Properties

Label 504.2.cx.a.185.7
Level $504$
Weight $2$
Character 504.185
Analytic conductor $4.024$
Analytic rank $0$
Dimension $48$
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] = [504,2,Mod(185,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.185");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.cx (of order \(6\), 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 185.7
Character \(\chi\) \(=\) 504.185
Dual form 504.2.cx.a.425.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.11231 - 1.32769i) q^{3} +3.53401 q^{5} +(1.30083 + 2.30388i) q^{7} +(-0.525517 + 2.95361i) q^{9} +O(q^{10})\) \(q+(-1.11231 - 1.32769i) q^{3} +3.53401 q^{5} +(1.30083 + 2.30388i) q^{7} +(-0.525517 + 2.95361i) q^{9} +1.81827i q^{11} +(2.78280 + 1.60665i) q^{13} +(-3.93093 - 4.69207i) q^{15} +(-2.93434 + 5.08242i) q^{17} +(-2.09143 + 1.20749i) q^{19} +(1.61190 - 4.28973i) q^{21} -3.84871i q^{23} +7.48923 q^{25} +(4.50602 - 2.58762i) q^{27} +(5.79110 - 3.34349i) q^{29} +(-4.21959 + 2.43618i) q^{31} +(2.41410 - 2.02249i) q^{33} +(4.59714 + 8.14192i) q^{35} +(-0.905385 - 1.56817i) q^{37} +(-0.962213 - 5.48178i) q^{39} +(5.03152 - 8.71485i) q^{41} +(-2.36232 - 4.09166i) q^{43} +(-1.85718 + 10.4381i) q^{45} +(3.96076 - 6.86023i) q^{47} +(-3.61569 + 5.99390i) q^{49} +(10.0118 - 1.75736i) q^{51} +(2.26517 + 1.30780i) q^{53} +6.42580i q^{55} +(3.92950 + 1.43366i) q^{57} +(4.40604 + 7.63149i) q^{59} +(-8.98447 - 5.18718i) q^{61} +(-7.48837 + 2.63142i) q^{63} +(9.83443 + 5.67791i) q^{65} +(3.92384 + 6.79630i) q^{67} +(-5.10990 + 4.28098i) q^{69} -7.62952i q^{71} +(-11.7672 - 6.79382i) q^{73} +(-8.33037 - 9.94337i) q^{75} +(-4.18907 + 2.36526i) q^{77} +(0.921268 - 1.59568i) q^{79} +(-8.44766 - 3.10435i) q^{81} +(-5.61748 - 9.72975i) q^{83} +(-10.3700 + 17.9613i) q^{85} +(-10.8806 - 3.96977i) q^{87} +(-6.89792 - 11.9475i) q^{89} +(-0.0815748 + 8.50119i) q^{91} +(7.92800 + 2.89251i) q^{93} +(-7.39114 + 4.26728i) q^{95} +(-13.7982 + 7.96638i) q^{97} +(-5.37048 - 0.955533i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 2 q^{9} + 8 q^{15} - 10 q^{21} + 48 q^{25} + 18 q^{27} + 18 q^{29} + 18 q^{31} + 12 q^{33} - 4 q^{39} - 6 q^{41} - 6 q^{43} - 18 q^{45} + 18 q^{47} - 12 q^{49} + 6 q^{51} - 12 q^{53} + 4 q^{57} + 18 q^{61} - 32 q^{63} - 36 q^{65} - 12 q^{77} + 6 q^{79} + 6 q^{81} - 54 q^{87} - 18 q^{89} + 6 q^{91} + 4 q^{93} - 54 q^{95} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(e\left(\frac{1}{6}\right)\) \(1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −1.11231 1.32769i −0.642195 0.766542i
\(4\) 0 0
\(5\) 3.53401 1.58046 0.790229 0.612812i \(-0.209962\pi\)
0.790229 + 0.612812i \(0.209962\pi\)
\(6\) 0 0
\(7\) 1.30083 + 2.30388i 0.491667 + 0.870783i
\(8\) 0 0
\(9\) −0.525517 + 2.95361i −0.175172 + 0.984538i
\(10\) 0 0
\(11\) 1.81827i 0.548230i 0.961697 + 0.274115i \(0.0883849\pi\)
−0.961697 + 0.274115i \(0.911615\pi\)
\(12\) 0 0
\(13\) 2.78280 + 1.60665i 0.771809 + 0.445604i 0.833519 0.552490i \(-0.186322\pi\)
−0.0617107 + 0.998094i \(0.519656\pi\)
\(14\) 0 0
\(15\) −3.93093 4.69207i −1.01496 1.21149i
\(16\) 0 0
\(17\) −2.93434 + 5.08242i −0.711682 + 1.23267i 0.252544 + 0.967585i \(0.418733\pi\)
−0.964225 + 0.265083i \(0.914600\pi\)
\(18\) 0 0
\(19\) −2.09143 + 1.20749i −0.479807 + 0.277017i −0.720336 0.693625i \(-0.756012\pi\)
0.240529 + 0.970642i \(0.422679\pi\)
\(20\) 0 0
\(21\) 1.61190 4.28973i 0.351746 0.936096i
\(22\) 0 0
\(23\) 3.84871i 0.802512i −0.915966 0.401256i \(-0.868574\pi\)
0.915966 0.401256i \(-0.131426\pi\)
\(24\) 0 0
\(25\) 7.48923 1.49785
\(26\) 0 0
\(27\) 4.50602 2.58762i 0.867184 0.497988i
\(28\) 0 0
\(29\) 5.79110 3.34349i 1.07538 0.620871i 0.145734 0.989324i \(-0.453446\pi\)
0.929647 + 0.368452i \(0.120112\pi\)
\(30\) 0 0
\(31\) −4.21959 + 2.43618i −0.757861 + 0.437551i −0.828527 0.559949i \(-0.810821\pi\)
0.0706663 + 0.997500i \(0.477487\pi\)
\(32\) 0 0
\(33\) 2.41410 2.02249i 0.420241 0.352070i
\(34\) 0 0
\(35\) 4.59714 + 8.14192i 0.777059 + 1.37624i
\(36\) 0 0
\(37\) −0.905385 1.56817i −0.148844 0.257806i 0.781956 0.623333i \(-0.214222\pi\)
−0.930801 + 0.365527i \(0.880889\pi\)
\(38\) 0 0
\(39\) −0.962213 5.48178i −0.154077 0.877788i
\(40\) 0 0
\(41\) 5.03152 8.71485i 0.785792 1.36103i −0.142733 0.989761i \(-0.545589\pi\)
0.928525 0.371270i \(-0.121078\pi\)
\(42\) 0 0
\(43\) −2.36232 4.09166i −0.360251 0.623973i 0.627751 0.778414i \(-0.283976\pi\)
−0.988002 + 0.154441i \(0.950642\pi\)
\(44\) 0 0
\(45\) −1.85718 + 10.4381i −0.276852 + 1.55602i
\(46\) 0 0
\(47\) 3.96076 6.86023i 0.577736 1.00067i −0.418003 0.908446i \(-0.637270\pi\)
0.995738 0.0922218i \(-0.0293969\pi\)
\(48\) 0 0
\(49\) −3.61569 + 5.99390i −0.516527 + 0.856271i
\(50\) 0 0
\(51\) 10.0118 1.75736i 1.40193 0.246080i
\(52\) 0 0
\(53\) 2.26517 + 1.30780i 0.311146 + 0.179640i 0.647439 0.762117i \(-0.275840\pi\)
−0.336293 + 0.941757i \(0.609173\pi\)
\(54\) 0 0
\(55\) 6.42580i 0.866454i
\(56\) 0 0
\(57\) 3.92950 + 1.43366i 0.520474 + 0.189893i
\(58\) 0 0
\(59\) 4.40604 + 7.63149i 0.573618 + 0.993535i 0.996190 + 0.0872059i \(0.0277938\pi\)
−0.422573 + 0.906329i \(0.638873\pi\)
\(60\) 0 0
\(61\) −8.98447 5.18718i −1.15034 0.664151i −0.201372 0.979515i \(-0.564540\pi\)
−0.948971 + 0.315364i \(0.897873\pi\)
\(62\) 0 0
\(63\) −7.48837 + 2.63142i −0.943445 + 0.331528i
\(64\) 0 0
\(65\) 9.83443 + 5.67791i 1.21981 + 0.704258i
\(66\) 0 0
\(67\) 3.92384 + 6.79630i 0.479374 + 0.830300i 0.999720 0.0236554i \(-0.00753044\pi\)
