Properties

Label 3150.2.bf.f.1601.14
Level $3150$
Weight $2$
Character 3150.1601
Analytic conductor $25.153$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3150,2,Mod(1151,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.1151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.bf (of order \(6\), degree \(2\), 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: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1601.14
Character \(\chi\) \(=\) 3150.1601
Dual form 3150.2.bf.f.1151.16

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(2.24547 + 1.39924i) q^{7} +1.00000i q^{8} +O(q^{10})\) \(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(2.24547 + 1.39924i) q^{7} +1.00000i q^{8} +(1.37897 - 0.796151i) q^{11} -0.925091i q^{13} +(1.24501 + 2.33451i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-1.97883 - 3.42743i) q^{17} +(-0.541679 - 0.312739i) q^{19} +1.59230 q^{22} +(6.74256 + 3.89282i) q^{23} +(0.462546 - 0.801153i) q^{26} +(-0.0890445 + 2.64425i) q^{28} +9.34805i q^{29} +(8.94618 - 5.16508i) q^{31} +(-0.866025 + 0.500000i) q^{32} -3.95765i q^{34} +(0.213192 - 0.369259i) q^{37} +(-0.312739 - 0.541679i) q^{38} -8.35463 q^{41} +6.27133 q^{43} +(1.37897 + 0.796151i) q^{44} +(3.89282 + 6.74256i) q^{46} +(1.39065 - 2.40868i) q^{47} +(3.08425 + 6.28390i) q^{49} +(0.801153 - 0.462546i) q^{52} +(2.90217 - 1.67557i) q^{53} +(-1.39924 + 2.24547i) q^{56} +(-4.67403 + 8.09565i) q^{58} +(-3.10680 - 5.38113i) q^{59} +(9.52671 + 5.50025i) q^{61} +10.3302 q^{62} -1.00000 q^{64} +(-0.178089 - 0.308459i) q^{67} +(1.97883 - 3.42743i) q^{68} +9.07975i q^{71} +(-5.91515 + 3.41511i) q^{73} +(0.369259 - 0.213192i) q^{74} -0.625477i q^{76} +(4.21045 + 0.141786i) q^{77} +(4.52582 - 7.83895i) q^{79} +(-7.23532 - 4.17731i) q^{82} -0.809898 q^{83} +(5.43113 + 3.13566i) q^{86} +(0.796151 + 1.37897i) q^{88} +(-2.00721 + 3.47659i) q^{89} +(1.29443 - 2.07726i) q^{91} +7.78564i q^{92} +(2.40868 - 1.39065i) q^{94} +7.87721i q^{97} +(-0.470912 + 6.98414i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 16 q^{4} - 16 q^{16} - 48 q^{19} + 24 q^{31} - 16 q^{46} + 56 q^{49} + 48 q^{61} - 32 q^{64} - 8 q^{79} - 56 q^{91} + 120 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.866025 + 0.500000i 0.612372 + 0.353553i
\(3\) 0 0
\(4\) 0.500000 + 0.866025i 0.250000 + 0.433013i
\(5\) 0 0
\(6\) 0 0
\(7\) 2.24547 + 1.39924i 0.848707 + 0.528863i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.37897 0.796151i 0.415776 0.240049i −0.277492 0.960728i \(-0.589503\pi\)
0.693269 + 0.720679i \(0.256170\pi\)
\(12\) 0 0
\(13\) 0.925091i 0.256574i −0.991737 0.128287i \(-0.959052\pi\)
0.991737 0.128287i \(-0.0409479\pi\)
\(14\) 1.24501 + 2.33451i 0.332743 + 0.623925i
\(15\) 0 0
\(16\) −0.500000 + 0.866025i −0.125000 + 0.216506i
\(17\) −1.97883 3.42743i −0.479936 0.831273i 0.519799 0.854288i \(-0.326007\pi\)
−0.999735 + 0.0230153i \(0.992673\pi\)
\(18\) 0 0
\(19\) −0.541679 0.312739i −0.124270 0.0717472i 0.436577 0.899667i \(-0.356191\pi\)
−0.560846 + 0.827920i \(0.689524\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.59230 0.339480
\(23\) 6.74256 + 3.89282i 1.40592 + 0.811709i 0.994992 0.0999578i \(-0.0318708\pi\)
0.410930 + 0.911667i \(0.365204\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.462546 0.801153i 0.0907127 0.157119i
\(27\) 0 0
\(28\) −0.0890445 + 2.64425i −0.0168278 + 0.499717i
\(29\) 9.34805i 1.73589i 0.496661 + 0.867945i \(0.334559\pi\)
−0.496661 + 0.867945i \(0.665441\pi\)
\(30\) 0 0
\(31\) 8.94618 5.16508i 1.60678 0.927675i 0.616696 0.787201i \(-0.288471\pi\)
0.990084 0.140474i \(-0.0448626\pi\)
\(32\) −0.866025 + 0.500000i −0.153093 + 0.0883883i
\(33\) 0 0
\(34\) 3.95765i 0.678732i
\(35\) 0 0
\(36\) 0 0
\(37\) 0.213192 0.369259i 0.0350485 0.0607058i −0.847969 0.530046i \(-0.822175\pi\)
0.883018 + 0.469340i \(0.155508\pi\)
\(38\) −0.312739 0.541679i −0.0507329 0.0878720i
\(39\) 0 0
\(40\) 0 0
\(41\) −8.35463 −1.30477 −0.652387 0.757886i \(-0.726232\pi\)
−0.652387 + 0.757886i \(0.726232\pi\)
\(42\) 0 0
\(43\) 6.27133 0.956369 0.478184 0.878259i \(-0.341295\pi\)
0.478184 + 0.878259i \(0.341295\pi\)
\(44\) 1.37897 + 0.796151i 0.207888 + 0.120024i
\(45\) 0 0
\(46\) 3.89282 + 6.74256i 0.573965 + 0.994137i
\(47\) 1.39065 2.40868i 0.202847 0.351342i −0.746597 0.665276i \(-0.768314\pi\)
0.949445 + 0.313934i \(0.101647\pi\)
\(48\) 0 0
\(49\) 3.08425 + 6.28390i 0.440607 + 0.897700i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.801153 0.462546i 0.111100 0.0641435i
\(53\) 2.90217 1.67557i 0.398644 0.230157i −0.287255 0.957854i \(-0.592743\pi\)
0.685899 + 0.727697i \(0.259409\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.39924 + 2.24547i −0.186981 + 0.300063i
\(57\) 0 0
\(58\) −4.67403 + 8.09565i −0.613730 + 1.06301i
\(59\) −3.10680 5.38113i −0.404471 0.700564i 0.589789 0.807557i \(-0.299211\pi\)
−0.994260 + 0.106994i \(0.965878\pi\)
\(60\) 0 0
\(61\) 9.52671 + 5.50025i 1.21977 + 0.704235i 0.964869 0.262732i \(-0.0846236\pi\)
0.254902 + 0.966967i \(0.417957\pi\)
\(62\) 10.3302 1.31193
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −0.178089 0.308459i −0.0217570 0.0376843i 0.854942 0.518724i \(-0.173593\pi\)
−0.876699 + 0.481039i \(0.840259\pi\)
\(68\) 1.97883 3.42743i 0.239968 0.415637i
\(69\) 0 0
\(70\) 0 0
\(71\) 9.07975i 1.07757i 0.842444 + 0.538784i \(0.181116\pi\)
−0.842444 + 0.538784i \(0.818884\pi\)
\(72\) 0 0
\(73\) −5.91515 + 3.41511i −0.692316 + 0.399709i −0.804479 0.593981i \(-0.797555\pi\)
0.112163 + 0.993690i \(0.464222\pi\)
\(74\) 0.369259 0.213192i 0.0429255 0.0247830i
\(75\) 0 0
\(76\) 0.625477i 0.0717472i
