Properties

Label 2475.2.a.bf.1.1
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
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] = [2475,2,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, 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("2475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(19.7629745003\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.48704.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 495)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.69696\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.69696 q^{2} +5.27358 q^{4} +4.13176 q^{7} -8.82872 q^{8} +O(q^{10})\) \(q-2.69696 q^{2} +5.27358 q^{4} +4.13176 q^{7} -8.82872 q^{8} -1.00000 q^{11} -2.73785 q^{13} -11.1432 q^{14} +13.2635 q^{16} -2.41541 q^{17} -1.15325 q^{19} +2.69696 q^{22} +8.54717 q^{23} +7.38386 q^{26} +21.7892 q^{28} -1.67756 q^{29} -10.2635 q^{31} -18.1137 q^{32} +6.51425 q^{34} -5.71636 q^{37} +3.11028 q^{38} -9.11028 q^{41} -8.13176 q^{43} -5.27358 q^{44} -23.0514 q^{46} -1.47569 q^{47} +10.0715 q^{49} -14.4383 q^{52} +2.54717 q^{53} -36.4782 q^{56} +4.52431 q^{58} -8.24066 q^{59} +3.47569 q^{61} +27.6803 q^{62} +22.3249 q^{64} -15.3350 q^{67} -12.7378 q^{68} +2.54717 q^{71} -2.73785 q^{73} +15.4168 q^{74} -6.08178 q^{76} -4.13176 q^{77} +5.15325 q^{79} +24.5700 q^{82} -9.28502 q^{83} +21.9310 q^{86} +8.82872 q^{88} +4.78783 q^{89} -11.3121 q^{91} +45.0742 q^{92} +3.97989 q^{94} -4.24066 q^{97} -27.1624 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 8 q^{4} - 4 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 8 q^{4} - 4 q^{7} - 6 q^{8} - 4 q^{11} - 8 q^{13} - 8 q^{14} + 12 q^{16} - 4 q^{17} + 4 q^{19} + 2 q^{22} + 8 q^{23} - 16 q^{26} + 8 q^{28} - 4 q^{29} - 14 q^{32} + 4 q^{34} - 8 q^{37} - 20 q^{38} - 4 q^{41} - 12 q^{43} - 8 q^{44} - 16 q^{46} + 20 q^{49} - 20 q^{52} - 16 q^{53} - 48 q^{56} + 24 q^{58} - 24 q^{59} + 8 q^{61} + 20 q^{62} - 48 q^{68} - 16 q^{71} - 8 q^{73} + 12 q^{74} - 36 q^{76} + 4 q^{77} + 12 q^{79} + 40 q^{82} - 8 q^{83} + 16 q^{86} + 6 q^{88} - 16 q^{89} - 16 q^{91} + 72 q^{92} - 40 q^{94} - 8 q^{97} - 62 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.69696 −1.90704 −0.953519 0.301334i \(-0.902568\pi\)
−0.953519 + 0.301334i \(0.902568\pi\)
\(3\) 0 0
\(4\) 5.27358 2.63679
\(5\) 0 0
\(6\) 0 0
\(7\) 4.13176 1.56166 0.780830 0.624744i \(-0.214796\pi\)
0.780830 + 0.624744i \(0.214796\pi\)
\(8\) −8.82872 −3.12142
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −2.73785 −0.759342 −0.379671 0.925122i \(-0.623963\pi\)
−0.379671 + 0.925122i \(0.623963\pi\)
\(14\) −11.1432 −2.97814
\(15\) 0 0
\(16\) 13.2635 3.31588
\(17\) −2.41541 −0.585822 −0.292911 0.956140i \(-0.594624\pi\)
−0.292911 + 0.956140i \(0.594624\pi\)
\(18\) 0 0
\(19\) −1.15325 −0.264574 −0.132287 0.991211i \(-0.542232\pi\)
−0.132287 + 0.991211i \(0.542232\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.69696 0.574993
\(23\) 8.54717 1.78221 0.891104 0.453799i \(-0.149932\pi\)
0.891104 + 0.453799i \(0.149932\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 7.38386 1.44809
\(27\) 0 0
\(28\) 21.7892 4.11777
\(29\) −1.67756 −0.311515 −0.155757 0.987795i \(-0.549782\pi\)
−0.155757 + 0.987795i \(0.549782\pi\)
\(30\) 0 0
\(31\) −10.2635 −1.84338 −0.921692 0.387922i \(-0.873193\pi\)
−0.921692 + 0.387922i \(0.873193\pi\)
\(32\) −18.1137 −3.20209
\(33\) 0 0
\(34\) 6.51425 1.11718
\(35\) 0 0
\(36\) 0 0
\(37\) −5.71636 −0.939764 −0.469882 0.882729i \(-0.655703\pi\)
−0.469882 + 0.882729i \(0.655703\pi\)
\(38\) 3.11028 0.504553
\(39\) 0 0
\(40\) 0 0
\(41\) −9.11028 −1.42279 −0.711393 0.702794i \(-0.751935\pi\)
−0.711393 + 0.702794i \(0.751935\pi\)
\(42\) 0 0
\(43\) −8.13176 −1.24008 −0.620041 0.784569i \(-0.712884\pi\)
−0.620041 + 0.784569i \(0.712884\pi\)
\(44\) −5.27358 −0.795023
\(45\) 0 0
\(46\) −23.0514 −3.39874
\(47\) −1.47569 −0.215252 −0.107626 0.994191i \(-0.534325\pi\)
−0.107626 + 0.994191i \(0.534325\pi\)
\(48\) 0 0
\(49\) 10.0715 1.43878
\(50\) 0 0
\(51\) 0 0
\(52\) −14.4383 −2.00223
\(53\) 2.54717 0.349881 0.174940 0.984579i \(-0.444027\pi\)
0.174940 + 0.984579i \(0.444027\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −36.4782 −4.87460
\(57\) 0 0
\(58\) 4.52431 0.594070
\(59\) −8.24066 −1.07284 −0.536422 0.843950i \(-0.680224\pi\)
−0.536422 + 0.843950i \(0.680224\pi\)
\(60\) 0 0
\(61\) 3.47569 0.445017 0.222509 0.974931i \(-0.428575\pi\)
