Properties

Label 1148.2.k.a.729.3
Level $1148$
Weight $2$
Character 1148.729
Analytic conductor $9.167$
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] = [1148,2,Mod(337,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1148.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.k (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: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.110166016.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 729.3
Root \(0.360409i\) of defining polynomial
Character \(\chi\) \(=\) 1148.729
Dual form 1148.2.k.a.337.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.254848 - 0.254848i) q^{3} -1.56350i q^{5} +(0.707107 - 0.707107i) q^{7} +2.87011i q^{9} +O(q^{10})\) \(q+(0.254848 - 0.254848i) q^{3} -1.56350i q^{5} +(0.707107 - 0.707107i) q^{7} +2.87011i q^{9} +(4.48173 - 4.48173i) q^{11} +(-0.972034 + 0.972034i) q^{13} +(-0.398455 - 0.398455i) q^{15} +(-4.04523 - 4.04523i) q^{17} +(-1.81835 - 1.81835i) q^{19} -0.360409i q^{21} +2.75523 q^{23} +2.55547 q^{25} +(1.49598 + 1.49598i) q^{27} +(-1.30297 + 1.30297i) q^{29} +4.14929 q^{31} -2.28432i q^{33} +(-1.10556 - 1.10556i) q^{35} -5.41421 q^{37} +0.495442i q^{39} +(1.29857 - 6.27006i) q^{41} +4.26203i q^{43} +4.48741 q^{45} +(0.0142542 + 0.0142542i) q^{47} -1.00000i q^{49} -2.06184 q^{51} +(10.1645 - 10.1645i) q^{53} +(-7.00718 - 7.00718i) q^{55} -0.926804 q^{57} -14.7696 q^{59} -13.4610i q^{61} +(2.02947 + 2.02947i) q^{63} +(1.51978 + 1.51978i) q^{65} +(0.159366 + 0.159366i) q^{67} +(0.702165 - 0.702165i) q^{69} +(9.37123 - 9.37123i) q^{71} +2.11564i q^{73} +(0.651256 - 0.651256i) q^{75} -6.33812i q^{77} +(3.39193 - 3.39193i) q^{79} -7.84782 q^{81} -9.91178 q^{83} +(-6.32472 + 6.32472i) q^{85} +0.664120i q^{87} +(3.39791 - 3.39791i) q^{89} +1.37466i q^{91} +(1.05744 - 1.05744i) q^{93} +(-2.84299 + 2.84299i) q^{95} +(11.5392 + 11.5392i) q^{97} +(12.8630 + 12.8630i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 12 q^{11} + 4 q^{13} - 8 q^{15} + 8 q^{17} + 8 q^{19} + 28 q^{23} - 4 q^{25} + 8 q^{27} - 16 q^{29} + 28 q^{31} - 8 q^{35} - 32 q^{37} - 4 q^{45} + 20 q^{47} - 20 q^{51} + 32 q^{53} - 4 q^{55} - 36 q^{57} - 20 q^{59} - 8 q^{63} - 4 q^{67} - 44 q^{69} + 8 q^{71} + 12 q^{75} - 12 q^{79} - 16 q^{81} - 64 q^{83} - 56 q^{85} + 4 q^{89} - 4 q^{93} - 52 q^{95} + 56 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) 0.254848 0.254848i 0.147136 0.147136i −0.629701 0.776838i \(-0.716823\pi\)
0.776838 + 0.629701i \(0.216823\pi\)
\(4\) 0 0
\(5\) 1.56350i 0.699218i −0.936896 0.349609i \(-0.886314\pi\)
0.936896 0.349609i \(-0.113686\pi\)
\(6\) 0 0
\(7\) 0.707107 0.707107i 0.267261 0.267261i
\(8\) 0 0
\(9\) 2.87011i 0.956702i
\(10\) 0 0
\(11\) 4.48173 4.48173i 1.35129 1.35129i 0.467074 0.884218i \(-0.345308\pi\)
0.884218 0.467074i \(-0.154692\pi\)
\(12\) 0 0
\(13\) −0.972034 + 0.972034i −0.269594 + 0.269594i −0.828936 0.559343i \(-0.811054\pi\)
0.559343 + 0.828936i \(0.311054\pi\)
\(14\) 0 0
\(15\) −0.398455 0.398455i −0.102881 0.102881i
\(16\) 0 0
\(17\) −4.04523 4.04523i −0.981112 0.981112i 0.0187126 0.999825i \(-0.494043\pi\)
−0.999825 + 0.0187126i \(0.994043\pi\)
\(18\) 0 0
\(19\) −1.81835 1.81835i −0.417158 0.417158i 0.467065 0.884223i \(-0.345311\pi\)
−0.884223 + 0.467065i \(0.845311\pi\)
\(20\) 0 0
\(21\) 0.360409i 0.0786478i
\(22\) 0 0
\(23\) 2.75523 0.574505 0.287253 0.957855i \(-0.407258\pi\)
0.287253 + 0.957855i \(0.407258\pi\)
\(24\) 0 0
\(25\) 2.55547 0.511094
\(26\) 0 0
\(27\) 1.49598 + 1.49598i 0.287902 + 0.287902i
\(28\) 0 0
\(29\) −1.30297 + 1.30297i −0.241956 + 0.241956i −0.817659 0.575703i \(-0.804728\pi\)
0.575703 + 0.817659i \(0.304728\pi\)
\(30\) 0 0
\(31\) 4.14929 0.745234 0.372617 0.927985i \(-0.378461\pi\)
0.372617 + 0.927985i \(0.378461\pi\)
\(32\) 0 0
\(33\) 2.28432i 0.397649i
\(34\) 0 0
\(35\) −1.10556 1.10556i −0.186874 0.186874i
\(36\) 0 0
\(37\) −5.41421 −0.890091 −0.445046 0.895508i \(-0.646813\pi\)
−0.445046 + 0.895508i \(0.646813\pi\)
\(38\) 0 0
\(39\) 0.495442i 0.0793341i
\(40\) 0 0
\(41\) 1.29857 6.27006i 0.202803 0.979220i
\(42\) 0 0
\(43\) 4.26203i 0.649954i 0.945722 + 0.324977i \(0.105357\pi\)
−0.945722 + 0.324977i \(0.894643\pi\)
\(44\) 0 0
\(45\) 4.48741 0.668943
\(46\) 0 0
\(47\) 0.0142542 + 0.0142542i 0.00207919 + 0.00207919i 0.708146 0.706066i \(-0.249532\pi\)
−0.706066 + 0.708146i \(0.749532\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) −2.06184 −0.288715
\(52\) 0 0
\(53\) 10.1645 10.1645i 1.39620 1.39620i 0.585606 0.810596i \(-0.300857\pi\)
0.810596 0.585606i \(-0.199143\pi\)
\(54\) 0 0
\(55\) −7.00718 7.00718i −0.944849 0.944849i
\(56\) 0 0
\(57\) −0.926804 −0.122758
\(58\) 0 0
\(59\) −14.7696 −1.92284 −0.961419 0.275088i \(-0.911293\pi\)
−0.961419 + 0.275088i \(0.911293\pi\)
\(60\) 0 0
\(61\) 13.4610i 1.72351i −0.507326 0.861754i \(-0.669366\pi\)
0.507326 0.861754i \(-0.330634\pi\)
\(62\) 0 0
\(63\) 2.02947 + 2.02947i 0.255689 + 0.255689i
\(64\) 0 0
\(65\) 1.51978 + 1.51978i 0.188505 + 0.188505i
