Properties

Label 1815.2.c.c.364.2
Level $1815$
Weight $2$
Character 1815.364
Analytic conductor $14.493$
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] = [1815,2,Mod(364,1815)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1815, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1815.364");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.c (of order \(2\), degree \(1\), 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: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 364.2
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 1815.364
Dual form 1815.2.c.c.364.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23607i q^{2} +1.00000i q^{3} -3.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} +2.23607 q^{6} +2.23607i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-2.23607i q^{2} +1.00000i q^{3} -3.00000 q^{4} +(-1.00000 + 2.00000i) q^{5} +2.23607 q^{6} +2.23607i q^{8} -1.00000 q^{9} +(4.47214 + 2.23607i) q^{10} -3.00000i q^{12} -4.47214i q^{13} +(-2.00000 - 1.00000i) q^{15} -1.00000 q^{16} +4.47214i q^{17} +2.23607i q^{18} +(3.00000 - 6.00000i) q^{20} -4.00000i q^{23} -2.23607 q^{24} +(-3.00000 - 4.00000i) q^{25} -10.0000 q^{26} -1.00000i q^{27} +8.94427 q^{29} +(-2.23607 + 4.47214i) q^{30} +6.70820i q^{32} +10.0000 q^{34} +3.00000 q^{36} +8.00000i q^{37} +4.47214 q^{39} +(-4.47214 - 2.23607i) q^{40} +8.94427 q^{41} +8.94427i q^{43} +(1.00000 - 2.00000i) q^{45} -8.94427 q^{46} +12.0000i q^{47} -1.00000i q^{48} +7.00000 q^{49} +(-8.94427 + 6.70820i) q^{50} -4.47214 q^{51} +13.4164i q^{52} -4.00000i q^{53} -2.23607 q^{54} -20.0000i q^{58} +(6.00000 + 3.00000i) q^{60} +13.0000 q^{64} +(8.94427 + 4.47214i) q^{65} +12.0000i q^{67} -13.4164i q^{68} +4.00000 q^{69} +12.0000 q^{71} -2.23607i q^{72} +13.4164i q^{73} +17.8885 q^{74} +(4.00000 - 3.00000i) q^{75} -10.0000i q^{78} +(1.00000 - 2.00000i) q^{80} +1.00000 q^{81} -20.0000i q^{82} +(-8.94427 - 4.47214i) q^{85} +20.0000 q^{86} +8.94427i q^{87} -6.00000 q^{89} +(-4.47214 - 2.23607i) q^{90} +12.0000i q^{92} +26.8328 q^{94} -6.70820 q^{96} +8.00000i q^{97} -15.6525i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{4} - 4 q^{5} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{4} - 4 q^{5} - 4 q^{9} - 8 q^{15} - 4 q^{16} + 12 q^{20} - 12 q^{25} - 40 q^{26} + 40 q^{34} + 12 q^{36} + 4 q^{45} + 28 q^{49} + 24 q^{60} + 52 q^{64} + 16 q^{69} + 48 q^{71} + 16 q^{75} + 4 q^{80} + 4 q^{81} + 80 q^{86} - 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1815\mathbb{Z}\right)^\times\).

\(n\) \(727\) \(1211\) \(1696\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.23607i 1.58114i −0.612372 0.790569i \(-0.709785\pi\)
0.612372 0.790569i \(-0.290215\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −3.00000 −1.50000
\(5\) −1.00000 + 2.00000i −0.447214 + 0.894427i
\(6\) 2.23607 0.912871
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 2.23607i 0.790569i
\(9\) −1.00000 −0.333333
\(10\) 4.47214 + 2.23607i 1.41421 + 0.707107i
\(11\) 0 0
\(12\) 3.00000i 0.866025i
\(13\) 4.47214i 1.24035i −0.784465 0.620174i \(-0.787062\pi\)
0.784465 0.620174i \(-0.212938\pi\)
\(14\) 0 0
\(15\) −2.00000 1.00000i −0.516398 0.258199i
\(16\) −1.00000 −0.250000
\(17\) 4.47214i 1.08465i 0.840168 + 0.542326i \(0.182456\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 2.23607i 0.527046i
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 3.00000 6.00000i 0.670820 1.34164i
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000i 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) −2.23607 −0.456435
\(25\) −3.00000 4.00000i −0.600000 0.800000i
\(26\) −10.0000 −1.96116
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 8.94427 1.66091 0.830455 0.557086i \(-0.188081\pi\)
0.830455 + 0.557086i \(0.188081\pi\)
\(30\) −2.23607 + 4.47214i −0.408248 + 0.816497i
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 6.70820i 1.18585i
\(33\) 0 0
\(34\) 10.0000 1.71499
\(35\) 0 0
\(36\) 3.00000 0.500000
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 0 0
\(39\) 4.47214 0.716115
\(40\) −4.47214 2.23607i −0.707107 0.353553i
\(41\) 8.94427 1.39686 0.698430 0.715678i \(-0.253882\pi\)
0.698430 + 0.715678i \(0.253882\pi\)
\(42\) 0 0
\(43\) 8.94427i 1.36399i 0.731357 + 0.681994i \(0.238887\pi\)
−0.731357 + 0.681994i \(0.761113\pi\)
\(44\) 0 0
\(45\) 1.00000 2.00000i 0.149071 0.298142i
\(46\) −8.94427 −1.31876
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 7.00000 1.00000
\(50\) −8.94427 + 6.70820i −1.26491 + 0.948683i
\(51\) −4.47214 −0.626224
\(52\) 13.4164i 1.86052i
\(53\) 4.00000i 0.549442i −0.961524 0.274721i \(-0.911414\pi\)
0.961524 0.274721i \(-0.0885855\pi\)
\(54\) −2.23607 −0.304290