−0.520346 + 0.853955i \(0.674197\pi\)
\(68\) 0 0
\(69\) −5.10990 + 4.28098i −0.615159 + 0.515369i
\(70\) 0 0
\(71\) 7.62952i 0.905458i −0.891648 0.452729i \(-0.850451\pi\)
0.891648 0.452729i \(-0.149549\pi\)
\(72\) 0 0
\(73\) −11.7672 6.79382i −1.37725 0.795156i −0.385423 0.922740i \(-0.625945\pi\)
−0.991828 + 0.127584i \(0.959278\pi\)
\(74\) 0 0
\(75\) −8.33037 9.94337i −0.961909 1.14816i
\(76\) 0 0
\(77\) −4.18907 + 2.36526i −0.477389 + 0.269547i
\(78\) 0 0
\(79\) 0.921268 1.59568i 0.103651 0.179528i −0.809535 0.587071i \(-0.800281\pi\)
0.913186 + 0.407543i \(0.133614\pi\)
\(80\) 0 0
\(81\) −8.44766 3.10435i −0.938629 0.344927i
\(82\) 0 0
\(83\) −5.61748 9.72975i −0.616598 1.06798i −0.990102 0.140351i \(-0.955177\pi\)
0.373504 0.927629i \(-0.378156\pi\)
\(84\) 0 0
\(85\) −10.3700 + 17.9613i −1.12478 + 1.94818i
\(86\) 0 0
\(87\) −10.8806 3.96977i −1.16653 0.425604i
\(88\) 0 0
\(89\) −6.89792 11.9475i −0.731178 1.26644i −0.956380 0.292125i \(-0.905638\pi\)
0.225202 0.974312i \(-0.427696\pi\)
\(90\) 0 0
\(91\) −0.0815748 + 8.50119i −0.00855137 + 0.891167i
\(92\) 0 0
\(93\) 7.92800 + 2.89251i 0.822095 + 0.299939i
\(94\) 0 0
\(95\) −7.39114 + 4.26728i −0.758315 + 0.437813i
\(96\) 0 0
\(97\) −13.7982 + 7.96638i −1.40099 + 0.808863i −0.994494 0.104789i \(-0.966583\pi\)
−0.406497 + 0.913652i \(0.633250\pi\)
\(98\) 0 0
\(99\) −5.37048 0.955533i −0.539753 0.0960346i
\(100\) 0 0
\(101\) 11.1250 1.10698 0.553489 0.832856i \(-0.313296\pi\)
0.553489 + 0.832856i \(0.313296\pi\)
\(102\) 0 0
\(103\) 0.603028i 0.0594181i −0.999559 0.0297091i \(-0.990542\pi\)
0.999559 0.0297091i \(-0.00945808\pi\)
\(104\) 0 0
\(105\) 5.69647 15.1599i 0.555919 1.47946i
\(106\) 0 0
\(107\) −11.3342 + 6.54381i −1.09572 + 0.632614i −0.935094 0.354401i \(-0.884685\pi\)
−0.160627 + 0.987015i \(0.551351\pi\)
\(108\) 0 0
\(109\) −1.37061 + 2.37396i −0.131280 + 0.227384i −0.924170 0.381981i \(-0.875242\pi\)
0.792890 + 0.609365i \(0.208575\pi\)
\(110\) 0 0
\(111\) −1.07497 + 2.94637i −0.102032 + 0.279657i
\(112\) 0 0
\(113\) 6.65125 + 3.84010i 0.625697 + 0.361246i 0.779084 0.626920i \(-0.215685\pi\)
−0.153387 + 0.988166i \(0.549018\pi\)
\(114\) 0 0
\(115\) 13.6014i 1.26834i
\(116\) 0 0
\(117\) −6.20782 + 7.37498i −0.573913 + 0.681817i
\(118\) 0 0
\(119\) −15.5263 0.148986i −1.42330 0.0136575i
\(120\) 0 0
\(121\) 7.69388 0.699444
\(122\) 0 0
\(123\) −17.1672 + 3.01335i −1.54792 + 0.271705i
\(124\) 0 0
\(125\) 8.79697 0.786825
\(126\) 0 0
\(127\) 17.9246 1.59055 0.795277 0.606246i \(-0.207325\pi\)
0.795277 + 0.606246i \(0.207325\pi\)
\(128\) 0 0
\(129\) −2.80481 + 7.68765i −0.246950 + 0.676859i
\(130\) 0 0
\(131\) 16.9707 1.48273 0.741367 0.671100i \(-0.234178\pi\)
0.741367 + 0.671100i \(0.234178\pi\)
\(132\) 0 0
\(133\) −5.50250 3.24766i −0.477127 0.281608i
\(134\) 0 0
\(135\) 15.9243 9.14468i 1.37055 0.787049i
\(136\) 0 0
\(137\) 5.56139i 0.475142i −0.971370 0.237571i \(-0.923649\pi\)
0.971370 0.237571i \(-0.0763512\pi\)
\(138\) 0 0
\(139\) −12.4503 7.18817i −1.05602 0.609693i −0.131690 0.991291i \(-0.542040\pi\)
−0.924328 + 0.381598i \(0.875374\pi\)
\(140\) 0 0
\(141\) −13.5139 + 2.37208i −1.13807 + 0.199765i
\(142\) 0 0
\(143\) −2.92132 + 5.05988i −0.244293 + 0.423129i
\(144\) 0 0
\(145\) 20.4658 11.8159i 1.69959 0.981261i
\(146\) 0 0
\(147\) 11.9798 1.86659i 0.988078 0.153954i
\(148\) 0 0
\(149\) 6.42112i 0.526039i −0.964791 0.263019i \(-0.915282\pi\)
0.964791 0.263019i \(-0.0847184\pi\)
\(150\) 0 0
\(151\) 15.9925 1.30145 0.650727 0.759312i \(-0.274464\pi\)
0.650727 + 0.759312i \(0.274464\pi\)
\(152\) 0 0
\(153\) −13.4695 11.3378i −1.08894 0.916607i
\(154\) 0 0
\(155\) −14.9121 + 8.60949i −1.19777 + 0.691531i
\(156\) 0 0
\(157\) −12.0612 + 6.96356i −0.962592 + 0.555753i −0.896970 0.442092i \(-0.854236\pi\)
−0.0656219 + 0.997845i \(0.520903\pi\)
\(158\) 0 0
\(159\) −0.783234 4.46213i −0.0621145 0.353870i
\(160\) 0 0
\(161\) 8.86696 5.00652i 0.698814 0.394569i
\(162\) 0 0
\(163\) 7.79498 + 13.5013i 0.610550 + 1.05750i 0.991148 + 0.132762i \(0.0423847\pi\)
−0.380598 + 0.924740i \(0.624282\pi\)
\(164\) 0 0
\(165\) 8.53146 7.14750i 0.664173 0.556432i
\(166\) 0 0
\(167\) −6.58728 + 11.4095i −0.509739 + 0.882893i 0.490198 + 0.871611i \(0.336925\pi\)
−0.999936 + 0.0112821i \(0.996409\pi\)
\(168\) 0 0
\(169\) −1.33736 2.31638i −0.102874 0.178183i
\(170\) 0 0
\(171\) −2.46737 6.81183i −0.188685 0.520914i
\(172\) 0 0
\(173\) −5.60402 + 9.70645i −0.426066 + 0.737968i −0.996519 0.0833623i \(-0.973434\pi\)
0.570453 + 0.821330i \(0.306767\pi\)
\(174\) 0 0
\(175\) 9.74221 + 17.2543i 0.736442 + 1.30430i
\(176\) 0 0
\(177\) 5.23134 14.3385i 0.393212 1.07774i
\(178\) 0 0
\(179\) 9.22783 + 5.32769i 0.689721 + 0.398210i 0.803507 0.595295i \(-0.202965\pi\)
−0.113787 + 0.993505i \(0.536298\pi\)
\(180\) 0 0
\(181\) 11.8897i 0.883754i 0.897076 + 0.441877i \(0.145687\pi\)
−0.897076 + 0.441877i \(0.854313\pi\)
\(182\) 0 0
\(183\) 3.10658 + 17.6984i 0.229645 + 1.30830i
\(184\) 0 0
\(185\) −3.19964 5.54194i −0.235242 0.407451i
\(186\) 0 0
\(187\) −9.24123 5.33543i −0.675786 0.390165i
\(188\) 0 0
\(189\) 11.8231 + 7.01525i 0.860006 + 0.510285i
\(190\) 0 0
\(191\) 0.193961 + 0.111983i 0.0140345 + 0.00810283i 0.507001 0.861946i \(-0.330754\pi\)
−0.492966 + 0.870048i \(0.664087\pi\)
\(192\) 0 0
\(193\) −13.0373 22.5812i −0.938442 1.62543i −0.768378 0.639996i \(-0.778936\pi\)
−0.170064 0.985433i \(-0.554398\pi\)
\(194\) 0 0
\(195\) −3.40047 19.3727i −0.243513 1.38731i
\(196\) 0 0
