Properties

Label 1456.2.r.p.417.5
Level $1456$
Weight $2$
Character 1456.417
Analytic conductor $11.626$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1456,2,Mod(417,1456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1456.417");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1456.r (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.6262185343\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 8x^{8} + 7x^{7} + 41x^{6} + 18x^{5} + 58x^{4} + 28x^{3} + 64x^{2} + 16x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 91)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 417.5
Root \(-0.606661 - 1.05077i\) of defining polynomial
Character \(\chi\) \(=\) 1456.417
Dual form 1456.2.r.p.625.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.23721 - 2.14292i) q^{3} +(1.06140 + 1.83839i) q^{5} +(-2.63169 - 0.272389i) q^{7} +(-1.56140 - 2.70442i) q^{9} +O(q^{10})\) \(q+(1.23721 - 2.14292i) q^{3} +(1.06140 + 1.83839i) q^{5} +(-2.63169 - 0.272389i) q^{7} +(-1.56140 - 2.70442i) q^{9} +(2.39448 - 4.14736i) q^{11} +1.00000 q^{13} +5.25271 q^{15} +(1.88914 - 3.27208i) q^{17} +(-1.78362 - 3.08931i) q^{19} +(-3.83967 + 5.30250i) q^{21} +(2.23721 + 3.87497i) q^{23} +(0.246870 - 0.427591i) q^{25} -0.303848 q^{27} -5.90107 q^{29} +(-1.88558 + 3.26592i) q^{31} +(-5.92496 - 10.2623i) q^{33} +(-2.29251 - 5.12720i) q^{35} +(-2.81285 - 4.87200i) q^{37} +(1.23721 - 2.14292i) q^{39} +10.3948 q^{41} -3.40733 q^{43} +(3.31453 - 5.74093i) q^{45} +(-3.55438 - 6.15636i) q^{47} +(6.85161 + 1.43369i) q^{49} +(-4.67454 - 8.09654i) q^{51} +(6.19003 - 10.7214i) q^{53} +10.1660 q^{55} -8.82686 q^{57} +(2.39448 - 4.14736i) q^{59} +(-1.60348 - 2.77732i) q^{61} +(3.37246 + 7.54251i) q^{63} +(1.06140 + 1.83839i) q^{65} +(-1.44978 + 2.51109i) q^{67} +11.0717 q^{69} +2.53876 q^{71} +(-3.85035 + 6.66901i) q^{73} +(-0.610862 - 1.05804i) q^{75} +(-7.43122 + 10.2623i) q^{77} +(-2.58925 - 4.48471i) q^{79} +(4.30827 - 7.46214i) q^{81} -3.46731 q^{83} +8.02051 q^{85} +(-7.30089 + 12.6455i) q^{87} +(-1.83216 - 3.17339i) q^{89} +(-2.63169 - 0.272389i) q^{91} +(4.66574 + 8.08129i) q^{93} +(3.78625 - 6.55798i) q^{95} -5.40733 q^{97} -14.9549 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{5} - q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 2 q^{5} - q^{7} - 3 q^{9} + 11 q^{11} + 10 q^{13} + 5 q^{17} + 9 q^{19} + 2 q^{21} + 10 q^{23} - 9 q^{25} - 6 q^{29} - 6 q^{31} - 8 q^{33} + 4 q^{35} - 4 q^{37} + 28 q^{41} - 4 q^{43} + 32 q^{45} + q^{47} - 11 q^{49} - 8 q^{51} - 17 q^{53} - 32 q^{57} + 11 q^{59} + 11 q^{61} - 5 q^{63} - 2 q^{65} + 13 q^{67} + 36 q^{69} - 30 q^{71} - 20 q^{75} - 46 q^{77} + 2 q^{79} + 19 q^{81} - 12 q^{83} - 44 q^{85} - 8 q^{87} + 4 q^{89} - q^{91} - 18 q^{93} - 12 q^{95} - 24 q^{97} - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1456\mathbb{Z}\right)^\times\).

\(n\) \(561\) \(911\) \(1093\) \(1249\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.23721 2.14292i 0.714306 1.23721i −0.248921 0.968524i \(-0.580076\pi\)
0.963227 0.268690i \(-0.0865908\pi\)
\(4\) 0 0
\(5\) 1.06140 + 1.83839i 0.474671 + 0.822155i 0.999579 0.0290040i \(-0.00923354\pi\)
−0.524908 + 0.851159i \(0.675900\pi\)
\(6\) 0 0
\(7\) −2.63169 0.272389i −0.994686 0.102953i
\(8\) 0 0
\(9\) −1.56140 2.70442i −0.520466 0.901473i
\(10\) 0 0
\(11\) 2.39448 4.14736i 0.721962 1.25048i −0.238250 0.971204i \(-0.576574\pi\)
0.960212 0.279272i \(-0.0900930\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 5.25271 1.35624
\(16\) 0 0
\(17\) 1.88914 3.27208i 0.458183 0.793597i −0.540682 0.841227i \(-0.681834\pi\)
0.998865 + 0.0476304i \(0.0151670\pi\)
\(18\) 0 0
\(19\) −1.78362 3.08931i −0.409190 0.708737i 0.585609 0.810593i \(-0.300855\pi\)
−0.994799 + 0.101856i \(0.967522\pi\)
\(20\) 0 0
\(21\) −3.83967 + 5.30250i −0.837886 + 1.15710i
\(22\) 0 0
\(23\) 2.23721 + 3.87497i 0.466491 + 0.807987i 0.999267 0.0382695i \(-0.0121845\pi\)
−0.532776 + 0.846256i \(0.678851\pi\)
\(24\) 0 0
\(25\) 0.246870 0.427591i 0.0493740 0.0855182i
\(26\) 0 0
\(27\) −0.303848 −0.0584757
\(28\) 0 0
\(29\) −5.90107 −1.09580 −0.547901 0.836543i \(-0.684573\pi\)
−0.547901 + 0.836543i \(0.684573\pi\)
\(30\) 0 0
\(31\) −1.88558 + 3.26592i −0.338660 + 0.586577i −0.984181 0.177166i \(-0.943307\pi\)
0.645521 + 0.763743i \(0.276640\pi\)
\(32\) 0 0
\(33\) −5.92496 10.2623i −1.03140 1.78644i
\(34\) 0 0
\(35\) −2.29251 5.12720i −0.387505 0.866655i
\(36\) 0 0
\(37\) −2.81285 4.87200i −0.462429 0.800951i 0.536652 0.843804i \(-0.319689\pi\)
−0.999081 + 0.0428524i \(0.986355\pi\)
\(38\) 0 0
\(39\) 1.23721 2.14292i 0.198113 0.343141i
\(40\) 0 0
\(41\) 10.3948 1.62340 0.811698 0.584077i \(-0.198543\pi\)
0.811698 + 0.584077i \(0.198543\pi\)
\(42\) 0 0
\(43\) −3.40733 −0.519613 −0.259807 0.965661i \(-0.583659\pi\)
−0.259807 + 0.965661i \(0.583659\pi\)
\(44\) 0 0
\(45\) 3.31453 5.74093i 0.494101 0.855807i
\(46\) 0 0
\(47\) −3.55438 6.15636i −0.518459 0.897998i −0.999770 0.0214479i \(-0.993172\pi\)
0.481311 0.876550i \(-0.340161\pi\)
\(48\) 0 0
\(49\) 6.85161 + 1.43369i 0.978801 + 0.204813i
\(50\) 0 0
\(51\) −4.67454 8.09654i −0.654566 1.13374i
\(52\) 0 0
\(53\) 6.19003 10.7214i 0.850266 1.47270i −0.0307027 0.999529i \(-0.509774\pi\)
0.880968 0.473175i \(-0.156892\pi\)
\(54\) 0 0
\(55\) 10.1660 1.37078
\(56\) 0 0
\(57\) −8.82686 −1.16915
\(58\) 0 0
