Properties

Label 1900.2.l.b.493.3
Level $1900$
Weight $2$
Character 1900.493
Analytic conductor $15.172$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1900,2,Mod(493,1900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1900.493");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1900.l (of order \(4\), 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: \(15.1715763840\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 28 x^{10} - 64 x^{9} + 236 x^{8} - 420 x^{7} + 946 x^{6} - 1216 x^{5} + 1896 x^{4} - 1564 x^{3} + 2284 x^{2} - 1088 x + 1370 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 493.3
Root \(0.344446 - 1.15131i\) of defining polynomial
Character \(\chi\) \(=\) 1900.493
Dual form 1900.2.l.b.1557.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.49576 + 1.49576i) q^{3} +(-0.311108 + 0.311108i) q^{7} -1.47457i q^{9} +O(q^{10})\) \(q+(-1.49576 + 1.49576i) q^{3} +(-0.311108 + 0.311108i) q^{7} -1.47457i q^{9} +2.90321 q^{11} +(-2.84674 + 2.84674i) q^{13} +(2.52543 - 2.52543i) q^{17} +(4.34250 - 0.377784i) q^{19} -0.930683i q^{21} +(4.11753 + 4.11753i) q^{23} +(-2.28167 - 2.28167i) q^{27} -2.99151 q^{29} -0.930683i q^{31} +(-4.34250 + 4.34250i) q^{33} +(8.11992 + 8.11992i) q^{37} -8.51606i q^{39} +2.06083i q^{41} +(-2.11753 - 2.11753i) q^{43} +(2.73975 - 2.73975i) q^{47} +6.80642i q^{49} +7.55485i q^{51} +(-0.565073 + 0.565073i) q^{53} +(-5.93024 + 7.06039i) q^{57} -9.32613 q^{59} +3.52543 q^{61} +(0.458751 + 0.458751i) q^{63} +(-0.144771 - 0.144771i) q^{67} -12.3176 q^{69} +9.61568i q^{71} +(4.09679 + 4.09679i) q^{73} +(-0.903212 + 0.903212i) q^{77} -12.6072 q^{79} +11.2494 q^{81} +(-9.21432 - 9.21432i) q^{83} +(4.47457 - 4.47457i) q^{87} +7.55485 q^{89} -1.77129i q^{91} +(1.39207 + 1.39207i) q^{93} +(12.0421 + 12.0421i) q^{97} -4.28100i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{7} + 8 q^{11} + 4 q^{17} - 4 q^{23} + 28 q^{43} - 20 q^{47} - 24 q^{57} + 16 q^{61} - 20 q^{63} + 76 q^{73} + 16 q^{77} + 4 q^{81} - 84 q^{83} + 80 q^{87} - 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\) \(951\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.49576 + 1.49576i −0.863575 + 0.863575i −0.991751 0.128176i \(-0.959088\pi\)
0.128176 + 0.991751i \(0.459088\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.311108 + 0.311108i −0.117588 + 0.117588i −0.763452 0.645864i \(-0.776497\pi\)
0.645864 + 0.763452i \(0.276497\pi\)
\(8\) 0 0
\(9\) 1.47457i 0.491524i
\(10\) 0 0
\(11\) 2.90321 0.875351 0.437676 0.899133i \(-0.355802\pi\)
0.437676 + 0.899133i \(0.355802\pi\)
\(12\) 0 0
\(13\) −2.84674 + 2.84674i −0.789544 + 0.789544i −0.981419 0.191875i \(-0.938543\pi\)
0.191875 + 0.981419i \(0.438543\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.52543 2.52543i 0.612506 0.612506i −0.331092 0.943598i \(-0.607417\pi\)
0.943598 + 0.331092i \(0.107417\pi\)
\(18\) 0 0
\(19\) 4.34250 0.377784i 0.996237 0.0866697i
\(20\) 0 0
\(21\) 0.930683i 0.203092i
\(22\) 0 0
\(23\) 4.11753 + 4.11753i 0.858565 + 0.858565i 0.991169 0.132604i \(-0.0423340\pi\)
−0.132604 + 0.991169i \(0.542334\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.28167 2.28167i −0.439107 0.439107i
\(28\) 0 0
\(29\) −2.99151 −0.555510 −0.277755 0.960652i \(-0.589590\pi\)
−0.277755 + 0.960652i \(0.589590\pi\)
\(30\) 0 0
\(31\) 0.930683i 0.167156i −0.996501 0.0835778i \(-0.973365\pi\)
0.996501 0.0835778i \(-0.0266347\pi\)
\(32\) 0 0
\(33\) −4.34250 + 4.34250i −0.755932 + 0.755932i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.11992 + 8.11992i 1.33491 + 1.33491i 0.900919 + 0.433988i \(0.142894\pi\)
0.433988 + 0.900919i \(0.357106\pi\)
\(38\) 0 0
\(39\) 8.51606i 1.36366i
\(40\) 0 0
\(41\) 2.06083i 0.321847i 0.986967 + 0.160924i \(0.0514473\pi\)
−0.986967 + 0.160924i \(0.948553\pi\)
\(42\) 0 0
\(43\) −2.11753 2.11753i −0.322921 0.322921i 0.526966 0.849886i \(-0.323330\pi\)
−0.849886 + 0.526966i \(0.823330\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.73975 2.73975i 0.399633 0.399633i −0.478470 0.878104i \(-0.658809\pi\)
0.878104 + 0.478470i \(0.158809\pi\)
\(48\) 0 0
\(49\) 6.80642i 0.972346i
\(50\) 0 0
\(51\) 7.55485i 1.05789i
\(52\) 0 0
\(53\) −0.565073 + 0.565073i −0.0776188 + 0.0776188i −0.744850 0.667232i \(-0.767479\pi\)
0.667232 + 0.744850i \(0.267479\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.93024 + 7.06039i −0.785480 + 0.935171i
\(58\) 0 0
\(59\) −9.32613 −1.21416 −0.607080 0.794641i \(-0.707659\pi\)
−0.607080 + 0.794641i \(0.707659\pi\)
\(60\) 0 0
\(61\) 3.52543 0.451385 0.225692 0.974199i \(-0.427536\pi\)
0.225692 + 0.974199i \(0.427536\pi\)
\(62\) 0 0
\(63\) 0.458751 + 0.458751i 0.0577972 + 0.0577972i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.144771 0.144771i −0.0176866 0.0176866i 0.698208 0.715895i \(-0.253981\pi\)
