Properties

Label 525.2.g.c.524.4
Level $525$
Weight $2$
Character 525.524
Analytic conductor $4.192$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,2,Mod(524,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.524");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.g (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: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 524.4
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 525.524
Dual form 525.2.g.c.524.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{2} +1.73205 q^{3} +1.00000 q^{4} +3.00000 q^{6} +(1.73205 + 2.00000i) q^{7} -1.73205 q^{8} +3.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{2} +1.73205 q^{3} +1.00000 q^{4} +3.00000 q^{6} +(1.73205 + 2.00000i) q^{7} -1.73205 q^{8} +3.00000 q^{9} +3.46410i q^{11} +1.73205 q^{12} +(3.00000 + 3.46410i) q^{14} -5.00000 q^{16} -6.00000i q^{17} +5.19615 q^{18} -3.46410i q^{19} +(3.00000 + 3.46410i) q^{21} +6.00000i q^{22} -3.46410 q^{23} -3.00000 q^{24} +5.19615 q^{27} +(1.73205 + 2.00000i) q^{28} -6.92820i q^{29} -3.46410i q^{31} -5.19615 q^{32} +6.00000i q^{33} -10.3923i q^{34} +3.00000 q^{36} -2.00000i q^{37} -6.00000i q^{38} -6.00000 q^{41} +(5.19615 + 6.00000i) q^{42} +8.00000i q^{43} +3.46410i q^{44} -6.00000 q^{46} +12.0000i q^{47} -8.66025 q^{48} +(-1.00000 + 6.92820i) q^{49} -10.3923i q^{51} +9.00000 q^{54} +(-3.00000 - 3.46410i) q^{56} -6.00000i q^{57} -12.0000i q^{58} -12.0000 q^{59} -6.92820i q^{61} -6.00000i q^{62} +(5.19615 + 6.00000i) q^{63} +1.00000 q^{64} +10.3923i q^{66} +8.00000i q^{67} -6.00000i q^{68} -6.00000 q^{69} -3.46410i q^{71} -5.19615 q^{72} +6.92820 q^{73} -3.46410i q^{74} -3.46410i q^{76} +(-6.92820 + 6.00000i) q^{77} -8.00000 q^{79} +9.00000 q^{81} -10.3923 q^{82} +(3.00000 + 3.46410i) q^{84} +13.8564i q^{86} -12.0000i q^{87} -6.00000i q^{88} +6.00000 q^{89} -3.46410 q^{92} -6.00000i q^{93} +20.7846i q^{94} -9.00000 q^{96} -6.92820 q^{97} +(-1.73205 + 12.0000i) q^{98} +10.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} + 12 q^{6} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} + 12 q^{6} + 12 q^{9} + 12 q^{14} - 20 q^{16} + 12 q^{21} - 12 q^{24} + 12 q^{36} - 24 q^{41} - 24 q^{46} - 4 q^{49} + 36 q^{54} - 12 q^{56} - 48 q^{59} + 4 q^{64} - 24 q^{69} - 32 q^{79} + 36 q^{81} + 12 q^{84} + 24 q^{89} - 36 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205 1.22474 0.612372 0.790569i \(-0.290215\pi\)
0.612372 + 0.790569i \(0.290215\pi\)
\(3\) 1.73205 1.00000
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 3.00000 1.22474
\(7\) 1.73205 + 2.00000i 0.654654 + 0.755929i
\(8\) −1.73205 −0.612372
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 1.73205 0.500000
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 3.00000 + 3.46410i 0.801784 + 0.925820i
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 6.00000i 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 5.19615 1.22474
\(19\) 3.46410i 0.794719i −0.917663 0.397360i \(-0.869927\pi\)
0.917663 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 3.00000 + 3.46410i 0.654654 + 0.755929i
\(22\) 6.00000i 1.27920i
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 1.73205 + 2.00000i 0.327327 + 0.377964i
\(29\) 6.92820i 1.28654i −0.765641 0.643268i \(-0.777578\pi\)
0.765641 0.643268i \(-0.222422\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i −0.950382 0.311086i \(-0.899307\pi\)
0.950382 0.311086i \(-0.100693\pi\)
\(32\) −5.19615 −0.918559
\(33\) 6.00000i 1.04447i
\(34\) 10.3923i 1.78227i
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) 2.00000i 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 6.00000i 0.973329i
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 5.19615 + 6.00000i 0.801784 + 0.925820i
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) 3.46410i 0.522233i
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) −8.66025 −1.25000
\(49\) −1.00000 + 6.92820i −0.142857 + 0.989743i
\(50\) 0 0
\(51\) 10.3923i 1.45521i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) −3.00000 3.46410i −0.400892 0.462910i
\(57\) 6.00000i 0.794719i
\(58\) 12.0000i 1.57568i
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 6.92820i 0.887066i −0.896258 0.443533i \(-0.853725\pi\)
0.896258 0.443533i \(-0.146275\pi\)
\(62\) 6.00000i 0.762001i
\(63\) 5.19615 + 6.00000i 0.654654 + 0.755929i
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 10.3923i 1.27920i
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 3.46410i 0.411113i −0.978645 0.205557i \(-0.934100\pi\)
0.978645 0.205557i \(-0.0659005\pi\)
\(72\) −5.19615 −0.612372
