Properties

Label 525.2.b.a.251.2
Level $525$
Weight $2$
Character 525.251
Analytic conductor $4.192$
Analytic rank $0$
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] = [525,2,Mod(251,525)]
 
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, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 525.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.19214610612\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 525.251
Dual form 525.2.b.a.251.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{2} +1.73205i q^{3} -1.00000 q^{4} -3.00000 q^{6} +(-2.00000 - 1.73205i) q^{7} +1.73205i q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{2} +1.73205i q^{3} -1.00000 q^{4} -3.00000 q^{6} +(-2.00000 - 1.73205i) q^{7} +1.73205i q^{8} -3.00000 q^{9} +3.46410i q^{11} -1.73205i q^{12} +(3.00000 - 3.46410i) q^{14} -5.00000 q^{16} -6.00000 q^{17} -5.19615i q^{18} -3.46410i q^{19} +(3.00000 - 3.46410i) q^{21} -6.00000 q^{22} +3.46410i q^{23} -3.00000 q^{24} -5.19615i q^{27} +(2.00000 + 1.73205i) q^{28} +6.92820i q^{29} +3.46410i q^{31} -5.19615i q^{32} -6.00000 q^{33} -10.3923i q^{34} +3.00000 q^{36} +2.00000 q^{37} +6.00000 q^{38} +6.00000 q^{41} +(6.00000 + 5.19615i) q^{42} +8.00000 q^{43} -3.46410i q^{44} -6.00000 q^{46} +12.0000 q^{47} -8.66025i q^{48} +(1.00000 + 6.92820i) q^{49} -10.3923i q^{51} +9.00000 q^{54} +(3.00000 - 3.46410i) q^{56} +6.00000 q^{57} -12.0000 q^{58} -12.0000 q^{59} +6.92820i q^{61} -6.00000 q^{62} +(6.00000 + 5.19615i) q^{63} -1.00000 q^{64} -10.3923i q^{66} -8.00000 q^{67} +6.00000 q^{68} -6.00000 q^{69} -3.46410i q^{71} -5.19615i q^{72} +6.92820i q^{73} +3.46410i q^{74} +3.46410i q^{76} +(6.00000 - 6.92820i) q^{77} +8.00000 q^{79} +9.00000 q^{81} +10.3923i q^{82} +(-3.00000 + 3.46410i) q^{84} +13.8564i q^{86} -12.0000 q^{87} -6.00000 q^{88} +6.00000 q^{89} -3.46410i q^{92} -6.00000 q^{93} +20.7846i q^{94} +9.00000 q^{96} +6.92820i q^{97} +(-12.0000 + 1.73205i) q^{98} -10.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 6 q^{6} - 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 6 q^{6} - 4 q^{7} - 6 q^{9} + 6 q^{14} - 10 q^{16} - 12 q^{17} + 6 q^{21} - 12 q^{22} - 6 q^{24} + 4 q^{28} - 12 q^{33} + 6 q^{36} + 4 q^{37} + 12 q^{38} + 12 q^{41} + 12 q^{42} + 16 q^{43} - 12 q^{46} + 24 q^{47} + 2 q^{49} + 18 q^{54} + 6 q^{56} + 12 q^{57} - 24 q^{58} - 24 q^{59} - 12 q^{62} + 12 q^{63} - 2 q^{64} - 16 q^{67} + 12 q^{68} - 12 q^{69} + 12 q^{77} + 16 q^{79} + 18 q^{81} - 6 q^{84} - 24 q^{87} - 12 q^{88} + 12 q^{89} - 12 q^{93} + 18 q^{96} - 24 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205i 1.22474i 0.790569 + 0.612372i \(0.209785\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) 1.73205i 1.00000i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) −2.00000 1.73205i −0.755929 0.654654i
\(8\) 1.73205i 0.612372i
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 1.73205i 0.500000i
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 3.00000 3.46410i 0.801784 0.925820i
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 5.19615i 1.22474i
\(19\) 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
\(22\) −6.00000 −1.27920
\(23\) 3.46410i 0.722315i 0.932505 + 0.361158i \(0.117618\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615i 1.00000i
\(28\) 2.00000 + 1.73205i 0.377964 + 0.327327i
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 5.19615i 0.918559i
\(33\) −6.00000 −1.04447
\(34\) 10.3923i 1.78227i
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 6.00000 + 5.19615i 0.925820 + 0.801784i
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 3.46410i 0.522233i
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 12.0000 1.75038 0.875190 0.483779i \(-0.160736\pi\)
0.875190 + 0.483779i \(0.160736\pi\)
\(48\) 8.66025i 1.25000i
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) 10.3923i 1.45521i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) 3.00000 3.46410i 0.400892 0.462910i
\(57\) 6.00000 0.794719
\(58\) −12.0000 −1.57568
\(59\) −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 \(0.146275\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −6.00000 −0.762001
\(63\) 6.00000 + 5.19615i 0.755929 + 0.654654i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 10.3923i 1.27920i
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 6.00000 0.727607
