Properties

Label 672.2.h.e.575.8
Level $672$
Weight $2$
Character 672.575
Analytic conductor $5.366$
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] = [672,2,Mod(575,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.h (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: \(5.36594701583\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 575.8
Root \(-1.14412 + 1.14412i\) of defining polynomial
Character \(\chi\) \(=\) 672.575
Dual form 672.2.h.e.575.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.707107 + 1.58114i) q^{3} +1.41421i q^{5} -1.00000i q^{7} +(-2.00000 + 2.23607i) q^{9} +O(q^{10})\) \(q+(0.707107 + 1.58114i) q^{3} +1.41421i q^{5} -1.00000i q^{7} +(-2.00000 + 2.23607i) q^{9} -4.47214 q^{11} -7.16228 q^{13} +(-2.23607 + 1.00000i) q^{15} +7.30056i q^{17} -0.837722i q^{19} +(1.58114 - 0.707107i) q^{21} +5.65685 q^{23} +3.00000 q^{25} +(-4.94975 - 1.58114i) q^{27} +1.64371i q^{29} -6.32456i q^{31} +(-3.16228 - 7.07107i) q^{33} +1.41421 q^{35} +4.32456 q^{37} +(-5.06450 - 11.3246i) q^{39} +10.1290i q^{41} +8.32456i q^{43} +(-3.16228 - 2.82843i) q^{45} -8.94427 q^{47} -1.00000 q^{49} +(-11.5432 + 5.16228i) q^{51} +1.18472i q^{53} -6.32456i q^{55} +(1.32456 - 0.592359i) q^{57} -1.41421 q^{59} +3.16228 q^{61} +(2.23607 + 2.00000i) q^{63} -10.1290i q^{65} -2.00000i q^{67} +(4.00000 + 8.94427i) q^{69} +10.1290 q^{71} +4.32456 q^{73} +(2.12132 + 4.74342i) q^{75} +4.47214i q^{77} -4.00000i q^{79} +(-1.00000 - 8.94427i) q^{81} +7.53006 q^{83} -10.3246 q^{85} +(-2.59893 + 1.16228i) q^{87} +1.18472i q^{89} +7.16228i q^{91} +(10.0000 - 4.47214i) q^{93} +1.18472 q^{95} -10.6491 q^{97} +(8.94427 - 10.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{9} - 32 q^{13} + 24 q^{25} - 16 q^{37} - 8 q^{49} - 40 q^{57} + 32 q^{69} - 16 q^{73} - 8 q^{81} - 32 q^{85} + 80 q^{93} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.707107 + 1.58114i 0.408248 + 0.912871i
\(4\) 0 0
\(5\) 1.41421i 0.632456i 0.948683 + 0.316228i \(0.102416\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −2.00000 + 2.23607i −0.666667 + 0.745356i
\(10\) 0 0
\(11\) −4.47214 −1.34840 −0.674200 0.738549i \(-0.735511\pi\)
−0.674200 + 0.738549i \(0.735511\pi\)
\(12\) 0 0
\(13\) −7.16228 −1.98646 −0.993229 0.116171i \(-0.962938\pi\)
−0.993229 + 0.116171i \(0.962938\pi\)
\(14\) 0 0
\(15\) −2.23607 + 1.00000i −0.577350 + 0.258199i
\(16\) 0 0
\(17\) 7.30056i 1.77065i 0.464976 + 0.885323i \(0.346063\pi\)
−0.464976 + 0.885323i \(0.653937\pi\)
\(18\) 0 0
\(19\) 0.837722i 0.192187i −0.995372 0.0960933i \(-0.969365\pi\)
0.995372 0.0960933i \(-0.0306347\pi\)
\(20\) 0 0
\(21\) 1.58114 0.707107i 0.345033 0.154303i
\(22\) 0 0
\(23\) 5.65685 1.17954 0.589768 0.807573i \(-0.299219\pi\)
0.589768 + 0.807573i \(0.299219\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) −4.94975 1.58114i −0.952579 0.304290i
\(28\) 0 0
\(29\) 1.64371i 0.305229i 0.988286 + 0.152615i \(0.0487693\pi\)
−0.988286 + 0.152615i \(0.951231\pi\)
\(30\) 0 0
\(31\) 6.32456i 1.13592i −0.823055 0.567962i \(-0.807732\pi\)
0.823055 0.567962i \(-0.192268\pi\)
\(32\) 0 0
\(33\) −3.16228 7.07107i −0.550482 1.23091i
\(34\) 0 0
\(35\) 1.41421 0.239046
\(36\) 0 0
\(37\) 4.32456 0.710953 0.355476 0.934685i \(-0.384319\pi\)
0.355476 + 0.934685i \(0.384319\pi\)
\(38\) 0 0
\(39\) −5.06450 11.3246i −0.810968 1.81338i
\(40\) 0 0
\(41\) 10.1290i 1.58188i 0.611892 + 0.790941i \(0.290409\pi\)
−0.611892 + 0.790941i \(0.709591\pi\)
\(42\) 0 0
\(43\) 8.32456i 1.26948i 0.772725 + 0.634741i \(0.218893\pi\)
−0.772725 + 0.634741i \(0.781107\pi\)
\(44\) 0 0
\(45\) −3.16228 2.82843i −0.471405 0.421637i
\(46\) 0 0
\(47\) −8.94427 −1.30466 −0.652328 0.757937i \(-0.726208\pi\)
−0.652328 + 0.757937i \(0.726208\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −11.5432 + 5.16228i −1.61637 + 0.722863i
\(52\) 0 0
\(53\) 1.18472i 0.162734i 0.996684 + 0.0813668i \(0.0259285\pi\)
−0.996684 + 0.0813668i \(0.974071\pi\)
\(54\) 0 0
\(55\) 6.32456i 0.852803i
\(56\) 0 0
\(57\) 1.32456 0.592359i 0.175442 0.0784599i
\(58\) 0 0
\(59\) −1.41421 −0.184115 −0.0920575 0.995754i \(-0.529344\pi\)
−0.0920575 + 0.995754i \(0.529344\pi\)
\(60\) 0 0
\(61\) 3.16228 0.404888 0.202444 0.979294i \(-0.435112\pi\)
0.202444 + 0.979294i \(0.435112\pi\)
\(62\) 0 0
\(63\) 2.23607 + 2.00000i 0.281718 + 0.251976i
\(64\) 0 0
\(65\) 10.1290i 1.25635i
\(66\) 0 0
\(67\) 2.00000i 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) 0 0
\(69\) 4.00000 + 8.94427i 0.481543 + 1.07676i
\(70\) 0 0
\(71\) 10.1290 1.20209 0.601045 0.799215i \(-0.294751\pi\)
0.601045 + 0.799215i \(0.294751\pi\)
\(72\) 0 0
\(73\) 4.32456 0.506151 0.253075 0.967447i \(-0.418558\pi\)
