Properties

Label 2352.3.m.k.1471.3
Level $2352$
Weight $3$
Character 2352.1471
Analytic conductor $64.087$
Analytic rank $0$
Dimension $4$
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] = [2352,3,Mod(1471,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.1471");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2352.m (of order \(2\), degree \(1\), 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: \(64.0873581775\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 336)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1471.3
Root \(1.39564 - 0.228425i\) of defining polynomial
Character \(\chi\) \(=\) 2352.1471
Dual form 2352.3.m.k.1471.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} +0.417424 q^{5} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{3} +0.417424 q^{5} -3.00000 q^{9} -18.4249i q^{11} +1.16515 q^{13} +0.723000i q^{15} -0.417424 q^{17} +21.1660i q^{19} -16.9789i q^{23} -24.8258 q^{25} -5.19615i q^{27} +4.33030 q^{29} +20.7846i q^{31} +31.9129 q^{33} -61.1652 q^{37} +2.01810i q^{39} +9.07803 q^{41} -7.30960i q^{43} -1.25227 q^{45} +19.3386i q^{47} -0.723000i q^{51} +92.1561 q^{53} -7.69100i q^{55} -36.6606 q^{57} +99.2036i q^{59} -78.6606 q^{61} +0.486363 q^{65} +77.6562i q^{67} +29.4083 q^{69} +43.9289i q^{71} +53.8258 q^{73} -42.9995i q^{75} +74.7642i q^{79} +9.00000 q^{81} +32.5118i q^{83} -0.174243 q^{85} +7.50030i q^{87} -81.9129 q^{89} -36.0000 q^{93} +8.83521i q^{95} +30.1742 q^{97} +55.2747i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 20 q^{5} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 20 q^{5} - 12 q^{9} - 32 q^{13} - 20 q^{17} + 84 q^{25} - 56 q^{29} + 36 q^{33} - 208 q^{37} - 92 q^{41} - 60 q^{45} + 112 q^{53} - 168 q^{61} - 328 q^{65} - 84 q^{69} + 32 q^{73} + 36 q^{81} - 184 q^{85} - 236 q^{89} - 144 q^{93} + 304 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) 1.73205i 0.577350i
\(4\) 0 0
\(5\) 0.417424 0.0834849 0.0417424 0.999128i \(-0.486709\pi\)
0.0417424 + 0.999128i \(0.486709\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) − 18.4249i − 1.67499i −0.546444 0.837496i \(-0.684019\pi\)
0.546444 0.837496i \(-0.315981\pi\)
\(12\) 0 0
\(13\) 1.16515 0.0896270 0.0448135 0.998995i \(-0.485731\pi\)
0.0448135 + 0.998995i \(0.485731\pi\)
\(14\) 0 0
\(15\) 0.723000i 0.0482000i
\(16\) 0 0
\(17\) −0.417424 −0.0245544 −0.0122772 0.999925i \(-0.503908\pi\)
−0.0122772 + 0.999925i \(0.503908\pi\)
\(18\) 0 0
\(19\) 21.1660i 1.11400i 0.830512 + 0.557000i \(0.188048\pi\)
−0.830512 + 0.557000i \(0.811952\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 16.9789i − 0.738213i −0.929387 0.369107i \(-0.879664\pi\)
0.929387 0.369107i \(-0.120336\pi\)
\(24\) 0 0
\(25\) −24.8258 −0.993030
\(26\) 0 0
\(27\) − 5.19615i − 0.192450i
\(28\) 0 0
\(29\) 4.33030 0.149321 0.0746604 0.997209i \(-0.476213\pi\)
0.0746604 + 0.997209i \(0.476213\pi\)
\(30\) 0 0
\(31\) 20.7846i 0.670471i 0.942134 + 0.335236i \(0.108816\pi\)
−0.942134 + 0.335236i \(0.891184\pi\)
\(32\) 0 0
\(33\) 31.9129 0.967057
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −61.1652 −1.65311 −0.826556 0.562854i \(-0.809703\pi\)
−0.826556 + 0.562854i \(0.809703\pi\)
\(38\) 0 0
\(39\) 2.01810i 0.0517462i
\(40\) 0 0
\(41\) 9.07803 0.221415 0.110708 0.993853i \(-0.464688\pi\)
0.110708 + 0.993853i \(0.464688\pi\)
\(42\) 0 0
\(43\) − 7.30960i − 0.169991i −0.996381 0.0849954i \(-0.972912\pi\)
0.996381 0.0849954i \(-0.0270876\pi\)
\(44\) 0 0
\(45\) −1.25227 −0.0278283
\(46\) 0 0
\(47\) 19.3386i 0.411460i 0.978609 + 0.205730i \(0.0659568\pi\)
−0.978609 + 0.205730i \(0.934043\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) − 0.723000i − 0.0141765i
\(52\) 0 0
\(53\) 92.1561 1.73879 0.869397 0.494115i \(-0.164508\pi\)
0.869397 + 0.494115i \(0.164508\pi\)
\(54\) 0 0
\(55\) − 7.69100i − 0.139836i
\(56\) 0 0
\(57\) −36.6606 −0.643169
\(58\) 0 0
\(59\) 99.2036i 1.68142i 0.541487 + 0.840709i \(0.317861\pi\)
−0.541487 + 0.840709i \(0.682139\pi\)
\(60\) 0 0
\(61\) −78.6606 −1.28952 −0.644759 0.764386i \(-0.723042\pi\)
−0.644759 + 0.764386i \(0.723042\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.486363 0.00748250
\(66\) 0 0
\(67\) 77.6562i 1.15905i 0.814955 + 0.579524i \(0.196762\pi\)
−0.814955 + 0.579524i \(0.803238\pi\)