\(197\) 1.56289i 0.111352i 0.998449 + 0.0556758i \(0.0177313\pi\)
−0.998449 + 0.0556758i \(0.982269\pi\)
\(198\) 0 0
\(199\) 13.4725 + 7.77832i 0.955037 + 0.551391i 0.894642 0.446784i \(-0.147431\pi\)
0.0603949 + 0.998175i \(0.480764\pi\)
\(200\) 0 0
\(201\) 4.65883 12.7693i 0.328608 0.900674i
\(202\) 0 0
\(203\) 15.2362 + 8.99266i 1.06937 + 0.631161i
\(204\) 0 0
\(205\) 17.7815 30.7984i 1.24191 2.15105i
\(206\) 0 0
\(207\) 11.3676 + 2.02256i 0.790104 + 0.140578i
\(208\) 0 0
\(209\) −2.19554 3.80279i −0.151869 0.263045i
\(210\) 0 0
\(211\) 6.98216 12.0935i 0.480672 0.832548i −0.519082 0.854724i \(-0.673726\pi\)
0.999754 + 0.0221763i \(0.00705950\pi\)
\(212\) 0 0
\(213\) −10.1296 + 8.48642i −0.694071 + 0.581480i
\(214\) 0 0
\(215\) −8.34848 14.4600i −0.569361 0.986163i
\(216\) 0 0
\(217\) −11.1016 6.55236i −0.753628 0.444803i
\(218\) 0 0
\(219\) 4.06878 + 23.1801i 0.274943 + 1.56637i
\(220\) 0 0
\(221\) −16.3313 + 9.42890i −1.09856 + 0.634256i
\(222\) 0 0
\(223\) −21.4242 + 12.3693i −1.43467 + 0.828307i −0.997472 0.0710586i \(-0.977362\pi\)
−0.437198 + 0.899366i \(0.644029\pi\)
\(224\) 0 0
\(225\) −3.93571 + 22.1203i −0.262381 + 1.47469i
\(226\) 0 0
\(227\) −6.24153 −0.414265 −0.207132 0.978313i \(-0.566413\pi\)
−0.207132 + 0.978313i \(0.566413\pi\)
\(228\) 0 0
\(229\) 15.8574i 1.04789i 0.851753 + 0.523943i \(0.175540\pi\)
−0.851753 + 0.523943i \(0.824460\pi\)
\(230\) 0 0
\(231\) 7.79990 + 2.93087i 0.513196 + 0.192837i
\(232\) 0 0
\(233\) 2.94574 1.70072i 0.192982 0.111418i −0.400396 0.916342i \(-0.631127\pi\)
0.593378 + 0.804924i \(0.297794\pi\)
\(234\) 0 0
\(235\) 13.9974 24.2441i 0.913087 1.58151i
\(236\) 0 0
\(237\) −3.14331 + 0.551742i −0.204180 + 0.0358395i
\(238\) 0 0
\(239\) −16.3482 9.43864i −1.05748 0.610535i −0.132744 0.991150i \(-0.542379\pi\)
−0.924733 + 0.380615i \(0.875712\pi\)
\(240\) 0 0
\(241\) 11.1845i 0.720460i −0.932864 0.360230i \(-0.882698\pi\)
0.932864 0.360230i \(-0.117302\pi\)
\(242\) 0 0
\(243\) 5.27485 + 14.6689i 0.338382 + 0.941009i
\(244\) 0 0
\(245\) −12.7779 + 21.1825i −0.816348 + 1.35330i
\(246\) 0 0
\(247\) −7.76003 −0.493759
\(248\) 0 0
\(249\) −6.66969 + 18.2808i −0.422675 + 1.15850i
\(250\) 0 0
\(251\) 15.4537 0.975431 0.487716 0.873003i \(-0.337830\pi\)
0.487716 + 0.873003i \(0.337830\pi\)
\(252\) 0 0
\(253\) 6.99801 0.439961
\(254\) 0 0
\(255\) 35.3818 6.21053i 2.21569 0.388918i
\(256\) 0 0
\(257\) −10.6448 −0.664002 −0.332001 0.943279i \(-0.607724\pi\)
−0.332001 + 0.943279i \(0.607724\pi\)
\(258\) 0 0
\(259\) 2.43512 4.12582i 0.151311 0.256366i
\(260\) 0 0
\(261\) 6.83207 + 18.8617i 0.422895 + 1.16751i
\(262\) 0 0
\(263\) 11.0875i 0.683685i −0.939757 0.341842i \(-0.888949\pi\)
0.939757 0.341842i \(-0.111051\pi\)
\(264\) 0 0
\(265\) 8.00515 + 4.62178i 0.491753 + 0.283913i
\(266\) 0 0
\(267\) −8.18998 + 22.4477i −0.501218 + 1.37378i
\(268\) 0 0
\(269\) −3.14147 + 5.44119i −0.191539 + 0.331755i −0.945760 0.324865i \(-0.894681\pi\)
0.754221 + 0.656620i \(0.228014\pi\)
\(270\) 0 0
\(271\) 0.954533 0.551100i 0.0579838 0.0334769i −0.470728 0.882278i \(-0.656009\pi\)
0.528712 + 0.848802i \(0.322675\pi\)
\(272\) 0 0
\(273\) 11.3777 9.34768i 0.688608 0.565748i
\(274\) 0 0
\(275\) 13.6175i 0.821164i
\(276\) 0 0
\(277\) −15.3616 −0.922990 −0.461495 0.887143i \(-0.652687\pi\)
−0.461495 + 0.887143i \(0.652687\pi\)
\(278\) 0 0
\(279\) −4.97807 13.7433i −0.298030 0.822790i
\(280\) 0 0
\(281\) 18.9133 10.9196i 1.12827 0.651407i 0.184771 0.982782i \(-0.440846\pi\)
0.943499 + 0.331374i \(0.107512\pi\)
\(282\) 0 0
\(283\) −9.59426 + 5.53925i −0.570320 + 0.329274i −0.757277 0.653094i \(-0.773471\pi\)
0.186957 + 0.982368i \(0.440137\pi\)
\(284\) 0 0
\(285\) 13.8869 + 5.06658i 0.822588 + 0.300119i
\(286\) 0 0
\(287\) 26.6231 + 0.255467i 1.57151 + 0.0150797i
\(288\) 0 0
\(289\) −8.72069 15.1047i −0.512982 0.888511i
\(290\) 0 0
\(291\) 25.9248 + 9.45857i 1.51974 + 0.554471i
\(292\) 0 0
\(293\) 12.6330 21.8810i 0.738027 1.27830i −0.215355 0.976536i \(-0.569091\pi\)
0.953382 0.301765i \(-0.0975757\pi\)
\(294\) 0 0
\(295\) 15.5710 + 26.9698i 0.906578 + 1.57024i
\(296\) 0 0
\(297\) 4.70500 + 8.19317i 0.273012 + 0.475416i
\(298\) 0 0
\(299\) 6.18353 10.7102i 0.357603 0.619386i
\(300\) 0 0
\(301\) 6.35371 10.7651i 0.366222 0.620488i
\(302\) 0 0
\(303\) −12.3745 14.7705i −0.710896 0.848545i
\(304\) 0 0
\(305\) −31.7512 18.3316i −1.81807 1.04966i
\(306\) 0 0
\(307\) 12.2163i 0.697219i −0.937268 0.348610i \(-0.886654\pi\)
0.937268 0.348610i \(-0.113346\pi\)
\(308\) 0 0
\(309\) −0.800634 + 0.670756i −0.0455465 + 0.0381580i
\(310\) 0 0
\(311\) −4.74484 8.21830i −0.269055 0.466017i 0.699563 0.714571i \(-0.253378\pi\)
−0.968618 + 0.248554i \(0.920045\pi\)
\(312\) 0 0
\(313\) 11.7358 + 6.77564i 0.663344 + 0.382982i 0.793550 0.608505i \(-0.208231\pi\)
−0.130206 + 0.991487i \(0.541564\pi\)
\(314\) 0 0
\(315\) −26.4640 + 9.29947i −1.49108 + 0.523966i
\(316\) 0 0
\(317\) 13.7559 + 7.94197i 0.772608 + 0.446065i 0.833804 0.552060i \(-0.186158\pi\)
−0.0611961 + 0.998126i \(0.519492\pi\)
\(318\) 0 0
\(319\) 6.07939 + 10.5298i 0.340380 + 0.589556i
\(320\) 0 0
\(321\) 21.2954 + 7.76954i 1.18859 + 0.433654i
\(322\) 0 0
\(323\) 14.1727i 0.788591i
\(324\) 0 0
\(325\) 20.8410 + 12.0326i 1.15605 + 0.667446i
\(326\) 0 0
\(327\) 4.67642 0.820849i 0.258607 0.0453930i
\(328\) 0 0
\(329\) 20.9574 + 0.201101i 1.15542 + 0.0110870i
\(330\) 0 0
\(331\) −5.48697 + 9.50372i −0.301591 + 0.522371i −0.976497 0.215533i \(-0.930851\pi\)
0.674905 + 0.737904i \(0.264184\pi\)
