Properties

Label 560.2.ci.e.33.1
Level $560$
Weight $2$
Character 560.33
Analytic conductor $4.472$
Analytic rank $0$
Dimension $48$
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] = [560,2,Mod(17,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.ci (of order \(12\), degree \(4\), 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: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 33.1
Character \(\chi\) \(=\) 560.33
Dual form 560.2.ci.e.17.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.19510 - 0.856125i) q^{3} +(0.672461 - 2.13256i) q^{5} +(2.52104 - 0.802724i) q^{7} +(6.87765 + 3.97081i) q^{9} +O(q^{10})\) \(q+(-3.19510 - 0.856125i) q^{3} +(0.672461 - 2.13256i) q^{5} +(2.52104 - 0.802724i) q^{7} +(6.87765 + 3.97081i) q^{9} +(1.05351 + 1.82473i) q^{11} +(-1.20510 + 1.20510i) q^{13} +(-3.97432 + 6.23802i) q^{15} +(0.850985 - 3.17592i) q^{17} +(2.36491 - 4.09614i) q^{19} +(-8.74221 + 0.406461i) q^{21} +(4.00492 - 1.07312i) q^{23} +(-4.09559 - 2.86812i) q^{25} +(-11.5583 - 11.5583i) q^{27} +3.65910i q^{29} +(1.63858 - 0.946036i) q^{31} +(-1.80387 - 6.73212i) q^{33} +(-0.0165540 - 5.91606i) q^{35} +(-2.65625 - 9.91327i) q^{37} +(4.88214 - 2.81870i) q^{39} +0.826081i q^{41} +(-4.70172 - 4.70172i) q^{43} +(13.0929 - 11.9968i) q^{45} +(-3.90561 + 1.04650i) q^{47} +(5.71127 - 4.04740i) q^{49} +(-5.43796 + 9.41883i) q^{51} +(-0.645746 + 2.40996i) q^{53} +(4.59977 - 1.01960i) q^{55} +(-11.0629 + 11.0629i) q^{57} +(1.15566 + 2.00166i) q^{59} +(1.44974 + 0.837008i) q^{61} +(20.5263 + 4.48972i) q^{63} +(1.75956 + 3.38033i) q^{65} +(12.2602 + 3.28512i) q^{67} -13.7149 q^{69} -10.7915 q^{71} +(-12.6133 - 3.37973i) q^{73} +(10.6304 + 12.6703i) q^{75} +(4.12068 + 3.75453i) q^{77} +(-8.29280 - 4.78785i) q^{79} +(15.1223 + 26.1925i) q^{81} +(3.47691 - 3.47691i) q^{83} +(-6.20057 - 3.95045i) q^{85} +(3.13265 - 11.6912i) q^{87} +(2.86698 - 4.96576i) q^{89} +(-2.07074 + 4.00547i) q^{91} +(-6.04536 + 1.61985i) q^{93} +(-7.14494 - 7.79779i) q^{95} +(-1.02714 - 1.02714i) q^{97} +16.7331i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 4 q^{7} - 4 q^{11} - 8 q^{15} - 4 q^{21} + 4 q^{23} - 8 q^{25} - 36 q^{33} - 24 q^{35} + 8 q^{37} + 16 q^{43} + 48 q^{45} - 24 q^{51} + 16 q^{53} - 96 q^{57} - 36 q^{61} + 68 q^{63} + 12 q^{65} + 16 q^{67} + 64 q^{71} - 48 q^{73} + 48 q^{75} + 4 q^{77} - 40 q^{85} + 12 q^{87} + 80 q^{91} - 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\) \(1\)

