Properties

Label 650.6.b.a.599.2
Level $650$
Weight $6$
Character 650.599
Analytic conductor $104.249$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,6,Mod(599,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.599");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 650.b (of order \(2\), degree \(1\), 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: \(104.249482878\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 650.599
Dual form 650.6.b.a.599.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000i q^{2} -16.0000 q^{4} +170.000i q^{7} -64.0000i q^{8} +243.000 q^{9} +O(q^{10})\) \(q+4.00000i q^{2} -16.0000 q^{4} +170.000i q^{7} -64.0000i q^{8} +243.000 q^{9} -250.000 q^{11} -169.000i q^{13} -680.000 q^{14} +256.000 q^{16} -1062.00i q^{17} +972.000i q^{18} +78.0000 q^{19} -1000.00i q^{22} +1576.00i q^{23} +676.000 q^{26} -2720.00i q^{28} -2578.00 q^{29} -8654.00 q^{31} +1024.00i q^{32} +4248.00 q^{34} -3888.00 q^{36} -10986.0i q^{37} +312.000i q^{38} +1050.00 q^{41} -5900.00i q^{43} +4000.00 q^{44} -6304.00 q^{46} +5962.00i q^{47} -12093.0 q^{49} +2704.00i q^{52} +29046.0i q^{53} +10880.0 q^{56} -10312.0i q^{58} +13922.0 q^{59} -32882.0 q^{61} -34616.0i q^{62} +41310.0i q^{63} -4096.00 q^{64} +69566.0i q^{67} +16992.0i q^{68} -50542.0 q^{71} -15552.0i q^{72} -46750.0i q^{73} +43944.0 q^{74} -1248.00 q^{76} -42500.0i q^{77} +19348.0 q^{79} +59049.0 q^{81} +4200.00i q^{82} -87438.0i q^{83} +23600.0 q^{86} +16000.0i q^{88} -94170.0 q^{89} +28730.0 q^{91} -25216.0i q^{92} -23848.0 q^{94} -182786. i q^{97} -48372.0i q^{98} -60750.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} + 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} + 486 q^{9} - 500 q^{11} - 1360 q^{14} + 512 q^{16} + 156 q^{19} + 1352 q^{26} - 5156 q^{29} - 17308 q^{31} + 8496 q^{34} - 7776 q^{36} + 2100 q^{41} + 8000 q^{44} - 12608 q^{46} - 24186 q^{49} + 21760 q^{56} + 27844 q^{59} - 65764 q^{61} - 8192 q^{64} - 101084 q^{71} + 87888 q^{74} - 2496 q^{76} + 38696 q^{79} + 118098 q^{81} + 47200 q^{86} - 188340 q^{89} + 57460 q^{91} - 47696 q^{94} - 121500 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000i 0.707107i
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 170.000i 1.31131i 0.755063 + 0.655653i \(0.227606\pi\)
−0.755063 + 0.655653i \(0.772394\pi\)
\(8\) − 64.0000i − 0.353553i
\(9\) 243.000 1.00000
\(10\) 0 0
\(11\) −250.000 −0.622957 −0.311479 0.950253i \(-0.600824\pi\)
−0.311479 + 0.950253i \(0.600824\pi\)
\(12\) 0 0
\(13\) − 169.000i − 0.277350i
\(14\) −680.000 −0.927233
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) − 1062.00i − 0.891255i −0.895218 0.445628i \(-0.852981\pi\)
0.895218 0.445628i \(-0.147019\pi\)
\(18\) 972.000i 0.707107i
\(19\) 78.0000 0.0495691 0.0247845 0.999693i \(-0.492110\pi\)
0.0247845 + 0.999693i \(0.492110\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 1000.00i − 0.440497i
\(23\) 1576.00i 0.621207i 0.950539 + 0.310604i \(0.100531\pi\)
−0.950539 + 0.310604i \(0.899469\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 676.000 0.196116
\(27\) 0 0
\(28\) − 2720.00i − 0.655653i
\(29\) −2578.00 −0.569230 −0.284615 0.958642i \(-0.591866\pi\)
−0.284615 + 0.958642i \(0.591866\pi\)
\(30\) 0 0
\(31\) −8654.00 −1.61738 −0.808691 0.588234i \(-0.799824\pi\)
−0.808691 + 0.588234i \(0.799824\pi\)
\(32\) 1024.00i 0.176777i
\(33\) 0 0
\(34\) 4248.00 0.630213
\(35\) 0 0
\(36\) −3888.00 −0.500000
\(37\) − 10986.0i − 1.31927i −0.751584 0.659637i \(-0.770710\pi\)
0.751584 0.659637i \(-0.229290\pi\)
\(38\) 312.000i 0.0350506i
\(39\) 0 0
\(40\) 0 0
\(41\) 1050.00 0.0975505 0.0487753 0.998810i \(-0.484468\pi\)
0.0487753 + 0.998810i \(0.484468\pi\)
\(42\) 0 0
\(43\) − 5900.00i − 0.486610i −0.969950 0.243305i \(-0.921768\pi\)
0.969950 0.243305i \(-0.0782316\pi\)
\(44\) 4000.00 0.311479
\(45\) 0 0
\(46\) −6304.00 −0.439260
\(47\) 5962.00i 0.393684i 0.980435 + 0.196842i \(0.0630685\pi\)
−0.980435 + 0.196842i \(0.936931\pi\)
\(48\) 0 0
\(49\) −12093.0 −0.719522
\(50\) 0 0
\(51\) 0 0
\(52\) 2704.00i 0.138675i
\(53\) 29046.0i 1.42035i 0.704023 + 0.710177i \(0.251385\pi\)
−0.704023 + 0.710177i \(0.748615\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 10880.0 0.463616
\(57\) 0 0
\(58\) − 10312.0i − 0.402507i
\(59\) 13922.0 0.520681 0.260340 0.965517i \(-0.416165\pi\)
0.260340 + 0.965517i \(0.416165\pi\)
\(60\) 0 0
\(61\) −32882.0 −1.13145 −0.565723 0.824596i \(-0.691403\pi\)
−0.565723 + 0.824596i \(0.691403\pi\)
\(62\) − 34616.0i − 1.14366i
\(63\) 41310.0i 1.31131i
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 69566.0i 1.89326i 0.322324 + 0.946629i \(0.395536\pi\)
−0.322324 + 0.946629i \(0.604464\pi\)
\(68\) 16992.0i 0.445628i
\(69\) 0 0
\(70\) 0 0
