Properties

Label 2175.2.c.f.349.3
Level $2175$
Weight $2$
Character 2175.349
Analytic conductor $17.367$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,2,Mod(349,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.c (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: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 11x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.3
Root \(1.79129i\) of defining polynomial
Character \(\chi\) \(=\) 2175.349
Dual form 2175.2.c.f.349.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.79129i q^{2} -1.00000i q^{3} -1.20871 q^{4} +1.79129 q^{6} +1.00000i q^{7} +1.41742i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.79129i q^{2} -1.00000i q^{3} -1.20871 q^{4} +1.79129 q^{6} +1.00000i q^{7} +1.41742i q^{8} -1.00000 q^{9} +5.00000 q^{11} +1.20871i q^{12} +4.58258i q^{13} -1.79129 q^{14} -4.95644 q^{16} -3.00000i q^{17} -1.79129i q^{18} -3.58258 q^{19} +1.00000 q^{21} +8.95644i q^{22} +4.00000i q^{23} +1.41742 q^{24} -8.20871 q^{26} +1.00000i q^{27} -1.20871i q^{28} -1.00000 q^{29} +4.00000 q^{31} -6.04356i q^{32} -5.00000i q^{33} +5.37386 q^{34} +1.20871 q^{36} -4.00000i q^{37} -6.41742i q^{38} +4.58258 q^{39} -9.16515 q^{41} +1.79129i q^{42} +9.58258i q^{43} -6.04356 q^{44} -7.16515 q^{46} +10.5826i q^{47} +4.95644i q^{48} +6.00000 q^{49} -3.00000 q^{51} -5.53901i q^{52} -0.417424i q^{53} -1.79129 q^{54} -1.41742 q^{56} +3.58258i q^{57} -1.79129i q^{58} +7.58258 q^{59} +12.7477 q^{61} +7.16515i q^{62} -1.00000i q^{63} +0.912878 q^{64} +8.95644 q^{66} -4.16515i q^{67} +3.62614i q^{68} +4.00000 q^{69} -9.58258 q^{71} -1.41742i q^{72} -4.00000i q^{73} +7.16515 q^{74} +4.33030 q^{76} +5.00000i q^{77} +8.20871i q^{78} -7.58258 q^{79} +1.00000 q^{81} -16.4174i q^{82} +11.5826i q^{83} -1.20871 q^{84} -17.1652 q^{86} +1.00000i q^{87} +7.08712i q^{88} -1.41742 q^{89} -4.58258 q^{91} -4.83485i q^{92} -4.00000i q^{93} -18.9564 q^{94} -6.04356 q^{96} +11.5826i q^{97} +10.7477i q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 14 q^{4} - 2 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 14 q^{4} - 2 q^{6} - 4 q^{9} + 20 q^{11} + 2 q^{14} + 26 q^{16} + 4 q^{19} + 4 q^{21} + 24 q^{24} - 42 q^{26} - 4 q^{29} + 16 q^{31} - 6 q^{34} + 14 q^{36} - 70 q^{44} + 8 q^{46} + 24 q^{49} - 12 q^{51} + 2 q^{54} - 24 q^{56} + 12 q^{59} - 4 q^{61} - 88 q^{64} - 10 q^{66} + 16 q^{69} - 20 q^{71} - 8 q^{74} - 56 q^{76} - 12 q^{79} + 4 q^{81} - 14 q^{84} - 32 q^{86} - 24 q^{89} - 30 q^{94} - 70 q^{96} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.79129i 1.26663i 0.773893 + 0.633316i \(0.218307\pi\)
−0.773893 + 0.633316i \(0.781693\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −1.20871 −0.604356
\(5\) 0 0
\(6\) 1.79129 0.731290
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 1.41742i 0.501135i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 1.20871i 0.348925i
\(13\) 4.58258i 1.27098i 0.772110 + 0.635489i \(0.219201\pi\)
−0.772110 + 0.635489i \(0.780799\pi\)
\(14\) −1.79129 −0.478742
\(15\) 0 0
\(16\) −4.95644 −1.23911
\(17\) − 3.00000i − 0.727607i −0.931476 0.363803i \(-0.881478\pi\)
0.931476 0.363803i \(-0.118522\pi\)
\(18\) − 1.79129i − 0.422211i
\(19\) −3.58258 −0.821899 −0.410950 0.911658i \(-0.634803\pi\)
−0.410950 + 0.911658i \(0.634803\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 8.95644i 1.90952i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 1.41742 0.289331
\(25\) 0 0
\(26\) −8.20871 −1.60986
\(27\) 1.00000i 0.192450i
\(28\) − 1.20871i − 0.228425i
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) − 6.04356i − 1.06836i
\(33\) − 5.00000i − 0.870388i
\(34\) 5.37386 0.921610
\(35\) 0 0
\(36\) 1.20871 0.201452
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) − 6.41742i − 1.04104i
\(39\) 4.58258 0.733799
\(40\) 0 0
\(41\) −9.16515 −1.43136 −0.715678 0.698430i \(-0.753882\pi\)
−0.715678 + 0.698430i \(0.753882\pi\)
\(42\) 1.79129i 0.276402i
\(43\) 9.58258i 1.46133i 0.682737 + 0.730665i \(0.260790\pi\)
−0.682737 + 0.730665i \(0.739210\pi\)
\(44\) −6.04356 −0.911101
\(45\) 0 0
\(46\) −7.16515 −1.05644
\(47\) 10.5826i 1.54363i 0.635849 + 0.771814i \(0.280650\pi\)
−0.635849 + 0.771814i \(0.719350\pi\)
\(48\) 4.95644i 0.715400i
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) − 5.53901i − 0.768123i
\(53\) − 0.417424i − 0.0573376i −0.999589 0.0286688i \(-0.990873\pi\)
0.999589 0.0286688i \(-0.00912682\pi\)
\(54\) −1.79129 −0.243763
\(55\) 0 0
\(56\) −1.41742 −0.189411
\(57\) 3.58258i 0.474524i
\(58\) − 1.79129i − 0.235208i
\(59\) 7.58258 0.987167 0.493584 0.869698i \(-0.335687\pi\)
0.493584 + 0.869698i \(0.335687\pi\)
\(60\) 0 0
\(61\) 12.7477 1.63218 0.816090 0.577925i \(-0.196138\pi\)
0.816090 + 0.577925i \(0.196138\pi\)
\(62\) 7.16515i 0.909975i
\(63\) − 1.00000i − 0.125988i
\(64\) 0.912878 0.114110
\(65\) 0 0
\(66\) 8.95644 1.10246