−0.715895 + 0.698208i \(0.753981\pi\)
\(68\) 0 0
\(69\) −12.3176 −1.48287
\(70\) 0 0
\(71\) 9.61568i 1.14117i 0.821238 + 0.570585i \(0.193284\pi\)
−0.821238 + 0.570585i \(0.806716\pi\)
\(72\) 0 0
\(73\) 4.09679 + 4.09679i 0.479493 + 0.479493i 0.904969 0.425477i \(-0.139894\pi\)
−0.425477 + 0.904969i \(0.639894\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.903212 + 0.903212i −0.102931 + 0.102931i
\(78\) 0 0
\(79\) −12.6072 −1.41842 −0.709210 0.704998i \(-0.750948\pi\)
−0.709210 + 0.704998i \(0.750948\pi\)
\(80\) 0 0
\(81\) 11.2494 1.24993
\(82\) 0 0
\(83\) −9.21432 9.21432i −1.01140 1.01140i −0.999934 0.0114688i \(-0.996349\pi\)
−0.0114688 0.999934i \(-0.503651\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 4.47457 4.47457i 0.479725 0.479725i
\(88\) 0 0
\(89\) 7.55485 0.800812 0.400406 0.916338i \(-0.368869\pi\)
0.400406 + 0.916338i \(0.368869\pi\)
\(90\) 0 0
\(91\) 1.77129i 0.185681i
\(92\) 0 0
\(93\) 1.39207 + 1.39207i 0.144351 + 0.144351i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0421 + 12.0421i 1.22269 + 1.22269i 0.966671 + 0.256020i \(0.0824113\pi\)
0.256020 + 0.966671i \(0.417589\pi\)
\(98\) 0 0
\(99\) 4.28100i 0.430256i
\(100\) 0 0
\(101\) −8.76986 −0.872634 −0.436317 0.899793i \(-0.643717\pi\)
−0.436317 + 0.899793i \(0.643717\pi\)
\(102\) 0 0
\(103\) −8.54022 + 8.54022i −0.841493 + 0.841493i −0.989053 0.147560i \(-0.952858\pi\)
0.147560 + 0.989053i \(0.452858\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.77742 + 3.77742i 0.365177 + 0.365177i 0.865715 0.500538i \(-0.166864\pi\)
−0.500538 + 0.865715i \(0.666864\pi\)
\(108\) 0 0
\(109\) −13.7373 −1.31580 −0.657899 0.753106i \(-0.728555\pi\)
−0.657899 + 0.753106i \(0.728555\pi\)
\(110\) 0 0
\(111\) −24.2908 −2.30558
\(112\) 0 0
\(113\) 1.91606 1.91606i 0.180248 0.180248i −0.611216 0.791464i \(-0.709319\pi\)
0.791464 + 0.611216i \(0.209319\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 4.19773 + 4.19773i 0.388080 + 0.388080i
\(118\) 0 0
\(119\) 1.57136i 0.144046i
\(120\) 0 0
\(121\) −2.57136 −0.233760
\(122\) 0 0
\(123\) −3.08250 3.08250i −0.277939 0.277939i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −5.12841 5.12841i −0.455073 0.455073i 0.441961 0.897034i \(-0.354283\pi\)
−0.897034 + 0.441961i \(0.854283\pi\)
\(128\) 0 0
\(129\) 6.33462 0.557732
\(130\) 0 0
\(131\) −10.3684 −0.905893 −0.452946 0.891538i \(-0.649627\pi\)
−0.452946 + 0.891538i \(0.649627\pi\)
\(132\) 0 0
\(133\) −1.23345 + 1.46852i −0.106954 + 0.127337i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.99063 + 7.99063i −0.682686 + 0.682686i −0.960605 0.277919i \(-0.910355\pi\)
0.277919 + 0.960605i \(0.410355\pi\)
\(138\) 0 0
\(139\) 10.1891i 0.864231i 0.901818 + 0.432115i \(0.142233\pi\)
−0.901818 + 0.432115i \(0.857767\pi\)
\(140\) 0 0
\(141\) 8.19599i 0.690227i
\(142\) 0 0
\(143\) −8.26469 + 8.26469i −0.691128 + 0.691128i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −10.1807 10.1807i −0.839694 0.839694i
\(148\) 0 0
\(149\) 18.0874i 1.48178i 0.671627 + 0.740890i \(0.265596\pi\)
−0.671627 + 0.740890i \(0.734404\pi\)
\(150\) 0 0
\(151\) 13.5379i 1.10170i −0.834605 0.550848i \(-0.814304\pi\)
0.834605 0.550848i \(-0.185696\pi\)
\(152\) 0 0
\(153\) −3.72393 3.72393i −0.301062 0.301062i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −15.1891 + 15.1891i −1.21222 + 1.21222i −0.241931 + 0.970293i \(0.577781\pi\)
−0.970293 + 0.241931i \(0.922219\pi\)
\(158\) 0 0
\(159\) 1.69042i 0.134059i
\(160\) 0 0
\(161\) −2.56199 −0.201913
\(162\) 0 0
\(163\) −5.44938 5.44938i −0.426829 0.426829i 0.460718 0.887547i \(-0.347592\pi\)
−0.887547 + 0.460718i \(0.847592\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.42644 2.42644i −0.187763 0.187763i 0.606965 0.794729i \(-0.292387\pi\)
−0.794729 + 0.606965i \(0.792387\pi\)
\(168\) 0 0
\(169\) 3.20787i 0.246759i
\(170\) 0 0
\(171\) −0.557070 6.40333i −0.0426002 0.489675i
\(172\) 0 0
\(173\) 5.83825 5.83825i 0.443874 0.443874i −0.449438 0.893312i \(-0.648376\pi\)
0.893312 + 0.449438i \(0.148376\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 13.9496 13.9496i 1.04852 1.04852i
\(178\) 0 0
\(179\) 20.4516 1.52862 0.764311 0.644847i \(-0.223079\pi\)
0.764311 + 0.644847i \(0.223079\pi\)
\(180\) 0 0
\(181\) 11.9660i 0.889429i 0.895672 + 0.444715i \(0.146695\pi\)
−0.895672 + 0.444715i \(0.853305\pi\)
\(182\) 0 0
\(183\) −5.27318 + 5.27318i −0.389805 + 0.389805i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 7.33185 7.33185i 0.536158 0.536158i
\(188\) 0 0
\(189\) 1.41969 0.103267
\(190\) 0 0
\(191\) 23.0923 1.67090 0.835452 0.549564i \(-0.185206\pi\)
0.835452 + 0.549564i \(0.185206\pi\)
\(192\) 0 0
\(193\) −2.42644 + 2.42644i −0.174659 + 0.174659i −0.789023 0.614364i \(-0.789413\pi\)
