Properties

Label 1512.2.c.d.757.11
Level $1512$
Weight $2$
Character 1512.757
Analytic conductor $12.073$
Analytic rank $0$
Dimension $16$
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] = [1512,2,Mod(757,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.757");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.c (of order \(2\), degree \(1\), 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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 757.11
Root \(0.987688 - 0.156434i\) of defining polynomial
Character \(\chi\) \(=\) 1512.757
Dual form 1512.2.c.d.757.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.437016 + 1.34500i) q^{2} +(-1.61803 + 1.17557i) q^{4} +0.0549306i q^{5} -1.00000 q^{7} +(-2.28825 - 1.66251i) q^{8} +O(q^{10})\) \(q+(0.437016 + 1.34500i) q^{2} +(-1.61803 + 1.17557i) q^{4} +0.0549306i q^{5} -1.00000 q^{7} +(-2.28825 - 1.66251i) q^{8} +(-0.0738814 + 0.0240055i) q^{10} -2.63506i q^{11} -3.67853i q^{13} +(-0.437016 - 1.34500i) q^{14} +(1.23607 - 3.80423i) q^{16} +3.16228 q^{17} +3.07768i q^{19} +(-0.0645747 - 0.0888795i) q^{20} +(3.54415 - 1.15156i) q^{22} +2.86834 q^{23} +4.99698 q^{25} +(4.94761 - 1.60758i) q^{26} +(1.61803 - 1.17557i) q^{28} +10.1898i q^{29} +9.32437 q^{31} +5.65685 q^{32} +(1.38197 + 4.25325i) q^{34} -0.0549306i q^{35} +0.774554i q^{37} +(-4.13948 + 1.34500i) q^{38} +(0.0913225 - 0.125695i) q^{40} +6.36183 q^{41} -9.98927i q^{43} +(3.09770 + 4.26362i) q^{44} +(1.25351 + 3.85790i) q^{46} -12.3126 q^{47} +1.00000 q^{49} +(2.18376 + 6.72093i) q^{50} +(4.32437 + 5.95199i) q^{52} -3.39291i q^{53} +0.144746 q^{55} +(2.28825 + 1.66251i) q^{56} +(-13.7052 + 4.45309i) q^{58} +6.93263i q^{59} -8.35114i q^{61} +(4.07490 + 12.5413i) q^{62} +(2.47214 + 7.60845i) q^{64} +0.202064 q^{65} -8.93584i q^{67} +(-5.11667 + 3.71748i) q^{68} +(0.0738814 - 0.0240055i) q^{70} +4.28255 q^{71} +8.38081 q^{73} +(-1.04177 + 0.338492i) q^{74} +(-3.61803 - 4.97980i) q^{76} +2.63506i q^{77} -3.03186 q^{79} +(0.208968 + 0.0678979i) q^{80} +(2.78022 + 8.55664i) q^{82} +10.3255i q^{83} +0.173706i q^{85} +(13.4355 - 4.36547i) q^{86} +(-4.38081 + 6.02967i) q^{88} +12.8155 q^{89} +3.67853i q^{91} +(-4.64107 + 3.37193i) q^{92} +(-5.38081 - 16.5604i) q^{94} -0.169059 q^{95} -10.1803 q^{97} +(0.437016 + 1.34500i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} - 16 q^{7} - 20 q^{10} - 16 q^{16} + 20 q^{22} - 32 q^{25} + 8 q^{28} + 40 q^{31} + 40 q^{34} + 40 q^{40} + 4 q^{46} + 16 q^{49} - 40 q^{52} - 72 q^{55} - 32 q^{64} + 20 q^{70} + 24 q^{73} - 40 q^{76} + 24 q^{79} - 28 q^{82} + 40 q^{88} + 24 q^{94} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.437016 + 1.34500i 0.309017 + 0.951057i
\(3\) 0 0
\(4\) −1.61803 + 1.17557i −0.809017 + 0.587785i
\(5\) 0.0549306i 0.0245657i 0.999925 + 0.0122828i \(0.00390985\pi\)
−0.999925 + 0.0122828i \(0.996090\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.28825 1.66251i −0.809017 0.587785i
\(9\) 0 0
\(10\) −0.0738814 + 0.0240055i −0.0233634 + 0.00759122i
\(11\) 2.63506i 0.794502i −0.917710 0.397251i \(-0.869964\pi\)
0.917710 0.397251i \(-0.130036\pi\)
\(12\) 0 0
\(13\) 3.67853i 1.02024i −0.860103 0.510121i \(-0.829601\pi\)
0.860103 0.510121i \(-0.170399\pi\)
\(14\) −0.437016 1.34500i −0.116797 0.359466i
\(15\) 0 0
\(16\) 1.23607 3.80423i 0.309017 0.951057i
\(17\) 3.16228 0.766965 0.383482 0.923548i \(-0.374725\pi\)
0.383482 + 0.923548i \(0.374725\pi\)
\(18\) 0 0
\(19\) 3.07768i 0.706069i 0.935610 + 0.353035i \(0.114850\pi\)
−0.935610 + 0.353035i \(0.885150\pi\)
\(20\) −0.0645747 0.0888795i −0.0144394 0.0198741i
\(21\) 0 0
\(22\) 3.54415 1.15156i 0.755616 0.245514i
\(23\) 2.86834 0.598090 0.299045 0.954239i \(-0.403332\pi\)
0.299045 + 0.954239i \(0.403332\pi\)
\(24\) 0 0
\(25\) 4.99698 0.999397
\(26\) 4.94761 1.60758i 0.970307 0.315272i
\(27\) 0 0
\(28\) 1.61803 1.17557i 0.305780 0.222162i
\(29\) 10.1898i 1.89219i 0.323890 + 0.946095i \(0.395009\pi\)
−0.323890 + 0.946095i \(0.604991\pi\)
\(30\) 0 0
\(31\) 9.32437 1.67471 0.837353 0.546662i \(-0.184102\pi\)
0.837353 + 0.546662i \(0.184102\pi\)
\(32\) 5.65685 1.00000
\(33\) 0 0
\(34\) 1.38197 + 4.25325i 0.237005 + 0.729427i
\(35\) 0.0549306i 0.00928496i
\(36\) 0 0
\(37\) 0.774554i 0.127336i 0.997971 + 0.0636679i \(0.0202798\pi\)
−0.997971 + 0.0636679i \(0.979720\pi\)
\(38\) −4.13948 + 1.34500i −0.671512 + 0.218187i
\(39\) 0 0
\(40\) 0.0913225 0.125695i 0.0144394 0.0198741i
\(41\) 6.36183 0.993551 0.496775 0.867879i \(-0.334517\pi\)
0.496775 + 0.867879i \(0.334517\pi\)
\(42\) 0 0
\(43\) 9.98927i 1.52335i −0.647960 0.761674i \(-0.724378\pi\)
0.647960 0.761674i \(-0.275622\pi\)
\(44\) 3.09770 + 4.26362i 0.466996 + 0.642765i
\(45\) 0 0
\(46\) 1.25351 + 3.85790i 0.184820 + 0.568817i
\(47\) −12.3126 −1.79598 −0.897990 0.440015i \(-0.854973\pi\)
−0.897990 + 0.440015i \(0.854973\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 2.18376 + 6.72093i 0.308831 + 0.950483i
\(51\) 0 0
\(52\) 4.32437 + 5.95199i 0.599683 + 0.825392i
\(53\) 3.39291i 0.466052i −0.972470 0.233026i \(-0.925137\pi\)
0.972470 0.233026i \(-0.0748628\pi\)
\(54\) 0 0
\(55\) 0.144746 0.0195175
\(56\) 2.28825 + 1.66251i 0.305780 + 0.222162i
\(57\) 0 0
\(58\) −13.7052 + 4.45309i −1.79958 + 0.584719i
\(59\) 6.93263i 0.902552i 0.892384 + 0.451276i \(0.149031\pi\)
−0.892384 + 0.451276i \(0.850969\pi\)
\(60\) 0 0
\(61\) 8.35114i 1.06925i −0.845088 0.534627i \(-0.820452\pi\)
0.845088 0.534627i \(-0.179548\pi\)
\(62\) 4.07490 + 12.5413i 0.517513 + 1.59274i
\(63\) 0 0
\(64\) 2.47214 + 7.60845i 0.309017 + 0.951057i
\(65\) 0.202064 0.0250629
