Properties

Label 1575.2.d.d.1324.2
Level $1575$
Weight $2$
Character 1575.1324
Analytic conductor $12.576$
Analytic rank $0$
Dimension $4$
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] = [1575,2,Mod(1324,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.1324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.d (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: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1324.2
Root \(0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 1575.1324
Dual form 1575.2.d.d.1324.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23607i q^{2} -3.00000 q^{4} +1.00000i q^{7} +2.23607i q^{8} +O(q^{10})\) \(q-2.23607i q^{2} -3.00000 q^{4} +1.00000i q^{7} +2.23607i q^{8} +2.47214 q^{11} +4.47214i q^{13} +2.23607 q^{14} -1.00000 q^{16} +2.00000i q^{17} -6.47214 q^{19} -5.52786i q^{22} +4.00000i q^{23} +10.0000 q^{26} -3.00000i q^{28} -2.00000 q^{29} +10.4721 q^{31} +6.70820i q^{32} +4.47214 q^{34} +10.9443i q^{37} +14.4721i q^{38} +2.00000 q^{41} +8.94427i q^{43} -7.41641 q^{44} +8.94427 q^{46} +4.94427i q^{47} -1.00000 q^{49} -13.4164i q^{52} -12.4721i q^{53} -2.23607 q^{56} +4.47214i q^{58} +8.94427 q^{59} -2.00000 q^{61} -23.4164i q^{62} +13.0000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -14.4721 q^{71} +3.52786i q^{73} +24.4721 q^{74} +19.4164 q^{76} +2.47214i q^{77} +4.94427 q^{79} -4.47214i q^{82} +0.944272i q^{83} +20.0000 q^{86} +5.52786i q^{88} -2.00000 q^{89} -4.47214 q^{91} -12.0000i q^{92} +11.0557 q^{94} -0.472136i q^{97} +2.23607i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{4} - 8 q^{11} - 4 q^{16} - 8 q^{19} + 40 q^{26} - 8 q^{29} + 24 q^{31} + 8 q^{41} + 24 q^{44} - 4 q^{49} - 8 q^{61} + 52 q^{64} - 40 q^{71} + 80 q^{74} + 24 q^{76} - 16 q^{79} + 80 q^{86} - 8 q^{89} + 80 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.23607i − 1.58114i −0.612372 0.790569i \(-0.709785\pi\)
0.612372 0.790569i \(-0.290215\pi\)
\(3\) 0 0
\(4\) −3.00000 −1.50000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 2.23607i 0.790569i
\(9\) 0 0
\(10\) 0 0
\(11\) 2.47214 0.745377 0.372689 0.927957i \(-0.378436\pi\)
0.372689 + 0.927957i \(0.378436\pi\)
\(12\) 0 0
\(13\) 4.47214i 1.24035i 0.784465 + 0.620174i \(0.212938\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 2.23607 0.597614
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 2.00000i 0.485071i 0.970143 + 0.242536i \(0.0779791\pi\)
−0.970143 + 0.242536i \(0.922021\pi\)
\(18\) 0 0
\(19\) −6.47214 −1.48481 −0.742405 0.669951i \(-0.766315\pi\)
−0.742405 + 0.669951i \(0.766315\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 5.52786i − 1.17854i
\(23\) 4.00000i 0.834058i 0.908893 + 0.417029i \(0.136929\pi\)
−0.908893 + 0.417029i \(0.863071\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 10.0000 1.96116
\(27\) 0 0
\(28\) − 3.00000i − 0.566947i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 10.4721 1.88085 0.940426 0.340000i \(-0.110427\pi\)
0.940426 + 0.340000i \(0.110427\pi\)
\(32\) 6.70820i 1.18585i
\(33\) 0 0
\(34\) 4.47214 0.766965
\(35\) 0 0
\(36\) 0 0
\(37\) 10.9443i 1.79923i 0.436687 + 0.899614i \(0.356152\pi\)
−0.436687 + 0.899614i \(0.643848\pi\)
\(38\) 14.4721i 2.34769i
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 8.94427i 1.36399i 0.731357 + 0.681994i \(0.238887\pi\)
−0.731357 + 0.681994i \(0.761113\pi\)
\(44\) −7.41641 −1.11807
\(45\) 0 0
\(46\) 8.94427 1.31876
\(47\) 4.94427i 0.721196i 0.932721 + 0.360598i \(0.117427\pi\)
−0.932721 + 0.360598i \(0.882573\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) − 13.4164i − 1.86052i
\(53\) − 12.4721i − 1.71318i −0.515998 0.856590i \(-0.672579\pi\)
0.515998 0.856590i \(-0.327421\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.23607 −0.298807
\(57\) 0 0
\(58\) 4.47214i 0.587220i
\(59\) 8.94427 1.16445 0.582223 0.813029i \(-0.302183\pi\)
0.582223 + 0.813029i \(0.302183\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) − 23.4164i − 2.97389i
\(63\) 0 0
\(64\) 13.0000 1.62500
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) −14.4721 −1.71753 −0.858763 0.512373i \(-0.828767\pi\)
−0.858763 + 0.512373i \(0.828767\pi\)
\(72\) 0 0
\(73\) 3.52786i 0.412905i 0.978457 + 0.206453i \(0.0661919\pi\)
−0.978457 + 0.206453i \(0.933808\pi\)
\(74\) 24.4721 2.84483
\(75\) 0 0
\(76\) 19.4164 2.22721
\(77\) 2.47214i 0.281726i
\(78\) 0 0
