Properties

Label 325.2.b.b.274.2
Level $325$
Weight $2$
Character 325.274
Analytic conductor $2.595$
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] = [325,2,Mod(274,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("325.274");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.b (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: \(2.59513806569\)
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 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 325.274
Dual form 325.2.b.b.274.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.00000i q^{3} +1.00000 q^{4} +2.00000 q^{6} +4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -2.00000i q^{3} +1.00000 q^{4} +2.00000 q^{6} +4.00000i q^{7} +3.00000i q^{8} -1.00000 q^{9} +2.00000 q^{11} -2.00000i q^{12} -1.00000i q^{13} -4.00000 q^{14} -1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +6.00000 q^{19} +8.00000 q^{21} +2.00000i q^{22} -6.00000i q^{23} +6.00000 q^{24} +1.00000 q^{26} -4.00000i q^{27} +4.00000i q^{28} -2.00000 q^{29} -10.0000 q^{31} +5.00000i q^{32} -4.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} +2.00000i q^{37} +6.00000i q^{38} -2.00000 q^{39} -6.00000 q^{41} +8.00000i q^{42} +10.0000i q^{43} +2.00000 q^{44} +6.00000 q^{46} -4.00000i q^{47} +2.00000i q^{48} -9.00000 q^{49} -4.00000 q^{51} -1.00000i q^{52} +2.00000i q^{53} +4.00000 q^{54} -12.0000 q^{56} -12.0000i q^{57} -2.00000i q^{58} -6.00000 q^{59} +2.00000 q^{61} -10.0000i q^{62} -4.00000i q^{63} -7.00000 q^{64} +4.00000 q^{66} +4.00000i q^{67} -2.00000i q^{68} -12.0000 q^{69} +6.00000 q^{71} -3.00000i q^{72} -6.00000i q^{73} -2.00000 q^{74} +6.00000 q^{76} +8.00000i q^{77} -2.00000i q^{78} +12.0000 q^{79} -11.0000 q^{81} -6.00000i q^{82} -16.0000i q^{83} +8.00000 q^{84} -10.0000 q^{86} +4.00000i q^{87} +6.00000i q^{88} -2.00000 q^{89} +4.00000 q^{91} -6.00000i q^{92} +20.0000i q^{93} +4.00000 q^{94} +10.0000 q^{96} +2.00000i q^{97} -9.00000i q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 4 q^{6} - 2 q^{9} + 4 q^{11} - 8 q^{14} - 2 q^{16} + 12 q^{19} + 16 q^{21} + 12 q^{24} + 2 q^{26} - 4 q^{29} - 20 q^{31} + 4 q^{34} - 2 q^{36} - 4 q^{39} - 12 q^{41} + 4 q^{44} + 12 q^{46} - 18 q^{49} - 8 q^{51} + 8 q^{54} - 24 q^{56} - 12 q^{59} + 4 q^{61} - 14 q^{64} + 8 q^{66} - 24 q^{69} + 12 q^{71} - 4 q^{74} + 12 q^{76} + 24 q^{79} - 22 q^{81} + 16 q^{84} - 20 q^{86} - 4 q^{89} + 8 q^{91} + 8 q^{94} + 20 q^{96} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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 \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) − 2.00000i − 1.15470i −0.816497 0.577350i \(-0.804087\pi\)
0.816497 0.577350i \(-0.195913\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) 4.00000i 1.51186i 0.654654 + 0.755929i \(0.272814\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 3.00000i 1.06066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) − 2.00000i − 0.577350i
\(13\) − 1.00000i − 0.277350i
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 8.00000 1.74574
\(22\) 2.00000i 0.426401i
\(23\) − 6.00000i − 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 6.00000 1.22474
\(25\) 0 0
\(26\) 1.00000 0.196116
\(27\) − 4.00000i − 0.769800i
\(28\) 4.00000i 0.755929i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 5.00000i 0.883883i
\(33\) − 4.00000i − 0.696311i
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 6.00000i 0.973329i
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 8.00000i 1.23443i
\(43\) 10.0000i 1.52499i 0.646997 + 0.762493i \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 2.00000 0.301511
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) − 4.00000i − 0.583460i −0.956501 0.291730i \(-0.905769\pi\)
0.956501 0.291730i \(-0.0942309\pi\)
\(48\) 2.00000i 0.288675i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) − 1.00000i − 0.138675i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 4.00000 0.544331
\(55\) 0 0
\(56\) −12.0000 −1.60357
\(57\) − 12.0000i − 1.58944i
\(58\) − 2.00000i − 0.262613i
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) − 10.0000i − 1.27000i
\(63\) − 4.00000i − 0.503953i
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 4.00000 0.492366
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) 6.00000 0.688247
\(77\) 8.00000i 0.911685i
\(78\) − 2.00000i − 0.226455i
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) − 6.00000i − 0.662589i