\(59\) 2.39448 4.14736i 0.311734 0.539940i −0.667003 0.745055i \(-0.732423\pi\)
0.978738 + 0.205115i \(0.0657567\pi\)
\(60\) 0 0
\(61\) −1.60348 2.77732i −0.205305 0.355599i 0.744925 0.667148i \(-0.232485\pi\)
−0.950230 + 0.311550i \(0.899152\pi\)
\(62\) 0 0
\(63\) 3.37246 + 7.54251i 0.424890 + 0.950267i
\(64\) 0 0
\(65\) 1.06140 + 1.83839i 0.131650 + 0.228025i
\(66\) 0 0
\(67\) −1.44978 + 2.51109i −0.177118 + 0.306778i −0.940892 0.338706i \(-0.890011\pi\)
0.763774 + 0.645484i \(0.223344\pi\)
\(68\) 0 0
\(69\) 11.0717 1.33287
\(70\) 0 0
\(71\) 2.53876 0.301295 0.150648 0.988588i \(-0.451864\pi\)
0.150648 + 0.988588i \(0.451864\pi\)
\(72\) 0 0
\(73\) −3.85035 + 6.66901i −0.450650 + 0.780548i −0.998426 0.0560762i \(-0.982141\pi\)
0.547777 + 0.836625i \(0.315474\pi\)
\(74\) 0 0
\(75\) −0.610862 1.05804i −0.0705362 0.122172i
\(76\) 0 0
\(77\) −7.43122 + 10.2623i −0.846867 + 1.16950i
\(78\) 0 0
\(79\) −2.58925 4.48471i −0.291313 0.504569i 0.682807 0.730598i \(-0.260759\pi\)
−0.974120 + 0.226029i \(0.927425\pi\)
\(80\) 0 0
\(81\) 4.30827 7.46214i 0.478696 0.829126i
\(82\) 0 0
\(83\) −3.46731 −0.380587 −0.190294 0.981727i \(-0.560944\pi\)
−0.190294 + 0.981727i \(0.560944\pi\)
\(84\) 0 0
\(85\) 8.02051 0.869946
\(86\) 0 0
\(87\) −7.30089 + 12.6455i −0.782738 + 1.35574i
\(88\) 0 0
\(89\) −1.83216 3.17339i −0.194209 0.336379i 0.752432 0.658670i \(-0.228881\pi\)
−0.946641 + 0.322291i \(0.895547\pi\)
\(90\) 0 0
\(91\) −2.63169 0.272389i −0.275876 0.0285541i
\(92\) 0 0
\(93\) 4.66574 + 8.08129i 0.483814 + 0.837991i
\(94\) 0 0
\(95\) 3.78625 6.55798i 0.388461 0.672835i
\(96\) 0 0
\(97\) −5.40733 −0.549031 −0.274516 0.961583i \(-0.588518\pi\)
−0.274516 + 0.961583i \(0.588518\pi\)
\(98\) 0 0
\(99\) −14.9549 −1.50303
\(100\) 0 0
\(101\) −4.65862 + 8.06897i −0.463550 + 0.802892i −0.999135 0.0415891i \(-0.986758\pi\)
0.535585 + 0.844482i \(0.320091\pi\)
\(102\) 0 0
\(103\) 3.65318 + 6.32749i 0.359958 + 0.623466i 0.987953 0.154751i \(-0.0494576\pi\)
−0.627995 + 0.778217i \(0.716124\pi\)
\(104\) 0 0
\(105\) −13.8235 1.43078i −1.34904 0.139630i
\(106\) 0 0
\(107\) 3.37365 + 5.84333i 0.326143 + 0.564896i 0.981743 0.190212i \(-0.0609176\pi\)
−0.655600 + 0.755108i \(0.727584\pi\)
\(108\) 0 0
\(109\) −2.08822 + 3.61691i −0.200016 + 0.346437i −0.948533 0.316678i \(-0.897433\pi\)
0.748518 + 0.663115i \(0.230766\pi\)
\(110\) 0 0
\(111\) −13.9204 −1.32126
\(112\) 0 0
\(113\) 5.90107 0.555126 0.277563 0.960707i \(-0.410473\pi\)
0.277563 + 0.960707i \(0.410473\pi\)
\(114\) 0 0
\(115\) −4.74915 + 8.22577i −0.442860 + 0.767057i
\(116\) 0 0
\(117\) −1.56140 2.70442i −0.144351 0.250024i
\(118\) 0 0
\(119\) −5.86291 + 8.09654i −0.537452 + 0.742208i
\(120\) 0 0
\(121\) −5.96705 10.3352i −0.542459 0.939567i
\(122\) 0 0
\(123\) 12.8606 22.2752i 1.15960 2.00849i
\(124\) 0 0
\(125\) 11.6621 1.04309
\(126\) 0 0
\(127\) 10.5268 0.934100 0.467050 0.884231i \(-0.345317\pi\)
0.467050 + 0.884231i \(0.345317\pi\)
\(128\) 0 0
\(129\) −4.21560 + 7.30163i −0.371163 + 0.642873i
\(130\) 0 0
\(131\) 2.71204 + 4.69740i 0.236952 + 0.410413i 0.959838 0.280554i \(-0.0905182\pi\)
−0.722886 + 0.690967i \(0.757185\pi\)
\(132\) 0 0
\(133\) 3.85243 + 8.61596i 0.334048 + 0.747099i
\(134\) 0 0
\(135\) −0.322504 0.558593i −0.0277567 0.0480761i
\(136\) 0 0
\(137\) −11.1224 + 19.2645i −0.950248 + 1.64588i −0.205363 + 0.978686i \(0.565837\pi\)
−0.744886 + 0.667192i \(0.767496\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −17.5901 −1.48135
\(142\) 0 0
\(143\) 2.39448 4.14736i 0.200236 0.346819i
\(144\) 0 0
\(145\) −6.26338 10.8485i −0.520146 0.900919i
\(146\) 0 0
\(147\) 11.5492 12.9087i 0.952561 1.06469i
\(148\) 0 0
\(149\) −1.47736 2.55887i −0.121030 0.209630i 0.799144 0.601140i \(-0.205286\pi\)
−0.920174 + 0.391509i \(0.871953\pi\)
\(150\) 0 0
\(151\) −9.27736 + 16.0689i −0.754981 + 1.30766i 0.190403 + 0.981706i \(0.439020\pi\)
−0.945384 + 0.325959i \(0.894313\pi\)
\(152\) 0 0
\(153\) −11.7988 −0.953875
\(154\) 0 0
\(155\) −8.00541 −0.643010
\(156\) 0 0
\(157\) 4.89982 8.48673i 0.391048 0.677315i −0.601540 0.798843i \(-0.705446\pi\)
0.992588 + 0.121528i \(0.0387793\pi\)
\(158\) 0 0
\(159\) −15.3168 26.5294i −1.21470 2.10392i
\(160\) 0 0
\(161\) −4.83216 10.8071i −0.380828 0.851720i
\(162\) 0 0
\(163\) 6.91709 + 11.9808i 0.541788 + 0.938405i 0.998801 + 0.0489451i \(0.0155859\pi\)
−0.457013 + 0.889460i \(0.651081\pi\)
\(164\) 0 0
\(165\) 12.5775 21.7848i 0.979156 1.69595i
\(166\) 0 0
\(167\) −17.3534 −1.34285 −0.671424 0.741073i \(-0.734317\pi\)
−0.671424 + 0.741073i \(0.734317\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −5.56987 + 9.64730i −0.425939 + 0.737747i
\(172\) 0 0
\(173\) 1.48069 + 2.56463i 0.112575 + 0.194985i 0.916808 0.399329i \(-0.130757\pi\)
−0.804233 + 0.594314i \(0.797424\pi\)
\(174\) 0 0
\(175\) −0.766156 + 1.05804i −0.0579160 + 0.0799806i
\(176\) 0 0
\(177\) −5.92496 10.2623i −0.445348 0.771365i
\(178\) 0 0
\(179\) −2.83444 + 4.90939i −0.211856 + 0.366945i −0.952295 0.305178i \(-0.901284\pi\)
0.740440 + 0.672123i \(0.234617\pi\)
\(180\) 0 0
\(181\) 7.17645 0.533421 0.266711 0.963777i \(-0.414063\pi\)
0.266711 + 0.963777i \(0.414063\pi\)
\(182\) 0 0
\(183\) −7.93541 −0.586603
\(184\) 0 0
\(185\) 5.97110 10.3423i 0.439004 0.760377i
\(186\) 0 0
\(187\) −9.04700 15.6699i −0.661582 1.14589i
\(188\) 0 0
\(189\) 0.799636 + 0.0827650i 0.0581649 + 0.00602027i
\(190\) 0 0
