Properties

Label 1520.2.q.l.961.2
Level $1520$
Weight $2$
Character 1520.961
Analytic conductor $12.137$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1520,2,Mod(881,1520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1520.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1520.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.1372611072\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{6} - 2x^{5} + 48x^{4} - 7x^{3} + 8x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 760)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.2
Root \(0.232322 + 0.402393i\) of defining polynomial
Character \(\chi\) \(=\) 1520.961
Dual form 1520.2.q.l.881.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.232322 + 0.402393i) q^{3} +(0.500000 - 0.866025i) q^{5} +4.83973 q^{7} +(1.39205 + 2.41111i) q^{9} +O(q^{10})\) \(q+(-0.232322 + 0.402393i) q^{3} +(0.500000 - 0.866025i) q^{5} +4.83973 q^{7} +(1.39205 + 2.41111i) q^{9} +2.46464 q^{11} +(2.23583 + 3.87256i) q^{13} +(0.232322 + 0.402393i) q^{15} +(-1.30842 + 2.26624i) q^{17} +(0.851316 - 4.27496i) q^{19} +(-1.12437 + 1.94747i) q^{21} +(-0.927410 - 1.60632i) q^{23} +(-0.500000 - 0.866025i) q^{25} -2.68755 q^{27} +(2.19696 + 3.80525i) q^{29} -8.61683 q^{31} +(-0.572590 + 0.991755i) q^{33} +(2.41987 - 4.19133i) q^{35} -5.76902 q^{37} -2.07772 q^{39} +(-2.62034 + 4.53856i) q^{41} +(2.74337 - 4.75165i) q^{43} +2.78411 q^{45} +(5.20397 + 9.01355i) q^{47} +16.4230 q^{49} +(-0.607947 - 1.05299i) q^{51} +(-2.94033 - 5.09281i) q^{53} +(1.23232 - 2.13444i) q^{55} +(1.52243 + 1.33573i) q^{57} +(-0.531852 + 0.921195i) q^{59} +(-7.07556 - 12.2552i) q^{61} +(6.73717 + 11.6691i) q^{63} +4.47165 q^{65} +(-5.98458 - 10.3656i) q^{67} +0.861830 q^{69} +(0.201003 - 0.348147i) q^{71} +(-5.49596 + 9.51929i) q^{73} +0.464643 q^{75} +11.9282 q^{77} +(5.76902 - 9.99224i) q^{79} +(-3.55179 + 6.15187i) q^{81} +15.6050 q^{83} +(1.30842 + 2.26624i) q^{85} -2.04161 q^{87} +(3.30438 + 5.72335i) q^{89} +(10.8208 + 18.7422i) q^{91} +(2.00188 - 3.46735i) q^{93} +(-3.27656 - 2.87474i) q^{95} +(4.60741 - 7.98027i) q^{97} +(3.43091 + 5.94252i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5} + 4 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{5} + 4 q^{7} - 2 q^{9} + 16 q^{11} - 3 q^{13} + q^{17} + 7 q^{19} + 6 q^{21} + 2 q^{23} - 4 q^{25} - 6 q^{27} + 12 q^{29} - 46 q^{31} - 14 q^{33} + 2 q^{35} - 4 q^{37} + 14 q^{39} + 21 q^{41} + 9 q^{43} - 4 q^{45} - 2 q^{47} + 60 q^{49} - 18 q^{51} - 5 q^{53} + 8 q^{55} + 25 q^{57} - 27 q^{59} - q^{61} - 11 q^{63} - 6 q^{65} + 3 q^{67} - 50 q^{69} + 23 q^{71} - 17 q^{73} - 4 q^{77} + 4 q^{79} - 4 q^{81} + 44 q^{83} - q^{85} - 84 q^{87} - 12 q^{89} + 20 q^{91} + 18 q^{93} + 8 q^{95} + 4 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(401\) \(1141\) \(1217\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

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
\(3\) −0.232322 + 0.402393i −0.134131 + 0.232322i −0.925265 0.379321i \(-0.876158\pi\)
0.791134 + 0.611643i \(0.209491\pi\)
\(4\) 0 0
\(5\) 0.500000 0.866025i 0.223607 0.387298i
\(6\) 0 0
\(7\) 4.83973 1.82925 0.914624 0.404306i \(-0.132487\pi\)
0.914624 + 0.404306i \(0.132487\pi\)
\(8\) 0 0
\(9\) 1.39205 + 2.41111i 0.464018 + 0.803702i
\(10\) 0 0
\(11\) 2.46464 0.743118 0.371559 0.928409i \(-0.378823\pi\)
0.371559 + 0.928409i \(0.378823\pi\)
\(12\) 0 0
\(13\) 2.23583 + 3.87256i 0.620107 + 1.07406i 0.989465 + 0.144769i \(0.0462440\pi\)
−0.369359 + 0.929287i \(0.620423\pi\)
\(14\) 0 0
\(15\) 0.232322 + 0.402393i 0.0599852 + 0.103897i
\(16\) 0 0
\(17\) −1.30842 + 2.26624i −0.317338 + 0.549645i −0.979932 0.199334i \(-0.936122\pi\)
0.662594 + 0.748979i \(0.269455\pi\)
\(18\) 0 0
\(19\) 0.851316 4.27496i 0.195305 0.980743i
\(20\) 0 0
\(21\) −1.12437 + 1.94747i −0.245359 + 0.424974i
\(22\) 0 0
\(23\) −0.927410 1.60632i −0.193378 0.334941i 0.752989 0.658033i \(-0.228611\pi\)
−0.946368 + 0.323092i \(0.895278\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) −2.68755 −0.517218
\(28\) 0 0
\(29\) 2.19696 + 3.80525i 0.407966 + 0.706618i 0.994662 0.103190i \(-0.0329048\pi\)
−0.586696 + 0.809808i \(0.699571\pi\)
\(30\) 0 0
\(31\) −8.61683 −1.54763 −0.773814 0.633412i \(-0.781654\pi\)
−0.773814 + 0.633412i \(0.781654\pi\)
\(32\) 0 0
\(33\) −0.572590 + 0.991755i −0.0996751 + 0.172642i
\(34\) 0 0
\(35\) 2.41987 4.19133i 0.409032 0.708465i
\(36\) 0 0
\(37\) −5.76902 −0.948421 −0.474211 0.880411i \(-0.657266\pi\)
−0.474211 + 0.880411i \(0.657266\pi\)
\(38\) 0 0
\(39\) −2.07772 −0.332702
\(40\) 0 0
\(41\) −2.62034 + 4.53856i −0.409228 + 0.708803i −0.994803 0.101814i \(-0.967535\pi\)
0.585576 + 0.810618i \(0.300869\pi\)
\(42\) 0 0
\(43\) 2.74337 4.75165i 0.418360 0.724621i −0.577415 0.816451i \(-0.695938\pi\)
0.995775 + 0.0918304i \(0.0292718\pi\)
\(44\) 0 0
\(45\) 2.78411 0.415030
\(46\) 0 0
\(47\) 5.20397 + 9.01355i 0.759078 + 1.31476i 0.943321 + 0.331880i \(0.107683\pi\)
−0.184244 + 0.982881i \(0.558984\pi\)
\(48\) 0 0
\(49\) 16.4230 2.34615
\(50\) 0 0
\(51\) −0.607947 1.05299i −0.0851296 0.147449i
\(52\) 0 0
\(53\) −2.94033 5.09281i −0.403886 0.699551i 0.590305 0.807180i \(-0.299007\pi\)
−0.994191 + 0.107629i \(0.965674\pi\)
\(54\) 0 0
\(55\) 1.23232 2.13444i 0.166166 0.287808i
