Properties

Label 5376.2.c.p.2689.1
Level $5376$
Weight $2$
Character 5376.2689
Analytic conductor $42.928$
Analytic rank $1$
Dimension $2$
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] = [5376,2,Mod(2689,5376)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5376, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5376.2689");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5376.c (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: \(42.9275761266\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2689.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5376.2689
Dual form 5376.2.c.p.2689.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +4.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +4.00000i q^{5} -1.00000 q^{7} -1.00000 q^{9} -2.00000i q^{11} +6.00000i q^{13} +4.00000 q^{15} -4.00000 q^{17} -4.00000i q^{19} +1.00000i q^{21} +2.00000 q^{23} -11.0000 q^{25} +1.00000i q^{27} +2.00000i q^{29} -2.00000 q^{33} -4.00000i q^{35} +2.00000i q^{37} +6.00000 q^{39} +4.00000i q^{43} -4.00000i q^{45} -12.0000 q^{47} +1.00000 q^{49} +4.00000i q^{51} -6.00000i q^{53} +8.00000 q^{55} -4.00000 q^{57} +8.00000i q^{59} -6.00000i q^{61} +1.00000 q^{63} -24.0000 q^{65} -8.00000i q^{67} -2.00000i q^{69} +14.0000 q^{71} +2.00000 q^{73} +11.0000i q^{75} +2.00000i q^{77} -12.0000 q^{79} +1.00000 q^{81} -4.00000i q^{83} -16.0000i q^{85} +2.00000 q^{87} -6.00000i q^{91} +16.0000 q^{95} -2.00000 q^{97} +2.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} - 2 q^{9} + 8 q^{15} - 8 q^{17} + 4 q^{23} - 22 q^{25} - 4 q^{33} + 12 q^{39} - 24 q^{47} + 2 q^{49} + 16 q^{55} - 8 q^{57} + 2 q^{63} - 48 q^{65} + 28 q^{71} + 4 q^{73} - 24 q^{79} + 2 q^{81} + 4 q^{87} + 32 q^{95} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5376\mathbb{Z}\right)^\times\).

\(n\) \(1793\) \(2815\) \(4609\) \(5125\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 4.00000i 1.78885i 0.447214 + 0.894427i \(0.352416\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 2.00000i − 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 1.00000i 0.218218i
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) −11.0000 −2.20000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 2.00000i 0.371391i 0.982607 + 0.185695i \(0.0594537\pi\)
−0.982607 + 0.185695i \(0.940546\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) − 4.00000i − 0.676123i
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) − 4.00000i − 0.596285i
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.00000i 0.560112i
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 8.00000i 1.04151i 0.853706 + 0.520756i \(0.174350\pi\)
−0.853706 + 0.520756i \(0.825650\pi\)
\(60\) 0 0
\(61\) − 6.00000i − 0.768221i −0.923287 0.384111i \(-0.874508\pi\)
0.923287 0.384111i \(-0.125492\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −24.0000 −2.97683
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) − 2.00000i − 0.240772i
\(70\) 0 0
\(71\) 14.0000 1.66149 0.830747 0.556650i \(-0.187914\pi\)
0.830747 + 0.556650i \(0.187914\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) 11.0000i 1.27017i
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) − 16.0000i − 1.73544i
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) − 6.00000i − 0.628971i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 16.0000 1.64157
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 2.00000i 0.201008i
\(100\) 0 0
\(101\) 16.0000i 1.59206i 0.605257 + 0.796030i \(0.293070\pi\)
−0.605257 + 0.796030i \(0.706930\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) −4.00000 −0.390360
