Properties

Label 2100.2.bc.e.1549.4
Level $2100$
Weight $2$
Character 2100.1549
Analytic conductor $16.769$
Analytic rank $0$
Dimension $8$
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(949,2100)]
 
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, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.949");
 
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.bc (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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1549.4
Root \(0.228425 - 1.39564i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1549
Dual form 2100.2.bc.e.949.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 - 0.500000i) q^{3} +(2.29129 + 1.32288i) q^{7} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(0.866025 - 0.500000i) q^{3} +(2.29129 + 1.32288i) q^{7} +(0.500000 - 0.866025i) q^{9} +(0.822876 + 1.42526i) q^{11} -2.64575i q^{13} +(1.42526 - 0.822876i) q^{17} +(4.14575 - 7.18065i) q^{19} +2.64575 q^{21} +(-1.42526 - 0.822876i) q^{23} -1.00000i q^{27} -7.64575 q^{29} +(2.14575 + 3.71655i) q^{31} +(1.42526 + 0.822876i) q^{33} +(0.559237 + 0.322876i) q^{37} +(-1.32288 - 2.29129i) q^{39} +4.93725 q^{41} -5.93725i q^{43} +(5.19615 + 3.00000i) q^{47} +(3.50000 + 6.06218i) q^{49} +(0.822876 - 1.42526i) q^{51} +(2.85052 - 1.64575i) q^{53} -8.29150i q^{57} +(5.46863 + 9.47194i) q^{59} +(-4.00000 + 6.92820i) q^{61} +(2.29129 - 1.32288i) q^{63} +(-0.559237 + 0.322876i) q^{67} -1.64575 q^{69} +13.6458 q^{71} +(11.4564 - 6.61438i) q^{73} +4.35425i q^{77} +(1.14575 - 1.98450i) q^{79} +(-0.500000 - 0.866025i) q^{81} -10.9373i q^{83} +(-6.62141 + 3.82288i) q^{87} +(-7.11438 + 12.3225i) q^{89} +(3.50000 - 6.06218i) q^{91} +(3.71655 + 2.14575i) q^{93} -8.00000i q^{97} +1.64575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 4 q^{11} + 12 q^{19} - 40 q^{29} - 4 q^{31} - 24 q^{41} + 28 q^{49} - 4 q^{51} + 12 q^{59} - 32 q^{61} + 8 q^{69} + 88 q^{71} - 12 q^{79} - 4 q^{81} - 4 q^{89} + 28 q^{91} - 8 q^{99}+O(q^{100}) \) 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
\(3\) 0.866025 0.500000i 0.500000 0.288675i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.29129 + 1.32288i 0.866025 + 0.500000i
\(8\) 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
\(13\) 2.64575i 0.733799i −0.930261 0.366900i \(-0.880419\pi\)
0.930261 0.366900i \(-0.119581\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.42526 0.822876i 0.345677 0.199577i −0.317103 0.948391i \(-0.602710\pi\)
0.662780 + 0.748815i \(0.269377\pi\)
\(18\) 0 0
\(19\) 4.14575 7.18065i 0.951101 1.64735i 0.208051 0.978118i \(-0.433288\pi\)
0.743049 0.669237i \(-0.233379\pi\)
\(20\) 0 0
\(21\) 2.64575 0.577350
\(22\) 0 0
\(23\) −1.42526 0.822876i −0.297188 0.171581i 0.343991 0.938973i \(-0.388221\pi\)
−0.641179 + 0.767391i \(0.721554\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −7.64575 −1.41978 −0.709890 0.704312i \(-0.751255\pi\)
−0.709890 + 0.704312i \(0.751255\pi\)
\(30\) 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
\(33\) 1.42526 + 0.822876i 0.248106 + 0.143244i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.559237 + 0.322876i 0.0919380 + 0.0530804i 0.545264 0.838264i \(-0.316429\pi\)
−0.453326 + 0.891345i \(0.649763\pi\)
\(38\) 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
\(43\) 5.93725i 0.905423i −0.891657 0.452711i \(-0.850457\pi\)
0.891657 0.452711i \(-0.149543\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.19615 + 3.00000i 0.757937 + 0.437595i 0.828554 0.559908i \(-0.189164\pi\)
−0.0706177 + 0.997503i \(0.522497\pi\)
\(48\) 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
\(53\) 2.85052 1.64575i 0.391550 0.226061i −0.291282 0.956637i \(-0.594082\pi\)
0.682831 + 0.730576i \(0.260748\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.29150i 1.09824i
\(58\) 0 0
\(59\) 5.46863 + 9.47194i 0.711955 + 1.23314i 0.964122 + 0.265458i \(0.0855232\pi\)
−0.252168 + 0.967684i \(0.581144\pi\)
\(60\) 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
\(63\) 2.29129 1.32288i 0.288675 0.166667i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.559237 + 0.322876i −0.0683217 + 0.0394455i −0.533772 0.845629i \(-0.679226\pi\)
0.465450 + 0.885074i \(0.345892\pi\)
