Properties

Label 124.4.e.b.25.1
Level $124$
Weight $4$
Character 124.25
Analytic conductor $7.316$
Analytic rank $0$
Dimension $6$
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] = [124,4,Mod(5,124)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(124, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("124.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 124 = 2^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 124.e (of order \(3\), 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: \(7.31623684071\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.250722553392.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 46x^{4} - 24x^{3} + 2116x^{2} - 552x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 25.1
Root \(-3.45459 + 5.98353i\) of defining polynomial
Character \(\chi\) \(=\) 124.25
Dual form 124.4.e.b.5.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-4.95459 - 8.58160i) q^{3} +(-7.40918 + 12.8331i) q^{5} +(4.04541 + 7.00685i) q^{7} +(-35.5960 + 61.6540i) q^{9} +O(q^{10})\) \(q+(-4.95459 - 8.58160i) q^{3} +(-7.40918 + 12.8331i) q^{5} +(4.04541 + 7.00685i) q^{7} +(-35.5960 + 61.6540i) q^{9} +(18.9045 - 32.7436i) q^{11} +(13.5500 - 23.4694i) q^{13} +146.838 q^{15} +(39.1960 + 67.8895i) q^{17} +(72.0742 + 124.836i) q^{19} +(40.0867 - 69.4322i) q^{21} -114.058 q^{23} +(-47.2920 - 81.9122i) q^{25} +437.906 q^{27} -86.1021 q^{29} +(-157.260 + 71.1346i) q^{31} -374.657 q^{33} -119.893 q^{35} +(170.656 + 295.585i) q^{37} -268.540 q^{39} +(-157.996 + 273.657i) q^{41} +(163.129 + 282.548i) q^{43} +(-527.474 - 913.612i) q^{45} +58.1466 q^{47} +(138.769 - 240.356i) q^{49} +(388.401 - 672.730i) q^{51} +(-133.540 + 231.298i) q^{53} +(280.135 + 485.207i) q^{55} +(714.197 - 1237.03i) q^{57} +(-279.244 - 483.665i) q^{59} -234.587 q^{61} -576.001 q^{63} +(200.790 + 347.778i) q^{65} +(-107.555 + 186.292i) q^{67} +(565.109 + 978.798i) q^{69} +(-399.541 + 692.026i) q^{71} +(193.956 - 335.942i) q^{73} +(-468.625 + 811.683i) q^{75} +305.906 q^{77} +(-73.5118 - 127.326i) q^{79} +(-1208.55 - 2093.28i) q^{81} +(429.550 - 744.003i) q^{83} -1161.64 q^{85} +(426.601 + 738.894i) q^{87} -284.219 q^{89} +219.262 q^{91} +(1389.61 + 997.104i) q^{93} -2136.04 q^{95} -59.2991 q^{97} +(1345.85 + 2331.08i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{3} - 3 q^{5} + 45 q^{7} - 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 9 q^{3} - 3 q^{5} + 45 q^{7} - 38 q^{9} + 61 q^{11} + 113 q^{13} + 386 q^{15} + 123 q^{17} + 63 q^{19} + 43 q^{21} + 96 q^{23} + 4 q^{25} + 918 q^{27} + 332 q^{29} - 668 q^{31} - 1214 q^{33} + 278 q^{35} + 129 q^{37} - 14 q^{39} - 709 q^{41} + 107 q^{43} - 1214 q^{45} - 568 q^{47} + 262 q^{49} + 293 q^{51} - 1267 q^{53} + 909 q^{55} + 681 q^{57} - 989 q^{59} - 2500 q^{61} + 36 q^{63} - 551 q^{65} + 741 q^{67} + 440 q^{69} - 1089 q^{71} - 197 q^{73} - 500 q^{75} + 982 q^{77} + 677 q^{79} - 2423 q^{81} + q^{83} + 58 q^{85} + 1194 q^{87} - 564 q^{89} + 4054 q^{91} + 1439 q^{93} - 2094 q^{95} - 2092 q^{97} + 2086 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/124\mathbb{Z}\right)^\times\).

\(n\) \(63\) \(65\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −4.95459 8.58160i −0.953512 1.65153i −0.737738 0.675087i \(-0.764106\pi\)
−0.215774 0.976443i \(-0.569227\pi\)
\(4\) 0 0
\(5\) −7.40918 + 12.8331i −0.662698 + 1.14783i 0.317207 + 0.948356i \(0.397255\pi\)
−0.979904 + 0.199469i \(0.936078\pi\)
\(6\) 0 0
\(7\) 4.04541 + 7.00685i 0.218432 + 0.378334i 0.954329 0.298759i \(-0.0965726\pi\)
−0.735897 + 0.677093i \(0.763239\pi\)
\(8\) 0 0
\(9\) −35.5960 + 61.6540i −1.31837 + 2.28348i
\(10\) 0 0
\(11\) 18.9045 32.7436i 0.518176 0.897507i −0.481601 0.876391i \(-0.659945\pi\)
0.999777 0.0211166i \(-0.00672212\pi\)
\(12\) 0 0
\(13\) 13.5500 23.4694i 0.289085 0.500710i −0.684507 0.729007i \(-0.739982\pi\)
0.973592 + 0.228297i \(0.0733156\pi\)
\(14\) 0 0
\(15\) 146.838 2.52756
\(16\) 0 0
\(17\) 39.1960 + 67.8895i 0.559202 + 0.968567i 0.997563 + 0.0697676i \(0.0222258\pi\)
−0.438361 + 0.898799i \(0.644441\pi\)
\(18\) 0 0
\(19\) 72.0742 + 124.836i 0.870261 + 1.50734i 0.861727 + 0.507373i \(0.169383\pi\)
0.00853467 + 0.999964i \(0.497283\pi\)
\(20\) 0 0
\(21\) 40.0867 69.4322i 0.416554 0.721493i
\(22\) 0 0
\(23\) −114.058 −1.03403 −0.517014 0.855977i \(-0.672957\pi\)
−0.517014 + 0.855977i \(0.672957\pi\)
\(24\) 0 0
\(25\) −47.2920 81.9122i −0.378336 0.655297i
\(26\) 0 0
\(27\) 437.906 3.12130
\(28\) 0 0
\(29\) −86.1021 −0.551337 −0.275668 0.961253i \(-0.588899\pi\)
−0.275668 + 0.961253i \(0.588899\pi\)
\(30\) 0 0
\(31\) −157.260 + 71.1346i −0.911123 + 0.412134i
\(32\) 0 0
\(33\) −374.657 −1.97635
\(34\) 0 0
\(35\) −119.893 −0.579016
\(36\) 0 0
\(37\) 170.656 + 295.585i 0.758261 + 1.31335i 0.943737 + 0.330698i \(0.107284\pi\)
−0.185476 + 0.982649i \(0.559383\pi\)
\(38\) 0 0
\(39\) −268.540 −1.10258
\(40\) 0 0
\(41\) −157.996 + 273.657i −0.601825 + 1.04239i 0.390720 + 0.920510i \(0.372226\pi\)
−0.992545 + 0.121881i \(0.961107\pi\)
\(42\) 0 0
\(43\) 163.129 + 282.548i 0.578533 + 1.00205i 0.995648 + 0.0931953i \(0.0297081\pi\)
−0.417114 + 0.908854i \(0.636959\pi\)
\(44\) 0 0
\(45\) −527.474 913.612i −1.74736 3.02652i
\(46\) 0 0
\(47\) 58.1466 0.180459 0.0902293 0.995921i \(-0.471240\pi\)
