Properties

Label 7500.2.d.c.1249.4
Level $7500$
Weight $2$
Character 7500.1249
Analytic conductor $59.888$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7500,2,Mod(1249,7500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7500, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7500.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7500 = 2^{2} \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7500.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: \(59.8878015160\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.6724000000.12
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 16x^{6} + 86x^{4} + 181x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.4
Root \(1.12233i\) of defining polynomial
Character \(\chi\) \(=\) 7500.1249
Dual form 7500.2.d.c.1249.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.74037i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +1.74037i q^{7} -1.00000 q^{9} +2.32027 q^{11} +4.93831i q^{13} -3.74037i q^{17} +1.69364 q^{19} +1.74037 q^{21} +8.67867i q^{23} +1.00000i q^{27} -3.12233 q^{29} +9.56166 q^{31} -2.32027i q^{33} -5.36700i q^{37} +4.93831 q^{39} -8.72540 q^{41} +2.86270i q^{43} +8.46248i q^{47} +3.97112 q^{49} -3.74037 q^{51} -1.34449i q^{53} -1.69364i q^{57} -4.38793 q^{59} -15.3316 q^{61} -1.74037i q^{63} -9.23075i q^{67} +8.67867 q^{69} +0.0235645 q^{71} -1.02251i q^{73} +4.03813i q^{77} +0.798776 q^{79} +1.00000 q^{81} +2.37008i q^{83} +3.12233i q^{87} +4.86205 q^{89} -8.59447 q^{91} -9.56166i q^{93} +3.46151i q^{97} -2.32027 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} - 2 q^{11} + 10 q^{19} - 8 q^{21} - 12 q^{29} + 22 q^{31} + 10 q^{39} + 8 q^{49} - 8 q^{51} - 2 q^{59} - 44 q^{61} + 18 q^{69} + 40 q^{71} + 6 q^{79} + 8 q^{81} + 30 q^{89} - 20 q^{91} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7500\mathbb{Z}\right)^\times\).

\(n\) \(2501\) \(3751\) \(6877\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.74037i 0.657797i 0.944365 + 0.328899i \(0.106677\pi\)
−0.944365 + 0.328899i \(0.893323\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.32027 0.699589 0.349794 0.936827i \(-0.386251\pi\)
0.349794 + 0.936827i \(0.386251\pi\)
\(12\) 0 0
\(13\) 4.93831i 1.36964i 0.728712 + 0.684820i \(0.240119\pi\)
−0.728712 + 0.684820i \(0.759881\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.74037i − 0.907172i −0.891212 0.453586i \(-0.850144\pi\)
0.891212 0.453586i \(-0.149856\pi\)
\(18\) 0 0
\(19\) 1.69364 0.388548 0.194274 0.980947i \(-0.437765\pi\)
0.194274 + 0.980947i \(0.437765\pi\)
\(20\) 0 0
\(21\) 1.74037 0.379779
\(22\) 0 0
\(23\) 8.67867i 1.80963i 0.425806 + 0.904814i \(0.359991\pi\)
−0.425806 + 0.904814i \(0.640009\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −3.12233 −0.579803 −0.289901 0.957057i \(-0.593623\pi\)
−0.289901 + 0.957057i \(0.593623\pi\)
\(30\) 0 0
\(31\) 9.56166 1.71732 0.858662 0.512542i \(-0.171296\pi\)
0.858662 + 0.512542i \(0.171296\pi\)
\(32\) 0 0
\(33\) − 2.32027i − 0.403908i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 5.36700i − 0.882329i −0.897426 0.441165i \(-0.854565\pi\)
0.897426 0.441165i \(-0.145435\pi\)
\(38\) 0 0
\(39\) 4.93831 0.790762
\(40\) 0 0
\(41\) −8.72540 −1.36268 −0.681339 0.731968i \(-0.738602\pi\)
−0.681339 + 0.731968i \(0.738602\pi\)
\(42\) 0 0
\(43\) 2.86270i 0.436558i 0.975886 + 0.218279i \(0.0700442\pi\)
−0.975886 + 0.218279i \(0.929956\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.46248i 1.23438i 0.786814 + 0.617190i \(0.211729\pi\)
−0.786814 + 0.617190i \(0.788271\pi\)
\(48\) 0 0
\(49\) 3.97112 0.567303
\(50\) 0 0
\(51\) −3.74037 −0.523756
\(52\) 0 0
\(53\) − 1.34449i − 0.184680i −0.995728 0.0923398i \(-0.970565\pi\)
0.995728 0.0923398i \(-0.0294346\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.69364i − 0.224328i
\(58\) 0 0
\(59\) −4.38793 −0.571260 −0.285630 0.958340i \(-0.592203\pi\)
−0.285630 + 0.958340i \(0.592203\pi\)
\(60\) 0 0
\(61\) −15.3316 −1.96300 −0.981502 0.191452i \(-0.938681\pi\)
−0.981502 + 0.191452i \(0.938681\pi\)
\(62\) 0 0
\(63\) − 1.74037i − 0.219266i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 9.23075i − 1.12772i −0.825872 0.563858i \(-0.809317\pi\)
0.825872 0.563858i \(-0.190683\pi\)
\(68\) 0 0
\(69\) 8.67867 1.04479
