Properties

Label 224.3.o.c.79.5
Level $224$
Weight $3$
Character 224.79
Analytic conductor $6.104$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [224,3,Mod(79,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.79");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.o (of order \(6\), 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: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 2x^{10} - 12x^{9} + 12x^{8} - 12x^{7} + 148x^{6} - 48x^{5} + 192x^{4} - 768x^{3} + 512x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 79.5
Root \(-0.685878 - 1.87872i\) of defining polynomial
Character \(\chi\) \(=\) 224.79
Dual form 224.3.o.c.207.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.46995 + 2.54604i) q^{3} +(-7.59793 - 4.38667i) q^{5} +(3.55324 - 6.03112i) q^{7} +(0.178469 - 0.309118i) q^{9} +O(q^{10})\) \(q+(1.46995 + 2.54604i) q^{3} +(-7.59793 - 4.38667i) q^{5} +(3.55324 - 6.03112i) q^{7} +(0.178469 - 0.309118i) q^{9} +(-1.64842 - 2.85515i) q^{11} -4.10293i q^{13} -25.7928i q^{15} +(-11.6184 - 20.1236i) q^{17} +(2.76861 - 4.79537i) q^{19} +(20.5786 + 0.181210i) q^{21} +(12.8739 + 7.43273i) q^{23} +(25.9857 + 45.0085i) q^{25} +27.5085 q^{27} -43.1747i q^{29} +(-26.6958 + 15.4128i) q^{31} +(4.84621 - 8.39389i) q^{33} +(-53.4538 + 30.2371i) q^{35} +(11.3647 + 6.56141i) q^{37} +(10.4462 - 6.03112i) q^{39} -8.56863 q^{41} +11.3076 q^{43} +(-2.71199 + 1.56577i) q^{45} +(-31.9084 - 18.4223i) q^{47} +(-23.7489 - 42.8601i) q^{49} +(34.1570 - 59.1616i) q^{51} +(-53.2193 + 30.7262i) q^{53} +28.9243i q^{55} +16.2789 q^{57} +(-31.3354 - 54.2746i) q^{59} +(34.3623 + 19.8391i) q^{61} +(-1.23018 - 2.17474i) q^{63} +(-17.9982 + 31.1738i) q^{65} +(21.3147 + 36.9182i) q^{67} +43.7031i q^{69} +107.155i q^{71} +(-18.0085 - 31.1917i) q^{73} +(-76.3955 + 132.321i) q^{75} +(-23.0770 - 0.203211i) q^{77} +(36.3029 + 20.9595i) q^{79} +(38.8301 + 67.2557i) q^{81} +87.8010 q^{83} +203.864i q^{85} +(109.924 - 63.4648i) q^{87} +(-1.04794 + 1.81508i) q^{89} +(-24.7453 - 14.5787i) q^{91} +(-78.4831 - 45.3122i) q^{93} +(-42.0713 + 24.2899i) q^{95} -40.6387 q^{97} -1.17677 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} - 8 q^{9} + 14 q^{11} - 82 q^{17} + 94 q^{19} + 116 q^{25} + 60 q^{27} + 146 q^{33} - 270 q^{35} + 120 q^{41} - 40 q^{43} - 204 q^{49} + 106 q^{51} - 372 q^{57} - 62 q^{59} - 64 q^{65} + 178 q^{67} + 54 q^{73} - 140 q^{75} + 206 q^{81} + 392 q^{83} - 26 q^{89} + 88 q^{91} - 184 q^{97} - 872 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

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

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
\(3\) 1.46995 + 2.54604i 0.489985 + 0.848678i 0.999934 0.0115263i \(-0.00366901\pi\)
−0.509949 + 0.860205i \(0.670336\pi\)
\(4\) 0 0
\(5\) −7.59793 4.38667i −1.51959 0.877333i −0.999734 0.0230765i \(-0.992654\pi\)
−0.519852 0.854257i \(-0.674013\pi\)
\(6\) 0 0
\(7\) 3.55324 6.03112i 0.507606 0.861589i
\(8\) 0 0
\(9\) 0.178469 0.309118i 0.0198299 0.0343464i
\(10\) 0 0
\(11\) −1.64842 2.85515i −0.149857 0.259559i 0.781318 0.624134i \(-0.214548\pi\)
−0.931174 + 0.364574i \(0.881215\pi\)
\(12\) 0 0
\(13\) 4.10293i 0.315610i −0.987470 0.157805i \(-0.949558\pi\)
0.987470 0.157805i \(-0.0504418\pi\)
\(14\) 0 0
\(15\) 25.7928i 1.71952i
\(16\) 0 0
\(17\) −11.6184 20.1236i −0.683434 1.18374i −0.973926 0.226865i \(-0.927152\pi\)
0.290492 0.956877i \(-0.406181\pi\)
\(18\) 0 0
\(19\) 2.76861 4.79537i 0.145716 0.252388i −0.783924 0.620857i \(-0.786785\pi\)
0.929640 + 0.368469i \(0.120118\pi\)
\(20\) 0 0
\(21\) 20.5786 + 0.181210i 0.979931 + 0.00862905i
\(22\) 0 0
\(23\) 12.8739 + 7.43273i 0.559734 + 0.323162i 0.753039 0.657976i \(-0.228587\pi\)
−0.193305 + 0.981139i \(0.561921\pi\)
\(24\) 0 0
\(25\) 25.9857 + 45.0085i 1.03943 + 1.80034i
\(26\) 0 0
\(27\) 27.5085 1.01883
\(28\) 0 0
\(29\) 43.1747i 1.48878i −0.667744 0.744391i \(-0.732740\pi\)
0.667744 0.744391i \(-0.267260\pi\)
\(30\) 0 0
\(31\) −26.6958 + 15.4128i −0.861153 + 0.497187i −0.864398 0.502808i \(-0.832300\pi\)
0.00324492 + 0.999995i \(0.498967\pi\)
\(32\) 0 0
\(33\) 4.84621 8.39389i 0.146855 0.254360i
\(34\) 0 0
\(35\) −53.4538 + 30.2371i −1.52725 + 0.863918i
\(36\) 0 0
\(37\) 11.3647 + 6.56141i 0.307154 + 0.177335i 0.645652 0.763632i \(-0.276586\pi\)
−0.338498 + 0.940967i \(0.609919\pi\)
\(38\) 0 0
\(39\) 10.4462 6.03112i 0.267852 0.154644i
\(40\) 0 0
\(41\) −8.56863 −0.208991 −0.104496 0.994525i \(-0.533323\pi\)
−0.104496 + 0.994525i \(0.533323\pi\)
\(42\) 0 0
\(43\) 11.3076 0.262967 0.131483 0.991318i \(-0.458026\pi\)
0.131483 + 0.991318i \(0.458026\pi\)
\(44\) 0 0
\(45\) −2.71199 + 1.56577i −0.0602665 + 0.0347949i
\(46\) 0 0
\(47\) −31.9084 18.4223i −0.678902 0.391964i 0.120539 0.992709i \(-0.461538\pi\)
−0.799441 + 0.600744i \(0.794871\pi\)
\(48\) 0 0
\(49\) −23.7489 42.8601i −0.484672 0.874696i
\(50\) 0 0
\(51\) 34.1570 59.1616i 0.669744 1.16003i
\(52\) 0 0
\(53\) −53.2193 + 30.7262i −1.00414 + 0.579739i −0.909470 0.415770i \(-0.863512\pi\)
−0.0946675 + 0.995509i \(0.530179\pi\)
\(54\) 0 0
\(55\) 28.9243i 0.525897i
\(56\) 0 0
\(57\) 16.2789 0.285595
\(58\) 0 0
\(59\) −31.3354 54.2746i −0.531109 0.919908i −0.999341 0.0363022i \(-0.988442\pi\)
0.468232 0.883606i \(-0.344891\pi\)
\(60\) 0 0
\(61\) 34.3623 + 19.8391i 0.563317 + 0.325231i 0.754476 0.656328i \(-0.227891\pi\)
−0.191159 + 0.981559i \(0.561225\pi\)
\(62\) 0 0