\(66\) 0 0
\(67\) 8.93584i 1.09169i −0.837887 0.545843i \(-0.816209\pi\)
0.837887 0.545843i \(-0.183791\pi\)
\(68\) −5.11667 + 3.71748i −0.620488 + 0.450811i
\(69\) 0 0
\(70\) 0.0738814 0.0240055i 0.00883052 0.00286921i
\(71\) 4.28255 0.508245 0.254123 0.967172i \(-0.418213\pi\)
0.254123 + 0.967172i \(0.418213\pi\)
\(72\) 0 0
\(73\) 8.38081 0.980900 0.490450 0.871469i \(-0.336832\pi\)
0.490450 + 0.871469i \(0.336832\pi\)
\(74\) −1.04177 + 0.338492i −0.121104 + 0.0393489i
\(75\) 0 0
\(76\) −3.61803 4.97980i −0.415017 0.571222i
\(77\) 2.63506i 0.300293i
\(78\) 0 0
\(79\) −3.03186 −0.341111 −0.170556 0.985348i \(-0.554556\pi\)
−0.170556 + 0.985348i \(0.554556\pi\)
\(80\) 0.208968 + 0.0678979i 0.0233634 + 0.00759122i
\(81\) 0 0
\(82\) 2.78022 + 8.55664i 0.307024 + 0.944923i
\(83\) 10.3255i 1.13338i 0.823932 + 0.566688i \(0.191776\pi\)
−0.823932 + 0.566688i \(0.808224\pi\)
\(84\) 0 0
\(85\) 0.173706i 0.0188410i
\(86\) 13.4355 4.36547i 1.44879 0.470741i
\(87\) 0 0
\(88\) −4.38081 + 6.02967i −0.466996 + 0.642765i
\(89\) 12.8155 1.35844 0.679222 0.733933i \(-0.262317\pi\)
0.679222 + 0.733933i \(0.262317\pi\)
\(90\) 0 0
\(91\) 3.67853i 0.385615i
\(92\) −4.64107 + 3.37193i −0.483865 + 0.351548i
\(93\) 0 0
\(94\) −5.38081 16.5604i −0.554989 1.70808i
\(95\) −0.169059 −0.0173451
\(96\) 0 0
\(97\) −10.1803 −1.03366 −0.516828 0.856089i \(-0.672887\pi\)
−0.516828 + 0.856089i \(0.672887\pi\)
\(98\) 0.437016 + 1.34500i 0.0441453 + 0.135865i
\(99\) 0 0
\(100\) −8.08529 + 5.87431i −0.808529 + 0.587431i
\(101\) 1.02749i 0.102239i 0.998693 + 0.0511194i \(0.0162789\pi\)
−0.998693 + 0.0511194i \(0.983721\pi\)
\(102\) 0 0
\(103\) 1.72905 0.170368 0.0851842 0.996365i \(-0.472852\pi\)
0.0851842 + 0.996365i \(0.472852\pi\)
\(104\) −6.11559 + 8.41738i −0.599683 + 0.825392i
\(105\) 0 0
\(106\) 4.56346 1.48276i 0.443242 0.144018i
\(107\) 6.11985i 0.591629i 0.955245 + 0.295814i \(0.0955910\pi\)
−0.955245 + 0.295814i \(0.904409\pi\)
\(108\) 0 0
\(109\) 8.85597i 0.848248i −0.905604 0.424124i \(-0.860582\pi\)
0.905604 0.424124i \(-0.139418\pi\)
\(110\) 0.0632561 + 0.194682i 0.00603123 + 0.0185622i
\(111\) 0 0
\(112\) −1.23607 + 3.80423i −0.116797 + 0.359466i
\(113\) 9.02383 0.848891 0.424445 0.905454i \(-0.360469\pi\)
0.424445 + 0.905454i \(0.360469\pi\)
\(114\) 0 0
\(115\) 0.157559i 0.0146925i
\(116\) −11.9788 16.4874i −1.11220 1.53081i
\(117\) 0 0
\(118\) −9.32437 + 3.02967i −0.858378 + 0.278904i
\(119\) −3.16228 −0.289886
\(120\) 0 0
\(121\) 4.05644 0.368767
\(122\) 11.2323 3.64958i 1.01692 0.330418i
\(123\) 0 0
\(124\) −15.0872 + 10.9615i −1.35487 + 0.984368i
\(125\) 0.549140i 0.0491166i
\(126\) 0 0
\(127\) 18.2656 1.62081 0.810406 0.585868i \(-0.199246\pi\)
0.810406 + 0.585868i \(0.199246\pi\)
\(128\) −9.15298 + 6.65003i −0.809017 + 0.587785i
\(129\) 0 0
\(130\) 0.0883051 + 0.271775i 0.00774487 + 0.0238363i
\(131\) 17.4470i 1.52435i −0.647373 0.762174i \(-0.724132\pi\)
0.647373 0.762174i \(-0.275868\pi\)
\(132\) 0 0
\(133\) 3.07768i 0.266869i
\(134\) 12.0187 3.90511i 1.03826 0.337350i
\(135\) 0 0
\(136\) −7.23607 5.25731i −0.620488 0.450811i
\(137\) 7.73613 0.660942 0.330471 0.943816i \(-0.392792\pi\)
0.330471 + 0.943816i \(0.392792\pi\)
\(138\) 0 0
\(139\) 0.802034i 0.0680276i 0.999421 + 0.0340138i \(0.0108290\pi\)
−0.999421 + 0.0340138i \(0.989171\pi\)
\(140\) 0.0645747 + 0.0888795i 0.00545756 + 0.00751169i
\(141\) 0 0
\(142\) 1.87154 + 5.76002i 0.157056 + 0.483370i
\(143\) −9.69316 −0.810583
\(144\) 0 0
\(145\) −0.559729 −0.0464829
\(146\) 3.66255 + 11.2722i 0.303115 + 0.932891i
\(147\) 0 0
\(148\) −0.910542 1.25325i −0.0748461 0.103017i
\(149\) 15.7165i 1.28755i 0.765215 + 0.643775i \(0.222633\pi\)
−0.765215 + 0.643775i \(0.777367\pi\)
\(150\) 0 0
\(151\) 2.58129 0.210062 0.105031 0.994469i \(-0.466506\pi\)
0.105031 + 0.994469i \(0.466506\pi\)
\(152\) 5.11667 7.04250i 0.415017 0.571222i
\(153\) 0 0
\(154\) −3.54415 + 1.15156i −0.285596 + 0.0927957i
\(155\) 0.512193i 0.0411403i
\(156\) 0 0
\(157\) 0.680724i 0.0543277i −0.999631 0.0271638i \(-0.991352\pi\)
0.999631 0.0271638i \(-0.00864758\pi\)
\(158\) −1.32497 4.07785i −0.105409 0.324416i
\(159\) 0 0
\(160\) 0.310734i 0.0245657i
\(161\) −2.86834 −0.226057
\(162\) 0 0
\(163\) 21.8339i 1.71016i −0.518494 0.855081i \(-0.673507\pi\)
0.518494 0.855081i \(-0.326493\pi\)
\(164\) −10.2937 + 7.47878i −0.803799 + 0.583994i
\(165\) 0 0
\(166\) −13.8878 + 4.51243i −1.07790 + 0.350232i
\(167\) −17.2203 −1.33255 −0.666274 0.745707i \(-0.732112\pi\)
−0.666274 + 0.745707i \(0.732112\pi\)
\(168\) 0 0
\(169\) −0.531594 −0.0408918
\(170\) −0.233634 + 0.0759122i −0.0179189 + 0.00582220i
\(171\) 0 0
\(172\) 11.7431 + 16.1630i 0.895402 + 1.23242i
\(173\) 2.57218i 0.195559i 0.995208 + 0.0977797i \(0.0311741\pi\)
−0.995208 + 0.0977797i \(0.968826\pi\)
\(174\) 0 0
\(175\) −4.99698 −0.377736
\(176\) −10.0244 3.25712i −0.755616 0.245514i
\(177\) 0 0
\(178\) 5.60059 + 17.2369i 0.419782 + 1.29196i
\(179\) 5.26703i 0.393676i −0.980436 0.196838i \(-0.936933\pi\)
0.980436 0.196838i \(-0.0630673\pi\)
\(180\) 0 0
\(181\) 16.0189i 1.19068i 0.803475 + 0.595339i \(0.202982\pi\)
−0.803475 + 0.595339i \(0.797018\pi\)
\(182\) −4.94761 + 1.60758i −0.366742 + 0.119162i
\(183\) 0 0
\(184\) −6.56346 4.76863i −0.483865 0.351548i
\(185\) −0.0425467 −0.00312809
\(186\) 0 0
\(187\) 8.33280i 0.609355i
\(188\) 19.9222 14.4744i 1.45298 1.05565i
\(189\) 0 0
\(190\) −0.0738814 0.227384i −0.00535992 0.0164961i
\(191\) −1.83730 −0.132943 −0.0664713 0.997788i \(-0.521174\pi\)
−0.0664713 + 0.997788i \(0.521174\pi\)
\(192\) 0 0
