Properties

Label 945.2.cs.a.524.2
Level $945$
Weight $2$
Character 945.524
Analytic conductor $7.546$
Analytic rank $0$
Dimension $24$
CM discriminant -35
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(104,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([11, 9, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.104");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.cs (of order \(18\), degree \(6\), 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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(4\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{18}]$

Embedding invariants

Embedding label 524.2
Character \(\chi\) \(=\) 945.524
Dual form 945.2.cs.a.734.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.306808 - 1.70466i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(0.764780 - 2.10122i) q^{5} +(-0.459430 + 2.60556i) q^{7} +(-2.81174 + 1.04601i) q^{9} +O(q^{10})\) \(q+(-0.306808 - 1.70466i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(0.764780 - 2.10122i) q^{5} +(-0.459430 + 2.60556i) q^{7} +(-2.81174 + 1.04601i) q^{9} +(-0.890490 - 2.44660i) q^{11} +(-3.25097 + 1.19632i) q^{12} +(-0.146967 + 0.123320i) q^{13} +(-3.81650 - 0.659021i) q^{15} +(-3.75877 + 1.36808i) q^{16} +(-7.10570 + 4.10248i) q^{17} +(-4.40419 - 0.776578i) q^{20} +(4.58255 - 0.0162334i) q^{21} +(-3.83022 - 3.21394i) q^{25} +(2.64575 + 4.47214i) q^{27} +5.29150 q^{28} +(6.91142 - 8.23672i) q^{29} +(-3.89741 + 2.26862i) q^{33} +(5.12348 + 2.95804i) q^{35} +(3.03674 + 5.17477i) q^{36} +(0.255310 + 0.212694i) q^{39} +(-4.50960 + 2.60362i) q^{44} +(0.0475262 + 6.70804i) q^{45} +(-7.67301 - 1.35296i) q^{47} +(3.48533 + 5.98769i) q^{48} +(-6.57785 - 2.39414i) q^{49} +(9.17342 + 10.8541i) q^{51} +(0.293934 + 0.246640i) q^{52} -5.82187 q^{55} +(0.0274394 + 7.74592i) q^{60} +(-1.43363 - 7.80671i) q^{63} +(4.00000 + 6.92820i) q^{64} +(0.146725 + 0.403123i) q^{65} +(10.5481 + 12.5707i) q^{68} +(8.45506 - 4.88153i) q^{71} +(-7.52515 + 13.0339i) q^{73} +(-4.30353 + 7.51529i) q^{75} +(6.78387 - 1.19618i) q^{77} +(-9.32613 - 7.82555i) q^{79} +8.94427i q^{80} +(6.81174 - 5.88220i) q^{81} +(10.9632 - 13.0654i) q^{83} +(-1.62348 - 9.02022i) q^{84} +(3.18590 + 18.0681i) q^{85} +(-16.1613 - 9.25455i) q^{87} +(-0.253796 - 0.439588i) q^{91} +(3.18582 - 1.15954i) q^{97} +(5.06298 + 5.94774i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 6 q^{9} - 18 q^{11} - 24 q^{39} + 96 q^{64} - 180 q^{65} - 12 q^{79} + 102 q^{81} + 84 q^{84} + 60 q^{85} + 228 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/945\mathbb{Z}\right)^\times\).

\(n\) \(136\) \(596\) \(757\)
\(\chi(n)\) \(-1\) \(e\left(\frac{13}{18}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(3\) −0.306808 1.70466i −0.177136 0.984186i
\(4\) −0.347296 1.96962i −0.173648 0.984808i
\(5\) 0.764780 2.10122i 0.342020 0.939693i
\(6\) 0 0
\(7\) −0.459430 + 2.60556i −0.173648 + 0.984808i
\(8\) 0 0
\(9\) −2.81174 + 1.04601i −0.937246 + 0.348669i
\(10\) 0 0
\(11\) −0.890490 2.44660i −0.268493 0.737678i −0.998526 0.0542666i \(-0.982718\pi\)
0.730034 0.683411i \(-0.239504\pi\)
\(12\) −3.25097 + 1.19632i −0.938475 + 0.345347i
\(13\) −0.146967 + 0.123320i −0.0407614 + 0.0342029i −0.662941 0.748672i \(-0.730692\pi\)
0.622179 + 0.782875i \(0.286247\pi\)
\(14\) 0 0
\(15\) −3.81650 0.659021i −0.985417 0.170158i
\(16\) −3.75877 + 1.36808i −0.939693 + 0.342020i
\(17\) −7.10570 + 4.10248i −1.72339 + 0.994997i −0.811742 + 0.584016i \(0.801481\pi\)
−0.911644 + 0.410982i \(0.865186\pi\)
\(18\) 0 0
\(19\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) −4.40419 0.776578i −0.984808 0.173648i
\(21\) 4.58255 0.0162334i 0.999994 0.00354242i
\(22\) 0 0
\(23\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(24\) 0 0
\(25\) −3.83022 3.21394i −0.766044 0.642788i
\(26\) 0 0
\(27\) 2.64575 + 4.47214i 0.509175 + 0.860663i
\(28\) 5.29150 1.00000
\(29\) 6.91142 8.23672i 1.28342 1.52952i 0.597365 0.801970i \(-0.296214\pi\)
0.686055 0.727550i \(-0.259341\pi\)
\(30\) 0 0
\(31\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(32\) 0 0
\(33\) −3.89741 + 2.26862i −0.678453 + 0.394916i
\(34\) 0 0
\(35\) 5.12348 + 2.95804i 0.866025 + 0.500000i
\(36\) 3.03674 + 5.17477i 0.506123 + 0.862461i
\(37\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(38\) 0 0
\(39\) 0.255310 + 0.212694i 0.0408823 + 0.0340583i
\(40\) 0 0
\(41\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(42\) 0 0
\(43\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(44\) −4.50960 + 2.60362i −0.679847 + 0.392510i