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 3.46410i 0.411113i −0.978645 0.205557i \(-0.934100\pi\)
0.978645 0.205557i \(-0.0659005\pi\)
\(72\) 5.19615i 0.612372i
\(73\) 6.92820i 0.810885i 0.914121 + 0.405442i \(0.132883\pi\)
−0.914121 + 0.405442i \(0.867117\pi\)
\(74\) 3.46410i 0.402694i
\(75\) 0 0
\(76\) 3.46410i 0.397360i
\(77\) 6.00000 6.92820i 0.683763 0.789542i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 10.3923i 1.14764i
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −3.00000 + 3.46410i −0.327327 + 0.377964i
\(85\) 0 0
\(86\) 13.8564i 1.49417i
\(87\) −12.0000 −1.28654
\(88\) −6.00000 −0.639602
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3.46410i 0.361158i
\(93\) −6.00000 −0.622171
\(94\) 20.7846i 2.14377i
\(95\) 0 0
\(96\) 9.00000 0.918559
\(97\) 6.92820i 0.703452i 0.936103 + 0.351726i \(0.114405\pi\)
−0.936103 + 0.351726i \(0.885595\pi\)
\(98\) −12.0000 + 1.73205i −1.21218 + 0.174964i
\(99\) 10.3923i 1.04447i
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 18.0000 1.78227
\(103\) 3.46410i 0.341328i −0.985329 0.170664i \(-0.945409\pi\)
0.985329 0.170664i \(-0.0545913\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.3923i 1.00466i 0.864675 + 0.502331i \(0.167524\pi\)
−0.864675 + 0.502331i \(0.832476\pi\)
\(108\) 5.19615i 0.500000i
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 3.46410i 0.328798i
\(112\) 10.0000 + 8.66025i 0.944911 + 0.818317i
\(113\) 6.92820i 0.651751i −0.945413 0.325875i \(-0.894341\pi\)
0.945413 0.325875i \(-0.105659\pi\)
\(114\) 10.3923i 0.973329i
\(115\) 0 0
\(116\) 6.92820i 0.643268i
\(117\) 0 0
\(118\) 20.7846i 1.91338i
\(119\) 12.0000 + 10.3923i 1.10004 + 0.952661i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) −12.0000 −1.08643
\(123\) 10.3923i 0.937043i
\(124\) 3.46410i 0.311086i
\(125\) 0 0
\(126\) −9.00000 + 10.3923i −0.801784 + 0.925820i
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 12.1244i 1.07165i
\(129\) 13.8564i 1.21999i
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 6.00000 0.522233
\(133\) −6.00000 + 6.92820i −0.520266 + 0.600751i
\(134\) 13.8564i 1.19701i
\(135\) 0 0
\(136\) 10.3923i 0.891133i
\(137\) 20.7846i 1.77575i 0.460086 + 0.887875i \(0.347819\pi\)
−0.460086 + 0.887875i \(0.652181\pi\)
\(138\) 10.3923i 0.884652i
\(139\) 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
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 15.0000 1.25000
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) −12.0000 + 1.73205i −0.989743 + 0.142857i
\(148\) −2.00000 −0.164399
\(149\) 6.92820i 0.567581i −0.958886 0.283790i \(-0.908408\pi\)
0.958886 0.283790i \(-0.0915919\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 6.00000 0.486664
\(153\) 18.0000 1.45521
\(154\) 12.0000 + 10.3923i 0.966988 + 0.837436i
\(155\) 0 0
\(156\) 0 0
\(157\) 13.8564i 1.10586i 0.833227 + 0.552931i \(0.186491\pi\)
−0.833227 + 0.552931i \(0.813509\pi\)
\(158\) 13.8564i 1.10236i
\(159\) 0 0
\(160\) 0 0
\(161\) 6.00000 6.92820i 0.472866 0.546019i
\(162\) 15.5885i 1.22474i
\(163\) 16.0000 1.25322 0.626608 0.779334i \(-0.284443\pi\)
0.626608 + 0.779334i \(0.284443\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 6.00000 + 5.19615i 0.462910 + 0.400892i
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 10.3923i 0.794719i
\(172\) −8.00000 −0.609994
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 20.7846i 1.57568i
\(175\) 0 0
\(176\) 17.3205i 1.30558i
\(177\) 20.7846i 1.56227i
\(178\) 10.3923i 0.778936i
\(179\) 10.3923i 0.776757i −0.921500 0.388379i \(-0.873035\pi\)
0.921500 0.388379i \(-0.126965\pi\)
\(180\) 0 0
\(181\) 20.7846i 1.54491i 0.635071 + 0.772454i \(0.280971\pi\)
−0.635071 + 0.772454i \(0.719029\pi\)
\(182\) 0 0
\(183\) −12.0000 −0.887066
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 10.3923i 0.762001i
\(187\) 20.7846i 1.51992i
\(188\) −12.0000 −0.875190
\(189\) −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 \(0.877306\pi\)
\(192\) 1.73205i 0.125000i
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) −1.00000 6.92820i −0.0714286 0.494872i
\(197\) 13.8564i 0.987228i −0.869681 0.493614i \(-0.835676\pi\)
0.869681 0.493614i \(-0.164324\pi\)
\(198\) 18.0000 1.27920
\(199\) 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
\(202\) 10.3923i 0.731200i
\(203\) 12.0000 13.8564i 0.842235 0.972529i
\(204\) 10.3923i 0.727607i
\(205\) 0 0
\(206\) 6.00000 0.418040
\(207\) 10.3923i 0.722315i
