Properties

Label 2025.2.b.k.649.4
Level $2025$
Weight $2$
Character 2025.649
Analytic conductor $16.170$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2025,2,Mod(649,2025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2025.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.1697064093\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 81)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.4
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2025.649
Dual form 2025.2.b.k.649.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} +O(q^{10})\) \(q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} -3.46410 q^{11} +1.00000i q^{13} -3.46410 q^{14} -5.00000 q^{16} +5.19615i q^{17} -2.00000 q^{19} -6.00000i q^{22} -3.46410i q^{23} -1.73205 q^{26} -2.00000i q^{28} -1.73205 q^{29} +8.00000 q^{31} -5.19615i q^{32} -9.00000 q^{34} -7.00000i q^{37} -3.46410i q^{38} -6.92820 q^{41} -2.00000i q^{43} +3.46410 q^{44} +6.00000 q^{46} +6.92820i q^{47} +3.00000 q^{49} -1.00000i q^{52} -3.46410 q^{56} -3.00000i q^{58} -13.8564 q^{59} -7.00000 q^{61} +13.8564i q^{62} -1.00000 q^{64} -10.0000i q^{67} -5.19615i q^{68} -10.3923 q^{71} +7.00000i q^{73} +12.1244 q^{74} +2.00000 q^{76} -6.92820i q^{77} -2.00000 q^{79} -12.0000i q^{82} +13.8564i q^{83} +3.46410 q^{86} -6.00000i q^{88} +5.19615 q^{89} -2.00000 q^{91} +3.46410i q^{92} -12.0000 q^{94} +2.00000i q^{97} +5.19615i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 20 q^{16} - 8 q^{19} + 32 q^{31} - 36 q^{34} + 24 q^{46} + 12 q^{49} - 28 q^{61} - 4 q^{64} + 8 q^{76} - 8 q^{79} - 8 q^{91} - 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(1702\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205i 1.22474i 0.790569 + 0.612372i \(0.209785\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 1.73205i 0.612372i
\(9\) 0 0
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) −3.46410 −0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 5.19615i 1.26025i 0.776493 + 0.630126i \(0.216997\pi\)
−0.776493 + 0.630126i \(0.783003\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 6.00000i − 1.27920i
\(23\) − 3.46410i − 0.722315i −0.932505 0.361158i \(-0.882382\pi\)
0.932505 0.361158i \(-0.117618\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.73205 −0.339683
\(27\) 0 0
\(28\) − 2.00000i − 0.377964i
\(29\) −1.73205 −0.321634 −0.160817 0.986984i \(-0.551413\pi\)
−0.160817 + 0.986984i \(0.551413\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) − 5.19615i − 0.918559i
\(33\) 0 0
\(34\) −9.00000 −1.54349
\(35\) 0 0
\(36\) 0 0
\(37\) − 7.00000i − 1.15079i −0.817875 0.575396i \(-0.804848\pi\)
0.817875 0.575396i \(-0.195152\pi\)
\(38\) − 3.46410i − 0.561951i
\(39\) 0 0
\(40\) 0 0
\(41\) −6.92820 −1.08200 −0.541002 0.841021i \(-0.681955\pi\)
−0.541002 + 0.841021i \(0.681955\pi\)
\(42\) 0 0
\(43\) − 2.00000i − 0.304997i −0.988304 0.152499i \(-0.951268\pi\)
0.988304 0.152499i \(-0.0487319\pi\)
\(44\) 3.46410 0.522233
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 6.92820i 1.01058i 0.862949 + 0.505291i \(0.168615\pi\)
−0.862949 + 0.505291i \(0.831385\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) − 1.00000i − 0.138675i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.46410 −0.462910
\(57\) 0 0
\(58\) − 3.00000i − 0.393919i
\(59\) −13.8564 −1.80395 −0.901975 0.431788i \(-0.857883\pi\)
−0.901975 + 0.431788i \(0.857883\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 13.8564i 1.75977i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 10.0000i − 1.22169i −0.791748 0.610847i \(-0.790829\pi\)
0.791748 0.610847i \(-0.209171\pi\)
\(68\) − 5.19615i − 0.630126i
\(69\) 0 0
\(70\) 0 0
\(71\) −10.3923 −1.23334 −0.616670 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(72\) 0 0
\(73\) 7.00000i 0.819288i 0.912245 + 0.409644i \(0.134347\pi\)
−0.912245 + 0.409644i \(0.865653\pi\)
\(74\) 12.1244 1.40943
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) − 6.92820i − 0.789542i
\(78\) 0 0
\(79\) −2.00000 −0.225018 −0.112509 0.993651i \(-0.535889\pi\)
−0.112509 + 0.993651i \(0.535889\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 12.0000i − 1.32518i
\(83\) 13.8564i 1.52094i 0.649374 + 0.760469i \(0.275031\pi\)
−0.649374 + 0.760469i \(0.724969\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.46410 0.373544
