(394\) 24.4797 1.23327
\(395\) 0 0
\(396\) −0.584098 −0.0293520
\(397\) 29.0214i 1.45654i 0.685288 + 0.728272i \(0.259676\pi\)
−0.685288 + 0.728272i \(0.740324\pi\)
\(398\) 2.07022i 0.103771i
\(399\) 0.905670 0.0453402
\(400\) 0 0
\(401\) 23.3926 1.16817 0.584084 0.811693i \(-0.301454\pi\)
0.584084 + 0.811693i \(0.301454\pi\)
\(402\) − 20.5541i − 1.02515i
\(403\) 7.02187i 0.349784i
\(404\) 2.30898 0.114876
\(405\) 0 0
\(406\) 6.00041 0.297795
\(407\) − 11.8425i − 0.587009i
\(408\) 12.0364i 0.595892i
\(409\) 15.9918 0.790742 0.395371 0.918521i \(-0.370616\pi\)
0.395371 + 0.918521i \(0.370616\pi\)
\(410\) 0 0
\(411\) −4.87244 −0.240340
\(412\) − 1.90542i − 0.0938731i
\(413\) − 1.57026i − 0.0772675i
\(414\) 8.51563 0.418521
\(415\) 0 0
\(416\) −6.97004 −0.341734
\(417\) − 0.185784i − 0.00909787i
\(418\) 2.55504i 0.124971i
\(419\) −32.3769 −1.58172 −0.790858 0.611999i \(-0.790366\pi\)
−0.790858 + 0.611999i \(0.790366\pi\)
\(420\) 0 0
\(421\) 18.0520 0.879801 0.439900 0.898047i \(-0.355014\pi\)
0.439900 + 0.898047i \(0.355014\pi\)
\(422\) 15.3586i 0.747646i
\(423\) − 7.64173i − 0.371554i
\(424\) −1.53865 −0.0747235
\(425\) 0 0
\(426\) −14.4052 −0.697932
\(427\) 16.9373i 0.819654i
\(428\) 1.64259i 0.0793977i
\(429\) 23.5004 1.13461
\(430\) 0 0
\(431\) 33.1353 1.59607 0.798035 0.602612i \(-0.205873\pi\)
0.798035 + 0.602612i \(0.205873\pi\)
\(432\) 3.61877i 0.174108i
\(433\) − 22.6653i − 1.08922i −0.838688 0.544612i \(-0.816677\pi\)
0.838688 0.544612i \(-0.183323\pi\)
\(434\) −2.14665 −0.103042
\(435\) 0 0
\(436\) 2.73228 0.130852
\(437\) 3.57774i 0.171147i
\(438\) − 7.50834i − 0.358762i
\(439\) 8.50436 0.405891 0.202945 0.979190i \(-0.434949\pi\)
0.202945 + 0.979190i \(0.434949\pi\)
\(440\) 0 0
\(441\) −4.45343 −0.212068
\(442\) − 39.0173i − 1.85586i
\(443\) 6.35768i 0.302063i 0.988529 + 0.151031i \(0.0482594\pi\)
−0.988529 + 0.151031i \(0.951741\pi\)
\(444\) 0.622755 0.0295547
\(445\) 0 0
\(446\) 22.8975 1.08423
\(447\) − 3.88889i − 0.183938i
\(448\) − 13.6804i − 0.646340i
\(449\) 6.25726 0.295298 0.147649 0.989040i \(-0.452829\pi\)
0.147649 + 0.989040i \(0.452829\pi\)
\(450\) 0 0
\(451\) 3.87576 0.182502
\(452\) 1.76102i 0.0828313i
\(453\) − 22.1146i − 1.03903i
\(454\) −19.0625 −0.894648
\(455\) 0 0
\(456\) −1.66765 −0.0780947
\(457\) 11.0441i 0.516620i 0.966062 + 0.258310i \(0.0831657\pi\)
−0.966062 + 0.258310i \(0.916834\pi\)
\(458\) 0.0837440i 0.00391310i
\(459\) 4.09625 0.191196
\(460\) 0 0
\(461\) −23.6622 −1.10206 −0.551029 0.834486i \(-0.685765\pi\)
−0.551029 + 0.834486i \(0.685765\pi\)
\(462\) 7.18428i 0.334243i
\(463\) − 6.22442i − 0.289273i −0.989485 0.144637i \(-0.953799\pi\)
0.989485 0.144637i \(-0.0462013\pi\)
\(464\) −10.0731 −0.467632
\(465\) 0 0
\(466\) −35.1473 −1.62817
\(467\) − 4.94679i − 0.228910i −0.993428 0.114455i \(-0.963488\pi\)
0.993428 0.114455i \(-0.0365122\pi\)
