Properties

Label 136.4.k.a.89.5
Level $136$
Weight $4$
Character 136.89
Analytic conductor $8.024$
Analytic rank $0$
Dimension $14$
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] = [136,4,Mod(81,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.81");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.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: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 142x^{12} + 7785x^{10} + 208489x^{8} + 2822686x^{6} + 17977041x^{4} + 45020416x^{2} + 31719424 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 89.5
Root \(3.07347i\) of defining polynomial
Character \(\chi\) \(=\) 136.89
Dual form 136.4.k.a.81.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.07347 - 3.07347i) q^{3} +(-8.67626 + 8.67626i) q^{5} +(9.02315 + 9.02315i) q^{7} +8.10760i q^{9} +O(q^{10})\) \(q+(3.07347 - 3.07347i) q^{3} +(-8.67626 + 8.67626i) q^{5} +(9.02315 + 9.02315i) q^{7} +8.10760i q^{9} +(39.0636 + 39.0636i) q^{11} -0.868393 q^{13} +53.3324i q^{15} +(69.8166 - 6.21656i) q^{17} -50.0440i q^{19} +55.4647 q^{21} +(-34.6437 - 34.6437i) q^{23} -25.5549i q^{25} +(107.902 + 107.902i) q^{27} +(100.912 - 100.912i) q^{29} +(-208.341 + 208.341i) q^{31} +240.121 q^{33} -156.574 q^{35} +(-92.7068 + 92.7068i) q^{37} +(-2.66898 + 2.66898i) q^{39} +(127.219 + 127.219i) q^{41} +274.357i q^{43} +(-70.3436 - 70.3436i) q^{45} -92.6014 q^{47} -180.166i q^{49} +(195.473 - 233.685i) q^{51} -460.648i q^{53} -677.851 q^{55} +(-153.808 - 153.808i) q^{57} -539.280i q^{59} +(-456.343 - 456.343i) q^{61} +(-73.1561 + 73.1561i) q^{63} +(7.53440 - 7.53440i) q^{65} -105.099 q^{67} -212.952 q^{69} +(-57.5002 + 57.5002i) q^{71} +(636.819 - 636.819i) q^{73} +(-78.5421 - 78.5421i) q^{75} +704.953i q^{77} +(448.664 + 448.664i) q^{79} +444.362 q^{81} -264.124i q^{83} +(-551.810 + 659.683i) q^{85} -620.297i q^{87} -1573.69 q^{89} +(-7.83564 - 7.83564i) q^{91} +1280.66i q^{93} +(434.194 + 434.194i) q^{95} +(224.583 - 224.583i) q^{97} +(-316.712 + 316.712i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{3} + 8 q^{5} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4 q^{3} + 8 q^{5} - 10 q^{7} + 16 q^{11} + 72 q^{13} - 40 q^{17} - 60 q^{21} + 246 q^{23} + 572 q^{27} + 260 q^{29} + 566 q^{31} - 904 q^{33} - 804 q^{35} - 28 q^{37} + 476 q^{39} - 786 q^{41} - 604 q^{45} + 448 q^{47} - 380 q^{51} + 1612 q^{55} + 1004 q^{57} + 660 q^{61} - 2250 q^{63} - 796 q^{65} - 284 q^{67} - 2532 q^{69} + 326 q^{71} + 730 q^{73} - 864 q^{75} + 342 q^{79} - 950 q^{81} + 4148 q^{85} - 4516 q^{89} - 3112 q^{91} + 3356 q^{95} - 1194 q^{97} + 2876 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 3.07347 3.07347i 0.591489 0.591489i −0.346544 0.938034i \(-0.612645\pi\)
0.938034 + 0.346544i \(0.112645\pi\)
\(4\) 0 0
\(5\) −8.67626 + 8.67626i −0.776028 + 0.776028i −0.979153 0.203125i \(-0.934890\pi\)
0.203125 + 0.979153i \(0.434890\pi\)
\(6\) 0 0
\(7\) 9.02315 + 9.02315i 0.487204 + 0.487204i 0.907423 0.420219i \(-0.138047\pi\)
−0.420219 + 0.907423i \(0.638047\pi\)
\(8\) 0 0
\(9\) 8.10760i 0.300281i
\(10\) 0 0
\(11\) 39.0636 + 39.0636i 1.07074 + 1.07074i 0.997300 + 0.0734374i \(0.0233969\pi\)
0.0734374 + 0.997300i \(0.476603\pi\)
\(12\) 0 0
\(13\) −0.868393 −0.0185268 −0.00926342 0.999957i \(-0.502949\pi\)
−0.00926342 + 0.999957i \(0.502949\pi\)
\(14\) 0 0
\(15\) 53.3324i 0.918024i
\(16\) 0 0
\(17\) 69.8166 6.21656i 0.996059 0.0886904i
\(18\) 0 0
\(19\) 50.0440i 0.604256i −0.953267 0.302128i \(-0.902303\pi\)
0.953267 0.302128i \(-0.0976971\pi\)
\(20\) 0 0
\(21\) 55.4647 0.576352
\(22\) 0 0
\(23\) −34.6437 34.6437i −0.314074 0.314074i 0.532412 0.846486i \(-0.321286\pi\)
−0.846486 + 0.532412i \(0.821286\pi\)
\(24\) 0 0
\(25\) 25.5549i 0.204439i
\(26\) 0 0
\(27\) 107.902 + 107.902i 0.769102 + 0.769102i
\(28\) 0 0
\(29\) 100.912 100.912i 0.646166 0.646166i −0.305898 0.952064i \(-0.598957\pi\)
0.952064 + 0.305898i \(0.0989567\pi\)
\(30\) 0 0
\(31\) −208.341 + 208.341i −1.20707 + 1.20707i −0.235095 + 0.971972i \(0.575540\pi\)
−0.971972 + 0.235095i \(0.924460\pi\)
\(32\) 0 0
\(33\) 240.121 1.26666
\(34\) 0 0
\(35\) −156.574 −0.756168
\(36\) 0 0
\(37\) −92.7068 + 92.7068i −0.411916 + 0.411916i −0.882406 0.470489i \(-0.844077\pi\)
0.470489 + 0.882406i \(0.344077\pi\)
\(38\) 0 0
\(39\) −2.66898 + 2.66898i −0.0109584 + 0.0109584i
\(40\) 0 0
\(41\) 127.219 + 127.219i 0.484590 + 0.484590i 0.906594 0.422004i \(-0.138673\pi\)
−0.422004 + 0.906594i \(0.638673\pi\)
\(42\) 0 0
\(43\) 274.357i 0.973001i 0.873680 + 0.486500i \(0.161727\pi\)
−0.873680 + 0.486500i \(0.838273\pi\)
\(44\) 0 0
\(45\) −70.3436 70.3436i −0.233027 0.233027i
\(46\) 0 0
\(47\) −92.6014 −0.287389 −0.143695 0.989622i \(-0.545898\pi\)
−0.143695 + 0.989622i \(0.545898\pi\)
\(48\) 0 0
\(49\) 180.166i 0.525264i
\(50\) 0 0
\(51\) 195.473 233.685i 0.536699 0.641618i
\(52\) 0 0
\(53\) 460.648i 1.19386i −0.802292 0.596932i \(-0.796386\pi\)
0.802292 0.596932i \(-0.203614\pi\)
\(54\) 0 0
\(55\) −677.851 −1.66184
\(56\) 0 0
\(57\) −153.808 153.808i −0.357411 0.357411i
\(58\) 0 0