\(66\) 0 0
\(67\) 0.159366 + 0.159366i 0.0194696 + 0.0194696i 0.716775 0.697305i \(-0.245618\pi\)
−0.697305 + 0.716775i \(0.745618\pi\)
\(68\) 0 0
\(69\) 0.702165 0.702165i 0.0845307 0.0845307i
\(70\) 0 0
\(71\) 9.37123 9.37123i 1.11216 1.11216i 0.119302 0.992858i \(-0.461934\pi\)
0.992858 0.119302i \(-0.0380656\pi\)
\(72\) 0 0
\(73\) 2.11564i 0.247617i 0.992306 + 0.123809i \(0.0395109\pi\)
−0.992306 + 0.123809i \(0.960489\pi\)
\(74\) 0 0
\(75\) 0.651256 0.651256i 0.0752005 0.0752005i
\(76\) 0 0
\(77\) 6.33812i 0.722296i
\(78\) 0 0
\(79\) 3.39193 3.39193i 0.381622 0.381622i −0.490065 0.871686i \(-0.663027\pi\)
0.871686 + 0.490065i \(0.163027\pi\)
\(80\) 0 0
\(81\) −7.84782 −0.871980
\(82\) 0 0
\(83\) −9.91178 −1.08796 −0.543980 0.839098i \(-0.683083\pi\)
−0.543980 + 0.839098i \(0.683083\pi\)
\(84\) 0 0
\(85\) −6.32472 + 6.32472i −0.686012 + 0.686012i
\(86\) 0 0
\(87\) 0.664120i 0.0712011i
\(88\) 0 0
\(89\) 3.39791 3.39791i 0.360178 0.360178i −0.503700 0.863878i \(-0.668028\pi\)
0.863878 + 0.503700i \(0.168028\pi\)
\(90\) 0 0
\(91\) 1.37466i 0.144104i
\(92\) 0 0
\(93\) 1.05744 1.05744i 0.109651 0.109651i
\(94\) 0 0
\(95\) −2.84299 + 2.84299i −0.291684 + 0.291684i
\(96\) 0 0
\(97\) 11.5392 + 11.5392i 1.17162 + 1.17162i 0.981822 + 0.189803i \(0.0607848\pi\)
0.189803 + 0.981822i \(0.439215\pi\)
\(98\) 0 0
\(99\) 12.8630 + 12.8630i 1.29278 + 1.29278i
\(100\) 0 0
\(101\) 12.3010 + 12.3010i 1.22400 + 1.22400i 0.966198 + 0.257801i \(0.0829980\pi\)
0.257801 + 0.966198i \(0.417002\pi\)
\(102\) 0 0
\(103\) 14.5076i 1.42947i 0.699394 + 0.714737i \(0.253453\pi\)
−0.699394 + 0.714737i \(0.746547\pi\)
\(104\) 0 0
\(105\) −0.563500 −0.0549920
\(106\) 0 0
\(107\) −14.5830 −1.40979 −0.704896 0.709310i \(-0.749006\pi\)
−0.704896 + 0.709310i \(0.749006\pi\)
\(108\) 0 0
\(109\) −8.08123 8.08123i −0.774041 0.774041i 0.204769 0.978810i \(-0.434356\pi\)
−0.978810 + 0.204769i \(0.934356\pi\)
\(110\) 0 0
\(111\) −1.37980 + 1.37980i −0.130965 + 0.130965i
\(112\) 0 0
\(113\) 16.9353 1.59314 0.796568 0.604549i \(-0.206647\pi\)
0.796568 + 0.604549i \(0.206647\pi\)
\(114\) 0 0
\(115\) 4.30780i 0.401705i
\(116\) 0 0
\(117\) −2.78984 2.78984i −0.257921 0.257921i
\(118\) 0 0
\(119\) −5.72082 −0.524427
\(120\) 0 0
\(121\) 29.1718i 2.65198i
\(122\) 0 0
\(123\) −1.26697 1.92885i −0.114239 0.173919i
\(124\) 0 0
\(125\) 11.8130i 1.05658i
\(126\) 0 0
\(127\) −2.10985 −0.187219 −0.0936095 0.995609i \(-0.529841\pi\)
−0.0936095 + 0.995609i \(0.529841\pi\)
\(128\) 0 0
\(129\) 1.08617 + 1.08617i 0.0956319 + 0.0956319i
\(130\) 0 0
\(131\) 17.4703i 1.52638i 0.646172 + 0.763192i \(0.276369\pi\)
−0.646172 + 0.763192i \(0.723631\pi\)
\(132\) 0 0
\(133\) −2.57153 −0.222980
\(134\) 0 0
\(135\) 2.33897 2.33897i 0.201307 0.201307i
\(136\) 0 0
\(137\) 13.8853 + 13.8853i 1.18630 + 1.18630i 0.978082 + 0.208221i \(0.0667674\pi\)
0.208221 + 0.978082i \(0.433233\pi\)
\(138\) 0 0
\(139\) 2.29826 0.194935 0.0974677 0.995239i \(-0.468926\pi\)
0.0974677 + 0.995239i \(0.468926\pi\)
\(140\) 0 0
\(141\) 0.00726530 0.000611849
\(142\) 0 0
\(143\) 8.71279i 0.728600i
\(144\) 0 0
\(145\) 2.03720 + 2.03720i 0.169180 + 0.169180i
\(146\) 0 0
\(147\) −0.254848 0.254848i −0.0210195 0.0210195i
\(148\) 0 0
\(149\) −7.30448 7.30448i −0.598406 0.598406i 0.341482 0.939888i \(-0.389071\pi\)
−0.939888 + 0.341482i \(0.889071\pi\)
\(150\) 0 0
\(151\) −8.04736 + 8.04736i −0.654885 + 0.654885i −0.954165 0.299280i \(-0.903253\pi\)
0.299280 + 0.954165i \(0.403253\pi\)
\(152\) 0 0
\(153\) 11.6102 11.6102i 0.938632 0.938632i
\(154\) 0 0
\(155\) 6.48741i 0.521081i
\(156\) 0 0
\(157\) −12.6499 + 12.6499i −1.00957 + 1.00957i −0.00961539 + 0.999954i \(0.503061\pi\)
−0.999954 + 0.00961539i \(0.996939\pi\)
\(158\) 0 0
\(159\) 5.18080i 0.410865i
\(160\) 0 0
\(161\) 1.94824 1.94824i 0.153543 0.153543i
\(162\) 0 0
\(163\) 17.4762 1.36884 0.684419 0.729089i \(-0.260056\pi\)
0.684419 + 0.729089i \(0.260056\pi\)
\(164\) 0 0
\(165\) −3.57153 −0.278043
\(166\) 0 0
\(167\) 5.47037 5.47037i 0.423310 0.423310i −0.463032 0.886342i \(-0.653238\pi\)
0.886342 + 0.463032i \(0.153238\pi\)
\(168\) 0 0
\(169\) 11.1103i 0.854638i
\(170\) 0 0
\(171\) 5.21885 5.21885i 0.399095 0.399095i
\(172\) 0 0
\(173\) 4.42858i 0.336699i −0.985727 0.168349i \(-0.946156\pi\)
0.985727 0.168349i \(-0.0538437\pi\)
\(174\) 0 0
\(175\) 1.80699 1.80699i 0.136595 0.136595i
\(176\) 0 0
\(177\) −3.76400 + 3.76400i −0.282920 + 0.282920i
\(178\) 0 0
\(179\) 13.3504 + 13.3504i 0.997859 + 0.997859i 0.999998 0.00213863i \(-0.000680749\pi\)
−0.00213863 + 0.999998i \(0.500681\pi\)
\(180\) 0 0
\(181\) −9.94844 9.94844i −0.739462 0.739462i 0.233012 0.972474i \(-0.425142\pi\)
−0.972474 + 0.233012i \(0.925142\pi\)
\(182\) 0 0
\(183\) −3.43051 3.43051i −0.253591 0.253591i
\(184\) 0 0
\(185\) 8.46512i 0.622368i
\(186\) 0 0
\(187\) −36.2593 −2.65154
\(188\) 0 0
\(189\) 2.11564 0.153890
\(190\) 0 0
\(191\) 5.99711 + 5.99711i 0.433935 + 0.433935i 0.889965 0.456029i \(-0.150729\pi\)
