Properties

Label 2352.2.bl.r.607.1
Level $2352$
Weight $2$
Character 2352.607
Analytic conductor $18.781$
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] = [2352,2,Mod(31,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 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("2352.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.bl (of order \(6\), degree \(2\), 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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 607.1
Root \(-0.662827 - 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 2352.607
Dual form 2352.2.bl.r.31.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(-3.72153 - 2.14862i) q^{5} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{3} +(-3.72153 - 2.14862i) q^{5} +(-0.500000 + 0.866025i) q^{9} +(-4.38435 + 2.53131i) q^{11} +3.37849i q^{13} -4.29725i q^{15} +(2.39587 - 1.38326i) q^{17} +(2.35159 - 4.07308i) q^{19} +(-4.18394 - 2.41560i) q^{23} +(6.73317 + 11.6622i) q^{25} -1.00000 q^{27} +2.46054 q^{29} +(-2.84882 - 4.93430i) q^{31} +(-4.38435 - 2.53131i) q^{33} +(1.16765 - 2.02243i) q^{37} +(-2.92586 + 1.68925i) q^{39} -5.14822i q^{41} +13.0199i q^{43} +(3.72153 - 2.14862i) q^{45} +(2.67725 - 4.63713i) q^{47} +(2.39587 + 1.38326i) q^{51} +(2.11185 + 3.65784i) q^{53} +21.7553 q^{55} +4.70319 q^{57} +(4.80249 + 8.31815i) q^{59} +(3.35757 + 1.93849i) q^{61} +(7.25911 - 12.5732i) q^{65} +(4.12524 - 2.38171i) q^{67} -4.83120i q^{69} -12.3181i q^{71} +(9.96809 - 5.75508i) q^{73} +(-6.73317 + 11.6622i) q^{75} +(12.0521 + 6.95830i) q^{79} +(-0.500000 - 0.866025i) q^{81} -7.32191 q^{83} -11.8884 q^{85} +(1.23027 + 2.13089i) q^{87} +(12.1631 + 7.02239i) q^{89} +(2.84882 - 4.93430i) q^{93} +(-17.5030 + 10.1054i) q^{95} -14.5716i q^{97} -5.06262i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 4 q^{9} - 24 q^{23} + 12 q^{25} - 8 q^{27} + 16 q^{29} - 16 q^{31} - 8 q^{47} - 8 q^{53} + 64 q^{55} + 24 q^{59} + 48 q^{61} + 8 q^{65} + 48 q^{67} + 48 q^{73} - 12 q^{75} + 24 q^{79} - 4 q^{81} - 64 q^{85} + 8 q^{87} + 48 q^{89} + 16 q^{93} - 72 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) −3.72153 2.14862i −1.66432 0.960894i −0.970617 0.240630i \(-0.922646\pi\)
−0.693701 0.720264i \(-0.744021\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −4.38435 + 2.53131i −1.32193 + 0.763218i −0.984037 0.177966i \(-0.943048\pi\)
−0.337896 + 0.941184i \(0.609715\pi\)
\(12\) 0 0
\(13\) 3.37849i 0.937025i 0.883457 + 0.468513i \(0.155210\pi\)
−0.883457 + 0.468513i \(0.844790\pi\)
\(14\) 0 0
\(15\) 4.29725i 1.10954i
\(16\) 0 0
\(17\) 2.39587 1.38326i 0.581084 0.335489i −0.180480 0.983579i \(-0.557765\pi\)
0.761564 + 0.648089i \(0.224432\pi\)
\(18\) 0 0
\(19\) 2.35159 4.07308i 0.539492 0.934428i −0.459439 0.888209i \(-0.651950\pi\)
0.998931 0.0462188i \(-0.0147172\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.18394 2.41560i −0.872412 0.503687i −0.00426301 0.999991i \(-0.501357\pi\)
−0.868149 + 0.496304i \(0.834690\pi\)
\(24\) 0 0
\(25\) 6.73317 + 11.6622i 1.34663 + 2.33244i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.46054 0.456912 0.228456 0.973554i \(-0.426632\pi\)
0.228456 + 0.973554i \(0.426632\pi\)
\(30\) 0 0
\(31\) −2.84882 4.93430i −0.511663 0.886227i −0.999909 0.0135202i \(-0.995696\pi\)
0.488245 0.872706i \(-0.337637\pi\)
\(32\) 0 0
\(33\) −4.38435 2.53131i −0.763218 0.440644i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.16765 2.02243i 0.191961 0.332486i −0.753939 0.656944i \(-0.771849\pi\)
0.945900 + 0.324458i \(0.105182\pi\)
\(38\) 0 0
\(39\) −2.92586 + 1.68925i −0.468513 + 0.270496i
\(40\) 0 0
\(41\) 5.14822i 0.804018i −0.915636 0.402009i \(-0.868312\pi\)
0.915636 0.402009i \(-0.131688\pi\)
\(42\) 0 0
\(43\) 13.0199i 1.98552i 0.120117 + 0.992760i \(0.461673\pi\)
−0.120117 + 0.992760i \(0.538327\pi\)
\(44\) 0 0
\(45\) 3.72153 2.14862i 0.554772 0.320298i
\(46\) 0 0
\(47\) 2.67725 4.63713i 0.390517 0.676395i −0.602001 0.798495i \(-0.705630\pi\)
0.992518 + 0.122101i \(0.0389631\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.39587 + 1.38326i 0.335489 + 0.193695i
\(52\) 0 0
\(53\) 2.11185 + 3.65784i 0.290085 + 0.502443i 0.973830 0.227279i \(-0.0729830\pi\)
−0.683744 + 0.729722i \(0.739650\pi\)
\(54\) 0 0
\(55\) 21.7553 2.93349
\(56\) 0 0
\(57\) 4.70319 0.622952
\(58\) 0 0
\(59\) 4.80249 + 8.31815i 0.625231 + 1.08293i 0.988496 + 0.151246i \(0.0483285\pi\)
−0.363265 + 0.931686i \(0.618338\pi\)
\(60\) 0 0
\(61\) 3.35757 + 1.93849i 0.429892 + 0.248198i 0.699301 0.714828i \(-0.253495\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.25911 12.5732i 0.900382 1.55951i
\(66\) 0 0
\(67\) 4.12524 2.38171i 0.503978 0.290972i −0.226377 0.974040i \(-0.572688\pi\)
0.730355 + 0.683068i \(0.239355\pi\)
\(68\) 0 0
\(69\) 4.83120i 0.581608i
\(70\) 0 0
\(71\) 12.3181i 1.46189i −0.682437 0.730944i \(-0.739080\pi\)
0.682437 0.730944i \(-0.260920\pi\)
\(72\) 0 0
\(73\) 9.96809 5.75508i 1.16668 0.673581i 0.213782 0.976881i \(-0.431422\pi\)
0.952895 + 0.303300i \(0.0980886\pi\)