\(63\) −1.23018 2.17474i −0.0195267 0.0345197i
\(64\) 0 0
\(65\) −17.9982 + 31.1738i −0.276895 + 0.479597i
\(66\) 0 0
\(67\) 21.3147 + 36.9182i 0.318130 + 0.551018i 0.980098 0.198514i \(-0.0636117\pi\)
−0.661968 + 0.749533i \(0.730278\pi\)
\(68\) 0 0
\(69\) 43.7031i 0.633378i
\(70\) 0 0
\(71\) 107.155i 1.50922i 0.656174 + 0.754610i \(0.272174\pi\)
−0.656174 + 0.754610i \(0.727826\pi\)
\(72\) 0 0
\(73\) −18.0085 31.1917i −0.246692 0.427284i 0.715914 0.698189i \(-0.246010\pi\)
−0.962606 + 0.270905i \(0.912677\pi\)
\(74\) 0 0
\(75\) −76.3955 + 132.321i −1.01861 + 1.76428i
\(76\) 0 0
\(77\) −23.0770 0.203211i −0.299702 0.00263910i
\(78\) 0 0
\(79\) 36.3029 + 20.9595i 0.459530 + 0.265310i 0.711847 0.702335i \(-0.247859\pi\)
−0.252317 + 0.967645i \(0.581192\pi\)
\(80\) 0 0
\(81\) 38.8301 + 67.2557i 0.479384 + 0.830317i
\(82\) 0 0
\(83\) 87.8010 1.05784 0.528922 0.848671i \(-0.322596\pi\)
0.528922 + 0.848671i \(0.322596\pi\)
\(84\) 0 0
\(85\) 203.864i 2.39840i
\(86\) 0 0
\(87\) 109.924 63.4648i 1.26350 0.729480i
\(88\) 0 0
\(89\) −1.04794 + 1.81508i −0.0117746 + 0.0203942i −0.871853 0.489768i \(-0.837081\pi\)
0.860078 + 0.510162i \(0.170415\pi\)
\(90\) 0 0
\(91\) −24.7453 14.5787i −0.271926 0.160206i
\(92\) 0 0
\(93\) −78.4831 45.3122i −0.843904 0.487228i
\(94\) 0 0
\(95\) −42.0713 + 24.2899i −0.442856 + 0.255683i
\(96\) 0 0
\(97\) −40.6387 −0.418956 −0.209478 0.977813i \(-0.567176\pi\)
−0.209478 + 0.977813i \(0.567176\pi\)
\(98\) 0 0
\(99\) −1.17677 −0.0118866
\(100\) 0 0
\(101\) 24.4300 14.1047i 0.241882 0.139650i −0.374160 0.927364i \(-0.622069\pi\)
0.616041 + 0.787714i \(0.288735\pi\)
\(102\) 0 0
\(103\) 74.6380 + 43.0923i 0.724641 + 0.418372i 0.816458 0.577404i \(-0.195934\pi\)
−0.0918176 + 0.995776i \(0.529268\pi\)
\(104\) 0 0
\(105\) −155.559 91.6481i −1.48152 0.872839i
\(106\) 0 0
\(107\) 32.1677 55.7161i 0.300633 0.520711i −0.675647 0.737225i \(-0.736136\pi\)
0.976279 + 0.216515i \(0.0694689\pi\)
\(108\) 0 0
\(109\) 99.8826 57.6672i 0.916354 0.529057i 0.0338838 0.999426i \(-0.489212\pi\)
0.882470 + 0.470369i \(0.155879\pi\)
\(110\) 0 0
\(111\) 38.5799i 0.347566i
\(112\) 0 0
\(113\) 201.331 1.78169 0.890845 0.454308i \(-0.150113\pi\)
0.890845 + 0.454308i \(0.150113\pi\)
\(114\) 0 0
\(115\) −65.2098 112.947i −0.567042 0.982146i
\(116\) 0 0
\(117\) −1.26829 0.732248i −0.0108401 0.00625853i
\(118\) 0 0
\(119\) −162.651 1.43227i −1.36681 0.0120359i
\(120\) 0 0
\(121\) 55.0654 95.3761i 0.455086 0.788232i
\(122\) 0 0
\(123\) −12.5955 21.8160i −0.102402 0.177366i
\(124\) 0 0
\(125\) 236.628i 1.89303i
\(126\) 0 0
\(127\) 105.675i 0.832088i −0.909344 0.416044i \(-0.863416\pi\)
0.909344 0.416044i \(-0.136584\pi\)
\(128\) 0 0
\(129\) 16.6216 + 28.7895i 0.128850 + 0.223174i
\(130\) 0 0
\(131\) −81.8153 + 141.708i −0.624544 + 1.08174i 0.364084 + 0.931366i \(0.381382\pi\)
−0.988629 + 0.150377i \(0.951951\pi\)
\(132\) 0 0
\(133\) −19.0839 33.7369i −0.143488 0.253661i
\(134\) 0 0
\(135\) −209.008 120.671i −1.54821 0.893858i
\(136\) 0 0
\(137\) 16.0139 + 27.7368i 0.116889 + 0.202458i 0.918533 0.395343i \(-0.129374\pi\)
−0.801644 + 0.597802i \(0.796041\pi\)
\(138\) 0 0
\(139\) 27.2303 0.195901 0.0979507 0.995191i \(-0.468771\pi\)
0.0979507 + 0.995191i \(0.468771\pi\)
\(140\) 0 0
\(141\) 108.320i 0.768226i
\(142\) 0 0
\(143\) −11.7145 + 6.76337i −0.0819196 + 0.0472963i
\(144\) 0 0
\(145\) −189.393 + 328.038i −1.30616 + 2.26233i
\(146\) 0 0
\(147\) 74.2136 123.468i 0.504854 0.839918i
\(148\) 0 0
\(149\) 107.716 + 62.1901i 0.722930 + 0.417384i 0.815830 0.578292i \(-0.196280\pi\)
−0.0929004 + 0.995675i \(0.529614\pi\)
\(150\) 0 0
\(151\) 92.9010 53.6364i 0.615238 0.355208i −0.159774 0.987154i \(-0.551077\pi\)
0.775013 + 0.631945i \(0.217743\pi\)
\(152\) 0 0
\(153\) −8.29410 −0.0542098
\(154\) 0 0
\(155\) 270.443 1.74480
\(156\) 0 0
\(157\) 92.7573 53.5534i 0.590811 0.341105i −0.174607 0.984638i \(-0.555866\pi\)
0.765418 + 0.643533i \(0.222532\pi\)
\(158\) 0 0
\(159\) −156.460 90.3321i −0.984024 0.568126i
\(160\) 0 0
\(161\) 90.5718 51.2336i 0.562558 0.318221i
\(162\) 0 0
\(163\) −100.378 + 173.859i −0.615813 + 1.06662i 0.374428 + 0.927256i \(0.377839\pi\)
−0.990241 + 0.139364i \(0.955494\pi\)
\(164\) 0 0
\(165\) −73.6424 + 42.5174i −0.446317 + 0.257681i
\(166\) 0 0
\(167\) 207.929i 1.24508i −0.782587 0.622541i \(-0.786100\pi\)
0.782587 0.622541i \(-0.213900\pi\)
\(168\) 0 0
\(169\) 152.166 0.900390
\(170\) 0 0
\(171\) −0.988223 1.71165i −0.00577908 0.0100097i
\(172\) 0 0
\(173\) −188.386 108.765i −1.08894 0.628699i −0.155645 0.987813i \(-0.549746\pi\)
−0.933294 + 0.359114i \(0.883079\pi\)
\(174\) 0 0
\(175\) 363.785 + 3.20341i 2.07877 + 0.0183052i
\(176\) 0 0
\(177\) 92.1233 159.562i 0.520471 0.901482i
\(178\) 0 0
\(179\) −23.2072 40.1960i −0.129649 0.224559i 0.793892 0.608059i \(-0.208052\pi\)
−0.923541 + 0.383501i \(0.874718\pi\)
\(180\) 0 0
\(181\) 247.523i 1.36753i 0.729702 + 0.683765i \(0.239659\pi\)
−0.729702 + 0.683765i \(0.760341\pi\)
\(182\) 0 0
\(183\) 116.650i 0.637433i
\(184\) 0 0
\(185\) −57.5654 99.7062i −0.311164 0.538952i
\(186\) 0 0
\(187\) −38.3040 + 66.3445i −0.204834 + 0.354783i
\(188\) 0 0
\(189\) 97.7446 165.907i 0.517167 0.877817i
\(190\) 0 0
\(191\) 276.177 + 159.451i 1.44595 + 0.834822i 0.998237 0.0593533i \(-0.0189039\pi\)
0.447717 + 0.894175i \(0.352237\pi\)
\(192\) 0 0
\(193\) −154.875 268.251i −0.802461 1.38990i −0.917992 0.396599i \(-0.870190\pi\)