\(193\) 15.3429 1.10441 0.552204 0.833709i \(-0.313787\pi\)
0.552204 + 0.833709i \(0.313787\pi\)
\(194\) −4.44897 13.6925i −0.319418 0.983066i
\(195\) 0 0
\(196\) −1.61803 + 1.17557i −0.115574 + 0.0839693i
\(197\) 5.02447i 0.357979i −0.983851 0.178989i \(-0.942717\pi\)
0.983851 0.178989i \(-0.0572828\pi\)
\(198\) 0 0
\(199\) −22.0333 −1.56190 −0.780949 0.624595i \(-0.785264\pi\)
−0.780949 + 0.624595i \(0.785264\pi\)
\(200\) −11.4343 8.30752i −0.808529 0.587431i
\(201\) 0 0
\(202\) −1.38197 + 0.449028i −0.0972348 + 0.0315935i
\(203\) 10.1898i 0.715180i
\(204\) 0 0
\(205\) 0.349459i 0.0244073i
\(206\) 0.755623 + 2.32557i 0.0526467 + 0.162030i
\(207\) 0 0
\(208\) −13.9940 4.54691i −0.970307 0.315272i
\(209\) 8.10989 0.560973
\(210\) 0 0
\(211\) 8.50651i 0.585612i 0.956172 + 0.292806i \(0.0945890\pi\)
−0.956172 + 0.292806i \(0.905411\pi\)
\(212\) 3.98861 + 5.48985i 0.273939 + 0.377044i
\(213\) 0 0
\(214\) −8.23119 + 2.67447i −0.562672 + 0.182823i
\(215\) 0.548716 0.0374221
\(216\) 0 0
\(217\) −9.32437 −0.632980
\(218\) 11.9112 3.87020i 0.806732 0.262123i
\(219\) 0 0
\(220\) −0.234203 + 0.170159i −0.0157900 + 0.0114721i
\(221\) 11.6325i 0.782489i
\(222\) 0 0
\(223\) −13.5574 −0.907872 −0.453936 0.891034i \(-0.649981\pi\)
−0.453936 + 0.891034i \(0.649981\pi\)
\(224\) −5.65685 −0.377964
\(225\) 0 0
\(226\) 3.94356 + 12.1370i 0.262322 + 0.807343i
\(227\) 16.0721i 1.06674i −0.845882 0.533370i \(-0.820925\pi\)
0.845882 0.533370i \(-0.179075\pi\)
\(228\) 0 0
\(229\) 5.50870i 0.364025i −0.983296 0.182013i \(-0.941739\pi\)
0.983296 0.182013i \(-0.0582612\pi\)
\(230\) −0.211917 + 0.0688559i −0.0139734 + 0.00454023i
\(231\) 0 0
\(232\) 16.9405 23.3167i 1.11220 1.53081i
\(233\) 28.6599 1.87757 0.938787 0.344498i \(-0.111951\pi\)
0.938787 + 0.344498i \(0.111951\pi\)
\(234\) 0 0
\(235\) 0.676339i 0.0441195i
\(236\) −8.14980 11.2172i −0.530507 0.730180i
\(237\) 0 0
\(238\) −1.38197 4.25325i −0.0895796 0.275698i
\(239\) 3.82998 0.247741 0.123870 0.992298i \(-0.460469\pi\)
0.123870 + 0.992298i \(0.460469\pi\)
\(240\) 0 0
\(241\) −1.72834 −0.111332 −0.0556660 0.998449i \(-0.517728\pi\)
−0.0556660 + 0.998449i \(0.517728\pi\)
\(242\) 1.77273 + 5.45590i 0.113955 + 0.350719i
\(243\) 0 0
\(244\) 9.81736 + 13.5124i 0.628492 + 0.865045i
\(245\) 0.0549306i 0.00350938i
\(246\) 0 0
\(247\) 11.3214 0.720361
\(248\) −21.3365 15.5018i −1.35487 0.984368i
\(249\) 0 0
\(250\) −0.738591 + 0.239983i −0.0467126 + 0.0151779i
\(251\) 0.245657i 0.0155057i 0.999970 + 0.00775286i \(0.00246784\pi\)
−0.999970 + 0.00775286i \(0.997532\pi\)
\(252\) 0 0
\(253\) 7.55825i 0.475183i
\(254\) 7.98237 + 24.5672i 0.500859 + 1.54148i
\(255\) 0 0
\(256\) −12.9443 9.40456i −0.809017 0.587785i
\(257\) −16.2120 −1.01128 −0.505638 0.862746i \(-0.668743\pi\)
−0.505638 + 0.862746i \(0.668743\pi\)
\(258\) 0 0
\(259\) 0.774554i 0.0481284i
\(260\) −0.326946 + 0.237540i −0.0202763 + 0.0147316i
\(261\) 0 0
\(262\) 23.4661 7.62460i 1.44974 0.471049i
\(263\) −23.3265 −1.43837 −0.719187 0.694817i \(-0.755485\pi\)
−0.719187 + 0.694817i \(0.755485\pi\)
\(264\) 0 0
\(265\) 0.186375 0.0114489
\(266\) 4.13948 1.34500i 0.253808 0.0824671i
\(267\) 0 0
\(268\) 10.5047 + 14.4585i 0.641677 + 0.883193i
\(269\) 23.6947i 1.44469i −0.691534 0.722344i \(-0.743065\pi\)
0.691534 0.722344i \(-0.256935\pi\)
\(270\) 0 0
\(271\) 3.52786 0.214302 0.107151 0.994243i \(-0.465827\pi\)
0.107151 + 0.994243i \(0.465827\pi\)
\(272\) 3.90879 12.0300i 0.237005 0.729427i
\(273\) 0 0
\(274\) 3.38081 + 10.4051i 0.204242 + 0.628594i
\(275\) 13.1674i 0.794022i
\(276\) 0 0
\(277\) 28.6014i 1.71849i −0.511561 0.859247i \(-0.670932\pi\)
0.511561 0.859247i \(-0.329068\pi\)
\(278\) −1.07873 + 0.350502i −0.0646981 + 0.0210217i
\(279\) 0 0
\(280\) −0.0913225 + 0.125695i −0.00545756 + 0.00751169i
\(281\) −25.6546 −1.53043 −0.765214 0.643776i \(-0.777367\pi\)
−0.765214 + 0.643776i \(0.777367\pi\)
\(282\) 0 0
\(283\) 14.8539i 0.882974i 0.897268 + 0.441487i \(0.145549\pi\)
−0.897268 + 0.441487i \(0.854451\pi\)
\(284\) −6.92931 + 5.03444i −0.411179 + 0.298739i
\(285\) 0 0
\(286\) −4.23607 13.0373i −0.250484 0.770910i
\(287\) −6.36183 −0.375527
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) −0.244610 0.752834i −0.0143640 0.0442079i
\(291\) 0 0
\(292\) −13.5604 + 9.85224i −0.793565 + 0.576559i
\(293\) 21.6946i 1.26741i 0.773574 + 0.633706i \(0.218467\pi\)
−0.773574 + 0.633706i \(0.781533\pi\)
\(294\) 0 0
\(295\) −0.380813 −0.0221718
\(296\) 1.28770 1.77237i 0.0748461 0.103017i
\(297\) 0 0
\(298\) −21.1387 + 6.86838i −1.22453 + 0.397875i
\(299\) 10.5513i 0.610195i
\(300\) 0 0
\(301\) 9.98927i 0.575772i
\(302\) 1.12806 + 3.47182i 0.0649128 + 0.199781i
\(303\) 0 0
\(304\) 11.7082 + 3.80423i 0.671512 + 0.218187i
\(305\) 0.458733 0.0262670
\(306\) 0 0
\(307\) 0.422903i 0.0241363i −0.999927 0.0120682i \(-0.996158\pi\)
0.999927 0.0120682i \(-0.00384151\pi\)
\(308\) −3.09770 4.26362i −0.176508 0.242942i
\(309\) 0 0
\(310\) −0.688898 + 0.223837i −0.0391268 + 0.0127131i
\(311\) −13.6732 −0.775338 −0.387669 0.921799i \(-0.626720\pi\)
−0.387669 + 0.921799i \(0.626720\pi\)
\(312\) 0 0
\(313\) −7.68665 −0.434475 −0.217237 0.976119i \(-0.569705\pi\)
−0.217237 + 0.976119i \(0.569705\pi\)
\(314\) 0.915571 0.297487i 0.0516687 0.0167882i
\(315\) 0 0
\(316\) 4.90566 3.56417i 0.275965 0.200500i
\(317\) 24.2068i 1.35959i 0.733401 + 0.679796i \(0.237932\pi\)
−0.733401 + 0.679796i \(0.762068\pi\)
\(318\) 0 0
\(319\) 26.8506 1.50335
\(320\) −0.417937 + 0.135796i −0.0233634 + 0.00759122i
\(321\) 0 0
\(322\) −1.25351 3.85790i −0.0698553 0.214993i
