Properties

Label 4096.2.a.s.1.1
Level $4096$
Weight $2$
Character 4096.1
Self dual yes
Analytic conductor $32.707$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4096,2,Mod(1,4096)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4096, 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("4096.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4096 = 2^{12} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4096.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: \(32.7067246679\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{48})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 20x^{4} - 16x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1024)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.261052\) of defining polynomial
Character \(\chi\) \(=\) 4096.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.931852 q^{3} -0.956470 q^{5} -3.32633 q^{7} -2.13165 q^{9} +O(q^{10})\) \(q-0.931852 q^{3} -0.956470 q^{5} -3.32633 q^{7} -2.13165 q^{9} -1.96713 q^{11} -2.21694 q^{13} +0.891289 q^{15} -6.00997 q^{17} -4.04284 q^{19} +3.09965 q^{21} +4.11870 q^{23} -4.08516 q^{25} +4.78194 q^{27} -10.4925 q^{29} -7.52140 q^{31} +1.83307 q^{33} +3.18154 q^{35} +2.59915 q^{37} +2.06586 q^{39} +1.95011 q^{41} +11.7603 q^{43} +2.03886 q^{45} -3.33173 q^{47} +4.06450 q^{49} +5.60040 q^{51} -8.69153 q^{53} +1.88150 q^{55} +3.76733 q^{57} +12.3061 q^{59} +0.179920 q^{61} +7.09059 q^{63} +2.12044 q^{65} -9.08808 q^{67} -3.83802 q^{69} -11.2039 q^{71} -8.26670 q^{73} +3.80677 q^{75} +6.54332 q^{77} -1.80100 q^{79} +1.93890 q^{81} +5.18519 q^{83} +5.74836 q^{85} +9.77748 q^{87} -9.02458 q^{89} +7.37429 q^{91} +7.00883 q^{93} +3.86686 q^{95} -0.874915 q^{97} +4.19323 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 8 q^{11} + 8 q^{17} - 8 q^{25} + 8 q^{27} + 40 q^{33} - 8 q^{35} + 32 q^{41} + 56 q^{43} + 8 q^{49} + 48 q^{51} + 8 q^{57} + 32 q^{59} - 16 q^{65} + 24 q^{67} - 8 q^{73} + 16 q^{81} + 48 q^{83} - 8 q^{89} + 8 q^{91} + 8 q^{97} + 64 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.931852 −0.538005 −0.269002 0.963140i \(-0.586694\pi\)
−0.269002 + 0.963140i \(0.586694\pi\)
\(4\) 0 0
\(5\) −0.956470 −0.427747 −0.213873 0.976861i \(-0.568608\pi\)
−0.213873 + 0.976861i \(0.568608\pi\)
\(6\) 0 0
\(7\) −3.32633 −1.25724 −0.628618 0.777714i \(-0.716379\pi\)
−0.628618 + 0.777714i \(0.716379\pi\)
\(8\) 0 0
\(9\) −2.13165 −0.710551
\(10\) 0 0
\(11\) −1.96713 −0.593111 −0.296556 0.955016i \(-0.595838\pi\)
−0.296556 + 0.955016i \(0.595838\pi\)
\(12\) 0 0
\(13\) −2.21694 −0.614869 −0.307435 0.951569i \(-0.599471\pi\)
−0.307435 + 0.951569i \(0.599471\pi\)
\(14\) 0 0
\(15\) 0.891289 0.230130
\(16\) 0 0
\(17\) −6.00997 −1.45763 −0.728816 0.684710i \(-0.759929\pi\)
−0.728816 + 0.684710i \(0.759929\pi\)
\(18\) 0 0
\(19\) −4.04284 −0.927491 −0.463746 0.885968i \(-0.653495\pi\)
−0.463746 + 0.885968i \(0.653495\pi\)
\(20\) 0 0
\(21\) 3.09965 0.676399
\(22\) 0 0
\(23\) 4.11870 0.858808 0.429404 0.903112i \(-0.358724\pi\)
0.429404 + 0.903112i \(0.358724\pi\)
\(24\) 0 0
\(25\) −4.08516 −0.817033
\(26\) 0 0
\(27\) 4.78194 0.920285
\(28\) 0 0
\(29\) −10.4925 −1.94841 −0.974207 0.225658i \(-0.927547\pi\)
−0.974207 + 0.225658i \(0.927547\pi\)
\(30\) 0 0
\(31\) −7.52140 −1.35088 −0.675442 0.737413i \(-0.736047\pi\)
−0.675442 + 0.737413i \(0.736047\pi\)
\(32\) 0 0
\(33\) 1.83307 0.319097
\(34\) 0 0
\(35\) 3.18154 0.537779
\(36\) 0 0
\(37\) 2.59915 0.427298 0.213649 0.976911i \(-0.431465\pi\)
0.213649 + 0.976911i \(0.431465\pi\)
\(38\) 0 0
\(39\) 2.06586 0.330803
\(40\) 0 0
\(41\) 1.95011 0.304556 0.152278 0.988338i \(-0.451339\pi\)
0.152278 + 0.988338i \(0.451339\pi\)
\(42\) 0 0
\(43\) 11.7603 1.79343 0.896713 0.442613i \(-0.145948\pi\)
0.896713 + 0.442613i \(0.145948\pi\)
\(44\) 0 0
\(45\) 2.03886 0.303936
\(46\) 0 0
\(47\) −3.33173 −0.485983 −0.242991 0.970028i \(-0.578129\pi\)
−0.242991 + 0.970028i \(0.578129\pi\)
\(48\) 0 0
\(49\) 4.06450 0.580643
\(50\) 0 0
\(51\) 5.60040 0.784213
\(52\) 0 0
\(53\) −8.69153 −1.19387 −0.596937 0.802288i \(-0.703616\pi\)
−0.596937 + 0.802288i \(0.703616\pi\)
\(54\) 0 0
\(55\) 1.88150 0.253701
\(56\) 0 0
\(57\) 3.76733 0.498995
\(58\) 0 0
\(59\) 12.3061 1.60212 0.801062 0.598582i \(-0.204269\pi\)
0.801062 + 0.598582i \(0.204269\pi\)
\(60\) 0 0
\(61\) 0.179920 0.0230364 0.0115182 0.999934i \(-0.496334\pi\)
0.0115182 + 0.999934i \(0.496334\pi\)
