Properties

Label 900.2.s.b.49.2
Level $900$
Weight $2$
Character 900.49
Analytic conductor $7.187$
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] = [900,2,Mod(49,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 49.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 900.49
Dual form 900.2.s.b.349.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205 q^{3} +(0.866025 - 0.500000i) q^{7} +3.00000 q^{9} +O(q^{10})\) \(q+1.73205 q^{3} +(0.866025 - 0.500000i) q^{7} +3.00000 q^{9} +(-1.50000 - 2.59808i) q^{11} +(0.866025 + 0.500000i) q^{13} -6.00000i q^{17} +4.00000 q^{19} +(1.50000 - 0.866025i) q^{21} +(2.59808 + 1.50000i) q^{23} +5.19615 q^{27} +(1.50000 + 2.59808i) q^{29} +(-2.50000 + 4.33013i) q^{31} +(-2.59808 - 4.50000i) q^{33} -2.00000i q^{37} +(1.50000 + 0.866025i) q^{39} +(-1.50000 + 2.59808i) q^{41} +(-0.866025 + 0.500000i) q^{43} +(7.79423 - 4.50000i) q^{47} +(-3.00000 + 5.19615i) q^{49} -10.3923i q^{51} -6.00000i q^{53} +6.92820 q^{57} +(-1.50000 + 2.59808i) q^{59} +(6.50000 + 11.2583i) q^{61} +(2.59808 - 1.50000i) q^{63} +(-6.06218 - 3.50000i) q^{67} +(4.50000 + 2.59808i) q^{69} -12.0000 q^{71} -10.0000i q^{73} +(-2.59808 - 1.50000i) q^{77} +(5.50000 + 9.52628i) q^{79} +9.00000 q^{81} +(-7.79423 + 4.50000i) q^{83} +(2.59808 + 4.50000i) q^{87} -6.00000 q^{89} +1.00000 q^{91} +(-4.33013 + 7.50000i) q^{93} +(-9.52628 + 5.50000i) q^{97} +(-4.50000 - 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{9} - 6 q^{11} + 16 q^{19} + 6 q^{21} + 6 q^{29} - 10 q^{31} + 6 q^{39} - 6 q^{41} - 12 q^{49} - 6 q^{59} + 26 q^{61} + 18 q^{69} - 48 q^{71} + 22 q^{79} + 36 q^{81} - 24 q^{89} + 4 q^{91} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.73205 1.00000
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.866025 0.500000i 0.327327 0.188982i −0.327327 0.944911i \(-0.606148\pi\)
0.654654 + 0.755929i \(0.272814\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −1.50000 2.59808i −0.452267 0.783349i 0.546259 0.837616i \(-0.316051\pi\)
−0.998526 + 0.0542666i \(0.982718\pi\)
\(12\) 0 0
\(13\) 0.866025 + 0.500000i 0.240192 + 0.138675i 0.615265 0.788320i \(-0.289049\pi\)
−0.375073 + 0.926995i \(0.622382\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000i 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 1.50000 0.866025i 0.327327 0.188982i
\(22\) 0 0
\(23\) 2.59808 + 1.50000i 0.541736 + 0.312772i 0.745782 0.666190i \(-0.232076\pi\)
−0.204046 + 0.978961i \(0.565409\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 1.50000 + 2.59808i 0.278543 + 0.482451i 0.971023 0.238987i \(-0.0768152\pi\)
−0.692480 + 0.721437i \(0.743482\pi\)
\(30\) 0 0
\(31\) −2.50000 + 4.33013i −0.449013 + 0.777714i −0.998322 0.0579057i \(-0.981558\pi\)
0.549309 + 0.835619i \(0.314891\pi\)
\(32\) 0 0
\(33\) −2.59808 4.50000i −0.452267 0.783349i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 1.50000 + 0.866025i 0.240192 + 0.138675i
\(40\) 0 0
\(41\) −1.50000 + 2.59808i −0.234261 + 0.405751i −0.959058 0.283211i \(-0.908600\pi\)
0.724797 + 0.688963i \(0.241934\pi\)
\(42\) 0 0
\(43\) −0.866025 + 0.500000i −0.132068 + 0.0762493i −0.564578 0.825380i \(-0.690961\pi\)
0.432511 + 0.901629i \(0.357628\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.79423 4.50000i 1.13691 0.656392i 0.191243 0.981543i \(-0.438748\pi\)
0.945662 + 0.325150i \(0.105415\pi\)
\(48\) 0 0
\(49\) −3.00000 + 5.19615i −0.428571 + 0.742307i
\(50\) 0 0
\(51\) 10.3923i 1.45521i
\(52\) 0 0
\(53\) 6.00000i 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 6.92820 0.917663
\(58\) 0 0
\(59\) −1.50000 + 2.59808i −0.195283 + 0.338241i −0.946993 0.321253i \(-0.895896\pi\)
0.751710 + 0.659494i \(0.229229\pi\)
\(60\) 0 0
\(61\) 6.50000 + 11.2583i 0.832240 + 1.44148i 0.896258 + 0.443533i \(0.146275\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 2.59808 1.50000i 0.327327 0.188982i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.06218 3.50000i −0.740613 0.427593i 0.0816792 0.996659i \(-0.473972\pi\)
−0.822292 + 0.569066i \(0.807305\pi\)
\(68\) 0 0
\(69\) 4.50000 + 2.59808i 0.541736 + 0.312772i
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.59808 1.50000i −0.296078 0.170941i
\(78\) 0 0
\(79\) 5.50000 + 9.52628i 0.618798 + 1.07179i 0.989705 + 0.143120i \(0.0457135\pi\)
−0.370907 + 0.928670i \(0.620953\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −7.79423 + 4.50000i −0.855528 + 0.493939i −0.862512 0.506036i \(-0.831110\pi\)
0.00698436 + 0.999976i \(0.497777\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.59808 + 4.50000i 0.278543 + 0.482451i
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) −4.33013 + 7.50000i −0.449013 + 0.777714i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.52628 + 5.50000i −0.967247 + 0.558440i −0.898396 0.439187i \(-0.855267\pi\)
−0.0688512 + 0.997627i \(0.521933\pi\)
