Properties

Label 1575.2.d.d.1324.4
Level $1575$
Weight $2$
Character 1575.1324
Analytic conductor $12.576$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,2,Mod(1324,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, 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("1575.1324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.d (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: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1324.4
Root \(-1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1575.1324
Dual form 1575.2.d.d.1324.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.23607i q^{2} -3.00000 q^{4} +1.00000i q^{7} -2.23607i q^{8} +O(q^{10})\) \(q+2.23607i q^{2} -3.00000 q^{4} +1.00000i q^{7} -2.23607i q^{8} -6.47214 q^{11} -4.47214i q^{13} -2.23607 q^{14} -1.00000 q^{16} +2.00000i q^{17} +2.47214 q^{19} -14.4721i q^{22} +4.00000i q^{23} +10.0000 q^{26} -3.00000i q^{28} -2.00000 q^{29} +1.52786 q^{31} -6.70820i q^{32} -4.47214 q^{34} -6.94427i q^{37} +5.52786i q^{38} +2.00000 q^{41} -8.94427i q^{43} +19.4164 q^{44} -8.94427 q^{46} -12.9443i q^{47} -1.00000 q^{49} +13.4164i q^{52} -3.52786i q^{53} +2.23607 q^{56} -4.47214i q^{58} -8.94427 q^{59} -2.00000 q^{61} +3.41641i q^{62} +13.0000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -5.52786 q^{71} +12.4721i q^{73} +15.5279 q^{74} -7.41641 q^{76} -6.47214i q^{77} -12.9443 q^{79} +4.47214i q^{82} -16.9443i q^{83} +20.0000 q^{86} +14.4721i q^{88} -2.00000 q^{89} +4.47214 q^{91} -12.0000i q^{92} +28.9443 q^{94} +8.47214i q^{97} -2.23607i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{4} - 8 q^{11} - 4 q^{16} - 8 q^{19} + 40 q^{26} - 8 q^{29} + 24 q^{31} + 8 q^{41} + 24 q^{44} - 4 q^{49} - 8 q^{61} + 52 q^{64} - 40 q^{71} + 80 q^{74} + 24 q^{76} - 16 q^{79} + 80 q^{86} - 8 q^{89} + 80 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607i 1.58114i 0.612372 + 0.790569i \(0.290215\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 0 0
\(4\) −3.00000 −1.50000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) − 2.23607i − 0.790569i
\(9\) 0 0
\(10\) 0 0
\(11\) −6.47214 −1.95142 −0.975711 0.219061i \(-0.929701\pi\)
−0.975711 + 0.219061i \(0.929701\pi\)
\(12\) 0 0
\(13\) − 4.47214i − 1.24035i −0.784465 0.620174i \(-0.787062\pi\)
0.784465 0.620174i \(-0.212938\pi\)
\(14\) −2.23607 −0.597614
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) 2.47214 0.567147 0.283573 0.958951i \(-0.408480\pi\)
0.283573 + 0.958951i \(0.408480\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 14.4721i − 3.08547i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 10.0000 1.96116
\(27\) 0 0
\(28\) − 3.00000i − 0.566947i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 1.52786 0.274412 0.137206 0.990543i \(-0.456188\pi\)
0.137206 + 0.990543i \(0.456188\pi\)
\(32\) − 6.70820i − 1.18585i
\(33\) 0 0
\(34\) −4.47214 −0.766965
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.94427i − 1.14163i −0.821078 0.570816i \(-0.806627\pi\)
0.821078 0.570816i \(-0.193373\pi\)
\(38\) 5.52786i 0.896738i
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) − 8.94427i − 1.36399i −0.731357 0.681994i \(-0.761113\pi\)
0.731357 0.681994i \(-0.238887\pi\)
\(44\) 19.4164 2.92713
\(45\) 0 0
\(46\) −8.94427 −1.31876
\(47\) − 12.9443i − 1.88812i −0.329779 0.944058i \(-0.606974\pi\)
0.329779 0.944058i \(-0.393026\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 13.4164i 1.86052i
\(53\) − 3.52786i − 0.484589i −0.970203 0.242295i \(-0.922100\pi\)
0.970203 0.242295i \(-0.0779001\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.23607 0.298807
\(57\) 0 0
\(58\) − 4.47214i − 0.587220i
\(59\) −8.94427 −1.16445 −0.582223 0.813029i \(-0.697817\pi\)
−0.582223 + 0.813029i \(0.697817\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 3.41641i 0.433884i
\(63\) 0 0
\(64\) 13.0000 1.62500
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) −5.52786 −0.656037 −0.328018 0.944671i \(-0.606381\pi\)
−0.328018 + 0.944671i \(0.606381\pi\)
\(72\) 0 0
\(73\) 12.4721i 1.45975i 0.683579 + 0.729877i \(0.260422\pi\)
−0.683579 + 0.729877i \(0.739578\pi\)
\(74\) 15.5279 1.80508
\(75\) 0 0
\(76\) −7.41641 −0.850720
\(77\) − 6.47214i − 0.737568i
\(78\) 0 0
\(79\) −12.9443 −1.45634 −0.728172 0.685394i \(-0.759630\pi\)