\(70\) 0 0
\(71\) 0.0235645 0.00279659 0.00139830 0.999999i \(-0.499555\pi\)
0.00139830 + 0.999999i \(0.499555\pi\)
\(72\) 0 0
\(73\) − 1.02251i − 0.119676i −0.998208 0.0598380i \(-0.980942\pi\)
0.998208 0.0598380i \(-0.0190584\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.03813i 0.460187i
\(78\) 0 0
\(79\) 0.798776 0.0898693 0.0449346 0.998990i \(-0.485692\pi\)
0.0449346 + 0.998990i \(0.485692\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.37008i 0.260150i 0.991504 + 0.130075i \(0.0415219\pi\)
−0.991504 + 0.130075i \(0.958478\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.12233i 0.334749i
\(88\) 0 0
\(89\) 4.86205 0.515376 0.257688 0.966228i \(-0.417039\pi\)
0.257688 + 0.966228i \(0.417039\pi\)
\(90\) 0 0
\(91\) −8.59447 −0.900945
\(92\) 0 0
\(93\) − 9.56166i − 0.991498i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.46151i 0.351463i 0.984438 + 0.175731i \(0.0562290\pi\)
−0.984438 + 0.175731i \(0.943771\pi\)
\(98\) 0 0
\(99\) −2.32027 −0.233196
\(100\) 0 0
\(101\) −7.58056 −0.754294 −0.377147 0.926154i \(-0.623095\pi\)
−0.377147 + 0.926154i \(0.623095\pi\)
\(102\) 0 0
\(103\) 10.7699i 1.06119i 0.847626 + 0.530595i \(0.178031\pi\)
−0.847626 + 0.530595i \(0.821969\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.34344i 0.419896i 0.977713 + 0.209948i \(0.0673294\pi\)
−0.977713 + 0.209948i \(0.932671\pi\)
\(108\) 0 0
\(109\) −1.07127 −0.102609 −0.0513045 0.998683i \(-0.516338\pi\)
−0.0513045 + 0.998683i \(0.516338\pi\)
\(110\) 0 0
\(111\) −5.36700 −0.509413
\(112\) 0 0
\(113\) 18.5617i 1.74613i 0.487601 + 0.873067i \(0.337872\pi\)
−0.487601 + 0.873067i \(0.662128\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 4.93831i − 0.456547i
\(118\) 0 0
\(119\) 6.50961 0.596735
\(120\) 0 0
\(121\) −5.61633 −0.510576
\(122\) 0 0
\(123\) 8.72540i 0.786743i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 19.0096i − 1.68683i −0.537265 0.843414i \(-0.680542\pi\)
0.537265 0.843414i \(-0.319458\pi\)
\(128\) 0 0
\(129\) 2.86270 0.252047
\(130\) 0 0
\(131\) −0.745032 −0.0650937 −0.0325469 0.999470i \(-0.510362\pi\)
−0.0325469 + 0.999470i \(0.510362\pi\)
\(132\) 0 0
\(133\) 2.94756i 0.255586i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.1326i 0.951119i 0.879684 + 0.475559i \(0.157754\pi\)
−0.879684 + 0.475559i \(0.842246\pi\)
\(138\) 0 0
\(139\) −13.6889 −1.16108 −0.580539 0.814233i \(-0.697158\pi\)
−0.580539 + 0.814233i \(0.697158\pi\)
\(140\) 0 0
\(141\) 8.46248 0.712670
\(142\) 0 0
\(143\) 11.4582i 0.958185i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 3.97112i − 0.327533i
\(148\) 0 0
\(149\) 10.6283 0.870701 0.435351 0.900261i \(-0.356624\pi\)
0.435351 + 0.900261i \(0.356624\pi\)
\(150\) 0 0
\(151\) −10.4689 −0.851943 −0.425972 0.904737i \(-0.640068\pi\)
−0.425972 + 0.904737i \(0.640068\pi\)
\(152\) 0 0
\(153\) 3.74037i 0.302391i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 17.2987i 1.38059i 0.723528 + 0.690295i \(0.242519\pi\)
−0.723528 + 0.690295i \(0.757481\pi\)
\(158\) 0 0
\(159\) −1.34449 −0.106625
\(160\) 0 0
\(161\) −15.1041 −1.19037
\(162\) 0 0
\(163\) − 6.14688i − 0.481460i −0.970592 0.240730i \(-0.922613\pi\)
0.970592 0.240730i \(-0.0773869\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.90018i 0.147040i 0.997294 + 0.0735201i \(0.0234233\pi\)
−0.997294 + 0.0735201i \(0.976577\pi\)
\(168\) 0 0
\(169\) −11.3869 −0.875914
\(170\) 0 0
\(171\) −1.69364 −0.129516
\(172\) 0 0
\(173\) 11.5859i 0.880857i 0.897788 + 0.440429i \(0.145174\pi\)
−0.897788 + 0.440429i \(0.854826\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 4.38793i 0.329817i
\(178\) 0 0
\(179\) −22.0477 −1.64792 −0.823961 0.566646i \(-0.808241\pi\)
−0.823961 + 0.566646i \(0.808241\pi\)
\(180\) 0 0
\(181\) 20.6571 1.53543 0.767717 0.640790i \(-0.221393\pi\)
0.767717 + 0.640790i \(0.221393\pi\)
\(182\) 0 0
\(183\) 15.3316i 1.13334i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 8.67867i − 0.634648i
\(188\) 0 0
\(189\) −1.74037 −0.126593
\(190\) 0 0
\(191\) 20.8321 1.50736 0.753680 0.657242i \(-0.228277\pi\)
0.753680 + 0.657242i \(0.228277\pi\)
\(192\) 0 0
\(193\) 4.78053i 0.344110i 0.985087 + 0.172055i \(0.0550406\pi\)
−0.985087 + 0.172055i \(0.944959\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 17.3520i − 1.23628i −0.786068 0.618141i \(-0.787886\pi\)
