Properties

Label 1500.2.m.c.1201.1
Level $1500$
Weight $2$
Character 1500.1201
Analytic conductor $11.978$
Analytic rank $0$
Dimension $24$
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] = [1500,2,Mod(301,1500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1500, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1500.301");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1500.m (of order \(5\), degree \(4\), 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: \(11.9775603032\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{5})\)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 1201.1
Character \(\chi\) \(=\) 1500.1201
Dual form 1500.2.m.c.301.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.309017 + 0.951057i) q^{3} -3.54704 q^{7} +(-0.809017 + 0.587785i) q^{9} +O(q^{10})\) \(q+(0.309017 + 0.951057i) q^{3} -3.54704 q^{7} +(-0.809017 + 0.587785i) q^{9} +(1.78482 + 1.29675i) q^{11} +(5.80689 - 4.21895i) q^{13} +(1.96699 - 6.05378i) q^{17} +(-0.715151 + 2.20101i) q^{19} +(-1.09610 - 3.37344i) q^{21} +(-1.76038 - 1.27899i) q^{23} +(-0.809017 - 0.587785i) q^{27} +(0.262008 + 0.806379i) q^{29} +(-1.32905 + 4.09040i) q^{31} +(-0.681742 + 2.09819i) q^{33} +(5.84918 - 4.24968i) q^{37} +(5.80689 + 4.21895i) q^{39} +(1.08778 - 0.790317i) q^{41} +8.18973 q^{43} +(1.87095 + 5.75820i) q^{47} +5.58150 q^{49} +6.36532 q^{51} +(3.69530 + 11.3730i) q^{53} -2.31428 q^{57} +(10.0896 - 7.33050i) q^{59} +(5.59873 + 4.06772i) q^{61} +(2.86962 - 2.08490i) q^{63} +(1.46291 - 4.50239i) q^{67} +(0.672404 - 2.06945i) q^{69} +(4.25799 + 13.1047i) q^{71} +(-0.881155 - 0.640196i) q^{73} +(-6.33085 - 4.59963i) q^{77} +(-1.80542 - 5.55650i) q^{79} +(0.309017 - 0.951057i) q^{81} +(3.87127 - 11.9145i) q^{83} +(-0.685947 + 0.498370i) q^{87} +(5.68424 + 4.12984i) q^{89} +(-20.5973 + 14.9648i) q^{91} -4.30090 q^{93} +(-5.60695 - 17.2564i) q^{97} -2.20616 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 6 q^{3} + 16 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 6 q^{3} + 16 q^{7} - 6 q^{9} - 6 q^{11} - 4 q^{17} - 10 q^{19} - 4 q^{21} - 14 q^{23} - 6 q^{27} - 4 q^{29} + 6 q^{31} + 4 q^{33} + 8 q^{37} - 10 q^{41} + 56 q^{43} - 26 q^{47} + 56 q^{49} + 16 q^{51} + 32 q^{53} + 20 q^{57} + 36 q^{59} - 12 q^{61} - 4 q^{63} - 36 q^{67} - 4 q^{69} + 40 q^{71} - 32 q^{73} + 46 q^{77} - 8 q^{79} - 6 q^{81} + 6 q^{83} - 4 q^{87} - 30 q^{91} - 4 q^{93} - 48 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1500\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(877\) \(1001\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{5}\right)\) \(1\)

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
\(3\) 0.309017 + 0.951057i 0.178411 + 0.549093i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.54704 −1.34066 −0.670328 0.742065i \(-0.733846\pi\)
−0.670328 + 0.742065i \(0.733846\pi\)
\(8\) 0 0
\(9\) −0.809017 + 0.587785i −0.269672 + 0.195928i
\(10\) 0 0
\(11\) 1.78482 + 1.29675i 0.538145 + 0.390985i 0.823396 0.567468i \(-0.192077\pi\)
−0.285251 + 0.958453i \(0.592077\pi\)
\(12\) 0 0
\(13\) 5.80689 4.21895i 1.61054 1.17013i 0.748773 0.662827i \(-0.230643\pi\)
0.861769 0.507301i \(-0.169357\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.96699 6.05378i 0.477066 1.46826i −0.366086 0.930581i \(-0.619302\pi\)
0.843151 0.537676i \(-0.180698\pi\)
\(18\) 0 0
\(19\) −0.715151 + 2.20101i −0.164067 + 0.504946i −0.998966 0.0454566i \(-0.985526\pi\)
0.834899 + 0.550402i \(0.185526\pi\)
\(20\) 0 0
\(21\) −1.09610 3.37344i −0.239188 0.736144i
\(22\) 0 0
\(23\) −1.76038 1.27899i −0.367064 0.266687i 0.388929 0.921268i \(-0.372845\pi\)
−0.755992 + 0.654580i \(0.772845\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.809017 0.587785i −0.155695 0.113119i
\(28\) 0 0
\(29\) 0.262008 + 0.806379i 0.0486538 + 0.149741i 0.972432 0.233188i \(-0.0749156\pi\)
−0.923778 + 0.382928i \(0.874916\pi\)
\(30\) 0 0
\(31\) −1.32905 + 4.09040i −0.238705 + 0.734657i 0.757904 + 0.652367i \(0.226224\pi\)
−0.996608 + 0.0822910i \(0.973776\pi\)
\(32\) 0 0
\(33\) −0.681742 + 2.09819i −0.118676 + 0.365248i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.84918 4.24968i 0.961599 0.698643i 0.00807756 0.999967i \(-0.497429\pi\)
0.953522 + 0.301325i \(0.0974288\pi\)
\(38\) 0 0
\(39\) 5.80689 + 4.21895i 0.929847 + 0.675573i
\(40\) 0 0
\(41\) 1.08778 0.790317i 0.169882 0.123427i −0.499595 0.866259i \(-0.666518\pi\)
0.669478 + 0.742832i \(0.266518\pi\)
\(42\) 0 0
\(43\) 8.18973 1.24892 0.624461 0.781056i \(-0.285319\pi\)
0.624461 + 0.781056i \(0.285319\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.87095 + 5.75820i 0.272907 + 0.839920i 0.989766 + 0.142702i \(0.0455790\pi\)
−0.716859 + 0.697218i \(0.754421\pi\)
\(48\) 0 0
\(49\) 5.58150 0.797357
\(50\) 0 0
\(51\) 6.36532 0.891323
\(52\) 0 0
\(53\) 3.69530 + 11.3730i 0.507589 + 1.56220i 0.796374 + 0.604805i \(0.206749\pi\)
−0.288785 + 0.957394i \(0.593251\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.31428 −0.306533
\(58\) 0 0
\(59\) 10.0896 7.33050i 1.31355 0.954350i 0.313561 0.949568i \(-0.398478\pi\)
0.999989 0.00478202i \(-0.00152217\pi\)
\(60\) 0 0
