Properties

Label 225.3.g.b.82.1
Level $225$
Weight $3$
Character 225.82
Analytic conductor $6.131$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,3,Mod(82,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.82");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 225.g (of order \(4\), degree \(2\), 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: \(6.13080594811\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 82.1
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 225.82
Dual form 225.3.g.b.118.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.44949 - 2.44949i) q^{2} +8.00000i q^{4} +(-6.12372 - 6.12372i) q^{7} +(9.79796 - 9.79796i) q^{8} -6.00000 q^{11} +(-3.67423 + 3.67423i) q^{13} +30.0000i q^{14} -16.0000 q^{16} +(17.1464 + 17.1464i) q^{17} +23.0000i q^{19} +(14.6969 + 14.6969i) q^{22} +(-12.2474 + 12.2474i) q^{23} +18.0000 q^{26} +(48.9898 - 48.9898i) q^{28} -6.00000i q^{29} +25.0000 q^{31} -84.0000i q^{34} +(24.4949 + 24.4949i) q^{37} +(56.3383 - 56.3383i) q^{38} +60.0000 q^{41} +(-60.0125 + 60.0125i) q^{43} -48.0000i q^{44} +60.0000 q^{46} +(7.34847 + 7.34847i) q^{47} +26.0000i q^{49} +(-29.3939 - 29.3939i) q^{52} +(-24.4949 + 24.4949i) q^{53} -120.000 q^{56} +(-14.6969 + 14.6969i) q^{58} +18.0000i q^{59} -37.0000 q^{61} +(-61.2372 - 61.2372i) q^{62} +64.0000i q^{64} +(25.7196 + 25.7196i) q^{67} +(-137.171 + 137.171i) q^{68} -132.000 q^{71} +(24.4949 - 24.4949i) q^{73} -120.000i q^{74} -184.000 q^{76} +(36.7423 + 36.7423i) q^{77} +10.0000i q^{79} +(-146.969 - 146.969i) q^{82} +(2.44949 - 2.44949i) q^{83} +294.000 q^{86} +(-58.7878 + 58.7878i) q^{88} +132.000i q^{89} +45.0000 q^{91} +(-97.9796 - 97.9796i) q^{92} -36.0000i q^{94} +(-23.2702 - 23.2702i) q^{97} +(63.6867 - 63.6867i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 24 q^{11} - 64 q^{16} + 72 q^{26} + 100 q^{31} + 240 q^{41} + 240 q^{46} - 480 q^{56} - 148 q^{61} - 528 q^{71} - 736 q^{76} + 1176 q^{86} + 180 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.44949 2.44949i −1.22474 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(3\) 0 0
\(4\) 8.00000i 2.00000i
\(5\) 0 0
\(6\) 0 0
\(7\) −6.12372 6.12372i −0.874818 0.874818i 0.118175 0.992993i \(-0.462296\pi\)
−0.992993 + 0.118175i \(0.962296\pi\)
\(8\) 9.79796 9.79796i 1.22474 1.22474i
\(9\) 0 0
\(10\) 0 0
\(11\) −6.00000 −0.545455 −0.272727 0.962091i \(-0.587926\pi\)
−0.272727 + 0.962091i \(0.587926\pi\)
\(12\) 0 0
\(13\) −3.67423 + 3.67423i −0.282633 + 0.282633i −0.834158 0.551525i \(-0.814046\pi\)
0.551525 + 0.834158i \(0.314046\pi\)
\(14\) 30.0000i 2.14286i
\(15\) 0 0
\(16\) −16.0000 −1.00000
\(17\) 17.1464 + 17.1464i 1.00861 + 1.00861i 0.999963 + 0.00865084i \(0.00275368\pi\)
0.00865084 + 0.999963i \(0.497246\pi\)
\(18\) 0 0
\(19\) 23.0000i 1.21053i 0.796025 + 0.605263i \(0.206932\pi\)
−0.796025 + 0.605263i \(0.793068\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 14.6969 + 14.6969i 0.668043 + 0.668043i
\(23\) −12.2474 + 12.2474i −0.532498 + 0.532498i −0.921315 0.388817i \(-0.872884\pi\)
0.388817 + 0.921315i \(0.372884\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 18.0000 0.692308
\(27\) 0 0
\(28\) 48.9898 48.9898i 1.74964 1.74964i
\(29\) 6.00000i 0.206897i −0.994635 0.103448i \(-0.967012\pi\)
0.994635 0.103448i \(-0.0329876\pi\)
\(30\) 0 0
\(31\) 25.0000 0.806452 0.403226 0.915101i \(-0.367889\pi\)
0.403226 + 0.915101i \(0.367889\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 84.0000i 2.47059i
\(35\) 0 0
\(36\) 0 0
\(37\) 24.4949 + 24.4949i 0.662024 + 0.662024i 0.955857 0.293833i \(-0.0949308\pi\)
−0.293833 + 0.955857i \(0.594931\pi\)
\(38\) 56.3383 56.3383i 1.48259 1.48259i
\(39\) 0 0
\(40\) 0 0
\(41\) 60.0000 1.46341 0.731707 0.681619i \(-0.238724\pi\)
0.731707 + 0.681619i \(0.238724\pi\)
\(42\) 0 0
\(43\) −60.0125 + 60.0125i −1.39564 + 1.39564i −0.583594 + 0.812046i \(0.698354\pi\)
−0.812046 + 0.583594i \(0.801646\pi\)
\(44\) 48.0000i 1.09091i
\(45\) 0 0
\(46\) 60.0000 1.30435
\(47\) 7.34847 + 7.34847i 0.156350 + 0.156350i 0.780947 0.624597i \(-0.214737\pi\)
−0.624597 + 0.780947i \(0.714737\pi\)
\(48\) 0 0
\(49\) 26.0000i 0.530612i
\(50\) 0 0
\(51\) 0 0
\(52\) −29.3939 29.3939i −0.565267 0.565267i
\(53\) −24.4949 + 24.4949i −0.462168 + 0.462168i −0.899365 0.437198i \(-0.855971\pi\)
0.437198 + 0.899365i \(0.355971\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −120.000 −2.14286
\(57\) 0 0
\(58\) −14.6969 + 14.6969i −0.253395 + 0.253395i
\(59\) 18.0000i 0.305085i 0.988297 + 0.152542i \(0.0487461\pi\)
−0.988297 + 0.152542i \(0.951254\pi\)
\(60\) 0 0
\(61\) −37.0000 −0.606557 −0.303279 0.952902i \(-0.598081\pi\)
−0.303279 + 0.952902i \(0.598081\pi\)
\(62\) −61.2372 61.2372i −0.987697 0.987697i
\(63\) 0 0
\(64\) 64.0000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 25.7196 + 25.7196i 0.383875 + 0.383875i 0.872496 0.488621i \(-0.162500\pi\)
−0.488621 + 0.872496i \(0.662500\pi\)
\(68\) −137.171 + 137.171i −2.01723 + 2.01723i
\(69\) 0 0
\(70\) 0 0
\(71\) −132.000 −1.85915 −0.929577 0.368627i \(-0.879828\pi\)
−0.929577 + 0.368627i \(0.879828\pi\)
\(72\) 0 0