\(68\) 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
\(73\) 11.4564 6.61438i 1.34087 0.774154i 0.353939 0.935269i \(-0.384842\pi\)
0.986936 + 0.161114i \(0.0515087\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.35425i 0.496213i
\(78\) 0 0
\(79\) 1.14575 1.98450i 0.128907 0.223274i −0.794346 0.607465i \(-0.792186\pi\)
0.923253 + 0.384192i \(0.125520\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 10.9373i 1.20052i −0.799805 0.600260i \(-0.795064\pi\)
0.799805 0.600260i \(-0.204936\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.62141 + 3.82288i −0.709890 + 0.409855i
\(88\) 0 0
\(89\) −7.11438 + 12.3225i −0.754123 + 1.30618i 0.191687 + 0.981456i \(0.438604\pi\)
−0.945809 + 0.324722i \(0.894729\pi\)
\(90\) 0 0
\(91\) 3.50000 6.06218i 0.366900 0.635489i
\(92\) 0 0
\(93\) 3.71655 + 2.14575i 0.385388 + 0.222504i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000i 0.812277i −0.913812 0.406138i \(-0.866875\pi\)
0.913812 0.406138i \(-0.133125\pi\)
\(98\) 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
\(103\) −8.10102 4.67712i −0.798217 0.460851i 0.0446304 0.999004i \(-0.485789\pi\)
−0.842847 + 0.538153i \(0.819122\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.6681 8.46863i −1.41802 0.818693i −0.421893 0.906645i \(-0.638634\pi\)
−0.996125 + 0.0879524i \(0.971968\pi\)
\(108\) 0 0
\(109\) −6.50000 11.2583i −0.622587 1.07835i −0.989002 0.147901i \(-0.952748\pi\)
0.366415 0.930451i \(-0.380585\pi\)
\(110\) 0 0
\(111\) 0.645751 0.0612920
\(112\) 0 0
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −2.29129 1.32288i −0.211830 0.122300i
\(118\) 0 0
\(119\) 4.35425 0.399153
\(120\) 0 0
\(121\) 4.14575 7.18065i 0.376886 0.652787i
\(122\) 0 0
\(123\) 4.27579 2.46863i 0.385535 0.222589i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.06275i 0.183039i 0.995803 + 0.0915196i \(0.0291724\pi\)
−0.995803 + 0.0915196i \(0.970828\pi\)
\(128\) 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
\(133\) 18.9982 10.9686i 1.64735 0.951101i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 17.5186 10.1144i 1.49672 0.864130i 0.496724 0.867909i \(-0.334536\pi\)
0.999993 + 0.00377913i \(0.00120294\pi\)
\(138\) 0 0
\(139\) −5.00000 −0.424094 −0.212047 0.977259i \(-0.568013\pi\)
−0.212047 + 0.977259i \(0.568013\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 3.77089 2.17712i 0.315338 0.182060i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.06218 + 3.50000i 0.500000 + 0.288675i
\(148\) 0 0
\(149\) −3.29150 + 5.70105i −0.269650 + 0.467048i −0.968772 0.247955i \(-0.920241\pi\)
0.699121 + 0.715003i \(0.253575\pi\)
\(150\) 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
\(153\) 1.64575i 0.133051i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.11847 0.645751i 0.0892639 0.0515366i −0.454704 0.890643i \(-0.650255\pi\)
0.543968 + 0.839106i \(0.316921\pi\)
\(158\) 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
\(163\) 13.8564 + 8.00000i 1.08532 + 0.626608i 0.932326 0.361619i \(-0.117776\pi\)
0.152992 + 0.988227i \(0.451109\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 19.6458i 1.52023i −0.649786 0.760117i \(-0.725142\pi\)
0.649786 0.760117i \(-0.274858\pi\)
\(168\) 0 0
\(169\) 6.00000 0.461538
\(170\) 0 0
\(171\) −4.14575 7.18065i −0.317034 0.549118i
\(172\) 0 0
\(173\) −8.55157 4.93725i −0.650164 0.375372i 0.138355 0.990383i \(-0.455819\pi\)
−0.788519 + 0.615010i \(0.789152\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 9.47194 + 5.46863i 0.711955 + 0.411047i
\(178\) 0 0
\(179\) 0.291503 + 0.504897i 0.0217879 + 0.0377378i 0.876714 0.481012i \(-0.159731\pi\)
−0.854926 + 0.518750i \(0.826397\pi\)
\(180\) 0 0
\(181\) −4.29150 −0.318985 −0.159492 0.987199i \(-0.550986\pi\)
−0.159492 + 0.987199i \(0.550986\pi\)
\(182\) 0 0
\(183\) 8.00000i 0.591377i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.34563 + 1.35425i 0.171529 + 0.0990325i
\(188\) 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
\(193\) −6.98254 + 4.03137i −0.502614 + 0.290185i −0.729793 0.683669i \(-0.760383\pi\)
0.227178 + 0.973853i \(0.427050\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.93725i 0.351765i 0.984411 + 0.175882i \(0.0562778\pi\)