\(61\) 5.59873 + 4.06772i 0.716844 + 0.520818i 0.885374 0.464879i \(-0.153902\pi\)
−0.168530 + 0.985696i \(0.553902\pi\)
\(62\) 0 0
\(63\) 2.86962 2.08490i 0.361538 0.262672i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.46291 4.50239i 0.178723 0.550054i −0.821060 0.570841i \(-0.806617\pi\)
0.999784 + 0.0207870i \(0.00661719\pi\)
\(68\) 0 0
\(69\) 0.672404 2.06945i 0.0809479 0.249132i
\(70\) 0 0
\(71\) 4.25799 + 13.1047i 0.505330 + 1.55525i 0.800214 + 0.599714i \(0.204719\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(72\) 0 0
\(73\) −0.881155 0.640196i −0.103131 0.0749293i 0.535024 0.844837i \(-0.320302\pi\)
−0.638156 + 0.769907i \(0.720302\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.33085 4.59963i −0.721467 0.524176i
\(78\) 0 0
\(79\) −1.80542 5.55650i −0.203125 0.625156i −0.999785 0.0207276i \(-0.993402\pi\)
0.796660 0.604428i \(-0.206598\pi\)
\(80\) 0 0
\(81\) 0.309017 0.951057i 0.0343352 0.105673i
\(82\) 0 0
\(83\) 3.87127 11.9145i 0.424927 1.30779i −0.478137 0.878285i \(-0.658688\pi\)
0.903064 0.429505i \(-0.141312\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.685947 + 0.498370i −0.0735412 + 0.0534308i
\(88\) 0 0
\(89\) 5.68424 + 4.12984i 0.602528 + 0.437762i 0.846775 0.531951i \(-0.178541\pi\)
−0.244247 + 0.969713i \(0.578541\pi\)
\(90\) 0 0
\(91\) −20.5973 + 14.9648i −2.15918 + 1.56874i
\(92\) 0 0
\(93\) −4.30090 −0.445983
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −5.60695 17.2564i −0.569299 1.75212i −0.654818 0.755787i \(-0.727255\pi\)
0.0855183 0.996337i \(-0.472745\pi\)
\(98\) 0 0
\(99\) −2.20616 −0.221728
\(100\) 0 0
\(101\) 5.97473 0.594508 0.297254 0.954798i \(-0.403929\pi\)
0.297254 + 0.954798i \(0.403929\pi\)
\(102\) 0 0
\(103\) −0.142014 0.437076i −0.0139931 0.0430663i 0.943816 0.330471i \(-0.107208\pi\)
−0.957809 + 0.287405i \(0.907208\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.47862 0.239617 0.119809 0.992797i \(-0.461772\pi\)
0.119809 + 0.992797i \(0.461772\pi\)
\(108\) 0 0
\(109\) 2.70314 1.96394i 0.258914 0.188112i −0.450754 0.892648i \(-0.648845\pi\)
0.709668 + 0.704536i \(0.248845\pi\)
\(110\) 0 0
\(111\) 5.84918 + 4.24968i 0.555180 + 0.403362i
\(112\) 0 0
\(113\) −14.2370 + 10.3438i −1.33930 + 0.973059i −0.339832 + 0.940486i \(0.610370\pi\)
−0.999469 + 0.0325728i \(0.989630\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.21804 + 6.82641i −0.205057 + 0.631102i
\(118\) 0 0
\(119\) −6.97700 + 21.4730i −0.639581 + 1.96843i
\(120\) 0 0
\(121\) −1.89515 5.83267i −0.172286 0.530243i
\(122\) 0 0
\(123\) 1.08778 + 0.790317i 0.0980816 + 0.0712605i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −10.0415 7.29555i −0.891036 0.647376i 0.0451118 0.998982i \(-0.485636\pi\)
−0.936148 + 0.351606i \(0.885636\pi\)
\(128\) 0 0
\(129\) 2.53077 + 7.78890i 0.222822 + 0.685774i
\(130\) 0 0
\(131\) 4.88963 15.0487i 0.427209 1.31481i −0.473654 0.880711i \(-0.657065\pi\)
0.900863 0.434104i \(-0.142935\pi\)
\(132\) 0 0
\(133\) 2.53667 7.80706i 0.219957 0.676958i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.98262 + 4.34663i −0.511130 + 0.371358i −0.813252 0.581912i \(-0.802305\pi\)
0.302122 + 0.953269i \(0.402305\pi\)
\(138\) 0 0
\(139\) −3.18667 2.31525i −0.270290 0.196377i 0.444381 0.895838i \(-0.353424\pi\)
−0.714671 + 0.699461i \(0.753424\pi\)
\(140\) 0 0
\(141\) −4.89822 + 3.55876i −0.412504 + 0.299702i
\(142\) 0 0
\(143\) 15.8352 1.32421
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.72478 + 5.30832i 0.142257 + 0.437823i
\(148\) 0 0
\(149\) 3.92892 0.321870 0.160935 0.986965i \(-0.448549\pi\)
0.160935 + 0.986965i \(0.448549\pi\)
\(150\) 0 0
\(151\) 7.93418 0.645674 0.322837 0.946455i \(-0.395363\pi\)
0.322837 + 0.946455i \(0.395363\pi\)
\(152\) 0 0
\(153\) 1.96699 + 6.05378i 0.159022 + 0.489419i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −6.09738 −0.486624 −0.243312 0.969948i \(-0.578234\pi\)
−0.243312 + 0.969948i \(0.578234\pi\)
\(158\) 0 0
\(159\) −9.67443 + 7.02889i −0.767232 + 0.557427i
\(160\) 0 0
\(161\) 6.24413 + 4.53662i 0.492106 + 0.357536i
\(162\) 0 0
\(163\) 0.00206811 0.00150257i 0.000161987 0.000117690i −0.587704 0.809076i \(-0.699968\pi\)
0.587866 + 0.808958i \(0.299968\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.12128 + 9.60630i −0.241532 + 0.743358i 0.754656 + 0.656121i \(0.227804\pi\)
−0.996188 + 0.0872372i \(0.972196\pi\)
\(168\) 0 0
\(169\) 11.9032 36.6343i 0.915631 2.81802i
\(170\) 0 0
\(171\) −0.715151 2.20101i −0.0546889 0.168315i
\(172\) 0 0
\(173\) 4.99963 + 3.63244i 0.380115 + 0.276169i 0.761393 0.648291i \(-0.224516\pi\)
−0.381278 + 0.924460i \(0.624516\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 10.0896 + 7.33050i 0.758378 + 0.550994i
\(178\) 0 0
\(179\) 4.45991 + 13.7262i 0.333349 + 1.02594i 0.967530 + 0.252758i \(0.0813376\pi\)
−0.634181 + 0.773185i \(0.718662\pi\)
\(180\) 0 0
\(181\) 2.08366 6.41283i 0.154877 0.476662i −0.843272 0.537488i \(-0.819373\pi\)
0.998148 + 0.0608258i \(0.0193734\pi\)
\(182\) 0 0
\(183\) −2.13853 + 6.58170i −0.158084 + 0.486534i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 11.3610 8.25424i 0.830798 0.603610i
\(188\) 0 0
\(189\) 2.86962 + 2.08490i 0.208734 + 0.151654i
