Properties

Label 1512.2.s.n.1297.2
Level $1512$
Weight $2$
Character 1512.1297
Analytic conductor $12.073$
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] = [1512,2,Mod(865,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.s (of order \(3\), degree \(2\), 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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 4x^{6} + 28x^{5} + 14x^{4} - 52x^{3} + 306x^{2} + 1052x + 1051 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1297.2
Root \(2.45973 + 2.20662i\) of defining polynomial
Character \(\chi\) \(=\) 1512.1297
Dual form 1512.2.s.n.865.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.26913 + 2.19820i) q^{5} +(2.64086 - 0.160857i) q^{7} +O(q^{10})\) \(q+(-1.26913 + 2.19820i) q^{5} +(2.64086 - 0.160857i) q^{7} +(-1.19060 - 2.06219i) q^{11} -0.461738 q^{13} +(-3.64086 - 6.30615i) q^{17} +(4.32198 - 7.48589i) q^{19} +(1.49052 - 2.58165i) q^{23} +(-0.721387 - 1.24948i) q^{25} +3.55723 q^{29} +(4.15034 + 7.18860i) q^{31} +(-2.99800 + 6.00928i) q^{35} +(1.10259 - 1.90975i) q^{37} +6.81998 q^{41} +2.38121 q^{43} +(-4.21938 + 7.30819i) q^{47} +(6.94825 - 0.849602i) q^{49} +(-0.181122 - 0.313712i) q^{53} +6.04413 q^{55} +(3.73397 + 6.46743i) q^{59} +(-3.08801 + 5.34859i) q^{61} +(0.586006 - 1.01499i) q^{65} +(-5.87173 - 10.1701i) q^{67} +9.72048 q^{71} +(-4.67912 - 8.10447i) q^{73} +(-3.47593 - 5.25442i) q^{77} +(4.97903 - 8.62394i) q^{79} -3.20118 q^{83} +18.4829 q^{85} +(3.29320 - 5.70399i) q^{89} +(-1.21938 + 0.0742739i) q^{91} +(10.9703 + 19.0011i) q^{95} +15.6400 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{5} - 4 q^{7} - q^{11} - 20 q^{13} - 4 q^{17} + q^{19} + 12 q^{23} - 14 q^{25} + 12 q^{29} + 8 q^{31} + 9 q^{35} - 12 q^{41} + 2 q^{43} - 9 q^{47} + 6 q^{49} + 7 q^{53} - 36 q^{55} - 4 q^{59} - 25 q^{61} - 28 q^{65} - 30 q^{67} - 22 q^{71} + 4 q^{73} - 37 q^{77} + 7 q^{79} + 58 q^{83} + 14 q^{85} + 9 q^{89} + 15 q^{91} - 4 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.26913 + 2.19820i −0.567573 + 0.983065i 0.429233 + 0.903194i \(0.358784\pi\)
−0.996805 + 0.0798707i \(0.974549\pi\)
\(6\) 0 0
\(7\) 2.64086 0.160857i 0.998150 0.0607983i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.19060 2.06219i −0.358981 0.621773i 0.628810 0.777559i \(-0.283542\pi\)
−0.987791 + 0.155786i \(0.950209\pi\)
\(12\) 0 0
\(13\) −0.461738 −0.128063 −0.0640315 0.997948i \(-0.520396\pi\)
−0.0640315 + 0.997948i \(0.520396\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.64086 6.30615i −0.883037 1.52947i −0.847947 0.530082i \(-0.822161\pi\)
−0.0350909 0.999384i \(-0.511172\pi\)
\(18\) 0 0
\(19\) 4.32198 7.48589i 0.991530 1.71738i 0.383287 0.923629i \(-0.374792\pi\)
0.608243 0.793751i \(-0.291875\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.49052 2.58165i 0.310794 0.538312i −0.667740 0.744394i \(-0.732738\pi\)
0.978535 + 0.206083i \(0.0660716\pi\)
\(24\) 0 0
\(25\) −0.721387 1.24948i −0.144277 0.249896i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.55723 0.660560 0.330280 0.943883i \(-0.392857\pi\)
0.330280 + 0.943883i \(0.392857\pi\)
\(30\) 0 0
\(31\) 4.15034 + 7.18860i 0.745423 + 1.29111i 0.949997 + 0.312259i \(0.101086\pi\)
−0.204574 + 0.978851i \(0.565581\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.99800 + 6.00928i −0.506754 + 1.01575i
\(36\) 0 0
\(37\) 1.10259 1.90975i 0.181265 0.313961i −0.761046 0.648698i \(-0.775314\pi\)
0.942312 + 0.334737i \(0.108647\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.81998 1.06510 0.532551 0.846398i \(-0.321234\pi\)
0.532551 + 0.846398i \(0.321234\pi\)
\(42\) 0 0
\(43\) 2.38121 0.363131 0.181565 0.983379i \(-0.441884\pi\)
0.181565 + 0.983379i \(0.441884\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.21938 + 7.30819i −0.615460 + 1.06601i 0.374843 + 0.927088i \(0.377697\pi\)
−0.990304 + 0.138920i \(0.955637\pi\)
\(48\) 0 0
\(49\) 6.94825 0.849602i 0.992607 0.121372i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.181122 0.313712i −0.0248790 0.0430917i 0.853318 0.521391i \(-0.174587\pi\)
−0.878197 + 0.478299i \(0.841253\pi\)
\(54\) 0 0
\(55\) 6.04413 0.814990
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.73397 + 6.46743i 0.486121 + 0.841987i 0.999873 0.0159521i \(-0.00507793\pi\)
−0.513751 + 0.857939i \(0.671745\pi\)
\(60\) 0 0
\(61\) −3.08801 + 5.34859i −0.395379 + 0.684817i −0.993149 0.116851i \(-0.962720\pi\)
0.597770 + 0.801667i \(0.296053\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.586006 1.01499i 0.0726851 0.125894i
\(66\) 0 0
\(67\) −5.87173 10.1701i −0.717345 1.24248i −0.962048 0.272881i \(-0.912024\pi\)
0.244702 0.969598i \(-0.421310\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.72048 1.15361 0.576804 0.816882i \(-0.304299\pi\)