0.222509 + 0.974931i \(0.428575\pi\)
\(62\) 27.6803 3.51540
\(63\) 0 0
\(64\) 22.3249 2.79062
\(65\) 0 0
\(66\) 0 0
\(67\) −15.3350 −1.87347 −0.936734 0.350041i \(-0.886168\pi\)
−0.936734 + 0.350041i \(0.886168\pi\)
\(68\) −12.7378 −1.54469
\(69\) 0 0
\(70\) 0 0
\(71\) 2.54717 0.302293 0.151147 0.988511i \(-0.451703\pi\)
0.151147 + 0.988511i \(0.451703\pi\)
\(72\) 0 0
\(73\) −2.73785 −0.320441 −0.160220 0.987081i \(-0.551220\pi\)
−0.160220 + 0.987081i \(0.551220\pi\)
\(74\) 15.4168 1.79216
\(75\) 0 0
\(76\) −6.08178 −0.697628
\(77\) −4.13176 −0.470858
\(78\) 0 0
\(79\) 5.15325 0.579786 0.289893 0.957059i \(-0.406380\pi\)
0.289893 + 0.957059i \(0.406380\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 24.5700 2.71331
\(83\) −9.28502 −1.01916 −0.509582 0.860422i \(-0.670200\pi\)
−0.509582 + 0.860422i \(0.670200\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 21.9310 2.36488
\(87\) 0 0
\(88\) 8.82872 0.941145
\(89\) 4.78783 0.507509 0.253755 0.967269i \(-0.418334\pi\)
0.253755 + 0.967269i \(0.418334\pi\)
\(90\) 0 0
\(91\) −11.3121 −1.18583
\(92\) 45.0742 4.69931
\(93\) 0 0
\(94\) 3.97989 0.410494
\(95\) 0 0
\(96\) 0 0
\(97\) −4.24066 −0.430574 −0.215287 0.976551i \(-0.569069\pi\)
−0.215287 + 0.976551i \(0.569069\pi\)
\(98\) −27.1624 −2.74381
\(99\) 0 0
\(100\) 0 0
\(101\) −13.9411 −1.38719 −0.693595 0.720365i \(-0.743974\pi\)
−0.693595 + 0.720365i \(0.743974\pi\)
\(102\) 0 0
\(103\) 18.5042 1.82327 0.911636 0.410998i \(-0.134820\pi\)
0.911636 + 0.410998i \(0.134820\pi\)
\(104\) 24.1717 2.37023
\(105\) 0 0
\(106\) −6.86961 −0.667236
\(107\) −13.7663 −1.33084 −0.665421 0.746468i \(-0.731748\pi\)
−0.665421 + 0.746468i \(0.731748\pi\)
\(108\) 0 0
\(109\) 12.7878 1.22485 0.612426 0.790528i \(-0.290194\pi\)
0.612426 + 0.790528i \(0.290194\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 54.8018 5.17828
\(113\) 14.8107 1.39327 0.696637 0.717424i \(-0.254679\pi\)
0.696637 + 0.717424i \(0.254679\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −8.84675 −0.821400
\(117\) 0 0
\(118\) 22.2247 2.04595
\(119\) −9.97989 −0.914855
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −9.37380 −0.848664
\(123\) 0 0
\(124\) −54.1256 −4.86062
\(125\) 0 0
\(126\) 0 0
\(127\) −12.9626 −1.15024 −0.575121 0.818068i \(-0.695045\pi\)
−0.575121 + 0.818068i \(0.695045\pi\)
\(128\) −23.9820 −2.11973
\(129\) 0 0
\(130\) 0 0
\(131\) −1.47569 −0.128932 −0.0644660 0.997920i \(-0.520534\pi\)
−0.0644660 + 0.997920i \(0.520534\pi\)
\(132\) 0 0
\(133\) −4.76497 −0.413175
\(134\) 41.3579 3.57278
\(135\) 0 0
\(136\) 21.3249 1.82860
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −0.889725 −0.0754655 −0.0377327 0.999288i \(-0.512014\pi\)
−0.0377327 + 0.999288i \(0.512014\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.86961 −0.576485
\(143\) 2.73785 0.228950
\(144\) 0 0
\(145\) 0 0
\(146\) 7.38386 0.611093
\(147\) 0 0
\(148\) −30.1457 −2.47796
\(149\) 12.7289 1.04279 0.521397 0.853314i \(-0.325411\pi\)
0.521397 + 0.853314i \(0.325411\pi\)
\(150\) 0 0
\(151\) −1.15325 −0.0938504 −0.0469252 0.998898i \(-0.514942\pi\)
−0.0469252 + 0.998898i \(0.514942\pi\)
\(152\) 10.1818 0.825849
\(153\) 0 0
\(154\) 11.1432 0.897944
\(155\) 0 0
\(156\) 0 0
\(157\) 3.95702 0.315805 0.157902 0.987455i \(-0.449527\pi\)
0.157902 + 0.987455i \(0.449527\pi\)
\(158\) −13.8981 −1.10567
\(159\) 0 0
\(160\) 0 0
\(161\) 35.3149 2.78320
\(162\) 0 0
\(163\) −3.07148 −0.240577 −0.120288 0.992739i \(-0.538382\pi\)
−0.120288 + 0.992739i \(0.538382\pi\)
\(164\) −48.0438 −3.75159
\(165\) 0 0
\(166\) 25.0413 1.94358
\(167\) −2.71498 −0.210092 −0.105046 0.994467i \(-0.533499\pi\)
−0.105046 + 0.994467i \(0.533499\pi\)
\(168\) 0 0
\(169\) −5.50419 −0.423399
\(170\) 0 0
\(171\) 0 0
\(172\) −42.8835 −3.26984
\(173\) −14.4154 −1.09598 −0.547992 0.836484i \(-0.684607\pi\)
−0.547992 + 0.836484i \(0.684607\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −13.2635 −0.999776
\(177\) 0 0
\(178\) −12.9126 −0.967839
\(179\) 10.7878 0.806321 0.403160 0.915129i \(-0.367912\pi\)
0.403160 + 0.915129i \(0.367912\pi\)
\(180\) 0 0
\(181\) −6.50419 −0.483453 −0.241726 0.970344i \(-0.577714\pi\)
−0.241726 + 0.970344i \(0.577714\pi\)
\(182\) 30.5084 2.26143
\(183\) 0 0
\(184\) −75.4606 −5.56303
\(185\) 0 0
\(186\) 0 0
\(187\) 2.41541 0.176632
\(188\) −7.78220 −0.567575
\(189\) 0 0
\(190\) 0 0
\(191\) 15.8822 1.14919 0.574597 0.818437i \(-0.305159\pi\)