\(87\) 0 0
\(88\) − 6.00000i − 0.639602i
\(89\) 5.19615 0.550791 0.275396 0.961331i \(-0.411191\pi\)
0.275396 + 0.961331i \(0.411191\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 3.46410i 0.361158i
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 5.19615i 0.524891i
\(99\) 0 0
\(100\) 0 0
\(101\) 6.92820 0.689382 0.344691 0.938716i \(-0.387984\pi\)
0.344691 + 0.938716i \(0.387984\pi\)
\(102\) 0 0
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) −1.73205 −0.169842
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −11.0000 −1.05361 −0.526804 0.849987i \(-0.676610\pi\)
−0.526804 + 0.849987i \(0.676610\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 10.0000i − 0.944911i
\(113\) 1.73205i 0.162938i 0.996676 + 0.0814688i \(0.0259611\pi\)
−0.996676 + 0.0814688i \(0.974039\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.73205 0.160817
\(117\) 0 0
\(118\) − 24.0000i − 2.20938i
\(119\) −10.3923 −0.952661
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 12.1244i − 1.09769i
\(123\) 0 0
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) − 12.1244i − 1.07165i
\(129\) 0 0
\(130\) 0 0
\(131\) 3.46410 0.302660 0.151330 0.988483i \(-0.451644\pi\)
0.151330 + 0.988483i \(0.451644\pi\)
\(132\) 0 0
\(133\) − 4.00000i − 0.346844i
\(134\) 17.3205 1.49626
\(135\) 0 0
\(136\) −9.00000 −0.771744
\(137\) 1.73205i 0.147979i 0.997259 + 0.0739895i \(0.0235731\pi\)
−0.997259 + 0.0739895i \(0.976427\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 18.0000i − 1.51053i
\(143\) − 3.46410i − 0.289683i
\(144\) 0 0
\(145\) 0 0
\(146\) −12.1244 −1.00342
\(147\) 0 0
\(148\) 7.00000i 0.575396i
\(149\) −8.66025 −0.709476 −0.354738 0.934966i \(-0.615430\pi\)
−0.354738 + 0.934966i \(0.615430\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) − 3.46410i − 0.280976i
\(153\) 0 0
\(154\) 12.0000 0.966988
\(155\) 0 0
\(156\) 0 0
\(157\) 17.0000i 1.35675i 0.734717 + 0.678374i \(0.237315\pi\)
−0.734717 + 0.678374i \(0.762685\pi\)
\(158\) − 3.46410i − 0.275589i
\(159\) 0 0
\(160\) 0 0
\(161\) 6.92820 0.546019
\(162\) 0 0
\(163\) 16.0000i 1.25322i 0.779334 + 0.626608i \(0.215557\pi\)
−0.779334 + 0.626608i \(0.784443\pi\)
\(164\) 6.92820 0.541002
\(165\) 0 0
\(166\) −24.0000 −1.86276
\(167\) − 17.3205i − 1.34030i −0.742225 0.670151i \(-0.766230\pi\)
0.742225 0.670151i \(-0.233770\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 2.00000i 0.152499i
\(173\) 19.0526i 1.44854i 0.689517 + 0.724270i \(0.257823\pi\)
−0.689517 + 0.724270i \(0.742177\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 17.3205 1.30558
\(177\) 0 0
\(178\) 9.00000i 0.674579i
\(179\) −20.7846 −1.55351 −0.776757 0.629800i \(-0.783137\pi\)
−0.776757 + 0.629800i \(0.783137\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) − 3.46410i − 0.256776i
\(183\) 0 0
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) 0 0
\(187\) − 18.0000i − 1.31629i
\(188\) − 6.92820i − 0.505291i
\(189\) 0 0
\(190\) 0 0
\(191\) 17.3205 1.25327 0.626634 0.779314i \(-0.284432\pi\)
0.626634 + 0.779314i \(0.284432\pi\)
\(192\) 0 0
\(193\) 1.00000i 0.0719816i 0.999352 + 0.0359908i \(0.0114587\pi\)
−0.999352 + 0.0359908i \(0.988541\pi\)
\(194\) −3.46410 −0.248708
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 5.19615i − 0.370211i −0.982719 0.185105i \(-0.940737\pi\)
0.982719 0.185105i \(-0.0592626\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 12.0000i 0.844317i
\(203\) − 3.46410i − 0.243132i
\(204\) 0 0
\(205\) 0 0
\(206\) 13.8564 0.965422
\(207\) 0 0
\(208\) − 5.00000i − 0.346688i
\(209\) 6.92820 0.479234
\(210\) 0 0
\(211\) −10.0000 −0.688428 −0.344214 0.938891i \(-0.611855\pi\)
−0.344214 + 0.938891i \(0.611855\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 16.0000i 1.08615i
\(218\) − 19.0526i − 1.29040i
\(219\) 0 0
\(220\) 0 0
\(221\) −5.19615 −0.349531
\(222\) 0 0
\(223\) − 2.00000i − 0.133930i −0.997755 0.0669650i \(-0.978668\pi\)
0.997755 0.0669650i \(-0.0213316\pi\)
\(224\) 10.3923 0.694365
\(225\) 0 0
\(226\) −3.00000 −0.199557
\(227\) − 3.46410i − 0.229920i −0.993370 0.114960i \(-0.963326\pi\)
0.993370 0.114960i \(-0.0366741\pi\)
\(228\) 0 0
\(229\) 1.00000 0.0660819 0.0330409 0.999454i \(-0.489481\pi\)
0.0330409 + 0.999454i \(0.489481\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 3.00000i − 0.196960i