0.576804 + 0.816882i \(0.304299\pi\)
\(72\) 0 0
\(73\) −4.67912 8.10447i −0.547649 0.948557i −0.998435 0.0559244i \(-0.982189\pi\)
0.450786 0.892632i \(-0.351144\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.47593 5.25442i −0.396119 0.598797i
\(78\) 0 0
\(79\) 4.97903 8.62394i 0.560185 0.970269i −0.437295 0.899318i \(-0.644063\pi\)
0.997480 0.0709506i \(-0.0226033\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.20118 −0.351376 −0.175688 0.984446i \(-0.556215\pi\)
−0.175688 + 0.984446i \(0.556215\pi\)
\(84\) 0 0
\(85\) 18.4829 2.00475
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.29320 5.70399i 0.349078 0.604621i −0.637008 0.770858i \(-0.719828\pi\)
0.986086 + 0.166236i \(0.0531614\pi\)
\(90\) 0 0
\(91\) −1.21938 + 0.0742739i −0.127826 + 0.00778602i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 10.9703 + 19.0011i 1.12553 + 1.94948i
\(96\) 0 0
\(97\) 15.6400 1.58800 0.793998 0.607920i \(-0.207996\pi\)
0.793998 + 0.607920i \(0.207996\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.08911 15.7428i −0.904400 1.56647i −0.821721 0.569890i \(-0.806986\pi\)
−0.0826792 0.996576i \(-0.526348\pi\)
\(102\) 0 0
\(103\) −3.91199 + 6.77577i −0.385460 + 0.667636i −0.991833 0.127544i \(-0.959291\pi\)
0.606373 + 0.795180i \(0.292624\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.27223 + 7.39972i −0.413012 + 0.715358i −0.995218 0.0976837i \(-0.968857\pi\)
0.582205 + 0.813042i \(0.302190\pi\)
\(108\) 0 0
\(109\) 3.28171 + 5.68409i 0.314331 + 0.544438i 0.979295 0.202438i \(-0.0648864\pi\)
−0.664964 + 0.746875i \(0.731553\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 15.9770 1.50299 0.751496 0.659737i \(-0.229332\pi\)
0.751496 + 0.659737i \(0.229332\pi\)
\(114\) 0 0
\(115\) 3.78333 + 6.55291i 0.352797 + 0.611062i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −10.6294 16.0680i −0.974393 1.47295i
\(120\) 0 0
\(121\) 2.66492 4.61578i 0.242266 0.419617i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.02917 −0.807594
\(126\) 0 0
\(127\) 19.9730 1.77232 0.886160 0.463380i \(-0.153364\pi\)
0.886160 + 0.463380i \(0.153364\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −8.73634 + 15.1318i −0.763298 + 1.32207i 0.177844 + 0.984059i \(0.443088\pi\)
−0.941142 + 0.338012i \(0.890245\pi\)
\(132\) 0 0
\(133\) 10.2096 20.4644i 0.885282 1.77449i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −3.74345 6.48385i −0.319825 0.553953i 0.660627 0.750715i \(-0.270291\pi\)
−0.980451 + 0.196762i \(0.936957\pi\)
\(138\) 0 0
\(139\) −13.3772 −1.13464 −0.567320 0.823498i \(-0.692020\pi\)
−0.567320 + 0.823498i \(0.692020\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0.549747 + 0.952190i 0.0459722 + 0.0796261i
\(144\) 0 0
\(145\) −4.51459 + 7.81949i −0.374916 + 0.649373i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.19571 5.53513i 0.261803 0.453455i −0.704918 0.709288i \(-0.749016\pi\)
0.966721 + 0.255833i \(0.0823497\pi\)
\(150\) 0 0
\(151\) −2.89741 5.01845i −0.235787 0.408396i 0.723714 0.690100i \(-0.242434\pi\)
−0.959501 + 0.281704i \(0.909100\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −21.0693 −1.69233
\(156\) 0 0
\(157\) −1.83818 3.18381i −0.146702 0.254096i 0.783304 0.621638i \(-0.213533\pi\)
−0.930007 + 0.367542i \(0.880199\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.52097 7.05754i 0.277491 0.556212i
\(162\) 0 0
\(163\) −4.42147 + 7.65822i −0.346316 + 0.599838i −0.985592 0.169140i \(-0.945901\pi\)
0.639276 + 0.768978i \(0.279234\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.91071 0.612149 0.306075 0.952008i \(-0.400984\pi\)
0.306075 + 0.952008i \(0.400984\pi\)
\(168\) 0 0
\(169\) −12.7868 −0.983600
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 5.52206 9.56450i 0.419835 0.727175i −0.576088 0.817388i \(-0.695421\pi\)
0.995923 + 0.0902127i \(0.0287547\pi\)
\(174\) 0 0
\(175\) −2.10607 3.18365i −0.159204 0.240662i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.28809 2.23105i −0.0962767 0.166756i 0.813864 0.581055i \(-0.197360\pi\)
−0.910141 + 0.414299i \(0.864027\pi\)
\(180\) 0 0
\(181\) −20.8641 −1.55082 −0.775408 0.631460i \(-0.782456\pi\)
−0.775408 + 0.631460i \(0.782456\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.79867 + 4.84745i 0.205763 + 0.356391i
\(186\) 0 0
\(187\) −8.66964 + 15.0163i −0.633987 + 1.09810i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.19060 3.79424i 0.158507 0.274541i −0.775824 0.630950i \(-0.782665\pi\)
0.934330 + 0.356408i \(0.115999\pi\)
\(192\) 0 0
\(193\) −7.40999 12.8345i −0.533383 0.923846i −0.999240 0.0389858i \(-0.987587\pi\)
