Properties

Label 224.3.o.c.79.2
Level $224$
Weight $3$
Character 224.79
Analytic conductor $6.104$
Analytic rank $0$
Dimension $12$
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] = [224,3,Mod(79,224)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(224, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("224.79");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 224.o (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: \(6.10355792167\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 2x^{10} - 12x^{9} + 12x^{8} - 12x^{7} + 148x^{6} - 48x^{5} + 192x^{4} - 768x^{3} + 512x^{2} + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 79.2
Root \(0.0421483 + 1.99956i\) of defining polynomial
Character \(\chi\) \(=\) 224.79
Dual form 224.3.o.c.207.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.25274 - 3.90186i) q^{3} +(6.07099 + 3.50509i) q^{5} +(2.51181 + 6.53382i) q^{7} +(-5.64968 + 9.78553i) q^{9} +O(q^{10})\) \(q+(-2.25274 - 3.90186i) q^{3} +(6.07099 + 3.50509i) q^{5} +(2.51181 + 6.53382i) q^{7} +(-5.64968 + 9.78553i) q^{9} +(7.90242 + 13.6874i) q^{11} -2.90039i q^{13} -31.5842i q^{15} +(1.65516 + 2.86682i) q^{17} +(8.10854 - 14.0444i) q^{19} +(19.8356 - 24.5197i) q^{21} +(16.4804 + 9.51498i) q^{23} +(12.0713 + 20.9081i) q^{25} +10.3597 q^{27} -21.1392i q^{29} +(23.6995 - 13.6829i) q^{31} +(35.6042 - 61.6683i) q^{33} +(-7.65245 + 48.4709i) q^{35} +(15.2932 + 8.82951i) q^{37} +(-11.3169 + 6.53382i) q^{39} +1.13482 q^{41} -50.2084 q^{43} +(-68.5983 + 39.6053i) q^{45} +(-0.657646 - 0.379692i) q^{47} +(-36.3816 + 32.8234i) q^{49} +(7.45728 - 12.9164i) q^{51} +(-38.9677 + 22.4980i) q^{53} +110.795i q^{55} -73.0658 q^{57} +(1.19239 + 2.06529i) q^{59} +(86.1653 + 49.7476i) q^{61} +(-78.1278 - 12.3346i) q^{63} +(10.1661 - 17.6082i) q^{65} +(-33.2440 - 57.5804i) q^{67} -85.7391i q^{69} -44.4376i q^{71} +(-0.859703 - 1.48905i) q^{73} +(54.3870 - 94.2011i) q^{75} +(-69.5816 + 86.0131i) q^{77} +(-62.9171 - 36.3252i) q^{79} +(27.5094 + 47.6476i) q^{81} +102.081 q^{83} +23.2059i q^{85} +(-82.4821 + 47.6211i) q^{87} +(30.6865 - 53.1505i) q^{89} +(18.9506 - 7.28522i) q^{91} +(-106.778 - 61.6480i) q^{93} +(98.4538 - 56.8423i) q^{95} -102.826 q^{97} -178.584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{3} - 8 q^{9} + 14 q^{11} - 82 q^{17} + 94 q^{19} + 116 q^{25} + 60 q^{27} + 146 q^{33} - 270 q^{35} + 120 q^{41} - 40 q^{43} - 204 q^{49} + 106 q^{51} - 372 q^{57} - 62 q^{59} - 64 q^{65} + 178 q^{67} + 54 q^{73} - 140 q^{75} + 206 q^{81} + 392 q^{83} - 26 q^{89} + 88 q^{91} - 184 q^{97} - 872 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(-1\)

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
\(3\) −2.25274 3.90186i −0.750913 1.30062i −0.947380 0.320110i \(-0.896280\pi\)
0.196467 0.980510i \(-0.437053\pi\)
\(4\) 0 0
\(5\) 6.07099 + 3.50509i 1.21420 + 0.701018i 0.963671 0.267092i \(-0.0860626\pi\)
0.250528 + 0.968109i \(0.419396\pi\)
\(6\) 0 0
\(7\) 2.51181 + 6.53382i 0.358830 + 0.933403i
\(8\) 0 0
\(9\) −5.64968 + 9.78553i −0.627742 + 1.08728i
\(10\) 0 0
\(11\) 7.90242 + 13.6874i 0.718402 + 1.24431i 0.961633 + 0.274340i \(0.0884593\pi\)
−0.243231 + 0.969968i \(0.578207\pi\)
\(12\) 0 0
\(13\) 2.90039i 0.223107i −0.993758 0.111553i \(-0.964417\pi\)
0.993758 0.111553i \(-0.0355826\pi\)
\(14\) 0 0
\(15\) 31.5842i 2.10562i
\(16\) 0 0
\(17\) 1.65516 + 2.86682i 0.0973623 + 0.168636i 0.910592 0.413306i \(-0.135626\pi\)
−0.813230 + 0.581943i \(0.802293\pi\)
\(18\) 0 0
\(19\) 8.10854 14.0444i 0.426765 0.739179i −0.569818 0.821771i \(-0.692986\pi\)
0.996583 + 0.0825915i \(0.0263197\pi\)
\(20\) 0 0
\(21\) 19.8356 24.5197i 0.944553 1.16761i
\(22\) 0 0
\(23\) 16.4804 + 9.51498i 0.716540 + 0.413695i 0.813478 0.581596i \(-0.197571\pi\)
−0.0969377 + 0.995290i \(0.530905\pi\)
\(24\) 0 0
\(25\) 12.0713 + 20.9081i 0.482852 + 0.836325i
\(26\) 0 0
\(27\) 10.3597 0.383693
\(28\) 0 0
\(29\) 21.1392i 0.728937i −0.931216 0.364468i \(-0.881251\pi\)
0.931216 0.364468i \(-0.118749\pi\)
\(30\) 0 0
\(31\) 23.6995 13.6829i 0.764499 0.441384i −0.0664095 0.997792i \(-0.521154\pi\)
0.830909 + 0.556409i \(0.187821\pi\)
\(32\) 0 0
\(33\) 35.6042 61.6683i 1.07891 1.86874i
\(34\) 0 0
\(35\) −7.65245 + 48.4709i −0.218641 + 1.38488i
\(36\) 0 0
\(37\) 15.2932 + 8.82951i 0.413328 + 0.238635i 0.692219 0.721688i \(-0.256633\pi\)
−0.278890 + 0.960323i \(0.589967\pi\)
\(38\) 0 0
\(39\) −11.3169 + 6.53382i −0.290177 + 0.167534i
\(40\) 0 0
\(41\) 1.13482 0.0276785 0.0138392 0.999904i \(-0.495595\pi\)
0.0138392 + 0.999904i \(0.495595\pi\)
\(42\) 0 0
\(43\) −50.2084 −1.16764 −0.583818 0.811884i \(-0.698442\pi\)
−0.583818 + 0.811884i \(0.698442\pi\)
\(44\) 0 0
\(45\) −68.5983 + 39.6053i −1.52441 + 0.880117i
\(46\) 0 0
\(47\) −0.657646 0.379692i −0.0139925 0.00807855i 0.492987 0.870036i \(-0.335905\pi\)
−0.506980 + 0.861958i \(0.669238\pi\)
\(48\) 0 0
\(49\) −36.3816 + 32.8234i −0.742482 + 0.669866i
\(50\) 0 0
\(51\) 7.45728 12.9164i 0.146221 0.253263i
\(52\) 0 0
\(53\) −38.9677 + 22.4980i −0.735240 + 0.424491i −0.820336 0.571882i \(-0.806214\pi\)
0.0850958 + 0.996373i \(0.472880\pi\)
\(54\) 0 0
\(55\) 110.795i 2.01445i
\(56\) 0 0
\(57\) −73.0658 −1.28186
\(58\) 0 0
\(59\) 1.19239 + 2.06529i 0.0202101 + 0.0350048i 0.875954 0.482395i \(-0.160233\pi\)
−0.855743 + 0.517400i \(0.826900\pi\)
\(60\) 0 0
\(61\) 86.1653 + 49.7476i 1.41255 + 0.815534i 0.995628 0.0934097i \(-0.0297767\pi\)
