Properties

Label 725.2.b.a.349.2
Level $725$
Weight $2$
Character 725.349
Analytic conductor $5.789$
Analytic rank $0$
Dimension $2$
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] = [725,2,Mod(349,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("725.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 725.349
Dual form 725.2.b.a.349.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{4} +2.00000i q^{7} +3.00000i q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{4} +2.00000i q^{7} +3.00000i q^{8} +3.00000 q^{9} -6.00000 q^{11} +2.00000i q^{13} -2.00000 q^{14} -1.00000 q^{16} +2.00000i q^{17} +3.00000i q^{18} +2.00000 q^{19} -6.00000i q^{22} +2.00000i q^{23} -2.00000 q^{26} +2.00000i q^{28} +1.00000 q^{29} +2.00000 q^{31} +5.00000i q^{32} -2.00000 q^{34} +3.00000 q^{36} -10.0000i q^{37} +2.00000i q^{38} +2.00000 q^{41} +8.00000i q^{43} -6.00000 q^{44} -2.00000 q^{46} +12.0000i q^{47} +3.00000 q^{49} +2.00000i q^{52} -6.00000i q^{53} -6.00000 q^{56} +1.00000i q^{58} +8.00000 q^{59} -6.00000 q^{61} +2.00000i q^{62} +6.00000i q^{63} -7.00000 q^{64} -2.00000i q^{67} +2.00000i q^{68} -12.0000 q^{71} +9.00000i q^{72} -6.00000i q^{73} +10.0000 q^{74} +2.00000 q^{76} -12.0000i q^{77} +10.0000 q^{79} +9.00000 q^{81} +2.00000i q^{82} -14.0000i q^{83} -8.00000 q^{86} -18.0000i q^{88} -18.0000 q^{89} -4.00000 q^{91} +2.00000i q^{92} -12.0000 q^{94} -2.00000i q^{97} +3.00000i q^{98} -18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 6 q^{9} - 12 q^{11} - 4 q^{14} - 2 q^{16} + 4 q^{19} - 4 q^{26} + 2 q^{29} + 4 q^{31} - 4 q^{34} + 6 q^{36} + 4 q^{41} - 12 q^{44} - 4 q^{46} + 6 q^{49} - 12 q^{56} + 16 q^{59} - 12 q^{61} - 14 q^{64} - 24 q^{71} + 20 q^{74} + 4 q^{76} + 20 q^{79} + 18 q^{81} - 16 q^{86} - 36 q^{89} - 8 q^{91} - 24 q^{94} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).

\(n\) \(176\) \(552\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 3.00000i 1.06066i
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 3.00000i 0.707107i
\(19\) 2.00000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 6.00000i − 1.27920i
\(23\) 2.00000i 0.417029i 0.978019 + 0.208514i \(0.0668628\pi\)
−0.978019 + 0.208514i \(0.933137\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 2.00000i 0.377964i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 0 0
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) − 10.0000i − 1.64399i −0.569495 0.821995i \(-0.692861\pi\)
0.569495 0.821995i \(-0.307139\pi\)
\(38\) 2.00000i 0.324443i
\(39\) 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
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) −6.00000 −0.904534
\(45\) 0 0
\(46\) −2.00000 −0.294884
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −6.00000 −0.801784
\(57\) 0 0
\(58\) 1.00000i 0.131306i
\(59\) 8.00000 1.04151 0.520756 0.853706i \(-0.325650\pi\)
0.520756 + 0.853706i \(0.325650\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) 2.00000i 0.254000i
\(63\) 6.00000i 0.755929i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.00000i − 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) 2.00000i 0.242536i
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 9.00000i 1.06066i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 10.0000 1.16248
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) − 12.0000i − 1.36753i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 2.00000i 0.220863i
\(83\) − 14.0000i − 1.53670i −0.640030 0.768350i \(-0.721078\pi\)
0.640030 0.768350i \(-0.278922\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) − 18.0000i − 1.91881i
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 2.00000i 0.208514i
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) − 2.00000i − 0.203069i −0.994832 0.101535i \(-0.967625\pi\)
0.994832 0.101535i \(-0.0323753\pi\)
\(98\) 3.00000i 0.303046i