0.465857 0.884860i \(-0.345746\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −6.18623 −0.440750 −0.220375 0.975415i \(-0.570728\pi\)
−0.220375 + 0.975415i \(0.570728\pi\)
\(198\) 0 0
\(199\) 2.41947 + 4.19064i 0.171512 + 0.297067i 0.938949 0.344058i \(-0.111802\pi\)
−0.767437 + 0.641125i \(0.778468\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.39412 0.572206i 0.659338 0.0401610i
\(204\) 0 0
\(205\) −8.65544 + 14.9917i −0.604522 + 1.04706i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −20.5831 −1.42376
\(210\) 0 0
\(211\) 10.7183 0.737877 0.368939 0.929454i \(-0.379721\pi\)
0.368939 + 0.929454i \(0.379721\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.02206 + 5.23437i −0.206103 + 0.356981i
\(216\) 0 0
\(217\) 12.1168 + 18.3164i 0.822541 + 1.24340i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.68112 + 2.91179i 0.113084 + 0.195868i
\(222\) 0 0
\(223\) −21.2398 −1.42232 −0.711160 0.703030i \(-0.751830\pi\)
−0.711160 + 0.703030i \(0.751830\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.45226 + 5.97948i 0.229134 + 0.396872i 0.957552 0.288261i \(-0.0930772\pi\)
−0.728418 + 0.685134i \(0.759744\pi\)
\(228\) 0 0
\(229\) −0.392301 + 0.679486i −0.0259240 + 0.0449017i −0.878696 0.477381i \(-0.841586\pi\)
0.852772 + 0.522283i \(0.174919\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.73197 6.46396i 0.244489 0.423468i −0.717499 0.696560i \(-0.754713\pi\)
0.961988 + 0.273092i \(0.0880464\pi\)
\(234\) 0 0
\(235\) −10.7099 18.5501i −0.698637 1.21007i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 15.0314 0.972298 0.486149 0.873876i \(-0.338401\pi\)
0.486149 + 0.873876i \(0.338401\pi\)
\(240\) 0 0
\(241\) 6.93877 + 12.0183i 0.446965 + 0.774167i 0.998187 0.0601923i \(-0.0191714\pi\)
−0.551221 + 0.834359i \(0.685838\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.95064 + 16.3519i −0.444060 + 1.04468i
\(246\) 0 0
\(247\) −1.99562 + 3.45652i −0.126978 + 0.219933i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 3.38121 0.213420 0.106710 0.994290i \(-0.465968\pi\)
0.106710 + 0.994290i \(0.465968\pi\)
\(252\) 0 0
\(253\) −7.09847 −0.446277
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.39269 + 5.87632i −0.211630 + 0.366555i −0.952225 0.305398i \(-0.901211\pi\)
0.740595 + 0.671952i \(0.234544\pi\)
\(258\) 0 0
\(259\) 2.60460 5.22074i 0.161842 0.324401i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.883211 + 1.52977i 0.0544611 + 0.0943294i 0.891971 0.452093i \(-0.149323\pi\)
−0.837510 + 0.546423i \(0.815989\pi\)
\(264\) 0 0
\(265\) 0.919470 0.0564826
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.88431 + 17.1201i 0.602657 + 1.04383i 0.992417 + 0.122916i \(0.0392246\pi\)
−0.389760 + 0.920916i \(0.627442\pi\)
\(270\) 0 0
\(271\) −5.38321 + 9.32399i −0.327007 + 0.566392i −0.981916 0.189315i \(-0.939373\pi\)
0.654910 + 0.755707i \(0.272707\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.71777 + 2.97527i −0.103586 + 0.179415i
\(276\) 0 0
\(277\) −8.41476 14.5748i −0.505594 0.875714i −0.999979 0.00647123i \(-0.997940\pi\)
0.494385 0.869243i \(-0.335393\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.08053 0.482044 0.241022 0.970520i \(-0.422517\pi\)
0.241022 + 0.970520i \(0.422517\pi\)
\(282\) 0 0
\(283\) −0.217047 0.375936i −0.0129021 0.0223471i 0.859502 0.511132i \(-0.170774\pi\)
−0.872404 + 0.488785i \(0.837440\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 18.0106 1.09704i 1.06313 0.0647564i
\(288\) 0 0
\(289\) −18.0117 + 31.1971i −1.05951 + 1.83513i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 29.1458 1.70272 0.851358 0.524584i \(-0.175779\pi\)
0.851358 + 0.524584i \(0.175779\pi\)
\(294\) 0 0
\(295\) −18.9556 −1.10364
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −0.688229 + 1.19205i −0.0398013 + 0.0689379i
\(300\) 0 0
\(301\) 6.28843 0.383035i 0.362459 0.0220777i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.83818 13.5761i −0.448813 0.777366i
\(306\) 0 0
\(307\) −2.73324 −0.155994 −0.0779972 0.996954i \(-0.524853\pi\)
−0.0779972 + 0.996954i \(0.524853\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −0.423810 0.734061i −0.0240321 0.0416248i 0.853759 0.520668i \(-0.174317\pi\)
−0.877791 + 0.479043i \(0.840984\pi\)
\(312\) 0 0
\(313\) −2.96174 + 5.12988i −0.167407 + 0.289958i −0.937508 0.347965i \(-0.886873\pi\)
0.770100 + 0.637923i \(0.220206\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.15034 8.92065i 0.289272 0.501034i −0.684364 0.729140i \(-0.739920\pi\)
0.973636 + 0.228107i \(0.0732535\pi\)
\(318\) 0 0
\(319\) −4.23525 7.33566i −0.237128 0.410718i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −62.9428 −3.50223
\(324\) 0 0