\(71\) −50542.0 −1.18989 −0.594945 0.803767i \(-0.702826\pi\)
−0.594945 + 0.803767i \(0.702826\pi\)
\(72\) − 15552.0i − 0.353553i
\(73\) − 46750.0i − 1.02677i −0.858157 0.513387i \(-0.828391\pi\)
0.858157 0.513387i \(-0.171609\pi\)
\(74\) 43944.0 0.932868
\(75\) 0 0
\(76\) −1248.00 −0.0247845
\(77\) − 42500.0i − 0.816887i
\(78\) 0 0
\(79\) 19348.0 0.348793 0.174397 0.984675i \(-0.444202\pi\)
0.174397 + 0.984675i \(0.444202\pi\)
\(80\) 0 0
\(81\) 59049.0 1.00000
\(82\) 4200.00i 0.0689786i
\(83\) − 87438.0i − 1.39317i −0.717473 0.696586i \(-0.754701\pi\)
0.717473 0.696586i \(-0.245299\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 23600.0 0.344085
\(87\) 0 0
\(88\) 16000.0i 0.220249i
\(89\) −94170.0 −1.26019 −0.630097 0.776516i \(-0.716985\pi\)
−0.630097 + 0.776516i \(0.716985\pi\)
\(90\) 0 0
\(91\) 28730.0 0.363691
\(92\) − 25216.0i − 0.310604i
\(93\) 0 0
\(94\) −23848.0 −0.278376
\(95\) 0 0
\(96\) 0 0
\(97\) − 182786.i − 1.97248i −0.165307 0.986242i \(-0.552861\pi\)
0.165307 0.986242i \(-0.447139\pi\)
\(98\) − 48372.0i − 0.508779i
\(99\) −60750.0 −0.622957
\(100\) 0 0
\(101\) −18514.0 −0.180591 −0.0902957 0.995915i \(-0.528781\pi\)
−0.0902957 + 0.995915i \(0.528781\pi\)
\(102\) 0 0
\(103\) 116056.i 1.07789i 0.842341 + 0.538945i \(0.181177\pi\)
−0.842341 + 0.538945i \(0.818823\pi\)
\(104\) −10816.0 −0.0980581
\(105\) 0 0
\(106\) −116184. −1.00434
\(107\) − 153520.i − 1.29630i −0.761513 0.648150i \(-0.775543\pi\)
0.761513 0.648150i \(-0.224457\pi\)
\(108\) 0 0
\(109\) 178622. 1.44002 0.720010 0.693963i \(-0.244137\pi\)
0.720010 + 0.693963i \(0.244137\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 43520.0i 0.327826i
\(113\) − 244754.i − 1.80316i −0.432615 0.901579i \(-0.642409\pi\)
0.432615 0.901579i \(-0.357591\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 41248.0 0.284615
\(117\) − 41067.0i − 0.277350i
\(118\) 55688.0i 0.368177i
\(119\) 180540. 1.16871
\(120\) 0 0
\(121\) −98551.0 −0.611924
\(122\) − 131528.i − 0.800053i
\(123\) 0 0
\(124\) 138464. 0.808691
\(125\) 0 0
\(126\) −165240. −0.927233
\(127\) − 256600.i − 1.41172i −0.708353 0.705858i \(-0.750562\pi\)
0.708353 0.705858i \(-0.249438\pi\)
\(128\) − 16384.0i − 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) −262736. −1.33765 −0.668823 0.743421i \(-0.733202\pi\)
−0.668823 + 0.743421i \(0.733202\pi\)
\(132\) 0 0
\(133\) 13260.0i 0.0650002i
\(134\) −278264. −1.33874
\(135\) 0 0
\(136\) −67968.0 −0.315106
\(137\) 38286.0i 0.174276i 0.996196 + 0.0871382i \(0.0277722\pi\)
−0.996196 + 0.0871382i \(0.972228\pi\)
\(138\) 0 0
\(139\) 57776.0 0.253636 0.126818 0.991926i \(-0.459524\pi\)
0.126818 + 0.991926i \(0.459524\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 202168.i − 0.841379i
\(143\) 42250.0i 0.172777i
\(144\) 62208.0 0.250000
\(145\) 0 0
\(146\) 187000. 0.726038
\(147\) 0 0
\(148\) 175776.i 0.659637i
\(149\) −28866.0 −0.106517 −0.0532587 0.998581i \(-0.516961\pi\)
−0.0532587 + 0.998581i \(0.516961\pi\)
\(150\) 0 0
\(151\) 39870.0 0.142300 0.0711498 0.997466i \(-0.477333\pi\)
0.0711498 + 0.997466i \(0.477333\pi\)
\(152\) − 4992.00i − 0.0175253i
\(153\) − 258066.i − 0.891255i
\(154\) 170000. 0.577627
\(155\) 0 0
\(156\) 0 0
\(157\) − 161042.i − 0.521423i −0.965417 0.260711i \(-0.916043\pi\)
0.965417 0.260711i \(-0.0839571\pi\)
\(158\) 77392.0i 0.246634i
\(159\) 0 0
\(160\) 0 0
\(161\) −267920. −0.814593
\(162\) 236196.i 0.707107i
\(163\) 312830.i 0.922230i 0.887340 + 0.461115i \(0.152550\pi\)
−0.887340 + 0.461115i \(0.847450\pi\)
\(164\) −16800.0 −0.0487753
\(165\) 0 0
\(166\) 349752. 0.985122
\(167\) − 532926.i − 1.47869i −0.673329 0.739343i \(-0.735136\pi\)
0.673329 0.739343i \(-0.264864\pi\)
\(168\) 0 0
\(169\) −28561.0 −0.0769231
\(170\) 0 0
\(171\) 18954.0 0.0495691
\(172\) 94400.0i 0.243305i
\(173\) − 630458.i − 1.60155i −0.598964 0.800776i \(-0.704421\pi\)
0.598964 0.800776i \(-0.295579\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −64000.0 −0.155739
\(177\) 0 0
\(178\) − 376680.i − 0.891092i
\(179\) 674916. 1.57441 0.787204 0.616693i \(-0.211528\pi\)
0.787204 + 0.616693i \(0.211528\pi\)
\(180\) 0 0
\(181\) 186282. 0.422644 0.211322 0.977417i \(-0.432223\pi\)
0.211322 + 0.977417i \(0.432223\pi\)
\(182\) 114920.i 0.257168i
\(183\) 0 0
\(184\) 100864. 0.219630
\(185\) 0 0
\(186\) 0 0
\(187\) 265500.i 0.555214i
\(188\) − 95392.0i − 0.196842i
\(189\) 0 0
\(190\) 0 0
\(191\) 812180. 1.61090 0.805451 0.592663i \(-0.201923\pi\)
0.805451 + 0.592663i \(0.201923\pi\)
\(192\) 0 0
\(193\) − 150142.i − 0.290141i −0.989421 0.145070i \(-0.953659\pi\)
0.989421 0.145070i \(-0.0463409\pi\)
\(194\) 731144. 1.39476
\(195\) 0 0
\(196\) 193488. 0.359761
\(197\) − 236394.i − 0.433981i −0.976174 0.216991i \(-0.930376\pi\)
