Properties

Label 725.2.b.e.349.4
Level $725$
Weight $2$
Character 725.349
Analytic conductor $5.789$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(349,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.4
Root \(1.45161 - 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 725.349
Dual form 725.2.b.e.349.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.311108i q^{2} -2.90321i q^{3} +1.90321 q^{4} +0.903212 q^{6} -0.903212i q^{7} +1.21432i q^{8} -5.42864 q^{9} +O(q^{10})\) \(q+0.311108i q^{2} -2.90321i q^{3} +1.90321 q^{4} +0.903212 q^{6} -0.903212i q^{7} +1.21432i q^{8} -5.42864 q^{9} -1.52543 q^{11} -5.52543i q^{12} +0.622216i q^{13} +0.280996 q^{14} +3.42864 q^{16} -7.95407i q^{17} -1.68889i q^{18} +1.09679 q^{19} -2.62222 q^{21} -0.474572i q^{22} -7.52543i q^{23} +3.52543 q^{24} -0.193576 q^{26} +7.05086i q^{27} -1.71900i q^{28} +1.00000 q^{29} -6.90321 q^{31} +3.49532i q^{32} +4.42864i q^{33} +2.47457 q^{34} -10.3319 q^{36} +3.95407i q^{37} +0.341219i q^{38} +1.80642 q^{39} +3.67307 q^{41} -0.815792i q^{42} +10.5161i q^{43} -2.90321 q^{44} +2.34122 q^{46} +6.90321i q^{47} -9.95407i q^{48} +6.18421 q^{49} -23.0923 q^{51} +1.18421i q^{52} -6.42864i q^{53} -2.19358 q^{54} +1.09679 q^{56} -3.18421i q^{57} +0.311108i q^{58} +1.67307 q^{59} -1.86665 q^{61} -2.14764i q^{62} +4.90321i q^{63} +5.76986 q^{64} -1.37778 q^{66} +11.5254i q^{67} -15.1383i q^{68} -21.8479 q^{69} +13.6731 q^{71} -6.59210i q^{72} -10.1891i q^{73} -1.23014 q^{74} +2.08742 q^{76} +1.37778i q^{77} +0.561993i q^{78} -9.13828 q^{79} +4.18421 q^{81} +1.14272i q^{82} -10.7096i q^{83} -4.99063 q^{84} -3.27163 q^{86} -2.90321i q^{87} -1.85236i q^{88} +7.80642 q^{89} +0.561993 q^{91} -14.3225i q^{92} +20.0415i q^{93} -2.14764 q^{94} +10.1476 q^{96} -4.08742i q^{97} +1.92396i q^{98} +8.28100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9} + 4 q^{11} - 12 q^{14} - 6 q^{16} + 20 q^{19} - 16 q^{21} + 8 q^{24} - 28 q^{26} + 6 q^{29} - 28 q^{31} + 28 q^{34} - 22 q^{36} - 16 q^{39} - 4 q^{41} - 4 q^{44} + 28 q^{46} + 10 q^{49} - 32 q^{51} - 40 q^{54} + 20 q^{56} - 16 q^{59} - 12 q^{61} + 22 q^{64} - 8 q^{66} - 24 q^{69} + 56 q^{71} - 20 q^{74} - 28 q^{76} + 12 q^{79} - 2 q^{81} + 24 q^{84} + 48 q^{86} + 20 q^{89} - 24 q^{91} + 48 q^{96} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).

\(n\) \(176\) \(552\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.311108i 0.219986i 0.993932 + 0.109993i \(0.0350829\pi\)
−0.993932 + 0.109993i \(0.964917\pi\)
\(3\) − 2.90321i − 1.67617i −0.545540 0.838085i \(-0.683675\pi\)
0.545540 0.838085i \(-0.316325\pi\)
\(4\) 1.90321 0.951606
\(5\) 0 0
\(6\) 0.903212 0.368735
\(7\) − 0.903212i − 0.341382i −0.985325 0.170691i \(-0.945400\pi\)
0.985325 0.170691i \(-0.0546000\pi\)
\(8\) 1.21432i 0.429327i
\(9\) −5.42864 −1.80955
\(10\) 0 0
\(11\) −1.52543 −0.459934 −0.229967 0.973198i \(-0.573862\pi\)
−0.229967 + 0.973198i \(0.573862\pi\)
\(12\) − 5.52543i − 1.59505i
\(13\) 0.622216i 0.172572i 0.996270 + 0.0862858i \(0.0274998\pi\)
−0.996270 + 0.0862858i \(0.972500\pi\)
\(14\) 0.280996 0.0750994
\(15\) 0 0
\(16\) 3.42864 0.857160
\(17\) − 7.95407i − 1.92914i −0.263819 0.964572i \(-0.584982\pi\)
0.263819 0.964572i \(-0.415018\pi\)
\(18\) − 1.68889i − 0.398076i
\(19\) 1.09679 0.251620 0.125810 0.992054i \(-0.459847\pi\)
0.125810 + 0.992054i \(0.459847\pi\)
\(20\) 0 0
\(21\) −2.62222 −0.572214
\(22\) − 0.474572i − 0.101179i
\(23\) − 7.52543i − 1.56916i −0.620028 0.784580i \(-0.712879\pi\)
0.620028 0.784580i \(-0.287121\pi\)
\(24\) 3.52543 0.719625
\(25\) 0 0
\(26\) −0.193576 −0.0379634
\(27\) 7.05086i 1.35694i
\(28\) − 1.71900i − 0.324861i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −6.90321 −1.23985 −0.619927 0.784660i \(-0.712838\pi\)
−0.619927 + 0.784660i \(0.712838\pi\)
\(32\) 3.49532i 0.617890i
\(33\) 4.42864i 0.770927i
\(34\) 2.47457 0.424386
\(35\) 0 0
\(36\) −10.3319 −1.72198
\(37\) 3.95407i 0.650045i 0.945706 + 0.325022i \(0.105372\pi\)
−0.945706 + 0.325022i \(0.894628\pi\)
\(38\) 0.341219i 0.0553531i
\(39\) 1.80642 0.289259
\(40\) 0 0
\(41\) 3.67307 0.573637 0.286819 0.957985i \(-0.407402\pi\)
0.286819 + 0.957985i \(0.407402\pi\)
\(42\) − 0.815792i − 0.125879i
\(43\) 10.5161i 1.60368i 0.597536 + 0.801842i \(0.296146\pi\)
−0.597536 + 0.801842i \(0.703854\pi\)
\(44\) −2.90321 −0.437676
\(45\) 0 0
\(46\) 2.34122 0.345194
\(47\) 6.90321i 1.00694i 0.864014 + 0.503468i \(0.167943\pi\)
−0.864014 + 0.503468i \(0.832057\pi\)
\(48\) − 9.95407i − 1.43675i
\(49\) 6.18421 0.883458
\(50\) 0 0
\(51\) −23.0923 −3.23357
\(52\) 1.18421i 0.164220i
\(53\) − 6.42864i − 0.883042i −0.897251 0.441521i \(-0.854439\pi\)
0.897251 0.441521i \(-0.145561\pi\)
\(54\) −2.19358 −0.298508
\(55\) 0 0
\(56\) 1.09679 0.146564
\(57\) − 3.18421i − 0.421759i
\(58\) 0.311108i 0.0408505i
\(59\) 1.67307 0.217815 0.108908 0.994052i \(-0.465265\pi\)
0.108908 + 0.994052i \(0.465265\pi\)
\(60\) 0 0
\(61\) −1.86665 −0.239000 −0.119500 0.992834i \(-0.538129\pi\)
−0.119500 + 0.992834i \(0.538129\pi\)