\(74\) 0 0
\(75\) −6.73317 + 11.6622i −0.777480 + 1.34663i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 12.0521 + 6.95830i 1.35597 + 0.782870i 0.989078 0.147393i \(-0.0470882\pi\)
0.366893 + 0.930263i \(0.380422\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) −7.32191 −0.803685 −0.401842 0.915709i \(-0.631630\pi\)
−0.401842 + 0.915709i \(0.631630\pi\)
\(84\) 0 0
\(85\) −11.8884 −1.28948
\(86\) 0 0
\(87\) 1.23027 + 2.13089i 0.131899 + 0.228456i
\(88\) 0 0
\(89\) 12.1631 + 7.02239i 1.28929 + 0.744372i 0.978528 0.206116i \(-0.0660823\pi\)
0.310762 + 0.950488i \(0.399416\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 2.84882 4.93430i 0.295409 0.511663i
\(94\) 0 0
\(95\) −17.5030 + 10.1054i −1.79577 + 1.03679i
\(96\) 0 0
\(97\) 14.5716i 1.47952i −0.672868 0.739762i \(-0.734938\pi\)
0.672868 0.739762i \(-0.265062\pi\)
\(98\) 0 0
\(99\) 5.06262i 0.508812i
\(100\) 0 0
\(101\) 11.3524 6.55434i 1.12961 0.652181i 0.185773 0.982593i \(-0.440521\pi\)
0.943837 + 0.330412i \(0.107188\pi\)
\(102\) 0 0
\(103\) 0.104849 0.181604i 0.0103311 0.0178939i −0.860814 0.508920i \(-0.830045\pi\)
0.871145 + 0.491026i \(0.163378\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.56558 + 3.21329i 0.538044 + 0.310640i 0.744286 0.667861i \(-0.232790\pi\)
−0.206242 + 0.978501i \(0.566123\pi\)
\(108\) 0 0
\(109\) −0.470012 0.814084i −0.0450190 0.0779751i 0.842638 0.538481i \(-0.181001\pi\)
−0.887657 + 0.460506i \(0.847668\pi\)
\(110\) 0 0
\(111\) 2.33530 0.221657
\(112\) 0 0
\(113\) 1.09821 0.103311 0.0516554 0.998665i \(-0.483550\pi\)
0.0516554 + 0.998665i \(0.483550\pi\)
\(114\) 0 0
\(115\) 10.3804 + 17.9794i 0.967980 + 1.67659i
\(116\) 0 0
\(117\) −2.92586 1.68925i −0.270496 0.156171i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 7.31504 12.6700i 0.665004 1.15182i
\(122\) 0 0
\(123\) 4.45849 2.57411i 0.402009 0.232100i
\(124\) 0 0
\(125\) 36.3820i 3.25411i
\(126\) 0 0
\(127\) 5.22625i 0.463755i −0.972745 0.231877i \(-0.925513\pi\)
0.972745 0.231877i \(-0.0744868\pi\)
\(128\) 0 0
\(129\) −11.2756 + 6.50996i −0.992760 + 0.573170i
\(130\) 0 0
\(131\) 0.437508 0.757785i 0.0382252 0.0662080i −0.846280 0.532739i \(-0.821163\pi\)
0.884505 + 0.466531i \(0.154496\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.72153 + 2.14862i 0.320298 + 0.184924i
\(136\) 0 0
\(137\) −0.118419 0.205108i −0.0101172 0.0175235i 0.860922 0.508736i \(-0.169887\pi\)
−0.871040 + 0.491213i \(0.836554\pi\)
\(138\) 0 0
\(139\) 3.00555 0.254927 0.127464 0.991843i \(-0.459316\pi\)
0.127464 + 0.991843i \(0.459316\pi\)
\(140\) 0 0
\(141\) 5.35449 0.450930
\(142\) 0 0
\(143\) −8.55201 14.8125i −0.715155 1.23868i
\(144\) 0 0
\(145\) −9.15698 5.28679i −0.760446 0.439044i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.40083 12.8186i 0.606299 1.05014i −0.385545 0.922689i \(-0.625987\pi\)
0.991845 0.127452i \(-0.0406799\pi\)
\(150\) 0 0
\(151\) 10.7087 6.18269i 0.871464 0.503140i 0.00362965 0.999993i \(-0.498845\pi\)
0.867835 + 0.496853i \(0.165511\pi\)
\(152\) 0 0
\(153\) 2.76652i 0.223659i
\(154\) 0 0
\(155\) 24.4842i 1.96662i
\(156\) 0 0
\(157\) 9.43153 5.44530i 0.752718 0.434582i −0.0739569 0.997261i \(-0.523563\pi\)
0.826675 + 0.562679i \(0.190229\pi\)
\(158\) 0 0
\(159\) −2.11185 + 3.65784i −0.167481 + 0.290085i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.30791 + 0.755120i 0.102443 + 0.0591455i 0.550346 0.834937i \(-0.314496\pi\)
−0.447903 + 0.894082i \(0.647829\pi\)
\(164\) 0 0
\(165\) 10.8777 + 18.8407i 0.846825 + 1.46674i
\(166\) 0 0
\(167\) −10.3333 −0.799612 −0.399806 0.916600i \(-0.630922\pi\)
−0.399806 + 0.916600i \(0.630922\pi\)
\(168\) 0 0
\(169\) 1.58579 0.121984
\(170\) 0 0
\(171\) 2.35159 + 4.07308i 0.179831 + 0.311476i
\(172\) 0 0
\(173\) 2.90674 + 1.67821i 0.220996 + 0.127592i 0.606411 0.795151i \(-0.292609\pi\)
−0.385416 + 0.922743i \(0.625942\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −4.80249 + 8.31815i −0.360977 + 0.625231i
\(178\) 0 0
\(179\) −11.4826 + 6.62946i −0.858247 + 0.495509i −0.863425 0.504477i \(-0.831685\pi\)
0.00517789 + 0.999987i \(0.498352\pi\)
\(180\) 0 0
\(181\) 11.8519i 0.880946i 0.897766 + 0.440473i \(0.145189\pi\)
−0.897766 + 0.440473i \(0.854811\pi\)
\(182\) 0 0
\(183\) 3.87698i 0.286595i
\(184\) 0 0
\(185\) −8.69089 + 5.01769i −0.638967 + 0.368908i
\(186\) 0 0
\(187\) −7.00290 + 12.1294i −0.512103 + 0.886988i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.16619 0.673302i −0.0843828 0.0487184i 0.457215 0.889356i \(-0.348847\pi\)
−0.541598 + 0.840638i \(0.682180\pi\)
\(192\) 0 0
\(193\) 13.0577 + 22.6166i 0.939912 + 1.62798i 0.765631 + 0.643280i \(0.222427\pi\)
0.174281 + 0.984696i \(0.444240\pi\)
\(194\) 0 0
\(195\) 14.5182 1.03967
\(196\) 0 0
\(197\) −7.49083 −0.533699 −0.266850 0.963738i \(-0.585983\pi\)
−0.266850 + 0.963738i \(0.585983\pi\)
\(198\) 0 0
\(199\) 2.24674 + 3.89147i 0.159267 + 0.275859i 0.934605 0.355688i \(-0.115753\pi\)
−0.775337 + 0.631547i \(0.782420\pi\)
\(200\) 0 0