\(59\) 539.280i 1.18997i −0.803736 0.594986i \(-0.797158\pi\)
0.803736 0.594986i \(-0.202842\pi\)
\(60\) 0 0
\(61\) −456.343 456.343i −0.957849 0.957849i 0.0412984 0.999147i \(-0.486851\pi\)
−0.999147 + 0.0412984i \(0.986851\pi\)
\(62\) 0 0
\(63\) −73.1561 + 73.1561i −0.146298 + 0.146298i
\(64\) 0 0
\(65\) 7.53440 7.53440i 0.0143773 0.0143773i
\(66\) 0 0
\(67\) −105.099 −0.191640 −0.0958200 0.995399i \(-0.530547\pi\)
−0.0958200 + 0.995399i \(0.530547\pi\)
\(68\) 0 0
\(69\) −212.952 −0.371543
\(70\) 0 0
\(71\) −57.5002 + 57.5002i −0.0961129 + 0.0961129i −0.753528 0.657415i \(-0.771650\pi\)
0.657415 + 0.753528i \(0.271650\pi\)
\(72\) 0 0
\(73\) 636.819 636.819i 1.02101 1.02101i 0.0212391 0.999774i \(-0.493239\pi\)
0.999774 0.0212391i \(-0.00676113\pi\)
\(74\) 0 0
\(75\) −78.5421 78.5421i −0.120923 0.120923i
\(76\) 0 0
\(77\) 704.953i 1.04334i
\(78\) 0 0
\(79\) 448.664 + 448.664i 0.638970 + 0.638970i 0.950301 0.311332i \(-0.100775\pi\)
−0.311332 + 0.950301i \(0.600775\pi\)
\(80\) 0 0
\(81\) 444.362 0.609550
\(82\) 0 0
\(83\) 264.124i 0.349293i −0.984631 0.174647i \(-0.944122\pi\)
0.984631 0.174647i \(-0.0558783\pi\)
\(84\) 0 0
\(85\) −551.810 + 659.683i −0.704144 + 0.841796i
\(86\) 0 0
\(87\) 620.297i 0.764400i
\(88\) 0 0
\(89\) −1573.69 −1.87428 −0.937141 0.348951i \(-0.886538\pi\)
−0.937141 + 0.348951i \(0.886538\pi\)
\(90\) 0 0
\(91\) −7.83564 7.83564i −0.00902636 0.00902636i
\(92\) 0 0
\(93\) 1280.66i 1.42793i
\(94\) 0 0
\(95\) 434.194 + 434.194i 0.468920 + 0.468920i
\(96\) 0 0
\(97\) 224.583 224.583i 0.235082 0.235082i −0.579728 0.814810i \(-0.696841\pi\)
0.814810 + 0.579728i \(0.196841\pi\)
\(98\) 0 0
\(99\) −316.712 + 316.712i −0.321522 + 0.321522i
\(100\) 0 0
\(101\) 1692.20 1.66713 0.833564 0.552423i \(-0.186296\pi\)
0.833564 + 0.552423i \(0.186296\pi\)
\(102\) 0 0
\(103\) 642.624 0.614753 0.307377 0.951588i \(-0.400549\pi\)
0.307377 + 0.951588i \(0.400549\pi\)
\(104\) 0 0
\(105\) −481.226 + 481.226i −0.447265 + 0.447265i
\(106\) 0 0
\(107\) 1188.35 1188.35i 1.07366 1.07366i 0.0765994 0.997062i \(-0.475594\pi\)
0.997062 0.0765994i \(-0.0244062\pi\)
\(108\) 0 0
\(109\) −526.566 526.566i −0.462714 0.462714i 0.436830 0.899544i \(-0.356101\pi\)
−0.899544 + 0.436830i \(0.856101\pi\)
\(110\) 0 0
\(111\) 569.862i 0.487288i
\(112\) 0 0
\(113\) 204.185 + 204.185i 0.169983 + 0.169983i 0.786972 0.616989i \(-0.211648\pi\)
−0.616989 + 0.786972i \(0.711648\pi\)
\(114\) 0 0
\(115\) 601.155 0.487460
\(116\) 0 0
\(117\) 7.04058i 0.00556327i
\(118\) 0 0
\(119\) 686.058 + 573.873i 0.528495 + 0.442074i
\(120\) 0 0
\(121\) 1720.93i 1.29296i
\(122\) 0 0
\(123\) 782.004 0.573260
\(124\) 0 0
\(125\) −862.811 862.811i −0.617378 0.617378i
\(126\) 0 0
\(127\) 955.662i 0.667726i 0.942622 + 0.333863i \(0.108352\pi\)
−0.942622 + 0.333863i \(0.891648\pi\)
\(128\) 0 0
\(129\) 843.227 + 843.227i 0.575519 + 0.575519i
\(130\) 0 0
\(131\) 775.731 775.731i 0.517374 0.517374i −0.399402 0.916776i \(-0.630782\pi\)
0.916776 + 0.399402i \(0.130782\pi\)
\(132\) 0 0
\(133\) 451.554 451.554i 0.294396 0.294396i
\(134\) 0 0
\(135\) −1872.37 −1.19369
\(136\) 0 0
\(137\) −535.781 −0.334123 −0.167061 0.985946i \(-0.553428\pi\)
−0.167061 + 0.985946i \(0.553428\pi\)
\(138\) 0 0
\(139\) 555.349 555.349i 0.338878 0.338878i −0.517067 0.855945i \(-0.672976\pi\)
0.855945 + 0.517067i \(0.172976\pi\)
\(140\) 0 0
\(141\) −284.607 + 284.607i −0.169988 + 0.169988i
\(142\) 0 0
\(143\) −33.9225 33.9225i −0.0198374 0.0198374i
\(144\) 0 0
\(145\) 1751.07i 1.00289i
\(146\) 0 0
\(147\) −553.733 553.733i −0.310688 0.310688i
\(148\) 0 0
\(149\) 2846.15 1.56487 0.782435 0.622733i \(-0.213978\pi\)
0.782435 + 0.622733i \(0.213978\pi\)
\(150\) 0 0
\(151\) 311.072i 0.167647i 0.996481 + 0.0838235i \(0.0267132\pi\)
−0.996481 + 0.0838235i \(0.973287\pi\)
\(152\) 0 0
\(153\) 50.4013 + 566.045i 0.0266321 + 0.299098i
\(154\) 0 0
\(155\) 3615.24i 1.87344i
\(156\) 0 0
\(157\) −294.137 −0.149521 −0.0747603 0.997202i \(-0.523819\pi\)
−0.0747603 + 0.997202i \(0.523819\pi\)
\(158\) 0 0
\(159\) −1415.79 1415.79i −0.706158 0.706158i
\(160\) 0 0
\(161\) 625.190i 0.306036i
\(162\) 0 0
\(163\) 1717.64 + 1717.64i 0.825372 + 0.825372i 0.986873 0.161500i \(-0.0516333\pi\)
−0.161500 + 0.986873i \(0.551633\pi\)
\(164\) 0 0
\(165\) −2083.35 + 2083.35i −0.982963 + 0.982963i
\(166\) 0 0
\(167\) −1974.22 + 1974.22i −0.914790 + 0.914790i −0.996644 0.0818543i \(-0.973916\pi\)
0.0818543 + 0.996644i \(0.473916\pi\)
\(168\) 0 0
\(169\) −2196.25 −0.999657
\(170\) 0 0
\(171\) 405.736 0.181447
\(172\) 0 0
\(173\) 3015.57 3015.57i 1.32526 1.32526i 0.415804 0.909454i \(-0.363500\pi\)
0.909454 0.415804i \(-0.136500\pi\)
\(174\) 0 0
\(175\) 230.585 230.585i 0.0996036 0.0996036i
\(176\) 0 0
\(177\) −1657.46 1657.46i −0.703855 0.703855i
\(178\) 0 0
\(179\) 891.459i 0.372239i −0.982527 0.186120i \(-0.940409\pi\)
0.982527 0.186120i \(-0.0595912\pi\)
\(180\) 0 0
\(181\) −1545.68 1545.68i −0.634748 0.634748i 0.314507 0.949255i \(-0.398161\pi\)
−0.949255 + 0.314507i \(0.898161\pi\)
\(182\) 0 0
\(183\) −2805.11 −1.13311
