Properties

Label 2100.2.q.h.1801.2
Level $2100$
Weight $2$
Character 2100.1801
Analytic conductor $16.769$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(1201,2100)]
 
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, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1201");
 
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.q (of order \(3\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1801.2
Root \(1.32288 + 2.29129i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1801
Dual form 2100.2.q.h.1201.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(1.32288 - 2.29129i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{3} +(1.32288 - 2.29129i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(0.822876 + 1.42526i) q^{11} +2.64575 q^{13} +(-0.822876 - 1.42526i) q^{17} +(-4.14575 + 7.18065i) q^{19} +2.64575 q^{21} +(0.822876 - 1.42526i) q^{23} -1.00000 q^{27} +7.64575 q^{29} +(2.14575 + 3.71655i) q^{31} +(-0.822876 + 1.42526i) q^{33} +(0.322876 - 0.559237i) q^{37} +(1.32288 + 2.29129i) q^{39} +4.93725 q^{41} +5.93725 q^{43} +(3.00000 - 5.19615i) q^{47} +(-3.50000 - 6.06218i) q^{49} +(0.822876 - 1.42526i) q^{51} +(1.64575 + 2.85052i) q^{53} -8.29150 q^{57} +(-5.46863 - 9.47194i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(1.32288 + 2.29129i) q^{63} +(0.322876 + 0.559237i) q^{67} +1.64575 q^{69} +13.6458 q^{71} +(6.61438 + 11.4564i) q^{73} +4.35425 q^{77} +(-1.14575 + 1.98450i) q^{79} +(-0.500000 - 0.866025i) q^{81} +10.9373 q^{83} +(3.82288 + 6.62141i) q^{87} +(7.11438 - 12.3225i) q^{89} +(3.50000 - 6.06218i) q^{91} +(-2.14575 + 3.71655i) q^{93} -8.00000 q^{97} -1.64575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{9} - 2 q^{11} + 2 q^{17} - 6 q^{19} - 2 q^{23} - 4 q^{27} + 20 q^{29} - 2 q^{31} + 2 q^{33} - 4 q^{37} - 12 q^{41} - 8 q^{43} + 12 q^{47} - 14 q^{49} - 2 q^{51} - 4 q^{53} - 12 q^{57} - 6 q^{59} - 16 q^{61} - 4 q^{67} - 4 q^{69} + 44 q^{71} + 28 q^{77} + 6 q^{79} - 2 q^{81} + 12 q^{83} + 10 q^{87} + 2 q^{89} + 14 q^{91} + 2 q^{93} - 32 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.32288 2.29129i 0.500000 0.866025i
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 0.822876 + 1.42526i 0.248106 + 0.429733i 0.963000 0.269500i \(-0.0868584\pi\)
−0.714894 + 0.699233i \(0.753525\pi\)
\(12\) 0 0
\(13\) 2.64575 0.733799 0.366900 0.930261i \(-0.380419\pi\)
0.366900 + 0.930261i \(0.380419\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.822876 1.42526i −0.199577 0.345677i 0.748815 0.662780i \(-0.230623\pi\)
−0.948391 + 0.317103i \(0.897290\pi\)
\(18\) 0 0
\(19\) −4.14575 + 7.18065i −0.951101 + 1.64735i −0.208051 + 0.978118i \(0.566712\pi\)
−0.743049 + 0.669237i \(0.766621\pi\)
\(20\) 0 0
\(21\) 2.64575 0.577350
\(22\) 0 0
\(23\) 0.822876 1.42526i 0.171581 0.297188i −0.767391 0.641179i \(-0.778446\pi\)
0.938973 + 0.343991i \(0.111779\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.64575 1.41978 0.709890 0.704312i \(-0.248745\pi\)
0.709890 + 0.704312i \(0.248745\pi\)
\(30\) 0 0
\(31\) 2.14575 + 3.71655i 0.385388 + 0.667512i 0.991823 0.127621i \(-0.0407342\pi\)
−0.606435 + 0.795133i \(0.707401\pi\)
\(32\) 0 0
\(33\) −0.822876 + 1.42526i −0.143244 + 0.248106i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.322876 0.559237i 0.0530804 0.0919380i −0.838264 0.545264i \(-0.816429\pi\)
0.891345 + 0.453326i \(0.149763\pi\)
\(38\) 0 0
\(39\) 1.32288 + 2.29129i 0.211830 + 0.366900i
\(40\) 0 0
\(41\) 4.93725 0.771070 0.385535 0.922693i \(-0.374017\pi\)
0.385535 + 0.922693i \(0.374017\pi\)
\(42\) 0 0
\(43\) 5.93725 0.905423 0.452711 0.891657i \(-0.350457\pi\)
0.452711 + 0.891657i \(0.350457\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000 5.19615i 0.437595 0.757937i −0.559908 0.828554i \(-0.689164\pi\)
0.997503 + 0.0706177i \(0.0224970\pi\)
\(48\) 0 0
\(49\) −3.50000 6.06218i −0.500000 0.866025i
\(50\) 0 0
\(51\) 0.822876 1.42526i 0.115226 0.199577i
\(52\) 0 0
\(53\) 1.64575 + 2.85052i 0.226061 + 0.391550i 0.956637 0.291282i \(-0.0940817\pi\)
−0.730576 + 0.682831i \(0.760748\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −8.29150 −1.09824
\(58\) 0 0
\(59\) −5.46863 9.47194i −0.711955 1.23314i −0.964122 0.265458i \(-0.914477\pi\)
0.252168 0.967684i \(-0.418856\pi\)
\(60\) 0 0
\(61\) −4.00000 + 6.92820i −0.512148 + 0.887066i 0.487753 + 0.872982i \(0.337817\pi\)
−0.999901 + 0.0140840i \(0.995517\pi\)
\(62\) 0 0
\(63\) 1.32288 + 2.29129i 0.166667 + 0.288675i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0.322876 + 0.559237i 0.0394455 + 0.0683217i 0.885074 0.465450i \(-0.154108\pi\)
−0.845629 + 0.533772i \(0.820774\pi\)
\(68\) 0 0
\(69\) 1.64575 0.198125
\(70\) 0 0
\(71\) 13.6458 1.61945 0.809726 0.586808i \(-0.199616\pi\)
0.809726 + 0.586808i \(0.199616\pi\)
\(72\) 0 0
\(73\) 6.61438 + 11.4564i 0.774154 + 1.34087i 0.935269 + 0.353939i \(0.115158\pi\)