0.416919 + 0.908944i \(0.363110\pi\)
\(62\) 0 0
\(63\) −78.1278 12.3346i −1.24012 0.195787i
\(64\) 0 0
\(65\) 10.1661 17.6082i 0.156402 0.270896i
\(66\) 0 0
\(67\) −33.2440 57.5804i −0.496180 0.859408i 0.503811 0.863814i \(-0.331931\pi\)
−0.999990 + 0.00440572i \(0.998598\pi\)
\(68\) 0 0
\(69\) 85.7391i 1.24260i
\(70\) 0 0
\(71\) 44.4376i 0.625882i −0.949773 0.312941i \(-0.898686\pi\)
0.949773 0.312941i \(-0.101314\pi\)
\(72\) 0 0
\(73\) −0.859703 1.48905i −0.0117768 0.0203979i 0.860077 0.510164i \(-0.170415\pi\)
−0.871854 + 0.489766i \(0.837082\pi\)
\(74\) 0 0
\(75\) 54.3870 94.2011i 0.725161 1.25602i
\(76\) 0 0
\(77\) −69.5816 + 86.0131i −0.903657 + 1.11705i
\(78\) 0 0
\(79\) −62.9171 36.3252i −0.796418 0.459812i 0.0457989 0.998951i \(-0.485417\pi\)
−0.842217 + 0.539138i \(0.818750\pi\)
\(80\) 0 0
\(81\) 27.5094 + 47.6476i 0.339622 + 0.588243i
\(82\) 0 0
\(83\) 102.081 1.22990 0.614948 0.788568i \(-0.289177\pi\)
0.614948 + 0.788568i \(0.289177\pi\)
\(84\) 0 0
\(85\) 23.2059i 0.273011i
\(86\) 0 0
\(87\) −82.4821 + 47.6211i −0.948070 + 0.547369i
\(88\) 0 0
\(89\) 30.6865 53.1505i 0.344792 0.597197i −0.640524 0.767938i \(-0.721283\pi\)
0.985316 + 0.170741i \(0.0546162\pi\)
\(90\) 0 0
\(91\) 18.9506 7.28522i 0.208249 0.0800574i
\(92\) 0 0
\(93\) −106.778 61.6480i −1.14815 0.662882i
\(94\) 0 0
\(95\) 98.4538 56.8423i 1.03636 0.598340i
\(96\) 0 0
\(97\) −102.826 −1.06006 −0.530030 0.847979i \(-0.677819\pi\)
−0.530030 + 0.847979i \(0.677819\pi\)
\(98\) 0 0
\(99\) −178.584 −1.80388
\(100\) 0 0
\(101\) 157.651 91.0197i 1.56090 0.901185i 0.563731 0.825958i \(-0.309365\pi\)
0.997166 0.0752264i \(-0.0239679\pi\)
\(102\) 0 0
\(103\) 39.7104 + 22.9268i 0.385538 + 0.222591i 0.680225 0.733003i \(-0.261882\pi\)
−0.294687 + 0.955594i \(0.595215\pi\)
\(104\) 0 0
\(105\) 206.366 79.3336i 1.96539 0.755558i
\(106\) 0 0
\(107\) −36.9463 + 63.9928i −0.345292 + 0.598064i −0.985407 0.170216i \(-0.945554\pi\)
0.640115 + 0.768279i \(0.278887\pi\)
\(108\) 0 0
\(109\) −104.724 + 60.4622i −0.960767 + 0.554699i −0.896409 0.443228i \(-0.853833\pi\)
−0.0643577 + 0.997927i \(0.520500\pi\)
\(110\) 0 0
\(111\) 79.5623i 0.716778i
\(112\) 0 0
\(113\) −97.7663 −0.865188 −0.432594 0.901589i \(-0.642402\pi\)
−0.432594 + 0.901589i \(0.642402\pi\)
\(114\) 0 0
\(115\) 66.7017 + 115.531i 0.580015 + 1.00462i
\(116\) 0 0
\(117\) 28.3818 + 16.3863i 0.242580 + 0.140053i
\(118\) 0 0
\(119\) −14.5738 + 18.0154i −0.122469 + 0.151390i
\(120\) 0 0
\(121\) −64.3964 + 111.538i −0.532202 + 0.921801i
\(122\) 0 0
\(123\) −2.55645 4.42790i −0.0207841 0.0359992i
\(124\) 0 0
\(125\) 6.01041i 0.0480833i
\(126\) 0 0
\(127\) 175.050i 1.37835i 0.724595 + 0.689175i \(0.242027\pi\)
−0.724595 + 0.689175i \(0.757973\pi\)
\(128\) 0 0
\(129\) 113.106 + 195.906i 0.876794 + 1.51865i
\(130\) 0 0
\(131\) −4.30993 + 7.46501i −0.0329002 + 0.0569848i −0.882007 0.471237i \(-0.843808\pi\)
0.849106 + 0.528222i \(0.177141\pi\)
\(132\) 0 0
\(133\) 112.131 + 17.7029i 0.843088 + 0.133104i
\(134\) 0 0
\(135\) 62.8937 + 36.3117i 0.465879 + 0.268975i
\(136\) 0 0
\(137\) −128.042 221.776i −0.934615 1.61880i −0.775320 0.631569i \(-0.782411\pi\)
−0.159295 0.987231i \(-0.550922\pi\)
\(138\) 0 0
\(139\) −218.834 −1.57434 −0.787171 0.616735i \(-0.788455\pi\)
−0.787171 + 0.616735i \(0.788455\pi\)
\(140\) 0 0
\(141\) 3.42139i 0.0242652i
\(142\) 0 0
\(143\) 39.6987 22.9201i 0.277614 0.160280i
\(144\) 0 0
\(145\) 74.0947 128.336i 0.510998 0.885074i
\(146\) 0 0
\(147\) 210.031 + 68.0134i 1.42878 + 0.462676i
\(148\) 0 0
\(149\) −20.6792 11.9391i −0.138786 0.0801284i 0.428999 0.903305i \(-0.358866\pi\)
−0.567785 + 0.823177i \(0.692200\pi\)
\(150\) 0 0
\(151\) 158.916 91.7504i 1.05243 0.607619i 0.129099 0.991632i \(-0.458792\pi\)
0.923328 + 0.384013i \(0.125458\pi\)
\(152\) 0 0
\(153\) −37.4044 −0.244474
\(154\) 0 0
\(155\) 191.839 1.23767
\(156\) 0 0
\(157\) −143.703 + 82.9671i −0.915307 + 0.528453i −0.882135 0.470997i \(-0.843894\pi\)
−0.0331720 + 0.999450i \(0.510561\pi\)
\(158\) 0 0
\(159\) 175.568 + 101.364i 1.10420 + 0.637512i
\(160\) 0 0
\(161\) −20.7735 + 131.580i −0.129028 + 0.817267i
\(162\) 0 0
\(163\) −5.05216 + 8.75059i −0.0309948 + 0.0536846i −0.881107 0.472917i \(-0.843201\pi\)
0.850112 + 0.526602i \(0.176534\pi\)
\(164\) 0 0
\(165\) 432.306 249.592i 2.62003 1.51268i
\(166\) 0 0
\(167\) 285.049i 1.70688i −0.521190 0.853441i \(-0.674512\pi\)
0.521190 0.853441i \(-0.325488\pi\)
\(168\) 0 0
\(169\) 160.588 0.950223
\(170\) 0 0
\(171\) 91.6213 + 158.693i 0.535797 + 0.928028i
\(172\) 0 0
\(173\) −84.3266 48.6860i −0.487437 0.281422i 0.236074 0.971735i \(-0.424139\pi\)
−0.723511 + 0.690313i \(0.757473\pi\)
\(174\) 0 0
\(175\) −106.289 + 131.389i −0.607366 + 0.750794i
\(176\) 0 0
\(177\) 5.37230 9.30510i 0.0303520 0.0525712i
\(178\) 0 0
\(179\) −150.908 261.380i −0.843061 1.46022i −0.887295 0.461203i \(-0.847418\pi\)
0.0442343 0.999021i \(-0.485915\pi\)
\(180\) 0 0
\(181\) 60.4535i 0.333997i 0.985957 + 0.166999i \(0.0534076\pi\)
−0.985957 + 0.166999i \(0.946592\pi\)
\(182\) 0 0
\(183\) 448.273i 2.44958i
\(184\) 0 0
\(185\) 61.8964 + 107.208i 0.334575 + 0.579501i
\(186\) 0 0
\(187\) −26.1595 + 45.3096i −0.139890 + 0.242297i
\(188\) 0 0
\(189\) 26.0216 + 67.6884i 0.137680 + 0.358140i
\(190\) 0 0
\(191\) 63.5769 + 36.7062i 0.332864 + 0.192179i 0.657112 0.753793i \(-0.271778\pi\)
−0.324248 + 0.945972i \(0.605111\pi\)
\(192\) 0 0