\(62\) 0 0
\(63\) 7.09059 0.893330
\(64\) 0 0
\(65\) 2.12044 0.263008
\(66\) 0 0
\(67\) −9.08808 −1.11029 −0.555143 0.831755i \(-0.687336\pi\)
−0.555143 + 0.831755i \(0.687336\pi\)
\(68\) 0 0
\(69\) −3.83802 −0.462043
\(70\) 0 0
\(71\) −11.2039 −1.32966 −0.664829 0.746996i \(-0.731496\pi\)
−0.664829 + 0.746996i \(0.731496\pi\)
\(72\) 0 0
\(73\) −8.26670 −0.967545 −0.483772 0.875194i \(-0.660734\pi\)
−0.483772 + 0.875194i \(0.660734\pi\)
\(74\) 0 0
\(75\) 3.80677 0.439568
\(76\) 0 0
\(77\) 6.54332 0.745681
\(78\) 0 0
\(79\) −1.80100 −0.202628 −0.101314 0.994855i \(-0.532305\pi\)
−0.101314 + 0.994855i \(0.532305\pi\)
\(80\) 0 0
\(81\) 1.93890 0.215433
\(82\) 0 0
\(83\) 5.18519 0.569148 0.284574 0.958654i \(-0.408148\pi\)
0.284574 + 0.958654i \(0.408148\pi\)
\(84\) 0 0
\(85\) 5.74836 0.623497
\(86\) 0 0
\(87\) 9.77748 1.04826
\(88\) 0 0
\(89\) −9.02458 −0.956604 −0.478302 0.878196i \(-0.658747\pi\)
−0.478302 + 0.878196i \(0.658747\pi\)
\(90\) 0 0
\(91\) 7.37429 0.773036
\(92\) 0 0
\(93\) 7.00883 0.726782
\(94\) 0 0
\(95\) 3.86686 0.396731
\(96\) 0 0
\(97\) −0.874915 −0.0888342 −0.0444171 0.999013i \(-0.514143\pi\)
−0.0444171 + 0.999013i \(0.514143\pi\)
\(98\) 0 0
\(99\) 4.19323 0.421436
\(100\) 0 0
\(101\) 0.00183995 0.000183082 0 9.15410e−5 1.00000i \(-0.499971\pi\)
9.15410e−5 1.00000i \(0.499971\pi\)
\(102\) 0 0
\(103\) −12.6303 −1.24450 −0.622251 0.782818i \(-0.713782\pi\)
−0.622251 + 0.782818i \(0.713782\pi\)
\(104\) 0 0
\(105\) −2.96472 −0.289327
\(106\) 0 0
\(107\) 3.33145 0.322064 0.161032 0.986949i \(-0.448518\pi\)
0.161032 + 0.986949i \(0.448518\pi\)
\(108\) 0 0
\(109\) 8.34751 0.799546 0.399773 0.916614i \(-0.369089\pi\)
0.399773 + 0.916614i \(0.369089\pi\)
\(110\) 0 0
\(111\) −2.42202 −0.229888
\(112\) 0 0
\(113\) 7.03528 0.661823 0.330912 0.943662i \(-0.392644\pi\)
0.330912 + 0.943662i \(0.392644\pi\)
\(114\) 0 0
\(115\) −3.93942 −0.367352
\(116\) 0 0
\(117\) 4.72575 0.436896
\(118\) 0 0
\(119\) 19.9912 1.83259
\(120\) 0 0
\(121\) −7.13041 −0.648219
\(122\) 0 0
\(123\) −1.81722 −0.163853
\(124\) 0 0
\(125\) 8.68969 0.777230
\(126\) 0 0
\(127\) 1.09821 0.0974502 0.0487251 0.998812i \(-0.484484\pi\)
0.0487251 + 0.998812i \(0.484484\pi\)
\(128\) 0 0
\(129\) −10.9588 −0.964872
\(130\) 0 0
\(131\) 18.4665 1.61343 0.806713 0.590943i \(-0.201244\pi\)
0.806713 + 0.590943i \(0.201244\pi\)
\(132\) 0 0
\(133\) 13.4478 1.16608
\(134\) 0 0
\(135\) −4.57378 −0.393649
\(136\) 0 0
\(137\) 8.24745 0.704627 0.352314 0.935882i \(-0.385395\pi\)
0.352314 + 0.935882i \(0.385395\pi\)
\(138\) 0 0
\(139\) 13.4312 1.13922 0.569611 0.821914i \(-0.307094\pi\)
0.569611 + 0.821914i \(0.307094\pi\)
\(140\) 0 0
\(141\) 3.10468 0.261461
\(142\) 0 0
\(143\) 4.36101 0.364686
\(144\) 0 0
\(145\) 10.0358 0.833427
\(146\) 0 0
\(147\) −3.78751 −0.312389
\(148\) 0 0
\(149\) −17.9879 −1.47362 −0.736812 0.676097i \(-0.763670\pi\)
−0.736812 + 0.676097i \(0.763670\pi\)
\(150\) 0 0
\(151\) 12.0898 0.983853 0.491927 0.870637i \(-0.336293\pi\)
0.491927 + 0.870637i \(0.336293\pi\)
\(152\) 0 0
\(153\) 12.8112 1.03572
\(154\) 0 0
\(155\) 7.19400 0.577836
\(156\) 0 0
\(157\) 5.91874 0.472367 0.236184 0.971708i \(-0.424103\pi\)
0.236184 + 0.971708i \(0.424103\pi\)
\(158\) 0 0
\(159\) 8.09922 0.642310
\(160\) 0 0
\(161\) −13.7002 −1.07973
\(162\) 0 0
\(163\) −7.75298 −0.607260 −0.303630 0.952790i \(-0.598199\pi\)
−0.303630 + 0.952790i \(0.598199\pi\)
\(164\) 0 0
\(165\) −1.75328 −0.136493
\(166\) 0 0
\(167\) −20.6401 −1.59718 −0.798591 0.601874i \(-0.794421\pi\)
−0.798591 + 0.601874i \(0.794421\pi\)
\(168\) 0 0
\(169\) −8.08516 −0.621936
\(170\) 0 0
\(171\) 8.61793 0.659030
\(172\) 0 0
\(173\) −7.18056 −0.545928 −0.272964 0.962024i \(-0.588004\pi\)
−0.272964 + 0.962024i \(0.588004\pi\)
\(174\) 0 0
\(175\) 13.5886 1.02720
\(176\) 0 0
\(177\) −11.4675 −0.861950
\(178\) 0 0
\(179\) −0.871267 −0.0651216 −0.0325608 0.999470i \(-0.510366\pi\)
−0.0325608 + 0.999470i \(0.510366\pi\)
\(180\) 0 0
\(181\) 17.2495 1.28215 0.641073 0.767480i \(-0.278490\pi\)
0.641073 + 0.767480i \(0.278490\pi\)
\(182\) 0 0
\(183\) −0.167659 −0.0123937
\(184\) 0 0
\(185\) −2.48601 −0.182775
\(186\) 0 0
\(187\) 11.8224 0.864538
\(188\) 0 0
\(189\) −15.9063 −1.15702
\(190\) 0 0
\(191\) −21.7629 −1.57471 −0.787356 0.616499i \(-0.788551\pi\)
−0.787356 + 0.616499i \(0.788551\pi\)
\(192\) 0 0
\(193\) 0.640834 0.0461283 0.0230641 0.999734i \(-0.492658\pi\)