\(73\) 24.4949 24.4949i 0.335547 0.335547i −0.519142 0.854688i \(-0.673748\pi\)
0.854688 + 0.519142i \(0.173748\pi\)
\(74\) 120.000i 1.62162i
\(75\) 0 0
\(76\) −184.000 −2.42105
\(77\) 36.7423 + 36.7423i 0.477173 + 0.477173i
\(78\) 0 0
\(79\) 10.0000i 0.126582i 0.997995 + 0.0632911i \(0.0201597\pi\)
−0.997995 + 0.0632911i \(0.979840\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −146.969 146.969i −1.79231 1.79231i
\(83\) 2.44949 2.44949i 0.0295119 0.0295119i −0.692197 0.721709i \(-0.743357\pi\)
0.721709 + 0.692197i \(0.243357\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 294.000 3.41860
\(87\) 0 0
\(88\) −58.7878 + 58.7878i −0.668043 + 0.668043i
\(89\) 132.000i 1.48315i 0.670872 + 0.741573i \(0.265920\pi\)
−0.670872 + 0.741573i \(0.734080\pi\)
\(90\) 0 0
\(91\) 45.0000 0.494505
\(92\) −97.9796 97.9796i −1.06500 1.06500i
\(93\) 0 0
\(94\) 36.0000i 0.382979i
\(95\) 0 0
\(96\) 0 0
\(97\) −23.2702 23.2702i −0.239898 0.239898i 0.576910 0.816808i \(-0.304259\pi\)
−0.816808 + 0.576910i \(0.804259\pi\)
\(98\) 63.6867 63.6867i 0.649865 0.649865i
\(99\) 0 0
\(100\) 0 0
\(101\) −96.0000 −0.950495 −0.475248 0.879852i \(-0.657641\pi\)
−0.475248 + 0.879852i \(0.657641\pi\)
\(102\) 0 0
\(103\) 19.5959 19.5959i 0.190252 0.190252i −0.605553 0.795805i \(-0.707048\pi\)
0.795805 + 0.605553i \(0.207048\pi\)
\(104\) 72.0000i 0.692308i
\(105\) 0 0
\(106\) 120.000 1.13208
\(107\) −29.3939 29.3939i −0.274709 0.274709i 0.556283 0.830993i \(-0.312227\pi\)
−0.830993 + 0.556283i \(0.812227\pi\)
\(108\) 0 0
\(109\) 167.000i 1.53211i −0.642775 0.766055i \(-0.722217\pi\)
0.642775 0.766055i \(-0.277783\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 97.9796 + 97.9796i 0.874818 + 0.874818i
\(113\) −58.7878 + 58.7878i −0.520246 + 0.520246i −0.917646 0.397400i \(-0.869913\pi\)
0.397400 + 0.917646i \(0.369913\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 48.0000 0.413793
\(117\) 0 0
\(118\) 44.0908 44.0908i 0.373651 0.373651i
\(119\) 210.000i 1.76471i
\(120\) 0 0
\(121\) −85.0000 −0.702479
\(122\) 90.6311 + 90.6311i 0.742878 + 0.742878i
\(123\) 0 0
\(124\) 200.000i 1.61290i
\(125\) 0 0
\(126\) 0 0
\(127\) 44.0908 + 44.0908i 0.347172 + 0.347172i 0.859055 0.511883i \(-0.171052\pi\)
−0.511883 + 0.859055i \(0.671052\pi\)
\(128\) 156.767 156.767i 1.22474 1.22474i
\(129\) 0 0
\(130\) 0 0
\(131\) −108.000 −0.824427 −0.412214 0.911087i \(-0.635244\pi\)
−0.412214 + 0.911087i \(0.635244\pi\)
\(132\) 0 0
\(133\) 140.846 140.846i 1.05899 1.05899i
\(134\) 126.000i 0.940299i
\(135\) 0 0
\(136\) 336.000 2.47059
\(137\) 120.025 + 120.025i 0.876095 + 0.876095i 0.993128 0.117033i \(-0.0373384\pi\)
−0.117033 + 0.993128i \(0.537338\pi\)
\(138\) 0 0
\(139\) 58.0000i 0.417266i −0.977994 0.208633i \(-0.933099\pi\)
0.977994 0.208633i \(-0.0669014\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 323.333 + 323.333i 2.27699 + 2.27699i
\(143\) 22.0454 22.0454i 0.154164 0.154164i
\(144\) 0 0
\(145\) 0 0
\(146\) −120.000 −0.821918
\(147\) 0 0
\(148\) −195.959 + 195.959i −1.32405 + 1.32405i
\(149\) 186.000i 1.24832i 0.781296 + 0.624161i \(0.214559\pi\)
−0.781296 + 0.624161i \(0.785441\pi\)
\(150\) 0 0
\(151\) −83.0000 −0.549669 −0.274834 0.961492i \(-0.588623\pi\)
−0.274834 + 0.961492i \(0.588623\pi\)
\(152\) 225.353 + 225.353i 1.48259 + 1.48259i
\(153\) 0 0
\(154\) 180.000i 1.16883i
\(155\) 0 0
\(156\) 0 0
\(157\) 45.3156 + 45.3156i 0.288634 + 0.288634i 0.836540 0.547906i \(-0.184575\pi\)
−0.547906 + 0.836540i \(0.684575\pi\)
\(158\) 24.4949 24.4949i 0.155031 0.155031i
\(159\) 0 0
\(160\) 0 0
\(161\) 150.000 0.931677
\(162\) 0 0
\(163\) −99.2043 + 99.2043i −0.608616 + 0.608616i −0.942584 0.333969i \(-0.891612\pi\)
0.333969 + 0.942584i \(0.391612\pi\)
\(164\) 480.000i 2.92683i
\(165\) 0 0
\(166\) −12.0000 −0.0722892
\(167\) −97.9796 97.9796i −0.586704 0.586704i 0.350033 0.936737i \(-0.386170\pi\)
−0.936737 + 0.350033i \(0.886170\pi\)
\(168\) 0 0
\(169\) 142.000i 0.840237i
\(170\) 0 0
\(171\) 0 0
\(172\) −480.100 480.100i −2.79128 2.79128i
\(173\) 173.914 173.914i 1.00528 1.00528i 0.00529594 0.999986i \(-0.498314\pi\)
0.999986 0.00529594i \(-0.00168576\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 96.0000 0.545455
\(177\) 0 0
\(178\) 323.333 323.333i 1.81648 1.81648i
\(179\) 150.000i 0.837989i −0.907989 0.418994i \(-0.862383\pi\)
0.907989 0.418994i \(-0.137617\pi\)
\(180\) 0 0
\(181\) −215.000 −1.18785 −0.593923 0.804522i \(-0.702421\pi\)
−0.593923 + 0.804522i \(0.702421\pi\)
\(182\) −110.227 110.227i −0.605643 0.605643i
\(183\) 0 0
\(184\) 240.000i 1.30435i
\(185\) 0 0
\(186\) 0 0
\(187\) −102.879 102.879i −0.550153 0.550153i
\(188\) −58.7878 + 58.7878i −0.312701 + 0.312701i
\(189\) 0 0
\(190\) 0 0
\(191\) 234.000 1.22513 0.612565 0.790420i \(-0.290138\pi\)
0.612565 + 0.790420i \(0.290138\pi\)
\(192\) 0 0
\(193\) 64.9115 64.9115i 0.336329 0.336329i −0.518655 0.854984i \(-0.673567\pi\)
0.854984 + 0.518655i \(0.173567\pi\)
\(194\) 114.000i 0.587629i
\(195\) 0 0
\(196\) −208.000 −1.06122
\(197\) −193.510 193.510i −0.982283 0.982283i 0.0175631 0.999846i \(-0.494409\pi\)
−0.999846 + 0.0175631i \(0.994409\pi\)
\(198\) 0 0