\(77\) 4.21045 + 0.141786i 0.479825 + 0.0161580i
\(78\) 0 0
\(79\) 4.52582 7.83895i 0.509195 0.881951i −0.490749 0.871301i \(-0.663277\pi\)
0.999943 0.0106498i \(-0.00339001\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7.23532 4.17731i −0.799007 0.461307i
\(83\) −0.809898 −0.0888978 −0.0444489 0.999012i \(-0.514153\pi\)
−0.0444489 + 0.999012i \(0.514153\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 5.43113 + 3.13566i 0.585654 + 0.338127i
\(87\) 0 0
\(88\) 0.796151 + 1.37897i 0.0848700 + 0.146999i
\(89\) −2.00721 + 3.47659i −0.212764 + 0.368518i −0.952579 0.304293i \(-0.901580\pi\)
0.739815 + 0.672811i \(0.234913\pi\)
\(90\) 0 0
\(91\) 1.29443 2.07726i 0.135693 0.217756i
\(92\) 7.78564i 0.811709i
\(93\) 0 0
\(94\) 2.40868 1.39065i 0.248436 0.143435i
\(95\) 0 0
\(96\) 0 0
\(97\) 7.87721i 0.799809i 0.916557 + 0.399905i \(0.130957\pi\)
−0.916557 + 0.399905i \(0.869043\pi\)
\(98\) −0.470912 + 6.98414i −0.0475693 + 0.705505i
\(99\) 0 0
\(100\) 0 0
\(101\) 3.76411 + 6.51962i 0.374543 + 0.648727i 0.990258 0.139241i \(-0.0444664\pi\)
−0.615716 + 0.787968i \(0.711133\pi\)
\(102\) 0 0
\(103\) 1.44073 + 0.831805i 0.141959 + 0.0819601i 0.569297 0.822132i \(-0.307215\pi\)
−0.427338 + 0.904092i \(0.640549\pi\)
\(104\) 0.925091 0.0907127
\(105\) 0 0
\(106\) 3.35114 0.325492
\(107\) −16.8033 9.70139i −1.62444 0.937869i −0.985713 0.168433i \(-0.946129\pi\)
−0.638724 0.769436i \(-0.720537\pi\)
\(108\) 0 0
\(109\) 1.00000 + 1.73205i 0.0957826 + 0.165900i 0.909935 0.414751i \(-0.136131\pi\)
−0.814152 + 0.580651i \(0.802798\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.33451 + 1.24501i −0.220591 + 0.117643i
\(113\) 16.0750i 1.51221i −0.654451 0.756104i \(-0.727100\pi\)
0.654451 0.756104i \(-0.272900\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −8.09565 + 4.67403i −0.751662 + 0.433972i
\(117\) 0 0
\(118\) 6.21360i 0.572008i
\(119\) 0.352407 10.4650i 0.0323051 0.959328i
\(120\) 0 0
\(121\) −4.23229 + 7.33053i −0.384753 + 0.666412i
\(122\) 5.50025 + 9.52671i 0.497969 + 0.862508i
\(123\) 0 0
\(124\) 8.94618 + 5.16508i 0.803390 + 0.463838i
\(125\) 0 0
\(126\) 0 0
\(127\) −13.2173 −1.17285 −0.586424 0.810004i \(-0.699465\pi\)
−0.586424 + 0.810004i \(0.699465\pi\)
\(128\) −0.866025 0.500000i −0.0765466 0.0441942i
\(129\) 0 0
\(130\) 0 0
\(131\) −4.70080 + 8.14203i −0.410711 + 0.711372i −0.994968 0.100197i \(-0.968053\pi\)
0.584257 + 0.811569i \(0.301386\pi\)
\(132\) 0 0
\(133\) −0.778726 1.46018i −0.0675241 0.126614i
\(134\) 0.356178i 0.0307691i
\(135\) 0 0
\(136\) 3.42743 1.97883i 0.293899 0.169683i
\(137\) −8.11701 + 4.68636i −0.693483 + 0.400382i −0.804915 0.593390i \(-0.797789\pi\)
0.111433 + 0.993772i \(0.464456\pi\)
\(138\) 0 0
\(139\) 14.1106i 1.19685i 0.801180 + 0.598423i \(0.204206\pi\)
−0.801180 + 0.598423i \(0.795794\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4.53988 + 7.86330i −0.380978 + 0.659873i
\(143\) −0.736513 1.27568i −0.0615903 0.106678i
\(144\) 0 0
\(145\) 0 0
\(146\) −6.83023 −0.565274
\(147\) 0 0
\(148\) 0.426383 0.0350485
\(149\) −2.14103 1.23612i −0.175400 0.101267i 0.409730 0.912207i \(-0.365623\pi\)
−0.585130 + 0.810940i \(0.698956\pi\)
\(150\) 0 0
\(151\) 10.4425 + 18.0869i 0.849796 + 1.47189i 0.881391 + 0.472388i \(0.156608\pi\)
−0.0315949 + 0.999501i \(0.510059\pi\)
\(152\) 0.312739 0.541679i 0.0253664 0.0439360i
\(153\) 0 0
\(154\) 3.57546 + 2.22802i 0.288119 + 0.179539i
\(155\) 0 0
\(156\) 0 0
\(157\) 21.1722 12.2238i 1.68972 0.975563i 0.735002 0.678065i \(-0.237181\pi\)
0.954723 0.297498i \(-0.0961521\pi\)
\(158\) 7.83895 4.52582i 0.623634 0.360055i
\(159\) 0 0
\(160\) 0 0
\(161\) 9.69321 + 18.1757i 0.763932 + 1.43244i
\(162\) 0 0
\(163\) 2.95758 5.12267i 0.231655 0.401239i −0.726640 0.687018i \(-0.758919\pi\)
0.958295 + 0.285780i \(0.0922526\pi\)
\(164\) −4.17731 7.23532i −0.326193 0.564983i
\(165\) 0 0
\(166\) −0.701392 0.404949i −0.0544386 0.0314301i
\(167\) 12.1440 0.939733 0.469867 0.882737i \(-0.344302\pi\)
0.469867 + 0.882737i \(0.344302\pi\)
\(168\) 0 0
\(169\) 12.1442 0.934170
\(170\) 0 0
\(171\) 0 0
\(172\) 3.13566 + 5.43113i 0.239092 + 0.414120i
\(173\) 8.19918 14.2014i 0.623372 1.07971i −0.365481 0.930819i \(-0.619096\pi\)
0.988853 0.148893i \(-0.0475711\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.59230i 0.120024i
\(177\) 0 0
\(178\) −3.47659 + 2.00721i −0.260581 + 0.150447i
\(179\) −2.73398 + 1.57846i −0.204347 + 0.117980i −0.598682 0.800987i \(-0.704309\pi\)
0.394334 + 0.918967i \(0.370975\pi\)
\(180\) 0 0
\(181\) 8.17916i 0.607952i −0.952680 0.303976i \(-0.901686\pi\)
0.952680 0.303976i \(-0.0983143\pi\)
\(182\) 2.15964 1.15175i 0.160083 0.0853733i
\(183\) 0 0
\(184\) −3.89282 + 6.74256i −0.286983 + 0.497068i
\(185\) 0 0
\(186\) 0 0
\(187\) −5.45750 3.15089i −0.399092 0.230416i
\(188\) 2.78130 0.202847
\(189\) 0 0
\(190\) 0 0
\(191\) 2.44949 + 1.41421i 0.177239 + 0.102329i 0.585995 0.810315i \(-0.300704\pi\)
−0.408756 + 0.912644i \(0.634037\pi\)
\(192\) 0 0
\(193\) 8.85772 + 15.3420i 0.637593 + 1.10434i 0.985959 + 0.166985i \(0.0534031\pi\)
−0.348367 + 0.937358i \(0.613264\pi\)
\(194\) −3.93860 + 6.82186i −0.282775 + 0.489781i
\(195\) 0 0
\(196\) −3.89989 + 5.81299i −0.278564 + 0.415213i
\(197\) 4.30350i 0.306612i 0.988179 + 0.153306i \(0.0489919\pi\)
−0.988179 + 0.153306i \(0.951008\pi\)
\(198\) 0 0
\(199\) 3.00000 1.73205i 0.212664 0.122782i −0.389885 0.920864i \(-0.627485\pi\)
0.602549 + 0.798082i \(0.294152\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 7.52821i 0.529683i
\(203\) −13.0802 + 20.9907i −0.918048 + 1.47326i
\(204\) 0 0
\(205\) 0 0
\(206\) 0.831805 + 1.44073i 0.0579546 + 0.100380i
\(207\) 0 0