0.574597 + 0.818437i \(0.305159\pi\)
\(192\) 0 0
\(193\) 3.83219 0.275847 0.137923 0.990443i \(-0.455957\pi\)
0.137923 + 0.990443i \(0.455957\pi\)
\(194\) 11.4369 0.821121
\(195\) 0 0
\(196\) 53.1128 3.79377
\(197\) 2.67893 0.190866 0.0954331 0.995436i \(-0.469576\pi\)
0.0954331 + 0.995436i \(0.469576\pi\)
\(198\) 0 0
\(199\) 6.78783 0.481177 0.240588 0.970627i \(-0.422660\pi\)
0.240588 + 0.970627i \(0.422660\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 37.5985 2.64542
\(203\) −6.93128 −0.486480
\(204\) 0 0
\(205\) 0 0
\(206\) −49.9050 −3.47705
\(207\) 0 0
\(208\) −36.3135 −2.51789
\(209\) 1.15325 0.0797722
\(210\) 0 0
\(211\) 17.6803 1.21716 0.608581 0.793491i \(-0.291739\pi\)
0.608581 + 0.793491i \(0.291739\pi\)
\(212\) 13.4327 0.922563
\(213\) 0 0
\(214\) 37.1273 2.53797
\(215\) 0 0
\(216\) 0 0
\(217\) −42.4065 −2.87874
\(218\) −34.4883 −2.33584
\(219\) 0 0
\(220\) 0 0
\(221\) 6.61301 0.444839
\(222\) 0 0
\(223\) 8.83081 0.591355 0.295677 0.955288i \(-0.404455\pi\)
0.295677 + 0.955288i \(0.404455\pi\)
\(224\) −74.8417 −5.00057
\(225\) 0 0
\(226\) −39.9438 −2.65702
\(227\) −10.4972 −0.696723 −0.348361 0.937360i \(-0.613262\pi\)
−0.348361 + 0.937360i \(0.613262\pi\)
\(228\) 0 0
\(229\) 4.95139 0.327197 0.163598 0.986527i \(-0.447690\pi\)
0.163598 + 0.986527i \(0.447690\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 14.8107 0.972370
\(233\) 3.89110 0.254914 0.127457 0.991844i \(-0.459318\pi\)
0.127457 + 0.991844i \(0.459318\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −43.4578 −2.82886
\(237\) 0 0
\(238\) 26.9153 1.74466
\(239\) −15.6186 −1.01029 −0.505143 0.863036i \(-0.668560\pi\)
−0.505143 + 0.863036i \(0.668560\pi\)
\(240\) 0 0
\(241\) −10.2635 −0.661132 −0.330566 0.943783i \(-0.607240\pi\)
−0.330566 + 0.943783i \(0.607240\pi\)
\(242\) −2.69696 −0.173367
\(243\) 0 0
\(244\) 18.3294 1.17342
\(245\) 0 0
\(246\) 0 0
\(247\) 3.15743 0.200902
\(248\) 90.6138 5.75398
\(249\) 0 0
\(250\) 0 0
\(251\) −19.6415 −1.23976 −0.619881 0.784696i \(-0.712819\pi\)
−0.619881 + 0.784696i \(0.712819\pi\)
\(252\) 0 0
\(253\) −8.54717 −0.537356
\(254\) 34.9595 2.19356
\(255\) 0 0
\(256\) 20.0285 1.25178
\(257\) 1.16919 0.0729320 0.0364660 0.999335i \(-0.488390\pi\)
0.0364660 + 0.999335i \(0.488390\pi\)
\(258\) 0 0
\(259\) −23.6186 −1.46759
\(260\) 0 0
\(261\) 0 0
\(262\) 3.97989 0.245878
\(263\) −8.93553 −0.550989 −0.275494 0.961303i \(-0.588842\pi\)
−0.275494 + 0.961303i \(0.588842\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 12.8509 0.787941
\(267\) 0 0
\(268\) −80.8704 −4.93995
\(269\) −14.9514 −0.911602 −0.455801 0.890082i \(-0.650647\pi\)
−0.455801 + 0.890082i \(0.650647\pi\)
\(270\) 0 0
\(271\) −22.7289 −1.38068 −0.690342 0.723483i \(-0.742540\pi\)
−0.690342 + 0.723483i \(0.742540\pi\)
\(272\) −32.0368 −1.94252
\(273\) 0 0
\(274\) −16.1818 −0.977575
\(275\) 0 0
\(276\) 0 0
\(277\) 22.1387 1.33019 0.665093 0.746761i \(-0.268392\pi\)
0.665093 + 0.746761i \(0.268392\pi\)
\(278\) 2.39955 0.143915
\(279\) 0 0
\(280\) 0 0
\(281\) −19.9841 −1.19215 −0.596075 0.802929i \(-0.703274\pi\)
−0.596075 + 0.802929i \(0.703274\pi\)
\(282\) 0 0
\(283\) −24.2775 −1.44315 −0.721573 0.692339i \(-0.756580\pi\)
−0.721573 + 0.692339i \(0.756580\pi\)
\(284\) 13.4327 0.797085
\(285\) 0 0
\(286\) −7.38386 −0.436617
\(287\) −37.6415 −2.22191
\(288\) 0 0
\(289\) −11.1658 −0.656813
\(290\) 0 0
\(291\) 0 0
\(292\) −14.4383 −0.844936
\(293\) 16.4182 0.959159 0.479579 0.877498i \(-0.340789\pi\)
0.479579 + 0.877498i \(0.340789\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 50.4681 2.93340
\(297\) 0 0
\(298\) −34.3294 −1.98865
\(299\) −23.4008 −1.35331
\(300\) 0 0
\(301\) −33.5985 −1.93659
\(302\) 3.11028 0.178976
\(303\) 0 0
\(304\) −15.2962 −0.877297
\(305\) 0 0
\(306\) 0 0
\(307\) −5.44390 −0.310700 −0.155350 0.987859i \(-0.549651\pi\)
−0.155350 + 0.987859i \(0.549651\pi\)
\(308\) −21.7892 −1.24156
\(309\) 0 0
\(310\) 0 0
\(311\) 28.2864 1.60397 0.801987 0.597341i \(-0.203776\pi\)
0.801987 + 0.597341i \(0.203776\pi\)
\(312\) 0 0
\(313\) −22.8107 −1.28934 −0.644668 0.764462i \(-0.723005\pi\)
−0.644668 + 0.764462i \(0.723005\pi\)
\(314\) −10.6719 −0.602252
\(315\) 0 0
\(316\) 27.1761 1.52878
\(317\) −23.8593 −1.34007 −0.670036 0.742328i \(-0.733721\pi\)
−0.670036 + 0.742328i \(0.733721\pi\)
\(318\) 0 0
\(319\) 1.67756 0.0939252
\(320\) 0 0
\(321\) 0 0
\(322\) −95.2428 −5.30767