\(62\) − 2.14764i − 0.272751i
\(63\) 4.90321i 0.617747i
\(64\) 5.76986 0.721232
\(65\) 0 0
\(66\) −1.37778 −0.169594
\(67\) 11.5254i 1.40806i 0.710173 + 0.704028i \(0.248617\pi\)
−0.710173 + 0.704028i \(0.751383\pi\)
\(68\) − 15.1383i − 1.83579i
\(69\) −21.8479 −2.63018
\(70\) 0 0
\(71\) 13.6731 1.62269 0.811347 0.584564i \(-0.198734\pi\)
0.811347 + 0.584564i \(0.198734\pi\)
\(72\) − 6.59210i − 0.776887i
\(73\) − 10.1891i − 1.19255i −0.802781 0.596274i \(-0.796647\pi\)
0.802781 0.596274i \(-0.203353\pi\)
\(74\) −1.23014 −0.143001
\(75\) 0 0
\(76\) 2.08742 0.239444
\(77\) 1.37778i 0.157013i
\(78\) 0.561993i 0.0636331i
\(79\) −9.13828 −1.02814 −0.514068 0.857749i \(-0.671862\pi\)
−0.514068 + 0.857749i \(0.671862\pi\)
\(80\) 0 0
\(81\) 4.18421 0.464912
\(82\) 1.14272i 0.126192i
\(83\) − 10.7096i − 1.17554i −0.809030 0.587768i \(-0.800007\pi\)
0.809030 0.587768i \(-0.199993\pi\)
\(84\) −4.99063 −0.544523
\(85\) 0 0
\(86\) −3.27163 −0.352789
\(87\) − 2.90321i − 0.311257i
\(88\) − 1.85236i − 0.197462i
\(89\) 7.80642 0.827479 0.413740 0.910395i \(-0.364222\pi\)
0.413740 + 0.910395i \(0.364222\pi\)
\(90\) 0 0
\(91\) 0.561993 0.0589128
\(92\) − 14.3225i − 1.49322i
\(93\) 20.0415i 2.07821i
\(94\) −2.14764 −0.221512
\(95\) 0 0
\(96\) 10.1476 1.03569
\(97\) − 4.08742i − 0.415015i −0.978233 0.207507i \(-0.933465\pi\)
0.978233 0.207507i \(-0.0665351\pi\)
\(98\) 1.92396i 0.194349i
\(99\) 8.28100 0.832271
\(100\) 0 0
\(101\) 13.9081 1.38391 0.691956 0.721940i \(-0.256749\pi\)
0.691956 + 0.721940i \(0.256749\pi\)
\(102\) − 7.18421i − 0.711343i
\(103\) 12.9447i 1.27548i 0.770252 + 0.637740i \(0.220130\pi\)
−0.770252 + 0.637740i \(0.779870\pi\)
\(104\) −0.755569 −0.0740896
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) 11.0049i 1.06389i 0.846780 + 0.531943i \(0.178538\pi\)
−0.846780 + 0.531943i \(0.821462\pi\)
\(108\) 13.4193i 1.29127i
\(109\) 18.0415 1.72806 0.864031 0.503439i \(-0.167932\pi\)
0.864031 + 0.503439i \(0.167932\pi\)
\(110\) 0 0
\(111\) 11.4795 1.08959
\(112\) − 3.09679i − 0.292619i
\(113\) 10.2810i 0.967155i 0.875302 + 0.483577i \(0.160663\pi\)
−0.875302 + 0.483577i \(0.839337\pi\)
\(114\) 0.990632 0.0927812
\(115\) 0 0
\(116\) 1.90321 0.176709
\(117\) − 3.37778i − 0.312276i
\(118\) 0.520505i 0.0479164i
\(119\) −7.18421 −0.658575
\(120\) 0 0
\(121\) −8.67307 −0.788461
\(122\) − 0.580728i − 0.0525767i
\(123\) − 10.6637i − 0.961514i
\(124\) −13.1383 −1.17985
\(125\) 0 0
\(126\) −1.52543 −0.135896
\(127\) 6.22077i 0.552004i 0.961157 + 0.276002i \(0.0890097\pi\)
−0.961157 + 0.276002i \(0.910990\pi\)
\(128\) 8.78568i 0.776552i
\(129\) 30.5303 2.68805
\(130\) 0 0
\(131\) −11.7605 −1.02752 −0.513759 0.857934i \(-0.671748\pi\)
−0.513759 + 0.857934i \(0.671748\pi\)
\(132\) 8.42864i 0.733619i
\(133\) − 0.990632i − 0.0858987i
\(134\) −3.58565 −0.309753
\(135\) 0 0
\(136\) 9.65878 0.828234
\(137\) − 3.56691i − 0.304742i −0.988323 0.152371i \(-0.951309\pi\)
0.988323 0.152371i \(-0.0486909\pi\)
\(138\) − 6.79706i − 0.578604i
\(139\) 8.56199 0.726219 0.363109 0.931747i \(-0.381715\pi\)
0.363109 + 0.931747i \(0.381715\pi\)
\(140\) 0 0
\(141\) 20.0415 1.68780
\(142\) 4.25380i 0.356971i
\(143\) − 0.949145i − 0.0793715i
\(144\) −18.6128 −1.55107
\(145\) 0 0
\(146\) 3.16992 0.262344
\(147\) − 17.9541i − 1.48083i
\(148\) 7.52543i 0.618586i
\(149\) 5.61285 0.459822 0.229911 0.973212i \(-0.426156\pi\)
0.229911 + 0.973212i \(0.426156\pi\)
\(150\) 0 0
\(151\) 10.7971 0.878652 0.439326 0.898328i \(-0.355217\pi\)
0.439326 + 0.898328i \(0.355217\pi\)
\(152\) 1.33185i 0.108027i
\(153\) 43.1798i 3.49088i
\(154\) −0.428639 −0.0345408
\(155\) 0 0
\(156\) 3.43801 0.275261
\(157\) 2.28100i 0.182043i 0.995849 + 0.0910217i \(0.0290133\pi\)
−0.995849 + 0.0910217i \(0.970987\pi\)
\(158\) − 2.84299i − 0.226176i
\(159\) −18.6637 −1.48013
\(160\) 0 0
\(161\) −6.79706 −0.535683
\(162\) 1.30174i 0.102274i
\(163\) − 16.3225i − 1.27848i −0.769009 0.639238i \(-0.779250\pi\)
0.769009 0.639238i \(-0.220750\pi\)
\(164\) 6.99063 0.545877
\(165\) 0 0
\(166\) 3.33185 0.258602
\(167\) − 4.76986i − 0.369103i −0.982823 0.184551i \(-0.940917\pi\)
0.982823 0.184551i \(-0.0590832\pi\)
\(168\) − 3.18421i − 0.245667i
\(169\) 12.6128 0.970219
\(170\) 0 0
\(171\) −5.95407 −0.455319
\(172\) 20.0143i 1.52608i
\(173\) − 4.23506i − 0.321986i −0.986956 0.160993i \(-0.948530\pi\)
0.986956 0.160993i \(-0.0514696\pi\)
\(174\) 0.903212 0.0684723
\(175\) 0 0
\(176\) −5.23014 −0.394237
\(177\) − 4.85728i − 0.365095i
\(178\) 2.42864i 0.182034i
\(179\) −9.71456 −0.726100 −0.363050 0.931770i \(-0.618265\pi\)
−0.363050 + 0.931770i \(0.618265\pi\)
\(180\) 0 0
\(181\) 0.326929 0.0243005 0.0121502 0.999926i \(-0.496132\pi\)
0.0121502 + 0.999926i \(0.496132\pi\)
\(182\) 0.174840i 0.0129600i
\(183\) 5.41927i 0.400604i
\(184\) 9.13828 0.673683
\(185\) 0 0
\(186\) −6.23506 −0.457177
\(187\) 12.1334i 0.887279i
\(188\) 13.1383i 0.958207i
\(189\) 6.36842 0.463234
\(190\) 0 0
\(191\) −14.9447 −1.08136 −0.540680 0.841228i \(-0.681833\pi\)