\(106\) 0 0
\(107\) − 18.0000i − 1.74013i −0.492941 0.870063i \(-0.664078\pi\)
0.492941 0.870063i \(-0.335922\pi\)
\(108\) 0 0
\(109\) 2.00000i 0.191565i 0.995402 + 0.0957826i \(0.0305354\pi\)
−0.995402 + 0.0957826i \(0.969465\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 8.00000i 0.746004i
\(116\) 0 0
\(117\) − 6.00000i − 0.554700i
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 24.0000i − 2.14663i
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) 0 0
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 4.00000i 0.349482i 0.984614 + 0.174741i \(0.0559088\pi\)
−0.984614 + 0.174741i \(0.944091\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) − 4.00000i − 0.339276i −0.985506 0.169638i \(-0.945740\pi\)
0.985506 0.169638i \(-0.0542598\pi\)
\(140\) 0 0
\(141\) 12.0000i 1.01058i
\(142\) 0 0
\(143\) 12.0000 1.00349
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 0 0
\(147\) − 1.00000i − 0.0824786i
\(148\) 0 0
\(149\) − 6.00000i − 0.491539i −0.969328 0.245770i \(-0.920959\pi\)
0.969328 0.245770i \(-0.0790407\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) −2.00000 −0.157622
\(162\) 0 0
\(163\) − 16.0000i − 1.25322i −0.779334 0.626608i \(-0.784443\pi\)
0.779334 0.626608i \(-0.215557\pi\)
\(164\) 0 0
\(165\) − 8.00000i − 0.622799i
\(166\) 0 0
\(167\) 4.00000 0.309529 0.154765 0.987951i \(-0.450538\pi\)
0.154765 + 0.987951i \(0.450538\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) 4.00000i 0.305888i
\(172\) 0 0
\(173\) − 16.0000i − 1.21646i −0.793762 0.608229i \(-0.791880\pi\)
0.793762 0.608229i \(-0.208120\pi\)
\(174\) 0 0
\(175\) 11.0000 0.831522
\(176\) 0 0
\(177\) 8.00000 0.601317
\(178\) 0 0
\(179\) − 18.0000i − 1.34538i −0.739923 0.672692i \(-0.765138\pi\)
0.739923 0.672692i \(-0.234862\pi\)
\(180\) 0 0
\(181\) − 6.00000i − 0.445976i −0.974821 0.222988i \(-0.928419\pi\)
0.974821 0.222988i \(-0.0715812\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 8.00000i 0.585018i
\(188\) 0 0
\(189\) − 1.00000i − 0.0727393i
\(190\) 0 0
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 0 0
\(195\) 24.0000i 1.71868i
\(196\) 0 0
\(197\) − 22.0000i − 1.56744i −0.621117 0.783718i \(-0.713321\pi\)
0.621117 0.783718i \(-0.286679\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −8.00000 −0.564276
\(202\) 0 0
\(203\) − 2.00000i − 0.140372i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −2.00000 −0.139010
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) 0 0
\(211\) − 4.00000i − 0.275371i −0.990476 0.137686i \(-0.956034\pi\)
0.990476 0.137686i \(-0.0439664\pi\)
\(212\) 0 0
\(213\) − 14.0000i − 0.959264i
\(214\) 0 0
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 2.00000i − 0.135147i
\(220\) 0 0
\(221\) − 24.0000i − 1.61441i
\(222\) 0 0
\(223\) −8.00000 −0.535720 −0.267860 0.963458i \(-0.586316\pi\)
−0.267860 + 0.963458i \(0.586316\pi\)
\(224\) 0 0
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 10.0000i 0.660819i 0.943838 + 0.330409i \(0.107187\pi\)
−0.943838 + 0.330409i \(0.892813\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) 0 0
\(235\) − 48.0000i − 3.13117i
\(236\) 0 0
\(237\) 12.0000i 0.779484i
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 4.00000i 0.255551i
\(246\) 0 0
\(247\) 24.0000 1.52708
\(248\) 0 0
\(249\) −4.00000 −0.253490
\(250\) 0 0
\(251\) 24.0000i 1.51487i 0.652913 + 0.757433i \(0.273547\pi\)
−0.652913 + 0.757433i \(0.726453\pi\)
\(252\) 0 0
\(253\) − 4.00000i − 0.251478i
\(254\) 0 0
\(255\) −16.0000 −1.00196
\(256\) 0 0
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 0 0
\(259\) − 2.00000i − 0.124274i
\(260\) 0 0