−0.728172 + 0.685394i \(0.759630\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 4.47214i 0.493865i
\(83\) − 16.9443i − 1.85988i −0.367717 0.929938i \(-0.619860\pi\)
0.367717 0.929938i \(-0.380140\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 20.0000 2.15666
\(87\) 0 0
\(88\) 14.4721i 1.54273i
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 4.47214 0.468807
\(92\) − 12.0000i − 1.25109i
\(93\) 0 0
\(94\) 28.9443 2.98537
\(95\) 0 0
\(96\) 0 0
\(97\) 8.47214i 0.860215i 0.902778 + 0.430108i \(0.141524\pi\)
−0.902778 + 0.430108i \(0.858476\pi\)
\(98\) − 2.23607i − 0.225877i
\(99\) 0 0
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −10.0000 −0.980581
\(105\) 0 0
\(106\) 7.88854 0.766203
\(107\) 12.9443i 1.25137i 0.780076 + 0.625685i \(0.215180\pi\)
−0.780076 + 0.625685i \(0.784820\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.00000i − 0.0944911i
\(113\) 0.472136i 0.0444148i 0.999753 + 0.0222074i \(0.00706942\pi\)
−0.999753 + 0.0222074i \(0.992931\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) − 20.0000i − 1.84115i
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 30.8885 2.80805
\(122\) − 4.47214i − 0.404888i
\(123\) 0 0
\(124\) −4.58359 −0.411619
\(125\) 0 0
\(126\) 0 0
\(127\) − 4.94427i − 0.438733i −0.975643 0.219367i \(-0.929601\pi\)
0.975643 0.219367i \(-0.0703991\pi\)
\(128\) 15.6525i 1.38350i
\(129\) 0 0
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 2.47214i 0.214361i
\(134\) 8.94427 0.772667
\(135\) 0 0
\(136\) 4.47214 0.383482
\(137\) − 3.52786i − 0.301406i −0.988579 0.150703i \(-0.951846\pi\)
0.988579 0.150703i \(-0.0481537\pi\)
\(138\) 0 0
\(139\) −7.41641 −0.629052 −0.314526 0.949249i \(-0.601845\pi\)
−0.314526 + 0.949249i \(0.601845\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 12.3607i − 1.03729i
\(143\) 28.9443i 2.42044i
\(144\) 0 0
\(145\) 0 0
\(146\) −27.8885 −2.30807
\(147\) 0 0
\(148\) 20.8328i 1.71245i
\(149\) −14.9443 −1.22428 −0.612141 0.790748i \(-0.709692\pi\)
−0.612141 + 0.790748i \(0.709692\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) − 5.52786i − 0.448369i
\(153\) 0 0
\(154\) 14.4721 1.16620
\(155\) 0 0
\(156\) 0 0
\(157\) − 0.472136i − 0.0376806i −0.999823 0.0188403i \(-0.994003\pi\)
0.999823 0.0188403i \(-0.00599740\pi\)
\(158\) − 28.9443i − 2.30268i
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 16.9443i 1.32718i 0.748097 + 0.663589i \(0.230968\pi\)
−0.748097 + 0.663589i \(0.769032\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 37.8885 2.94072
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) 26.8328i 2.04598i
\(173\) − 2.94427i − 0.223849i −0.993717 0.111924i \(-0.964299\pi\)
0.993717 0.111924i \(-0.0357015\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.47214 0.487856
\(177\) 0 0
\(178\) − 4.47214i − 0.335201i
\(179\) 6.47214 0.483750 0.241875 0.970307i \(-0.422238\pi\)
0.241875 + 0.970307i \(0.422238\pi\)
\(180\) 0 0
\(181\) 1.05573 0.0784717 0.0392358 0.999230i \(-0.487508\pi\)
0.0392358 + 0.999230i \(0.487508\pi\)
\(182\) 10.0000i 0.741249i
\(183\) 0 0
\(184\) 8.94427 0.659380
\(185\) 0 0
\(186\) 0 0
\(187\) − 12.9443i − 0.946579i
\(188\) 38.8328i 2.83217i
\(189\) 0 0
\(190\) 0 0
\(191\) −0.583592 −0.0422272 −0.0211136 0.999777i \(-0.506721\pi\)
−0.0211136 + 0.999777i \(0.506721\pi\)
\(192\) 0 0
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −18.9443 −1.36012
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) − 15.5279i − 1.10631i −0.833077 0.553157i \(-0.813423\pi\)
0.833077 0.553157i \(-0.186577\pi\)
\(198\) 0 0
\(199\) −27.4164 −1.94350 −0.971749 0.236017i \(-0.924158\pi\)
−0.971749 + 0.236017i \(0.924158\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 31.3050i 2.20261i
\(203\) − 2.00000i − 0.140372i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 4.47214i 0.310087i
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −16.9443 −1.16649 −0.583246 0.812296i \(-0.698218\pi\)
−0.583246 + 0.812296i \(0.698218\pi\)
\(212\) 10.5836i 0.726884i
\(213\) 0 0
\(214\) −28.9443 −1.97859
\(215\) 0 0
\(216\) 0 0
\(217\) 1.52786i 0.103718i
\(218\) 4.47214i 0.302891i
\(219\) 0 0
\(220\) 0 0
\(221\) 8.94427 0.601657
\(222\) 0 0
\(223\) 12.9443i 0.866813i 0.901199 + 0.433406i \(0.142688\pi\)
−0.901199 + 0.433406i \(0.857312\pi\)
\(224\) 6.70820 0.448211
\(225\) 0 0
\(226\) −1.05573 −0.0702260