\(323\) 2.78557 0.154993
\(324\) 0 0
\(325\) 0 0
\(326\) 8.28364 0.458788
\(327\) 0 0
\(328\) 80.4321 4.44112
\(329\) −6.09722 −0.336151
\(330\) 0 0
\(331\) −18.9084 −1.03930 −0.519650 0.854379i \(-0.673938\pi\)
−0.519650 + 0.854379i \(0.673938\pi\)
\(332\) −48.9653 −2.68732
\(333\) 0 0
\(334\) 7.32220 0.400653
\(335\) 0 0
\(336\) 0 0
\(337\) −24.6630 −1.34348 −0.671740 0.740787i \(-0.734453\pi\)
−0.671740 + 0.740787i \(0.734453\pi\)
\(338\) 14.8446 0.807439
\(339\) 0 0
\(340\) 0 0
\(341\) 10.2635 0.555801
\(342\) 0 0
\(343\) 12.6906 0.685229
\(344\) 71.7931 3.87082
\(345\) 0 0
\(346\) 38.8778 2.09008
\(347\) 15.5056 0.832383 0.416191 0.909277i \(-0.363365\pi\)
0.416191 + 0.909277i \(0.363365\pi\)
\(348\) 0 0
\(349\) 20.4841 1.09649 0.548244 0.836319i \(-0.315297\pi\)
0.548244 + 0.836319i \(0.315297\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 18.1137 0.965466
\(353\) 21.9799 1.16987 0.584936 0.811080i \(-0.301120\pi\)
0.584936 + 0.811080i \(0.301120\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 25.2490 1.33820
\(357\) 0 0
\(358\) −29.0943 −1.53768
\(359\) −19.6962 −1.03953 −0.519764 0.854310i \(-0.673980\pi\)
−0.519764 + 0.854310i \(0.673980\pi\)
\(360\) 0 0
\(361\) −17.6700 −0.930000
\(362\) 17.5415 0.921963
\(363\) 0 0
\(364\) −59.6555 −3.12680
\(365\) 0 0
\(366\) 0 0
\(367\) 28.2635 1.47534 0.737672 0.675159i \(-0.235925\pi\)
0.737672 + 0.675159i \(0.235925\pi\)
\(368\) 113.366 5.90959
\(369\) 0 0
\(370\) 0 0
\(371\) 10.5243 0.546395
\(372\) 0 0
\(373\) 9.78921 0.506866 0.253433 0.967353i \(-0.418440\pi\)
0.253433 + 0.967353i \(0.418440\pi\)
\(374\) −6.51425 −0.336844
\(375\) 0 0
\(376\) 13.0285 0.671893
\(377\) 4.59290 0.236546
\(378\) 0 0
\(379\) 5.66162 0.290818 0.145409 0.989372i \(-0.453550\pi\)
0.145409 + 0.989372i \(0.453550\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −42.8336 −2.19156
\(383\) 26.5042 1.35430 0.677150 0.735845i \(-0.263215\pi\)
0.677150 + 0.735845i \(0.263215\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10.3352 −0.526050
\(387\) 0 0
\(388\) −22.3635 −1.13533
\(389\) −15.3551 −0.778535 −0.389268 0.921125i \(-0.627272\pi\)
−0.389268 + 0.921125i \(0.627272\pi\)
\(390\) 0 0
\(391\) −20.6449 −1.04406
\(392\) −88.9183 −4.49105
\(393\) 0 0
\(394\) −7.22497 −0.363989
\(395\) 0 0
\(396\) 0 0
\(397\) −16.7677 −0.841548 −0.420774 0.907166i \(-0.638241\pi\)
−0.420774 + 0.907166i \(0.638241\pi\)
\(398\) −18.3065 −0.917622
\(399\) 0 0
\(400\) 0 0
\(401\) −26.0457 −1.30066 −0.650331 0.759651i \(-0.725370\pi\)
−0.650331 + 0.759651i \(0.725370\pi\)
\(402\) 0 0
\(403\) 28.1000 1.39976
\(404\) −73.5195 −3.65773
\(405\) 0 0
\(406\) 18.6934 0.927736
\(407\) 5.71636 0.283349
\(408\) 0 0
\(409\) 22.4495 1.11005 0.555027 0.831832i \(-0.312708\pi\)
0.555027 + 0.831832i \(0.312708\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 97.5834 4.80759
\(413\) −34.0485 −1.67542
\(414\) 0 0
\(415\) 0 0
\(416\) 49.5927 2.43148
\(417\) 0 0
\(418\) −3.11028 −0.152129
\(419\) 27.8822 1.36213 0.681067 0.732221i \(-0.261516\pi\)
0.681067 + 0.732221i \(0.261516\pi\)
\(420\) 0 0
\(421\) 15.5985 0.760226 0.380113 0.924940i \(-0.375885\pi\)
0.380113 + 0.924940i \(0.375885\pi\)
\(422\) −47.6831 −2.32118
\(423\) 0 0
\(424\) −22.4883 −1.09213
\(425\) 0 0
\(426\) 0 0
\(427\) 14.3608 0.694965
\(428\) −72.5980 −3.50916
\(429\) 0 0
\(430\) 0 0
\(431\) 2.95139 0.142163 0.0710817 0.997470i \(-0.477355\pi\)
0.0710817 + 0.997470i \(0.477355\pi\)
\(432\) 0 0
\(433\) −24.2864 −1.16713 −0.583565 0.812067i \(-0.698343\pi\)
−0.583565 + 0.812067i \(0.698343\pi\)
\(434\) 114.369 5.48986
\(435\) 0 0
\(436\) 67.4377 3.22968
\(437\) −9.85705 −0.471527
\(438\) 0 0
\(439\) −19.1103 −0.912084 −0.456042 0.889958i \(-0.650733\pi\)
−0.456042 + 0.889958i \(0.650733\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −17.8350 −0.848325
\(443\) 1.97714 0.0939365 0.0469683 0.998896i \(-0.485044\pi\)
0.0469683 + 0.998896i \(0.485044\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −23.8163 −1.12774
\(447\) 0 0
\(448\) 92.2414 4.35800
\(449\) 15.1719 0.716008 0.358004 0.933720i \(-0.383457\pi\)
0.358004 + 0.933720i \(0.383457\pi\)
\(450\) 0 0
\(451\) 9.11028 0.428986
\(452\) 78.1055 3.67377
\(453\) 0 0
\(454\) 28.3105 1.32868
\(455\) 0 0
\(456\) 0 0
\(457\) 6.17056 0.288647 0.144323 0.989531i \(-0.453899\pi\)
0.144323 + 0.989531i \(0.453899\pi\)
\(458\) −13.3537 −0.623977
