Properties

Label 2520.2.bi.h.1801.1
Level $2520$
Weight $2$
Character 2520.1801
Analytic conductor $20.122$
Analytic rank $0$
Dimension $2$
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] = [2520,2,Mod(361,2520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("2520.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.bi (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: \(20.1223013094\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1801.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2520.1801
Dual form 2520.2.bi.h.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +(-0.500000 - 2.59808i) q^{7} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{5} +(-0.500000 - 2.59808i) q^{7} +(2.00000 + 3.46410i) q^{11} -5.00000 q^{13} +(3.00000 + 5.19615i) q^{17} +(2.50000 - 4.33013i) q^{19} +(1.00000 - 1.73205i) q^{23} +(-0.500000 - 0.866025i) q^{25} +2.00000 q^{29} +(4.50000 + 7.79423i) q^{31} +(-2.50000 - 0.866025i) q^{35} +(5.50000 - 9.52628i) q^{37} +8.00000 q^{41} +1.00000 q^{43} +(3.00000 - 5.19615i) q^{47} +(-6.50000 + 2.59808i) q^{49} +(-2.00000 - 3.46410i) q^{53} +4.00000 q^{55} +(3.00000 + 5.19615i) q^{59} +(7.00000 - 12.1244i) q^{61} +(-2.50000 + 4.33013i) q^{65} +(0.500000 + 0.866025i) q^{67} -12.0000 q^{71} +(-5.50000 - 9.52628i) q^{73} +(8.00000 - 6.92820i) q^{77} +(-0.500000 + 0.866025i) q^{79} +4.00000 q^{83} +6.00000 q^{85} +(9.00000 - 15.5885i) q^{89} +(2.50000 + 12.9904i) q^{91} +(-2.50000 - 4.33013i) q^{95} -18.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{5} - q^{7} + 4 q^{11} - 10 q^{13} + 6 q^{17} + 5 q^{19} + 2 q^{23} - q^{25} + 4 q^{29} + 9 q^{31} - 5 q^{35} + 11 q^{37} + 16 q^{41} + 2 q^{43} + 6 q^{47} - 13 q^{49} - 4 q^{53} + 8 q^{55} + 6 q^{59} + 14 q^{61} - 5 q^{65} + q^{67} - 24 q^{71} - 11 q^{73} + 16 q^{77} - q^{79} + 8 q^{83} + 12 q^{85} + 18 q^{89} + 5 q^{91} - 5 q^{95} - 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0.500000 0.866025i 0.223607 0.387298i
\(6\) 0 0
\(7\) −0.500000 2.59808i −0.188982 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 + 3.46410i 0.603023 + 1.04447i 0.992361 + 0.123371i \(0.0393705\pi\)
−0.389338 + 0.921095i \(0.627296\pi\)
\(12\) 0 0
\(13\) −5.00000 −1.38675 −0.693375 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000 + 5.19615i 0.727607 + 1.26025i 0.957892 + 0.287129i \(0.0927008\pi\)
−0.230285 + 0.973123i \(0.573966\pi\)
\(18\) 0 0
\(19\) 2.50000 4.33013i 0.573539 0.993399i −0.422659 0.906289i \(-0.638903\pi\)
0.996199 0.0871106i \(-0.0277634\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 1.73205i 0.208514 0.361158i −0.742732 0.669588i \(-0.766471\pi\)
0.951247 + 0.308431i \(0.0998038\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 4.50000 + 7.79423i 0.808224 + 1.39988i 0.914093 + 0.405505i \(0.132904\pi\)
−0.105869 + 0.994380i \(0.533762\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.50000 0.866025i −0.422577 0.146385i
\(36\) 0 0
\(37\) 5.50000 9.52628i 0.904194 1.56611i 0.0821995 0.996616i \(-0.473806\pi\)
0.821995 0.569495i \(-0.192861\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000 5.19615i 0.437595 0.757937i −0.559908 0.828554i \(-0.689164\pi\)
0.997503 + 0.0706177i \(0.0224970\pi\)
\(48\) 0 0
\(49\) −6.50000 + 2.59808i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 3.46410i −0.274721 0.475831i 0.695344 0.718677i \(-0.255252\pi\)
−0.970065 + 0.242846i \(0.921919\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.00000 + 5.19615i 0.390567 + 0.676481i 0.992524 0.122047i \(-0.0389457\pi\)
−0.601958 + 0.798528i \(0.705612\pi\)
\(60\) 0 0
\(61\) 7.00000 12.1244i 0.896258 1.55236i 0.0640184 0.997949i \(-0.479608\pi\)
0.832240 0.554416i \(-0.187058\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.50000 + 4.33013i −0.310087 + 0.537086i
\(66\) 0 0
\(67\) 0.500000 + 0.866025i 0.0610847 + 0.105802i 0.894951 0.446165i \(-0.147211\pi\)
−0.833866 + 0.551967i \(0.813877\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −5.50000 9.52628i −0.643726 1.11497i −0.984594 0.174855i \(-0.944054\pi\)
0.340868 0.940111i \(-0.389279\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.00000 6.92820i 0.911685 0.789542i
\(78\) 0 0