−0.161114 + 0.986936i \(0.551509\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.35425 0.496213
\(78\) 0 0
\(79\) −1.14575 + 1.98450i −0.128907 + 0.223274i −0.923253 0.384192i \(-0.874480\pi\)
0.794346 + 0.607465i \(0.207814\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 10.9373 1.20052 0.600260 0.799805i \(-0.295064\pi\)
0.600260 + 0.799805i \(0.295064\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 3.82288 + 6.62141i 0.409855 + 0.709890i
\(88\) 0 0
\(89\) 7.11438 12.3225i 0.754123 1.30618i −0.191687 0.981456i \(-0.561396\pi\)
0.945809 0.324722i \(-0.105271\pi\)
\(90\) 0 0
\(91\) 3.50000 6.06218i 0.366900 0.635489i
\(92\) 0 0
\(93\) −2.14575 + 3.71655i −0.222504 + 0.385388i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 0 0
\(99\) −1.64575 −0.165404
\(100\) 0 0
\(101\) 2.46863 + 4.27579i 0.245638 + 0.425457i 0.962311 0.271953i \(-0.0876694\pi\)
−0.716673 + 0.697409i \(0.754336\pi\)
\(102\) 0 0
\(103\) 4.67712 8.10102i 0.460851 0.798217i −0.538153 0.842847i \(-0.680878\pi\)
0.999004 + 0.0446304i \(0.0142110\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.46863 + 14.6681i −0.818693 + 1.41802i 0.0879524 + 0.996125i \(0.471968\pi\)
−0.906645 + 0.421893i \(0.861366\pi\)
\(108\) 0 0
\(109\) 6.50000 + 11.2583i 0.622587 + 1.07835i 0.989002 + 0.147901i \(0.0472517\pi\)
−0.366415 + 0.930451i \(0.619415\pi\)
\(110\) 0 0
\(111\) 0.645751 0.0612920
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.32288 + 2.29129i −0.122300 + 0.211830i
\(118\) 0 0
\(119\) −4.35425 −0.399153
\(120\) 0 0
\(121\) 4.14575 7.18065i 0.376886 0.652787i
\(122\) 0 0
\(123\) 2.46863 + 4.27579i 0.222589 + 0.385535i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.06275 0.183039 0.0915196 0.995803i \(-0.470828\pi\)
0.0915196 + 0.995803i \(0.470828\pi\)
\(128\) 0 0
\(129\) 2.96863 + 5.14181i 0.261373 + 0.452711i
\(130\) 0 0
\(131\) −10.6458 + 18.4390i −0.930124 + 1.61102i −0.147017 + 0.989134i \(0.546967\pi\)
−0.783107 + 0.621887i \(0.786366\pi\)
\(132\) 0 0
\(133\) 10.9686 + 18.9982i 0.951101 + 1.64735i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −10.1144 17.5186i −0.864130 1.49672i −0.867909 0.496724i \(-0.834536\pi\)
0.00377913 0.999993i \(-0.498797\pi\)
\(138\) 0 0
\(139\) 5.00000 0.424094 0.212047 0.977259i \(-0.431987\pi\)
0.212047 + 0.977259i \(0.431987\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 2.17712 + 3.77089i 0.182060 + 0.315338i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 3.50000 6.06218i 0.288675 0.500000i
\(148\) 0 0
\(149\) 3.29150 5.70105i 0.269650 0.467048i −0.699121 0.715003i \(-0.746425\pi\)
0.968772 + 0.247955i \(0.0797585\pi\)
\(150\) 0 0
\(151\) 2.29150 + 3.96900i 0.186480 + 0.322993i 0.944074 0.329733i \(-0.106959\pi\)
−0.757594 + 0.652726i \(0.773625\pi\)
\(152\) 0 0
\(153\) 1.64575 0.133051
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.645751 1.11847i −0.0515366 0.0892639i 0.839106 0.543968i \(-0.183079\pi\)
−0.890643 + 0.454704i \(0.849745\pi\)
\(158\) 0 0
\(159\) −1.64575 + 2.85052i −0.130517 + 0.226061i
\(160\) 0 0
\(161\) −2.17712 3.77089i −0.171581 0.297188i
\(162\) 0 0
\(163\) −8.00000 + 13.8564i −0.626608 + 1.08532i 0.361619 + 0.932326i \(0.382224\pi\)
−0.988227 + 0.152992i \(0.951109\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.6458 −1.52023 −0.760117 0.649786i \(-0.774858\pi\)
−0.760117 + 0.649786i \(0.774858\pi\)
\(168\) 0 0
\(169\) −6.00000 −0.461538
\(170\) 0 0
\(171\) −4.14575 7.18065i −0.317034 0.549118i
\(172\) 0 0
\(173\) 4.93725 8.55157i 0.375372 0.650164i −0.615010 0.788519i \(-0.710848\pi\)
0.990383 + 0.138355i \(0.0441815\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.46863 9.47194i 0.411047 0.711955i
\(178\) 0 0
\(179\) −0.291503 0.504897i −0.0217879 0.0377378i 0.854926 0.518750i \(-0.173603\pi\)
−0.876714 + 0.481012i \(0.840269\pi\)
\(180\) 0 0
\(181\) −4.29150 −0.318985 −0.159492 0.987199i \(-0.550986\pi\)
−0.159492 + 0.987199i \(0.550986\pi\)
\(182\) 0 0
\(183\) −8.00000 −0.591377
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.35425 2.34563i 0.0990325 0.171529i
\(188\) 0 0
\(189\) −1.32288 + 2.29129i −0.0962250 + 0.166667i
\(190\) 0 0
\(191\) −9.58301 + 16.5983i −0.693402 + 1.20101i 0.277315 + 0.960779i \(0.410556\pi\)
−0.970716 + 0.240228i \(0.922778\pi\)
\(192\) 0 0
\(193\) −4.03137 6.98254i −0.290185 0.502614i 0.683669 0.729793i \(-0.260383\pi\)
−0.973853 + 0.227178i \(0.927050\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.93725 0.351765 0.175882 0.984411i \(-0.443722\pi\)
0.175882 + 0.984411i \(0.443722\pi\)
\(198\) 0 0
\(199\) 11.5830 + 20.0624i 0.821097 + 1.42218i 0.904866 + 0.425697i \(0.139971\pi\)
−0.0837682 + 0.996485i \(0.526696\pi\)
\(200\) 0 0
\(201\) −0.322876 + 0.559237i −0.0227739 + 0.0394455i
\(202\) 0 0
\(203\) 10.1144 17.5186i 0.709890 1.22957i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.822876 + 1.42526i 0.0571938 + 0.0990626i