0.0902293 + 0.995921i \(0.471240\pi\)
\(48\) 0 0
\(49\) 138.769 240.356i 0.404575 0.700745i
\(50\) 0 0
\(51\) 388.401 672.730i 1.06641 1.84708i
\(52\) 0 0
\(53\) −133.540 + 231.298i −0.346096 + 0.599456i −0.985552 0.169371i \(-0.945826\pi\)
0.639456 + 0.768828i \(0.279160\pi\)
\(54\) 0 0
\(55\) 280.135 + 485.207i 0.686788 + 1.18955i
\(56\) 0 0
\(57\) 714.197 1237.03i 1.65961 2.87453i
\(58\) 0 0
\(59\) −279.244 483.665i −0.616177 1.06725i −0.990177 0.139821i \(-0.955347\pi\)
0.373999 0.927429i \(-0.377986\pi\)
\(60\) 0 0
\(61\) −234.587 −0.492391 −0.246195 0.969220i \(-0.579180\pi\)
−0.246195 + 0.969220i \(0.579180\pi\)
\(62\) 0 0
\(63\) −576.001 −1.15189
\(64\) 0 0
\(65\) 200.790 + 347.778i 0.383152 + 0.663639i
\(66\) 0 0
\(67\) −107.555 + 186.292i −0.196119 + 0.339689i −0.947267 0.320446i \(-0.896167\pi\)
0.751148 + 0.660134i \(0.229501\pi\)
\(68\) 0 0
\(69\) 565.109 + 978.798i 0.985959 + 1.70773i
\(70\) 0 0
\(71\) −399.541 + 692.026i −0.667843 + 1.15674i 0.310663 + 0.950520i \(0.399449\pi\)
−0.978506 + 0.206218i \(0.933884\pi\)
\(72\) 0 0
\(73\) 193.956 335.942i 0.310971 0.538617i −0.667602 0.744518i \(-0.732679\pi\)
0.978573 + 0.205901i \(0.0660125\pi\)
\(74\) 0 0
\(75\) −468.625 + 811.683i −0.721496 + 1.24967i
\(76\) 0 0
\(77\) 305.906 0.452744
\(78\) 0 0
\(79\) −73.5118 127.326i −0.104693 0.181333i 0.808920 0.587919i \(-0.200053\pi\)
−0.913613 + 0.406586i \(0.866719\pi\)
\(80\) 0 0
\(81\) −1208.55 2093.28i −1.65782 2.87144i
\(82\) 0 0
\(83\) 429.550 744.003i 0.568063 0.983914i −0.428694 0.903450i \(-0.641026\pi\)
0.996757 0.0804647i \(-0.0256404\pi\)
\(84\) 0 0
\(85\) −1161.64 −1.48233
\(86\) 0 0
\(87\) 426.601 + 738.894i 0.525706 + 0.910549i
\(88\) 0 0
\(89\) −284.219 −0.338507 −0.169254 0.985573i \(-0.554136\pi\)
−0.169254 + 0.985573i \(0.554136\pi\)
\(90\) 0 0
\(91\) 219.262 0.252581
\(92\) 0 0
\(93\) 1389.61 + 997.104i 1.54942 + 1.11177i
\(94\) 0 0
\(95\) −2136.04 −2.30688
\(96\) 0 0
\(97\) −59.2991 −0.0620712 −0.0310356 0.999518i \(-0.509881\pi\)
−0.0310356 + 0.999518i \(0.509881\pi\)
\(98\) 0 0
\(99\) 1345.85 + 2331.08i 1.36629 + 2.36649i
\(100\) 0 0
\(101\) 163.729 0.161303 0.0806517 0.996742i \(-0.474300\pi\)
0.0806517 + 0.996742i \(0.474300\pi\)
\(102\) 0 0
\(103\) 650.644 1126.95i 0.622426 1.07807i −0.366607 0.930376i \(-0.619481\pi\)
0.989033 0.147697i \(-0.0471861\pi\)
\(104\) 0 0
\(105\) 594.019 + 1028.87i 0.552099 + 0.956263i
\(106\) 0 0
\(107\) 770.368 + 1334.32i 0.696021 + 1.20554i 0.969835 + 0.243761i \(0.0783814\pi\)
−0.273814 + 0.961783i \(0.588285\pi\)
\(108\) 0 0
\(109\) −592.520 −0.520671 −0.260335 0.965518i \(-0.583833\pi\)
−0.260335 + 0.965518i \(0.583833\pi\)
\(110\) 0 0
\(111\) 1691.06 2929.00i 1.44602 2.50458i
\(112\) 0 0
\(113\) −775.271 + 1342.81i −0.645410 + 1.11788i 0.338797 + 0.940860i \(0.389980\pi\)
−0.984207 + 0.177023i \(0.943353\pi\)
\(114\) 0 0
\(115\) 845.074 1463.71i 0.685248 1.18688i
\(116\) 0 0
\(117\) 964.654 + 1670.83i 0.762241 + 1.32024i
\(118\) 0 0
\(119\) −317.128 + 549.282i −0.244295 + 0.423131i
\(120\) 0 0
\(121\) −49.2640 85.3278i −0.0370128 0.0641080i
\(122\) 0 0
\(123\) 3131.22 2.29539
\(124\) 0 0
\(125\) −450.715 −0.322506
\(126\) 0 0
\(127\) −74.1212 128.382i −0.0517889 0.0897011i 0.838969 0.544180i \(-0.183159\pi\)
−0.890758 + 0.454478i \(0.849826\pi\)
\(128\) 0 0
\(129\) 1616.48 2799.82i 1.10328 1.91093i
\(130\) 0 0
\(131\) −132.170 228.924i −0.0881504 0.152681i 0.818579 0.574394i \(-0.194762\pi\)
−0.906729 + 0.421713i \(0.861429\pi\)
\(132\) 0 0
\(133\) −583.139 + 1010.03i −0.380185 + 0.658500i
\(134\) 0 0
\(135\) −3244.52 + 5619.68i −2.06848 + 3.58271i
\(136\) 0 0
\(137\) −112.868 + 195.494i −0.0703868 + 0.121913i −0.899071 0.437803i \(-0.855757\pi\)
0.828684 + 0.559717i \(0.189090\pi\)
\(138\) 0 0
\(139\) 2986.61 1.82245 0.911225 0.411909i \(-0.135138\pi\)
0.911225 + 0.411909i \(0.135138\pi\)
\(140\) 0 0
\(141\) −288.093 498.991i −0.172069 0.298033i
\(142\) 0 0
\(143\) −512.315 887.355i −0.299594 0.518912i
\(144\) 0 0
\(145\) 637.946 1104.96i 0.365369 0.632838i
\(146\) 0 0
\(147\) −2750.18 −1.54307
\(148\) 0 0
\(149\) −962.109 1666.42i −0.528987 0.916232i −0.999429 0.0338012i \(-0.989239\pi\)
0.470442 0.882431i \(-0.344095\pi\)
\(150\) 0 0
\(151\) −2227.25 −1.20034 −0.600169 0.799874i \(-0.704900\pi\)
−0.600169 + 0.799874i \(0.704900\pi\)
\(152\) 0 0
\(153\) −5580.88 −2.94894
\(154\) 0 0
\(155\) 252.295 2545.19i 0.130741 1.31893i
\(156\) 0 0
\(157\) 428.582 0.217863 0.108932 0.994049i \(-0.465257\pi\)
0.108932 + 0.994049i \(0.465257\pi\)
\(158\) 0 0
\(159\) 2646.54 1.32003
\(160\) 0 0
\(161\) −461.410 799.185i −0.225864 0.391209i
\(162\) 0 0
\(163\) 2992.70 1.43807 0.719037 0.694972i \(-0.244583\pi\)
0.719037 + 0.694972i \(0.244583\pi\)
\(164\) 0 0
\(165\) 2775.90 4808.01i 1.30972 2.26850i
\(166\) 0 0
\(167\) 500.005 + 866.034i 0.231686 + 0.401292i 0.958304 0.285749i \(-0.0922425\pi\)
−0.726618 + 0.687041i \(0.758909\pi\)
\(168\) 0 0
\(169\) 731.293 + 1266.64i 0.332860 + 0.576530i
\(170\) 0 0
\(171\) −10262.2 −4.58930
\(172\) 0 0
\(173\) −981.140 + 1699.38i −0.431183 + 0.746831i −0.996975 0.0777167i \(-0.975237\pi\)
0.565792 + 0.824548i \(0.308570\pi\)
\(174\) 0 0
\(175\) 382.631 662.736i 0.165281 0.286275i
\(176\) 0 0
\(177\) −2767.08 + 4792.72i −1.17506 + 2.03527i