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 20.0000i 2.62613i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 6.00000 + 3.00000i 0.774597 + 0.387298i
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 13.0000 1.62500
\(65\) 8.94427 + 4.47214i 1.10940 + 0.554700i
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 13.4164i 1.62698i
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 2.23607i 0.263523i
\(73\) 13.4164i 1.57027i 0.619324 + 0.785136i \(0.287407\pi\)
−0.619324 + 0.785136i \(0.712593\pi\)
\(74\) 17.8885 2.07950
\(75\) 4.00000 3.00000i 0.461880 0.346410i
\(76\) 0 0
\(77\) 0 0
\(78\) 10.0000i 1.13228i
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 2.00000i 0.111803 0.223607i
\(81\) 1.00000 0.111111
\(82\) 20.0000i 2.20863i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) −8.94427 4.47214i −0.970143 0.485071i
\(86\) 20.0000 2.15666
\(87\) 8.94427i 0.958927i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) −4.47214 2.23607i −0.471405 0.235702i
\(91\) 0 0
\(92\) 12.0000i 1.25109i
\(93\) 0 0
\(94\) 26.8328 2.76759
\(95\) 0 0
\(96\) −6.70820 −0.684653
\(97\) 8.00000i 0.812277i 0.913812 + 0.406138i \(0.133125\pi\)
−0.913812 + 0.406138i \(0.866875\pi\)
\(98\) 15.6525i 1.58114i
\(99\) 0 0
\(100\) 9.00000 + 12.0000i 0.900000 + 1.20000i
\(101\) 8.94427 0.889988 0.444994 0.895533i \(-0.353206\pi\)
0.444994 + 0.895533i \(0.353206\pi\)
\(102\) 10.0000i 0.990148i
\(103\) 4.00000i 0.394132i −0.980390 0.197066i \(-0.936859\pi\)
0.980390 0.197066i \(-0.0631413\pi\)
\(104\) 10.0000 0.980581
\(105\) 0 0
\(106\) −8.94427 −0.868744
\(107\) 17.8885i 1.72935i −0.502331 0.864675i \(-0.667524\pi\)
0.502331 0.864675i \(-0.332476\pi\)
\(108\) 3.00000i 0.288675i
\(109\) −17.8885 −1.71341 −0.856706 0.515805i \(-0.827493\pi\)
−0.856706 + 0.515805i \(0.827493\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) 4.00000i 0.376288i −0.982141 0.188144i \(-0.939753\pi\)
0.982141 0.188144i \(-0.0602472\pi\)
\(114\) 0 0
\(115\) 8.00000 + 4.00000i 0.746004 + 0.373002i
\(116\) −26.8328 −2.49136
\(117\) 4.47214i 0.413449i
\(118\) 0 0
\(119\) 0 0
\(120\) 2.23607 4.47214i 0.204124 0.408248i
\(121\) 0 0
\(122\) 0 0
\(123\) 8.94427i 0.806478i
\(124\) 0 0
\(125\) 11.0000 2.00000i 0.983870 0.178885i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 15.6525i 1.38350i
\(129\) −8.94427 −0.787499
\(130\) 10.0000 20.0000i 0.877058 1.75412i
\(131\) 17.8885 1.56293 0.781465 0.623949i \(-0.214473\pi\)
0.781465 + 0.623949i \(0.214473\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 26.8328 2.31800
\(135\) 2.00000 + 1.00000i 0.172133 + 0.0860663i
\(136\) −10.0000 −0.857493
\(137\) 12.0000i 1.02523i −0.858619 0.512615i \(-0.828677\pi\)
0.858619 0.512615i \(-0.171323\pi\)
\(138\) 8.94427i 0.761387i
\(139\) 17.8885 1.51729 0.758643 0.651506i \(-0.225863\pi\)
0.758643 + 0.651506i \(0.225863\pi\)
\(140\) 0 0
\(141\) −12.0000 −1.01058
\(142\) 26.8328i 2.25176i
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −8.94427 + 17.8885i −0.742781 + 1.48556i
\(146\) 30.0000 2.48282
\(147\) 7.00000i 0.577350i
\(148\) 24.0000i 1.97279i
\(149\) 8.94427 0.732743 0.366372 0.930469i \(-0.380600\pi\)
0.366372 + 0.930469i \(0.380600\pi\)
\(150\) −6.70820 8.94427i −0.547723 0.730297i
\(151\) −17.8885 −1.45575 −0.727875 0.685710i \(-0.759492\pi\)
−0.727875 + 0.685710i \(0.759492\pi\)
\(152\) 0 0
\(153\) 4.47214i 0.361551i
\(154\) 0 0
\(155\) 0 0
\(156\) −13.4164 −1.07417
\(157\) 8.00000i 0.638470i 0.947676 + 0.319235i \(0.103426\pi\)
−0.947676 + 0.319235i \(0.896574\pi\)
\(158\) 0 0
\(159\) 4.00000 0.317221
\(160\) −13.4164 6.70820i −1.06066 0.530330i
\(161\) 0 0
\(162\) 2.23607i 0.175682i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) −26.8328 −2.09529
\(165\) 0 0
\(166\) 0 0
\(167\) 8.94427i 0.692129i 0.938211 + 0.346064i \(0.112482\pi\)
−0.938211 + 0.346064i \(0.887518\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.538462
\(170\) −10.0000 + 20.0000i −0.766965 + 1.53393i
\(171\) 0 0
\(172\) 26.8328i 2.04598i
\(173\) 13.4164i 1.02003i −0.860165 0.510015i \(-0.829640\pi\)
0.860165 0.510015i \(-0.170360\pi\)
\(174\) 20.0000 1.51620
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 13.4164i 1.00560i
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) −3.00000 + 6.00000i −0.223607 + 0.447214i
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 8.94427 0.659380
\(185\) −16.0000 8.00000i −1.17634 0.588172i
\(186\) 0 0
\(187\) 0 0
\(188\) 36.0000i 2.62557i
\(189\) 0 0
\(190\) 0 0
\(191\) −20.0000 −1.44715 −0.723575 0.690246i \(-0.757502\pi\)
−0.723575 + 0.690246i \(0.757502\pi\)
\(192\) 13.0000i 0.938194i
\(193\) 4.47214i 0.321911i −0.986962 0.160956i \(-0.948542\pi\)
0.986962 0.160956i \(-0.0514576\pi\)
