Properties

Label 1120.2.h.b.111.10
Level $1120$
Weight $2$
Character 1120.111
Analytic conductor $8.943$
Analytic rank $0$
Dimension $16$
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] = [1120,2,Mod(111,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1120.111");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.94324502638\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} - 2x^{12} + 6x^{11} - 12x^{9} + 8x^{8} - 24x^{7} + 48x^{5} - 32x^{4} - 128x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 111.10
Root \(1.14218 + 0.833926i\) of defining polynomial
Character \(\chi\) \(=\) 1120.111
Dual form 1120.2.h.b.111.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.586834i q^{3} +1.00000 q^{5} +(2.52442 + 0.792014i) q^{7} +2.65563 q^{9} +O(q^{10})\) \(q+0.586834i q^{3} +1.00000 q^{5} +(2.52442 + 0.792014i) q^{7} +2.65563 q^{9} +0.580189 q^{11} +1.14766 q^{13} +0.586834i q^{15} -1.82880i q^{17} -4.72462i q^{19} +(-0.464781 + 1.48142i) q^{21} -2.79961i q^{23} +1.00000 q^{25} +3.31891i q^{27} +7.40518i q^{29} -4.73007 q^{31} +0.340475i q^{33} +(2.52442 + 0.792014i) q^{35} +4.35175i q^{37} +0.673485i q^{39} -8.46264i q^{41} +4.32314 q^{43} +2.65563 q^{45} -3.56830 q^{47} +(5.74543 + 3.99876i) q^{49} +1.07320 q^{51} +5.48977i q^{53} +0.580189 q^{55} +2.77257 q^{57} -13.2957i q^{59} +0.275184 q^{61} +(6.70392 + 2.10329i) q^{63} +1.14766 q^{65} +7.71818 q^{67} +1.64291 q^{69} +1.13454i q^{71} +1.78121i q^{73} +0.586834i q^{75} +(1.46464 + 0.459518i) q^{77} +10.0108i q^{79} +6.01923 q^{81} +15.6230i q^{83} -1.82880i q^{85} -4.34561 q^{87} +15.3393i q^{89} +(2.89717 + 0.908961i) q^{91} -2.77577i q^{93} -4.72462i q^{95} -14.9759i q^{97} +1.54077 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{5} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 16 q^{5} - 16 q^{9} + 4 q^{11} - 4 q^{21} + 16 q^{25} + 16 q^{31} + 4 q^{43} - 16 q^{45} - 8 q^{49} + 40 q^{51} + 4 q^{55} - 16 q^{57} - 8 q^{61} - 28 q^{63} - 20 q^{67} - 40 q^{69} - 4 q^{77} + 24 q^{81} - 72 q^{87} + 32 q^{91} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(421\) \(801\) \(897\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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
\(3\) 0.586834i 0.338809i 0.985547 + 0.169404i \(0.0541844\pi\)
−0.985547 + 0.169404i \(0.945816\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.52442 + 0.792014i 0.954142 + 0.299353i
\(8\) 0 0
\(9\) 2.65563 0.885209
\(10\) 0 0
\(11\) 0.580189 0.174934 0.0874668 0.996167i \(-0.472123\pi\)
0.0874668 + 0.996167i \(0.472123\pi\)
\(12\) 0 0
\(13\) 1.14766 0.318303 0.159152 0.987254i \(-0.449124\pi\)
0.159152 + 0.987254i \(0.449124\pi\)
\(14\) 0 0
\(15\) 0.586834i 0.151520i
\(16\) 0 0
\(17\) 1.82880i 0.443550i −0.975098 0.221775i \(-0.928815\pi\)
0.975098 0.221775i \(-0.0711851\pi\)
\(18\) 0 0
\(19\) 4.72462i 1.08390i −0.840410 0.541951i \(-0.817686\pi\)
0.840410 0.541951i \(-0.182314\pi\)
\(20\) 0 0
\(21\) −0.464781 + 1.48142i −0.101423 + 0.323272i
\(22\) 0 0
\(23\) 2.79961i 0.583759i −0.956455 0.291879i \(-0.905719\pi\)
0.956455 0.291879i \(-0.0942806\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 3.31891i 0.638725i
\(28\) 0 0
\(29\) 7.40518i 1.37511i 0.726133 + 0.687554i \(0.241316\pi\)
−0.726133 + 0.687554i \(0.758684\pi\)
\(30\) 0 0
\(31\) −4.73007 −0.849546 −0.424773 0.905300i \(-0.639646\pi\)
−0.424773 + 0.905300i \(0.639646\pi\)
\(32\) 0 0
\(33\) 0.340475i 0.0592691i
\(34\) 0 0
\(35\) 2.52442 + 0.792014i 0.426705 + 0.133875i
\(36\) 0 0
\(37\) 4.35175i 0.715423i 0.933832 + 0.357712i \(0.116443\pi\)
−0.933832 + 0.357712i \(0.883557\pi\)
\(38\) 0 0
\(39\) 0.673485i 0.107844i
\(40\) 0 0
\(41\) 8.46264i 1.32164i −0.750544 0.660821i \(-0.770208\pi\)
0.750544 0.660821i \(-0.229792\pi\)
\(42\) 0 0
\(43\) 4.32314 0.659273 0.329637 0.944108i \(-0.393074\pi\)
0.329637 + 0.944108i \(0.393074\pi\)
\(44\) 0 0
\(45\) 2.65563 0.395877
\(46\) 0 0
\(47\) −3.56830 −0.520490 −0.260245 0.965543i \(-0.583803\pi\)
−0.260245 + 0.965543i \(0.583803\pi\)
\(48\) 0 0
\(49\) 5.74543 + 3.99876i 0.820775 + 0.571251i
\(50\) 0 0
\(51\) 1.07320 0.150279
\(52\) 0 0
\(53\) 5.48977i 0.754079i 0.926197 + 0.377039i \(0.123058\pi\)
−0.926197 + 0.377039i \(0.876942\pi\)
\(54\) 0 0
\(55\) 0.580189 0.0782327
\(56\) 0 0
\(57\) 2.77257 0.367235
\(58\) 0 0
\(59\) 13.2957i 1.73095i −0.500948 0.865477i \(-0.667015\pi\)
0.500948 0.865477i \(-0.332985\pi\)
\(60\) 0 0
\(61\) 0.275184 0.0352337 0.0176168 0.999845i \(-0.494392\pi\)
0.0176168 + 0.999845i \(0.494392\pi\)