\(325\) 0.333092 + 0.576932i 0.0184766 + 0.0320024i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.96721 + 19.9786i −0.549510 + 1.10146i
\(330\) 0 0
\(331\) 7.41437 12.8421i 0.407530 0.705863i −0.587082 0.809528i \(-0.699723\pi\)
0.994612 + 0.103664i \(0.0330567\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 29.8080 1.62858
\(336\) 0 0
\(337\) −24.4701 −1.33297 −0.666487 0.745517i \(-0.732203\pi\)
−0.666487 + 0.745517i \(0.732203\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 9.88282 17.1175i 0.535185 0.926967i
\(342\) 0 0
\(343\) 18.2127 3.36135i 0.983392 0.181496i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.04975 + 7.01437i 0.217402 + 0.376551i 0.954013 0.299766i \(-0.0969085\pi\)
−0.736611 + 0.676317i \(0.763575\pi\)
\(348\) 0 0
\(349\) −9.77262 −0.523117 −0.261558 0.965188i \(-0.584236\pi\)
−0.261558 + 0.965188i \(0.584236\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.8247 23.9452i −0.735817 1.27447i −0.954364 0.298646i \(-0.903465\pi\)
0.218547 0.975826i \(-0.429868\pi\)
\(354\) 0 0
\(355\) −12.3366 + 21.3676i −0.654757 + 1.13407i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 7.65854 13.2650i 0.404202 0.700099i −0.590026 0.807384i \(-0.700882\pi\)
0.994228 + 0.107285i \(0.0342158\pi\)
\(360\) 0 0
\(361\) −27.8590 48.2532i −1.46626 2.53964i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 23.7537 1.24332
\(366\) 0 0
\(367\) 14.9503 + 25.8946i 0.780397 + 1.35169i 0.931711 + 0.363201i \(0.118316\pi\)
−0.151314 + 0.988486i \(0.548350\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −0.528780 0.799335i −0.0274529 0.0414994i
\(372\) 0 0
\(373\) −6.83894 + 11.8454i −0.354107 + 0.613331i −0.986965 0.160937i \(-0.948548\pi\)
0.632858 + 0.774268i \(0.281882\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.64251 −0.0845934
\(378\) 0 0
\(379\) 35.6036 1.82883 0.914416 0.404776i \(-0.132651\pi\)
0.914416 + 0.404776i \(0.132651\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.62266 6.27463i 0.185109 0.320618i −0.758504 0.651668i \(-0.774069\pi\)
0.943613 + 0.331050i \(0.107403\pi\)
\(384\) 0 0
\(385\) 15.9617 0.972242i 0.813483 0.0495501i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.11518 10.5918i −0.310052 0.537025i 0.668321 0.743873i \(-0.267013\pi\)
−0.978373 + 0.206847i \(0.933680\pi\)
\(390\) 0 0
\(391\) −21.7071 −1.09777
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 12.6381 + 21.8898i 0.635891 + 1.10140i
\(396\) 0 0
\(397\) −18.3826 + 31.8397i −0.922598 + 1.59799i −0.127219 + 0.991875i \(0.540605\pi\)
−0.795379 + 0.606112i \(0.792728\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −19.3870 + 33.5793i −0.968141 + 1.67687i −0.267215 + 0.963637i \(0.586103\pi\)
−0.700927 + 0.713233i \(0.747230\pi\)
\(402\) 0 0
\(403\) −1.91637 3.31925i −0.0954611 0.165344i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.25101 −0.260283
\(408\) 0 0
\(409\) 7.45973 + 12.9206i 0.368860 + 0.638885i 0.989388 0.145300i \(-0.0464148\pi\)
−0.620527 + 0.784185i \(0.713081\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.9012 + 16.4789i 0.536414 + 0.810874i
\(414\) 0 0
\(415\) 4.06272 7.03684i 0.199431 0.345425i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −29.1418 −1.42367 −0.711835 0.702346i \(-0.752136\pi\)
−0.711835 + 0.702346i \(0.752136\pi\)
\(420\) 0 0
\(421\) 12.7049 0.619197 0.309598 0.950867i \(-0.399805\pi\)
0.309598 + 0.950867i \(0.399805\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5.25293 + 9.09835i −0.254805 + 0.441335i
\(426\) 0 0
\(427\) −7.29463 + 14.6216i −0.353012 + 0.707588i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −15.7466 27.2738i −0.758485 1.31373i −0.943623 0.331022i \(-0.892607\pi\)
0.185138 0.982712i \(-0.440727\pi\)
\(432\) 0 0
\(433\) −37.4599 −1.80021 −0.900105 0.435674i \(-0.856510\pi\)
−0.900105 + 0.435674i \(0.856510\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.8840 22.3157i −0.616324 1.06750i
\(438\) 0 0
\(439\) −11.4652 + 19.8583i −0.547205 + 0.947786i 0.451260 + 0.892393i \(0.350975\pi\)
−0.998465 + 0.0553937i \(0.982359\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.6561 + 18.4570i −0.506289 + 0.876918i 0.493685 + 0.869641i \(0.335650\pi\)
−0.999974 + 0.00727709i \(0.997684\pi\)
\(444\) 0 0
\(445\) 8.35900 + 14.4782i 0.396255 + 0.686333i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −11.5729 −0.546157 −0.273078 0.961992i \(-0.588042\pi\)
−0.273078 + 0.961992i \(0.588042\pi\)
\(450\) 0 0
\(451\) −8.11989 14.0641i −0.382351 0.662251i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.38429 2.77471i 0.0648965 0.130081i
\(456\) 0 0
\(457\) 2.18422 3.78318i 0.102174 0.176970i −0.810406 0.585868i \(-0.800754\pi\)