\(193\) −32.5446 56.3690i −0.168625 0.292067i 0.769312 0.638874i \(-0.220599\pi\)
−0.937937 + 0.346807i \(0.887266\pi\)
\(194\) 0 0
\(195\) −91.6065 −0.469777
\(196\) 0 0
\(197\) 348.324i 1.76814i −0.467355 0.884070i \(-0.654793\pi\)
0.467355 0.884070i \(-0.345207\pi\)
\(198\) 0 0
\(199\) −225.265 + 130.057i −1.13198 + 0.653551i −0.944433 0.328704i \(-0.893388\pi\)
−0.187550 + 0.982255i \(0.560055\pi\)
\(200\) 0 0
\(201\) −149.780 + 259.427i −0.745176 + 1.29068i
\(202\) 0 0
\(203\) 138.120 53.0976i 0.680392 0.261564i
\(204\) 0 0
\(205\) 6.88946 + 3.97763i 0.0336071 + 0.0194031i
\(206\) 0 0
\(207\) −186.218 + 107.513i −0.899605 + 0.519387i
\(208\) 0 0
\(209\) 256.308 1.22636
\(210\) 0 0
\(211\) −4.73025 −0.0224182 −0.0112091 0.999937i \(-0.503568\pi\)
−0.0112091 + 0.999937i \(0.503568\pi\)
\(212\) 0 0
\(213\) −173.390 + 100.106i −0.814035 + 0.469983i
\(214\) 0 0
\(215\) −304.815 175.985i −1.41774 0.818534i
\(216\) 0 0
\(217\) 148.930 + 120.479i 0.686314 + 0.555204i
\(218\) 0 0
\(219\) −3.87337 + 6.70888i −0.0176866 + 0.0306342i
\(220\) 0 0
\(221\) 8.31489 4.80060i 0.0376239 0.0217222i
\(222\) 0 0
\(223\) 109.315i 0.490200i 0.969498 + 0.245100i \(0.0788209\pi\)
−0.969498 + 0.245100i \(0.921179\pi\)
\(224\) 0 0
\(225\) −272.796 −1.21243
\(226\) 0 0
\(227\) −60.7205 105.171i −0.267491 0.463308i 0.700722 0.713434i \(-0.252861\pi\)
−0.968213 + 0.250126i \(0.919528\pi\)
\(228\) 0 0
\(229\) 124.878 + 72.0983i 0.545319 + 0.314840i 0.747232 0.664564i \(-0.231383\pi\)
−0.201913 + 0.979403i \(0.564716\pi\)
\(230\) 0 0
\(231\) 492.360 + 77.7325i 2.13143 + 0.336504i
\(232\) 0 0
\(233\) −97.1309 + 168.236i −0.416871 + 0.722042i −0.995623 0.0934621i \(-0.970207\pi\)
0.578752 + 0.815504i \(0.303540\pi\)
\(234\) 0 0
\(235\) −2.66171 4.61021i −0.0113264 0.0196179i
\(236\) 0 0
\(237\) 327.325i 1.38112i
\(238\) 0 0
\(239\) 315.567i 1.32036i 0.751106 + 0.660181i \(0.229521\pi\)
−0.751106 + 0.660181i \(0.770479\pi\)
\(240\) 0 0
\(241\) −181.356 314.117i −0.752513 1.30339i −0.946601 0.322406i \(-0.895508\pi\)
0.194089 0.980984i \(-0.437825\pi\)
\(242\) 0 0
\(243\) 170.562 295.421i 0.701900 1.21573i
\(244\) 0 0
\(245\) −335.922 + 71.7499i −1.37111 + 0.292857i
\(246\) 0 0
\(247\) −40.7342 23.5179i −0.164916 0.0952143i
\(248\) 0 0
\(249\) −229.963 398.307i −0.923545 1.59963i
\(250\) 0 0
\(251\) 9.04237 0.0360254 0.0180127 0.999838i \(-0.494266\pi\)
0.0180127 + 0.999838i \(0.494266\pi\)
\(252\) 0 0
\(253\) 300.765i 1.18880i
\(254\) 0 0
\(255\) 90.5463 52.2769i 0.355083 0.205007i
\(256\) 0 0
\(257\) 185.539 321.363i 0.721942 1.25044i −0.238279 0.971197i \(-0.576583\pi\)
0.960221 0.279243i \(-0.0900834\pi\)
\(258\) 0 0
\(259\) −19.2769 + 122.101i −0.0744283 + 0.471432i
\(260\) 0 0
\(261\) 206.858 + 119.430i 0.792559 + 0.457584i
\(262\) 0 0
\(263\) −410.392 + 236.940i −1.56043 + 0.900912i −0.563212 + 0.826313i \(0.690434\pi\)
−0.997214 + 0.0745993i \(0.976232\pi\)
\(264\) 0 0
\(265\) −315.431 −1.19030
\(266\) 0 0
\(267\) −276.515 −1.03563
\(268\) 0 0
\(269\) −81.5026 + 47.0556i −0.302984 + 0.174928i −0.643782 0.765209i \(-0.722636\pi\)
0.340799 + 0.940136i \(0.389303\pi\)
\(270\) 0 0
\(271\) 137.021 + 79.1093i 0.505614 + 0.291916i 0.731029 0.682347i \(-0.239041\pi\)
−0.225415 + 0.974263i \(0.572374\pi\)
\(272\) 0 0
\(273\) −71.1167 57.5309i −0.260501 0.210736i
\(274\) 0 0
\(275\) −190.785 + 330.449i −0.693764 + 1.20163i
\(276\) 0 0
\(277\) −205.022 + 118.370i −0.740153 + 0.427328i −0.822125 0.569307i \(-0.807212\pi\)
0.0819717 + 0.996635i \(0.473878\pi\)
\(278\) 0 0
\(279\) 309.216i 1.10830i
\(280\) 0 0
\(281\) 241.948 0.861025 0.430512 0.902585i \(-0.358333\pi\)
0.430512 + 0.902585i \(0.358333\pi\)
\(282\) 0 0
\(283\) −116.737 202.194i −0.412498 0.714468i 0.582664 0.812713i \(-0.302010\pi\)
−0.995162 + 0.0982453i \(0.968677\pi\)
\(284\) 0 0
\(285\) −443.582 256.102i −1.55643 0.898604i
\(286\) 0 0
\(287\) 2.85044 + 7.41469i 0.00993186 + 0.0258351i
\(288\) 0 0
\(289\) 139.021 240.791i 0.481041 0.833188i
\(290\) 0 0
\(291\) 231.640 + 401.212i 0.796014 + 1.37874i
\(292\) 0 0
\(293\) 216.492i 0.738882i −0.929254 0.369441i \(-0.879549\pi\)
0.929254 0.369441i \(-0.120451\pi\)
\(294\) 0 0
\(295\) 16.7178i 0.0566704i
\(296\) 0 0
\(297\) 81.8667 + 141.797i 0.275645 + 0.477432i
\(298\) 0 0
\(299\) 27.5971 47.7996i 0.0922981 0.159865i
\(300\) 0 0
\(301\) −126.114 328.053i −0.418983 1.08988i
\(302\) 0 0
\(303\) −710.292 410.087i −2.34420 1.35342i
\(304\) 0 0
\(305\) 348.739 + 604.034i 1.14341 + 1.98044i
\(306\) 0 0
\(307\) −173.487 −0.565106 −0.282553 0.959252i \(-0.591181\pi\)
−0.282553 + 0.959252i \(0.591181\pi\)
\(308\) 0 0
\(309\) 206.593i 0.668585i
\(310\) 0 0
\(311\) 100.205 57.8536i 0.322204 0.186024i −0.330171 0.943921i \(-0.607106\pi\)
0.652374 + 0.757897i \(0.273773\pi\)
\(312\) 0 0
\(313\) −105.261 + 182.317i −0.336297 + 0.582483i −0.983733 0.179637i \(-0.942508\pi\)
0.647436 + 0.762120i \(0.275841\pi\)
\(314\) 0 0
\(315\) −431.080 348.728i −1.36851 1.10707i
\(316\) 0 0
\(317\) 302.997 + 174.936i 0.955828 + 0.551847i 0.894886 0.446294i \(-0.147256\pi\)
0.0609413 + 0.998141i \(0.480590\pi\)
\(318\) 0 0
\(319\) 289.340 167.051i 0.907022 0.523669i
\(320\) 0 0
\(321\) 332.921 1.03714
\(322\) 0 0
\(323\) 53.6837 0.166203
\(324\) 0 0
\(325\) 60.6417 35.0115i 0.186590 0.107728i
\(326\) 0 0
\(327\) 471.830 + 272.411i 1.44291 + 0.833062i
\(328\) 0 0
\(329\) 0.828958 5.25065i 0.00251963 0.0159594i