\(68\) 0 0
\(69\) 29.4083 0.426208
\(70\) 0 0
\(71\) 43.9289i 0.618717i 0.950945 + 0.309359i \(0.100114\pi\)
−0.950945 + 0.309359i \(0.899886\pi\)
\(72\) 0 0
\(73\) 53.8258 0.737339 0.368670 0.929561i \(-0.379813\pi\)
0.368670 + 0.929561i \(0.379813\pi\)
\(74\) 0 0
\(75\) − 42.9995i − 0.573326i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 74.7642i 0.946383i 0.880960 + 0.473191i \(0.156898\pi\)
−0.880960 + 0.473191i \(0.843102\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 32.5118i 0.391709i 0.980633 + 0.195854i \(0.0627480\pi\)
−0.980633 + 0.195854i \(0.937252\pi\)
\(84\) 0 0
\(85\) −0.174243 −0.00204992
\(86\) 0 0
\(87\) 7.50030i 0.0862104i
\(88\) 0 0
\(89\) −81.9129 −0.920369 −0.460185 0.887823i \(-0.652217\pi\)
−0.460185 + 0.887823i \(0.652217\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −36.0000 −0.387097
\(94\) 0 0
\(95\) 8.83521i 0.0930022i
\(96\) 0 0
\(97\) 30.1742 0.311075 0.155537 0.987830i \(-0.450289\pi\)
0.155537 + 0.987830i \(0.450289\pi\)
\(98\) 0 0
\(99\) 55.2747i 0.558331i
\(100\) 0 0
\(101\) −183.060 −1.81247 −0.906237 0.422770i \(-0.861058\pi\)
−0.906237 + 0.422770i \(0.861058\pi\)
\(102\) 0 0
\(103\) 30.9862i 0.300837i 0.988622 + 0.150419i \(0.0480621\pi\)
−0.988622 + 0.150419i \(0.951938\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 118.010i 1.10290i 0.834209 + 0.551448i \(0.185925\pi\)
−0.834209 + 0.551448i \(0.814075\pi\)
\(108\) 0 0
\(109\) 85.3394 0.782930 0.391465 0.920193i \(-0.371968\pi\)
0.391465 + 0.920193i \(0.371968\pi\)
\(110\) 0 0
\(111\) − 105.941i − 0.954425i
\(112\) 0 0
\(113\) 70.6606 0.625315 0.312658 0.949866i \(-0.398781\pi\)
0.312658 + 0.949866i \(0.398781\pi\)
\(114\) 0 0
\(115\) − 7.08741i − 0.0616296i
\(116\) 0 0
\(117\) −3.49545 −0.0298757
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −218.477 −1.80560
\(122\) 0 0
\(123\) 15.7236i 0.127834i
\(124\) 0 0
\(125\) −20.7985 −0.166388
\(126\) 0 0
\(127\) 133.241i 1.04914i 0.851367 + 0.524571i \(0.175774\pi\)
−0.851367 + 0.524571i \(0.824226\pi\)
\(128\) 0 0
\(129\) 12.6606 0.0981442
\(130\) 0 0
\(131\) 41.1878i 0.314411i 0.987566 + 0.157205i \(0.0502485\pi\)
−0.987566 + 0.157205i \(0.949752\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) − 2.16900i − 0.0160667i
\(136\) 0 0
\(137\) −52.5045 −0.383245 −0.191622 0.981469i \(-0.561375\pi\)
−0.191622 + 0.981469i \(0.561375\pi\)
\(138\) 0 0
\(139\) − 265.179i − 1.90776i −0.300187 0.953880i \(-0.597049\pi\)
0.300187 0.953880i \(-0.402951\pi\)
\(140\) 0 0
\(141\) −33.4955 −0.237556
\(142\) 0 0
\(143\) − 21.4678i − 0.150125i
\(144\) 0 0
\(145\) 1.80757 0.0124660
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −121.826 −0.817623 −0.408811 0.912619i \(-0.634057\pi\)
−0.408811 + 0.912619i \(0.634057\pi\)
\(150\) 0 0
\(151\) − 142.378i − 0.942901i −0.881892 0.471451i \(-0.843731\pi\)
0.881892 0.471451i \(-0.156269\pi\)
\(152\) 0 0
\(153\) 1.25227 0.00818479
\(154\) 0 0
\(155\) 8.67600i 0.0559742i
\(156\) 0 0
\(157\) −241.652 −1.53918 −0.769591 0.638537i \(-0.779540\pi\)
−0.769591 + 0.638537i \(0.779540\pi\)
\(158\) 0 0
\(159\) 159.619i 1.00389i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 32.4322i 0.198971i 0.995039 + 0.0994853i \(0.0317196\pi\)
−0.995039 + 0.0994853i \(0.968280\pi\)
\(164\) 0 0
\(165\) 13.3212 0.0807346
\(166\) 0 0
\(167\) 28.7774i 0.172320i 0.996281 + 0.0861599i \(0.0274596\pi\)
−0.996281 + 0.0861599i \(0.972540\pi\)
\(168\) 0 0
\(169\) −167.642 −0.991967
\(170\) 0 0
\(171\) − 63.4980i − 0.371334i
\(172\) 0 0
\(173\) −210.399 −1.21618 −0.608090 0.793868i \(-0.708064\pi\)
−0.608090 + 0.793868i \(0.708064\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −171.826 −0.970767
\(178\) 0 0
\(179\) − 286.575i − 1.60098i −0.599347 0.800489i \(-0.704573\pi\)
0.599347 0.800489i \(-0.295427\pi\)
\(180\) 0 0
\(181\) 182.174 1.00649 0.503244 0.864144i \(-0.332140\pi\)
0.503244 + 0.864144i \(0.332140\pi\)
\(182\) 0 0
\(183\) − 136.244i − 0.744504i
\(184\) 0 0
\(185\) −25.5318 −0.138010
\(186\) 0 0
\(187\) 7.69100i 0.0411284i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 192.695i 1.00887i 0.863449 + 0.504436i \(0.168300\pi\)
−0.863449 + 0.504436i \(0.831700\pi\)
\(192\) 0 0
\(193\) 141.826 0.734848 0.367424 0.930053i \(-0.380240\pi\)
