Properties

Label 2100.2.bi.m.101.8
Level $2100$
Weight $2$
Character 2100.101
Analytic conductor $16.769$
Analytic rank $0$
Dimension $16$
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] = [2100,2,Mod(101,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 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("2100.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.bi (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 9 x^{14} - 18 x^{13} + 32 x^{12} - 36 x^{11} + 51 x^{9} - 167 x^{8} + 153 x^{7} - 972 x^{5} + 2592 x^{4} - 4374 x^{3} + 6561 x^{2} - 6561 x + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 101.8
Root \(1.72685 + 0.134063i\) of defining polynomial
Character \(\chi\) \(=\) 2100.101
Dual form 2100.2.bi.m.1601.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.72685 - 0.134063i) q^{3} +(-0.786875 - 2.52603i) q^{7} +(2.96405 - 0.463016i) q^{9} +O(q^{10})\) \(q+(1.72685 - 0.134063i) q^{3} +(-0.786875 - 2.52603i) q^{7} +(2.96405 - 0.463016i) q^{9} +(2.34474 + 1.35373i) q^{11} -1.12489i q^{13} +(3.69121 - 6.39336i) q^{17} +(-0.412264 + 0.238021i) q^{19} +(-1.69747 - 4.25660i) q^{21} +(-4.84706 + 2.79845i) q^{23} +(5.05642 - 1.19693i) q^{27} -2.20372i q^{29} +(-2.07315 - 1.19693i) q^{31} +(4.23051 + 2.02336i) q^{33} +(-4.34879 - 7.53233i) q^{37} +(-0.150806 - 1.94251i) q^{39} +5.42336 q^{41} -4.16209 q^{43} +(6.21227 + 10.7600i) q^{47} +(-5.76166 + 3.97534i) q^{49} +(5.51707 - 11.5353i) q^{51} +(4.25321 + 2.45559i) q^{53} +(-0.680011 + 0.466297i) q^{57} +(1.15586 - 2.00200i) q^{59} +(-2.26895 + 1.30998i) q^{61} +(-3.50193 - 7.12295i) q^{63} +(7.34210 - 12.7169i) q^{67} +(-7.99501 + 5.48234i) q^{69} -9.89729i q^{71} +(-6.29046 - 3.63180i) q^{73} +(1.57456 - 6.98810i) q^{77} +(3.47478 + 6.01850i) q^{79} +(8.57123 - 2.74481i) q^{81} +11.3005 q^{83} +(-0.295438 - 3.80550i) q^{87} +(3.48186 + 6.03075i) q^{89} +(-2.84150 + 0.885144i) q^{91} +(-3.74049 - 1.78899i) q^{93} -8.79691i q^{97} +(7.57673 + 2.92689i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 3 q^{3} + 6 q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 3 q^{3} + 6 q^{7} - 9 q^{9} - 18 q^{19} - 11 q^{21} - 18 q^{31} + 12 q^{33} - 6 q^{37} + 12 q^{39} + 4 q^{43} - 18 q^{49} - q^{51} + 6 q^{57} + 36 q^{61} + 19 q^{63} + 30 q^{67} + 54 q^{73} + 7 q^{81} + 81 q^{87} + 20 q^{91} - 34 q^{93} - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.72685 0.134063i 0.997000 0.0774015i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.786875 2.52603i −0.297411 0.954750i
\(8\) 0 0
\(9\) 2.96405 0.463016i 0.988018 0.154339i
\(10\) 0 0
\(11\) 2.34474 + 1.35373i 0.706965 + 0.408166i 0.809936 0.586518i \(-0.199502\pi\)
−0.102971 + 0.994684i \(0.532835\pi\)
\(12\) 0 0
\(13\) 1.12489i 0.311987i −0.987758 0.155994i \(-0.950142\pi\)
0.987758 0.155994i \(-0.0498579\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.69121 6.39336i 0.895250 1.55062i 0.0617544 0.998091i \(-0.480330\pi\)
0.833495 0.552527i \(-0.186336\pi\)
\(18\) 0 0
\(19\) −0.412264 + 0.238021i −0.0945799 + 0.0546057i −0.546544 0.837430i \(-0.684057\pi\)
0.451964 + 0.892036i \(0.350724\pi\)
\(20\) 0 0
\(21\) −1.69747 4.25660i −0.370417 0.928865i
\(22\) 0 0
\(23\) −4.84706 + 2.79845i −1.01068 + 0.583518i −0.911392 0.411540i \(-0.864991\pi\)
−0.0992913 + 0.995058i \(0.531658\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.05642 1.19693i 0.973108 0.230350i
\(28\) 0 0
\(29\) 2.20372i 0.409220i −0.978844 0.204610i \(-0.934407\pi\)
0.978844 0.204610i \(-0.0655926\pi\)
\(30\) 0 0
\(31\) −2.07315 1.19693i −0.372348 0.214975i 0.302136 0.953265i \(-0.402300\pi\)
−0.674484 + 0.738290i \(0.735634\pi\)
\(32\) 0 0
\(33\) 4.23051 + 2.02336i 0.736437 + 0.352222i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.34879 7.53233i −0.714937 1.23831i −0.962984 0.269559i \(-0.913122\pi\)
0.248047 0.968748i \(-0.420211\pi\)
\(38\) 0 0
\(39\) −0.150806 1.94251i −0.0241483 0.311051i
\(40\) 0 0
\(41\) 5.42336 0.846986 0.423493 0.905899i \(-0.360804\pi\)
0.423493 + 0.905899i \(0.360804\pi\)
\(42\) 0 0
\(43\) −4.16209 −0.634712 −0.317356 0.948306i \(-0.602795\pi\)
−0.317356 + 0.948306i \(0.602795\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.21227 + 10.7600i 0.906153 + 1.56950i 0.819362 + 0.573277i \(0.194328\pi\)
0.0867912 + 0.996227i \(0.472339\pi\)
\(48\) 0 0
\(49\) −5.76166 + 3.97534i −0.823094 + 0.567905i
\(50\) 0 0
\(51\) 5.51707 11.5353i 0.772544 1.61526i
\(52\) 0 0
\(53\) 4.25321 + 2.45559i 0.584224 + 0.337302i 0.762810 0.646622i \(-0.223819\pi\)
−0.178586 + 0.983924i \(0.557152\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.680011 + 0.466297i −0.0900696 + 0.0617626i
\(58\) 0 0
\(59\) 1.15586 2.00200i 0.150480 0.260638i −0.780924 0.624626i \(-0.785252\pi\)
0.931404 + 0.363987i \(0.118585\pi\)
\(60\) 0 0
\(61\) −2.26895 + 1.30998i −0.290509 + 0.167726i −0.638172 0.769894i \(-0.720309\pi\)
0.347662 + 0.937620i \(0.386976\pi\)
\(62\) 0 0
\(63\) −3.50193 7.12295i −0.441202 0.897408i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.34210 12.7169i 0.896980 1.55361i 0.0656452 0.997843i \(-0.479089\pi\)
0.831335 0.555772i \(-0.187577\pi\)
\(68\) 0 0
