Properties

Label 384.3.l.a.223.2
Level $384$
Weight $3$
Character 384.223
Analytic conductor $10.463$
Analytic rank $0$
Dimension $16$
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] = [384,3,Mod(31,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 384.l (of order \(4\), degree \(2\), 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: \(10.4632421514\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 223.2
Root \(1.78012 + 0.911682i\) of defining polynomial
Character \(\chi\) \(=\) 384.223
Dual form 384.3.l.a.31.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.22474 + 1.22474i) q^{3} +(-1.00772 + 1.00772i) q^{5} +10.0236 q^{7} -3.00000i q^{9} +O(q^{10})\) \(q+(-1.22474 + 1.22474i) q^{3} +(-1.00772 + 1.00772i) q^{5} +10.0236 q^{7} -3.00000i q^{9} +(-2.26517 - 2.26517i) q^{11} +(6.88229 + 6.88229i) q^{13} -2.46840i q^{15} -22.3801 q^{17} +(16.8918 - 16.8918i) q^{19} +(-12.2763 + 12.2763i) q^{21} +33.2007 q^{23} +22.9690i q^{25} +(3.67423 + 3.67423i) q^{27} +(24.6412 + 24.6412i) q^{29} +41.3761i q^{31} +5.54852 q^{33} +(-10.1010 + 10.1010i) q^{35} +(6.60031 - 6.60031i) q^{37} -16.8581 q^{39} +47.1477i q^{41} +(48.8218 + 48.8218i) q^{43} +(3.02316 + 3.02316i) q^{45} -45.6048i q^{47} +51.4717 q^{49} +(27.4100 - 27.4100i) q^{51} +(-25.1401 + 25.1401i) q^{53} +4.56532 q^{55} +41.3762i q^{57} +(-6.23974 - 6.23974i) q^{59} +(-35.9513 - 35.9513i) q^{61} -30.0707i q^{63} -13.8709 q^{65} +(-10.2045 + 10.2045i) q^{67} +(-40.6624 + 40.6624i) q^{69} +11.9529 q^{71} -111.332i q^{73} +(-28.1312 - 28.1312i) q^{75} +(-22.7051 - 22.7051i) q^{77} -4.46031i q^{79} -9.00000 q^{81} +(-10.1751 + 10.1751i) q^{83} +(22.5530 - 22.5530i) q^{85} -60.3583 q^{87} +21.9364i q^{89} +(68.9850 + 68.9850i) q^{91} +(-50.6752 - 50.6752i) q^{93} +34.0444i q^{95} +107.309 q^{97} +(-6.79552 + 6.79552i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 32 q^{11} + 32 q^{19} - 128 q^{23} - 32 q^{29} - 96 q^{35} + 96 q^{37} - 160 q^{43} + 112 q^{49} + 96 q^{51} + 160 q^{53} - 256 q^{55} + 128 q^{59} + 32 q^{61} - 32 q^{65} - 320 q^{67} - 96 q^{69} + 512 q^{71} - 192 q^{75} - 224 q^{77} - 144 q^{81} + 160 q^{83} - 160 q^{85} + 480 q^{91} - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.22474 + 1.22474i −0.408248 + 0.408248i
\(4\) 0 0
\(5\) −1.00772 + 1.00772i −0.201544 + 0.201544i −0.800661 0.599117i \(-0.795518\pi\)
0.599117 + 0.800661i \(0.295518\pi\)
\(6\) 0 0
\(7\) 10.0236 1.43194 0.715969 0.698133i \(-0.245985\pi\)
0.715969 + 0.698133i \(0.245985\pi\)
\(8\) 0 0
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) −2.26517 2.26517i −0.205925 0.205925i 0.596608 0.802533i \(-0.296515\pi\)
−0.802533 + 0.596608i \(0.796515\pi\)
\(12\) 0 0
\(13\) 6.88229 + 6.88229i 0.529407 + 0.529407i 0.920395 0.390989i \(-0.127867\pi\)
−0.390989 + 0.920395i \(0.627867\pi\)
\(14\) 0 0
\(15\) 2.46840i 0.164560i
\(16\) 0 0
\(17\) −22.3801 −1.31648 −0.658240 0.752809i \(-0.728699\pi\)
−0.658240 + 0.752809i \(0.728699\pi\)
\(18\) 0 0
\(19\) 16.8918 16.8918i 0.889041 0.889041i −0.105390 0.994431i \(-0.533609\pi\)
0.994431 + 0.105390i \(0.0336092\pi\)
\(20\) 0 0
\(21\) −12.2763 + 12.2763i −0.584586 + 0.584586i
\(22\) 0 0
\(23\) 33.2007 1.44351 0.721755 0.692149i \(-0.243336\pi\)
0.721755 + 0.692149i \(0.243336\pi\)
\(24\) 0 0
\(25\) 22.9690i 0.918760i
\(26\) 0 0
\(27\) 3.67423 + 3.67423i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 24.6412 + 24.6412i 0.849696 + 0.849696i 0.990095 0.140399i \(-0.0448385\pi\)
−0.140399 + 0.990095i \(0.544839\pi\)
\(30\) 0 0
\(31\) 41.3761i 1.33471i 0.744738 + 0.667357i \(0.232574\pi\)
−0.744738 + 0.667357i \(0.767426\pi\)
\(32\) 0 0
\(33\) 5.54852 0.168137
\(34\) 0 0
\(35\) −10.1010 + 10.1010i −0.288599 + 0.288599i
\(36\) 0 0
\(37\) 6.60031 6.60031i 0.178387 0.178387i −0.612266 0.790652i \(-0.709742\pi\)
0.790652 + 0.612266i \(0.209742\pi\)
\(38\) 0 0
\(39\) −16.8581 −0.432259
\(40\) 0 0
\(41\) 47.1477i 1.14994i 0.818173 + 0.574972i \(0.194987\pi\)
−0.818173 + 0.574972i \(0.805013\pi\)
\(42\) 0 0
\(43\) 48.8218 + 48.8218i 1.13539 + 1.13539i 0.989266 + 0.146124i \(0.0466799\pi\)
0.146124 + 0.989266i \(0.453320\pi\)
\(44\) 0 0
\(45\) 3.02316 + 3.02316i 0.0671814 + 0.0671814i
\(46\) 0 0
\(47\) 45.6048i 0.970315i −0.874427 0.485157i \(-0.838762\pi\)
0.874427 0.485157i \(-0.161238\pi\)
\(48\) 0 0
\(49\) 51.4717 1.05044
\(50\) 0 0
\(51\) 27.4100 27.4100i 0.537450 0.537450i
\(52\) 0 0
\(53\) −25.1401 + 25.1401i −0.474341 + 0.474341i −0.903316 0.428975i \(-0.858875\pi\)
0.428975 + 0.903316i \(0.358875\pi\)
\(54\) 0 0
\(55\) 4.56532 0.0830059
\(56\) 0 0
\(57\) 41.3762i 0.725899i
\(58\) 0 0
\(59\) −6.23974 6.23974i −0.105758 0.105758i 0.652248 0.758006i \(-0.273826\pi\)
−0.758006 + 0.652248i \(0.773826\pi\)
\(60\) 0 0
\(61\) −35.9513 35.9513i −0.589366 0.589366i 0.348093 0.937460i \(-0.386829\pi\)
−0.937460 + 0.348093i \(0.886829\pi\)
\(62\) 0 0
\(63\) 30.0707i 0.477312i
\(64\) 0 0
\(65\) −13.8709 −0.213398
\(66\) 0 0
\(67\) −10.2045 + 10.2045i −0.152307 + 0.152307i −0.779147 0.626841i \(-0.784348\pi\)