−0.456029 + 0.889965i \(0.650729\pi\)
\(192\) 0 0
\(193\) −14.3295 + 14.3295i −1.03146 + 1.03146i −0.0319748 + 0.999489i \(0.510180\pi\)
−0.999489 + 0.0319748i \(0.989820\pi\)
\(194\) 0 0
\(195\) 0.774623 0.0554719
\(196\) 0 0
\(197\) 19.3067i 1.37555i 0.725925 + 0.687773i \(0.241412\pi\)
−0.725925 + 0.687773i \(0.758588\pi\)
\(198\) 0 0
\(199\) −0.292893 0.292893i −0.0207626 0.0207626i 0.696649 0.717412i \(-0.254673\pi\)
−0.717412 + 0.696649i \(0.754673\pi\)
\(200\) 0 0
\(201\) 0.0812280 0.00572938
\(202\) 0 0
\(203\) 1.84268i 0.129331i
\(204\) 0 0
\(205\) −9.80325 2.03032i −0.684688 0.141804i
\(206\) 0 0
\(207\) 7.90780i 0.549630i
\(208\) 0 0
\(209\) −16.2987 −1.12740
\(210\) 0 0
\(211\) 9.80251 + 9.80251i 0.674832 + 0.674832i 0.958826 0.283994i \(-0.0916595\pi\)
−0.283994 + 0.958826i \(0.591659\pi\)
\(212\) 0 0
\(213\) 4.77647i 0.327279i
\(214\) 0 0
\(215\) 6.66369 0.454460
\(216\) 0 0
\(217\) 2.93399 2.93399i 0.199172 0.199172i
\(218\) 0 0
\(219\) 0.539167 + 0.539167i 0.0364335 + 0.0364335i
\(220\) 0 0
\(221\) 7.86420 0.529003
\(222\) 0 0
\(223\) −3.52619 −0.236131 −0.118066 0.993006i \(-0.537669\pi\)
−0.118066 + 0.993006i \(0.537669\pi\)
\(224\) 0 0
\(225\) 7.33446i 0.488964i
\(226\) 0 0
\(227\) 11.9973 + 11.9973i 0.796291 + 0.796291i 0.982508 0.186218i \(-0.0596230\pi\)
−0.186218 + 0.982508i \(0.559623\pi\)
\(228\) 0 0
\(229\) −20.7088 20.7088i −1.36848 1.36848i −0.862614 0.505862i \(-0.831174\pi\)
−0.505862 0.862614i \(-0.668826\pi\)
\(230\) 0 0
\(231\) −1.61526 1.61526i −0.106276 0.106276i
\(232\) 0 0
\(233\) −12.8698 + 12.8698i −0.843129 + 0.843129i −0.989265 0.146136i \(-0.953316\pi\)
0.146136 + 0.989265i \(0.453316\pi\)
\(234\) 0 0
\(235\) 0.0222864 0.0222864i 0.00145381 0.00145381i
\(236\) 0 0
\(237\) 1.72885i 0.112301i
\(238\) 0 0
\(239\) −14.7955 + 14.7955i −0.957043 + 0.957043i −0.999115 0.0420720i \(-0.986604\pi\)
0.0420720 + 0.999115i \(0.486604\pi\)
\(240\) 0 0
\(241\) 5.11973i 0.329791i −0.986311 0.164896i \(-0.947271\pi\)
0.986311 0.164896i \(-0.0527287\pi\)
\(242\) 0 0
\(243\) −6.48795 + 6.48795i −0.416202 + 0.416202i
\(244\) 0 0
\(245\) −1.56350 −0.0998884
\(246\) 0 0
\(247\) 3.53499 0.224926
\(248\) 0 0
\(249\) −2.52600 + 2.52600i −0.160079 + 0.160079i
\(250\) 0 0
\(251\) 17.4051i 1.09860i 0.835625 + 0.549300i \(0.185106\pi\)
−0.835625 + 0.549300i \(0.814894\pi\)
\(252\) 0 0
\(253\) 12.3482 12.3482i 0.776325 0.776325i
\(254\) 0 0
\(255\) 3.22368i 0.201875i
\(256\) 0 0
\(257\) −0.694981 + 0.694981i −0.0433517 + 0.0433517i −0.728450 0.685099i \(-0.759759\pi\)
0.685099 + 0.728450i \(0.259759\pi\)
\(258\) 0 0
\(259\) −3.82843 + 3.82843i −0.237887 + 0.237887i
\(260\) 0 0
\(261\) −3.73967 3.73967i −0.231480 0.231480i
\(262\) 0 0
\(263\) 0.0938966 + 0.0938966i 0.00578991 + 0.00578991i 0.709996 0.704206i \(-0.248697\pi\)
−0.704206 + 0.709996i \(0.748697\pi\)
\(264\) 0 0
\(265\) −15.8922 15.8922i −0.976250 0.976250i
\(266\) 0 0
\(267\) 1.73190i 0.105991i
\(268\) 0 0
\(269\) 12.4227 0.757424 0.378712 0.925515i \(-0.376367\pi\)
0.378712 + 0.925515i \(0.376367\pi\)
\(270\) 0 0
\(271\) 11.2913 0.685898 0.342949 0.939354i \(-0.388574\pi\)
0.342949 + 0.939354i \(0.388574\pi\)
\(272\) 0 0
\(273\) 0.350330 + 0.350330i 0.0212029 + 0.0212029i
\(274\) 0 0
\(275\) 11.4529 11.4529i 0.690637 0.690637i
\(276\) 0 0
\(277\) −1.58880 −0.0954615 −0.0477307 0.998860i \(-0.515199\pi\)
−0.0477307 + 0.998860i \(0.515199\pi\)
\(278\) 0 0
\(279\) 11.9089i 0.712966i
\(280\) 0 0
\(281\) 7.72951 + 7.72951i 0.461104 + 0.461104i 0.899017 0.437913i \(-0.144282\pi\)
−0.437913 + 0.899017i \(0.644282\pi\)
\(282\) 0 0
\(283\) 11.2764 0.670312 0.335156 0.942163i \(-0.391211\pi\)
0.335156 + 0.942163i \(0.391211\pi\)
\(284\) 0 0
\(285\) 1.44906i 0.0858348i
\(286\) 0 0
\(287\) −3.51538 5.35183i −0.207506 0.315909i
\(288\) 0 0
\(289\) 15.7278i 0.925163i
\(290\) 0 0
\(291\) 5.88146 0.344778
\(292\) 0 0
\(293\) 6.41476 + 6.41476i 0.374754 + 0.374754i 0.869205 0.494451i \(-0.164631\pi\)
−0.494451 + 0.869205i \(0.664631\pi\)
\(294\) 0 0
\(295\) 23.0923i 1.34448i
\(296\) 0 0
\(297\) 13.4092 0.778080
\(298\) 0 0
\(299\) −2.67818 + 2.67818i −0.154883 + 0.154883i
\(300\) 0 0
\(301\) 3.01371 + 3.01371i 0.173707 + 0.173707i
\(302\) 0 0
\(303\) 6.26979 0.360190
\(304\) 0 0
\(305\) −21.0463 −1.20511
\(306\) 0 0
\(307\) 21.1776i 1.20867i −0.796731 0.604334i \(-0.793439\pi\)
0.796731 0.604334i \(-0.206561\pi\)
\(308\) 0 0
\(309\) 3.69722 + 3.69722i 0.210328 + 0.210328i
\(310\) 0 0
\(311\) 1.89100 + 1.89100i 0.107229 + 0.107229i 0.758686 0.651457i \(-0.225842\pi\)
−0.651457 + 0.758686i \(0.725842\pi\)
\(312\) 0 0
\(313\) 1.31197 + 1.31197i 0.0741568 + 0.0741568i 0.743212 0.669056i \(-0.233301\pi\)
−0.669056 + 0.743212i \(0.733301\pi\)
\(314\) 0 0
\(315\) 3.17308 3.17308i 0.178783 0.178783i
\(316\) 0 0
\(317\) 23.5752 23.5752i 1.32412 1.32412i 0.413703 0.910412i \(-0.364235\pi\)
0.910412 0.413703i \(-0.135765\pi\)
\(318\) 0 0
\(319\) 11.6791i 0.653906i
\(320\) 0 0