\(208\) 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
\(213\) 6.00000 0.411113
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) 9.00000 0.612372
\(217\) 6.00000 6.92820i 0.407307 0.470317i
\(218\) 3.46410i 0.234619i
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) 0 0
\(222\) −6.00000 −0.402694
\(223\) 17.3205i 1.15987i −0.814664 0.579934i \(-0.803079\pi\)
0.814664 0.579934i \(-0.196921\pi\)
\(224\) −9.00000 + 10.3923i −0.601338 + 0.694365i
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) 24.0000 1.59294 0.796468 0.604681i \(-0.206699\pi\)
0.796468 + 0.604681i \(0.206699\pi\)
\(228\) −6.00000 −0.397360
\(229\) 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
\(232\) −12.0000 −0.787839
\(233\) 6.92820i 0.453882i 0.973909 + 0.226941i \(0.0728724\pi\)
−0.973909 + 0.226941i \(0.927128\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) 13.8564i 0.900070i
\(238\) −18.0000 + 20.7846i −1.16677 + 1.34727i
\(239\) 10.3923i 0.672222i 0.941822 + 0.336111i \(0.109112\pi\)
−0.941822 + 0.336111i \(0.890888\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 1.73205i 0.111340i
\(243\) 15.5885i 1.00000i
\(244\) 6.92820i 0.443533i
\(245\) 0 0
\(246\) −18.0000 −1.14764
\(247\) 0 0
\(248\) −6.00000 −0.381000
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −6.00000 5.19615i −0.377964 0.327327i
\(253\) −12.0000 −0.754434
\(254\) 6.92820i 0.434714i
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −6.00000 −0.374270 −0.187135 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) −24.0000 −1.49417
\(259\) −4.00000 3.46410i −0.248548 0.215249i
\(260\) 0 0
\(261\) 20.7846i 1.28654i
\(262\) 20.7846i 1.28408i
\(263\) 24.2487i 1.49524i −0.664127 0.747620i \(-0.731197\pi\)
0.664127 0.747620i \(-0.268803\pi\)
\(264\) 10.3923i 0.639602i
\(265\) 0 0
\(266\) −12.0000 10.3923i −0.735767 0.637193i
\(267\) 10.3923i 0.635999i
\(268\) 8.00000 0.488678
\(269\) 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 \(-0.736476\pi\)
0.676435 0.736502i \(-0.263524\pi\)
\(272\) 30.0000 1.81902
\(273\) 0 0
\(274\) −36.0000 −2.17484
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) −14.0000 −0.841178 −0.420589 0.907251i \(-0.638177\pi\)
−0.420589 + 0.907251i \(0.638177\pi\)
\(278\) 30.0000 1.79928
\(279\) 10.3923i 0.622171i
\(280\) 0 0
\(281\) 13.8564i 0.826604i −0.910594 0.413302i \(-0.864375\pi\)
0.910594 0.413302i \(-0.135625\pi\)
\(282\) −36.0000 −2.14377
\(283\) 17.3205i 1.02960i 0.857311 + 0.514799i \(0.172133\pi\)
−0.857311 + 0.514799i \(0.827867\pi\)
\(284\) 3.46410i 0.205557i
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 10.3923i −0.708338 0.613438i
\(288\) 15.5885i 0.918559i
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −12.0000 −0.703452
\(292\) 6.92820i 0.405442i
\(293\) 6.00000 0.350524 0.175262 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) −3.00000 20.7846i −0.174964 1.21218i
\(295\) 0 0
\(296\) 3.46410i 0.201347i
\(297\) 18.0000 1.04447
\(298\) 12.0000 0.695141
\(299\) 0 0
\(300\) 0 0
\(301\) −16.0000 13.8564i −0.922225 0.798670i
\(302\) 13.8564i 0.797347i
\(303\) 10.3923i 0.597022i
\(304\) 17.3205i 0.993399i
\(305\) 0 0
\(306\) 31.1769i 1.78227i
\(307\) 24.2487i 1.38395i −0.721923 0.691974i \(-0.756741\pi\)
0.721923 0.691974i \(-0.243259\pi\)
\(308\) −6.00000 + 6.92820i −0.341882 + 0.394771i
\(309\) 6.00000 0.341328
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 20.7846i 1.17482i −0.809291 0.587408i \(-0.800148\pi\)
0.809291 0.587408i \(-0.199852\pi\)
\(314\) −24.0000 −1.35440
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 27.7128i 1.55651i 0.627950 + 0.778253i \(0.283894\pi\)
−0.627950 + 0.778253i \(0.716106\pi\)
\(318\) 0 0
\(319\) −24.0000 −1.34374
\(320\) 0 0
\(321\) −18.0000 −1.00466
\(322\) 12.0000 + 10.3923i 0.668734 + 0.579141i
\(323\) 20.7846i 1.15649i
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) 27.7128i 1.53487i
\(327\) 3.46410i 0.191565i
\(328\) 10.3923i 0.573819i
\(329\) −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
\(333\) −6.00000 −0.328798
\(334\) 20.7846i 1.13728i
\(335\) 0 0
\(336\) −15.0000 + 17.3205i −0.818317 + 0.944911i
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 22.5167i 1.22474i
\(339\) 12.0000 0.651751
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) −18.0000 −0.973329
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 13.8564i 0.747087i
\(345\) 0 0
\(346\) 31.1769i 1.67608i