0.115531 0.993304i \(-0.463143\pi\)
\(194\) 0 0
\(195\) −105.826 −0.542698
\(196\) 0 0
\(197\) 157.969i 0.801872i −0.916106 0.400936i \(-0.868685\pi\)
0.916106 0.400936i \(-0.131315\pi\)
\(198\) 0 0
\(199\) 16.8606 9.73449i 0.0847268 0.0489171i −0.457038 0.889447i \(-0.651090\pi\)
0.541765 + 0.840530i \(0.317756\pi\)
\(200\) 0 0
\(201\) −62.6634 + 108.536i −0.311758 + 0.539981i
\(202\) 0 0
\(203\) −260.392 153.410i −1.28272 0.755715i
\(204\) 0 0
\(205\) 65.1039 + 37.5877i 0.317580 + 0.183355i
\(206\) 0 0
\(207\) 4.59518 2.65303i 0.0221990 0.0128166i
\(208\) 0 0
\(209\) −18.2553 −0.0873462
\(210\) 0 0
\(211\) 197.272 0.934937 0.467468 0.884010i \(-0.345166\pi\)
0.467468 + 0.884010i \(0.345166\pi\)
\(212\) 0 0
\(213\) −272.819 + 157.512i −1.28084 + 0.739494i
\(214\) 0 0
\(215\) −85.9141 49.6025i −0.399600 0.230709i
\(216\) 0 0
\(217\) −1.90003 + 215.771i −0.00875589 + 0.994336i
\(218\) 0 0
\(219\) 52.9435 91.7008i 0.241751 0.418725i
\(220\) 0 0
\(221\) −82.5659 + 47.6694i −0.373601 + 0.215699i
\(222\) 0 0
\(223\) 263.559i 1.18188i 0.806716 + 0.590940i \(0.201243\pi\)
−0.806716 + 0.590940i \(0.798757\pi\)
\(224\) 0 0
\(225\) 18.5506 0.0824470
\(226\) 0 0
\(227\) 48.5497 + 84.0905i 0.213875 + 0.370443i 0.952924 0.303209i \(-0.0980580\pi\)
−0.739049 + 0.673652i \(0.764725\pi\)
\(228\) 0 0
\(229\) 119.051 + 68.7339i 0.519872 + 0.300148i 0.736882 0.676021i \(-0.236297\pi\)
−0.217011 + 0.976169i \(0.569631\pi\)
\(230\) 0 0
\(231\) −33.4048 59.0537i −0.144610 0.255644i
\(232\) 0 0
\(233\) −68.6614 + 118.925i −0.294684 + 0.510408i −0.974911 0.222593i \(-0.928548\pi\)
0.680227 + 0.733001i \(0.261881\pi\)
\(234\) 0 0
\(235\) 161.625 + 279.943i 0.687766 + 1.19125i
\(236\) 0 0
\(237\) 123.238i 0.519991i
\(238\) 0 0
\(239\) 325.305i 1.36111i −0.732697 0.680555i \(-0.761739\pi\)
0.732697 0.680555i \(-0.238261\pi\)
\(240\) 0 0
\(241\) −126.919 219.830i −0.526635 0.912159i −0.999518 0.0310335i \(-0.990120\pi\)
0.472883 0.881125i \(-0.343213\pi\)
\(242\) 0 0
\(243\) 9.63158 16.6824i 0.0396361 0.0686518i
\(244\) 0 0
\(245\) −7.57049 + 429.827i −0.0309000 + 1.75439i
\(246\) 0 0
\(247\) −19.6751 11.3594i −0.0796562 0.0459895i
\(248\) 0 0
\(249\) 129.063 + 223.545i 0.518327 + 0.897769i
\(250\) 0 0
\(251\) −192.899 −0.768521 −0.384261 0.923225i \(-0.625544\pi\)
−0.384261 + 0.923225i \(0.625544\pi\)
\(252\) 0 0
\(253\) 49.0092i 0.193712i
\(254\) 0 0
\(255\) −519.044 + 299.670i −2.03547 + 1.17518i
\(256\) 0 0
\(257\) −16.7985 + 29.0959i −0.0653640 + 0.113214i −0.896855 0.442324i \(-0.854154\pi\)
0.831491 + 0.555538i \(0.187488\pi\)
\(258\) 0 0
\(259\) 79.9542 45.2276i 0.308703 0.174624i
\(260\) 0 0
\(261\) −13.3461 7.70535i −0.0511344 0.0295224i
\(262\) 0 0
\(263\) −317.791 + 183.477i −1.20833 + 0.697630i −0.962394 0.271656i \(-0.912429\pi\)
−0.245936 + 0.969286i \(0.579095\pi\)
\(264\) 0 0
\(265\) 539.142 2.03450
\(266\) 0 0
\(267\) −6.16168 −0.0230775
\(268\) 0 0
\(269\) 90.9729 52.5232i 0.338189 0.195254i −0.321282 0.946984i \(-0.604114\pi\)
0.659471 + 0.751730i \(0.270780\pi\)
\(270\) 0 0
\(271\) 228.342 + 131.833i 0.842592 + 0.486470i 0.858144 0.513409i \(-0.171617\pi\)
−0.0155528 + 0.999879i \(0.504951\pi\)
\(272\) 0 0
\(273\) 0.743493 84.4325i 0.00272342 0.309276i
\(274\) 0 0
\(275\) 85.6708 148.386i 0.311530 0.539586i
\(276\) 0 0
\(277\) 240.983 139.132i 0.869974 0.502280i 0.00263452 0.999997i \(-0.499161\pi\)
0.867340 + 0.497717i \(0.165828\pi\)
\(278\) 0 0
\(279\) 11.0029i 0.0394367i
\(280\) 0 0
\(281\) −400.248 −1.42437 −0.712184 0.701993i \(-0.752294\pi\)
−0.712184 + 0.701993i \(0.752294\pi\)
\(282\) 0 0
\(283\) −95.8351 165.991i −0.338640 0.586541i 0.645537 0.763729i \(-0.276633\pi\)
−0.984177 + 0.177187i \(0.943300\pi\)
\(284\) 0 0
\(285\) −123.686 71.4101i −0.433986 0.250562i
\(286\) 0 0
\(287\) −30.4465 + 51.6785i −0.106085 + 0.180064i
\(288\) 0 0
\(289\) −125.473 + 217.326i −0.434164 + 0.751994i
\(290\) 0 0
\(291\) −59.7370 103.468i −0.205282 0.355559i
\(292\) 0 0
\(293\) 33.7800i 0.115290i 0.998337 + 0.0576451i \(0.0183592\pi\)
−0.998337 + 0.0576451i \(0.981641\pi\)
\(294\) 0 0
\(295\) 549.832i 1.86384i
\(296\) 0 0
\(297\) −45.3457 78.5411i −0.152679 0.264448i
\(298\) 0 0
\(299\) 30.4960 52.8206i 0.101993 0.176658i
\(300\) 0 0
\(301\) 40.1786 68.1974i 0.133484 0.226569i
\(302\) 0 0
\(303\) 71.8221 + 41.4665i 0.237037 + 0.136853i
\(304\) 0 0
\(305\) −174.055 301.472i −0.570672 0.988433i
\(306\) 0 0
\(307\) −493.884 −1.60874 −0.804371 0.594128i \(-0.797497\pi\)
−0.804371 + 0.594128i \(0.797497\pi\)
\(308\) 0 0
\(309\) 253.375i 0.819983i
\(310\) 0 0
\(311\) 452.989 261.533i 1.45656 0.840943i 0.457717 0.889098i \(-0.348667\pi\)
0.998840 + 0.0481545i \(0.0153340\pi\)
\(312\) 0 0
\(313\) 75.6865 131.093i 0.241810 0.418827i −0.719420 0.694575i \(-0.755592\pi\)
0.961230 + 0.275748i \(0.0889256\pi\)
\(314\) 0 0
\(315\) −0.193022 + 21.9199i −0.000612768 + 0.0695871i
\(316\) 0 0
\(317\) −0.375088 0.216557i −0.00118324 0.000683145i 0.499408 0.866367i \(-0.333551\pi\)
−0.500592 + 0.865684i \(0.666884\pi\)
\(318\) 0 0
\(319\) −123.270 + 71.1701i −0.386427 + 0.223104i
\(320\) 0 0
\(321\) 189.140 0.589222
\(322\) 0 0
\(323\) −128.667 −0.398349
\(324\) 0 0
\(325\) 184.667 106.617i 0.568206 0.328054i
\(326\) 0 0
\(327\) 293.646 + 169.536i 0.897999 + 0.518460i
\(328\) 0 0
\(329\) −224.486 + 126.984i −0.682327 + 0.385971i
\(330\) 0 0