\(45\) 0.0475262 + 6.70804i 0.00708479 + 0.999975i
\(46\) 0 0
\(47\) −7.67301 1.35296i −1.11922 0.197349i −0.416724 0.909033i \(-0.636822\pi\)
−0.702500 + 0.711684i \(0.747933\pi\)
\(48\) 3.48533 + 5.98769i 0.503065 + 0.864249i
\(49\) −6.57785 2.39414i −0.939693 0.342020i
\(50\) 0 0
\(51\) 9.17342 + 10.8541i 1.28454 + 1.51988i
\(52\) 0.293934 + 0.246640i 0.0407614 + 0.0342029i
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −5.82187 −0.785020
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(60\) 0.0274394 + 7.74592i 0.00354242 + 0.999994i
\(61\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(62\) 0 0
\(63\) −1.43363 7.80671i −0.180621 0.983553i
\(64\) 4.00000 + 6.92820i 0.500000 + 0.866025i
\(65\) 0.146725 + 0.403123i 0.0181990 + 0.0500012i
\(66\) 0 0
\(67\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(68\) 10.5481 + 12.5707i 1.27914 + 1.52442i
\(69\) 0 0
\(70\) 0 0
\(71\) 8.45506 4.88153i 1.00343 0.579331i 0.0941692 0.995556i \(-0.469981\pi\)
0.909262 + 0.416225i \(0.136647\pi\)
\(72\) 0 0
\(73\) −7.52515 + 13.0339i −0.880752 + 1.52551i −0.0302463 + 0.999542i \(0.509629\pi\)
−0.850506 + 0.525965i \(0.823704\pi\)
\(74\) 0 0
\(75\) −4.30353 + 7.51529i −0.496929 + 0.867791i
\(76\) 0 0
\(77\) 6.78387 1.19618i 0.773094 0.136317i
\(78\) 0 0
\(79\) −9.32613 7.82555i −1.04927 0.880443i −0.0562544 0.998416i \(-0.517916\pi\)
−0.993017 + 0.117973i \(0.962360\pi\)
\(80\) 8.94427i 1.00000i
\(81\) 6.81174 5.88220i 0.756860 0.653577i
\(82\) 0 0
\(83\) 10.9632 13.0654i 1.20336 1.43411i 0.332136 0.943231i \(-0.392231\pi\)
0.871227 0.490881i \(-0.163325\pi\)
\(84\) −1.62348 9.02022i −0.177136 0.984186i
\(85\) 3.18590 + 18.0681i 0.345559 + 1.95976i
\(86\) 0 0
\(87\) −16.1613 9.25455i −1.73267 0.992191i
\(88\) 0 0
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) −0.253796 0.439588i −0.0266051 0.0460814i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.18582 1.15954i 0.323471 0.117734i −0.175180 0.984536i \(-0.556051\pi\)
0.498652 + 0.866802i \(0.333829\pi\)
\(98\) 0 0
\(99\) 5.06298 + 5.94774i 0.508849 + 0.597770i
\(100\) −5.00000 + 8.66025i −0.500000 + 0.866025i
\(101\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(102\) 0 0
\(103\) −15.1343 5.50842i −1.49122 0.542761i −0.537452 0.843294i \(-0.680613\pi\)
−0.953771 + 0.300533i \(0.902835\pi\)
\(104\) 0 0
\(105\) 3.47053 9.64134i 0.338689 0.940898i
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 7.88953 6.76427i 0.759170 0.650892i
\(109\) −16.4069 −1.57149 −0.785747 0.618548i \(-0.787721\pi\)
−0.785747 + 0.618548i \(0.787721\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.83772 10.4222i −0.173648 0.984808i
\(113\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −18.6235 10.7523i −1.72915 0.998323i
\(117\) 0.284240 0.500473i 0.0262780 0.0462687i
\(118\) 0 0
\(119\) −7.42467 20.3991i −0.680618 1.86998i
\(120\) 0 0
\(121\) 3.23361 2.71332i 0.293964 0.246666i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.68246 + 5.59017i −0.866025 + 0.500000i
\(126\) 0 0
\(127\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(132\) 5.82187 + 6.88852i 0.506728 + 0.599569i
\(133\) 0 0
\(134\) 0 0
\(135\) 11.4203 2.13910i 0.982907 0.184104i
\(136\) 0 0
\(137\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(138\) 0 0
\(139\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(140\) 4.04684 11.1186i 0.342020 0.939693i
\(141\) 0.0478051 + 13.4950i 0.00402592 + 1.13648i
\(142\) 0 0
\(143\) 0.432588 + 0.249755i 0.0361748 + 0.0208855i
\(144\) 9.13765 7.77838i 0.761471 0.648199i
\(145\) −12.0214 20.8217i −0.998323 1.72915i
\(146\) 0 0
\(147\) −2.06306 + 11.9475i −0.170158 + 0.985417i
\(148\) 0 0
\(149\) −12.9747 15.4627i −1.06293 1.26675i −0.962348 0.271821i \(-0.912374\pi\)
−0.100582 0.994929i \(-0.532070\pi\)
\(150\) 0 0
\(151\) −1.51705 + 0.552160i −0.123455 + 0.0449341i −0.403009 0.915196i \(-0.632036\pi\)
0.279554 + 0.960130i \(0.409814\pi\)
\(152\) 0 0
\(153\) 15.6881 18.9677i 1.26831 1.53345i
\(154\) 0 0
\(155\) 0 0
\(156\) 0.330257 0.576730i 0.0264417 0.0461753i
\(157\) 23.3401 + 8.49510i 1.86274 + 0.677983i 0.976788 + 0.214210i \(0.0687178\pi\)
0.885954 + 0.463772i \(0.153504\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 1.78619 + 9.92431i 0.139055 + 0.772606i
\(166\) 0 0
\(167\) 8.18939 22.5002i 0.633714 1.74111i −0.0369115 0.999319i \(-0.511752\pi\)
0.670625 0.741796i \(-0.266026\pi\)
\(168\) 0 0