0.253075 + 0.967447i \(0.418558\pi\)
\(74\) 0 0
\(75\) 2.12132 + 4.74342i 0.244949 + 0.547723i
\(76\) 0 0
\(77\) 4.47214i 0.509647i
\(78\) 0 0
\(79\) 4.00000i 0.450035i −0.974355 0.225018i \(-0.927756\pi\)
0.974355 0.225018i \(-0.0722440\pi\)
\(80\) 0 0
\(81\) −1.00000 8.94427i −0.111111 0.993808i
\(82\) 0 0
\(83\) 7.53006 0.826531 0.413266 0.910611i \(-0.364388\pi\)
0.413266 + 0.910611i \(0.364388\pi\)
\(84\) 0 0
\(85\) −10.3246 −1.11986
\(86\) 0 0
\(87\) −2.59893 + 1.16228i −0.278635 + 0.124609i
\(88\) 0 0
\(89\) 1.18472i 0.125580i 0.998027 + 0.0627899i \(0.0199998\pi\)
−0.998027 + 0.0627899i \(0.980000\pi\)
\(90\) 0 0
\(91\) 7.16228i 0.750811i
\(92\) 0 0
\(93\) 10.0000 4.47214i 1.03695 0.463739i
\(94\) 0 0
\(95\) 1.18472 0.121550
\(96\) 0 0
\(97\) −10.6491 −1.08125 −0.540627 0.841263i \(-0.681813\pi\)
−0.540627 + 0.841263i \(0.681813\pi\)
\(98\) 0 0
\(99\) 8.94427 10.0000i 0.898933 1.00504i
\(100\) 0 0
\(101\) 1.41421i 0.140720i 0.997522 + 0.0703598i \(0.0224147\pi\)
−0.997522 + 0.0703598i \(0.977585\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 0 0
\(105\) 1.00000 + 2.23607i 0.0975900 + 0.218218i
\(106\) 0 0
\(107\) 9.67000 0.934834 0.467417 0.884037i \(-0.345185\pi\)
0.467417 + 0.884037i \(0.345185\pi\)
\(108\) 0 0
\(109\) −4.32456 −0.414217 −0.207109 0.978318i \(-0.566405\pi\)
−0.207109 + 0.978318i \(0.566405\pi\)
\(110\) 0 0
\(111\) 3.05792 + 6.83772i 0.290245 + 0.649008i
\(112\) 0 0
\(113\) 6.84157i 0.643601i 0.946807 + 0.321801i \(0.104288\pi\)
−0.946807 + 0.321801i \(0.895712\pi\)
\(114\) 0 0
\(115\) 8.00000i 0.746004i
\(116\) 0 0
\(117\) 14.3246 16.0153i 1.32431 1.48062i
\(118\) 0 0
\(119\) 7.30056 0.669242
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0 0
\(123\) −16.0153 + 7.16228i −1.44405 + 0.645801i
\(124\) 0 0
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) 6.00000i 0.532414i 0.963916 + 0.266207i \(0.0857705\pi\)
−0.963916 + 0.266207i \(0.914230\pi\)
\(128\) 0 0
\(129\) −13.1623 + 5.88635i −1.15887 + 0.518264i
\(130\) 0 0
\(131\) 4.70163 0.410783 0.205392 0.978680i \(-0.434153\pi\)
0.205392 + 0.978680i \(0.434153\pi\)
\(132\) 0 0
\(133\) −0.837722 −0.0726397
\(134\) 0 0
\(135\) 2.23607 7.00000i 0.192450 0.602464i
\(136\) 0 0
\(137\) 17.8885i 1.52832i −0.645026 0.764161i \(-0.723153\pi\)
0.645026 0.764161i \(-0.276847\pi\)
\(138\) 0 0
\(139\) 7.16228i 0.607496i 0.952752 + 0.303748i \(0.0982382\pi\)
−0.952752 + 0.303748i \(0.901762\pi\)
\(140\) 0 0
\(141\) −6.32456 14.1421i −0.532624 1.19098i
\(142\) 0 0
\(143\) 32.0307 2.67854
\(144\) 0 0
\(145\) −2.32456 −0.193044
\(146\) 0 0
\(147\) −0.707107 1.58114i −0.0583212 0.130410i
\(148\) 0 0
\(149\) 4.47214i 0.366372i −0.983078 0.183186i \(-0.941359\pi\)
0.983078 0.183186i \(-0.0586410\pi\)
\(150\) 0 0
\(151\) 10.0000i 0.813788i −0.913475 0.406894i \(-0.866612\pi\)
0.913475 0.406894i \(-0.133388\pi\)
\(152\) 0 0
\(153\) −16.3246 14.6011i −1.31976 1.18043i
\(154\) 0 0
\(155\) 8.94427 0.718421
\(156\) 0 0
\(157\) −13.4868 −1.07637 −0.538183 0.842828i \(-0.680889\pi\)
−0.538183 + 0.842828i \(0.680889\pi\)
\(158\) 0 0
\(159\) −1.87320 + 0.837722i −0.148555 + 0.0664357i
\(160\) 0 0
\(161\) 5.65685i 0.445823i
\(162\) 0 0
\(163\) 4.32456i 0.338725i 0.985554 + 0.169363i \(0.0541709\pi\)
−0.985554 + 0.169363i \(0.945829\pi\)
\(164\) 0 0
\(165\) 10.0000 4.47214i 0.778499 0.348155i
\(166\) 0 0
\(167\) 0.458991 0.0355178 0.0177589 0.999842i \(-0.494347\pi\)
0.0177589 + 0.999842i \(0.494347\pi\)
\(168\) 0 0
\(169\) 38.2982 2.94602
\(170\) 0 0
\(171\) 1.87320 + 1.67544i 0.143247 + 0.128124i
\(172\) 0 0
\(173\) 3.78365i 0.287666i 0.989602 + 0.143833i \(0.0459427\pi\)
−0.989602 + 0.143833i \(0.954057\pi\)
\(174\) 0 0
\(175\) 3.00000i 0.226779i
\(176\) 0 0
\(177\) −1.00000 2.23607i −0.0751646 0.168073i
\(178\) 0 0
\(179\) −19.5323 −1.45991 −0.729955 0.683496i \(-0.760459\pi\)
−0.729955 + 0.683496i \(0.760459\pi\)
\(180\) 0 0
\(181\) 18.1359 1.34803 0.674017 0.738716i \(-0.264567\pi\)
0.674017 + 0.738716i \(0.264567\pi\)
\(182\) 0 0
\(183\) 2.23607 + 5.00000i 0.165295 + 0.369611i
\(184\) 0 0
\(185\) 6.11584i 0.449646i
\(186\) 0 0
\(187\) 32.6491i 2.38754i
\(188\) 0 0
\(189\) −1.58114 + 4.94975i −0.115011 + 0.360041i
\(190\) 0 0
\(191\) −12.4984 −0.904354 −0.452177 0.891928i \(-0.649353\pi\)
−0.452177 + 0.891928i \(0.649353\pi\)
\(192\) 0 0
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) 0 0
\(195\) 16.0153 7.16228i 1.14688 0.512901i
\(196\) 0 0
\(197\) 10.1290i 0.721661i 0.932631 + 0.360830i \(0.117507\pi\)
−0.932631 + 0.360830i \(0.882493\pi\)
\(198\) 0 0
\(199\) 24.6491i 1.74733i 0.486529 + 0.873665i \(0.338263\pi\)