\(208\) 0 0
\(209\) −13.6458 −0.943896
\(210\) 0 0
\(211\) 8.58301 0.590878 0.295439 0.955362i \(-0.404534\pi\)
0.295439 + 0.955362i \(0.404534\pi\)
\(212\) 0 0
\(213\) 6.82288 + 11.8176i 0.467496 + 0.809726i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 11.3542 0.770777
\(218\) 0 0
\(219\) −6.61438 + 11.4564i −0.446958 + 0.774154i
\(220\) 0 0
\(221\) −2.17712 3.77089i −0.146449 0.253658i
\(222\) 0 0
\(223\) −17.8745 −1.19697 −0.598483 0.801136i \(-0.704230\pi\)
−0.598483 + 0.801136i \(0.704230\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.82288 + 17.0137i 0.651967 + 1.12924i 0.982645 + 0.185497i \(0.0593893\pi\)
−0.330678 + 0.943744i \(0.607277\pi\)
\(228\) 0 0
\(229\) 3.50000 6.06218i 0.231287 0.400600i −0.726900 0.686743i \(-0.759040\pi\)
0.958187 + 0.286143i \(0.0923732\pi\)
\(230\) 0 0
\(231\) 2.17712 + 3.77089i 0.143244 + 0.248106i
\(232\) 0 0
\(233\) 10.6458 18.4390i 0.697426 1.20798i −0.271930 0.962317i \(-0.587662\pi\)
0.969356 0.245661i \(-0.0790049\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −2.29150 −0.148849
\(238\) 0 0
\(239\) 6.00000 0.388108 0.194054 0.980991i \(-0.437836\pi\)
0.194054 + 0.980991i \(0.437836\pi\)
\(240\) 0 0
\(241\) 6.35425 + 11.0059i 0.409313 + 0.708951i 0.994813 0.101721i \(-0.0324350\pi\)
−0.585500 + 0.810673i \(0.699102\pi\)
\(242\) 0 0
\(243\) 0.500000 0.866025i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −10.9686 + 18.9982i −0.697917 + 1.20883i
\(248\) 0 0
\(249\) 5.46863 + 9.47194i 0.346560 + 0.600260i
\(250\) 0 0
\(251\) −10.9373 −0.690353 −0.345177 0.938538i \(-0.612181\pi\)
−0.345177 + 0.938538i \(0.612181\pi\)
\(252\) 0 0
\(253\) 2.70850 0.170282
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.76013 4.78068i 0.172172 0.298211i −0.767007 0.641639i \(-0.778255\pi\)
0.939179 + 0.343428i \(0.111588\pi\)
\(258\) 0 0
\(259\) −0.854249 1.47960i −0.0530804 0.0919380i
\(260\) 0 0
\(261\) −3.82288 + 6.62141i −0.236630 + 0.409855i
\(262\) 0 0
\(263\) −14.2288 24.6449i −0.877383 1.51967i −0.854203 0.519940i \(-0.825954\pi\)
−0.0231800 0.999731i \(-0.507379\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 14.2288 0.870786
\(268\) 0 0
\(269\) −13.9373 24.1400i −0.849769 1.47184i −0.881414 0.472345i \(-0.843408\pi\)
0.0316446 0.999499i \(-0.489926\pi\)
\(270\) 0 0
\(271\) −3.70850 + 6.42331i −0.225275 + 0.390188i −0.956402 0.292054i \(-0.905661\pi\)
0.731127 + 0.682242i \(0.238995\pi\)
\(272\) 0 0
\(273\) 7.00000 0.423659
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −10.3229 17.8797i −0.620241 1.07429i −0.989441 0.144939i \(-0.953701\pi\)
0.369199 0.929350i \(-0.379632\pi\)
\(278\) 0 0
\(279\) −4.29150 −0.256926
\(280\) 0 0
\(281\) 24.0000 1.43172 0.715860 0.698244i \(-0.246035\pi\)
0.715860 + 0.698244i \(0.246035\pi\)
\(282\) 0 0
\(283\) −7.32288 12.6836i −0.435300 0.753961i 0.562020 0.827123i \(-0.310024\pi\)
−0.997320 + 0.0731621i \(0.976691\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.53137 11.3127i 0.385535 0.667766i
\(288\) 0 0
\(289\) 7.14575 12.3768i 0.420338 0.728047i
\(290\) 0 0
\(291\) −4.00000 6.92820i −0.234484 0.406138i
\(292\) 0 0
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −0.822876 1.42526i −0.0477481 0.0827021i
\(298\) 0 0
\(299\) 2.17712 3.77089i 0.125906 0.218076i
\(300\) 0 0
\(301\) 7.85425 13.6040i 0.452711 0.784119i
\(302\) 0 0
\(303\) −2.46863 + 4.27579i −0.141819 + 0.245638i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −27.9373 −1.59446 −0.797232 0.603674i \(-0.793703\pi\)
−0.797232 + 0.603674i \(0.793703\pi\)
\(308\) 0 0
\(309\) 9.35425 0.532145
\(310\) 0 0
\(311\) −6.82288 11.8176i −0.386890 0.670113i 0.605140 0.796119i \(-0.293117\pi\)
−0.992029 + 0.126007i \(0.959784\pi\)
\(312\) 0 0
\(313\) −10.6144 + 18.3846i −0.599960 + 1.03916i 0.392866 + 0.919596i \(0.371484\pi\)
−0.992826 + 0.119566i \(0.961850\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.7601 20.3691i 0.660515 1.14404i −0.319966 0.947429i \(-0.603671\pi\)
0.980481 0.196616i \(-0.0629952\pi\)
\(318\) 0 0
\(319\) 6.29150 + 10.8972i 0.352257 + 0.610126i
\(320\) 0 0
\(321\) −16.9373 −0.945345
\(322\) 0 0
\(323\) 13.6458 0.759270
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −6.50000 + 11.2583i −0.359451 + 0.622587i
\(328\) 0 0
\(329\) −7.93725 13.7477i −0.437595 0.757937i
\(330\) 0 0
\(331\) 10.0830 17.4643i 0.554212 0.959923i −0.443752 0.896149i \(-0.646353\pi\)
0.997964 0.0637740i \(-0.0203137\pi\)
\(332\) 0 0
\(333\) 0.322876 + 0.559237i 0.0176935 + 0.0306460i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 33.2288 1.81009 0.905043 0.425320i \(-0.139839\pi\)
0.905043 + 0.425320i \(0.139839\pi\)
\(338\) 0 0
\(339\) −3.00000 5.19615i −0.162938 0.282216i
\(340\) 0 0
\(341\) −3.53137 + 6.11652i −0.191235 + 0.331228i
\(342\) 0 0
