Properties

Label 3150.2.b.d.251.6
Level $3150$
Weight $2$
Character 3150.251
Analytic conductor $25.153$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3150,2,Mod(251,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3150.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3150.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.6
Root \(-1.14412 + 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 3150.251
Dual form 3150.2.b.d.251.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +(-1.41421 + 2.23607i) q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +(-1.41421 + 2.23607i) q^{7} -1.00000i q^{8} -1.41421i q^{11} +0.926210i q^{13} +(-2.23607 - 1.41421i) q^{14} +1.00000 q^{16} -2.23607 q^{17} -7.63441i q^{19} +1.41421 q^{22} -1.00000i q^{23} -0.926210 q^{26} +(1.41421 - 2.23607i) q^{28} -0.757359i q^{29} +4.08849i q^{31} +1.00000i q^{32} -2.23607i q^{34} +2.82843 q^{37} +7.63441 q^{38} +8.56062 q^{41} +3.58579 q^{43} +1.41421i q^{44} +1.00000 q^{46} +1.30986 q^{47} +(-3.00000 - 6.32456i) q^{49} -0.926210i q^{52} -8.07107i q^{53} +(2.23607 + 1.41421i) q^{56} +0.757359 q^{58} +7.25077 q^{59} +0.926210i q^{61} -4.08849 q^{62} -1.00000 q^{64} +2.23607 q^{68} -15.6569i q^{71} +13.9590i q^{73} +2.82843i q^{74} +7.63441i q^{76} +(3.16228 + 2.00000i) q^{77} -13.0711 q^{79} +8.56062i q^{82} +14.3426 q^{83} +3.58579i q^{86} -1.41421 q^{88} -2.61972 q^{89} +(-2.07107 - 1.30986i) q^{91} +1.00000i q^{92} +1.30986i q^{94} +0.542561i q^{97} +(6.32456 - 3.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 8 q^{16} + 40 q^{43} + 8 q^{46} - 24 q^{49} + 40 q^{58} - 8 q^{64} - 48 q^{79} + 40 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −1.41421 + 2.23607i −0.534522 + 0.845154i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421i 0.426401i −0.977008 0.213201i \(-0.931611\pi\)
0.977008 0.213201i \(-0.0683888\pi\)
\(12\) 0 0
\(13\) 0.926210i 0.256884i 0.991717 + 0.128442i \(0.0409977\pi\)
−0.991717 + 0.128442i \(0.959002\pi\)
\(14\) −2.23607 1.41421i −0.597614 0.377964i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.23607 −0.542326 −0.271163 0.962533i \(-0.587408\pi\)
−0.271163 + 0.962533i \(0.587408\pi\)
\(18\) 0 0
\(19\) 7.63441i 1.75145i −0.482806 0.875727i \(-0.660382\pi\)
0.482806 0.875727i \(-0.339618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.41421 0.301511
\(23\) 1.00000i 0.208514i −0.994550 0.104257i \(-0.966753\pi\)
0.994550 0.104257i \(-0.0332465\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −0.926210 −0.181645
\(27\) 0 0
\(28\) 1.41421 2.23607i 0.267261 0.422577i
\(29\) 0.757359i 0.140638i −0.997525 0.0703190i \(-0.977598\pi\)
0.997525 0.0703190i \(-0.0224017\pi\)
\(30\) 0 0
\(31\) 4.08849i 0.734314i 0.930159 + 0.367157i \(0.119669\pi\)
−0.930159 + 0.367157i \(0.880331\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 2.23607i 0.383482i
\(35\) 0 0
\(36\) 0 0
\(37\) 2.82843 0.464991 0.232495 0.972598i \(-0.425311\pi\)
0.232495 + 0.972598i \(0.425311\pi\)
\(38\) 7.63441 1.23847
\(39\) 0 0
\(40\) 0 0
\(41\) 8.56062 1.33694 0.668472 0.743737i \(-0.266948\pi\)
0.668472 + 0.743737i \(0.266948\pi\)
\(42\) 0 0
\(43\) 3.58579 0.546827 0.273414 0.961897i \(-0.411847\pi\)
0.273414 + 0.961897i \(0.411847\pi\)
\(44\) 1.41421i 0.213201i
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 1.30986 0.191062 0.0955312 0.995426i \(-0.469545\pi\)
0.0955312 + 0.995426i \(0.469545\pi\)
\(48\) 0 0
\(49\) −3.00000 6.32456i −0.428571 0.903508i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.926210i 0.128442i
\(53\) 8.07107i 1.10865i −0.832301 0.554323i \(-0.812977\pi\)
0.832301 0.554323i \(-0.187023\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.23607 + 1.41421i 0.298807 + 0.188982i
\(57\) 0 0
\(58\) 0.757359 0.0994461
\(59\) 7.25077 0.943969 0.471985 0.881607i \(-0.343538\pi\)
0.471985 + 0.881607i \(0.343538\pi\)
\(60\) 0 0
\(61\) 0.926210i 0.118589i 0.998241 + 0.0592945i \(0.0188851\pi\)
−0.998241 + 0.0592945i \(0.981115\pi\)
\(62\) −4.08849 −0.519238
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 2.23607 0.271163
\(69\) 0 0
\(70\) 0 0
\(71\) 15.6569i 1.85813i −0.369921 0.929063i \(-0.620615\pi\)
0.369921 0.929063i \(-0.379385\pi\)
\(72\) 0 0
\(73\) 13.9590i 1.63377i 0.576798 + 0.816887i \(0.304302\pi\)
−0.576798 + 0.816887i \(0.695698\pi\)
\(74\) 2.82843i 0.328798i
\(75\) 0 0
\(76\) 7.63441i 0.875727i
\(77\) 3.16228 + 2.00000i 0.360375 + 0.227921i
\(78\) 0 0
\(79\) −13.0711 −1.47061 −0.735305 0.677736i \(-0.762961\pi\)
−0.735305 + 0.677736i \(0.762961\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 8.56062i 0.945363i