0.912580 + 0.408898i \(0.134087\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −23.7118 −1.10437 −0.552184 0.833722i \(-0.686206\pi\)
−0.552184 + 0.833722i \(0.686206\pi\)
\(462\) 0 0
\(463\) −17.9888 −0.836009 −0.418004 0.908445i \(-0.637270\pi\)
−0.418004 + 0.908445i \(0.637270\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 19.0603 33.0135i 0.882007 1.52768i 0.0329003 0.999459i \(-0.489526\pi\)
0.849106 0.528222i \(-0.177141\pi\)
\(468\) 0 0
\(469\) −17.1423 25.9133i −0.791559 1.19657i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.83508 4.91049i −0.130357 0.225785i
\(474\) 0 0
\(475\) −12.4713 −0.572221
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 12.5599 + 21.7544i 0.573878 + 0.993986i 0.996163 + 0.0875226i \(0.0278950\pi\)
−0.422284 + 0.906463i \(0.638772\pi\)
\(480\) 0 0
\(481\) −0.509110 + 0.881804i −0.0232134 + 0.0402068i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −19.8491 + 34.3797i −0.901303 + 1.56110i
\(486\) 0 0
\(487\) −9.03388 15.6471i −0.409364 0.709040i 0.585454 0.810705i \(-0.300916\pi\)
−0.994819 + 0.101666i \(0.967583\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0.327654 0.0147868 0.00739341 0.999973i \(-0.497647\pi\)
0.00739341 + 0.999973i \(0.497647\pi\)
\(492\) 0 0
\(493\) −12.9514 22.4324i −0.583299 1.01030i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 25.6704 1.56361i 1.15147 0.0701375i
\(498\) 0 0
\(499\) −15.8610 + 27.4721i −0.710036 + 1.22982i 0.254807 + 0.966992i \(0.417988\pi\)
−0.964843 + 0.262827i \(0.915345\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.18155 0.231034 0.115517 0.993306i \(-0.463148\pi\)
0.115517 + 0.993306i \(0.463148\pi\)
\(504\) 0 0
\(505\) 46.1411 2.05325
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 11.6104 20.1098i 0.514622 0.891352i −0.485234 0.874384i \(-0.661265\pi\)
0.999856 0.0169675i \(-0.00540117\pi\)
\(510\) 0 0
\(511\) −13.6605 20.6501i −0.604307 0.913506i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −9.92966 17.1987i −0.437553 0.757864i
\(516\) 0 0
\(517\) 20.0945 0.883753
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.26713 + 12.5870i 0.318379 + 0.551448i 0.980150 0.198258i \(-0.0635283\pi\)
−0.661771 + 0.749706i \(0.730195\pi\)
\(522\) 0 0
\(523\) −3.12827 + 5.41833i −0.136790 + 0.236927i −0.926280 0.376837i \(-0.877012\pi\)
0.789490 + 0.613764i \(0.210345\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 30.2216 52.3453i 1.31647 2.28020i
\(528\) 0 0
\(529\) 7.05671 + 12.2226i 0.306814 + 0.531417i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −3.14904 −0.136400
\(534\) 0 0
\(535\) −10.8440 18.7824i −0.468829 0.812035i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −10.0247 13.3170i −0.431792 0.573606i
\(540\) 0 0
\(541\) 2.45463 4.25155i 0.105533 0.182788i −0.808423 0.588602i \(-0.799679\pi\)
0.913956 + 0.405814i \(0.133012\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −16.6597 −0.713623
\(546\) 0 0
\(547\) 8.88087 0.379719 0.189859 0.981811i \(-0.439197\pi\)
0.189859 + 0.981811i \(0.439197\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 15.3743 26.6290i 0.654965 1.13443i
\(552\) 0 0
\(553\) 11.7617 23.5755i 0.500158 1.00253i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.4047 + 28.4138i 0.695090 + 1.20393i 0.970150 + 0.242504i \(0.0779687\pi\)
−0.275061 + 0.961427i \(0.588698\pi\)
\(558\) 0 0
\(559\) −1.09949 −0.0465036
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.56981 9.64719i −0.234739 0.406581i 0.724457 0.689319i \(-0.242090\pi\)
−0.959197 + 0.282739i \(0.908757\pi\)
\(564\) 0 0
\(565\) −20.2769 + 35.1207i −0.853057 + 1.47754i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10.6416 + 18.4318i −0.446120 + 0.772702i −0.998129 0.0611354i \(-0.980528\pi\)
0.552010 + 0.833838i \(0.313861\pi\)
\(570\) 0 0
\(571\) 2.00310 + 3.46947i 0.0838272 + 0.145193i 0.904891 0.425644i \(-0.139952\pi\)
−0.821064 + 0.570837i \(0.806619\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.30096 −0.179362
\(576\) 0 0
\(577\) 13.9834 + 24.2200i 0.582137 + 1.00829i 0.995226 + 0.0976001i \(0.0311166\pi\)
−0.413089 + 0.910691i \(0.635550\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −8.45387 + 0.514934i −0.350726 + 0.0213630i
\(582\) 0 0
\(583\) −0.431289 + 0.747014i −0.0178622 + 0.0309382i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −32.7133 −1.35022 −0.675110 0.737717i \(-0.735904\pi\)
−0.675110 + 0.737717i \(0.735904\pi\)
\(588\) 0 0
\(589\) 71.7507 2.95644
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15.7076 27.2063i 0.645032 1.11723i −0.339262 0.940692i \(-0.610177\pi\)
0.984294 0.176537i \(-0.0564895\pi\)
\(594\) 0 0
\(595\) 48.8107 2.97311i 2.00104 0.121886i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 5.70680 + 9.88447i 0.233174 + 0.403868i 0.958740 0.284283i \(-0.0917556\pi\)