\(227\) − 0.944272i − 0.0626735i −0.999509 0.0313368i \(-0.990024\pi\)
0.999509 0.0313368i \(-0.00997644\pi\)
\(228\) 0 0
\(229\) −23.8885 −1.57860 −0.789300 0.614008i \(-0.789556\pi\)
−0.789300 + 0.614008i \(0.789556\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 4.47214i 0.293610i
\(233\) − 9.41641i − 0.616889i −0.951242 0.308445i \(-0.900192\pi\)
0.951242 0.308445i \(-0.0998085\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 26.8328 1.74667
\(237\) 0 0
\(238\) − 4.47214i − 0.289886i
\(239\) −10.4721 −0.677386 −0.338693 0.940897i \(-0.609985\pi\)
−0.338693 + 0.940897i \(0.609985\pi\)
\(240\) 0 0
\(241\) −18.9443 −1.22031 −0.610154 0.792283i \(-0.708892\pi\)
−0.610154 + 0.792283i \(0.708892\pi\)
\(242\) 69.0689i 4.43992i
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) 0 0
\(247\) − 11.0557i − 0.703459i
\(248\) − 3.41641i − 0.216942i
\(249\) 0 0
\(250\) 0 0
\(251\) −16.9443 −1.06951 −0.534756 0.845006i \(-0.679597\pi\)
−0.534756 + 0.845006i \(0.679597\pi\)
\(252\) 0 0
\(253\) − 25.8885i − 1.62760i
\(254\) 11.0557 0.693698
\(255\) 0 0
\(256\) −9.00000 −0.562500
\(257\) − 18.9443i − 1.18171i −0.806777 0.590856i \(-0.798790\pi\)
0.806777 0.590856i \(-0.201210\pi\)
\(258\) 0 0
\(259\) 6.94427 0.431496
\(260\) 0 0
\(261\) 0 0
\(262\) − 8.94427i − 0.552579i
\(263\) 7.05573i 0.435075i 0.976052 + 0.217537i \(0.0698024\pi\)
−0.976052 + 0.217537i \(0.930198\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −5.52786 −0.338935
\(267\) 0 0
\(268\) 12.0000i 0.733017i
\(269\) 11.8885 0.724857 0.362429 0.932012i \(-0.381948\pi\)
0.362429 + 0.932012i \(0.381948\pi\)
\(270\) 0 0
\(271\) −1.52786 −0.0928111 −0.0464056 0.998923i \(-0.514777\pi\)
−0.0464056 + 0.998923i \(0.514777\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 0 0
\(274\) 7.88854 0.476564
\(275\) 0 0
\(276\) 0 0
\(277\) 18.9443i 1.13825i 0.822251 + 0.569125i \(0.192718\pi\)
−0.822251 + 0.569125i \(0.807282\pi\)
\(278\) − 16.5836i − 0.994618i
\(279\) 0 0
\(280\) 0 0
\(281\) 10.9443 0.652881 0.326440 0.945218i \(-0.394151\pi\)
0.326440 + 0.945218i \(0.394151\pi\)
\(282\) 0 0
\(283\) − 12.0000i − 0.713326i −0.934233 0.356663i \(-0.883914\pi\)
0.934233 0.356663i \(-0.116086\pi\)
\(284\) 16.5836 0.984055
\(285\) 0 0
\(286\) −64.7214 −3.82705
\(287\) 2.00000i 0.118056i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) − 37.4164i − 2.18963i
\(293\) 5.05573i 0.295359i 0.989035 + 0.147679i \(0.0471804\pi\)
−0.989035 + 0.147679i \(0.952820\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −15.5279 −0.902539
\(297\) 0 0
\(298\) − 33.4164i − 1.93576i
\(299\) 17.8885 1.03452
\(300\) 0 0
\(301\) 8.94427 0.515539
\(302\) − 35.7771i − 2.05874i
\(303\) 0 0
\(304\) −2.47214 −0.141787
\(305\) 0 0
\(306\) 0 0
\(307\) − 15.0557i − 0.859276i −0.903001 0.429638i \(-0.858641\pi\)
0.903001 0.429638i \(-0.141359\pi\)
\(308\) 19.4164i 1.10635i
\(309\) 0 0
\(310\) 0 0
\(311\) −25.8885 −1.46800 −0.734002 0.679147i \(-0.762350\pi\)
−0.734002 + 0.679147i \(0.762350\pi\)
\(312\) 0 0
\(313\) 17.4164i 0.984434i 0.870473 + 0.492217i \(0.163813\pi\)
−0.870473 + 0.492217i \(0.836187\pi\)
\(314\) 1.05573 0.0595782
\(315\) 0 0
\(316\) 38.8328 2.18452
\(317\) − 14.3607i − 0.806576i −0.915073 0.403288i \(-0.867867\pi\)
0.915073 0.403288i \(-0.132133\pi\)
\(318\) 0 0
\(319\) 12.9443 0.724740
\(320\) 0 0
\(321\) 0 0
\(322\) − 8.94427i − 0.498445i
\(323\) 4.94427i 0.275107i
\(324\) 0 0
\(325\) 0 0
\(326\) −37.8885 −2.09845
\(327\) 0 0
\(328\) − 4.47214i − 0.246932i
\(329\) 12.9443 0.713641
\(330\) 0 0
\(331\) 0.944272 0.0519019 0.0259509 0.999663i \(-0.491739\pi\)
0.0259509 + 0.999663i \(0.491739\pi\)
\(332\) 50.8328i 2.78981i
\(333\) 0 0
\(334\) −17.8885 −0.978818
\(335\) 0 0
\(336\) 0 0
\(337\) − 23.8885i − 1.30129i −0.759381 0.650646i \(-0.774498\pi\)
0.759381 0.650646i \(-0.225502\pi\)
\(338\) − 15.6525i − 0.851382i
\(339\) 0 0
\(340\) 0 0
\(341\) −9.88854 −0.535495
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) −20.0000 −1.07833
\(345\) 0 0
\(346\) 6.58359 0.353936
\(347\) 8.00000i 0.429463i 0.976673 + 0.214731i \(0.0688876\pi\)
−0.976673 + 0.214731i \(0.931112\pi\)
\(348\) 0 0
\(349\) 11.8885 0.636379 0.318190 0.948027i \(-0.396925\pi\)
0.318190 + 0.948027i \(0.396925\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 43.4164i 2.31410i
\(353\) 7.88854i 0.419865i 0.977716 + 0.209932i \(0.0673244\pi\)