\(184\) 0 0
\(185\) 1608.70i 0.639317i
\(186\) 0 0
\(187\) 2970.13 + 2484.44i 1.16148 + 0.971554i
\(188\) 0 0
\(189\) 1947.23i 0.749420i
\(190\) 0 0
\(191\) 1937.35 0.733937 0.366968 0.930233i \(-0.380396\pi\)
0.366968 + 0.930233i \(0.380396\pi\)
\(192\) 0 0
\(193\) −1909.84 1909.84i −0.712298 0.712298i 0.254718 0.967015i \(-0.418017\pi\)
−0.967015 + 0.254718i \(0.918017\pi\)
\(194\) 0 0
\(195\) 46.3135i 0.0170081i
\(196\) 0 0
\(197\) 399.215 + 399.215i 0.144380 + 0.144380i 0.775602 0.631222i \(-0.217446\pi\)
−0.631222 + 0.775602i \(0.717446\pi\)
\(198\) 0 0
\(199\) −3648.28 + 3648.28i −1.29960 + 1.29960i −0.370941 + 0.928656i \(0.620965\pi\)
−0.928656 + 0.370941i \(0.879035\pi\)
\(200\) 0 0
\(201\) −323.018 + 323.018i −0.113353 + 0.113353i
\(202\) 0 0
\(203\) 1821.08 0.629630
\(204\) 0 0
\(205\) −2207.56 −0.752111
\(206\) 0 0
\(207\) 280.877 280.877i 0.0943106 0.0943106i
\(208\) 0 0
\(209\) 1954.90 1954.90i 0.647000 0.647000i
\(210\) 0 0
\(211\) −3555.86 3555.86i −1.16017 1.16017i −0.984439 0.175728i \(-0.943772\pi\)
−0.175728 0.984439i \(-0.556228\pi\)
\(212\) 0 0
\(213\) 353.450i 0.113699i
\(214\) 0 0
\(215\) −2380.39 2380.39i −0.755076 0.755076i
\(216\) 0 0
\(217\) −3759.78 −1.17618
\(218\) 0 0
\(219\) 3914.48i 1.20784i
\(220\) 0 0
\(221\) −60.6282 + 5.39842i −0.0184538 + 0.00164315i
\(222\) 0 0
\(223\) 485.861i 0.145900i −0.997336 0.0729499i \(-0.976759\pi\)
0.997336 0.0729499i \(-0.0232413\pi\)
\(224\) 0 0
\(225\) 207.189 0.0613892
\(226\) 0 0
\(227\) 1690.16 + 1690.16i 0.494183 + 0.494183i 0.909621 0.415438i \(-0.136372\pi\)
−0.415438 + 0.909621i \(0.636372\pi\)
\(228\) 0 0
\(229\) 187.996i 0.0542493i 0.999632 + 0.0271247i \(0.00863511\pi\)
−0.999632 + 0.0271247i \(0.991365\pi\)
\(230\) 0 0
\(231\) 2166.65 + 2166.65i 0.617122 + 0.617122i
\(232\) 0 0
\(233\) 3789.41 3789.41i 1.06546 1.06546i 0.0677606 0.997702i \(-0.478415\pi\)
0.997702 0.0677606i \(-0.0215854\pi\)
\(234\) 0 0
\(235\) 803.433 803.433i 0.223022 0.223022i
\(236\) 0 0
\(237\) 2757.91 0.755887
\(238\) 0 0
\(239\) −4249.16 −1.15002 −0.575011 0.818146i \(-0.695002\pi\)
−0.575011 + 0.818146i \(0.695002\pi\)
\(240\) 0 0
\(241\) 4536.47 4536.47i 1.21253 1.21253i 0.242338 0.970192i \(-0.422086\pi\)
0.970192 0.242338i \(-0.0779143\pi\)
\(242\) 0 0
\(243\) −1547.62 + 1547.62i −0.408560 + 0.408560i
\(244\) 0 0
\(245\) 1563.16 + 1563.16i 0.407620 + 0.407620i
\(246\) 0 0
\(247\) 43.4578i 0.0111950i
\(248\) 0 0
\(249\) −811.775 811.775i −0.206603 0.206603i
\(250\) 0 0
\(251\) 529.038 0.133038 0.0665191 0.997785i \(-0.478811\pi\)
0.0665191 + 0.997785i \(0.478811\pi\)
\(252\) 0 0
\(253\) 2706.61i 0.672582i
\(254\) 0 0
\(255\) 331.544 + 3723.48i 0.0814199 + 0.914407i
\(256\) 0 0
\(257\) 4391.33i 1.06585i 0.846162 + 0.532926i \(0.178907\pi\)
−0.846162 + 0.532926i \(0.821093\pi\)
\(258\) 0 0
\(259\) −1673.01 −0.401375
\(260\) 0 0
\(261\) 818.151 + 818.151i 0.194032 + 0.194032i
\(262\) 0 0
\(263\) 6530.80i 1.53120i 0.643315 + 0.765601i \(0.277558\pi\)
−0.643315 + 0.765601i \(0.722442\pi\)
\(264\) 0 0
\(265\) 3996.70 + 3996.70i 0.926472 + 0.926472i
\(266\) 0 0
\(267\) −4836.69 + 4836.69i −1.10862 + 1.10862i
\(268\) 0 0
\(269\) 5185.75 5185.75i 1.17539 1.17539i 0.194488 0.980905i \(-0.437695\pi\)
0.980905 0.194488i \(-0.0623047\pi\)
\(270\) 0 0
\(271\) −3156.92 −0.707635 −0.353818 0.935314i \(-0.615117\pi\)
−0.353818 + 0.935314i \(0.615117\pi\)
\(272\) 0 0
\(273\) −48.1652 −0.0106780
\(274\) 0 0
\(275\) 998.265 998.265i 0.218900 0.218900i
\(276\) 0 0
\(277\) −2386.93 + 2386.93i −0.517749 + 0.517749i −0.916890 0.399141i \(-0.869308\pi\)
0.399141 + 0.916890i \(0.369308\pi\)
\(278\) 0 0
\(279\) −1689.14 1689.14i −0.362460 0.362460i
\(280\) 0 0
\(281\) 5791.03i 1.22941i 0.788758 + 0.614704i \(0.210725\pi\)
−0.788758 + 0.614704i \(0.789275\pi\)
\(282\) 0 0
\(283\) 5454.87 + 5454.87i 1.14579 + 1.14579i 0.987372 + 0.158416i \(0.0506386\pi\)
0.158416 + 0.987372i \(0.449361\pi\)
\(284\) 0 0
\(285\) 2668.96 0.554722
\(286\) 0 0
\(287\) 2295.82i 0.472189i
\(288\) 0 0
\(289\) 4835.71 868.038i 0.984268 0.176682i
\(290\) 0 0
\(291\) 1380.50i 0.278097i
\(292\) 0 0
\(293\) −4992.55 −0.995453 −0.497727 0.867334i \(-0.665832\pi\)
−0.497727 + 0.867334i \(0.665832\pi\)
\(294\) 0 0
\(295\) 4678.94 + 4678.94i 0.923451 + 0.923451i
\(296\) 0 0
\(297\) 8430.08i 1.64701i
\(298\) 0 0
\(299\) 30.0843 + 30.0843i 0.00581880 + 0.00581880i
\(300\) 0 0
\(301\) −2475.56 + 2475.56i −0.474050 + 0.474050i
\(302\) 0 0
\(303\) 5200.92 5200.92i 0.986088 0.986088i
\(304\) 0 0
\(305\) 7918.70 1.48663
\(306\) 0 0
\(307\) −7425.89 −1.38051 −0.690257 0.723564i \(-0.742503\pi\)
−0.690257 + 0.723564i \(0.742503\pi\)
\(308\) 0 0
\(309\) 1975.08 1975.08i 0.363620 0.363620i
\(310\) 0 0
\(311\) 6145.92 6145.92i 1.12059 1.12059i 0.128935 0.991653i \(-0.458844\pi\)
0.991653 0.128935i \(-0.0411559\pi\)
\(312\) 0 0
\(313\) −2545.72 2545.72i −0.459720 0.459720i 0.438843 0.898564i \(-0.355388\pi\)
−0.898564 + 0.438843i \(0.855388\pi\)
\(314\) 0 0
\(315\) 1269.44i 0.227063i
\(316\) 0 0