\(199\) 11.0000i 0.0552764i −0.999618 0.0276382i \(-0.991201\pi\)
0.999618 0.0276382i \(-0.00879863\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 235.151 + 235.151i 1.16411 + 1.16411i
\(203\) −36.7423 + 36.7423i −0.180997 + 0.180997i
\(204\) 0 0
\(205\) 0 0
\(206\) −96.0000 −0.466019
\(207\) 0 0
\(208\) 58.7878 58.7878i 0.282633 0.282633i
\(209\) 138.000i 0.660287i
\(210\) 0 0
\(211\) −85.0000 −0.402844 −0.201422 0.979505i \(-0.564556\pi\)
−0.201422 + 0.979505i \(0.564556\pi\)
\(212\) −195.959 195.959i −0.924336 0.924336i
\(213\) 0 0
\(214\) 144.000i 0.672897i
\(215\) 0 0
\(216\) 0 0
\(217\) −153.093 153.093i −0.705498 0.705498i
\(218\) −409.065 + 409.065i −1.87644 + 1.87644i
\(219\) 0 0
\(220\) 0 0
\(221\) −126.000 −0.570136
\(222\) 0 0
\(223\) −101.654 + 101.654i −0.455847 + 0.455847i −0.897289 0.441443i \(-0.854467\pi\)
0.441443 + 0.897289i \(0.354467\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 288.000 1.27434
\(227\) 195.959 + 195.959i 0.863256 + 0.863256i 0.991715 0.128459i \(-0.0410029\pi\)
−0.128459 + 0.991715i \(0.541003\pi\)
\(228\) 0 0
\(229\) 227.000i 0.991266i 0.868532 + 0.495633i \(0.165064\pi\)
−0.868532 + 0.495633i \(0.834936\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −58.7878 58.7878i −0.253395 0.253395i
\(233\) 102.879 102.879i 0.441539 0.441539i −0.450990 0.892529i \(-0.648929\pi\)
0.892529 + 0.450990i \(0.148929\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −144.000 −0.610169
\(237\) 0 0
\(238\) −514.393 + 514.393i −2.16131 + 2.16131i
\(239\) 228.000i 0.953975i −0.878910 0.476987i \(-0.841729\pi\)
0.878910 0.476987i \(-0.158271\pi\)
\(240\) 0 0
\(241\) 191.000 0.792531 0.396266 0.918136i \(-0.370306\pi\)
0.396266 + 0.918136i \(0.370306\pi\)
\(242\) 208.207 + 208.207i 0.860358 + 0.860358i
\(243\) 0 0
\(244\) 296.000i 1.21311i
\(245\) 0 0
\(246\) 0 0
\(247\) −84.5074 84.5074i −0.342135 0.342135i
\(248\) 244.949 244.949i 0.987697 0.987697i
\(249\) 0 0
\(250\) 0 0
\(251\) −192.000 −0.764940 −0.382470 0.923968i \(-0.624927\pi\)
−0.382470 + 0.923968i \(0.624927\pi\)
\(252\) 0 0
\(253\) 73.4847 73.4847i 0.290453 0.290453i
\(254\) 216.000i 0.850394i
\(255\) 0 0
\(256\) −512.000 −2.00000
\(257\) 142.070 + 142.070i 0.552803 + 0.552803i 0.927249 0.374446i \(-0.122167\pi\)
−0.374446 + 0.927249i \(0.622167\pi\)
\(258\) 0 0
\(259\) 300.000i 1.15830i
\(260\) 0 0
\(261\) 0 0
\(262\) 264.545 + 264.545i 1.00971 + 1.00971i
\(263\) −249.848 + 249.848i −0.949992 + 0.949992i −0.998808 0.0488156i \(-0.984455\pi\)
0.0488156 + 0.998808i \(0.484455\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −690.000 −2.59398
\(267\) 0 0
\(268\) −205.757 + 205.757i −0.767751 + 0.767751i
\(269\) 462.000i 1.71747i 0.512418 + 0.858736i \(0.328750\pi\)
−0.512418 + 0.858736i \(0.671250\pi\)
\(270\) 0 0
\(271\) 446.000 1.64576 0.822878 0.568218i \(-0.192367\pi\)
0.822878 + 0.568218i \(0.192367\pi\)
\(272\) −274.343 274.343i −1.00861 1.00861i
\(273\) 0 0
\(274\) 588.000i 2.14599i
\(275\) 0 0
\(276\) 0 0
\(277\) 175.139 + 175.139i 0.632269 + 0.632269i 0.948637 0.316368i \(-0.102463\pi\)
−0.316368 + 0.948637i \(0.602463\pi\)
\(278\) −142.070 + 142.070i −0.511045 + 0.511045i
\(279\) 0 0
\(280\) 0 0
\(281\) 546.000 1.94306 0.971530 0.236916i \(-0.0761365\pi\)
0.971530 + 0.236916i \(0.0761365\pi\)
\(282\) 0 0
\(283\) 192.285 192.285i 0.679452 0.679452i −0.280424 0.959876i \(-0.590475\pi\)
0.959876 + 0.280424i \(0.0904751\pi\)
\(284\) 1056.00i 3.71831i
\(285\) 0 0
\(286\) −108.000 −0.377622
\(287\) −367.423 367.423i −1.28022 1.28022i
\(288\) 0 0
\(289\) 299.000i 1.03460i
\(290\) 0 0
\(291\) 0 0
\(292\) 195.959 + 195.959i 0.671093 + 0.671093i
\(293\) 276.792 276.792i 0.944684 0.944684i −0.0538645 0.998548i \(-0.517154\pi\)
0.998548 + 0.0538645i \(0.0171539\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 480.000 1.62162
\(297\) 0 0
\(298\) 455.605 455.605i 1.52888 1.52888i
\(299\) 90.0000i 0.301003i
\(300\) 0 0
\(301\) 735.000 2.44186
\(302\) 203.308 + 203.308i 0.673204 + 0.673204i
\(303\) 0 0
\(304\) 368.000i 1.21053i
\(305\) 0 0
\(306\) 0 0
\(307\) −86.9569 86.9569i −0.283247 0.283247i 0.551155 0.834403i \(-0.314187\pi\)
−0.834403 + 0.551155i \(0.814187\pi\)
\(308\) −293.939 + 293.939i −0.954347 + 0.954347i
\(309\) 0 0
\(310\) 0 0
\(311\) 294.000 0.945338 0.472669 0.881240i \(-0.343291\pi\)
0.472669 + 0.881240i \(0.343291\pi\)
\(312\) 0 0
\(313\) −393.143 + 393.143i −1.25605 + 1.25605i −0.303085 + 0.952964i \(0.598016\pi\)
−0.952964 + 0.303085i \(0.901984\pi\)
\(314\) 222.000i 0.707006i
\(315\) 0 0
\(316\) −80.0000 −0.253165
\(317\) 93.0806 + 93.0806i 0.293630 + 0.293630i 0.838512 0.544883i \(-0.183426\pi\)
−0.544883 + 0.838512i \(0.683426\pi\)
\(318\) 0 0
\(319\) 36.0000i 0.112853i
\(320\) 0 0
\(321\) 0 0
\(322\) −367.423 367.423i −1.14107 1.14107i
\(323\) −394.368 + 394.368i −1.22095 + 1.22095i
\(324\) 0 0
\(325\) 0 0
\(326\) 486.000 1.49080
\(327\) 0 0
\(328\) 587.878 587.878i 1.79231 1.79231i
\(329\) 90.0000i 0.273556i
\(330\) 0 0
\(331\) −178.000 −0.537764 −0.268882 0.963173i \(-0.586654\pi\)
−0.268882 + 0.963173i \(0.586654\pi\)
\(332\) 19.5959 + 19.5959i 0.0590238 + 0.0590238i