0.614364 + 0.789023i \(0.289413\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −13.0874 + 13.0874i −0.932440 + 0.932440i −0.997858 0.0654179i \(-0.979162\pi\)
0.0654179 + 0.997858i \(0.479162\pi\)
\(198\) 0 0
\(199\) 10.2953i 0.729814i −0.931044 0.364907i \(-0.881101\pi\)
0.931044 0.364907i \(-0.118899\pi\)
\(200\) 0 0
\(201\) 0.433085 0.0305475
\(202\) 0 0
\(203\) 0.930683 0.930683i 0.0653211 0.0653211i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.07160 6.07160i 0.422005 0.422005i
\(208\) 0 0
\(209\) 12.6072 1.09679i 0.872057 0.0758664i
\(210\) 0 0
\(211\) 14.9576i 1.02972i −0.857274 0.514860i \(-0.827844\pi\)
0.857274 0.514860i \(-0.172156\pi\)
\(212\) 0 0
\(213\) −14.3827 14.3827i −0.985487 0.985487i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.289543 + 0.289543i 0.0196554 + 0.0196554i
\(218\) 0 0
\(219\) −12.2556 −0.828156
\(220\) 0 0
\(221\) 14.3785i 0.967201i
\(222\) 0 0
\(223\) −3.55659 + 3.55659i −0.238167 + 0.238167i −0.816091 0.577924i \(-0.803863\pi\)
0.577924 + 0.816091i \(0.303863\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.83825 + 5.83825i 0.387498 + 0.387498i 0.873794 0.486296i \(-0.161652\pi\)
−0.486296 + 0.873794i \(0.661652\pi\)
\(228\) 0 0
\(229\) 12.5763i 0.831064i 0.909579 + 0.415532i \(0.136405\pi\)
−0.909579 + 0.415532i \(0.863595\pi\)
\(230\) 0 0
\(231\) 2.70197i 0.177777i
\(232\) 0 0
\(233\) 8.74620 + 8.74620i 0.572983 + 0.572983i 0.932961 0.359978i \(-0.117216\pi\)
−0.359978 + 0.932961i \(0.617216\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 18.8573 18.8573i 1.22491 1.22491i
\(238\) 0 0
\(239\) 17.0923i 1.10561i −0.833310 0.552806i \(-0.813557\pi\)
0.833310 0.552806i \(-0.186443\pi\)
\(240\) 0 0
\(241\) 25.8555i 1.66550i −0.553649 0.832750i \(-0.686765\pi\)
0.553649 0.832750i \(-0.313235\pi\)
\(242\) 0 0
\(243\) −9.98129 + 9.98129i −0.640300 + 0.640300i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −11.2865 + 13.4374i −0.718143 + 0.855002i
\(248\) 0 0
\(249\) 27.5647 1.74685
\(250\) 0 0
\(251\) 5.37778 0.339443 0.169721 0.985492i \(-0.445713\pi\)
0.169721 + 0.985492i \(0.445713\pi\)
\(252\) 0 0
\(253\) 11.9541 + 11.9541i 0.751546 + 0.751546i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.48727 4.48727i −0.279908 0.279908i 0.553164 0.833072i \(-0.313420\pi\)
−0.833072 + 0.553164i \(0.813420\pi\)
\(258\) 0 0
\(259\) −5.05234 −0.313937
\(260\) 0 0
\(261\) 4.41120i 0.273047i
\(262\) 0 0
\(263\) 11.8365 + 11.8365i 0.729872 + 0.729872i 0.970594 0.240722i \(-0.0773842\pi\)
−0.240722 + 0.970594i \(0.577384\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −11.3002 + 11.3002i −0.691562 + 0.691562i
\(268\) 0 0
\(269\) 6.33462 0.386229 0.193114 0.981176i \(-0.438141\pi\)
0.193114 + 0.981176i \(0.438141\pi\)
\(270\) 0 0
\(271\) 13.5669 0.824131 0.412066 0.911154i \(-0.364807\pi\)
0.412066 + 0.911154i \(0.364807\pi\)
\(272\) 0 0
\(273\) 2.64941 + 2.64941i 0.160350 + 0.160350i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 11.9032 11.9032i 0.715195 0.715195i −0.252422 0.967617i \(-0.581227\pi\)
0.967617 + 0.252422i \(0.0812273\pi\)
\(278\) 0 0
\(279\) −1.37236 −0.0821610
\(280\) 0 0
\(281\) 26.6961i 1.59256i 0.604930 + 0.796279i \(0.293201\pi\)
−0.604930 + 0.796279i \(0.706799\pi\)
\(282\) 0 0
\(283\) 13.1684 + 13.1684i 0.782779 + 0.782779i 0.980299 0.197520i \(-0.0632886\pi\)
−0.197520 + 0.980299i \(0.563289\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.641140 0.641140i −0.0378453 0.0378453i
\(288\) 0 0
\(289\) 4.24443i 0.249672i
\(290\) 0 0
\(291\) −36.0241 −2.11177
\(292\) 0 0
\(293\) 22.7879 22.7879i 1.33129 1.33129i 0.427064 0.904221i \(-0.359548\pi\)
0.904221 0.427064i \(-0.140452\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −6.62416 6.62416i −0.384373 0.384373i
\(298\) 0 0
\(299\) −23.4431 −1.35575
\(300\) 0 0
\(301\) 1.31756 0.0759430
\(302\) 0 0
\(303\) 13.1176 13.1176i 0.753585 0.753585i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −19.5756 19.5756i −1.11724 1.11724i −0.992145 0.125093i \(-0.960077\pi\)
−0.125093 0.992145i \(-0.539923\pi\)
\(308\) 0 0
\(309\) 25.5482i 1.45339i
\(310\) 0 0
\(311\) 30.1289 1.70845 0.854227 0.519901i \(-0.174031\pi\)
0.854227 + 0.519901i \(0.174031\pi\)
\(312\) 0 0
\(313\) −11.5620 11.5620i −0.653522 0.653522i 0.300317 0.953839i \(-0.402908\pi\)
−0.953839 + 0.300317i \(0.902908\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.6011 + 10.6011i 0.595414 + 0.595414i 0.939089 0.343674i \(-0.111672\pi\)
−0.343674 + 0.939089i \(0.611672\pi\)
\(318\) 0 0
\(319\) −8.68499 −0.486266
\(320\) 0 0
\(321\) −11.3002 −0.630716
\(322\) 0 0
\(323\) 10.0126 11.9207i 0.557116 0.663287i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 20.5477 20.5477i 1.13629 1.13629i
\(328\) 0 0
\(329\) 1.70471i 0.0939839i
\(330\) 0 0
\(331\) 26.7862i 1.47230i −0.676817 0.736151i \(-0.736641\pi\)