\(79\) 4.94427 0.556274 0.278137 0.960541i \(-0.410283\pi\)
0.278137 + 0.960541i \(0.410283\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 4.47214i − 0.493865i
\(83\) 0.944272i 0.103647i 0.998656 + 0.0518237i \(0.0165034\pi\)
−0.998656 + 0.0518237i \(0.983497\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 20.0000 2.15666
\(87\) 0 0
\(88\) 5.52786i 0.589272i
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −4.47214 −0.468807
\(92\) − 12.0000i − 1.25109i
\(93\) 0 0
\(94\) 11.0557 1.14031
\(95\) 0 0
\(96\) 0 0
\(97\) − 0.472136i − 0.0479381i −0.999713 0.0239691i \(-0.992370\pi\)
0.999713 0.0239691i \(-0.00763032\pi\)
\(98\) 2.23607i 0.225877i
\(99\) 0 0
\(100\) 0 0
\(101\) 14.0000 1.39305 0.696526 0.717532i \(-0.254728\pi\)
0.696526 + 0.717532i \(0.254728\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −10.0000 −0.980581
\(105\) 0 0
\(106\) −27.8885 −2.70877
\(107\) − 4.94427i − 0.477981i −0.971022 0.238990i \(-0.923184\pi\)
0.971022 0.238990i \(-0.0768164\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 1.00000i − 0.0944911i
\(113\) − 8.47214i − 0.796992i −0.917170 0.398496i \(-0.869532\pi\)
0.917170 0.398496i \(-0.130468\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) − 20.0000i − 1.84115i
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −4.88854 −0.444413
\(122\) 4.47214i 0.404888i
\(123\) 0 0
\(124\) −31.4164 −2.82128
\(125\) 0 0
\(126\) 0 0
\(127\) 12.9443i 1.14862i 0.818638 + 0.574309i \(0.194729\pi\)
−0.818638 + 0.574309i \(0.805271\pi\)
\(128\) − 15.6525i − 1.38350i
\(129\) 0 0
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) − 6.47214i − 0.561205i
\(134\) −8.94427 −0.772667
\(135\) 0 0
\(136\) −4.47214 −0.383482
\(137\) − 12.4721i − 1.06557i −0.846252 0.532783i \(-0.821146\pi\)
0.846252 0.532783i \(-0.178854\pi\)
\(138\) 0 0
\(139\) 19.4164 1.64688 0.823439 0.567405i \(-0.192052\pi\)
0.823439 + 0.567405i \(0.192052\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 32.3607i 2.71565i
\(143\) 11.0557i 0.924526i
\(144\) 0 0
\(145\) 0 0
\(146\) 7.88854 0.652861
\(147\) 0 0
\(148\) − 32.8328i − 2.69884i
\(149\) 2.94427 0.241204 0.120602 0.992701i \(-0.461517\pi\)
0.120602 + 0.992701i \(0.461517\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) − 14.4721i − 1.17385i
\(153\) 0 0
\(154\) 5.52786 0.445448
\(155\) 0 0
\(156\) 0 0
\(157\) 8.47214i 0.676150i 0.941119 + 0.338075i \(0.109776\pi\)
−0.941119 + 0.338075i \(0.890224\pi\)
\(158\) − 11.0557i − 0.879547i
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) − 0.944272i − 0.0739611i −0.999316 0.0369805i \(-0.988226\pi\)
0.999316 0.0369805i \(-0.0117740\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 2.11146 0.163881
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) − 26.8328i − 2.04598i
\(173\) 14.9443i 1.13619i 0.822962 + 0.568096i \(0.192320\pi\)
−0.822962 + 0.568096i \(0.807680\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.47214 −0.186344
\(177\) 0 0
\(178\) 4.47214i 0.335201i
\(179\) −2.47214 −0.184776 −0.0923881 0.995723i \(-0.529450\pi\)
−0.0923881 + 0.995723i \(0.529450\pi\)
\(180\) 0 0
\(181\) 18.9443 1.40812 0.704058 0.710142i \(-0.251369\pi\)
0.704058 + 0.710142i \(0.251369\pi\)
\(182\) 10.0000i 0.741249i
\(183\) 0 0
\(184\) −8.94427 −0.659380
\(185\) 0 0
\(186\) 0 0
\(187\) 4.94427i 0.361561i
\(188\) − 14.8328i − 1.08179i
\(189\) 0 0
\(190\) 0 0
\(191\) −27.4164 −1.98378 −0.991891 0.127093i \(-0.959435\pi\)
−0.991891 + 0.127093i \(0.959435\pi\)
\(192\) 0 0
\(193\) 14.0000i 1.00774i 0.863779 + 0.503871i \(0.168091\pi\)
−0.863779 + 0.503871i \(0.831909\pi\)
\(194\) −1.05573 −0.0757969
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) − 24.4721i − 1.74357i −0.489891 0.871784i \(-0.662963\pi\)
0.489891 0.871784i \(-0.337037\pi\)
\(198\) 0 0
\(199\) −0.583592 −0.0413697 −0.0206849 0.999786i \(-0.506585\pi\)
−0.0206849 + 0.999786i \(0.506585\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 31.3050i − 2.20261i
\(203\) − 2.00000i − 0.140372i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) − 4.47214i − 0.310087i
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) 0.944272 0.0650064 0.0325032 0.999472i \(-0.489652\pi\)
0.0325032 + 0.999472i \(0.489652\pi\)
\(212\) 37.4164i 2.56977i
\(213\) 0 0
\(214\) −11.0557 −0.755754
\(215\) 0 0
\(216\) 0 0
\(217\) 10.4721i 0.710895i
\(218\) − 4.47214i − 0.302891i
\(219\) 0 0
\(220\) 0 0
\(221\) −8.94427 −0.601657