\(69\) −7.99501 + 5.48234i −0.962486 + 0.659996i
\(70\) 0 0
\(71\) 9.89729i 1.17459i −0.809372 0.587296i \(-0.800193\pi\)
0.809372 0.587296i \(-0.199807\pi\)
\(72\) 0 0
\(73\) −6.29046 3.63180i −0.736242 0.425070i 0.0844591 0.996427i \(-0.473084\pi\)
−0.820701 + 0.571357i \(0.806417\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.57456 6.98810i 0.179438 0.796368i
\(78\) 0 0
\(79\) 3.47478 + 6.01850i 0.390944 + 0.677134i 0.992574 0.121640i \(-0.0388154\pi\)
−0.601631 + 0.798774i \(0.705482\pi\)
\(80\) 0 0
\(81\) 8.57123 2.74481i 0.952359 0.304979i
\(82\) 0 0
\(83\) 11.3005 1.24040 0.620198 0.784446i \(-0.287052\pi\)
0.620198 + 0.784446i \(0.287052\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.295438 3.80550i −0.0316743 0.407992i
\(88\) 0 0
\(89\) 3.48186 + 6.03075i 0.369076 + 0.639258i 0.989421 0.145070i \(-0.0463409\pi\)
−0.620345 + 0.784329i \(0.713008\pi\)
\(90\) 0 0
\(91\) −2.84150 + 0.885144i −0.297870 + 0.0927883i
\(92\) 0 0
\(93\) −3.74049 1.78899i −0.387871 0.185510i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.79691i 0.893190i −0.894736 0.446595i \(-0.852636\pi\)
0.894736 0.446595i \(-0.147364\pi\)
\(98\) 0 0
\(99\) 7.57673 + 2.92689i 0.761490 + 0.294164i
\(100\) 0 0
\(101\) −5.23274 + 9.06338i −0.520677 + 0.901840i 0.479034 + 0.877797i \(0.340987\pi\)
−0.999711 + 0.0240431i \(0.992346\pi\)
\(102\) 0 0
\(103\) 2.83087 1.63440i 0.278934 0.161042i −0.354007 0.935243i \(-0.615181\pi\)
0.632941 + 0.774200i \(0.281848\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.31809 2.49305i 0.417445 0.241012i −0.276538 0.961003i \(-0.589187\pi\)
0.693984 + 0.719991i \(0.255854\pi\)
\(108\) 0 0
\(109\) −2.40043 + 4.15767i −0.229920 + 0.398232i −0.957784 0.287489i \(-0.907180\pi\)
0.727864 + 0.685721i \(0.240513\pi\)
\(110\) 0 0
\(111\) −8.51954 12.4242i −0.808639 1.17925i
\(112\) 0 0
\(113\) 18.8874i 1.77678i −0.459094 0.888388i \(-0.651826\pi\)
0.459094 0.888388i \(-0.348174\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.520840 3.33422i −0.0481517 0.308249i
\(118\) 0 0
\(119\) −19.0543 4.29333i −1.74671 0.393569i
\(120\) 0 0
\(121\) −1.83480 3.17797i −0.166800 0.288907i
\(122\) 0 0
\(123\) 9.36535 0.727074i 0.844445 0.0655580i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.77289 0.689732 0.344866 0.938652i \(-0.387924\pi\)
0.344866 + 0.938652i \(0.387924\pi\)
\(128\) 0 0
\(129\) −7.18732 + 0.557983i −0.632808 + 0.0491277i
\(130\) 0 0
\(131\) −2.32600 4.02875i −0.203224 0.351994i 0.746342 0.665563i \(-0.231809\pi\)
−0.949565 + 0.313569i \(0.898475\pi\)
\(132\) 0 0
\(133\) 0.925648 + 0.854100i 0.0802639 + 0.0740598i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.53654 + 5.50592i 0.814762 + 0.470403i 0.848607 0.529024i \(-0.177442\pi\)
−0.0338451 + 0.999427i \(0.510775\pi\)
\(138\) 0 0
\(139\) 20.5928i 1.74665i 0.487134 + 0.873327i \(0.338042\pi\)
−0.487134 + 0.873327i \(0.661958\pi\)
\(140\) 0 0
\(141\) 12.1702 + 17.7481i 1.02492 + 1.49466i
\(142\) 0 0
\(143\) 1.52280 2.63756i 0.127343 0.220564i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −9.41660 + 7.63726i −0.776668 + 0.629910i
\(148\) 0 0
\(149\) −14.8906 + 8.59708i −1.21988 + 0.704300i −0.964893 0.262643i \(-0.915406\pi\)
−0.254991 + 0.966943i \(0.582072\pi\)
\(150\) 0 0
\(151\) −1.00060 + 1.73309i −0.0814279 + 0.141037i −0.903863 0.427821i \(-0.859281\pi\)
0.822436 + 0.568858i \(0.192615\pi\)
\(152\) 0 0
\(153\) 7.98072 20.6594i 0.645203 1.67021i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −15.7938 9.11856i −1.26048 0.727741i −0.287316 0.957836i \(-0.592763\pi\)
−0.973168 + 0.230095i \(0.926096\pi\)
\(158\) 0 0
\(159\) 7.67389 + 3.67025i 0.608579 + 0.291070i
\(160\) 0 0
\(161\) 10.8830 + 10.0418i 0.857701 + 0.791405i
\(162\) 0 0
\(163\) 9.90522 + 17.1563i 0.775837 + 1.34379i 0.934323 + 0.356427i \(0.116005\pi\)
−0.158487 + 0.987361i \(0.550661\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −12.5257 −0.969269 −0.484634 0.874717i \(-0.661047\pi\)
−0.484634 + 0.874717i \(0.661047\pi\)
\(168\) 0 0
\(169\) 11.7346 0.902664
\(170\) 0 0
\(171\) −1.11177 + 0.896392i −0.0850189 + 0.0685488i
\(172\) 0 0
\(173\) 7.36813 + 12.7620i 0.560188 + 0.970275i 0.997480 + 0.0709548i \(0.0226046\pi\)
−0.437291 + 0.899320i \(0.644062\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.72760 3.61212i 0.129854 0.271504i
\(178\) 0 0
\(179\) 1.18770 + 0.685720i 0.0887730 + 0.0512531i 0.543729 0.839261i \(-0.317012\pi\)
−0.454956 + 0.890514i \(0.650345\pi\)
\(180\) 0 0
\(181\) 21.3843i 1.58949i 0.606947 + 0.794743i \(0.292394\pi\)
−0.606947 + 0.794743i \(0.707606\pi\)
\(182\) 0 0
\(183\) −3.74253 + 2.56633i −0.276656 + 0.189708i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 17.3098 9.99384i 1.26582 0.730822i
\(188\) 0 0
\(189\) −7.00225 11.8308i −0.509339 0.860566i
\(190\) 0 0
\(191\) 14.9775 8.64724i 1.08373 0.625692i 0.151830 0.988407i \(-0.451483\pi\)
0.931900 + 0.362714i \(0.118150\pi\)
\(192\) 0 0
\(193\) −6.40401 + 11.0921i −0.460971 + 0.798425i −0.999010 0.0444950i \(-0.985832\pi\)
0.538039 + 0.842920i \(0.319165\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.17207i 0.653483i 0.945114 + 0.326741i \(0.105951\pi\)