\(459\) 0 0
\(460\) 0 0
\(461\) −23.1990 −1.08048 −0.540242 0.841510i \(-0.681667\pi\)
−0.540242 + 0.841510i \(0.681667\pi\)
\(462\) 0 0
\(463\) 1.39809 0.0649750 0.0324875 0.999472i \(-0.489657\pi\)
0.0324875 + 0.999472i \(0.489657\pi\)
\(464\) −22.2503 −1.03295
\(465\) 0 0
\(466\) −10.4941 −0.486131
\(467\) 22.6901 1.04997 0.524987 0.851110i \(-0.324070\pi\)
0.524987 + 0.851110i \(0.324070\pi\)
\(468\) 0 0
\(469\) −63.3606 −2.92572
\(470\) 0 0
\(471\) 0 0
\(472\) 72.7545 3.34880
\(473\) 8.13176 0.373899
\(474\) 0 0
\(475\) 0 0
\(476\) −52.6298 −2.41228
\(477\) 0 0
\(478\) 42.1228 1.92665
\(479\) −24.7906 −1.13271 −0.566355 0.824161i \(-0.691647\pi\)
−0.566355 + 0.824161i \(0.691647\pi\)
\(480\) 0 0
\(481\) 15.6505 0.713602
\(482\) 27.6803 1.26080
\(483\) 0 0
\(484\) 5.27358 0.239708
\(485\) 0 0
\(486\) 0 0
\(487\) 33.7191 1.52796 0.763979 0.645241i \(-0.223243\pi\)
0.763979 + 0.645241i \(0.223243\pi\)
\(488\) −30.6859 −1.38909
\(489\) 0 0
\(490\) 0 0
\(491\) 6.90566 0.311648 0.155824 0.987785i \(-0.450197\pi\)
0.155824 + 0.987785i \(0.450197\pi\)
\(492\) 0 0
\(493\) 4.05198 0.182492
\(494\) −8.51546 −0.383129
\(495\) 0 0
\(496\) −136.131 −6.11244
\(497\) 10.5243 0.472080
\(498\) 0 0
\(499\) −2.26078 −0.101206 −0.0506031 0.998719i \(-0.516114\pi\)
−0.0506031 + 0.998719i \(0.516114\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 52.9723 2.36427
\(503\) −20.5999 −0.918504 −0.459252 0.888306i \(-0.651883\pi\)
−0.459252 + 0.888306i \(0.651883\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 23.0514 1.02476
\(507\) 0 0
\(508\) −68.3592 −3.03295
\(509\) −7.43272 −0.329449 −0.164725 0.986340i \(-0.552674\pi\)
−0.164725 + 0.986340i \(0.552674\pi\)
\(510\) 0 0
\(511\) −11.3121 −0.500420
\(512\) −6.05208 −0.267466
\(513\) 0 0
\(514\) −3.15325 −0.139084
\(515\) 0 0
\(516\) 0 0
\(517\) 1.47569 0.0649010
\(518\) 63.6985 2.79875
\(519\) 0 0
\(520\) 0 0
\(521\) −35.0084 −1.53375 −0.766873 0.641799i \(-0.778188\pi\)
−0.766873 + 0.641799i \(0.778188\pi\)
\(522\) 0 0
\(523\) 1.52986 0.0668960 0.0334480 0.999440i \(-0.489351\pi\)
0.0334480 + 0.999440i \(0.489351\pi\)
\(524\) −7.78220 −0.339967
\(525\) 0 0
\(526\) 24.0988 1.05076
\(527\) 24.7906 1.07989
\(528\) 0 0
\(529\) 50.0541 2.17627
\(530\) 0 0
\(531\) 0 0
\(532\) −25.1285 −1.08946
\(533\) 24.9425 1.08038
\(534\) 0 0
\(535\) 0 0
\(536\) 135.388 5.84789
\(537\) 0 0
\(538\) 40.3233 1.73846
\(539\) −10.0715 −0.433809
\(540\) 0 0
\(541\) 16.6700 0.716700 0.358350 0.933587i \(-0.383340\pi\)
0.358350 + 0.933587i \(0.383340\pi\)
\(542\) 61.2990 2.63302
\(543\) 0 0
\(544\) 43.7520 1.87585
\(545\) 0 0
\(546\) 0 0
\(547\) 2.25234 0.0963032 0.0481516 0.998840i \(-0.484667\pi\)
0.0481516 + 0.998840i \(0.484667\pi\)
\(548\) 31.6415 1.35166
\(549\) 0 0
\(550\) 0 0
\(551\) 1.93465 0.0824188
\(552\) 0 0
\(553\) 21.2920 0.905429
\(554\) −59.7071 −2.53671
\(555\) 0 0
\(556\) −4.69204 −0.198987
\(557\) 11.9911 0.508078 0.254039 0.967194i \(-0.418241\pi\)
0.254039 + 0.967194i \(0.418241\pi\)
\(558\) 0 0
\(559\) 22.2635 0.941647
\(560\) 0 0
\(561\) 0 0
\(562\) 53.8962 2.27347
\(563\) 19.4598 0.820134 0.410067 0.912055i \(-0.365505\pi\)
0.410067 + 0.912055i \(0.365505\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 65.4753 2.75213
\(567\) 0 0
\(568\) −22.4883 −0.943586
\(569\) 19.7205 0.826728 0.413364 0.910566i \(-0.364354\pi\)
0.413364 + 0.910566i \(0.364354\pi\)
\(570\) 0 0
\(571\) 5.41678 0.226685 0.113343 0.993556i \(-0.463844\pi\)
0.113343 + 0.993556i \(0.463844\pi\)
\(572\) 14.4383 0.603694
\(573\) 0 0
\(574\) 101.518 4.23726
\(575\) 0 0
\(576\) 0 0
\(577\) −6.42708 −0.267563 −0.133781 0.991011i \(-0.542712\pi\)
−0.133781 + 0.991011i \(0.542712\pi\)
\(578\) 30.1137 1.25257
\(579\) 0 0
\(580\) 0 0
\(581\) −38.3635 −1.59159
\(582\) 0 0
\(583\) −2.54717 −0.105493
\(584\) 24.1717 1.00023
\(585\) 0 0
\(586\) −44.2791 −1.82915
\(587\) −44.9883 −1.85686 −0.928432 0.371502i \(-0.878843\pi\)
−0.928432 + 0.371502i \(0.878843\pi\)
\(588\) 0 0
\(589\) 11.8364 0.487712
\(590\) 0 0
\(591\) 0 0
\(592\) −75.8191 −3.11615
\(593\) −21.3892 −0.878348 −0.439174 0.898402i \(-0.644729\pi\)
−0.439174 + 0.898402i \(0.644729\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 67.1270 2.74963
\(597\) 0 0
\(598\) 63.1111 2.58081
\(599\) −26.9514 −1.10120 −0.550602 0.834768i \(-0.685602\pi\)
−0.550602 + 0.834768i \(0.685602\pi\)
\(600\) 0 0
