Properties

Label 448.4.i.c.65.1
Level $448$
Weight $4$
Character 448.65
Analytic conductor $26.433$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,4,Mod(65,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(26.4328556826\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 65.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 448.65
Dual form 448.4.i.c.193.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{3} +(3.50000 - 6.06218i) q^{5} +(-10.0000 + 15.5885i) q^{7} +(13.0000 - 22.5167i) q^{9} +(17.5000 + 30.3109i) q^{11} -66.0000 q^{13} -7.00000 q^{15} +(-29.5000 - 51.0955i) q^{17} +(68.5000 - 118.645i) q^{19} +(18.5000 + 0.866025i) q^{21} +(3.50000 - 6.06218i) q^{23} +(38.0000 + 65.8179i) q^{25} -53.0000 q^{27} -106.000 q^{29} +(-37.5000 - 64.9519i) q^{31} +(17.5000 - 30.3109i) q^{33} +(59.5000 + 115.181i) q^{35} +(5.50000 - 9.52628i) q^{37} +(33.0000 + 57.1577i) q^{39} -498.000 q^{41} -260.000 q^{43} +(-91.0000 - 157.617i) q^{45} +(85.5000 - 148.090i) q^{47} +(-143.000 - 311.769i) q^{49} +(-29.5000 + 51.0955i) q^{51} +(-208.500 - 361.133i) q^{53} +245.000 q^{55} -137.000 q^{57} +(-8.50000 - 14.7224i) q^{59} +(25.5000 - 44.1673i) q^{61} +(221.000 + 427.817i) q^{63} +(-231.000 + 400.104i) q^{65} +(219.500 + 380.185i) q^{67} -7.00000 q^{69} -784.000 q^{71} +(-147.500 - 255.477i) q^{73} +(38.0000 - 65.8179i) q^{75} +(-647.500 - 30.3109i) q^{77} +(247.500 - 428.683i) q^{79} +(-324.500 - 562.050i) q^{81} -932.000 q^{83} -413.000 q^{85} +(53.0000 + 91.7987i) q^{87} +(436.500 - 756.040i) q^{89} +(660.000 - 1028.84i) q^{91} +(-37.5000 + 64.9519i) q^{93} +(-479.500 - 830.518i) q^{95} -290.000 q^{97} +910.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 7 q^{5} - 20 q^{7} + 26 q^{9} + 35 q^{11} - 132 q^{13} - 14 q^{15} - 59 q^{17} + 137 q^{19} + 37 q^{21} + 7 q^{23} + 76 q^{25} - 106 q^{27} - 212 q^{29} - 75 q^{31} + 35 q^{33} + 119 q^{35}+ \cdots + 1820 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.500000 0.866025i −0.0962250 0.166667i 0.813894 0.581013i \(-0.197344\pi\)
−0.910119 + 0.414346i \(0.864010\pi\)
\(4\) 0 0
\(5\) 3.50000 6.06218i 0.313050 0.542218i −0.665971 0.745977i \(-0.731983\pi\)
0.979021 + 0.203760i \(0.0653161\pi\)
\(6\) 0 0
\(7\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(8\) 0 0
\(9\) 13.0000 22.5167i 0.481481 0.833950i
\(10\) 0 0
\(11\) 17.5000 + 30.3109i 0.479677 + 0.830825i 0.999728 0.0233099i \(-0.00742046\pi\)
−0.520051 + 0.854135i \(0.674087\pi\)
\(12\) 0 0
\(13\) −66.0000 −1.40809 −0.704043 0.710158i \(-0.748624\pi\)
−0.704043 + 0.710158i \(0.748624\pi\)
\(14\) 0 0
\(15\) −7.00000 −0.120493
\(16\) 0 0
\(17\) −29.5000 51.0955i −0.420871 0.728969i 0.575154 0.818045i \(-0.304942\pi\)
−0.996025 + 0.0890757i \(0.971609\pi\)
\(18\) 0 0
\(19\) 68.5000 118.645i 0.827104 1.43259i −0.0731965 0.997318i \(-0.523320\pi\)
0.900301 0.435269i \(-0.143347\pi\)
\(20\) 0 0
\(21\) 18.5000 + 0.866025i 0.192240 + 0.00899915i
\(22\) 0 0
\(23\) 3.50000 6.06218i 0.0317305 0.0549588i −0.849724 0.527228i \(-0.823232\pi\)
0.881455 + 0.472269i \(0.156565\pi\)
\(24\) 0 0
\(25\) 38.0000 + 65.8179i 0.304000 + 0.526543i
\(26\) 0 0
\(27\) −53.0000 −0.377772
\(28\) 0 0
\(29\) −106.000 −0.678748 −0.339374 0.940651i \(-0.610215\pi\)
−0.339374 + 0.940651i \(0.610215\pi\)
\(30\) 0 0
\(31\) −37.5000 64.9519i −0.217264 0.376313i 0.736706 0.676213i \(-0.236380\pi\)
−0.953971 + 0.299900i \(0.903047\pi\)
\(32\) 0 0
\(33\) 17.5000 30.3109i 0.0923139 0.159892i
\(34\) 0 0
\(35\) 59.5000 + 115.181i 0.287352 + 0.556263i
\(36\) 0 0
\(37\) 5.50000 9.52628i 0.0244377 0.0423273i −0.853548 0.521014i \(-0.825554\pi\)
0.877986 + 0.478687i \(0.158887\pi\)
\(38\) 0 0
\(39\) 33.0000 + 57.1577i 0.135493 + 0.234681i
\(40\) 0 0
\(41\) −498.000 −1.89694 −0.948470 0.316867i \(-0.897369\pi\)
−0.948470 + 0.316867i \(0.897369\pi\)
\(42\) 0 0
\(43\) −260.000 −0.922084 −0.461042 0.887378i \(-0.652524\pi\)
−0.461042 + 0.887378i \(0.652524\pi\)
\(44\) 0 0
\(45\) −91.0000 157.617i −0.301455 0.522136i
\(46\) 0 0
\(47\) 85.5000 148.090i 0.265350 0.459600i −0.702305 0.711876i \(-0.747846\pi\)
0.967655 + 0.252276i \(0.0811791\pi\)
\(48\) 0 0
\(49\) −143.000 311.769i −0.416910 0.908948i
\(50\) 0 0
\(51\) −29.5000 + 51.0955i −0.0809966 + 0.140290i
\(52\) 0 0
\(53\) −208.500 361.133i −0.540371 0.935951i −0.998883 0.0472619i \(-0.984950\pi\)
0.458511 0.888689i \(-0.348383\pi\)
\(54\) 0 0
\(55\) 245.000 0.600651
\(56\) 0 0
\(57\) −137.000 −0.318353
\(58\) 0 0
\(59\) −8.50000 14.7224i −0.0187560 0.0324864i 0.856495 0.516155i \(-0.172637\pi\)
−0.875251 + 0.483669i \(0.839304\pi\)
\(60\) 0 0
\(61\) 25.5000 44.1673i 0.0535236 0.0927056i −0.838022 0.545636i \(-0.816288\pi\)
0.891546 + 0.452930i \(0.149621\pi\)
\(62\) 0 0
\(63\) 221.000 + 427.817i 0.441958 + 0.855553i
\(64\) 0 0
\(65\) −231.000 + 400.104i −0.440800 + 0.763489i
\(66\) 0 0
\(67\) 219.500 + 380.185i 0.400242 + 0.693239i 0.993755 0.111585i \(-0.0355928\pi\)
−0.593513 + 0.804824i \(0.702260\pi\)
\(68\) 0 0
\(69\) −7.00000 −0.0122131
\(70\) 0 0
\(71\) −784.000 −1.31047 −0.655237 0.755423i \(-0.727431\pi\)
−0.655237 + 0.755423i \(0.727431\pi\)
\(72\) 0 0
\(73\) −147.500 255.477i −0.236487 0.409608i 0.723217 0.690621i \(-0.242663\pi\)
−0.959704 + 0.281013i \(0.909329\pi\)
\(74\) 0 0
\(75\) 38.0000 65.8179i 0.0585048 0.101333i
\(76\) 0 0
\(77\) −647.500 30.3109i −0.958305 0.0448603i
\(78\) 0 0