\(208\) 0.801153 + 0.462546i 0.0555499 + 0.0320718i
\(209\) −0.995949 −0.0688912
\(210\) 0 0
\(211\) 10.3886 0.715183 0.357592 0.933878i \(-0.383598\pi\)
0.357592 + 0.933878i \(0.383598\pi\)
\(212\) 2.90217 + 1.67557i 0.199322 + 0.115079i
\(213\) 0 0
\(214\) −9.70139 16.8033i −0.663173 1.14865i
\(215\) 0 0
\(216\) 0 0
\(217\) 27.3155 + 0.919844i 1.85430 + 0.0624431i
\(218\) 2.00000i 0.135457i
\(219\) 0 0
\(220\) 0 0
\(221\) −3.17068 + 1.83059i −0.213283 + 0.123139i
\(222\) 0 0
\(223\) 18.3555i 1.22918i −0.788848 0.614589i \(-0.789322\pi\)
0.788848 0.614589i \(-0.210678\pi\)
\(224\) −2.64425 0.0890445i −0.176677 0.00594954i
\(225\) 0 0
\(226\) 8.03750 13.9214i 0.534646 0.926035i
\(227\) −1.51152 2.61803i −0.100323 0.173765i 0.811495 0.584360i \(-0.198654\pi\)
−0.911818 + 0.410595i \(0.865321\pi\)
\(228\) 0 0
\(229\) −9.22014 5.32325i −0.609284 0.351770i 0.163401 0.986560i \(-0.447754\pi\)
−0.772685 + 0.634789i \(0.781087\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −9.34805 −0.613730
\(233\) −3.45185 1.99293i −0.226138 0.130561i 0.382651 0.923893i \(-0.375011\pi\)
−0.608789 + 0.793332i \(0.708345\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 3.10680 5.38113i 0.202235 0.350282i
\(237\) 0 0
\(238\) 5.53771 8.88678i 0.358956 0.576044i
\(239\) 8.24699i 0.533453i 0.963772 + 0.266727i \(0.0859421\pi\)
−0.963772 + 0.266727i \(0.914058\pi\)
\(240\) 0 0
\(241\) −9.48785 + 5.47782i −0.611166 + 0.352857i −0.773422 0.633892i \(-0.781456\pi\)
0.162255 + 0.986749i \(0.448123\pi\)
\(242\) −7.33053 + 4.23229i −0.471225 + 0.272062i
\(243\) 0 0
\(244\) 11.0005i 0.704235i
\(245\) 0 0
\(246\) 0 0
\(247\) −0.289312 + 0.501103i −0.0184085 + 0.0318844i
\(248\) 5.16508 + 8.94618i 0.327983 + 0.568083i
\(249\) 0 0
\(250\) 0 0
\(251\) −21.7369 −1.37202 −0.686012 0.727590i \(-0.740640\pi\)
−0.686012 + 0.727590i \(0.740640\pi\)
\(252\) 0 0
\(253\) 12.3971 0.779399
\(254\) −11.4465 6.60867i −0.718220 0.414665i
\(255\) 0 0
\(256\) −0.500000 0.866025i −0.0312500 0.0541266i
\(257\) 8.71758 15.0993i 0.543787 0.941868i −0.454895 0.890545i \(-0.650323\pi\)
0.998682 0.0513223i \(-0.0163436\pi\)
\(258\) 0 0
\(259\) 0.995397 0.530852i 0.0618510 0.0329856i
\(260\) 0 0
\(261\) 0 0
\(262\) −8.14203 + 4.70080i −0.503016 + 0.290417i
\(263\) −15.8926 + 9.17557i −0.979977 + 0.565790i −0.902263 0.431186i \(-0.858095\pi\)
−0.0777137 + 0.996976i \(0.524762\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0.0556953 1.65392i 0.00341490 0.101408i
\(267\) 0 0
\(268\) 0.178089 0.308459i 0.0108785 0.0188422i
\(269\) 12.4185 + 21.5095i 0.757171 + 1.31146i 0.944288 + 0.329121i \(0.106752\pi\)
−0.187117 + 0.982338i \(0.559914\pi\)
\(270\) 0 0
\(271\) 21.1663 + 12.2204i 1.28576 + 0.742335i 0.977895 0.209094i \(-0.0670516\pi\)
0.307867 + 0.951430i \(0.400385\pi\)
\(272\) 3.95765 0.239968
\(273\) 0 0
\(274\) −9.37271 −0.566226
\(275\) 0 0
\(276\) 0 0
\(277\) −11.5985 20.0892i −0.696888 1.20704i −0.969540 0.244932i \(-0.921234\pi\)
0.272653 0.962113i \(-0.412099\pi\)
\(278\) −7.05530 + 12.2201i −0.423149 + 0.732915i
\(279\) 0 0
\(280\) 0 0
\(281\) 14.0801i 0.839949i −0.907536 0.419974i \(-0.862039\pi\)
0.907536 0.419974i \(-0.137961\pi\)
\(282\) 0 0
\(283\) 2.42331 1.39910i 0.144051 0.0831677i −0.426242 0.904609i \(-0.640163\pi\)
0.570293 + 0.821441i \(0.306830\pi\)
\(284\) −7.86330 + 4.53988i −0.466601 + 0.269392i
\(285\) 0 0
\(286\) 1.47303i 0.0871018i
\(287\) −18.7600 11.6901i −1.10737 0.690047i
\(288\) 0 0
\(289\) 0.668498 1.15787i 0.0393234 0.0681101i
\(290\) 0 0
\(291\) 0 0
\(292\) −5.91515 3.41511i −0.346158 0.199854i
\(293\) −25.5598 −1.49322 −0.746609 0.665263i \(-0.768319\pi\)
−0.746609 + 0.665263i \(0.768319\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.369259 + 0.213192i 0.0214627 + 0.0123915i
\(297\) 0 0
\(298\) −1.23612 2.14103i −0.0716068 0.124027i
\(299\) 3.60121 6.23749i 0.208264 0.360723i
\(300\) 0 0
\(301\) 14.0821 + 8.77510i 0.811677 + 0.505788i
\(302\) 20.8849i 1.20179i
\(303\) 0 0
\(304\) 0.541679 0.312739i 0.0310674 0.0179368i
\(305\) 0 0
\(306\) 0 0
\(307\) 34.2860i 1.95681i −0.206704 0.978403i \(-0.566274\pi\)
0.206704 0.978403i \(-0.433726\pi\)
\(308\) 1.98243 + 3.71725i 0.112960 + 0.211810i
\(309\) 0 0
\(310\) 0 0
\(311\) −4.34021 7.51746i −0.246110 0.426276i 0.716333 0.697759i \(-0.245819\pi\)
−0.962443 + 0.271483i \(0.912486\pi\)
\(312\) 0 0
\(313\) −25.9992 15.0106i −1.46956 0.848452i −0.470145 0.882589i \(-0.655798\pi\)
−0.999417 + 0.0341376i \(0.989132\pi\)
\(314\) 24.4475 1.37965
\(315\) 0 0
\(316\) 9.05164 0.509195
\(317\) −24.0111 13.8628i −1.34860 0.778613i −0.360547 0.932741i \(-0.617410\pi\)
−0.988051 + 0.154128i \(0.950743\pi\)
\(318\) 0 0
\(319\) 7.44246 + 12.8907i 0.416698 + 0.721742i
\(320\) 0 0
\(321\) 0 0
\(322\) −0.693269 + 20.5872i −0.0386344 + 1.14728i
\(323\) 2.47542i 0.137736i
\(324\) 0 0
\(325\) 0 0
\(326\) 5.12267 2.95758i 0.283719 0.163805i
\(327\) 0 0
\(328\) 8.35463i 0.461307i
\(329\) 6.49298 3.46275i 0.357970 0.190908i
\(330\) 0 0
\(331\) −3.15175 + 5.45899i −0.173236 + 0.300053i −0.939549 0.342414i \(-0.888756\pi\)
0.766314 + 0.642467i \(0.222089\pi\)
\(332\) −0.404949 0.701392i −0.0222245 0.0384939i
\(333\) 0 0
\(334\) 10.5170 + 6.07201i 0.575467 + 0.332246i
\(335\) 0 0
\(336\) 0 0
\(337\) 27.4097 1.49310 0.746550 0.665329i \(-0.231709\pi\)
0.746550 + 0.665329i \(0.231709\pi\)
\(338\) 10.5172 + 6.07210i 0.572060 + 0.330279i
\(339\) 0 0
\(340\) 0 0
\(341\) 8.22437 14.2450i 0.445374 0.771411i
\(342\) 0 0
\(343\) −1.86711 + 18.4259i −0.100815 + 0.994905i
\(344\) 6.27133i 0.338127i
\(345\) 0 0
\(346\) 14.2014 8.19918i 0.763472 0.440791i