\(222\) 0 0
\(223\) − 4.94427i − 0.331093i −0.986202 0.165546i \(-0.947061\pi\)
0.986202 0.165546i \(-0.0529388\pi\)
\(224\) −6.70820 −0.448211
\(225\) 0 0
\(226\) −18.9443 −1.26015
\(227\) 16.9443i 1.12463i 0.826923 + 0.562315i \(0.190089\pi\)
−0.826923 + 0.562315i \(0.809911\pi\)
\(228\) 0 0
\(229\) 11.8885 0.785617 0.392809 0.919620i \(-0.371504\pi\)
0.392809 + 0.919620i \(0.371504\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 4.47214i − 0.293610i
\(233\) 17.4164i 1.14099i 0.821302 + 0.570493i \(0.193248\pi\)
−0.821302 + 0.570493i \(0.806752\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −26.8328 −1.74667
\(237\) 0 0
\(238\) 4.47214i 0.289886i
\(239\) −1.52786 −0.0988293 −0.0494147 0.998778i \(-0.515736\pi\)
−0.0494147 + 0.998778i \(0.515736\pi\)
\(240\) 0 0
\(241\) −1.05573 −0.0680054 −0.0340027 0.999422i \(-0.510825\pi\)
−0.0340027 + 0.999422i \(0.510825\pi\)
\(242\) 10.9311i 0.702679i
\(243\) 0 0
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) 0 0
\(247\) − 28.9443i − 1.84168i
\(248\) 23.4164i 1.48694i
\(249\) 0 0
\(250\) 0 0
\(251\) 0.944272 0.0596019 0.0298010 0.999556i \(-0.490513\pi\)
0.0298010 + 0.999556i \(0.490513\pi\)
\(252\) 0 0
\(253\) 9.88854i 0.621687i
\(254\) 28.9443 1.81613
\(255\) 0 0
\(256\) −9.00000 −0.562500
\(257\) − 1.05573i − 0.0658545i −0.999458 0.0329273i \(-0.989517\pi\)
0.999458 0.0329273i \(-0.0104830\pi\)
\(258\) 0 0
\(259\) −10.9443 −0.680044
\(260\) 0 0
\(261\) 0 0
\(262\) 8.94427i 0.552579i
\(263\) 24.9443i 1.53813i 0.639171 + 0.769065i \(0.279278\pi\)
−0.639171 + 0.769065i \(0.720722\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −14.4721 −0.887344
\(267\) 0 0
\(268\) 12.0000i 0.733017i
\(269\) −23.8885 −1.45651 −0.728255 0.685306i \(-0.759668\pi\)
−0.728255 + 0.685306i \(0.759668\pi\)
\(270\) 0 0
\(271\) −10.4721 −0.636137 −0.318068 0.948068i \(-0.603034\pi\)
−0.318068 + 0.948068i \(0.603034\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 0 0
\(274\) −27.8885 −1.68481
\(275\) 0 0
\(276\) 0 0
\(277\) 1.05573i 0.0634326i 0.999497 + 0.0317163i \(0.0100973\pi\)
−0.999497 + 0.0317163i \(0.989903\pi\)
\(278\) − 43.4164i − 2.60394i
\(279\) 0 0
\(280\) 0 0
\(281\) −6.94427 −0.414261 −0.207130 0.978313i \(-0.566412\pi\)
−0.207130 + 0.978313i \(0.566412\pi\)
\(282\) 0 0
\(283\) − 12.0000i − 0.713326i −0.934233 0.356663i \(-0.883914\pi\)
0.934233 0.356663i \(-0.116086\pi\)
\(284\) 43.4164 2.57629
\(285\) 0 0
\(286\) 24.7214 1.46180
\(287\) 2.00000i 0.118056i
\(288\) 0 0
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) − 10.5836i − 0.619358i
\(293\) 22.9443i 1.34042i 0.742172 + 0.670209i \(0.233796\pi\)
−0.742172 + 0.670209i \(0.766204\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −24.4721 −1.42241
\(297\) 0 0
\(298\) − 6.58359i − 0.381377i
\(299\) −17.8885 −1.03452
\(300\) 0 0
\(301\) −8.94427 −0.515539
\(302\) 35.7771i 2.05874i
\(303\) 0 0
\(304\) 6.47214 0.371202
\(305\) 0 0
\(306\) 0 0
\(307\) − 32.9443i − 1.88023i −0.340859 0.940114i \(-0.610718\pi\)
0.340859 0.940114i \(-0.389282\pi\)
\(308\) − 7.41641i − 0.422589i
\(309\) 0 0
\(310\) 0 0
\(311\) 9.88854 0.560728 0.280364 0.959894i \(-0.409545\pi\)
0.280364 + 0.959894i \(0.409545\pi\)
\(312\) 0 0
\(313\) − 9.41641i − 0.532247i −0.963939 0.266123i \(-0.914257\pi\)
0.963939 0.266123i \(-0.0857429\pi\)
\(314\) 18.9443 1.06909
\(315\) 0 0
\(316\) −14.8328 −0.834411
\(317\) 30.3607i 1.70523i 0.522543 + 0.852613i \(0.324983\pi\)
−0.522543 + 0.852613i \(0.675017\pi\)
\(318\) 0 0
\(319\) −4.94427 −0.276826
\(320\) 0 0
\(321\) 0 0
\(322\) 8.94427i 0.498445i
\(323\) − 12.9443i − 0.720239i
\(324\) 0 0
\(325\) 0 0
\(326\) −2.11146 −0.116943
\(327\) 0 0
\(328\) 4.47214i 0.246932i
\(329\) −4.94427 −0.272587
\(330\) 0 0
\(331\) −16.9443 −0.931341 −0.465671 0.884958i \(-0.654187\pi\)
−0.465671 + 0.884958i \(0.654187\pi\)
\(332\) − 2.83282i − 0.155471i
\(333\) 0 0
\(334\) 17.8885 0.978818
\(335\) 0 0
\(336\) 0 0
\(337\) 11.8885i 0.647610i 0.946124 + 0.323805i \(0.104962\pi\)
−0.946124 + 0.323805i \(0.895038\pi\)
\(338\) 15.6525i 0.851382i
\(339\) 0 0
\(340\) 0 0
\(341\) 25.8885 1.40194
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) −20.0000 −1.07833
\(345\) 0 0
\(346\) 33.4164 1.79648
\(347\) 8.00000i 0.429463i 0.976673 + 0.214731i \(0.0688876\pi\)
−0.976673 + 0.214731i \(0.931112\pi\)
\(348\) 0 0
\(349\) −23.8885 −1.27872 −0.639362 0.768906i \(-0.720802\pi\)