\(56\) 0 0
\(57\) 1.52243 + 1.33573i 0.201651 + 0.176922i
\(58\) 0 0
\(59\) −0.531852 + 0.921195i −0.0692412 + 0.119929i −0.898568 0.438835i \(-0.855391\pi\)
0.829326 + 0.558765i \(0.188724\pi\)
\(60\) 0 0
\(61\) −7.07556 12.2552i −0.905933 1.56912i −0.819660 0.572850i \(-0.805838\pi\)
−0.0862726 0.996272i \(-0.527496\pi\)
\(62\) 0 0
\(63\) 6.73717 + 11.6691i 0.848804 + 1.47017i
\(64\) 0 0
\(65\) 4.47165 0.554640
\(66\) 0 0
\(67\) −5.98458 10.3656i −0.731132 1.26636i −0.956400 0.292061i \(-0.905659\pi\)
0.225267 0.974297i \(-0.427674\pi\)
\(68\) 0 0
\(69\) 0.861830 0.103752
\(70\) 0 0
\(71\) 0.201003 0.348147i 0.0238546 0.0413175i −0.853852 0.520516i \(-0.825739\pi\)
0.877706 + 0.479199i \(0.159073\pi\)
\(72\) 0 0
\(73\) −5.49596 + 9.51929i −0.643254 + 1.11415i 0.341448 + 0.939901i \(0.389083\pi\)
−0.984702 + 0.174247i \(0.944251\pi\)
\(74\) 0 0
\(75\) 0.464643 0.0536524
\(76\) 0 0
\(77\) 11.9282 1.35935
\(78\) 0 0
\(79\) 5.76902 9.99224i 0.649066 1.12421i −0.334281 0.942474i \(-0.608493\pi\)
0.983346 0.181741i \(-0.0581733\pi\)
\(80\) 0 0
\(81\) −3.55179 + 6.15187i −0.394643 + 0.683541i
\(82\) 0 0
\(83\) 15.6050 1.71287 0.856436 0.516253i \(-0.172674\pi\)
0.856436 + 0.516253i \(0.172674\pi\)
\(84\) 0 0
\(85\) 1.30842 + 2.26624i 0.141918 + 0.245809i
\(86\) 0 0
\(87\) −2.04161 −0.218884
\(88\) 0 0
\(89\) 3.30438 + 5.72335i 0.350263 + 0.606674i 0.986295 0.164989i \(-0.0527587\pi\)
−0.636032 + 0.771663i \(0.719425\pi\)
\(90\) 0 0
\(91\) 10.8208 + 18.7422i 1.13433 + 1.96471i
\(92\) 0 0
\(93\) 2.00188 3.46735i 0.207585 0.359548i
\(94\) 0 0
\(95\) −3.27656 2.87474i −0.336168 0.294942i
\(96\) 0 0
\(97\) 4.60741 7.98027i 0.467812 0.810274i −0.531512 0.847051i \(-0.678376\pi\)
0.999323 + 0.0367770i \(0.0117091\pi\)
\(98\) 0 0
\(99\) 3.43091 + 5.94252i 0.344820 + 0.597246i
\(100\) 0 0
\(101\) 6.46061 + 11.1901i 0.642854 + 1.11346i 0.984793 + 0.173734i \(0.0555832\pi\)
−0.341938 + 0.939722i \(0.611083\pi\)
\(102\) 0 0
\(103\) −1.32715 −0.130768 −0.0653840 0.997860i \(-0.520827\pi\)
−0.0653840 + 0.997860i \(0.520827\pi\)
\(104\) 0 0
\(105\) 1.12437 + 1.94747i 0.109728 + 0.190054i
\(106\) 0 0
\(107\) −2.30813 −0.223135 −0.111568 0.993757i \(-0.535587\pi\)
−0.111568 + 0.993757i \(0.535587\pi\)
\(108\) 0 0
\(109\) 0.888549 1.53901i 0.0851075 0.147411i −0.820330 0.571891i \(-0.806210\pi\)
0.905437 + 0.424481i \(0.139543\pi\)
\(110\) 0 0
\(111\) 1.34027 2.32141i 0.127213 0.220339i
\(112\) 0 0
\(113\) 0.0328566 0.00309089 0.00154544 0.999999i \(-0.499508\pi\)
0.00154544 + 0.999999i \(0.499508\pi\)
\(114\) 0 0
\(115\) −1.85482 −0.172963
\(116\) 0 0
\(117\) −6.22478 + 10.7816i −0.575481 + 0.996762i
\(118\) 0 0
\(119\) −6.33239 + 10.9680i −0.580489 + 1.00544i
\(120\) 0 0
\(121\) −4.92553 −0.447776
\(122\) 0 0
\(123\) −1.21752 2.10881i −0.109780 0.190145i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 1.31245 + 2.27324i 0.116461 + 0.201717i 0.918363 0.395739i \(-0.129512\pi\)
−0.801902 + 0.597456i \(0.796178\pi\)
\(128\) 0 0
\(129\) 1.27469 + 2.20782i 0.112230 + 0.194388i
\(130\) 0 0
\(131\) −3.24525 + 5.62093i −0.283538 + 0.491103i −0.972254 0.233929i \(-0.924842\pi\)
0.688715 + 0.725032i \(0.258175\pi\)
\(132\) 0 0
\(133\) 4.12014 20.6897i 0.357262 1.79402i
\(134\) 0 0
\(135\) −1.34377 + 2.32748i −0.115654 + 0.200318i
\(136\) 0 0
\(137\) 1.93683 + 3.35469i 0.165474 + 0.286610i 0.936824 0.349802i \(-0.113751\pi\)
−0.771349 + 0.636412i \(0.780418\pi\)
\(138\) 0 0
\(139\) −0.596561 1.03327i −0.0505996 0.0876411i 0.839616 0.543180i \(-0.182780\pi\)
−0.890216 + 0.455539i \(0.849447\pi\)
\(140\) 0 0
\(141\) −4.83598 −0.407263
\(142\) 0 0
\(143\) 5.51051 + 9.54449i 0.460812 + 0.798150i
\(144\) 0 0
\(145\) 4.39393 0.364896
\(146\) 0 0
\(147\) −3.81543 + 6.60851i −0.314691 + 0.545061i
\(148\) 0 0
\(149\) 3.71152 6.42854i 0.304059 0.526646i −0.672992 0.739650i \(-0.734991\pi\)
0.977051 + 0.213003i \(0.0683246\pi\)
\(150\) 0 0
\(151\) 4.16727 0.339128 0.169564 0.985519i \(-0.445764\pi\)
0.169564 + 0.985519i \(0.445764\pi\)
\(152\) 0 0
\(153\) −7.28554 −0.589001
\(154\) 0 0
\(155\) −4.30842 + 7.46240i −0.346060 + 0.599394i
\(156\) 0 0
\(157\) 4.51830 7.82593i 0.360600 0.624577i −0.627460 0.778649i \(-0.715905\pi\)
0.988060 + 0.154072i \(0.0492386\pi\)
\(158\) 0 0
\(159\) 2.73241 0.216694
\(160\) 0 0
\(161\) −4.48842 7.77417i −0.353737 0.612690i
\(162\) 0 0
\(163\) 5.15920 0.404100 0.202050 0.979375i \(-0.435240\pi\)
0.202050 + 0.979375i \(0.435240\pi\)
\(164\) 0 0
\(165\) 0.572590 + 0.991755i 0.0445761 + 0.0772080i
\(166\) 0 0
\(167\) 4.49058 + 7.77791i 0.347492 + 0.601873i 0.985803 0.167905i \(-0.0537002\pi\)
−0.638312 + 0.769778i \(0.720367\pi\)
\(168\) 0 0
\(169\) −3.49784 + 6.05843i −0.269064 + 0.466033i
\(170\) 0 0
\(171\) 11.4925 3.89836i 0.878850 0.298115i
\(172\) 0 0
\(173\) 12.0437 20.8603i 0.915666 1.58598i 0.109743 0.993960i \(-0.464997\pi\)
0.805923 0.592020i \(-0.201670\pi\)
\(174\) 0 0
\(175\) −2.41987 4.19133i −0.182925 0.316835i
\(176\) 0 0
\(177\) −0.247121 0.428027i −0.0185748 0.0321725i
\(178\) 0 0
\(179\) −21.7938 −1.62895 −0.814473 0.580202i \(-0.802974\pi\)
−0.814473 + 0.580202i \(0.802974\pi\)
\(180\) 0 0
\(181\) −10.1014 17.4962i −0.750832 1.30048i −0.947420 0.319993i \(-0.896320\pi\)