−0.486529 + 0.873665i \(0.661737\pi\)
\(200\) 0 0
\(201\) 3.16228 1.41421i 0.223050 0.0997509i
\(202\) 0 0
\(203\) 1.64371 0.115366
\(204\) 0 0
\(205\) −14.3246 −1.00047
\(206\) 0 0
\(207\) −11.3137 + 12.6491i −0.786357 + 0.879174i
\(208\) 0 0
\(209\) 3.74641i 0.259144i
\(210\) 0 0
\(211\) 1.35089i 0.0929991i 0.998918 + 0.0464995i \(0.0148066\pi\)
−0.998918 + 0.0464995i \(0.985193\pi\)
\(212\) 0 0
\(213\) 7.16228 + 16.0153i 0.490751 + 1.09735i
\(214\) 0 0
\(215\) −11.7727 −0.802891
\(216\) 0 0
\(217\) −6.32456 −0.429339
\(218\) 0 0
\(219\) 3.05792 + 6.83772i 0.206635 + 0.462050i
\(220\) 0 0
\(221\) 52.2887i 3.51732i
\(222\) 0 0
\(223\) 2.32456i 0.155664i 0.996966 + 0.0778319i \(0.0247997\pi\)
−0.996966 + 0.0778319i \(0.975200\pi\)
\(224\) 0 0
\(225\) −6.00000 + 6.70820i −0.400000 + 0.447214i
\(226\) 0 0
\(227\) 13.1869 0.875246 0.437623 0.899159i \(-0.355820\pi\)
0.437623 + 0.899159i \(0.355820\pi\)
\(228\) 0 0
\(229\) 3.81139 0.251864 0.125932 0.992039i \(-0.459808\pi\)
0.125932 + 0.992039i \(0.459808\pi\)
\(230\) 0 0
\(231\) −7.07107 + 3.16228i −0.465242 + 0.208063i
\(232\) 0 0
\(233\) 5.65685i 0.370593i 0.982683 + 0.185296i \(0.0593245\pi\)
−0.982683 + 0.185296i \(0.940675\pi\)
\(234\) 0 0
\(235\) 12.6491i 0.825137i
\(236\) 0 0
\(237\) 6.32456 2.82843i 0.410824 0.183726i
\(238\) 0 0
\(239\) −25.9148 −1.67629 −0.838146 0.545446i \(-0.816360\pi\)
−0.838146 + 0.545446i \(0.816360\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 13.4350 7.90569i 0.861858 0.507151i
\(244\) 0 0
\(245\) 1.41421i 0.0903508i
\(246\) 0 0
\(247\) 6.00000i 0.381771i
\(248\) 0 0
\(249\) 5.32456 + 11.9061i 0.337430 + 0.754516i
\(250\) 0 0
\(251\) 18.8438 1.18941 0.594704 0.803945i \(-0.297269\pi\)
0.594704 + 0.803945i \(0.297269\pi\)
\(252\) 0 0
\(253\) −25.2982 −1.59049
\(254\) 0 0
\(255\) −7.30056 16.3246i −0.457179 1.02228i
\(256\) 0 0
\(257\) 16.2448i 1.01332i −0.862144 0.506662i \(-0.830879\pi\)
0.862144 0.506662i \(-0.169121\pi\)
\(258\) 0 0
\(259\) 4.32456i 0.268715i
\(260\) 0 0
\(261\) −3.67544 3.28742i −0.227504 0.203486i
\(262\) 0 0
\(263\) −2.10270 −0.129658 −0.0648290 0.997896i \(-0.520650\pi\)
−0.0648290 + 0.997896i \(0.520650\pi\)
\(264\) 0 0
\(265\) −1.67544 −0.102922
\(266\) 0 0
\(267\) −1.87320 + 0.837722i −0.114638 + 0.0512678i
\(268\) 0 0
\(269\) 6.61208i 0.403145i −0.979474 0.201573i \(-0.935395\pi\)
0.979474 0.201573i \(-0.0646052\pi\)
\(270\) 0 0
\(271\) 1.67544i 0.101776i −0.998704 0.0508880i \(-0.983795\pi\)
0.998704 0.0508880i \(-0.0162052\pi\)
\(272\) 0 0
\(273\) −11.3246 + 5.06450i −0.685393 + 0.306517i
\(274\) 0 0
\(275\) −13.4164 −0.809040
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 14.1421 + 12.6491i 0.846668 + 0.757282i
\(280\) 0 0
\(281\) 23.5454i 1.40460i 0.711881 + 0.702300i \(0.247844\pi\)
−0.711881 + 0.702300i \(0.752156\pi\)
\(282\) 0 0
\(283\) 6.51317i 0.387168i −0.981084 0.193584i \(-0.937989\pi\)
0.981084 0.193584i \(-0.0620111\pi\)
\(284\) 0 0
\(285\) 0.837722 + 1.87320i 0.0496224 + 0.110959i
\(286\) 0 0
\(287\) 10.1290 0.597895
\(288\) 0 0
\(289\) −36.2982 −2.13519
\(290\) 0 0
\(291\) −7.53006 16.8377i −0.441420 0.987045i
\(292\) 0 0
\(293\) 22.1312i 1.29292i 0.762949 + 0.646459i \(0.223751\pi\)
−0.762949 + 0.646459i \(0.776249\pi\)
\(294\) 0 0
\(295\) 2.00000i 0.116445i
\(296\) 0 0
\(297\) 22.1359 + 7.07107i 1.28446 + 0.410305i
\(298\) 0 0
\(299\) −40.5160 −2.34310
\(300\) 0 0
\(301\) 8.32456 0.479819
\(302\) 0 0
\(303\) −2.23607 + 1.00000i −0.128459 + 0.0574485i
\(304\) 0 0
\(305\) 4.47214i 0.256074i
\(306\) 0 0
\(307\) 2.51317i 0.143434i 0.997425 + 0.0717170i \(0.0228478\pi\)
−0.997425 + 0.0717170i \(0.977152\pi\)
\(308\) 0 0
\(309\) −12.6491 + 5.65685i −0.719583 + 0.321807i
\(310\) 0 0
\(311\) −17.4296 −0.988339 −0.494170 0.869365i \(-0.664528\pi\)
−0.494170 + 0.869365i \(0.664528\pi\)
\(312\) 0 0
\(313\) −4.97367 −0.281128 −0.140564 0.990072i \(-0.544892\pi\)
−0.140564 + 0.990072i \(0.544892\pi\)
\(314\) 0 0
\(315\) −2.82843 + 3.16228i −0.159364 + 0.178174i
\(316\) 0 0
\(317\) 7.75955i 0.435820i 0.975969 + 0.217910i \(0.0699239\pi\)
−0.975969 + 0.217910i \(0.930076\pi\)
\(318\) 0 0
\(319\) 7.35089i 0.411571i
\(320\) 0 0
\(321\) 6.83772 + 15.2896i 0.381644 + 0.853383i
\(322\) 0 0
\(323\) 6.11584 0.340295
\(324\) 0 0
\(325\) −21.4868 −1.19188
\(326\) 0 0
\(327\) −3.05792 6.83772i −0.169103 0.378127i
\(328\) 0 0
\(329\) 8.94427i 0.493114i
\(330\) 0 0
\(331\) 28.9737i 1.59254i −0.604944 0.796268i \(-0.706804\pi\)
0.604944 0.796268i \(-0.293196\pi\)
\(332\) 0 0
\(333\) −8.64911 + 9.67000i −0.473968 + 0.529913i
\(334\) 0 0
\(335\) 2.82843 0.154533
\(336\) 0 0
\(337\) −7.35089 −0.400428 −0.200214 0.979752i \(-0.564164\pi\)
