Properties

Label 1792.2.m.d.449.4
Level $1792$
Weight $2$
Character 1792.449
Analytic conductor $14.309$
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] = [1792,2,Mod(449,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.m (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.18939904.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 449.4
Root \(0.500000 - 0.691860i\) of defining polynomial
Character \(\chi\) \(=\) 1792.449
Dual form 1792.2.m.d.1345.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.89897 - 1.89897i) q^{3} +(0.372364 + 0.372364i) q^{5} +1.00000i q^{7} -4.21215i q^{9} +O(q^{10})\) \(q+(1.89897 - 1.89897i) q^{3} +(0.372364 + 0.372364i) q^{5} +1.00000i q^{7} -4.21215i q^{9} +(0.158942 + 0.158942i) q^{11} +(-1.48475 + 1.48475i) q^{13} +1.41421 q^{15} +3.79793 q^{17} +(2.94552 - 2.94552i) q^{19} +(1.89897 + 1.89897i) q^{21} -1.95687i q^{23} -4.72269i q^{25} +(-2.30182 - 2.30182i) q^{27} +(0.0304945 - 0.0304945i) q^{29} +9.16902 q^{31} +0.603650 q^{33} +(-0.372364 + 0.372364i) q^{35} +(-5.78530 - 5.78530i) q^{37} +5.63899i q^{39} -10.9442i q^{41} +(2.62636 + 2.62636i) q^{43} +(1.56845 - 1.56845i) q^{45} +7.79793 q^{47} -1.00000 q^{49} +(7.21215 - 7.21215i) q^{51} +(2.21215 + 2.21215i) q^{53} +0.118368i q^{55} -11.1869i q^{57} +(-9.17030 - 9.17030i) q^{59} +(4.97601 - 4.97601i) q^{61} +4.21215 q^{63} -1.10574 q^{65} +(-4.42429 + 4.42429i) q^{67} +(-3.71604 - 3.71604i) q^{69} +14.5174i q^{71} +12.0202i q^{73} +(-8.96823 - 8.96823i) q^{75} +(-0.158942 + 0.158942i) q^{77} -11.4165 q^{79} +3.89426 q^{81} +(0.123744 - 0.123744i) q^{83} +(1.41421 + 1.41421i) q^{85} -0.115816i q^{87} +6.79956i q^{89} +(-1.48475 - 1.48475i) q^{91} +(17.4117 - 17.4117i) q^{93} +2.19361 q^{95} +8.80572 q^{97} +(0.669485 - 0.669485i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 4 q^{5} - 8 q^{11} - 12 q^{13} + 8 q^{17} + 4 q^{19} + 4 q^{21} - 8 q^{27} + 8 q^{31} - 16 q^{33} - 4 q^{35} + 8 q^{37} - 24 q^{43} - 12 q^{45} + 40 q^{47} - 8 q^{49} + 24 q^{51} - 16 q^{53} - 52 q^{59} + 20 q^{61} - 24 q^{65} + 32 q^{67} - 8 q^{69} - 28 q^{75} + 8 q^{77} + 16 q^{81} - 12 q^{83} - 12 q^{91} + 40 q^{93} + 80 q^{95} + 72 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) 1.89897 1.89897i 1.09637 1.09637i 0.101537 0.994832i \(-0.467624\pi\)
0.994832 0.101537i \(-0.0323760\pi\)
\(4\) 0 0
\(5\) 0.372364 + 0.372364i 0.166526 + 0.166526i 0.785451 0.618924i \(-0.212431\pi\)
−0.618924 + 0.785451i \(0.712431\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 4.21215i 1.40405i
\(10\) 0 0
\(11\) 0.158942 + 0.158942i 0.0479227 + 0.0479227i 0.730662 0.682739i \(-0.239212\pi\)
−0.682739 + 0.730662i \(0.739212\pi\)
\(12\) 0 0
\(13\) −1.48475 + 1.48475i −0.411796 + 0.411796i −0.882364 0.470568i \(-0.844049\pi\)
0.470568 + 0.882364i \(0.344049\pi\)
\(14\) 0 0
\(15\) 1.41421 0.365148
\(16\) 0 0
\(17\) 3.79793 0.921134 0.460567 0.887625i \(-0.347646\pi\)
0.460567 + 0.887625i \(0.347646\pi\)
\(18\) 0 0
\(19\) 2.94552 2.94552i 0.675748 0.675748i −0.283287 0.959035i \(-0.591425\pi\)
0.959035 + 0.283287i \(0.0914248\pi\)
\(20\) 0 0
\(21\) 1.89897 + 1.89897i 0.414388 + 0.414388i
\(22\) 0 0
\(23\) 1.95687i 0.408036i −0.978967 0.204018i \(-0.934600\pi\)
0.978967 0.204018i \(-0.0654002\pi\)
\(24\) 0 0
\(25\) 4.72269i 0.944538i
\(26\) 0 0
\(27\) −2.30182 2.30182i −0.442986 0.442986i
\(28\) 0 0
\(29\) 0.0304945 0.0304945i 0.00566268 0.00566268i −0.704270 0.709932i \(-0.748725\pi\)
0.709932 + 0.704270i \(0.248725\pi\)
\(30\) 0 0
\(31\) 9.16902 1.64680 0.823402 0.567458i \(-0.192073\pi\)
0.823402 + 0.567458i \(0.192073\pi\)
\(32\) 0 0
\(33\) 0.603650 0.105082
\(34\) 0 0
\(35\) −0.372364 + 0.372364i −0.0629410 + 0.0629410i
\(36\) 0 0
\(37\) −5.78530 5.78530i −0.951098 0.951098i 0.0477611 0.998859i \(-0.484791\pi\)
−0.998859 + 0.0477611i \(0.984791\pi\)
\(38\) 0 0
\(39\) 5.63899i 0.902961i
\(40\) 0 0
\(41\) 10.9442i 1.70920i −0.519284 0.854602i \(-0.673801\pi\)
0.519284 0.854602i \(-0.326199\pi\)
\(42\) 0 0
\(43\) 2.62636 + 2.62636i 0.400516 + 0.400516i 0.878415 0.477899i \(-0.158601\pi\)
−0.477899 + 0.878415i \(0.658601\pi\)
\(44\) 0 0
\(45\) 1.56845 1.56845i 0.233811 0.233811i
\(46\) 0 0
\(47\) 7.79793 1.13745 0.568723 0.822529i \(-0.307438\pi\)
0.568723 + 0.822529i \(0.307438\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 7.21215 7.21215i 1.00990 1.00990i
\(52\) 0 0
\(53\) 2.21215 + 2.21215i 0.303862 + 0.303862i 0.842523 0.538661i \(-0.181070\pi\)
−0.538661 + 0.842523i \(0.681070\pi\)
\(54\) 0 0
\(55\) 0.118368i 0.0159608i
\(56\) 0 0
\(57\) 11.1869i 1.48174i
\(58\) 0 0
\(59\) −9.17030 9.17030i −1.19387 1.19387i −0.975971 0.217900i \(-0.930079\pi\)
−0.217900 0.975971i \(-0.569921\pi\)
\(60\) 0 0
\(61\) 4.97601 4.97601i 0.637113 0.637113i −0.312729 0.949842i \(-0.601243\pi\)
0.949842 + 0.312729i \(0.101243\pi\)
\(62\) 0 0
\(63\) 4.21215 0.530681
\(64\) 0 0
\(65\) −1.10574 −0.137150
\(66\) 0 0
\(67\) −4.42429 + 4.42429i −0.540513 + 0.540513i −0.923679 0.383166i \(-0.874834\pi\)
0.383166 + 0.923679i \(0.374834\pi\)