\(468\) 1.23581i 0.0571252i
\(469\) 24.2815 1.12122
\(470\) 0 0
\(471\) 13.6058 0.626923
\(472\) 2.89138i 0.133087i
\(473\) − 0.390006i − 0.0179325i
\(474\) 19.7179 0.905671
\(475\) 0 0
\(476\) −1.14563 −0.0525099
\(477\) 0.523635i 0.0239756i
\(478\) 26.3372i 1.20463i
\(479\) 30.0898 1.37484 0.687419 0.726261i \(-0.258744\pi\)
0.687419 + 0.726261i \(0.258744\pi\)
\(480\) 0 0
\(481\) −25.0557 −1.14244
\(482\) 5.53651i 0.252181i
\(483\) 10.0599i 0.457742i
\(484\) 0.0188135 0.000855159 0
\(485\) 0 0
\(486\) 1.35083 0.0612749
\(487\) 34.2499i 1.55201i 0.630726 + 0.776006i \(0.282757\pi\)
−0.630726 + 0.776006i \(0.717243\pi\)
\(488\) − 31.1874i − 1.41179i
\(489\) −8.62895 −0.390215
\(490\) 0 0
\(491\) −10.4193 −0.470218 −0.235109 0.971969i \(-0.575545\pi\)
−0.235109 + 0.971969i \(0.575545\pi\)
\(492\) 0.203813i 0.00918861i
\(493\) 11.4022i 0.513530i
\(494\) 5.40584 0.243220
\(495\) 0 0
\(496\) 3.60365 0.161809
\(497\) − 17.0175i − 0.763338i
\(498\) − 6.78654i − 0.304112i
\(499\) −8.83514 −0.395515 −0.197757 0.980251i \(-0.563366\pi\)
−0.197757 + 0.980251i \(0.563366\pi\)
\(500\) 0 0
\(501\) 6.59891 0.294818
\(502\) 1.39044i 0.0620585i
\(503\) 21.3734i 0.952994i 0.879176 + 0.476497i \(0.158094\pi\)
−0.879176 + 0.476497i \(0.841906\pi\)
\(504\) −4.68910 −0.208869
\(505\) 0 0
\(506\) −28.3806 −1.26167
\(507\) − 36.7211i − 1.63084i
\(508\) 0.171190i 0.00759535i
\(509\) 16.7800 0.743759 0.371879 0.928281i \(-0.378714\pi\)
0.371879 + 0.928281i \(0.378714\pi\)
\(510\) 0 0
\(511\) 8.86994 0.392383
\(512\) 24.8438i 1.09795i
\(513\) 0.567535i 0.0250573i
\(514\) 25.0710 1.10584
\(515\) 0 0
\(516\) 0.0205091 0.000902863 0
\(517\) 25.4681i 1.12009i
\(518\) − 7.65976i − 0.336550i
\(519\) −13.6595 −0.599586
\(520\) 0 0
\(521\) −4.60508 −0.201752 −0.100876 0.994899i \(-0.532165\pi\)
−0.100876 + 0.994899i \(0.532165\pi\)
\(522\) 3.76013i 0.164577i
\(523\) 12.9634i 0.566852i 0.958994 + 0.283426i \(0.0914710\pi\)
−0.958994 + 0.283426i \(0.908529\pi\)
\(524\) 1.76339 0.0770340
\(525\) 0 0
\(526\) 16.7418 0.729979
\(527\) − 4.07914i − 0.177690i
\(528\) − 12.0605i − 0.524866i
\(529\) −16.7405 −0.727846
\(530\) 0 0
\(531\) 0.983998 0.0427019
\(532\) − 0.158727i − 0.00688169i
\(533\) − 8.20015i − 0.355188i
\(534\) −3.81407 −0.165051
\(535\) 0 0
\(536\) −44.7106 −1.93120
\(537\) − 9.82880i − 0.424144i
\(538\) − 7.15983i − 0.308682i
\(539\) 14.8423 0.639301
\(540\) 0 0
\(541\) −27.3641 −1.17647 −0.588237 0.808688i \(-0.700178\pi\)
−0.588237 + 0.808688i \(0.700178\pi\)
\(542\) − 1.07719i − 0.0462692i
\(543\) 17.8687i 0.766819i
\(544\) 4.04903 0.173601
\(545\) 0 0
\(546\) 15.2002 0.650507
\(547\) − 27.6453i − 1.18203i −0.806661 0.591015i \(-0.798728\pi\)
0.806661 0.591015i \(-0.201272\pi\)
\(548\) 0.853941i 0.0364785i
\(549\) −10.6137 −0.452982
\(550\) 0 0
\(551\) −1.57978 −0.0673007
\(552\) − 18.5237i − 0.788422i
\(553\) 23.2936i 0.990545i