\(347\) 7.07258 4.08336i 0.379676 0.219206i −0.298001 0.954566i \(-0.596320\pi\)
0.677677 + 0.735359i \(0.262987\pi\)
\(348\) 0 0
\(349\) 27.5885i 1.47678i −0.674374 0.738390i \(-0.735587\pi\)
0.674374 0.738390i \(-0.264413\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −0.796151 + 1.37897i −0.0424350 + 0.0734996i
\(353\) −3.52142 6.09929i −0.187426 0.324632i 0.756965 0.653455i \(-0.226681\pi\)
−0.944391 + 0.328823i \(0.893348\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −4.01442 −0.212764
\(357\) 0 0
\(358\) −3.15693 −0.166849
\(359\) 14.1545 + 8.17213i 0.747048 + 0.431308i 0.824626 0.565678i \(-0.191385\pi\)
−0.0775782 + 0.996986i \(0.524719\pi\)
\(360\) 0 0
\(361\) −9.30439 16.1157i −0.489705 0.848193i
\(362\) 4.08958 7.08336i 0.214943 0.372293i
\(363\) 0 0
\(364\) 2.44618 + 0.0823743i 0.128214 + 0.00431759i
\(365\) 0 0
\(366\) 0 0
\(367\) 2.60498 1.50399i 0.135979 0.0785076i −0.430467 0.902606i \(-0.641651\pi\)
0.566446 + 0.824099i \(0.308318\pi\)
\(368\) −6.74256 + 3.89282i −0.351480 + 0.202927i
\(369\) 0 0
\(370\) 0 0
\(371\) 8.86126 + 0.298401i 0.460054 + 0.0154922i
\(372\) 0 0
\(373\) −11.5985 + 20.0892i −0.600549 + 1.04018i 0.392189 + 0.919884i \(0.371718\pi\)
−0.992738 + 0.120296i \(0.961616\pi\)
\(374\) −3.15089 5.45750i −0.162929 0.282201i
\(375\) 0 0
\(376\) 2.40868 + 1.39065i 0.124218 + 0.0717174i
\(377\) 8.64780 0.445384
\(378\) 0 0
\(379\) −27.2718 −1.40086 −0.700429 0.713722i \(-0.747008\pi\)
−0.700429 + 0.713722i \(0.747008\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.41421 + 2.44949i 0.0723575 + 0.125327i
\(383\) −2.13098 + 3.69096i −0.108888 + 0.188599i −0.915320 0.402727i \(-0.868062\pi\)
0.806432 + 0.591327i \(0.201396\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.7154i 0.901692i
\(387\) 0 0
\(388\) −6.82186 + 3.93860i −0.346327 + 0.199952i
\(389\) −1.80282 + 1.04086i −0.0914066 + 0.0527736i −0.545007 0.838432i \(-0.683473\pi\)
0.453600 + 0.891205i \(0.350140\pi\)
\(390\) 0 0
\(391\) 30.8129i 1.55827i
\(392\) −6.28390 + 3.08425i −0.317385 + 0.155778i
\(393\) 0 0
\(394\) −2.15175 + 3.72694i −0.108404 + 0.187760i
\(395\) 0 0
\(396\) 0 0
\(397\) −19.3791 11.1885i −0.972610 0.561537i −0.0725790 0.997363i \(-0.523123\pi\)
−0.900031 + 0.435826i \(0.856456\pi\)
\(398\) 3.46410 0.173640
\(399\) 0 0
\(400\) 0 0
\(401\) −17.0295 9.83196i −0.850411 0.490985i 0.0103787 0.999946i \(-0.496696\pi\)
−0.860789 + 0.508961i \(0.830030\pi\)
\(402\) 0 0
\(403\) −4.77817 8.27603i −0.238017 0.412258i
\(404\) −3.76411 + 6.51962i −0.187271 + 0.324363i
\(405\) 0 0
\(406\) −21.8231 + 11.6384i −1.08306 + 0.577606i
\(407\) 0.678932i 0.0336534i
\(408\) 0 0
\(409\) −2.19932 + 1.26978i −0.108750 + 0.0627866i −0.553388 0.832923i \(-0.686665\pi\)
0.444639 + 0.895710i \(0.353332\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.66361i 0.0819601i
\(413\) 0.553287 16.4303i 0.0272255 0.808483i
\(414\) 0 0
\(415\) 0 0
\(416\) 0.462546 + 0.801153i 0.0226782 + 0.0392797i
\(417\) 0 0
\(418\) −0.862517 0.497974i −0.0421871 0.0243567i
\(419\) −32.0568 −1.56608 −0.783039 0.621973i \(-0.786331\pi\)
−0.783039 + 0.621973i \(0.786331\pi\)
\(420\) 0 0
\(421\) −25.2201 −1.22915 −0.614577 0.788857i \(-0.710673\pi\)
−0.614577 + 0.788857i \(0.710673\pi\)
\(422\) 8.99682 + 5.19432i 0.437959 + 0.252855i
\(423\) 0 0
\(424\) 1.67557 + 2.90217i 0.0813729 + 0.140942i
\(425\) 0 0
\(426\) 0 0
\(427\) 13.6957 + 25.6808i 0.662784 + 1.24278i
\(428\) 19.4028i 0.937869i
\(429\) 0 0
\(430\) 0 0
\(431\) 29.1599 16.8355i 1.40458 0.810937i 0.409726 0.912209i \(-0.365624\pi\)
0.994859 + 0.101271i \(0.0322910\pi\)
\(432\) 0 0
\(433\) 22.7610i 1.09382i −0.837190 0.546912i \(-0.815803\pi\)
0.837190 0.546912i \(-0.184197\pi\)
\(434\) 23.1960 + 14.4544i 1.11344 + 0.693832i
\(435\) 0 0
\(436\) −1.00000 + 1.73205i −0.0478913 + 0.0829502i
\(437\) −2.43487 4.21732i −0.116476 0.201742i
\(438\) 0 0
\(439\) −1.86282 1.07550i −0.0889074 0.0513307i 0.454887 0.890549i \(-0.349680\pi\)
−0.543795 + 0.839218i \(0.683013\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.66119 −0.174145
\(443\) −6.63790 3.83239i −0.315376 0.182082i 0.333954 0.942590i \(-0.391617\pi\)
−0.649330 + 0.760507i \(0.724950\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 9.17777 15.8964i 0.434580 0.752715i
\(447\) 0 0
\(448\) −2.24547 1.39924i −0.106088 0.0661079i
\(449\) 18.8337i 0.888817i 0.895824 + 0.444408i \(0.146586\pi\)
−0.895824 + 0.444408i \(0.853414\pi\)
\(450\) 0 0
\(451\) −11.5208 + 6.65155i −0.542494 + 0.313209i
\(452\) 13.9214 8.03750i 0.654805 0.378052i
\(453\) 0 0
\(454\) 3.02304i 0.141879i
\(455\) 0 0
\(456\) 0 0
\(457\) −14.6325 + 25.3442i −0.684478 + 1.18555i 0.289123 + 0.957292i \(0.406636\pi\)
−0.973601 + 0.228258i \(0.926697\pi\)
\(458\) −5.32325 9.22014i −0.248739 0.430829i
\(459\) 0 0
\(460\) 0 0
\(461\) −14.1963 −0.661188 −0.330594 0.943773i \(-0.607249\pi\)
−0.330594 + 0.943773i \(0.607249\pi\)
\(462\) 0 0
\(463\) −7.65787 −0.355891 −0.177946 0.984040i \(-0.556945\pi\)
−0.177946 + 0.984040i \(0.556945\pi\)
\(464\) −8.09565 4.67403i −0.375831 0.216986i
\(465\) 0 0
\(466\) −1.99293 3.45185i −0.0923206 0.159904i
\(467\) −8.85713 + 15.3410i −0.409859 + 0.709897i −0.994874 0.101126i \(-0.967755\pi\)
0.585015 + 0.811023i \(0.301089\pi\)
\(468\) 0 0
\(469\) 0.0317157 0.941825i 0.00146450 0.0434894i
\(470\) 0 0
\(471\) 0 0
\(472\) 5.38113 3.10680i 0.247687 0.143002i
\(473\) 8.64800 4.99293i 0.397636 0.229575i
\(474\) 0 0
\(475\) 0 0
\(476\) 9.23918 4.92732i 0.423477 0.225843i
\(477\) 0 0
\(478\) −4.12349 + 7.14210i −0.188604 + 0.326672i
\(479\) −6.41996 11.1197i −0.293336 0.508072i 0.681261 0.732041i \(-0.261432\pi\)