\(79\) −0.500000 + 0.866025i −0.0562544 + 0.0974355i −0.892781 0.450490i \(-0.851249\pi\)
0.836527 + 0.547926i \(0.184582\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.00000 15.5885i 0.953998 1.65237i 0.217354 0.976093i \(-0.430258\pi\)
0.736644 0.676280i \(-0.236409\pi\)
\(90\) 0 0
\(91\) 2.50000 + 12.9904i 0.262071 + 1.36176i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.50000 4.33013i −0.256495 0.444262i
\(96\) 0 0
\(97\) −18.0000 −1.82762 −0.913812 0.406138i \(-0.866875\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 10.3923i −0.597022 1.03407i −0.993258 0.115924i \(-0.963017\pi\)
0.396236 0.918149i \(-0.370316\pi\)
\(102\) 0 0
\(103\) 5.50000 9.52628i 0.541931 0.938652i −0.456862 0.889538i \(-0.651027\pi\)
0.998793 0.0491146i \(-0.0156400\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.00000 + 8.66025i −0.483368 + 0.837218i −0.999818 0.0190994i \(-0.993920\pi\)
0.516449 + 0.856318i \(0.327253\pi\)
\(108\) 0 0
\(109\) −2.50000 4.33013i −0.239457 0.414751i 0.721102 0.692829i \(-0.243636\pi\)
−0.960558 + 0.278078i \(0.910303\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −1.00000 1.73205i −0.0932505 0.161515i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.0000 10.3923i 1.10004 0.952661i
\(120\) 0 0
\(121\) −2.50000 + 4.33013i −0.227273 + 0.393648i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −7.00000 −0.621150 −0.310575 0.950549i \(-0.600522\pi\)
−0.310575 + 0.950549i \(0.600522\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.00000 + 1.73205i −0.0873704 + 0.151330i −0.906399 0.422423i \(-0.861180\pi\)
0.819028 + 0.573753i \(0.194513\pi\)
\(132\) 0 0
\(133\) −12.5000 4.33013i −1.08389 0.375470i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.00000 + 1.73205i 0.0854358 + 0.147979i 0.905577 0.424182i \(-0.139438\pi\)
−0.820141 + 0.572161i \(0.806105\pi\)
\(138\) 0 0
\(139\) 19.0000 1.61156 0.805779 0.592216i \(-0.201747\pi\)
0.805779 + 0.592216i \(0.201747\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −10.0000 17.3205i −0.836242 1.44841i
\(144\) 0 0
\(145\) 1.00000 1.73205i 0.0830455 0.143839i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(150\) 0 0
\(151\) 4.00000 + 6.92820i 0.325515 + 0.563809i 0.981617 0.190864i \(-0.0611289\pi\)
−0.656101 + 0.754673i \(0.727796\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.00000 0.722897
\(156\) 0 0
\(157\) 9.00000 + 15.5885i 0.718278 + 1.24409i 0.961681 + 0.274169i \(0.0884028\pi\)
−0.243403 + 0.969925i \(0.578264\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.00000 1.73205i −0.394055 0.136505i
\(162\) 0 0
\(163\) 2.00000 3.46410i 0.156652 0.271329i −0.777007 0.629492i \(-0.783263\pi\)
0.933659 + 0.358162i \(0.116597\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.00000 3.46410i 0.152057 0.263371i −0.779926 0.625871i \(-0.784744\pi\)
0.931984 + 0.362500i \(0.118077\pi\)
\(174\) 0 0
\(175\) −2.00000 + 1.73205i −0.151186 + 0.130931i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.00000 + 15.5885i 0.672692 + 1.16514i 0.977138 + 0.212607i \(0.0681952\pi\)
−0.304446 + 0.952529i \(0.598471\pi\)
\(180\) 0 0
\(181\) 5.00000 0.371647 0.185824 0.982583i \(-0.440505\pi\)
0.185824 + 0.982583i \(0.440505\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −5.50000 9.52628i −0.404368 0.700386i
\(186\) 0 0
\(187\) −12.0000 + 20.7846i −0.877527 + 1.51992i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.00000 1.73205i 0.0723575 0.125327i −0.827577 0.561353i \(-0.810281\pi\)
0.899934 + 0.436026i \(0.143614\pi\)
\(192\) 0 0
\(193\) 1.50000 + 2.59808i 0.107972 + 0.187014i 0.914949 0.403570i \(-0.132231\pi\)
−0.806976 + 0.590584i \(0.798898\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) 4.00000 + 6.92820i 0.283552 + 0.491127i 0.972257 0.233915i \(-0.0751537\pi\)
−0.688705 + 0.725042i \(0.741820\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.00000 5.19615i −0.0701862 0.364698i
\(204\) 0 0
\(205\) 4.00000 6.92820i 0.279372 0.483887i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 20.0000 1.38343
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.500000 0.866025i 0.0340997 0.0590624i
\(216\) 0 0
\(217\) 18.0000 15.5885i 1.22192 1.05821i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −15.0000 25.9808i −1.00901 1.74766i