\(73\) 6.92820 0.810885 0.405442 0.914121i \(-0.367117\pi\)
0.405442 + 0.914121i \(0.367117\pi\)
\(74\) 3.46410i 0.402694i
\(75\) 0 0
\(76\) 3.46410i 0.397360i
\(77\) −6.92820 + 6.00000i −0.789542 + 0.683763i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −10.3923 −1.14764
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 3.00000 + 3.46410i 0.327327 + 0.377964i
\(85\) 0 0
\(86\) 13.8564i 1.49417i
\(87\) 12.0000i 1.28654i
\(88\) 6.00000i 0.639602i
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −3.46410 −0.361158
\(93\) 6.00000i 0.622171i
\(94\) 20.7846i 2.14377i
\(95\) 0 0
\(96\) −9.00000 −0.918559
\(97\) −6.92820 −0.703452 −0.351726 0.936103i \(-0.614405\pi\)
−0.351726 + 0.936103i \(0.614405\pi\)
\(98\) −1.73205 + 12.0000i −0.174964 + 1.21218i
\(99\) 10.3923i 1.04447i
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 18.0000i 1.78227i
\(103\) −3.46410 −0.341328 −0.170664 0.985329i \(-0.554591\pi\)
−0.170664 + 0.985329i \(0.554591\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.3923 1.00466 0.502331 0.864675i \(-0.332476\pi\)
0.502331 + 0.864675i \(0.332476\pi\)
\(108\) 5.19615 0.500000
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 3.46410i 0.328798i
\(112\) −8.66025 10.0000i −0.818317 0.944911i
\(113\) 6.92820 0.651751 0.325875 0.945413i \(-0.394341\pi\)
0.325875 + 0.945413i \(0.394341\pi\)
\(114\) 10.3923i 0.973329i
\(115\) 0 0
\(116\) 6.92820i 0.643268i
\(117\) 0 0
\(118\) −20.7846 −1.91338
\(119\) 12.0000 10.3923i 1.10004 0.952661i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 12.0000i 1.08643i
\(123\) −10.3923 −0.937043
\(124\) 3.46410i 0.311086i
\(125\) 0 0
\(126\) 9.00000 + 10.3923i 0.801784 + 0.925820i
\(127\) 4.00000i 0.354943i −0.984126 0.177471i \(-0.943208\pi\)
0.984126 0.177471i \(-0.0567917\pi\)
\(128\) 12.1244 1.07165
\(129\) 13.8564i 1.21999i
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 6.00000i 0.522233i
\(133\) 6.92820 6.00000i 0.600751 0.520266i
\(134\) 13.8564i 1.19701i
\(135\) 0 0
\(136\) 10.3923i 0.891133i
\(137\) 20.7846 1.77575 0.887875 0.460086i \(-0.152181\pi\)
0.887875 + 0.460086i \(0.152181\pi\)
\(138\) −10.3923 −0.884652
\(139\) 17.3205i 1.46911i −0.678551 0.734553i \(-0.737392\pi\)
0.678551 0.734553i \(-0.262608\pi\)
\(140\) 0 0
\(141\) 20.7846i 1.75038i
\(142\) 6.00000i 0.503509i
\(143\) 0 0
\(144\) −15.0000 −1.25000
\(145\) 0 0
\(146\) 12.0000 0.993127
\(147\) −1.73205 + 12.0000i −0.142857 + 0.989743i
\(148\) 2.00000i 0.164399i
\(149\) 6.92820i 0.567581i 0.958886 + 0.283790i \(0.0915919\pi\)
−0.958886 + 0.283790i \(0.908408\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 6.00000i 0.486664i
\(153\) 18.0000i 1.45521i
\(154\) −12.0000 + 10.3923i −0.966988 + 0.837436i
\(155\) 0 0
\(156\) 0 0
\(157\) −13.8564 −1.10586 −0.552931 0.833227i \(-0.686491\pi\)
−0.552931 + 0.833227i \(0.686491\pi\)
\(158\) −13.8564 −1.10236
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00000 6.92820i −0.472866 0.546019i
\(162\) 15.5885 1.22474
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000i 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) −5.19615 6.00000i −0.400892 0.462910i
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 10.3923i 0.794719i
\(172\) 8.00000i 0.609994i
\(173\) 18.0000i 1.36851i 0.729241 + 0.684257i \(0.239873\pi\)
−0.729241 + 0.684257i \(0.760127\pi\)
\(174\) 20.7846i 1.57568i
\(175\) 0 0
\(176\) 17.3205i 1.30558i
\(177\) −20.7846 −1.56227
\(178\) 10.3923 0.778936
\(179\) 10.3923i 0.776757i 0.921500 + 0.388379i \(0.126965\pi\)
−0.921500 + 0.388379i \(0.873035\pi\)
\(180\) 0 0
\(181\) 20.7846i 1.54491i −0.635071 0.772454i \(-0.719029\pi\)
0.635071 0.772454i \(-0.280971\pi\)
\(182\) 0 0
\(183\) 12.0000i 0.887066i
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) 10.3923i 0.762001i
\(187\) 20.7846 1.51992
\(188\) 12.0000i 0.875190i
\(189\) 9.00000 + 10.3923i 0.654654 + 0.755929i
\(190\) 0 0
\(191\) 10.3923i 0.751961i 0.926628 + 0.375980i \(0.122694\pi\)
−0.926628 + 0.375980i \(0.877306\pi\)
\(192\) 1.73205 0.125000
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) −1.00000 + 6.92820i −0.0714286 + 0.494872i
\(197\) −13.8564 −0.987228 −0.493614 0.869681i \(-0.664324\pi\)
−0.493614 + 0.869681i \(0.664324\pi\)
\(198\) 18.0000i 1.27920i
\(199\) 10.3923i 0.736691i −0.929689 0.368345i \(-0.879924\pi\)
0.929689 0.368345i \(-0.120076\pi\)
\(200\) 0 0
\(201\) 13.8564i 0.977356i
\(202\) 10.3923 0.731200
\(203\) 13.8564 12.0000i 0.972529 0.842235i
\(204\) 10.3923i 0.727607i
\(205\) 0 0
\(206\) −6.00000 −0.418040
\(207\) −10.3923 −0.722315
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) 6.00000i 0.411113i