\(99\) −18.0000 −1.80907
\(100\) 0 0
\(101\) 10.0000 0.995037 0.497519 0.867453i \(-0.334245\pi\)
0.497519 + 0.867453i \(0.334245\pi\)
\(102\) 0 0
\(103\) − 6.00000i − 0.591198i −0.955312 0.295599i \(-0.904481\pi\)
0.955312 0.295599i \(-0.0955191\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) − 6.00000i − 0.580042i −0.957020 0.290021i \(-0.906338\pi\)
0.957020 0.290021i \(-0.0936623\pi\)
\(108\) 0 0
\(109\) 14.0000 1.34096 0.670478 0.741929i \(-0.266089\pi\)
0.670478 + 0.741929i \(0.266089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 2.00000i − 0.188982i
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.00000 0.0928477
\(117\) 6.00000i 0.554700i
\(118\) 8.00000i 0.736460i
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) 25.0000 2.27273
\(122\) − 6.00000i − 0.543214i
\(123\) 0 0
\(124\) 2.00000 0.179605
\(125\) 0 0
\(126\) −6.00000 −0.534522
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 0 0
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) − 6.00000i − 0.512615i −0.966595 0.256307i \(-0.917494\pi\)
0.966595 0.256307i \(-0.0825059\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 12.0000i − 1.00702i
\(143\) − 12.0000i − 1.00349i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) 0 0
\(148\) − 10.0000i − 0.821995i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) −4.00000 −0.325515 −0.162758 0.986666i \(-0.552039\pi\)
−0.162758 + 0.986666i \(0.552039\pi\)
\(152\) 6.00000i 0.486664i
\(153\) 6.00000i 0.485071i
\(154\) 12.0000 0.966988
\(155\) 0 0
\(156\) 0 0
\(157\) − 22.0000i − 1.75579i −0.478852 0.877896i \(-0.658947\pi\)
0.478852 0.877896i \(-0.341053\pi\)
\(158\) 10.0000i 0.795557i
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 9.00000i 0.707107i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) 14.0000 1.08661
\(167\) − 18.0000i − 1.39288i −0.717614 0.696441i \(-0.754766\pi\)
0.717614 0.696441i \(-0.245234\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 6.00000 0.458831
\(172\) 8.00000i 0.609994i
\(173\) − 14.0000i − 1.06440i −0.846619 0.532200i \(-0.821365\pi\)
0.846619 0.532200i \(-0.178635\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) 0 0
\(178\) − 18.0000i − 1.34916i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) − 4.00000i − 0.296500i
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 0 0
\(187\) − 12.0000i − 0.877527i
\(188\) 12.0000i 0.875190i
\(189\) 0 0
\(190\) 0 0
\(191\) −22.0000 −1.59186 −0.795932 0.605386i \(-0.793019\pi\)
−0.795932 + 0.605386i \(0.793019\pi\)
\(192\) 0 0
\(193\) − 10.0000i − 0.719816i −0.932988 0.359908i \(-0.882808\pi\)
0.932988 0.359908i \(-0.117192\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) − 18.0000i − 1.27920i
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 10.0000i 0.703598i
\(203\) 2.00000i 0.140372i
\(204\) 0 0
\(205\) 0 0
\(206\) 6.00000 0.418040
\(207\) 6.00000i 0.417029i
\(208\) − 2.00000i − 0.138675i
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) 0 0
\(214\) 6.00000 0.410152
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000i 0.271538i
\(218\) 14.0000i 0.948200i
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) − 14.0000i − 0.937509i −0.883328 0.468755i \(-0.844703\pi\)
0.883328 0.468755i \(-0.155297\pi\)
\(224\) −10.0000 −0.668153
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) − 22.0000i − 1.46019i −0.683345 0.730096i \(-0.739475\pi\)
0.683345 0.730096i \(-0.260525\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 3.00000i 0.196960i
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 8.00000 0.520756
\(237\) 0 0
\(238\) − 4.00000i − 0.259281i
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 25.0000i 1.60706i
\(243\) 0 0
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 0 0
\(247\) 4.00000i 0.254514i
\(248\) 6.00000i 0.381000i
\(249\) 0 0
\(250\) 0 0