\(554\) 6.40868 0.272279
\(555\) 0 0
\(556\) −0.0325603 −0.00138086
\(557\) 6.17333i 0.261572i 0.991411 + 0.130786i \(0.0417501\pi\)
−0.991411 + 0.130786i \(0.958250\pi\)
\(558\) − 1.34519i − 0.0569464i
\(559\) −0.825156 −0.0349004
\(560\) 0 0
\(561\) −13.6518 −0.576381
\(562\) 25.3143i 1.06782i
\(563\) 5.69934i 0.240198i 0.992762 + 0.120099i \(0.0383213\pi\)
−0.992762 + 0.120099i \(0.961679\pi\)
\(564\) −1.33928 −0.0563940
\(565\) 0 0
\(566\) 15.8874 0.667796
\(567\) 1.59580i 0.0670172i
\(568\) 31.3350i 1.31479i
\(569\) −20.1708 −0.845605 −0.422802 0.906222i \(-0.638954\pi\)
−0.422802 + 0.906222i \(0.638954\pi\)
\(570\) 0 0
\(571\) −15.7554 −0.659341 −0.329671 0.944096i \(-0.606938\pi\)
−0.329671 + 0.944096i \(0.606938\pi\)
\(572\) − 4.11866i − 0.172210i
\(573\) 0.325813i 0.0136111i
\(574\) 2.50686 0.104634
\(575\) 0 0
\(576\) 8.57279 0.357200
\(577\) 16.1062i 0.670509i 0.942128 + 0.335255i \(0.108822\pi\)
−0.942128 + 0.335255i \(0.891178\pi\)
\(578\) − 0.298195i − 0.0124033i
\(579\) 2.90187 0.120598
\(580\) 0 0
\(581\) 8.01725 0.332611
\(582\) − 2.30441i − 0.0955207i
\(583\) − 1.74515i − 0.0722769i
\(584\) −16.3326 −0.675847
\(585\) 0 0
\(586\) 30.0990 1.24338
\(587\) − 2.00072i − 0.0825786i −0.999147 0.0412893i \(-0.986853\pi\)
0.999147 0.0412893i \(-0.0131465\pi\)
\(588\) 0.780505i 0.0321875i
\(589\) 0.565165 0.0232872
\(590\) 0 0
\(591\) −18.1220 −0.745438
\(592\) 12.8587i 0.528489i
\(593\) − 26.8231i − 1.10149i −0.834672 0.550747i \(-0.814343\pi\)
0.834672 0.550747i \(-0.185657\pi\)
\(594\) −4.50200 −0.184719
\(595\) 0 0
\(596\) −0.681563 −0.0279179
\(597\) − 1.53256i − 0.0627234i
\(598\) 60.0464i 2.45548i
\(599\) 44.8025 1.83058 0.915290 0.402796i \(-0.131962\pi\)
0.915290 + 0.402796i \(0.131962\pi\)
\(600\) 0 0
\(601\) −14.2298 −0.580446 −0.290223 0.956959i \(-0.593729\pi\)
−0.290223 + 0.956959i \(0.593729\pi\)
\(602\) − 0.252258i − 0.0102813i
\(603\) 15.2159i 0.619641i
\(604\) −3.87578 −0.157703
\(605\) 0 0
\(606\) 17.7967 0.722943
\(607\) 20.3346i 0.825356i 0.910877 + 0.412678i \(0.135406\pi\)
−0.910877 + 0.412678i \(0.864594\pi\)
\(608\) 0.560993i 0.0227513i
\(609\) −4.44202 −0.180000
\(610\) 0 0
\(611\) 53.8843 2.17993
\(612\) − 0.717905i − 0.0290196i
\(613\) − 20.7921i − 0.839783i −0.907574 0.419892i \(-0.862068\pi\)
0.907574 0.419892i \(-0.137932\pi\)
\(614\) 20.6761 0.834421
\(615\) 0 0
\(616\) 15.6277 0.629657
\(617\) − 4.94634i − 0.199132i −0.995031 0.0995660i \(-0.968255\pi\)
0.995031 0.0995660i \(-0.0317454\pi\)
\(618\) − 14.6862i − 0.590766i
\(619\) 45.9527 1.84700 0.923498 0.383603i \(-0.125317\pi\)
0.923498 + 0.383603i \(0.125317\pi\)
\(620\) 0 0
\(621\) −6.30400 −0.252971
\(622\) 17.3643i 0.696243i
\(623\) − 4.50574i − 0.180519i
\(624\) −25.5171 −1.02150
\(625\) 0 0
\(626\) −14.1934 −0.567283
\(627\) − 1.89146i − 0.0755377i
\(628\) − 2.38454i − 0.0951537i
\(629\) 14.5554 0.580360