−0.945114 + 0.326741i \(0.894049\pi\)
\(198\) 0 0
\(199\) 7.16818 + 4.13855i 0.508139 + 0.293374i 0.732068 0.681231i \(-0.238555\pi\)
−0.223930 + 0.974605i \(0.571889\pi\)
\(200\) 0 0
\(201\) 10.9739 22.9445i 0.774037 1.61838i
\(202\) 0 0
\(203\) −5.56666 + 1.73405i −0.390703 + 0.121706i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −13.0712 + 10.5390i −0.908513 + 0.732514i
\(208\) 0 0
\(209\) −1.28887 −0.0891529
\(210\) 0 0
\(211\) −8.50872 −0.585765 −0.292882 0.956148i \(-0.594614\pi\)
−0.292882 + 0.956148i \(0.594614\pi\)
\(212\) 0 0
\(213\) −1.32686 17.0912i −0.0909152 1.17107i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.39218 + 6.17867i −0.0945073 + 0.419435i
\(218\) 0 0
\(219\) −11.3496 5.42827i −0.766935 0.366808i
\(220\) 0 0
\(221\) −7.19180 4.15219i −0.483773 0.279306i
\(222\) 0 0
\(223\) 5.54451i 0.371288i −0.982617 0.185644i \(-0.940563\pi\)
0.982617 0.185644i \(-0.0594371\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.09736 + 5.36478i −0.205579 + 0.356073i −0.950317 0.311284i \(-0.899241\pi\)
0.744738 + 0.667357i \(0.232574\pi\)
\(228\) 0 0
\(229\) −0.307553 + 0.177566i −0.0203237 + 0.0117339i −0.510127 0.860099i \(-0.670402\pi\)
0.489804 + 0.871833i \(0.337068\pi\)
\(230\) 0 0
\(231\) 1.78219 12.2785i 0.117259 0.807867i
\(232\) 0 0
\(233\) −16.1652 + 9.33296i −1.05901 + 0.611422i −0.925160 0.379578i \(-0.876069\pi\)
−0.133855 + 0.991001i \(0.542736\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 6.80730 + 9.92723i 0.442182 + 0.644843i
\(238\) 0 0
\(239\) 21.1914i 1.37076i 0.728186 + 0.685379i \(0.240364\pi\)
−0.728186 + 0.685379i \(0.759636\pi\)
\(240\) 0 0
\(241\) −1.92350 1.11053i −0.123903 0.0715356i 0.436767 0.899574i \(-0.356123\pi\)
−0.560671 + 0.828039i \(0.689457\pi\)
\(242\) 0 0
\(243\) 14.4333 5.88897i 0.925896 0.377778i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.267746 + 0.463750i 0.0170363 + 0.0295077i
\(248\) 0 0
\(249\) 19.5144 1.51499i 1.23667 0.0960084i
\(250\) 0 0
\(251\) −18.3017 −1.15519 −0.577597 0.816322i \(-0.696010\pi\)
−0.577597 + 0.816322i \(0.696010\pi\)
\(252\) 0 0
\(253\) −15.1535 −0.952690
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.59968 9.69894i −0.349299 0.605003i 0.636826 0.771007i \(-0.280247\pi\)
−0.986125 + 0.166004i \(0.946913\pi\)
\(258\) 0 0
\(259\) −15.6049 + 16.9122i −0.969643 + 1.05087i
\(260\) 0 0
\(261\) −1.02036 6.53194i −0.0631585 0.404317i
\(262\) 0 0
\(263\) −11.4450 6.60778i −0.705730 0.407453i 0.103748 0.994604i \(-0.466916\pi\)
−0.809478 + 0.587150i \(0.800250\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.82116 + 9.94744i 0.417448 + 0.608774i
\(268\) 0 0
\(269\) 12.6657 21.9377i 0.772244 1.33757i −0.164086 0.986446i \(-0.552468\pi\)
0.936330 0.351120i \(-0.114199\pi\)
\(270\) 0 0
\(271\) −23.9303 + 13.8162i −1.45366 + 0.839273i −0.998687 0.0512313i \(-0.983685\pi\)
−0.454976 + 0.890504i \(0.650352\pi\)
\(272\) 0 0
\(273\) −4.78818 + 1.90946i −0.289794 + 0.115566i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 11.8309 20.4917i 0.710848 1.23122i −0.253691 0.967285i \(-0.581645\pi\)
0.964539 0.263939i \(-0.0850219\pi\)
\(278\) 0 0
\(279\) −6.69912 2.58787i −0.401066 0.154932i
\(280\) 0 0
\(281\) 17.3694i 1.03617i 0.855329 + 0.518085i \(0.173355\pi\)
−0.855329 + 0.518085i \(0.826645\pi\)
\(282\) 0 0
\(283\) 11.1897 + 6.46037i 0.665158 + 0.384029i 0.794239 0.607605i \(-0.207870\pi\)
−0.129082 + 0.991634i \(0.541203\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.26750 13.6996i −0.251903 0.808660i
\(288\) 0 0
\(289\) −18.7500 32.4760i −1.10294 1.91036i
\(290\) 0 0
\(291\) −1.17934 15.1910i −0.0691343 0.890511i
\(292\) 0 0
\(293\) 11.2345 0.656326 0.328163 0.944621i \(-0.393571\pi\)
0.328163 + 0.944621i \(0.393571\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 13.4763 + 4.03856i 0.781974 + 0.234341i
\(298\) 0 0
\(299\) 3.14794 + 5.45239i 0.182050 + 0.315320i
\(300\) 0 0
\(301\) 3.27504 + 10.5136i 0.188770 + 0.605991i
\(302\) 0 0
\(303\) −7.82112 + 16.3527i −0.449312 + 0.939435i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 31.0165i 1.77021i 0.465395 + 0.885103i \(0.345912\pi\)
−0.465395 + 0.885103i \(0.654088\pi\)
\(308\) 0 0
\(309\) 4.66938 3.20189i 0.265632 0.182149i
\(310\) 0 0
\(311\) 0.771357 1.33603i 0.0437396 0.0757593i −0.843327 0.537401i \(-0.819406\pi\)
0.887066 + 0.461642i \(0.152739\pi\)
\(312\) 0 0
\(313\) −22.2646 + 12.8545i −1.25847 + 0.726579i −0.972777 0.231742i \(-0.925558\pi\)
−0.285694 + 0.958321i \(0.592224\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.69412 + 1.55545i −0.151317 + 0.0873629i −0.573747 0.819033i \(-0.694511\pi\)
0.422430 + 0.906396i \(0.361177\pi\)
\(318\) 0 0
\(319\) 2.98325 5.16714i 0.167030 0.289304i
\(320\) 0 0
\(321\) 7.12248 4.88403i 0.397538 0.272600i
\(322\) 0 0
\(323\) 3.51434i 0.195543i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −3.58780 + 7.50150i −0.198406 + 0.414834i
\(328\) 0 0
\(329\) 22.2917 24.1591i 1.22898 1.33194i
\(330\) 0 0
\(331\) 0.139978 + 0.242450i 0.00769391 + 0.0133262i 0.869847 0.493322i \(-0.164218\pi\)
−0.862153 + 0.506648i \(0.830884\pi\)
\(332\) 0 0