0.786068 0.618141i \(-0.212114\pi\)
\(198\) 0 0
\(199\) −11.0756 −0.785129 −0.392564 0.919724i \(-0.628412\pi\)
−0.392564 + 0.919724i \(0.628412\pi\)
\(200\) 0 0
\(201\) −9.23075 −0.651087
\(202\) 0 0
\(203\) − 5.43401i − 0.381393i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 8.67867i − 0.603210i
\(208\) 0 0
\(209\) 3.92971 0.271824
\(210\) 0 0
\(211\) 5.66902 0.390272 0.195136 0.980776i \(-0.437485\pi\)
0.195136 + 0.980776i \(0.437485\pi\)
\(212\) 0 0
\(213\) − 0.0235645i − 0.00161461i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.6408i 1.12965i
\(218\) 0 0
\(219\) −1.02251 −0.0690950
\(220\) 0 0
\(221\) 18.4711 1.24250
\(222\) 0 0
\(223\) − 21.3805i − 1.43174i −0.698231 0.715872i \(-0.746029\pi\)
0.698231 0.715872i \(-0.253971\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 22.9624i 1.52407i 0.647535 + 0.762036i \(0.275800\pi\)
−0.647535 + 0.762036i \(0.724200\pi\)
\(228\) 0 0
\(229\) 28.1047 1.85721 0.928604 0.371072i \(-0.121010\pi\)
0.928604 + 0.371072i \(0.121010\pi\)
\(230\) 0 0
\(231\) 4.03813 0.265689
\(232\) 0 0
\(233\) − 1.33195i − 0.0872592i −0.999048 0.0436296i \(-0.986108\pi\)
0.999048 0.0436296i \(-0.0138921\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 0.798776i − 0.0518861i
\(238\) 0 0
\(239\) −3.44529 −0.222857 −0.111429 0.993772i \(-0.535543\pi\)
−0.111429 + 0.993772i \(0.535543\pi\)
\(240\) 0 0
\(241\) −24.0938 −1.55202 −0.776008 0.630722i \(-0.782759\pi\)
−0.776008 + 0.630722i \(0.782759\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.36372i 0.532171i
\(248\) 0 0
\(249\) 2.37008 0.150198
\(250\) 0 0
\(251\) −30.1621 −1.90381 −0.951907 0.306387i \(-0.900880\pi\)
−0.951907 + 0.306387i \(0.900880\pi\)
\(252\) 0 0
\(253\) 20.1369i 1.26600i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 10.6862i − 0.666588i −0.942823 0.333294i \(-0.891840\pi\)
0.942823 0.333294i \(-0.108160\pi\)
\(258\) 0 0
\(259\) 9.34055 0.580394
\(260\) 0 0
\(261\) 3.12233 0.193268
\(262\) 0 0
\(263\) 9.23278i 0.569318i 0.958629 + 0.284659i \(0.0918803\pi\)
−0.958629 + 0.284659i \(0.908120\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 4.86205i − 0.297553i
\(268\) 0 0
\(269\) −26.7381 −1.63025 −0.815124 0.579286i \(-0.803331\pi\)
−0.815124 + 0.579286i \(0.803331\pi\)
\(270\) 0 0
\(271\) 24.1840 1.46907 0.734535 0.678571i \(-0.237400\pi\)
0.734535 + 0.678571i \(0.237400\pi\)
\(272\) 0 0
\(273\) 8.59447i 0.520161i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 31.6079i 1.89914i 0.313563 + 0.949568i \(0.398477\pi\)
−0.313563 + 0.949568i \(0.601523\pi\)
\(278\) 0 0
\(279\) −9.56166 −0.572441
\(280\) 0 0
\(281\) 23.2625 1.38773 0.693863 0.720107i \(-0.255907\pi\)
0.693863 + 0.720107i \(0.255907\pi\)
\(282\) 0 0
\(283\) 24.8321i 1.47612i 0.674737 + 0.738058i \(0.264257\pi\)
−0.674737 + 0.738058i \(0.735743\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 15.1854i − 0.896366i
\(288\) 0 0
\(289\) 3.00965 0.177038
\(290\) 0 0
\(291\) 3.46151 0.202917
\(292\) 0 0
\(293\) 29.2758i 1.71031i 0.518371 + 0.855156i \(0.326539\pi\)
−0.518371 + 0.855156i \(0.673461\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.32027i 0.134636i
\(298\) 0 0
\(299\) −42.8580 −2.47854
\(300\) 0 0
\(301\) −4.98215 −0.287166
\(302\) 0 0
\(303\) 7.58056i 0.435492i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 8.73972i − 0.498802i −0.968400 0.249401i \(-0.919766\pi\)
0.968400 0.249401i \(-0.0802337\pi\)
\(308\) 0 0
\(309\) 10.7699 0.612678
\(310\) 0 0
\(311\) −15.4561 −0.876436 −0.438218 0.898869i \(-0.644390\pi\)
−0.438218 + 0.898869i \(0.644390\pi\)
\(312\) 0 0
\(313\) − 1.55898i − 0.0881185i −0.999029 0.0440593i \(-0.985971\pi\)
0.999029 0.0440593i \(-0.0140290\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.46256i 0.306808i 0.988164 + 0.153404i \(0.0490236\pi\)
−0.988164 + 0.153404i \(0.950976\pi\)
\(318\) 0 0
\(319\) −7.24467 −0.405623
\(320\) 0 0
\(321\) 4.34344 0.242427
\(322\) 0 0
\(323\) − 6.33484i − 0.352480i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.07127i 0.0592414i
\(328\) 0 0
\(329\) −14.7278 −0.811972
\(330\) 0 0
\(331\) 0.855804 0.0470392 0.0235196 0.999723i \(-0.492513\pi\)
0.0235196 + 0.999723i \(0.492513\pi\)
\(332\) 0 0
