Properties

Label 435.2.a.j.1.1
Level $435$
Weight $2$
Character 435.1
Self dual yes
Analytic conductor $3.473$
Analytic rank $1$
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] = [435,2,Mod(1,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("435.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 435.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: \(3.47349248793\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.2225.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.75660\) of defining polynomial
Character \(\chi\) \(=\) 435.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.75660 q^{2} -1.00000 q^{3} +5.59883 q^{4} -1.00000 q^{5} +2.75660 q^{6} +0.393832 q^{7} -9.92054 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.75660 q^{2} -1.00000 q^{3} +5.59883 q^{4} -1.00000 q^{5} +2.75660 q^{6} +0.393832 q^{7} -9.92054 q^{8} +1.00000 q^{9} +2.75660 q^{10} -0.393832 q^{11} -5.59883 q^{12} -2.56511 q^{13} -1.08564 q^{14} +1.00000 q^{15} +16.1493 q^{16} +2.07830 q^{17} -2.75660 q^{18} -0.958939 q^{19} -5.59883 q^{20} -0.393832 q^{21} +1.08564 q^{22} +6.15661 q^{23} +9.92054 q^{24} +1.00000 q^{25} +7.07097 q^{26} -1.00000 q^{27} +2.20500 q^{28} -1.00000 q^{29} -2.75660 q^{30} -10.1566 q^{31} -24.6760 q^{32} +0.393832 q^{33} -5.72905 q^{34} -0.393832 q^{35} +5.59883 q^{36} -7.34192 q^{37} +2.64341 q^{38} +2.56511 q^{39} +9.92054 q^{40} -1.65745 q^{41} +1.08564 q^{42} +10.3279 q^{43} -2.20500 q^{44} -1.00000 q^{45} -16.9713 q^{46} -11.5915 q^{47} -16.1493 q^{48} -6.84490 q^{49} -2.75660 q^{50} -2.07830 q^{51} -14.3616 q^{52} -12.3279 q^{53} +2.75660 q^{54} +0.393832 q^{55} -3.90703 q^{56} +0.958939 q^{57} +2.75660 q^{58} -9.54022 q^{59} +5.59883 q^{60} -6.25340 q^{61} +27.9977 q^{62} +0.393832 q^{63} +35.7232 q^{64} +2.56511 q^{65} -1.08564 q^{66} -7.42023 q^{67} +11.6361 q^{68} -6.15661 q^{69} +1.08564 q^{70} +5.98533 q^{71} -9.92054 q^{72} +3.34192 q^{73} +20.2387 q^{74} -1.00000 q^{75} -5.36894 q^{76} -0.155104 q^{77} -7.07097 q^{78} -2.06745 q^{79} -16.1493 q^{80} +1.00000 q^{81} +4.56892 q^{82} +6.41000 q^{83} -2.20500 q^{84} -2.07830 q^{85} -28.4698 q^{86} +1.00000 q^{87} +3.90703 q^{88} +15.8302 q^{89} +2.75660 q^{90} -1.01022 q^{91} +34.4698 q^{92} +10.1566 q^{93} +31.9531 q^{94} +0.958939 q^{95} +24.6760 q^{96} -18.4575 q^{97} +18.8686 q^{98} -0.393832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{2} - 4 q^{3} + 5 q^{4} - 4 q^{5} + 3 q^{6} + 2 q^{7} - 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 3 q^{2} - 4 q^{3} + 5 q^{4} - 4 q^{5} + 3 q^{6} + 2 q^{7} - 12 q^{8} + 4 q^{9} + 3 q^{10} - 2 q^{11} - 5 q^{12} - 8 q^{13} - 3 q^{14} + 4 q^{15} + 11 q^{16} - 10 q^{17} - 3 q^{18} - 2 q^{19} - 5 q^{20} - 2 q^{21} + 3 q^{22} - 12 q^{23} + 12 q^{24} + 4 q^{25} - 7 q^{26} - 4 q^{27} - 9 q^{28} - 4 q^{29} - 3 q^{30} - 4 q^{31} - 17 q^{32} + 2 q^{33} - q^{34} - 2 q^{35} + 5 q^{36} - 16 q^{37} - 10 q^{38} + 8 q^{39} + 12 q^{40} - 12 q^{41} + 3 q^{42} + 2 q^{43} + 9 q^{44} - 4 q^{45} - 8 q^{46} - 12 q^{47} - 11 q^{48} + 6 q^{49} - 3 q^{50} + 10 q^{51} - 3 q^{52} - 10 q^{53} + 3 q^{54} + 2 q^{55} + 2 q^{57} + 3 q^{58} + 2 q^{59} + 5 q^{60} - 26 q^{61} + 20 q^{62} + 2 q^{63} + 34 q^{64} + 8 q^{65} - 3 q^{66} + 2 q^{67} + 9 q^{68} + 12 q^{69} + 3 q^{70} - 10 q^{71} - 12 q^{72} + 48 q^{74} - 4 q^{75} + 16 q^{76} - 34 q^{77} + 7 q^{78} + 22 q^{79} - 11 q^{80} + 4 q^{81} + 38 q^{82} - 10 q^{83} + 9 q^{84} + 10 q^{85} - 4 q^{86} + 4 q^{87} - 4 q^{89} + 3 q^{90} - 8 q^{91} + 28 q^{92} + 4 q^{93} + 39 q^{94} + 2 q^{95} + 17 q^{96} - 22 q^{97} + 34 q^{98} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.75660 −1.94921 −0.974605 0.223932i \(-0.928110\pi\)
−0.974605 + 0.223932i \(0.928110\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.59883 2.79942
\(5\) −1.00000 −0.447214
\(6\) 2.75660 1.12538
\(7\) 0.393832 0.148855 0.0744273 0.997226i \(-0.476287\pi\)
0.0744273 + 0.997226i \(0.476287\pi\)
\(8\) −9.92054 −3.50744
\(9\) 1.00000 0.333333
\(10\) 2.75660 0.871713
\(11\) −0.393832 −0.118745 −0.0593725 0.998236i \(-0.518910\pi\)
−0.0593725 + 0.998236i \(0.518910\pi\)
\(12\) −5.59883 −1.61624
\(13\) −2.56511 −0.711433 −0.355716 0.934594i \(-0.615763\pi\)
−0.355716 + 0.934594i \(0.615763\pi\)
\(14\) −1.08564 −0.290149
\(15\) 1.00000 0.258199
\(16\) 16.1493 4.03732
\(17\) 2.07830 0.504063 0.252031 0.967719i \(-0.418901\pi\)
0.252031 + 0.967719i \(0.418901\pi\)
\(18\) −2.75660 −0.649736
\(19\) −0.958939 −0.219996 −0.109998 0.993932i \(-0.535084\pi\)
−0.109998 + 0.993932i \(0.535084\pi\)
\(20\) −5.59883 −1.25194
\(21\) −0.393832 −0.0859413
\(22\) 1.08564 0.231459
\(23\) 6.15661 1.28374 0.641871 0.766813i \(-0.278159\pi\)
0.641871 + 0.766813i \(0.278159\pi\)
\(24\) 9.92054 2.02502
\(25\) 1.00000 0.200000
\(26\) 7.07097 1.38673
\(27\) −1.00000 −0.192450
\(28\) 2.20500 0.416706
\(29\) −1.00000 −0.185695
\(30\) −2.75660 −0.503284
\(31\) −10.1566 −1.82418 −0.912090 0.409989i \(-0.865532\pi\)
−0.912090 + 0.409989i \(0.865532\pi\)
\(32\) −24.6760 −4.36214
\(33\) 0.393832 0.0685574
\(34\) −5.72905 −0.982524
\(35\) −0.393832 −0.0665698
\(36\) 5.59883 0.933139
\(37\) −7.34192 −1.20700 −0.603502 0.797361i \(-0.706229\pi\)
−0.603502 + 0.797361i \(0.706229\pi\)
\(38\) 2.64341 0.428818
\(39\) 2.56511 0.410746
\(40\) 9.92054 1.56857