\(333\) −16.3776 20.3127i −0.897489 1.11313i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 5.96891 0.325147 0.162574 0.986696i \(-0.448021\pi\)
0.162574 + 0.986696i \(0.448021\pi\)
\(338\) 0 0
\(339\) −2.53211 32.6158i −0.137525 1.77145i
\(340\) 0 0
\(341\) −3.24066 5.61298i −0.175491 0.303960i
\(342\) 0 0
\(343\) 14.5755 + 11.4260i 0.787004 + 0.616947i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.41426 + 4.28063i 0.398019 + 0.229796i 0.685629 0.727951i \(-0.259527\pi\)
−0.287610 + 0.957748i \(0.592861\pi\)
\(348\) 0 0
\(349\) 7.23833i 0.387459i 0.981055 + 0.193729i \(0.0620584\pi\)
−0.981055 + 0.193729i \(0.937942\pi\)
\(350\) 0 0
\(351\) −1.34641 5.68789i −0.0718661 0.303597i
\(352\) 0 0
\(353\) 11.8813 20.5790i 0.632377 1.09531i −0.354688 0.934985i \(-0.615413\pi\)
0.987065 0.160324i \(-0.0512538\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −33.4797 4.85947i −1.77193 0.257190i
\(358\) 0 0
\(359\) 31.3993 18.1284i 1.65719 0.956780i 0.683188 0.730242i \(-0.260593\pi\)
0.974003 0.226537i \(-0.0727405\pi\)
\(360\) 0 0
\(361\) −9.38669 + 16.2582i −0.494036 + 0.855696i
\(362\) 0 0
\(363\) −3.59449 5.24192i −0.188662 0.275129i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.26512 0.730418i −0.0660387 0.0381275i 0.466617 0.884459i \(-0.345473\pi\)
−0.532656 + 0.846332i \(0.678806\pi\)
\(368\) 0 0
\(369\) 16.0751 2.51110i 0.836838 0.130723i
\(370\) 0 0
\(371\) 2.85616 12.6760i 0.148284 0.658105i
\(372\) 0 0
\(373\) 10.5919 + 18.3457i 0.548429 + 0.949907i 0.998382 + 0.0568547i \(0.0181072\pi\)
−0.449954 + 0.893052i \(0.648559\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.47893 −0.127671
\(378\) 0 0
\(379\) 24.3581 1.25119 0.625596 0.780147i \(-0.284856\pi\)
0.625596 + 0.780147i \(0.284856\pi\)
\(380\) 0 0
\(381\) 13.4226 1.04206i 0.687663 0.0533863i
\(382\) 0 0
\(383\) 12.5964 + 21.8176i 0.643647 + 1.11483i 0.984612 + 0.174754i \(0.0559129\pi\)
−0.340965 + 0.940076i \(0.610754\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −12.3367 + 1.92711i −0.627107 + 0.0979606i
\(388\) 0 0
\(389\) 10.9380 + 6.31508i 0.554581 + 0.320187i 0.750967 0.660339i \(-0.229587\pi\)
−0.196387 + 0.980527i \(0.562921\pi\)
\(390\) 0 0
\(391\) 41.3187i 2.08958i
\(392\) 0 0
\(393\) −4.55677 6.64524i −0.229859 0.335208i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −14.3269 + 8.27166i −0.719048 + 0.415143i −0.814402 0.580301i \(-0.802935\pi\)
0.0953542 + 0.995443i \(0.469602\pi\)
\(398\) 0 0
\(399\) 1.71296 + 1.35081i 0.0857555 + 0.0676251i
\(400\) 0 0
\(401\) −9.72834 + 5.61666i −0.485810 + 0.280482i −0.722835 0.691021i \(-0.757161\pi\)
0.237025 + 0.971504i \(0.423828\pi\)
\(402\) 0 0
\(403\) −1.34641 + 2.33205i −0.0670695 + 0.116168i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 23.5484i 1.16725i
\(408\) 0 0
\(409\) −31.6764 18.2884i −1.56630 0.904303i −0.996595 0.0824520i \(-0.973725\pi\)
−0.569703 0.821851i \(-0.692942\pi\)
\(410\) 0 0
\(411\) 17.2064 + 8.22943i 0.848727 + 0.405928i
\(412\) 0 0
\(413\) −5.96663 1.34440i −0.293599 0.0661537i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.76073 + 35.5607i 0.135194 + 1.74141i
\(418\) 0 0
\(419\) −8.29006 −0.404996 −0.202498 0.979283i \(-0.564906\pi\)
−0.202498 + 0.979283i \(0.564906\pi\)
\(420\) 0 0
\(421\) 23.8826 1.16397 0.581984 0.813200i \(-0.302277\pi\)
0.581984 + 0.813200i \(0.302277\pi\)
\(422\) 0 0
\(423\) 23.3955 + 29.0168i 1.13753 + 1.41084i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 5.09443 + 4.70065i 0.246537 + 0.227480i
\(428\) 0 0
\(429\) 2.27605 4.75884i 0.109889 0.229759i
\(430\) 0 0
\(431\) 9.93859 + 5.73805i 0.478725 + 0.276392i 0.719885 0.694093i \(-0.244194\pi\)
−0.241160 + 0.970485i \(0.577528\pi\)
\(432\) 0 0
\(433\) 3.31657i 0.159384i 0.996820 + 0.0796920i \(0.0253937\pi\)
−0.996820 + 0.0796920i \(0.974606\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.33218 2.30741i 0.0637269 0.110378i
\(438\) 0 0
\(439\) 23.6144 13.6338i 1.12705 0.650705i 0.183862 0.982952i \(-0.441140\pi\)
0.943192 + 0.332247i \(0.107807\pi\)
\(440\) 0 0
\(441\) −15.2372 + 14.4509i −0.725582 + 0.688136i
\(442\) 0 0
\(443\) −35.4300 + 20.4555i −1.68333 + 0.971872i −0.723911 + 0.689893i \(0.757657\pi\)
−0.959421 + 0.281979i \(0.909009\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −24.5613 + 16.8422i −1.16171 + 0.796608i
\(448\) 0 0
\(449\) 19.6734i 0.928446i 0.885718 + 0.464223i \(0.153666\pi\)
−0.885718 + 0.464223i \(0.846334\pi\)
\(450\) 0 0
\(451\) 12.7164 + 7.34179i 0.598790 + 0.345711i
\(452\) 0 0
\(453\) −1.49555 + 3.12695i −0.0702671 + 0.146917i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.59563 7.95987i −0.214975 0.372347i 0.738290 0.674483i \(-0.235634\pi\)
−0.953265 + 0.302136i \(0.902300\pi\)
\(458\) 0 0
\(459\) 11.0119 36.7456i 0.513990 1.71514i
\(460\) 0 0
\(461\) −28.7958 −1.34115 −0.670577 0.741840i \(-0.733953\pi\)
−0.670577 + 0.741840i \(0.733953\pi\)
\(462\) 0 0
\(463\) −25.0825 −1.16568 −0.582841 0.812586i \(-0.698059\pi\)
−0.582841 + 0.812586i \(0.698059\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11.6675 + 20.2087i 0.539907 + 0.935145i 0.998908 + 0.0467103i \(0.0148737\pi\)
−0.459002 + 0.888435i \(0.651793\pi\)
\(468\) 0 0
