Properties

Label 320.2.l.a.81.7
Level $320$
Weight $2$
Character 320.81
Analytic conductor $2.555$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,2,Mod(81,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 320.l (of order \(4\), 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: \(2.55521286468\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 4 x^{14} + 7 x^{12} - 8 x^{11} - 28 x^{10} + 28 x^{9} + 17 x^{8} + 56 x^{7} - 112 x^{6} - 64 x^{5} + 112 x^{4} + 256 x^{2} - 512 x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 81.7
Root \(1.32070 - 0.505727i\) of defining polynomial
Character \(\chi\) \(=\) 320.81
Dual form 320.2.l.a.241.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.66366 + 1.66366i) q^{3} +(-0.707107 + 0.707107i) q^{5} +2.89402i q^{7} +2.53555i q^{9} +O(q^{10})\) \(q+(1.66366 + 1.66366i) q^{3} +(-0.707107 + 0.707107i) q^{5} +2.89402i q^{7} +2.53555i q^{9} +(-1.84462 + 1.84462i) q^{11} +(-3.08011 - 3.08011i) q^{13} -2.35278 q^{15} +7.29875 q^{17} +(1.23593 + 1.23593i) q^{19} +(-4.81468 + 4.81468i) q^{21} -4.60490i q^{23} -1.00000i q^{25} +(0.772683 - 0.772683i) q^{27} +(4.24680 + 4.24680i) q^{29} -2.06299 q^{31} -6.13767 q^{33} +(-2.04638 - 2.04638i) q^{35} +(-1.17899 + 1.17899i) q^{37} -10.2485i q^{39} -4.61484i q^{41} +(-3.03019 + 3.03019i) q^{43} +(-1.79291 - 1.79291i) q^{45} +11.7111 q^{47} -1.37537 q^{49} +(12.1427 + 12.1427i) q^{51} +(2.73048 - 2.73048i) q^{53} -2.60869i q^{55} +4.11235i q^{57} +(-3.11306 + 3.11306i) q^{59} +(2.34962 + 2.34962i) q^{61} -7.33795 q^{63} +4.35593 q^{65} +(-8.24311 - 8.24311i) q^{67} +(7.66101 - 7.66101i) q^{69} -3.25937i q^{71} -12.6877i q^{73} +(1.66366 - 1.66366i) q^{75} +(-5.33839 - 5.33839i) q^{77} +0.113885 q^{79} +10.1776 q^{81} +(-9.76813 - 9.76813i) q^{83} +(-5.16100 + 5.16100i) q^{85} +14.1305i q^{87} +3.74593i q^{89} +(8.91390 - 8.91390i) q^{91} +(-3.43212 - 3.43212i) q^{93} -1.74787 q^{95} -13.9853 q^{97} +(-4.67714 - 4.67714i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{11} + 8 q^{15} + 8 q^{19} - 24 q^{27} - 16 q^{29} - 16 q^{37} - 8 q^{43} + 40 q^{47} - 16 q^{49} + 32 q^{51} + 16 q^{53} + 8 q^{59} + 16 q^{61} - 40 q^{63} - 40 q^{67} + 16 q^{69} + 16 q^{77} - 16 q^{79} - 16 q^{81} - 40 q^{83} - 16 q^{85} - 32 q^{91} - 48 q^{93} - 32 q^{95} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 1.66366 + 1.66366i 0.960517 + 0.960517i 0.999250 0.0387330i \(-0.0123322\pi\)
−0.0387330 + 0.999250i \(0.512332\pi\)
\(4\) 0 0
\(5\) −0.707107 + 0.707107i −0.316228 + 0.316228i
\(6\) 0 0
\(7\) 2.89402i 1.09384i 0.837186 + 0.546919i \(0.184199\pi\)
−0.837186 + 0.546919i \(0.815801\pi\)
\(8\) 0 0
\(9\) 2.53555i 0.845184i
\(10\) 0 0
\(11\) −1.84462 + 1.84462i −0.556175 + 0.556175i −0.928216 0.372041i \(-0.878658\pi\)
0.372041 + 0.928216i \(0.378658\pi\)
\(12\) 0 0
\(13\) −3.08011 3.08011i −0.854268 0.854268i 0.136388 0.990656i \(-0.456451\pi\)
−0.990656 + 0.136388i \(0.956451\pi\)
\(14\) 0 0
\(15\) −2.35278 −0.607484
\(16\) 0 0
\(17\) 7.29875 1.77021 0.885104 0.465393i \(-0.154087\pi\)
0.885104 + 0.465393i \(0.154087\pi\)
\(18\) 0 0
\(19\) 1.23593 + 1.23593i 0.283542 + 0.283542i 0.834520 0.550978i \(-0.185745\pi\)
−0.550978 + 0.834520i \(0.685745\pi\)
\(20\) 0 0
\(21\) −4.81468 + 4.81468i −1.05065 + 1.05065i
\(22\) 0 0
\(23\) 4.60490i 0.960189i −0.877217 0.480094i \(-0.840602\pi\)
0.877217 0.480094i \(-0.159398\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) 0 0
\(27\) 0.772683 0.772683i 0.148703 0.148703i
\(28\) 0 0
\(29\) 4.24680 + 4.24680i 0.788611 + 0.788611i 0.981266 0.192656i \(-0.0617101\pi\)
−0.192656 + 0.981266i \(0.561710\pi\)
\(30\) 0 0
\(31\) −2.06299 −0.370524 −0.185262 0.982689i \(-0.559313\pi\)
−0.185262 + 0.982689i \(0.559313\pi\)
\(32\) 0 0
\(33\) −6.13767 −1.06843
\(34\) 0 0
\(35\) −2.04638 2.04638i −0.345902 0.345902i
\(36\) 0 0
\(37\) −1.17899 + 1.17899i −0.193825 + 0.193825i −0.797346 0.603522i \(-0.793764\pi\)
0.603522 + 0.797346i \(0.293764\pi\)
\(38\) 0 0
\(39\) 10.2485i 1.64108i
\(40\) 0 0
\(41\) 4.61484i 0.720717i −0.932814 0.360359i \(-0.882654\pi\)
0.932814 0.360359i \(-0.117346\pi\)
\(42\) 0 0
\(43\) −3.03019 + 3.03019i −0.462099 + 0.462099i −0.899343 0.437244i \(-0.855955\pi\)
0.437244 + 0.899343i \(0.355955\pi\)
\(44\) 0 0
\(45\) −1.79291 1.79291i −0.267271 0.267271i
\(46\) 0 0
\(47\) 11.7111 1.70823 0.854117 0.520081i \(-0.174098\pi\)
0.854117 + 0.520081i \(0.174098\pi\)
\(48\) 0 0
\(49\) −1.37537 −0.196481
\(50\) 0 0
\(51\) 12.1427 + 12.1427i 1.70031 + 1.70031i
\(52\) 0 0
\(53\) 2.73048 2.73048i 0.375061 0.375061i −0.494256 0.869316i \(-0.664559\pi\)
0.869316 + 0.494256i \(0.164559\pi\)
\(54\) 0 0
\(55\) 2.60869i 0.351756i
\(56\) 0 0
\(57\) 4.11235i 0.544694i
\(58\) 0 0
\(59\) −3.11306 + 3.11306i −0.405285 + 0.405285i −0.880091 0.474805i \(-0.842518\pi\)
0.474805 + 0.880091i \(0.342518\pi\)
\(60\) 0 0
\(61\) 2.34962 + 2.34962i 0.300838 + 0.300838i 0.841342 0.540503i \(-0.181766\pi\)
−0.540503 + 0.841342i \(0.681766\pi\)
\(62\) 0 0
\(63\) −7.33795 −0.924495
\(64\) 0 0
\(65\) 4.35593 0.540286
\(66\) 0 0
\(67\) −8.24311 8.24311i −1.00706 1.00706i −0.999975 0.00708173i \(-0.997746\pi\)