−0.639362 + 0.768906i \(0.720802\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 16.5836i 0.883908i
\(353\) − 27.8885i − 1.48436i −0.670202 0.742179i \(-0.733793\pi\)
0.670202 0.742179i \(-0.266207\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.00000 0.317999
\(357\) 0 0
\(358\) 5.52786i 0.292157i
\(359\) 9.52786 0.502861 0.251431 0.967875i \(-0.419099\pi\)
0.251431 + 0.967875i \(0.419099\pi\)
\(360\) 0 0
\(361\) 22.8885 1.20466
\(362\) − 42.3607i − 2.22643i
\(363\) 0 0
\(364\) 13.4164 0.703211
\(365\) 0 0
\(366\) 0 0
\(367\) 20.9443i 1.09328i 0.837367 + 0.546641i \(0.184094\pi\)
−0.837367 + 0.546641i \(0.815906\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) 0 0
\(370\) 0 0
\(371\) 12.4721 0.647521
\(372\) 0 0
\(373\) − 6.00000i − 0.310668i −0.987862 0.155334i \(-0.950355\pi\)
0.987862 0.155334i \(-0.0496454\pi\)
\(374\) 11.0557 0.571678
\(375\) 0 0
\(376\) −11.0557 −0.570156
\(377\) − 8.94427i − 0.460653i
\(378\) 0 0
\(379\) 2.11146 0.108458 0.0542291 0.998529i \(-0.482730\pi\)
0.0542291 + 0.998529i \(0.482730\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 61.3050i 3.13663i
\(383\) − 8.00000i − 0.408781i −0.978889 0.204390i \(-0.934479\pi\)
0.978889 0.204390i \(-0.0655212\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 31.3050 1.59338
\(387\) 0 0
\(388\) 1.41641i 0.0719072i
\(389\) 10.9443 0.554897 0.277448 0.960741i \(-0.410511\pi\)
0.277448 + 0.960741i \(0.410511\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) − 2.23607i − 0.112938i
\(393\) 0 0
\(394\) −54.7214 −2.75682
\(395\) 0 0
\(396\) 0 0
\(397\) 13.4164i 0.673350i 0.941621 + 0.336675i \(0.109302\pi\)
−0.941621 + 0.336675i \(0.890698\pi\)
\(398\) 1.30495i 0.0654113i
\(399\) 0 0
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) 46.8328i 2.33291i
\(404\) −42.0000 −2.08958
\(405\) 0 0
\(406\) −4.47214 −0.221948
\(407\) 27.0557i 1.34110i
\(408\) 0 0
\(409\) 23.8885 1.18121 0.590606 0.806960i \(-0.298889\pi\)
0.590606 + 0.806960i \(0.298889\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.94427i 0.440119i
\(414\) 0 0
\(415\) 0 0
\(416\) −30.0000 −1.47087
\(417\) 0 0
\(418\) 35.7771i 1.74991i
\(419\) 5.88854 0.287674 0.143837 0.989601i \(-0.454056\pi\)
0.143837 + 0.989601i \(0.454056\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) − 2.11146i − 0.102784i
\(423\) 0 0
\(424\) 27.8885 1.35439
\(425\) 0 0
\(426\) 0 0
\(427\) − 2.00000i − 0.0967868i
\(428\) 14.8328i 0.716971i
\(429\) 0 0
\(430\) 0 0
\(431\) 9.52786 0.458941 0.229471 0.973316i \(-0.426301\pi\)
0.229471 + 0.973316i \(0.426301\pi\)
\(432\) 0 0
\(433\) − 7.52786i − 0.361766i −0.983505 0.180883i \(-0.942104\pi\)
0.983505 0.180883i \(-0.0578955\pi\)
\(434\) 23.4164 1.12402
\(435\) 0 0
\(436\) −6.00000 −0.287348
\(437\) − 25.8885i − 1.23842i
\(438\) 0 0
\(439\) −10.4721 −0.499808 −0.249904 0.968271i \(-0.580399\pi\)
−0.249904 + 0.968271i \(0.580399\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 20.0000i 0.951303i
\(443\) − 8.00000i − 0.380091i −0.981775 0.190046i \(-0.939136\pi\)
0.981775 0.190046i \(-0.0608636\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −11.0557 −0.523504
\(447\) 0 0
\(448\) 13.0000i 0.614192i
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) 4.94427 0.232817
\(452\) 25.4164i 1.19549i
\(453\) 0 0
\(454\) 37.8885 1.77820
\(455\) 0 0
\(456\) 0 0
\(457\) − 10.9443i − 0.511951i −0.966683 0.255976i \(-0.917603\pi\)
0.966683 0.255976i \(-0.0823967\pi\)
\(458\) − 26.5836i − 1.24217i
\(459\) 0 0
\(460\) 0 0
\(461\) 31.8885 1.48520 0.742599 0.669737i \(-0.233593\pi\)
0.742599 + 0.669737i \(0.233593\pi\)
\(462\) 0 0
\(463\) − 3.05573i − 0.142012i −0.997476 0.0710059i \(-0.977379\pi\)
0.997476 0.0710059i \(-0.0226209\pi\)
\(464\) 2.00000 0.0928477
\(465\) 0 0
\(466\) 38.9443 1.80406
\(467\) − 8.94427i − 0.413892i −0.978352 0.206946i \(-0.933648\pi\)
0.978352 0.206946i \(-0.0663524\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 20.0000i 0.920575i
\(473\) 22.1115i 1.01669i
\(474\) 0 0
\(475\) 0 0
\(476\) 6.00000 0.275010
\(477\) 0 0
\(478\) 3.41641i 0.156263i
\(479\) 17.8885 0.817348 0.408674 0.912680i \(-0.365991\pi\)
0.408674 + 0.912680i \(0.365991\pi\)
\(480\) 0 0
\(481\) −48.9443 −2.23167
\(482\) 2.36068i 0.107526i
\(483\) 0 0
\(484\) 14.6656 0.666620
\(485\) 0 0
\(486\) 0 0
\(487\) − 3.05573i − 0.138468i −0.997600 0.0692341i \(-0.977944\pi\)