\(347\) 17.3205i 0.929814i −0.885360 0.464907i \(-0.846088\pi\)
0.885360 0.464907i \(-0.153912\pi\)
\(348\) 12.0000 0.643268
\(349\) 6.92820i 0.370858i −0.982658 0.185429i \(-0.940632\pi\)
0.982658 0.185429i \(-0.0593675\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 18.0000 0.959403
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 36.0000 1.91338
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) −18.0000 + 20.7846i −0.952661 + 1.10004i
\(358\) 18.0000 0.951330
\(359\) 3.46410i 0.182828i −0.995813 0.0914141i \(-0.970861\pi\)
0.995813 0.0914141i \(-0.0291387\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) −36.0000 −1.89212
\(363\) 1.73205i 0.0909091i
\(364\) 0 0
\(365\) 0 0
\(366\) 20.7846i 1.08643i
\(367\) 10.3923i 0.542474i 0.962513 + 0.271237i \(0.0874327\pi\)
−0.962513 + 0.271237i \(0.912567\pi\)
\(368\) 17.3205i 0.902894i
\(369\) −18.0000 −0.937043
\(370\) 0 0
\(371\) 0 0
\(372\) 6.00000 0.311086
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 36.0000 1.86152
\(375\) 0 0
\(376\) 20.7846i 1.07188i
\(377\) 0 0
\(378\) −18.0000 15.5885i −0.925820 0.801784i
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 6.92820i 0.354943i
\(382\) −18.0000 −0.920960
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 21.0000 1.07165
\(385\) 0 0
\(386\) 24.2487i 1.23423i
\(387\) −24.0000 −1.21999
\(388\) 6.92820i 0.351726i
\(389\) 6.92820i 0.351274i −0.984455 0.175637i \(-0.943802\pi\)
0.984455 0.175637i \(-0.0561985\pi\)
\(390\) 0 0
\(391\) 20.7846i 1.05112i
\(392\) −12.0000 + 1.73205i −0.606092 + 0.0874818i
\(393\) 20.7846i 1.04844i
\(394\) 24.0000 1.20910
\(395\) 0 0
\(396\) 10.3923i 0.522233i
\(397\) 13.8564i 0.695433i 0.937600 + 0.347717i \(0.113043\pi\)
−0.937600 + 0.347717i \(0.886957\pi\)
\(398\) 18.0000 0.902258
\(399\) −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 \(0.756749\pi\)
\(402\) 24.0000 1.19701
\(403\) 0 0
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 24.0000 + 20.7846i 1.19110 + 1.03152i
\(407\) 6.92820i 0.343418i
\(408\) 18.0000 0.891133
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −36.0000 −1.77575
\(412\) 3.46410i 0.170664i
\(413\) 24.0000 + 20.7846i 1.18096 + 1.02274i
\(414\) 18.0000 0.884652
\(415\) 0 0
\(416\) 0 0
\(417\) 30.0000 1.46911
\(418\) 20.7846i 1.01661i
\(419\) 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 \(0.578356\pi\)
\(422\) 6.92820i 0.337260i
\(423\) −36.0000 −1.75038
\(424\) 0 0
\(425\) 0 0
\(426\) 10.3923i 0.503509i
\(427\) 12.0000 13.8564i 0.580721 0.670559i
\(428\) 10.3923i 0.502331i
\(429\) 0 0
\(430\) 0 0
\(431\) 10.3923i 0.500580i 0.968171 + 0.250290i \(0.0805259\pi\)
−0.968171 + 0.250290i \(0.919474\pi\)
\(432\) 25.9808i 1.25000i
\(433\) 6.92820i 0.332948i −0.986046 0.166474i \(-0.946762\pi\)
0.986046 0.166474i \(-0.0532382\pi\)
\(434\) 12.0000 + 10.3923i 0.576018 + 0.498847i
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 12.0000 0.574038
\(438\) 20.7846i 0.993127i
\(439\) 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
\(443\) 10.3923i 0.493753i 0.969047 + 0.246877i \(0.0794043\pi\)
−0.969047 + 0.246877i \(0.920596\pi\)
\(444\) 3.46410i 0.164399i
\(445\) 0 0
\(446\) 30.0000 1.42054
\(447\) 12.0000 0.567581
\(448\) 2.00000 + 1.73205i 0.0944911 + 0.0818317i
\(449\) 13.8564i 0.653924i 0.945037 + 0.326962i \(0.106025\pi\)
−0.945037 + 0.326962i \(0.893975\pi\)
\(450\) 0 0
\(451\) 20.7846i 0.978709i
\(452\) 6.92820i 0.325875i
\(453\) 13.8564i 0.651031i
\(454\) 41.5692i 1.95094i
\(455\) 0 0
\(456\) 10.3923i 0.486664i
\(457\) 38.0000 1.77757 0.888783 0.458329i \(-0.151552\pi\)
0.888783 + 0.458329i \(0.151552\pi\)
\(458\) −12.0000 −0.560723
\(459\) 31.1769i 1.45521i
\(460\) 0 0
\(461\) 18.0000 0.838344 0.419172 0.907907i \(-0.362320\pi\)
0.419172 + 0.907907i \(0.362320\pi\)
\(462\) −18.0000 + 20.7846i −0.837436 + 0.966988i
\(463\) −20.0000 −0.929479 −0.464739 0.885448i \(-0.653852\pi\)
−0.464739 + 0.885448i \(0.653852\pi\)
\(464\) 34.6410i 1.60817i
\(465\) 0 0
\(466\) −12.0000 −0.555889
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) 16.0000 + 13.8564i 0.738811 + 0.639829i
\(470\) 0 0
\(471\) −24.0000 −1.10586
\(472\) 20.7846i 0.956689i
\(473\) 27.7128i 1.27424i
\(474\) −24.0000 −1.10236
\(475\) 0 0
\(476\) −12.0000 10.3923i −0.550019 0.476331i
\(477\) 0 0
\(478\) −18.0000 −0.823301
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 12.0000 + 10.3923i 0.546019 + 0.472866i