0.976174 0.216991i \(-0.0696241\pi\)
\(198\) − 243000.i − 0.440497i
\(199\) 39376.0 0.0704854 0.0352427 0.999379i \(-0.488780\pi\)
0.0352427 + 0.999379i \(0.488780\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 74056.0i − 0.127697i
\(203\) − 438260.i − 0.746435i
\(204\) 0 0
\(205\) 0 0
\(206\) −464224. −0.762183
\(207\) 382968.i 0.621207i
\(208\) − 43264.0i − 0.0693375i
\(209\) −19500.0 −0.0308794
\(210\) 0 0
\(211\) −410776. −0.635183 −0.317592 0.948228i \(-0.602874\pi\)
−0.317592 + 0.948228i \(0.602874\pi\)
\(212\) − 464736.i − 0.710177i
\(213\) 0 0
\(214\) 614080. 0.916623
\(215\) 0 0
\(216\) 0 0
\(217\) − 1.47118e6i − 2.12088i
\(218\) 714488.i 1.01825i
\(219\) 0 0
\(220\) 0 0
\(221\) −179478. −0.247190
\(222\) 0 0
\(223\) 1.08688e6i 1.46359i 0.681523 + 0.731796i \(0.261318\pi\)
−0.681523 + 0.731796i \(0.738682\pi\)
\(224\) −174080. −0.231808
\(225\) 0 0
\(226\) 979016. 1.27502
\(227\) 256470.i 0.330348i 0.986264 + 0.165174i \(0.0528186\pi\)
−0.986264 + 0.165174i \(0.947181\pi\)
\(228\) 0 0
\(229\) 298110. 0.375654 0.187827 0.982202i \(-0.439856\pi\)
0.187827 + 0.982202i \(0.439856\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 164992.i 0.201253i
\(233\) − 611926.i − 0.738430i −0.929344 0.369215i \(-0.879627\pi\)
0.929344 0.369215i \(-0.120373\pi\)
\(234\) 164268. 0.196116
\(235\) 0 0
\(236\) −222752. −0.260340
\(237\) 0 0
\(238\) 722160.i 0.826401i
\(239\) −36570.0 −0.0414124 −0.0207062 0.999786i \(-0.506591\pi\)
−0.0207062 + 0.999786i \(0.506591\pi\)
\(240\) 0 0
\(241\) 380922. 0.422468 0.211234 0.977436i \(-0.432252\pi\)
0.211234 + 0.977436i \(0.432252\pi\)
\(242\) − 394204.i − 0.432696i
\(243\) 0 0
\(244\) 526112. 0.565723
\(245\) 0 0
\(246\) 0 0
\(247\) − 13182.0i − 0.0137480i
\(248\) 553856.i 0.571831i
\(249\) 0 0
\(250\) 0 0
\(251\) −1.22807e6 −1.23038 −0.615188 0.788380i \(-0.710920\pi\)
−0.615188 + 0.788380i \(0.710920\pi\)
\(252\) − 660960.i − 0.655653i
\(253\) − 394000.i − 0.386986i
\(254\) 1.02640e6 0.998234
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 439278.i 0.414865i 0.978249 + 0.207432i \(0.0665107\pi\)
−0.978249 + 0.207432i \(0.933489\pi\)
\(258\) 0 0
\(259\) 1.86762e6 1.72997
\(260\) 0 0
\(261\) −626454. −0.569230
\(262\) − 1.05094e6i − 0.945859i
\(263\) − 1.67987e6i − 1.49757i −0.662816 0.748783i \(-0.730639\pi\)
0.662816 0.748783i \(-0.269361\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −53040.0 −0.0459621
\(267\) 0 0
\(268\) − 1.11306e6i − 0.946629i
\(269\) −1.93840e6 −1.63329 −0.816645 0.577141i \(-0.804168\pi\)
−0.816645 + 0.577141i \(0.804168\pi\)
\(270\) 0 0
\(271\) −695498. −0.575271 −0.287636 0.957740i \(-0.592869\pi\)
−0.287636 + 0.957740i \(0.592869\pi\)
\(272\) − 271872.i − 0.222814i
\(273\) 0 0
\(274\) −153144. −0.123232
\(275\) 0 0
\(276\) 0 0
\(277\) 1.13138e6i 0.885948i 0.896534 + 0.442974i \(0.146077\pi\)
−0.896534 + 0.442974i \(0.853923\pi\)
\(278\) 231104.i 0.179348i
\(279\) −2.10292e6 −1.61738
\(280\) 0 0
\(281\) 1.73122e6 1.30793 0.653967 0.756523i \(-0.273103\pi\)
0.653967 + 0.756523i \(0.273103\pi\)
\(282\) 0 0
\(283\) − 1.47124e6i − 1.09199i −0.837788 0.545995i \(-0.816152\pi\)
0.837788 0.545995i \(-0.183848\pi\)
\(284\) 808672. 0.594945
\(285\) 0 0
\(286\) −169000. −0.122172
\(287\) 178500.i 0.127919i
\(288\) 248832.i 0.176777i
\(289\) 292013. 0.205664
\(290\) 0 0
\(291\) 0 0
\(292\) 748000.i 0.513387i
\(293\) 2.88855e6i 1.96567i 0.184491 + 0.982834i \(0.440936\pi\)
−0.184491 + 0.982834i \(0.559064\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −703104. −0.466434
\(297\) 0 0
\(298\) − 115464.i − 0.0753192i
\(299\) 266344. 0.172292
\(300\) 0 0
\(301\) 1.00300e6 0.638094
\(302\) 159480.i 0.100621i
\(303\) 0 0
\(304\) 19968.0 0.0123923
\(305\) 0 0
\(306\) 1.03226e6 0.630213
\(307\) − 874118.i − 0.529327i −0.964341 0.264664i \(-0.914739\pi\)
0.964341 0.264664i \(-0.0852609\pi\)
\(308\) 680000.i 0.408444i
\(309\) 0 0
\(310\) 0 0
\(311\) 2.68224e6 1.57252 0.786261 0.617895i \(-0.212014\pi\)
0.786261 + 0.617895i \(0.212014\pi\)
\(312\) 0 0
\(313\) − 1.34459e6i − 0.775761i −0.921710 0.387880i \(-0.873207\pi\)
0.921710 0.387880i \(-0.126793\pi\)
\(314\) 644168. 0.368702
\(315\) 0 0
\(316\) −309568. −0.174397
\(317\) − 1.32074e6i − 0.738191i −0.929392 0.369095i \(-0.879668\pi\)
0.929392 0.369095i \(-0.120332\pi\)
\(318\) 0 0
\(319\) 644500. 0.354606
\(320\) 0 0
\(321\) 0 0
\(322\) − 1.07168e6i − 0.576004i
\(323\) − 82836.0i − 0.0441787i
\(324\) −944784. −0.500000
\(325\) 0 0
\(326\) −1.25132e6 −0.652115
\(327\) 0 0
\(328\) − 67200.0i − 0.0344893i
\(329\) −1.01354e6 −0.516239
\(330\) 0 0
\(331\) −2.05728e6 −1.03210 −0.516051 0.856558i \(-0.672599\pi\)
−0.516051 + 0.856558i \(0.672599\pi\)
\(332\) 1.39901e6i 0.696586i