−0.540680 + 0.841228i \(0.681833\pi\)
\(192\) − 16.7511i − 1.20891i
\(193\) 14.1476i 1.01837i 0.860657 + 0.509185i \(0.170053\pi\)
−0.860657 + 0.509185i \(0.829947\pi\)
\(194\) 1.27163 0.0912976
\(195\) 0 0
\(196\) 11.7699 0.840704
\(197\) − 5.70471i − 0.406444i −0.979133 0.203222i \(-0.934859\pi\)
0.979133 0.203222i \(-0.0651413\pi\)
\(198\) 2.57628i 0.183088i
\(199\) 22.1432 1.56969 0.784845 0.619692i \(-0.212743\pi\)
0.784845 + 0.619692i \(0.212743\pi\)
\(200\) 0 0
\(201\) 33.4608 2.36014
\(202\) 4.32693i 0.304442i
\(203\) − 0.903212i − 0.0633930i
\(204\) −43.9496 −3.07709
\(205\) 0 0
\(206\) −4.02720 −0.280588
\(207\) 40.8528i 2.83947i
\(208\) 2.13335i 0.147921i
\(209\) −1.67307 −0.115729
\(210\) 0 0
\(211\) −20.8430 −1.43489 −0.717445 0.696615i \(-0.754689\pi\)
−0.717445 + 0.696615i \(0.754689\pi\)
\(212\) − 12.2351i − 0.840308i
\(213\) − 39.6958i − 2.71991i
\(214\) −3.42372 −0.234040
\(215\) 0 0
\(216\) −8.56199 −0.582570
\(217\) 6.23506i 0.423264i
\(218\) 5.61285i 0.380150i
\(219\) −29.5812 −1.99891
\(220\) 0 0
\(221\) 4.94914 0.332916
\(222\) 3.57136i 0.239694i
\(223\) − 9.03657i − 0.605133i −0.953128 0.302567i \(-0.902157\pi\)
0.953128 0.302567i \(-0.0978435\pi\)
\(224\) 3.15701 0.210937
\(225\) 0 0
\(226\) −3.19850 −0.212761
\(227\) 19.4050i 1.28795i 0.765045 + 0.643977i \(0.222717\pi\)
−0.765045 + 0.643977i \(0.777283\pi\)
\(228\) − 6.06022i − 0.401348i
\(229\) 25.6128 1.69254 0.846272 0.532751i \(-0.178842\pi\)
0.846272 + 0.532751i \(0.178842\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) 1.21432i 0.0797240i
\(233\) 3.12399i 0.204659i 0.994751 + 0.102330i \(0.0326296\pi\)
−0.994751 + 0.102330i \(0.967370\pi\)
\(234\) 1.05086 0.0686965
\(235\) 0 0
\(236\) 3.18421 0.207274
\(237\) 26.5303i 1.72333i
\(238\) − 2.23506i − 0.144878i
\(239\) −13.9398 −0.901689 −0.450845 0.892602i \(-0.648877\pi\)
−0.450845 + 0.892602i \(0.648877\pi\)
\(240\) 0 0
\(241\) −18.4701 −1.18977 −0.594883 0.803813i \(-0.702802\pi\)
−0.594883 + 0.803813i \(0.702802\pi\)
\(242\) − 2.69826i − 0.173451i
\(243\) 9.00492i 0.577666i
\(244\) −3.55262 −0.227433
\(245\) 0 0
\(246\) 3.31756 0.211520
\(247\) 0.682439i 0.0434225i
\(248\) − 8.38271i − 0.532302i
\(249\) −31.0923 −1.97040
\(250\) 0 0
\(251\) −13.7921 −0.870552 −0.435276 0.900297i \(-0.643349\pi\)
−0.435276 + 0.900297i \(0.643349\pi\)
\(252\) 9.33185i 0.587851i
\(253\) 11.4795i 0.721710i
\(254\) −1.93533 −0.121433
\(255\) 0 0
\(256\) 8.80642 0.550401
\(257\) − 1.47949i − 0.0922883i −0.998935 0.0461442i \(-0.985307\pi\)
0.998935 0.0461442i \(-0.0146934\pi\)
\(258\) 9.49823i 0.591334i
\(259\) 3.57136 0.221914
\(260\) 0 0
\(261\) −5.42864 −0.336024
\(262\) − 3.65878i − 0.226040i
\(263\) 0.442930i 0.0273122i 0.999907 + 0.0136561i \(0.00434701\pi\)
−0.999907 + 0.0136561i \(0.995653\pi\)
\(264\) −5.37778 −0.330980
\(265\) 0 0
\(266\) 0.308193 0.0188965
\(267\) − 22.6637i − 1.38700i
\(268\) 21.9353i 1.33991i
\(269\) −3.93978 −0.240212 −0.120106 0.992761i \(-0.538324\pi\)
−0.120106 + 0.992761i \(0.538324\pi\)
\(270\) 0 0
\(271\) 6.20787 0.377101 0.188551 0.982063i \(-0.439621\pi\)
0.188551 + 0.982063i \(0.439621\pi\)
\(272\) − 27.2716i − 1.65359i
\(273\) − 1.63158i − 0.0987479i
\(274\) 1.10970 0.0670391
\(275\) 0 0
\(276\) −41.5812 −2.50289
\(277\) 5.57136i 0.334751i 0.985893 + 0.167375i \(0.0535292\pi\)
−0.985893 + 0.167375i \(0.946471\pi\)
\(278\) 2.66370i 0.159758i
\(279\) 37.4750 2.24357
\(280\) 0 0
\(281\) 6.69535 0.399411 0.199705 0.979856i \(-0.436001\pi\)
0.199705 + 0.979856i \(0.436001\pi\)
\(282\) 6.23506i 0.371293i
\(283\) − 25.8020i − 1.53377i −0.641785 0.766884i \(-0.721806\pi\)
0.641785 0.766884i \(-0.278194\pi\)
\(284\) 26.0228 1.54417
\(285\) 0 0
\(286\) 0.295286 0.0174607
\(287\) − 3.31756i − 0.195829i
\(288\) − 18.9748i − 1.11810i
\(289\) −46.2672 −2.72160
\(290\) 0 0
\(291\) −11.8666 −0.695635
\(292\) − 19.3921i − 1.13484i
\(293\) 18.8430i 1.10082i 0.834895 + 0.550410i \(0.185528\pi\)
−0.834895 + 0.550410i \(0.814472\pi\)
\(294\) 5.58565 0.325762
\(295\) 0 0
\(296\) −4.80150 −0.279082
\(297\) − 10.7556i − 0.624101i
\(298\) 1.74620i 0.101155i
\(299\) 4.68244 0.270792
\(300\) 0 0
\(301\) 9.49823 0.547469
\(302\) 3.35905i 0.193292i
\(303\) − 40.3783i − 2.31967i
\(304\) 3.76049 0.215679
\(305\) 0 0
\(306\) −13.4336 −0.767946
\(307\) − 1.65878i − 0.0946716i −0.998879 0.0473358i \(-0.984927\pi\)
0.998879 0.0473358i \(-0.0150731\pi\)
\(308\) 2.62222i 0.149415i
\(309\) 37.5812 2.13792
\(310\) 0 0
\(311\) 21.3002 1.20782 0.603912 0.797051i \(-0.293608\pi\)
0.603912 + 0.797051i \(0.293608\pi\)
\(312\) 2.19358i 0.124187i
\(313\) − 8.62222i − 0.487356i −0.969856 0.243678i \(-0.921646\pi\)
0.969856 0.243678i \(-0.0783541\pi\)
\(314\) −0.709636 −0.0400471
\(315\) 0 0
\(316\) −17.3921 −0.978381
\(317\) − 27.5955i − 1.54992i −0.632012 0.774959i \(-0.717771\pi\)
0.632012 0.774959i \(-0.282229\pi\)
\(318\) − 5.80642i − 0.325608i
\(319\) −1.52543 −0.0854075
\(320\) 0 0
\(321\) 31.9496 1.78325
\(322\) − 2.11462i − 0.117843i
\(323\) − 8.72393i − 0.485412i