\(601\) 6.42708 0.262166 0.131083 0.991371i \(-0.458155\pi\)
0.131083 + 0.991371i \(0.458155\pi\)
\(602\) 90.6138 3.69314
\(603\) 0 0
\(604\) −6.08178 −0.247464
\(605\) 0 0
\(606\) 0 0
\(607\) −12.9626 −0.526135 −0.263067 0.964777i \(-0.584734\pi\)
−0.263067 + 0.964777i \(0.584734\pi\)
\(608\) 20.8897 0.847190
\(609\) 0 0
\(610\) 0 0
\(611\) 4.04023 0.163450
\(612\) 0 0
\(613\) 24.4771 0.988620 0.494310 0.869286i \(-0.335421\pi\)
0.494310 + 0.869286i \(0.335421\pi\)
\(614\) 14.6820 0.592517
\(615\) 0 0
\(616\) 36.4782 1.46975
\(617\) 11.9453 0.480898 0.240449 0.970662i \(-0.422705\pi\)
0.240449 + 0.970662i \(0.422705\pi\)
\(618\) 0 0
\(619\) −18.9084 −0.759993 −0.379997 0.924988i \(-0.624075\pi\)
−0.379997 + 0.924988i \(0.624075\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −76.2872 −3.05884
\(623\) 19.7822 0.792557
\(624\) 0 0
\(625\) 0 0
\(626\) 61.5195 2.45881
\(627\) 0 0
\(628\) 20.8677 0.832712
\(629\) 13.8073 0.550534
\(630\) 0 0
\(631\) −11.8364 −0.471201 −0.235601 0.971850i \(-0.575706\pi\)
−0.235601 + 0.971850i \(0.575706\pi\)
\(632\) −45.4966 −1.80976
\(633\) 0 0
\(634\) 64.3476 2.55557
\(635\) 0 0
\(636\) 0 0
\(637\) −27.5742 −1.09253
\(638\) −4.52431 −0.179119
\(639\) 0 0
\(640\) 0 0
\(641\) −12.2206 −0.482683 −0.241341 0.970440i \(-0.577587\pi\)
−0.241341 + 0.970440i \(0.577587\pi\)
\(642\) 0 0
\(643\) 8.84257 0.348717 0.174358 0.984682i \(-0.444215\pi\)
0.174358 + 0.984682i \(0.444215\pi\)
\(644\) 186.236 7.33873
\(645\) 0 0
\(646\) −7.51258 −0.295578
\(647\) −9.40985 −0.369939 −0.184970 0.982744i \(-0.559219\pi\)
−0.184970 + 0.982744i \(0.559219\pi\)
\(648\) 0 0
\(649\) 8.24066 0.323474
\(650\) 0 0
\(651\) 0 0
\(652\) −16.1977 −0.634350
\(653\) −22.7449 −0.890075 −0.445038 0.895512i \(-0.646810\pi\)
−0.445038 + 0.895512i \(0.646810\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −120.834 −4.71779
\(657\) 0 0
\(658\) 16.4440 0.641052
\(659\) −20.3635 −0.793249 −0.396625 0.917981i \(-0.629819\pi\)
−0.396625 + 0.917981i \(0.629819\pi\)
\(660\) 0 0
\(661\) 11.6616 0.453585 0.226792 0.973943i \(-0.427176\pi\)
0.226792 + 0.973943i \(0.427176\pi\)
\(662\) 50.9952 1.98198
\(663\) 0 0
\(664\) 81.9748 3.18124
\(665\) 0 0
\(666\) 0 0
\(667\) −14.3384 −0.555184
\(668\) −14.3177 −0.553968
\(669\) 0 0
\(670\) 0 0
\(671\) −3.47569 −0.134178
\(672\) 0 0
\(673\) −30.0070 −1.15669 −0.578343 0.815794i \(-0.696300\pi\)
−0.578343 + 0.815794i \(0.696300\pi\)
\(674\) 66.5151 2.56207
\(675\) 0 0
\(676\) −29.0268 −1.11642
\(677\) −51.4695 −1.97813 −0.989067 0.147466i \(-0.952888\pi\)
−0.989067 + 0.147466i \(0.952888\pi\)
\(678\) 0 0
\(679\) −17.5214 −0.672411
\(680\) 0 0
\(681\) 0 0
\(682\) −27.6803 −1.05993
\(683\) 27.6186 1.05680 0.528399 0.848996i \(-0.322793\pi\)
0.528399 + 0.848996i \(0.322793\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −34.2261 −1.30676
\(687\) 0 0
\(688\) −107.856 −4.11197
\(689\) −6.97376 −0.265679
\(690\) 0 0
\(691\) −34.2344 −1.30234 −0.651169 0.758933i \(-0.725721\pi\)
−0.651169 + 0.758933i \(0.725721\pi\)
\(692\) −76.0209 −2.88988
\(693\) 0 0
\(694\) −41.8179 −1.58738
\(695\) 0 0
\(696\) 0 0
\(697\) 22.0050 0.833499
\(698\) −55.2447 −2.09104
\(699\) 0 0
\(700\) 0 0
\(701\) 11.8524 0.447658 0.223829 0.974628i \(-0.428144\pi\)
0.223829 + 0.974628i \(0.428144\pi\)
\(702\) 0 0
\(703\) 6.59241 0.248637
\(704\) −22.3249 −0.841403
\(705\) 0 0
\(706\) −59.2788 −2.23099
\(707\) −57.6013 −2.16632
\(708\) 0 0
\(709\) −28.1658 −1.05779 −0.528895 0.848687i \(-0.677393\pi\)
−0.528895 + 0.848687i \(0.677393\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −42.2705 −1.58415
\(713\) −87.7241 −3.28529
\(714\) 0 0
\(715\) 0 0
\(716\) 56.8906 2.12610
\(717\) 0 0
\(718\) 53.1200 1.98242
\(719\) 26.2663 0.979567 0.489783 0.871844i \(-0.337076\pi\)
0.489783 + 0.871844i \(0.337076\pi\)
\(720\) 0 0
\(721\) 76.4550 2.84733
\(722\) 47.6553 1.77355
\(723\) 0 0
\(724\) −34.3004 −1.27476
\(725\) 0 0
\(726\) 0 0
\(727\) −21.1401 −0.784042 −0.392021 0.919956i \(-0.628224\pi\)
−0.392021 + 0.919956i \(0.628224\pi\)
\(728\) 99.8717 3.70149
\(729\) 0 0
\(730\) 0 0
\(731\) 19.6415 0.726467
\(732\) 0 0
\(733\) −3.86406 −0.142722 −0.0713611 0.997451i \(-0.522734\pi\)
−0.0713611 + 0.997451i \(0.522734\pi\)
\(734\) −76.2256 −2.81354
\(735\) 0 0
\(736\) −154.821 −5.70679
\(737\) 15.3350 0.564872
\(738\) 0 0
\(739\) −14.0298 −0.516094 −0.258047 0.966132i \(-0.583079\pi\)