\(333\) 0 0
\(334\) 480.000i 1.43713i
\(335\) 0 0
\(336\) 0 0
\(337\) 150.644 + 150.644i 0.447014 + 0.447014i 0.894361 0.447347i \(-0.147631\pi\)
−0.447347 + 0.894361i \(0.647631\pi\)
\(338\) 347.828 347.828i 1.02908 1.02908i
\(339\) 0 0
\(340\) 0 0
\(341\) −150.000 −0.439883
\(342\) 0 0
\(343\) −140.846 + 140.846i −0.410629 + 0.410629i
\(344\) 1176.00i 3.41860i
\(345\) 0 0
\(346\) −852.000 −2.46243
\(347\) −374.772 374.772i −1.08003 1.08003i −0.996505 0.0835290i \(-0.973381\pi\)
−0.0835290 0.996505i \(-0.526619\pi\)
\(348\) 0 0
\(349\) 514.000i 1.47278i −0.676558 0.736390i \(-0.736529\pi\)
0.676558 0.736390i \(-0.263471\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 242.499 242.499i 0.686967 0.686967i −0.274593 0.961561i \(-0.588543\pi\)
0.961561 + 0.274593i \(0.0885432\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1056.00 −2.96629
\(357\) 0 0
\(358\) −367.423 + 367.423i −1.02632 + 1.02632i
\(359\) 648.000i 1.80501i 0.430675 + 0.902507i \(0.358275\pi\)
−0.430675 + 0.902507i \(0.641725\pi\)
\(360\) 0 0
\(361\) −168.000 −0.465374
\(362\) 526.640 + 526.640i 1.45481 + 1.45481i
\(363\) 0 0
\(364\) 360.000i 0.989011i
\(365\) 0 0
\(366\) 0 0
\(367\) −143.295 143.295i −0.390450 0.390450i 0.484398 0.874848i \(-0.339039\pi\)
−0.874848 + 0.484398i \(0.839039\pi\)
\(368\) 195.959 195.959i 0.532498 0.532498i
\(369\) 0 0
\(370\) 0 0
\(371\) 300.000 0.808625
\(372\) 0 0
\(373\) −35.5176 + 35.5176i −0.0952215 + 0.0952215i −0.753113 0.657891i \(-0.771449\pi\)
0.657891 + 0.753113i \(0.271449\pi\)
\(374\) 504.000i 1.34759i
\(375\) 0 0
\(376\) 144.000 0.382979
\(377\) 22.0454 + 22.0454i 0.0584759 + 0.0584759i
\(378\) 0 0
\(379\) 215.000i 0.567282i 0.958930 + 0.283641i \(0.0915425\pi\)
−0.958930 + 0.283641i \(0.908458\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −573.181 573.181i −1.50047 1.50047i
\(383\) 19.5959 19.5959i 0.0511643 0.0511643i −0.681062 0.732226i \(-0.738481\pi\)
0.732226 + 0.681062i \(0.238481\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −318.000 −0.823834
\(387\) 0 0
\(388\) 186.161 186.161i 0.479797 0.479797i
\(389\) 192.000i 0.493573i −0.969070 0.246787i \(-0.920625\pi\)
0.969070 0.246787i \(-0.0793747\pi\)
\(390\) 0 0
\(391\) −420.000 −1.07417
\(392\) 254.747 + 254.747i 0.649865 + 0.649865i
\(393\) 0 0
\(394\) 948.000i 2.40609i
\(395\) 0 0
\(396\) 0 0
\(397\) 366.199 + 366.199i 0.922415 + 0.922415i 0.997200 0.0747848i \(-0.0238270\pi\)
−0.0747848 + 0.997200i \(0.523827\pi\)
\(398\) −26.9444 + 26.9444i −0.0676995 + 0.0676995i
\(399\) 0 0
\(400\) 0 0
\(401\) −228.000 −0.568579 −0.284289 0.958739i \(-0.591758\pi\)
−0.284289 + 0.958739i \(0.591758\pi\)
\(402\) 0 0
\(403\) −91.8559 + 91.8559i −0.227930 + 0.227930i
\(404\) 768.000i 1.90099i
\(405\) 0 0
\(406\) 180.000 0.443350
\(407\) −146.969 146.969i −0.361104 0.361104i
\(408\) 0 0
\(409\) 707.000i 1.72861i 0.502971 + 0.864303i \(0.332240\pi\)
−0.502971 + 0.864303i \(0.667760\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 156.767 + 156.767i 0.380503 + 0.380503i
\(413\) 110.227 110.227i 0.266894 0.266894i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) −338.030 + 338.030i −0.808683 + 0.808683i
\(419\) 48.0000i 0.114558i −0.998358 0.0572792i \(-0.981757\pi\)
0.998358 0.0572792i \(-0.0182425\pi\)
\(420\) 0 0
\(421\) 514.000 1.22090 0.610451 0.792054i \(-0.290988\pi\)
0.610451 + 0.792054i \(0.290988\pi\)
\(422\) 208.207 + 208.207i 0.493381 + 0.493381i
\(423\) 0 0
\(424\) 480.000i 1.13208i
\(425\) 0 0
\(426\) 0 0
\(427\) 226.578 + 226.578i 0.530627 + 0.530627i
\(428\) 235.151 235.151i 0.549418 0.549418i
\(429\) 0 0
\(430\) 0 0
\(431\) −30.0000 −0.0696056 −0.0348028 0.999394i \(-0.511080\pi\)
−0.0348028 + 0.999394i \(0.511080\pi\)
\(432\) 0 0
\(433\) 353.951 353.951i 0.817439 0.817439i −0.168297 0.985736i \(-0.553827\pi\)
0.985736 + 0.168297i \(0.0538267\pi\)
\(434\) 750.000i 1.72811i
\(435\) 0 0
\(436\) 1336.00 3.06422
\(437\) −281.691 281.691i −0.644603 0.644603i
\(438\) 0 0
\(439\) 575.000i 1.30979i −0.755718 0.654897i \(-0.772712\pi\)
0.755718 0.654897i \(-0.227288\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 308.636 + 308.636i 0.698271 + 0.698271i
\(443\) −112.677 + 112.677i −0.254349 + 0.254349i −0.822751 0.568402i \(-0.807562\pi\)
0.568402 + 0.822751i \(0.307562\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 498.000 1.11659
\(447\) 0 0
\(448\) 391.918 391.918i 0.874818 0.874818i
\(449\) 204.000i 0.454343i −0.973855 0.227171i \(-0.927052\pi\)
0.973855 0.227171i \(-0.0729478\pi\)
\(450\) 0 0
\(451\) −360.000 −0.798226
\(452\) −470.302 470.302i −1.04049 1.04049i
\(453\) 0 0
\(454\) 960.000i 2.11454i
\(455\) 0 0
\(456\) 0 0
\(457\) −440.908 440.908i −0.964788 0.964788i 0.0346127 0.999401i \(-0.488980\pi\)
−0.999401 + 0.0346127i \(0.988980\pi\)
\(458\) 556.034 556.034i 1.21405 1.21405i
\(459\) 0 0
\(460\) 0 0
\(461\) −132.000 −0.286334 −0.143167 0.989699i \(-0.545729\pi\)
−0.143167 + 0.989699i \(0.545729\pi\)
\(462\) 0 0
\(463\) 436.009 436.009i 0.941704 0.941704i −0.0566875 0.998392i \(-0.518054\pi\)
0.998392 + 0.0566875i \(0.0180539\pi\)