\(630\) 0 0
\(631\) −21.6636 −0.862414 −0.431207 0.902253i \(-0.641912\pi\)
−0.431207 + 0.902253i \(0.641912\pi\)
\(632\) − 42.8915i − 1.70613i
\(633\) − 11.3698i − 0.451908i
\(634\) 26.3150 1.04510
\(635\) 0 0
\(636\) 0.0917718 0.00363899
\(637\) − 31.4026i − 1.24421i
\(638\) − 12.5317i − 0.496133i
\(639\) 10.6639 0.421859
\(640\) 0 0
\(641\) 36.3870 1.43720 0.718600 0.695424i \(-0.244783\pi\)
0.718600 + 0.695424i \(0.244783\pi\)
\(642\) 12.6605i 0.499668i
\(643\) − 1.01349i − 0.0399682i −0.999800 0.0199841i \(-0.993638\pi\)
0.999800 0.0199841i \(-0.00636156\pi\)
\(644\) 1.76309 0.0694755
\(645\) 0 0
\(646\) −3.14036 −0.123556
\(647\) − 13.7007i − 0.538629i −0.963052 0.269315i \(-0.913203\pi\)
0.963052 0.269315i \(-0.0867972\pi\)
\(648\) − 2.93840i − 0.115431i
\(649\) −3.27944 −0.128729
\(650\) 0 0
\(651\) 1.58913 0.0622830
\(652\) 1.51230i 0.0592263i
\(653\) − 23.8372i − 0.932822i −0.884568 0.466411i \(-0.845547\pi\)
0.884568 0.466411i \(-0.154453\pi\)
\(654\) 21.0593 0.823485
\(655\) 0 0
\(656\) −4.20835 −0.164309
\(657\) 5.55832i 0.216851i
\(658\) 16.4729i 0.642181i
\(659\) −14.7758 −0.575583 −0.287791 0.957693i \(-0.592921\pi\)
−0.287791 + 0.957693i \(0.592921\pi\)
\(660\) 0 0
\(661\) 7.55471 0.293844 0.146922 0.989148i \(-0.453063\pi\)
0.146922 + 0.989148i \(0.453063\pi\)
\(662\) 19.5769i 0.760877i
\(663\) 28.8839i 1.12176i
\(664\) −14.7625 −0.572895
\(665\) 0 0
\(666\) 4.79996 0.185995
\(667\) − 17.5477i − 0.679448i
\(668\) − 1.15652i − 0.0447471i
\(669\) −16.9507 −0.655351
\(670\) 0 0
\(671\) 35.3730 1.36556
\(672\) 1.57740i 0.0608496i
\(673\) − 12.1947i − 0.470069i −0.971987 0.235035i \(-0.924480\pi\)
0.971987 0.235035i \(-0.0755204\pi\)
\(674\) −12.6063 −0.485576
\(675\) 0 0
\(676\) −6.43571 −0.247527
\(677\) − 17.9507i − 0.689900i −0.938621 0.344950i \(-0.887896\pi\)
0.938621 0.344950i \(-0.112104\pi\)
\(678\) 13.5732i 0.521277i
\(679\) 2.72230 0.104472
\(680\) 0 0
\(681\) 14.1117 0.540762
\(682\) 4.48320i 0.171671i
\(683\) 10.0273i 0.383685i 0.981426 + 0.191842i \(0.0614462\pi\)
−0.981426 + 0.191842i \(0.938554\pi\)
\(684\) 0.0994657 0.00380317
\(685\) 0 0
\(686\) 24.6896 0.942653
\(687\) − 0.0619945i − 0.00236524i
\(688\) 0.423474i 0.0161448i
\(689\) −3.69231 −0.140666
\(690\) 0 0
\(691\) 19.1296 0.727722 0.363861 0.931453i \(-0.381458\pi\)
0.363861 + 0.931453i \(0.381458\pi\)
\(692\) 2.39396i 0.0910045i
\(693\) − 5.31842i − 0.202030i
\(694\) 1.42297 0.0540154
\(695\) 0 0
\(696\) 8.17927 0.310034
\(697\) 4.76363i 0.180435i
\(698\) − 17.6573i − 0.668340i
\(699\) 26.0191 0.984131
\(700\) 0 0
\(701\) −3.81920 −0.144249 −0.0721246 0.997396i \(-0.522978\pi\)
−0.0721246 + 0.997396i \(0.522978\pi\)
\(702\) 9.52513i 0.359503i
\(703\) 2.01664i 0.0760592i
\(704\) −28.5711 −1.07681
\(705\) 0 0
\(706\) 45.8570 1.72585
\(707\) 21.0241i 0.790692i
\(708\) − 0.172455i − 0.00648124i