\(331\) −71.4511 + 123.757i −0.215864 + 0.373888i −0.953540 0.301268i \(-0.902590\pi\)
0.737675 + 0.675156i \(0.235924\pi\)
\(332\) 0 0
\(333\) 4.05650 2.34202i 0.0121817 0.00703309i
\(334\) 0 0
\(335\) 374.003i 1.11643i
\(336\) 0 0
\(337\) 14.0248 0.0416167 0.0208084 0.999783i \(-0.493376\pi\)
0.0208084 + 0.999783i \(0.493376\pi\)
\(338\) 0 0
\(339\) 295.947 + 512.596i 0.873001 + 1.51208i
\(340\) 0 0
\(341\) 88.0118 + 50.8137i 0.258099 + 0.149014i
\(342\) 0 0
\(343\) −342.880 9.05985i −0.999651 0.0264135i
\(344\) 0 0
\(345\) 191.711 332.053i 0.555684 0.962473i
\(346\) 0 0
\(347\) 187.840 + 325.348i 0.541325 + 0.937602i 0.998828 + 0.0483939i \(0.0154103\pi\)
−0.457504 + 0.889208i \(0.651256\pi\)
\(348\) 0 0
\(349\) 203.347i 0.582657i 0.956623 + 0.291328i \(0.0940972\pi\)
−0.956623 + 0.291328i \(0.905903\pi\)
\(350\) 0 0
\(351\) 112.866i 0.321555i
\(352\) 0 0
\(353\) 259.114 + 448.799i 0.734035 + 1.27139i 0.955145 + 0.296137i \(0.0956986\pi\)
−0.221110 + 0.975249i \(0.570968\pi\)
\(354\) 0 0
\(355\) 470.051 814.152i 1.32409 2.29339i
\(356\) 0 0
\(357\) −235.443 416.221i −0.659504 1.16588i
\(358\) 0 0
\(359\) −136.713 78.9312i −0.380816 0.219864i 0.297357 0.954766i \(-0.403895\pi\)
−0.678173 + 0.734902i \(0.737228\pi\)
\(360\) 0 0
\(361\) 165.170 + 286.082i 0.457534 + 0.792471i
\(362\) 0 0
\(363\) 323.774 0.891941
\(364\) 0 0
\(365\) 315.990i 0.865725i
\(366\) 0 0
\(367\) 269.087 155.358i 0.733208 0.423318i −0.0863866 0.996262i \(-0.527532\pi\)
0.819595 + 0.572944i \(0.194199\pi\)
\(368\) 0 0
\(369\) −1.52924 + 2.64872i −0.00414428 + 0.00717810i
\(370\) 0 0
\(371\) −3.78780 + 430.150i −0.0102097 + 1.15943i
\(372\) 0 0
\(373\) −348.944 201.463i −0.935506 0.540115i −0.0469575 0.998897i \(-0.514953\pi\)
−0.888549 + 0.458782i \(0.848286\pi\)
\(374\) 0 0
\(375\) 602.464 347.833i 1.60657 0.927554i
\(376\) 0 0
\(377\) −177.143 −0.469875
\(378\) 0 0
\(379\) −400.615 −1.05703 −0.528516 0.848923i \(-0.677251\pi\)
−0.528516 + 0.848923i \(0.677251\pi\)
\(380\) 0 0
\(381\) 269.053 155.338i 0.706176 0.407711i
\(382\) 0 0
\(383\) −375.665 216.890i −0.980849 0.566293i −0.0783224 0.996928i \(-0.524956\pi\)
−0.902526 + 0.430635i \(0.858290\pi\)
\(384\) 0 0
\(385\) 174.446 + 102.775i 0.453107 + 0.266949i
\(386\) 0 0
\(387\) 2.01806 3.49537i 0.00521461 0.00903197i
\(388\) 0 0
\(389\) −280.337 + 161.853i −0.720660 + 0.416073i −0.814996 0.579467i \(-0.803261\pi\)
0.0943355 + 0.995540i \(0.469927\pi\)
\(390\) 0 0
\(391\) 345.425i 0.883441i
\(392\) 0 0
\(393\) −481.059 −1.22407
\(394\) 0 0
\(395\) −183.884 318.497i −0.465530 0.806322i
\(396\) 0 0
\(397\) 71.0133 + 40.9996i 0.178875 + 0.103273i 0.586764 0.809758i \(-0.300402\pi\)
−0.407889 + 0.913031i \(0.633735\pi\)
\(398\) 0 0
\(399\) 57.8429 98.1801i 0.144970 0.246065i
\(400\) 0 0
\(401\) −34.3758 + 59.5405i −0.0857251 + 0.148480i −0.905700 0.423919i \(-0.860654\pi\)
0.819975 + 0.572400i \(0.193987\pi\)
\(402\) 0 0
\(403\) 63.2377 + 109.531i 0.156917 + 0.271789i
\(404\) 0 0
\(405\) 681.338i 1.68232i
\(406\) 0 0
\(407\) 43.2639i 0.106300i
\(408\) 0 0
\(409\) −168.427 291.724i −0.411802 0.713263i 0.583285 0.812268i \(-0.301767\pi\)
−0.995087 + 0.0990053i \(0.968434\pi\)
\(410\) 0 0
\(411\) −47.0793 + 81.5437i −0.114548 + 0.198403i
\(412\) 0 0
\(413\) −438.679 3.86291i −1.06218 0.00935328i
\(414\) 0 0
\(415\) −667.106 385.154i −1.60748 0.928081i
\(416\) 0 0
\(417\) 40.0273 + 69.3293i 0.0959887 + 0.166257i
\(418\) 0 0
\(419\) 619.580 1.47871 0.739356 0.673315i \(-0.235130\pi\)
0.739356 + 0.673315i \(0.235130\pi\)
\(420\) 0 0
\(421\) 606.552i 1.44074i −0.693589 0.720371i \(-0.743972\pi\)
0.693589 0.720371i \(-0.256028\pi\)
\(422\) 0 0
\(423\) −11.3893 + 6.57564i −0.0269251 + 0.0155452i
\(424\) 0 0
\(425\) 603.823 1045.85i 1.42076 2.46083i
\(426\) 0 0
\(427\) 241.750 136.750i 0.566159 0.320258i
\(428\) 0 0
\(429\) −34.4396 19.8837i −0.0802787 0.0463489i
\(430\) 0 0
\(431\) 232.207 134.065i 0.538763 0.311055i −0.205815 0.978591i \(-0.565984\pi\)
0.744577 + 0.667536i \(0.232651\pi\)
\(432\) 0 0
\(433\) −397.578 −0.918194 −0.459097 0.888386i \(-0.651827\pi\)
−0.459097 + 0.888386i \(0.651827\pi\)
\(434\) 0 0
\(435\) −1113.59 −2.55999
\(436\) 0 0
\(437\) 71.2854 41.1566i 0.163124 0.0941799i
\(438\) 0 0
\(439\) 168.717 + 97.4090i 0.384322 + 0.221888i 0.679697 0.733493i \(-0.262111\pi\)
−0.295375 + 0.955381i \(0.595445\pi\)
\(440\) 0 0
\(441\) −17.4873 0.308002i −0.0396537 0.000698417i
\(442\) 0 0
\(443\) −47.3631 + 82.0352i −0.106914 + 0.185181i −0.914519 0.404544i \(-0.867430\pi\)
0.807604 + 0.589725i \(0.200764\pi\)
\(444\) 0 0
\(445\) 15.9243 9.19391i 0.0357850 0.0206605i
\(446\) 0 0
\(447\) 365.667i 0.818046i
\(448\) 0 0
\(449\) 212.378 0.473003 0.236502 0.971631i \(-0.423999\pi\)
0.236502 + 0.971631i \(0.423999\pi\)
\(450\) 0 0
\(451\) 14.1247 + 24.4648i 0.0313187 + 0.0542456i
\(452\) 0 0
\(453\) 273.120 + 157.686i 0.602915 + 0.348093i
\(454\) 0 0
\(455\) 124.061 + 219.317i 0.272662 + 0.482016i
\(456\) 0 0
\(457\) −109.467 + 189.602i −0.239534 + 0.414884i −0.960581 0.278002i \(-0.910328\pi\)
0.721047 + 0.692886i \(0.243661\pi\)
\(458\) 0 0
\(459\) −319.605 553.571i −0.696306 1.20604i
\(460\) 0 0
\(461\) 311.878i 0.676525i −0.941052 0.338263i \(-0.890161\pi\)
0.941052 0.338263i \(-0.109839\pi\)
\(462\) 0 0
\(463\) 219.733i 0.474586i −0.971438 0.237293i \(-0.923740\pi\)
0.971438 0.237293i \(-0.0762601\pi\)
\(464\) 0 0
\(465\) 397.539 + 688.558i 0.854923 + 1.48077i