\(261\) − 2.00000i − 0.123797i
\(262\) 0 0
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 24.0000 1.47431
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 20.0000i 1.21942i 0.792624 + 0.609711i \(0.208714\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) −6.00000 −0.363137
\(274\) 0 0
\(275\) 22.0000i 1.32665i
\(276\) 0 0
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) − 16.0000i − 0.947758i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 2.00000i 0.117242i
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) −32.0000 −1.86311
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) 12.0000i 0.693978i
\(300\) 0 0
\(301\) − 4.00000i − 0.230556i
\(302\) 0 0
\(303\) 16.0000 0.919176
\(304\) 0 0
\(305\) 24.0000 1.37424
\(306\) 0 0
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) 16.0000i 0.910208i
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 0 0
\(315\) 4.00000i 0.225374i
\(316\) 0 0
\(317\) − 18.0000i − 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 0 0
\(323\) 16.0000i 0.890264i
\(324\) 0 0
\(325\) − 66.0000i − 3.66102i
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 28.0000i 1.53902i 0.638635 + 0.769510i \(0.279499\pi\)
−0.638635 + 0.769510i \(0.720501\pi\)
\(332\) 0 0
\(333\) − 2.00000i − 0.109599i
\(334\) 0 0
\(335\) 32.0000 1.74835
\(336\) 0 0
\(337\) −6.00000 −0.326841 −0.163420 0.986557i \(-0.552253\pi\)
−0.163420 + 0.986557i \(0.552253\pi\)
\(338\) 0 0
\(339\) − 10.0000i − 0.543125i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 8.00000 0.430706
\(346\) 0 0
\(347\) − 6.00000i − 0.322097i −0.986947 0.161048i \(-0.948512\pi\)
0.986947 0.161048i \(-0.0514875\pi\)
\(348\) 0 0
\(349\) − 22.0000i − 1.17763i −0.808267 0.588817i \(-0.799594\pi\)
0.808267 0.588817i \(-0.200406\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) 0 0
\(353\) 12.0000 0.638696 0.319348 0.947638i \(-0.396536\pi\)
0.319348 + 0.947638i \(0.396536\pi\)
\(354\) 0 0
\(355\) 56.0000i 2.97217i
\(356\) 0 0
\(357\) − 4.00000i − 0.211702i
\(358\) 0 0
\(359\) −2.00000 −0.105556 −0.0527780 0.998606i \(-0.516808\pi\)
−0.0527780 + 0.998606i \(0.516808\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) − 7.00000i − 0.367405i
\(364\) 0 0
\(365\) 8.00000i 0.418739i
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.00000i 0.311504i
\(372\) 0 0
\(373\) 22.0000i 1.13912i 0.821951 + 0.569558i \(0.192886\pi\)
−0.821951 + 0.569558i \(0.807114\pi\)
\(374\) 0 0
\(375\) −24.0000 −1.23935
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) − 4.00000i − 0.205466i −0.994709 0.102733i \(-0.967241\pi\)
0.994709 0.102733i \(-0.0327588\pi\)
\(380\) 0 0
\(381\) 12.0000i 0.614779i
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) −8.00000 −0.407718
\(386\) 0 0
\(387\) − 4.00000i − 0.203331i
\(388\) 0 0
\(389\) 22.0000i 1.11544i 0.830028 + 0.557722i \(0.188325\pi\)
−0.830028 + 0.557722i \(0.811675\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) − 48.0000i − 2.41514i
\(396\) 0 0
\(397\) 18.0000i 0.903394i 0.892171 + 0.451697i \(0.149181\pi\)
−0.892171 + 0.451697i \(0.850819\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 4.00000i 0.198762i
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) −38.0000 −1.87898 −0.939490 0.342578i \(-0.888700\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) − 2.00000i − 0.0986527i
\(412\) 0 0
\(413\) − 8.00000i − 0.393654i
\(414\) 0 0
\(415\) 16.0000 0.785409
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) − 6.00000i − 0.292422i −0.989253 0.146211i \(-0.953292\pi\)
0.989253 0.146211i \(-0.0467079\pi\)
\(422\) 0 0
\(423\) 12.0000 0.583460