0.196588 0.980486i \(-0.437014\pi\)
\(182\) 0 0
\(183\) 6.57522 0.486054
\(184\) 0 0
\(185\) −2.88451 + 4.99612i −0.212073 + 0.367322i
\(186\) 0 0
\(187\) −3.22478 + 5.58548i −0.235819 + 0.408451i
\(188\) 0 0
\(189\) −13.0070 −0.946121
\(190\) 0 0
\(191\) −24.5926 −1.77946 −0.889729 0.456489i \(-0.849107\pi\)
−0.889729 + 0.456489i \(0.849107\pi\)
\(192\) 0 0
\(193\) −2.71743 + 4.70673i −0.195605 + 0.338798i −0.947099 0.320943i \(-0.896000\pi\)
0.751494 + 0.659740i \(0.229334\pi\)
\(194\) 0 0
\(195\) −1.03886 + 1.79936i −0.0743944 + 0.128855i
\(196\) 0 0
\(197\) 9.30706 0.663101 0.331550 0.943438i \(-0.392428\pi\)
0.331550 + 0.943438i \(0.392428\pi\)
\(198\) 0 0
\(199\) −6.11145 10.5853i −0.433229 0.750375i 0.563920 0.825829i \(-0.309293\pi\)
−0.997149 + 0.0754543i \(0.975959\pi\)
\(200\) 0 0
\(201\) 5.56139 0.392270
\(202\) 0 0
\(203\) 10.6327 + 18.4164i 0.746271 + 1.29258i
\(204\) 0 0
\(205\) 2.62034 + 4.53856i 0.183012 + 0.316987i
\(206\) 0 0
\(207\) 2.58201 4.47217i 0.179462 0.310837i
\(208\) 0 0
\(209\) 2.09819 10.5362i 0.145135 0.728807i
\(210\) 0 0
\(211\) −7.56821 + 13.1085i −0.521017 + 0.902428i 0.478684 + 0.877987i \(0.341114\pi\)
−0.999701 + 0.0244412i \(0.992219\pi\)
\(212\) 0 0
\(213\) 0.0933946 + 0.161764i 0.00639929 + 0.0110839i
\(214\) 0 0
\(215\) −2.74337 4.75165i −0.187096 0.324060i
\(216\) 0 0
\(217\) −41.7032 −2.83100
\(218\) 0 0
\(219\) −2.55366 4.42307i −0.172560 0.298883i
\(220\) 0 0
\(221\) −11.7016 −0.787132
\(222\) 0 0
\(223\) 13.1532 22.7820i 0.880803 1.52560i 0.0303532 0.999539i \(-0.490337\pi\)
0.850450 0.526056i \(-0.176330\pi\)
\(224\) 0 0
\(225\) 1.39205 2.41111i 0.0928036 0.160740i
\(226\) 0 0
\(227\) 0.528347 0.0350677 0.0175338 0.999846i \(-0.494419\pi\)
0.0175338 + 0.999846i \(0.494419\pi\)
\(228\) 0 0
\(229\) −7.83906 −0.518020 −0.259010 0.965875i \(-0.583396\pi\)
−0.259010 + 0.965875i \(0.583396\pi\)
\(230\) 0 0
\(231\) −2.77118 + 4.79983i −0.182330 + 0.315806i
\(232\) 0 0
\(233\) 13.0680 22.6345i 0.856114 1.48283i −0.0194930 0.999810i \(-0.506205\pi\)
0.875607 0.483024i \(-0.160461\pi\)
\(234\) 0 0
\(235\) 10.4079 0.678940
\(236\) 0 0
\(237\) 2.68054 + 4.64283i 0.174120 + 0.301584i
\(238\) 0 0
\(239\) −17.8785 −1.15647 −0.578233 0.815872i \(-0.696258\pi\)
−0.578233 + 0.815872i \(0.696258\pi\)
\(240\) 0 0
\(241\) 15.2631 + 26.4365i 0.983183 + 1.70292i 0.649747 + 0.760150i \(0.274875\pi\)
0.333436 + 0.942773i \(0.391792\pi\)
\(242\) 0 0
\(243\) −5.68163 9.84087i −0.364477 0.631292i
\(244\) 0 0
\(245\) 8.21152 14.2228i 0.524615 0.908659i
\(246\) 0 0
\(247\) 18.4584 6.26129i 1.17448 0.398396i
\(248\) 0 0
\(249\) −3.62538 + 6.27934i −0.229749 + 0.397937i
\(250\) 0 0
\(251\) −6.16862 10.6844i −0.389360 0.674391i 0.603004 0.797738i \(-0.293970\pi\)
−0.992364 + 0.123348i \(0.960637\pi\)
\(252\) 0 0
\(253\) −2.28574 3.95901i −0.143703 0.248901i
\(254\) 0 0
\(255\) −1.21589 −0.0761422
\(256\) 0 0
\(257\) 6.82331 + 11.8183i 0.425626 + 0.737206i 0.996479 0.0838465i \(-0.0267205\pi\)
−0.570853 + 0.821053i \(0.693387\pi\)
\(258\) 0 0
\(259\) −27.9205 −1.73490
\(260\) 0 0
\(261\) −6.11658 + 10.5942i −0.378607 + 0.655767i
\(262\) 0 0
\(263\) 11.2407 19.4694i 0.693130 1.20054i −0.277678 0.960674i \(-0.589565\pi\)
0.970807 0.239861i \(-0.0771020\pi\)
\(264\) 0 0
\(265\) −5.88067 −0.361247
\(266\) 0 0
\(267\) −3.07071 −0.187925
\(268\) 0 0
\(269\) 2.52944 4.38112i 0.154223 0.267122i −0.778553 0.627579i \(-0.784046\pi\)
0.932776 + 0.360457i \(0.117379\pi\)
\(270\) 0 0
\(271\) 4.76715 8.25694i 0.289583 0.501573i −0.684127 0.729363i \(-0.739817\pi\)
0.973710 + 0.227790i \(0.0731499\pi\)
\(272\) 0 0
\(273\) −10.0556 −0.608594
\(274\) 0 0
\(275\) −1.23232 2.13444i −0.0743118 0.128712i
\(276\) 0 0
\(277\) −9.70099 −0.582876 −0.291438 0.956590i \(-0.594134\pi\)
−0.291438 + 0.956590i \(0.594134\pi\)
\(278\) 0 0
\(279\) −11.9951 20.7761i −0.718127 1.24383i
\(280\) 0 0
\(281\) −12.4619 21.5847i −0.743417 1.28764i −0.950930 0.309405i \(-0.899870\pi\)
0.207513 0.978232i \(-0.433463\pi\)
\(282\) 0 0
\(283\) 4.26922 7.39450i 0.253779 0.439557i −0.710784 0.703410i \(-0.751660\pi\)
0.964563 + 0.263852i \(0.0849932\pi\)
\(284\) 0 0
\(285\) 1.91799 0.650602i 0.113612 0.0385383i
\(286\) 0 0
\(287\) −12.6817 + 21.9654i −0.748579 + 1.29658i
\(288\) 0 0
\(289\) 5.07609 + 8.79205i 0.298594 + 0.517180i
\(290\) 0 0
\(291\) 2.14080 + 3.70798i 0.125496 + 0.217366i
\(292\) 0 0
\(293\) 25.5698 1.49381 0.746903 0.664933i \(-0.231540\pi\)
0.746903 + 0.664933i \(0.231540\pi\)
\(294\) 0 0
\(295\) 0.531852 + 0.921195i 0.0309656 + 0.0536340i
\(296\) 0 0
\(297\) −6.62384 −0.384354
\(298\) 0 0
\(299\) 4.14706 7.18291i 0.239830 0.415399i
\(300\) 0 0
\(301\) 13.2772 22.9967i 0.765284 1.32551i
\(302\) 0 0
\(303\) −6.00375 −0.344907
\(304\) 0 0
\(305\) −14.1511 −0.810291
\(306\) 0 0
\(307\) −5.86424 + 10.1572i −0.334690 + 0.579700i −0.983425 0.181314i \(-0.941965\pi\)
0.648735 + 0.761014i \(0.275298\pi\)
\(308\) 0 0
\(309\) 0.308326 0.534035i 0.0175400 0.0303802i
\(310\) 0 0
\(311\) −25.2238 −1.43031 −0.715157 0.698964i \(-0.753645\pi\)
−0.715157 + 0.698964i \(0.753645\pi\)
\(312\) 0 0
\(313\) 3.95360 + 6.84783i 0.223470 + 0.387062i 0.955859 0.293824i \(-0.0949280\pi\)
−0.732389 + 0.680886i \(0.761595\pi\)
\(314\) 0 0
\(315\) 13.4743 0.759193