\(67\) − 4.16515i − 0.508854i −0.967092 0.254427i \(-0.918113\pi\)
0.967092 0.254427i \(-0.0818869\pi\)
\(68\) 3.62614i 0.439734i
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −9.58258 −1.13724 −0.568621 0.822599i \(-0.692523\pi\)
−0.568621 + 0.822599i \(0.692523\pi\)
\(72\) − 1.41742i − 0.167045i
\(73\) − 4.00000i − 0.468165i −0.972217 0.234082i \(-0.924791\pi\)
0.972217 0.234082i \(-0.0752085\pi\)
\(74\) 7.16515 0.832932
\(75\) 0 0
\(76\) 4.33030 0.496720
\(77\) 5.00000i 0.569803i
\(78\) 8.20871i 0.929454i
\(79\) −7.58258 −0.853106 −0.426553 0.904462i \(-0.640272\pi\)
−0.426553 + 0.904462i \(0.640272\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 16.4174i − 1.81300i
\(83\) 11.5826i 1.27135i 0.771956 + 0.635676i \(0.219279\pi\)
−0.771956 + 0.635676i \(0.780721\pi\)
\(84\) −1.20871 −0.131881
\(85\) 0 0
\(86\) −17.1652 −1.85097
\(87\) 1.00000i 0.107211i
\(88\) 7.08712i 0.755490i
\(89\) −1.41742 −0.150247 −0.0751233 0.997174i \(-0.523935\pi\)
−0.0751233 + 0.997174i \(0.523935\pi\)
\(90\) 0 0
\(91\) −4.58258 −0.480384
\(92\) − 4.83485i − 0.504068i
\(93\) − 4.00000i − 0.414781i
\(94\) −18.9564 −1.95521
\(95\) 0 0
\(96\) −6.04356 −0.616818
\(97\) 11.5826i 1.17603i 0.808849 + 0.588016i \(0.200091\pi\)
−0.808849 + 0.588016i \(0.799909\pi\)
\(98\) 10.7477i 1.08568i
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) −0.582576 −0.0579684 −0.0289842 0.999580i \(-0.509227\pi\)
−0.0289842 + 0.999580i \(0.509227\pi\)
\(102\) − 5.37386i − 0.532092i
\(103\) − 15.1652i − 1.49427i −0.664674 0.747133i \(-0.731430\pi\)
0.664674 0.747133i \(-0.268570\pi\)
\(104\) −6.49545 −0.636932
\(105\) 0 0
\(106\) 0.747727 0.0726257
\(107\) 5.16515i 0.499334i 0.968332 + 0.249667i \(0.0803212\pi\)
−0.968332 + 0.249667i \(0.919679\pi\)
\(108\) − 1.20871i − 0.116308i
\(109\) −14.1652 −1.35678 −0.678388 0.734704i \(-0.737321\pi\)
−0.678388 + 0.734704i \(0.737321\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) − 4.95644i − 0.468339i
\(113\) 14.1652i 1.33255i 0.745708 + 0.666273i \(0.232111\pi\)
−0.745708 + 0.666273i \(0.767889\pi\)
\(114\) −6.41742 −0.601047
\(115\) 0 0
\(116\) 1.20871 0.112226
\(117\) − 4.58258i − 0.423659i
\(118\) 13.5826i 1.25038i
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 22.8348i 2.06737i
\(123\) 9.16515i 0.826394i
\(124\) −4.83485 −0.434182
\(125\) 0 0
\(126\) 1.79129 0.159581
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) − 10.4519i − 0.923826i
\(129\) 9.58258 0.843699
\(130\) 0 0
\(131\) −15.0000 −1.31056 −0.655278 0.755388i \(-0.727449\pi\)
−0.655278 + 0.755388i \(0.727449\pi\)
\(132\) 6.04356i 0.526024i
\(133\) − 3.58258i − 0.310649i
\(134\) 7.46099 0.644531
\(135\) 0 0
\(136\) 4.25227 0.364629
\(137\) 16.3303i 1.39519i 0.716491 + 0.697596i \(0.245747\pi\)
−0.716491 + 0.697596i \(0.754253\pi\)
\(138\) 7.16515i 0.609938i
\(139\) 9.41742 0.798776 0.399388 0.916782i \(-0.369223\pi\)
0.399388 + 0.916782i \(0.369223\pi\)
\(140\) 0 0
\(141\) 10.5826 0.891214
\(142\) − 17.1652i − 1.44047i
\(143\) 22.9129i 1.91607i
\(144\) 4.95644 0.413037
\(145\) 0 0
\(146\) 7.16515 0.592992
\(147\) − 6.00000i − 0.494872i
\(148\) 4.83485i 0.397422i
\(149\) −16.7477 −1.37203 −0.686014 0.727589i \(-0.740641\pi\)
−0.686014 + 0.727589i \(0.740641\pi\)
\(150\) 0 0
\(151\) 7.16515 0.583092 0.291546 0.956557i \(-0.405830\pi\)
0.291546 + 0.956557i \(0.405830\pi\)
\(152\) − 5.07803i − 0.411883i
\(153\) 3.00000i 0.242536i
\(154\) −8.95644 −0.721730
\(155\) 0 0
\(156\) −5.53901 −0.443476
\(157\) 10.7477i 0.857762i 0.903361 + 0.428881i \(0.141092\pi\)
−0.903361 + 0.428881i \(0.858908\pi\)
\(158\) − 13.5826i − 1.08057i
\(159\) −0.417424 −0.0331039
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 1.79129i 0.140737i
\(163\) − 7.58258i − 0.593913i −0.954891 0.296957i \(-0.904028\pi\)
0.954891 0.296957i \(-0.0959717\pi\)
\(164\) 11.0780 0.865049
\(165\) 0 0
\(166\) −20.7477 −1.61034
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 1.41742i 0.109357i
\(169\) −8.00000 −0.615385
\(170\) 0 0
\(171\) 3.58258 0.273966
\(172\) − 11.5826i − 0.883163i
\(173\) 3.16515i 0.240642i 0.992735 + 0.120321i \(0.0383924\pi\)
−0.992735 + 0.120321i \(0.961608\pi\)
\(174\) −1.79129 −0.135797
\(175\) 0 0
\(176\) −24.7822 −1.86803
\(177\) − 7.58258i − 0.569941i
\(178\) − 2.53901i − 0.190307i
\(179\) −4.74773 −0.354862 −0.177431 0.984133i \(-0.556779\pi\)
−0.177431 + 0.984133i \(0.556779\pi\)
\(180\) 0 0
\(181\) 16.1652 1.20155 0.600773 0.799420i \(-0.294860\pi\)
0.600773 + 0.799420i \(0.294860\pi\)
\(182\) − 8.20871i − 0.608470i
\(183\) − 12.7477i − 0.942339i
\(184\) −5.66970 −0.417976
\(185\) 0 0
\(186\) 7.16515 0.525374
\(187\) − 15.0000i − 1.09691i
\(188\) − 12.7913i − 0.932901i
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −4.00000 −0.289430 −0.144715 0.989473i \(-0.546227\pi\)
−0.144715 + 0.989473i \(0.546227\pi\)
\(192\) − 0.912878i − 0.0658813i
\(193\) 20.3303i 1.46341i 0.681623 + 0.731704i \(0.261274\pi\)