\(317\) −2469.84 2469.84i −0.437603 0.437603i 0.453602 0.891204i \(-0.350139\pi\)
−0.891204 + 0.453602i \(0.850139\pi\)
\(318\) 0 0
\(319\) 7883.94 1.38375
\(320\) 0 0
\(321\) 7304.69i 1.27012i
\(322\) 0 0
\(323\) −311.101 3493.90i −0.0535917 0.601875i
\(324\) 0 0
\(325\) 22.1917i 0.00378761i
\(326\) 0 0
\(327\) −3236.76 −0.547381
\(328\) 0 0
\(329\) −835.556 835.556i −0.140017 0.140017i
\(330\) 0 0
\(331\) 5092.63i 0.845669i 0.906207 + 0.422834i \(0.138965\pi\)
−0.906207 + 0.422834i \(0.861035\pi\)
\(332\) 0 0
\(333\) −751.629 751.629i −0.123691 0.123691i
\(334\) 0 0
\(335\) 911.865 911.865i 0.148718 0.148718i
\(336\) 0 0
\(337\) −6369.59 + 6369.59i −1.02960 + 1.02960i −0.0300466 + 0.999548i \(0.509566\pi\)
−0.999548 + 0.0300466i \(0.990434\pi\)
\(338\) 0 0
\(339\) 1255.11 0.201086
\(340\) 0 0
\(341\) −16277.1 −2.58490
\(342\) 0 0
\(343\) 4720.60 4720.60i 0.743115 0.743115i
\(344\) 0 0
\(345\) 1847.63 1847.63i 0.288328 0.288328i
\(346\) 0 0
\(347\) −3438.43 3438.43i −0.531943 0.531943i 0.389207 0.921150i \(-0.372749\pi\)
−0.921150 + 0.389207i \(0.872749\pi\)
\(348\) 0 0
\(349\) 2937.29i 0.450514i −0.974299 0.225257i \(-0.927678\pi\)
0.974299 0.225257i \(-0.0723221\pi\)
\(350\) 0 0
\(351\) −93.7014 93.7014i −0.0142490 0.0142490i
\(352\) 0 0
\(353\) −11318.3 −1.70655 −0.853274 0.521462i \(-0.825387\pi\)
−0.853274 + 0.521462i \(0.825387\pi\)
\(354\) 0 0
\(355\) 997.772i 0.149173i
\(356\) 0 0
\(357\) 3872.36 344.800i 0.574081 0.0511169i
\(358\) 0 0
\(359\) 10060.7i 1.47906i −0.673124 0.739530i \(-0.735048\pi\)
0.673124 0.739530i \(-0.264952\pi\)
\(360\) 0 0
\(361\) 4354.60 0.634874
\(362\) 0 0
\(363\) 5289.21 + 5289.21i 0.764770 + 0.764770i
\(364\) 0 0
\(365\) 11050.4i 1.58467i
\(366\) 0 0
\(367\) 1489.62 + 1489.62i 0.211873 + 0.211873i 0.805063 0.593190i \(-0.202132\pi\)
−0.593190 + 0.805063i \(0.702132\pi\)
\(368\) 0 0
\(369\) −1031.44 + 1031.44i −0.145513 + 0.145513i
\(370\) 0 0
\(371\) 4156.49 4156.49i 0.581656 0.581656i
\(372\) 0 0
\(373\) 1001.78 0.139062 0.0695312 0.997580i \(-0.477850\pi\)
0.0695312 + 0.997580i \(0.477850\pi\)
\(374\) 0 0
\(375\) −5303.65 −0.730344
\(376\) 0 0
\(377\) −87.6310 + 87.6310i −0.0119714 + 0.0119714i
\(378\) 0 0
\(379\) 2198.89 2198.89i 0.298020 0.298020i −0.542218 0.840238i \(-0.682415\pi\)
0.840238 + 0.542218i \(0.182415\pi\)
\(380\) 0 0
\(381\) 2937.19 + 2937.19i 0.394953 + 0.394953i
\(382\) 0 0
\(383\) 3039.84i 0.405558i 0.979225 + 0.202779i \(0.0649972\pi\)
−0.979225 + 0.202779i \(0.935003\pi\)
\(384\) 0 0
\(385\) −6116.35 6116.35i −0.809658 0.809658i
\(386\) 0 0
\(387\) −2224.37 −0.292174
\(388\) 0 0
\(389\) 7047.01i 0.918503i 0.888306 + 0.459252i \(0.151882\pi\)
−0.888306 + 0.459252i \(0.848118\pi\)
\(390\) 0 0
\(391\) −2634.07 2203.34i −0.340692 0.284981i
\(392\) 0 0
\(393\) 4768.37i 0.612042i
\(394\) 0 0
\(395\) −7785.44 −0.991717
\(396\) 0 0
\(397\) −2938.84 2938.84i −0.371527 0.371527i 0.496506 0.868033i \(-0.334616\pi\)
−0.868033 + 0.496506i \(0.834616\pi\)
\(398\) 0 0
\(399\) 2775.67i 0.348264i
\(400\) 0 0
\(401\) 7877.17 + 7877.17i 0.980965 + 0.980965i 0.999822 0.0188573i \(-0.00600283\pi\)
−0.0188573 + 0.999822i \(0.506003\pi\)
\(402\) 0 0
\(403\) 180.922 180.922i 0.0223632 0.0223632i
\(404\) 0 0
\(405\) −3855.40 + 3855.40i −0.473028 + 0.473028i
\(406\) 0 0
\(407\) −7242.91 −0.882108
\(408\) 0 0
\(409\) −6227.41 −0.752874 −0.376437 0.926442i \(-0.622851\pi\)
−0.376437 + 0.926442i \(0.622851\pi\)
\(410\) 0 0
\(411\) −1646.70 + 1646.70i −0.197630 + 0.197630i
\(412\) 0 0
\(413\) 4866.01 4866.01i 0.579759 0.579759i
\(414\) 0 0
\(415\) 2291.60 + 2291.60i 0.271061 + 0.271061i
\(416\) 0 0
\(417\) 3413.69i 0.400886i
\(418\) 0 0
\(419\) −1855.27 1855.27i −0.216315 0.216315i 0.590629 0.806943i \(-0.298880\pi\)
−0.806943 + 0.590629i \(0.798880\pi\)
\(420\) 0 0
\(421\) −13594.4 −1.57376 −0.786878 0.617109i \(-0.788304\pi\)
−0.786878 + 0.617109i \(0.788304\pi\)
\(422\) 0 0
\(423\) 750.774i 0.0862976i
\(424\) 0 0
\(425\) −158.863 1784.15i −0.0181318 0.203633i
\(426\) 0 0
\(427\) 8235.31i 0.933336i
\(428\) 0 0
\(429\) −208.520 −0.0234672
\(430\) 0 0
\(431\) −5094.94 5094.94i −0.569408 0.569408i 0.362555 0.931962i \(-0.381905\pi\)
−0.931962 + 0.362555i \(0.881905\pi\)
\(432\) 0 0
\(433\) 7430.53i 0.824684i −0.911029 0.412342i \(-0.864711\pi\)
0.911029 0.412342i \(-0.135289\pi\)
\(434\) 0 0
\(435\) 5381.86 + 5381.86i 0.593196 + 0.593196i
\(436\) 0 0
\(437\) −1733.71 + 1733.71i −0.189781 + 0.189781i
\(438\) 0 0
\(439\) 10536.6 10536.6i 1.14553 1.14553i 0.158103 0.987423i \(-0.449462\pi\)
0.987423 0.158103i \(-0.0505380\pi\)
\(440\) 0 0
\(441\) 1460.71 0.157727
\(442\) 0 0
\(443\) −17733.7 −1.90193 −0.950964 0.309302i \(-0.899904\pi\)
−0.950964 + 0.309302i \(0.899904\pi\)
\(444\) 0 0
\(445\) 13653.8 13653.8i 1.45450 1.45450i
\(446\) 0 0
\(447\) 8747.54 8747.54i 0.925603 0.925603i
\(448\) 0 0
\(449\) 10516.0 + 10516.0i 1.10530 + 1.10530i 0.993759 + 0.111544i \(0.0355797\pi\)
0.111544 + 0.993759i \(0.464420\pi\)
\(450\) 0 0
\(451\) 9939.22i 1.03774i
\(452\) 0 0
\(453\) 956.070 + 956.070i 0.0991614 + 0.0991614i