0.367424 + 0.930053i \(0.380240\pi\)
\(194\) 0 0
\(195\) 0.842405i 0.00432002i
\(196\) 0 0
\(197\) 31.8439 0.161644 0.0808222 0.996729i \(-0.474245\pi\)
0.0808222 + 0.996729i \(0.474245\pi\)
\(198\) 0 0
\(199\) 146.796i 0.737667i 0.929496 + 0.368833i \(0.120243\pi\)
−0.929496 + 0.368833i \(0.879757\pi\)
\(200\) 0 0
\(201\) −134.505 −0.669177
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 3.78939 0.0184848
\(206\) 0 0
\(207\) 50.9367i 0.246071i
\(208\) 0 0
\(209\) 389.982 1.86594
\(210\) 0 0
\(211\) − 274.999i − 1.30331i −0.758514 0.651656i \(-0.774074\pi\)
0.758514 0.651656i \(-0.225926\pi\)
\(212\) 0 0
\(213\) −76.0871 −0.357217
\(214\) 0 0
\(215\) − 3.05121i − 0.0141917i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 93.2289i 0.425703i
\(220\) 0 0
\(221\) −0.486363 −0.00220074
\(222\) 0 0
\(223\) 257.647i 1.15537i 0.816261 + 0.577684i \(0.196043\pi\)
−0.816261 + 0.577684i \(0.803957\pi\)
\(224\) 0 0
\(225\) 74.4773 0.331010
\(226\) 0 0
\(227\) 291.064i 1.28222i 0.767449 + 0.641110i \(0.221526\pi\)
−0.767449 + 0.641110i \(0.778474\pi\)
\(228\) 0 0
\(229\) 227.459 0.993271 0.496636 0.867959i \(-0.334569\pi\)
0.496636 + 0.867959i \(0.334569\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −442.138 −1.89759 −0.948794 0.315896i \(-0.897695\pi\)
−0.948794 + 0.315896i \(0.897695\pi\)
\(234\) 0 0
\(235\) 8.07241i 0.0343507i
\(236\) 0 0
\(237\) −129.495 −0.546394
\(238\) 0 0
\(239\) 182.191i 0.762306i 0.924512 + 0.381153i \(0.124473\pi\)
−0.924512 + 0.381153i \(0.875527\pi\)
\(240\) 0 0
\(241\) −354.450 −1.47075 −0.735373 0.677662i \(-0.762993\pi\)
−0.735373 + 0.677662i \(0.762993\pi\)
\(242\) 0 0
\(243\) 15.5885i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 24.6616i 0.0998446i
\(248\) 0 0
\(249\) −56.3121 −0.226153
\(250\) 0 0
\(251\) 388.822i 1.54909i 0.632518 + 0.774545i \(0.282021\pi\)
−0.632518 + 0.774545i \(0.717979\pi\)
\(252\) 0 0
\(253\) −312.835 −1.23650
\(254\) 0 0
\(255\) − 0.301798i − 0.00118352i
\(256\) 0 0
\(257\) −302.922 −1.17868 −0.589342 0.807883i \(-0.700613\pi\)
−0.589342 + 0.807883i \(0.700613\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −12.9909 −0.0497736
\(262\) 0 0
\(263\) 46.6783i 0.177484i 0.996055 + 0.0887421i \(0.0282847\pi\)
−0.996055 + 0.0887421i \(0.971715\pi\)
\(264\) 0 0
\(265\) 38.4682 0.145163
\(266\) 0 0
\(267\) − 141.877i − 0.531376i
\(268\) 0 0
\(269\) 260.033 0.966664 0.483332 0.875437i \(-0.339426\pi\)
0.483332 + 0.875437i \(0.339426\pi\)
\(270\) 0 0
\(271\) 115.650i 0.426754i 0.976970 + 0.213377i \(0.0684462\pi\)
−0.976970 + 0.213377i \(0.931554\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 457.412i 1.66332i
\(276\) 0 0
\(277\) 221.826 0.800815 0.400408 0.916337i \(-0.368869\pi\)
0.400408 + 0.916337i \(0.368869\pi\)
\(278\) 0 0
\(279\) − 62.3538i − 0.223490i
\(280\) 0 0
\(281\) −376.468 −1.33974 −0.669872 0.742476i \(-0.733651\pi\)
−0.669872 + 0.742476i \(0.733651\pi\)
\(282\) 0 0
\(283\) 27.7128i 0.0979251i 0.998801 + 0.0489626i \(0.0155915\pi\)
−0.998801 + 0.0489626i \(0.984408\pi\)
\(284\) 0 0
\(285\) −15.3030 −0.0536948
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −288.826 −0.999397
\(290\) 0 0
\(291\) 52.2633i 0.179599i
\(292\) 0 0
\(293\) 547.161 1.86744 0.933722 0.357998i \(-0.116540\pi\)
0.933722 + 0.357998i \(0.116540\pi\)
\(294\) 0 0
\(295\) 41.4100i 0.140373i
\(296\) 0 0
\(297\) −95.7386 −0.322352
\(298\) 0 0
\(299\) − 19.7830i − 0.0661639i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 317.069i − 1.04643i
\(304\) 0 0
\(305\) −32.8348 −0.107655
\(306\) 0 0
\(307\) − 564.013i − 1.83718i −0.395215 0.918589i \(-0.629330\pi\)
0.395215 0.918589i \(-0.370670\pi\)
\(308\) 0 0
\(309\) −53.6697 −0.173688
\(310\) 0 0
\(311\) − 397.435i − 1.27793i −0.769238 0.638963i \(-0.779364\pi\)
0.769238 0.638963i \(-0.220636\pi\)
\(312\) 0 0
\(313\) −519.909 −1.66105 −0.830526 0.556980i \(-0.811960\pi\)
−0.830526 + 0.556980i \(0.811960\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 100.156 0.315950 0.157975 0.987443i \(-0.449504\pi\)
0.157975 + 0.987443i \(0.449504\pi\)
\(318\) 0 0
\(319\) − 79.7854i − 0.250111i
\(320\) 0 0
\(321\) −204.399 −0.636758
\(322\) 0 0
\(323\) − 8.83521i − 0.0273536i
\(324\) 0 0
\(325\) −28.9258 −0.0890024
\(326\) 0 0