\(68\) 0 0
\(69\) −3.71604 3.71604i −0.447358 0.447358i
\(70\) 0 0
\(71\) 14.5174i 1.72290i 0.507844 + 0.861449i \(0.330443\pi\)
−0.507844 + 0.861449i \(0.669557\pi\)
\(72\) 0 0
\(73\) 12.0202i 1.40685i 0.710768 + 0.703427i \(0.248348\pi\)
−0.710768 + 0.703427i \(0.751652\pi\)
\(74\) 0 0
\(75\) −8.96823 8.96823i −1.03556 1.03556i
\(76\) 0 0
\(77\) −0.158942 + 0.158942i −0.0181131 + 0.0181131i
\(78\) 0 0
\(79\) −11.4165 −1.28446 −0.642229 0.766513i \(-0.721990\pi\)
−0.642229 + 0.766513i \(0.721990\pi\)
\(80\) 0 0
\(81\) 3.89426 0.432696
\(82\) 0 0
\(83\) 0.123744 0.123744i 0.0135826 0.0135826i −0.700283 0.713865i \(-0.746943\pi\)
0.713865 + 0.700283i \(0.246943\pi\)
\(84\) 0 0
\(85\) 1.41421 + 1.41421i 0.153393 + 0.153393i
\(86\) 0 0
\(87\) 0.115816i 0.0124168i
\(88\) 0 0
\(89\) 6.79956i 0.720751i 0.932807 + 0.360376i \(0.117352\pi\)
−0.932807 + 0.360376i \(0.882648\pi\)
\(90\) 0 0
\(91\) −1.48475 1.48475i −0.155644 0.155644i
\(92\) 0 0
\(93\) 17.4117 17.4117i 1.80551 1.80551i
\(94\) 0 0
\(95\) 2.19361 0.225060
\(96\) 0 0
\(97\) 8.80572 0.894085 0.447043 0.894513i \(-0.352477\pi\)
0.447043 + 0.894513i \(0.352477\pi\)
\(98\) 0 0
\(99\) 0.669485 0.669485i 0.0672858 0.0672858i
\(100\) 0 0
\(101\) −4.82372 4.82372i −0.479978 0.479978i 0.425146 0.905125i \(-0.360223\pi\)
−0.905125 + 0.425146i \(0.860223\pi\)
\(102\) 0 0
\(103\) 12.1158i 1.19381i 0.802313 + 0.596903i \(0.203602\pi\)
−0.802313 + 0.596903i \(0.796398\pi\)
\(104\) 0 0
\(105\) 1.41421i 0.138013i
\(106\) 0 0
\(107\) 13.4344 + 13.4344i 1.29875 + 1.29875i 0.929218 + 0.369531i \(0.120482\pi\)
0.369531 + 0.929218i \(0.379518\pi\)
\(108\) 0 0
\(109\) −0.0963303 + 0.0963303i −0.00922677 + 0.00922677i −0.711705 0.702478i \(-0.752077\pi\)
0.702478 + 0.711705i \(0.252077\pi\)
\(110\) 0 0
\(111\) −21.9722 −2.08551
\(112\) 0 0
\(113\) −8.89426 −0.836702 −0.418351 0.908285i \(-0.637392\pi\)
−0.418351 + 0.908285i \(0.637392\pi\)
\(114\) 0 0
\(115\) 0.728670 0.728670i 0.0679488 0.0679488i
\(116\) 0 0
\(117\) 6.25400 + 6.25400i 0.578182 + 0.578182i
\(118\) 0 0
\(119\) 3.79793i 0.348156i
\(120\) 0 0
\(121\) 10.9495i 0.995407i
\(122\) 0 0
\(123\) −20.7827 20.7827i −1.87392 1.87392i
\(124\) 0 0
\(125\) 3.62038 3.62038i 0.323817 0.323817i
\(126\) 0 0
\(127\) −4.61790 −0.409773 −0.204886 0.978786i \(-0.565682\pi\)
−0.204886 + 0.978786i \(0.565682\pi\)
\(128\) 0 0
\(129\) 9.97474 0.878227
\(130\) 0 0
\(131\) −12.6411 + 12.6411i −1.10446 + 1.10446i −0.110596 + 0.993865i \(0.535276\pi\)
−0.993865 + 0.110596i \(0.964724\pi\)
\(132\) 0 0
\(133\) 2.94552 + 2.94552i 0.255409 + 0.255409i
\(134\) 0 0
\(135\) 1.71423i 0.147538i
\(136\) 0 0
\(137\) 14.6812i 1.25430i 0.778899 + 0.627149i \(0.215778\pi\)
−0.778899 + 0.627149i \(0.784222\pi\)
\(138\) 0 0
\(139\) −0.688625 0.688625i −0.0584084 0.0584084i 0.677299 0.735708i \(-0.263150\pi\)
−0.735708 + 0.677299i \(0.763150\pi\)
\(140\) 0 0
\(141\) 14.8080 14.8080i 1.24706 1.24706i
\(142\) 0 0
\(143\) −0.471978 −0.0394688
\(144\) 0 0
\(145\) 0.0227101 0.00188597
\(146\) 0 0
\(147\) −1.89897 + 1.89897i −0.156624 + 0.156624i
\(148\) 0 0
\(149\) −15.1133 15.1133i −1.23813 1.23813i −0.960766 0.277361i \(-0.910540\pi\)
−0.277361 0.960766i \(-0.589460\pi\)
\(150\) 0 0
\(151\) 9.55274i 0.777391i 0.921366 + 0.388695i \(0.127074\pi\)
−0.921366 + 0.388695i \(0.872926\pi\)
\(152\) 0 0
\(153\) 15.9974i 1.29332i
\(154\) 0 0
\(155\) 3.41421 + 3.41421i 0.274236 + 0.274236i
\(156\) 0 0
\(157\) −4.77214 + 4.77214i −0.380858 + 0.380858i −0.871411 0.490553i \(-0.836795\pi\)
0.490553 + 0.871411i \(0.336795\pi\)
\(158\) 0 0
\(159\) 8.40158 0.666289
\(160\) 0 0
\(161\) 1.95687 0.154223
\(162\) 0 0
\(163\) 3.33051 3.33051i 0.260866 0.260866i −0.564540 0.825406i \(-0.690946\pi\)
0.825406 + 0.564540i \(0.190946\pi\)
\(164\) 0 0
\(165\) 0.224777 + 0.224777i 0.0174989 + 0.0174989i
\(166\) 0 0
\(167\) 0.202067i 0.0156364i −0.999969 0.00781822i \(-0.997511\pi\)
0.999969 0.00781822i \(-0.00248864\pi\)
\(168\) 0 0
\(169\) 8.59102i 0.660848i
\(170\) 0 0
\(171\) −12.4070 12.4070i −0.948784 0.948784i
\(172\) 0 0
\(173\) −4.48998 + 4.48998i −0.341367 + 0.341367i −0.856881 0.515514i \(-0.827601\pi\)
0.515514 + 0.856881i \(0.327601\pi\)
\(174\) 0 0
\(175\) 4.72269 0.357002
\(176\) 0 0
\(177\) −34.8282 −2.61785
\(178\) 0 0
\(179\) −8.10641 + 8.10641i −0.605901 + 0.605901i −0.941872 0.335971i \(-0.890936\pi\)
0.335971 + 0.941872i \(0.390936\pi\)
\(180\) 0 0
\(181\) −15.7889 15.7889i −1.17358 1.17358i −0.981350 0.192227i \(-0.938429\pi\)
−0.192227 0.981350i \(-0.561571\pi\)
\(182\) 0 0
\(183\) 18.8986i 1.39702i
\(184\) 0 0
\(185\) 4.30848i 0.316765i
\(186\) 0 0
\(187\) 0.603650 + 0.603650i 0.0441432 + 0.0441432i
\(188\) 0 0
\(189\) 2.30182 2.30182i 0.167433 0.167433i
\(190\) 0 0
\(191\) 5.37470 0.388899 0.194450 0.980912i \(-0.437708\pi\)
0.194450 + 0.980912i \(0.437708\pi\)
\(192\) 0 0
\(193\) 16.4194 1.18190 0.590949 0.806709i \(-0.298754\pi\)
0.590949 + 0.806709i \(0.298754\pi\)
\(194\) 0 0
\(195\) −2.09976 + 2.09976i −0.150367 + 0.150367i