\(321\) −3.71645 + 3.71645i −0.207432 + 0.207432i
\(322\) 0 0
\(323\) 14.7113i 0.818557i
\(324\) 0 0
\(325\) −2.48400 + 2.48400i −0.137788 + 0.137788i
\(326\) 0 0
\(327\) −4.11897 −0.227779
\(328\) 0 0
\(329\) 0.0201585 0.00111137
\(330\) 0 0
\(331\) −17.3078 + 17.3078i −0.951323 + 0.951323i −0.998869 0.0475461i \(-0.984860\pi\)
0.0475461 + 0.998869i \(0.484860\pi\)
\(332\) 0 0
\(333\) 15.5394i 0.851552i
\(334\) 0 0
\(335\) 0.249168 0.249168i 0.0136135 0.0136135i
\(336\) 0 0
\(337\) 29.9402i 1.63095i 0.578795 + 0.815473i \(0.303523\pi\)
−0.578795 + 0.815473i \(0.696477\pi\)
\(338\) 0 0
\(339\) 4.31592 4.31592i 0.234408 0.234408i
\(340\) 0 0
\(341\) 18.5960 18.5960i 1.00703 1.00703i
\(342\) 0 0
\(343\) −0.707107 0.707107i −0.0381802 0.0381802i
\(344\) 0 0
\(345\) −1.09783 1.09783i −0.0591054 0.0591054i
\(346\) 0 0
\(347\) 20.8256 + 20.8256i 1.11798 + 1.11798i 0.992038 + 0.125942i \(0.0401952\pi\)
0.125942 + 0.992038i \(0.459805\pi\)
\(348\) 0 0
\(349\) 19.5168i 1.04471i 0.852728 + 0.522355i \(0.174946\pi\)
−0.852728 + 0.522355i \(0.825054\pi\)
\(350\) 0 0
\(351\) −2.90829 −0.155233
\(352\) 0 0
\(353\) 3.97149 0.211381 0.105691 0.994399i \(-0.466295\pi\)
0.105691 + 0.994399i \(0.466295\pi\)
\(354\) 0 0
\(355\) −14.6519 14.6519i −0.777643 0.777643i
\(356\) 0 0
\(357\) −1.45794 + 1.45794i −0.0771623 + 0.0771623i
\(358\) 0 0
\(359\) −18.8844 −0.996678 −0.498339 0.866982i \(-0.666057\pi\)
−0.498339 + 0.866982i \(0.666057\pi\)
\(360\) 0 0
\(361\) 12.3872i 0.651959i
\(362\) 0 0
\(363\) −7.43437 7.43437i −0.390203 0.390203i
\(364\) 0 0
\(365\) 3.30780 0.173138
\(366\) 0 0
\(367\) 26.2146i 1.36839i 0.729300 + 0.684195i \(0.239846\pi\)
−0.729300 + 0.684195i \(0.760154\pi\)
\(368\) 0 0
\(369\) 17.9957 + 3.72704i 0.936821 + 0.194022i
\(370\) 0 0
\(371\) 14.3748i 0.746301i
\(372\) 0 0
\(373\) −15.0195 −0.777681 −0.388840 0.921305i \(-0.627124\pi\)
−0.388840 + 0.921305i \(0.627124\pi\)
\(374\) 0 0
\(375\) −3.01051 3.01051i −0.155462 0.155462i
\(376\) 0 0
\(377\) 2.53307i 0.130460i
\(378\) 0 0
\(379\) 5.80687 0.298279 0.149139 0.988816i \(-0.452350\pi\)
0.149139 + 0.988816i \(0.452350\pi\)
\(380\) 0 0
\(381\) −0.537691 + 0.537691i −0.0275467 + 0.0275467i
\(382\) 0 0
\(383\) −24.4288 24.4288i −1.24825 1.24825i −0.956491 0.291763i \(-0.905758\pi\)
−0.291763 0.956491i \(-0.594242\pi\)
\(384\) 0 0
\(385\) −9.90966 −0.505043
\(386\) 0 0
\(387\) −12.2325 −0.621812
\(388\) 0 0
\(389\) 28.3372i 1.43675i −0.695656 0.718375i \(-0.744886\pi\)
0.695656 0.718375i \(-0.255114\pi\)
\(390\) 0 0
\(391\) −11.1455 11.1455i −0.563654 0.563654i
\(392\) 0 0
\(393\) 4.45226 + 4.45226i 0.224587 + 0.224587i
\(394\) 0 0
\(395\) −5.30328 5.30328i −0.266837 0.266837i
\(396\) 0 0
\(397\) 19.8306 19.8306i 0.995269 0.995269i −0.00472015 0.999989i \(-0.501502\pi\)
0.999989 + 0.00472015i \(0.00150248\pi\)
\(398\) 0 0
\(399\) −0.655350 + 0.655350i −0.0328085 + 0.0328085i
\(400\) 0 0
\(401\) 19.5160i 0.974584i 0.873239 + 0.487292i \(0.162015\pi\)
−0.873239 + 0.487292i \(0.837985\pi\)
\(402\) 0 0
\(403\) −4.03325 + 4.03325i −0.200910 + 0.200910i
\(404\) 0 0
\(405\) 12.2701i 0.609704i
\(406\) 0 0
\(407\) −24.2650 + 24.2650i −1.20277 + 1.20277i
\(408\) 0 0
\(409\) 6.13126 0.303171 0.151586 0.988444i \(-0.451562\pi\)
0.151586 + 0.988444i \(0.451562\pi\)
\(410\) 0 0
\(411\) 7.07729 0.349097
\(412\) 0 0
\(413\) −10.4437 + 10.4437i −0.513900 + 0.513900i
\(414\) 0 0
\(415\) 15.4971i 0.760721i
\(416\) 0 0
\(417\) 0.585705 0.585705i 0.0286821 0.0286821i
\(418\) 0 0
\(419\) 3.75478i 0.183433i 0.995785 + 0.0917166i \(0.0292354\pi\)
−0.995785 + 0.0917166i \(0.970765\pi\)
\(420\) 0 0
\(421\) 1.66007 1.66007i 0.0809067 0.0809067i −0.665495 0.746402i \(-0.731780\pi\)
0.746402 + 0.665495i \(0.231780\pi\)
\(422\) 0 0
\(423\) −0.0409110 + 0.0409110i −0.00198916 + 0.00198916i
\(424\) 0 0
\(425\) −10.3375 10.3375i −0.501440 0.501440i
\(426\) 0 0
\(427\) −9.51838 9.51838i −0.460627 0.460627i
\(428\) 0 0
\(429\) 2.22044 + 2.22044i 0.107204 + 0.107204i
\(430\) 0 0
\(431\) 8.73186i 0.420599i 0.977637 + 0.210299i \(0.0674439\pi\)
−0.977637 + 0.210299i \(0.932556\pi\)
\(432\) 0 0
\(433\) −3.23449 −0.155440 −0.0777199 0.996975i \(-0.524764\pi\)
−0.0777199 + 0.996975i \(0.524764\pi\)
\(434\) 0 0
\(435\) 1.03835 0.0497851
\(436\) 0 0
\(437\) −5.00997 5.00997i −0.239659 0.239659i
\(438\) 0 0
\(439\) −11.2187 + 11.2187i −0.535441 + 0.535441i −0.922186 0.386746i \(-0.873599\pi\)
0.386746 + 0.922186i \(0.373599\pi\)
\(440\) 0 0
\(441\) 2.87011 0.136672
\(442\) 0 0
\(443\) 12.5388i 0.595735i 0.954607 + 0.297867i \(0.0962753\pi\)
−0.954607 + 0.297867i \(0.903725\pi\)
\(444\) 0 0
\(445\) −5.31264 5.31264i −0.251843 0.251843i
\(446\) 0 0
\(447\) −3.72306 −0.176095
\(448\) 0 0
\(449\) 23.1930i 1.09455i −0.836954 0.547274i \(-0.815666\pi\)
0.836954 0.547274i \(-0.184334\pi\)
\(450\) 0 0
\(451\) −22.2809 33.9206i −1.04917 1.59726i
\(452\) 0 0
\(453\) 4.10170i 0.192715i
\(454\) 0 0
\(455\) 2.14929 0.100760
\(456\) 0 0