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
\(3\) −3.19510 0.856125i −1.84469 0.494284i −0.845482 0.534004i \(-0.820687\pi\)
−0.999211 + 0.0397200i \(0.987353\pi\)
\(4\) 0 0
\(5\) 0.672461 2.13256i 0.300734 0.953708i
\(6\) 0 0
\(7\) 2.52104 0.802724i 0.952863 0.303401i
\(8\) 0 0
\(9\) 6.87765 + 3.97081i 2.29255 + 1.32360i
\(10\) 0 0
\(11\) 1.05351 + 1.82473i 0.317644 + 0.550175i 0.979996 0.199017i \(-0.0637750\pi\)
−0.662352 + 0.749193i \(0.730442\pi\)
\(12\) 0 0
\(13\) −1.20510 + 1.20510i −0.334235 + 0.334235i −0.854192 0.519957i \(-0.825948\pi\)
0.519957 + 0.854192i \(0.325948\pi\)
\(14\) 0 0
\(15\) −3.97432 + 6.23802i −1.02616 + 1.61065i
\(16\) 0 0
\(17\) 0.850985 3.17592i 0.206394 0.770273i −0.782626 0.622492i \(-0.786120\pi\)
0.989020 0.147781i \(-0.0472131\pi\)
\(18\) 0 0
\(19\) 2.36491 4.09614i 0.542547 0.939718i −0.456210 0.889872i \(-0.650794\pi\)
0.998757 0.0498463i \(-0.0158731\pi\)
\(20\) 0 0
\(21\) −8.74221 + 0.406461i −1.90771 + 0.0886970i
\(22\) 0 0
\(23\) 4.00492 1.07312i 0.835084 0.223760i 0.184154 0.982897i \(-0.441046\pi\)
0.650931 + 0.759137i \(0.274379\pi\)
\(24\) 0 0
\(25\) −4.09559 2.86812i −0.819118 0.573624i
\(26\) 0 0
\(27\) −11.5583 11.5583i −2.22440 2.22440i
\(28\) 0 0
\(29\) 3.65910i 0.679478i 0.940520 + 0.339739i \(0.110339\pi\)
−0.940520 + 0.339739i \(0.889661\pi\)
\(30\) 0 0
\(31\) 1.63858 0.946036i 0.294298 0.169913i −0.345581 0.938389i \(-0.612318\pi\)
0.639879 + 0.768476i \(0.278985\pi\)
\(32\) 0 0
\(33\) −1.80387 6.73212i −0.314013 1.17191i
\(34\) 0 0
\(35\) −0.0165540 5.91606i −0.00279813 0.999996i
\(36\) 0 0
\(37\) −2.65625 9.91327i −0.436685 1.62973i −0.737001 0.675891i \(-0.763759\pi\)
0.300316 0.953840i \(-0.402908\pi\)
\(38\) 0 0
\(39\) 4.88214 2.81870i 0.781768 0.451354i
\(40\) 0 0
\(41\) 0.826081i 0.129012i 0.997917 + 0.0645061i \(0.0205472\pi\)
−0.997917 + 0.0645061i \(0.979453\pi\)
\(42\) 0 0
\(43\) −4.70172 4.70172i −0.717005 0.717005i 0.250985 0.967991i \(-0.419245\pi\)
−0.967991 + 0.250985i \(0.919245\pi\)
\(44\) 0 0
\(45\) 13.0929 11.9968i 1.95178 1.78837i
\(46\) 0 0
\(47\) −3.90561 + 1.04650i −0.569692 + 0.152648i −0.532155 0.846647i \(-0.678618\pi\)
−0.0375365 + 0.999295i \(0.511951\pi\)
\(48\) 0 0
\(49\) 5.71127 4.04740i 0.815896 0.578199i
\(50\) 0 0
\(51\) −5.43796 + 9.41883i −0.761467 + 1.31890i
\(52\) 0 0
\(53\) −0.645746 + 2.40996i −0.0887000 + 0.331033i −0.995989 0.0894744i \(-0.971481\pi\)