0.676817 0.736151i \(-0.263359\pi\)
\(332\) 0 0
\(333\) 11.9734 11.9734i 0.656139 0.656139i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −10.0500 10.0500i −0.547458 0.547458i 0.378247 0.925705i \(-0.376527\pi\)
−0.925705 + 0.378247i \(0.876527\pi\)
\(338\) 0 0
\(339\) 5.73191i 0.311315i
\(340\) 0 0
\(341\) 2.70197i 0.146320i
\(342\) 0 0
\(343\) −4.29529 4.29529i −0.231924 0.231924i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −18.0573 + 18.0573i −0.969367 + 0.969367i −0.999545 0.0301774i \(-0.990393\pi\)
0.0301774 + 0.999545i \(0.490393\pi\)
\(348\) 0 0
\(349\) 29.1941i 1.56272i −0.624079 0.781361i \(-0.714526\pi\)
0.624079 0.781361i \(-0.285474\pi\)
\(350\) 0 0
\(351\) 12.9906 0.693389
\(352\) 0 0
\(353\) 8.89384 + 8.89384i 0.473372 + 0.473372i 0.903004 0.429632i \(-0.141357\pi\)
−0.429632 + 0.903004i \(0.641357\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −2.35037 2.35037i −0.124395 0.124395i
\(358\) 0 0
\(359\) 27.9037i 1.47270i 0.676601 + 0.736350i \(0.263452\pi\)
−0.676601 + 0.736350i \(0.736548\pi\)
\(360\) 0 0
\(361\) 18.7146 3.28105i 0.984977 0.172687i
\(362\) 0 0
\(363\) 3.84613 3.84613i 0.201869 0.201869i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.89676 + 1.89676i −0.0990100 + 0.0990100i −0.754877 0.655867i \(-0.772303\pi\)
0.655867 + 0.754877i \(0.272303\pi\)
\(368\) 0 0
\(369\) 3.03884 0.158196
\(370\) 0 0
\(371\) 0.351597i 0.0182540i
\(372\) 0 0
\(373\) 6.05909 6.05909i 0.313728 0.313728i −0.532624 0.846352i \(-0.678794\pi\)
0.846352 + 0.532624i \(0.178794\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.51606 8.51606i 0.438599 0.438599i
\(378\) 0 0
\(379\) 25.2144 1.29518 0.647588 0.761991i \(-0.275778\pi\)
0.647588 + 0.761991i \(0.275778\pi\)
\(380\) 0 0
\(381\) 15.3417 0.785979
\(382\) 0 0
\(383\) 11.3323 11.3323i 0.579052 0.579052i −0.355590 0.934642i \(-0.615720\pi\)
0.934642 + 0.355590i \(0.115720\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.12245 + 3.12245i −0.158723 + 0.158723i
\(388\) 0 0
\(389\) 19.1338i 0.970124i 0.874480 + 0.485062i \(0.161203\pi\)
−0.874480 + 0.485062i \(0.838797\pi\)
\(390\) 0 0
\(391\) 20.7971 1.05175
\(392\) 0 0
\(393\) 15.5086 15.5086i 0.782306 0.782306i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 6.71900 6.71900i 0.337217 0.337217i −0.518102 0.855319i \(-0.673361\pi\)
0.855319 + 0.518102i \(0.173361\pi\)
\(398\) 0 0
\(399\) −0.351597 4.04149i −0.0176019 0.202327i
\(400\) 0 0
\(401\) 4.56334i 0.227882i 0.993488 + 0.113941i \(0.0363475\pi\)
−0.993488 + 0.113941i \(0.963653\pi\)
\(402\) 0 0
\(403\) 2.64941 + 2.64941i 0.131977 + 0.131977i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 23.5738 + 23.5738i 1.16851 + 1.16851i
\(408\) 0 0
\(409\) −21.5817 −1.06715 −0.533574 0.845754i \(-0.679151\pi\)
−0.533574 + 0.845754i \(0.679151\pi\)
\(410\) 0 0
\(411\) 23.9041i 1.17910i
\(412\) 0 0
\(413\) 2.90143 2.90143i 0.142770 0.142770i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −15.2405 15.2405i −0.746328 0.746328i
\(418\) 0 0
\(419\) 7.58120i 0.370366i −0.982704 0.185183i \(-0.940712\pi\)
0.982704 0.185183i \(-0.0592878\pi\)
\(420\) 0 0
\(421\) 25.0149i 1.21915i 0.792728 + 0.609576i \(0.208660\pi\)
−0.792728 + 0.609576i \(0.791340\pi\)
\(422\) 0 0
\(423\) −4.03996 4.03996i −0.196429 0.196429i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.09679 + 1.09679i −0.0530773 + 0.0530773i
\(428\) 0 0
\(429\) 24.7239i 1.19368i
\(430\) 0 0
\(431\) 21.2021i 1.02127i 0.859798 + 0.510635i \(0.170590\pi\)
−0.859798 + 0.510635i \(0.829410\pi\)
\(432\) 0 0
\(433\) 19.0866 19.0866i 0.917243 0.917243i −0.0795855 0.996828i \(-0.525360\pi\)
0.996828 + 0.0795855i \(0.0253597\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 19.4359 + 16.3248i 0.929746 + 0.780922i
\(438\) 0 0
\(439\) 38.6622 1.84524 0.922622 0.385705i \(-0.126042\pi\)
0.922622 + 0.385705i \(0.126042\pi\)
\(440\) 0 0
\(441\) 10.0366 0.477932
\(442\) 0 0
\(443\) −0.725457 0.725457i −0.0344675 0.0344675i 0.689663 0.724130i \(-0.257759\pi\)
−0.724130 + 0.689663i \(0.757759\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −27.0544 27.0544i −1.27963 1.27963i
\(448\) 0 0
\(449\) 41.0746 1.93843 0.969215 0.246216i \(-0.0791872\pi\)
0.969215 + 0.246216i \(0.0791872\pi\)
\(450\) 0 0
\(451\) 5.98302i 0.281730i
\(452\) 0 0
\(453\) 20.2494 + 20.2494i 0.951398 + 0.951398i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −15.3684 + 15.3684i −0.718904 + 0.718904i −0.968381 0.249477i \(-0.919741\pi\)
0.249477 + 0.968381i \(0.419741\pi\)
\(458\) 0 0
\(459\) −11.5244 −0.537912
\(460\) 0 0
\(461\) −16.9304 −0.788528 −0.394264 0.918997i \(-0.629000\pi\)
−0.394264 + 0.918997i \(0.629000\pi\)
\(462\) 0 0
\(463\) 1.49532 + 1.49532i 0.0694932 + 0.0694932i 0.740999 0.671506i \(-0.234352\pi\)