\(194\) 17.8885 1.28432
\(195\) −4.47214 + 8.94427i −0.320256 + 0.640513i
\(196\) −21.0000 −1.50000
\(197\) 4.47214i 0.318626i −0.987228 0.159313i \(-0.949072\pi\)
0.987228 0.159313i \(-0.0509280\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 8.94427 6.70820i 0.632456 0.474342i
\(201\) −12.0000 −0.846415
\(202\) 20.0000i 1.40720i
\(203\) 0 0
\(204\) 13.4164 0.939336
\(205\) −8.94427 + 17.8885i −0.624695 + 1.24939i
\(206\) −8.94427 −0.623177
\(207\) 4.00000i 0.278019i
\(208\) 4.47214i 0.310087i
\(209\) 0 0
\(210\) 0 0
\(211\) 17.8885 1.23150 0.615749 0.787942i \(-0.288854\pi\)
0.615749 + 0.787942i \(0.288854\pi\)
\(212\) 12.0000i 0.824163i
\(213\) 12.0000i 0.822226i
\(214\) −40.0000 −2.73434
\(215\) −17.8885 8.94427i −1.21999 0.609994i
\(216\) 2.23607 0.152145
\(217\) 0 0
\(218\) 40.0000i 2.70914i
\(219\) −13.4164 −0.906597
\(220\) 0 0
\(221\) 20.0000 1.34535
\(222\) 17.8885i 1.20060i
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 0 0
\(225\) 3.00000 + 4.00000i 0.200000 + 0.266667i
\(226\) −8.94427 −0.594964
\(227\) 17.8885i 1.18730i 0.804722 + 0.593652i \(0.202314\pi\)
−0.804722 + 0.593652i \(0.797686\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 8.94427 17.8885i 0.589768 1.17954i
\(231\) 0 0
\(232\) 20.0000i 1.31306i
\(233\) 22.3607i 1.46490i 0.680823 + 0.732448i \(0.261622\pi\)
−0.680823 + 0.732448i \(0.738378\pi\)
\(234\) 10.0000 0.653720
\(235\) −24.0000 12.0000i −1.56559 0.782794i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 17.8885 1.15711 0.578557 0.815642i \(-0.303616\pi\)
0.578557 + 0.815642i \(0.303616\pi\)
\(240\) 2.00000 + 1.00000i 0.129099 + 0.0645497i
\(241\) 17.8885 1.15230 0.576151 0.817343i \(-0.304554\pi\)
0.576151 + 0.817343i \(0.304554\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) −7.00000 + 14.0000i −0.447214 + 0.894427i
\(246\) 20.0000 1.27515
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −4.47214 24.5967i −0.282843 1.55563i
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 4.47214 8.94427i 0.280056 0.560112i
\(256\) −9.00000 −0.562500
\(257\) 12.0000i 0.748539i 0.927320 + 0.374270i \(0.122107\pi\)
−0.927320 + 0.374270i \(0.877893\pi\)
\(258\) 20.0000i 1.24515i
\(259\) 0 0
\(260\) −26.8328 13.4164i −1.66410 0.832050i
\(261\) −8.94427 −0.553637
\(262\) 40.0000i 2.47121i
\(263\) 8.94427i 0.551527i −0.961225 0.275764i \(-0.911069\pi\)
0.961225 0.275764i \(-0.0889307\pi\)
\(264\) 0 0
\(265\) 8.00000 + 4.00000i 0.491436 + 0.245718i
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) 36.0000i 2.19905i
\(269\) 10.0000 0.609711 0.304855 0.952399i \(-0.401392\pi\)
0.304855 + 0.952399i \(0.401392\pi\)
\(270\) 2.23607 4.47214i 0.136083 0.272166i
\(271\) −17.8885 −1.08665 −0.543326 0.839522i \(-0.682835\pi\)
−0.543326 + 0.839522i \(0.682835\pi\)
\(272\) 4.47214i 0.271163i
\(273\) 0 0
\(274\) −26.8328 −1.62103
\(275\) 0 0
\(276\) −12.0000 −0.722315
\(277\) 13.4164i 0.806114i −0.915175 0.403057i \(-0.867948\pi\)
0.915175 0.403057i \(-0.132052\pi\)
\(278\) 40.0000i 2.39904i
\(279\) 0 0
\(280\) 0 0
\(281\) 8.94427 0.533571 0.266785 0.963756i \(-0.414039\pi\)
0.266785 + 0.963756i \(0.414039\pi\)
\(282\) 26.8328i 1.59787i
\(283\) 8.94427i 0.531682i −0.964017 0.265841i \(-0.914350\pi\)
0.964017 0.265841i \(-0.0856496\pi\)
\(284\) −36.0000 −2.13621
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 6.70820i 0.395285i
\(289\) −3.00000 −0.176471
\(290\) 40.0000 + 20.0000i 2.34888 + 1.17444i
\(291\) −8.00000 −0.468968
\(292\) 40.2492i 2.35541i
\(293\) 4.47214i 0.261265i −0.991431 0.130632i \(-0.958299\pi\)
0.991431 0.130632i \(-0.0417008\pi\)
\(294\) 15.6525 0.912871
\(295\) 0 0
\(296\) −17.8885 −1.03975
\(297\) 0 0
\(298\) 20.0000i 1.15857i
\(299\) −17.8885 −1.03452
\(300\) −12.0000 + 9.00000i −0.692820 + 0.519615i
\(301\) 0 0
\(302\) 40.0000i 2.30174i
\(303\) 8.94427i 0.513835i
\(304\) 0 0
\(305\) 0 0
\(306\) −10.0000 −0.571662
\(307\) 8.94427i 0.510477i −0.966878 0.255238i \(-0.917846\pi\)
0.966878 0.255238i \(-0.0821539\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 10.0000i 0.566139i
\(313\) 16.0000i 0.904373i −0.891923 0.452187i \(-0.850644\pi\)
0.891923 0.452187i \(-0.149356\pi\)
\(314\) 17.8885 1.00951
\(315\) 0 0
\(316\) 0 0
\(317\) 12.0000i 0.673987i 0.941507 + 0.336994i \(0.109410\pi\)
−0.941507 + 0.336994i \(0.890590\pi\)
\(318\) 8.94427i 0.501570i
\(319\) 0 0
\(320\) −13.0000 + 26.0000i −0.726722 + 1.45344i
\(321\) 17.8885 0.998441
\(322\) 0 0
\(323\) 0 0
\(324\) −3.00000 −0.166667
\(325\) −17.8885 + 13.4164i −0.992278 + 0.744208i
\(326\) 8.94427 0.495377
\(327\) 17.8885i 0.989239i
\(328\) 20.0000i 1.10432i
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) 8.00000i 0.438397i
\(334\) 20.0000 1.09435