\(196\) 0 0
\(197\) −12.0685 12.0685i −0.859846 0.859846i 0.131474 0.991320i \(-0.458029\pi\)
−0.991320 + 0.131474i \(0.958029\pi\)
\(198\) 0 0
\(199\) 5.00162i 0.354556i 0.984161 + 0.177278i \(0.0567291\pi\)
−0.984161 + 0.177278i \(0.943271\pi\)
\(200\) 0 0
\(201\) 16.8032i 1.18520i
\(202\) 0 0
\(203\) 0.0304945 + 0.0304945i 0.00214029 + 0.00214029i
\(204\) 0 0
\(205\) 4.07524 4.07524i 0.284627 0.284627i
\(206\) 0 0
\(207\) −8.24264 −0.572903
\(208\) 0 0
\(209\) 0.936331 0.0647674
\(210\) 0 0
\(211\) 7.59587 7.59587i 0.522921 0.522921i −0.395531 0.918452i \(-0.629440\pi\)
0.918452 + 0.395531i \(0.129440\pi\)
\(212\) 0 0
\(213\) 27.5681 + 27.5681i 1.88893 + 1.88893i
\(214\) 0 0
\(215\) 1.95592i 0.133393i
\(216\) 0 0
\(217\) 9.16902i 0.622434i
\(218\) 0 0
\(219\) 22.8259 + 22.8259i 1.54243 + 1.54243i
\(220\) 0 0
\(221\) −5.63899 + 5.63899i −0.379320 + 0.379320i
\(222\) 0 0
\(223\) −19.5995 −1.31248 −0.656239 0.754553i \(-0.727854\pi\)
−0.656239 + 0.754553i \(0.727854\pi\)
\(224\) 0 0
\(225\) −19.8927 −1.32618
\(226\) 0 0
\(227\) −12.2008 + 12.2008i −0.809795 + 0.809795i −0.984603 0.174808i \(-0.944070\pi\)
0.174808 + 0.984603i \(0.444070\pi\)
\(228\) 0 0
\(229\) 8.97166 + 8.97166i 0.592864 + 0.592864i 0.938404 0.345540i \(-0.112304\pi\)
−0.345540 + 0.938404i \(0.612304\pi\)
\(230\) 0 0
\(231\) 0.603650i 0.0397172i
\(232\) 0 0
\(233\) 20.7013i 1.35619i −0.734974 0.678095i \(-0.762806\pi\)
0.734974 0.678095i \(-0.237194\pi\)
\(234\) 0 0
\(235\) 2.90367 + 2.90367i 0.189415 + 0.189415i
\(236\) 0 0
\(237\) −21.6796 + 21.6796i −1.40824 + 1.40824i
\(238\) 0 0
\(239\) 17.9926 1.16384 0.581922 0.813244i \(-0.302301\pi\)
0.581922 + 0.813244i \(0.302301\pi\)
\(240\) 0 0
\(241\) 6.45896 0.416058 0.208029 0.978123i \(-0.433295\pi\)
0.208029 + 0.978123i \(0.433295\pi\)
\(242\) 0 0
\(243\) 14.3005 14.3005i 0.917381 0.917381i
\(244\) 0 0
\(245\) −0.372364 0.372364i −0.0237895 0.0237895i
\(246\) 0 0
\(247\) 8.74674i 0.556541i
\(248\) 0 0
\(249\) 0.469970i 0.0297832i
\(250\) 0 0
\(251\) −1.50099 1.50099i −0.0947419 0.0947419i 0.658147 0.752889i \(-0.271340\pi\)
−0.752889 + 0.658147i \(0.771340\pi\)
\(252\) 0 0
\(253\) 0.311029 0.311029i 0.0195542 0.0195542i
\(254\) 0 0
\(255\) 5.37109 0.336351
\(256\) 0 0
\(257\) 11.0124 0.686933 0.343466 0.939165i \(-0.388399\pi\)
0.343466 + 0.939165i \(0.388399\pi\)
\(258\) 0 0
\(259\) 5.78530 5.78530i 0.359481 0.359481i
\(260\) 0 0
\(261\) −0.128447 0.128447i −0.00795068 0.00795068i
\(262\) 0 0
\(263\) 27.9018i 1.72050i −0.509874 0.860249i \(-0.670308\pi\)
0.509874 0.860249i \(-0.329692\pi\)
\(264\) 0 0
\(265\) 1.64745i 0.101202i
\(266\) 0 0
\(267\) 12.9121 + 12.9121i 0.790209 + 0.790209i
\(268\) 0 0
\(269\) −17.1069 + 17.1069i −1.04303 + 1.04303i −0.0439967 + 0.999032i \(0.514009\pi\)
−0.999032 + 0.0439967i \(0.985991\pi\)
\(270\) 0 0
\(271\) 21.3660 1.29789 0.648946 0.760835i \(-0.275210\pi\)
0.648946 + 0.760835i \(0.275210\pi\)
\(272\) 0 0
\(273\) −5.63899 −0.341287
\(274\) 0 0
\(275\) 0.750632 0.750632i 0.0452648 0.0452648i
\(276\) 0 0
\(277\) 9.57839 + 9.57839i 0.575510 + 0.575510i 0.933663 0.358153i \(-0.116593\pi\)
−0.358153 + 0.933663i \(0.616593\pi\)
\(278\) 0 0
\(279\) 38.6213i 2.31219i
\(280\) 0 0
\(281\) 8.95365i 0.534130i −0.963679 0.267065i \(-0.913946\pi\)
0.963679 0.267065i \(-0.0860538\pi\)
\(282\) 0 0
\(283\) 5.86685 + 5.86685i 0.348748 + 0.348748i 0.859643 0.510895i \(-0.170686\pi\)
−0.510895 + 0.859643i \(0.670686\pi\)
\(284\) 0 0
\(285\) 4.16559 4.16559i 0.246748 0.246748i
\(286\) 0 0
\(287\) 10.9442 0.646018
\(288\) 0 0
\(289\) −2.57571 −0.151512
\(290\) 0 0
\(291\) 16.7218 16.7218i 0.980247 0.980247i
\(292\) 0 0
\(293\) −7.63885 7.63885i −0.446266 0.446266i 0.447845 0.894111i \(-0.352192\pi\)
−0.894111 + 0.447845i \(0.852192\pi\)
\(294\) 0 0
\(295\) 6.82938i 0.397622i
\(296\) 0 0
\(297\) 0.731712i 0.0424582i
\(298\) 0 0
\(299\) 2.90547 + 2.90547i 0.168028 + 0.168028i
\(300\) 0 0
\(301\) −2.62636 + 2.62636i −0.151381 + 0.151381i
\(302\) 0 0
\(303\) −18.3202 −1.05247
\(304\) 0 0
\(305\) 3.70578 0.212192
\(306\) 0 0
\(307\) 6.58271 6.58271i 0.375695 0.375695i −0.493851 0.869546i \(-0.664411\pi\)
0.869546 + 0.493851i \(0.164411\pi\)
\(308\) 0 0
\(309\) 23.0075 + 23.0075i 1.30885 + 1.30885i
\(310\) 0 0
\(311\) 23.6229i 1.33953i 0.742573 + 0.669765i \(0.233605\pi\)
−0.742573 + 0.669765i \(0.766395\pi\)
\(312\) 0 0
\(313\) 7.51578i 0.424817i 0.977181 + 0.212408i \(0.0681307\pi\)
−0.977181 + 0.212408i \(0.931869\pi\)
\(314\) 0 0
\(315\) 1.56845 + 1.56845i 0.0883722 + 0.0883722i
\(316\) 0 0
\(317\) −0.661701 + 0.661701i −0.0371648 + 0.0371648i −0.725445 0.688280i \(-0.758366\pi\)
0.688280 + 0.725445i \(0.258366\pi\)
\(318\) 0 0
\(319\) 0.00969368 0.000542742
\(320\) 0 0
\(321\) 51.0228 2.84782
\(322\) 0 0
\(323\) 11.1869 11.1869i 0.622455 0.622455i
\(324\) 0 0
\(325\) 7.01203 + 7.01203i 0.388957 + 0.388957i
\(326\) 0 0
\(327\) 0.365856i 0.0202319i
\(328\) 0 0
\(329\) 7.79793i 0.429914i
\(330\) 0 0
\(331\) 23.5612 + 23.5612i 1.29504 + 1.29504i 0.931629 + 0.363412i \(0.118388\pi\)