\(327\) 147.812i 0.452025i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 72.9368i − 0.220353i −0.993912 0.110176i \(-0.964858\pi\)
0.993912 0.110176i \(-0.0351416\pi\)
\(332\) 0 0
\(333\) 183.495 0.551037
\(334\) 0 0
\(335\) 32.4156i 0.0967630i
\(336\) 0 0
\(337\) 358.936 1.06509 0.532547 0.846401i \(-0.321235\pi\)
0.532547 + 0.846401i \(0.321235\pi\)
\(338\) 0 0
\(339\) 122.388i 0.361026i
\(340\) 0 0
\(341\) 382.955 1.12303
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 12.2758 0.0355819
\(346\) 0 0
\(347\) − 357.827i − 1.03120i −0.856829 0.515601i \(-0.827569\pi\)
0.856829 0.515601i \(-0.172431\pi\)
\(348\) 0 0
\(349\) −482.624 −1.38288 −0.691439 0.722435i \(-0.743023\pi\)
−0.691439 + 0.722435i \(0.743023\pi\)
\(350\) 0 0
\(351\) − 6.05430i − 0.0172487i
\(352\) 0 0
\(353\) −535.023 −1.51565 −0.757824 0.652459i \(-0.773737\pi\)
−0.757824 + 0.652459i \(0.773737\pi\)
\(354\) 0 0
\(355\) 18.3370i 0.0516535i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 572.920i 1.59588i 0.602739 + 0.797939i \(0.294076\pi\)
−0.602739 + 0.797939i \(0.705924\pi\)
\(360\) 0 0
\(361\) −87.0000 −0.240997
\(362\) 0 0
\(363\) − 378.414i − 1.04246i
\(364\) 0 0
\(365\) 22.4682 0.0615567
\(366\) 0 0
\(367\) − 272.107i − 0.741436i −0.928746 0.370718i \(-0.879112\pi\)
0.928746 0.370718i \(-0.120888\pi\)
\(368\) 0 0
\(369\) −27.2341 −0.0738051
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −297.339 −0.797157 −0.398578 0.917134i \(-0.630496\pi\)
−0.398578 + 0.917134i \(0.630496\pi\)
\(374\) 0 0
\(375\) − 36.0240i − 0.0960641i
\(376\) 0 0
\(377\) 5.04546 0.0133832
\(378\) 0 0
\(379\) − 147.240i − 0.388496i −0.980952 0.194248i \(-0.937773\pi\)
0.980952 0.194248i \(-0.0622267\pi\)
\(380\) 0 0
\(381\) −230.780 −0.605723
\(382\) 0 0
\(383\) − 339.101i − 0.885380i −0.896675 0.442690i \(-0.854024\pi\)
0.896675 0.442690i \(-0.145976\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 21.9288i 0.0566636i
\(388\) 0 0
\(389\) −457.579 −1.17630 −0.588148 0.808754i \(-0.700143\pi\)
−0.588148 + 0.808754i \(0.700143\pi\)
\(390\) 0 0
\(391\) 7.08741i 0.0181264i
\(392\) 0 0
\(393\) −71.3394 −0.181525
\(394\) 0 0
\(395\) 31.2084i 0.0790086i
\(396\) 0 0
\(397\) −174.312 −0.439073 −0.219537 0.975604i \(-0.570455\pi\)
−0.219537 + 0.975604i \(0.570455\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 260.468 0.649547 0.324773 0.945792i \(-0.394712\pi\)
0.324773 + 0.945792i \(0.394712\pi\)
\(402\) 0 0
\(403\) 24.2172i 0.0600923i
\(404\) 0 0
\(405\) 3.75682 0.00927610
\(406\) 0 0
\(407\) 1126.96i 2.76895i
\(408\) 0 0
\(409\) 134.174 0.328054 0.164027 0.986456i \(-0.447552\pi\)
0.164027 + 0.986456i \(0.447552\pi\)
\(410\) 0 0
\(411\) − 90.9405i − 0.221267i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 13.5712i 0.0327017i
\(416\) 0 0
\(417\) 459.303 1.10145
\(418\) 0 0
\(419\) 342.835i 0.818222i 0.912485 + 0.409111i \(0.134161\pi\)
−0.912485 + 0.409111i \(0.865839\pi\)
\(420\) 0 0
\(421\) 186.486 0.442960 0.221480 0.975165i \(-0.428911\pi\)
0.221480 + 0.975165i \(0.428911\pi\)
\(422\) 0 0
\(423\) − 58.0158i − 0.137153i
\(424\) 0 0
\(425\) 10.3629 0.0243832
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 37.1833 0.0866744
\(430\) 0 0
\(431\) 114.577i 0.265841i 0.991127 + 0.132920i \(0.0424355\pi\)
−0.991127 + 0.132920i \(0.957565\pi\)
\(432\) 0 0
\(433\) −289.027 −0.667499 −0.333750 0.942662i \(-0.608314\pi\)
−0.333750 + 0.942662i \(0.608314\pi\)
\(434\) 0 0
\(435\) 3.13081i 0.00719726i
\(436\) 0 0
\(437\) 359.376 0.822370
\(438\) 0 0
\(439\) 730.894i 1.66491i 0.554095 + 0.832453i \(0.313064\pi\)
−0.554095 + 0.832453i \(0.686936\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 186.848i − 0.421778i −0.977510 0.210889i \(-0.932364\pi\)
0.977510 0.210889i \(-0.0676358\pi\)
\(444\) 0 0
\(445\) −34.1924 −0.0768369
\(446\) 0 0
\(447\) − 211.008i − 0.472055i
\(448\) 0 0
\(449\) 673.652 1.50034 0.750169 0.661246i \(-0.229972\pi\)
0.750169 + 0.661246i \(0.229972\pi\)
\(450\) 0 0
\(451\) − 167.262i − 0.370869i
\(452\) 0 0
\(453\) 246.606 0.544384
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 199.982 0.437597 0.218798 0.975770i \(-0.429786\pi\)
0.218798 + 0.975770i \(0.429786\pi\)
\(458\) 0 0
\(459\) 2.16900i 0.00472549i
\(460\) 0 0
\(461\) 348.904 0.756841 0.378421 0.925634i \(-0.376467\pi\)