\(222\) 0 0
\(223\) −4.00000 −0.267860 −0.133930 0.990991i \(-0.542760\pi\)
−0.133930 + 0.990991i \(0.542760\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.00000 6.92820i −0.265489 0.459841i 0.702202 0.711977i \(-0.252200\pi\)
−0.967692 + 0.252136i \(0.918867\pi\)
\(228\) 0 0
\(229\) −0.500000 + 0.866025i −0.0330409 + 0.0572286i −0.882073 0.471113i \(-0.843853\pi\)
0.849032 + 0.528341i \(0.177186\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3.00000 + 5.19615i −0.196537 + 0.340411i −0.947403 0.320043i \(-0.896303\pi\)
0.750867 + 0.660454i \(0.229636\pi\)
\(234\) 0 0
\(235\) −3.00000 5.19615i −0.195698 0.338960i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −30.0000 −1.94054 −0.970269 0.242028i \(-0.922188\pi\)
−0.970269 + 0.242028i \(0.922188\pi\)
\(240\) 0 0
\(241\) 11.0000 + 19.0526i 0.708572 + 1.22728i 0.965387 + 0.260822i \(0.0839937\pi\)
−0.256814 + 0.966461i \(0.582673\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00000 + 6.92820i −0.0638877 + 0.442627i
\(246\) 0 0
\(247\) −12.5000 + 21.6506i −0.795356 + 1.37760i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.00000 0.378717 0.189358 0.981908i \(-0.439359\pi\)
0.189358 + 0.981908i \(0.439359\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6.00000 + 10.3923i −0.374270 + 0.648254i −0.990217 0.139533i \(-0.955440\pi\)
0.615948 + 0.787787i \(0.288773\pi\)
\(258\) 0 0
\(259\) −27.5000 9.52628i −1.70877 0.591934i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.00000 + 3.46410i 0.123325 + 0.213606i 0.921077 0.389380i \(-0.127311\pi\)
−0.797752 + 0.602986i \(0.793977\pi\)
\(264\) 0 0
\(265\) −4.00000 −0.245718
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.00000 + 5.19615i 0.182913 + 0.316815i 0.942871 0.333157i \(-0.108114\pi\)
−0.759958 + 0.649972i \(0.774781\pi\)
\(270\) 0 0
\(271\) −8.00000 + 13.8564i −0.485965 + 0.841717i −0.999870 0.0161307i \(-0.994865\pi\)
0.513905 + 0.857847i \(0.328199\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.00000 3.46410i 0.120605 0.208893i
\(276\) 0 0
\(277\) −9.50000 16.4545i −0.570800 0.988654i −0.996484 0.0837823i \(-0.973300\pi\)
0.425684 0.904872i \(-0.360033\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) 0 0
\(283\) −2.50000 4.33013i −0.148610 0.257399i 0.782104 0.623148i \(-0.214146\pi\)
−0.930714 + 0.365748i \(0.880813\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.00000 20.7846i −0.236113 1.22688i
\(288\) 0 0
\(289\) −9.50000 + 16.4545i −0.558824 + 0.967911i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 20.0000 1.16841 0.584206 0.811605i \(-0.301406\pi\)
0.584206 + 0.811605i \(0.301406\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.00000 + 8.66025i −0.289157 + 0.500835i
\(300\) 0 0
\(301\) −0.500000 2.59808i −0.0288195 0.149751i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −7.00000 12.1244i −0.400819 0.694239i
\(306\) 0 0
\(307\) 7.00000 0.399511 0.199756 0.979846i \(-0.435985\pi\)
0.199756 + 0.979846i \(0.435985\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.00000 + 13.8564i 0.453638 + 0.785725i 0.998609 0.0527306i \(-0.0167924\pi\)
−0.544970 + 0.838455i \(0.683459\pi\)
\(312\) 0 0
\(313\) 15.5000 26.8468i 0.876112 1.51747i 0.0205381 0.999789i \(-0.493462\pi\)
0.855574 0.517681i \(-0.173205\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −11.0000 + 19.0526i −0.617822 + 1.07010i 0.372061 + 0.928208i \(0.378651\pi\)
−0.989882 + 0.141890i \(0.954682\pi\)
\(318\) 0 0
\(319\) 4.00000 + 6.92820i 0.223957 + 0.387905i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 30.0000 1.66924
\(324\) 0 0
\(325\) 2.50000 + 4.33013i 0.138675 + 0.240192i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −15.0000 5.19615i −0.826977 0.286473i
\(330\) 0 0
\(331\) 15.5000 26.8468i 0.851957 1.47563i −0.0274825 0.999622i \(-0.508749\pi\)
0.879440 0.476011i \(-0.157918\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.00000 0.0546358
\(336\) 0 0
\(337\) −5.00000 −0.272367 −0.136184 0.990684i \(-0.543484\pi\)
−0.136184 + 0.990684i \(0.543484\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −18.0000 + 31.1769i −0.974755 + 1.68832i
\(342\) 0 0