−0.681623 + 0.731704i \(0.738726\pi\)
\(194\) −20.7477 −1.48960
\(195\) 0 0
\(196\) −7.25227 −0.518019
\(197\) − 16.3303i − 1.16349i −0.813373 0.581743i \(-0.802371\pi\)
0.813373 0.581743i \(-0.197629\pi\)
\(198\) − 8.95644i − 0.636506i
\(199\) −13.4174 −0.951136 −0.475568 0.879679i \(-0.657757\pi\)
−0.475568 + 0.879679i \(0.657757\pi\)
\(200\) 0 0
\(201\) −4.16515 −0.293787
\(202\) − 1.04356i − 0.0734247i
\(203\) − 1.00000i − 0.0701862i
\(204\) 3.62614 0.253880
\(205\) 0 0
\(206\) 27.1652 1.89269
\(207\) − 4.00000i − 0.278019i
\(208\) − 22.7133i − 1.57488i
\(209\) −17.9129 −1.23906
\(210\) 0 0
\(211\) 20.3303 1.39960 0.699798 0.714341i \(-0.253273\pi\)
0.699798 + 0.714341i \(0.253273\pi\)
\(212\) 0.504546i 0.0346523i
\(213\) 9.58258i 0.656587i
\(214\) −9.25227 −0.632472
\(215\) 0 0
\(216\) −1.41742 −0.0964435
\(217\) 4.00000i 0.271538i
\(218\) − 25.3739i − 1.71853i
\(219\) −4.00000 −0.270295
\(220\) 0 0
\(221\) 13.7477 0.924772
\(222\) − 7.16515i − 0.480893i
\(223\) − 7.00000i − 0.468755i −0.972146 0.234377i \(-0.924695\pi\)
0.972146 0.234377i \(-0.0753051\pi\)
\(224\) 6.04356 0.403802
\(225\) 0 0
\(226\) −25.3739 −1.68784
\(227\) − 7.58258i − 0.503273i −0.967822 0.251637i \(-0.919031\pi\)
0.967822 0.251637i \(-0.0809688\pi\)
\(228\) − 4.33030i − 0.286781i
\(229\) 17.1652 1.13431 0.567153 0.823613i \(-0.308045\pi\)
0.567153 + 0.823613i \(0.308045\pi\)
\(230\) 0 0
\(231\) 5.00000 0.328976
\(232\) − 1.41742i − 0.0930585i
\(233\) 13.1652i 0.862478i 0.902238 + 0.431239i \(0.141923\pi\)
−0.902238 + 0.431239i \(0.858077\pi\)
\(234\) 8.20871 0.536620
\(235\) 0 0
\(236\) −9.16515 −0.596601
\(237\) 7.58258i 0.492541i
\(238\) 5.37386i 0.348336i
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 0 0
\(241\) 29.3303 1.88933 0.944665 0.328035i \(-0.106387\pi\)
0.944665 + 0.328035i \(0.106387\pi\)
\(242\) 25.0780i 1.61208i
\(243\) − 1.00000i − 0.0641500i
\(244\) −15.4083 −0.986417
\(245\) 0 0
\(246\) −16.4174 −1.04674
\(247\) − 16.4174i − 1.04462i
\(248\) 5.66970i 0.360026i
\(249\) 11.5826 0.734016
\(250\) 0 0
\(251\) 16.1652 1.02034 0.510168 0.860075i \(-0.329583\pi\)
0.510168 + 0.860075i \(0.329583\pi\)
\(252\) 1.20871i 0.0761417i
\(253\) 20.0000i 1.25739i
\(254\) −3.58258 −0.224791
\(255\) 0 0
\(256\) 20.5481 1.28426
\(257\) 12.7477i 0.795181i 0.917563 + 0.397591i \(0.130154\pi\)
−0.917563 + 0.397591i \(0.869846\pi\)
\(258\) 17.1652i 1.06866i
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) − 26.8693i − 1.65999i
\(263\) − 30.3303i − 1.87025i −0.354322 0.935123i \(-0.615288\pi\)
0.354322 0.935123i \(-0.384712\pi\)
\(264\) 7.08712 0.436182
\(265\) 0 0
\(266\) 6.41742 0.393478
\(267\) 1.41742i 0.0867450i
\(268\) 5.03447i 0.307529i
\(269\) −22.5826 −1.37688 −0.688442 0.725291i \(-0.741705\pi\)
−0.688442 + 0.725291i \(0.741705\pi\)
\(270\) 0 0
\(271\) 1.16515 0.0707779 0.0353890 0.999374i \(-0.488733\pi\)
0.0353890 + 0.999374i \(0.488733\pi\)
\(272\) 14.8693i 0.901585i
\(273\) 4.58258i 0.277350i
\(274\) −29.2523 −1.76719
\(275\) 0 0
\(276\) −4.83485 −0.291024
\(277\) − 24.9129i − 1.49687i −0.663208 0.748435i \(-0.730806\pi\)
0.663208 0.748435i \(-0.269194\pi\)
\(278\) 16.8693i 1.01175i
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −0.417424 −0.0249014 −0.0124507 0.999922i \(-0.503963\pi\)
−0.0124507 + 0.999922i \(0.503963\pi\)
\(282\) 18.9564i 1.12884i
\(283\) 0.834849i 0.0496266i 0.999692 + 0.0248133i \(0.00789913\pi\)
−0.999692 + 0.0248133i \(0.992101\pi\)
\(284\) 11.5826 0.687299
\(285\) 0 0
\(286\) −41.0436 −2.42696
\(287\) − 9.16515i − 0.541002i
\(288\) 6.04356i 0.356120i
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 11.5826 0.678983
\(292\) 4.83485i 0.282938i
\(293\) − 30.1652i − 1.76227i −0.472868 0.881133i \(-0.656781\pi\)
0.472868 0.881133i \(-0.343219\pi\)
\(294\) 10.7477 0.626820
\(295\) 0 0
\(296\) 5.66970 0.329544
\(297\) 5.00000i 0.290129i
\(298\) − 30.0000i − 1.73785i
\(299\) −18.3303 −1.06007
\(300\) 0 0
\(301\) −9.58258 −0.552330
\(302\) 12.8348i 0.738563i
\(303\) 0.582576i 0.0334681i
\(304\) 17.7568 1.01842
\(305\) 0 0
\(306\) −5.37386 −0.307203
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) − 6.04356i − 0.344364i
\(309\) −15.1652 −0.862715
\(310\) 0 0
\(311\) −3.00000 −0.170114 −0.0850572 0.996376i \(-0.527107\pi\)
−0.0850572 + 0.996376i \(0.527107\pi\)
\(312\) 6.49545i 0.367733i
\(313\) − 10.5826i − 0.598163i −0.954228 0.299081i \(-0.903320\pi\)
0.954228 0.299081i \(-0.0966802\pi\)
\(314\) −19.2523 −1.08647
\(315\) 0 0
\(316\) 9.16515 0.515580
\(317\) − 25.0000i − 1.40414i −0.712108 0.702070i \(-0.752259\pi\)
0.712108 0.702070i \(-0.247741\pi\)
\(318\) − 0.747727i − 0.0419305i
\(319\) −5.00000 −0.279946
\(320\) 0 0
\(321\) 5.16515 0.288291
\(322\) − 7.16515i − 0.399298i
\(323\) 10.7477i 0.598020i
\(324\) −1.20871 −0.0671507
\(325\) 0 0
\(326\) 13.5826 0.752269
\(327\) 14.1652i 0.783335i
\(328\) − 12.9909i − 0.717303i