0.626841 + 0.779147i \(0.284348\pi\)
\(68\) 0 0
\(69\) −40.6624 + 40.6624i −0.589310 + 0.589310i
\(70\) 0 0
\(71\) 11.9529 0.168350 0.0841752 0.996451i \(-0.473174\pi\)
0.0841752 + 0.996451i \(0.473174\pi\)
\(72\) 0 0
\(73\) 111.332i 1.52510i −0.646929 0.762550i \(-0.723947\pi\)
0.646929 0.762550i \(-0.276053\pi\)
\(74\) 0 0
\(75\) −28.1312 28.1312i −0.375082 0.375082i
\(76\) 0 0
\(77\) −22.7051 22.7051i −0.294871 0.294871i
\(78\) 0 0
\(79\) 4.46031i 0.0564596i −0.999601 0.0282298i \(-0.991013\pi\)
0.999601 0.0282298i \(-0.00898702\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) 0 0
\(83\) −10.1751 + 10.1751i −0.122592 + 0.122592i −0.765741 0.643149i \(-0.777627\pi\)
0.643149 + 0.765741i \(0.277627\pi\)
\(84\) 0 0
\(85\) 22.5530 22.5530i 0.265329 0.265329i
\(86\) 0 0
\(87\) −60.3583 −0.693774
\(88\) 0 0
\(89\) 21.9364i 0.246476i 0.992377 + 0.123238i \(0.0393279\pi\)
−0.992377 + 0.123238i \(0.960672\pi\)
\(90\) 0 0
\(91\) 68.9850 + 68.9850i 0.758077 + 0.758077i
\(92\) 0 0
\(93\) −50.6752 50.6752i −0.544895 0.544895i
\(94\) 0 0
\(95\) 34.0444i 0.358362i
\(96\) 0 0
\(97\) 107.309 1.10628 0.553140 0.833088i \(-0.313429\pi\)
0.553140 + 0.833088i \(0.313429\pi\)
\(98\) 0 0
\(99\) −6.79552 + 6.79552i −0.0686416 + 0.0686416i
\(100\) 0 0
\(101\) 100.780 100.780i 0.997824 0.997824i −0.00217389 0.999998i \(-0.500692\pi\)
0.999998 + 0.00217389i \(0.000691973\pi\)
\(102\) 0 0
\(103\) 58.0562 0.563653 0.281826 0.959465i \(-0.409060\pi\)
0.281826 + 0.959465i \(0.409060\pi\)
\(104\) 0 0
\(105\) 24.7422i 0.235640i
\(106\) 0 0
\(107\) −112.747 112.747i −1.05371 1.05371i −0.998473 0.0552381i \(-0.982408\pi\)
−0.0552381 0.998473i \(-0.517592\pi\)
\(108\) 0 0
\(109\) 81.1384 + 81.1384i 0.744389 + 0.744389i 0.973419 0.229030i \(-0.0735554\pi\)
−0.229030 + 0.973419i \(0.573555\pi\)
\(110\) 0 0
\(111\) 16.1674i 0.145652i
\(112\) 0 0
\(113\) −171.844 −1.52074 −0.760371 0.649489i \(-0.774983\pi\)
−0.760371 + 0.649489i \(0.774983\pi\)
\(114\) 0 0
\(115\) −33.4571 + 33.4571i −0.290931 + 0.290931i
\(116\) 0 0
\(117\) 20.6469 20.6469i 0.176469 0.176469i
\(118\) 0 0
\(119\) −224.329 −1.88512
\(120\) 0 0
\(121\) 110.738i 0.915190i
\(122\) 0 0
\(123\) −57.7439 57.7439i −0.469463 0.469463i
\(124\) 0 0
\(125\) −48.3394 48.3394i −0.386715 0.386715i
\(126\) 0 0
\(127\) 36.8333i 0.290026i 0.989430 + 0.145013i \(0.0463224\pi\)
−0.989430 + 0.145013i \(0.953678\pi\)
\(128\) 0 0
\(129\) −119.588 −0.927042
\(130\) 0 0
\(131\) 12.3686 12.3686i 0.0944170 0.0944170i −0.658321 0.752738i \(-0.728733\pi\)
0.752738 + 0.658321i \(0.228733\pi\)
\(132\) 0 0
\(133\) 169.316 169.316i 1.27305 1.27305i
\(134\) 0 0
\(135\) −7.40521 −0.0548534
\(136\) 0 0
\(137\) 145.679i 1.06335i −0.846949 0.531674i \(-0.821563\pi\)
0.846949 0.531674i \(-0.178437\pi\)
\(138\) 0 0
\(139\) −82.5709 82.5709i −0.594035 0.594035i 0.344684 0.938719i \(-0.387986\pi\)
−0.938719 + 0.344684i \(0.887986\pi\)
\(140\) 0 0
\(141\) 55.8542 + 55.8542i 0.396129 + 0.396129i
\(142\) 0 0
\(143\) 31.1791i 0.218036i
\(144\) 0 0
\(145\) −49.6629 −0.342503
\(146\) 0 0
\(147\) −63.0398 + 63.0398i −0.428842 + 0.428842i
\(148\) 0 0
\(149\) −196.248 + 196.248i −1.31710 + 1.31710i −0.401043 + 0.916059i \(0.631352\pi\)
−0.916059 + 0.401043i \(0.868648\pi\)
\(150\) 0 0
\(151\) 64.5007 0.427157 0.213578 0.976926i \(-0.431488\pi\)
0.213578 + 0.976926i \(0.431488\pi\)
\(152\) 0 0
\(153\) 67.1404i 0.438826i
\(154\) 0 0
\(155\) −41.6956 41.6956i −0.269004 0.269004i
\(156\) 0 0
\(157\) −54.4202 54.4202i −0.346625 0.346625i 0.512226 0.858851i \(-0.328821\pi\)
−0.858851 + 0.512226i \(0.828821\pi\)
\(158\) 0 0
\(159\) 61.5803i 0.387298i
\(160\) 0 0
\(161\) 332.789 2.06701
\(162\) 0 0
\(163\) −104.803 + 104.803i −0.642961 + 0.642961i −0.951282 0.308321i \(-0.900233\pi\)
0.308321 + 0.951282i \(0.400233\pi\)
\(164\) 0 0
\(165\) −5.59136 + 5.59136i −0.0338870 + 0.0338870i
\(166\) 0 0
\(167\) 53.3110 0.319228 0.159614 0.987180i \(-0.448975\pi\)
0.159614 + 0.987180i \(0.448975\pi\)
\(168\) 0 0
\(169\) 74.2683i 0.439457i
\(170\) 0 0
\(171\) −50.6753 50.6753i −0.296347 0.296347i
\(172\) 0 0
\(173\) 41.5780 + 41.5780i 0.240335 + 0.240335i 0.816989 0.576654i \(-0.195642\pi\)
−0.576654 + 0.816989i \(0.695642\pi\)
\(174\) 0 0
\(175\) 230.231i 1.31561i
\(176\) 0 0
\(177\) 15.2842 0.0863513
\(178\) 0 0
\(179\) −53.0709 + 53.0709i −0.296486 + 0.296486i −0.839636 0.543150i \(-0.817231\pi\)
0.543150 + 0.839636i \(0.317231\pi\)
\(180\) 0 0
\(181\) 66.6042 66.6042i 0.367979 0.367979i −0.498761 0.866740i \(-0.666211\pi\)
0.866740 + 0.498761i \(0.166211\pi\)
\(182\) 0 0
\(183\) 88.0625 0.481216
\(184\) 0 0
\(185\) 13.3025i 0.0719056i
\(186\) 0 0
\(187\) 50.6949 + 50.6949i 0.271096 + 0.271096i
\(188\) 0 0
\(189\) 36.8289 + 36.8289i 0.194862 + 0.194862i
\(190\) 0 0
\(191\) 113.753i 0.595567i −0.954633 0.297784i \(-0.903753\pi\)
0.954633 0.297784i \(-0.0962474\pi\)
\(192\) 0 0
\(193\) −26.5596 −0.137615 −0.0688073 0.997630i \(-0.521919\pi\)
−0.0688073 + 0.997630i \(0.521919\pi\)
\(194\) 0 0
\(195\) 16.9883 16.9883i 0.0871193 0.0871193i