\(191\) 5.94088 + 10.2899i 0.429867 + 0.744552i 0.996861 0.0791703i \(-0.0252271\pi\)
−0.566994 + 0.823722i \(0.691894\pi\)
\(192\) 0 0
\(193\) −11.4851 + 19.8927i −0.826714 + 1.43191i 0.0738876 + 0.997267i \(0.476459\pi\)
−0.900602 + 0.434645i \(0.856874\pi\)
\(194\) 0 0
\(195\) 5.25271 0.376154
\(196\) 0 0
\(197\) 16.9216 1.20561 0.602806 0.797888i \(-0.294049\pi\)
0.602806 + 0.797888i \(0.294049\pi\)
\(198\) 0 0
\(199\) 5.02953 8.71140i 0.356534 0.617535i −0.630845 0.775909i \(-0.717292\pi\)
0.987379 + 0.158374i \(0.0506251\pi\)
\(200\) 0 0
\(201\) 3.58737 + 6.21351i 0.253034 + 0.438267i
\(202\) 0 0
\(203\) 15.5298 + 1.60739i 1.08998 + 0.112817i
\(204\) 0 0
\(205\) 11.0330 + 19.1098i 0.770580 + 1.33468i
\(206\) 0 0
\(207\) 6.98636 12.1007i 0.485586 0.841059i
\(208\) 0 0
\(209\) −17.0833 −1.18168
\(210\) 0 0
\(211\) 24.4609 1.68396 0.841978 0.539512i \(-0.181391\pi\)
0.841978 + 0.539512i \(0.181391\pi\)
\(212\) 0 0
\(213\) 3.14099 5.44035i 0.215217 0.372767i
\(214\) 0 0
\(215\) −3.61654 6.26402i −0.246646 0.427203i
\(216\) 0 0
\(217\) 5.85187 8.08129i 0.397251 0.548594i
\(218\) 0 0
\(219\) 9.52742 + 16.5020i 0.643804 + 1.11510i
\(220\) 0 0
\(221\) 1.88914 3.27208i 0.127077 0.220104i
\(222\) 0 0
\(223\) 29.2625 1.95956 0.979780 0.200076i \(-0.0641188\pi\)
0.979780 + 0.200076i \(0.0641188\pi\)
\(224\) 0 0
\(225\) −1.54185 −0.102790
\(226\) 0 0
\(227\) −5.03685 + 8.72408i −0.334307 + 0.579038i −0.983352 0.181713i \(-0.941836\pi\)
0.649044 + 0.760751i \(0.275169\pi\)
\(228\) 0 0
\(229\) 5.56997 + 9.64748i 0.368074 + 0.637523i 0.989264 0.146137i \(-0.0466839\pi\)
−0.621190 + 0.783660i \(0.713351\pi\)
\(230\) 0 0
\(231\) 12.7973 + 28.6212i 0.842003 + 1.88314i
\(232\) 0 0
\(233\) 8.54166 + 14.7946i 0.559583 + 0.969226i 0.997531 + 0.0702257i \(0.0223720\pi\)
−0.437948 + 0.899000i \(0.644295\pi\)
\(234\) 0 0
\(235\) 7.54522 13.0687i 0.492196 0.852508i
\(236\) 0 0
\(237\) −12.8138 −0.832347
\(238\) 0 0
\(239\) −6.92142 −0.447710 −0.223855 0.974622i \(-0.571864\pi\)
−0.223855 + 0.974622i \(0.571864\pi\)
\(240\) 0 0
\(241\) −3.24812 + 5.62592i −0.209230 + 0.362397i −0.951472 0.307735i \(-0.900429\pi\)
0.742242 + 0.670132i \(0.233762\pi\)
\(242\) 0 0
\(243\) −11.1163 19.2539i −0.713109 1.23514i
\(244\) 0 0
\(245\) 4.63660 + 14.1177i 0.296221 + 0.901945i
\(246\) 0 0
\(247\) −1.78362 3.08931i −0.113489 0.196568i
\(248\) 0 0
\(249\) −4.28981 + 7.43017i −0.271856 + 0.470868i
\(250\) 0 0
\(251\) 9.86804 0.622865 0.311433 0.950268i \(-0.399191\pi\)
0.311433 + 0.950268i \(0.399191\pi\)
\(252\) 0 0
\(253\) 21.4278 1.34716
\(254\) 0 0
\(255\) 9.92309 17.1873i 0.621408 1.07631i
\(256\) 0 0
\(257\) −3.43234 5.94499i −0.214104 0.370838i 0.738891 0.673825i \(-0.235350\pi\)
−0.952995 + 0.302986i \(0.902016\pi\)
\(258\) 0 0
\(259\) 6.07547 + 13.5878i 0.377511 + 0.844304i
\(260\) 0 0
\(261\) 9.21392 + 15.9590i 0.570327 + 0.987836i
\(262\) 0 0
\(263\) −0.0632753 + 0.109596i −0.00390172 + 0.00675798i −0.867970 0.496617i \(-0.834575\pi\)
0.864068 + 0.503375i \(0.167909\pi\)
\(264\) 0 0
\(265\) 26.2803 1.61439
\(266\) 0 0
\(267\) −9.06710 −0.554897
\(268\) 0 0
\(269\) 2.12154 3.67462i 0.129353 0.224045i −0.794073 0.607822i \(-0.792043\pi\)
0.923426 + 0.383777i \(0.125377\pi\)
\(270\) 0 0
\(271\) 0.783616 + 1.35726i 0.0476013 + 0.0824479i 0.888844 0.458209i \(-0.151509\pi\)
−0.841243 + 0.540657i \(0.818176\pi\)
\(272\) 0 0
\(273\) −3.83967 + 5.30250i −0.232388 + 0.320922i
\(274\) 0 0
\(275\) −1.18225 2.04771i −0.0712923 0.123482i
\(276\) 0 0
\(277\) 6.37260 11.0377i 0.382892 0.663189i −0.608582 0.793491i \(-0.708261\pi\)
0.991474 + 0.130302i \(0.0415947\pi\)
\(278\) 0 0
\(279\) 11.7766 0.705045
\(280\) 0 0
\(281\) 4.62986 0.276194 0.138097 0.990419i \(-0.455901\pi\)
0.138097 + 0.990419i \(0.455901\pi\)
\(282\) 0 0
\(283\) 1.82416 3.15954i 0.108435 0.187815i −0.806701 0.590959i \(-0.798749\pi\)
0.915136 + 0.403144i \(0.132083\pi\)
\(284\) 0 0
\(285\) −9.36881 16.2273i −0.554960 0.961220i
\(286\) 0 0
\(287\) −27.3559 2.83143i −1.61477 0.167134i
\(288\) 0 0
\(289\) 1.36231 + 2.35959i 0.0801360 + 0.138800i
\(290\) 0 0
\(291\) −6.69003 + 11.5875i −0.392176 + 0.679269i
\(292\) 0 0
\(293\) −21.0415 −1.22926 −0.614630 0.788816i \(-0.710695\pi\)
−0.614630 + 0.788816i \(0.710695\pi\)
\(294\) 0 0
\(295\) 10.1660 0.591886
\(296\) 0 0
\(297\) −0.727559 + 1.26017i −0.0422172 + 0.0731224i
\(298\) 0 0
\(299\) 2.23721 + 3.87497i 0.129381 + 0.224095i
\(300\) 0 0
\(301\) 8.96705 + 0.928120i 0.516852 + 0.0534960i
\(302\) 0 0
\(303\) 11.5274 + 19.9661i 0.662233 + 1.14702i
\(304\) 0 0
\(305\) 3.40387 5.89567i 0.194905 0.337585i
\(306\) 0 0
\(307\) 4.95861 0.283003 0.141502 0.989938i \(-0.454807\pi\)
0.141502 + 0.989938i \(0.454807\pi\)
\(308\) 0 0
\(309\) 18.0791 1.02848
\(310\) 0 0
\(311\) −1.21079 + 2.09715i −0.0686575 + 0.118918i −0.898311 0.439361i \(-0.855205\pi\)
0.829653 + 0.558279i \(0.188538\pi\)
\(312\) 0 0
\(313\) −6.98026 12.0902i −0.394548 0.683377i 0.598496 0.801126i \(-0.295765\pi\)
−0.993043 + 0.117749i \(0.962432\pi\)
\(314\) 0 0
\(315\) −10.2866 + 14.2055i −0.579583 + 0.800390i
\(316\) 0 0
\(317\) −1.53431 2.65750i −0.0861753 0.149260i 0.819716 0.572770i \(-0.194131\pi\)
−0.905891 + 0.423510i \(0.860798\pi\)
\(318\) 0 0
\(319\) −14.1300 + 24.4739i −0.791127 + 1.37027i
\(320\) 0 0
\(321\) 16.6957 0.931863
\(322\) 0 0
\(323\) −13.4780 −0.749936