\(41\) −1.65745 −0.258850 −0.129425 0.991589i \(-0.541313\pi\)
−0.129425 + 0.991589i \(0.541313\pi\)
\(42\) 1.08564 0.167517
\(43\) 10.3279 1.57499 0.787494 0.616323i \(-0.211378\pi\)
0.787494 + 0.616323i \(0.211378\pi\)
\(44\) −2.20500 −0.332417
\(45\) −1.00000 −0.149071
\(46\) −16.9713 −2.50228
\(47\) −11.5915 −1.69079 −0.845397 0.534138i \(-0.820636\pi\)
−0.845397 + 0.534138i \(0.820636\pi\)
\(48\) −16.1493 −2.33095
\(49\) −6.84490 −0.977842
\(50\) −2.75660 −0.389842
\(51\) −2.07830 −0.291021
\(52\) −14.3616 −1.99160
\(53\) −12.3279 −1.69336 −0.846682 0.532099i \(-0.821404\pi\)
−0.846682 + 0.532099i \(0.821404\pi\)
\(54\) 2.75660 0.375126
\(55\) 0.393832 0.0531043
\(56\) −3.90703 −0.522099
\(57\) 0.958939 0.127015
\(58\) 2.75660 0.361959
\(59\) −9.54022 −1.24203 −0.621015 0.783798i \(-0.713280\pi\)
−0.621015 + 0.783798i \(0.713280\pi\)
\(60\) 5.59883 0.722806
\(61\) −6.25340 −0.800665 −0.400333 0.916370i \(-0.631105\pi\)
−0.400333 + 0.916370i \(0.631105\pi\)
\(62\) 27.9977 3.55571
\(63\) 0.393832 0.0496182
\(64\) 35.7232 4.46540
\(65\) 2.56511 0.318162
\(66\) −1.08564 −0.133633
\(67\) −7.42023 −0.906525 −0.453262 0.891377i \(-0.649740\pi\)
−0.453262 + 0.891377i \(0.649740\pi\)
\(68\) 11.6361 1.41108
\(69\) −6.15661 −0.741168
\(70\) 1.08564 0.129758
\(71\) 5.98533 0.710328 0.355164 0.934804i \(-0.384425\pi\)
0.355164 + 0.934804i \(0.384425\pi\)
\(72\) −9.92054 −1.16915
\(73\) 3.34192 0.391142 0.195571 0.980690i \(-0.437344\pi\)
0.195571 + 0.980690i \(0.437344\pi\)
\(74\) 20.2387 2.35270
\(75\) −1.00000 −0.115470
\(76\) −5.36894 −0.615860
\(77\) −0.155104 −0.0176757
\(78\) −7.07097 −0.800630
\(79\) −2.06745 −0.232607 −0.116303 0.993214i \(-0.537104\pi\)
−0.116303 + 0.993214i \(0.537104\pi\)
\(80\) −16.1493 −1.80554
\(81\) 1.00000 0.111111
\(82\) 4.56892 0.504553
\(83\) 6.41000 0.703589 0.351795 0.936077i \(-0.385572\pi\)
0.351795 + 0.936077i \(0.385572\pi\)
\(84\) −2.20500 −0.240585
\(85\) −2.07830 −0.225424
\(86\) −28.4698 −3.06998
\(87\) 1.00000 0.107211
\(88\) 3.90703 0.416491
\(89\) 15.8302 1.67800 0.839000 0.544131i \(-0.183140\pi\)
0.839000 + 0.544131i \(0.183140\pi\)
\(90\) 2.75660 0.290571
\(91\) −1.01022 −0.105900
\(92\) 34.4698 3.59373
\(93\) 10.1566 1.05319
\(94\) 31.9531 3.29571
\(95\) 0.958939 0.0983851
\(96\) 24.6760 2.51848
\(97\) −18.4575 −1.87407 −0.937036 0.349233i \(-0.886442\pi\)
−0.937036 + 0.349233i \(0.886442\pi\)
\(98\) 18.8686 1.90602
\(99\) −0.393832 −0.0395816
\(100\) 5.59883 0.559883
\(101\) −12.8038 −1.27403 −0.637015 0.770852i \(-0.719831\pi\)
−0.637015 + 0.770852i \(0.719831\pi\)
\(102\) 5.72905 0.567260
\(103\) 4.86979 0.479834 0.239917 0.970793i \(-0.422880\pi\)
0.239917 + 0.970793i \(0.422880\pi\)
\(104\) 25.4472 2.49531
\(105\) 0.393832 0.0384341
\(106\) 33.9830 3.30072
\(107\) −2.34255 −0.226463 −0.113231 0.993569i \(-0.536120\pi\)
−0.113231 + 0.993569i \(0.536120\pi\)
\(108\) −5.59883 −0.538748
\(109\) 8.55044 0.818984 0.409492 0.912314i \(-0.365706\pi\)
0.409492 + 0.912314i \(0.365706\pi\)
\(110\) −1.08564 −0.103511
\(111\) 7.34192 0.696864
\(112\) 6.36011 0.600973
\(113\) −11.2085 −1.05441 −0.527204 0.849739i \(-0.676760\pi\)
−0.527204 + 0.849739i \(0.676760\pi\)
\(114\) −2.64341 −0.247578
\(115\) −6.15661 −0.574107
\(116\) −5.59883 −0.519839
\(117\) −2.56511 −0.237144
\(118\) 26.2985 2.42098
\(119\) 0.818503 0.0750321
\(120\) −9.92054 −0.905617
\(121\) −10.8449 −0.985900
\(122\) 17.2381 1.56066
\(123\) 1.65745 0.149447
\(124\) −56.8652 −5.10664
\(125\) −1.00000 −0.0894427
\(126\) −1.08564 −0.0967163
\(127\) 20.1566 1.78861 0.894305 0.447458i \(-0.147671\pi\)
0.894305 + 0.447458i \(0.147671\pi\)
\(128\) −49.1226 −4.34187
\(129\) −10.3279 −0.909319
\(130\) −7.07097 −0.620165
\(131\) −5.50235 −0.480742 −0.240371 0.970681i \(-0.577269\pi\)
−0.240371 + 0.970681i \(0.577269\pi\)
\(132\) 2.20500 0.191921
\(133\) −0.377661 −0.0327474
\(134\) 20.4546 1.76701
\(135\) 1.00000 0.0860663
\(136\) −20.6179 −1.76797
\(137\) 7.49853 0.640643 0.320321 0.947309i \(-0.396209\pi\)
0.320321 + 0.947309i \(0.396209\pi\)
\(138\) 16.9713 1.44469
\(139\) −9.35277 −0.793292 −0.396646 0.917972i \(-0.629826\pi\)
−0.396646 + 0.917972i \(0.629826\pi\)
\(140\) −2.20500 −0.186357
\(141\) 11.5915 0.976180
\(142\) −16.4992 −1.38458
\(143\) 1.01022 0.0844790
\(144\) 16.1493 1.34577
\(145\) 1.00000 0.0830455
\(146\) −9.21234 −0.762418
\(147\) 6.84490 0.564558
\(148\) −41.1062 −3.37891
\(149\) −11.5402 −0.945411 −0.472706 0.881220i \(-0.656723\pi\)
−0.472706 + 0.881220i \(0.656723\pi\)
\(150\) 2.75660 0.225075
\(151\) 6.08212 0.494956 0.247478 0.968894i \(-0.420398\pi\)
0.247478 + 0.968894i \(0.420398\pi\)
\(152\) 9.51320 0.771622
\(153\) 2.07830 0.168021
\(154\) 0.427559 0.0344537
\(155\) 10.1566 0.815798
\(156\) 14.3616 1.14985
\(157\) −11.5953 −0.925407 −0.462704 0.886513i \(-0.653121\pi\)
−0.462704 + 0.886513i \(0.653121\pi\)
\(158\) 5.69914 0.453399
\(159\) 12.3279 0.977665
\(160\) 24.6760 1.95081
\(161\) 2.42467 0.191091
\(162\) −2.75660 −0.216579
\(163\) −0.855118 −0.0669780 −0.0334890 0.999439i \(-0.510662\pi\)
−0.0334890 + 0.999439i \(0.510662\pi\)
\(164\) −9.27979 −0.724630
\(165\) −0.393832 −0.0306598
\(166\) −17.6698 −1.37144
\(167\) 7.73194 0.598315 0.299158 0.954204i \(-0.403294\pi\)
0.299158 + 0.954204i \(0.403294\pi\)