\(424\) 0 0
\(425\) 44.0000 2.13431
\(426\) 0 0
\(427\) 6.00000i 0.290360i
\(428\) 0 0
\(429\) − 12.0000i − 0.579365i
\(430\) 0 0
\(431\) −30.0000 −1.44505 −0.722525 0.691345i \(-0.757018\pi\)
−0.722525 + 0.691345i \(0.757018\pi\)
\(432\) 0 0
\(433\) 26.0000 1.24948 0.624740 0.780833i \(-0.285205\pi\)
0.624740 + 0.780833i \(0.285205\pi\)
\(434\) 0 0
\(435\) 8.00000i 0.383571i
\(436\) 0 0
\(437\) − 8.00000i − 0.382692i
\(438\) 0 0
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) −1.00000 −0.0476190
\(442\) 0 0
\(443\) − 38.0000i − 1.80543i −0.430234 0.902717i \(-0.641569\pi\)
0.430234 0.902717i \(-0.358431\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 8.00000i 0.375873i
\(454\) 0 0
\(455\) 24.0000 1.12514
\(456\) 0 0
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) 0 0
\(459\) − 4.00000i − 0.186704i
\(460\) 0 0
\(461\) − 40.0000i − 1.86299i −0.363760 0.931493i \(-0.618507\pi\)
0.363760 0.931493i \(-0.381493\pi\)
\(462\) 0 0
\(463\) −4.00000 −0.185896 −0.0929479 0.995671i \(-0.529629\pi\)
−0.0929479 + 0.995671i \(0.529629\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 8.00000i − 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) 0 0
\(469\) 8.00000i 0.369406i
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 44.0000i 2.01886i
\(476\) 0 0
\(477\) 6.00000i 0.274721i
\(478\) 0 0
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) 0 0
\(483\) 2.00000i 0.0910032i
\(484\) 0 0
\(485\) − 8.00000i − 0.363261i
\(486\) 0 0
\(487\) −16.0000 −0.725029 −0.362515 0.931978i \(-0.618082\pi\)
−0.362515 + 0.931978i \(0.618082\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 42.0000i 1.89543i 0.319113 + 0.947717i \(0.396615\pi\)
−0.319113 + 0.947717i \(0.603385\pi\)
\(492\) 0 0
\(493\) − 8.00000i − 0.360302i
\(494\) 0 0
\(495\) −8.00000 −0.359573
\(496\) 0 0
\(497\) −14.0000 −0.627986
\(498\) 0 0
\(499\) − 20.0000i − 0.895323i −0.894203 0.447661i \(-0.852257\pi\)
0.894203 0.447661i \(-0.147743\pi\)
\(500\) 0 0
\(501\) − 4.00000i − 0.178707i
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −64.0000 −2.84796
\(506\) 0 0
\(507\) 23.0000i 1.02147i
\(508\) 0 0
\(509\) 36.0000i 1.59567i 0.602875 + 0.797836i \(0.294022\pi\)
−0.602875 + 0.797836i \(0.705978\pi\)
\(510\) 0 0
\(511\) −2.00000 −0.0884748
\(512\) 0 0
\(513\) 4.00000 0.176604
\(514\) 0 0
\(515\) − 64.0000i − 2.82018i
\(516\) 0 0
\(517\) 24.0000i 1.05552i
\(518\) 0 0
\(519\) −16.0000 −0.702322
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) 28.0000i 1.22435i 0.790721 + 0.612177i \(0.209706\pi\)
−0.790721 + 0.612177i \(0.790294\pi\)
\(524\) 0 0
\(525\) − 11.0000i − 0.480079i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) − 8.00000i − 0.347170i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 72.0000 3.11283
\(536\) 0 0
\(537\) −18.0000 −0.776757
\(538\) 0 0
\(539\) − 2.00000i − 0.0861461i
\(540\) 0 0
\(541\) 30.0000i 1.28980i 0.764267 + 0.644900i \(0.223101\pi\)
−0.764267 + 0.644900i \(0.776899\pi\)
\(542\) 0 0
\(543\) −6.00000 −0.257485
\(544\) 0 0
\(545\) −8.00000 −0.342682
\(546\) 0 0
\(547\) 24.0000i 1.02617i 0.858339 + 0.513083i \(0.171497\pi\)
−0.858339 + 0.513083i \(0.828503\pi\)
\(548\) 0 0
\(549\) 6.00000i 0.256074i
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 0 0
\(555\) 8.00000i 0.339581i
\(556\) 0 0
\(557\) 38.0000i 1.61011i 0.593199 + 0.805056i \(0.297865\pi\)
−0.593199 + 0.805056i \(0.702135\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) 8.00000i 0.337160i 0.985688 + 0.168580i \(0.0539181\pi\)
−0.985688 + 0.168580i \(0.946082\pi\)
\(564\) 0 0
\(565\) 40.0000i 1.68281i