0.0230641 + 0.999734i \(0.492658\pi\)
\(194\) 0 0
\(195\) −1.97594 −0.141500
\(196\) 0 0
\(197\) −20.6906 −1.47414 −0.737072 0.675815i \(-0.763792\pi\)
−0.737072 + 0.675815i \(0.763792\pi\)
\(198\) 0 0
\(199\) −11.2579 −0.798051 −0.399025 0.916940i \(-0.630651\pi\)
−0.399025 + 0.916940i \(0.630651\pi\)
\(200\) 0 0
\(201\) 8.46875 0.597339
\(202\) 0 0
\(203\) 34.9017 2.44962
\(204\) 0 0
\(205\) −1.86522 −0.130273
\(206\) 0 0
\(207\) −8.77964 −0.610227
\(208\) 0 0
\(209\) 7.95278 0.550105
\(210\) 0 0
\(211\) 5.94986 0.409605 0.204803 0.978803i \(-0.434345\pi\)
0.204803 + 0.978803i \(0.434345\pi\)
\(212\) 0 0
\(213\) 10.4404 0.715362
\(214\) 0 0
\(215\) −11.2484 −0.767132
\(216\) 0 0
\(217\) 25.0187 1.69838
\(218\) 0 0
\(219\) 7.70334 0.520544
\(220\) 0 0
\(221\) 13.3238 0.896253
\(222\) 0 0
\(223\) 15.3054 1.02493 0.512464 0.858709i \(-0.328733\pi\)
0.512464 + 0.858709i \(0.328733\pi\)
\(224\) 0 0
\(225\) 8.70815 0.580543
\(226\) 0 0
\(227\) −17.4512 −1.15827 −0.579137 0.815230i \(-0.696610\pi\)
−0.579137 + 0.815230i \(0.696610\pi\)
\(228\) 0 0
\(229\) −15.2781 −1.00960 −0.504801 0.863235i \(-0.668434\pi\)
−0.504801 + 0.863235i \(0.668434\pi\)
\(230\) 0 0
\(231\) −6.09741 −0.401180
\(232\) 0 0
\(233\) 11.6189 0.761181 0.380591 0.924744i \(-0.375721\pi\)
0.380591 + 0.924744i \(0.375721\pi\)
\(234\) 0 0
\(235\) 3.18670 0.207877
\(236\) 0 0
\(237\) 1.67826 0.109015
\(238\) 0 0
\(239\) 4.21394 0.272577 0.136289 0.990669i \(-0.456483\pi\)
0.136289 + 0.990669i \(0.456483\pi\)
\(240\) 0 0
\(241\) −20.9382 −1.34875 −0.674373 0.738391i \(-0.735586\pi\)
−0.674373 + 0.738391i \(0.735586\pi\)
\(242\) 0 0
\(243\) −16.1526 −1.03619
\(244\) 0 0
\(245\) −3.88757 −0.248368
\(246\) 0 0
\(247\) 8.96275 0.570286
\(248\) 0 0
\(249\) −4.83183 −0.306205
\(250\) 0 0
\(251\) 18.2424 1.15145 0.575725 0.817644i \(-0.304720\pi\)
0.575725 + 0.817644i \(0.304720\pi\)
\(252\) 0 0
\(253\) −8.10201 −0.509369
\(254\) 0 0
\(255\) −5.35662 −0.335444
\(256\) 0 0
\(257\) 15.7839 0.984570 0.492285 0.870434i \(-0.336162\pi\)
0.492285 + 0.870434i \(0.336162\pi\)
\(258\) 0 0
\(259\) −8.64564 −0.537214
\(260\) 0 0
\(261\) 22.3664 1.38445
\(262\) 0 0
\(263\) 22.2230 1.37033 0.685163 0.728389i \(-0.259731\pi\)
0.685163 + 0.728389i \(0.259731\pi\)
\(264\) 0 0
\(265\) 8.31319 0.510675
\(266\) 0 0
\(267\) 8.40957 0.514657
\(268\) 0 0
\(269\) −9.47367 −0.577620 −0.288810 0.957386i \(-0.593260\pi\)
−0.288810 + 0.957386i \(0.593260\pi\)
\(270\) 0 0
\(271\) 6.69345 0.406598 0.203299 0.979117i \(-0.434834\pi\)
0.203299 + 0.979117i \(0.434834\pi\)
\(272\) 0 0
\(273\) −6.87175 −0.415897
\(274\) 0 0
\(275\) 8.03604 0.484591
\(276\) 0 0
\(277\) −25.2893 −1.51949 −0.759744 0.650222i \(-0.774676\pi\)
−0.759744 + 0.650222i \(0.774676\pi\)
\(278\) 0 0
\(279\) 16.0330 0.959871
\(280\) 0 0
\(281\) −19.3123 −1.15208 −0.576038 0.817423i \(-0.695402\pi\)
−0.576038 + 0.817423i \(0.695402\pi\)
\(282\) 0 0
\(283\) 13.7995 0.820293 0.410147 0.912020i \(-0.365478\pi\)
0.410147 + 0.912020i \(0.365478\pi\)
\(284\) 0 0
\(285\) −3.60334 −0.213443
\(286\) 0 0
\(287\) −6.48672 −0.382899
\(288\) 0 0
\(289\) 19.1197 1.12469
\(290\) 0 0
\(291\) 0.815291 0.0477932
\(292\) 0 0
\(293\) 16.0812 0.939475 0.469737 0.882806i \(-0.344349\pi\)
0.469737 + 0.882806i \(0.344349\pi\)
\(294\) 0 0
\(295\) −11.7705 −0.685303
\(296\) 0 0
\(297\) −9.40669 −0.545831
\(298\) 0 0
\(299\) −9.13092 −0.528055
\(300\) 0 0
\(301\) −39.1186 −2.25476
\(302\) 0 0
\(303\) −0.00171456 −9.84990e−5 0
\(304\) 0 0
\(305\) −0.172088 −0.00985375
\(306\) 0 0
\(307\) −6.93701 −0.395916 −0.197958 0.980210i \(-0.563431\pi\)
−0.197958 + 0.980210i \(0.563431\pi\)
\(308\) 0 0
\(309\) 11.7696 0.669548
\(310\) 0 0
\(311\) 30.7625 1.74438 0.872192 0.489165i \(-0.162698\pi\)
0.872192 + 0.489165i \(0.162698\pi\)
\(312\) 0 0
\(313\) 3.94082 0.222749 0.111374 0.993779i \(-0.464475\pi\)
0.111374 + 0.993779i \(0.464475\pi\)
\(314\) 0 0
\(315\) −6.78194 −0.382119
\(316\) 0 0
\(317\) 4.23285 0.237741 0.118870 0.992910i \(-0.462073\pi\)
0.118870 + 0.992910i \(0.462073\pi\)
\(318\) 0 0
\(319\) 20.6401 1.15563
\(320\) 0 0
\(321\) −3.10442 −0.173272
\(322\) 0 0
\(323\) 24.2973 1.35194
\(324\) 0 0
\(325\) 9.05658 0.502368
\(326\) 0 0
\(327\) −7.77864 −0.430160
\(328\) 0 0
\(329\) 11.0824 0.610995
\(330\) 0 0
\(331\) −16.1959 −0.890207 −0.445104 0.895479i \(-0.646833\pi\)
−0.445104 + 0.895479i \(0.646833\pi\)