\(233\) 25.9808i 1.70206i 0.525120 + 0.851028i \(0.324020\pi\)
−0.525120 + 0.851028i \(0.675980\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 13.8564 0.901975
\(237\) 0 0
\(238\) − 18.0000i − 1.16677i
\(239\) 27.7128 1.79259 0.896296 0.443455i \(-0.146248\pi\)
0.896296 + 0.443455i \(0.146248\pi\)
\(240\) 0 0
\(241\) 29.0000 1.86805 0.934027 0.357202i \(-0.116269\pi\)
0.934027 + 0.357202i \(0.116269\pi\)
\(242\) 1.73205i 0.111340i
\(243\) 0 0
\(244\) 7.00000 0.448129
\(245\) 0 0
\(246\) 0 0
\(247\) − 2.00000i − 0.127257i
\(248\) 13.8564i 0.879883i
\(249\) 0 0
\(250\) 0 0
\(251\) 10.3923 0.655956 0.327978 0.944685i \(-0.393633\pi\)
0.327978 + 0.944685i \(0.393633\pi\)
\(252\) 0 0
\(253\) 12.0000i 0.754434i
\(254\) −3.46410 −0.217357
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 8.66025i 0.540212i 0.962831 + 0.270106i \(0.0870587\pi\)
−0.962831 + 0.270106i \(0.912941\pi\)
\(258\) 0 0
\(259\) 14.0000 0.869918
\(260\) 0 0
\(261\) 0 0
\(262\) 6.00000i 0.370681i
\(263\) − 6.92820i − 0.427211i −0.976920 0.213606i \(-0.931479\pi\)
0.976920 0.213606i \(-0.0685208\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 6.92820 0.424795
\(267\) 0 0
\(268\) 10.0000i 0.610847i
\(269\) 15.5885 0.950445 0.475223 0.879866i \(-0.342368\pi\)
0.475223 + 0.879866i \(0.342368\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) − 25.9808i − 1.57532i
\(273\) 0 0
\(274\) −3.00000 −0.181237
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) − 13.8564i − 0.831052i
\(279\) 0 0
\(280\) 0 0
\(281\) 12.1244 0.723278 0.361639 0.932318i \(-0.382217\pi\)
0.361639 + 0.932318i \(0.382217\pi\)
\(282\) 0 0
\(283\) 28.0000i 1.66443i 0.554455 + 0.832214i \(0.312927\pi\)
−0.554455 + 0.832214i \(0.687073\pi\)
\(284\) 10.3923 0.616670
\(285\) 0 0
\(286\) 6.00000 0.354787
\(287\) − 13.8564i − 0.817918i
\(288\) 0 0
\(289\) −10.0000 −0.588235
\(290\) 0 0
\(291\) 0 0
\(292\) − 7.00000i − 0.409644i
\(293\) − 19.0526i − 1.11306i −0.830827 0.556531i \(-0.812132\pi\)
0.830827 0.556531i \(-0.187868\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 12.1244 0.704714
\(297\) 0 0
\(298\) − 15.0000i − 0.868927i
\(299\) 3.46410 0.200334
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 34.6410i 1.99337i
\(303\) 0 0
\(304\) 10.0000 0.573539
\(305\) 0 0
\(306\) 0 0
\(307\) − 16.0000i − 0.913168i −0.889680 0.456584i \(-0.849073\pi\)
0.889680 0.456584i \(-0.150927\pi\)
\(308\) 6.92820i 0.394771i
\(309\) 0 0
\(310\) 0 0
\(311\) −6.92820 −0.392862 −0.196431 0.980518i \(-0.562935\pi\)
−0.196431 + 0.980518i \(0.562935\pi\)
\(312\) 0 0
\(313\) 25.0000i 1.41308i 0.707671 + 0.706542i \(0.249746\pi\)
−0.707671 + 0.706542i \(0.750254\pi\)
\(314\) −29.4449 −1.66167
\(315\) 0 0
\(316\) 2.00000 0.112509
\(317\) − 8.66025i − 0.486408i −0.969975 0.243204i \(-0.921801\pi\)
0.969975 0.243204i \(-0.0781985\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) 0 0
\(321\) 0 0
\(322\) 12.0000i 0.668734i
\(323\) − 10.3923i − 0.578243i
\(324\) 0 0
\(325\) 0 0
\(326\) −27.7128 −1.53487
\(327\) 0 0
\(328\) − 12.0000i − 0.662589i
\(329\) −13.8564 −0.763928
\(330\) 0 0
\(331\) 2.00000 0.109930 0.0549650 0.998488i \(-0.482495\pi\)
0.0549650 + 0.998488i \(0.482495\pi\)
\(332\) − 13.8564i − 0.760469i
\(333\) 0 0
\(334\) 30.0000 1.64153
\(335\) 0 0
\(336\) 0 0
\(337\) 26.0000i 1.41631i 0.706057 + 0.708155i \(0.250472\pi\)
−0.706057 + 0.708155i \(0.749528\pi\)
\(338\) 20.7846i 1.13053i
\(339\) 0 0
\(340\) 0 0
\(341\) −27.7128 −1.50073
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 3.46410 0.186772
\(345\) 0 0
\(346\) −33.0000 −1.77409
\(347\) 3.46410i 0.185963i 0.995668 + 0.0929814i \(0.0296397\pi\)
−0.995668 + 0.0929814i \(0.970360\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 18.0000i 0.959403i
\(353\) 13.8564i 0.737502i 0.929528 + 0.368751i \(0.120215\pi\)
−0.929528 + 0.368751i \(0.879785\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −5.19615 −0.275396
\(357\) 0 0
\(358\) − 36.0000i − 1.90266i
\(359\) −10.3923 −0.548485 −0.274242 0.961661i \(-0.588427\pi\)
−0.274242 + 0.961661i \(0.588427\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 3.46410i 0.182069i
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) 20.0000i 1.04399i 0.852948 + 0.521996i \(0.174812\pi\)
−0.852948 + 0.521996i \(0.825188\pi\)