\(323\) 9.73249i 0.541530i
\(324\) 0 0
\(325\) 18.3816i 1.01963i
\(326\) 29.3665 9.54176i 1.62646 0.528469i
\(327\) 0 0
\(328\) −14.5574 10.5766i −0.803799 0.583994i
\(329\) 12.3126 0.678817
\(330\) 0 0
\(331\) 9.15756i 0.503345i 0.967812 + 0.251672i \(0.0809806\pi\)
−0.967812 + 0.251672i \(0.919019\pi\)
\(332\) −12.1384 16.7071i −0.666182 0.916921i
\(333\) 0 0
\(334\) −7.52556 23.1613i −0.411780 1.26733i
\(335\) 0.490851 0.0268180
\(336\) 0 0
\(337\) 33.2441 1.81092 0.905460 0.424432i \(-0.139526\pi\)
0.905460 + 0.424432i \(0.139526\pi\)
\(338\) −0.232315 0.714992i −0.0126363 0.0388904i
\(339\) 0 0
\(340\) −0.204203 0.281062i −0.0110745 0.0152427i
\(341\) 24.5703i 1.33056i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −16.6072 + 22.8579i −0.895402 + 1.23242i
\(345\) 0 0
\(346\) −3.45958 + 1.12408i −0.185988 + 0.0604312i
\(347\) 8.03418i 0.431297i −0.976471 0.215649i \(-0.930813\pi\)
0.976471 0.215649i \(-0.0691866\pi\)
\(348\) 0 0
\(349\) 5.46493i 0.292531i 0.989245 + 0.146265i \(0.0467254\pi\)
−0.989245 + 0.146265i \(0.953275\pi\)
\(350\) −2.18376 6.72093i −0.116727 0.359249i
\(351\) 0 0
\(352\) 14.9062i 0.794502i
\(353\) −7.85159 −0.417898 −0.208949 0.977927i \(-0.567004\pi\)
−0.208949 + 0.977927i \(0.567004\pi\)
\(354\) 0 0
\(355\) 0.235243i 0.0124854i
\(356\) −20.7360 + 15.0656i −1.09900 + 0.798473i
\(357\) 0 0
\(358\) 7.08414 2.30178i 0.374408 0.121653i
\(359\) −30.4616 −1.60770 −0.803851 0.594831i \(-0.797219\pi\)
−0.803851 + 0.594831i \(0.797219\pi\)
\(360\) 0 0
\(361\) 9.52786 0.501467
\(362\) −21.5454 + 7.00053i −1.13240 + 0.367940i
\(363\) 0 0
\(364\) −4.32437 5.95199i −0.226659 0.311969i
\(365\) 0.460363i 0.0240965i
\(366\) 0 0
\(367\) 8.97241 0.468356 0.234178 0.972194i \(-0.424760\pi\)
0.234178 + 0.972194i \(0.424760\pi\)
\(368\) 3.54546 10.9118i 0.184820 0.568817i
\(369\) 0 0
\(370\) −0.0185936 0.0572251i −0.000966634 0.00297499i
\(371\) 3.39291i 0.176151i
\(372\) 0 0
\(373\) 19.6614i 1.01803i 0.860759 + 0.509013i \(0.169990\pi\)
−0.860759 + 0.509013i \(0.830010\pi\)
\(374\) 11.2076 3.64157i 0.579531 0.188301i
\(375\) 0 0
\(376\) 28.1743 + 20.4698i 1.45298 + 1.05565i
\(377\) 37.4833 1.93049
\(378\) 0 0
\(379\) 24.9871i 1.28350i 0.766914 + 0.641750i \(0.221791\pi\)
−0.766914 + 0.641750i \(0.778209\pi\)
\(380\) 0.273543 0.198741i 0.0140325 0.0101952i
\(381\) 0 0
\(382\) −0.802931 2.47117i −0.0410815 0.126436i
\(383\) −20.3323 −1.03893 −0.519465 0.854492i \(-0.673869\pi\)
−0.519465 + 0.854492i \(0.673869\pi\)
\(384\) 0 0
\(385\) −0.144746 −0.00737691
\(386\) 6.70510 + 20.6362i 0.341281 + 1.05035i
\(387\) 0 0
\(388\) 16.4721 11.9677i 0.836246 0.607568i
\(389\) 2.92753i 0.148432i 0.997242 + 0.0742159i \(0.0236454\pi\)
−0.997242 + 0.0742159i \(0.976355\pi\)
\(390\) 0 0
\(391\) 9.07048 0.458714
\(392\) −2.28825 1.66251i −0.115574 0.0839693i
\(393\) 0 0
\(394\) 6.75790 2.19577i 0.340458 0.110621i
\(395\) 0.166542i 0.00837964i
\(396\) 0 0
\(397\) 8.20287i 0.411690i 0.978585 + 0.205845i \(0.0659943\pi\)
−0.978585 + 0.205845i \(0.934006\pi\)
\(398\) −9.62890 29.6347i −0.482653 1.48545i
\(399\) 0 0
\(400\) 6.17661 19.0097i 0.308831 0.950483i
\(401\) 36.5590 1.82567 0.912834 0.408332i \(-0.133889\pi\)
0.912834 + 0.408332i \(0.133889\pi\)
\(402\) 0 0
\(403\) 34.3000i 1.70860i
\(404\) −1.20788 1.66251i −0.0600944 0.0827129i
\(405\) 0 0
\(406\) 13.7052 4.45309i 0.680177 0.221003i
\(407\) 2.04100 0.101169
\(408\) 0 0
\(409\) −6.01783 −0.297562 −0.148781 0.988870i \(-0.547535\pi\)
−0.148781 + 0.988870i \(0.547535\pi\)
\(410\) −0.470021 + 0.152719i −0.0232127 + 0.00754226i
\(411\) 0 0
\(412\) −2.79766 + 2.03262i −0.137831 + 0.100140i
\(413\) 6.93263i 0.341133i
\(414\) 0 0
\(415\) −0.567188 −0.0278422
\(416\) 20.8089i 1.02024i
\(417\) 0 0
\(418\) 3.54415 + 10.9078i 0.173350 + 0.533517i
\(419\) 2.25247i 0.110040i 0.998485 + 0.0550201i \(0.0175223\pi\)
−0.998485 + 0.0550201i \(0.982478\pi\)
\(420\) 0 0
\(421\) 2.87431i 0.140085i −0.997544 0.0700425i \(-0.977686\pi\)
0.997544 0.0700425i \(-0.0223135\pi\)
\(422\) −11.4412 + 3.71748i −0.556950 + 0.180964i
\(423\) 0 0
\(424\) −5.64074 + 7.76382i −0.273939 + 0.377044i
\(425\) 15.8018 0.766502
\(426\) 0 0
\(427\) 8.35114i 0.404140i
\(428\) −7.19432 9.90213i −0.347751 0.478638i
\(429\) 0 0
\(430\) 0.239798 + 0.738021i 0.0115641 + 0.0355905i
\(431\) −15.7211 −0.757261 −0.378630 0.925548i \(-0.623605\pi\)
−0.378630 + 0.925548i \(0.623605\pi\)
\(432\) 0 0
\(433\) −3.44027 −0.165329 −0.0826644 0.996577i \(-0.526343\pi\)
−0.0826644 + 0.996577i \(0.526343\pi\)
\(434\) −4.07490 12.5413i −0.195601 0.601999i
\(435\) 0 0
\(436\) 10.4108 + 14.3293i 0.498588 + 0.686247i
\(437\) 8.82783i 0.422292i
\(438\) 0 0
\(439\) −12.2955 −0.586833 −0.293417 0.955985i \(-0.594792\pi\)
−0.293417 + 0.955985i \(0.594792\pi\)
\(440\) −0.331213 0.240641i −0.0157900 0.0114721i
\(441\) 0 0
\(442\) 15.6457 5.08361i 0.744191 0.241802i
\(443\) 15.8932i 0.755107i 0.925988 + 0.377553i \(0.123235\pi\)
−0.925988 + 0.377553i \(0.876765\pi\)
\(444\) 0 0
\(445\) 0.703964i 0.0333711i
\(446\) −5.92481 18.2347i −0.280548 0.863438i
\(447\) 0 0
\(448\) −2.47214 7.60845i −0.116797 0.359466i
\(449\) 8.97450 0.423533 0.211766 0.977320i \(-0.432078\pi\)
0.211766 + 0.977320i \(0.432078\pi\)
\(450\) 0 0
\(451\) 16.7638i 0.789377i
\(452\) −14.6009 + 10.6082i −0.686767 + 0.498965i
\(453\) 0 0
\(454\) 21.6169 7.02375i 1.01453 0.329641i
\(455\) −0.202064 −0.00947290
\(456\) 0 0
\(457\) −15.0037 −0.701845 −0.350922 0.936405i \(-0.614132\pi\)
−0.350922 + 0.936405i \(0.614132\pi\)
\(458\) 7.40919 2.40739i 0.346209 0.112490i
\(459\) 0 0