\(333\) 5.36700i 0.294110i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 14.8246i − 0.807546i −0.914859 0.403773i \(-0.867699\pi\)
0.914859 0.403773i \(-0.132301\pi\)
\(338\) 0 0
\(339\) 18.5617 1.00813
\(340\) 0 0
\(341\) 22.1857 1.20142
\(342\) 0 0
\(343\) 19.0938i 1.03097i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.17069i 0.0628459i 0.999506 + 0.0314229i \(0.0100039\pi\)
−0.999506 + 0.0314229i \(0.989996\pi\)
\(348\) 0 0
\(349\) 21.4346 1.14737 0.573683 0.819077i \(-0.305514\pi\)
0.573683 + 0.819077i \(0.305514\pi\)
\(350\) 0 0
\(351\) −4.93831 −0.263587
\(352\) 0 0
\(353\) − 3.56271i − 0.189624i −0.995495 0.0948119i \(-0.969775\pi\)
0.995495 0.0948119i \(-0.0302250\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 6.50961i − 0.344525i
\(358\) 0 0
\(359\) −13.3912 −0.706761 −0.353381 0.935480i \(-0.614968\pi\)
−0.353381 + 0.935480i \(0.614968\pi\)
\(360\) 0 0
\(361\) −16.1316 −0.849031
\(362\) 0 0
\(363\) 5.61633i 0.294781i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 1.42436i 0.0743508i 0.999309 + 0.0371754i \(0.0118360\pi\)
−0.999309 + 0.0371754i \(0.988164\pi\)
\(368\) 0 0
\(369\) 8.72540 0.454226
\(370\) 0 0
\(371\) 2.33990 0.121482
\(372\) 0 0
\(373\) 25.5327i 1.32203i 0.750371 + 0.661017i \(0.229875\pi\)
−0.750371 + 0.661017i \(0.770125\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 15.4190i − 0.794121i
\(378\) 0 0
\(379\) 38.0791 1.95599 0.977995 0.208628i \(-0.0668997\pi\)
0.977995 + 0.208628i \(0.0668997\pi\)
\(380\) 0 0
\(381\) −19.0096 −0.973890
\(382\) 0 0
\(383\) 13.6545i 0.697710i 0.937177 + 0.348855i \(0.113429\pi\)
−0.937177 + 0.348855i \(0.886571\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 2.86270i − 0.145519i
\(388\) 0 0
\(389\) −35.7899 −1.81462 −0.907310 0.420462i \(-0.861868\pi\)
−0.907310 + 0.420462i \(0.861868\pi\)
\(390\) 0 0
\(391\) 32.4614 1.64165
\(392\) 0 0
\(393\) 0.745032i 0.0375819i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 24.5424i − 1.23175i −0.787846 0.615873i \(-0.788804\pi\)
0.787846 0.615873i \(-0.211196\pi\)
\(398\) 0 0
\(399\) 2.94756 0.147562
\(400\) 0 0
\(401\) −25.7068 −1.28373 −0.641867 0.766816i \(-0.721840\pi\)
−0.641867 + 0.766816i \(0.721840\pi\)
\(402\) 0 0
\(403\) 47.2184i 2.35212i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 12.4529i − 0.617268i
\(408\) 0 0
\(409\) 18.5559 0.917532 0.458766 0.888557i \(-0.348292\pi\)
0.458766 + 0.888557i \(0.348292\pi\)
\(410\) 0 0
\(411\) 11.1326 0.549129
\(412\) 0 0
\(413\) − 7.63661i − 0.375773i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 13.6889i 0.670348i
\(418\) 0 0
\(419\) −4.69324 −0.229280 −0.114640 0.993407i \(-0.536571\pi\)
−0.114640 + 0.993407i \(0.536571\pi\)
\(420\) 0 0
\(421\) 9.22050 0.449380 0.224690 0.974430i \(-0.427863\pi\)
0.224690 + 0.974430i \(0.427863\pi\)
\(422\) 0 0
\(423\) − 8.46248i − 0.411460i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 26.6825i − 1.29126i
\(428\) 0 0
\(429\) 11.4582 0.553208
\(430\) 0 0
\(431\) 28.5809 1.37669 0.688346 0.725382i \(-0.258337\pi\)
0.688346 + 0.725382i \(0.258337\pi\)
\(432\) 0 0
\(433\) 3.63365i 0.174622i 0.996181 + 0.0873110i \(0.0278274\pi\)
−0.996181 + 0.0873110i \(0.972173\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14.6986i 0.703127i
\(438\) 0 0
\(439\) 17.6073 0.840352 0.420176 0.907443i \(-0.361968\pi\)
0.420176 + 0.907443i \(0.361968\pi\)
\(440\) 0 0
\(441\) −3.97112 −0.189101
\(442\) 0 0
\(443\) 7.11807i 0.338190i 0.985600 + 0.169095i \(0.0540844\pi\)
−0.985600 + 0.169095i \(0.945916\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 10.6283i − 0.502700i
\(448\) 0 0
\(449\) 20.4121 0.963305 0.481653 0.876362i \(-0.340037\pi\)
0.481653 + 0.876362i \(0.340037\pi\)
\(450\) 0 0
\(451\) −20.2453 −0.953315
\(452\) 0 0
\(453\) 10.4689i 0.491870i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16.8375i 0.787624i 0.919191 + 0.393812i \(0.128844\pi\)
−0.919191 + 0.393812i \(0.871156\pi\)
\(458\) 0 0
\(459\) 3.74037 0.174585
\(460\) 0 0
\(461\) 17.3249 0.806903 0.403451 0.915001i \(-0.367810\pi\)
0.403451 + 0.915001i \(0.367810\pi\)
\(462\) 0 0
\(463\) 8.60412i 0.399867i 0.979809 + 0.199934i \(0.0640727\pi\)
−0.979809 + 0.199934i \(0.935927\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 0.835852i − 0.0386786i −0.999813 0.0193393i \(-0.993844\pi\)