\(214\) 18.0000 1.23045
\(215\) 0 0
\(216\) −9.00000 −0.612372
\(217\) 6.92820 6.00000i 0.470317 0.407307i
\(218\) 3.46410 0.234619
\(219\) 12.0000 0.810885
\(220\) 0 0
\(221\) 0 0
\(222\) 6.00000i 0.402694i
\(223\) −17.3205 −1.15987 −0.579934 0.814664i \(-0.696921\pi\)
−0.579934 + 0.814664i \(0.696921\pi\)
\(224\) −9.00000 10.3923i −0.601338 0.694365i
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) 24.0000i 1.59294i 0.604681 + 0.796468i \(0.293301\pi\)
−0.604681 + 0.796468i \(0.706699\pi\)
\(228\) 6.00000i 0.397360i
\(229\) 6.92820i 0.457829i 0.973447 + 0.228914i \(0.0735176\pi\)
−0.973447 + 0.228914i \(0.926482\pi\)
\(230\) 0 0
\(231\) −12.0000 + 10.3923i −0.789542 + 0.683763i
\(232\) 12.0000i 0.787839i
\(233\) −6.92820 −0.453882 −0.226941 0.973909i \(-0.572872\pi\)
−0.226941 + 0.973909i \(0.572872\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) −13.8564 −0.900070
\(238\) 20.7846 18.0000i 1.34727 1.16677i
\(239\) 10.3923i 0.672222i −0.941822 0.336111i \(-0.890888\pi\)
0.941822 0.336111i \(-0.109112\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −1.73205 −0.111340
\(243\) 15.5885 1.00000
\(244\) 6.92820i 0.443533i
\(245\) 0 0
\(246\) −18.0000 −1.14764
\(247\) 0 0
\(248\) 6.00000i 0.381000i
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 5.19615 + 6.00000i 0.327327 + 0.377964i
\(253\) 12.0000i 0.754434i
\(254\) 6.92820i 0.434714i
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 6.00000i 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) 24.0000i 1.49417i
\(259\) 4.00000 3.46410i 0.248548 0.215249i
\(260\) 0 0
\(261\) 20.7846i 1.28654i
\(262\) 20.7846 1.28408
\(263\) 24.2487 1.49524 0.747620 0.664127i \(-0.231197\pi\)
0.747620 + 0.664127i \(0.231197\pi\)
\(264\) 10.3923i 0.639602i
\(265\) 0 0
\(266\) 12.0000 10.3923i 0.735767 0.637193i
\(267\) 10.3923 0.635999
\(268\) 8.00000i 0.488678i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) 24.2487i 1.47300i 0.676435 + 0.736502i \(0.263524\pi\)
−0.676435 + 0.736502i \(0.736476\pi\)
\(272\) 30.0000i 1.81902i
\(273\) 0 0
\(274\) 36.0000 2.17484
\(275\) 0 0
\(276\) −6.00000 −0.361158
\(277\) 14.0000i 0.841178i 0.907251 + 0.420589i \(0.138177\pi\)
−0.907251 + 0.420589i \(0.861823\pi\)
\(278\) 30.0000i 1.79928i
\(279\) 10.3923i 0.622171i
\(280\) 0 0
\(281\) 13.8564i 0.826604i −0.910594 0.413302i \(-0.864375\pi\)
0.910594 0.413302i \(-0.135625\pi\)
\(282\) 36.0000i 2.14377i
\(283\) 17.3205 1.02960 0.514799 0.857311i \(-0.327867\pi\)
0.514799 + 0.857311i \(0.327867\pi\)
\(284\) 3.46410i 0.205557i
\(285\) 0 0
\(286\) 0 0
\(287\) −10.3923 12.0000i −0.613438 0.708338i
\(288\) −15.5885 −0.918559
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) 6.92820 0.405442
\(293\) 6.00000i 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) −3.00000 + 20.7846i −0.174964 + 1.21218i
\(295\) 0 0
\(296\) 3.46410i 0.201347i
\(297\) 18.0000i 1.04447i
\(298\) 12.0000i 0.695141i
\(299\) 0 0
\(300\) 0 0
\(301\) −16.0000 + 13.8564i −0.922225 + 0.798670i
\(302\) −13.8564 −0.797347
\(303\) 10.3923 0.597022
\(304\) 17.3205i 0.993399i
\(305\) 0 0
\(306\) 31.1769i 1.78227i
\(307\) 24.2487 1.38395 0.691974 0.721923i \(-0.256741\pi\)
0.691974 + 0.721923i \(0.256741\pi\)
\(308\) −6.92820 + 6.00000i −0.394771 + 0.341882i
\(309\) −6.00000 −0.341328
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −20.7846 −1.17482 −0.587408 0.809291i \(-0.699852\pi\)
−0.587408 + 0.809291i \(0.699852\pi\)
\(314\) −24.0000 −1.35440
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 27.7128 1.55651 0.778253 0.627950i \(-0.216106\pi\)
0.778253 + 0.627950i \(0.216106\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 18.0000 1.00466
\(322\) −10.3923 12.0000i −0.579141 0.668734i
\(323\) −20.7846 −1.15649
\(324\) 9.00000 0.500000
\(325\) 0 0
\(326\) 27.7128i 1.53487i
\(327\) 3.46410 0.191565
\(328\) 10.3923 0.573819
\(329\) −24.0000 + 20.7846i −1.32316 + 1.14589i
\(330\) 0 0
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 20.7846i 1.13728i
\(335\) 0 0
\(336\) −15.0000 17.3205i −0.818317 0.944911i
\(337\) 14.0000i 0.762629i −0.924445 0.381314i \(-0.875472\pi\)
0.924445 0.381314i \(-0.124528\pi\)
\(338\) −22.5167 −1.22474
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) 12.0000 0.649836
\(342\) 18.0000i 0.973329i
\(343\) −15.5885 + 10.0000i −0.841698 + 0.539949i
\(344\) 13.8564i 0.747087i
\(345\) 0 0
\(346\) 31.1769i 1.67608i
\(347\) −17.3205 −0.929814 −0.464907 0.885360i \(-0.653912\pi\)
−0.464907 + 0.885360i \(0.653912\pi\)
\(348\) 12.0000i 0.643268i
\(349\) 6.92820i 0.370858i −0.982658 0.185429i \(-0.940632\pi\)