−0.725567 + 0.688152i \(0.758422\pi\)
\(600\) 0 0
\(601\) 7.69610 0.313930 0.156965 0.987604i \(-0.449829\pi\)
0.156965 + 0.987604i \(0.449829\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 6.76428 + 11.7161i 0.275007 + 0.476326i
\(606\) 0 0
\(607\) −0.228866 + 0.396407i −0.00928938 + 0.0160897i −0.870633 0.491934i \(-0.836290\pi\)
0.861343 + 0.508023i \(0.169624\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.94825 3.37447i 0.0788178 0.136516i
\(612\) 0 0
\(613\) −5.34094 9.25078i −0.215719 0.373636i 0.737776 0.675046i \(-0.235876\pi\)
−0.953495 + 0.301410i \(0.902543\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 28.0379 1.12876 0.564382 0.825514i \(-0.309114\pi\)
0.564382 + 0.825514i \(0.309114\pi\)
\(618\) 0 0
\(619\) 17.6447 + 30.5616i 0.709201 + 1.22837i 0.965154 + 0.261683i \(0.0842776\pi\)
−0.255953 + 0.966689i \(0.582389\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.77934 15.5931i 0.311673 0.624726i
\(624\) 0 0
\(625\) 15.0661 26.0953i 0.602645 1.04381i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16.0576 −0.640257
\(630\) 0 0
\(631\) −4.12466 −0.164200 −0.0821001 0.996624i \(-0.526163\pi\)
−0.0821001 + 0.996624i \(0.526163\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −25.3484 + 43.9047i −1.00592 + 1.74230i
\(636\) 0 0
\(637\) −3.20827 + 0.392294i −0.127116 + 0.0155432i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16.4998 + 28.5785i 0.651704 + 1.12878i 0.982709 + 0.185155i \(0.0592788\pi\)
−0.331006 + 0.943629i \(0.607388\pi\)
\(642\) 0 0
\(643\) 3.45306 0.136175 0.0680876 0.997679i \(-0.478310\pi\)
0.0680876 + 0.997679i \(0.478310\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 15.0623 + 26.0887i 0.592161 + 1.02565i 0.993941 + 0.109917i \(0.0350585\pi\)
−0.401779 + 0.915736i \(0.631608\pi\)
\(648\) 0 0
\(649\) 8.89136 15.4003i 0.349016 0.604514i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −23.7003 + 41.0501i −0.927464 + 1.60641i −0.139913 + 0.990164i \(0.544682\pi\)
−0.787550 + 0.616250i \(0.788651\pi\)
\(654\) 0 0
\(655\) −22.1751 38.4085i −0.866454 1.50074i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 33.8623 1.31909 0.659544 0.751666i \(-0.270749\pi\)
0.659544 + 0.751666i \(0.270749\pi\)
\(660\) 0 0
\(661\) 1.57142 + 2.72178i 0.0611212 + 0.105865i 0.894967 0.446133i \(-0.147199\pi\)
−0.833846 + 0.551998i \(0.813866\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 32.0275 + 48.4146i 1.24197 + 1.87744i
\(666\) 0 0
\(667\) 5.30211 9.18352i 0.205298 0.355587i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 14.7064 0.567734
\(672\) 0 0
\(673\) 4.50186 0.173534 0.0867670 0.996229i \(-0.472346\pi\)
0.0867670 + 0.996229i \(0.472346\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −10.5827 + 18.3298i −0.406727 + 0.704472i −0.994521 0.104539i \(-0.966663\pi\)
0.587794 + 0.809011i \(0.299997\pi\)
\(678\) 0 0
\(679\) 41.3029 2.51580i 1.58506 0.0965475i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9.44153 + 16.3532i 0.361270 + 0.625738i 0.988170 0.153361i \(-0.0490098\pi\)
−0.626900 + 0.779100i \(0.715677\pi\)
\(684\) 0 0
\(685\) 19.0037 0.726095
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.0836309 + 0.144853i 0.00318608 + 0.00551846i
\(690\) 0 0
\(691\) 13.3286 23.0857i 0.507042 0.878223i −0.492925 0.870072i \(-0.664072\pi\)
0.999967 0.00815061i \(-0.00259445\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.9774 29.4058i 0.643990 1.11542i
\(696\) 0 0
\(697\) −24.8306 43.0078i −0.940524 1.62904i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 7.77329 0.293593 0.146797 0.989167i \(-0.453104\pi\)
0.146797 + 0.989167i \(0.453104\pi\)
\(702\) 0 0
\(703\) −9.53078 16.5078i −0.359460 0.622603i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −26.5354 40.1124i −0.997965 1.50858i
\(708\) 0 0
\(709\) 20.1123 34.8355i 0.755332 1.30827i −0.189877 0.981808i \(-0.560809\pi\)
0.945209 0.326466i \(-0.105858\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 24.7446 0.926693
\(714\) 0 0
\(715\) −2.79080 −0.104370
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 16.1822 28.0284i 0.603495 1.04528i −0.388792 0.921325i \(-0.627108\pi\)
0.992287 0.123959i \(-0.0395590\pi\)
\(720\) 0 0
\(721\) −9.24108 + 18.5231i −0.344156 + 0.689836i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.56614 4.44468i −0.0953039 0.165071i
\(726\) 0 0
\(727\) 35.4282 1.31396 0.656980 0.753908i \(-0.271834\pi\)
0.656980 + 0.753908i \(0.271834\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −8.66964 15.0163i −0.320658 0.555396i
\(732\) 0 0
\(733\) −2.49927 + 4.32887i −0.0923128 + 0.159890i −0.908484 0.417920i \(-0.862759\pi\)