−0.974596 + 0.223969i \(0.928099\pi\)
\(480\) 0 0
\(481\) −0.341598 0.197222i −0.0155755 0.00899254i
\(482\) −10.9556 −0.499015
\(483\) 0 0
\(484\) −8.46457 −0.384753
\(485\) 0 0
\(486\) 0 0
\(487\) −1.88929 3.27235i −0.0856119 0.148284i 0.820040 0.572306i \(-0.193951\pi\)
−0.905652 + 0.424022i \(0.860618\pi\)
\(488\) −5.50025 + 9.52671i −0.248985 + 0.431254i
\(489\) 0 0
\(490\) 0 0
\(491\) 32.1664i 1.45165i 0.687880 + 0.725824i \(0.258541\pi\)
−0.687880 + 0.725824i \(0.741459\pi\)
\(492\) 0 0
\(493\) 32.0398 18.4982i 1.44300 0.833115i
\(494\) −0.501103 + 0.289312i −0.0225457 + 0.0130168i
\(495\) 0 0
\(496\) 10.3302i 0.463838i
\(497\) −12.7048 + 20.3883i −0.569886 + 0.914540i
\(498\) 0 0
\(499\) 15.9683 27.6579i 0.714839 1.23814i −0.248183 0.968713i \(-0.579833\pi\)
0.963022 0.269424i \(-0.0868332\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −18.8247 10.8685i −0.840190 0.485084i
\(503\) −5.29834 −0.236241 −0.118121 0.992999i \(-0.537687\pi\)
−0.118121 + 0.992999i \(0.537687\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 10.7362 + 6.19855i 0.477282 + 0.275559i
\(507\) 0 0
\(508\) −6.60867 11.4465i −0.293212 0.507858i
\(509\) 18.6321 32.2718i 0.825854 1.43042i −0.0754100 0.997153i \(-0.524027\pi\)
0.901265 0.433269i \(-0.142640\pi\)
\(510\) 0 0
\(511\) −18.0608 0.608195i −0.798965 0.0269049i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 15.0993 8.71758i 0.666001 0.384516i
\(515\) 0 0
\(516\) 0 0
\(517\) 4.42867i 0.194773i
\(518\) 1.12747 + 0.0379671i 0.0495380 + 0.00166818i
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00676 10.4040i −0.263161 0.455808i 0.703919 0.710280i \(-0.251432\pi\)
−0.967080 + 0.254472i \(0.918098\pi\)
\(522\) 0 0
\(523\) −23.4136 13.5178i −1.02380 0.591093i −0.108600 0.994085i \(-0.534637\pi\)
−0.915203 + 0.402992i \(0.867970\pi\)
\(524\) −9.40160 −0.410711
\(525\) 0 0
\(526\) −18.3511 −0.800148
\(527\) −35.4058 20.4416i −1.54230 0.890449i
\(528\) 0 0
\(529\) 18.8081 + 32.5766i 0.817744 + 1.41637i
\(530\) 0 0
\(531\) 0 0
\(532\) 0.875193 1.40449i 0.0379444 0.0608923i
\(533\) 7.72879i 0.334771i
\(534\) 0 0
\(535\) 0 0
\(536\) 0.308459 0.178089i 0.0133234 0.00769228i
\(537\) 0 0
\(538\) 24.8371i 1.07080i
\(539\) 9.25604 + 6.20981i 0.398686 + 0.267475i
\(540\) 0 0
\(541\) 5.85450 10.1403i 0.251705 0.435965i −0.712291 0.701885i \(-0.752342\pi\)
0.963995 + 0.265919i \(0.0856755\pi\)
\(542\) 12.2204 + 21.1663i 0.524910 + 0.909171i
\(543\) 0 0
\(544\) 3.42743 + 1.97883i 0.146950 + 0.0848415i
\(545\) 0 0
\(546\) 0 0
\(547\) 17.6050 0.752737 0.376369 0.926470i \(-0.377173\pi\)
0.376369 + 0.926470i \(0.377173\pi\)
\(548\) −8.11701 4.68636i −0.346741 0.200191i
\(549\) 0 0
\(550\) 0 0
\(551\) 2.92350 5.06364i 0.124545 0.215718i
\(552\) 0 0
\(553\) 21.1312 11.2694i 0.898589 0.479224i
\(554\) 23.1970i 0.985548i
\(555\) 0 0
\(556\) −12.2201 + 7.05530i −0.518249 + 0.299211i
\(557\) 17.3913 10.0409i 0.736892 0.425445i −0.0840462 0.996462i \(-0.526784\pi\)
0.820938 + 0.571017i \(0.193451\pi\)
\(558\) 0 0
\(559\) 5.80155i 0.245380i
\(560\) 0 0
\(561\) 0 0
\(562\) 7.04005 12.1937i 0.296967 0.514361i
\(563\) 11.9038 + 20.6180i 0.501687 + 0.868947i 0.999998 + 0.00194851i \(0.000620229\pi\)
−0.498312 + 0.866998i \(0.666046\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2.79820 0.117617
\(567\) 0 0
\(568\) −9.07975 −0.380978
\(569\) −35.8766 20.7134i −1.50403 0.868350i −0.999989 0.00466765i \(-0.998514\pi\)
−0.504037 0.863682i \(-0.668152\pi\)
\(570\) 0 0
\(571\) −20.5784 35.6428i −0.861177 1.49160i −0.870793 0.491649i \(-0.836394\pi\)
0.00961607 0.999954i \(-0.496939\pi\)
\(572\) 0.736513 1.27568i 0.0307951 0.0533388i
\(573\) 0 0
\(574\) −10.4016 19.5040i −0.434155 0.814080i
\(575\) 0 0
\(576\) 0 0
\(577\) 10.7532 6.20835i 0.447661 0.258457i −0.259181 0.965829i \(-0.583453\pi\)
0.706842 + 0.707372i \(0.250119\pi\)
\(578\) 1.15787 0.668498i 0.0481611 0.0278058i
\(579\) 0 0
\(580\) 0 0
\(581\) −1.81860 1.13324i −0.0754482 0.0470148i
\(582\) 0 0
\(583\) 2.66802 4.62114i 0.110498 0.191388i
\(584\) −3.41511 5.91515i −0.141318 0.244771i
\(585\) 0 0
\(586\) −22.1354 12.7799i −0.914406 0.527932i
\(587\) 4.01980 0.165915 0.0829575 0.996553i \(-0.473563\pi\)
0.0829575 + 0.996553i \(0.473563\pi\)
\(588\) 0 0
\(589\) −6.46127 −0.266232
\(590\) 0 0
\(591\) 0 0
\(592\) 0.213192 + 0.369259i 0.00876213 + 0.0151764i
\(593\) 21.7667 37.7010i 0.893850 1.54819i 0.0586292 0.998280i \(-0.481327\pi\)
0.835221 0.549914i \(-0.185340\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.47225i 0.101267i
\(597\) 0 0
\(598\) 6.23749 3.60121i 0.255070 0.147265i
\(599\) 27.8218 16.0629i 1.13677 0.656314i 0.191141 0.981563i \(-0.438781\pi\)
0.945629 + 0.325249i \(0.105448\pi\)
\(600\) 0 0
\(601\) 8.21681i 0.335171i −0.985858 0.167585i \(-0.946403\pi\)
0.985858 0.167585i \(-0.0535970\pi\)
\(602\) 7.80788 + 14.6405i 0.318225 + 0.596702i
\(603\) 0 0
\(604\) −10.4425 + 18.0869i −0.424898 + 0.735945i
\(605\) 0 0
\(606\) 0 0
\(607\) −24.4532 14.1180i −0.992523 0.573034i −0.0864957 0.996252i \(-0.527567\pi\)
−0.906028 + 0.423219i \(0.860900\pi\)
\(608\) 0.625477 0.0253664
\(609\) 0 0
\(610\) 0 0
\(611\) −2.22825 1.28648i −0.0901452 0.0520454i
\(612\) 0 0
\(613\) 15.3962 + 26.6670i 0.621846 + 1.07707i 0.989142 + 0.146965i \(0.0469504\pi\)
−0.367296 + 0.930104i \(0.619716\pi\)
\(614\) 17.1430 29.6926i 0.691836 1.19829i
\(615\) 0 0
\(616\) −0.141786 + 4.21045i −0.00571272 + 0.169644i
\(617\) 4.42613i 0.178189i −0.996023 0.0890947i \(-0.971603\pi\)
0.996023 0.0890947i \(-0.0283974\pi\)
\(618\) 0 0
\(619\) −9.47047 + 5.46778i −0.380650 + 0.219768i −0.678101 0.734969i \(-0.737197\pi\)