\(466\) 0 0
\(467\) 43.0338 74.5368i 0.0921496 0.159608i −0.816266 0.577676i \(-0.803960\pi\)
0.908415 + 0.418069i \(0.137293\pi\)
\(468\) 0 0
\(469\) 298.395 + 2.62759i 0.636236 + 0.00560255i
\(470\) 0 0
\(471\) 272.698 + 157.442i 0.578976 + 0.334272i
\(472\) 0 0
\(473\) −18.6397 32.2849i −0.0394073 0.0682555i
\(474\) 0 0
\(475\) 287.776 0.605845
\(476\) 0 0
\(477\) 21.9347i 0.0459847i
\(478\) 0 0
\(479\) −125.523 + 72.4708i −0.262052 + 0.151296i −0.625270 0.780408i \(-0.715011\pi\)
0.363218 + 0.931704i \(0.381678\pi\)
\(480\) 0 0
\(481\) 26.9210 46.6286i 0.0559689 0.0969409i
\(482\) 0 0
\(483\) 263.579 + 155.288i 0.545712 + 0.321507i
\(484\) 0 0
\(485\) 308.770 + 178.268i 0.636639 + 0.367564i
\(486\) 0 0
\(487\) −410.200 + 236.829i −0.842300 + 0.486302i −0.858045 0.513574i \(-0.828321\pi\)
0.0157454 + 0.999876i \(0.494988\pi\)
\(488\) 0 0
\(489\) −590.202 −1.20696
\(490\) 0 0
\(491\) 895.083 1.82298 0.911490 0.411322i \(-0.134933\pi\)
0.911490 + 0.411322i \(0.134933\pi\)
\(492\) 0 0
\(493\) −868.831 + 501.620i −1.76233 + 1.01748i
\(494\) 0 0
\(495\) 8.94103 + 5.16211i 0.0180627 + 0.0104285i
\(496\) 0 0
\(497\) 646.262 + 380.746i 1.30033 + 0.766089i
\(498\) 0 0
\(499\) −379.515 + 657.339i −0.760551 + 1.31731i 0.182016 + 0.983296i \(0.441738\pi\)
−0.942567 + 0.334017i \(0.891596\pi\)
\(500\) 0 0
\(501\) 529.394 305.646i 1.05667 0.610072i
\(502\) 0 0
\(503\) 802.733i 1.59589i 0.602730 + 0.797945i \(0.294080\pi\)
−0.602730 + 0.797945i \(0.705920\pi\)
\(504\) 0 0
\(505\) −247.490 −0.490080
\(506\) 0 0
\(507\) 223.677 + 387.420i 0.441177 + 0.764142i
\(508\) 0 0
\(509\) 617.270 + 356.381i 1.21271 + 0.700159i 0.963349 0.268251i \(-0.0864458\pi\)
0.249362 + 0.968410i \(0.419779\pi\)
\(510\) 0 0
\(511\) −252.110 2.22002i −0.493366 0.00434446i
\(512\) 0 0
\(513\) 76.1603 131.914i 0.148461 0.257141i
\(514\) 0 0
\(515\) −378.063 654.824i −0.734102 1.27150i
\(516\) 0 0
\(517\) 121.471i 0.234954i
\(518\) 0 0
\(519\) 639.518i 1.23221i
\(520\) 0 0
\(521\) 410.672 + 711.305i 0.788238 + 1.36527i 0.927046 + 0.374949i \(0.122340\pi\)
−0.138808 + 0.990319i \(0.544327\pi\)
\(522\) 0 0
\(523\) −237.335 + 411.077i −0.453796 + 0.785998i −0.998618 0.0525541i \(-0.983264\pi\)
0.544822 + 0.838552i \(0.316597\pi\)
\(524\) 0 0
\(525\) 526.592 + 930.919i 1.00303 + 1.77318i
\(526\) 0 0
\(527\) 620.323 + 358.144i 1.17708 + 0.679589i
\(528\) 0 0
\(529\) −154.009 266.751i −0.291132 0.504256i
\(530\) 0 0
\(531\) −22.3697 −0.0421274
\(532\) 0 0
\(533\) 35.1565i 0.0659597i
\(534\) 0 0
\(535\) −488.816 + 282.218i −0.913674 + 0.527510i
\(536\) 0 0
\(537\) 68.2270 118.173i 0.127052 0.220061i
\(538\) 0 0
\(539\) −83.2239 + 138.458i −0.154404 + 0.256880i
\(540\) 0 0
\(541\) 349.210 + 201.616i 0.645489 + 0.372673i 0.786726 0.617303i \(-0.211775\pi\)
−0.141237 + 0.989976i \(0.545108\pi\)
\(542\) 0 0
\(543\) −630.202 + 363.847i −1.16059 + 0.670069i
\(544\) 0 0
\(545\) −1011.87 −1.85664
\(546\) 0 0
\(547\) −623.273 −1.13944 −0.569719 0.821839i \(-0.692948\pi\)
−0.569719 + 0.821839i \(0.692948\pi\)
\(548\) 0 0
\(549\) 12.2652 7.08134i 0.0223411 0.0128986i
\(550\) 0 0
\(551\) −207.038 119.534i −0.375750 0.216940i
\(552\) 0 0
\(553\) 255.402 144.473i 0.461848 0.261253i
\(554\) 0 0
\(555\) 169.237 293.127i 0.304932 0.528157i
\(556\) 0 0
\(557\) 19.3784 11.1881i 0.0347906 0.0200864i −0.482504 0.875894i \(-0.660273\pi\)
0.517294 + 0.855808i \(0.326939\pi\)
\(558\) 0 0
\(559\) 46.3942i 0.0829950i
\(560\) 0 0
\(561\) −225.221 −0.401463
\(562\) 0 0
\(563\) −155.749 269.766i −0.276642 0.479157i 0.693906 0.720065i \(-0.255888\pi\)
−0.970548 + 0.240908i \(0.922555\pi\)
\(564\) 0 0
\(565\) −1529.70 883.171i −2.70743 1.56314i
\(566\) 0 0
\(567\) 543.600 + 4.78681i 0.958730 + 0.00844235i
\(568\) 0 0
\(569\) −245.668 + 425.509i −0.431754 + 0.747820i −0.997024 0.0770859i \(-0.975438\pi\)
0.565271 + 0.824906i \(0.308772\pi\)
\(570\) 0 0
\(571\) −332.451 575.821i −0.582225 1.00844i −0.995215 0.0977080i \(-0.968849\pi\)
0.412990 0.910736i \(-0.364484\pi\)
\(572\) 0 0
\(573\) 937.543i 1.63620i
\(574\) 0 0
\(575\) 772.578i 1.34361i
\(576\) 0 0
\(577\) −381.078 660.047i −0.660448 1.14393i −0.980498 0.196528i \(-0.937033\pi\)
0.320051 0.947400i \(-0.396300\pi\)
\(578\) 0 0
\(579\) 455.318 788.634i 0.786387 1.36206i
\(580\) 0 0
\(581\) 311.978 529.539i 0.536968 0.911427i
\(582\) 0 0
\(583\) 175.456 + 101.299i 0.300953 + 0.173756i
\(584\) 0 0
\(585\) 6.42425 + 11.1271i 0.0109816 + 0.0190207i
\(586\) 0 0
\(587\) −551.907 −0.940216 −0.470108 0.882609i \(-0.655785\pi\)
−0.470108 + 0.882609i \(0.655785\pi\)
\(588\) 0 0
\(589\) 170.688i 0.289793i
\(590\) 0 0
\(591\) 402.194 232.207i 0.680532 0.392905i
\(592\) 0 0
\(593\) 417.977 723.958i 0.704852 1.22084i −0.261892 0.965097i \(-0.584347\pi\)
0.966745 0.255743i \(-0.0823201\pi\)
\(594\) 0 0
\(595\) 1229.53 + 724.378i 2.06643 + 1.21744i
\(596\) 0 0
\(597\) 49.5687 + 28.6185i 0.0830297 + 0.0479372i
\(598\) 0 0
\(599\) 226.579 130.816i 0.378263 0.218390i −0.298799 0.954316i \(-0.596586\pi\)
0.677062 + 0.735926i \(0.263253\pi\)
\(600\) 0 0
\(601\) 731.253 1.21673 0.608364 0.793658i \(-0.291826\pi\)
0.608364 + 0.793658i \(0.291826\pi\)
\(602\) 0 0
\(603\) 15.2161 0.0252340
\(604\) 0 0
\(605\) −836.766 + 483.107i −1.38308 + 0.798524i
\(606\) 0 0
\(607\) 10.5201 + 6.07378i 0.0173313 + 0.0100062i 0.508641 0.860979i \(-0.330148\pi\)
−0.491309 + 0.870985i \(0.663482\pi\)