\(201\) 4.12524 + 2.38171i 0.290972 + 0.167993i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −11.0616 + 19.1593i −0.772576 + 1.33814i
\(206\) 0 0
\(207\) 4.18394 2.41560i 0.290804 0.167896i
\(208\) 0 0
\(209\) 23.8104i 1.64700i
\(210\) 0 0
\(211\) 5.93122i 0.408322i −0.978937 0.204161i \(-0.934553\pi\)
0.978937 0.204161i \(-0.0654466\pi\)
\(212\) 0 0
\(213\) 10.6678 6.15905i 0.730944 0.422011i
\(214\) 0 0
\(215\) 27.9749 48.4540i 1.90787 3.30453i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 9.96809 + 5.75508i 0.673581 + 0.388892i
\(220\) 0 0
\(221\) 4.67333 + 8.09444i 0.314362 + 0.544491i
\(222\) 0 0
\(223\) −20.6163 −1.38057 −0.690286 0.723537i \(-0.742515\pi\)
−0.690286 + 0.723537i \(0.742515\pi\)
\(224\) 0 0
\(225\) −13.4663 −0.897756
\(226\) 0 0
\(227\) 10.2455 + 17.7458i 0.680021 + 1.17783i 0.974974 + 0.222318i \(0.0713625\pi\)
−0.294954 + 0.955512i \(0.595304\pi\)
\(228\) 0 0
\(229\) −25.1502 14.5205i −1.66197 0.959539i −0.971773 0.235917i \(-0.924191\pi\)
−0.690197 0.723622i \(-0.742476\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −8.86819 + 15.3602i −0.580975 + 1.00628i 0.414390 + 0.910100i \(0.363995\pi\)
−0.995364 + 0.0961779i \(0.969338\pi\)
\(234\) 0 0
\(235\) −19.9269 + 11.5048i −1.29989 + 0.750490i
\(236\) 0 0
\(237\) 13.9166i 0.903981i
\(238\) 0 0
\(239\) 20.4248i 1.32117i 0.750750 + 0.660586i \(0.229692\pi\)
−0.750750 + 0.660586i \(0.770308\pi\)
\(240\) 0 0
\(241\) −2.82101 + 1.62871i −0.181717 + 0.104915i −0.588099 0.808789i \(-0.700124\pi\)
0.406382 + 0.913703i \(0.366790\pi\)
\(242\) 0 0
\(243\) 0.500000 0.866025i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 13.7609 + 7.94484i 0.875583 + 0.505518i
\(248\) 0 0
\(249\) −3.66096 6.34096i −0.232004 0.401842i
\(250\) 0 0
\(251\) −3.95633 −0.249721 −0.124861 0.992174i \(-0.539848\pi\)
−0.124861 + 0.992174i \(0.539848\pi\)
\(252\) 0 0
\(253\) 24.4585 1.53769
\(254\) 0 0
\(255\) −5.94420 10.2957i −0.372240 0.644739i
\(256\) 0 0
\(257\) −4.45849 2.57411i −0.278113 0.160569i 0.354456 0.935073i \(-0.384666\pi\)
−0.632569 + 0.774504i \(0.717999\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −1.23027 + 2.13089i −0.0761519 + 0.131899i
\(262\) 0 0
\(263\) −0.0679848 + 0.0392511i −0.00419212 + 0.00242032i −0.502095 0.864813i \(-0.667437\pi\)
0.497902 + 0.867233i \(0.334104\pi\)
\(264\) 0 0
\(265\) 18.1503i 1.11497i
\(266\) 0 0
\(267\) 14.0448i 0.859527i
\(268\) 0 0
\(269\) 0.764956 0.441648i 0.0466402 0.0269277i −0.476499 0.879175i \(-0.658094\pi\)
0.523139 + 0.852247i \(0.324761\pi\)
\(270\) 0 0
\(271\) −9.65421 + 16.7216i −0.586451 + 1.01576i 0.408241 + 0.912874i \(0.366142\pi\)
−0.994693 + 0.102890i \(0.967191\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −59.0412 34.0875i −3.56032 2.05555i
\(276\) 0 0
\(277\) −8.46635 14.6641i −0.508694 0.881083i −0.999949 0.0100677i \(-0.996795\pi\)
0.491256 0.871015i \(-0.336538\pi\)
\(278\) 0 0
\(279\) 5.69764 0.341109
\(280\) 0 0
\(281\) 5.05417 0.301507 0.150753 0.988571i \(-0.451830\pi\)
0.150753 + 0.988571i \(0.451830\pi\)
\(282\) 0 0
\(283\) −9.55781 16.5546i −0.568153 0.984069i −0.996749 0.0805731i \(-0.974325\pi\)
0.428596 0.903496i \(-0.359008\pi\)
\(284\) 0 0
\(285\) −17.5030 10.1054i −1.03679 0.598591i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4.67320 + 8.09421i −0.274894 + 0.476130i
\(290\) 0 0
\(291\) 12.6194 7.28581i 0.739762 0.427102i
\(292\) 0 0
\(293\) 2.37837i 0.138946i −0.997584 0.0694730i \(-0.977868\pi\)
0.997584 0.0694730i \(-0.0221318\pi\)
\(294\) 0 0
\(295\) 41.2750i 2.40312i
\(296\) 0 0
\(297\) 4.38435 2.53131i 0.254406 0.146881i
\(298\) 0 0
\(299\) 8.16109 14.1354i 0.471968 0.817472i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 11.3524 + 6.55434i 0.652181 + 0.376537i
\(304\) 0 0
\(305\) −8.33018 14.4283i −0.476985 0.826162i
\(306\) 0 0
\(307\) −4.89599 −0.279429 −0.139714 0.990192i \(-0.544618\pi\)
−0.139714 + 0.990192i \(0.544618\pi\)
\(308\) 0 0
\(309\) 0.209698 0.0119293
\(310\) 0 0
\(311\) 5.24000 + 9.07594i 0.297133 + 0.514649i 0.975479 0.220093i \(-0.0706362\pi\)
−0.678346 + 0.734743i \(0.737303\pi\)
\(312\) 0 0
\(313\) −8.31534 4.80086i −0.470011 0.271361i 0.246234 0.969211i \(-0.420807\pi\)
−0.716244 + 0.697850i \(0.754140\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.39672 14.5435i 0.471607 0.816847i −0.527865 0.849328i \(-0.677007\pi\)
0.999472 + 0.0324809i \(0.0103408\pi\)
\(318\) 0 0
\(319\) −10.7879 + 6.22840i −0.604006 + 0.348723i
\(320\) 0 0
\(321\) 6.42657i 0.358696i
\(322\) 0 0
\(323\) 13.0114i 0.723976i
\(324\) 0 0
\(325\) −39.4007 + 22.7480i −2.18556 + 1.26183i
\(326\) 0 0
\(327\) 0.470012 0.814084i 0.0259917 0.0450190i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −23.2409 13.4181i −1.27743 0.737526i −0.301057 0.953606i \(-0.597340\pi\)
−0.976376 + 0.216080i \(0.930673\pi\)
\(332\) 0 0
\(333\) 1.16765 + 2.02243i 0.0639869 + 0.110829i
\(334\) 0 0
\(335\) −20.4696 −1.11837
\(336\) 0 0
\(337\) 23.0827 1.25739 0.628697 0.777651i \(-0.283589\pi\)
0.628697 + 0.777651i \(0.283589\pi\)
\(338\) 0 0