0.999813 0.0193393i \(-0.00615628\pi\)
\(468\) 0 0
\(469\) 16.0649 0.741808
\(470\) 0 0
\(471\) 17.2987 0.797084
\(472\) 0 0
\(473\) 6.64225i 0.305411i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.34449i 0.0615599i
\(478\) 0 0
\(479\) 33.4495 1.52835 0.764173 0.645012i \(-0.223147\pi\)
0.764173 + 0.645012i \(0.223147\pi\)
\(480\) 0 0
\(481\) 26.5039 1.20847
\(482\) 0 0
\(483\) 15.1041i 0.687260i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 24.4158i 1.10638i 0.833054 + 0.553192i \(0.186590\pi\)
−0.833054 + 0.553192i \(0.813410\pi\)
\(488\) 0 0
\(489\) −6.14688 −0.277971
\(490\) 0 0
\(491\) −14.6475 −0.661032 −0.330516 0.943800i \(-0.607223\pi\)
−0.330516 + 0.943800i \(0.607223\pi\)
\(492\) 0 0
\(493\) 11.6787i 0.525981i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.0410109i 0.00183959i
\(498\) 0 0
\(499\) 13.5655 0.607276 0.303638 0.952787i \(-0.401799\pi\)
0.303638 + 0.952787i \(0.401799\pi\)
\(500\) 0 0
\(501\) 1.90018 0.0848937
\(502\) 0 0
\(503\) − 40.9282i − 1.82490i −0.409191 0.912449i \(-0.634189\pi\)
0.409191 0.912449i \(-0.365811\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 11.3869i 0.505709i
\(508\) 0 0
\(509\) −12.1412 −0.538147 −0.269074 0.963120i \(-0.586718\pi\)
−0.269074 + 0.963120i \(0.586718\pi\)
\(510\) 0 0
\(511\) 1.77955 0.0787226
\(512\) 0 0
\(513\) 1.69364i 0.0747760i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 19.6353i 0.863559i
\(518\) 0 0
\(519\) 11.5859 0.508563
\(520\) 0 0
\(521\) 26.2250 1.14894 0.574470 0.818526i \(-0.305208\pi\)
0.574470 + 0.818526i \(0.305208\pi\)
\(522\) 0 0
\(523\) 15.3245i 0.670095i 0.942201 + 0.335048i \(0.108752\pi\)
−0.942201 + 0.335048i \(0.891248\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 35.7641i − 1.55791i
\(528\) 0 0
\(529\) −52.3194 −2.27476
\(530\) 0 0
\(531\) 4.38793 0.190420
\(532\) 0 0
\(533\) − 43.0887i − 1.86638i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 22.0477i 0.951429i
\(538\) 0 0
\(539\) 9.21409 0.396879
\(540\) 0 0
\(541\) 9.47745 0.407467 0.203734 0.979026i \(-0.434692\pi\)
0.203734 + 0.979026i \(0.434692\pi\)
\(542\) 0 0
\(543\) − 20.6571i − 0.886483i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 0.645935i − 0.0276182i −0.999905 0.0138091i \(-0.995604\pi\)
0.999905 0.0138091i \(-0.00439571\pi\)
\(548\) 0 0
\(549\) 15.3316 0.654335
\(550\) 0 0
\(551\) −5.28811 −0.225281
\(552\) 0 0
\(553\) 1.39016i 0.0591158i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 12.0297i 0.509716i 0.966978 + 0.254858i \(0.0820287\pi\)
−0.966978 + 0.254858i \(0.917971\pi\)
\(558\) 0 0
\(559\) −14.1369 −0.597927
\(560\) 0 0
\(561\) −8.67867 −0.366414
\(562\) 0 0
\(563\) 15.8832i 0.669398i 0.942325 + 0.334699i \(0.108635\pi\)
−0.942325 + 0.334699i \(0.891365\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.74037i 0.0730886i
\(568\) 0 0
\(569\) −21.9304 −0.919368 −0.459684 0.888082i \(-0.652037\pi\)
−0.459684 + 0.888082i \(0.652037\pi\)
\(570\) 0 0
\(571\) −6.80835 −0.284921 −0.142460 0.989801i \(-0.545501\pi\)
−0.142460 + 0.989801i \(0.545501\pi\)
\(572\) 0 0
\(573\) − 20.8321i − 0.870274i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 18.6273i − 0.775464i −0.921772 0.387732i \(-0.873259\pi\)
0.921772 0.387732i \(-0.126741\pi\)
\(578\) 0 0
\(579\) 4.78053 0.198672
\(580\) 0 0
\(581\) −4.12481 −0.171126
\(582\) 0 0
\(583\) − 3.11958i − 0.129200i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 14.2102i 0.586518i 0.956033 + 0.293259i \(0.0947398\pi\)
−0.956033 + 0.293259i \(0.905260\pi\)
\(588\) 0 0
\(589\) 16.1940 0.667262
\(590\) 0 0
\(591\) −17.3520 −0.713767
\(592\) 0 0
\(593\) 40.7850i 1.67484i 0.546559 + 0.837420i \(0.315937\pi\)
−0.546559 + 0.837420i \(0.684063\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 11.0756i 0.453294i
\(598\) 0 0
\(599\) −34.8928 −1.42568 −0.712841 0.701326i \(-0.752592\pi\)
−0.712841 + 0.701326i \(0.752592\pi\)
\(600\) 0 0
\(601\) 10.6287 0.433552 0.216776 0.976221i \(-0.430446\pi\)
0.216776 + 0.976221i \(0.430446\pi\)
\(602\) 0 0
\(603\) 9.23075i 0.375905i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 30.5033i − 1.23809i −0.785355 0.619045i \(-0.787520\pi\)
0.785355 0.619045i \(-0.212480\pi\)