\(83\) − 16.0000i − 1.75623i −0.478451 0.878114i \(-0.658802\pi\)
0.478451 0.878114i \(-0.341198\pi\)
\(84\) 8.00000 0.872872
\(85\) 0 0
\(86\) −10.0000 −1.07833
\(87\) 4.00000i 0.428845i
\(88\) 6.00000i 0.639602i
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) − 6.00000i − 0.625543i
\(93\) 20.0000i 2.07390i
\(94\) 4.00000 0.412568
\(95\) 0 0
\(96\) 10.0000 1.02062
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −18.0000 −1.79107 −0.895533 0.444994i \(-0.853206\pi\)
−0.895533 + 0.444994i \(0.853206\pi\)
\(102\) − 4.00000i − 0.396059i
\(103\) 2.00000i 0.197066i 0.995134 + 0.0985329i \(0.0314150\pi\)
−0.995134 + 0.0985329i \(0.968585\pi\)
\(104\) 3.00000 0.294174
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) − 10.0000i − 0.966736i −0.875417 0.483368i \(-0.839413\pi\)
0.875417 0.483368i \(-0.160587\pi\)
\(108\) − 4.00000i − 0.384900i
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) − 4.00000i − 0.377964i
\(113\) − 14.0000i − 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 12.0000 1.12390
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) 1.00000i 0.0924500i
\(118\) − 6.00000i − 0.552345i
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 2.00000i 0.181071i
\(123\) 12.0000i 1.08200i
\(124\) −10.0000 −0.898027
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 20.0000 1.76090
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) 24.0000i 2.08106i
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 2.00000i 0.170872i 0.996344 + 0.0854358i \(0.0272282\pi\)
−0.996344 + 0.0854358i \(0.972772\pi\)
\(138\) − 12.0000i − 1.02151i
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 6.00000i 0.503509i
\(143\) − 2.00000i − 0.167248i
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) 18.0000i 1.48461i
\(148\) 2.00000i 0.164399i
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 18.0000i 1.45999i
\(153\) 2.00000i 0.161690i
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 6.00000i 0.478852i 0.970915 + 0.239426i \(0.0769593\pi\)
−0.970915 + 0.239426i \(0.923041\pi\)
\(158\) 12.0000i 0.954669i
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) 24.0000 1.89146
\(162\) − 11.0000i − 0.864242i
\(163\) − 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
\(166\) 16.0000 1.24184
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 24.0000i 1.85164i
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 10.0000i 0.762493i
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 12.0000i 0.901975i
\(178\) − 2.00000i − 0.149906i
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 4.00000i 0.296500i
\(183\) − 4.00000i − 0.295689i
\(184\) 18.0000 1.32698
\(185\) 0 0
\(186\) −20.0000 −1.46647
\(187\) − 4.00000i − 0.292509i
\(188\) − 4.00000i − 0.291730i
\(189\) 16.0000 1.16383
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 14.0000i 1.01036i
\(193\) − 2.00000i − 0.143963i −0.997406 0.0719816i \(-0.977068\pi\)
0.997406 0.0719816i \(-0.0229323\pi\)
\(194\) −2.00000 −0.143592
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) − 2.00000i − 0.142134i
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) − 18.0000i − 1.26648i
\(203\) − 8.00000i − 0.561490i
\(204\) −4.00000 −0.280056
\(205\) 0 0
\(206\) −2.00000 −0.139347
\(207\) 6.00000i 0.417029i
\(208\) 1.00000i 0.0693375i
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 2.00000i 0.137361i
\(213\) − 12.0000i − 0.822226i
\(214\) 10.0000 0.683586
\(215\) 0 0
\(216\) 12.0000 0.816497
\(217\) − 40.0000i − 2.71538i
\(218\) − 10.0000i − 0.677285i
\(219\) −12.0000 −0.810885
\(220\) 0 0
\(221\) −2.00000 −0.134535
\(222\) 4.00000i 0.268462i
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) −20.0000 −1.33631
\(225\) 0 0
\(226\) 14.0000 0.931266
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) − 12.0000i − 0.794719i
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 16.0000 1.05272
\(232\) − 6.00000i − 0.393919i
\(233\) 10.0000i 0.655122i 0.944830 + 0.327561i \(0.106227\pi\)
−0.944830 + 0.327561i \(0.893773\pi\)
\(234\) −1.00000 −0.0653720
\(235\) 0 0
\(236\) −6.00000 −0.390567
\(237\) − 24.0000i − 1.55897i
\(238\) 8.00000i 0.518563i
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) − 7.00000i − 0.449977i
\(243\) 10.0000i 0.641500i
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −12.0000 −0.765092
\(247\) − 6.00000i − 0.381771i