\(168\) 3.90703 0.301434
\(169\) −6.42023 −0.493863
\(170\) 5.72905 0.439398
\(171\) −0.958939 −0.0733319
\(172\) 57.8241 4.40905
\(173\) 8.07449 0.613892 0.306946 0.951727i \(-0.400693\pi\)
0.306946 + 0.951727i \(0.400693\pi\)
\(174\) −2.75660 −0.208977
\(175\) 0.393832 0.0297709
\(176\) −6.36011 −0.479411
\(177\) 9.54022 0.717087
\(178\) −43.6376 −3.27077
\(179\) 12.4100 0.927567 0.463784 0.885949i \(-0.346492\pi\)
0.463784 + 0.885949i \(0.346492\pi\)
\(180\) −5.59883 −0.417312
\(181\) −2.49765 −0.185649 −0.0928246 0.995682i \(-0.529590\pi\)
−0.0928246 + 0.995682i \(0.529590\pi\)
\(182\) 2.78478 0.206421
\(183\) 6.25340 0.462264
\(184\) −61.0769 −4.50265
\(185\) 7.34192 0.539789
\(186\) −27.9977 −2.05289
\(187\) −0.818503 −0.0598549
\(188\) −64.8989 −4.73324
\(189\) −0.393832 −0.0286471
\(190\) −2.64341 −0.191773
\(191\) −6.32085 −0.457361 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(192\) −35.7232 −2.57810
\(193\) 25.2921 1.82057 0.910284 0.413984i \(-0.135863\pi\)
0.910284 + 0.413984i \(0.135863\pi\)
\(194\) 50.8798 3.65296
\(195\) −2.56511 −0.183691
\(196\) −38.3234 −2.73739
\(197\) −21.2047 −1.51077 −0.755386 0.655280i \(-0.772551\pi\)
−0.755386 + 0.655280i \(0.772551\pi\)
\(198\) 1.08564 0.0771529
\(199\) −2.64723 −0.187657 −0.0938285 0.995588i \(-0.529911\pi\)
−0.0938285 + 0.995588i \(0.529911\pi\)
\(200\) −9.92054 −0.701488
\(201\) 7.42023 0.523382
\(202\) 35.2950 2.48335
\(203\) −0.393832 −0.0276416
\(204\) −11.6361 −0.814688
\(205\) 1.65745 0.115761
\(206\) −13.4240 −0.935297
\(207\) 6.15661 0.427914
\(208\) −41.4246 −2.87228
\(209\) 0.377661 0.0261234
\(210\) −1.08564 −0.0749161
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) −69.0218 −4.74043
\(213\) −5.98533 −0.410108
\(214\) 6.45747 0.441423
\(215\) −10.3279 −0.704356
\(216\) 9.92054 0.675007
\(217\) −4.00000 −0.271538
\(218\) −23.5701 −1.59637
\(219\) −3.34192 −0.225826
\(220\) 2.20500 0.148661
\(221\) −5.33107 −0.358607
\(222\) −20.2387 −1.35833
\(223\) 12.5504 0.840440 0.420220 0.907422i \(-0.361953\pi\)
0.420220 + 0.907422i \(0.361953\pi\)
\(224\) −9.71820 −0.649324
\(225\) 1.00000 0.0666667
\(226\) 30.8974 2.05526
\(227\) 1.30149 0.0863829 0.0431914 0.999067i \(-0.486247\pi\)
0.0431914 + 0.999067i \(0.486247\pi\)
\(228\) 5.36894 0.355567
\(229\) 3.28682 0.217199 0.108600 0.994086i \(-0.465363\pi\)
0.108600 + 0.994086i \(0.465363\pi\)
\(230\) 16.9713 1.11905
\(231\) 0.155104 0.0102051
\(232\) 9.92054 0.651315
\(233\) 17.7115 1.16032 0.580159 0.814503i \(-0.302990\pi\)
0.580159 + 0.814503i \(0.302990\pi\)
\(234\) 7.07097 0.462244
\(235\) 11.5915 0.756146
\(236\) −53.4141 −3.47696
\(237\) 2.06745 0.134296
\(238\) −2.25628 −0.146253
\(239\) 21.2651 1.37553 0.687763 0.725935i \(-0.258593\pi\)
0.687763 + 0.725935i \(0.258593\pi\)
\(240\) 16.1493 1.04243
\(241\) 15.3177 0.986697 0.493349 0.869832i \(-0.335773\pi\)
0.493349 + 0.869832i \(0.335773\pi\)
\(242\) 29.8950 1.92172
\(243\) −1.00000 −0.0641500
\(244\) −35.0117 −2.24140
\(245\) 6.84490 0.437304
\(246\) −4.56892 −0.291304
\(247\) 2.45978 0.156512
\(248\) 100.759 6.39820
\(249\) −6.41000 −0.406217
\(250\) 2.75660 0.174343
\(251\) 26.8917 1.69739 0.848696 0.528882i \(-0.177388\pi\)
0.848696 + 0.528882i \(0.177388\pi\)
\(252\) 2.20500 0.138902
\(253\) −2.42467 −0.152438
\(254\) −55.5637 −3.48637
\(255\) 2.07830 0.130148
\(256\) 63.9648 3.99780
\(257\) −6.85512 −0.427611 −0.213805 0.976876i \(-0.568586\pi\)
−0.213805 + 0.976876i \(0.568586\pi\)
\(258\) 28.4698 1.77245
\(259\) −2.89149 −0.179668
\(260\) 14.3616 0.890669
\(261\) −1.00000 −0.0618984
\(262\) 15.1678 0.937067
\(263\) −13.6294 −0.840423 −0.420212 0.907426i \(-0.638044\pi\)
−0.420212 + 0.907426i \(0.638044\pi\)
\(264\) −3.90703 −0.240461
\(265\) 12.3279 0.757296
\(266\) 1.04106 0.0638315
\(267\) −15.8302 −0.968794
\(268\) −41.5446 −2.53774
\(269\) −8.46129 −0.515894 −0.257947 0.966159i \(-0.583046\pi\)
−0.257947 + 0.966159i \(0.583046\pi\)
\(270\) −2.75660 −0.167761
\(271\) 6.34255 0.385282 0.192641 0.981269i \(-0.438295\pi\)
0.192641 + 0.981269i \(0.438295\pi\)
\(272\) 33.5631 2.03506
\(273\) 1.01022 0.0611414
\(274\) −20.6704 −1.24875
\(275\) −0.393832 −0.0237490
\(276\) −34.4698 −2.07484
\(277\) −17.1464 −1.03023 −0.515113 0.857122i \(-0.672250\pi\)
−0.515113 + 0.857122i \(0.672250\pi\)
\(278\) 25.7818 1.54629
\(279\) −10.1566 −0.608060
\(280\) 3.90703 0.233490
\(281\) −12.2985 −0.733670 −0.366835 0.930286i \(-0.619559\pi\)
−0.366835 + 0.930286i \(0.619559\pi\)
\(282\) −31.9531 −1.90278
\(283\) 25.1830 1.49697 0.748487 0.663149i \(-0.230781\pi\)
0.748487 + 0.663149i \(0.230781\pi\)
\(284\) 33.5109 1.98851
\(285\) −0.958939 −0.0568027
\(286\) −2.78478 −0.164667
\(287\) −0.652757 −0.0385311
\(288\) −24.6760 −1.45405
\(289\) −12.6807 −0.745921
\(290\) −2.75660 −0.161873
\(291\) 18.4575 1.08200
\(292\) 18.7109 1.09497
\(293\) 12.3170 0.719569 0.359784 0.933035i \(-0.382850\pi\)
0.359784 + 0.933035i \(0.382850\pi\)
\(294\) −18.8686 −1.10044
\(295\) 9.54022 0.555453
\(296\) 72.8358 4.23350
\(297\) 0.393832 0.0228525
\(298\) 31.8117 1.84280
\(299\) −15.7924 −0.913296
\(300\) −5.59883 −0.323249
\(301\) 4.06745 0.234444
\(302\) −16.7660 −0.964773
\(303\) 12.8038 0.735561
\(304\) −15.4862 −0.888193
\(305\) 6.25340 0.358068
\(306\) −5.72905 −0.327508