\(79\) 247.500 428.683i 0.352480 0.610513i −0.634203 0.773166i \(-0.718672\pi\)
0.986683 + 0.162653i \(0.0520051\pi\)
\(80\) 0 0
\(81\) −324.500 562.050i −0.445130 0.770988i
\(82\) 0 0
\(83\) −932.000 −1.23253 −0.616267 0.787537i \(-0.711356\pi\)
−0.616267 + 0.787537i \(0.711356\pi\)
\(84\) 0 0
\(85\) −413.000 −0.527013
\(86\) 0 0
\(87\) 53.0000 + 91.7987i 0.0653126 + 0.113125i
\(88\) 0 0
\(89\) 436.500 756.040i 0.519875 0.900451i −0.479858 0.877346i \(-0.659312\pi\)
0.999733 0.0231042i \(-0.00735495\pi\)
\(90\) 0 0
\(91\) 660.000 1028.84i 0.760294 1.18518i
\(92\) 0 0
\(93\) −37.5000 + 64.9519i −0.0418126 + 0.0724215i
\(94\) 0 0
\(95\) −479.500 830.518i −0.517849 0.896941i
\(96\) 0 0
\(97\) −290.000 −0.303557 −0.151779 0.988415i \(-0.548500\pi\)
−0.151779 + 0.988415i \(0.548500\pi\)
\(98\) 0 0
\(99\) 910.000 0.923823
\(100\) 0 0
\(101\) −542.500 939.638i −0.534463 0.925717i −0.999189 0.0402627i \(-0.987181\pi\)
0.464726 0.885454i \(-0.346153\pi\)
\(102\) 0 0
\(103\) −776.500 + 1344.94i −0.742823 + 1.28661i 0.208381 + 0.978048i \(0.433181\pi\)
−0.951205 + 0.308560i \(0.900153\pi\)
\(104\) 0 0
\(105\) 70.0000 109.119i 0.0650600 0.101419i
\(106\) 0 0
\(107\) 64.5000 111.717i 0.0582752 0.100936i −0.835416 0.549618i \(-0.814773\pi\)
0.893691 + 0.448682i \(0.148107\pi\)
\(108\) 0 0
\(109\) −482.500 835.715i −0.423992 0.734376i 0.572334 0.820021i \(-0.306038\pi\)
−0.996326 + 0.0856452i \(0.972705\pi\)
\(110\) 0 0
\(111\) −11.0000 −0.00940607
\(112\) 0 0
\(113\) −50.0000 −0.0416248 −0.0208124 0.999783i \(-0.506625\pi\)
−0.0208124 + 0.999783i \(0.506625\pi\)
\(114\) 0 0
\(115\) −24.5000 42.4352i −0.0198664 0.0344096i
\(116\) 0 0
\(117\) −858.000 + 1486.10i −0.677967 + 1.17427i
\(118\) 0 0
\(119\) 1091.50 + 51.0955i 0.840821 + 0.0393606i
\(120\) 0 0
\(121\) 53.0000 91.7987i 0.0398197 0.0689697i
\(122\) 0 0
\(123\) 249.000 + 431.281i 0.182533 + 0.316157i
\(124\) 0 0
\(125\) 1407.00 1.00677
\(126\) 0 0
\(127\) 936.000 0.653989 0.326994 0.945026i \(-0.393964\pi\)
0.326994 + 0.945026i \(0.393964\pi\)
\(128\) 0 0
\(129\) 130.000 + 225.167i 0.0887276 + 0.153681i
\(130\) 0 0
\(131\) −377.500 + 653.849i −0.251773 + 0.436084i −0.964014 0.265851i \(-0.914347\pi\)
0.712241 + 0.701935i \(0.247680\pi\)
\(132\) 0 0
\(133\) 1164.50 + 2254.26i 0.759210 + 1.46970i
\(134\) 0 0
\(135\) −185.500 + 321.295i −0.118261 + 0.204835i
\(136\) 0 0
\(137\) 1178.50 + 2041.22i 0.734935 + 1.27294i 0.954752 + 0.297403i \(0.0961205\pi\)
−0.219817 + 0.975541i \(0.570546\pi\)
\(138\) 0 0
\(139\) −28.0000 −0.0170858 −0.00854291 0.999964i \(-0.502719\pi\)
−0.00854291 + 0.999964i \(0.502719\pi\)
\(140\) 0 0
\(141\) −171.000 −0.102133
\(142\) 0 0
\(143\) −1155.00 2000.52i −0.675426 1.16987i
\(144\) 0 0
\(145\) −371.000 + 642.591i −0.212482 + 0.368029i
\(146\) 0 0
\(147\) −198.500 + 279.726i −0.111374 + 0.156948i
\(148\) 0 0
\(149\) 1147.50 1987.53i 0.630919 1.09278i −0.356446 0.934316i \(-0.616012\pi\)
0.987364 0.158467i \(-0.0506551\pi\)
\(150\) 0 0
\(151\) 554.500 + 960.422i 0.298838 + 0.517603i 0.975870 0.218350i \(-0.0700676\pi\)
−0.677032 + 0.735953i \(0.736734\pi\)
\(152\) 0 0
\(153\) −1534.00 −0.810566
\(154\) 0 0
\(155\) −525.000 −0.272058
\(156\) 0 0
\(157\) 779.500 + 1350.13i 0.396248 + 0.686321i 0.993260 0.115911i \(-0.0369789\pi\)
−0.597012 + 0.802232i \(0.703646\pi\)
\(158\) 0 0
\(159\) −208.500 + 361.133i −0.103995 + 0.180124i
\(160\) 0 0
\(161\) 59.5000 + 115.181i 0.0291258 + 0.0563824i
\(162\) 0 0
\(163\) −1125.50 + 1949.42i −0.540834 + 0.936752i 0.458022 + 0.888941i \(0.348558\pi\)
−0.998856 + 0.0478115i \(0.984775\pi\)
\(164\) 0 0
\(165\) −122.500 212.176i −0.0577976 0.100108i
\(166\) 0 0
\(167\) 2788.00 1.29187 0.645934 0.763393i \(-0.276468\pi\)
0.645934 + 0.763393i \(0.276468\pi\)
\(168\) 0 0
\(169\) 2159.00 0.982704
\(170\) 0 0
\(171\) −1781.00 3084.78i −0.796471 1.37953i
\(172\) 0 0
\(173\) 789.500 1367.45i 0.346963 0.600957i −0.638746 0.769418i \(-0.720546\pi\)
0.985708 + 0.168461i \(0.0538797\pi\)
\(174\) 0 0
\(175\) −1406.00 65.8179i −0.607335 0.0284307i
\(176\) 0 0
\(177\) −8.50000 + 14.7224i −0.00360960 + 0.00625201i
\(178\) 0 0
\(179\) 1225.50 + 2122.63i 0.511722 + 0.886328i 0.999908 + 0.0135883i \(0.00432541\pi\)
−0.488186 + 0.872740i \(0.662341\pi\)
\(180\) 0 0
\(181\) 1170.00 0.480472 0.240236 0.970715i \(-0.422775\pi\)
0.240236 + 0.970715i \(0.422775\pi\)
\(182\) 0 0
\(183\) −51.0000 −0.0206012
\(184\) 0 0
\(185\) −38.5000 66.6840i −0.0153004 0.0265011i
\(186\) 0 0
\(187\) 1032.50 1788.34i 0.403764 0.699340i
\(188\) 0 0
\(189\) 530.000 826.188i 0.203978 0.317970i
\(190\) 0 0
\(191\) 637.500 1104.18i 0.241507 0.418303i −0.719637 0.694351i \(-0.755692\pi\)
0.961144 + 0.276048i \(0.0890249\pi\)
\(192\) 0 0
\(193\) −17.5000 30.3109i −0.00652683 0.0113048i 0.862744 0.505642i \(-0.168744\pi\)
−0.869270 + 0.494337i \(0.835411\pi\)
\(194\) 0 0
\(195\) 462.000 0.169664
\(196\) 0 0
\(197\) 2734.00 0.988779 0.494389 0.869241i \(-0.335392\pi\)
0.494389 + 0.869241i \(0.335392\pi\)
\(198\) 0 0
\(199\) −1121.50 1942.49i −0.399503 0.691959i 0.594162 0.804345i \(-0.297484\pi\)
−0.993665 + 0.112387i \(0.964151\pi\)
\(200\) 0 0
\(201\) 219.500 380.185i 0.0770265 0.133414i
\(202\) 0 0
\(203\) 1060.00 1652.38i 0.366490 0.571301i
\(204\) 0 0
\(205\) −1743.00 + 3018.96i −0.593836 + 1.02855i
\(206\) 0 0
\(207\) −91.0000 157.617i −0.0305553 0.0529232i
\(208\) 0 0