0.907289 + 0.420507i \(0.138148\pi\)
\(54\) 0 0
\(55\) 4.59977 1.01960i 0.620233 0.137483i
\(56\) 0 0
\(57\) −11.0629 + 11.0629i −1.46532 + 1.46532i
\(58\) 0 0
\(59\) 1.15566 + 2.00166i 0.150454 + 0.260594i 0.931394 0.364011i \(-0.118593\pi\)
−0.780940 + 0.624606i \(0.785260\pi\)
\(60\) 0 0
\(61\) 1.44974 + 0.837008i 0.185620 + 0.107168i 0.589931 0.807454i \(-0.299155\pi\)
−0.404310 + 0.914622i \(0.632488\pi\)
\(62\) 0 0
\(63\) 20.5263 + 4.48972i 2.58607 + 0.565651i
\(64\) 0 0
\(65\) 1.75956 + 3.38033i 0.218247 + 0.419278i
\(66\) 0 0
\(67\) 12.2602 + 3.28512i 1.49783 + 0.401342i 0.912372 0.409363i \(-0.134249\pi\)
0.585456 + 0.810704i \(0.300916\pi\)
\(68\) 0 0
\(69\) −13.7149 −1.65108
\(70\) 0 0
\(71\) −10.7915 −1.28071 −0.640356 0.768079i \(-0.721213\pi\)
−0.640356 + 0.768079i \(0.721213\pi\)
\(72\) 0 0
\(73\) −12.6133 3.37973i −1.47628 0.395567i −0.571198 0.820812i \(-0.693521\pi\)
−0.905078 + 0.425245i \(0.860188\pi\)
\(74\) 0 0
\(75\) 10.6304 + 12.6703i 1.22749 + 1.46304i
\(76\) 0 0
\(77\) 4.12068 + 3.75453i 0.469595 + 0.427868i
\(78\) 0 0
\(79\) −8.29280 4.78785i −0.933013 0.538675i −0.0452496 0.998976i \(-0.514408\pi\)
−0.887763 + 0.460301i \(0.847742\pi\)
\(80\) 0 0
\(81\) 15.1223 + 26.1925i 1.68025 + 2.91028i
\(82\) 0 0
\(83\) 3.47691 3.47691i 0.381641 0.381641i −0.490052 0.871693i \(-0.663022\pi\)
0.871693 + 0.490052i \(0.163022\pi\)
\(84\) 0 0
\(85\) −6.20057 3.95045i −0.672546 0.428487i
\(86\) 0 0
\(87\) 3.13265 11.6912i 0.335855 1.25343i
\(88\) 0 0
\(89\) 2.86698 4.96576i 0.303900 0.526370i −0.673116 0.739537i \(-0.735045\pi\)
0.977016 + 0.213167i \(0.0683779\pi\)
\(90\) 0 0
\(91\) −2.07074 + 4.00547i −0.217073 + 0.419887i
\(92\) 0 0
\(93\) −6.04536 + 1.61985i −0.626875 + 0.167971i
\(94\) 0 0
\(95\) −7.14494 7.79779i −0.733055 0.800036i
\(96\) 0 0
\(97\) −1.02714 1.02714i −0.104290 0.104290i 0.653036 0.757327i \(-0.273495\pi\)
−0.757327 + 0.653036i \(0.773495\pi\)
\(98\) 0 0
\(99\) 16.7331i 1.68174i
\(100\) 0 0
\(101\) 1.11869 0.645878i 0.111314 0.0642672i −0.443309 0.896369i \(-0.646196\pi\)
0.554623 + 0.832101i \(0.312862\pi\)
\(102\) 0 0
\(103\) 2.82954 + 10.5600i 0.278803 + 1.04051i 0.953250 + 0.302184i \(0.0977155\pi\)
−0.674447 + 0.738324i \(0.735618\pi\)
\(104\) 0 0
\(105\) −5.01199 + 18.9166i −0.489120 + 1.84607i
\(106\) 0 0
\(107\) 0.770212 + 2.87447i 0.0744592 + 0.277885i 0.993110 0.117185i \(-0.0373870\pi\)
−0.918651 + 0.395070i \(0.870720\pi\)