0.363412 + 0.931629i \(0.381612\pi\)
\(332\) 0 0
\(333\) −24.3685 + 24.3685i −1.33539 + 1.33539i
\(334\) 0 0
\(335\) −3.29489 −0.180019
\(336\) 0 0
\(337\) 17.8071 0.970014 0.485007 0.874510i \(-0.338817\pi\)
0.485007 + 0.874510i \(0.338817\pi\)
\(338\) 0 0
\(339\) −16.8899 + 16.8899i −0.917334 + 0.917334i
\(340\) 0 0
\(341\) 1.45734 + 1.45734i 0.0789193 + 0.0789193i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 2.76744i 0.148994i
\(346\) 0 0
\(347\) 1.31265 + 1.31265i 0.0704668 + 0.0704668i 0.741462 0.670995i \(-0.234133\pi\)
−0.670995 + 0.741462i \(0.734133\pi\)
\(348\) 0 0
\(349\) −13.9134 + 13.9134i −0.744767 + 0.744767i −0.973491 0.228724i \(-0.926545\pi\)
0.228724 + 0.973491i \(0.426545\pi\)
\(350\) 0 0
\(351\) 6.83528 0.364840
\(352\) 0 0
\(353\) −18.8807 −1.00492 −0.502459 0.864601i \(-0.667571\pi\)
−0.502459 + 0.864601i \(0.667571\pi\)
\(354\) 0 0
\(355\) −5.40576 + 5.40576i −0.286908 + 0.286908i
\(356\) 0 0
\(357\) 7.21215 + 7.21215i 0.381707 + 0.381707i
\(358\) 0 0
\(359\) 21.8252i 1.15189i −0.817488 0.575945i \(-0.804634\pi\)
0.817488 0.575945i \(-0.195366\pi\)
\(360\) 0 0
\(361\) 1.64783i 0.0867280i
\(362\) 0 0
\(363\) −20.7927 20.7927i −1.09133 1.09133i
\(364\) 0 0
\(365\) −4.47587 + 4.47587i −0.234278 + 0.234278i
\(366\) 0 0
\(367\) −7.61788 −0.397650 −0.198825 0.980035i \(-0.563713\pi\)
−0.198825 + 0.980035i \(0.563713\pi\)
\(368\) 0 0
\(369\) −46.0988 −2.39981
\(370\) 0 0
\(371\) −2.21215 + 2.21215i −0.114849 + 0.114849i
\(372\) 0 0
\(373\) 15.7101 + 15.7101i 0.813436 + 0.813436i 0.985147 0.171711i \(-0.0549296\pi\)
−0.171711 + 0.985147i \(0.554930\pi\)
\(374\) 0 0
\(375\) 13.7500i 0.710045i
\(376\) 0 0
\(377\) 0.0905535i 0.00466374i
\(378\) 0 0
\(379\) 10.8259 + 10.8259i 0.556088 + 0.556088i 0.928191 0.372103i \(-0.121363\pi\)
−0.372103 + 0.928191i \(0.621363\pi\)
\(380\) 0 0
\(381\) −8.76924 + 8.76924i −0.449262 + 0.449262i
\(382\) 0 0
\(383\) 3.76582 0.192424 0.0962121 0.995361i \(-0.469327\pi\)
0.0962121 + 0.995361i \(0.469327\pi\)
\(384\) 0 0
\(385\) −0.118368 −0.00603261
\(386\) 0 0
\(387\) 11.0626 11.0626i 0.562344 0.562344i
\(388\) 0 0
\(389\) −25.0059 25.0059i −1.26785 1.26785i −0.947197 0.320653i \(-0.896098\pi\)
−0.320653 0.947197i \(-0.603902\pi\)
\(390\) 0 0
\(391\) 7.43208i 0.375856i
\(392\) 0 0
\(393\) 48.0102i 2.42179i
\(394\) 0 0
\(395\) −4.25110 4.25110i −0.213896 0.213896i
\(396\) 0 0
\(397\) −11.8246 + 11.8246i −0.593460 + 0.593460i −0.938564 0.345105i \(-0.887843\pi\)
0.345105 + 0.938564i \(0.387843\pi\)
\(398\) 0 0
\(399\) 11.1869 0.560045
\(400\) 0 0
\(401\) −5.39511 −0.269419 −0.134710 0.990885i \(-0.543010\pi\)
−0.134710 + 0.990885i \(0.543010\pi\)
\(402\) 0 0
\(403\) −13.6137 + 13.6137i −0.678148 + 0.678148i
\(404\) 0 0
\(405\) 1.45008 + 1.45008i 0.0720552 + 0.0720552i
\(406\) 0 0
\(407\) 1.83905i 0.0911584i
\(408\) 0 0
\(409\) 3.95203i 0.195415i −0.995215 0.0977076i \(-0.968849\pi\)
0.995215 0.0977076i \(-0.0311510\pi\)
\(410\) 0 0
\(411\) 27.8791 + 27.8791i 1.37517 + 1.37517i
\(412\) 0 0
\(413\) 9.17030 9.17030i 0.451241 0.451241i
\(414\) 0 0
\(415\) 0.0921555 0.00452373
\(416\) 0 0
\(417\) −2.61535 −0.128074
\(418\) 0 0
\(419\) −14.2684 + 14.2684i −0.697059 + 0.697059i −0.963775 0.266717i \(-0.914061\pi\)
0.266717 + 0.963775i \(0.414061\pi\)
\(420\) 0 0
\(421\) −23.3092 23.3092i −1.13602 1.13602i −0.989157 0.146862i \(-0.953083\pi\)
−0.146862 0.989157i \(-0.546917\pi\)
\(422\) 0 0
\(423\) 32.8460i 1.59703i
\(424\) 0 0
\(425\) 17.9365i 0.870046i
\(426\) 0 0
\(427\) 4.97601 + 4.97601i 0.240806 + 0.240806i
\(428\) 0 0
\(429\) −0.896271 + 0.896271i −0.0432723 + 0.0432723i
\(430\) 0 0
\(431\) 14.1697 0.682530 0.341265 0.939967i \(-0.389145\pi\)
0.341265 + 0.939967i \(0.389145\pi\)
\(432\) 0 0
\(433\) −10.4860 −0.503924 −0.251962 0.967737i \(-0.581076\pi\)
−0.251962 + 0.967737i \(0.581076\pi\)
\(434\) 0 0
\(435\) 0.0431257 0.0431257i 0.00206772 0.00206772i
\(436\) 0 0
\(437\) −5.76401 5.76401i −0.275730 0.275730i
\(438\) 0 0
\(439\) 10.2381i 0.488637i −0.969695 0.244318i \(-0.921436\pi\)
0.969695 0.244318i \(-0.0785642\pi\)
\(440\) 0 0
\(441\) 4.21215i 0.200578i
\(442\) 0 0
\(443\) −1.33536 1.33536i −0.0634449 0.0634449i 0.674672 0.738117i \(-0.264285\pi\)
−0.738117 + 0.674672i \(0.764285\pi\)
\(444\) 0 0
\(445\) −2.53191 + 2.53191i −0.120024 + 0.120024i
\(446\) 0 0
\(447\) −57.3992 −2.71489
\(448\) 0 0
\(449\) −35.7127 −1.68539 −0.842694 0.538393i \(-0.819032\pi\)
−0.842694 + 0.538393i \(0.819032\pi\)
\(450\) 0 0
\(451\) 1.73950 1.73950i 0.0819097 0.0819097i
\(452\) 0 0
\(453\) 18.1403 + 18.1403i 0.852307 + 0.852307i
\(454\) 0 0
\(455\) 1.10574i 0.0518378i
\(456\) 0 0
\(457\) 24.3144i 1.13738i 0.822552 + 0.568689i \(0.192549\pi\)
−0.822552 + 0.568689i \(0.807451\pi\)
\(458\) 0 0
\(459\) −8.74218 8.74218i −0.408050 0.408050i
\(460\) 0 0
\(461\) −30.3124 + 30.3124i −1.41179 + 1.41179i −0.664522 + 0.747269i \(0.731365\pi\)
−0.747269 + 0.664522i \(0.768635\pi\)
\(462\) 0 0
\(463\) 4.33480 0.201455 0.100728 0.994914i \(-0.467883\pi\)