\(368\) 17.3205i 0.902894i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) 31.1769 1.61212
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) − 1.73205i − 0.0892052i
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 30.0000i 1.53493i
\(383\) 17.3205i 0.885037i 0.896759 + 0.442518i \(0.145915\pi\)
−0.896759 + 0.442518i \(0.854085\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.73205 −0.0881591
\(387\) 0 0
\(388\) − 2.00000i − 0.101535i
\(389\) −27.7128 −1.40510 −0.702548 0.711637i \(-0.747954\pi\)
−0.702548 + 0.711637i \(0.747954\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 5.19615i 0.262445i
\(393\) 0 0
\(394\) 9.00000 0.453413
\(395\) 0 0
\(396\) 0 0
\(397\) 29.0000i 1.45547i 0.685859 + 0.727734i \(0.259427\pi\)
−0.685859 + 0.727734i \(0.740573\pi\)
\(398\) − 34.6410i − 1.73640i
\(399\) 0 0
\(400\) 0 0
\(401\) −12.1244 −0.605461 −0.302731 0.953076i \(-0.597898\pi\)
−0.302731 + 0.953076i \(0.597898\pi\)
\(402\) 0 0
\(403\) 8.00000i 0.398508i
\(404\) −6.92820 −0.344691
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 24.2487i 1.20196i
\(408\) 0 0
\(409\) 19.0000 0.939490 0.469745 0.882802i \(-0.344346\pi\)
0.469745 + 0.882802i \(0.344346\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.00000i 0.394132i
\(413\) − 27.7128i − 1.36366i
\(414\) 0 0
\(415\) 0 0
\(416\) 5.19615 0.254762
\(417\) 0 0
\(418\) 12.0000i 0.586939i
\(419\) 6.92820 0.338465 0.169232 0.985576i \(-0.445871\pi\)
0.169232 + 0.985576i \(0.445871\pi\)
\(420\) 0 0
\(421\) −25.0000 −1.21843 −0.609213 0.793007i \(-0.708514\pi\)
−0.609213 + 0.793007i \(0.708514\pi\)
\(422\) − 17.3205i − 0.843149i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 14.0000i − 0.677507i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) − 11.0000i − 0.528626i −0.964437 0.264313i \(-0.914855\pi\)
0.964437 0.264313i \(-0.0851452\pi\)
\(434\) −27.7128 −1.33026
\(435\) 0 0
\(436\) 11.0000 0.526804
\(437\) 6.92820i 0.331421i
\(438\) 0 0
\(439\) −20.0000 −0.954548 −0.477274 0.878755i \(-0.658375\pi\)
−0.477274 + 0.878755i \(0.658375\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 9.00000i − 0.428086i
\(443\) 34.6410i 1.64584i 0.568154 + 0.822922i \(0.307658\pi\)
−0.568154 + 0.822922i \(0.692342\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.46410 0.164030
\(447\) 0 0
\(448\) − 2.00000i − 0.0944911i
\(449\) −20.7846 −0.980886 −0.490443 0.871473i \(-0.663165\pi\)
−0.490443 + 0.871473i \(0.663165\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) − 1.73205i − 0.0814688i
\(453\) 0 0
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) 29.0000i 1.35656i 0.734802 + 0.678281i \(0.237275\pi\)
−0.734802 + 0.678281i \(0.762725\pi\)
\(458\) 1.73205i 0.0809334i
\(459\) 0 0
\(460\) 0 0
\(461\) −13.8564 −0.645357 −0.322679 0.946509i \(-0.604583\pi\)
−0.322679 + 0.946509i \(0.604583\pi\)
\(462\) 0 0
\(463\) − 8.00000i − 0.371792i −0.982569 0.185896i \(-0.940481\pi\)
0.982569 0.185896i \(-0.0595187\pi\)
\(464\) 8.66025 0.402042
\(465\) 0 0
\(466\) −45.0000 −2.08458
\(467\) − 20.7846i − 0.961797i −0.876776 0.480899i \(-0.840311\pi\)
0.876776 0.480899i \(-0.159689\pi\)
\(468\) 0 0
\(469\) 20.0000 0.923514
\(470\) 0 0
\(471\) 0 0
\(472\) − 24.0000i − 1.10469i
\(473\) 6.92820i 0.318559i
\(474\) 0 0
\(475\) 0 0
\(476\) 10.3923 0.476331
\(477\) 0 0
\(478\) 48.0000i 2.19547i
\(479\) 24.2487 1.10795 0.553976 0.832533i \(-0.313110\pi\)
0.553976 + 0.832533i \(0.313110\pi\)
\(480\) 0 0
\(481\) 7.00000 0.319173
\(482\) 50.2295i 2.28789i
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) − 16.0000i − 0.725029i −0.931978 0.362515i \(-0.881918\pi\)
0.931978 0.362515i \(-0.118082\pi\)
\(488\) − 12.1244i − 0.548844i
\(489\) 0 0
\(490\) 0 0
\(491\) −17.3205 −0.781664 −0.390832 0.920462i \(-0.627813\pi\)
−0.390832 + 0.920462i \(0.627813\pi\)
\(492\) 0 0
\(493\) − 9.00000i − 0.405340i
\(494\) 3.46410 0.155857
\(495\) 0 0
\(496\) −40.0000 −1.79605
\(497\) − 20.7846i − 0.932317i
\(498\) 0 0
\(499\) 10.0000 0.447661 0.223831 0.974628i \(-0.428144\pi\)
0.223831 + 0.974628i \(0.428144\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18.0000i 0.803379i
\(503\) − 20.7846i − 0.926740i −0.886165 0.463370i \(-0.846640\pi\)
0.886165 0.463370i \(-0.153360\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −20.7846 −0.923989
\(507\) 0 0