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −27.0000 −1.22474
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) −12.0000 −0.543214
\(489\) 27.7128i 1.25322i
\(490\) 0 0
\(491\) 24.2487i 1.09433i −0.837025 0.547165i \(-0.815707\pi\)
0.837025 0.547165i \(-0.184293\pi\)
\(492\) 10.3923i 0.468521i
\(493\) 41.5692i 1.87218i
\(494\) 0 0
\(495\) 0 0
\(496\) 17.3205i 0.777714i
\(497\) −6.00000 + 6.92820i −0.269137 + 0.310772i
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 20.7846i 0.928588i
\(502\) 20.7846i 0.927663i
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) −9.00000 + 10.3923i −0.400892 + 0.462910i
\(505\) 0 0
\(506\) 20.7846i 0.923989i
\(507\) 22.5167i 1.00000i
\(508\) −4.00000 −0.177471
\(509\) −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
\(512\) 8.66025i 0.382733i
\(513\) −18.0000 −0.794719
\(514\) 10.3923i 0.458385i
\(515\) 0 0
\(516\) 13.8564i 0.609994i
\(517\) 41.5692i 1.82821i
\(518\) 6.00000 6.92820i 0.263625 0.304408i
\(519\) 31.1769i 1.36851i
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 36.0000 1.57568
\(523\) 17.3205i 0.757373i 0.925525 + 0.378686i \(0.123624\pi\)
−0.925525 + 0.378686i \(0.876376\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 42.0000 1.83129
\(527\) 20.7846i 0.905392i
\(528\) 30.0000 1.30558
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) 36.0000 1.56227
\(532\) 6.00000 6.92820i 0.260133 0.300376i
\(533\) 0 0
\(534\) −18.0000 −0.778936
\(535\) 0 0
\(536\) 13.8564i 0.598506i
\(537\) 18.0000 0.776757
\(538\) 31.1769i 1.34413i
\(539\) −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 \(0.760893\pi\)
\(542\) 42.0000 1.80405
\(543\) −36.0000 −1.54491
\(544\) 31.1769i 1.33670i
\(545\) 0 0
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 20.7846i 0.887875i
\(549\) 20.7846i 0.887066i
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 10.3923i 0.442326i
\(553\) −16.0000 13.8564i −0.680389 0.589234i
\(554\) 24.2487i 1.03023i
\(555\) 0 0
\(556\) 17.3205i 0.734553i
\(557\) 27.7128i 1.17423i −0.809504 0.587115i \(-0.800264\pi\)
0.809504 0.587115i \(-0.199736\pi\)
\(558\) 18.0000 0.762001
\(559\) 0 0
\(560\) 0 0
\(561\) 36.0000 1.51992
\(562\) 24.0000 1.01238
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 20.7846i 0.875190i
\(565\) 0 0
\(566\) −30.0000 −1.26099
\(567\) −18.0000 15.5885i −0.755929 0.654654i
\(568\) 6.00000 0.251754
\(569\) 27.7128i 1.16178i 0.813982 + 0.580891i \(0.197296\pi\)
−0.813982 + 0.580891i \(0.802704\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) −18.0000 −0.751961
\(574\) 18.0000 20.7846i 0.751305 0.867533i
\(575\) 0 0
\(576\) 3.00000 0.125000
\(577\) 34.6410i 1.44212i 0.692870 + 0.721062i \(0.256346\pi\)
−0.692870 + 0.721062i \(0.743654\pi\)
\(578\) 32.9090i 1.36883i
\(579\) 24.2487i 1.00774i
\(580\) 0 0
\(581\) 0 0
\(582\) 20.7846i 0.861550i
\(583\) 0 0
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) 10.3923i 0.429302i
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 12.0000 1.73205i 0.494872 0.0714286i
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) −10.0000 −0.410997
\(593\) −6.00000 −0.246390 −0.123195 0.992382i \(-0.539314\pi\)
−0.123195 + 0.992382i \(0.539314\pi\)
\(594\) 31.1769i 1.27920i
\(595\) 0 0
\(596\) 6.92820i 0.283790i
\(597\) 18.0000 0.736691
\(598\) 0 0
\(599\) 24.2487i 0.990775i 0.868672 + 0.495388i \(0.164974\pi\)
−0.868672 + 0.495388i \(0.835026\pi\)
\(600\) 0 0
\(601\) 13.8564i 0.565215i −0.959236 0.282607i \(-0.908801\pi\)
0.959236 0.282607i \(-0.0911993\pi\)
\(602\) 24.0000 27.7128i 0.978167 1.12949i
\(603\) 24.0000 0.977356
\(604\) 8.00000 0.325515
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) 17.3205i 0.703018i −0.936185 0.351509i \(-0.885669\pi\)
0.936185 0.351509i \(-0.114331\pi\)
\(608\) −18.0000 −0.729996
\(609\) 24.0000 + 20.7846i 0.972529 + 0.842235i
\(610\) 0 0
\(611\) 0 0
\(612\) −18.0000 −0.727607
\(613\) 34.0000 1.37325 0.686624 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 42.0000 1.69498
\(615\) 0 0
\(616\) 12.0000 + 10.3923i 0.483494 + 0.418718i
\(617\) 6.92820i 0.278919i −0.990228 0.139459i \(-0.955464\pi\)
0.990228 0.139459i \(-0.0445365\pi\)
\(618\) 10.3923i 0.418040i
\(619\) 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
\(623\) −12.0000 10.3923i −0.480770 0.416359i
\(624\) 0 0
\(625\) 0 0
\(626\) 36.0000 1.43885
\(627\) 20.7846i 0.830057i