\(709\) −38.0535 −1.42913 −0.714565 0.699569i \(-0.753375\pi\)
−0.714565 + 0.699569i \(0.753375\pi\)
\(710\) 0 0
\(711\) −14.5969 −0.547425
\(712\) 8.29660i 0.310928i
\(713\) 6.27768i 0.235101i
\(714\) −8.83008 −0.330457
\(715\) 0 0
\(716\) −1.72259 −0.0643761
\(717\) − 19.4970i − 0.728130i
\(718\) − 8.23264i − 0.307239i
\(719\) −19.4340 −0.724766 −0.362383 0.932029i \(-0.618037\pi\)
−0.362383 + 0.932029i \(0.618037\pi\)
\(720\) 0 0
\(721\) 17.3495 0.646129
\(722\) 25.2307i 0.938988i
\(723\) − 4.09860i − 0.152429i
\(724\) 3.13165 0.116387
\(725\) 0 0
\(726\) 0.145007 0.00538172
\(727\) 38.0739i 1.41208i 0.708170 + 0.706042i \(0.249521\pi\)
−0.708170 + 0.706042i \(0.750479\pi\)
\(728\) − 33.0643i − 1.22544i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0.479350 0.0177294
\(732\) 1.86015i 0.0687531i
\(733\) 16.9591i 0.626400i 0.949687 + 0.313200i \(0.101401\pi\)
−0.949687 + 0.313200i \(0.898599\pi\)
\(734\) 29.5341 1.09012
\(735\) 0 0
\(736\) −6.23134 −0.229690
\(737\) − 50.7112i − 1.86797i
\(738\) 1.57091i 0.0578261i
\(739\) −13.2040 −0.485716 −0.242858 0.970062i \(-0.578085\pi\)
−0.242858 + 0.970062i \(0.578085\pi\)
\(740\) 0 0
\(741\) −4.00187 −0.147012
\(742\) − 1.12877i − 0.0414386i
\(743\) 42.2364i 1.54950i 0.632265 + 0.774752i \(0.282125\pi\)
−0.632265 + 0.774752i \(0.717875\pi\)
\(744\) −2.92613 −0.107277
\(745\) 0 0
\(746\) −33.0358 −1.20953
\(747\) 5.02398i 0.183818i
\(748\) 2.39261i 0.0874825i
\(749\) −14.9564 −0.546494
\(750\) 0 0
\(751\) −1.04801 −0.0382426 −0.0191213 0.999817i \(-0.506087\pi\)
−0.0191213 + 0.999817i \(0.506087\pi\)
\(752\) − 27.6536i − 1.00842i
\(753\) − 1.02933i − 0.0375107i
\(754\) −26.5139 −0.965579
\(755\) 0 0
\(756\) 0.279678 0.0101718
\(757\) − 17.0074i − 0.618146i −0.951038 0.309073i \(-0.899981\pi\)
0.951038 0.309073i \(-0.100019\pi\)
\(758\) − 8.48079i − 0.308036i
\(759\) 21.0098 0.762607
\(760\) 0 0
\(761\) −24.1244 −0.874508 −0.437254 0.899338i \(-0.644049\pi\)
−0.437254 + 0.899338i \(0.644049\pi\)
\(762\) 1.31947i 0.0477993i
\(763\) 24.8783i 0.900657i
\(764\) 0.0571018 0.00206587
\(765\) 0 0
\(766\) 33.3487 1.20494
\(767\) 6.93848i 0.250534i
\(768\) − 4.17298i − 0.150579i
\(769\) −49.7819 −1.79518 −0.897591 0.440830i \(-0.854684\pi\)
−0.897591 + 0.440830i \(0.854684\pi\)
\(770\) 0 0
\(771\) −18.5597 −0.668412
\(772\) − 0.508580i − 0.0183042i
\(773\) − 31.4743i − 1.13205i −0.824388 0.566025i \(-0.808480\pi\)
0.824388 0.566025i \(-0.191520\pi\)
\(774\) 0.158076 0.00568193
\(775\) 0 0
\(776\) −5.01268 −0.179945
\(777\) 5.67041i 0.203425i
\(778\) − 17.4575i − 0.625883i
\(779\) −0.660001 −0.0236470
\(780\) 0 0
\(781\) −35.5404 −1.27174
\(782\) − 34.8822i − 1.24738i
\(783\) − 2.78357i − 0.0994768i
\(784\) −16.1159 −0.575569
\(785\) 0 0
\(786\) 13.5915 0.484793
\(787\) 19.1415i 0.682321i 0.940005 + 0.341161i \(0.110820\pi\)
−0.940005 + 0.341161i \(0.889180\pi\)
\(788\) 3.17604i 0.113142i