\(307\) −22.7379 −1.29772 −0.648860 0.760908i \(-0.724754\pi\)
−0.648860 + 0.760908i \(0.724754\pi\)
\(308\) −0.868401 −0.0494817
\(309\) −4.86979 −0.277032
\(310\) −27.9977 −1.59016
\(311\) −5.33810 −0.302696 −0.151348 0.988481i \(-0.548361\pi\)
−0.151348 + 0.988481i \(0.548361\pi\)
\(312\) −25.4472 −1.44067
\(313\) −1.96338 −0.110977 −0.0554885 0.998459i \(-0.517672\pi\)
−0.0554885 + 0.998459i \(0.517672\pi\)
\(314\) 31.9636 1.80381
\(315\) −0.393832 −0.0221899
\(316\) −11.5753 −0.651163
\(317\) 1.29064 0.0724895 0.0362448 0.999343i \(-0.488460\pi\)
0.0362448 + 0.999343i \(0.488460\pi\)
\(318\) −33.9830 −1.90567
\(319\) 0.393832 0.0220504
\(320\) −35.7232 −1.99699
\(321\) 2.34255 0.130748
\(322\) −6.68384 −0.372476
\(323\) −1.99297 −0.110892
\(324\) 5.59883 0.311046
\(325\) −2.56511 −0.142287
\(326\) 2.35722 0.130554
\(327\) −8.55044 −0.472840
\(328\) 16.4428 0.907902
\(329\) −4.56511 −0.251683
\(330\) 1.08564 0.0597624
\(331\) 28.9971 1.59382 0.796911 0.604096i \(-0.206466\pi\)
0.796911 + 0.604096i \(0.206466\pi\)
\(332\) 35.8885 1.96964
\(333\) −7.34192 −0.402335
\(334\) −21.3138 −1.16624
\(335\) 7.42023 0.405410
\(336\) −6.36011 −0.346972
\(337\) −16.7854 −0.914356 −0.457178 0.889375i \(-0.651140\pi\)
−0.457178 + 0.889375i \(0.651140\pi\)
\(338\) 17.6980 0.962643
\(339\) 11.2085 0.608763
\(340\) −11.6361 −0.631055
\(341\) 4.00000 0.216612
\(342\) 2.64341 0.142939
\(343\) −5.45257 −0.294411
\(344\) −102.458 −5.52417
\(345\) 6.15661 0.331461
\(346\) −22.2581 −1.19660
\(347\) −17.8681 −0.959210 −0.479605 0.877485i \(-0.659220\pi\)
−0.479605 + 0.877485i \(0.659220\pi\)
\(348\) 5.59883 0.300129
\(349\) 8.73789 0.467728 0.233864 0.972269i \(-0.424863\pi\)
0.233864 + 0.972269i \(0.424863\pi\)
\(350\) −1.08564 −0.0580298
\(351\) 2.56511 0.136915
\(352\) 9.71820 0.517982
\(353\) −22.6017 −1.20297 −0.601484 0.798885i \(-0.705424\pi\)
−0.601484 + 0.798885i \(0.705424\pi\)
\(354\) −26.2985 −1.39775
\(355\) −5.98533 −0.317668
\(356\) 88.6308 4.69742
\(357\) −0.818503 −0.0433198
\(358\) −34.2094 −1.80802
\(359\) −13.1830 −0.695772 −0.347886 0.937537i \(-0.613100\pi\)
−0.347886 + 0.937537i \(0.613100\pi\)
\(360\) 9.92054 0.522858
\(361\) −18.0804 −0.951602
\(362\) 6.88503 0.361869
\(363\) 10.8449 0.569209
\(364\) −5.65607 −0.296458
\(365\) −3.34192 −0.174924
\(366\) −17.2381 −0.901050
\(367\) 17.4511 0.910938 0.455469 0.890252i \(-0.349472\pi\)
0.455469 + 0.890252i \(0.349472\pi\)
\(368\) 99.4247 5.18287
\(369\) −1.65745 −0.0862834
\(370\) −20.2387 −1.05216
\(371\) −4.85512 −0.252065
\(372\) 56.8652 2.94832
\(373\) −32.2018 −1.66734 −0.833672 0.552260i \(-0.813766\pi\)
−0.833672 + 0.552260i \(0.813766\pi\)
\(374\) 2.25628 0.116670
\(375\) 1.00000 0.0516398
\(376\) 114.994 5.93036
\(377\) 2.56511 0.132110
\(378\) 1.08564 0.0558392
\(379\) 32.3660 1.66253 0.831265 0.555876i \(-0.187617\pi\)
0.831265 + 0.555876i \(0.187617\pi\)
\(380\) 5.36894 0.275421
\(381\) −20.1566 −1.03265
\(382\) 17.4240 0.891492
\(383\) −25.8739 −1.32209 −0.661047 0.750345i \(-0.729887\pi\)
−0.661047 + 0.750345i \(0.729887\pi\)
\(384\) 49.1226 2.50678
\(385\) 0.155104 0.00790483
\(386\) −69.7203 −3.54867
\(387\) 10.3279 0.524996
\(388\) −103.340 −5.24631
\(389\) 32.6103 1.65341 0.826703 0.562639i \(-0.190214\pi\)
0.826703 + 0.562639i \(0.190214\pi\)
\(390\) 7.07097 0.358052
\(391\) 12.7953 0.647086
\(392\) 67.9051 3.42972
\(393\) 5.50235 0.277557
\(394\) 58.4528 2.94481
\(395\) 2.06745 0.104025
\(396\) −2.20500 −0.110806
\(397\) 4.39534 0.220596 0.110298 0.993899i \(-0.464820\pi\)
0.110298 + 0.993899i \(0.464820\pi\)
\(398\) 7.29734 0.365783
\(399\) 0.377661 0.0189067
\(400\) 16.1493 0.807464
\(401\) −19.0658 −0.952099 −0.476049 0.879418i \(-0.657932\pi\)
−0.476049 + 0.879418i \(0.657932\pi\)
\(402\) −20.4546 −1.02018
\(403\) 26.0528 1.29778
\(404\) −71.6865 −3.56654
\(405\) −1.00000 −0.0496904
\(406\) 1.08564 0.0538793
\(407\) 2.89149 0.143326
\(408\) 20.6179 1.02074
\(409\) −1.38235 −0.0683530 −0.0341765 0.999416i \(-0.510881\pi\)
−0.0341765 + 0.999416i \(0.510881\pi\)
\(410\) −4.56892 −0.225643
\(411\) −7.49853 −0.369875
\(412\) 27.2651 1.34326
\(413\) −3.75725 −0.184882
\(414\) −16.9713 −0.834094
\(415\) −6.41000 −0.314655
\(416\) 63.2965 3.10337
\(417\) 9.35277 0.458007
\(418\) −1.04106 −0.0509199
\(419\) 2.86979 0.140198 0.0700991 0.997540i \(-0.477668\pi\)
0.0700991 + 0.997540i \(0.477668\pi\)
\(420\) 2.20500 0.107593
\(421\) 13.6440 0.664970 0.332485 0.943109i \(-0.392113\pi\)
0.332485 + 0.943109i \(0.392113\pi\)
\(422\) −5.51320 −0.268378
\(423\) −11.5915 −0.563598
\(424\) 122.299 5.93938
\(425\) 2.07830 0.100813
\(426\) 16.4992 0.799387
\(427\) −2.46279 −0.119183
\(428\) −13.1155 −0.633964
\(429\) −1.01022 −0.0487740
\(430\) 28.4698 1.37294
\(431\) −11.9707 −0.576607 −0.288303 0.957539i \(-0.593091\pi\)
−0.288303 + 0.957539i \(0.593091\pi\)
\(432\) −16.1493 −0.776982
\(433\) 17.9907 0.864576 0.432288 0.901736i \(-0.357706\pi\)
0.432288 + 0.901736i \(0.357706\pi\)
\(434\) 11.0264 0.529284
\(435\) −1.00000 −0.0479463
\(436\) 47.8725 2.29268
\(437\) −5.90381 −0.282418
\(438\) 9.21234 0.440182
\(439\) −16.2226 −0.774260 −0.387130 0.922025i \(-0.626534\pi\)
−0.387130 + 0.922025i \(0.626534\pi\)
\(440\) −3.90703 −0.186260
\(441\) −6.84490 −0.325947
\(442\) 14.6956 0.698999