\(324\) 0 0
\(325\) 0.246870 0.427591i 0.0136939 0.0237185i
\(326\) 0 0
\(327\) 5.16716 + 8.94978i 0.285745 + 0.494924i
\(328\) 0 0
\(329\) 7.67710 + 17.1698i 0.423252 + 0.946603i
\(330\) 0 0
\(331\) 6.80261 + 11.7825i 0.373905 + 0.647623i 0.990162 0.139922i \(-0.0446853\pi\)
−0.616257 + 0.787545i \(0.711352\pi\)
\(332\) 0 0
\(333\) −8.78395 + 15.2142i −0.481358 + 0.833736i
\(334\) 0 0
\(335\) −6.15516 −0.336292
\(336\) 0 0
\(337\) −35.1646 −1.91554 −0.957769 0.287538i \(-0.907163\pi\)
−0.957769 + 0.287538i \(0.907163\pi\)
\(338\) 0 0
\(339\) 7.30089 12.6455i 0.396530 0.686810i
\(340\) 0 0
\(341\) 9.02997 + 15.6404i 0.489000 + 0.846973i
\(342\) 0 0
\(343\) −17.6408 5.63933i −0.952514 0.304495i
\(344\) 0 0
\(345\) 11.7514 + 20.3541i 0.632676 + 1.09583i
\(346\) 0 0
\(347\) −2.73551 + 4.73804i −0.146850 + 0.254351i −0.930062 0.367404i \(-0.880247\pi\)
0.783212 + 0.621755i \(0.213580\pi\)
\(348\) 0 0
\(349\) 4.34196 0.232420 0.116210 0.993225i \(-0.462925\pi\)
0.116210 + 0.993225i \(0.462925\pi\)
\(350\) 0 0
\(351\) −0.303848 −0.0162182
\(352\) 0 0
\(353\) −13.7996 + 23.9016i −0.734479 + 1.27216i 0.220472 + 0.975393i \(0.429240\pi\)
−0.954951 + 0.296762i \(0.904093\pi\)
\(354\) 0 0
\(355\) 2.69463 + 4.66724i 0.143016 + 0.247712i
\(356\) 0 0
\(357\) 10.0965 + 22.5809i 0.534365 + 1.19511i
\(358\) 0 0
\(359\) 3.31427 + 5.74049i 0.174921 + 0.302971i 0.940134 0.340806i \(-0.110700\pi\)
−0.765213 + 0.643777i \(0.777366\pi\)
\(360\) 0 0
\(361\) 3.13742 5.43418i 0.165128 0.286009i
\(362\) 0 0
\(363\) −29.5301 −1.54993
\(364\) 0 0
\(365\) −16.3470 −0.855643
\(366\) 0 0
\(367\) 15.6037 27.0264i 0.814506 1.41077i −0.0951768 0.995460i \(-0.530342\pi\)
0.909682 0.415305i \(-0.136325\pi\)
\(368\) 0 0
\(369\) −16.2304 28.1119i −0.844923 1.46345i
\(370\) 0 0
\(371\) −19.2107 + 26.5294i −0.997368 + 1.37734i
\(372\) 0 0
\(373\) 7.88730 + 13.6612i 0.408389 + 0.707350i 0.994709 0.102729i \(-0.0327574\pi\)
−0.586321 + 0.810079i \(0.699424\pi\)
\(374\) 0 0
\(375\) 14.4285 24.9909i 0.745084 1.29052i
\(376\) 0 0
\(377\) −5.90107 −0.303921
\(378\) 0 0
\(379\) −31.6512 −1.62581 −0.812907 0.582393i \(-0.802116\pi\)
−0.812907 + 0.582393i \(0.802116\pi\)
\(380\) 0 0
\(381\) 13.0239 22.5580i 0.667233 1.15568i
\(382\) 0 0
\(383\) 6.19675 + 10.7331i 0.316639 + 0.548435i 0.979785 0.200055i \(-0.0641122\pi\)
−0.663145 + 0.748491i \(0.730779\pi\)
\(384\) 0 0
\(385\) −26.7537 2.76910i −1.36350 0.141126i
\(386\) 0 0
\(387\) 5.32020 + 9.21486i 0.270441 + 0.468418i
\(388\) 0 0
\(389\) 7.03705 12.1885i 0.356792 0.617983i −0.630631 0.776083i \(-0.717204\pi\)
0.987423 + 0.158100i \(0.0505370\pi\)
\(390\) 0 0
\(391\) 16.9056 0.854954
\(392\) 0 0
\(393\) 13.4215 0.677026
\(394\) 0 0
\(395\) 5.49644 9.52012i 0.276556 0.479009i
\(396\) 0 0
\(397\) 3.48652 + 6.03884i 0.174984 + 0.303081i 0.940156 0.340745i \(-0.110679\pi\)
−0.765172 + 0.643826i \(0.777346\pi\)
\(398\) 0 0
\(399\) 23.2296 + 2.40434i 1.16293 + 0.120368i
\(400\) 0 0
\(401\) −1.36841 2.37016i −0.0683352 0.118360i 0.829833 0.558011i \(-0.188435\pi\)
−0.898169 + 0.439651i \(0.855102\pi\)
\(402\) 0 0
\(403\) −1.88558 + 3.26592i −0.0939275 + 0.162687i
\(404\) 0 0
\(405\) 18.2911 0.908894
\(406\) 0 0
\(407\) −26.9412 −1.33543
\(408\) 0 0
\(409\) 12.2577 21.2309i 0.606104 1.04980i −0.385772 0.922594i \(-0.626065\pi\)
0.991876 0.127208i \(-0.0406017\pi\)
\(410\) 0 0
\(411\) 27.5215 + 47.6686i 1.35754 + 2.35132i
\(412\) 0 0
\(413\) −7.43122 + 10.2623i −0.365667 + 0.504977i
\(414\) 0 0
\(415\) −3.68020 6.37429i −0.180654 0.312902i
\(416\) 0 0
\(417\) 4.94886 8.57167i 0.242347 0.419757i
\(418\) 0 0
\(419\) 3.01252 0.147171 0.0735856 0.997289i \(-0.476556\pi\)
0.0735856 + 0.997289i \(0.476556\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) −11.0996 + 19.2251i −0.539681 + 0.934755i
\(424\) 0 0
\(425\) −0.932742 1.61556i −0.0452447 0.0783660i
\(426\) 0 0
\(427\) 3.46337 + 7.74581i 0.167604 + 0.374846i
\(428\) 0 0
\(429\) −5.92496 10.2623i −0.286060 0.495470i
\(430\) 0 0
\(431\) −9.39711 + 16.2763i −0.452643 + 0.784001i −0.998549 0.0538455i \(-0.982852\pi\)
0.545906 + 0.837846i \(0.316185\pi\)
\(432\) 0 0
\(433\) −7.76911 −0.373360 −0.186680 0.982421i \(-0.559773\pi\)
−0.186680 + 0.982421i \(0.559773\pi\)
\(434\) 0 0
\(435\) −30.9966 −1.48617
\(436\) 0 0
\(437\) 7.98066 13.8229i 0.381767 0.661240i
\(438\) 0 0
\(439\) −18.9841 32.8814i −0.906060 1.56934i −0.819488 0.573096i \(-0.805742\pi\)
−0.0865713 0.996246i \(-0.527591\pi\)
\(440\) 0 0
\(441\) −6.82079 20.7682i −0.324799 0.988961i
\(442\) 0 0
\(443\) 17.8135 + 30.8539i 0.846344 + 1.46591i 0.884449 + 0.466637i \(0.154535\pi\)
−0.0381050 + 0.999274i \(0.512132\pi\)
\(444\) 0 0
\(445\) 3.88930 6.73647i 0.184371 0.319339i
\(446\) 0 0
\(447\) −7.31125 −0.345810
\(448\) 0 0
\(449\) −8.05285 −0.380038 −0.190019 0.981780i \(-0.560855\pi\)
−0.190019 + 0.981780i \(0.560855\pi\)
\(450\) 0 0
\(451\) 24.8901 43.1110i 1.17203 2.03002i
\(452\) 0 0
\(453\) 22.9562 + 39.7612i 1.07857 + 1.86815i
\(454\) 0 0
\(455\) −2.29251 5.12720i −0.107475 0.240367i
\(456\) 0 0
\(457\) −7.79881 13.5079i −0.364813 0.631875i 0.623933 0.781478i \(-0.285534\pi\)
−0.988746 + 0.149603i \(0.952200\pi\)
\(458\) 0 0
\(459\) −0.574012 + 0.994218i −0.0267926 + 0.0464061i
\(460\) 0 0
\(461\) −25.6991 −1.19692 −0.598462 0.801151i \(-0.704221\pi\)
−0.598462 + 0.801151i \(0.704221\pi\)
