Properties

Label 115.2.a.c.1.1
Level $115$
Weight $2$
Character 115.1
Self dual yes
Analytic conductor $0.918$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [115,2,Mod(1,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("115.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.a (trivial)

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: yes
Analytic conductor: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.15317.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.69353\) of defining polynomial
Character \(\chi\) \(=\) 115.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.69353 q^{2} -2.56155 q^{3} +0.868028 q^{4} +1.00000 q^{5} +4.33805 q^{6} -0.819031 q^{7} +1.91702 q^{8} +3.56155 q^{9} +O(q^{10})\) \(q-1.69353 q^{2} -2.56155 q^{3} +0.868028 q^{4} +1.00000 q^{5} +4.33805 q^{6} -0.819031 q^{7} +1.91702 q^{8} +3.56155 q^{9} -1.69353 q^{10} +5.38705 q^{11} -2.22350 q^{12} +2.46356 q^{13} +1.38705 q^{14} -2.56155 q^{15} -4.98258 q^{16} +4.20608 q^{17} -6.03158 q^{18} -5.38705 q^{19} +0.868028 q^{20} +2.09799 q^{21} -9.12311 q^{22} -1.00000 q^{23} -4.91056 q^{24} +1.00000 q^{25} -4.17210 q^{26} -1.43845 q^{27} -0.710942 q^{28} +2.35547 q^{29} +4.33805 q^{30} +6.66964 q^{31} +4.60408 q^{32} -13.7992 q^{33} -7.12311 q^{34} -0.819031 q^{35} +3.09153 q^{36} +7.42718 q^{37} +9.12311 q^{38} -6.31054 q^{39} +1.91702 q^{40} +5.64453 q^{41} -3.55300 q^{42} -1.90201 q^{43} +4.67611 q^{44} +3.56155 q^{45} +1.69353 q^{46} -9.33565 q^{47} +12.7632 q^{48} -6.32919 q^{49} -1.69353 q^{50} -10.7741 q^{51} +2.13844 q^{52} +13.8571 q^{53} +2.43605 q^{54} +5.38705 q^{55} -1.57010 q^{56} +13.7992 q^{57} -3.98905 q^{58} +5.27896 q^{59} -2.22350 q^{60} -8.51016 q^{61} -11.2952 q^{62} -2.91702 q^{63} +2.16804 q^{64} +2.46356 q^{65} +23.3693 q^{66} +4.10809 q^{67} +3.65100 q^{68} +2.56155 q^{69} +1.38705 q^{70} -11.7927 q^{71} +6.82758 q^{72} -10.4636 q^{73} -12.5781 q^{74} -2.56155 q^{75} -4.67611 q^{76} -4.41216 q^{77} +10.6871 q^{78} +13.7992 q^{79} -4.98258 q^{80} -7.00000 q^{81} -9.55915 q^{82} +3.32919 q^{83} +1.82112 q^{84} +4.20608 q^{85} +3.22110 q^{86} -6.03366 q^{87} +10.3271 q^{88} -3.58304 q^{89} -6.03158 q^{90} -2.01773 q^{91} -0.868028 q^{92} -17.0846 q^{93} +15.8102 q^{94} -5.38705 q^{95} -11.7936 q^{96} -9.02511 q^{97} +10.7186 q^{98} +19.1863 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 2 q^{3} + 4 q^{4} + 4 q^{5} - q^{6} - 3 q^{7} + 9 q^{8} + 6 q^{9} + 2 q^{10} + 4 q^{11} - 19 q^{12} - 12 q^{14} - 2 q^{15} + 8 q^{16} - q^{17} + 3 q^{18} - 4 q^{19} + 4 q^{20} + 10 q^{21} - 20 q^{22} - 4 q^{23} - 30 q^{24} + 4 q^{25} - q^{26} - 14 q^{27} - 22 q^{28} + 19 q^{29} - q^{30} - q^{31} + 20 q^{32} - 2 q^{33} - 12 q^{34} - 3 q^{35} + 23 q^{36} - 3 q^{37} + 20 q^{38} + 9 q^{40} + 13 q^{41} + 6 q^{42} - 6 q^{43} - 18 q^{44} + 6 q^{45} - 2 q^{46} + 6 q^{47} - 21 q^{48} + 9 q^{49} + 2 q^{50} - 8 q^{51} - q^{52} + 19 q^{53} - 7 q^{54} + 4 q^{55} - 10 q^{56} + 2 q^{57} + 21 q^{58} + 23 q^{59} - 19 q^{60} - 13 q^{62} - 13 q^{63} + 27 q^{64} + 44 q^{66} - 3 q^{67} - 4 q^{68} + 2 q^{69} - 12 q^{70} - 3 q^{71} + 39 q^{72} - 32 q^{73} - 12 q^{74} - 2 q^{75} + 18 q^{76} + 18 q^{77} + 43 q^{78} + 2 q^{79} + 8 q^{80} - 28 q^{81} - 5 q^{82} - 21 q^{83} + 28 q^{84} - q^{85} - 2 q^{86} - 18 q^{87} - 14 q^{88} + 3 q^{90} - 40 q^{91} - 4 q^{92} - 8 q^{93} + 47 q^{94} - 4 q^{95} - 61 q^{96} - 18 q^{97} + 16 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.69353 −1.19750 −0.598752 0.800935i \(-0.704336\pi\)
−0.598752 + 0.800935i \(0.704336\pi\)
\(3\) −2.56155 −1.47891 −0.739457 0.673204i \(-0.764917\pi\)
−0.739457 + 0.673204i \(0.764917\pi\)
\(4\) 0.868028 0.434014
\(5\) 1.00000 0.447214
\(6\) 4.33805 1.77100
\(7\) −0.819031 −0.309565 −0.154782 0.987949i \(-0.549468\pi\)
−0.154782 + 0.987949i \(0.549468\pi\)
\(8\) 1.91702 0.677770
\(9\) 3.56155 1.18718
\(10\) −1.69353 −0.535540
\(11\) 5.38705 1.62426 0.812128 0.583479i \(-0.198309\pi\)
0.812128 + 0.583479i \(0.198309\pi\)
\(12\) −2.22350 −0.641869
\(13\) 2.46356 0.683269 0.341634 0.939833i \(-0.389020\pi\)
0.341634 + 0.939833i \(0.389020\pi\)
\(14\) 1.38705 0.370705
\(15\) −2.56155 −0.661390
\(16\) −4.98258 −1.24565
\(17\) 4.20608 1.02012 0.510062 0.860137i \(-0.329622\pi\)
0.510062 + 0.860137i \(0.329622\pi\)
\(18\) −6.03158 −1.42166
\(19\) −5.38705 −1.23587 −0.617937 0.786228i \(-0.712031\pi\)
−0.617937 + 0.786228i \(0.712031\pi\)
\(20\) 0.868028 0.194097
\(21\) 2.09799 0.457819
\(22\) −9.12311 −1.94505
\(23\) −1.00000 −0.208514
\(24\) −4.91056 −1.00236
\(25\) 1.00000 0.200000
\(26\) −4.17210 −0.818216
\(27\) −1.43845 −0.276829
\(28\) −0.710942 −0.134355
\(29\) 2.35547 0.437400 0.218700 0.975792i \(-0.429818\pi\)
0.218700 + 0.975792i \(0.429818\pi\)
\(30\) 4.33805 0.792017
\(31\) 6.66964 1.19790 0.598952 0.800785i \(-0.295584\pi\)
0.598952 + 0.800785i \(0.295584\pi\)
\(32\) 4.60408 0.813895
\(33\) −13.7992 −2.40213
\(34\) −7.12311 −1.22160
\(35\) −0.819031 −0.138442
\(36\) 3.09153 0.515254
\(37\) 7.42718 1.22102 0.610510 0.792008i \(-0.290964\pi\)
0.610510 + 0.792008i \(0.290964\pi\)
\(38\) 9.12311 1.47996
\(39\) −6.31054 −1.01050
\(40\) 1.91702 0.303108
\(41\) 5.64453 0.881527 0.440764 0.897623i \(-0.354708\pi\)
0.440764 + 0.897623i \(0.354708\pi\)
\(42\) −3.55300 −0.548240
\(43\) −1.90201 −0.290053 −0.145027 0.989428i \(-0.546327\pi\)