\(443\) −35.3762 −1.68078 −0.840388 0.541986i \(-0.817673\pi\)
−0.840388 + 0.541986i \(0.817673\pi\)
\(444\) 41.1062 1.95081
\(445\) −15.8302 −0.750425
\(446\) −34.5965 −1.63819
\(447\) 11.5402 0.545834
\(448\) 14.0690 0.664696
\(449\) 41.6971 1.96781 0.983903 0.178702i \(-0.0571898\pi\)
0.983903 + 0.178702i \(0.0571898\pi\)
\(450\) −2.75660 −0.129947
\(451\) 0.652757 0.0307371
\(452\) −62.7546 −2.95173
\(453\) −6.08212 −0.285763
\(454\) −3.58768 −0.168378
\(455\) 1.01022 0.0473599
\(456\) −9.51320 −0.445496
\(457\) −16.1111 −0.753646 −0.376823 0.926285i \(-0.622983\pi\)
−0.376823 + 0.926285i \(0.622983\pi\)
\(458\) −9.06045 −0.423367
\(459\) −2.07830 −0.0970069
\(460\) −34.4698 −1.60716
\(461\) 17.3396 0.807586 0.403793 0.914850i \(-0.367692\pi\)
0.403793 + 0.914850i \(0.367692\pi\)
\(462\) −0.427559 −0.0198918
\(463\) 19.3177 0.897768 0.448884 0.893590i \(-0.351822\pi\)
0.448884 + 0.893590i \(0.351822\pi\)
\(464\) −16.1493 −0.749711
\(465\) −10.1566 −0.471001
\(466\) −48.8235 −2.26170
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) −14.3616 −0.663866
\(469\) −2.92232 −0.134940
\(470\) −31.9531 −1.47389
\(471\) 11.5953 0.534284
\(472\) 94.6441 4.35635
\(473\) −4.06745 −0.187022
\(474\) −5.69914 −0.261770
\(475\) −0.958939 −0.0439992
\(476\) 4.58266 0.210046
\(477\) −12.3279 −0.564455
\(478\) −58.6194 −2.68119
\(479\) −40.3877 −1.84536 −0.922681 0.385565i \(-0.874006\pi\)
−0.922681 + 0.385565i \(0.874006\pi\)
\(480\) −24.6760 −1.12630
\(481\) 18.8328 0.858702
\(482\) −42.2246 −1.92328
\(483\) −2.42467 −0.110326
\(484\) −60.7188 −2.75994
\(485\) 18.4575 0.838110
\(486\) 2.75660 0.125042
\(487\) 2.84171 0.128770 0.0643850 0.997925i \(-0.479491\pi\)
0.0643850 + 0.997925i \(0.479491\pi\)
\(488\) 62.0371 2.80829
\(489\) 0.855118 0.0386698
\(490\) −18.8686 −0.852398
\(491\) 0.157863 0.00712428 0.00356214 0.999994i \(-0.498866\pi\)
0.00356214 + 0.999994i \(0.498866\pi\)
\(492\) 9.27979 0.418365
\(493\) −2.07830 −0.0936021
\(494\) −6.78063 −0.305075
\(495\) 0.393832 0.0177014
\(496\) −164.022 −7.36480
\(497\) 2.35722 0.105736
\(498\) 17.6698 0.791803
\(499\) 2.64723 0.118506 0.0592531 0.998243i \(-0.481128\pi\)
0.0592531 + 0.998243i \(0.481128\pi\)
\(500\) −5.59883 −0.250387
\(501\) −7.73194 −0.345437
\(502\) −74.1297 −3.30857
\(503\) 13.9545 0.622200 0.311100 0.950377i \(-0.399303\pi\)
0.311100 + 0.950377i \(0.399303\pi\)
\(504\) −3.90703 −0.174033
\(505\) 12.8038 0.569763
\(506\) 6.68384 0.297133
\(507\) 6.42023 0.285132
\(508\) 112.853 5.00706
\(509\) −16.4921 −0.731001 −0.365500 0.930811i \(-0.619102\pi\)
−0.365500 + 0.930811i \(0.619102\pi\)
\(510\) −5.72905 −0.253687
\(511\) 1.31616 0.0582233
\(512\) −78.0802 −3.45069
\(513\) 0.958939 0.0423382
\(514\) 18.8968 0.833502
\(515\) −4.86979 −0.214588
\(516\) −57.8241 −2.54556
\(517\) 4.56511 0.200773
\(518\) 7.97066 0.350211
\(519\) −8.07449 −0.354431
\(520\) −25.4472 −1.11594
\(521\) −12.2253 −0.535601 −0.267800 0.963474i \(-0.586297\pi\)
−0.267800 + 0.963474i \(0.586297\pi\)
\(522\) 2.75660 0.120653
\(523\) 1.66659 0.0728748 0.0364374 0.999336i \(-0.488399\pi\)
0.0364374 + 0.999336i \(0.488399\pi\)
\(524\) −30.8067 −1.34580
\(525\) −0.393832 −0.0171883
\(526\) 37.5707 1.63816
\(527\) −21.1085 −0.919501
\(528\) 6.36011 0.276788
\(529\) 14.9038 0.647992
\(530\) −33.9830 −1.47613
\(531\) −9.54022 −0.414010
\(532\) −2.11446 −0.0916736
\(533\) 4.25154 0.184155
\(534\) 43.6376 1.88838
\(535\) 2.34255 0.101277
\(536\) 73.6126 3.17958
\(537\) −12.4100 −0.535531
\(538\) 23.3244 1.00558
\(539\) 2.69574 0.116114
\(540\) 5.59883 0.240935
\(541\) −34.6558 −1.48997 −0.744984 0.667083i \(-0.767543\pi\)
−0.744984 + 0.667083i \(0.767543\pi\)
\(542\) −17.4839 −0.750996
\(543\) 2.49765 0.107185
\(544\) −51.2842 −2.19879
\(545\) −8.55044 −0.366261
\(546\) −2.78478 −0.119177
\(547\) 38.2530 1.63558 0.817791 0.575515i \(-0.195199\pi\)
0.817791 + 0.575515i \(0.195199\pi\)
\(548\) 41.9830 1.79343
\(549\) −6.25340 −0.266888
\(550\) 1.08564 0.0462917
\(551\) 0.958939 0.0408522
\(552\) 61.0769 2.59960
\(553\) −0.814230 −0.0346246
\(554\) 47.2657 2.00813
\(555\) −7.34192 −0.311647
\(556\) −52.3646 −2.22075
\(557\) −28.7760 −1.21928 −0.609639 0.792679i \(-0.708686\pi\)
−0.609639 + 0.792679i \(0.708686\pi\)
\(558\) 27.9977 1.18524
\(559\) −26.4921 −1.12050
\(560\) −6.36011 −0.268764
\(561\) 0.818503 0.0345572
\(562\) 33.9022 1.43008
\(563\) −32.9809 −1.38998 −0.694989 0.719020i \(-0.744591\pi\)
−0.694989 + 0.719020i \(0.744591\pi\)
\(564\) 64.8989 2.73274
\(565\) 11.2085 0.471546
\(566\) −69.4194 −2.91792
\(567\) 0.393832 0.0165394
\(568\) −59.3777 −2.49143
\(569\) 16.8038 0.704453 0.352227 0.935915i \(-0.385425\pi\)
0.352227 + 0.935915i \(0.385425\pi\)
\(570\) 2.64341 0.110720
\(571\) −6.21703 −0.260175 −0.130087 0.991503i \(-0.541526\pi\)
−0.130087 + 0.991503i \(0.541526\pi\)
\(572\) 5.65607 0.236492
\(573\) 6.32085 0.264057
\(574\) 1.79939 0.0751051
\(575\) 6.15661 0.256748
\(576\) 35.7232 1.48847
\(577\) 11.9977 0.499470 0.249735 0.968314i \(-0.419656\pi\)
0.249735 + 0.968314i \(0.419656\pi\)
\(578\) 34.9555 1.45396
\(579\) −25.2921 −1.05111
\(580\) 5.59883 0.232479
\(581\) 2.52447 0.104733
\(582\) −50.8798 −2.10904
\(583\) 4.85512 0.201078
\(584\) −33.1537 −1.37191
\(585\) 2.56511 0.106054
\(586\) −33.9531 −1.40259