\(608\) 0 0
\(609\) −5.43401 −0.220197
\(610\) 0 0
\(611\) −41.7904 −1.69066
\(612\) 0 0
\(613\) − 12.7732i − 0.515904i −0.966158 0.257952i \(-0.916952\pi\)
0.966158 0.257952i \(-0.0830476\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11.9327i 0.480391i 0.970725 + 0.240196i \(0.0772116\pi\)
−0.970725 + 0.240196i \(0.922788\pi\)
\(618\) 0 0
\(619\) 14.5045 0.582987 0.291494 0.956573i \(-0.405848\pi\)
0.291494 + 0.956573i \(0.405848\pi\)
\(620\) 0 0
\(621\) −8.67867 −0.348263
\(622\) 0 0
\(623\) 8.46176i 0.339013i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 3.92971i − 0.156937i
\(628\) 0 0
\(629\) −20.0746 −0.800425
\(630\) 0 0
\(631\) −3.66780 −0.146013 −0.0730064 0.997331i \(-0.523259\pi\)
−0.0730064 + 0.997331i \(0.523259\pi\)
\(632\) 0 0
\(633\) − 5.66902i − 0.225323i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 19.6106i 0.777001i
\(638\) 0 0
\(639\) −0.0235645 −0.000932198 0
\(640\) 0 0
\(641\) −13.8519 −0.547116 −0.273558 0.961856i \(-0.588201\pi\)
−0.273558 + 0.961856i \(0.588201\pi\)
\(642\) 0 0
\(643\) − 27.1641i − 1.07125i −0.844456 0.535624i \(-0.820076\pi\)
0.844456 0.535624i \(-0.179924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5.14988i 0.202463i 0.994863 + 0.101231i \(0.0322782\pi\)
−0.994863 + 0.101231i \(0.967722\pi\)
\(648\) 0 0
\(649\) −10.1812 −0.399647
\(650\) 0 0
\(651\) 16.6408 0.652204
\(652\) 0 0
\(653\) − 29.6734i − 1.16121i −0.814186 0.580604i \(-0.802816\pi\)
0.814186 0.580604i \(-0.197184\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.02251i 0.0398920i
\(658\) 0 0
\(659\) −15.7740 −0.614466 −0.307233 0.951634i \(-0.599403\pi\)
−0.307233 + 0.951634i \(0.599403\pi\)
\(660\) 0 0
\(661\) −24.0557 −0.935657 −0.467828 0.883819i \(-0.654963\pi\)
−0.467828 + 0.883819i \(0.654963\pi\)
\(662\) 0 0
\(663\) − 18.4711i − 0.717357i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 27.0977i − 1.04923i
\(668\) 0 0
\(669\) −21.3805 −0.826618
\(670\) 0 0
\(671\) −35.5734 −1.37330
\(672\) 0 0
\(673\) 32.0035i 1.23365i 0.787102 + 0.616823i \(0.211580\pi\)
−0.787102 + 0.616823i \(0.788420\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 24.3809i 0.937035i 0.883454 + 0.468517i \(0.155212\pi\)
−0.883454 + 0.468517i \(0.844788\pi\)
\(678\) 0 0
\(679\) −6.02429 −0.231191
\(680\) 0 0
\(681\) 22.9624 0.879923
\(682\) 0 0
\(683\) − 11.1528i − 0.426752i −0.976970 0.213376i \(-0.931554\pi\)
0.976970 0.213376i \(-0.0684459\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 28.1047i − 1.07226i
\(688\) 0 0
\(689\) 6.63949 0.252945
\(690\) 0 0
\(691\) 10.8286 0.411941 0.205970 0.978558i \(-0.433965\pi\)
0.205970 + 0.978558i \(0.433965\pi\)
\(692\) 0 0
\(693\) − 4.03813i − 0.153396i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 32.6362i 1.23618i
\(698\) 0 0
\(699\) −1.33195 −0.0503791
\(700\) 0 0
\(701\) 43.2512 1.63357 0.816787 0.576939i \(-0.195753\pi\)
0.816787 + 0.576939i \(0.195753\pi\)
\(702\) 0 0
\(703\) − 9.08977i − 0.342827i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 13.1930i − 0.496172i
\(708\) 0 0
\(709\) 44.6052 1.67518 0.837592 0.546296i \(-0.183963\pi\)
0.837592 + 0.546296i \(0.183963\pi\)
\(710\) 0 0
\(711\) −0.798776 −0.0299564
\(712\) 0 0
\(713\) 82.9825i 3.10772i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.44529i 0.128667i
\(718\) 0 0
\(719\) 32.0868 1.19664 0.598318 0.801259i \(-0.295836\pi\)
0.598318 + 0.801259i \(0.295836\pi\)
\(720\) 0 0
\(721\) −18.7436 −0.698047
\(722\) 0 0
\(723\) 24.0938i 0.896057i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 13.6216i 0.505196i 0.967571 + 0.252598i \(0.0812850\pi\)
−0.967571 + 0.252598i \(0.918715\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 10.7076 0.396033
\(732\) 0 0
\(733\) 28.4242i 1.04987i 0.851142 + 0.524935i \(0.175910\pi\)
−0.851142 + 0.524935i \(0.824090\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 21.4179i − 0.788937i
\(738\) 0 0
\(739\) −9.71457 −0.357356 −0.178678 0.983908i \(-0.557182\pi\)
−0.178678 + 0.983908i \(0.557182\pi\)
\(740\) 0 0
\(741\) 8.36372 0.307249
\(742\) 0 0
\(743\) 34.2029i 1.25478i 0.778705 + 0.627390i \(0.215877\pi\)
−0.778705 + 0.627390i \(0.784123\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 2.37008i − 0.0867168i
\(748\) 0 0