\(464\) 96.0000i 0.206897i
\(465\) 0 0
\(466\) −504.000 −1.08155
\(467\) 276.792 + 276.792i 0.592703 + 0.592703i 0.938361 0.345658i \(-0.112344\pi\)
−0.345658 + 0.938361i \(0.612344\pi\)
\(468\) 0 0
\(469\) 315.000i 0.671642i
\(470\) 0 0
\(471\) 0 0
\(472\) 176.363 + 176.363i 0.373651 + 0.373651i
\(473\) 360.075 360.075i 0.761258 0.761258i
\(474\) 0 0
\(475\) 0 0
\(476\) 1680.00 3.52941
\(477\) 0 0
\(478\) −558.484 + 558.484i −1.16838 + 1.16838i
\(479\) 810.000i 1.69102i −0.533957 0.845511i \(-0.679296\pi\)
0.533957 0.845511i \(-0.320704\pi\)
\(480\) 0 0
\(481\) −180.000 −0.374220
\(482\) −467.853 467.853i −0.970648 0.970648i
\(483\) 0 0
\(484\) 680.000i 1.40496i
\(485\) 0 0
\(486\) 0 0
\(487\) −488.673 488.673i −1.00344 1.00344i −0.999994 0.00344166i \(-0.998904\pi\)
−0.00344166 0.999994i \(-0.501096\pi\)
\(488\) −362.524 + 362.524i −0.742878 + 0.742878i
\(489\) 0 0
\(490\) 0 0
\(491\) −348.000 −0.708758 −0.354379 0.935102i \(-0.615308\pi\)
−0.354379 + 0.935102i \(0.615308\pi\)
\(492\) 0 0
\(493\) 102.879 102.879i 0.208679 0.208679i
\(494\) 414.000i 0.838057i
\(495\) 0 0
\(496\) −400.000 −0.806452
\(497\) 808.332 + 808.332i 1.62642 + 1.62642i
\(498\) 0 0
\(499\) 227.000i 0.454910i −0.973789 0.227455i \(-0.926960\pi\)
0.973789 0.227455i \(-0.0730404\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 470.302 + 470.302i 0.936857 + 0.936857i
\(503\) −137.171 + 137.171i −0.272707 + 0.272707i −0.830189 0.557482i \(-0.811767\pi\)
0.557482 + 0.830189i \(0.311767\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −360.000 −0.711462
\(507\) 0 0
\(508\) −352.727 + 352.727i −0.694344 + 0.694344i
\(509\) 618.000i 1.21415i 0.794646 + 0.607073i \(0.207656\pi\)
−0.794646 + 0.607073i \(0.792344\pi\)
\(510\) 0 0
\(511\) −300.000 −0.587084
\(512\) 627.069 + 627.069i 1.22474 + 1.22474i
\(513\) 0 0
\(514\) 696.000i 1.35409i
\(515\) 0 0
\(516\) 0 0
\(517\) −44.0908 44.0908i −0.0852820 0.0852820i
\(518\) −734.847 + 734.847i −1.41862 + 1.41862i
\(519\) 0 0
\(520\) 0 0
\(521\) −690.000 −1.32438 −0.662188 0.749338i \(-0.730372\pi\)
−0.662188 + 0.749338i \(0.730372\pi\)
\(522\) 0 0
\(523\) −255.972 + 255.972i −0.489430 + 0.489430i −0.908126 0.418697i \(-0.862487\pi\)
0.418697 + 0.908126i \(0.362487\pi\)
\(524\) 864.000i 1.64885i
\(525\) 0 0
\(526\) 1224.00 2.32700
\(527\) 428.661 + 428.661i 0.813398 + 0.813398i
\(528\) 0 0
\(529\) 229.000i 0.432892i
\(530\) 0 0
\(531\) 0 0
\(532\) 1126.77 + 1126.77i 2.11798 + 2.11798i
\(533\) −220.454 + 220.454i −0.413610 + 0.413610i
\(534\) 0 0
\(535\) 0 0
\(536\) 504.000 0.940299
\(537\) 0 0
\(538\) 1131.66 1131.66i 2.10347 2.10347i
\(539\) 156.000i 0.289425i
\(540\) 0 0
\(541\) −325.000 −0.600739 −0.300370 0.953823i \(-0.597110\pi\)
−0.300370 + 0.953823i \(0.597110\pi\)
\(542\) −1092.47 1092.47i −2.01563 2.01563i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 489.898 + 489.898i 0.895609 + 0.895609i 0.995044 0.0994353i \(-0.0317036\pi\)
−0.0994353 + 0.995044i \(0.531704\pi\)
\(548\) −960.200 + 960.200i −1.75219 + 1.75219i
\(549\) 0 0
\(550\) 0 0
\(551\) 138.000 0.250454
\(552\) 0 0
\(553\) 61.2372 61.2372i 0.110736 0.110736i
\(554\) 858.000i 1.54874i
\(555\) 0 0
\(556\) 464.000 0.834532
\(557\) −154.318 154.318i −0.277052 0.277052i 0.554879 0.831931i \(-0.312764\pi\)
−0.831931 + 0.554879i \(0.812764\pi\)
\(558\) 0 0
\(559\) 441.000i 0.788909i
\(560\) 0 0
\(561\) 0 0
\(562\) −1337.42 1337.42i −2.37975 2.37975i
\(563\) −609.923 + 609.923i −1.08334 + 1.08334i −0.0871492 + 0.996195i \(0.527776\pi\)
−0.996195 + 0.0871492i \(0.972224\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −942.000 −1.66431
\(567\) 0 0
\(568\) −1293.33 + 1293.33i −2.27699 + 2.27699i
\(569\) 198.000i 0.347979i −0.984748 0.173989i \(-0.944334\pi\)
0.984748 0.173989i \(-0.0556659\pi\)
\(570\) 0 0
\(571\) −169.000 −0.295972 −0.147986 0.988989i \(-0.547279\pi\)
−0.147986 + 0.988989i \(0.547279\pi\)
\(572\) 176.363 + 176.363i 0.308327 + 0.308327i
\(573\) 0 0
\(574\) 1800.00i 3.13589i
\(575\) 0 0
\(576\) 0 0
\(577\) −243.724 243.724i −0.422399 0.422399i 0.463630 0.886029i \(-0.346547\pi\)
−0.886029 + 0.463630i \(0.846547\pi\)
\(578\) 732.397 732.397i 1.26712 1.26712i
\(579\) 0 0
\(580\) 0 0
\(581\) −30.0000 −0.0516351
\(582\) 0 0
\(583\) 146.969 146.969i 0.252092 0.252092i
\(584\) 480.000i 0.821918i
\(585\) 0 0
\(586\) −1356.00 −2.31399
\(587\) 289.040 + 289.040i 0.492402 + 0.492402i 0.909062 0.416660i \(-0.136800\pi\)
−0.416660 + 0.909062i \(0.636800\pi\)
\(588\) 0 0
\(589\) 575.000i 0.976231i
\(590\) 0 0
\(591\) 0 0
\(592\) −391.918 391.918i −0.662024 0.662024i
\(593\) 421.312 421.312i 0.710476 0.710476i −0.256159 0.966635i \(-0.582457\pi\)
0.966635 + 0.256159i \(0.0824570\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1488.00 −2.49664
\(597\) 0 0
\(598\) −220.454 + 220.454i −0.368652 + 0.368652i
\(599\) 144.000i 0.240401i −0.992750 0.120200i \(-0.961646\pi\)
0.992750 0.120200i \(-0.0383537\pi\)
\(600\) 0 0
\(601\) −899.000 −1.49584 −0.747920 0.663789i \(-0.768947\pi\)
−0.747920 + 0.663789i \(0.768947\pi\)
\(602\) −1800.37 1800.37i −2.99066 2.99066i
\(603\) 0 0
\(604\) 664.000i 1.09934i
