Properties

Label 2960.2.a.z.1.4
Level $2960$
Weight $2$
Character 2960.1
Self dual yes
Analytic conductor $23.636$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2960,2,Mod(1,2960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2960.a (trivial)

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: yes
Analytic conductor: \(23.6357189983\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.935504.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 8x^{3} + 4x^{2} + 8x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1480)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.43118\) of defining polynomial
Character \(\chi\) \(=\) 2960.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.43118 q^{3} -1.00000 q^{5} +1.96628 q^{7} -0.951735 q^{9} +O(q^{10})\) \(q+1.43118 q^{3} -1.00000 q^{5} +1.96628 q^{7} -0.951735 q^{9} -5.12053 q^{11} +1.62611 q^{13} -1.43118 q^{15} +5.16879 q^{17} +6.46621 q^{19} +2.81409 q^{21} -0.862352 q^{23} +1.00000 q^{25} -5.65563 q^{27} +1.93255 q^{29} +7.62962 q^{31} -7.32838 q^{33} -1.96628 q^{35} +1.00000 q^{37} +2.32725 q^{39} -10.1907 q^{41} +9.83602 q^{43} +0.951735 q^{45} -8.66465 q^{47} -3.13376 q^{49} +7.39745 q^{51} -2.89752 q^{53} +5.12053 q^{55} +9.25429 q^{57} +14.4920 q^{59} +7.24119 q^{61} -1.87137 q^{63} -1.62611 q^{65} +11.3286 q^{67} -1.23418 q^{69} +9.89739 q^{71} +9.03504 q^{73} +1.43118 q^{75} -10.0684 q^{77} +11.9492 q^{79} -5.23899 q^{81} +7.24915 q^{83} -5.16879 q^{85} +2.76582 q^{87} +7.85407 q^{89} +3.19738 q^{91} +10.9193 q^{93} -6.46621 q^{95} +2.24947 q^{97} +4.87339 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{3} - 5 q^{5} + q^{7} + 2 q^{9} + 7 q^{11} + 4 q^{13} - q^{15} + 10 q^{19} - 5 q^{21} + 8 q^{23} + 5 q^{25} + 7 q^{27} - 8 q^{29} + 2 q^{31} - 11 q^{33} - q^{35} + 5 q^{37} + 2 q^{39} - 13 q^{41} + 18 q^{43} - 2 q^{45} + 9 q^{47} - 6 q^{49} + 22 q^{51} - 13 q^{53} - 7 q^{55} + 4 q^{57} + 24 q^{59} - 2 q^{61} + 20 q^{63} - 4 q^{65} + 22 q^{67} - 32 q^{69} + 21 q^{71} + 29 q^{73} + q^{75} + 11 q^{77} + 6 q^{79} + 5 q^{81} + 33 q^{83} - 12 q^{87} - 26 q^{89} + 38 q^{91} + 14 q^{93} - 10 q^{95} + 26 q^{97} + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 1.43118 0.826290 0.413145 0.910665i \(-0.364430\pi\)
0.413145 + 0.910665i \(0.364430\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.96628 0.743183 0.371591 0.928396i \(-0.378812\pi\)
0.371591 + 0.928396i \(0.378812\pi\)
\(8\) 0 0
\(9\) −0.951735 −0.317245
\(10\) 0 0
\(11\) −5.12053 −1.54390 −0.771949 0.635685i \(-0.780718\pi\)
−0.771949 + 0.635685i \(0.780718\pi\)
\(12\) 0 0
\(13\) 1.62611 0.451002 0.225501 0.974243i \(-0.427598\pi\)
0.225501 + 0.974243i \(0.427598\pi\)
\(14\) 0 0
\(15\) −1.43118 −0.369528
\(16\) 0 0
\(17\) 5.16879 1.25362 0.626808 0.779174i \(-0.284361\pi\)
0.626808 + 0.779174i \(0.284361\pi\)
\(18\) 0 0
\(19\) 6.46621 1.48345 0.741725 0.670704i \(-0.234008\pi\)
0.741725 + 0.670704i \(0.234008\pi\)
\(20\) 0 0
\(21\) 2.81409 0.614084
\(22\) 0 0
\(23\) −0.862352 −0.179813 −0.0899064 0.995950i \(-0.528657\pi\)
−0.0899064 + 0.995950i \(0.528657\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.65563 −1.08843
\(28\) 0 0
\(29\) 1.93255 0.358866 0.179433 0.983770i \(-0.442574\pi\)
0.179433 + 0.983770i \(0.442574\pi\)
\(30\) 0 0
\(31\) 7.62962 1.37032 0.685160 0.728393i \(-0.259732\pi\)
0.685160 + 0.728393i \(0.259732\pi\)
\(32\) 0 0
\(33\) −7.32838 −1.27571
\(34\) 0 0
\(35\) −1.96628 −0.332361
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 0 0
\(39\) 2.32725 0.372659
\(40\) 0 0
\(41\) −10.1907 −1.59152 −0.795762 0.605609i \(-0.792930\pi\)
−0.795762 + 0.605609i \(0.792930\pi\)
\(42\) 0 0
\(43\) 9.83602 1.49998 0.749990 0.661449i \(-0.230058\pi\)
0.749990 + 0.661449i \(0.230058\pi\)
\(44\) 0 0
\(45\) 0.951735 0.141876
\(46\) 0 0
\(47\) −8.66465 −1.26387 −0.631935 0.775022i \(-0.717739\pi\)
−0.631935 + 0.775022i \(0.717739\pi\)
\(48\) 0 0
\(49\) −3.13376 −0.447680
\(50\) 0 0
\(51\) 7.39745 1.03585
\(52\) 0 0
\(53\) −2.89752 −0.398005 −0.199002 0.979999i \(-0.563770\pi\)
−0.199002 + 0.979999i \(0.563770\pi\)
\(54\) 0 0
\(55\) 5.12053 0.690452
\(56\) 0 0
\(57\) 9.25429 1.22576
\(58\) 0 0
\(59\) 14.4920 1.88669 0.943347 0.331808i \(-0.107659\pi\)