\(457\) 5.50478 + 5.50478i 0.257503 + 0.257503i 0.824038 0.566535i \(-0.191716\pi\)
−0.566535 + 0.824038i \(0.691716\pi\)
\(458\) 0 0
\(459\) 12.1032i 0.564929i
\(460\) 0 0
\(461\) −16.9113 −0.787638 −0.393819 0.919188i \(-0.628846\pi\)
−0.393819 + 0.919188i \(0.628846\pi\)
\(462\) 0 0
\(463\) 2.01712 2.01712i 0.0937435 0.0937435i −0.658680 0.752423i \(-0.728885\pi\)
0.752423 + 0.658680i \(0.228885\pi\)
\(464\) 0 0
\(465\) −1.65330 1.65330i −0.0766701 0.0766701i
\(466\) 0 0
\(467\) 34.8616 1.61320 0.806600 0.591097i \(-0.201305\pi\)
0.806600 + 0.591097i \(0.201305\pi\)
\(468\) 0 0
\(469\) 0.225377 0.0104069
\(470\) 0 0
\(471\) 6.44758i 0.297089i
\(472\) 0 0
\(473\) 19.1013 + 19.1013i 0.878278 + 0.878278i
\(474\) 0 0
\(475\) −4.64673 4.64673i −0.213207 0.213207i
\(476\) 0 0
\(477\) 29.1732 + 29.1732i 1.33575 + 1.33575i
\(478\) 0 0
\(479\) 30.4086 30.4086i 1.38941 1.38941i 0.562838 0.826567i \(-0.309709\pi\)
0.826567 0.562838i \(-0.190291\pi\)
\(480\) 0 0
\(481\) 5.26280 5.26280i 0.239963 0.239963i
\(482\) 0 0
\(483\) 0.993011i 0.0451836i
\(484\) 0 0
\(485\) 18.0415 18.0415i 0.819222 0.819222i
\(486\) 0 0
\(487\) 15.7055i 0.711685i −0.934546 0.355843i \(-0.884194\pi\)
0.934546 0.355843i \(-0.115806\pi\)
\(488\) 0 0
\(489\) 4.45376 4.45376i 0.201406 0.201406i
\(490\) 0 0
\(491\) −11.4699 −0.517631 −0.258816 0.965927i \(-0.583332\pi\)
−0.258816 + 0.965927i \(0.583332\pi\)
\(492\) 0 0
\(493\) 10.5416 0.474772
\(494\) 0 0
\(495\) 20.1114 20.1114i 0.903938 0.903938i
\(496\) 0 0
\(497\) 13.2529i 0.594475i
\(498\) 0 0
\(499\) 4.31496 4.31496i 0.193164 0.193164i −0.603898 0.797062i \(-0.706387\pi\)
0.797062 + 0.603898i \(0.206387\pi\)
\(500\) 0 0
\(501\) 2.78822i 0.124569i
\(502\) 0 0
\(503\) 7.95360 7.95360i 0.354634 0.354634i −0.507197 0.861830i \(-0.669318\pi\)
0.861830 + 0.507197i \(0.169318\pi\)
\(504\) 0 0
\(505\) 19.2327 19.2327i 0.855843 0.855843i
\(506\) 0 0
\(507\) 2.83144 + 2.83144i 0.125749 + 0.125749i
\(508\) 0 0
\(509\) −0.886631 0.886631i −0.0392992 0.0392992i 0.687184 0.726483i \(-0.258847\pi\)
−0.726483 + 0.687184i \(0.758847\pi\)
\(510\) 0 0
\(511\) 1.49598 + 1.49598i 0.0661784 + 0.0661784i
\(512\) 0 0
\(513\) 5.44044i 0.240201i
\(514\) 0 0
\(515\) 22.6826 0.999514
\(516\) 0 0
\(517\) 0.127767 0.00561918
\(518\) 0 0
\(519\) −1.12861 1.12861i −0.0495407 0.0495407i
\(520\) 0 0
\(521\) −7.46171 + 7.46171i −0.326904 + 0.326904i −0.851408 0.524504i \(-0.824251\pi\)
0.524504 + 0.851408i \(0.324251\pi\)
\(522\) 0 0
\(523\) −11.9101 −0.520792 −0.260396 0.965502i \(-0.583853\pi\)
−0.260396 + 0.965502i \(0.583853\pi\)
\(524\) 0 0
\(525\) 0.921014i 0.0401964i
\(526\) 0 0
\(527\) −16.7848 16.7848i −0.731158 0.731158i
\(528\) 0 0
\(529\) −15.4087 −0.669943
\(530\) 0 0
\(531\) 42.3903i 1.83958i
\(532\) 0 0
\(533\) 4.83246 + 7.35697i 0.209317 + 0.318666i
\(534\) 0 0
\(535\) 22.8005i 0.985753i
\(536\) 0 0
\(537\) 6.80467 0.293643
\(538\) 0 0
\(539\) −4.48173 4.48173i −0.193042 0.193042i
\(540\) 0 0
\(541\) 29.4589i 1.26654i −0.773932 0.633269i \(-0.781713\pi\)
0.773932 0.633269i \(-0.218287\pi\)
\(542\) 0 0
\(543\) −5.07068 −0.217604
\(544\) 0 0
\(545\) −12.6350 + 12.6350i −0.541224 + 0.541224i
\(546\) 0 0
\(547\) 11.2371 + 11.2371i 0.480463 + 0.480463i 0.905279 0.424817i \(-0.139661\pi\)
−0.424817 + 0.905279i \(0.639661\pi\)
\(548\) 0 0
\(549\) 38.6346 1.64888
\(550\) 0 0
\(551\) 4.73851 0.201868
\(552\) 0 0
\(553\) 4.79691i 0.203985i
\(554\) 0 0
\(555\) 2.15732 + 2.15732i 0.0915731 + 0.0915731i
\(556\) 0 0
\(557\) 13.4187 + 13.4187i 0.568570 + 0.568570i 0.931728 0.363158i \(-0.118301\pi\)
−0.363158 + 0.931728i \(0.618301\pi\)
\(558\) 0 0
\(559\) −4.14284 4.14284i −0.175223 0.175223i
\(560\) 0 0
\(561\) −9.24059 + 9.24059i −0.390138 + 0.390138i
\(562\) 0 0
\(563\) −22.7624 + 22.7624i −0.959320 + 0.959320i −0.999204 0.0398841i \(-0.987301\pi\)
0.0398841 + 0.999204i \(0.487301\pi\)
\(564\) 0 0
\(565\) 26.4783i 1.11395i
\(566\) 0 0
\(567\) −5.54925 + 5.54925i −0.233046 + 0.233046i
\(568\) 0 0
\(569\) 6.44661i 0.270256i −0.990828 0.135128i \(-0.956855\pi\)
0.990828 0.135128i \(-0.0431446\pi\)
\(570\) 0 0
\(571\) −10.3652 + 10.3652i −0.433771 + 0.433771i −0.889909 0.456138i \(-0.849232\pi\)
0.456138 + 0.889909i \(0.349232\pi\)
\(572\) 0 0
\(573\) 3.05670 0.127695
\(574\) 0 0
\(575\) 7.04090 0.293626
\(576\) 0 0
\(577\) 0.989459 0.989459i 0.0411917 0.0411917i −0.686211 0.727403i \(-0.740727\pi\)
0.727403 + 0.686211i \(0.240727\pi\)
\(578\) 0 0
\(579\) 7.30371i 0.303532i
\(580\) 0 0
\(581\) −7.00869 + 7.00869i −0.290769 + 0.290769i
\(582\) 0 0
\(583\) 91.1091i 3.77335i
\(584\) 0 0
\(585\) −4.36191 + 4.36191i −0.180343 + 0.180343i
\(586\) 0 0
\(587\) 11.5081 11.5081i 0.474989 0.474989i −0.428536 0.903525i \(-0.640970\pi\)
0.903525 + 0.428536i \(0.140970\pi\)
\(588\) 0 0
\(589\) −7.54485 7.54485i −0.310880 0.310880i
\(590\) 0 0
\(591\) 4.92028 + 4.92028i 0.202393 + 0.202393i
\(592\) 0 0
\(593\) −31.3684 31.3684i −1.28815 1.28815i −0.935912 0.352233i \(-0.885422\pi\)