\(98\) 0 0
\(99\) −4.50000 7.79423i −0.452267 0.783349i
\(100\) 0 0
\(101\) −7.50000 12.9904i −0.746278 1.29259i −0.949595 0.313478i \(-0.898506\pi\)
0.203317 0.979113i \(-0.434828\pi\)
\(102\) 0 0
\(103\) 6.06218 + 3.50000i 0.597324 + 0.344865i 0.767988 0.640464i \(-0.221258\pi\)
−0.170664 + 0.985329i \(0.554591\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 3.46410i 0.328798i
\(112\) 0 0
\(113\) 7.79423 + 4.50000i 0.733219 + 0.423324i 0.819599 0.572938i \(-0.194196\pi\)
−0.0863794 + 0.996262i \(0.527530\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.59808 + 1.50000i 0.240192 + 0.138675i
\(118\) 0 0
\(119\) −3.00000 5.19615i −0.275010 0.476331i
\(120\) 0 0
\(121\) 1.00000 1.73205i 0.0909091 0.157459i
\(122\) 0 0
\(123\) −2.59808 + 4.50000i −0.234261 + 0.405751i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 0 0
\(129\) −1.50000 + 0.866025i −0.132068 + 0.0762493i
\(130\) 0 0
\(131\) −10.5000 + 18.1865i −0.917389 + 1.58896i −0.114024 + 0.993478i \(0.536374\pi\)
−0.803365 + 0.595487i \(0.796959\pi\)
\(132\) 0 0
\(133\) 3.46410 2.00000i 0.300376 0.173422i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.59808 + 1.50000i −0.221969 + 0.128154i −0.606861 0.794808i \(-0.707572\pi\)
0.384893 + 0.922961i \(0.374238\pi\)
\(138\) 0 0
\(139\) 2.50000 4.33013i 0.212047 0.367277i −0.740308 0.672268i \(-0.765320\pi\)
0.952355 + 0.304991i \(0.0986536\pi\)
\(140\) 0 0
\(141\) 13.5000 7.79423i 1.13691 0.656392i
\(142\) 0 0
\(143\) 3.00000i 0.250873i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −5.19615 + 9.00000i −0.428571 + 0.742307i
\(148\) 0 0
\(149\) 7.50000 12.9904i 0.614424 1.06421i −0.376061 0.926595i \(-0.622722\pi\)
0.990485 0.137619i \(-0.0439449\pi\)
\(150\) 0 0
\(151\) 6.50000 + 11.2583i 0.528962 + 0.916190i 0.999430 + 0.0337724i \(0.0107521\pi\)
−0.470467 + 0.882418i \(0.655915\pi\)
\(152\) 0 0
\(153\) 18.0000i 1.45521i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −11.2583 6.50000i −0.898513 0.518756i −0.0217953 0.999762i \(-0.506938\pi\)
−0.876717 + 0.481006i \(0.840272\pi\)
\(158\) 0 0
\(159\) 10.3923i 0.824163i
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) 20.0000i 1.56652i 0.621694 + 0.783260i \(0.286445\pi\)
−0.621694 + 0.783260i \(0.713555\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.79423 + 4.50000i 0.603136 + 0.348220i 0.770274 0.637713i \(-0.220119\pi\)
−0.167139 + 0.985933i \(0.553453\pi\)
\(168\) 0 0
\(169\) −6.00000 10.3923i −0.461538 0.799408i
\(170\) 0 0
\(171\) 12.0000 0.917663
\(172\) 0 0
\(173\) −7.79423 + 4.50000i −0.592584 + 0.342129i −0.766119 0.642699i \(-0.777815\pi\)
0.173534 + 0.984828i \(0.444481\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2.59808 + 4.50000i −0.195283 + 0.338241i
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 11.2583 + 19.5000i 0.832240 + 1.44148i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −15.5885 + 9.00000i −1.13994 + 0.658145i
\(188\) 0 0
\(189\) 4.50000 2.59808i 0.327327 0.188982i
\(190\) 0 0
\(191\) −7.50000 12.9904i −0.542681 0.939951i −0.998749 0.0500060i \(-0.984076\pi\)
0.456068 0.889945i \(-0.349257\pi\)
\(192\) 0 0
\(193\) −9.52628 5.50000i −0.685717 0.395899i 0.116289 0.993215i \(-0.462900\pi\)
−0.802005 + 0.597317i \(0.796234\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.00000i 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) −10.5000 6.06218i −0.740613 0.427593i
\(202\) 0 0
\(203\) 2.59808 + 1.50000i 0.182349 + 0.105279i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 7.79423 + 4.50000i 0.541736 + 0.312772i
\(208\) 0 0
\(209\) −6.00000 10.3923i −0.415029 0.718851i
\(210\) 0 0
\(211\) −8.50000 + 14.7224i −0.585164 + 1.01353i 0.409691 + 0.912224i \(0.365637\pi\)
−0.994855 + 0.101310i \(0.967697\pi\)
\(212\) 0 0
\(213\) −20.7846 −1.42414
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.00000i 0.339422i
\(218\) 0 0
\(219\) 17.3205i 1.17041i
\(220\) 0 0
\(221\) 3.00000 5.19615i 0.201802 0.349531i
\(222\) 0 0
\(223\) −0.866025 + 0.500000i −0.0579934 + 0.0334825i −0.528716 0.848799i \(-0.677326\pi\)
0.470723 + 0.882281i \(0.343993\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −23.3827 + 13.5000i −1.55196 + 0.896026i −0.553981 + 0.832529i \(0.686892\pi\)
−0.997982 + 0.0634974i \(0.979775\pi\)
\(228\) 0 0
\(229\) −6.50000 + 11.2583i −0.429532 + 0.743971i −0.996832 0.0795401i \(-0.974655\pi\)
0.567300 + 0.823511i \(0.307988\pi\)
\(230\) 0 0
\(231\) −4.50000 2.59808i −0.296078 0.170941i
\(232\) 0 0
\(233\) 6.00000i 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 9.52628 + 16.5000i 0.618798 + 1.07179i
\(238\) 0 0
\(239\) −13.5000 + 23.3827i −0.873242 + 1.51250i −0.0146191 + 0.999893i \(0.504654\pi\)
−0.858623 + 0.512607i \(0.828680\pi\)
\(240\) 0 0
\(241\) 0.500000 + 0.866025i 0.0322078 + 0.0557856i 0.881680 0.471848i \(-0.156413\pi\)