\(339\) 0.549104 + 0.951076i 0.0298232 + 0.0516554i
\(340\) 0 0
\(341\) 24.9805 + 14.4225i 1.35277 + 0.781021i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −10.3804 + 17.9794i −0.558864 + 0.967980i
\(346\) 0 0
\(347\) −1.13181 + 0.653449i −0.0607586 + 0.0350790i −0.530072 0.847953i \(-0.677835\pi\)
0.469313 + 0.883032i \(0.344502\pi\)
\(348\) 0 0
\(349\) 26.5489i 1.42113i 0.703633 + 0.710564i \(0.251560\pi\)
−0.703633 + 0.710564i \(0.748440\pi\)
\(350\) 0 0
\(351\) 3.37849i 0.180331i
\(352\) 0 0
\(353\) 14.6266 8.44466i 0.778494 0.449464i −0.0574020 0.998351i \(-0.518282\pi\)
0.835896 + 0.548887i \(0.184948\pi\)
\(354\) 0 0
\(355\) −26.4670 + 45.8421i −1.40472 + 2.43305i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 29.0899 + 16.7950i 1.53530 + 0.886408i 0.999104 + 0.0423171i \(0.0134740\pi\)
0.536200 + 0.844091i \(0.319859\pi\)
\(360\) 0 0
\(361\) −1.55998 2.70196i −0.0821040 0.142208i
\(362\) 0 0
\(363\) 14.6301 0.767880
\(364\) 0 0
\(365\) −49.4620 −2.58896
\(366\) 0 0
\(367\) −3.67989 6.37376i −0.192089 0.332708i 0.753854 0.657043i \(-0.228193\pi\)
−0.945942 + 0.324335i \(0.894859\pi\)
\(368\) 0 0
\(369\) 4.45849 + 2.57411i 0.232100 + 0.134003i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.81504 + 3.14374i −0.0939791 + 0.162777i −0.909182 0.416399i \(-0.863292\pi\)
0.815203 + 0.579175i \(0.196625\pi\)
\(374\) 0 0
\(375\) 31.5077 18.1910i 1.62705 0.939379i
\(376\) 0 0
\(377\) 8.31293i 0.428138i
\(378\) 0 0
\(379\) 13.6647i 0.701908i −0.936393 0.350954i \(-0.885857\pi\)
0.936393 0.350954i \(-0.114143\pi\)
\(380\) 0 0
\(381\) 4.52607 2.61313i 0.231877 0.133874i
\(382\) 0 0
\(383\) −6.82843 + 11.8272i −0.348916 + 0.604341i −0.986057 0.166407i \(-0.946784\pi\)
0.637141 + 0.770747i \(0.280117\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −11.2756 6.50996i −0.573170 0.330920i
\(388\) 0 0
\(389\) −13.4540 23.3030i −0.682144 1.18151i −0.974325 0.225145i \(-0.927714\pi\)
0.292181 0.956363i \(-0.405619\pi\)
\(390\) 0 0
\(391\) −13.3656 −0.675927
\(392\) 0 0
\(393\) 0.875015 0.0441387
\(394\) 0 0
\(395\) −29.9016 51.7910i −1.50451 2.60589i
\(396\) 0 0
\(397\) 31.8041 + 18.3621i 1.59620 + 0.921568i 0.992210 + 0.124579i \(0.0397581\pi\)
0.603994 + 0.796989i \(0.293575\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.8408 23.9730i 0.691176 1.19715i −0.280276 0.959919i \(-0.590426\pi\)
0.971453 0.237233i \(-0.0762405\pi\)
\(402\) 0 0
\(403\) 16.6705 9.62472i 0.830417 0.479441i
\(404\) 0 0
\(405\) 4.29725i 0.213532i
\(406\) 0 0
\(407\) 11.8227i 0.586032i
\(408\) 0 0
\(409\) −7.29903 + 4.21410i −0.360914 + 0.208374i −0.669482 0.742829i \(-0.733484\pi\)
0.308568 + 0.951202i \(0.400150\pi\)
\(410\) 0 0
\(411\) 0.118419 0.205108i 0.00584118 0.0101172i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 27.2487 + 15.7320i 1.33759 + 0.772256i
\(416\) 0 0
\(417\) 1.50277 + 2.60288i 0.0735911 + 0.127464i
\(418\) 0 0
\(419\) 18.3029 0.894154 0.447077 0.894495i \(-0.352465\pi\)
0.447077 + 0.894495i \(0.352465\pi\)
\(420\) 0 0
\(421\) 22.6274 1.10279 0.551396 0.834243i \(-0.314095\pi\)
0.551396 + 0.834243i \(0.314095\pi\)
\(422\) 0 0
\(423\) 2.67725 + 4.63713i 0.130172 + 0.225465i
\(424\) 0 0
\(425\) 32.2636 + 18.6274i 1.56502 + 0.903563i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 8.55201 14.8125i 0.412895 0.715155i
\(430\) 0 0
\(431\) 3.85300 2.22453i 0.185592 0.107152i −0.404325 0.914615i \(-0.632494\pi\)
0.589918 + 0.807464i \(0.299160\pi\)
\(432\) 0 0
\(433\) 1.66205i 0.0798730i −0.999202 0.0399365i \(-0.987284\pi\)
0.999202 0.0399365i \(-0.0127156\pi\)
\(434\) 0 0
\(435\) 10.5736i 0.506964i
\(436\) 0 0
\(437\) −19.6779 + 11.3610i −0.941319 + 0.543471i
\(438\) 0 0
\(439\) 9.37753 16.2424i 0.447565 0.775206i −0.550662 0.834728i \(-0.685625\pi\)
0.998227 + 0.0595229i \(0.0189579\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −28.6060 16.5157i −1.35911 0.784684i −0.369609 0.929187i \(-0.620508\pi\)
−0.989504 + 0.144503i \(0.953842\pi\)
\(444\) 0 0
\(445\) −30.1770 52.2680i −1.43053 2.47774i
\(446\) 0 0
\(447\) 14.8017 0.700094
\(448\) 0 0
\(449\) 9.29441 0.438630 0.219315 0.975654i \(-0.429618\pi\)
0.219315 + 0.975654i \(0.429618\pi\)
\(450\) 0 0
\(451\) 13.0317 + 22.5716i 0.613641 + 1.06286i
\(452\) 0 0
\(453\) 10.7087 + 6.18269i 0.503140 + 0.290488i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13.7224 23.7679i 0.641906 1.11181i −0.343101 0.939298i \(-0.611477\pi\)
0.985007 0.172515i \(-0.0551893\pi\)
\(458\) 0 0
\(459\) −2.39587 + 1.38326i −0.111830 + 0.0645649i
\(460\) 0 0
\(461\) 4.28209i 0.199437i −0.995016 0.0997183i \(-0.968206\pi\)
0.995016 0.0997183i \(-0.0317942\pi\)
\(462\) 0 0
\(463\) 16.1278i 0.749521i 0.927122 + 0.374760i \(0.122275\pi\)
−0.927122 + 0.374760i \(0.877725\pi\)
\(464\) 0 0
\(465\) −21.2039 + 12.2421i −0.983308 + 0.567713i
\(466\) 0 0
\(467\) 13.7702 23.8506i 0.637207 1.10368i −0.348836 0.937184i \(-0.613423\pi\)
0.986043 0.166491i \(-0.0532438\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 9.43153 + 5.44530i 0.434582 + 0.250906i