\(508\) − 2.00000i − 0.0887357i
\(509\) 27.7128 1.22835 0.614174 0.789170i \(-0.289489\pi\)
0.614174 + 0.789170i \(0.289489\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 8.66025i 0.382733i
\(513\) 0 0
\(514\) −15.0000 −0.661622
\(515\) 0 0
\(516\) 0 0
\(517\) − 24.0000i − 1.05552i
\(518\) 24.2487i 1.06543i
\(519\) 0 0
\(520\) 0 0
\(521\) −20.7846 −0.910590 −0.455295 0.890341i \(-0.650466\pi\)
−0.455295 + 0.890341i \(0.650466\pi\)
\(522\) 0 0
\(523\) − 38.0000i − 1.66162i −0.556553 0.830812i \(-0.687876\pi\)
0.556553 0.830812i \(-0.312124\pi\)
\(524\) −3.46410 −0.151330
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 41.5692i 1.81078i
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000i 0.173422i
\(533\) − 6.92820i − 0.300094i
\(534\) 0 0
\(535\) 0 0
\(536\) 17.3205 0.748132
\(537\) 0 0
\(538\) 27.0000i 1.16405i
\(539\) −10.3923 −0.447628
\(540\) 0 0
\(541\) 11.0000 0.472927 0.236463 0.971640i \(-0.424012\pi\)
0.236463 + 0.971640i \(0.424012\pi\)
\(542\) 3.46410i 0.148796i
\(543\) 0 0
\(544\) 27.0000 1.15762
\(545\) 0 0
\(546\) 0 0
\(547\) 20.0000i 0.855138i 0.903983 + 0.427569i \(0.140630\pi\)
−0.903983 + 0.427569i \(0.859370\pi\)
\(548\) − 1.73205i − 0.0739895i
\(549\) 0 0
\(550\) 0 0
\(551\) 3.46410 0.147576
\(552\) 0 0
\(553\) − 4.00000i − 0.170097i
\(554\) −3.46410 −0.147176
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 36.3731i 1.54118i 0.637333 + 0.770588i \(0.280037\pi\)
−0.637333 + 0.770588i \(0.719963\pi\)
\(558\) 0 0
\(559\) 2.00000 0.0845910
\(560\) 0 0
\(561\) 0 0
\(562\) 21.0000i 0.885832i
\(563\) − 34.6410i − 1.45994i −0.683477 0.729972i \(-0.739533\pi\)
0.683477 0.729972i \(-0.260467\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −48.4974 −2.03850
\(567\) 0 0
\(568\) − 18.0000i − 0.755263i
\(569\) −32.9090 −1.37962 −0.689808 0.723993i \(-0.742305\pi\)
−0.689808 + 0.723993i \(0.742305\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) 3.46410i 0.144841i
\(573\) 0 0
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 0 0
\(577\) 11.0000i 0.457936i 0.973434 + 0.228968i \(0.0735351\pi\)
−0.973434 + 0.228968i \(0.926465\pi\)
\(578\) − 17.3205i − 0.720438i
\(579\) 0 0
\(580\) 0 0
\(581\) −27.7128 −1.14972
\(582\) 0 0
\(583\) 0 0
\(584\) −12.1244 −0.501709
\(585\) 0 0
\(586\) 33.0000 1.36322
\(587\) 38.1051i 1.57277i 0.617739 + 0.786383i \(0.288049\pi\)
−0.617739 + 0.786383i \(0.711951\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) 35.0000i 1.43849i
\(593\) − 15.5885i − 0.640141i −0.947394 0.320071i \(-0.896293\pi\)
0.947394 0.320071i \(-0.103707\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 8.66025 0.354738
\(597\) 0 0
\(598\) 6.00000i 0.245358i
\(599\) −13.8564 −0.566157 −0.283079 0.959097i \(-0.591356\pi\)
−0.283079 + 0.959097i \(0.591356\pi\)
\(600\) 0 0
\(601\) −25.0000 −1.01977 −0.509886 0.860242i \(-0.670312\pi\)
−0.509886 + 0.860242i \(0.670312\pi\)
\(602\) 6.92820i 0.282372i
\(603\) 0 0
\(604\) −20.0000 −0.813788
\(605\) 0 0
\(606\) 0 0
\(607\) 26.0000i 1.05531i 0.849460 + 0.527654i \(0.176928\pi\)
−0.849460 + 0.527654i \(0.823072\pi\)
\(608\) 10.3923i 0.421464i
\(609\) 0 0
\(610\) 0 0
\(611\) −6.92820 −0.280285
\(612\) 0 0
\(613\) 34.0000i 1.37325i 0.727013 + 0.686624i \(0.240908\pi\)
−0.727013 + 0.686624i \(0.759092\pi\)
\(614\) 27.7128 1.11840
\(615\) 0 0
\(616\) 12.0000 0.483494
\(617\) − 12.1244i − 0.488108i −0.969762 0.244054i \(-0.921523\pi\)
0.969762 0.244054i \(-0.0784774\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 12.0000i − 0.481156i
\(623\) 10.3923i 0.416359i
\(624\) 0 0
\(625\) 0 0
\(626\) −43.3013 −1.73067
\(627\) 0 0
\(628\) − 17.0000i − 0.678374i
\(629\) 36.3731 1.45029
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) − 3.46410i − 0.137795i
\(633\) 0 0
\(634\) 15.0000 0.595726
\(635\) 0 0
\(636\) 0 0
\(637\) 3.00000i 0.118864i
\(638\) 10.3923i 0.411435i
\(639\) 0 0
\(640\) 0 0
\(641\) 22.5167 0.889355 0.444677 0.895691i \(-0.353318\pi\)
0.444677 + 0.895691i \(0.353318\pi\)
\(642\) 0 0
\(643\) − 8.00000i − 0.315489i −0.987480 0.157745i \(-0.949578\pi\)
0.987480 0.157745i \(-0.0504223\pi\)
\(644\) −6.92820 −0.273009
\(645\) 0 0
\(646\) 18.0000 0.708201
\(647\) − 31.1769i − 1.22569i −0.790203 0.612845i \(-0.790025\pi\)
0.790203 0.612845i \(-0.209975\pi\)
\(648\) 0 0