\(248\) − 30.0000i − 1.90500i
\(249\) −32.0000 −2.02792
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) − 4.00000i − 0.251976i
\(253\) − 12.0000i − 0.754434i
\(254\) −2.00000 −0.125491
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) − 2.00000i − 0.124757i −0.998053 0.0623783i \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) 20.0000i 1.24515i
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 20.0000i 1.23560i
\(263\) 14.0000i 0.863277i 0.902047 + 0.431638i \(0.142064\pi\)
−0.902047 + 0.431638i \(0.857936\pi\)
\(264\) 12.0000 0.738549
\(265\) 0 0
\(266\) −24.0000 −1.47153
\(267\) 4.00000i 0.244796i
\(268\) 4.00000i 0.244339i
\(269\) −6.00000 −0.365826 −0.182913 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 2.00000i 0.121268i
\(273\) − 8.00000i − 0.484182i
\(274\) −2.00000 −0.120824
\(275\) 0 0
\(276\) −12.0000 −0.722315
\(277\) 14.0000i 0.841178i 0.907251 + 0.420589i \(0.138177\pi\)
−0.907251 + 0.420589i \(0.861823\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) −6.00000 −0.357930 −0.178965 0.983855i \(-0.557275\pi\)
−0.178965 + 0.983855i \(0.557275\pi\)
\(282\) − 8.00000i − 0.476393i
\(283\) 2.00000i 0.118888i 0.998232 + 0.0594438i \(0.0189327\pi\)
−0.998232 + 0.0594438i \(0.981067\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) − 24.0000i − 1.41668i
\(288\) − 5.00000i − 0.294628i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 4.00000 0.234484
\(292\) − 6.00000i − 0.351123i
\(293\) 22.0000i 1.28525i 0.766179 + 0.642627i \(0.222155\pi\)
−0.766179 + 0.642627i \(0.777845\pi\)
\(294\) −18.0000 −1.04978
\(295\) 0 0
\(296\) −6.00000 −0.348743
\(297\) − 8.00000i − 0.464207i
\(298\) − 18.0000i − 1.04271i
\(299\) −6.00000 −0.346989
\(300\) 0 0
\(301\) −40.0000 −2.30556
\(302\) 10.0000i 0.575435i
\(303\) 36.0000i 2.06815i
\(304\) −6.00000 −0.344124
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) − 8.00000i − 0.456584i −0.973593 0.228292i \(-0.926686\pi\)
0.973593 0.228292i \(-0.0733141\pi\)
\(308\) 8.00000i 0.455842i
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −4.00000 −0.226819 −0.113410 0.993548i \(-0.536177\pi\)
−0.113410 + 0.993548i \(0.536177\pi\)
\(312\) − 6.00000i − 0.339683i
\(313\) − 22.0000i − 1.24351i −0.783210 0.621757i \(-0.786419\pi\)
0.783210 0.621757i \(-0.213581\pi\)
\(314\) −6.00000 −0.338600
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 4.00000i 0.224309i
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) 24.0000i 1.33747i
\(323\) − 12.0000i − 0.667698i
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) 20.0000i 1.10600i
\(328\) − 18.0000i − 0.993884i
\(329\) 16.0000 0.882109
\(330\) 0 0
\(331\) −18.0000 −0.989369 −0.494685 0.869072i \(-0.664716\pi\)
−0.494685 + 0.869072i \(0.664716\pi\)
\(332\) − 16.0000i − 0.878114i
\(333\) − 2.00000i − 0.109599i
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) −8.00000 −0.436436
\(337\) − 26.0000i − 1.41631i −0.706057 0.708155i \(-0.749528\pi\)
0.706057 0.708155i \(-0.250472\pi\)
\(338\) − 1.00000i − 0.0543928i
\(339\) −28.0000 −1.52075
\(340\) 0 0
\(341\) −20.0000 −1.08306
\(342\) − 6.00000i − 0.324443i
\(343\) − 8.00000i − 0.431959i
\(344\) −30.0000 −1.61749
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 22.0000i 1.18102i 0.807030 + 0.590511i \(0.201074\pi\)
−0.807030 + 0.590511i \(0.798926\pi\)
\(348\) 4.00000i 0.214423i
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 10.0000i 0.533002i
\(353\) 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
\(356\) −2.00000 −0.106000
\(357\) − 16.0000i − 0.846810i
\(358\) − 12.0000i − 0.634220i
\(359\) 10.0000 0.527780 0.263890 0.964553i \(-0.414994\pi\)
0.263890 + 0.964553i \(0.414994\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) − 22.0000i − 1.15629i
\(363\) 14.0000i 0.734809i
\(364\) 4.00000 0.209657
\(365\) 0 0
\(366\) 4.00000 0.209083
\(367\) 14.0000i 0.730794i 0.930852 + 0.365397i \(0.119067\pi\)
−0.930852 + 0.365397i \(0.880933\pi\)
\(368\) 6.00000i 0.312772i
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −8.00000 −0.415339
\(372\) 20.0000i 1.03695i
\(373\) 34.0000i 1.76045i 0.474554 + 0.880227i \(0.342610\pi\)
−0.474554 + 0.880227i \(0.657390\pi\)
\(374\) 4.00000 0.206835
\(375\) 0 0
\(376\) 12.0000 0.618853
\(377\) 2.00000i 0.103005i
\(378\) 16.0000i 0.822951i
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) 4.00000 0.204926
\(382\) 0 0