−0.352233 0.935912i \(-0.614578\pi\)
\(594\) 0 0
\(595\) 8.94450i 0.366689i
\(596\) 0 0
\(597\) −0.149286 −0.00610989
\(598\) 0 0
\(599\) 25.2661 1.03235 0.516173 0.856484i \(-0.327356\pi\)
0.516173 + 0.856484i \(0.327356\pi\)
\(600\) 0 0
\(601\) 3.12731 + 3.12731i 0.127565 + 0.127565i 0.768007 0.640442i \(-0.221249\pi\)
−0.640442 + 0.768007i \(0.721249\pi\)
\(602\) 0 0
\(603\) −0.457396 + 0.457396i −0.0186266 + 0.0186266i
\(604\) 0 0
\(605\) −45.6101 −1.85431
\(606\) 0 0
\(607\) 19.6679i 0.798295i −0.916887 0.399148i \(-0.869306\pi\)
0.916887 0.399148i \(-0.130694\pi\)
\(608\) 0 0
\(609\) 0.469603 + 0.469603i 0.0190293 + 0.0190293i
\(610\) 0 0
\(611\) −0.0277111 −0.00112107
\(612\) 0 0
\(613\) 3.92464i 0.158515i −0.996854 0.0792573i \(-0.974745\pi\)
0.996854 0.0792573i \(-0.0252549\pi\)
\(614\) 0 0
\(615\) −3.01576 + 1.98091i −0.121607 + 0.0798782i
\(616\) 0 0
\(617\) 14.9036i 0.599996i −0.953940 0.299998i \(-0.903014\pi\)
0.953940 0.299998i \(-0.0969860\pi\)
\(618\) 0 0
\(619\) −26.3485 −1.05904 −0.529518 0.848299i \(-0.677627\pi\)
−0.529518 + 0.848299i \(0.677627\pi\)
\(620\) 0 0
\(621\) 4.12178 + 4.12178i 0.165401 + 0.165401i
\(622\) 0 0
\(623\) 4.80537i 0.192523i
\(624\) 0 0
\(625\) −5.69225 −0.227690
\(626\) 0 0
\(627\) −4.15369 + 4.15369i −0.165882 + 0.165882i
\(628\) 0 0
\(629\) 21.9017 + 21.9017i 0.873279 + 0.873279i
\(630\) 0 0
\(631\) 20.2937 0.807880 0.403940 0.914785i \(-0.367640\pi\)
0.403940 + 0.914785i \(0.367640\pi\)
\(632\) 0 0
\(633\) 4.99630 0.198585
\(634\) 0 0
\(635\) 3.29875i 0.130907i
\(636\) 0 0
\(637\) 0.972034 + 0.972034i 0.0385134 + 0.0385134i
\(638\) 0 0
\(639\) 26.8964 + 26.8964i 1.06401 + 1.06401i
\(640\) 0 0
\(641\) −8.35473 8.35473i −0.329992 0.329992i 0.522591 0.852583i \(-0.324965\pi\)
−0.852583 + 0.522591i \(0.824965\pi\)
\(642\) 0 0
\(643\) 3.44242 3.44242i 0.135756 0.135756i −0.635963 0.771719i \(-0.719397\pi\)
0.771719 + 0.635963i \(0.219397\pi\)
\(644\) 0 0
\(645\) 1.69823 1.69823i 0.0668676 0.0668676i
\(646\) 0 0
\(647\) 27.9293i 1.09801i 0.835817 + 0.549007i \(0.184994\pi\)
−0.835817 + 0.549007i \(0.815006\pi\)
\(648\) 0 0
\(649\) −66.1934 + 66.1934i −2.59832 + 2.59832i
\(650\) 0 0
\(651\) 1.49544i 0.0586110i
\(652\) 0 0
\(653\) −1.04040 + 1.04040i −0.0407139 + 0.0407139i −0.727171 0.686457i \(-0.759165\pi\)
0.686457 + 0.727171i \(0.259165\pi\)
\(654\) 0 0
\(655\) 27.3148 1.06728
\(656\) 0 0
\(657\) −6.07211 −0.236896
\(658\) 0 0
\(659\) 25.2088 25.2088i 0.981994 0.981994i −0.0178465 0.999841i \(-0.505681\pi\)
0.999841 + 0.0178465i \(0.00568101\pi\)
\(660\) 0 0
\(661\) 7.62851i 0.296715i 0.988934 + 0.148357i \(0.0473986\pi\)
−0.988934 + 0.148357i \(0.952601\pi\)
\(662\) 0 0
\(663\) 2.00417 2.00417i 0.0778357 0.0778357i
\(664\) 0 0
\(665\) 4.02059i 0.155912i
\(666\) 0 0
\(667\) −3.58999 + 3.58999i −0.139005 + 0.139005i
\(668\) 0 0
\(669\) −0.898643 + 0.898643i −0.0347435 + 0.0347435i
\(670\) 0 0
\(671\) −60.3287 60.3287i −2.32896 2.32896i
\(672\) 0 0
\(673\) −31.2048 31.2048i −1.20286 1.20286i −0.973294 0.229562i \(-0.926271\pi\)
−0.229562 0.973294i \(-0.573729\pi\)
\(674\) 0 0
\(675\) 3.82294 + 3.82294i 0.147145 + 0.147145i
\(676\) 0 0
\(677\) 33.8516i 1.30102i −0.759497 0.650511i \(-0.774555\pi\)
0.759497 0.650511i \(-0.225445\pi\)
\(678\) 0 0
\(679\) 16.3188 0.626260
\(680\) 0 0
\(681\) 6.11499 0.234327
\(682\) 0 0
\(683\) 8.24531 + 8.24531i 0.315498 + 0.315498i 0.847035 0.531537i \(-0.178385\pi\)
−0.531537 + 0.847035i \(0.678385\pi\)
\(684\) 0 0
\(685\) 21.7097 21.7097i 0.829485 0.829485i
\(686\) 0 0
\(687\) −10.5552 −0.402706
\(688\) 0 0
\(689\) 19.7605i 0.752814i
\(690\) 0 0
\(691\) −18.3595 18.3595i −0.698427 0.698427i 0.265644 0.964071i \(-0.414415\pi\)
−0.964071 + 0.265644i \(0.914415\pi\)
\(692\) 0 0
\(693\) 18.1911 0.691022
\(694\) 0 0
\(695\) 3.59332i 0.136302i
\(696\) 0 0
\(697\) −30.6169 + 20.1108i −1.15970 + 0.761752i
\(698\) 0 0
\(699\) 6.55968i 0.248110i
\(700\) 0 0
\(701\) 22.6033 0.853714 0.426857 0.904319i \(-0.359621\pi\)
0.426857 + 0.904319i \(0.359621\pi\)
\(702\) 0 0
\(703\) 9.84492 + 9.84492i 0.371308 + 0.371308i
\(704\) 0 0
\(705\) 0.0113593i 0.000427816i
\(706\) 0 0
\(707\) 17.3963 0.654255
\(708\) 0 0
\(709\) 0.664120 0.664120i 0.0249415 0.0249415i −0.694526 0.719468i \(-0.744386\pi\)
0.719468 + 0.694526i \(0.244386\pi\)
\(710\) 0 0
\(711\) 9.73519 + 9.73519i 0.365098 + 0.365098i
\(712\) 0 0
\(713\) 11.4322 0.428141
\(714\) 0 0
\(715\) 13.6224 0.509450
\(716\) 0 0
\(717\) 7.54121i 0.281632i
\(718\) 0 0
\(719\) 8.92610 + 8.92610i 0.332887 + 0.332887i 0.853682 0.520795i \(-0.174364\pi\)
−0.520795 + 0.853682i \(0.674364\pi\)
\(720\) 0 0
\(721\) 10.2584 + 10.2584i 0.382043 + 0.382043i
\(722\) 0 0
\(723\) −1.30475 1.30475i −0.0485243 0.0485243i
\(724\) 0 0
\(725\) −3.32970 + 3.32970i −0.123662 + 0.123662i
\(726\) 0 0
\(727\) 19.8758 19.8758i 0.737153 0.737153i −0.234873 0.972026i \(-0.575467\pi\)
0.972026 + 0.234873i \(0.0754674\pi\)
\(728\) 0 0
\(729\) 20.2366i 0.749503i