0.816171 + 0.577810i \(0.196093\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.9818 + 24.2172i −0.515026 + 0.892052i
\(738\) 0 0
\(739\) 16.5002 + 28.5792i 0.606969 + 1.05130i 0.991737 + 0.128288i \(0.0409483\pi\)
−0.384768 + 0.923014i \(0.625718\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −45.0817 −1.65389 −0.826944 0.562285i \(-0.809922\pi\)
−0.826944 + 0.562285i \(0.809922\pi\)
\(744\) 0 0
\(745\) 8.11154 + 14.0496i 0.297184 + 0.514738i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −10.0921 + 20.2288i −0.368756 + 0.739145i
\(750\) 0 0
\(751\) −13.2833 + 23.0074i −0.484715 + 0.839552i −0.999846 0.0175601i \(-0.994410\pi\)
0.515130 + 0.857112i \(0.327743\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 14.7087 0.535306
\(756\) 0 0
\(757\) −15.6833 −0.570021 −0.285010 0.958524i \(-0.591997\pi\)
−0.285010 + 0.958524i \(0.591997\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −22.3282 + 38.6735i −0.809396 + 1.40191i 0.103887 + 0.994589i \(0.466872\pi\)
−0.913283 + 0.407326i \(0.866461\pi\)
\(762\) 0 0
\(763\) 9.58086 + 14.4830i 0.346851 + 0.524320i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.72412 2.98626i −0.0622542 0.107827i
\(768\) 0 0
\(769\) −7.15705 −0.258090 −0.129045 0.991639i \(-0.541191\pi\)
−0.129045 + 0.991639i \(0.541191\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.9545 36.2942i −0.753679 1.30541i −0.946028 0.324084i \(-0.894944\pi\)
0.192349 0.981327i \(-0.438389\pi\)
\(774\) 0 0
\(775\) 5.98800 10.3715i 0.215095 0.372556i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 29.4758 51.0536i 1.05608 1.82918i
\(780\) 0 0
\(781\) −11.5732 20.0454i −0.414123 0.717282i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 9.33155 0.333057
\(786\) 0 0
\(787\) −4.62627 8.01294i −0.164909 0.285630i 0.771714 0.635970i \(-0.219400\pi\)
−0.936623 + 0.350339i \(0.886066\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 42.1930 2.57002i 1.50021 0.0913794i
\(792\) 0 0
\(793\) 1.42585 2.46965i 0.0506335 0.0876997i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0.389964 0.0138132 0.00690662 0.999976i \(-0.497802\pi\)
0.00690662 + 0.999976i \(0.497802\pi\)
\(798\) 0 0
\(799\) 61.4487 2.17390
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −11.1420 + 19.2984i −0.393191 + 0.681027i
\(804\) 0 0
\(805\) 11.0453 + 16.6967i 0.389296 + 0.588482i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −25.3440 43.8972i −0.891049 1.54334i −0.838619 0.544718i \(-0.816637\pi\)
−0.0524297 0.998625i \(-0.516697\pi\)
\(810\) 0 0
\(811\) 22.5296 0.791121 0.395560 0.918440i \(-0.370550\pi\)
0.395560 + 0.918440i \(0.370550\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −11.2229 19.4386i −0.393120 0.680903i
\(816\) 0 0
\(817\) 10.2915 17.8255i 0.360055 0.623634i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.4991 21.6491i 0.436221 0.755558i −0.561173 0.827699i \(-0.689650\pi\)
0.997394 + 0.0721409i \(0.0229831\pi\)
\(822\) 0 0
\(823\) −11.9706 20.7338i −0.417271 0.722734i 0.578393 0.815758i \(-0.303680\pi\)
−0.995664 + 0.0930242i \(0.970347\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −42.7292 −1.48584 −0.742921 0.669380i \(-0.766560\pi\)
−0.742921 + 0.669380i \(0.766560\pi\)
\(828\) 0 0
\(829\) 19.4971 + 33.7700i 0.677162 + 1.17288i 0.975832 + 0.218522i \(0.0701237\pi\)
−0.298670 + 0.954356i \(0.596543\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −30.6553 40.7234i −1.06214 1.41098i
\(834\) 0 0
\(835\) −10.0397 + 17.3893i −0.347439 + 0.601782i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 56.4891 1.95022 0.975110 0.221721i \(-0.0711674\pi\)
0.975110 + 0.221721i \(0.0711674\pi\)
\(840\) 0 0
\(841\) −16.3461 −0.563660
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 16.2281 28.1079i 0.558264 0.966942i
\(846\) 0 0
\(847\) 6.29520 12.6183i 0.216306 0.433570i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −3.28687 5.69303i −0.112673 0.195155i
\(852\) 0 0
\(853\) −19.6327 −0.672212 −0.336106 0.941824i \(-0.609110\pi\)
−0.336106 + 0.941824i \(0.609110\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.4020 + 19.7488i 0.389485 + 0.674607i 0.992380 0.123213i \(-0.0393198\pi\)
−0.602896 + 0.797820i \(0.705986\pi\)
\(858\) 0 0
\(859\) 1.26314 2.18782i 0.0430978 0.0746476i −0.843672 0.536859i \(-0.819611\pi\)
0.886770 + 0.462212i \(0.152944\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −15.2946 + 26.4911i −0.520635 + 0.901767i 0.479077 + 0.877773i \(0.340972\pi\)
−0.999712 + 0.0239938i \(0.992362\pi\)
\(864\) 0 0
\(865\) 14.0164 + 24.2772i 0.476573 + 0.825450i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −23.7122 −0.804382
\(870\) 0 0