\(196\) 0 0
\(197\) −51.8935 + 51.8935i −0.263419 + 0.263419i −0.826442 0.563023i \(-0.809638\pi\)
0.563023 + 0.826442i \(0.309638\pi\)
\(198\) 0 0
\(199\) −136.741 −0.687140 −0.343570 0.939127i \(-0.611636\pi\)
−0.343570 + 0.939127i \(0.611636\pi\)
\(200\) 0 0
\(201\) 24.9959i 0.124358i
\(202\) 0 0
\(203\) 246.992 + 246.992i 1.21671 + 1.21671i
\(204\) 0 0
\(205\) −47.5118 47.5118i −0.231765 0.231765i
\(206\) 0 0
\(207\) 99.6022i 0.481170i
\(208\) 0 0
\(209\) −76.5255 −0.366151
\(210\) 0 0
\(211\) 141.171 141.171i 0.669057 0.669057i −0.288441 0.957498i \(-0.593137\pi\)
0.957498 + 0.288441i \(0.0931368\pi\)
\(212\) 0 0
\(213\) −14.6392 + 14.6392i −0.0687288 + 0.0687288i
\(214\) 0 0
\(215\) −98.3975 −0.457663
\(216\) 0 0
\(217\) 414.736i 1.91123i
\(218\) 0 0
\(219\) 136.354 + 136.354i 0.622620 + 0.622620i
\(220\) 0 0
\(221\) −154.027 154.027i −0.696953 0.696953i
\(222\) 0 0
\(223\) 122.607i 0.549806i −0.961472 0.274903i \(-0.911354\pi\)
0.961472 0.274903i \(-0.0886457\pi\)
\(224\) 0 0
\(225\) 68.9070 0.306253
\(226\) 0 0
\(227\) 295.844 295.844i 1.30328 1.30328i 0.377112 0.926168i \(-0.376917\pi\)
0.926168 0.377112i \(-0.123083\pi\)
\(228\) 0 0
\(229\) −73.3817 + 73.3817i −0.320444 + 0.320444i −0.848937 0.528493i \(-0.822757\pi\)
0.528493 + 0.848937i \(0.322757\pi\)
\(230\) 0 0
\(231\) 55.6159 0.240761
\(232\) 0 0
\(233\) 156.229i 0.670509i 0.942128 + 0.335255i \(0.108822\pi\)
−0.942128 + 0.335255i \(0.891178\pi\)
\(234\) 0 0
\(235\) 45.9569 + 45.9569i 0.195561 + 0.195561i
\(236\) 0 0
\(237\) 5.46274 + 5.46274i 0.0230495 + 0.0230495i
\(238\) 0 0
\(239\) 13.1716i 0.0551113i −0.999620 0.0275557i \(-0.991228\pi\)
0.999620 0.0275557i \(-0.00877235\pi\)
\(240\) 0 0
\(241\) −189.519 −0.786386 −0.393193 0.919456i \(-0.628630\pi\)
−0.393193 + 0.919456i \(0.628630\pi\)
\(242\) 0 0
\(243\) 11.0227 11.0227i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) −51.8692 + 51.8692i −0.211711 + 0.211711i
\(246\) 0 0
\(247\) 232.508 0.941328
\(248\) 0 0
\(249\) 24.9238i 0.100096i
\(250\) 0 0
\(251\) 27.4434 + 27.4434i 0.109336 + 0.109336i 0.759658 0.650322i \(-0.225366\pi\)
−0.650322 + 0.759658i \(0.725366\pi\)
\(252\) 0 0
\(253\) −75.2053 75.2053i −0.297254 0.297254i
\(254\) 0 0
\(255\) 55.2432i 0.216640i
\(256\) 0 0
\(257\) 135.375 0.526752 0.263376 0.964693i \(-0.415164\pi\)
0.263376 + 0.964693i \(0.415164\pi\)
\(258\) 0 0
\(259\) 66.1586 66.1586i 0.255438 0.255438i
\(260\) 0 0
\(261\) 73.9236 73.9236i 0.283232 0.283232i
\(262\) 0 0
\(263\) 31.6123 0.120199 0.0600994 0.998192i \(-0.480858\pi\)
0.0600994 + 0.998192i \(0.480858\pi\)
\(264\) 0 0
\(265\) 50.6684i 0.191201i
\(266\) 0 0
\(267\) −26.8665 26.8665i −0.100624 0.100624i
\(268\) 0 0
\(269\) 194.213 + 194.213i 0.721981 + 0.721981i 0.969008 0.247028i \(-0.0794538\pi\)
−0.247028 + 0.969008i \(0.579454\pi\)
\(270\) 0 0
\(271\) 291.647i 1.07619i −0.842884 0.538095i \(-0.819144\pi\)
0.842884 0.538095i \(-0.180856\pi\)
\(272\) 0 0
\(273\) −168.978 −0.618967
\(274\) 0 0
\(275\) 52.0287 52.0287i 0.189195 0.189195i
\(276\) 0 0
\(277\) −305.166 + 305.166i −1.10168 + 1.10168i −0.107475 + 0.994208i \(0.534277\pi\)
−0.994208 + 0.107475i \(0.965723\pi\)
\(278\) 0 0
\(279\) 124.128 0.444905
\(280\) 0 0
\(281\) 211.861i 0.753955i 0.926222 + 0.376978i \(0.123037\pi\)
−0.926222 + 0.376978i \(0.876963\pi\)
\(282\) 0 0
\(283\) −105.325 105.325i −0.372175 0.372175i 0.496094 0.868269i \(-0.334767\pi\)
−0.868269 + 0.496094i \(0.834767\pi\)
\(284\) 0 0
\(285\) −41.6957 41.6957i −0.146301 0.146301i
\(286\) 0 0
\(287\) 472.588i 1.64665i
\(288\) 0 0
\(289\) 211.871 0.733117
\(290\) 0 0
\(291\) −131.426 + 131.426i −0.451637 + 0.451637i
\(292\) 0 0
\(293\) 171.289 171.289i 0.584603 0.584603i −0.351562 0.936165i \(-0.614349\pi\)
0.936165 + 0.351562i \(0.114349\pi\)
\(294\) 0 0
\(295\) 12.5758 0.0426300
\(296\) 0 0
\(297\) 16.6455i 0.0560456i
\(298\) 0 0
\(299\) 228.497 + 228.497i 0.764204 + 0.764204i
\(300\) 0 0
\(301\) 489.368 + 489.368i 1.62581 + 1.62581i
\(302\) 0 0
\(303\) 246.860i 0.814720i
\(304\) 0 0
\(305\) 72.4579 0.237567
\(306\) 0 0
\(307\) 27.1124 27.1124i 0.0883140 0.0883140i −0.661570 0.749884i \(-0.730109\pi\)
0.749884 + 0.661570i \(0.230109\pi\)
\(308\) 0 0
\(309\) −71.1041 + 71.1041i −0.230110 + 0.230110i
\(310\) 0 0
\(311\) 371.124 1.19333 0.596663 0.802492i \(-0.296493\pi\)
0.596663 + 0.802492i \(0.296493\pi\)
\(312\) 0 0
\(313\) 374.501i 1.19649i 0.801313 + 0.598245i \(0.204135\pi\)
−0.801313 + 0.598245i \(0.795865\pi\)
\(314\) 0 0
\(315\) 30.3029 + 30.3029i 0.0961996 + 0.0961996i
\(316\) 0 0
\(317\) 48.5840 + 48.5840i 0.153262 + 0.153262i 0.779573 0.626311i \(-0.215436\pi\)
−0.626311 + 0.779573i \(0.715436\pi\)
\(318\) 0 0
\(319\) 111.633i 0.349947i
\(320\) 0 0
\(321\) 276.173 0.860352
\(322\) 0 0
\(323\) −378.040 + 378.040i −1.17040 + 1.17040i
\(324\) 0 0
\(325\) −158.079 + 158.079i −0.486398 + 0.486398i
\(326\) 0 0
\(327\) −198.748 −0.607791
\(328\) 0 0
\(329\) 457.122i 1.38943i
\(330\) 0 0
\(331\) 1.88883 + 1.88883i 0.00570644 + 0.00570644i 0.709954 0.704248i \(-0.248716\pi\)