\(178\) 0 0
\(179\) −1791.98 3103.80i −0.748262 1.29603i −0.948655 0.316313i \(-0.897555\pi\)
0.200392 0.979716i \(-0.435778\pi\)
\(180\) 0 0
\(181\) 1922.87 3330.51i 0.789646 1.36771i −0.136537 0.990635i \(-0.543597\pi\)
0.926184 0.377073i \(-0.123069\pi\)
\(182\) 0 0
\(183\) 1162.28 + 2013.14i 0.469500 + 0.813198i
\(184\) 0 0
\(185\) −5057.68 −2.00999
\(186\) 0 0
\(187\) 2963.93 1.15906
\(188\) 0 0
\(189\) 1771.51 + 3068.34i 0.681790 + 1.18089i
\(190\) 0 0
\(191\) 1676.89 2904.46i 0.635265 1.10031i −0.351194 0.936303i \(-0.614224\pi\)
0.986459 0.164009i \(-0.0524425\pi\)
\(192\) 0 0
\(193\) 2084.12 + 3609.80i 0.777297 + 1.34632i 0.933495 + 0.358592i \(0.116743\pi\)
−0.156198 + 0.987726i \(0.549924\pi\)
\(194\) 0 0
\(195\) 1989.66 3446.19i 0.730680 1.26557i
\(196\) 0 0
\(197\) −1913.93 + 3315.02i −0.692191 + 1.19891i 0.278927 + 0.960312i \(0.410021\pi\)
−0.971118 + 0.238598i \(0.923312\pi\)
\(198\) 0 0
\(199\) −874.649 + 1514.94i −0.311569 + 0.539653i −0.978702 0.205285i \(-0.934188\pi\)
0.667133 + 0.744938i \(0.267521\pi\)
\(200\) 0 0
\(201\) 2131.57 0.748008
\(202\) 0 0
\(203\) −348.318 603.305i −0.120429 0.208590i
\(204\) 0 0
\(205\) −2341.24 4055.15i −0.797655 1.38158i
\(206\) 0 0
\(207\) 4059.99 7032.11i 1.36323 2.36119i
\(208\) 0 0
\(209\) 5450.12 1.80379
\(210\) 0 0
\(211\) −176.325 305.404i −0.0575295 0.0996440i 0.835826 0.548994i \(-0.184989\pi\)
−0.893356 + 0.449350i \(0.851656\pi\)
\(212\) 0 0
\(213\) 7918.26 2.54718
\(214\) 0 0
\(215\) −4834.61 −1.53357
\(216\) 0 0
\(217\) −1134.61 814.133i −0.354943 0.254686i
\(218\) 0 0
\(219\) −3843.90 −1.18606
\(220\) 0 0
\(221\) 2124.43 0.646628
\(222\) 0 0
\(223\) −2738.81 4743.75i −0.822439 1.42451i −0.903861 0.427827i \(-0.859279\pi\)
0.0814212 0.996680i \(-0.474054\pi\)
\(224\) 0 0
\(225\) 6733.62 1.99515
\(226\) 0 0
\(227\) −385.602 + 667.883i −0.112746 + 0.195282i −0.916876 0.399171i \(-0.869298\pi\)
0.804131 + 0.594453i \(0.202631\pi\)
\(228\) 0 0
\(229\) 2169.44 + 3757.58i 0.626028 + 1.08431i 0.988341 + 0.152256i \(0.0486539\pi\)
−0.362313 + 0.932057i \(0.618013\pi\)
\(230\) 0 0
\(231\) −1515.64 2625.17i −0.431697 0.747720i
\(232\) 0 0
\(233\) −5612.89 −1.57816 −0.789082 0.614287i \(-0.789444\pi\)
−0.789082 + 0.614287i \(0.789444\pi\)
\(234\) 0 0
\(235\) −430.819 + 746.200i −0.119589 + 0.207135i
\(236\) 0 0
\(237\) −728.442 + 1261.70i −0.199651 + 0.345806i
\(238\) 0 0
\(239\) −2631.53 + 4557.95i −0.712217 + 1.23360i 0.251807 + 0.967778i \(0.418975\pi\)
−0.964023 + 0.265818i \(0.914358\pi\)
\(240\) 0 0
\(241\) 1598.96 + 2769.48i 0.427378 + 0.740240i 0.996639 0.0819163i \(-0.0261040\pi\)
−0.569261 + 0.822157i \(0.692771\pi\)
\(242\) 0 0
\(243\) −6064.05 + 10503.2i −1.60086 + 2.77277i
\(244\) 0 0
\(245\) 2056.34 + 3561.68i 0.536222 + 0.928764i
\(246\) 0 0
\(247\) 3906.44 1.00632
\(248\) 0 0
\(249\) −8512.98 −2.16662
\(250\) 0 0
\(251\) −1007.35 1744.78i −0.253320 0.438763i 0.711118 0.703073i \(-0.248189\pi\)
−0.964438 + 0.264310i \(0.914856\pi\)
\(252\) 0 0
\(253\) −2156.21 + 3734.66i −0.535809 + 0.928048i
\(254\) 0 0
\(255\) 5755.47 + 9968.76i 1.41342 + 2.44811i
\(256\) 0 0
\(257\) 1632.81 2828.10i 0.396310 0.686428i −0.596958 0.802273i \(-0.703624\pi\)
0.993267 + 0.115844i \(0.0369573\pi\)
\(258\) 0 0
\(259\) −1380.75 + 2391.52i −0.331256 + 0.573752i
\(260\) 0 0
\(261\) 3064.89 5308.54i 0.726865 1.25897i
\(262\) 0 0
\(263\) 1694.90 0.397384 0.198692 0.980062i \(-0.436331\pi\)
0.198692 + 0.980062i \(0.436331\pi\)
\(264\) 0 0
\(265\) −1978.84 3427.45i −0.458714 0.794516i
\(266\) 0 0
\(267\) 1408.19 + 2439.06i 0.322771 + 0.559055i
\(268\) 0 0
\(269\) 2938.67 5089.93i 0.666075 1.15368i −0.312918 0.949780i \(-0.601307\pi\)
0.978993 0.203895i \(-0.0653601\pi\)
\(270\) 0 0
\(271\) −6034.41 −1.35264 −0.676318 0.736610i \(-0.736425\pi\)
−0.676318 + 0.736610i \(0.736425\pi\)
\(272\) 0 0
\(273\) −1086.35 1881.62i −0.240839 0.417145i
\(274\) 0 0
\(275\) −3576.14 −0.784179
\(276\) 0 0
\(277\) 5538.88 1.20144 0.600720 0.799460i \(-0.294881\pi\)
0.600720 + 0.799460i \(0.294881\pi\)
\(278\) 0 0
\(279\) 1212.10 12227.8i 0.260096 2.62388i
\(280\) 0 0
\(281\) 981.401 0.208347 0.104173 0.994559i \(-0.466780\pi\)
0.104173 + 0.994559i \(0.466780\pi\)
\(282\) 0 0
\(283\) 1032.18 0.216809 0.108404 0.994107i \(-0.465426\pi\)
0.108404 + 0.994107i \(0.465426\pi\)
\(284\) 0 0
\(285\) 10583.2 + 18330.7i 2.19964 + 3.80988i
\(286\) 0 0
\(287\) −2556.63 −0.525830
\(288\) 0 0
\(289\) −616.160 + 1067.22i −0.125414 + 0.217224i
\(290\) 0 0
\(291\) 293.803 + 508.881i 0.0591856 + 0.102512i
\(292\) 0 0
\(293\) 3222.20 + 5581.02i 0.642468 + 1.11279i 0.984880 + 0.173238i \(0.0554228\pi\)
−0.342412 + 0.939550i \(0.611244\pi\)
\(294\) 0 0
\(295\) 8275.88 1.63336
\(296\) 0 0
\(297\) 8278.41 14338.6i 1.61738 2.80139i
\(298\) 0 0
\(299\) −1545.49 + 2676.86i −0.298922 + 0.517749i
\(300\) 0 0
\(301\) −1319.85 + 2286.04i −0.252740 + 0.437758i
\(302\) 0 0
\(303\) −811.210 1405.06i −0.153805 0.266398i
\(304\) 0 0
\(305\) 1738.10 3010.48i 0.326306 0.565179i
\(306\) 0 0
\(307\) −2353.34 4076.10i −0.437499 0.757770i 0.559997 0.828495i \(-0.310802\pi\)
−0.997496 + 0.0707242i \(0.977469\pi\)
\(308\) 0 0
\(309\) −12894.7 −2.37396
\(310\) 0 0
\(311\) 6549.04 1.19409 0.597045 0.802208i \(-0.296341\pi\)