−0.849472 + 0.527633i \(0.823079\pi\)
\(242\) 0 0
\(243\) 15.5885 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 3.46410 + 2.00000i 0.220416 + 0.127257i
\(248\) 0 0
\(249\) −13.5000 + 7.79423i −0.855528 + 0.493939i
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 9.00000i 0.565825i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.79423 4.50000i −0.486191 0.280702i 0.236802 0.971558i \(-0.423901\pi\)
−0.722993 + 0.690856i \(0.757234\pi\)
\(258\) 0 0
\(259\) −1.00000 1.73205i −0.0621370 0.107624i
\(260\) 0 0
\(261\) 4.50000 + 7.79423i 0.278543 + 0.482451i
\(262\) 0 0
\(263\) −18.1865 + 10.5000i −1.12143 + 0.647458i −0.941766 0.336270i \(-0.890834\pi\)
−0.179664 + 0.983728i \(0.557501\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −10.3923 −0.635999
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 0 0
\(273\) 1.73205 0.104828
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.866025 0.500000i 0.0520344 0.0300421i −0.473757 0.880656i \(-0.657103\pi\)
0.525792 + 0.850613i \(0.323769\pi\)
\(278\) 0 0
\(279\) −7.50000 + 12.9904i −0.449013 + 0.777714i
\(280\) 0 0
\(281\) −1.50000 2.59808i −0.0894825 0.154988i 0.817810 0.575488i \(-0.195188\pi\)
−0.907293 + 0.420500i \(0.861855\pi\)
\(282\) 0 0
\(283\) −4.33013 2.50000i −0.257399 0.148610i 0.365748 0.930714i \(-0.380813\pi\)
−0.623148 + 0.782104i \(0.714146\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.00000i 0.177084i
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −16.5000 + 9.52628i −0.967247 + 0.558440i
\(292\) 0 0
\(293\) 18.1865 + 10.5000i 1.06247 + 0.613417i 0.926114 0.377244i \(-0.123128\pi\)
0.136355 + 0.990660i \(0.456461\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −7.79423 13.5000i −0.452267 0.783349i
\(298\) 0 0
\(299\) 1.50000 + 2.59808i 0.0867472 + 0.150251i
\(300\) 0 0
\(301\) −0.500000 + 0.866025i −0.0288195 + 0.0499169i
\(302\) 0 0
\(303\) −12.9904 22.5000i −0.746278 1.29259i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 20.0000i 1.14146i −0.821138 0.570730i \(-0.806660\pi\)
0.821138 0.570730i \(-0.193340\pi\)
\(308\) 0 0
\(309\) 10.5000 + 6.06218i 0.597324 + 0.344865i
\(310\) 0 0
\(311\) −10.5000 + 18.1865i −0.595400 + 1.03126i 0.398090 + 0.917346i \(0.369673\pi\)
−0.993490 + 0.113917i \(0.963660\pi\)
\(312\) 0 0
\(313\) −0.866025 + 0.500000i −0.0489506 + 0.0282617i −0.524276 0.851549i \(-0.675664\pi\)
0.475325 + 0.879810i \(0.342331\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.1865 10.5000i 1.02146 0.589739i 0.106932 0.994266i \(-0.465897\pi\)
0.914526 + 0.404528i \(0.132564\pi\)
\(318\) 0 0
\(319\) 4.50000 7.79423i 0.251952 0.436393i
\(320\) 0 0
\(321\) 20.7846i 1.16008i
\(322\) 0 0
\(323\) 24.0000i 1.33540i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −3.46410 −0.191565
\(328\) 0 0
\(329\) 4.50000 7.79423i 0.248093 0.429710i
\(330\) 0 0
\(331\) −5.50000 9.52628i −0.302307 0.523612i 0.674351 0.738411i \(-0.264424\pi\)
−0.976658 + 0.214799i \(0.931090\pi\)
\(332\) 0 0
\(333\) 6.00000i 0.328798i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 19.9186 + 11.5000i 1.08503 + 0.626445i 0.932250 0.361815i \(-0.117843\pi\)
0.152784 + 0.988260i \(0.451176\pi\)
\(338\) 0 0
\(339\) 13.5000 + 7.79423i 0.733219 + 0.423324i
\(340\) 0 0
\(341\) 15.0000 0.812296
\(342\) 0 0
\(343\) 13.0000i 0.701934i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.79423 + 4.50000i 0.418416 + 0.241573i 0.694399 0.719590i \(-0.255670\pi\)
−0.275983 + 0.961162i \(0.589003\pi\)
\(348\) 0 0
\(349\) −0.500000 0.866025i −0.0267644 0.0463573i 0.852333 0.523000i \(-0.175187\pi\)
−0.879097 + 0.476642i \(0.841854\pi\)
\(350\) 0 0
\(351\) 4.50000 + 2.59808i 0.240192 + 0.138675i
\(352\) 0 0
\(353\) 2.59808 1.50000i 0.138282 0.0798369i −0.429263 0.903179i \(-0.641227\pi\)
0.567545 + 0.823343i \(0.307893\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −5.19615 9.00000i −0.275010 0.476331i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 1.73205 3.00000i 0.0909091 0.157459i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 11.2583 6.50000i 0.587680 0.339297i −0.176500 0.984301i \(-0.556477\pi\)
0.764180 + 0.645003i \(0.223144\pi\)
\(368\) 0 0
\(369\) −4.50000 + 7.79423i −0.234261 + 0.405751i
\(370\) 0 0
\(371\) −3.00000 5.19615i −0.155752 0.269771i
\(372\) 0 0
\(373\) 0.866025 + 0.500000i 0.0448411 + 0.0258890i 0.522253 0.852791i \(-0.325092\pi\)
−0.477412 + 0.878680i \(0.658425\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.00000i 0.154508i
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 27.7128i 1.41977i
\(382\) 0 0
\(383\) 12.9904 + 7.50000i 0.663777 + 0.383232i 0.793715 0.608290i \(-0.208144\pi\)
−0.129937 + 0.991522i \(0.541478\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.59808 + 1.50000i −0.132068 + 0.0762493i
\(388\) 0 0