\(454\) 0 0
\(455\) 135.968 0.0140094
\(456\) 0 0
\(457\) 6104.30i 0.624829i 0.949946 + 0.312415i \(0.101138\pi\)
−0.949946 + 0.312415i \(0.898862\pi\)
\(458\) 0 0
\(459\) 8204.13 + 6862.57i 0.834283 + 0.697859i
\(460\) 0 0
\(461\) 4495.32i 0.454160i 0.973876 + 0.227080i \(0.0729179\pi\)
−0.973876 + 0.227080i \(0.927082\pi\)
\(462\) 0 0
\(463\) −4964.24 −0.498289 −0.249144 0.968466i \(-0.580149\pi\)
−0.249144 + 0.968466i \(0.580149\pi\)
\(464\) 0 0
\(465\) −11111.3 11111.3i −1.10812 1.10812i
\(466\) 0 0
\(467\) 789.079i 0.0781889i 0.999236 + 0.0390944i \(0.0124473\pi\)
−0.999236 + 0.0390944i \(0.987553\pi\)
\(468\) 0 0
\(469\) −948.323 948.323i −0.0933678 0.0933678i
\(470\) 0 0
\(471\) −904.022 + 904.022i −0.0884398 + 0.0884398i
\(472\) 0 0
\(473\) −10717.4 + 10717.4i −1.04183 + 1.04183i
\(474\) 0 0
\(475\) −1278.87 −0.123534
\(476\) 0 0
\(477\) 3734.74 0.358495
\(478\) 0 0
\(479\) 2956.32 2956.32i 0.281999 0.281999i −0.551907 0.833906i \(-0.686100\pi\)
0.833906 + 0.551907i \(0.186100\pi\)
\(480\) 0 0
\(481\) 80.5059 80.5059i 0.00763151 0.00763151i
\(482\) 0 0
\(483\) −1921.50 1921.50i −0.181017 0.181017i
\(484\) 0 0
\(485\) 3897.08i 0.364861i
\(486\) 0 0
\(487\) 1159.43 + 1159.43i 0.107883 + 0.107883i 0.758988 0.651105i \(-0.225694\pi\)
−0.651105 + 0.758988i \(0.725694\pi\)
\(488\) 0 0
\(489\) 10558.2 0.976398
\(490\) 0 0
\(491\) 765.124i 0.0703249i −0.999382 0.0351625i \(-0.988805\pi\)
0.999382 0.0351625i \(-0.0111949\pi\)
\(492\) 0 0
\(493\) 6417.98 7672.63i 0.586311 0.700928i
\(494\) 0 0
\(495\) 5495.74i 0.499021i
\(496\) 0 0
\(497\) −1037.67 −0.0936532
\(498\) 0 0
\(499\) 11036.9 + 11036.9i 0.990142 + 0.990142i 0.999952 0.00980984i \(-0.00312262\pi\)
−0.00980984 + 0.999952i \(0.503123\pi\)
\(500\) 0 0
\(501\) 12135.4i 1.08218i
\(502\) 0 0
\(503\) 14656.0 + 14656.0i 1.29917 + 1.29917i 0.928943 + 0.370224i \(0.120719\pi\)
0.370224 + 0.928943i \(0.379281\pi\)
\(504\) 0 0
\(505\) −14681.9 + 14681.9i −1.29374 + 1.29374i
\(506\) 0 0
\(507\) −6750.09 + 6750.09i −0.591286 + 0.591286i
\(508\) 0 0
\(509\) 20332.9 1.77061 0.885304 0.465012i \(-0.153950\pi\)
0.885304 + 0.465012i \(0.153950\pi\)
\(510\) 0 0
\(511\) 11492.2 0.994884
\(512\) 0 0
\(513\) 5399.85 5399.85i 0.464735 0.464735i
\(514\) 0 0
\(515\) −5575.57 + 5575.57i −0.477066 + 0.477066i
\(516\) 0 0
\(517\) −3617.34 3617.34i −0.307718 0.307718i
\(518\) 0 0
\(519\) 18536.5i 1.56775i
\(520\) 0 0
\(521\) 1124.38 + 1124.38i 0.0945490 + 0.0945490i 0.752799 0.658250i \(-0.228703\pi\)
−0.658250 + 0.752799i \(0.728703\pi\)
\(522\) 0 0
\(523\) 8009.53 0.669660 0.334830 0.942278i \(-0.391321\pi\)
0.334830 + 0.942278i \(0.391321\pi\)
\(524\) 0 0
\(525\) 1417.39i 0.117829i
\(526\) 0 0
\(527\) −13250.5 + 15840.8i −1.09526 + 1.30937i
\(528\) 0 0
\(529\) 9766.63i 0.802715i
\(530\) 0 0
\(531\) 4372.27 0.357326
\(532\) 0 0
\(533\) −110.476 110.476i −0.00897793 0.00897793i
\(534\) 0 0
\(535\) 20620.8i 1.66638i
\(536\) 0 0
\(537\) −2739.87 2739.87i −0.220175 0.220175i
\(538\) 0 0
\(539\) 7037.91 7037.91i 0.562420 0.562420i
\(540\) 0 0
\(541\) 4328.20 4328.20i 0.343963 0.343963i −0.513892 0.857855i \(-0.671797\pi\)
0.857855 + 0.513892i \(0.171797\pi\)
\(542\) 0 0
\(543\) −9501.18 −0.750893
\(544\) 0 0
\(545\) 9137.24 0.718158
\(546\) 0 0
\(547\) −12109.7 + 12109.7i −0.946573 + 0.946573i −0.998643 0.0520704i \(-0.983418\pi\)
0.0520704 + 0.998643i \(0.483418\pi\)
\(548\) 0 0
\(549\) 3699.85 3699.85i 0.287624 0.287624i
\(550\) 0 0
\(551\) −5050.02 5050.02i −0.390450 0.390450i
\(552\) 0 0
\(553\) 8096.72i 0.622618i
\(554\) 0 0
\(555\) −4944.27 4944.27i −0.378149 0.378149i
\(556\) 0 0
\(557\) −1702.39 −0.129502 −0.0647509 0.997901i \(-0.520625\pi\)
−0.0647509 + 0.997901i \(0.520625\pi\)
\(558\) 0 0
\(559\) 238.250i 0.0180266i
\(560\) 0 0
\(561\) 16764.4 1492.73i 1.26167 0.112340i
\(562\) 0 0
\(563\) 19262.1i 1.44192i 0.692976 + 0.720961i \(0.256299\pi\)
−0.692976 + 0.720961i \(0.743701\pi\)
\(564\) 0 0
\(565\) −3543.12 −0.263823
\(566\) 0 0
\(567\) 4009.54 + 4009.54i 0.296975 + 0.296975i
\(568\) 0 0
\(569\) 4320.59i 0.318328i −0.987252 0.159164i \(-0.949120\pi\)
0.987252 0.159164i \(-0.0508799\pi\)
\(570\) 0 0
\(571\) −3305.73 3305.73i −0.242277 0.242277i 0.575514 0.817792i \(-0.304802\pi\)
−0.817792 + 0.575514i \(0.804802\pi\)
\(572\) 0 0
\(573\) 5954.39 5954.39i 0.434116 0.434116i
\(574\) 0 0
\(575\) −885.314 + 885.314i −0.0642090 + 0.0642090i
\(576\) 0 0
\(577\) −4858.52 −0.350542 −0.175271 0.984520i \(-0.556080\pi\)
−0.175271 + 0.984520i \(0.556080\pi\)
\(578\) 0 0
\(579\) −11739.7 −0.842633
\(580\) 0 0
\(581\) 2383.23 2383.23i 0.170177 0.170177i
\(582\) 0 0
\(583\) 17994.5 17994.5i 1.27832 1.27832i
\(584\) 0 0
\(585\) 61.0859 + 61.0859i 0.00431725 + 0.00431725i
\(586\) 0 0
\(587\) 27354.2i 1.92339i 0.274125 + 0.961694i \(0.411612\pi\)
−0.274125 + 0.961694i \(0.588388\pi\)
\(588\) 0 0
\(589\) 10426.2 + 10426.2i 0.729378 + 0.729378i
\(590\) 0 0
\(591\) 2453.95 0.170799
\(592\) 0 0
\(593\) 13606.3i 0.942233i −0.882071 0.471117i \(-0.843851\pi\)
0.882071 0.471117i \(-0.156149\pi\)