\(62\) 0 0
\(63\) 6.70392 + 2.10329i 0.844615 + 0.264990i
\(64\) 0 0
\(65\) 1.14766 0.142349
\(66\) 0 0
\(67\) 7.71818 0.942925 0.471463 0.881886i \(-0.343726\pi\)
0.471463 + 0.881886i \(0.343726\pi\)
\(68\) 0 0
\(69\) 1.64291 0.197783
\(70\) 0 0
\(71\) 1.13454i 0.134645i 0.997731 + 0.0673224i \(0.0214456\pi\)
−0.997731 + 0.0673224i \(0.978554\pi\)
\(72\) 0 0
\(73\) 1.78121i 0.208474i 0.994552 + 0.104237i \(0.0332401\pi\)
−0.994552 + 0.104237i \(0.966760\pi\)
\(74\) 0 0
\(75\) 0.586834i 0.0677617i
\(76\) 0 0
\(77\) 1.46464 + 0.459518i 0.166912 + 0.0523669i
\(78\) 0 0
\(79\) 10.0108i 1.12630i 0.826355 + 0.563149i \(0.190410\pi\)
−0.826355 + 0.563149i \(0.809590\pi\)
\(80\) 0 0
\(81\) 6.01923 0.668803
\(82\) 0 0
\(83\) 15.6230i 1.71485i 0.514607 + 0.857426i \(0.327938\pi\)
−0.514607 + 0.857426i \(0.672062\pi\)
\(84\) 0 0
\(85\) 1.82880i 0.198362i
\(86\) 0 0
\(87\) −4.34561 −0.465899
\(88\) 0 0
\(89\) 15.3393i 1.62596i 0.582293 + 0.812979i \(0.302156\pi\)
−0.582293 + 0.812979i \(0.697844\pi\)
\(90\) 0 0
\(91\) 2.89717 + 0.908961i 0.303706 + 0.0952850i
\(92\) 0 0
\(93\) 2.77577i 0.287833i
\(94\) 0 0
\(95\) 4.72462i 0.484736i
\(96\) 0 0
\(97\) 14.9759i 1.52058i −0.649585 0.760289i \(-0.725057\pi\)
0.649585 0.760289i \(-0.274943\pi\)
\(98\) 0 0
\(99\) 1.54077 0.154853
\(100\) 0 0
\(101\) −12.7803 −1.27169 −0.635845 0.771817i \(-0.719348\pi\)
−0.635845 + 0.771817i \(0.719348\pi\)
\(102\) 0 0
\(103\) 15.1439 1.49217 0.746085 0.665851i \(-0.231931\pi\)
0.746085 + 0.665851i \(0.231931\pi\)
\(104\) 0 0
\(105\) −0.464781 + 1.48142i −0.0453579 + 0.144572i
\(106\) 0 0
\(107\) −6.70224 −0.647930 −0.323965 0.946069i \(-0.605016\pi\)
−0.323965 + 0.946069i \(0.605016\pi\)
\(108\) 0 0
\(109\) 11.2152i 1.07422i −0.843511 0.537112i \(-0.819515\pi\)
0.843511 0.537112i \(-0.180485\pi\)
\(110\) 0 0
\(111\) −2.55376 −0.242392
\(112\) 0 0
\(113\) 1.41012 0.132653 0.0663265 0.997798i \(-0.478872\pi\)
0.0663265 + 0.997798i \(0.478872\pi\)
\(114\) 0 0
\(115\) 2.79961i 0.261065i
\(116\) 0 0
\(117\) 3.04775 0.281765
\(118\) 0 0
\(119\) 1.44844 4.61668i 0.132778 0.423210i
\(120\) 0 0
\(121\) −10.6634 −0.969398
\(122\) 0 0
\(123\) 4.96616 0.447784
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 15.7103i 1.39406i 0.717041 + 0.697031i \(0.245496\pi\)
−0.717041 + 0.697031i \(0.754504\pi\)
\(128\) 0 0
\(129\) 2.53697i 0.223368i
\(130\) 0 0
\(131\) 5.01749i 0.438380i −0.975682 0.219190i \(-0.929658\pi\)
0.975682 0.219190i \(-0.0703415\pi\)
\(132\) 0 0
\(133\) 3.74196 11.9269i 0.324469 1.03420i
\(134\) 0 0
\(135\) 3.31891i 0.285647i
\(136\) 0 0
\(137\) −14.8703 −1.27045 −0.635226 0.772326i \(-0.719093\pi\)
−0.635226 + 0.772326i \(0.719093\pi\)
\(138\) 0 0
\(139\) 0.148572i 0.0126017i −0.999980 0.00630086i \(-0.997994\pi\)
0.999980 0.00630086i \(-0.00200564\pi\)
\(140\) 0 0
\(141\) 2.09400i 0.176346i
\(142\) 0 0
\(143\) 0.665859 0.0556819
\(144\) 0 0
\(145\) 7.40518i 0.614967i
\(146\) 0 0
\(147\) −2.34661 + 3.37161i −0.193545 + 0.278086i
\(148\) 0 0
\(149\) 8.85493i 0.725424i −0.931901 0.362712i \(-0.881851\pi\)
0.931901 0.362712i \(-0.118149\pi\)
\(150\) 0 0
\(151\) 12.3932i 1.00855i 0.863544 + 0.504273i \(0.168239\pi\)
−0.863544 + 0.504273i \(0.831761\pi\)
\(152\) 0 0
\(153\) 4.85662i 0.392635i
\(154\) 0 0
\(155\) −4.73007 −0.379928
\(156\) 0 0
\(157\) −15.6090 −1.24574 −0.622869 0.782326i \(-0.714033\pi\)
−0.622869 + 0.782326i \(0.714033\pi\)
\(158\) 0 0
\(159\) −3.22159 −0.255488
\(160\) 0 0
\(161\) 2.21733 7.06740i 0.174750 0.556989i
\(162\) 0 0
\(163\) 5.60110 0.438712 0.219356 0.975645i \(-0.429604\pi\)
0.219356 + 0.975645i \(0.429604\pi\)
\(164\) 0 0
\(165\) 0.340475i 0.0265059i
\(166\) 0 0
\(167\) −9.34952 −0.723488 −0.361744 0.932278i \(-0.617819\pi\)
−0.361744 + 0.932278i \(0.617819\pi\)
\(168\) 0 0
\(169\) −11.6829 −0.898683
\(170\) 0 0
\(171\) 12.5468i 0.959479i
\(172\) 0 0
\(173\) 23.5706 1.79204 0.896018 0.444018i \(-0.146447\pi\)
0.896018 + 0.444018i \(0.146447\pi\)
\(174\) 0 0
\(175\) 2.52442 + 0.792014i 0.190828 + 0.0598706i
\(176\) 0 0
\(177\) 7.80238 0.586463
\(178\) 0 0
\(179\) −19.7086 −1.47309 −0.736546 0.676387i \(-0.763545\pi\)
−0.736546 + 0.676387i \(0.763545\pi\)
\(180\) 0 0
\(181\) −21.0102 −1.56168 −0.780839 0.624732i \(-0.785208\pi\)
−0.780839 + 0.624732i \(0.785208\pi\)
\(182\) 0 0
\(183\) 0.161487i 0.0119375i
\(184\) 0 0
\(185\) 4.35175i 0.319947i
\(186\) 0 0
\(187\) 1.06105i 0.0775919i
\(188\) 0 0
\(189\) −2.62862 + 8.37834i −0.191204 + 0.609435i
\(190\) 0 0