0.982658 0.185429i \(-0.0593675\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 18.0000i 0.959403i
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) −36.0000 −1.91338
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 20.7846 18.0000i 1.10004 0.952661i
\(358\) 18.0000i 0.951330i
\(359\) 3.46410i 0.182828i 0.995813 + 0.0914141i \(0.0291387\pi\)
−0.995813 + 0.0914141i \(0.970861\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 36.0000i 1.89212i
\(363\) −1.73205 −0.0909091
\(364\) 0 0
\(365\) 0 0
\(366\) 20.7846i 1.08643i
\(367\) −10.3923 −0.542474 −0.271237 0.962513i \(-0.587433\pi\)
−0.271237 + 0.962513i \(0.587433\pi\)
\(368\) 17.3205 0.902894
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 0 0
\(372\) 6.00000i 0.311086i
\(373\) 14.0000i 0.724893i −0.932005 0.362446i \(-0.881942\pi\)
0.932005 0.362446i \(-0.118058\pi\)
\(374\) 36.0000 1.86152
\(375\) 0 0
\(376\) 20.7846i 1.07188i
\(377\) 0 0
\(378\) 15.5885 + 18.0000i 0.801784 + 0.925820i
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 6.92820i 0.354943i
\(382\) 18.0000i 0.920960i
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 21.0000 1.07165
\(385\) 0 0
\(386\) 24.2487i 1.23423i
\(387\) 24.0000i 1.21999i
\(388\) −6.92820 −0.351726
\(389\) 6.92820i 0.351274i 0.984455 + 0.175637i \(0.0561985\pi\)
−0.984455 + 0.175637i \(0.943802\pi\)
\(390\) 0 0
\(391\) 20.7846i 1.05112i
\(392\) 1.73205 12.0000i 0.0874818 0.606092i
\(393\) 20.7846 1.04844
\(394\) −24.0000 −1.20910
\(395\) 0 0
\(396\) 10.3923i 0.522233i
\(397\) −13.8564 −0.695433 −0.347717 0.937600i \(-0.613043\pi\)
−0.347717 + 0.937600i \(0.613043\pi\)
\(398\) 18.0000i 0.902258i
\(399\) 12.0000 10.3923i 0.600751 0.520266i
\(400\) 0 0
\(401\) 27.7128i 1.38391i 0.721940 + 0.691956i \(0.243251\pi\)
−0.721940 + 0.691956i \(0.756749\pi\)
\(402\) 24.0000i 1.19701i
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 24.0000 20.7846i 1.19110 1.03152i
\(407\) 6.92820 0.343418
\(408\) 18.0000i 0.891133i
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 36.0000 1.77575
\(412\) −3.46410 −0.170664
\(413\) −20.7846 24.0000i −1.02274 1.18096i
\(414\) −18.0000 −0.884652
\(415\) 0 0
\(416\) 0 0
\(417\) 30.0000i 1.46911i
\(418\) 20.7846 1.01661
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 6.92820 0.337260
\(423\) 36.0000i 1.75038i
\(424\) 0 0
\(425\) 0 0
\(426\) 10.3923i 0.503509i
\(427\) 13.8564 12.0000i 0.670559 0.580721i
\(428\) 10.3923 0.502331
\(429\) 0 0
\(430\) 0 0
\(431\) 10.3923i 0.500580i 0.968171 + 0.250290i \(0.0805259\pi\)
−0.968171 + 0.250290i \(0.919474\pi\)
\(432\) −25.9808 −1.25000
\(433\) −6.92820 −0.332948 −0.166474 0.986046i \(-0.553238\pi\)
−0.166474 + 0.986046i \(0.553238\pi\)
\(434\) 12.0000 10.3923i 0.576018 0.498847i
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 12.0000i 0.574038i
\(438\) 20.7846 0.993127
\(439\) 17.3205i 0.826663i 0.910581 + 0.413331i \(0.135635\pi\)
−0.910581 + 0.413331i \(0.864365\pi\)
\(440\) 0 0
\(441\) −3.00000 + 20.7846i −0.142857 + 0.989743i
\(442\) 0 0
\(443\) −10.3923 −0.493753 −0.246877 0.969047i \(-0.579404\pi\)
−0.246877 + 0.969047i \(0.579404\pi\)
\(444\) 3.46410i 0.164399i
\(445\) 0 0
\(446\) −30.0000 −1.42054
\(447\) 12.0000i 0.567581i
\(448\) 1.73205 + 2.00000i 0.0818317 + 0.0944911i
\(449\) 13.8564i 0.653924i −0.945037 0.326962i \(-0.893975\pi\)
0.945037 0.326962i \(-0.106025\pi\)
\(450\) 0 0
\(451\) 20.7846i 0.978709i
\(452\) 6.92820 0.325875
\(453\) −13.8564 −0.651031
\(454\) 41.5692i 1.95094i
\(455\) 0 0
\(456\) 10.3923i 0.486664i
\(457\) 38.0000i 1.77757i −0.458329 0.888783i \(-0.651552\pi\)
0.458329 0.888783i \(-0.348448\pi\)
\(458\) 12.0000i 0.560723i
\(459\) 31.1769i 1.45521i
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) −20.7846 + 18.0000i −0.966988 + 0.837436i
\(463\) 20.0000i 0.929479i −0.885448 0.464739i \(-0.846148\pi\)
0.885448 0.464739i \(-0.153852\pi\)
\(464\) 34.6410i 1.60817i
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) 24.0000i 1.11059i −0.831654 0.555294i \(-0.812606\pi\)
0.831654 0.555294i \(-0.187394\pi\)
\(468\) 0 0
\(469\) −16.0000 + 13.8564i −0.738811 + 0.639829i
\(470\) 0 0
\(471\) −24.0000 −1.10586
\(472\) 20.7846 0.956689
\(473\) −27.7128 −1.27424
\(474\) −24.0000 −1.10236
\(475\) 0 0
\(476\) 12.0000 10.3923i 0.550019 0.476331i
\(477\) 0 0
\(478\) 18.0000i 0.823301i
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −10.3923 12.0000i −0.472866 0.546019i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 27.0000 1.22474
\(487\) 4.00000i 0.181257i 0.995885 + 0.0906287i \(0.0288876\pi\)
−0.995885 + 0.0906287i \(0.971112\pi\)
\(488\) 12.0000i 0.543214i
\(489\) 27.7128i 1.25322i
\(490\) 0 0