\(587\) −11.2194 −0.463073 −0.231536 0.972826i \(-0.574375\pi\)
−0.231536 + 0.972826i \(0.574375\pi\)
\(588\) 38.3234 1.58043
\(589\) 9.73957 0.401312
\(590\) −26.2985 −1.08269
\(591\) 21.2047 0.872245
\(592\) −118.567 −4.87306
\(593\) −7.69682 −0.316071 −0.158035 0.987433i \(-0.550516\pi\)
−0.158035 + 0.987433i \(0.550516\pi\)
\(594\) −1.08564 −0.0445442
\(595\) −0.818503 −0.0335554
\(596\) −64.6118 −2.64660
\(597\) 2.64723 0.108344
\(598\) 43.5332 1.78020
\(599\) 20.0543 0.819396 0.409698 0.912221i \(-0.365634\pi\)
0.409698 + 0.912221i \(0.365634\pi\)
\(600\) 9.92054 0.405004
\(601\) 3.10851 0.126799 0.0633995 0.997988i \(-0.479806\pi\)
0.0633995 + 0.997988i \(0.479806\pi\)
\(602\) −11.2123 −0.456981
\(603\) −7.42023 −0.302175
\(604\) 34.0528 1.38559
\(605\) 10.8449 0.440908
\(606\) −35.2950 −1.43376
\(607\) 37.0441 1.50357 0.751786 0.659407i \(-0.229193\pi\)
0.751786 + 0.659407i \(0.229193\pi\)
\(608\) 23.6628 0.959652
\(609\) 0.393832 0.0159589
\(610\) −17.2381 −0.697950
\(611\) 29.7334 1.20289
\(612\) 11.6361 0.470361
\(613\) −13.3839 −0.540569 −0.270284 0.962781i \(-0.587118\pi\)
−0.270284 + 0.962781i \(0.587118\pi\)
\(614\) 62.6792 2.52953
\(615\) −1.65745 −0.0668348
\(616\) 1.53871 0.0619966
\(617\) −12.6364 −0.508721 −0.254361 0.967109i \(-0.581865\pi\)
−0.254361 + 0.967109i \(0.581865\pi\)
\(618\) 13.4240 0.539994
\(619\) −41.2447 −1.65776 −0.828882 0.559424i \(-0.811022\pi\)
−0.828882 + 0.559424i \(0.811022\pi\)
\(620\) 56.8652 2.28376
\(621\) −6.15661 −0.247056
\(622\) 14.7150 0.590018
\(623\) 6.23446 0.249778
\(624\) 41.4246 1.65831
\(625\) 1.00000 0.0400000
\(626\) 5.41226 0.216318
\(627\) −0.377661 −0.0150823
\(628\) −64.9203 −2.59060
\(629\) −15.2587 −0.608406
\(630\) 1.08564 0.0432528
\(631\) 29.8009 1.18635 0.593177 0.805072i \(-0.297873\pi\)
0.593177 + 0.805072i \(0.297873\pi\)
\(632\) 20.5103 0.815854
\(633\) −2.00000 −0.0794929
\(634\) −3.55777 −0.141297
\(635\) −20.1566 −0.799891
\(636\) 69.0218 2.73689
\(637\) 17.5579 0.695669
\(638\) −1.08564 −0.0429808
\(639\) 5.98533 0.236776
\(640\) 49.1226 1.94174
\(641\) 17.7774 0.702167 0.351083 0.936344i \(-0.385813\pi\)
0.351083 + 0.936344i \(0.385813\pi\)
\(642\) −6.45747 −0.254856
\(643\) −44.5534 −1.75702 −0.878508 0.477727i \(-0.841461\pi\)
−0.878508 + 0.477727i \(0.841461\pi\)
\(644\) 13.5753 0.534943
\(645\) 10.3279 0.406660
\(646\) 5.49381 0.216151
\(647\) 42.9339 1.68790 0.843952 0.536418i \(-0.180223\pi\)
0.843952 + 0.536418i \(0.180223\pi\)
\(648\) −9.92054 −0.389716
\(649\) 3.75725 0.147485
\(650\) 7.07097 0.277346
\(651\) 4.00000 0.156772
\(652\) −4.78766 −0.187499
\(653\) −19.8426 −0.776500 −0.388250 0.921554i \(-0.626920\pi\)
−0.388250 + 0.921554i \(0.626920\pi\)
\(654\) 23.5701 0.921665
\(655\) 5.50235 0.214994
\(656\) −26.7666 −1.04506
\(657\) 3.34192 0.130381
\(658\) 12.5842 0.490582
\(659\) 23.0483 0.897836 0.448918 0.893573i \(-0.351810\pi\)
0.448918 + 0.893573i \(0.351810\pi\)
\(660\) −2.20500 −0.0858296
\(661\) −36.2079 −1.40832 −0.704162 0.710040i \(-0.748677\pi\)
−0.704162 + 0.710040i \(0.748677\pi\)
\(662\) −79.9332 −3.10669
\(663\) 5.33107 0.207042
\(664\) −63.5907 −2.46780
\(665\) 0.377661 0.0146451
\(666\) 20.2387 0.784235
\(667\) −6.15661 −0.238385
\(668\) 43.2898 1.67493
\(669\) −12.5504 −0.485228
\(670\) −20.4546 −0.790229
\(671\) 2.46279 0.0950749
\(672\) 9.71820 0.374888
\(673\) 22.9000 0.882731 0.441365 0.897327i \(-0.354494\pi\)
0.441365 + 0.897327i \(0.354494\pi\)
\(674\) 46.2705 1.78227
\(675\) −1.00000 −0.0384900
\(676\) −35.9458 −1.38253
\(677\) 36.0068 1.38385 0.691927 0.721967i \(-0.256762\pi\)
0.691927 + 0.721967i \(0.256762\pi\)
\(678\) −30.8974 −1.18661
\(679\) −7.26915 −0.278964
\(680\) 20.6179 0.790660
\(681\) −1.30149 −0.0498732
\(682\) −11.0264 −0.422222
\(683\) 9.12896 0.349310 0.174655 0.984630i \(-0.444119\pi\)
0.174655 + 0.984630i \(0.444119\pi\)
\(684\) −5.36894 −0.205287
\(685\) −7.49853 −0.286504
\(686\) 15.0305 0.573869
\(687\) −3.28682 −0.125400
\(688\) 166.788 6.35873
\(689\) 31.6223 1.20472
\(690\) −16.9713 −0.646086
\(691\) 5.85956 0.222908 0.111454 0.993770i \(-0.464449\pi\)
0.111454 + 0.993770i \(0.464449\pi\)
\(692\) 45.2077 1.71854
\(693\) −0.155104 −0.00589191
\(694\) 49.2552 1.86970
\(695\) 9.35277 0.354771
\(696\) −9.92054 −0.376037
\(697\) −3.44469 −0.130477
\(698\) −24.0868 −0.911700
\(699\) −17.7115 −0.669910
\(700\) 2.20500 0.0833412
\(701\) −7.56955 −0.285898 −0.142949 0.989730i \(-0.545658\pi\)
−0.142949 + 0.989730i \(0.545658\pi\)
\(702\) −7.07097 −0.266877
\(703\) 7.04046 0.265536
\(704\) −14.0690 −0.530244
\(705\) −11.5915 −0.436561
\(706\) 62.3039 2.34484
\(707\) −5.04256 −0.189645
\(708\) 53.4141 2.00742
\(709\) 14.5209 0.545342 0.272671 0.962107i \(-0.412093\pi\)
0.272671 + 0.962107i \(0.412093\pi\)
\(710\) 16.4992 0.619202
\(711\) −2.06745 −0.0775356
\(712\) −157.044 −5.88549
\(713\) −62.5302 −2.34178
\(714\) 2.25628 0.0844393
\(715\) −1.01022 −0.0377802
\(716\) 69.4815 2.59665
\(717\) −21.2651 −0.794161
\(718\) 36.3402 1.35621
\(719\) 12.2164 0.455596 0.227798 0.973708i \(-0.426847\pi\)
0.227798 + 0.973708i \(0.426847\pi\)
\(720\) −16.1493 −0.601848
\(721\) 1.91788 0.0714255
\(722\) 49.8405 1.85487
\(723\) −15.3177 −0.569670
\(724\) −13.9839 −0.519709
\(725\) −1.00000 −0.0371391
\(726\) −29.8950 −1.10951