0.997600 0.0692341i \(-0.0220556\pi\)
\(488\) − 4.47214i − 0.202444i
\(489\) 0 0
\(490\) 0 0
\(491\) −41.3050 −1.86407 −0.932033 0.362373i \(-0.881967\pi\)
−0.932033 + 0.362373i \(0.881967\pi\)
\(492\) 0 0
\(493\) − 4.00000i − 0.180151i
\(494\) −64.7214 −2.91195
\(495\) 0 0
\(496\) −10.4721 −0.470213
\(497\) − 14.4721i − 0.649164i
\(498\) 0 0
\(499\) −21.8885 −0.979866 −0.489933 0.871760i \(-0.662979\pi\)
−0.489933 + 0.871760i \(0.662979\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 2.11146i − 0.0942389i
\(503\) 32.0000i 1.42681i 0.700752 + 0.713405i \(0.252848\pi\)
−0.700752 + 0.713405i \(0.747152\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 22.1115 0.982974
\(507\) 0 0
\(508\) − 38.8328i − 1.72293i
\(509\) 11.8885 0.526950 0.263475 0.964666i \(-0.415131\pi\)
0.263475 + 0.964666i \(0.415131\pi\)
\(510\) 0 0
\(511\) −3.52786 −0.156064
\(512\) − 11.1803i − 0.494106i
\(513\) 0 0
\(514\) −2.36068 −0.104125
\(515\) 0 0
\(516\) 0 0
\(517\) 12.2229i 0.537563i
\(518\) 24.4721i 1.07524i
\(519\) 0 0
\(520\) 0 0
\(521\) −15.8885 −0.696090 −0.348045 0.937478i \(-0.613154\pi\)
−0.348045 + 0.937478i \(0.613154\pi\)
\(522\) 0 0
\(523\) − 8.94427i − 0.391106i −0.980693 0.195553i \(-0.937350\pi\)
0.980693 0.195553i \(-0.0626501\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) 55.7771 2.43200
\(527\) 20.9443i 0.912347i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 19.4164i 0.841808i
\(533\) 8.94427i 0.387419i
\(534\) 0 0
\(535\) 0 0
\(536\) 8.94427 0.386334
\(537\) 0 0
\(538\) 53.4164i 2.30294i
\(539\) −2.47214 −0.106482
\(540\) 0 0
\(541\) 23.8885 1.02705 0.513524 0.858075i \(-0.328340\pi\)
0.513524 + 0.858075i \(0.328340\pi\)
\(542\) 23.4164i 1.00582i
\(543\) 0 0
\(544\) −13.4164 −0.575224
\(545\) 0 0
\(546\) 0 0
\(547\) − 29.8885i − 1.27794i −0.769231 0.638971i \(-0.779360\pi\)
0.769231 0.638971i \(-0.220640\pi\)
\(548\) 37.4164i 1.59835i
\(549\) 0 0
\(550\) 0 0
\(551\) 12.9443 0.551445
\(552\) 0 0
\(553\) 4.94427i 0.210252i
\(554\) 2.36068 0.100296
\(555\) 0 0
\(556\) −58.2492 −2.47032
\(557\) − 11.5279i − 0.488451i −0.969718 0.244226i \(-0.921466\pi\)
0.969718 0.244226i \(-0.0785338\pi\)
\(558\) 0 0
\(559\) −40.0000 −1.69182
\(560\) 0 0
\(561\) 0 0
\(562\) 15.5279i 0.655003i
\(563\) − 21.8885i − 0.922492i −0.887272 0.461246i \(-0.847403\pi\)
0.887272 0.461246i \(-0.152597\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −26.8328 −1.12787
\(567\) 0 0
\(568\) − 32.3607i − 1.35782i
\(569\) −4.11146 −0.172361 −0.0861806 0.996280i \(-0.527466\pi\)
−0.0861806 + 0.996280i \(0.527466\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) − 33.1672i − 1.38679i
\(573\) 0 0
\(574\) 4.47214 0.186663
\(575\) 0 0
\(576\) 0 0
\(577\) − 34.3607i − 1.43045i −0.698892 0.715227i \(-0.746323\pi\)
0.698892 0.715227i \(-0.253677\pi\)
\(578\) − 29.0689i − 1.20911i
\(579\) 0 0
\(580\) 0 0
\(581\) −0.944272 −0.0391750
\(582\) 0 0
\(583\) − 30.8328i − 1.27696i
\(584\) −7.88854 −0.326430
\(585\) 0 0
\(586\) 51.3050 2.11939
\(587\) 4.00000i 0.165098i 0.996587 + 0.0825488i \(0.0263060\pi\)
−0.996587 + 0.0825488i \(0.973694\pi\)
\(588\) 0 0
\(589\) −67.7771 −2.79271
\(590\) 0 0
\(591\) 0 0
\(592\) − 10.9443i − 0.449807i
\(593\) − 11.8885i − 0.488204i −0.969750 0.244102i \(-0.921507\pi\)
0.969750 0.244102i \(-0.0784932\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −8.83282 −0.361806
\(597\) 0 0
\(598\) 40.0000i 1.63572i
\(599\) −32.3607 −1.32222 −0.661111 0.750288i \(-0.729915\pi\)
−0.661111 + 0.750288i \(0.729915\pi\)
\(600\) 0 0
\(601\) 21.0557 0.858881 0.429441 0.903095i \(-0.358711\pi\)
0.429441 + 0.903095i \(0.358711\pi\)
\(602\) 20.0000i 0.815139i
\(603\) 0 0
\(604\) 48.0000 1.95309
\(605\) 0 0
\(606\) 0 0
\(607\) − 14.8328i − 0.602045i −0.953617 0.301023i \(-0.902672\pi\)
0.953617 0.301023i \(-0.0973280\pi\)
\(608\) − 43.4164i − 1.76077i
\(609\) 0 0
\(610\) 0 0
\(611\) −22.1115 −0.894534
\(612\) 0 0
\(613\) − 10.9443i − 0.442035i −0.975270 0.221017i \(-0.929062\pi\)
0.975270 0.221017i \(-0.0709378\pi\)
\(614\) −73.6656 −2.97290
\(615\) 0 0
\(616\) −5.52786 −0.222724
\(617\) − 7.52786i − 0.303060i −0.988453 0.151530i \(-0.951580\pi\)
0.988453 0.151530i \(-0.0484201\pi\)
\(618\) 0 0
\(619\) −12.5836 −0.505777 −0.252889 0.967495i \(-0.581381\pi\)
−0.252889 + 0.967495i \(0.581381\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 22.1115i − 0.886589i