\(324\) 7.96343 0.442413
\(325\) 0 0
\(326\) 5.07805 0.281247
\(327\) − 52.3783i − 2.89652i
\(328\) 4.46028i 0.246278i
\(329\) 6.23506 0.343750
\(330\) 0 0
\(331\) 16.9131 0.929626 0.464813 0.885409i \(-0.346122\pi\)
0.464813 + 0.885409i \(0.346122\pi\)
\(332\) − 20.3827i − 1.11865i
\(333\) − 21.4652i − 1.17629i
\(334\) 1.48394 0.0811976
\(335\) 0 0
\(336\) −8.99063 −0.490479
\(337\) 11.9956i 0.653439i 0.945121 + 0.326720i \(0.105943\pi\)
−0.945121 + 0.326720i \(0.894057\pi\)
\(338\) 3.92396i 0.213435i
\(339\) 29.8479 1.62112
\(340\) 0 0
\(341\) 10.5303 0.570250
\(342\) − 1.85236i − 0.100164i
\(343\) − 11.9081i − 0.642979i
\(344\) −12.7699 −0.688505
\(345\) 0 0
\(346\) 1.31756 0.0708325
\(347\) 6.14764i 0.330023i 0.986292 + 0.165011i \(0.0527661\pi\)
−0.986292 + 0.165011i \(0.947234\pi\)
\(348\) − 5.52543i − 0.296194i
\(349\) 7.12399 0.381338 0.190669 0.981654i \(-0.438934\pi\)
0.190669 + 0.981654i \(0.438934\pi\)
\(350\) 0 0
\(351\) −4.38715 −0.234169
\(352\) − 5.33185i − 0.284189i
\(353\) 16.9175i 0.900428i 0.892921 + 0.450214i \(0.148652\pi\)
−0.892921 + 0.450214i \(0.851348\pi\)
\(354\) 1.51114 0.0803160
\(355\) 0 0
\(356\) 14.8573 0.787434
\(357\) 20.8573i 1.10388i
\(358\) − 3.02227i − 0.159732i
\(359\) −36.7096 −1.93746 −0.968730 0.248116i \(-0.920188\pi\)
−0.968730 + 0.248116i \(0.920188\pi\)
\(360\) 0 0
\(361\) −17.7971 −0.936687
\(362\) 0.101710i 0.00534577i
\(363\) 25.1798i 1.32159i
\(364\) 1.06959 0.0560618
\(365\) 0 0
\(366\) −1.68598 −0.0881275
\(367\) 8.41435i 0.439225i 0.975587 + 0.219613i \(0.0704794\pi\)
−0.975587 + 0.219613i \(0.929521\pi\)
\(368\) − 25.8020i − 1.34502i
\(369\) −19.9398 −1.03802
\(370\) 0 0
\(371\) −5.80642 −0.301455
\(372\) 38.1432i 1.97763i
\(373\) 8.66370i 0.448590i 0.974521 + 0.224295i \(0.0720078\pi\)
−0.974521 + 0.224295i \(0.927992\pi\)
\(374\) −3.77478 −0.195189
\(375\) 0 0
\(376\) −8.38271 −0.432305
\(377\) 0.622216i 0.0320457i
\(378\) 1.98126i 0.101905i
\(379\) 2.76986 0.142278 0.0711390 0.997466i \(-0.477337\pi\)
0.0711390 + 0.997466i \(0.477337\pi\)
\(380\) 0 0
\(381\) 18.0602 0.925253
\(382\) − 4.64941i − 0.237885i
\(383\) − 1.67752i − 0.0857171i −0.999081 0.0428585i \(-0.986354\pi\)
0.999081 0.0428585i \(-0.0136465\pi\)
\(384\) 25.5067 1.30163
\(385\) 0 0
\(386\) −4.40144 −0.224028
\(387\) − 57.0879i − 2.90194i
\(388\) − 7.77923i − 0.394930i
\(389\) 5.77478 0.292793 0.146397 0.989226i \(-0.453232\pi\)
0.146397 + 0.989226i \(0.453232\pi\)
\(390\) 0 0
\(391\) −59.8578 −3.02714
\(392\) 7.50961i 0.379292i
\(393\) 34.1432i 1.72230i
\(394\) 1.77478 0.0894122
\(395\) 0 0
\(396\) 15.7605 0.791994
\(397\) 29.9081i 1.50105i 0.660844 + 0.750523i \(0.270198\pi\)
−0.660844 + 0.750523i \(0.729802\pi\)
\(398\) 6.88892i 0.345310i
\(399\) −2.87601 −0.143981
\(400\) 0 0
\(401\) 8.53035 0.425985 0.212993 0.977054i \(-0.431679\pi\)
0.212993 + 0.977054i \(0.431679\pi\)
\(402\) 10.4099i 0.519199i
\(403\) − 4.29529i − 0.213963i
\(404\) 26.4701 1.31694
\(405\) 0 0
\(406\) 0.280996 0.0139456
\(407\) − 6.03164i − 0.298977i
\(408\) − 28.0415i − 1.38826i
\(409\) −5.09234 −0.251800 −0.125900 0.992043i \(-0.540182\pi\)
−0.125900 + 0.992043i \(0.540182\pi\)
\(410\) 0 0
\(411\) −10.3555 −0.510800
\(412\) 24.6365i 1.21375i
\(413\) − 1.51114i − 0.0743582i
\(414\) −12.7096 −0.624645
\(415\) 0 0
\(416\) −2.17484 −0.106630
\(417\) − 24.8573i − 1.21727i
\(418\) − 0.520505i − 0.0254588i
\(419\) 24.3368 1.18893 0.594465 0.804122i \(-0.297364\pi\)
0.594465 + 0.804122i \(0.297364\pi\)
\(420\) 0 0
\(421\) 24.5018 1.19414 0.597072 0.802188i \(-0.296331\pi\)
0.597072 + 0.802188i \(0.296331\pi\)
\(422\) − 6.48442i − 0.315656i
\(423\) − 37.4750i − 1.82210i
\(424\) 7.80642 0.379113
\(425\) 0 0
\(426\) 12.3497 0.598344
\(427\) 1.68598i 0.0815902i
\(428\) 20.9447i 1.01240i
\(429\) −2.75557 −0.133040
\(430\) 0 0
\(431\) 4.26671 0.205520 0.102760 0.994706i \(-0.467233\pi\)
0.102760 + 0.994706i \(0.467233\pi\)
\(432\) 24.1748i 1.16311i
\(433\) − 27.0049i − 1.29777i −0.760885 0.648887i \(-0.775235\pi\)
0.760885 0.648887i \(-0.224765\pi\)
\(434\) −1.93978 −0.0931123
\(435\) 0 0
\(436\) 34.3368 1.64443
\(437\) − 8.25380i − 0.394833i
\(438\) − 9.20294i − 0.439734i
\(439\) 2.03164 0.0969650 0.0484825 0.998824i \(-0.484561\pi\)
0.0484825 + 0.998824i \(0.484561\pi\)
\(440\) 0 0
\(441\) −33.5718 −1.59866
\(442\) 1.53972i 0.0732369i
\(443\) 3.46520i 0.164637i 0.996606 + 0.0823184i \(0.0262324\pi\)
−0.996606 + 0.0823184i \(0.973768\pi\)
\(444\) 21.8479 1.03686
\(445\) 0 0
\(446\) 2.81135 0.133121
\(447\) − 16.2953i − 0.770741i
\(448\) − 5.21141i − 0.246216i
\(449\) −37.3590 −1.76308 −0.881541 0.472107i \(-0.843494\pi\)
−0.881541 + 0.472107i \(0.843494\pi\)
\(450\) 0 0
\(451\) −5.60300 −0.263835
\(452\) 19.5669i 0.920350i
\(453\) − 31.3461i − 1.47277i
\(454\) −6.03704 −0.283332
\(455\) 0 0
\(456\) 3.86665 0.181072
\(457\) 13.4509i 0.629207i 0.949223 + 0.314604i \(0.101872\pi\)
−0.949223 + 0.314604i \(0.898128\pi\)
\(458\) 7.96836i 0.372337i
\(459\) 56.0830 2.61773
\(460\) 0 0