\(251\) −6.00000 −0.378717 −0.189358 0.981908i \(-0.560641\pi\)
−0.189358 + 0.981908i \(0.560641\pi\)
\(252\) 6.00000i 0.377964i
\(253\) − 12.0000i − 0.754434i
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 30.0000i 1.87135i 0.352865 + 0.935674i \(0.385208\pi\)
−0.352865 + 0.935674i \(0.614792\pi\)
\(258\) 0 0
\(259\) 20.0000 1.24274
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 14.0000i 0.864923i
\(263\) 12.0000i 0.739952i 0.929041 + 0.369976i \(0.120634\pi\)
−0.929041 + 0.369976i \(0.879366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.00000 −0.245256
\(267\) 0 0
\(268\) − 2.00000i − 0.122169i
\(269\) −26.0000 −1.58525 −0.792624 0.609711i \(-0.791286\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) −2.00000 −0.121491 −0.0607457 0.998153i \(-0.519348\pi\)
−0.0607457 + 0.998153i \(0.519348\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) − 18.0000i − 1.08152i −0.841178 0.540758i \(-0.818138\pi\)
0.841178 0.540758i \(-0.181862\pi\)
\(278\) 0 0
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) 22.0000i 1.30776i 0.756596 + 0.653882i \(0.226861\pi\)
−0.756596 + 0.653882i \(0.773139\pi\)
\(284\) −12.0000 −0.712069
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 4.00000i 0.236113i
\(288\) 15.0000i 0.883883i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) − 6.00000i − 0.351123i
\(293\) − 2.00000i − 0.116841i −0.998292 0.0584206i \(-0.981394\pi\)
0.998292 0.0584206i \(-0.0186065\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 30.0000 1.74371
\(297\) 0 0
\(298\) 10.0000i 0.579284i
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) − 4.00000i − 0.230174i
\(303\) 0 0
\(304\) −2.00000 −0.114708
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) − 12.0000i − 0.683763i
\(309\) 0 0
\(310\) 0 0
\(311\) −22.0000 −1.24751 −0.623753 0.781622i \(-0.714393\pi\)
−0.623753 + 0.781622i \(0.714393\pi\)
\(312\) 0 0
\(313\) 2.00000i 0.113047i 0.998401 + 0.0565233i \(0.0180015\pi\)
−0.998401 + 0.0565233i \(0.981998\pi\)
\(314\) 22.0000 1.24153
\(315\) 0 0
\(316\) 10.0000 0.562544
\(317\) 14.0000i 0.786318i 0.919470 + 0.393159i \(0.128618\pi\)
−0.919470 + 0.393159i \(0.871382\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) 0 0
\(322\) − 4.00000i − 0.222911i
\(323\) 4.00000i 0.222566i
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) −4.00000 −0.221540
\(327\) 0 0
\(328\) 6.00000i 0.331295i
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) −18.0000 −0.989369 −0.494685 0.869072i \(-0.664716\pi\)
−0.494685 + 0.869072i \(0.664716\pi\)
\(332\) − 14.0000i − 0.768350i
\(333\) − 30.0000i − 1.64399i
\(334\) 18.0000 0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) − 2.00000i − 0.108947i −0.998515 0.0544735i \(-0.982652\pi\)
0.998515 0.0544735i \(-0.0173480\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 0 0
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 6.00000i 0.324443i
\(343\) 20.0000i 1.07990i
\(344\) −24.0000 −1.29399
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 6.00000i 0.322097i 0.986947 + 0.161048i \(0.0514875\pi\)
−0.986947 + 0.161048i \(0.948512\pi\)
\(348\) 0 0
\(349\) −34.0000 −1.81998 −0.909989 0.414632i \(-0.863910\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 30.0000i − 1.59901i
\(353\) − 14.0000i − 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) 12.0000i 0.634220i
\(359\) −22.0000 −1.16112 −0.580558 0.814219i \(-0.697165\pi\)
−0.580558 + 0.814219i \(0.697165\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 6.00000i 0.315353i
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) 0 0
\(366\) 0 0
\(367\) − 24.0000i − 1.25279i −0.779506 0.626395i \(-0.784530\pi\)
0.779506 0.626395i \(-0.215470\pi\)
\(368\) − 2.00000i − 0.104257i
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) 18.0000i 0.932005i 0.884783 + 0.466002i \(0.154306\pi\)