\(330\) 0 0
\(331\) 107.055 185.425i 0.323430 0.560196i −0.657764 0.753224i \(-0.728497\pi\)
0.981193 + 0.193028i \(0.0618308\pi\)
\(332\) 0 0
\(333\) −172.803 + 99.7677i −0.518927 + 0.299603i
\(334\) 0 0
\(335\) 466.093i 1.39132i
\(336\) 0 0
\(337\) −437.275 −1.29755 −0.648777 0.760979i \(-0.724719\pi\)
−0.648777 + 0.760979i \(0.724719\pi\)
\(338\) 0 0
\(339\) 220.242 + 381.470i 0.649682 + 1.12528i
\(340\) 0 0
\(341\) 374.566 + 216.256i 1.09844 + 0.634182i
\(342\) 0 0
\(343\) −305.846 155.265i −0.891680 0.452667i
\(344\) 0 0
\(345\) 300.523 520.522i 0.871082 1.50876i
\(346\) 0 0
\(347\) 188.694 + 326.827i 0.543786 + 0.941865i 0.998682 + 0.0513208i \(0.0163431\pi\)
−0.454896 + 0.890545i \(0.650324\pi\)
\(348\) 0 0
\(349\) 211.146i 0.605003i −0.953149 0.302502i \(-0.902178\pi\)
0.953149 0.302502i \(-0.0978218\pi\)
\(350\) 0 0
\(351\) 30.0472i 0.0856044i
\(352\) 0 0
\(353\) −149.375 258.725i −0.423159 0.732933i 0.573087 0.819494i \(-0.305746\pi\)
−0.996247 + 0.0865610i \(0.972412\pi\)
\(354\) 0 0
\(355\) 155.758 269.781i 0.438755 0.759946i
\(356\) 0 0
\(357\) 103.125 + 16.2810i 0.288865 + 0.0456051i
\(358\) 0 0
\(359\) 191.258 + 110.423i 0.532751 + 0.307584i 0.742136 0.670249i \(-0.233813\pi\)
−0.209385 + 0.977833i \(0.567146\pi\)
\(360\) 0 0
\(361\) 49.0031 + 84.8758i 0.135743 + 0.235113i
\(362\) 0 0
\(363\) 580.274 1.59855
\(364\) 0 0
\(365\) 12.0533i 0.0330229i
\(366\) 0 0
\(367\) 186.496 107.674i 0.508165 0.293389i −0.223914 0.974609i \(-0.571884\pi\)
0.732079 + 0.681220i \(0.238550\pi\)
\(368\) 0 0
\(369\) −6.41135 + 11.1048i −0.0173749 + 0.0300943i
\(370\) 0 0
\(371\) −244.878 198.097i −0.660048 0.533955i
\(372\) 0 0
\(373\) −193.884 111.939i −0.519796 0.300104i 0.217055 0.976159i \(-0.430355\pi\)
−0.736851 + 0.676055i \(0.763688\pi\)
\(374\) 0 0
\(375\) −23.4518 + 13.5399i −0.0625381 + 0.0361064i
\(376\) 0 0
\(377\) −61.3118 −0.162631
\(378\) 0 0
\(379\) 507.051 1.33787 0.668933 0.743323i \(-0.266751\pi\)
0.668933 + 0.743323i \(0.266751\pi\)
\(380\) 0 0
\(381\) 683.023 394.343i 1.79271 1.03502i
\(382\) 0 0
\(383\) 128.681 + 74.2939i 0.335981 + 0.193979i 0.658493 0.752586i \(-0.271194\pi\)
−0.322512 + 0.946565i \(0.604527\pi\)
\(384\) 0 0
\(385\) −723.913 + 278.295i −1.88029 + 0.722845i
\(386\) 0 0
\(387\) 283.661 491.316i 0.732975 1.26955i
\(388\) 0 0
\(389\) 137.107 79.1586i 0.352459 0.203493i −0.313309 0.949651i \(-0.601437\pi\)
0.665768 + 0.746159i \(0.268104\pi\)
\(390\) 0 0
\(391\) 62.9952i 0.161113i
\(392\) 0 0
\(393\) 38.8366 0.0988208
\(394\) 0 0
\(395\) −254.646 441.060i −0.644673 1.11661i
\(396\) 0 0
\(397\) 240.531 + 138.871i 0.605872 + 0.349801i 0.771348 0.636413i \(-0.219583\pi\)
−0.165476 + 0.986214i \(0.552916\pi\)
\(398\) 0 0
\(399\) −183.527 477.399i −0.459968 1.19649i
\(400\) 0 0
\(401\) −311.899 + 540.225i −0.777803 + 1.34719i 0.155402 + 0.987851i \(0.450333\pi\)
−0.933205 + 0.359344i \(0.883001\pi\)
\(402\) 0 0
\(403\) −39.6857 68.7377i −0.0984757 0.170565i
\(404\) 0 0
\(405\) 385.691i 0.952324i
\(406\) 0 0
\(407\) 279.098i 0.685744i
\(408\) 0 0
\(409\) 247.569 + 428.801i 0.605302 + 1.04841i 0.992004 + 0.126209i \(0.0402810\pi\)
−0.386701 + 0.922205i \(0.626386\pi\)
\(410\) 0 0
\(411\) −576.892 + 999.206i −1.40363 + 2.43116i
\(412\) 0 0
\(413\) −10.4991 + 12.9785i −0.0254217 + 0.0314249i
\(414\) 0 0
\(415\) 619.735 + 357.804i 1.49334 + 0.862179i
\(416\) 0 0
\(417\) 492.975 + 853.858i 1.18219 + 2.04762i
\(418\) 0 0
\(419\) −112.631 −0.268810 −0.134405 0.990927i \(-0.542912\pi\)
−0.134405 + 0.990927i \(0.542912\pi\)
\(420\) 0 0
\(421\) 821.936i 1.95234i 0.217003 + 0.976171i \(0.430372\pi\)
−0.217003 + 0.976171i \(0.569628\pi\)
\(422\) 0 0
\(423\) 7.43097 4.29027i 0.0175673 0.0101425i
\(424\) 0 0
\(425\) −39.9599 + 69.2125i −0.0940232 + 0.162853i
\(426\) 0 0
\(427\) −108.611 + 687.945i −0.254358 + 1.61111i
\(428\) 0 0
\(429\) −178.862 103.266i −0.416928 0.240713i
\(430\) 0 0
\(431\) −289.087 + 166.904i −0.670735 + 0.387249i −0.796355 0.604830i \(-0.793241\pi\)
0.125620 + 0.992078i \(0.459908\pi\)
\(432\) 0 0
\(433\) 604.681 1.39649 0.698246 0.715858i \(-0.253964\pi\)
0.698246 + 0.715858i \(0.253964\pi\)
\(434\) 0 0
\(435\) −667.664 −1.53486
\(436\) 0 0
\(437\) 267.265 154.305i 0.611589 0.353101i
\(438\) 0 0
\(439\) −416.657 240.557i −0.949105 0.547966i −0.0563019 0.998414i \(-0.517931\pi\)
−0.892803 + 0.450448i \(0.851264\pi\)
\(440\) 0 0
\(441\) −115.650 541.455i −0.262245 1.22779i
\(442\) 0 0
\(443\) 92.3599 159.972i 0.208487 0.361111i −0.742751 0.669568i \(-0.766479\pi\)
0.951238 + 0.308457i \(0.0998127\pi\)
\(444\) 0 0
\(445\) 372.595 215.118i 0.837291 0.483410i
\(446\) 0 0
\(447\) 107.583i 0.240678i
\(448\) 0 0
\(449\) 115.894 0.258115 0.129057 0.991637i \(-0.458805\pi\)
0.129057 + 0.991637i \(0.458805\pi\)
\(450\) 0 0
\(451\) 8.96779 + 15.5327i 0.0198842 + 0.0344405i
\(452\) 0 0
\(453\) −715.995 413.380i −1.58056 0.912538i
\(454\) 0 0
\(455\) 140.584 + 22.1951i 0.308977 + 0.0487804i
\(456\) 0 0
\(457\) 79.2311 137.232i 0.173372 0.300290i −0.766224 0.642573i \(-0.777867\pi\)
0.939597 + 0.342283i \(0.111200\pi\)
\(458\) 0 0
\(459\) 17.1470 + 29.6994i 0.0373572 + 0.0647046i
\(460\) 0 0
\(461\) 42.3829i 0.0919368i 0.998943 + 0.0459684i \(0.0146374\pi\)
−0.998943 + 0.0459684i \(0.985363\pi\)
\(462\) 0 0
\(463\) 390.038i 0.842416i −0.906964 0.421208i \(-0.861606\pi\)
0.906964 0.421208i \(-0.138394\pi\)