\(316\) 0 0
\(317\) −12.6387 21.8909i −0.709862 1.22952i −0.964908 0.262588i \(-0.915424\pi\)
0.255046 0.966929i \(-0.417909\pi\)
\(318\) 0 0
\(319\) 5.41473 + 9.37860i 0.303167 + 0.525101i
\(320\) 0 0
\(321\) 0.536229 0.928775i 0.0299294 0.0518392i
\(322\) 0 0
\(323\) 8.57422 + 7.52271i 0.477082 + 0.418575i
\(324\) 0 0
\(325\) 2.23583 3.87256i 0.124021 0.214811i
\(326\) 0 0
\(327\) 0.412858 + 0.715091i 0.0228311 + 0.0395446i
\(328\) 0 0
\(329\) 25.1859 + 43.6232i 1.38854 + 2.40502i
\(330\) 0 0
\(331\) 18.9915 1.04387 0.521935 0.852986i \(-0.325211\pi\)
0.521935 + 0.852986i \(0.325211\pi\)
\(332\) 0 0
\(333\) −8.03079 13.9097i −0.440084 0.762248i
\(334\) 0 0
\(335\) −11.9692 −0.653945
\(336\) 0 0
\(337\) −1.75091 + 3.03267i −0.0953782 + 0.165200i −0.909766 0.415121i \(-0.863739\pi\)
0.814388 + 0.580321i \(0.197073\pi\)
\(338\) 0 0
\(339\) −0.00763330 + 0.0132213i −0.000414584 + 0.000718080i
\(340\) 0 0
\(341\) −21.2374 −1.15007
\(342\) 0 0
\(343\) 45.6050 2.46244
\(344\) 0 0
\(345\) 0.430915 0.746366i 0.0231997 0.0401830i
\(346\) 0 0
\(347\) 1.83219 3.17345i 0.0983572 0.170360i −0.812648 0.582755i \(-0.801975\pi\)
0.911005 + 0.412396i \(0.135308\pi\)
\(348\) 0 0
\(349\) 2.66438 0.142621 0.0713106 0.997454i \(-0.477282\pi\)
0.0713106 + 0.997454i \(0.477282\pi\)
\(350\) 0 0
\(351\) −6.00889 10.4077i −0.320731 0.555522i
\(352\) 0 0
\(353\) 19.2866 1.02652 0.513261 0.858232i \(-0.328437\pi\)
0.513261 + 0.858232i \(0.328437\pi\)
\(354\) 0 0
\(355\) −0.201003 0.348147i −0.0106681 0.0184777i
\(356\) 0 0
\(357\) −2.94230 5.09621i −0.155723 0.269720i
\(358\) 0 0
\(359\) −9.26355 + 16.0449i −0.488911 + 0.846819i −0.999919 0.0127570i \(-0.995939\pi\)
0.511007 + 0.859576i \(0.329273\pi\)
\(360\) 0 0
\(361\) −17.5505 7.27868i −0.923712 0.383088i
\(362\) 0 0
\(363\) 1.14431 1.98200i 0.0600606 0.104028i
\(364\) 0 0
\(365\) 5.49596 + 9.51929i 0.287672 + 0.498262i
\(366\) 0 0
\(367\) −10.7782 18.6684i −0.562617 0.974481i −0.997267 0.0738819i \(-0.976461\pi\)
0.434650 0.900599i \(-0.356872\pi\)
\(368\) 0 0
\(369\) −14.5906 −0.759556
\(370\) 0 0
\(371\) −14.2304 24.6478i −0.738807 1.27965i
\(372\) 0 0
\(373\) −16.1257 −0.834955 −0.417478 0.908687i \(-0.637086\pi\)
−0.417478 + 0.908687i \(0.637086\pi\)
\(374\) 0 0
\(375\) 0.232322 0.402393i 0.0119970 0.0207795i
\(376\) 0 0
\(377\) −9.82406 + 17.0158i −0.505965 + 0.876357i
\(378\) 0 0
\(379\) −14.3342 −0.736296 −0.368148 0.929767i \(-0.620008\pi\)
−0.368148 + 0.929767i \(0.620008\pi\)
\(380\) 0 0
\(381\) −1.21965 −0.0624843
\(382\) 0 0
\(383\) −6.86163 + 11.8847i −0.350613 + 0.607280i −0.986357 0.164620i \(-0.947360\pi\)
0.635744 + 0.771900i \(0.280693\pi\)
\(384\) 0 0
\(385\) 5.96411 10.3301i 0.303959 0.526473i
\(386\) 0 0
\(387\) 15.2757 0.776506
\(388\) 0 0
\(389\) 9.97516 + 17.2775i 0.505761 + 0.876003i 0.999978 + 0.00666453i \(0.00212140\pi\)
−0.494217 + 0.869338i \(0.664545\pi\)
\(390\) 0 0
\(391\) 4.85375 0.245465
\(392\) 0 0
\(393\) −1.50788 2.61173i −0.0760625 0.131744i
\(394\) 0 0
\(395\) −5.76902 9.99224i −0.290271 0.502764i
\(396\) 0 0
\(397\) 3.70647 6.41980i 0.186023 0.322201i −0.757898 0.652373i \(-0.773774\pi\)
0.943921 + 0.330172i \(0.107107\pi\)
\(398\) 0 0
\(399\) 7.36817 + 6.46457i 0.368870 + 0.323633i
\(400\) 0 0
\(401\) −15.8241 + 27.4082i −0.790219 + 1.36870i 0.135613 + 0.990762i \(0.456700\pi\)
−0.925831 + 0.377937i \(0.876634\pi\)
\(402\) 0 0
\(403\) −19.2657 33.3692i −0.959695 1.66224i
\(404\) 0 0
\(405\) 3.55179 + 6.15187i 0.176490 + 0.305689i
\(406\) 0 0
\(407\) −14.2186 −0.704789
\(408\) 0 0
\(409\) 2.95685 + 5.12142i 0.146207 + 0.253238i 0.929823 0.368008i \(-0.119960\pi\)
−0.783616 + 0.621246i \(0.786627\pi\)
\(410\) 0 0
\(411\) −1.79987 −0.0887810
\(412\) 0 0
\(413\) −2.57402 + 4.45834i −0.126659 + 0.219381i
\(414\) 0 0
\(415\) 7.80250 13.5143i 0.383010 0.663392i
\(416\) 0 0
\(417\) 0.554376 0.0271479
\(418\) 0 0
\(419\) 20.3620 0.994750 0.497375 0.867536i \(-0.334297\pi\)
0.497375 + 0.867536i \(0.334297\pi\)
\(420\) 0 0
\(421\) −16.1203 + 27.9212i −0.785657 + 1.36080i 0.142949 + 0.989730i \(0.454342\pi\)
−0.928606 + 0.371068i \(0.878992\pi\)
\(422\) 0 0
\(423\) −14.4884 + 25.0947i −0.704451 + 1.22015i
\(424\) 0 0
\(425\) 2.61683 0.126935
\(426\) 0 0
\(427\) −34.2438 59.3121i −1.65718 2.87031i
\(428\) 0 0
\(429\) −5.12085 −0.247237
\(430\) 0 0
\(431\) 3.08858 + 5.34957i 0.148771 + 0.257680i 0.930774 0.365596i \(-0.119135\pi\)
−0.782002 + 0.623276i \(0.785801\pi\)
\(432\) 0 0
\(433\) 16.5651 + 28.6916i 0.796067 + 1.37883i 0.922160 + 0.386810i \(0.126423\pi\)
−0.126093 + 0.992018i \(0.540244\pi\)
\(434\) 0 0
\(435\) −1.02080 + 1.76809i −0.0489438 + 0.0847732i
\(436\) 0 0
\(437\) −7.65647 + 2.59715i −0.366259 + 0.124239i
\(438\) 0 0
\(439\) 14.6735 25.4152i 0.700326 1.21300i −0.268026 0.963412i \(-0.586371\pi\)
0.968352 0.249589i \(-0.0802955\pi\)
\(440\) 0 0
\(441\) 22.8617 + 39.5977i 1.08865 + 1.88560i
\(442\) 0 0
\(443\) −1.66520 2.88422i −0.0791162 0.137033i 0.823753 0.566949i \(-0.191876\pi\)
−0.902869 + 0.429916i \(0.858543\pi\)
\(444\) 0 0
\(445\) 6.60876 0.313285
\(446\) 0 0
\(447\) 1.72453 + 2.98698i 0.0815675 + 0.141279i
\(448\) 0 0
\(449\) −23.8153 −1.12391 −0.561957 0.827166i \(-0.689951\pi\)
−0.561957 + 0.827166i \(0.689951\pi\)