\(389\) 7.50000 + 12.9904i 0.380265 + 0.658638i 0.991100 0.133120i \(-0.0424994\pi\)
−0.610835 + 0.791758i \(0.709166\pi\)
\(390\) 0 0
\(391\) 9.00000 15.5885i 0.455150 0.788342i
\(392\) 0 0
\(393\) −18.1865 + 31.5000i −0.917389 + 1.58896i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.00000i 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 0 0
\(399\) 6.00000 3.46410i 0.300376 0.173422i
\(400\) 0 0
\(401\) −1.50000 + 2.59808i −0.0749064 + 0.129742i −0.901046 0.433724i \(-0.857199\pi\)
0.826139 + 0.563466i \(0.190532\pi\)
\(402\) 0 0
\(403\) −4.33013 + 2.50000i −0.215699 + 0.124534i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.19615 + 3.00000i −0.257564 + 0.148704i
\(408\) 0 0
\(409\) 11.5000 19.9186i 0.568638 0.984911i −0.428063 0.903749i \(-0.640804\pi\)
0.996701 0.0811615i \(-0.0258630\pi\)
\(410\) 0 0
\(411\) −4.50000 + 2.59808i −0.221969 + 0.128154i
\(412\) 0 0
\(413\) 3.00000i 0.147620i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.33013 7.50000i 0.212047 0.367277i
\(418\) 0 0
\(419\) 4.50000 7.79423i 0.219839 0.380773i −0.734919 0.678155i \(-0.762780\pi\)
0.954759 + 0.297382i \(0.0961133\pi\)
\(420\) 0 0
\(421\) −17.5000 30.3109i −0.852898 1.47726i −0.878582 0.477592i \(-0.841510\pi\)
0.0256838 0.999670i \(-0.491824\pi\)
\(422\) 0 0
\(423\) 23.3827 13.5000i 1.13691 0.656392i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 11.2583 + 6.50000i 0.544829 + 0.314557i
\(428\) 0 0
\(429\) 5.19615i 0.250873i
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) 34.0000i 1.63394i −0.576683 0.816968i \(-0.695653\pi\)
0.576683 0.816968i \(-0.304347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 10.3923 + 6.00000i 0.497131 + 0.287019i
\(438\) 0 0
\(439\) 17.5000 + 30.3109i 0.835229 + 1.44666i 0.893843 + 0.448379i \(0.147999\pi\)
−0.0586141 + 0.998281i \(0.518668\pi\)
\(440\) 0 0
\(441\) −9.00000 + 15.5885i −0.428571 + 0.742307i
\(442\) 0 0
\(443\) −7.79423 + 4.50000i −0.370315 + 0.213801i −0.673596 0.739100i \(-0.735251\pi\)
0.303281 + 0.952901i \(0.401918\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 12.9904 22.5000i 0.614424 1.06421i
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 9.00000 0.423793
\(452\) 0 0
\(453\) 11.2583 + 19.5000i 0.528962 + 0.916190i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 32.0429 18.5000i 1.49891 0.865393i 0.498906 0.866656i \(-0.333735\pi\)
0.999999 + 0.00126243i \(0.000401844\pi\)
\(458\) 0 0
\(459\) 31.1769i 1.45521i
\(460\) 0 0
\(461\) −1.50000 2.59808i −0.0698620 0.121004i 0.828978 0.559281i \(-0.188923\pi\)
−0.898840 + 0.438276i \(0.855589\pi\)
\(462\) 0 0
\(463\) 16.4545 + 9.50000i 0.764705 + 0.441502i 0.830982 0.556299i \(-0.187779\pi\)
−0.0662777 + 0.997801i \(0.521112\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000i 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) 0 0
\(469\) −7.00000 −0.323230
\(470\) 0 0
\(471\) −19.5000 11.2583i −0.898513 0.518756i
\(472\) 0 0
\(473\) 2.59808 + 1.50000i 0.119460 + 0.0689701i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 18.0000i 0.824163i
\(478\) 0 0
\(479\) 13.5000 + 23.3827i 0.616831 + 1.06838i 0.990060 + 0.140643i \(0.0449170\pi\)
−0.373230 + 0.927739i \(0.621750\pi\)
\(480\) 0 0
\(481\) 1.00000 1.73205i 0.0455961 0.0789747i
\(482\) 0 0
\(483\) 5.19615 0.236433
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 32.0000i 1.45006i −0.688718 0.725029i \(-0.741826\pi\)
0.688718 0.725029i \(-0.258174\pi\)
\(488\) 0 0
\(489\) 34.6410i 1.56652i
\(490\) 0 0
\(491\) 1.50000 2.59808i 0.0676941 0.117250i −0.830192 0.557478i \(-0.811769\pi\)
0.897886 + 0.440228i \(0.145102\pi\)
\(492\) 0 0
\(493\) 15.5885 9.00000i 0.702069 0.405340i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −10.3923 + 6.00000i −0.466159 + 0.269137i
\(498\) 0 0
\(499\) 2.50000 4.33013i 0.111915 0.193843i −0.804627 0.593780i \(-0.797635\pi\)
0.916542 + 0.399937i \(0.130968\pi\)
\(500\) 0 0
\(501\) 13.5000 + 7.79423i 0.603136 + 0.348220i
\(502\) 0 0
\(503\) 36.0000i 1.60516i −0.596544 0.802580i \(-0.703460\pi\)
0.596544 0.802580i \(-0.296540\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −10.3923 18.0000i −0.461538 0.799408i
\(508\) 0 0
\(509\) 19.5000 33.7750i 0.864322 1.49705i −0.00339621 0.999994i \(-0.501081\pi\)
0.867719 0.497056i \(-0.165586\pi\)
\(510\) 0 0
\(511\) −5.00000 8.66025i −0.221187 0.383107i
\(512\) 0 0
\(513\) 20.7846 0.917663
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −23.3827 13.5000i −1.02837 0.593729i
\(518\) 0 0
\(519\) −13.5000 + 7.79423i −0.592584 + 0.342129i
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) 8.00000i 0.349816i 0.984585 + 0.174908i \(0.0559627\pi\)
−0.984585 + 0.174908i \(0.944037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 25.9808 + 15.0000i 1.13174 + 0.653410i
\(528\) 0 0
\(529\) −7.00000 12.1244i −0.304348 0.527146i