0.943347 + 0.331808i \(0.107659\pi\)
\(60\) 0 0
\(61\) 7.24119 0.927139 0.463569 0.886061i \(-0.346568\pi\)
0.463569 + 0.886061i \(0.346568\pi\)
\(62\) 0 0
\(63\) −1.87137 −0.235771
\(64\) 0 0
\(65\) −1.62611 −0.201694
\(66\) 0 0
\(67\) 11.3286 1.38400 0.692002 0.721895i \(-0.256729\pi\)
0.692002 + 0.721895i \(0.256729\pi\)
\(68\) 0 0
\(69\) −1.23418 −0.148577
\(70\) 0 0
\(71\) 9.89739 1.17460 0.587302 0.809368i \(-0.300190\pi\)
0.587302 + 0.809368i \(0.300190\pi\)
\(72\) 0 0
\(73\) 9.03504 1.05747 0.528735 0.848787i \(-0.322666\pi\)
0.528735 + 0.848787i \(0.322666\pi\)
\(74\) 0 0
\(75\) 1.43118 0.165258
\(76\) 0 0
\(77\) −10.0684 −1.14740
\(78\) 0 0
\(79\) 11.9492 1.34439 0.672193 0.740376i \(-0.265353\pi\)
0.672193 + 0.740376i \(0.265353\pi\)
\(80\) 0 0
\(81\) −5.23899 −0.582110
\(82\) 0 0
\(83\) 7.24915 0.795698 0.397849 0.917451i \(-0.369757\pi\)
0.397849 + 0.917451i \(0.369757\pi\)
\(84\) 0 0
\(85\) −5.16879 −0.560634
\(86\) 0 0
\(87\) 2.76582 0.296527
\(88\) 0 0
\(89\) 7.85407 0.832530 0.416265 0.909243i \(-0.363339\pi\)
0.416265 + 0.909243i \(0.363339\pi\)
\(90\) 0 0
\(91\) 3.19738 0.335177
\(92\) 0 0
\(93\) 10.9193 1.13228
\(94\) 0 0
\(95\) −6.46621 −0.663419
\(96\) 0 0
\(97\) 2.24947 0.228399 0.114200 0.993458i \(-0.463570\pi\)
0.114200 + 0.993458i \(0.463570\pi\)
\(98\) 0 0
\(99\) 4.87339 0.489794
\(100\) 0 0
\(101\) 10.0218 0.997207 0.498603 0.866830i \(-0.333846\pi\)
0.498603 + 0.866830i \(0.333846\pi\)
\(102\) 0 0
\(103\) −18.0207 −1.77563 −0.887815 0.460201i \(-0.847777\pi\)
−0.887815 + 0.460201i \(0.847777\pi\)
\(104\) 0 0
\(105\) −2.81409 −0.274627
\(106\) 0 0
\(107\) 1.76451 0.170582 0.0852909 0.996356i \(-0.472818\pi\)
0.0852909 + 0.996356i \(0.472818\pi\)
\(108\) 0 0
\(109\) −14.7839 −1.41604 −0.708019 0.706194i \(-0.750411\pi\)
−0.708019 + 0.706194i \(0.750411\pi\)
\(110\) 0 0
\(111\) 1.43118 0.135841
\(112\) 0 0
\(113\) −14.1687 −1.33288 −0.666438 0.745561i \(-0.732182\pi\)
−0.666438 + 0.745561i \(0.732182\pi\)
\(114\) 0 0
\(115\) 0.862352 0.0804147
\(116\) 0 0
\(117\) −1.54763 −0.143078
\(118\) 0 0
\(119\) 10.1633 0.931666
\(120\) 0 0
\(121\) 15.2198 1.38362
\(122\) 0 0
\(123\) −14.5847 −1.31506
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.7134 −1.12813 −0.564064 0.825731i \(-0.690763\pi\)
−0.564064 + 0.825731i \(0.690763\pi\)
\(128\) 0 0
\(129\) 14.0771 1.23942
\(130\) 0 0
\(131\) 13.9569 1.21942 0.609709 0.792626i \(-0.291286\pi\)
0.609709 + 0.792626i \(0.291286\pi\)
\(132\) 0 0
\(133\) 12.7144 1.10247
\(134\) 0 0
\(135\) 5.65563 0.486759
\(136\) 0 0
\(137\) −3.38974 −0.289605 −0.144803 0.989461i \(-0.546255\pi\)
−0.144803 + 0.989461i \(0.546255\pi\)
\(138\) 0 0
\(139\) −11.6982 −0.992232 −0.496116 0.868256i \(-0.665241\pi\)
−0.496116 + 0.868256i \(0.665241\pi\)
\(140\) 0 0
\(141\) −12.4006 −1.04432
\(142\) 0 0
\(143\) −8.32655 −0.696301
\(144\) 0 0
\(145\) −1.93255 −0.160490
\(146\) 0 0
\(147\) −4.48496 −0.369913
\(148\) 0 0
\(149\) −13.6610 −1.11915 −0.559577 0.828778i \(-0.689036\pi\)
−0.559577 + 0.828778i \(0.689036\pi\)
\(150\) 0 0
\(151\) 4.31954 0.351519 0.175760 0.984433i \(-0.443762\pi\)
0.175760 + 0.984433i \(0.443762\pi\)
\(152\) 0 0
\(153\) −4.91932 −0.397704
\(154\) 0 0
\(155\) −7.62962 −0.612826
\(156\) 0 0
\(157\) −7.27609 −0.580695 −0.290348 0.956921i \(-0.593771\pi\)
−0.290348 + 0.956921i \(0.593771\pi\)
\(158\) 0 0
\(159\) −4.14686 −0.328867
\(160\) 0 0
\(161\) −1.69562 −0.133634
\(162\) 0 0
\(163\) 24.0800 1.88609 0.943044 0.332668i \(-0.107949\pi\)
0.943044 + 0.332668i \(0.107949\pi\)
\(164\) 0 0
\(165\) 7.32838 0.570513
\(166\) 0 0
\(167\) 22.8045 1.76467 0.882334 0.470624i \(-0.155971\pi\)
0.882334 + 0.470624i \(0.155971\pi\)
\(168\) 0 0
\(169\) −10.3558 −0.796597
\(170\) 0 0
\(171\) −6.15412 −0.470617
\(172\) 0 0
\(173\) −7.38804 −0.561703 −0.280851 0.959751i \(-0.590617\pi\)
−0.280851 + 0.959751i \(0.590617\pi\)
\(174\) 0 0
\(175\) 1.96628 0.148637
\(176\) 0 0
\(177\) 20.7406 1.55896
\(178\) 0 0
\(179\) 3.08751 0.230771 0.115386 0.993321i \(-0.463190\pi\)
0.115386 + 0.993321i \(0.463190\pi\)
\(180\) 0 0
\(181\) 17.8320 1.32544 0.662721 0.748866i \(-0.269401\pi\)
0.662721 + 0.748866i \(0.269401\pi\)
\(182\) 0 0
\(183\) 10.3634 0.766085