\(83\) 14.3426 1.57431 0.787153 0.616757i \(-0.211554\pi\)
0.787153 + 0.616757i \(0.211554\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 3.58579i 0.386665i
\(87\) 0 0
\(88\) −1.41421 −0.150756
\(89\) −2.61972 −0.277689 −0.138845 0.990314i \(-0.544339\pi\)
−0.138845 + 0.990314i \(0.544339\pi\)
\(90\) 0 0
\(91\) −2.07107 1.30986i −0.217107 0.137310i
\(92\) 1.00000i 0.104257i
\(93\) 0 0
\(94\) 1.30986i 0.135102i
\(95\) 0 0
\(96\) 0 0
\(97\) 0.542561i 0.0550887i 0.999621 + 0.0275444i \(0.00876875\pi\)
−0.999621 + 0.0275444i \(0.991231\pi\)
\(98\) 6.32456 3.00000i 0.638877 0.303046i
\(99\) 0 0
\(100\) 0 0
\(101\) 12.1065 1.20465 0.602323 0.798252i \(-0.294242\pi\)
0.602323 + 0.798252i \(0.294242\pi\)
\(102\) 0 0
\(103\) 10.4130i 1.02603i 0.858380 + 0.513014i \(0.171471\pi\)
−0.858380 + 0.513014i \(0.828529\pi\)
\(104\) 0.926210 0.0908223
\(105\) 0 0
\(106\) 8.07107 0.783931
\(107\) 2.00000i 0.193347i −0.995316 0.0966736i \(-0.969180\pi\)
0.995316 0.0966736i \(-0.0308203\pi\)
\(108\) 0 0
\(109\) −7.07107 −0.677285 −0.338643 0.940915i \(-0.609968\pi\)
−0.338643 + 0.940915i \(0.609968\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.41421 + 2.23607i −0.133631 + 0.211289i
\(113\) 1.07107i 0.100758i −0.998730 0.0503788i \(-0.983957\pi\)
0.998730 0.0503788i \(-0.0160429\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.757359i 0.0703190i
\(117\) 0 0
\(118\) 7.25077i 0.667487i
\(119\) 3.16228 5.00000i 0.289886 0.458349i
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) −0.926210 −0.0838551
\(123\) 0 0
\(124\) 4.08849i 0.367157i
\(125\) 0 0
\(126\) 0 0
\(127\) 10.0000 0.887357 0.443678 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 12.6491 1.10516 0.552579 0.833461i \(-0.313644\pi\)
0.552579 + 0.833461i \(0.313644\pi\)
\(132\) 0 0
\(133\) 17.0711 + 10.7967i 1.48025 + 0.936192i
\(134\) 0 0
\(135\) 0 0
\(136\) 2.23607i 0.191741i
\(137\) 12.1421i 1.03737i −0.854965 0.518686i \(-0.826421\pi\)
0.854965 0.518686i \(-0.173579\pi\)
\(138\) 0 0
\(139\) 10.7967i 0.915763i −0.889013 0.457882i \(-0.848608\pi\)
0.889013 0.457882i \(-0.151392\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 15.6569 1.31389
\(143\) 1.30986 0.109536
\(144\) 0 0
\(145\) 0 0
\(146\) −13.9590 −1.15525
\(147\) 0 0
\(148\) −2.82843 −0.232495
\(149\) 0.757359i 0.0620453i 0.999519 + 0.0310226i \(0.00987640\pi\)
−0.999519 + 0.0310226i \(0.990124\pi\)
\(150\) 0 0
\(151\) −14.1421 −1.15087 −0.575435 0.817847i \(-0.695167\pi\)
−0.575435 + 0.817847i \(0.695167\pi\)
\(152\) −7.63441 −0.619233
\(153\) 0 0
\(154\) −2.00000 + 3.16228i −0.161165 + 0.254824i
\(155\) 0 0
\(156\) 0 0
\(157\) 10.7967i 0.861670i 0.902431 + 0.430835i \(0.141781\pi\)
−0.902431 + 0.430835i \(0.858219\pi\)
\(158\) 13.0711i 1.03988i
\(159\) 0 0
\(160\) 0 0
\(161\) 2.23607 + 1.41421i 0.176227 + 0.111456i
\(162\) 0 0
\(163\) 7.92893 0.621042 0.310521 0.950567i \(-0.399497\pi\)
0.310521 + 0.950567i \(0.399497\pi\)
\(164\) −8.56062 −0.668472
\(165\) 0 0
\(166\) 14.3426i 1.11320i
\(167\) 12.1065 0.936833 0.468416 0.883508i \(-0.344825\pi\)
0.468416 + 0.883508i \(0.344825\pi\)
\(168\) 0 0
\(169\) 12.1421 0.934010
\(170\) 0 0
\(171\) 0 0
\(172\) −3.58579 −0.273414
\(173\) 10.2541 0.779607 0.389804 0.920898i \(-0.372543\pi\)
0.389804 + 0.920898i \(0.372543\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.41421i 0.106600i
\(177\) 0 0
\(178\) 2.61972i 0.196356i
\(179\) 11.3137i 0.845626i 0.906217 + 0.422813i \(0.138957\pi\)
−0.906217 + 0.422813i \(0.861043\pi\)
\(180\) 0 0
\(181\) 2.61972i 0.194722i −0.995249 0.0973610i \(-0.968960\pi\)
0.995249 0.0973610i \(-0.0310401\pi\)
\(182\) 1.30986 2.07107i 0.0970932 0.153518i
\(183\) 0 0
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 0 0
\(187\) 3.16228i 0.231249i
\(188\) −1.30986 −0.0955312
\(189\) 0 0
\(190\) 0 0
\(191\) 0.656854i 0.0475283i 0.999718 + 0.0237642i \(0.00756508\pi\)
−0.999718 + 0.0237642i \(0.992435\pi\)
\(192\) 0 0
\(193\) 8.48528 0.610784 0.305392 0.952227i \(-0.401213\pi\)
0.305392 + 0.952227i \(0.401213\pi\)
\(194\) −0.542561 −0.0389536
\(195\) 0 0
\(196\) 3.00000 + 6.32456i 0.214286 + 0.451754i
\(197\) 5.92893i 0.422419i −0.977441 0.211209i \(-0.932260\pi\)
0.977441 0.211209i \(-0.0677402\pi\)
\(198\) 0 0
\(199\) 21.5934i 1.53071i −0.643606 0.765357i \(-0.722562\pi\)
0.643606 0.765357i \(-0.277438\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 12.1065i 0.851814i
\(203\) 1.69351 + 1.07107i 0.118861 + 0.0751742i
\(204\) 0 0
\(205\) 0 0
\(206\) −10.4130 −0.725511
\(207\) 0 0
\(208\) 0.926210i 0.0642211i
\(209\) −10.7967 −0.746823
\(210\) 0 0