\(749\) −7.55917 −0.276206
\(750\) 0 0
\(751\) −12.7840 −0.466495 −0.233248 0.972417i \(-0.574935\pi\)
−0.233248 + 0.972417i \(0.574935\pi\)
\(752\) 0 0
\(753\) 30.1621i 1.09917i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 26.6200i − 0.967520i −0.875201 0.483760i \(-0.839271\pi\)
0.875201 0.483760i \(-0.160729\pi\)
\(758\) 0 0
\(759\) 20.1369 0.730923
\(760\) 0 0
\(761\) 25.7137 0.932122 0.466061 0.884753i \(-0.345673\pi\)
0.466061 + 0.884753i \(0.345673\pi\)
\(762\) 0 0
\(763\) − 1.86440i − 0.0674959i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 21.6689i − 0.782420i
\(768\) 0 0
\(769\) 38.3717 1.38372 0.691860 0.722032i \(-0.256792\pi\)
0.691860 + 0.722032i \(0.256792\pi\)
\(770\) 0 0
\(771\) −10.6862 −0.384855
\(772\) 0 0
\(773\) − 1.39884i − 0.0503126i −0.999684 0.0251563i \(-0.991992\pi\)
0.999684 0.0251563i \(-0.00800835\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 9.34055i − 0.335090i
\(778\) 0 0
\(779\) −14.7777 −0.529466
\(780\) 0 0
\(781\) 0.0546761 0.00195647
\(782\) 0 0
\(783\) − 3.12233i − 0.111583i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 4.21023i 0.150078i 0.997181 + 0.0750392i \(0.0239082\pi\)
−0.997181 + 0.0750392i \(0.976092\pi\)
\(788\) 0 0
\(789\) 9.23278 0.328696
\(790\) 0 0
\(791\) −32.3041 −1.14860
\(792\) 0 0
\(793\) − 75.7119i − 2.68861i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 31.6196i 1.12002i 0.828485 + 0.560012i \(0.189203\pi\)
−0.828485 + 0.560012i \(0.810797\pi\)
\(798\) 0 0
\(799\) 31.6528 1.11980
\(800\) 0 0
\(801\) −4.86205 −0.171792
\(802\) 0 0
\(803\) − 2.37251i − 0.0837240i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 26.7381i 0.941224i
\(808\) 0 0
\(809\) −42.1098 −1.48050 −0.740252 0.672330i \(-0.765294\pi\)
−0.740252 + 0.672330i \(0.765294\pi\)
\(810\) 0 0
\(811\) 15.1530 0.532095 0.266048 0.963960i \(-0.414282\pi\)
0.266048 + 0.963960i \(0.414282\pi\)
\(812\) 0 0
\(813\) − 24.1840i − 0.848168i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4.84839i 0.169624i
\(818\) 0 0
\(819\) 8.59447 0.300315
\(820\) 0 0
\(821\) −0.402320 −0.0140411 −0.00702053 0.999975i \(-0.502235\pi\)
−0.00702053 + 0.999975i \(0.502235\pi\)
\(822\) 0 0
\(823\) − 0.609361i − 0.0212410i −0.999944 0.0106205i \(-0.996619\pi\)
0.999944 0.0106205i \(-0.00338067\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 27.4672i − 0.955128i −0.878597 0.477564i \(-0.841520\pi\)
0.878597 0.477564i \(-0.158480\pi\)
\(828\) 0 0
\(829\) −31.1543 −1.08204 −0.541018 0.841011i \(-0.681961\pi\)
−0.541018 + 0.841011i \(0.681961\pi\)
\(830\) 0 0
\(831\) 31.6079 1.09647
\(832\) 0 0
\(833\) − 14.8535i − 0.514642i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 9.56166i 0.330499i
\(838\) 0 0
\(839\) −39.1176 −1.35049 −0.675245 0.737594i \(-0.735962\pi\)
−0.675245 + 0.737594i \(0.735962\pi\)
\(840\) 0 0
\(841\) −19.2510 −0.663829
\(842\) 0 0
\(843\) − 23.2625i − 0.801204i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 9.77448i − 0.335855i
\(848\) 0 0
\(849\) 24.8321 0.852236
\(850\) 0 0
\(851\) 46.5785 1.59669
\(852\) 0 0
\(853\) − 52.5757i − 1.80016i −0.435727 0.900079i \(-0.643509\pi\)
0.435727 0.900079i \(-0.356491\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 31.0291i 1.05993i 0.848018 + 0.529967i \(0.177796\pi\)
−0.848018 + 0.529967i \(0.822204\pi\)
\(858\) 0 0
\(859\) −38.8450 −1.32537 −0.662687 0.748897i \(-0.730584\pi\)
−0.662687 + 0.748897i \(0.730584\pi\)
\(860\) 0 0
\(861\) −15.1854 −0.517517
\(862\) 0 0
\(863\) − 3.60246i − 0.122629i −0.998118 0.0613147i \(-0.980471\pi\)
0.998118 0.0613147i \(-0.0195293\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 3.00965i − 0.102213i
\(868\) 0 0
\(869\) 1.85338 0.0628715
\(870\) 0 0
\(871\) 45.5843 1.54456
\(872\) 0 0
\(873\) − 3.46151i − 0.117154i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 31.5597i 1.06569i 0.846211 + 0.532847i \(0.178878\pi\)
−0.846211 + 0.532847i \(0.821122\pi\)
\(878\) 0 0
\(879\) 29.2758 0.987449
\(880\) 0 0
\(881\) 6.02783 0.203083 0.101541 0.994831i \(-0.467623\pi\)
0.101541 + 0.994831i \(0.467623\pi\)
\(882\) 0 0
\(883\) − 3.96581i − 0.133460i −0.997771 0.0667300i \(-0.978743\pi\)
0.997771 0.0667300i \(-0.0212566\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 55.7151i − 1.87073i −0.353684 0.935365i \(-0.615071\pi\)
