Properties

Label 224.3.o.c.207.1
Level $224$
Weight $3$
Character 224.207
Analytic conductor $6.104$
Analytic rank $0$
Dimension $12$
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] = [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 207.1
Root \(1.71059 - 1.03628i\) of defining polynomial
Character \(\chi\) \(=\) 224.207
Dual form 224.3.o.c.79.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.25274 + 3.90186i) q^{3} +(-6.07099 + 3.50509i) q^{5} +(-2.51181 + 6.53382i) q^{7} +(-5.64968 - 9.78553i) q^{9} +O(q^{10})\) \(q+(-2.25274 + 3.90186i) q^{3} +(-6.07099 + 3.50509i) q^{5} +(-2.51181 + 6.53382i) q^{7} +(-5.64968 - 9.78553i) q^{9} +(7.90242 - 13.6874i) q^{11} -2.90039i q^{13} -31.5842i q^{15} +(1.65516 - 2.86682i) q^{17} +(8.10854 + 14.0444i) q^{19} +(-19.8356 - 24.5197i) q^{21} +(-16.4804 + 9.51498i) q^{23} +(12.0713 - 20.9081i) q^{25} +10.3597 q^{27} -21.1392i q^{29} +(-23.6995 - 13.6829i) q^{31} +(35.6042 + 61.6683i) q^{33} +(-7.65245 - 48.4709i) q^{35} +(-15.2932 + 8.82951i) q^{37} +(11.3169 + 6.53382i) q^{39} +1.13482 q^{41} -50.2084 q^{43} +(68.5983 + 39.6053i) q^{45} +(0.657646 - 0.379692i) q^{47} +(-36.3816 - 32.8234i) q^{49} +(7.45728 + 12.9164i) q^{51} +(38.9677 + 22.4980i) q^{53} +110.795i q^{55} -73.0658 q^{57} +(1.19239 - 2.06529i) q^{59} +(-86.1653 + 49.7476i) q^{61} +(78.1278 - 12.3346i) q^{63} +(10.1661 + 17.6082i) q^{65} +(-33.2440 + 57.5804i) q^{67} -85.7391i q^{69} -44.4376i q^{71} +(-0.859703 + 1.48905i) q^{73} +(54.3870 + 94.2011i) q^{75} +(69.5816 + 86.0131i) q^{77} +(62.9171 - 36.3252i) q^{79} +(27.5094 - 47.6476i) q^{81} +102.081 q^{83} +23.2059i q^{85} +(82.4821 + 47.6211i) q^{87} +(30.6865 + 53.1505i) q^{89} +(18.9506 + 7.28522i) q^{91} +(106.778 - 61.6480i) q^{93} +(-98.4538 - 56.8423i) q^{95} -102.826 q^{97} -178.584 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{3} - 8 q^{9}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/224\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(-1\)

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 0.196467 + 0.980510i \(0.437053\pi\)
−0.947380 + 0.320110i \(0.896280\pi\)
\(4\) 0 0
\(5\) −6.07099 + 3.50509i −1.21420 + 0.701018i −0.963671 0.267092i \(-0.913937\pi\)
−0.250528 + 0.968109i \(0.580604\pi\)
\(6\) 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 −0.243231 0.969968i \(-0.578207\pi\)
0.961633 0.274340i \(-0.0884593\pi\)
\(12\) 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 −0.813230 0.581943i \(-0.802293\pi\)
0.910592 + 0.413306i \(0.135626\pi\)
\(18\) 0 0
\(19\) 8.10854 + 14.0444i 0.426765 + 0.739179i 0.996583 0.0825915i \(-0.0263197\pi\)
−0.569818 + 0.821771i \(0.692986\pi\)
\(20\) 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 \(-0.802429\pi\)
0.0969377 + 0.995290i \(0.469095\pi\)
\(24\) 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 \(-0.478846\pi\)
−0.830909 + 0.556409i \(0.812179\pi\)
\(32\) 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 \(-0.743367\pi\)
0.278890 + 0.960323i \(0.410033\pi\)
\(38\) 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 \(-0.664095\pi\)
0.506980 + 0.861958i \(0.330762\pi\)
\(48\) 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 \(-0.193786\pi\)
−0.0850958 + 0.996373i \(0.527120\pi\)
\(54\) 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 −0.855743 0.517400i \(-0.826900\pi\)
0.875954 + 0.482395i \(0.160233\pi\)
\(60\) 0 0
\(61\) −86.1653 + 49.7476i −1.41255 + 0.815534i −0.995628 0.0934097i \(-0.970223\pi\)
−0.416919 + 0.908944i \(0.636890\pi\)