\(190\) 0 0
\(191\) −12.4512 + 9.04634i −0.900938 + 0.654570i −0.938707 0.344717i \(-0.887975\pi\)
0.0377687 + 0.999287i \(0.487975\pi\)
\(192\) 0 0
\(193\) 16.3875 1.17960 0.589799 0.807550i \(-0.299207\pi\)
0.589799 + 0.807550i \(0.299207\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.04392 + 9.36824i 0.216871 + 0.667459i 0.999015 + 0.0443625i \(0.0141257\pi\)
−0.782145 + 0.623097i \(0.785874\pi\)
\(198\) 0 0
\(199\) −4.96275 −0.351800 −0.175900 0.984408i \(-0.556284\pi\)
−0.175900 + 0.984408i \(0.556284\pi\)
\(200\) 0 0
\(201\) 4.73409 0.333917
\(202\) 0 0
\(203\) −0.929355 2.86026i −0.0652279 0.200751i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 2.17594 0.151239
\(208\) 0 0
\(209\) −4.13058 + 3.00104i −0.285718 + 0.207586i
\(210\) 0 0
\(211\) −2.83140 2.05713i −0.194921 0.141619i 0.486044 0.873934i \(-0.338440\pi\)
−0.680965 + 0.732316i \(0.738440\pi\)
\(212\) 0 0
\(213\) −11.1476 + 8.09918i −0.763818 + 0.554947i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.71420 14.5088i 0.320021 0.984923i
\(218\) 0 0
\(219\) 0.336571 1.03586i 0.0227434 0.0699969i
\(220\) 0 0
\(221\) −14.1185 43.4523i −0.949714 2.92292i
\(222\) 0 0
\(223\) −18.6710 13.5653i −1.25030 0.908397i −0.252062 0.967711i \(-0.581109\pi\)
−0.998239 + 0.0593138i \(0.981109\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.73323 + 3.43889i 0.314155 + 0.228247i 0.733677 0.679498i \(-0.237802\pi\)
−0.419522 + 0.907745i \(0.637802\pi\)
\(228\) 0 0
\(229\) −1.97484 6.07793i −0.130501 0.401641i 0.864362 0.502870i \(-0.167723\pi\)
−0.994863 + 0.101229i \(0.967723\pi\)
\(230\) 0 0
\(231\) 2.41817 7.44236i 0.159104 0.489671i
\(232\) 0 0
\(233\) 1.40390 4.32076i 0.0919725 0.283062i −0.894480 0.447107i \(-0.852454\pi\)
0.986453 + 0.164045i \(0.0524543\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 4.72664 3.43411i 0.307029 0.223069i
\(238\) 0 0
\(239\) −18.6407 13.5432i −1.20576 0.876040i −0.210926 0.977502i \(-0.567648\pi\)
−0.994839 + 0.101463i \(0.967648\pi\)
\(240\) 0 0
\(241\) −13.3064 + 9.66766i −0.857140 + 0.622749i −0.927105 0.374801i \(-0.877711\pi\)
0.0699655 + 0.997549i \(0.477711\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.13315 + 15.7982i 0.326614 + 1.00522i
\(248\) 0 0
\(249\) 12.5277 0.793910
\(250\) 0 0
\(251\) 4.66327 0.294343 0.147171 0.989111i \(-0.452983\pi\)
0.147171 + 0.989111i \(0.452983\pi\)
\(252\) 0 0
\(253\) −1.48343 4.56554i −0.0932627 0.287033i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.78734 −0.361004 −0.180502 0.983575i \(-0.557772\pi\)
−0.180502 + 0.983575i \(0.557772\pi\)
\(258\) 0 0
\(259\) −20.7473 + 15.0738i −1.28917 + 0.936639i
\(260\) 0 0
\(261\) −0.685947 0.498370i −0.0424591 0.0308483i
\(262\) 0 0
\(263\) −18.5668 + 13.4896i −1.14488 + 0.831803i −0.987792 0.155781i \(-0.950210\pi\)
−0.157087 + 0.987585i \(0.550210\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −2.17119 + 6.68222i −0.132874 + 0.408945i
\(268\) 0 0
\(269\) −1.39366 + 4.28923i −0.0849727 + 0.261519i −0.984511 0.175323i \(-0.943903\pi\)
0.899538 + 0.436842i \(0.143903\pi\)
\(270\) 0 0
\(271\) −3.06895 9.44525i −0.186425 0.573758i 0.813545 0.581502i \(-0.197535\pi\)
−0.999970 + 0.00774433i \(0.997535\pi\)
\(272\) 0 0
\(273\) −20.5973 14.9648i −1.24660 0.905711i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 18.6197 + 13.5280i 1.11875 + 0.812821i 0.984019 0.178061i \(-0.0569825\pi\)
0.134732 + 0.990882i \(0.456983\pi\)
\(278\) 0 0
\(279\) −1.32905 4.09040i −0.0795682 0.244886i
\(280\) 0 0
\(281\) 0.226820 0.698080i 0.0135309 0.0416439i −0.944063 0.329765i \(-0.893031\pi\)
0.957594 + 0.288121i \(0.0930305\pi\)
\(282\) 0 0
\(283\) 6.64144 20.4402i 0.394793 1.21505i −0.534330 0.845276i \(-0.679436\pi\)
0.929123 0.369771i \(-0.120564\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.85839 + 2.80329i −0.227754 + 0.165473i
\(288\) 0 0
\(289\) −19.0259 13.8231i −1.11917 0.813126i
\(290\) 0 0
\(291\) 14.6792 10.6651i 0.860509 0.625196i
\(292\) 0 0
\(293\) −26.5961 −1.55376 −0.776880 0.629649i \(-0.783199\pi\)
−0.776880 + 0.629649i \(0.783199\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.681742 2.09819i −0.0395587 0.121749i
\(298\) 0 0
\(299\) −15.6183 −0.903230
\(300\) 0 0
\(301\) −29.0493 −1.67437
\(302\) 0 0
\(303\) 1.84629 + 5.68231i 0.106067 + 0.326440i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.1736 0.694785 0.347393 0.937720i \(-0.387067\pi\)
0.347393 + 0.937720i \(0.387067\pi\)
\(308\) 0 0
\(309\) 0.371799 0.270128i 0.0211509 0.0153670i
\(310\) 0 0
\(311\) 20.0330 + 14.5548i 1.13597 + 0.825327i 0.986552 0.163447i \(-0.0522614\pi\)
0.149414 + 0.988775i \(0.452261\pi\)
\(312\) 0 0
\(313\) −12.0102 + 8.72589i −0.678854 + 0.493216i −0.872977 0.487761i \(-0.837814\pi\)
0.194123 + 0.980977i \(0.437814\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0.600207 1.84725i 0.0337110 0.103752i −0.932785 0.360433i \(-0.882629\pi\)
0.966496 + 0.256681i \(0.0826290\pi\)
\(318\) 0 0
\(319\) −0.578034 + 1.77901i −0.0323637 + 0.0996052i
\(320\) 0 0
\(321\) 0.765935 + 2.35731i 0.0427503 + 0.131572i
\(322\) 0 0