\(789\) −12.3938 −0.441229
\(790\) 0 0
\(791\) −16.0347 −0.570128
\(792\) 9.79302i 0.347980i
\(793\) − 74.8406i − 2.65767i
\(794\) 39.2030 1.39126
\(795\) 0 0
\(796\) −0.268595 −0.00952009
\(797\) 7.21437i 0.255546i 0.991803 + 0.127773i \(0.0407829\pi\)
−0.991803 + 0.127773i \(0.959217\pi\)
\(798\) − 1.22341i − 0.0433081i
\(799\) −31.3024 −1.10740
\(800\) 0 0
\(801\) 2.82350 0.0997636
\(802\) − 31.5994i − 1.11581i
\(803\) − 18.5246i − 0.653718i
\(804\) 2.66673 0.0940484
\(805\) 0 0
\(806\) 9.48535 0.334107
\(807\) 5.30032i 0.186580i
\(808\) − 38.7125i − 1.36190i
\(809\) −1.05941 −0.0372469 −0.0186234 0.999827i \(-0.505928\pi\)
−0.0186234 + 0.999827i \(0.505928\pi\)
\(810\) 0 0
\(811\) 21.6400 0.759883 0.379941 0.925011i \(-0.375944\pi\)
0.379941 + 0.925011i \(0.375944\pi\)
\(812\) 0.778504i 0.0273202i
\(813\) 0.797428i 0.0279670i
\(814\) −15.9971 −0.560700
\(815\) 0 0
\(816\) 14.8234 0.518922
\(817\) 0.0664138i 0.00232353i
\(818\) − 21.6022i − 0.755302i
\(819\) −11.2525 −0.393193
\(820\) 0 0
\(821\) −23.1828 −0.809085 −0.404543 0.914519i \(-0.632569\pi\)
−0.404543 + 0.914519i \(0.632569\pi\)
\(822\) 6.58184i 0.229568i
\(823\) − 27.9769i − 0.975214i −0.873063 0.487607i \(-0.837870\pi\)
0.873063 0.487607i \(-0.162130\pi\)
\(824\) −31.9463 −1.11290
\(825\) 0 0
\(826\) −2.12116 −0.0738044
\(827\) 41.3045i 1.43630i 0.695889 + 0.718150i \(0.255011\pi\)
−0.695889 + 0.718150i \(0.744989\pi\)
\(828\) 1.10483i 0.0383956i
\(829\) −47.5687 −1.65213 −0.826064 0.563577i \(-0.809425\pi\)
−0.826064 + 0.563577i \(0.809425\pi\)
\(830\) 0 0
\(831\) −4.74425 −0.164576
\(832\) 60.4495i 2.09571i
\(833\) 18.2424i 0.632060i
\(834\) −0.250962 −0.00869011
\(835\) 0 0
\(836\) −0.331496 −0.0114650
\(837\) 0.995824i 0.0344207i
\(838\) 43.7357i 1.51083i
\(839\) −49.0322 −1.69278 −0.846389 0.532565i \(-0.821228\pi\)
−0.846389 + 0.532565i \(0.821228\pi\)
\(840\) 0 0
\(841\) −21.2517 −0.732818
\(842\) − 24.3852i − 0.840369i
\(843\) − 18.7398i − 0.645433i
\(844\) −1.99266 −0.0685900
\(845\) 0 0
\(846\) −10.3227 −0.354901
\(847\) 0.171304i 0.00588606i
\(848\) 1.89491i 0.0650715i
\(849\) −11.7612 −0.403643
\(850\) 0 0
\(851\) −22.4003 −0.767871
\(852\) − 1.86895i − 0.0640293i
\(853\) − 32.5376i − 1.11407i −0.830490 0.557033i \(-0.811939\pi\)
0.830490 0.557033i \(-0.188061\pi\)
\(854\) 22.8794 0.782918
\(855\) 0 0
\(856\) 27.5398 0.941290
\(857\) 30.4813i 1.04122i 0.853794 + 0.520610i \(0.174296\pi\)
−0.853794 + 0.520610i \(0.825704\pi\)
\(858\) − 31.7451i − 1.08376i
\(859\) −4.12215 −0.140646 −0.0703231 0.997524i \(-0.522403\pi\)
−0.0703231 + 0.997524i \(0.522403\pi\)
\(860\) 0 0
\(861\) −1.85579 −0.0632452
\(862\) − 44.7601i − 1.52453i
\(863\) − 18.4184i − 0.626969i −0.949593 0.313485i \(-0.898504\pi\)
0.949593 0.313485i \(-0.101496\pi\)
\(864\) −0.988473 −0.0336285
\(865\) 0 0
\(866\) −30.6169 −1.04041
\(867\) 0.220750i 0.00749705i