\(332\) 0 0
\(333\) 5.10757 1.85006i 0.279893 0.101382i
\(334\) 0 0
\(335\) 13.8669 + 24.0182i 0.757630 + 1.31225i
\(336\) 0 0
\(337\) 2.80751 4.86275i 0.152935 0.264891i −0.779370 0.626564i \(-0.784461\pi\)
0.932305 + 0.361673i \(0.117794\pi\)
\(338\) 0 0
\(339\) −2.29982 13.1022i −0.124909 0.711613i
\(340\) 0 0
\(341\) −4.42964 7.67237i −0.239879 0.415482i
\(342\) 0 0
\(343\) −18.5126 0.533054i −0.999586 0.0287822i
\(344\) 0 0
\(345\) −18.0584 + 15.1290i −0.972233 + 0.814519i
\(346\) 0 0
\(347\) −4.18003 + 2.41334i −0.224396 + 0.129555i −0.607984 0.793949i \(-0.708022\pi\)
0.383588 + 0.923504i \(0.374688\pi\)
\(348\) 0 0
\(349\) 22.7167 13.1155i 1.21600 0.702056i 0.251937 0.967744i \(-0.418932\pi\)
0.964059 + 0.265688i \(0.0855991\pi\)
\(350\) 0 0
\(351\) 16.6967 + 0.0387633i 0.891206 + 0.00206903i
\(352\) 0 0
\(353\) −34.9914 −1.86240 −0.931201 0.364505i \(-0.881238\pi\)
−0.931201 + 0.364505i \(0.881238\pi\)
\(354\) 0 0
\(355\) 26.9628i 1.43104i
\(356\) 0 0
\(357\) 17.0724 + 20.7799i 0.903565 + 1.09979i
\(358\) 0 0
\(359\) −21.6396 + 12.4936i −1.14209 + 0.659388i −0.946948 0.321386i \(-0.895851\pi\)
−0.195146 + 0.980774i \(0.562518\pi\)
\(360\) 0 0
\(361\) −6.58394 + 11.4037i −0.346523 + 0.600196i
\(362\) 0 0
\(363\) −8.55801 10.2151i −0.449179 0.536153i
\(364\) 0 0
\(365\) −41.5855 24.0094i −2.17669 1.25671i
\(366\) 0 0
\(367\) 24.6067i 1.28446i −0.766512 0.642230i \(-0.778009\pi\)
0.766512 0.642230i \(-0.221991\pi\)
\(368\) 0 0
\(369\) 23.0962 + 19.4410i 1.20234 + 1.01206i
\(370\) 0 0
\(371\) −0.0664013 + 6.91990i −0.00344738 + 0.359264i
\(372\) 0 0
\(373\) −10.4374 −0.540428 −0.270214 0.962800i \(-0.587094\pi\)
−0.270214 + 0.962800i \(0.587094\pi\)
\(374\) 0 0
\(375\) −9.78499 11.6796i −0.505295 0.603134i
\(376\) 0 0
\(377\) 21.4873 1.10665
\(378\) 0 0
\(379\) 31.8728 1.63719 0.818597 0.574368i \(-0.194752\pi\)
0.818597 + 0.574368i \(0.194752\pi\)
\(380\) 0 0
\(381\) −19.9378 23.7983i −1.02145 1.21923i
\(382\) 0 0
\(383\) −24.7719 −1.26579 −0.632893 0.774239i \(-0.718133\pi\)
−0.632893 + 0.774239i \(0.718133\pi\)
\(384\) 0 0
\(385\) −14.8042 + 8.35886i −0.754494 + 0.426007i
\(386\) 0 0
\(387\) 13.3266 4.82715i 0.677431 0.245378i
\(388\) 0 0
\(389\) 23.0227i 1.16729i −0.812007 0.583647i \(-0.801625\pi\)
0.812007 0.583647i \(-0.198375\pi\)
\(390\) 0 0
\(391\) 19.5608 + 11.2934i 0.989232 + 0.571133i
\(392\) 0 0
\(393\) −18.8767 22.5318i −0.952204 1.13658i
\(394\) 0 0
\(395\) 3.25577 5.63916i 0.163816 0.283737i
\(396\) 0 0
\(397\) 23.9247 13.8129i 1.20075 0.693251i 0.240025 0.970767i \(-0.422844\pi\)
0.960721 + 0.277515i \(0.0895109\pi\)
\(398\) 0 0
\(399\) 1.80862 + 10.9180i 0.0905442 + 0.546585i
\(400\) 0 0
\(401\) 5.32384i 0.265860i 0.991125 + 0.132930i \(0.0424385\pi\)
−0.991125 + 0.132930i \(0.957561\pi\)
\(402\) 0 0
\(403\) −15.6563 −0.779898
\(404\) 0 0
\(405\) −29.8541 10.9708i −1.48346 0.545143i
\(406\) 0 0
\(407\) 2.85137 1.64624i 0.141337 0.0816009i
\(408\) 0 0
\(409\) 6.11912 3.53288i 0.302571 0.174690i −0.341026 0.940054i \(-0.610774\pi\)
0.643597 + 0.765364i \(0.277441\pi\)
\(410\) 0 0
\(411\) −7.38380 + 6.18602i −0.364216 + 0.305134i
\(412\) 0 0
\(413\) −11.8505 + 20.0782i −0.583124 + 0.987985i
\(414\) 0 0
\(415\) −19.8522 34.3851i −0.974507 1.68790i
\(416\) 0 0
\(417\) 4.30496 + 24.5256i 0.210815 + 1.20102i
\(418\) 0 0
\(419\) −1.94504 + 3.36891i −0.0950215 + 0.164582i −0.909618 0.415447i \(-0.863625\pi\)
0.814596 + 0.580029i \(0.196959\pi\)
\(420\) 0 0
\(421\) −2.18533 3.78510i −0.106506 0.184475i 0.807846 0.589393i \(-0.200633\pi\)
−0.914353 + 0.404919i \(0.867300\pi\)
\(422\) 0 0
\(423\) 18.1810 + 15.3037i 0.883992 + 0.744092i
\(424\) 0 0
\(425\) −21.9759 + 38.0634i −1.06599 + 1.84635i
\(426\) 0 0
\(427\) 0.263371 27.4467i 0.0127454 1.32824i
\(428\) 0 0
\(429\) 9.96738 1.74957i 0.481230 0.0844698i
\(430\) 0 0
\(431\) 10.7505 + 6.20681i 0.517834 + 0.298971i 0.736048 0.676929i \(-0.236690\pi\)
−0.218214 + 0.975901i \(0.570023\pi\)
\(432\) 0 0
\(433\) 25.2979i 1.21574i 0.794036 + 0.607870i \(0.207976\pi\)
−0.794036 + 0.607870i \(0.792024\pi\)
\(434\) 0 0
\(435\) −38.4523 14.0292i −1.84365 0.672649i
\(436\) 0 0
\(437\) 4.64728 + 8.04932i 0.222309 + 0.385051i
\(438\) 0 0
\(439\) −23.2408 13.4181i −1.10922 0.640410i −0.170595 0.985341i \(-0.554569\pi\)
−0.938628 + 0.344931i \(0.887902\pi\)
\(440\) 0 0
\(441\) −15.8036 13.8292i −0.752550 0.658535i
\(442\) 0 0
\(443\) −11.3051 6.52699i −0.537120 0.310107i 0.206791 0.978385i \(-0.433698\pi\)
−0.743911 + 0.668279i \(0.767031\pi\)
\(444\) 0 0
\(445\) −24.3773 42.2227i −1.15560 2.00155i
\(446\) 0 0
\(447\) −8.52526 + 7.14230i −0.403231 + 0.337819i
\(448\) 0 0
\(449\) 0.468645i 0.0221167i −0.999939 0.0110584i \(-0.996480\pi\)
0.999939 0.0110584i \(-0.00352006\pi\)
\(450\) 0 0
\(451\) 15.8460 + 9.14868i 0.746158 + 0.430795i
\(452\) 0 0
\(453\) −17.7887 21.2331i −0.835787 0.997618i
\(454\) 0 0
\(455\) −0.288286 + 30.0433i −0.0135151 + 1.40845i
\(456\) 0 0
\(457\) 2.06486 3.57644i 0.0965901 0.167299i −0.813681 0.581312i \(-0.802540\pi\)
0.910271 + 0.414013i \(0.135873\pi\)
\(458\) 0 0
\(459\) −0.0707962 + 30.4945i −0.00330448 + 1.42336i
\(460\) 0 0
\(461\) 12.9115 + 22.3633i 0.601347 + 1.04156i 0.992617 + 0.121288i \(0.0387023\pi\)
−0.391271 + 0.920276i \(0.627964\pi\)
\(462\) 0 0
\(463\) −2.56402 + 4.44101i −0.119160 + 0.206391i −0.919435 0.393242i \(-0.871353\pi\)
0.800275 + 0.599633i \(0.204687\pi\)
\(464\) 0 0
\(465\) 28.0176 + 10.2221i 1.29929 + 0.474041i
\(466\) 0 0