\(323\) 11.9177 + 8.65873i 0.663120 + 0.481785i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2.70314 + 1.96394i 0.149484 + 0.108606i
\(328\) 0 0
\(329\) −6.63635 20.4246i −0.365874 1.12604i
\(330\) 0 0
\(331\) −8.56924 + 26.3734i −0.471008 + 1.44961i 0.380258 + 0.924880i \(0.375835\pi\)
−0.851266 + 0.524734i \(0.824165\pi\)
\(332\) 0 0
\(333\) −2.23419 + 6.87612i −0.122433 + 0.376809i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8.81048 + 6.40119i −0.479937 + 0.348695i −0.801301 0.598261i \(-0.795859\pi\)
0.321364 + 0.946956i \(0.395859\pi\)
\(338\) 0 0
\(339\) −14.2370 10.3438i −0.773246 0.561796i
\(340\) 0 0
\(341\) −7.67636 + 5.57720i −0.415698 + 0.302022i
\(342\) 0 0
\(343\) 5.03148 0.271675
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.06224 + 21.7353i 0.379121 + 1.16681i 0.940656 + 0.339361i \(0.110211\pi\)
−0.561536 + 0.827453i \(0.689789\pi\)
\(348\) 0 0
\(349\) −28.8539 −1.54451 −0.772256 0.635311i \(-0.780872\pi\)
−0.772256 + 0.635311i \(0.780872\pi\)
\(350\) 0 0
\(351\) −7.17771 −0.383118
\(352\) 0 0
\(353\) −8.41799 25.9079i −0.448045 1.37894i −0.879110 0.476619i \(-0.841862\pi\)
0.431065 0.902321i \(-0.358138\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −22.5781 −1.19496
\(358\) 0 0
\(359\) 19.3648 14.0694i 1.02204 0.742553i 0.0553373 0.998468i \(-0.482377\pi\)
0.966699 + 0.255915i \(0.0823766\pi\)
\(360\) 0 0
\(361\) 11.0383 + 8.01981i 0.580965 + 0.422096i
\(362\) 0 0
\(363\) 4.96157 3.60479i 0.260415 0.189202i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2.07450 + 6.38465i −0.108288 + 0.333276i −0.990488 0.137599i \(-0.956062\pi\)
0.882200 + 0.470875i \(0.156062\pi\)
\(368\) 0 0
\(369\) −0.415494 + 1.27876i −0.0216298 + 0.0665696i
\(370\) 0 0
\(371\) −13.1074 40.3404i −0.680502 2.09437i
\(372\) 0 0
\(373\) −11.0100 7.99923i −0.570076 0.414184i 0.265057 0.964233i \(-0.414609\pi\)
−0.835133 + 0.550048i \(0.814609\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.92353 + 3.57715i 0.253575 + 0.184233i
\(378\) 0 0
\(379\) 0.137272 + 0.422481i 0.00705120 + 0.0217014i 0.954520 0.298146i \(-0.0963683\pi\)
−0.947469 + 0.319848i \(0.896368\pi\)
\(380\) 0 0
\(381\) 3.83550 11.8045i 0.196499 0.604760i
\(382\) 0 0
\(383\) 3.16140 9.72980i 0.161540 0.497170i −0.837224 0.546859i \(-0.815823\pi\)
0.998765 + 0.0496897i \(0.0158232\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −6.62563 + 4.81380i −0.336800 + 0.244699i
\(388\) 0 0
\(389\) 19.5834 + 14.2282i 0.992919 + 0.721398i 0.960558 0.278078i \(-0.0896975\pi\)
0.0323607 + 0.999476i \(0.489697\pi\)
\(390\) 0 0
\(391\) −11.2054 + 8.14117i −0.566679 + 0.411717i
\(392\) 0 0
\(393\) 15.8232 0.798174
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.841124 + 2.58871i 0.0422148 + 0.129924i 0.969943 0.243333i \(-0.0782409\pi\)
−0.927728 + 0.373257i \(0.878241\pi\)
\(398\) 0 0
\(399\) 8.20883 0.410956
\(400\) 0 0
\(401\) 31.3538 1.56573 0.782867 0.622189i \(-0.213756\pi\)
0.782867 + 0.622189i \(0.213756\pi\)
\(402\) 0 0
\(403\) 9.53955 + 29.3597i 0.475199 + 1.46251i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.9505 0.790639
\(408\) 0 0
\(409\) −2.68805 + 1.95298i −0.132916 + 0.0965688i −0.652256 0.757998i \(-0.726177\pi\)
0.519341 + 0.854567i \(0.326177\pi\)
\(410\) 0 0
\(411\) −5.98262 4.34663i −0.295101 0.214404i
\(412\) 0 0
\(413\) −35.7881 + 26.0016i −1.76102 + 1.27945i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.21720 3.74615i 0.0596065 0.183450i
\(418\) 0 0
\(419\) −3.41347 + 10.5056i −0.166759 + 0.513231i −0.999162 0.0409399i \(-0.986965\pi\)
0.832403 + 0.554171i \(0.186965\pi\)
\(420\) 0 0
\(421\) −7.56044 23.2686i −0.368473 1.13404i −0.947777 0.318932i \(-0.896676\pi\)
0.579304 0.815111i \(-0.303324\pi\)
\(422\) 0 0
\(423\) −4.89822 3.55876i −0.238160 0.173033i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −19.8589 14.4284i −0.961041 0.698237i
\(428\) 0 0
\(429\) 4.89335 + 15.0602i 0.236253 + 0.727113i
\(430\) 0 0
\(431\) 0.354621 1.09141i 0.0170815 0.0525715i −0.942152 0.335185i \(-0.891201\pi\)
0.959234 + 0.282613i \(0.0912013\pi\)
\(432\) 0 0
\(433\) 1.80916 5.56802i 0.0869427 0.267582i −0.898128 0.439735i \(-0.855072\pi\)
0.985070 + 0.172153i \(0.0550723\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.07400 2.95993i 0.194886 0.141593i
\(438\) 0 0
\(439\) 16.4473 + 11.9497i 0.784988 + 0.570328i 0.906472 0.422266i \(-0.138765\pi\)
−0.121484 + 0.992593i \(0.538765\pi\)
\(440\) 0 0
\(441\) −4.51553 + 3.28072i −0.215025 + 0.156225i
\(442\) 0 0
\(443\) −17.5912 −0.835783 −0.417891 0.908497i \(-0.637231\pi\)
−0.417891 + 0.908497i \(0.637231\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1.21410 + 3.73662i 0.0574251 + 0.176736i
\(448\) 0 0
\(449\) −2.83956 −0.134007 −0.0670037 0.997753i \(-0.521344\pi\)
−0.0670037 + 0.997753i \(0.521344\pi\)
\(450\) 0 0
\(451\) 2.96634 0.139679
\(452\) 0 0
\(453\) 2.45180 + 7.54585i 0.115195 + 0.354535i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.12004 0.379840 0.189920 0.981800i \(-0.439177\pi\)
0.189920 + 0.981800i \(0.439177\pi\)
\(458\) 0 0
\(459\) −5.14965 + 3.74144i −0.240365 + 0.174636i
\(460\) 0 0
\(461\) −14.5629 10.5806i −0.678261 0.492786i 0.194519 0.980899i \(-0.437685\pi\)