\(491\) 24.2487i 1.09433i −0.837025 0.547165i \(-0.815707\pi\)
0.837025 0.547165i \(-0.184293\pi\)
\(492\) −10.3923 −0.468521
\(493\) −41.5692 −1.87218
\(494\) 0 0
\(495\) 0 0
\(496\) 17.3205i 0.777714i
\(497\) 6.92820 6.00000i 0.310772 0.269137i
\(498\) 0 0
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 20.7846i 0.928588i
\(502\) −20.7846 −0.927663
\(503\) 12.0000i 0.535054i −0.963550 0.267527i \(-0.913794\pi\)
0.963550 0.267527i \(-0.0862064\pi\)
\(504\) −9.00000 10.3923i −0.400892 0.462910i
\(505\) 0 0
\(506\) 20.7846i 0.923989i
\(507\) −22.5167 −1.00000
\(508\) 4.00000i 0.177471i
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 12.0000 + 13.8564i 0.530849 + 0.612971i
\(512\) 8.66025 0.382733
\(513\) 18.0000i 0.794719i
\(514\) 10.3923i 0.458385i
\(515\) 0 0
\(516\) 13.8564i 0.609994i
\(517\) −41.5692 −1.82821
\(518\) 6.92820 6.00000i 0.304408 0.263625i
\(519\) 31.1769i 1.36851i
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 36.0000i 1.57568i
\(523\) 17.3205 0.757373 0.378686 0.925525i \(-0.376376\pi\)
0.378686 + 0.925525i \(0.376376\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 42.0000 1.83129
\(527\) −20.7846 −0.905392
\(528\) 30.0000i 1.30558i
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) −36.0000 −1.56227
\(532\) 6.92820 6.00000i 0.300376 0.260133i
\(533\) 0 0
\(534\) 18.0000 0.778936
\(535\) 0 0
\(536\) 13.8564i 0.598506i
\(537\) 18.0000i 0.776757i
\(538\) 31.1769 1.34413
\(539\) −24.0000 3.46410i −1.03375 0.149209i
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 42.0000i 1.80405i
\(543\) 36.0000i 1.54491i
\(544\) 31.1769i 1.33670i
\(545\) 0 0
\(546\) 0 0
\(547\) 8.00000i 0.342055i 0.985266 + 0.171028i \(0.0547087\pi\)
−0.985266 + 0.171028i \(0.945291\pi\)
\(548\) 20.7846 0.887875
\(549\) 20.7846i 0.887066i
\(550\) 0 0
\(551\) −24.0000 −1.02243
\(552\) 10.3923 0.442326
\(553\) −13.8564 16.0000i −0.589234 0.680389i
\(554\) 24.2487i 1.03023i
\(555\) 0 0
\(556\) 17.3205i 0.734553i
\(557\) −27.7128 −1.17423 −0.587115 0.809504i \(-0.699736\pi\)
−0.587115 + 0.809504i \(0.699736\pi\)
\(558\) 18.0000i 0.762001i
\(559\) 0 0
\(560\) 0 0
\(561\) 36.0000 1.51992
\(562\) 24.0000i 1.01238i
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 20.7846i 0.875190i
\(565\) 0 0
\(566\) 30.0000 1.26099
\(567\) 15.5885 + 18.0000i 0.654654 + 0.755929i
\(568\) 6.00000i 0.251754i
\(569\) 27.7128i 1.16178i −0.813982 0.580891i \(-0.802704\pi\)
0.813982 0.580891i \(-0.197296\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) 18.0000i 0.751961i
\(574\) −18.0000 20.7846i −0.751305 0.867533i
\(575\) 0 0
\(576\) 3.00000 0.125000
\(577\) −34.6410 −1.44212 −0.721062 0.692870i \(-0.756346\pi\)
−0.721062 + 0.692870i \(0.756346\pi\)
\(578\) −32.9090 −1.36883
\(579\) 24.2487i 1.00774i
\(580\) 0 0
\(581\) 0 0
\(582\) −20.7846 −0.861550
\(583\) 0 0
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) 10.3923i 0.429302i
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −1.73205 + 12.0000i −0.0714286 + 0.494872i
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) 10.0000i 0.410997i
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 31.1769i 1.27920i
\(595\) 0 0
\(596\) 6.92820i 0.283790i
\(597\) 18.0000i 0.736691i
\(598\) 0 0
\(599\) 24.2487i 0.990775i −0.868672 0.495388i \(-0.835026\pi\)
0.868672 0.495388i \(-0.164974\pi\)
\(600\) 0 0
\(601\) 13.8564i 0.565215i 0.959236 + 0.282607i \(0.0911993\pi\)
−0.959236 + 0.282607i \(0.908801\pi\)
\(602\) −27.7128 + 24.0000i −1.12949 + 0.978167i
\(603\) 24.0000i 0.977356i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) 17.3205 0.703018 0.351509 0.936185i \(-0.385669\pi\)
0.351509 + 0.936185i \(0.385669\pi\)
\(608\) 18.0000i 0.729996i
\(609\) 24.0000 20.7846i 0.972529 0.842235i
\(610\) 0 0
\(611\) 0 0
\(612\) 18.0000i 0.727607i
\(613\) 34.0000i 1.37325i 0.727013 + 0.686624i \(0.240908\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) 42.0000 1.69498
\(615\) 0 0
\(616\) 12.0000 10.3923i 0.483494 0.418718i
\(617\) −6.92820 −0.278919 −0.139459 0.990228i \(-0.544536\pi\)
−0.139459 + 0.990228i \(0.544536\pi\)
\(618\) −10.3923 −0.418040
\(619\) 17.3205i 0.696170i −0.937463 0.348085i \(-0.886832\pi\)
0.937463 0.348085i \(-0.113168\pi\)
\(620\) 0 0
\(621\) −18.0000 −0.722315
\(622\) 0 0
\(623\) 10.3923 + 12.0000i 0.416359 + 0.480770i
\(624\) 0 0
\(625\) 0 0
\(626\) −36.0000 −1.43885
\(627\) 20.7846 0.830057
\(628\) −13.8564 −0.552931
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 13.8564 0.551178
\(633\) 6.92820 0.275371
\(634\) 48.0000 1.90632
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 41.5692 1.64574
\(639\) 10.3923i 0.411113i