\(462\) 0 0
\(463\) 20.5209 0.953685 0.476842 0.878989i \(-0.341781\pi\)
0.476842 + 0.878989i \(0.341781\pi\)
\(464\) 0 0
\(465\) −9.90440 + 17.1549i −0.459306 + 0.795541i
\(466\) 0 0
\(467\) 5.91241 + 10.2406i 0.273594 + 0.473878i 0.969779 0.243984i \(-0.0784543\pi\)
−0.696186 + 0.717862i \(0.745121\pi\)
\(468\) 0 0
\(469\) 4.49936 6.21351i 0.207761 0.286913i
\(470\) 0 0
\(471\) −12.1242 20.9998i −0.558656 0.967620i
\(472\) 0 0
\(473\) −8.15878 + 14.1314i −0.375141 + 0.649764i
\(474\) 0 0
\(475\) −1.76128 −0.0808133
\(476\) 0 0
\(477\) −38.6604 −1.77014
\(478\) 0 0
\(479\) 11.3276 19.6200i 0.517571 0.896459i −0.482221 0.876050i \(-0.660170\pi\)
0.999792 0.0204092i \(-0.00649690\pi\)
\(480\) 0 0
\(481\) −2.81285 4.87200i −0.128255 0.222144i
\(482\) 0 0
\(483\) −29.1372 3.01580i −1.32579 0.137224i
\(484\) 0 0
\(485\) −5.73933 9.94081i −0.260610 0.451389i
\(486\) 0 0
\(487\) −16.3584 + 28.3335i −0.741268 + 1.28391i 0.210650 + 0.977562i \(0.432442\pi\)
−0.951918 + 0.306353i \(0.900891\pi\)
\(488\) 0 0
\(489\) 34.2317 1.54801
\(490\) 0 0
\(491\) −6.17281 −0.278575 −0.139288 0.990252i \(-0.544481\pi\)
−0.139288 + 0.990252i \(0.544481\pi\)
\(492\) 0 0
\(493\) −11.1479 + 19.3088i −0.502078 + 0.869625i
\(494\) 0 0
\(495\) −15.8731 27.4931i −0.713444 1.23572i
\(496\) 0 0
\(497\) −6.68123 0.691531i −0.299694 0.0310194i
\(498\) 0 0
\(499\) −7.31934 12.6775i −0.327659 0.567521i 0.654388 0.756159i \(-0.272926\pi\)
−0.982047 + 0.188637i \(0.939593\pi\)
\(500\) 0 0
\(501\) −21.4699 + 37.1870i −0.959205 + 1.66139i
\(502\) 0 0
\(503\) −12.7787 −0.569774 −0.284887 0.958561i \(-0.591956\pi\)
−0.284887 + 0.958561i \(0.591956\pi\)
\(504\) 0 0
\(505\) −19.7786 −0.880136
\(506\) 0 0
\(507\) 1.23721 2.14292i 0.0549466 0.0951703i
\(508\) 0 0
\(509\) −5.84263 10.1197i −0.258970 0.448549i 0.706996 0.707217i \(-0.250050\pi\)
−0.965966 + 0.258668i \(0.916716\pi\)
\(510\) 0 0
\(511\) 11.9495 16.5020i 0.528615 0.730005i
\(512\) 0 0
\(513\) 0.541949 + 0.938683i 0.0239276 + 0.0414439i
\(514\) 0 0
\(515\) −7.75495 + 13.4320i −0.341724 + 0.591883i
\(516\) 0 0
\(517\) −34.0435 −1.49723
\(518\) 0 0
\(519\) 7.32772 0.321651
\(520\) 0 0
\(521\) −4.23838 + 7.34108i −0.185687 + 0.321619i −0.943808 0.330495i \(-0.892784\pi\)
0.758121 + 0.652114i \(0.226118\pi\)
\(522\) 0 0
\(523\) 16.3554 + 28.3284i 0.715172 + 1.23871i 0.962893 + 0.269883i \(0.0869849\pi\)
−0.247721 + 0.968831i \(0.579682\pi\)
\(524\) 0 0
\(525\) 1.31940 + 2.95084i 0.0575833 + 0.128785i
\(526\) 0 0
\(527\) 7.12425 + 12.3396i 0.310337 + 0.537520i
\(528\) 0 0
\(529\) 1.48975 2.58032i 0.0647716 0.112188i
\(530\) 0 0
\(531\) −14.9549 −0.648989
\(532\) 0 0
\(533\) 10.3948 0.450249
\(534\) 0 0
\(535\) −7.16156 + 12.4042i −0.309621 + 0.536280i
\(536\) 0 0
\(537\) 7.01361 + 12.1479i 0.302660 + 0.524222i
\(538\) 0 0
\(539\) 22.3520 24.9831i 0.962771 1.07610i
\(540\) 0 0
\(541\) 14.0853 + 24.3964i 0.605573 + 1.04888i 0.991961 + 0.126547i \(0.0403893\pi\)
−0.386388 + 0.922336i \(0.626277\pi\)
\(542\) 0 0
\(543\) 8.87880 15.3785i 0.381026 0.659956i
\(544\) 0 0
\(545\) −8.86574 −0.379767
\(546\) 0 0
\(547\) 18.5377 0.792615 0.396307 0.918118i \(-0.370291\pi\)
0.396307 + 0.918118i \(0.370291\pi\)
\(548\) 0 0
\(549\) −5.00735 + 8.67299i −0.213709 + 0.370154i
\(550\) 0 0
\(551\) 10.5252 + 18.2303i 0.448391 + 0.776635i
\(552\) 0 0
\(553\) 5.59252 + 12.5077i 0.237818 + 0.531880i
\(554\) 0 0
\(555\) −14.7751 25.5912i −0.627167 1.08628i
\(556\) 0 0
\(557\) −2.00142 + 3.46655i −0.0848027 + 0.146883i −0.905307 0.424758i \(-0.860359\pi\)
0.820504 + 0.571640i \(0.193693\pi\)
\(558\) 0 0
\(559\) −3.40733 −0.144115
\(560\) 0 0
\(561\) −44.7723 −1.89029
\(562\) 0 0
\(563\) −8.93100 + 15.4689i −0.376397 + 0.651938i −0.990535 0.137260i \(-0.956170\pi\)
0.614138 + 0.789199i \(0.289504\pi\)
\(564\) 0 0
\(565\) 6.26338 + 10.8485i 0.263503 + 0.456400i
\(566\) 0 0
\(567\) −13.3706 + 18.4645i −0.561514 + 0.775437i
\(568\) 0 0
\(569\) 18.7336 + 32.4475i 0.785353 + 1.36027i 0.928788 + 0.370612i \(0.120852\pi\)
−0.143434 + 0.989660i \(0.545815\pi\)
\(570\) 0 0
\(571\) 8.78514 15.2163i 0.367646 0.636782i −0.621551 0.783374i \(-0.713497\pi\)
0.989197 + 0.146592i \(0.0468303\pi\)
\(572\) 0 0
\(573\) 29.4006 1.22823
\(574\) 0 0
\(575\) 2.20920 0.0921301
\(576\) 0 0
\(577\) 17.1247 29.6608i 0.712910 1.23480i −0.250850 0.968026i \(-0.580710\pi\)
0.963760 0.266770i \(-0.0859565\pi\)
\(578\) 0 0
\(579\) 28.4190 + 49.2232i 1.18105 + 2.04565i
\(580\) 0 0
\(581\) 9.12490 + 0.944459i 0.378565 + 0.0391828i
\(582\) 0 0
\(583\) −29.6438 51.3445i −1.22772 2.12647i
\(584\) 0 0
\(585\) 3.31453 5.74093i 0.137039 0.237358i
\(586\) 0 0
\(587\) 29.4494 1.21551 0.607754 0.794126i \(-0.292071\pi\)
0.607754 + 0.794126i \(0.292071\pi\)
\(588\) 0 0
\(589\) 13.4526 0.554305
\(590\) 0 0
\(591\) 20.9356 36.2616i 0.861176 1.49160i
\(592\) 0 0
\(593\) 17.0001 + 29.4450i 0.698109 + 1.20916i 0.969121 + 0.246584i \(0.0793081\pi\)
−0.271013 + 0.962576i \(0.587359\pi\)
\(594\) 0 0
\(595\) −21.1075 2.18470i −0.865324 0.0895640i
\(596\) 0 0
\(597\) −12.4452 21.5557i −0.509349 0.882218i
\(598\) 0 0
\(599\) 10.7209 18.5691i 0.438043 0.758713i −0.559495 0.828834i \(-0.689005\pi\)
0.997539 + 0.0701203i \(0.0223383\pi\)
\(600\) 0 0
\(601\) 40.4039 1.64811 0.824054 0.566511i \(-0.191707\pi\)
0.824054 + 0.566511i \(0.191707\pi\)
\(602\) 0 0
\(603\) 9.05472 0.368737