\(727\) −40.8856 −1.51636 −0.758182 0.652044i \(-0.773912\pi\)
−0.758182 + 0.652044i \(0.773912\pi\)
\(728\) 10.0219 0.371438
\(729\) 1.00000 0.0370370
\(730\) 9.21234 0.340964
\(731\) 21.4645 0.793892
\(732\) 35.0117 1.29407
\(733\) 25.4915 0.941550 0.470775 0.882253i \(-0.343974\pi\)
0.470775 + 0.882253i \(0.343974\pi\)
\(734\) −48.1056 −1.77561
\(735\) −6.84490 −0.252478
\(736\) −151.920 −5.59986
\(737\) 2.92232 0.107645
\(738\) 4.56892 0.168184
\(739\) 9.56830 0.351975 0.175988 0.984392i \(-0.443688\pi\)
0.175988 + 0.984392i \(0.443688\pi\)
\(740\) 41.1062 1.51109
\(741\) −2.45978 −0.0903624
\(742\) 13.3836 0.491328
\(743\) −18.0455 −0.662025 −0.331013 0.943626i \(-0.607390\pi\)
−0.331013 + 0.943626i \(0.607390\pi\)
\(744\) −100.759 −3.69400
\(745\) 11.5402 0.422801
\(746\) 88.7673 3.25000
\(747\) 6.41000 0.234530
\(748\) −4.58266 −0.167559
\(749\) −0.922572 −0.0337100
\(750\) −2.75660 −0.100657
\(751\) 11.9255 0.435168 0.217584 0.976042i \(-0.430182\pi\)
0.217584 + 0.976042i \(0.430182\pi\)
\(752\) −187.194 −6.82627
\(753\) −26.8917 −0.979989
\(754\) −7.07097 −0.257510
\(755\) −6.08212 −0.221351
\(756\) −2.20500 −0.0801951
\(757\) 5.86152 0.213041 0.106520 0.994311i \(-0.466029\pi\)
0.106520 + 0.994311i \(0.466029\pi\)
\(758\) −89.2201 −3.24062
\(759\) 2.42467 0.0880100
\(760\) −9.51320 −0.345080
\(761\) 49.8739 1.80793 0.903963 0.427610i \(-0.140644\pi\)
0.903963 + 0.427610i \(0.140644\pi\)
\(762\) 55.5637 2.01286
\(763\) 3.36744 0.121909
\(764\) −35.3894 −1.28034
\(765\) −2.07830 −0.0751412
\(766\) 71.3239 2.57704
\(767\) 24.4717 0.883621
\(768\) −63.9648 −2.30813
\(769\) 29.5121 1.06423 0.532117 0.846671i \(-0.321396\pi\)
0.532117 + 0.846671i \(0.321396\pi\)
\(770\) −0.427559 −0.0154082
\(771\) 6.85512 0.246881
\(772\) 141.607 5.09653
\(773\) −2.41935 −0.0870180 −0.0435090 0.999053i \(-0.513854\pi\)
−0.0435090 + 0.999053i \(0.513854\pi\)
\(774\) −28.4698 −1.02333
\(775\) −10.1566 −0.364836
\(776\) 183.108 6.57320
\(777\) 2.89149 0.103731
\(778\) −89.8934 −3.22283
\(779\) 1.58939 0.0569460
\(780\) −14.3616 −0.514228
\(781\) −2.35722 −0.0843479
\(782\) −35.2715 −1.26131
\(783\) 1.00000 0.0357371
\(784\) −110.540 −3.94786
\(785\) 11.5953 0.413855
\(786\) −15.1678 −0.541016
\(787\) −51.1549 −1.82348 −0.911738 0.410772i \(-0.865259\pi\)
−0.911738 + 0.410772i \(0.865259\pi\)
\(788\) −118.722 −4.22928
\(789\) 13.6294 0.485218
\(790\) −5.69914 −0.202766
\(791\) −4.41428 −0.156954
\(792\) 3.90703 0.138830
\(793\) 16.0406 0.569619
\(794\) −12.1162 −0.429987
\(795\) −12.3279 −0.437225
\(796\) −14.8214 −0.525330
\(797\) 28.8662 1.02249 0.511247 0.859434i \(-0.329184\pi\)
0.511247 + 0.859434i \(0.329184\pi\)
\(798\) −1.04106 −0.0368531
\(799\) −24.0907 −0.852266
\(800\) −24.6760 −0.872428
\(801\) 15.8302 0.559334
\(802\) 52.5567 1.85584
\(803\) −1.31616 −0.0464462
\(804\) 41.5446 1.46517
\(805\) −2.42467 −0.0854584
\(806\) −71.8171 −2.52965
\(807\) 8.46129 0.297851
\(808\) 127.021 4.46858
\(809\) 19.0426 0.669501 0.334750 0.942307i \(-0.391348\pi\)
0.334750 + 0.942307i \(0.391348\pi\)
\(810\) 2.75660 0.0968570
\(811\) −20.8783 −0.733137 −0.366569 0.930391i \(-0.619467\pi\)
−0.366569 + 0.930391i \(0.619467\pi\)
\(812\) −2.20500 −0.0773804
\(813\) −6.34255 −0.222443
\(814\) −7.97066 −0.279372
\(815\) 0.855118 0.0299535
\(816\) −33.5631 −1.17494
\(817\) −9.90381 −0.346491
\(818\) 3.81060 0.133234
\(819\) −1.01022 −0.0353000
\(820\) 9.27979 0.324064
\(821\) 3.60466 0.125804 0.0629018 0.998020i \(-0.479965\pi\)
0.0629018 + 0.998020i \(0.479965\pi\)
\(822\) 20.6704 0.720964
\(823\) 27.2288 0.949135 0.474567 0.880219i \(-0.342605\pi\)
0.474567 + 0.880219i \(0.342605\pi\)
\(824\) −48.3109 −1.68299
\(825\) 0.393832 0.0137115
\(826\) 10.3572 0.360374
\(827\) 47.3936 1.64804 0.824019 0.566562i \(-0.191727\pi\)
0.824019 + 0.566562i \(0.191727\pi\)
\(828\) 34.4698 1.19791
\(829\) −41.8388 −1.45312 −0.726560 0.687103i \(-0.758882\pi\)
−0.726560 + 0.687103i \(0.758882\pi\)
\(830\) 17.6698 0.613328
\(831\) 17.1464 0.594802
\(832\) −91.6339 −3.17683
\(833\) −14.2258 −0.492894
\(834\) −25.7818 −0.892752
\(835\) −7.73194 −0.267575
\(836\) 2.11446 0.0731302
\(837\) 10.1566 0.351064
\(838\) −7.91085 −0.273276
\(839\) −31.6385 −1.09228 −0.546141 0.837693i \(-0.683904\pi\)
−0.546141 + 0.837693i \(0.683904\pi\)
\(840\) −3.90703 −0.134805
\(841\) 1.00000 0.0344828
\(842\) −37.6111 −1.29617
\(843\) 12.2985 0.423584
\(844\) 11.1977 0.385440
\(845\) 6.42023 0.220862
\(846\) 31.9531 1.09857
\(847\) −4.27107 −0.146756
\(848\) −199.086 −6.83665
\(849\) −25.1830 −0.864278
\(850\) −5.72905 −0.196505
\(851\) −45.2013 −1.54948
\(852\) −33.5109 −1.14806
\(853\) 44.2000 1.51338 0.756690 0.653773i \(-0.226815\pi\)
0.756690 + 0.653773i \(0.226815\pi\)
\(854\) 6.78892 0.232312
\(855\) 0.958939 0.0327950
\(856\) 23.2394 0.794305
\(857\) 14.4775 0.494541 0.247270 0.968947i \(-0.420466\pi\)
0.247270 + 0.968947i \(0.420466\pi\)
\(858\) 2.78478 0.0950707
\(859\) 22.5883 0.770703 0.385352 0.922770i \(-0.374080\pi\)
0.385352 + 0.922770i \(0.374080\pi\)
\(860\) −57.8241 −1.97179
\(861\) 0.652757 0.0222459
\(862\) 32.9983 1.12393
\(863\) −22.9900 −0.782590 −0.391295 0.920265i \(-0.627973\pi\)
−0.391295 + 0.920265i \(0.627973\pi\)
\(864\) 24.6760 0.839494
\(865\) −8.07449 −0.274541
\(866\) −49.5930 −1.68524