\(209\) 4795.00 1.58697
\(210\) 0 0
\(211\) −1172.00 −0.382388 −0.191194 0.981552i \(-0.561236\pi\)
−0.191194 + 0.981552i \(0.561236\pi\)
\(212\) 0 0
\(213\) 392.000 + 678.964i 0.126100 + 0.218412i
\(214\) 0 0
\(215\) −910.000 + 1576.17i −0.288658 + 0.499970i
\(216\) 0 0
\(217\) 1387.50 + 64.9519i 0.434054 + 0.0203190i
\(218\) 0 0
\(219\) −147.500 + 255.477i −0.0455120 + 0.0788291i
\(220\) 0 0
\(221\) 1947.00 + 3372.30i 0.592622 + 1.02645i
\(222\) 0 0
\(223\) 2024.00 0.607790 0.303895 0.952706i \(-0.401713\pi\)
0.303895 + 0.952706i \(0.401713\pi\)
\(224\) 0 0
\(225\) 1976.00 0.585481
\(226\) 0 0
\(227\) 1285.50 + 2226.55i 0.375866 + 0.651019i 0.990456 0.137827i \(-0.0440119\pi\)
−0.614590 + 0.788847i \(0.710679\pi\)
\(228\) 0 0
\(229\) 447.500 775.093i 0.129134 0.223666i −0.794207 0.607647i \(-0.792114\pi\)
0.923341 + 0.383980i \(0.125447\pi\)
\(230\) 0 0
\(231\) 297.500 + 575.907i 0.0847362 + 0.164034i
\(232\) 0 0
\(233\) −893.500 + 1547.59i −0.251224 + 0.435132i −0.963863 0.266398i \(-0.914166\pi\)
0.712639 + 0.701531i \(0.247500\pi\)
\(234\) 0 0
\(235\) −598.500 1036.63i −0.166135 0.287755i
\(236\) 0 0
\(237\) −495.000 −0.135670
\(238\) 0 0
\(239\) −5100.00 −1.38030 −0.690150 0.723667i \(-0.742455\pi\)
−0.690150 + 0.723667i \(0.742455\pi\)
\(240\) 0 0
\(241\) 2088.50 + 3617.39i 0.558225 + 0.966873i 0.997645 + 0.0685917i \(0.0218506\pi\)
−0.439420 + 0.898282i \(0.644816\pi\)
\(242\) 0 0
\(243\) −1040.00 + 1801.33i −0.274552 + 0.475537i
\(244\) 0 0
\(245\) −2390.50 224.301i −0.623361 0.0584900i
\(246\) 0 0
\(247\) −4521.00 + 7830.60i −1.16463 + 2.01720i
\(248\) 0 0
\(249\) 466.000 + 807.136i 0.118601 + 0.205422i
\(250\) 0 0
\(251\) 4680.00 1.17689 0.588444 0.808538i \(-0.299741\pi\)
0.588444 + 0.808538i \(0.299741\pi\)
\(252\) 0 0
\(253\) 245.000 0.0608815
\(254\) 0 0
\(255\) 206.500 + 357.668i 0.0507119 + 0.0878356i
\(256\) 0 0
\(257\) 874.500 1514.68i 0.212256 0.367638i −0.740164 0.672426i \(-0.765252\pi\)
0.952420 + 0.304788i \(0.0985856\pi\)
\(258\) 0 0
\(259\) 93.5000 + 180.999i 0.0224317 + 0.0434237i
\(260\) 0 0
\(261\) −1378.00 + 2386.77i −0.326805 + 0.566043i
\(262\) 0 0
\(263\) 2236.50 + 3873.73i 0.524367 + 0.908230i 0.999598 + 0.0283689i \(0.00903130\pi\)
−0.475231 + 0.879861i \(0.657635\pi\)
\(264\) 0 0
\(265\) −2919.00 −0.676652
\(266\) 0 0
\(267\) −873.000 −0.200100
\(268\) 0 0
\(269\) 987.500 + 1710.40i 0.223825 + 0.387676i 0.955966 0.293476i \(-0.0948122\pi\)
−0.732141 + 0.681153i \(0.761479\pi\)
\(270\) 0 0
\(271\) 4219.50 7308.39i 0.945817 1.63820i 0.191710 0.981452i \(-0.438597\pi\)
0.754107 0.656751i \(-0.228070\pi\)
\(272\) 0 0
\(273\) −1221.00 57.1577i −0.270690 0.0126716i
\(274\) 0 0
\(275\) −1330.00 + 2303.63i −0.291644 + 0.505142i
\(276\) 0 0
\(277\) 263.500 + 456.395i 0.0571559 + 0.0989969i 0.893188 0.449684i \(-0.148463\pi\)
−0.836032 + 0.548681i \(0.815130\pi\)
\(278\) 0 0
\(279\) −1950.00 −0.418435
\(280\) 0 0
\(281\) −202.000 −0.0428837 −0.0214418 0.999770i \(-0.506826\pi\)
−0.0214418 + 0.999770i \(0.506826\pi\)
\(282\) 0 0
\(283\) −3974.50 6884.04i −0.834839 1.44598i −0.894161 0.447745i \(-0.852227\pi\)
0.0593220 0.998239i \(-0.481106\pi\)
\(284\) 0 0
\(285\) −479.500 + 830.518i −0.0996601 + 0.172616i
\(286\) 0 0
\(287\) 4980.00 7763.05i 1.02425 1.59665i
\(288\) 0 0
\(289\) 716.000 1240.15i 0.145736 0.252422i
\(290\) 0 0
\(291\) 145.000 + 251.147i 0.0292098 + 0.0505929i
\(292\) 0 0
\(293\) −318.000 −0.0634053 −0.0317027 0.999497i \(-0.510093\pi\)
−0.0317027 + 0.999497i \(0.510093\pi\)
\(294\) 0 0
\(295\) −119.000 −0.0234863
\(296\) 0 0
\(297\) −927.500 1606.48i −0.181209 0.313863i
\(298\) 0 0
\(299\) −231.000 + 400.104i −0.0446792 + 0.0773866i
\(300\) 0 0
\(301\) 2600.00 4053.00i 0.497879 0.776116i
\(302\) 0 0
\(303\) −542.500 + 939.638i −0.102857 + 0.178154i
\(304\) 0 0
\(305\) −178.500 309.171i −0.0335111 0.0580429i
\(306\) 0 0
\(307\) 8132.00 1.51178 0.755892 0.654696i \(-0.227203\pi\)
0.755892 + 0.654696i \(0.227203\pi\)
\(308\) 0 0
\(309\) 1553.00 0.285913
\(310\) 0 0
\(311\) 464.500 + 804.538i 0.0846925 + 0.146692i 0.905260 0.424858i \(-0.139676\pi\)
−0.820568 + 0.571549i \(0.806343\pi\)
\(312\) 0 0
\(313\) 104.500 180.999i 0.0188712 0.0326859i −0.856436 0.516254i \(-0.827326\pi\)
0.875307 + 0.483568i \(0.160659\pi\)
\(314\) 0 0
\(315\) 3367.00 + 157.617i 0.602251 + 0.0281927i
\(316\) 0 0
\(317\) 3565.50 6175.63i 0.631730 1.09419i −0.355468 0.934689i \(-0.615678\pi\)
0.987198 0.159500i \(-0.0509882\pi\)
\(318\) 0 0
\(319\) −1855.00 3212.95i −0.325580 0.563921i
\(320\) 0 0
\(321\) −129.000 −0.0224301
\(322\) 0 0
\(323\) −8083.00 −1.39242
\(324\) 0 0
\(325\) −2508.00 4343.98i −0.428058 0.741418i
\(326\) 0 0
\(327\) −482.500 + 835.715i −0.0815973 + 0.141331i
\(328\) 0 0
\(329\) 1453.50 + 2813.72i 0.243569 + 0.471505i
\(330\) 0 0
\(331\) −3285.50 + 5690.65i −0.545581 + 0.944975i 0.452989 + 0.891516i \(0.350358\pi\)
−0.998570 + 0.0534583i \(0.982976\pi\)
\(332\) 0 0
\(333\) −143.000 247.683i −0.0235326 0.0407596i
\(334\) 0 0
\(335\) 3073.00 0.501182
\(336\) 0 0
\(337\) −11466.0 −1.85339 −0.926696 0.375813i \(-0.877364\pi\)
−0.926696 + 0.375813i \(0.877364\pi\)
\(338\) 0 0
\(339\) 25.0000 + 43.3013i 0.00400535 + 0.00693747i
\(340\) 0 0
\(341\) 1312.50 2273.32i 0.208434 0.361018i
\(342\) 0 0
\(343\) 6290.00 + 888.542i 0.990169 + 0.139874i
\(344\) 0 0
\(345\) −24.5000 + 42.4352i −0.00382329 + 0.00662214i
\(346\) 0 0