\(460\) −0.185222 0.254936i −0.00863602 0.0118865i
\(461\) 8.69884i 0.405145i −0.979267 0.202573i \(-0.935070\pi\)
0.979267 0.202573i \(-0.0649303\pi\)
\(462\) 0 0
\(463\) −9.34522 −0.434309 −0.217155 0.976137i \(-0.569678\pi\)
−0.217155 + 0.976137i \(0.569678\pi\)
\(464\) 38.7641 + 12.5952i 1.79958 + 0.584719i
\(465\) 0 0
\(466\) 12.5248 + 38.5475i 0.580202 + 1.78568i
\(467\) 9.23326i 0.427265i 0.976914 + 0.213632i \(0.0685294\pi\)
−0.976914 + 0.213632i \(0.931471\pi\)
\(468\) 0 0
\(469\) 8.93584i 0.412619i
\(470\) 0.909674 0.295571i 0.0419601 0.0136337i
\(471\) 0 0
\(472\) 11.5256 15.8636i 0.530507 0.730180i
\(473\) −26.3223 −1.21030
\(474\) 0 0
\(475\) 15.3791i 0.705643i
\(476\) 5.11667 3.71748i 0.234522 0.170390i
\(477\) 0 0
\(478\) 1.67376 + 5.15131i 0.0765561 + 0.235615i
\(479\) 8.66050 0.395708 0.197854 0.980231i \(-0.436603\pi\)
0.197854 + 0.980231i \(0.436603\pi\)
\(480\) 0 0
\(481\) 2.84922 0.129913
\(482\) −0.755311 2.32461i −0.0344035 0.105883i
\(483\) 0 0
\(484\) −6.56346 + 4.76863i −0.298339 + 0.216756i
\(485\) 0.559212i 0.0253925i
\(486\) 0 0
\(487\) 11.1424 0.504912 0.252456 0.967608i \(-0.418762\pi\)
0.252456 + 0.967608i \(0.418762\pi\)
\(488\) −13.8838 + 19.1095i −0.628492 + 0.865045i
\(489\) 0 0
\(490\) −0.0738814 + 0.0240055i −0.00333762 + 0.00108446i
\(491\) 25.4479i 1.14845i −0.818698 0.574224i \(-0.805304\pi\)
0.818698 0.574224i \(-0.194696\pi\)
\(492\) 0 0
\(493\) 32.2228i 1.45124i
\(494\) 4.94761 + 15.2272i 0.222604 + 0.685104i
\(495\) 0 0
\(496\) 11.5256 35.4720i 0.517513 1.59274i
\(497\) −4.28255 −0.192099
\(498\) 0 0
\(499\) 4.17422i 0.186864i −0.995626 0.0934318i \(-0.970216\pi\)
0.995626 0.0934318i \(-0.0297837\pi\)
\(500\) −0.645553 0.888527i −0.0288700 0.0397361i
\(501\) 0 0
\(502\) −0.330408 + 0.107356i −0.0147468 + 0.00479153i
\(503\) −10.6106 −0.473105 −0.236552 0.971619i \(-0.576017\pi\)
−0.236552 + 0.971619i \(0.576017\pi\)
\(504\) 0 0
\(505\) −0.0564404 −0.00251156
\(506\) 10.1658 3.30308i 0.451926 0.146840i
\(507\) 0 0
\(508\) −29.5544 + 21.4725i −1.31126 + 0.952690i
\(509\) 24.1111i 1.06871i 0.845262 + 0.534353i \(0.179445\pi\)
−0.845262 + 0.534353i \(0.820555\pi\)
\(510\) 0 0
\(511\) −8.38081 −0.370745
\(512\) 6.99226 21.5200i 0.309017 0.951057i
\(513\) 0 0
\(514\) −7.08490 21.8051i −0.312502 0.961781i
\(515\) 0.0949777i 0.00418522i
\(516\) 0 0
\(517\) 32.4445i 1.42691i
\(518\) 1.04177 0.338492i 0.0457728 0.0148725i
\(519\) 0 0
\(520\) −0.462372 0.335933i −0.0202763 0.0147316i
\(521\) −32.5642 −1.42666 −0.713331 0.700827i \(-0.752815\pi\)
−0.713331 + 0.700827i \(0.752815\pi\)
\(522\) 0 0
\(523\) 18.6166i 0.814046i −0.913418 0.407023i \(-0.866567\pi\)
0.913418 0.407023i \(-0.133433\pi\)
\(524\) 20.5101 + 28.2298i 0.895989 + 1.23322i
\(525\) 0 0
\(526\) −10.1941 31.3741i −0.444482 1.36797i
\(527\) 29.4863 1.28444
\(528\) 0 0
\(529\) −14.7726 −0.642289
\(530\) 0.0814487 + 0.250673i 0.00353791 + 0.0108886i
\(531\) 0 0
\(532\) 3.61803 + 4.97980i 0.156862 + 0.215902i
\(533\) 23.4022i 1.01366i
\(534\) 0 0
\(535\) −0.336167 −0.0145338
\(536\) −14.8559 + 20.4474i −0.641677 + 0.883193i
\(537\) 0 0
\(538\) 31.8692 10.3549i 1.37398 0.446433i
\(539\) 2.63506i 0.113500i
\(540\) 0 0
\(541\) 41.3556i 1.77802i −0.457891 0.889008i \(-0.651395\pi\)
0.457891 0.889008i \(-0.348605\pi\)
\(542\) 1.54173 + 4.74497i 0.0662231 + 0.203814i
\(543\) 0 0
\(544\) 17.8885 0.766965
\(545\) 0.486463 0.0208378
\(546\) 0 0
\(547\) 18.8135i 0.804408i −0.915550 0.402204i \(-0.868244\pi\)
0.915550 0.402204i \(-0.131756\pi\)
\(548\) −12.5173 + 9.09437i −0.534714 + 0.388492i
\(549\) 0 0
\(550\) 17.7101 5.75435i 0.755160 0.245366i
\(551\) −31.3608 −1.33602
\(552\) 0 0
\(553\) 3.03186 0.128928
\(554\) 38.4689 12.4993i 1.63439 0.531044i
\(555\) 0 0
\(556\) −0.942847 1.29772i −0.0399856 0.0550355i
\(557\) 4.05808i 0.171946i −0.996297 0.0859731i \(-0.972600\pi\)
0.996297 0.0859731i \(-0.0273999\pi\)
\(558\) 0 0
\(559\) −36.7458 −1.55418
\(560\) −0.208968 0.0678979i −0.00883052 0.00286921i
\(561\) 0 0
\(562\) −11.2115 34.5054i −0.472928 1.45552i
\(563\) 45.2454i 1.90687i −0.301605 0.953433i \(-0.597522\pi\)
0.301605 0.953433i \(-0.402478\pi\)
\(564\) 0 0
\(565\) 0.495684i 0.0208536i
\(566\) −19.9785 + 6.49140i −0.839758 + 0.272854i
\(567\) 0 0
\(568\) −9.79953 7.11977i −0.411179 0.298739i
\(569\) 13.6255 0.571212 0.285606 0.958347i \(-0.407805\pi\)
0.285606 + 0.958347i \(0.407805\pi\)
\(570\) 0 0
\(571\) 30.9574i 1.29553i 0.761842 + 0.647763i \(0.224295\pi\)
−0.761842 + 0.647763i \(0.775705\pi\)
\(572\) 15.6839 11.3950i 0.655775 0.476449i
\(573\) 0 0
\(574\) −2.78022 8.55664i −0.116044 0.357147i
\(575\) 14.3330 0.597729
\(576\) 0 0
\(577\) −38.4675 −1.60142 −0.800712 0.599049i \(-0.795545\pi\)
−0.800712 + 0.599049i \(0.795545\pi\)
\(578\) −3.05911 9.41498i −0.127242 0.391612i
\(579\) 0 0
\(580\) 0.905660 0.658001i 0.0376055 0.0273220i
\(581\) 10.3255i 0.428376i
\(582\) 0 0
\(583\) −8.94054 −0.370279
\(584\) −19.1774 13.9332i −0.793565 0.576559i
\(585\) 0 0
\(586\) −29.1792 + 9.48089i −1.20538 + 0.391652i
\(587\) 28.6131i 1.18099i −0.807041 0.590495i \(-0.798932\pi\)
0.807041 0.590495i \(-0.201068\pi\)
\(588\) 0 0
\(589\) 28.6975i 1.18246i
\(590\) −0.166422 0.512193i −0.00685147 0.0210867i
\(591\) 0 0
\(592\) 2.94658 + 0.957401i 0.121104 + 0.0393489i
\(593\) 22.5554 0.926239 0.463119 0.886296i \(-0.346730\pi\)
0.463119 + 0.886296i \(0.346730\pi\)
\(594\) 0 0
\(595\) 0.173706i 0.00712124i
\(596\) −18.4759 25.4299i −0.756803 1.04165i
\(597\) 0 0
\(598\) 14.1914 4.61107i 0.580330 0.188561i