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) − 16.0000i − 0.626608i
\(653\) − 13.8564i − 0.542243i −0.962545 0.271122i \(-0.912605\pi\)
0.962545 0.271122i \(-0.0873945\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 34.6410 1.35250
\(657\) 0 0
\(658\) − 24.0000i − 0.935617i
\(659\) 3.46410 0.134942 0.0674711 0.997721i \(-0.478507\pi\)
0.0674711 + 0.997721i \(0.478507\pi\)
\(660\) 0 0
\(661\) 17.0000 0.661223 0.330612 0.943767i \(-0.392745\pi\)
0.330612 + 0.943767i \(0.392745\pi\)
\(662\) 3.46410i 0.134636i
\(663\) 0 0
\(664\) −24.0000 −0.931381
\(665\) 0 0
\(666\) 0 0
\(667\) 6.00000i 0.232321i
\(668\) 17.3205i 0.670151i
\(669\) 0 0
\(670\) 0 0
\(671\) 24.2487 0.936111
\(672\) 0 0
\(673\) 25.0000i 0.963679i 0.876259 + 0.481840i \(0.160031\pi\)
−0.876259 + 0.481840i \(0.839969\pi\)
\(674\) −45.0333 −1.73462
\(675\) 0 0
\(676\) −12.0000 −0.461538
\(677\) − 13.8564i − 0.532545i −0.963898 0.266272i \(-0.914208\pi\)
0.963898 0.266272i \(-0.0857921\pi\)
\(678\) 0 0
\(679\) −4.00000 −0.153506
\(680\) 0 0
\(681\) 0 0
\(682\) − 48.0000i − 1.83801i
\(683\) 20.7846i 0.795301i 0.917537 + 0.397650i \(0.130174\pi\)
−0.917537 + 0.397650i \(0.869826\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −34.6410 −1.32260
\(687\) 0 0
\(688\) 10.0000i 0.381246i
\(689\) 0 0
\(690\) 0 0
\(691\) 2.00000 0.0760836 0.0380418 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) − 19.0526i − 0.724270i
\(693\) 0 0
\(694\) −6.00000 −0.227757
\(695\) 0 0
\(696\) 0 0
\(697\) − 36.0000i − 1.36360i
\(698\) − 3.46410i − 0.131118i
\(699\) 0 0
\(700\) 0 0
\(701\) 46.7654 1.76630 0.883152 0.469087i \(-0.155417\pi\)
0.883152 + 0.469087i \(0.155417\pi\)
\(702\) 0 0
\(703\) 14.0000i 0.528020i
\(704\) 3.46410 0.130558
\(705\) 0 0
\(706\) −24.0000 −0.903252
\(707\) 13.8564i 0.521124i
\(708\) 0 0
\(709\) 25.0000 0.938895 0.469447 0.882960i \(-0.344453\pi\)
0.469447 + 0.882960i \(0.344453\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 9.00000i 0.337289i
\(713\) − 27.7128i − 1.03785i
\(714\) 0 0
\(715\) 0 0
\(716\) 20.7846 0.776757
\(717\) 0 0
\(718\) − 18.0000i − 0.671754i
\(719\) 10.3923 0.387568 0.193784 0.981044i \(-0.437924\pi\)
0.193784 + 0.981044i \(0.437924\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) − 25.9808i − 0.966904i
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) − 34.0000i − 1.26099i −0.776193 0.630495i \(-0.782852\pi\)
0.776193 0.630495i \(-0.217148\pi\)
\(728\) − 3.46410i − 0.128388i
\(729\) 0 0
\(730\) 0 0
\(731\) 10.3923 0.384373
\(732\) 0 0
\(733\) 46.0000i 1.69905i 0.527549 + 0.849524i \(0.323111\pi\)
−0.527549 + 0.849524i \(0.676889\pi\)
\(734\) −34.6410 −1.27862
\(735\) 0 0
\(736\) −18.0000 −0.663489
\(737\) 34.6410i 1.27602i
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 6.92820i 0.254171i 0.991892 + 0.127086i \(0.0405623\pi\)
−0.991892 + 0.127086i \(0.959438\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −17.3205 −0.634149
\(747\) 0 0
\(748\) 18.0000i 0.658145i
\(749\) 0 0
\(750\) 0 0
\(751\) −10.0000 −0.364905 −0.182453 0.983215i \(-0.558404\pi\)
−0.182453 + 0.983215i \(0.558404\pi\)
\(752\) − 34.6410i − 1.26323i
\(753\) 0 0
\(754\) 3.00000 0.109254
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 27.7128i 1.00657i
\(759\) 0 0
\(760\) 0 0
\(761\) 29.4449 1.06738 0.533688 0.845682i \(-0.320806\pi\)
0.533688 + 0.845682i \(0.320806\pi\)
\(762\) 0 0
\(763\) − 22.0000i − 0.796453i
\(764\) −17.3205 −0.626634
\(765\) 0 0
\(766\) −30.0000 −1.08394
\(767\) − 13.8564i − 0.500326i
\(768\) 0 0
\(769\) 1.00000 0.0360609 0.0180305 0.999837i \(-0.494260\pi\)
0.0180305 + 0.999837i \(0.494260\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 1.00000i − 0.0359908i
\(773\) 25.9808i 0.934463i 0.884135 + 0.467232i \(0.154749\pi\)
−0.884135 + 0.467232i \(0.845251\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −3.46410 −0.124354
\(777\) 0 0
\(778\) − 48.0000i − 1.72088i
\(779\) 13.8564 0.496457
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) 31.1769i 1.11488i
\(783\) 0 0
\(784\) −15.0000 −0.535714
\(785\) 0 0
\(786\) 0 0
\(787\) 26.0000i 0.926800i 0.886149 + 0.463400i \(0.153371\pi\)
−0.886149 + 0.463400i \(0.846629\pi\)
\(788\) 5.19615i 0.185105i
\(789\) 0 0
\(790\) 0 0
\(791\) −3.46410 −0.123169
\(792\) 0 0
\(793\) − 7.00000i − 0.248577i
\(794\) −50.2295 −1.78258