−0.00708173 0.999975i \(-0.502254\pi\)
\(68\) 0 0
\(69\) 7.66101 7.66101i 0.922277 0.922277i
\(70\) 0 0
\(71\) 3.25937i 0.386816i −0.981118 0.193408i \(-0.938046\pi\)
0.981118 0.193408i \(-0.0619541\pi\)
\(72\) 0 0
\(73\) 12.6877i 1.48499i −0.669853 0.742494i \(-0.733643\pi\)
0.669853 0.742494i \(-0.266357\pi\)
\(74\) 0 0
\(75\) 1.66366 1.66366i 0.192103 0.192103i
\(76\) 0 0
\(77\) −5.33839 5.33839i −0.608365 0.608365i
\(78\) 0 0
\(79\) 0.113885 0.0128130 0.00640652 0.999979i \(-0.497961\pi\)
0.00640652 + 0.999979i \(0.497961\pi\)
\(80\) 0 0
\(81\) 10.1776 1.13085
\(82\) 0 0
\(83\) −9.76813 9.76813i −1.07219 1.07219i −0.997183 0.0750089i \(-0.976101\pi\)
−0.0750089 0.997183i \(-0.523899\pi\)
\(84\) 0 0
\(85\) −5.16100 + 5.16100i −0.559789 + 0.559789i
\(86\) 0 0
\(87\) 14.1305i 1.51495i
\(88\) 0 0
\(89\) 3.74593i 0.397068i 0.980094 + 0.198534i \(0.0636180\pi\)
−0.980094 + 0.198534i \(0.936382\pi\)
\(90\) 0 0
\(91\) 8.91390 8.91390i 0.934430 0.934430i
\(92\) 0 0
\(93\) −3.43212 3.43212i −0.355894 0.355894i
\(94\) 0 0
\(95\) −1.74787 −0.179328
\(96\) 0 0
\(97\) −13.9853 −1.41999 −0.709995 0.704206i \(-0.751303\pi\)
−0.709995 + 0.704206i \(0.751303\pi\)
\(98\) 0 0
\(99\) −4.67714 4.67714i −0.470071 0.470071i
\(100\) 0 0
\(101\) 3.52228 3.52228i 0.350480 0.350480i −0.509808 0.860288i \(-0.670284\pi\)
0.860288 + 0.509808i \(0.170284\pi\)
\(102\) 0 0
\(103\) 0.150216i 0.0148013i 0.999973 + 0.00740063i \(0.00235572\pi\)
−0.999973 + 0.00740063i \(0.997644\pi\)
\(104\) 0 0
\(105\) 6.80899i 0.664489i
\(106\) 0 0
\(107\) 2.75062 2.75062i 0.265912 0.265912i −0.561539 0.827451i \(-0.689790\pi\)
0.827451 + 0.561539i \(0.189790\pi\)
\(108\) 0 0
\(109\) −6.90778 6.90778i −0.661646 0.661646i 0.294122 0.955768i \(-0.404973\pi\)
−0.955768 + 0.294122i \(0.904973\pi\)
\(110\) 0 0
\(111\) −3.92288 −0.372344
\(112\) 0 0
\(113\) −3.49507 −0.328788 −0.164394 0.986395i \(-0.552567\pi\)
−0.164394 + 0.986395i \(0.552567\pi\)
\(114\) 0 0
\(115\) 3.25616 + 3.25616i 0.303638 + 0.303638i
\(116\) 0 0
\(117\) 7.80977 7.80977i 0.722014 0.722014i
\(118\) 0 0
\(119\) 21.1228i 1.93632i
\(120\) 0 0
\(121\) 4.19472i 0.381338i
\(122\) 0 0
\(123\) 7.67755 7.67755i 0.692261 0.692261i
\(124\) 0 0
\(125\) 0.707107 + 0.707107i 0.0632456 + 0.0632456i
\(126\) 0 0
\(127\) −6.25357 −0.554915 −0.277458 0.960738i \(-0.589492\pi\)
−0.277458 + 0.960738i \(0.589492\pi\)
\(128\) 0 0
\(129\) −10.0824 −0.887708
\(130\) 0 0
\(131\) 5.16490 + 5.16490i 0.451259 + 0.451259i 0.895772 0.444513i \(-0.146623\pi\)
−0.444513 + 0.895772i \(0.646623\pi\)
\(132\) 0 0
\(133\) −3.57681 + 3.57681i −0.310149 + 0.310149i
\(134\) 0 0
\(135\) 1.09274i 0.0940480i
\(136\) 0 0
\(137\) 18.9408i 1.61823i 0.587654 + 0.809113i \(0.300052\pi\)
−0.587654 + 0.809113i \(0.699948\pi\)
\(138\) 0 0
\(139\) −2.79057 + 2.79057i −0.236693 + 0.236693i −0.815479 0.578786i \(-0.803527\pi\)
0.578786 + 0.815479i \(0.303527\pi\)
\(140\) 0 0
\(141\) 19.4833 + 19.4833i 1.64079 + 1.64079i
\(142\) 0 0
\(143\) 11.3633 0.950245
\(144\) 0 0
\(145\) −6.00588 −0.498761
\(146\) 0 0
\(147\) −2.28815 2.28815i −0.188723 0.188723i
\(148\) 0 0
\(149\) −1.60372 + 1.60372i −0.131382 + 0.131382i −0.769740 0.638358i \(-0.779614\pi\)
0.638358 + 0.769740i \(0.279614\pi\)
\(150\) 0 0
\(151\) 2.53754i 0.206502i 0.994655 + 0.103251i \(0.0329245\pi\)
−0.994655 + 0.103251i \(0.967076\pi\)
\(152\) 0 0
\(153\) 18.5064i 1.49615i
\(154\) 0 0
\(155\) 1.45875 1.45875i 0.117170 0.117170i
\(156\) 0 0
\(157\) 10.2405 + 10.2405i 0.817278 + 0.817278i 0.985713 0.168435i \(-0.0538712\pi\)
−0.168435 + 0.985713i \(0.553871\pi\)
\(158\) 0 0
\(159\) 9.08521 0.720504
\(160\) 0 0
\(161\) 13.3267 1.05029
\(162\) 0 0
\(163\) −8.02607 8.02607i −0.628650 0.628650i 0.319078 0.947728i \(-0.396627\pi\)
−0.947728 + 0.319078i \(0.896627\pi\)
\(164\) 0 0
\(165\) 4.33999 4.33999i 0.337868 0.337868i
\(166\) 0 0
\(167\) 6.82611i 0.528221i −0.964492 0.264110i \(-0.914922\pi\)
0.964492 0.264110i \(-0.0850783\pi\)
\(168\) 0 0
\(169\) 5.97411i 0.459547i
\(170\) 0 0
\(171\) −3.13377 + 3.13377i −0.239645 + 0.239645i
\(172\) 0 0
\(173\) 5.08901 + 5.08901i 0.386910 + 0.386910i 0.873584 0.486674i \(-0.161790\pi\)
−0.486674 + 0.873584i \(0.661790\pi\)
\(174\) 0 0
\(175\) 2.89402 0.218768
\(176\) 0 0
\(177\) −10.3582 −0.778567
\(178\) 0 0
\(179\) −1.63797 1.63797i −0.122428 0.122428i 0.643238 0.765666i \(-0.277590\pi\)
−0.765666 + 0.643238i \(0.777590\pi\)
\(180\) 0 0
\(181\) −16.7757 + 16.7757i −1.24693 + 1.24693i −0.289855 + 0.957071i \(0.593607\pi\)
−0.957071 + 0.289855i \(0.906393\pi\)
\(182\) 0 0
\(183\) 7.81797i 0.577921i
\(184\) 0 0
\(185\) 1.66734i 0.122585i
\(186\) 0 0
\(187\) −13.4635 + 13.4635i −0.984546 + 0.984546i
\(188\) 0 0
\(189\) 2.23616 + 2.23616i 0.162657 + 0.162657i
\(190\) 0 0
\(191\) −5.85815 −0.423881 −0.211940 0.977283i \(-0.567978\pi\)
−0.211940 + 0.977283i \(0.567978\pi\)
\(192\) 0 0
\(193\) −0.0241155 −0.00173587 −0.000867935 1.00000i \(-0.500276\pi\)
−0.000867935 1.00000i \(0.500276\pi\)
\(194\) 0 0
\(195\) 7.24680 + 7.24680i 0.518954 + 0.518954i
\(196\) 0 0
\(197\) 14.9086 14.9086i 1.06219 1.06219i 0.0642576 0.997933i \(-0.479532\pi\)