\(594\) 0 0
\(595\) −10931.5 + 973.353i −0.753189 + 0.0670649i
\(596\) 0 0
\(597\) 22425.8i 1.53740i
\(598\) 0 0
\(599\) 17513.3 1.19461 0.597307 0.802013i \(-0.296238\pi\)
0.597307 + 0.802013i \(0.296238\pi\)
\(600\) 0 0
\(601\) −13737.7 13737.7i −0.932399 0.932399i 0.0654564 0.997855i \(-0.479150\pi\)
−0.997855 + 0.0654564i \(0.979150\pi\)
\(602\) 0 0
\(603\) 852.099i 0.0575459i
\(604\) 0 0
\(605\) −14931.2 14931.2i −1.00337 1.00337i
\(606\) 0 0
\(607\) 4673.11 4673.11i 0.312480 0.312480i −0.533389 0.845870i \(-0.679082\pi\)
0.845870 + 0.533389i \(0.179082\pi\)
\(608\) 0 0
\(609\) 5597.03 5597.03i 0.372419 0.372419i
\(610\) 0 0
\(611\) 80.4144 0.00532442
\(612\) 0 0
\(613\) 24289.4 1.60039 0.800196 0.599739i \(-0.204729\pi\)
0.800196 + 0.599739i \(0.204729\pi\)
\(614\) 0 0
\(615\) −6784.87 + 6784.87i −0.444866 + 0.444866i
\(616\) 0 0
\(617\) −19700.6 + 19700.6i −1.28544 + 1.28544i −0.347915 + 0.937526i \(0.613110\pi\)
−0.937526 + 0.347915i \(0.886890\pi\)
\(618\) 0 0
\(619\) −21044.4 21044.4i −1.36647 1.36647i −0.865415 0.501055i \(-0.832945\pi\)
−0.501055 0.865415i \(-0.667055\pi\)
\(620\) 0 0
\(621\) 7476.24i 0.483110i
\(622\) 0 0
\(623\) −14199.7 14199.7i −0.913158 0.913158i
\(624\) 0 0
\(625\) 18166.3 1.16264
\(626\) 0 0
\(627\) 12016.6i 0.765387i
\(628\) 0 0
\(629\) −5896.15 + 7048.79i −0.373760 + 0.446826i
\(630\) 0 0
\(631\) 3117.99i 0.196712i −0.995151 0.0983559i \(-0.968642\pi\)
0.995151 0.0983559i \(-0.0313584\pi\)
\(632\) 0 0
\(633\) −21857.6 −1.37245
\(634\) 0 0
\(635\) −8291.56 8291.56i −0.518174 0.518174i
\(636\) 0 0
\(637\) 156.455i 0.00973148i
\(638\) 0 0
\(639\) −466.188 466.188i −0.0288609 0.0288609i
\(640\) 0 0
\(641\) −1391.32 + 1391.32i −0.0857314 + 0.0857314i −0.748672 0.662941i \(-0.769308\pi\)
0.662941 + 0.748672i \(0.269308\pi\)
\(642\) 0 0
\(643\) 3638.34 3638.34i 0.223145 0.223145i −0.586676 0.809821i \(-0.699564\pi\)
0.809821 + 0.586676i \(0.199564\pi\)
\(644\) 0 0
\(645\) −14632.1 −0.893238
\(646\) 0 0
\(647\) −3564.67 −0.216602 −0.108301 0.994118i \(-0.534541\pi\)
−0.108301 + 0.994118i \(0.534541\pi\)
\(648\) 0 0
\(649\) 21066.2 21066.2i 1.27415 1.27415i
\(650\) 0 0
\(651\) −11555.6 + 11555.6i −0.695696 + 0.695696i
\(652\) 0 0
\(653\) −19594.4 19594.4i −1.17425 1.17425i −0.981185 0.193070i \(-0.938156\pi\)
−0.193070 0.981185i \(-0.561844\pi\)
\(654\) 0 0
\(655\) 13460.9i 0.802993i
\(656\) 0 0
\(657\) 5163.07 + 5163.07i 0.306591 + 0.306591i
\(658\) 0 0
\(659\) 3093.29 0.182849 0.0914245 0.995812i \(-0.470858\pi\)
0.0914245 + 0.995812i \(0.470858\pi\)
\(660\) 0 0
\(661\) 26602.2i 1.56537i −0.622420 0.782683i \(-0.713851\pi\)
0.622420 0.782683i \(-0.286149\pi\)
\(662\) 0 0
\(663\) −169.747 + 202.931i −0.00994333 + 0.0118871i
\(664\) 0 0
\(665\) 7835.60i 0.456920i
\(666\) 0 0
\(667\) −6991.90 −0.405888
\(668\) 0 0
\(669\) −1493.28 1493.28i −0.0862981 0.0862981i
\(670\) 0 0
\(671\) 35652.8i 2.05121i
\(672\) 0 0
\(673\) −11937.4 11937.4i −0.683736 0.683736i 0.277104 0.960840i \(-0.410625\pi\)
−0.960840 + 0.277104i \(0.910625\pi\)
\(674\) 0 0
\(675\) 2757.42 2757.42i 0.157234 0.157234i
\(676\) 0 0
\(677\) 2726.21 2726.21i 0.154766 0.154766i −0.625477 0.780243i \(-0.715095\pi\)
0.780243 + 0.625477i \(0.215095\pi\)
\(678\) 0 0
\(679\) 4052.89 0.229066
\(680\) 0 0
\(681\) 10389.3 0.584608
\(682\) 0 0
\(683\) −16602.4 + 16602.4i −0.930123 + 0.930123i −0.997713 0.0675904i \(-0.978469\pi\)
0.0675904 + 0.997713i \(0.478469\pi\)
\(684\) 0 0
\(685\) 4648.57 4648.57i 0.259289 0.259289i
\(686\) 0 0
\(687\) 577.798 + 577.798i 0.0320879 + 0.0320879i
\(688\) 0 0
\(689\) 400.023i 0.0221185i
\(690\) 0 0
\(691\) 10570.1 + 10570.1i 0.581920 + 0.581920i 0.935431 0.353511i \(-0.115012\pi\)
−0.353511 + 0.935431i \(0.615012\pi\)
\(692\) 0 0
\(693\) −5715.47 −0.313294
\(694\) 0 0
\(695\) 9636.70i 0.525958i
\(696\) 0 0
\(697\) 9672.83 + 8091.10i 0.525659 + 0.439702i
\(698\) 0 0
\(699\) 23293.3i 1.26042i
\(700\) 0 0
\(701\) −8204.92 −0.442076 −0.221038 0.975265i \(-0.570945\pi\)
−0.221038 + 0.975265i \(0.570945\pi\)
\(702\) 0 0
\(703\) 4639.41 + 4639.41i 0.248903 + 0.248903i
\(704\) 0 0
\(705\) 4938.65i 0.263830i
\(706\) 0 0
\(707\) 15269.0 + 15269.0i 0.812232 + 0.812232i
\(708\) 0 0
\(709\) −15750.5 + 15750.5i −0.834304 + 0.834304i −0.988102 0.153798i \(-0.950849\pi\)
0.153798 + 0.988102i \(0.450849\pi\)
\(710\) 0 0
\(711\) −3637.58 + 3637.58i −0.191871 + 0.191871i
\(712\) 0 0
\(713\) 14435.4 0.758217
\(714\) 0 0
\(715\) 588.641 0.0307887
\(716\) 0 0
\(717\) −13059.6 + 13059.6i −0.680225 + 0.680225i
\(718\) 0 0
\(719\) −10899.1 + 10899.1i −0.565326 + 0.565326i −0.930816 0.365489i \(-0.880902\pi\)
0.365489 + 0.930816i \(0.380902\pi\)
\(720\) 0 0
\(721\) 5798.49 + 5798.49i 0.299511 + 0.299511i
\(722\) 0 0
\(723\) 27885.4i 1.43440i
\(724\) 0 0
\(725\) −2578.78 2578.78i −0.132102 0.132102i
\(726\) 0 0
\(727\) −597.129 −0.0304626 −0.0152313 0.999884i \(-0.504848\pi\)
−0.0152313 + 0.999884i \(0.504848\pi\)
\(728\) 0 0
\(729\) 21510.9i 1.09287i
\(730\) 0 0
\(731\) 1705.56 + 19154.7i 0.0862958 + 0.969166i