\(640\) 0 0
\(641\) 27.7128i 1.09459i −0.836940 0.547295i \(-0.815658\pi\)
0.836940 0.547295i \(-0.184342\pi\)
\(642\) 31.1769 1.23045
\(643\) 31.1769 1.22950 0.614749 0.788723i \(-0.289257\pi\)
0.614749 + 0.788723i \(0.289257\pi\)
\(644\) −6.00000 6.92820i −0.236433 0.273009i
\(645\) 0 0
\(646\) −36.0000 −1.41640
\(647\) 12.0000i 0.471769i −0.971781 0.235884i \(-0.924201\pi\)
0.971781 0.235884i \(-0.0757987\pi\)
\(648\) −15.5885 −0.612372
\(649\) 41.5692i 1.63173i
\(650\) 0 0
\(651\) 12.0000 10.3923i 0.470317 0.407307i
\(652\) 16.0000i 0.626608i
\(653\) 41.5692 1.62673 0.813365 0.581754i \(-0.197633\pi\)
0.813365 + 0.581754i \(0.197633\pi\)
\(654\) 6.00000 0.234619
\(655\) 0 0
\(656\) 30.0000 1.17130
\(657\) 20.7846 0.810885
\(658\) −41.5692 + 36.0000i −1.62054 + 1.40343i
\(659\) 10.3923i 0.404827i 0.979300 + 0.202413i \(0.0648785\pi\)
−0.979300 + 0.202413i \(0.935122\pi\)
\(660\) 0 0
\(661\) 48.4974i 1.88633i −0.332323 0.943166i \(-0.607832\pi\)
0.332323 0.943166i \(-0.392168\pi\)
\(662\) 48.4974 1.88491
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 10.3923i 0.402694i
\(667\) 24.0000i 0.929284i
\(668\) 12.0000i 0.464294i
\(669\) −30.0000 −1.15987
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) −15.5885 18.0000i −0.601338 0.694365i
\(673\) 14.0000i 0.539660i 0.962908 + 0.269830i \(0.0869676\pi\)
−0.962908 + 0.269830i \(0.913032\pi\)
\(674\) 24.2487i 0.934025i
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) 20.7846 0.798228
\(679\) −12.0000 13.8564i −0.460518 0.531760i
\(680\) 0 0
\(681\) 41.5692i 1.59294i
\(682\) 20.7846 0.795884
\(683\) 17.3205 0.662751 0.331375 0.943499i \(-0.392487\pi\)
0.331375 + 0.943499i \(0.392487\pi\)
\(684\) 10.3923i 0.397360i
\(685\) 0 0
\(686\) −27.0000 + 17.3205i −1.03086 + 0.661300i
\(687\) 12.0000i 0.457829i
\(688\) 40.0000i 1.52499i
\(689\) 0 0
\(690\) 0 0
\(691\) 31.1769i 1.18603i 0.805193 + 0.593013i \(0.202062\pi\)
−0.805193 + 0.593013i \(0.797938\pi\)
\(692\) 18.0000i 0.684257i
\(693\) −20.7846 + 18.0000i −0.789542 + 0.683763i
\(694\) −30.0000 −1.13878
\(695\) 0 0
\(696\) 20.7846i 0.787839i
\(697\) 36.0000i 1.36360i
\(698\) 12.0000i 0.454207i
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 20.7846i 0.785024i 0.919747 + 0.392512i \(0.128394\pi\)
−0.919747 + 0.392512i \(0.871606\pi\)
\(702\) 0 0
\(703\) −6.92820 −0.261302
\(704\) 3.46410i 0.130558i
\(705\) 0 0
\(706\) 10.3923i 0.391120i
\(707\) 10.3923 + 12.0000i 0.390843 + 0.451306i
\(708\) −20.7846 −0.781133
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) −24.0000 −0.900070
\(712\) −10.3923 −0.389468
\(713\) 12.0000i 0.449404i
\(714\) 36.0000 31.1769i 1.34727 1.16677i
\(715\) 0 0
\(716\) 10.3923i 0.388379i
\(717\) 18.0000i 0.672222i
\(718\) 6.00000i 0.223918i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −6.00000 6.92820i −0.223452 0.258020i
\(722\) 12.1244 0.451222
\(723\) 0 0
\(724\) 20.7846i 0.772454i
\(725\) 0 0
\(726\) −3.00000 −0.111340
\(727\) 3.46410 0.128476 0.0642382 0.997935i \(-0.479538\pi\)
0.0642382 + 0.997935i \(0.479538\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 48.0000 1.77534
\(732\) 12.0000i 0.443533i
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) −18.0000 −0.664392
\(735\) 0 0
\(736\) 18.0000 0.663489
\(737\) −27.7128 −1.02081
\(738\) −31.1769 −1.14764
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −31.1769 −1.14377 −0.571885 0.820334i \(-0.693788\pi\)
−0.571885 + 0.820334i \(0.693788\pi\)
\(744\) 10.3923i 0.381000i
\(745\) 0 0
\(746\) 24.2487i 0.887808i
\(747\) 0 0
\(748\) 20.7846 0.759961
\(749\) 18.0000 + 20.7846i 0.657706 + 0.759453i
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 60.0000i 2.18797i
\(753\) −20.7846 −0.757433
\(754\) 0 0
\(755\) 0 0
\(756\) 9.00000 + 10.3923i 0.327327 + 0.377964i
\(757\) 2.00000i 0.0726912i −0.999339 0.0363456i \(-0.988428\pi\)
0.999339 0.0363456i \(-0.0115717\pi\)
\(758\) −6.92820 −0.251644
\(759\) 20.7846i 0.754434i
\(760\) 0 0
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 12.0000i 0.434714i
\(763\) 3.46410 + 4.00000i 0.125409 + 0.144810i
\(764\) 10.3923i 0.375980i
\(765\) 0 0
\(766\) 20.7846i 0.750978i
\(767\) 0 0
\(768\) 32.9090 1.18750
\(769\) 41.5692i 1.49902i 0.661991 + 0.749512i \(0.269712\pi\)
−0.661991 + 0.749512i \(0.730288\pi\)
\(770\) 0 0
\(771\) 10.3923i 0.374270i
\(772\) 14.0000i 0.503871i
\(773\) 6.00000i 0.215805i −0.994161 0.107903i \(-0.965587\pi\)
0.994161 0.107903i \(-0.0344134\pi\)
\(774\) 41.5692i 1.49417i
\(775\) 0 0
\(776\) 12.0000 0.430775
\(777\) 6.92820 6.00000i 0.248548 0.215249i
\(778\) 12.0000i 0.430221i
\(779\) 20.7846i 0.744686i