−0.671506 + 0.740999i \(0.734352\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 20.2192 20.2192i 0.935635 0.935635i −0.0624153 0.998050i \(-0.519880\pi\)
0.998050 + 0.0624153i \(0.0198803\pi\)
\(468\) 0 0
\(469\) 0.0900790 0.00415946
\(470\) 0 0
\(471\) 45.4385i 2.09369i
\(472\) 0 0
\(473\) −6.14764 6.14764i −0.282669 0.282669i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.833241 + 0.833241i 0.0381515 + 0.0381515i
\(478\) 0 0
\(479\) 11.2587i 0.514424i −0.966355 0.257212i \(-0.917196\pi\)
0.966355 0.257212i \(-0.0828039\pi\)
\(480\) 0 0
\(481\) −46.2306 −2.10793
\(482\) 0 0
\(483\) 3.83212 3.83212i 0.174367 0.174367i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −15.4539 15.4539i −0.700284 0.700284i 0.264187 0.964471i \(-0.414896\pi\)
−0.964471 + 0.264187i \(0.914896\pi\)
\(488\) 0 0
\(489\) 16.3019 0.737197
\(490\) 0 0
\(491\) −20.9906 −0.947294 −0.473647 0.880715i \(-0.657063\pi\)
−0.473647 + 0.880715i \(0.657063\pi\)
\(492\) 0 0
\(493\) −7.55485 + 7.55485i −0.340253 + 0.340253i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.99151 2.99151i −0.134188 0.134188i
\(498\) 0 0
\(499\) 24.6780i 1.10474i 0.833599 + 0.552369i \(0.186276\pi\)
−0.833599 + 0.552369i \(0.813724\pi\)
\(500\) 0 0
\(501\) 7.25872 0.324296
\(502\) 0 0
\(503\) 15.0065 + 15.0065i 0.669105 + 0.669105i 0.957509 0.288404i \(-0.0931246\pi\)
−0.288404 + 0.957509i \(0.593125\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 4.79819 + 4.79819i 0.213095 + 0.213095i
\(508\) 0 0
\(509\) −34.6306 −1.53497 −0.767487 0.641065i \(-0.778493\pi\)
−0.767487 + 0.641065i \(0.778493\pi\)
\(510\) 0 0
\(511\) −2.54909 −0.112765
\(512\) 0 0
\(513\) −10.7701 9.04616i −0.475512 0.399398i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 7.95407 7.95407i 0.349819 0.349819i
\(518\) 0 0
\(519\) 17.4652i 0.766637i
\(520\) 0 0
\(521\) 39.0418i 1.71045i −0.518255 0.855226i \(-0.673418\pi\)
0.518255 0.855226i \(-0.326582\pi\)
\(522\) 0 0
\(523\) −25.2904 + 25.2904i −1.10587 + 1.10587i −0.112187 + 0.993687i \(0.535785\pi\)
−0.993687 + 0.112187i \(0.964215\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.35037 2.35037i −0.102384 0.102384i
\(528\) 0 0
\(529\) 10.9081i 0.474267i
\(530\) 0 0
\(531\) 13.7521i 0.596789i
\(532\) 0 0
\(533\) −5.86665 5.86665i −0.254113 0.254113i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −30.5906 + 30.5906i −1.32008 + 1.32008i
\(538\) 0 0
\(539\) 19.7605i 0.851145i
\(540\) 0 0
\(541\) −8.85283 −0.380613 −0.190307 0.981725i \(-0.560948\pi\)
−0.190307 + 0.981725i \(0.560948\pi\)
\(542\) 0 0
\(543\) −17.8983 17.8983i −0.768089 0.768089i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −11.9520 11.9520i −0.511032 0.511032i 0.403810 0.914843i \(-0.367686\pi\)
−0.914843 + 0.403810i \(0.867686\pi\)
\(548\) 0 0
\(549\) 5.19850i 0.221867i
\(550\) 0 0
\(551\) −12.9906 + 1.13015i −0.553420 + 0.0481459i
\(552\) 0 0
\(553\) 3.92219 3.92219i 0.166789 0.166789i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −19.8430 + 19.8430i −0.840774 + 0.840774i −0.988960 0.148185i \(-0.952657\pi\)
0.148185 + 0.988960i \(0.452657\pi\)
\(558\) 0 0
\(559\) 12.0561 0.509920
\(560\) 0 0
\(561\) 21.9333i 0.926026i
\(562\) 0 0
\(563\) 10.4703 10.4703i 0.441270 0.441270i −0.451169 0.892439i \(-0.648993\pi\)
0.892439 + 0.451169i \(0.148993\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −3.49976 + 3.49976i −0.146976 + 0.146976i
\(568\) 0 0
\(569\) 42.6744 1.78901 0.894503 0.447062i \(-0.147530\pi\)
0.894503 + 0.447062i \(0.147530\pi\)
\(570\) 0 0
\(571\) −23.2815 −0.974299 −0.487150 0.873319i \(-0.661963\pi\)
−0.487150 + 0.873319i \(0.661963\pi\)
\(572\) 0 0
\(573\) −34.5405 + 34.5405i −1.44295 + 1.44295i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 14.5526 14.5526i 0.605834 0.605834i −0.336021 0.941855i \(-0.609081\pi\)
0.941855 + 0.336021i \(0.109081\pi\)
\(578\) 0 0
\(579\) 7.25872i 0.301662i
\(580\) 0 0
\(581\) 5.73329 0.237857
\(582\) 0 0
\(583\) −1.64053 + 1.64053i −0.0679437 + 0.0679437i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.20495 6.20495i 0.256106 0.256106i −0.567362 0.823468i \(-0.692036\pi\)
0.823468 + 0.567362i \(0.192036\pi\)
\(588\) 0 0
\(589\) −0.351597 4.04149i −0.0144873 0.166527i
\(590\) 0 0
\(591\) 39.1512i 1.61046i
\(592\) 0 0
\(593\) −32.5210 32.5210i −1.33548 1.33548i −0.900387 0.435089i \(-0.856717\pi\)
−0.435089 0.900387i \(-0.643283\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 15.3992 + 15.3992i 0.630249 + 0.630249i
\(598\) 0 0
\(599\) −20.8505 −0.851929 −0.425964 0.904740i \(-0.640065\pi\)
−0.425964 + 0.904740i \(0.640065\pi\)
\(600\) 0 0
\(601\) 7.64493i 0.311843i 0.987769 + 0.155922i \(0.0498347\pi\)
−0.987769 + 0.155922i \(0.950165\pi\)
\(602\) 0 0