\(169\) −2.25103 + 12.7663i −0.173157 + 0.982019i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.14105 14.1249i −0.390867 1.07390i −0.966607 0.256265i \(-0.917508\pi\)
0.575740 0.817633i \(-0.304714\pi\)
\(174\) 0 0
\(175\) 10.1338 8.50328i 0.766044 0.642788i
\(176\) 6.69429 + 7.97795i 0.504601 + 0.601360i
\(177\) 0 0
\(178\) 0 0
\(179\) 16.7377 9.66354i 1.25104 0.722287i 0.279722 0.960081i \(-0.409758\pi\)
0.971316 + 0.237794i \(0.0764244\pi\)
\(180\) 13.1957 2.42328i 0.983553 0.180621i
\(181\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 16.3647 + 13.7316i 1.19670 + 1.00415i
\(188\) 15.5828i 1.13649i
\(189\) −12.8679 + 4.83902i −0.936005 + 0.351987i
\(190\) 0 0
\(191\) −3.80278 + 4.53198i −0.275160 + 0.327923i −0.885872 0.463930i \(-0.846439\pi\)
0.610712 + 0.791853i \(0.290883\pi\)
\(192\) 10.5830 8.94427i 0.763763 0.645497i
\(193\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(194\) 0 0
\(195\) 0.642171 0.373797i 0.0459869 0.0267682i
\(196\) −2.43107 + 13.7873i −0.173648 + 0.984808i
\(197\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(198\) 0 0
\(199\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 18.2859 + 21.7923i 1.28342 + 1.52952i
\(204\) 18.1926 21.8377i 1.27374 1.52895i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.383704 0.664595i 0.0266051 0.0460814i
\(209\) 0 0
\(210\) 0 0
\(211\) −20.1776 7.34403i −1.38908 0.505584i −0.464162 0.885750i \(-0.653645\pi\)
−0.924918 + 0.380166i \(0.875867\pi\)
\(212\) 0 0
\(213\) −10.9154 12.9153i −0.747913 0.884943i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 24.5272 + 8.82891i 1.65740 + 0.596603i
\(220\) 2.02191 + 11.4668i 0.136317 + 0.773094i
\(221\) 0.538387 1.47921i 0.0362158 0.0995022i
\(222\) 0 0
\(223\) −2.02474 + 11.4829i −0.135587 + 0.768950i 0.838863 + 0.544343i \(0.183221\pi\)
−0.974449 + 0.224607i \(0.927890\pi\)
\(224\) 0 0
\(225\) 14.1314 + 5.03031i 0.942092 + 0.335354i
\(226\) 0 0
\(227\) −10.2710 28.2195i −0.681713 1.87299i −0.418306 0.908306i \(-0.637376\pi\)
−0.263407 0.964685i \(-0.584846\pi\)
\(228\) 0 0
\(229\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(230\) 0 0
\(231\) −4.12043 11.1972i −0.271104 0.736722i
\(232\) 0 0
\(233\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(234\) 0 0
\(235\) −8.71103 + 15.0879i −0.568245 + 0.984229i
\(236\) 0 0
\(237\) −10.4786 + 18.2988i −0.680657 + 1.18864i
\(238\) 0 0
\(239\) −6.88969 + 1.21484i −0.445657 + 0.0785813i −0.391973 0.919977i \(-0.628207\pi\)
−0.0536837 + 0.998558i \(0.517096\pi\)
\(240\) 15.2470 2.74417i 0.984186 0.177136i
\(241\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(242\) 0 0
\(243\) −12.1170 9.80700i −0.777309 0.629119i
\(244\) 0 0
\(245\) −10.0612 + 11.9905i −0.642788 + 0.766044i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −25.6356 14.6799i −1.62459 0.930301i
\(250\) 0 0
\(251\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(252\) −14.8783 + 5.53495i −0.937246 + 0.348669i
\(253\) 0 0
\(254\) 0 0
\(255\) 29.8226 10.9743i 1.86756 0.687238i
\(256\) 12.2567 10.2846i 0.766044 0.642788i
\(257\) 4.27863 + 5.09908i 0.266894 + 0.318072i 0.882801 0.469747i \(-0.155655\pi\)
−0.615907 + 0.787819i \(0.711210\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0.743040 0.428994i 0.0460814 0.0266051i
\(261\) −10.8174 + 30.3889i −0.669583 + 1.88102i
\(262\) 0 0
\(263\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 21.0962 25.1415i 1.27914 1.52442i
\(273\) −0.671482 + 0.567506i −0.0406400 + 0.0343470i
\(274\) 0 0
\(275\) −4.45245 + 12.2330i −0.268493 + 0.737678i
\(276\) 0 0
\(277\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.7638 + 29.5734i 0.642116 + 1.76420i 0.644974 + 0.764204i \(0.276868\pi\)
−0.00285781 + 0.999996i \(0.500910\pi\)
\(282\) 0 0
\(283\) 25.6555 21.5275i 1.52506 1.27968i 0.701036 0.713126i \(-0.252721\pi\)
0.824025 0.566553i \(-0.191723\pi\)
\(284\) −12.5511 14.9579i −0.744774 0.887587i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 25.1607 43.5796i 1.48004 2.56350i
\(290\) 0 0
\(291\) −2.95407 5.07499i −0.173170 0.297501i
\(292\) 28.2853 + 10.2950i 1.65527 + 0.602470i
\(293\) −8.19677 + 1.44531i −0.478860 + 0.0844360i −0.407868 0.913041i \(-0.633728\pi\)
−0.0709922 + 0.997477i \(0.522617\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 8.58551 10.4555i 0.498182 0.606689i
\(298\) 0 0
\(299\) 0 0
\(300\) 16.2968 + 5.86627i 0.940898 + 0.338689i
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −15.3299 26.5522i −0.874926 1.51542i −0.856842 0.515579i \(-0.827577\pi\)