\(184\) 0 0
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −26.4670 −1.93546
\(188\) 0 0
\(189\) −11.1205 −0.808899
\(190\) 0 0
\(191\) −17.7442 −1.28392 −0.641962 0.766736i \(-0.721879\pi\)
−0.641962 + 0.766736i \(0.721879\pi\)
\(192\) 0 0
\(193\) −20.4702 −1.47348 −0.736739 0.676177i \(-0.763635\pi\)
−0.736739 + 0.676177i \(0.763635\pi\)
\(194\) 0 0
\(195\) −2.32725 −0.166658
\(196\) 0 0
\(197\) −2.85627 −0.203501 −0.101750 0.994810i \(-0.532444\pi\)
−0.101750 + 0.994810i \(0.532444\pi\)
\(198\) 0 0
\(199\) −5.00902 −0.355080 −0.177540 0.984114i \(-0.556814\pi\)
−0.177540 + 0.984114i \(0.556814\pi\)
\(200\) 0 0
\(201\) 16.2132 1.14359
\(202\) 0 0
\(203\) 3.79993 0.266703
\(204\) 0 0
\(205\) 10.1907 0.711751
\(206\) 0 0
\(207\) 0.820731 0.0570447
\(208\) 0 0
\(209\) −33.1104 −2.29030
\(210\) 0 0
\(211\) 9.45224 0.650719 0.325359 0.945590i \(-0.394515\pi\)
0.325359 + 0.945590i \(0.394515\pi\)
\(212\) 0 0
\(213\) 14.1649 0.970563
\(214\) 0 0
\(215\) −9.83602 −0.670811
\(216\) 0 0
\(217\) 15.0019 1.01840
\(218\) 0 0
\(219\) 12.9307 0.873777
\(220\) 0 0
\(221\) 8.40503 0.565384
\(222\) 0 0
\(223\) 7.04681 0.471889 0.235945 0.971766i \(-0.424182\pi\)
0.235945 + 0.971766i \(0.424182\pi\)
\(224\) 0 0
\(225\) −0.951735 −0.0634490
\(226\) 0 0
\(227\) 3.65437 0.242549 0.121275 0.992619i \(-0.461302\pi\)
0.121275 + 0.992619i \(0.461302\pi\)
\(228\) 0 0
\(229\) 9.48132 0.626543 0.313272 0.949664i \(-0.398575\pi\)
0.313272 + 0.949664i \(0.398575\pi\)
\(230\) 0 0
\(231\) −14.4096 −0.948083
\(232\) 0 0
\(233\) 20.4379 1.33893 0.669465 0.742844i \(-0.266524\pi\)
0.669465 + 0.742844i \(0.266524\pi\)
\(234\) 0 0
\(235\) 8.66465 0.565220
\(236\) 0 0
\(237\) 17.1013 1.11085
\(238\) 0 0
\(239\) −23.6661 −1.53084 −0.765418 0.643533i \(-0.777468\pi\)
−0.765418 + 0.643533i \(0.777468\pi\)
\(240\) 0 0
\(241\) 27.5265 1.77314 0.886568 0.462597i \(-0.153082\pi\)
0.886568 + 0.462597i \(0.153082\pi\)
\(242\) 0 0
\(243\) 9.46896 0.607434
\(244\) 0 0
\(245\) 3.13376 0.200208
\(246\) 0 0
\(247\) 10.5148 0.669039
\(248\) 0 0
\(249\) 10.3748 0.657477
\(250\) 0 0
\(251\) 17.9382 1.13225 0.566124 0.824320i \(-0.308442\pi\)
0.566124 + 0.824320i \(0.308442\pi\)
\(252\) 0 0
\(253\) 4.41570 0.277613
\(254\) 0 0
\(255\) −7.39745 −0.463246
\(256\) 0 0
\(257\) 19.3166 1.20493 0.602467 0.798143i \(-0.294184\pi\)
0.602467 + 0.798143i \(0.294184\pi\)
\(258\) 0 0
\(259\) 1.96628 0.122178
\(260\) 0 0
\(261\) −1.83928 −0.113848
\(262\) 0 0
\(263\) −2.13733 −0.131794 −0.0658968 0.997826i \(-0.520991\pi\)
−0.0658968 + 0.997826i \(0.520991\pi\)
\(264\) 0 0
\(265\) 2.89752 0.177993
\(266\) 0 0
\(267\) 11.2406 0.687911
\(268\) 0 0
\(269\) −15.0542 −0.917872 −0.458936 0.888469i \(-0.651769\pi\)
−0.458936 + 0.888469i \(0.651769\pi\)
\(270\) 0 0
\(271\) −11.6188 −0.705794 −0.352897 0.935662i \(-0.614803\pi\)
−0.352897 + 0.935662i \(0.614803\pi\)
\(272\) 0 0
\(273\) 4.57602 0.276953
\(274\) 0 0
\(275\) −5.12053 −0.308779
\(276\) 0 0
\(277\) −14.8551 −0.892558 −0.446279 0.894894i \(-0.647251\pi\)
−0.446279 + 0.894894i \(0.647251\pi\)
\(278\) 0 0
\(279\) −7.26138 −0.434727
\(280\) 0 0
\(281\) −21.5081 −1.28306 −0.641532 0.767096i \(-0.721701\pi\)
−0.641532 + 0.767096i \(0.721701\pi\)
\(282\) 0 0
\(283\) −31.5556 −1.87578 −0.937892 0.346928i \(-0.887225\pi\)
−0.937892 + 0.346928i \(0.887225\pi\)
\(284\) 0 0
\(285\) −9.25429 −0.548177
\(286\) 0 0
\(287\) −20.0378 −1.18279
\(288\) 0 0
\(289\) 9.71642 0.571554
\(290\) 0 0
\(291\) 3.21939 0.188724
\(292\) 0 0
\(293\) −9.70954 −0.567237 −0.283619 0.958937i \(-0.591535\pi\)
−0.283619 + 0.958937i \(0.591535\pi\)
\(294\) 0 0
\(295\) −14.4920 −0.843755
\(296\) 0 0
\(297\) 28.9598 1.68042
\(298\) 0 0
\(299\) −1.40228 −0.0810960
\(300\) 0 0
\(301\) 19.3403 1.11476
\(302\) 0 0
\(303\) 14.3430 0.823982
\(304\) 0 0
\(305\) −7.24119 −0.414629
\(306\) 0 0
\(307\) −1.28306 −0.0732283 −0.0366142 0.999329i \(-0.511657\pi\)
−0.0366142 + 0.999329i \(0.511657\pi\)
\(308\) 0 0
\(309\) −25.7907 −1.46718
\(310\) 0 0
\(311\) 6.93386 0.393183 0.196592 0.980485i \(-0.437013\pi\)
0.196592 + 0.980485i \(0.437013\pi\)
\(312\) 0 0
\(313\) 24.0619 1.36006 0.680030 0.733184i \(-0.261967\pi\)
0.680030 + 0.733184i \(0.261967\pi\)
\(314\) 0 0