\(332\) 0 0
\(333\) −5.54048 −0.303617
\(334\) 0 0
\(335\) 8.69248 0.474921
\(336\) 0 0
\(337\) −2.16071 −0.117702 −0.0588508 0.998267i \(-0.518744\pi\)
−0.0588508 + 0.998267i \(0.518744\pi\)
\(338\) 0 0
\(339\) −6.55583 −0.356064
\(340\) 0 0
\(341\) 14.7956 0.801224
\(342\) 0 0
\(343\) 9.76445 0.527231
\(344\) 0 0
\(345\) 3.67095 0.197637
\(346\) 0 0
\(347\) −8.79143 −0.471948 −0.235974 0.971759i \(-0.575828\pi\)
−0.235974 + 0.971759i \(0.575828\pi\)
\(348\) 0 0
\(349\) 16.0209 0.857581 0.428791 0.903404i \(-0.358940\pi\)
0.428791 + 0.903404i \(0.358940\pi\)
\(350\) 0 0
\(351\) −10.6013 −0.565855
\(352\) 0 0
\(353\) 2.59235 0.137977 0.0689885 0.997617i \(-0.478023\pi\)
0.0689885 + 0.997617i \(0.478023\pi\)
\(354\) 0 0
\(355\) 10.7162 0.568757
\(356\) 0 0
\(357\) −18.6288 −0.985940
\(358\) 0 0
\(359\) −6.99581 −0.369225 −0.184612 0.982811i \(-0.559103\pi\)
−0.184612 + 0.982811i \(0.559103\pi\)
\(360\) 0 0
\(361\) −2.65545 −0.139760
\(362\) 0 0
\(363\) 6.64448 0.348745
\(364\) 0 0
\(365\) 7.90686 0.413864
\(366\) 0 0
\(367\) −19.5663 −1.02135 −0.510675 0.859774i \(-0.670605\pi\)
−0.510675 + 0.859774i \(0.670605\pi\)
\(368\) 0 0
\(369\) −4.15696 −0.216403
\(370\) 0 0
\(371\) 28.9109 1.50098
\(372\) 0 0
\(373\) −22.0445 −1.14142 −0.570710 0.821151i \(-0.693332\pi\)
−0.570710 + 0.821151i \(0.693332\pi\)
\(374\) 0 0
\(375\) −8.09750 −0.418153
\(376\) 0 0
\(377\) 23.2613 1.19802
\(378\) 0 0
\(379\) 4.20637 0.216067 0.108033 0.994147i \(-0.465545\pi\)
0.108033 + 0.994147i \(0.465545\pi\)
\(380\) 0 0
\(381\) −1.02337 −0.0524287
\(382\) 0 0
\(383\) −20.6568 −1.05552 −0.527758 0.849395i \(-0.676967\pi\)
−0.527758 + 0.849395i \(0.676967\pi\)
\(384\) 0 0
\(385\) −6.25850 −0.318963
\(386\) 0 0
\(387\) −25.0688 −1.27432
\(388\) 0 0
\(389\) −17.2094 −0.872552 −0.436276 0.899813i \(-0.643703\pi\)
−0.436276 + 0.899813i \(0.643703\pi\)
\(390\) 0 0
\(391\) −24.7533 −1.25183
\(392\) 0 0
\(393\) −17.2080 −0.868031
\(394\) 0 0
\(395\) 1.72260 0.0866733
\(396\) 0 0
\(397\) −19.2902 −0.968148 −0.484074 0.875027i \(-0.660843\pi\)
−0.484074 + 0.875027i \(0.660843\pi\)
\(398\) 0 0
\(399\) −12.5314 −0.627354
\(400\) 0 0
\(401\) 0.119032 0.00594418 0.00297209 0.999996i \(-0.499054\pi\)
0.00297209 + 0.999996i \(0.499054\pi\)
\(402\) 0 0
\(403\) 16.6745 0.830617
\(404\) 0 0
\(405\) −1.85450 −0.0921509
\(406\) 0 0
\(407\) −5.11286 −0.253435
\(408\) 0 0
\(409\) 14.9207 0.737780 0.368890 0.929473i \(-0.379738\pi\)
0.368890 + 0.929473i \(0.379738\pi\)
\(410\) 0 0
\(411\) −7.68540 −0.379093
\(412\) 0 0
\(413\) −40.9344 −2.01425
\(414\) 0 0
\(415\) −4.95948 −0.243451
\(416\) 0 0
\(417\) −12.5159 −0.612907
\(418\) 0 0
\(419\) 9.13973 0.446505 0.223253 0.974761i \(-0.428333\pi\)
0.223253 + 0.974761i \(0.428333\pi\)
\(420\) 0 0
\(421\) 27.3008 1.33056 0.665281 0.746593i \(-0.268312\pi\)
0.665281 + 0.746593i \(0.268312\pi\)
\(422\) 0 0
\(423\) 7.10209 0.345315
\(424\) 0 0
\(425\) 24.5517 1.19093
\(426\) 0 0
\(427\) −0.598475 −0.0289622
\(428\) 0 0
\(429\) −4.06381 −0.196203
\(430\) 0 0
\(431\) 3.31726 0.159787 0.0798934 0.996803i \(-0.474542\pi\)
0.0798934 + 0.996803i \(0.474542\pi\)
\(432\) 0 0
\(433\) 22.3224 1.07275 0.536374 0.843981i \(-0.319794\pi\)
0.536374 + 0.843981i \(0.319794\pi\)
\(434\) 0 0
\(435\) −9.35187 −0.448388
\(436\) 0 0
\(437\) −16.6512 −0.796537
\(438\) 0 0
\(439\) −29.8766 −1.42593 −0.712967 0.701198i \(-0.752649\pi\)
−0.712967 + 0.701198i \(0.752649\pi\)
\(440\) 0 0
\(441\) −8.66410 −0.412576
\(442\) 0 0
\(443\) 0.239370 0.0113728 0.00568640 0.999984i \(-0.498190\pi\)
0.00568640 + 0.999984i \(0.498190\pi\)
\(444\) 0 0
\(445\) 8.63174 0.409184
\(446\) 0 0
\(447\) 16.7620 0.792817
\(448\) 0 0
\(449\) 21.5081 1.01503 0.507515 0.861643i \(-0.330564\pi\)
0.507515 + 0.861643i \(0.330564\pi\)
\(450\) 0 0
\(451\) −3.83612 −0.180636
\(452\) 0 0
\(453\) −11.2659 −0.529318
\(454\) 0 0
\(455\) −7.05329 −0.330664
\(456\) 0 0
\(457\) −13.7705 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(458\) 0 0
\(459\) −28.7393 −1.34144
\(460\) 0 0
\(461\) 29.5412 1.37587 0.687935 0.725772i \(-0.258517\pi\)
0.687935 + 0.725772i \(0.258517\pi\)
\(462\) 0 0
\(463\) −39.6338 −1.84194 −0.920969 0.389635i \(-0.872601\pi\)
−0.920969 + 0.389635i \(0.872601\pi\)
\(464\) 0 0
\(465\) −6.70374 −0.310878
\(466\) 0 0
\(467\) −21.3341 −0.987226 −0.493613 0.869682i \(-0.664324\pi\)
−0.493613 + 0.869682i \(0.664324\pi\)