−0.0180837 0.999836i \(-0.505757\pi\)
\(308\) −4.71203 12.9462i −0.268493 0.737678i
\(309\) −4.74668 + 27.4888i −0.270029 + 1.56378i
\(310\) 0 0
\(311\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(312\) 0 0
\(313\) −30.0514 + 10.9378i −1.69861 + 0.618242i −0.995666 0.0930055i \(-0.970353\pi\)
−0.702941 + 0.711248i \(0.748130\pi\)
\(314\) 0 0
\(315\) −17.5000 2.95804i −0.986013 0.166667i
\(316\) −12.1744 + 21.0867i −0.684863 + 1.18622i
\(317\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(318\) 0 0
\(319\) −26.3065 9.57478i −1.47288 0.536085i
\(320\) 17.6168 3.10631i 0.984808 0.173648i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −13.9514 11.3736i −0.775075 0.631869i
\(325\) 0.959261 0.0532102
\(326\) 0 0
\(327\) 5.03376 + 27.9682i 0.278368 + 1.54664i
\(328\) 0 0
\(329\) 7.05042 19.3709i 0.388702 1.06795i
\(330\) 0 0
\(331\) −5.82903 + 33.0580i −0.320392 + 1.81703i 0.219860 + 0.975531i \(0.429440\pi\)
−0.540252 + 0.841503i \(0.681671\pi\)
\(332\) −29.5412 17.0556i −1.62129 0.936050i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −17.2025 + 6.33031i −0.938475 + 0.345347i
\(337\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 34.4808 12.5500i 1.86998 0.680618i
\(341\) 0 0
\(342\) 0 0
\(343\) 9.26013 16.0390i 0.500000 0.866025i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(348\) −12.6151 + 35.0456i −0.676243 + 1.87864i
\(349\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(350\) 0 0
\(351\) −0.940343 0.330983i −0.0501918 0.0176666i
\(352\) 0 0
\(353\) −22.5979 + 26.9311i −1.20276 + 1.43340i −0.330887 + 0.943670i \(0.607348\pi\)
−0.871876 + 0.489726i \(0.837097\pi\)
\(354\) 0 0
\(355\) −3.79089 21.4992i −0.201200 1.14106i
\(356\) 0 0
\(357\) −32.4956 + 18.9152i −1.71985 + 1.00110i
\(358\) 0 0
\(359\) 21.8765 + 12.6304i 1.15460 + 0.666608i 0.950004 0.312239i \(-0.101079\pi\)
0.204595 + 0.978847i \(0.434412\pi\)
\(360\) 0 0
\(361\) −9.50000 16.4545i −0.500000 0.866025i
\(362\) 0 0
\(363\) −5.61739 4.67974i −0.294836 0.245623i
\(364\) −0.777678 + 0.652549i −0.0407614 + 0.0342029i
\(365\) 21.6321 + 25.7801i 1.13227 + 1.34939i
\(366\) 0 0
\(367\) 6.92780 2.52151i 0.361628 0.131622i −0.154815 0.987944i \(-0.549478\pi\)
0.516443 + 0.856322i \(0.327256\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(374\) 0 0
\(375\) 12.5000 + 14.7902i 0.645497 + 0.763763i
\(376\) 0 0
\(377\) 2.06285i 0.106242i
\(378\) 0 0
\(379\) 10.5600 0.542430 0.271215 0.962519i \(-0.412575\pi\)
0.271215 + 0.962519i \(0.412575\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.22207 8.85257i 0.164640 0.452345i −0.829748 0.558138i \(-0.811516\pi\)
0.994388 + 0.105793i \(0.0337381\pi\)
\(384\) 0 0
\(385\) 2.67474 15.1692i 0.136317 0.773094i
\(386\) 0 0
\(387\) 0 0
\(388\) −3.39028 5.87214i −0.172115 0.298113i
\(389\) 6.46352 + 17.7584i 0.327714 + 0.900386i 0.988689 + 0.149979i \(0.0479205\pi\)
−0.660976 + 0.750407i \(0.729857\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −23.5756 + 13.6114i −1.18622 + 0.684863i
\(396\) 9.95640 12.0378i 0.500328 0.604920i
\(397\) −0.115521 + 0.200088i −0.00579782 + 0.0100421i −0.868910 0.494971i \(-0.835179\pi\)
0.863112 + 0.505013i \(0.168512\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 18.7939 + 6.84040i 0.939693 + 0.342020i
\(401\) 29.1310 5.13658i 1.45473 0.256509i 0.610300 0.792170i \(-0.291049\pi\)
0.844433 + 0.535662i \(0.179938\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −7.15028 18.8115i −0.355301 0.934752i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −5.59340 + 31.7217i −0.275567 + 1.56282i
\(413\) 0 0
\(414\) 0 0
\(415\) −19.0688 33.0281i −0.936050 1.62129i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(420\) −20.1950 3.48721i −0.985417 0.170158i
\(421\) 37.3547 13.5960i 1.82055 0.662628i 0.825372 0.564590i \(-0.190966\pi\)
0.995183 0.0980380i \(-0.0312567\pi\)
\(422\) 0 0
\(423\) 22.9897 4.22186i 1.11780 0.205274i
\(424\) 0 0
\(425\) 40.4015 + 7.12388i 1.95976 + 0.345559i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.293026 0.814042i 0.0141474 0.0393023i
\(430\) 0 0
\(431\) 39.9411i 1.92390i −0.273231 0.961948i \(-0.588092\pi\)
0.273231 0.961948i \(-0.411908\pi\)
\(432\) −16.0630 13.1901i −0.772832 0.634611i
\(433\) −40.1484 −1.92941 −0.964703 0.263339i \(-0.915176\pi\)
−0.964703 + 0.263339i \(0.915176\pi\)
\(434\) 0 0
\(435\) −31.8056 + 26.8807i −1.52496 + 1.28883i
\(436\) 5.69805 + 32.3152i 0.272887 + 1.54762i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(440\) 0 0