\(450\) 0 0
\(451\) −6.45820 + 11.1859i −0.304105 + 0.526725i
\(452\) 0 0
\(453\) −0.968148 + 1.67688i −0.0454876 + 0.0787868i
\(454\) 0 0
\(455\) 21.6416 1.01457
\(456\) 0 0
\(457\) 14.1100 0.660038 0.330019 0.943974i \(-0.392945\pi\)
0.330019 + 0.943974i \(0.392945\pi\)
\(458\) 0 0
\(459\) 3.51643 6.09063i 0.164133 0.284286i
\(460\) 0 0
\(461\) −11.0028 + 19.0574i −0.512450 + 0.887590i 0.487445 + 0.873153i \(0.337929\pi\)
−0.999896 + 0.0144366i \(0.995405\pi\)
\(462\) 0 0
\(463\) −21.5860 −1.00319 −0.501593 0.865104i \(-0.667252\pi\)
−0.501593 + 0.865104i \(0.667252\pi\)
\(464\) 0 0
\(465\) −2.00188 3.46735i −0.0928348 0.160795i
\(466\) 0 0
\(467\) −28.8705 −1.33597 −0.667983 0.744177i \(-0.732842\pi\)
−0.667983 + 0.744177i \(0.732842\pi\)
\(468\) 0 0
\(469\) −28.9638 50.1667i −1.33742 2.31648i
\(470\) 0 0
\(471\) 2.09940 + 3.63627i 0.0967352 + 0.167550i
\(472\) 0 0
\(473\) 6.76143 11.7111i 0.310891 0.538478i
\(474\) 0 0
\(475\) −4.12788 + 1.40022i −0.189400 + 0.0642464i
\(476\) 0 0
\(477\) 8.18620 14.1789i 0.374820 0.649208i
\(478\) 0 0
\(479\) 11.0121 + 19.0734i 0.503153 + 0.871487i 0.999993 + 0.00364517i \(0.00116030\pi\)
−0.496840 + 0.867842i \(0.665506\pi\)
\(480\) 0 0
\(481\) −12.8985 22.3409i −0.588122 1.01866i
\(482\) 0 0
\(483\) 4.17103 0.189788
\(484\) 0 0
\(485\) −4.60741 7.98027i −0.209212 0.362366i
\(486\) 0 0
\(487\) 4.28228 0.194049 0.0970244 0.995282i \(-0.469068\pi\)
0.0970244 + 0.995282i \(0.469068\pi\)
\(488\) 0 0
\(489\) −1.19859 + 2.07602i −0.0542023 + 0.0938811i
\(490\) 0 0
\(491\) 4.78598 8.28957i 0.215988 0.374103i −0.737589 0.675249i \(-0.764036\pi\)
0.953578 + 0.301147i \(0.0973693\pi\)
\(492\) 0 0
\(493\) −11.4982 −0.517852
\(494\) 0 0
\(495\) 6.86183 0.308416
\(496\) 0 0
\(497\) 0.972800 1.68494i 0.0436361 0.0755799i
\(498\) 0 0
\(499\) −5.23045 + 9.05940i −0.234147 + 0.405554i −0.959024 0.283323i \(-0.908563\pi\)
0.724878 + 0.688878i \(0.241896\pi\)
\(500\) 0 0
\(501\) −4.17304 −0.186437
\(502\) 0 0
\(503\) −19.4398 33.6707i −0.866778 1.50130i −0.865271 0.501304i \(-0.832854\pi\)
−0.00150688 0.999999i \(-0.500480\pi\)
\(504\) 0 0
\(505\) 12.9212 0.574986
\(506\) 0 0
\(507\) −1.62525 2.81501i −0.0721797 0.125019i
\(508\) 0 0
\(509\) −12.0656 20.8982i −0.534799 0.926299i −0.999173 0.0406595i \(-0.987054\pi\)
0.464374 0.885639i \(-0.346279\pi\)
\(510\) 0 0
\(511\) −26.5990 + 46.0708i −1.17667 + 2.03805i
\(512\) 0 0
\(513\) −2.28795 + 11.4891i −0.101015 + 0.507258i
\(514\) 0 0
\(515\) −0.663575 + 1.14935i −0.0292406 + 0.0506462i
\(516\) 0 0
\(517\) 12.8259 + 22.2152i 0.564084 + 0.977022i
\(518\) 0 0
\(519\) 5.59603 + 9.69260i 0.245638 + 0.425458i
\(520\) 0 0
\(521\) −29.9350 −1.31148 −0.655739 0.754988i \(-0.727643\pi\)
−0.655739 + 0.754988i \(0.727643\pi\)
\(522\) 0 0
\(523\) 12.8964 + 22.3372i 0.563919 + 0.976736i 0.997149 + 0.0754529i \(0.0240402\pi\)
−0.433231 + 0.901283i \(0.642626\pi\)
\(524\) 0 0
\(525\) 2.24875 0.0981435
\(526\) 0 0
\(527\) 11.2744 19.5278i 0.491121 0.850646i
\(528\) 0 0
\(529\) 9.77982 16.9391i 0.425210 0.736485i
\(530\) 0 0
\(531\) −2.96147 −0.128517
\(532\) 0 0
\(533\) −23.4345 −1.01506
\(534\) 0 0
\(535\) −1.15407 + 1.99890i −0.0498946 + 0.0864200i
\(536\) 0 0
\(537\) 5.06317 8.76967i 0.218492 0.378439i
\(538\) 0 0
\(539\) 40.4769 1.74346
\(540\) 0 0
\(541\) −3.61703 6.26487i −0.155508 0.269348i 0.777736 0.628591i \(-0.216368\pi\)
−0.933244 + 0.359243i \(0.883035\pi\)
\(542\) 0 0
\(543\) 9.38710 0.402839
\(544\) 0 0
\(545\) −0.888549 1.53901i −0.0380612 0.0659240i
\(546\) 0 0
\(547\) 16.2966 + 28.2265i 0.696792 + 1.20688i 0.969573 + 0.244803i \(0.0787232\pi\)
−0.272781 + 0.962076i \(0.587943\pi\)
\(548\) 0 0
\(549\) 19.6991 34.1199i 0.840738 1.45620i
\(550\) 0 0
\(551\) 18.1376 6.15246i 0.772688 0.262104i
\(552\) 0 0
\(553\) 27.9205 48.3598i 1.18730 2.05647i
\(554\) 0 0
\(555\) −1.34027 2.32141i −0.0568912 0.0985385i
\(556\) 0 0
\(557\) 4.09637 + 7.09511i 0.173569 + 0.300630i 0.939665 0.342096i \(-0.111137\pi\)
−0.766096 + 0.642726i \(0.777804\pi\)
\(558\) 0 0
\(559\) 24.5348 1.03771
\(560\) 0 0
\(561\) −1.49837 2.59526i −0.0632613 0.109572i
\(562\) 0 0
\(563\) −15.8105 −0.666334 −0.333167 0.942868i \(-0.608117\pi\)
−0.333167 + 0.942868i \(0.608117\pi\)
\(564\) 0 0
\(565\) 0.0164283 0.0284546i 0.000691144 0.00119710i
\(566\) 0 0
\(567\) −17.1897 + 29.7734i −0.721899 + 1.25037i
\(568\) 0 0
\(569\) 9.25519 0.387998 0.193999 0.981002i \(-0.437854\pi\)
0.193999 + 0.981002i \(0.437854\pi\)
\(570\) 0 0
\(571\) −27.4297 −1.14790 −0.573948 0.818892i \(-0.694589\pi\)
−0.573948 + 0.818892i \(0.694589\pi\)
\(572\) 0 0
\(573\) 5.71339 9.89589i 0.238680 0.413407i
\(574\) 0 0
\(575\) −0.927410 + 1.60632i −0.0386757 + 0.0669882i
\(576\) 0 0
\(577\) −28.0723 −1.16867 −0.584333 0.811514i \(-0.698644\pi\)
−0.584333 + 0.811514i \(0.698644\pi\)
\(578\) 0 0
\(579\) −1.26264 2.18695i −0.0524734 0.0908865i
\(580\) 0 0
\(581\) 75.5241 3.13327
\(582\) 0 0
\(583\) −7.24687 12.5520i −0.300135 0.519849i
\(584\) 0 0
\(585\) 6.22478 + 10.7816i 0.257363 + 0.445766i
\(586\) 0 0
\(587\) 8.79684 15.2366i 0.363084 0.628880i −0.625383 0.780318i \(-0.715057\pi\)
0.988467 + 0.151438i \(0.0483904\pi\)
\(588\) 0 0
\(589\) −7.33564 + 36.8366i −0.302260 + 1.51783i
\(590\) 0 0