\(628\) 13.8564i 0.552931i
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 13.8564i 0.551178i
\(633\) 6.92820i 0.275371i
\(634\) −48.0000 −1.90632
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 41.5692i 1.64574i
\(639\) 10.3923i 0.411113i
\(640\) 0 0
\(641\) 27.7128i 1.09459i −0.836940 0.547295i \(-0.815658\pi\)
0.836940 0.547295i \(-0.184342\pi\)
\(642\) 31.1769i 1.23045i
\(643\) 31.1769i 1.22950i 0.788723 + 0.614749i \(0.210743\pi\)
−0.788723 + 0.614749i \(0.789257\pi\)
\(644\) −6.00000 + 6.92820i −0.236433 + 0.273009i
\(645\) 0 0
\(646\) −36.0000 −1.41640
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 15.5885i 0.612372i
\(649\) 41.5692i 1.63173i
\(650\) 0 0
\(651\) 12.0000 + 10.3923i 0.470317 + 0.407307i
\(652\) −16.0000 −0.626608
\(653\) 41.5692i 1.62673i −0.581754 0.813365i \(-0.697633\pi\)
0.581754 0.813365i \(-0.302367\pi\)
\(654\) 6.00000 0.234619
\(655\) 0 0
\(656\) −30.0000 −1.17130
\(657\) 20.7846i 0.810885i
\(658\) 36.0000 41.5692i 1.40343 1.62054i
\(659\) 10.3923i 0.404827i −0.979300 0.202413i \(-0.935122\pi\)
0.979300 0.202413i \(-0.0648785\pi\)
\(660\) 0 0
\(661\) 48.4974i 1.88633i 0.332323 + 0.943166i \(0.392168\pi\)
−0.332323 + 0.943166i \(0.607832\pi\)
\(662\) 48.4974i 1.88491i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 10.3923i 0.402694i
\(667\) −24.0000 −0.929284
\(668\) 12.0000 0.464294
\(669\) 30.0000 1.15987
\(670\) 0 0
\(671\) −24.0000 −0.926510
\(672\) −18.0000 15.5885i −0.694365 0.601338i
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 24.2487i 0.934025i
\(675\) 0 0
\(676\) −13.0000 −0.500000
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 20.7846i 0.798228i
\(679\) 12.0000 13.8564i 0.460518 0.531760i
\(680\) 0 0
\(681\) 41.5692i 1.59294i
\(682\) 20.7846i 0.795884i
\(683\) 17.3205i 0.662751i −0.943499 0.331375i \(-0.892487\pi\)
0.943499 0.331375i \(-0.107513\pi\)
\(684\) 10.3923i 0.397360i
\(685\) 0 0
\(686\) 27.0000 + 17.3205i 1.03086 + 0.661300i
\(687\) −12.0000 −0.457829
\(688\) −40.0000 −1.52499
\(689\) 0 0
\(690\) 0 0
\(691\) 31.1769i 1.18603i −0.805193 0.593013i \(-0.797938\pi\)
0.805193 0.593013i \(-0.202062\pi\)
\(692\) 18.0000 0.684257
\(693\) −18.0000 + 20.7846i −0.683763 + 0.789542i
\(694\) 30.0000 1.13878
\(695\) 0 0
\(696\) 20.7846i 0.787839i
\(697\) −36.0000 −1.36360
\(698\) 12.0000 0.454207
\(699\) −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
\(703\) 6.92820i 0.261302i
\(704\) 3.46410i 0.130558i
\(705\) 0 0
\(706\) 10.3923i 0.391120i
\(707\) 12.0000 + 10.3923i 0.451306 + 0.390843i
\(708\) 20.7846i 0.781133i
\(709\) 22.0000 0.826227 0.413114 0.910679i \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 0 0
\(711\) −24.0000 −0.900070
\(712\) 10.3923i 0.389468i
\(713\) −12.0000 −0.449404
\(714\) −36.0000 31.1769i −1.34727 1.16677i
\(715\) 0 0
\(716\) 10.3923i 0.388379i
\(717\) −18.0000 −0.672222
\(718\) 6.00000 0.223918
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −6.00000 + 6.92820i −0.223452 + 0.258020i
\(722\) 12.1244i 0.451222i
\(723\) 0 0
\(724\) 20.7846i 0.772454i
\(725\) 0 0
\(726\) 3.00000 0.111340
\(727\) 3.46410i 0.128476i −0.997935 0.0642382i \(-0.979538\pi\)
0.997935 0.0642382i \(-0.0204617\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) 12.0000 0.443533
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) −18.0000 −0.664392
\(735\) 0 0
\(736\) 18.0000 0.663489
\(737\) 27.7128i 1.02081i
\(738\) 31.1769i 1.14764i
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 31.1769i 1.14377i 0.820334 + 0.571885i \(0.193788\pi\)
−0.820334 + 0.571885i \(0.806212\pi\)
\(744\) 10.3923i 0.381000i
\(745\) 0 0
\(746\) 24.2487i 0.887808i
\(747\) 0 0
\(748\) 20.7846i 0.759961i
\(749\) 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 \(0.405705\pi\)
\(752\) −60.0000 −2.18797
\(753\) 20.7846i 0.757433i
\(754\) 0 0
\(755\) 0 0
\(756\) 9.00000 10.3923i 0.327327 0.377964i
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 6.92820i 0.251644i
\(759\) 20.7846i 0.754434i
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) −12.0000 −0.434714
\(763\) 4.00000 + 3.46410i 0.144810 + 0.125409i
\(764\) 10.3923i 0.375980i
\(765\) 0 0
\(766\) 20.7846i 0.750978i
\(767\) 0 0
\(768\) 32.9090i 1.18750i
\(769\) 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
\(772\) −14.0000 −0.503871
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 41.5692i 1.49417i
\(775\) 0 0