\(347\) −4888.50 8467.13i −0.756278 1.30991i −0.944737 0.327831i \(-0.893682\pi\)
0.188459 0.982081i \(-0.439651\pi\)
\(348\) 0 0
\(349\) −11914.0 −1.82734 −0.913670 0.406456i \(-0.866764\pi\)
−0.913670 + 0.406456i \(0.866764\pi\)
\(350\) 0 0
\(351\) 3498.00 0.531936
\(352\) 0 0
\(353\) −4561.50 7900.75i −0.687774 1.19126i −0.972556 0.232667i \(-0.925255\pi\)
0.284783 0.958592i \(-0.408079\pi\)
\(354\) 0 0
\(355\) −2744.00 + 4752.75i −0.410243 + 0.710562i
\(356\) 0 0
\(357\) −501.500 970.814i −0.0743479 0.143924i
\(358\) 0 0
\(359\) −4074.50 + 7057.24i −0.599008 + 1.03751i 0.393960 + 0.919128i \(0.371105\pi\)
−0.992968 + 0.118385i \(0.962228\pi\)
\(360\) 0 0
\(361\) −5955.00 10314.4i −0.868202 1.50377i
\(362\) 0 0
\(363\) −106.000 −0.0153266
\(364\) 0 0
\(365\) −2065.00 −0.296129
\(366\) 0 0
\(367\) −4835.50 8375.33i −0.687769 1.19125i −0.972558 0.232660i \(-0.925257\pi\)
0.284790 0.958590i \(-0.408076\pi\)
\(368\) 0 0
\(369\) −6474.00 + 11213.3i −0.913341 + 1.58195i
\(370\) 0 0
\(371\) 7714.50 + 361.133i 1.07956 + 0.0505366i
\(372\) 0 0
\(373\) −2054.50 + 3558.50i −0.285196 + 0.493973i −0.972657 0.232248i \(-0.925392\pi\)
0.687461 + 0.726221i \(0.258725\pi\)
\(374\) 0 0
\(375\) −703.500 1218.50i −0.0968762 0.167795i
\(376\) 0 0
\(377\) 6996.00 0.955736
\(378\) 0 0
\(379\) 3488.00 0.472735 0.236367 0.971664i \(-0.424043\pi\)
0.236367 + 0.971664i \(0.424043\pi\)
\(380\) 0 0
\(381\) −468.000 810.600i −0.0629301 0.108998i
\(382\) 0 0
\(383\) −4358.50 + 7549.14i −0.581485 + 1.00716i 0.413818 + 0.910360i \(0.364195\pi\)
−0.995304 + 0.0968028i \(0.969138\pi\)
\(384\) 0 0
\(385\) −2450.00 + 3819.17i −0.324321 + 0.505566i
\(386\) 0 0
\(387\) −3380.00 + 5854.33i −0.443967 + 0.768973i
\(388\) 0 0
\(389\) 81.5000 + 141.162i 0.0106227 + 0.0183990i 0.871288 0.490772i \(-0.163285\pi\)
−0.860665 + 0.509171i \(0.829952\pi\)
\(390\) 0 0
\(391\) −413.000 −0.0534177
\(392\) 0 0
\(393\) 755.000 0.0969077
\(394\) 0 0
\(395\) −1732.50 3000.78i −0.220687 0.382242i
\(396\) 0 0
\(397\) 499.500 865.159i 0.0631466 0.109373i −0.832724 0.553689i \(-0.813220\pi\)
0.895870 + 0.444316i \(0.146553\pi\)
\(398\) 0 0
\(399\) 1370.00 2135.62i 0.171894 0.267957i
\(400\) 0 0
\(401\) 7378.50 12779.9i 0.918865 1.59152i 0.117722 0.993047i \(-0.462441\pi\)
0.801143 0.598474i \(-0.204226\pi\)
\(402\) 0 0
\(403\) 2475.00 + 4286.83i 0.305927 + 0.529881i
\(404\) 0 0
\(405\) −4543.00 −0.557391
\(406\) 0 0
\(407\) 385.000 0.0468888
\(408\) 0 0
\(409\) 66.5000 + 115.181i 0.00803964 + 0.0139251i 0.870017 0.493021i \(-0.164108\pi\)
−0.861978 + 0.506946i \(0.830774\pi\)
\(410\) 0 0
\(411\) 1178.50 2041.22i 0.141438 0.244978i
\(412\) 0 0
\(413\) 314.500 + 14.7224i 0.0374710 + 0.00175410i
\(414\) 0 0
\(415\) −3262.00 + 5649.95i −0.385844 + 0.668302i
\(416\) 0 0
\(417\) 14.0000 + 24.2487i 0.00164408 + 0.00284764i
\(418\) 0 0
\(419\) 6420.00 0.748538 0.374269 0.927320i \(-0.377894\pi\)
0.374269 + 0.927320i \(0.377894\pi\)
\(420\) 0 0
\(421\) −10266.0 −1.18844 −0.594221 0.804302i \(-0.702540\pi\)
−0.594221 + 0.804302i \(0.702540\pi\)
\(422\) 0 0
\(423\) −2223.00 3850.35i −0.255522 0.442578i
\(424\) 0 0
\(425\) 2242.00 3883.26i 0.255889 0.443213i
\(426\) 0 0
\(427\) 433.500 + 839.179i 0.0491301 + 0.0951070i
\(428\) 0 0
\(429\) −1155.00 + 2000.52i −0.129986 + 0.225142i
\(430\) 0 0
\(431\) 7606.50 + 13174.8i 0.850098 + 1.47241i 0.881119 + 0.472894i \(0.156791\pi\)
−0.0310213 + 0.999519i \(0.509876\pi\)
\(432\) 0 0
\(433\) −1378.00 −0.152939 −0.0764693 0.997072i \(-0.524365\pi\)
−0.0764693 + 0.997072i \(0.524365\pi\)
\(434\) 0 0
\(435\) 742.000 0.0817843
\(436\) 0 0
\(437\) −479.500 830.518i −0.0524888 0.0909132i
\(438\) 0 0
\(439\) 1381.50 2392.83i 0.150195 0.260145i −0.781104 0.624401i \(-0.785343\pi\)
0.931299 + 0.364256i \(0.118677\pi\)
\(440\) 0 0
\(441\) −8879.00 833.116i −0.958752 0.0899597i
\(442\) 0 0
\(443\) 2924.50 5065.38i 0.313651 0.543259i −0.665499 0.746399i \(-0.731781\pi\)
0.979150 + 0.203140i \(0.0651146\pi\)
\(444\) 0 0
\(445\) −3055.50 5292.28i −0.325493 0.563771i
\(446\) 0 0
\(447\) −2295.00 −0.242841
\(448\) 0 0
\(449\) 4582.00 0.481599 0.240799 0.970575i \(-0.422590\pi\)
0.240799 + 0.970575i \(0.422590\pi\)
\(450\) 0 0
\(451\) −8715.00 15094.8i −0.909919 1.57603i
\(452\) 0 0
\(453\) 554.500 960.422i 0.0575114 0.0996127i
\(454\) 0 0
\(455\) −3927.00 7601.97i −0.404617 0.783266i
\(456\) 0 0
\(457\) −5775.50 + 10003.5i −0.591174 + 1.02394i 0.402901 + 0.915244i \(0.368002\pi\)
−0.994075 + 0.108700i \(0.965331\pi\)
\(458\) 0 0
\(459\) 1563.50 + 2708.06i 0.158993 + 0.275384i
\(460\) 0 0
\(461\) 9494.00 0.959175 0.479587 0.877494i \(-0.340786\pi\)
0.479587 + 0.877494i \(0.340786\pi\)
\(462\) 0 0
\(463\) −10160.0 −1.01982 −0.509908 0.860229i \(-0.670321\pi\)
−0.509908 + 0.860229i \(0.670321\pi\)
\(464\) 0 0
\(465\) 262.500 + 454.663i 0.0261788 + 0.0453430i
\(466\) 0 0
\(467\) −653.500 + 1131.90i −0.0647545 + 0.112158i −0.896585 0.442872i \(-0.853960\pi\)
0.831831 + 0.555030i \(0.187293\pi\)
\(468\) 0 0
\(469\) −8121.50 380.185i −0.799608 0.0374314i
\(470\) 0 0
\(471\) 779.500 1350.13i 0.0762579 0.132083i
\(472\) 0 0
\(473\) −4550.00 7880.83i −0.442303 0.766091i
\(474\) 0 0
\(475\) 10412.0 1.00576
\(476\) 0 0
\(477\) −10842.0 −1.04072
\(478\) 0 0
\(479\) −9143.50 15837.0i −0.872186 1.51067i −0.859730 0.510748i \(-0.829368\pi\)
−0.0124559 0.999922i \(-0.503965\pi\)
\(480\) 0 0