\(343\) −18.5203 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.58301 + 11.4021i 0.353394 + 0.612097i 0.986842 0.161689i \(-0.0516941\pi\)
−0.633448 + 0.773786i \(0.718361\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) −2.64575 −0.141220
\(352\) 0 0
\(353\) −18.0516 31.2663i −0.960791 1.66414i −0.720520 0.693434i \(-0.756097\pi\)
−0.240271 0.970706i \(-0.577236\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −2.17712 3.77089i −0.115226 0.199577i
\(358\) 0 0
\(359\) 10.4059 18.0235i 0.549201 0.951245i −0.449128 0.893467i \(-0.648265\pi\)
0.998329 0.0577773i \(-0.0184013\pi\)
\(360\) 0 0
\(361\) −24.8745 43.0839i −1.30918 2.26757i
\(362\) 0 0
\(363\) 8.29150 0.435191
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.61438 + 6.26029i 0.188669 + 0.326784i 0.944807 0.327628i \(-0.106249\pi\)
−0.756138 + 0.654413i \(0.772916\pi\)
\(368\) 0 0
\(369\) −2.46863 + 4.27579i −0.128512 + 0.222589i
\(370\) 0 0
\(371\) 8.70850 0.452123
\(372\) 0 0
\(373\) 0.0313730 0.0543397i 0.00162443 0.00281360i −0.865212 0.501406i \(-0.832816\pi\)
0.866836 + 0.498593i \(0.166150\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.2288 1.04183
\(378\) 0 0
\(379\) −15.7085 −0.806891 −0.403446 0.915004i \(-0.632188\pi\)
−0.403446 + 0.915004i \(0.632188\pi\)
\(380\) 0 0
\(381\) 1.03137 + 1.78639i 0.0528388 + 0.0915196i
\(382\) 0 0
\(383\) −13.6458 + 23.6351i −0.697265 + 1.20770i 0.272146 + 0.962256i \(0.412267\pi\)
−0.969411 + 0.245443i \(0.921067\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.96863 + 5.14181i −0.150904 + 0.261373i
\(388\) 0 0
\(389\) 0.239870 + 0.415468i 0.0121619 + 0.0210651i 0.872042 0.489430i \(-0.162795\pi\)
−0.859880 + 0.510496i \(0.829462\pi\)
\(390\) 0 0
\(391\) −2.70850 −0.136975
\(392\) 0 0
\(393\) −21.2915 −1.07401
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −5.67712 + 9.83307i −0.284927 + 0.493508i −0.972591 0.232521i \(-0.925303\pi\)
0.687665 + 0.726028i \(0.258636\pi\)
\(398\) 0 0
\(399\) −10.9686 + 18.9982i −0.549118 + 0.951101i
\(400\) 0 0
\(401\) −4.35425 + 7.54178i −0.217441 + 0.376619i −0.954025 0.299727i \(-0.903104\pi\)
0.736584 + 0.676346i \(0.236438\pi\)
\(402\) 0 0
\(403\) 5.67712 + 9.83307i 0.282798 + 0.489820i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.06275 0.0526784
\(408\) 0 0
\(409\) −3.08301 5.33992i −0.152445 0.264042i 0.779681 0.626177i \(-0.215381\pi\)
−0.932126 + 0.362135i \(0.882048\pi\)
\(410\) 0 0
\(411\) 10.1144 17.5186i 0.498905 0.864130i
\(412\) 0 0
\(413\) −28.9373 −1.42391
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.50000 + 4.33013i 0.122426 + 0.212047i
\(418\) 0 0
\(419\) 21.2915 1.04016 0.520079 0.854118i \(-0.325903\pi\)
0.520079 + 0.854118i \(0.325903\pi\)
\(420\) 0 0
\(421\) −19.5830 −0.954417 −0.477209 0.878790i \(-0.658351\pi\)
−0.477209 + 0.878790i \(0.658351\pi\)
\(422\) 0 0
\(423\) 3.00000 + 5.19615i 0.145865 + 0.252646i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 10.5830 + 18.3303i 0.512148 + 0.887066i
\(428\) 0 0
\(429\) −2.17712 + 3.77089i −0.105113 + 0.182060i
\(430\) 0 0
\(431\) −19.9373 34.5323i −0.960344 1.66336i −0.721636 0.692273i \(-0.756610\pi\)
−0.238708 0.971091i \(-0.576724\pi\)
\(432\) 0 0
\(433\) −3.35425 −0.161195 −0.0805975 0.996747i \(-0.525683\pi\)
−0.0805975 + 0.996747i \(0.525683\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6.82288 + 11.8176i 0.326382 + 0.565311i
\(438\) 0 0
\(439\) −2.64575 + 4.58258i −0.126275 + 0.218714i −0.922231 0.386640i \(-0.873635\pi\)
0.795956 + 0.605355i \(0.206969\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) 4.93725 8.55157i 0.234576 0.406298i −0.724573 0.689198i \(-0.757963\pi\)
0.959149 + 0.282900i \(0.0912965\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.58301 0.311365
\(448\) 0 0
\(449\) −12.5830 −0.593829 −0.296914 0.954904i \(-0.595958\pi\)
−0.296914 + 0.954904i \(0.595958\pi\)
\(450\) 0 0
\(451\) 4.06275 + 7.03688i 0.191307 + 0.331354i
\(452\) 0 0
\(453\) −2.29150 + 3.96900i −0.107664 + 0.186480i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7.32288 + 12.6836i −0.342550 + 0.593313i −0.984905 0.173093i \(-0.944624\pi\)
0.642356 + 0.766407i \(0.277957\pi\)
\(458\) 0 0
\(459\) 0.822876 + 1.42526i 0.0384085 + 0.0665256i
\(460\) 0 0
\(461\) −19.6458 −0.914994 −0.457497 0.889211i \(-0.651254\pi\)
−0.457497 + 0.889211i \(0.651254\pi\)
\(462\) 0 0
\(463\) −10.5203 −0.488918 −0.244459 0.969660i \(-0.578610\pi\)
−0.244459 + 0.969660i \(0.578610\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −18.8745 + 32.6916i −0.873408 + 1.51279i −0.0149591 + 0.999888i \(0.504762\pi\)
−0.858449 + 0.512899i \(0.828572\pi\)
\(468\) 0 0
\(469\) 1.70850 0.0788911
\(470\) 0 0
\(471\) 0.645751 1.11847i 0.0297546 0.0515366i
\(472\) 0 0
\(473\) 4.88562 + 8.46215i 0.224641 + 0.389090i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −3.29150 −0.150708
\(478\) 0 0
\(479\) 3.29150 + 5.70105i 0.150393 + 0.260488i 0.931372 0.364069i \(-0.118613\pi\)