\(108\) 0 0
\(109\) 13.0694 7.54561i 1.25182 0.722739i 0.280349 0.959898i \(-0.409550\pi\)
0.971471 + 0.237160i \(0.0762164\pi\)
\(110\) 0 0
\(111\) 33.9480i 3.22220i
\(112\) 0 0
\(113\) 4.23171 + 4.23171i 0.398086 + 0.398086i 0.877558 0.479471i \(-0.159172\pi\)
−0.479471 + 0.877558i \(0.659172\pi\)
\(114\) 0 0
\(115\) 0.404675 9.26236i 0.0377361 0.863719i
\(116\) 0 0
\(117\) −13.0735 + 3.50303i −1.20865 + 0.323855i
\(118\) 0 0
\(119\) −0.404020 8.68972i −0.0370365 0.796585i
\(120\) 0 0
\(121\) 3.28025 5.68156i 0.298205 0.516506i
\(122\) 0 0
\(123\) 0.707229 2.63941i 0.0637687 0.237988i
\(124\) 0 0
\(125\) −8.87056 + 6.80538i −0.793407 + 0.608692i
\(126\) 0 0
\(127\) 4.53114 4.53114i 0.402073 0.402073i −0.476890 0.878963i \(-0.658236\pi\)
0.878963 + 0.476890i \(0.158236\pi\)
\(128\) 0 0
\(129\) 10.9972 + 19.0477i 0.968250 + 1.67706i
\(130\) 0 0
\(131\) 2.40936 + 1.39105i 0.210507 + 0.121536i 0.601547 0.798837i \(-0.294551\pi\)
−0.391040 + 0.920374i \(0.627885\pi\)
\(132\) 0 0
\(133\) 2.67395 12.2249i 0.231861 1.06003i
\(134\) 0 0
\(135\) −32.4214 + 16.8763i −2.79039 + 1.45248i
\(136\) 0 0
\(137\) 21.7199 + 5.81982i 1.85565 + 0.497221i 0.999800 0.0199984i \(-0.00636610\pi\)
0.855853 + 0.517219i \(0.173033\pi\)
\(138\) 0 0
\(139\) 8.33499 0.706965 0.353482 0.935441i \(-0.384997\pi\)
0.353482 + 0.935441i \(0.384997\pi\)
\(140\) 0 0
\(141\) 13.3748 1.12636
\(142\) 0 0
\(143\) −3.46856 0.929398i −0.290056 0.0777202i
\(144\) 0 0
\(145\) 7.80324 + 2.46060i 0.648024 + 0.204342i
\(146\) 0 0
\(147\) −21.7132 + 8.04228i −1.79087 + 0.663316i
\(148\) 0 0
\(149\) 7.44844 + 4.30036i 0.610200 + 0.352299i 0.773044 0.634353i \(-0.218733\pi\)
−0.162844 + 0.986652i \(0.552067\pi\)
\(150\) 0 0
\(151\) −5.13200 8.88888i −0.417636 0.723366i 0.578065 0.815990i \(-0.303808\pi\)
−0.995701 + 0.0926240i \(0.970475\pi\)
\(152\) 0 0
\(153\) 18.4637 18.4637i 1.49271 1.49271i
\(154\) 0 0
\(155\) −0.915592 4.13054i −0.0735421 0.331773i
\(156\) 0 0
\(157\) 2.63385 9.82966i 0.210204 0.784492i −0.777596 0.628764i \(-0.783561\pi\)
0.987800 0.155728i \(-0.0497723\pi\)
\(158\) 0 0
\(159\) 4.12645 7.14722i 0.327249 0.566811i
\(160\) 0 0
\(161\) 9.23515 5.92022i 0.727832 0.466578i
\(162\) 0 0
\(163\) −2.79779 + 0.749665i −0.219140 + 0.0587183i −0.366718 0.930332i \(-0.619519\pi\)
0.147579 + 0.989050i \(0.452852\pi\)
\(164\) 0 0
\(165\) −15.5696 0.680242i −1.21210 0.0529567i
\(166\) 0 0