0.297451 + 0.954737i \(0.403864\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 8.68041i 0.348053i
\(623\) −9.37171 + 4.99800i −0.375470 + 0.200241i
\(624\) 0 0
\(625\) 0 0
\(626\) −15.0106 25.9992i −0.599946 1.03914i
\(627\) 0 0
\(628\) 21.1722 + 12.2238i 0.844862 + 0.487781i
\(629\) −1.68748 −0.0672841
\(630\) 0 0
\(631\) −38.2293 −1.52189 −0.760943 0.648819i \(-0.775263\pi\)
−0.760943 + 0.648819i \(0.775263\pi\)
\(632\) 7.83895 + 4.52582i 0.311817 + 0.180028i
\(633\) 0 0
\(634\) −13.8628 24.0111i −0.550563 0.953603i
\(635\) 0 0
\(636\) 0 0
\(637\) 5.81318 2.85321i 0.230327 0.113048i
\(638\) 14.8849i 0.589300i
\(639\) 0 0
\(640\) 0 0
\(641\) 29.0339 16.7627i 1.14677 0.662088i 0.198671 0.980066i \(-0.436337\pi\)
0.948098 + 0.317979i \(0.103004\pi\)
\(642\) 0 0
\(643\) 17.4072i 0.686474i 0.939249 + 0.343237i \(0.111523\pi\)
−0.939249 + 0.343237i \(0.888477\pi\)
\(644\) −10.8940 + 17.4824i −0.429283 + 0.688903i
\(645\) 0 0
\(646\) −1.23771 + 2.14378i −0.0486971 + 0.0843458i
\(647\) 19.1301 + 33.1343i 0.752082 + 1.30264i 0.946812 + 0.321787i \(0.104284\pi\)
−0.194730 + 0.980857i \(0.562383\pi\)
\(648\) 0 0
\(649\) −8.56839 4.94696i −0.336339 0.194185i
\(650\) 0 0
\(651\) 0 0
\(652\) 5.91515 0.231655
\(653\) −28.0023 16.1671i −1.09581 0.632669i −0.160696 0.987004i \(-0.551374\pi\)
−0.935119 + 0.354335i \(0.884707\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 4.17731 7.23532i 0.163097 0.282492i
\(657\) 0 0
\(658\) 7.35446 + 0.247660i 0.286707 + 0.00965478i
\(659\) 16.4516i 0.640865i −0.947271 0.320433i \(-0.896172\pi\)
0.947271 0.320433i \(-0.103828\pi\)
\(660\) 0 0
\(661\) −2.14550 + 1.23870i −0.0834502 + 0.0481800i −0.541144 0.840930i \(-0.682009\pi\)
0.457694 + 0.889110i \(0.348675\pi\)
\(662\) −5.45899 + 3.15175i −0.212170 + 0.122496i
\(663\) 0 0
\(664\) 0.809898i 0.0314301i
\(665\) 0 0
\(666\) 0 0
\(667\) −36.3903 + 63.0298i −1.40904 + 2.44052i
\(668\) 6.07201 + 10.5170i 0.234933 + 0.406916i
\(669\) 0 0
\(670\) 0 0
\(671\) 17.5161 0.676202
\(672\) 0 0
\(673\) 17.0784 0.658326 0.329163 0.944273i \(-0.393233\pi\)
0.329163 + 0.944273i \(0.393233\pi\)
\(674\) 23.7375 + 13.7048i 0.914333 + 0.527891i
\(675\) 0 0
\(676\) 6.07210 + 10.5172i 0.233542 + 0.404507i
\(677\) 23.3041 40.3639i 0.895650 1.55131i 0.0626524 0.998035i \(-0.480044\pi\)
0.832998 0.553276i \(-0.186623\pi\)
\(678\) 0 0
\(679\) −11.0221 + 17.6880i −0.422990 + 0.678803i
\(680\) 0 0
\(681\) 0 0
\(682\) 14.2450 8.22437i 0.545470 0.314927i
\(683\) −2.04994 + 1.18353i −0.0784388 + 0.0452867i −0.538706 0.842494i \(-0.681087\pi\)
0.460268 + 0.887780i \(0.347753\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −10.8299 + 15.0237i −0.413488 + 0.573609i
\(687\) 0 0
\(688\) −3.13566 + 5.43113i −0.119546 + 0.207060i
\(689\) −1.55006 2.68478i −0.0590524 0.102282i
\(690\) 0 0
\(691\) −19.0534 11.0005i −0.724826 0.418479i 0.0917001 0.995787i \(-0.470770\pi\)
−0.816527 + 0.577308i \(0.804103\pi\)
\(692\) 16.3984 0.623372
\(693\) 0 0
\(694\) 8.16672 0.310004
\(695\) 0 0
\(696\) 0 0
\(697\) 16.5323 + 28.6349i 0.626207 + 1.08462i
\(698\) 13.7943 23.8924i 0.522120 0.904339i
\(699\) 0 0
\(700\) 0 0
\(701\) 28.7909i 1.08742i 0.839274 + 0.543708i \(0.182980\pi\)
−0.839274 + 0.543708i \(0.817020\pi\)
\(702\) 0 0
\(703\) −0.230963 + 0.133347i −0.00871094 + 0.00502926i
\(704\) −1.37897 + 0.796151i −0.0519721 + 0.0300061i
\(705\) 0 0
\(706\) 7.04285i 0.265061i
\(707\) −0.670346 + 19.9065i −0.0252110 + 0.748661i
\(708\) 0 0
\(709\) −7.64049 + 13.2337i −0.286945 + 0.497003i −0.973079 0.230472i \(-0.925973\pi\)
0.686134 + 0.727475i \(0.259306\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −3.47659 2.00721i −0.130291 0.0752234i
\(713\) 80.4269 3.01201
\(714\) 0 0
\(715\) 0 0
\(716\) −2.73398 1.57846i −0.102174 0.0589900i
\(717\) 0 0
\(718\) 8.17213 + 14.1545i 0.304981 + 0.528243i
\(719\) −8.33730 + 14.4406i −0.310929 + 0.538545i −0.978564 0.205944i \(-0.933973\pi\)
0.667635 + 0.744489i \(0.267307\pi\)
\(720\) 0 0
\(721\) 2.07121 + 3.88372i 0.0771360 + 0.144637i
\(722\) 18.6088i 0.692547i
\(723\) 0 0
\(724\) 7.08336 4.08958i 0.263251 0.151988i
\(725\) 0 0
\(726\) 0 0
\(727\) 32.4228i 1.20250i −0.799062 0.601248i \(-0.794670\pi\)
0.799062 0.601248i \(-0.205330\pi\)
\(728\) 2.07726 + 1.29443i 0.0769885 + 0.0479746i
\(729\) 0 0
\(730\) 0 0
\(731\) −12.4099 21.4945i −0.458996 0.795004i
\(732\) 0 0
\(733\) 40.1207 + 23.1637i 1.48189 + 0.855570i 0.999789 0.0205452i \(-0.00654019\pi\)
0.482102 + 0.876115i \(0.339874\pi\)
\(734\) 3.00798 0.111026
\(735\) 0 0
\(736\) −7.78564 −0.286983
\(737\) −0.491161 0.283572i −0.0180921 0.0104455i
\(738\) 0 0
\(739\) −8.23689 14.2667i −0.302999 0.524809i 0.673815 0.738900i \(-0.264655\pi\)
−0.976814 + 0.214091i \(0.931321\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 7.52488 + 4.68905i 0.276247 + 0.172141i
\(743\) 38.4778i 1.41161i 0.708405 + 0.705806i \(0.249415\pi\)
−0.708405 + 0.705806i \(0.750585\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −20.0892 + 11.5985i −0.735519 + 0.424652i
\(747\) 0 0
\(748\) 6.30178i 0.230416i
\(749\) −24.1567 45.2960i −0.882666 1.65508i
\(750\) 0 0
\(751\) −18.9145 + 32.7608i −0.690198 + 1.19546i 0.281574 + 0.959539i \(0.409143\pi\)
−0.971773 + 0.235919i \(0.924190\pi\)
\(752\) 1.39065 + 2.40868i 0.0507118 + 0.0878355i
\(753\) 0 0
\(754\) 7.48922 + 4.32390i 0.272741 + 0.157467i
\(755\) 0 0
\(756\) 0 0
\(757\) −11.1485 −0.405197 −0.202599 0.979262i \(-0.564939\pi\)
−0.202599 + 0.979262i \(0.564939\pi\)
\(758\) −23.6181 13.6359i −0.857846 0.495278i
\(759\) 0 0
\(760\) 0 0
\(761\) 14.6239 25.3294i 0.530117 0.918189i −0.469266 0.883057i \(-0.655481\pi\)