\(329\) −10.5826 −0.583436
\(330\) 0 0
\(331\) −8.33030 −0.457875 −0.228937 0.973441i \(-0.573525\pi\)
−0.228937 + 0.973441i \(0.573525\pi\)
\(332\) − 14.0000i − 0.768350i
\(333\) 4.00000i 0.219199i
\(334\) 0 0
\(335\) 0 0
\(336\) −4.95644 −0.270396
\(337\) − 21.1652i − 1.15294i −0.817119 0.576470i \(-0.804430\pi\)
0.817119 0.576470i \(-0.195570\pi\)
\(338\) − 14.3303i − 0.779466i
\(339\) 14.1652 0.769345
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) 6.41742i 0.347015i
\(343\) 13.0000i 0.701934i
\(344\) −13.5826 −0.732323
\(345\) 0 0
\(346\) −5.66970 −0.304805
\(347\) 7.16515i 0.384645i 0.981332 + 0.192323i \(0.0616020\pi\)
−0.981332 + 0.192323i \(0.938398\pi\)
\(348\) − 1.20871i − 0.0647938i
\(349\) 4.33030 0.231796 0.115898 0.993261i \(-0.463025\pi\)
0.115898 + 0.993261i \(0.463025\pi\)
\(350\) 0 0
\(351\) −4.58258 −0.244600
\(352\) − 30.2178i − 1.61061i
\(353\) 14.8348i 0.789579i 0.918772 + 0.394790i \(0.129183\pi\)
−0.918772 + 0.394790i \(0.870817\pi\)
\(354\) 13.5826 0.721906
\(355\) 0 0
\(356\) 1.71326 0.0908025
\(357\) − 3.00000i − 0.158777i
\(358\) − 8.50455i − 0.449479i
\(359\) 27.1652 1.43372 0.716861 0.697216i \(-0.245578\pi\)
0.716861 + 0.697216i \(0.245578\pi\)
\(360\) 0 0
\(361\) −6.16515 −0.324482
\(362\) 28.9564i 1.52192i
\(363\) − 14.0000i − 0.734809i
\(364\) 5.53901 0.290323
\(365\) 0 0
\(366\) 22.8348 1.19360
\(367\) − 37.4955i − 1.95725i −0.205661 0.978623i \(-0.565934\pi\)
0.205661 0.978623i \(-0.434066\pi\)
\(368\) − 19.8258i − 1.03349i
\(369\) 9.16515 0.477119
\(370\) 0 0
\(371\) 0.417424 0.0216716
\(372\) 4.83485i 0.250675i
\(373\) − 28.3303i − 1.46689i −0.679750 0.733444i \(-0.737912\pi\)
0.679750 0.733444i \(-0.262088\pi\)
\(374\) 26.8693 1.38938
\(375\) 0 0
\(376\) −15.0000 −0.773566
\(377\) − 4.58258i − 0.236015i
\(378\) − 1.79129i − 0.0921339i
\(379\) 26.0000 1.33553 0.667765 0.744372i \(-0.267251\pi\)
0.667765 + 0.744372i \(0.267251\pi\)
\(380\) 0 0
\(381\) 2.00000 0.102463
\(382\) − 7.16515i − 0.366601i
\(383\) − 3.58258i − 0.183061i −0.995802 0.0915305i \(-0.970824\pi\)
0.995802 0.0915305i \(-0.0291759\pi\)
\(384\) −10.4519 −0.533371
\(385\) 0 0
\(386\) −36.4174 −1.85360
\(387\) − 9.58258i − 0.487110i
\(388\) − 14.0000i − 0.710742i
\(389\) 6.58258 0.333750 0.166875 0.985978i \(-0.446632\pi\)
0.166875 + 0.985978i \(0.446632\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 8.50455i 0.429544i
\(393\) 15.0000i 0.756650i
\(394\) 29.2523 1.47371
\(395\) 0 0
\(396\) 6.04356 0.303700
\(397\) 10.8348i 0.543785i 0.962328 + 0.271893i \(0.0876496\pi\)
−0.962328 + 0.271893i \(0.912350\pi\)
\(398\) − 24.0345i − 1.20474i
\(399\) −3.58258 −0.179353
\(400\) 0 0
\(401\) 12.4174 0.620097 0.310048 0.950721i \(-0.399655\pi\)
0.310048 + 0.950721i \(0.399655\pi\)
\(402\) − 7.46099i − 0.372120i
\(403\) 18.3303i 0.913097i
\(404\) 0.704166 0.0350336
\(405\) 0 0
\(406\) 1.79129 0.0889001
\(407\) − 20.0000i − 0.991363i
\(408\) − 4.25227i − 0.210519i
\(409\) 2.74773 0.135866 0.0679332 0.997690i \(-0.478360\pi\)
0.0679332 + 0.997690i \(0.478360\pi\)
\(410\) 0 0
\(411\) 16.3303 0.805514
\(412\) 18.3303i 0.903069i
\(413\) 7.58258i 0.373114i
\(414\) 7.16515 0.352148
\(415\) 0 0
\(416\) 27.6951 1.35786
\(417\) − 9.41742i − 0.461173i
\(418\) − 32.0871i − 1.56943i
\(419\) 37.1652 1.81564 0.907818 0.419364i \(-0.137747\pi\)
0.907818 + 0.419364i \(0.137747\pi\)
\(420\) 0 0
\(421\) 4.41742 0.215292 0.107646 0.994189i \(-0.465669\pi\)
0.107646 + 0.994189i \(0.465669\pi\)
\(422\) 36.4174i 1.77277i
\(423\) − 10.5826i − 0.514542i
\(424\) 0.591667 0.0287339
\(425\) 0 0
\(426\) −17.1652 −0.831654
\(427\) 12.7477i 0.616906i
\(428\) − 6.24318i − 0.301776i
\(429\) 22.9129 1.10624
\(430\) 0 0
\(431\) −39.1652 −1.88652 −0.943259 0.332057i \(-0.892258\pi\)
−0.943259 + 0.332057i \(0.892258\pi\)
\(432\) − 4.95644i − 0.238467i
\(433\) − 25.0780i − 1.20517i −0.798053 0.602587i \(-0.794137\pi\)
0.798053 0.602587i \(-0.205863\pi\)
\(434\) −7.16515 −0.343938
\(435\) 0 0
\(436\) 17.1216 0.819975
\(437\) − 14.3303i − 0.685511i
\(438\) − 7.16515i − 0.342364i
\(439\) 16.9129 0.807208 0.403604 0.914934i \(-0.367757\pi\)
0.403604 + 0.914934i \(0.367757\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 24.6261i 1.17135i
\(443\) − 0.582576i − 0.0276790i −0.999904 0.0138395i \(-0.995595\pi\)
0.999904 0.0138395i \(-0.00440539\pi\)
\(444\) 4.83485 0.229452
\(445\) 0 0
\(446\) 12.5390 0.593740
\(447\) 16.7477i 0.792140i
\(448\) 0.912878i 0.0431295i
\(449\) 30.0780 1.41947 0.709735 0.704469i \(-0.248815\pi\)
0.709735 + 0.704469i \(0.248815\pi\)
\(450\) 0 0
\(451\) −45.8258 −2.15785
\(452\) − 17.1216i − 0.805332i
\(453\) − 7.16515i − 0.336648i
\(454\) 13.5826 0.637462
\(455\) 0 0
\(456\) −5.07803 −0.237801
\(457\) − 3.74773i − 0.175311i −0.996151 0.0876556i \(-0.972062\pi\)
0.996151 0.0876556i \(-0.0279375\pi\)
\(458\) 30.7477i 1.43675i
\(459\) 3.00000 0.140028
\(460\) 0 0