−0.977716 + 0.209932i \(0.932676\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 14.4721i 0.764876i
\(359\) 18.4721 0.974922 0.487461 0.873145i \(-0.337923\pi\)
0.487461 + 0.873145i \(0.337923\pi\)
\(360\) 0 0
\(361\) −12.8885 −0.678344
\(362\) 2.36068i 0.124075i
\(363\) 0 0
\(364\) −13.4164 −0.703211
\(365\) 0 0
\(366\) 0 0
\(367\) 3.05573i 0.159508i 0.996815 + 0.0797539i \(0.0254134\pi\)
−0.996815 + 0.0797539i \(0.974587\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) 0 0
\(370\) 0 0
\(371\) 3.52786 0.183158
\(372\) 0 0
\(373\) − 6.00000i − 0.310668i −0.987862 0.155334i \(-0.950355\pi\)
0.987862 0.155334i \(-0.0496454\pi\)
\(374\) 28.9443 1.49667
\(375\) 0 0
\(376\) −28.9443 −1.49269
\(377\) 8.94427i 0.460653i
\(378\) 0 0
\(379\) 37.8885 1.94620 0.973102 0.230375i \(-0.0739953\pi\)
0.973102 + 0.230375i \(0.0739953\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 1.30495i − 0.0667671i
\(383\) − 8.00000i − 0.408781i −0.978889 0.204390i \(-0.934479\pi\)
0.978889 0.204390i \(-0.0655212\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −31.3050 −1.59338
\(387\) 0 0
\(388\) − 25.4164i − 1.29032i
\(389\) −6.94427 −0.352089 −0.176044 0.984382i \(-0.556330\pi\)
−0.176044 + 0.984382i \(0.556330\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 2.23607i 0.112938i
\(393\) 0 0
\(394\) 34.7214 1.74924
\(395\) 0 0
\(396\) 0 0
\(397\) − 13.4164i − 0.673350i −0.941621 0.336675i \(-0.890698\pi\)
0.941621 0.336675i \(-0.109302\pi\)
\(398\) − 61.3050i − 3.07294i
\(399\) 0 0
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) − 6.83282i − 0.340367i
\(404\) −42.0000 −2.08958
\(405\) 0 0
\(406\) 4.47214 0.221948
\(407\) 44.9443i 2.22780i
\(408\) 0 0
\(409\) −11.8885 −0.587851 −0.293925 0.955828i \(-0.594962\pi\)
−0.293925 + 0.955828i \(0.594962\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 8.94427i − 0.440119i
\(414\) 0 0
\(415\) 0 0
\(416\) −30.0000 −1.47087
\(417\) 0 0
\(418\) − 35.7771i − 1.74991i
\(419\) −29.8885 −1.46015 −0.730075 0.683367i \(-0.760515\pi\)
−0.730075 + 0.683367i \(0.760515\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) − 37.8885i − 1.84439i
\(423\) 0 0
\(424\) −7.88854 −0.383102
\(425\) 0 0
\(426\) 0 0
\(427\) − 2.00000i − 0.0967868i
\(428\) − 38.8328i − 1.87705i
\(429\) 0 0
\(430\) 0 0
\(431\) 18.4721 0.889771 0.444886 0.895587i \(-0.353244\pi\)
0.444886 + 0.895587i \(0.353244\pi\)
\(432\) 0 0
\(433\) − 16.4721i − 0.791600i −0.918337 0.395800i \(-0.870467\pi\)
0.918337 0.395800i \(-0.129533\pi\)
\(434\) −3.41641 −0.163993
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) 9.88854i 0.473033i
\(438\) 0 0
\(439\) −1.52786 −0.0729210 −0.0364605 0.999335i \(-0.511608\pi\)
−0.0364605 + 0.999335i \(0.511608\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 20.0000i 0.951303i
\(443\) − 8.00000i − 0.380091i −0.981775 0.190046i \(-0.939136\pi\)
0.981775 0.190046i \(-0.0608636\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −28.9443 −1.37055
\(447\) 0 0
\(448\) 13.0000i 0.614192i
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) −12.9443 −0.609522
\(452\) − 1.41641i − 0.0666222i
\(453\) 0 0
\(454\) 2.11146 0.0990955
\(455\) 0 0
\(456\) 0 0
\(457\) 6.94427i 0.324839i 0.986722 + 0.162420i \(0.0519298\pi\)
−0.986722 + 0.162420i \(0.948070\pi\)
\(458\) − 53.4164i − 2.49598i
\(459\) 0 0
\(460\) 0 0
\(461\) −3.88854 −0.181108 −0.0905538 0.995892i \(-0.528864\pi\)
−0.0905538 + 0.995892i \(0.528864\pi\)
\(462\) 0 0
\(463\) − 20.9443i − 0.973363i −0.873580 0.486681i \(-0.838207\pi\)
0.873580 0.486681i \(-0.161793\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 21.0557 0.975388
\(467\) 8.94427i 0.413892i 0.978352 + 0.206946i \(0.0663524\pi\)
−0.978352 + 0.206946i \(0.933648\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 20.0000i 0.920575i
\(473\) 57.8885i 2.66172i
\(474\) 0 0
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) 0 0
\(478\) − 23.4164i − 1.07104i
\(479\) −17.8885 −0.817348 −0.408674 0.912680i \(-0.634009\pi\)
−0.408674 + 0.912680i \(0.634009\pi\)
\(480\) 0 0
\(481\) −31.0557 −1.41602
\(482\) − 42.3607i − 1.92948i
\(483\) 0 0
\(484\) −92.6656 −4.21207
\(485\) 0 0
\(486\) 0 0
\(487\) − 20.9443i − 0.949076i −0.880235 0.474538i \(-0.842615\pi\)
0.880235 0.474538i \(-0.157385\pi\)
\(488\) 4.47214i 0.202444i