\(467\) −1.26091 2.18396i −0.0583480 0.101062i 0.835376 0.549679i \(-0.185250\pi\)
−0.893724 + 0.448617i \(0.851917\pi\)
\(468\) 0 0
\(469\) −10.5536 + 17.8809i −0.487319 + 0.825662i
\(470\) 0 0
\(471\) 22.6613 + 8.26791i 1.04418 + 0.380965i
\(472\) 0 0
\(473\) 7.43976 4.29535i 0.342081 0.197500i
\(474\) 0 0
\(475\) −15.6632 + 9.04316i −0.718677 + 0.414929i
\(476\) 0 0
\(477\) −5.05312 + 6.00318i −0.231366 + 0.274867i
\(478\) 0 0
\(479\) −11.4924 −0.525103 −0.262552 0.964918i \(-0.584564\pi\)
−0.262552 + 0.964918i \(0.584564\pi\)
\(480\) 0 0
\(481\) 5.81854i 0.265303i
\(482\) 0 0
\(483\) −16.5099 6.20374i −0.751228 0.282280i
\(484\) 0 0
\(485\) −48.7629 + 28.1533i −2.21421 + 1.27837i
\(486\) 0 0
\(487\) −10.8463 + 18.7864i −0.491493 + 0.851292i −0.999952 0.00979483i \(-0.996882\pi\)
0.508459 + 0.861086i \(0.330215\pi\)
\(488\) 0 0
\(489\) 9.25506 25.3670i 0.418528 1.14713i
\(490\) 0 0
\(491\) −19.3592 11.1771i −0.873669 0.504413i −0.00510349 0.999987i \(-0.501624\pi\)
−0.868566 + 0.495574i \(0.834958\pi\)
\(492\) 0 0
\(493\) 39.2438i 1.76745i
\(494\) 0 0
\(495\) −18.9793 3.37686i −0.853057 0.151779i
\(496\) 0 0
\(497\) 17.5775 9.92470i 0.788457 0.445184i
\(498\) 0 0
\(499\) 17.5246 0.784510 0.392255 0.919857i \(-0.371695\pi\)
0.392255 + 0.919857i \(0.371695\pi\)
\(500\) 0 0
\(501\) 22.4754 3.94508i 1.00413 0.176253i
\(502\) 0 0
\(503\) −7.60926 −0.339280 −0.169640 0.985506i \(-0.554261\pi\)
−0.169640 + 0.985506i \(0.554261\pi\)
\(504\) 0 0
\(505\) 39.3159 1.74953
\(506\) 0 0
\(507\) −1.58787 + 4.35215i −0.0705197 + 0.193286i
\(508\) 0 0
\(509\) −6.78746 −0.300849 −0.150424 0.988622i \(-0.548064\pi\)
−0.150424 + 0.988622i \(0.548064\pi\)
\(510\) 0 0
\(511\) 0.344945 35.9479i 0.0152595 1.59024i
\(512\) 0 0
\(513\) −6.29951 + 10.8528i −0.278130 + 0.479163i
\(514\) 0 0
\(515\) 2.13111i 0.0939078i
\(516\) 0 0
\(517\) 12.4738 + 7.20174i 0.548596 + 0.316732i
\(518\) 0 0
\(519\) 19.1206 3.35622i 0.839300 0.147322i
\(520\) 0 0
\(521\) −13.8435 + 23.9777i −0.606497 + 1.05048i 0.385316 + 0.922785i \(0.374092\pi\)
−0.991813 + 0.127698i \(0.959241\pi\)
\(522\) 0 0
\(523\) −0.834923 + 0.482043i −0.0365086 + 0.0210783i −0.518143 0.855294i \(-0.673377\pi\)
0.481635 + 0.876372i \(0.340043\pi\)
\(524\) 0 0
\(525\) 12.0719 32.1268i 0.526861 1.40213i
\(526\) 0 0
\(527\) 28.5943i 1.24559i
\(528\) 0 0
\(529\) 8.18740 0.355974
\(530\) 0 0
\(531\) −24.8559 + 9.00327i −1.07865 + 0.390709i
\(532\) 0 0
\(533\) 28.0034 16.1678i 1.21296 0.700304i
\(534\) 0 0
\(535\) −40.0552 + 23.1259i −1.73174 + 0.999820i
\(536\) 0 0
\(537\) −3.19073 18.1778i −0.137690 0.784428i
\(538\) 0 0
\(539\) −10.8985 6.57430i −0.469433 0.283175i
\(540\) 0 0
\(541\) 2.36867 + 4.10266i 0.101837 + 0.176387i 0.912442 0.409207i \(-0.134195\pi\)
−0.810604 + 0.585594i \(0.800861\pi\)
\(542\) 0 0
\(543\) 15.7858 13.2251i 0.677435 0.567542i
\(544\) 0 0
\(545\) −4.84374 + 8.38960i −0.207483 + 0.359371i
\(546\) 0 0
\(547\) −0.840875 1.45644i −0.0359532 0.0622728i 0.847489 0.530813i \(-0.178113\pi\)
−0.883442 + 0.468540i \(0.844780\pi\)
\(548\) 0 0
\(549\) 20.0424 23.8107i 0.855390 1.01622i
\(550\) 0 0
\(551\) −8.07446 + 13.9854i −0.343984 + 0.595797i
\(552\) 0 0
\(553\) 4.87467 + 0.0467758i 0.207292 + 0.00198911i
\(554\) 0 0
\(555\) −3.79897 + 10.4125i −0.161257 + 0.441986i
\(556\) 0 0
\(557\) 8.14983 + 4.70530i 0.345319 + 0.199370i 0.662622 0.748954i \(-0.269444\pi\)
−0.317303 + 0.948324i \(0.602777\pi\)
\(558\) 0 0
\(559\) 15.1817i 0.642117i
\(560\) 0 0
\(561\) 3.19536 + 18.2042i 0.134908 + 0.768580i
\(562\) 0 0
\(563\) −15.5319 26.9020i −0.654591 1.13378i −0.981996 0.188901i \(-0.939508\pi\)
0.327405 0.944884i \(-0.393826\pi\)
\(564\) 0 0
\(565\) 23.5056 + 13.5710i 0.988888 + 0.570935i
\(566\) 0 0
\(567\) −3.83694 23.5006i −0.161136 0.986932i
\(568\) 0 0
\(569\) 17.4099 + 10.0516i 0.729859 + 0.421384i 0.818371 0.574691i \(-0.194878\pi\)
−0.0885114 + 0.996075i \(0.528211\pi\)
\(570\) 0 0
\(571\) 3.93707 + 6.81920i 0.164761 + 0.285375i 0.936570 0.350479i \(-0.113981\pi\)
−0.771809 + 0.635854i \(0.780648\pi\)
\(572\) 0 0
\(573\) −0.0670662 0.382080i −0.00280173 0.0159616i
\(574\) 0 0
\(575\) 28.8239i 1.20204i
\(576\) 0 0
\(577\) −2.77842 1.60412i −0.115667 0.0667805i 0.441050 0.897482i \(-0.354606\pi\)
−0.556717 + 0.830702i \(0.687939\pi\)
\(578\) 0 0
\(579\) −15.4793 + 42.4268i −0.643297 + 1.76320i
\(580\) 0 0
\(581\) 15.1088 25.5987i 0.626817 1.06201i
\(582\) 0 0
\(583\) −2.37794 + 4.11871i −0.0984840 + 0.170579i
\(584\) 0 0
\(585\) −21.9385 + 26.0633i −0.907046 + 1.07758i
\(586\) 0 0
\(587\) −2.37708 4.11722i −0.0981125 0.169936i 0.812791 0.582556i \(-0.197947\pi\)
−0.910903 + 0.412620i \(0.864614\pi\)
\(588\) 0 0
\(589\) 5.88332 10.1902i 0.242418 0.419880i
\(590\) 0 0
\(591\) 2.07504 1.73843i 0.0853556 0.0715094i
\(592\) 0 0
\(593\) 13.0354 + 22.5780i 0.535300 + 0.927166i 0.999149 + 0.0412519i \(0.0131346\pi\)
−0.463849 + 0.885914i \(0.653532\pi\)
\(594\) 0 0
\(595\) −54.8703 0.526519i −2.24946 0.0215852i
\(596\) 0 0
\(597\) −4.65840 26.5392i −0.190656 1.08618i
\(598\) 0 0
\(599\) −23.8178 + 13.7512i −0.973168 + 0.561859i −0.900200 0.435476i \(-0.856580\pi\)
−0.0729672 + 0.997334i \(0.523247\pi\)
\(600\) 0 0
\(601\) −26.8647 + 15.5104i −1.09584 + 0.632681i −0.935124 0.354320i \(-0.884712\pi\)
−0.160712 + 0.987001i \(0.551379\pi\)
\(602\) 0 0
\(603\) −22.1357 + 8.01795i −0.901435 + 0.326516i
\(604\) 0 0
\(605\) 27.1903 1.10544
\(606\) 0 0
\(607\) 39.2304i 1.59231i 0.605092 + 0.796156i \(0.293136\pi\)