\(604\) 0 0
\(605\) 12.6668 21.9396i 0.514980 0.891971i
\(606\) 0 0
\(607\) −21.9456 38.0110i −0.890746 1.54282i −0.838983 0.544158i \(-0.816849\pi\)
−0.0517636 0.998659i \(-0.516484\pi\)
\(608\) 0 0
\(609\) 22.6582 31.2904i 0.918156 1.26795i
\(610\) 0 0
\(611\) −3.55438 6.15636i −0.143795 0.249060i
\(612\) 0 0
\(613\) 7.15777 12.3976i 0.289100 0.500735i −0.684496 0.729017i \(-0.739977\pi\)
0.973595 + 0.228282i \(0.0733108\pi\)
\(614\) 0 0
\(615\) 54.6009 2.20172
\(616\) 0 0
\(617\) −36.9097 −1.48593 −0.742965 0.669330i \(-0.766581\pi\)
−0.742965 + 0.669330i \(0.766581\pi\)
\(618\) 0 0
\(619\) 7.14646 12.3780i 0.287240 0.497515i −0.685910 0.727687i \(-0.740595\pi\)
0.973150 + 0.230172i \(0.0739288\pi\)
\(620\) 0 0
\(621\) −0.679774 1.17740i −0.0272784 0.0472476i
\(622\) 0 0
\(623\) 3.95728 + 8.85046i 0.158545 + 0.354586i
\(624\) 0 0
\(625\) 11.1438 + 19.3016i 0.445750 + 0.772062i
\(626\) 0 0
\(627\) −21.1357 + 36.6082i −0.844080 + 1.46199i
\(628\) 0 0
\(629\) −21.2554 −0.847510
\(630\) 0 0
\(631\) 0.0431064 0.00171604 0.000858019 1.00000i \(-0.499727\pi\)
0.000858019 1.00000i \(0.499727\pi\)
\(632\) 0 0
\(633\) 30.2633 52.4176i 1.20286 2.08341i
\(634\) 0 0
\(635\) 11.1731 + 19.3524i 0.443390 + 0.767975i
\(636\) 0 0
\(637\) 6.85161 + 1.43369i 0.271471 + 0.0568048i
\(638\) 0 0
\(639\) −3.96401 6.86587i −0.156814 0.271610i
\(640\) 0 0
\(641\) −21.3328 + 36.9494i −0.842594 + 1.45942i 0.0451008 + 0.998982i \(0.485639\pi\)
−0.887695 + 0.460433i \(0.847694\pi\)
\(642\) 0 0
\(643\) 5.49737 0.216795 0.108398 0.994108i \(-0.465428\pi\)
0.108398 + 0.994108i \(0.465428\pi\)
\(644\) 0 0
\(645\) −17.8977 −0.704722
\(646\) 0 0
\(647\) −19.0933 + 33.0706i −0.750637 + 1.30014i 0.196877 + 0.980428i \(0.436920\pi\)
−0.947514 + 0.319713i \(0.896413\pi\)
\(648\) 0 0
\(649\) −11.4671 19.8615i −0.450121 0.779633i
\(650\) 0 0
\(651\) −10.0775 22.5384i −0.394969 0.883348i
\(652\) 0 0
\(653\) −19.2510 33.3437i −0.753349 1.30484i −0.946191 0.323608i \(-0.895104\pi\)
0.192843 0.981230i \(-0.438229\pi\)
\(654\) 0 0
\(655\) −5.75711 + 9.97161i −0.224949 + 0.389623i
\(656\) 0 0
\(657\) 24.0477 0.938192
\(658\) 0 0
\(659\) −19.4843 −0.759002 −0.379501 0.925191i \(-0.623904\pi\)
−0.379501 + 0.925191i \(0.623904\pi\)
\(660\) 0 0
\(661\) 20.8334 36.0844i 0.810324 1.40352i −0.102314 0.994752i \(-0.532625\pi\)
0.912638 0.408770i \(-0.134042\pi\)
\(662\) 0 0
\(663\) −4.67454 8.09654i −0.181544 0.314443i
\(664\) 0 0
\(665\) −11.7506 + 16.2273i −0.455668 + 0.629266i
\(666\) 0 0
\(667\) −13.2020 22.8665i −0.511182 0.885393i
\(668\) 0 0
\(669\) 36.2040 62.7071i 1.39973 2.42440i
\(670\) 0 0
\(671\) −15.3580 −0.592890
\(672\) 0 0
\(673\) −14.3157 −0.551830 −0.275915 0.961182i \(-0.588981\pi\)
−0.275915 + 0.961182i \(0.588981\pi\)
\(674\) 0 0
\(675\) −0.0750110 + 0.129923i −0.00288718 + 0.00500073i
\(676\) 0 0
\(677\) 14.7641 + 25.5721i 0.567429 + 0.982815i 0.996819 + 0.0796963i \(0.0253950\pi\)
−0.429391 + 0.903119i \(0.641272\pi\)
\(678\) 0 0
\(679\) 14.2304 + 1.47290i 0.546114 + 0.0565247i
\(680\) 0 0
\(681\) 12.4633 + 21.5871i 0.477596 + 0.827220i
\(682\) 0 0
\(683\) 23.5349 40.7637i 0.900539 1.55978i 0.0737441 0.997277i \(-0.476505\pi\)
0.826795 0.562503i \(-0.190161\pi\)
\(684\) 0 0
\(685\) −47.2210 −1.80422
\(686\) 0 0
\(687\) 27.5650 1.05167
\(688\) 0 0
\(689\) 6.19003 10.7214i 0.235821 0.408454i
\(690\) 0 0
\(691\) −15.4334 26.7314i −0.587113 1.01691i −0.994608 0.103703i \(-0.966931\pi\)
0.407495 0.913207i \(-0.366402\pi\)
\(692\) 0 0
\(693\) 39.3568 + 4.07356i 1.49504 + 0.154742i
\(694\) 0 0
\(695\) 4.24559 + 7.35358i 0.161044 + 0.278937i
\(696\) 0 0
\(697\) 19.6372 34.0127i 0.743813 1.28832i
\(698\) 0 0
\(699\) 42.2715 1.59885
\(700\) 0 0
\(701\) 6.48958 0.245108 0.122554 0.992462i \(-0.460892\pi\)
0.122554 + 0.992462i \(0.460892\pi\)
\(702\) 0 0
\(703\) −10.0341 + 17.3795i −0.378443 + 0.655482i
\(704\) 0 0
\(705\) −18.6701 32.3376i −0.703157 1.21790i
\(706\) 0 0
\(707\) 14.4580 19.9661i 0.543747 0.750902i
\(708\) 0 0
\(709\) 6.68689 + 11.5820i 0.251131 + 0.434972i 0.963838 0.266490i \(-0.0858641\pi\)
−0.712706 + 0.701463i \(0.752531\pi\)
\(710\) 0 0
\(711\) −8.08569 + 14.0048i −0.303237 + 0.525222i
\(712\) 0 0
\(713\) −16.8738 −0.631929
\(714\) 0 0
\(715\) 10.1660 0.380186
\(716\) 0 0
\(717\) −8.56328 + 14.8320i −0.319802 + 0.553913i
\(718\) 0 0
\(719\) −8.37048 14.4981i −0.312166 0.540688i 0.666665 0.745358i \(-0.267721\pi\)
−0.978831 + 0.204670i \(0.934388\pi\)
\(720\) 0 0
\(721\) −7.89050 17.6471i −0.293858 0.657212i
\(722\) 0 0
\(723\) 8.03725 + 13.9209i 0.298909 + 0.517725i
\(724\) 0 0
\(725\) −1.45680 + 2.52325i −0.0541041 + 0.0937110i
\(726\) 0 0
\(727\) −38.8138 −1.43952 −0.719761 0.694221i \(-0.755749\pi\)
−0.719761 + 0.694221i \(0.755749\pi\)
\(728\) 0 0
\(729\) −29.1632 −1.08012
\(730\) 0 0
\(731\) −6.43692 + 11.1491i −0.238078 + 0.412364i
\(732\) 0 0
\(733\) −18.8639 32.6733i −0.696756 1.20682i −0.969585 0.244754i \(-0.921293\pi\)
0.272830 0.962062i \(-0.412040\pi\)
\(734\) 0 0
\(735\) 35.9895 + 7.53074i 1.32749 + 0.277776i
\(736\) 0 0
\(737\) 6.94292 + 12.0255i 0.255746 + 0.442965i
\(738\) 0 0
\(739\) −4.61476 + 7.99300i −0.169757 + 0.294027i −0.938334 0.345729i \(-0.887632\pi\)
0.768578 + 0.639757i \(0.220965\pi\)
\(740\) 0 0
\(741\) −8.82686 −0.324263
\(742\) 0 0
\(743\) 3.56327 0.130724 0.0653619 0.997862i \(-0.479180\pi\)
