Properties

Label 2175.2.c.b.349.1
Level $2175$
Weight $2$
Character 2175.349
Analytic conductor $17.367$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,2,Mod(349,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2175.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.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: \(17.3674624396\)
Analytic rank: \(0\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 435)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2175.349
Dual form 2175.2.c.b.349.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} -4.00000i q^{7} -3.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +4.00000 q^{19} +4.00000 q^{21} +4.00000i q^{22} -4.00000i q^{23} +3.00000 q^{24} +6.00000 q^{26} -1.00000i q^{27} -4.00000i q^{28} -1.00000 q^{29} -8.00000 q^{31} -5.00000i q^{32} -4.00000i q^{33} -6.00000 q^{34} -1.00000 q^{36} -2.00000i q^{37} -4.00000i q^{38} -6.00000 q^{39} -6.00000 q^{41} -4.00000i q^{42} +4.00000i q^{43} -4.00000 q^{44} -4.00000 q^{46} -1.00000i q^{48} -9.00000 q^{49} +6.00000 q^{51} +6.00000i q^{52} -10.0000i q^{53} -1.00000 q^{54} -12.0000 q^{56} +4.00000i q^{57} +1.00000i q^{58} +12.0000 q^{59} -10.0000 q^{61} +8.00000i q^{62} +4.00000i q^{63} -7.00000 q^{64} -4.00000 q^{66} -8.00000i q^{67} -6.00000i q^{68} +4.00000 q^{69} -8.00000 q^{71} +3.00000i q^{72} -2.00000i q^{73} -2.00000 q^{74} +4.00000 q^{76} +16.0000i q^{77} +6.00000i q^{78} +1.00000 q^{81} +6.00000i q^{82} +8.00000i q^{83} +4.00000 q^{84} +4.00000 q^{86} -1.00000i q^{87} +12.0000i q^{88} +6.00000 q^{89} +24.0000 q^{91} -4.00000i q^{92} -8.00000i q^{93} +5.00000 q^{96} +2.00000i q^{97} +9.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} - 8 q^{14} - 2 q^{16} + 8 q^{19} + 8 q^{21} + 6 q^{24} + 12 q^{26} - 2 q^{29} - 16 q^{31} - 12 q^{34} - 2 q^{36} - 12 q^{39} - 12 q^{41} - 8 q^{44} - 8 q^{46} - 18 q^{49} + 12 q^{51} - 2 q^{54} - 24 q^{56} + 24 q^{59} - 20 q^{61} - 14 q^{64} - 8 q^{66} + 8 q^{69} - 16 q^{71} - 4 q^{74} + 8 q^{76} + 2 q^{81} + 8 q^{84} + 8 q^{86} + 12 q^{89} + 48 q^{91} + 10 q^{96} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i −0.935414 0.353553i \(-0.884973\pi\)
0.935414 0.353553i \(-0.115027\pi\)
\(3\) 1.00000i 0.577350i
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 4.00000 0.872872
\(22\) 4.00000i 0.852803i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) − 1.00000i − 0.192450i
\(28\) − 4.00000i − 0.755929i
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) − 5.00000i − 0.883883i
\(33\) − 4.00000i − 0.696311i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) −6.00000 −0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) − 4.00000i − 0.617213i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −4.00000 −0.589768
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 6.00000i 0.832050i
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −12.0000 −1.60357
\(57\) 4.00000i 0.529813i
\(58\) 1.00000i 0.131306i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 4.00000i 0.503953i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 3.00000i 0.353553i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 16.0000i 1.82337i
\(78\) 6.00000i 0.679366i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000i 0.662589i
\(83\) 8.00000i 0.878114i 0.898459 + 0.439057i \(0.144687\pi\)
−0.898459 + 0.439057i \(0.855313\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) − 1.00000i − 0.107211i
\(88\) 12.0000i 1.27920i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 24.0000 2.51588
\(92\) − 4.00000i − 0.417029i
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) 5.00000 0.510310
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) − 12.0000i − 1.18240i −0.806527 0.591198i \(-0.798655\pi\)
0.806527 0.591198i \(-0.201345\pi\)
\(104\) 18.0000 1.76505
\(105\) 0 0
\(106\) −10.0000 −0.971286