0.597045 + 0.802208i \(0.296341\pi\)
\(312\) 0 0
\(313\) 16.0379 + 27.7785i 0.00289622 + 0.00501640i 0.867470 0.497490i \(-0.165745\pi\)
−0.864574 + 0.502506i \(0.832411\pi\)
\(314\) 0 0
\(315\) 4267.70 7391.87i 0.763357 1.32217i
\(316\) 0 0
\(317\) 1624.24 + 2813.26i 0.287780 + 0.498450i 0.973280 0.229623i \(-0.0737494\pi\)
−0.685499 + 0.728073i \(0.740416\pi\)
\(318\) 0 0
\(319\) −1627.72 + 2819.30i −0.285689 + 0.494829i
\(320\) 0 0
\(321\) 7633.72 13222.0i 1.32733 2.29900i
\(322\) 0 0
\(323\) −5650.05 + 9786.17i −0.973304 + 1.68581i
\(324\) 0 0
\(325\) −2563.23 −0.437485
\(326\) 0 0
\(327\) 2935.69 + 5084.77i 0.496465 + 0.859903i
\(328\) 0 0
\(329\) 235.227 + 407.425i 0.0394178 + 0.0682737i
\(330\) 0 0
\(331\) 1217.92 2109.50i 0.202245 0.350298i −0.747007 0.664816i \(-0.768510\pi\)
0.949251 + 0.314519i \(0.101843\pi\)
\(332\) 0 0
\(333\) −24298.6 −3.99867
\(334\) 0 0
\(335\) −1593.80 2760.54i −0.259936 0.450222i
\(336\) 0 0
\(337\) −2602.95 −0.420746 −0.210373 0.977621i \(-0.567468\pi\)
−0.210373 + 0.977621i \(0.567468\pi\)
\(338\) 0 0
\(339\) 15364.6 2.46162
\(340\) 0 0
\(341\) −643.732 + 6494.05i −0.102229 + 1.03130i
\(342\) 0 0
\(343\) 5020.66 0.790351
\(344\) 0 0
\(345\) −16748.0 −2.61357
\(346\) 0 0
\(347\) −1041.96 1804.72i −0.161197 0.279201i 0.774101 0.633062i \(-0.218202\pi\)
−0.935298 + 0.353861i \(0.884869\pi\)
\(348\) 0 0
\(349\) −8972.63 −1.37620 −0.688100 0.725616i \(-0.741555\pi\)
−0.688100 + 0.725616i \(0.741555\pi\)
\(350\) 0 0
\(351\) 5933.64 10277.4i 0.902320 1.56286i
\(352\) 0 0
\(353\) −161.687 280.051i −0.0243789 0.0422255i 0.853579 0.520964i \(-0.174427\pi\)
−0.877957 + 0.478739i \(0.841094\pi\)
\(354\) 0 0
\(355\) −5920.55 10254.7i −0.885156 1.53313i
\(356\) 0 0
\(357\) 6284.96 0.931752
\(358\) 0 0
\(359\) −3176.90 + 5502.56i −0.467049 + 0.808953i −0.999291 0.0376394i \(-0.988016\pi\)
0.532242 + 0.846592i \(0.321350\pi\)
\(360\) 0 0
\(361\) −6959.89 + 12054.9i −1.01471 + 1.75753i
\(362\) 0 0
\(363\) −488.166 + 845.528i −0.0705842 + 0.122255i
\(364\) 0 0
\(365\) 2874.12 + 4978.11i 0.412159 + 0.713881i
\(366\) 0 0
\(367\) −530.123 + 918.199i −0.0754010 + 0.130598i −0.901261 0.433277i \(-0.857357\pi\)
0.825860 + 0.563876i \(0.190690\pi\)
\(368\) 0 0
\(369\) −11248.0 19482.2i −1.58685 2.74851i
\(370\) 0 0
\(371\) −2160.89 −0.302393
\(372\) 0 0
\(373\) −1475.27 −0.204790 −0.102395 0.994744i \(-0.532651\pi\)
−0.102395 + 0.994744i \(0.532651\pi\)
\(374\) 0 0
\(375\) 2233.11 + 3867.86i 0.307513 + 0.532628i
\(376\) 0 0
\(377\) −1166.69 + 2020.76i −0.159383 + 0.276060i
\(378\) 0 0
\(379\) −1803.47 3123.70i −0.244427 0.423361i 0.717543 0.696514i \(-0.245267\pi\)
−0.961970 + 0.273154i \(0.911933\pi\)
\(380\) 0 0
\(381\) −734.481 + 1272.16i −0.0987627 + 0.171062i
\(382\) 0 0
\(383\) 1888.86 3271.61i 0.252001 0.436478i −0.712076 0.702103i \(-0.752245\pi\)
0.964077 + 0.265624i \(0.0855781\pi\)
\(384\) 0 0
\(385\) −2266.52 + 3925.72i −0.300032 + 0.519671i
\(386\) 0 0
\(387\) −23226.9 −3.05088
\(388\) 0 0
\(389\) −3208.07 5556.54i −0.418138 0.724236i 0.577614 0.816310i \(-0.303984\pi\)
−0.995752 + 0.0920738i \(0.970650\pi\)
\(390\) 0 0
\(391\) −4470.61 7743.32i −0.578231 1.00153i
\(392\) 0 0
\(393\) −1309.69 + 2268.45i −0.168105 + 0.291166i
\(394\) 0 0
\(395\) 2178.65 0.277518
\(396\) 0 0
\(397\) −2568.18 4448.22i −0.324668 0.562342i 0.656777 0.754085i \(-0.271919\pi\)
−0.981445 + 0.191743i \(0.938586\pi\)
\(398\) 0 0
\(399\) 11556.9 1.45004
\(400\) 0 0
\(401\) 2245.96 0.279695 0.139847 0.990173i \(-0.455339\pi\)
0.139847 + 0.990173i \(0.455339\pi\)
\(402\) 0 0
\(403\) −461.402 + 4654.68i −0.0570324 + 0.575350i
\(404\) 0 0
\(405\) 35817.6 4.39454
\(406\) 0 0
\(407\) 12904.7 1.57165
\(408\) 0 0
\(409\) 2770.52 + 4798.68i 0.334947 + 0.580145i 0.983475 0.181046i \(-0.0579483\pi\)
−0.648528 + 0.761191i \(0.724615\pi\)
\(410\) 0 0
\(411\) 2236.86 0.268458
\(412\) 0 0
\(413\) 2259.31 3913.24i 0.269185 0.466242i
\(414\) 0 0
\(415\) 6365.23 + 11024.9i 0.752908 + 1.30408i
\(416\) 0 0
\(417\) −14797.4 25629.9i −1.73773 3.00983i
\(418\) 0 0
\(419\) 9559.85 1.11463 0.557314 0.830302i \(-0.311832\pi\)
0.557314 + 0.830302i \(0.311832\pi\)
\(420\) 0 0
\(421\) 798.547 1383.12i 0.0924437 0.160117i −0.816095 0.577918i \(-0.803866\pi\)
0.908539 + 0.417800i \(0.137199\pi\)
\(422\) 0 0
\(423\) −2069.78 + 3584.97i −0.237911 + 0.412074i
\(424\) 0 0
\(425\) 3707.32 6421.26i 0.423133 0.732887i
\(426\) 0 0
\(427\) −949.002 1643.72i −0.107554 0.186288i
\(428\) 0 0
\(429\) −5076.62 + 8792.97i −0.571332 + 0.989577i
\(430\) 0 0
\(431\) 2431.24 + 4211.03i 0.271714 + 0.470622i 0.969301 0.245878i \(-0.0790763\pi\)
−0.697587 + 0.716500i \(0.745743\pi\)
\(432\) 0 0
\(433\) 8991.51 0.997932 0.498966 0.866622i \(-0.333713\pi\)
0.498966 + 0.866622i \(0.333713\pi\)
\(434\) 0 0
\(435\) −12643.1 −1.39354
\(436\) 0 0
\(437\) −8220.62 14238.5i −0.899875 1.55863i
\(438\) 0 0
\(439\) −3439.44 + 5957.29i −0.373931 + 0.647667i −0.990166 0.139895i \(-0.955324\pi\)
0.616236 + 0.787562i \(0.288657\pi\)
\(440\) 0 0
\(441\) 9879.26 + 17111.4i 1.06676 + 1.84768i
\(442\) 0 0
\(443\) 7587.15 13141.3i 0.813716 1.40940i −0.0965298 0.995330i \(-0.530774\pi\)
0.910246 0.414068i \(-0.135892\pi\)
\(444\) 0 0
\(445\) 2105.83 3647.41i 0.224328 0.388548i
\(446\) 0 0