\(335\) −24.0000 12.0000i −1.31126 0.655630i
\(336\) 0 0
\(337\) 31.3050i 1.70529i −0.522491 0.852645i \(-0.674997\pi\)
0.522491 0.852645i \(-0.325003\pi\)
\(338\) 15.6525i 0.851382i
\(339\) 4.00000 0.217250
\(340\) 26.8328 + 13.4164i 1.45521 + 0.727607i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −20.0000 −1.07833
\(345\) −4.00000 + 8.00000i −0.215353 + 0.430706i
\(346\) −30.0000 −1.61281
\(347\) 17.8885i 0.960307i 0.877184 + 0.480154i \(0.159419\pi\)
−0.877184 + 0.480154i \(0.840581\pi\)
\(348\) 26.8328i 1.43839i
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) −4.47214 −0.238705
\(352\) 0 0
\(353\) 4.00000i 0.212899i −0.994318 0.106449i \(-0.966052\pi\)
0.994318 0.106449i \(-0.0339482\pi\)
\(354\) 0 0
\(355\) −12.0000 + 24.0000i −0.636894 + 1.27379i
\(356\) 18.0000 0.953998
\(357\) 0 0
\(358\) 35.7771i 1.89088i
\(359\) 17.8885 0.944121 0.472061 0.881566i \(-0.343510\pi\)
0.472061 + 0.881566i \(0.343510\pi\)
\(360\) 4.47214 + 2.23607i 0.235702 + 0.117851i
\(361\) −19.0000 −1.00000
\(362\) 22.3607i 1.17525i
\(363\) 0 0
\(364\) 0 0
\(365\) −26.8328 13.4164i −1.40449 0.702247i
\(366\) 0 0
\(367\) 28.0000i 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) 4.00000i 0.208514i
\(369\) −8.94427 −0.465620
\(370\) −17.8885 + 35.7771i −0.929981 + 1.85996i
\(371\) 0 0
\(372\) 0 0
\(373\) 22.3607i 1.15779i 0.815401 + 0.578896i \(0.196516\pi\)
−0.815401 + 0.578896i \(0.803484\pi\)
\(374\) 0 0
\(375\) 2.00000 + 11.0000i 0.103280 + 0.568038i
\(376\) −26.8328 −1.38380
\(377\) 40.0000i 2.06010i
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 44.7214i 2.28814i
\(383\) 4.00000i 0.204390i 0.994764 + 0.102195i \(0.0325866\pi\)
−0.994764 + 0.102195i \(0.967413\pi\)
\(384\) 15.6525 0.798762
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 8.94427i 0.454663i
\(388\) 24.0000i 1.21842i
\(389\) 30.0000 1.52106 0.760530 0.649303i \(-0.224939\pi\)
0.760530 + 0.649303i \(0.224939\pi\)
\(390\) 20.0000 + 10.0000i 1.01274 + 0.506370i
\(391\) 17.8885 0.904663
\(392\) 15.6525i 0.790569i
\(393\) 17.8885i 0.902358i
\(394\) −10.0000 −0.503793
\(395\) 0 0
\(396\) 0 0
\(397\) 8.00000i 0.401508i 0.979642 + 0.200754i \(0.0643393\pi\)
−0.979642 + 0.200754i \(0.935661\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 3.00000 + 4.00000i 0.150000 + 0.200000i
\(401\) 30.0000 1.49813 0.749064 0.662497i \(-0.230503\pi\)
0.749064 + 0.662497i \(0.230503\pi\)
\(402\) 26.8328i 1.33830i
\(403\) 0 0
\(404\) −26.8328 −1.33498
\(405\) −1.00000 + 2.00000i −0.0496904 + 0.0993808i
\(406\) 0 0
\(407\) 0 0
\(408\) 10.0000i 0.495074i
\(409\) 17.8885 0.884532 0.442266 0.896884i \(-0.354175\pi\)
0.442266 + 0.896884i \(0.354175\pi\)
\(410\) 40.0000 + 20.0000i 1.97546 + 0.987730i
\(411\) 12.0000 0.591916
\(412\) 12.0000i 0.591198i
\(413\) 0 0
\(414\) 8.94427 0.439587
\(415\) 0 0
\(416\) 30.0000 1.47087
\(417\) 17.8885i 0.876006i
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 40.0000i 1.94717i
\(423\) 12.0000i 0.583460i
\(424\) 8.94427 0.434372
\(425\) 17.8885 13.4164i 0.867722 0.650791i
\(426\) 26.8328 1.30005
\(427\) 0 0
\(428\) 53.6656i 2.59403i
\(429\) 0 0
\(430\) −20.0000 + 40.0000i −0.964486 + 1.92897i
\(431\) −17.8885 −0.861661 −0.430830 0.902433i \(-0.641779\pi\)
−0.430830 + 0.902433i \(0.641779\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 16.0000i 0.768911i −0.923144 0.384455i \(-0.874389\pi\)
0.923144 0.384455i \(-0.125611\pi\)
\(434\) 0 0
\(435\) −17.8885 8.94427i −0.857690 0.428845i
\(436\) 53.6656 2.57012
\(437\) 0 0
\(438\) 30.0000i 1.43346i
\(439\) 17.8885 0.853774 0.426887 0.904305i \(-0.359610\pi\)
0.426887 + 0.904305i \(0.359610\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 44.7214i 2.12718i
\(443\) 36.0000i 1.71041i 0.518289 + 0.855206i \(0.326569\pi\)
−0.518289 + 0.855206i \(0.673431\pi\)
\(444\) 24.0000 1.13899
\(445\) 6.00000 12.0000i 0.284427 0.568855i
\(446\) 8.94427 0.423524
\(447\) 8.94427i 0.423050i
\(448\) 0 0
\(449\) −34.0000 −1.60456 −0.802280 0.596948i \(-0.796380\pi\)
−0.802280 + 0.596948i \(0.796380\pi\)
\(450\) 8.94427 6.70820i 0.421637 0.316228i
\(451\) 0 0
\(452\) 12.0000i 0.564433i
\(453\) 17.8885i 0.840477i
\(454\) 40.0000 1.87729
\(455\) 0 0
\(456\) 0 0
\(457\) 13.4164i 0.627593i 0.949490 + 0.313797i \(0.101601\pi\)
−0.949490 + 0.313797i \(0.898399\pi\)
\(458\) 22.3607i 1.04485i
\(459\) 4.47214 0.208741
\(460\) −24.0000 12.0000i −1.11901 0.559503i
\(461\) 26.8328 1.24973 0.624864 0.780733i \(-0.285154\pi\)
0.624864 + 0.780733i \(0.285154\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) −8.94427 −0.415227
\(465\) 0 0
\(466\) 50.0000 2.31621
\(467\) 28.0000i 1.29569i 0.761774 + 0.647843i \(0.224329\pi\)
−0.761774 + 0.647843i \(0.775671\pi\)
\(468\) 13.4164i 0.620174i