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 36.0000i 1.28736i
\(783\) 36.0000i 1.28654i
\(784\) 5.00000 34.6410i 0.178571 1.23718i
\(785\) 0 0
\(786\) 36.0000 1.28408
\(787\) 24.2487 0.864373 0.432187 0.901784i \(-0.357742\pi\)
0.432187 + 0.901784i \(0.357742\pi\)
\(788\) −13.8564 −0.493614
\(789\) 42.0000 1.49524
\(790\) 0 0
\(791\) 12.0000 + 13.8564i 0.426671 + 0.492677i
\(792\) 18.0000i 0.639602i
\(793\) 0 0
\(794\) −24.0000 −0.851728
\(795\) 0 0
\(796\) 10.3923i 0.368345i
\(797\) 18.0000i 0.637593i −0.947823 0.318796i \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) 20.7846 18.0000i 0.735767 0.637193i
\(799\) 72.0000 2.54718
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 48.0000i 1.69494i
\(803\) 24.0000i 0.846942i
\(804\) 13.8564i 0.488678i
\(805\) 0 0
\(806\) 0 0
\(807\) 31.1769 1.09748
\(808\) −10.3923 −0.365600
\(809\) 55.4256i 1.94866i 0.225122 + 0.974331i \(0.427722\pi\)
−0.225122 + 0.974331i \(0.572278\pi\)
\(810\) 0 0
\(811\) 38.1051i 1.33805i −0.743239 0.669026i \(-0.766712\pi\)
0.743239 0.669026i \(-0.233288\pi\)
\(812\) 13.8564 12.0000i 0.486265 0.421117i
\(813\) 42.0000i 1.47300i
\(814\) 12.0000 0.420600
\(815\) 0 0
\(816\) 51.9615i 1.81902i
\(817\) 27.7128 0.969549
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 20.7846i 0.725388i −0.931908 0.362694i \(-0.881857\pi\)
0.931908 0.362694i \(-0.118143\pi\)
\(822\) 62.3538 2.17484
\(823\) 44.0000i 1.53374i −0.641800 0.766872i \(-0.721812\pi\)
0.641800 0.766872i \(-0.278188\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) −36.0000 41.5692i −1.25260 1.44638i
\(827\) 38.1051 1.32504 0.662522 0.749042i \(-0.269486\pi\)
0.662522 + 0.749042i \(0.269486\pi\)
\(828\) −10.3923 −0.361158
\(829\) 34.6410i 1.20313i −0.798823 0.601566i \(-0.794544\pi\)
0.798823 0.601566i \(-0.205456\pi\)
\(830\) 0 0
\(831\) 24.2487i 0.841178i
\(832\) 0 0
\(833\) 41.5692 + 6.00000i 1.44029 + 0.207888i
\(834\) 51.9615i 1.79928i
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) 18.0000i 0.622171i
\(838\) 20.7846 0.717992
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) −19.0000 −0.655172
\(842\) −17.3205 −0.596904
\(843\) 24.0000i 0.826604i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 62.3538i 2.14377i
\(847\) −1.73205 2.00000i −0.0595140 0.0687208i
\(848\) 0 0
\(849\) 30.0000 1.02960
\(850\) 0 0
\(851\) 6.92820i 0.237496i
\(852\) 6.00000i 0.205557i
\(853\) 41.5692 1.42330 0.711651 0.702533i \(-0.247948\pi\)
0.711651 + 0.702533i \(0.247948\pi\)
\(854\) 24.0000 20.7846i 0.821263 0.711235i
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) 30.0000i 1.02478i −0.858753 0.512390i \(-0.828760\pi\)
0.858753 0.512390i \(-0.171240\pi\)
\(858\) 0 0
\(859\) 38.1051i 1.30013i 0.759879 + 0.650065i \(0.225258\pi\)
−0.759879 + 0.650065i \(0.774742\pi\)
\(860\) 0 0
\(861\) −18.0000 20.7846i −0.613438 0.708338i
\(862\) 18.0000i 0.613082i
\(863\) −17.3205 −0.589597 −0.294798 0.955559i \(-0.595253\pi\)
−0.294798 + 0.955559i \(0.595253\pi\)
\(864\) −27.0000 −0.918559
\(865\) 0 0
\(866\) −12.0000 −0.407777
\(867\) −32.9090 −1.11765
\(868\) 6.92820 6.00000i 0.235159 0.203653i
\(869\) 27.7128i 0.940093i
\(870\) 0 0
\(871\) 0 0
\(872\) −3.46410 −0.117309
\(873\) −20.7846 −0.703452
\(874\) 20.7846i 0.703050i
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) 38.0000i 1.28317i 0.767052 + 0.641584i \(0.221723\pi\)
−0.767052 + 0.641584i \(0.778277\pi\)
\(878\) 30.0000i 1.01245i
\(879\) 10.3923i 0.350524i
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) −5.19615 + 36.0000i −0.174964 + 1.21218i
\(883\) 16.0000i 0.538443i −0.963078 0.269221i \(-0.913234\pi\)
0.963078 0.269221i \(-0.0867663\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −18.0000 −0.604722
\(887\) 36.0000i 1.20876i 0.796696 + 0.604381i \(0.206579\pi\)
−0.796696 + 0.604381i \(0.793421\pi\)
\(888\) 6.00000i 0.201347i
\(889\) 8.00000 6.92820i 0.268311 0.232364i
\(890\) 0 0
\(891\) 31.1769i 1.04447i
\(892\) −17.3205 −0.579934
\(893\) 41.5692 1.39106
\(894\) 20.7846i 0.695141i
\(895\) 0 0
\(896\) 21.0000 + 24.2487i 0.701561 + 0.810093i
\(897\) 0 0
\(898\) 24.0000i 0.800890i
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 0 0
\(902\) 36.0000i 1.19867i
\(903\) −27.7128 + 24.0000i −0.922225 + 0.798670i
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) −24.0000 −0.797347
\(907\) 32.0000i 1.06254i −0.847202 0.531271i \(-0.821714\pi\)
0.847202 0.531271i \(-0.178286\pi\)
\(908\) 24.0000i 0.796468i
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 17.3205i 0.573854i −0.957952 0.286927i \(-0.907366\pi\)
0.957952 0.286927i \(-0.0926337\pi\)
\(912\) 30.0000i 0.993399i
\(913\) 0 0