\(776\) −12.0000 −0.430775
\(777\) 6.00000 6.92820i 0.215249 0.248548i
\(778\) 12.0000 0.430221
\(779\) 20.7846i 0.744686i
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 36.0000 1.28736
\(783\) 36.0000 1.28654
\(784\) −5.00000 34.6410i −0.178571 1.23718i
\(785\) 0 0
\(786\) 36.0000 1.28408
\(787\) 24.2487i 0.864373i −0.901784 0.432187i \(-0.857742\pi\)
0.901784 0.432187i \(-0.142258\pi\)
\(788\) 13.8564i 0.493614i
\(789\) 42.0000 1.49524
\(790\) 0 0
\(791\) −12.0000 + 13.8564i −0.426671 + 0.492677i
\(792\) 18.0000 0.639602
\(793\) 0 0
\(794\) −24.0000 −0.851728
\(795\) 0 0
\(796\) 10.3923i 0.368345i
\(797\) −18.0000 −0.637593 −0.318796 0.947823i \(-0.603279\pi\)
−0.318796 + 0.947823i \(0.603279\pi\)
\(798\) 18.0000 20.7846i 0.637193 0.735767i
\(799\) −72.0000 −2.54718
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) −48.0000 −1.69494
\(803\) −24.0000 −0.846942
\(804\) 13.8564i 0.488678i
\(805\) 0 0
\(806\) 0 0
\(807\) 31.1769i 1.09748i
\(808\) 10.3923i 0.365600i
\(809\) 55.4256i 1.94866i −0.225122 0.974331i \(-0.572278\pi\)
0.225122 0.974331i \(-0.427722\pi\)
\(810\) 0 0
\(811\) 38.1051i 1.33805i 0.743239 + 0.669026i \(0.233288\pi\)
−0.743239 + 0.669026i \(0.766712\pi\)
\(812\) −12.0000 + 13.8564i −0.421117 + 0.486265i
\(813\) 42.0000 1.47300
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 51.9615i 1.81902i
\(817\) 27.7128i 0.969549i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 20.7846i 0.725388i −0.931908 0.362694i \(-0.881857\pi\)
0.931908 0.362694i \(-0.118143\pi\)
\(822\) 62.3538i 2.17484i
\(823\) −44.0000 −1.53374 −0.766872 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) −36.0000 + 41.5692i −1.25260 + 1.44638i
\(827\) 38.1051i 1.32504i 0.749042 + 0.662522i \(0.230514\pi\)
−0.749042 + 0.662522i \(0.769486\pi\)
\(828\) 10.3923i 0.361158i
\(829\) 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
\(833\) −6.00000 41.5692i −0.207888 1.44029i
\(834\) 51.9615i 1.79928i
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) 18.0000 0.622171
\(838\) 20.7846i 0.717992i
\(839\) −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
\(842\) 17.3205i 0.596904i
\(843\) 24.0000 0.826604
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 62.3538i 2.14377i
\(847\) 2.00000 + 1.73205i 0.0687208 + 0.0595140i
\(848\) 0 0
\(849\) −30.0000 −1.02960
\(850\) 0 0
\(851\) 6.92820i 0.237496i
\(852\) −6.00000 −0.205557
\(853\) 41.5692i 1.42330i 0.702533 + 0.711651i \(0.252052\pi\)
−0.702533 + 0.711651i \(0.747948\pi\)
\(854\) 24.0000 + 20.7846i 0.821263 + 0.711235i
\(855\) 0 0
\(856\) −18.0000 −0.615227
\(857\) −30.0000 −1.02478 −0.512390 0.858753i \(-0.671240\pi\)
−0.512390 + 0.858753i \(0.671240\pi\)
\(858\) 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
\(862\) −18.0000 −0.613082
\(863\) 17.3205i 0.589597i 0.955559 + 0.294798i \(0.0952525\pi\)
−0.955559 + 0.294798i \(0.904747\pi\)
\(864\) −27.0000 −0.918559
\(865\) 0 0
\(866\) 12.0000 0.407777
\(867\) 32.9090i 1.11765i
\(868\) −6.00000 + 6.92820i −0.203653 + 0.235159i
\(869\) 27.7128i 0.940093i
\(870\) 0 0
\(871\) 0 0
\(872\) 3.46410i 0.117309i
\(873\) 20.7846i 0.703452i
\(874\) 20.7846i 0.703050i
\(875\) 0 0
\(876\) 12.0000 0.405442
\(877\) −38.0000 −1.28317 −0.641584 0.767052i \(-0.721723\pi\)
−0.641584 + 0.767052i \(0.721723\pi\)
\(878\) −30.0000 −1.01245
\(879\) 10.3923i 0.350524i
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 36.0000 5.19615i 1.21218 0.174964i
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −18.0000 −0.604722
\(887\) 36.0000 1.20876 0.604381 0.796696i \(-0.293421\pi\)
0.604381 + 0.796696i \(0.293421\pi\)
\(888\) −6.00000 −0.201347
\(889\) −8.00000 6.92820i −0.268311 0.232364i
\(890\) 0 0
\(891\) 31.1769i 1.04447i
\(892\) 17.3205i 0.579934i
\(893\) 41.5692i 1.39106i
\(894\) 20.7846i 0.695141i
\(895\) 0 0
\(896\) −21.0000 + 24.2487i −0.701561 + 0.810093i
\(897\) 0 0
\(898\) −24.0000 −0.800890
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 0 0
\(902\) −36.0000 −1.19867
\(903\) 24.0000 27.7128i 0.798670 0.922225i
\(904\) 12.0000 0.399114
\(905\) 0 0
\(906\) 24.0000 0.797347
\(907\) 32.0000 1.06254 0.531271 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) −24.0000 −0.796468
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 17.3205i 0.573854i −0.957952 0.286927i \(-0.907366\pi\)
0.957952 0.286927i \(-0.0926337\pi\)
\(912\) −30.0000 −0.993399
\(913\) 0 0