\(469\) 0 0
\(470\) −26.8328 + 53.6656i −1.23771 + 2.47541i
\(471\) −8.00000 −0.368621
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 4.00000i 0.183147i
\(478\) 40.0000i 1.82956i
\(479\) 17.8885 0.817348 0.408674 0.912680i \(-0.365991\pi\)
0.408674 + 0.912680i \(0.365991\pi\)
\(480\) 6.70820 13.4164i 0.306186 0.612372i
\(481\) 35.7771 1.63129
\(482\) 40.0000i 1.82195i
\(483\) 0 0
\(484\) 0 0
\(485\) −16.0000 8.00000i −0.726523 0.363261i
\(486\) 2.23607 0.101430
\(487\) 28.0000i 1.26880i −0.773004 0.634401i \(-0.781247\pi\)
0.773004 0.634401i \(-0.218753\pi\)
\(488\) 0 0
\(489\) −4.00000 −0.180886
\(490\) 31.3050 + 15.6525i 1.41421 + 0.707107i
\(491\) −35.7771 −1.61460 −0.807299 0.590143i \(-0.799071\pi\)
−0.807299 + 0.590143i \(0.799071\pi\)
\(492\) 26.8328i 1.20972i
\(493\) 40.0000i 1.80151i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −33.0000 + 6.00000i −1.47580 + 0.268328i
\(501\) −8.94427 −0.399601
\(502\) 17.8885i 0.798405i
\(503\) 26.8328i 1.19642i 0.801341 + 0.598208i \(0.204120\pi\)
−0.801341 + 0.598208i \(0.795880\pi\)
\(504\) 0 0
\(505\) −8.94427 + 17.8885i −0.398015 + 0.796030i
\(506\) 0 0
\(507\) 7.00000i 0.310881i
\(508\) 0 0
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) −20.0000 10.0000i −0.885615 0.442807i
\(511\) 0 0
\(512\) 11.1803i 0.494106i
\(513\) 0 0
\(514\) 26.8328 1.18354
\(515\) 8.00000 + 4.00000i 0.352522 + 0.176261i
\(516\) 26.8328 1.18125
\(517\) 0 0
\(518\) 0 0
\(519\) 13.4164 0.588915
\(520\) −10.0000 + 20.0000i −0.438529 + 0.877058i
\(521\) −10.0000 −0.438108 −0.219054 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 20.0000i 0.875376i
\(523\) 8.94427i 0.391106i −0.980693 0.195553i \(-0.937350\pi\)
0.980693 0.195553i \(-0.0626501\pi\)
\(524\) −53.6656 −2.34439
\(525\) 0 0
\(526\) −20.0000 −0.872041
\(527\) 0 0
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 8.94427 17.8885i 0.388514 0.777029i
\(531\) 0 0
\(532\) 0 0
\(533\) 40.0000i 1.73259i
\(534\) −13.4164 −0.580585
\(535\) 35.7771 + 17.8885i 1.54678 + 0.773389i
\(536\) −26.8328 −1.15900
\(537\) 16.0000i 0.690451i
\(538\) 22.3607i 0.964037i
\(539\) 0 0
\(540\) −6.00000 3.00000i −0.258199 0.129099i
\(541\) 17.8885 0.769089 0.384544 0.923107i \(-0.374359\pi\)
0.384544 + 0.923107i \(0.374359\pi\)
\(542\) 40.0000i 1.71815i
\(543\) 10.0000i 0.429141i
\(544\) −30.0000 −1.28624
\(545\) 17.8885 35.7771i 0.766261 1.53252i
\(546\) 0 0
\(547\) 44.7214i 1.91215i 0.293127 + 0.956074i \(0.405304\pi\)
−0.293127 + 0.956074i \(0.594696\pi\)
\(548\) 36.0000i 1.53784i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 8.94427i 0.380693i
\(553\) 0 0
\(554\) −30.0000 −1.27458
\(555\) 8.00000 16.0000i 0.339581 0.679162i
\(556\) −53.6656 −2.27593
\(557\) 13.4164i 0.568471i 0.958754 + 0.284236i \(0.0917398\pi\)
−0.958754 + 0.284236i \(0.908260\pi\)
\(558\) 0 0
\(559\) 40.0000 1.69182
\(560\) 0 0
\(561\) 0 0
\(562\) 20.0000i 0.843649i
\(563\) 17.8885i 0.753912i −0.926231 0.376956i \(-0.876971\pi\)
0.926231 0.376956i \(-0.123029\pi\)
\(564\) 36.0000 1.51587
\(565\) 8.00000 + 4.00000i 0.336563 + 0.168281i
\(566\) −20.0000 −0.840663
\(567\) 0 0
\(568\) 26.8328i 1.12588i
\(569\) −8.94427 −0.374963 −0.187482 0.982268i \(-0.560033\pi\)
−0.187482 + 0.982268i \(0.560033\pi\)
\(570\) 0 0
\(571\) −35.7771 −1.49722 −0.748612 0.663008i \(-0.769280\pi\)
−0.748612 + 0.663008i \(0.769280\pi\)
\(572\) 0 0
\(573\) 20.0000i 0.835512i
\(574\) 0 0
\(575\) −16.0000 + 12.0000i −0.667246 + 0.500435i
\(576\) −13.0000 −0.541667
\(577\) 32.0000i 1.33218i 0.745873 + 0.666089i \(0.232033\pi\)
−0.745873 + 0.666089i \(0.767967\pi\)
\(578\) 6.70820i 0.279024i
\(579\) 4.47214 0.185856
\(580\) 26.8328 53.6656i 1.11417 2.22834i
\(581\) 0 0
\(582\) 17.8885i 0.741504i
\(583\) 0 0
\(584\) −30.0000 −1.24141
\(585\) −8.94427 4.47214i −0.369800 0.184900i
\(586\) −10.0000 −0.413096
\(587\) 28.0000i 1.15568i 0.816149 + 0.577842i \(0.196105\pi\)
−0.816149 + 0.577842i \(0.803895\pi\)
\(588\) 21.0000i 0.866025i
\(589\) 0 0
\(590\) 0 0
\(591\) 4.47214 0.183959
\(592\) 8.00000i 0.328798i
\(593\) 4.47214i 0.183649i −0.995775 0.0918243i \(-0.970730\pi\)
0.995775 0.0918243i \(-0.0292698\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −26.8328 −1.09911
\(597\) 0 0
\(598\) 40.0000i 1.63572i
\(599\) −4.00000 −0.163436 −0.0817178 0.996656i \(-0.526041\pi\)
−0.0817178 + 0.996656i \(0.526041\pi\)
\(600\) 6.70820 + 8.94427i 0.273861 + 0.365148i
\(601\) −35.7771 −1.45938 −0.729689 0.683779i \(-0.760335\pi\)
−0.729689 + 0.683779i \(0.760335\pi\)
\(602\) 0 0
\(603\) 12.0000i 0.488678i
\(604\) 53.6656 2.18362
\(605\) 0 0
\(606\) 20.0000 0.812444
\(607\) 17.8885i 0.726074i −0.931775 0.363037i \(-0.881740\pi\)