\(730\) 0 0
\(731\) 17.2409 17.2409i 0.637678 0.637678i
\(732\) 0 0
\(733\) 46.9920i 1.73569i 0.496834 + 0.867845i \(0.334496\pi\)
−0.496834 + 0.867845i \(0.665504\pi\)
\(734\) 0 0
\(735\) −0.398455 + 0.398455i −0.0146972 + 0.0146972i
\(736\) 0 0
\(737\) 1.42847 0.0526183
\(738\) 0 0
\(739\) 17.7510 0.652980 0.326490 0.945201i \(-0.394134\pi\)
0.326490 + 0.945201i \(0.394134\pi\)
\(740\) 0 0
\(741\) 0.900885 0.900885i 0.0330948 0.0330948i
\(742\) 0 0
\(743\) 23.6333i 0.867022i 0.901148 + 0.433511i \(0.142726\pi\)
−0.901148 + 0.433511i \(0.857274\pi\)
\(744\) 0 0
\(745\) −11.4206 + 11.4206i −0.418417 + 0.418417i
\(746\) 0 0
\(747\) 28.4479i 1.04085i
\(748\) 0 0
\(749\) −10.3117 + 10.3117i −0.376783 + 0.376783i
\(750\) 0 0
\(751\) 13.9429 13.9429i 0.508783 0.508783i −0.405370 0.914153i \(-0.632857\pi\)
0.914153 + 0.405370i \(0.132857\pi\)
\(752\) 0 0
\(753\) 4.43565 + 4.43565i 0.161644 + 0.161644i
\(754\) 0 0
\(755\) 12.5820 + 12.5820i 0.457907 + 0.457907i
\(756\) 0 0
\(757\) 3.25553 + 3.25553i 0.118324 + 0.118324i 0.763790 0.645465i \(-0.223336\pi\)
−0.645465 + 0.763790i \(0.723336\pi\)
\(758\) 0 0
\(759\) 6.29383i 0.228451i
\(760\) 0 0
\(761\) 4.17017 0.151169 0.0755843 0.997139i \(-0.475918\pi\)
0.0755843 + 0.997139i \(0.475918\pi\)
\(762\) 0 0
\(763\) −11.4286 −0.413742
\(764\) 0 0
\(765\) −18.1526 18.1526i −0.656309 0.656309i
\(766\) 0 0
\(767\) 14.3566 14.3566i 0.518385 0.518385i
\(768\) 0 0
\(769\) 7.37265 0.265865 0.132932 0.991125i \(-0.457561\pi\)
0.132932 + 0.991125i \(0.457561\pi\)
\(770\) 0 0
\(771\) 0.354229i 0.0127572i
\(772\) 0 0
\(773\) 6.31710 + 6.31710i 0.227210 + 0.227210i 0.811526 0.584316i \(-0.198637\pi\)
−0.584316 + 0.811526i \(0.698637\pi\)
\(774\) 0 0
\(775\) 10.6034 0.380884
\(776\) 0 0
\(777\) 1.95133i 0.0700037i
\(778\) 0 0
\(779\) −13.7624 + 9.03990i −0.493090 + 0.323888i
\(780\) 0 0
\(781\) 83.9986i 3.00571i
\(782\) 0 0
\(783\) −3.89845 −0.139319
\(784\) 0 0
\(785\) 19.7781 + 19.7781i 0.705909 + 0.705909i
\(786\) 0 0
\(787\) 29.5407i 1.05301i −0.850171 0.526507i \(-0.823501\pi\)
0.850171 0.526507i \(-0.176499\pi\)
\(788\) 0 0
\(789\) 0.0478587 0.00170381
\(790\) 0 0
\(791\) 11.9750 11.9750i 0.425784 0.425784i
\(792\) 0 0
\(793\) 13.0846 + 13.0846i 0.464647 + 0.464647i
\(794\) 0 0
\(795\) −8.10019 −0.287284
\(796\) 0 0
\(797\) 25.1515 0.890911 0.445456 0.895304i \(-0.353042\pi\)
0.445456 + 0.895304i \(0.353042\pi\)
\(798\) 0 0
\(799\) 0.115323i 0.00407984i
\(800\) 0 0
\(801\) 9.75237 + 9.75237i 0.344583 + 0.344583i
\(802\) 0 0
\(803\) 9.48173 + 9.48173i 0.334603 + 0.334603i
\(804\) 0 0
\(805\) −3.04608 3.04608i −0.107360 0.107360i
\(806\) 0 0
\(807\) 3.16589 3.16589i 0.111445 0.111445i
\(808\) 0 0
\(809\) −22.9213 + 22.9213i −0.805868 + 0.805868i −0.984006 0.178137i \(-0.942993\pi\)
0.178137 + 0.984006i \(0.442993\pi\)
\(810\) 0 0
\(811\) 3.09001i 0.108505i −0.998527 0.0542525i \(-0.982722\pi\)
0.998527 0.0542525i \(-0.0172776\pi\)
\(812\) 0 0
\(813\) 2.87757 2.87757i 0.100921 0.100921i
\(814\) 0 0
\(815\) 27.3240i 0.957117i
\(816\) 0 0
\(817\) 7.74986 7.74986i 0.271133 0.271133i
\(818\) 0 0
\(819\) −3.94543 −0.137864
\(820\) 0 0
\(821\) 12.8825 0.449603 0.224801 0.974405i \(-0.427827\pi\)
0.224801 + 0.974405i \(0.427827\pi\)
\(822\) 0 0
\(823\) −27.0156 + 27.0156i −0.941703 + 0.941703i −0.998392 0.0566888i \(-0.981946\pi\)
0.0566888 + 0.998392i \(0.481946\pi\)
\(824\) 0 0
\(825\) 5.83750i 0.203236i
\(826\) 0 0
\(827\) −25.2793 + 25.2793i −0.879049 + 0.879049i −0.993436 0.114387i \(-0.963510\pi\)
0.114387 + 0.993436i \(0.463510\pi\)
\(828\) 0 0
\(829\) 0.638230i 0.0221666i 0.999939 + 0.0110833i \(0.00352800\pi\)
−0.999939 + 0.0110833i \(0.996472\pi\)
\(830\) 0 0
\(831\) −0.404901 + 0.404901i −0.0140459 + 0.0140459i
\(832\) 0 0
\(833\) −4.04523 + 4.04523i −0.140159 + 0.140159i
\(834\) 0 0
\(835\) −8.55292 8.55292i −0.295986 0.295986i
\(836\) 0 0
\(837\) 6.20727 + 6.20727i 0.214554 + 0.214554i
\(838\) 0 0
\(839\) 11.5218 + 11.5218i 0.397777 + 0.397777i 0.877448 0.479671i \(-0.159244\pi\)
−0.479671 + 0.877448i \(0.659244\pi\)
\(840\) 0 0
\(841\) 25.6045i 0.882915i
\(842\) 0 0
\(843\) 3.93970 0.135690
\(844\) 0 0
\(845\) 17.3710 0.597579
\(846\) 0 0
\(847\) −20.6276 20.6276i −0.708772 0.708772i
\(848\) 0 0
\(849\) 2.87377 2.87377i 0.0986274 0.0986274i
\(850\) 0 0
\(851\) −14.9174 −0.511362
\(852\) 0 0
\(853\) 27.1139i 0.928364i −0.885740 0.464182i \(-0.846348\pi\)
0.885740 0.464182i \(-0.153652\pi\)
\(854\) 0 0
\(855\) −8.15967 8.15967i −0.279055 0.279055i
\(856\) 0 0
\(857\) 16.1588 0.551974 0.275987 0.961161i \(-0.410995\pi\)
0.275987 + 0.961161i \(0.410995\pi\)
\(858\) 0 0
\(859\) 12.8872i 0.439705i 0.975533 + 0.219853i \(0.0705577\pi\)
−0.975533 + 0.219853i \(0.929442\pi\)
\(860\) 0 0
\(861\) −2.25979 0.468018i −0.0770134 0.0159500i
\(862\) 0 0
\(863\) 18.8684i 0.642289i −0.947030 0.321144i \(-0.895933\pi\)
0.947030 0.321144i \(-0.104067\pi\)
\(864\) 0 0
\(865\) −6.92409 −0.235426
\(866\) 0 0