0.997933 0.0642576i \(-0.0204679\pi\)
\(198\) 0 0
\(199\) 13.6525i 0.967801i 0.875123 + 0.483900i \(0.160780\pi\)
−0.875123 + 0.483900i \(0.839220\pi\)
\(200\) 0 0
\(201\) 27.4275i 1.93459i
\(202\) 0 0
\(203\) −12.2903 + 12.2903i −0.862612 + 0.862612i
\(204\) 0 0
\(205\) 3.26319 + 3.26319i 0.227911 + 0.227911i
\(206\) 0 0
\(207\) 11.6760 0.811537
\(208\) 0 0
\(209\) −4.55966 −0.315398
\(210\) 0 0
\(211\) 2.45103 + 2.45103i 0.168736 + 0.168736i 0.786424 0.617688i \(-0.211930\pi\)
−0.617688 + 0.786424i \(0.711930\pi\)
\(212\) 0 0
\(213\) 5.42249 5.42249i 0.371543 0.371543i
\(214\) 0 0
\(215\) 4.28533i 0.292257i
\(216\) 0 0
\(217\) 5.97033i 0.405293i
\(218\) 0 0
\(219\) 21.1081 21.1081i 1.42636 1.42636i
\(220\) 0 0
\(221\) −22.4809 22.4809i −1.51223 1.51223i
\(222\) 0 0
\(223\) −13.9483 −0.934045 −0.467023 0.884245i \(-0.654673\pi\)
−0.467023 + 0.884245i \(0.654673\pi\)
\(224\) 0 0
\(225\) 2.53555 0.169037
\(226\) 0 0
\(227\) −4.43883 4.43883i −0.294616 0.294616i 0.544285 0.838901i \(-0.316801\pi\)
−0.838901 + 0.544285i \(0.816801\pi\)
\(228\) 0 0
\(229\) −5.35068 + 5.35068i −0.353583 + 0.353583i −0.861441 0.507858i \(-0.830438\pi\)
0.507858 + 0.861441i \(0.330438\pi\)
\(230\) 0 0
\(231\) 17.7626i 1.16869i
\(232\) 0 0
\(233\) 11.9370i 0.782019i 0.920387 + 0.391010i \(0.127874\pi\)
−0.920387 + 0.391010i \(0.872126\pi\)
\(234\) 0 0
\(235\) −8.28097 + 8.28097i −0.540191 + 0.540191i
\(236\) 0 0
\(237\) 0.189466 + 0.189466i 0.0123071 + 0.0123071i
\(238\) 0 0
\(239\) 16.7720 1.08489 0.542445 0.840091i \(-0.317499\pi\)
0.542445 + 0.840091i \(0.317499\pi\)
\(240\) 0 0
\(241\) −22.0294 −1.41904 −0.709519 0.704686i \(-0.751088\pi\)
−0.709519 + 0.704686i \(0.751088\pi\)
\(242\) 0 0
\(243\) 14.6141 + 14.6141i 0.937495 + 0.937495i
\(244\) 0 0
\(245\) 0.972532 0.972532i 0.0621328 0.0621328i
\(246\) 0 0
\(247\) 7.61360i 0.484442i
\(248\) 0 0
\(249\) 32.5018i 2.05972i
\(250\) 0 0
\(251\) −6.63925 + 6.63925i −0.419066 + 0.419066i −0.884882 0.465816i \(-0.845761\pi\)
0.465816 + 0.884882i \(0.345761\pi\)
\(252\) 0 0
\(253\) 8.49432 + 8.49432i 0.534033 + 0.534033i
\(254\) 0 0
\(255\) −17.1723 −1.07537
\(256\) 0 0
\(257\) −7.25821 −0.452755 −0.226377 0.974040i \(-0.572688\pi\)
−0.226377 + 0.974040i \(0.572688\pi\)
\(258\) 0 0
\(259\) −3.41202 3.41202i −0.212013 0.212013i
\(260\) 0 0
\(261\) −10.7680 + 10.7680i −0.666521 + 0.666521i
\(262\) 0 0
\(263\) 9.27431i 0.571878i −0.958248 0.285939i \(-0.907694\pi\)
0.958248 0.285939i \(-0.0923055\pi\)
\(264\) 0 0
\(265\) 3.86149i 0.237209i
\(266\) 0 0
\(267\) −6.23197 + 6.23197i −0.381390 + 0.381390i
\(268\) 0 0
\(269\) −13.4195 13.4195i −0.818199 0.818199i 0.167648 0.985847i \(-0.446383\pi\)
−0.985847 + 0.167648i \(0.946383\pi\)
\(270\) 0 0
\(271\) −22.5999 −1.37285 −0.686423 0.727202i \(-0.740820\pi\)
−0.686423 + 0.727202i \(0.740820\pi\)
\(272\) 0 0
\(273\) 29.6595 1.79507
\(274\) 0 0
\(275\) 1.84462 + 1.84462i 0.111235 + 0.111235i
\(276\) 0 0
\(277\) 16.2015 16.2015i 0.973451 0.973451i −0.0262056 0.999657i \(-0.508342\pi\)
0.999657 + 0.0262056i \(0.00834245\pi\)
\(278\) 0 0
\(279\) 5.23082i 0.313161i
\(280\) 0 0
\(281\) 8.84793i 0.527824i 0.964547 + 0.263912i \(0.0850128\pi\)
−0.964547 + 0.263912i \(0.914987\pi\)
\(282\) 0 0
\(283\) 20.3062 20.3062i 1.20708 1.20708i 0.235109 0.971969i \(-0.424455\pi\)
0.971969 0.235109i \(-0.0755447\pi\)
\(284\) 0 0
\(285\) −2.90787 2.90787i −0.172247 0.172247i
\(286\) 0 0
\(287\) 13.3555 0.788348
\(288\) 0 0
\(289\) 36.2718 2.13364
\(290\) 0 0
\(291\) −23.2668 23.2668i −1.36392 1.36392i
\(292\) 0 0
\(293\) −7.16936 + 7.16936i −0.418839 + 0.418839i −0.884803 0.465965i \(-0.845707\pi\)
0.465965 + 0.884803i \(0.345707\pi\)
\(294\) 0 0
\(295\) 4.40253i 0.256325i
\(296\) 0 0
\(297\) 2.85062i 0.165410i
\(298\) 0 0
\(299\) −14.1836 + 14.1836i −0.820258 + 0.820258i
\(300\) 0 0
\(301\) −8.76943 8.76943i −0.505461 0.505461i
\(302\) 0 0
\(303\) 11.7198 0.673284
\(304\) 0 0
\(305\) −3.32287 −0.190267
\(306\) 0 0
\(307\) 18.4308 + 18.4308i 1.05190 + 1.05190i 0.998577 + 0.0533241i \(0.0169816\pi\)
0.0533241 + 0.998577i \(0.483018\pi\)
\(308\) 0 0
\(309\) −0.249910 + 0.249910i −0.0142169 + 0.0142169i
\(310\) 0 0
\(311\) 7.08961i 0.402015i 0.979590 + 0.201007i \(0.0644215\pi\)
−0.979590 + 0.201007i \(0.935578\pi\)
\(312\) 0 0
\(313\) 22.0477i 1.24621i −0.782139 0.623104i \(-0.785871\pi\)
0.782139 0.623104i \(-0.214129\pi\)
\(314\) 0 0
\(315\) 5.18871 5.18871i 0.292351 0.292351i
\(316\) 0 0
\(317\) −6.19670 6.19670i −0.348042 0.348042i 0.511338 0.859380i \(-0.329150\pi\)
−0.859380 + 0.511338i \(0.829150\pi\)
\(318\) 0 0
\(319\) −15.6675 −0.877211
\(320\) 0 0
\(321\) 9.15220 0.510826
\(322\) 0 0
\(323\) 9.02076 + 9.02076i 0.501929 + 0.501929i
\(324\) 0 0
\(325\) −3.08011 + 3.08011i −0.170854 + 0.170854i
\(326\) 0 0
\(327\) 22.9845i 1.27104i
\(328\) 0 0
\(329\) 33.8921i 1.86853i
\(330\) 0 0
\(331\) 18.6174 18.6174i 1.02330 1.02330i 0.0235823 0.999722i \(-0.492493\pi\)
0.999722 0.0235823i \(-0.00750716\pi\)
\(332\) 0 0
\(333\) −2.98939 2.98939i −0.163818 0.163818i
\(334\) 0 0
\(335\) 11.6575 0.636919
\(336\) 0 0
\(337\) 14.2577 0.776666 0.388333 0.921519i \(-0.373051\pi\)