\(333\) − 2.66960e6i − 1.31927i
\(334\) 2.13170e6 1.04559
\(335\) 0 0
\(336\) 0 0
\(337\) − 453398.i − 0.217473i −0.994071 0.108736i \(-0.965320\pi\)
0.994071 0.108736i \(-0.0346804\pi\)
\(338\) − 114244.i − 0.0543928i
\(339\) 0 0
\(340\) 0 0
\(341\) 2.16350e6 1.00756
\(342\) 75816.0i 0.0350506i
\(343\) 801380.i 0.367793i
\(344\) −377600. −0.172043
\(345\) 0 0
\(346\) 2.52183e6 1.13247
\(347\) 1.23065e6i 0.548669i 0.961634 + 0.274334i \(0.0884575\pi\)
−0.961634 + 0.274334i \(0.911543\pi\)
\(348\) 0 0
\(349\) 2.43825e6 1.07155 0.535777 0.844360i \(-0.320019\pi\)
0.535777 + 0.844360i \(0.320019\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 256000.i − 0.110124i
\(353\) − 2.68315e6i − 1.14606i −0.819534 0.573031i \(-0.805767\pi\)
0.819534 0.573031i \(-0.194233\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1.50672e6 0.630097
\(357\) 0 0
\(358\) 2.69966e6i 1.11327i
\(359\) −1.58693e6 −0.649864 −0.324932 0.945737i \(-0.605341\pi\)
−0.324932 + 0.945737i \(0.605341\pi\)
\(360\) 0 0
\(361\) −2.47002e6 −0.997543
\(362\) 745128.i 0.298854i
\(363\) 0 0
\(364\) −459680. −0.181845
\(365\) 0 0
\(366\) 0 0
\(367\) 60052.0i 0.0232735i 0.999932 + 0.0116368i \(0.00370418\pi\)
−0.999932 + 0.0116368i \(0.996296\pi\)
\(368\) 403456.i 0.155302i
\(369\) 255150. 0.0975505
\(370\) 0 0
\(371\) −4.93782e6 −1.86252
\(372\) 0 0
\(373\) − 4.01853e6i − 1.49553i −0.663963 0.747766i \(-0.731127\pi\)
0.663963 0.747766i \(-0.268873\pi\)
\(374\) −1.06200e6 −0.392596
\(375\) 0 0
\(376\) 381568. 0.139188
\(377\) 435682.i 0.157876i
\(378\) 0 0
\(379\) −1.67581e6 −0.599276 −0.299638 0.954053i \(-0.596866\pi\)
−0.299638 + 0.954053i \(0.596866\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 3.24872e6i 1.13908i
\(383\) 687258.i 0.239399i 0.992810 + 0.119700i \(0.0381932\pi\)
−0.992810 + 0.119700i \(0.961807\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 600568. 0.205161
\(387\) − 1.43370e6i − 0.486610i
\(388\) 2.92458e6i 0.986242i
\(389\) −1.37611e6 −0.461082 −0.230541 0.973063i \(-0.574050\pi\)
−0.230541 + 0.973063i \(0.574050\pi\)
\(390\) 0 0
\(391\) 1.67371e6 0.553655
\(392\) 773952.i 0.254389i
\(393\) 0 0
\(394\) 945576. 0.306871
\(395\) 0 0
\(396\) 972000. 0.311479
\(397\) 721198.i 0.229656i 0.993385 + 0.114828i \(0.0366317\pi\)
−0.993385 + 0.114828i \(0.963368\pi\)
\(398\) 157504.i 0.0498407i
\(399\) 0 0
\(400\) 0 0
\(401\) 2.22681e6 0.691548 0.345774 0.938318i \(-0.387616\pi\)
0.345774 + 0.938318i \(0.387616\pi\)
\(402\) 0 0
\(403\) 1.46253e6i 0.448581i
\(404\) 296224. 0.0902957
\(405\) 0 0
\(406\) 1.75304e6 0.527809
\(407\) 2.74650e6i 0.821852i
\(408\) 0 0
\(409\) −2.00783e6 −0.593496 −0.296748 0.954956i \(-0.595902\pi\)
−0.296748 + 0.954956i \(0.595902\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 1.85690e6i − 0.538945i
\(413\) 2.36674e6i 0.682772i
\(414\) −1.53187e6 −0.439260
\(415\) 0 0
\(416\) 173056. 0.0490290
\(417\) 0 0
\(418\) − 78000.0i − 0.0218350i
\(419\) −5.99378e6 −1.66788 −0.833942 0.551852i \(-0.813921\pi\)
−0.833942 + 0.551852i \(0.813921\pi\)
\(420\) 0 0
\(421\) −5.32737e6 −1.46490 −0.732449 0.680822i \(-0.761623\pi\)
−0.732449 + 0.680822i \(0.761623\pi\)
\(422\) − 1.64310e6i − 0.449142i
\(423\) 1.44877e6i 0.393684i
\(424\) 1.85894e6 0.502171
\(425\) 0 0
\(426\) 0 0
\(427\) − 5.58994e6i − 1.48367i
\(428\) 2.45632e6i 0.648150i
\(429\) 0 0
\(430\) 0 0
\(431\) −5.42972e6 −1.40794 −0.703970 0.710230i \(-0.748591\pi\)
−0.703970 + 0.710230i \(0.748591\pi\)
\(432\) 0 0
\(433\) 7.43979e6i 1.90696i 0.301459 + 0.953479i \(0.402526\pi\)
−0.301459 + 0.953479i \(0.597474\pi\)
\(434\) 5.88472e6 1.49969
\(435\) 0 0
\(436\) −2.85795e6 −0.720010
\(437\) 122928.i 0.0307927i
\(438\) 0 0
\(439\) −6.86418e6 −1.69991 −0.849957 0.526852i \(-0.823372\pi\)
−0.849957 + 0.526852i \(0.823372\pi\)
\(440\) 0 0
\(441\) −2.93860e6 −0.719522
\(442\) − 717912.i − 0.174790i
\(443\) − 3.46630e6i − 0.839182i −0.907713 0.419591i \(-0.862173\pi\)
0.907713 0.419591i \(-0.137827\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4.34753e6 −1.03492
\(447\) 0 0
\(448\) − 696320.i − 0.163913i
\(449\) 1.40426e6 0.328725 0.164362 0.986400i \(-0.447443\pi\)
0.164362 + 0.986400i \(0.447443\pi\)
\(450\) 0 0
\(451\) −262500. −0.0607698
\(452\) 3.91606e6i 0.901579i
\(453\) 0 0
\(454\) −1.02588e6 −0.233591
\(455\) 0 0
\(456\) 0 0
\(457\) 5.95072e6i 1.33284i 0.745575 + 0.666421i \(0.232175\pi\)
−0.745575 + 0.666421i \(0.767825\pi\)
\(458\) 1.19244e6i 0.265627i
\(459\) 0 0
\(460\) 0 0
\(461\) −6.25465e6 −1.37073 −0.685363 0.728202i \(-0.740356\pi\)
−0.685363 + 0.728202i \(0.740356\pi\)
\(462\) 0 0
\(463\) − 1.55055e6i − 0.336149i −0.985774 0.168075i \(-0.946245\pi\)
0.985774 0.168075i \(-0.0537550\pi\)
\(464\) −659968. −0.142308