\(489\) 0 0
\(490\) 0 0
\(491\) 21.3050 0.961479 0.480740 0.876863i \(-0.340368\pi\)
0.480740 + 0.876863i \(0.340368\pi\)
\(492\) 0 0
\(493\) − 4.00000i − 0.180151i
\(494\) 24.7214 1.11227
\(495\) 0 0
\(496\) −1.52786 −0.0686031
\(497\) − 5.52786i − 0.247959i
\(498\) 0 0
\(499\) 13.8885 0.621737 0.310868 0.950453i \(-0.399380\pi\)
0.310868 + 0.950453i \(0.399380\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 37.8885i − 1.69105i
\(503\) 32.0000i 1.42681i 0.700752 + 0.713405i \(0.252848\pi\)
−0.700752 + 0.713405i \(0.747152\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 57.8885 2.57346
\(507\) 0 0
\(508\) 14.8328i 0.658100i
\(509\) −23.8885 −1.05884 −0.529421 0.848360i \(-0.677591\pi\)
−0.529421 + 0.848360i \(0.677591\pi\)
\(510\) 0 0
\(511\) −12.4721 −0.551735
\(512\) 11.1803i 0.494106i
\(513\) 0 0
\(514\) 42.3607 1.86845
\(515\) 0 0
\(516\) 0 0
\(517\) 83.7771i 3.68451i
\(518\) 15.5279i 0.682255i
\(519\) 0 0
\(520\) 0 0
\(521\) 19.8885 0.871333 0.435666 0.900108i \(-0.356513\pi\)
0.435666 + 0.900108i \(0.356513\pi\)
\(522\) 0 0
\(523\) 8.94427i 0.391106i 0.980693 + 0.195553i \(0.0626501\pi\)
−0.980693 + 0.195553i \(0.937350\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −15.7771 −0.687914
\(527\) 3.05573i 0.133110i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) − 7.41641i − 0.321542i
\(533\) − 8.94427i − 0.387419i
\(534\) 0 0
\(535\) 0 0
\(536\) −8.94427 −0.386334
\(537\) 0 0
\(538\) 26.5836i 1.14610i
\(539\) 6.47214 0.278775
\(540\) 0 0
\(541\) −11.8885 −0.511128 −0.255564 0.966792i \(-0.582261\pi\)
−0.255564 + 0.966792i \(0.582261\pi\)
\(542\) − 3.41641i − 0.146747i
\(543\) 0 0
\(544\) 13.4164 0.575224
\(545\) 0 0
\(546\) 0 0
\(547\) 5.88854i 0.251776i 0.992044 + 0.125888i \(0.0401780\pi\)
−0.992044 + 0.125888i \(0.959822\pi\)
\(548\) 10.5836i 0.452109i
\(549\) 0 0
\(550\) 0 0
\(551\) −4.94427 −0.210633
\(552\) 0 0
\(553\) − 12.9443i − 0.550446i
\(554\) −42.3607 −1.79973
\(555\) 0 0
\(556\) 22.2492 0.943577
\(557\) − 20.4721i − 0.867432i −0.901050 0.433716i \(-0.857202\pi\)
0.901050 0.433716i \(-0.142798\pi\)
\(558\) 0 0
\(559\) −40.0000 −1.69182
\(560\) 0 0
\(561\) 0 0
\(562\) 24.4721i 1.03229i
\(563\) 13.8885i 0.585332i 0.956215 + 0.292666i \(0.0945425\pi\)
−0.956215 + 0.292666i \(0.905458\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 26.8328 1.12787
\(567\) 0 0
\(568\) 12.3607i 0.518643i
\(569\) −39.8885 −1.67221 −0.836107 0.548566i \(-0.815174\pi\)
−0.836107 + 0.548566i \(0.815174\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) − 86.8328i − 3.63066i
\(573\) 0 0
\(574\) −4.47214 −0.186663
\(575\) 0 0
\(576\) 0 0
\(577\) 10.3607i 0.431321i 0.976468 + 0.215660i \(0.0691904\pi\)
−0.976468 + 0.215660i \(0.930810\pi\)
\(578\) 29.0689i 1.20911i
\(579\) 0 0
\(580\) 0 0
\(581\) 16.9443 0.702967
\(582\) 0 0
\(583\) 22.8328i 0.945639i
\(584\) 27.8885 1.15404
\(585\) 0 0
\(586\) −11.3050 −0.467003
\(587\) 4.00000i 0.165098i 0.996587 + 0.0825488i \(0.0263060\pi\)
−0.996587 + 0.0825488i \(0.973694\pi\)
\(588\) 0 0
\(589\) 3.77709 0.155632
\(590\) 0 0
\(591\) 0 0
\(592\) 6.94427i 0.285408i
\(593\) 23.8885i 0.980985i 0.871445 + 0.490492i \(0.163183\pi\)
−0.871445 + 0.490492i \(0.836817\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 44.8328 1.83642
\(597\) 0 0
\(598\) 40.0000i 1.63572i
\(599\) 12.3607 0.505044 0.252522 0.967591i \(-0.418740\pi\)
0.252522 + 0.967591i \(0.418740\pi\)
\(600\) 0 0
\(601\) 38.9443 1.58857 0.794285 0.607545i \(-0.207846\pi\)
0.794285 + 0.607545i \(0.207846\pi\)
\(602\) 20.0000i 0.815139i
\(603\) 0 0
\(604\) 48.0000 1.95309
\(605\) 0 0
\(606\) 0 0
\(607\) 38.8328i 1.57618i 0.615563 + 0.788088i \(0.288929\pi\)
−0.615563 + 0.788088i \(0.711071\pi\)
\(608\) − 16.5836i − 0.672553i
\(609\) 0 0
\(610\) 0 0
\(611\) −57.8885 −2.34192
\(612\) 0 0
\(613\) 6.94427i 0.280477i 0.990118 + 0.140238i \(0.0447868\pi\)
−0.990118 + 0.140238i \(0.955213\pi\)
\(614\) 33.6656 1.35863
\(615\) 0 0
\(616\) −14.4721 −0.583099
\(617\) − 16.4721i − 0.663143i −0.943430 0.331572i \(-0.892421\pi\)
0.943430 0.331572i \(-0.107579\pi\)
\(618\) 0 0
\(619\) −39.4164 −1.58428 −0.792140 0.610340i \(-0.791033\pi\)
−0.792140 + 0.610340i \(0.791033\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 57.8885i − 2.32112i
\(623\) − 2.00000i − 0.0801283i
\(624\) 0 0
\(625\) 0 0