\(62\) 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 −0.999990 0.00440572i \(-0.998598\pi\)
0.503811 + 0.863814i \(0.331931\pi\)
\(68\) 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 −0.871854 0.489766i \(-0.837082\pi\)
0.860077 + 0.510164i \(0.170415\pi\)
\(74\) 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 \(-0.514583\pi\)
0.842217 + 0.539138i \(0.181250\pi\)
\(80\) 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 0.985316 0.170741i \(-0.0546162\pi\)
−0.640524 + 0.767938i \(0.721283\pi\)
\(90\) 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 −0.997166 0.0752264i \(-0.976032\pi\)
−0.563731 0.825958i \(-0.690635\pi\)
\(102\) 0 0
\(103\) −39.7104 + 22.9268i −0.385538 + 0.222591i −0.680225 0.733003i \(-0.738118\pi\)
0.294687 + 0.955594i \(0.404785\pi\)
\(104\) 0 0
\(105\) 206.366 + 79.3336i 1.96539 + 0.755558i
\(106\) 0 0
\(107\) −36.9463 63.9928i −0.345292 0.598064i 0.640115 0.768279i \(-0.278887\pi\)
−0.985407 + 0.170216i \(0.945554\pi\)
\(108\) 0 0
\(109\) 104.724 + 60.4622i 0.960767 + 0.554699i 0.896409 0.443228i \(-0.146167\pi\)
0.0643577 + 0.997927i \(0.479500\pi\)
\(110\) 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 0.849106 0.528222i \(-0.177141\pi\)
−0.882007 + 0.471237i \(0.843808\pi\)
\(132\) 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 −0.159295 + 0.987231i \(0.550922\pi\)
−0.775320 + 0.631569i \(0.782411\pi\)
\(138\) 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 \(-0.641134\pi\)
0.567785 + 0.823177i \(0.307800\pi\)
\(150\) 0 0
\(151\) −158.916 91.7504i −1.05243 0.607619i −0.129099 0.991632i \(-0.541208\pi\)
−0.923328 + 0.384013i \(0.874542\pi\)
\(152\) 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 \(-0.156106\pi\)
0.0331720 + 0.999450i \(0.489439\pi\)
\(158\) 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 0.850112 0.526602i \(-0.176534\pi\)
−0.881107 + 0.472917i \(0.843201\pi\)
\(164\) 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 \(-0.575861\pi\)
0.723511 + 0.690313i \(0.242527\pi\)
\(174\) 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 0.0442343 + 0.999021i \(0.485915\pi\)
−0.887295 + 0.461203i \(0.847418\pi\)
\(180\) 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 \(-0.728222\pi\)
0.324248 + 0.945972i \(0.394889\pi\)
\(192\) 0 0
\(193\) −32.5446 + 56.3690i −0.168625 + 0.292067i −0.937937 0.346807i \(-0.887266\pi\)
0.769312 + 0.638874i \(0.220599\pi\)
\(194\) 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 \(-0.106612\pi\)
0.187550 + 0.982255i \(0.439945\pi\)
\(200\) 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 −0.968213 0.250126i \(-0.919528\pi\)
0.700722 + 0.713434i \(0.252861\pi\)
\(228\) 0 0
\(229\) −124.878 + 72.0983i −0.545319 + 0.314840i −0.747232 0.664564i \(-0.768617\pi\)
0.201913 + 0.979403i \(0.435284\pi\)
\(230\) 0 0
\(231\) −492.360 + 77.7325i −2.13143 + 0.336504i
\(232\) 0 0
\(233\) −97.1309 168.236i −0.416871 0.722042i 0.578752 0.815504i \(-0.303540\pi\)
−0.995623 + 0.0934621i \(0.970207\pi\)
\(234\) 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 0.194089 + 0.980984i \(0.437825\pi\)
−0.946601 + 0.322406i \(0.895508\pi\)
\(242\) 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 0.960221 + 0.279243i \(0.0900834\pi\)
−0.238279 + 0.971197i \(0.576583\pi\)
\(258\) 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 0.997214 + 0.0745993i \(0.0237678\pi\)
0.563212 + 0.826313i \(0.309566\pi\)
\(264\) 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 \(-0.277364\pi\)
−0.340799 + 0.940136i \(0.610697\pi\)
\(270\) 0 0
\(271\) −137.021 + 79.1093i −0.505614 + 0.291916i −0.731029 0.682347i \(-0.760959\pi\)