\(599\) 42.1743 1.72319 0.861597 0.507593i \(-0.169465\pi\)
0.861597 + 0.507593i \(0.169465\pi\)
\(600\) 0 0
\(601\) −14.0155 −0.571705 −0.285853 0.958274i \(-0.592277\pi\)
−0.285853 + 0.958274i \(0.592277\pi\)
\(602\) −13.4355 + 4.36547i −0.547591 + 0.177923i
\(603\) 0 0
\(604\) −4.17661 + 3.03448i −0.169944 + 0.123471i
\(605\) 0.222823i 0.00905902i
\(606\) 0 0
\(607\) −21.5893 −0.876282 −0.438141 0.898906i \(-0.644363\pi\)
−0.438141 + 0.898906i \(0.644363\pi\)
\(608\) 17.4100i 0.706069i
\(609\) 0 0
\(610\) 0.200474 + 0.616994i 0.00811694 + 0.0249814i
\(611\) 45.2924i 1.83233i
\(612\) 0 0
\(613\) 0.788508i 0.0318475i −0.999873 0.0159238i \(-0.994931\pi\)
0.999873 0.0159238i \(-0.00506891\pi\)
\(614\) 0.568803 0.184815i 0.0229550 0.00745854i
\(615\) 0 0
\(616\) 4.38081 6.02967i 0.176508 0.242942i
\(617\) −34.4936 −1.38866 −0.694331 0.719656i \(-0.744299\pi\)
−0.694331 + 0.719656i \(0.744299\pi\)
\(618\) 0 0
\(619\) 47.8005i 1.92127i 0.277822 + 0.960633i \(0.410388\pi\)
−0.277822 + 0.960633i \(0.589612\pi\)
\(620\) −0.602119 0.828746i −0.0241817 0.0332832i
\(621\) 0 0
\(622\) −5.97542 18.3905i −0.239593 0.737390i
\(623\) −12.8155 −0.513443
\(624\) 0 0
\(625\) 24.9547 0.998190
\(626\) −3.35919 10.3385i −0.134260 0.413210i
\(627\) 0 0
\(628\) 0.800239 + 1.10143i 0.0319330 + 0.0439520i
\(629\) 2.44935i 0.0976621i
\(630\) 0 0
\(631\) −21.7200 −0.864659 −0.432330 0.901716i \(-0.642308\pi\)
−0.432330 + 0.901716i \(0.642308\pi\)
\(632\) 6.93765 + 5.04050i 0.275965 + 0.200500i
\(633\) 0 0
\(634\) −32.5581 + 10.5788i −1.29305 + 0.420137i
\(635\) 1.00334i 0.0398164i
\(636\) 0 0
\(637\) 3.67853i 0.145749i
\(638\) 11.7342 + 36.1140i 0.464560 + 1.42977i
\(639\) 0 0
\(640\) −0.365290 0.502778i −0.0144394 0.0198741i
\(641\) 17.0166 0.672117 0.336058 0.941841i \(-0.390906\pi\)
0.336058 + 0.941841i \(0.390906\pi\)
\(642\) 0 0
\(643\) 38.4170i 1.51502i 0.652824 + 0.757509i \(0.273584\pi\)
−0.652824 + 0.757509i \(0.726416\pi\)
\(644\) 4.64107 3.37193i 0.182884 0.132873i
\(645\) 0 0
\(646\) −13.0902 + 4.25325i −0.515026 + 0.167342i
\(647\) 22.7497 0.894382 0.447191 0.894439i \(-0.352425\pi\)
0.447191 + 0.894439i \(0.352425\pi\)
\(648\) 0 0
\(649\) 18.2679 0.717079
\(650\) 24.7231 8.03304i 0.969721 0.315082i
\(651\) 0 0
\(652\) 25.6673 + 35.3280i 1.00521 + 1.38355i
\(653\) 12.9198i 0.505591i 0.967520 + 0.252795i \(0.0813499\pi\)
−0.967520 + 0.252795i \(0.918650\pi\)
\(654\) 0 0
\(655\) 0.958371 0.0374466
\(656\) 7.86365 24.2018i 0.307024 0.944923i
\(657\) 0 0
\(658\) 5.38081 + 16.5604i 0.209766 + 0.645593i
\(659\) 47.1905i 1.83828i 0.393932 + 0.919140i \(0.371115\pi\)
−0.393932 + 0.919140i \(0.628885\pi\)
\(660\) 0 0
\(661\) 31.5648i 1.22773i 0.789412 + 0.613864i \(0.210386\pi\)
−0.789412 + 0.613864i \(0.789614\pi\)
\(662\) −12.3169 + 4.00200i −0.478710 + 0.155542i
\(663\) 0 0
\(664\) 17.1663 23.6274i 0.666182 0.916921i
\(665\) 0.169059 0.00655582
\(666\) 0 0
\(667\) 29.2276i 1.13170i
\(668\) 27.8631 20.2437i 1.07805 0.783253i
\(669\) 0 0
\(670\) 0.214510 + 0.660193i 0.00828723 + 0.0255055i
\(671\) −22.0058 −0.849524
\(672\) 0 0
\(673\) −37.1186 −1.43082 −0.715408 0.698707i \(-0.753759\pi\)
−0.715408 + 0.698707i \(0.753759\pi\)
\(674\) 14.5282 + 44.7132i 0.559605 + 1.72229i
\(675\) 0 0
\(676\) 0.860137 0.624926i 0.0330822 0.0240356i
\(677\) 15.8912i 0.610750i 0.952232 + 0.305375i \(0.0987818\pi\)
−0.952232 + 0.305375i \(0.901218\pi\)
\(678\) 0 0
\(679\) 10.1803 0.390686
\(680\) 0.288787 0.397481i 0.0110745 0.0152427i
\(681\) 0 0
\(682\) 33.0470 10.7376i 1.26544 0.411165i
\(683\) 27.4319i 1.04965i 0.851210 + 0.524826i \(0.175869\pi\)
−0.851210 + 0.524826i \(0.824131\pi\)
\(684\) 0 0
\(685\) 0.424950i 0.0162365i
\(686\) −0.437016 1.34500i −0.0166853 0.0513522i
\(687\) 0 0
\(688\) −38.0014 12.3474i −1.44879 0.470741i
\(689\) −12.4809 −0.475486
\(690\) 0 0
\(691\) 6.57770i 0.250227i 0.992142 + 0.125114i \(0.0399296\pi\)
−0.992142 + 0.125114i \(0.960070\pi\)
\(692\) −3.02378 4.16188i −0.114947 0.158211i
\(693\) 0 0
\(694\) 10.8059 3.51107i 0.410188 0.133278i
\(695\) −0.0440562 −0.00167115
\(696\) 0 0
\(697\) 20.1179 0.762018
\(698\) −7.35031 + 2.38826i −0.278213 + 0.0903970i
\(699\) 0 0
\(700\) 8.08529 5.87431i 0.305595 0.222028i
\(701\) 9.07153i 0.342627i −0.985217 0.171314i \(-0.945199\pi\)
0.985217 0.171314i \(-0.0548011\pi\)
\(702\) 0 0
\(703\) −2.38383 −0.0899079
\(704\) 20.0488 6.51424i 0.755616 0.245514i
\(705\) 0 0
\(706\) −3.43127 10.5604i −0.129138 0.397445i
\(707\) 1.02749i 0.0386426i
\(708\) 0 0
\(709\) 6.16889i 0.231678i 0.993268 + 0.115839i \(0.0369556\pi\)
−0.993268 + 0.115839i \(0.963044\pi\)
\(710\) −0.316401 + 0.102805i −0.0118743 + 0.00385820i
\(711\) 0 0
\(712\) −29.3251 21.3059i −1.09900 0.798473i
\(713\) 26.7454 1.00162
\(714\) 0 0
\(715\) 0.532451i 0.0199125i
\(716\) 6.19176 + 8.52223i 0.231397 + 0.318491i
\(717\) 0 0
\(718\) −13.3122 40.9707i −0.496807 1.52901i
\(719\) −30.5440 −1.13910 −0.569550 0.821956i \(-0.692883\pi\)
−0.569550 + 0.821956i \(0.692883\pi\)
\(720\) 0 0
\(721\) −1.72905 −0.0643932
\(722\) 4.16383 + 12.8149i 0.154962 + 0.476923i
\(723\) 0 0
\(724\) −18.8314 25.9192i −0.699863 0.963279i
\(725\) 50.9180i 1.89105i
\(726\) 0 0
\(727\) −21.3429 −0.791565 −0.395782 0.918344i \(-0.629527\pi\)
−0.395782 + 0.918344i \(0.629527\pi\)
\(728\) 6.11559 8.41738i 0.226659 0.311969i
\(729\) 0 0
\(730\) −0.619187 + 0.201186i −0.0229171 + 0.00744622i
\(731\) 31.5888i 1.16836i
\(732\) 0 0
\(733\) 15.1235i 0.558599i −0.960204 0.279300i \(-0.909898\pi\)
0.960204 0.279300i \(-0.0901023\pi\)
\(734\) 3.92109 + 12.0679i 0.144730 + 0.445433i
\(735\) 0 0