0.100728 + 0.994914i \(0.467883\pi\)
\(464\) 0 0
\(465\) 12.9670 0.601328
\(466\) 0 0
\(467\) −19.0292 + 19.0292i −0.880567 + 0.880567i −0.993592 0.113025i \(-0.963946\pi\)
0.113025 + 0.993592i \(0.463946\pi\)
\(468\) 0 0
\(469\) −4.42429 4.42429i −0.204295 0.204295i
\(470\) 0 0
\(471\) 18.1243i 0.835122i
\(472\) 0 0
\(473\) 0.834876i 0.0383876i
\(474\) 0 0
\(475\) −13.9108 13.9108i −0.638270 0.638270i
\(476\) 0 0
\(477\) 9.31788 9.31788i 0.426637 0.426637i
\(478\) 0 0
\(479\) 4.69245 0.214404 0.107202 0.994237i \(-0.465811\pi\)
0.107202 + 0.994237i \(0.465811\pi\)
\(480\) 0 0
\(481\) 17.1795 0.783317
\(482\) 0 0
\(483\) 3.71604 3.71604i 0.169086 0.169086i
\(484\) 0 0
\(485\) 3.27893 + 3.27893i 0.148889 + 0.148889i
\(486\) 0 0
\(487\) 0.300020i 0.0135952i −0.999977 0.00679760i \(-0.997836\pi\)
0.999977 0.00679760i \(-0.00216376\pi\)
\(488\) 0 0
\(489\) 12.6491i 0.572011i
\(490\) 0 0
\(491\) 6.00361 + 6.00361i 0.270939 + 0.270939i 0.829478 0.558539i \(-0.188638\pi\)
−0.558539 + 0.829478i \(0.688638\pi\)
\(492\) 0 0
\(493\) 0.115816 0.115816i 0.00521609 0.00521609i
\(494\) 0 0
\(495\) 0.498585 0.0224097
\(496\) 0 0
\(497\) −14.5174 −0.651194
\(498\) 0 0
\(499\) −5.25367 + 5.25367i −0.235187 + 0.235187i −0.814854 0.579667i \(-0.803183\pi\)
0.579667 + 0.814854i \(0.303183\pi\)
\(500\) 0 0
\(501\) −0.383719 0.383719i −0.0171433 0.0171433i
\(502\) 0 0
\(503\) 0.645460i 0.0287797i 0.999896 + 0.0143898i \(0.00458058\pi\)
−0.999896 + 0.0143898i \(0.995419\pi\)
\(504\) 0 0
\(505\) 3.59236i 0.159858i
\(506\) 0 0
\(507\) 16.3141 + 16.3141i 0.724533 + 0.724533i
\(508\) 0 0
\(509\) 22.9125 22.9125i 1.01558 1.01558i 0.0157007 0.999877i \(-0.495002\pi\)
0.999877 0.0157007i \(-0.00499791\pi\)
\(510\) 0 0
\(511\) −12.0202 −0.531740
\(512\) 0 0
\(513\) −13.5601 −0.598695
\(514\) 0 0
\(515\) −4.51149 + 4.51149i −0.198800 + 0.198800i
\(516\) 0 0
\(517\) 1.23942 + 1.23942i 0.0545095 + 0.0545095i
\(518\) 0 0
\(519\) 17.0527i 0.748529i
\(520\) 0 0
\(521\) 35.9433i 1.57471i 0.616503 + 0.787353i \(0.288549\pi\)
−0.616503 + 0.787353i \(0.711451\pi\)
\(522\) 0 0
\(523\) −0.198239 0.198239i −0.00866838 0.00866838i 0.702759 0.711428i \(-0.251951\pi\)
−0.711428 + 0.702759i \(0.751951\pi\)
\(524\) 0 0
\(525\) 8.96823 8.96823i 0.391406 0.391406i
\(526\) 0 0
\(527\) 34.8233 1.51693
\(528\) 0 0
\(529\) 19.1706 0.833506
\(530\) 0 0
\(531\) −38.6266 + 38.6266i −1.67625 + 1.67625i
\(532\) 0 0
\(533\) 16.2495 + 16.2495i 0.703844 + 0.703844i
\(534\) 0 0
\(535\) 10.0050i 0.432552i
\(536\) 0 0
\(537\) 30.7876i 1.32858i
\(538\) 0 0
\(539\) −0.158942 0.158942i −0.00684610 0.00684610i
\(540\) 0 0
\(541\) 19.4243 19.4243i 0.835116 0.835116i −0.153096 0.988211i \(-0.548924\pi\)
0.988211 + 0.153096i \(0.0489243\pi\)
\(542\) 0 0
\(543\) −59.9651 −2.57335
\(544\) 0 0
\(545\) −0.0717399 −0.00307300
\(546\) 0 0
\(547\) −6.42914 + 6.42914i −0.274890 + 0.274890i −0.831065 0.556175i \(-0.812268\pi\)
0.556175 + 0.831065i \(0.312268\pi\)
\(548\) 0 0
\(549\) −20.9597 20.9597i −0.894538 0.894538i
\(550\) 0 0
\(551\) 0.179644i 0.00765310i
\(552\) 0 0
\(553\) 11.4165i 0.485479i
\(554\) 0 0
\(555\) −8.18165 8.18165i −0.347292 0.347292i
\(556\) 0 0
\(557\) 14.3961 14.3961i 0.609982 0.609982i −0.332959 0.942941i \(-0.608047\pi\)
0.942941 + 0.332959i \(0.108047\pi\)
\(558\) 0 0
\(559\) −7.79899 −0.329862
\(560\) 0 0
\(561\) 2.29262 0.0967945
\(562\) 0 0
\(563\) 16.5187 16.5187i 0.696179 0.696179i −0.267405 0.963584i \(-0.586166\pi\)
0.963584 + 0.267405i \(0.0861662\pi\)
\(564\) 0 0
\(565\) −3.31190 3.31190i −0.139333 0.139333i
\(566\) 0 0
\(567\) 3.89426i 0.163544i
\(568\) 0 0
\(569\) 34.1860i 1.43315i 0.697510 + 0.716575i \(0.254291\pi\)
−0.697510 + 0.716575i \(0.745709\pi\)
\(570\) 0 0
\(571\) 2.70390 + 2.70390i 0.113155 + 0.113155i 0.761417 0.648262i \(-0.224504\pi\)
−0.648262 + 0.761417i \(0.724504\pi\)
\(572\) 0 0
\(573\) 10.2064 10.2064i 0.426377 0.426377i
\(574\) 0 0
\(575\) −9.24171 −0.385406
\(576\) 0 0
\(577\) 12.3564 0.514406 0.257203 0.966357i \(-0.417199\pi\)
0.257203 + 0.966357i \(0.417199\pi\)
\(578\) 0 0
\(579\) 31.1800 31.1800i 1.29580 1.29580i
\(580\) 0 0
\(581\) 0.123744 + 0.123744i 0.00513376 + 0.00513376i
\(582\) 0 0
\(583\) 0.703204i 0.0291237i
\(584\) 0 0
\(585\) 4.65753i 0.192565i
\(586\) 0 0
\(587\) −9.74183 9.74183i −0.402088 0.402088i 0.476880 0.878968i \(-0.341768\pi\)
−0.878968 + 0.476880i \(0.841768\pi\)
\(588\) 0 0
\(589\) 27.0075 27.0075i 1.11283 1.11283i
\(590\) 0 0
\(591\) −45.8354 −1.88542
\(592\) 0 0
\(593\) −32.0335 −1.31546 −0.657728 0.753255i \(-0.728482\pi\)
−0.657728 + 0.753255i \(0.728482\pi\)
\(594\) 0 0
\(595\) −1.41421 + 1.41421i −0.0579771 + 0.0579771i
\(596\) 0 0
\(597\) 9.49791 + 9.49791i 0.388724 + 0.388724i
\(598\) 0 0
\(599\) 4.42105i 0.180639i −0.995913 0.0903196i \(-0.971211\pi\)
0.995913 0.0903196i \(-0.0287888\pi\)
\(600\) 0 0
\(601\) 17.7279i 0.723138i 0.932345 + 0.361569i \(0.117759\pi\)
−0.932345 + 0.361569i \(0.882241\pi\)
\(602\) 0 0
\(603\) 18.6358 + 18.6358i 0.758907 + 0.758907i