\(461\) −9.16515 −0.426864 −0.213432 0.976958i \(-0.568464\pi\)
−0.213432 + 0.976958i \(0.568464\pi\)
\(462\) 8.95644i 0.416691i
\(463\) 0.165151i 0.00767524i 0.999993 + 0.00383762i \(0.00122155\pi\)
−0.999993 + 0.00383762i \(0.998778\pi\)
\(464\) 4.95644 0.230097
\(465\) 0 0
\(466\) −23.5826 −1.09244
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) 5.53901i 0.256041i
\(469\) 4.16515 0.192329
\(470\) 0 0
\(471\) 10.7477 0.495229
\(472\) 10.7477i 0.494704i
\(473\) 47.9129i 2.20304i
\(474\) −13.5826 −0.623868
\(475\) 0 0
\(476\) −3.62614 −0.166204
\(477\) 0.417424i 0.0191125i
\(478\) 46.5735i 2.13022i
\(479\) 19.1652 0.875678 0.437839 0.899053i \(-0.355744\pi\)
0.437839 + 0.899053i \(0.355744\pi\)
\(480\) 0 0
\(481\) 18.3303 0.835790
\(482\) 52.5390i 2.39309i
\(483\) 4.00000i 0.182006i
\(484\) −16.9220 −0.769180
\(485\) 0 0
\(486\) 1.79129 0.0812545
\(487\) 2.33030i 0.105596i 0.998605 + 0.0527980i \(0.0168140\pi\)
−0.998605 + 0.0527980i \(0.983186\pi\)
\(488\) 18.0689i 0.817942i
\(489\) −7.58258 −0.342896
\(490\) 0 0
\(491\) 16.0000 0.722070 0.361035 0.932552i \(-0.382424\pi\)
0.361035 + 0.932552i \(0.382424\pi\)
\(492\) − 11.0780i − 0.499436i
\(493\) 3.00000i 0.135113i
\(494\) 29.4083 1.32314
\(495\) 0 0
\(496\) −19.8258 −0.890203
\(497\) − 9.58258i − 0.429837i
\(498\) 20.7477i 0.929728i
\(499\) 13.4174 0.600646 0.300323 0.953837i \(-0.402905\pi\)
0.300323 + 0.953837i \(0.402905\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 28.9564i 1.29239i
\(503\) 22.9129i 1.02163i 0.859689 + 0.510817i \(0.170657\pi\)
−0.859689 + 0.510817i \(0.829343\pi\)
\(504\) 1.41742 0.0631371
\(505\) 0 0
\(506\) −35.8258 −1.59265
\(507\) 8.00000i 0.355292i
\(508\) − 2.41742i − 0.107256i
\(509\) −26.7477 −1.18557 −0.592786 0.805360i \(-0.701972\pi\)
−0.592786 + 0.805360i \(0.701972\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 15.9038i 0.702855i
\(513\) − 3.58258i − 0.158175i
\(514\) −22.8348 −1.00720
\(515\) 0 0
\(516\) −11.5826 −0.509894
\(517\) 52.9129i 2.32711i
\(518\) 7.16515i 0.314819i
\(519\) 3.16515 0.138935
\(520\) 0 0
\(521\) 43.0780 1.88728 0.943641 0.330970i \(-0.107376\pi\)
0.943641 + 0.330970i \(0.107376\pi\)
\(522\) 1.79129i 0.0784025i
\(523\) 33.3303i 1.45743i 0.684816 + 0.728716i \(0.259883\pi\)
−0.684816 + 0.728716i \(0.740117\pi\)
\(524\) 18.1307 0.792043
\(525\) 0 0
\(526\) 54.3303 2.36891
\(527\) − 12.0000i − 0.522728i
\(528\) 24.7822i 1.07851i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −7.58258 −0.329056
\(532\) 4.33030i 0.187742i
\(533\) − 42.0000i − 1.81922i
\(534\) −2.53901 −0.109874
\(535\) 0 0
\(536\) 5.90379 0.255005
\(537\) 4.74773i 0.204880i
\(538\) − 40.4519i − 1.74400i
\(539\) 30.0000 1.29219
\(540\) 0 0
\(541\) −31.4955 −1.35410 −0.677048 0.735939i \(-0.736741\pi\)
−0.677048 + 0.735939i \(0.736741\pi\)
\(542\) 2.08712i 0.0896495i
\(543\) − 16.1652i − 0.693713i
\(544\) −18.1307 −0.777347
\(545\) 0 0
\(546\) −8.20871 −0.351300
\(547\) 4.16515i 0.178089i 0.996028 + 0.0890445i \(0.0283813\pi\)
−0.996028 + 0.0890445i \(0.971619\pi\)
\(548\) − 19.7386i − 0.843193i
\(549\) −12.7477 −0.544060
\(550\) 0 0
\(551\) 3.58258 0.152623
\(552\) 5.66970i 0.241318i
\(553\) − 7.58258i − 0.322444i
\(554\) 44.6261 1.89598
\(555\) 0 0
\(556\) −11.3830 −0.482745
\(557\) − 22.7477i − 0.963852i −0.876212 0.481926i \(-0.839937\pi\)
0.876212 0.481926i \(-0.160063\pi\)
\(558\) − 7.16515i − 0.303325i
\(559\) −43.9129 −1.85732
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) − 0.747727i − 0.0315410i
\(563\) 14.5826i 0.614582i 0.951616 + 0.307291i \(0.0994225\pi\)
−0.951616 + 0.307291i \(0.900577\pi\)
\(564\) −12.7913 −0.538610
\(565\) 0 0
\(566\) −1.49545 −0.0628586
\(567\) 1.00000i 0.0419961i
\(568\) − 13.5826i − 0.569912i
\(569\) −28.5826 −1.19824 −0.599122 0.800658i \(-0.704484\pi\)
−0.599122 + 0.800658i \(0.704484\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) − 27.6951i − 1.15799i
\(573\) 4.00000i 0.167102i
\(574\) 16.4174 0.685250
\(575\) 0 0
\(576\) −0.912878 −0.0380366
\(577\) − 19.1652i − 0.797856i −0.916982 0.398928i \(-0.869382\pi\)
0.916982 0.398928i \(-0.130618\pi\)
\(578\) 14.3303i 0.596062i
\(579\) 20.3303 0.844899
\(580\) 0 0
\(581\) −11.5826 −0.480526
\(582\) 20.7477i 0.860021i
\(583\) − 2.08712i − 0.0864397i
\(584\) 5.66970 0.234614
\(585\) 0 0
\(586\) 54.0345 2.23214
\(587\) − 11.5826i − 0.478064i −0.971012 0.239032i \(-0.923170\pi\)
0.971012 0.239032i \(-0.0768301\pi\)
\(588\) 7.25227i 0.299079i
\(589\) −14.3303 −0.590470
\(590\) 0 0
\(591\) −16.3303 −0.671739
\(592\) 19.8258i 0.814834i
\(593\) − 8.41742i − 0.345662i −0.984951 0.172831i \(-0.944709\pi\)
0.984951 0.172831i \(-0.0552915\pi\)
\(594\) −8.95644 −0.367487
\(595\) 0 0
\(596\) 20.2432 0.829193
\(597\) 13.4174i 0.549139i
\(598\) − 32.8348i − 1.34272i
\(599\) −0.165151 −0.00674790 −0.00337395 0.999994i \(-0.501074\pi\)
−0.00337395 + 0.999994i \(0.501074\pi\)