\(914\) 65.8179i 2.17706i
\(915\) 0 0
\(916\) 6.92820i 0.228914i
\(917\) 20.7846 + 24.0000i 0.686368 + 0.792550i
\(918\) 54.0000i 1.78227i
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 42.0000 1.38395
\(922\) −31.1769 −1.02676
\(923\) 0 0
\(924\) −12.0000 + 10.3923i −0.394771 + 0.341882i
\(925\) 0 0
\(926\) 34.6410i 1.13837i
\(927\) −10.3923 −0.341328
\(928\) 36.0000i 1.18176i
\(929\) 30.0000 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(930\) 0 0
\(931\) 24.0000 + 3.46410i 0.786568 + 0.113531i
\(932\) −6.92820 −0.226941
\(933\) 0 0
\(934\) 41.5692i 1.36019i
\(935\) 0 0
\(936\) 0 0
\(937\) 6.92820 0.226335 0.113167 0.993576i \(-0.463900\pi\)
0.113167 + 0.993576i \(0.463900\pi\)
\(938\) −27.7128 + 24.0000i −0.904855 + 0.783628i
\(939\) −36.0000 −1.17482
\(940\) 0 0
\(941\) −18.0000 −0.586783 −0.293392 0.955992i \(-0.594784\pi\)
−0.293392 + 0.955992i \(0.594784\pi\)
\(942\) −41.5692 −1.35440
\(943\) 20.7846 0.676840
\(944\) 60.0000 1.95283
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) −31.1769 −1.01311 −0.506557 0.862207i \(-0.669082\pi\)
−0.506557 + 0.862207i \(0.669082\pi\)
\(948\) −13.8564 −0.450035
\(949\) 0 0
\(950\) 0 0
\(951\) 48.0000 1.55651
\(952\) −20.7846 + 18.0000i −0.673633 + 0.583383i
\(953\) −34.6410 −1.12213 −0.561066 0.827771i \(-0.689609\pi\)
−0.561066 + 0.827771i \(0.689609\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 10.3923i 0.336111i
\(957\) 41.5692 1.34374
\(958\) 0 0
\(959\) 36.0000 + 41.5692i 1.16250 + 1.34234i
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) 31.1769 1.00466
\(964\) 0 0
\(965\) 0 0
\(966\) −18.0000 20.7846i −0.579141 0.668734i
\(967\) 20.0000i 0.643157i 0.946883 + 0.321578i \(0.104213\pi\)
−0.946883 + 0.321578i \(0.895787\pi\)
\(968\) 1.73205 0.0556702
\(969\) −36.0000 −1.15649
\(970\) 0 0
\(971\) −60.0000 −1.92549 −0.962746 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 15.5885 0.500000
\(973\) 34.6410 30.0000i 1.11054 0.961756i
\(974\) 6.92820i 0.221994i
\(975\) 0 0
\(976\) 34.6410i 1.10883i
\(977\) 6.92820 0.221653 0.110826 0.993840i \(-0.464650\pi\)
0.110826 + 0.993840i \(0.464650\pi\)
\(978\) 48.0000i 1.53487i
\(979\) 20.7846i 0.664279i
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 42.0000i 1.34027i
\(983\) 60.0000i 1.91370i −0.290578 0.956851i \(-0.593847\pi\)
0.290578 0.956851i \(-0.406153\pi\)
\(984\) 18.0000 0.573819
\(985\) 0 0
\(986\) −72.0000 −2.29295
\(987\) −41.5692 + 36.0000i −1.32316 + 1.14589i
\(988\) 0 0
\(989\) 27.7128i 0.881216i
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 18.0000i 0.571501i
\(993\) 48.4974 1.53902
\(994\) 12.0000 10.3923i 0.380617 0.329624i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 34.6410 1.09654
\(999\) 10.3923i 0.328798i
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 525.2.g.c.524.4 4
3.2 odd 2 525.2.g.b.524.2 4
5.2 odd 4 525.2.b.b.251.2 2
5.3 odd 4 105.2.b.b.41.1 yes 2
5.4 even 2 inner 525.2.g.c.524.1 4
7.6 odd 2 525.2.g.b.524.4 4
15.2 even 4 525.2.b.a.251.1 2
15.8 even 4 105.2.b.a.41.2 yes 2
15.14 odd 2 525.2.g.b.524.3 4
20.3 even 4 1680.2.f.c.881.1 2
21.20 even 2 inner 525.2.g.c.524.2 4
35.3 even 12 735.2.s.d.656.1 2
35.13 even 4 105.2.b.a.41.1 2
35.18 odd 12 735.2.s.f.656.1 2
35.23 odd 12 735.2.s.a.521.1 2
35.27 even 4 525.2.b.a.251.2 2
35.33 even 12 735.2.s.b.521.1 2
35.34 odd 2 525.2.g.b.524.1 4
60.23 odd 4 1680.2.f.b.881.1 2
105.23 even 12 735.2.s.d.521.1 2
105.38 odd 12 735.2.s.a.656.1 2
105.53 even 12 735.2.s.b.656.1 2
105.62 odd 4 525.2.b.b.251.1 2
105.68 odd 12 735.2.s.f.521.1 2
105.83 odd 4 105.2.b.b.41.2 yes 2
105.104 even 2 inner 525.2.g.c.524.3 4
140.83 odd 4 1680.2.f.b.881.2 2
420.83 even 4 1680.2.f.c.881.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.b.a.41.1 2 35.13 even 4
105.2.b.a.41.2 yes 2 15.8 even 4
105.2.b.b.41.1 yes 2 5.3 odd 4
105.2.b.b.41.2 yes 2 105.83 odd 4
525.2.b.a.251.1 2 15.2 even 4
525.2.b.a.251.2 2 35.27 even 4
525.2.b.b.251.1 2 105.62 odd 4
525.2.b.b.251.2 2 5.2 odd 4
525.2.g.b.524.1 4 35.34 odd 2
525.2.g.b.524.2 4 3.2 odd 2
525.2.g.b.524.3 4 15.14 odd 2
525.2.g.b.524.4 4 7.6 odd 2
525.2.g.c.524.1 4 5.4 even 2 inner
525.2.g.c.524.2 4 21.20 even 2 inner
525.2.g.c.524.3 4 105.104 even 2 inner
525.2.g.c.524.4 4 1.1 even 1 trivial
735.2.s.a.521.1 2 35.23 odd 12
735.2.s.a.656.1 2 105.38 odd 12
735.2.s.b.521.1 2 35.33 even 12
735.2.s.b.656.1 2 105.53 even 12
735.2.s.d.521.1 2 105.23 even 12
735.2.s.d.656.1 2 35.3 even 12
735.2.s.f.521.1 2 105.68 odd 12
735.2.s.f.656.1 2 35.18 odd 12
1680.2.f.b.881.1 2 60.23 odd 4
1680.2.f.b.881.2 2 140.83 odd 4
1680.2.f.c.881.1 2 20.3 even 4
1680.2.f.c.881.2 2 420.83 even 4