\(211\) 16.2132 1.11616 0.558081 0.829786i \(-0.311538\pi\)
0.558081 + 0.829786i \(0.311538\pi\)
\(212\) 8.07107i 0.554323i
\(213\) 0 0
\(214\) 2.00000 0.136717
\(215\) 0 0
\(216\) 0 0
\(217\) −9.14214 5.78199i −0.620609 0.392507i
\(218\) 7.07107i 0.478913i
\(219\) 0 0
\(220\) 0 0
\(221\) 2.07107i 0.139315i
\(222\) 0 0
\(223\) 13.0328i 0.872738i 0.899768 + 0.436369i \(0.143736\pi\)
−0.899768 + 0.436369i \(0.856264\pi\)
\(224\) −2.23607 1.41421i −0.149404 0.0944911i
\(225\) 0 0
\(226\) 1.07107 0.0712464
\(227\) 25.1393 1.66855 0.834277 0.551345i \(-0.185885\pi\)
0.834277 + 0.551345i \(0.185885\pi\)
\(228\) 0 0
\(229\) 10.7967i 0.713465i 0.934206 + 0.356733i \(0.116109\pi\)
−0.934206 + 0.356733i \(0.883891\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.757359 −0.0497231
\(233\) 23.0711i 1.51144i 0.654897 + 0.755718i \(0.272712\pi\)
−0.654897 + 0.755718i \(0.727288\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.25077 −0.471985
\(237\) 0 0
\(238\) 5.00000 + 3.16228i 0.324102 + 0.204980i
\(239\) 7.17157i 0.463890i −0.972729 0.231945i \(-0.925491\pi\)
0.972729 0.231945i \(-0.0745090\pi\)
\(240\) 0 0
\(241\) 4.47214i 0.288076i −0.989572 0.144038i \(-0.953991\pi\)
0.989572 0.144038i \(-0.0460087\pi\)
\(242\) 9.00000i 0.578542i
\(243\) 0 0
\(244\) 0.926210i 0.0592945i
\(245\) 0 0
\(246\) 0 0
\(247\) 7.07107 0.449921
\(248\) 4.08849 0.259619
\(249\) 0 0
\(250\) 0 0
\(251\) 18.8148 1.18758 0.593788 0.804621i \(-0.297632\pi\)
0.593788 + 0.804621i \(0.297632\pi\)
\(252\) 0 0
\(253\) −1.41421 −0.0889108
\(254\) 10.0000i 0.627456i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 21.2097 1.32303 0.661513 0.749933i \(-0.269914\pi\)
0.661513 + 0.749933i \(0.269914\pi\)
\(258\) 0 0
\(259\) −4.00000 + 6.32456i −0.248548 + 0.392989i
\(260\) 0 0
\(261\) 0 0
\(262\) 12.6491i 0.781465i
\(263\) 29.2843i 1.80575i −0.429908 0.902873i \(-0.641454\pi\)
0.429908 0.902873i \(-0.358546\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −10.7967 + 17.0711i −0.661988 + 1.04669i
\(267\) 0 0
\(268\) 0 0
\(269\) −27.9179 −1.70219 −0.851093 0.525014i \(-0.824060\pi\)
−0.851093 + 0.525014i \(0.824060\pi\)
\(270\) 0 0
\(271\) 7.09185i 0.430799i −0.976526 0.215400i \(-0.930895\pi\)
0.976526 0.215400i \(-0.0691054\pi\)
\(272\) −2.23607 −0.135582
\(273\) 0 0
\(274\) 12.1421 0.733533
\(275\) 0 0
\(276\) 0 0
\(277\) −21.2132 −1.27458 −0.637289 0.770625i \(-0.719944\pi\)
−0.637289 + 0.770625i \(0.719944\pi\)
\(278\) 10.7967 0.647543
\(279\) 0 0
\(280\) 0 0
\(281\) 28.4853i 1.69929i −0.527356 0.849645i \(-0.676816\pi\)
0.527356 0.849645i \(-0.323184\pi\)
\(282\) 0 0
\(283\) 7.63441i 0.453819i −0.973916 0.226909i \(-0.927138\pi\)
0.973916 0.226909i \(-0.0728621\pi\)
\(284\) 15.6569i 0.929063i
\(285\) 0 0
\(286\) 1.30986i 0.0774535i
\(287\) −12.1065 + 19.1421i −0.714627 + 1.12992i
\(288\) 0 0
\(289\) −12.0000 −0.705882
\(290\) 0 0
\(291\) 0 0
\(292\) 13.9590i 0.816887i
\(293\) −25.2982 −1.47794 −0.738969 0.673740i \(-0.764687\pi\)
−0.738969 + 0.673740i \(0.764687\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.82843i 0.164399i
\(297\) 0 0
\(298\) −0.757359 −0.0438726
\(299\) 0.926210 0.0535641
\(300\) 0 0
\(301\) −5.07107 + 8.01806i −0.292291 + 0.462153i
\(302\) 14.1421i 0.813788i
\(303\) 0 0
\(304\) 7.63441i 0.437864i
\(305\) 0 0
\(306\) 0 0
\(307\) 3.70484i 0.211446i −0.994396 0.105723i \(-0.966284\pi\)
0.994396 0.105723i \(-0.0337157\pi\)
\(308\) −3.16228 2.00000i −0.180187 0.113961i
\(309\) 0 0
\(310\) 0 0
\(311\) −5.23943 −0.297101 −0.148550 0.988905i \(-0.547461\pi\)
−0.148550 + 0.988905i \(0.547461\pi\)
\(312\) 0 0
\(313\) 16.5787i 0.937083i 0.883442 + 0.468541i \(0.155220\pi\)
−0.883442 + 0.468541i \(0.844780\pi\)
\(314\) −10.7967 −0.609293
\(315\) 0 0
\(316\) 13.0711 0.735305
\(317\) 30.0711i 1.68896i −0.535588 0.844480i \(-0.679910\pi\)
0.535588 0.844480i \(-0.320090\pi\)
\(318\) 0 0
\(319\) −1.07107 −0.0599683
\(320\) 0 0
\(321\) 0 0
\(322\) −1.41421 + 2.23607i −0.0788110 + 0.124611i
\(323\) 17.0711i 0.949860i
\(324\) 0 0
\(325\) 0 0
\(326\) 7.92893i 0.439143i
\(327\) 0 0
\(328\) 8.56062i 0.472681i
\(329\) −1.85242 + 2.92893i −0.102127 + 0.161477i
\(330\) 0 0
\(331\) 7.92893 0.435814 0.217907 0.975970i \(-0.430077\pi\)
0.217907 + 0.975970i \(0.430077\pi\)
\(332\) −14.3426 −0.787153
\(333\) 0 0
\(334\) 12.1065i 0.662441i
\(335\) 0 0
\(336\) 0 0
\(337\) 11.9706 0.652078 0.326039 0.945356i \(-0.394286\pi\)
0.326039 + 0.945356i \(0.394286\pi\)
\(338\) 12.1421i 0.660445i
\(339\) 0 0
\(340\) 0 0
\(341\) 5.78199 0.313113
\(342\) 0 0
\(343\) 18.3848 + 2.23607i 0.992685 + 0.120736i
\(344\) 3.58579i 0.193333i
\(345\) 0 0