−0.872780 + 0.488113i \(0.837685\pi\)
\(462\) 0 0
\(463\) 23.1960 16.8529i 1.07801 0.783219i 0.100674 0.994919i \(-0.467900\pi\)
0.977335 + 0.211700i \(0.0679001\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.19421 + 19.0638i −0.286634 + 0.882169i 0.699270 + 0.714858i \(0.253509\pi\)
−0.985904 + 0.167311i \(0.946491\pi\)
\(468\) 0 0
\(469\) −5.18902 + 15.9702i −0.239607 + 0.737433i
\(470\) 0 0
\(471\) −1.88419 5.79896i −0.0868191 0.267202i
\(472\) 0 0
\(473\) 14.6172 + 10.6200i 0.672101 + 0.488310i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −9.67443 7.02889i −0.442962 0.321831i
\(478\) 0 0
\(479\) −4.78232 14.7185i −0.218510 0.672505i −0.998886 0.0471937i \(-0.984972\pi\)
0.780376 0.625311i \(-0.215028\pi\)
\(480\) 0 0
\(481\) 16.0364 49.3548i 0.731195 2.25039i
\(482\) 0 0
\(483\) −2.38504 + 7.34041i −0.108523 + 0.334000i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −30.3110 + 22.0222i −1.37352 + 0.997923i −0.376070 + 0.926591i \(0.622725\pi\)
−0.997453 + 0.0713312i \(0.977275\pi\)
\(488\) 0 0
\(489\) 0.00206811 + 0.00150257i 9.35232e−5 + 6.79486e-5i
\(490\) 0 0
\(491\) −32.3517 + 23.5049i −1.46001 + 1.06076i −0.476651 + 0.879093i \(0.658149\pi\)
−0.983360 + 0.181667i \(0.941851\pi\)
\(492\) 0 0
\(493\) 5.39701 0.243069
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −15.1033 46.4831i −0.677474 2.08505i
\(498\) 0 0
\(499\) 28.3040 1.26706 0.633530 0.773718i \(-0.281605\pi\)
0.633530 + 0.773718i \(0.281605\pi\)
\(500\) 0 0
\(501\) −10.1007 −0.451264
\(502\) 0 0
\(503\) 10.0998 + 31.0839i 0.450327 + 1.38596i 0.876535 + 0.481338i \(0.159849\pi\)
−0.426208 + 0.904625i \(0.640151\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 38.5196 1.71071
\(508\) 0 0
\(509\) −13.8620 + 10.0713i −0.614422 + 0.446404i −0.850969 0.525217i \(-0.823984\pi\)
0.236547 + 0.971620i \(0.423984\pi\)
\(510\) 0 0
\(511\) 3.12549 + 2.27080i 0.138264 + 0.100454i
\(512\) 0 0
\(513\) 1.87229 1.36030i 0.0826636 0.0600586i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −4.12763 + 12.7035i −0.181533 + 0.558701i
\(518\) 0 0
\(519\) −1.90969 + 5.87741i −0.0838260 + 0.257990i
\(520\) 0 0
\(521\) 6.22259 + 19.1512i 0.272617 + 0.839028i 0.989840 + 0.142185i \(0.0454128\pi\)
−0.717223 + 0.696843i \(0.754587\pi\)
\(522\) 0 0
\(523\) −22.5777 16.4036i −0.987253 0.717281i −0.0279349 0.999610i \(-0.508893\pi\)
−0.959318 + 0.282329i \(0.908893\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.1482 + 16.0916i 0.964789 + 0.700960i
\(528\) 0 0
\(529\) −5.64428 17.3713i −0.245403 0.755274i
\(530\) 0 0
\(531\) −3.85387 + 11.8610i −0.167244 + 0.514724i
\(532\) 0 0
\(533\) 2.98230 9.17857i 0.129178 0.397568i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −11.6762 + 8.48325i −0.503865 + 0.366079i
\(538\) 0 0
\(539\) 9.96200 + 7.23781i 0.429094 + 0.311755i
\(540\) 0 0
\(541\) 9.00089 6.53953i 0.386979 0.281156i −0.377238 0.926116i \(-0.623126\pi\)
0.764216 + 0.644960i \(0.223126\pi\)
\(542\) 0 0
\(543\) 6.74285 0.289363
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −8.60009 26.4684i −0.367713 1.13171i −0.948264 0.317482i \(-0.897163\pi\)
0.580551 0.814224i \(-0.302837\pi\)
\(548\) 0 0
\(549\) −6.92041 −0.295356
\(550\) 0 0
\(551\) −1.96222 −0.0835935
\(552\) 0 0
\(553\) 6.40389 + 19.7091i 0.272321 + 0.838118i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 6.36580 0.269728 0.134864 0.990864i \(-0.456940\pi\)
0.134864 + 0.990864i \(0.456940\pi\)
\(558\) 0 0
\(559\) 47.5569 34.5521i 2.01144 1.46140i
\(560\) 0 0
\(561\) 11.3610 + 8.25424i 0.479661 + 0.348494i
\(562\) 0 0
\(563\) 13.8569 10.0676i 0.584000 0.424301i −0.256164 0.966633i \(-0.582459\pi\)
0.840164 + 0.542333i \(0.182459\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.09610 + 3.37344i −0.0460317 + 0.141671i
\(568\) 0 0
\(569\) 4.75461 14.6332i 0.199324 0.613455i −0.800575 0.599232i \(-0.795473\pi\)
0.999899 0.0142229i \(-0.00452743\pi\)
\(570\) 0 0
\(571\) 2.63760 + 8.11770i 0.110380 + 0.339715i 0.990955 0.134191i \(-0.0428435\pi\)
−0.880575 + 0.473906i \(0.842844\pi\)
\(572\) 0 0
\(573\) −12.4512 9.04634i −0.520157 0.377916i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.86491 1.35494i −0.0776374 0.0564068i 0.548290 0.836289i \(-0.315279\pi\)
−0.625927 + 0.779882i \(0.715279\pi\)
\(578\) 0 0
\(579\) 5.06402 + 15.5854i 0.210453 + 0.647709i
\(580\) 0 0
\(581\) −13.7315 + 42.2614i −0.569681 + 1.75330i
\(582\) 0 0
\(583\) −8.15245 + 25.0907i −0.337640 + 1.03915i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.743663 0.540303i 0.0306943 0.0223007i −0.572332 0.820022i \(-0.693961\pi\)
0.603027 + 0.797721i \(0.293961\pi\)
\(588\) 0 0
\(589\) −8.05253 5.85051i −0.331799 0.241066i
\(590\) 0 0
\(591\) −7.96910 + 5.78989i −0.327805 + 0.238164i
\(592\) 0 0
\(593\) 25.5925 1.05096 0.525478 0.850807i \(-0.323886\pi\)
0.525478 + 0.850807i \(0.323886\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.53357 4.71985i −0.0627650 0.193171i
\(598\) 0 0
\(599\) −37.0204 −1.51261 −0.756307 0.654217i \(-0.772998\pi\)
−0.756307 + 0.654217i \(0.772998\pi\)
\(600\) 0 0
\(601\) 33.4191 1.36319 0.681597 0.731728i \(-0.261286\pi\)
0.681597 + 0.731728i \(0.261286\pi\)