−0.704248 + 0.709954i \(0.748716\pi\)
\(332\) 0 0
\(333\) −19.8009 19.8009i −0.0594622 0.0594622i
\(334\) 0 0
\(335\) 20.5667i 0.0613931i
\(336\) 0 0
\(337\) −386.980 −1.14831 −0.574154 0.818747i \(-0.694669\pi\)
−0.574154 + 0.818747i \(0.694669\pi\)
\(338\) 0 0
\(339\) 210.465 210.465i 0.620840 0.620840i
\(340\) 0 0
\(341\) 93.7240 93.7240i 0.274851 0.274851i
\(342\) 0 0
\(343\) 24.7757 0.0722325
\(344\) 0 0
\(345\) 81.9528i 0.237544i
\(346\) 0 0
\(347\) −441.887 441.887i −1.27345 1.27345i −0.944266 0.329183i \(-0.893227\pi\)
−0.329183 0.944266i \(-0.606773\pi\)
\(348\) 0 0
\(349\) −119.382 119.382i −0.342068 0.342068i 0.515076 0.857144i \(-0.327764\pi\)
−0.857144 + 0.515076i \(0.827764\pi\)
\(350\) 0 0
\(351\) 50.5743i 0.144086i
\(352\) 0 0
\(353\) −515.642 −1.46074 −0.730371 0.683050i \(-0.760653\pi\)
−0.730371 + 0.683050i \(0.760653\pi\)
\(354\) 0 0
\(355\) −12.0452 + 12.0452i −0.0339301 + 0.0339301i
\(356\) 0 0
\(357\) 274.745 274.745i 0.769595 0.769595i
\(358\) 0 0
\(359\) 428.264 1.19294 0.596468 0.802637i \(-0.296570\pi\)
0.596468 + 0.802637i \(0.296570\pi\)
\(360\) 0 0
\(361\) 209.664i 0.580786i
\(362\) 0 0
\(363\) 135.626 + 135.626i 0.373625 + 0.373625i
\(364\) 0 0
\(365\) 112.192 + 112.192i 0.307375 + 0.307375i
\(366\) 0 0
\(367\) 219.482i 0.598043i −0.954246 0.299021i \(-0.903340\pi\)
0.954246 0.299021i \(-0.0966602\pi\)
\(368\) 0 0
\(369\) 141.443 0.383315
\(370\) 0 0
\(371\) −251.993 + 251.993i −0.679226 + 0.679226i
\(372\) 0 0
\(373\) −425.005 + 425.005i −1.13942 + 1.13942i −0.150870 + 0.988554i \(0.548207\pi\)
−0.988554 + 0.150870i \(0.951793\pi\)
\(374\) 0 0
\(375\) 118.407 0.315752
\(376\) 0 0
\(377\) 339.175i 0.899669i
\(378\) 0 0
\(379\) −365.916 365.916i −0.965476 0.965476i 0.0339473 0.999424i \(-0.489192\pi\)
−0.999424 + 0.0339473i \(0.989192\pi\)
\(380\) 0 0
\(381\) −45.1114 45.1114i −0.118403 0.118403i
\(382\) 0 0
\(383\) 213.276i 0.556857i −0.960457 0.278428i \(-0.910187\pi\)
0.960457 0.278428i \(-0.0898135\pi\)
\(384\) 0 0
\(385\) 45.7608 0.118859
\(386\) 0 0
\(387\) 146.465 146.465i 0.378464 0.378464i
\(388\) 0 0
\(389\) 210.798 210.798i 0.541898 0.541898i −0.382187 0.924085i \(-0.624829\pi\)
0.924085 + 0.382187i \(0.124829\pi\)
\(390\) 0 0
\(391\) −743.037 −1.90035
\(392\) 0 0
\(393\) 30.2968i 0.0770912i
\(394\) 0 0
\(395\) 4.49475 + 4.49475i 0.0113791 + 0.0113791i
\(396\) 0 0
\(397\) −392.907 392.907i −0.989690 0.989690i 0.0102579 0.999947i \(-0.496735\pi\)
−0.999947 + 0.0102579i \(0.996735\pi\)
\(398\) 0 0
\(399\) 414.737i 1.03944i
\(400\) 0 0
\(401\) 29.3290 0.0731396 0.0365698 0.999331i \(-0.488357\pi\)
0.0365698 + 0.999331i \(0.488357\pi\)
\(402\) 0 0
\(403\) −284.762 + 284.762i −0.706606 + 0.706606i
\(404\) 0 0
\(405\) 9.06949 9.06949i 0.0223938 0.0223938i
\(406\) 0 0
\(407\) −29.9017 −0.0734684
\(408\) 0 0
\(409\) 601.115i 1.46972i −0.678219 0.734860i \(-0.737248\pi\)
0.678219 0.734860i \(-0.262752\pi\)
\(410\) 0 0
\(411\) 178.419 + 178.419i 0.434110 + 0.434110i
\(412\) 0 0
\(413\) −62.5444 62.5444i −0.151439 0.151439i
\(414\) 0 0
\(415\) 20.5073i 0.0494153i
\(416\) 0 0
\(417\) 202.257 0.485028
\(418\) 0 0
\(419\) 518.885 518.885i 1.23839 1.23839i 0.277729 0.960659i \(-0.410418\pi\)
0.960659 0.277729i \(-0.0895819\pi\)
\(420\) 0 0
\(421\) 411.213 411.213i 0.976754 0.976754i −0.0229817 0.999736i \(-0.507316\pi\)
0.999736 + 0.0229817i \(0.00731596\pi\)
\(422\) 0 0
\(423\) −136.814 −0.323438
\(424\) 0 0
\(425\) 514.049i 1.20953i
\(426\) 0 0
\(427\) −360.360 360.360i −0.843936 0.843936i
\(428\) 0 0
\(429\) 38.1865 + 38.1865i 0.0890128 + 0.0890128i
\(430\) 0 0
\(431\) 41.1083i 0.0953789i −0.998862 0.0476895i \(-0.984814\pi\)
0.998862 0.0476895i \(-0.0151858\pi\)
\(432\) 0 0
\(433\) −351.682 −0.812199 −0.406100 0.913829i \(-0.633111\pi\)
−0.406100 + 0.913829i \(0.633111\pi\)
\(434\) 0 0
\(435\) 60.8244 60.8244i 0.139826 0.139826i
\(436\) 0 0
\(437\) 560.819 560.819i 1.28334 1.28334i
\(438\) 0 0
\(439\) −775.613 −1.76677 −0.883386 0.468646i \(-0.844742\pi\)
−0.883386 + 0.468646i \(0.844742\pi\)
\(440\) 0 0
\(441\) 154.415i 0.350148i
\(442\) 0 0
\(443\) 241.372 + 241.372i 0.544858 + 0.544858i 0.924949 0.380091i \(-0.124107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 0 0
\(445\) −22.1058 22.1058i −0.0496759 0.0496759i
\(446\) 0 0
\(447\) 480.708i 1.07541i
\(448\) 0 0
\(449\) 266.360 0.593228 0.296614 0.954997i \(-0.404142\pi\)
0.296614 + 0.954997i \(0.404142\pi\)
\(450\) 0 0
\(451\) 106.798 106.798i 0.236802 0.236802i
\(452\) 0 0
\(453\) −78.9969 + 78.9969i −0.174386 + 0.174386i
\(454\) 0 0
\(455\) −139.035 −0.305572
\(456\) 0 0
\(457\) 515.244i 1.12745i 0.825963 + 0.563725i \(0.190632\pi\)
−0.825963 + 0.563725i \(0.809368\pi\)
\(458\) 0 0
\(459\) −82.2299 82.2299i −0.179150 0.179150i
\(460\) 0 0
\(461\) −5.67717 5.67717i −0.0123149 0.0123149i 0.700923 0.713237i \(-0.252772\pi\)
−0.713237 + 0.700923i \(0.752772\pi\)
\(462\) 0 0
\(463\) 464.510i 1.00326i 0.865082 + 0.501631i \(0.167267\pi\)
−0.865082 + 0.501631i \(0.832733\pi\)
\(464\) 0 0