\(468\) 0 0
\(469\) 30.2300 1.39589
\(470\) 0 0
\(471\) −5.51539 −0.254136
\(472\) 0 0
\(473\) −23.1340 −1.06370
\(474\) 0 0
\(475\) 16.5157 0.757791
\(476\) 0 0
\(477\) 18.5273 0.848308
\(478\) 0 0
\(479\) 3.07863 0.140666 0.0703331 0.997524i \(-0.477594\pi\)
0.0703331 + 0.997524i \(0.477594\pi\)
\(480\) 0 0
\(481\) −5.76217 −0.262732
\(482\) 0 0
\(483\) 12.7665 0.580897
\(484\) 0 0
\(485\) 0.836831 0.0379985
\(486\) 0 0
\(487\) 29.7895 1.34989 0.674945 0.737868i \(-0.264167\pi\)
0.674945 + 0.737868i \(0.264167\pi\)
\(488\) 0 0
\(489\) 7.22463 0.326709
\(490\) 0 0
\(491\) 17.4068 0.785559 0.392779 0.919633i \(-0.371514\pi\)
0.392779 + 0.919633i \(0.371514\pi\)
\(492\) 0 0
\(493\) 63.0597 2.84007
\(494\) 0 0
\(495\) −4.01070 −0.180268
\(496\) 0 0
\(497\) 37.2679 1.67169
\(498\) 0 0
\(499\) 20.1192 0.900660 0.450330 0.892862i \(-0.351306\pi\)
0.450330 + 0.892862i \(0.351306\pi\)
\(500\) 0 0
\(501\) 19.2335 0.859292
\(502\) 0 0
\(503\) −15.6866 −0.699430 −0.349715 0.936856i \(-0.613722\pi\)
−0.349715 + 0.936856i \(0.613722\pi\)
\(504\) 0 0
\(505\) −0.00175986 −7.83127e−5 0
\(506\) 0 0
\(507\) 7.53417 0.334604
\(508\) 0 0
\(509\) 36.3256 1.61010 0.805051 0.593205i \(-0.202138\pi\)
0.805051 + 0.593205i \(0.202138\pi\)
\(510\) 0 0
\(511\) 27.4978 1.21643
\(512\) 0 0
\(513\) −19.3326 −0.853556
\(514\) 0 0
\(515\) 12.0805 0.532331
\(516\) 0 0
\(517\) 6.55394 0.288242
\(518\) 0 0
\(519\) 6.69122 0.293712
\(520\) 0 0
\(521\) 7.45856 0.326766 0.163383 0.986563i \(-0.447759\pi\)
0.163383 + 0.986563i \(0.447759\pi\)
\(522\) 0 0
\(523\) −6.14674 −0.268778 −0.134389 0.990929i \(-0.542907\pi\)
−0.134389 + 0.990929i \(0.542907\pi\)
\(524\) 0 0
\(525\) −12.6626 −0.552640
\(526\) 0 0
\(527\) 45.2034 1.96909
\(528\) 0 0
\(529\) −6.03631 −0.262448
\(530\) 0 0
\(531\) −26.2324 −1.13839
\(532\) 0 0
\(533\) −4.32329 −0.187262
\(534\) 0 0
\(535\) −3.18644 −0.137762
\(536\) 0 0
\(537\) 0.811892 0.0350357
\(538\) 0 0
\(539\) −7.99539 −0.344386
\(540\) 0 0
\(541\) −21.1218 −0.908096 −0.454048 0.890977i \(-0.650020\pi\)
−0.454048 + 0.890977i \(0.650020\pi\)
\(542\) 0 0
\(543\) −16.0740 −0.689801
\(544\) 0 0
\(545\) −7.98414 −0.342003
\(546\) 0 0
\(547\) 3.90338 0.166896 0.0834481 0.996512i \(-0.473407\pi\)
0.0834481 + 0.996512i \(0.473407\pi\)
\(548\) 0 0
\(549\) −0.383527 −0.0163685
\(550\) 0 0
\(551\) 42.4196 1.80714
\(552\) 0 0
\(553\) 5.99071 0.254751
\(554\) 0 0
\(555\) 2.31659 0.0983339
\(556\) 0 0
\(557\) −19.1224 −0.810241 −0.405120 0.914263i \(-0.632770\pi\)
−0.405120 + 0.914263i \(0.632770\pi\)
\(558\) 0 0
\(559\) −26.0719 −1.10272
\(560\) 0 0
\(561\) −11.0167 −0.465125
\(562\) 0 0
\(563\) 2.94285 0.124026 0.0620132 0.998075i \(-0.480248\pi\)
0.0620132 + 0.998075i \(0.480248\pi\)
\(564\) 0 0
\(565\) −6.72903 −0.283093
\(566\) 0 0
\(567\) −6.44943 −0.270851
\(568\) 0 0
\(569\) 14.7138 0.616836 0.308418 0.951251i \(-0.400201\pi\)
0.308418 + 0.951251i \(0.400201\pi\)
\(570\) 0 0
\(571\) −40.9101 −1.71203 −0.856017 0.516947i \(-0.827068\pi\)
−0.856017 + 0.516947i \(0.827068\pi\)
\(572\) 0 0
\(573\) 20.2798 0.847203
\(574\) 0 0
\(575\) −16.8256 −0.701675
\(576\) 0 0
\(577\) −30.1981 −1.25716 −0.628582 0.777744i \(-0.716364\pi\)
−0.628582 + 0.777744i \(0.716364\pi\)
\(578\) 0 0
\(579\) −0.597162 −0.0248172
\(580\) 0 0
\(581\) −17.2477 −0.715554
\(582\) 0 0
\(583\) 17.0974 0.708100
\(584\) 0 0
\(585\) −4.52004 −0.186881
\(586\) 0 0
\(587\) −21.7558 −0.897958 −0.448979 0.893542i \(-0.648212\pi\)
−0.448979 + 0.893542i \(0.648212\pi\)
\(588\) 0 0
\(589\) 30.4078 1.25293
\(590\) 0 0
\(591\) 19.2806 0.793096
\(592\) 0 0
\(593\) −38.2715 −1.57162 −0.785810 0.618468i \(-0.787754\pi\)
−0.785810 + 0.618468i \(0.787754\pi\)
\(594\) 0 0
\(595\) −19.1210 −0.783883
\(596\) 0 0
\(597\) 10.4907 0.429355
\(598\) 0 0
\(599\) −3.90639 −0.159611 −0.0798053 0.996810i \(-0.525430\pi\)
−0.0798053 + 0.996810i \(0.525430\pi\)
\(600\) 0 0
\(601\) −29.4101 −1.19966 −0.599831 0.800127i \(-0.704766\pi\)
−0.599831 + 0.800127i \(0.704766\pi\)
\(602\) 0 0
\(603\) 19.3726 0.788915
\(604\) 0 0
\(605\) 6.82002 0.277273
\(606\) 0 0
\(607\) −18.7402 −0.760642 −0.380321 0.924855i \(-0.624187\pi\)
−0.380321 + 0.924855i \(0.624187\pi\)
\(608\) 0 0
\(609\) −32.5232 −1.31790
\(610\) 0 0
\(611\) 7.38625 0.298816
\(612\) 0 0
\(613\) −16.1882 −0.653837 −0.326918 0.945053i \(-0.606010\pi\)
−0.326918 + 0.945053i \(0.606010\pi\)