\(441\) 20.9995 0.148781i 0.999975 0.00708479i
\(442\) 0 0
\(443\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −22.3778 + 26.8615i −1.05844 + 1.27051i
\(448\) −19.8895 + 7.23920i −0.939693 + 0.342020i
\(449\) −24.6822 + 14.2503i −1.16482 + 0.672511i −0.952455 0.304679i \(-0.901451\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 1.40669 + 2.41664i 0.0660919 + 0.113544i
\(454\) 0 0
\(455\) −1.11777 + 0.197093i −0.0524018 + 0.00923985i
\(456\) 0 0
\(457\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(458\) 0 0
\(459\) −37.1468 20.9235i −1.73386 0.976627i
\(460\) 0 0
\(461\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(462\) 0 0
\(463\) 0 0 −0.173648 0.984808i \(-0.555556\pi\)
0.173648 + 0.984808i \(0.444444\pi\)
\(464\) −14.7100 + 40.4153i −0.682893 + 1.87623i
\(465\) 0 0
\(466\) 0 0
\(467\) 13.1585 + 7.59708i 0.608904 + 0.351551i 0.772536 0.634970i \(-0.218988\pi\)
−0.163632 + 0.986521i \(0.552321\pi\)
\(468\) −1.08445 0.386030i −0.0501289 0.0178443i
\(469\) 0 0
\(470\) 0 0
\(471\) 7.32034 42.3933i 0.337303 1.95338i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −37.5998 + 21.7083i −1.72339 + 0.994997i
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.984808 0.173648i \(-0.944444\pi\)
0.984808 + 0.173648i \(0.0555556\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −6.46722 5.42664i −0.293964 0.246666i
\(485\) 7.58090i 0.344231i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13.3672 36.7261i 0.603254 1.65743i −0.141380 0.989955i \(-0.545154\pi\)
0.744635 0.667472i \(-0.232624\pi\)
\(492\) 0 0
\(493\) −15.3196 + 86.8816i −0.689959 + 3.91295i
\(494\) 0 0
\(495\) 16.3696 6.08971i 0.735757 0.273712i
\(496\) 0 0
\(497\) 8.83460 + 24.2729i 0.396286 + 1.08879i
\(498\) 0 0
\(499\) 18.8433 15.8114i 0.843542 0.707816i −0.114816 0.993387i \(-0.536628\pi\)
0.958358 + 0.285571i \(0.0921833\pi\)
\(500\) 14.3732 + 17.1293i 0.642788 + 0.766044i
\(501\) −40.8677 7.05690i −1.82583 0.315279i
\(502\) 0 0
\(503\) 1.39864 0.807506i 0.0623624 0.0360049i −0.468495 0.883466i \(-0.655203\pi\)
0.530857 + 0.847461i \(0.321870\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 22.4528 0.0795375i 0.997162 0.00353239i
\(508\) 0 0
\(509\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(510\) 0 0
\(511\) −30.5034 25.5954i −1.34939 1.13227i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −23.1488 + 27.5876i −1.02006 + 1.21566i
\(516\) 0 0
\(517\) 3.52259 + 19.9776i 0.154923 + 0.878613i
\(518\) 0 0
\(519\) −22.5009 + 13.0974i −0.987679 + 0.574911i
\(520\) 0 0
\(521\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(522\) 0 0
\(523\) −21.9920 38.0913i −0.961644 1.66562i −0.718372 0.695659i \(-0.755112\pi\)
−0.243272 0.969958i \(-0.578221\pi\)
\(524\) 0 0
\(525\) −17.6043 14.6658i −0.768317 0.640070i
\(526\) 0 0
\(527\) 0 0
\(528\) 11.5458 13.8592i 0.502468 0.603144i
\(529\) 21.6129 7.86646i 0.939693 0.342020i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −21.6083 25.5673i −0.932468 1.10331i
\(538\) 0 0
\(539\) 18.2253i 0.785020i
\(540\) −8.17944 21.7508i −0.351987 0.936005i
\(541\) 34.3365 1.47624 0.738121 0.674668i \(-0.235713\pi\)
0.738121 + 0.674668i \(0.235713\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −12.5477 + 34.4744i −0.537482 + 1.47672i
\(546\) 0 0
\(547\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 24.6746 20.7045i 1.04927 0.880443i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −23.3048 4.10927i −0.984808 0.173648i
\(561\) 18.3869 32.1092i 0.776296 1.35565i
\(562\) 0 0
\(563\) −23.0436 + 4.06321i −0.971173 + 0.171244i −0.636658 0.771146i \(-0.719684\pi\)
−0.334515 + 0.942390i \(0.608573\pi\)
\(564\) 26.5633 4.78092i 1.11852 0.201313i
\(565\) 0 0
\(566\) 0 0
\(567\) 12.1969 + 20.4508i 0.512221 + 0.858854i
\(568\) 0 0
\(569\) 9.08090 10.8222i 0.380691 0.453690i −0.541341 0.840803i \(-0.682083\pi\)
0.922032 + 0.387113i \(0.126528\pi\)
\(570\) 0 0
\(571\) −3.11345 17.6573i −0.130294 0.738933i −0.978022 0.208502i \(-0.933141\pi\)
0.847728 0.530431i \(-0.177970\pi\)
\(572\) 0.341685 0.938771i 0.0142866 0.0392520i
\(573\) 8.89221 + 5.09201i 0.371478 + 0.212722i
\(574\) 0 0
\(575\) 0 0
\(576\) −18.4939 15.2963i −0.770579 0.637344i
\(577\) 21.8961 + 37.9252i 0.911547 + 1.57885i 0.811880 + 0.583825i \(0.198444\pi\)
0.0996670 + 0.995021i \(0.468222\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) −36.8357 + 30.9088i −1.52952 + 1.28342i