\(464\) 0 0
\(465\) −432.164 748.530i −0.929385 1.60974i
\(466\) 0 0
\(467\) 19.1723 33.2074i 0.0410541 0.0711078i −0.844768 0.535132i \(-0.820262\pi\)
0.885822 + 0.464025i \(0.153595\pi\)
\(468\) 0 0
\(469\) 292.717 361.841i 0.624130 0.771517i
\(470\) 0 0
\(471\) 647.452 + 373.807i 1.37463 + 0.793644i
\(472\) 0 0
\(473\) −396.768 687.222i −0.838832 1.45290i
\(474\) 0 0
\(475\) 391.523 0.824259
\(476\) 0 0
\(477\) 508.427i 1.06588i
\(478\) 0 0
\(479\) 561.326 324.082i 1.17187 0.676580i 0.217751 0.976004i \(-0.430128\pi\)
0.954120 + 0.299424i \(0.0967945\pi\)
\(480\) 0 0
\(481\) 25.6090 44.3561i 0.0532412 0.0922164i
\(482\) 0 0
\(483\) 560.204 215.360i 1.15984 0.445881i
\(484\) 0 0
\(485\) −624.255 360.414i −1.28712 0.743121i
\(486\) 0 0
\(487\) 303.304 175.113i 0.622800 0.359574i −0.155158 0.987890i \(-0.549589\pi\)
0.777959 + 0.628316i \(0.216255\pi\)
\(488\) 0 0
\(489\) 45.5248 0.0930977
\(490\) 0 0
\(491\) 453.547 0.923722 0.461861 0.886952i \(-0.347182\pi\)
0.461861 + 0.886952i \(0.347182\pi\)
\(492\) 0 0
\(493\) 60.6022 34.9887i 0.122925 0.0709710i
\(494\) 0 0
\(495\) −1084.19 625.955i −2.19027 1.26455i
\(496\) 0 0
\(497\) 290.348 111.619i 0.584200 0.224585i
\(498\) 0 0
\(499\) −264.783 + 458.617i −0.530626 + 0.919072i 0.468735 + 0.883339i \(0.344710\pi\)
−0.999361 + 0.0357329i \(0.988623\pi\)
\(500\) 0 0
\(501\) −1112.22 + 642.142i −2.22000 + 1.28172i
\(502\) 0 0
\(503\) 321.597i 0.639358i 0.947526 + 0.319679i \(0.103575\pi\)
−0.947526 + 0.319679i \(0.896425\pi\)
\(504\) 0 0
\(505\) 1276.13 2.52699
\(506\) 0 0
\(507\) −361.762 626.591i −0.713536 1.23588i
\(508\) 0 0
\(509\) 623.464 + 359.957i 1.22488 + 0.707185i 0.965954 0.258712i \(-0.0832981\pi\)
0.258926 + 0.965897i \(0.416631\pi\)
\(510\) 0 0
\(511\) 7.56977 9.35735i 0.0148136 0.0183118i
\(512\) 0 0
\(513\) 84.0021 145.496i 0.163747 0.283618i
\(514\) 0 0
\(515\) 160.721 + 278.377i 0.312080 + 0.540538i
\(516\) 0 0
\(517\) 12.0019i 0.0232146i
\(518\) 0 0
\(519\) 438.708i 0.845294i
\(520\) 0 0
\(521\) −206.990 358.517i −0.397294 0.688133i 0.596097 0.802912i \(-0.296717\pi\)
−0.993391 + 0.114779i \(0.963384\pi\)
\(522\) 0 0
\(523\) −174.324 + 301.938i −0.333316 + 0.577320i −0.983160 0.182747i \(-0.941501\pi\)
0.649844 + 0.760068i \(0.274834\pi\)
\(524\) 0 0
\(525\) 752.103 + 118.740i 1.43258 + 0.226171i
\(526\) 0 0
\(527\) 78.4528 + 45.2947i 0.148867 + 0.0859483i
\(528\) 0 0
\(529\) −83.4303 144.506i −0.157713 0.273167i
\(530\) 0 0
\(531\) −26.9466 −0.0507468
\(532\) 0 0
\(533\) 3.29141i 0.00617525i
\(534\) 0 0
\(535\) −448.601 + 259.000i −0.838507 + 0.484112i
\(536\) 0 0
\(537\) −679.912 + 1177.64i −1.26613 + 2.19300i
\(538\) 0 0
\(539\) −736.770 238.585i −1.36692 0.442644i
\(540\) 0 0
\(541\) −202.840 117.110i −0.374936 0.216469i 0.300677 0.953726i \(-0.402787\pi\)
−0.675613 + 0.737257i \(0.736121\pi\)
\(542\) 0 0
\(543\) 235.881 136.186i 0.434404 0.250803i
\(544\) 0 0
\(545\) −847.701 −1.55542
\(546\) 0 0
\(547\) 577.082 1.05499 0.527497 0.849557i \(-0.323130\pi\)
0.527497 + 0.849557i \(0.323130\pi\)
\(548\) 0 0
\(549\) −973.613 + 562.116i −1.77343 + 1.02389i
\(550\) 0 0
\(551\) −296.887 171.408i −0.538815 0.311085i
\(552\) 0 0
\(553\) 79.3065 502.331i 0.143411 0.908374i
\(554\) 0 0
\(555\) 278.873 483.022i 0.502474 0.870311i
\(556\) 0 0
\(557\) 36.7252 21.2033i 0.0659339 0.0380670i −0.466671 0.884431i \(-0.654547\pi\)
0.532605 + 0.846364i \(0.321213\pi\)
\(558\) 0 0
\(559\) 145.624i 0.260508i
\(560\) 0 0
\(561\) 235.722 0.420182
\(562\) 0 0
\(563\) −123.648 214.165i −0.219624 0.380399i 0.735069 0.677992i \(-0.237150\pi\)
−0.954693 + 0.297593i \(0.903816\pi\)
\(564\) 0 0
\(565\) −593.539 342.680i −1.05051 0.606513i
\(566\) 0 0
\(567\) −242.223 + 299.423i −0.427201 + 0.528083i
\(568\) 0 0
\(569\) −312.737 + 541.677i −0.549626 + 0.951980i 0.448674 + 0.893696i \(0.351896\pi\)
−0.998300 + 0.0582847i \(0.981437\pi\)
\(570\) 0 0
\(571\) −291.287 504.524i −0.510135 0.883579i −0.999931 0.0117423i \(-0.996262\pi\)
0.489796 0.871837i \(-0.337071\pi\)
\(572\) 0 0
\(573\) 330.758i 0.577239i
\(574\) 0 0
\(575\) 459.433i 0.799014i
\(576\) 0 0
\(577\) 198.380 + 343.605i 0.343814 + 0.595503i 0.985137 0.171768i \(-0.0549478\pi\)
−0.641324 + 0.767270i \(0.721615\pi\)
\(578\) 0 0
\(579\) −146.629 + 253.969i −0.253246 + 0.438634i
\(580\) 0 0
\(581\) 256.409 + 666.981i 0.441324 + 1.14799i
\(582\) 0 0
\(583\) −615.879 355.578i −1.05640 0.609910i
\(584\) 0 0
\(585\) 114.871 + 198.962i 0.196360 + 0.340106i
\(586\) 0 0
\(587\) −355.763 −0.606070 −0.303035 0.952979i \(-0.598000\pi\)
−0.303035 + 0.952979i \(0.598000\pi\)
\(588\) 0 0
\(589\) 443.794i 0.753470i
\(590\) 0 0
\(591\) −1359.11 + 784.682i −2.29968 + 1.32772i
\(592\) 0 0
\(593\) 89.8611 155.644i 0.151536 0.262469i −0.780256 0.625460i \(-0.784911\pi\)
0.931792 + 0.362991i \(0.118245\pi\)
\(594\) 0 0
\(595\) −151.623 + 58.2888i −0.254829 + 0.0979644i
\(596\) 0 0
\(597\) 1014.93 + 585.967i 1.70004 + 0.981520i
\(598\) 0 0
\(599\) −35.2310 + 20.3406i −0.0588163 + 0.0339576i −0.529120 0.848547i \(-0.677478\pi\)
0.470303 + 0.882505i \(0.344144\pi\)
\(600\) 0 0
\(601\) −1020.79 −1.69849 −0.849245 0.527998i \(-0.822943\pi\)
−0.849245 + 0.527998i \(0.822943\pi\)
\(602\) 0 0
\(603\) 751.272 1.24589
\(604\) 0 0
\(605\) −781.901 + 451.431i −1.29240 + 0.746166i
\(606\) 0 0
\(607\) 273.220 + 157.744i 0.450116 + 0.259874i 0.707879 0.706334i \(-0.249652\pi\)