\(461\) −16.2766 −0.758075 −0.379037 0.925381i \(-0.623745\pi\)
−0.379037 + 0.925381i \(0.623745\pi\)
\(462\) 1.24443i 0.0578962i
\(463\) 30.3926i 1.41246i 0.707982 + 0.706231i \(0.249606\pi\)
−0.707982 + 0.706231i \(0.750394\pi\)
\(464\) 3.42864 0.159171
\(465\) 0 0
\(466\) −0.971896 −0.0450222
\(467\) − 1.18865i − 0.0550043i −0.999622 0.0275022i \(-0.991245\pi\)
0.999622 0.0275022i \(-0.00875532\pi\)
\(468\) − 6.42864i − 0.297164i
\(469\) 10.4099 0.480685
\(470\) 0 0
\(471\) 6.62222 0.305136
\(472\) 2.03164i 0.0935139i
\(473\) − 16.0415i − 0.737588i
\(474\) −8.25380 −0.379110
\(475\) 0 0
\(476\) −13.6731 −0.626704
\(477\) 34.8988i 1.59790i
\(478\) − 4.33677i − 0.198359i
\(479\) −41.0464 −1.87546 −0.937729 0.347367i \(-0.887076\pi\)
−0.937729 + 0.347367i \(0.887076\pi\)
\(480\) 0 0
\(481\) −2.46028 −0.112179
\(482\) − 5.74620i − 0.261732i
\(483\) 19.7333i 0.897896i
\(484\) −16.5067 −0.750304
\(485\) 0 0
\(486\) −2.80150 −0.127079
\(487\) − 10.1476i − 0.459834i −0.973210 0.229917i \(-0.926155\pi\)
0.973210 0.229917i \(-0.0738454\pi\)
\(488\) − 2.26671i − 0.102609i
\(489\) −47.3876 −2.14294
\(490\) 0 0
\(491\) −29.2083 −1.31815 −0.659077 0.752075i \(-0.729053\pi\)
−0.659077 + 0.752075i \(0.729053\pi\)
\(492\) − 20.2953i − 0.914982i
\(493\) − 7.95407i − 0.358233i
\(494\) −0.212312 −0.00955237
\(495\) 0 0
\(496\) −23.6686 −1.06275
\(497\) − 12.3497i − 0.553959i
\(498\) − 9.67307i − 0.433461i
\(499\) −21.9813 −0.984017 −0.492008 0.870591i \(-0.663737\pi\)
−0.492008 + 0.870591i \(0.663737\pi\)
\(500\) 0 0
\(501\) −13.8479 −0.618679
\(502\) − 4.29084i − 0.191510i
\(503\) − 5.77923i − 0.257683i −0.991665 0.128841i \(-0.958874\pi\)
0.991665 0.128841i \(-0.0411258\pi\)
\(504\) −5.95407 −0.265215
\(505\) 0 0
\(506\) −3.57136 −0.158766
\(507\) − 36.6178i − 1.62625i
\(508\) 11.8394i 0.525291i
\(509\) −13.6543 −0.605218 −0.302609 0.953115i \(-0.597858\pi\)
−0.302609 + 0.953115i \(0.597858\pi\)
\(510\) 0 0
\(511\) −9.20294 −0.407114
\(512\) 20.3111i 0.897633i
\(513\) 7.73329i 0.341433i
\(514\) 0.460282 0.0203022
\(515\) 0 0
\(516\) 58.1057 2.55796
\(517\) − 10.5303i − 0.463124i
\(518\) 1.11108i 0.0488180i
\(519\) −12.2953 −0.539703
\(520\) 0 0
\(521\) −19.6731 −0.861893 −0.430946 0.902378i \(-0.641820\pi\)
−0.430946 + 0.902378i \(0.641820\pi\)
\(522\) − 1.68889i − 0.0739208i
\(523\) 15.1383i 0.661951i 0.943639 + 0.330975i \(0.107378\pi\)
−0.943639 + 0.330975i \(0.892622\pi\)
\(524\) −22.3827 −0.977793
\(525\) 0 0
\(526\) −0.137799 −0.00600832
\(527\) 54.9086i 2.39186i
\(528\) 15.1842i 0.660808i
\(529\) −33.6321 −1.46226
\(530\) 0 0
\(531\) −9.08250 −0.394147
\(532\) − 1.88538i − 0.0817417i
\(533\) 2.28544i 0.0989935i
\(534\) 7.05086 0.305120
\(535\) 0 0
\(536\) −13.9956 −0.604516
\(537\) 28.2034i 1.21707i
\(538\) − 1.22570i − 0.0528435i
\(539\) −9.43356 −0.406332
\(540\) 0 0
\(541\) 2.68244 0.115327 0.0576635 0.998336i \(-0.481635\pi\)
0.0576635 + 0.998336i \(0.481635\pi\)
\(542\) 1.93132i 0.0829571i
\(543\) − 0.949145i − 0.0407317i
\(544\) 27.8020 1.19200
\(545\) 0 0
\(546\) 0.507598 0.0217232
\(547\) − 15.3635i − 0.656896i −0.944522 0.328448i \(-0.893475\pi\)
0.944522 0.328448i \(-0.106525\pi\)
\(548\) − 6.78859i − 0.289994i
\(549\) 10.1334 0.432481
\(550\) 0 0
\(551\) 1.09679 0.0467247
\(552\) − 26.5303i − 1.12921i
\(553\) 8.25380i 0.350987i
\(554\) −1.73329 −0.0736406
\(555\) 0 0
\(556\) 16.2953 0.691074
\(557\) − 9.87955i − 0.418610i −0.977850 0.209305i \(-0.932880\pi\)
0.977850 0.209305i \(-0.0671202\pi\)
\(558\) 11.6588i 0.493556i
\(559\) −6.54326 −0.276750
\(560\) 0 0
\(561\) 35.2257 1.48723
\(562\) 2.08297i 0.0878650i
\(563\) 27.4938i 1.15872i 0.815070 + 0.579362i \(0.196698\pi\)
−0.815070 + 0.579362i \(0.803302\pi\)
\(564\) 38.1432 1.60612
\(565\) 0 0
\(566\) 8.02720 0.337408
\(567\) − 3.77923i − 0.158713i
\(568\) 16.6035i 0.696667i
\(569\) −17.3590 −0.727729 −0.363865 0.931452i \(-0.618543\pi\)
−0.363865 + 0.931452i \(0.618543\pi\)
\(570\) 0 0
\(571\) −25.4479 −1.06496 −0.532480 0.846443i \(-0.678740\pi\)
−0.532480 + 0.846443i \(0.678740\pi\)
\(572\) − 1.80642i − 0.0755304i
\(573\) 43.3876i 1.81254i
\(574\) 1.03212 0.0430798
\(575\) 0 0
\(576\) −31.3225 −1.30510
\(577\) − 10.6178i − 0.442024i −0.975271 0.221012i \(-0.929064\pi\)
0.975271 0.221012i \(-0.0709359\pi\)
\(578\) − 14.3941i − 0.598715i
\(579\) 41.0736 1.70696
\(580\) 0 0
\(581\) −9.67307 −0.401307
\(582\) − 3.69181i − 0.153030i
\(583\) 9.80642i 0.406141i
\(584\) 12.3729 0.511993
\(585\) 0 0
\(586\) −5.86220 −0.242165
\(587\) − 8.94470i − 0.369187i −0.982815 0.184594i \(-0.940903\pi\)
0.982815 0.184594i \(-0.0590969\pi\)
\(588\) − 34.1704i − 1.40916i
\(589\) −7.57136 −0.311972
\(590\) 0 0
\(591\) −16.5620 −0.681269
\(592\) 13.5571i 0.557192i
\(593\) − 14.1619i − 0.581561i −0.956790 0.290780i \(-0.906085\pi\)
0.956790 0.290780i \(-0.0939149\pi\)
\(594\) 3.34614 0.137294
\(595\) 0 0
\(596\) 10.6824 0.437570
\(597\) − 64.2864i − 2.63107i
\(598\) 1.45674i 0.0595707i
\(599\) 22.5575 0.921676 0.460838 0.887484i \(-0.347549\pi\)