\(447\) −9533.72 + 16512.9i −1.00879 + 1.74728i
\(448\) 0 0
\(449\) 7703.39 0.809678 0.404839 0.914388i \(-0.367328\pi\)
0.404839 + 0.914388i \(0.367328\pi\)
\(450\) 0 0
\(451\) 5973.68 + 10346.7i 0.623702 + 1.08028i
\(452\) 0 0
\(453\) 11035.1 + 19113.4i 1.14454 + 1.98239i
\(454\) 0 0
\(455\) −1624.55 + 2813.80i −0.167385 + 0.289919i
\(456\) 0 0
\(457\) −1834.21 −0.187747 −0.0938736 0.995584i \(-0.529925\pi\)
−0.0938736 + 0.995584i \(0.529925\pi\)
\(458\) 0 0
\(459\) 17164.2 + 29729.2i 1.74544 + 3.02318i
\(460\) 0 0
\(461\) 6434.20 0.650045 0.325022 0.945706i \(-0.394628\pi\)
0.325022 + 0.945706i \(0.394628\pi\)
\(462\) 0 0
\(463\) 2211.28 0.221959 0.110979 0.993823i \(-0.464601\pi\)
0.110979 + 0.993823i \(0.464601\pi\)
\(464\) 0 0
\(465\) −23091.8 + 10445.3i −2.30292 + 1.04169i
\(466\) 0 0
\(467\) −3041.83 −0.301411 −0.150706 0.988579i \(-0.548155\pi\)
−0.150706 + 0.988579i \(0.548155\pi\)
\(468\) 0 0
\(469\) −1740.42 −0.171355
\(470\) 0 0
\(471\) −2123.45 3677.92i −0.207735 0.359808i
\(472\) 0 0
\(473\) 12335.5 1.19913
\(474\) 0 0
\(475\) 6817.07 11807.5i 0.658502 1.14056i
\(476\) 0 0
\(477\) −9506.95 16466.5i −0.912565 1.58061i
\(478\) 0 0
\(479\) 489.900 + 848.532i 0.0467309 + 0.0809404i 0.888445 0.458984i \(-0.151786\pi\)
−0.841714 + 0.539924i \(0.818453\pi\)
\(480\) 0 0
\(481\) 9249.58 0.876807
\(482\) 0 0
\(483\) −4572.19 + 7919.27i −0.430729 + 0.746044i
\(484\) 0 0
\(485\) 439.358 760.990i 0.0411344 0.0712469i
\(486\) 0 0
\(487\) 1499.08 2596.47i 0.139486 0.241596i −0.787816 0.615910i \(-0.788788\pi\)
0.927302 + 0.374314i \(0.122122\pi\)
\(488\) 0 0
\(489\) −14827.6 25682.1i −1.37122 2.37502i
\(490\) 0 0
\(491\) −2986.00 + 5171.90i −0.274452 + 0.475366i −0.969997 0.243118i \(-0.921830\pi\)
0.695544 + 0.718483i \(0.255163\pi\)
\(492\) 0 0
\(493\) −3374.86 5845.43i −0.308309 0.534006i
\(494\) 0 0
\(495\) −39886.6 −3.62176
\(496\) 0 0
\(497\) −6465.23 −0.583512
\(498\) 0 0
\(499\) 1881.74 + 3259.28i 0.168815 + 0.292395i 0.938003 0.346626i \(-0.112673\pi\)
−0.769189 + 0.639022i \(0.779339\pi\)
\(500\) 0 0
\(501\) 4954.64 8581.69i 0.441831 0.765273i
\(502\) 0 0
\(503\) −6698.12 11601.5i −0.593747 1.02840i −0.993722 0.111874i \(-0.964315\pi\)
0.399976 0.916526i \(-0.369019\pi\)
\(504\) 0 0
\(505\) −1213.10 + 2101.15i −0.106895 + 0.185148i
\(506\) 0 0
\(507\) 7246.51 12551.3i 0.634771 1.09946i
\(508\) 0 0
\(509\) 6735.53 11666.3i 0.586536 1.01591i −0.408146 0.912917i \(-0.633824\pi\)
0.994682 0.102994i \(-0.0328422\pi\)
\(510\) 0 0
\(511\) 3138.53 0.271703
\(512\) 0 0
\(513\) 31561.7 + 54666.5i 2.71634 + 4.70485i
\(514\) 0 0
\(515\) 9641.48 + 16699.5i 0.824960 + 1.42887i
\(516\) 0 0
\(517\) 1099.24 1903.93i 0.0935093 0.161963i
\(518\) 0 0
\(519\) 19444.6 1.64455
\(520\) 0 0
\(521\) −4236.78 7338.33i −0.356270 0.617078i 0.631064 0.775731i \(-0.282618\pi\)
−0.987335 + 0.158652i \(0.949285\pi\)
\(522\) 0 0
\(523\) −20087.3 −1.67946 −0.839730 0.543004i \(-0.817287\pi\)
−0.839730 + 0.543004i \(0.817287\pi\)
\(524\) 0 0
\(525\) −7583.12 −0.630390
\(526\) 0 0
\(527\) −10993.3 7888.15i −0.908681 0.652017i
\(528\) 0 0
\(529\) 842.147 0.0692157
\(530\) 0 0
\(531\) 39759.8 3.24940
\(532\) 0 0
\(533\) 4281.70 + 7416.12i 0.347957 + 0.602679i
\(534\) 0 0
\(535\) −22831.2 −1.84501
\(536\) 0 0
\(537\) −17757.1 + 30756.2i −1.42695 + 2.47156i
\(538\) 0 0
\(539\) −5246.74 9087.63i −0.419282 0.726219i
\(540\) 0 0
\(541\) 11473.0 + 19871.8i 0.911760 + 1.57922i 0.811576 + 0.584247i \(0.198610\pi\)
0.100184 + 0.994969i \(0.468057\pi\)
\(542\) 0 0
\(543\) −38108.2 −3.01175
\(544\) 0 0
\(545\) 4390.09 7603.85i 0.345047 0.597639i
\(546\) 0 0
\(547\) −5347.88 + 9262.80i −0.418023 + 0.724038i −0.995741 0.0921994i \(-0.970610\pi\)
0.577717 + 0.816237i \(0.303944\pi\)
\(548\) 0 0
\(549\) 8350.36 14463.3i 0.649153 1.12437i
\(550\) 0 0
\(551\) −6205.74 10748.7i −0.479807 0.831050i
\(552\) 0 0
\(553\) 594.771 1030.17i 0.0457364 0.0792177i
\(554\) 0 0
\(555\) 25058.8 + 43403.0i 1.91655 + 3.31956i
\(556\) 0 0
\(557\) 21326.4 1.62231 0.811157 0.584828i \(-0.198838\pi\)
0.811157 + 0.584828i \(0.198838\pi\)
\(558\) 0 0
\(559\) 8841.62 0.668981
\(560\) 0 0
\(561\) −14685.1 25435.3i −1.10518 1.91422i
\(562\) 0 0
\(563\) −284.862 + 493.396i −0.0213242 + 0.0369346i −0.876491 0.481419i \(-0.840122\pi\)
0.855166 + 0.518354i \(0.173455\pi\)
\(564\) 0 0
\(565\) −11488.2 19898.2i −0.855423 1.48164i
\(566\) 0 0
\(567\) 9778.19 16936.3i 0.724242 1.25442i
\(568\) 0 0
\(569\) 8017.16 13886.1i 0.590680 1.02309i −0.403461 0.914997i \(-0.632193\pi\)
0.994141 0.108091i \(-0.0344736\pi\)
\(570\) 0 0
\(571\) 12821.7 22207.8i 0.939702 1.62761i 0.173675 0.984803i \(-0.444436\pi\)
0.766027 0.642809i \(-0.222231\pi\)
\(572\) 0 0
\(573\) −33233.3 −2.42293
\(574\) 0 0
\(575\) 5394.01 + 9342.71i 0.391210 + 0.677596i
\(576\) 0 0
\(577\) −320.446 555.029i −0.0231202 0.0400454i 0.854234 0.519889i \(-0.174027\pi\)
−0.877354 + 0.479844i \(0.840693\pi\)
\(578\) 0 0
\(579\) 20651.9 35770.2i 1.48232 2.56746i
\(580\) 0 0
\(581\) 6950.82 0.496332
\(582\) 0 0
\(583\) 5049.02 + 8745.16i 0.358677 + 0.621248i
\(584\) 0 0
\(585\) −28589.2 −2.02054
\(586\) 0 0
\(587\) −5647.21 −0.397079 −0.198540 0.980093i \(-0.563620\pi\)
−0.198540 + 0.980093i \(0.563620\pi\)
\(588\) 0 0
\(589\) −20214.6 14504.8i −1.41414 1.01471i