\(736\) 16.2258 0.598090
\(737\) −23.5465 −0.867347
\(738\) 0 0
\(739\) 38.7212i 1.42438i −0.701985 0.712192i \(-0.747703\pi\)
0.701985 0.712192i \(-0.252297\pi\)
\(740\) 0.0688419 0.0500166i 0.00253068 0.00183865i
\(741\) 0 0
\(742\) −4.56346 + 1.48276i −0.167530 + 0.0544337i
\(743\) 39.7258 1.45740 0.728698 0.684835i \(-0.240126\pi\)
0.728698 + 0.684835i \(0.240126\pi\)
\(744\) 0 0
\(745\) −0.863319 −0.0316296
\(746\) −26.4445 + 8.59233i −0.968201 + 0.314588i
\(747\) 0 0
\(748\) 9.79580 + 13.4828i 0.358170 + 0.492978i
\(749\) 6.11985i 0.223615i
\(750\) 0 0
\(751\) −2.96814 −0.108309 −0.0541544 0.998533i \(-0.517246\pi\)
−0.0541544 + 0.998533i \(0.517246\pi\)
\(752\) −15.2192 + 46.8400i −0.554989 + 1.70808i
\(753\) 0 0
\(754\) 16.3808 + 50.4150i 0.596554 + 1.83600i
\(755\) 0.141792i 0.00516032i
\(756\) 0 0
\(757\) 47.1588i 1.71401i 0.515305 + 0.857007i \(0.327679\pi\)
−0.515305 + 0.857007i \(0.672321\pi\)
\(758\) −33.6075 + 10.9198i −1.22068 + 0.396623i
\(759\) 0 0
\(760\) 0.386848 + 0.281062i 0.0140325 + 0.0101952i
\(761\) 40.8000 1.47900 0.739499 0.673158i \(-0.235063\pi\)
0.739499 + 0.673158i \(0.235063\pi\)
\(762\) 0 0
\(763\) 8.85597i 0.320608i
\(764\) 2.97282 2.15988i 0.107553 0.0781417i
\(765\) 0 0
\(766\) −8.88553 27.3468i −0.321047 0.988081i
\(767\) 25.5019 0.920821
\(768\) 0 0
\(769\) −52.9121 −1.90806 −0.954029 0.299714i \(-0.903109\pi\)
−0.954029 + 0.299714i \(0.903109\pi\)
\(770\) −0.0632561 0.194682i −0.00227959 0.00701586i
\(771\) 0 0
\(772\) −24.8254 + 18.0367i −0.893484 + 0.649154i
\(773\) 39.0168i 1.40334i −0.712503 0.701669i \(-0.752439\pi\)
0.712503 0.701669i \(-0.247561\pi\)
\(774\) 0 0
\(775\) 46.5937 1.67370
\(776\) 23.2951 + 16.9249i 0.836246 + 0.607568i
\(777\) 0 0
\(778\) −3.93752 + 1.27938i −0.141167 + 0.0458680i
\(779\) 19.5797i 0.701515i
\(780\) 0 0
\(781\) 11.2848i 0.403802i
\(782\) 3.96394 + 12.1998i 0.141750 + 0.436263i
\(783\) 0 0
\(784\) 1.23607 3.80423i 0.0441453 0.135865i
\(785\) 0.0373925 0.00133460
\(786\) 0 0
\(787\) 1.01675i 0.0362431i 0.999836 + 0.0181215i \(0.00576858\pi\)
−0.999836 + 0.0181215i \(0.994231\pi\)
\(788\) 5.90662 + 8.12976i 0.210415 + 0.289611i
\(789\) 0 0
\(790\) 0.223999 0.0727815i 0.00796951 0.00258945i
\(791\) −9.02383 −0.320851
\(792\) 0 0
\(793\) −30.7199 −1.09090
\(794\) −11.0328 + 3.58478i −0.391541 + 0.127219i
\(795\) 0 0
\(796\) 35.6506 25.9017i 1.26360 0.918061i
\(797\) 50.9667i 1.80533i 0.430339 + 0.902667i \(0.358394\pi\)
−0.430339 + 0.902667i \(0.641606\pi\)
\(798\) 0 0
\(799\) −38.9359 −1.37745
\(800\) 28.2672 0.999397
\(801\) 0 0
\(802\) 15.9768 + 49.1717i 0.564162 + 1.73631i
\(803\) 22.0840i 0.779327i
\(804\) 0 0
\(805\) 0.157559i 0.00555324i
\(806\) 46.1334 14.9896i 1.62498 0.527988i
\(807\) 0 0
\(808\) 1.70820 2.35114i 0.0600944 0.0827129i
\(809\) −20.3800 −0.716522 −0.358261 0.933622i \(-0.616630\pi\)
−0.358261 + 0.933622i \(0.616630\pi\)
\(810\) 0 0
\(811\) 40.7638i 1.43141i 0.698402 + 0.715706i \(0.253895\pi\)
−0.698402 + 0.715706i \(0.746105\pi\)
\(812\) 11.9788 + 16.4874i 0.420372 + 0.578593i
\(813\) 0 0
\(814\) 0.891949 + 2.74514i 0.0312628 + 0.0962170i
\(815\) 1.19935 0.0420113
\(816\) 0 0
\(817\) 30.7438 1.07559
\(818\) −2.62989 8.09396i −0.0919519 0.282999i
\(819\) 0 0
\(820\) −0.410813 0.565436i −0.0143462 0.0197459i
\(821\) 35.8394i 1.25080i 0.780303 + 0.625402i \(0.215065\pi\)
−0.780303 + 0.625402i \(0.784935\pi\)
\(822\) 0 0
\(823\) 34.8745 1.21565 0.607824 0.794071i \(-0.292042\pi\)
0.607824 + 0.794071i \(0.292042\pi\)
\(824\) −3.95649 2.87456i −0.137831 0.100140i
\(825\) 0 0
\(826\) 9.32437 3.02967i 0.324436 0.105416i
\(827\) 42.4727i 1.47692i 0.674296 + 0.738461i \(0.264447\pi\)
−0.674296 + 0.738461i \(0.735553\pi\)
\(828\) 0 0
\(829\) 33.1210i 1.15034i 0.818034 + 0.575169i \(0.195064\pi\)
−0.818034 + 0.575169i \(0.804936\pi\)
\(830\) −0.247870 0.762866i −0.00860370 0.0264795i
\(831\) 0 0
\(832\) 27.9879 9.09383i 0.970307 0.315272i
\(833\) 3.16228 0.109566
\(834\) 0 0
\(835\) 0.945922i 0.0327350i
\(836\) −13.1221 + 9.53375i −0.453837 + 0.329732i
\(837\) 0 0
\(838\) −3.02956 + 0.984364i −0.104654 + 0.0340043i
\(839\) 43.5817 1.50461 0.752304 0.658817i \(-0.228943\pi\)
0.752304 + 0.658817i \(0.228943\pi\)
\(840\) 0 0
\(841\) −74.8311 −2.58038
\(842\) 3.86593 1.25612i 0.133229 0.0432887i
\(843\) 0 0
\(844\) −10.0000 13.7638i −0.344214 0.473770i
\(845\) 0.0292007i 0.00100454i
\(846\) 0 0
\(847\) −4.05644 −0.139381
\(848\) −12.9074 4.19387i −0.443242 0.144018i
\(849\) 0 0
\(850\) 6.90566 + 21.2534i 0.236862 + 0.728987i
\(851\) 2.22168i 0.0761582i
\(852\) 0 0
\(853\) 36.8928i 1.26319i −0.775300 0.631593i \(-0.782401\pi\)
0.775300 0.631593i \(-0.217599\pi\)
\(854\) −11.2323 + 3.64958i −0.384360 + 0.124886i
\(855\) 0 0
\(856\) 10.1743 14.0037i 0.347751 0.478638i
\(857\) −19.2607 −0.657933 −0.328967 0.944342i \(-0.606700\pi\)
−0.328967 + 0.944342i \(0.606700\pi\)
\(858\) 0 0
\(859\) 19.4667i 0.664195i −0.943245 0.332098i \(-0.892244\pi\)
0.943245 0.332098i \(-0.107756\pi\)
\(860\) −0.887841 + 0.645054i −0.0302751 + 0.0219962i
\(861\) 0 0
\(862\) −6.87039 21.1449i −0.234006 0.720198i
\(863\) 21.4715 0.730898 0.365449 0.930831i \(-0.380915\pi\)
0.365449 + 0.930831i \(0.380915\pi\)
\(864\) 0 0
\(865\) −0.141291 −0.00480405
\(866\) −1.50345 4.62715i −0.0510894 0.157237i
\(867\) 0 0
\(868\) 15.0872 10.9615i 0.512091 0.372056i
\(869\) 7.98916i 0.271014i
\(870\) 0 0
\(871\) −32.8708 −1.11378
\(872\) −14.7231 + 20.2646i −0.498588 + 0.686247i
\(873\) 0 0
\(874\) −11.8734 + 3.85790i −0.401624 + 0.130496i
\(875\) 0.549140i 0.0185643i