\(605\) 0 0
\(606\) 0 0
\(607\) −4.89898 4.89898i −0.00807081 0.00807081i 0.703060 0.711131i \(-0.251817\pi\)
−0.711131 + 0.703060i \(0.751817\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −54.0000 −0.0883797
\(612\) 0 0
\(613\) −259.646 + 259.646i −0.423566 + 0.423566i −0.886430 0.462864i \(-0.846822\pi\)
0.462864 + 0.886430i \(0.346822\pi\)
\(614\) 426.000i 0.693811i
\(615\) 0 0
\(616\) 720.000 1.16883
\(617\) 9.79796 + 9.79796i 0.0158800 + 0.0158800i 0.715002 0.699122i \(-0.246426\pi\)
−0.699122 + 0.715002i \(0.746426\pi\)
\(618\) 0 0
\(619\) 637.000i 1.02908i 0.857467 + 0.514540i \(0.172037\pi\)
−0.857467 + 0.514540i \(0.827963\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −720.150 720.150i −1.15780 1.15780i
\(623\) 808.332 808.332i 1.29748 1.29748i
\(624\) 0 0
\(625\) 0 0
\(626\) 1926.00 3.07668
\(627\) 0 0
\(628\) −362.524 + 362.524i −0.577268 + 0.577268i
\(629\) 840.000i 1.33545i
\(630\) 0 0
\(631\) 287.000 0.454834 0.227417 0.973798i \(-0.426972\pi\)
0.227417 + 0.973798i \(0.426972\pi\)
\(632\) 97.9796 + 97.9796i 0.155031 + 0.155031i
\(633\) 0 0
\(634\) 456.000i 0.719243i
\(635\) 0 0
\(636\) 0 0
\(637\) −95.5301 95.5301i −0.149969 0.149969i
\(638\) 88.1816 88.1816i 0.138216 0.138216i
\(639\) 0 0
\(640\) 0 0
\(641\) 60.0000 0.0936037 0.0468019 0.998904i \(-0.485097\pi\)
0.0468019 + 0.998904i \(0.485097\pi\)
\(642\) 0 0
\(643\) 191.060 191.060i 0.297139 0.297139i −0.542753 0.839892i \(-0.682618\pi\)
0.839892 + 0.542753i \(0.182618\pi\)
\(644\) 1200.00i 1.86335i
\(645\) 0 0
\(646\) 1932.00 2.99071
\(647\) −656.463 656.463i −1.01463 1.01463i −0.999891 0.0147349i \(-0.995310\pi\)
−0.0147349 0.999891i \(-0.504690\pi\)
\(648\) 0 0
\(649\) 108.000i 0.166410i
\(650\) 0 0
\(651\) 0 0
\(652\) −793.635 793.635i −1.21723 1.21723i
\(653\) −751.993 + 751.993i −1.15160 + 1.15160i −0.165365 + 0.986232i \(0.552880\pi\)
−0.986232 + 0.165365i \(0.947120\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −960.000 −1.46341
\(657\) 0 0
\(658\) −220.454 + 220.454i −0.335037 + 0.335037i
\(659\) 420.000i 0.637329i −0.947868 0.318665i \(-0.896766\pi\)
0.947868 0.318665i \(-0.103234\pi\)
\(660\) 0 0
\(661\) −158.000 −0.239032 −0.119516 0.992832i \(-0.538134\pi\)
−0.119516 + 0.992832i \(0.538134\pi\)
\(662\) 436.009 + 436.009i 0.658624 + 0.658624i
\(663\) 0 0
\(664\) 48.0000i 0.0722892i
\(665\) 0 0
\(666\) 0 0
\(667\) 73.4847 + 73.4847i 0.110172 + 0.110172i
\(668\) 783.837 783.837i 1.17341 1.17341i
\(669\) 0 0
\(670\) 0 0
\(671\) 222.000 0.330849
\(672\) 0 0
\(673\) −122.474 + 122.474i −0.181983 + 0.181983i −0.792219 0.610236i \(-0.791074\pi\)
0.610236 + 0.792219i \(0.291074\pi\)
\(674\) 738.000i 1.09496i
\(675\) 0 0
\(676\) −1136.00 −1.68047
\(677\) −242.499 242.499i −0.358197 0.358197i 0.504951 0.863148i \(-0.331511\pi\)
−0.863148 + 0.504951i \(0.831511\pi\)
\(678\) 0 0
\(679\) 285.000i 0.419735i
\(680\) 0 0
\(681\) 0 0
\(682\) 367.423 + 367.423i 0.538744 + 0.538744i
\(683\) 19.5959 19.5959i 0.0286909 0.0286909i −0.692616 0.721307i \(-0.743542\pi\)
0.721307 + 0.692616i \(0.243542\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 690.000 1.00583
\(687\) 0 0
\(688\) 960.200 960.200i 1.39564 1.39564i
\(689\) 180.000i 0.261248i
\(690\) 0 0
\(691\) −74.0000 −0.107091 −0.0535456 0.998565i \(-0.517052\pi\)
−0.0535456 + 0.998565i \(0.517052\pi\)
\(692\) 1391.31 + 1391.31i 2.01056 + 2.01056i
\(693\) 0 0
\(694\) 1836.00i 2.64553i
\(695\) 0 0
\(696\) 0 0
\(697\) 1028.79 + 1028.79i 1.47602 + 1.47602i
\(698\) −1259.04 + 1259.04i −1.80378 + 1.80378i
\(699\) 0 0
\(700\) 0 0
\(701\) 102.000 0.145506 0.0727532 0.997350i \(-0.476821\pi\)
0.0727532 + 0.997350i \(0.476821\pi\)
\(702\) 0 0
\(703\) −563.383 + 563.383i −0.801398 + 0.801398i
\(704\) 384.000i 0.545455i
\(705\) 0 0
\(706\) −1188.00 −1.68272
\(707\) 587.878 + 587.878i 0.831510 + 0.831510i
\(708\) 0 0
\(709\) 817.000i 1.15233i −0.817334 0.576164i \(-0.804549\pi\)
0.817334 0.576164i \(-0.195451\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1293.33 + 1293.33i 1.81648 + 1.81648i
\(713\) −306.186 + 306.186i −0.429434 + 0.429434i
\(714\) 0 0
\(715\) 0 0
\(716\) 1200.00 1.67598
\(717\) 0 0
\(718\) 1587.27 1587.27i 2.21068 2.21068i
\(719\) 6.00000i 0.00834492i 0.999991 + 0.00417246i \(0.00132814\pi\)
−0.999991 + 0.00417246i \(0.998672\pi\)
\(720\) 0 0
\(721\) −240.000 −0.332871
\(722\) 411.514 + 411.514i 0.569964 + 0.569964i
\(723\) 0 0
\(724\) 1720.00i 2.37569i
\(725\) 0 0
\(726\) 0 0
\(727\) 45.3156 + 45.3156i 0.0623323 + 0.0623323i 0.737586 0.675253i \(-0.235966\pi\)
−0.675253 + 0.737586i \(0.735966\pi\)
\(728\) 440.908 440.908i 0.605643 0.605643i
\(729\) 0 0
\(730\) 0 0
\(731\) −2058.00 −2.81532
\(732\) 0 0
\(733\) 484.999 484.999i 0.661663 0.661663i −0.294109 0.955772i \(-0.595023\pi\)
0.955772 + 0.294109i \(0.0950228\pi\)
\(734\) 702.000i 0.956403i
\(735\) 0 0
\(736\) 0 0
\(737\) −154.318 154.318i −0.209387 0.209387i
\(738\) 0 0
\(739\) 298.000i 0.403248i −0.979463 0.201624i \(-0.935378\pi\)
0.979463 0.201624i \(-0.0646218\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −734.847 734.847i −0.990360 0.990360i