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −30.0000 −1.25767 −0.628833 0.777541i \(-0.716467\pi\)
−0.628833 + 0.777541i \(0.716467\pi\)
\(570\) 0 0
\(571\) 24.0000i 1.00437i 0.864761 + 0.502184i \(0.167470\pi\)
−0.864761 + 0.502184i \(0.832530\pi\)
\(572\) 0 0
\(573\) − 18.0000i − 0.751961i
\(574\) 0 0
\(575\) −22.0000 −0.917463
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 0 0
\(579\) 10.0000i 0.415586i
\(580\) 0 0
\(581\) 4.00000i 0.165948i
\(582\) 0 0
\(583\) −12.0000 −0.496989
\(584\) 0 0
\(585\) 24.0000 0.992278
\(586\) 0 0
\(587\) − 32.0000i − 1.32078i −0.750922 0.660391i \(-0.770391\pi\)
0.750922 0.660391i \(-0.229609\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −22.0000 −0.904959
\(592\) 0 0
\(593\) 12.0000 0.492781 0.246390 0.969171i \(-0.420755\pi\)
0.246390 + 0.969171i \(0.420755\pi\)
\(594\) 0 0
\(595\) 16.0000i 0.655936i
\(596\) 0 0
\(597\) 16.0000i 0.654836i
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) 46.0000 1.87638 0.938190 0.346122i \(-0.112502\pi\)
0.938190 + 0.346122i \(0.112502\pi\)
\(602\) 0 0
\(603\) 8.00000i 0.325785i
\(604\) 0 0
\(605\) 28.0000i 1.13836i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) −2.00000 −0.0810441
\(610\) 0 0
\(611\) − 72.0000i − 2.91281i
\(612\) 0 0
\(613\) 6.00000i 0.242338i 0.992632 + 0.121169i \(0.0386643\pi\)
−0.992632 + 0.121169i \(0.961336\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −38.0000 −1.52982 −0.764911 0.644136i \(-0.777217\pi\)
−0.764911 + 0.644136i \(0.777217\pi\)
\(618\) 0 0
\(619\) 20.0000i 0.803868i 0.915669 + 0.401934i \(0.131662\pi\)
−0.915669 + 0.401934i \(0.868338\pi\)
\(620\) 0 0
\(621\) 2.00000i 0.0802572i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 0 0
\(627\) 8.00000i 0.319489i
\(628\) 0 0
\(629\) − 8.00000i − 0.318981i
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) − 48.0000i − 1.90482i
\(636\) 0 0
\(637\) 6.00000i 0.237729i
\(638\) 0 0
\(639\) −14.0000 −0.553831
\(640\) 0 0
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 0 0
\(643\) 44.0000i 1.73519i 0.497271 + 0.867595i \(0.334335\pi\)
−0.497271 + 0.867595i \(0.665665\pi\)
\(644\) 0 0
\(645\) 16.0000i 0.629999i
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 16.0000 0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 6.00000i − 0.234798i −0.993085 0.117399i \(-0.962544\pi\)
0.993085 0.117399i \(-0.0374557\pi\)
\(654\) 0 0
\(655\) −16.0000 −0.625172
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 2.00000i 0.0779089i 0.999241 + 0.0389545i \(0.0124027\pi\)
−0.999241 + 0.0389545i \(0.987597\pi\)
\(660\) 0 0
\(661\) 22.0000i 0.855701i 0.903850 + 0.427850i \(0.140729\pi\)
−0.903850 + 0.427850i \(0.859271\pi\)
\(662\) 0 0
\(663\) −24.0000 −0.932083
\(664\) 0 0
\(665\) −16.0000 −0.620453
\(666\) 0 0
\(667\) 4.00000i 0.154881i
\(668\) 0 0
\(669\) 8.00000i 0.309298i
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) − 11.0000i − 0.423390i
\(676\) 0 0
\(677\) − 24.0000i − 0.922395i −0.887298 0.461197i \(-0.847420\pi\)
0.887298 0.461197i \(-0.152580\pi\)
\(678\) 0 0
\(679\) 2.00000 0.0767530
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 30.0000i − 1.14792i −0.818884 0.573959i \(-0.805407\pi\)
0.818884 0.573959i \(-0.194593\pi\)
\(684\) 0 0
\(685\) 8.00000i 0.305664i
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 0 0
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) − 28.0000i − 1.06517i −0.846376 0.532585i \(-0.821221\pi\)
0.846376 0.532585i \(-0.178779\pi\)
\(692\) 0 0
\(693\) − 2.00000i − 0.0759737i
\(694\) 0 0
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 14.0000i 0.529529i