\(626\) −38.9443 −1.55653
\(627\) 0 0
\(628\) 1.41641i 0.0565208i
\(629\) 13.8885 0.553773
\(630\) 0 0
\(631\) 30.8328 1.22744 0.613718 0.789526i \(-0.289673\pi\)
0.613718 + 0.789526i \(0.289673\pi\)
\(632\) 28.9443i 1.15134i
\(633\) 0 0
\(634\) 32.1115 1.27531
\(635\) 0 0
\(636\) 0 0
\(637\) 4.47214i 0.177192i
\(638\) 28.9443i 1.14591i
\(639\) 0 0
\(640\) 0 0
\(641\) −16.8328 −0.664856 −0.332428 0.943129i \(-0.607868\pi\)
−0.332428 + 0.943129i \(0.607868\pi\)
\(642\) 0 0
\(643\) 15.0557i 0.593740i 0.954918 + 0.296870i \(0.0959428\pi\)
−0.954918 + 0.296870i \(0.904057\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) −11.0557 −0.434982
\(647\) 1.88854i 0.0742463i 0.999311 + 0.0371232i \(0.0118194\pi\)
−0.999311 + 0.0371232i \(0.988181\pi\)
\(648\) 0 0
\(649\) 57.8885 2.27232
\(650\) 0 0
\(651\) 0 0
\(652\) − 50.8328i − 1.99077i
\(653\) − 22.5836i − 0.883764i −0.897073 0.441882i \(-0.854311\pi\)
0.897073 0.441882i \(-0.145689\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 28.9443i 1.12837i
\(659\) 21.3050 0.829923 0.414962 0.909839i \(-0.363795\pi\)
0.414962 + 0.909839i \(0.363795\pi\)
\(660\) 0 0
\(661\) −35.8885 −1.39590 −0.697951 0.716145i \(-0.745905\pi\)
−0.697951 + 0.716145i \(0.745905\pi\)
\(662\) 2.11146i 0.0820641i
\(663\) 0 0
\(664\) −37.8885 −1.47036
\(665\) 0 0
\(666\) 0 0
\(667\) − 8.00000i − 0.309761i
\(668\) − 24.0000i − 0.928588i
\(669\) 0 0
\(670\) 0 0
\(671\) 12.9443 0.499708
\(672\) 0 0
\(673\) − 8.83282i − 0.340480i −0.985403 0.170240i \(-0.945546\pi\)
0.985403 0.170240i \(-0.0544543\pi\)
\(674\) 53.4164 2.05752
\(675\) 0 0
\(676\) 21.0000 0.807692
\(677\) − 21.0557i − 0.809237i −0.914485 0.404619i \(-0.867404\pi\)
0.914485 0.404619i \(-0.132596\pi\)
\(678\) 0 0
\(679\) −8.47214 −0.325131
\(680\) 0 0
\(681\) 0 0
\(682\) − 22.1115i − 0.846691i
\(683\) − 1.88854i − 0.0722631i −0.999347 0.0361316i \(-0.988496\pi\)
0.999347 0.0361316i \(-0.0115035\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 2.23607 0.0853735
\(687\) 0 0
\(688\) 8.94427i 0.340997i
\(689\) −15.7771 −0.601059
\(690\) 0 0
\(691\) 44.3607 1.68756 0.843780 0.536689i \(-0.180325\pi\)
0.843780 + 0.536689i \(0.180325\pi\)
\(692\) 8.83282i 0.335773i
\(693\) 0 0
\(694\) −17.8885 −0.679040
\(695\) 0 0
\(696\) 0 0
\(697\) 4.00000i 0.151511i
\(698\) 26.5836i 1.00620i
\(699\) 0 0
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 0 0
\(703\) − 17.1672i − 0.647473i
\(704\) −84.1378 −3.17106
\(705\) 0 0
\(706\) −17.6393 −0.663865
\(707\) 14.0000i 0.526524i
\(708\) 0 0
\(709\) −25.7771 −0.968079 −0.484039 0.875046i \(-0.660831\pi\)
−0.484039 + 0.875046i \(0.660831\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 4.47214i 0.167600i
\(713\) 6.11146i 0.228876i
\(714\) 0 0
\(715\) 0 0
\(716\) −19.4164 −0.725625
\(717\) 0 0
\(718\) 41.3050i 1.54149i
\(719\) −6.83282 −0.254821 −0.127411 0.991850i \(-0.540667\pi\)
−0.127411 + 0.991850i \(0.540667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 28.8197i − 1.07256i
\(723\) 0 0
\(724\) −3.16718 −0.117707
\(725\) 0 0
\(726\) 0 0
\(727\) − 38.8328i − 1.44023i −0.693855 0.720115i \(-0.744089\pi\)
0.693855 0.720115i \(-0.255911\pi\)
\(728\) − 10.0000i − 0.370625i
\(729\) 0 0
\(730\) 0 0
\(731\) 17.8885 0.661632
\(732\) 0 0
\(733\) − 10.5836i − 0.390914i −0.980712 0.195457i \(-0.937381\pi\)
0.980712 0.195457i \(-0.0626190\pi\)
\(734\) −6.83282 −0.252204
\(735\) 0 0
\(736\) 26.8328 0.989071
\(737\) 25.8885i 0.953617i
\(738\) 0 0
\(739\) 5.88854 0.216614 0.108307 0.994118i \(-0.465457\pi\)
0.108307 + 0.994118i \(0.465457\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 7.88854i 0.289598i
\(743\) 34.8328i 1.27789i 0.769252 + 0.638946i \(0.220629\pi\)
−0.769252 + 0.638946i \(0.779371\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 13.4164 0.491210
\(747\) 0 0
\(748\) 38.8328i 1.41987i
\(749\) −12.9443 −0.472973
\(750\) 0 0
\(751\) −20.9443 −0.764267 −0.382134 0.924107i \(-0.624811\pi\)
−0.382134 + 0.924107i \(0.624811\pi\)
\(752\) 12.9443i 0.472029i
\(753\) 0 0
\(754\) −20.0000 −0.728357
\(755\) 0 0
\(756\) 0 0
\(757\) 31.8885i 1.15901i 0.814969 + 0.579504i \(0.196754\pi\)
−0.814969 + 0.579504i \(0.803246\pi\)
\(758\) 84.7214i 3.07722i
\(759\) 0 0
\(760\) 0 0
\(761\) 27.8885 1.01096 0.505479 0.862839i \(-0.331316\pi\)
0.505479 + 0.862839i \(0.331316\pi\)
\(762\) 0 0