−0.984411 + 0.175882i \(0.943722\pi\)
\(198\) 0 0
\(199\) −11.5830 20.0624i −0.821097 1.42218i −0.904866 0.425697i \(-0.860029\pi\)
0.0837682 0.996485i \(-0.473304\pi\)
\(200\) 0 0
\(201\) −0.322876 + 0.559237i −0.0227739 + 0.0394455i
\(202\) 0 0
\(203\) −17.5186 10.1144i −1.22957 0.709890i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1.42526 + 0.822876i −0.0990626 + 0.0571938i
\(208\) 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
\(213\) 11.8176 6.82288i 0.809726 0.467496i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 11.3542i 0.770777i
\(218\) 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
\(223\) 17.8745i 1.19697i 0.801136 + 0.598483i \(0.204230\pi\)
−0.801136 + 0.598483i \(0.795770\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −17.0137 + 9.82288i −1.12924 + 0.651967i −0.943744 0.330678i \(-0.892723\pi\)
−0.185497 + 0.982645i \(0.559389\pi\)
\(228\) 0 0
\(229\) −3.50000 + 6.06218i −0.231287 + 0.400600i −0.958187 0.286143i \(-0.907627\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 2.17712 + 3.77089i 0.143244 + 0.248106i
\(232\) 0 0
\(233\) −18.4390 10.6458i −1.20798 0.697426i −0.245661 0.969356i \(-0.579005\pi\)
−0.962317 + 0.271930i \(0.912338\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 2.29150i 0.148849i
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 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
\(243\) −0.866025 0.500000i −0.0555556 0.0320750i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −18.9982 10.9686i −1.20883 0.697917i
\(248\) 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
\(253\) 2.70850i 0.170282i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.78068 + 2.76013i 0.298211 + 0.172172i 0.641639 0.767007i \(-0.278255\pi\)
−0.343428 + 0.939179i \(0.611588\pi\)
\(258\) 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
\(263\) −24.6449 + 14.2288i −1.51967 + 0.877383i −0.519940 + 0.854203i \(0.674046\pi\)
−0.999731 + 0.0231800i \(0.992621\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 14.2288i 0.870786i
\(268\) 0 0
\(269\) 13.9373 + 24.1400i 0.849769 + 1.47184i 0.881414 + 0.472345i \(0.156592\pi\)
−0.0316446 + 0.999499i \(0.510074\pi\)
\(270\) 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
\(273\) 7.00000i 0.423659i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 17.8797 10.3229i 1.07429 0.620241i 0.144939 0.989441i \(-0.453701\pi\)
0.929350 + 0.369199i \(0.120368\pi\)
\(278\) 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
\(283\) −12.6836 + 7.32288i −0.753961 + 0.435300i −0.827123 0.562020i \(-0.810024\pi\)
0.0731621 + 0.997320i \(0.476691\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 11.3127 + 6.53137i 0.667766 + 0.385535i
\(288\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.42526 0.822876i 0.0827021 0.0477481i
\(298\) 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
\(303\) 4.27579 + 2.46863i 0.245638 + 0.141819i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 27.9373i 1.59446i −0.603674 0.797232i \(-0.706297\pi\)
0.603674 0.797232i \(-0.293703\pi\)
\(308\) 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
\(313\) 18.3846 + 10.6144i 1.03916 + 0.599960i 0.919596 0.392866i \(-0.128516\pi\)
0.119566 + 0.992826i \(0.461850\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.3691 + 11.7601i 1.14404 + 0.660515i 0.947429 0.319966i \(-0.103671\pi\)
0.196616 + 0.980481i \(0.437005\pi\)
\(318\) 0 0
\(319\) −6.29150 10.8972i −0.352257 0.610126i
\(320\) 0 0
\(321\) −16.9373 −0.945345
\(322\) 0 0
\(323\) 13.6458i 0.759270i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −11.2583 6.50000i −0.622587 0.359451i
\(328\) 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
\(333\) 0.559237 0.322876i 0.0306460 0.0176935i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 33.2288i 1.81009i 0.425320 + 0.905043i \(0.360161\pi\)
−0.425320 + 0.905043i \(0.639839\pi\)
\(338\) 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
\(343\) 18.5203i 1.00000i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −11.4021 + 6.58301i −0.612097 + 0.353394i −0.773786 0.633448i \(-0.781639\pi\)
0.161689 + 0.986842i \(0.448306\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) −2.64575 −0.141220