−0.257763 + 0.966208i \(0.582986\pi\)
\(608\) 0 0
\(609\) −518.327 419.308i −0.851111 0.688519i
\(610\) 0 0
\(611\) −1.10125 + 1.90743i −0.00180238 + 0.00312181i
\(612\) 0 0
\(613\) −358.417 + 206.932i −0.584693 + 0.337572i −0.762996 0.646403i \(-0.776272\pi\)
0.178303 + 0.983976i \(0.442939\pi\)
\(614\) 0 0
\(615\) 35.8423i 0.0582802i
\(616\) 0 0
\(617\) 139.167 0.225554 0.112777 0.993620i \(-0.464025\pi\)
0.112777 + 0.993620i \(0.464025\pi\)
\(618\) 0 0
\(619\) 167.006 + 289.264i 0.269800 + 0.467308i 0.968810 0.247804i \(-0.0797090\pi\)
−0.699010 + 0.715112i \(0.746376\pi\)
\(620\) 0 0
\(621\) 170.732 + 98.5724i 0.274931 + 0.158732i
\(622\) 0 0
\(623\) 424.354 + 66.9959i 0.681147 + 0.107538i
\(624\) 0 0
\(625\) 322.850 559.192i 0.516560 0.894707i
\(626\) 0 0
\(627\) −577.396 1000.08i −0.920887 1.59502i
\(628\) 0 0
\(629\) 58.4569i 0.0929363i
\(630\) 0 0
\(631\) 284.343i 0.450623i −0.974287 0.225311i \(-0.927660\pi\)
0.974287 0.225311i \(-0.0723399\pi\)
\(632\) 0 0
\(633\) 10.6560 + 18.4568i 0.0168342 + 0.0291576i
\(634\) 0 0
\(635\) −613.568 + 1062.73i −0.966248 + 1.67359i
\(636\) 0 0
\(637\) 95.2007 + 105.521i 0.149452 + 0.165653i
\(638\) 0 0
\(639\) 434.846 + 251.058i 0.680510 + 0.392893i
\(640\) 0 0
\(641\) −32.4545 56.2128i −0.0506310 0.0876955i 0.839599 0.543206i \(-0.182790\pi\)
−0.890230 + 0.455511i \(0.849457\pi\)
\(642\) 0 0
\(643\) −683.365 −1.06278 −0.531388 0.847128i \(-0.678329\pi\)
−0.531388 + 0.847128i \(0.678329\pi\)
\(644\) 0 0
\(645\) 1585.79i 2.45859i
\(646\) 0 0
\(647\) 1103.61 637.171i 1.70574 0.984808i 0.766047 0.642785i \(-0.222221\pi\)
0.939691 0.342024i \(-0.111112\pi\)
\(648\) 0 0
\(649\) −18.8456 + 32.6415i −0.0290379 + 0.0502951i
\(650\) 0 0
\(651\) 134.592 852.514i 0.206747 1.30954i
\(652\) 0 0
\(653\) 472.891 + 273.024i 0.724182 + 0.418107i 0.816290 0.577642i \(-0.196027\pi\)
−0.0921079 + 0.995749i \(0.529360\pi\)
\(654\) 0 0
\(655\) −52.3311 + 30.2134i −0.0798948 + 0.0461273i
\(656\) 0 0
\(657\) 19.4282 0.0295710
\(658\) 0 0
\(659\) 398.883 0.605285 0.302642 0.953104i \(-0.402131\pi\)
0.302642 + 0.953104i \(0.402131\pi\)
\(660\) 0 0
\(661\) 718.146 414.622i 1.08645 0.627264i 0.153824 0.988098i \(-0.450841\pi\)
0.932630 + 0.360834i \(0.117508\pi\)
\(662\) 0 0
\(663\) −37.4626 21.6290i −0.0565046 0.0326230i
\(664\) 0 0
\(665\) 618.695 + 500.502i 0.930368 + 0.752635i
\(666\) 0 0
\(667\) 201.139 348.383i 0.301557 0.522313i
\(668\) 0 0
\(669\) 426.531 246.258i 0.637565 0.368098i
\(670\) 0 0
\(671\) 1572.50i 2.34352i
\(672\) 0 0
\(673\) 946.218 1.40597 0.702985 0.711205i \(-0.251850\pi\)
0.702985 + 0.711205i \(0.251850\pi\)
\(674\) 0 0
\(675\) 125.055 + 216.602i 0.185267 + 0.320892i
\(676\) 0 0
\(677\) −987.793 570.303i −1.45907 0.842397i −0.460108 0.887863i \(-0.652189\pi\)
−0.998966 + 0.0454660i \(0.985523\pi\)
\(678\) 0 0
\(679\) −258.279 671.846i −0.380381 0.989463i
\(680\) 0 0
\(681\) −273.575 + 473.846i −0.401725 + 0.695809i
\(682\) 0 0
\(683\) 58.7041 + 101.678i 0.0859504 + 0.148870i 0.905796 0.423715i \(-0.139274\pi\)
−0.819845 + 0.572585i \(0.805941\pi\)
\(684\) 0 0
\(685\) 1795.20i 2.62073i
\(686\) 0 0
\(687\) 649.675i 0.945670i
\(688\) 0 0
\(689\) 65.2530 + 113.022i 0.0947069 + 0.164037i
\(690\) 0 0
\(691\) 477.961 827.853i 0.691695 1.19805i −0.279588 0.960120i \(-0.590198\pi\)
0.971282 0.237930i \(-0.0764690\pi\)
\(692\) 0 0
\(693\) −448.570 1166.84i −0.647287 1.68375i
\(694\) 0 0
\(695\) −1328.54 767.031i −1.91156 1.10364i
\(696\) 0 0
\(697\) 1.87830 + 3.25331i 0.00269484 + 0.00466759i
\(698\) 0 0
\(699\) 875.243 1.25214
\(700\) 0 0
\(701\) 824.950i 1.17682i 0.808563 + 0.588410i \(0.200246\pi\)
−0.808563 + 0.588410i \(0.799754\pi\)
\(702\) 0 0
\(703\) 248.010 143.189i 0.352789 0.203683i
\(704\) 0 0
\(705\) −11.9923 + 20.7712i −0.0170103 + 0.0294627i
\(706\) 0 0
\(707\) 990.695 + 801.437i 1.40127 + 1.13357i
\(708\) 0 0
\(709\) 112.576 + 64.9959i 0.158782 + 0.0916726i 0.577285 0.816542i \(-0.304112\pi\)
−0.418504 + 0.908215i \(0.637445\pi\)
\(710\) 0 0
\(711\) 710.922 410.451i 0.999890 0.577287i
\(712\) 0 0
\(713\) 520.770 0.730393
\(714\) 0 0
\(715\) 321.348 0.449437
\(716\) 0 0
\(717\) 1231.30 710.890i 1.71729 0.991478i
\(718\) 0 0
\(719\) −438.623 253.239i −0.610046 0.352210i 0.162937 0.986636i \(-0.447903\pi\)
−0.772984 + 0.634426i \(0.781237\pi\)
\(720\) 0 0
\(721\) −50.0547 + 317.049i −0.0694240 + 0.439735i
\(722\) 0 0
\(723\) −817.094 + 1415.25i −1.13014 + 1.95747i
\(724\) 0 0
\(725\) 441.980 255.177i 0.609628 0.351969i
\(726\) 0 0
\(727\) 1272.41i 1.75022i −0.483924 0.875110i \(-0.660789\pi\)
0.483924 0.875110i \(-0.339211\pi\)
\(728\) 0 0
\(729\) −1041.76 −1.42902
\(730\) 0 0
\(731\) −83.1028 143.938i −0.113684 0.196906i
\(732\) 0 0
\(733\) −615.475 355.345i −0.839666 0.484781i 0.0174847 0.999847i \(-0.494434\pi\)
−0.857151 + 0.515066i \(0.827767\pi\)
\(734\) 0 0
\(735\) 1036.70 + 1149.09i 1.41048 + 1.56338i
\(736\) 0 0
\(737\) 525.417 910.048i 0.712913 1.23480i
\(738\) 0 0
\(739\) −105.509 182.748i −0.142773 0.247291i 0.785767 0.618523i \(-0.212269\pi\)
−0.928540 + 0.371232i \(0.878935\pi\)
\(740\) 0 0
\(741\) 211.919i 0.285991i
\(742\) 0 0
\(743\) 244.355i 0.328876i 0.986387 + 0.164438i \(0.0525811\pi\)
−0.986387 + 0.164438i \(0.947419\pi\)
\(744\) 0 0
\(745\) −83.6954 144.965i −0.112343 0.194583i
\(746\) 0 0
\(747\) −576.727 + 998.920i −0.772057 + 1.33724i
\(748\) 0 0
\(749\) −510.919 80.6625i −0.682136 0.107694i