\(867\) 12.6807 0.430658
\(868\) −22.3953 −0.760147
\(869\) 0.814230 0.0276209
\(870\) 2.75660 0.0934574
\(871\) 19.0337 0.644931
\(872\) −84.8250 −2.87254
\(873\) −18.4575 −0.624691
\(874\) 16.2744 0.550491
\(875\) −0.393832 −0.0133140
\(876\) −18.7109 −0.632182
\(877\) 13.5741 0.458364 0.229182 0.973384i \(-0.426395\pi\)
0.229182 + 0.973384i \(0.426395\pi\)
\(878\) 44.7191 1.50920
\(879\) −12.3170 −0.415443
\(880\) 6.36011 0.214399
\(881\) 9.91235 0.333956 0.166978 0.985961i \(-0.446599\pi\)
0.166978 + 0.985961i \(0.446599\pi\)
\(882\) 18.8686 0.635340
\(883\) 39.3192 1.32320 0.661598 0.749859i \(-0.269879\pi\)
0.661598 + 0.749859i \(0.269879\pi\)
\(884\) −29.8478 −1.00389
\(885\) −9.54022 −0.320691
\(886\) 97.5180 3.27618
\(887\) −59.2520 −1.98949 −0.994743 0.102403i \(-0.967347\pi\)
−0.994743 + 0.102403i \(0.967347\pi\)
\(888\) −72.8358 −2.44421
\(889\) 7.93832 0.266243
\(890\) 43.6376 1.46274
\(891\) −0.393832 −0.0131939
\(892\) 70.2678 2.35274
\(893\) 11.1155 0.371968
\(894\) −31.8117 −1.06394
\(895\) −12.4100 −0.414821
\(896\) −19.3461 −0.646307
\(897\) 15.7924 0.527291
\(898\) −114.942 −3.83567
\(899\) 10.1566 0.338742
\(900\) 5.59883 0.186628
\(901\) −25.6211 −0.853562
\(902\) −1.79939 −0.0599131
\(903\) −4.06745 −0.135356
\(904\) 111.195 3.69828
\(905\) 2.49765 0.0830248
\(906\) 16.7660 0.557012
\(907\) −5.91210 −0.196308 −0.0981541 0.995171i \(-0.531294\pi\)
−0.0981541 + 0.995171i \(0.531294\pi\)
\(908\) 7.28682 0.241822
\(909\) −12.8038 −0.424676
\(910\) −2.78478 −0.0923144
\(911\) −12.4185 −0.411445 −0.205722 0.978610i \(-0.565954\pi\)
−0.205722 + 0.978610i \(0.565954\pi\)
\(912\) 15.4862 0.512799
\(913\) −2.52447 −0.0835476
\(914\) 44.4118 1.46901
\(915\) −6.25340 −0.206731
\(916\) 18.4024 0.608031
\(917\) −2.16700 −0.0715607
\(918\) 5.72905 0.189087
\(919\) 16.4332 0.542081 0.271041 0.962568i \(-0.412632\pi\)
0.271041 + 0.962568i \(0.412632\pi\)
\(920\) 61.0769 2.01364
\(921\) 22.7379 0.749239
\(922\) −47.7983 −1.57415
\(923\) −15.3530 −0.505351
\(924\) 0.868401 0.0285683
\(925\) −7.34192 −0.241401
\(926\) −53.2510 −1.74994
\(927\) 4.86979 0.159945
\(928\) 24.6760 0.810029
\(929\) 13.8358 0.453936 0.226968 0.973902i \(-0.427119\pi\)
0.226968 + 0.973902i \(0.427119\pi\)
\(930\) 27.9977 0.918080
\(931\) 6.56384 0.215121
\(932\) 99.1637 3.24822
\(933\) 5.33810 0.174762
\(934\) 22.0528 0.721589
\(935\) 0.818503 0.0267679
\(936\) 25.4472 0.831769
\(937\) 20.7528 0.677964 0.338982 0.940793i \(-0.389917\pi\)
0.338982 + 0.940793i \(0.389917\pi\)
\(938\) 8.05567 0.263027
\(939\) 1.96338 0.0640726
\(940\) 64.8989 2.11677
\(941\) 40.1666 1.30940 0.654698 0.755891i \(-0.272796\pi\)
0.654698 + 0.755891i \(0.272796\pi\)
\(942\) −31.9636 −1.04143
\(943\) −10.2043 −0.332297
\(944\) −154.068 −5.01447
\(945\) 0.393832 0.0128114
\(946\) 11.2123 0.364544
\(947\) −17.4144 −0.565894 −0.282947 0.959136i \(-0.591312\pi\)
−0.282947 + 0.959136i \(0.591312\pi\)
\(948\) 11.5753 0.375949
\(949\) −8.57239 −0.278271
\(950\) 2.64341 0.0857636
\(951\) −1.29064 −0.0418518
\(952\) −8.11999 −0.263170
\(953\) 13.5121 0.437701 0.218851 0.975758i \(-0.429769\pi\)
0.218851 + 0.975758i \(0.429769\pi\)
\(954\) 33.9830 1.10024
\(955\) 6.32085 0.204538
\(956\) 119.060 3.85067
\(957\) −0.393832 −0.0127308
\(958\) 111.333 3.59700
\(959\) 2.95316 0.0953626
\(960\) 35.7232 1.15296
\(961\) 72.1567 2.32763
\(962\) −51.9145 −1.67379
\(963\) −2.34255 −0.0754876
\(964\) 85.7610 2.76218
\(965\) −25.2921 −0.814183
\(966\) 6.68384 0.215049
\(967\) −24.0598 −0.773712 −0.386856 0.922140i \(-0.626439\pi\)
−0.386856 + 0.922140i \(0.626439\pi\)
\(968\) 107.587 3.45798
\(969\) 1.99297 0.0640233
\(970\) −50.8798 −1.63365
\(971\) 34.7877 1.11639 0.558195 0.829710i \(-0.311494\pi\)
0.558195 + 0.829710i \(0.311494\pi\)
\(972\) −5.59883 −0.179583
\(973\) −3.68342 −0.118085
\(974\) −7.83344 −0.251000
\(975\) 2.56511 0.0821492
\(976\) −100.988 −3.23254
\(977\) 35.7566 1.14396 0.571978 0.820269i \(-0.306176\pi\)
0.571978 + 0.820269i \(0.306176\pi\)
\(978\) −2.35722 −0.0753755
\(979\) −6.23446 −0.199254
\(980\) 38.3234 1.22420
\(981\) 8.55044 0.272995
\(982\) −0.435166 −0.0138867
\(983\) −7.19064 −0.229346 −0.114673 0.993403i \(-0.536582\pi\)
−0.114673 + 0.993403i \(0.536582\pi\)
\(984\) −16.4428 −0.524177
\(985\) 21.2047 0.675638
\(986\) 5.72905 0.182450
\(987\) 4.56511 0.145309
\(988\) 13.7719 0.438143
\(989\) 63.5847 2.02188
\(990\) −1.08564 −0.0345038
\(991\) −19.9123 −0.632537 −0.316268 0.948670i \(-0.602430\pi\)
−0.316268 + 0.948670i \(0.602430\pi\)
\(992\) 250.624 7.95733
\(993\) −28.9971 −0.920194
\(994\) −6.49790 −0.206101
\(995\) 2.64723 0.0839228
\(996\) −35.8885 −1.13717
\(997\) −1.57302 −0.0498179 −0.0249089 0.999690i \(-0.507930\pi\)
−0.0249089 + 0.999690i \(0.507930\pi\)
\(998\) −7.29734 −0.230993
\(999\) 7.34192 0.232288
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 435.2.a.j.1.1 4
3.2 odd 2 1305.2.a.r.1.4 4
4.3 odd 2 6960.2.a.co.1.2 4
5.2 odd 4 2175.2.c.n.349.1 8
5.3 odd 4 2175.2.c.n.349.8 8
5.4 even 2 2175.2.a.v.1.4 4
15.14 odd 2 6525.2.a.bi.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.1 4 1.1 even 1 trivial
1305.2.a.r.1.4 4 3.2 odd 2
2175.2.a.v.1.4 4 5.4 even 2
2175.2.c.n.349.1 8 5.2 odd 4
2175.2.c.n.349.8 8 5.3 odd 4
6525.2.a.bi.1.1 4 15.14 odd 2
6960.2.a.co.1.2 4 4.3 odd 2