\(600\) 0 0
\(601\) −32.3303 −1.31878 −0.659390 0.751801i \(-0.729185\pi\)
−0.659390 + 0.751801i \(0.729185\pi\)
\(602\) − 17.1652i − 0.699599i
\(603\) 4.16515i 0.169618i
\(604\) −8.66061 −0.352395
\(605\) 0 0
\(606\) −1.04356 −0.0423918
\(607\) 3.58258i 0.145412i 0.997353 + 0.0727061i \(0.0231635\pi\)
−0.997353 + 0.0727061i \(0.976836\pi\)
\(608\) 21.6515i 0.878085i
\(609\) −1.00000 −0.0405220
\(610\) 0 0
\(611\) −48.4955 −1.96192
\(612\) − 3.62614i − 0.146578i
\(613\) 43.7477i 1.76695i 0.468474 + 0.883477i \(0.344804\pi\)
−0.468474 + 0.883477i \(0.655196\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −7.08712 −0.285548
\(617\) 15.4955i 0.623823i 0.950111 + 0.311912i \(0.100969\pi\)
−0.950111 + 0.311912i \(0.899031\pi\)
\(618\) − 27.1652i − 1.09274i
\(619\) −13.1652 −0.529152 −0.264576 0.964365i \(-0.585232\pi\)
−0.264576 + 0.964365i \(0.585232\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) − 5.37386i − 0.215472i
\(623\) − 1.41742i − 0.0567879i
\(624\) −22.7133 −0.909258
\(625\) 0 0
\(626\) 18.9564 0.757652
\(627\) 17.9129i 0.715371i
\(628\) − 12.9909i − 0.518394i
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 10.9129 0.434435 0.217217 0.976123i \(-0.430302\pi\)
0.217217 + 0.976123i \(0.430302\pi\)
\(632\) − 10.7477i − 0.427522i
\(633\) − 20.3303i − 0.808057i
\(634\) 44.7822 1.77853
\(635\) 0 0
\(636\) 0.504546 0.0200065
\(637\) 27.4955i 1.08941i
\(638\) − 8.95644i − 0.354589i
\(639\) 9.58258 0.379081
\(640\) 0 0
\(641\) −6.58258 −0.259996 −0.129998 0.991514i \(-0.541497\pi\)
−0.129998 + 0.991514i \(0.541497\pi\)
\(642\) 9.25227i 0.365158i
\(643\) − 43.6606i − 1.72181i −0.508769 0.860903i \(-0.669899\pi\)
0.508769 0.860903i \(-0.330101\pi\)
\(644\) 4.83485 0.190520
\(645\) 0 0
\(646\) −19.2523 −0.757471
\(647\) 24.7477i 0.972934i 0.873699 + 0.486467i \(0.161715\pi\)
−0.873699 + 0.486467i \(0.838285\pi\)
\(648\) 1.41742i 0.0556817i
\(649\) 37.9129 1.48821
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 9.16515i 0.358935i
\(653\) − 26.1652i − 1.02392i −0.859009 0.511961i \(-0.828919\pi\)
0.859009 0.511961i \(-0.171081\pi\)
\(654\) −25.3739 −0.992197
\(655\) 0 0
\(656\) 45.4265 1.77361
\(657\) 4.00000i 0.156055i
\(658\) − 18.9564i − 0.738999i
\(659\) −18.1652 −0.707614 −0.353807 0.935318i \(-0.615113\pi\)
−0.353807 + 0.935318i \(0.615113\pi\)
\(660\) 0 0
\(661\) 25.0000 0.972387 0.486194 0.873851i \(-0.338385\pi\)
0.486194 + 0.873851i \(0.338385\pi\)
\(662\) − 14.9220i − 0.579959i
\(663\) − 13.7477i − 0.533917i
\(664\) −16.4174 −0.637120
\(665\) 0 0
\(666\) −7.16515 −0.277644
\(667\) − 4.00000i − 0.154881i
\(668\) 0 0
\(669\) −7.00000 −0.270636
\(670\) 0 0
\(671\) 63.7386 2.46060
\(672\) − 6.04356i − 0.233135i
\(673\) − 1.74773i − 0.0673699i −0.999433 0.0336850i \(-0.989276\pi\)
0.999433 0.0336850i \(-0.0107243\pi\)
\(674\) 37.9129 1.46035
\(675\) 0 0
\(676\) 9.66970 0.371911
\(677\) 44.8258i 1.72279i 0.507932 + 0.861397i \(0.330410\pi\)
−0.507932 + 0.861397i \(0.669590\pi\)
\(678\) 25.3739i 0.974477i
\(679\) −11.5826 −0.444498
\(680\) 0 0
\(681\) −7.58258 −0.290565
\(682\) 35.8258i 1.37184i
\(683\) − 7.16515i − 0.274167i −0.990560 0.137083i \(-0.956227\pi\)
0.990560 0.137083i \(-0.0437728\pi\)
\(684\) −4.33030 −0.165573
\(685\) 0 0
\(686\) −23.2867 −0.889092
\(687\) − 17.1652i − 0.654891i
\(688\) − 47.4955i − 1.81075i
\(689\) 1.91288 0.0728749
\(690\) 0 0
\(691\) 42.9129 1.63248 0.816241 0.577711i \(-0.196054\pi\)
0.816241 + 0.577711i \(0.196054\pi\)
\(692\) − 3.82576i − 0.145433i
\(693\) − 5.00000i − 0.189934i
\(694\) −12.8348 −0.487204
\(695\) 0 0
\(696\) −1.41742 −0.0537273
\(697\) 27.4955i 1.04146i
\(698\) 7.75682i 0.293600i
\(699\) 13.1652 0.497952
\(700\) 0 0
\(701\) −23.0780 −0.871645 −0.435823 0.900033i \(-0.643542\pi\)
−0.435823 + 0.900033i \(0.643542\pi\)
\(702\) − 8.20871i − 0.309818i
\(703\) 14.3303i 0.540478i
\(704\) 4.56439 0.172027
\(705\) 0 0
\(706\) −26.5735 −1.00011
\(707\) − 0.582576i − 0.0219100i
\(708\) 9.16515i 0.344447i
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 7.58258 0.284369
\(712\) − 2.00909i − 0.0752939i
\(713\) 16.0000i 0.599205i
\(714\) 5.37386 0.201112
\(715\) 0 0
\(716\) 5.73864 0.214463
\(717\) − 26.0000i − 0.970988i
\(718\) 48.6606i 1.81600i
\(719\) −7.91288 −0.295101 −0.147550 0.989055i \(-0.547139\pi\)
−0.147550 + 0.989055i \(0.547139\pi\)
\(720\) 0 0
\(721\) 15.1652 0.564780
\(722\) − 11.0436i − 0.410999i
\(723\) − 29.3303i − 1.09081i
\(724\) −19.5390 −0.726162
\(725\) 0 0
\(726\) 25.0780 0.930733
\(727\) 9.66970i 0.358629i 0.983792 + 0.179315i \(0.0573880\pi\)
−0.983792 + 0.179315i \(0.942612\pi\)
\(728\) − 6.49545i − 0.240738i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 28.7477 1.06327
\(732\) 15.4083i 0.569508i
\(733\) 38.4174i 1.41898i 0.704716 + 0.709490i \(0.251075\pi\)
−0.704716 + 0.709490i \(0.748925\pi\)
\(734\) 67.1652 2.47911
\(735\) 0 0
\(736\) 24.1742 0.891074