\(614\) 0 0
\(615\) 1.73811 0.0700875
\(616\) 0 0
\(617\) −36.4721 −1.46831 −0.734156 0.678981i \(-0.762422\pi\)
−0.734156 + 0.678981i \(0.762422\pi\)
\(618\) 0 0
\(619\) 5.78117 0.232365 0.116183 0.993228i \(-0.462934\pi\)
0.116183 + 0.993228i \(0.462934\pi\)
\(620\) 0 0
\(621\) 19.6954 0.790348
\(622\) 0 0
\(623\) 30.0188 1.20268
\(624\) 0 0
\(625\) 12.1144 0.484576
\(626\) 0 0
\(627\) −7.41081 −0.295959
\(628\) 0 0
\(629\) −15.6208 −0.622842
\(630\) 0 0
\(631\) −7.68906 −0.306097 −0.153048 0.988219i \(-0.548909\pi\)
−0.153048 + 0.988219i \(0.548909\pi\)
\(632\) 0 0
\(633\) −5.54439 −0.220370
\(634\) 0 0
\(635\) −1.05040 −0.0416840
\(636\) 0 0
\(637\) −9.01076 −0.357019
\(638\) 0 0
\(639\) 23.8828 0.944789
\(640\) 0 0
\(641\) 11.1732 0.441315 0.220658 0.975351i \(-0.429180\pi\)
0.220658 + 0.975351i \(0.429180\pi\)
\(642\) 0 0
\(643\) −38.7449 −1.52795 −0.763976 0.645245i \(-0.776755\pi\)
−0.763976 + 0.645245i \(0.776755\pi\)
\(644\) 0 0
\(645\) 10.4818 0.412721
\(646\) 0 0
\(647\) −4.31385 −0.169595 −0.0847975 0.996398i \(-0.527024\pi\)
−0.0847975 + 0.996398i \(0.527024\pi\)
\(648\) 0 0
\(649\) −24.2078 −0.950238
\(650\) 0 0
\(651\) −23.3137 −0.913736
\(652\) 0 0
\(653\) −16.8647 −0.659967 −0.329983 0.943987i \(-0.607043\pi\)
−0.329983 + 0.943987i \(0.607043\pi\)
\(654\) 0 0
\(655\) −17.6627 −0.690138
\(656\) 0 0
\(657\) 17.6217 0.687490
\(658\) 0 0
\(659\) 41.9416 1.63381 0.816906 0.576770i \(-0.195687\pi\)
0.816906 + 0.576770i \(0.195687\pi\)
\(660\) 0 0
\(661\) −24.5399 −0.954492 −0.477246 0.878770i \(-0.658365\pi\)
−0.477246 + 0.878770i \(0.658365\pi\)
\(662\) 0 0
\(663\) −12.4158 −0.482188
\(664\) 0 0
\(665\) −12.8625 −0.498785
\(666\) 0 0
\(667\) −43.2156 −1.67331
\(668\) 0 0
\(669\) −14.2624 −0.551416
\(670\) 0 0
\(671\) −0.353926 −0.0136632
\(672\) 0 0
\(673\) 19.5003 0.751680 0.375840 0.926685i \(-0.377354\pi\)
0.375840 + 0.926685i \(0.377354\pi\)
\(674\) 0 0
\(675\) −19.5350 −0.751903
\(676\) 0 0
\(677\) 16.8407 0.647241 0.323620 0.946187i \(-0.395100\pi\)
0.323620 + 0.946187i \(0.395100\pi\)
\(678\) 0 0
\(679\) 2.91026 0.111686
\(680\) 0 0
\(681\) 16.2619 0.623157
\(682\) 0 0
\(683\) −49.0773 −1.87789 −0.938946 0.344064i \(-0.888196\pi\)
−0.938946 + 0.344064i \(0.888196\pi\)
\(684\) 0 0
\(685\) −7.88844 −0.301402
\(686\) 0 0
\(687\) 14.2369 0.543171
\(688\) 0 0
\(689\) 19.2686 0.734076
\(690\) 0 0
\(691\) 8.79239 0.334478 0.167239 0.985916i \(-0.446515\pi\)
0.167239 + 0.985916i \(0.446515\pi\)
\(692\) 0 0
\(693\) −13.9481 −0.529844
\(694\) 0 0
\(695\) −12.8466 −0.487298
\(696\) 0 0
\(697\) −11.7201 −0.443931
\(698\) 0 0
\(699\) −10.8271 −0.409519
\(700\) 0 0
\(701\) −27.7148 −1.04678 −0.523388 0.852095i \(-0.675332\pi\)
−0.523388 + 0.852095i \(0.675332\pi\)
\(702\) 0 0
\(703\) −10.5079 −0.396315
\(704\) 0 0
\(705\) −2.96953 −0.111839
\(706\) 0 0
\(707\) −0.00612030 −0.000230177 0
\(708\) 0 0
\(709\) −22.2864 −0.836982 −0.418491 0.908221i \(-0.637441\pi\)
−0.418491 + 0.908221i \(0.637441\pi\)
\(710\) 0 0
\(711\) 3.83910 0.143977
\(712\) 0 0
\(713\) −30.9784 −1.16015
\(714\) 0 0
\(715\) −4.17118 −0.155993
\(716\) 0 0
\(717\) −3.92677 −0.146648
\(718\) 0 0
\(719\) 33.6036 1.25320 0.626601 0.779340i \(-0.284445\pi\)
0.626601 + 0.779340i \(0.284445\pi\)
\(720\) 0 0
\(721\) 42.0126 1.56463
\(722\) 0 0
\(723\) 19.5113 0.725632
\(724\) 0 0
\(725\) 42.8637 1.59192
\(726\) 0 0
\(727\) −9.18854 −0.340784 −0.170392 0.985376i \(-0.554503\pi\)
−0.170392 + 0.985376i \(0.554503\pi\)
\(728\) 0 0
\(729\) 9.23511 0.342041
\(730\) 0 0
\(731\) −70.6789 −2.61415
\(732\) 0 0
\(733\) 15.5832 0.575579 0.287790 0.957694i \(-0.407080\pi\)
0.287790 + 0.957694i \(0.407080\pi\)
\(734\) 0 0
\(735\) 3.62264 0.133623
\(736\) 0 0
\(737\) 17.8774 0.658523
\(738\) 0 0
\(739\) −9.94470 −0.365822 −0.182911 0.983129i \(-0.558552\pi\)
−0.182911 + 0.983129i \(0.558552\pi\)
\(740\) 0 0
\(741\) −8.35195 −0.306817
\(742\) 0 0
\(743\) −4.64206 −0.170301 −0.0851503 0.996368i \(-0.527137\pi\)
−0.0851503 + 0.996368i \(0.527137\pi\)
\(744\) 0 0
\(745\) 17.2049 0.630338
\(746\) 0 0
\(747\) −11.0530 −0.404409
\(748\) 0 0
\(749\) −11.0815 −0.404910
\(750\) 0 0
\(751\) −5.40568 −0.197256 −0.0986280 0.995124i \(-0.531445\pi\)
−0.0986280 + 0.995124i \(0.531445\pi\)
\(752\) 0 0
\(753\) −16.9992 −0.619485
\(754\) 0 0
\(755\) −11.5635 −0.420840
\(756\) 0 0
\(757\) 0.576865 0.0209665 0.0104833 0.999945i \(-0.496663\pi\)