\(481\) −363.000 + 628.734i −0.0344103 + 0.0596005i
\(482\) 0 0
\(483\) 70.0000 109.119i 0.00659443 0.0102797i
\(484\) 0 0
\(485\) −1015.00 + 1758.03i −0.0950284 + 0.164594i
\(486\) 0 0
\(487\) 7476.50 + 12949.7i 0.695673 + 1.20494i 0.969953 + 0.243291i \(0.0782269\pi\)
−0.274281 + 0.961650i \(0.588440\pi\)
\(488\) 0 0
\(489\) 2251.00 0.208167
\(490\) 0 0
\(491\) −14352.0 −1.31914 −0.659569 0.751644i \(-0.729261\pi\)
−0.659569 + 0.751644i \(0.729261\pi\)
\(492\) 0 0
\(493\) 3127.00 + 5416.12i 0.285665 + 0.494787i
\(494\) 0 0
\(495\) 3185.00 5516.58i 0.289202 0.500913i
\(496\) 0 0
\(497\) 7840.00 12221.4i 0.707590 1.10302i
\(498\) 0 0
\(499\) −2765.50 + 4789.99i −0.248098 + 0.429718i −0.962998 0.269509i \(-0.913139\pi\)
0.714900 + 0.699226i \(0.246472\pi\)
\(500\) 0 0
\(501\) −1394.00 2414.48i −0.124310 0.215311i
\(502\) 0 0
\(503\) 8400.00 0.744607 0.372304 0.928111i \(-0.378568\pi\)
0.372304 + 0.928111i \(0.378568\pi\)
\(504\) 0 0
\(505\) −7595.00 −0.669254
\(506\) 0 0
\(507\) −1079.50 1869.75i −0.0945607 0.163784i
\(508\) 0 0
\(509\) −1192.50 + 2065.47i −0.103844 + 0.179863i −0.913265 0.407365i \(-0.866448\pi\)
0.809421 + 0.587228i \(0.199781\pi\)
\(510\) 0 0
\(511\) 5457.50 + 255.477i 0.472457 + 0.0221167i
\(512\) 0 0
\(513\) −3630.50 + 6288.21i −0.312457 + 0.541192i
\(514\) 0 0
\(515\) 5435.50 + 9414.56i 0.465081 + 0.805544i
\(516\) 0 0
\(517\) 5985.00 0.509130
\(518\) 0 0
\(519\) −1579.00 −0.133546
\(520\) 0 0
\(521\) 4576.50 + 7926.73i 0.384837 + 0.666557i 0.991747 0.128214i \(-0.0409243\pi\)
−0.606910 + 0.794771i \(0.707591\pi\)
\(522\) 0 0
\(523\) −6903.50 + 11957.2i −0.577187 + 0.999718i 0.418613 + 0.908165i \(0.362516\pi\)
−0.995800 + 0.0915530i \(0.970817\pi\)
\(524\) 0 0
\(525\) 646.000 + 1250.54i 0.0537024 + 0.103958i
\(526\) 0 0
\(527\) −2212.50 + 3832.16i −0.182880 + 0.316758i
\(528\) 0 0
\(529\) 6059.00 + 10494.5i 0.497986 + 0.862538i
\(530\) 0 0
\(531\) −442.000 −0.0361227
\(532\) 0 0
\(533\) 32868.0 2.67105
\(534\) 0 0
\(535\) −451.500 782.021i −0.0364861 0.0631957i
\(536\) 0 0
\(537\) 1225.50 2122.63i 0.0984809 0.170574i
\(538\) 0 0
\(539\) 6947.50 9790.42i 0.555195 0.782381i
\(540\) 0 0
\(541\) 4087.50 7079.76i 0.324834 0.562629i −0.656645 0.754200i \(-0.728025\pi\)
0.981479 + 0.191571i \(0.0613581\pi\)
\(542\) 0 0
\(543\) −585.000 1013.25i −0.0462334 0.0800787i
\(544\) 0 0
\(545\) −6755.00 −0.530922
\(546\) 0 0
\(547\) −4656.00 −0.363942 −0.181971 0.983304i \(-0.558248\pi\)
−0.181971 + 0.983304i \(0.558248\pi\)
\(548\) 0 0
\(549\) −663.000 1148.35i −0.0515413 0.0892721i
\(550\) 0 0
\(551\) −7261.00 + 12576.4i −0.561396 + 0.972366i
\(552\) 0 0
\(553\) 4207.50 + 8144.97i 0.323546 + 0.626328i
\(554\) 0 0
\(555\) −38.5000 + 66.6840i −0.00294457 + 0.00510014i
\(556\) 0 0
\(557\) 3501.50 + 6064.78i 0.266361 + 0.461352i 0.967919 0.251261i \(-0.0808452\pi\)
−0.701558 + 0.712612i \(0.747512\pi\)
\(558\) 0 0
\(559\) 17160.0 1.29837
\(560\) 0 0
\(561\) −2065.00 −0.155409
\(562\) 0 0
\(563\) −9876.50 17106.6i −0.739334 1.28056i −0.952796 0.303612i \(-0.901807\pi\)
0.213462 0.976951i \(-0.431526\pi\)
\(564\) 0 0
\(565\) −175.000 + 303.109i −0.0130306 + 0.0225697i
\(566\) 0 0
\(567\) 12006.5 + 562.050i 0.889287 + 0.0416295i
\(568\) 0 0
\(569\) 3448.50 5972.98i 0.254075 0.440071i −0.710569 0.703628i \(-0.751562\pi\)
0.964644 + 0.263557i \(0.0848957\pi\)
\(570\) 0 0
\(571\) 12457.5 + 21577.0i 0.913013 + 1.58138i 0.809785 + 0.586726i \(0.199584\pi\)
0.103227 + 0.994658i \(0.467083\pi\)
\(572\) 0 0
\(573\) −1275.00 −0.0929562
\(574\) 0 0
\(575\) 532.000 0.0385842
\(576\) 0 0
\(577\) −63.5000 109.985i −0.00458152 0.00793543i 0.863726 0.503962i \(-0.168125\pi\)
−0.868307 + 0.496027i \(0.834792\pi\)
\(578\) 0 0
\(579\) −17.5000 + 30.3109i −0.00125609 + 0.00217561i
\(580\) 0 0
\(581\) 9320.00 14528.4i 0.665506 1.03742i
\(582\) 0 0
\(583\) 7297.50 12639.6i 0.518407 0.897908i
\(584\) 0 0
\(585\) 6006.00 + 10402.7i 0.424474 + 0.735211i
\(586\) 0 0
\(587\) −9044.00 −0.635921 −0.317961 0.948104i \(-0.602998\pi\)
−0.317961 + 0.948104i \(0.602998\pi\)
\(588\) 0 0
\(589\) −10275.0 −0.718801
\(590\) 0 0
\(591\) −1367.00 2367.71i −0.0951453 0.164796i
\(592\) 0 0
\(593\) 5350.50 9267.34i 0.370521 0.641760i −0.619125 0.785292i \(-0.712513\pi\)
0.989646 + 0.143532i \(0.0458460\pi\)
\(594\) 0 0
\(595\) 4130.00 6438.03i 0.284560 0.443586i
\(596\) 0 0
\(597\) −1121.50 + 1942.49i −0.0768843 + 0.133168i
\(598\) 0 0
\(599\) −10399.5 18012.5i −0.709369 1.22866i −0.965091 0.261913i \(-0.915647\pi\)
0.255722 0.966750i \(-0.417687\pi\)
\(600\) 0 0
\(601\) −1402.00 −0.0951560 −0.0475780 0.998868i \(-0.515150\pi\)
−0.0475780 + 0.998868i \(0.515150\pi\)
\(602\) 0 0
\(603\) 11414.0 0.770836
\(604\) 0 0
\(605\) −371.000 642.591i −0.0249311 0.0431819i
\(606\) 0 0
\(607\) −3262.50 + 5650.82i −0.218156 + 0.377858i −0.954244 0.299028i \(-0.903338\pi\)
0.736088 + 0.676886i \(0.236671\pi\)
\(608\) 0 0
\(609\) −1961.00 91.7987i −0.130482 0.00610816i
\(610\) 0 0
\(611\) −5643.00 + 9773.96i −0.373636 + 0.647156i
\(612\) 0 0
\(613\) 7525.50 + 13034.5i 0.495844 + 0.858826i 0.999989 0.00479285i \(-0.00152562\pi\)
−0.504145 + 0.863619i \(0.668192\pi\)
\(614\) 0 0
\(615\) 3486.00 0.228568
\(616\) 0 0
\(617\) 11150.0 0.727524 0.363762 0.931492i \(-0.381492\pi\)
0.363762 + 0.931492i \(0.381492\pi\)
\(618\) 0 0
\(619\) 1707.50 + 2957.48i 0.110873 + 0.192037i 0.916122 0.400899i \(-0.131302\pi\)