\(591\) −2.16223 + 3.74510i −0.0889423 + 0.154053i
\(592\) 0 0
\(593\) 23.8011 + 41.2248i 0.977396 + 1.69290i 0.671790 + 0.740742i \(0.265526\pi\)
0.305606 + 0.952158i \(0.401141\pi\)
\(594\) 0 0
\(595\) 6.33239 + 10.9680i 0.259603 + 0.449645i
\(596\) 0 0
\(597\) 5.67929 0.232438
\(598\) 0 0
\(599\) 0.863420 + 1.49549i 0.0352784 + 0.0611040i 0.883126 0.469137i \(-0.155435\pi\)
−0.847847 + 0.530241i \(0.822102\pi\)
\(600\) 0 0
\(601\) 39.8558 1.62575 0.812875 0.582438i \(-0.197901\pi\)
0.812875 + 0.582438i \(0.197901\pi\)
\(602\) 0 0
\(603\) 16.6617 28.8589i 0.678517 1.17523i
\(604\) 0 0
\(605\) −2.46277 + 4.26564i −0.100126 + 0.173423i
\(606\) 0 0
\(607\) −37.9801 −1.54156 −0.770782 0.637099i \(-0.780134\pi\)
−0.770782 + 0.637099i \(0.780134\pi\)
\(608\) 0 0
\(609\) −9.88085 −0.400392
\(610\) 0 0
\(611\) −23.2704 + 40.3055i −0.941418 + 1.63058i
\(612\) 0 0
\(613\) −14.7374 + 25.5259i −0.595237 + 1.03098i 0.398277 + 0.917265i \(0.369608\pi\)
−0.993513 + 0.113715i \(0.963725\pi\)
\(614\) 0 0
\(615\) −2.43504 −0.0981904
\(616\) 0 0
\(617\) −14.4949 25.1059i −0.583543 1.01073i −0.995055 0.0993213i \(-0.968333\pi\)
0.411513 0.911404i \(-0.365001\pi\)
\(618\) 0 0
\(619\) 3.61577 0.145330 0.0726649 0.997356i \(-0.476850\pi\)
0.0726649 + 0.997356i \(0.476850\pi\)
\(620\) 0 0
\(621\) 2.49246 + 4.31706i 0.100019 + 0.173238i
\(622\) 0 0
\(623\) 15.9923 + 27.6995i 0.640719 + 1.10976i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 3.75225 + 3.29209i 0.149851 + 0.131474i
\(628\) 0 0
\(629\) 7.54828 13.0740i 0.300970 0.521295i
\(630\) 0 0
\(631\) 19.7603 + 34.2258i 0.786644 + 1.36251i 0.928012 + 0.372550i \(0.121517\pi\)
−0.141368 + 0.989957i \(0.545150\pi\)
\(632\) 0 0
\(633\) −3.51652 6.09079i −0.139769 0.242087i
\(634\) 0 0
\(635\) 2.62491 0.104166
\(636\) 0 0
\(637\) 36.7191 + 63.5993i 1.45486 + 2.51989i
\(638\) 0 0
\(639\) 1.11923 0.0442759
\(640\) 0 0
\(641\) 4.02540 6.97221i 0.158994 0.275386i −0.775512 0.631333i \(-0.782508\pi\)
0.934506 + 0.355947i \(0.115842\pi\)
\(642\) 0 0
\(643\) −0.170779 + 0.295798i −0.00673487 + 0.0116651i −0.869373 0.494156i \(-0.835477\pi\)
0.862638 + 0.505821i \(0.168810\pi\)
\(644\) 0 0
\(645\) 2.54938 0.100382
\(646\) 0 0
\(647\) 19.2796 0.757959 0.378980 0.925405i \(-0.376275\pi\)
0.378980 + 0.925405i \(0.376275\pi\)
\(648\) 0 0
\(649\) −1.31083 + 2.27042i −0.0514544 + 0.0891217i
\(650\) 0 0
\(651\) 9.68855 16.7811i 0.379724 0.657702i
\(652\) 0 0
\(653\) −24.7026 −0.966689 −0.483344 0.875430i \(-0.660578\pi\)
−0.483344 + 0.875430i \(0.660578\pi\)
\(654\) 0 0
\(655\) 3.24525 + 5.62093i 0.126802 + 0.219628i
\(656\) 0 0
\(657\) −30.6027 −1.19392
\(658\) 0 0
\(659\) 7.96834 + 13.8016i 0.310403 + 0.537633i 0.978450 0.206486i \(-0.0662028\pi\)
−0.668047 + 0.744119i \(0.732869\pi\)
\(660\) 0 0
\(661\) −11.7533 20.3574i −0.457152 0.791811i 0.541657 0.840600i \(-0.317797\pi\)
−0.998809 + 0.0487887i \(0.984464\pi\)
\(662\) 0 0
\(663\) 2.71853 4.70863i 0.105579 0.182868i
\(664\) 0 0
\(665\) −15.8577 13.9130i −0.614935 0.539522i
\(666\) 0 0
\(667\) 4.07497 7.05806i 0.157784 0.273289i
\(668\) 0 0
\(669\) 6.11154 + 10.5855i 0.236286 + 0.409259i
\(670\) 0 0
\(671\) −17.4387 30.2048i −0.673215 1.16604i
\(672\) 0 0
\(673\) 12.8634 0.495850 0.247925 0.968779i \(-0.420251\pi\)
0.247925 + 0.968779i \(0.420251\pi\)
\(674\) 0 0
\(675\) 1.34377 + 2.32748i 0.0517218 + 0.0895849i
\(676\) 0 0
\(677\) 43.7369 1.68095 0.840473 0.541853i \(-0.182277\pi\)
0.840473 + 0.541853i \(0.182277\pi\)
\(678\) 0 0
\(679\) 22.2987 38.6224i 0.855744 1.48219i
\(680\) 0 0
\(681\) −0.122747 + 0.212603i −0.00470366 + 0.00814697i
\(682\) 0 0
\(683\) −14.6409 −0.560220 −0.280110 0.959968i \(-0.590371\pi\)
−0.280110 + 0.959968i \(0.590371\pi\)
\(684\) 0 0
\(685\) 3.87366 0.148005
\(686\) 0 0
\(687\) 1.82118 3.15438i 0.0694824 0.120347i
\(688\) 0 0
\(689\) 13.1482 22.7733i 0.500905 0.867592i
\(690\) 0 0
\(691\) −9.83609 −0.374182 −0.187091 0.982343i \(-0.559906\pi\)
−0.187091 + 0.982343i \(0.559906\pi\)
\(692\) 0 0
\(693\) 16.6047 + 28.7602i 0.630761 + 1.09251i
\(694\) 0 0
\(695\) −1.19312 −0.0452577
\(696\) 0 0
\(697\) −6.85698 11.8766i −0.259727 0.449860i
\(698\) 0 0
\(699\) 6.07197 + 10.5170i 0.229663 + 0.397788i
\(700\) 0 0
\(701\) 7.19212 12.4571i 0.271643 0.470499i −0.697640 0.716448i \(-0.745767\pi\)
0.969283 + 0.245950i \(0.0790998\pi\)
\(702\) 0 0
\(703\) −4.91126 + 24.6623i −0.185232 + 0.930157i
\(704\) 0 0
\(705\) −2.41799 + 4.18808i −0.0910668 + 0.157732i
\(706\) 0 0
\(707\) 31.2676 + 54.1571i 1.17594 + 2.03679i
\(708\) 0 0
\(709\) −22.3849 38.7717i −0.840682 1.45610i −0.889319 0.457287i \(-0.848821\pi\)
0.0486375 0.998816i \(-0.484512\pi\)
\(710\) 0 0
\(711\) 32.1231 1.20471
\(712\) 0 0
\(713\) 7.99134 + 13.8414i 0.299278 + 0.518365i
\(714\) 0 0
\(715\) 11.0210 0.412163
\(716\) 0 0
\(717\) 4.15357 7.19419i 0.155118 0.268672i
\(718\) 0 0
\(719\) −8.55798 + 14.8228i −0.319159 + 0.552799i −0.980313 0.197451i \(-0.936734\pi\)
0.661154 + 0.750250i \(0.270067\pi\)
\(720\) 0 0
\(721\) −6.42305 −0.239207
\(722\) 0 0
\(723\) −14.1838 −0.527501
\(724\) 0 0
\(725\) 2.19696 3.80525i 0.0815932 0.141324i
\(726\) 0 0
\(727\) 7.77998 13.4753i 0.288543 0.499772i −0.684919 0.728619i \(-0.740162\pi\)
0.973462 + 0.228847i \(0.0734957\pi\)