\(191\) 16.9750i 1.22827i −0.789201 0.614135i \(-0.789505\pi\)
0.789201 0.614135i \(-0.210495\pi\)
\(192\) 0 0
\(193\) 14.8445 1.06853 0.534266 0.845317i \(-0.320588\pi\)
0.534266 + 0.845317i \(0.320588\pi\)
\(194\) 0 0
\(195\) 0.673485i 0.0482292i
\(196\) 0 0
\(197\) 6.30785i 0.449416i −0.974426 0.224708i \(-0.927857\pi\)
0.974426 0.224708i \(-0.0721428\pi\)
\(198\) 0 0
\(199\) −11.9770 −0.849030 −0.424515 0.905421i \(-0.639555\pi\)
−0.424515 + 0.905421i \(0.639555\pi\)
\(200\) 0 0
\(201\) 4.52929i 0.319471i
\(202\) 0 0
\(203\) −5.86501 + 18.6938i −0.411643 + 1.31205i
\(204\) 0 0
\(205\) 8.46264i 0.591056i
\(206\) 0 0
\(207\) 7.43471i 0.516748i
\(208\) 0 0
\(209\) 2.74117i 0.189611i
\(210\) 0 0
\(211\) −10.8525 −0.747120 −0.373560 0.927606i \(-0.621863\pi\)
−0.373560 + 0.927606i \(0.621863\pi\)
\(212\) 0 0
\(213\) −0.665785 −0.0456189
\(214\) 0 0
\(215\) 4.32314 0.294836
\(216\) 0 0
\(217\) −11.9407 3.74628i −0.810588 0.254314i
\(218\) 0 0
\(219\) −1.04527 −0.0706329
\(220\) 0 0
\(221\) 2.09884i 0.141183i
\(222\) 0 0
\(223\) 14.0737 0.942445 0.471223 0.882014i \(-0.343813\pi\)
0.471223 + 0.882014i \(0.343813\pi\)
\(224\) 0 0
\(225\) 2.65563 0.177042
\(226\) 0 0
\(227\) 14.9683i 0.993478i −0.867900 0.496739i \(-0.834531\pi\)
0.867900 0.496739i \(-0.165469\pi\)
\(228\) 0 0
\(229\) 21.5684 1.42528 0.712641 0.701529i \(-0.247499\pi\)
0.712641 + 0.701529i \(0.247499\pi\)
\(230\) 0 0
\(231\) −0.269661 + 0.859503i −0.0177424 + 0.0565511i
\(232\) 0 0
\(233\) −19.5888 −1.28331 −0.641653 0.766995i \(-0.721751\pi\)
−0.641653 + 0.766995i \(0.721751\pi\)
\(234\) 0 0
\(235\) −3.56830 −0.232770
\(236\) 0 0
\(237\) −5.87465 −0.381600
\(238\) 0 0
\(239\) 23.2562i 1.50432i −0.658981 0.752160i \(-0.729012\pi\)
0.658981 0.752160i \(-0.270988\pi\)
\(240\) 0 0
\(241\) 20.9056i 1.34665i −0.739347 0.673324i \(-0.764866\pi\)
0.739347 0.673324i \(-0.235134\pi\)
\(242\) 0 0
\(243\) 13.4890i 0.865321i
\(244\) 0 0
\(245\) 5.74543 + 3.99876i 0.367062 + 0.255471i
\(246\) 0 0
\(247\) 5.42225i 0.345009i
\(248\) 0 0
\(249\) −9.16813 −0.581007
\(250\) 0 0
\(251\) 3.33668i 0.210610i 0.994440 + 0.105305i \(0.0335818\pi\)
−0.994440 + 0.105305i \(0.966418\pi\)
\(252\) 0 0
\(253\) 1.62430i 0.102119i
\(254\) 0 0
\(255\) 1.07320 0.0672067
\(256\) 0 0
\(257\) 24.4200i 1.52328i 0.648002 + 0.761639i \(0.275605\pi\)
−0.648002 + 0.761639i \(0.724395\pi\)
\(258\) 0 0
\(259\) −3.44665 + 10.9857i −0.214164 + 0.682616i
\(260\) 0 0
\(261\) 19.6654i 1.21726i
\(262\) 0 0
\(263\) 27.1113i 1.67176i −0.548915 0.835878i \(-0.684959\pi\)
0.548915 0.835878i \(-0.315041\pi\)
\(264\) 0 0
\(265\) 5.48977i 0.337234i
\(266\) 0 0
\(267\) −9.00160 −0.550889
\(268\) 0 0
\(269\) −18.5701 −1.13224 −0.566121 0.824322i \(-0.691556\pi\)
−0.566121 + 0.824322i \(0.691556\pi\)
\(270\) 0 0
\(271\) −12.9119 −0.784344 −0.392172 0.919892i \(-0.628276\pi\)
−0.392172 + 0.919892i \(0.628276\pi\)
\(272\) 0 0
\(273\) −0.533409 + 1.70016i −0.0322834 + 0.102898i
\(274\) 0 0
\(275\) 0.580189 0.0349867
\(276\) 0 0
\(277\) 0.557017i 0.0334679i −0.999860 0.0167340i \(-0.994673\pi\)
0.999860 0.0167340i \(-0.00532683\pi\)
\(278\) 0 0
\(279\) −12.5613 −0.752025
\(280\) 0 0
\(281\) −27.3862 −1.63372 −0.816860 0.576835i \(-0.804287\pi\)
−0.816860 + 0.576835i \(0.804287\pi\)
\(282\) 0 0
\(283\) 5.24149i 0.311574i 0.987791 + 0.155787i \(0.0497914\pi\)
−0.987791 + 0.155787i \(0.950209\pi\)
\(284\) 0 0
\(285\) 2.77257 0.164233
\(286\) 0 0
\(287\) 6.70253 21.3633i 0.395638 1.26103i
\(288\) 0 0
\(289\) 13.6555 0.803263
\(290\) 0 0
\(291\) 8.78840 0.515185
\(292\) 0 0
\(293\) 6.66098 0.389138 0.194569 0.980889i \(-0.437669\pi\)
0.194569 + 0.980889i \(0.437669\pi\)
\(294\) 0 0
\(295\) 13.2957i 0.774107i
\(296\) 0 0
\(297\) 1.92560i 0.111735i
\(298\) 0 0
\(299\) 3.21299i 0.185812i
\(300\) 0 0
\(301\) 10.9134 + 3.42399i 0.629041 + 0.197355i
\(302\) 0 0
\(303\) 7.49993i 0.430860i
\(304\) 0 0
\(305\) 0.275184 0.0157570
\(306\) 0 0
\(307\) 17.0826i 0.974957i −0.873135 0.487479i \(-0.837917\pi\)
0.873135 0.487479i \(-0.162083\pi\)
\(308\) 0 0
\(309\) 8.88694i 0.505560i
\(310\) 0 0
\(311\) −29.4721 −1.67121 −0.835604 0.549332i \(-0.814882\pi\)
−0.835604 + 0.549332i \(0.814882\pi\)
\(312\) 0 0
\(313\) 26.1624i 1.47879i 0.673275 + 0.739393i \(0.264887\pi\)
−0.673275 + 0.739393i \(0.735113\pi\)
\(314\) 0 0
\(315\) 6.70392 + 2.10329i 0.377723 + 0.118507i
\(316\) 0 0
\(317\) 5.72675i 0.321646i 0.986983 + 0.160823i \(0.0514149\pi\)
−0.986983 + 0.160823i \(0.948585\pi\)
\(318\) 0 0
\(319\) 4.29641i 0.240553i
\(320\) 0 0