0.378421 + 0.925634i \(0.376467\pi\)
\(462\) 0 0
\(463\) − 5.80061i − 0.0125283i −0.999980 0.00626416i \(-0.998006\pi\)
0.999980 0.00626416i \(-0.00199396\pi\)
\(464\) 0 0
\(465\) −15.0273 −0.0323167
\(466\) 0 0
\(467\) − 756.716i − 1.62038i −0.586169 0.810189i \(-0.699365\pi\)
0.586169 0.810189i \(-0.300635\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) − 418.553i − 0.888647i
\(472\) 0 0
\(473\) −134.679 −0.284733
\(474\) 0 0
\(475\) − 525.462i − 1.10624i
\(476\) 0 0
\(477\) −276.468 −0.579598
\(478\) 0 0
\(479\) 810.076i 1.69118i 0.533832 + 0.845591i \(0.320751\pi\)
−0.533832 + 0.845591i \(0.679249\pi\)
\(480\) 0 0
\(481\) −71.2667 −0.148164
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.5955 0.0259700
\(486\) 0 0
\(487\) 739.888i 1.51928i 0.650345 + 0.759639i \(0.274624\pi\)
−0.650345 + 0.759639i \(0.725376\pi\)
\(488\) 0 0
\(489\) −56.1742 −0.114876
\(490\) 0 0
\(491\) − 332.244i − 0.676667i −0.941026 0.338334i \(-0.890137\pi\)
0.941026 0.338334i \(-0.109863\pi\)
\(492\) 0 0
\(493\) −1.80757 −0.00366648
\(494\) 0 0
\(495\) 23.0730i 0.0466122i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 156.075i 0.312776i 0.987696 + 0.156388i \(0.0499850\pi\)
−0.987696 + 0.156388i \(0.950015\pi\)
\(500\) 0 0
\(501\) −49.8439 −0.0994889
\(502\) 0 0
\(503\) 812.348i 1.61501i 0.589864 + 0.807503i \(0.299181\pi\)
−0.589864 + 0.807503i \(0.700819\pi\)
\(504\) 0 0
\(505\) −76.4136 −0.151314
\(506\) 0 0
\(507\) − 290.365i − 0.572712i
\(508\) 0 0
\(509\) 261.078 0.512923 0.256462 0.966554i \(-0.417443\pi\)
0.256462 + 0.966554i \(0.417443\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 109.982 0.214390
\(514\) 0 0
\(515\) 12.9344i 0.0251153i
\(516\) 0 0
\(517\) 356.312 0.689192
\(518\) 0 0
\(519\) − 364.422i − 0.702162i
\(520\) 0 0
\(521\) 752.831 1.44497 0.722487 0.691385i \(-0.242999\pi\)
0.722487 + 0.691385i \(0.242999\pi\)
\(522\) 0 0
\(523\) − 282.531i − 0.540212i −0.962831 0.270106i \(-0.912941\pi\)
0.962831 0.270106i \(-0.0870587\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 8.67600i − 0.0164630i
\(528\) 0 0
\(529\) 240.717 0.455041
\(530\) 0 0
\(531\) − 297.611i − 0.560473i
\(532\) 0 0
\(533\) 10.5773 0.0198448
\(534\) 0 0
\(535\) 49.2602i 0.0920752i
\(536\) 0 0
\(537\) 496.363 0.924326
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −254.138 −0.469756 −0.234878 0.972025i \(-0.575469\pi\)
−0.234878 + 0.972025i \(0.575469\pi\)
\(542\) 0 0
\(543\) 315.535i 0.581096i
\(544\) 0 0
\(545\) 35.6227 0.0653628
\(546\) 0 0
\(547\) − 696.777i − 1.27382i −0.770940 0.636908i \(-0.780213\pi\)
0.770940 0.636908i \(-0.219787\pi\)
\(548\) 0 0
\(549\) 235.982 0.429839
\(550\) 0 0
\(551\) 91.6552i 0.166343i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) − 44.2224i − 0.0796800i
\(556\) 0 0
\(557\) 640.395 1.14972 0.574861 0.818251i \(-0.305056\pi\)
0.574861 + 0.818251i \(0.305056\pi\)
\(558\) 0 0
\(559\) − 8.51680i − 0.0152358i
\(560\) 0 0
\(561\) −13.3212 −0.0237455
\(562\) 0 0
\(563\) 734.025i 1.30377i 0.758316 + 0.651887i \(0.226022\pi\)
−0.758316 + 0.651887i \(0.773978\pi\)
\(564\) 0 0
\(565\) 29.4955 0.0522043
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1129.13 1.98441 0.992205 0.124620i \(-0.0397711\pi\)
0.992205 + 0.124620i \(0.0397711\pi\)
\(570\) 0 0
\(571\) 345.664i 0.605366i 0.953091 + 0.302683i \(0.0978823\pi\)
−0.953091 + 0.302683i \(0.902118\pi\)
\(572\) 0 0
\(573\) −333.757 −0.582473
\(574\) 0 0
\(575\) 421.514i 0.733068i
\(576\) 0 0
\(577\) 820.330 1.42172 0.710858 0.703335i \(-0.248307\pi\)
0.710858 + 0.703335i \(0.248307\pi\)
\(578\) 0 0
\(579\) 245.649i 0.424265i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 1697.97i − 2.91246i
\(584\) 0 0
\(585\) −1.45909 −0.00249417
\(586\) 0 0
\(587\) 736.154i 1.25410i 0.778981 + 0.627048i \(0.215737\pi\)
−0.778981 + 0.627048i \(0.784263\pi\)
\(588\) 0 0
\(589\) −439.927 −0.746905
\(590\) 0 0
\(591\) 55.1553i 0.0933254i
\(592\) 0 0
\(593\) −381.354 −0.643092 −0.321546 0.946894i \(-0.604203\pi\)
−0.321546 + 0.946894i \(0.604203\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −254.258 −0.425892
\(598\) 0 0
\(599\) − 539.281i − 0.900301i −0.892953 0.450151i \(-0.851370\pi\)
0.892953 0.450151i \(-0.148630\pi\)
\(600\) 0 0