0.353684 0.935365i \(-0.384929\pi\)
\(888\) 0 0
\(889\) 33.0837 1.10959
\(890\) 0 0
\(891\) 2.32027 0.0777321
\(892\) 0 0
\(893\) 14.3324i 0.479616i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 42.8580i 1.43099i
\(898\) 0 0
\(899\) −29.8547 −0.995709
\(900\) 0 0
\(901\) −5.02888 −0.167536
\(902\) 0 0
\(903\) 4.98215i 0.165796i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 21.4063i 0.710785i 0.934717 + 0.355392i \(0.115653\pi\)
−0.934717 + 0.355392i \(0.884347\pi\)
\(908\) 0 0
\(909\) 7.58056 0.251431
\(910\) 0 0
\(911\) 31.5890 1.04659 0.523295 0.852151i \(-0.324702\pi\)
0.523295 + 0.852151i \(0.324702\pi\)
\(912\) 0 0
\(913\) 5.49924i 0.181998i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1.29663i − 0.0428185i
\(918\) 0 0
\(919\) 18.0775 0.596322 0.298161 0.954516i \(-0.403627\pi\)
0.298161 + 0.954516i \(0.403627\pi\)
\(920\) 0 0
\(921\) −8.73972 −0.287983
\(922\) 0 0
\(923\) 0.116369i 0.00383033i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 10.7699i − 0.353730i
\(928\) 0 0
\(929\) −8.93003 −0.292985 −0.146492 0.989212i \(-0.546798\pi\)
−0.146492 + 0.989212i \(0.546798\pi\)
\(930\) 0 0
\(931\) 6.72565 0.220424
\(932\) 0 0
\(933\) 15.4561i 0.506011i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.0607147i 0.00198346i 1.00000 0.000991732i \(0.000315678\pi\)
−1.00000 0.000991732i \(0.999684\pi\)
\(938\) 0 0
\(939\) −1.55898 −0.0508752
\(940\) 0 0
\(941\) 20.6449 0.673005 0.336503 0.941682i \(-0.390756\pi\)
0.336503 + 0.941682i \(0.390756\pi\)
\(942\) 0 0
\(943\) − 75.7249i − 2.46594i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 51.3845i − 1.66977i −0.550423 0.834886i \(-0.685534\pi\)
0.550423 0.834886i \(-0.314466\pi\)
\(948\) 0 0
\(949\) 5.04948 0.163913
\(950\) 0 0
\(951\) 5.46256 0.177136
\(952\) 0 0
\(953\) 24.5442i 0.795064i 0.917588 + 0.397532i \(0.130133\pi\)
−0.917588 + 0.397532i \(0.869867\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 7.24467i 0.234187i
\(958\) 0 0
\(959\) −19.3747 −0.625643
\(960\) 0 0
\(961\) 60.4253 1.94920
\(962\) 0 0
\(963\) − 4.34344i − 0.139965i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 38.1844i − 1.22793i −0.789334 0.613964i \(-0.789574\pi\)
0.789334 0.613964i \(-0.210426\pi\)
\(968\) 0 0
\(969\) −6.33484 −0.203504
\(970\) 0 0
\(971\) 25.8172 0.828512 0.414256 0.910160i \(-0.364042\pi\)
0.414256 + 0.910160i \(0.364042\pi\)
\(972\) 0 0
\(973\) − 23.8237i − 0.763753i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 20.6259i − 0.659881i −0.944002 0.329941i \(-0.892971\pi\)
0.944002 0.329941i \(-0.107029\pi\)
\(978\) 0 0
\(979\) 11.2813 0.360551
\(980\) 0 0
\(981\) 1.07127 0.0342030
\(982\) 0 0
\(983\) − 22.1336i − 0.705953i −0.935632 0.352976i \(-0.885170\pi\)
0.935632 0.352976i \(-0.114830\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 14.7278i 0.468792i
\(988\) 0 0
\(989\) −24.8445 −0.790008
\(990\) 0 0
\(991\) −1.66509 −0.0528933 −0.0264466 0.999650i \(-0.508419\pi\)
−0.0264466 + 0.999650i \(0.508419\pi\)
\(992\) 0 0
\(993\) − 0.855804i − 0.0271581i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 23.4762i 0.743497i 0.928333 + 0.371749i \(0.121242\pi\)
−0.928333 + 0.371749i \(0.878758\pi\)
\(998\) 0 0
\(999\) 5.36700 0.169804
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 7500.2.d.c.1249.4 8
5.2 odd 4 7500.2.a.e.1.1 4
5.3 odd 4 7500.2.a.f.1.4 4
5.4 even 2 inner 7500.2.d.c.1249.5 8
25.3 odd 20 1500.2.m.a.1201.2 8
25.4 even 10 1500.2.o.b.49.1 16
25.6 even 5 1500.2.o.b.949.2 16
25.8 odd 20 1500.2.m.a.301.2 8
25.17 odd 20 300.2.m.b.61.1 8
25.19 even 10 1500.2.o.b.949.3 16
25.21 even 5 1500.2.o.b.49.4 16
25.22 odd 20 300.2.m.b.241.1 yes 8
75.17 even 20 900.2.n.b.361.2 8
75.47 even 20 900.2.n.b.541.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.m.b.61.1 8 25.17 odd 20
300.2.m.b.241.1 yes 8 25.22 odd 20
900.2.n.b.361.2 8 75.17 even 20
900.2.n.b.541.2 8 75.47 even 20
1500.2.m.a.301.2 8 25.8 odd 20
1500.2.m.a.1201.2 8 25.3 odd 20
1500.2.o.b.49.1 16 25.4 even 10
1500.2.o.b.49.4 16 25.21 even 5
1500.2.o.b.949.2 16 25.6 even 5
1500.2.o.b.949.3 16 25.19 even 10
7500.2.a.e.1.1 4 5.2 odd 4
7500.2.a.f.1.4 4 5.3 odd 4
7500.2.d.c.1249.4 8 1.1 even 1 trivial
7500.2.d.c.1249.5 8 5.4 even 2 inner