\(321\) 3.93310i 0.219524i
\(322\) 0 0
\(323\) −8.64040 −0.480765
\(324\) 0 0
\(325\) 1.14766 0.0636606
\(326\) 0 0
\(327\) 6.58148 0.363957
\(328\) 0 0
\(329\) −9.00789 2.82614i −0.496621 0.155810i
\(330\) 0 0
\(331\) −17.6047 −0.967641 −0.483820 0.875167i \(-0.660751\pi\)
−0.483820 + 0.875167i \(0.660751\pi\)
\(332\) 0 0
\(333\) 11.5566i 0.633299i
\(334\) 0 0
\(335\) 7.71818 0.421689
\(336\) 0 0
\(337\) 24.3443 1.32612 0.663059 0.748567i \(-0.269258\pi\)
0.663059 + 0.748567i \(0.269258\pi\)
\(338\) 0 0
\(339\) 0.827507i 0.0449440i
\(340\) 0 0
\(341\) −2.74434 −0.148614
\(342\) 0 0
\(343\) 11.3368 + 14.6450i 0.612131 + 0.790756i
\(344\) 0 0
\(345\) 1.64291 0.0884510
\(346\) 0 0
\(347\) −16.7400 −0.898650 −0.449325 0.893368i \(-0.648336\pi\)
−0.449325 + 0.893368i \(0.648336\pi\)
\(348\) 0 0
\(349\) −14.2317 −0.761807 −0.380903 0.924615i \(-0.624387\pi\)
−0.380903 + 0.924615i \(0.624387\pi\)
\(350\) 0 0
\(351\) 3.80898i 0.203308i
\(352\) 0 0
\(353\) 6.08004i 0.323608i −0.986823 0.161804i \(-0.948269\pi\)
0.986823 0.161804i \(-0.0517312\pi\)
\(354\) 0 0
\(355\) 1.13454i 0.0602150i
\(356\) 0 0
\(357\) 2.70922 + 0.849993i 0.143387 + 0.0449864i
\(358\) 0 0
\(359\) 16.3854i 0.864787i 0.901685 + 0.432394i \(0.142331\pi\)
−0.901685 + 0.432394i \(0.857669\pi\)
\(360\) 0 0
\(361\) −3.32202 −0.174843
\(362\) 0 0
\(363\) 6.25763i 0.328441i
\(364\) 0 0
\(365\) 1.78121i 0.0932326i
\(366\) 0 0
\(367\) −13.1245 −0.685092 −0.342546 0.939501i \(-0.611289\pi\)
−0.342546 + 0.939501i \(0.611289\pi\)
\(368\) 0 0
\(369\) 22.4736i 1.16993i
\(370\) 0 0
\(371\) −4.34798 + 13.8585i −0.225736 + 0.719498i
\(372\) 0 0
\(373\) 0.552362i 0.0286002i −0.999898 0.0143001i \(-0.995448\pi\)
0.999898 0.0143001i \(-0.00455202\pi\)
\(374\) 0 0
\(375\) 0.586834i 0.0303040i
\(376\) 0 0
\(377\) 8.49862i 0.437701i
\(378\) 0 0
\(379\) 29.1600 1.49785 0.748925 0.662655i \(-0.230570\pi\)
0.748925 + 0.662655i \(0.230570\pi\)
\(380\) 0 0
\(381\) −9.21933 −0.472320
\(382\) 0 0
\(383\) 5.89451 0.301195 0.150598 0.988595i \(-0.451880\pi\)
0.150598 + 0.988595i \(0.451880\pi\)
\(384\) 0 0
\(385\) 1.46464 + 0.459518i 0.0746451 + 0.0234192i
\(386\) 0 0
\(387\) 11.4807 0.583594
\(388\) 0 0
\(389\) 13.5376i 0.686385i −0.939265 0.343192i \(-0.888492\pi\)
0.939265 0.343192i \(-0.111508\pi\)
\(390\) 0 0
\(391\) −5.11994 −0.258926
\(392\) 0 0
\(393\) 2.94444 0.148527
\(394\) 0 0
\(395\) 10.0108i 0.503696i
\(396\) 0 0
\(397\) −4.22501 −0.212047 −0.106024 0.994364i \(-0.533812\pi\)
−0.106024 + 0.994364i \(0.533812\pi\)
\(398\) 0 0
\(399\) 6.99913 + 2.19591i 0.350395 + 0.109933i
\(400\) 0 0
\(401\) 8.09225 0.404108 0.202054 0.979374i \(-0.435238\pi\)
0.202054 + 0.979374i \(0.435238\pi\)
\(402\) 0 0
\(403\) −5.42850 −0.270413
\(404\) 0 0
\(405\) 6.01923 0.299098
\(406\) 0 0
\(407\) 2.52484i 0.125152i
\(408\) 0 0
\(409\) 27.6608i 1.36774i −0.729605 0.683868i \(-0.760296\pi\)
0.729605 0.683868i \(-0.239704\pi\)
\(410\) 0 0
\(411\) 8.72637i 0.430440i
\(412\) 0 0
\(413\) 10.5304 33.5640i 0.518167 1.65158i
\(414\) 0 0
\(415\) 15.6230i 0.766905i
\(416\) 0 0
\(417\) 0.0871872 0.00426957
\(418\) 0 0
\(419\) 10.5982i 0.517755i 0.965910 + 0.258877i \(0.0833526\pi\)
−0.965910 + 0.258877i \(0.916647\pi\)
\(420\) 0 0
\(421\) 25.5951i 1.24743i −0.781652 0.623715i \(-0.785623\pi\)
0.781652 0.623715i \(-0.214377\pi\)
\(422\) 0 0
\(423\) −9.47606 −0.460742
\(424\) 0 0
\(425\) 1.82880i 0.0887101i
\(426\) 0 0
\(427\) 0.694680 + 0.217949i 0.0336179 + 0.0105473i
\(428\) 0 0
\(429\) 0.390749i 0.0188655i
\(430\) 0 0
\(431\) 32.7406i 1.57706i 0.614997 + 0.788529i \(0.289157\pi\)
−0.614997 + 0.788529i \(0.710843\pi\)
\(432\) 0 0
\(433\) 16.8933i 0.811840i 0.913909 + 0.405920i \(0.133049\pi\)
−0.913909 + 0.405920i \(0.866951\pi\)
\(434\) 0 0
\(435\) −4.34561 −0.208356
\(436\) 0 0
\(437\) −13.2271 −0.632737
\(438\) 0 0
\(439\) 31.6749 1.51176 0.755881 0.654709i \(-0.227209\pi\)
0.755881 + 0.654709i \(0.227209\pi\)
\(440\) 0 0
\(441\) 15.2577 + 10.6192i 0.726558 + 0.505676i
\(442\) 0 0
\(443\) 12.8303 0.609587 0.304794 0.952418i \(-0.401413\pi\)
0.304794 + 0.952418i \(0.401413\pi\)
\(444\) 0 0
\(445\) 15.3393i 0.727151i
\(446\) 0 0
\(447\) 5.19637 0.245780
\(448\) 0 0
\(449\) 9.10148 0.429525 0.214763 0.976666i \(-0.431102\pi\)
0.214763 + 0.976666i \(0.431102\pi\)
\(450\) 0 0
\(451\) 4.90993i 0.231200i
\(452\) 0 0
\(453\) −7.27276 −0.341704
\(454\) 0 0
\(455\) 2.89717 + 0.908961i 0.135822 + 0.0426127i
\(456\) 0 0
\(457\) −5.29969 −0.247909 −0.123954 0.992288i \(-0.539558\pi\)