\(315\) 1.87137 0.105440
\(316\) 0 0
\(317\) 10.8481 0.609288 0.304644 0.952466i \(-0.401463\pi\)
0.304644 + 0.952466i \(0.401463\pi\)
\(318\) 0 0
\(319\) −9.89569 −0.554052
\(320\) 0 0
\(321\) 2.52533 0.140950
\(322\) 0 0
\(323\) 33.4225 1.85968
\(324\) 0 0
\(325\) 1.62611 0.0902004
\(326\) 0 0
\(327\) −21.1583 −1.17006
\(328\) 0 0
\(329\) −17.0371 −0.939286
\(330\) 0 0
\(331\) 2.14379 0.117833 0.0589166 0.998263i \(-0.481235\pi\)
0.0589166 + 0.998263i \(0.481235\pi\)
\(332\) 0 0
\(333\) −0.951735 −0.0521548
\(334\) 0 0
\(335\) −11.3286 −0.618946
\(336\) 0 0
\(337\) 20.2817 1.10482 0.552408 0.833574i \(-0.313709\pi\)
0.552408 + 0.833574i \(0.313709\pi\)
\(338\) 0 0
\(339\) −20.2778 −1.10134
\(340\) 0 0
\(341\) −39.0677 −2.11563
\(342\) 0 0
\(343\) −19.9258 −1.07589
\(344\) 0 0
\(345\) 1.23418 0.0664459
\(346\) 0 0
\(347\) −22.7426 −1.22089 −0.610444 0.792060i \(-0.709009\pi\)
−0.610444 + 0.792060i \(0.709009\pi\)
\(348\) 0 0
\(349\) −32.3742 −1.73295 −0.866477 0.499218i \(-0.833621\pi\)
−0.866477 + 0.499218i \(0.833621\pi\)
\(350\) 0 0
\(351\) −9.19668 −0.490883
\(352\) 0 0
\(353\) 20.0358 1.06640 0.533200 0.845989i \(-0.320989\pi\)
0.533200 + 0.845989i \(0.320989\pi\)
\(354\) 0 0
\(355\) −9.89739 −0.525299
\(356\) 0 0
\(357\) 14.5454 0.769826
\(358\) 0 0
\(359\) −35.2849 −1.86226 −0.931132 0.364683i \(-0.881177\pi\)
−0.931132 + 0.364683i \(0.881177\pi\)
\(360\) 0 0
\(361\) 22.8119 1.20063
\(362\) 0 0
\(363\) 21.7822 1.14327
\(364\) 0 0
\(365\) −9.03504 −0.472915
\(366\) 0 0
\(367\) 6.45843 0.337127 0.168564 0.985691i \(-0.446087\pi\)
0.168564 + 0.985691i \(0.446087\pi\)
\(368\) 0 0
\(369\) 9.69888 0.504903
\(370\) 0 0
\(371\) −5.69732 −0.295790
\(372\) 0 0
\(373\) −29.6727 −1.53639 −0.768197 0.640213i \(-0.778846\pi\)
−0.768197 + 0.640213i \(0.778846\pi\)
\(374\) 0 0
\(375\) −1.43118 −0.0739056
\(376\) 0 0
\(377\) 3.14255 0.161849
\(378\) 0 0
\(379\) −8.66321 −0.444999 −0.222500 0.974933i \(-0.571422\pi\)
−0.222500 + 0.974933i \(0.571422\pi\)
\(380\) 0 0
\(381\) −18.1950 −0.932160
\(382\) 0 0
\(383\) 14.7713 0.754780 0.377390 0.926054i \(-0.376822\pi\)
0.377390 + 0.926054i \(0.376822\pi\)
\(384\) 0 0
\(385\) 10.0684 0.513132
\(386\) 0 0
\(387\) −9.36129 −0.475861
\(388\) 0 0
\(389\) 10.4077 0.527689 0.263845 0.964565i \(-0.415009\pi\)
0.263845 + 0.964565i \(0.415009\pi\)
\(390\) 0 0
\(391\) −4.45732 −0.225416
\(392\) 0 0
\(393\) 19.9747 1.00759
\(394\) 0 0
\(395\) −11.9492 −0.601227
\(396\) 0 0
\(397\) −18.6548 −0.936257 −0.468129 0.883660i \(-0.655072\pi\)
−0.468129 + 0.883660i \(0.655072\pi\)
\(398\) 0 0
\(399\) 18.1965 0.910964
\(400\) 0 0
\(401\) −27.4491 −1.37075 −0.685373 0.728193i \(-0.740361\pi\)
−0.685373 + 0.728193i \(0.740361\pi\)
\(402\) 0 0
\(403\) 12.4066 0.618017
\(404\) 0 0
\(405\) 5.23899 0.260328
\(406\) 0 0
\(407\) −5.12053 −0.253815
\(408\) 0 0
\(409\) −25.0540 −1.23884 −0.619420 0.785059i \(-0.712632\pi\)
−0.619420 + 0.785059i \(0.712632\pi\)
\(410\) 0 0
\(411\) −4.85132 −0.239298
\(412\) 0 0
\(413\) 28.4952 1.40216
\(414\) 0 0
\(415\) −7.24915 −0.355847
\(416\) 0 0
\(417\) −16.7422 −0.819871
\(418\) 0 0
\(419\) 11.0041 0.537583 0.268792 0.963198i \(-0.413376\pi\)
0.268792 + 0.963198i \(0.413376\pi\)
\(420\) 0 0
\(421\) −5.32255 −0.259405 −0.129703 0.991553i \(-0.541402\pi\)
−0.129703 + 0.991553i \(0.541402\pi\)
\(422\) 0 0
\(423\) 8.24646 0.400956
\(424\) 0 0
\(425\) 5.16879 0.250723
\(426\) 0 0
\(427\) 14.2382 0.689034
\(428\) 0 0
\(429\) −11.9168 −0.575346
\(430\) 0 0
\(431\) 5.75785 0.277346 0.138673 0.990338i \(-0.455716\pi\)
0.138673 + 0.990338i \(0.455716\pi\)
\(432\) 0 0
\(433\) −35.0962 −1.68661 −0.843307 0.537431i \(-0.819395\pi\)
−0.843307 + 0.537431i \(0.819395\pi\)
\(434\) 0 0
\(435\) −2.76582 −0.132611
\(436\) 0 0
\(437\) −5.57615 −0.266743
\(438\) 0 0
\(439\) 27.2085 1.29859 0.649294 0.760537i \(-0.275064\pi\)
0.649294 + 0.760537i \(0.275064\pi\)
\(440\) 0 0
\(441\) 2.98251 0.142024
\(442\) 0 0
\(443\) 2.64372 0.125607 0.0628035 0.998026i \(-0.479996\pi\)
0.0628035 + 0.998026i \(0.479996\pi\)
\(444\) 0 0
\(445\) −7.85407 −0.372319
\(446\) 0 0
\(447\) −19.5513 −0.924746
\(448\) 0 0
\(449\) −17.7067 −0.835629 −0.417814 0.908532i \(-0.637204\pi\)
−0.417814 + 0.908532i \(0.637204\pi\)