−0.145027 + 0.989428i \(0.546327\pi\)
\(44\) 4.67611 0.704950
\(45\) 3.56155 0.530925
\(46\) 1.69353 0.249697
\(47\) −9.33565 −1.36175 −0.680873 0.732402i \(-0.738399\pi\)
−0.680873 + 0.732402i \(0.738399\pi\)
\(48\) 12.7632 1.84220
\(49\) −6.32919 −0.904170
\(50\) −1.69353 −0.239501
\(51\) −10.7741 −1.50868
\(52\) 2.13844 0.296548
\(53\) 13.8571 1.90342 0.951708 0.307005i \(-0.0993268\pi\)
0.951708 + 0.307005i \(0.0993268\pi\)
\(54\) 2.43605 0.331504
\(55\) 5.38705 0.726390
\(56\) −1.57010 −0.209814
\(57\) 13.7992 1.82775
\(58\) −3.98905 −0.523788
\(59\) 5.27896 0.687262 0.343631 0.939105i \(-0.388343\pi\)
0.343631 + 0.939105i \(0.388343\pi\)
\(60\) −2.22350 −0.287052
\(61\) −8.51016 −1.08961 −0.544807 0.838562i \(-0.683397\pi\)
−0.544807 + 0.838562i \(0.683397\pi\)
\(62\) −11.2952 −1.43449
\(63\) −2.91702 −0.367510
\(64\) 2.16804 0.271005
\(65\) 2.46356 0.305567
\(66\) 23.3693 2.87656
\(67\) 4.10809 0.501883 0.250942 0.968002i \(-0.419260\pi\)
0.250942 + 0.968002i \(0.419260\pi\)
\(68\) 3.65100 0.442748
\(69\) 2.56155 0.308375
\(70\) 1.38705 0.165784
\(71\) −11.7927 −1.39954 −0.699771 0.714367i \(-0.746715\pi\)
−0.699771 + 0.714367i \(0.746715\pi\)
\(72\) 6.82758 0.804638
\(73\) −10.4636 −1.22467 −0.612334 0.790600i \(-0.709769\pi\)
−0.612334 + 0.790600i \(0.709769\pi\)
\(74\) −12.5781 −1.46218
\(75\) −2.56155 −0.295783
\(76\) −4.67611 −0.536386
\(77\) −4.41216 −0.502813
\(78\) 10.6871 1.21007
\(79\) 13.7992 1.55253 0.776266 0.630405i \(-0.217111\pi\)
0.776266 + 0.630405i \(0.217111\pi\)
\(80\) −4.98258 −0.557070
\(81\) −7.00000 −0.777778
\(82\) −9.55915 −1.05563
\(83\) 3.32919 0.365426 0.182713 0.983166i \(-0.441512\pi\)
0.182713 + 0.983166i \(0.441512\pi\)
\(84\) 1.82112 0.198700
\(85\) 4.20608 0.456214
\(86\) 3.22110 0.347340
\(87\) −6.03366 −0.646877
\(88\) 10.3271 1.10087
\(89\) −3.58304 −0.379801 −0.189900 0.981803i \(-0.560817\pi\)
−0.189900 + 0.981803i \(0.560817\pi\)
\(90\) −6.03158 −0.635784
\(91\) −2.01773 −0.211516
\(92\) −0.868028 −0.0904981
\(93\) −17.0846 −1.77159
\(94\) 15.8102 1.63069
\(95\) −5.38705 −0.552700
\(96\) −11.7936 −1.20368
\(97\) −9.02511 −0.916361 −0.458181 0.888859i \(-0.651499\pi\)
−0.458181 + 0.888859i \(0.651499\pi\)
\(98\) 10.7186 1.08275
\(99\) 19.1863 1.92829
\(100\) 0.868028 0.0868028
\(101\) −12.1033 −1.20432 −0.602161 0.798375i \(-0.705694\pi\)
−0.602161 + 0.798375i \(0.705694\pi\)
\(102\) 18.2462 1.80664
\(103\) −3.80402 −0.374821 −0.187410 0.982282i \(-0.560009\pi\)
−0.187410 + 0.982282i \(0.560009\pi\)
\(104\) 4.72270 0.463099
\(105\) 2.09799 0.204743
\(106\) −23.4673 −2.27935
\(107\) −2.47003 −0.238786 −0.119393 0.992847i \(-0.538095\pi\)
−0.119393 + 0.992847i \(0.538095\pi\)
\(108\) −1.24861 −0.120148
\(109\) −4.49722 −0.430756 −0.215378 0.976531i \(-0.569098\pi\)
−0.215378 + 0.976531i \(0.569098\pi\)
\(110\) −9.12311 −0.869854
\(111\) −19.0251 −1.80578
\(112\) 4.08089 0.385608
\(113\) 6.30407 0.593037 0.296519 0.955027i \(-0.404174\pi\)
0.296519 + 0.955027i \(0.404174\pi\)
\(114\) −23.3693 −2.18874
\(115\) −1.00000 −0.0932505
\(116\) 2.04461 0.189838
\(117\) 8.77410 0.811166
\(118\) −8.94005 −0.822999
\(119\) −3.44491 −0.315795
\(120\) −4.91056 −0.448271
\(121\) 18.0203 1.63821
\(122\) 14.4122 1.30482
\(123\) −14.4588 −1.30370
\(124\) 5.78943 0.519906
\(125\) 1.00000 0.0894427
\(126\) 4.94005 0.440095
\(127\) −0.910558 −0.0807989 −0.0403995 0.999184i \(-0.512863\pi\)
−0.0403995 + 0.999184i \(0.512863\pi\)
\(128\) −12.8798 −1.13842
\(129\) 4.87209 0.428964
\(130\) −4.17210 −0.365918
\(131\) 18.6070 1.62570 0.812850 0.582474i \(-0.197915\pi\)
0.812850 + 0.582474i \(0.197915\pi\)
\(132\) −11.9781 −1.04256
\(133\) 4.41216 0.382583
\(134\) −6.95715 −0.601006
\(135\) −1.43845 −0.123802
\(136\) 8.06316 0.691410
\(137\) −14.4251 −1.23242 −0.616210 0.787582i \(-0.711333\pi\)
−0.616210 + 0.787582i \(0.711333\pi\)
\(138\) −4.33805 −0.369280
\(139\) 18.3507 1.55648 0.778242 0.627965i \(-0.216112\pi\)
0.778242 + 0.627965i \(0.216112\pi\)
\(140\) −0.710942 −0.0600856
\(141\) 23.9138 2.01390
\(142\) 19.9713 1.67596
\(143\) 13.2713 1.10980
\(144\) −17.7457 −1.47881
\(145\) 2.35547 0.195611
\(146\) 17.7203 1.46654
\(147\) 16.2125 1.33719
\(148\) 6.44700 0.529940
\(149\) 7.63806 0.625734 0.312867 0.949797i \(-0.398711\pi\)
0.312867 + 0.949797i \(0.398711\pi\)
\(150\) 4.33805 0.354201
\(151\) −0.659545 −0.0536730 −0.0268365 0.999640i \(-0.508543\pi\)
−0.0268365 + 0.999640i \(0.508543\pi\)
\(152\) −10.3271 −0.837639
\(153\) 14.9802 1.21108
\(154\) 7.47211 0.602120
\(155\) 6.66964 0.535719
\(156\) −5.47772 −0.438569
\(157\) 3.94214 0.314617 0.157308 0.987550i \(-0.449718\pi\)
0.157308 + 0.987550i \(0.449718\pi\)
\(158\) −23.3693 −1.85916
\(159\) −35.4956 −2.81499
\(160\) 4.60408 0.363985
\(161\) 0.819031 0.0645487
\(162\) 11.8547 0.931391
\(163\) −3.80038 −0.297669 −0.148835 0.988862i \(-0.547552\pi\)
−0.148835 + 0.988862i \(0.547552\pi\)
\(164\) 4.89961 0.382595
\(165\) −13.7992 −1.07427
\(166\) −5.63806 −0.437599
\(167\) −2.24621 −0.173817 −0.0869085 0.996216i \(-0.527699\pi\)
−0.0869085 + 0.996216i \(0.527699\pi\)
\(168\) 4.02190 0.310296
\(169\) −6.93087 −0.533144
\(170\) −7.12311 −0.546317
\(171\) −19.1863 −1.46721
\(172\) −1.65100 −0.125887
\(173\) −13.3822 −1.01743 −0.508717 0.860934i \(-0.669880\pi\)
−0.508717 + 0.860934i \(0.669880\pi\)