−0.123954 + 0.992288i \(0.539558\pi\)
\(458\) 0 0
\(459\) 6.06964 0.283307
\(460\) 0 0
\(461\) 0.755725 0.0351976 0.0175988 0.999845i \(-0.494398\pi\)
0.0175988 + 0.999845i \(0.494398\pi\)
\(462\) 0 0
\(463\) 3.99447i 0.185639i 0.995683 + 0.0928193i \(0.0295879\pi\)
−0.995683 + 0.0928193i \(0.970412\pi\)
\(464\) 0 0
\(465\) 2.77577i 0.128723i
\(466\) 0 0
\(467\) 3.90589i 0.180743i 0.995908 + 0.0903716i \(0.0288055\pi\)
−0.995908 + 0.0903716i \(0.971195\pi\)
\(468\) 0 0
\(469\) 19.4839 + 6.11290i 0.899685 + 0.282268i
\(470\) 0 0
\(471\) 9.15992i 0.422067i
\(472\) 0 0
\(473\) 2.50824 0.115329
\(474\) 0 0
\(475\) 4.72462i 0.216780i
\(476\) 0 0
\(477\) 14.5788i 0.667517i
\(478\) 0 0
\(479\) −9.21055 −0.420841 −0.210420 0.977611i \(-0.567483\pi\)
−0.210420 + 0.977611i \(0.567483\pi\)
\(480\) 0 0
\(481\) 4.99432i 0.227721i
\(482\) 0 0
\(483\) 4.14739 + 1.30120i 0.188713 + 0.0592068i
\(484\) 0 0
\(485\) 14.9759i 0.680023i
\(486\) 0 0
\(487\) 21.4403i 0.971552i 0.874083 + 0.485776i \(0.161463\pi\)
−0.874083 + 0.485776i \(0.838537\pi\)
\(488\) 0 0
\(489\) 3.28692i 0.148640i
\(490\) 0 0
\(491\) −19.6672 −0.887570 −0.443785 0.896133i \(-0.646365\pi\)
−0.443785 + 0.896133i \(0.646365\pi\)
\(492\) 0 0
\(493\) 13.5426 0.609930
\(494\) 0 0
\(495\) 1.54077 0.0692523
\(496\) 0 0
\(497\) −0.898570 + 2.86405i −0.0403064 + 0.128470i
\(498\) 0 0
\(499\) 12.2344 0.547686 0.273843 0.961774i \(-0.411705\pi\)
0.273843 + 0.961774i \(0.411705\pi\)
\(500\) 0 0
\(501\) 5.48662i 0.245124i
\(502\) 0 0
\(503\) −5.42742 −0.241997 −0.120998 0.992653i \(-0.538610\pi\)
−0.120998 + 0.992653i \(0.538610\pi\)
\(504\) 0 0
\(505\) −12.7803 −0.568717
\(506\) 0 0
\(507\) 6.85591i 0.304482i
\(508\) 0 0
\(509\) −17.3838 −0.770522 −0.385261 0.922808i \(-0.625889\pi\)
−0.385261 + 0.922808i \(0.625889\pi\)
\(510\) 0 0
\(511\) −1.41074 + 4.49652i −0.0624074 + 0.198914i
\(512\) 0 0
\(513\) 15.6806 0.692315
\(514\) 0 0
\(515\) 15.1439 0.667319
\(516\) 0 0
\(517\) −2.07029 −0.0910511
\(518\) 0 0
\(519\) 13.8320i 0.607157i
\(520\) 0 0
\(521\) 12.8139i 0.561388i −0.959797 0.280694i \(-0.909435\pi\)
0.959797 0.280694i \(-0.0905646\pi\)
\(522\) 0 0
\(523\) 8.05393i 0.352174i −0.984375 0.176087i \(-0.943656\pi\)
0.984375 0.176087i \(-0.0563439\pi\)
\(524\) 0 0
\(525\) −0.464781 + 1.48142i −0.0202847 + 0.0646544i
\(526\) 0 0
\(527\) 8.65037i 0.376816i
\(528\) 0 0
\(529\) 15.1622 0.659226
\(530\) 0 0
\(531\) 35.3085i 1.53226i
\(532\) 0 0
\(533\) 9.71221i 0.420683i
\(534\) 0 0
\(535\) −6.70224 −0.289763
\(536\) 0 0
\(537\) 11.5657i 0.499096i
\(538\) 0 0
\(539\) 3.33344 + 2.32004i 0.143581 + 0.0999310i
\(540\) 0 0
\(541\) 15.9099i 0.684018i 0.939697 + 0.342009i \(0.111107\pi\)
−0.939697 + 0.342009i \(0.888893\pi\)
\(542\) 0 0
\(543\) 12.3295i 0.529110i
\(544\) 0 0
\(545\) 11.2152i 0.480408i
\(546\) 0 0
\(547\) 28.6893 1.22667 0.613333 0.789825i \(-0.289828\pi\)
0.613333 + 0.789825i \(0.289828\pi\)
\(548\) 0 0
\(549\) 0.730785 0.0311891
\(550\) 0 0
\(551\) 34.9867 1.49048
\(552\) 0 0
\(553\) −7.92866 + 25.2714i −0.337161 + 1.07465i
\(554\) 0 0
\(555\) −2.55376 −0.108401
\(556\) 0 0
\(557\) 29.6910i 1.25805i −0.777386 0.629024i \(-0.783455\pi\)
0.777386 0.629024i \(-0.216545\pi\)
\(558\) 0 0
\(559\) 4.96149 0.209849
\(560\) 0 0
\(561\) 0.622662 0.0262888
\(562\) 0 0
\(563\) 25.7978i 1.08725i −0.839329 0.543624i \(-0.817052\pi\)
0.839329 0.543624i \(-0.182948\pi\)
\(564\) 0 0
\(565\) 1.41012 0.0593242
\(566\) 0 0
\(567\) 15.1951 + 4.76731i 0.638133 + 0.200208i
\(568\) 0 0
\(569\) −6.17604 −0.258913 −0.129457 0.991585i \(-0.541323\pi\)
−0.129457 + 0.991585i \(0.541323\pi\)
\(570\) 0 0
\(571\) 44.7626 1.87326 0.936628 0.350325i \(-0.113929\pi\)
0.936628 + 0.350325i \(0.113929\pi\)
\(572\) 0 0
\(573\) 9.96153 0.416149
\(574\) 0 0
\(575\) 2.79961i 0.116752i
\(576\) 0 0
\(577\) 45.0609i 1.87591i −0.346760 0.937954i \(-0.612718\pi\)
0.346760 0.937954i \(-0.387282\pi\)
\(578\) 0 0
\(579\) 8.71126i 0.362028i
\(580\) 0 0
\(581\) −12.3737 + 39.4392i −0.513346 + 1.63621i
\(582\) 0 0
\(583\) 3.18511i 0.131914i
\(584\) 0 0
\(585\) 3.04775 0.126009
\(586\) 0 0
\(587\) 0.738708i 0.0304897i −0.999884 0.0152449i \(-0.995147\pi\)
0.999884 0.0152449i \(-0.00485278\pi\)
\(588\) 0 0
\(589\) 22.3478i 0.920824i
\(590\) 0 0
\(591\) 3.70166 0.152266
\(592\) 0 0
\(593\) 5.51961i 0.226663i 0.993557 + 0.113332i \(0.0361522\pi\)
−0.993557 + 0.113332i \(0.963848\pi\)
\(594\) 0 0
\(595\) 1.44844 4.61668i 0.0593802 0.189265i
\(596\) 0 0
\(597\) 7.02853i 0.287659i
\(598\) 0 0