\(352\) 0 0
\(353\) −31.2663 + 18.0516i −1.66414 + 0.960791i −0.693434 + 0.720520i \(0.743903\pi\)
−0.970706 + 0.240271i \(0.922764\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3.77089 2.17712i 0.199577 0.115226i
\(358\) 0 0
\(359\) −10.4059 + 18.0235i −0.549201 + 0.951245i 0.449128 + 0.893467i \(0.351735\pi\)
−0.998329 + 0.0577773i \(0.981599\pi\)
\(360\) 0 0
\(361\) −24.8745 43.0839i −1.30918 2.26757i
\(362\) 0 0
\(363\) 8.29150i 0.435191i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −6.26029 + 3.61438i −0.326784 + 0.188669i −0.654413 0.756138i \(-0.727084\pi\)
0.327628 + 0.944807i \(0.393751\pi\)
\(368\) 0 0
\(369\) 2.46863 4.27579i 0.128512 0.222589i
\(370\) 0 0
\(371\) 8.70850 0.452123
\(372\) 0 0
\(373\) −0.0543397 0.0313730i −0.00281360 0.00162443i 0.498593 0.866836i \(-0.333850\pi\)
−0.501406 + 0.865212i \(0.667184\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.2288i 1.04183i
\(378\) 0 0
\(379\) 15.7085 0.806891 0.403446 0.915004i \(-0.367812\pi\)
0.403446 + 0.915004i \(0.367812\pi\)
\(380\) 0 0
\(381\) 1.03137 + 1.78639i 0.0528388 + 0.0915196i
\(382\) 0 0
\(383\) 23.6351 + 13.6458i 1.20770 + 0.697265i 0.962256 0.272146i \(-0.0877333\pi\)
0.245443 + 0.969411i \(0.421067\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.14181 2.96863i −0.261373 0.150904i
\(388\) 0 0
\(389\) −0.239870 0.415468i −0.0121619 0.0210651i 0.859880 0.510496i \(-0.170538\pi\)
−0.872042 + 0.489430i \(0.837205\pi\)
\(390\) 0 0
\(391\) −2.70850 −0.136975
\(392\) 0 0
\(393\) 21.2915i 1.07401i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −9.83307 5.67712i −0.493508 0.284927i 0.232521 0.972591i \(-0.425303\pi\)
−0.726028 + 0.687665i \(0.758636\pi\)
\(398\) 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
\(403\) 9.83307 5.67712i 0.489820 0.282798i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.06275i 0.0526784i
\(408\) 0 0
\(409\) 3.08301 + 5.33992i 0.152445 + 0.264042i 0.932126 0.362135i \(-0.117952\pi\)
−0.779681 + 0.626177i \(0.784619\pi\)
\(410\) 0 0
\(411\) 10.1144 17.5186i 0.498905 0.864130i
\(412\) 0 0
\(413\) 28.9373i 1.42391i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −4.33013 + 2.50000i −0.212047 + 0.122426i
\(418\) 0 0
\(419\) −21.2915 −1.04016 −0.520079 0.854118i \(-0.674097\pi\)
−0.520079 + 0.854118i \(0.674097\pi\)
\(420\) 0 0
\(421\) −19.5830 −0.954417 −0.477209 0.878790i \(-0.658351\pi\)
−0.477209 + 0.878790i \(0.658351\pi\)
\(422\) 0 0
\(423\) 5.19615 3.00000i 0.252646 0.145865i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −18.3303 + 10.5830i −0.887066 + 0.512148i
\(428\) 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
\(433\) 3.35425i 0.161195i 0.996747 + 0.0805975i \(0.0256828\pi\)
−0.996747 + 0.0805975i \(0.974317\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −11.8176 + 6.82288i −0.565311 + 0.326382i
\(438\) 0 0
\(439\) 2.64575 4.58258i 0.126275 0.218714i −0.795956 0.605355i \(-0.793031\pi\)
0.922231 + 0.386640i \(0.126365\pi\)
\(440\) 0 0
\(441\) 7.00000 0.333333
\(442\) 0 0
\(443\) −8.55157 4.93725i −0.406298 0.234576i 0.282900 0.959149i \(-0.408704\pi\)
−0.689198 + 0.724573i \(0.742037\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.58301i 0.311365i
\(448\) 0 0
\(449\) 12.5830 0.593829 0.296914 0.954904i \(-0.404042\pi\)
0.296914 + 0.954904i \(0.404042\pi\)
\(450\) 0 0
\(451\) 4.06275 + 7.03688i 0.191307 + 0.331354i
\(452\) 0 0
\(453\) 3.96900 + 2.29150i 0.186480 + 0.107664i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −12.6836 7.32288i −0.593313 0.342550i 0.173093 0.984905i \(-0.444624\pi\)
−0.766407 + 0.642356i \(0.777957\pi\)
\(458\) 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
\(463\) 10.5203i 0.488918i 0.969660 + 0.244459i \(0.0786103\pi\)
−0.969660 + 0.244459i \(0.921390\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −32.6916 18.8745i −1.51279 0.873408i −0.999888 0.0149591i \(-0.995238\pi\)
−0.512899 0.858449i \(-0.671428\pi\)
\(468\) 0 0
\(469\) −1.70850 −0.0788911
\(470\) 0 0
\(471\) 0.645751 1.11847i 0.0297546 0.0515366i
\(472\) 0 0
\(473\) 8.46215 4.88562i 0.389090 0.224641i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.29150i 0.150708i
\(478\) 0 0