\(623\) − 2.00000i − 0.0801283i
\(624\) 0 0
\(625\) 0 0
\(626\) −21.0557 −0.841556
\(627\) 0 0
\(628\) − 25.4164i − 1.01423i
\(629\) −21.8885 −0.872753
\(630\) 0 0
\(631\) −22.8328 −0.908960 −0.454480 0.890757i \(-0.650175\pi\)
−0.454480 + 0.890757i \(0.650175\pi\)
\(632\) 11.0557i 0.439773i
\(633\) 0 0
\(634\) 67.8885 2.69620
\(635\) 0 0
\(636\) 0 0
\(637\) − 4.47214i − 0.177192i
\(638\) 11.0557i 0.437700i
\(639\) 0 0
\(640\) 0 0
\(641\) 36.8328 1.45481 0.727404 0.686209i \(-0.240726\pi\)
0.727404 + 0.686209i \(0.240726\pi\)
\(642\) 0 0
\(643\) 32.9443i 1.29920i 0.760278 + 0.649598i \(0.225063\pi\)
−0.760278 + 0.649598i \(0.774937\pi\)
\(644\) 12.0000 0.472866
\(645\) 0 0
\(646\) −28.9443 −1.13880
\(647\) − 33.8885i − 1.33230i −0.745820 0.666148i \(-0.767942\pi\)
0.745820 0.666148i \(-0.232058\pi\)
\(648\) 0 0
\(649\) 22.1115 0.867951
\(650\) 0 0
\(651\) 0 0
\(652\) 2.83282i 0.110942i
\(653\) − 49.4164i − 1.93381i −0.255130 0.966907i \(-0.582118\pi\)
0.255130 0.966907i \(-0.417882\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.00000 −0.0780869
\(657\) 0 0
\(658\) 11.0557i 0.430997i
\(659\) −41.3050 −1.60901 −0.804506 0.593944i \(-0.797570\pi\)
−0.804506 + 0.593944i \(0.797570\pi\)
\(660\) 0 0
\(661\) −0.111456 −0.00433514 −0.00216757 0.999998i \(-0.500690\pi\)
−0.00216757 + 0.999998i \(0.500690\pi\)
\(662\) 37.8885i 1.47258i
\(663\) 0 0
\(664\) −2.11146 −0.0819404
\(665\) 0 0
\(666\) 0 0
\(667\) − 8.00000i − 0.309761i
\(668\) − 24.0000i − 0.928588i
\(669\) 0 0
\(670\) 0 0
\(671\) −4.94427 −0.190872
\(672\) 0 0
\(673\) 44.8328i 1.72818i 0.503339 + 0.864089i \(0.332105\pi\)
−0.503339 + 0.864089i \(0.667895\pi\)
\(674\) 26.5836 1.02396
\(675\) 0 0
\(676\) 21.0000 0.807692
\(677\) − 38.9443i − 1.49675i −0.663276 0.748375i \(-0.730834\pi\)
0.663276 0.748375i \(-0.269166\pi\)
\(678\) 0 0
\(679\) 0.472136 0.0181189
\(680\) 0 0
\(681\) 0 0
\(682\) − 57.8885i − 2.21667i
\(683\) 33.8885i 1.29671i 0.761339 + 0.648355i \(0.224543\pi\)
−0.761339 + 0.648355i \(0.775457\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.23607 −0.0853735
\(687\) 0 0
\(688\) − 8.94427i − 0.340997i
\(689\) 55.7771 2.12494
\(690\) 0 0
\(691\) −0.360680 −0.0137209 −0.00686045 0.999976i \(-0.502184\pi\)
−0.00686045 + 0.999976i \(0.502184\pi\)
\(692\) − 44.8328i − 1.70429i
\(693\) 0 0
\(694\) 17.8885 0.679040
\(695\) 0 0
\(696\) 0 0
\(697\) 4.00000i 0.151511i
\(698\) 53.4164i 2.02184i
\(699\) 0 0
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 0 0
\(703\) − 70.8328i − 2.67151i
\(704\) 32.1378 1.21124
\(705\) 0 0
\(706\) −62.3607 −2.34698
\(707\) 14.0000i 0.526524i
\(708\) 0 0
\(709\) 45.7771 1.71919 0.859597 0.510972i \(-0.170714\pi\)
0.859597 + 0.510972i \(0.170714\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 4.47214i − 0.167600i
\(713\) 41.8885i 1.56874i
\(714\) 0 0
\(715\) 0 0
\(716\) 7.41641 0.277164
\(717\) 0 0
\(718\) − 21.3050i − 0.795094i
\(719\) 46.8328 1.74657 0.873285 0.487210i \(-0.161985\pi\)
0.873285 + 0.487210i \(0.161985\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 51.1803i − 1.90474i
\(723\) 0 0
\(724\) −56.8328 −2.11217
\(725\) 0 0
\(726\) 0 0
\(727\) 14.8328i 0.550119i 0.961427 + 0.275059i \(0.0886975\pi\)
−0.961427 + 0.275059i \(0.911302\pi\)
\(728\) − 10.0000i − 0.370625i
\(729\) 0 0
\(730\) 0 0
\(731\) −17.8885 −0.661632
\(732\) 0 0
\(733\) − 37.4164i − 1.38201i −0.722852 0.691003i \(-0.757169\pi\)
0.722852 0.691003i \(-0.242831\pi\)
\(734\) 46.8328 1.72863
\(735\) 0 0
\(736\) −26.8328 −0.989071
\(737\) − 9.88854i − 0.364249i
\(738\) 0 0
\(739\) −29.8885 −1.09947 −0.549734 0.835340i \(-0.685271\pi\)
−0.549734 + 0.835340i \(0.685271\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 27.8885i − 1.02382i
\(743\) − 18.8328i − 0.690909i −0.938436 0.345455i \(-0.887725\pi\)
0.938436 0.345455i \(-0.112275\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −13.4164 −0.491210
\(747\) 0 0
\(748\) − 14.8328i − 0.542341i
\(749\) 4.94427 0.180660
\(750\) 0 0
\(751\) −3.05573 −0.111505 −0.0557526 0.998445i \(-0.517756\pi\)
−0.0557526 + 0.998445i \(0.517756\pi\)
\(752\) − 4.94427i − 0.180299i
\(753\) 0 0
\(754\) −20.0000 −0.728357
\(755\) 0 0
\(756\) 0 0
\(757\) − 3.88854i − 0.141332i −0.997500 0.0706658i \(-0.977488\pi\)
0.997500 0.0706658i \(-0.0225124\pi\)
\(758\) − 4.72136i − 0.171488i
\(759\) 0 0
\(760\) 0 0
\(761\) −7.88854 −0.285959 −0.142980 0.989726i \(-0.545668\pi\)