\(602\) 0 0
\(603\) 1.46291 + 4.50239i 0.0595745 + 0.183351i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −4.34502 −0.176359 −0.0881795 0.996105i \(-0.528105\pi\)
−0.0881795 + 0.996105i \(0.528105\pi\)
\(608\) 0 0
\(609\) 2.43308 1.76774i 0.0985935 0.0716323i
\(610\) 0 0
\(611\) 35.1580 + 25.5438i 1.42234 + 1.03339i
\(612\) 0 0
\(613\) 3.20367 2.32761i 0.129395 0.0940111i −0.521205 0.853431i \(-0.674517\pi\)
0.650600 + 0.759420i \(0.274517\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −6.26189 + 19.2721i −0.252094 + 0.775866i 0.742294 + 0.670074i \(0.233738\pi\)
−0.994388 + 0.105792i \(0.966262\pi\)
\(618\) 0 0
\(619\) −4.65930 + 14.3398i −0.187273 + 0.576367i −0.999980 0.00630532i \(-0.997993\pi\)
0.812707 + 0.582672i \(0.197993\pi\)
\(620\) 0 0
\(621\) 0.672404 + 2.06945i 0.0269826 + 0.0830440i
\(622\) 0 0
\(623\) −20.1622 14.6487i −0.807782 0.586888i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4.13058 3.00104i −0.164959 0.119850i
\(628\) 0 0
\(629\) −14.2213 43.7687i −0.567041 1.74517i
\(630\) 0 0
\(631\) −0.755761 + 2.32599i −0.0300864 + 0.0925963i −0.964972 0.262352i \(-0.915502\pi\)
0.934886 + 0.354949i \(0.115502\pi\)
\(632\) 0 0
\(633\) 1.08150 3.32851i 0.0429857 0.132296i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 32.4112 23.5481i 1.28418 0.933009i
\(638\) 0 0
\(639\) −11.1476 8.09918i −0.440991 0.320399i
\(640\) 0 0
\(641\) −10.8353 + 7.87228i −0.427967 + 0.310936i −0.780836 0.624737i \(-0.785206\pi\)
0.352868 + 0.935673i \(0.385206\pi\)
\(642\) 0 0
\(643\) −35.4836 −1.39934 −0.699669 0.714467i \(-0.746669\pi\)
−0.699669 + 0.714467i \(0.746669\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.91707 21.2886i −0.271938 0.836940i −0.990013 0.140974i \(-0.954977\pi\)
0.718075 0.695966i \(-0.245023\pi\)
\(648\) 0 0
\(649\) 27.5140 1.08002
\(650\) 0 0
\(651\) 15.2555 0.597909
\(652\) 0 0
\(653\) −5.84923 18.0021i −0.228898 0.704476i −0.997872 0.0651961i \(-0.979233\pi\)
0.768974 0.639279i \(-0.220767\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.08917 0.0424925
\(658\) 0 0
\(659\) −4.58371 + 3.33026i −0.178556 + 0.129729i −0.673473 0.739212i \(-0.735198\pi\)
0.494917 + 0.868940i \(0.335198\pi\)
\(660\) 0 0
\(661\) 20.4225 + 14.8378i 0.794342 + 0.577123i 0.909249 0.416253i \(-0.136657\pi\)
−0.114907 + 0.993376i \(0.536657\pi\)
\(662\) 0 0
\(663\) 36.9627 26.8550i 1.43551 1.04296i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.570116 1.75464i 0.0220750 0.0679398i
\(668\) 0 0
\(669\) 7.13168 21.9491i 0.275727 0.848600i
\(670\) 0 0
\(671\) 4.71794 + 14.5203i 0.182134 + 0.560551i
\(672\) 0 0
\(673\) −10.6538 7.74044i −0.410674 0.298372i 0.363201 0.931711i \(-0.381684\pi\)
−0.773875 + 0.633339i \(0.781684\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −21.8139 15.8487i −0.838375 0.609115i 0.0835409 0.996504i \(-0.473377\pi\)
−0.921916 + 0.387389i \(0.873377\pi\)
\(678\) 0 0
\(679\) 19.8881 + 61.2092i 0.763234 + 2.34899i
\(680\) 0 0
\(681\) −1.80793 + 5.56424i −0.0692801 + 0.213222i
\(682\) 0 0
\(683\) 16.0008 49.2454i 0.612254 1.88432i 0.176356 0.984326i \(-0.443569\pi\)
0.435897 0.899996i \(-0.356431\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 5.17019 3.75637i 0.197255 0.143314i
\(688\) 0 0
\(689\) 69.4403 + 50.4513i 2.64546 + 1.92204i
\(690\) 0 0
\(691\) 3.00647 2.18432i 0.114371 0.0830956i −0.529129 0.848541i \(-0.677481\pi\)
0.643501 + 0.765446i \(0.277481\pi\)
\(692\) 0 0
\(693\) 7.82536 0.297261
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −2.64475 8.13972i −0.100177 0.308314i
\(698\) 0 0
\(699\) 4.54311 0.171836
\(700\) 0 0
\(701\) −28.4299 −1.07378 −0.536890 0.843652i \(-0.680401\pi\)
−0.536890 + 0.843652i \(0.680401\pi\)
\(702\) 0 0
\(703\) 5.17053 + 15.9132i 0.195010 + 0.600180i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −21.1926 −0.797030
\(708\) 0 0
\(709\) −33.1126 + 24.0577i −1.24357 + 0.903507i −0.997831 0.0658288i \(-0.979031\pi\)
−0.245740 + 0.969336i \(0.579031\pi\)
\(710\) 0 0
\(711\) 4.72664 + 3.43411i 0.177263 + 0.128789i
\(712\) 0 0
\(713\) 7.57120 5.50080i 0.283544 0.206007i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 7.12010 21.9134i 0.265905 0.818372i
\(718\) 0 0
\(719\) 1.46368 4.50475i 0.0545861 0.167999i −0.920047 0.391809i \(-0.871849\pi\)
0.974633 + 0.223810i \(0.0718495\pi\)
\(720\) 0 0
\(721\) 0.503731 + 1.55032i 0.0187599 + 0.0577371i
\(722\) 0 0
\(723\) −13.3064 9.66766i −0.494870 0.359544i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 4.50692 + 3.27447i 0.167153 + 0.121443i 0.668217 0.743967i \(-0.267058\pi\)
−0.501064 + 0.865410i \(0.667058\pi\)
\(728\) 0 0
\(729\) 0.309017 + 0.951057i 0.0114451 + 0.0352243i
\(730\) 0 0
\(731\) 16.1091 49.5788i 0.595818 1.83374i
\(732\) 0 0
\(733\) −9.38166 + 28.8738i −0.346520 + 1.06648i 0.614246 + 0.789115i \(0.289460\pi\)
−0.960765 + 0.277363i \(0.910540\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.44952 6.13894i 0.311242 0.226131i
\(738\) 0 0
\(739\) −39.6127 28.7803i −1.45718 1.05870i −0.984087 0.177686i \(-0.943139\pi\)
−0.473089 0.881015i \(-0.656861\pi\)
\(740\) 0 0
\(741\) −13.4388 + 9.76383i −0.493685 + 0.358683i