\(479\) −3.29150 5.70105i −0.150393 0.260488i 0.780979 0.624557i \(-0.214720\pi\)
−0.931372 + 0.364069i \(0.881387\pi\)
\(480\) 0 0
\(481\) 0.854249 1.47960i 0.0389504 0.0674641i
\(482\) 0 0
\(483\) −3.77089 2.17712i −0.171581 0.0990626i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2.29129 1.32288i 0.103828 0.0599452i −0.447187 0.894441i \(-0.647574\pi\)
0.551015 + 0.834495i \(0.314241\pi\)
\(488\) 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
\(493\) −10.8972 + 6.29150i −0.490785 + 0.283355i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 31.2663 + 18.0516i 1.40249 + 0.809726i
\(498\) 0 0
\(499\) 13.7288 23.7789i 0.614584 1.06449i −0.375874 0.926671i \(-0.622657\pi\)
0.990457 0.137819i \(-0.0440093\pi\)
\(500\) 0 0
\(501\) −9.82288 17.0137i −0.438854 0.760117i
\(502\) 0 0
\(503\) 22.4575i 1.00133i 0.865641 + 0.500666i \(0.166911\pi\)
−0.865641 + 0.500666i \(0.833089\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 5.19615 3.00000i 0.230769 0.133235i
\(508\) 0 0
\(509\) −13.9373 + 24.1400i −0.617758 + 1.06999i 0.372136 + 0.928178i \(0.378626\pi\)
−0.989894 + 0.141810i \(0.954708\pi\)
\(510\) 0 0
\(511\) 35.0000 1.54831
\(512\) 0 0
\(513\) −7.18065 4.14575i −0.317034 0.183039i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 9.87451i 0.434280i
\(518\) 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
\(523\) −27.0449 15.6144i −1.18259 0.682769i −0.225978 0.974132i \(-0.572558\pi\)
−0.956612 + 0.291363i \(0.905891\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.11652 + 3.53137i 0.266440 + 0.153829i
\(528\) 0 0
\(529\) −10.1458 17.5730i −0.441120 0.764042i
\(530\) 0 0
\(531\) 10.9373 0.474636
\(532\) 0 0
\(533\) 13.0627i 0.565810i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.504897 + 0.291503i 0.0217879 + 0.0125793i
\(538\) 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
\(543\) −3.71655 + 2.14575i −0.159492 + 0.0920830i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 43.2915i 1.85101i 0.378734 + 0.925505i \(0.376359\pi\)
−0.378734 + 0.925505i \(0.623641\pi\)
\(548\) 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
\(553\) 5.25049 3.03137i 0.223274 0.128907i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.5885 9.00000i 0.660504 0.381342i −0.131965 0.991254i \(-0.542129\pi\)
0.792469 + 0.609912i \(0.208795\pi\)
\(558\) 0 0
\(559\) −15.7085 −0.664399
\(560\) 0 0
\(561\) 2.70850 0.114353
\(562\) 0 0
\(563\) −31.6818 + 18.2915i −1.33523 + 0.770895i −0.986096 0.166178i \(-0.946857\pi\)
−0.349133 + 0.937073i \(0.613524\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.64575i 0.111111i
\(568\) 0 0
\(569\) −1.88562 + 3.26599i −0.0790494 + 0.136918i −0.902840 0.429977i \(-0.858522\pi\)
0.823791 + 0.566894i \(0.191855\pi\)
\(570\) 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
\(573\) 19.1660i 0.800672i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 36.8236 21.2601i 1.53299 0.885071i 0.533766 0.845633i \(-0.320776\pi\)
0.999222 0.0394383i \(-0.0125568\pi\)
\(578\) 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
\(583\) 4.69126 + 2.70850i 0.194292 + 0.112174i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 16.9373i 0.699075i −0.936922 0.349538i \(-0.886339\pi\)
0.936922 0.349538i \(-0.113661\pi\)
\(588\) 0 0
\(589\) 35.5830 1.46617
\(590\) 0 0
\(591\) 2.46863 + 4.27579i 0.101546 + 0.175882i
\(592\) 0 0
\(593\) 21.7050 + 12.5314i 0.891316 + 0.514602i 0.874373 0.485255i \(-0.161273\pi\)
0.0169436 + 0.999856i \(0.494606\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −20.0624 11.5830i −0.821097 0.474061i
\(598\) 0 0
\(599\) 1.06275 + 1.84073i 0.0434226 + 0.0752102i 0.886920 0.461923i \(-0.152840\pi\)
−0.843497 + 0.537133i \(0.819507\pi\)
\(600\) 0 0
\(601\) −37.5830 −1.53304 −0.766521 0.642219i \(-0.778014\pi\)
−0.766521 + 0.642219i \(0.778014\pi\)
\(602\) 0 0
\(603\) 0.645751i 0.0262970i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −14.5243 8.38562i −0.589524 0.340362i 0.175385 0.984500i \(-0.443883\pi\)
−0.764909 + 0.644138i \(0.777216\pi\)
\(608\) 0 0
\(609\) −20.2288 −0.819711
\(610\) 0 0
\(611\) 7.93725 13.7477i 0.321107 0.556174i
\(612\) 0 0
\(613\) −35.6508 + 20.5830i −1.43992 + 0.831340i −0.997844 0.0656363i \(-0.979092\pi\)