\(174\) 10.2182 0.774637
\(175\) −0.819031 −0.0619130
\(176\) −26.8414 −2.02325
\(177\) −13.5223 −1.01640
\(178\) 6.06796 0.454813
\(179\) 18.2077 1.36091 0.680455 0.732789i \(-0.261782\pi\)
0.680455 + 0.732789i \(0.261782\pi\)
\(180\) 3.09153 0.230429
\(181\) −1.01218 −0.0752348 −0.0376174 0.999292i \(-0.511977\pi\)
−0.0376174 + 0.999292i \(0.511977\pi\)
\(182\) 3.41708 0.253291
\(183\) 21.7992 1.61144
\(184\) −1.91702 −0.141325
\(185\) 7.42718 0.546057
\(186\) 28.9333 2.12149
\(187\) 22.6584 1.65694
\(188\) −8.10361 −0.591016
\(189\) 1.17813 0.0856966
\(190\) 9.12311 0.661860
\(191\) −13.1231 −0.949555 −0.474777 0.880106i \(-0.657471\pi\)
−0.474777 + 0.880106i \(0.657471\pi\)
\(192\) −5.55354 −0.400792
\(193\) −21.3179 −1.53450 −0.767249 0.641350i \(-0.778375\pi\)
−0.767249 + 0.641350i \(0.778375\pi\)
\(194\) 15.2843 1.09735
\(195\) −6.31054 −0.451907
\(196\) −5.49391 −0.392422
\(197\) 26.3608 1.87813 0.939063 0.343744i \(-0.111695\pi\)
0.939063 + 0.343744i \(0.111695\pi\)
\(198\) −32.4924 −2.30914
\(199\) −4.16115 −0.294976 −0.147488 0.989064i \(-0.547119\pi\)
−0.147488 + 0.989064i \(0.547119\pi\)
\(200\) 1.91702 0.135554
\(201\) −10.5231 −0.742241
\(202\) 20.4972 1.44218
\(203\) −1.92920 −0.135404
\(204\) −9.35222 −0.654786
\(205\) 5.64453 0.394231
\(206\) 6.44220 0.448849
\(207\) −3.56155 −0.247545
\(208\) −12.2749 −0.851111
\(209\) −29.0203 −2.00738
\(210\) −3.55300 −0.245180
\(211\) −15.7414 −1.08368 −0.541840 0.840482i \(-0.682272\pi\)
−0.541840 + 0.840482i \(0.682272\pi\)
\(212\) 12.0283 0.826109
\(213\) 30.2077 2.06980
\(214\) 4.18305 0.285948
\(215\) −1.90201 −0.129716
\(216\) −2.75754 −0.187627
\(217\) −5.46265 −0.370829
\(218\) 7.61616 0.515832
\(219\) 26.8030 1.81118
\(220\) 4.67611 0.315263
\(221\) 10.3619 0.697019
\(222\) 32.2195 2.16243
\(223\) 0.0802588 0.00537453 0.00268726 0.999996i \(-0.499145\pi\)
0.00268726 + 0.999996i \(0.499145\pi\)
\(224\) −3.77089 −0.251953
\(225\) 3.56155 0.237437
\(226\) −10.6761 −0.710164
\(227\) −28.4705 −1.88965 −0.944827 0.327568i \(-0.893771\pi\)
−0.944827 + 0.327568i \(0.893771\pi\)
\(228\) 11.9781 0.793269
\(229\) −0.612950 −0.0405048 −0.0202524 0.999795i \(-0.506447\pi\)
−0.0202524 + 0.999795i \(0.506447\pi\)
\(230\) 1.69353 0.111668
\(231\) 11.3020 0.743616
\(232\) 4.51549 0.296457
\(233\) 3.79558 0.248657 0.124328 0.992241i \(-0.460322\pi\)
0.124328 + 0.992241i \(0.460322\pi\)
\(234\) −14.8592 −0.971374
\(235\) −9.33565 −0.608991
\(236\) 4.58228 0.298281
\(237\) −35.3474 −2.29606
\(238\) 5.83405 0.378165
\(239\) 21.1118 1.36561 0.682806 0.730600i \(-0.260760\pi\)
0.682806 + 0.730600i \(0.260760\pi\)
\(240\) 12.7632 0.823858
\(241\) −0.774101 −0.0498642 −0.0249321 0.999689i \(-0.507937\pi\)
−0.0249321 + 0.999689i \(0.507937\pi\)
\(242\) −30.5179 −1.96176
\(243\) 22.2462 1.42710
\(244\) −7.38705 −0.472907
\(245\) −6.32919 −0.404357
\(246\) 24.4863 1.56119
\(247\) −13.2713 −0.844434
\(248\) 12.7859 0.811903
\(249\) −8.52789 −0.540433
\(250\) −1.69353 −0.107108
\(251\) −16.8041 −1.06067 −0.530334 0.847789i \(-0.677933\pi\)
−0.530334 + 0.847789i \(0.677933\pi\)
\(252\) −2.53206 −0.159505
\(253\) −5.38705 −0.338681
\(254\) 1.54205 0.0967570
\(255\) −10.7741 −0.674700
\(256\) 17.4762 1.09226
\(257\) −5.12428 −0.319644 −0.159822 0.987146i \(-0.551092\pi\)
−0.159822 + 0.987146i \(0.551092\pi\)
\(258\) −8.25101 −0.513686
\(259\) −6.08309 −0.377985
\(260\) 2.13844 0.132620
\(261\) 8.38913 0.519274
\(262\) −31.5114 −1.94678
\(263\) 6.97205 0.429915 0.214958 0.976623i \(-0.431039\pi\)
0.214958 + 0.976623i \(0.431039\pi\)
\(264\) −26.4534 −1.62810
\(265\) 13.8571 0.851233
\(266\) −7.47211 −0.458144
\(267\) 9.17813 0.561693
\(268\) 3.56594 0.217824
\(269\) −21.3328 −1.30068 −0.650342 0.759641i \(-0.725375\pi\)
−0.650342 + 0.759641i \(0.725375\pi\)
\(270\) 2.43605 0.148253
\(271\) −0.751071 −0.0456243 −0.0228122 0.999740i \(-0.507262\pi\)
−0.0228122 + 0.999740i \(0.507262\pi\)
\(272\) −20.9572 −1.27071
\(273\) 5.16853 0.312814
\(274\) 24.4293 1.47583
\(275\) 5.38705 0.324851
\(276\) 2.22350 0.133839
\(277\) −8.20775 −0.493156 −0.246578 0.969123i \(-0.579306\pi\)
−0.246578 + 0.969123i \(0.579306\pi\)
\(278\) −31.0773 −1.86389
\(279\) 23.7543 1.42213
\(280\) −1.57010 −0.0938316
\(281\) 14.9749 0.893327 0.446663 0.894702i \(-0.352612\pi\)
0.446663 + 0.894702i \(0.352612\pi\)
\(282\) −40.4986 −2.41165
\(283\) −23.4224 −1.39232 −0.696158 0.717889i \(-0.745109\pi\)
−0.696158 + 0.717889i \(0.745109\pi\)
\(284\) −10.2364 −0.607420
\(285\) 13.7992 0.817395
\(286\) −22.4753 −1.32899
\(287\) −4.62305 −0.272890
\(288\) 16.3977 0.966243
\(289\) 0.691125 0.0406544
\(290\) −3.98905 −0.234245
\(291\) 23.1183 1.35522
\(292\) −9.08266 −0.531522
\(293\) 14.9600 0.873972 0.436986 0.899468i \(-0.356046\pi\)
0.436986 + 0.899468i \(0.356046\pi\)
\(294\) −27.4564 −1.60129
\(295\) 5.27896 0.307353
\(296\) 14.2381 0.827572
\(297\) −7.74899 −0.449642
\(298\) −12.9353 −0.749319
\(299\) −2.46356 −0.142471
\(300\) −2.22350 −0.128374
\(301\) 1.55780 0.0897903
\(302\) 1.11696 0.0642736
\(303\) 31.0032 1.78109
\(304\) 26.8414 1.53946
\(305\) −8.51016 −0.487290
\(306\) −25.3693 −1.45027
\(307\) −19.1863 −1.09502 −0.547509 0.836800i \(-0.684424\pi\)
−0.547509 + 0.836800i \(0.684424\pi\)
\(308\) −3.82988 −0.218228
\(309\) 9.74419 0.554327
\(310\) −11.2952 −0.641525