\(604\) 0 0
\(605\) 4.07719 4.07719i 0.165761 0.165761i
\(606\) 0 0
\(607\) −32.3546 −1.31323 −0.656616 0.754225i \(-0.728013\pi\)
−0.656616 + 0.754225i \(0.728013\pi\)
\(608\) 0 0
\(609\) 0.115816 0.00469310
\(610\) 0 0
\(611\) −11.5780 + 11.5780i −0.468396 + 0.468396i
\(612\) 0 0
\(613\) 18.7860 + 18.7860i 0.758758 + 0.758758i 0.976096 0.217338i \(-0.0697374\pi\)
−0.217338 + 0.976096i \(0.569737\pi\)
\(614\) 0 0
\(615\) 15.4775i 0.624113i
\(616\) 0 0
\(617\) 7.06491i 0.284422i 0.989836 + 0.142211i \(0.0454212\pi\)
−0.989836 + 0.142211i \(0.954579\pi\)
\(618\) 0 0
\(619\) −15.3058 15.3058i −0.615191 0.615191i 0.329103 0.944294i \(-0.393254\pi\)
−0.944294 + 0.329103i \(0.893254\pi\)
\(620\) 0 0
\(621\) −4.50438 + 4.50438i −0.180755 + 0.180755i
\(622\) 0 0
\(623\) −6.79956 −0.272418
\(624\) 0 0
\(625\) −20.9173 −0.836690
\(626\) 0 0
\(627\) 1.77806 1.77806i 0.0710089 0.0710089i
\(628\) 0 0
\(629\) −21.9722 21.9722i −0.876088 0.876088i
\(630\) 0 0
\(631\) 43.3637i 1.72628i 0.504963 + 0.863141i \(0.331506\pi\)
−0.504963 + 0.863141i \(0.668494\pi\)
\(632\) 0 0
\(633\) 28.8486i 1.14663i
\(634\) 0 0
\(635\) −1.71954 1.71954i −0.0682379 0.0682379i
\(636\) 0 0
\(637\) 1.48475 1.48475i 0.0588280 0.0588280i
\(638\) 0 0
\(639\) 61.1494 2.41903
\(640\) 0 0
\(641\) 15.9439 0.629745 0.314872 0.949134i \(-0.398038\pi\)
0.314872 + 0.949134i \(0.398038\pi\)
\(642\) 0 0
\(643\) 15.1476 15.1476i 0.597363 0.597363i −0.342247 0.939610i \(-0.611188\pi\)
0.939610 + 0.342247i \(0.111188\pi\)
\(644\) 0 0
\(645\) 3.71423 + 3.71423i 0.146248 + 0.146248i
\(646\) 0 0
\(647\) 41.6849i 1.63880i −0.573220 0.819402i \(-0.694306\pi\)
0.573220 0.819402i \(-0.305694\pi\)
\(648\) 0 0
\(649\) 2.91508i 0.114427i
\(650\) 0 0
\(651\) 17.4117 + 17.4117i 0.682417 + 0.682417i
\(652\) 0 0
\(653\) 4.38694 4.38694i 0.171674 0.171674i −0.616040 0.787715i \(-0.711264\pi\)
0.787715 + 0.616040i \(0.211264\pi\)
\(654\) 0 0
\(655\) −9.41421 −0.367844
\(656\) 0 0
\(657\) 50.6307 1.97529
\(658\) 0 0
\(659\) −33.6974 + 33.6974i −1.31267 + 1.31267i −0.393222 + 0.919444i \(0.628640\pi\)
−0.919444 + 0.393222i \(0.871360\pi\)
\(660\) 0 0
\(661\) −21.0875 21.0875i −0.820208 0.820208i 0.165930 0.986138i \(-0.446937\pi\)
−0.986138 + 0.165930i \(0.946937\pi\)
\(662\) 0 0
\(663\) 21.4165i 0.831748i
\(664\) 0 0
\(665\) 2.19361i 0.0850646i
\(666\) 0 0
\(667\) −0.0596739 0.0596739i −0.00231058 0.00231058i
\(668\) 0 0
\(669\) −37.2187 + 37.2187i −1.43896 + 1.43896i
\(670\) 0 0
\(671\) 1.58179 0.0610644
\(672\) 0 0
\(673\) −24.6656 −0.950790 −0.475395 0.879773i \(-0.657695\pi\)
−0.475395 + 0.879773i \(0.657695\pi\)
\(674\) 0 0
\(675\) −10.8708 + 10.8708i −0.418417 + 0.418417i
\(676\) 0 0
\(677\) −16.1104 16.1104i −0.619172 0.619172i 0.326147 0.945319i \(-0.394249\pi\)
−0.945319 + 0.326147i \(0.894249\pi\)
\(678\) 0 0
\(679\) 8.80572i 0.337932i
\(680\) 0 0
\(681\) 46.3378i 1.77567i
\(682\) 0 0
\(683\) −14.3105 14.3105i −0.547575 0.547575i 0.378163 0.925739i \(-0.376556\pi\)
−0.925739 + 0.378163i \(0.876556\pi\)
\(684\) 0 0
\(685\) −5.46675 + 5.46675i −0.208874 + 0.208874i
\(686\) 0 0
\(687\) 34.0737 1.29999
\(688\) 0 0
\(689\) −6.56898 −0.250258
\(690\) 0 0
\(691\) 14.0209 14.0209i 0.533382 0.533382i −0.388195 0.921577i \(-0.626901\pi\)
0.921577 + 0.388195i \(0.126901\pi\)
\(692\) 0 0
\(693\) 0.669485 + 0.669485i 0.0254317 + 0.0254317i
\(694\) 0 0
\(695\) 0.512838i 0.0194531i
\(696\) 0 0
\(697\) 41.5655i 1.57441i
\(698\) 0 0
\(699\) −39.3112 39.3112i −1.48688 1.48688i
\(700\) 0 0
\(701\) −26.3024 + 26.3024i −0.993430 + 0.993430i −0.999979 0.00654904i \(-0.997915\pi\)
0.00654904 + 0.999979i \(0.497915\pi\)
\(702\) 0 0
\(703\) −34.0814 −1.28541
\(704\) 0 0
\(705\) 11.0279 0.415336
\(706\) 0 0
\(707\) 4.82372 4.82372i 0.181415 0.181415i
\(708\) 0 0
\(709\) 18.4665 + 18.4665i 0.693524 + 0.693524i 0.963005 0.269482i \(-0.0868524\pi\)
−0.269482 + 0.963005i \(0.586852\pi\)
\(710\) 0 0
\(711\) 48.0880i 1.80344i
\(712\) 0 0
\(713\) 17.9426i 0.671956i
\(714\) 0 0
\(715\) −0.175748 0.175748i −0.00657259 0.00657259i
\(716\) 0 0
\(717\) 34.1673 34.1673i 1.27600 1.27600i
\(718\) 0 0
\(719\) 12.0026 0.447620 0.223810 0.974633i \(-0.428151\pi\)
0.223810 + 0.974633i \(0.428151\pi\)
\(720\) 0 0
\(721\) −12.1158 −0.451217
\(722\) 0 0
\(723\) 12.2654 12.2654i 0.456153 0.456153i
\(724\) 0 0
\(725\) −0.144016 0.144016i −0.00534862 0.00534862i
\(726\) 0 0
\(727\) 19.8622i 0.736647i 0.929698 + 0.368323i \(0.120068\pi\)
−0.929698 + 0.368323i \(0.879932\pi\)
\(728\) 0 0
\(729\) 42.6297i 1.57888i
\(730\) 0 0
\(731\) 9.97474 + 9.97474i 0.368929 + 0.368929i
\(732\) 0 0
\(733\) 29.1704 29.1704i 1.07743 1.07743i 0.0806954 0.996739i \(-0.474286\pi\)
0.996739 0.0806954i \(-0.0257141\pi\)
\(734\) 0 0
\(735\) −1.41421 −0.0521641
\(736\) 0 0
\(737\) −1.40641 −0.0518057
\(738\) 0 0
\(739\) −16.4158 + 16.4158i −0.603866 + 0.603866i −0.941336 0.337470i \(-0.890429\pi\)
0.337470 + 0.941336i \(0.390429\pi\)
\(740\) 0 0
\(741\) 16.6098 + 16.6098i 0.610175 + 0.610175i