\(465\) 0 0
\(466\) 2.44770e6 0.522149
\(467\) 1.80480e6i 0.382945i 0.981498 + 0.191472i \(0.0613262\pi\)
−0.981498 + 0.191472i \(0.938674\pi\)
\(468\) 657072.i 0.138675i
\(469\) −1.18262e7 −2.48264
\(470\) 0 0
\(471\) 0 0
\(472\) − 891008.i − 0.184088i
\(473\) 1.47500e6i 0.303137i
\(474\) 0 0
\(475\) 0 0
\(476\) −2.88864e6 −0.584354
\(477\) 7.05818e6i 1.42035i
\(478\) − 146280.i − 0.0292830i
\(479\) −2.21809e6 −0.441712 −0.220856 0.975306i \(-0.570885\pi\)
−0.220856 + 0.975306i \(0.570885\pi\)
\(480\) 0 0
\(481\) −1.85663e6 −0.365901
\(482\) 1.52369e6i 0.298730i
\(483\) 0 0
\(484\) 1.57682e6 0.305962
\(485\) 0 0
\(486\) 0 0
\(487\) 6.14268e6i 1.17364i 0.809717 + 0.586821i \(0.199621\pi\)
−0.809717 + 0.586821i \(0.800379\pi\)
\(488\) 2.10445e6i 0.400026i
\(489\) 0 0
\(490\) 0 0
\(491\) 6.44486e6 1.20645 0.603226 0.797571i \(-0.293882\pi\)
0.603226 + 0.797571i \(0.293882\pi\)
\(492\) 0 0
\(493\) 2.73784e6i 0.507330i
\(494\) 52728.0 0.00972129
\(495\) 0 0
\(496\) −2.21542e6 −0.404346
\(497\) − 8.59214e6i − 1.56031i
\(498\) 0 0
\(499\) −4.25838e6 −0.765584 −0.382792 0.923835i \(-0.625037\pi\)
−0.382792 + 0.923835i \(0.625037\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 4.91227e6i − 0.870008i
\(503\) − 3.56242e6i − 0.627806i −0.949455 0.313903i \(-0.898363\pi\)
0.949455 0.313903i \(-0.101637\pi\)
\(504\) 2.64384e6 0.463616
\(505\) 0 0
\(506\) 1.57600e6 0.273640
\(507\) 0 0
\(508\) 4.10560e6i 0.705858i
\(509\) −4.23936e6 −0.725281 −0.362640 0.931929i \(-0.618125\pi\)
−0.362640 + 0.931929i \(0.618125\pi\)
\(510\) 0 0
\(511\) 7.94750e6 1.34641
\(512\) 262144.i 0.0441942i
\(513\) 0 0
\(514\) −1.75711e6 −0.293354
\(515\) 0 0
\(516\) 0 0
\(517\) − 1.49050e6i − 0.245248i
\(518\) 7.47048e6i 1.22328i
\(519\) 0 0
\(520\) 0 0
\(521\) 2.38657e6 0.385194 0.192597 0.981278i \(-0.438309\pi\)
0.192597 + 0.981278i \(0.438309\pi\)
\(522\) − 2.50582e6i − 0.402507i
\(523\) − 8.84129e6i − 1.41339i −0.707519 0.706694i \(-0.750186\pi\)
0.707519 0.706694i \(-0.249814\pi\)
\(524\) 4.20378e6 0.668823
\(525\) 0 0
\(526\) 6.71947e6 1.05894
\(527\) 9.19055e6i 1.44150i
\(528\) 0 0
\(529\) 3.95257e6 0.614101
\(530\) 0 0
\(531\) 3.38305e6 0.520681
\(532\) − 212160.i − 0.0325001i
\(533\) − 177450.i − 0.0270557i
\(534\) 0 0
\(535\) 0 0
\(536\) 4.45222e6 0.669368
\(537\) 0 0
\(538\) − 7.75361e6i − 1.15491i
\(539\) 3.02325e6 0.448231
\(540\) 0 0
\(541\) 70058.0 0.0102912 0.00514558 0.999987i \(-0.498362\pi\)
0.00514558 + 0.999987i \(0.498362\pi\)
\(542\) − 2.78199e6i − 0.406778i
\(543\) 0 0
\(544\) 1.08749e6 0.157553
\(545\) 0 0
\(546\) 0 0
\(547\) 6.60752e6i 0.944213i 0.881541 + 0.472107i \(0.156506\pi\)
−0.881541 + 0.472107i \(0.843494\pi\)
\(548\) − 612576.i − 0.0871382i
\(549\) −7.99033e6 −1.13145
\(550\) 0 0
\(551\) −201084. −0.0282162
\(552\) 0 0
\(553\) 3.28916e6i 0.457375i
\(554\) −4.52551e6 −0.626460
\(555\) 0 0
\(556\) −924416. −0.126818
\(557\) 1.10726e7i 1.51221i 0.654448 + 0.756107i \(0.272901\pi\)
−0.654448 + 0.756107i \(0.727099\pi\)
\(558\) − 8.41169e6i − 1.14366i
\(559\) −997100. −0.134961
\(560\) 0 0
\(561\) 0 0
\(562\) 6.92487e6i 0.924849i
\(563\) − 1.43532e6i − 0.190843i −0.995437 0.0954216i \(-0.969580\pi\)
0.995437 0.0954216i \(-0.0304199\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 5.88498e6 0.772153
\(567\) 1.00383e7i 1.31131i
\(568\) 3.23469e6i 0.420689i
\(569\) 1.17051e7 1.51564 0.757818 0.652466i \(-0.226266\pi\)
0.757818 + 0.652466i \(0.226266\pi\)
\(570\) 0 0
\(571\) 4.81885e6 0.618519 0.309260 0.950978i \(-0.399919\pi\)
0.309260 + 0.950978i \(0.399919\pi\)
\(572\) − 676000.i − 0.0863886i
\(573\) 0 0
\(574\) −714000. −0.0904521
\(575\) 0 0
\(576\) −995328. −0.125000
\(577\) 1.35572e6i 0.169523i 0.996401 + 0.0847617i \(0.0270129\pi\)
−0.996401 + 0.0847617i \(0.972987\pi\)
\(578\) 1.16805e6i 0.145426i
\(579\) 0 0
\(580\) 0 0
\(581\) 1.48645e7 1.82687
\(582\) 0 0
\(583\) − 7.26150e6i − 0.884820i
\(584\) −2.99200e6 −0.363019
\(585\) 0 0
\(586\) −1.15542e7 −1.38994
\(587\) − 5.03941e6i − 0.603649i −0.953364 0.301824i \(-0.902404\pi\)
0.953364 0.301824i \(-0.0975957\pi\)
\(588\) 0 0
\(589\) −675012. −0.0801721
\(590\) 0 0
\(591\) 0 0
\(592\) − 2.81242e6i − 0.329819i
\(593\) 9.16124e6i 1.06984i 0.844904 + 0.534919i \(0.179658\pi\)
−0.844904 + 0.534919i \(0.820342\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 461856. 0.0532587
\(597\) 0 0
\(598\) 1.06538e6i 0.121829i
\(599\) 6.46635e6 0.736363 0.368182 0.929754i \(-0.379980\pi\)
0.368182 + 0.929754i \(0.379980\pi\)
\(600\) 0 0
\(601\) −1.18021e7 −1.33282 −0.666411 0.745585i \(-0.732170\pi\)
−0.666411 + 0.745585i \(0.732170\pi\)
\(602\) 4.01200e6i 0.451201i
\(603\) 1.69045e7i 1.89326i
\(604\) −637920. −0.0711498
\(605\) 0 0
\(606\) 0 0