\(311\) −19.4336 −1.10198 −0.550990 0.834512i \(-0.685750\pi\)
−0.550990 + 0.834512i \(0.685750\pi\)
\(312\) −12.0975 −0.684884
\(313\) −20.9300 −1.18303 −0.591516 0.806294i \(-0.701470\pi\)
−0.591516 + 0.806294i \(0.701470\pi\)
\(314\) −6.67611 −0.376755
\(315\) −2.91702 −0.164356
\(316\) 11.9781 0.673821
\(317\) 10.8041 0.606821 0.303410 0.952860i \(-0.401875\pi\)
0.303410 + 0.952860i \(0.401875\pi\)
\(318\) 60.1128 3.37096
\(319\) 12.6890 0.710450
\(320\) 2.16804 0.121197
\(321\) 6.32710 0.353144
\(322\) −1.38705 −0.0772973
\(323\) −22.6584 −1.26075
\(324\) −6.07619 −0.337566
\(325\) 2.46356 0.136654
\(326\) 6.43605 0.356460
\(327\) 11.5199 0.637051
\(328\) 10.8207 0.597473
\(329\) 7.64619 0.421548
\(330\) 23.3693 1.28644
\(331\) 22.7628 1.25116 0.625579 0.780161i \(-0.284863\pi\)
0.625579 + 0.780161i \(0.284863\pi\)
\(332\) 2.88983 0.158600
\(333\) 26.4523 1.44958
\(334\) 3.80402 0.208146
\(335\) 4.10809 0.224449
\(336\) −10.4534 −0.570281
\(337\) −2.97489 −0.162052 −0.0810262 0.996712i \(-0.525820\pi\)
−0.0810262 + 0.996712i \(0.525820\pi\)
\(338\) 11.7376 0.638441
\(339\) −16.1482 −0.877051
\(340\) 3.65100 0.198003
\(341\) 35.9297 1.94570
\(342\) 32.4924 1.75699
\(343\) 10.9170 0.589464
\(344\) −3.64619 −0.196590
\(345\) 2.56155 0.137909
\(346\) 22.6632 1.21838
\(347\) 35.4024 1.90050 0.950251 0.311484i \(-0.100826\pi\)
0.950251 + 0.311484i \(0.100826\pi\)
\(348\) −5.23739 −0.280753
\(349\) 22.4818 1.20342 0.601711 0.798714i \(-0.294486\pi\)
0.601711 + 0.798714i \(0.294486\pi\)
\(350\) 1.38705 0.0741410
\(351\) −3.54370 −0.189149
\(352\) 24.8024 1.32197
\(353\) 6.28051 0.334278 0.167139 0.985933i \(-0.446547\pi\)
0.167139 + 0.985933i \(0.446547\pi\)
\(354\) 22.9004 1.21714
\(355\) −11.7927 −0.625894
\(356\) −3.11017 −0.164839
\(357\) 8.82433 0.467033
\(358\) −30.8353 −1.62970
\(359\) 7.35702 0.388289 0.194144 0.980973i \(-0.437807\pi\)
0.194144 + 0.980973i \(0.437807\pi\)
\(360\) 6.82758 0.359845
\(361\) 10.0203 0.527385
\(362\) 1.71415 0.0900939
\(363\) −46.1600 −2.42277
\(364\) −1.75145 −0.0918008
\(365\) −10.4636 −0.547688
\(366\) −36.9175 −1.92971
\(367\) −0.886992 −0.0463006 −0.0231503 0.999732i \(-0.507370\pi\)
−0.0231503 + 0.999732i \(0.507370\pi\)
\(368\) 4.98258 0.259735
\(369\) 20.1033 1.04654
\(370\) −12.5781 −0.653905
\(371\) −11.3494 −0.589231
\(372\) −14.8299 −0.768897
\(373\) −14.6283 −0.757427 −0.378713 0.925514i \(-0.623633\pi\)
−0.378713 + 0.925514i \(0.623633\pi\)
\(374\) −38.3725 −1.98420
\(375\) −2.56155 −0.132278
\(376\) −17.8967 −0.922950
\(377\) 5.80285 0.298862
\(378\) −1.99520 −0.102622
\(379\) −32.5183 −1.67035 −0.835176 0.549983i \(-0.814634\pi\)
−0.835176 + 0.549983i \(0.814634\pi\)
\(380\) −4.67611 −0.239879
\(381\) 2.33244 0.119495
\(382\) 22.2243 1.13709
\(383\) 21.8393 1.11594 0.557969 0.829862i \(-0.311581\pi\)
0.557969 + 0.829862i \(0.311581\pi\)
\(384\) 32.9923 1.68363
\(385\) −4.41216 −0.224865
\(386\) 36.1024 1.83757
\(387\) −6.77410 −0.344347
\(388\) −7.83405 −0.397714
\(389\) −15.1490 −0.768083 −0.384042 0.923316i \(-0.625468\pi\)
−0.384042 + 0.923316i \(0.625468\pi\)
\(390\) 10.6871 0.541160
\(391\) −4.20608 −0.212711
\(392\) −12.1332 −0.612819
\(393\) −47.6628 −2.40427
\(394\) −44.6426 −2.24906
\(395\) 13.7992 0.694314
\(396\) 16.6542 0.836905
\(397\) −7.70239 −0.386572 −0.193286 0.981142i \(-0.561915\pi\)
−0.193286 + 0.981142i \(0.561915\pi\)
\(398\) 7.04701 0.353235
\(399\) −11.3020 −0.565807
\(400\) −4.98258 −0.249129
\(401\) −23.7442 −1.18573 −0.592864 0.805303i \(-0.702003\pi\)
−0.592864 + 0.805303i \(0.702003\pi\)
\(402\) 17.8211 0.888836
\(403\) 16.4311 0.818490
\(404\) −10.5060 −0.522692
\(405\) −7.00000 −0.347833
\(406\) 3.26716 0.162146
\(407\) 40.0106 1.98325
\(408\) −20.6542 −1.02254
\(409\) 10.2397 0.506323 0.253161 0.967424i \(-0.418530\pi\)
0.253161 + 0.967424i \(0.418530\pi\)
\(410\) −9.55915 −0.472093
\(411\) 36.9506 1.82264
\(412\) −3.30199 −0.162677
\(413\) −4.32363 −0.212752
\(414\) 6.03158 0.296436
\(415\) 3.32919 0.163423
\(416\) 11.3424 0.556109
\(417\) −47.0062 −2.30190
\(418\) 49.1466 2.40384
\(419\) −9.98227 −0.487666 −0.243833 0.969817i \(-0.578405\pi\)
−0.243833 + 0.969817i \(0.578405\pi\)
\(420\) 1.82112 0.0888613
\(421\) −17.2470 −0.840566 −0.420283 0.907393i \(-0.638069\pi\)
−0.420283 + 0.907393i \(0.638069\pi\)
\(422\) 26.6584 1.29771
\(423\) −33.2494 −1.61664
\(424\) 26.5643 1.29008
\(425\) 4.20608 0.204025
\(426\) −51.1576 −2.47859
\(427\) 6.97009 0.337306
\(428\) −2.14405 −0.103637
\(429\) −33.9952 −1.64130
\(430\) 3.22110 0.155335
\(431\) −3.15377 −0.151912 −0.0759559 0.997111i \(-0.524201\pi\)
−0.0759559 + 0.997111i \(0.524201\pi\)
\(432\) 7.16718 0.344831
\(433\) −26.1333 −1.25589 −0.627944 0.778259i \(-0.716103\pi\)
−0.627944 + 0.778259i \(0.716103\pi\)
\(434\) 9.25113 0.444068
\(435\) −6.03366 −0.289292
\(436\) −3.90371 −0.186954
\(437\) 5.38705 0.257698
\(438\) −45.3915 −2.16889
\(439\) 2.08143 0.0993412 0.0496706 0.998766i \(-0.484183\pi\)
0.0496706 + 0.998766i \(0.484183\pi\)
\(440\) 10.3271 0.492325
\(441\) −22.5417 −1.07342
\(442\) −17.5482 −0.834683
\(443\) −12.2799 −0.583434 −0.291717 0.956505i \(-0.594227\pi\)
−0.291717 + 0.956505i \(0.594227\pi\)
\(444\) −16.5143 −0.783735
\(445\) −3.58304 −0.169852
\(446\) −0.135920 −0.00643601
\(447\) −19.5653 −0.925407
\(448\) −1.77569 −0.0838935