\(590\) 0 0
\(591\) 37930.9 2.64005
\(592\) 0 0
\(593\) 27731.1 1.92037 0.960184 0.279370i \(-0.0901255\pi\)
0.960184 + 0.279370i \(0.0901255\pi\)
\(594\) 0 0
\(595\) −4699.32 8139.46i −0.323787 0.560816i
\(596\) 0 0
\(597\) 17334.1 1.18834
\(598\) 0 0
\(599\) −4433.20 + 7678.53i −0.302397 + 0.523767i −0.976678 0.214708i \(-0.931120\pi\)
0.674281 + 0.738475i \(0.264453\pi\)
\(600\) 0 0
\(601\) 3540.20 + 6131.81i 0.240279 + 0.416176i 0.960794 0.277264i \(-0.0894276\pi\)
−0.720514 + 0.693440i \(0.756094\pi\)
\(602\) 0 0
\(603\) −7657.08 13262.5i −0.517115 0.895670i
\(604\) 0 0
\(605\) 1460.02 0.0981131
\(606\) 0 0
\(607\) −1928.06 + 3339.51i −0.128925 + 0.223305i −0.923261 0.384174i \(-0.874486\pi\)
0.794335 + 0.607480i \(0.207819\pi\)
\(608\) 0 0
\(609\) −3451.55 + 5978.26i −0.229661 + 0.397785i
\(610\) 0 0
\(611\) 787.889 1364.66i 0.0521679 0.0903574i
\(612\) 0 0
\(613\) −3122.69 5408.66i −0.205749 0.356368i 0.744622 0.667487i \(-0.232630\pi\)
−0.950371 + 0.311118i \(0.899296\pi\)
\(614\) 0 0
\(615\) −23199.8 + 40183.2i −1.52115 + 2.63470i
\(616\) 0 0
\(617\) 1722.12 + 2982.80i 0.112366 + 0.194624i 0.916724 0.399521i \(-0.130824\pi\)
−0.804358 + 0.594146i \(0.797490\pi\)
\(618\) 0 0
\(619\) −12428.1 −0.806990 −0.403495 0.914982i \(-0.632205\pi\)
−0.403495 + 0.914982i \(0.632205\pi\)
\(620\) 0 0
\(621\) −49946.5 −3.22751
\(622\) 0 0
\(623\) −1149.78 1991.48i −0.0739407 0.128069i
\(624\) 0 0
\(625\) 9250.93 16023.1i 0.592060 1.02548i
\(626\) 0 0
\(627\) −27003.1 46770.8i −1.71994 2.97902i
\(628\) 0 0
\(629\) −13378.1 + 23171.5i −0.848042 + 1.46885i
\(630\) 0 0
\(631\) −2140.54 + 3707.53i −0.135045 + 0.233905i −0.925615 0.378467i \(-0.876451\pi\)
0.790569 + 0.612372i \(0.209785\pi\)
\(632\) 0 0
\(633\) −1747.24 + 3026.31i −0.109710 + 0.190023i
\(634\) 0 0
\(635\) 2196.71 0.137282
\(636\) 0 0
\(637\) −3760.66 6513.66i −0.233913 0.405150i
\(638\) 0 0
\(639\) −28444.1 49266.7i −1.76093 3.05001i
\(640\) 0 0
\(641\) −4000.08 + 6928.34i −0.246480 + 0.426915i −0.962547 0.271116i \(-0.912607\pi\)
0.716067 + 0.698032i \(0.245941\pi\)
\(642\) 0 0
\(643\) 5658.50 0.347044 0.173522 0.984830i \(-0.444485\pi\)
0.173522 + 0.984830i \(0.444485\pi\)
\(644\) 0 0
\(645\) 23953.5 + 41488.7i 1.46228 + 2.53274i
\(646\) 0 0
\(647\) 28240.7 1.71600 0.858002 0.513646i \(-0.171706\pi\)
0.858002 + 0.513646i \(0.171706\pi\)
\(648\) 0 0
\(649\) −21115.9 −1.27715
\(650\) 0 0
\(651\) −1365.02 + 13770.5i −0.0821803 + 0.829045i
\(652\) 0 0
\(653\) 21480.5 1.28729 0.643643 0.765326i \(-0.277422\pi\)
0.643643 + 0.765326i \(0.277422\pi\)
\(654\) 0 0
\(655\) 3917.07 0.233668
\(656\) 0 0
\(657\) 13808.1 + 23916.4i 0.819949 + 1.42019i
\(658\) 0 0
\(659\) −9927.81 −0.586847 −0.293424 0.955982i \(-0.594795\pi\)
−0.293424 + 0.955982i \(0.594795\pi\)
\(660\) 0 0
\(661\) 3150.45 5456.74i 0.185383 0.321093i −0.758322 0.651880i \(-0.773981\pi\)
0.943706 + 0.330786i \(0.107314\pi\)
\(662\) 0 0
\(663\) −10525.7 18231.0i −0.616567 1.06793i
\(664\) 0 0
\(665\) −8641.17 14967.0i −0.503895 0.872772i
\(666\) 0 0
\(667\) 9820.61 0.570098
\(668\) 0 0
\(669\) −27139.3 + 47006.7i −1.56841 + 2.71657i
\(670\) 0 0
\(671\) −4434.77 + 7681.24i −0.255145 + 0.441924i
\(672\) 0 0
\(673\) −1488.29 + 2577.79i −0.0852441 + 0.147647i −0.905495 0.424356i \(-0.860500\pi\)
0.820251 + 0.572004i \(0.193834\pi\)
\(674\) 0 0
\(675\) −20709.4 35869.8i −1.18090 2.04538i
\(676\) 0 0
\(677\) 6191.39 10723.8i 0.351484 0.608788i −0.635026 0.772491i \(-0.719011\pi\)
0.986510 + 0.163703i \(0.0523439\pi\)
\(678\) 0 0
\(679\) −239.889 415.500i −0.0135583 0.0234837i
\(680\) 0 0
\(681\) 7642.01 0.430018
\(682\) 0 0
\(683\) 1543.16 0.0864531 0.0432265 0.999065i \(-0.486236\pi\)
0.0432265 + 0.999065i \(0.486236\pi\)
\(684\) 0 0
\(685\) −1672.52 2896.90i −0.0932903 0.161583i
\(686\) 0 0
\(687\) 21497.4 37234.5i 1.19385 2.06781i
\(688\) 0 0
\(689\) 3618.94 + 6268.19i 0.200102 + 0.346588i
\(690\) 0 0
\(691\) 13856.7 24000.5i 0.762856 1.32130i −0.178517 0.983937i \(-0.557130\pi\)
0.941373 0.337368i \(-0.109537\pi\)
\(692\) 0 0
\(693\) −10889.0 + 18860.4i −0.596883 + 1.03383i
\(694\) 0 0
\(695\) −22128.3 + 38327.4i −1.20773 + 2.09186i
\(696\) 0 0
\(697\) −24771.3 −1.34617
\(698\) 0 0
\(699\) 27809.6 + 48167.6i 1.50480 + 2.60639i
\(700\) 0 0
\(701\) −8589.04 14876.6i −0.462772 0.801545i 0.536326 0.844011i \(-0.319812\pi\)
−0.999098 + 0.0424659i \(0.986479\pi\)
\(702\) 0 0
\(703\) −24599.8 + 42608.1i −1.31977 + 2.28591i
\(704\) 0 0
\(705\) 8538.13 0.456120
\(706\) 0 0
\(707\) 662.351 + 1147.23i 0.0352338 + 0.0610266i
\(708\) 0 0
\(709\) −16812.4 −0.890556 −0.445278 0.895392i \(-0.646895\pi\)
−0.445278 + 0.895392i \(0.646895\pi\)
\(710\) 0 0
\(711\) 10466.9 0.552095
\(712\) 0 0
\(713\) 17936.8 8113.45i 0.942128 0.426159i
\(714\) 0 0
\(715\) 15183.3 0.794160
\(716\) 0 0
\(717\) 52152.7 2.71643
\(718\) 0 0
\(719\) 4488.07 + 7773.56i 0.232791 + 0.403206i 0.958628 0.284660i \(-0.0918809\pi\)
−0.725837 + 0.687866i \(0.758548\pi\)
\(720\) 0 0
\(721\) 10528.5 0.543830
\(722\) 0 0
\(723\) 15844.4 27443.3i 0.815020 1.41166i
\(724\) 0 0
\(725\) 4071.94 + 7052.81i 0.208591 + 0.361289i
\(726\) 0 0
\(727\) 7195.10 + 12462.3i 0.367058 + 0.635764i 0.989104 0.147217i \(-0.0470316\pi\)
−0.622046 + 0.782981i \(0.713698\pi\)
\(728\) 0 0