\(868\) − 0.278510i − 0.00945325i
\(869\) 48.6479 1.65027
\(870\) 0 0
\(871\) −107.292 −3.63546
\(872\) − 45.8095i − 1.55131i
\(873\) 1.70592i 0.0577366i
\(874\) 4.83292 0.163476
\(875\) 0 0
\(876\) 0.974146 0.0329133
\(877\) 18.9126i 0.638634i 0.947648 + 0.319317i \(0.103453\pi\)
−0.947648 + 0.319317i \(0.896547\pi\)
\(878\) − 11.4879i − 0.387699i
\(879\) −22.2819 −0.751548
\(880\) 0 0
\(881\) 34.6336 1.16684 0.583418 0.812172i \(-0.301715\pi\)
0.583418 + 0.812172i \(0.301715\pi\)
\(882\) 6.01583i 0.202563i
\(883\) − 32.6874i − 1.10002i −0.835158 0.550009i \(-0.814624\pi\)
0.835158 0.550009i \(-0.185376\pi\)
\(884\) 5.06218 0.170259
\(885\) 0 0
\(886\) 8.58814 0.288524
\(887\) − 48.7918i − 1.63827i −0.573602 0.819134i \(-0.694455\pi\)
0.573602 0.819134i \(-0.305545\pi\)
\(888\) − 10.4411i − 0.350382i
\(889\) −1.55875 −0.0522788
\(890\) 0 0
\(891\) 3.33277 0.111652
\(892\) 2.97076i 0.0994684i
\(893\) − 4.33695i − 0.145131i
\(894\) −5.25322 −0.175694
\(895\) 0 0
\(896\) −15.3251 −0.511977
\(897\) − 44.4515i − 1.48419i
\(898\) − 8.45249i − 0.282063i
\(899\) −2.77195 −0.0924497
\(900\) 0 0
\(901\) 2.14494 0.0714582
\(902\) − 5.23549i − 0.174323i
\(903\) 0.186743i 0.00621441i
\(904\) 29.5253 0.981997
\(905\) 0 0
\(906\) −29.8730 −0.992465
\(907\) − 45.0367i − 1.49542i −0.664025 0.747710i \(-0.731153\pi\)
0.664025 0.747710i \(-0.268847\pi\)
\(908\) − 2.47321i − 0.0820762i
\(909\) −13.1747 −0.436976
\(910\) 0 0
\(911\) 12.9449 0.428882 0.214441 0.976737i \(-0.431207\pi\)
0.214441 + 0.976737i \(0.431207\pi\)
\(912\) 2.05378i 0.0680073i
\(913\) − 16.7438i − 0.554137i
\(914\) 14.9187 0.493466
\(915\) 0 0
\(916\) −0.0108651 −0.000358993 0
\(917\) 16.0563i 0.530225i
\(918\) − 5.53333i − 0.182627i
\(919\) −14.5719 −0.480682 −0.240341 0.970689i \(-0.577259\pi\)
−0.240341 + 0.970689i \(0.577259\pi\)
\(920\) 0 0
\(921\) −15.3063 −0.504358
\(922\) 31.9636i 1.05266i
\(923\) 75.1948i 2.47507i
\(924\) −0.932102 −0.0306639
\(925\) 0 0
\(926\) −8.40813 −0.276308
\(927\) 10.8720i 0.357083i
\(928\) − 2.75149i − 0.0903220i
\(929\) 31.3591 1.02886 0.514430 0.857532i \(-0.328004\pi\)
0.514430 + 0.857532i \(0.328004\pi\)
\(930\) 0 0
\(931\) −2.52748 −0.0828347
\(932\) − 4.56008i − 0.149370i
\(933\) − 12.8545i − 0.420838i
\(934\) −6.68227 −0.218651
\(935\) 0 0
\(936\) 20.7196 0.677242
\(937\) 48.7726i 1.59333i 0.604420 + 0.796666i \(0.293405\pi\)
−0.604420 + 0.796666i \(0.706595\pi\)
\(938\) − 32.8002i − 1.07097i
\(939\) 10.5072 0.342889
\(940\) 0 0
\(941\) −32.2359 −1.05086 −0.525431 0.850836i \(-0.676096\pi\)
−0.525431 + 0.850836i \(0.676096\pi\)
\(942\) − 18.3791i − 0.598825i
\(943\) − 7.33108i − 0.238733i
\(944\) 3.56086 0.115896
\(945\) 0 0
\(946\) −0.526832 −0.0171288
\(947\) 51.9235i 1.68729i 0.536903 + 0.843644i \(0.319594\pi\)
−0.536903 + 0.843644i \(0.680406\pi\)
\(948\) 2.55823i 0.0830875i
\(949\) −39.1935 −1.27227