−0.200214 + 0.979752i \(0.564164\pi\)
\(338\) 0 0
\(339\) −10.8175 + 4.83772i −0.587525 + 0.262749i
\(340\) 0 0
\(341\) 28.2843i 1.53168i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) −12.6491 + 5.65685i −0.681005 + 0.304555i
\(346\) 0 0
\(347\) 21.4427 1.15110 0.575552 0.817765i \(-0.304787\pi\)
0.575552 + 0.817765i \(0.304787\pi\)
\(348\) 0 0
\(349\) −16.8377 −0.901303 −0.450651 0.892700i \(-0.648808\pi\)
−0.450651 + 0.892700i \(0.648808\pi\)
\(350\) 0 0
\(351\) 35.4515 + 11.3246i 1.89226 + 0.604460i
\(352\) 0 0
\(353\) 12.9574i 0.689654i −0.938666 0.344827i \(-0.887938\pi\)
0.938666 0.344827i \(-0.112062\pi\)
\(354\) 0 0
\(355\) 14.3246i 0.760268i
\(356\) 0 0
\(357\) 5.16228 + 11.5432i 0.273217 + 0.610931i
\(358\) 0 0
\(359\) −12.2317 −0.645564 −0.322782 0.946473i \(-0.604618\pi\)
−0.322782 + 0.946473i \(0.604618\pi\)
\(360\) 0 0
\(361\) 18.2982 0.963064
\(362\) 0 0
\(363\) 6.36396 + 14.2302i 0.334021 + 0.746894i
\(364\) 0 0
\(365\) 6.11584i 0.320118i
\(366\) 0 0
\(367\) 18.3246i 0.956534i 0.878214 + 0.478267i \(0.158735\pi\)
−0.878214 + 0.478267i \(0.841265\pi\)
\(368\) 0 0
\(369\) −22.6491 20.2580i −1.17907 1.05459i
\(370\) 0 0
\(371\) 1.18472 0.0615075
\(372\) 0 0
\(373\) 30.6491 1.58695 0.793475 0.608602i \(-0.208270\pi\)
0.793475 + 0.608602i \(0.208270\pi\)
\(374\) 0 0
\(375\) −17.8885 + 8.00000i −0.923760 + 0.413118i
\(376\) 0 0
\(377\) 11.7727i 0.606325i
\(378\) 0 0
\(379\) 24.3246i 1.24947i −0.780837 0.624734i \(-0.785207\pi\)
0.780837 0.624734i \(-0.214793\pi\)
\(380\) 0 0
\(381\) −9.48683 + 4.24264i −0.486025 + 0.217357i
\(382\) 0 0
\(383\) −22.6274 −1.15621 −0.578103 0.815963i \(-0.696207\pi\)
−0.578103 + 0.815963i \(0.696207\pi\)
\(384\) 0 0
\(385\) −6.32456 −0.322329
\(386\) 0 0
\(387\) −18.6143 16.6491i −0.946217 0.846322i
\(388\) 0 0
\(389\) 6.38258i 0.323610i −0.986823 0.161805i \(-0.948268\pi\)
0.986823 0.161805i \(-0.0517315\pi\)
\(390\) 0 0
\(391\) 41.2982i 2.08854i
\(392\) 0 0
\(393\) 3.32456 + 7.43393i 0.167702 + 0.374992i
\(394\) 0 0
\(395\) 5.65685 0.284627
\(396\) 0 0
\(397\) 23.1623 1.16248 0.581241 0.813732i \(-0.302567\pi\)
0.581241 + 0.813732i \(0.302567\pi\)
\(398\) 0 0
\(399\) −0.592359 1.32456i −0.0296550 0.0663107i
\(400\) 0 0
\(401\) 21.4427i 1.07080i −0.844599 0.535399i \(-0.820161\pi\)
0.844599 0.535399i \(-0.179839\pi\)
\(402\) 0 0
\(403\) 45.2982i 2.25647i
\(404\) 0 0
\(405\) 12.6491 1.41421i 0.628539 0.0702728i
\(406\) 0 0
\(407\) −19.3400 −0.958648
\(408\) 0 0
\(409\) −11.6754 −0.577314 −0.288657 0.957433i \(-0.593209\pi\)
−0.288657 + 0.957433i \(0.593209\pi\)
\(410\) 0 0
\(411\) 28.2843 12.6491i 1.39516 0.623935i
\(412\) 0 0
\(413\) 1.41421i 0.0695889i
\(414\) 0 0
\(415\) 10.6491i 0.522744i
\(416\) 0 0
\(417\) −11.3246 + 5.06450i −0.554566 + 0.248009i
\(418\) 0 0
\(419\) −27.7880 −1.35753 −0.678767 0.734353i \(-0.737485\pi\)
−0.678767 + 0.734353i \(0.737485\pi\)
\(420\) 0 0
\(421\) −22.6491 −1.10385 −0.551925 0.833894i \(-0.686107\pi\)
−0.551925 + 0.833894i \(0.686107\pi\)
\(422\) 0 0
\(423\) 17.8885 20.0000i 0.869771 0.972433i
\(424\) 0 0
\(425\) 21.9017i 1.06239i
\(426\) 0 0
\(427\) 3.16228i 0.153033i
\(428\) 0 0
\(429\) 22.6491 + 50.6450i 1.09351 + 2.44516i
\(430\) 0 0
\(431\) −14.6011 −0.703311 −0.351656 0.936129i \(-0.614381\pi\)
−0.351656 + 0.936129i \(0.614381\pi\)
\(432\) 0 0
\(433\) 6.00000 0.288342 0.144171 0.989553i \(-0.453949\pi\)
0.144171 + 0.989553i \(0.453949\pi\)
\(434\) 0 0
\(435\) −1.64371 3.67544i −0.0788098 0.176224i
\(436\) 0 0
\(437\) 4.73887i 0.226691i
\(438\) 0 0
\(439\) 0.649111i 0.0309804i −0.999880 0.0154902i \(-0.995069\pi\)
0.999880 0.0154902i \(-0.00493087\pi\)
\(440\) 0 0
\(441\) 2.00000 2.23607i 0.0952381 0.106479i
\(442\) 0 0
\(443\) −4.01315 −0.190670 −0.0953351 0.995445i \(-0.530392\pi\)
−0.0953351 + 0.995445i \(0.530392\pi\)
\(444\) 0 0
\(445\) −1.67544 −0.0794237
\(446\) 0 0
\(447\) 7.07107 3.16228i 0.334450 0.149571i
\(448\) 0 0
\(449\) 16.9706i 0.800890i 0.916321 + 0.400445i \(0.131145\pi\)
−0.916321 + 0.400445i \(0.868855\pi\)
\(450\) 0 0
\(451\) 45.2982i 2.13301i
\(452\) 0 0
\(453\) 15.8114 7.07107i 0.742884 0.332228i
\(454\) 0 0
\(455\) −10.1290 −0.474854
\(456\) 0 0
\(457\) 24.6491 1.15304 0.576518 0.817084i \(-0.304411\pi\)
0.576518 + 0.817084i \(0.304411\pi\)
\(458\) 0 0
\(459\) 11.5432 36.1359i 0.538791 1.68668i
\(460\) 0 0
\(461\) 32.9859i 1.53631i −0.640266 0.768153i \(-0.721176\pi\)
0.640266 0.768153i \(-0.278824\pi\)
\(462\) 0 0
\(463\) 40.6491i 1.88912i −0.328333 0.944562i \(-0.606487\pi\)
0.328333 0.944562i \(-0.393513\pi\)
\(464\) 0 0
\(465\) 6.32456 + 14.1421i 0.293294 + 0.655826i
\(466\) 0 0