0.460838 + 0.887484i \(0.347549\pi\)
\(600\) 0 0
\(601\) −40.6133 −1.65665 −0.828326 0.560246i \(-0.810706\pi\)
−0.828326 + 0.560246i \(0.810706\pi\)
\(602\) 2.95497i 0.120436i
\(603\) − 62.5674i − 2.54794i
\(604\) 20.5491 0.836130
\(605\) 0 0
\(606\) 12.5620 0.510296
\(607\) 13.5955i 0.551824i 0.961183 + 0.275912i \(0.0889799\pi\)
−0.961183 + 0.275912i \(0.911020\pi\)
\(608\) 3.83362i 0.155474i
\(609\) −2.62222 −0.106258
\(610\) 0 0
\(611\) −4.29529 −0.173769
\(612\) 82.1802i 3.32194i
\(613\) 42.0830i 1.69972i 0.527012 + 0.849858i \(0.323312\pi\)
−0.527012 + 0.849858i \(0.676688\pi\)
\(614\) 0.516060 0.0208265
\(615\) 0 0
\(616\) −1.67307 −0.0674099
\(617\) 33.5067i 1.34893i 0.738307 + 0.674464i \(0.235625\pi\)
−0.738307 + 0.674464i \(0.764375\pi\)
\(618\) 11.6918i 0.470313i
\(619\) 14.6780 0.589958 0.294979 0.955504i \(-0.404687\pi\)
0.294979 + 0.955504i \(0.404687\pi\)
\(620\) 0 0
\(621\) 53.0607 2.12925
\(622\) 6.62666i 0.265705i
\(623\) − 7.05086i − 0.282487i
\(624\) 6.19358 0.247941
\(625\) 0 0
\(626\) 2.68244 0.107212
\(627\) 4.85728i 0.193981i
\(628\) 4.34122i 0.173234i
\(629\) 31.4509 1.25403
\(630\) 0 0
\(631\) 11.3176 0.450545 0.225273 0.974296i \(-0.427673\pi\)
0.225273 + 0.974296i \(0.427673\pi\)
\(632\) − 11.0968i − 0.441407i
\(633\) 60.5116i 2.40512i
\(634\) 8.58517 0.340961
\(635\) 0 0
\(636\) −35.5210 −1.40850
\(637\) 3.84791i 0.152460i
\(638\) − 0.474572i − 0.0187885i
\(639\) −74.2262 −2.93634
\(640\) 0 0
\(641\) 34.8988 1.37842 0.689209 0.724562i \(-0.257958\pi\)
0.689209 + 0.724562i \(0.257958\pi\)
\(642\) 9.93978i 0.392292i
\(643\) 41.9768i 1.65540i 0.561168 + 0.827702i \(0.310352\pi\)
−0.561168 + 0.827702i \(0.689648\pi\)
\(644\) −12.9362 −0.509759
\(645\) 0 0
\(646\) 2.71408 0.106784
\(647\) 5.46520i 0.214859i 0.994213 + 0.107430i \(0.0342620\pi\)
−0.994213 + 0.107430i \(0.965738\pi\)
\(648\) 5.08097i 0.199599i
\(649\) −2.55215 −0.100181
\(650\) 0 0
\(651\) 18.1017 0.709462
\(652\) − 31.0651i − 1.21660i
\(653\) 8.76986i 0.343191i 0.985167 + 0.171596i \(0.0548922\pi\)
−0.985167 + 0.171596i \(0.945108\pi\)
\(654\) 16.2953 0.637196
\(655\) 0 0
\(656\) 12.5936 0.491699
\(657\) 55.3131i 2.15797i
\(658\) 1.93978i 0.0756204i
\(659\) −3.29036 −0.128174 −0.0640872 0.997944i \(-0.520414\pi\)
−0.0640872 + 0.997944i \(0.520414\pi\)
\(660\) 0 0
\(661\) 19.7560 0.768421 0.384211 0.923246i \(-0.374474\pi\)
0.384211 + 0.923246i \(0.374474\pi\)
\(662\) 5.26178i 0.204505i
\(663\) − 14.3684i − 0.558023i
\(664\) 13.0049 0.504689
\(665\) 0 0
\(666\) 6.67799 0.258767
\(667\) − 7.52543i − 0.291386i
\(668\) − 9.07805i − 0.351240i
\(669\) −26.2351 −1.01431
\(670\) 0 0
\(671\) 2.84743 0.109924
\(672\) − 9.16547i − 0.353566i
\(673\) − 44.3970i − 1.71138i −0.517490 0.855689i \(-0.673134\pi\)
0.517490 0.855689i \(-0.326866\pi\)
\(674\) −3.73191 −0.143748
\(675\) 0 0
\(676\) 24.0049 0.923266
\(677\) 6.09726i 0.234337i 0.993112 + 0.117168i \(0.0373817\pi\)
−0.993112 + 0.117168i \(0.962618\pi\)
\(678\) 9.28592i 0.356624i
\(679\) −3.69181 −0.141679
\(680\) 0 0
\(681\) 56.3368 2.15883
\(682\) 3.27607i 0.125447i
\(683\) − 37.9224i − 1.45106i −0.688190 0.725531i \(-0.741594\pi\)
0.688190 0.725531i \(-0.258406\pi\)
\(684\) −11.3319 −0.433284
\(685\) 0 0
\(686\) 3.70471 0.141447
\(687\) − 74.3595i − 2.83699i
\(688\) 36.0558i 1.37461i
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) 13.3145 0.506507 0.253254 0.967400i \(-0.418499\pi\)
0.253254 + 0.967400i \(0.418499\pi\)
\(692\) − 8.06022i − 0.306404i
\(693\) − 7.47949i − 0.284123i
\(694\) −1.91258 −0.0726005
\(695\) 0 0
\(696\) 3.52543 0.133631
\(697\) − 29.2159i − 1.10663i
\(698\) 2.21633i 0.0838892i
\(699\) 9.06959 0.343043
\(700\) 0 0
\(701\) −23.4893 −0.887180 −0.443590 0.896230i \(-0.646295\pi\)
−0.443590 + 0.896230i \(0.646295\pi\)
\(702\) − 1.36488i − 0.0515140i
\(703\) 4.33677i 0.163565i
\(704\) −8.80150 −0.331719
\(705\) 0 0
\(706\) −5.26317 −0.198082
\(707\) − 12.5620i − 0.472442i
\(708\) − 9.24443i − 0.347427i
\(709\) −11.6731 −0.438391 −0.219196 0.975681i \(-0.570343\pi\)
−0.219196 + 0.975681i \(0.570343\pi\)
\(710\) 0 0
\(711\) 49.6084 1.86046
\(712\) 9.47949i 0.355259i
\(713\) 51.9496i 1.94553i
\(714\) −6.48886 −0.242840
\(715\) 0 0
\(716\) −18.4889 −0.690961
\(717\) 40.4701i 1.51138i
\(718\) − 11.4207i − 0.426215i
\(719\) −29.5526 −1.10213 −0.551063 0.834463i \(-0.685778\pi\)
−0.551063 + 0.834463i \(0.685778\pi\)
\(720\) 0 0
\(721\) 11.6918 0.435426
\(722\) − 5.53680i − 0.206058i
\(723\) 53.6227i 1.99425i
\(724\) 0.622216 0.0231245
\(725\) 0 0
\(726\) −7.83362 −0.290733
\(727\) 3.88094i 0.143936i 0.997407 + 0.0719680i \(0.0229279\pi\)
−0.997407 + 0.0719680i \(0.977072\pi\)
\(728\) 0.682439i 0.0252929i
\(729\) 38.6958 1.43318
\(730\) 0 0
\(731\) 83.6454 3.09374
\(732\) 10.3140i 0.381217i
\(733\) − 14.8845i − 0.549771i −0.961477 0.274885i \(-0.911360\pi\)
0.961477 0.274885i \(-0.0886399\pi\)
\(734\) −2.61777 −0.0966236
\(735\) 0 0
\(736\) 26.3037 0.969569
\(737\) − 17.5812i − 0.647612i
\(738\) − 6.20342i − 0.228351i