\(871\) 2.71120 + 4.69593i 0.0918655 + 0.159116i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −23.8447 + 1.45241i −0.806100 + 0.0491004i
\(876\) 0 0
\(877\) 10.8010 18.7079i 0.364724 0.631721i −0.624008 0.781418i \(-0.714497\pi\)
0.988732 + 0.149697i \(0.0478299\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −43.2366 −1.45668 −0.728338 0.685218i \(-0.759707\pi\)
−0.728338 + 0.685218i \(0.759707\pi\)
\(882\) 0 0
\(883\) −30.7820 −1.03590 −0.517949 0.855411i \(-0.673304\pi\)
−0.517949 + 0.855411i \(0.673304\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −4.61259 + 7.98924i −0.154876 + 0.268252i −0.933014 0.359841i \(-0.882831\pi\)
0.778138 + 0.628093i \(0.216164\pi\)
\(888\) 0 0
\(889\) 52.7459 3.21281i 1.76904 0.107754i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 36.4722 + 63.1717i 1.22049 + 2.11396i
\(894\) 0 0
\(895\) 6.53905 0.218576
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 14.7637 + 25.5715i 0.492397 + 0.852856i
\(900\) 0 0
\(901\) −1.31888 + 2.28436i −0.0439382 + 0.0761032i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 26.4793 45.8635i 0.880201 1.52455i
\(906\) 0 0
\(907\) −22.6210 39.1807i −0.751118 1.30097i −0.947281 0.320403i \(-0.896182\pi\)
0.196164 0.980571i \(-0.437152\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −22.3678 −0.741078 −0.370539 0.928817i \(-0.620827\pi\)
−0.370539 + 0.928817i \(0.620827\pi\)
\(912\) 0 0
\(913\) 3.81134 + 6.60144i 0.126137 + 0.218476i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −20.6374 + 41.3662i −0.681506 + 1.36603i
\(918\) 0 0
\(919\) −5.36191 + 9.28710i −0.176873 + 0.306353i −0.940808 0.338940i \(-0.889932\pi\)
0.763935 + 0.645293i \(0.223265\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −4.48832 −0.147735
\(924\) 0 0
\(925\) −3.18159 −0.104610
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −8.63065 + 14.9487i −0.283162 + 0.490452i −0.972162 0.234310i \(-0.924717\pi\)
0.688999 + 0.724762i \(0.258050\pi\)
\(930\) 0 0
\(931\) 23.6702 55.6858i 0.775758 1.82503i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −22.0058 38.1152i −0.719667 1.24650i
\(936\) 0 0
\(937\) −35.1775 −1.14920 −0.574600 0.818434i \(-0.694842\pi\)
−0.574600 + 0.818434i \(0.694842\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −20.2711 35.1106i −0.660820 1.14457i −0.980401 0.197015i \(-0.936875\pi\)
0.319580 0.947559i \(-0.396458\pi\)
\(942\) 0 0
\(943\) 10.1653 17.6068i 0.331028 0.573357i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −14.5133 + 25.1378i −0.471619 + 0.816867i −0.999473 0.0324677i \(-0.989663\pi\)
0.527854 + 0.849335i \(0.322997\pi\)
\(948\) 0 0
\(949\) 2.16053 + 3.74214i 0.0701337 + 0.121475i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 57.3207 1.85680 0.928400 0.371583i \(-0.121185\pi\)
0.928400 + 0.371583i \(0.121185\pi\)
\(954\) 0 0
\(955\) 5.56033 + 9.63077i 0.179928 + 0.311644i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −10.9289 16.5208i −0.352912 0.533483i
\(960\) 0 0
\(961\) −18.9506 + 32.8234i −0.611310 + 1.05882i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 37.6170 1.21093
\(966\) 0 0
\(967\) 13.7030 0.440660 0.220330 0.975425i \(-0.429287\pi\)
0.220330 + 0.975425i \(0.429287\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −15.4734 + 26.8007i −0.496566 + 0.860077i −0.999992 0.00396126i \(-0.998739\pi\)
0.503427 + 0.864038i \(0.332072\pi\)
\(972\) 0 0
\(973\) −35.3273 + 2.15182i −1.13254 + 0.0689842i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −24.0622 41.6769i −0.769818 1.33336i −0.937662 0.347550i \(-0.887014\pi\)
0.167844 0.985814i \(-0.446320\pi\)
\(978\) 0 0
\(979\) −15.6836 −0.501249
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 11.3437 + 19.6478i 0.361806 + 0.626667i 0.988258 0.152793i \(-0.0488268\pi\)
−0.626452 + 0.779460i \(0.715494\pi\)
\(984\) 0 0
\(985\) 7.85113 13.5986i 0.250158 0.433286i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.54923 6.14745i 0.112859 0.195478i
\(990\) 0 0
\(991\) 22.8326 + 39.5472i 0.725300 + 1.25626i 0.958850 + 0.283912i \(0.0916323\pi\)
−0.233550 + 0.972345i \(0.575034\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −12.2825 −0.389381
\(996\) 0 0
\(997\) −20.5374 35.5717i −0.650425 1.12657i −0.983020 0.183499i \(-0.941258\pi\)
0.332595 0.943070i \(-0.392076\pi\)
\(998\) 0 0
\(999\) 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 1512.2.s.n.1297.2 yes 8
3.2 odd 2 1512.2.s.o.1297.3 yes 8
7.4 even 3 inner 1512.2.s.n.865.2 8
21.11 odd 6 1512.2.s.o.865.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.s.n.865.2 8 7.4 even 3 inner
1512.2.s.n.1297.2 yes 8 1.1 even 1 trivial
1512.2.s.o.865.3 yes 8 21.11 odd 6
1512.2.s.o.1297.3 yes 8 3.2 odd 2