−0.884783 + 0.466002i \(0.845694\pi\)
\(374\) 12.0000 0.620505
\(375\) 0 0
\(376\) −36.0000 −1.85656
\(377\) 2.00000i 0.103005i
\(378\) 0 0
\(379\) −6.00000 −0.308199 −0.154100 0.988055i \(-0.549248\pi\)
−0.154100 + 0.988055i \(0.549248\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 22.0000i − 1.12562i
\(383\) 14.0000i 0.715367i 0.933843 + 0.357683i \(0.116433\pi\)
−0.933843 + 0.357683i \(0.883567\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 10.0000 0.508987
\(387\) 24.0000i 1.21999i
\(388\) − 2.00000i − 0.101535i
\(389\) 14.0000 0.709828 0.354914 0.934899i \(-0.384510\pi\)
0.354914 + 0.934899i \(0.384510\pi\)
\(390\) 0 0
\(391\) −4.00000 −0.202289
\(392\) 9.00000i 0.454569i
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) −18.0000 −0.904534
\(397\) 30.0000i 1.50566i 0.658217 + 0.752828i \(0.271311\pi\)
−0.658217 + 0.752828i \(0.728689\pi\)
\(398\) 4.00000i 0.200502i
\(399\) 0 0
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) 0 0
\(403\) 4.00000i 0.199254i
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) −2.00000 −0.0992583
\(407\) 60.0000i 2.97409i
\(408\) 0 0
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 6.00000i − 0.295599i
\(413\) 16.0000i 0.787309i
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) −10.0000 −0.490290
\(417\) 0 0
\(418\) − 12.0000i − 0.586939i
\(419\) −4.00000 −0.195413 −0.0977064 0.995215i \(-0.531151\pi\)
−0.0977064 + 0.995215i \(0.531151\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 14.0000i 0.681509i
\(423\) 36.0000i 1.75038i
\(424\) 18.0000 0.874157
\(425\) 0 0
\(426\) 0 0
\(427\) − 12.0000i − 0.580721i
\(428\) − 6.00000i − 0.290021i
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) − 34.0000i − 1.63394i −0.576683 0.816968i \(-0.695653\pi\)
0.576683 0.816968i \(-0.304347\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) 4.00000i 0.191346i
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) − 4.00000i − 0.190261i
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 14.0000 0.662919
\(447\) 0 0
\(448\) − 14.0000i − 0.661438i
\(449\) −10.0000 −0.471929 −0.235965 0.971762i \(-0.575825\pi\)
−0.235965 + 0.971762i \(0.575825\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) 2.00000i 0.0940721i
\(453\) 0 0
\(454\) 22.0000 1.03251
\(455\) 0 0
\(456\) 0 0
\(457\) 38.0000i 1.77757i 0.458329 + 0.888783i \(0.348448\pi\)
−0.458329 + 0.888783i \(0.651552\pi\)
\(458\) 6.00000i 0.280362i
\(459\) 0 0
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) 0 0
\(463\) 22.0000i 1.02243i 0.859454 + 0.511213i \(0.170804\pi\)
−0.859454 + 0.511213i \(0.829196\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) − 36.0000i − 1.66588i −0.553362 0.832941i \(-0.686655\pi\)
0.553362 0.832941i \(-0.313345\pi\)
\(468\) 6.00000i 0.277350i
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 24.0000i 1.10469i
\(473\) − 48.0000i − 2.20704i
\(474\) 0 0
\(475\) 0 0
\(476\) −4.00000 −0.183340
\(477\) − 18.0000i − 0.824163i
\(478\) 12.0000i 0.548867i
\(479\) 14.0000 0.639676 0.319838 0.947472i \(-0.396371\pi\)
0.319838 + 0.947472i \(0.396371\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) − 26.0000i − 1.18427i
\(483\) 0 0
\(484\) 25.0000 1.13636
\(485\) 0 0
\(486\) 0 0
\(487\) 34.0000i 1.54069i 0.637629 + 0.770344i \(0.279915\pi\)
−0.637629 + 0.770344i \(0.720085\pi\)
\(488\) − 18.0000i − 0.814822i
\(489\) 0 0
\(490\) 0 0
\(491\) −2.00000 −0.0902587 −0.0451294 0.998981i \(-0.514370\pi\)
−0.0451294 + 0.998981i \(0.514370\pi\)
\(492\) 0 0
\(493\) 2.00000i 0.0900755i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) −2.00000 −0.0898027
\(497\) − 24.0000i − 1.07655i
\(498\) 0 0
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 6.00000i − 0.267793i
\(503\) 16.0000i 0.713405i 0.934218 + 0.356702i \(0.116099\pi\)
−0.934218 + 0.356702i \(0.883901\pi\)
\(504\) −18.0000 −0.801784
\(505\) 0 0