−0.780979 + 0.624557i \(0.785280\pi\)
\(480\) 0 0
\(481\) 0.854249 1.47960i 0.0389504 0.0674641i
\(482\) 0 0
\(483\) 2.17712 3.77089i 0.0990626 0.171581i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.32288 2.29129i −0.0599452 0.103828i 0.834495 0.551015i \(-0.185759\pi\)
−0.894441 + 0.447187i \(0.852426\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) 2.12549 0.0959221 0.0479611 0.998849i \(-0.484728\pi\)
0.0479611 + 0.998849i \(0.484728\pi\)
\(492\) 0 0
\(493\) −6.29150 10.8972i −0.283355 0.490785i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 18.0516 31.2663i 0.809726 1.40249i
\(498\) 0 0
\(499\) −13.7288 + 23.7789i −0.614584 + 1.06449i 0.375874 + 0.926671i \(0.377343\pi\)
−0.990457 + 0.137819i \(0.955991\pi\)
\(500\) 0 0
\(501\) −9.82288 17.0137i −0.438854 0.760117i
\(502\) 0 0
\(503\) −22.4575 −1.00133 −0.500666 0.865641i \(-0.666911\pi\)
−0.500666 + 0.865641i \(0.666911\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −3.00000 5.19615i −0.133235 0.230769i
\(508\) 0 0
\(509\) 13.9373 24.1400i 0.617758 1.06999i −0.372136 0.928178i \(-0.621374\pi\)
0.989894 0.141810i \(-0.0452922\pi\)
\(510\) 0 0
\(511\) 35.0000 1.54831
\(512\) 0 0
\(513\) 4.14575 7.18065i 0.183039 0.317034i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 9.87451 0.434280
\(518\) 0 0
\(519\) 9.87451 0.433443
\(520\) 0 0
\(521\) 1.06275 + 1.84073i 0.0465598 + 0.0806439i 0.888366 0.459136i \(-0.151841\pi\)
−0.841806 + 0.539780i \(0.818508\pi\)
\(522\) 0 0
\(523\) 15.6144 27.0449i 0.682769 1.18259i −0.291363 0.956612i \(-0.594109\pi\)
0.974132 0.225978i \(-0.0725578\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.53137 6.11652i 0.153829 0.266440i
\(528\) 0 0
\(529\) 10.1458 + 17.5730i 0.441120 + 0.764042i
\(530\) 0 0
\(531\) 10.9373 0.474636
\(532\) 0 0
\(533\) 13.0627 0.565810
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.291503 0.504897i 0.0125793 0.0217879i
\(538\) 0 0
\(539\) 5.76013 9.97684i 0.248106 0.429733i
\(540\) 0 0
\(541\) −18.3745 + 31.8256i −0.789982 + 1.36829i 0.135996 + 0.990709i \(0.456577\pi\)
−0.925977 + 0.377579i \(0.876757\pi\)
\(542\) 0 0
\(543\) −2.14575 3.71655i −0.0920830 0.159492i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 43.2915 1.85101 0.925505 0.378734i \(-0.123641\pi\)
0.925505 + 0.378734i \(0.123641\pi\)
\(548\) 0 0
\(549\) −4.00000 6.92820i −0.170716 0.295689i
\(550\) 0 0
\(551\) −31.6974 + 54.9015i −1.35035 + 2.33888i
\(552\) 0 0
\(553\) 3.03137 + 5.25049i 0.128907 + 0.223274i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.00000 15.5885i −0.381342 0.660504i 0.609912 0.792469i \(-0.291205\pi\)
−0.991254 + 0.131965i \(0.957871\pi\)
\(558\) 0 0
\(559\) 15.7085 0.664399
\(560\) 0 0
\(561\) 2.70850 0.114353
\(562\) 0 0
\(563\) −18.2915 31.6818i −0.770895 1.33523i −0.937073 0.349133i \(-0.886476\pi\)
0.166178 0.986096i \(-0.446857\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.64575 −0.111111
\(568\) 0 0
\(569\) 1.88562 3.26599i 0.0790494 0.136918i −0.823791 0.566894i \(-0.808145\pi\)
0.902840 + 0.429977i \(0.141478\pi\)
\(570\) 0 0
\(571\) −22.1458 38.3576i −0.926771 1.60521i −0.788688 0.614793i \(-0.789239\pi\)
−0.138083 0.990421i \(-0.544094\pi\)
\(572\) 0 0
\(573\) −19.1660 −0.800672
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −21.2601 36.8236i −0.885071 1.53299i −0.845633 0.533766i \(-0.820776\pi\)
−0.0394383 0.999222i \(-0.512557\pi\)
\(578\) 0 0
\(579\) 4.03137 6.98254i 0.167538 0.290185i
\(580\) 0 0
\(581\) 14.4686 25.0604i 0.600260 1.03968i
\(582\) 0 0
\(583\) −2.70850 + 4.69126i −0.112174 + 0.194292i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −16.9373 −0.699075 −0.349538 0.936922i \(-0.613661\pi\)
−0.349538 + 0.936922i \(0.613661\pi\)
\(588\) 0 0
\(589\) −35.5830 −1.46617
\(590\) 0 0
\(591\) 2.46863 + 4.27579i 0.101546 + 0.175882i
\(592\) 0 0
\(593\) −12.5314 + 21.7050i −0.514602 + 0.891316i 0.485255 + 0.874373i \(0.338727\pi\)
−0.999856 + 0.0169436i \(0.994606\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −11.5830 + 20.0624i −0.474061 + 0.821097i
\(598\) 0 0
\(599\) −1.06275 1.84073i −0.0434226 0.0752102i 0.843497 0.537133i \(-0.180493\pi\)
−0.886920 + 0.461923i \(0.847160\pi\)
\(600\) 0 0
\(601\) −37.5830 −1.53304 −0.766521 0.642219i \(-0.778014\pi\)
−0.766521 + 0.642219i \(0.778014\pi\)
\(602\) 0 0
\(603\) −0.645751 −0.0262970
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −8.38562 + 14.5243i −0.340362 + 0.589524i −0.984500 0.175385i \(-0.943883\pi\)
0.644138 + 0.764909i \(0.277216\pi\)
\(608\) 0 0
\(609\) 20.2288 0.819711
\(610\) 0 0
\(611\) 7.93725 13.7477i 0.321107 0.556174i
\(612\) 0 0
\(613\) −20.5830 35.6508i −0.831340 1.43992i −0.896976 0.442079i \(-0.854241\pi\)
0.0656363 0.997844i \(-0.479092\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −9.29150 −0.374062 −0.187031 0.982354i \(-0.559886\pi\)