\(450\) 0 0
\(451\) 52.1819 2.45715
\(452\) 0 0
\(453\) 6.18202 0.290457
\(454\) 0 0
\(455\) −3.19738 −0.149896
\(456\) 0 0
\(457\) 21.5575 1.00842 0.504208 0.863582i \(-0.331784\pi\)
0.504208 + 0.863582i \(0.331784\pi\)
\(458\) 0 0
\(459\) −29.2328 −1.36447
\(460\) 0 0
\(461\) 19.7163 0.918279 0.459140 0.888364i \(-0.348158\pi\)
0.459140 + 0.888364i \(0.348158\pi\)
\(462\) 0 0
\(463\) −19.8570 −0.922831 −0.461415 0.887184i \(-0.652658\pi\)
−0.461415 + 0.887184i \(0.652658\pi\)
\(464\) 0 0
\(465\) −10.9193 −0.506371
\(466\) 0 0
\(467\) 10.4068 0.481569 0.240784 0.970579i \(-0.422595\pi\)
0.240784 + 0.970579i \(0.422595\pi\)
\(468\) 0 0
\(469\) 22.2751 1.02857
\(470\) 0 0
\(471\) −10.4134 −0.479823
\(472\) 0 0
\(473\) −50.3656 −2.31581
\(474\) 0 0
\(475\) 6.46621 0.296690
\(476\) 0 0
\(477\) 2.75767 0.126265
\(478\) 0 0
\(479\) 29.4266 1.34453 0.672267 0.740309i \(-0.265321\pi\)
0.672267 + 0.740309i \(0.265321\pi\)
\(480\) 0 0
\(481\) 1.62611 0.0741443
\(482\) 0 0
\(483\) −2.42673 −0.110420
\(484\) 0 0
\(485\) −2.24947 −0.102143
\(486\) 0 0
\(487\) −30.5370 −1.38376 −0.691882 0.722010i \(-0.743218\pi\)
−0.691882 + 0.722010i \(0.743218\pi\)
\(488\) 0 0
\(489\) 34.4627 1.55846
\(490\) 0 0
\(491\) 2.73261 0.123321 0.0616606 0.998097i \(-0.480360\pi\)
0.0616606 + 0.998097i \(0.480360\pi\)
\(492\) 0 0
\(493\) 9.98896 0.449880
\(494\) 0 0
\(495\) −4.87339 −0.219043
\(496\) 0 0
\(497\) 19.4610 0.872945
\(498\) 0 0
\(499\) −24.3826 −1.09152 −0.545758 0.837943i \(-0.683758\pi\)
−0.545758 + 0.837943i \(0.683758\pi\)
\(500\) 0 0
\(501\) 32.6373 1.45813
\(502\) 0 0
\(503\) −17.3183 −0.772183 −0.386092 0.922460i \(-0.626175\pi\)
−0.386092 + 0.922460i \(0.626175\pi\)
\(504\) 0 0
\(505\) −10.0218 −0.445965
\(506\) 0 0
\(507\) −14.8209 −0.658220
\(508\) 0 0
\(509\) 31.4419 1.39364 0.696819 0.717247i \(-0.254598\pi\)
0.696819 + 0.717247i \(0.254598\pi\)
\(510\) 0 0
\(511\) 17.7654 0.785894
\(512\) 0 0
\(513\) −36.5705 −1.61463
\(514\) 0 0
\(515\) 18.0207 0.794086
\(516\) 0 0
\(517\) 44.3676 1.95128
\(518\) 0 0
\(519\) −10.5736 −0.464129
\(520\) 0 0
\(521\) −10.3564 −0.453722 −0.226861 0.973927i \(-0.572846\pi\)
−0.226861 + 0.973927i \(0.572846\pi\)
\(522\) 0 0
\(523\) −12.7283 −0.556571 −0.278285 0.960498i \(-0.589766\pi\)
−0.278285 + 0.960498i \(0.589766\pi\)
\(524\) 0 0
\(525\) 2.81409 0.122817
\(526\) 0 0
\(527\) 39.4359 1.71786
\(528\) 0 0
\(529\) −22.2563 −0.967667
\(530\) 0 0
\(531\) −13.7925 −0.598544
\(532\) 0 0
\(533\) −16.5713 −0.717781
\(534\) 0 0
\(535\) −1.76451 −0.0762865
\(536\) 0 0
\(537\) 4.41876 0.190684
\(538\) 0 0
\(539\) 16.0465 0.691171
\(540\) 0 0
\(541\) −16.9313 −0.727933 −0.363966 0.931412i \(-0.618578\pi\)
−0.363966 + 0.931412i \(0.618578\pi\)
\(542\) 0 0
\(543\) 25.5207 1.09520
\(544\) 0 0
\(545\) 14.7839 0.633271
\(546\) 0 0
\(547\) 29.2625 1.25117 0.625587 0.780155i \(-0.284860\pi\)
0.625587 + 0.780155i \(0.284860\pi\)
\(548\) 0 0
\(549\) −6.89169 −0.294130
\(550\) 0 0
\(551\) 12.4963 0.532360
\(552\) 0 0
\(553\) 23.4953 0.999124
\(554\) 0 0
\(555\) −1.43118 −0.0607500
\(556\) 0 0
\(557\) 0.254156 0.0107689 0.00538447 0.999986i \(-0.498286\pi\)
0.00538447 + 0.999986i \(0.498286\pi\)
\(558\) 0 0
\(559\) 15.9945 0.676494
\(560\) 0 0
\(561\) −37.8789 −1.59925
\(562\) 0 0
\(563\) 10.8152 0.455807 0.227904 0.973684i \(-0.426813\pi\)
0.227904 + 0.973684i \(0.426813\pi\)
\(564\) 0 0
\(565\) 14.1687 0.596080
\(566\) 0 0
\(567\) −10.3013 −0.432614
\(568\) 0 0
\(569\) −34.5266 −1.44743 −0.723716 0.690098i \(-0.757567\pi\)
−0.723716 + 0.690098i \(0.757567\pi\)
\(570\) 0 0
\(571\) 10.9437 0.457978 0.228989 0.973429i \(-0.426458\pi\)
0.228989 + 0.973429i \(0.426458\pi\)
\(572\) 0 0
\(573\) −25.3951 −1.06089
\(574\) 0 0
\(575\) −0.862352 −0.0359626
\(576\) 0 0
\(577\) −8.82017 −0.367189 −0.183594 0.983002i \(-0.558773\pi\)
−0.183594 + 0.983002i \(0.558773\pi\)
\(578\) 0 0
\(579\) −29.2965 −1.21752
\(580\) 0 0
\(581\) 14.2538 0.591349
\(582\) 0 0
\(583\) 14.8368 0.614478
\(584\) 0 0
\(585\) 1.54763 0.0639865
\(586\) 0 0
\(587\) 7.95988 0.328539 0.164270 0.986415i \(-0.447473\pi\)
0.164270 + 0.986415i \(0.447473\pi\)
\(588\) 0 0
\(589\) 49.3347 2.03280
\(590\) 0 0
\(591\) −4.08782 −0.168151