0.931775 0.363037i \(-0.118260\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 53.6656 2.17108
\(612\) 13.4164i 0.542326i
\(613\) 13.4164i 0.541884i −0.962596 0.270942i \(-0.912665\pi\)
0.962596 0.270942i \(-0.0873351\pi\)
\(614\) −20.0000 −0.807134
\(615\) −17.8885 8.94427i −0.721336 0.360668i
\(616\) 0 0
\(617\) 12.0000i 0.483102i 0.970388 + 0.241551i \(0.0776561\pi\)
−0.970388 + 0.241551i \(0.922344\pi\)
\(618\) 8.94427i 0.359791i
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) −4.00000 −0.160514
\(622\) 26.8328i 1.07590i
\(623\) 0 0
\(624\) −4.47214 −0.179029
\(625\) −7.00000 + 24.0000i −0.280000 + 0.960000i
\(626\) −35.7771 −1.42994
\(627\) 0 0
\(628\) 24.0000i 0.957704i
\(629\) −35.7771 −1.42653
\(630\) 0 0
\(631\) −8.00000 −0.318475 −0.159237 0.987240i \(-0.550904\pi\)
−0.159237 + 0.987240i \(0.550904\pi\)
\(632\) 0 0
\(633\) 17.8885i 0.711006i
\(634\) 26.8328 1.06567
\(635\) 0 0
\(636\) −12.0000 −0.475831
\(637\) 31.3050i 1.24035i
\(638\) 0 0
\(639\) −12.0000 −0.474713
\(640\) 31.3050 + 15.6525i 1.23744 + 0.618718i
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 40.0000i 1.57867i
\(643\) 36.0000i 1.41970i 0.704352 + 0.709851i \(0.251238\pi\)
−0.704352 + 0.709851i \(0.748762\pi\)
\(644\) 0 0
\(645\) 8.94427 17.8885i 0.352180 0.704361i
\(646\) 0 0
\(647\) 28.0000i 1.10079i −0.834903 0.550397i \(-0.814476\pi\)
0.834903 0.550397i \(-0.185524\pi\)
\(648\) 2.23607i 0.0878410i
\(649\) 0 0
\(650\) 30.0000 + 40.0000i 1.17670 + 1.56893i
\(651\) 0 0
\(652\) 12.0000i 0.469956i
\(653\) 4.00000i 0.156532i −0.996933 0.0782660i \(-0.975062\pi\)
0.996933 0.0782660i \(-0.0249384\pi\)
\(654\) −40.0000 −1.56412
\(655\) −17.8885 + 35.7771i −0.698963 + 1.39793i
\(656\) −8.94427 −0.349215
\(657\) 13.4164i 0.523424i
\(658\) 0 0
\(659\) 17.8885 0.696839 0.348419 0.937339i \(-0.386719\pi\)
0.348419 + 0.937339i \(0.386719\pi\)
\(660\) 0 0
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 44.7214i 1.73814i
\(663\) 20.0000i 0.776736i
\(664\) 0 0
\(665\) 0 0
\(666\) −17.8885 −0.693167
\(667\) 35.7771i 1.38529i
\(668\) 26.8328i 1.03819i
\(669\) −4.00000 −0.154649
\(670\) −26.8328 + 53.6656i −1.03664 + 2.07328i
\(671\) 0 0
\(672\) 0 0
\(673\) 4.47214i 0.172388i −0.996278 0.0861941i \(-0.972529\pi\)
0.996278 0.0861941i \(-0.0274705\pi\)
\(674\) −70.0000 −2.69630
\(675\) −4.00000 + 3.00000i −0.153960 + 0.115470i
\(676\) 21.0000 0.807692
\(677\) 31.3050i 1.20315i −0.798817 0.601574i \(-0.794541\pi\)
0.798817 0.601574i \(-0.205459\pi\)
\(678\) 8.94427i 0.343503i
\(679\) 0 0
\(680\) 10.0000 20.0000i 0.383482 0.766965i
\(681\) −17.8885 −0.685490
\(682\) 0 0
\(683\) 36.0000i 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 0 0
\(685\) 24.0000 + 12.0000i 0.916993 + 0.458496i
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) 8.94427i 0.340997i
\(689\) −17.8885 −0.681499
\(690\) 17.8885 + 8.94427i 0.681005 + 0.340503i
\(691\) 12.0000 0.456502 0.228251 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 40.2492i 1.53005i
\(693\) 0 0
\(694\) 40.0000 1.51838
\(695\) −17.8885 + 35.7771i −0.678551 + 1.35710i
\(696\) −20.0000 −0.758098
\(697\) 40.0000i 1.51511i
\(698\) 0 0
\(699\) −22.3607 −0.845759
\(700\) 0 0
\(701\) 8.94427 0.337820 0.168910 0.985631i \(-0.445975\pi\)
0.168910 + 0.985631i \(0.445975\pi\)
\(702\) 10.0000i 0.377426i
\(703\) 0 0
\(704\) 0 0
\(705\) 12.0000 24.0000i 0.451946 0.903892i
\(706\) −8.94427 −0.336622
\(707\) 0 0
\(708\) 0 0
\(709\) 6.00000 0.225335 0.112667 0.993633i \(-0.464061\pi\)
0.112667 + 0.993633i \(0.464061\pi\)
\(710\) 53.6656 + 26.8328i 2.01404 + 1.00702i
\(711\) 0 0
\(712\) 13.4164i 0.502801i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 48.0000 1.79384
\(717\) 17.8885i 0.668060i
\(718\) 40.0000i 1.49279i
\(719\) −36.0000 −1.34257 −0.671287 0.741198i \(-0.734258\pi\)
−0.671287 + 0.741198i \(0.734258\pi\)
\(720\) −1.00000 + 2.00000i −0.0372678 + 0.0745356i
\(721\) 0 0
\(722\) 42.4853i 1.58114i
\(723\) 17.8885i 0.665282i
\(724\) 30.0000 1.11494
\(725\) −26.8328 35.7771i −0.996546 1.32873i
\(726\) 0 0
\(727\) 28.0000i 1.03846i 0.854634 + 0.519231i \(0.173782\pi\)
−0.854634 + 0.519231i \(0.826218\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) −30.0000 + 60.0000i −1.11035 + 2.22070i
\(731\) −40.0000 −1.47945
\(732\) 0 0
\(733\) 31.3050i 1.15627i −0.815939 0.578137i \(-0.803780\pi\)
0.815939 0.578137i \(-0.196220\pi\)
\(734\) −62.6099 −2.31097
\(735\) −14.0000 7.00000i −0.516398 0.258199i
\(736\) 26.8328 0.989071
\(737\) 0 0
\(738\) 20.0000i 0.736210i
\(739\) −17.8885 −0.658041 −0.329020 0.944323i \(-0.606718\pi\)
−0.329020 + 0.944323i \(0.606718\pi\)
\(740\) 48.0000 + 24.0000i 1.76452 + 0.882258i
\(741\) 0 0
\(742\) 0 0
\(743\) 26.8328i 0.984401i −0.870482 0.492200i \(-0.836193\pi\)