0.225415 + 0.974263i \(0.427626\pi\)
\(272\) 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 \(-0.192788\pi\)
−0.0819717 + 0.996635i \(0.526122\pi\)
\(278\) 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 −0.995162 0.0982453i \(-0.968677\pi\)
0.582664 + 0.812713i \(0.302010\pi\)
\(284\) 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 \(-0.392894\pi\)
−0.652374 + 0.757897i \(0.726227\pi\)
\(312\) 0 0
\(313\) −105.261 182.317i −0.336297 0.582483i 0.647436 0.762120i \(-0.275841\pi\)
−0.983733 + 0.179637i \(0.942508\pi\)
\(314\) 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 \(-0.852744\pi\)
−0.0609413 + 0.998141i \(0.519410\pi\)
\(318\) 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 0.981193 0.193028i \(-0.0618308\pi\)
−0.657764 + 0.753224i \(0.728497\pi\)
\(332\) 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 −0.454896 0.890545i \(-0.650324\pi\)
0.998682 0.0513208i \(-0.0163431\pi\)
\(348\) 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 −0.996247 0.0865610i \(-0.972412\pi\)
0.573087 + 0.819494i \(0.305746\pi\)
\(354\) 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 \(-0.766187\pi\)
0.209385 + 0.977833i \(0.432854\pi\)
\(360\) 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 \(-0.428116\pi\)
−0.732079 + 0.681220i \(0.761450\pi\)
\(368\) 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 \(-0.569645\pi\)
0.736851 + 0.676055i \(0.236312\pi\)
\(374\) 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 \(-0.728806\pi\)
0.322512 + 0.946565i \(0.395473\pi\)
\(384\) 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 \(-0.398563\pi\)
−0.665768 + 0.746159i \(0.731896\pi\)
\(390\) 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 \(-0.780417\pi\)
0.165476 + 0.986214i \(0.447084\pi\)
\(398\) 0 0
\(399\) 183.527 477.399i 0.459968 1.19649i
\(400\) 0 0
\(401\) −311.899 540.225i −0.777803 1.34719i −0.933205 0.359344i \(-0.883001\pi\)
0.155402 0.987851i \(-0.450333\pi\)
\(402\) 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 −0.386701 0.922205i \(-0.626386\pi\)
0.992004 0.126209i \(-0.0402810\pi\)
\(410\) 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 \(-0.206759\pi\)
−0.125620 + 0.992078i \(0.540092\pi\)
\(432\) 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 \(-0.482069\pi\)
0.892803 + 0.450448i \(0.148736\pi\)
\(440\) 0 0
\(441\) −115.650 + 541.455i −0.262245 + 1.22779i
\(442\) 0 0
\(443\) 92.3599 + 159.972i 0.208487 + 0.361111i 0.951238 0.308457i \(-0.0998127\pi\)
−0.742751 + 0.669568i \(0.766479\pi\)
\(444\) 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 0.939597 0.342283i \(-0.111200\pi\)
−0.766224 + 0.642573i \(0.777867\pi\)
\(458\) 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 0.885822 0.464025i \(-0.153595\pi\)
−0.844768 + 0.535132i \(0.820262\pi\)
\(468\) 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 \(-0.569872\pi\)
−0.954120 + 0.299424i \(0.903205\pi\)
\(480\) 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 \(-0.450411\pi\)
−0.777959 + 0.628316i \(0.783745\pi\)
\(488\) 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 −0.999361 0.0357329i \(-0.988623\pi\)
0.468735 0.883339i \(-0.344710\pi\)
\(500\) 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 \(-0.916702\pi\)
−0.258926 + 0.965897i \(0.583369\pi\)
\(510\) 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 −0.993391 0.114779i \(-0.963384\pi\)
0.596097 + 0.802912i \(0.296717\pi\)
\(522\) 0 0
\(523\) −174.324 301.938i −0.333316 0.577320i 0.649844 0.760068i \(-0.274834\pi\)