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 4.00000i 0.377964i
\(113\) − 2.00000i − 0.188144i −0.995565 0.0940721i \(-0.970012\pi\)
0.995565 0.0940721i \(-0.0299884\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) − 6.00000i − 0.554700i
\(118\) − 12.0000i − 1.10469i
\(119\) −24.0000 −2.20008
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 10.0000i 0.905357i
\(123\) − 6.00000i − 0.541002i
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) − 3.00000i − 0.265165i
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) − 16.0000i − 1.38738i
\(134\) −8.00000 −0.691095
\(135\) 0 0
\(136\) −18.0000 −1.54349
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 8.00000i 0.671345i
\(143\) − 24.0000i − 2.00698i
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) − 9.00000i − 0.742307i
\(148\) − 2.00000i − 0.164399i
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) − 12.0000i − 0.973329i
\(153\) 6.00000i 0.485071i
\(154\) 16.0000 1.28932
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) − 18.0000i − 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) − 1.00000i − 0.0785674i
\(163\) − 12.0000i − 0.939913i −0.882690 0.469956i \(-0.844270\pi\)
0.882690 0.469956i \(-0.155730\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) − 12.0000i − 0.925820i
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 4.00000i 0.304997i
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 12.0000i 0.901975i
\(178\) − 6.00000i − 0.449719i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) − 24.0000i − 1.77900i
\(183\) − 10.0000i − 0.739221i
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 24.0000i 1.75505i
\(188\) 0 0
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) − 7.00000i − 0.505181i
\(193\) 22.0000i 1.58359i 0.610784 + 0.791797i \(0.290854\pi\)
−0.610784 + 0.791797i \(0.709146\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) − 14.0000i − 0.997459i −0.866758 0.498729i \(-0.833800\pi\)
0.866758 0.498729i \(-0.166200\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) 24.0000 1.70131 0.850657 0.525720i \(-0.176204\pi\)
0.850657 + 0.525720i \(0.176204\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 2.00000i 0.140720i
\(203\) 4.00000i 0.280745i
\(204\) 6.00000 0.420084
\(205\) 0 0
\(206\) −12.0000 −0.836080
\(207\) 4.00000i 0.278019i
\(208\) − 6.00000i − 0.416025i
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) − 10.0000i − 0.686803i
\(213\) − 8.00000i − 0.548151i
\(214\) −8.00000 −0.546869
\(215\) 0 0
\(216\) −3.00000 −0.204124
\(217\) 32.0000i 2.17230i
\(218\) 14.0000i 0.948200i
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 36.0000 2.42162
\(222\) − 2.00000i − 0.134231i
\(223\) 28.0000i 1.87502i 0.347960 + 0.937509i \(0.386874\pi\)
−0.347960 + 0.937509i \(0.613126\pi\)
\(224\) −20.0000 −1.33631
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) − 24.0000i − 1.59294i −0.604681 0.796468i \(-0.706699\pi\)
0.604681 0.796468i \(-0.293301\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) −16.0000 −1.05272
\(232\) 3.00000i 0.196960i
\(233\) − 22.0000i − 1.44127i −0.693316 0.720634i \(-0.743851\pi\)
0.693316 0.720634i \(-0.256149\pi\)
\(234\) −6.00000 −0.392232
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 0 0
\(238\) 24.0000i 1.55569i
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) − 5.00000i − 0.321412i
\(243\) 1.00000i 0.0641500i
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 24.0000i 1.52708i
\(248\) 24.0000i 1.52400i
\(249\) −8.00000 −0.506979
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 4.00000i 0.251976i
\(253\) 16.0000i 1.00591i
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 22.0000i 1.37232i 0.727450 + 0.686161i \(0.240706\pi\)
−0.727450 + 0.686161i \(0.759294\pi\)
\(258\) 4.00000i 0.249029i
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) − 12.0000i − 0.741362i