\(346\) 10.2541i 0.551265i
\(347\) 9.07107i 0.486960i −0.969906 0.243480i \(-0.921711\pi\)
0.969906 0.243480i \(-0.0782891\pi\)
\(348\) 0 0
\(349\) 6.16564i 0.330039i 0.986290 + 0.165020i \(0.0527688\pi\)
−0.986290 + 0.165020i \(0.947231\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.41421 0.0753778
\(353\) −18.9737 −1.00987 −0.504933 0.863158i \(-0.668483\pi\)
−0.504933 + 0.863158i \(0.668483\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 2.61972 0.138845
\(357\) 0 0
\(358\) −11.3137 −0.597948
\(359\) 16.3137i 0.861005i −0.902589 0.430502i \(-0.858336\pi\)
0.902589 0.430502i \(-0.141664\pi\)
\(360\) 0 0
\(361\) −39.2843 −2.06759
\(362\) 2.61972 0.137689
\(363\) 0 0
\(364\) 2.07107 + 1.30986i 0.108553 + 0.0686552i
\(365\) 0 0
\(366\) 0 0
\(367\) 24.5967i 1.28394i −0.766730 0.641970i \(-0.778117\pi\)
0.766730 0.641970i \(-0.221883\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 0 0
\(370\) 0 0
\(371\) 18.0475 + 11.4142i 0.936977 + 0.592596i
\(372\) 0 0
\(373\) −7.07107 −0.366126 −0.183063 0.983101i \(-0.558601\pi\)
−0.183063 + 0.983101i \(0.558601\pi\)
\(374\) −3.16228 −0.163517
\(375\) 0 0
\(376\) 1.30986i 0.0675508i
\(377\) 0.701474 0.0361277
\(378\) 0 0
\(379\) 22.0711 1.13371 0.566857 0.823816i \(-0.308159\pi\)
0.566857 + 0.823816i \(0.308159\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −0.656854 −0.0336076
\(383\) −16.0361 −0.819408 −0.409704 0.912219i \(-0.634368\pi\)
−0.409704 + 0.912219i \(0.634368\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.48528i 0.431889i
\(387\) 0 0
\(388\) 0.542561i 0.0275444i
\(389\) 22.8284i 1.15745i 0.815524 + 0.578724i \(0.196449\pi\)
−0.815524 + 0.578724i \(0.803551\pi\)
\(390\) 0 0
\(391\) 2.23607i 0.113083i
\(392\) −6.32456 + 3.00000i −0.319438 + 0.151523i
\(393\) 0 0
\(394\) 5.92893 0.298695
\(395\) 0 0
\(396\) 0 0
\(397\) 29.6114i 1.48616i −0.669205 0.743078i \(-0.733365\pi\)
0.669205 0.743078i \(-0.266635\pi\)
\(398\) 21.5934 1.08238
\(399\) 0 0
\(400\) 0 0
\(401\) 9.79899i 0.489338i 0.969607 + 0.244669i \(0.0786793\pi\)
−0.969607 + 0.244669i \(0.921321\pi\)
\(402\) 0 0
\(403\) −3.78680 −0.188634
\(404\) −12.1065 −0.602323
\(405\) 0 0
\(406\) −1.07107 + 1.69351i −0.0531562 + 0.0840473i
\(407\) 4.00000i 0.198273i
\(408\) 0 0
\(409\) 22.9032i 1.13249i 0.824236 + 0.566246i \(0.191605\pi\)
−0.824236 + 0.566246i \(0.808395\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 10.4130i 0.513014i
\(413\) −10.2541 + 16.2132i −0.504573 + 0.797800i
\(414\) 0 0
\(415\) 0 0
\(416\) −0.926210 −0.0454112
\(417\) 0 0
\(418\) 10.7967i 0.528083i
\(419\) 28.8441 1.40913 0.704564 0.709640i \(-0.251142\pi\)
0.704564 + 0.709640i \(0.251142\pi\)
\(420\) 0 0
\(421\) 33.3553 1.62564 0.812820 0.582515i \(-0.197931\pi\)
0.812820 + 0.582515i \(0.197931\pi\)
\(422\) 16.2132i 0.789246i
\(423\) 0 0
\(424\) −8.07107 −0.391966
\(425\) 0 0
\(426\) 0 0
\(427\) −2.07107 1.30986i −0.100226 0.0633885i
\(428\) 2.00000i 0.0966736i
\(429\) 0 0
\(430\) 0 0
\(431\) 4.79899i 0.231159i −0.993298 0.115580i \(-0.963127\pi\)
0.993298 0.115580i \(-0.0368725\pi\)
\(432\) 0 0
\(433\) 9.71157i 0.466708i 0.972392 + 0.233354i \(0.0749701\pi\)
−0.972392 + 0.233354i \(0.925030\pi\)
\(434\) 5.78199 9.14214i 0.277545 0.438837i
\(435\) 0 0
\(436\) 7.07107 0.338643
\(437\) −7.63441 −0.365204
\(438\) 0 0
\(439\) 4.08849i 0.195133i −0.995229 0.0975664i \(-0.968894\pi\)
0.995229 0.0975664i \(-0.0311058\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.07107 0.0985106
\(443\) 23.0711i 1.09614i 0.836433 + 0.548070i \(0.184637\pi\)
−0.836433 + 0.548070i \(0.815363\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −13.0328 −0.617119
\(447\) 0 0
\(448\) 1.41421 2.23607i 0.0668153 0.105644i
\(449\) 18.3848i 0.867631i 0.901002 + 0.433816i \(0.142833\pi\)
−0.901002 + 0.433816i \(0.857167\pi\)
\(450\) 0 0
\(451\) 12.1065i 0.570075i
\(452\) 1.07107i 0.0503788i
\(453\) 0 0
\(454\) 25.1393i 1.17985i
\(455\) 0 0
\(456\) 0 0
\(457\) −17.8284 −0.833979 −0.416989 0.908911i \(-0.636915\pi\)
−0.416989 + 0.908911i \(0.636915\pi\)
\(458\) −10.7967 −0.504496
\(459\) 0 0
\(460\) 0 0
\(461\) −38.9394 −1.81359 −0.906794 0.421575i \(-0.861477\pi\)
−0.906794 + 0.421575i \(0.861477\pi\)
\(462\) 0 0
\(463\) −38.2843 −1.77922 −0.889610 0.456720i \(-0.849024\pi\)
−0.889610 + 0.456720i \(0.849024\pi\)
\(464\) 0.757359i 0.0351595i
\(465\) 0 0
\(466\) −23.0711 −1.06875
\(467\) −29.6114 −1.37025 −0.685127 0.728424i \(-0.740253\pi\)
−0.685127 + 0.728424i \(0.740253\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 7.25077i 0.333744i
\(473\) 5.07107i 0.233168i
\(474\) 0 0
\(475\) 0 0