\(465\) 102.133 0.219641
\(466\) 0 0
\(467\) −495.985 + 495.985i −1.06207 + 1.06207i −0.0641248 + 0.997942i \(0.520426\pi\)
−0.997942 + 0.0641248i \(0.979574\pi\)
\(468\) 0 0
\(469\) −102.286 + 102.286i −0.218094 + 0.218094i
\(470\) 0 0
\(471\) 133.302 0.283018
\(472\) 0 0
\(473\) 221.180i 0.467610i
\(474\) 0 0
\(475\) 387.987 + 387.987i 0.816815 + 0.816815i
\(476\) 0 0
\(477\) 75.4202 + 75.4202i 0.158114 + 0.158114i
\(478\) 0 0
\(479\) 378.802i 0.790818i 0.918505 + 0.395409i \(0.129397\pi\)
−0.918505 + 0.395409i \(0.870603\pi\)
\(480\) 0 0
\(481\) 90.8504 0.188878
\(482\) 0 0
\(483\) −407.582 + 407.582i −0.843855 + 0.843855i
\(484\) 0 0
\(485\) −108.138 + 108.138i −0.222964 + 0.222964i
\(486\) 0 0
\(487\) 147.446 0.302764 0.151382 0.988475i \(-0.451628\pi\)
0.151382 + 0.988475i \(0.451628\pi\)
\(488\) 0 0
\(489\) 256.713i 0.524976i
\(490\) 0 0
\(491\) −109.547 109.547i −0.223110 0.223110i 0.586697 0.809807i \(-0.300428\pi\)
−0.809807 + 0.586697i \(0.800428\pi\)
\(492\) 0 0
\(493\) −551.473 551.473i −1.11861 1.11861i
\(494\) 0 0
\(495\) 13.6960i 0.0276686i
\(496\) 0 0
\(497\) 119.810 0.241067
\(498\) 0 0
\(499\) 360.523 360.523i 0.722491 0.722491i −0.246621 0.969112i \(-0.579320\pi\)
0.969112 + 0.246621i \(0.0793202\pi\)
\(500\) 0 0
\(501\) −65.2924 + 65.2924i −0.130324 + 0.130324i
\(502\) 0 0
\(503\) −927.420 −1.84378 −0.921889 0.387454i \(-0.873355\pi\)
−0.921889 + 0.387454i \(0.873355\pi\)
\(504\) 0 0
\(505\) 203.117i 0.402211i
\(506\) 0 0
\(507\) 90.9597 + 90.9597i 0.179408 + 0.179408i
\(508\) 0 0
\(509\) −677.931 677.931i −1.33189 1.33189i −0.903680 0.428208i \(-0.859145\pi\)
−0.428208 0.903680i \(-0.640855\pi\)
\(510\) 0 0
\(511\) 1115.95i 2.18385i
\(512\) 0 0
\(513\) 124.129 0.241966
\(514\) 0 0
\(515\) −58.5045 + 58.5045i −0.113601 + 0.113601i
\(516\) 0 0
\(517\) −103.303 + 103.303i −0.199812 + 0.199812i
\(518\) 0 0
\(519\) −101.845 −0.196233
\(520\) 0 0
\(521\) 143.173i 0.274804i 0.990515 + 0.137402i \(0.0438753\pi\)
−0.990515 + 0.137402i \(0.956125\pi\)
\(522\) 0 0
\(523\) −226.187 226.187i −0.432481 0.432481i 0.456991 0.889471i \(-0.348927\pi\)
−0.889471 + 0.456991i \(0.848927\pi\)
\(524\) 0 0
\(525\) −281.974 281.974i −0.537094 0.537094i
\(526\) 0 0
\(527\) 926.004i 1.75712i
\(528\) 0 0
\(529\) 573.288 1.08372
\(530\) 0 0
\(531\) −18.7192 + 18.7192i −0.0352528 + 0.0352528i
\(532\) 0 0
\(533\) −324.484 + 324.484i −0.608788 + 0.608788i
\(534\) 0 0
\(535\) 227.235 0.424739
\(536\) 0 0
\(537\) 129.997i 0.242080i
\(538\) 0 0
\(539\) −116.592 116.592i −0.216312 0.216312i
\(540\) 0 0
\(541\) 156.708 + 156.708i 0.289663 + 0.289663i 0.836947 0.547284i \(-0.184338\pi\)
−0.547284 + 0.836947i \(0.684338\pi\)
\(542\) 0 0
\(543\) 163.146i 0.300454i
\(544\) 0 0
\(545\) −163.530 −0.300055
\(546\) 0 0
\(547\) −247.357 + 247.357i −0.452207 + 0.452207i −0.896086 0.443880i \(-0.853602\pi\)
0.443880 + 0.896086i \(0.353602\pi\)
\(548\) 0 0
\(549\) −107.854 + 107.854i −0.196455 + 0.196455i
\(550\) 0 0
\(551\) 832.466 1.51083
\(552\) 0 0
\(553\) 44.7082i 0.0808466i
\(554\) 0 0
\(555\) −16.2922 16.2922i −0.0293553 0.0293553i
\(556\) 0 0
\(557\) 661.193 + 661.193i 1.18706 + 1.18706i 0.977876 + 0.209184i \(0.0670808\pi\)
0.209184 + 0.977876i \(0.432919\pi\)
\(558\) 0 0
\(559\) 672.011i 1.20217i
\(560\) 0 0
\(561\) −124.177 −0.221349
\(562\) 0 0
\(563\) −246.685 + 246.685i −0.438162 + 0.438162i −0.891393 0.453231i \(-0.850271\pi\)
0.453231 + 0.891393i \(0.350271\pi\)
\(564\) 0 0
\(565\) 173.171 173.171i 0.306497 0.306497i
\(566\) 0 0
\(567\) −90.2120 −0.159104
\(568\) 0 0
\(569\) 243.567i 0.428061i 0.976827 + 0.214030i \(0.0686592\pi\)
−0.976827 + 0.214030i \(0.931341\pi\)
\(570\) 0 0
\(571\) −59.9229 59.9229i −0.104944 0.104944i 0.652685 0.757629i \(-0.273642\pi\)
−0.757629 + 0.652685i \(0.773642\pi\)
\(572\) 0 0
\(573\) 139.319 + 139.319i 0.243139 + 0.243139i
\(574\) 0 0
\(575\) 762.587i 1.32624i
\(576\) 0 0
\(577\) 136.609 0.236757 0.118378 0.992969i \(-0.462230\pi\)
0.118378 + 0.992969i \(0.462230\pi\)
\(578\) 0 0
\(579\) 32.5287 32.5287i 0.0561809 0.0561809i
\(580\) 0 0
\(581\) −101.991 + 101.991i −0.175543 + 0.175543i
\(582\) 0 0
\(583\) 113.893 0.195357
\(584\) 0 0
\(585\) 41.6126i 0.0711326i
\(586\) 0 0
\(587\) 331.817 + 331.817i 0.565276 + 0.565276i 0.930801 0.365525i \(-0.119111\pi\)
−0.365525 + 0.930801i \(0.619111\pi\)
\(588\) 0 0
\(589\) 698.916 + 698.916i 1.18661 + 1.18661i
\(590\) 0 0
\(591\) 127.113i 0.215081i
\(592\) 0 0
\(593\) 131.285 0.221391 0.110695 0.993854i \(-0.464692\pi\)
0.110695 + 0.993854i \(0.464692\pi\)
\(594\) 0 0
\(595\) 226.061 226.061i 0.379934 0.379934i
\(596\) 0 0
\(597\) 167.473 167.473i 0.280524 0.280524i
\(598\) 0 0
\(599\) −136.119 −0.227243 −0.113621 0.993524i \(-0.536245\pi\)
−0.113621 + 0.993524i \(0.536245\pi\)
\(600\) 0 0
\(601\) 498.566i 0.829561i 0.909922 + 0.414780i \(0.136142\pi\)
−0.909922 + 0.414780i \(0.863858\pi\)
\(602\) 0 0
\(603\) 30.6136 + 30.6136i 0.0507689 + 0.0507689i
\(604\) 0 0
\(605\) 111.593 + 111.593i 0.184451 + 0.184451i
\(606\) 0 0