\(603\) −0.213476 + 0.213476i −0.00869341 + 0.00869341i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −16.0951 16.0951i −0.653279 0.653279i 0.300502 0.953781i \(-0.402846\pi\)
−0.953781 + 0.300502i \(0.902846\pi\)
\(608\) 0 0
\(609\) 2.78415i 0.112819i
\(610\) 0 0
\(611\) 15.5987i 0.631056i
\(612\) 0 0
\(613\) −9.69535 9.69535i −0.391591 0.391591i 0.483663 0.875254i \(-0.339306\pi\)
−0.875254 + 0.483663i \(0.839306\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.3319 26.3319i 1.06008 1.06008i 0.0620046 0.998076i \(-0.480251\pi\)
0.998076 0.0620046i \(-0.0197493\pi\)
\(618\) 0 0
\(619\) 38.0054i 1.52757i 0.645473 + 0.763783i \(0.276660\pi\)
−0.645473 + 0.763783i \(0.723340\pi\)
\(620\) 0 0
\(621\) 18.7897i 0.754004i
\(622\) 0 0
\(623\) −2.35037 + 2.35037i −0.0941657 + 0.0941657i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −17.2168 + 20.4978i −0.687571 + 0.818604i
\(628\) 0 0
\(629\) 41.0125 1.63528
\(630\) 0 0
\(631\) 25.0968 0.999087 0.499544 0.866289i \(-0.333501\pi\)
0.499544 + 0.866289i \(0.333501\pi\)
\(632\) 0 0
\(633\) 22.3729 + 22.3729i 0.889241 + 0.889241i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −19.3761 19.3761i −0.767710 0.767710i
\(638\) 0 0
\(639\) 14.1790 0.560913
\(640\) 0 0
\(641\) 10.1667i 0.401562i −0.979636 0.200781i \(-0.935652\pi\)
0.979636 0.200781i \(-0.0643480\pi\)
\(642\) 0 0
\(643\) −31.1082 31.1082i −1.22679 1.22679i −0.965173 0.261614i \(-0.915745\pi\)
−0.261614 0.965173i \(-0.584255\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.6336 + 24.6336i −0.968446 + 0.968446i −0.999517 0.0310708i \(-0.990108\pi\)
0.0310708 + 0.999517i \(0.490108\pi\)
\(648\) 0 0
\(649\) −27.0757 −1.06282
\(650\) 0 0
\(651\) −0.866170 −0.0339479
\(652\) 0 0
\(653\) −4.09234 4.09234i −0.160146 0.160146i 0.622486 0.782631i \(-0.286123\pi\)
−0.782631 + 0.622486i \(0.786123\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.04101 6.04101i 0.235682 0.235682i
\(658\) 0 0
\(659\) 9.90522 0.385853 0.192926 0.981213i \(-0.438202\pi\)
0.192926 + 0.981213i \(0.438202\pi\)
\(660\) 0 0
\(661\) 22.1328i 0.860866i 0.902623 + 0.430433i \(0.141639\pi\)
−0.902623 + 0.430433i \(0.858361\pi\)
\(662\) 0 0
\(663\) −21.5067 21.5067i −0.835251 0.835251i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.3176 12.3176i −0.476941 0.476941i
\(668\) 0 0
\(669\) 10.6396i 0.411349i
\(670\) 0 0
\(671\) 10.2351 0.395120
\(672\) 0 0
\(673\) 27.1304 27.1304i 1.04580 1.04580i 0.0469019 0.998899i \(-0.485065\pi\)
0.998899 0.0469019i \(-0.0149348\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21.5677 + 21.5677i 0.828915 + 0.828915i 0.987367 0.158452i \(-0.0506503\pi\)
−0.158452 + 0.987367i \(0.550650\pi\)
\(678\) 0 0
\(679\) −7.49279 −0.287547
\(680\) 0 0
\(681\) −17.4652 −0.669268
\(682\) 0 0
\(683\) −18.0872 + 18.0872i −0.692087 + 0.692087i −0.962691 0.270604i \(-0.912777\pi\)
0.270604 + 0.962691i \(0.412777\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −18.8111 18.8111i −0.717686 0.717686i
\(688\) 0 0
\(689\) 3.21723i 0.122567i
\(690\) 0 0
\(691\) 9.09679 0.346058 0.173029 0.984917i \(-0.444645\pi\)
0.173029 + 0.984917i \(0.444645\pi\)
\(692\) 0 0
\(693\) 1.33185 + 1.33185i 0.0505929 + 0.0505929i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 5.20448 + 5.20448i 0.197134 + 0.197134i
\(698\) 0 0
\(699\) −26.1644 −0.989627
\(700\) 0 0
\(701\) −14.0687 −0.531367 −0.265683 0.964060i \(-0.585598\pi\)
−0.265683 + 0.964060i \(0.585598\pi\)
\(702\) 0 0
\(703\) 38.3283 + 32.1932i 1.44558 + 1.21419i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.72837 2.72837i 0.102611 0.102611i
\(708\) 0 0
\(709\) 40.6035i 1.52490i 0.647050 + 0.762448i \(0.276003\pi\)
−0.647050 + 0.762448i \(0.723997\pi\)
\(710\) 0 0
\(711\) 18.5902i 0.697187i
\(712\) 0 0
\(713\) 3.83212 3.83212i 0.143514 0.143514i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 25.5660 + 25.5660i 0.954779 + 0.954779i
\(718\) 0 0
\(719\) 33.5067i 1.24959i −0.780789 0.624794i \(-0.785183\pi\)
0.780789 0.624794i \(-0.214817\pi\)
\(720\) 0 0
\(721\) 5.31386i 0.197898i
\(722\) 0 0
\(723\) 38.6735 + 38.6735i 1.43828 + 1.43828i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −2.59210 + 2.59210i −0.0961358 + 0.0961358i −0.753539 0.657403i \(-0.771655\pi\)
0.657403 + 0.753539i \(0.271655\pi\)
\(728\) 0 0
\(729\) 3.88892i 0.144034i
\(730\) 0 0
\(731\) −10.6953 −0.395582
\(732\) 0 0
\(733\) −15.1334 15.1334i −0.558963 0.558963i 0.370049 0.929012i \(-0.379341\pi\)
−0.929012 + 0.370049i \(0.879341\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.420302 0.420302i −0.0154820 0.0154820i
\(738\) 0 0
\(739\) 13.4291i 0.493998i −0.969016 0.246999i \(-0.920556\pi\)
0.969016 0.246999i \(-0.0794444\pi\)
\(740\) 0 0
\(741\) −3.21723 36.9810i −0.118188 1.35853i
\(742\) 0 0
\(743\) 23.7186 23.7186i 0.870152 0.870152i −0.122337 0.992489i \(-0.539039\pi\)