\(469\) −37.9005 8.53977i −1.75008 0.394330i
\(470\) 0 0
\(471\) −28.4961 13.6291i −1.31303 0.627994i
\(472\) 0 0
\(473\) −9.75900 5.63436i −0.448719 0.259068i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 13.7437 + 5.30921i 0.629282 + 0.243092i
\(478\) 0 0
\(479\) −8.12075 + 14.0655i −0.371046 + 0.642671i −0.989727 0.142971i \(-0.954334\pi\)
0.618680 + 0.785643i \(0.287668\pi\)
\(480\) 0 0
\(481\) −8.47301 + 4.89189i −0.386336 + 0.223051i
\(482\) 0 0
\(483\) 20.1396 + 15.8817i 0.916384 + 0.722643i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −5.07733 + 8.79420i −0.230076 + 0.398503i −0.957830 0.287335i \(-0.907231\pi\)
0.727754 + 0.685838i \(0.240564\pi\)
\(488\) 0 0
\(489\) 19.4049 + 28.2986i 0.877520 + 1.27971i
\(490\) 0 0
\(491\) 25.5734i 1.15411i 0.816704 + 0.577057i \(0.195799\pi\)
−0.816704 + 0.577057i \(0.804201\pi\)
\(492\) 0 0
\(493\) −14.0892 8.13438i −0.634544 0.366354i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −25.0008 + 7.78792i −1.12144 + 0.349336i
\(498\) 0 0
\(499\) −4.80838 8.32836i −0.215253 0.372829i 0.738098 0.674693i \(-0.235724\pi\)
−0.953351 + 0.301865i \(0.902391\pi\)
\(500\) 0 0
\(501\) −21.6301 + 1.67924i −0.966361 + 0.0750229i
\(502\) 0 0
\(503\) −6.82743 −0.304420 −0.152210 0.988348i \(-0.548639\pi\)
−0.152210 + 0.988348i \(0.548639\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 20.2640 1.57318i 0.899956 0.0698675i
\(508\) 0 0
\(509\) −7.02977 12.1759i −0.311589 0.539688i 0.667118 0.744952i \(-0.267528\pi\)
−0.978707 + 0.205265i \(0.934195\pi\)
\(510\) 0 0
\(511\) −4.22423 + 18.7477i −0.186869 + 0.829347i
\(512\) 0 0
\(513\) −1.79969 + 1.69699i −0.0794581 + 0.0749237i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 33.6391i 1.47945i
\(518\) 0 0
\(519\) 14.4346 + 21.0503i 0.633609 + 0.924005i
\(520\) 0 0
\(521\) 13.6122 23.5771i 0.596363 1.03293i −0.396990 0.917823i \(-0.629945\pi\)
0.993353 0.115109i \(-0.0367216\pi\)
\(522\) 0 0
\(523\) 0.916202 0.528969i 0.0400627 0.0231302i −0.479835 0.877359i \(-0.659303\pi\)
0.519898 + 0.854229i \(0.325970\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −15.3048 + 8.83625i −0.666689 + 0.384913i
\(528\) 0 0
\(529\) 4.16269 7.20999i 0.180987 0.313478i
\(530\) 0 0
\(531\) 2.49906 6.46922i 0.108450 0.280740i
\(532\) 0 0
\(533\) 6.10066i 0.264249i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 2.14292 + 1.02491i 0.0924738 + 0.0442282i
\(538\) 0 0
\(539\) −18.8911 + 1.52137i −0.813698 + 0.0655300i
\(540\) 0 0
\(541\) 13.1032 + 22.6953i 0.563349 + 0.975748i 0.997201 + 0.0747644i \(0.0238205\pi\)
−0.433853 + 0.900984i \(0.642846\pi\)
\(542\) 0 0
\(543\) 2.86686 + 36.9276i 0.123029 + 1.58472i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 36.6385 1.56655 0.783275 0.621675i \(-0.213548\pi\)
0.783275 + 0.621675i \(0.213548\pi\)
\(548\) 0 0
\(549\) −6.11875 + 4.93341i −0.261142 + 0.210553i
\(550\) 0 0
\(551\) 0.524531 + 0.908514i 0.0223458 + 0.0387040i
\(552\) 0 0
\(553\) 12.4687 13.5132i 0.530223 0.574640i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −29.6236 17.1032i −1.25519 0.724685i −0.283055 0.959104i \(-0.591348\pi\)
−0.972136 + 0.234419i \(0.924681\pi\)
\(558\) 0 0
\(559\) 4.68187i 0.198022i
\(560\) 0 0
\(561\) 28.5518 19.5785i 1.20546 0.826606i
\(562\) 0 0
\(563\) −11.9319 + 20.6666i −0.502868 + 0.870993i 0.497127 + 0.867678i \(0.334388\pi\)
−0.999995 + 0.00331483i \(0.998945\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −13.6780 19.4914i −0.574420 0.818561i
\(568\) 0 0
\(569\) 27.8326 16.0692i 1.16680 0.673655i 0.213879 0.976860i \(-0.431390\pi\)
0.952925 + 0.303205i \(0.0980567\pi\)
\(570\) 0 0
\(571\) 9.85333 17.0665i 0.412349 0.714209i −0.582797 0.812618i \(-0.698042\pi\)
0.995146 + 0.0984083i \(0.0313751\pi\)
\(572\) 0 0
\(573\) 24.7046 16.9405i 1.03205 0.707697i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −10.7514 6.20731i −0.447586 0.258414i 0.259224 0.965817i \(-0.416533\pi\)
−0.706810 + 0.707403i \(0.749866\pi\)
\(578\) 0 0
\(579\) −9.57176 + 20.0129i −0.397789 + 0.831710i
\(580\) 0 0
\(581\) −8.89211 28.5455i −0.368907 1.18427i
\(582\) 0 0
\(583\) 6.64845 + 11.5154i 0.275351 + 0.476921i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −30.6227 −1.26394 −0.631968 0.774995i \(-0.717753\pi\)
−0.631968 + 0.774995i \(0.717753\pi\)
\(588\) 0 0
\(589\) 1.13958 0.0469556
\(590\) 0 0
\(591\) 1.22964 + 15.8388i 0.0505805 + 0.651522i
\(592\) 0 0
\(593\) −21.7660 37.6999i −0.893823 1.54815i −0.835254 0.549864i \(-0.814679\pi\)
−0.0585693 0.998283i \(-0.518654\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 12.9332 + 6.18568i 0.529322 + 0.253163i
\(598\) 0 0
\(599\) 22.8060 + 13.1670i 0.931827 + 0.537991i 0.887389 0.461022i \(-0.152517\pi\)
0.0444381 + 0.999012i \(0.485850\pi\)
\(600\) 0 0
\(601\) 18.8846i 0.770317i 0.922850 + 0.385159i \(0.125853\pi\)
−0.922850 + 0.385159i \(0.874147\pi\)
\(602\) 0 0
\(603\) 15.8743 41.0930i 0.646450 1.67344i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −12.4123 + 7.16622i −0.503798 + 0.290868i −0.730281 0.683147i \(-0.760611\pi\)
0.226482 + 0.974015i \(0.427277\pi\)
\(608\) 0 0
\(609\) −9.38034 + 3.74074i −0.380110 + 0.151582i