0.870482 0.492200i \(-0.163807\pi\)
\(744\) 0 0
\(745\) −8.94427 + 17.8885i −0.327693 + 0.655386i
\(746\) 50.0000 1.83063
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 24.5967 4.47214i 0.898146 0.163299i
\(751\) −48.0000 −1.75154 −0.875772 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) 12.0000i 0.437595i
\(753\) 8.00000i 0.291536i
\(754\) −89.4427 −3.25731
\(755\) 17.8885 35.7771i 0.651031 1.30206i
\(756\) 0 0
\(757\) 48.0000i 1.74459i 0.488980 + 0.872295i \(0.337369\pi\)
−0.488980 + 0.872295i \(0.662631\pi\)
\(758\) 44.7214i 1.62435i
\(759\) 0 0
\(760\) 0 0
\(761\) 8.94427 0.324230 0.162115 0.986772i \(-0.448169\pi\)
0.162115 + 0.986772i \(0.448169\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 60.0000 2.17072
\(765\) 8.94427 + 4.47214i 0.323381 + 0.161690i
\(766\) 8.94427 0.323170
\(767\) 0 0
\(768\) 9.00000i 0.324760i
\(769\) −35.7771 −1.29015 −0.645077 0.764117i \(-0.723175\pi\)
−0.645077 + 0.764117i \(0.723175\pi\)
\(770\) 0 0
\(771\) −12.0000 −0.432169
\(772\) 13.4164i 0.482867i
\(773\) 36.0000i 1.29483i −0.762138 0.647415i \(-0.775850\pi\)
0.762138 0.647415i \(-0.224150\pi\)
\(774\) −20.0000 −0.718885
\(775\) 0 0
\(776\) −17.8885 −0.642161
\(777\) 0 0
\(778\) 67.0820i 2.40501i
\(779\) 0 0
\(780\) 13.4164 26.8328i 0.480384 0.960769i
\(781\) 0 0
\(782\) 40.0000i 1.43040i
\(783\) 8.94427i 0.319642i
\(784\) −7.00000 −0.250000
\(785\) −16.0000 8.00000i −0.571064 0.285532i
\(786\) 40.0000 1.42675
\(787\) 26.8328i 0.956487i −0.878227 0.478243i \(-0.841274\pi\)
0.878227 0.478243i \(-0.158726\pi\)
\(788\) 13.4164i 0.477940i
\(789\) 8.94427 0.318425
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 17.8885 0.634841
\(795\) −4.00000 + 8.00000i −0.141865 + 0.283731i
\(796\) 0 0
\(797\) 12.0000i 0.425062i −0.977154 0.212531i \(-0.931829\pi\)
0.977154 0.212531i \(-0.0681706\pi\)
\(798\) 0 0
\(799\) −53.6656 −1.89855
\(800\) 26.8328 20.1246i 0.948683 0.711512i
\(801\) 6.00000 0.212000
\(802\) 67.0820i 2.36875i
\(803\) 0 0
\(804\) 36.0000 1.26962
\(805\) 0 0
\(806\) 0 0
\(807\) 10.0000i 0.352017i
\(808\) 20.0000i 0.703598i
\(809\) −26.8328 −0.943392 −0.471696 0.881761i \(-0.656358\pi\)
−0.471696 + 0.881761i \(0.656358\pi\)
\(810\) 4.47214 + 2.23607i 0.157135 + 0.0785674i
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 17.8885i 0.627379i
\(814\) 0 0
\(815\) −8.00000 4.00000i −0.280228 0.140114i
\(816\) 4.47214 0.156556
\(817\) 0 0
\(818\) 40.0000i 1.39857i
\(819\) 0 0
\(820\) 26.8328 53.6656i 0.937043 1.87409i
\(821\) −8.94427 −0.312157 −0.156079 0.987745i \(-0.549885\pi\)
−0.156079 + 0.987745i \(0.549885\pi\)
\(822\) 26.8328i 0.935902i
\(823\) 4.00000i 0.139431i −0.997567 0.0697156i \(-0.977791\pi\)
0.997567 0.0697156i \(-0.0222092\pi\)
\(824\) 8.94427 0.311588
\(825\) 0 0
\(826\) 0 0
\(827\) 17.8885i 0.622046i 0.950402 + 0.311023i \(0.100672\pi\)
−0.950402 + 0.311023i \(0.899328\pi\)
\(828\) 12.0000i 0.417029i
\(829\) 46.0000 1.59765 0.798823 0.601566i \(-0.205456\pi\)
0.798823 + 0.601566i \(0.205456\pi\)
\(830\) 0 0
\(831\) 13.4164 0.465410
\(832\) 58.1378i 2.01556i
\(833\) 31.3050i 1.08465i
\(834\) 40.0000 1.38509
\(835\) −17.8885 8.94427i −0.619059 0.309529i
\(836\) 0 0
\(837\) 0 0
\(838\) 53.6656i 1.85385i
\(839\) 20.0000 0.690477 0.345238 0.938515i \(-0.387798\pi\)
0.345238 + 0.938515i \(0.387798\pi\)
\(840\) 0 0
\(841\) 51.0000 1.75862
\(842\) 22.3607i 0.770600i
\(843\) 8.94427i 0.308057i
\(844\) −53.6656 −1.84725
\(845\) 7.00000 14.0000i 0.240807 0.481615i
\(846\) −26.8328 −0.922531
\(847\) 0 0
\(848\) 4.00000i 0.137361i
\(849\) 8.94427 0.306967
\(850\) −30.0000 40.0000i −1.02899 1.37199i
\(851\) 32.0000 1.09695
\(852\) 36.0000i 1.23334i
\(853\) 49.1935i 1.68435i −0.539202 0.842177i \(-0.681274\pi\)
0.539202 0.842177i \(-0.318726\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 40.0000 1.36717
\(857\) 13.4164i 0.458296i −0.973392 0.229148i \(-0.926406\pi\)
0.973392 0.229148i \(-0.0735939\pi\)
\(858\) 0 0
\(859\) −44.0000 −1.50126 −0.750630 0.660722i \(-0.770250\pi\)
−0.750630 + 0.660722i \(0.770250\pi\)
\(860\) 53.6656 + 26.8328i 1.82998 + 0.914991i
\(861\) 0 0
\(862\) 40.0000i 1.36241i
\(863\) 4.00000i 0.136162i 0.997680 + 0.0680808i \(0.0216876\pi\)
−0.997680 + 0.0680808i \(0.978312\pi\)
\(864\) 6.70820 0.228218
\(865\) 26.8328 + 13.4164i 0.912343 + 0.456172i
\(866\) −35.7771 −1.21575
\(867\) 3.00000i 0.101885i
\(868\) 0 0
\(869\) 0 0
\(870\) −20.0000 + 40.0000i −0.678064 + 1.35613i
\(871\) 53.6656 1.81839
\(872\) 40.0000i 1.35457i
\(873\) 8.00000i 0.270759i
\(874\) 0 0
\(875\) 0 0
\(876\) 40.2492 1.35990
\(877\) 31.3050i 1.05709i 0.848904 + 0.528547i \(0.177263\pi\)
−0.848904 + 0.528547i \(0.822737\pi\)
\(878\) 40.0000i 1.34993i