0.992489 + 0.122337i \(0.0390388\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −13.5872 + 13.5872i −0.497129 + 0.497129i
\(748\) 0 0
\(749\) −2.35037 −0.0858807
\(750\) 0 0
\(751\) 20.3615i 0.743002i 0.928433 + 0.371501i \(0.121157\pi\)
−0.928433 + 0.371501i \(0.878843\pi\)
\(752\) 0 0
\(753\) −8.04385 + 8.04385i −0.293134 + 0.293134i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3.67799 3.67799i 0.133679 0.133679i −0.637101 0.770780i \(-0.719867\pi\)
0.770780 + 0.637101i \(0.219867\pi\)
\(758\) 0 0
\(759\) −35.7607 −1.29803
\(760\) 0 0
\(761\) −15.9126 −0.576831 −0.288415 0.957505i \(-0.593128\pi\)
−0.288415 + 0.957505i \(0.593128\pi\)
\(762\) 0 0
\(763\) 4.27379 4.27379i 0.154722 0.154722i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 26.5491 26.5491i 0.958632 0.958632i
\(768\) 0 0
\(769\) 27.0277i 0.974643i −0.873223 0.487321i \(-0.837974\pi\)
0.873223 0.487321i \(-0.162026\pi\)
\(770\) 0 0
\(771\) 13.4237 0.483443
\(772\) 0 0
\(773\) −8.89182 + 8.89182i −0.319817 + 0.319817i −0.848697 0.528880i \(-0.822612\pi\)
0.528880 + 0.848697i \(0.322612\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 7.55707 7.55707i 0.271108 0.271108i
\(778\) 0 0
\(779\) 0.778549 + 8.94914i 0.0278944 + 0.320636i
\(780\) 0 0
\(781\) 27.9163i 0.998925i
\(782\) 0 0
\(783\) 6.82564 + 6.82564i 0.243928 + 0.243928i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 20.9266 + 20.9266i 0.745952 + 0.745952i 0.973716 0.227764i \(-0.0731416\pi\)
−0.227764 + 0.973716i \(0.573142\pi\)
\(788\) 0 0
\(789\) −35.4091 −1.26060
\(790\) 0 0
\(791\) 1.19220i 0.0423898i
\(792\) 0 0
\(793\) −10.0360 + 10.0360i −0.356388 + 0.356388i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 35.3951 + 35.3951i 1.25376 + 1.25376i 0.954021 + 0.299738i \(0.0968993\pi\)
0.299738 + 0.954021i \(0.403101\pi\)
\(798\) 0 0
\(799\) 13.8381i 0.489556i
\(800\) 0 0
\(801\) 11.1402i 0.393619i
\(802\) 0 0
\(803\) 11.8938 + 11.8938i 0.419725 + 0.419725i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −9.47505 + 9.47505i −0.333538 + 0.333538i
\(808\) 0 0
\(809\) 18.3555i 0.645345i −0.946511 0.322673i \(-0.895419\pi\)
0.946511 0.322673i \(-0.104581\pi\)
\(810\) 0 0
\(811\) 1.77129i 0.0621983i −0.999516 0.0310991i \(-0.990099\pi\)
0.999516 0.0310991i \(-0.00990076\pi\)
\(812\) 0 0
\(813\) −20.2928 + 20.2928i −0.711699 + 0.711699i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −9.99534 8.39540i −0.349693 0.293718i
\(818\) 0 0
\(819\) −2.61189 −0.0912668
\(820\) 0 0
\(821\) 18.7654 0.654917 0.327459 0.944865i \(-0.393808\pi\)
0.327459 + 0.944865i \(0.393808\pi\)
\(822\) 0 0
\(823\) 25.6844 + 25.6844i 0.895304 + 0.895304i 0.995016 0.0997127i \(-0.0317924\pi\)
−0.0997127 + 0.995016i \(0.531792\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 19.7063 + 19.7063i 0.685257 + 0.685257i 0.961180 0.275923i \(-0.0889835\pi\)
−0.275923 + 0.961180i \(0.588983\pi\)
\(828\) 0 0
\(829\) −14.1790 −0.492457 −0.246229 0.969212i \(-0.579191\pi\)
−0.246229 + 0.969212i \(0.579191\pi\)
\(830\) 0 0
\(831\) 35.6086i 1.23525i
\(832\) 0 0
\(833\) 17.1891 + 17.1891i 0.595568 + 0.595568i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.12351 + 2.12351i −0.0733992 + 0.0733992i
\(838\) 0 0
\(839\) −19.0319 −0.657054 −0.328527 0.944495i \(-0.606552\pi\)
−0.328527 + 0.944495i \(0.606552\pi\)
\(840\) 0 0
\(841\) −20.0509 −0.691409
\(842\) 0 0
\(843\) −39.9309 39.9309i −1.37529 1.37529i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0.799970 0.799970i 0.0274873 0.0274873i
\(848\) 0 0
\(849\) −39.3934 −1.35198
\(850\) 0 0
\(851\) 66.8681i 2.29221i
\(852\) 0 0
\(853\) 11.9906 + 11.9906i 0.410551 + 0.410551i 0.881931 0.471379i \(-0.156244\pi\)
−0.471379 + 0.881931i \(0.656244\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 26.5107 + 26.5107i 0.905587 + 0.905587i 0.995912 0.0903251i \(-0.0287906\pi\)
−0.0903251 + 0.995912i \(0.528791\pi\)
\(858\) 0 0
\(859\) 44.9719i 1.53442i −0.641395 0.767211i \(-0.721644\pi\)
0.641395 0.767211i \(-0.278356\pi\)
\(860\) 0 0
\(861\) 1.91798 0.0653645
\(862\) 0 0
\(863\) −29.4121 + 29.4121i −1.00120 + 1.00120i −0.00120035 + 0.999999i \(0.500382\pi\)
−0.999999 + 0.00120035i \(0.999618\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −6.34863 6.34863i −0.215611 0.215611i
\(868\) 0 0
\(869\) −36.6013 −1.24162
\(870\) 0 0
\(871\) 0.824253 0.0279287
\(872\) 0 0
\(873\) 17.7570 17.7570i 0.600982 0.600982i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6.98977 6.98977i −0.236028 0.236028i 0.579175 0.815203i \(-0.303375\pi\)
−0.815203 + 0.579175i \(0.803375\pi\)
\(878\) 0 0
\(879\) 68.1704i 2.29933i
\(880\) 0 0
\(881\) 15.5254 0.523065 0.261532 0.965195i \(-0.415772\pi\)
0.261532 + 0.965195i \(0.415772\pi\)
\(882\) 0 0
\(883\) −3.40790 3.40790i −0.114685 0.114685i 0.647435 0.762120i \(-0.275842\pi\)