\(476\) −3.16228 + 5.00000i −0.144943 + 0.229175i
\(477\) 0 0
\(478\) 7.17157 0.328020
\(479\) 33.6999 1.53979 0.769895 0.638171i \(-0.220309\pi\)
0.769895 + 0.638171i \(0.220309\pi\)
\(480\) 0 0
\(481\) 2.61972i 0.119449i
\(482\) 4.47214 0.203700
\(483\) 0 0
\(484\) −9.00000 −0.409091
\(485\) 0 0
\(486\) 0 0
\(487\) −29.8995 −1.35488 −0.677438 0.735580i \(-0.736910\pi\)
−0.677438 + 0.735580i \(0.736910\pi\)
\(488\) 0.926210 0.0419275
\(489\) 0 0
\(490\) 0 0
\(491\) 21.5147i 0.970946i −0.874252 0.485473i \(-0.838647\pi\)
0.874252 0.485473i \(-0.161353\pi\)
\(492\) 0 0
\(493\) 1.69351i 0.0762717i
\(494\) 7.07107i 0.318142i
\(495\) 0 0
\(496\) 4.08849i 0.183579i
\(497\) 35.0098 + 22.1421i 1.57040 + 0.993211i
\(498\) 0 0
\(499\) −22.0711 −0.988037 −0.494018 0.869451i \(-0.664472\pi\)
−0.494018 + 0.869451i \(0.664472\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18.8148i 0.839744i
\(503\) −21.8181 −0.972822 −0.486411 0.873730i \(-0.661694\pi\)
−0.486411 + 0.873730i \(0.661694\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.41421i 0.0628695i
\(507\) 0 0
\(508\) −10.0000 −0.443678
\(509\) 0.224736 0.00996125 0.00498063 0.999988i \(-0.498415\pi\)
0.00498063 + 0.999988i \(0.498415\pi\)
\(510\) 0 0
\(511\) −31.2132 19.7410i −1.38079 0.873289i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 21.2097i 0.935521i
\(515\) 0 0
\(516\) 0 0
\(517\) 1.85242i 0.0814693i
\(518\) −6.32456 4.00000i −0.277885 0.175750i
\(519\) 0 0
\(520\) 0 0
\(521\) 17.5049 0.766903 0.383452 0.923561i \(-0.374735\pi\)
0.383452 + 0.923561i \(0.374735\pi\)
\(522\) 0 0
\(523\) 28.4605i 1.24449i −0.782822 0.622245i \(-0.786221\pi\)
0.782822 0.622245i \(-0.213779\pi\)
\(524\) −12.6491 −0.552579
\(525\) 0 0
\(526\) 29.2843 1.27685
\(527\) 9.14214i 0.398238i
\(528\) 0 0
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) −17.0711 10.7967i −0.740125 0.468096i
\(533\) 7.92893i 0.343440i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 27.9179i 1.20363i
\(539\) −8.94427 + 4.24264i −0.385257 + 0.182743i
\(540\) 0 0
\(541\) 5.07107 0.218022 0.109011 0.994041i \(-0.465232\pi\)
0.109011 + 0.994041i \(0.465232\pi\)
\(542\) 7.09185 0.304621
\(543\) 0 0
\(544\) 2.23607i 0.0958706i
\(545\) 0 0
\(546\) 0 0
\(547\) 16.2132 0.693227 0.346613 0.938008i \(-0.387332\pi\)
0.346613 + 0.938008i \(0.387332\pi\)
\(548\) 12.1421i 0.518686i
\(549\) 0 0
\(550\) 0 0
\(551\) −5.78199 −0.246321
\(552\) 0 0
\(553\) 18.4853 29.2278i 0.786074 1.24289i
\(554\) 21.2132i 0.901263i
\(555\) 0 0
\(556\) 10.7967i 0.457882i
\(557\) 26.1421i 1.10768i 0.832624 + 0.553839i \(0.186838\pi\)
−0.832624 + 0.553839i \(0.813162\pi\)
\(558\) 0 0
\(559\) 3.32119i 0.140471i
\(560\) 0 0
\(561\) 0 0
\(562\) 28.4853 1.20158
\(563\) 24.3720 1.02716 0.513579 0.858042i \(-0.328319\pi\)
0.513579 + 0.858042i \(0.328319\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 7.63441 0.320898
\(567\) 0 0
\(568\) −15.6569 −0.656947
\(569\) 22.5269i 0.944377i −0.881498 0.472189i \(-0.843464\pi\)
0.881498 0.472189i \(-0.156536\pi\)
\(570\) 0 0
\(571\) −38.2132 −1.59917 −0.799586 0.600551i \(-0.794948\pi\)
−0.799586 + 0.600551i \(0.794948\pi\)
\(572\) −1.30986 −0.0547679
\(573\) 0 0
\(574\) −19.1421 12.1065i −0.798977 0.505318i
\(575\) 0 0
\(576\) 0 0
\(577\) 23.4458i 0.976062i −0.872826 0.488031i \(-0.837715\pi\)
0.872826 0.488031i \(-0.162285\pi\)
\(578\) 12.0000i 0.499134i
\(579\) 0 0
\(580\) 0 0
\(581\) −20.2835 + 32.0711i −0.841502 + 1.33053i
\(582\) 0 0
\(583\) −11.4142 −0.472728
\(584\) 13.9590 0.577626
\(585\) 0 0
\(586\) 25.2982i 1.04506i
\(587\) −15.4277 −0.636771 −0.318385 0.947961i \(-0.603141\pi\)
−0.318385 + 0.947961i \(0.603141\pi\)
\(588\) 0 0
\(589\) 31.2132 1.28612
\(590\) 0 0
\(591\) 0 0
\(592\) 2.82843 0.116248
\(593\) −46.8916 −1.92561 −0.962804 0.270202i \(-0.912910\pi\)
−0.962804 + 0.270202i \(0.912910\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.757359i 0.0310226i
\(597\) 0 0
\(598\) 0.926210i 0.0378755i
\(599\) 3.68629i 0.150618i −0.997160 0.0753089i \(-0.976006\pi\)
0.997160 0.0753089i \(-0.0239943\pi\)
\(600\) 0 0
\(601\) 12.1065i 0.493836i −0.969036 0.246918i \(-0.920582\pi\)
0.969036 0.246918i \(-0.0794179\pi\)
\(602\) −8.01806 5.07107i −0.326792 0.206681i
\(603\) 0 0
\(604\) 14.1421 0.575435
\(605\) 0 0
\(606\) 0 0
\(607\) 43.9541i 1.78404i 0.451996 + 0.892020i \(0.350712\pi\)
−0.451996 + 0.892020i \(0.649288\pi\)
\(608\) 7.63441 0.309616
\(609\) 0 0
\(610\) 0 0
\(611\) 1.21320i 0.0490810i
\(612\) 0 0
\(613\) −15.8579 −0.640493 −0.320247 0.947334i \(-0.603766\pi\)
−0.320247 + 0.947334i \(0.603766\pi\)
\(614\) 3.70484 0.149515