\(763\) 2.00000i 0.0724049i
\(764\) 1.75078 0.0633409
\(765\) 0 0
\(766\) 17.8885 0.646339
\(767\) 40.0000i 1.44432i
\(768\) 0 0
\(769\) 52.8328 1.90520 0.952600 0.304226i \(-0.0983976\pi\)
0.952600 + 0.304226i \(0.0983976\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 42.0000i − 1.51161i
\(773\) − 42.9443i − 1.54460i −0.635259 0.772299i \(-0.719107\pi\)
0.635259 0.772299i \(-0.280893\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 18.9443 0.680060
\(777\) 0 0
\(778\) − 15.5279i − 0.556701i
\(779\) 4.94427 0.177147
\(780\) 0 0
\(781\) 35.7771 1.28020
\(782\) − 17.8885i − 0.639693i
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 31.0557i 1.10702i 0.832843 + 0.553509i \(0.186711\pi\)
−0.832843 + 0.553509i \(0.813289\pi\)
\(788\) 46.5836i 1.65947i
\(789\) 0 0
\(790\) 0 0
\(791\) −0.472136 −0.0167872
\(792\) 0 0
\(793\) 8.94427i 0.317620i
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) 82.2492 2.91525
\(797\) 18.9443i 0.671041i 0.942033 + 0.335520i \(0.108912\pi\)
−0.942033 + 0.335520i \(0.891088\pi\)
\(798\) 0 0
\(799\) 25.8885 0.915871
\(800\) 0 0
\(801\) 0 0
\(802\) − 22.3607i − 0.789583i
\(803\) − 80.7214i − 2.84859i
\(804\) 0 0
\(805\) 0 0
\(806\) 15.2786 0.538167
\(807\) 0 0
\(808\) − 31.3050i − 1.10130i
\(809\) 38.9443 1.36921 0.684604 0.728915i \(-0.259975\pi\)
0.684604 + 0.728915i \(0.259975\pi\)
\(810\) 0 0
\(811\) 55.4164 1.94593 0.972967 0.230946i \(-0.0741820\pi\)
0.972967 + 0.230946i \(0.0741820\pi\)
\(812\) 6.00000i 0.210559i
\(813\) 0 0
\(814\) −100.498 −3.52247
\(815\) 0 0
\(816\) 0 0
\(817\) − 22.1115i − 0.773582i
\(818\) − 26.5836i − 0.929474i
\(819\) 0 0
\(820\) 0 0
\(821\) −33.7771 −1.17883 −0.589414 0.807831i \(-0.700641\pi\)
−0.589414 + 0.807831i \(0.700641\pi\)
\(822\) 0 0
\(823\) − 44.9443i − 1.56666i −0.621607 0.783329i \(-0.713520\pi\)
0.621607 0.783329i \(-0.286480\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 20.0000 0.695889
\(827\) − 12.9443i − 0.450116i −0.974345 0.225058i \(-0.927743\pi\)
0.974345 0.225058i \(-0.0722572\pi\)
\(828\) 0 0
\(829\) 13.0557 0.453444 0.226722 0.973959i \(-0.427199\pi\)
0.226722 + 0.973959i \(0.427199\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 58.1378i − 2.01556i
\(833\) − 2.00000i − 0.0692959i
\(834\) 0 0
\(835\) 0 0
\(836\) 48.0000 1.66011
\(837\) 0 0
\(838\) − 66.8328i − 2.30870i
\(839\) 54.8328 1.89304 0.946520 0.322647i \(-0.104573\pi\)
0.946520 + 0.322647i \(0.104573\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 49.1935i 1.69532i
\(843\) 0 0
\(844\) 50.8328 1.74974
\(845\) 0 0
\(846\) 0 0
\(847\) 30.8885i 1.06134i
\(848\) 3.52786i 0.121147i
\(849\) 0 0
\(850\) 0 0
\(851\) 27.7771 0.952186
\(852\) 0 0
\(853\) 31.3050i 1.07186i 0.844262 + 0.535931i \(0.180039\pi\)
−0.844262 + 0.535931i \(0.819961\pi\)
\(854\) 4.47214 0.153033
\(855\) 0 0
\(856\) 28.9443 0.989295
\(857\) − 36.8328i − 1.25819i −0.777330 0.629093i \(-0.783427\pi\)
0.777330 0.629093i \(-0.216573\pi\)
\(858\) 0 0
\(859\) −50.4721 −1.72209 −0.861044 0.508531i \(-0.830189\pi\)
−0.861044 + 0.508531i \(0.830189\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 41.3050i 1.40685i
\(863\) 21.8885i 0.745095i 0.928013 + 0.372547i \(0.121516\pi\)
−0.928013 + 0.372547i \(0.878484\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 36.8328 1.25163
\(867\) 0 0
\(868\) − 4.58359i − 0.155577i
\(869\) 83.7771 2.84194
\(870\) 0 0
\(871\) −17.8885 −0.606130
\(872\) − 4.47214i − 0.151446i
\(873\) 0 0
\(874\) −22.1115 −0.747931
\(875\) 0 0
\(876\) 0 0
\(877\) − 56.8328i − 1.91911i −0.281525 0.959554i \(-0.590840\pi\)
0.281525 0.959554i \(-0.409160\pi\)
\(878\) − 3.41641i − 0.115298i
\(879\) 0 0
\(880\) 0 0
\(881\) 27.8885 0.939589 0.469794 0.882776i \(-0.344328\pi\)
0.469794 + 0.882776i \(0.344328\pi\)
\(882\) 0 0
\(883\) 37.8885i 1.27505i 0.770429 + 0.637526i \(0.220042\pi\)
−0.770429 + 0.637526i \(0.779958\pi\)
\(884\) −26.8328 −0.902485
\(885\) 0 0
\(886\) 17.8885 0.600977
\(887\) 30.8328i 1.03526i 0.855603 + 0.517632i \(0.173186\pi\)
−0.855603 + 0.517632i \(0.826814\pi\)
\(888\) 0 0
\(889\) 4.94427 0.165826
\(890\) 0 0
\(891\) 0 0
\(892\) − 38.8328i − 1.30022i
\(893\) − 32.0000i − 1.07084i
\(894\) 0 0
\(895\) 0 0
\(896\) −15.6525 −0.522913
\(897\) 0 0
\(898\) − 31.3050i − 1.04466i
\(899\) −3.05573 −0.101914
\(900\) 0 0
\(901\) 7.05573 0.235060