−0.142980 + 0.989726i \(0.545668\pi\)
\(762\) 0 0
\(763\) 2.00000i 0.0724049i
\(764\) 82.2492 2.97567
\(765\) 0 0
\(766\) −17.8885 −0.646339
\(767\) 40.0000i 1.44432i
\(768\) 0 0
\(769\) −0.832816 −0.0300321 −0.0150161 0.999887i \(-0.504780\pi\)
−0.0150161 + 0.999887i \(0.504780\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 42.0000i − 1.51161i
\(773\) − 25.0557i − 0.901192i −0.892728 0.450596i \(-0.851212\pi\)
0.892728 0.450596i \(-0.148788\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.05573 0.0378984
\(777\) 0 0
\(778\) − 24.4721i − 0.877369i
\(779\) −12.9443 −0.463777
\(780\) 0 0
\(781\) −35.7771 −1.28020
\(782\) 17.8885i 0.639693i
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 0 0
\(787\) 48.9443i 1.74467i 0.488904 + 0.872337i \(0.337397\pi\)
−0.488904 + 0.872337i \(0.662603\pi\)
\(788\) 73.4164i 2.61535i
\(789\) 0 0
\(790\) 0 0
\(791\) 8.47214 0.301234
\(792\) 0 0
\(793\) − 8.94427i − 0.317620i
\(794\) 30.0000 1.06466
\(795\) 0 0
\(796\) 1.75078 0.0620546
\(797\) 1.05573i 0.0373958i 0.999825 + 0.0186979i \(0.00595207\pi\)
−0.999825 + 0.0186979i \(0.994048\pi\)
\(798\) 0 0
\(799\) −9.88854 −0.349832
\(800\) 0 0
\(801\) 0 0
\(802\) 22.3607i 0.789583i
\(803\) 8.72136i 0.307770i
\(804\) 0 0
\(805\) 0 0
\(806\) 104.721 3.68865
\(807\) 0 0
\(808\) 31.3050i 1.10130i
\(809\) 21.0557 0.740280 0.370140 0.928976i \(-0.379310\pi\)
0.370140 + 0.928976i \(0.379310\pi\)
\(810\) 0 0
\(811\) 28.5836 1.00371 0.501853 0.864953i \(-0.332652\pi\)
0.501853 + 0.864953i \(0.332652\pi\)
\(812\) 6.00000i 0.210559i
\(813\) 0 0
\(814\) 60.4984 2.12047
\(815\) 0 0
\(816\) 0 0
\(817\) − 57.8885i − 2.02526i
\(818\) − 53.4164i − 1.86766i
\(819\) 0 0
\(820\) 0 0
\(821\) 37.7771 1.31843 0.659215 0.751955i \(-0.270889\pi\)
0.659215 + 0.751955i \(0.270889\pi\)
\(822\) 0 0
\(823\) − 27.0557i − 0.943103i −0.881838 0.471552i \(-0.843694\pi\)
0.881838 0.471552i \(-0.156306\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 20.0000 0.695889
\(827\) 4.94427i 0.171929i 0.996298 + 0.0859646i \(0.0273972\pi\)
−0.996298 + 0.0859646i \(0.972603\pi\)
\(828\) 0 0
\(829\) 30.9443 1.07474 0.537369 0.843347i \(-0.319418\pi\)
0.537369 + 0.843347i \(0.319418\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 58.1378i 2.01556i
\(833\) − 2.00000i − 0.0692959i
\(834\) 0 0
\(835\) 0 0
\(836\) 48.0000 1.66011
\(837\) 0 0
\(838\) − 13.1672i − 0.454853i
\(839\) 1.16718 0.0402957 0.0201478 0.999797i \(-0.493586\pi\)
0.0201478 + 0.999797i \(0.493586\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) − 49.1935i − 1.69532i
\(843\) 0 0
\(844\) −2.83282 −0.0975095
\(845\) 0 0
\(846\) 0 0
\(847\) − 4.88854i − 0.167972i
\(848\) 12.4721i 0.428295i
\(849\) 0 0
\(850\) 0 0
\(851\) −43.7771 −1.50066
\(852\) 0 0
\(853\) − 31.3050i − 1.07186i −0.844262 0.535931i \(-0.819961\pi\)
0.844262 0.535931i \(-0.180039\pi\)
\(854\) −4.47214 −0.153033
\(855\) 0 0
\(856\) 11.0557 0.377877
\(857\) 16.8328i 0.574998i 0.957781 + 0.287499i \(0.0928238\pi\)
−0.957781 + 0.287499i \(0.907176\pi\)
\(858\) 0 0
\(859\) −41.5279 −1.41691 −0.708456 0.705755i \(-0.750608\pi\)
−0.708456 + 0.705755i \(0.750608\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 21.3050i − 0.725650i
\(863\) − 13.8885i − 0.472772i −0.971659 0.236386i \(-0.924037\pi\)
0.971659 0.236386i \(-0.0759629\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −16.8328 −0.572002
\(867\) 0 0
\(868\) − 31.4164i − 1.06634i
\(869\) 12.2229 0.414634
\(870\) 0 0
\(871\) 17.8885 0.606130
\(872\) 4.47214i 0.151446i
\(873\) 0 0
\(874\) −57.8885 −1.95811
\(875\) 0 0
\(876\) 0 0
\(877\) − 3.16718i − 0.106948i −0.998569 0.0534741i \(-0.982971\pi\)
0.998569 0.0534741i \(-0.0170295\pi\)
\(878\) 23.4164i 0.790265i
\(879\) 0 0
\(880\) 0 0
\(881\) −7.88854 −0.265772 −0.132886 0.991131i \(-0.542424\pi\)
−0.132886 + 0.991131i \(0.542424\pi\)
\(882\) 0 0
\(883\) 2.11146i 0.0710562i 0.999369 + 0.0355281i \(0.0113113\pi\)
−0.999369 + 0.0355281i \(0.988689\pi\)
\(884\) 26.8328 0.902485
\(885\) 0 0
\(886\) −17.8885 −0.600977
\(887\) − 22.8328i − 0.766651i −0.923613 0.383325i \(-0.874779\pi\)
0.923613 0.383325i \(-0.125221\pi\)
\(888\) 0 0
\(889\) −12.9443 −0.434137
\(890\) 0 0
\(891\) 0 0
\(892\) 14.8328i 0.496639i
\(893\) − 32.0000i − 1.07084i
\(894\) 0 0
\(895\) 0 0
\(896\) 15.6525 0.522913
\(897\) 0 0
\(898\) 31.3050i 1.04466i
\(899\) −20.9443 −0.698531