−0.442079 + 0.896976i \(0.645759\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.29150i 0.374062i −0.982354 0.187031i \(-0.940114\pi\)
0.982354 0.187031i \(-0.0598865\pi\)
\(618\) 0 0
\(619\) −9.50000 16.4545i −0.381837 0.661361i 0.609488 0.792796i \(-0.291375\pi\)
−0.991325 + 0.131434i \(0.958042\pi\)
\(620\) 0 0
\(621\) −0.822876 + 1.42526i −0.0330209 + 0.0571938i
\(622\) 0 0
\(623\) −32.6022 + 18.8229i −1.30618 + 0.754123i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 11.8176 6.82288i 0.471948 0.272479i
\(628\) 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
\(633\) 7.43310 4.29150i 0.295439 0.170572i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 16.0390 9.26013i 0.635489 0.366900i
\(638\) 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
\(643\) 21.8118i 0.860172i −0.902788 0.430086i \(-0.858483\pi\)
0.902788 0.430086i \(-0.141517\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −19.8642 + 11.4686i −0.780944 + 0.450878i −0.836765 0.547563i \(-0.815556\pi\)
0.0558207 + 0.998441i \(0.482222\pi\)
\(648\) 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
\(653\) −24.5555 14.1771i −0.960931 0.554794i −0.0644715 0.997920i \(-0.520536\pi\)
−0.896459 + 0.443126i \(0.853869\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 13.2288i 0.516103i
\(658\) 0 0
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 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
\(663\) −3.77089 2.17712i −0.146449 0.0845525i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 10.8972 + 6.29150i 0.421941 + 0.243608i
\(668\) 0 0
\(669\) 8.93725 + 15.4798i 0.345534 + 0.598483i
\(670\) 0 0
\(671\) −13.1660 −0.508268
\(672\) 0 0
\(673\) 5.35425i 0.206391i −0.994661 0.103196i \(-0.967093\pi\)
0.994661 0.103196i \(-0.0329067\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −19.3593 11.1771i −0.744040 0.429572i 0.0794963 0.996835i \(-0.474669\pi\)
−0.823536 + 0.567263i \(0.808002\pi\)
\(678\) 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
\(683\) 4.27579 2.46863i 0.163608 0.0944594i −0.415960 0.909383i \(-0.636554\pi\)
0.579569 + 0.814923i \(0.303221\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7.00000i 0.267067i
\(688\) 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
\(693\) 3.77089 + 2.17712i 0.143244 + 0.0827021i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 7.03688 4.06275i 0.266541 0.153887i
\(698\) 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
\(703\) 4.63692 2.67712i 0.174885 0.100970i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.0627i 0.491275i
\(708\) 0 0
\(709\) −11.2915 + 19.5575i −0.424061 + 0.734496i −0.996332 0.0855689i \(-0.972729\pi\)
0.572271 + 0.820065i \(0.306063\pi\)
\(710\) 0 0
\(711\) −1.14575 1.98450i −0.0429690 0.0744245i
\(712\) 0 0
\(713\) 7.06275i 0.264502i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −5.19615 + 3.00000i −0.194054 + 0.112037i
\(718\) 0 0
\(719\) −7.35425 + 12.7379i −0.274267 + 0.475045i −0.969950 0.243304i \(-0.921769\pi\)
0.695683 + 0.718349i \(0.255102\pi\)
\(720\) 0 0
\(721\) −12.3745 21.4333i −0.460851 0.798217i
\(722\) 0 0
\(723\) 11.0059 + 6.35425i 0.409313 + 0.236317i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 33.8118i 1.25401i 0.779016 + 0.627004i \(0.215719\pi\)
−0.779016 + 0.627004i \(0.784281\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −4.88562 8.46215i −0.180701 0.312984i
\(732\) 0 0
\(733\) −6.26029 3.61438i −0.231229 0.133500i 0.379910 0.925023i \(-0.375955\pi\)
−0.611139 + 0.791523i \(0.709288\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.920365 0.531373i −0.0339021 0.0195734i
\(738\) 0 0
\(739\) 14.5000 + 25.1147i 0.533391 + 0.923861i 0.999239 + 0.0389959i \(0.0124159\pi\)
−0.465848 + 0.884865i \(0.654251\pi\)
\(740\) 0 0
\(741\) −21.9373 −0.805885
\(742\) 0 0
\(743\) 13.0627i 0.479226i −0.970869 0.239613i \(-0.922979\pi\)
0.970869 0.239613i \(-0.0770205\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −9.47194 5.46863i −0.346560 0.200087i
\(748\) 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
\(753\) −9.47194 + 5.46863i −0.345177 + 0.199288i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.83399i 0.321077i 0.987030 + 0.160538i \(0.0513230\pi\)