\(729\) 54917.7 2.79011
\(730\) 0 0
\(731\) −12788.0 + 22149.5i −0.647034 + 1.12070i
\(732\) 0 0
\(733\) 1958.83 3392.80i 0.0987056 0.170963i −0.812444 0.583040i \(-0.801863\pi\)
0.911149 + 0.412077i \(0.135196\pi\)
\(734\) 0 0
\(735\) 20376.6 35293.3i 1.02259 1.77117i
\(736\) 0 0
\(737\) 4066.58 + 7043.52i 0.203249 + 0.352037i
\(738\) 0 0
\(739\) −5695.58 + 9865.03i −0.283512 + 0.491057i −0.972247 0.233956i \(-0.924833\pi\)
0.688735 + 0.725013i \(0.258166\pi\)
\(740\) 0 0
\(741\) −19354.8 33523.5i −0.959536 1.66196i
\(742\) 0 0
\(743\) −22717.2 −1.12169 −0.560844 0.827921i \(-0.689523\pi\)
−0.560844 + 0.827921i \(0.689523\pi\)
\(744\) 0 0
\(745\) 28513.8 1.40223
\(746\) 0 0
\(747\) 30580.5 + 52967.0i 1.49783 + 2.59432i
\(748\) 0 0
\(749\) −6232.90 + 10795.7i −0.304066 + 0.526658i
\(750\) 0 0
\(751\) −5162.14 8941.09i −0.250824 0.434441i 0.712929 0.701237i \(-0.247368\pi\)
−0.963753 + 0.266796i \(0.914035\pi\)
\(752\) 0 0
\(753\) −9982.01 + 17289.3i −0.483087 + 0.836731i
\(754\) 0 0
\(755\) 16502.1 28582.5i 0.795460 1.37778i
\(756\) 0 0
\(757\) 2235.86 3872.62i 0.107350 0.185935i −0.807346 0.590078i \(-0.799097\pi\)
0.914696 + 0.404143i \(0.132430\pi\)
\(758\) 0 0
\(759\) 42732.5 2.04360
\(760\) 0 0
\(761\) −8215.06 14228.9i −0.391322 0.677789i 0.601302 0.799021i \(-0.294649\pi\)
−0.992624 + 0.121232i \(0.961315\pi\)
\(762\) 0 0
\(763\) −2396.98 4151.70i −0.113731 0.196988i
\(764\) 0 0
\(765\) 41349.8 71619.9i 1.95425 3.38487i
\(766\) 0 0
\(767\) −15135.1 −0.712511
\(768\) 0 0
\(769\) 5951.89 + 10309.0i 0.279104 + 0.483422i 0.971162 0.238420i \(-0.0766294\pi\)
−0.692059 + 0.721841i \(0.743296\pi\)
\(770\) 0 0
\(771\) −32359.5 −1.51154
\(772\) 0 0
\(773\) 487.871 0.0227005 0.0113503 0.999936i \(-0.496387\pi\)
0.0113503 + 0.999936i \(0.496387\pi\)
\(774\) 0 0
\(775\) 13264.0 + 9517.45i 0.614781 + 0.441131i
\(776\) 0 0
\(777\) 27364.1 1.26343
\(778\) 0 0
\(779\) −45549.7 −2.09498
\(780\) 0 0
\(781\) 15106.3 + 26164.9i 0.692120 + 1.19879i
\(782\) 0 0
\(783\) −37704.6 −1.72089
\(784\) 0 0
\(785\) −3175.44 + 5500.02i −0.144377 + 0.250069i
\(786\) 0 0
\(787\) 9922.08 + 17185.5i 0.449408 + 0.778397i 0.998348 0.0574648i \(-0.0183017\pi\)
−0.548940 + 0.835862i \(0.684968\pi\)
\(788\) 0 0
\(789\) −8397.53 14545.0i −0.378910 0.656292i
\(790\) 0 0
\(791\) −12545.1 −0.563911
\(792\) 0 0
\(793\) −3178.67 + 5505.61i −0.142343 + 0.246545i
\(794\) 0 0
\(795\) −19608.7 + 33963.3i −0.874779 + 1.51516i
\(796\) 0 0
\(797\) 10767.3 18649.5i 0.478540 0.828856i −0.521157 0.853461i \(-0.674499\pi\)
0.999697 + 0.0246049i \(0.00783278\pi\)
\(798\) 0 0
\(799\) 2279.12 + 3947.55i 0.100913 + 0.174786i
\(800\) 0 0
\(801\) 10117.1 17523.2i 0.446278 0.772976i
\(802\) 0 0
\(803\) −7333.31 12701.7i −0.322275 0.558197i
\(804\) 0 0
\(805\) 13674.7 0.598719
\(806\) 0 0
\(807\) −58239.7 −2.54044
\(808\) 0 0
\(809\) 4950.09 + 8573.81i 0.215125 + 0.372607i 0.953311 0.301990i \(-0.0976508\pi\)
−0.738186 + 0.674597i \(0.764317\pi\)
\(810\) 0 0
\(811\) −1926.18 + 3336.23i −0.0833997 + 0.144453i −0.904708 0.426032i \(-0.859911\pi\)
0.821308 + 0.570484i \(0.193244\pi\)
\(812\) 0 0
\(813\) 29898.0 + 51784.9i 1.28975 + 2.23392i
\(814\) 0 0
\(815\) −22173.4 + 38405.5i −0.953008 + 1.65066i
\(816\) 0 0
\(817\) −23514.8 + 40728.8i −1.00695 + 1.74409i
\(818\) 0 0
\(819\) −7804.83 + 13518.4i −0.332995 + 0.576764i
\(820\) 0 0
\(821\) −2385.49 −0.101406 −0.0507029 0.998714i \(-0.516146\pi\)
−0.0507029 + 0.998714i \(0.516146\pi\)
\(822\) 0 0
\(823\) 20672.1 + 35805.2i 0.875559 + 1.51651i 0.856167 + 0.516700i \(0.172840\pi\)
0.0193921 + 0.999812i \(0.493827\pi\)
\(824\) 0 0
\(825\) 17718.3 + 30689.0i 0.747723 + 1.29510i
\(826\) 0 0
\(827\) −339.085 + 587.312i −0.0142577 + 0.0246951i −0.873066 0.487602i \(-0.837872\pi\)
0.858809 + 0.512297i \(0.171205\pi\)
\(828\) 0 0
\(829\) −32565.9 −1.36437 −0.682183 0.731182i \(-0.738969\pi\)
−0.682183 + 0.731182i \(0.738969\pi\)
\(830\) 0 0
\(831\) −27442.9 47532.4i −1.14559 1.98421i
\(832\) 0 0
\(833\) 21756.8 0.904958
\(834\) 0 0
\(835\) −14818.5 −0.614151
\(836\) 0 0
\(837\) −68865.3 + 31150.3i −2.84389 + 1.28639i
\(838\) 0 0
\(839\) −3007.55 −0.123757 −0.0618785 0.998084i \(-0.519709\pi\)
−0.0618785 + 0.998084i \(0.519709\pi\)
\(840\) 0 0
\(841\) −16975.4 −0.696028
\(842\) 0 0
\(843\) −4862.44 8421.99i −0.198661 0.344091i
\(844\) 0 0
\(845\) −21673.1 −0.882341
\(846\) 0 0
\(847\) 398.586 690.371i 0.0161695 0.0280064i
\(848\) 0 0
\(849\) −5114.04 8857.77i −0.206730 0.358066i
\(850\) 0 0
\(851\) −19464.6 33713.7i −0.784064 1.35804i
\(852\) 0 0
\(853\) 14097.7 0.565882 0.282941 0.959137i \(-0.408690\pi\)
0.282941 + 0.959137i \(0.408690\pi\)
\(854\) 0 0
\(855\) 76034.6 131696.i 3.04132 5.26772i
\(856\) 0 0
\(857\) 3126.51 5415.28i 0.124620 0.215849i −0.796964 0.604027i \(-0.793562\pi\)
0.921584 + 0.388178i \(0.126895\pi\)
\(858\) 0 0
\(859\) −2224.24 + 3852.49i −0.0883469 + 0.153021i −0.906812 0.421534i \(-0.861492\pi\)
0.818466 + 0.574556i \(0.194825\pi\)
\(860\) 0 0
\(861\) 12667.1 + 21940.0i 0.501385 + 0.868424i
\(862\) 0 0
\(863\) 949.854 1645.20i 0.0374663 0.0648935i −0.846684 0.532096i \(-0.821405\pi\)
0.884151 + 0.467202i \(0.154738\pi\)
\(864\) 0 0
\(865\) −14538.9 25182.1i −0.571488 0.989846i
\(866\) 0 0
\(867\) 12211.3 0.478335