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) −12.0000 −0.738549
\(265\) 0 0
\(266\) −16.0000 −0.981023
\(267\) 6.00000i 0.367194i
\(268\) − 8.00000i − 0.488678i
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 6.00000i 0.363803i
\(273\) 24.0000i 1.45255i
\(274\) 2.00000 0.120824
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) 26.0000i 1.56219i 0.624413 + 0.781094i \(0.285338\pi\)
−0.624413 + 0.781094i \(0.714662\pi\)
\(278\) 4.00000i 0.239904i
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 8.00000i 0.475551i 0.971320 + 0.237775i \(0.0764182\pi\)
−0.971320 + 0.237775i \(0.923582\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −24.0000 −1.41915
\(287\) 24.0000i 1.41668i
\(288\) 5.00000i 0.294628i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) − 2.00000i − 0.117041i
\(293\) 18.0000i 1.05157i 0.850617 + 0.525786i \(0.176229\pi\)
−0.850617 + 0.525786i \(0.823771\pi\)
\(294\) −9.00000 −0.524891
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) 4.00000i 0.232104i
\(298\) 6.00000i 0.347571i
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) − 16.0000i − 0.920697i
\(303\) − 2.00000i − 0.114897i
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 16.0000i 0.911685i
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 18.0000i 1.01905i
\(313\) − 30.0000i − 1.69570i −0.530236 0.847850i \(-0.677897\pi\)
0.530236 0.847850i \(-0.322103\pi\)
\(314\) −18.0000 −1.01580
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.00000i − 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) − 10.0000i − 0.560772i
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) 8.00000 0.446516
\(322\) 16.0000i 0.891645i
\(323\) − 24.0000i − 1.33540i
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) − 14.0000i − 0.774202i
\(328\) 18.0000i 0.993884i
\(329\) 0 0
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 8.00000i 0.439057i
\(333\) 2.00000i 0.109599i
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) − 30.0000i − 1.63420i −0.576493 0.817102i \(-0.695579\pi\)
0.576493 0.817102i \(-0.304421\pi\)
\(338\) 23.0000i 1.25104i
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) 32.0000 1.73290
\(342\) 4.00000i 0.216295i
\(343\) 8.00000i 0.431959i
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) − 16.0000i − 0.858925i −0.903085 0.429463i \(-0.858703\pi\)
0.903085 0.429463i \(-0.141297\pi\)
\(348\) − 1.00000i − 0.0536056i
\(349\) 34.0000 1.81998 0.909989 0.414632i \(-0.136090\pi\)
0.909989 + 0.414632i \(0.136090\pi\)
\(350\) 0 0
\(351\) 6.00000 0.320256
\(352\) 20.0000i 1.06600i
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) − 24.0000i − 1.27021i
\(358\) − 12.0000i − 0.634220i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 22.0000i − 1.15629i
\(363\) 5.00000i 0.262432i
\(364\) 24.0000 1.25794
\(365\) 0 0
\(366\) −10.0000 −0.522708
\(367\) − 8.00000i − 0.417597i −0.977959 0.208798i \(-0.933045\pi\)
0.977959 0.208798i \(-0.0669552\pi\)
\(368\) 4.00000i 0.208514i
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −40.0000 −2.07670
\(372\) − 8.00000i − 0.414781i
\(373\) − 2.00000i − 0.103556i −0.998659 0.0517780i \(-0.983511\pi\)
0.998659 0.0517780i \(-0.0164888\pi\)
\(374\) 24.0000 1.24101
\(375\) 0 0
\(376\) 0 0
\(377\) − 6.00000i − 0.309016i
\(378\) 4.00000i 0.205738i
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 16.0000i 0.818631i
\(383\) 20.0000i 1.02195i 0.859595 + 0.510976i \(0.170716\pi\)
−0.859595 + 0.510976i \(0.829284\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) − 4.00000i − 0.203331i
\(388\) 2.00000i 0.101535i
\(389\) 2.00000 0.101404 0.0507020 0.998714i \(-0.483854\pi\)
0.0507020 + 0.998714i \(0.483854\pi\)
\(390\) 0 0
\(391\) −24.0000 −1.21373
\(392\) 27.0000i 1.36371i
\(393\) 12.0000i 0.605320i
\(394\) −14.0000 −0.705310
\(395\) 0 0