\(167\) −8.39226 8.39226i −0.649412 0.649412i 0.303439 0.952851i \(-0.401865\pi\)
−0.952851 + 0.303439i \(0.901865\pi\)
\(168\) 0 0
\(169\) 10.0955i 0.776574i
\(170\) 0 0
\(171\) 32.5300 18.7812i 2.48763 1.43623i
\(172\) 0 0
\(173\) 0.0653257 + 0.243799i 0.00496662 + 0.0185357i 0.968365 0.249539i \(-0.0802792\pi\)
−0.963398 + 0.268075i \(0.913613\pi\)
\(174\) 0 0
\(175\) −12.6275 3.94301i −0.954546 0.298064i
\(176\) 0 0
\(177\) −1.97878 7.38490i −0.148734 0.555083i
\(178\) 0 0
\(179\) −18.8244 + 10.8683i −1.40700 + 0.812331i −0.995098 0.0988972i \(-0.968469\pi\)
−0.411901 + 0.911228i \(0.635135\pi\)
\(180\) 0 0
\(181\) 3.06050i 0.227485i 0.993510 + 0.113743i \(0.0362839\pi\)
−0.993510 + 0.113743i \(0.963716\pi\)
\(182\) 0 0
\(183\) −3.91548 3.91548i −0.289441 0.289441i
\(184\) 0 0
\(185\) −22.9268 1.00168i −1.68561 0.0736449i
\(186\) 0 0
\(187\) 6.69170 1.79303i 0.489345 0.131120i
\(188\) 0 0
\(189\) −38.4172 19.8609i −2.79444 1.44467i
\(190\) 0 0
\(191\) 10.5499 18.2729i 0.763361 1.32218i −0.177748 0.984076i \(-0.556881\pi\)
0.941109 0.338103i \(-0.109785\pi\)
\(192\) 0 0
\(193\) −5.39085 + 20.1189i −0.388042 + 1.44819i 0.445273 + 0.895395i \(0.353106\pi\)
−0.833315 + 0.552798i \(0.813560\pi\)
\(194\) 0 0
\(195\) −2.72800 12.3069i −0.195356 0.881316i
\(196\) 0 0
\(197\) −9.29732 + 9.29732i −0.662406 + 0.662406i −0.955947 0.293540i \(-0.905167\pi\)
0.293540 + 0.955947i \(0.405167\pi\)
\(198\) 0 0
\(199\) −3.77827 6.54415i −0.267834 0.463902i 0.700468 0.713684i \(-0.252975\pi\)
−0.968302 + 0.249781i \(0.919641\pi\)
\(200\) 0 0
\(201\) −36.3603 20.9926i −2.56465 1.48070i
\(202\) 0 0
\(203\) 2.93725 + 9.22474i 0.206154 + 0.647450i
\(204\) 0 0
\(205\) 1.76167 + 0.555508i 0.123040 + 0.0387983i
\(206\) 0 0
\(207\) 31.8056 + 8.52229i 2.21064 + 0.592340i
\(208\) 0 0
\(209\) 9.96577 0.689347
\(210\) 0 0
\(211\) 1.32737 0.0913799 0.0456900 0.998956i \(-0.485451\pi\)
0.0456900 + 0.998956i \(0.485451\pi\)
\(212\) 0 0
\(213\) 34.4798 + 9.23884i 2.36252 + 0.633035i
\(214\) 0 0
\(215\) −13.1884 + 6.86496i −0.899442 + 0.468186i
\(216\) 0 0
\(217\) 3.37152 3.70032i 0.228874 0.251194i
\(218\) 0 0
\(219\) 37.4074 + 21.5971i 2.52775 + 1.45940i
\(220\) 0 0
\(221\) 2.80178 + 4.85282i 0.188468 + 0.326436i
\(222\) 0 0
\(223\) 7.24966 7.24966i 0.485473 0.485473i −0.421401 0.906874i \(-0.638462\pi\)
0.906874 + 0.421401i \(0.138462\pi\)
\(224\) 0 0
\(225\) −16.7793 35.9888i −1.11862 2.39925i