\(728\) 0 0
\(729\) −16.0308 −0.593735
\(730\) 0 0
\(731\) 7.17894 + 12.4343i 0.265523 + 0.459899i
\(732\) 0 0
\(733\) −14.3218 −0.528986 −0.264493 0.964388i \(-0.585205\pi\)
−0.264493 + 0.964388i \(0.585205\pi\)
\(734\) 0 0
\(735\) 3.81543 + 6.60851i 0.140734 + 0.243759i
\(736\) 0 0
\(737\) −14.7498 25.5475i −0.543318 0.941054i
\(738\) 0 0
\(739\) 7.93692 13.7471i 0.291964 0.505697i −0.682310 0.731063i \(-0.739024\pi\)
0.974274 + 0.225366i \(0.0723578\pi\)
\(740\) 0 0
\(741\) −1.76880 + 8.88218i −0.0649784 + 0.326295i
\(742\) 0 0
\(743\) −7.86539 + 13.6233i −0.288553 + 0.499789i −0.973465 0.228838i \(-0.926508\pi\)
0.684912 + 0.728626i \(0.259841\pi\)
\(744\) 0 0
\(745\) −3.71152 6.42854i −0.135979 0.235523i
\(746\) 0 0
\(747\) 21.7230 + 37.6253i 0.794803 + 1.37664i
\(748\) 0 0
\(749\) −11.1707 −0.408170
\(750\) 0 0
\(751\) 1.66593 + 2.88548i 0.0607907 + 0.105293i 0.894819 0.446429i \(-0.147304\pi\)
−0.834028 + 0.551722i \(0.813971\pi\)
\(752\) 0 0
\(753\) 5.73241 0.208901
\(754\) 0 0
\(755\) 2.08364 3.60897i 0.0758313 0.131344i
\(756\) 0 0
\(757\) 13.3582 23.1371i 0.485513 0.840934i −0.514348 0.857581i \(-0.671966\pi\)
0.999861 + 0.0166478i \(0.00529941\pi\)
\(758\) 0 0
\(759\) 2.12410 0.0771000
\(760\) 0 0
\(761\) 12.8029 0.464106 0.232053 0.972703i \(-0.425456\pi\)
0.232053 + 0.972703i \(0.425456\pi\)
\(762\) 0 0
\(763\) 4.30034 7.44841i 0.155683 0.269650i
\(764\) 0 0
\(765\) −3.64277 + 6.30946i −0.131705 + 0.228119i
\(766\) 0 0
\(767\) −4.75652 −0.171748
\(768\) 0 0
\(769\) 4.95216 + 8.57740i 0.178580 + 0.309309i 0.941394 0.337308i \(-0.109516\pi\)
−0.762815 + 0.646617i \(0.776183\pi\)
\(770\) 0 0
\(771\) −6.34081 −0.228359
\(772\) 0 0
\(773\) 15.5442 + 26.9233i 0.559084 + 0.968362i 0.997573 + 0.0696258i \(0.0221805\pi\)
−0.438489 + 0.898737i \(0.644486\pi\)
\(774\) 0 0
\(775\) 4.30842 + 7.46240i 0.154763 + 0.268057i
\(776\) 0 0
\(777\) 6.48654 11.2350i 0.232703 0.403054i
\(778\) 0 0
\(779\) 17.1714 + 15.0656i 0.615229 + 0.539780i
\(780\) 0 0
\(781\) 0.495400 0.858058i 0.0177268 0.0307037i
\(782\) 0 0
\(783\) −5.90444 10.2268i −0.211008 0.365476i
\(784\) 0 0
\(785\) −4.51830 7.82593i −0.161265 0.279320i
\(786\) 0 0
\(787\) 29.2780 1.04365 0.521823 0.853054i \(-0.325252\pi\)
0.521823 + 0.853054i \(0.325252\pi\)
\(788\) 0 0
\(789\) 5.22290 + 9.04633i 0.185940 + 0.322058i
\(790\) 0 0
\(791\) 0.159017 0.00565400
\(792\) 0 0
\(793\) 31.6395 54.8011i 1.12355 1.94605i
\(794\) 0 0
\(795\) 1.36621 2.36634i 0.0484543 0.0839254i
\(796\) 0 0
\(797\) −36.8850 −1.30653 −0.653266 0.757128i \(-0.726602\pi\)
−0.653266 + 0.757128i \(0.726602\pi\)
\(798\) 0 0
\(799\) −27.2359 −0.963535
\(800\) 0 0
\(801\) −9.19974 + 15.9344i −0.325057 + 0.563015i
\(802\) 0 0
\(803\) −13.5456 + 23.4616i −0.478013 + 0.827943i
\(804\) 0 0
\(805\) −8.97684 −0.316392
\(806\) 0 0
\(807\) 1.17529 + 2.03566i 0.0413721 + 0.0716586i
\(808\) 0 0
\(809\) 29.3738 1.03273 0.516365 0.856369i \(-0.327285\pi\)
0.516365 + 0.856369i \(0.327285\pi\)
\(810\) 0 0
\(811\) −14.8261 25.6796i −0.520616 0.901733i −0.999713 0.0239712i \(-0.992369\pi\)
0.479097 0.877762i \(-0.340964\pi\)
\(812\) 0 0
\(813\) 2.21502 + 3.83653i 0.0776842 + 0.134553i
\(814\) 0 0
\(815\) 2.57960 4.46800i 0.0903594 0.156507i
\(816\) 0 0
\(817\) −17.9776 15.7729i −0.628958 0.551825i
\(818\) 0 0
\(819\) −30.1263 + 52.1802i −1.05270 + 1.82333i
\(820\) 0 0
\(821\) 2.61919 + 4.53657i 0.0914103 + 0.158327i 0.908105 0.418743i \(-0.137529\pi\)
−0.816694 + 0.577070i \(0.804196\pi\)
\(822\) 0 0
\(823\) −15.3302 26.5527i −0.534378 0.925570i −0.999193 0.0401622i \(-0.987213\pi\)
0.464815 0.885408i \(-0.346121\pi\)
\(824\) 0 0
\(825\) 1.14518 0.0398700
\(826\) 0 0
\(827\) 14.0898 + 24.4043i 0.489951 + 0.848621i 0.999933 0.0115646i \(-0.00368121\pi\)
−0.509982 + 0.860185i \(0.670348\pi\)
\(828\) 0 0
\(829\) −12.5194 −0.434817 −0.217409 0.976081i \(-0.569760\pi\)
−0.217409 + 0.976081i \(0.569760\pi\)
\(830\) 0 0
\(831\) 2.25375 3.90361i 0.0781817 0.135415i
\(832\) 0 0
\(833\) −21.4882 + 37.2186i −0.744521 + 1.28955i
\(834\) 0 0
\(835\) 8.98116 0.310806
\(836\) 0 0
\(837\) 23.1581 0.800462
\(838\) 0 0
\(839\) 8.93658 15.4786i 0.308525 0.534381i −0.669515 0.742799i \(-0.733498\pi\)
0.978040 + 0.208418i \(0.0668313\pi\)
\(840\) 0 0
\(841\) 4.84669 8.39472i 0.167127 0.289473i
\(842\) 0 0
\(843\) 11.5807 0.398861
\(844\) 0 0
\(845\) 3.49784 + 6.05843i 0.120329 + 0.208416i
\(846\) 0 0
\(847\) −23.8383 −0.819093
\(848\) 0 0
\(849\) 1.98366 + 3.43580i 0.0680791 + 0.117916i
\(850\) 0 0
\(851\) 5.35025 + 9.26690i 0.183404 + 0.317665i
\(852\) 0 0
\(853\) 22.5959 39.1373i 0.773670 1.34004i −0.161869 0.986812i \(-0.551752\pi\)
0.935539 0.353224i \(-0.114915\pi\)
\(854\) 0 0
\(855\) 2.37015 11.9019i 0.0810575 0.407038i
\(856\) 0 0
\(857\) −3.30144 + 5.71827i −0.112775 + 0.195332i −0.916888 0.399144i \(-0.869307\pi\)
0.804113 + 0.594476i \(0.202641\pi\)
\(858\) 0 0
\(859\) 21.0508 + 36.4610i 0.718243 + 1.24403i 0.961695 + 0.274121i \(0.0883867\pi\)
−0.243452 + 0.969913i \(0.578280\pi\)
\(860\) 0 0
\(861\) −5.89248 10.2061i −0.200815 0.347822i
\(862\) 0 0
\(863\) −15.6952 −0.534272 −0.267136 0.963659i \(-0.586077\pi\)
−0.267136 + 0.963659i \(0.586077\pi\)
\(864\) 0 0
\(865\) −12.0437 20.8603i −0.409498 0.709272i