\(396\) 4.00000 0.201008
\(397\) 10.0000i 0.501886i 0.968002 + 0.250943i \(0.0807406\pi\)
−0.968002 + 0.250943i \(0.919259\pi\)
\(398\) − 24.0000i − 1.20301i
\(399\) 16.0000 0.801002
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) − 8.00000i − 0.399004i
\(403\) − 48.0000i − 2.39105i
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 4.00000 0.198517
\(407\) 8.00000i 0.396545i
\(408\) − 18.0000i − 0.891133i
\(409\) 22.0000 1.08783 0.543915 0.839140i \(-0.316941\pi\)
0.543915 + 0.839140i \(0.316941\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) − 12.0000i − 0.591198i
\(413\) − 48.0000i − 2.36193i
\(414\) 4.00000 0.196589
\(415\) 0 0
\(416\) 30.0000 1.47087
\(417\) − 4.00000i − 0.195881i
\(418\) 16.0000i 0.782586i
\(419\) 4.00000 0.195413 0.0977064 0.995215i \(-0.468849\pi\)
0.0977064 + 0.995215i \(0.468849\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) − 20.0000i − 0.973585i
\(423\) 0 0
\(424\) −30.0000 −1.45693
\(425\) 0 0
\(426\) −8.00000 −0.387601
\(427\) 40.0000i 1.93574i
\(428\) − 8.00000i − 0.386695i
\(429\) 24.0000 1.15873
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 34.0000i − 1.63394i −0.576683 0.816968i \(-0.695653\pi\)
0.576683 0.816968i \(-0.304347\pi\)
\(434\) 32.0000 1.53605
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) − 16.0000i − 0.765384i
\(438\) − 2.00000i − 0.0955637i
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) − 36.0000i − 1.71235i
\(443\) 20.0000i 0.950229i 0.879924 + 0.475114i \(0.157593\pi\)
−0.879924 + 0.475114i \(0.842407\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 28.0000 1.32584
\(447\) − 6.00000i − 0.283790i
\(448\) 28.0000i 1.32288i
\(449\) −26.0000 −1.22702 −0.613508 0.789689i \(-0.710242\pi\)
−0.613508 + 0.789689i \(0.710242\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) − 2.00000i − 0.0940721i
\(453\) 16.0000i 0.751746i
\(454\) −24.0000 −1.12638
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) − 2.00000i − 0.0934539i
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 6.00000 0.279448 0.139724 0.990190i \(-0.455378\pi\)
0.139724 + 0.990190i \(0.455378\pi\)
\(462\) 16.0000i 0.744387i
\(463\) 36.0000i 1.67306i 0.547920 + 0.836531i \(0.315420\pi\)
−0.547920 + 0.836531i \(0.684580\pi\)
\(464\) 1.00000 0.0464238
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) 20.0000i 0.925490i 0.886492 + 0.462745i \(0.153135\pi\)
−0.886492 + 0.462745i \(0.846865\pi\)
\(468\) − 6.00000i − 0.277350i
\(469\) −32.0000 −1.47762
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) − 36.0000i − 1.65703i
\(473\) − 16.0000i − 0.735681i
\(474\) 0 0
\(475\) 0 0
\(476\) −24.0000 −1.10004
\(477\) 10.0000i 0.457869i
\(478\) − 8.00000i − 0.365911i
\(479\) 16.0000 0.731059 0.365529 0.930800i \(-0.380888\pi\)
0.365529 + 0.930800i \(0.380888\pi\)
\(480\) 0 0
\(481\) 12.0000 0.547153
\(482\) − 2.00000i − 0.0910975i
\(483\) − 16.0000i − 0.728025i
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 28.0000i 1.26880i 0.773004 + 0.634401i \(0.218753\pi\)
−0.773004 + 0.634401i \(0.781247\pi\)
\(488\) 30.0000i 1.35804i
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −44.0000 −1.98569 −0.992846 0.119401i \(-0.961903\pi\)
−0.992846 + 0.119401i \(0.961903\pi\)
\(492\) − 6.00000i − 0.270501i
\(493\) 6.00000i 0.270226i
\(494\) 24.0000 1.07981
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 32.0000i 1.43540i
\(498\) 8.00000i 0.358489i
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 20.0000i 0.892644i
\(503\) 24.0000i 1.07011i 0.844818 + 0.535054i \(0.179709\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(504\) 12.0000 0.534522
\(505\) 0 0
\(506\) 16.0000 0.711287
\(507\) − 23.0000i − 1.02147i
\(508\) 16.0000i 0.709885i
\(509\) 2.00000 0.0886484 0.0443242 0.999017i \(-0.485887\pi\)
0.0443242 + 0.999017i \(0.485887\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) 11.0000i 0.486136i