−0.983160 + 0.182747i \(0.941501\pi\)
\(524\) 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 \(-0.597213\pi\)
0.675613 + 0.737257i \(0.263879\pi\)
\(542\) 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 \(-0.345453\pi\)
−0.532605 + 0.846364i \(0.678787\pi\)
\(558\) 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 −0.954693 0.297593i \(-0.903816\pi\)
0.735069 + 0.677992i \(0.237150\pi\)
\(564\) 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 −0.998300 0.0582847i \(-0.981437\pi\)
0.448674 0.893696i \(-0.351896\pi\)
\(570\) 0 0
\(571\) −291.287 + 504.524i −0.510135 + 0.883579i 0.489796 + 0.871837i \(0.337071\pi\)
−0.999931 + 0.0117423i \(0.996262\pi\)
\(572\) 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 −0.641324 0.767270i \(-0.721615\pi\)
0.985137 + 0.171768i \(0.0549478\pi\)
\(578\) 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 0.931792 0.362991i \(-0.118245\pi\)
−0.780256 + 0.625460i \(0.784911\pi\)
\(594\) 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 \(-0.322522\pi\)
−0.470303 + 0.882505i \(0.655856\pi\)
\(600\) 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 \(-0.750348\pi\)
0.257763 + 0.966208i \(0.417014\pi\)
\(608\) 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 \(-0.223728\pi\)
−0.178303 + 0.983976i \(0.557061\pi\)
\(614\) 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 −0.699010 0.715112i \(-0.746376\pi\)
0.968810 + 0.247804i \(0.0797090\pi\)
\(620\) 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 −0.890230 0.455511i \(-0.849457\pi\)
0.839599 + 0.543206i \(0.182790\pi\)
\(642\) 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 −0.939691 0.342024i \(-0.888888\pi\)
−0.766047 0.642785i \(-0.777779\pi\)
\(648\) 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 \(-0.803973\pi\)
0.0921079 + 0.995749i \(0.470640\pi\)
\(654\) 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 \(-0.549159\pi\)
−0.932630 + 0.360834i \(0.882492\pi\)
\(662\) 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 \(-0.347811\pi\)
0.998966 + 0.0454660i \(0.0144773\pi\)
\(678\) 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 −0.819845 0.572585i \(-0.805941\pi\)
0.905796 + 0.423715i \(0.139274\pi\)
\(684\) 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 0.971282 + 0.237930i \(0.0764690\pi\)
−0.279588 + 0.960120i \(0.590198\pi\)
\(692\) 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 \(-0.695888\pi\)
0.418504 + 0.908215i \(0.362555\pi\)
\(710\) 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 \(-0.552097\pi\)
0.772984 + 0.634426i \(0.218763\pi\)
\(720\) 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 \(-0.505566\pi\)
0.857151 + 0.515066i \(0.172233\pi\)
\(734\) 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 −0.928540 0.371232i \(-0.878935\pi\)
0.785767 + 0.618523i \(0.212269\pi\)
\(740\) 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 \(-0.731406\pi\)
0.314769 + 0.949168i \(0.398073\pi\)
\(752\) 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 0.834643 0.550792i \(-0.185674\pi\)
−0.894321 + 0.447426i \(0.852341\pi\)
\(762\) 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 \(-0.209430\pi\)
−0.133942 + 0.990989i \(0.542763\pi\)
\(774\) 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 −0.00107958 0.999999i \(-0.500344\pi\)
0.866565 0.499065i \(-0.166323\pi\)
\(788\) 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 −0.848240 0.529613i \(-0.822337\pi\)
0.882778 + 0.469791i \(0.155671\pi\)
\(810\) 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 \(-0.355877\pi\)