\(610\) 0 0
\(611\) 12.1037 6.98810i 0.489665 0.282708i
\(612\) 0 0
\(613\) −9.26188 + 16.0421i −0.374084 + 0.647932i −0.990189 0.139731i \(-0.955376\pi\)
0.616106 + 0.787664i \(0.288709\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 46.6900i 1.87967i 0.341633 + 0.939833i \(0.389020\pi\)
−0.341633 + 0.939833i \(0.610980\pi\)
\(618\) 0 0
\(619\) 33.0066 + 19.0564i 1.32665 + 0.765940i 0.984780 0.173808i \(-0.0556073\pi\)
0.341868 + 0.939748i \(0.388941\pi\)
\(620\) 0 0
\(621\) −21.1592 + 19.9518i −0.849090 + 0.800636i
\(622\) 0 0
\(623\) 12.4941 13.5407i 0.500565 0.542498i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −2.22569 + 0.172790i −0.0888855 + 0.00690057i
\(628\) 0 0
\(629\) −64.2092 −2.56019
\(630\) 0 0
\(631\) −17.1097 −0.681128 −0.340564 0.940221i \(-0.610618\pi\)
−0.340564 + 0.940221i \(0.610618\pi\)
\(632\) 0 0
\(633\) −14.6933 + 1.14071i −0.584007 + 0.0453391i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 4.47180 + 6.48121i 0.177179 + 0.256795i
\(638\) 0 0
\(639\) −4.58260 29.3361i −0.181285 1.16052i
\(640\) 0 0
\(641\) −18.2874 10.5583i −0.722310 0.417026i 0.0932922 0.995639i \(-0.470261\pi\)
−0.815602 + 0.578613i \(0.803594\pi\)
\(642\) 0 0
\(643\) 6.86538i 0.270744i −0.990795 0.135372i \(-0.956777\pi\)
0.990795 0.135372i \(-0.0432230\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −8.90522 + 15.4243i −0.350100 + 0.606391i −0.986267 0.165161i \(-0.947186\pi\)
0.636167 + 0.771552i \(0.280519\pi\)
\(648\) 0 0
\(649\) 5.42036 3.12944i 0.212768 0.122841i
\(650\) 0 0
\(651\) −1.57576 + 10.8563i −0.0617589 + 0.425492i
\(652\) 0 0
\(653\) 39.5964 22.8610i 1.54953 0.894619i 0.551348 0.834275i \(-0.314114\pi\)
0.998178 0.0603437i \(-0.0192197\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −20.3268 7.85226i −0.793025 0.306346i
\(658\) 0 0
\(659\) 6.74688i 0.262821i 0.991328 + 0.131411i \(0.0419506\pi\)
−0.991328 + 0.131411i \(0.958049\pi\)
\(660\) 0 0
\(661\) −43.2369 24.9628i −1.68172 0.970942i −0.960518 0.278218i \(-0.910256\pi\)
−0.721203 0.692724i \(-0.756410\pi\)
\(662\) 0 0
\(663\) −12.9759 6.20607i −0.503940 0.241024i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6.16700 + 10.6816i 0.238787 + 0.413592i
\(668\) 0 0
\(669\) −0.743316 9.57457i −0.0287383 0.370174i
\(670\) 0 0
\(671\) −7.09346 −0.273840
\(672\) 0 0
\(673\) −20.0600 −0.773257 −0.386628 0.922236i \(-0.626360\pi\)
−0.386628 + 0.922236i \(0.626360\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −12.4751 21.6076i −0.479458 0.830446i 0.520264 0.854005i \(-0.325833\pi\)
−0.999722 + 0.0235594i \(0.992500\pi\)
\(678\) 0 0
\(679\) −22.2212 + 6.92206i −0.852773 + 0.265644i
\(680\) 0 0
\(681\) −4.62947 + 9.67944i −0.177402 + 0.370917i
\(682\) 0 0
\(683\) 26.7156 + 15.4243i 1.02225 + 0.590194i 0.914754 0.404012i \(-0.132385\pi\)
0.107492 + 0.994206i \(0.465718\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −0.507294 + 0.347862i −0.0193545 + 0.0132718i
\(688\) 0 0
\(689\) 2.76226 4.78438i 0.105234 0.182270i
\(690\) 0 0
\(691\) 15.5472 8.97619i 0.591444 0.341470i −0.174224 0.984706i \(-0.555742\pi\)
0.765668 + 0.643236i \(0.222408\pi\)
\(692\) 0 0
\(693\) 1.43148 21.4421i 0.0543776 0.814520i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 20.0188 34.6735i 0.758264 1.31335i
\(698\) 0 0
\(699\) −26.6637 + 18.2838i −1.00851 + 0.691557i
\(700\) 0 0
\(701\) 31.7744i 1.20010i 0.799961 + 0.600052i \(0.204853\pi\)
−0.799961 + 0.600052i \(0.795147\pi\)
\(702\) 0 0
\(703\) 3.58570 + 2.07021i 0.135237 + 0.0780793i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 27.0119 + 6.08633i 1.01589 + 0.228900i
\(708\) 0 0
\(709\) −13.2201 22.8978i −0.496490 0.859946i 0.503502 0.863994i \(-0.332045\pi\)
−0.999992 + 0.00404829i \(0.998711\pi\)
\(710\) 0 0
\(711\) 13.0861 + 16.2303i 0.490767 + 0.608683i
\(712\) 0 0
\(713\) 13.3982 0.501768
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 2.84099 + 36.5945i 0.106099 + 1.36665i
\(718\) 0 0
\(719\) 12.2625 + 21.2393i 0.457314 + 0.792092i 0.998818 0.0486066i \(-0.0154781\pi\)
−0.541504 + 0.840698i \(0.682145\pi\)
\(720\) 0 0
\(721\) −6.35608 5.86478i −0.236713 0.218416i
\(722\) 0 0
\(723\) −3.47048 1.65986i −0.129069 0.0617307i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 49.4804i 1.83513i 0.397589 + 0.917564i \(0.369847\pi\)
−0.397589 + 0.917564i \(0.630153\pi\)
\(728\) 0 0
\(729\) 24.1347 12.1044i 0.893878 0.448310i
\(730\) 0 0
\(731\) −15.3631 + 26.6097i −0.568226 + 0.984196i
\(732\) 0 0
\(733\) 14.3352 8.27642i 0.529482 0.305697i −0.211324 0.977416i \(-0.567777\pi\)
0.740805 + 0.671720i \(0.234444\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 34.4306 19.8785i 1.26827 0.732234i
\(738\) 0 0
\(739\) −8.41010 + 14.5667i −0.309371 + 0.535846i −0.978225 0.207548i \(-0.933452\pi\)
0.668854 + 0.743394i \(0.266785\pi\)
\(740\) 0 0
\(741\) 0.524531 + 0.764935i 0.0192691 + 0.0281006i
\(742\) 0 0
\(743\) 4.76463i 0.174797i −0.996173 0.0873987i \(-0.972145\pi\)
0.996173 0.0873987i \(-0.0278554\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 33.4954 5.23233i 1.22553 0.191441i
\(748\) 0 0
\(749\) −9.69531 8.94590i −0.354259 0.326876i
\(750\) 0 0