\(950\) 0 0
\(951\) −19.4806 −0.631703
\(952\) 19.2077i 0.622525i
\(953\) − 39.7788i − 1.28856i −0.764789 0.644281i \(-0.777157\pi\)
0.764789 0.644281i \(-0.222843\pi\)
\(954\) 0.707341 0.0229010
\(955\) 0 0
\(956\) −3.41703 −0.110515
\(957\) 9.27701i 0.299883i
\(958\) − 40.6462i − 1.31322i
\(959\) −7.77543 −0.251082
\(960\) 0 0
\(961\) −30.0083 −0.968011
\(962\) 33.8460i 1.09124i
\(963\) − 9.37236i − 0.302020i
\(964\) −0.718317 −0.0231354
\(965\) 0 0
\(966\) 13.5892 0.437226
\(967\) 19.8817i 0.639353i 0.947527 + 0.319677i \(0.103574\pi\)
−0.947527 + 0.319677i \(0.896426\pi\)
\(968\) − 0.315428i − 0.0101382i
\(969\) 2.32476 0.0746822
\(970\) 0 0
\(971\) −41.3841 −1.32808 −0.664039 0.747698i \(-0.731159\pi\)
−0.664039 + 0.747698i \(0.731159\pi\)
\(972\) 0.175259i 0.00562144i
\(973\) − 0.296473i − 0.00950449i
\(974\) 46.2658 1.48245
\(975\) 0 0
\(976\) −38.4085 −1.22943
\(977\) 3.61991i 0.115811i 0.998322 + 0.0579056i \(0.0184422\pi\)
−0.998322 + 0.0579056i \(0.981558\pi\)
\(978\) 11.6562i 0.372725i
\(979\) −9.41009 −0.300748
\(980\) 0 0
\(981\) −15.5899 −0.497748
\(982\) 14.0747i 0.449143i
\(983\) 20.1099i 0.641405i 0.947180 + 0.320702i \(0.103919\pi\)
−0.947180 + 0.320702i \(0.896081\pi\)
\(984\) 3.41714 0.108935
\(985\) 0 0
\(986\) 15.4024 0.490514
\(987\) − 12.1947i − 0.388160i
\(988\) 0.701364i 0.0223134i
\(989\) −0.737705 −0.0234576
\(990\) 0 0
\(991\) 7.43990 0.236336 0.118168 0.992994i \(-0.462298\pi\)
0.118168 + 0.992994i \(0.462298\pi\)
\(992\) 0.984345i 0.0312530i
\(993\) − 14.4925i − 0.459905i
\(994\) −22.9877 −0.729126
\(995\) 0 0
\(996\) 0.880498 0.0278996
\(997\) − 46.1472i − 1.46150i −0.682648 0.730748i \(-0.739172\pi\)
0.682648 0.730748i \(-0.260828\pi\)
\(998\) 11.9348i 0.377788i
\(999\) −3.55334 −0.112423
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.h.1249.5 16
5.2 odd 4 1875.2.a.m.1.7 8
5.3 odd 4 1875.2.a.p.1.2 8
5.4 even 2 inner 1875.2.b.h.1249.12 16
15.2 even 4 5625.2.a.bd.1.2 8
15.8 even 4 5625.2.a.t.1.7 8
25.2 odd 20 375.2.g.e.226.4 16
25.9 even 10 375.2.i.c.349.2 16
25.11 even 5 375.2.i.c.274.2 16
25.12 odd 20 375.2.g.e.151.4 16
25.13 odd 20 375.2.g.d.151.1 16
25.14 even 10 75.2.i.a.4.3 16
25.16 even 5 75.2.i.a.19.3 yes 16
25.23 odd 20 375.2.g.d.226.1 16
75.14 odd 10 225.2.m.b.154.2 16
75.41 odd 10 225.2.m.b.19.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.i.a.4.3 16 25.14 even 10
75.2.i.a.19.3 yes 16 25.16 even 5
225.2.m.b.19.2 16 75.41 odd 10
225.2.m.b.154.2 16 75.14 odd 10
375.2.g.d.151.1 16 25.13 odd 20
375.2.g.d.226.1 16 25.23 odd 20
375.2.g.e.151.4 16 25.12 odd 20
375.2.g.e.226.4 16 25.2 odd 20
375.2.i.c.274.2 16 25.11 even 5
375.2.i.c.349.2 16 25.9 even 10
1875.2.a.m.1.7 8 5.2 odd 4
1875.2.a.p.1.2 8 5.3 odd 4
1875.2.b.h.1249.5 16 1.1 even 1 trivial
1875.2.b.h.1249.12 16 5.4 even 2 inner
5625.2.a.t.1.7 8 15.8 even 4
5625.2.a.bd.1.2 8 15.2 even 4