\(607\) 568.740i 0.936969i 0.883472 + 0.468484i \(0.155200\pi\)
−0.883472 + 0.468484i \(0.844800\pi\)
\(608\) 0 0
\(609\) −605.005 −0.993441
\(610\) 0 0
\(611\) 313.865 313.865i 0.513691 0.513691i
\(612\) 0 0
\(613\) 168.441 168.441i 0.274782 0.274782i −0.556240 0.831022i \(-0.687756\pi\)
0.831022 + 0.556240i \(0.187756\pi\)
\(614\) 0 0
\(615\) 116.380 0.189235
\(616\) 0 0
\(617\) 599.157i 0.971081i −0.874214 0.485541i \(-0.838623\pi\)
0.874214 0.485541i \(-0.161377\pi\)
\(618\) 0 0
\(619\) −126.719 126.719i −0.204715 0.204715i 0.597301 0.802017i \(-0.296240\pi\)
−0.802017 + 0.597301i \(0.796240\pi\)
\(620\) 0 0
\(621\) 121.987 + 121.987i 0.196437 + 0.196437i
\(622\) 0 0
\(623\) 219.881i 0.352939i
\(624\) 0 0
\(625\) −476.800 −0.762879
\(626\) 0 0
\(627\) 93.7242 93.7242i 0.149480 0.149480i
\(628\) 0 0
\(629\) −147.716 + 147.716i −0.234842 + 0.234842i
\(630\) 0 0
\(631\) 668.283 1.05909 0.529543 0.848283i \(-0.322363\pi\)
0.529543 + 0.848283i \(0.322363\pi\)
\(632\) 0 0
\(633\) 345.797i 0.546283i
\(634\) 0 0
\(635\) −37.1177 37.1177i −0.0584531 0.0584531i
\(636\) 0 0
\(637\) 354.243 + 354.243i 0.556112 + 0.556112i
\(638\) 0 0
\(639\) 35.8586i 0.0561168i
\(640\) 0 0
\(641\) 484.574 0.755966 0.377983 0.925813i \(-0.376618\pi\)
0.377983 + 0.925813i \(0.376618\pi\)
\(642\) 0 0
\(643\) −75.2980 + 75.2980i −0.117104 + 0.117104i −0.763230 0.646126i \(-0.776388\pi\)
0.646126 + 0.763230i \(0.276388\pi\)
\(644\) 0 0
\(645\) 120.512 120.512i 0.186840 0.186840i
\(646\) 0 0
\(647\) −582.307 −0.900011 −0.450006 0.893026i \(-0.648578\pi\)
−0.450006 + 0.893026i \(0.648578\pi\)
\(648\) 0 0
\(649\) 28.2682i 0.0435565i
\(650\) 0 0
\(651\) −507.946 507.946i −0.780255 0.780255i
\(652\) 0 0
\(653\) −457.453 457.453i −0.700541 0.700541i 0.263986 0.964527i \(-0.414963\pi\)
−0.964527 + 0.263986i \(0.914963\pi\)
\(654\) 0 0
\(655\) 24.9283i 0.0380584i
\(656\) 0 0
\(657\) −333.997 −0.508367
\(658\) 0 0
\(659\) 430.079 430.079i 0.652623 0.652623i −0.301001 0.953624i \(-0.597321\pi\)
0.953624 + 0.301001i \(0.0973207\pi\)
\(660\) 0 0
\(661\) 513.622 513.622i 0.777038 0.777038i −0.202288 0.979326i \(-0.564838\pi\)
0.979326 + 0.202288i \(0.0648376\pi\)
\(662\) 0 0
\(663\) 377.287 0.569060
\(664\) 0 0
\(665\) 341.246i 0.513152i
\(666\) 0 0
\(667\) 818.105 + 818.105i 1.22654 + 1.22654i
\(668\) 0 0
\(669\) 150.162 + 150.162i 0.224457 + 0.224457i
\(670\) 0 0
\(671\) 162.872i 0.242730i
\(672\) 0 0
\(673\) −1112.68 −1.65332 −0.826659 0.562703i \(-0.809761\pi\)
−0.826659 + 0.562703i \(0.809761\pi\)
\(674\) 0 0
\(675\) −84.3935 + 84.3935i −0.125027 + 0.125027i
\(676\) 0 0
\(677\) −633.271 + 633.271i −0.935408 + 0.935408i −0.998037 0.0626291i \(-0.980051\pi\)
0.0626291 + 0.998037i \(0.480051\pi\)
\(678\) 0 0
\(679\) 1075.62 1.58412
\(680\) 0 0
\(681\) 724.668i 1.06412i
\(682\) 0 0
\(683\) 429.651 + 429.651i 0.629065 + 0.629065i 0.947833 0.318768i \(-0.103269\pi\)
−0.318768 + 0.947833i \(0.603269\pi\)
\(684\) 0 0
\(685\) 146.803 + 146.803i 0.214312 + 0.214312i
\(686\) 0 0
\(687\) 179.748i 0.261642i
\(688\) 0 0
\(689\) −346.042 −0.502239
\(690\) 0 0
\(691\) 151.617 151.617i 0.219417 0.219417i −0.588836 0.808253i \(-0.700414\pi\)
0.808253 + 0.588836i \(0.200414\pi\)
\(692\) 0 0
\(693\) −68.1153 + 68.1153i −0.0982904 + 0.0982904i
\(694\) 0 0
\(695\) 166.417 0.239449
\(696\) 0 0
\(697\) 1055.17i 1.51388i
\(698\) 0 0
\(699\) −191.340 191.340i −0.273734 0.273734i
\(700\) 0 0
\(701\) 920.704 + 920.704i 1.31341 + 1.31341i 0.918882 + 0.394533i \(0.129094\pi\)
0.394533 + 0.918882i \(0.370906\pi\)
\(702\) 0 0
\(703\) 222.982i 0.317186i
\(704\) 0 0
\(705\) −112.571 −0.159675
\(706\) 0 0
\(707\) 1010.18 1010.18i 1.42882 1.42882i
\(708\) 0 0
\(709\) −405.348 + 405.348i −0.571718 + 0.571718i −0.932608 0.360890i \(-0.882473\pi\)
0.360890 + 0.932608i \(0.382473\pi\)
\(710\) 0 0
\(711\) −13.3809 −0.0188199
\(712\) 0 0
\(713\) 1373.72i 1.92667i
\(714\) 0 0
\(715\) 31.4199 + 31.4199i 0.0439439 + 0.0439439i
\(716\) 0 0
\(717\) 16.1319 + 16.1319i 0.0224991 + 0.0224991i
\(718\) 0 0
\(719\) 880.704i 1.22490i −0.790509 0.612450i \(-0.790184\pi\)
0.790509 0.612450i \(-0.209816\pi\)
\(720\) 0 0
\(721\) 581.930 0.807115
\(722\) 0 0
\(723\) 232.112 232.112i 0.321041 0.321041i
\(724\) 0 0
\(725\) −565.983 + 565.983i −0.780667 + 0.780667i
\(726\) 0 0
\(727\) 1000.46 1.37615 0.688077 0.725637i \(-0.258455\pi\)
0.688077 + 0.725637i \(0.258455\pi\)
\(728\) 0 0
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) −1092.64 1092.64i −1.49472 1.49472i
\(732\) 0 0
\(733\) −540.306 540.306i −0.737116 0.737116i 0.234903 0.972019i \(-0.424523\pi\)
−0.972019 + 0.234903i \(0.924523\pi\)
\(734\) 0 0
\(735\) 127.053i 0.172861i
\(736\) 0 0
\(737\) 46.2301 0.0627274
\(738\) 0 0
\(739\) −893.726 + 893.726i −1.20937 + 1.20937i −0.238142 + 0.971230i \(0.576538\pi\)
−0.971230 + 0.238142i \(0.923462\pi\)
\(740\) 0 0
\(741\) −284.763 + 284.763i −0.384296 + 0.384296i
\(742\) 0 0
\(743\) 1295.75 1.74394 0.871969 0.489561i \(-0.162843\pi\)
0.871969 + 0.489561i \(0.162843\pi\)
\(744\) 0 0
\(745\) 395.527i 0.530909i