\(449\) −19.3794 −0.914571 −0.457286 0.889320i \(-0.651178\pi\)
−0.457286 + 0.889320i \(0.651178\pi\)
\(450\) −6.03158 −0.284331
\(451\) 30.4074 1.43183
\(452\) 5.47211 0.257386
\(453\) 1.68946 0.0793777
\(454\) 48.2155 2.26287
\(455\) −2.01773 −0.0945928
\(456\) 26.4534 1.23879
\(457\) −2.65788 −0.124330 −0.0621652 0.998066i \(-0.519801\pi\)
−0.0621652 + 0.998066i \(0.519801\pi\)
\(458\) 1.03805 0.0485047
\(459\) −6.05023 −0.282400
\(460\) −0.868028 −0.0404720
\(461\) 8.49423 0.395616 0.197808 0.980241i \(-0.436618\pi\)
0.197808 + 0.980241i \(0.436618\pi\)
\(462\) −19.1402 −0.890483
\(463\) 15.1604 0.704564 0.352282 0.935894i \(-0.385406\pi\)
0.352282 + 0.935894i \(0.385406\pi\)
\(464\) −11.7363 −0.544845
\(465\) −17.0846 −0.792281
\(466\) −6.42792 −0.297767
\(467\) −12.7039 −0.587868 −0.293934 0.955826i \(-0.594965\pi\)
−0.293934 + 0.955826i \(0.594965\pi\)
\(468\) 7.61616 0.352057
\(469\) −3.36465 −0.155365
\(470\) 15.8102 0.729269
\(471\) −10.0980 −0.465291
\(472\) 10.1199 0.465806
\(473\) −10.2462 −0.471121
\(474\) 59.8617 2.74954
\(475\) −5.38705 −0.247175
\(476\) −2.99028 −0.137059
\(477\) 49.3527 2.25971
\(478\) −35.7534 −1.63532
\(479\) −11.6681 −0.533129 −0.266564 0.963817i \(-0.585889\pi\)
−0.266564 + 0.963817i \(0.585889\pi\)
\(480\) −11.7936 −0.538302
\(481\) 18.2973 0.834285
\(482\) 1.31096 0.0597126
\(483\) −2.09799 −0.0954620
\(484\) 15.6421 0.711006
\(485\) −9.02511 −0.409809
\(486\) −37.6745 −1.70895
\(487\) 31.7778 1.43999 0.719996 0.693978i \(-0.244144\pi\)
0.719996 + 0.693978i \(0.244144\pi\)
\(488\) −16.3142 −0.738508
\(489\) 9.73489 0.440227
\(490\) 10.7186 0.484219
\(491\) −6.17722 −0.278774 −0.139387 0.990238i \(-0.544513\pi\)
−0.139387 + 0.990238i \(0.544513\pi\)
\(492\) −12.5506 −0.565825
\(493\) 9.90730 0.446203
\(494\) 22.4753 1.01121
\(495\) 19.1863 0.862358
\(496\) −33.2320 −1.49216
\(497\) 9.65863 0.433249
\(498\) 14.4422 0.647170
\(499\) −37.3409 −1.67161 −0.835805 0.549026i \(-0.814999\pi\)
−0.835805 + 0.549026i \(0.814999\pi\)
\(500\) 0.868028 0.0388194
\(501\) 5.75379 0.257060
\(502\) 28.4582 1.27015
\(503\) 9.36648 0.417631 0.208815 0.977955i \(-0.433039\pi\)
0.208815 + 0.977955i \(0.433039\pi\)
\(504\) −5.59200 −0.249088
\(505\) −12.1033 −0.538589
\(506\) 9.12311 0.405572
\(507\) 17.7538 0.788473
\(508\) −0.790389 −0.0350678
\(509\) −21.0798 −0.934347 −0.467174 0.884166i \(-0.654728\pi\)
−0.467174 + 0.884166i \(0.654728\pi\)
\(510\) 18.2462 0.807956
\(511\) 8.56999 0.379114
\(512\) −3.83676 −0.169563
\(513\) 7.74899 0.342126
\(514\) 8.67809 0.382774
\(515\) −3.80402 −0.167625
\(516\) 4.22911 0.186176
\(517\) −50.2916 −2.21182
\(518\) 10.3019 0.452638
\(519\) 34.2793 1.50470
\(520\) 4.72270 0.207104
\(521\) −8.87689 −0.388904 −0.194452 0.980912i \(-0.562293\pi\)
−0.194452 + 0.980912i \(0.562293\pi\)
\(522\) −14.2072 −0.621833
\(523\) −2.82187 −0.123392 −0.0616958 0.998095i \(-0.519651\pi\)
−0.0616958 + 0.998095i \(0.519651\pi\)
\(524\) 16.1514 0.705576
\(525\) 2.09799 0.0915639
\(526\) −11.8073 −0.514825
\(527\) 28.0531 1.22201
\(528\) 68.7557 2.99221
\(529\) 1.00000 0.0434783
\(530\) −23.4673 −1.01935
\(531\) 18.8013 0.815907
\(532\) 3.82988 0.166046
\(533\) 13.9056 0.602320
\(534\) −15.5434 −0.672629
\(535\) −2.47003 −0.106789
\(536\) 7.87531 0.340161
\(537\) −46.6401 −2.01267
\(538\) 36.1277 1.55757
\(539\) −34.0957 −1.46860
\(540\) −1.24861 −0.0537317
\(541\) 21.3786 0.919139 0.459569 0.888142i \(-0.348004\pi\)
0.459569 + 0.888142i \(0.348004\pi\)
\(542\) 1.27196 0.0546353
\(543\) 2.59276 0.111266
\(544\) 19.3651 0.830274
\(545\) −4.49722 −0.192640
\(546\) −8.75304 −0.374595
\(547\) 7.28543 0.311502 0.155751 0.987796i \(-0.450220\pi\)
0.155751 + 0.987796i \(0.450220\pi\)
\(548\) −12.5214 −0.534887
\(549\) −30.3094 −1.29357
\(550\) −9.12311 −0.389011
\(551\) −12.6890 −0.540571
\(552\) 4.91056 0.209007
\(553\) −11.3020 −0.480610
\(554\) 13.9000 0.590556
\(555\) −19.0251 −0.807571
\(556\) 15.9289 0.675535
\(557\) 41.4101 1.75460 0.877301 0.479941i \(-0.159342\pi\)
0.877301 + 0.479941i \(0.159342\pi\)
\(558\) −40.2285 −1.70301
\(559\) −4.68571 −0.198184
\(560\) 4.08089 0.172449
\(561\) −58.0406 −2.45048
\(562\) −25.3603 −1.06976
\(563\) 21.5210 0.907002 0.453501 0.891256i \(-0.350175\pi\)
0.453501 + 0.891256i \(0.350175\pi\)
\(564\) 20.7578 0.874062
\(565\) 6.30407 0.265214
\(566\) 39.6664 1.66730
\(567\) 5.73322 0.240773
\(568\) −22.6070 −0.948568
\(569\) 21.9645 0.920801 0.460401 0.887711i \(-0.347706\pi\)
0.460401 + 0.887711i \(0.347706\pi\)
\(570\) −23.3693 −0.978833
\(571\) −24.4924 −1.02498 −0.512488 0.858694i \(-0.671276\pi\)
−0.512488 + 0.858694i \(0.671276\pi\)
\(572\) 11.5199 0.481670
\(573\) 33.6155 1.40431
\(574\) 7.82925 0.326786
\(575\) −1.00000 −0.0417029
\(576\) 7.72158 0.321732
\(577\) −10.0159 −0.416969 −0.208484 0.978026i \(-0.566853\pi\)
−0.208484 + 0.978026i \(0.566853\pi\)
\(578\) −1.17044 −0.0486838
\(579\) 54.6070 2.26939
\(580\) 2.04461 0.0848980
\(581\) −2.72671 −0.113123
\(582\) −39.1514 −1.62288
\(583\) 74.6488 3.09164
\(584\) −20.0589 −0.830043
\(585\) 8.77410 0.362764
\(586\) −25.3351 −1.04658
\(587\) −5.64736 −0.233092 −0.116546 0.993185i \(-0.537182\pi\)
−0.116546 + 0.993185i \(0.537182\pi\)
\(588\) 14.0729 0.580358
\(589\) −35.9297 −1.48046
\(590\) −8.94005 −0.368056
\(591\) −67.5245 −2.77759