\(383\) 12.0000i 0.613171i 0.951843 + 0.306586i \(0.0991866\pi\)
−0.951843 + 0.306586i \(0.900813\pi\)
\(384\) 6.00000 0.306186
\(385\) 0 0
\(386\) 2.00000 0.101797
\(387\) − 10.0000i − 0.508329i
\(388\) 2.00000i 0.101535i
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) − 27.0000i − 1.36371i
\(393\) − 40.0000i − 2.01773i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −2.00000 −0.100504
\(397\) − 6.00000i − 0.301131i −0.988600 0.150566i \(-0.951890\pi\)
0.988600 0.150566i \(-0.0481095\pi\)
\(398\) 16.0000i 0.802008i
\(399\) 48.0000 2.40301
\(400\) 0 0
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 8.00000i 0.399004i
\(403\) 10.0000i 0.498135i
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) 8.00000 0.397033
\(407\) 4.00000i 0.198273i
\(408\) − 12.0000i − 0.594089i
\(409\) −18.0000 −0.890043 −0.445021 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 0 0
\(411\) 4.00000 0.197305
\(412\) 2.00000i 0.0985329i
\(413\) − 24.0000i − 1.18096i
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) 5.00000 0.245145
\(417\) 0 0
\(418\) 12.0000i 0.586939i
\(419\) 40.0000 1.95413 0.977064 0.212946i \(-0.0683059\pi\)
0.977064 + 0.212946i \(0.0683059\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 12.0000i 0.584151i
\(423\) 4.00000i 0.194487i
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 12.0000 0.581402
\(427\) 8.00000i 0.387147i
\(428\) − 10.0000i − 0.483368i
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 14.0000 0.674356 0.337178 0.941441i \(-0.390528\pi\)
0.337178 + 0.941441i \(0.390528\pi\)
\(432\) 4.00000i 0.192450i
\(433\) 10.0000i 0.480569i 0.970702 + 0.240285i \(0.0772408\pi\)
−0.970702 + 0.240285i \(0.922759\pi\)
\(434\) 40.0000 1.92006
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) − 36.0000i − 1.72211i
\(438\) − 12.0000i − 0.573382i
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) − 2.00000i − 0.0951303i
\(443\) 14.0000i 0.665160i 0.943075 + 0.332580i \(0.107919\pi\)
−0.943075 + 0.332580i \(0.892081\pi\)
\(444\) 4.00000 0.189832
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) 36.0000i 1.70274i
\(448\) − 28.0000i − 1.32288i
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) − 14.0000i − 0.658505i
\(453\) − 20.0000i − 0.939682i
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 36.0000 1.68585
\(457\) − 38.0000i − 1.77757i −0.458329 0.888783i \(-0.651552\pi\)
0.458329 0.888783i \(-0.348448\pi\)
\(458\) 22.0000i 1.02799i
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) 10.0000 0.465746 0.232873 0.972507i \(-0.425187\pi\)
0.232873 + 0.972507i \(0.425187\pi\)
\(462\) 16.0000i 0.744387i
\(463\) − 24.0000i − 1.11537i −0.830051 0.557687i \(-0.811689\pi\)
0.830051 0.557687i \(-0.188311\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 10.0000i 0.462745i 0.972865 + 0.231372i \(0.0743216\pi\)
−0.972865 + 0.231372i \(0.925678\pi\)
\(468\) 1.00000i 0.0462250i
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 12.0000 0.552931
\(472\) − 18.0000i − 0.828517i
\(473\) 20.0000i 0.919601i
\(474\) 24.0000 1.10236
\(475\) 0 0
\(476\) 8.00000 0.366679
\(477\) − 2.00000i − 0.0915737i
\(478\) 6.00000i 0.274434i
\(479\) −30.0000 −1.37073 −0.685367 0.728197i \(-0.740358\pi\)
−0.685367 + 0.728197i \(0.740358\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 10.0000i 0.455488i
\(483\) − 48.0000i − 2.18408i
\(484\) −7.00000 −0.318182
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) 40.0000i 1.81257i 0.422664 + 0.906287i \(0.361095\pi\)
−0.422664 + 0.906287i \(0.638905\pi\)
\(488\) 6.00000i 0.271607i
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) −24.0000 −1.08310 −0.541552 0.840667i \(-0.682163\pi\)
−0.541552 + 0.840667i \(0.682163\pi\)
\(492\) 12.0000i 0.541002i
\(493\) 4.00000i 0.180151i
\(494\) 6.00000 0.269953
\(495\) 0 0
\(496\) 10.0000 0.449013
\(497\) 24.0000i 1.07655i
\(498\) − 32.0000i − 1.43395i
\(499\) −10.0000 −0.447661 −0.223831 0.974628i \(-0.571856\pi\)
−0.223831 + 0.974628i \(0.571856\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) 24.0000i 1.07117i
\(503\) − 18.0000i − 0.802580i −0.915951 0.401290i \(-0.868562\pi\)
0.915951 0.401290i \(-0.131438\pi\)
\(504\) 12.0000 0.534522
\(505\) 0 0
\(506\) 12.0000 0.533465
\(507\) 2.00000i 0.0888231i
\(508\) 2.00000i 0.0887357i
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) − 11.0000i − 0.486136i
\(513\) − 24.0000i − 1.05963i
\(514\) 2.00000 0.0882162