−0.805250 + 0.592936i \(0.797969\pi\)
\(620\) 0 0
\(621\) −185.500 + 321.295i −0.0119869 + 0.0207619i
\(622\) 0 0
\(623\) 7420.50 + 14364.8i 0.477201 + 0.923775i
\(624\) 0 0
\(625\) 174.500 302.243i 0.0111680 0.0193435i
\(626\) 0 0
\(627\) −2397.50 4152.59i −0.152706 0.264495i
\(628\) 0 0
\(629\) −649.000 −0.0411404
\(630\) 0 0
\(631\) −21184.0 −1.33648 −0.668242 0.743944i \(-0.732953\pi\)
−0.668242 + 0.743944i \(0.732953\pi\)
\(632\) 0 0
\(633\) 586.000 + 1014.98i 0.0367953 + 0.0637313i
\(634\) 0 0
\(635\) 3276.00 5674.20i 0.204731 0.354604i
\(636\) 0 0
\(637\) 9438.00 + 20576.8i 0.587044 + 1.27988i
\(638\) 0 0
\(639\) −10192.0 + 17653.1i −0.630969 + 1.09287i
\(640\) 0 0
\(641\) 5352.50 + 9270.80i 0.329814 + 0.571255i 0.982475 0.186395i \(-0.0596805\pi\)
−0.652660 + 0.757651i \(0.726347\pi\)
\(642\) 0 0
\(643\) −6860.00 −0.420734 −0.210367 0.977622i \(-0.567466\pi\)
−0.210367 + 0.977622i \(0.567466\pi\)
\(644\) 0 0
\(645\) 1820.00 0.111105
\(646\) 0 0
\(647\) −7231.50 12525.3i −0.439412 0.761084i 0.558232 0.829685i \(-0.311480\pi\)
−0.997644 + 0.0686008i \(0.978147\pi\)
\(648\) 0 0
\(649\) 297.500 515.285i 0.0179937 0.0311660i
\(650\) 0 0
\(651\) −637.500 1234.09i −0.0383803 0.0742975i
\(652\) 0 0
\(653\) 2989.50 5177.97i 0.179155 0.310305i −0.762436 0.647063i \(-0.775997\pi\)
0.941591 + 0.336758i \(0.109330\pi\)
\(654\) 0 0
\(655\) 2642.50 + 4576.94i 0.157635 + 0.273032i
\(656\) 0 0
\(657\) −7670.00 −0.455457
\(658\) 0 0
\(659\) 6940.00 0.410234 0.205117 0.978737i \(-0.434243\pi\)
0.205117 + 0.978737i \(0.434243\pi\)
\(660\) 0 0
\(661\) 6699.50 + 11603.9i 0.394221 + 0.682812i 0.993001 0.118102i \(-0.0376810\pi\)
−0.598780 + 0.800914i \(0.704348\pi\)
\(662\) 0 0
\(663\) 1947.00 3372.30i 0.114050 0.197541i
\(664\) 0 0
\(665\) 17741.5 + 830.518i 1.03457 + 0.0484303i
\(666\) 0 0
\(667\) −371.000 + 642.591i −0.0215370 + 0.0373032i
\(668\) 0 0
\(669\) −1012.00 1752.84i −0.0584846 0.101298i
\(670\) 0 0
\(671\) 1785.00 0.102696
\(672\) 0 0
\(673\) 29510.0 1.69023 0.845117 0.534582i \(-0.179531\pi\)
0.845117 + 0.534582i \(0.179531\pi\)
\(674\) 0 0
\(675\) −2014.00 3488.35i −0.114843 0.198914i
\(676\) 0 0
\(677\) −13000.5 + 22517.5i −0.738035 + 1.27831i 0.215344 + 0.976538i \(0.430913\pi\)
−0.953379 + 0.301776i \(0.902421\pi\)
\(678\) 0 0
\(679\) 2900.00 4520.65i 0.163905 0.255503i
\(680\) 0 0
\(681\) 1285.50 2226.55i 0.0723355 0.125289i
\(682\) 0 0
\(683\) −4402.50 7625.35i −0.246643 0.427198i 0.715949 0.698152i \(-0.245994\pi\)
−0.962592 + 0.270954i \(0.912661\pi\)
\(684\) 0 0
\(685\) 16499.0 0.920284
\(686\) 0 0
\(687\) −895.000 −0.0497036
\(688\) 0 0
\(689\) 13761.0 + 23834.8i 0.760889 + 1.31790i
\(690\) 0 0
\(691\) 14342.5 24841.9i 0.789601 1.36763i −0.136610 0.990625i \(-0.543621\pi\)
0.926211 0.377004i \(-0.123046\pi\)
\(692\) 0 0
\(693\) −9100.00 + 14185.5i −0.498817 + 0.777579i
\(694\) 0 0
\(695\) −98.0000 + 169.741i −0.00534871 + 0.00926423i
\(696\) 0 0
\(697\) 14691.0 + 25445.6i 0.798366 + 1.38281i
\(698\) 0 0
\(699\) 1787.00 0.0966961
\(700\) 0 0
\(701\) 3146.00 0.169505 0.0847523 0.996402i \(-0.472990\pi\)
0.0847523 + 0.996402i \(0.472990\pi\)
\(702\) 0 0
\(703\) −753.500 1305.10i −0.0404250 0.0700182i
\(704\) 0 0
\(705\) −598.500 + 1036.63i −0.0319728 + 0.0553785i
\(706\) 0 0
\(707\) 20072.5 + 939.638i 1.06776 + 0.0499840i
\(708\) 0 0
\(709\) 629.500 1090.33i 0.0333447 0.0577547i −0.848871 0.528599i \(-0.822717\pi\)
0.882216 + 0.470845i \(0.156051\pi\)
\(710\) 0 0
\(711\) −6435.00 11145.7i −0.339425 0.587902i
\(712\) 0 0
\(713\) −525.000 −0.0275756
\(714\) 0 0
\(715\) −16170.0 −0.845767
\(716\) 0 0
\(717\) 2550.00 + 4416.73i 0.132819 + 0.230050i
\(718\) 0 0
\(719\) −8212.50 + 14224.5i −0.425973 + 0.737807i −0.996511 0.0834645i \(-0.973401\pi\)
0.570538 + 0.821271i \(0.306735\pi\)
\(720\) 0 0
\(721\) −13200.5 25553.8i −0.681848 1.31994i
\(722\) 0 0
\(723\) 2088.50 3617.39i 0.107430 0.186075i
\(724\) 0 0
\(725\) −4028.00 6976.70i −0.206340 0.357391i
\(726\) 0 0
\(727\) −6032.00 −0.307723 −0.153861 0.988092i \(-0.549171\pi\)
−0.153861 + 0.988092i \(0.549171\pi\)
\(728\) 0 0
\(729\) −15443.0 −0.784586
\(730\) 0 0
\(731\) 7670.00 + 13284.8i 0.388078 + 0.672171i
\(732\) 0 0
\(733\) 7621.50 13200.8i 0.384047 0.665189i −0.607589 0.794251i \(-0.707863\pi\)
0.991636 + 0.129062i \(0.0411967\pi\)
\(734\) 0 0
\(735\) 1001.00 + 2182.38i 0.0502346 + 0.109522i
\(736\) 0 0
\(737\) −7682.50 + 13306.5i −0.383974 + 0.665062i
\(738\) 0 0
\(739\) −5026.50 8706.15i −0.250207 0.433371i 0.713376 0.700782i \(-0.247165\pi\)
−0.963583 + 0.267411i \(0.913832\pi\)
\(740\) 0 0
\(741\) 9042.00 0.448267
\(742\) 0 0
\(743\) 24384.0 1.20399 0.601993 0.798501i \(-0.294373\pi\)
0.601993 + 0.798501i \(0.294373\pi\)
\(744\) 0 0
\(745\) −8032.50 13912.7i −0.395017 0.684190i
\(746\) 0 0
\(747\) −12116.0 + 20985.5i −0.593442 + 1.02787i
\(748\) 0 0
\(749\) 1096.50 + 2122.63i 0.0534916 + 0.103550i
\(750\) 0 0
\(751\) −5794.50 + 10036.4i −0.281550 + 0.487660i −0.971767 0.235943i \(-0.924182\pi\)
0.690216 + 0.723603i \(0.257515\pi\)
\(752\) 0 0
\(753\) −2340.00 4053.00i −0.113246 0.196148i
\(754\) 0 0
\(755\) 7763.00 0.374205
\(756\) 0 0
\(757\) −14562.0 −0.699161 −0.349581 0.936906i \(-0.613676\pi\)
−0.349581 + 0.936906i \(0.613676\pi\)
\(758\) 0 0
\(759\) −122.500 212.176i −0.00585832 0.0101469i
\(760\) 0 0