0.388333 + 0.921519i \(0.373051\pi\)
\(338\) 0 0
\(339\) −5.81462 5.81462i −0.315807 0.315807i
\(340\) 0 0
\(341\) 3.80544 3.80544i 0.206076 0.206076i
\(342\) 0 0
\(343\) 16.2778i 0.878919i
\(344\) 0 0
\(345\) 10.8343i 0.583299i
\(346\) 0 0
\(347\) 23.5395 23.5395i 1.26367 1.26367i 0.314363 0.949303i \(-0.398209\pi\)
0.949303 0.314363i \(-0.101791\pi\)
\(348\) 0 0
\(349\) −1.56682 1.56682i −0.0838701 0.0838701i 0.663927 0.747797i \(-0.268888\pi\)
−0.747797 + 0.663927i \(0.768888\pi\)
\(350\) 0 0
\(351\) −4.75989 −0.254064
\(352\) 0 0
\(353\) 9.44678 0.502801 0.251401 0.967883i \(-0.419109\pi\)
0.251401 + 0.967883i \(0.419109\pi\)
\(354\) 0 0
\(355\) 2.30472 + 2.30472i 0.122322 + 0.122322i
\(356\) 0 0
\(357\) −35.1412 + 35.1412i −1.85987 + 1.85987i
\(358\) 0 0
\(359\) 18.0452i 0.952392i −0.879339 0.476196i \(-0.842015\pi\)
0.879339 0.476196i \(-0.157985\pi\)
\(360\) 0 0
\(361\) 15.9449i 0.839208i
\(362\) 0 0
\(363\) −6.97860 + 6.97860i −0.366282 + 0.366282i
\(364\) 0 0
\(365\) 8.97159 + 8.97159i 0.469594 + 0.469594i
\(366\) 0 0
\(367\) −29.1329 −1.52073 −0.760363 0.649498i \(-0.774979\pi\)
−0.760363 + 0.649498i \(0.774979\pi\)
\(368\) 0 0
\(369\) 11.7012 0.609139
\(370\) 0 0
\(371\) 7.90208 + 7.90208i 0.410255 + 0.410255i
\(372\) 0 0
\(373\) 3.35598 3.35598i 0.173766 0.173766i −0.614866 0.788632i \(-0.710790\pi\)
0.788632 + 0.614866i \(0.210790\pi\)
\(374\) 0 0
\(375\) 2.35278i 0.121497i
\(376\) 0 0
\(377\) 26.1612i 1.34737i
\(378\) 0 0
\(379\) −11.6507 + 11.6507i −0.598457 + 0.598457i −0.939902 0.341445i \(-0.889084\pi\)
0.341445 + 0.939902i \(0.389084\pi\)
\(380\) 0 0
\(381\) −10.4038 10.4038i −0.533005 0.533005i
\(382\) 0 0
\(383\) 21.8044 1.11415 0.557077 0.830461i \(-0.311923\pi\)
0.557077 + 0.830461i \(0.311923\pi\)
\(384\) 0 0
\(385\) 7.54962 0.384764
\(386\) 0 0
\(387\) −7.68320 7.68320i −0.390559 0.390559i
\(388\) 0 0
\(389\) −11.8899 + 11.8899i −0.602842 + 0.602842i −0.941066 0.338224i \(-0.890174\pi\)
0.338224 + 0.941066i \(0.390174\pi\)
\(390\) 0 0
\(391\) 33.6101i 1.69973i
\(392\) 0 0
\(393\) 17.1853i 0.866884i
\(394\) 0 0
\(395\) −0.0805287 + 0.0805287i −0.00405184 + 0.00405184i
\(396\) 0 0
\(397\) −9.23905 9.23905i −0.463694 0.463694i 0.436170 0.899864i \(-0.356335\pi\)
−0.899864 + 0.436170i \(0.856335\pi\)
\(398\) 0 0
\(399\) −11.9012 −0.595807
\(400\) 0 0
\(401\) −14.4744 −0.722818 −0.361409 0.932407i \(-0.617704\pi\)
−0.361409 + 0.932407i \(0.617704\pi\)
\(402\) 0 0
\(403\) 6.35422 + 6.35422i 0.316526 + 0.316526i
\(404\) 0 0
\(405\) −7.19667 + 7.19667i −0.357605 + 0.357605i
\(406\) 0 0
\(407\) 4.34958i 0.215601i
\(408\) 0 0
\(409\) 9.54117i 0.471781i 0.971780 + 0.235890i \(0.0758006\pi\)
−0.971780 + 0.235890i \(0.924199\pi\)
\(410\) 0 0
\(411\) −31.5112 + 31.5112i −1.55433 + 1.55433i
\(412\) 0 0
\(413\) −9.00925 9.00925i −0.443316 0.443316i
\(414\) 0 0
\(415\) 13.8142 0.678114
\(416\) 0 0
\(417\) −9.28514 −0.454695
\(418\) 0 0
\(419\) −0.837667 0.837667i −0.0409227 0.0409227i 0.686349 0.727272i \(-0.259212\pi\)
−0.727272 + 0.686349i \(0.759212\pi\)
\(420\) 0 0
\(421\) 17.9679 17.9679i 0.875702 0.875702i −0.117385 0.993087i \(-0.537451\pi\)
0.993087 + 0.117385i \(0.0374511\pi\)
\(422\) 0 0
\(423\) 29.6940i 1.44377i
\(424\) 0 0
\(425\) 7.29875i 0.354042i
\(426\) 0 0
\(427\) −6.79986 + 6.79986i −0.329068 + 0.329068i
\(428\) 0 0
\(429\) 18.9047 + 18.9047i 0.912726 + 0.912726i
\(430\) 0 0
\(431\) 3.85473 0.185676 0.0928380 0.995681i \(-0.470406\pi\)
0.0928380 + 0.995681i \(0.470406\pi\)
\(432\) 0 0
\(433\) −25.5651 −1.22858 −0.614289 0.789081i \(-0.710557\pi\)
−0.614289 + 0.789081i \(0.710557\pi\)
\(434\) 0 0
\(435\) −9.99176 9.99176i −0.479068 0.479068i
\(436\) 0 0
\(437\) 5.69135 5.69135i 0.272254 0.272254i
\(438\) 0 0
\(439\) 30.1311i 1.43808i −0.694970 0.719039i \(-0.744582\pi\)
0.694970 0.719039i \(-0.255418\pi\)
\(440\) 0 0
\(441\) 3.48732i 0.166063i
\(442\) 0 0
\(443\) −20.1625 + 20.1625i −0.957948 + 0.957948i −0.999151 0.0412027i \(-0.986881\pi\)
0.0412027 + 0.999151i \(0.486881\pi\)
\(444\) 0 0
\(445\) −2.64877 2.64877i −0.125564 0.125564i
\(446\) 0 0
\(447\) −5.33610 −0.252389
\(448\) 0 0
\(449\) −36.5827 −1.72644 −0.863221 0.504826i \(-0.831557\pi\)
−0.863221 + 0.504826i \(0.831557\pi\)
\(450\) 0 0
\(451\) 8.51265 + 8.51265i 0.400845 + 0.400845i
\(452\) 0 0
\(453\) −4.22161 + 4.22161i −0.198349 + 0.198349i
\(454\) 0 0
\(455\) 12.6062i 0.590986i
\(456\) 0 0
\(457\) 16.7340i 0.782785i 0.920224 + 0.391392i \(0.128006\pi\)
−0.920224 + 0.391392i \(0.871994\pi\)
\(458\) 0 0
\(459\) 5.63962 5.63962i 0.263235 0.263235i
\(460\) 0 0
\(461\) 11.8377 + 11.8377i 0.551335 + 0.551335i 0.926826 0.375491i \(-0.122526\pi\)
−0.375491 + 0.926826i \(0.622526\pi\)
\(462\) 0 0
\(463\) 32.2711 1.49976 0.749882 0.661572i \(-0.230110\pi\)
0.749882 + 0.661572i \(0.230110\pi\)
\(464\) 0 0
\(465\) 4.85375 0.225087
\(466\) 0 0
\(467\) −1.22565 1.22565i −0.0567163 0.0567163i 0.678180 0.734896i \(-0.262769\pi\)
−0.734896 + 0.678180i \(0.762769\pi\)
\(468\) 0 0
\(469\) 23.8558 23.8558i 1.10156 1.10156i
\(470\) 0 0
\(471\) 34.0734i 1.57002i
\(472\) 0 0
\(473\) 11.1791i 0.514016i
\(474\) 0 0