\(515\) 0 0
\(516\) 20.0000 0.880451
\(517\) − 8.00000i − 0.351840i
\(518\) − 8.00000i − 0.351500i
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) 6.00000i 0.262362i 0.991358 + 0.131181i \(0.0418769\pi\)
−0.991358 + 0.131181i \(0.958123\pi\)
\(524\) 20.0000 0.873704
\(525\) 0 0
\(526\) −14.0000 −0.610429
\(527\) 20.0000i 0.871214i
\(528\) 4.00000i 0.174078i
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 6.00000 0.260378
\(532\) 24.0000i 1.04053i
\(533\) 6.00000i 0.259889i
\(534\) −4.00000 −0.173097
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 24.0000i 1.03568i
\(538\) − 6.00000i − 0.258678i
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 2.00000i 0.0859074i
\(543\) 44.0000i 1.88822i
\(544\) 10.0000 0.428746
\(545\) 0 0
\(546\) 8.00000 0.342368
\(547\) 6.00000i 0.256541i 0.991739 + 0.128271i \(0.0409426\pi\)
−0.991739 + 0.128271i \(0.959057\pi\)
\(548\) 2.00000i 0.0854358i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) − 36.0000i − 1.53226i
\(553\) 48.0000i 2.04117i
\(554\) −14.0000 −0.594803
\(555\) 0 0
\(556\) 0 0
\(557\) 26.0000i 1.10166i 0.834619 + 0.550828i \(0.185688\pi\)
−0.834619 + 0.550828i \(0.814312\pi\)
\(558\) 10.0000i 0.423334i
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) − 6.00000i − 0.253095i
\(563\) − 22.0000i − 0.927189i −0.886047 0.463595i \(-0.846559\pi\)
0.886047 0.463595i \(-0.153441\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) −2.00000 −0.0840663
\(567\) − 44.0000i − 1.84783i
\(568\) 18.0000i 0.755263i
\(569\) 26.0000 1.08998 0.544988 0.838444i \(-0.316534\pi\)
0.544988 + 0.838444i \(0.316534\pi\)
\(570\) 0 0
\(571\) 24.0000 1.00437 0.502184 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) − 2.00000i − 0.0836242i
\(573\) 0 0
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 13.0000i 0.540729i
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 64.0000 2.65517
\(582\) 4.00000i 0.165805i
\(583\) 4.00000i 0.165663i
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) −22.0000 −0.908812
\(587\) 44.0000i 1.81607i 0.418890 + 0.908037i \(0.362419\pi\)
−0.418890 + 0.908037i \(0.637581\pi\)
\(588\) 18.0000i 0.742307i
\(589\) −60.0000 −2.47226
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) − 2.00000i − 0.0821995i
\(593\) 14.0000i 0.574911i 0.957794 + 0.287456i \(0.0928094\pi\)
−0.957794 + 0.287456i \(0.907191\pi\)
\(594\) 8.00000 0.328244
\(595\) 0 0
\(596\) −18.0000 −0.737309
\(597\) − 32.0000i − 1.30967i
\(598\) − 6.00000i − 0.245358i
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) − 40.0000i − 1.63028i
\(603\) − 4.00000i − 0.162893i
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) −36.0000 −1.46240
\(607\) − 34.0000i − 1.38002i −0.723801 0.690009i \(-0.757607\pi\)
0.723801 0.690009i \(-0.242393\pi\)
\(608\) 30.0000i 1.21666i
\(609\) −16.0000 −0.648353
\(610\) 0 0
\(611\) −4.00000 −0.161823
\(612\) 2.00000i 0.0808452i
\(613\) − 6.00000i − 0.242338i −0.992632 0.121169i \(-0.961336\pi\)
0.992632 0.121169i \(-0.0386643\pi\)
\(614\) 8.00000 0.322854
\(615\) 0 0
\(616\) −24.0000 −0.966988
\(617\) 42.0000i 1.69086i 0.534089 + 0.845428i \(0.320655\pi\)
−0.534089 + 0.845428i \(0.679345\pi\)
\(618\) 4.00000i 0.160904i
\(619\) 2.00000 0.0803868 0.0401934 0.999192i \(-0.487203\pi\)
0.0401934 + 0.999192i \(0.487203\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) − 4.00000i − 0.160385i
\(623\) − 8.00000i − 0.320513i
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) 22.0000 0.879297
\(627\) − 24.0000i − 0.958468i
\(628\) 6.00000i 0.239426i
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) −14.0000 −0.557331 −0.278666 0.960388i \(-0.589892\pi\)
−0.278666 + 0.960388i \(0.589892\pi\)
\(632\) 36.0000i 1.43200i
\(633\) − 24.0000i − 0.953914i
\(634\) −18.0000 −0.714871
\(635\) 0 0
\(636\) 4.00000 0.158610
\(637\) 9.00000i 0.356593i
\(638\) − 4.00000i − 0.158362i
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −46.0000 −1.81689 −0.908445 0.418004i \(-0.862730\pi\)
−0.908445 + 0.418004i \(0.862730\pi\)
\(642\) − 20.0000i − 0.789337i
\(643\) − 16.0000i − 0.630978i −0.948929 0.315489i \(-0.897831\pi\)
0.948929 0.315489i \(-0.102169\pi\)
\(644\) 24.0000 0.945732
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) − 38.0000i − 1.49393i −0.664861 0.746967i \(-0.731509\pi\)
0.664861 0.746967i \(-0.268491\pi\)
\(648\) − 33.0000i − 1.29636i