\(467\) −27.3290 −1.26464 −0.632319 0.774708i \(-0.717897\pi\)
−0.632319 + 0.774708i \(0.717897\pi\)
\(468\) 0 0
\(469\) −2.00000 −0.0923514
\(470\) 0 0
\(471\) −9.53663 21.3246i −0.439425 0.982584i
\(472\) 0 0
\(473\) 37.2285i 1.71177i
\(474\) 0 0
\(475\) 2.51317i 0.115312i
\(476\) 0 0
\(477\) −2.64911 2.36944i −0.121294 0.108489i
\(478\) 0 0
\(479\) −5.65685 −0.258468 −0.129234 0.991614i \(-0.541252\pi\)
−0.129234 + 0.991614i \(0.541252\pi\)
\(480\) 0 0
\(481\) −30.9737 −1.41228
\(482\) 0 0
\(483\) 8.94427 4.00000i 0.406978 0.182006i
\(484\) 0 0
\(485\) 15.0601i 0.683845i
\(486\) 0 0
\(487\) 23.2982i 1.05574i −0.849324 0.527872i \(-0.822990\pi\)
0.849324 0.527872i \(-0.177010\pi\)
\(488\) 0 0
\(489\) −6.83772 + 3.05792i −0.309212 + 0.138284i
\(490\) 0 0
\(491\) −29.9280 −1.35063 −0.675315 0.737529i \(-0.735992\pi\)
−0.675315 + 0.737529i \(0.735992\pi\)
\(492\) 0 0
\(493\) −12.0000 −0.540453
\(494\) 0 0
\(495\) 14.1421 + 12.6491i 0.635642 + 0.568535i
\(496\) 0 0
\(497\) 10.1290i 0.454347i
\(498\) 0 0
\(499\) 2.00000i 0.0895323i 0.998997 + 0.0447661i \(0.0142543\pi\)
−0.998997 + 0.0447661i \(0.985746\pi\)
\(500\) 0 0
\(501\) 0.324555 + 0.725728i 0.0145001 + 0.0324231i
\(502\) 0 0
\(503\) 11.7727 0.524919 0.262459 0.964943i \(-0.415466\pi\)
0.262459 + 0.964943i \(0.415466\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) 27.0809 + 60.5548i 1.20271 + 2.68933i
\(508\) 0 0
\(509\) 4.70163i 0.208396i 0.994557 + 0.104198i \(0.0332276\pi\)
−0.994557 + 0.104198i \(0.966772\pi\)
\(510\) 0 0
\(511\) 4.32456i 0.191307i
\(512\) 0 0
\(513\) −1.32456 + 4.14651i −0.0584805 + 0.183073i
\(514\) 0 0
\(515\) −11.3137 −0.498542
\(516\) 0 0
\(517\) 40.0000 1.75920
\(518\) 0 0
\(519\) −5.98248 + 2.67544i −0.262602 + 0.117439i
\(520\) 0 0
\(521\) 2.10270i 0.0921209i −0.998939 0.0460605i \(-0.985333\pi\)
0.998939 0.0460605i \(-0.0146667\pi\)
\(522\) 0 0
\(523\) 39.8114i 1.74083i 0.492318 + 0.870415i \(0.336150\pi\)
−0.492318 + 0.870415i \(0.663850\pi\)
\(524\) 0 0
\(525\) 4.74342 2.12132i 0.207020 0.0925820i
\(526\) 0 0
\(527\) 46.1728 2.01132
\(528\) 0 0
\(529\) 9.00000 0.391304
\(530\) 0 0
\(531\) 2.82843 3.16228i 0.122743 0.137231i
\(532\) 0 0
\(533\) 72.5466i 3.14234i
\(534\) 0 0
\(535\) 13.6754i 0.591241i
\(536\) 0 0
\(537\) −13.8114 30.8832i −0.596005 1.33271i
\(538\) 0 0
\(539\) 4.47214 0.192629
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 12.8240 + 28.6754i 0.550333 + 1.23058i
\(544\) 0 0
\(545\) 6.11584i 0.261974i
\(546\) 0 0
\(547\) 36.3246i 1.55313i 0.630040 + 0.776563i \(0.283039\pi\)
−0.630040 + 0.776563i \(0.716961\pi\)
\(548\) 0 0
\(549\) −6.32456 + 7.07107i −0.269925 + 0.301786i
\(550\) 0 0
\(551\) 1.37697 0.0586610
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 0 0
\(555\) −9.67000 + 4.32456i −0.410469 + 0.183567i
\(556\) 0 0
\(557\) 32.7564i 1.38793i 0.720007 + 0.693967i \(0.244139\pi\)
−0.720007 + 0.693967i \(0.755861\pi\)
\(558\) 0 0
\(559\) 59.6228i 2.52177i
\(560\) 0 0
\(561\) 51.6228 23.0864i 2.17952 0.974709i
\(562\) 0 0
\(563\) 22.1312 0.932718 0.466359 0.884595i \(-0.345565\pi\)
0.466359 + 0.884595i \(0.345565\pi\)
\(564\) 0 0
\(565\) −9.67544 −0.407049
\(566\) 0 0
\(567\) −8.94427 + 1.00000i −0.375624 + 0.0419961i
\(568\) 0 0
\(569\) 29.4690i 1.23540i −0.786412 0.617702i \(-0.788064\pi\)
0.786412 0.617702i \(-0.211936\pi\)
\(570\) 0 0
\(571\) 12.3246i 0.515767i −0.966176 0.257883i \(-0.916975\pi\)
0.966176 0.257883i \(-0.0830250\pi\)
\(572\) 0 0
\(573\) −8.83772 19.7617i −0.369201 0.825559i
\(574\) 0 0
\(575\) 16.9706 0.707721
\(576\) 0 0
\(577\) 35.2982 1.46948 0.734742 0.678347i \(-0.237303\pi\)
0.734742 + 0.678347i \(0.237303\pi\)
\(578\) 0 0
\(579\) 11.3137 + 25.2982i 0.470182 + 1.05136i
\(580\) 0 0
\(581\) 7.53006i 0.312399i
\(582\) 0 0
\(583\) 5.29822i 0.219430i
\(584\) 0 0
\(585\) 22.6491 + 20.2580i 0.936425 + 0.837564i
\(586\) 0 0
\(587\) −9.44050 −0.389651 −0.194826 0.980838i \(-0.562414\pi\)
−0.194826 + 0.980838i \(0.562414\pi\)
\(588\) 0 0
\(589\) −5.29822 −0.218309
\(590\) 0 0
\(591\) −16.0153 + 7.16228i −0.658783 + 0.294617i
\(592\) 0 0
\(593\) 24.2711i 0.996696i 0.866977 + 0.498348i \(0.166060\pi\)
−0.866977 + 0.498348i \(0.833940\pi\)
\(594\) 0 0
\(595\) 10.3246i 0.423266i
\(596\) 0 0
\(597\) −38.9737 + 17.4296i −1.59509 + 0.713344i
\(598\) 0 0
\(599\) 24.7301 1.01045 0.505223 0.862989i \(-0.331410\pi\)
0.505223 + 0.862989i \(0.331410\pi\)
\(600\) 0 0
\(601\) −29.6228 −1.20834 −0.604169 0.796856i \(-0.706495\pi\)
−0.604169 + 0.796856i \(0.706495\pi\)
\(602\) 0 0
\(603\) 4.47214 + 4.00000i 0.182119 + 0.162893i
\(604\) 0 0
\(605\) 12.7279i 0.517464i
\(606\) 0 0