\(615\) 0 0
\(616\) 2.00000 3.16228i 0.0805823 0.127412i
\(617\) 32.0000i 1.28827i −0.764911 0.644136i \(-0.777217\pi\)
0.764911 0.644136i \(-0.222783\pi\)
\(618\) 0 0
\(619\) 18.2064i 0.731776i 0.930659 + 0.365888i \(0.119235\pi\)
−0.930659 + 0.365888i \(0.880765\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 5.23943i 0.210082i
\(623\) 3.70484 5.85786i 0.148431 0.234690i
\(624\) 0 0
\(625\) 0 0
\(626\) −16.5787 −0.662618
\(627\) 0 0
\(628\) 10.7967i 0.430835i
\(629\) −6.32456 −0.252177
\(630\) 0 0
\(631\) 10.2843 0.409410 0.204705 0.978824i \(-0.434376\pi\)
0.204705 + 0.978824i \(0.434376\pi\)
\(632\) 13.0711i 0.519939i
\(633\) 0 0
\(634\) 30.0711 1.19427
\(635\) 0 0
\(636\) 0 0
\(637\) 5.85786 2.77863i 0.232097 0.110093i
\(638\) 1.07107i 0.0424040i
\(639\) 0 0
\(640\) 0 0
\(641\) 28.4853i 1.12510i 0.826763 + 0.562550i \(0.190180\pi\)
−0.826763 + 0.562550i \(0.809820\pi\)
\(642\) 0 0
\(643\) 29.9951i 1.18289i −0.806345 0.591446i \(-0.798557\pi\)
0.806345 0.591446i \(-0.201443\pi\)
\(644\) −2.23607 1.41421i −0.0881134 0.0557278i
\(645\) 0 0
\(646\) −17.0711 −0.671652
\(647\) −37.0869 −1.45804 −0.729019 0.684493i \(-0.760023\pi\)
−0.729019 + 0.684493i \(0.760023\pi\)
\(648\) 0 0
\(649\) 10.2541i 0.402510i
\(650\) 0 0
\(651\) 0 0
\(652\) −7.92893 −0.310521
\(653\) 8.14214i 0.318626i 0.987228 + 0.159313i \(0.0509280\pi\)
−0.987228 + 0.159313i \(0.949072\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 8.56062 0.334236
\(657\) 0 0
\(658\) −2.92893 1.85242i −0.114182 0.0722148i
\(659\) 48.3848i 1.88480i −0.334483 0.942402i \(-0.608562\pi\)
0.334483 0.942402i \(-0.391438\pi\)
\(660\) 0 0
\(661\) 18.2064i 0.708146i −0.935218 0.354073i \(-0.884796\pi\)
0.935218 0.354073i \(-0.115204\pi\)
\(662\) 7.92893i 0.308167i
\(663\) 0 0
\(664\) 14.3426i 0.556602i
\(665\) 0 0
\(666\) 0 0
\(667\) −0.757359 −0.0293251
\(668\) −12.1065 −0.468416
\(669\) 0 0
\(670\) 0 0
\(671\) 1.30986 0.0505665
\(672\) 0 0
\(673\) −34.7990 −1.34140 −0.670701 0.741728i \(-0.734007\pi\)
−0.670701 + 0.741728i \(0.734007\pi\)
\(674\) 11.9706i 0.461089i
\(675\) 0 0
\(676\) −12.1421 −0.467005
\(677\) 22.9032 0.880243 0.440122 0.897938i \(-0.354935\pi\)
0.440122 + 0.897938i \(0.354935\pi\)
\(678\) 0 0
\(679\) −1.21320 0.767297i −0.0465585 0.0294462i
\(680\) 0 0
\(681\) 0 0
\(682\) 5.78199i 0.221404i
\(683\) 36.0000i 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −2.23607 + 18.3848i −0.0853735 + 0.701934i
\(687\) 0 0
\(688\) 3.58579 0.136707
\(689\) 7.47550 0.284794
\(690\) 0 0
\(691\) 23.9884i 0.912560i −0.889836 0.456280i \(-0.849181\pi\)
0.889836 0.456280i \(-0.150819\pi\)
\(692\) −10.2541 −0.389804
\(693\) 0 0
\(694\) 9.07107 0.344333
\(695\) 0 0
\(696\) 0 0
\(697\) −19.1421 −0.725060
\(698\) −6.16564 −0.233373
\(699\) 0 0
\(700\) 0 0
\(701\) 41.8701i 1.58141i −0.612197 0.790705i \(-0.709714\pi\)
0.612197 0.790705i \(-0.290286\pi\)
\(702\) 0 0
\(703\) 21.5934i 0.814410i
\(704\) 1.41421i 0.0533002i
\(705\) 0 0
\(706\) 18.9737i 0.714083i
\(707\) −17.1212 + 27.0711i −0.643911 + 1.01811i
\(708\) 0 0
\(709\) 24.1421 0.906677 0.453338 0.891338i \(-0.350233\pi\)
0.453338 + 0.891338i \(0.350233\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 2.61972i 0.0981780i
\(713\) 4.08849 0.153115
\(714\) 0 0
\(715\) 0 0
\(716\) 11.3137i 0.422813i
\(717\) 0 0
\(718\) 16.3137 0.608822
\(719\) 23.9884 0.894615 0.447307 0.894380i \(-0.352383\pi\)
0.447307 + 0.894380i \(0.352383\pi\)
\(720\) 0 0
\(721\) −23.2843 14.7263i −0.867152 0.548435i
\(722\) 39.2843i 1.46201i
\(723\) 0 0
\(724\) 2.61972i 0.0973610i
\(725\) 0 0
\(726\) 0 0
\(727\) 6.70820i 0.248794i −0.992233 0.124397i \(-0.960300\pi\)
0.992233 0.124397i \(-0.0396996\pi\)
\(728\) −1.30986 + 2.07107i −0.0485466 + 0.0767589i
\(729\) 0 0
\(730\) 0 0
\(731\) −8.01806 −0.296559
\(732\) 0 0
\(733\) 39.6408i 1.46417i −0.681214 0.732084i \(-0.738548\pi\)
0.681214 0.732084i \(-0.261452\pi\)
\(734\) 24.5967 0.907883
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 0 0
\(738\) 0 0
\(739\) −26.3553 −0.969497 −0.484748 0.874654i \(-0.661089\pi\)
−0.484748 + 0.874654i \(0.661089\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −11.4142 + 18.0475i −0.419029 + 0.662543i
\(743\) 35.1421i 1.28924i 0.764503 + 0.644620i \(0.222984\pi\)
−0.764503 + 0.644620i \(0.777016\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 7.07107i 0.258890i
\(747\) 0 0
\(748\) 3.16228i 0.115624i
\(749\) 4.47214 + 2.82843i 0.163408 + 0.103348i
\(750\) 0 0
\(751\) 31.2132 1.13899 0.569493 0.821996i \(-0.307140\pi\)
0.569493 + 0.821996i \(0.307140\pi\)
\(752\) 1.30986 0.0477656
\(753\) 0 0