\(914\) 65.8179i 2.17706i
\(915\) 0 0
\(916\) 6.92820i 0.228914i
\(917\) 24.0000 + 20.7846i 0.792550 + 0.686368i
\(918\) −54.0000 −1.78227
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 42.0000 1.38395
\(922\) 31.1769i 1.02676i
\(923\) 0 0
\(924\) −12.0000 10.3923i −0.394771 0.341882i
\(925\) 0 0
\(926\) 34.6410i 1.13837i
\(927\) 10.3923i 0.341328i
\(928\) 36.0000 1.18176
\(929\) 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
\(932\) 6.92820i 0.226941i
\(933\) 0 0
\(934\) 41.5692i 1.36019i
\(935\) 0 0
\(936\) 0 0
\(937\) 6.92820i 0.226335i −0.993576 0.113167i \(-0.963900\pi\)
0.993576 0.113167i \(-0.0360996\pi\)
\(938\) −24.0000 + 27.7128i −0.783628 + 0.904855i
\(939\) 36.0000 1.17482
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 41.5692i 1.35440i
\(943\) 20.7846i 0.676840i
\(944\) 60.0000 1.95283
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) 31.1769i 1.01311i −0.862207 0.506557i \(-0.830918\pi\)
0.862207 0.506557i \(-0.169082\pi\)
\(948\) 13.8564i 0.450035i
\(949\) 0 0
\(950\) 0 0
\(951\) −48.0000 −1.55651
\(952\) −18.0000 + 20.7846i −0.583383 + 0.673633i
\(953\) 34.6410i 1.12213i 0.827771 + 0.561066i \(0.189609\pi\)
−0.827771 + 0.561066i \(0.810391\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 10.3923i 0.336111i
\(957\) 41.5692i 1.34374i
\(958\) 0 0
\(959\) 36.0000 41.5692i 1.16250 1.34234i
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) 31.1769i 1.00466i
\(964\) 0 0
\(965\) 0 0
\(966\) −18.0000 + 20.7846i −0.579141 + 0.668734i
\(967\) −20.0000 −0.643157 −0.321578 0.946883i \(-0.604213\pi\)
−0.321578 + 0.946883i \(0.604213\pi\)
\(968\) 1.73205i 0.0556702i
\(969\) −36.0000 −1.15649
\(970\) 0 0
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) 15.5885i 0.500000i
\(973\) −30.0000 + 34.6410i −0.961756 + 1.11054i
\(974\) 6.92820i 0.221994i
\(975\) 0 0
\(976\) 34.6410i 1.10883i
\(977\) 6.92820i 0.221653i 0.993840 + 0.110826i \(0.0353498\pi\)
−0.993840 + 0.110826i \(0.964650\pi\)
\(978\) −48.0000 −1.53487
\(979\) 20.7846i 0.664279i
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 42.0000 1.34027
\(983\) 60.0000 1.91370 0.956851 0.290578i \(-0.0938475\pi\)
0.956851 + 0.290578i \(0.0938475\pi\)
\(984\) −18.0000 −0.573819
\(985\) 0 0
\(986\) 72.0000 2.29295
\(987\) 36.0000 41.5692i 1.14589 1.32316i
\(988\) 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 \(0.418211\pi\)
\(992\) 18.0000 0.571501
\(993\) 48.4974i 1.53902i
\(994\) −12.0000 10.3923i −0.380617 0.329624i
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 34.6410i 1.09654i
\(999\) 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.b.a.251.2 2
3.2 odd 2 525.2.b.b.251.1 2
5.2 odd 4 525.2.g.b.524.1 4
5.3 odd 4 525.2.g.b.524.4 4
5.4 even 2 105.2.b.a.41.1 2
7.6 odd 2 525.2.b.b.251.2 2
15.2 even 4 525.2.g.c.524.3 4
15.8 even 4 525.2.g.c.524.2 4
15.14 odd 2 105.2.b.b.41.2 yes 2
20.19 odd 2 1680.2.f.b.881.2 2
21.20 even 2 inner 525.2.b.a.251.1 2
35.4 even 6 735.2.s.d.656.1 2
35.9 even 6 735.2.s.b.521.1 2
35.13 even 4 525.2.g.c.524.4 4
35.19 odd 6 735.2.s.a.521.1 2
35.24 odd 6 735.2.s.f.656.1 2
35.27 even 4 525.2.g.c.524.1 4
35.34 odd 2 105.2.b.b.41.1 yes 2
60.59 even 2 1680.2.f.c.881.2 2
105.44 odd 6 735.2.s.f.521.1 2
105.59 even 6 735.2.s.b.656.1 2
105.62 odd 4 525.2.g.b.524.3 4
105.74 odd 6 735.2.s.a.656.1 2
105.83 odd 4 525.2.g.b.524.2 4
105.89 even 6 735.2.s.d.521.1 2
105.104 even 2 105.2.b.a.41.2 yes 2
140.139 even 2 1680.2.f.c.881.1 2
420.419 odd 2 1680.2.f.b.881.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.b.a.41.1 2 5.4 even 2
105.2.b.a.41.2 yes 2 105.104 even 2
105.2.b.b.41.1 yes 2 35.34 odd 2
105.2.b.b.41.2 yes 2 15.14 odd 2
525.2.b.a.251.1 2 21.20 even 2 inner
525.2.b.a.251.2 2 1.1 even 1 trivial
525.2.b.b.251.1 2 3.2 odd 2
525.2.b.b.251.2 2 7.6 odd 2
525.2.g.b.524.1 4 5.2 odd 4
525.2.g.b.524.2 4 105.83 odd 4
525.2.g.b.524.3 4 105.62 odd 4
525.2.g.b.524.4 4 5.3 odd 4
525.2.g.c.524.1 4 35.27 even 4
525.2.g.c.524.2 4 15.8 even 4
525.2.g.c.524.3 4 15.2 even 4
525.2.g.c.524.4 4 35.13 even 4
735.2.s.a.521.1 2 35.19 odd 6
735.2.s.a.656.1 2 105.74 odd 6
735.2.s.b.521.1 2 35.9 even 6
735.2.s.b.656.1 2 105.59 even 6
735.2.s.d.521.1 2 105.89 even 6
735.2.s.d.656.1 2 35.4 even 6
735.2.s.f.521.1 2 105.44 odd 6
735.2.s.f.656.1 2 35.24 odd 6
1680.2.f.b.881.1 2 420.419 odd 2
1680.2.f.b.881.2 2 20.19 odd 2
1680.2.f.c.881.1 2 140.139 even 2
1680.2.f.c.881.2 2 60.59 even 2