\(581\) 29.0058 + 34.5678i 1.20336 + 1.43411i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −0.834221 0.980001i −0.0344908 0.0405180i
\(586\) 0 0
\(587\) −36.2124 6.38523i −1.49465 0.263546i −0.634232 0.773142i \(-0.718684\pi\)
−0.860414 + 0.509596i \(0.829795\pi\)
\(588\) 24.2486 0.0858990i 0.999994 0.00354242i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 41.8642i 1.71916i 0.511003 + 0.859579i \(0.329274\pi\)
−0.511003 + 0.859579i \(0.670726\pi\)
\(594\) 0 0
\(595\) −48.5412 −1.98999
\(596\) −25.9494 + 30.9253i −1.06293 + 1.26675i
\(597\) 0 0
\(598\) 0 0
\(599\) −16.3241 + 44.8502i −0.666987 + 1.83253i −0.124975 + 0.992160i \(0.539885\pi\)
−0.542012 + 0.840371i \(0.682337\pi\)
\(600\) 0 0
\(601\) 0 0 0.173648 0.984808i \(-0.444444\pi\)
−0.173648 + 0.984808i \(0.555556\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.61441 + 2.79624i 0.0656893 + 0.113777i
\(605\) −3.22827 8.86961i −0.131248 0.360601i
\(606\) 0 0
\(607\) 30.5784 25.6584i 1.24114 1.04144i 0.243706 0.969849i \(-0.421637\pi\)
0.997434 0.0715913i \(-0.0228077\pi\)
\(608\) 0 0
\(609\) 31.5382 37.8573i 1.27799 1.53406i
\(610\) 0 0
\(611\) 1.29453 0.747397i 0.0523710 0.0302364i
\(612\) −42.8075 24.3122i −1.73039 0.982763i
\(613\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(618\) 0 0
\(619\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −1.25063 0.450182i −0.0500654 0.0180217i
\(625\) 4.34120 + 24.6202i 0.173648 + 0.984808i
\(626\) 0 0
\(627\) 0 0
\(628\) 8.62615 48.9213i 0.344221 1.95217i
\(629\) 0 0
\(630\) 0 0
\(631\) 25.1179 + 43.5055i 0.999928 + 1.73193i 0.510357 + 0.859962i \(0.329513\pi\)
0.489570 + 0.871964i \(0.337154\pi\)
\(632\) 0 0
\(633\) −6.32845 + 36.6491i −0.251533 + 1.45667i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.26197 0.459321i 0.0500012 0.0181990i
\(638\) 0 0
\(639\) −18.6673 + 22.5696i −0.738467 + 0.892841i
\(640\) 0 0
\(641\) 38.6564 + 6.81617i 1.52684 + 0.269223i 0.873116 0.487513i \(-0.162096\pi\)
0.653722 + 0.756735i \(0.273207\pi\)
\(642\) 0 0
\(643\) 42.2329 + 15.3715i 1.66550 + 0.606194i 0.991213 0.132273i \(-0.0422275\pi\)
0.674290 + 0.738467i \(0.264450\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 50.1875i 1.97307i −0.163543 0.986536i \(-0.552292\pi\)
0.163543 0.986536i \(-0.447708\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 7.52515 44.5194i 0.293584 1.73687i
\(658\) 0 0
\(659\) −12.7622 35.0638i −0.497144 1.36589i −0.894023 0.448022i \(-0.852129\pi\)
0.396878 0.917871i \(-0.370093\pi\)
\(660\) 18.9267 6.96479i 0.736722 0.271104i
\(661\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(662\) 0 0
\(663\) −2.68673 0.463935i −0.104344 0.0180177i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −47.1608 8.31572i −1.82471 0.321745i
\(669\) 20.1956 0.0715417i 0.780808 0.00276596i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.766044 0.642788i \(-0.777778\pi\)
0.766044 + 0.642788i \(0.222222\pi\)
\(674\) 0 0
\(675\) 4.23935 25.6326i 0.163173 0.986598i
\(676\) 25.9264 0.997169
\(677\) 29.3197 34.9418i 1.12685 1.34292i 0.194690 0.980865i \(-0.437630\pi\)
0.932156 0.362058i \(-0.117926\pi\)
\(678\) 0 0
\(679\) 1.55760 + 8.83357i 0.0597751 + 0.339001i
\(680\) 0 0
\(681\) −44.9534 + 26.1666i −1.72262 + 1.00271i
\(682\) 0 0
\(683\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(692\) −26.0352 + 15.0314i −0.989709 + 0.571409i
\(693\) −17.8233 + 10.4593i −0.677049 + 0.397317i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −20.2676 17.0066i −0.766044 0.642788i
\(701\) 52.9359i 1.99936i 0.0253058 + 0.999680i \(0.491944\pi\)
−0.0253058 + 0.999680i \(0.508056\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 13.3886 15.9559i 0.504601 0.601360i
\(705\) 28.3924 + 10.2202i 1.06932 + 0.384917i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −7.92032 + 44.9183i −0.297454 + 1.68694i 0.359605 + 0.933105i \(0.382911\pi\)
−0.657059 + 0.753839i \(0.728200\pi\)
\(710\) 0 0
\(711\) 34.4082 + 12.2482i 1.29041 + 0.459343i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0.855623 0.717953i 0.0319985 0.0268499i
\(716\) −24.8464 29.6108i −0.928554 1.10661i
\(717\) 4.18470 + 11.3719i 0.156280 + 0.424690i
\(718\) 0 0
\(719\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(720\) −9.35577 25.1489i −0.348669 0.937246i
\(721\) 21.3056 36.9024i 0.793463 1.37432i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −52.9446 + 9.33556i −1.96631 + 0.346714i
\(726\) 0 0
\(727\) 39.7818 + 33.3809i 1.47543 + 1.23803i 0.910909 + 0.412608i \(0.135382\pi\)
0.564517 + 0.825421i \(0.309062\pi\)