\(750\) 0 0
\(751\) 262.737 + 151.692i 0.349850 + 0.201986i 0.664619 0.747182i \(-0.268594\pi\)
−0.314769 + 0.949168i \(0.601927\pi\)
\(752\) 0 0
\(753\) −20.3701 35.2821i −0.0270519 0.0468553i
\(754\) 0 0
\(755\) 1286.37 1.70381
\(756\) 0 0
\(757\) 668.225i 0.882728i −0.897328 0.441364i \(-0.854495\pi\)
0.897328 0.441364i \(-0.145505\pi\)
\(758\) 0 0
\(759\) 1173.54 677.546i 1.54617 0.892683i
\(760\) 0 0
\(761\) −45.4152 + 78.6615i −0.0596784 + 0.103366i −0.894321 0.447426i \(-0.852341\pi\)
0.834643 + 0.550792i \(0.185674\pi\)
\(762\) 0 0
\(763\) −658.095 532.375i −0.862509 0.697740i
\(764\) 0 0
\(765\) −227.082 131.106i −0.296839 0.171380i
\(766\) 0 0
\(767\) 5.99013 3.45840i 0.00780982 0.00450900i
\(768\) 0 0
\(769\) 421.091 0.547582 0.273791 0.961789i \(-0.411722\pi\)
0.273791 + 0.961789i \(0.411722\pi\)
\(770\) 0 0
\(771\) −1671.88 −2.16846
\(772\) 0 0
\(773\) −508.100 + 293.352i −0.657309 + 0.379498i −0.791251 0.611492i \(-0.790570\pi\)
0.133942 + 0.990989i \(0.457237\pi\)
\(774\) 0 0
\(775\) 572.167 + 330.341i 0.738281 + 0.426247i
\(776\) 0 0
\(777\) 519.846 199.845i 0.669043 0.257201i
\(778\) 0 0
\(779\) 9.20171 15.9378i 0.0118122 0.0204593i
\(780\) 0 0
\(781\) 608.235 351.165i 0.778791 0.449635i
\(782\) 0 0
\(783\) 218.996i 0.279688i
\(784\) 0 0
\(785\) −1163.23 −1.48182
\(786\) 0 0
\(787\) 681.137 + 1179.76i 0.865485 + 1.49906i 0.866565 + 0.499065i \(0.166323\pi\)
−0.00107958 + 0.999999i \(0.500344\pi\)
\(788\) 0 0
\(789\) 1849.01 + 1067.53i 2.34349 + 1.35301i
\(790\) 0 0
\(791\) −245.570 638.787i −0.310456 0.807569i
\(792\) 0 0
\(793\) 144.287 249.913i 0.181951 0.315149i
\(794\) 0 0
\(795\) 710.583 + 1230.77i 0.893815 + 1.54813i
\(796\) 0 0
\(797\) 874.598i 1.09736i 0.836032 + 0.548681i \(0.184870\pi\)
−0.836032 + 0.548681i \(0.815130\pi\)
\(798\) 0 0
\(799\) 2.51380i 0.00314618i
\(800\) 0 0
\(801\) 346.737 + 600.567i 0.432880 + 0.749771i
\(802\) 0 0
\(803\) 13.5875 23.5342i 0.0169209 0.0293078i
\(804\) 0 0
\(805\) −587.315 + 726.008i −0.729584 + 0.901874i
\(806\) 0 0
\(807\) 367.208 + 212.008i 0.455029 + 0.262711i
\(808\) 0 0
\(809\) 27.9415 + 48.3961i 0.0345383 + 0.0598222i 0.882778 0.469791i \(-0.155671\pi\)
−0.848240 + 0.529613i \(0.822337\pi\)
\(810\) 0 0
\(811\) −917.924 −1.13184 −0.565921 0.824459i \(-0.691479\pi\)
−0.565921 + 0.824459i \(0.691479\pi\)
\(812\) 0 0
\(813\) 712.850i 0.876815i
\(814\) 0 0
\(815\) −61.3432 + 35.4165i −0.0752677 + 0.0434559i
\(816\) 0 0
\(817\) −407.117 + 705.147i −0.498307 + 0.863093i
\(818\) 0 0
\(819\) −35.7751 + 226.601i −0.0436815 + 0.276680i
\(820\) 0 0
\(821\) −1178.10 680.176i −1.43496 0.828472i −0.437463 0.899237i \(-0.644123\pi\)
−0.997493 + 0.0707646i \(0.977456\pi\)
\(822\) 0 0
\(823\) −488.057 + 281.780i −0.593022 + 0.342382i −0.766292 0.642493i \(-0.777900\pi\)
0.173269 + 0.984874i \(0.444567\pi\)
\(824\) 0 0
\(825\) 1719.16 2.08383
\(826\) 0 0
\(827\) −63.4180 −0.0766844 −0.0383422 0.999265i \(-0.512208\pi\)
−0.0383422 + 0.999265i \(0.512208\pi\)
\(828\) 0 0
\(829\) −252.620 + 145.850i −0.304728 + 0.175935i −0.644565 0.764549i \(-0.722962\pi\)
0.339837 + 0.940484i \(0.389628\pi\)
\(830\) 0 0
\(831\) 923.725 + 533.313i 1.11158 + 0.641772i
\(832\) 0 0
\(833\) −154.316 49.9715i −0.185254 0.0599899i
\(834\) 0 0
\(835\) 999.123 1730.53i 1.19655 2.07249i
\(836\) 0 0
\(837\) 245.520 141.751i 0.293333 0.169356i
\(838\) 0 0
\(839\) 463.917i 0.552940i 0.961023 + 0.276470i \(0.0891646\pi\)
−0.961023 + 0.276470i \(0.910835\pi\)
\(840\) 0 0
\(841\) 394.135 0.468651
\(842\) 0 0
\(843\) −545.046 944.047i −0.646555 1.11987i
\(844\) 0 0
\(845\) 974.927 + 562.874i 1.15376 + 0.666124i
\(846\) 0 0
\(847\) −890.520 140.593i −1.05138 0.165989i
\(848\) 0 0
\(849\) −525.956 + 910.983i −0.619501 + 1.07301i
\(850\) 0 0
\(851\) 168.025 + 291.028i 0.197444 + 0.341984i
\(852\) 0 0
\(853\) 649.909i 0.761909i 0.924594 + 0.380955i \(0.124405\pi\)
−0.924594 + 0.380955i \(0.875595\pi\)
\(854\) 0 0
\(855\) 1284.56i 1.50241i
\(856\) 0 0
\(857\) −584.140 1011.76i −0.681610 1.18058i −0.974489 0.224433i \(-0.927947\pi\)
0.292880 0.956149i \(-0.405386\pi\)
\(858\) 0 0
\(859\) 733.208 1269.95i 0.853560 1.47841i −0.0244139 0.999702i \(-0.507772\pi\)
0.877974 0.478708i \(-0.158895\pi\)
\(860\) 0 0
\(861\) 22.5098 27.8254i 0.0261438 0.0323175i
\(862\) 0 0
\(863\) 36.9921 + 21.3574i 0.0428646 + 0.0247479i 0.521279 0.853386i \(-0.325455\pi\)
−0.478415 + 0.878134i \(0.658788\pi\)
\(864\) 0 0
\(865\) −341.298 591.145i −0.394564 0.683404i
\(866\) 0 0
\(867\) −1252.71 −1.44488
\(868\) 0 0
\(869\) 1148.23i 1.32132i
\(870\) 0 0
\(871\) −167.005 + 96.4206i −0.191740 + 0.110701i
\(872\) 0 0
\(873\) 580.933 1006.21i 0.665444 1.15258i
\(874\) 0 0
\(875\) 39.2709 15.0970i 0.0448811 0.0172537i
\(876\) 0 0
\(877\) 842.217 + 486.254i 0.960338 + 0.554452i 0.896277 0.443494i \(-0.146261\pi\)
0.0640612 + 0.997946i \(0.479595\pi\)
\(878\) 0 0
\(879\) −844.723 + 487.701i −0.961005 + 0.554836i
\(880\) 0 0
\(881\) −110.722 −0.125678 −0.0628390 0.998024i \(-0.520015\pi\)
−0.0628390 + 0.998024i \(0.520015\pi\)
\(882\) 0 0
\(883\) −1283.66 −1.45375 −0.726874 0.686770i \(-0.759028\pi\)
−0.726874 + 0.686770i \(0.759028\pi\)
\(884\) 0 0
\(885\) 65.2305 37.6608i 0.0737067 0.0425546i
\(886\) 0 0
\(887\) 429.755 + 248.119i 0.484504 + 0.279729i 0.722292 0.691589i \(-0.243089\pi\)
−0.237788 + 0.971317i \(0.576422\pi\)
\(888\) 0 0