0.999383 0.0351321i \(-0.0111852\pi\)
\(762\) 0 0
\(763\) −0.178089 + 5.28850i −0.00644726 + 0.191457i
\(764\) 2.82843i 0.102329i
\(765\) 0 0
\(766\) −3.69096 + 2.13098i −0.133360 + 0.0769953i
\(767\) −4.97804 + 2.87407i −0.179747 + 0.103777i
\(768\) 0 0
\(769\) 0.892823i 0.0321960i 0.999870 + 0.0160980i \(0.00512438\pi\)
−0.999870 + 0.0160980i \(0.994876\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8.85772 + 15.3420i −0.318796 + 0.552172i
\(773\) 9.89011 + 17.1302i 0.355723 + 0.616130i 0.987241 0.159231i \(-0.0509014\pi\)
−0.631519 + 0.775361i \(0.717568\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −7.87721 −0.282775
\(777\) 0 0
\(778\) −2.08172 −0.0746332
\(779\) 4.52553 + 2.61281i 0.162144 + 0.0936138i
\(780\) 0 0
\(781\) 7.22886 + 12.5207i 0.258669 + 0.448028i
\(782\) 15.4064 26.6847i 0.550933 0.954243i
\(783\) 0 0
\(784\) −6.98414 0.470912i −0.249434 0.0168183i
\(785\) 0 0
\(786\) 0 0
\(787\) 23.0325 13.2978i 0.821021 0.474017i −0.0297473 0.999557i \(-0.509470\pi\)
0.850769 + 0.525541i \(0.176137\pi\)
\(788\) −3.72694 + 2.15175i −0.132767 + 0.0766529i
\(789\) 0 0
\(790\) 0 0
\(791\) 22.4928 36.0959i 0.799752 1.28342i
\(792\) 0 0
\(793\) 5.08823 8.81308i 0.180688 0.312962i
\(794\) −11.1885 19.3791i −0.397066 0.687739i
\(795\) 0 0
\(796\) 3.00000 + 1.73205i 0.106332 + 0.0613909i
\(797\) −21.4698 −0.760499 −0.380250 0.924884i \(-0.624162\pi\)
−0.380250 + 0.924884i \(0.624162\pi\)
\(798\) 0 0
\(799\) −11.0074 −0.389415
\(800\) 0 0
\(801\) 0 0
\(802\) −9.83196 17.0295i −0.347179 0.601331i
\(803\) −5.43790 + 9.41871i −0.191899 + 0.332379i
\(804\) 0 0
\(805\) 0 0
\(806\) 9.55634i 0.336608i
\(807\) 0 0
\(808\) −6.51962 + 3.76411i −0.229360 + 0.132421i
\(809\) 18.2930 10.5615i 0.643149 0.371322i −0.142677 0.989769i \(-0.545571\pi\)
0.785827 + 0.618447i \(0.212238\pi\)
\(810\) 0 0
\(811\) 13.3784i 0.469779i −0.972022 0.234889i \(-0.924527\pi\)
0.972022 0.234889i \(-0.0754728\pi\)
\(812\) −24.7186 0.832393i −0.867453 0.0292113i
\(813\) 0 0
\(814\) 0.339466 0.587972i 0.0118983 0.0206084i
\(815\) 0 0
\(816\) 0 0
\(817\) −3.39705 1.96129i −0.118848 0.0686167i
\(818\) −2.53956 −0.0887936
\(819\) 0 0
\(820\) 0 0
\(821\) −5.67591 3.27699i −0.198091 0.114368i 0.397674 0.917527i \(-0.369818\pi\)
−0.595765 + 0.803159i \(0.703151\pi\)
\(822\) 0 0
\(823\) −21.9344 37.9915i −0.764585 1.32430i −0.940466 0.339889i \(-0.889611\pi\)
0.175880 0.984412i \(-0.443723\pi\)
\(824\) −0.831805 + 1.44073i −0.0289773 + 0.0501901i
\(825\) 0 0
\(826\) 8.69432 13.9524i 0.302514 0.485467i
\(827\) 19.1611i 0.666296i 0.942875 + 0.333148i \(0.108111\pi\)
−0.942875 + 0.333148i \(0.891889\pi\)
\(828\) 0 0
\(829\) −14.6635 + 8.46597i −0.509284 + 0.294035i −0.732539 0.680725i \(-0.761665\pi\)
0.223255 + 0.974760i \(0.428332\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.925091i 0.0320718i
\(833\) 15.4344 23.0058i 0.534771 0.797103i
\(834\) 0 0
\(835\) 0 0
\(836\) −0.497974 0.862517i −0.0172228 0.0298308i
\(837\) 0 0
\(838\) −27.7620 16.0284i −0.959023 0.553692i
\(839\) −10.5028 −0.362596 −0.181298 0.983428i \(-0.558030\pi\)
−0.181298 + 0.983428i \(0.558030\pi\)
\(840\) 0 0
\(841\) −58.3861 −2.01331
\(842\) −21.8413 12.6101i −0.752700 0.434572i
\(843\) 0 0
\(844\) 5.19432 + 8.99682i 0.178796 + 0.309683i
\(845\) 0 0
\(846\) 0 0
\(847\) −19.7606 + 10.5385i −0.678984 + 0.362107i
\(848\) 3.35114i 0.115079i
\(849\) 0 0
\(850\) 0 0
\(851\) 2.87492 1.65983i 0.0985509 0.0568984i
\(852\) 0 0
\(853\) 30.2419i 1.03546i 0.855544 + 0.517731i \(0.173223\pi\)
−0.855544 + 0.517731i \(0.826777\pi\)
\(854\) −0.979534 + 29.0881i −0.0335190 + 0.995374i
\(855\) 0 0
\(856\) 9.70139 16.8033i 0.331587 0.574325i
\(857\) 11.1322 + 19.2816i 0.380270 + 0.658646i 0.991101 0.133115i \(-0.0424979\pi\)
−0.610831 + 0.791761i \(0.709165\pi\)
\(858\) 0 0
\(859\) −26.7059 15.4187i −0.911195 0.526078i −0.0303793 0.999538i \(-0.509672\pi\)
−0.880815 + 0.473460i \(0.843005\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 33.6710 1.14684
\(863\) 20.0190 + 11.5580i 0.681454 + 0.393437i 0.800403 0.599463i \(-0.204619\pi\)
−0.118949 + 0.992900i \(0.537952\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 11.3805 19.7116i 0.386725 0.669828i
\(867\) 0 0
\(868\) 12.8612 + 24.1159i 0.436536 + 0.818546i
\(869\) 14.4130i 0.488926i
\(870\) 0 0
\(871\) −0.285353 + 0.164749i −0.00966882 + 0.00558230i
\(872\) −1.73205 + 1.00000i −0.0586546 + 0.0338643i
\(873\) 0 0
\(874\) 4.86974i 0.164721i
\(875\) 0 0
\(876\) 0 0
\(877\) −12.6074 + 21.8366i −0.425720 + 0.737369i −0.996487 0.0837427i \(-0.973313\pi\)
0.570767 + 0.821112i \(0.306646\pi\)
\(878\) −1.07550 1.86282i −0.0362963 0.0628670i
\(879\) 0 0
\(880\) 0 0
\(881\) 20.5142 0.691140 0.345570 0.938393i \(-0.387686\pi\)
0.345570 + 0.938393i \(0.387686\pi\)
\(882\) 0 0
\(883\) −43.0491 −1.44872 −0.724358 0.689424i \(-0.757864\pi\)
−0.724358 + 0.689424i \(0.757864\pi\)
\(884\) −3.17068 1.83059i −0.106642 0.0615696i
\(885\) 0 0
\(886\) −3.83239 6.63790i −0.128752 0.223005i
\(887\) 16.6952 28.9170i 0.560571 0.970938i −0.436876 0.899522i \(-0.643915\pi\)
0.997447 0.0714156i \(-0.0227517\pi\)
\(888\) 0 0
\(889\) −29.6791 18.4942i −0.995405 0.620277i
\(890\) 0 0
\(891\) 0 0
\(892\) 15.8964 9.17777i 0.532250 0.307294i
\(893\) −1.50657 + 0.869820i −0.0504156 + 0.0291074i
\(894\) 0 0
\(895\) 0 0
\(896\) −1.24501 2.33451i −0.0415929 0.0779906i
\(897\) 0 0
\(898\) −9.41684 + 16.3104i −0.314244 + 0.544287i
\(899\) 48.2834 + 83.6293i 1.61034 + 2.78919i
\(900\) 0 0
\(901\) −11.4858 6.63132i −0.382647 0.220921i
\(902\) −13.3031 −0.442945
\(903\) 0 0
\(904\) 16.0750 0.534646