\(751\) −4.11749 7.13171i −0.150249 0.260240i 0.781070 0.624444i \(-0.214674\pi\)
−0.931319 + 0.364204i \(0.881341\pi\)
\(752\) 0 0
\(753\) −31.6044 + 2.45359i −1.15173 + 0.0894138i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 3.01624 0.109627 0.0548136 0.998497i \(-0.482544\pi\)
0.0548136 + 0.998497i \(0.482544\pi\)
\(758\) 0 0
\(759\) −26.1678 + 2.03152i −0.949832 + 0.0737396i
\(760\) 0 0
\(761\) 7.59504 + 13.1550i 0.275320 + 0.476868i 0.970216 0.242242i \(-0.0778829\pi\)
−0.694896 + 0.719110i \(0.744550\pi\)
\(762\) 0 0
\(763\) 12.3912 + 2.79200i 0.448593 + 0.101077i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.25202 1.30021i −0.0813158 0.0469477i
\(768\) 0 0
\(769\) 33.1611i 1.19582i 0.801563 + 0.597910i \(0.204002\pi\)
−0.801563 + 0.597910i \(0.795998\pi\)
\(770\) 0 0
\(771\) −10.9701 15.9979i −0.395079 0.576152i
\(772\) 0 0
\(773\) −4.84706 + 8.39536i −0.174337 + 0.301960i −0.939932 0.341363i \(-0.889111\pi\)
0.765595 + 0.643323i \(0.222445\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −24.6801 + 31.2969i −0.885395 + 1.12277i
\(778\) 0 0
\(779\) −2.23586 + 1.29087i −0.0801079 + 0.0462503i
\(780\) 0 0
\(781\) 13.3983 23.2065i 0.479429 0.830395i
\(782\) 0 0
\(783\) −2.63770 11.1429i −0.0942637 0.398215i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −36.3526 20.9882i −1.29583 0.748148i −0.316149 0.948710i \(-0.602390\pi\)
−0.979681 + 0.200562i \(0.935723\pi\)
\(788\) 0 0
\(789\) −20.6497 9.87632i −0.735150 0.351606i
\(790\) 0 0
\(791\) −47.7101 + 14.8620i −1.69638 + 0.528432i
\(792\) 0 0
\(793\) 1.47358 + 2.55231i 0.0523283 + 0.0906352i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 29.8186 1.05623 0.528115 0.849173i \(-0.322899\pi\)
0.528115 + 0.849173i \(0.322899\pi\)
\(798\) 0 0
\(799\) 91.7232 3.24493
\(800\) 0 0
\(801\) 13.1127 + 16.2633i 0.463316 + 0.574636i
\(802\) 0 0
\(803\) −9.83298 17.0312i −0.346998 0.601019i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 18.9309 39.5813i 0.666398 1.39333i
\(808\) 0 0
\(809\) −15.5493 8.97739i −0.546684 0.315628i 0.201099 0.979571i \(-0.435549\pi\)
−0.747784 + 0.663942i \(0.768882\pi\)
\(810\) 0 0
\(811\) 4.42584i 0.155412i −0.996976 0.0777061i \(-0.975240\pi\)
0.996976 0.0777061i \(-0.0247596\pi\)
\(812\) 0 0
\(813\) −39.4719 + 27.0667i −1.38434 + 0.949270i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.71588 0.990664i 0.0600311 0.0346589i
\(818\) 0 0
\(819\) −8.01251 + 3.93927i −0.279980 + 0.137649i
\(820\) 0 0
\(821\) 32.5257 18.7787i 1.13515 0.655381i 0.189927 0.981798i \(-0.439175\pi\)
0.945226 + 0.326417i \(0.105841\pi\)
\(822\) 0 0
\(823\) 18.4164 31.8982i 0.641957 1.11190i −0.343038 0.939321i \(-0.611456\pi\)
0.984995 0.172581i \(-0.0552106\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 14.9737i 0.520686i 0.965516 + 0.260343i \(0.0838357\pi\)
−0.965516 + 0.260343i \(0.916164\pi\)
\(828\) 0 0
\(829\) −2.95565 1.70645i −0.102654 0.0592674i 0.447794 0.894137i \(-0.352210\pi\)
−0.550448 + 0.834869i \(0.685543\pi\)
\(830\) 0 0
\(831\) 17.6830 36.9722i 0.613417 1.28255i
\(832\) 0 0
\(833\) 4.14829 + 51.5102i 0.143730 + 1.78472i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −11.9153 3.57077i −0.411854 0.123424i
\(838\) 0 0
\(839\) 3.71578 0.128283 0.0641416 0.997941i \(-0.479569\pi\)
0.0641416 + 0.997941i \(0.479569\pi\)
\(840\) 0 0
\(841\) 24.1436 0.832539
\(842\) 0 0
\(843\) 2.32860 + 29.9944i 0.0802012 + 1.03306i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −6.58390 + 7.13544i −0.226225 + 0.245176i
\(848\) 0 0
\(849\) 20.1891 + 9.65599i 0.692887 + 0.331393i
\(850\) 0 0
\(851\) 42.1577 + 24.3398i 1.44515 + 0.834357i
\(852\) 0 0
\(853\) 10.5549i 0.361393i −0.983539 0.180696i \(-0.942165\pi\)
0.983539 0.180696i \(-0.0578351\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.2967 33.4229i 0.659163 1.14170i −0.321669 0.946852i \(-0.604244\pi\)
0.980833 0.194852i \(-0.0624228\pi\)
\(858\) 0 0
\(859\) −25.9876 + 15.0039i −0.886685 + 0.511928i −0.872857 0.487977i \(-0.837735\pi\)
−0.0138282 + 0.999904i \(0.504402\pi\)
\(860\) 0 0
\(861\) −9.20597 23.0850i −0.313739 0.786736i
\(862\) 0 0
\(863\) 33.0204 19.0643i 1.12403 0.648958i 0.181602 0.983372i \(-0.441872\pi\)
0.942426 + 0.334414i \(0.108538\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −36.7325 53.5677i −1.24750 1.81925i
\(868\) 0 0
\(869\) 18.8157i 0.638280i
\(870\) 0 0
\(871\) −14.3050 8.25902i −0.484708 0.279846i
\(872\) 0 0
\(873\) −4.07311 26.0745i −0.137854 0.882488i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −16.5855 28.7268i −0.560051 0.970037i −0.997491 0.0707893i \(-0.977448\pi\)
0.437440 0.899247i \(-0.355885\pi\)
\(878\) 0 0
\(879\) 19.4003 1.50613i 0.654357 0.0508006i
\(880\) 0 0
\(881\) 19.8093 0.667393 0.333697 0.942681i \(-0.391704\pi\)
0.333697 + 0.942681i \(0.391704\pi\)
\(882\) 0 0
\(883\) −3.37029 −0.113419 −0.0567096 0.998391i \(-0.518061\pi\)
−0.0567096 + 0.998391i \(0.518061\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −18.0715 31.3008i −0.606783 1.05098i −0.991767 0.128056i \(-0.959126\pi\)
0.384984 0.922923i \(-0.374207\pi\)
\(888\) 0 0
\(889\) −6.11629 19.6346i −0.205134 0.658522i
\(890\) 0 0