0.0653619 + 0.997862i \(0.479180\pi\)
\(744\) 0 0
\(745\) 3.13614 5.43195i 0.114899 0.199011i
\(746\) 0 0
\(747\) 5.41386 + 9.37707i 0.198083 + 0.343089i
\(748\) 0 0
\(749\) −7.28674 16.2968i −0.266252 0.595472i
\(750\) 0 0
\(751\) 25.6053 + 44.3496i 0.934350 + 1.61834i 0.775789 + 0.630992i \(0.217352\pi\)
0.158561 + 0.987349i \(0.449315\pi\)
\(752\) 0 0
\(753\) 12.2089 21.1464i 0.444916 0.770618i
\(754\) 0 0
\(755\) −39.3879 −1.43347
\(756\) 0 0
\(757\) 25.2305 0.917019 0.458509 0.888690i \(-0.348384\pi\)
0.458509 + 0.888690i \(0.348384\pi\)
\(758\) 0 0
\(759\) 26.5108 45.9181i 0.962282 1.66672i
\(760\) 0 0
\(761\) −1.82372 3.15878i −0.0661099 0.114506i 0.831076 0.556159i \(-0.187725\pi\)
−0.897186 + 0.441653i \(0.854392\pi\)
\(762\) 0 0
\(763\) 6.48077 8.94978i 0.234620 0.324004i
\(764\) 0 0
\(765\) −12.5232 21.6908i −0.452777 0.784233i
\(766\) 0 0
\(767\) 2.39448 4.14736i 0.0864596 0.149752i
\(768\) 0 0
\(769\) 21.9882 0.792914 0.396457 0.918053i \(-0.370240\pi\)
0.396457 + 0.918053i \(0.370240\pi\)
\(770\) 0 0
\(771\) −16.9862 −0.611742
\(772\) 0 0
\(773\) −10.9295 + 18.9305i −0.393108 + 0.680882i −0.992858 0.119305i \(-0.961933\pi\)
0.599750 + 0.800187i \(0.295267\pi\)
\(774\) 0 0
\(775\) 0.930986 + 1.61252i 0.0334420 + 0.0579233i
\(776\) 0 0
\(777\) 36.6342 + 3.79176i 1.31424 + 0.136029i
\(778\) 0 0
\(779\) −18.5404 32.1128i −0.664277 1.15056i
\(780\) 0 0
\(781\) 6.07900 10.5291i 0.217524 0.376762i
\(782\) 0 0
\(783\) 1.79303 0.0640777
\(784\) 0 0
\(785\) 20.8026 0.742477
\(786\) 0 0
\(787\) 19.9336 34.5261i 0.710557 1.23072i −0.254091 0.967180i \(-0.581776\pi\)
0.964648 0.263541i \(-0.0848905\pi\)
\(788\) 0 0
\(789\) 0.156570 + 0.271188i 0.00557405 + 0.00965454i
\(790\) 0 0
\(791\) −15.5298 1.60739i −0.552176 0.0571521i
\(792\) 0 0
\(793\) −1.60348 2.77732i −0.0569414 0.0986254i
\(794\) 0 0
\(795\) 32.5144 56.3166i 1.15317 1.99734i
\(796\) 0 0
\(797\) −40.1971 −1.42385 −0.711927 0.702253i \(-0.752178\pi\)
−0.711927 + 0.702253i \(0.752178\pi\)
\(798\) 0 0
\(799\) −26.8589 −0.950198
\(800\) 0 0
\(801\) −5.72146 + 9.90986i −0.202158 + 0.350148i
\(802\) 0 0
\(803\) 18.4392 + 31.9376i 0.650704 + 1.12705i
\(804\) 0 0
\(805\) 14.7389 20.3541i 0.519478 0.717387i
\(806\) 0 0
\(807\) −5.24960 9.09258i −0.184795 0.320074i
\(808\) 0 0
\(809\) −1.26924 + 2.19840i −0.0446243 + 0.0772915i −0.887475 0.460856i \(-0.847542\pi\)
0.842851 + 0.538148i \(0.180876\pi\)
\(810\) 0 0
\(811\) 41.7062 1.46450 0.732251 0.681035i \(-0.238470\pi\)
0.732251 + 0.681035i \(0.238470\pi\)
\(812\) 0 0
\(813\) 3.87801 0.136008
\(814\) 0 0
\(815\) −14.6836 + 25.4327i −0.514343 + 0.890868i
\(816\) 0 0
\(817\) 6.07737 + 10.5263i 0.212620 + 0.368269i
\(818\) 0 0
\(819\) 3.37246 + 7.54251i 0.117843 + 0.263557i
\(820\) 0 0
\(821\) −15.9652 27.6525i −0.557189 0.965079i −0.997730 0.0673467i \(-0.978547\pi\)
0.440541 0.897733i \(-0.354787\pi\)
\(822\) 0 0
\(823\) −17.1266 + 29.6641i −0.596995 + 1.03402i 0.396267 + 0.918135i \(0.370305\pi\)
−0.993262 + 0.115890i \(0.963028\pi\)
\(824\) 0 0
\(825\) −5.85078 −0.203698
\(826\) 0 0
\(827\) −36.9755 −1.28576 −0.642882 0.765965i \(-0.722261\pi\)
−0.642882 + 0.765965i \(0.722261\pi\)
\(828\) 0 0
\(829\) 9.99473 17.3114i 0.347131 0.601249i −0.638607 0.769533i \(-0.720489\pi\)
0.985739 + 0.168284i \(0.0538225\pi\)
\(830\) 0 0
\(831\) −15.7685 27.3119i −0.547005 0.947440i
\(832\) 0 0
\(833\) 17.6348 19.7106i 0.611009 0.682932i
\(834\) 0 0
\(835\) −18.4189 31.9024i −0.637412 1.10403i
\(836\) 0 0
\(837\) 0.572931 0.992346i 0.0198034 0.0343005i
\(838\) 0 0
\(839\) −12.8147 −0.442411 −0.221206 0.975227i \(-0.570999\pi\)
−0.221206 + 0.975227i \(0.570999\pi\)
\(840\) 0 0
\(841\) 5.82265 0.200781
\(842\) 0 0
\(843\) 5.72812 9.92140i 0.197287 0.341711i
\(844\) 0 0
\(845\) 1.06140 + 1.83839i 0.0365132 + 0.0632427i
\(846\) 0 0
\(847\) 12.8882 + 28.8245i 0.442845 + 0.990422i
\(848\) 0 0
\(849\) −4.51375 7.81805i −0.154912 0.268315i
\(850\) 0 0
\(851\) 12.5859 21.7994i 0.431439 0.747274i
\(852\) 0 0
\(853\) −30.1839 −1.03348 −0.516739 0.856143i \(-0.672854\pi\)
−0.516739 + 0.856143i \(0.672854\pi\)
\(854\) 0 0
\(855\) −23.6474 −0.808724
\(856\) 0 0
\(857\) 26.6164 46.1009i 0.909197 1.57478i 0.0940154 0.995571i \(-0.470030\pi\)
0.815182 0.579205i \(-0.196637\pi\)
\(858\) 0 0
\(859\) −6.13597 10.6278i −0.209357 0.362616i 0.742155 0.670228i \(-0.233804\pi\)
−0.951512 + 0.307611i \(0.900470\pi\)
\(860\) 0 0
\(861\) −39.9127 + 55.1185i −1.36022 + 1.87843i
\(862\) 0 0
\(863\) 12.2226 + 21.1702i 0.416064 + 0.720643i 0.995539 0.0943460i \(-0.0300760\pi\)
−0.579476 + 0.814989i \(0.696743\pi\)
\(864\) 0 0
\(865\) −3.14320 + 5.44418i −0.106872 + 0.185108i
\(866\) 0 0
\(867\) 6.74189 0.228967
\(868\) 0 0
\(869\) −24.7996 −0.841268
\(870\) 0 0
\(871\) −1.44978 + 2.51109i −0.0491238 + 0.0850850i
\(872\) 0 0
\(873\) 8.44300 + 14.6237i 0.285752 + 0.494937i
\(874\) 0 0
\(875\) −30.6910 3.17663i −1.03755 0.107390i
\(876\) 0 0
\(877\) 26.4376 + 45.7913i 0.892736 + 1.54626i 0.836582 + 0.547842i \(0.184550\pi\)
0.0561539 + 0.998422i \(0.482116\pi\)
\(878\) 0 0
\(879\) −26.0329 + 45.0903i −0.878068 + 1.52086i
\(880\) 0 0
\(881\) 55.0118 1.85339 0.926697 0.375809i \(-0.122635\pi\)
0.926697 + 0.375809i \(0.122635\pi\)
\(882\) 0 0
\(883\) −44.1730 −1.48654 −0.743269 0.668992i \(-0.766726\pi\)
−0.743269 + 0.668992i \(0.766726\pi\)