\(226\) 0 0
\(227\) −7.76791 + 28.9902i −0.515574 + 1.92415i −0.171618 + 0.985164i \(0.554899\pi\)
−0.343956 + 0.938986i \(0.611767\pi\)
\(228\) 0 0
\(229\) 10.3484 17.9239i 0.683841 1.18445i −0.289958 0.957039i \(-0.593641\pi\)
0.973800 0.227408i \(-0.0730252\pi\)
\(230\) 0 0
\(231\) −9.95164 15.5239i −0.654770 1.02140i
\(232\) 0 0
\(233\) 3.82659 1.02533i 0.250688 0.0671717i −0.131287 0.991344i \(-0.541911\pi\)
0.381975 + 0.924173i \(0.375244\pi\)
\(234\) 0 0
\(235\) −0.394639 + 9.03266i −0.0257434 + 0.589226i
\(236\) 0 0
\(237\) 22.3973 + 22.3973i 1.45486 + 1.45486i
\(238\) 0 0
\(239\) 14.2588i 0.922325i −0.887316 0.461163i \(-0.847432\pi\)
0.887316 0.461163i \(-0.152568\pi\)
\(240\) 0 0
\(241\) −4.21711 + 2.43475i −0.271648 + 0.156836i −0.629636 0.776890i \(-0.716796\pi\)
0.357989 + 0.933726i \(0.383463\pi\)
\(242\) 0 0
\(243\) −13.2012 49.2674i −0.846854 3.16050i
\(244\) 0 0
\(245\) −4.79069 14.9013i −0.306066 0.952010i
\(246\) 0 0
\(247\) 2.08631 + 7.78621i 0.132749 + 0.495425i
\(248\) 0 0
\(249\) −14.0858 + 8.13241i −0.892648 + 0.515371i
\(250\) 0 0
\(251\) 16.5118i 1.04222i 0.853491 + 0.521108i \(0.174481\pi\)
−0.853491 + 0.521108i \(0.825519\pi\)
\(252\) 0 0
\(253\) 6.17735 + 6.17735i 0.388367 + 0.388367i
\(254\) 0 0
\(255\) 16.4294 + 17.9306i 1.02885 + 1.12286i
\(256\) 0 0
\(257\) −9.47367 + 2.53846i −0.590951 + 0.158345i −0.541888 0.840451i \(-0.682290\pi\)
−0.0490634 + 0.998796i \(0.515624\pi\)
\(258\) 0 0
\(259\) −14.6541 22.8595i −0.910563 1.42042i
\(260\) 0 0
\(261\) −14.5296 + 25.1660i −0.899360 + 1.55774i
\(262\) 0 0
\(263\) −3.21180 + 11.9866i −0.198048 + 0.739125i 0.793409 + 0.608689i \(0.208304\pi\)
−0.991457 + 0.130436i \(0.958362\pi\)
\(264\) 0 0
\(265\) 4.70513 + 2.99769i 0.289034 + 0.184147i
\(266\) 0 0
\(267\) −13.4116 + 13.4116i −0.820778 + 0.820778i
\(268\) 0 0
\(269\) 1.86530 + 3.23079i 0.113729 + 0.196985i 0.917271 0.398264i \(-0.130387\pi\)
−0.803542 + 0.595248i \(0.797054\pi\)
\(270\) 0 0
\(271\) 1.43539 + 0.828725i 0.0871940 + 0.0503415i 0.542963 0.839757i \(-0.317302\pi\)
−0.455769 + 0.890098i \(0.650636\pi\)
\(272\) 0 0
\(273\) 10.0454 11.0251i 0.607976 0.667268i
\(274\) 0 0
\(275\) 0.918805 10.4949i 0.0554060 0.632867i
\(276\) 0 0
\(277\) −14.7235 3.94516i −0.884652 0.237042i −0.212239 0.977218i \(-0.568076\pi\)
−0.672413 + 0.740176i \(0.734742\pi\)
\(278\) 0 0
\(279\) 15.0261 0.899590
\(280\) 0 0
\(281\) −2.25722 −0.134655 −0.0673273 0.997731i \(-0.521447\pi\)