\(506\) 12.0000 0.533465
\(507\) 0 0
\(508\) 16.0000i 0.709885i
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 12.0000 0.530849
\(512\) − 11.0000i − 0.486136i
\(513\) 0 0
\(514\) −30.0000 −1.32324
\(515\) 0 0
\(516\) 0 0
\(517\) − 72.0000i − 3.16656i
\(518\) 20.0000i 0.878750i
\(519\) 0 0
\(520\) 0 0
\(521\) 30.0000 1.31432 0.657162 0.753749i \(-0.271757\pi\)
0.657162 + 0.753749i \(0.271757\pi\)
\(522\) 3.00000i 0.131306i
\(523\) 42.0000i 1.83653i 0.395964 + 0.918266i \(0.370410\pi\)
−0.395964 + 0.918266i \(0.629590\pi\)
\(524\) 14.0000 0.611593
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 4.00000i 0.174243i
\(528\) 0 0
\(529\) 19.0000 0.826087
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) 4.00000i 0.173422i
\(533\) 4.00000i 0.173259i
\(534\) 0 0
\(535\) 0 0
\(536\) 6.00000 0.259161
\(537\) 0 0
\(538\) − 26.0000i − 1.12094i
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −14.0000 −0.601907 −0.300954 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) − 2.00000i − 0.0859074i
\(543\) 0 0
\(544\) −10.0000 −0.428746
\(545\) 0 0
\(546\) 0 0
\(547\) 2.00000i 0.0855138i 0.999086 + 0.0427569i \(0.0136141\pi\)
−0.999086 + 0.0427569i \(0.986386\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) −18.0000 −0.768221
\(550\) 0 0
\(551\) 2.00000 0.0852029
\(552\) 0 0
\(553\) 20.0000i 0.850487i
\(554\) 18.0000 0.764747
\(555\) 0 0
\(556\) 0 0
\(557\) 22.0000i 0.932170i 0.884740 + 0.466085i \(0.154336\pi\)
−0.884740 + 0.466085i \(0.845664\pi\)
\(558\) 6.00000i 0.254000i
\(559\) −16.0000 −0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 22.0000i 0.928014i
\(563\) 20.0000i 0.842900i 0.906852 + 0.421450i \(0.138479\pi\)
−0.906852 + 0.421450i \(0.861521\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −22.0000 −0.924729
\(567\) 18.0000i 0.755929i
\(568\) − 36.0000i − 1.51053i
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) − 12.0000i − 0.501745i
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) 0 0
\(576\) −21.0000 −0.875000
\(577\) − 22.0000i − 0.915872i −0.888985 0.457936i \(-0.848589\pi\)
0.888985 0.457936i \(-0.151411\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 0 0
\(580\) 0 0
\(581\) 28.0000 1.16164
\(582\) 0 0
\(583\) 36.0000i 1.49097i
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) 2.00000 0.0826192
\(587\) 18.0000i 0.742940i 0.928445 + 0.371470i \(0.121146\pi\)
−0.928445 + 0.371470i \(0.878854\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 10.0000i 0.410997i
\(593\) − 14.0000i − 0.574911i −0.957794 0.287456i \(-0.907191\pi\)
0.957794 0.287456i \(-0.0928094\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 0 0
\(598\) − 4.00000i − 0.163572i
\(599\) 18.0000 0.735460 0.367730 0.929933i \(-0.380135\pi\)
0.367730 + 0.929933i \(0.380135\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) − 16.0000i − 0.652111i
\(603\) − 6.00000i − 0.244339i
\(604\) −4.00000 −0.162758
\(605\) 0 0
\(606\) 0 0
\(607\) − 28.0000i − 1.13648i −0.822861 0.568242i \(-0.807624\pi\)
0.822861 0.568242i \(-0.192376\pi\)
\(608\) 10.0000i 0.405554i
\(609\) 0 0
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 6.00000i 0.242536i
\(613\) − 6.00000i − 0.242338i −0.992632 0.121169i \(-0.961336\pi\)
0.992632 0.121169i \(-0.0386643\pi\)
\(614\) 12.0000 0.484281
\(615\) 0 0
\(616\) 36.0000 1.45048
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 22.0000i − 0.882120i
\(623\) − 36.0000i − 1.44231i
\(624\) 0 0
\(625\) 0 0
\(626\) −2.00000 −0.0799361
\(627\) 0 0
\(628\) − 22.0000i − 0.877896i
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 30.0000i 1.19334i
\(633\) 0 0
\(634\) −14.0000 −0.556011
\(635\) 0 0
\(636\) 0 0
\(637\) 6.00000i 0.237729i
\(638\) − 6.00000i − 0.237542i
\(639\) −36.0000 −1.42414
\(640\) 0 0
\(641\) 26.0000 1.02694 0.513469 0.858108i \(-0.328360\pi\)
0.513469 + 0.858108i \(0.328360\pi\)