\(343\) 10.0000 + 15.5885i 0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.00000 + 10.3923i 0.322097 + 0.557888i 0.980921 0.194409i \(-0.0622790\pi\)
−0.658824 + 0.752297i \(0.728946\pi\)
\(348\) 0 0
\(349\) −30.0000 −1.60586 −0.802932 0.596071i \(-0.796728\pi\)
−0.802932 + 0.596071i \(0.796728\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.0000 20.7846i −0.638696 1.10625i −0.985719 0.168397i \(-0.946141\pi\)
0.347024 0.937856i \(-0.387192\pi\)
\(354\) 0 0
\(355\) −6.00000 + 10.3923i −0.318447 + 0.551566i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.0000 + 25.9808i −0.791670 + 1.37121i 0.133263 + 0.991081i \(0.457455\pi\)
−0.924932 + 0.380131i \(0.875879\pi\)
\(360\) 0 0
\(361\) −3.00000 5.19615i −0.157895 0.273482i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −11.0000 −0.575766
\(366\) 0 0
\(367\) 8.50000 + 14.7224i 0.443696 + 0.768505i 0.997960 0.0638362i \(-0.0203335\pi\)
−0.554264 + 0.832341i \(0.687000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −8.00000 + 6.92820i −0.415339 + 0.359694i
\(372\) 0 0
\(373\) 9.50000 16.4545i 0.491891 0.851981i −0.508065 0.861319i \(-0.669639\pi\)
0.999956 + 0.00933789i \(0.00297238\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10.0000 −0.515026
\(378\) 0 0
\(379\) −15.0000 −0.770498 −0.385249 0.922813i \(-0.625884\pi\)
−0.385249 + 0.922813i \(0.625884\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.00000 3.46410i 0.102195 0.177007i −0.810394 0.585886i \(-0.800747\pi\)
0.912589 + 0.408879i \(0.134080\pi\)
\(384\) 0 0
\(385\) −2.00000 10.3923i −0.101929 0.529641i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10.0000 17.3205i −0.507020 0.878185i −0.999967 0.00812520i \(-0.997414\pi\)
0.492947 0.870059i \(-0.335920\pi\)
\(390\) 0 0
\(391\) 12.0000 0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0.500000 + 0.866025i 0.0251577 + 0.0435745i
\(396\) 0 0
\(397\) 3.50000 6.06218i 0.175660 0.304252i −0.764730 0.644351i \(-0.777127\pi\)
0.940389 + 0.340099i \(0.110461\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −6.00000 + 10.3923i −0.299626 + 0.518967i −0.976050 0.217545i \(-0.930195\pi\)
0.676425 + 0.736512i \(0.263528\pi\)
\(402\) 0 0
\(403\) −22.5000 38.9711i −1.12080 1.94129i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 44.0000 2.18100
\(408\) 0 0
\(409\) −10.5000 18.1865i −0.519192 0.899266i −0.999751 0.0223042i \(-0.992900\pi\)
0.480560 0.876962i \(-0.340434\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 12.0000 10.3923i 0.590481 0.511372i
\(414\) 0 0
\(415\) 2.00000 3.46410i 0.0981761 0.170046i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) 5.00000 0.243685 0.121843 0.992549i \(-0.461120\pi\)
0.121843 + 0.992549i \(0.461120\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.00000 5.19615i 0.145521 0.252050i
\(426\) 0 0
\(427\) −35.0000 12.1244i −1.69377 0.586739i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.0000 + 25.9808i 0.722525 + 1.25145i 0.959985 + 0.280052i \(0.0903517\pi\)
−0.237460 + 0.971397i \(0.576315\pi\)
\(432\) 0 0
\(433\) −19.0000 −0.913082 −0.456541 0.889702i \(-0.650912\pi\)
−0.456541 + 0.889702i \(0.650912\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.00000 8.66025i −0.239182 0.414276i
\(438\) 0 0
\(439\) 14.0000 24.2487i 0.668184 1.15733i −0.310228 0.950662i \(-0.600405\pi\)
0.978412 0.206666i \(-0.0662612\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −20.0000 + 34.6410i −0.950229 + 1.64584i −0.205301 + 0.978699i \(0.565817\pi\)
−0.744927 + 0.667146i \(0.767516\pi\)
\(444\) 0 0
\(445\) −9.00000 15.5885i −0.426641 0.738964i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 16.0000 + 27.7128i 0.753411 + 1.30495i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 12.5000 + 4.33013i 0.586009 + 0.202999i
\(456\) 0 0
\(457\) −1.50000 + 2.59808i −0.0701670 + 0.121533i −0.898974 0.438001i \(-0.855687\pi\)
0.828807 + 0.559534i \(0.189020\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 0 0
\(463\) −13.0000 −0.604161 −0.302081 0.953282i \(-0.597681\pi\)
−0.302081 + 0.953282i \(0.597681\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.00000 12.1244i 0.323921 0.561048i −0.657372 0.753566i \(-0.728332\pi\)
0.981293 + 0.192518i \(0.0616653\pi\)
\(468\) 0 0