\(472\) 0 0
\(473\) −32.9574 57.0839i −1.51538 2.62472i
\(474\) 0 0
\(475\) 63.3347 2.90600
\(476\) 0 0
\(477\) −4.22371 −0.193390
\(478\) 0 0
\(479\) −9.77545 16.9316i −0.446652 0.773624i 0.551514 0.834166i \(-0.314050\pi\)
−0.998166 + 0.0605420i \(0.980717\pi\)
\(480\) 0 0
\(481\) 6.83277 + 3.94490i 0.311548 + 0.179872i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −31.3089 + 54.2287i −1.42167 + 2.46240i
\(486\) 0 0
\(487\) 32.7285 18.8958i 1.48307 0.856252i 0.483257 0.875479i \(-0.339454\pi\)
0.999815 + 0.0192268i \(0.00612046\pi\)
\(488\) 0 0
\(489\) 1.51024i 0.0682954i
\(490\) 0 0
\(491\) 29.2605i 1.32051i 0.751042 + 0.660254i \(0.229551\pi\)
−0.751042 + 0.660254i \(0.770449\pi\)
\(492\) 0 0
\(493\) 5.89515 3.40357i 0.265504 0.153289i
\(494\) 0 0
\(495\) −10.8777 + 18.8407i −0.488914 + 0.846825i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −10.9485 6.32111i −0.490122 0.282972i 0.234503 0.972115i \(-0.424654\pi\)
−0.724625 + 0.689143i \(0.757987\pi\)
\(500\) 0 0
\(501\) −5.16663 8.94887i −0.230828 0.399806i
\(502\) 0 0
\(503\) −27.0714 −1.20706 −0.603528 0.797342i \(-0.706239\pi\)
−0.603528 + 0.797342i \(0.706239\pi\)
\(504\) 0 0
\(505\) −56.3312 −2.50671
\(506\) 0 0
\(507\) 0.792893 + 1.37333i 0.0352136 + 0.0609918i
\(508\) 0 0
\(509\) 10.1311 + 5.84917i 0.449051 + 0.259260i 0.707429 0.706784i \(-0.249855\pi\)
−0.258378 + 0.966044i \(0.583188\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2.35159 + 4.07308i −0.103825 + 0.179831i
\(514\) 0 0
\(515\) −0.780396 + 0.450562i −0.0343884 + 0.0198541i
\(516\) 0 0
\(517\) 27.1077i 1.19220i
\(518\) 0 0
\(519\) 3.35642i 0.147330i
\(520\) 0 0
\(521\) 6.24095 3.60322i 0.273421 0.157860i −0.357020 0.934097i \(-0.616207\pi\)
0.630441 + 0.776237i \(0.282874\pi\)
\(522\) 0 0
\(523\) −8.85727 + 15.3412i −0.387301 + 0.670825i −0.992086 0.125564i \(-0.959926\pi\)
0.604784 + 0.796389i \(0.293259\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −13.6508 7.88130i −0.594639 0.343315i
\(528\) 0 0
\(529\) 0.170243 + 0.294869i 0.00740185 + 0.0128204i
\(530\) 0 0
\(531\) −9.60498 −0.416821
\(532\) 0 0
\(533\) 17.3932 0.753385
\(534\) 0 0
\(535\) −13.8083 23.9167i −0.596984 1.03401i
\(536\) 0 0
\(537\) −11.4826 6.62946i −0.495509 0.286082i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −11.5590 + 20.0208i −0.496961 + 0.860761i −0.999994 0.00350600i \(-0.998884\pi\)
0.503033 + 0.864267i \(0.332217\pi\)
\(542\) 0 0
\(543\) −10.2641 + 5.92596i −0.440473 + 0.254307i
\(544\) 0 0
\(545\) 4.03951i 0.173034i
\(546\) 0 0
\(547\) 16.0524i 0.686351i −0.939271 0.343176i \(-0.888497\pi\)
0.939271 0.343176i \(-0.111503\pi\)
\(548\) 0 0
\(549\) −3.35757 + 1.93849i −0.143297 + 0.0827328i
\(550\) 0 0
\(551\) 5.78620 10.0220i 0.246500 0.426951i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −8.69089 5.01769i −0.368908 0.212989i
\(556\) 0 0
\(557\) 6.29041 + 10.8953i 0.266533 + 0.461649i 0.967964 0.251088i \(-0.0807884\pi\)
−0.701431 + 0.712738i \(0.747455\pi\)
\(558\) 0 0
\(559\) −43.9877 −1.86048
\(560\) 0 0
\(561\) −14.0058 −0.591325
\(562\) 0 0
\(563\) 18.2974 + 31.6921i 0.771144 + 1.33566i 0.936936 + 0.349500i \(0.113649\pi\)
−0.165792 + 0.986161i \(0.553018\pi\)
\(564\) 0 0
\(565\) −4.08701 2.35964i −0.171942 0.0992707i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12.2495 21.2167i 0.513524 0.889450i −0.486353 0.873763i \(-0.661673\pi\)
0.999877 0.0156875i \(-0.00499368\pi\)
\(570\) 0 0
\(571\) 28.3990 16.3962i 1.18846 0.686159i 0.230506 0.973071i \(-0.425962\pi\)
0.957957 + 0.286912i \(0.0926286\pi\)
\(572\) 0 0
\(573\) 1.34660i 0.0562552i
\(574\) 0 0
\(575\) 65.0586i 2.71313i
\(576\) 0 0
\(577\) 16.6398 9.60699i 0.692724 0.399944i −0.111908 0.993719i \(-0.535696\pi\)
0.804632 + 0.593774i \(0.202363\pi\)
\(578\) 0 0
\(579\) −13.0577 + 22.6166i −0.542659 + 0.939912i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −18.5182 10.6915i −0.766946 0.442797i
\(584\) 0 0
\(585\) 7.25911 + 12.5732i 0.300127 + 0.519836i
\(586\) 0 0
\(587\) −24.1451 −0.996573 −0.498286 0.867012i \(-0.666037\pi\)
−0.498286 + 0.867012i \(0.666037\pi\)
\(588\) 0 0
\(589\) −26.7971 −1.10415
\(590\) 0 0
\(591\) −3.74541 6.48725i −0.154066 0.266850i
\(592\) 0 0
\(593\) 2.20475 + 1.27291i 0.0905380 + 0.0522722i 0.544585 0.838705i \(-0.316687\pi\)
−0.454047 + 0.890978i \(0.650020\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −2.24674 + 3.89147i −0.0919531 + 0.159267i
\(598\) 0 0
\(599\) 12.9816 7.49493i 0.530414 0.306234i −0.210771 0.977535i \(-0.567598\pi\)
0.741185 + 0.671301i \(0.234264\pi\)
\(600\) 0 0
\(601\) 24.3343i 0.992618i 0.868146 + 0.496309i \(0.165312\pi\)
−0.868146 + 0.496309i \(0.834688\pi\)
\(602\) 0 0
\(603\) 4.76342i 0.193981i
\(604\) 0 0
\(605\) −54.4462 + 31.4345i −2.21355 + 1.27800i
\(606\) 0 0
\(607\) 4.45644 7.71878i 0.180881 0.313296i −0.761300 0.648400i \(-0.775438\pi\)
0.942181 + 0.335105i \(0.108772\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 15.6665 + 9.04506i 0.633799 + 0.365924i
\(612\) 0 0