−0.187031 + 0.982354i \(0.559886\pi\)
\(618\) 0 0
\(619\) 9.50000 + 16.4545i 0.381837 + 0.661361i 0.991325 0.131434i \(-0.0419582\pi\)
−0.609488 + 0.792796i \(0.708625\pi\)
\(620\) 0 0
\(621\) −0.822876 + 1.42526i −0.0330209 + 0.0571938i
\(622\) 0 0
\(623\) −18.8229 32.6022i −0.754123 1.30618i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −6.82288 11.8176i −0.272479 0.471948i
\(628\) 0 0
\(629\) −1.06275 −0.0423745
\(630\) 0 0
\(631\) −35.7490 −1.42315 −0.711573 0.702612i \(-0.752017\pi\)
−0.711573 + 0.702612i \(0.752017\pi\)
\(632\) 0 0
\(633\) 4.29150 + 7.43310i 0.170572 + 0.295439i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −9.26013 16.0390i −0.366900 0.635489i
\(638\) 0 0
\(639\) −6.82288 + 11.8176i −0.269909 + 0.467496i
\(640\) 0 0
\(641\) 6.05163 + 10.4817i 0.239025 + 0.414004i 0.960435 0.278505i \(-0.0898388\pi\)
−0.721410 + 0.692509i \(0.756506\pi\)
\(642\) 0 0
\(643\) 21.8118 0.860172 0.430086 0.902788i \(-0.358483\pi\)
0.430086 + 0.902788i \(0.358483\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 11.4686 + 19.8642i 0.450878 + 0.780944i 0.998441 0.0558207i \(-0.0177775\pi\)
−0.547563 + 0.836765i \(0.684444\pi\)
\(648\) 0 0
\(649\) 9.00000 15.5885i 0.353281 0.611900i
\(650\) 0 0
\(651\) 5.67712 + 9.83307i 0.222504 + 0.385388i
\(652\) 0 0
\(653\) 14.1771 24.5555i 0.554794 0.960931i −0.443126 0.896459i \(-0.646131\pi\)
0.997920 0.0644715i \(-0.0205362\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −13.2288 −0.516103
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) 20.4373 + 35.3984i 0.794917 + 1.37684i 0.922892 + 0.385059i \(0.125819\pi\)
−0.127975 + 0.991777i \(0.540848\pi\)
\(662\) 0 0
\(663\) 2.17712 3.77089i 0.0845525 0.146449i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6.29150 10.8972i 0.243608 0.421941i
\(668\) 0 0
\(669\) −8.93725 15.4798i −0.345534 0.598483i
\(670\) 0 0
\(671\) −13.1660 −0.508268
\(672\) 0 0
\(673\) 5.35425 0.206391 0.103196 0.994661i \(-0.467093\pi\)
0.103196 + 0.994661i \(0.467093\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −11.1771 + 19.3593i −0.429572 + 0.744040i −0.996835 0.0794963i \(-0.974669\pi\)
0.567263 + 0.823536i \(0.308002\pi\)
\(678\) 0 0
\(679\) −10.5830 + 18.3303i −0.406138 + 0.703452i
\(680\) 0 0
\(681\) −9.82288 + 17.0137i −0.376413 + 0.651967i
\(682\) 0 0
\(683\) 2.46863 + 4.27579i 0.0944594 + 0.163608i 0.909383 0.415960i \(-0.136554\pi\)
−0.814923 + 0.579569i \(0.803221\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7.00000 0.267067
\(688\) 0 0
\(689\) 4.35425 + 7.54178i 0.165884 + 0.287319i
\(690\) 0 0
\(691\) −11.7915 + 20.4235i −0.448570 + 0.776946i −0.998293 0.0584009i \(-0.981400\pi\)
0.549723 + 0.835347i \(0.314733\pi\)
\(692\) 0 0
\(693\) −2.17712 + 3.77089i −0.0827021 + 0.143244i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −4.06275 7.03688i −0.153887 0.266541i
\(698\) 0 0
\(699\) 21.2915 0.805319
\(700\) 0 0
\(701\) 34.9373 1.31956 0.659781 0.751458i \(-0.270649\pi\)
0.659781 + 0.751458i \(0.270649\pi\)
\(702\) 0 0
\(703\) 2.67712 + 4.63692i 0.100970 + 0.174885i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.0627 0.491275
\(708\) 0 0
\(709\) 11.2915 19.5575i 0.424061 0.734496i −0.572271 0.820065i \(-0.693937\pi\)
0.996332 + 0.0855689i \(0.0272708\pi\)
\(710\) 0 0
\(711\) −1.14575 1.98450i −0.0429690 0.0744245i
\(712\) 0 0
\(713\) 7.06275 0.264502
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.00000 + 5.19615i 0.112037 + 0.194054i
\(718\) 0 0
\(719\) 7.35425 12.7379i 0.274267 0.475045i −0.695683 0.718349i \(-0.744898\pi\)
0.969950 + 0.243304i \(0.0782314\pi\)
\(720\) 0 0
\(721\) −12.3745 21.4333i −0.460851 0.798217i
\(722\) 0 0
\(723\) −6.35425 + 11.0059i −0.236317 + 0.409313i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 33.8118 1.25401 0.627004 0.779016i \(-0.284281\pi\)
0.627004 + 0.779016i \(0.284281\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.88562 8.46215i −0.180701 0.312984i
\(732\) 0 0
\(733\) 3.61438 6.26029i 0.133500 0.231229i −0.791523 0.611139i \(-0.790712\pi\)
0.925023 + 0.379910i \(0.124045\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.531373 + 0.920365i −0.0195734 + 0.0339021i
\(738\) 0 0
\(739\) −14.5000 25.1147i −0.533391 0.923861i −0.999239 0.0389959i \(-0.987584\pi\)
0.465848 0.884865i \(-0.345749\pi\)
\(740\) 0 0
\(741\) −21.9373 −0.805885
\(742\) 0 0
\(743\) 13.0627 0.479226 0.239613 0.970869i \(-0.422979\pi\)
0.239613 + 0.970869i \(0.422979\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −5.46863 + 9.47194i −0.200087 + 0.346560i
\(748\) 0 0
\(749\) 22.4059 + 38.8081i 0.818693 + 1.41802i
\(750\) 0 0
\(751\) −12.0830 + 20.9284i −0.440915 + 0.763687i −0.997758 0.0669307i \(-0.978679\pi\)
0.556843 + 0.830618i \(0.312013\pi\)
\(752\) 0 0
\(753\) −5.46863 9.47194i −0.199288 0.345177i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.83399 0.321077 0.160538 0.987030i \(-0.448677\pi\)