\(601\) −938.276 −1.56119 −0.780595 0.625037i \(-0.785084\pi\)
−0.780595 + 0.625037i \(0.785084\pi\)
\(602\) 0 0
\(603\) − 232.969i − 0.386349i
\(604\) 0 0
\(605\) −91.1977 −0.150740
\(606\) 0 0
\(607\) 274.077i 0.451527i 0.974182 + 0.225764i \(0.0724877\pi\)
−0.974182 + 0.225764i \(0.927512\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 22.5324i 0.0368779i
\(612\) 0 0
\(613\) −907.836 −1.48097 −0.740486 0.672071i \(-0.765405\pi\)
−0.740486 + 0.672071i \(0.765405\pi\)
\(614\) 0 0
\(615\) 6.56342i 0.0106722i
\(616\) 0 0
\(617\) 289.753 0.469616 0.234808 0.972042i \(-0.424554\pi\)
0.234808 + 0.972042i \(0.424554\pi\)
\(618\) 0 0
\(619\) 908.391i 1.46751i 0.679412 + 0.733757i \(0.262235\pi\)
−0.679412 + 0.733757i \(0.737765\pi\)
\(620\) 0 0
\(621\) −88.2250 −0.142069
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 611.962 0.979139
\(626\) 0 0
\(627\) 675.468i 1.07730i
\(628\) 0 0
\(629\) 25.5318 0.0405911
\(630\) 0 0
\(631\) 437.319i 0.693057i 0.938039 + 0.346529i \(0.112640\pi\)
−0.938039 + 0.346529i \(0.887360\pi\)
\(632\) 0 0
\(633\) 476.312 0.752468
\(634\) 0 0
\(635\) 55.6181i 0.0875875i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) − 131.787i − 0.206239i
\(640\) 0 0
\(641\) −766.450 −1.19571 −0.597855 0.801604i \(-0.703980\pi\)
−0.597855 + 0.801604i \(0.703980\pi\)
\(642\) 0 0
\(643\) 191.257i 0.297445i 0.988879 + 0.148722i \(0.0475161\pi\)
−0.988879 + 0.148722i \(0.952484\pi\)
\(644\) 0 0
\(645\) 5.28484 0.00819356
\(646\) 0 0
\(647\) − 1017.92i − 1.57330i −0.617402 0.786648i \(-0.711815\pi\)
0.617402 0.786648i \(-0.288185\pi\)
\(648\) 0 0
\(649\) 1827.82 2.81636
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 548.330 0.839709 0.419855 0.907591i \(-0.362081\pi\)
0.419855 + 0.907591i \(0.362081\pi\)
\(654\) 0 0
\(655\) 17.1928i 0.0262485i
\(656\) 0 0
\(657\) −161.477 −0.245780
\(658\) 0 0
\(659\) − 569.789i − 0.864627i −0.901723 0.432313i \(-0.857697\pi\)
0.901723 0.432313i \(-0.142303\pi\)
\(660\) 0 0
\(661\) 171.633 0.259657 0.129829 0.991536i \(-0.458557\pi\)
0.129829 + 0.991536i \(0.458557\pi\)
\(662\) 0 0
\(663\) − 0.842405i − 0.00127060i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 73.5238i − 0.110231i
\(668\) 0 0
\(669\) −446.258 −0.667052
\(670\) 0 0
\(671\) 1449.31i 2.15993i
\(672\) 0 0
\(673\) 279.423 0.415190 0.207595 0.978215i \(-0.433436\pi\)
0.207595 + 0.978215i \(0.433436\pi\)
\(674\) 0 0
\(675\) 128.998i 0.191109i
\(676\) 0 0
\(677\) −719.960 −1.06346 −0.531728 0.846915i \(-0.678457\pi\)
−0.531728 + 0.846915i \(0.678457\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −504.138 −0.740291
\(682\) 0 0
\(683\) − 1066.95i − 1.56215i −0.624434 0.781077i \(-0.714670\pi\)
0.624434 0.781077i \(-0.285330\pi\)
\(684\) 0 0
\(685\) −21.9167 −0.0319951
\(686\) 0 0
\(687\) 393.971i 0.573465i
\(688\) 0 0
\(689\) 107.376 0.155843
\(690\) 0 0
\(691\) − 716.497i − 1.03690i −0.855108 0.518449i \(-0.826509\pi\)
0.855108 0.518449i \(-0.173491\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 110.692i − 0.159269i
\(696\) 0 0
\(697\) −3.78939 −0.00543671
\(698\) 0 0
\(699\) − 765.805i − 1.09557i
\(700\) 0 0
\(701\) 1080.02 1.54068 0.770341 0.637632i \(-0.220086\pi\)
0.770341 + 0.637632i \(0.220086\pi\)
\(702\) 0 0
\(703\) − 1294.62i − 1.84157i
\(704\) 0 0
\(705\) −13.9818 −0.0198324
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 773.891 1.09152 0.545762 0.837940i \(-0.316240\pi\)
0.545762 + 0.837940i \(0.316240\pi\)
\(710\) 0 0
\(711\) − 224.293i − 0.315461i
\(712\) 0 0
\(713\) 352.900 0.494951
\(714\) 0 0
\(715\) − 8.96118i − 0.0125331i
\(716\) 0 0
\(717\) −315.564 −0.440118
\(718\) 0 0
\(719\) 680.204i 0.946042i 0.881051 + 0.473021i \(0.156837\pi\)
−0.881051 + 0.473021i \(0.843163\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) − 613.925i − 0.849136i
\(724\) 0 0
\(725\) −107.503 −0.148280
\(726\) 0 0
\(727\) − 984.982i − 1.35486i −0.735588 0.677429i \(-0.763094\pi\)
0.735588 0.677429i \(-0.236906\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) 3.05121i 0.00417402i
\(732\) 0 0
\(733\) 893.056 1.21836 0.609179 0.793033i \(-0.291499\pi\)
0.609179 + 0.793033i \(0.291499\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1430.81 1.94140
\(738\) 0 0
\(739\) − 467.035i − 0.631983i −0.948762 0.315991i \(-0.897663\pi\)