\(642\) 0 0
\(643\) − 26.0000i − 1.02534i −0.858586 0.512670i \(-0.828656\pi\)
0.858586 0.512670i \(-0.171344\pi\)
\(644\) −4.00000 −0.157622
\(645\) 0 0
\(646\) −4.00000 −0.157378
\(647\) 6.00000i 0.235884i 0.993020 + 0.117942i \(0.0376297\pi\)
−0.993020 + 0.117942i \(0.962370\pi\)
\(648\) 27.0000i 1.06066i
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) − 34.0000i − 1.33052i −0.746611 0.665261i \(-0.768320\pi\)
0.746611 0.665261i \(-0.231680\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) − 18.0000i − 0.702247i
\(658\) − 24.0000i − 0.935617i
\(659\) 26.0000 1.01282 0.506408 0.862294i \(-0.330973\pi\)
0.506408 + 0.862294i \(0.330973\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) − 18.0000i − 0.699590i
\(663\) 0 0
\(664\) 42.0000 1.62992
\(665\) 0 0
\(666\) 30.0000 1.16248
\(667\) 2.00000i 0.0774403i
\(668\) − 18.0000i − 0.696441i
\(669\) 0 0
\(670\) 0 0
\(671\) 36.0000 1.38976
\(672\) 0 0
\(673\) − 14.0000i − 0.539660i −0.962908 0.269830i \(-0.913032\pi\)
0.962908 0.269830i \(-0.0869676\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) 18.0000i 0.691796i 0.938272 + 0.345898i \(0.112426\pi\)
−0.938272 + 0.345898i \(0.887574\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) 0 0
\(681\) 0 0
\(682\) − 12.0000i − 0.459504i
\(683\) 18.0000i 0.688751i 0.938832 + 0.344375i \(0.111909\pi\)
−0.938832 + 0.344375i \(0.888091\pi\)
\(684\) 6.00000 0.229416
\(685\) 0 0
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) − 8.00000i − 0.304997i
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) −20.0000 −0.760836 −0.380418 0.924815i \(-0.624220\pi\)
−0.380418 + 0.924815i \(0.624220\pi\)
\(692\) − 14.0000i − 0.532200i
\(693\) − 36.0000i − 1.36753i
\(694\) −6.00000 −0.227757
\(695\) 0 0
\(696\) 0 0
\(697\) 4.00000i 0.151511i
\(698\) − 34.0000i − 1.28692i
\(699\) 0 0
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) − 20.0000i − 0.754314i
\(704\) 42.0000 1.58293
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) 20.0000i 0.752177i
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 0 0
\(711\) 30.0000 1.12509
\(712\) − 54.0000i − 2.02374i
\(713\) 4.00000i 0.149801i
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) − 22.0000i − 0.821033i
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) − 15.0000i − 0.558242i
\(723\) 0 0
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) 0 0
\(727\) 32.0000i 1.18681i 0.804902 + 0.593407i \(0.202218\pi\)
−0.804902 + 0.593407i \(0.797782\pi\)
\(728\) − 12.0000i − 0.444750i
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) −16.0000 −0.591781
\(732\) 0 0
\(733\) − 30.0000i − 1.10808i −0.832492 0.554038i \(-0.813086\pi\)
0.832492 0.554038i \(-0.186914\pi\)
\(734\) 24.0000 0.885856
\(735\) 0 0
\(736\) −10.0000 −0.368605
\(737\) 12.0000i 0.442026i
\(738\) 6.00000i 0.220863i
\(739\) −46.0000 −1.69214 −0.846069 0.533074i \(-0.821037\pi\)
−0.846069 + 0.533074i \(0.821037\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 12.0000i 0.440534i
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −18.0000 −0.659027
\(747\) − 42.0000i − 1.53670i
\(748\) − 12.0000i − 0.438763i
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −26.0000 −0.948753 −0.474377 0.880322i \(-0.657327\pi\)
−0.474377 + 0.880322i \(0.657327\pi\)
\(752\) − 12.0000i − 0.437595i
\(753\) 0 0
\(754\) −2.00000 −0.0728357
\(755\) 0 0
\(756\) 0 0
\(757\) 26.0000i 0.944986i 0.881334 + 0.472493i \(0.156646\pi\)
−0.881334 + 0.472493i \(0.843354\pi\)
\(758\) − 6.00000i − 0.217930i
\(759\) 0 0
\(760\) 0 0
\(761\) −10.0000 −0.362500 −0.181250 0.983437i \(-0.558014\pi\)
−0.181250 + 0.983437i \(0.558014\pi\)
\(762\) 0 0
\(763\) 28.0000i 1.01367i
\(764\) −22.0000 −0.795932
\(765\) 0 0
\(766\) −14.0000 −0.505841
\(767\) 16.0000i 0.577727i
\(768\) 0 0
\(769\) 46.0000 1.65880 0.829401 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 10.0000i − 0.359908i