\(469\) 2.00000 1.73205i 0.0923514 0.0799787i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.00000 + 3.46410i 0.0919601 + 0.159280i
\(474\) 0 0
\(475\) −5.00000 −0.229416
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 6.00000 + 10.3923i 0.274147 + 0.474837i 0.969920 0.243426i \(-0.0782712\pi\)
−0.695773 + 0.718262i \(0.744938\pi\)
\(480\) 0 0
\(481\) −27.5000 + 47.6314i −1.25389 + 2.17180i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −9.00000 + 15.5885i −0.408669 + 0.707835i
\(486\) 0 0
\(487\) −1.50000 2.59808i −0.0679715 0.117730i 0.830037 0.557709i \(-0.188319\pi\)
−0.898008 + 0.439979i \(0.854986\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) 6.00000 + 10.3923i 0.270226 + 0.468046i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 6.00000 + 31.1769i 0.269137 + 1.39848i
\(498\) 0 0
\(499\) −7.50000 + 12.9904i −0.335746 + 0.581529i −0.983628 0.180212i \(-0.942322\pi\)
0.647882 + 0.761741i \(0.275655\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −38.0000 −1.69434 −0.847168 0.531325i \(-0.821694\pi\)
−0.847168 + 0.531325i \(0.821694\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.00000 + 5.19615i −0.132973 + 0.230315i −0.924821 0.380402i \(-0.875786\pi\)
0.791849 + 0.610718i \(0.209119\pi\)
\(510\) 0 0
\(511\) −22.0000 + 19.0526i −0.973223 + 0.842836i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −5.50000 9.52628i −0.242359 0.419778i
\(516\) 0 0
\(517\) 24.0000 1.05552
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 20.0000 + 34.6410i 0.876216 + 1.51765i 0.855462 + 0.517866i \(0.173273\pi\)
0.0207541 + 0.999785i \(0.493393\pi\)
\(522\) 0 0
\(523\) 10.5000 18.1865i 0.459133 0.795242i −0.539782 0.841805i \(-0.681493\pi\)
0.998915 + 0.0465630i \(0.0148268\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −27.0000 + 46.7654i −1.17614 + 2.03713i
\(528\) 0 0
\(529\) 9.50000 + 16.4545i 0.413043 + 0.715412i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −40.0000 −1.73259
\(534\) 0 0
\(535\) 5.00000 + 8.66025i 0.216169 + 0.374415i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −22.0000 17.3205i −0.947607 0.746047i
\(540\) 0 0
\(541\) −12.5000 + 21.6506i −0.537417 + 0.930834i 0.461625 + 0.887075i \(0.347267\pi\)
−0.999042 + 0.0437584i \(0.986067\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.00000 −0.214176
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5.00000 8.66025i 0.213007 0.368939i
\(552\) 0 0
\(553\) 2.50000 + 0.866025i 0.106311 + 0.0368271i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.00000 + 1.73205i 0.0423714 + 0.0733893i 0.886433 0.462856i \(-0.153175\pi\)
−0.844062 + 0.536246i \(0.819842\pi\)
\(558\) 0 0
\(559\) −5.00000 −0.211477
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 15.0000 + 25.9808i 0.632175 + 1.09496i 0.987106 + 0.160066i \(0.0511708\pi\)
−0.354932 + 0.934892i \(0.615496\pi\)
\(564\) 0 0
\(565\) 1.00000 1.73205i 0.0420703 0.0728679i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18.0000 + 31.1769i −0.754599 + 1.30700i 0.190974 + 0.981595i \(0.438835\pi\)
−0.945573 + 0.325409i \(0.894498\pi\)
\(570\) 0 0
\(571\) 16.5000 + 28.5788i 0.690504 + 1.19599i 0.971673 + 0.236329i \(0.0759443\pi\)
−0.281170 + 0.959658i \(0.590722\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −2.00000 −0.0834058
\(576\) 0 0
\(577\) 7.50000 + 12.9904i 0.312229 + 0.540797i 0.978845 0.204605i \(-0.0655910\pi\)
−0.666616 + 0.745402i \(0.732258\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.00000 10.3923i −0.0829740 0.431145i
\(582\) 0 0
\(583\) 8.00000 13.8564i 0.331326 0.573874i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −30.0000 −1.23823 −0.619116 0.785299i \(-0.712509\pi\)
−0.619116 + 0.785299i \(0.712509\pi\)
\(588\) 0 0
\(589\) 45.0000 1.85419
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 0 0
\(595\) −3.00000 15.5885i −0.122988 0.639064i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 2.00000 + 3.46410i 0.0817178 + 0.141539i 0.903988 0.427558i \(-0.140626\pi\)
−0.822270 + 0.569097i \(0.807293\pi\)
\(600\) 0 0
\(601\) 7.00000 0.285536 0.142768 0.989756i \(-0.454400\pi\)
0.142768 + 0.989756i \(0.454400\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.50000 + 4.33013i 0.101639 + 0.176045i