0.948762 0.315991i \(-0.102337\pi\)
\(740\) 0 0
\(741\) −42.7152 −0.0576453
\(742\) 0 0
\(743\) 1012.07i 1.36214i 0.732220 + 0.681068i \(0.238484\pi\)
−0.732220 + 0.681068i \(0.761516\pi\)
\(744\) 0 0
\(745\) −50.8530 −0.0682591
\(746\) 0 0
\(747\) − 97.5355i − 0.130570i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 450.715i − 0.600153i −0.953915 0.300076i \(-0.902988\pi\)
0.953915 0.300076i \(-0.0970121\pi\)
\(752\) 0 0
\(753\) −673.459 −0.894368
\(754\) 0 0
\(755\) − 59.4321i − 0.0787180i
\(756\) 0 0
\(757\) −429.303 −0.567111 −0.283556 0.958956i \(-0.591514\pi\)
−0.283556 + 0.958956i \(0.591514\pi\)
\(758\) 0 0
\(759\) − 541.846i − 0.713894i
\(760\) 0 0
\(761\) 341.390 0.448607 0.224304 0.974519i \(-0.427989\pi\)
0.224304 + 0.974519i \(0.427989\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.522729 0.000683306 0
\(766\) 0 0
\(767\) 115.587i 0.150700i
\(768\) 0 0
\(769\) −181.064 −0.235453 −0.117727 0.993046i \(-0.537561\pi\)
−0.117727 + 0.993046i \(0.537561\pi\)
\(770\) 0 0
\(771\) − 524.676i − 0.680514i
\(772\) 0 0
\(773\) 1211.23 1.56693 0.783463 0.621438i \(-0.213451\pi\)
0.783463 + 0.621438i \(0.213451\pi\)
\(774\) 0 0
\(775\) − 515.994i − 0.665798i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 192.146i 0.246657i
\(780\) 0 0
\(781\) 809.386 1.03635
\(782\) 0 0
\(783\) − 22.5009i − 0.0287368i
\(784\) 0 0
\(785\) −100.871 −0.128498
\(786\) 0 0
\(787\) − 1070.85i − 1.36068i −0.732898 0.680339i \(-0.761833\pi\)
0.732898 0.680339i \(-0.238167\pi\)
\(788\) 0 0
\(789\) −80.8492 −0.102471
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −91.6515 −0.115576
\(794\) 0 0
\(795\) 66.6288i 0.0838099i
\(796\) 0 0
\(797\) −1498.33 −1.87996 −0.939979 0.341232i \(-0.889156\pi\)
−0.939979 + 0.341232i \(0.889156\pi\)
\(798\) 0 0
\(799\) − 8.07241i − 0.0101031i
\(800\) 0 0
\(801\) 245.739 0.306790
\(802\) 0 0
\(803\) − 991.735i − 1.23504i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 450.390i 0.558104i
\(808\) 0 0
\(809\) −416.221 −0.514489 −0.257244 0.966346i \(-0.582814\pi\)
−0.257244 + 0.966346i \(0.582814\pi\)
\(810\) 0 0
\(811\) − 326.103i − 0.402100i −0.979581 0.201050i \(-0.935565\pi\)
0.979581 0.201050i \(-0.0644354\pi\)
\(812\) 0 0
\(813\) −200.312 −0.246386
\(814\) 0 0
\(815\) 13.5380i 0.0166110i
\(816\) 0 0
\(817\) 154.715 0.189370
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 952.468 1.16013 0.580066 0.814570i \(-0.303027\pi\)
0.580066 + 0.814570i \(0.303027\pi\)
\(822\) 0 0
\(823\) 486.483i 0.591110i 0.955326 + 0.295555i \(0.0955045\pi\)
−0.955326 + 0.295555i \(0.904496\pi\)
\(824\) 0 0
\(825\) −792.261 −0.960317
\(826\) 0 0
\(827\) − 542.110i − 0.655513i −0.944762 0.327757i \(-0.893707\pi\)
0.944762 0.327757i \(-0.106293\pi\)
\(828\) 0 0
\(829\) −161.826 −0.195206 −0.0976030 0.995225i \(-0.531118\pi\)
−0.0976030 + 0.995225i \(0.531118\pi\)
\(830\) 0 0
\(831\) 384.213i 0.462351i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 12.0124i 0.0143861i
\(836\) 0 0
\(837\) 108.000 0.129032
\(838\) 0 0
\(839\) 1103.95i 1.31580i 0.753107 + 0.657898i \(0.228554\pi\)
−0.753107 + 0.657898i \(0.771446\pi\)
\(840\) 0 0
\(841\) −822.248 −0.977703
\(842\) 0 0
\(843\) − 652.062i − 0.773502i
\(844\) 0 0
\(845\) −69.9780 −0.0828142
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −48.0000 −0.0565371
\(850\) 0 0
\(851\) 1038.52i 1.22035i
\(852\) 0 0
\(853\) −851.873 −0.998678 −0.499339 0.866407i \(-0.666424\pi\)
−0.499339 + 0.866407i \(0.666424\pi\)
\(854\) 0 0
\(855\) − 26.5056i − 0.0310007i
\(856\) 0 0
\(857\) −1331.79 −1.55401 −0.777004 0.629495i \(-0.783262\pi\)
−0.777004 + 0.629495i \(0.783262\pi\)
\(858\) 0 0
\(859\) − 569.479i − 0.662956i −0.943463 0.331478i \(-0.892453\pi\)
0.943463 0.331478i \(-0.107547\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 161.931i − 0.187637i −0.995589 0.0938184i \(-0.970093\pi\)
0.995589 0.0938184i \(-0.0299073\pi\)
\(864\) 0 0
\(865\) −87.8258 −0.101533
\(866\) 0 0
\(867\) − 500.261i − 0.577002i
\(868\) 0 0
\(869\) 1377.52 1.58518
\(870\) 0 0
\(871\) 90.4813i 0.103882i
\(872\) 0 0
\(873\) −90.5227 −0.103692
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1299.18 1.48139 0.740693 0.671844i \(-0.234497\pi\)
0.740693 + 0.671844i \(0.234497\pi\)