\(868\) 0 0
\(869\) −5558.83 −0.216997
\(870\) 0 0
\(871\) 2914.76 + 5048.52i 0.113390 + 0.196398i
\(872\) 0 0
\(873\) 2110.81 3656.02i 0.0818327 0.141738i
\(874\) 0 0
\(875\) −1823.33 3158.10i −0.0704454 0.122015i
\(876\) 0 0
\(877\) 24776.5 42914.1i 0.953983 1.65235i 0.217303 0.976104i \(-0.430274\pi\)
0.736680 0.676242i \(-0.236393\pi\)
\(878\) 0 0
\(879\) 31929.4 55303.4i 1.22520 2.12211i
\(880\) 0 0
\(881\) −8861.87 + 15349.2i −0.338892 + 0.586979i −0.984225 0.176923i \(-0.943386\pi\)
0.645332 + 0.763902i \(0.276719\pi\)
\(882\) 0 0
\(883\) 26593.2 1.01352 0.506758 0.862089i \(-0.330844\pi\)
0.506758 + 0.862089i \(0.330844\pi\)
\(884\) 0 0
\(885\) −41003.6 71020.3i −1.55742 2.69754i
\(886\) 0 0
\(887\) 11380.2 + 19711.1i 0.430789 + 0.746149i 0.996941 0.0781518i \(-0.0249019\pi\)
−0.566152 + 0.824301i \(0.691569\pi\)
\(888\) 0 0
\(889\) 599.701 1038.71i 0.0226247 0.0391871i
\(890\) 0 0
\(891\) −91388.7 −3.43618
\(892\) 0 0
\(893\) 4190.87 + 7258.80i 0.157046 + 0.272012i
\(894\) 0 0
\(895\) 53108.5 1.98349
\(896\) 0 0
\(897\) 30629.0 1.14010
\(898\) 0 0
\(899\) 13540.5 6124.84i 0.502336 0.227225i
\(900\) 0 0
\(901\) −20936.9 −0.774151
\(902\) 0 0
\(903\) 26157.2 0.963962
\(904\) 0 0
\(905\) 28493.8 + 49352.8i 1.04659 + 1.81275i
\(906\) 0 0
\(907\) −15569.5 −0.569986 −0.284993 0.958530i \(-0.591991\pi\)
−0.284993 + 0.958530i \(0.591991\pi\)
\(908\) 0 0
\(909\) −5828.09 + 10094.6i −0.212657 + 0.368333i
\(910\) 0 0
\(911\) −4802.10 8317.49i −0.174644 0.302492i 0.765394 0.643562i \(-0.222544\pi\)
−0.940038 + 0.341070i \(0.889211\pi\)
\(912\) 0 0
\(913\) −16240.9 28130.1i −0.588713 1.01968i
\(914\) 0 0
\(915\) −34446.3 −1.24455
\(916\) 0 0
\(917\) 1069.36 1852.18i 0.0385097 0.0667007i
\(918\) 0 0
\(919\) 9386.88 16258.6i 0.336937 0.583591i −0.646918 0.762559i \(-0.723943\pi\)
0.983855 + 0.178968i \(0.0572759\pi\)
\(920\) 0 0
\(921\) −23319.7 + 40390.9i −0.834321 + 1.44509i
\(922\) 0 0
\(923\) 10827.6 + 18754.0i 0.386127 + 0.668791i
\(924\) 0 0
\(925\) 16141.3 27957.6i 0.573755 0.993773i
\(926\) 0 0
\(927\) 46320.6 + 80229.6i 1.64117 + 2.84260i
\(928\) 0 0
\(929\) −16277.5 −0.574864 −0.287432 0.957801i \(-0.592802\pi\)
−0.287432 + 0.957801i \(0.592802\pi\)
\(930\) 0 0
\(931\) 40006.8 1.40834
\(932\) 0 0
\(933\) −32447.8 56201.2i −1.13858 1.97208i
\(934\) 0 0
\(935\) −21960.3 + 38036.4i −0.768107 + 1.33040i
\(936\) 0 0
\(937\) 8870.66 + 15364.4i 0.309276 + 0.535682i 0.978204 0.207645i \(-0.0665798\pi\)
−0.668928 + 0.743327i \(0.733247\pi\)
\(938\) 0 0
\(939\) 158.923 275.262i 0.00552316 0.00956639i
\(940\) 0 0
\(941\) 2598.71 4501.09i 0.0900270 0.155931i −0.817495 0.575935i \(-0.804638\pi\)
0.907522 + 0.420004i \(0.137971\pi\)
\(942\) 0 0
\(943\) 18020.6 31212.7i 0.622304 1.07786i
\(944\) 0 0
\(945\) −52501.7 −1.80728
\(946\) 0 0
\(947\) −9149.42 15847.3i −0.313956 0.543787i 0.665259 0.746613i \(-0.268321\pi\)
−0.979215 + 0.202825i \(0.934988\pi\)
\(948\) 0 0
\(949\) −5256.23 9104.06i −0.179794 0.311412i
\(950\) 0 0
\(951\) 16094.9 27877.2i 0.548804 0.950556i
\(952\) 0 0
\(953\) −9410.33 −0.319864 −0.159932 0.987128i \(-0.551128\pi\)
−0.159932 + 0.987128i \(0.551128\pi\)
\(954\) 0 0
\(955\) 24848.8 + 43039.4i 0.841977 + 1.45835i
\(956\) 0 0
\(957\) 32258.8 1.08963
\(958\) 0 0
\(959\) −1826.39 −0.0614987
\(960\) 0 0
\(961\) 19670.7 22373.3i 0.660291 0.751010i
\(962\) 0 0
\(963\) −109688. −3.67045
\(964\) 0 0
\(965\) −61766.5 −2.06045
\(966\) 0 0
\(967\) 2.15592 + 3.73416i 7.16956e−5 + 0.000124180i 0.866061 0.499938i \(-0.166644\pi\)
−0.865990 + 0.500062i \(0.833311\pi\)
\(968\) 0 0
\(969\) 111975. 3.71223
\(970\) 0 0
\(971\) 6426.53 11131.1i 0.212397 0.367882i −0.740067 0.672533i \(-0.765206\pi\)
0.952464 + 0.304651i \(0.0985397\pi\)
\(972\) 0 0
\(973\) 12082.0 + 20926.7i 0.398081 + 0.689496i
\(974\) 0 0
\(975\) 12699.8 + 21996.7i 0.417147 + 0.722520i
\(976\) 0 0
\(977\) −38951.1 −1.27549 −0.637746 0.770247i \(-0.720133\pi\)
−0.637746 + 0.770247i \(0.720133\pi\)
\(978\) 0 0
\(979\) −5373.03 + 9306.37i −0.175406 + 0.303813i
\(980\) 0 0
\(981\) 21091.3 36531.2i 0.686436 1.18894i
\(982\) 0 0
\(983\) 6573.07 11384.9i 0.213274 0.369401i −0.739463 0.673197i \(-0.764921\pi\)
0.952737 + 0.303795i \(0.0982540\pi\)
\(984\) 0 0
\(985\) −28361.3 49123.2i −0.917427 1.58903i
\(986\) 0 0
\(987\) 2330.90 4037.25i 0.0751707 0.130200i
\(988\) 0 0
\(989\) −18606.1 32226.7i −0.598220 1.03615i
\(990\) 0 0
\(991\) −23070.3 −0.739509 −0.369755 0.929129i \(-0.620558\pi\)
−0.369755 + 0.929129i \(0.620558\pi\)
\(992\) 0 0
\(993\) −24137.2 −0.771370
\(994\) 0 0
\(995\) −12960.9 22448.9i −0.412952 0.715253i
\(996\) 0 0
\(997\) −21000.5 + 36374.0i −0.667094 + 1.15544i 0.311618 + 0.950207i \(0.399129\pi\)
−0.978713 + 0.205234i \(0.934204\pi\)
\(998\) 0 0
\(999\) 74731.2 + 129438.i 2.36676 + 4.09934i
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 124.4.e.b.25.1 yes 6
3.2 odd 2 1116.4.i.d.397.3 6
4.3 odd 2 496.4.i.b.273.3 6
31.5 even 3 inner 124.4.e.b.5.1 6
93.5 odd 6 1116.4.i.d.253.3 6
124.67 odd 6 496.4.i.b.129.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
124.4.e.b.5.1 6 31.5 even 3 inner
124.4.e.b.25.1 yes 6 1.1 even 1 trivial
496.4.i.b.129.3 6 124.67 odd 6
496.4.i.b.273.3 6 4.3 odd 2
1116.4.i.d.253.3 6 93.5 odd 6
1116.4.i.d.397.3 6 3.2 odd 2