\(475\) 1.23593 1.23593i 0.0567084 0.0567084i
\(476\) 0 0
\(477\) 6.92328 + 6.92328i 0.316995 + 0.316995i
\(478\) 0 0
\(479\) −28.8399 −1.31773 −0.658865 0.752261i \(-0.728963\pi\)
−0.658865 + 0.752261i \(0.728963\pi\)
\(480\) 0 0
\(481\) 7.26282 0.331156
\(482\) 0 0
\(483\) 22.1711 + 22.1711i 1.00882 + 1.00882i
\(484\) 0 0
\(485\) 9.88909 9.88909i 0.449041 0.449041i
\(486\) 0 0
\(487\) 32.1668i 1.45762i 0.684718 + 0.728808i \(0.259925\pi\)
−0.684718 + 0.728808i \(0.740075\pi\)
\(488\) 0 0
\(489\) 26.7054i 1.20766i
\(490\) 0 0
\(491\) 5.43607 5.43607i 0.245326 0.245326i −0.573723 0.819049i \(-0.694501\pi\)
0.819049 + 0.573723i \(0.194501\pi\)
\(492\) 0 0
\(493\) 30.9963 + 30.9963i 1.39600 + 1.39600i
\(494\) 0 0
\(495\) 6.61448 0.297299
\(496\) 0 0
\(497\) 9.43268 0.423114
\(498\) 0 0
\(499\) 17.1282 + 17.1282i 0.766762 + 0.766762i 0.977535 0.210773i \(-0.0675981\pi\)
−0.210773 + 0.977535i \(0.567598\pi\)
\(500\) 0 0
\(501\) 11.3564 11.3564i 0.507365 0.507365i
\(502\) 0 0
\(503\) 23.5180i 1.04862i −0.851529 0.524308i \(-0.824324\pi\)
0.851529 0.524308i \(-0.175676\pi\)
\(504\) 0 0
\(505\) 4.98126i 0.221663i
\(506\) 0 0
\(507\) −9.93890 + 9.93890i −0.441402 + 0.441402i
\(508\) 0 0
\(509\) 20.3147 + 20.3147i 0.900434 + 0.900434i 0.995474 0.0950391i \(-0.0302976\pi\)
−0.0950391 + 0.995474i \(0.530298\pi\)
\(510\) 0 0
\(511\) 36.7186 1.62434
\(512\) 0 0
\(513\) 1.90997 0.0843271
\(514\) 0 0
\(515\) −0.106219 0.106219i −0.00468057 0.00468057i
\(516\) 0 0
\(517\) −21.6025 + 21.6025i −0.950077 + 0.950077i
\(518\) 0 0
\(519\) 16.9328i 0.743268i
\(520\) 0 0
\(521\) 35.5082i 1.55564i 0.628487 + 0.777820i \(0.283675\pi\)
−0.628487 + 0.777820i \(0.716325\pi\)
\(522\) 0 0
\(523\) −0.677766 + 0.677766i −0.0296366 + 0.0296366i −0.721770 0.692133i \(-0.756671\pi\)
0.692133 + 0.721770i \(0.256671\pi\)
\(524\) 0 0
\(525\) 4.81468 + 4.81468i 0.210130 + 0.210130i
\(526\) 0 0
\(527\) −15.0572 −0.655904
\(528\) 0 0
\(529\) 1.79485 0.0780371
\(530\) 0 0
\(531\) −7.89332 7.89332i −0.342541 0.342541i
\(532\) 0 0
\(533\) −14.2142 + 14.2142i −0.615686 + 0.615686i
\(534\) 0 0
\(535\) 3.88996i 0.168178i
\(536\) 0 0
\(537\) 5.45007i 0.235188i
\(538\) 0 0
\(539\) 2.53704 2.53704i 0.109278 0.109278i
\(540\) 0 0
\(541\) 5.37099 + 5.37099i 0.230917 + 0.230917i 0.813075 0.582158i \(-0.197792\pi\)
−0.582158 + 0.813075i \(0.697792\pi\)
\(542\) 0 0
\(543\) −55.8181 −2.39539
\(544\) 0 0
\(545\) 9.76908 0.418461
\(546\) 0 0
\(547\) −8.86782 8.86782i −0.379161 0.379161i 0.491639 0.870799i \(-0.336398\pi\)
−0.870799 + 0.491639i \(0.836398\pi\)
\(548\) 0 0
\(549\) −5.95760 + 5.95760i −0.254264 + 0.254264i
\(550\) 0 0
\(551\) 10.4975i 0.447209i
\(552\) 0 0
\(553\) 0.329585i 0.0140154i
\(554\) 0 0
\(555\) 2.77390 2.77390i 0.117745 0.117745i
\(556\) 0 0
\(557\) −22.8089 22.8089i −0.966446 0.966446i 0.0330091 0.999455i \(-0.489491\pi\)
−0.999455 + 0.0330091i \(0.989491\pi\)
\(558\) 0 0
\(559\) 18.6666 0.789513
\(560\) 0 0
\(561\) −44.7973 −1.89135
\(562\) 0 0
\(563\) −20.9711 20.9711i −0.883826 0.883826i 0.110095 0.993921i \(-0.464884\pi\)
−0.993921 + 0.110095i \(0.964884\pi\)
\(564\) 0 0
\(565\) 2.47139 2.47139i 0.103972 0.103972i
\(566\) 0 0
\(567\) 29.4543i 1.23696i
\(568\) 0 0
\(569\) 8.05295i 0.337597i 0.985651 + 0.168799i \(0.0539888\pi\)
−0.985651 + 0.168799i \(0.946011\pi\)
\(570\) 0 0
\(571\) 22.5040 22.5040i 0.941762 0.941762i −0.0566333 0.998395i \(-0.518037\pi\)
0.998395 + 0.0566333i \(0.0180366\pi\)
\(572\) 0 0
\(573\) −9.74599 9.74599i −0.407144 0.407144i
\(574\) 0 0
\(575\) −4.60490 −0.192038
\(576\) 0 0
\(577\) 15.9819 0.665334 0.332667 0.943044i \(-0.392051\pi\)
0.332667 + 0.943044i \(0.392051\pi\)
\(578\) 0 0
\(579\) −0.0401200 0.0401200i −0.00166733 0.00166733i
\(580\) 0 0
\(581\) 28.2692 28.2692i 1.17280 1.17280i
\(582\) 0 0
\(583\) 10.0734i 0.417199i
\(584\) 0 0
\(585\) 11.0447i 0.456642i
\(586\) 0 0
\(587\) 5.25752 5.25752i 0.217001 0.217001i −0.590232 0.807233i \(-0.700964\pi\)
0.807233 + 0.590232i \(0.200964\pi\)
\(588\) 0 0
\(589\) −2.54971 2.54971i −0.105059 0.105059i
\(590\) 0 0
\(591\) 49.6057 2.04050
\(592\) 0 0
\(593\) 3.96571 0.162852 0.0814260 0.996679i \(-0.474053\pi\)
0.0814260 + 0.996679i \(0.474053\pi\)
\(594\) 0 0
\(595\) −14.9360 14.9360i −0.612318 0.612318i
\(596\) 0 0
\(597\) −22.7132 + 22.7132i −0.929589 + 0.929589i
\(598\) 0 0
\(599\) 8.31600i 0.339783i 0.985463 + 0.169891i \(0.0543417\pi\)
−0.985463 + 0.169891i \(0.945658\pi\)
\(600\) 0 0
\(601\) 46.0550i 1.87862i 0.343068 + 0.939310i \(0.388534\pi\)
−0.343068 + 0.939310i \(0.611466\pi\)
\(602\) 0 0
\(603\) 20.9009 20.9009i 0.851149 0.851149i
\(604\) 0 0
\(605\) −2.96612 2.96612i −0.120590 0.120590i
\(606\) 0 0
\(607\) 5.05760 0.205282 0.102641 0.994718i \(-0.467271\pi\)
0.102641 + 0.994718i \(0.467271\pi\)
\(608\) 0 0
\(609\) −40.8940 −1.65711
\(610\) 0 0
\(611\) −36.0713 36.0713i −1.45929 1.45929i
\(612\) 0 0
\(613\) 31.2000 31.2000i 1.26016 1.26016i 0.309141 0.951016i \(-0.399958\pi\)
0.951016 0.309141i \(-0.100042\pi\)
\(614\) 0 0
\(615\) 10.8577i 0.437824i
\(616\) 0 0
\(617\) 30.7412i 1.23759i −0.785551 0.618796i \(-0.787621\pi\)
0.785551 0.618796i \(-0.212379\pi\)