\(607\) 17.6754i 0.717424i −0.933448 0.358712i \(-0.883216\pi\)
0.933448 0.358712i \(-0.116784\pi\)
\(608\) 0 0
\(609\) 1.16228 + 2.59893i 0.0470979 + 0.105314i
\(610\) 0 0
\(611\) 64.0614 2.59165
\(612\) 0 0
\(613\) 19.6754 0.794684 0.397342 0.917671i \(-0.369933\pi\)
0.397342 + 0.917671i \(0.369933\pi\)
\(614\) 0 0
\(615\) −10.1290 22.6491i −0.408440 0.913300i
\(616\) 0 0
\(617\) 27.0996i 1.09099i 0.838115 + 0.545493i \(0.183658\pi\)
−0.838115 + 0.545493i \(0.816342\pi\)
\(618\) 0 0
\(619\) 8.18861i 0.329128i −0.986366 0.164564i \(-0.947378\pi\)
0.986366 0.164564i \(-0.0526217\pi\)
\(620\) 0 0
\(621\) −28.0000 8.94427i −1.12360 0.358921i
\(622\) 0 0
\(623\) 1.18472 0.0474647
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) −5.92359 + 2.64911i −0.236565 + 0.105795i
\(628\) 0 0
\(629\) 31.5717i 1.25885i
\(630\) 0 0
\(631\) 37.9473i 1.51066i 0.655345 + 0.755330i \(0.272523\pi\)
−0.655345 + 0.755330i \(0.727477\pi\)
\(632\) 0 0
\(633\) −2.13594 + 0.955223i −0.0848961 + 0.0379667i
\(634\) 0 0
\(635\) −8.48528 −0.336728
\(636\) 0 0
\(637\) 7.16228 0.283780
\(638\) 0 0
\(639\) −20.2580 + 22.6491i −0.801393 + 0.895985i
\(640\) 0 0
\(641\) 12.4984i 0.493658i −0.969059 0.246829i \(-0.920611\pi\)
0.969059 0.246829i \(-0.0793886\pi\)
\(642\) 0 0
\(643\) 4.83772i 0.190781i 0.995440 + 0.0953906i \(0.0304100\pi\)
−0.995440 + 0.0953906i \(0.969590\pi\)
\(644\) 0 0
\(645\) −8.32456 18.6143i −0.327779 0.732936i
\(646\) 0 0
\(647\) 13.2242 0.519895 0.259948 0.965623i \(-0.416295\pi\)
0.259948 + 0.965623i \(0.416295\pi\)
\(648\) 0 0
\(649\) 6.32456 0.248261
\(650\) 0 0
\(651\) −4.47214 10.0000i −0.175277 0.391931i
\(652\) 0 0
\(653\) 47.8165i 1.87121i −0.353055 0.935603i \(-0.614857\pi\)
0.353055 0.935603i \(-0.385143\pi\)
\(654\) 0 0
\(655\) 6.64911i 0.259802i
\(656\) 0 0
\(657\) −8.64911 + 9.67000i −0.337434 + 0.377263i
\(658\) 0 0
\(659\) −16.7038 −0.650689 −0.325344 0.945596i \(-0.605480\pi\)
−0.325344 + 0.945596i \(0.605480\pi\)
\(660\) 0 0
\(661\) 22.5132 0.875661 0.437830 0.899058i \(-0.355747\pi\)
0.437830 + 0.899058i \(0.355747\pi\)
\(662\) 0 0
\(663\) 82.6756 36.9737i 3.21086 1.43594i
\(664\) 0 0
\(665\) 1.18472i 0.0459414i
\(666\) 0 0
\(667\) 9.29822i 0.360029i
\(668\) 0 0
\(669\) −3.67544 + 1.64371i −0.142101 + 0.0635495i
\(670\) 0 0
\(671\) −14.1421 −0.545951
\(672\) 0 0
\(673\) 27.2982 1.05227 0.526135 0.850401i \(-0.323641\pi\)
0.526135 + 0.850401i \(0.323641\pi\)
\(674\) 0 0
\(675\) −14.8492 4.74342i −0.571548 0.182574i
\(676\) 0 0
\(677\) 8.44804i 0.324685i −0.986735 0.162342i \(-0.948095\pi\)
0.986735 0.162342i \(-0.0519049\pi\)
\(678\) 0 0
\(679\) 10.6491i 0.408675i
\(680\) 0 0
\(681\) 9.32456 + 20.8503i 0.357318 + 0.798987i
\(682\) 0 0
\(683\) 51.1039 1.95544 0.977719 0.209918i \(-0.0673197\pi\)
0.977719 + 0.209918i \(0.0673197\pi\)
\(684\) 0 0
\(685\) 25.2982 0.966595
\(686\) 0 0
\(687\) 2.69506 + 6.02633i 0.102823 + 0.229919i
\(688\) 0 0
\(689\) 8.48528i 0.323263i
\(690\) 0 0
\(691\) 15.1623i 0.576800i 0.957510 + 0.288400i \(0.0931233\pi\)
−0.957510 + 0.288400i \(0.906877\pi\)
\(692\) 0 0
\(693\) −10.0000 8.94427i −0.379869 0.339765i
\(694\) 0 0
\(695\) −10.1290 −0.384214
\(696\) 0 0
\(697\) −73.9473 −2.80095
\(698\) 0 0
\(699\) −8.94427 + 4.00000i −0.338303 + 0.151294i
\(700\) 0 0
\(701\) 26.6406i 1.00620i 0.864228 + 0.503100i \(0.167807\pi\)
−0.864228 + 0.503100i \(0.832193\pi\)
\(702\) 0 0
\(703\) 3.62278i 0.136636i
\(704\) 0 0
\(705\) 20.0000 8.94427i 0.753244 0.336861i
\(706\) 0 0
\(707\) 1.41421 0.0531870
\(708\) 0 0
\(709\) −3.02633 −0.113656 −0.0568282 0.998384i \(-0.518099\pi\)
−0.0568282 + 0.998384i \(0.518099\pi\)
\(710\) 0 0
\(711\) 8.94427 + 8.00000i 0.335436 + 0.300023i
\(712\) 0 0
\(713\) 35.7771i 1.33986i
\(714\) 0 0
\(715\) 45.2982i 1.69406i
\(716\) 0 0
\(717\) −18.3246 40.9750i −0.684343 1.53024i
\(718\) 0 0
\(719\) 12.2317 0.456165 0.228083 0.973642i \(-0.426754\pi\)
0.228083 + 0.973642i \(0.426754\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) 9.89949 + 22.1359i 0.368166 + 0.823245i
\(724\) 0 0
\(725\) 4.93113i 0.183137i
\(726\) 0 0
\(727\) 16.6491i 0.617481i 0.951146 + 0.308741i \(0.0999075\pi\)
−0.951146 + 0.308741i \(0.900092\pi\)
\(728\) 0 0
\(729\) 22.0000 + 15.6525i 0.814815 + 0.579721i
\(730\) 0 0
\(731\) −60.7739 −2.24781
\(732\) 0 0
\(733\) 46.1359 1.70407 0.852035 0.523485i \(-0.175368\pi\)
0.852035 + 0.523485i \(0.175368\pi\)
\(734\) 0 0
\(735\) 2.23607 1.00000i 0.0824786 0.0368856i
\(736\) 0 0
\(737\) 8.94427i 0.329466i
\(738\) 0 0
\(739\) 32.9737i 1.21296i 0.795100 + 0.606478i \(0.207418\pi\)
−0.795100 + 0.606478i \(0.792582\pi\)
\(740\) 0 0
\(741\) −9.48683 + 4.24264i −0.348508 + 0.155857i