\(879\) 4.47214 0.150841
\(880\) 0 0
\(881\) 2.00000 0.0673817 0.0336909 0.999432i \(-0.489274\pi\)
0.0336909 + 0.999432i \(0.489274\pi\)
\(882\) 15.6525i 0.527046i
\(883\) 4.00000i 0.134611i 0.997732 + 0.0673054i \(0.0214402\pi\)
−0.997732 + 0.0673054i \(0.978560\pi\)
\(884\) −60.0000 −2.01802
\(885\) 0 0
\(886\) 80.4984 2.70440
\(887\) 44.7214i 1.50160i −0.660532 0.750798i \(-0.729669\pi\)
0.660532 0.750798i \(-0.270331\pi\)
\(888\) 17.8885i 0.600300i
\(889\) 0 0
\(890\) −26.8328 13.4164i −0.899438 0.449719i
\(891\) 0 0
\(892\) 12.0000i 0.401790i
\(893\) 0 0
\(894\) 20.0000 0.668900
\(895\) 16.0000 32.0000i 0.534821 1.06964i
\(896\) 0 0
\(897\) 17.8885i 0.597281i
\(898\) 76.0263i 2.53703i
\(899\) 0 0
\(900\) −9.00000 12.0000i −0.300000 0.400000i
\(901\) 17.8885 0.595954
\(902\) 0 0
\(903\) 0 0
\(904\) 8.94427 0.297482
\(905\) 10.0000 20.0000i 0.332411 0.664822i
\(906\) −40.0000 −1.32891
\(907\) 12.0000i 0.398453i 0.979953 + 0.199227i \(0.0638430\pi\)
−0.979953 + 0.199227i \(0.936157\pi\)
\(908\) 53.6656i 1.78096i
\(909\) −8.94427 −0.296663
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 30.0000 0.992312
\(915\) 0 0
\(916\) 30.0000 0.991228
\(917\) 0 0
\(918\) 10.0000i 0.330049i
\(919\) −53.6656 −1.77027 −0.885133 0.465338i \(-0.845933\pi\)
−0.885133 + 0.465338i \(0.845933\pi\)
\(920\) −8.94427 + 17.8885i −0.294884 + 0.589768i
\(921\) 8.94427 0.294724
\(922\) 60.0000i 1.97599i
\(923\) 53.6656i 1.76643i
\(924\) 0 0
\(925\) 32.0000 24.0000i 1.05215 0.789115i
\(926\) 8.94427 0.293927
\(927\) 4.00000i 0.131377i
\(928\) 60.0000i 1.96960i
\(929\) 50.0000 1.64045 0.820223 0.572043i \(-0.193849\pi\)
0.820223 + 0.572043i \(0.193849\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 67.0820i 2.19735i
\(933\) 12.0000i 0.392862i
\(934\) 62.6099 2.04866
\(935\) 0 0
\(936\) −10.0000 −0.326860
\(937\) 22.3607i 0.730492i −0.930911 0.365246i \(-0.880985\pi\)
0.930911 0.365246i \(-0.119015\pi\)
\(938\) 0 0
\(939\) 16.0000 0.522140
\(940\) 72.0000 + 36.0000i 2.34838 + 1.17419i
\(941\) 8.94427 0.291575 0.145787 0.989316i \(-0.453428\pi\)
0.145787 + 0.989316i \(0.453428\pi\)
\(942\) 17.8885i 0.582840i
\(943\) 35.7771i 1.16506i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.0000i 0.909878i 0.890523 + 0.454939i \(0.150339\pi\)
−0.890523 + 0.454939i \(0.849661\pi\)
\(948\) 0 0
\(949\) 60.0000 1.94768
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) 0 0
\(953\) 13.4164i 0.434600i −0.976105 0.217300i \(-0.930275\pi\)
0.976105 0.217300i \(-0.0697250\pi\)
\(954\) 8.94427 0.289581
\(955\) 20.0000 40.0000i 0.647185 1.29437i
\(956\) −53.6656 −1.73567
\(957\) 0 0
\(958\) 40.0000i 1.29234i
\(959\) 0 0
\(960\) −26.0000 13.0000i −0.839146 0.419573i
\(961\) −31.0000 −1.00000
\(962\) 80.0000i 2.57930i
\(963\) 17.8885i 0.576450i
\(964\) −53.6656 −1.72845
\(965\) 8.94427 + 4.47214i 0.287926 + 0.143963i
\(966\) 0 0
\(967\) 17.8885i 0.575257i −0.957742 0.287628i \(-0.907133\pi\)
0.957742 0.287628i \(-0.0928668\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −17.8885 + 35.7771i −0.574367 + 1.14873i
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 3.00000i 0.0962250i
\(973\) 0 0
\(974\) −62.6099 −2.00615
\(975\) −13.4164 17.8885i −0.429669 0.572892i
\(976\) 0 0
\(977\) 12.0000i 0.383914i 0.981403 + 0.191957i \(0.0614834\pi\)
−0.981403 + 0.191957i \(0.938517\pi\)
\(978\) 8.94427i 0.286006i
\(979\) 0 0
\(980\) 21.0000 42.0000i 0.670820 1.34164i
\(981\) 17.8885 0.571137
\(982\) 80.0000i 2.55290i
\(983\) 36.0000i 1.14822i 0.818778 + 0.574111i \(0.194652\pi\)
−0.818778 + 0.574111i \(0.805348\pi\)
\(984\) −20.0000 −0.637577
\(985\) 8.94427 + 4.47214i 0.284988 + 0.142494i
\(986\) 89.4427 2.84844
\(987\) 0 0
\(988\) 0 0
\(989\) 35.7771 1.13765
\(990\) 0 0
\(991\) −40.0000 −1.27064 −0.635321 0.772248i \(-0.719132\pi\)
−0.635321 + 0.772248i \(0.719132\pi\)
\(992\) 0 0
\(993\) 20.0000i 0.634681i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 13.4164i 0.424902i 0.977172 + 0.212451i \(0.0681446\pi\)
−0.977172 + 0.212451i \(0.931855\pi\)
\(998\) 44.7214i 1.41563i
\(999\) 8.00000 0.253109
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1815.2.c.c.364.2 yes 4
5.2 odd 4 9075.2.a.bq.1.2 2
5.3 odd 4 9075.2.a.bj.1.1 2
5.4 even 2 inner 1815.2.c.c.364.3 yes 4
11.10 odd 2 inner 1815.2.c.c.364.4 yes 4
55.32 even 4 9075.2.a.bq.1.1 2
55.43 even 4 9075.2.a.bj.1.2 2
55.54 odd 2 inner 1815.2.c.c.364.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.c.c.364.1 4 55.54 odd 2 inner
1815.2.c.c.364.2 yes 4 1.1 even 1 trivial
1815.2.c.c.364.3 yes 4 5.4 even 2 inner
1815.2.c.c.364.4 yes 4 11.10 odd 2 inner
9075.2.a.bj.1.1 2 5.3 odd 4
9075.2.a.bj.1.2 2 55.43 even 4
9075.2.a.bq.1.1 2 55.32 even 4
9075.2.a.bq.1.2 2 5.2 odd 4