−0.987030 + 0.160538i \(0.948677\pi\)
\(758\) 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
\(763\) 34.3948i 1.24517i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 25.0604 14.4686i 0.904878 0.522432i
\(768\) 0 0
\(769\) 3.70850 0.133732 0.0668659 0.997762i \(-0.478700\pi\)
0.0668659 + 0.997762i \(0.478700\pi\)
\(770\) 0 0
\(771\) 5.52026 0.198807
\(772\) 0 0
\(773\) 23.7246 13.6974i 0.853313 0.492661i −0.00845413 0.999964i \(-0.502691\pi\)
0.861767 + 0.507304i \(0.169358\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.47960 + 0.854249i 0.0530804 + 0.0306460i
\(778\) 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
\(783\) 7.64575i 0.273237i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −5.08747 + 2.93725i −0.181349 + 0.104702i −0.587926 0.808915i \(-0.700055\pi\)
0.406577 + 0.913616i \(0.366722\pi\)
\(788\) 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
\(793\) 18.3303 + 10.5830i 0.650928 + 0.375814i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4.93725i 0.174887i −0.996170 0.0874433i \(-0.972130\pi\)
0.996170 0.0874433i \(-0.0278696\pi\)
\(798\) 0 0
\(799\) 9.87451 0.349335
\(800\) 0 0
\(801\) 7.11438 + 12.3225i 0.251374 + 0.435393i
\(802\) 0 0
\(803\) 18.8544 + 10.8856i 0.665359 + 0.384145i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 24.1400 + 13.9373i 0.849769 + 0.490615i
\(808\) 0 0
\(809\) 9.58301 + 16.5983i 0.336921 + 0.583563i 0.983852 0.178985i \(-0.0572813\pi\)
−0.646931 + 0.762548i \(0.723948\pi\)
\(810\) 0 0
\(811\) 17.2915 0.607187 0.303593 0.952802i \(-0.401814\pi\)
0.303593 + 0.952802i \(0.401814\pi\)
\(812\) 0 0
\(813\) 7.41699i 0.260125i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −42.6334 24.6144i −1.49155 0.861148i
\(818\) 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
\(823\) 15.9847 9.22876i 0.557191 0.321694i −0.194826 0.980838i \(-0.562414\pi\)
0.752017 + 0.659144i \(0.229081\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 22.4575i 0.780924i −0.920619 0.390462i \(-0.872315\pi\)
0.920619 0.390462i \(-0.127685\pi\)
\(828\) 0 0
\(829\) 9.85425 + 17.0681i 0.342252 + 0.592798i 0.984851 0.173405i \(-0.0554769\pi\)
−0.642598 + 0.766203i \(0.722144\pi\)
\(830\) 0 0
\(831\) 10.3229 17.8797i 0.358097 0.620241i
\(832\) 0 0
\(833\) 9.97684 + 5.76013i 0.345677 + 0.199577i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.71655 2.14575i 0.128463 0.0741680i
\(838\) 0 0
\(839\) −16.3542 −0.564611 −0.282306 0.959325i \(-0.591099\pi\)
−0.282306 + 0.959325i \(0.591099\pi\)
\(840\) 0 0
\(841\) 29.4575 1.01578
\(842\) 0 0
\(843\) 20.7846 12.0000i 0.715860 0.413302i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 18.9982 10.9686i 0.652787 0.376886i
\(848\) 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
\(853\) 43.2288i 1.48012i 0.672538 + 0.740062i \(0.265204\pi\)
−0.672538 + 0.740062i \(0.734796\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.8176 6.82288i 0.403680 0.233065i −0.284390 0.958709i \(-0.591791\pi\)
0.688071 + 0.725644i \(0.258458\pi\)
\(858\) 0 0
\(859\) −13.2288 + 22.9129i −0.451359 + 0.781777i −0.998471 0.0552825i \(-0.982394\pi\)
0.547111 + 0.837060i \(0.315727\pi\)
\(860\) 0 0
\(861\) 13.0627 0.445177
\(862\) 0 0
\(863\) 29.8411 + 17.2288i 1.01580 + 0.586474i 0.912885 0.408216i \(-0.133849\pi\)
0.102917 + 0.994690i \(0.467182\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 14.2915i 0.485365i
\(868\) 0 0
\(869\) 3.77124 0.127931
\(870\) 0 0
\(871\) 0.854249 + 1.47960i 0.0289451 + 0.0501344i
\(872\) 0 0
\(873\) −6.92820 4.00000i −0.234484 0.135379i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 12.1244 + 7.00000i 0.409410 + 0.236373i 0.690536 0.723298i \(-0.257375\pi\)
−0.281126 + 0.959671i \(0.590708\pi\)
\(878\) 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
\(883\) 21.8118i 0.734024i −0.930216 0.367012i \(-0.880381\pi\)
0.930216 0.367012i \(-0.119619\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −17.0137 9.82288i −0.571265 0.329820i 0.186390 0.982476i \(-0.440321\pi\)
−0.757654 + 0.652656i \(0.773655\pi\)
\(888\) 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