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) − 53.6936i − 1.90192i −0.309308 0.950962i \(-0.600097\pi\)
0.309308 0.950962i \(-0.399903\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) 0 0
\(801\) 0 0
\(802\) − 21.0000i − 0.741536i
\(803\) − 24.2487i − 0.855718i
\(804\) 0 0
\(805\) 0 0
\(806\) −13.8564 −0.488071
\(807\) 0 0
\(808\) 12.0000i 0.422159i
\(809\) 46.7654 1.64418 0.822091 0.569355i \(-0.192807\pi\)
0.822091 + 0.569355i \(0.192807\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 3.46410i 0.121566i
\(813\) 0 0
\(814\) −42.0000 −1.47210
\(815\) 0 0
\(816\) 0 0
\(817\) 4.00000i 0.139942i
\(818\) 32.9090i 1.15063i
\(819\) 0 0
\(820\) 0 0
\(821\) 12.1244 0.423143 0.211571 0.977363i \(-0.432142\pi\)
0.211571 + 0.977363i \(0.432142\pi\)
\(822\) 0 0
\(823\) 28.0000i 0.976019i 0.872838 + 0.488009i \(0.162277\pi\)
−0.872838 + 0.488009i \(0.837723\pi\)
\(824\) 13.8564 0.482711
\(825\) 0 0
\(826\) 48.0000 1.67013
\(827\) − 10.3923i − 0.361376i −0.983540 0.180688i \(-0.942168\pi\)
0.983540 0.180688i \(-0.0578324\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 1.00000i − 0.0346688i
\(833\) 15.5885i 0.540108i
\(834\) 0 0
\(835\) 0 0
\(836\) −6.92820 −0.239617
\(837\) 0 0
\(838\) 12.0000i 0.414533i
\(839\) 45.0333 1.55472 0.777361 0.629054i \(-0.216558\pi\)
0.777361 + 0.629054i \(0.216558\pi\)
\(840\) 0 0
\(841\) −26.0000 −0.896552
\(842\) − 43.3013i − 1.49226i
\(843\) 0 0
\(844\) 10.0000 0.344214
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −24.2487 −0.831235
\(852\) 0 0
\(853\) 34.0000i 1.16414i 0.813139 + 0.582069i \(0.197757\pi\)
−0.813139 + 0.582069i \(0.802243\pi\)
\(854\) 24.2487 0.829774
\(855\) 0 0
\(856\) 0 0
\(857\) 22.5167i 0.769154i 0.923093 + 0.384577i \(0.125653\pi\)
−0.923093 + 0.384577i \(0.874347\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 31.1769i − 1.06127i −0.847599 0.530637i \(-0.821953\pi\)
0.847599 0.530637i \(-0.178047\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 19.0526 0.647432
\(867\) 0 0
\(868\) − 16.0000i − 0.543075i
\(869\) 6.92820 0.235023
\(870\) 0 0
\(871\) 10.0000 0.338837
\(872\) − 19.0526i − 0.645201i
\(873\) 0 0
\(874\) −12.0000 −0.405906
\(875\) 0 0
\(876\) 0 0
\(877\) 53.0000i 1.78968i 0.446384 + 0.894841i \(0.352711\pi\)
−0.446384 + 0.894841i \(0.647289\pi\)
\(878\) − 34.6410i − 1.16908i
\(879\) 0 0
\(880\) 0 0
\(881\) 20.7846 0.700251 0.350126 0.936703i \(-0.386139\pi\)
0.350126 + 0.936703i \(0.386139\pi\)
\(882\) 0 0
\(883\) − 56.0000i − 1.88455i −0.334840 0.942275i \(-0.608682\pi\)
0.334840 0.942275i \(-0.391318\pi\)
\(884\) 5.19615 0.174766
\(885\) 0 0
\(886\) −60.0000 −2.01574
\(887\) 3.46410i 0.116313i 0.998307 + 0.0581566i \(0.0185223\pi\)
−0.998307 + 0.0581566i \(0.981478\pi\)
\(888\) 0 0
\(889\) −4.00000 −0.134156
\(890\) 0 0
\(891\) 0 0
\(892\) 2.00000i 0.0669650i
\(893\) − 13.8564i − 0.463687i
\(894\) 0 0
\(895\) 0 0
\(896\) 24.2487 0.810093
\(897\) 0 0
\(898\) − 36.0000i − 1.20134i
\(899\) −13.8564 −0.462137
\(900\) 0 0
\(901\) 0 0
\(902\) 41.5692i 1.38410i
\(903\) 0 0
\(904\) −3.00000 −0.0997785
\(905\) 0 0
\(906\) 0 0
\(907\) − 52.0000i − 1.72663i −0.504664 0.863316i \(-0.668384\pi\)
0.504664 0.863316i \(-0.331616\pi\)
\(908\) 3.46410i 0.114960i
\(909\) 0 0
\(910\) 0 0
\(911\) −24.2487 −0.803396 −0.401698 0.915772i \(-0.631580\pi\)
−0.401698 + 0.915772i \(0.631580\pi\)
\(912\) 0 0
\(913\) − 48.0000i − 1.58857i
\(914\) −50.2295 −1.66144
\(915\) 0 0
\(916\) −1.00000 −0.0330409
\(917\) 6.92820i 0.228789i
\(918\) 0 0
\(919\) −2.00000 −0.0659739 −0.0329870 0.999456i \(-0.510502\pi\)
−0.0329870 + 0.999456i \(0.510502\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 24.0000i − 0.790398i
\(923\) − 10.3923i − 0.342067i
\(924\) 0 0
\(925\) 0 0
\(926\) 13.8564 0.455350
\(927\) 0 0
\(928\) 9.00000i 0.295439i
\(929\) 50.2295 1.64798 0.823988 0.566608i \(-0.191744\pi\)
0.823988 + 0.566608i \(0.191744\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) − 25.9808i − 0.851028i
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) − 25.0000i − 0.816714i −0.912822 0.408357i \(-0.866102\pi\)
0.912822 0.408357i \(-0.133898\pi\)
\(938\) 34.6410i 1.13107i
\(939\) 0 0
\(940\) 0 0
\(941\) 50.2295 1.63743 0.818717 0.574197i \(-0.194686\pi\)
0.818717 + 0.574197i \(0.194686\pi\)
\(942\) 0 0
\(943\) 24.0000i 0.781548i