\(876\) 0 0
\(877\) 23.8421i 0.805091i −0.915400 0.402545i \(-0.868126\pi\)
0.915400 0.402545i \(-0.131874\pi\)
\(878\) −5.37334 16.5374i −0.181342 0.558112i
\(879\) 0 0
\(880\) 0.178915 0.550645i 0.00603123 0.0185622i
\(881\) −8.81134 −0.296862 −0.148431 0.988923i \(-0.547422\pi\)
−0.148431 + 0.988923i \(0.547422\pi\)
\(882\) 0 0
\(883\) 30.0935i 1.01273i 0.862320 + 0.506363i \(0.169011\pi\)
−0.862320 + 0.506363i \(0.830989\pi\)
\(884\) 13.6749 + 18.8218i 0.459936 + 0.633047i
\(885\) 0 0
\(886\) −21.3763 + 6.94557i −0.718149 + 0.233341i
\(887\) 13.0549 0.438340 0.219170 0.975687i \(-0.429665\pi\)
0.219170 + 0.975687i \(0.429665\pi\)
\(888\) 0 0
\(889\) −18.2656 −0.612609
\(890\) −0.946830 + 0.307644i −0.0317378 + 0.0103122i
\(891\) 0 0
\(892\) 21.9364 15.9377i 0.734484 0.533634i
\(893\) 37.8944i 1.26809i
\(894\) 0 0
\(895\) 0.289321 0.00967092
\(896\) 9.15298 6.65003i 0.305780 0.222162i
\(897\) 0 0
\(898\) 3.92200 + 12.0707i 0.130879 + 0.402804i
\(899\) 95.0131i 3.16886i
\(900\) 0 0
\(901\) 10.7293i 0.357446i
\(902\) 22.5473 7.32606i 0.750743 0.243931i
\(903\) 0 0
\(904\) −20.6487 15.0022i −0.686767 0.498965i
\(905\) −0.879929 −0.0292498
\(906\) 0 0
\(907\) 35.8987i 1.19200i −0.802985 0.595999i \(-0.796756\pi\)
0.802985 0.595999i \(-0.203244\pi\)
\(908\) 18.8938 + 26.0051i 0.627014 + 0.863011i
\(909\) 0 0
\(910\) −0.0883051 0.271775i −0.00292729 0.00900926i
\(911\) 46.3926 1.53705 0.768527 0.639817i \(-0.220990\pi\)
0.768527 + 0.639817i \(0.220990\pi\)
\(912\) 0 0
\(913\) 27.2085 0.900469
\(914\) −6.55687 20.1800i −0.216882 0.667494i
\(915\) 0 0
\(916\) 6.47587 + 8.91326i 0.213969 + 0.294503i
\(917\) 17.4470i 0.576149i
\(918\) 0 0
\(919\) −9.87878 −0.325871 −0.162935 0.986637i \(-0.552096\pi\)
−0.162935 + 0.986637i \(0.552096\pi\)
\(920\) 0.261944 0.360534i 0.00863602 0.0118865i
\(921\) 0 0
\(922\) 11.6999 3.80153i 0.385316 0.125197i
\(923\) 15.7535i 0.518533i
\(924\) 0 0
\(925\) 3.87043i 0.127259i
\(926\) −4.08401 12.5693i −0.134209 0.413053i
\(927\) 0 0
\(928\) 57.6419i 1.89219i
\(929\) −3.21688 −0.105543 −0.0527713 0.998607i \(-0.516805\pi\)
−0.0527713 + 0.998607i \(0.516805\pi\)
\(930\) 0 0
\(931\) 3.07768i 0.100867i
\(932\) −46.3727 + 33.6918i −1.51899 + 1.10361i
\(933\) 0 0
\(934\) −12.4187 + 4.03508i −0.406353 + 0.132032i
\(935\) 0.457725 0.0149692
\(936\) 0 0
\(937\) 54.7874 1.78983 0.894913 0.446241i \(-0.147237\pi\)
0.894913 + 0.446241i \(0.147237\pi\)
\(938\) −12.0187 + 3.90511i −0.392424 + 0.127506i
\(939\) 0 0
\(940\) 0.795084 + 1.09434i 0.0259328 + 0.0356934i
\(941\) 8.31449i 0.271045i 0.990774 + 0.135522i \(0.0432713\pi\)
−0.990774 + 0.135522i \(0.956729\pi\)
\(942\) 0 0
\(943\) 18.2479 0.594232
\(944\) 26.3733 + 8.56921i 0.858378 + 0.278904i
\(945\) 0 0
\(946\) −11.5033 35.4035i −0.374004 1.15107i
\(947\) 12.2157i 0.396958i −0.980105 0.198479i \(-0.936400\pi\)
0.980105 0.198479i \(-0.0636001\pi\)
\(948\) 0 0
\(949\) 30.8291i 1.00075i
\(950\) −20.6849 + 6.72093i −0.671106 + 0.218056i
\(951\) 0 0
\(952\) 7.23607 + 5.25731i 0.234522 + 0.170390i
\(953\) −50.2955 −1.62923 −0.814615 0.580002i \(-0.803052\pi\)
−0.814615 + 0.580002i \(0.803052\pi\)
\(954\) 0 0
\(955\) 0.100924i 0.00326583i
\(956\) −6.19704 + 4.50241i −0.200426 + 0.145618i
\(957\) 0 0
\(958\) 3.78478 + 11.6483i 0.122281 + 0.376341i
\(959\) −7.73613 −0.249813
\(960\) 0 0
\(961\) 55.9439 1.80464
\(962\) 1.24515 + 3.83219i 0.0401454 + 0.123555i
\(963\) 0 0
\(964\) 2.79651 2.03178i 0.0900695 0.0654393i
\(965\) 0.842795i 0.0271305i
\(966\) 0 0
\(967\) 6.17661 0.198626 0.0993132 0.995056i \(-0.468335\pi\)
0.0993132 + 0.995056i \(0.468335\pi\)
\(968\) −9.28213 6.74386i −0.298339 0.216756i
\(969\) 0 0
\(970\) 0.752138 0.244384i 0.0241497 0.00784671i
\(971\) 3.74342i 0.120132i 0.998194 + 0.0600660i \(0.0191311\pi\)
−0.998194 + 0.0600660i \(0.980869\pi\)
\(972\) 0 0
\(973\) 0.802034i 0.0257120i
\(974\) 4.86943 + 14.9865i 0.156026 + 0.480200i
\(975\) 0 0
\(976\) −31.7696 10.3226i −1.01692 0.330418i
\(977\) −43.1012 −1.37893 −0.689464 0.724320i \(-0.742154\pi\)
−0.689464 + 0.724320i \(0.742154\pi\)
\(978\) 0 0
\(979\) 33.7697i 1.07929i
\(980\) −0.0645747 0.0888795i −0.00206276 0.00283915i
\(981\) 0 0
\(982\) 34.2273 11.1211i 1.09224 0.354890i
\(983\) 46.8079 1.49294 0.746470 0.665419i \(-0.231747\pi\)
0.746470 + 0.665419i \(0.231747\pi\)
\(984\) 0 0
\(985\) 0.275997 0.00879399
\(986\) −43.3396 + 14.0819i −1.38021 + 0.448459i
\(987\) 0 0
\(988\) −18.3183 + 13.3091i −0.582784 + 0.423417i
\(989\) 28.6526i 0.911099i
\(990\) 0 0
\(991\) −44.6923 −1.41970 −0.709850 0.704353i \(-0.751237\pi\)
−0.709850 + 0.704353i \(0.751237\pi\)
\(992\) 52.7466 1.67471
\(993\) 0 0
\(994\) −1.87154 5.76002i −0.0593617 0.182697i
\(995\) 1.21030i 0.0383691i
\(996\) 0 0
\(997\) 41.1361i 1.30279i 0.758737 + 0.651397i \(0.225817\pi\)
−0.758737 + 0.651397i \(0.774183\pi\)
\(998\) 5.61431 1.82420i 0.177718 0.0577440i
\(999\) 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 1512.2.c.d.757.11 yes 16
3.2 odd 2 inner 1512.2.c.d.757.6 16
4.3 odd 2 6048.2.c.d.3025.9 16
8.3 odd 2 6048.2.c.d.3025.8 16
8.5 even 2 inner 1512.2.c.d.757.10 yes 16
12.11 even 2 6048.2.c.d.3025.7 16
24.5 odd 2 inner 1512.2.c.d.757.7 yes 16
24.11 even 2 6048.2.c.d.3025.10 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.d.757.6 16 3.2 odd 2 inner
1512.2.c.d.757.7 yes 16 24.5 odd 2 inner
1512.2.c.d.757.10 yes 16 8.5 even 2 inner
1512.2.c.d.757.11 yes 16 1.1 even 1 trivial
6048.2.c.d.3025.7 16 12.11 even 2
6048.2.c.d.3025.8 16 8.3 odd 2
6048.2.c.d.3025.9 16 4.3 odd 2
6048.2.c.d.3025.10 16 24.11 even 2