\(608\) 0 0
\(609\) 7.82368 888.472i 0.0128468 1.45890i
\(610\) 0 0
\(611\) −75.5855 + 130.918i −0.123708 + 0.214268i
\(612\) 0 0
\(613\) −1045.72 + 603.744i −1.70590 + 0.984900i −0.766388 + 0.642378i \(0.777948\pi\)
−0.939510 + 0.342522i \(0.888719\pi\)
\(614\) 0 0
\(615\) 221.009i 0.359364i
\(616\) 0 0
\(617\) 48.0515 0.0778792 0.0389396 0.999242i \(-0.487602\pi\)
0.0389396 + 0.999242i \(0.487602\pi\)
\(618\) 0 0
\(619\) 73.8783 + 127.961i 0.119351 + 0.206722i 0.919511 0.393065i \(-0.128585\pi\)
−0.800160 + 0.599787i \(0.795252\pi\)
\(620\) 0 0
\(621\) 354.141 + 204.464i 0.570276 + 0.329249i
\(622\) 0 0
\(623\) 7.22340 + 12.7697i 0.0115945 + 0.0204971i
\(624\) 0 0
\(625\) −388.368 + 672.673i −0.621389 + 1.07628i
\(626\) 0 0
\(627\) −26.8345 46.4788i −0.0427983 0.0741288i
\(628\) 0 0
\(629\) 304.932i 0.484788i
\(630\) 0 0
\(631\) 912.286i 1.44578i −0.690964 0.722889i \(-0.742814\pi\)
0.690964 0.722889i \(-0.257186\pi\)
\(632\) 0 0
\(633\) 289.980 + 502.261i 0.458105 + 0.793461i
\(634\) 0 0
\(635\) −463.562 + 802.913i −0.730019 + 1.26443i
\(636\) 0 0
\(637\) −175.852 + 97.4402i −0.276063 + 0.152967i
\(638\) 0 0
\(639\) 33.1234 + 19.1238i 0.0518363 + 0.0299277i
\(640\) 0 0
\(641\) −286.819 496.784i −0.447455 0.775015i 0.550765 0.834661i \(-0.314336\pi\)
−0.998220 + 0.0596460i \(0.981003\pi\)
\(642\) 0 0
\(643\) 197.633 0.307360 0.153680 0.988121i \(-0.450887\pi\)
0.153680 + 0.988121i \(0.450887\pi\)
\(644\) 0 0
\(645\) 291.654i 0.452176i
\(646\) 0 0
\(647\) −869.817 + 502.189i −1.34439 + 0.776181i −0.987448 0.157947i \(-0.949513\pi\)
−0.356938 + 0.934128i \(0.616179\pi\)
\(648\) 0 0
\(649\) −103.308 + 178.935i −0.159180 + 0.275709i
\(650\) 0 0
\(651\) −552.153 + 312.336i −0.848162 + 0.479778i
\(652\) 0 0
\(653\) −374.151 216.016i −0.572973 0.330806i 0.185363 0.982670i \(-0.440654\pi\)
−0.758336 + 0.651864i \(0.773987\pi\)
\(654\) 0 0
\(655\) 1243.25 717.793i 1.89810 1.09587i
\(656\) 0 0
\(657\) −12.8559 −0.0195676
\(658\) 0 0
\(659\) −222.867 −0.338190 −0.169095 0.985600i \(-0.554084\pi\)
−0.169095 + 0.985600i \(0.554084\pi\)
\(660\) 0 0
\(661\) 635.382 366.838i 0.961243 0.554974i 0.0646877 0.997906i \(-0.479395\pi\)
0.896555 + 0.442932i \(0.146062\pi\)
\(662\) 0 0
\(663\) −242.736 140.144i −0.366118 0.211378i
\(664\) 0 0
\(665\) −2.99436 + 340.045i −0.00450280 + 0.511347i
\(666\) 0 0
\(667\) 320.906 555.825i 0.481118 0.833321i
\(668\) 0 0
\(669\) −671.031 + 387.420i −1.00304 + 0.579103i
\(670\) 0 0
\(671\) 130.813i 0.194952i
\(672\) 0 0
\(673\) 331.486 0.492550 0.246275 0.969200i \(-0.420793\pi\)
0.246275 + 0.969200i \(0.420793\pi\)
\(674\) 0 0
\(675\) 714.828 + 1238.12i 1.05900 + 1.83425i
\(676\) 0 0
\(677\) 893.931 + 516.111i 1.32043 + 0.762350i 0.983797 0.179286i \(-0.0573786\pi\)
0.336633 + 0.941636i \(0.390712\pi\)
\(678\) 0 0
\(679\) −144.399 + 245.097i −0.212665 + 0.360968i
\(680\) 0 0
\(681\) −142.732 + 247.218i −0.209591 + 0.363023i
\(682\) 0 0
\(683\) 82.4745 + 142.850i 0.120753 + 0.209151i 0.920065 0.391766i \(-0.128136\pi\)
−0.799312 + 0.600917i \(0.794802\pi\)
\(684\) 0 0
\(685\) 280.990i 0.410204i
\(686\) 0 0
\(687\) 404.143i 0.588272i
\(688\) 0 0
\(689\) 126.067 + 218.355i 0.182972 + 0.316916i
\(690\) 0 0
\(691\) 302.018 523.111i 0.437074 0.757034i −0.560389 0.828230i \(-0.689348\pi\)
0.997462 + 0.0711958i \(0.0226815\pi\)
\(692\) 0 0
\(693\) −4.18136 + 7.09726i −0.00603371 + 0.0102414i
\(694\) 0 0
\(695\) −206.894 119.450i −0.297689 0.171871i
\(696\) 0 0
\(697\) 99.5536 + 172.432i 0.142832 + 0.247392i
\(698\) 0 0
\(699\) −403.716 −0.577563
\(700\) 0 0
\(701\) 307.326i 0.438410i −0.975679 0.219205i \(-0.929654\pi\)
0.975679 0.219205i \(-0.0703464\pi\)
\(702\) 0 0
\(703\) 62.9287 36.3319i 0.0895145 0.0516812i
\(704\) 0 0
\(705\) −475.163 + 823.006i −0.673990 + 1.16738i
\(706\) 0 0
\(707\) 1.73877 197.458i 0.00245936 0.279290i
\(708\) 0 0
\(709\) −568.779 328.385i −0.802228 0.463166i 0.0420218 0.999117i \(-0.486620\pi\)
−0.844250 + 0.535950i \(0.819953\pi\)
\(710\) 0 0
\(711\) 12.9579 7.48125i 0.0182249 0.0105222i
\(712\) 0 0
\(713\) −458.237 −0.642689
\(714\) 0 0
\(715\) 118.675 0.165978
\(716\) 0 0
\(717\) 828.239 478.184i 1.15514 0.666923i
\(718\) 0 0
\(719\) 1047.32 + 604.668i 1.45663 + 0.840984i 0.998844 0.0480788i \(-0.0153099\pi\)
0.457784 + 0.889063i \(0.348643\pi\)
\(720\) 0 0
\(721\) 525.102 297.034i 0.728297 0.411975i
\(722\) 0 0
\(723\) 373.130 646.281i 0.516086 0.893887i
\(724\) 0 0
\(725\) 1943.23 1121.92i 2.68031 1.54748i
\(726\) 0 0
\(727\) 1199.28i 1.64963i −0.565404 0.824814i \(-0.691280\pi\)
0.565404 0.824814i \(-0.308720\pi\)
\(728\) 0 0
\(729\) 755.573 1.03645
\(730\) 0 0
\(731\) −131.376 227.549i −0.179720 0.311285i
\(732\) 0 0
\(733\) 495.767 + 286.231i 0.676354 + 0.390493i 0.798480 0.602022i \(-0.205638\pi\)
−0.122126 + 0.992515i \(0.538971\pi\)
\(734\) 0 0
\(735\) −1105.48 + 612.551i −1.50406 + 0.833402i
\(736\) 0 0
\(737\) 70.2714 121.714i 0.0953479 0.165147i
\(738\) 0 0
\(739\) −437.226 757.297i −0.591645 1.02476i −0.994011 0.109280i \(-0.965145\pi\)
0.402366 0.915479i \(-0.368188\pi\)
\(740\) 0 0
\(741\) 66.7912i 0.0901366i
\(742\) 0 0
\(743\) 63.9768i 0.0861061i 0.999073 + 0.0430530i \(0.0137084\pi\)
−0.999073 + 0.0430530i \(0.986292\pi\)
\(744\) 0 0
\(745\) −545.615 945.032i −0.732369 1.26850i
\(746\) 0 0
\(747\) 15.6698 27.1409i 0.0209770 0.0363332i
\(748\) 0 0
\(749\) −221.731 391.980i −0.296036 0.523338i