\(742\) 0 0
\(743\) 6.10225i 0.223870i −0.993716 0.111935i \(-0.964295\pi\)
0.993716 0.111935i \(-0.0357048\pi\)
\(744\) 0 0
\(745\) 11.2553i 0.412361i
\(746\) 0 0
\(747\) −0.521227 0.521227i −0.0190707 0.0190707i
\(748\) 0 0
\(749\) −13.4344 + 13.4344i −0.490881 + 0.490881i
\(750\) 0 0
\(751\) 37.6119 1.37248 0.686238 0.727377i \(-0.259261\pi\)
0.686238 + 0.727377i \(0.259261\pi\)
\(752\) 0 0
\(753\) −5.70067 −0.207744
\(754\) 0 0
\(755\) −3.55710 + 3.55710i −0.129456 + 0.129456i
\(756\) 0 0
\(757\) 33.5029 + 33.5029i 1.21768 + 1.21768i 0.968441 + 0.249242i \(0.0801814\pi\)
0.249242 + 0.968441i \(0.419819\pi\)
\(758\) 0 0
\(759\) 1.18127i 0.0428773i
\(760\) 0 0
\(761\) 31.2942i 1.13442i −0.823575 0.567208i \(-0.808024\pi\)
0.823575 0.567208i \(-0.191976\pi\)
\(762\) 0 0
\(763\) −0.0963303 0.0963303i −0.00348739 0.00348739i
\(764\) 0 0
\(765\) 5.95687 5.95687i 0.215371 0.215371i
\(766\) 0 0
\(767\) 27.2312 0.983263
\(768\) 0 0
\(769\) 16.5011 0.595046 0.297523 0.954715i \(-0.403839\pi\)
0.297523 + 0.954715i \(0.403839\pi\)
\(770\) 0 0
\(771\) 20.9121 20.9121i 0.753132 0.753132i
\(772\) 0 0
\(773\) −33.1931 33.1931i −1.19387 1.19387i −0.975970 0.217904i \(-0.930078\pi\)
−0.217904 0.975970i \(-0.569922\pi\)
\(774\) 0 0
\(775\) 43.3024i 1.55547i
\(776\) 0 0
\(777\) 21.9722i 0.788248i
\(778\) 0 0
\(779\) −32.2365 32.2365i −1.15499 1.15499i
\(780\) 0 0
\(781\) −2.30742 + 2.30742i −0.0825660 + 0.0825660i
\(782\) 0 0
\(783\) −0.140386 −0.00501698
\(784\) 0 0
\(785\) −3.55395 −0.126846
\(786\) 0 0
\(787\) 37.4484 37.4484i 1.33489 1.33489i 0.433959 0.900933i \(-0.357116\pi\)
0.900933 0.433959i \(-0.142884\pi\)
\(788\) 0 0
\(789\) −52.9846 52.9846i −1.88630 1.88630i
\(790\) 0 0
\(791\) 8.89426i 0.316244i
\(792\) 0 0
\(793\) 14.7763i 0.524722i
\(794\) 0 0
\(795\) 3.12845 + 3.12845i 0.110955 + 0.110955i
\(796\) 0 0
\(797\) −36.2951 + 36.2951i −1.28564 + 1.28564i −0.348227 + 0.937410i \(0.613216\pi\)
−0.937410 + 0.348227i \(0.886784\pi\)
\(798\) 0 0
\(799\) 29.6160 1.04774
\(800\) 0 0
\(801\) 28.6407 1.01197
\(802\) 0 0
\(803\) −1.91050 + 1.91050i −0.0674202 + 0.0674202i
\(804\) 0 0
\(805\) 0.728670 + 0.728670i 0.0256822 + 0.0256822i
\(806\) 0 0
\(807\) 64.9710i 2.28709i
\(808\) 0 0
\(809\) 53.8799i 1.89432i −0.320768 0.947158i \(-0.603941\pi\)
0.320768 0.947158i \(-0.396059\pi\)
\(810\) 0 0
\(811\) −3.37216 3.37216i −0.118413 0.118413i 0.645417 0.763830i \(-0.276683\pi\)
−0.763830 + 0.645417i \(0.776683\pi\)
\(812\) 0 0
\(813\) 40.5733 40.5733i 1.42297 1.42297i
\(814\) 0 0
\(815\) 2.48033 0.0868821
\(816\) 0 0
\(817\) 15.4720 0.541296
\(818\) 0 0
\(819\) −6.25400 + 6.25400i −0.218532 + 0.218532i
\(820\) 0 0
\(821\) 10.9339 + 10.9339i 0.381596 + 0.381596i 0.871677 0.490081i \(-0.163033\pi\)
−0.490081 + 0.871677i \(0.663033\pi\)
\(822\) 0 0
\(823\) 51.6379i 1.79999i −0.435905 0.899993i \(-0.643572\pi\)
0.435905 0.899993i \(-0.356428\pi\)
\(824\) 0 0
\(825\) 2.85085i 0.0992539i
\(826\) 0 0
\(827\) −31.2096 31.2096i −1.08526 1.08526i −0.996009 0.0892551i \(-0.971551\pi\)
−0.0892551 0.996009i \(-0.528449\pi\)
\(828\) 0 0
\(829\) −15.3646 + 15.3646i −0.533634 + 0.533634i −0.921652 0.388018i \(-0.873160\pi\)
0.388018 + 0.921652i \(0.373160\pi\)
\(830\) 0 0
\(831\) 36.3781 1.26194
\(832\) 0 0
\(833\) −3.79793 −0.131591
\(834\) 0 0
\(835\) 0.0752426 0.0752426i 0.00260388 0.00260388i
\(836\) 0 0
\(837\) −21.1055 21.1055i −0.729512 0.729512i
\(838\) 0 0
\(839\) 35.4882i 1.22519i 0.790397 + 0.612595i \(0.209874\pi\)
−0.790397 + 0.612595i \(0.790126\pi\)
\(840\) 0 0
\(841\) 28.9981i 0.999936i
\(842\) 0 0
\(843\) −17.0027 17.0027i −0.585603 0.585603i
\(844\) 0 0
\(845\) −3.19899 + 3.19899i −0.110048 + 0.110048i
\(846\) 0 0
\(847\) 10.9495 0.376228
\(848\) 0 0
\(849\) 22.2819 0.764713
\(850\) 0 0
\(851\) −11.3211 + 11.3211i −0.388083 + 0.388083i
\(852\) 0 0
\(853\) 35.3684 + 35.3684i 1.21099 + 1.21099i 0.970701 + 0.240291i \(0.0772428\pi\)
0.240291 + 0.970701i \(0.422757\pi\)
\(854\) 0 0
\(855\) 9.23981i 0.315995i
\(856\) 0 0
\(857\) 26.2359i 0.896202i −0.893983 0.448101i \(-0.852100\pi\)
0.893983 0.448101i \(-0.147900\pi\)
\(858\) 0 0
\(859\) 14.4164 + 14.4164i 0.491880 + 0.491880i 0.908898 0.417018i \(-0.136925\pi\)
−0.417018 + 0.908898i \(0.636925\pi\)
\(860\) 0 0
\(861\) 20.7827 20.7827i 0.708274 0.708274i
\(862\) 0 0
\(863\) 42.9719 1.46278 0.731390 0.681959i \(-0.238872\pi\)
0.731390 + 0.681959i \(0.238872\pi\)
\(864\) 0 0
\(865\) −3.34382 −0.113693
\(866\) 0 0
\(867\) −4.89118 + 4.89118i −0.166113 + 0.166113i
\(868\) 0 0
\(869\) −1.81456 1.81456i −0.0615547 0.0615547i
\(870\) 0 0
\(871\) 13.1380i 0.445163i
\(872\) 0 0
\(873\) 37.0910i 1.25534i
\(874\) 0 0
\(875\) 3.62038 + 3.62038i 0.122391 + 0.122391i
\(876\) 0 0
\(877\) −2.33766 + 2.33766i −0.0789370 + 0.0789370i −0.745473 0.666536i \(-0.767776\pi\)
0.666536 + 0.745473i \(0.267776\pi\)
\(878\) 0 0
\(879\) −29.0118 −0.978545
\(880\) 0 0
\(881\) −0.183946 −0.00619731 −0.00309865 0.999995i \(-0.500986\pi\)
−0.00309865 + 0.999995i \(0.500986\pi\)
\(882\) 0 0