\(867\) 4.00819 + 4.00819i 0.136125 + 0.136125i
\(868\) 0 0
\(869\) 30.4034i 1.03136i
\(870\) 0 0
\(871\) −0.309818 −0.0104978
\(872\) 0 0
\(873\) −33.1186 + 33.1186i −1.12090 + 1.12090i
\(874\) 0 0
\(875\) −8.35303 8.35303i −0.282384 0.282384i
\(876\) 0 0
\(877\) −29.0659 −0.981485 −0.490742 0.871305i \(-0.663274\pi\)
−0.490742 + 0.871305i \(0.663274\pi\)
\(878\) 0 0
\(879\) 3.26957 0.110280
\(880\) 0 0
\(881\) 32.3751i 1.09075i 0.838194 + 0.545373i \(0.183612\pi\)
−0.838194 + 0.545373i \(0.816388\pi\)
\(882\) 0 0
\(883\) −16.2872 16.2872i −0.548110 0.548110i 0.377784 0.925894i \(-0.376686\pi\)
−0.925894 + 0.377784i \(0.876686\pi\)
\(884\) 0 0
\(885\) 5.88502 + 5.88502i 0.197823 + 0.197823i
\(886\) 0 0
\(887\) 34.3493 + 34.3493i 1.15334 + 1.15334i 0.985879 + 0.167456i \(0.0535554\pi\)
0.167456 + 0.985879i \(0.446445\pi\)
\(888\) 0 0
\(889\) −1.49189 + 1.49189i −0.0500364 + 0.0500364i
\(890\) 0 0
\(891\) −35.1718 + 35.1718i −1.17830 + 1.17830i
\(892\) 0 0
\(893\) 0.0518382i 0.00173470i
\(894\) 0 0
\(895\) 20.8734 20.8734i 0.697721 0.697721i
\(896\) 0 0
\(897\) 1.36506i 0.0455779i
\(898\) 0 0
\(899\) −5.40641 + 5.40641i −0.180314 + 0.180314i
\(900\) 0 0
\(901\) −82.2355 −2.73966
\(902\) 0 0
\(903\) 1.53608 0.0511174
\(904\) 0 0
\(905\) −15.5544 + 15.5544i −0.517045 + 0.517045i
\(906\) 0 0
\(907\) 33.9735i 1.12807i 0.825751 + 0.564035i \(0.190752\pi\)
−0.825751 + 0.564035i \(0.809248\pi\)
\(908\) 0 0
\(909\) −35.3053 + 35.3053i −1.17100 + 1.17100i
\(910\) 0 0
\(911\) 24.2317i 0.802831i −0.915896 0.401416i \(-0.868518\pi\)
0.915896 0.401416i \(-0.131482\pi\)
\(912\) 0 0
\(913\) −44.4219 + 44.4219i −1.47015 + 1.47015i
\(914\) 0 0
\(915\) −5.36361 + 5.36361i −0.177315 + 0.177315i
\(916\) 0 0
\(917\) 12.3533 + 12.3533i 0.407943 + 0.407943i
\(918\) 0 0
\(919\) 26.1571 + 26.1571i 0.862843 + 0.862843i 0.991667 0.128825i \(-0.0411205\pi\)
−0.128825 + 0.991667i \(0.541120\pi\)
\(920\) 0 0
\(921\) −5.39706 5.39706i −0.177839 0.177839i
\(922\) 0 0
\(923\) 18.2183i 0.599663i
\(924\) 0 0
\(925\) −13.8358 −0.454920
\(926\) 0 0
\(927\) −41.6382 −1.36758
\(928\) 0 0
\(929\) 9.22797 + 9.22797i 0.302760 + 0.302760i 0.842093 0.539333i \(-0.181324\pi\)
−0.539333 + 0.842093i \(0.681324\pi\)
\(930\) 0 0
\(931\) −1.81835 + 1.81835i −0.0595939 + 0.0595939i
\(932\) 0 0
\(933\) 0.963835 0.0315546
\(934\) 0 0
\(935\) 56.6913i 1.85401i
\(936\) 0 0
\(937\) 1.55404 + 1.55404i 0.0507684 + 0.0507684i 0.732035 0.681267i \(-0.238571\pi\)
−0.681267 + 0.732035i \(0.738571\pi\)
\(938\) 0 0
\(939\) 0.668704 0.0218223
\(940\) 0 0
\(941\) 34.2396i 1.11618i −0.829781 0.558089i \(-0.811535\pi\)
0.829781 0.558089i \(-0.188465\pi\)
\(942\) 0 0
\(943\) 3.57787 17.2755i 0.116511 0.562567i
\(944\) 0 0
\(945\) 3.30780i 0.107603i
\(946\) 0 0
\(947\) −42.0556 −1.36662 −0.683311 0.730127i \(-0.739461\pi\)
−0.683311 + 0.730127i \(0.739461\pi\)
\(948\) 0 0
\(949\) −2.05647 2.05647i −0.0667560 0.0667560i
\(950\) 0 0
\(951\) 12.0162i 0.389651i
\(952\) 0 0
\(953\) 5.29357 0.171475 0.0857377 0.996318i \(-0.472675\pi\)
0.0857377 + 0.996318i \(0.472675\pi\)
\(954\) 0 0
\(955\) 9.37647 9.37647i 0.303416 0.303416i
\(956\) 0 0
\(957\) 2.97640 + 2.97640i 0.0962135 + 0.0962135i
\(958\) 0 0
\(959\) 19.6368 0.634106
\(960\) 0 0
\(961\) −13.7834 −0.444627
\(962\) 0 0
\(963\) 41.8548i 1.34875i
\(964\) 0 0
\(965\) 22.4042 + 22.4042i 0.721218 + 0.721218i
\(966\) 0 0
\(967\) 1.26902 + 1.26902i 0.0408090 + 0.0408090i 0.727217 0.686408i \(-0.240813\pi\)
−0.686408 + 0.727217i \(0.740813\pi\)
\(968\) 0 0
\(969\) 3.74914 + 3.74914i 0.120440 + 0.120440i
\(970\) 0 0
\(971\) 3.24821 3.24821i 0.104240 0.104240i −0.653063 0.757303i \(-0.726516\pi\)
0.757303 + 0.653063i \(0.226516\pi\)
\(972\) 0 0
\(973\) 1.62511 1.62511i 0.0520987 0.0520987i
\(974\) 0 0
\(975\) 1.26608i 0.0405472i
\(976\) 0 0
\(977\) 3.99196 3.99196i 0.127714 0.127714i −0.640361 0.768074i \(-0.721215\pi\)
0.768074 + 0.640361i \(0.221215\pi\)
\(978\) 0 0
\(979\) 30.4570i 0.973412i
\(980\) 0 0
\(981\) 23.1940 23.1940i 0.740527 0.740527i
\(982\) 0 0
\(983\) −18.8434 −0.601011 −0.300506 0.953780i \(-0.597155\pi\)
−0.300506 + 0.953780i \(0.597155\pi\)
\(984\) 0 0
\(985\) 30.1861 0.961808
\(986\) 0 0
\(987\) 0.00513735 0.00513735i 0.000163524 0.000163524i
\(988\) 0 0
\(989\) 11.7429i 0.373402i
\(990\) 0 0
\(991\) −17.6867 + 17.6867i −0.561838 + 0.561838i −0.929829 0.367991i \(-0.880046\pi\)
0.367991 + 0.929829i \(0.380046\pi\)
\(992\) 0 0
\(993\) 8.82171i 0.279949i
\(994\) 0 0
\(995\) −0.457939 + 0.457939i −0.0145176 + 0.0145176i
\(996\) 0 0
\(997\) −29.0055 + 29.0055i −0.918615 + 0.918615i −0.996929 0.0783141i \(-0.975046\pi\)
0.0783141 + 0.996929i \(0.475046\pi\)
\(998\) 0 0
\(999\) −8.09958 8.09958i −0.256259 0.256259i
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 1148.2.k.a.729.3 yes 8
41.9 even 4 inner 1148.2.k.a.337.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1148.2.k.a.337.3 8 41.9 even 4 inner
1148.2.k.a.729.3 yes 8 1.1 even 1 trivial