0.0104833 + 0.999945i \(0.496663\pi\)
\(758\) 0 0
\(759\) 7.54987 0.274043
\(760\) 0 0
\(761\) −5.29201 −0.191835 −0.0959177 0.995389i \(-0.530579\pi\)
−0.0959177 + 0.995389i \(0.530579\pi\)
\(762\) 0 0
\(763\) −27.7666 −1.00522
\(764\) 0 0
\(765\) −12.2535 −0.443026
\(766\) 0 0
\(767\) −27.2820 −0.985097
\(768\) 0 0
\(769\) 43.7699 1.57838 0.789192 0.614146i \(-0.210499\pi\)
0.789192 + 0.614146i \(0.210499\pi\)
\(770\) 0 0
\(771\) −14.7082 −0.529704
\(772\) 0 0
\(773\) 47.7295 1.71671 0.858355 0.513057i \(-0.171487\pi\)
0.858355 + 0.513057i \(0.171487\pi\)
\(774\) 0 0
\(775\) 30.7262 1.10372
\(776\) 0 0
\(777\) 8.05646 0.289024
\(778\) 0 0
\(779\) −7.88399 −0.282473
\(780\) 0 0
\(781\) 22.0395 0.788635
\(782\) 0 0
\(783\) −50.1746 −1.79309
\(784\) 0 0
\(785\) −5.66110 −0.202053
\(786\) 0 0
\(787\) −38.7247 −1.38038 −0.690192 0.723626i \(-0.742474\pi\)
−0.690192 + 0.723626i \(0.742474\pi\)
\(788\) 0 0
\(789\) −20.7085 −0.737242
\(790\) 0 0
\(791\) −23.4017 −0.832068
\(792\) 0 0
\(793\) −0.398873 −0.0141644
\(794\) 0 0
\(795\) −7.74666 −0.274746
\(796\) 0 0
\(797\) −28.3788 −1.00523 −0.502614 0.864511i \(-0.667628\pi\)
−0.502614 + 0.864511i \(0.667628\pi\)
\(798\) 0 0
\(799\) 20.0236 0.708383
\(800\) 0 0
\(801\) 19.2373 0.679715
\(802\) 0 0
\(803\) 16.2617 0.573862
\(804\) 0 0
\(805\) 13.1038 0.461849
\(806\) 0 0
\(807\) 8.82805 0.310762
\(808\) 0 0
\(809\) −51.1815 −1.79945 −0.899723 0.436462i \(-0.856231\pi\)
−0.899723 + 0.436462i \(0.856231\pi\)
\(810\) 0 0
\(811\) −4.89583 −0.171916 −0.0859579 0.996299i \(-0.527395\pi\)
−0.0859579 + 0.996299i \(0.527395\pi\)
\(812\) 0 0
\(813\) −6.23731 −0.218752
\(814\) 0 0
\(815\) 7.41550 0.259754
\(816\) 0 0
\(817\) −47.5449 −1.66339
\(818\) 0 0
\(819\) −15.7194 −0.549281
\(820\) 0 0
\(821\) 29.6353 1.03428 0.517139 0.855901i \(-0.326997\pi\)
0.517139 + 0.855901i \(0.326997\pi\)
\(822\) 0 0
\(823\) −24.8933 −0.867727 −0.433864 0.900979i \(-0.642850\pi\)
−0.433864 + 0.900979i \(0.642850\pi\)
\(824\) 0 0
\(825\) −7.48840 −0.260713
\(826\) 0 0
\(827\) −23.7571 −0.826116 −0.413058 0.910705i \(-0.635539\pi\)
−0.413058 + 0.910705i \(0.635539\pi\)
\(828\) 0 0
\(829\) −49.8269 −1.73056 −0.865279 0.501291i \(-0.832859\pi\)
−0.865279 + 0.501291i \(0.832859\pi\)
\(830\) 0 0
\(831\) 23.5659 0.817492
\(832\) 0 0
\(833\) −24.4275 −0.846363
\(834\) 0 0
\(835\) 19.7417 0.683189
\(836\) 0 0
\(837\) −35.9669 −1.24320
\(838\) 0 0
\(839\) 46.1823 1.59439 0.797196 0.603721i \(-0.206316\pi\)
0.797196 + 0.603721i \(0.206316\pi\)
\(840\) 0 0
\(841\) 81.0931 2.79631
\(842\) 0 0
\(843\) 17.9962 0.619822
\(844\) 0 0
\(845\) 7.73322 0.266031
\(846\) 0 0
\(847\) 23.7181 0.814964
\(848\) 0 0
\(849\) −12.8591 −0.441322
\(850\) 0 0
\(851\) 10.7051 0.366967
\(852\) 0 0
\(853\) 0.275353 0.00942791 0.00471395 0.999989i \(-0.498499\pi\)
0.00471395 + 0.999989i \(0.498499\pi\)
\(854\) 0 0
\(855\) −8.24280 −0.281898
\(856\) 0 0
\(857\) 22.2182 0.758960 0.379480 0.925200i \(-0.376103\pi\)
0.379480 + 0.925200i \(0.376103\pi\)
\(858\) 0 0
\(859\) −0.537410 −0.0183362 −0.00916809 0.999958i \(-0.502918\pi\)
−0.00916809 + 0.999958i \(0.502918\pi\)
\(860\) 0 0
\(861\) 6.04466 0.206002
\(862\) 0 0
\(863\) 11.3841 0.387520 0.193760 0.981049i \(-0.437932\pi\)
0.193760 + 0.981049i \(0.437932\pi\)
\(864\) 0 0
\(865\) 6.86800 0.233519
\(866\) 0 0
\(867\) −17.8167 −0.605088
\(868\) 0 0
\(869\) 3.54279 0.120181
\(870\) 0 0
\(871\) 20.1478 0.682681
\(872\) 0 0
\(873\) 1.86502 0.0631212
\(874\) 0 0
\(875\) −28.9048 −0.977161
\(876\) 0 0
\(877\) −4.60167 −0.155387 −0.0776937 0.996977i \(-0.524756\pi\)
−0.0776937 + 0.996977i \(0.524756\pi\)
\(878\) 0 0
\(879\) −14.9853 −0.505442
\(880\) 0 0
\(881\) −41.1185 −1.38532 −0.692658 0.721266i \(-0.743561\pi\)
−0.692658 + 0.721266i \(0.743561\pi\)
\(882\) 0 0
\(883\) −31.1779 −1.04922 −0.524610 0.851343i \(-0.675789\pi\)
−0.524610 + 0.851343i \(0.675789\pi\)
\(884\) 0 0
\(885\) 10.9683 0.368696
\(886\) 0 0
\(887\) −12.6870 −0.425988 −0.212994 0.977054i \(-0.568321\pi\)
−0.212994 + 0.977054i \(0.568321\pi\)
\(888\) 0 0
\(889\) −3.65301 −0.122518
\(890\) 0 0
\(891\) −3.81406 −0.127776
\(892\) 0 0
\(893\) 13.4696 0.450745
\(894\) 0 0
\(895\) 0.833341 0.0278555
\(896\) 0 0
\(897\) 8.50867 0.284096
\(898\) 0 0
\(899\) 78.9185 2.63208
\(900\) 0 0
\(901\) 52.2358 1.74023
\(902\) 0 0
\(903\) 36.4527 1.21307
\(904\) 0 0
\(905\) −16.4986 −0.548434