\(607\) − 2.25748e6i − 0.248686i −0.992239 0.124343i \(-0.960318\pi\)
0.992239 0.124343i \(-0.0396823\pi\)
\(608\) 79872.0i 0.00876265i
\(609\) 0 0
\(610\) 0 0
\(611\) 1.00758e6 0.109188
\(612\) 4.12906e6i 0.445628i
\(613\) − 2.75378e6i − 0.295991i −0.988988 0.147995i \(-0.952718\pi\)
0.988988 0.147995i \(-0.0472821\pi\)
\(614\) 3.49647e6 0.374291
\(615\) 0 0
\(616\) −2.72000e6 −0.288813
\(617\) − 3.41607e6i − 0.361255i −0.983552 0.180627i \(-0.942187\pi\)
0.983552 0.180627i \(-0.0578128\pi\)
\(618\) 0 0
\(619\) −9.43169e6 −0.989379 −0.494690 0.869070i \(-0.664718\pi\)
−0.494690 + 0.869070i \(0.664718\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 1.07290e7i 1.11194i
\(623\) − 1.60089e7i − 1.65250i
\(624\) 0 0
\(625\) 0 0
\(626\) 5.37834e6 0.548546
\(627\) 0 0
\(628\) 2.57667e6i 0.260711i
\(629\) −1.16671e7 −1.17581
\(630\) 0 0
\(631\) −4.87474e6 −0.487391 −0.243696 0.969852i \(-0.578360\pi\)
−0.243696 + 0.969852i \(0.578360\pi\)
\(632\) − 1.23827e6i − 0.123317i
\(633\) 0 0
\(634\) 5.28295e6 0.521980
\(635\) 0 0
\(636\) 0 0
\(637\) 2.04372e6i 0.199559i
\(638\) 2.57800e6i 0.250744i
\(639\) −1.22817e7 −1.18989
\(640\) 0 0
\(641\) 9.74279e6 0.936566 0.468283 0.883579i \(-0.344873\pi\)
0.468283 + 0.883579i \(0.344873\pi\)
\(642\) 0 0
\(643\) 1.63894e6i 0.156327i 0.996941 + 0.0781637i \(0.0249057\pi\)
−0.996941 + 0.0781637i \(0.975094\pi\)
\(644\) 4.28672e6 0.407296
\(645\) 0 0
\(646\) 331344. 0.0312390
\(647\) 1.59069e6i 0.149391i 0.997206 + 0.0746955i \(0.0237985\pi\)
−0.997206 + 0.0746955i \(0.976202\pi\)
\(648\) − 3.77914e6i − 0.353553i
\(649\) −3.48050e6 −0.324362
\(650\) 0 0
\(651\) 0 0
\(652\) − 5.00528e6i − 0.461115i
\(653\) 1.59778e7i 1.46634i 0.680045 + 0.733170i \(0.261960\pi\)
−0.680045 + 0.733170i \(0.738040\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 268800. 0.0243876
\(657\) − 1.13602e7i − 1.02677i
\(658\) − 4.05416e6i − 0.365036i
\(659\) 6.02458e6 0.540397 0.270199 0.962805i \(-0.412911\pi\)
0.270199 + 0.962805i \(0.412911\pi\)
\(660\) 0 0
\(661\) −2.00705e7 −1.78671 −0.893355 0.449352i \(-0.851655\pi\)
−0.893355 + 0.449352i \(0.851655\pi\)
\(662\) − 8.22911e6i − 0.729807i
\(663\) 0 0
\(664\) −5.59603e6 −0.492561
\(665\) 0 0
\(666\) 1.06784e7 0.932868
\(667\) − 4.06293e6i − 0.353610i
\(668\) 8.52682e6i 0.739343i
\(669\) 0 0
\(670\) 0 0
\(671\) 8.22050e6 0.704842
\(672\) 0 0
\(673\) − 5.48575e6i − 0.466873i −0.972372 0.233436i \(-0.925003\pi\)
0.972372 0.233436i \(-0.0749971\pi\)
\(674\) 1.81359e6 0.153776
\(675\) 0 0
\(676\) 456976. 0.0384615
\(677\) 4.74926e6i 0.398248i 0.979974 + 0.199124i \(0.0638097\pi\)
−0.979974 + 0.199124i \(0.936190\pi\)
\(678\) 0 0
\(679\) 3.10736e7 2.58653
\(680\) 0 0
\(681\) 0 0
\(682\) 8.65400e6i 0.712453i
\(683\) − 6.13964e6i − 0.503606i −0.967778 0.251803i \(-0.918976\pi\)
0.967778 0.251803i \(-0.0810236\pi\)
\(684\) −303264. −0.0247845
\(685\) 0 0
\(686\) −3.20552e6 −0.260069
\(687\) 0 0
\(688\) − 1.51040e6i − 0.121652i
\(689\) 4.90877e6 0.393935
\(690\) 0 0
\(691\) 1.57617e7 1.25577 0.627883 0.778308i \(-0.283922\pi\)
0.627883 + 0.778308i \(0.283922\pi\)
\(692\) 1.00873e7i 0.800776i
\(693\) − 1.03275e7i − 0.816887i
\(694\) −4.92259e6 −0.387967
\(695\) 0 0
\(696\) 0 0
\(697\) − 1.11510e6i − 0.0869424i
\(698\) 9.75298e6i 0.757703i
\(699\) 0 0
\(700\) 0 0
\(701\) −1.42036e7 −1.09170 −0.545851 0.837882i \(-0.683793\pi\)
−0.545851 + 0.837882i \(0.683793\pi\)
\(702\) 0 0
\(703\) − 856908.i − 0.0653952i
\(704\) 1.02400e6 0.0778697
\(705\) 0 0
\(706\) 1.07326e7 0.810388
\(707\) − 3.14738e6i − 0.236810i
\(708\) 0 0
\(709\) −1.60718e7 −1.20074 −0.600369 0.799723i \(-0.704980\pi\)
−0.600369 + 0.799723i \(0.704980\pi\)
\(710\) 0 0
\(711\) 4.70156e6 0.348793
\(712\) 6.02688e6i 0.445546i
\(713\) − 1.36387e7i − 1.00473i
\(714\) 0 0
\(715\) 0 0
\(716\) −1.07987e7 −0.787204
\(717\) 0 0
\(718\) − 6.34774e6i − 0.459524i
\(719\) 2.07078e7 1.49387 0.746933 0.664900i \(-0.231526\pi\)
0.746933 + 0.664900i \(0.231526\pi\)
\(720\) 0 0
\(721\) −1.97295e7 −1.41344
\(722\) − 9.88006e6i − 0.705369i
\(723\) 0 0
\(724\) −2.98051e6 −0.211322
\(725\) 0 0
\(726\) 0 0
\(727\) − 5.04803e6i − 0.354231i −0.984190 0.177115i \(-0.943323\pi\)
0.984190 0.177115i \(-0.0566766\pi\)
\(728\) − 1.83872e6i − 0.128584i
\(729\) 1.43489e7 1.00000
\(730\) 0 0
\(731\) −6.26580e6 −0.433694
\(732\) 0 0
\(733\) − 2.10377e7i − 1.44623i −0.690728 0.723115i \(-0.742710\pi\)
0.690728 0.723115i \(-0.257290\pi\)
\(734\) −240208. −0.0164569
\(735\) 0 0
\(736\) −1.61382e6 −0.109815
\(737\) − 1.73915e7i − 1.17942i
\(738\) 1.02060e6i 0.0689786i
\(739\) −1.38992e7 −0.936218 −0.468109 0.883671i \(-0.655065\pi\)
−0.468109 + 0.883671i \(0.655065\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 1.97513e7i − 1.31700i