\(599\) 0.145815i 0.00595785i 0.999996 + 0.00297893i \(0.000948223\pi\)
−0.999996 + 0.00297893i \(0.999052\pi\)
\(600\) 0 0
\(601\) 21.7269i 0.886257i 0.896458 + 0.443128i \(0.146131\pi\)
−0.896458 + 0.443128i \(0.853869\pi\)
\(602\) 0 0
\(603\) 20.4966 0.834686
\(604\) 0 0
\(605\) −10.6634 −0.433528
\(606\) 0 0
\(607\) −18.6456 −0.756804 −0.378402 0.925641i \(-0.623526\pi\)
−0.378402 + 0.925641i \(0.623526\pi\)
\(608\) 0 0
\(609\) −10.9702 3.44179i −0.444534 0.139468i
\(610\) 0 0
\(611\) −4.09518 −0.165673
\(612\) 0 0
\(613\) 47.1735i 1.90532i 0.304041 + 0.952659i \(0.401664\pi\)
−0.304041 + 0.952659i \(0.598336\pi\)
\(614\) 0 0
\(615\) 4.96616 0.200255
\(616\) 0 0
\(617\) 16.2076 0.652493 0.326246 0.945285i \(-0.394216\pi\)
0.326246 + 0.945285i \(0.394216\pi\)
\(618\) 0 0
\(619\) 30.2319i 1.21512i 0.794273 + 0.607561i \(0.207852\pi\)
−0.794273 + 0.607561i \(0.792148\pi\)
\(620\) 0 0
\(621\) 9.29166 0.372861
\(622\) 0 0
\(623\) −12.1489 + 38.7228i −0.486736 + 1.55140i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.60861 0.0642418
\(628\) 0 0
\(629\) 7.95850 0.317326
\(630\) 0 0
\(631\) 6.27504i 0.249806i −0.992169 0.124903i \(-0.960138\pi\)
0.992169 0.124903i \(-0.0398619\pi\)
\(632\) 0 0
\(633\) 6.36864i 0.253131i
\(634\) 0 0
\(635\) 15.7103i 0.623443i
\(636\) 0 0
\(637\) 6.59379 + 4.58920i 0.261255 + 0.181831i
\(638\) 0 0
\(639\) 3.01291i 0.119189i
\(640\) 0 0
\(641\) −0.954180 −0.0376878 −0.0188439 0.999822i \(-0.505999\pi\)
−0.0188439 + 0.999822i \(0.505999\pi\)
\(642\) 0 0
\(643\) 29.7150i 1.17184i −0.810367 0.585922i \(-0.800732\pi\)
0.810367 0.585922i \(-0.199268\pi\)
\(644\) 0 0
\(645\) 2.53697i 0.0998930i
\(646\) 0 0
\(647\) −22.3760 −0.879693 −0.439846 0.898073i \(-0.644967\pi\)
−0.439846 + 0.898073i \(0.644967\pi\)
\(648\) 0 0
\(649\) 7.71403i 0.302802i
\(650\) 0 0
\(651\) 2.19844 7.00721i 0.0861638 0.274634i
\(652\) 0 0
\(653\) 3.18538i 0.124653i 0.998056 + 0.0623267i \(0.0198521\pi\)
−0.998056 + 0.0623267i \(0.980148\pi\)
\(654\) 0 0
\(655\) 5.01749i 0.196050i
\(656\) 0 0
\(657\) 4.73022i 0.184543i
\(658\) 0 0
\(659\) 1.50751 0.0587242 0.0293621 0.999569i \(-0.490652\pi\)
0.0293621 + 0.999569i \(0.490652\pi\)
\(660\) 0 0
\(661\) 11.0607 0.430210 0.215105 0.976591i \(-0.430991\pi\)
0.215105 + 0.976591i \(0.430991\pi\)
\(662\) 0 0
\(663\) 1.23167 0.0478342
\(664\) 0 0
\(665\) 3.74196 11.9269i 0.145107 0.462507i
\(666\) 0 0
\(667\) 20.7316 0.802731
\(668\) 0 0
\(669\) 8.25893i 0.319309i
\(670\) 0 0
\(671\) 0.159659 0.00616355
\(672\) 0 0
\(673\) 31.5781 1.21725 0.608624 0.793459i \(-0.291722\pi\)
0.608624 + 0.793459i \(0.291722\pi\)
\(674\) 0 0
\(675\) 3.31891i 0.127745i
\(676\) 0 0
\(677\) 5.96254 0.229159 0.114580 0.993414i \(-0.463448\pi\)
0.114580 + 0.993414i \(0.463448\pi\)
\(678\) 0 0
\(679\) 11.8612 37.8056i 0.455189 1.45085i
\(680\) 0 0
\(681\) 8.78388 0.336599
\(682\) 0 0
\(683\) −32.0466 −1.22623 −0.613114 0.789994i \(-0.710084\pi\)
−0.613114 + 0.789994i \(0.710084\pi\)
\(684\) 0 0
\(685\) −14.8703 −0.568163
\(686\) 0 0
\(687\) 12.6571i 0.482898i
\(688\) 0 0
\(689\) 6.30038i 0.240026i
\(690\) 0 0
\(691\) 17.0740i 0.649526i 0.945795 + 0.324763i \(0.105285\pi\)
−0.945795 + 0.324763i \(0.894715\pi\)
\(692\) 0 0
\(693\) 3.88955 + 1.22031i 0.147752 + 0.0463557i
\(694\) 0 0
\(695\) 0.148572i 0.00563566i
\(696\) 0 0
\(697\) −15.4765 −0.586215
\(698\) 0 0
\(699\) 11.4954i 0.434796i
\(700\) 0 0
\(701\) 32.7289i 1.23615i −0.786118 0.618076i \(-0.787912\pi\)
0.786118 0.618076i \(-0.212088\pi\)
\(702\) 0 0
\(703\) 20.5604 0.775449
\(704\) 0 0
\(705\) 2.09400i 0.0788645i
\(706\) 0 0
\(707\) −32.2629 10.1222i −1.21337 0.380684i
\(708\) 0 0
\(709\) 20.9447i 0.786594i −0.919412 0.393297i \(-0.871334\pi\)
0.919412 0.393297i \(-0.128666\pi\)
\(710\) 0 0
\(711\) 26.5848i 0.997009i
\(712\) 0 0
\(713\) 13.2423i 0.495930i
\(714\) 0 0
\(715\) 0.665859 0.0249017
\(716\) 0 0
\(717\) 13.6475 0.509677
\(718\) 0 0
\(719\) 0.417030 0.0155526 0.00777630 0.999970i \(-0.497525\pi\)
0.00777630 + 0.999970i \(0.497525\pi\)
\(720\) 0 0
\(721\) 38.2295 + 11.9942i 1.42374 + 0.446686i
\(722\) 0 0
\(723\) 12.2681 0.456256
\(724\) 0 0
\(725\) 7.40518i 0.275022i
\(726\) 0 0
\(727\) −4.48987 −0.166520 −0.0832601 0.996528i \(-0.526533\pi\)
−0.0832601 + 0.996528i \(0.526533\pi\)
\(728\) 0 0
\(729\) 10.1419 0.375624
\(730\) 0 0
\(731\) 7.90619i 0.292421i
\(732\) 0 0
\(733\) 22.8948 0.845640 0.422820 0.906214i \(-0.361040\pi\)
0.422820 + 0.906214i \(0.361040\pi\)
\(734\) 0 0
\(735\) −2.34661 + 3.37161i −0.0865559 + 0.124364i