\(592\) 0 0
\(593\) −8.27685 −0.339890 −0.169945 0.985454i \(-0.554359\pi\)
−0.169945 + 0.985454i \(0.554359\pi\)
\(594\) 0 0
\(595\) −10.1633 −0.416654
\(596\) 0 0
\(597\) −7.16879 −0.293399
\(598\) 0 0
\(599\) −37.8285 −1.54563 −0.772816 0.634630i \(-0.781152\pi\)
−0.772816 + 0.634630i \(0.781152\pi\)
\(600\) 0 0
\(601\) 6.98714 0.285012 0.142506 0.989794i \(-0.454484\pi\)
0.142506 + 0.989794i \(0.454484\pi\)
\(602\) 0 0
\(603\) −10.7818 −0.439069
\(604\) 0 0
\(605\) −15.2198 −0.618773
\(606\) 0 0
\(607\) −28.1934 −1.14434 −0.572168 0.820137i \(-0.693897\pi\)
−0.572168 + 0.820137i \(0.693897\pi\)
\(608\) 0 0
\(609\) 5.43837 0.220374
\(610\) 0 0
\(611\) −14.0897 −0.570008
\(612\) 0 0
\(613\) −5.45523 −0.220335 −0.110167 0.993913i \(-0.535139\pi\)
−0.110167 + 0.993913i \(0.535139\pi\)
\(614\) 0 0
\(615\) 14.5847 0.588113
\(616\) 0 0
\(617\) −31.8893 −1.28382 −0.641908 0.766781i \(-0.721857\pi\)
−0.641908 + 0.766781i \(0.721857\pi\)
\(618\) 0 0
\(619\) 37.6809 1.51452 0.757261 0.653112i \(-0.226537\pi\)
0.757261 + 0.653112i \(0.226537\pi\)
\(620\) 0 0
\(621\) 4.87714 0.195713
\(622\) 0 0
\(623\) 15.4433 0.618722
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −47.3868 −1.89245
\(628\) 0 0
\(629\) 5.16879 0.206093
\(630\) 0 0
\(631\) −10.7753 −0.428960 −0.214480 0.976728i \(-0.568806\pi\)
−0.214480 + 0.976728i \(0.568806\pi\)
\(632\) 0 0
\(633\) 13.5278 0.537682
\(634\) 0 0
\(635\) 12.7134 0.504514
\(636\) 0 0
\(637\) −5.09584 −0.201905
\(638\) 0 0
\(639\) −9.41969 −0.372637
\(640\) 0 0
\(641\) 10.9725 0.433389 0.216695 0.976239i \(-0.430472\pi\)
0.216695 + 0.976239i \(0.430472\pi\)
\(642\) 0 0
\(643\) −17.5820 −0.693368 −0.346684 0.937982i \(-0.612692\pi\)
−0.346684 + 0.937982i \(0.612692\pi\)
\(644\) 0 0
\(645\) −14.0771 −0.554284
\(646\) 0 0
\(647\) 28.1582 1.10701 0.553506 0.832845i \(-0.313289\pi\)
0.553506 + 0.832845i \(0.313289\pi\)
\(648\) 0 0
\(649\) −74.2065 −2.91286
\(650\) 0 0
\(651\) 21.4704 0.841492
\(652\) 0 0
\(653\) 12.7639 0.499491 0.249745 0.968312i \(-0.419653\pi\)
0.249745 + 0.968312i \(0.419653\pi\)
\(654\) 0 0
\(655\) −13.9569 −0.545340
\(656\) 0 0
\(657\) −8.59896 −0.335477
\(658\) 0 0
\(659\) −2.64289 −0.102952 −0.0514762 0.998674i \(-0.516393\pi\)
−0.0514762 + 0.998674i \(0.516393\pi\)
\(660\) 0 0
\(661\) −11.1897 −0.435228 −0.217614 0.976035i \(-0.569827\pi\)
−0.217614 + 0.976035i \(0.569827\pi\)
\(662\) 0 0
\(663\) 12.0291 0.467171
\(664\) 0 0
\(665\) −12.7144 −0.493042
\(666\) 0 0
\(667\) −1.66654 −0.0645287
\(668\) 0 0
\(669\) 10.0852 0.389917
\(670\) 0 0
\(671\) −37.0787 −1.43141
\(672\) 0 0
\(673\) 14.2846 0.550632 0.275316 0.961354i \(-0.411217\pi\)
0.275316 + 0.961354i \(0.411217\pi\)
\(674\) 0 0
\(675\) −5.65563 −0.217685
\(676\) 0 0
\(677\) 13.0898 0.503082 0.251541 0.967847i \(-0.419063\pi\)
0.251541 + 0.967847i \(0.419063\pi\)
\(678\) 0 0
\(679\) 4.42308 0.169742
\(680\) 0 0
\(681\) 5.23005 0.200416
\(682\) 0 0
\(683\) 36.0870 1.38083 0.690415 0.723414i \(-0.257428\pi\)
0.690415 + 0.723414i \(0.257428\pi\)
\(684\) 0 0
\(685\) 3.38974 0.129515
\(686\) 0 0
\(687\) 13.5694 0.517706
\(688\) 0 0
\(689\) −4.71169 −0.179501
\(690\) 0 0
\(691\) 12.0837 0.459687 0.229844 0.973228i \(-0.426179\pi\)
0.229844 + 0.973228i \(0.426179\pi\)
\(692\) 0 0
\(693\) 9.58243 0.364006
\(694\) 0 0
\(695\) 11.6982 0.443740
\(696\) 0 0
\(697\) −52.6738 −1.99516
\(698\) 0 0
\(699\) 29.2502 1.10634
\(700\) 0 0
\(701\) 7.45578 0.281601 0.140801 0.990038i \(-0.455032\pi\)
0.140801 + 0.990038i \(0.455032\pi\)
\(702\) 0 0
\(703\) 6.46621 0.243878
\(704\) 0 0
\(705\) 12.4006 0.467035
\(706\) 0 0
\(707\) 19.7056 0.741107
\(708\) 0 0
\(709\) −5.69787 −0.213988 −0.106994 0.994260i \(-0.534123\pi\)
−0.106994 + 0.994260i \(0.534123\pi\)
\(710\) 0 0
\(711\) −11.3724 −0.426500
\(712\) 0 0
\(713\) −6.57941 −0.246401
\(714\) 0 0
\(715\) 8.32655 0.311395
\(716\) 0 0
\(717\) −33.8704 −1.26491
\(718\) 0 0
\(719\) −22.1129 −0.824671 −0.412336 0.911032i \(-0.635287\pi\)
−0.412336 + 0.911032i \(0.635287\pi\)
\(720\) 0 0
\(721\) −35.4336 −1.31962
\(722\) 0 0
\(723\) 39.3952 1.46513
\(724\) 0 0
\(725\) 1.93255 0.0717732
\(726\) 0 0
\(727\) 2.34486 0.0869659 0.0434829 0.999054i \(-0.486155\pi\)