\(513\) − 4.00000i − 0.176604i
\(514\) 22.0000 0.970378
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) 8.00000i 0.351500i
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) − 1.00000i − 0.0437688i
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 0 0
\(527\) 48.0000i 2.09091i
\(528\) 4.00000i 0.174078i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) − 16.0000i − 0.693688i
\(533\) − 36.0000i − 1.55933i
\(534\) 6.00000 0.259645
\(535\) 0 0
\(536\) −24.0000 −1.03664
\(537\) 12.0000i 0.517838i
\(538\) 6.00000i 0.258678i
\(539\) 36.0000 1.55063
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) − 16.0000i − 0.687259i
\(543\) 22.0000i 0.944110i
\(544\) −30.0000 −1.28624
\(545\) 0 0
\(546\) 24.0000 1.02711
\(547\) − 40.0000i − 1.71028i −0.518400 0.855138i \(-0.673472\pi\)
0.518400 0.855138i \(-0.326528\pi\)
\(548\) 2.00000i 0.0854358i
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −4.00000 −0.170406
\(552\) − 12.0000i − 0.510754i
\(553\) 0 0
\(554\) 26.0000 1.10463
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) − 30.0000i − 1.27114i −0.772043 0.635570i \(-0.780765\pi\)
0.772043 0.635570i \(-0.219235\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) −24.0000 −1.01328
\(562\) − 10.0000i − 0.421825i
\(563\) − 28.0000i − 1.18006i −0.807382 0.590030i \(-0.799116\pi\)
0.807382 0.590030i \(-0.200884\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 8.00000 0.336265
\(567\) − 4.00000i − 0.167984i
\(568\) 24.0000i 1.00702i
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) − 24.0000i − 1.00349i
\(573\) − 16.0000i − 0.668410i
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) 10.0000i 0.416305i 0.978096 + 0.208153i \(0.0667451\pi\)
−0.978096 + 0.208153i \(0.933255\pi\)
\(578\) 19.0000i 0.790296i
\(579\) −22.0000 −0.914289
\(580\) 0 0
\(581\) 32.0000 1.32758
\(582\) 2.00000i 0.0829027i
\(583\) 40.0000i 1.65663i
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 18.0000 0.743573
\(587\) 16.0000i 0.660391i 0.943913 + 0.330195i \(0.107115\pi\)
−0.943913 + 0.330195i \(0.892885\pi\)
\(588\) − 9.00000i − 0.371154i
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 14.0000 0.575883
\(592\) 2.00000i 0.0821995i
\(593\) 34.0000i 1.39621i 0.715994 + 0.698106i \(0.245974\pi\)
−0.715994 + 0.698106i \(0.754026\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 24.0000i 0.982255i
\(598\) − 24.0000i − 0.981433i
\(599\) −24.0000 −0.980613 −0.490307 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) − 16.0000i − 0.652111i
\(603\) 8.00000i 0.325785i
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) − 32.0000i − 1.29884i −0.760430 0.649420i \(-0.775012\pi\)
0.760430 0.649420i \(-0.224988\pi\)
\(608\) − 20.0000i − 0.811107i
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000i 0.242536i
\(613\) 6.00000i 0.242338i 0.992632 + 0.121169i \(0.0386643\pi\)
−0.992632 + 0.121169i \(0.961336\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) 48.0000 1.93398
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) − 12.0000i − 0.482711i
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 0 0
\(623\) − 24.0000i − 0.961540i
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) −30.0000 −1.19904
\(627\) − 16.0000i − 0.638978i
\(628\) − 18.0000i − 0.718278i
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 0 0
\(633\) 20.0000i 0.794929i
\(634\) −2.00000 −0.0794301
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) − 54.0000i − 2.13956i
\(638\) − 4.00000i − 0.158362i
\(639\) 8.00000 0.316475
\(640\) 0 0
\(641\) 34.0000 1.34292 0.671460 0.741041i \(-0.265668\pi\)
0.671460 + 0.741041i \(0.265668\pi\)
\(642\) − 8.00000i − 0.315735i
\(643\) 16.0000i 0.630978i 0.948929 + 0.315489i \(0.102169\pi\)
−0.948929 + 0.315489i \(0.897831\pi\)
\(644\) −16.0000 −0.630488