0.160538 + 0.987030i \(0.448677\pi\)
\(758\) 0 0
\(759\) 1.35425 + 2.34563i 0.0491561 + 0.0851409i
\(760\) 0 0
\(761\) 6.05163 10.4817i 0.219371 0.379963i −0.735244 0.677802i \(-0.762933\pi\)
0.954616 + 0.297839i \(0.0962660\pi\)
\(762\) 0 0
\(763\) 34.3948 1.24517
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −14.4686 25.0604i −0.522432 0.904878i
\(768\) 0 0
\(769\) −3.70850 −0.133732 −0.0668659 0.997762i \(-0.521300\pi\)
−0.0668659 + 0.997762i \(0.521300\pi\)
\(770\) 0 0
\(771\) 5.52026 0.198807
\(772\) 0 0
\(773\) 13.6974 + 23.7246i 0.492661 + 0.853313i 0.999964 0.00845413i \(-0.00269106\pi\)
−0.507304 + 0.861767i \(0.669358\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0.854249 1.47960i 0.0306460 0.0530804i
\(778\) 0 0
\(779\) −20.4686 + 35.4527i −0.733365 + 1.27022i
\(780\) 0 0
\(781\) 11.2288 + 19.4488i 0.401796 + 0.695932i
\(782\) 0 0
\(783\) −7.64575 −0.273237
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 2.93725 + 5.08747i 0.104702 + 0.181349i 0.913616 0.406577i \(-0.133278\pi\)
−0.808915 + 0.587926i \(0.799945\pi\)
\(788\) 0 0
\(789\) 14.2288 24.6449i 0.506557 0.877383i
\(790\) 0 0
\(791\) −7.93725 + 13.7477i −0.282216 + 0.488813i
\(792\) 0 0
\(793\) −10.5830 + 18.3303i −0.375814 + 0.650928i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −4.93725 −0.174887 −0.0874433 0.996170i \(-0.527870\pi\)
−0.0874433 + 0.996170i \(0.527870\pi\)
\(798\) 0 0
\(799\) −9.87451 −0.349335
\(800\) 0 0
\(801\) 7.11438 + 12.3225i 0.251374 + 0.435393i
\(802\) 0 0
\(803\) −10.8856 + 18.8544i −0.384145 + 0.665359i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 13.9373 24.1400i 0.490615 0.849769i
\(808\) 0 0
\(809\) −9.58301 16.5983i −0.336921 0.583563i 0.646931 0.762548i \(-0.276052\pi\)
−0.983852 + 0.178985i \(0.942719\pi\)
\(810\) 0 0
\(811\) 17.2915 0.607187 0.303593 0.952802i \(-0.401814\pi\)
0.303593 + 0.952802i \(0.401814\pi\)
\(812\) 0 0
\(813\) −7.41699 −0.260125
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −24.6144 + 42.6334i −0.861148 + 1.49155i
\(818\) 0 0
\(819\) 3.50000 + 6.06218i 0.122300 + 0.211830i
\(820\) 0 0
\(821\) 25.9889 45.0141i 0.907018 1.57100i 0.0888337 0.996046i \(-0.471686\pi\)
0.818185 0.574955i \(-0.194981\pi\)
\(822\) 0 0
\(823\) 9.22876 + 15.9847i 0.321694 + 0.557191i 0.980838 0.194826i \(-0.0624144\pi\)
−0.659144 + 0.752017i \(0.729081\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −22.4575 −0.780924 −0.390462 0.920619i \(-0.627685\pi\)
−0.390462 + 0.920619i \(0.627685\pi\)
\(828\) 0 0
\(829\) −9.85425 17.0681i −0.342252 0.592798i 0.642598 0.766203i \(-0.277856\pi\)
−0.984851 + 0.173405i \(0.944523\pi\)
\(830\) 0 0
\(831\) 10.3229 17.8797i 0.358097 0.620241i
\(832\) 0 0
\(833\) −5.76013 + 9.97684i −0.199577 + 0.345677i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −2.14575 3.71655i −0.0741680 0.128463i
\(838\) 0 0
\(839\) 16.3542 0.564611 0.282306 0.959325i \(-0.408901\pi\)
0.282306 + 0.959325i \(0.408901\pi\)
\(840\) 0 0
\(841\) 29.4575 1.01578
\(842\) 0 0
\(843\) 12.0000 + 20.7846i 0.413302 + 0.715860i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −10.9686 18.9982i −0.376886 0.652787i
\(848\) 0 0
\(849\) 7.32288 12.6836i 0.251320 0.435300i
\(850\) 0 0
\(851\) −0.531373 0.920365i −0.0182152 0.0315497i
\(852\) 0 0
\(853\) −43.2288 −1.48012 −0.740062 0.672538i \(-0.765204\pi\)
−0.740062 + 0.672538i \(0.765204\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −6.82288 11.8176i −0.233065 0.403680i 0.725644 0.688071i \(-0.241542\pi\)
−0.958709 + 0.284390i \(0.908209\pi\)
\(858\) 0 0
\(859\) 13.2288 22.9129i 0.451359 0.781777i −0.547111 0.837060i \(-0.684273\pi\)
0.998471 + 0.0552825i \(0.0176059\pi\)
\(860\) 0 0
\(861\) 13.0627 0.445177
\(862\) 0 0
\(863\) −17.2288 + 29.8411i −0.586474 + 1.01580i 0.408216 + 0.912885i \(0.366151\pi\)
−0.994690 + 0.102917i \(0.967182\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 14.2915 0.485365
\(868\) 0 0
\(869\) −3.77124 −0.127931
\(870\) 0 0
\(871\) 0.854249 + 1.47960i 0.0289451 + 0.0501344i
\(872\) 0 0
\(873\) 4.00000 6.92820i 0.135379 0.234484i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.00000 12.1244i 0.236373 0.409410i −0.723298 0.690536i \(-0.757375\pi\)
0.959671 + 0.281126i \(0.0907079\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.16601 0.0392839 0.0196419 0.999807i \(-0.493747\pi\)
0.0196419 + 0.999807i \(0.493747\pi\)
\(882\) 0 0
\(883\) 21.8118 0.734024 0.367012 0.930216i \(-0.380381\pi\)
0.367012 + 0.930216i \(0.380381\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −9.82288 + 17.0137i −0.329820 + 0.571265i −0.982476 0.186390i \(-0.940321\pi\)
0.652656 + 0.757654i \(0.273655\pi\)
\(888\) 0 0
\(889\) 2.72876 4.72634i 0.0915196 0.158517i
\(890\) 0 0
\(891\) 0.822876 1.42526i 0.0275674 0.0477481i
\(892\) 0 0