−0.0673273 + 0.997731i \(0.521447\pi\)
\(282\) 0 0
\(283\) 5.87101 + 1.57313i 0.348995 + 0.0935130i 0.429058 0.903277i \(-0.358846\pi\)
−0.0800631 + 0.996790i \(0.525512\pi\)
\(284\) 0 0
\(285\) 16.1529 + 31.0317i 0.956816 + 1.83816i
\(286\) 0 0
\(287\) 0.663115 + 2.08258i 0.0391425 + 0.122931i
\(288\) 0 0
\(289\) 5.36015 + 3.09469i 0.315303 + 0.182040i
\(290\) 0 0
\(291\) 2.40246 + 4.16118i 0.140834 + 0.243932i
\(292\) 0 0
\(293\) −17.5770 + 17.5770i −1.02686 + 1.02686i −0.0272289 + 0.999629i \(0.508668\pi\)
−0.999629 + 0.0272289i \(0.991332\pi\)
\(294\) 0 0
\(295\) 5.04579 1.11847i 0.293777 0.0651198i
\(296\) 0 0
\(297\) 8.91403 33.2676i 0.517244 1.93038i
\(298\) 0 0
\(299\) −3.53313 + 6.11955i −0.204326 + 0.353903i
\(300\) 0 0
\(301\) −15.6274 8.07903i −0.900748 0.465668i
\(302\) 0 0
\(303\) −4.12729 + 1.10590i −0.237107 + 0.0635325i
\(304\) 0 0
\(305\) 2.75986 2.52880i 0.158029 0.144798i
\(306\) 0 0
\(307\) 21.1121 + 21.1121i 1.20493 + 1.20493i 0.972648 + 0.232283i \(0.0746195\pi\)
0.232283 + 0.972648i \(0.425380\pi\)
\(308\) 0 0
\(309\) 36.1627i 2.05722i
\(310\) 0 0
\(311\) −16.3420 + 9.43506i −0.926670 + 0.535013i −0.885757 0.464150i \(-0.846360\pi\)
−0.0409129 + 0.999163i \(0.513027\pi\)
\(312\) 0 0
\(313\) 6.71594 + 25.0642i 0.379607 + 1.41671i 0.846495 + 0.532397i \(0.178709\pi\)
−0.466888 + 0.884317i \(0.654625\pi\)
\(314\) 0 0
\(315\) 23.3777 40.7543i 1.31718 2.29624i
\(316\) 0 0
\(317\) 0.370421 + 1.38243i 0.0208049 + 0.0776451i 0.975548 0.219787i \(-0.0705364\pi\)
−0.954743 + 0.297432i \(0.903870\pi\)
\(318\) 0 0
\(319\) −6.67686 + 3.85489i −0.373832 + 0.215832i
\(320\) 0 0
\(321\) 9.84362i 0.549417i
\(322\) 0 0
\(323\) −10.9965 10.9965i −0.611861 0.611861i
\(324\) 0 0
\(325\) 8.39198 1.47923i 0.465503 0.0820527i
\(326\) 0 0
\(327\) −48.2180 + 12.9200i −2.66646 + 0.714476i
\(328\) 0 0
\(329\) −9.00614 + 5.77340i −0.496524 + 0.318298i
\(330\) 0 0
\(331\) 8.21858 14.2350i 0.451734 0.782426i −0.546760 0.837290i \(-0.684139\pi\)
0.998494 + 0.0548631i \(0.0174722\pi\)
\(332\) 0 0
\(333\) 21.0950 78.7275i 1.15600 4.31424i
\(334\) 0 0
\(335\) 15.2502 23.9365i 0.833210 1.30779i
\(336\) 0 0
\(337\) 16.4453 16.4453i 0.895835 0.895835i −0.0992293 0.995065i \(-0.531638\pi\)
0.995065 + 0.0992293i \(0.0316377\pi\)
\(338\) 0 0
\(339\) −9.89788 17.1436i −0.537579 0.931115i
\(340\) 0 0
\(341\) 3.45251 + 1.99331i 0.186964 + 0.107944i
\(342\) 0 0