\(530\) 0 0
\(531\) −4.50000 + 7.79423i −0.195283 + 0.338241i
\(532\) 0 0
\(533\) −2.59808 + 1.50000i −0.112535 + 0.0649722i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −20.7846 −0.896922
\(538\) 0 0
\(539\) 18.0000 0.775315
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) 3.46410 0.148659
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 11.2583 6.50000i 0.481371 0.277920i −0.239616 0.970868i \(-0.577022\pi\)
0.720988 + 0.692948i \(0.243688\pi\)
\(548\) 0 0
\(549\) 19.5000 + 33.7750i 0.832240 + 1.44148i
\(550\) 0 0
\(551\) 6.00000 + 10.3923i 0.255609 + 0.442727i
\(552\) 0 0
\(553\) 9.52628 + 5.50000i 0.405099 + 0.233884i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 0 0
\(559\) −1.00000 −0.0422955
\(560\) 0 0
\(561\) −27.0000 + 15.5885i −1.13994 + 0.658145i
\(562\) 0 0
\(563\) −7.79423 4.50000i −0.328488 0.189652i 0.326682 0.945134i \(-0.394069\pi\)
−0.655169 + 0.755482i \(0.727403\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 7.79423 4.50000i 0.327327 0.188982i
\(568\) 0 0
\(569\) 7.50000 + 12.9904i 0.314416 + 0.544585i 0.979313 0.202350i \(-0.0648579\pi\)
−0.664897 + 0.746935i \(0.731525\pi\)
\(570\) 0 0
\(571\) 15.5000 26.8468i 0.648655 1.12350i −0.334790 0.942293i \(-0.608665\pi\)
0.983444 0.181210i \(-0.0580014\pi\)
\(572\) 0 0
\(573\) −12.9904 22.5000i −0.542681 0.939951i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 10.0000i 0.416305i 0.978096 + 0.208153i \(0.0667451\pi\)
−0.978096 + 0.208153i \(0.933255\pi\)
\(578\) 0 0
\(579\) −16.5000 9.52628i −0.685717 0.395899i
\(580\) 0 0
\(581\) −4.50000 + 7.79423i −0.186691 + 0.323359i
\(582\) 0 0
\(583\) −15.5885 + 9.00000i −0.645608 + 0.372742i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.9904 + 7.50000i −0.536170 + 0.309558i −0.743525 0.668708i \(-0.766848\pi\)
0.207355 + 0.978266i \(0.433514\pi\)
\(588\) 0 0
\(589\) −10.0000 + 17.3205i −0.412043 + 0.713679i
\(590\) 0 0
\(591\) 10.3923i 0.427482i
\(592\) 0 0
\(593\) 6.00000i 0.246390i 0.992382 + 0.123195i \(0.0393141\pi\)
−0.992382 + 0.123195i \(0.960686\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.92820 0.283552
\(598\) 0 0
\(599\) −19.5000 + 33.7750i −0.796748 + 1.38001i 0.124975 + 0.992160i \(0.460115\pi\)
−0.921723 + 0.387849i \(0.873218\pi\)
\(600\) 0 0
\(601\) −17.5000 30.3109i −0.713840 1.23641i −0.963405 0.268049i \(-0.913621\pi\)
0.249565 0.968358i \(-0.419712\pi\)
\(602\) 0 0
\(603\) −18.1865 10.5000i −0.740613 0.427593i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 35.5070 + 20.5000i 1.44119 + 0.832069i 0.997929 0.0643251i \(-0.0204895\pi\)
0.443257 + 0.896394i \(0.353823\pi\)
\(608\) 0 0
\(609\) 4.50000 + 2.59808i 0.182349 + 0.105279i
\(610\) 0 0
\(611\) 9.00000 0.364101
\(612\) 0 0
\(613\) 26.0000i 1.05013i 0.851062 + 0.525065i \(0.175959\pi\)
−0.851062 + 0.525065i \(0.824041\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.59808 + 1.50000i 0.104595 + 0.0603877i 0.551385 0.834251i \(-0.314100\pi\)
−0.446790 + 0.894639i \(0.647433\pi\)
\(618\) 0 0
\(619\) −6.50000 11.2583i −0.261257 0.452510i 0.705319 0.708890i \(-0.250804\pi\)
−0.966576 + 0.256379i \(0.917470\pi\)
\(620\) 0 0
\(621\) 13.5000 + 7.79423i 0.541736 + 0.312772i
\(622\) 0 0
\(623\) −5.19615 + 3.00000i −0.208179 + 0.120192i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −10.3923 18.0000i −0.415029 0.718851i
\(628\) 0 0
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) −14.7224 + 25.5000i −0.585164 + 1.01353i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −5.19615 + 3.00000i −0.205879 + 0.118864i
\(638\) 0 0
\(639\) −36.0000 −1.42414
\(640\) 0 0
\(641\) 16.5000 + 28.5788i 0.651711 + 1.12880i 0.982708 + 0.185164i \(0.0592817\pi\)
−0.330997 + 0.943632i \(0.607385\pi\)
\(642\) 0 0
\(643\) −35.5070 20.5000i −1.40026 0.808441i −0.405842 0.913943i \(-0.633022\pi\)
−0.994419 + 0.105502i \(0.966355\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 9.00000 0.353281
\(650\) 0 0
\(651\) 8.66025i 0.339422i
\(652\) 0 0
\(653\) 18.1865 + 10.5000i 0.711694 + 0.410897i 0.811688 0.584091i \(-0.198549\pi\)
−0.0999939 + 0.994988i \(0.531882\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 30.0000i 1.17041i
\(658\) 0 0
\(659\) −10.5000 18.1865i −0.409022 0.708447i 0.585758 0.810486i \(-0.300797\pi\)
−0.994780 + 0.102039i \(0.967463\pi\)
\(660\) 0 0
\(661\) −5.50000 + 9.52628i −0.213925 + 0.370529i −0.952940 0.303160i \(-0.901958\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 0 0
\(663\) 5.19615 9.00000i 0.201802 0.349531i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.00000i 0.348481i
\(668\) 0 0
\(669\) −1.50000 + 0.866025i −0.0579934 + 0.0334825i
\(670\) 0 0
\(671\) 19.5000 33.7750i 0.752789 1.30387i
\(672\) 0 0
\(673\) 9.52628 5.50000i 0.367211 0.212009i −0.305028 0.952343i \(-0.598666\pi\)