\(592\) −37.0065 −1.52096
\(593\) 12.4624 0.511769 0.255885 0.966707i \(-0.417633\pi\)
0.255885 + 0.966707i \(0.417633\pi\)
\(594\) 13.1231 0.538448
\(595\) −3.44491 −0.141228
\(596\) 6.63005 0.271577
\(597\) 10.6590 0.436244
\(598\) 4.17210 0.170610
\(599\) 24.7985 1.01324 0.506619 0.862170i \(-0.330895\pi\)
0.506619 + 0.862170i \(0.330895\pi\)
\(600\) −4.91056 −0.200473
\(601\) 9.10371 0.371348 0.185674 0.982611i \(-0.440553\pi\)
0.185674 + 0.982611i \(0.440553\pi\)
\(602\) −2.63818 −0.107524
\(603\) 14.6312 0.595828
\(604\) −0.572503 −0.0232948
\(605\) 18.0203 0.732630
\(606\) −52.5047 −2.13286
\(607\) 20.6283 0.837279 0.418639 0.908153i \(-0.362507\pi\)
0.418639 + 0.908153i \(0.362507\pi\)
\(608\) −24.8024 −1.00587
\(609\) 4.94176 0.200250
\(610\) 14.4122 0.583531
\(611\) −22.9989 −0.930438
\(612\) 13.0032 0.525624
\(613\) −11.7514 −0.474637 −0.237318 0.971432i \(-0.576268\pi\)
−0.237318 + 0.971432i \(0.576268\pi\)
\(614\) 32.4924 1.31129
\(615\) −14.4588 −0.583033
\(616\) −8.45822 −0.340792
\(617\) 0.462764 0.0186302 0.00931510 0.999957i \(-0.497035\pi\)
0.00931510 + 0.999957i \(0.497035\pi\)
\(618\) −16.5020 −0.663809
\(619\) −21.7183 −0.872933 −0.436467 0.899720i \(-0.643770\pi\)
−0.436467 + 0.899720i \(0.643770\pi\)
\(620\) 5.78943 0.232509
\(621\) 1.43845 0.0577229
\(622\) 32.9114 1.31963
\(623\) 2.93462 0.117573
\(624\) 31.4428 1.25872
\(625\) 1.00000 0.0400000
\(626\) 35.4454 1.41668
\(627\) 74.3371 2.96874
\(628\) 3.42188 0.136548
\(629\) 31.2393 1.24559
\(630\) 4.94005 0.196816
\(631\) −4.18369 −0.166550 −0.0832750 0.996527i \(-0.526538\pi\)
−0.0832750 + 0.996527i \(0.526538\pi\)
\(632\) 26.4534 1.05226
\(633\) 40.3223 1.60267
\(634\) −18.2971 −0.726670
\(635\) −0.910558 −0.0361344
\(636\) −30.8112 −1.22174
\(637\) −15.5923 −0.617791
\(638\) −21.4892 −0.850766
\(639\) −42.0005 −1.66151
\(640\) −12.8798 −0.509118
\(641\) −13.6535 −0.539279 −0.269640 0.962961i \(-0.586905\pi\)
−0.269640 + 0.962961i \(0.586905\pi\)
\(642\) −10.7151 −0.422892
\(643\) 39.5583 1.56003 0.780014 0.625763i \(-0.215212\pi\)
0.780014 + 0.625763i \(0.215212\pi\)
\(644\) 0.710942 0.0280150
\(645\) 4.87209 0.191838
\(646\) 38.3725 1.50975
\(647\) 9.12674 0.358809 0.179405 0.983775i \(-0.442583\pi\)
0.179405 + 0.983775i \(0.442583\pi\)
\(648\) −13.4192 −0.527155
\(649\) 28.4380 1.11629
\(650\) −4.17210 −0.163643
\(651\) 13.9929 0.548423
\(652\) −3.29884 −0.129192
\(653\) 0.745239 0.0291634 0.0145817 0.999894i \(-0.495358\pi\)
0.0145817 + 0.999894i \(0.495358\pi\)
\(654\) −19.5092 −0.762870
\(655\) 18.6070 0.727035
\(656\) −28.1243 −1.09807
\(657\) −37.2665 −1.45391
\(658\) −12.9490 −0.504805
\(659\) −33.6285 −1.30998 −0.654989 0.755638i \(-0.727327\pi\)
−0.654989 + 0.755638i \(0.727327\pi\)
\(660\) −11.9781 −0.466247
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) −38.5494 −1.49827
\(663\) −26.5426 −1.03083
\(664\) 6.38213 0.247675
\(665\) 4.41216 0.171096
\(666\) −44.7976 −1.73587
\(667\) −2.35547 −0.0912042
\(668\) −1.94977 −0.0754390
\(669\) −0.205587 −0.00794846
\(670\) −6.95715 −0.268778
\(671\) −45.8446 −1.76981
\(672\) 9.65933 0.372617
\(673\) 19.5641 0.754142 0.377071 0.926184i \(-0.376931\pi\)
0.377071 + 0.926184i \(0.376931\pi\)
\(674\) 5.03805 0.194058
\(675\) −1.43845 −0.0553659
\(676\) −6.01619 −0.231392
\(677\) −12.7412 −0.489685 −0.244843 0.969563i \(-0.578736\pi\)
−0.244843 + 0.969563i \(0.578736\pi\)
\(678\) 27.3474 1.05027
\(679\) 7.39185 0.283673
\(680\) 8.06316 0.309208
\(681\) 72.9287 2.79464
\(682\) −60.8479 −2.32998
\(683\) −12.5186 −0.479010 −0.239505 0.970895i \(-0.576985\pi\)
−0.239505 + 0.970895i \(0.576985\pi\)
\(684\) −16.6542 −0.636790
\(685\) −14.4251 −0.551155
\(686\) −18.4883 −0.705885
\(687\) 1.57010 0.0599032
\(688\) 9.47691 0.361304
\(689\) 34.1377 1.30054
\(690\) −4.33805 −0.165147
\(691\) 3.70356 0.140890 0.0704451 0.997516i \(-0.477558\pi\)
0.0704451 + 0.997516i \(0.477558\pi\)
\(692\) −11.6162 −0.441580
\(693\) −15.7142 −0.596931
\(694\) −59.9549 −2.27586
\(695\) 18.3507 0.696081
\(696\) −11.5667 −0.438434
\(697\) 23.7414 0.899268
\(698\) −38.0735 −1.44110
\(699\) −9.72259 −0.367742
\(700\) −0.710942 −0.0268711
\(701\) 12.6388 0.477361 0.238681 0.971098i \(-0.423285\pi\)
0.238681 + 0.971098i \(0.423285\pi\)
\(702\) 6.00135 0.226506
\(703\) −40.0106 −1.50903
\(704\) 11.6793 0.440181
\(705\) 23.9138 0.900645
\(706\) −10.6362 −0.400299
\(707\) 9.91297 0.372816
\(708\) −11.7378 −0.441132
\(709\) 21.5978 0.811122 0.405561 0.914068i \(-0.367076\pi\)
0.405561 + 0.914068i \(0.367076\pi\)
\(710\) 19.9713 0.749510
\(711\) 49.1466 1.84314
\(712\) −6.86876 −0.257418
\(713\) −6.66964 −0.249780
\(714\) −14.9442 −0.559273
\(715\) 13.2713 0.496319
\(716\) 15.8048 0.590654
\(717\) −54.0791 −2.01962
\(718\) −12.4593 −0.464977
\(719\) −31.8418 −1.18750 −0.593749 0.804650i \(-0.702353\pi\)
−0.593749 + 0.804650i \(0.702353\pi\)
\(720\) −17.7457 −0.661344
\(721\) 3.11561 0.116031
\(722\) −16.9697 −0.631545
\(723\) 1.98290 0.0737448
\(724\) −0.878601 −0.0326530
\(725\) 2.35547 0.0874800
\(726\) 78.1731 2.90128
\(727\) −32.0507 −1.18870 −0.594348 0.804208i \(-0.702590\pi\)
−0.594348 + 0.804208i \(0.702590\pi\)
\(728\) −3.86804 −0.143359
\(729\) −35.9848 −1.33277
\(730\) 17.7203 0.655858
\(731\) −8.00000 −0.295891
\(732\) 18.9223 0.699389
\(733\) −35.9202 −1.32674 −0.663372 0.748290i \(-0.730875\pi\)