\(773\) 10.0000i 0.359675i 0.983696 + 0.179838i \(0.0575572\pi\)
−0.983696 + 0.179838i \(0.942443\pi\)
\(774\) −24.0000 −0.862662
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) 0 0
\(778\) 14.0000i 0.501924i
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) 72.0000 2.57636
\(782\) − 4.00000i − 0.143040i
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) − 18.0000i − 0.641631i −0.947142 0.320815i \(-0.896043\pi\)
0.947142 0.320815i \(-0.103957\pi\)
\(788\) − 2.00000i − 0.0712470i
\(789\) 0 0
\(790\) 0 0
\(791\) −4.00000 −0.142224
\(792\) − 54.0000i − 1.91881i
\(793\) − 12.0000i − 0.426132i
\(794\) −30.0000 −1.06466
\(795\) 0 0
\(796\) 4.00000 0.141776
\(797\) 34.0000i 1.20434i 0.798367 + 0.602171i \(0.205697\pi\)
−0.798367 + 0.602171i \(0.794303\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) −54.0000 −1.90800
\(802\) − 14.0000i − 0.494357i
\(803\) 36.0000i 1.27041i
\(804\) 0 0
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) 0 0
\(808\) 30.0000i 1.05540i
\(809\) 14.0000 0.492214 0.246107 0.969243i \(-0.420849\pi\)
0.246107 + 0.969243i \(0.420849\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) 2.00000i 0.0701862i
\(813\) 0 0
\(814\) −60.0000 −2.10300
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000i 0.559769i
\(818\) 22.0000i 0.769212i
\(819\) −12.0000 −0.419314
\(820\) 0 0
\(821\) −2.00000 −0.0698005 −0.0349002 0.999391i \(-0.511111\pi\)
−0.0349002 + 0.999391i \(0.511111\pi\)
\(822\) 0 0
\(823\) 4.00000i 0.139431i 0.997567 + 0.0697156i \(0.0222092\pi\)
−0.997567 + 0.0697156i \(0.977791\pi\)
\(824\) 18.0000 0.627060
\(825\) 0 0
\(826\) −16.0000 −0.556711
\(827\) 48.0000i 1.66912i 0.550914 + 0.834562i \(0.314279\pi\)
−0.550914 + 0.834562i \(0.685721\pi\)
\(828\) 6.00000i 0.208514i
\(829\) −10.0000 −0.347314 −0.173657 0.984806i \(-0.555558\pi\)
−0.173657 + 0.984806i \(0.555558\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 14.0000i − 0.485363i
\(833\) 6.00000i 0.207888i
\(834\) 0 0
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 0 0
\(838\) − 4.00000i − 0.138178i
\(839\) 14.0000 0.483334 0.241667 0.970359i \(-0.422306\pi\)
0.241667 + 0.970359i \(0.422306\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 2.00000i 0.0689246i
\(843\) 0 0
\(844\) 14.0000 0.481900
\(845\) 0 0
\(846\) −36.0000 −1.23771
\(847\) 50.0000i 1.71802i
\(848\) 6.00000i 0.206041i
\(849\) 0 0
\(850\) 0 0
\(851\) 20.0000 0.685591
\(852\) 0 0
\(853\) − 22.0000i − 0.753266i −0.926363 0.376633i \(-0.877082\pi\)
0.926363 0.376633i \(-0.122918\pi\)
\(854\) 12.0000 0.410632
\(855\) 0 0
\(856\) 18.0000 0.615227
\(857\) − 58.0000i − 1.98124i −0.136637 0.990621i \(-0.543630\pi\)
0.136637 0.990621i \(-0.456370\pi\)
\(858\) 0 0
\(859\) −18.0000 −0.614152 −0.307076 0.951685i \(-0.599351\pi\)
−0.307076 + 0.951685i \(0.599351\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 24.0000i − 0.817443i
\(863\) − 42.0000i − 1.42970i −0.699280 0.714848i \(-0.746496\pi\)
0.699280 0.714848i \(-0.253504\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 34.0000 1.15537
\(867\) 0 0
\(868\) 4.00000i 0.135769i
\(869\) −60.0000 −2.03536
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 42.0000i 1.42230i
\(873\) − 6.00000i − 0.203069i
\(874\) −4.00000 −0.135302
\(875\) 0 0
\(876\) 0 0
\(877\) 38.0000i 1.28317i 0.767052 + 0.641584i \(0.221723\pi\)
−0.767052 + 0.641584i \(0.778277\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 42.0000 1.41502 0.707508 0.706705i \(-0.249819\pi\)
0.707508 + 0.706705i \(0.249819\pi\)
\(882\) 9.00000i 0.303046i
\(883\) − 50.0000i − 1.68263i −0.540542 0.841317i \(-0.681781\pi\)
0.540542 0.841317i \(-0.318219\pi\)
\(884\) −4.00000 −0.134535
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 8.00000i 0.268614i 0.990940 + 0.134307i \(0.0428808\pi\)