\(750\) 0 0
\(751\) −498.130 287.595i −0.663289 0.382950i 0.130240 0.991482i \(-0.458425\pi\)
−0.793529 + 0.608533i \(0.791758\pi\)
\(752\) 0 0
\(753\) −283.552 491.127i −0.376564 0.652227i
\(754\) 0 0
\(755\) −941.140 −1.24654
\(756\) 0 0
\(757\) 130.657i 0.172598i 0.996269 + 0.0862989i \(0.0275040\pi\)
−0.996269 + 0.0862989i \(0.972496\pi\)
\(758\) 0 0
\(759\) 124.779 72.0412i 0.164399 0.0949160i
\(760\) 0 0
\(761\) −59.2992 + 102.709i −0.0779228 + 0.134966i −0.902354 0.430997i \(-0.858162\pi\)
0.824431 + 0.565963i \(0.191495\pi\)
\(762\) 0 0
\(763\) 7.10898 807.310i 0.00931715 1.05807i
\(764\) 0 0
\(765\) 63.0179 + 36.3834i 0.0823764 + 0.0475600i
\(766\) 0 0
\(767\) −222.685 + 128.567i −0.290332 + 0.167623i
\(768\) 0 0
\(769\) 1145.31 1.48935 0.744674 0.667429i \(-0.232605\pi\)
0.744674 + 0.667429i \(0.232605\pi\)
\(770\) 0 0
\(771\) −98.7724 −0.128109
\(772\) 0 0
\(773\) 548.098 316.445i 0.709053 0.409372i −0.101657 0.994819i \(-0.532415\pi\)
0.810710 + 0.585447i \(0.199081\pi\)
\(774\) 0 0
\(775\) −1387.41 801.024i −1.79021 1.03358i
\(776\) 0 0
\(777\) 232.680 + 137.084i 0.299459 + 0.176427i
\(778\) 0 0
\(779\) −23.7232 + 41.0898i −0.0304534 + 0.0527468i
\(780\) 0 0
\(781\) 305.943 176.636i 0.391732 0.226167i
\(782\) 0 0
\(783\) 1187.67i 1.51682i
\(784\) 0 0
\(785\) −939.684 −1.19705
\(786\) 0 0
\(787\) −46.8023 81.0639i −0.0594692 0.103004i 0.834758 0.550617i \(-0.185607\pi\)
−0.894227 + 0.447613i \(0.852274\pi\)
\(788\) 0 0
\(789\) −934.276 539.405i −1.18413 0.683656i
\(790\) 0 0
\(791\) 715.378 1214.25i 0.904397 1.53508i
\(792\) 0 0
\(793\) 81.3985 140.986i 0.102646 0.177789i
\(794\) 0 0
\(795\) 792.513 + 1372.67i 0.996872 + 1.72663i
\(796\) 0 0
\(797\) 214.365i 0.268965i 0.990916 + 0.134483i \(0.0429372\pi\)
−0.990916 + 0.134483i \(0.957063\pi\)
\(798\) 0 0
\(799\) 856.150i 1.07153i
\(800\) 0 0
\(801\) 0.374050 + 0.647873i 0.000466978 + 0.000808830i
\(802\) 0 0
\(803\) −59.3714 + 102.834i −0.0739370 + 0.128063i
\(804\) 0 0
\(805\) −912.902 8.03880i −1.13404 0.00998609i
\(806\) 0 0
\(807\) 267.452 + 154.413i 0.331415 + 0.191343i
\(808\) 0 0
\(809\) 136.449 + 236.336i 0.168663 + 0.292133i 0.937950 0.346770i \(-0.112722\pi\)
−0.769287 + 0.638904i \(0.779388\pi\)
\(810\) 0 0
\(811\) 1339.86 1.65211 0.826054 0.563592i \(-0.190581\pi\)
0.826054 + 0.563592i \(0.190581\pi\)
\(812\) 0 0
\(813\) 775.157i 0.953452i
\(814\) 0 0
\(815\) 1525.32 880.646i 1.87156 1.08055i
\(816\) 0 0
\(817\) 31.3062 54.2240i 0.0383185 0.0663696i
\(818\) 0 0
\(819\) −8.92282 + 5.04736i −0.0108948 + 0.00616283i
\(820\) 0 0
\(821\) −586.088 338.378i −0.713871 0.412154i 0.0986214 0.995125i \(-0.468557\pi\)
−0.812493 + 0.582971i \(0.801890\pi\)
\(822\) 0 0
\(823\) −984.303 + 568.288i −1.19599 + 0.690507i −0.959660 0.281164i \(-0.909279\pi\)
−0.236334 + 0.971672i \(0.575946\pi\)
\(824\) 0 0
\(825\) 503.728 0.610580
\(826\) 0 0
\(827\) 1140.05 1.37854 0.689269 0.724506i \(-0.257932\pi\)
0.689269 + 0.724506i \(0.257932\pi\)
\(828\) 0 0
\(829\) 898.011 518.467i 1.08325 0.625413i 0.151476 0.988461i \(-0.451597\pi\)
0.931770 + 0.363048i \(0.118264\pi\)
\(830\) 0 0
\(831\) 708.467 + 409.034i 0.852548 + 0.492219i
\(832\) 0 0
\(833\) −586.577 + 975.879i −0.704174 + 1.17152i
\(834\) 0 0
\(835\) −912.114 + 1579.83i −1.09235 + 1.89201i
\(836\) 0 0
\(837\) −734.361 + 423.984i −0.877373 + 0.506552i
\(838\) 0 0
\(839\) 1330.82i 1.58620i 0.609091 + 0.793100i \(0.291534\pi\)
−0.609091 + 0.793100i \(0.708466\pi\)
\(840\) 0 0
\(841\) −1023.05 −1.21647
\(842\) 0 0
\(843\) −588.346 1019.04i −0.697919 1.20883i
\(844\) 0 0
\(845\) −1156.15 667.501i −1.36822 0.789942i
\(846\) 0 0
\(847\) −379.564 671.001i −0.448128 0.792209i
\(848\) 0 0
\(849\) 281.746 487.999i 0.331857 0.574793i
\(850\) 0 0
\(851\) 97.5384 + 168.941i 0.114616 + 0.198521i
\(852\) 0 0
\(853\) 1325.15i 1.55352i −0.629798 0.776759i \(-0.716862\pi\)
0.629798 0.776759i \(-0.283138\pi\)
\(854\) 0 0
\(855\) 17.3400i 0.0202807i
\(856\) 0 0
\(857\) −18.5345 32.1027i −0.0216272 0.0374594i 0.855009 0.518613i \(-0.173551\pi\)
−0.876636 + 0.481153i \(0.840218\pi\)
\(858\) 0 0
\(859\) −125.970 + 218.187i −0.146648 + 0.254001i −0.929986 0.367594i \(-0.880182\pi\)
0.783339 + 0.621595i \(0.213515\pi\)
\(860\) 0 0
\(861\) −176.330 1.55272i −0.204797 0.00180339i
\(862\) 0 0
\(863\) −1443.94 833.661i −1.67317 0.966004i −0.965848 0.259111i \(-0.916570\pi\)
−0.707321 0.706893i \(-0.750096\pi\)
\(864\) 0 0
\(865\) 954.231 + 1652.78i 1.10316 + 1.91072i
\(866\) 0 0
\(867\) −737.761 −0.850935
\(868\) 0 0
\(869\) 138.200i 0.159034i
\(870\) 0 0
\(871\) 151.473 87.4530i 0.173907 0.100405i
\(872\) 0 0
\(873\) −7.25276 + 12.5622i −0.00830786 + 0.0143896i
\(874\) 0 0
\(875\) −1427.14 840.798i −1.63101 0.960913i
\(876\) 0 0
\(877\) 214.659 + 123.934i 0.244765 + 0.141315i 0.617365 0.786677i \(-0.288200\pi\)
−0.372600 + 0.927992i \(0.621533\pi\)
\(878\) 0 0
\(879\) −86.0052 + 49.6551i −0.0978443 + 0.0564905i
\(880\) 0 0
\(881\) 1130.22 1.28288 0.641440 0.767173i \(-0.278337\pi\)
0.641440 + 0.767173i \(0.278337\pi\)
\(882\) 0 0
\(883\) −10.7432 −0.0121667 −0.00608335 0.999981i \(-0.501936\pi\)
−0.00608335 + 0.999981i \(0.501936\pi\)
\(884\) 0 0
\(885\) −1399.89 + 808.228i −1.58180 + 0.913252i
\(886\) 0 0
\(887\) 776.894 + 448.540i 0.875866 + 0.505682i 0.869293 0.494297i \(-0.164574\pi\)
0.00657316 + 0.999978i \(0.497908\pi\)
\(888\) 0 0
\(889\) −637.340 375.490i −0.716918 0.422373i