\(944\) 69.2820 2.25494
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) 17.3205i 0.562841i 0.959585 + 0.281420i \(0.0908056\pi\)
−0.959585 + 0.281420i \(0.909194\pi\)
\(948\) 0 0
\(949\) −7.00000 −0.227230
\(950\) 0 0
\(951\) 0 0
\(952\) − 18.0000i − 0.583383i
\(953\) 5.19615i 0.168320i 0.996452 + 0.0841599i \(0.0268207\pi\)
−0.996452 + 0.0841599i \(0.973179\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −27.7128 −0.896296
\(957\) 0 0
\(958\) 42.0000i 1.35696i
\(959\) −3.46410 −0.111862
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 12.1244i 0.390905i
\(963\) 0 0
\(964\) −29.0000 −0.934027
\(965\) 0 0
\(966\) 0 0
\(967\) − 46.0000i − 1.47926i −0.673014 0.739630i \(-0.735000\pi\)
0.673014 0.739630i \(-0.265000\pi\)
\(968\) 1.73205i 0.0556702i
\(969\) 0 0
\(970\) 0 0
\(971\) 31.1769 1.00051 0.500257 0.865877i \(-0.333239\pi\)
0.500257 + 0.865877i \(0.333239\pi\)
\(972\) 0 0
\(973\) − 16.0000i − 0.512936i
\(974\) 27.7128 0.887976
\(975\) 0 0
\(976\) 35.0000 1.12032
\(977\) − 48.4974i − 1.55157i −0.630997 0.775785i \(-0.717354\pi\)
0.630997 0.775785i \(-0.282646\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) − 30.0000i − 0.957338i
\(983\) 34.6410i 1.10488i 0.833554 + 0.552438i \(0.186303\pi\)
−0.833554 + 0.552438i \(0.813697\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 15.5885 0.496438
\(987\) 0 0
\(988\) 2.00000i 0.0636285i
\(989\) −6.92820 −0.220304
\(990\) 0 0
\(991\) −34.0000 −1.08005 −0.540023 0.841650i \(-0.681584\pi\)
−0.540023 + 0.841650i \(0.681584\pi\)
\(992\) − 41.5692i − 1.31982i
\(993\) 0 0
\(994\) 36.0000 1.14185
\(995\) 0 0
\(996\) 0 0
\(997\) − 7.00000i − 0.221692i −0.993838 0.110846i \(-0.964644\pi\)
0.993838 0.110846i \(-0.0353561\pi\)
\(998\) 17.3205i 0.548271i
\(999\) 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 2025.2.b.k.649.4 4
3.2 odd 2 inner 2025.2.b.k.649.2 4
5.2 odd 4 2025.2.a.j.1.1 2
5.3 odd 4 81.2.a.a.1.2 yes 2
5.4 even 2 inner 2025.2.b.k.649.1 4
15.2 even 4 2025.2.a.j.1.2 2
15.8 even 4 81.2.a.a.1.1 2
15.14 odd 2 inner 2025.2.b.k.649.3 4
20.3 even 4 1296.2.a.o.1.1 2
35.13 even 4 3969.2.a.i.1.2 2
40.3 even 4 5184.2.a.bq.1.2 2
40.13 odd 4 5184.2.a.br.1.2 2
45.13 odd 12 81.2.c.b.55.1 4
45.23 even 12 81.2.c.b.55.2 4
45.38 even 12 81.2.c.b.28.2 4
45.43 odd 12 81.2.c.b.28.1 4
55.43 even 4 9801.2.a.v.1.1 2
60.23 odd 4 1296.2.a.o.1.2 2
105.83 odd 4 3969.2.a.i.1.1 2
120.53 even 4 5184.2.a.br.1.1 2
120.83 odd 4 5184.2.a.bq.1.1 2
135.13 odd 36 729.2.e.o.163.1 12
135.23 even 36 729.2.e.o.406.1 12
135.38 even 36 729.2.e.o.82.1 12
135.43 odd 36 729.2.e.o.82.2 12
135.58 odd 36 729.2.e.o.406.2 12
135.68 even 36 729.2.e.o.163.2 12
135.83 even 36 729.2.e.o.568.2 12
135.88 odd 36 729.2.e.o.325.2 12
135.103 odd 36 729.2.e.o.649.2 12
135.113 even 36 729.2.e.o.649.1 12
135.128 even 36 729.2.e.o.325.1 12
135.133 odd 36 729.2.e.o.568.1 12
165.98 odd 4 9801.2.a.v.1.2 2
180.23 odd 12 1296.2.i.s.865.1 4
180.43 even 12 1296.2.i.s.433.2 4
180.83 odd 12 1296.2.i.s.433.1 4
180.103 even 12 1296.2.i.s.865.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
81.2.a.a.1.1 2 15.8 even 4
81.2.a.a.1.2 yes 2 5.3 odd 4
81.2.c.b.28.1 4 45.43 odd 12
81.2.c.b.28.2 4 45.38 even 12
81.2.c.b.55.1 4 45.13 odd 12
81.2.c.b.55.2 4 45.23 even 12
729.2.e.o.82.1 12 135.38 even 36
729.2.e.o.82.2 12 135.43 odd 36
729.2.e.o.163.1 12 135.13 odd 36
729.2.e.o.163.2 12 135.68 even 36
729.2.e.o.325.1 12 135.128 even 36
729.2.e.o.325.2 12 135.88 odd 36
729.2.e.o.406.1 12 135.23 even 36
729.2.e.o.406.2 12 135.58 odd 36
729.2.e.o.568.1 12 135.133 odd 36
729.2.e.o.568.2 12 135.83 even 36
729.2.e.o.649.1 12 135.113 even 36
729.2.e.o.649.2 12 135.103 odd 36
1296.2.a.o.1.1 2 20.3 even 4
1296.2.a.o.1.2 2 60.23 odd 4
1296.2.i.s.433.1 4 180.83 odd 12
1296.2.i.s.433.2 4 180.43 even 12
1296.2.i.s.865.1 4 180.23 odd 12
1296.2.i.s.865.2 4 180.103 even 12
2025.2.a.j.1.1 2 5.2 odd 4
2025.2.a.j.1.2 2 15.2 even 4
2025.2.b.k.649.1 4 5.4 even 2 inner
2025.2.b.k.649.2 4 3.2 odd 2 inner
2025.2.b.k.649.3 4 15.14 odd 2 inner
2025.2.b.k.649.4 4 1.1 even 1 trivial
3969.2.a.i.1.1 2 105.83 odd 4
3969.2.a.i.1.2 2 35.13 even 4
5184.2.a.bq.1.1 2 120.83 odd 4
5184.2.a.bq.1.2 2 40.3 even 4
5184.2.a.br.1.1 2 120.53 even 4
5184.2.a.br.1.2 2 40.13 odd 4
9801.2.a.v.1.1 2 55.43 even 4
9801.2.a.v.1.2 2 165.98 odd 4