\(743\) 249.848 249.848i 0.336269 0.336269i −0.518692 0.854961i \(-0.673581\pi\)
0.854961 + 0.518692i \(0.173581\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 174.000 0.233244
\(747\) 0 0
\(748\) 823.029 823.029i 1.10031 1.10031i
\(749\) 360.000i 0.480641i
\(750\) 0 0
\(751\) 1186.00 1.57923 0.789614 0.613604i \(-0.210281\pi\)
0.789614 + 0.613604i \(0.210281\pi\)
\(752\) −117.576 117.576i −0.156350 0.156350i
\(753\) 0 0
\(754\) 108.000i 0.143236i
\(755\) 0 0
\(756\) 0 0
\(757\) 743.420 + 743.420i 0.982061 + 0.982061i 0.999842 0.0177810i \(-0.00566015\pi\)
−0.0177810 + 0.999842i \(0.505660\pi\)
\(758\) 526.640 526.640i 0.694776 0.694776i
\(759\) 0 0
\(760\) 0 0
\(761\) 576.000 0.756899 0.378449 0.925622i \(-0.376457\pi\)
0.378449 + 0.925622i \(0.376457\pi\)
\(762\) 0 0
\(763\) −1022.66 + 1022.66i −1.34032 + 1.34032i
\(764\) 1872.00i 2.45026i
\(765\) 0 0
\(766\) −96.0000 −0.125326
\(767\) −66.1362 66.1362i −0.0862271 0.0862271i
\(768\) 0 0
\(769\) 491.000i 0.638492i −0.947672 0.319246i \(-0.896570\pi\)
0.947672 0.319246i \(-0.103430\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 519.292 + 519.292i 0.672658 + 0.672658i
\(773\) −519.292 + 519.292i −0.671788 + 0.671788i −0.958128 0.286340i \(-0.907561\pi\)
0.286340 + 0.958128i \(0.407561\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −456.000 −0.587629
\(777\) 0 0
\(778\) −470.302 + 470.302i −0.604501 + 0.604501i
\(779\) 1380.00i 1.77150i
\(780\) 0 0
\(781\) 792.000 1.01408
\(782\) 1028.79 + 1028.79i 1.31558 + 1.31558i
\(783\) 0 0
\(784\) 416.000i 0.530612i
\(785\) 0 0
\(786\) 0 0
\(787\) −123.699 123.699i −0.157178 0.157178i 0.624137 0.781315i \(-0.285451\pi\)
−0.781315 + 0.624137i \(0.785451\pi\)
\(788\) 1548.08 1548.08i 1.96457 1.96457i
\(789\) 0 0
\(790\) 0 0
\(791\) 720.000 0.910240
\(792\) 0 0
\(793\) 135.947 135.947i 0.171433 0.171433i
\(794\) 1794.00i 2.25945i
\(795\) 0 0
\(796\) 88.0000 0.110553
\(797\) 460.504 + 460.504i 0.577797 + 0.577797i 0.934296 0.356499i \(-0.116030\pi\)
−0.356499 + 0.934296i \(0.616030\pi\)
\(798\) 0 0
\(799\) 252.000i 0.315394i
\(800\) 0 0
\(801\) 0 0
\(802\) 558.484 + 558.484i 0.696364 + 0.696364i
\(803\) −146.969 + 146.969i −0.183025 + 0.183025i
\(804\) 0 0
\(805\) 0 0
\(806\) 450.000 0.558313
\(807\) 0 0
\(808\) −940.604 + 940.604i −1.16411 + 1.16411i
\(809\) 900.000i 1.11248i −0.831020 0.556242i \(-0.812243\pi\)
0.831020 0.556242i \(-0.187757\pi\)
\(810\) 0 0
\(811\) 1477.00 1.82121 0.910604 0.413280i \(-0.135617\pi\)
0.910604 + 0.413280i \(0.135617\pi\)
\(812\) −293.939 293.939i −0.361994 0.361994i
\(813\) 0 0
\(814\) 720.000i 0.884521i
\(815\) 0 0
\(816\) 0 0
\(817\) −1380.29 1380.29i −1.68946 1.68946i
\(818\) 1731.79 1731.79i 2.11710 2.11710i
\(819\) 0 0
\(820\) 0 0
\(821\) 786.000 0.957369 0.478685 0.877987i \(-0.341114\pi\)
0.478685 + 0.877987i \(0.341114\pi\)
\(822\) 0 0
\(823\) 371.098 371.098i 0.450909 0.450909i −0.444747 0.895656i \(-0.646707\pi\)
0.895656 + 0.444747i \(0.146707\pi\)
\(824\) 384.000i 0.466019i
\(825\) 0 0
\(826\) −540.000 −0.653753
\(827\) 788.736 + 788.736i 0.953731 + 0.953731i 0.998976 0.0452447i \(-0.0144068\pi\)
−0.0452447 + 0.998976i \(0.514407\pi\)
\(828\) 0 0
\(829\) 170.000i 0.205066i 0.994730 + 0.102533i \(0.0326948\pi\)
−0.994730 + 0.102533i \(0.967305\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −235.151 235.151i −0.282633 0.282633i
\(833\) −445.807 + 445.807i −0.535183 + 0.535183i
\(834\) 0 0
\(835\) 0 0
\(836\) 1104.00 1.32057
\(837\) 0 0
\(838\) −117.576 + 117.576i −0.140305 + 0.140305i
\(839\) 990.000i 1.17998i 0.807412 + 0.589988i \(0.200868\pi\)
−0.807412 + 0.589988i \(0.799132\pi\)
\(840\) 0 0
\(841\) 805.000 0.957194
\(842\) −1259.04 1259.04i −1.49529 1.49529i
\(843\) 0 0
\(844\) 680.000i 0.805687i
\(845\) 0 0
\(846\) 0 0
\(847\) 520.517 + 520.517i 0.614541 + 0.614541i
\(848\) 391.918 391.918i 0.462168 0.462168i
\(849\) 0 0
\(850\) 0 0
\(851\) −600.000 −0.705053
\(852\) 0 0
\(853\) −723.824 + 723.824i −0.848563 + 0.848563i −0.989954 0.141391i \(-0.954843\pi\)
0.141391 + 0.989954i \(0.454843\pi\)
\(854\) 1110.00i 1.29977i
\(855\) 0 0
\(856\) −576.000 −0.672897
\(857\) −739.746 739.746i −0.863181 0.863181i 0.128525 0.991706i \(-0.458976\pi\)
−0.991706 + 0.128525i \(0.958976\pi\)
\(858\) 0 0
\(859\) 1510.00i 1.75786i 0.476953 + 0.878929i \(0.341741\pi\)
−0.476953 + 0.878929i \(0.658259\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 73.4847 + 73.4847i 0.0852491 + 0.0852491i
\(863\) 372.322 372.322i 0.431428 0.431428i −0.457686 0.889114i \(-0.651322\pi\)
0.889114 + 0.457686i \(0.151322\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −1734.00 −2.00231
\(867\) 0 0
\(868\) 1224.74 1224.74i 1.41100 1.41100i
\(869\) 60.0000i 0.0690449i
\(870\) 0 0
\(871\) −189.000 −0.216992
\(872\) −1636.26 1636.26i −1.87644 1.87644i
\(873\) 0 0
\(874\) 1380.00i 1.57895i
\(875\) 0 0
\(876\) 0 0
\(877\) −300.062 300.062i −0.342147 0.342147i 0.515027 0.857174i \(-0.327782\pi\)
−0.857174 + 0.515027i \(0.827782\pi\)
\(878\) −1408.46 + 1408.46i −1.60416 + 1.60416i
\(879\) 0 0
\(880\) 0 0
\(881\) 216.000 0.245176 0.122588 0.992458i \(-0.460881\pi\)