\(606\) 0 0
\(607\) −11.5000 + 19.9186i −0.466771 + 0.808470i −0.999279 0.0379540i \(-0.987916\pi\)
0.532509 + 0.846424i \(0.321249\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −15.0000 + 25.9808i −0.606835 + 1.05107i
\(612\) 0 0
\(613\) −23.0000 39.8372i −0.928961 1.60901i −0.785063 0.619416i \(-0.787370\pi\)
−0.143898 0.989593i \(-0.545964\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −22.0000 −0.885687 −0.442843 0.896599i \(-0.646030\pi\)
−0.442843 + 0.896599i \(0.646030\pi\)
\(618\) 0 0
\(619\) −10.5000 18.1865i −0.422031 0.730978i 0.574107 0.818780i \(-0.305349\pi\)
−0.996138 + 0.0878015i \(0.972016\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −45.0000 15.5885i −1.80289 0.624538i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 66.0000 2.63159
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.50000 + 6.06218i −0.138893 + 0.240570i
\(636\) 0 0
\(637\) 32.5000 12.9904i 1.28770 0.514698i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11.0000 19.0526i −0.434474 0.752531i 0.562779 0.826608i \(-0.309732\pi\)
−0.997253 + 0.0740768i \(0.976399\pi\)
\(642\) 0 0
\(643\) 1.00000 0.0394362 0.0197181 0.999806i \(-0.493723\pi\)
0.0197181 + 0.999806i \(0.493723\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −18.0000 31.1769i −0.707653 1.22569i −0.965726 0.259565i \(-0.916421\pi\)
0.258073 0.966126i \(-0.416913\pi\)
\(648\) 0 0
\(649\) −12.0000 + 20.7846i −0.471041 + 0.815867i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.0000 31.1769i 0.704394 1.22005i −0.262515 0.964928i \(-0.584552\pi\)
0.966910 0.255119i \(-0.0821147\pi\)
\(654\) 0 0
\(655\) 1.00000 + 1.73205i 0.0390732 + 0.0676768i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −40.0000 −1.55818 −0.779089 0.626913i \(-0.784318\pi\)
−0.779089 + 0.626913i \(0.784318\pi\)
\(660\) 0 0
\(661\) 16.5000 + 28.5788i 0.641776 + 1.11159i 0.985036 + 0.172348i \(0.0551353\pi\)
−0.343261 + 0.939240i \(0.611531\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −10.0000 + 8.66025i −0.387783 + 0.335830i
\(666\) 0 0
\(667\) 2.00000 3.46410i 0.0774403 0.134131i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 56.0000 2.16186
\(672\) 0 0
\(673\) −23.0000 −0.886585 −0.443292 0.896377i \(-0.646190\pi\)
−0.443292 + 0.896377i \(0.646190\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 9.00000 15.5885i 0.345898 0.599113i −0.639618 0.768693i \(-0.720908\pi\)
0.985517 + 0.169580i \(0.0542410\pi\)
\(678\) 0 0
\(679\) 9.00000 + 46.7654i 0.345388 + 1.79469i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −13.0000 22.5167i −0.497431 0.861576i 0.502564 0.864540i \(-0.332390\pi\)
−0.999996 + 0.00296369i \(0.999057\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 10.0000 + 17.3205i 0.380970 + 0.659859i
\(690\) 0 0
\(691\) 13.5000 23.3827i 0.513564 0.889519i −0.486312 0.873785i \(-0.661658\pi\)
0.999876 0.0157341i \(-0.00500851\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.50000 16.4545i 0.360356 0.624154i
\(696\) 0 0
\(697\) 24.0000 + 41.5692i 0.909065 + 1.57455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) −27.5000 47.6314i −1.03718 1.79645i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −24.0000 + 20.7846i −0.902613 + 0.781686i
\(708\) 0 0
\(709\) 5.00000 8.66025i 0.187779 0.325243i −0.756730 0.653727i \(-0.773204\pi\)
0.944509 + 0.328484i \(0.106538\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 18.0000 0.674105
\(714\) 0 0
\(715\) −20.0000 −0.747958
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 17.0000 29.4449i 0.633993 1.09811i −0.352735 0.935723i \(-0.614748\pi\)
0.986728 0.162385i \(-0.0519185\pi\)
\(720\) 0 0
\(721\) −27.5000 9.52628i −1.02415 0.354777i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.00000 1.73205i −0.0371391 0.0643268i
\(726\) 0 0
\(727\) 19.0000 0.704671 0.352335 0.935874i \(-0.385388\pi\)
0.352335 + 0.935874i \(0.385388\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.00000 + 5.19615i 0.110959 + 0.192187i
\(732\) 0 0
\(733\) 0.500000 0.866025i 0.0184679 0.0319874i −0.856644 0.515908i \(-0.827454\pi\)
0.875112 + 0.483921i \(0.160788\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.00000 + 3.46410i −0.0736709 + 0.127602i