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) −80.0000 −3.13545
\(652\) − 12.0000i − 0.469956i
\(653\) − 6.00000i − 0.234798i −0.993085 0.117399i \(-0.962544\pi\)
0.993085 0.117399i \(-0.0374557\pi\)
\(654\) −20.0000 −0.782062
\(655\) 0 0
\(656\) 6.00000 0.234261
\(657\) 6.00000i 0.234082i
\(658\) 16.0000i 0.623745i
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) − 18.0000i − 0.699590i
\(663\) 4.00000i 0.155347i
\(664\) 48.0000 1.86276
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 12.0000i 0.464642i
\(668\) 12.0000i 0.464294i
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 4.00000 0.154418
\(672\) 40.0000i 1.54303i
\(673\) − 30.0000i − 1.15642i −0.815890 0.578208i \(-0.803752\pi\)
0.815890 0.578208i \(-0.196248\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) 6.00000i 0.230599i 0.993331 + 0.115299i \(0.0367827\pi\)
−0.993331 + 0.115299i \(0.963217\pi\)
\(678\) − 28.0000i − 1.07533i
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) −8.00000 −0.306561
\(682\) − 20.0000i − 0.765840i
\(683\) − 20.0000i − 0.765279i −0.923898 0.382639i \(-0.875015\pi\)
0.923898 0.382639i \(-0.124985\pi\)
\(684\) −6.00000 −0.229416
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) − 44.0000i − 1.67870i
\(688\) − 10.0000i − 0.381246i
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) 22.0000 0.836919 0.418460 0.908235i \(-0.362570\pi\)
0.418460 + 0.908235i \(0.362570\pi\)
\(692\) − 6.00000i − 0.228086i
\(693\) − 8.00000i − 0.303895i
\(694\) −22.0000 −0.835109
\(695\) 0 0
\(696\) −12.0000 −0.454859
\(697\) 12.0000i 0.454532i
\(698\) 30.0000i 1.13552i
\(699\) 20.0000 0.756469
\(700\) 0 0
\(701\) −26.0000 −0.982006 −0.491003 0.871158i \(-0.663370\pi\)
−0.491003 + 0.871158i \(0.663370\pi\)
\(702\) − 4.00000i − 0.150970i
\(703\) 12.0000i 0.452589i
\(704\) −14.0000 −0.527645
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) − 72.0000i − 2.70784i
\(708\) 12.0000i 0.450988i
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) − 6.00000i − 0.224860i
\(713\) 60.0000i 2.24702i
\(714\) 16.0000 0.598785
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) − 12.0000i − 0.448148i
\(718\) 10.0000i 0.373197i
\(719\) 32.0000 1.19340 0.596699 0.802465i \(-0.296479\pi\)
0.596699 + 0.802465i \(0.296479\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 17.0000i 0.632674i
\(723\) − 20.0000i − 0.743808i
\(724\) −22.0000 −0.817624
\(725\) 0 0
\(726\) −14.0000 −0.519589
\(727\) 18.0000i 0.667583i 0.942647 + 0.333792i \(0.108328\pi\)
−0.942647 + 0.333792i \(0.891672\pi\)
\(728\) 12.0000i 0.444750i
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 20.0000 0.739727
\(732\) − 4.00000i − 0.147844i
\(733\) 34.0000i 1.25582i 0.778287 + 0.627909i \(0.216089\pi\)
−0.778287 + 0.627909i \(0.783911\pi\)
\(734\) −14.0000 −0.516749
\(735\) 0 0
\(736\) 30.0000 1.10581
\(737\) 8.00000i 0.294684i
\(738\) 6.00000i 0.220863i
\(739\) −6.00000 −0.220714 −0.110357 0.993892i \(-0.535199\pi\)
−0.110357 + 0.993892i \(0.535199\pi\)
\(740\) 0 0
\(741\) −12.0000 −0.440831
\(742\) − 8.00000i − 0.293689i
\(743\) − 12.0000i − 0.440237i −0.975473 0.220119i \(-0.929356\pi\)
0.975473 0.220119i \(-0.0706445\pi\)
\(744\) −60.0000 −2.19971
\(745\) 0 0
\(746\) −34.0000 −1.24483
\(747\) 16.0000i 0.585409i
\(748\) − 4.00000i − 0.146254i
\(749\) 40.0000 1.46157
\(750\) 0 0
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) 4.00000i 0.145865i
\(753\) − 48.0000i − 1.74922i
\(754\) −2.00000 −0.0728357
\(755\) 0 0
\(756\) 16.0000 0.581914
\(757\) − 2.00000i − 0.0726912i −0.999339 0.0363456i \(-0.988428\pi\)
0.999339 0.0363456i \(-0.0115717\pi\)
\(758\) − 10.0000i − 0.363216i
\(759\) −24.0000 −0.871145
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 4.00000i 0.144905i
\(763\) − 40.0000i − 1.44810i
\(764\) 0 0
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 6.00000i 0.216647i
\(768\) 34.0000i 1.22687i
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) −4.00000 −0.144056
\(772\) − 2.00000i − 0.0719816i
\(773\) − 2.00000i − 0.0719350i −0.999353 0.0359675i \(-0.988549\pi\)
0.999353 0.0359675i \(-0.0114513\pi\)
\(774\) 10.0000 0.359443
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 16.0000i 0.573997i
\(778\) 10.0000i 0.358517i
\(779\) −36.0000 −1.28983
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) − 12.0000i − 0.429119i
\(783\) 8.00000i 0.285897i
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 40.0000 1.42675