\(893\) 43.0839 24.8745i 1.44175 0.832394i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.35425i 0.145384i
\(898\) 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
\(903\) 15.7085i 0.522746i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 17.8797 10.3229i 0.593687 0.342765i −0.172867 0.984945i \(-0.555303\pi\)
0.766554 + 0.642180i \(0.221970\pi\)
\(908\) 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
\(913\) 15.5885 9.00000i 0.515903 0.297857i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −48.7850 + 28.1660i −1.61102 + 0.930124i
\(918\) 0 0
\(919\) 5.20850 9.02138i 0.171812 0.297588i −0.767241 0.641359i \(-0.778371\pi\)
0.939054 + 0.343771i \(0.111704\pi\)
\(920\) 0 0
\(921\) −13.9686 24.1944i −0.460282 0.797232i
\(922\) 0 0
\(923\) 36.1033i 1.18835i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −8.10102 + 4.67712i −0.266072 + 0.153617i
\(928\) 0 0
\(929\) 17.7601 30.7614i 0.582691 1.00925i −0.412468 0.910972i \(-0.635333\pi\)
0.995159 0.0982783i \(-0.0313335\pi\)
\(930\) 0 0
\(931\) 58.0405 1.90220
\(932\) 0 0
\(933\) −11.8176 6.82288i −0.386890 0.223371i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 38.9778i 1.27335i −0.771133 0.636674i \(-0.780310\pi\)
0.771133 0.636674i \(-0.219690\pi\)
\(938\) 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
\(943\) −7.03688 4.06275i −0.229152 0.132301i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −34.9478 20.1771i −1.13565 0.655668i −0.190301 0.981726i \(-0.560946\pi\)
−0.945350 + 0.326057i \(0.894280\pi\)
\(948\) 0 0
\(949\) −17.5000 30.3109i −0.568074 0.983933i
\(950\) 0 0
\(951\) 23.5203 0.762697
\(952\) 0 0
\(953\) 52.4575i 1.69927i −0.527375 0.849633i \(-0.676824\pi\)
0.527375 0.849633i \(-0.323176\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −10.8972 6.29150i −0.352257 0.203375i
\(958\) 0 0
\(959\) 53.5203 1.72826
\(960\) 0 0
\(961\) 6.29150 10.8972i 0.202952 0.351523i
\(962\) 0 0
\(963\) −14.6681 + 8.46863i −0.472673 + 0.272898i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 28.3948i 0.913114i 0.889694 + 0.456557i \(0.150918\pi\)
−0.889694 + 0.456557i \(0.849082\pi\)
\(968\) 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
\(973\) −11.4564 6.61438i −0.367277 0.212047i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 21.7050 12.5314i 0.694404 0.400914i −0.110856 0.993836i \(-0.535359\pi\)
0.805260 + 0.592922i \(0.202026\pi\)
\(978\) 0 0
\(979\) −23.4170 −0.748410
\(980\) 0 0
\(981\) −13.0000 −0.415058
\(982\) 0 0
\(983\) 21.3789 12.3431i 0.681882 0.393685i −0.118682 0.992932i \(-0.537867\pi\)
0.800564 + 0.599247i \(0.204533\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 13.7477 + 7.93725i 0.437595 + 0.252646i
\(988\) 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
\(993\) 20.1660i 0.639949i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −37.4372 + 21.6144i −1.18565 + 0.684534i −0.957314 0.289049i \(-0.906661\pi\)
−0.228334 + 0.973583i \(0.573328\pi\)
\(998\) 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.bc.e.1549.4 8
5.2 odd 4 420.2.q.c.121.1 4
5.3 odd 4 2100.2.q.h.1801.2 4
5.4 even 2 inner 2100.2.bc.e.1549.1 8
7.4 even 3 inner 2100.2.bc.e.949.1 8
15.2 even 4 1260.2.s.f.541.1 4
20.7 even 4 1680.2.bg.q.961.2 4
35.2 odd 12 2940.2.a.s.1.1 2
35.4 even 6 inner 2100.2.bc.e.949.4 8
35.12 even 12 2940.2.a.m.1.1 2
35.17 even 12 2940.2.q.t.361.2 4
35.18 odd 12 2100.2.q.h.1201.2 4
35.27 even 4 2940.2.q.t.961.2 4
35.32 odd 12 420.2.q.c.361.1 yes 4
105.2 even 12 8820.2.a.be.1.2 2
105.32 even 12 1260.2.s.f.361.1 4
105.47 odd 12 8820.2.a.bj.1.2 2
140.67 even 12 1680.2.bg.q.1201.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.q.c.121.1 4 5.2 odd 4
420.2.q.c.361.1 yes 4 35.32 odd 12
1260.2.s.f.361.1 4 105.32 even 12
1260.2.s.f.541.1 4 15.2 even 4
1680.2.bg.q.961.2 4 20.7 even 4
1680.2.bg.q.1201.2 4 140.67 even 12
2100.2.q.h.1201.2 4 35.18 odd 12
2100.2.q.h.1801.2 4 5.3 odd 4
2100.2.bc.e.949.1 8 7.4 even 3 inner
2100.2.bc.e.949.4 8 35.4 even 6 inner
2100.2.bc.e.1549.1 8 5.4 even 2 inner
2100.2.bc.e.1549.4 8 1.1 even 1 trivial
2940.2.a.m.1.1 2 35.12 even 12
2940.2.a.s.1.1 2 35.2 odd 12
2940.2.q.t.361.2 4 35.17 even 12
2940.2.q.t.961.2 4 35.27 even 4
8820.2.a.be.1.2 2 105.2 even 12
8820.2.a.bj.1.2 2 105.47 odd 12