−0.990940 + 0.134307i \(0.957119\pi\)
\(888\) 0 0
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) −54.0000 −1.80907
\(892\) − 14.0000i − 0.468755i
\(893\) 24.0000i 0.803129i
\(894\) 0 0
\(895\) 0 0
\(896\) −6.00000 −0.200446
\(897\) 0 0
\(898\) − 10.0000i − 0.333704i
\(899\) 2.00000 0.0667037
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) − 12.0000i − 0.399556i
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) − 24.0000i − 0.796907i −0.917189 0.398453i \(-0.869547\pi\)
0.917189 0.398453i \(-0.130453\pi\)
\(908\) − 22.0000i − 0.730096i
\(909\) 30.0000 0.995037
\(910\) 0 0
\(911\) 14.0000 0.463841 0.231920 0.972735i \(-0.425499\pi\)
0.231920 + 0.972735i \(0.425499\pi\)
\(912\) 0 0
\(913\) 84.0000i 2.77999i
\(914\) −38.0000 −1.25693
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 28.0000i 0.924641i
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 18.0000i 0.592798i
\(923\) − 24.0000i − 0.789970i
\(924\) 0 0
\(925\) 0 0
\(926\) −22.0000 −0.722965
\(927\) − 18.0000i − 0.591198i
\(928\) 5.00000i 0.164133i
\(929\) −2.00000 −0.0656179 −0.0328089 0.999462i \(-0.510445\pi\)
−0.0328089 + 0.999462i \(0.510445\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 18.0000i 0.589610i
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) −18.0000 −0.588348
\(937\) − 18.0000i − 0.588034i −0.955800 0.294017i \(-0.905008\pi\)
0.955800 0.294017i \(-0.0949923\pi\)
\(938\) 4.00000i 0.130605i
\(939\) 0 0
\(940\) 0 0
\(941\) −26.0000 −0.847576 −0.423788 0.905761i \(-0.639300\pi\)
−0.423788 + 0.905761i \(0.639300\pi\)
\(942\) 0 0
\(943\) 4.00000i 0.130258i
\(944\) −8.00000 −0.260378
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) − 60.0000i − 1.94974i −0.222779 0.974869i \(-0.571513\pi\)
0.222779 0.974869i \(-0.428487\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) − 12.0000i − 0.388922i
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) 18.0000 0.582772
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) 14.0000i 0.452319i
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 20.0000i 0.644826i
\(963\) − 18.0000i − 0.580042i
\(964\) −26.0000 −0.837404
\(965\) 0 0
\(966\) 0 0
\(967\) 20.0000i 0.643157i 0.946883 + 0.321578i \(0.104213\pi\)
−0.946883 + 0.321578i \(0.895787\pi\)
\(968\) 75.0000i 2.41059i
\(969\) 0 0
\(970\) 0 0
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −34.0000 −1.08943
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) − 42.0000i − 1.34370i −0.740688 0.671850i \(-0.765500\pi\)
0.740688 0.671850i \(-0.234500\pi\)
\(978\) 0 0
\(979\) 108.000 3.45169
\(980\) 0 0
\(981\) 42.0000 1.34096
\(982\) − 2.00000i − 0.0638226i
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.00000 −0.0636930
\(987\) 0 0
\(988\) 4.00000i 0.127257i
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) 28.0000 0.889449 0.444725 0.895667i \(-0.353302\pi\)
0.444725 + 0.895667i \(0.353302\pi\)
\(992\) 10.0000i 0.317500i
\(993\) 0 0
\(994\) 24.0000 0.761234
\(995\) 0 0
\(996\) 0 0
\(997\) 58.0000i 1.83688i 0.395562 + 0.918439i \(0.370550\pi\)
−0.395562 + 0.918439i \(0.629450\pi\)
\(998\) 12.0000i 0.379853i
\(999\) 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 725.2.b.a.349.2 2
5.2 odd 4 145.2.a.a.1.1 1
5.3 odd 4 725.2.a.a.1.1 1
5.4 even 2 inner 725.2.b.a.349.1 2
15.2 even 4 1305.2.a.f.1.1 1
15.8 even 4 6525.2.a.d.1.1 1
20.7 even 4 2320.2.a.e.1.1 1
35.27 even 4 7105.2.a.b.1.1 1
40.27 even 4 9280.2.a.o.1.1 1
40.37 odd 4 9280.2.a.l.1.1 1
145.57 odd 4 4205.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.a.1.1 1 5.2 odd 4
725.2.a.a.1.1 1 5.3 odd 4
725.2.b.a.349.1 2 5.4 even 2 inner
725.2.b.a.349.2 2 1.1 even 1 trivial
1305.2.a.f.1.1 1 15.2 even 4
2320.2.a.e.1.1 1 20.7 even 4
4205.2.a.a.1.1 1 145.57 odd 4
6525.2.a.d.1.1 1 15.8 even 4
7105.2.a.b.1.1 1 35.27 even 4
9280.2.a.l.1.1 1 40.37 odd 4
9280.2.a.o.1.1 1 40.27 even 4