\(739\) −2.24935 −0.0827438 −0.0413719 0.999144i \(-0.513173\pi\)
−0.0413719 + 0.999144i \(0.513173\pi\)
\(740\) 0 0
\(741\) 1.98126 0.0727836
\(742\) − 1.80642i − 0.0663159i
\(743\) 3.46520i 0.127126i 0.997978 + 0.0635630i \(0.0202464\pi\)
−0.997978 + 0.0635630i \(0.979754\pi\)
\(744\) −24.3368 −0.892229
\(745\) 0 0
\(746\) −2.69535 −0.0986836
\(747\) 58.1388i 2.12719i
\(748\) 23.0923i 0.844340i
\(749\) 9.93978 0.363192
\(750\) 0 0
\(751\) 3.16992 0.115672 0.0578360 0.998326i \(-0.481580\pi\)
0.0578360 + 0.998326i \(0.481580\pi\)
\(752\) 23.6686i 0.863106i
\(753\) 40.0415i 1.45919i
\(754\) −0.193576 −0.00704963
\(755\) 0 0
\(756\) 12.1204 0.440816
\(757\) − 52.0785i − 1.89283i −0.322958 0.946413i \(-0.604677\pi\)
0.322958 0.946413i \(-0.395323\pi\)
\(758\) 0.861725i 0.0312993i
\(759\) 33.3274 1.20971
\(760\) 0 0
\(761\) −14.9777 −0.542942 −0.271471 0.962447i \(-0.587510\pi\)
−0.271471 + 0.962447i \(0.587510\pi\)
\(762\) 5.61868i 0.203543i
\(763\) − 16.2953i − 0.589929i
\(764\) −28.4429 −1.02903
\(765\) 0 0
\(766\) 0.521889 0.0188566
\(767\) 1.04101i 0.0375887i
\(768\) − 25.5669i − 0.922567i
\(769\) 1.90813 0.0688091 0.0344045 0.999408i \(-0.489047\pi\)
0.0344045 + 0.999408i \(0.489047\pi\)
\(770\) 0 0
\(771\) −4.29529 −0.154691
\(772\) 26.9260i 0.969087i
\(773\) − 21.7891i − 0.783698i −0.920029 0.391849i \(-0.871835\pi\)
0.920029 0.391849i \(-0.128165\pi\)
\(774\) 17.7605 0.638388
\(775\) 0 0
\(776\) 4.96343 0.178177
\(777\) − 10.3684i − 0.371965i
\(778\) 1.79658i 0.0644105i
\(779\) 4.02858 0.144339
\(780\) 0 0
\(781\) −20.8573 −0.746332
\(782\) − 18.6222i − 0.665929i
\(783\) 7.05086i 0.251977i
\(784\) 21.2034 0.757265
\(785\) 0 0
\(786\) −10.6222 −0.378882
\(787\) − 18.1388i − 0.646577i −0.946301 0.323288i \(-0.895212\pi\)
0.946301 0.323288i \(-0.104788\pi\)
\(788\) − 10.8573i − 0.386775i
\(789\) 1.28592 0.0457799
\(790\) 0 0
\(791\) 9.28592 0.330169
\(792\) 10.0558i 0.357316i
\(793\) − 1.16146i − 0.0412445i
\(794\) −9.30465 −0.330210
\(795\) 0 0
\(796\) 42.1432 1.49373
\(797\) 2.96343i 0.104970i 0.998622 + 0.0524851i \(0.0167142\pi\)
−0.998622 + 0.0524851i \(0.983286\pi\)
\(798\) − 0.894751i − 0.0316738i
\(799\) 54.9086 1.94253
\(800\) 0 0
\(801\) −42.3783 −1.49736
\(802\) 2.65386i 0.0937110i
\(803\) 15.5428i 0.548493i
\(804\) 63.6829 2.24592
\(805\) 0 0
\(806\) 1.33630 0.0470691
\(807\) 11.4380i 0.402637i
\(808\) 16.8889i 0.594150i
\(809\) 26.2953 0.924493 0.462247 0.886751i \(-0.347044\pi\)
0.462247 + 0.886751i \(0.347044\pi\)
\(810\) 0 0
\(811\) 24.3783 0.856037 0.428018 0.903770i \(-0.359212\pi\)
0.428018 + 0.903770i \(0.359212\pi\)
\(812\) − 1.71900i − 0.0603252i
\(813\) − 18.0228i − 0.632085i
\(814\) 1.87649 0.0657710
\(815\) 0 0
\(816\) −79.1753 −2.77169
\(817\) 11.5339i 0.403520i
\(818\) − 1.58427i − 0.0553926i
\(819\) −3.05086 −0.106606
\(820\) 0 0
\(821\) 1.52987 0.0533929 0.0266965 0.999644i \(-0.491501\pi\)
0.0266965 + 0.999644i \(0.491501\pi\)
\(822\) − 3.22168i − 0.112369i
\(823\) − 46.7195i − 1.62854i −0.580487 0.814269i \(-0.697138\pi\)
0.580487 0.814269i \(-0.302862\pi\)
\(824\) −15.7190 −0.547597
\(825\) 0 0
\(826\) 0.470127 0.0163578
\(827\) − 29.6499i − 1.03103i −0.856881 0.515514i \(-0.827601\pi\)
0.856881 0.515514i \(-0.172399\pi\)
\(828\) 77.7516i 2.70205i
\(829\) 8.79706 0.305534 0.152767 0.988262i \(-0.451182\pi\)
0.152767 + 0.988262i \(0.451182\pi\)
\(830\) 0 0
\(831\) 16.1748 0.561099
\(832\) 3.59010i 0.124464i
\(833\) − 49.1896i − 1.70432i
\(834\) 7.73329 0.267782
\(835\) 0 0
\(836\) −3.18421 −0.110128
\(837\) − 48.6735i − 1.68240i
\(838\) 7.57136i 0.261548i
\(839\) −11.3319 −0.391219 −0.195609 0.980682i \(-0.562668\pi\)
−0.195609 + 0.980682i \(0.562668\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 7.62269i 0.262695i
\(843\) − 19.4380i − 0.669481i
\(844\) −39.6686 −1.36545
\(845\) 0 0
\(846\) 11.6588 0.400837
\(847\) 7.83362i 0.269166i
\(848\) − 22.0415i − 0.756908i
\(849\) −74.9086 −2.57086
\(850\) 0 0
\(851\) 29.7560 1.02002
\(852\) − 75.5496i − 2.58829i
\(853\) − 54.8845i − 1.87921i −0.342263 0.939604i \(-0.611193\pi\)
0.342263 0.939604i \(-0.388807\pi\)
\(854\) −0.524521 −0.0179487
\(855\) 0 0
\(856\) −13.3635 −0.456755
\(857\) 36.4385i 1.24471i 0.782733 + 0.622357i \(0.213825\pi\)
−0.782733 + 0.622357i \(0.786175\pi\)
\(858\) − 0.857279i − 0.0292670i
\(859\) −1.72885 −0.0589875 −0.0294938 0.999565i \(-0.509390\pi\)
−0.0294938 + 0.999565i \(0.509390\pi\)
\(860\) 0 0
\(861\) −9.63158 −0.328243
\(862\) 1.32741i 0.0452116i
\(863\) 9.40192i 0.320045i 0.987113 + 0.160023i \(0.0511567\pi\)
−0.987113 + 0.160023i \(0.948843\pi\)
\(864\) −24.6450 −0.838439
\(865\) 0 0
\(866\) 8.40144 0.285493
\(867\) 134.323i 4.56186i
\(868\) 11.8666i 0.402780i
\(869\) 13.9398 0.472875
\(870\) 0 0
\(871\) −7.17130 −0.242990
\(872\) 21.9081i 0.741903i
\(873\) 22.1891i 0.750988i
\(874\) 2.56782 0.0868579
\(875\) 0 0
\(876\) −56.2993 −1.90218
\(877\) 8.91750i 0.301123i 0.988601 + 0.150561i \(0.0481081\pi\)
−0.988601 + 0.150561i \(0.951892\pi\)
\(878\) 0.632060i 0.0213310i