\(884\) 0 0
\(885\) 12.5775 21.7848i 0.422788 0.732290i
\(886\) 0 0
\(887\) −2.54330 4.40512i −0.0853955 0.147909i 0.820164 0.572128i \(-0.193882\pi\)
−0.905560 + 0.424219i \(0.860549\pi\)
\(888\) 0 0
\(889\) −27.7032 2.86738i −0.929136 0.0961688i
\(890\) 0 0
\(891\) −20.6321 35.7359i −0.691202 1.19720i
\(892\) 0 0
\(893\) −12.6793 + 21.9612i −0.424296 + 0.734903i
\(894\) 0 0
\(895\) −12.0339 −0.402248
\(896\) 0 0
\(897\) 11.0717 0.369672
\(898\) 0 0
\(899\) 11.1270 19.2724i 0.371105 0.642772i
\(900\) 0 0
\(901\) −23.3876 40.5086i −0.779155 1.34954i
\(902\) 0 0
\(903\) 13.0830 18.0674i 0.435377 0.601244i
\(904\) 0 0
\(905\) 7.61706 + 13.1931i 0.253200 + 0.438555i
\(906\) 0 0
\(907\) −9.06264 + 15.6969i −0.300920 + 0.521209i −0.976345 0.216220i \(-0.930627\pi\)
0.675425 + 0.737429i \(0.263960\pi\)
\(908\) 0 0
\(909\) 29.0958 0.965048
\(910\) 0 0
\(911\) 9.65804 0.319985 0.159993 0.987118i \(-0.448853\pi\)
0.159993 + 0.987118i \(0.448853\pi\)
\(912\) 0 0
\(913\) −8.30241 + 14.3802i −0.274770 + 0.475915i
\(914\) 0 0
\(915\) −8.42263 14.5884i −0.278444 0.482278i
\(916\) 0 0
\(917\) −5.85774 13.1008i −0.193440 0.432628i
\(918\) 0 0
\(919\) 23.8801 + 41.3616i 0.787733 + 1.36439i 0.927353 + 0.374188i \(0.122078\pi\)
−0.139620 + 0.990205i \(0.544588\pi\)
\(920\) 0 0
\(921\) 6.13487 10.6259i 0.202151 0.350135i
\(922\) 0 0
\(923\) 2.53876 0.0835643
\(924\) 0 0
\(925\) −2.77763 −0.0913279
\(926\) 0 0
\(927\) 11.4081 19.7595i 0.374692 0.648986i
\(928\) 0 0
\(929\) −16.9905 29.4285i −0.557442 0.965517i −0.997709 0.0676505i \(-0.978450\pi\)
0.440267 0.897867i \(-0.354884\pi\)
\(930\) 0 0
\(931\) −7.79153 23.7239i −0.255357 0.777520i
\(932\) 0 0
\(933\) 2.99601 + 5.18924i 0.0980850 + 0.169888i
\(934\) 0 0
\(935\) 19.2049 33.2639i 0.628068 1.08785i
\(936\) 0 0
\(937\) −24.7948 −0.810012 −0.405006 0.914314i \(-0.632731\pi\)
−0.405006 + 0.914314i \(0.632731\pi\)
\(938\) 0 0
\(939\) −34.5443 −1.12731
\(940\) 0 0
\(941\) 4.12098 7.13774i 0.134340 0.232684i −0.791005 0.611810i \(-0.790442\pi\)
0.925345 + 0.379126i \(0.123775\pi\)
\(942\) 0 0
\(943\) 23.2554 + 40.2796i 0.757301 + 1.31168i
\(944\) 0 0
\(945\) 0.696577 + 1.55789i 0.0226596 + 0.0506783i
\(946\) 0 0
\(947\) 9.98643 + 17.2970i 0.324515 + 0.562077i 0.981414 0.191902i \(-0.0614655\pi\)
−0.656899 + 0.753979i \(0.728132\pi\)
\(948\) 0 0
\(949\) −3.85035 + 6.66901i −0.124988 + 0.216485i
\(950\) 0 0
\(951\) −7.59307 −0.246222
\(952\) 0 0
\(953\) −21.5341 −0.697557 −0.348778 0.937205i \(-0.613403\pi\)
−0.348778 + 0.937205i \(0.613403\pi\)
\(954\) 0 0
\(955\) −12.6113 + 21.8434i −0.408091 + 0.706835i
\(956\) 0 0
\(957\) 34.9636 + 60.5588i 1.13021 + 1.95759i
\(958\) 0 0
\(959\) 34.5181 47.6686i 1.11465 1.53930i
\(960\) 0 0
\(961\) 8.38917 + 14.5305i 0.270618 + 0.468725i
\(962\) 0 0
\(963\) 10.5352 18.2475i 0.339492 0.588018i
\(964\) 0 0
\(965\) −48.7610 −1.56967
\(966\) 0 0
\(967\) −43.2887 −1.39207 −0.696036 0.718007i \(-0.745055\pi\)
−0.696036 + 0.718007i \(0.745055\pi\)
\(968\) 0 0
\(969\) −16.6752 + 28.8822i −0.535683 + 0.927831i
\(970\) 0 0
\(971\) −26.3356 45.6147i −0.845151 1.46384i −0.885490 0.464658i \(-0.846177\pi\)
0.0403390 0.999186i \(-0.487156\pi\)
\(972\) 0 0
\(973\) −10.5268 1.08956i −0.337473 0.0349296i
\(974\) 0 0
\(975\) −0.610862 1.05804i −0.0195632 0.0338845i
\(976\) 0 0
\(977\) −7.70305 + 13.3421i −0.246442 + 0.426851i −0.962536 0.271153i \(-0.912595\pi\)
0.716094 + 0.698004i \(0.245928\pi\)
\(978\) 0 0
\(979\) −17.5483 −0.560845
\(980\) 0 0
\(981\) 13.0422 0.416405
\(982\) 0 0
\(983\) 3.79073 6.56574i 0.120906 0.209415i −0.799219 0.601039i \(-0.794754\pi\)
0.920125 + 0.391625i \(0.128087\pi\)
\(984\) 0 0
\(985\) 17.9605 + 31.1085i 0.572270 + 0.991201i
\(986\) 0 0
\(987\) 46.2918 + 4.79136i 1.47348 + 0.152511i
\(988\) 0 0
\(989\) −7.62293 13.2033i −0.242395 0.419841i
\(990\) 0 0
\(991\) −9.50923 + 16.4705i −0.302071 + 0.523202i −0.976605 0.215042i \(-0.931011\pi\)
0.674534 + 0.738244i \(0.264345\pi\)
\(992\) 0 0
\(993\) 33.6651 1.06833
\(994\) 0 0
\(995\) 21.3533 0.676946
\(996\) 0 0
\(997\) −23.0499 + 39.9236i −0.729998 + 1.26439i 0.226885 + 0.973922i \(0.427146\pi\)
−0.956883 + 0.290473i \(0.906187\pi\)
\(998\) 0 0
\(999\) 0.854680 + 1.48035i 0.0270409 + 0.0468362i
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 1456.2.r.p.417.5 10
4.3 odd 2 91.2.e.c.53.2 10
7.2 even 3 inner 1456.2.r.p.625.5 10
12.11 even 2 819.2.j.h.235.4 10
28.3 even 6 637.2.a.k.1.4 5
28.11 odd 6 637.2.a.l.1.4 5
28.19 even 6 637.2.e.m.79.2 10
28.23 odd 6 91.2.e.c.79.2 yes 10
28.27 even 2 637.2.e.m.508.2 10
52.51 odd 2 1183.2.e.f.508.4 10
84.11 even 6 5733.2.a.bl.1.2 5
84.23 even 6 819.2.j.h.352.4 10
84.59 odd 6 5733.2.a.bm.1.2 5
364.51 odd 6 1183.2.e.f.170.4 10
364.207 odd 6 8281.2.a.bw.1.2 5
364.311 even 6 8281.2.a.bx.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.e.c.53.2 10 4.3 odd 2
91.2.e.c.79.2 yes 10 28.23 odd 6
637.2.a.k.1.4 5 28.3 even 6
637.2.a.l.1.4 5 28.11 odd 6
637.2.e.m.79.2 10 28.19 even 6
637.2.e.m.508.2 10 28.27 even 2
819.2.j.h.235.4 10 12.11 even 2
819.2.j.h.352.4 10 84.23 even 6
1183.2.e.f.170.4 10 364.51 odd 6
1183.2.e.f.508.4 10 52.51 odd 2
1456.2.r.p.417.5 10 1.1 even 1 trivial
1456.2.r.p.625.5 10 7.2 even 3 inner
5733.2.a.bl.1.2 5 84.11 even 6
5733.2.a.bm.1.2 5 84.59 odd 6
8281.2.a.bw.1.2 5 364.207 odd 6
8281.2.a.bx.1.2 5 364.311 even 6