\(732\) 0 0
\(733\) 16993.0i 0.856274i 0.903714 + 0.428137i \(0.140830\pi\)
−0.903714 + 0.428137i \(0.859170\pi\)
\(734\) 0 0
\(735\) 9608.66 0.482205
\(736\) 0 0
\(737\) −4105.54 4105.54i −0.205196 0.205196i
\(738\) 0 0
\(739\) 37983.6i 1.89073i −0.326017 0.945364i \(-0.605707\pi\)
0.326017 0.945364i \(-0.394293\pi\)
\(740\) 0 0
\(741\) 133.566 + 133.566i 0.00662170 + 0.00662170i
\(742\) 0 0
\(743\) 2340.76 2340.76i 0.115578 0.115578i −0.646953 0.762530i \(-0.723957\pi\)
0.762530 + 0.646953i \(0.223957\pi\)
\(744\) 0 0
\(745\) −24693.9 + 24693.9i −1.21438 + 1.21438i
\(746\) 0 0
\(747\) 2141.41 0.104886
\(748\) 0 0
\(749\) 21445.3 1.04618
\(750\) 0 0
\(751\) −13570.6 + 13570.6i −0.659383 + 0.659383i −0.955234 0.295851i \(-0.904397\pi\)
0.295851 + 0.955234i \(0.404397\pi\)
\(752\) 0 0
\(753\) 1625.98 1625.98i 0.0786907 0.0786907i
\(754\) 0 0
\(755\) −2698.94 2698.94i −0.130099 0.130099i
\(756\) 0 0
\(757\) 2949.36i 0.141607i 0.997490 + 0.0708035i \(0.0225563\pi\)
−0.997490 + 0.0708035i \(0.977444\pi\)
\(758\) 0 0
\(759\) −8318.68 8318.68i −0.397825 0.397825i
\(760\) 0 0
\(761\) −21295.1 −1.01438 −0.507192 0.861833i \(-0.669317\pi\)
−0.507192 + 0.861833i \(0.669317\pi\)
\(762\) 0 0
\(763\) 9502.56i 0.450873i
\(764\) 0 0
\(765\) −5348.44 4473.85i −0.252776 0.211441i
\(766\) 0 0
\(767\) 468.308i 0.0220464i
\(768\) 0 0
\(769\) 27668.1 1.29745 0.648724 0.761024i \(-0.275303\pi\)
0.648724 + 0.761024i \(0.275303\pi\)
\(770\) 0 0
\(771\) 13496.6 + 13496.6i 0.630439 + 0.630439i
\(772\) 0 0
\(773\) 19775.2i 0.920133i 0.887884 + 0.460067i \(0.152174\pi\)
−0.887884 + 0.460067i \(0.847826\pi\)
\(774\) 0 0
\(775\) 5324.12 + 5324.12i 0.246772 + 0.246772i
\(776\) 0 0
\(777\) −5141.95 + 5141.95i −0.237409 + 0.237409i
\(778\) 0 0
\(779\) 6366.52 6366.52i 0.292817 0.292817i
\(780\) 0 0
\(781\) −4492.32 −0.205823
\(782\) 0 0
\(783\) 21777.1 0.993936
\(784\) 0 0
\(785\) 2552.01 2552.01i 0.116032 0.116032i
\(786\) 0 0
\(787\) −3947.52 + 3947.52i −0.178798 + 0.178798i −0.790832 0.612034i \(-0.790352\pi\)
0.612034 + 0.790832i \(0.290352\pi\)
\(788\) 0 0
\(789\) 20072.2 + 20072.2i 0.905690 + 0.905690i
\(790\) 0 0
\(791\) 3684.78i 0.165633i
\(792\) 0 0
\(793\) 396.285 + 396.285i 0.0177459 + 0.0177459i
\(794\) 0 0
\(795\) 24567.4 1.09600
\(796\) 0 0
\(797\) 16897.8i 0.751006i −0.926821 0.375503i \(-0.877470\pi\)
0.926821 0.375503i \(-0.122530\pi\)
\(798\) 0 0
\(799\) −6465.11 + 575.662i −0.286257 + 0.0254887i
\(800\) 0 0
\(801\) 12758.9i 0.562812i
\(802\) 0 0
\(803\) 49752.8 2.18647
\(804\) 0 0
\(805\) 5424.31 + 5424.31i 0.237493 + 0.237493i
\(806\) 0 0
\(807\) 31876.5i 1.39046i
\(808\) 0 0
\(809\) −21479.4 21479.4i −0.933467 0.933467i 0.0644539 0.997921i \(-0.479469\pi\)
−0.997921 + 0.0644539i \(0.979469\pi\)
\(810\) 0 0
\(811\) 29914.5 29914.5i 1.29524 1.29524i 0.363740 0.931501i \(-0.381500\pi\)
0.931501 0.363740i \(-0.118500\pi\)
\(812\) 0 0
\(813\) −9702.69 + 9702.69i −0.418559 + 0.418559i
\(814\) 0 0
\(815\) −29805.3 −1.28102
\(816\) 0 0
\(817\) 13729.9 0.587942
\(818\) 0 0
\(819\) 63.5282 63.5282i 0.00271045 0.00271045i
\(820\) 0 0
\(821\) −9116.42 + 9116.42i −0.387534 + 0.387534i −0.873807 0.486273i \(-0.838356\pi\)
0.486273 + 0.873807i \(0.338356\pi\)
\(822\) 0 0
\(823\) 26462.2 + 26462.2i 1.12079 + 1.12079i 0.991622 + 0.129171i \(0.0412316\pi\)
0.129171 + 0.991622i \(0.458768\pi\)
\(824\) 0 0
\(825\) 6136.27i 0.258954i
\(826\) 0 0
\(827\) 9620.86 + 9620.86i 0.404535 + 0.404535i 0.879828 0.475293i \(-0.157658\pi\)
−0.475293 + 0.879828i \(0.657658\pi\)
\(828\) 0 0
\(829\) −14740.5 −0.617562 −0.308781 0.951133i \(-0.599921\pi\)
−0.308781 + 0.951133i \(0.599921\pi\)
\(830\) 0 0
\(831\) 14672.3i 0.612486i
\(832\) 0 0
\(833\) −1120.01 12578.5i −0.0465859 0.523194i
\(834\) 0 0
\(835\) 34257.7i 1.41981i
\(836\) 0 0
\(837\) −44960.8 −1.85672
\(838\) 0 0
\(839\) −8464.19 8464.19i −0.348291 0.348291i 0.511182 0.859473i \(-0.329208\pi\)
−0.859473 + 0.511182i \(0.829208\pi\)
\(840\) 0 0
\(841\) 4022.69i 0.164939i
\(842\) 0 0
\(843\) 17798.5 + 17798.5i 0.727182 + 0.727182i
\(844\) 0 0
\(845\) 19055.2 19055.2i 0.775762 0.775762i
\(846\) 0 0
\(847\) −15528.2 + 15528.2i −0.629934 + 0.629934i
\(848\) 0 0
\(849\) 33530.7 1.35544
\(850\) 0 0
\(851\) 6423.40 0.258744
\(852\) 0 0
\(853\) −19959.8 + 19959.8i −0.801185 + 0.801185i −0.983281 0.182096i \(-0.941712\pi\)
0.182096 + 0.983281i \(0.441712\pi\)
\(854\) 0 0
\(855\) −3520.27 + 3520.27i −0.140808 + 0.140808i
\(856\) 0 0
\(857\) 2185.79 + 2185.79i 0.0871238 + 0.0871238i 0.749326 0.662202i \(-0.230378\pi\)
−0.662202 + 0.749326i \(0.730378\pi\)
\(858\) 0 0
\(859\) 21827.6i 0.866994i 0.901155 + 0.433497i \(0.142720\pi\)
−0.901155 + 0.433497i \(0.857280\pi\)
\(860\) 0 0
\(861\) 7056.14 + 7056.14i 0.279295 + 0.279295i
\(862\) 0 0
\(863\) 12268.5 0.483921 0.241960 0.970286i \(-0.422210\pi\)
0.241960 + 0.970286i \(0.422210\pi\)
\(864\) 0 0
\(865\) 52327.7i 2.05687i
\(866\) 0 0
\(867\) 12194.5 17530.3i 0.477678 0.686689i
\(868\) 0 0
\(869\) 35052.8i 1.36834i
\(870\) 0 0
\(871\) 91.2672 0.00355048