0.672239 + 0.740334i \(0.265333\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −12.9904 + 7.50000i −0.499261 + 0.288248i −0.728408 0.685143i \(-0.759740\pi\)
0.229147 + 0.973392i \(0.426406\pi\)
\(678\) 0 0
\(679\) −5.50000 + 9.52628i −0.211071 + 0.365585i
\(680\) 0 0
\(681\) −40.5000 + 23.3827i −1.55196 + 0.896026i
\(682\) 0 0
\(683\) 36.0000i 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −11.2583 + 19.5000i −0.429532 + 0.743971i
\(688\) 0 0
\(689\) 3.00000 5.19615i 0.114291 0.197958i
\(690\) 0 0
\(691\) 0.500000 + 0.866025i 0.0190209 + 0.0329452i 0.875379 0.483437i \(-0.160612\pi\)
−0.856358 + 0.516382i \(0.827278\pi\)
\(692\) 0 0
\(693\) −7.79423 4.50000i −0.296078 0.170941i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 15.5885 + 9.00000i 0.590455 + 0.340899i
\(698\) 0 0
\(699\) 10.3923i 0.393073i
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12.9904 7.50000i −0.488554 0.282067i
\(708\) 0 0
\(709\) −12.5000 21.6506i −0.469447 0.813107i 0.529943 0.848034i \(-0.322213\pi\)
−0.999390 + 0.0349269i \(0.988880\pi\)
\(710\) 0 0
\(711\) 16.5000 + 28.5788i 0.618798 + 1.07179i
\(712\) 0 0
\(713\) −12.9904 + 7.50000i −0.486494 + 0.280877i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −23.3827 + 40.5000i −0.873242 + 1.51250i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 7.00000 0.260694
\(722\) 0 0
\(723\) 0.866025 + 1.50000i 0.0322078 + 0.0557856i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 32.0429 18.5000i 1.18841 0.686127i 0.230463 0.973081i \(-0.425976\pi\)
0.957944 + 0.286954i \(0.0926427\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 3.00000 + 5.19615i 0.110959 + 0.192187i
\(732\) 0 0
\(733\) −19.9186 11.5000i −0.735710 0.424762i 0.0847976 0.996398i \(-0.472976\pi\)
−0.820507 + 0.571636i \(0.806309\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 21.0000i 0.773545i
\(738\) 0 0
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 6.00000 + 3.46410i 0.220416 + 0.127257i
\(742\) 0 0
\(743\) −7.79423 4.50000i −0.285943 0.165089i 0.350168 0.936687i \(-0.386124\pi\)
−0.636111 + 0.771598i \(0.719458\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −23.3827 + 13.5000i −0.855528 + 0.493939i
\(748\) 0 0
\(749\) −6.00000 10.3923i −0.219235 0.379727i
\(750\) 0 0
\(751\) 15.5000 26.8468i 0.565603 0.979653i −0.431390 0.902165i \(-0.641977\pi\)
0.996993 0.0774878i \(-0.0246899\pi\)
\(752\) 0 0
\(753\) 20.7846 0.757433
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 10.0000i 0.363456i 0.983349 + 0.181728i \(0.0581691\pi\)
−0.983349 + 0.181728i \(0.941831\pi\)
\(758\) 0 0
\(759\) 15.5885i 0.565825i
\(760\) 0 0
\(761\) −13.5000 + 23.3827i −0.489375 + 0.847622i −0.999925 0.0122260i \(-0.996108\pi\)
0.510551 + 0.859848i \(0.329442\pi\)
\(762\) 0 0
\(763\) −1.73205 + 1.00000i −0.0627044 + 0.0362024i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.59808 + 1.50000i −0.0938111 + 0.0541619i
\(768\) 0 0
\(769\) −0.500000 + 0.866025i −0.0180305 + 0.0312297i −0.874900 0.484304i \(-0.839073\pi\)
0.856869 + 0.515534i \(0.172406\pi\)
\(770\) 0 0
\(771\) −13.5000 7.79423i −0.486191 0.280702i
\(772\) 0 0
\(773\) 18.0000i 0.647415i 0.946157 + 0.323708i \(0.104929\pi\)
−0.946157 + 0.323708i \(0.895071\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.73205 3.00000i −0.0621370 0.107624i
\(778\) 0 0
\(779\) −6.00000 + 10.3923i −0.214972 + 0.372343i
\(780\) 0 0
\(781\) 18.0000 + 31.1769i 0.644091 + 1.11560i
\(782\) 0 0
\(783\) 7.79423 + 13.5000i 0.278543 + 0.482451i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −37.2391 21.5000i −1.32743 0.766392i −0.342529 0.939507i \(-0.611283\pi\)
−0.984902 + 0.173115i \(0.944617\pi\)
\(788\) 0 0
\(789\) −31.5000 + 18.1865i −1.12143 + 0.647458i
\(790\) 0 0
\(791\) 9.00000 0.320003
\(792\) 0 0
\(793\) 13.0000i 0.461644i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −7.79423 4.50000i −0.276086 0.159398i 0.355564 0.934652i \(-0.384289\pi\)
−0.631650 + 0.775254i \(0.717622\pi\)
\(798\) 0 0
\(799\) −27.0000 46.7654i −0.955191 1.65444i
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) 0 0
\(803\) −25.9808 + 15.0000i −0.916841 + 0.529339i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 10.3923 0.365826
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) 13.8564 0.485965
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −3.46410 + 2.00000i −0.121194 + 0.0699711i
\(818\) 0 0
\(819\) 3.00000 0.104828
\(820\) 0 0
\(821\) −25.5000 44.1673i −0.889956 1.54145i −0.839926 0.542702i \(-0.817401\pi\)
−0.0500305 0.998748i \(-0.515932\pi\)
\(822\) 0 0
\(823\) 16.4545 + 9.50000i 0.573567 + 0.331149i 0.758573 0.651588i \(-0.225897\pi\)
−0.185006 + 0.982737i \(0.559230\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 0 0
\(831\) 1.50000 0.866025i 0.0520344 0.0300421i
\(832\) 0 0