\(742\) 0 0
\(743\) 23.5454 0.863797 0.431898 0.901922i \(-0.357844\pi\)
0.431898 + 0.901922i \(0.357844\pi\)
\(744\) 0 0
\(745\) 6.32456 0.231714
\(746\) 0 0
\(747\) −15.0601 + 16.8377i −0.551021 + 0.616060i
\(748\) 0 0
\(749\) 9.67000i 0.353334i
\(750\) 0 0
\(751\) 17.3509i 0.633143i 0.948569 + 0.316571i \(0.102532\pi\)
−0.948569 + 0.316571i \(0.897468\pi\)
\(752\) 0 0
\(753\) 13.3246 + 29.7946i 0.485574 + 1.08578i
\(754\) 0 0
\(755\) 14.1421 0.514685
\(756\) 0 0
\(757\) −20.3246 −0.738709 −0.369354 0.929289i \(-0.620421\pi\)
−0.369354 + 0.929289i \(0.620421\pi\)
\(758\) 0 0
\(759\) −17.8885 40.0000i −0.649313 1.45191i
\(760\) 0 0
\(761\) 4.47214i 0.162115i −0.996709 0.0810574i \(-0.974170\pi\)
0.996709 0.0810574i \(-0.0258297\pi\)
\(762\) 0 0
\(763\) 4.32456i 0.156559i
\(764\) 0 0
\(765\) 20.6491 23.0864i 0.746570 0.834691i
\(766\) 0 0
\(767\) 10.1290 0.365737
\(768\) 0 0
\(769\) −27.2982 −0.984399 −0.492200 0.870482i \(-0.663807\pi\)
−0.492200 + 0.870482i \(0.663807\pi\)
\(770\) 0 0
\(771\) 25.6853 11.4868i 0.925035 0.413688i
\(772\) 0 0
\(773\) 45.6766i 1.64287i −0.570300 0.821436i \(-0.693173\pi\)
0.570300 0.821436i \(-0.306827\pi\)
\(774\) 0 0
\(775\) 18.9737i 0.681554i
\(776\) 0 0
\(777\) 6.83772 3.05792i 0.245302 0.109702i
\(778\) 0 0
\(779\) 8.48528 0.304017
\(780\) 0 0
\(781\) −45.2982 −1.62090
\(782\) 0 0
\(783\) 2.59893 8.13594i 0.0928782 0.290755i
\(784\) 0 0
\(785\) 19.0733i 0.680754i
\(786\) 0 0
\(787\) 39.1623i 1.39598i −0.716105 0.697992i \(-0.754077\pi\)
0.716105 0.697992i \(-0.245923\pi\)
\(788\) 0 0
\(789\) −1.48683 3.32466i −0.0529327 0.118361i
\(790\) 0 0
\(791\) 6.84157 0.243258
\(792\) 0 0
\(793\) −22.6491 −0.804294
\(794\) 0 0
\(795\) −1.18472 2.64911i −0.0420176 0.0939543i
\(796\) 0 0
\(797\) 17.9258i 0.634964i 0.948264 + 0.317482i \(0.102837\pi\)
−0.948264 + 0.317482i \(0.897163\pi\)
\(798\) 0 0
\(799\) 65.2982i 2.31009i
\(800\) 0 0
\(801\) −2.64911 2.36944i −0.0936017 0.0837199i
\(802\) 0 0
\(803\) −19.3400 −0.682494
\(804\) 0 0
\(805\) 8.00000 0.281963
\(806\) 0 0
\(807\) 10.4546 4.67544i 0.368020 0.164583i
\(808\) 0 0
\(809\) 44.9881i 1.58170i 0.612012 + 0.790849i \(0.290360\pi\)
−0.612012 + 0.790849i \(0.709640\pi\)
\(810\) 0 0
\(811\) 33.4868i 1.17588i 0.808904 + 0.587941i \(0.200061\pi\)
−0.808904 + 0.587941i \(0.799939\pi\)
\(812\) 0 0
\(813\) 2.64911 1.18472i 0.0929084 0.0415499i
\(814\) 0 0
\(815\) −6.11584 −0.214229
\(816\) 0 0
\(817\) 6.97367 0.243978
\(818\) 0 0
\(819\) −16.0153 14.3246i −0.559621 0.500540i
\(820\) 0 0
\(821\) 22.3607i 0.780393i 0.920732 + 0.390197i \(0.127593\pi\)
−0.920732 + 0.390197i \(0.872407\pi\)
\(822\) 0 0
\(823\) 21.9473i 0.765036i −0.923948 0.382518i \(-0.875057\pi\)
0.923948 0.382518i \(-0.124943\pi\)
\(824\) 0 0
\(825\) −9.48683 21.2132i −0.330289 0.738549i
\(826\) 0 0
\(827\) −4.93113 −0.171472 −0.0857360 0.996318i \(-0.527324\pi\)
−0.0857360 + 0.996318i \(0.527324\pi\)
\(828\) 0 0
\(829\) 35.1623 1.22124 0.610618 0.791925i \(-0.290921\pi\)
0.610618 + 0.791925i \(0.290921\pi\)
\(830\) 0 0
\(831\) 1.41421 + 3.16228i 0.0490585 + 0.109698i
\(832\) 0 0
\(833\) 7.30056i 0.252950i
\(834\) 0 0
\(835\) 0.649111i 0.0224634i
\(836\) 0 0
\(837\) −10.0000 + 31.3050i −0.345651 + 1.08206i
\(838\) 0 0
\(839\) −49.0012 −1.69171 −0.845855 0.533412i \(-0.820909\pi\)
−0.845855 + 0.533412i \(0.820909\pi\)
\(840\) 0 0
\(841\) 26.2982 0.906835
\(842\) 0 0
\(843\) −37.2285 + 16.6491i −1.28222 + 0.573426i
\(844\) 0 0
\(845\) 54.1619i 1.86322i
\(846\) 0 0
\(847\) 9.00000i 0.309244i
\(848\) 0 0
\(849\) 10.2982 4.60550i 0.353434 0.158061i
\(850\) 0 0
\(851\) 24.4634 0.838594
\(852\) 0 0
\(853\) −26.1359 −0.894878 −0.447439 0.894315i \(-0.647664\pi\)
−0.447439 + 0.894315i \(0.647664\pi\)
\(854\) 0 0
\(855\) −2.36944 + 2.64911i −0.0810330 + 0.0905977i
\(856\) 0 0
\(857\) 19.0733i 0.651530i −0.945451 0.325765i \(-0.894378\pi\)
0.945451 0.325765i \(-0.105622\pi\)
\(858\) 0 0
\(859\) 48.4605i 1.65345i −0.562606 0.826725i \(-0.690201\pi\)
0.562606 0.826725i \(-0.309799\pi\)
\(860\) 0 0
\(861\) 7.16228 + 16.0153i 0.244090 + 0.545801i
\(862\) 0 0
\(863\) −11.0470 −0.376043 −0.188022 0.982165i \(-0.560208\pi\)
−0.188022 + 0.982165i \(0.560208\pi\)
\(864\) 0 0
\(865\) −5.35089 −0.181936
\(866\) 0 0
\(867\) −25.6667 57.3925i −0.871687 1.94915i
\(868\) 0 0
\(869\) 17.8885i 0.606827i
\(870\) 0 0
\(871\) 14.3246i 0.485369i
\(872\) 0 0
\(873\) 21.2982 23.8121i 0.720836 0.805919i
\(874\) 0 0
\(875\) 11.3137 0.382473
\(876\) 0 0
\(877\) 40.9737 1.38358 0.691791 0.722098i \(-0.256822\pi\)
0.691791 + 0.722098i \(0.256822\pi\)
\(878\) 0 0
\(879\) −34.9925 + 15.6491i −1.18027 + 0.527831i