\(746\) 0 0
\(747\) 30.5253 + 30.5253i 0.0408639 + 0.0408639i
\(748\) 0 0
\(749\) −1130.13 1130.13i −1.50885 1.50885i
\(750\) 0 0
\(751\) 229.818i 0.306016i 0.988225 + 0.153008i \(0.0488961\pi\)
−0.988225 + 0.153008i \(0.951104\pi\)
\(752\) 0 0
\(753\) −67.2223 −0.0892726
\(754\) 0 0
\(755\) −64.9987 + 64.9987i −0.0860910 + 0.0860910i
\(756\) 0 0
\(757\) 373.678 373.678i 0.493630 0.493630i −0.415818 0.909448i \(-0.636505\pi\)
0.909448 + 0.415818i \(0.136505\pi\)
\(758\) 0 0
\(759\) 184.215 0.242707
\(760\) 0 0
\(761\) 384.012i 0.504615i −0.967647 0.252307i \(-0.918811\pi\)
0.967647 0.252307i \(-0.0811894\pi\)
\(762\) 0 0
\(763\) 813.296 + 813.296i 1.06592 + 1.06592i
\(764\) 0 0
\(765\) −67.6589 67.6589i −0.0884429 0.0884429i
\(766\) 0 0
\(767\) 85.8874i 0.111978i
\(768\) 0 0
\(769\) 865.026 1.12487 0.562436 0.826841i \(-0.309864\pi\)
0.562436 + 0.826841i \(0.309864\pi\)
\(770\) 0 0
\(771\) −165.800 + 165.800i −0.215045 + 0.215045i
\(772\) 0 0
\(773\) 1.78859 1.78859i 0.00231383 0.00231383i −0.705949 0.708263i \(-0.749479\pi\)
0.708263 + 0.705949i \(0.249479\pi\)
\(774\) 0 0
\(775\) −950.368 −1.22628
\(776\) 0 0
\(777\) 162.055i 0.208565i
\(778\) 0 0
\(779\) 796.409 + 796.409i 1.02235 + 1.02235i
\(780\) 0 0
\(781\) −27.0753 27.0753i −0.0346675 0.0346675i
\(782\) 0 0
\(783\) 181.075i 0.231258i
\(784\) 0 0
\(785\) 109.681 0.139721
\(786\) 0 0
\(787\) −143.702 + 143.702i −0.182595 + 0.182595i −0.792485 0.609891i \(-0.791213\pi\)
0.609891 + 0.792485i \(0.291213\pi\)
\(788\) 0 0
\(789\) −38.7170 + 38.7170i −0.0490710 + 0.0490710i
\(790\) 0 0
\(791\) −1722.49 −2.17761
\(792\) 0 0
\(793\) 494.855i 0.624029i
\(794\) 0 0
\(795\) 62.0558 + 62.0558i 0.0780577 + 0.0780577i
\(796\) 0 0
\(797\) 477.929 + 477.929i 0.599660 + 0.599660i 0.940222 0.340562i \(-0.110617\pi\)
−0.340562 + 0.940222i \(0.610617\pi\)
\(798\) 0 0
\(799\) 1020.64i 1.27740i
\(800\) 0 0
\(801\) 65.8092 0.0821588
\(802\) 0 0
\(803\) −252.187 + 252.187i −0.314056 + 0.314056i
\(804\) 0 0
\(805\) −335.359 + 335.359i −0.416595 + 0.416595i
\(806\) 0 0
\(807\) −475.722 −0.589495
\(808\) 0 0
\(809\) 1524.96i 1.88500i 0.334212 + 0.942498i \(0.391530\pi\)
−0.334212 + 0.942498i \(0.608470\pi\)
\(810\) 0 0
\(811\) 576.427 + 576.427i 0.710761 + 0.710761i 0.966694 0.255933i \(-0.0823828\pi\)
−0.255933 + 0.966694i \(0.582383\pi\)
\(812\) 0 0
\(813\) 357.194 + 357.194i 0.439353 + 0.439353i
\(814\) 0 0
\(815\) 211.224i 0.259170i
\(816\) 0 0
\(817\) 1649.37 2.01882
\(818\) 0 0
\(819\) 206.955 206.955i 0.252692 0.252692i
\(820\) 0 0
\(821\) 386.324 386.324i 0.470552 0.470552i −0.431541 0.902093i \(-0.642030\pi\)
0.902093 + 0.431541i \(0.142030\pi\)
\(822\) 0 0
\(823\) 377.870 0.459138 0.229569 0.973292i \(-0.426268\pi\)
0.229569 + 0.973292i \(0.426268\pi\)
\(824\) 0 0
\(825\) 127.444i 0.154477i
\(826\) 0 0
\(827\) −140.900 140.900i −0.170375 0.170375i 0.616769 0.787144i \(-0.288441\pi\)
−0.787144 + 0.616769i \(0.788441\pi\)
\(828\) 0 0
\(829\) 522.203 + 522.203i 0.629919 + 0.629919i 0.948048 0.318128i \(-0.103054\pi\)
−0.318128 + 0.948048i \(0.603054\pi\)
\(830\) 0 0
\(831\) 747.501i 0.899520i
\(832\) 0 0
\(833\) −1151.95 −1.38289
\(834\) 0 0
\(835\) −53.7226 + 53.7226i −0.0643385 + 0.0643385i
\(836\) 0 0
\(837\) −152.026 + 152.026i −0.181632 + 0.181632i
\(838\) 0 0
\(839\) −442.133 −0.526976 −0.263488 0.964663i \(-0.584873\pi\)
−0.263488 + 0.964663i \(0.584873\pi\)
\(840\) 0 0
\(841\) 373.376i 0.443967i
\(842\) 0 0
\(843\) −259.476 259.476i −0.307801 0.307801i
\(844\) 0 0
\(845\) 74.8417 + 74.8417i 0.0885701 + 0.0885701i
\(846\) 0 0
\(847\) 1109.99i 1.31049i
\(848\) 0 0
\(849\) 257.994 0.303879
\(850\) 0 0
\(851\) 219.135 219.135i 0.257503 0.257503i
\(852\) 0 0
\(853\) 494.617 494.617i 0.579856 0.579856i −0.355007 0.934863i \(-0.615522\pi\)
0.934863 + 0.355007i \(0.115522\pi\)
\(854\) 0 0
\(855\) 102.133 0.119454
\(856\) 0 0
\(857\) 1676.75i 1.95654i −0.207340 0.978269i \(-0.566481\pi\)
0.207340 0.978269i \(-0.433519\pi\)
\(858\) 0 0
\(859\) −228.948 228.948i −0.266529 0.266529i 0.561171 0.827700i \(-0.310351\pi\)
−0.827700 + 0.561171i \(0.810351\pi\)
\(860\) 0 0
\(861\) −578.800 578.800i −0.672241 0.672241i
\(862\) 0 0
\(863\) 1603.23i 1.85774i 0.370401 + 0.928872i \(0.379220\pi\)
−0.370401 + 0.928872i \(0.620780\pi\)
\(864\) 0 0
\(865\) −83.7981 −0.0968764
\(866\) 0 0
\(867\) −259.488 + 259.488i −0.299294 + 0.299294i
\(868\) 0 0
\(869\) −10.1034 + 10.1034i −0.0116264 + 0.0116264i
\(870\) 0 0
\(871\) −140.461 −0.161264
\(872\) 0 0
\(873\) 321.927i 0.368760i
\(874\) 0 0
\(875\) −484.533 484.533i −0.553752 0.553752i
\(876\) 0 0
\(877\) −289.017 289.017i −0.329552 0.329552i 0.522864 0.852416i \(-0.324864\pi\)
−0.852416 + 0.522864i \(0.824864\pi\)
\(878\) 0 0
\(879\) 419.570i 0.477326i
\(880\) 0 0
\(881\) 937.450 1.06408 0.532038 0.846721i \(-0.321426\pi\)
0.532038 + 0.846721i \(0.321426\pi\)
\(882\) 0 0
\(883\) 485.966 485.966i 0.550357 0.550357i −0.376187 0.926544i \(-0.622765\pi\)
0.926544 + 0.376187i \(0.122765\pi\)
\(884\) 0 0
\(885\) −15.4022 + 15.4022i −0.0174036 + 0.0174036i