\(900\) 0 0
\(901\) 24.9443 0.831014
\(902\) − 11.0557i − 0.368115i
\(903\) 0 0
\(904\) 18.9443 0.630077
\(905\) 0 0
\(906\) 0 0
\(907\) 18.1115i 0.601381i 0.953722 + 0.300691i \(0.0972171\pi\)
−0.953722 + 0.300691i \(0.902783\pi\)
\(908\) − 50.8328i − 1.68695i
\(909\) 0 0
\(910\) 0 0
\(911\) 34.2492 1.13473 0.567364 0.823467i \(-0.307963\pi\)
0.567364 + 0.823467i \(0.307963\pi\)
\(912\) 0 0
\(913\) 2.33437i 0.0772563i
\(914\) −24.4721 −0.809466
\(915\) 0 0
\(916\) −35.6656 −1.17843
\(917\) − 4.00000i − 0.132092i
\(918\) 0 0
\(919\) 52.9443 1.74647 0.873235 0.487299i \(-0.162018\pi\)
0.873235 + 0.487299i \(0.162018\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 71.3050i − 2.34830i
\(923\) − 64.7214i − 2.13033i
\(924\) 0 0
\(925\) 0 0
\(926\) −6.83282 −0.224540
\(927\) 0 0
\(928\) − 13.4164i − 0.440415i
\(929\) −51.8885 −1.70241 −0.851204 0.524835i \(-0.824127\pi\)
−0.851204 + 0.524835i \(0.824127\pi\)
\(930\) 0 0
\(931\) 6.47214 0.212116
\(932\) − 52.2492i − 1.71148i
\(933\) 0 0
\(934\) −20.0000 −0.654420
\(935\) 0 0
\(936\) 0 0
\(937\) − 43.5279i − 1.42199i −0.703195 0.710997i \(-0.748244\pi\)
0.703195 0.710997i \(-0.251756\pi\)
\(938\) − 8.94427i − 0.292041i
\(939\) 0 0
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) 8.00000i 0.260516i
\(944\) −8.94427 −0.291111
\(945\) 0 0
\(946\) 49.4427 1.60752
\(947\) − 17.8885i − 0.581300i −0.956830 0.290650i \(-0.906129\pi\)
0.956830 0.290650i \(-0.0938715\pi\)
\(948\) 0 0
\(949\) −15.7771 −0.512146
\(950\) 0 0
\(951\) 0 0
\(952\) − 4.47214i − 0.144943i
\(953\) − 6.58359i − 0.213263i −0.994299 0.106632i \(-0.965993\pi\)
0.994299 0.106632i \(-0.0340066\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 4.58359 0.148244
\(957\) 0 0
\(958\) − 40.0000i − 1.29234i
\(959\) 12.4721 0.402746
\(960\) 0 0
\(961\) 78.6656 2.53760
\(962\) 109.443i 3.52857i
\(963\) 0 0
\(964\) 3.16718 0.102008
\(965\) 0 0
\(966\) 0 0
\(967\) − 9.88854i − 0.317994i −0.987279 0.158997i \(-0.949174\pi\)
0.987279 0.158997i \(-0.0508260\pi\)
\(968\) − 10.9311i − 0.351339i
\(969\) 0 0
\(970\) 0 0
\(971\) −23.0557 −0.739894 −0.369947 0.929053i \(-0.620624\pi\)
−0.369947 + 0.929053i \(0.620624\pi\)
\(972\) 0 0
\(973\) 19.4164i 0.622461i
\(974\) −6.83282 −0.218938
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) − 57.4164i − 1.83691i −0.395521 0.918457i \(-0.629436\pi\)
0.395521 0.918457i \(-0.370564\pi\)
\(978\) 0 0
\(979\) −4.94427 −0.158020
\(980\) 0 0
\(981\) 0 0
\(982\) 92.3607i 2.94735i
\(983\) 30.8328i 0.983414i 0.870761 + 0.491707i \(0.163627\pi\)
−0.870761 + 0.491707i \(0.836373\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −8.94427 −0.284844
\(987\) 0 0
\(988\) 86.8328i 2.76252i
\(989\) −35.7771 −1.13765
\(990\) 0 0
\(991\) −12.9443 −0.411188 −0.205594 0.978637i \(-0.565913\pi\)
−0.205594 + 0.978637i \(0.565913\pi\)
\(992\) 70.2492i 2.23042i
\(993\) 0 0
\(994\) −32.3607 −1.02642
\(995\) 0 0
\(996\) 0 0
\(997\) 21.4164i 0.678264i 0.940739 + 0.339132i \(0.110133\pi\)
−0.940739 + 0.339132i \(0.889867\pi\)
\(998\) 48.9443i 1.54930i
\(999\) 0 0
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 1575.2.d.d.1324.2 4
3.2 odd 2 525.2.d.c.274.4 4
5.2 odd 4 1575.2.a.r.1.2 2
5.3 odd 4 315.2.a.d.1.1 2
5.4 even 2 inner 1575.2.d.d.1324.3 4
15.2 even 4 525.2.a.g.1.1 2
15.8 even 4 105.2.a.b.1.2 2
15.14 odd 2 525.2.d.c.274.1 4
20.3 even 4 5040.2.a.bw.1.1 2
35.13 even 4 2205.2.a.w.1.1 2
60.23 odd 4 1680.2.a.v.1.2 2
60.47 odd 4 8400.2.a.cx.1.2 2
105.23 even 12 735.2.i.k.361.1 4
105.38 odd 12 735.2.i.i.226.1 4
105.53 even 12 735.2.i.k.226.1 4
105.62 odd 4 3675.2.a.y.1.1 2
105.68 odd 12 735.2.i.i.361.1 4
105.83 odd 4 735.2.a.k.1.2 2
120.53 even 4 6720.2.a.cx.1.2 2
120.83 odd 4 6720.2.a.cs.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.a.b.1.2 2 15.8 even 4
315.2.a.d.1.1 2 5.3 odd 4
525.2.a.g.1.1 2 15.2 even 4
525.2.d.c.274.1 4 15.14 odd 2
525.2.d.c.274.4 4 3.2 odd 2
735.2.a.k.1.2 2 105.83 odd 4
735.2.i.i.226.1 4 105.38 odd 12
735.2.i.i.361.1 4 105.68 odd 12
735.2.i.k.226.1 4 105.53 even 12
735.2.i.k.361.1 4 105.23 even 12
1575.2.a.r.1.2 2 5.2 odd 4
1575.2.d.d.1324.2 4 1.1 even 1 trivial
1575.2.d.d.1324.3 4 5.4 even 2 inner
1680.2.a.v.1.2 2 60.23 odd 4
2205.2.a.w.1.1 2 35.13 even 4
3675.2.a.y.1.1 2 105.62 odd 4
5040.2.a.bw.1.1 2 20.3 even 4
6720.2.a.cs.1.1 2 120.83 odd 4
6720.2.a.cx.1.2 2 120.53 even 4
8400.2.a.cx.1.2 2 60.47 odd 4