\(761\) 11382.5 19715.1i 0.542201 0.939120i −0.456576 0.889684i \(-0.650924\pi\)
0.998777 0.0494360i \(-0.0157424\pi\)
\(762\) 0 0
\(763\) 17852.5 + 835.715i 0.847056 + 0.0396526i
\(764\) 0 0
\(765\) −5369.00 + 9299.38i −0.253747 + 0.439503i
\(766\) 0 0
\(767\) 561.000 + 971.681i 0.0264101 + 0.0457436i
\(768\) 0 0
\(769\) 3766.00 0.176600 0.0883000 0.996094i \(-0.471857\pi\)
0.0883000 + 0.996094i \(0.471857\pi\)
\(770\) 0 0
\(771\) −1749.00 −0.0816974
\(772\) 0 0
\(773\) −13430.5 23262.3i −0.624918 1.08239i −0.988557 0.150849i \(-0.951799\pi\)
0.363639 0.931540i \(-0.381534\pi\)
\(774\) 0 0
\(775\) 2850.00 4936.34i 0.132097 0.228798i
\(776\) 0 0
\(777\) 110.000 171.473i 0.00507880 0.00791707i
\(778\) 0 0
\(779\) −34113.0 + 59085.4i −1.56897 + 2.71753i
\(780\) 0 0
\(781\) −13720.0 23763.7i −0.628605 1.08878i
\(782\) 0 0
\(783\) 5618.00 0.256412
\(784\) 0 0
\(785\) 10913.0 0.496180
\(786\) 0 0
\(787\) −1048.50 1816.06i −0.0474905 0.0822559i 0.841303 0.540564i \(-0.181789\pi\)
−0.888793 + 0.458308i \(0.848456\pi\)
\(788\) 0 0
\(789\) 2236.50 3873.73i 0.100914 0.174789i
\(790\) 0 0
\(791\) 500.000 779.423i 0.0224753 0.0350355i
\(792\) 0 0
\(793\) −1683.00 + 2915.04i −0.0753658 + 0.130537i
\(794\) 0 0
\(795\) 1459.50 + 2527.93i 0.0651109 + 0.112775i
\(796\) 0 0
\(797\) 35334.0 1.57038 0.785191 0.619254i \(-0.212565\pi\)
0.785191 + 0.619254i \(0.212565\pi\)
\(798\) 0 0
\(799\) −10089.0 −0.446712
\(800\) 0 0
\(801\) −11349.0 19657.0i −0.500621 0.867101i
\(802\) 0 0
\(803\) 5162.50 8941.71i 0.226875 0.392959i
\(804\) 0 0
\(805\) 906.500 + 42.4352i 0.0396894 + 0.00185795i
\(806\) 0 0
\(807\) 987.500 1710.40i 0.0430752 0.0746083i
\(808\) 0 0
\(809\) −21267.5 36836.4i −0.924259 1.60086i −0.792749 0.609549i \(-0.791351\pi\)
−0.131510 0.991315i \(-0.541983\pi\)
\(810\) 0 0
\(811\) −30676.0 −1.32821 −0.664106 0.747638i \(-0.731188\pi\)
−0.664106 + 0.747638i \(0.731188\pi\)
\(812\) 0 0
\(813\) −8439.00 −0.364045
\(814\) 0 0
\(815\) 7878.50 + 13646.0i 0.338616 + 0.586500i
\(816\) 0 0
\(817\) −17810.0 + 30847.8i −0.762660 + 1.32097i
\(818\) 0 0
\(819\) −14586.0 28235.9i −0.622315 1.20469i
\(820\) 0 0
\(821\) 18671.5 32340.0i 0.793715 1.37475i −0.129937 0.991522i \(-0.541478\pi\)
0.923652 0.383232i \(-0.125189\pi\)
\(822\) 0 0
\(823\) −1407.50 2437.86i −0.0596141 0.103255i 0.834678 0.550738i \(-0.185654\pi\)
−0.894292 + 0.447483i \(0.852320\pi\)
\(824\) 0 0
\(825\) 2660.00 0.112254
\(826\) 0 0
\(827\) 9276.00 0.390034 0.195017 0.980800i \(-0.437524\pi\)
0.195017 + 0.980800i \(0.437524\pi\)
\(828\) 0 0
\(829\) 9285.50 + 16083.0i 0.389021 + 0.673805i 0.992318 0.123712i \(-0.0394799\pi\)
−0.603297 + 0.797517i \(0.706147\pi\)
\(830\) 0 0
\(831\) 263.500 456.395i 0.0109997 0.0190520i
\(832\) 0 0
\(833\) −11711.5 + 16503.8i −0.487130 + 0.686464i
\(834\) 0 0
\(835\) 9758.00 16901.4i 0.404419 0.700474i
\(836\) 0 0
\(837\) 1987.50 + 3442.45i 0.0820765 + 0.142161i
\(838\) 0 0
\(839\) 29048.0 1.19529 0.597645 0.801761i \(-0.296103\pi\)
0.597645 + 0.801761i \(0.296103\pi\)
\(840\) 0 0
\(841\) −13153.0 −0.539301
\(842\) 0 0
\(843\) 101.000 + 174.937i 0.00412648 + 0.00714728i
\(844\) 0 0
\(845\) 7556.50 13088.2i 0.307635 0.532839i
\(846\) 0 0
\(847\) 901.000 + 1744.18i 0.0365510 + 0.0707563i
\(848\) 0 0
\(849\) −3974.50 + 6884.04i −0.160665 + 0.278280i
\(850\) 0 0
\(851\) −38.5000 66.6840i −0.00155084 0.00268613i
\(852\) 0 0
\(853\) −32090.0 −1.28809 −0.644045 0.764988i \(-0.722745\pi\)
−0.644045 + 0.764988i \(0.722745\pi\)
\(854\) 0 0
\(855\) −24934.0 −0.997339
\(856\) 0 0
\(857\) 12268.5 + 21249.7i 0.489013 + 0.846995i 0.999920 0.0126408i \(-0.00402379\pi\)
−0.510907 + 0.859636i \(0.670690\pi\)
\(858\) 0 0
\(859\) 10412.5 18035.0i 0.413585 0.716351i −0.581693 0.813408i \(-0.697610\pi\)
0.995279 + 0.0970571i \(0.0309430\pi\)
\(860\) 0 0
\(861\) −9213.00 431.281i −0.364667 0.0170709i
\(862\) 0 0
\(863\) 11423.5 19786.1i 0.450591 0.780447i −0.547831 0.836589i \(-0.684546\pi\)
0.998423 + 0.0561414i \(0.0178798\pi\)
\(864\) 0 0
\(865\) −5526.50 9572.18i −0.217233 0.376259i
\(866\) 0 0
\(867\) −1432.00 −0.0560937
\(868\) 0 0
\(869\) 17325.0 0.676307
\(870\) 0 0
\(871\) −14487.0 25092.2i −0.563574 0.976139i
\(872\) 0 0
\(873\) −3770.00 + 6529.83i −0.146157 + 0.253152i
\(874\) 0 0
\(875\) −14070.0 + 21933.0i −0.543603 + 0.847394i
\(876\) 0 0
\(877\) −21368.5 + 37011.3i −0.822763 + 1.42507i 0.0808543 + 0.996726i \(0.474235\pi\)
−0.903617 + 0.428341i \(0.859098\pi\)
\(878\) 0 0
\(879\) 159.000 + 275.396i 0.00610118 + 0.0105676i
\(880\) 0 0
\(881\) 6162.00 0.235645 0.117822 0.993035i \(-0.462409\pi\)
0.117822 + 0.993035i \(0.462409\pi\)
\(882\) 0 0
\(883\) −7748.00 −0.295290 −0.147645 0.989040i \(-0.547169\pi\)
−0.147645 + 0.989040i \(0.547169\pi\)
\(884\) 0 0
\(885\) 59.5000 + 103.057i 0.00225997 + 0.00391438i
\(886\) 0 0
\(887\) 12961.5 22450.0i 0.490648 0.849827i −0.509294 0.860592i \(-0.670094\pi\)
0.999942 + 0.0107656i \(0.00342685\pi\)
\(888\) 0 0
\(889\) −9360.00 + 14590.8i −0.353121 + 0.550461i
\(890\) 0 0
\(891\) 11357.5 19671.8i 0.427038 0.739651i
\(892\) 0 0
\(893\) −11713.5 20288.4i −0.438944 0.760274i
\(894\) 0 0
\(895\) 17157.0 0.640777
\(896\) 0 0
\(897\) 462.000 0.0171970
\(898\) 0 0
\(899\) 3975.00 + 6884.90i 0.147468 + 0.255422i
\(900\) 0 0
\(901\) −12301.5 + 21306.8i −0.454853 + 0.787828i
\(902\) 0 0
\(903\) −4810.00 225.167i −0.177261 0.00829798i
\(904\) 0 0
\(905\) 4095.00 7092.75i 0.150411 0.260520i