0.0434829 + 0.999054i \(0.486155\pi\)
\(728\) 0 0
\(729\) 29.2687 1.08403
\(730\) 0 0
\(731\) 50.8404 1.88040
\(732\) 0 0
\(733\) 28.9071 1.06771 0.533855 0.845576i \(-0.320743\pi\)
0.533855 + 0.845576i \(0.320743\pi\)
\(734\) 0 0
\(735\) 4.48496 0.165430
\(736\) 0 0
\(737\) −58.0082 −2.13676
\(738\) 0 0
\(739\) −37.5736 −1.38217 −0.691083 0.722776i \(-0.742866\pi\)
−0.691083 + 0.722776i \(0.742866\pi\)
\(740\) 0 0
\(741\) 15.0485 0.552820
\(742\) 0 0
\(743\) 25.1665 0.923268 0.461634 0.887070i \(-0.347263\pi\)
0.461634 + 0.887070i \(0.347263\pi\)
\(744\) 0 0
\(745\) 13.6610 0.500501
\(746\) 0 0
\(747\) −6.89928 −0.252431
\(748\) 0 0
\(749\) 3.46952 0.126773
\(750\) 0 0
\(751\) −25.3267 −0.924186 −0.462093 0.886832i \(-0.652901\pi\)
−0.462093 + 0.886832i \(0.652901\pi\)
\(752\) 0 0
\(753\) 25.6727 0.935565
\(754\) 0 0
\(755\) −4.31954 −0.157204
\(756\) 0 0
\(757\) 3.67645 0.133623 0.0668113 0.997766i \(-0.478717\pi\)
0.0668113 + 0.997766i \(0.478717\pi\)
\(758\) 0 0
\(759\) 6.31964 0.229388
\(760\) 0 0
\(761\) 25.1461 0.911545 0.455773 0.890096i \(-0.349363\pi\)
0.455773 + 0.890096i \(0.349363\pi\)
\(762\) 0 0
\(763\) −29.0692 −1.05237
\(764\) 0 0
\(765\) 4.91932 0.177859
\(766\) 0 0
\(767\) 23.5656 0.850903
\(768\) 0 0
\(769\) −33.5928 −1.21139 −0.605694 0.795698i \(-0.707104\pi\)
−0.605694 + 0.795698i \(0.707104\pi\)
\(770\) 0 0
\(771\) 27.6454 0.995625
\(772\) 0 0
\(773\) 1.23222 0.0443199 0.0221599 0.999754i \(-0.492946\pi\)
0.0221599 + 0.999754i \(0.492946\pi\)
\(774\) 0 0
\(775\) 7.62962 0.274064
\(776\) 0 0
\(777\) 2.81409 0.100955
\(778\) 0 0
\(779\) −65.8954 −2.36095
\(780\) 0 0
\(781\) −50.6799 −1.81347
\(782\) 0 0
\(783\) −10.9298 −0.390599
\(784\) 0 0
\(785\) 7.27609 0.259695
\(786\) 0 0
\(787\) 5.09991 0.181792 0.0908961 0.995860i \(-0.471027\pi\)
0.0908961 + 0.995860i \(0.471027\pi\)
\(788\) 0 0
\(789\) −3.05890 −0.108900
\(790\) 0 0
\(791\) −27.8595 −0.990570
\(792\) 0 0
\(793\) 11.7750 0.418142
\(794\) 0 0
\(795\) 4.14686 0.147074
\(796\) 0 0
\(797\) −5.94742 −0.210668 −0.105334 0.994437i \(-0.533591\pi\)
−0.105334 + 0.994437i \(0.533591\pi\)
\(798\) 0 0
\(799\) −44.7858 −1.58441
\(800\) 0 0
\(801\) −7.47500 −0.264116
\(802\) 0 0
\(803\) −46.2642 −1.63263
\(804\) 0 0
\(805\) 1.69562 0.0597628
\(806\) 0 0
\(807\) −21.5452 −0.758428
\(808\) 0 0
\(809\) −13.8869 −0.488238 −0.244119 0.969745i \(-0.578499\pi\)
−0.244119 + 0.969745i \(0.578499\pi\)
\(810\) 0 0
\(811\) −6.71735 −0.235878 −0.117939 0.993021i \(-0.537629\pi\)
−0.117939 + 0.993021i \(0.537629\pi\)
\(812\) 0 0
\(813\) −16.6286 −0.583190
\(814\) 0 0
\(815\) −24.0800 −0.843484
\(816\) 0 0
\(817\) 63.6018 2.22515
\(818\) 0 0
\(819\) −3.04306 −0.106333
\(820\) 0 0
\(821\) −12.3600 −0.431365 −0.215683 0.976464i \(-0.569198\pi\)
−0.215683 + 0.976464i \(0.569198\pi\)
\(822\) 0 0
\(823\) −4.75674 −0.165810 −0.0829048 0.996557i \(-0.526420\pi\)
−0.0829048 + 0.996557i \(0.526420\pi\)
\(824\) 0 0
\(825\) −7.32838 −0.255141
\(826\) 0 0
\(827\) −6.29371 −0.218854 −0.109427 0.993995i \(-0.534902\pi\)
−0.109427 + 0.993995i \(0.534902\pi\)
\(828\) 0 0
\(829\) 31.9655 1.11021 0.555104 0.831781i \(-0.312679\pi\)
0.555104 + 0.831781i \(0.312679\pi\)
\(830\) 0 0
\(831\) −21.2603 −0.737512
\(832\) 0 0
\(833\) −16.1977 −0.561219
\(834\) 0 0
\(835\) −22.8045 −0.789184
\(836\) 0 0
\(837\) −43.1503 −1.49149
\(838\) 0 0
\(839\) −14.6992 −0.507474 −0.253737 0.967273i \(-0.581660\pi\)
−0.253737 + 0.967273i \(0.581660\pi\)
\(840\) 0 0
\(841\) −25.2652 −0.871215
\(842\) 0 0
\(843\) −30.7818 −1.06018
\(844\) 0 0
\(845\) 10.3558 0.356249
\(846\) 0 0
\(847\) 29.9264 1.02828
\(848\) 0 0
\(849\) −45.1616 −1.54994
\(850\) 0 0
\(851\) −0.862352 −0.0295610
\(852\) 0 0
\(853\) 54.5264 1.86695 0.933473 0.358646i \(-0.116762\pi\)
0.933473 + 0.358646i \(0.116762\pi\)
\(854\) 0 0
\(855\) 6.15412 0.210467
\(856\) 0 0
\(857\) 38.2996 1.30829 0.654144 0.756370i \(-0.273029\pi\)
0.654144 + 0.756370i \(0.273029\pi\)
\(858\) 0 0
\(859\) 28.2599 0.964216 0.482108 0.876112i \(-0.339871\pi\)
0.482108 + 0.876112i \(0.339871\pi\)
\(860\) 0 0
\(861\) −28.6776 −0.977330
\(862\) 0 0
\(863\) −15.8707 −0.540246 −0.270123 0.962826i \(-0.587064\pi\)
−0.270123 + 0.962826i \(0.587064\pi\)
\(864\) 0 0