\(886\) 0 0
\(887\) −534.193 −0.602247 −0.301123 0.953585i \(-0.597362\pi\)
−0.301123 + 0.953585i \(0.597362\pi\)
\(888\) 0 0
\(889\) 369.201i 0.415299i
\(890\) 0 0
\(891\) 20.3865 + 20.3865i 0.0228805 + 0.0228805i
\(892\) 0 0
\(893\) −770.346 770.346i −0.862649 0.862649i
\(894\) 0 0
\(895\) 106.961i 0.119510i
\(896\) 0 0
\(897\) −559.701 −0.623970
\(898\) 0 0
\(899\) −1019.56 + 1019.56i −1.13410 + 1.13410i
\(900\) 0 0
\(901\) 562.638 562.638i 0.624460 0.624460i
\(902\) 0 0
\(903\) −1198.70 −1.32747
\(904\) 0 0
\(905\) 134.237i 0.148328i
\(906\) 0 0
\(907\) −368.669 368.669i −0.406471 0.406471i 0.474035 0.880506i \(-0.342797\pi\)
−0.880506 + 0.474035i \(0.842797\pi\)
\(908\) 0 0
\(909\) −302.341 302.341i −0.332608 0.332608i
\(910\) 0 0
\(911\) 1592.43i 1.74800i −0.485927 0.873999i \(-0.661518\pi\)
0.485927 0.873999i \(-0.338482\pi\)
\(912\) 0 0
\(913\) 46.0967 0.0504893
\(914\) 0 0
\(915\) −88.7424 + 88.7424i −0.0969863 + 0.0969863i
\(916\) 0 0
\(917\) 123.978 123.978i 0.135199 0.135199i
\(918\) 0 0
\(919\) −403.500 −0.439064 −0.219532 0.975605i \(-0.570453\pi\)
−0.219532 + 0.975605i \(0.570453\pi\)
\(920\) 0 0
\(921\) 66.4116i 0.0721081i
\(922\) 0 0
\(923\) 82.2632 + 82.2632i 0.0891258 + 0.0891258i
\(924\) 0 0
\(925\) 151.602 + 151.602i 0.163894 + 0.163894i
\(926\) 0 0
\(927\) 174.169i 0.187884i
\(928\) 0 0
\(929\) 348.546 0.375184 0.187592 0.982247i \(-0.439932\pi\)
0.187592 + 0.982247i \(0.439932\pi\)
\(930\) 0 0
\(931\) 869.449 869.449i 0.933887 0.933887i
\(932\) 0 0
\(933\) −454.533 + 454.533i −0.487173 + 0.487173i
\(934\) 0 0
\(935\) −102.173 −0.109276
\(936\) 0 0
\(937\) 248.875i 0.265609i 0.991142 + 0.132804i \(0.0423982\pi\)
−0.991142 + 0.132804i \(0.957602\pi\)
\(938\) 0 0
\(939\) −458.669 458.669i −0.488465 0.488465i
\(940\) 0 0
\(941\) −884.188 884.188i −0.939626 0.939626i 0.0586528 0.998278i \(-0.481319\pi\)
−0.998278 + 0.0586528i \(0.981319\pi\)
\(942\) 0 0
\(943\) 1565.34i 1.65996i
\(944\) 0 0
\(945\) −74.2266 −0.0785466
\(946\) 0 0
\(947\) −462.279 + 462.279i −0.488151 + 0.488151i −0.907722 0.419571i \(-0.862180\pi\)
0.419571 + 0.907722i \(0.362180\pi\)
\(948\) 0 0
\(949\) 766.221 766.221i 0.807398 0.807398i
\(950\) 0 0
\(951\) −119.006 −0.125138
\(952\) 0 0
\(953\) 707.656i 0.742556i −0.928522 0.371278i \(-0.878920\pi\)
0.928522 0.371278i \(-0.121080\pi\)
\(954\) 0 0
\(955\) 114.632 + 114.632i 0.120033 + 0.120033i
\(956\) 0 0
\(957\) 136.722 + 136.722i 0.142865 + 0.142865i
\(958\) 0 0
\(959\) 1460.22i 1.52265i
\(960\) 0 0
\(961\) −750.983 −0.781460
\(962\) 0 0
\(963\) −338.241 + 338.241i −0.351237 + 0.351237i
\(964\) 0 0
\(965\) 26.7647 26.7647i 0.0277354 0.0277354i
\(966\) 0 0
\(967\) −841.240 −0.869949 −0.434974 0.900443i \(-0.643243\pi\)
−0.434974 + 0.900443i \(0.643243\pi\)
\(968\) 0 0
\(969\) 926.006i 0.955630i
\(970\) 0 0
\(971\) −308.415 308.415i −0.317626 0.317626i 0.530229 0.847855i \(-0.322106\pi\)
−0.847855 + 0.530229i \(0.822106\pi\)
\(972\) 0 0
\(973\) −827.654 827.654i −0.850621 0.850621i
\(974\) 0 0
\(975\) 387.213i 0.397142i
\(976\) 0 0
\(977\) −1316.85 −1.34786 −0.673928 0.738797i \(-0.735394\pi\)
−0.673928 + 0.738797i \(0.735394\pi\)
\(978\) 0 0
\(979\) 49.6897 49.6897i 0.0507556 0.0507556i
\(980\) 0 0
\(981\) 243.415 243.415i 0.248130 0.248130i
\(982\) 0 0
\(983\) 504.538 0.513263 0.256632 0.966509i \(-0.417387\pi\)
0.256632 + 0.966509i \(0.417387\pi\)
\(984\) 0 0
\(985\) 104.588i 0.106181i
\(986\) 0 0
\(987\) 559.858 + 559.858i 0.567232 + 0.567232i
\(988\) 0 0
\(989\) 1620.92 + 1620.92i 1.63895 + 1.63895i
\(990\) 0 0
\(991\) 436.650i 0.440616i 0.975430 + 0.220308i \(0.0707062\pi\)
−0.975430 + 0.220308i \(0.929294\pi\)
\(992\) 0 0
\(993\) −4.62667 −0.00465929
\(994\) 0 0
\(995\) 137.797 137.797i 0.138489 0.138489i
\(996\) 0 0
\(997\) 383.801 383.801i 0.384956 0.384956i −0.487928 0.872884i \(-0.662247\pi\)
0.872884 + 0.487928i \(0.162247\pi\)
\(998\) 0 0
\(999\) 48.5021 0.0485507
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 384.3.l.a.223.2 16
3.2 odd 2 1152.3.m.f.991.5 16
4.3 odd 2 384.3.l.b.223.6 16
8.3 odd 2 192.3.l.a.175.3 16
8.5 even 2 48.3.l.a.19.3 16
12.11 even 2 1152.3.m.c.991.5 16
16.3 odd 4 48.3.l.a.43.3 yes 16
16.5 even 4 384.3.l.b.31.6 16
16.11 odd 4 inner 384.3.l.a.31.2 16
16.13 even 4 192.3.l.a.79.3 16
24.5 odd 2 144.3.m.c.19.6 16
24.11 even 2 576.3.m.c.559.4 16
48.5 odd 4 1152.3.m.c.415.5 16
48.11 even 4 1152.3.m.f.415.5 16
48.29 odd 4 576.3.m.c.271.4 16
48.35 even 4 144.3.m.c.91.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.3.l.a.19.3 16 8.5 even 2
48.3.l.a.43.3 yes 16 16.3 odd 4
144.3.m.c.19.6 16 24.5 odd 2
144.3.m.c.91.6 16 48.35 even 4
192.3.l.a.79.3 16 16.13 even 4
192.3.l.a.175.3 16 8.3 odd 2
384.3.l.a.31.2 16 16.11 odd 4 inner
384.3.l.a.223.2 16 1.1 even 1 trivial
384.3.l.b.31.6 16 16.5 even 4
384.3.l.b.223.6 16 4.3 odd 2
576.3.m.c.271.4 16 48.29 odd 4
576.3.m.c.559.4 16 24.11 even 2
1152.3.m.c.415.5 16 48.5 odd 4
1152.3.m.c.991.5 16 12.11 even 2
1152.3.m.f.415.5 16 48.11 even 4
1152.3.m.f.991.5 16 3.2 odd 2