\(891\) 23.8130 + 5.16732i 0.797767 + 0.173112i
\(892\) 0 0
\(893\) −5.12220 2.95730i −0.171408 0.0989623i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 6.16700 + 8.99347i 0.205910 + 0.300283i
\(898\) 0 0
\(899\) −2.63770 + 4.56863i −0.0879722 + 0.152372i
\(900\) 0 0
\(901\) 31.3990 18.1282i 1.04605 0.603939i
\(902\) 0 0
\(903\) 7.06500 + 17.7163i 0.235109 + 0.589562i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 23.0513 39.9260i 0.765405 1.32572i −0.174627 0.984635i \(-0.555872\pi\)
0.940032 0.341086i \(-0.110795\pi\)
\(908\) 0 0
\(909\) −11.3136 + 29.2872i −0.375250 + 0.971394i
\(910\) 0 0
\(911\) 58.0403i 1.92296i −0.274871 0.961481i \(-0.588635\pi\)
0.274871 0.961481i \(-0.411365\pi\)
\(912\) 0 0
\(913\) 26.4968 + 15.2979i 0.876916 + 0.506288i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −8.34648 + 9.04567i −0.275625 + 0.298714i
\(918\) 0 0
\(919\) −2.66148 4.60983i −0.0877943 0.152064i 0.818784 0.574101i \(-0.194648\pi\)
−0.906578 + 0.422037i \(0.861315\pi\)
\(920\) 0 0
\(921\) 4.15818 + 53.5610i 0.137017 + 1.76490i
\(922\) 0 0
\(923\) −11.1333 −0.366458
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 7.63409 6.15519i 0.250736 0.202163i
\(928\) 0 0
\(929\) 20.7113 + 35.8731i 0.679517 + 1.17696i 0.975127 + 0.221649i \(0.0711439\pi\)
−0.295610 + 0.955309i \(0.595523\pi\)
\(930\) 0 0
\(931\) 1.42911 3.01029i 0.0468373 0.0986581i
\(932\) 0 0
\(933\) 1.15291 2.41054i 0.0377445 0.0789175i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.523249i 0.0170938i −0.999963 0.00854690i \(-0.997279\pi\)
0.999963 0.00854690i \(-0.00272060\pi\)
\(938\) 0 0
\(939\) −36.7245 + 25.1827i −1.19846 + 0.821807i
\(940\) 0 0
\(941\) 9.58594 16.6033i 0.312493 0.541253i −0.666409 0.745587i \(-0.732169\pi\)
0.978901 + 0.204333i \(0.0655026\pi\)
\(942\) 0 0
\(943\) −26.2874 + 15.1770i −0.856035 + 0.494232i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26.0940 15.0654i 0.847940 0.489558i −0.0120156 0.999928i \(-0.503825\pi\)
0.859955 + 0.510370i \(0.170491\pi\)
\(948\) 0 0
\(949\) −4.08536 + 7.07605i −0.132616 + 0.229698i
\(950\) 0 0
\(951\) −4.44383 + 3.04722i −0.144101 + 0.0988130i
\(952\) 0 0
\(953\) 17.4015i 0.563691i 0.959460 + 0.281846i \(0.0909466\pi\)
−0.959460 + 0.281846i \(0.909053\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 4.45892 9.32285i 0.144136 0.301365i
\(958\) 0 0
\(959\) 6.40407 28.4221i 0.206798 0.917796i
\(960\) 0 0
\(961\) −12.6347 21.8840i −0.407571 0.705934i
\(962\) 0 0
\(963\) 11.6447 9.38887i 0.375246 0.302552i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −24.9154 −0.801224 −0.400612 0.916248i \(-0.631203\pi\)
−0.400612 + 0.916248i \(0.631203\pi\)
\(968\) 0 0
\(969\) 0.471144 + 6.06875i 0.0151353 + 0.194956i
\(970\) 0 0
\(971\) 9.45293 + 16.3730i 0.303359 + 0.525433i 0.976895 0.213722i \(-0.0685585\pi\)
−0.673536 + 0.739155i \(0.735225\pi\)
\(972\) 0 0
\(973\) 52.0179 16.2039i 1.66762 0.519474i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 35.0586 + 20.2411i 1.12163 + 0.647571i 0.941815 0.336131i \(-0.109118\pi\)
0.179810 + 0.983701i \(0.442452\pi\)
\(978\) 0 0
\(979\) 18.8540i 0.602578i
\(980\) 0 0
\(981\) −5.18994 + 13.4350i −0.165702 + 0.428946i
\(982\) 0 0
\(983\) −27.1552 + 47.0341i −0.866116 + 1.50016i −0.000180339 1.00000i \(0.500057\pi\)
−0.865935 + 0.500156i \(0.833276\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 35.2557 44.7078i 1.12220 1.42307i
\(988\) 0 0
\(989\) 20.1739 11.6474i 0.641493 0.370366i
\(990\) 0 0
\(991\) 30.8784 53.4830i 0.980886 1.69894i 0.321929 0.946764i \(-0.395669\pi\)
0.658957 0.752181i \(-0.270998\pi\)
\(992\) 0 0
\(993\) 0.274226 + 0.399910i 0.00870230 + 0.0126907i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −17.7948 10.2738i −0.563566 0.325375i 0.191009 0.981588i \(-0.438824\pi\)
−0.754576 + 0.656213i \(0.772157\pi\)
\(998\) 0 0
\(999\) −31.0050 32.8814i −0.980954 1.04032i
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 2100.2.bi.m.101.8 yes 16
3.2 odd 2 inner 2100.2.bi.m.101.5 yes 16
5.2 odd 4 2100.2.bo.i.1949.8 32
5.3 odd 4 2100.2.bo.i.1949.9 32
5.4 even 2 2100.2.bi.l.101.1 16
7.5 odd 6 inner 2100.2.bi.m.1601.5 yes 16
15.2 even 4 2100.2.bo.i.1949.13 32
15.8 even 4 2100.2.bo.i.1949.4 32
15.14 odd 2 2100.2.bi.l.101.4 yes 16
21.5 even 6 inner 2100.2.bi.m.1601.8 yes 16
35.12 even 12 2100.2.bo.i.1349.4 32
35.19 odd 6 2100.2.bi.l.1601.4 yes 16
35.33 even 12 2100.2.bo.i.1349.13 32
105.47 odd 12 2100.2.bo.i.1349.9 32
105.68 odd 12 2100.2.bo.i.1349.8 32
105.89 even 6 2100.2.bi.l.1601.1 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.bi.l.101.1 16 5.4 even 2
2100.2.bi.l.101.4 yes 16 15.14 odd 2
2100.2.bi.l.1601.1 yes 16 105.89 even 6
2100.2.bi.l.1601.4 yes 16 35.19 odd 6
2100.2.bi.m.101.5 yes 16 3.2 odd 2 inner
2100.2.bi.m.101.8 yes 16 1.1 even 1 trivial
2100.2.bi.m.1601.5 yes 16 7.5 odd 6 inner
2100.2.bi.m.1601.8 yes 16 21.5 even 6 inner
2100.2.bo.i.1349.4 32 35.12 even 12
2100.2.bo.i.1349.8 32 105.68 odd 12
2100.2.bo.i.1349.9 32 105.47 odd 12
2100.2.bo.i.1349.13 32 35.33 even 12
2100.2.bo.i.1949.4 32 15.8 even 4
2100.2.bo.i.1949.8 32 5.2 odd 4
2100.2.bo.i.1949.9 32 5.3 odd 4
2100.2.bo.i.1949.13 32 15.2 even 4