\(893\) 24.8745 + 43.0839i 0.832394 + 1.44175i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.35425 0.145384
\(898\) 0 0
\(899\) 16.4059 + 28.4158i 0.547167 + 0.947721i
\(900\) 0 0
\(901\) 2.70850 4.69126i 0.0902331 0.156288i
\(902\) 0 0
\(903\) 15.7085 0.522746
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −10.3229 17.8797i −0.342765 0.593687i 0.642180 0.766554i \(-0.278030\pi\)
−0.984945 + 0.172867i \(0.944697\pi\)
\(908\) 0 0
\(909\) −4.93725 −0.163758
\(910\) 0 0
\(911\) −5.52026 −0.182894 −0.0914472 0.995810i \(-0.529149\pi\)
−0.0914472 + 0.995810i \(0.529149\pi\)
\(912\) 0 0
\(913\) 9.00000 + 15.5885i 0.297857 + 0.515903i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 28.1660 + 48.7850i 0.930124 + 1.61102i
\(918\) 0 0
\(919\) −5.20850 + 9.02138i −0.171812 + 0.297588i −0.939054 0.343771i \(-0.888296\pi\)
0.767241 + 0.641359i \(0.221629\pi\)
\(920\) 0 0
\(921\) −13.9686 24.1944i −0.460282 0.797232i
\(922\) 0 0
\(923\) 36.1033 1.18835
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 4.67712 + 8.10102i 0.153617 + 0.266072i
\(928\) 0 0
\(929\) −17.7601 + 30.7614i −0.582691 + 1.00925i 0.412468 + 0.910972i \(0.364667\pi\)
−0.995159 + 0.0982783i \(0.968666\pi\)
\(930\) 0 0
\(931\) 58.0405 1.90220
\(932\) 0 0
\(933\) 6.82288 11.8176i 0.223371 0.386890i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −38.9778 −1.27335 −0.636674 0.771133i \(-0.719690\pi\)
−0.636674 + 0.771133i \(0.719690\pi\)
\(938\) 0 0
\(939\) −21.2288 −0.692774
\(940\) 0 0
\(941\) 5.46863 + 9.47194i 0.178272 + 0.308776i 0.941289 0.337602i \(-0.109616\pi\)
−0.763017 + 0.646379i \(0.776283\pi\)
\(942\) 0 0
\(943\) 4.06275 7.03688i 0.132301 0.229152i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −20.1771 + 34.9478i −0.655668 + 1.13565i 0.326057 + 0.945350i \(0.394280\pi\)
−0.981726 + 0.190301i \(0.939054\pi\)
\(948\) 0 0
\(949\) 17.5000 + 30.3109i 0.568074 + 0.983933i
\(950\) 0 0
\(951\) 23.5203 0.762697
\(952\) 0 0
\(953\) 52.4575 1.69927 0.849633 0.527375i \(-0.176824\pi\)
0.849633 + 0.527375i \(0.176824\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −6.29150 + 10.8972i −0.203375 + 0.352257i
\(958\) 0 0
\(959\) −53.5203 −1.72826
\(960\) 0 0
\(961\) 6.29150 10.8972i 0.202952 0.351523i
\(962\) 0 0
\(963\) −8.46863 14.6681i −0.272898 0.472673i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 28.3948 0.913114 0.456557 0.889694i \(-0.349082\pi\)
0.456557 + 0.889694i \(0.349082\pi\)
\(968\) 0 0
\(969\) 6.82288 + 11.8176i 0.219182 + 0.379635i
\(970\) 0 0
\(971\) −5.41699 + 9.38251i −0.173840 + 0.301099i −0.939759 0.341837i \(-0.888951\pi\)
0.765919 + 0.642937i \(0.222284\pi\)
\(972\) 0 0
\(973\) 6.61438 11.4564i 0.212047 0.367277i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12.5314 21.7050i −0.400914 0.694404i 0.592922 0.805260i \(-0.297974\pi\)
−0.993836 + 0.110856i \(0.964641\pi\)
\(978\) 0 0
\(979\) 23.4170 0.748410
\(980\) 0 0
\(981\) −13.0000 −0.415058
\(982\) 0 0
\(983\) 12.3431 + 21.3789i 0.393685 + 0.681882i 0.992932 0.118682i \(-0.0378668\pi\)
−0.599247 + 0.800564i \(0.704533\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 7.93725 13.7477i 0.252646 0.437595i
\(988\) 0 0
\(989\) 4.88562 8.46215i 0.155354 0.269081i
\(990\) 0 0
\(991\) −22.1458 38.3576i −0.703483 1.21847i −0.967236 0.253878i \(-0.918294\pi\)
0.263753 0.964590i \(-0.415040\pi\)
\(992\) 0 0
\(993\) 20.1660 0.639949
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 21.6144 + 37.4372i 0.684534 + 1.18565i 0.973583 + 0.228334i \(0.0733277\pi\)
−0.289049 + 0.957314i \(0.593339\pi\)
\(998\) 0 0
\(999\) −0.322876 + 0.559237i −0.0102153 + 0.0176935i
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.q.h.1801.2 4
5.2 odd 4 2100.2.bc.e.1549.4 8
5.3 odd 4 2100.2.bc.e.1549.1 8
5.4 even 2 420.2.q.c.121.1 4
7.4 even 3 inner 2100.2.q.h.1201.2 4
15.14 odd 2 1260.2.s.f.541.1 4
20.19 odd 2 1680.2.bg.q.961.2 4
35.4 even 6 420.2.q.c.361.1 yes 4
35.9 even 6 2940.2.a.s.1.1 2
35.18 odd 12 2100.2.bc.e.949.4 8
35.19 odd 6 2940.2.a.m.1.1 2
35.24 odd 6 2940.2.q.t.361.2 4
35.32 odd 12 2100.2.bc.e.949.1 8
35.34 odd 2 2940.2.q.t.961.2 4
105.44 odd 6 8820.2.a.be.1.2 2
105.74 odd 6 1260.2.s.f.361.1 4
105.89 even 6 8820.2.a.bj.1.2 2
140.39 odd 6 1680.2.bg.q.1201.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.q.c.121.1 4 5.4 even 2
420.2.q.c.361.1 yes 4 35.4 even 6
1260.2.s.f.361.1 4 105.74 odd 6
1260.2.s.f.541.1 4 15.14 odd 2
1680.2.bg.q.961.2 4 20.19 odd 2
1680.2.bg.q.1201.2 4 140.39 odd 6
2100.2.q.h.1201.2 4 7.4 even 3 inner
2100.2.q.h.1801.2 4 1.1 even 1 trivial
2100.2.bc.e.949.1 8 35.32 odd 12
2100.2.bc.e.949.4 8 35.18 odd 12
2100.2.bc.e.1549.1 8 5.3 odd 4
2100.2.bc.e.1549.4 8 5.2 odd 4
2940.2.a.m.1.1 2 35.19 odd 6
2940.2.a.s.1.1 2 35.9 even 6
2940.2.q.t.361.2 4 35.24 odd 6
2940.2.q.t.961.2 4 35.34 odd 2
8820.2.a.be.1.2 2 105.44 odd 6
8820.2.a.bj.1.2 2 105.89 even 6