\(618\) 0 0
\(619\) 16.8766 16.8766i 0.678329 0.678329i −0.281293 0.959622i \(-0.590763\pi\)
0.959622 + 0.281293i \(0.0907632\pi\)
\(620\) 0 0
\(621\) −3.55813 3.55813i −0.142783 0.142783i
\(622\) 0 0
\(623\) −10.8408 −0.434328
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) −7.58574 7.58574i −0.302945 0.302945i
\(628\) 0 0
\(629\) −8.60515 + 8.60515i −0.343110 + 0.343110i
\(630\) 0 0
\(631\) 30.7318i 1.22342i 0.791084 + 0.611708i \(0.209517\pi\)
−0.791084 + 0.611708i \(0.790483\pi\)
\(632\) 0 0
\(633\) 8.15539i 0.324147i
\(634\) 0 0
\(635\) 4.42195 4.42195i 0.175480 0.175480i
\(636\) 0 0
\(637\) 4.23628 + 4.23628i 0.167848 + 0.167848i
\(638\) 0 0
\(639\) 8.26430 0.326931
\(640\) 0 0
\(641\) 22.1658 0.875496 0.437748 0.899098i \(-0.355776\pi\)
0.437748 + 0.899098i \(0.355776\pi\)
\(642\) 0 0
\(643\) −0.975773 0.975773i −0.0384807 0.0384807i 0.687605 0.726085i \(-0.258662\pi\)
−0.726085 + 0.687605i \(0.758662\pi\)
\(644\) 0 0
\(645\) 7.12935 7.12935i 0.280718 0.280718i
\(646\) 0 0
\(647\) 23.2610i 0.914484i −0.889342 0.457242i \(-0.848837\pi\)
0.889342 0.457242i \(-0.151163\pi\)
\(648\) 0 0
\(649\) 11.4848i 0.450819i
\(650\) 0 0
\(651\) 9.93263 9.93263i 0.389290 0.389290i
\(652\) 0 0
\(653\) −23.9372 23.9372i −0.936735 0.936735i 0.0613792 0.998115i \(-0.480450\pi\)
−0.998115 + 0.0613792i \(0.980450\pi\)
\(654\) 0 0
\(655\) −7.30427 −0.285401
\(656\) 0 0
\(657\) 32.1704 1.25509
\(658\) 0 0
\(659\) 14.1064 + 14.1064i 0.549508 + 0.549508i 0.926299 0.376790i \(-0.122972\pi\)
−0.376790 + 0.926299i \(0.622972\pi\)
\(660\) 0 0
\(661\) −3.04121 + 3.04121i −0.118289 + 0.118289i −0.763774 0.645484i \(-0.776656\pi\)
0.645484 + 0.763774i \(0.276656\pi\)
\(662\) 0 0
\(663\) 74.8014i 2.90505i
\(664\) 0 0
\(665\) 5.05838i 0.196156i
\(666\) 0 0
\(667\) 19.5561 19.5561i 0.757215 0.757215i
\(668\) 0 0
\(669\) −23.2052 23.2052i −0.897166 0.897166i
\(670\) 0 0
\(671\) −8.66835 −0.334638
\(672\) 0 0
\(673\) −25.3628 −0.977662 −0.488831 0.872378i \(-0.662577\pi\)
−0.488831 + 0.872378i \(0.662577\pi\)
\(674\) 0 0
\(675\) −0.772683 0.772683i −0.0297406 0.0297406i
\(676\) 0 0
\(677\) 9.36526 9.36526i 0.359936 0.359936i −0.503853 0.863789i \(-0.668085\pi\)
0.863789 + 0.503853i \(0.168085\pi\)
\(678\) 0 0
\(679\) 40.4737i 1.55324i
\(680\) 0 0
\(681\) 14.7695i 0.565967i
\(682\) 0 0
\(683\) −4.20530 + 4.20530i −0.160911 + 0.160911i −0.782970 0.622059i \(-0.786296\pi\)
0.622059 + 0.782970i \(0.286296\pi\)
\(684\) 0 0
\(685\) −13.3932 13.3932i −0.511728 0.511728i
\(686\) 0 0
\(687\) −17.8035 −0.679245
\(688\) 0 0
\(689\) −16.8204 −0.640804
\(690\) 0 0
\(691\) −5.79295 5.79295i −0.220374 0.220374i 0.588282 0.808656i \(-0.299805\pi\)
−0.808656 + 0.588282i \(0.799805\pi\)
\(692\) 0 0
\(693\) 13.5358 13.5358i 0.514181 0.514181i
\(694\) 0 0
\(695\) 3.94646i 0.149698i
\(696\) 0 0
\(697\) 33.6826i 1.27582i
\(698\) 0 0
\(699\) −19.8592 + 19.8592i −0.751142 + 0.751142i
\(700\) 0 0
\(701\) −0.258991 0.258991i −0.00978196 0.00978196i 0.702199 0.711981i \(-0.252202\pi\)
−0.711981 + 0.702199i \(0.752202\pi\)
\(702\) 0 0
\(703\) −2.91430 −0.109915
\(704\) 0 0
\(705\) −27.5535 −1.03772
\(706\) 0 0
\(707\) 10.1936 + 10.1936i 0.383368 + 0.383368i
\(708\) 0 0
\(709\) −0.751674 + 0.751674i −0.0282297 + 0.0282297i −0.721081 0.692851i \(-0.756354\pi\)
0.692851 + 0.721081i \(0.256354\pi\)
\(710\) 0 0
\(711\) 0.288761i 0.0108294i
\(712\) 0 0
\(713\) 9.49986i 0.355773i
\(714\) 0 0
\(715\) −8.03505 + 8.03505i −0.300494 + 0.300494i
\(716\) 0 0
\(717\) 27.9029 + 27.9029i 1.04205 + 1.04205i
\(718\) 0 0
\(719\) −39.6557 −1.47891 −0.739455 0.673206i \(-0.764917\pi\)
−0.739455 + 0.673206i \(0.764917\pi\)
\(720\) 0 0
\(721\) −0.434730 −0.0161902
\(722\) 0 0
\(723\) −36.6495 36.6495i −1.36301 1.36301i
\(724\) 0 0
\(725\) 4.24680 4.24680i 0.157722 0.157722i
\(726\) 0 0
\(727\) 22.2952i 0.826881i 0.910531 + 0.413441i \(0.135673\pi\)
−0.910531 + 0.413441i \(0.864327\pi\)
\(728\) 0 0
\(729\) 18.0930i 0.670112i
\(730\) 0 0
\(731\) −22.1166 + 22.1166i −0.818011 + 0.818011i
\(732\) 0 0
\(733\) 28.2309 + 28.2309i 1.04273 + 1.04273i 0.999045 + 0.0436851i \(0.0139098\pi\)
0.0436851 + 0.999045i \(0.486090\pi\)
\(734\) 0 0
\(735\) 3.23593 0.119359
\(736\) 0 0
\(737\) 30.4109 1.12020
\(738\) 0 0
\(739\) 5.45140 + 5.45140i 0.200533 + 0.200533i 0.800228 0.599695i \(-0.204712\pi\)
−0.599695 + 0.800228i \(0.704712\pi\)
\(740\) 0 0
\(741\) 12.6665 12.6665i 0.465314 0.465314i
\(742\) 0 0
\(743\) 52.5667i 1.92849i 0.265020 + 0.964243i \(0.414621\pi\)
−0.265020 + 0.964243i \(0.585379\pi\)
\(744\) 0 0
\(745\) 2.26800i 0.0830931i
\(746\) 0 0
\(747\) 24.7676 24.7676i 0.906200 0.906200i
\(748\) 0 0
\(749\) 7.96035 + 7.96035i 0.290865 + 0.290865i
\(750\) 0 0
\(751\) −31.0189 −1.13190 −0.565948 0.824441i \(-0.691490\pi\)
−0.565948 + 0.824441i \(0.691490\pi\)
\(752\) 0 0
\(753\) −22.0910 −0.805040
\(754\) 0 0
\(755\) −1.79431 1.79431i −0.0653016 0.0653016i
\(756\) 0 0
\(757\) −2.47389 + 2.47389i −0.0899152 + 0.0899152i −0.750634 0.660719i \(-0.770252\pi\)
0.660719 + 0.750634i \(0.270252\pi\)
\(758\) 0 0
\(759\) 28.2634i 1.02590i
\(760\) 0 0
\(761\) 2.48375i 0.0900358i 0.998986 + 0.0450179i \(0.0143345\pi\)