\(883\) 32.7600 32.7600i 1.10246 1.10246i 0.108350 0.994113i \(-0.465443\pi\)
0.994113 0.108350i \(-0.0345568\pi\)
\(884\) 0 0
\(885\) −12.9688 12.9688i −0.435940 0.435940i
\(886\) 0 0
\(887\) 15.8506i 0.532210i 0.963944 + 0.266105i \(0.0857368\pi\)
−0.963944 + 0.266105i \(0.914263\pi\)
\(888\) 0 0
\(889\) 4.61790i 0.154880i
\(890\) 0 0
\(891\) 0.618961 + 0.618961i 0.0207360 + 0.0207360i
\(892\) 0 0
\(893\) 22.9690 22.9690i 0.768627 0.768627i
\(894\) 0 0
\(895\) −6.03707 −0.201797
\(896\) 0 0
\(897\) 11.0348 0.368441
\(898\) 0 0
\(899\) 0.279604 0.279604i 0.00932533 0.00932533i
\(900\) 0 0
\(901\) 8.40158 + 8.40158i 0.279897 + 0.279897i
\(902\) 0 0
\(903\) 9.97474i 0.331938i
\(904\) 0 0
\(905\) 11.7584i 0.390863i
\(906\) 0 0
\(907\) 23.6027 + 23.6027i 0.783715 + 0.783715i 0.980456 0.196740i \(-0.0630356\pi\)
−0.196740 + 0.980456i \(0.563036\pi\)
\(908\) 0 0
\(909\) −20.3182 + 20.3182i −0.673913 + 0.673913i
\(910\) 0 0
\(911\) 14.3105 0.474127 0.237064 0.971494i \(-0.423815\pi\)
0.237064 + 0.971494i \(0.423815\pi\)
\(912\) 0 0
\(913\) 0.0393361 0.00130183
\(914\) 0 0
\(915\) 7.03715 7.03715i 0.232641 0.232641i
\(916\) 0 0
\(917\) −12.6411 12.6411i −0.417447 0.417447i
\(918\) 0 0
\(919\) 10.5848i 0.349162i 0.984643 + 0.174581i \(0.0558570\pi\)
−0.984643 + 0.174581i \(0.944143\pi\)
\(920\) 0 0
\(921\) 25.0007i 0.823800i
\(922\) 0 0
\(923\) −21.5547 21.5547i −0.709483 0.709483i
\(924\) 0 0
\(925\) −27.3222 + 27.3222i −0.898348 + 0.898348i
\(926\) 0 0
\(927\) 51.0336 1.67616
\(928\) 0 0
\(929\) −37.3891 −1.22670 −0.613348 0.789813i \(-0.710178\pi\)
−0.613348 + 0.789813i \(0.710178\pi\)
\(930\) 0 0
\(931\) −2.94552 + 2.94552i −0.0965355 + 0.0965355i
\(932\) 0 0
\(933\) 44.8590 + 44.8590i 1.46862 + 1.46862i
\(934\) 0 0
\(935\) 0.449555i 0.0147020i
\(936\) 0 0
\(937\) 19.0986i 0.623925i −0.950094 0.311962i \(-0.899014\pi\)
0.950094 0.311962i \(-0.100986\pi\)
\(938\) 0 0
\(939\) 14.2722 + 14.2722i 0.465756 + 0.465756i
\(940\) 0 0
\(941\) 8.07900 8.07900i 0.263368 0.263368i −0.563053 0.826421i \(-0.690373\pi\)
0.826421 + 0.563053i \(0.190373\pi\)
\(942\) 0 0
\(943\) −21.4165 −0.697417
\(944\) 0 0
\(945\) 1.71423 0.0557640
\(946\) 0 0
\(947\) 30.2015 30.2015i 0.981418 0.981418i −0.0184129 0.999830i \(-0.505861\pi\)
0.999830 + 0.0184129i \(0.00586132\pi\)
\(948\) 0 0
\(949\) −17.8470 17.8470i −0.579337 0.579337i
\(950\) 0 0
\(951\) 2.51310i 0.0814927i
\(952\) 0 0
\(953\) 18.3192i 0.593418i −0.954968 0.296709i \(-0.904111\pi\)
0.954968 0.296709i \(-0.0958891\pi\)
\(954\) 0 0
\(955\) 2.00134 + 2.00134i 0.0647620 + 0.0647620i
\(956\) 0 0
\(957\) 0.0184080 0.0184080i 0.000595045 0.000595045i
\(958\) 0 0
\(959\) −14.6812 −0.474080
\(960\) 0 0
\(961\) 53.0709 1.71197
\(962\) 0 0
\(963\) 56.5875 56.5875i 1.82351 1.82351i
\(964\) 0 0
\(965\) 6.11401 + 6.11401i 0.196817 + 0.196817i
\(966\) 0 0
\(967\) 19.5495i 0.628669i 0.949312 + 0.314335i \(0.101781\pi\)
−0.949312 + 0.314335i \(0.898219\pi\)
\(968\) 0 0
\(969\) 42.4870i 1.36488i
\(970\) 0 0
\(971\) 37.1948 + 37.1948i 1.19364 + 1.19364i 0.976038 + 0.217598i \(0.0698222\pi\)
0.217598 + 0.976038i \(0.430178\pi\)
\(972\) 0 0
\(973\) 0.688625 0.688625i 0.0220763 0.0220763i
\(974\) 0 0
\(975\) 26.6312 0.852881
\(976\) 0 0
\(977\) 18.1324 0.580108 0.290054 0.957010i \(-0.406327\pi\)
0.290054 + 0.957010i \(0.406327\pi\)
\(978\) 0 0
\(979\) −1.08073 + 1.08073i −0.0345404 + 0.0345404i
\(980\) 0 0
\(981\) 0.405757 + 0.405757i 0.0129548 + 0.0129548i
\(982\) 0 0
\(983\) 1.72823i 0.0551220i −0.999620 0.0275610i \(-0.991226\pi\)
0.999620 0.0275610i \(-0.00877404\pi\)
\(984\) 0 0
\(985\) 8.98776i 0.286374i
\(986\) 0 0
\(987\) 14.8080 + 14.8080i 0.471344 + 0.471344i
\(988\) 0 0
\(989\) 5.13946 5.13946i 0.163425 0.163425i
\(990\) 0 0
\(991\) −48.5830 −1.54329 −0.771645 0.636054i \(-0.780566\pi\)
−0.771645 + 0.636054i \(0.780566\pi\)
\(992\) 0 0
\(993\) 89.4838 2.83968
\(994\) 0 0
\(995\) −1.86242 + 1.86242i −0.0590428 + 0.0590428i
\(996\) 0 0
\(997\) 10.1609 + 10.1609i 0.321799 + 0.321799i 0.849457 0.527658i \(-0.176930\pi\)
−0.527658 + 0.849457i \(0.676930\pi\)
\(998\) 0 0
\(999\) 26.6335i 0.842647i
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 1792.2.m.d.449.4 yes 8
4.3 odd 2 1792.2.m.b.449.1 yes 8
8.3 odd 2 1792.2.m.c.449.4 yes 8
8.5 even 2 1792.2.m.a.449.1 8
16.3 odd 4 1792.2.m.b.1345.1 yes 8
16.5 even 4 1792.2.m.a.1345.1 yes 8
16.11 odd 4 1792.2.m.c.1345.4 yes 8
16.13 even 4 inner 1792.2.m.d.1345.4 yes 8
32.3 odd 8 7168.2.a.w.1.1 4
32.13 even 8 7168.2.a.t.1.1 4
32.19 odd 8 7168.2.a.s.1.4 4
32.29 even 8 7168.2.a.x.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.a.449.1 8 8.5 even 2
1792.2.m.a.1345.1 yes 8 16.5 even 4
1792.2.m.b.449.1 yes 8 4.3 odd 2
1792.2.m.b.1345.1 yes 8 16.3 odd 4
1792.2.m.c.449.4 yes 8 8.3 odd 2
1792.2.m.c.1345.4 yes 8 16.11 odd 4
1792.2.m.d.449.4 yes 8 1.1 even 1 trivial
1792.2.m.d.1345.4 yes 8 16.13 even 4 inner
7168.2.a.s.1.4 4 32.19 odd 8
7168.2.a.t.1.1 4 32.13 even 8
7168.2.a.w.1.1 4 32.3 odd 8
7168.2.a.x.1.4 4 32.29 even 8