\(737\) − 20.8258i − 0.767127i
\(738\) 16.4174i 0.604334i
\(739\) −19.2523 −0.708206 −0.354103 0.935206i \(-0.615214\pi\)
−0.354103 + 0.935206i \(0.615214\pi\)
\(740\) 0 0
\(741\) −16.4174 −0.603109
\(742\) 0.747727i 0.0274499i
\(743\) − 29.7477i − 1.09134i −0.838001 0.545669i \(-0.816276\pi\)
0.838001 0.545669i \(-0.183724\pi\)
\(744\) 5.66970 0.207861
\(745\) 0 0
\(746\) 50.7477 1.85801
\(747\) − 11.5826i − 0.423784i
\(748\) 18.1307i 0.662923i
\(749\) −5.16515 −0.188731
\(750\) 0 0
\(751\) −17.4955 −0.638418 −0.319209 0.947684i \(-0.603417\pi\)
−0.319209 + 0.947684i \(0.603417\pi\)
\(752\) − 52.4519i − 1.91272i
\(753\) − 16.1652i − 0.589091i
\(754\) 8.20871 0.298944
\(755\) 0 0
\(756\) 1.20871 0.0439604
\(757\) − 2.33030i − 0.0846963i −0.999103 0.0423481i \(-0.986516\pi\)
0.999103 0.0423481i \(-0.0134839\pi\)
\(758\) 46.5735i 1.69163i
\(759\) 20.0000 0.725954
\(760\) 0 0
\(761\) 36.4174 1.32013 0.660065 0.751208i \(-0.270529\pi\)
0.660065 + 0.751208i \(0.270529\pi\)
\(762\) 3.58258i 0.129783i
\(763\) − 14.1652i − 0.512813i
\(764\) 4.83485 0.174919
\(765\) 0 0
\(766\) 6.41742 0.231871
\(767\) 34.7477i 1.25467i
\(768\) − 20.5481i − 0.741466i
\(769\) −25.5826 −0.922531 −0.461266 0.887262i \(-0.652604\pi\)
−0.461266 + 0.887262i \(0.652604\pi\)
\(770\) 0 0
\(771\) 12.7477 0.459098
\(772\) − 24.5735i − 0.884419i
\(773\) 5.16515i 0.185778i 0.995676 + 0.0928888i \(0.0296101\pi\)
−0.995676 + 0.0928888i \(0.970390\pi\)
\(774\) 17.1652 0.616989
\(775\) 0 0
\(776\) −16.4174 −0.589351
\(777\) − 4.00000i − 0.143499i
\(778\) 11.7913i 0.422738i
\(779\) 32.8348 1.17643
\(780\) 0 0
\(781\) −47.9129 −1.71446
\(782\) 21.4955i 0.768676i
\(783\) − 1.00000i − 0.0357371i
\(784\) −29.7386 −1.06209
\(785\) 0 0
\(786\) −26.8693 −0.958397
\(787\) − 24.0000i − 0.855508i −0.903895 0.427754i \(-0.859305\pi\)
0.903895 0.427754i \(-0.140695\pi\)
\(788\) 19.7386i 0.703160i
\(789\) −30.3303 −1.07979
\(790\) 0 0
\(791\) −14.1652 −0.503655
\(792\) − 7.08712i − 0.251830i
\(793\) 58.4174i 2.07446i
\(794\) −19.4083 −0.688776
\(795\) 0 0
\(796\) 16.2178 0.574825
\(797\) 32.3303i 1.14520i 0.819836 + 0.572599i \(0.194065\pi\)
−0.819836 + 0.572599i \(0.805935\pi\)
\(798\) − 6.41742i − 0.227174i
\(799\) 31.7477 1.12315
\(800\) 0 0
\(801\) 1.41742 0.0500822
\(802\) 22.2432i 0.785434i
\(803\) − 20.0000i − 0.705785i
\(804\) 5.03447 0.177552
\(805\) 0 0
\(806\) −32.8348 −1.15656
\(807\) 22.5826i 0.794944i
\(808\) − 0.825757i − 0.0290500i
\(809\) −44.0780 −1.54970 −0.774851 0.632145i \(-0.782175\pi\)
−0.774851 + 0.632145i \(0.782175\pi\)
\(810\) 0 0
\(811\) 32.5826 1.14413 0.572064 0.820209i \(-0.306143\pi\)
0.572064 + 0.820209i \(0.306143\pi\)
\(812\) 1.20871i 0.0424175i
\(813\) − 1.16515i − 0.0408636i
\(814\) 35.8258 1.25569
\(815\) 0 0
\(816\) 14.8693 0.520530
\(817\) − 34.3303i − 1.20107i
\(818\) 4.92197i 0.172093i
\(819\) 4.58258 0.160128
\(820\) 0 0
\(821\) −31.4955 −1.09920 −0.549599 0.835428i \(-0.685220\pi\)
−0.549599 + 0.835428i \(0.685220\pi\)
\(822\) 29.2523i 1.02029i
\(823\) − 51.0780i − 1.78047i −0.455503 0.890234i \(-0.650541\pi\)
0.455503 0.890234i \(-0.349459\pi\)
\(824\) 21.4955 0.748830
\(825\) 0 0
\(826\) −13.5826 −0.472598
\(827\) − 43.8258i − 1.52397i −0.647594 0.761985i \(-0.724225\pi\)
0.647594 0.761985i \(-0.275775\pi\)
\(828\) 4.83485i 0.168023i
\(829\) −4.00000 −0.138926 −0.0694629 0.997585i \(-0.522129\pi\)
−0.0694629 + 0.997585i \(0.522129\pi\)
\(830\) 0 0
\(831\) −24.9129 −0.864218
\(832\) 4.18333i 0.145031i
\(833\) − 18.0000i − 0.623663i
\(834\) 16.8693 0.584137
\(835\) 0 0
\(836\) 21.6515 0.748833
\(837\) 4.00000i 0.138260i
\(838\) 66.5735i 2.29974i
\(839\) 48.4955 1.67425 0.837125 0.547012i \(-0.184235\pi\)
0.837125 + 0.547012i \(0.184235\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 7.91288i 0.272696i
\(843\) 0.417424i 0.0143769i
\(844\) −24.5735 −0.845854
\(845\) 0 0
\(846\) 18.9564 0.651736
\(847\) 14.0000i 0.481046i
\(848\) 2.06894i 0.0710476i
\(849\) 0.834849 0.0286519
\(850\) 0 0
\(851\) 16.0000 0.548473
\(852\) − 11.5826i − 0.396813i
\(853\) − 20.7477i − 0.710389i −0.934792 0.355194i \(-0.884415\pi\)
0.934792 0.355194i \(-0.115585\pi\)
\(854\) −22.8348 −0.781392
\(855\) 0 0
\(856\) −7.32121 −0.250234
\(857\) − 46.6606i − 1.59390i −0.604048 0.796948i \(-0.706446\pi\)
0.604048 0.796948i \(-0.293554\pi\)
\(858\) 41.0436i 1.40120i
\(859\) 47.5826 1.62350 0.811748 0.584007i \(-0.198516\pi\)
0.811748 + 0.584007i \(0.198516\pi\)
\(860\) 0 0
\(861\) −9.16515 −0.312348
\(862\) − 70.1561i − 2.38952i
\(863\) 6.41742i 0.218452i 0.994017 + 0.109226i \(0.0348372\pi\)
−0.994017 + 0.109226i \(0.965163\pi\)
\(864\) 6.04356 0.205606
\(865\) 0 0
\(866\) 44.9220 1.52651
\(867\) − 8.00000i − 0.271694i
\(868\) − 4.83485i − 0.164105i
\(869\) −37.9129 −1.28611
\(870\) 0 0
\(871\) 19.0871 0.646742
\(872\) − 20.0780i − 0.679928i
\(873\) − 11.5826i − 0.392011i
\(874\) 25.6697 0.868290