0.997493 + 0.0707646i \(0.0225439\pi\)
\(822\) 0 0
\(823\) 488.057 + 281.780i 0.593022 + 0.342382i 0.766292 0.642493i \(-0.222100\pi\)
−0.173269 + 0.984874i \(0.555433\pi\)
\(824\) 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 \(-0.277038\pi\)
−0.339837 + 0.940484i \(0.610372\pi\)
\(830\) 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 0.292880 + 0.956149i \(0.405386\pi\)
−0.974489 + 0.224433i \(0.927947\pi\)
\(858\) 0 0
\(859\) 733.208 + 1269.95i 0.853560 + 1.47841i 0.877974 + 0.478708i \(0.158895\pi\)
−0.0244139 + 0.999702i \(0.507772\pi\)
\(860\) 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 \(-0.674545\pi\)
0.478415 + 0.878134i \(0.341212\pi\)
\(864\) 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 \(-0.853739\pi\)
−0.0640612 + 0.997946i \(0.520405\pi\)
\(878\) 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 \(-0.756911\pi\)
0.237788 + 0.971317i \(0.423578\pi\)
\(888\) 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 0.0239725 0.999713i \(-0.492369\pi\)
0.853790 0.520617i \(-0.174298\pi\)
\(908\) 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 \(-0.577232\pi\)
0.720530 + 0.693424i \(0.243898\pi\)
\(920\) 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 0.939627 0.342201i \(-0.111172\pi\)
−0.766168 + 0.642640i \(0.777839\pi\)
\(930\) 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 \(-0.345757\pi\)
−0.533411 + 0.845856i \(0.679090\pi\)
\(942\) 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 0.885591 0.464465i \(-0.153753\pi\)
−0.845034 + 0.534712i \(0.820420\pi\)
\(948\) 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 0.954657 0.297709i \(-0.0962225\pi\)
−0.735152 + 0.677902i \(0.762889\pi\)
\(972\) 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 −0.916784 0.399383i \(-0.869224\pi\)
0.804268 + 0.594267i \(0.202558\pi\)
\(978\) 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 \(-0.0108173\pi\)
0.470286 + 0.882514i \(0.344151\pi\)
\(984\) 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 \(-0.0909851\pi\)
0.235527 + 0.971868i \(0.424318\pi\)
\(992\) 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 \(-0.179321\pi\)
−0.0397440 + 0.999210i \(0.512654\pi\)
\(998\) 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.207.1 12
4.3 odd 2 56.3.k.c.11.5 yes 12
7.2 even 3 inner 224.3.o.c.79.2 12
7.3 odd 6 1568.3.g.i.687.2 6
7.4 even 3 1568.3.g.k.687.5 6
8.3 odd 2 inner 224.3.o.c.207.2 12
8.5 even 2 56.3.k.c.11.4 12
28.3 even 6 392.3.g.l.99.2 6
28.11 odd 6 392.3.g.k.99.2 6
28.19 even 6 392.3.k.k.275.4 12
28.23 odd 6 56.3.k.c.51.4 yes 12
28.27 even 2 392.3.k.k.67.5 12
56.3 even 6 1568.3.g.i.687.1 6
56.5 odd 6 392.3.k.k.275.5 12
56.11 odd 6 1568.3.g.k.687.6 6
56.13 odd 2 392.3.k.k.67.4 12
56.37 even 6 56.3.k.c.51.5 yes 12
56.45 odd 6 392.3.g.l.99.1 6
56.51 odd 6 inner 224.3.o.c.79.1 12
56.53 even 6 392.3.g.k.99.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.3.k.c.11.4 12 8.5 even 2
56.3.k.c.11.5 yes 12 4.3 odd 2
56.3.k.c.51.4 yes 12 28.23 odd 6
56.3.k.c.51.5 yes 12 56.37 even 6
224.3.o.c.79.1 12 56.51 odd 6 inner
224.3.o.c.79.2 12 7.2 even 3 inner
224.3.o.c.207.1 12 1.1 even 1 trivial
224.3.o.c.207.2 12 8.3 odd 2 inner
392.3.g.k.99.1 6 56.53 even 6
392.3.g.k.99.2 6 28.11 odd 6
392.3.g.l.99.1 6 56.45 odd 6
392.3.g.l.99.2 6 28.3 even 6
392.3.k.k.67.4 12 56.13 odd 2
392.3.k.k.67.5 12 28.27 even 2
392.3.k.k.275.4 12 28.19 even 6
392.3.k.k.275.5 12 56.5 odd 6
1568.3.g.i.687.1 6 56.3 even 6
1568.3.g.i.687.2 6 7.3 odd 6
1568.3.g.k.687.5 6 7.4 even 3
1568.3.g.k.687.6 6 56.11 odd 6