\(700\) 0 0
\(701\) 50.0000i 1.88847i 0.329267 + 0.944237i \(0.393198\pi\)
−0.329267 + 0.944237i \(0.606802\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) 0 0
\(705\) −48.0000 −1.80778
\(706\) 0 0
\(707\) − 16.0000i − 0.601742i
\(708\) 0 0
\(709\) 38.0000i 1.42712i 0.700594 + 0.713560i \(0.252918\pi\)
−0.700594 + 0.713560i \(0.747082\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 48.0000i 1.79510i
\(716\) 0 0
\(717\) − 6.00000i − 0.224074i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) − 14.0000i − 0.520666i
\(724\) 0 0
\(725\) − 22.0000i − 0.817059i
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) − 16.0000i − 0.591781i
\(732\) 0 0
\(733\) 6.00000i 0.221615i 0.993842 + 0.110808i \(0.0353437\pi\)
−0.993842 + 0.110808i \(0.964656\pi\)
\(734\) 0 0
\(735\) 4.00000 0.147542
\(736\) 0 0
\(737\) −16.0000 −0.589368
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) − 24.0000i − 0.881662i
\(742\) 0 0
\(743\) 30.0000 1.10059 0.550297 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) 0 0
\(745\) 24.0000 0.879292
\(746\) 0 0
\(747\) 4.00000i 0.146352i
\(748\) 0 0
\(749\) 18.0000i 0.657706i
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) − 32.0000i − 1.16460i
\(756\) 0 0
\(757\) − 42.0000i − 1.52652i −0.646094 0.763258i \(-0.723599\pi\)
0.646094 0.763258i \(-0.276401\pi\)
\(758\) 0 0
\(759\) −4.00000 −0.145191
\(760\) 0 0
\(761\) −32.0000 −1.16000 −0.580000 0.814617i \(-0.696947\pi\)
−0.580000 + 0.814617i \(0.696947\pi\)
\(762\) 0 0
\(763\) − 2.00000i − 0.0724049i
\(764\) 0 0
\(765\) 16.0000i 0.578481i
\(766\) 0 0
\(767\) −48.0000 −1.73318
\(768\) 0 0
\(769\) −38.0000 −1.37032 −0.685158 0.728395i \(-0.740267\pi\)
−0.685158 + 0.728395i \(0.740267\pi\)
\(770\) 0 0
\(771\) 24.0000i 0.864339i
\(772\) 0 0
\(773\) − 8.00000i − 0.287740i −0.989597 0.143870i \(-0.954045\pi\)
0.989597 0.143870i \(-0.0459547\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −2.00000 −0.0717496
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) − 28.0000i − 1.00192i
\(782\) 0 0
\(783\) −2.00000 −0.0714742
\(784\) 0 0
\(785\) 56.0000 1.99873
\(786\) 0 0
\(787\) − 12.0000i − 0.427754i −0.976861 0.213877i \(-0.931391\pi\)
0.976861 0.213877i \(-0.0686091\pi\)
\(788\) 0 0
\(789\) − 18.0000i − 0.640817i
\(790\) 0 0
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) 36.0000 1.27840
\(794\) 0 0
\(795\) − 24.0000i − 0.851192i
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 48.0000 1.69812
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 4.00000i − 0.141157i
\(804\) 0 0
\(805\) − 8.00000i − 0.281963i
\(806\) 0 0
\(807\) 20.0000 0.704033
\(808\) 0 0
\(809\) −10.0000 −0.351581 −0.175791 0.984428i \(-0.556248\pi\)
−0.175791 + 0.984428i \(0.556248\pi\)
\(810\) 0 0
\(811\) − 28.0000i − 0.983213i −0.870817 0.491606i \(-0.836410\pi\)
0.870817 0.491606i \(-0.163590\pi\)
\(812\) 0 0
\(813\) 24.0000i 0.841717i
\(814\) 0 0
\(815\) 64.0000 2.24182
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 6.00000i 0.209657i
\(820\) 0 0
\(821\) 2.00000i 0.0698005i 0.999391 + 0.0349002i \(0.0111113\pi\)
−0.999391 + 0.0349002i \(0.988889\pi\)
\(822\) 0 0
\(823\) −20.0000 −0.697156 −0.348578 0.937280i \(-0.613335\pi\)
−0.348578 + 0.937280i \(0.613335\pi\)
\(824\) 0 0
\(825\) 22.0000 0.765942
\(826\) 0 0
\(827\) − 18.0000i − 0.625921i −0.949766 0.312961i \(-0.898679\pi\)
0.949766 0.312961i \(-0.101321\pi\)
\(828\) 0 0
\(829\) 14.0000i 0.486240i 0.969996 + 0.243120i \(0.0781709\pi\)
−0.969996 + 0.243120i \(0.921829\pi\)
\(830\) 0 0
\(831\) −22.0000 −0.763172
\(832\) 0 0
\(833\) −4.00000 −0.138592
\(834\) 0 0