\(738\) 0 0
\(739\) 0.500000 + 0.866025i 0.0183928 + 0.0318573i 0.875075 0.483987i \(-0.160812\pi\)
−0.856683 + 0.515844i \(0.827478\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −36.0000 −1.32071 −0.660356 0.750953i \(-0.729595\pi\)
−0.660356 + 0.750953i \(0.729595\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 25.0000 + 8.66025i 0.913480 + 0.316439i
\(750\) 0 0
\(751\) 0.500000 0.866025i 0.0182453 0.0316017i −0.856759 0.515718i \(-0.827525\pi\)
0.875004 + 0.484116i \(0.160859\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −9.00000 + 15.5885i −0.326250 + 0.565081i −0.981764 0.190101i \(-0.939118\pi\)
0.655515 + 0.755182i \(0.272452\pi\)
\(762\) 0 0
\(763\) −10.0000 + 8.66025i −0.362024 + 0.313522i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −15.0000 25.9808i −0.541619 0.938111i
\(768\) 0 0
\(769\) −5.00000 −0.180305 −0.0901523 0.995928i \(-0.528735\pi\)
−0.0901523 + 0.995928i \(0.528735\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.00000 + 3.46410i 0.0719350 + 0.124595i 0.899749 0.436407i \(-0.143749\pi\)
−0.827814 + 0.561002i \(0.810416\pi\)
\(774\) 0 0
\(775\) 4.50000 7.79423i 0.161645 0.279977i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 20.0000 34.6410i 0.716574 1.24114i
\(780\) 0 0
\(781\) −24.0000 41.5692i −0.858788 1.48746i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 10.0000 + 17.3205i 0.356462 + 0.617409i 0.987367 0.158450i \(-0.0506498\pi\)
−0.630905 + 0.775860i \(0.717316\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.00000 5.19615i −0.0355559 0.184754i
\(792\) 0 0
\(793\) −35.0000 + 60.6218i −1.24289 + 2.15274i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 22.0000 38.1051i 0.776363 1.34470i
\(804\) 0 0
\(805\) −4.00000 + 3.46410i −0.140981 + 0.122094i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −17.0000 29.4449i −0.597688 1.03523i −0.993161 0.116749i \(-0.962753\pi\)
0.395473 0.918477i \(-0.370581\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.00000 3.46410i −0.0700569 0.121342i
\(816\) 0 0
\(817\) 2.50000 4.33013i 0.0874639 0.151492i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(822\) 0 0
\(823\) 20.0000 + 34.6410i 0.697156 + 1.20751i 0.969448 + 0.245295i \(0.0788849\pi\)
−0.272292 + 0.962215i \(0.587782\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −34.0000 −1.18230 −0.591148 0.806563i \(-0.701325\pi\)
−0.591148 + 0.806563i \(0.701325\pi\)
\(828\) 0 0
\(829\) −8.50000 14.7224i −0.295217 0.511331i 0.679818 0.733381i \(-0.262059\pi\)
−0.975035 + 0.222049i \(0.928725\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −33.0000 25.9808i −1.14338 0.900180i
\(834\) 0 0
\(835\) 10.0000 17.3205i 0.346064 0.599401i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −34.0000 −1.17381 −0.586905 0.809656i \(-0.699654\pi\)
−0.586905 + 0.809656i \(0.699654\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 6.00000 10.3923i 0.206406 0.357506i
\(846\) 0 0
\(847\) 12.5000 + 4.33013i 0.429505 + 0.148785i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −11.0000 19.0526i −0.377075 0.653113i
\(852\) 0 0
\(853\) 37.0000 1.26686 0.633428 0.773802i \(-0.281647\pi\)
0.633428 + 0.773802i \(0.281647\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.0000 + 32.9090i 0.649028 + 1.12415i 0.983355 + 0.181692i \(0.0581574\pi\)
−0.334328 + 0.942457i \(0.608509\pi\)
\(858\) 0 0
\(859\) 2.00000 3.46410i 0.0682391 0.118194i −0.829887 0.557931i \(-0.811595\pi\)
0.898126 + 0.439738i \(0.144929\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.00000 + 1.73205i −0.0340404 + 0.0589597i −0.882544 0.470230i \(-0.844171\pi\)
0.848503 + 0.529190i \(0.177504\pi\)
\(864\) 0 0
\(865\) −2.00000 3.46410i −0.0680020 0.117783i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) −2.50000 4.33013i −0.0847093 0.146721i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0.500000 + 2.59808i 0.0169031 + 0.0878310i
\(876\) 0 0
\(877\) −25.0000 + 43.3013i −0.844190 + 1.46218i 0.0421327 + 0.999112i \(0.486585\pi\)
−0.886323 + 0.463068i \(0.846749\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −16.0000 −0.539054 −0.269527 0.962993i \(-0.586867\pi\)
−0.269527 + 0.962993i \(0.586867\pi\)
\(882\) 0 0