\(906\) 0 0
\(907\) 15967.5 + 27656.5i 0.584556 + 1.01248i 0.994931 + 0.100563i \(0.0320645\pi\)
−0.410375 + 0.911917i \(0.634602\pi\)
\(908\) 0 0
\(909\) −28210.0 −1.02934
\(910\) 0 0
\(911\) 3408.00 0.123943 0.0619715 0.998078i \(-0.480261\pi\)
0.0619715 + 0.998078i \(0.480261\pi\)
\(912\) 0 0
\(913\) −16310.0 28249.7i −0.591218 1.02402i
\(914\) 0 0
\(915\) −178.500 + 309.171i −0.00644921 + 0.0111704i
\(916\) 0 0
\(917\) −6417.50 12423.1i −0.231106 0.447381i
\(918\) 0 0
\(919\) −6954.50 + 12045.5i −0.249628 + 0.432368i −0.963423 0.267987i \(-0.913642\pi\)
0.713795 + 0.700355i \(0.246975\pi\)
\(920\) 0 0
\(921\) −4066.00 7042.52i −0.145472 0.251964i
\(922\) 0 0
\(923\) 51744.0 1.84526
\(924\) 0 0
\(925\) 836.000 0.0297162
\(926\) 0 0
\(927\) 20189.0 + 34968.4i 0.715311 + 1.23896i
\(928\) 0 0
\(929\) 12268.5 21249.7i 0.433279 0.750462i −0.563874 0.825861i \(-0.690690\pi\)
0.997153 + 0.0753990i \(0.0240231\pi\)
\(930\) 0 0
\(931\) −46785.5 4389.88i −1.64697 0.154536i
\(932\) 0 0
\(933\) 464.500 804.538i 0.0162991 0.0282308i
\(934\) 0 0
\(935\) −7227.50 12518.4i −0.252796 0.437856i
\(936\) 0 0
\(937\) −32758.0 −1.14211 −0.571055 0.820912i \(-0.693466\pi\)
−0.571055 + 0.820912i \(0.693466\pi\)
\(938\) 0 0
\(939\) −209.000 −0.00726353
\(940\) 0 0
\(941\) −19280.5 33394.8i −0.667934 1.15690i −0.978481 0.206338i \(-0.933845\pi\)
0.310546 0.950558i \(-0.399488\pi\)
\(942\) 0 0
\(943\) −1743.00 + 3018.96i −0.0601908 + 0.104253i
\(944\) 0 0
\(945\) −3153.50 6104.61i −0.108554 0.210141i
\(946\) 0 0
\(947\) 19830.5 34347.4i 0.680470 1.17861i −0.294368 0.955692i \(-0.595109\pi\)
0.974838 0.222916i \(-0.0715575\pi\)
\(948\) 0 0
\(949\) 9735.00 + 16861.5i 0.332994 + 0.576763i
\(950\) 0 0
\(951\) −7131.00 −0.243153
\(952\) 0 0
\(953\) −46618.0 −1.58458 −0.792290 0.610144i \(-0.791111\pi\)
−0.792290 + 0.610144i \(0.791111\pi\)
\(954\) 0 0
\(955\) −4462.50 7729.28i −0.151207 0.261899i
\(956\) 0 0
\(957\) −1855.00 + 3212.95i −0.0626579 + 0.108527i
\(958\) 0 0
\(959\) −43604.5 2041.22i −1.46826 0.0687325i
\(960\) 0 0
\(961\) 12083.0 20928.4i 0.405592 0.702506i
\(962\) 0 0
\(963\) −1677.00 2904.65i −0.0561169 0.0971973i
\(964\) 0 0
\(965\) −245.000 −0.00817288
\(966\) 0 0
\(967\) 14816.0 0.492710 0.246355 0.969180i \(-0.420767\pi\)
0.246355 + 0.969180i \(0.420767\pi\)
\(968\) 0 0
\(969\) 4041.50 + 7000.08i 0.133985 + 0.232069i
\(970\) 0 0
\(971\) −8437.50 + 14614.2i −0.278859 + 0.482998i −0.971102 0.238667i \(-0.923290\pi\)
0.692242 + 0.721665i \(0.256623\pi\)
\(972\) 0 0
\(973\) 280.000 436.477i 0.00922548 0.0143811i
\(974\) 0 0
\(975\) −2508.00 + 4343.98i −0.0823798 + 0.142686i
\(976\) 0 0
\(977\) 7918.50 + 13715.2i 0.259299 + 0.449119i 0.966054 0.258339i \(-0.0831751\pi\)
−0.706755 + 0.707458i \(0.749842\pi\)
\(978\) 0 0
\(979\) 30555.0 0.997489
\(980\) 0 0
\(981\) −25090.0 −0.816577
\(982\) 0 0
\(983\) −4957.50 8586.64i −0.160854 0.278608i 0.774321 0.632793i \(-0.218092\pi\)
−0.935175 + 0.354185i \(0.884758\pi\)
\(984\) 0 0
\(985\) 9569.00 16574.0i 0.309537 0.536133i
\(986\) 0 0
\(987\) 1710.00 2665.63i 0.0551468 0.0859654i
\(988\) 0 0
\(989\) −910.000 + 1576.17i −0.0292582 + 0.0506766i
\(990\) 0 0
\(991\) 21840.5 + 37828.9i 0.700087 + 1.21259i 0.968435 + 0.249265i \(0.0801889\pi\)
−0.268348 + 0.963322i \(0.586478\pi\)
\(992\) 0 0
\(993\) 6571.00 0.209994
\(994\) 0 0
\(995\) −15701.0 −0.500256
\(996\) 0 0
\(997\) −23556.5 40801.1i −0.748287 1.29607i −0.948643 0.316348i \(-0.897543\pi\)
0.200357 0.979723i \(-0.435790\pi\)
\(998\) 0 0
\(999\) −291.500 + 504.893i −0.00923188 + 0.0159901i
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 448.4.i.c.65.1 2
4.3 odd 2 448.4.i.d.65.1 2
7.4 even 3 inner 448.4.i.c.193.1 2
8.3 odd 2 112.4.i.b.65.1 2
8.5 even 2 14.4.c.b.9.1 2
24.5 odd 2 126.4.g.c.37.1 2
28.11 odd 6 448.4.i.d.193.1 2
40.13 odd 4 350.4.j.d.149.1 4
40.29 even 2 350.4.e.b.51.1 2
40.37 odd 4 350.4.j.d.149.2 4
56.5 odd 6 98.4.a.c.1.1 1
56.11 odd 6 112.4.i.b.81.1 2
56.13 odd 2 98.4.c.e.79.1 2
56.19 even 6 784.4.a.j.1.1 1
56.37 even 6 98.4.a.b.1.1 1
56.45 odd 6 98.4.c.e.67.1 2
56.51 odd 6 784.4.a.l.1.1 1
56.53 even 6 14.4.c.b.11.1 yes 2
168.5 even 6 882.4.a.p.1.1 1
168.53 odd 6 126.4.g.c.109.1 2
168.101 even 6 882.4.g.d.361.1 2
168.125 even 2 882.4.g.d.667.1 2
168.149 odd 6 882.4.a.k.1.1 1
280.53 odd 12 350.4.j.d.249.2 4
280.109 even 6 350.4.e.b.151.1 2
280.149 even 6 2450.4.a.bh.1.1 1
280.229 odd 6 2450.4.a.bf.1.1 1
280.277 odd 12 350.4.j.d.249.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.4.c.b.9.1 2 8.5 even 2
14.4.c.b.11.1 yes 2 56.53 even 6
98.4.a.b.1.1 1 56.37 even 6
98.4.a.c.1.1 1 56.5 odd 6
98.4.c.e.67.1 2 56.45 odd 6
98.4.c.e.79.1 2 56.13 odd 2
112.4.i.b.65.1 2 8.3 odd 2
112.4.i.b.81.1 2 56.11 odd 6
126.4.g.c.37.1 2 24.5 odd 2
126.4.g.c.109.1 2 168.53 odd 6
350.4.e.b.51.1 2 40.29 even 2
350.4.e.b.151.1 2 280.109 even 6
350.4.j.d.149.1 4 40.13 odd 4
350.4.j.d.149.2 4 40.37 odd 4
350.4.j.d.249.1 4 280.277 odd 12
350.4.j.d.249.2 4 280.53 odd 12
448.4.i.c.65.1 2 1.1 even 1 trivial
448.4.i.c.193.1 2 7.4 even 3 inner
448.4.i.d.65.1 2 4.3 odd 2
448.4.i.d.193.1 2 28.11 odd 6
784.4.a.j.1.1 1 56.19 even 6
784.4.a.l.1.1 1 56.51 odd 6
882.4.a.k.1.1 1 168.149 odd 6
882.4.a.p.1.1 1 168.5 even 6
882.4.g.d.361.1 2 168.101 even 6
882.4.g.d.667.1 2 168.125 even 2
2450.4.a.bf.1.1 1 280.229 odd 6
2450.4.a.bh.1.1 1 280.149 even 6