−0.605092 + 0.796156i \(0.706864\pi\)
\(608\) 0 0
\(609\) −5.00801 30.2316i −0.202935 1.22505i
\(610\) 0 0
\(611\) 22.0440 12.7271i 0.891803 0.514883i
\(612\) 0 0
\(613\) −11.5364 + 19.9817i −0.465951 + 0.807051i −0.999244 0.0388795i \(-0.987621\pi\)
0.533293 + 0.845931i \(0.320954\pi\)
\(614\) 0 0
\(615\) −60.6692 + 10.6492i −2.44642 + 0.429418i
\(616\) 0 0
\(617\) 5.15715 + 2.97748i 0.207619 + 0.119869i 0.600204 0.799847i \(-0.295086\pi\)
−0.392585 + 0.919716i \(0.628419\pi\)
\(618\) 0 0
\(619\) 32.3650i 1.30086i −0.759567 0.650429i \(-0.774589\pi\)
0.759567 0.650429i \(-0.225411\pi\)
\(620\) 0 0
\(621\) −9.95902 17.3424i −0.399642 0.695926i
\(622\) 0 0
\(623\) 18.5526 31.4337i 0.743296 1.25936i
\(624\) 0 0
\(625\) −6.35758 −0.254303
\(626\) 0 0
\(627\) −2.60679 + 7.14490i −0.104105 + 0.285340i
\(628\) 0 0
\(629\) 10.6268 0.423719
\(630\) 0 0
\(631\) −35.6689 −1.41996 −0.709978 0.704224i \(-0.751295\pi\)
−0.709978 + 0.704224i \(0.751295\pi\)
\(632\) 0 0
\(633\) −23.8227 + 4.18158i −0.946868 + 0.166203i
\(634\) 0 0
\(635\) 63.3459 2.51380
\(636\) 0 0
\(637\) −19.6918 + 10.8707i −0.780218 + 0.430711i
\(638\) 0 0
\(639\) 22.5347 + 4.00944i 0.891457 + 0.158611i
\(640\) 0 0
\(641\) 24.8536i 0.981659i −0.871256 0.490829i \(-0.836694\pi\)
0.871256 0.490829i \(-0.163306\pi\)
\(642\) 0 0
\(643\) 14.8270 + 8.56038i 0.584720 + 0.337588i 0.763007 0.646390i \(-0.223722\pi\)
−0.178287 + 0.983979i \(0.557056\pi\)
\(644\) 0 0
\(645\) −9.91224 + 27.1682i −0.390294 + 1.06975i
\(646\) 0 0
\(647\) 16.7208 28.9613i 0.657363 1.13859i −0.323933 0.946080i \(-0.605005\pi\)
0.981296 0.192506i \(-0.0616614\pi\)
\(648\) 0 0
\(649\) −13.8761 + 8.01138i −0.544686 + 0.314474i
\(650\) 0 0
\(651\) 3.64900 + 22.0278i 0.143016 + 0.863337i
\(652\) 0 0
\(653\) 26.4238i 1.03404i 0.855972 + 0.517022i \(0.172959\pi\)
−0.855972 + 0.517022i \(0.827041\pi\)
\(654\) 0 0
\(655\) 59.9745 2.34340
\(656\) 0 0
\(657\) 26.2502 31.1856i 1.02412 1.21667i
\(658\) 0 0
\(659\) 25.8862 14.9454i 1.00838 0.582191i 0.0976658 0.995219i \(-0.468862\pi\)
0.910718 + 0.413029i \(0.135529\pi\)
\(660\) 0 0
\(661\) −3.92629 + 2.26684i −0.152715 + 0.0881699i −0.574410 0.818568i \(-0.694769\pi\)
0.421695 + 0.906738i \(0.361435\pi\)
\(662\) 0 0
\(663\) 30.6842 + 11.1950i 1.19168 + 0.434779i
\(664\) 0 0
\(665\) −19.4459 11.4773i −0.754079 0.445069i
\(666\) 0 0
\(667\) −12.8682 22.2883i −0.498257 0.863006i
\(668\) 0 0
\(669\) 40.2529 + 14.6862i 1.55627 + 0.567800i
\(670\) 0 0
\(671\) 9.43172 16.3362i 0.364107 0.630652i
\(672\) 0 0
\(673\) 4.12239 + 7.14019i 0.158907 + 0.275234i 0.934475 0.356030i \(-0.115870\pi\)
−0.775568 + 0.631264i \(0.782536\pi\)
\(674\) 0 0
\(675\) 33.7466 19.3793i 1.29891 0.745910i
\(676\) 0 0
\(677\) 2.75631 4.77407i 0.105934 0.183483i −0.808186 0.588928i \(-0.799550\pi\)
0.914119 + 0.405445i \(0.132884\pi\)
\(678\) 0 0
\(679\) −36.3026 21.4264i −1.39317 0.822268i
\(680\) 0 0
\(681\) 6.94253 + 8.28680i 0.266038 + 0.317551i
\(682\) 0 0
\(683\) 24.0097 + 13.8620i 0.918706 + 0.530415i 0.883222 0.468955i \(-0.155369\pi\)
0.0354840 + 0.999370i \(0.488703\pi\)
\(684\) 0 0
\(685\) 19.6540i 0.750942i
\(686\) 0 0
\(687\) 21.0537 17.6384i 0.803248 0.672947i
\(688\) 0 0
\(689\) 4.20235 + 7.27868i 0.160097 + 0.277296i
\(690\) 0 0
\(691\) 36.5312 + 21.0913i 1.38971 + 0.802351i 0.993283 0.115714i \(-0.0369157\pi\)
0.396430 + 0.918065i \(0.370249\pi\)
\(692\) 0 0
\(693\) −4.78464 13.6159i −0.181754 0.517225i
\(694\) 0 0
\(695\) −43.9994 25.4031i −1.66899 0.963593i
\(696\) 0 0
\(697\) 29.5284 + 51.1447i 1.11847 + 1.93724i
\(698\) 0 0
\(699\) −5.53461 2.01929i −0.209338 0.0763764i
\(700\) 0 0
\(701\) 1.07738i 0.0406922i 0.999793 + 0.0203461i \(0.00647681\pi\)
−0.999793 + 0.0203461i \(0.993523\pi\)
\(702\) 0 0
\(703\) 3.78710 + 2.18648i 0.142833 + 0.0824648i
\(704\) 0 0
\(705\) −47.7581 + 8.38294i −1.79867 + 0.315720i
\(706\) 0 0
\(707\) 14.4717 + 25.6306i 0.544265 + 0.963938i
\(708\) 0 0
\(709\) 10.9806 19.0189i 0.412385 0.714271i −0.582765 0.812640i \(-0.698029\pi\)
0.995150 + 0.0983694i \(0.0313627\pi\)
\(710\) 0 0
\(711\) 4.22889 + 3.55963i 0.158596 + 0.133496i
\(712\) 0 0
\(713\) 9.37617 + 16.2400i 0.351140 + 0.608193i
\(714\) 0 0
\(715\) −10.3240 + 17.8817i −0.386095 + 0.668737i
\(716\) 0 0
\(717\) 5.65275 + 32.2041i 0.211106 + 1.20268i
\(718\) 0 0
\(719\) −5.89267 10.2064i −0.219760 0.380635i 0.734975 0.678094i \(-0.237194\pi\)
−0.954734 + 0.297460i \(0.903861\pi\)
\(720\) 0 0
\(721\) 1.38930 0.784436i 0.0517403 0.0292139i
\(722\) 0 0
\(723\) −14.8496 + 12.4407i −0.552263 + 0.462676i
\(724\) 0 0
\(725\) 43.3709 25.0402i 1.61075 0.929970i
\(726\) 0 0
\(727\) −10.1963 + 5.88682i −0.378159 + 0.218330i −0.677017 0.735967i \(-0.736728\pi\)
0.298858 + 0.954298i \(0.403394\pi\)
\(728\) 0 0
\(729\) 13.6084 23.3198i 0.504016 0.863694i
\(730\) 0 0
\(731\) 27.7274 1.02554
\(732\) 0 0
\(733\) 43.7655i 1.61652i 0.588829 + 0.808258i \(0.299589\pi\)
−0.588829 + 0.808258i \(0.700411\pi\)
\(734\) 0 0
\(735\) 42.3368 6.59654i 1.56162 0.243317i
\(736\) 0 0
\(737\) −12.3575 + 7.13462i −0.455195 + 0.262807i
\(738\) 0 0
\(739\) 13.9728 24.2016i 0.513998 0.890270i −0.485870 0.874031i \(-0.661497\pi\)
0.999868 0.0162393i \(-0.00516935\pi\)
\(740\) 0 0
\(741\) 8.63159 + 10.3029i 0.317089 + 0.378487i
\(742\) 0 0
\(743\) 31.0951 + 17.9528i 1.14077 + 0.658623i 0.946621 0.322350i \(-0.104473\pi\)
0.194147 + 0.980972i \(0.437806\pi\)
\(744\) 0 0
\(745\) 22.6923i 0.831382i
\(746\) 0 0