\(736\) 0 0
\(737\) 4.47800 0.164949
\(738\) 0 0
\(739\) 0.354985 0.0130583 0.00652917 0.999979i \(-0.497922\pi\)
0.00652917 + 0.999979i \(0.497922\pi\)
\(740\) 0 0
\(741\) 3.18196 0.116892
\(742\) 0 0
\(743\) 6.75171i 0.247696i 0.992301 + 0.123848i \(0.0395235\pi\)
−0.992301 + 0.123848i \(0.960476\pi\)
\(744\) 0 0
\(745\) 8.85493i 0.324419i
\(746\) 0 0
\(747\) 41.4890i 1.51800i
\(748\) 0 0
\(749\) −16.9193 5.30827i −0.618218 0.193960i
\(750\) 0 0
\(751\) 27.4120i 1.00028i −0.865945 0.500139i \(-0.833282\pi\)
0.865945 0.500139i \(-0.166718\pi\)
\(752\) 0 0
\(753\) −1.95808 −0.0713564
\(754\) 0 0
\(755\) 12.3932i 0.451035i
\(756\) 0 0
\(757\) 27.6550i 1.00514i 0.864537 + 0.502570i \(0.167612\pi\)
−0.864537 + 0.502570i \(0.832388\pi\)
\(758\) 0 0
\(759\) 0.953196 0.0345988
\(760\) 0 0
\(761\) 14.1083i 0.511424i −0.966753 0.255712i \(-0.917690\pi\)
0.966753 0.255712i \(-0.0823099\pi\)
\(762\) 0 0
\(763\) 8.88262 28.3120i 0.321572 1.02496i
\(764\) 0 0
\(765\) 4.85662i 0.175592i
\(766\) 0 0
\(767\) 15.2589i 0.550968i
\(768\) 0 0
\(769\) 2.82610i 0.101912i −0.998701 0.0509558i \(-0.983773\pi\)
0.998701 0.0509558i \(-0.0162268\pi\)
\(770\) 0 0
\(771\) −14.3305 −0.516100
\(772\) 0 0
\(773\) −44.5587 −1.60266 −0.801332 0.598220i \(-0.795875\pi\)
−0.801332 + 0.598220i \(0.795875\pi\)
\(774\) 0 0
\(775\) −4.73007 −0.169909
\(776\) 0 0
\(777\) −6.44676 2.02261i −0.231276 0.0725607i
\(778\) 0 0
\(779\) −39.9827 −1.43253
\(780\) 0 0
\(781\) 0.658247i 0.0235539i
\(782\) 0 0
\(783\) −24.5772 −0.878316
\(784\) 0 0
\(785\) −15.6090 −0.557111
\(786\) 0 0
\(787\) 30.3744i 1.08273i 0.840788 + 0.541365i \(0.182092\pi\)
−0.840788 + 0.541365i \(0.817908\pi\)
\(788\) 0 0
\(789\) 15.9098 0.566406
\(790\) 0 0
\(791\) 3.55974 + 1.11684i 0.126570 + 0.0397101i
\(792\) 0 0
\(793\) 0.315817 0.0112150
\(794\) 0 0
\(795\) −3.22159 −0.114258
\(796\) 0 0
\(797\) 43.5064 1.54108 0.770538 0.637394i \(-0.219988\pi\)
0.770538 + 0.637394i \(0.219988\pi\)
\(798\) 0 0
\(799\) 6.52572i 0.230863i
\(800\) 0 0
\(801\) 40.7353i 1.43931i
\(802\) 0 0
\(803\) 1.03344i 0.0364692i
\(804\) 0 0
\(805\) 2.21733 7.06740i 0.0781506 0.249093i
\(806\) 0 0
\(807\) 10.8976i 0.383613i
\(808\) 0 0
\(809\) −17.7818 −0.625176 −0.312588 0.949889i \(-0.601196\pi\)
−0.312588 + 0.949889i \(0.601196\pi\)
\(810\) 0 0
\(811\) 7.03219i 0.246934i 0.992349 + 0.123467i \(0.0394012\pi\)
−0.992349 + 0.123467i \(0.960599\pi\)
\(812\) 0 0
\(813\) 7.57716i 0.265743i
\(814\) 0 0
\(815\) 5.60110 0.196198
\(816\) 0 0
\(817\) 20.4252i 0.714588i
\(818\) 0 0
\(819\) 7.69381 + 2.41386i 0.268844 + 0.0843471i
\(820\) 0 0
\(821\) 7.24453i 0.252836i 0.991977 + 0.126418i \(0.0403481\pi\)
−0.991977 + 0.126418i \(0.959652\pi\)
\(822\) 0 0
\(823\) 46.4665i 1.61972i −0.586624 0.809860i \(-0.699543\pi\)
0.586624 0.809860i \(-0.300457\pi\)
\(824\) 0 0
\(825\) 0.340475i 0.0118538i
\(826\) 0 0
\(827\) 51.0385 1.77478 0.887392 0.461016i \(-0.152515\pi\)
0.887392 + 0.461016i \(0.152515\pi\)
\(828\) 0 0
\(829\) −14.8970 −0.517393 −0.258696 0.965959i \(-0.583293\pi\)
−0.258696 + 0.965959i \(0.583293\pi\)
\(830\) 0 0
\(831\) 0.326876 0.0113392
\(832\) 0 0
\(833\) 7.31294 10.5073i 0.253379 0.364055i
\(834\) 0 0
\(835\) −9.34952 −0.323554
\(836\) 0 0
\(837\) 15.6987i 0.542626i
\(838\) 0 0
\(839\) −4.53355 −0.156516 −0.0782578 0.996933i \(-0.524936\pi\)
−0.0782578 + 0.996933i \(0.524936\pi\)
\(840\) 0 0
\(841\) −25.8367 −0.890922
\(842\) 0 0
\(843\) 16.0711i 0.553519i
\(844\) 0 0
\(845\) −11.6829 −0.401903
\(846\) 0 0
\(847\) −26.9189 8.44554i −0.924944 0.290192i
\(848\) 0 0
\(849\) −3.07588 −0.105564
\(850\) 0 0
\(851\) 12.1832 0.417635
\(852\) 0 0
\(853\) −39.0584 −1.33734 −0.668668 0.743561i \(-0.733135\pi\)
−0.668668 + 0.743561i \(0.733135\pi\)
\(854\) 0 0
\(855\) 12.5468i 0.429092i
\(856\) 0 0
\(857\) 52.9271i 1.80795i 0.427580 + 0.903977i \(0.359366\pi\)
−0.427580 + 0.903977i \(0.640634\pi\)
\(858\) 0 0
\(859\) 25.8556i 0.882182i 0.897462 + 0.441091i \(0.145408\pi\)
−0.897462 + 0.441091i \(0.854592\pi\)
\(860\) 0 0
\(861\) 12.5367 + 3.93327i 0.427250 + 0.134045i
\(862\) 0 0
\(863\) 31.9510i 1.08762i 0.839207 + 0.543812i \(0.183020\pi\)
−0.839207 + 0.543812i \(0.816980\pi\)
\(864\) 0 0
\(865\) 23.5706 0.801423
\(866\) 0 0
\(867\) 8.01350i 0.272153i
\(868\) 0 0
\(869\) 5.80813i 0.197027i
\(870\) 0 0
\(871\) 8.85783 0.300136
\(872\) 0 0
\(873\) 39.7705i 1.34603i
\(874\) 0 0
\(875\) 2.52442 + 0.792014i 0.0853411 + 0.0267750i
\(876\) 0 0
\(877\) 40.3185i 1.36146i 0.732534 + 0.680730i \(0.238337\pi\)