\(879\) 54.7052 1.84516
\(880\) 0 0
\(881\) −42.1245 −1.41921 −0.709605 0.704600i \(-0.751126\pi\)
−0.709605 + 0.704600i \(0.751126\pi\)
\(882\) − 10.4445i − 0.351683i
\(883\) 38.4340i 1.29341i 0.762741 + 0.646704i \(0.223853\pi\)
−0.762741 + 0.646704i \(0.776147\pi\)
\(884\) 9.41927 0.316804
\(885\) 0 0
\(886\) −1.07805 −0.0362179
\(887\) − 38.6365i − 1.29729i −0.761092 0.648643i \(-0.775337\pi\)
0.761092 0.648643i \(-0.224663\pi\)
\(888\) 13.9398i 0.467788i
\(889\) 5.61868 0.188444
\(890\) 0 0
\(891\) −6.38271 −0.213829
\(892\) − 17.1985i − 0.575848i
\(893\) 7.57136i 0.253366i
\(894\) 5.06959 0.169552
\(895\) 0 0
\(896\) 7.93533 0.265101
\(897\) − 13.5941i − 0.453894i
\(898\) − 11.6227i − 0.387854i
\(899\) −6.90321 −0.230235
\(900\) 0 0
\(901\) −51.1338 −1.70351
\(902\) − 1.74314i − 0.0580402i
\(903\) − 27.5754i − 0.917651i
\(904\) −12.4844 −0.415226
\(905\) 0 0
\(906\) 9.75203 0.323989
\(907\) 0.534795i 0.0177576i 0.999961 + 0.00887880i \(0.00282625\pi\)
−0.999961 + 0.00887880i \(0.997174\pi\)
\(908\) 36.9318i 1.22562i
\(909\) −75.5022 −2.50425
\(910\) 0 0
\(911\) −23.6686 −0.784177 −0.392088 0.919928i \(-0.628247\pi\)
−0.392088 + 0.919928i \(0.628247\pi\)
\(912\) − 10.9175i − 0.361515i
\(913\) 16.3368i 0.540668i
\(914\) −4.18468 −0.138417
\(915\) 0 0
\(916\) 48.7467 1.61064
\(917\) 10.6222i 0.350776i
\(918\) 17.4479i 0.575865i
\(919\) −35.7748 −1.18010 −0.590051 0.807366i \(-0.700892\pi\)
−0.590051 + 0.807366i \(0.700892\pi\)
\(920\) 0 0
\(921\) −4.81579 −0.158686
\(922\) − 5.06376i − 0.166766i
\(923\) 8.50760i 0.280031i
\(924\) 7.61285 0.250444
\(925\) 0 0
\(926\) −9.45536 −0.310722
\(927\) − 70.2721i − 2.30804i
\(928\) 3.49532i 0.114739i
\(929\) 52.7753 1.73150 0.865750 0.500477i \(-0.166842\pi\)
0.865750 + 0.500477i \(0.166842\pi\)
\(930\) 0 0
\(931\) 6.78277 0.222296
\(932\) 5.94561i 0.194755i
\(933\) − 61.8390i − 2.02452i
\(934\) 0.369800 0.0121002
\(935\) 0 0
\(936\) 4.10171 0.134069
\(937\) 42.1245i 1.37615i 0.725641 + 0.688073i \(0.241543\pi\)
−0.725641 + 0.688073i \(0.758457\pi\)
\(938\) 3.23860i 0.105744i
\(939\) −25.0321 −0.816892
\(940\) 0 0
\(941\) −3.89829 −0.127081 −0.0635403 0.997979i \(-0.520239\pi\)
−0.0635403 + 0.997979i \(0.520239\pi\)
\(942\) 2.06022i 0.0671257i
\(943\) − 27.6414i − 0.900129i
\(944\) 5.73636 0.186703
\(945\) 0 0
\(946\) 4.99063 0.162259
\(947\) 9.56691i 0.310883i 0.987845 + 0.155441i \(0.0496800\pi\)
−0.987845 + 0.155441i \(0.950320\pi\)
\(948\) 50.4929i 1.63993i
\(949\) 6.33984 0.205800
\(950\) 0 0
\(951\) −80.1156 −2.59793
\(952\) − 8.72393i − 0.282744i
\(953\) − 27.2070i − 0.881320i −0.897674 0.440660i \(-0.854744\pi\)
0.897674 0.440660i \(-0.145256\pi\)
\(954\) −10.8573 −0.351517
\(955\) 0 0
\(956\) −26.5303 −0.858053
\(957\) 4.42864i 0.143158i
\(958\) − 12.7699i − 0.412575i
\(959\) −3.22168 −0.104033
\(960\) 0 0
\(961\) 16.6543 0.537237
\(962\) − 0.765413i − 0.0246779i
\(963\) − 59.7418i − 1.92515i
\(964\) −35.1526 −1.13219
\(965\) 0 0
\(966\) −6.13918 −0.197525
\(967\) − 16.8015i − 0.540300i −0.962818 0.270150i \(-0.912927\pi\)
0.962818 0.270150i \(-0.0870733\pi\)
\(968\) − 10.5319i − 0.338507i
\(969\) −25.3274 −0.813633
\(970\) 0 0
\(971\) −17.4465 −0.559884 −0.279942 0.960017i \(-0.590315\pi\)
−0.279942 + 0.960017i \(0.590315\pi\)
\(972\) 17.1383i 0.549710i
\(973\) − 7.73329i − 0.247918i
\(974\) 3.15701 0.101157
\(975\) 0 0
\(976\) −6.40006 −0.204861
\(977\) 32.0513i 1.02541i 0.858564 + 0.512706i \(0.171357\pi\)
−0.858564 + 0.512706i \(0.828643\pi\)
\(978\) − 14.7427i − 0.471418i
\(979\) −11.9081 −0.380586
\(980\) 0 0
\(981\) −97.9407 −3.12701
\(982\) − 9.08694i − 0.289976i
\(983\) 16.5259i 0.527094i 0.964646 + 0.263547i \(0.0848925\pi\)
−0.964646 + 0.263547i \(0.915108\pi\)
\(984\) 12.9491 0.412804
\(985\) 0 0
\(986\) 2.47457 0.0788064
\(987\) − 18.1017i − 0.576184i
\(988\) 1.29883i 0.0413211i
\(989\) 79.1378 2.51644
\(990\) 0 0
\(991\) −9.34920 −0.296987 −0.148494 0.988913i \(-0.547442\pi\)
−0.148494 + 0.988913i \(0.547442\pi\)
\(992\) − 24.1289i − 0.766094i
\(993\) − 49.1022i − 1.55821i
\(994\) 3.84208 0.121863
\(995\) 0 0
\(996\) −59.1753 −1.87504
\(997\) − 15.9956i − 0.506584i −0.967390 0.253292i \(-0.918487\pi\)
0.967390 0.253292i \(-0.0815134\pi\)
\(998\) − 6.83854i − 0.216470i
\(999\) −27.8796 −0.882070
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 725.2.b.e.349.4 6
5.2 odd 4 725.2.a.e.1.2 3
5.3 odd 4 145.2.a.c.1.2 3
5.4 even 2 inner 725.2.b.e.349.3 6
15.2 even 4 6525.2.a.be.1.2 3
15.8 even 4 1305.2.a.p.1.2 3
20.3 even 4 2320.2.a.n.1.1 3
35.13 even 4 7105.2.a.o.1.2 3
40.3 even 4 9280.2.a.br.1.3 3
40.13 odd 4 9280.2.a.bj.1.1 3
145.28 odd 4 4205.2.a.f.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.2 3 5.3 odd 4
725.2.a.e.1.2 3 5.2 odd 4
725.2.b.e.349.3 6 5.4 even 2 inner
725.2.b.e.349.4 6 1.1 even 1 trivial
1305.2.a.p.1.2 3 15.8 even 4
2320.2.a.n.1.1 3 20.3 even 4
4205.2.a.f.1.2 3 145.28 odd 4
6525.2.a.be.1.2 3 15.2 even 4
7105.2.a.o.1.2 3 35.13 even 4
9280.2.a.bj.1.1 3 40.13 odd 4
9280.2.a.br.1.3 3 40.3 even 4