\(883\) −1.00000 −0.0336527 −0.0168263 0.999858i \(-0.505356\pi\)
−0.0168263 + 0.999858i \(0.505356\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(888\) 0 0
\(889\) 3.50000 + 18.1865i 0.117386 + 0.609957i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −15.0000 25.9808i −0.501956 0.869413i
\(894\) 0 0
\(895\) 18.0000 0.601674
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.00000 + 15.5885i 0.300167 + 0.519904i
\(900\) 0 0
\(901\) 12.0000 20.7846i 0.399778 0.692436i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.50000 4.33013i 0.0831028 0.143938i
\(906\) 0 0
\(907\) 4.50000 + 7.79423i 0.149420 + 0.258803i 0.931013 0.364985i \(-0.118926\pi\)
−0.781593 + 0.623788i \(0.785593\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 38.0000 1.25900 0.629498 0.777002i \(-0.283261\pi\)
0.629498 + 0.777002i \(0.283261\pi\)
\(912\) 0 0
\(913\) 8.00000 + 13.8564i 0.264761 + 0.458580i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.00000 + 1.73205i 0.165115 + 0.0571974i
\(918\) 0 0
\(919\) −18.5000 + 32.0429i −0.610259 + 1.05700i 0.380938 + 0.924601i \(0.375601\pi\)
−0.991197 + 0.132398i \(0.957732\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 60.0000 1.97492
\(924\) 0 0
\(925\) −11.0000 −0.361678
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −14.0000 + 24.2487i −0.459325 + 0.795574i −0.998925 0.0463469i \(-0.985242\pi\)
0.539600 + 0.841921i \(0.318575\pi\)
\(930\) 0 0
\(931\) −5.00000 + 34.6410i −0.163868 + 1.13531i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 12.0000 + 20.7846i 0.392442 + 0.679729i
\(936\) 0 0
\(937\) 37.0000 1.20874 0.604369 0.796705i \(-0.293425\pi\)
0.604369 + 0.796705i \(0.293425\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 9.00000 + 15.5885i 0.293392 + 0.508169i 0.974609 0.223912i \(-0.0718827\pi\)
−0.681218 + 0.732081i \(0.738549\pi\)
\(942\) 0 0
\(943\) 8.00000 13.8564i 0.260516 0.451227i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −6.00000 + 10.3923i −0.194974 + 0.337705i −0.946892 0.321552i \(-0.895796\pi\)
0.751918 + 0.659256i \(0.229129\pi\)
\(948\) 0 0
\(949\) 27.5000 + 47.6314i 0.892688 + 1.54618i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 12.0000 0.388718 0.194359 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) 0 0
\(955\) −1.00000 1.73205i −0.0323592 0.0560478i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.00000 3.46410i 0.129167 0.111862i
\(960\) 0 0
\(961\) −25.0000 + 43.3013i −0.806452 + 1.39682i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 3.00000 0.0965734
\(966\) 0 0
\(967\) −37.0000 −1.18984 −0.594920 0.803785i \(-0.702816\pi\)
−0.594920 + 0.803785i \(0.702816\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 6.00000 10.3923i 0.192549 0.333505i −0.753545 0.657396i \(-0.771658\pi\)
0.946094 + 0.323891i \(0.104991\pi\)
\(972\) 0 0
\(973\) −9.50000 49.3634i −0.304556 1.58252i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(978\) 0 0
\(979\) 72.0000 2.30113
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 23.0000 + 39.8372i 0.733586 + 1.27061i 0.955341 + 0.295506i \(0.0954882\pi\)
−0.221755 + 0.975102i \(0.571178\pi\)
\(984\) 0 0
\(985\) 9.00000 15.5885i 0.286764 0.496690i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.00000 1.73205i 0.0317982 0.0550760i
\(990\) 0 0
\(991\) −19.5000 33.7750i −0.619438 1.07290i −0.989588 0.143926i \(-0.954027\pi\)
0.370151 0.928972i \(-0.379306\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) −2.50000 4.33013i −0.0791758 0.137136i 0.823719 0.566999i \(-0.191896\pi\)
−0.902895 + 0.429862i \(0.858562\pi\)
\(998\) 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 2520.2.bi.h.1801.1 2
3.2 odd 2 840.2.bg.b.121.1 2
7.4 even 3 inner 2520.2.bi.h.361.1 2
12.11 even 2 1680.2.bg.c.961.1 2
21.2 odd 6 5880.2.a.q.1.1 1
21.5 even 6 5880.2.a.ba.1.1 1
21.11 odd 6 840.2.bg.b.361.1 yes 2
84.11 even 6 1680.2.bg.c.1201.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
840.2.bg.b.121.1 2 3.2 odd 2
840.2.bg.b.361.1 yes 2 21.11 odd 6
1680.2.bg.c.961.1 2 12.11 even 2
1680.2.bg.c.1201.1 2 84.11 even 6
2520.2.bi.h.361.1 2 7.4 even 3 inner
2520.2.bi.h.1801.1 2 1.1 even 1 trivial
5880.2.a.q.1.1 1 21.2 odd 6
5880.2.a.ba.1.1 1 21.5 even 6