−0.663372 + 0.748290i \(0.730875\pi\)
\(734\) 1.50214 0.0554451
\(735\) 16.2125 0.598009
\(736\) −4.60408 −0.169709
\(737\) 22.1305 0.815187
\(738\) −34.0454 −1.25323
\(739\) 1.74252 0.0640997 0.0320498 0.999486i \(-0.489796\pi\)
0.0320498 + 0.999486i \(0.489796\pi\)
\(740\) 6.44700 0.236996
\(741\) 33.9952 1.24884
\(742\) 19.2205 0.705605
\(743\) −3.80402 −0.139556 −0.0697779 0.997563i \(-0.522229\pi\)
−0.0697779 + 0.997563i \(0.522229\pi\)
\(744\) −32.7517 −1.20073
\(745\) 7.63806 0.279837
\(746\) 24.7735 0.907021
\(747\) 11.8571 0.433828
\(748\) 19.6681 0.719137
\(749\) 2.02303 0.0739199
\(750\) 4.33805 0.158403
\(751\) 22.3046 0.813905 0.406953 0.913449i \(-0.366591\pi\)
0.406953 + 0.913449i \(0.366591\pi\)
\(752\) 46.5157 1.69625
\(753\) 43.0447 1.56864
\(754\) −9.82726 −0.357888
\(755\) −0.659545 −0.0240033
\(756\) 1.02265 0.0371935
\(757\) −34.3973 −1.25019 −0.625095 0.780549i \(-0.714940\pi\)
−0.625095 + 0.780549i \(0.714940\pi\)
\(758\) 55.0705 2.00025
\(759\) 13.7992 0.500880
\(760\) −10.3271 −0.374603
\(761\) −28.1069 −1.01888 −0.509438 0.860508i \(-0.670147\pi\)
−0.509438 + 0.860508i \(0.670147\pi\)
\(762\) −3.95005 −0.143095
\(763\) 3.68337 0.133347
\(764\) −11.3912 −0.412120
\(765\) 14.9802 0.541610
\(766\) −36.9855 −1.33634
\(767\) 13.0050 0.469585
\(768\) −44.7661 −1.61536
\(769\) 5.49230 0.198058 0.0990288 0.995085i \(-0.468426\pi\)
0.0990288 + 0.995085i \(0.468426\pi\)
\(770\) 7.47211 0.269276
\(771\) 13.1261 0.472725
\(772\) −18.5045 −0.665993
\(773\) 27.5678 0.991543 0.495772 0.868453i \(-0.334885\pi\)
0.495772 + 0.868453i \(0.334885\pi\)
\(774\) 11.4721 0.412356
\(775\) 6.66964 0.239581
\(776\) −17.3014 −0.621083
\(777\) 15.5822 0.559007
\(778\) 25.6552 0.919782
\(779\) −30.4074 −1.08946
\(780\) −5.47772 −0.196134
\(781\) −63.5281 −2.27322
\(782\) 7.12311 0.254722
\(783\) −3.38822 −0.121085
\(784\) 31.5357 1.12628
\(785\) 3.94214 0.140701
\(786\) 80.7181 2.87912
\(787\) 26.7316 0.952880 0.476440 0.879207i \(-0.341927\pi\)
0.476440 + 0.879207i \(0.341927\pi\)
\(788\) 22.8819 0.815133
\(789\) −17.8593 −0.635807
\(790\) −23.3693 −0.831443
\(791\) −5.16324 −0.183584
\(792\) 36.7805 1.30694
\(793\) −20.9653 −0.744499
\(794\) 13.0442 0.462921
\(795\) −35.4956 −1.25890
\(796\) −3.61199 −0.128024
\(797\) −34.3017 −1.21503 −0.607515 0.794308i \(-0.707833\pi\)
−0.607515 + 0.794308i \(0.707833\pi\)
\(798\) 19.1402 0.677556
\(799\) −39.2665 −1.38915
\(800\) 4.60408 0.162779
\(801\) −12.7612 −0.450894
\(802\) 40.2114 1.41991
\(803\) −56.3677 −1.98917
\(804\) −9.13433 −0.322143
\(805\) 0.819031 0.0288671
\(806\) −27.8264 −0.980144
\(807\) 54.6451 1.92360
\(808\) −23.2023 −0.816254
\(809\) −25.9575 −0.912618 −0.456309 0.889821i \(-0.650829\pi\)
−0.456309 + 0.889821i \(0.650829\pi\)
\(810\) 11.8547 0.416531
\(811\) −25.2252 −0.885777 −0.442889 0.896577i \(-0.646046\pi\)
−0.442889 + 0.896577i \(0.646046\pi\)
\(812\) −1.67460 −0.0587670
\(813\) 1.92391 0.0674744
\(814\) −67.7589 −2.37495
\(815\) −3.80038 −0.133122
\(816\) 53.6829 1.87928
\(817\) 10.2462 0.358470
\(818\) −17.3413 −0.606323
\(819\) −7.18626 −0.251108
\(820\) 4.89961 0.171102
\(821\) 25.2761 0.882143 0.441071 0.897472i \(-0.354599\pi\)
0.441071 + 0.897472i \(0.354599\pi\)
\(822\) −62.5768 −2.18262
\(823\) −22.0337 −0.768045 −0.384023 0.923324i \(-0.625462\pi\)
−0.384023 + 0.923324i \(0.625462\pi\)
\(824\) −7.29239 −0.254042
\(825\) −13.7992 −0.480427
\(826\) 7.32218 0.254771
\(827\) −16.0531 −0.558220 −0.279110 0.960259i \(-0.590039\pi\)
−0.279110 + 0.960259i \(0.590039\pi\)
\(828\) −3.09153 −0.107438
\(829\) −20.5251 −0.712865 −0.356432 0.934321i \(-0.616007\pi\)
−0.356432 + 0.934321i \(0.616007\pi\)
\(830\) −5.63806 −0.195700
\(831\) 21.0246 0.729334
\(832\) 5.34109 0.185169
\(833\) −26.6211 −0.922366
\(834\) 79.6062 2.75654
\(835\) −2.24621 −0.0777333
\(836\) −25.1904 −0.871229
\(837\) −9.59393 −0.331615
\(838\) 16.9052 0.583981
\(839\) 43.0284 1.48551 0.742753 0.669565i \(-0.233519\pi\)
0.742753 + 0.669565i \(0.233519\pi\)
\(840\) 4.02190 0.138769
\(841\) −23.4518 −0.808681
\(842\) 29.2082 1.00658
\(843\) −38.3590 −1.32115
\(844\) −13.6639 −0.470332
\(845\) −6.93087 −0.238429
\(846\) 56.3087 1.93593
\(847\) −14.7592 −0.507132
\(848\) −69.0440 −2.37098
\(849\) 59.9977 2.05911
\(850\) −7.12311 −0.244321
\(851\) −7.42718 −0.254600
\(852\) 26.2212 0.898322
\(853\) 29.8203 1.02103 0.510513 0.859870i \(-0.329455\pi\)
0.510513 + 0.859870i \(0.329455\pi\)
\(854\) −11.8040 −0.403925
\(855\) −19.1863 −0.656156
\(856\) −4.73510 −0.161842
\(857\) −20.4465 −0.698438 −0.349219 0.937041i \(-0.613553\pi\)
−0.349219 + 0.937041i \(0.613553\pi\)
\(858\) 57.5717 1.96547
\(859\) 50.6000 1.72645 0.863224 0.504820i \(-0.168441\pi\)
0.863224 + 0.504820i \(0.168441\pi\)
\(860\) −1.65100 −0.0562985
\(861\) 11.8422 0.403580
\(862\) 5.34099 0.181915
\(863\) 28.1900 0.959599 0.479800 0.877378i \(-0.340709\pi\)
0.479800 + 0.877378i \(0.340709\pi\)
\(864\) −6.62273 −0.225310
\(865\) −13.3822 −0.455010
\(866\) 44.2574 1.50393
\(867\) −1.77035 −0.0601243
\(868\) −4.74173 −0.160945
\(869\) 74.3371 2.52171
\(870\) 10.2182 0.346428
\(871\) 10.1205 0.342921
\(872\) −8.62128 −0.291954
\(873\) −32.1434 −1.08789
\(874\) −9.12311 −0.308594
\(875\) −0.819031 −0.0276883
\(876\) 23.2657 0.786076