\(872\) 0 0
\(873\) 1820.83 + 1820.83i 0.0705908 + 0.0705908i
\(874\) 0 0
\(875\) 15570.6i 0.601578i
\(876\) 0 0
\(877\) −13528.2 13528.2i −0.520883 0.520883i 0.396955 0.917838i \(-0.370067\pi\)
−0.917838 + 0.396955i \(0.870067\pi\)
\(878\) 0 0
\(879\) −15344.4 + 15344.4i −0.588800 + 0.588800i
\(880\) 0 0
\(881\) −4803.46 + 4803.46i −0.183692 + 0.183692i −0.792962 0.609270i \(-0.791462\pi\)
0.609270 + 0.792962i \(0.291462\pi\)
\(882\) 0 0
\(883\) −28197.9 −1.07467 −0.537335 0.843369i \(-0.680569\pi\)
−0.537335 + 0.843369i \(0.680569\pi\)
\(884\) 0 0
\(885\) 28761.1 1.09242
\(886\) 0 0
\(887\) 16408.3 16408.3i 0.621126 0.621126i −0.324694 0.945819i \(-0.605261\pi\)
0.945819 + 0.324694i \(0.105261\pi\)
\(888\) 0 0
\(889\) −8623.08 + 8623.08i −0.325319 + 0.325319i
\(890\) 0 0
\(891\) 17358.4 + 17358.4i 0.652668 + 0.652668i
\(892\) 0 0
\(893\) 4634.14i 0.173657i
\(894\) 0 0
\(895\) 7734.53 + 7734.53i 0.288868 + 0.288868i
\(896\) 0 0
\(897\) 184.926 0.00688351
\(898\) 0 0
\(899\) 42048.0i 1.55993i
\(900\) 0 0
\(901\) −2863.64 32160.8i −0.105884 1.18916i
\(902\) 0 0
\(903\) 15217.1i 0.560791i
\(904\) 0 0
\(905\) 26821.4 0.985164
\(906\) 0 0
\(907\) −10692.8 10692.8i −0.391455 0.391455i 0.483751 0.875206i \(-0.339274\pi\)
−0.875206 + 0.483751i \(0.839274\pi\)
\(908\) 0 0
\(909\) 13719.7i 0.500608i
\(910\) 0 0
\(911\) 2425.45 + 2425.45i 0.0882095 + 0.0882095i 0.749835 0.661625i \(-0.230133\pi\)
−0.661625 + 0.749835i \(0.730133\pi\)
\(912\) 0 0
\(913\) 10317.6 10317.6i 0.374001 0.374001i
\(914\) 0 0
\(915\) 24337.9 24337.9i 0.879328 0.879328i
\(916\) 0 0
\(917\) 13999.1 0.504133
\(918\) 0 0
\(919\) −53382.8 −1.91614 −0.958071 0.286530i \(-0.907498\pi\)
−0.958071 + 0.286530i \(0.907498\pi\)
\(920\) 0 0
\(921\) −22823.2 + 22823.2i −0.816559 + 0.816559i
\(922\) 0 0
\(923\) 49.9328 49.9328i 0.00178067 0.00178067i
\(924\) 0 0
\(925\) 2369.11 + 2369.11i 0.0842117 + 0.0842117i
\(926\) 0 0
\(927\) 5210.13i 0.184599i
\(928\) 0 0
\(929\) −1215.07 1215.07i −0.0429119 0.0429119i 0.685325 0.728237i \(-0.259660\pi\)
−0.728237 + 0.685325i \(0.759660\pi\)
\(930\) 0 0
\(931\) −9016.20 −0.317394
\(932\) 0 0
\(933\) 37778.5i 1.32563i
\(934\) 0 0
\(935\) −47325.2 + 4213.90i −1.65530 + 0.147390i
\(936\) 0 0
\(937\) 11129.8i 0.388040i 0.980998 + 0.194020i \(0.0621527\pi\)
−0.980998 + 0.194020i \(0.937847\pi\)
\(938\) 0 0
\(939\) −15648.4 −0.543839
\(940\) 0 0
\(941\) 2133.89 + 2133.89i 0.0739242 + 0.0739242i 0.743102 0.669178i \(-0.233354\pi\)
−0.669178 + 0.743102i \(0.733354\pi\)
\(942\) 0 0
\(943\) 8814.63i 0.304394i
\(944\) 0 0
\(945\) −16894.7 16894.7i −0.581571 0.581571i
\(946\) 0 0
\(947\) 13784.2 13784.2i 0.472994 0.472994i −0.429888 0.902882i \(-0.641447\pi\)
0.902882 + 0.429888i \(0.141447\pi\)
\(948\) 0 0
\(949\) −553.009 + 553.009i −0.0189162 + 0.0189162i
\(950\) 0 0
\(951\) −15181.9 −0.517674
\(952\) 0 0
\(953\) −4535.38 −0.154161 −0.0770805 0.997025i \(-0.524560\pi\)
−0.0770805 + 0.997025i \(0.524560\pi\)
\(954\) 0 0
\(955\) −16809.0 + 16809.0i −0.569555 + 0.569555i
\(956\) 0 0
\(957\) 24231.0 24231.0i 0.818472 0.818472i
\(958\) 0 0
\(959\) −4834.43 4834.43i −0.162786 0.162786i
\(960\) 0 0
\(961\) 57020.7i 1.91402i
\(962\) 0 0
\(963\) 9634.63 + 9634.63i 0.322400 + 0.322400i
\(964\) 0 0
\(965\) 33140.6 1.10553
\(966\) 0 0
\(967\) 52772.2i 1.75495i 0.479620 + 0.877476i \(0.340774\pi\)
−0.479620 + 0.877476i \(0.659226\pi\)
\(968\) 0 0
\(969\) −11694.5 9782.22i −0.387701 0.324304i
\(970\) 0 0
\(971\) 16848.4i 0.556839i −0.960460 0.278420i \(-0.910189\pi\)
0.960460 0.278420i \(-0.0898106\pi\)
\(972\) 0 0
\(973\) 10022.0 0.330206
\(974\) 0 0
\(975\) 68.2054 + 68.2054i 0.00224033 + 0.00224033i
\(976\) 0 0
\(977\) 24152.4i 0.790893i −0.918489 0.395447i \(-0.870590\pi\)
0.918489 0.395447i \(-0.129410\pi\)
\(978\) 0 0
\(979\) −61474.1 61474.1i −2.00686 2.00686i
\(980\) 0 0
\(981\) 4269.18 4269.18i 0.138944 0.138944i
\(982\) 0 0
\(983\) 22057.4 22057.4i 0.715689 0.715689i −0.252030 0.967719i \(-0.581098\pi\)
0.967719 + 0.252030i \(0.0810982\pi\)
\(984\) 0 0
\(985\) −6927.38 −0.224086
\(986\) 0 0
\(987\) −5136.11 −0.165637
\(988\) 0 0
\(989\) 9504.73 9504.73i 0.305594 0.305594i
\(990\) 0 0
\(991\) 8193.28 8193.28i 0.262632 0.262632i −0.563491 0.826122i \(-0.690542\pi\)
0.826122 + 0.563491i \(0.190542\pi\)
\(992\) 0 0
\(993\) 15652.0 + 15652.0i 0.500204 + 0.500204i
\(994\) 0 0
\(995\) 63306.9i 2.01705i
\(996\) 0 0
\(997\) −347.648 347.648i −0.0110432 0.0110432i 0.701564 0.712607i \(-0.252486\pi\)
−0.712607 + 0.701564i \(0.752486\pi\)
\(998\) 0 0
\(999\) −20006.5 −0.633611
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 136.4.k.a.89.5 yes 14
4.3 odd 2 272.4.o.g.225.3 14
17.8 even 8 2312.4.a.m.1.5 14
17.9 even 8 2312.4.a.m.1.10 14
17.13 even 4 inner 136.4.k.a.81.5 14
68.47 odd 4 272.4.o.g.81.3 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.4.k.a.81.5 14 17.13 even 4 inner
136.4.k.a.89.5 yes 14 1.1 even 1 trivial
272.4.o.g.81.3 14 68.47 odd 4
272.4.o.g.225.3 14 4.3 odd 2
2312.4.a.m.1.5 14 17.8 even 8
2312.4.a.m.1.10 14 17.9 even 8