\(728\) 0 0
\(729\) −13.0000 + 23.6643i −0.481481 + 0.876456i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1.33817 7.58914i −0.0494264 0.280311i 0.950070 0.312036i \(-0.101011\pi\)
−0.999497 + 0.0317250i \(0.989900\pi\)
\(734\) 0 0
\(735\) 23.5266 + 13.4722i 0.867791 + 0.496929i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 25.8056 + 44.6967i 0.949276 + 1.64419i 0.746955 + 0.664875i \(0.231515\pi\)
0.202321 + 0.979319i \(0.435152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.642788 0.766044i \(-0.722222\pi\)
0.642788 + 0.766044i \(0.277778\pi\)
\(744\) 0 0
\(745\) −42.4132 + 15.4371i −1.55390 + 0.565573i
\(746\) 0 0
\(747\) −17.1590 + 48.2040i −0.627816 + 1.76369i
\(748\) 21.3626 37.0011i 0.781093 1.35289i
\(749\) 0 0
\(750\) 0 0
\(751\) 37.2224 + 13.5479i 1.35827 + 0.494368i 0.915518 0.402278i \(-0.131781\pi\)
0.442748 + 0.896646i \(0.354004\pi\)
\(752\) 30.6921 5.41184i 1.11922 0.197349i
\(753\) 0 0
\(754\) 0 0
\(755\) 3.60992i 0.131379i
\(756\) 14.0000 + 23.6643i 0.509175 + 0.860663i
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(762\) 0 0
\(763\) 7.53781 42.7490i 0.272887 1.54762i
\(764\) 10.2470 + 5.91608i 0.370722 + 0.214036i
\(765\) −27.8573 47.4703i −1.00718 1.71629i
\(766\) 0 0
\(767\) 0 0
\(768\) −21.2922 17.7381i −0.768317 0.640070i
\(769\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(770\) 0 0
\(771\) 7.37948 8.85805i 0.265765 0.319015i
\(772\) 0 0
\(773\) 18.2767 10.5521i 0.657367 0.379531i −0.133906 0.990994i \(-0.542752\pi\)
0.791273 + 0.611463i \(0.209419\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) −0.959261 1.13501i −0.0343470 0.0406400i
\(781\) −19.4723 16.3392i −0.696773 0.584662i
\(782\) 0 0
\(783\) 55.1216 + 9.11653i 1.96989 + 0.325798i
\(784\) 28.0000 1.00000
\(785\) 35.7001 42.5457i 1.27419 1.51852i
\(786\) 0 0
\(787\) 4.37961 + 24.8380i 0.156116 + 0.885378i 0.957758 + 0.287574i \(0.0928488\pi\)
−0.801642 + 0.597804i \(0.796040\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −24.2467 28.8961i −0.858863 1.02355i −0.999439 0.0334835i \(-0.989340\pi\)
0.140576 0.990070i \(-0.455105\pi\)
\(798\) 0 0
\(799\) 60.0726 21.8647i 2.12522 0.773516i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 38.5899 + 6.80444i 1.36181 + 0.240124i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 45.5987i 1.60317i 0.597884 + 0.801583i \(0.296008\pi\)
−0.597884 + 0.801583i \(0.703992\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 36.5718 43.5846i 1.28342 1.52952i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −49.3301 28.2482i −1.72690 0.988886i
\(817\) 0 0
\(818\) 0 0
\(819\) 1.17342 + 0.970534i 0.0410027 + 0.0339132i
\(820\) 0 0
\(821\) 10.9812 + 30.1706i 0.383247 + 1.05296i 0.969981 + 0.243182i \(0.0781913\pi\)
−0.586734 + 0.809780i \(0.699586\pi\)
\(822\) 0 0
\(823\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(824\) 0 0
\(825\) 22.2192 + 3.83673i 0.773572 + 0.133578i
\(826\) 0 0
\(827\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(828\) 0 0
\(829\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.44226 0.524938i −0.0500012 0.0181990i
\(833\) 56.5621 9.97343i 1.95976 0.345559i
\(834\) 0 0
\(835\) −41.0146 34.4154i −1.41937 1.19099i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(840\) 0 0
\(841\) −15.0399 85.2955i −0.518617 2.94122i
\(842\) 0 0
\(843\) 47.1102 27.4220i 1.62256 0.944465i
\(844\) −7.45733 + 42.2926i −0.256692 + 1.45577i
\(845\) 25.1031 + 14.4933i 0.863573 + 0.498584i
\(846\) 0 0
\(847\) 5.58409 + 9.67193i 0.191872 + 0.332332i
\(848\) 0 0
\(849\) −44.5684 37.1291i −1.52959 1.27427i
\(850\) 0 0
\(851\) 0 0
\(852\) −21.6473 + 25.9846i −0.741625 + 0.890219i
\(853\) 52.6443 19.1609i 1.80251 0.656058i 0.804430 0.594048i \(-0.202471\pi\)
0.998075 0.0620108i \(-0.0197513\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −39.2042 6.91277i −1.33919 0.236136i −0.542263 0.840209i \(-0.682432\pi\)
−0.796929 + 0.604073i \(0.793543\pi\)
\(858\) 0 0
\(859\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −33.6113 −1.14282
\(866\) 0 0
\(867\) −82.0079 29.5199i −2.78513 1.00255i
\(868\) 0 0
\(869\) −10.8412 + 29.7859i −0.367761 + 1.01042i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −7.74481 + 6.59273i −0.262122 + 0.223130i
\(874\) 0 0
\(875\) −10.1171 27.7965i −0.342020 0.939693i
\(876\) 8.87134 51.3755i 0.299735 1.73582i
\(877\) 0 0 0.766044 0.642788i \(-0.222222\pi\)
−0.766044 + 0.642788i \(0.777778\pi\)
\(878\) 0 0
\(879\) 4.97860 + 13.5293i 0.167924 + 0.456331i