\(880\) 0 0
\(881\) 24.2711i 0.817715i 0.912598 + 0.408858i \(0.134073\pi\)
−0.912598 + 0.408858i \(0.865927\pi\)
\(882\) 0 0
\(883\) 18.0000i 0.605748i −0.953031 0.302874i \(-0.902054\pi\)
0.953031 0.302874i \(-0.0979462\pi\)
\(884\) 0 0
\(885\) 3.16228 1.41421i 0.106299 0.0475383i
\(886\) 0 0
\(887\) 16.5116 0.554404 0.277202 0.960812i \(-0.410593\pi\)
0.277202 + 0.960812i \(0.410593\pi\)
\(888\) 0 0
\(889\) 6.00000 0.201234
\(890\) 0 0
\(891\) 4.47214 + 40.0000i 0.149822 + 1.34005i
\(892\) 0 0
\(893\) 7.49282i 0.250738i
\(894\) 0 0
\(895\) 27.6228i 0.923328i
\(896\) 0 0
\(897\) −28.6491 64.0614i −0.956566 2.13895i
\(898\) 0 0
\(899\) 10.3957 0.346717
\(900\) 0 0
\(901\) −8.64911 −0.288144
\(902\) 0 0
\(903\) 5.88635 + 13.1623i 0.195885 + 0.438013i
\(904\) 0 0
\(905\) 25.6481i 0.852572i
\(906\) 0 0
\(907\) 27.9473i 0.927976i −0.885842 0.463988i \(-0.846418\pi\)
0.885842 0.463988i \(-0.153582\pi\)
\(908\) 0 0
\(909\) −3.16228 2.82843i −0.104886 0.0938130i
\(910\) 0 0
\(911\) 13.6831 0.453343 0.226671 0.973971i \(-0.427216\pi\)
0.226671 + 0.973971i \(0.427216\pi\)
\(912\) 0 0
\(913\) −33.6754 −1.11449
\(914\) 0 0
\(915\) −7.07107 + 3.16228i −0.233762 + 0.104542i
\(916\) 0 0
\(917\) 4.70163i 0.155262i
\(918\) 0 0
\(919\) 20.6491i 0.681151i 0.940217 + 0.340576i \(0.110622\pi\)
−0.940217 + 0.340576i \(0.889378\pi\)
\(920\) 0 0
\(921\) −3.97367 + 1.77708i −0.130937 + 0.0585567i
\(922\) 0 0
\(923\) −72.5466 −2.38790
\(924\) 0 0
\(925\) 12.9737 0.426572
\(926\) 0 0
\(927\) −17.8885 16.0000i −0.587537 0.525509i
\(928\) 0 0
\(929\) 43.0777i 1.41333i 0.707547 + 0.706666i \(0.249802\pi\)
−0.707547 + 0.706666i \(0.750198\pi\)
\(930\) 0 0
\(931\) 0.837722i 0.0274552i
\(932\) 0 0
\(933\) −12.3246 27.5585i −0.403488 0.902226i
\(934\) 0 0
\(935\) 46.1728 1.51001
\(936\) 0 0
\(937\) −7.67544 −0.250746 −0.125373 0.992110i \(-0.540013\pi\)
−0.125373 + 0.992110i \(0.540013\pi\)
\(938\) 0 0
\(939\) −3.51691 7.86406i −0.114770 0.256634i
\(940\) 0 0
\(941\) 23.0492i 0.751381i −0.926745 0.375691i \(-0.877406\pi\)
0.926745 0.375691i \(-0.122594\pi\)
\(942\) 0 0
\(943\) 57.2982i 1.86589i
\(944\) 0 0
\(945\) −7.00000 2.23607i −0.227710 0.0727393i
\(946\) 0 0
\(947\) −4.47214 −0.145325 −0.0726624 0.997357i \(-0.523150\pi\)
−0.0726624 + 0.997357i \(0.523150\pi\)
\(948\) 0 0
\(949\) −30.9737 −1.00545
\(950\) 0 0
\(951\) −12.2689 + 5.48683i −0.397847 + 0.177923i
\(952\) 0 0
\(953\) 26.8328i 0.869200i −0.900624 0.434600i \(-0.856890\pi\)
0.900624 0.434600i \(-0.143110\pi\)
\(954\) 0 0
\(955\) 17.6754i 0.571964i
\(956\) 0 0
\(957\) 11.6228 5.19786i 0.375711 0.168023i
\(958\) 0 0
\(959\) −17.8885 −0.577651
\(960\) 0 0
\(961\) −9.00000 −0.290323
\(962\) 0 0
\(963\) −19.3400 + 21.6228i −0.623223 + 0.696784i
\(964\) 0 0
\(965\) 22.6274i 0.728402i
\(966\) 0 0
\(967\) 34.6491i 1.11424i −0.830432 0.557120i \(-0.811906\pi\)
0.830432 0.557120i \(-0.188094\pi\)
\(968\) 0 0
\(969\) 4.32456 + 9.67000i 0.138925 + 0.310645i
\(970\) 0 0
\(971\) 43.8406 1.40691 0.703456 0.710739i \(-0.251639\pi\)
0.703456 + 0.710739i \(0.251639\pi\)
\(972\) 0 0
\(973\) 7.16228 0.229612
\(974\) 0 0
\(975\) −15.1935 33.9737i −0.486581 1.08803i
\(976\) 0 0
\(977\) 49.4602i 1.58237i 0.611575 + 0.791187i \(0.290536\pi\)
−0.611575 + 0.791187i \(0.709464\pi\)
\(978\) 0 0
\(979\) 5.29822i 0.169332i
\(980\) 0 0
\(981\) 8.64911 9.67000i 0.276145 0.308739i
\(982\) 0 0
\(983\) −16.5116 −0.526637 −0.263319 0.964709i \(-0.584817\pi\)
−0.263319 + 0.964709i \(0.584817\pi\)
\(984\) 0 0
\(985\) −14.3246 −0.456418
\(986\) 0 0
\(987\) −14.1421 + 6.32456i −0.450149 + 0.201313i
\(988\) 0 0
\(989\) 47.0908i 1.49740i
\(990\) 0 0
\(991\) 46.5964i 1.48018i 0.672505 + 0.740092i \(0.265218\pi\)
−0.672505 + 0.740092i \(0.734782\pi\)
\(992\) 0 0
\(993\) 45.8114 20.4875i 1.45378 0.650150i
\(994\) 0 0
\(995\) −34.8591 −1.10511
\(996\) 0 0
\(997\) −25.4868 −0.807176 −0.403588 0.914941i \(-0.632237\pi\)
−0.403588 + 0.914941i \(0.632237\pi\)
\(998\) 0 0
\(999\) −21.4055 6.83772i −0.677239 0.216336i
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 672.2.h.e.575.8 yes 8
3.2 odd 2 inner 672.2.h.e.575.3 yes 8
4.3 odd 2 inner 672.2.h.e.575.2 8
8.3 odd 2 1344.2.h.f.575.7 8
8.5 even 2 1344.2.h.f.575.1 8
12.11 even 2 inner 672.2.h.e.575.5 yes 8
24.5 odd 2 1344.2.h.f.575.6 8
24.11 even 2 1344.2.h.f.575.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.2.h.e.575.2 8 4.3 odd 2 inner
672.2.h.e.575.3 yes 8 3.2 odd 2 inner
672.2.h.e.575.5 yes 8 12.11 even 2 inner
672.2.h.e.575.8 yes 8 1.1 even 1 trivial
1344.2.h.f.575.1 8 8.5 even 2
1344.2.h.f.575.4 8 24.11 even 2
1344.2.h.f.575.6 8 24.5 odd 2
1344.2.h.f.575.7 8 8.3 odd 2