\(754\) 0.701474i 0.0255462i
\(755\) 0 0
\(756\) 0 0
\(757\) 44.0416 1.60072 0.800360 0.599520i \(-0.204642\pi\)
0.800360 + 0.599520i \(0.204642\pi\)
\(758\) 22.0711i 0.801657i
\(759\) 0 0
\(760\) 0 0
\(761\) −31.3050 −1.13480 −0.567402 0.823441i \(-0.692051\pi\)
−0.567402 + 0.823441i \(0.692051\pi\)
\(762\) 0 0
\(763\) 10.0000 15.8114i 0.362024 0.572411i
\(764\) 0.656854i 0.0237642i
\(765\) 0 0
\(766\) 16.0361i 0.579409i
\(767\) 6.71573i 0.242491i
\(768\) 0 0
\(769\) 15.0441i 0.542504i 0.962508 + 0.271252i \(0.0874376\pi\)
−0.962508 + 0.271252i \(0.912562\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −8.48528 −0.305392
\(773\) −17.3460 −0.623892 −0.311946 0.950100i \(-0.600981\pi\)
−0.311946 + 0.950100i \(0.600981\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0.542561 0.0194768
\(777\) 0 0
\(778\) −22.8284 −0.818439
\(779\) 65.3553i 2.34160i
\(780\) 0 0
\(781\) −22.1421 −0.792308
\(782\) −2.23607 −0.0799616
\(783\) 0 0
\(784\) −3.00000 6.32456i −0.107143 0.225877i
\(785\) 0 0
\(786\) 0 0
\(787\) 33.9247i 1.20928i 0.796497 + 0.604642i \(0.206684\pi\)
−0.796497 + 0.604642i \(0.793316\pi\)
\(788\) 5.92893i 0.211209i
\(789\) 0 0
\(790\) 0 0
\(791\) 2.39498 + 1.51472i 0.0851557 + 0.0538572i
\(792\) 0 0
\(793\) −0.857864 −0.0304637
\(794\) 29.6114 1.05087
\(795\) 0 0
\(796\) 21.5934i 0.765357i
\(797\) 11.0214 0.390399 0.195199 0.980764i \(-0.437465\pi\)
0.195199 + 0.980764i \(0.437465\pi\)
\(798\) 0 0
\(799\) −2.92893 −0.103618
\(800\) 0 0
\(801\) 0 0
\(802\) −9.79899 −0.346014
\(803\) 19.7410 0.696643
\(804\) 0 0
\(805\) 0 0
\(806\) 3.78680i 0.133384i
\(807\) 0 0
\(808\) 12.1065i 0.425907i
\(809\) 19.8995i 0.699629i 0.936819 + 0.349814i \(0.113755\pi\)
−0.936819 + 0.349814i \(0.886245\pi\)
\(810\) 0 0
\(811\) 41.1096i 1.44355i −0.692126 0.721777i \(-0.743326\pi\)
0.692126 0.721777i \(-0.256674\pi\)
\(812\) −1.69351 1.07107i −0.0594304 0.0375871i
\(813\) 0 0
\(814\) 4.00000 0.140200
\(815\) 0 0
\(816\) 0 0
\(817\) 27.3754i 0.957743i
\(818\) −22.9032 −0.800793
\(819\) 0 0
\(820\) 0 0
\(821\) 43.9411i 1.53356i 0.641912 + 0.766778i \(0.278141\pi\)
−0.641912 + 0.766778i \(0.721859\pi\)
\(822\) 0 0
\(823\) −17.0711 −0.595060 −0.297530 0.954712i \(-0.596163\pi\)
−0.297530 + 0.954712i \(0.596163\pi\)
\(824\) 10.4130 0.362756
\(825\) 0 0
\(826\) −16.2132 10.2541i −0.564129 0.356787i
\(827\) 30.4264i 1.05803i 0.848613 + 0.529015i \(0.177438\pi\)
−0.848613 + 0.529015i \(0.822562\pi\)
\(828\) 0 0
\(829\) 21.4345i 0.744450i 0.928143 + 0.372225i \(0.121405\pi\)
−0.928143 + 0.372225i \(0.878595\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.926210i 0.0321105i
\(833\) 6.70820 + 14.1421i 0.232425 + 0.489996i
\(834\) 0 0
\(835\) 0 0
\(836\) 10.7967 0.373411
\(837\) 0 0
\(838\) 28.8441i 0.996405i
\(839\) 35.0098 1.20867 0.604336 0.796729i \(-0.293438\pi\)
0.604336 + 0.796729i \(0.293438\pi\)
\(840\) 0 0
\(841\) 28.4264 0.980221
\(842\) 33.3553i 1.14950i
\(843\) 0 0
\(844\) −16.2132 −0.558081
\(845\) 0 0
\(846\) 0 0
\(847\) −12.7279 + 20.1246i −0.437337 + 0.691490i
\(848\) 8.07107i 0.277162i
\(849\) 0 0
\(850\) 0 0
\(851\) 2.82843i 0.0969572i
\(852\) 0 0
\(853\) 9.10318i 0.311687i −0.987782 0.155844i \(-0.950190\pi\)
0.987782 0.155844i \(-0.0498096\pi\)
\(854\) 1.30986 2.07107i 0.0448224 0.0708705i
\(855\) 0 0
\(856\) −2.00000 −0.0683586
\(857\) 17.8885 0.611061 0.305531 0.952182i \(-0.401166\pi\)
0.305531 + 0.952182i \(0.401166\pi\)
\(858\) 0 0
\(859\) 2.61972i 0.0893836i −0.999001 0.0446918i \(-0.985769\pi\)
0.999001 0.0446918i \(-0.0142306\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 4.79899 0.163454
\(863\) 19.8579i 0.675970i −0.941152 0.337985i \(-0.890255\pi\)
0.941152 0.337985i \(-0.109745\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −9.71157 −0.330013
\(867\) 0 0
\(868\) 9.14214 + 5.78199i 0.310304 + 0.196254i
\(869\) 18.4853i 0.627070i
\(870\) 0 0
\(871\) 0 0
\(872\) 7.07107i 0.239457i
\(873\) 0 0
\(874\) 7.63441i 0.258238i
\(875\) 0 0
\(876\) 0 0
\(877\) −5.85786 −0.197806 −0.0989030 0.995097i \(-0.531533\pi\)
−0.0989030 + 0.995097i \(0.531533\pi\)
\(878\) 4.08849 0.137980
\(879\) 0 0
\(880\) 0 0
\(881\) 32.0064 1.07832 0.539162 0.842202i \(-0.318741\pi\)
0.539162 + 0.842202i \(0.318741\pi\)
\(882\) 0 0
\(883\) −36.0122 −1.21191 −0.605953 0.795500i \(-0.707208\pi\)
−0.605953 + 0.795500i \(0.707208\pi\)
\(884\) 2.07107i 0.0696575i
\(885\) 0 0
\(886\) −23.0711 −0.775088
\(887\) −16.3539 −0.549112 −0.274556 0.961571i \(-0.588531\pi\)
−0.274556 + 0.961571i \(0.588531\pi\)
\(888\) 0 0
\(889\) −14.1421 + 22.3607i −0.474312 + 0.749953i
\(890\) 0 0
\(891\) 0 0