\(613\) 8.58741 + 14.8738i 0.346842 + 0.600748i 0.985687 0.168588i \(-0.0539206\pi\)
−0.638844 + 0.769336i \(0.720587\pi\)
\(614\) 0 0
\(615\) −22.1232 −0.892094
\(616\) 0 0
\(617\) 35.9980 1.44922 0.724612 0.689157i \(-0.242019\pi\)
0.724612 + 0.689157i \(0.242019\pi\)
\(618\) 0 0
\(619\) −2.05562 3.56043i −0.0826222 0.143106i 0.821753 0.569843i \(-0.192996\pi\)
−0.904376 + 0.426738i \(0.859663\pi\)
\(620\) 0 0
\(621\) 4.18394 + 2.41560i 0.167896 + 0.0969347i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −44.5054 + 77.0856i −1.78022 + 3.08342i
\(626\) 0 0
\(627\) −20.6204 + 11.9052i −0.823501 + 0.475448i
\(628\) 0 0
\(629\) 6.46065i 0.257603i
\(630\) 0 0
\(631\) 34.8970i 1.38923i 0.719383 + 0.694613i \(0.244425\pi\)
−0.719383 + 0.694613i \(0.755575\pi\)
\(632\) 0 0
\(633\) 5.13659 2.96561i 0.204161 0.117872i
\(634\) 0 0
\(635\) −11.2293 + 19.4496i −0.445619 + 0.771835i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 10.6678 + 6.15905i 0.422011 + 0.243648i
\(640\) 0 0
\(641\) −3.66368 6.34567i −0.144707 0.250639i 0.784557 0.620057i \(-0.212890\pi\)
−0.929263 + 0.369418i \(0.879557\pi\)
\(642\) 0 0
\(643\) −9.24275 −0.364498 −0.182249 0.983252i \(-0.558338\pi\)
−0.182249 + 0.983252i \(0.558338\pi\)
\(644\) 0 0
\(645\) 55.9498 2.20302
\(646\) 0 0
\(647\) −7.59013 13.1465i −0.298399 0.516842i 0.677371 0.735642i \(-0.263119\pi\)
−0.975770 + 0.218800i \(0.929786\pi\)
\(648\) 0 0
\(649\) −42.1116 24.3132i −1.65303 0.954375i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.06425 1.84333i 0.0416471 0.0721349i −0.844450 0.535634i \(-0.820073\pi\)
0.886098 + 0.463499i \(0.153406\pi\)
\(654\) 0 0
\(655\) −3.25639 + 1.88008i −0.127238 + 0.0734608i
\(656\) 0 0
\(657\) 11.5102i 0.449054i
\(658\) 0 0
\(659\) 13.0520i 0.508436i −0.967147 0.254218i \(-0.918182\pi\)
0.967147 0.254218i \(-0.0818180\pi\)
\(660\) 0 0
\(661\) 16.6958 9.63931i 0.649390 0.374926i −0.138832 0.990316i \(-0.544335\pi\)
0.788223 + 0.615390i \(0.211002\pi\)
\(662\) 0 0
\(663\) −4.67333 + 8.09444i −0.181497 + 0.314362i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −10.2948 5.94369i −0.398615 0.230141i
\(668\) 0 0
\(669\) −10.3082 17.8543i −0.398537 0.690286i
\(670\) 0 0
\(671\) −19.6277 −0.757718
\(672\) 0 0
\(673\) −7.08216 −0.272997 −0.136499 0.990640i \(-0.543585\pi\)
−0.136499 + 0.990640i \(0.543585\pi\)
\(674\) 0 0
\(675\) −6.73317 11.6622i −0.259160 0.448878i
\(676\) 0 0
\(677\) −31.6135 18.2521i −1.21501 0.701485i −0.251161 0.967945i \(-0.580812\pi\)
−0.963846 + 0.266461i \(0.914146\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −10.2455 + 17.7458i −0.392610 + 0.680021i
\(682\) 0 0
\(683\) −26.0852 + 15.0603i −0.998124 + 0.576267i −0.907693 0.419636i \(-0.862158\pi\)
−0.0904313 + 0.995903i \(0.528825\pi\)
\(684\) 0 0
\(685\) 1.01775i 0.0388863i
\(686\) 0 0
\(687\) 29.0409i 1.10798i
\(688\) 0 0
\(689\) −12.3580 + 7.13488i −0.470801 + 0.271817i
\(690\) 0 0
\(691\) 19.9057 34.4776i 0.757247 1.31159i −0.187002 0.982359i \(-0.559877\pi\)
0.944249 0.329231i \(-0.106789\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −11.1852 6.45779i −0.424280 0.244958i
\(696\) 0 0
\(697\) −7.12132 12.3345i −0.269739 0.467202i
\(698\) 0 0
\(699\) −17.7364 −0.670852
\(700\) 0 0
\(701\) 28.0795 1.06055 0.530275 0.847826i \(-0.322089\pi\)
0.530275 + 0.847826i \(0.322089\pi\)
\(702\) 0 0
\(703\) −5.49168 9.51187i −0.207123 0.358747i
\(704\) 0 0
\(705\) −19.9269 11.5048i −0.750490 0.433296i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 18.4738 31.9975i 0.693797 1.20169i −0.276787 0.960931i \(-0.589270\pi\)
0.970585 0.240761i \(-0.0773969\pi\)
\(710\) 0 0
\(711\) −12.0521 + 6.95830i −0.451990 + 0.260957i
\(712\) 0 0
\(713\) 27.5264i 1.03087i
\(714\) 0 0
\(715\) 73.5002i 2.74875i
\(716\) 0 0
\(717\) −17.6884 + 10.2124i −0.660586 + 0.381389i
\(718\) 0 0
\(719\) 9.35449 16.2025i 0.348864 0.604250i −0.637184 0.770712i \(-0.719901\pi\)
0.986048 + 0.166462i \(0.0532343\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −2.82101 1.62871i −0.104915 0.0605724i
\(724\) 0 0
\(725\) 16.5673 + 28.6954i 0.615293 + 1.06572i
\(726\) 0 0
\(727\) 31.2078 1.15743 0.578716 0.815529i \(-0.303554\pi\)
0.578716 + 0.815529i \(0.303554\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 18.0099 + 31.1941i 0.666120 + 1.15375i
\(732\) 0 0
\(733\) 3.52383 + 2.03449i 0.130156 + 0.0751455i 0.563664 0.826004i \(-0.309391\pi\)
−0.433508 + 0.901150i \(0.642725\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.0577 + 20.8845i −0.444150 + 0.769291i
\(738\) 0 0
\(739\) 34.5566 19.9513i 1.27119 0.733919i 0.295974 0.955196i \(-0.404356\pi\)
0.975211 + 0.221277i \(0.0710224\pi\)
\(740\) 0 0
\(741\) 15.8897i 0.583722i
\(742\) 0 0
\(743\) 41.7530i 1.53177i 0.642979 + 0.765884i \(0.277698\pi\)
−0.642979 + 0.765884i \(0.722302\pi\)
\(744\) 0 0
\(745\) −55.0847 + 31.8032i −2.01815 + 1.16518i
\(746\) 0 0
\(747\) 3.66096 6.34096i 0.133947 0.232004i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 7.88020 + 4.54963i 0.287552 + 0.166019i 0.636838 0.770998i \(-0.280242\pi\)