\(833\) 31.1769 + 18.0000i 1.08022 + 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −12.9904 + 22.5000i −0.449013 + 0.777714i
\(838\) 0 0
\(839\) −4.50000 7.79423i −0.155357 0.269087i 0.777832 0.628473i \(-0.216320\pi\)
−0.933189 + 0.359386i \(0.882986\pi\)
\(840\) 0 0
\(841\) 10.0000 17.3205i 0.344828 0.597259i
\(842\) 0 0
\(843\) −2.59808 4.50000i −0.0894825 0.154988i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) 0 0
\(849\) −7.50000 4.33013i −0.257399 0.148610i
\(850\) 0 0
\(851\) 3.00000 5.19615i 0.102839 0.178122i
\(852\) 0 0
\(853\) −11.2583 + 6.50000i −0.385478 + 0.222556i −0.680199 0.733028i \(-0.738107\pi\)
0.294721 + 0.955583i \(0.404773\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −23.3827 + 13.5000i −0.798737 + 0.461151i −0.843029 0.537867i \(-0.819230\pi\)
0.0442921 + 0.999019i \(0.485897\pi\)
\(858\) 0 0
\(859\) 20.5000 35.5070i 0.699451 1.21148i −0.269206 0.963083i \(-0.586761\pi\)
0.968657 0.248402i \(-0.0799054\pi\)
\(860\) 0 0
\(861\) 5.19615i 0.177084i
\(862\) 0 0
\(863\) 24.0000i 0.816970i 0.912765 + 0.408485i \(0.133943\pi\)
−0.912765 + 0.408485i \(0.866057\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −32.9090 −1.11765
\(868\) 0 0
\(869\) 16.5000 28.5788i 0.559724 0.969471i
\(870\) 0 0
\(871\) −3.50000 6.06218i −0.118593 0.205409i
\(872\) 0 0
\(873\) −28.5788 + 16.5000i −0.967247 + 0.558440i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19.9186 + 11.5000i 0.672603 + 0.388327i 0.797062 0.603897i \(-0.206386\pi\)
−0.124459 + 0.992225i \(0.539720\pi\)
\(878\) 0 0
\(879\) 31.5000 + 18.1865i 1.06247 + 0.613417i
\(880\) 0 0
\(881\) 30.0000 1.01073 0.505363 0.862907i \(-0.331359\pi\)
0.505363 + 0.862907i \(0.331359\pi\)
\(882\) 0 0
\(883\) 4.00000i 0.134611i −0.997732 0.0673054i \(-0.978560\pi\)
0.997732 0.0673054i \(-0.0214402\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 18.1865 + 10.5000i 0.610644 + 0.352555i 0.773217 0.634141i \(-0.218646\pi\)
−0.162573 + 0.986696i \(0.551979\pi\)
\(888\) 0 0
\(889\) 8.00000 + 13.8564i 0.268311 + 0.464729i
\(890\) 0 0
\(891\) −13.5000 23.3827i −0.452267 0.783349i
\(892\) 0 0
\(893\) 31.1769 18.0000i 1.04330 0.602347i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.59808 + 4.50000i 0.0867472 + 0.150251i
\(898\) 0 0
\(899\) −15.0000 −0.500278
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) −0.866025 + 1.50000i −0.0288195 + 0.0499169i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −40.7032 + 23.5000i −1.35153 + 0.780305i −0.988463 0.151460i \(-0.951603\pi\)
−0.363064 + 0.931764i \(0.618269\pi\)
\(908\) 0 0
\(909\) −22.5000 38.9711i −0.746278 1.29259i
\(910\) 0 0
\(911\) 22.5000 + 38.9711i 0.745458 + 1.29117i 0.949980 + 0.312310i \(0.101103\pi\)
−0.204522 + 0.978862i \(0.565564\pi\)
\(912\) 0 0
\(913\) 23.3827 + 13.5000i 0.773854 + 0.446785i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 21.0000i 0.693481i
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 34.6410i 1.14146i
\(922\) 0 0
\(923\) −10.3923 6.00000i −0.342067 0.197492i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 18.1865 + 10.5000i 0.597324 + 0.344865i
\(928\) 0 0
\(929\) 13.5000 + 23.3827i 0.442921 + 0.767161i 0.997905 0.0646999i \(-0.0206090\pi\)
−0.554984 + 0.831861i \(0.687276\pi\)
\(930\) 0 0
\(931\) −12.0000 + 20.7846i −0.393284 + 0.681188i
\(932\) 0 0
\(933\) −18.1865 + 31.5000i −0.595400 + 1.03126i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 34.0000i 1.11073i 0.831606 + 0.555366i \(0.187422\pi\)
−0.831606 + 0.555366i \(0.812578\pi\)
\(938\) 0 0
\(939\) −1.50000 + 0.866025i −0.0489506 + 0.0282617i
\(940\) 0 0
\(941\) 10.5000 18.1865i 0.342290 0.592864i −0.642567 0.766229i \(-0.722131\pi\)
0.984858 + 0.173365i \(0.0554641\pi\)
\(942\) 0 0
\(943\) −7.79423 + 4.50000i −0.253815 + 0.146540i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23.3827 + 13.5000i −0.759835 + 0.438691i −0.829237 0.558898i \(-0.811224\pi\)
0.0694014 + 0.997589i \(0.477891\pi\)
\(948\) 0 0
\(949\) 5.00000 8.66025i 0.162307 0.281124i
\(950\) 0 0
\(951\) 31.5000 18.1865i 1.02146 0.589739i
\(952\) 0 0
\(953\) 54.0000i 1.74923i 0.484817 + 0.874616i \(0.338886\pi\)
−0.484817 + 0.874616i \(0.661114\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 7.79423 13.5000i 0.251952 0.436393i
\(958\) 0 0
\(959\) −1.50000 + 2.59808i −0.0484375 + 0.0838963i
\(960\) 0 0
\(961\) 3.00000 + 5.19615i 0.0967742 + 0.167618i
\(962\) 0 0
\(963\) 36.0000i 1.16008i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −37.2391 21.5000i −1.19753 0.691393i −0.237525 0.971381i \(-0.576336\pi\)
−0.960003 + 0.279988i \(0.909669\pi\)
\(968\) 0 0
\(969\) 41.5692i 1.33540i
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 5.00000i 0.160293i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −49.3634 28.5000i −1.57928 0.911796i −0.994960 0.100270i \(-0.968029\pi\)