\(892\) 13.0328i 0.436369i
\(893\) 10.0000i 0.334637i
\(894\) 0 0
\(895\) 0 0
\(896\) 2.23607 + 1.41421i 0.0747018 + 0.0472456i
\(897\) 0 0
\(898\) −18.3848 −0.613508
\(899\) 3.09645 0.103273
\(900\) 0 0
\(901\) 18.0475i 0.601248i
\(902\) 12.1065 0.403104
\(903\) 0 0
\(904\) −1.07107 −0.0356232
\(905\) 0 0
\(906\) 0 0
\(907\) −14.8995 −0.494730 −0.247365 0.968922i \(-0.579565\pi\)
−0.247365 + 0.968922i \(0.579565\pi\)
\(908\) −25.1393 −0.834277
\(909\) 0 0
\(910\) 0 0
\(911\) 12.3726i 0.409922i −0.978770 0.204961i \(-0.934293\pi\)
0.978770 0.204961i \(-0.0657067\pi\)
\(912\) 0 0
\(913\) 20.2835i 0.671287i
\(914\) 17.8284i 0.589712i
\(915\) 0 0
\(916\) 10.7967i 0.356733i
\(917\) −17.8885 + 28.2843i −0.590732 + 0.934029i
\(918\) 0 0
\(919\) 7.07107 0.233253 0.116627 0.993176i \(-0.462792\pi\)
0.116627 + 0.993176i \(0.462792\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 38.9394i 1.28240i
\(923\) 14.5015 0.477324
\(924\) 0 0
\(925\) 0 0
\(926\) 38.2843i 1.25810i
\(927\) 0 0
\(928\) 0.757359 0.0248615
\(929\) 25.3640 0.832167 0.416084 0.909326i \(-0.363402\pi\)
0.416084 + 0.909326i \(0.363402\pi\)
\(930\) 0 0
\(931\) −48.2843 + 22.9032i −1.58245 + 0.750623i
\(932\) 23.0711i 0.755718i
\(933\) 0 0
\(934\) 29.6114i 0.968916i
\(935\) 0 0
\(936\) 0 0
\(937\) 5.23943i 0.171165i −0.996331 0.0855824i \(-0.972725\pi\)
0.996331 0.0855824i \(-0.0272751\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −44.2719 −1.44322 −0.721611 0.692299i \(-0.756598\pi\)
−0.721611 + 0.692299i \(0.756598\pi\)
\(942\) 0 0
\(943\) 8.56062i 0.278772i
\(944\) 7.25077 0.235992
\(945\) 0 0
\(946\) 5.07107 0.164875
\(947\) 33.3553i 1.08390i −0.840410 0.541951i \(-0.817686\pi\)
0.840410 0.541951i \(-0.182314\pi\)
\(948\) 0 0
\(949\) −12.9289 −0.419691
\(950\) 0 0
\(951\) 0 0
\(952\) −5.00000 3.16228i −0.162051 0.102490i
\(953\) 36.0000i 1.16615i 0.812417 + 0.583077i \(0.198151\pi\)
−0.812417 + 0.583077i \(0.801849\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 7.17157i 0.231945i
\(957\) 0 0
\(958\) 33.6999i 1.08880i
\(959\) 27.1506 + 17.1716i 0.876740 + 0.554499i
\(960\) 0 0
\(961\) 14.2843 0.460783
\(962\) −2.61972 −0.0844631
\(963\) 0 0
\(964\) 4.47214i 0.144038i
\(965\) 0 0
\(966\) 0 0
\(967\) 49.4975 1.59173 0.795866 0.605473i \(-0.207016\pi\)
0.795866 + 0.605473i \(0.207016\pi\)
\(968\) 9.00000i 0.289271i
\(969\) 0 0
\(970\) 0 0
\(971\) 27.6001 0.885730 0.442865 0.896588i \(-0.353962\pi\)
0.442865 + 0.896588i \(0.353962\pi\)
\(972\) 0 0
\(973\) 24.1421 + 15.2688i 0.773961 + 0.489496i
\(974\) 29.8995i 0.958042i
\(975\) 0 0
\(976\) 0.926210i 0.0296472i
\(977\) 12.0000i 0.383914i −0.981403 0.191957i \(-0.938517\pi\)
0.981403 0.191957i \(-0.0614834\pi\)
\(978\) 0 0
\(979\) 3.70484i 0.118407i
\(980\) 0 0
\(981\) 0 0
\(982\) 21.5147 0.686562
\(983\) 6.32456 0.201722 0.100861 0.994901i \(-0.467840\pi\)
0.100861 + 0.994901i \(0.467840\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −1.69351 −0.0539322
\(987\) 0 0
\(988\) −7.07107 −0.224961
\(989\) 3.58579i 0.114021i
\(990\) 0 0
\(991\) −25.3553 −0.805439 −0.402719 0.915323i \(-0.631935\pi\)
−0.402719 + 0.915323i \(0.631935\pi\)
\(992\) −4.08849 −0.129810
\(993\) 0 0
\(994\) −22.1421 + 35.0098i −0.702306 + 1.11044i
\(995\) 0 0
\(996\) 0 0
\(997\) 1.85242i 0.0586667i 0.999570 + 0.0293334i \(0.00933844\pi\)
−0.999570 + 0.0293334i \(0.990662\pi\)
\(998\) 22.0711i 0.698647i
\(999\) 0 0
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 3150.2.b.d.251.6 yes 8
3.2 odd 2 inner 3150.2.b.d.251.2 yes 8
5.2 odd 4 3150.2.d.b.3149.2 8
5.3 odd 4 3150.2.d.e.3149.7 8
5.4 even 2 3150.2.b.a.251.3 8
7.6 odd 2 inner 3150.2.b.d.251.5 yes 8
15.2 even 4 3150.2.d.e.3149.2 8
15.8 even 4 3150.2.d.b.3149.7 8
15.14 odd 2 3150.2.b.a.251.7 yes 8
21.20 even 2 inner 3150.2.b.d.251.1 yes 8
35.13 even 4 3150.2.d.e.3149.4 8
35.27 even 4 3150.2.d.b.3149.5 8
35.34 odd 2 3150.2.b.a.251.4 yes 8
105.62 odd 4 3150.2.d.e.3149.5 8
105.83 odd 4 3150.2.d.b.3149.4 8
105.104 even 2 3150.2.b.a.251.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3150.2.b.a.251.3 8 5.4 even 2
3150.2.b.a.251.4 yes 8 35.34 odd 2
3150.2.b.a.251.7 yes 8 15.14 odd 2
3150.2.b.a.251.8 yes 8 105.104 even 2
3150.2.b.d.251.1 yes 8 21.20 even 2 inner
3150.2.b.d.251.2 yes 8 3.2 odd 2 inner
3150.2.b.d.251.5 yes 8 7.6 odd 2 inner
3150.2.b.d.251.6 yes 8 1.1 even 1 trivial
3150.2.d.b.3149.2 8 5.2 odd 4
3150.2.d.b.3149.4 8 105.83 odd 4
3150.2.d.b.3149.5 8 35.27 even 4
3150.2.d.b.3149.7 8 15.8 even 4
3150.2.d.e.3149.2 8 15.2 even 4
3150.2.d.e.3149.4 8 35.13 even 4
3150.2.d.e.3149.5 8 105.62 odd 4
3150.2.d.e.3149.7 8 5.3 odd 4