\(787\) − 8.00000i − 0.285169i −0.989783 0.142585i \(-0.954459\pi\)
0.989783 0.142585i \(-0.0455413\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 28.0000 0.996826
\(790\) 0 0
\(791\) 56.0000 1.99113
\(792\) − 6.00000i − 0.213201i
\(793\) − 2.00000i − 0.0710221i
\(794\) 6.00000 0.212932
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) − 42.0000i − 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) 48.0000i 1.69918i
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) 10.0000i 0.353112i
\(803\) − 12.0000i − 0.423471i
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −10.0000 −0.352235
\(807\) 12.0000i 0.422420i
\(808\) − 54.0000i − 1.89971i
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) −42.0000 −1.47482 −0.737410 0.675446i \(-0.763951\pi\)
−0.737410 + 0.675446i \(0.763951\pi\)
\(812\) − 8.00000i − 0.280745i
\(813\) − 4.00000i − 0.140286i
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) 4.00000 0.140028
\(817\) 60.0000i 2.09913i
\(818\) − 18.0000i − 0.629355i
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) 50.0000 1.74501 0.872506 0.488603i \(-0.162493\pi\)
0.872506 + 0.488603i \(0.162493\pi\)
\(822\) 4.00000i 0.139516i
\(823\) − 46.0000i − 1.60346i −0.597687 0.801730i \(-0.703913\pi\)
0.597687 0.801730i \(-0.296087\pi\)
\(824\) −6.00000 −0.209020
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 32.0000i 1.11275i 0.830932 + 0.556375i \(0.187808\pi\)
−0.830932 + 0.556375i \(0.812192\pi\)
\(828\) 6.00000i 0.208514i
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) 28.0000 0.971309
\(832\) 7.00000i 0.242681i
\(833\) 18.0000i 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) 40.0000i 1.38260i
\(838\) 40.0000i 1.38178i
\(839\) −38.0000 −1.31191 −0.655953 0.754802i \(-0.727733\pi\)
−0.655953 + 0.754802i \(0.727733\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 10.0000i 0.344623i
\(843\) 12.0000i 0.413302i
\(844\) 12.0000 0.413057
\(845\) 0 0
\(846\) −4.00000 −0.137523
\(847\) − 28.0000i − 0.962091i
\(848\) − 2.00000i − 0.0686803i
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) − 12.0000i − 0.411113i
\(853\) 6.00000i 0.205436i 0.994711 + 0.102718i \(0.0327539\pi\)
−0.994711 + 0.102718i \(0.967246\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 30.0000 1.02538
\(857\) − 18.0000i − 0.614868i −0.951569 0.307434i \(-0.900530\pi\)
0.951569 0.307434i \(-0.0994704\pi\)
\(858\) − 4.00000i − 0.136558i
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) −48.0000 −1.63584
\(862\) 14.0000i 0.476842i
\(863\) 12.0000i 0.408485i 0.978920 + 0.204242i \(0.0654731\pi\)
−0.978920 + 0.204242i \(0.934527\pi\)
\(864\) 20.0000 0.680414
\(865\) 0 0
\(866\) −10.0000 −0.339814
\(867\) − 26.0000i − 0.883006i
\(868\) − 40.0000i − 1.35769i
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) − 30.0000i − 1.01593i
\(873\) − 2.00000i − 0.0676897i
\(874\) 36.0000 1.21772
\(875\) 0 0
\(876\) −12.0000 −0.405442
\(877\) − 18.0000i − 0.607817i −0.952701 0.303908i \(-0.901708\pi\)
0.952701 0.303908i \(-0.0982917\pi\)
\(878\) 0 0
\(879\) 44.0000 1.48408
\(880\) 0 0
\(881\) 38.0000 1.28025 0.640126 0.768270i \(-0.278882\pi\)
0.640126 + 0.768270i \(0.278882\pi\)
\(882\) 9.00000i 0.303046i
\(883\) − 22.0000i − 0.740359i −0.928960 0.370179i \(-0.879296\pi\)
0.928960 0.370179i \(-0.120704\pi\)
\(884\) −2.00000 −0.0672673
\(885\) 0 0
\(886\) −14.0000 −0.470339
\(887\) − 58.0000i − 1.94745i −0.227728 0.973725i \(-0.573130\pi\)
0.227728 0.973725i \(-0.426870\pi\)
\(888\) 12.0000i 0.402694i
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) −22.0000 −0.737028
\(892\) 4.00000i 0.133930i
\(893\) − 24.0000i − 0.803129i
\(894\) −36.0000 −1.20402
\(895\) 0 0
\(896\) −12.0000 −0.400892
\(897\) 12.0000i 0.400668i
\(898\) 6.00000i 0.200223i
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) − 12.0000i − 0.399556i
\(903\) 80.0000i 2.66223i
\(904\) 42.0000 1.39690
\(905\) 0 0
\(906\) 20.0000 0.664455
\(907\) 34.0000i 1.12895i 0.825450 + 0.564476i \(0.190922\pi\)
−0.825450 + 0.564476i \(0.809078\pi\)
\(908\) − 4.00000i − 0.132745i
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) 12.0000i 0.397360i
\(913\) − 32.0000i − 1.05905i
\(914\) 38.0000 1.25693
\(915\) 0 0
\(916\) 22.0000 0.726900
\(917\) 80.0000i 2.64183i
\(918\) − 8.00000i − 0.264039i
\(919\) 52.0000 1.71532 0.857661 0.514216i \(-0.171917\pi\)