\(866\) 0 0
\(867\) −4.71715 −0.160203
\(868\) 0 0
\(869\) 14.2186 24.6273i 0.482332 0.835424i
\(870\) 0 0
\(871\) 26.7609 46.3513i 0.906760 1.57055i
\(872\) 0 0
\(873\) 25.6551 0.868292
\(874\) 0 0
\(875\) −4.83973 −0.163613
\(876\) 0 0
\(877\) −14.5532 + 25.2069i −0.491427 + 0.851177i −0.999951 0.00987081i \(-0.996858\pi\)
0.508524 + 0.861048i \(0.330191\pi\)
\(878\) 0 0
\(879\) −5.94042 + 10.2891i −0.200366 + 0.347043i
\(880\) 0 0
\(881\) −7.20417 −0.242715 −0.121357 0.992609i \(-0.538725\pi\)
−0.121357 + 0.992609i \(0.538725\pi\)
\(882\) 0 0
\(883\) −9.07393 15.7165i −0.305362 0.528903i 0.671980 0.740570i \(-0.265444\pi\)
−0.977342 + 0.211667i \(0.932111\pi\)
\(884\) 0 0
\(885\) −0.494243 −0.0166138
\(886\) 0 0
\(887\) −17.9630 31.1129i −0.603140 1.04467i −0.992342 0.123517i \(-0.960583\pi\)
0.389203 0.921152i \(-0.372751\pi\)
\(888\) 0 0
\(889\) 6.35193 + 11.0019i 0.213037 + 0.368991i
\(890\) 0 0
\(891\) −8.75388 + 15.1622i −0.293266 + 0.507952i
\(892\) 0 0
\(893\) 42.9628 14.5734i 1.43769 0.487680i
\(894\) 0 0
\(895\) −10.8969 + 18.8740i −0.364243 + 0.630888i
\(896\) 0 0
\(897\) 1.92690 + 3.33749i 0.0643374 + 0.111436i
\(898\) 0 0
\(899\) −18.9309 32.7892i −0.631380 1.09358i
\(900\) 0 0
\(901\) 15.3887 0.512673
\(902\) 0 0
\(903\) 6.16915 + 10.6853i 0.205296 + 0.355584i
\(904\) 0 0
\(905\) −20.2028 −0.671564
\(906\) 0 0
\(907\) 5.64896 9.78429i 0.187571 0.324882i −0.756869 0.653567i \(-0.773272\pi\)
0.944440 + 0.328685i \(0.106605\pi\)
\(908\) 0 0
\(909\) −17.9870 + 31.1544i −0.596592 + 1.03333i
\(910\) 0 0
\(911\) 39.0643 1.29426 0.647128 0.762381i \(-0.275970\pi\)
0.647128 + 0.762381i \(0.275970\pi\)
\(912\) 0 0
\(913\) 38.4608 1.27287
\(914\) 0 0
\(915\) 3.28761 5.69431i 0.108685 0.188248i
\(916\) 0 0
\(917\) −15.7061 + 27.2038i −0.518662 + 0.898349i
\(918\) 0 0
\(919\) 6.92535 0.228446 0.114223 0.993455i \(-0.463562\pi\)
0.114223 + 0.993455i \(0.463562\pi\)
\(920\) 0 0
\(921\) −2.72478 4.71946i −0.0897845 0.155511i
\(922\) 0 0
\(923\) 1.79763 0.0591697
\(924\) 0 0
\(925\) 2.88451 + 4.99612i 0.0948421 + 0.164271i
\(926\) 0 0
\(927\) −1.84746 3.19990i −0.0606787 0.105099i
\(928\) 0 0
\(929\) −8.16242 + 14.1377i −0.267800 + 0.463844i −0.968293 0.249815i \(-0.919630\pi\)
0.700493 + 0.713659i \(0.252963\pi\)
\(930\) 0 0
\(931\) 13.9812 70.2078i 0.458215 2.30097i
\(932\) 0 0
\(933\) 5.86004 10.1499i 0.191849 0.332293i
\(934\) 0 0
\(935\) 3.22478 + 5.58548i 0.105462 + 0.182665i
\(936\) 0 0
\(937\) 8.25845 + 14.3041i 0.269792 + 0.467293i 0.968808 0.247813i \(-0.0797117\pi\)
−0.699016 + 0.715106i \(0.746378\pi\)
\(938\) 0 0
\(939\) −3.67402 −0.119897
\(940\) 0 0
\(941\) −17.2963 29.9581i −0.563843 0.976605i −0.997156 0.0753618i \(-0.975989\pi\)
0.433313 0.901244i \(-0.357344\pi\)
\(942\) 0 0
\(943\) 9.72051 0.316543
\(944\) 0 0
\(945\) −6.50350 + 11.2644i −0.211559 + 0.366431i
\(946\) 0 0
\(947\) −30.1858 + 52.2833i −0.980906 + 1.69898i −0.322024 + 0.946731i \(0.604363\pi\)
−0.658881 + 0.752247i \(0.728970\pi\)
\(948\) 0 0
\(949\) −49.1521 −1.59554
\(950\) 0 0
\(951\) 11.7450 0.380858
\(952\) 0 0
\(953\) −14.3138 + 24.7922i −0.463669 + 0.803099i −0.999140 0.0414547i \(-0.986801\pi\)
0.535471 + 0.844554i \(0.320134\pi\)
\(954\) 0 0
\(955\) −12.2963 + 21.2978i −0.397899 + 0.689181i
\(956\) 0 0
\(957\) −5.03184 −0.162656
\(958\) 0 0
\(959\) 9.37374 + 16.2358i 0.302694 + 0.524281i
\(960\) 0 0
\(961\) 43.2498 1.39515
\(962\) 0 0
\(963\) −3.21304 5.56515i −0.103539 0.179334i
\(964\) 0 0
\(965\) 2.71743 + 4.70673i 0.0874772 + 0.151515i
\(966\) 0 0
\(967\) 12.4268 21.5238i 0.399619 0.692160i −0.594060 0.804421i \(-0.702476\pi\)
0.993679 + 0.112261i \(0.0358093\pi\)
\(968\) 0 0
\(969\) −5.01906 + 1.70252i −0.161235 + 0.0546927i
\(970\) 0 0
\(971\) −19.1635 + 33.1922i −0.614986 + 1.06519i 0.375401 + 0.926863i \(0.377505\pi\)
−0.990387 + 0.138325i \(0.955828\pi\)
\(972\) 0 0
\(973\) −2.88720 5.00077i −0.0925593 0.160317i
\(974\) 0 0
\(975\) 1.03886 + 1.79936i 0.0332702 + 0.0576257i
\(976\) 0 0
\(977\) −4.91834 −0.157352 −0.0786759 0.996900i \(-0.525069\pi\)
−0.0786759 + 0.996900i \(0.525069\pi\)
\(978\) 0 0
\(979\) 8.14411 + 14.1060i 0.260287 + 0.450830i
\(980\) 0 0
\(981\) 4.94763 0.157966
\(982\) 0 0
\(983\) 20.8685 36.1452i 0.665601 1.15285i −0.313522 0.949581i \(-0.601509\pi\)
0.979122 0.203273i \(-0.0651579\pi\)
\(984\) 0 0
\(985\) 4.65353 8.06015i 0.148274 0.256818i
\(986\) 0 0
\(987\) −23.4049 −0.744985
\(988\) 0 0
\(989\) −10.1769 −0.323607
\(990\) 0 0
\(991\) 4.20081 7.27601i 0.133443 0.231130i −0.791559 0.611093i \(-0.790730\pi\)
0.925002 + 0.379963i \(0.124063\pi\)
\(992\) 0 0
\(993\) −4.41214 + 7.64206i −0.140015 + 0.242513i
\(994\) 0 0
\(995\) −12.2229 −0.387492
\(996\) 0 0
\(997\) 24.6756 + 42.7393i 0.781483 + 1.35357i 0.931078 + 0.364821i \(0.118870\pi\)
−0.149595 + 0.988747i \(0.547797\pi\)
\(998\) 0 0
\(999\) 15.5045 0.490541
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 1520.2.q.l.961.2 8
4.3 odd 2 760.2.q.e.201.3 yes 8
19.7 even 3 inner 1520.2.q.l.881.2 8
76.7 odd 6 760.2.q.e.121.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.q.e.121.3 8 76.7 odd 6
760.2.q.e.201.3 yes 8 4.3 odd 2
1520.2.q.l.881.2 8 19.7 even 3 inner
1520.2.q.l.961.2 8 1.1 even 1 trivial