\(878\) 0 0
\(879\) 947.711i 1.07817i
\(880\) 0 0
\(881\) 725.492 0.823487 0.411743 0.911300i \(-0.364920\pi\)
0.411743 + 0.911300i \(0.364920\pi\)
\(882\) 0 0
\(883\) − 1016.70i − 1.15141i −0.817656 0.575707i \(-0.804727\pi\)
0.817656 0.575707i \(-0.195273\pi\)
\(884\) 0 0
\(885\) −71.7242 −0.0810443
\(886\) 0 0
\(887\) − 884.857i − 0.997584i −0.866722 0.498792i \(-0.833777\pi\)
0.866722 0.498792i \(-0.166223\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 165.824i − 0.186110i
\(892\) 0 0
\(893\) −409.321 −0.458366
\(894\) 0 0
\(895\) − 119.623i − 0.133658i
\(896\) 0 0
\(897\) 34.2652 0.0381997
\(898\) 0 0
\(899\) 90.0037i 0.100115i
\(900\) 0 0
\(901\) −38.4682 −0.0426950
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 76.0440 0.0840265
\(906\) 0 0
\(907\) 36.1036i 0.0398055i 0.999802 + 0.0199028i \(0.00633567\pi\)
−0.999802 + 0.0199028i \(0.993664\pi\)
\(908\) 0 0
\(909\) 549.180 0.604158
\(910\) 0 0
\(911\) 497.091i 0.545654i 0.962063 + 0.272827i \(0.0879587\pi\)
−0.962063 + 0.272827i \(0.912041\pi\)
\(912\) 0 0
\(913\) 599.027 0.656109
\(914\) 0 0
\(915\) − 56.8716i − 0.0621548i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 231.141i − 0.251514i −0.992061 0.125757i \(-0.959864\pi\)
0.992061 0.125757i \(-0.0401360\pi\)
\(920\) 0 0
\(921\) 976.900 1.06069
\(922\) 0 0
\(923\) 51.1838i 0.0554538i
\(924\) 0 0
\(925\) 1518.47 1.64159
\(926\) 0 0
\(927\) − 92.9586i − 0.100279i
\(928\) 0 0
\(929\) 249.775 0.268864 0.134432 0.990923i \(-0.457079\pi\)
0.134432 + 0.990923i \(0.457079\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 688.377 0.737811
\(934\) 0 0
\(935\) 3.21041i 0.00343360i
\(936\) 0 0
\(937\) 910.827 0.972068 0.486034 0.873940i \(-0.338443\pi\)
0.486034 + 0.873940i \(0.338443\pi\)
\(938\) 0 0
\(939\) − 900.509i − 0.959008i
\(940\) 0 0
\(941\) 777.042 0.825762 0.412881 0.910785i \(-0.364523\pi\)
0.412881 + 0.910785i \(0.364523\pi\)
\(942\) 0 0
\(943\) − 154.135i − 0.163452i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1245.72i 1.31544i 0.753264 + 0.657718i \(0.228478\pi\)
−0.753264 + 0.657718i \(0.771522\pi\)
\(948\) 0 0
\(949\) 62.7152 0.0660855
\(950\) 0 0
\(951\) 173.475i 0.182414i
\(952\) 0 0
\(953\) −510.000 −0.535152 −0.267576 0.963537i \(-0.586223\pi\)
−0.267576 + 0.963537i \(0.586223\pi\)
\(954\) 0 0
\(955\) 80.4354i 0.0842256i
\(956\) 0 0
\(957\) 138.192 0.144402
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 529.000 0.550468
\(962\) 0 0
\(963\) − 354.030i − 0.367632i
\(964\) 0 0
\(965\) 59.2015 0.0613487
\(966\) 0 0
\(967\) 171.315i 0.177161i 0.996069 + 0.0885805i \(0.0282331\pi\)
−0.996069 + 0.0885805i \(0.971767\pi\)
\(968\) 0 0
\(969\) 15.3030 0.0157926
\(970\) 0 0
\(971\) − 977.069i − 1.00625i −0.864214 0.503125i \(-0.832183\pi\)
0.864214 0.503125i \(-0.167817\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) − 50.1009i − 0.0513855i
\(976\) 0 0
\(977\) −448.468 −0.459026 −0.229513 0.973306i \(-0.573713\pi\)
−0.229513 + 0.973306i \(0.573713\pi\)
\(978\) 0 0
\(979\) 1509.24i 1.54161i
\(980\) 0 0
\(981\) −256.018 −0.260977
\(982\) 0 0
\(983\) − 563.377i − 0.573120i −0.958062 0.286560i \(-0.907488\pi\)
0.958062 0.286560i \(-0.0925117\pi\)
\(984\) 0 0
\(985\) 13.2924 0.0134949
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −124.109 −0.125489
\(990\) 0 0
\(991\) 1231.85i 1.24304i 0.783398 + 0.621521i \(0.213485\pi\)
−0.783398 + 0.621521i \(0.786515\pi\)
\(992\) 0 0
\(993\) 126.330 0.127221
\(994\) 0 0
\(995\) 61.2761i 0.0615840i
\(996\) 0 0
\(997\) −352.882 −0.353944 −0.176972 0.984216i \(-0.556630\pi\)
−0.176972 + 0.984216i \(0.556630\pi\)
\(998\) 0 0
\(999\) 317.823i 0.318142i
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 2352.3.m.k.1471.3 4
4.3 odd 2 inner 2352.3.m.k.1471.1 4
7.6 odd 2 336.3.m.a.127.2 4
21.20 even 2 1008.3.m.f.127.2 4
28.27 even 2 336.3.m.a.127.4 yes 4
56.13 odd 2 1344.3.m.c.127.3 4
56.27 even 2 1344.3.m.c.127.1 4
84.83 odd 2 1008.3.m.f.127.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
336.3.m.a.127.2 4 7.6 odd 2
336.3.m.a.127.4 yes 4 28.27 even 2
1008.3.m.f.127.1 4 84.83 odd 2
1008.3.m.f.127.2 4 21.20 even 2
1344.3.m.c.127.1 4 56.27 even 2
1344.3.m.c.127.3 4 56.13 odd 2
2352.3.m.k.1471.1 4 4.3 odd 2 inner
2352.3.m.k.1471.3 4 1.1 even 1 trivial