−0.349285 + 0.937017i \(0.613576\pi\)
\(752\) 0 0
\(753\) −1.97816 3.42628i −0.0720883 0.124861i
\(754\) 0 0
\(755\) −53.1371 −1.93386
\(756\) 0 0
\(757\) −0.902155 −0.0327894 −0.0163947 0.999866i \(-0.505219\pi\)
−0.0163947 + 0.999866i \(0.505219\pi\)
\(758\) 0 0
\(759\) 12.2293 + 21.1817i 0.443894 + 0.768847i
\(760\) 0 0
\(761\) −20.5924 11.8891i −0.746475 0.430978i 0.0779436 0.996958i \(-0.475165\pi\)
−0.824419 + 0.565980i \(0.808498\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 5.94420 10.2957i 0.214913 0.372240i
\(766\) 0 0
\(767\) −28.1028 + 16.2252i −1.01473 + 0.585857i
\(768\) 0 0
\(769\) 16.3028i 0.587895i 0.955822 + 0.293948i \(0.0949691\pi\)
−0.955822 + 0.293948i \(0.905031\pi\)
\(770\) 0 0
\(771\) 5.14822i 0.185409i
\(772\) 0 0
\(773\) 21.7270 12.5441i 0.781465 0.451179i −0.0554845 0.998460i \(-0.517670\pi\)
0.836949 + 0.547281i \(0.184337\pi\)
\(774\) 0 0
\(775\) 38.3632 66.4470i 1.37805 2.38685i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −20.9691 12.1065i −0.751297 0.433761i
\(780\) 0 0
\(781\) 31.1809 + 54.0069i 1.11574 + 1.93252i
\(782\) 0 0
\(783\) −2.46054 −0.0879327
\(784\) 0 0
\(785\) −46.7996 −1.67035
\(786\) 0 0
\(787\) −2.23565 3.87226i −0.0796924 0.138031i 0.823425 0.567425i \(-0.192060\pi\)
−0.903117 + 0.429394i \(0.858727\pi\)
\(788\) 0 0
\(789\) −0.0679848 0.0392511i −0.00242032 0.00139737i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −6.54918 + 11.3435i −0.232568 + 0.402820i
\(794\) 0 0
\(795\) 15.7186 9.07516i 0.557483 0.321863i
\(796\) 0 0
\(797\) 6.62320i 0.234606i 0.993096 + 0.117303i \(0.0374248\pi\)
−0.993096 + 0.117303i \(0.962575\pi\)
\(798\) 0 0
\(799\) 14.8133i 0.524056i
\(800\) 0 0
\(801\) −12.1631 + 7.02239i −0.429763 + 0.248124i
\(802\) 0 0
\(803\) −29.1358 + 50.4646i −1.02818 + 1.78086i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0.764956 + 0.441648i 0.0269277 + 0.0155467i
\(808\) 0 0
\(809\) 4.18522 + 7.24901i 0.147144 + 0.254862i 0.930171 0.367127i \(-0.119658\pi\)
−0.783027 + 0.621988i \(0.786325\pi\)
\(810\) 0 0
\(811\) 1.20944 0.0424692 0.0212346 0.999775i \(-0.493240\pi\)
0.0212346 + 0.999775i \(0.493240\pi\)
\(812\) 0 0
\(813\) −19.3084 −0.677176
\(814\) 0 0
\(815\) −3.24494 5.62040i −0.113665 0.196874i
\(816\) 0 0
\(817\) 53.0312 + 30.6176i 1.85533 + 1.07117i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −25.6260 + 44.3855i −0.894353 + 1.54906i −0.0597490 + 0.998213i \(0.519030\pi\)
−0.834604 + 0.550851i \(0.814303\pi\)
\(822\) 0 0
\(823\) −27.3271 + 15.7773i −0.952563 + 0.549962i −0.893876 0.448314i \(-0.852025\pi\)
−0.0586866 + 0.998276i \(0.518691\pi\)
\(824\) 0 0
\(825\) 68.1749i 2.37355i
\(826\) 0 0
\(827\) 23.4800i 0.816480i 0.912875 + 0.408240i \(0.133857\pi\)
−0.912875 + 0.408240i \(0.866143\pi\)
\(828\) 0 0
\(829\) 14.9510 8.63194i 0.519268 0.299800i −0.217367 0.976090i \(-0.569747\pi\)
0.736635 + 0.676290i \(0.236413\pi\)
\(830\) 0 0
\(831\) 8.46635 14.6641i 0.293694 0.508694i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 38.4555 + 22.2023i 1.33081 + 0.768342i
\(836\) 0 0
\(837\) 2.84882 + 4.93430i 0.0984696 + 0.170554i
\(838\) 0 0
\(839\) 37.4781 1.29389 0.646943 0.762538i \(-0.276047\pi\)
0.646943 + 0.762538i \(0.276047\pi\)
\(840\) 0 0
\(841\) −22.9457 −0.791232
\(842\) 0 0
\(843\) 2.52709 + 4.37704i 0.0870375 + 0.150753i
\(844\) 0 0
\(845\) −5.90155 3.40726i −0.203019 0.117213i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 9.55781 16.5546i 0.328023 0.568153i
\(850\) 0 0
\(851\) −9.77077 + 5.64116i −0.334938 + 0.193376i
\(852\) 0 0
\(853\) 9.50115i 0.325313i 0.986683 + 0.162657i \(0.0520063\pi\)
−0.986683 + 0.162657i \(0.947994\pi\)
\(854\) 0 0
\(855\) 20.2108i 0.691193i
\(856\) 0 0
\(857\) 28.3880 16.3898i 0.969717 0.559866i 0.0705668 0.997507i \(-0.477519\pi\)
0.899150 + 0.437641i \(0.144186\pi\)
\(858\) 0 0
\(859\) 2.68653 4.65321i 0.0916633 0.158765i −0.816548 0.577278i \(-0.804115\pi\)
0.908211 + 0.418512i \(0.137448\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25.7057 + 14.8412i 0.875031 + 0.505200i 0.869017 0.494782i \(-0.164752\pi\)
0.00601431 + 0.999982i \(0.498086\pi\)
\(864\) 0 0
\(865\) −7.21169 12.4910i −0.245205 0.424707i
\(866\) 0 0
\(867\) −9.34639 −0.317420
\(868\) 0 0
\(869\) −70.4544 −2.39000
\(870\) 0 0
\(871\) 8.04659 + 13.9371i 0.272648 + 0.472241i
\(872\) 0 0
\(873\) 12.6194 + 7.28581i 0.427102 + 0.246587i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 15.1621 26.2615i 0.511988 0.886789i −0.487915 0.872891i \(-0.662243\pi\)
0.999903 0.0138982i \(-0.00442409\pi\)
\(878\) 0 0
\(879\) 2.05973 1.18919i 0.0694730 0.0401103i
\(880\) 0 0
\(881\) 16.5012i 0.555939i 0.960590 + 0.277969i \(0.0896614\pi\)
−0.960590 + 0.277969i \(0.910339\pi\)
\(882\) 0 0
\(883\) 22.5909i 0.760245i 0.924936 + 0.380122i \(0.124118\pi\)
−0.924936 + 0.380122i \(0.875882\pi\)
\(884\) 0 0
\(885\) 35.7452 20.6375i 1.20156 0.693722i
\(886\) 0 0
\(887\) −19.3018 + 33.4317i −0.648090 + 1.12253i 0.335488 + 0.942044i \(0.391099\pi\)