−0.258047 + 0.966132i \(0.583079\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −28.3836 −1.04200
\(743\) 26.2934 0.964611 0.482306 0.876003i \(-0.339799\pi\)
0.482306 + 0.876003i \(0.339799\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −26.4011 −0.966613
\(747\) 0 0
\(748\) 12.7378 0.465742
\(749\) −56.8793 −2.07832
\(750\) 0 0
\(751\) 38.2206 1.39469 0.697344 0.716737i \(-0.254365\pi\)
0.697344 + 0.716737i \(0.254365\pi\)
\(752\) −19.5729 −0.713751
\(753\) 0 0
\(754\) −12.3869 −0.451103
\(755\) 0 0
\(756\) 0 0
\(757\) 36.6158 1.33082 0.665411 0.746477i \(-0.268256\pi\)
0.665411 + 0.746477i \(0.268256\pi\)
\(758\) −15.2692 −0.554601
\(759\) 0 0
\(760\) 0 0
\(761\) −51.6803 −1.87341 −0.936705 0.350120i \(-0.886141\pi\)
−0.936705 + 0.350120i \(0.886141\pi\)
\(762\) 0 0
\(763\) 52.8363 1.91280
\(764\) 83.7560 3.03019
\(765\) 0 0
\(766\) −71.4807 −2.58270
\(767\) 22.5617 0.814655
\(768\) 0 0
\(769\) 44.4841 1.60414 0.802068 0.597232i \(-0.203733\pi\)
0.802068 + 0.597232i \(0.203733\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 20.2094 0.727351
\(773\) 0.973762 0.0350238 0.0175119 0.999847i \(-0.494426\pi\)
0.0175119 + 0.999847i \(0.494426\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 37.4396 1.34400
\(777\) 0 0
\(778\) 41.4121 1.48470
\(779\) 10.5065 0.376433
\(780\) 0 0
\(781\) −2.54717 −0.0911449
\(782\) 55.6784 1.99106
\(783\) 0 0
\(784\) 133.583 4.77083
\(785\) 0 0
\(786\) 0 0
\(787\) 30.5382 1.08857 0.544285 0.838900i \(-0.316801\pi\)
0.544285 + 0.838900i \(0.316801\pi\)
\(788\) 14.1276 0.503274
\(789\) 0 0
\(790\) 0 0
\(791\) 61.1943 2.17582
\(792\) 0 0
\(793\) −9.51592 −0.337920
\(794\) 45.2218 1.60486
\(795\) 0 0
\(796\) 35.7962 1.26876
\(797\) 26.7247 0.946639 0.473319 0.880891i \(-0.343056\pi\)
0.473319 + 0.880891i \(0.343056\pi\)
\(798\) 0 0
\(799\) 3.56440 0.126099
\(800\) 0 0
\(801\) 0 0
\(802\) 70.2442 2.48041
\(803\) 2.73785 0.0966165
\(804\) 0 0
\(805\) 0 0
\(806\) −75.7845 −2.66939
\(807\) 0 0
\(808\) 123.082 4.33001
\(809\) 17.0327 0.598837 0.299418 0.954122i \(-0.403207\pi\)
0.299418 + 0.954122i \(0.403207\pi\)
\(810\) 0 0
\(811\) 23.1103 0.811512 0.405756 0.913982i \(-0.367008\pi\)
0.405756 + 0.913982i \(0.367008\pi\)
\(812\) −36.5527 −1.28275
\(813\) 0 0
\(814\) −15.4168 −0.540358
\(815\) 0 0
\(816\) 0 0
\(817\) 9.37798 0.328094
\(818\) −60.5453 −2.11692
\(819\) 0 0
\(820\) 0 0
\(821\) 28.3476 0.989337 0.494668 0.869082i \(-0.335290\pi\)
0.494668 + 0.869082i \(0.335290\pi\)
\(822\) 0 0
\(823\) 10.4585 0.364559 0.182280 0.983247i \(-0.441652\pi\)
0.182280 + 0.983247i \(0.441652\pi\)
\(824\) −163.368 −5.69121
\(825\) 0 0
\(826\) 91.8273 3.19508
\(827\) −45.2672 −1.57409 −0.787047 0.616893i \(-0.788391\pi\)
−0.787047 + 0.616893i \(0.788391\pi\)
\(828\) 0 0
\(829\) −22.9855 −0.798320 −0.399160 0.916881i \(-0.630698\pi\)
−0.399160 + 0.916881i \(0.630698\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −61.1223 −2.11903
\(833\) −24.3267 −0.842870
\(834\) 0 0
\(835\) 0 0
\(836\) 6.08178 0.210343
\(837\) 0 0
\(838\) −75.1971 −2.59764
\(839\) 20.1547 0.695818 0.347909 0.937528i \(-0.386892\pi\)
0.347909 + 0.937528i \(0.386892\pi\)
\(840\) 0 0
\(841\) −26.1858 −0.902959
\(842\) −42.0686 −1.44978
\(843\) 0 0
\(844\) 93.2386 3.20941
\(845\) 0 0
\(846\) 0 0
\(847\) 4.13176 0.141969
\(848\) 33.7845 1.16016
\(849\) 0 0
\(850\) 0 0
\(851\) −48.8587 −1.67485
\(852\) 0 0
\(853\) 38.2705 1.31036 0.655179 0.755474i \(-0.272593\pi\)
0.655179 + 0.755474i \(0.272593\pi\)
\(854\) −38.7303 −1.32533
\(855\) 0 0
\(856\) 121.539 4.15413
\(857\) −9.51250 −0.324941 −0.162470 0.986713i \(-0.551946\pi\)
−0.162470 + 0.986713i \(0.551946\pi\)
\(858\) 0 0
\(859\) −1.61865 −0.0552275 −0.0276137 0.999619i \(-0.508791\pi\)
−0.0276137 + 0.999619i \(0.508791\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −7.95977 −0.271111
\(863\) 46.0229 1.56664 0.783318 0.621621i \(-0.213526\pi\)
0.783318 + 0.621621i \(0.213526\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 65.4994 2.22576
\(867\) 0 0
\(868\) −223.634 −7.59064
\(869\) −5.15325 −0.174812
\(870\) 0 0
\(871\) 41.9849 1.42260
\(872\) −112.900 −3.82328
\(873\) 0 0
\(874\) 26.5841 0.899219
\(875\) 0 0
\(876\) 0 0
\(877\) 28.7630 0.971257 0.485628 0.874165i \(-0.338591\pi\)
0.485628 + 0.874165i \(0.338591\pi\)
\(878\) 51.5396 1.73938
\(879\) 0 0
\(880\) 0 0
\(881\) 23.3149 0.785499 0.392749 0.919646i \(-0.371524\pi\)