−0.998986 + 0.0450179i \(0.985666\pi\)
\(762\) 0 0
\(763\) 19.9913 19.9913i 0.723733 0.723733i
\(764\) 0 0
\(765\) −13.0860 13.0860i −0.473125 0.473125i
\(766\) 0 0
\(767\) 19.1771 0.692444
\(768\) 0 0
\(769\) 43.4690 1.56753 0.783767 0.621055i \(-0.213296\pi\)
0.783767 + 0.621055i \(0.213296\pi\)
\(770\) 0 0
\(771\) −12.0752 12.0752i −0.434879 0.434879i
\(772\) 0 0
\(773\) 0.297026 0.297026i 0.0106833 0.0106833i −0.701745 0.712428i \(-0.747595\pi\)
0.712428 + 0.701745i \(0.247595\pi\)
\(774\) 0 0
\(775\) 2.06299i 0.0741047i
\(776\) 0 0
\(777\) 11.3529i 0.407283i
\(778\) 0 0
\(779\) 5.70363 5.70363i 0.204354 0.204354i
\(780\) 0 0
\(781\) 6.01231 + 6.01231i 0.215137 + 0.215137i
\(782\) 0 0
\(783\) 6.56286 0.234537
\(784\) 0 0
\(785\) −14.4822 −0.516892
\(786\) 0 0
\(787\) 23.6931 + 23.6931i 0.844567 + 0.844567i 0.989449 0.144882i \(-0.0462802\pi\)
−0.144882 + 0.989449i \(0.546280\pi\)
\(788\) 0 0
\(789\) 15.4293 15.4293i 0.549299 0.549299i
\(790\) 0 0
\(791\) 10.1148i 0.359641i
\(792\) 0 0
\(793\) 14.4742i 0.513993i
\(794\) 0 0
\(795\) −6.42421 + 6.42421i −0.227843 + 0.227843i
\(796\) 0 0
\(797\) 38.2292 + 38.2292i 1.35415 + 1.35415i 0.880963 + 0.473186i \(0.156896\pi\)
0.473186 + 0.880963i \(0.343104\pi\)
\(798\) 0 0
\(799\) 85.4762 3.02393
\(800\) 0 0
\(801\) −9.49801 −0.335596
\(802\) 0 0
\(803\) 23.4041 + 23.4041i 0.825913 + 0.825913i
\(804\) 0 0
\(805\) −9.42340 + 9.42340i −0.332131 + 0.332131i
\(806\) 0 0
\(807\) 44.6509i 1.57179i
\(808\) 0 0
\(809\) 53.8310i 1.89260i −0.323296 0.946298i \(-0.604791\pi\)
0.323296 0.946298i \(-0.395209\pi\)
\(810\) 0 0
\(811\) −27.0549 + 27.0549i −0.950025 + 0.950025i −0.998809 0.0487847i \(-0.984465\pi\)
0.0487847 + 0.998809i \(0.484465\pi\)
\(812\) 0 0
\(813\) −37.5986 37.5986i −1.31864 1.31864i
\(814\) 0 0
\(815\) 11.3506 0.397593
\(816\) 0 0
\(817\) −7.49020 −0.262049
\(818\) 0 0
\(819\) 22.6017 + 22.6017i 0.789766 + 0.789766i
\(820\) 0 0
\(821\) −24.2170 + 24.2170i −0.845180 + 0.845180i −0.989527 0.144347i \(-0.953892\pi\)
0.144347 + 0.989527i \(0.453892\pi\)
\(822\) 0 0
\(823\) 41.3013i 1.43967i −0.694144 0.719836i \(-0.744217\pi\)
0.694144 0.719836i \(-0.255783\pi\)
\(824\) 0 0
\(825\) 6.13767i 0.213686i
\(826\) 0 0
\(827\) −15.7264 + 15.7264i −0.546862 + 0.546862i −0.925532 0.378670i \(-0.876382\pi\)
0.378670 + 0.925532i \(0.376382\pi\)
\(828\) 0 0
\(829\) −20.7323 20.7323i −0.720061 0.720061i 0.248556 0.968618i \(-0.420044\pi\)
−0.968618 + 0.248556i \(0.920044\pi\)
\(830\) 0 0
\(831\) 53.9075 1.87003
\(832\) 0 0
\(833\) −10.0385 −0.347813
\(834\) 0 0
\(835\) 4.82679 + 4.82679i 0.167038 + 0.167038i
\(836\) 0 0
\(837\) −1.59404 + 1.59404i −0.0550980 + 0.0550980i
\(838\) 0 0
\(839\) 43.6919i 1.50841i 0.656638 + 0.754206i \(0.271978\pi\)
−0.656638 + 0.754206i \(0.728022\pi\)
\(840\) 0 0
\(841\) 7.07060i 0.243814i
\(842\) 0 0
\(843\) −14.7200 + 14.7200i −0.506983 + 0.506983i
\(844\) 0 0
\(845\) −4.22433 4.22433i −0.145321 0.145321i
\(846\) 0 0
\(847\) −12.1396 −0.417122
\(848\) 0 0
\(849\) 67.5654 2.31884
\(850\) 0 0
\(851\) 5.42913 + 5.42913i 0.186108 + 0.186108i
\(852\) 0 0
\(853\) −35.0610 + 35.0610i −1.20046 + 1.20046i −0.226439 + 0.974025i \(0.572708\pi\)
−0.974025 + 0.226439i \(0.927292\pi\)
\(854\) 0 0
\(855\) 4.43182i 0.151565i
\(856\) 0 0
\(857\) 45.3397i 1.54878i 0.632711 + 0.774388i \(0.281942\pi\)
−0.632711 + 0.774388i \(0.718058\pi\)
\(858\) 0 0
\(859\) −32.1229 + 32.1229i −1.09602 + 1.09602i −0.101147 + 0.994871i \(0.532251\pi\)
−0.994871 + 0.101147i \(0.967749\pi\)
\(860\) 0 0
\(861\) 22.2190 + 22.2190i 0.757221 + 0.757221i
\(862\) 0 0
\(863\) −36.9142 −1.25657 −0.628287 0.777981i \(-0.716244\pi\)
−0.628287 + 0.777981i \(0.716244\pi\)
\(864\) 0 0
\(865\) −7.19695 −0.244704
\(866\) 0 0
\(867\) 60.3441 + 60.3441i 2.04939 + 2.04939i
\(868\) 0 0
\(869\) −0.210075 + 0.210075i −0.00712630 + 0.00712630i
\(870\) 0 0
\(871\) 50.7793i 1.72059i
\(872\) 0 0
\(873\) 35.4604i 1.20015i
\(874\) 0 0
\(875\) −2.04638 + 2.04638i −0.0691804 + 0.0691804i
\(876\) 0 0
\(877\) −15.7178 15.7178i −0.530753 0.530753i 0.390044 0.920796i \(-0.372460\pi\)
−0.920796 + 0.390044i \(0.872460\pi\)
\(878\) 0 0
\(879\) −23.8548 −0.804603
\(880\) 0 0
\(881\) 1.16748 0.0393335 0.0196667 0.999807i \(-0.493739\pi\)
0.0196667 + 0.999807i \(0.493739\pi\)
\(882\) 0 0
\(883\) −32.2410 32.2410i −1.08500 1.08500i −0.996035 0.0889621i \(-0.971645\pi\)
−0.0889621 0.996035i \(-0.528355\pi\)
\(884\) 0 0
\(885\) 7.32432 7.32432i 0.246204 0.246204i
\(886\) 0 0
\(887\) 42.7282i 1.43467i −0.696728 0.717336i \(-0.745361\pi\)
0.696728 0.717336i \(-0.254639\pi\)
\(888\) 0 0
\(889\) 18.0980i 0.606987i
\(890\) 0 0
\(891\) −18.7739 + 18.7739i −0.628949 + 0.628949i
\(892\) 0 0
\(893\) 14.4741 + 14.4741i 0.484356 + 0.484356i
\(894\) 0 0
\(895\) 2.31644 0.0774302
\(896\) 0 0
\(897\) −47.1935 −1.57574
\(898\) 0 0
\(899\) −8.76109 8.76109i −0.292199 0.292199i
\(900\) 0 0
\(901\) 19.9291 19.9291i 0.663935 0.663935i
\(902\) 0 0
\(903\) 29.1788i 0.971008i
\(904\) 0 0
\(905\) 23.7244i 0.788625i
\(906\) 0 0
\(907\) −1.23335 + 1.23335i −0.0409528 + 0.0409528i −0.727287 0.686334i \(-0.759219\pi\)