\(742\) 0 0
\(743\) 45.5953 1.67273 0.836364 0.548174i \(-0.184677\pi\)
0.836364 + 0.548174i \(0.184677\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3.87127 + 11.9145i 0.141642 + 0.435930i
\(748\) 0 0
\(749\) −8.79176 −0.321244
\(750\) 0 0
\(751\) −27.9100 −1.01845 −0.509225 0.860633i \(-0.670068\pi\)
−0.509225 + 0.860633i \(0.670068\pi\)
\(752\) 0 0
\(753\) 1.44103 + 4.43503i 0.0525140 + 0.161621i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 27.6680 1.00561 0.502804 0.864400i \(-0.332302\pi\)
0.502804 + 0.864400i \(0.332302\pi\)
\(758\) 0 0
\(759\) 3.88368 2.82166i 0.140969 0.102420i
\(760\) 0 0
\(761\) 9.80362 + 7.12275i 0.355381 + 0.258199i 0.751123 0.660162i \(-0.229513\pi\)
−0.395742 + 0.918362i \(0.629513\pi\)
\(762\) 0 0
\(763\) −9.58814 + 6.96619i −0.347114 + 0.252193i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 27.6620 85.1349i 0.998817 3.07404i
\(768\) 0 0
\(769\) −3.13398 + 9.64540i −0.113014 + 0.347822i −0.991528 0.129896i \(-0.958536\pi\)
0.878513 + 0.477718i \(0.158536\pi\)
\(770\) 0 0
\(771\) −1.78839 5.50409i −0.0644071 0.198225i
\(772\) 0 0
\(773\) 40.5906 + 29.4908i 1.45994 + 1.06071i 0.983378 + 0.181569i \(0.0581177\pi\)
0.476563 + 0.879140i \(0.341882\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −20.7473 15.0738i −0.744304 0.540769i
\(778\) 0 0
\(779\) 0.961568 + 2.95940i 0.0344518 + 0.106032i
\(780\) 0 0
\(781\) −9.39383 + 28.9112i −0.336138 + 1.03453i
\(782\) 0 0
\(783\) 0.262008 0.806379i 0.00936342 0.0288176i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −12.5059 + 9.08606i −0.445787 + 0.323883i −0.787930 0.615765i \(-0.788847\pi\)
0.342143 + 0.939648i \(0.388847\pi\)
\(788\) 0 0
\(789\) −18.5668 13.4896i −0.660996 0.480242i
\(790\) 0 0
\(791\) 50.4991 36.6897i 1.79554 1.30454i
\(792\) 0 0
\(793\) 49.6727 1.76393
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.77110 + 5.45087i 0.0627354 + 0.193080i 0.977512 0.210881i \(-0.0676332\pi\)
−0.914776 + 0.403961i \(0.867633\pi\)
\(798\) 0 0
\(799\) 38.5390 1.36341
\(800\) 0 0
\(801\) −7.02611 −0.248255
\(802\) 0 0
\(803\) −0.742531 2.28528i −0.0262034 0.0806457i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −4.50997 −0.158758
\(808\) 0 0
\(809\) 22.9907 16.7038i 0.808312 0.587273i −0.105029 0.994469i \(-0.533494\pi\)
0.913341 + 0.407196i \(0.133494\pi\)
\(810\) 0 0
\(811\) 9.86048 + 7.16406i 0.346248 + 0.251564i 0.747293 0.664494i \(-0.231353\pi\)
−0.401045 + 0.916058i \(0.631353\pi\)
\(812\) 0 0
\(813\) 8.03461 5.83748i 0.281786 0.204730i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −5.85689 + 18.0257i −0.204907 + 0.630638i
\(818\) 0 0
\(819\) 7.86746 24.2136i 0.274911 0.846090i
\(820\) 0 0
\(821\) 4.51854 + 13.9066i 0.157698 + 0.485345i 0.998424 0.0561157i \(-0.0178716\pi\)
−0.840726 + 0.541461i \(0.817872\pi\)
\(822\) 0 0
\(823\) 17.5077 + 12.7201i 0.610280 + 0.443394i 0.849513 0.527568i \(-0.176896\pi\)
−0.239233 + 0.970962i \(0.576896\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.03038 + 6.56096i 0.314017 + 0.228147i 0.733618 0.679562i \(-0.237830\pi\)
−0.419601 + 0.907709i \(0.637830\pi\)
\(828\) 0 0
\(829\) 8.60563 + 26.4854i 0.298886 + 0.919876i 0.981888 + 0.189460i \(0.0606738\pi\)
−0.683003 + 0.730416i \(0.739326\pi\)
\(830\) 0 0
\(831\) −7.11211 + 21.8888i −0.246716 + 0.759315i
\(832\) 0 0
\(833\) 10.9788 33.7892i 0.380392 1.17073i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.47950 2.52801i 0.120269 0.0873807i
\(838\) 0 0
\(839\) 15.3556 + 11.1565i 0.530133 + 0.385164i 0.820408 0.571779i \(-0.193747\pi\)
−0.290274 + 0.956943i \(0.593747\pi\)
\(840\) 0 0
\(841\) 22.8799 16.6232i 0.788962 0.573214i
\(842\) 0 0
\(843\) 0.734004 0.0252804
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 6.72218 + 20.6887i 0.230977 + 0.710873i
\(848\) 0 0
\(849\) 21.4921 0.737609
\(850\) 0 0
\(851\) −15.7320 −0.539287
\(852\) 0 0
\(853\) 5.10082 + 15.6987i 0.174649 + 0.537513i 0.999617 0.0276653i \(-0.00880727\pi\)
−0.824969 + 0.565179i \(0.808807\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −28.7615 −0.982475 −0.491237 0.871026i \(-0.663455\pi\)
−0.491237 + 0.871026i \(0.663455\pi\)
\(858\) 0 0
\(859\) 12.2446 8.89622i 0.417780 0.303535i −0.358964 0.933351i \(-0.616870\pi\)
0.776744 + 0.629816i \(0.216870\pi\)
\(860\) 0 0
\(861\) −3.85839 2.80329i −0.131494 0.0955357i
\(862\) 0 0
\(863\) −34.8258 + 25.3024i −1.18549 + 0.861305i −0.992780 0.119951i \(-0.961726\pi\)
−0.192705 + 0.981257i \(0.561726\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 7.26725 22.3663i 0.246809 0.759600i
\(868\) 0 0
\(869\) 3.98305 12.2586i 0.135116 0.415843i
\(870\) 0 0
\(871\) −10.5004 32.3168i −0.355792 1.09501i
\(872\) 0 0
\(873\) 14.6792 + 10.6651i 0.496815 + 0.360957i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 41.0134 + 29.7980i 1.38492 + 1.00621i 0.996401 + 0.0847668i \(0.0270145\pi\)
0.388523 + 0.921439i \(0.372985\pi\)
\(878\) 0 0
\(879\) −8.21864 25.2944i −0.277208 0.853158i
\(880\) 0 0
\(881\) −0.818966 + 2.52052i −0.0275917 + 0.0849184i −0.963904 0.266250i \(-0.914215\pi\)
0.936312 + 0.351168i \(0.114215\pi\)
\(882\) 0 0