\(902\) − 28.9443i − 0.963739i
\(903\) 0 0
\(904\) 1.05573 0.0351130
\(905\) 0 0
\(906\) 0 0
\(907\) 53.8885i 1.78934i 0.446728 + 0.894670i \(0.352589\pi\)
−0.446728 + 0.894670i \(0.647411\pi\)
\(908\) 2.83282i 0.0940103i
\(909\) 0 0
\(910\) 0 0
\(911\) −46.2492 −1.53231 −0.766153 0.642659i \(-0.777831\pi\)
−0.766153 + 0.642659i \(0.777831\pi\)
\(912\) 0 0
\(913\) 109.666i 3.62940i
\(914\) −15.5279 −0.513616
\(915\) 0 0
\(916\) 71.6656 2.36790
\(917\) − 4.00000i − 0.132092i
\(918\) 0 0
\(919\) 35.0557 1.15638 0.578191 0.815902i \(-0.303759\pi\)
0.578191 + 0.815902i \(0.303759\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 8.69505i − 0.286356i
\(923\) 24.7214i 0.813713i
\(924\) 0 0
\(925\) 0 0
\(926\) 46.8328 1.53902
\(927\) 0 0
\(928\) 13.4164i 0.440415i
\(929\) −16.1115 −0.528600 −0.264300 0.964441i \(-0.585141\pi\)
−0.264300 + 0.964441i \(0.585141\pi\)
\(930\) 0 0
\(931\) −2.47214 −0.0810210
\(932\) 28.2492i 0.925334i
\(933\) 0 0
\(934\) −20.0000 −0.654420
\(935\) 0 0
\(936\) 0 0
\(937\) − 52.4721i − 1.71419i −0.515158 0.857095i \(-0.672267\pi\)
0.515158 0.857095i \(-0.327733\pi\)
\(938\) 8.94427i 0.292041i
\(939\) 0 0
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) 8.00000i 0.260516i
\(944\) 8.94427 0.291111
\(945\) 0 0
\(946\) −129.443 −4.20855
\(947\) 17.8885i 0.581300i 0.956830 + 0.290650i \(0.0938715\pi\)
−0.956830 + 0.290650i \(0.906129\pi\)
\(948\) 0 0
\(949\) 55.7771 1.81060
\(950\) 0 0
\(951\) 0 0
\(952\) 4.47214i 0.144943i
\(953\) − 33.4164i − 1.08246i −0.840873 0.541232i \(-0.817958\pi\)
0.840873 0.541232i \(-0.182042\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 31.4164 1.01608
\(957\) 0 0
\(958\) − 40.0000i − 1.29234i
\(959\) 3.52786 0.113921
\(960\) 0 0
\(961\) −28.6656 −0.924698
\(962\) − 69.4427i − 2.23892i
\(963\) 0 0
\(964\) 56.8328 1.83046
\(965\) 0 0
\(966\) 0 0
\(967\) 25.8885i 0.832519i 0.909246 + 0.416260i \(0.136659\pi\)
−0.909246 + 0.416260i \(0.863341\pi\)
\(968\) − 69.0689i − 2.21996i
\(969\) 0 0
\(970\) 0 0
\(971\) −40.9443 −1.31396 −0.656982 0.753906i \(-0.728167\pi\)
−0.656982 + 0.753906i \(0.728167\pi\)
\(972\) 0 0
\(973\) − 7.41641i − 0.237759i
\(974\) 46.8328 1.50062
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) − 30.5836i − 0.978456i −0.872156 0.489228i \(-0.837279\pi\)
0.872156 0.489228i \(-0.162721\pi\)
\(978\) 0 0
\(979\) 12.9443 0.413701
\(980\) 0 0
\(981\) 0 0
\(982\) 47.6393i 1.52023i
\(983\) − 22.8328i − 0.728254i −0.931349 0.364127i \(-0.881367\pi\)
0.931349 0.364127i \(-0.118633\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 8.94427 0.284844
\(987\) 0 0
\(988\) 33.1672i 1.05519i
\(989\) 35.7771 1.13765
\(990\) 0 0
\(991\) 4.94427 0.157060 0.0785300 0.996912i \(-0.474977\pi\)
0.0785300 + 0.996912i \(0.474977\pi\)
\(992\) − 10.2492i − 0.325413i
\(993\) 0 0
\(994\) 12.3607 0.392057
\(995\) 0 0
\(996\) 0 0
\(997\) − 5.41641i − 0.171539i −0.996315 0.0857697i \(-0.972665\pi\)
0.996315 0.0857697i \(-0.0273349\pi\)
\(998\) 31.0557i 0.983052i
\(999\) 0 0
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 1575.2.d.d.1324.4 4
3.2 odd 2 525.2.d.c.274.2 4
5.2 odd 4 1575.2.a.r.1.1 2
5.3 odd 4 315.2.a.d.1.2 2
5.4 even 2 inner 1575.2.d.d.1324.1 4
15.2 even 4 525.2.a.g.1.2 2
15.8 even 4 105.2.a.b.1.1 2
15.14 odd 2 525.2.d.c.274.3 4
20.3 even 4 5040.2.a.bw.1.2 2
35.13 even 4 2205.2.a.w.1.2 2
60.23 odd 4 1680.2.a.v.1.1 2
60.47 odd 4 8400.2.a.cx.1.1 2
105.23 even 12 735.2.i.k.361.2 4
105.38 odd 12 735.2.i.i.226.2 4
105.53 even 12 735.2.i.k.226.2 4
105.62 odd 4 3675.2.a.y.1.2 2
105.68 odd 12 735.2.i.i.361.2 4
105.83 odd 4 735.2.a.k.1.1 2
120.53 even 4 6720.2.a.cx.1.1 2
120.83 odd 4 6720.2.a.cs.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.a.b.1.1 2 15.8 even 4
315.2.a.d.1.2 2 5.3 odd 4
525.2.a.g.1.2 2 15.2 even 4
525.2.d.c.274.2 4 3.2 odd 2
525.2.d.c.274.3 4 15.14 odd 2
735.2.a.k.1.1 2 105.83 odd 4
735.2.i.i.226.2 4 105.38 odd 12
735.2.i.i.361.2 4 105.68 odd 12
735.2.i.k.226.2 4 105.53 even 12
735.2.i.k.361.2 4 105.23 even 12
1575.2.a.r.1.1 2 5.2 odd 4
1575.2.d.d.1324.1 4 5.4 even 2 inner
1575.2.d.d.1324.4 4 1.1 even 1 trivial
1680.2.a.v.1.1 2 60.23 odd 4
2205.2.a.w.1.2 2 35.13 even 4
3675.2.a.y.1.2 2 105.62 odd 4
5040.2.a.bw.1.2 2 20.3 even 4
6720.2.a.cs.1.2 2 120.83 odd 4
6720.2.a.cx.1.1 2 120.53 even 4
8400.2.a.cx.1.1 2 60.47 odd 4