\(905\) 0 0
\(906\) 0 0
\(907\) −13.6898 23.7115i −0.454563 0.787327i 0.544100 0.839021i \(-0.316871\pi\)
−0.998663 + 0.0516937i \(0.983538\pi\)
\(908\) 1.51152 2.61803i 0.0501616 0.0868825i
\(909\) 0 0
\(910\) 0 0
\(911\) 33.0422i 1.09474i −0.836892 0.547368i \(-0.815630\pi\)
0.836892 0.547368i \(-0.184370\pi\)
\(912\) 0 0
\(913\) −1.11683 + 0.644801i −0.0369616 + 0.0213398i
\(914\) −25.3442 + 14.6325i −0.838311 + 0.483999i
\(915\) 0 0
\(916\) 10.6465i 0.351770i
\(917\) −21.9482 + 11.7051i −0.724792 + 0.386537i
\(918\) 0 0
\(919\) −13.2444 + 22.9400i −0.436893 + 0.756722i −0.997448 0.0713958i \(-0.977255\pi\)
0.560555 + 0.828117i \(0.310588\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −12.2944 7.09815i −0.404893 0.233765i
\(923\) 8.39960 0.276476
\(924\) 0 0
\(925\) 0 0
\(926\) −6.63191 3.82893i −0.217938 0.125827i
\(927\) 0 0
\(928\) −4.67403 8.09565i −0.153432 0.265753i
\(929\) −15.5952 + 27.0117i −0.511663 + 0.886226i 0.488246 + 0.872706i \(0.337637\pi\)
−0.999909 + 0.0135196i \(0.995696\pi\)
\(930\) 0 0
\(931\) 0.294545 4.36842i 0.00965332 0.143169i
\(932\) 3.98585i 0.130561i
\(933\) 0 0
\(934\) −15.3410 + 8.85713i −0.501973 + 0.289814i
\(935\) 0 0
\(936\) 0 0
\(937\) 6.40017i 0.209084i 0.994520 + 0.104542i \(0.0333377\pi\)
−0.994520 + 0.104542i \(0.966662\pi\)
\(938\) 0.498379 0.799787i 0.0162727 0.0261140i
\(939\) 0 0
\(940\) 0 0
\(941\) 2.56243 + 4.43825i 0.0835327 + 0.144683i 0.904765 0.425911i \(-0.140046\pi\)
−0.821232 + 0.570594i \(0.806713\pi\)
\(942\) 0 0
\(943\) −56.3316 32.5231i −1.83441 1.05910i
\(944\) 6.21360 0.202235
\(945\) 0 0
\(946\) 9.98585 0.324668
\(947\) 36.5679 + 21.1125i 1.18830 + 0.686064i 0.957919 0.287037i \(-0.0926704\pi\)
0.230378 + 0.973101i \(0.426004\pi\)
\(948\) 0 0
\(949\) 3.15929 + 5.47206i 0.102555 + 0.177630i
\(950\) 0 0
\(951\) 0 0
\(952\) 10.4650 + 0.352407i 0.339174 + 0.0114216i
\(953\) 8.12428i 0.263171i −0.991305 0.131586i \(-0.957993\pi\)
0.991305 0.131586i \(-0.0420068\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −7.14210 + 4.12349i −0.230992 + 0.133363i
\(957\) 0 0
\(958\) 12.8399i 0.414839i
\(959\) −24.7838 0.834589i −0.800311 0.0269503i
\(960\) 0 0
\(961\) 37.8560 65.5686i 1.22116 2.11512i
\(962\) −0.197222 0.341598i −0.00635869 0.0110136i
\(963\) 0 0
\(964\) −9.48785 5.47782i −0.305583 0.176429i
\(965\) 0 0
\(966\) 0 0
\(967\) 52.3097 1.68217 0.841084 0.540905i \(-0.181918\pi\)
0.841084 + 0.540905i \(0.181918\pi\)
\(968\) −7.33053 4.23229i −0.235612 0.136031i
\(969\) 0 0
\(970\) 0 0
\(971\) −8.55280 + 14.8139i −0.274473 + 0.475400i −0.970002 0.243097i \(-0.921837\pi\)
0.695529 + 0.718498i \(0.255170\pi\)
\(972\) 0 0
\(973\) −19.7441 + 31.6849i −0.632968 + 1.01577i
\(974\) 3.77858i 0.121073i
\(975\) 0 0
\(976\) −9.52671 + 5.50025i −0.304943 + 0.176059i
\(977\) 31.5215 18.1989i 1.00846 0.582235i 0.0977201 0.995214i \(-0.468845\pi\)
0.910741 + 0.412979i \(0.135512\pi\)
\(978\) 0 0
\(979\) 6.39217i 0.204295i
\(980\) 0 0
\(981\) 0 0
\(982\) −16.0832 + 27.8569i −0.513235 + 0.888950i
\(983\) −12.8641 22.2813i −0.410302 0.710664i 0.584621 0.811307i \(-0.301243\pi\)
−0.994923 + 0.100643i \(0.967910\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 36.9963 1.17820
\(987\) 0 0
\(988\) −0.578623 −0.0184085
\(989\) 42.2848 + 24.4132i 1.34458 + 0.776293i
\(990\) 0 0
\(991\) 6.07375 + 10.5200i 0.192939 + 0.334180i 0.946223 0.323515i \(-0.104865\pi\)
−0.753284 + 0.657696i \(0.771531\pi\)
\(992\) −5.16508 + 8.94618i −0.163991 + 0.284041i
\(993\) 0 0
\(994\) −21.1968 + 11.3044i −0.672321 + 0.358554i
\(995\) 0 0
\(996\) 0 0
\(997\) 14.8552 8.57663i 0.470468 0.271625i −0.245968 0.969278i \(-0.579106\pi\)
0.716435 + 0.697653i \(0.245772\pi\)
\(998\) 27.6579 15.9683i 0.875495 0.505467i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3150.2.bf.f.1601.14 32
3.2 odd 2 inner 3150.2.bf.f.1601.6 32
5.2 odd 4 630.2.bo.a.89.5 yes 16
5.3 odd 4 630.2.bo.b.89.7 yes 16
5.4 even 2 inner 3150.2.bf.f.1601.5 32
7.3 odd 6 inner 3150.2.bf.f.1151.6 32
15.2 even 4 630.2.bo.b.89.4 yes 16
15.8 even 4 630.2.bo.a.89.2 16
15.14 odd 2 inner 3150.2.bf.f.1601.13 32
21.17 even 6 inner 3150.2.bf.f.1151.16 32
35.2 odd 12 4410.2.d.b.4409.4 16
35.3 even 12 630.2.bo.b.269.4 yes 16
35.12 even 12 4410.2.d.b.4409.13 16
35.17 even 12 630.2.bo.a.269.2 yes 16
35.23 odd 12 4410.2.d.a.4409.3 16
35.24 odd 6 inner 3150.2.bf.f.1151.15 32
35.33 even 12 4410.2.d.a.4409.14 16
105.2 even 12 4410.2.d.a.4409.13 16
105.17 odd 12 630.2.bo.b.269.7 yes 16
105.23 even 12 4410.2.d.b.4409.14 16
105.38 odd 12 630.2.bo.a.269.5 yes 16
105.47 odd 12 4410.2.d.a.4409.4 16
105.59 even 6 inner 3150.2.bf.f.1151.5 32
105.68 odd 12 4410.2.d.b.4409.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
630.2.bo.a.89.2 16 15.8 even 4
630.2.bo.a.89.5 yes 16 5.2 odd 4
630.2.bo.a.269.2 yes 16 35.17 even 12
630.2.bo.a.269.5 yes 16 105.38 odd 12
630.2.bo.b.89.4 yes 16 15.2 even 4
630.2.bo.b.89.7 yes 16 5.3 odd 4
630.2.bo.b.269.4 yes 16 35.3 even 12
630.2.bo.b.269.7 yes 16 105.17 odd 12
3150.2.bf.f.1151.5 32 105.59 even 6 inner
3150.2.bf.f.1151.6 32 7.3 odd 6 inner
3150.2.bf.f.1151.15 32 35.24 odd 6 inner
3150.2.bf.f.1151.16 32 21.17 even 6 inner
3150.2.bf.f.1601.5 32 5.4 even 2 inner
3150.2.bf.f.1601.6 32 3.2 odd 2 inner
3150.2.bf.f.1601.13 32 15.14 odd 2 inner
3150.2.bf.f.1601.14 32 1.1 even 1 trivial
4410.2.d.a.4409.3 16 35.23 odd 12
4410.2.d.a.4409.4 16 105.47 odd 12
4410.2.d.a.4409.13 16 105.2 even 12
4410.2.d.a.4409.14 16 35.33 even 12
4410.2.d.b.4409.3 16 105.68 odd 12
4410.2.d.b.4409.4 16 35.2 odd 12
4410.2.d.b.4409.13 16 35.12 even 12
4410.2.d.b.4409.14 16 105.23 even 12