\(890\) 0 0
\(891\) 128.017 221.732i 0.143678 0.248857i
\(892\) 0 0
\(893\) −176.684 + 102.008i −0.197854 + 0.114231i
\(894\) 0 0
\(895\) 407.209i 0.454982i
\(896\) 0 0
\(897\) 179.311 0.199901
\(898\) 0 0
\(899\) 665.443 + 1152.58i 0.740203 + 1.28207i
\(900\) 0 0
\(901\) 1236.64 + 713.976i 1.37252 + 0.792427i
\(902\) 0 0
\(903\) 232.694 + 2.04905i 0.257689 + 0.00226915i
\(904\) 0 0
\(905\) 1085.80 1880.66i 1.19978 2.07808i
\(906\) 0 0
\(907\) 837.264 + 1450.18i 0.923114 + 1.59888i 0.794567 + 0.607176i \(0.207698\pi\)
0.128547 + 0.991703i \(0.458969\pi\)
\(908\) 0 0
\(909\) 10.0690i 0.0110770i
\(910\) 0 0
\(911\) 910.733i 0.999707i 0.866110 + 0.499853i \(0.166613\pi\)
−0.866110 + 0.499853i \(0.833387\pi\)
\(912\) 0 0
\(913\) −144.733 250.685i −0.158525 0.274573i
\(914\) 0 0
\(915\) 511.706 886.300i 0.559241 0.968634i
\(916\) 0 0
\(917\) 563.950 + 996.962i 0.614995 + 1.08720i
\(918\) 0 0
\(919\) 1107.77 + 639.571i 1.20541 + 0.695943i 0.961753 0.273919i \(-0.0883201\pi\)
0.243655 + 0.969862i \(0.421653\pi\)
\(920\) 0 0
\(921\) −725.986 1257.45i −0.788259 1.36530i
\(922\) 0 0
\(923\) 439.648 0.476325
\(924\) 0 0
\(925\) 682.010i 0.737308i
\(926\) 0 0
\(927\) 26.6412 15.3813i 0.0287392 0.0165926i
\(928\) 0 0
\(929\) −14.0674 + 24.3654i −0.0151425 + 0.0262276i −0.873497 0.486829i \(-0.838154\pi\)
0.858355 + 0.513056i \(0.171487\pi\)
\(930\) 0 0
\(931\) −271.281 4.77805i −0.291387 0.00513217i
\(932\) 0 0
\(933\) 1331.75 + 768.884i 1.42738 + 0.824099i
\(934\) 0 0
\(935\) 582.062 336.054i 0.622526 0.359416i
\(936\) 0 0
\(937\) −1165.92 −1.24431 −0.622154 0.782895i \(-0.713742\pi\)
−0.622154 + 0.782895i \(0.713742\pi\)
\(938\) 0 0
\(939\) 445.023 0.473933
\(940\) 0 0
\(941\) −766.535 + 442.559i −0.814596 + 0.470307i −0.848549 0.529116i \(-0.822524\pi\)
0.0339534 + 0.999423i \(0.489190\pi\)
\(942\) 0 0
\(943\) −110.312 63.6884i −0.116979 0.0675381i
\(944\) 0 0
\(945\) −1470.44 + 831.780i −1.55602 + 0.880190i
\(946\) 0 0
\(947\) 748.589 1296.59i 0.790485 1.36916i −0.135183 0.990821i \(-0.543162\pi\)
0.925667 0.378339i \(-0.123505\pi\)
\(948\) 0 0
\(949\) −127.978 + 73.8878i −0.134855 + 0.0778586i
\(950\) 0 0
\(951\) 1.27332i 0.00133892i
\(952\) 0 0
\(953\) 3.14896 0.00330426 0.00165213 0.999999i \(-0.499474\pi\)
0.00165213 + 0.999999i \(0.499474\pi\)
\(954\) 0 0
\(955\) −1398.92 2422.99i −1.46483 2.53717i
\(956\) 0 0
\(957\) −362.403 209.234i −0.378687 0.218635i
\(958\) 0 0
\(959\) 224.185 + 1.97412i 0.233770 + 0.00205852i
\(960\) 0 0
\(961\) −5.39103 + 9.33753i −0.00560981 + 0.00971647i
\(962\) 0 0
\(963\) −11.4819 19.8872i −0.0119230 0.0206513i
\(964\) 0 0
\(965\) 2717.54i 2.81610i
\(966\) 0 0
\(967\) 1451.63i 1.50117i −0.660775 0.750584i \(-0.729772\pi\)
0.660775 0.750584i \(-0.270228\pi\)
\(968\) 0 0
\(969\) −189.134 327.590i −0.195185 0.338071i
\(970\) 0 0
\(971\) −580.527 + 1005.50i −0.597865 + 1.03553i 0.395271 + 0.918565i \(0.370651\pi\)
−0.993136 + 0.116968i \(0.962683\pi\)
\(972\) 0 0
\(973\) 96.7559 164.229i 0.0994408 0.168786i
\(974\) 0 0
\(975\) 542.904 + 313.446i 0.556824 + 0.321483i
\(976\) 0 0
\(977\) 288.919 + 500.422i 0.295720 + 0.512202i 0.975152 0.221536i \(-0.0711071\pi\)
−0.679432 + 0.733738i \(0.737774\pi\)
\(978\) 0 0
\(979\) 6.90978 0.00705800
\(980\) 0 0
\(981\) 41.1673i 0.0419647i
\(982\) 0 0
\(983\) −1086.94 + 627.545i −1.10574 + 0.638398i −0.937722 0.347386i \(-0.887069\pi\)
−0.168016 + 0.985784i \(0.553736\pi\)
\(984\) 0 0
\(985\) −692.956 + 1200.24i −0.703509 + 1.21851i
\(986\) 0 0
\(987\) −653.290 384.887i −0.661895 0.389956i
\(988\) 0 0
\(989\) 145.572 + 84.0462i 0.147191 + 0.0849810i
\(990\) 0 0
\(991\) −814.150 + 470.049i −0.821543 + 0.474318i −0.850948 0.525249i \(-0.823972\pi\)
0.0294050 + 0.999568i \(0.490639\pi\)
\(992\) 0 0
\(993\) −420.120 −0.423081
\(994\) 0 0
\(995\) −170.808 −0.171666
\(996\) 0 0
\(997\) −195.863 + 113.082i −0.196452 + 0.113422i −0.595000 0.803726i \(-0.702848\pi\)
0.398547 + 0.917148i \(0.369514\pi\)
\(998\) 0 0
\(999\) 312.626 + 180.495i 0.312939 + 0.180675i
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 224.3.o.c.79.5 12
4.3 odd 2 56.3.k.c.51.3 yes 12
7.2 even 3 1568.3.g.k.687.2 6
7.4 even 3 inner 224.3.o.c.207.6 12
7.5 odd 6 1568.3.g.i.687.5 6
8.3 odd 2 inner 224.3.o.c.79.6 12
8.5 even 2 56.3.k.c.51.2 yes 12
28.3 even 6 392.3.k.k.67.2 12
28.11 odd 6 56.3.k.c.11.2 12
28.19 even 6 392.3.g.l.99.5 6
28.23 odd 6 392.3.g.k.99.5 6
28.27 even 2 392.3.k.k.275.3 12
56.5 odd 6 392.3.g.l.99.6 6
56.11 odd 6 inner 224.3.o.c.207.5 12
56.13 odd 2 392.3.k.k.275.2 12
56.19 even 6 1568.3.g.i.687.6 6
56.37 even 6 392.3.g.k.99.6 6
56.45 odd 6 392.3.k.k.67.3 12
56.51 odd 6 1568.3.g.k.687.1 6
56.53 even 6 56.3.k.c.11.3 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.k.c.11.2 12 28.11 odd 6
56.3.k.c.11.3 yes 12 56.53 even 6
56.3.k.c.51.2 yes 12 8.5 even 2
56.3.k.c.51.3 yes 12 4.3 odd 2
224.3.o.c.79.5 12 1.1 even 1 trivial
224.3.o.c.79.6 12 8.3 odd 2 inner
224.3.o.c.207.5 12 56.11 odd 6 inner
224.3.o.c.207.6 12 7.4 even 3 inner
392.3.g.k.99.5 6 28.23 odd 6
392.3.g.k.99.6 6 56.37 even 6
392.3.g.l.99.5 6 28.19 even 6
392.3.g.l.99.6 6 56.5 odd 6
392.3.k.k.67.2 12 28.3 even 6
392.3.k.k.67.3 12 56.45 odd 6
392.3.k.k.275.2 12 56.13 odd 2
392.3.k.k.275.3 12 28.27 even 2
1568.3.g.i.687.5 6 7.5 odd 6
1568.3.g.i.687.6 6 56.19 even 6
1568.3.g.k.687.1 6 56.51 odd 6
1568.3.g.k.687.2 6 7.2 even 3