\(743\) 1.23267e6i 0.0819169i 0.999161 + 0.0409584i \(0.0130411\pi\)
−0.999161 + 0.0409584i \(0.986959\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 1.60741e7 1.05750
\(747\) − 2.12474e7i − 1.39317i
\(748\) − 4.24800e6i − 0.277607i
\(749\) 2.60984e7 1.69985
\(750\) 0 0
\(751\) −1.62624e6 −0.105217 −0.0526084 0.998615i \(-0.516753\pi\)
−0.0526084 + 0.998615i \(0.516753\pi\)
\(752\) 1.52627e6i 0.0984209i
\(753\) 0 0
\(754\) −1.74273e6 −0.111635
\(755\) 0 0
\(756\) 0 0
\(757\) − 3.49882e6i − 0.221913i −0.993825 0.110956i \(-0.964609\pi\)
0.993825 0.110956i \(-0.0353914\pi\)
\(758\) − 6.70324e6i − 0.423752i
\(759\) 0 0
\(760\) 0 0
\(761\) 2.21713e7 1.38781 0.693905 0.720067i \(-0.255889\pi\)
0.693905 + 0.720067i \(0.255889\pi\)
\(762\) 0 0
\(763\) 3.03657e7i 1.88831i
\(764\) −1.29949e7 −0.805451
\(765\) 0 0
\(766\) −2.74903e6 −0.169281
\(767\) − 2.35282e6i − 0.144411i
\(768\) 0 0
\(769\) −1.08955e6 −0.0664400 −0.0332200 0.999448i \(-0.510576\pi\)
−0.0332200 + 0.999448i \(0.510576\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.40227e6i 0.145070i
\(773\) 1.95219e6i 0.117510i 0.998272 + 0.0587549i \(0.0187130\pi\)
−0.998272 + 0.0587549i \(0.981287\pi\)
\(774\) 5.73480e6 0.344085
\(775\) 0 0
\(776\) −1.16983e7 −0.697379
\(777\) 0 0
\(778\) − 5.50442e6i − 0.326034i
\(779\) 81900.0 0.00483549
\(780\) 0 0
\(781\) 1.26355e7 0.741250
\(782\) 6.69485e6i 0.391493i
\(783\) 0 0
\(784\) −3.09581e6 −0.179880
\(785\) 0 0
\(786\) 0 0
\(787\) 1.44531e7i 0.831809i 0.909408 + 0.415904i \(0.136535\pi\)
−0.909408 + 0.415904i \(0.863465\pi\)
\(788\) 3.78230e6i 0.216991i
\(789\) 0 0
\(790\) 0 0
\(791\) 4.16082e7 2.36449
\(792\) 3.88800e6i 0.220249i
\(793\) 5.55706e6i 0.313807i
\(794\) −2.88479e6 −0.162391
\(795\) 0 0
\(796\) −630016. −0.0352427
\(797\) − 1.23500e7i − 0.688685i −0.938844 0.344343i \(-0.888102\pi\)
0.938844 0.344343i \(-0.111898\pi\)
\(798\) 0 0
\(799\) 6.33164e6 0.350873
\(800\) 0 0
\(801\) −2.28833e7 −1.26019
\(802\) 8.90724e6i 0.488998i
\(803\) 1.16875e7i 0.639636i
\(804\) 0 0
\(805\) 0 0
\(806\) −5.85010e6 −0.317195
\(807\) 0 0
\(808\) 1.18490e6i 0.0638487i
\(809\) −1.15968e7 −0.622970 −0.311485 0.950251i \(-0.600826\pi\)
−0.311485 + 0.950251i \(0.600826\pi\)
\(810\) 0 0
\(811\) −2.47534e7 −1.32155 −0.660774 0.750585i \(-0.729772\pi\)
−0.660774 + 0.750585i \(0.729772\pi\)
\(812\) 7.01216e6i 0.373217i
\(813\) 0 0
\(814\) −1.09860e7 −0.581137
\(815\) 0 0
\(816\) 0 0
\(817\) − 460200.i − 0.0241208i
\(818\) − 8.03130e6i − 0.419665i
\(819\) 6.98139e6 0.363691
\(820\) 0 0
\(821\) −2.47470e6 −0.128134 −0.0640671 0.997946i \(-0.520407\pi\)
−0.0640671 + 0.997946i \(0.520407\pi\)
\(822\) 0 0
\(823\) − 7.84754e6i − 0.403863i −0.979400 0.201932i \(-0.935278\pi\)
0.979400 0.201932i \(-0.0647219\pi\)
\(824\) 7.42758e6 0.381092
\(825\) 0 0
\(826\) −9.46696e6 −0.482792
\(827\) 2.26192e7i 1.15004i 0.818140 + 0.575020i \(0.195006\pi\)
−0.818140 + 0.575020i \(0.804994\pi\)
\(828\) − 6.12749e6i − 0.310604i
\(829\) 1.73912e7 0.878907 0.439454 0.898265i \(-0.355172\pi\)
0.439454 + 0.898265i \(0.355172\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 692224.i 0.0346688i
\(833\) 1.28428e7i 0.641278i
\(834\) 0 0
\(835\) 0 0
\(836\) 312000. 0.0154397
\(837\) 0 0
\(838\) − 2.39751e7i − 1.17937i
\(839\) 3.43825e7 1.68629 0.843147 0.537684i \(-0.180701\pi\)
0.843147 + 0.537684i \(0.180701\pi\)
\(840\) 0 0
\(841\) −1.38651e7 −0.675977
\(842\) − 2.13095e7i − 1.03584i
\(843\) 0 0
\(844\) 6.57242e6 0.317592
\(845\) 0 0
\(846\) −5.79506e6 −0.278376
\(847\) − 1.67537e7i − 0.802419i
\(848\) 7.43578e6i 0.355089i
\(849\) 0 0
\(850\) 0 0
\(851\) 1.73139e7 0.819543
\(852\) 0 0
\(853\) 2.31007e7i 1.08706i 0.839391 + 0.543528i \(0.182912\pi\)
−0.839391 + 0.543528i \(0.817088\pi\)
\(854\) 2.23598e7 1.04911
\(855\) 0 0
\(856\) −9.82528e6 −0.458311
\(857\) 7.02305e6i 0.326643i 0.986573 + 0.163322i \(0.0522208\pi\)
−0.986573 + 0.163322i \(0.947779\pi\)
\(858\) 0 0
\(859\) −8.82135e6 −0.407899 −0.203949 0.978981i \(-0.565378\pi\)
−0.203949 + 0.978981i \(0.565378\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 2.17189e7i − 0.995564i
\(863\) 2.39560e7i 1.09493i 0.836828 + 0.547466i \(0.184408\pi\)
−0.836828 + 0.547466i \(0.815592\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −2.97592e7 −1.34842
\(867\) 0 0
\(868\) 2.35389e7i 1.06044i
\(869\) −4.83700e6 −0.217283
\(870\) 0 0
\(871\) 1.17567e7 0.525096
\(872\) − 1.14318e7i − 0.509124i
\(873\) − 4.44170e7i − 1.97248i
\(874\) −491712. −0.0217737
\(875\) 0 0
\(876\) 0 0
\(877\) 5.79805e6i 0.254556i 0.991867 + 0.127278i \(0.0406240\pi\)
−0.991867 + 0.127278i \(0.959376\pi\)
\(878\) − 2.74567e7i − 1.20202i
\(879\) 0 0
\(880\) 0 0