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 28.0000i 1.10079i 0.834903 + 0.550397i \(0.185524\pi\)
−0.834903 + 0.550397i \(0.814476\pi\)
\(648\) − 3.00000i − 0.117851i
\(649\) −48.0000 −1.88416
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) − 12.0000i − 0.469956i
\(653\) 50.0000i 1.95665i 0.207072 + 0.978326i \(0.433606\pi\)
−0.207072 + 0.978326i \(0.566394\pi\)
\(654\) −14.0000 −0.547443
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) − 4.00000i − 0.155464i
\(663\) 36.0000i 1.39812i
\(664\) 24.0000 0.931381
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 4.00000i 0.154881i
\(668\) − 12.0000i − 0.464294i
\(669\) −28.0000 −1.08254
\(670\) 0 0
\(671\) 40.0000 1.54418
\(672\) − 20.0000i − 0.771517i
\(673\) − 6.00000i − 0.231283i −0.993291 0.115642i \(-0.963108\pi\)
0.993291 0.115642i \(-0.0368924\pi\)
\(674\) −30.0000 −1.15556
\(675\) 0 0
\(676\) −23.0000 −0.884615
\(677\) − 50.0000i − 1.92166i −0.277145 0.960828i \(-0.589388\pi\)
0.277145 0.960828i \(-0.410612\pi\)
\(678\) − 2.00000i − 0.0768095i
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) − 32.0000i − 1.22534i
\(683\) 40.0000i 1.53056i 0.643699 + 0.765279i \(0.277399\pi\)
−0.643699 + 0.765279i \(0.722601\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) 2.00000i 0.0763048i
\(688\) − 4.00000i − 0.152499i
\(689\) 60.0000 2.28582
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 6.00000i 0.228086i
\(693\) − 16.0000i − 0.607790i
\(694\) −16.0000 −0.607352
\(695\) 0 0
\(696\) −3.00000 −0.113715
\(697\) 36.0000i 1.36360i
\(698\) − 34.0000i − 1.28692i
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) − 6.00000i − 0.226455i
\(703\) − 8.00000i − 0.301726i
\(704\) 28.0000 1.05529
\(705\) 0 0
\(706\) 18.0000 0.677439
\(707\) 8.00000i 0.300871i
\(708\) 12.0000i 0.450988i
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 18.0000i − 0.674579i
\(713\) 32.0000i 1.19841i
\(714\) −24.0000 −0.898177
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 8.00000i 0.298765i
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −48.0000 −1.78761
\(722\) 3.00000i 0.111648i
\(723\) 2.00000i 0.0743808i
\(724\) 22.0000 0.817624
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) − 16.0000i − 0.593407i −0.954970 0.296704i \(-0.904113\pi\)
0.954970 0.296704i \(-0.0958873\pi\)
\(728\) − 72.0000i − 2.66850i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) − 10.0000i − 0.369611i
\(733\) 2.00000i 0.0738717i 0.999318 + 0.0369358i \(0.0117597\pi\)
−0.999318 + 0.0369358i \(0.988240\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) −20.0000 −0.737210
\(737\) 32.0000i 1.17874i
\(738\) − 6.00000i − 0.220863i
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 0 0
\(741\) −24.0000 −0.881662
\(742\) 40.0000i 1.46845i
\(743\) − 8.00000i − 0.293492i −0.989174 0.146746i \(-0.953120\pi\)
0.989174 0.146746i \(-0.0468799\pi\)
\(744\) −24.0000 −0.879883
\(745\) 0 0
\(746\) −2.00000 −0.0732252
\(747\) − 8.00000i − 0.292705i
\(748\) 24.0000i 0.877527i
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) −8.00000 −0.291924 −0.145962 0.989290i \(-0.546628\pi\)
−0.145962 + 0.989290i \(0.546628\pi\)
\(752\) 0 0
\(753\) − 20.0000i − 0.728841i
\(754\) −6.00000 −0.218507
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 14.0000i 0.508839i 0.967094 + 0.254419i \(0.0818843\pi\)
−0.967094 + 0.254419i \(0.918116\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) −16.0000 −0.580763
\(760\) 0 0
\(761\) 26.0000 0.942499 0.471250 0.882000i \(-0.343803\pi\)
0.471250 + 0.882000i \(0.343803\pi\)
\(762\) 16.0000i 0.579619i
\(763\) 56.0000i 2.02734i
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) 20.0000 0.722629
\(767\) 72.0000i 2.59977i
\(768\) − 17.0000i − 0.613435i
\(769\) −42.0000 −1.51456 −0.757279 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) 22.0000i 0.791797i