\(906\) 0 0
\(907\) −6.38915 −0.212148 −0.106074 0.994358i \(-0.533828\pi\)
−0.106074 + 0.994358i \(0.533828\pi\)
\(908\) 0 0
\(909\) −0.00392214 −0.000130089 0
\(910\) 0 0
\(911\) 2.93353 0.0971921 0.0485961 0.998819i \(-0.484525\pi\)
0.0485961 + 0.998819i \(0.484525\pi\)
\(912\) 0 0
\(913\) −10.1999 −0.337568
\(914\) 0 0
\(915\) 0.160361 0.00530136
\(916\) 0 0
\(917\) −61.4258 −2.02846
\(918\) 0 0
\(919\) −43.8980 −1.44806 −0.724030 0.689768i \(-0.757712\pi\)
−0.724030 + 0.689768i \(0.757712\pi\)
\(920\) 0 0
\(921\) 6.46427 0.213005
\(922\) 0 0
\(923\) 24.8384 0.817566
\(924\) 0 0
\(925\) −10.6180 −0.349116
\(926\) 0 0
\(927\) 26.9234 0.884282
\(928\) 0 0
\(929\) 11.7583 0.385776 0.192888 0.981221i \(-0.438215\pi\)
0.192888 + 0.981221i \(0.438215\pi\)
\(930\) 0 0
\(931\) −16.4321 −0.538541
\(932\) 0 0
\(933\) −28.6661 −0.938486
\(934\) 0 0
\(935\) −11.3078 −0.369803
\(936\) 0 0
\(937\) 2.07498 0.0677867 0.0338934 0.999425i \(-0.489209\pi\)
0.0338934 + 0.999425i \(0.489209\pi\)
\(938\) 0 0
\(939\) −3.67226 −0.119840
\(940\) 0 0
\(941\) −10.1156 −0.329760 −0.164880 0.986314i \(-0.552724\pi\)
−0.164880 + 0.986314i \(0.552724\pi\)
\(942\) 0 0
\(943\) 8.03193 0.261556
\(944\) 0 0
\(945\) 15.2139 0.494909
\(946\) 0 0
\(947\) 35.7585 1.16199 0.580997 0.813906i \(-0.302663\pi\)
0.580997 + 0.813906i \(0.302663\pi\)
\(948\) 0 0
\(949\) 18.3268 0.594914
\(950\) 0 0
\(951\) −3.94439 −0.127906
\(952\) 0 0
\(953\) −15.9438 −0.516472 −0.258236 0.966082i \(-0.583141\pi\)
−0.258236 + 0.966082i \(0.583141\pi\)
\(954\) 0 0
\(955\) 20.8156 0.673578
\(956\) 0 0
\(957\) −19.2335 −0.621732
\(958\) 0 0
\(959\) −27.4338 −0.885883
\(960\) 0 0
\(961\) 25.5715 0.824886
\(962\) 0 0
\(963\) −7.10150 −0.228843
\(964\) 0 0
\(965\) −0.612939 −0.0197312
\(966\) 0 0
\(967\) 35.6191 1.14543 0.572717 0.819753i \(-0.305889\pi\)
0.572717 + 0.819753i \(0.305889\pi\)
\(968\) 0 0
\(969\) −22.6415 −0.727350
\(970\) 0 0
\(971\) 7.63473 0.245010 0.122505 0.992468i \(-0.460907\pi\)
0.122505 + 0.992468i \(0.460907\pi\)
\(972\) 0 0
\(973\) −44.6768 −1.43227
\(974\) 0 0
\(975\) −8.43939 −0.270277
\(976\) 0 0
\(977\) 21.0511 0.673484 0.336742 0.941597i \(-0.390675\pi\)
0.336742 + 0.941597i \(0.390675\pi\)
\(978\) 0 0
\(979\) 17.7525 0.567372
\(980\) 0 0
\(981\) −17.7940 −0.568118
\(982\) 0 0
\(983\) −33.4067 −1.06551 −0.532755 0.846270i \(-0.678843\pi\)
−0.532755 + 0.846270i \(0.678843\pi\)
\(984\) 0 0
\(985\) 19.7899 0.630560
\(986\) 0 0
\(987\) −10.3272 −0.328718
\(988\) 0 0
\(989\) 48.4371 1.54021
\(990\) 0 0
\(991\) 27.0358 0.858822 0.429411 0.903109i \(-0.358721\pi\)
0.429411 + 0.903109i \(0.358721\pi\)
\(992\) 0 0
\(993\) 15.0922 0.478936
\(994\) 0 0
\(995\) 10.7678 0.341364
\(996\) 0 0
\(997\) −35.0996 −1.11162 −0.555808 0.831311i \(-0.687591\pi\)
−0.555808 + 0.831311i \(0.687591\pi\)
\(998\) 0 0
\(999\) 12.4290 0.393235
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 4096.2.a.s.1.1 8
4.3 odd 2 4096.2.a.i.1.7 8
8.3 odd 2 inner 4096.2.a.s.1.2 8
8.5 even 2 4096.2.a.i.1.8 8
64.3 odd 16 1024.2.g.f.641.2 yes 16
64.5 even 16 1024.2.g.g.897.2 yes 16
64.11 odd 16 1024.2.g.a.385.3 yes 16
64.13 even 16 1024.2.g.g.129.2 yes 16
64.19 odd 16 1024.2.g.d.129.2 yes 16
64.21 even 16 1024.2.g.f.385.3 yes 16
64.27 odd 16 1024.2.g.d.897.2 yes 16
64.29 even 16 1024.2.g.a.641.2 yes 16
64.35 odd 16 1024.2.g.a.641.3 yes 16
64.37 even 16 1024.2.g.d.897.3 yes 16
64.43 odd 16 1024.2.g.f.385.2 yes 16
64.45 even 16 1024.2.g.d.129.3 yes 16
64.51 odd 16 1024.2.g.g.129.3 yes 16
64.53 even 16 1024.2.g.a.385.2 16
64.59 odd 16 1024.2.g.g.897.3 yes 16
64.61 even 16 1024.2.g.f.641.3 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1024.2.g.a.385.2 16 64.53 even 16
1024.2.g.a.385.3 yes 16 64.11 odd 16
1024.2.g.a.641.2 yes 16 64.29 even 16
1024.2.g.a.641.3 yes 16 64.35 odd 16
1024.2.g.d.129.2 yes 16 64.19 odd 16
1024.2.g.d.129.3 yes 16 64.45 even 16
1024.2.g.d.897.2 yes 16 64.27 odd 16
1024.2.g.d.897.3 yes 16 64.37 even 16
1024.2.g.f.385.2 yes 16 64.43 odd 16
1024.2.g.f.385.3 yes 16 64.21 even 16
1024.2.g.f.641.2 yes 16 64.3 odd 16
1024.2.g.f.641.3 yes 16 64.61 even 16
1024.2.g.g.129.2 yes 16 64.13 even 16
1024.2.g.g.129.3 yes 16 64.51 odd 16
1024.2.g.g.897.2 yes 16 64.5 even 16
1024.2.g.g.897.3 yes 16 64.59 odd 16
4096.2.a.i.1.7 8 4.3 odd 2
4096.2.a.i.1.8 8 8.5 even 2
4096.2.a.s.1.1 8 1.1 even 1 trivial
4096.2.a.s.1.2 8 8.3 odd 2 inner