0.857661 + 0.514216i \(0.171917\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) 10.0000i 0.329332i
\(923\) − 6.00000i − 0.197492i
\(924\) 16.0000 0.526361
\(925\) 0 0
\(926\) 24.0000 0.788689
\(927\) − 2.00000i − 0.0656886i
\(928\) − 10.0000i − 0.328266i
\(929\) 38.0000 1.24674 0.623370 0.781927i \(-0.285763\pi\)
0.623370 + 0.781927i \(0.285763\pi\)
\(930\) 0 0
\(931\) −54.0000 −1.76978
\(932\) 10.0000i 0.327561i
\(933\) 8.00000i 0.261908i
\(934\) −10.0000 −0.327210
\(935\) 0 0
\(936\) −3.00000 −0.0980581
\(937\) 30.0000i 0.980057i 0.871706 + 0.490029i \(0.163014\pi\)
−0.871706 + 0.490029i \(0.836986\pi\)
\(938\) − 16.0000i − 0.522419i
\(939\) −44.0000 −1.43589
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) 12.0000i 0.390981i
\(943\) 36.0000i 1.17232i
\(944\) 6.00000 0.195283
\(945\) 0 0
\(946\) −20.0000 −0.650256
\(947\) − 24.0000i − 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) − 24.0000i − 0.779484i
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) 36.0000 1.16738
\(952\) 24.0000i 0.777844i
\(953\) 18.0000i 0.583077i 0.956559 + 0.291539i \(0.0941672\pi\)
−0.956559 + 0.291539i \(0.905833\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) 6.00000 0.194054
\(957\) 8.00000i 0.258603i
\(958\) − 30.0000i − 0.969256i
\(959\) −8.00000 −0.258333
\(960\) 0 0
\(961\) 69.0000 2.22581
\(962\) 2.00000i 0.0644826i
\(963\) 10.0000i 0.322245i
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 48.0000 1.54437
\(967\) 8.00000i 0.257263i 0.991692 + 0.128631i \(0.0410584\pi\)
−0.991692 + 0.128631i \(0.958942\pi\)
\(968\) − 21.0000i − 0.674966i
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 10.0000i 0.320750i
\(973\) 0 0
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) − 18.0000i − 0.575871i −0.957650 0.287936i \(-0.907031\pi\)
0.957650 0.287936i \(-0.0929689\pi\)
\(978\) − 24.0000i − 0.767435i
\(979\) −4.00000 −0.127841
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) − 24.0000i − 0.765871i
\(983\) 56.0000i 1.78612i 0.449935 + 0.893061i \(0.351447\pi\)
−0.449935 + 0.893061i \(0.648553\pi\)
\(984\) −36.0000 −1.14764
\(985\) 0 0
\(986\) −4.00000 −0.127386
\(987\) − 32.0000i − 1.01857i
\(988\) − 6.00000i − 0.190885i
\(989\) 60.0000 1.90789
\(990\) 0 0
\(991\) 48.0000 1.52477 0.762385 0.647124i \(-0.224028\pi\)
0.762385 + 0.647124i \(0.224028\pi\)
\(992\) − 50.0000i − 1.58750i
\(993\) 36.0000i 1.14243i
\(994\) −24.0000 −0.761234
\(995\) 0 0
\(996\) −32.0000 −1.01396
\(997\) 22.0000i 0.696747i 0.937356 + 0.348373i \(0.113266\pi\)
−0.937356 + 0.348373i \(0.886734\pi\)
\(998\) − 10.0000i − 0.316544i
\(999\) 8.00000 0.253109
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 325.2.b.b.274.2 2
3.2 odd 2 2925.2.c.h.2224.1 2
5.2 odd 4 65.2.a.a.1.1 1
5.3 odd 4 325.2.a.d.1.1 1
5.4 even 2 inner 325.2.b.b.274.1 2
15.2 even 4 585.2.a.h.1.1 1
15.8 even 4 2925.2.a.f.1.1 1
15.14 odd 2 2925.2.c.h.2224.2 2
20.3 even 4 5200.2.a.d.1.1 1
20.7 even 4 1040.2.a.f.1.1 1
35.27 even 4 3185.2.a.e.1.1 1
40.27 even 4 4160.2.a.f.1.1 1
40.37 odd 4 4160.2.a.q.1.1 1
55.32 even 4 7865.2.a.c.1.1 1
60.47 odd 4 9360.2.a.ca.1.1 1
65.2 even 12 845.2.m.b.316.1 4
65.7 even 12 845.2.m.b.361.1 4
65.12 odd 4 845.2.a.a.1.1 1
65.17 odd 12 845.2.e.a.146.1 2
65.22 odd 12 845.2.e.b.146.1 2
65.32 even 12 845.2.m.b.361.2 4
65.37 even 12 845.2.m.b.316.2 4
65.38 odd 4 4225.2.a.g.1.1 1
65.42 odd 12 845.2.e.b.191.1 2
65.47 even 4 845.2.c.a.506.1 2
65.57 even 4 845.2.c.a.506.2 2
65.62 odd 12 845.2.e.a.191.1 2
195.77 even 4 7605.2.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.a.1.1 1 5.2 odd 4
325.2.a.d.1.1 1 5.3 odd 4
325.2.b.b.274.1 2 5.4 even 2 inner
325.2.b.b.274.2 2 1.1 even 1 trivial
585.2.a.h.1.1 1 15.2 even 4
845.2.a.a.1.1 1 65.12 odd 4
845.2.c.a.506.1 2 65.47 even 4
845.2.c.a.506.2 2 65.57 even 4
845.2.e.a.146.1 2 65.17 odd 12
845.2.e.a.191.1 2 65.62 odd 12
845.2.e.b.146.1 2 65.22 odd 12
845.2.e.b.191.1 2 65.42 odd 12
845.2.m.b.316.1 4 65.2 even 12
845.2.m.b.316.2 4 65.37 even 12
845.2.m.b.361.1 4 65.7 even 12
845.2.m.b.361.2 4 65.32 even 12
1040.2.a.f.1.1 1 20.7 even 4
2925.2.a.f.1.1 1 15.8 even 4
2925.2.c.h.2224.1 2 3.2 odd 2
2925.2.c.h.2224.2 2 15.14 odd 2
3185.2.a.e.1.1 1 35.27 even 4
4160.2.a.f.1.1 1 40.27 even 4
4160.2.a.q.1.1 1 40.37 odd 4
4225.2.a.g.1.1 1 65.38 odd 4
5200.2.a.d.1.1 1 20.3 even 4
7605.2.a.f.1.1 1 195.77 even 4
7865.2.a.c.1.1 1 55.32 even 4
9360.2.a.ca.1.1 1 60.47 odd 4