\(747\) 31.6900 11.4787i 1.15948 0.419984i
\(748\) 0 0
\(749\) −29.8200 17.6002i −1.08960 0.643099i
\(750\) 0 0
\(751\) −2.35031 −0.0857642 −0.0428821 0.999080i \(-0.513654\pi\)
−0.0428821 + 0.999080i \(0.513654\pi\)
\(752\) 0 0
\(753\) −17.1894 20.5178i −0.626417 0.747709i
\(754\) 0 0
\(755\) 56.5178 2.05689
\(756\) 0 0
\(757\) 8.88061 0.322771 0.161386 0.986891i \(-0.448404\pi\)
0.161386 + 0.986891i \(0.448404\pi\)
\(758\) 0 0
\(759\) −7.78399 9.29119i −0.282541 0.337249i
\(760\) 0 0
\(761\) 26.9766 0.977902 0.488951 0.872311i \(-0.337380\pi\)
0.488951 + 0.872311i \(0.337380\pi\)
\(762\) 0 0
\(763\) −7.25223 0.0695902i −0.262548 0.00251933i
\(764\) 0 0
\(765\) −47.6013 40.0679i −1.72103 1.44866i
\(766\) 0 0
\(767\) 28.3158i 1.02243i
\(768\) 0 0
\(769\) 7.88232 + 4.55086i 0.284244 + 0.164108i 0.635343 0.772230i \(-0.280859\pi\)
−0.351099 + 0.936338i \(0.614192\pi\)
\(770\) 0 0
\(771\) 11.8403 + 14.1329i 0.426419 + 0.508985i
\(772\) 0 0
\(773\) −6.46864 + 11.2040i −0.232661 + 0.402980i −0.958590 0.284789i \(-0.908077\pi\)
0.725929 + 0.687769i \(0.241410\pi\)
\(774\) 0 0
\(775\) −31.6015 + 18.2451i −1.13516 + 0.655384i
\(776\) 0 0
\(777\) −8.18643 + 1.35612i −0.293686 + 0.0486505i
\(778\) 0 0
\(779\) 24.3020i 0.870710i
\(780\) 0 0
\(781\) 13.8726 0.496399
\(782\) 0 0
\(783\) 17.4431 30.0510i 0.623366 1.07394i
\(784\) 0 0
\(785\) −42.6245 + 24.6093i −1.52134 + 0.878343i
\(786\) 0 0
\(787\) 21.5265 12.4283i 0.767336 0.443022i −0.0645875 0.997912i \(-0.520573\pi\)
0.831923 + 0.554890i \(0.187240\pi\)
\(788\) 0 0
\(789\) −14.7208 + 12.3328i −0.524073 + 0.439059i
\(790\) 0 0
\(791\) −0.194975 + 20.3190i −0.00693250 + 0.722460i
\(792\) 0 0
\(793\) −16.6680 28.8697i −0.591896 1.02519i
\(794\) 0 0
\(795\) −2.76796 15.7692i −0.0981693 0.559277i
\(796\) 0 0
\(797\) 6.71671 11.6337i 0.237918 0.412086i −0.722199 0.691686i \(-0.756868\pi\)
0.960117 + 0.279600i \(0.0902017\pi\)
\(798\) 0 0
\(799\) 23.2444 + 40.2605i 0.822328 + 1.42431i
\(800\) 0 0
\(801\) 38.9134 14.0951i 1.37494 0.498028i
\(802\) 0 0
\(803\) 12.3530 21.3961i 0.435928 0.755050i
\(804\) 0 0
\(805\) 31.3359 17.6931i 1.10445 0.623600i
\(806\) 0 0
\(807\) 10.7185 1.88141i 0.377309 0.0662288i
\(808\) 0 0
\(809\) −9.38524 5.41857i −0.329967 0.190507i 0.325859 0.945418i \(-0.394346\pi\)
−0.655827 + 0.754912i \(0.727680\pi\)
\(810\) 0 0
\(811\) 35.3334i 1.24072i 0.784316 + 0.620361i \(0.213014\pi\)
−0.784316 + 0.620361i \(0.786986\pi\)
\(812\) 0 0
\(813\) −1.79343 0.654327i −0.0628983 0.0229483i
\(814\) 0 0
\(815\) 27.5475 + 47.7137i 0.964948 + 1.67134i
\(816\) 0 0
\(817\) 9.88127 + 5.70496i 0.345702 + 0.199591i
\(818\) 0 0
\(819\) −25.0664 4.70846i −0.875890 0.164527i
\(820\) 0 0
\(821\) −16.6412 9.60782i −0.580783 0.335315i 0.180661 0.983545i \(-0.442176\pi\)
−0.761445 + 0.648230i \(0.775510\pi\)
\(822\) 0 0
\(823\) −14.9074 25.8204i −0.519639 0.900041i −0.999739 0.0228278i \(-0.992733\pi\)
0.480100 0.877214i \(-0.340600\pi\)
\(824\) 0 0
\(825\) 18.0798 15.1469i 0.629456 0.527347i
\(826\) 0 0
\(827\) 0.356137i 0.0123841i −0.999981 0.00619205i \(-0.998029\pi\)
0.999981 0.00619205i \(-0.00197100\pi\)
\(828\) 0 0
\(829\) −17.5571 10.1366i −0.609783 0.352058i 0.163098 0.986610i \(-0.447851\pi\)
−0.772880 + 0.634552i \(0.781185\pi\)
\(830\) 0 0
\(831\) 17.0869 + 20.3955i 0.592739 + 0.707510i
\(832\) 0 0
\(833\) −19.8539 35.9646i −0.687896 1.24610i
\(834\) 0 0
\(835\) −23.2795 + 40.3213i −0.805620 + 1.39538i
\(836\) 0 0
\(837\) −12.7096 + 21.8962i −0.439309 + 0.756843i
\(838\) 0 0
\(839\) −0.662663 1.14777i −0.0228777 0.0396253i 0.854360 0.519682i \(-0.173949\pi\)
−0.877238 + 0.480056i \(0.840616\pi\)
\(840\) 0 0
\(841\) 7.85791 13.6103i 0.270962 0.469321i
\(842\) 0 0
\(843\) −35.5353 12.9649i −1.22390 0.446536i
\(844\) 0 0
\(845\) −4.72626 8.18612i −0.162588 0.281611i
\(846\) 0 0
\(847\) 10.0084 + 17.7258i 0.343894 + 0.609064i
\(848\) 0 0
\(849\) 18.0262 + 6.57681i 0.618659 + 0.225716i
\(850\) 0 0
\(851\) −6.03545 + 3.48457i −0.206893 + 0.119449i
\(852\) 0 0
\(853\) −41.7558 + 24.1077i −1.42969 + 0.825432i −0.997096 0.0761571i \(-0.975735\pi\)
−0.432594 + 0.901589i \(0.642402\pi\)
\(854\) 0 0
\(855\) −8.71972 24.0731i −0.298208 0.823282i
\(856\) 0 0
\(857\) −20.8773 −0.713155 −0.356578 0.934266i \(-0.616056\pi\)
−0.356578 + 0.934266i \(0.616056\pi\)
\(858\) 0 0
\(859\) 13.7094i 0.467758i −0.972266 0.233879i \(-0.924858\pi\)
0.972266 0.233879i \(-0.0751419\pi\)
\(860\) 0 0
\(861\) −29.2740 35.6313i −0.997657 1.21431i
\(862\) 0 0
\(863\) 28.1125 16.2307i 0.956959 0.552501i 0.0617233 0.998093i \(-0.480340\pi\)
0.895236 + 0.445593i \(0.147007\pi\)
\(864\) 0 0
\(865\) −19.8047 + 34.3027i −0.673379 + 1.16633i
\(866\) 0 0
\(867\) −10.3542 + 28.3795i −0.351646 + 0.963819i
\(868\) 0 0
\(869\) 2.90139 + 1.67512i 0.0984228 + 0.0568244i
\(870\) 0 0
\(871\) 25.2169i 0.854444i
\(872\) 0 0
\(873\) −16.2784 44.9409i −0.550941 1.52102i
\(874\) 0 0
\(875\) 11.4434 + 20.2671i 0.386856 + 0.685154i
\(876\) 0 0
\(877\) 21.9338 0.740653 0.370326 0.928902i \(-0.379246\pi\)
0.370326 + 0.928902i \(0.379246\pi\)
\(878\) 0 0
\(879\) −43.1030 + 7.56583i −1.45383 + 0.255189i
\(880\) 0 0
\(881\) 21.8019 0.734526 0.367263 0.930117i \(-0.380295\pi\)
0.367263 + 0.930117i \(0.380295\pi\)
\(882\) 0 0
\(883\) −24.4085 −0.821411 −0.410705 0.911768i \(-0.634717\pi\)
−0.410705 + 0.911768i \(0.634717\pi\)
\(884\) 0 0
\(885\) 18.4876 50.6723i 0.621454 1.70333i
\(886\) 0 0
\(887\) −33.7444 −1.13303 −0.566513 0.824053i \(-0.691708\pi\)