\(881\) −1.30527e7 −0.566580 −0.283290 0.959034i \(-0.591426\pi\)
−0.283290 + 0.959034i \(0.591426\pi\)
\(882\) − 1.17544e7i − 0.508779i
\(883\) 4.73009e6i 0.204159i 0.994776 + 0.102079i \(0.0325496\pi\)
−0.994776 + 0.102079i \(0.967450\pi\)
\(884\) 2.87165e6 0.123595
\(885\) 0 0
\(886\) 1.38652e7 0.593392
\(887\) − 2.80737e7i − 1.19809i −0.800714 0.599046i \(-0.795547\pi\)
0.800714 0.599046i \(-0.204453\pi\)
\(888\) 0 0
\(889\) 4.36220e7 1.85119
\(890\) 0 0
\(891\) −1.47622e7 −0.622957
\(892\) − 1.73901e7i − 0.731796i
\(893\) 465036.i 0.0195145i
\(894\) 0 0
\(895\) 0 0
\(896\) 2.78528e6 0.115904
\(897\) 0 0
\(898\) 5.61705e6i 0.232443i
\(899\) 2.23100e7 0.920663
\(900\) 0 0
\(901\) 3.08469e7 1.26590
\(902\) − 1.05000e6i − 0.0429708i
\(903\) 0 0
\(904\) −1.56643e7 −0.637512
\(905\) 0 0
\(906\) 0 0
\(907\) − 2.28552e7i − 0.922500i −0.887270 0.461250i \(-0.847401\pi\)
0.887270 0.461250i \(-0.152599\pi\)
\(908\) − 4.10352e6i − 0.165174i
\(909\) −4.49890e6 −0.180591
\(910\) 0 0
\(911\) −3.27335e7 −1.30676 −0.653381 0.757029i \(-0.726650\pi\)
−0.653381 + 0.757029i \(0.726650\pi\)
\(912\) 0 0
\(913\) 2.18595e7i 0.867887i
\(914\) −2.38029e7 −0.942462
\(915\) 0 0
\(916\) −4.76976e6 −0.187827
\(917\) − 4.46651e7i − 1.75406i
\(918\) 0 0
\(919\) 1.27717e7 0.498839 0.249419 0.968396i \(-0.419760\pi\)
0.249419 + 0.968396i \(0.419760\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 2.50186e7i − 0.969249i
\(923\) 8.54160e6i 0.330016i
\(924\) 0 0
\(925\) 0 0
\(926\) 6.20218e6 0.237693
\(927\) 2.82016e7i 1.07789i
\(928\) − 2.63987e6i − 0.100627i
\(929\) −3.48297e7 −1.32407 −0.662034 0.749473i \(-0.730307\pi\)
−0.662034 + 0.749473i \(0.730307\pi\)
\(930\) 0 0
\(931\) −943254. −0.0356660
\(932\) 9.79082e6i 0.369215i
\(933\) 0 0
\(934\) −7.21918e6 −0.270783
\(935\) 0 0
\(936\) −2.62829e6 −0.0980581
\(937\) − 3.00172e7i − 1.11692i −0.829532 0.558459i \(-0.811393\pi\)
0.829532 0.558459i \(-0.188607\pi\)
\(938\) − 4.73049e7i − 1.75549i
\(939\) 0 0
\(940\) 0 0
\(941\) −4.50649e7 −1.65907 −0.829534 0.558457i \(-0.811394\pi\)
−0.829534 + 0.558457i \(0.811394\pi\)
\(942\) 0 0
\(943\) 1.65480e6i 0.0605991i
\(944\) 3.56403e6 0.130170
\(945\) 0 0
\(946\) −5.90000e6 −0.214350
\(947\) − 2.99276e7i − 1.08442i −0.840243 0.542210i \(-0.817588\pi\)
0.840243 0.542210i \(-0.182412\pi\)
\(948\) 0 0
\(949\) −7.90075e6 −0.284776
\(950\) 0 0
\(951\) 0 0
\(952\) − 1.15546e7i − 0.413201i
\(953\) − 4.25147e7i − 1.51638i −0.652036 0.758188i \(-0.726085\pi\)
0.652036 0.758188i \(-0.273915\pi\)
\(954\) −2.82327e7 −1.00434
\(955\) 0 0
\(956\) 585120. 0.0207062
\(957\) 0 0
\(958\) − 8.87234e6i − 0.312338i
\(959\) −6.50862e6 −0.228530
\(960\) 0 0
\(961\) 4.62626e7 1.61593
\(962\) − 7.42654e6i − 0.258731i
\(963\) − 3.73054e7i − 1.29630i
\(964\) −6.09475e6 −0.211234
\(965\) 0 0
\(966\) 0 0
\(967\) 3.00251e7i 1.03257i 0.856417 + 0.516284i \(0.172685\pi\)
−0.856417 + 0.516284i \(0.827315\pi\)
\(968\) 6.30726e6i 0.216348i
\(969\) 0 0
\(970\) 0 0
\(971\) −4.00864e7 −1.36442 −0.682211 0.731155i \(-0.738982\pi\)
−0.682211 + 0.731155i \(0.738982\pi\)
\(972\) 0 0
\(973\) 9.82192e6i 0.332594i
\(974\) −2.45707e7 −0.829890
\(975\) 0 0
\(976\) −8.41779e6 −0.282861
\(977\) − 5.12151e7i − 1.71657i −0.513174 0.858284i \(-0.671531\pi\)
0.513174 0.858284i \(-0.328469\pi\)
\(978\) 0 0
\(979\) 2.35425e7 0.785047
\(980\) 0 0
\(981\) 4.34051e7 1.44002
\(982\) 2.57794e7i 0.853090i
\(983\) 1.82382e7i 0.602004i 0.953624 + 0.301002i \(0.0973211\pi\)
−0.953624 + 0.301002i \(0.902679\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −1.09513e7 −0.358736
\(987\) 0 0
\(988\) 210912.i 0.00687399i
\(989\) 9.29840e6 0.302286
\(990\) 0 0
\(991\) −3.24103e7 −1.04833 −0.524166 0.851616i \(-0.675623\pi\)
−0.524166 + 0.851616i \(0.675623\pi\)
\(992\) − 8.86170e6i − 0.285915i
\(993\) 0 0
\(994\) 3.43686e7 1.10330
\(995\) 0 0
\(996\) 0 0
\(997\) 2.07867e7i 0.662289i 0.943580 + 0.331145i \(0.107435\pi\)
−0.943580 + 0.331145i \(0.892565\pi\)
\(998\) − 1.70335e7i − 0.541350i
\(999\) 0 0
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 650.6.b.a.599.2 2
5.2 odd 4 26.6.a.a.1.1 1
5.3 odd 4 650.6.a.a.1.1 1
5.4 even 2 inner 650.6.b.a.599.1 2
15.2 even 4 234.6.a.g.1.1 1
20.7 even 4 208.6.a.b.1.1 1
40.27 even 4 832.6.a.e.1.1 1
40.37 odd 4 832.6.a.d.1.1 1
65.12 odd 4 338.6.a.d.1.1 1
65.47 even 4 338.6.b.a.337.1 2
65.57 even 4 338.6.b.a.337.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.6.a.a.1.1 1 5.2 odd 4
208.6.a.b.1.1 1 20.7 even 4
234.6.a.g.1.1 1 15.2 even 4
338.6.a.d.1.1 1 65.12 odd 4
338.6.b.a.337.1 2 65.47 even 4
338.6.b.a.337.2 2 65.57 even 4
650.6.a.a.1.1 1 5.3 odd 4
650.6.b.a.599.1 2 5.4 even 2 inner
650.6.b.a.599.2 2 1.1 even 1 trivial
832.6.a.d.1.1 1 40.37 odd 4
832.6.a.e.1.1 1 40.27 even 4