\(880\) 21.8831 7.96478i 0.737678 0.268493i
\(881\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(882\) 0 0
\(883\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(884\) −3.10045 0.546693i −0.104279 0.0183873i
\(885\) 0 0
\(886\) 0 0
\(887\) 22.9998 4.05549i 0.772259 0.136170i 0.226385 0.974038i \(-0.427309\pi\)
0.545873 + 0.837868i \(0.316198\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −20.4572 11.4276i −0.685341 0.382838i
\(892\) 23.3200 0.780813
\(893\) 0 0
\(894\) 0 0
\(895\) −7.50449 42.5601i −0.250848 1.42263i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 5.00000 29.5804i 0.166667 0.986013i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 0.939693 0.342020i \(-0.111111\pi\)
−0.939693 + 0.342020i \(0.888889\pi\)
\(908\) −52.0144 + 30.0305i −1.72616 + 0.996598i
\(909\) 0 0
\(910\) 0 0
\(911\) −58.7903 10.3663i −1.94781 0.343451i −0.999673 0.0255863i \(-0.991855\pi\)
−0.948136 0.317865i \(-0.897034\pi\)
\(912\) 0 0
\(913\) −41.7283 15.1879i −1.38101 0.502645i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −60.6113 −1.99938 −0.999691 0.0248659i \(-0.992084\pi\)
−0.999691 + 0.0248659i \(0.992084\pi\)
\(920\) 0 0
\(921\) −40.5592 + 34.2788i −1.33647 + 1.12952i
\(922\) 0 0
\(923\) −0.640626 + 1.76010i −0.0210864 + 0.0579345i
\(924\) −20.6232 + 12.0044i −0.678453 + 0.394916i
\(925\) 0 0
\(926\) 0 0
\(927\) 48.3154 0.342313i 1.58689 0.0112430i
\(928\) 0 0
\(929\) 0 0 −0.342020 0.939693i \(-0.611111\pi\)
0.342020 + 0.939693i \(0.388889\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 41.3684 23.8841i 1.35289 0.781093i
\(936\) 0 0
\(937\) 21.1518 36.6360i 0.690999 1.19685i −0.280512 0.959851i \(-0.590504\pi\)
0.971511 0.236995i \(-0.0761625\pi\)
\(938\) 0 0
\(939\) 27.8653 + 47.8717i 0.909350 + 1.56223i
\(940\) 32.7428 + 11.9174i 1.06795 + 0.388702i
\(941\) 0 0 0.984808 0.173648i \(-0.0555556\pi\)
−0.984808 + 0.173648i \(0.944444\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0.326685 + 30.7391i 0.0106271 + 0.999944i
\(946\) 0 0
\(947\) 0 0 0.642788 0.766044i \(-0.277778\pi\)
−0.642788 + 0.766044i \(0.722222\pi\)
\(948\) 39.6808 + 14.2837i 1.28877 + 0.463911i
\(949\) −0.501397 2.84357i −0.0162760 0.0923060i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(954\) 0 0
\(955\) 6.61438 + 11.4564i 0.214036 + 0.370722i
\(956\) 4.78553 + 13.1481i 0.154775 + 0.425241i
\(957\) −8.25071 + 47.7813i −0.266708 + 1.54455i
\(958\) 0 0
\(959\) 0 0
\(960\) −10.7002 29.0776i −0.345347 0.938475i
\(961\) −29.1305 + 10.6026i −0.939693 + 0.342020i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 −0.939693 0.342020i \(-0.888889\pi\)
0.939693 + 0.342020i \(0.111111\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) −15.1078 + 27.2719i −0.484583 + 0.874745i
\(973\) 0 0
\(974\) 0 0
\(975\) −0.294309 1.63521i −0.00942543 0.0523688i
\(976\) 0 0
\(977\) 0 0 0.342020 0.939693i \(-0.388889\pi\)
−0.342020 + 0.939693i \(0.611111\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 27.1109 + 15.6525i 0.866025 + 0.500000i
\(981\) 46.1318 17.1617i 1.47288 0.547931i
\(982\) 0 0
\(983\) −4.59871 12.6348i −0.146676 0.402989i 0.844498 0.535559i \(-0.179899\pi\)
−0.991174 + 0.132570i \(0.957677\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −35.1839 6.07544i −1.11992 0.193383i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 8.50881 14.7377i 0.270291 0.468158i −0.698645 0.715468i \(-0.746213\pi\)
0.968936 + 0.247310i \(0.0795467\pi\)
\(992\) 0 0
\(993\) 58.1412 0.205961i 1.84505 0.00653599i
\(994\) 0 0
\(995\) 0 0
\(996\) −20.0106 + 55.5906i −0.634060 + 1.76146i
\(997\) 14.1873 + 11.9046i 0.449318 + 0.377022i 0.839183 0.543850i \(-0.183034\pi\)
−0.389865 + 0.920872i \(0.627478\pi\)
\(998\) 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 945.2.cs.a.524.2 24
5.4 even 2 inner 945.2.cs.a.524.3 yes 24
7.6 odd 2 inner 945.2.cs.a.524.3 yes 24
27.5 odd 18 inner 945.2.cs.a.734.2 yes 24
35.34 odd 2 CM 945.2.cs.a.524.2 24
135.59 odd 18 inner 945.2.cs.a.734.3 yes 24
189.167 even 18 inner 945.2.cs.a.734.3 yes 24
945.734 even 18 inner 945.2.cs.a.734.2 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
945.2.cs.a.524.2 24 1.1 even 1 trivial
945.2.cs.a.524.2 24 35.34 odd 2 CM
945.2.cs.a.524.3 yes 24 5.4 even 2 inner
945.2.cs.a.524.3 yes 24 7.6 odd 2 inner
945.2.cs.a.734.2 yes 24 27.5 odd 18 inner
945.2.cs.a.734.2 yes 24 945.734 even 18 inner
945.2.cs.a.734.3 yes 24 135.59 odd 18 inner
945.2.cs.a.734.3 yes 24 189.167 even 18 inner