\(773\) − 38.0000i − 1.36677i −0.730061 0.683383i \(-0.760508\pi\)
0.730061 0.683383i \(-0.239492\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 6.00000 0.215387
\(777\) − 8.00000i − 0.286998i
\(778\) − 2.00000i − 0.0717035i
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 24.0000i 0.858238i
\(783\) 1.00000i 0.0357371i
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 48.0000i 1.71102i 0.517790 + 0.855508i \(0.326755\pi\)
−0.517790 + 0.855508i \(0.673245\pi\)
\(788\) − 14.0000i − 0.498729i
\(789\) 0 0
\(790\) 0 0
\(791\) −8.00000 −0.284447
\(792\) − 12.0000i − 0.426401i
\(793\) − 60.0000i − 2.13066i
\(794\) 10.0000 0.354887
\(795\) 0 0
\(796\) 24.0000 0.850657
\(797\) − 50.0000i − 1.77109i −0.464553 0.885545i \(-0.653785\pi\)
0.464553 0.885545i \(-0.346215\pi\)
\(798\) − 16.0000i − 0.566394i
\(799\) 0 0
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) − 2.00000i − 0.0706225i
\(803\) 8.00000i 0.282314i
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −48.0000 −1.69073
\(807\) − 6.00000i − 0.211210i
\(808\) 6.00000i 0.211079i
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) −20.0000 −0.702295 −0.351147 0.936320i \(-0.614208\pi\)
−0.351147 + 0.936320i \(0.614208\pi\)
\(812\) 4.00000i 0.140372i
\(813\) 16.0000i 0.561144i
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) −6.00000 −0.210042
\(817\) 16.0000i 0.559769i
\(818\) − 22.0000i − 0.769212i
\(819\) −24.0000 −0.838628
\(820\) 0 0
\(821\) −10.0000 −0.349002 −0.174501 0.984657i \(-0.555831\pi\)
−0.174501 + 0.984657i \(0.555831\pi\)
\(822\) 2.00000i 0.0697580i
\(823\) 16.0000i 0.557725i 0.960331 + 0.278862i \(0.0899574\pi\)
−0.960331 + 0.278862i \(0.910043\pi\)
\(824\) −36.0000 −1.25412
\(825\) 0 0
\(826\) −48.0000 −1.67013
\(827\) − 20.0000i − 0.695468i −0.937593 0.347734i \(-0.886951\pi\)
0.937593 0.347734i \(-0.113049\pi\)
\(828\) 4.00000i 0.139010i
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) −26.0000 −0.901930
\(832\) − 42.0000i − 1.45609i
\(833\) 54.0000i 1.87099i
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) −16.0000 −0.553372
\(837\) 8.00000i 0.276520i
\(838\) − 4.00000i − 0.138178i
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 18.0000i 0.620321i
\(843\) 10.0000i 0.344418i
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) − 20.0000i − 0.687208i
\(848\) 10.0000i 0.343401i
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) − 8.00000i − 0.274075i
\(853\) − 38.0000i − 1.30110i −0.759465 0.650548i \(-0.774539\pi\)
0.759465 0.650548i \(-0.225461\pi\)
\(854\) 40.0000 1.36877
\(855\) 0 0
\(856\) −24.0000 −0.820303
\(857\) − 2.00000i − 0.0683187i −0.999416 0.0341593i \(-0.989125\pi\)
0.999416 0.0341593i \(-0.0108754\pi\)
\(858\) − 24.0000i − 0.819346i
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) −24.0000 −0.817918
\(862\) − 24.0000i − 0.817443i
\(863\) 4.00000i 0.136162i 0.997680 + 0.0680808i \(0.0216876\pi\)
−0.997680 + 0.0680808i \(0.978312\pi\)
\(864\) −5.00000 −0.170103
\(865\) 0 0
\(866\) −34.0000 −1.15537
\(867\) − 19.0000i − 0.645274i
\(868\) 32.0000i 1.08615i
\(869\) 0 0
\(870\) 0 0
\(871\) 48.0000 1.62642
\(872\) 42.0000i 1.42230i
\(873\) − 2.00000i − 0.0676897i
\(874\) −16.0000 −0.541208
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) − 14.0000i − 0.472746i −0.971662 0.236373i \(-0.924041\pi\)
0.971662 0.236373i \(-0.0759588\pi\)
\(878\) − 24.0000i − 0.809961i
\(879\) −18.0000 −0.607125
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) − 9.00000i − 0.303046i
\(883\) − 56.0000i − 1.88455i −0.334840 0.942275i \(-0.608682\pi\)
0.334840 0.942275i \(-0.391318\pi\)
\(884\) 36.0000 1.21081
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) − 8.00000i − 0.268614i −0.990940 0.134307i \(-0.957119\pi\)
0.990940 0.134307i \(-0.0428808\pi\)
\(888\) − 6.00000i − 0.201347i
\(889\) 64.0000 2.14649