0.122588 + 0.992458i \(0.460881\pi\)
\(882\) 0 0
\(883\) 846.299 846.299i 0.958436 0.958436i −0.0407343 0.999170i \(-0.512970\pi\)
0.999170 + 0.0407343i \(0.0129697\pi\)
\(884\) 1008.00i 1.14027i
\(885\) 0 0
\(886\) 552.000 0.623025
\(887\) −996.942 996.942i −1.12395 1.12395i −0.991142 0.132807i \(-0.957601\pi\)
−0.132807 0.991142i \(-0.542399\pi\)
\(888\) 0 0
\(889\) 540.000i 0.607424i
\(890\) 0 0
\(891\) 0 0
\(892\) −813.231 813.231i −0.911693 0.911693i
\(893\) −169.015 + 169.015i −0.189266 + 0.189266i
\(894\) 0 0
\(895\) 0 0
\(896\) −1920.00 −2.14286
\(897\) 0 0
\(898\) −499.696 + 499.696i −0.556454 + 0.556454i
\(899\) 150.000i 0.166852i
\(900\) 0 0
\(901\) −840.000 −0.932297
\(902\) 881.816 + 881.816i 0.977623 + 0.977623i
\(903\) 0 0
\(904\) 1152.00i 1.27434i
\(905\) 0 0
\(906\) 0 0
\(907\) 53.8888 + 53.8888i 0.0594143 + 0.0594143i 0.736190 0.676775i \(-0.236623\pi\)
−0.676775 + 0.736190i \(0.736623\pi\)
\(908\) −1567.67 + 1567.67i −1.72651 + 1.72651i
\(909\) 0 0
\(910\) 0 0
\(911\) 558.000 0.612514 0.306257 0.951949i \(-0.400923\pi\)
0.306257 + 0.951949i \(0.400923\pi\)
\(912\) 0 0
\(913\) −14.6969 + 14.6969i −0.0160974 + 0.0160974i
\(914\) 2160.00i 2.36324i
\(915\) 0 0
\(916\) −1816.00 −1.98253
\(917\) 661.362 + 661.362i 0.721224 + 0.721224i
\(918\) 0 0
\(919\) 755.000i 0.821545i 0.911738 + 0.410773i \(0.134741\pi\)
−0.911738 + 0.410773i \(0.865259\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 323.333 + 323.333i 0.350686 + 0.350686i
\(923\) 484.999 484.999i 0.525459 0.525459i
\(924\) 0 0
\(925\) 0 0
\(926\) −2136.00 −2.30670
\(927\) 0 0
\(928\) 0 0
\(929\) 762.000i 0.820237i 0.912032 + 0.410118i \(0.134513\pi\)
−0.912032 + 0.410118i \(0.865487\pi\)
\(930\) 0 0
\(931\) −598.000 −0.642320
\(932\) 823.029 + 823.029i 0.883078 + 0.883078i
\(933\) 0 0
\(934\) 1356.00i 1.45182i
\(935\) 0 0
\(936\) 0 0
\(937\) −388.244 388.244i −0.414348 0.414348i 0.468902 0.883250i \(-0.344650\pi\)
−0.883250 + 0.468902i \(0.844650\pi\)
\(938\) −771.589 + 771.589i −0.822590 + 0.822590i
\(939\) 0 0
\(940\) 0 0
\(941\) 690.000 0.733262 0.366631 0.930366i \(-0.380511\pi\)
0.366631 + 0.930366i \(0.380511\pi\)
\(942\) 0 0
\(943\) −734.847 + 734.847i −0.779265 + 0.779265i
\(944\) 288.000i 0.305085i
\(945\) 0 0
\(946\) −1764.00 −1.86469
\(947\) −100.429 100.429i −0.106050 0.106050i 0.652091 0.758141i \(-0.273892\pi\)
−0.758141 + 0.652091i \(0.773892\pi\)
\(948\) 0 0
\(949\) 180.000i 0.189673i
\(950\) 0 0
\(951\) 0 0
\(952\) −2057.57 2057.57i −2.16131 2.16131i
\(953\) −749.544 + 749.544i −0.786510 + 0.786510i −0.980920 0.194410i \(-0.937721\pi\)
0.194410 + 0.980920i \(0.437721\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1824.00 1.90795
\(957\) 0 0
\(958\) −1984.09 + 1984.09i −2.07107 + 2.07107i
\(959\) 1470.00i 1.53285i
\(960\) 0 0
\(961\) −336.000 −0.349636
\(962\) 440.908 + 440.908i 0.458324 + 0.458324i
\(963\) 0 0
\(964\) 1528.00i 1.58506i
\(965\) 0 0
\(966\) 0 0
\(967\) −4.89898 4.89898i −0.00506616 0.00506616i 0.704569 0.709635i \(-0.251140\pi\)
−0.709635 + 0.704569i \(0.751140\pi\)
\(968\) −832.827 + 832.827i −0.860358 + 0.860358i
\(969\) 0 0
\(970\) 0 0
\(971\) −90.0000 −0.0926880 −0.0463440 0.998926i \(-0.514757\pi\)
−0.0463440 + 0.998926i \(0.514757\pi\)
\(972\) 0 0
\(973\) −355.176 + 355.176i −0.365032 + 0.365032i
\(974\) 2394.00i 2.45791i
\(975\) 0 0
\(976\) 592.000 0.606557
\(977\) −301.287 301.287i −0.308380 0.308380i 0.535901 0.844281i \(-0.319972\pi\)
−0.844281 + 0.535901i \(0.819972\pi\)
\(978\) 0 0
\(979\) 792.000i 0.808989i
\(980\) 0 0
\(981\) 0 0
\(982\) 852.422 + 852.422i 0.868047 + 0.868047i
\(983\) 465.403 465.403i 0.473452 0.473452i −0.429578 0.903030i \(-0.641338\pi\)
0.903030 + 0.429578i \(0.141338\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −504.000 −0.511156
\(987\) 0 0
\(988\) 676.059 676.059i 0.684270 0.684270i
\(989\) 1470.00i 1.48635i
\(990\) 0 0
\(991\) 1067.00 1.07669 0.538345 0.842724i \(-0.319050\pi\)
0.538345 + 0.842724i \(0.319050\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 3960.00i 3.98390i
\(995\) 0 0
\(996\) 0 0
\(997\) −279.242 279.242i −0.280082 0.280082i 0.553060 0.833142i \(-0.313460\pi\)
−0.833142 + 0.553060i \(0.813460\pi\)
\(998\) −556.034 + 556.034i −0.557148 + 0.557148i
\(999\) 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 225.3.g.b.82.1 4
3.2 odd 2 75.3.f.b.7.2 yes 4
5.2 odd 4 inner 225.3.g.b.118.2 4
5.3 odd 4 inner 225.3.g.b.118.1 4
5.4 even 2 inner 225.3.g.b.82.2 4
12.11 even 2 1200.3.bg.g.1057.1 4
15.2 even 4 75.3.f.b.43.1 yes 4
15.8 even 4 75.3.f.b.43.2 yes 4
15.14 odd 2 75.3.f.b.7.1 4
60.23 odd 4 1200.3.bg.g.193.1 4
60.47 odd 4 1200.3.bg.g.193.2 4
60.59 even 2 1200.3.bg.g.1057.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.3.f.b.7.1 4 15.14 odd 2
75.3.f.b.7.2 yes 4 3.2 odd 2
75.3.f.b.43.1 yes 4 15.2 even 4
75.3.f.b.43.2 yes 4 15.8 even 4
225.3.g.b.82.1 4 1.1 even 1 trivial
225.3.g.b.82.2 4 5.4 even 2 inner
225.3.g.b.118.1 4 5.3 odd 4 inner
225.3.g.b.118.2 4 5.2 odd 4 inner
1200.3.bg.g.193.1 4 60.23 odd 4
1200.3.bg.g.193.2 4 60.47 odd 4
1200.3.bg.g.1057.1 4 12.11 even 2
1200.3.bg.g.1057.2 4 60.59 even 2