\(835\) 16.0000i 0.553703i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 44.0000 1.51905 0.759524 0.650479i \(-0.225432\pi\)
0.759524 + 0.650479i \(0.225432\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) − 10.0000i − 0.344418i
\(844\) 0 0
\(845\) − 92.0000i − 3.16490i
\(846\) 0 0
\(847\) −7.00000 −0.240523
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 4.00000i 0.137118i
\(852\) 0 0
\(853\) 14.0000i 0.479351i 0.970853 + 0.239675i \(0.0770410\pi\)
−0.970853 + 0.239675i \(0.922959\pi\)
\(854\) 0 0
\(855\) −16.0000 −0.547188
\(856\) 0 0
\(857\) 48.0000 1.63965 0.819824 0.572615i \(-0.194071\pi\)
0.819824 + 0.572615i \(0.194071\pi\)
\(858\) 0 0
\(859\) − 44.0000i − 1.50126i −0.660722 0.750630i \(-0.729750\pi\)
0.660722 0.750630i \(-0.270250\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 14.0000 0.476566 0.238283 0.971196i \(-0.423415\pi\)
0.238283 + 0.971196i \(0.423415\pi\)
\(864\) 0 0
\(865\) 64.0000 2.17607
\(866\) 0 0
\(867\) 1.00000i 0.0339618i
\(868\) 0 0
\(869\) 24.0000i 0.814144i
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 24.0000i 0.811348i
\(876\) 0 0
\(877\) − 46.0000i − 1.55331i −0.629926 0.776655i \(-0.716915\pi\)
0.629926 0.776655i \(-0.283085\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 36.0000 1.21287 0.606435 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) 0 0
\(883\) 44.0000i 1.48072i 0.672212 + 0.740359i \(0.265344\pi\)
−0.672212 + 0.740359i \(0.734656\pi\)
\(884\) 0 0
\(885\) 32.0000i 1.07567i
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) − 2.00000i − 0.0670025i
\(892\) 0 0
\(893\) 48.0000i 1.60626i
\(894\) 0 0
\(895\) 72.0000 2.40669
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 24.0000i 0.799556i
\(902\) 0 0
\(903\) −4.00000 −0.133112
\(904\) 0 0
\(905\) 24.0000 0.797787
\(906\) 0 0
\(907\) − 12.0000i − 0.398453i −0.979953 0.199227i \(-0.936157\pi\)
0.979953 0.199227i \(-0.0638430\pi\)
\(908\) 0 0
\(909\) − 16.0000i − 0.530687i
\(910\) 0 0
\(911\) 18.0000 0.596367 0.298183 0.954509i \(-0.403619\pi\)
0.298183 + 0.954509i \(0.403619\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) 0 0
\(915\) − 24.0000i − 0.793416i
\(916\) 0 0
\(917\) − 4.00000i − 0.132092i
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) 84.0000i 2.76489i
\(924\) 0 0
\(925\) − 22.0000i − 0.723356i
\(926\) 0 0
\(927\) 16.0000 0.525509
\(928\) 0 0
\(929\) 48.0000 1.57483 0.787414 0.616424i \(-0.211419\pi\)
0.787414 + 0.616424i \(0.211419\pi\)
\(930\) 0 0
\(931\) − 4.00000i − 0.131095i
\(932\) 0 0
\(933\) − 4.00000i − 0.130954i
\(934\) 0 0
\(935\) −32.0000 −1.04651
\(936\) 0 0
\(937\) −2.00000 −0.0653372 −0.0326686 0.999466i \(-0.510401\pi\)
−0.0326686 + 0.999466i \(0.510401\pi\)
\(938\) 0 0
\(939\) 10.0000i 0.326338i
\(940\) 0 0
\(941\) − 28.0000i − 0.912774i −0.889781 0.456387i \(-0.849143\pi\)
0.889781 0.456387i \(-0.150857\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 4.00000 0.130120
\(946\) 0 0
\(947\) 10.0000i 0.324956i 0.986712 + 0.162478i \(0.0519487\pi\)
−0.986712 + 0.162478i \(0.948051\pi\)
\(948\) 0 0
\(949\) 12.0000i 0.389536i
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 22.0000 0.712650 0.356325 0.934362i \(-0.384030\pi\)
0.356325 + 0.934362i \(0.384030\pi\)
\(954\) 0 0
\(955\) 72.0000i 2.32987i
\(956\) 0 0
\(957\) − 4.00000i − 0.129302i
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 18.0000i 0.580042i
\(964\) 0 0
\(965\) − 40.0000i − 1.28765i
\(966\) 0 0
\(967\) 20.0000 0.643157 0.321578 0.946883i \(-0.395787\pi\)
0.321578 + 0.946883i \(0.395787\pi\)
\(968\) 0 0
\(969\) 16.0000 0.513994
\(970\) 0 0