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 28.0000i 0.937509i
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) −12.0000 −0.400892
\(897\) 24.0000i 0.801337i
\(898\) 26.0000i 0.867631i
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) −60.0000 −1.99889
\(902\) − 24.0000i − 0.799113i
\(903\) 16.0000i 0.532447i
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 44.0000i 1.46100i 0.682915 + 0.730498i \(0.260712\pi\)
−0.682915 + 0.730498i \(0.739288\pi\)
\(908\) − 24.0000i − 0.796468i
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) −56.0000 −1.85536 −0.927681 0.373373i \(-0.878201\pi\)
−0.927681 + 0.373373i \(0.878201\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 32.0000i − 1.05905i
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) − 48.0000i − 1.58510i
\(918\) 6.00000i 0.198030i
\(919\) −8.00000 −0.263896 −0.131948 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) − 6.00000i − 0.197599i
\(923\) − 48.0000i − 1.57994i
\(924\) −16.0000 −0.526361
\(925\) 0 0
\(926\) 36.0000 1.18303
\(927\) 12.0000i 0.394132i
\(928\) 5.00000i 0.164133i
\(929\) −34.0000 −1.11550 −0.557752 0.830008i \(-0.688336\pi\)
−0.557752 + 0.830008i \(0.688336\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) − 22.0000i − 0.720634i
\(933\) 0 0
\(934\) 20.0000 0.654420
\(935\) 0 0
\(936\) −18.0000 −0.588348
\(937\) − 10.0000i − 0.326686i −0.986569 0.163343i \(-0.947772\pi\)
0.986569 0.163343i \(-0.0522277\pi\)
\(938\) 32.0000i 1.04484i
\(939\) 30.0000 0.979013
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) − 18.0000i − 0.586472i
\(943\) 24.0000i 0.781548i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) − 28.0000i − 0.909878i −0.890523 0.454939i \(-0.849661\pi\)
0.890523 0.454939i \(-0.150339\pi\)
\(948\) 0 0
\(949\) 12.0000 0.389536
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 72.0000i 2.33353i
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 8.00000 0.258738
\(957\) 4.00000i 0.129302i
\(958\) − 16.0000i − 0.516937i
\(959\) 8.00000 0.258333
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 12.0000i − 0.386896i
\(963\) 8.00000i 0.257796i
\(964\) 2.00000 0.0644157
\(965\) 0 0
\(966\) −16.0000 −0.514792
\(967\) − 32.0000i − 1.02905i −0.857475 0.514525i \(-0.827968\pi\)
0.857475 0.514525i \(-0.172032\pi\)
\(968\) − 15.0000i − 0.482118i
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 20.0000 0.641831 0.320915 0.947108i \(-0.396010\pi\)
0.320915 + 0.947108i \(0.396010\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 16.0000i 0.512936i
\(974\) 28.0000 0.897178
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) 6.00000i 0.191957i 0.995383 + 0.0959785i \(0.0305980\pi\)
−0.995383 + 0.0959785i \(0.969402\pi\)
\(978\) − 12.0000i − 0.383718i
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 14.0000 0.446986
\(982\) 44.0000i 1.40410i
\(983\) 40.0000i 1.27580i 0.770118 + 0.637901i \(0.220197\pi\)
−0.770118 + 0.637901i \(0.779803\pi\)
\(984\) −18.0000 −0.573819
\(985\) 0 0
\(986\) 6.00000 0.191079
\(987\) 0 0
\(988\) 24.0000i 0.763542i
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 40.0000i 1.27000i
\(993\) 4.00000i 0.126936i
\(994\) 32.0000 1.01498
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) − 50.0000i − 1.58352i −0.610835 0.791758i \(-0.709166\pi\)
0.610835 0.791758i \(-0.290834\pi\)
\(998\) − 36.0000i − 1.13956i
\(999\) −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 2175.2.c.b.349.1 2
5.2 odd 4 435.2.a.d.1.1 1
5.3 odd 4 2175.2.a.b.1.1 1
5.4 even 2 inner 2175.2.c.b.349.2 2
15.2 even 4 1305.2.a.b.1.1 1
15.8 even 4 6525.2.a.j.1.1 1
20.7 even 4 6960.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.d.1.1 1 5.2 odd 4
1305.2.a.b.1.1 1 15.2 even 4
2175.2.a.b.1.1 1 5.3 odd 4
2175.2.c.b.349.1 2 1.1 even 1 trivial
2175.2.c.b.349.2 2 5.4 even 2 inner
6525.2.a.j.1.1 1 15.8 even 4
6960.2.a.l.1.1 1 20.7 even 4