\(865\) 7.38804 0.251201
\(866\) 0 0
\(867\) 13.9059 0.472269
\(868\) 0 0
\(869\) −61.1860 −2.07559
\(870\) 0 0
\(871\) 18.4215 0.624189
\(872\) 0 0
\(873\) −2.14090 −0.0724585
\(874\) 0 0
\(875\) −1.96628 −0.0664723
\(876\) 0 0
\(877\) −40.0036 −1.35083 −0.675413 0.737440i \(-0.736035\pi\)
−0.675413 + 0.737440i \(0.736035\pi\)
\(878\) 0 0
\(879\) −13.8961 −0.468703
\(880\) 0 0
\(881\) 52.3990 1.76537 0.882684 0.469967i \(-0.155734\pi\)
0.882684 + 0.469967i \(0.155734\pi\)
\(882\) 0 0
\(883\) 18.8804 0.635376 0.317688 0.948195i \(-0.397094\pi\)
0.317688 + 0.948195i \(0.397094\pi\)
\(884\) 0 0
\(885\) −20.7406 −0.697186
\(886\) 0 0
\(887\) −9.60874 −0.322630 −0.161315 0.986903i \(-0.551574\pi\)
−0.161315 + 0.986903i \(0.551574\pi\)
\(888\) 0 0
\(889\) −24.9980 −0.838405
\(890\) 0 0
\(891\) 26.8264 0.898719
\(892\) 0 0
\(893\) −56.0275 −1.87489
\(894\) 0 0
\(895\) −3.08751 −0.103204
\(896\) 0 0
\(897\) −2.00691 −0.0670088
\(898\) 0 0
\(899\) 14.7446 0.491761
\(900\) 0 0
\(901\) −14.9767 −0.498945
\(902\) 0 0
\(903\) 27.6794 0.921114
\(904\) 0 0
\(905\) −17.8320 −0.592756
\(906\) 0 0
\(907\) −52.8613 −1.75523 −0.877616 0.479365i \(-0.840867\pi\)
−0.877616 + 0.479365i \(0.840867\pi\)
\(908\) 0 0
\(909\) −9.53811 −0.316359
\(910\) 0 0
\(911\) −42.2226 −1.39890 −0.699448 0.714684i \(-0.746571\pi\)
−0.699448 + 0.714684i \(0.746571\pi\)
\(912\) 0 0
\(913\) −37.1195 −1.22848
\(914\) 0 0
\(915\) −10.3634 −0.342604
\(916\) 0 0
\(917\) 27.4431 0.906250
\(918\) 0 0
\(919\) 3.11038 0.102602 0.0513010 0.998683i \(-0.483663\pi\)
0.0513010 + 0.998683i \(0.483663\pi\)
\(920\) 0 0
\(921\) −1.83629 −0.0605078
\(922\) 0 0
\(923\) 16.0943 0.529749
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 0 0
\(927\) 17.1509 0.563310
\(928\) 0 0
\(929\) −55.6513 −1.82586 −0.912930 0.408115i \(-0.866186\pi\)
−0.912930 + 0.408115i \(0.866186\pi\)
\(930\) 0 0
\(931\) −20.2635 −0.664111
\(932\) 0 0
\(933\) 9.92358 0.324883
\(934\) 0 0
\(935\) 26.4670 0.865562
\(936\) 0 0
\(937\) 12.1645 0.397398 0.198699 0.980061i \(-0.436328\pi\)
0.198699 + 0.980061i \(0.436328\pi\)
\(938\) 0 0
\(939\) 34.4368 1.12380
\(940\) 0 0
\(941\) −48.8595 −1.59277 −0.796387 0.604787i \(-0.793258\pi\)
−0.796387 + 0.604787i \(0.793258\pi\)
\(942\) 0 0
\(943\) 8.78799 0.286176
\(944\) 0 0
\(945\) 11.1205 0.361751
\(946\) 0 0
\(947\) 15.8474 0.514972 0.257486 0.966282i \(-0.417106\pi\)
0.257486 + 0.966282i \(0.417106\pi\)
\(948\) 0 0
\(949\) 14.6920 0.476922
\(950\) 0 0
\(951\) 15.5255 0.503448
\(952\) 0 0
\(953\) −12.5946 −0.407980 −0.203990 0.978973i \(-0.565391\pi\)
−0.203990 + 0.978973i \(0.565391\pi\)
\(954\) 0 0
\(955\) 17.7442 0.574189
\(956\) 0 0
\(957\) −14.1625 −0.457808
\(958\) 0 0
\(959\) −6.66517 −0.215229
\(960\) 0 0
\(961\) 27.2110 0.877776
\(962\) 0 0
\(963\) −1.67935 −0.0541162
\(964\) 0 0
\(965\) 20.4702 0.658959
\(966\) 0 0
\(967\) 8.89371 0.286002 0.143001 0.989723i \(-0.454325\pi\)
0.143001 + 0.989723i \(0.454325\pi\)
\(968\) 0 0
\(969\) 47.8335 1.53663
\(970\) 0 0
\(971\) −20.1917 −0.647981 −0.323991 0.946060i \(-0.605025\pi\)
−0.323991 + 0.946060i \(0.605025\pi\)
\(972\) 0 0
\(973\) −23.0020 −0.737410
\(974\) 0 0
\(975\) 2.32725 0.0745317
\(976\) 0 0
\(977\) −11.5456 −0.369375 −0.184688 0.982797i \(-0.559127\pi\)
−0.184688 + 0.982797i \(0.559127\pi\)
\(978\) 0 0
\(979\) −40.2170 −1.28534
\(980\) 0 0
\(981\) 14.0703 0.449231
\(982\) 0 0
\(983\) −30.5253 −0.973606 −0.486803 0.873512i \(-0.661837\pi\)
−0.486803 + 0.873512i \(0.661837\pi\)
\(984\) 0 0
\(985\) 2.85627 0.0910083
\(986\) 0 0
\(987\) −24.3831 −0.776122
\(988\) 0 0
\(989\) −8.48211 −0.269716
\(990\) 0 0
\(991\) 37.9540 1.20565 0.602824 0.797874i \(-0.294042\pi\)
0.602824 + 0.797874i \(0.294042\pi\)
\(992\) 0 0
\(993\) 3.06814 0.0973644
\(994\) 0 0
\(995\) 5.00902 0.158797
\(996\) 0 0
\(997\) −33.2156 −1.05195 −0.525974 0.850500i \(-0.676299\pi\)
−0.525974 + 0.850500i \(0.676299\pi\)
\(998\) 0 0
\(999\) −5.65563 −0.178936
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 2960.2.a.z.1.4 5
4.3 odd 2 1480.2.a.h.1.2 5
20.19 odd 2 7400.2.a.q.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1480.2.a.h.1.2 5 4.3 odd 2
2960.2.a.z.1.4 5 1.1 even 1 trivial
7400.2.a.q.1.4 5 20.19 odd 2