\(875\) 0 0
\(876\) 4.83485 0.163354
\(877\) 19.6697i 0.664198i 0.943244 + 0.332099i \(0.107757\pi\)
−0.943244 + 0.332099i \(0.892243\pi\)
\(878\) 30.2958i 1.02243i
\(879\) −30.1652 −1.01745
\(880\) 0 0
\(881\) −32.0780 −1.08074 −0.540368 0.841429i \(-0.681715\pi\)
−0.540368 + 0.841429i \(0.681715\pi\)
\(882\) − 10.7477i − 0.361895i
\(883\) − 16.0000i − 0.538443i −0.963078 0.269221i \(-0.913234\pi\)
0.963078 0.269221i \(-0.0867663\pi\)
\(884\) −16.6170 −0.558892
\(885\) 0 0
\(886\) 1.04356 0.0350591
\(887\) − 44.9129i − 1.50803i −0.656859 0.754013i \(-0.728115\pi\)
0.656859 0.754013i \(-0.271885\pi\)
\(888\) − 5.66970i − 0.190263i
\(889\) −2.00000 −0.0670778
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 8.46099i 0.283295i
\(893\) − 37.9129i − 1.26871i
\(894\) −30.0000 −1.00335
\(895\) 0 0
\(896\) 10.4519 0.349173
\(897\) 18.3303i 0.612031i
\(898\) 53.8784i 1.79795i
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) −1.25227 −0.0417193
\(902\) − 82.0871i − 2.73320i
\(903\) 9.58258i 0.318888i
\(904\) −20.0780 −0.667785
\(905\) 0 0
\(906\) 12.8348 0.426409
\(907\) 23.5826i 0.783047i 0.920168 + 0.391523i \(0.128052\pi\)
−0.920168 + 0.391523i \(0.871948\pi\)
\(908\) 9.16515i 0.304156i
\(909\) 0.582576 0.0193228
\(910\) 0 0
\(911\) −50.8258 −1.68393 −0.841966 0.539530i \(-0.818602\pi\)
−0.841966 + 0.539530i \(0.818602\pi\)
\(912\) − 17.7568i − 0.587987i
\(913\) 57.9129i 1.91664i
\(914\) 6.71326 0.222055
\(915\) 0 0
\(916\) −20.7477 −0.685524
\(917\) − 15.0000i − 0.495344i
\(918\) 5.37386i 0.177364i
\(919\) −18.9129 −0.623878 −0.311939 0.950102i \(-0.600979\pi\)
−0.311939 + 0.950102i \(0.600979\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 16.4174i − 0.540679i
\(923\) − 43.9129i − 1.44541i
\(924\) −6.04356 −0.198819
\(925\) 0 0
\(926\) −0.295834 −0.00972170
\(927\) 15.1652i 0.498089i
\(928\) 6.04356i 0.198390i
\(929\) −15.6697 −0.514106 −0.257053 0.966397i \(-0.582752\pi\)
−0.257053 + 0.966397i \(0.582752\pi\)
\(930\) 0 0
\(931\) −21.4955 −0.704485
\(932\) − 15.9129i − 0.521244i
\(933\) 3.00000i 0.0982156i
\(934\) 14.3303 0.468902
\(935\) 0 0
\(936\) 6.49545 0.212311
\(937\) − 18.5826i − 0.607066i −0.952821 0.303533i \(-0.901834\pi\)
0.952821 0.303533i \(-0.0981663\pi\)
\(938\) 7.46099i 0.243610i
\(939\) −10.5826 −0.345349
\(940\) 0 0
\(941\) 19.9129 0.649141 0.324571 0.945861i \(-0.394780\pi\)
0.324571 + 0.945861i \(0.394780\pi\)
\(942\) 19.2523i 0.627273i
\(943\) − 36.6606i − 1.19383i
\(944\) −37.5826 −1.22321
\(945\) 0 0
\(946\) −85.8258 −2.79044
\(947\) − 54.9129i − 1.78443i −0.451612 0.892214i \(-0.649151\pi\)
0.451612 0.892214i \(-0.350849\pi\)
\(948\) − 9.16515i − 0.297670i
\(949\) 18.3303 0.595027
\(950\) 0 0
\(951\) −25.0000 −0.810681
\(952\) 4.25227i 0.137817i
\(953\) 37.5826i 1.21742i 0.793393 + 0.608710i \(0.208313\pi\)
−0.793393 + 0.608710i \(0.791687\pi\)
\(954\) −0.747727 −0.0242086
\(955\) 0 0
\(956\) −31.4265 −1.01641
\(957\) 5.00000i 0.161627i
\(958\) 34.3303i 1.10916i
\(959\) −16.3303 −0.527333
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 32.8348i 1.05864i
\(963\) − 5.16515i − 0.166445i
\(964\) −35.4519 −1.14183
\(965\) 0 0
\(966\) −7.16515 −0.230535
\(967\) 9.25227i 0.297533i 0.988872 + 0.148767i \(0.0475303\pi\)
−0.988872 + 0.148767i \(0.952470\pi\)
\(968\) 19.8439i 0.637808i
\(969\) 10.7477 0.345267
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 1.20871i 0.0387695i
\(973\) 9.41742i 0.301909i
\(974\) −4.17424 −0.133751
\(975\) 0 0
\(976\) −63.1833 −2.02245
\(977\) − 2.50455i − 0.0801275i −0.999197 0.0400638i \(-0.987244\pi\)
0.999197 0.0400638i \(-0.0127561\pi\)
\(978\) − 13.5826i − 0.434323i
\(979\) −7.08712 −0.226505
\(980\) 0 0
\(981\) 14.1652 0.452258
\(982\) 28.6606i 0.914597i
\(983\) − 55.1652i − 1.75950i −0.475441 0.879748i \(-0.657712\pi\)
0.475441 0.879748i \(-0.342288\pi\)
\(984\) −12.9909 −0.414135
\(985\) 0 0
\(986\) −5.37386 −0.171139
\(987\) 10.5826i 0.336847i
\(988\) 19.8439i 0.631320i
\(989\) −38.3303 −1.21883
\(990\) 0 0
\(991\) 16.0780 0.510735 0.255368 0.966844i \(-0.417803\pi\)
0.255368 + 0.966844i \(0.417803\pi\)
\(992\) − 24.1742i − 0.767533i
\(993\) 8.33030i 0.264354i
\(994\) 17.1652 0.544446
\(995\) 0 0
\(996\) −14.0000 −0.443607
\(997\) − 0.834849i − 0.0264399i −0.999913 0.0132200i \(-0.995792\pi\)
0.999913 0.0132200i \(-0.00420817\pi\)
\(998\) 24.0345i 0.760798i
\(999\) 4.00000 0.126554
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 2175.2.c.f.349.3 4
5.2 odd 4 2175.2.a.r.1.1 2
5.3 odd 4 435.2.a.f.1.2 2
5.4 even 2 inner 2175.2.c.f.349.2 4
15.2 even 4 6525.2.a.t.1.2 2
15.8 even 4 1305.2.a.m.1.1 2
20.3 even 4 6960.2.a.bw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.f.1.2 2 5.3 odd 4
1305.2.a.m.1.1 2 15.8 even 4
2175.2.a.r.1.1 2 5.2 odd 4
2175.2.c.f.349.2 4 5.4 even 2 inner
2175.2.c.f.349.3 4 1.1 even 1 trivial
6525.2.a.t.1.2 2 15.2 even 4
6960.2.a.bw.1.1 2 20.3 even 4