−0.584316 0.811526i \(-0.698637\pi\)
\(978\) 0 0
\(979\) 9.00000 + 15.5885i 0.287641 + 0.498209i
\(980\) 0 0
\(981\) −6.00000 −0.191565
\(982\) 0 0
\(983\) 44.1673 25.5000i 1.40872 0.813324i 0.413453 0.910525i \(-0.364323\pi\)
0.995265 + 0.0972017i \(0.0309892\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 7.79423 13.5000i 0.248093 0.429710i
\(988\) 0 0
\(989\) −3.00000 −0.0953945
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) −9.52628 16.5000i −0.302307 0.523612i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.866025 0.500000i 0.0274273 0.0158352i −0.486224 0.873834i \(-0.661626\pi\)
0.513651 + 0.857999i \(0.328293\pi\)
\(998\) 0 0
\(999\) 10.3923i 0.328798i
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 900.2.s.b.49.2 4
3.2 odd 2 2700.2.s.b.1549.2 4
5.2 odd 4 36.2.e.a.13.1 2
5.3 odd 4 900.2.i.b.301.1 2
5.4 even 2 inner 900.2.s.b.49.1 4
9.2 odd 6 2700.2.s.b.2449.1 4
9.4 even 3 8100.2.d.h.649.2 2
9.5 odd 6 8100.2.d.c.649.2 2
9.7 even 3 inner 900.2.s.b.349.1 4
15.2 even 4 108.2.e.a.37.1 2
15.8 even 4 2700.2.i.b.901.1 2
15.14 odd 2 2700.2.s.b.1549.1 4
20.7 even 4 144.2.i.a.49.1 2
35.2 odd 12 1764.2.l.c.949.1 2
35.12 even 12 1764.2.l.a.949.1 2
35.17 even 12 1764.2.i.c.373.1 2
35.27 even 4 1764.2.j.b.589.1 2
35.32 odd 12 1764.2.i.a.373.1 2
40.27 even 4 576.2.i.e.193.1 2
40.37 odd 4 576.2.i.f.193.1 2
45.2 even 12 108.2.e.a.73.1 2
45.4 even 6 8100.2.d.h.649.1 2
45.7 odd 12 36.2.e.a.25.1 yes 2
45.13 odd 12 8100.2.a.j.1.1 1
45.14 odd 6 8100.2.d.c.649.1 2
45.22 odd 12 324.2.a.c.1.1 1
45.23 even 12 8100.2.a.g.1.1 1
45.29 odd 6 2700.2.s.b.2449.2 4
45.32 even 12 324.2.a.a.1.1 1
45.34 even 6 inner 900.2.s.b.349.2 4
45.38 even 12 2700.2.i.b.1801.1 2
45.43 odd 12 900.2.i.b.601.1 2
60.47 odd 4 432.2.i.c.145.1 2
105.2 even 12 5292.2.l.a.361.1 2
105.17 odd 12 5292.2.i.a.1549.1 2
105.32 even 12 5292.2.i.c.1549.1 2
105.47 odd 12 5292.2.l.c.361.1 2
105.62 odd 4 5292.2.j.a.1765.1 2
120.77 even 4 1728.2.i.d.577.1 2
120.107 odd 4 1728.2.i.c.577.1 2
180.7 even 12 144.2.i.a.97.1 2
180.47 odd 12 432.2.i.c.289.1 2
180.67 even 12 1296.2.a.k.1.1 1
180.167 odd 12 1296.2.a.b.1.1 1
315.2 even 12 5292.2.i.c.2125.1 2
315.47 odd 12 5292.2.i.a.2125.1 2
315.52 even 12 1764.2.l.a.961.1 2
315.97 even 12 1764.2.j.b.1177.1 2
315.137 even 12 5292.2.l.a.3313.1 2
315.142 odd 12 1764.2.i.a.1537.1 2
315.187 even 12 1764.2.i.c.1537.1 2
315.227 odd 12 5292.2.l.c.3313.1 2
315.272 odd 12 5292.2.j.a.3529.1 2
315.277 odd 12 1764.2.l.c.961.1 2
360.67 even 12 5184.2.a.f.1.1 1
360.77 even 12 5184.2.a.ba.1.1 1
360.157 odd 12 5184.2.a.e.1.1 1
360.187 even 12 576.2.i.e.385.1 2
360.227 odd 12 1728.2.i.c.1153.1 2
360.277 odd 12 576.2.i.f.385.1 2
360.317 even 12 1728.2.i.d.1153.1 2
360.347 odd 12 5184.2.a.bb.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.2.e.a.13.1 2 5.2 odd 4
36.2.e.a.25.1 yes 2 45.7 odd 12
108.2.e.a.37.1 2 15.2 even 4
108.2.e.a.73.1 2 45.2 even 12
144.2.i.a.49.1 2 20.7 even 4
144.2.i.a.97.1 2 180.7 even 12
324.2.a.a.1.1 1 45.32 even 12
324.2.a.c.1.1 1 45.22 odd 12
432.2.i.c.145.1 2 60.47 odd 4
432.2.i.c.289.1 2 180.47 odd 12
576.2.i.e.193.1 2 40.27 even 4
576.2.i.e.385.1 2 360.187 even 12
576.2.i.f.193.1 2 40.37 odd 4
576.2.i.f.385.1 2 360.277 odd 12
900.2.i.b.301.1 2 5.3 odd 4
900.2.i.b.601.1 2 45.43 odd 12
900.2.s.b.49.1 4 5.4 even 2 inner
900.2.s.b.49.2 4 1.1 even 1 trivial
900.2.s.b.349.1 4 9.7 even 3 inner
900.2.s.b.349.2 4 45.34 even 6 inner
1296.2.a.b.1.1 1 180.167 odd 12
1296.2.a.k.1.1 1 180.67 even 12
1728.2.i.c.577.1 2 120.107 odd 4
1728.2.i.c.1153.1 2 360.227 odd 12
1728.2.i.d.577.1 2 120.77 even 4
1728.2.i.d.1153.1 2 360.317 even 12
1764.2.i.a.373.1 2 35.32 odd 12
1764.2.i.a.1537.1 2 315.142 odd 12
1764.2.i.c.373.1 2 35.17 even 12
1764.2.i.c.1537.1 2 315.187 even 12
1764.2.j.b.589.1 2 35.27 even 4
1764.2.j.b.1177.1 2 315.97 even 12
1764.2.l.a.949.1 2 35.12 even 12
1764.2.l.a.961.1 2 315.52 even 12
1764.2.l.c.949.1 2 35.2 odd 12
1764.2.l.c.961.1 2 315.277 odd 12
2700.2.i.b.901.1 2 15.8 even 4
2700.2.i.b.1801.1 2 45.38 even 12
2700.2.s.b.1549.1 4 15.14 odd 2
2700.2.s.b.1549.2 4 3.2 odd 2
2700.2.s.b.2449.1 4 9.2 odd 6
2700.2.s.b.2449.2 4 45.29 odd 6
5184.2.a.e.1.1 1 360.157 odd 12
5184.2.a.f.1.1 1 360.67 even 12
5184.2.a.ba.1.1 1 360.77 even 12
5184.2.a.bb.1.1 1 360.347 odd 12
5292.2.i.a.1549.1 2 105.17 odd 12
5292.2.i.a.2125.1 2 315.47 odd 12
5292.2.i.c.1549.1 2 105.32 even 12
5292.2.i.c.2125.1 2 315.2 even 12
5292.2.j.a.1765.1 2 105.62 odd 4
5292.2.j.a.3529.1 2 315.272 odd 12
5292.2.l.a.361.1 2 105.2 even 12
5292.2.l.a.3313.1 2 315.137 even 12
5292.2.l.c.361.1 2 105.47 odd 12
5292.2.l.c.3313.1 2 315.227 odd 12
8100.2.a.g.1.1 1 45.23 even 12
8100.2.a.j.1.1 1 45.13 odd 12
8100.2.d.c.649.1 2 45.14 odd 6
8100.2.d.c.649.2 2 9.5 odd 6
8100.2.d.h.649.1 2 45.4 even 6
8100.2.d.h.649.2 2 9.4 even 3