−0.762120 + 0.647435i \(0.775842\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 23.4077 + 23.4077i 0.785954 + 0.785954i 0.980828 0.194874i \(-0.0624299\pi\)
−0.194874 + 0.980828i \(0.562430\pi\)
\(888\) 0 0
\(889\) 3.19098 0.107022
\(890\) 0 0
\(891\) 32.6593 1.09413
\(892\) 0 0
\(893\) 10.8623 12.9324i 0.363493 0.432766i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 35.0651 35.0651i 1.17079 1.17079i
\(898\) 0 0
\(899\) 2.78415i 0.0928566i
\(900\) 0 0
\(901\) 2.85410i 0.0950840i
\(902\) 0 0
\(903\) −1.97075 + 1.97075i −0.0655825 + 0.0655825i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −22.0781 22.0781i −0.733091 0.733091i 0.238140 0.971231i \(-0.423462\pi\)
−0.971231 + 0.238140i \(0.923462\pi\)
\(908\) 0 0
\(909\) 12.9318i 0.428920i
\(910\) 0 0
\(911\) 4.67272i 0.154814i 0.997000 + 0.0774071i \(0.0246641\pi\)
−0.997000 + 0.0774071i \(0.975336\pi\)
\(912\) 0 0
\(913\) −26.7511 26.7511i −0.885333 0.885333i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.22570 3.22570i 0.106522 0.106522i
\(918\) 0 0
\(919\) 16.5303i 0.545286i −0.962115 0.272643i \(-0.912102\pi\)
0.962115 0.272643i \(-0.0878978\pi\)
\(920\) 0 0
\(921\) 58.5606 1.92964
\(922\) 0 0
\(923\) −27.3733 27.3733i −0.901004 0.901004i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 12.5932 + 12.5932i 0.413614 + 0.413614i
\(928\) 0 0
\(929\) 33.1655i 1.08812i −0.839045 0.544062i \(-0.816886\pi\)
0.839045 0.544062i \(-0.183114\pi\)
\(930\) 0 0
\(931\) 2.57136 + 29.5569i 0.0842729 + 0.968687i
\(932\) 0 0
\(933\) −45.0655 + 45.0655i −1.47538 + 1.47538i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −32.6686 + 32.6686i −1.06724 + 1.06724i −0.0696672 + 0.997570i \(0.522194\pi\)
−0.997570 + 0.0696672i \(0.977806\pi\)
\(938\) 0 0
\(939\) 34.5878 1.12873
\(940\) 0 0
\(941\) 3.86014i 0.125837i 0.998019 + 0.0629185i \(0.0200408\pi\)
−0.998019 + 0.0629185i \(0.979959\pi\)
\(942\) 0 0
\(943\) −8.48553 + 8.48553i −0.276327 + 0.276327i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −36.1733 + 36.1733i −1.17547 + 1.17547i −0.194590 + 0.980885i \(0.562338\pi\)
−0.980885 + 0.194590i \(0.937662\pi\)
\(948\) 0 0
\(949\) −23.3250 −0.757161
\(950\) 0 0
\(951\) −31.7132 −1.02837
\(952\) 0 0
\(953\) 13.5239 13.5239i 0.438081 0.438081i −0.453285 0.891366i \(-0.649748\pi\)
0.891366 + 0.453285i \(0.149748\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 12.9906 12.9906i 0.419927 0.419927i
\(958\) 0 0
\(959\) 4.97190i 0.160551i
\(960\) 0 0
\(961\) 30.1338 0.972059
\(962\) 0 0
\(963\) 5.57008 5.57008i 0.179493 0.179493i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 21.0163 21.0163i 0.675839 0.675839i −0.283217 0.959056i \(-0.591402\pi\)
0.959056 + 0.283217i \(0.0914017\pi\)
\(968\) 0 0
\(969\) 2.85410 + 32.8069i 0.0916870 + 1.05391i
\(970\) 0 0
\(971\) 29.4261i 0.944329i −0.881510 0.472164i \(-0.843473\pi\)
0.881510 0.472164i \(-0.156527\pi\)
\(972\) 0 0
\(973\) −3.16992 3.16992i −0.101623 0.101623i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −10.4917 10.4917i −0.335658 0.335658i 0.519072 0.854730i \(-0.326278\pi\)
−0.854730 + 0.519072i \(0.826278\pi\)
\(978\) 0 0
\(979\) 21.9333 0.700992
\(980\) 0 0
\(981\) 20.2567i 0.646747i
\(982\) 0 0
\(983\) 4.06697 4.06697i 0.129716 0.129716i −0.639268 0.768984i \(-0.720763\pi\)
0.768984 + 0.639268i \(0.220763\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −2.54984 2.54984i −0.0811622 0.0811622i
\(988\) 0 0
\(989\) 17.4380i 0.554496i
\(990\) 0 0
\(991\) 16.8189i 0.534271i −0.963659 0.267135i \(-0.913923\pi\)
0.963659 0.267135i \(-0.0860771\pi\)
\(992\) 0 0
\(993\) 40.0656 + 40.0656i 1.27144 + 1.27144i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −17.1847 + 17.1847i −0.544244 + 0.544244i −0.924770 0.380526i \(-0.875743\pi\)
0.380526 + 0.924770i \(0.375743\pi\)
\(998\) 0 0
\(999\) 37.0539i 1.17233i
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 1900.2.l.b.493.3 12
5.2 odd 4 inner 1900.2.l.b.1557.4 12
5.3 odd 4 380.2.l.b.37.3 12
5.4 even 2 380.2.l.b.113.4 yes 12
15.8 even 4 3420.2.bb.d.37.3 12
15.14 odd 2 3420.2.bb.d.2773.4 12
19.18 odd 2 inner 1900.2.l.b.493.4 12
95.18 even 4 380.2.l.b.37.4 yes 12
95.37 even 4 inner 1900.2.l.b.1557.3 12
95.94 odd 2 380.2.l.b.113.3 yes 12
285.113 odd 4 3420.2.bb.d.37.4 12
285.284 even 2 3420.2.bb.d.2773.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.l.b.37.3 12 5.3 odd 4
380.2.l.b.37.4 yes 12 95.18 even 4
380.2.l.b.113.3 yes 12 95.94 odd 2
380.2.l.b.113.4 yes 12 5.4 even 2
1900.2.l.b.493.3 12 1.1 even 1 trivial
1900.2.l.b.493.4 12 19.18 odd 2 inner
1900.2.l.b.1557.3 12 95.37 even 4 inner
1900.2.l.b.1557.4 12 5.2 odd 4 inner
3420.2.bb.d.37.3 12 15.8 even 4
3420.2.bb.d.37.4 12 285.113 odd 4
3420.2.bb.d.2773.3 12 285.284 even 2
3420.2.bb.d.2773.4 12 15.14 odd 2