0.686334 + 0.727287i \(0.259219\pi\)
\(908\) 0 0
\(909\) 8.93093 + 8.93093i 0.296220 + 0.296220i
\(910\) 0 0
\(911\) −23.9284 −0.792785 −0.396392 0.918081i \(-0.629738\pi\)
−0.396392 + 0.918081i \(0.629738\pi\)
\(912\) 0 0
\(913\) 36.0371 1.19265
\(914\) 0 0
\(915\) −5.52814 5.52814i −0.182755 0.182755i
\(916\) 0 0
\(917\) −14.9473 + 14.9473i −0.493605 + 0.493605i
\(918\) 0 0
\(919\) 45.3844i 1.49709i −0.663082 0.748546i \(-0.730752\pi\)
0.663082 0.748546i \(-0.269248\pi\)
\(920\) 0 0
\(921\) 61.3253i 2.02074i
\(922\) 0 0
\(923\) −10.0392 + 10.0392i −0.330444 + 0.330444i
\(924\) 0 0
\(925\) 1.17899 + 1.17899i 0.0387649 + 0.0387649i
\(926\) 0 0
\(927\) −0.380882 −0.0125098
\(928\) 0 0
\(929\) −6.51036 −0.213598 −0.106799 0.994281i \(-0.534060\pi\)
−0.106799 + 0.994281i \(0.534060\pi\)
\(930\) 0 0
\(931\) −1.69986 1.69986i −0.0557107 0.0557107i
\(932\) 0 0
\(933\) −11.7947 + 11.7947i −0.386142 + 0.386142i
\(934\) 0 0
\(935\) 19.0402i 0.622681i
\(936\) 0 0
\(937\) 40.2986i 1.31650i 0.752801 + 0.658248i \(0.228702\pi\)
−0.752801 + 0.658248i \(0.771298\pi\)
\(938\) 0 0
\(939\) 36.6799 36.6799i 1.19700 1.19700i
\(940\) 0 0
\(941\) 1.10649 + 1.10649i 0.0360705 + 0.0360705i 0.724912 0.688841i \(-0.241880\pi\)
−0.688841 + 0.724912i \(0.741880\pi\)
\(942\) 0 0
\(943\) −21.2509 −0.692025
\(944\) 0 0
\(945\) −3.16241 −0.102873
\(946\) 0 0
\(947\) 8.83833 + 8.83833i 0.287207 + 0.287207i 0.835975 0.548768i \(-0.184903\pi\)
−0.548768 + 0.835975i \(0.684903\pi\)
\(948\) 0 0
\(949\) −39.0796 + 39.0796i −1.26858 + 1.26858i
\(950\) 0 0
\(951\) 20.6185i 0.668600i
\(952\) 0 0
\(953\) 14.9610i 0.484636i −0.970197 0.242318i \(-0.922092\pi\)
0.970197 0.242318i \(-0.0779077\pi\)
\(954\) 0 0
\(955\) 4.14234 4.14234i 0.134043 0.134043i
\(956\) 0 0
\(957\) −26.0654 26.0654i −0.842576 0.842576i
\(958\) 0 0
\(959\) −54.8152 −1.77008
\(960\) 0 0
\(961\) −26.7441 −0.862712
\(962\) 0 0
\(963\) 6.97433 + 6.97433i 0.224745 + 0.224745i
\(964\) 0 0
\(965\) 0.0170522 0.0170522i 0.000548930 0.000548930i
\(966\) 0 0
\(967\) 3.95287i 0.127116i −0.997978 0.0635578i \(-0.979755\pi\)
0.997978 0.0635578i \(-0.0202447\pi\)
\(968\) 0 0
\(969\) 30.0150i 0.964221i
\(970\) 0 0
\(971\) 29.0538 29.0538i 0.932380 0.932380i −0.0654740 0.997854i \(-0.520856\pi\)
0.997854 + 0.0654740i \(0.0208560\pi\)
\(972\) 0 0
\(973\) −8.07597 8.07597i −0.258904 0.258904i
\(974\) 0 0
\(975\) −10.2485 −0.328215
\(976\) 0 0
\(977\) 25.8962 0.828494 0.414247 0.910164i \(-0.364045\pi\)
0.414247 + 0.910164i \(0.364045\pi\)
\(978\) 0 0
\(979\) −6.90984 6.90984i −0.220839 0.220839i
\(980\) 0 0
\(981\) 17.5151 17.5151i 0.559213 0.559213i
\(982\) 0 0
\(983\) 22.0151i 0.702173i 0.936343 + 0.351087i \(0.114188\pi\)
−0.936343 + 0.351087i \(0.885812\pi\)
\(984\) 0 0
\(985\) 21.0839i 0.671789i
\(986\) 0 0
\(987\) −56.3850 + 56.3850i −1.79476 + 1.79476i
\(988\) 0 0
\(989\) 13.9537 + 13.9537i 0.443702 + 0.443702i
\(990\) 0 0
\(991\) 54.3207 1.72556 0.862778 0.505583i \(-0.168723\pi\)
0.862778 + 0.505583i \(0.168723\pi\)
\(992\) 0 0
\(993\) 61.9461 1.96580
\(994\) 0 0
\(995\) −9.65378 9.65378i −0.306045 0.306045i
\(996\) 0 0
\(997\) 8.14405 8.14405i 0.257925 0.257925i −0.566285 0.824210i \(-0.691620\pi\)
0.824210 + 0.566285i \(0.191620\pi\)
\(998\) 0 0
\(999\) 1.82197i 0.0576446i
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 320.2.l.a.81.7 16
3.2 odd 2 2880.2.t.c.721.7 16
4.3 odd 2 80.2.l.a.61.4 yes 16
5.2 odd 4 1600.2.q.g.849.7 16
5.3 odd 4 1600.2.q.h.849.2 16
5.4 even 2 1600.2.l.i.401.2 16
8.3 odd 2 640.2.l.b.161.7 16
8.5 even 2 640.2.l.a.161.2 16
12.11 even 2 720.2.t.c.541.5 16
16.3 odd 4 640.2.l.b.481.7 16
16.5 even 4 inner 320.2.l.a.241.7 16
16.11 odd 4 80.2.l.a.21.4 16
16.13 even 4 640.2.l.a.481.2 16
20.3 even 4 400.2.q.g.349.2 16
20.7 even 4 400.2.q.h.349.7 16
20.19 odd 2 400.2.l.h.301.5 16
32.5 even 8 5120.2.a.u.1.1 8
32.11 odd 8 5120.2.a.v.1.1 8
32.21 even 8 5120.2.a.t.1.8 8
32.27 odd 8 5120.2.a.s.1.8 8
48.5 odd 4 2880.2.t.c.2161.6 16
48.11 even 4 720.2.t.c.181.5 16
80.27 even 4 400.2.q.g.149.2 16
80.37 odd 4 1600.2.q.h.49.2 16
80.43 even 4 400.2.q.h.149.7 16
80.53 odd 4 1600.2.q.g.49.7 16
80.59 odd 4 400.2.l.h.101.5 16
80.69 even 4 1600.2.l.i.1201.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.2.l.a.21.4 16 16.11 odd 4
80.2.l.a.61.4 yes 16 4.3 odd 2
320.2.l.a.81.7 16 1.1 even 1 trivial
320.2.l.a.241.7 16 16.5 even 4 inner
400.2.l.h.101.5 16 80.59 odd 4
400.2.l.h.301.5 16 20.19 odd 2
400.2.q.g.149.2 16 80.27 even 4
400.2.q.g.349.2 16 20.3 even 4
400.2.q.h.149.7 16 80.43 even 4
400.2.q.h.349.7 16 20.7 even 4
640.2.l.a.161.2 16 8.5 even 2
640.2.l.a.481.2 16 16.13 even 4
640.2.l.b.161.7 16 8.3 odd 2
640.2.l.b.481.7 16 16.3 odd 4
720.2.t.c.181.5 16 48.11 even 4
720.2.t.c.541.5 16 12.11 even 2
1600.2.l.i.401.2 16 5.4 even 2
1600.2.l.i.1201.2 16 80.69 even 4
1600.2.q.g.49.7 16 80.53 odd 4
1600.2.q.g.849.7 16 5.2 odd 4
1600.2.q.h.49.2 16 80.37 odd 4
1600.2.q.h.849.2 16 5.3 odd 4
2880.2.t.c.721.7 16 3.2 odd 2
2880.2.t.c.2161.6 16 48.5 odd 4
5120.2.a.s.1.8 8 32.27 odd 8
5120.2.a.t.1.8 8 32.21 even 8
5120.2.a.u.1.1 8 32.5 even 8
5120.2.a.v.1.1 8 32.11 odd 8