0.392749 + 0.919646i \(0.371524\pi\)
\(882\) 0 0
\(883\) −29.6616 −0.998193 −0.499097 0.866546i \(-0.666335\pi\)
−0.499097 + 0.866546i \(0.666335\pi\)
\(884\) 34.8743 1.17295
\(885\) 0 0
\(886\) −5.33225 −0.179141
\(887\) 20.3364 0.682829 0.341414 0.939913i \(-0.389094\pi\)
0.341414 + 0.939913i \(0.389094\pi\)
\(888\) 0 0
\(889\) −53.5583 −1.79629
\(890\) 0 0
\(891\) 0 0
\(892\) 46.5700 1.55928
\(893\) 1.70185 0.0569502
\(894\) 0 0
\(895\) 0 0
\(896\) −99.0879 −3.31029
\(897\) 0 0
\(898\) −40.9181 −1.36545
\(899\) 17.2177 0.574241
\(900\) 0 0
\(901\) −6.15245 −0.204968
\(902\) −24.5700 −0.818093
\(903\) 0 0
\(904\) −130.760 −4.34900
\(905\) 0 0
\(906\) 0 0
\(907\) −9.87942 −0.328041 −0.164020 0.986457i \(-0.552446\pi\)
−0.164020 + 0.986457i \(0.552446\pi\)
\(908\) −55.3578 −1.83711
\(909\) 0 0
\(910\) 0 0
\(911\) −7.22343 −0.239323 −0.119662 0.992815i \(-0.538181\pi\)
−0.119662 + 0.992815i \(0.538181\pi\)
\(912\) 0 0
\(913\) 9.28502 0.307289
\(914\) −16.6418 −0.550460
\(915\) 0 0
\(916\) 26.1116 0.862751
\(917\) −6.09722 −0.201348
\(918\) 0 0
\(919\) 35.7233 1.17840 0.589201 0.807986i \(-0.299443\pi\)
0.589201 + 0.807986i \(0.299443\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 62.5667 2.06052
\(923\) −6.97376 −0.229544
\(924\) 0 0
\(925\) 0 0
\(926\) −3.77060 −0.123910
\(927\) 0 0
\(928\) 30.3869 0.997497
\(929\) 1.12621 0.0369498 0.0184749 0.999829i \(-0.494119\pi\)
0.0184749 + 0.999829i \(0.494119\pi\)
\(930\) 0 0
\(931\) −11.6150 −0.380665
\(932\) 20.5200 0.672157
\(933\) 0 0
\(934\) −61.1943 −2.00234
\(935\) 0 0
\(936\) 0 0
\(937\) 0.127587 0.00416807 0.00208404 0.999998i \(-0.499337\pi\)
0.00208404 + 0.999998i \(0.499337\pi\)
\(938\) 170.881 5.57946
\(939\) 0 0
\(940\) 0 0
\(941\) −1.41403 −0.0460961 −0.0230480 0.999734i \(-0.507337\pi\)
−0.0230480 + 0.999734i \(0.507337\pi\)
\(942\) 0 0
\(943\) −77.8671 −2.53570
\(944\) −109.300 −3.55742
\(945\) 0 0
\(946\) −21.9310 −0.713039
\(947\) −37.1943 −1.20865 −0.604326 0.796737i \(-0.706558\pi\)
−0.604326 + 0.796737i \(0.706558\pi\)
\(948\) 0 0
\(949\) 7.49581 0.243324
\(950\) 0 0
\(951\) 0 0
\(952\) 88.1097 2.85565
\(953\) −16.5041 −0.534621 −0.267310 0.963610i \(-0.586135\pi\)
−0.267310 + 0.963610i \(0.586135\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −82.3663 −2.66391
\(957\) 0 0
\(958\) 66.8592 2.16012
\(959\) 24.7906 0.800530
\(960\) 0 0
\(961\) 74.3400 2.39807
\(962\) −42.2088 −1.36087
\(963\) 0 0
\(964\) −54.1256 −1.74327
\(965\) 0 0
\(966\) 0 0
\(967\) 3.60471 0.115920 0.0579598 0.998319i \(-0.481540\pi\)
0.0579598 + 0.998319i \(0.481540\pi\)
\(968\) −8.82872 −0.283766
\(969\) 0 0
\(970\) 0 0
\(971\) −29.6214 −0.950596 −0.475298 0.879825i \(-0.657660\pi\)
−0.475298 + 0.879825i \(0.657660\pi\)
\(972\) 0 0
\(973\) −3.67613 −0.117851
\(974\) −90.9390 −2.91387
\(975\) 0 0
\(976\) 46.1000 1.47562
\(977\) −52.7157 −1.68653 −0.843263 0.537501i \(-0.819368\pi\)
−0.843263 + 0.537501i \(0.819368\pi\)
\(978\) 0 0
\(979\) −4.78783 −0.153020
\(980\) 0 0
\(981\) 0 0
\(982\) −18.6243 −0.594325
\(983\) 29.2602 0.933254 0.466627 0.884454i \(-0.345469\pi\)
0.466627 + 0.884454i \(0.345469\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −10.9280 −0.348019
\(987\) 0 0
\(988\) 16.6510 0.529738
\(989\) −69.5036 −2.21008
\(990\) 0 0
\(991\) 38.1776 1.21275 0.606375 0.795179i \(-0.292623\pi\)
0.606375 + 0.795179i \(0.292623\pi\)
\(992\) 185.911 5.90268
\(993\) 0 0
\(994\) −28.3836 −0.900274
\(995\) 0 0
\(996\) 0 0
\(997\) 40.1500 1.27156 0.635781 0.771870i \(-0.280678\pi\)
0.635781 + 0.771870i \(0.280678\pi\)
\(998\) 6.09722 0.193004
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2475.2.a.bf.1.1 4
3.2 odd 2 2475.2.a.bj.1.4 4
5.2 odd 4 2475.2.c.s.199.1 8
5.3 odd 4 2475.2.c.s.199.8 8
5.4 even 2 495.2.a.g.1.4 yes 4
15.2 even 4 2475.2.c.t.199.8 8
15.8 even 4 2475.2.c.t.199.1 8
15.14 odd 2 495.2.a.f.1.1 4
20.19 odd 2 7920.2.a.cn.1.4 4
55.54 odd 2 5445.2.a.bh.1.1 4
60.59 even 2 7920.2.a.cm.1.4 4
165.164 even 2 5445.2.a.bs.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
495.2.a.f.1.1 4 15.14 odd 2
495.2.a.g.1.4 yes 4 5.4 even 2
2475.2.a.bf.1.1 4 1.1 even 1 trivial
2475.2.a.bj.1.4 4 3.2 odd 2
2475.2.c.s.199.1 8 5.2 odd 4
2475.2.c.s.199.8 8 5.3 odd 4
2475.2.c.t.199.1 8 15.8 even 4
2475.2.c.t.199.8 8 15.2 even 4
5445.2.a.bh.1.1 4 55.54 odd 2
5445.2.a.bs.1.4 4 165.164 even 2
7920.2.a.cm.1.4 4 60.59 even 2
7920.2.a.cn.1.4 4 20.19 odd 2