−0.732534 + 0.680730i \(0.761663\pi\)
\(878\) 0 0
\(879\) 3.90889i 0.131843i
\(880\) 0 0
\(881\) 1.03841i 0.0349848i −0.999847 0.0174924i \(-0.994432\pi\)
0.999847 0.0174924i \(-0.00556829\pi\)
\(882\) 0 0
\(883\) −31.1810 −1.04932 −0.524662 0.851310i \(-0.675808\pi\)
−0.524662 + 0.851310i \(0.675808\pi\)
\(884\) 0 0
\(885\) 7.80238 0.262274
\(886\) 0 0
\(887\) 32.4273 1.08880 0.544402 0.838825i \(-0.316757\pi\)
0.544402 + 0.838825i \(0.316757\pi\)
\(888\) 0 0
\(889\) −12.4428 + 39.6594i −0.417317 + 1.33013i
\(890\) 0 0
\(891\) 3.49229 0.116996
\(892\) 0 0
\(893\) 16.8588i 0.564160i
\(894\) 0 0
\(895\) −19.7086 −0.658787
\(896\) 0 0
\(897\) 1.88549 0.0629548
\(898\) 0 0
\(899\) 35.0270i 1.16822i
\(900\) 0 0
\(901\) 10.0397 0.334472
\(902\) 0 0
\(903\) −2.00931 + 6.40438i −0.0668658 + 0.213124i
\(904\) 0 0
\(905\) −21.0102 −0.698404
\(906\) 0 0
\(907\) −48.3550 −1.60560 −0.802801 0.596248i \(-0.796658\pi\)
−0.802801 + 0.596248i \(0.796658\pi\)
\(908\) 0 0
\(909\) −33.9398 −1.12571
\(910\) 0 0
\(911\) 48.2821i 1.59966i −0.600229 0.799828i \(-0.704924\pi\)
0.600229 0.799828i \(-0.295076\pi\)
\(912\) 0 0
\(913\) 9.06432i 0.299985i
\(914\) 0 0
\(915\) 0.161487i 0.00533860i
\(916\) 0 0
\(917\) 3.97392 12.6663i 0.131231 0.418277i
\(918\) 0 0
\(919\) 6.72784i 0.221931i −0.993824 0.110965i \(-0.964606\pi\)
0.993824 0.110965i \(-0.0353943\pi\)
\(920\) 0 0
\(921\) 10.0247 0.330324
\(922\) 0 0
\(923\) 1.30206i 0.0428579i
\(924\) 0 0
\(925\) 4.35175i 0.143085i
\(926\) 0 0
\(927\) 40.2165 1.32088
\(928\) 0 0
\(929\) 15.7311i 0.516120i −0.966129 0.258060i \(-0.916917\pi\)
0.966129 0.258060i \(-0.0830832\pi\)
\(930\) 0 0
\(931\) 18.8926 27.1450i 0.619180 0.889640i
\(932\) 0 0
\(933\) 17.2952i 0.566220i
\(934\) 0 0
\(935\) 1.06105i 0.0347001i
\(936\) 0 0
\(937\) 32.8060i 1.07172i −0.844306 0.535862i \(-0.819987\pi\)
0.844306 0.535862i \(-0.180013\pi\)
\(938\) 0 0
\(939\) −15.3530 −0.501025
\(940\) 0 0
\(941\) 10.5674 0.344488 0.172244 0.985054i \(-0.444898\pi\)
0.172244 + 0.985054i \(0.444898\pi\)
\(942\) 0 0
\(943\) −23.6921 −0.771520
\(944\) 0 0
\(945\) −2.62862 + 8.37834i −0.0855092 + 0.272548i
\(946\) 0 0
\(947\) −34.2414 −1.11269 −0.556347 0.830950i \(-0.687798\pi\)
−0.556347 + 0.830950i \(0.687798\pi\)
\(948\) 0 0
\(949\) 2.04421i 0.0663580i
\(950\) 0 0
\(951\) −3.36065 −0.108977
\(952\) 0 0
\(953\) −3.54692 −0.114896 −0.0574480 0.998348i \(-0.518296\pi\)
−0.0574480 + 0.998348i \(0.518296\pi\)
\(954\) 0 0
\(955\) 16.9750i 0.549299i
\(956\) 0 0
\(957\) −2.52128 −0.0815013
\(958\) 0 0
\(959\) −37.5388 11.7775i −1.21219 0.380314i
\(960\) 0 0
\(961\) −8.62644 −0.278272
\(962\) 0 0
\(963\) −17.7986 −0.573553
\(964\) 0 0
\(965\) 14.8445 0.477862
\(966\) 0 0
\(967\) 13.7899i 0.443454i 0.975109 + 0.221727i \(0.0711693\pi\)
−0.975109 + 0.221727i \(0.928831\pi\)
\(968\) 0 0
\(969\) 5.07048i 0.162887i
\(970\) 0 0
\(971\) 39.0911i 1.25449i 0.778821 + 0.627246i \(0.215818\pi\)
−0.778821 + 0.627246i \(0.784182\pi\)
\(972\) 0 0
\(973\) 0.117671 0.375059i 0.00377236 0.0120238i
\(974\) 0 0
\(975\) 0.673485i 0.0215688i
\(976\) 0 0
\(977\) −18.1910 −0.581982 −0.290991 0.956726i \(-0.593985\pi\)
−0.290991 + 0.956726i \(0.593985\pi\)
\(978\) 0 0
\(979\) 8.89968i 0.284435i
\(980\) 0 0
\(981\) 29.7835i 0.950913i
\(982\) 0 0
\(983\) −44.1968 −1.40966 −0.704830 0.709376i \(-0.748977\pi\)
−0.704830 + 0.709376i \(0.748977\pi\)
\(984\) 0 0
\(985\) 6.30785i 0.200985i
\(986\) 0 0
\(987\) 1.65847 5.28614i 0.0527898 0.168260i
\(988\) 0 0
\(989\) 12.1031i 0.384857i
\(990\) 0 0
\(991\) 15.9016i 0.505132i 0.967580 + 0.252566i \(0.0812745\pi\)
−0.967580 + 0.252566i \(0.918726\pi\)
\(992\) 0 0
\(993\) 10.3310i 0.327845i
\(994\) 0 0
\(995\) −11.9770 −0.379698
\(996\) 0 0
\(997\) −13.5139 −0.427988 −0.213994 0.976835i \(-0.568647\pi\)
−0.213994 + 0.976835i \(0.568647\pi\)
\(998\) 0 0
\(999\) −14.4431 −0.456959
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 1120.2.h.b.111.10 16
4.3 odd 2 280.2.h.b.251.13 yes 16
7.6 odd 2 1120.2.h.a.111.7 16
8.3 odd 2 1120.2.h.a.111.10 16
8.5 even 2 280.2.h.a.251.14 yes 16
28.27 even 2 280.2.h.a.251.13 16
56.13 odd 2 280.2.h.b.251.14 yes 16
56.27 even 2 inner 1120.2.h.b.111.7 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.h.a.251.13 16 28.27 even 2
280.2.h.a.251.14 yes 16 8.5 even 2
280.2.h.b.251.13 yes 16 4.3 odd 2
280.2.h.b.251.14 yes 16 56.13 odd 2
1120.2.h.a.111.7 16 7.6 odd 2
1120.2.h.a.111.10 16 8.3 odd 2
1120.2.h.b.111.7 16 56.27 even 2 inner
1120.2.h.b.111.10 16 1.1 even 1 trivial