−0.983578 + 0.180481i \(0.942235\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4.38435 + 2.53131i 0.146881 + 0.0848020i
\(892\) 0 0
\(893\) −12.5916 21.8093i −0.421361 0.729819i
\(894\) 0 0
\(895\) 56.9769 1.90453
\(896\) 0 0
\(897\) 16.3222 0.544981
\(898\) 0 0
\(899\) −7.00965 12.1411i −0.233785 0.404927i
\(900\) 0 0
\(901\) 10.1195 + 5.84247i 0.337128 + 0.194641i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 25.4653 44.1072i 0.846496 1.46617i
\(906\) 0 0
\(907\) 31.7609 18.3371i 1.05460 0.608875i 0.130668 0.991426i \(-0.458288\pi\)
0.923934 + 0.382551i \(0.124954\pi\)
\(908\) 0 0
\(909\) 13.1087i 0.434787i
\(910\) 0 0
\(911\) 52.6835i 1.74548i −0.488185 0.872740i \(-0.662341\pi\)
0.488185 0.872740i \(-0.337659\pi\)
\(912\) 0 0
\(913\) 32.1019 18.5340i 1.06242 0.613387i
\(914\) 0 0
\(915\) 8.33018 14.4283i 0.275387 0.476985i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −4.43074 2.55809i −0.146157 0.0843836i 0.425138 0.905128i \(-0.360225\pi\)
−0.571295 + 0.820745i \(0.693559\pi\)
\(920\) 0 0
\(921\) −2.44799 4.24005i −0.0806642 0.139714i
\(922\) 0 0
\(923\) 41.6166 1.36983
\(924\) 0 0
\(925\) 31.4480 1.03400
\(926\) 0 0
\(927\) 0.104849 + 0.181604i 0.00344369 + 0.00596465i
\(928\) 0 0
\(929\) −30.6308 17.6847i −1.00496 0.580217i −0.0952512 0.995453i \(-0.530365\pi\)
−0.909713 + 0.415237i \(0.863699\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −5.24000 + 9.07594i −0.171550 + 0.297133i
\(934\) 0 0
\(935\) 52.1230 30.0932i 1.70460 0.984153i
\(936\) 0 0
\(937\) 23.6290i 0.771924i −0.922515 0.385962i \(-0.873870\pi\)
0.922515 0.385962i \(-0.126130\pi\)
\(938\) 0 0
\(939\) 9.60172i 0.313340i
\(940\) 0 0
\(941\) −16.6438 + 9.60933i −0.542574 + 0.313255i −0.746121 0.665810i \(-0.768086\pi\)
0.203548 + 0.979065i \(0.434753\pi\)
\(942\) 0 0
\(943\) −12.4360 + 21.5399i −0.404973 + 0.701435i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24.0892 + 13.9079i 0.782795 + 0.451947i 0.837420 0.546560i \(-0.184063\pi\)
−0.0546249 + 0.998507i \(0.517396\pi\)
\(948\) 0 0
\(949\) 19.4435 + 33.6771i 0.631162 + 1.09321i
\(950\) 0 0
\(951\) 16.7934 0.544565
\(952\) 0 0
\(953\) 56.8087 1.84021 0.920107 0.391668i \(-0.128102\pi\)
0.920107 + 0.391668i \(0.128102\pi\)
\(954\) 0 0
\(955\) 2.89335 + 5.01142i 0.0936264 + 0.162166i
\(956\) 0 0
\(957\) −10.7879 6.22840i −0.348723 0.201335i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.731549 + 1.26708i −0.0235984 + 0.0408736i
\(962\) 0 0
\(963\) −5.56558 + 3.21329i −0.179348 + 0.103547i
\(964\) 0 0
\(965\) 112.224i 3.61262i
\(966\) 0 0
\(967\) 20.3860i 0.655570i 0.944752 + 0.327785i \(0.106302\pi\)
−0.944752 + 0.327785i \(0.893698\pi\)
\(968\) 0 0
\(969\) 11.2682 6.50572i 0.361988 0.208994i
\(970\) 0 0
\(971\) 5.61437 9.72437i 0.180174 0.312070i −0.761766 0.647852i \(-0.775667\pi\)
0.941940 + 0.335782i \(0.109001\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −39.4007 22.7480i −1.26183 0.728518i
\(976\) 0 0
\(977\) 6.09164 + 10.5510i 0.194889 + 0.337557i 0.946864 0.321634i \(-0.104232\pi\)
−0.751975 + 0.659191i \(0.770899\pi\)
\(978\) 0 0
\(979\) −71.1033 −2.27247
\(980\) 0 0
\(981\) 0.940024 0.0300126
\(982\) 0 0
\(983\) −4.53187 7.84943i −0.144544 0.250358i 0.784659 0.619928i \(-0.212838\pi\)
−0.929203 + 0.369570i \(0.879505\pi\)
\(984\) 0 0
\(985\) 27.8773 + 16.0950i 0.888245 + 0.512829i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 31.4509 54.4746i 1.00008 1.73219i
\(990\) 0 0
\(991\) −4.65193 + 2.68579i −0.147773 + 0.0853170i −0.572064 0.820209i \(-0.693857\pi\)
0.424290 + 0.905526i \(0.360524\pi\)
\(992\) 0 0
\(993\) 26.8362i 0.851622i
\(994\) 0 0
\(995\) 19.3096i 0.612157i
\(996\) 0 0
\(997\) −12.4222 + 7.17199i −0.393417 + 0.227139i −0.683639 0.729820i \(-0.739604\pi\)
0.290223 + 0.956959i \(0.406271\pi\)
\(998\) 0 0
\(999\) −1.16765 + 2.02243i −0.0369429 + 0.0639869i
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 2352.2.bl.r.607.1 8
4.3 odd 2 2352.2.bl.q.607.1 8
7.2 even 3 2352.2.b.k.1567.8 yes 8
7.3 odd 6 2352.2.bl.q.31.1 8
7.4 even 3 2352.2.bl.t.31.4 8
7.5 odd 6 2352.2.b.l.1567.1 yes 8
7.6 odd 2 2352.2.bl.o.607.4 8
21.2 odd 6 7056.2.b.x.1567.1 8
21.5 even 6 7056.2.b.w.1567.8 8
28.3 even 6 inner 2352.2.bl.r.31.1 8
28.11 odd 6 2352.2.bl.o.31.4 8
28.19 even 6 2352.2.b.k.1567.1 8
28.23 odd 6 2352.2.b.l.1567.8 yes 8
28.27 even 2 2352.2.bl.t.607.4 8
84.23 even 6 7056.2.b.w.1567.1 8
84.47 odd 6 7056.2.b.x.1567.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2352.2.b.k.1567.1 8 28.19 even 6
2352.2.b.k.1567.8 yes 8 7.2 even 3
2352.2.b.l.1567.1 yes 8 7.5 odd 6
2352.2.b.l.1567.8 yes 8 28.23 odd 6
2352.2.bl.o.31.4 8 28.11 odd 6
2352.2.bl.o.607.4 8 7.6 odd 2
2352.2.bl.q.31.1 8 7.3 odd 6
2352.2.bl.q.607.1 8 4.3 odd 2
2352.2.bl.r.31.1 8 28.3 even 6 inner
2352.2.bl.r.607.1 8 1.1 even 1 trivial
2352.2.bl.t.31.4 8 7.4 even 3
2352.2.bl.t.607.4 8 28.27 even 2
7056.2.b.w.1567.1 8 84.23 even 6
7056.2.b.w.1567.8 8 21.5 even 6
7056.2.b.x.1567.1 8 21.2 odd 6
7056.2.b.x.1567.8 8 84.47 odd 6