\(877\) 3.25176 0.109804 0.0549021 0.998492i \(-0.482515\pi\)
0.0549021 + 0.998492i \(0.482515\pi\)
\(878\) −3.52495 −0.118961
\(879\) −38.3208 −1.29253
\(880\) −26.8414 −0.904824
\(881\) 40.4455 1.36264 0.681322 0.731984i \(-0.261405\pi\)
0.681322 + 0.731984i \(0.261405\pi\)
\(882\) 38.1750 1.28542
\(883\) −12.4178 −0.417893 −0.208947 0.977927i \(-0.567004\pi\)
−0.208947 + 0.977927i \(0.567004\pi\)
\(884\) 8.99445 0.302516
\(885\) −13.5223 −0.454548
\(886\) 20.7963 0.698665
\(887\) −13.2199 −0.443882 −0.221941 0.975060i \(-0.571239\pi\)
−0.221941 + 0.975060i \(0.571239\pi\)
\(888\) −36.4716 −1.22391
\(889\) 0.745775 0.0250125
\(890\) 6.06796 0.203398
\(891\) −37.7094 −1.26331
\(892\) 0.0696668 0.00233262
\(893\) 50.2916 1.68295
\(894\) 33.1343 1.10818
\(895\) 18.2077 0.608618
\(896\) 10.5490 0.352416
\(897\) 6.31054 0.210703
\(898\) 32.8195 1.09520
\(899\) 15.7101 0.523963
\(900\) 3.09153 0.103051
\(901\) 58.2840 1.94172
\(902\) −51.4956 −1.71462
\(903\) −3.99040 −0.132792
\(904\) 12.0851 0.401943
\(905\) −1.01218 −0.0336460
\(906\) −2.86114 −0.0950551
\(907\) −58.2386 −1.93378 −0.966890 0.255193i \(-0.917861\pi\)
−0.966890 + 0.255193i \(0.917861\pi\)
\(908\) −24.7132 −0.820136
\(909\) −43.1065 −1.42975
\(910\) 3.41708 0.113275
\(911\) −46.5977 −1.54385 −0.771925 0.635714i \(-0.780706\pi\)
−0.771925 + 0.635714i \(0.780706\pi\)
\(912\) −68.7557 −2.27673
\(913\) 17.9345 0.593545
\(914\) 4.50119 0.148886
\(915\) 21.7992 0.720660
\(916\) −0.532057 −0.0175797
\(917\) −15.2397 −0.503259
\(918\) 10.2462 0.338175
\(919\) 20.1386 0.664312 0.332156 0.943225i \(-0.392224\pi\)
0.332156 + 0.943225i \(0.392224\pi\)
\(920\) −1.91702 −0.0632024
\(921\) 49.1466 1.61944
\(922\) −14.3852 −0.473751
\(923\) −29.0521 −0.956263
\(924\) 9.81044 0.322740
\(925\) 7.42718 0.244204
\(926\) −25.6745 −0.843717
\(927\) −13.5482 −0.444981
\(928\) 10.8448 0.355997
\(929\) −1.01385 −0.0332632 −0.0166316 0.999862i \(-0.505294\pi\)
−0.0166316 + 0.999862i \(0.505294\pi\)
\(930\) 28.9333 0.948759
\(931\) 34.0957 1.11744
\(932\) 3.29467 0.107921
\(933\) 49.7803 1.62973
\(934\) 21.5144 0.703974
\(935\) 22.6584 0.741008
\(936\) 16.8202 0.549784
\(937\) 32.5904 1.06468 0.532341 0.846530i \(-0.321312\pi\)
0.532341 + 0.846530i \(0.321312\pi\)
\(938\) 5.69813 0.186050
\(939\) 53.6132 1.74960
\(940\) −8.10361 −0.264310
\(941\) −1.01848 −0.0332017 −0.0166008 0.999862i \(-0.505284\pi\)
−0.0166008 + 0.999862i \(0.505284\pi\)
\(942\) 17.1012 0.557187
\(943\) −5.64453 −0.183811
\(944\) −26.3029 −0.856085
\(945\) 1.17813 0.0383247
\(946\) 17.3522 0.564169
\(947\) 0.212548 0.00690688 0.00345344 0.999994i \(-0.498901\pi\)
0.00345344 + 0.999994i \(0.498901\pi\)
\(948\) −30.6825 −0.996522
\(949\) −25.7776 −0.836777
\(950\) 9.12311 0.295993
\(951\) −27.6754 −0.897435
\(952\) −6.60398 −0.214036
\(953\) 47.2067 1.52917 0.764587 0.644520i \(-0.222943\pi\)
0.764587 + 0.644520i \(0.222943\pi\)
\(954\) −83.5801 −2.70600
\(955\) −13.1231 −0.424654
\(956\) 18.3257 0.592694
\(957\) −32.5036 −1.05069
\(958\) 19.7602 0.638424
\(959\) 11.8146 0.381514
\(960\) −5.55354 −0.179240
\(961\) 13.4841 0.434972
\(962\) −30.9870 −0.999059
\(963\) −8.79713 −0.283484
\(964\) −0.671941 −0.0216418
\(965\) −21.3179 −0.686248
\(966\) 3.55300 0.114316
\(967\) −23.7851 −0.764878 −0.382439 0.923981i \(-0.624916\pi\)
−0.382439 + 0.923981i \(0.624916\pi\)
\(968\) 34.5454 1.11033
\(969\) 58.0406 1.86453
\(970\) 15.2843 0.490748
\(971\) −47.7012 −1.53081 −0.765403 0.643552i \(-0.777460\pi\)
−0.765403 + 0.643552i \(0.777460\pi\)
\(972\) 19.3103 0.619379
\(973\) −15.0298 −0.481832
\(974\) −53.8166 −1.72439
\(975\) −6.31054 −0.202099
\(976\) 42.4026 1.35727
\(977\) −5.04505 −0.161405 −0.0807027 0.996738i \(-0.525716\pi\)
−0.0807027 + 0.996738i \(0.525716\pi\)
\(978\) −16.4863 −0.527173
\(979\) −19.3020 −0.616894
\(980\) −5.49391 −0.175497
\(981\) −16.0171 −0.511387
\(982\) 10.4613 0.333833
\(983\) −3.52184 −0.112329 −0.0561647 0.998422i \(-0.517887\pi\)
−0.0561647 + 0.998422i \(0.517887\pi\)
\(984\) −27.7178 −0.883611
\(985\) 26.3608 0.839924
\(986\) −16.7783 −0.534329
\(987\) −19.5861 −0.623433
\(988\) −11.5199 −0.366496
\(989\) 1.90201 0.0604803
\(990\) −32.4924 −1.03268
\(991\) 53.3794 1.69565 0.847826 0.530274i \(-0.177911\pi\)
0.847826 + 0.530274i \(0.177911\pi\)
\(992\) 30.7076 0.974967
\(993\) −58.3082 −1.85035
\(994\) −16.3571 −0.518817
\(995\) −4.16115 −0.131917
\(996\) −7.40244 −0.234555
\(997\) 55.2269 1.74905 0.874527 0.484978i \(-0.161172\pi\)
0.874527 + 0.484978i \(0.161172\pi\)
\(998\) 63.2378 2.00176
\(999\) −10.6836 −0.338014
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 115.2.a.c.1.1 4
3.2 odd 2 1035.2.a.o.1.4 4
4.3 odd 2 1840.2.a.u.1.3 4
5.2 odd 4 575.2.b.e.24.2 8
5.3 odd 4 575.2.b.e.24.7 8
5.4 even 2 575.2.a.h.1.4 4
7.6 odd 2 5635.2.a.v.1.1 4
8.3 odd 2 7360.2.a.cg.1.1 4
8.5 even 2 7360.2.a.cj.1.4 4
15.14 odd 2 5175.2.a.bx.1.1 4
20.19 odd 2 9200.2.a.cl.1.2 4
23.22 odd 2 2645.2.a.m.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.a.c.1.1 4 1.1 even 1 trivial
575.2.a.h.1.4 4 5.4 even 2
575.2.b.e.24.2 8 5.2 odd 4
575.2.b.e.24.7 8 5.3 odd 4
1035.2.a.o.1.4 4 3.2 odd 2
1840.2.a.u.1.3 4 4.3 odd 2
2645.2.a.m.1.1 4 23.22 odd 2
5175.2.a.bx.1.1 4 15.14 odd 2
5635.2.a.v.1.1 4 7.6 odd 2
7360.2.a.cg.1.1 4 8.3 odd 2
7360.2.a.cj.1.4 4 8.5 even 2
9200.2.a.cl.1.2 4 20.19 odd 2