\(889\) −1143.75 + 439.693i −1.28656 + 0.494593i
\(890\) 0 0
\(891\) −434.781 + 753.063i −0.487970 + 0.845189i
\(892\) 0 0
\(893\) −10.6651 + 6.15750i −0.0119430 + 0.00689529i
\(894\) 0 0
\(895\) 2115.78i 2.36400i
\(896\) 0 0
\(897\) −248.677 −0.277232
\(898\) 0 0
\(899\) −289.245 500.987i −0.321741 0.557272i
\(900\) 0 0
\(901\) −128.996 74.4756i −0.143169 0.0826589i
\(902\) 0 0
\(903\) −995.914 + 1231.10i −1.10289 + 1.36334i
\(904\) 0 0
\(905\) −211.895 + 367.013i −0.234138 + 0.405539i
\(906\) 0 0
\(907\) 796.131 + 1378.94i 0.877763 + 1.52033i 0.853790 + 0.520617i \(0.174298\pi\)
0.0239725 + 0.999713i \(0.492369\pi\)
\(908\) 0 0
\(909\) 2056.93i 2.26285i
\(910\) 0 0
\(911\) 663.197i 0.727988i −0.931401 0.363994i \(-0.881413\pi\)
0.931401 0.363994i \(-0.118587\pi\)
\(912\) 0 0
\(913\) 806.690 + 1397.23i 0.883559 + 1.53037i
\(914\) 0 0
\(915\) 1571.24 2721.47i 1.71720 2.97428i
\(916\) 0 0
\(917\) −59.6008 9.40960i −0.0649954 0.0102613i
\(918\) 0 0
\(919\) −441.371 254.826i −0.480273 0.277286i 0.240257 0.970709i \(-0.422768\pi\)
−0.720530 + 0.693424i \(0.756102\pi\)
\(920\) 0 0
\(921\) 390.822 + 676.924i 0.424345 + 0.734988i
\(922\) 0 0
\(923\) −128.886 −0.139639
\(924\) 0 0
\(925\) 426.335i 0.460902i
\(926\) 0 0
\(927\) −448.702 + 259.058i −0.484037 + 0.279459i
\(928\) 0 0
\(929\) 161.143 279.108i 0.173459 0.300439i −0.766168 0.642640i \(-0.777839\pi\)
0.939627 + 0.342201i \(0.111172\pi\)
\(930\) 0 0
\(931\) 165.984 + 777.109i 0.178285 + 0.834703i
\(932\) 0 0
\(933\) −451.473 260.658i −0.483894 0.279376i
\(934\) 0 0
\(935\) −317.628 + 183.383i −0.339710 + 0.196131i
\(936\) 0 0
\(937\) −1077.79 −1.15026 −0.575130 0.818062i \(-0.695048\pi\)
−0.575130 + 0.818062i \(0.695048\pi\)
\(938\) 0 0
\(939\) 948.502 1.01012
\(940\) 0 0
\(941\) 63.5953 36.7168i 0.0675827 0.0390189i −0.465828 0.884875i \(-0.654243\pi\)
0.533411 + 0.845856i \(0.320910\pi\)
\(942\) 0 0
\(943\) 18.7023 + 10.7978i 0.0198327 + 0.0114504i
\(944\) 0 0
\(945\) −79.2771 + 502.144i −0.0838911 + 0.531369i
\(946\) 0 0
\(947\) 38.4077 66.5241i 0.0405572 0.0702472i −0.845034 0.534712i \(-0.820420\pi\)
0.885591 + 0.464465i \(0.153753\pi\)
\(948\) 0 0
\(949\) −4.31882 + 2.49347i −0.00455092 + 0.00262747i
\(950\) 0 0
\(951\) 1576.34i 1.65756i
\(952\) 0 0
\(953\) −1329.39 −1.39495 −0.697477 0.716607i \(-0.745694\pi\)
−0.697477 + 0.716607i \(0.745694\pi\)
\(954\) 0 0
\(955\) 257.317 + 445.686i 0.269442 + 0.466687i
\(956\) 0 0
\(957\) −1303.62 752.643i −1.36219 0.786461i
\(958\) 0 0
\(959\) 1127.42 1393.66i 1.17563 1.45325i
\(960\) 0 0
\(961\) −106.056 + 183.695i −0.110360 + 0.191150i
\(962\) 0 0
\(963\) −417.469 723.077i −0.433509 0.750859i
\(964\) 0 0
\(965\) 456.288i 0.472837i
\(966\) 0 0
\(967\) 601.743i 0.622279i 0.950364 + 0.311139i \(0.100711\pi\)
−0.950364 + 0.311139i \(0.899289\pi\)
\(968\) 0 0
\(969\) −120.935 209.466i −0.124804 0.216167i
\(970\) 0 0
\(971\) 213.139 369.168i 0.219505 0.380193i −0.735152 0.677902i \(-0.762889\pi\)
0.954657 + 0.297709i \(0.0962225\pi\)
\(972\) 0 0
\(973\) −549.668 1429.82i −0.564921 1.46950i
\(974\) 0 0
\(975\) −273.220 157.744i −0.280225 0.161788i
\(976\) 0 0
\(977\) −109.928 190.401i −0.112516 0.194884i 0.804268 0.594267i \(-0.202558\pi\)
−0.916784 + 0.399383i \(0.869224\pi\)
\(978\) 0 0
\(979\) 969.989 0.990796
\(980\) 0 0
\(981\) 1366.37i 1.39283i
\(982\) 0 0
\(983\) −1444.72 + 834.112i −1.46971 + 0.848537i −0.999423 0.0339771i \(-0.989183\pi\)
−0.470286 + 0.882514i \(0.655849\pi\)
\(984\) 0 0
\(985\) 1220.91 2114.67i 1.23950 2.14687i
\(986\) 0 0
\(987\) −22.3547 + 8.59388i −0.0226492 + 0.00870707i
\(988\) 0 0
\(989\) −827.456 477.732i −0.836659 0.483045i
\(990\) 0 0
\(991\) −1184.20 + 683.697i −1.19495 + 0.689906i −0.959426 0.281962i \(-0.909015\pi\)
−0.235527 + 0.971868i \(0.575682\pi\)
\(992\) 0 0
\(993\) −964.670 −0.971471
\(994\) 0 0
\(995\) −1823.44 −1.83260
\(996\) 0 0
\(997\) −803.308 + 463.790i −0.805725 + 0.465186i −0.845469 0.534024i \(-0.820679\pi\)
0.0397440 + 0.999210i \(0.487346\pi\)
\(998\) 0 0
\(999\) 158.433 + 91.4711i 0.158591 + 0.0915626i
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 224.3.o.c.79.2 12
4.3 odd 2 56.3.k.c.51.4 yes 12
7.2 even 3 1568.3.g.k.687.5 6
7.4 even 3 inner 224.3.o.c.207.1 12
7.5 odd 6 1568.3.g.i.687.2 6
8.3 odd 2 inner 224.3.o.c.79.1 12
8.5 even 2 56.3.k.c.51.5 yes 12
28.3 even 6 392.3.k.k.67.5 12
28.11 odd 6 56.3.k.c.11.5 yes 12
28.19 even 6 392.3.g.l.99.2 6
28.23 odd 6 392.3.g.k.99.2 6
28.27 even 2 392.3.k.k.275.4 12
56.5 odd 6 392.3.g.l.99.1 6
56.11 odd 6 inner 224.3.o.c.207.2 12
56.13 odd 2 392.3.k.k.275.5 12
56.19 even 6 1568.3.g.i.687.1 6
56.37 even 6 392.3.g.k.99.1 6
56.45 odd 6 392.3.k.k.67.4 12
56.51 odd 6 1568.3.g.k.687.6 6
56.53 even 6 56.3.k.c.11.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.k.c.11.4 12 56.53 even 6
56.3.k.c.11.5 yes 12 28.11 odd 6
56.3.k.c.51.4 yes 12 4.3 odd 2
56.3.k.c.51.5 yes 12 8.5 even 2
224.3.o.c.79.1 12 8.3 odd 2 inner
224.3.o.c.79.2 12 1.1 even 1 trivial
224.3.o.c.207.1 12 7.4 even 3 inner
224.3.o.c.207.2 12 56.11 odd 6 inner
392.3.g.k.99.1 6 56.37 even 6
392.3.g.k.99.2 6 28.23 odd 6
392.3.g.l.99.1 6 56.5 odd 6
392.3.g.l.99.2 6 28.19 even 6
392.3.k.k.67.4 12 56.45 odd 6
392.3.k.k.67.5 12 28.3 even 6
392.3.k.k.275.4 12 28.27 even 2
392.3.k.k.275.5 12 56.13 odd 2
1568.3.g.i.687.1 6 56.19 even 6
1568.3.g.i.687.2 6 7.5 odd 6
1568.3.g.k.687.5 6 7.2 even 3
1568.3.g.k.687.6 6 56.51 odd 6