−0.566513 + 0.824053i \(0.691708\pi\)
\(888\) 0 0
\(889\) 23.3169 + 41.2961i 0.782024 + 1.38503i
\(890\) 0 0
\(891\) 5.64455 15.3602i 0.189099 0.514585i
\(892\) 0 0
\(893\) 19.1303i 0.640170i
\(894\) 0 0
\(895\) 32.6113 + 18.8281i 1.09007 + 0.629355i
\(896\) 0 0
\(897\) −21.0978 + 3.70328i −0.704436 + 0.123649i
\(898\) 0 0
\(899\) −16.2907 + 28.2164i −0.543326 + 0.941068i
\(900\) 0 0
\(901\) −13.2936 + 7.67505i −0.442873 + 0.255693i
\(902\) 0 0
\(903\) −21.3600 + 3.53837i −0.710815 + 0.117750i
\(904\) 0 0
\(905\) 42.0183i 1.39674i
\(906\) 0 0
\(907\) 15.0566 0.499948 0.249974 0.968253i \(-0.419578\pi\)
0.249974 + 0.968253i \(0.419578\pi\)
\(908\) 0 0
\(909\) −5.84637 + 32.8589i −0.193912 + 1.08986i
\(910\) 0 0
\(911\) −21.5047 + 12.4157i −0.712481 + 0.411351i −0.811979 0.583686i \(-0.801610\pi\)
0.0994977 + 0.995038i \(0.468276\pi\)
\(912\) 0 0
\(913\) 17.6913 10.2141i 0.585498 0.338038i
\(914\) 0 0
\(915\) 10.9787 + 62.5462i 0.362944 + 2.06771i
\(916\) 0 0
\(917\) 22.0759 + 39.0983i 0.729012 + 1.29114i
\(918\) 0 0
\(919\) −12.0689 20.9039i −0.398115 0.689555i 0.595378 0.803445i \(-0.297002\pi\)
−0.993493 + 0.113890i \(0.963669\pi\)
\(920\) 0 0
\(921\) −16.2194 + 13.5883i −0.534447 + 0.447750i
\(922\) 0 0
\(923\) 12.2580 21.2314i 0.403476 0.698840i
\(924\) 0 0
\(925\) −6.78064 11.7444i −0.222946 0.386154i
\(926\) 0 0
\(927\) 1.78111 + 0.316901i 0.0584994 + 0.0104084i
\(928\) 0 0
\(929\) −13.3310 + 23.0900i −0.437376 + 0.757558i −0.997486 0.0708606i \(-0.977425\pi\)
0.560110 + 0.828418i \(0.310759\pi\)
\(930\) 0 0
\(931\) 0.324397 16.9017i 0.0106317 0.553932i
\(932\) 0 0
\(933\) −5.63360 + 15.4410i −0.184436 + 0.505515i
\(934\) 0 0
\(935\) −32.6586 18.8555i −1.06805 0.616640i
\(936\) 0 0
\(937\) 8.17165i 0.266956i 0.991052 + 0.133478i \(0.0426146\pi\)
−0.991052 + 0.133478i \(0.957385\pi\)
\(938\) 0 0
\(939\) −4.05790 23.1181i −0.132424 0.754430i
\(940\) 0 0
\(941\) 16.0136 + 27.7363i 0.522028 + 0.904179i 0.999672 + 0.0256254i \(0.00815770\pi\)
−0.477644 + 0.878554i \(0.658509\pi\)
\(942\) 0 0
\(943\) −33.5410 19.3649i −1.09224 0.630608i
\(944\) 0 0
\(945\) 41.7830 + 24.7920i 1.35920 + 0.806483i
\(946\) 0 0
\(947\) −15.5020 8.95009i −0.503748 0.290839i 0.226512 0.974008i \(-0.427268\pi\)
−0.730260 + 0.683169i \(0.760601\pi\)
\(948\) 0 0
\(949\) −21.8305 37.8116i −0.708650 1.22742i
\(950\) 0 0
\(951\) −4.75640 27.0975i −0.154237 0.878697i
\(952\) 0 0
\(953\) 11.2737i 0.365191i −0.983188 0.182596i \(-0.941550\pi\)
0.983188 0.182596i \(-0.0584499\pi\)
\(954\) 0 0
\(955\) 0.685459 + 0.395750i 0.0221809 + 0.0128062i
\(956\) 0 0
\(957\) 7.21812 19.7840i 0.233329 0.639525i
\(958\) 0 0
\(959\) 12.8128 7.23442i 0.413746 0.233612i
\(960\) 0 0
\(961\) −3.63003 + 6.28740i −0.117098 + 0.202819i
\(962\) 0 0
\(963\) −13.3716 36.9158i −0.430893 1.18959i
\(964\) 0 0
\(965\) −46.0738 79.8022i −1.48317 2.56892i
\(966\) 0 0
\(967\) 15.3313 26.5546i 0.493022 0.853939i −0.506946 0.861978i \(-0.669226\pi\)
0.999968 + 0.00803913i \(0.00255896\pi\)
\(968\) 0 0
\(969\) −18.8170 + 15.7645i −0.604488 + 0.506429i
\(970\) 0 0
\(971\) 19.0807 + 33.0488i 0.612330 + 1.06059i 0.990847 + 0.134992i \(0.0431010\pi\)
−0.378517 + 0.925595i \(0.623566\pi\)
\(972\) 0 0
\(973\) 0.364967 38.0345i 0.0117003 1.21933i
\(974\) 0 0
\(975\) −7.20624 41.0543i −0.230784 1.31479i
\(976\) 0 0
\(977\) 26.5862 15.3496i 0.850568 0.491076i −0.0102745 0.999947i \(-0.503271\pi\)
0.860842 + 0.508872i \(0.169937\pi\)
\(978\) 0 0
\(979\) 21.7239 12.5423i 0.694299 0.400854i
\(980\) 0 0
\(981\) −6.29148 5.29580i −0.200872 0.169082i
\(982\) 0 0
\(983\) 25.8057 0.823073 0.411536 0.911393i \(-0.364992\pi\)
0.411536 + 0.911393i \(0.364992\pi\)
\(984\) 0 0
\(985\) 5.52328i 0.175986i
\(986\) 0 0
\(987\) −23.0442 28.0486i −0.733505 0.892796i
\(988\) 0 0
\(989\) −15.7476 + 9.09191i −0.500746 + 0.289106i
\(990\) 0 0
\(991\) −11.4388 + 19.8126i −0.363366 + 0.629369i −0.988513 0.151139i \(-0.951706\pi\)
0.625146 + 0.780508i \(0.285039\pi\)
\(992\) 0 0
\(993\) 18.7212 3.28612i 0.594100 0.104282i
\(994\) 0 0
\(995\) 47.6118 + 27.4887i 1.50940 + 0.871450i
\(996\) 0 0
\(997\) 41.5407i 1.31561i −0.753189 0.657804i \(-0.771486\pi\)
0.753189 0.657804i \(-0.228514\pi\)
\(998\) 0 0
\(999\) −8.13752 4.72342i −0.257460 0.149443i
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 504.2.cx.a.185.7 yes 48
3.2 odd 2 1512.2.cx.a.17.3 48
4.3 odd 2 1008.2.df.e.689.18 48
7.5 odd 6 504.2.bs.a.257.1 48
9.2 odd 6 504.2.bs.a.353.1 yes 48
9.7 even 3 1512.2.bs.a.521.3 48
12.11 even 2 3024.2.df.e.17.3 48
21.5 even 6 1512.2.bs.a.1097.3 48
28.19 even 6 1008.2.ca.e.257.24 48
36.7 odd 6 3024.2.ca.e.2033.3 48
36.11 even 6 1008.2.ca.e.353.24 48
63.47 even 6 inner 504.2.cx.a.425.7 yes 48
63.61 odd 6 1512.2.cx.a.89.3 48
84.47 odd 6 3024.2.ca.e.2609.3 48
252.47 odd 6 1008.2.df.e.929.18 48
252.187 even 6 3024.2.df.e.1601.3 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.bs.a.257.1 48 7.5 odd 6
504.2.bs.a.353.1 yes 48 9.2 odd 6
504.2.cx.a.185.7 yes 48 1.1 even 1 trivial
504.2.cx.a.425.7 yes 48 63.47 even 6 inner
1008.2.ca.e.257.24 48 28.19 even 6
1008.2.ca.e.353.24 48 36.11 even 6
1008.2.df.e.689.18 48 4.3 odd 2
1008.2.df.e.929.18 48 252.47 odd 6
1512.2.bs.a.521.3 48 9.7 even 3
1512.2.bs.a.1097.3 48 21.5 even 6
1512.2.cx.a.17.3 48 3.2 odd 2
1512.2.cx.a.89.3 48 63.61 odd 6
3024.2.ca.e.2033.3 48 36.7 odd 6
3024.2.ca.e.2609.3 48 84.47 odd 6
3024.2.df.e.17.3 48 12.11 even 2
3024.2.df.e.1601.3 48 252.187 even 6