\(883\) −10.4361 + 32.1189i −0.351201 + 1.08089i 0.606978 + 0.794719i \(0.292382\pi\)
−0.958179 + 0.286168i \(0.907618\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.49811 2.54153i 0.117455 0.0853361i −0.527507 0.849551i \(-0.676873\pi\)
0.644962 + 0.764215i \(0.276873\pi\)
\(888\) 0 0
\(889\) 35.6175 + 25.8776i 1.19457 + 0.867908i
\(890\) 0 0
\(891\) 1.78482 1.29675i 0.0597939 0.0434428i
\(892\) 0 0
\(893\) −14.0119 −0.468889
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −4.82632 14.8539i −0.161146 0.495957i
\(898\) 0 0
\(899\) −3.64664 −0.121622
\(900\) 0 0
\(901\) 76.1182 2.53586
\(902\) 0 0
\(903\) −8.97673 27.6275i −0.298727 0.919387i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −47.2507 −1.56893 −0.784467 0.620171i \(-0.787063\pi\)
−0.784467 + 0.620171i \(0.787063\pi\)
\(908\) 0 0
\(909\) −4.83366 + 3.51186i −0.160322 + 0.116481i
\(910\) 0 0
\(911\) −41.1586 29.9035i −1.36365 0.990747i −0.998204 0.0599132i \(-0.980918\pi\)
−0.365443 0.930834i \(-0.619082\pi\)
\(912\) 0 0
\(913\) 22.3597 16.2453i 0.739999 0.537641i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −17.3437 + 53.3785i −0.572740 + 1.76271i
\(918\) 0 0
\(919\) −11.3396 + 34.8998i −0.374060 + 1.15124i 0.570051 + 0.821610i \(0.306924\pi\)
−0.944111 + 0.329629i \(0.893076\pi\)
\(920\) 0 0
\(921\) 3.76186 + 11.5778i 0.123957 + 0.381502i
\(922\) 0 0
\(923\) 80.0140 + 58.1336i 2.63369 + 1.91349i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0.371799 + 0.270128i 0.0122115 + 0.00887215i
\(928\) 0 0
\(929\) 14.9269 + 45.9401i 0.489734 + 1.50725i 0.825005 + 0.565125i \(0.191172\pi\)
−0.335271 + 0.942122i \(0.608828\pi\)
\(930\) 0 0
\(931\) −3.99161 + 12.2849i −0.130820 + 0.402622i
\(932\) 0 0
\(933\) −7.65191 + 23.5502i −0.250512 + 0.770998i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −7.67223 + 5.57420i −0.250641 + 0.182101i −0.706011 0.708201i \(-0.749507\pi\)
0.455370 + 0.890302i \(0.349507\pi\)
\(938\) 0 0
\(939\) −12.0102 8.72589i −0.391937 0.284759i
\(940\) 0 0
\(941\) 17.2272 12.5163i 0.561590 0.408019i −0.270450 0.962734i \(-0.587173\pi\)
0.832041 + 0.554715i \(0.187173\pi\)
\(942\) 0 0
\(943\) −2.92570 −0.0952740
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9.73768 29.9695i −0.316432 0.973877i −0.975161 0.221497i \(-0.928906\pi\)
0.658729 0.752380i \(-0.271094\pi\)
\(948\) 0 0
\(949\) −7.81773 −0.253774
\(950\) 0 0
\(951\) 1.94231 0.0629837
\(952\) 0 0
\(953\) −3.46513 10.6646i −0.112247 0.345460i 0.879116 0.476608i \(-0.158134\pi\)
−0.991363 + 0.131148i \(0.958134\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −1.87056 −0.0604665
\(958\) 0 0
\(959\) 21.2206 15.4177i 0.685249 0.497863i
\(960\) 0 0
\(961\) 10.1145 + 7.34864i 0.326275 + 0.237053i
\(962\) 0 0
\(963\) −2.00524 + 1.45689i −0.0646181 + 0.0469478i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 11.0077 33.8782i 0.353984 1.08945i −0.602612 0.798034i \(-0.705873\pi\)
0.956596 0.291417i \(-0.0941266\pi\)
\(968\) 0 0
\(969\) −4.55217 + 14.0101i −0.146237 + 0.450070i
\(970\) 0 0
\(971\) 13.2227 + 40.6953i 0.424337 + 1.30598i 0.903628 + 0.428319i \(0.140894\pi\)
−0.479290 + 0.877656i \(0.659106\pi\)
\(972\) 0 0
\(973\) 11.3032 + 8.21228i 0.362365 + 0.263274i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −32.9119 23.9119i −1.05294 0.765009i −0.0801740 0.996781i \(-0.525548\pi\)
−0.972770 + 0.231772i \(0.925548\pi\)
\(978\) 0 0
\(979\) 4.78999 + 14.7421i 0.153089 + 0.471159i
\(980\) 0 0
\(981\) −1.03251 + 3.17773i −0.0329654 + 0.101457i
\(982\) 0 0
\(983\) 15.1319 46.5713i 0.482634 1.48539i −0.352746 0.935719i \(-0.614752\pi\)
0.835379 0.549674i \(-0.185248\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 17.3742 12.6231i 0.553026 0.401797i
\(988\) 0 0
\(989\) −14.4170 10.4746i −0.458434 0.333072i
\(990\) 0 0
\(991\) 1.99334 1.44825i 0.0633206 0.0460051i −0.555675 0.831400i \(-0.687540\pi\)
0.618995 + 0.785395i \(0.287540\pi\)
\(992\) 0 0
\(993\) −27.7307 −0.880006
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 16.7650 + 51.5975i 0.530954 + 1.63411i 0.752232 + 0.658899i \(0.228977\pi\)
−0.221277 + 0.975211i \(0.571023\pi\)
\(998\) 0 0
\(999\) −7.22998 −0.228747
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 1500.2.m.c.1201.1 24
5.2 odd 4 300.2.o.a.109.4 24
5.3 odd 4 1500.2.o.c.49.3 24
5.4 even 2 1500.2.m.d.1201.6 24
15.2 even 4 900.2.w.c.109.5 24
25.2 odd 20 1500.2.o.c.949.3 24
25.6 even 5 7500.2.a.n.1.2 12
25.8 odd 20 7500.2.d.g.1249.23 24
25.11 even 5 inner 1500.2.m.c.301.1 24
25.14 even 10 1500.2.m.d.301.6 24
25.17 odd 20 7500.2.d.g.1249.2 24
25.19 even 10 7500.2.a.m.1.11 12
25.23 odd 20 300.2.o.a.289.4 yes 24
75.23 even 20 900.2.w.c.289.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.2.o.a.109.4 24 5.2 odd 4
300.2.o.a.289.4 yes 24 25.23 odd 20
900.2.w.c.109.5 24 15.2 even 4
900.2.w.c.289.5 24 75.23 even 20
1500.2.m.c.301.1 24 25.11 even 5 inner
1500.2.m.c.1201.1 24 1.1 even 1 trivial
1500.2.m.d.301.6 24 25.14 even 10
1500.2.m.d.1201.6 24 5.4 even 2
1500.2.o.c.49.3 24 5.3 odd 4
1500.2.o.c.949.3 24 25.2 odd 20
7500.2.a.m.1.11 12 25.19 even 10
7500.2.a.n.1.2 12 25.6 even 5
7500.2.d.g.1249.2 24 25.17 odd 20
7500.2.d.g.1249.23 24 25.8 odd 20