\(971\) − 36.0000i − 1.15529i −0.816286 0.577647i \(-0.803971\pi\)
0.816286 0.577647i \(-0.196029\pi\)
\(972\) 0 0
\(973\) 4.00000i 0.128234i
\(974\) 0 0
\(975\) −66.0000 −2.11369
\(976\) 0 0
\(977\) 30.0000 0.959785 0.479893 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) − 2.00000i − 0.0638551i
\(982\) 0 0
\(983\) −48.0000 −1.53096 −0.765481 0.643458i \(-0.777499\pi\)
−0.765481 + 0.643458i \(0.777499\pi\)
\(984\) 0 0
\(985\) 88.0000 2.80391
\(986\) 0 0
\(987\) − 12.0000i − 0.381964i
\(988\) 0 0
\(989\) 8.00000i 0.254385i
\(990\) 0 0
\(991\) −56.0000 −1.77890 −0.889449 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) 0 0
\(993\) 28.0000 0.888553
\(994\) 0 0
\(995\) − 64.0000i − 2.02894i
\(996\) 0 0
\(997\) 38.0000i 1.20347i 0.798695 + 0.601736i \(0.205524\pi\)
−0.798695 + 0.601736i \(0.794476\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
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 5376.2.c.p.2689.1 2
4.3 odd 2 5376.2.c.q.2689.2 2
8.3 odd 2 5376.2.c.q.2689.1 2
8.5 even 2 inner 5376.2.c.p.2689.2 2
16.3 odd 4 84.2.a.a.1.1 1
16.5 even 4 1344.2.a.a.1.1 1
16.11 odd 4 1344.2.a.k.1.1 1
16.13 even 4 336.2.a.f.1.1 1
48.5 odd 4 4032.2.a.bn.1.1 1
48.11 even 4 4032.2.a.bm.1.1 1
48.29 odd 4 1008.2.a.a.1.1 1
48.35 even 4 252.2.a.a.1.1 1
80.3 even 4 2100.2.k.i.1849.1 2
80.19 odd 4 2100.2.a.r.1.1 1
80.29 even 4 8400.2.a.e.1.1 1
80.67 even 4 2100.2.k.i.1849.2 2
112.3 even 12 588.2.i.d.373.1 2
112.13 odd 4 2352.2.a.a.1.1 1
112.19 even 12 588.2.i.d.361.1 2
112.27 even 4 9408.2.a.bn.1.1 1
112.45 odd 12 2352.2.q.z.961.1 2
112.51 odd 12 588.2.i.e.361.1 2
112.61 odd 12 2352.2.q.z.1537.1 2
112.67 odd 12 588.2.i.e.373.1 2
112.69 odd 4 9408.2.a.df.1.1 1
112.83 even 4 588.2.a.d.1.1 1
112.93 even 12 2352.2.q.b.1537.1 2
112.109 even 12 2352.2.q.b.961.1 2
144.67 odd 12 2268.2.j.a.1513.1 2
144.83 even 12 2268.2.j.n.757.1 2
144.115 odd 12 2268.2.j.a.757.1 2
144.131 even 12 2268.2.j.n.1513.1 2
240.83 odd 4 6300.2.k.g.6049.2 2
240.179 even 4 6300.2.a.w.1.1 1
240.227 odd 4 6300.2.k.g.6049.1 2
336.83 odd 4 1764.2.a.k.1.1 1
336.125 even 4 7056.2.a.cd.1.1 1
336.131 odd 12 1764.2.k.a.361.1 2
336.179 even 12 1764.2.k.k.1549.1 2
336.227 odd 12 1764.2.k.a.1549.1 2
336.275 even 12 1764.2.k.k.361.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.2.a.a.1.1 1 16.3 odd 4
252.2.a.a.1.1 1 48.35 even 4
336.2.a.f.1.1 1 16.13 even 4
588.2.a.d.1.1 1 112.83 even 4
588.2.i.d.361.1 2 112.19 even 12
588.2.i.d.373.1 2 112.3 even 12
588.2.i.e.361.1 2 112.51 odd 12
588.2.i.e.373.1 2 112.67 odd 12
1008.2.a.a.1.1 1 48.29 odd 4
1344.2.a.a.1.1 1 16.5 even 4
1344.2.a.k.1.1 1 16.11 odd 4
1764.2.a.k.1.1 1 336.83 odd 4
1764.2.k.a.361.1 2 336.131 odd 12
1764.2.k.a.1549.1 2 336.227 odd 12
1764.2.k.k.361.1 2 336.275 even 12
1764.2.k.k.1549.1 2 336.179 even 12
2100.2.a.r.1.1 1 80.19 odd 4
2100.2.k.i.1849.1 2 80.3 even 4
2100.2.k.i.1849.2 2 80.67 even 4
2268.2.j.a.757.1 2 144.115 odd 12
2268.2.j.a.1513.1 2 144.67 odd 12
2268.2.j.n.757.1 2 144.83 even 12
2268.2.j.n.1513.1 2 144.131 even 12
2352.2.a.a.1.1 1 112.13 odd 4
2352.2.q.b.961.1 2 112.109 even 12
2352.2.q.b.1537.1 2 112.93 even 12
2352.2.q.z.961.1 2 112.45 odd 12
2352.2.q.z.1537.1 2 112.61 odd 12
4032.2.a.bm.1.1 1 48.11 even 4
4032.2.a.bn.1.1 1 48.5 odd 4
5376.2.c.p.2689.1 2 1.1 even 1 trivial
5376.2.c.p.2689.2 2 8.5 even 2 inner
5376.2.c.q.2689.1 2 8.3 odd 2
5376.2.c.q.2689.2 2 4.3 odd 2
6300.2.a.w.1.1 1 240.179 even 4
6300.2.k.g.6049.1 2 240.227 odd 4
6300.2.k.g.6049.2 2 240.83 odd 4
7056.2.a.cd.1.1 1 336.125 even 4
8400.2.a.e.1.1 1 80.29 even 4
9408.2.a.bn.1.1 1 112.27 even 4
9408.2.a.df.1.1 1 112.69 odd 4