\(343\) 11.1494 14.7882i 0.602010 0.798488i
\(344\) 0 0
\(345\) −9.22271 + 29.2477i −0.496534 + 1.57464i
\(346\) 0 0
\(347\) 3.43767 + 0.921122i 0.184544 + 0.0494484i 0.349907 0.936784i \(-0.386213\pi\)
−0.165363 + 0.986233i \(0.552880\pi\)
\(348\) 0 0
\(349\) −8.16515 −0.437070 −0.218535 0.975829i \(-0.570128\pi\)
−0.218535 + 0.975829i \(0.570128\pi\)
\(350\) 0 0
\(351\) 27.8580 1.48695
\(352\) 0 0
\(353\) −0.779678 0.208914i −0.0414981 0.0111194i 0.238010 0.971263i \(-0.423505\pi\)
−0.279508 + 0.960143i \(0.590171\pi\)
\(354\) 0 0
\(355\) −7.25684 + 23.0134i −0.385153 + 1.22142i
\(356\) 0 0
\(357\) −6.14860 + 28.1104i −0.325418 + 1.48776i
\(358\) 0 0
\(359\) 23.7687 + 13.7228i 1.25446 + 0.724264i 0.971992 0.235012i \(-0.0755131\pi\)
0.282469 + 0.959276i \(0.408846\pi\)
\(360\) 0 0
\(361\) −1.68556 2.91947i −0.0887136 0.153656i
\(362\) 0 0
\(363\) −15.3449 + 15.3449i −0.805396 + 0.805396i
\(364\) 0 0
\(365\) −15.6894 + 24.6259i −0.821222 + 1.28898i
\(366\) 0 0
\(367\) 3.44136 12.8433i 0.179637 0.670416i −0.816078 0.577942i \(-0.803856\pi\)
0.995715 0.0924739i \(-0.0294775\pi\)
\(368\) 0 0
\(369\) −3.28021 + 5.68150i −0.170761 + 0.295767i
\(370\) 0 0
\(371\) 0.306579 + 6.59395i 0.0159168 + 0.342341i
\(372\) 0 0
\(373\) −12.3794 + 3.31706i −0.640982 + 0.171751i −0.564648 0.825332i \(-0.690988\pi\)
−0.0763341 + 0.997082i \(0.524322\pi\)
\(374\) 0 0
\(375\) 34.1686 14.1496i 1.76446 0.730681i
\(376\) 0 0
\(377\) −4.40959 4.40959i −0.227105 0.227105i
\(378\) 0 0
\(379\) 3.65754i 0.187875i −0.995578 0.0939375i \(-0.970055\pi\)
0.995578 0.0939375i \(-0.0299454\pi\)
\(380\) 0 0
\(381\) −18.3567 + 10.5982i −0.940441 + 0.542964i
\(382\) 0 0
\(383\) 7.80691 + 29.1358i 0.398915 + 1.48877i 0.815009 + 0.579448i \(0.196732\pi\)
−0.416095 + 0.909321i \(0.636601\pi\)
\(384\) 0 0
\(385\) 10.7777 6.26281i 0.549285 0.319182i
\(386\) 0 0
\(387\) −13.6671 51.0064i −0.694739 2.59280i
\(388\) 0 0
\(389\) −5.17854 + 2.98983i −0.262562 + 0.151591i −0.625503 0.780222i \(-0.715106\pi\)
0.362940 + 0.931812i \(0.381773\pi\)
\(390\) 0 0
\(391\) 13.6325i 0.689426i
\(392\) 0 0
\(393\) −6.50725 6.50725i −0.328247 0.328247i
\(394\) 0 0
\(395\) −15.7869 + 14.4652i −0.794327 + 0.727824i
\(396\) 0 0
\(397\) −1.45836 + 0.390768i −0.0731932 + 0.0196121i −0.295230 0.955426i \(-0.595396\pi\)
0.222037 + 0.975038i \(0.428730\pi\)
\(398\) 0 0
\(399\) −19.0096 + 36.7705i −0.951669 + 1.84083i
\(400\) 0 0
\(401\) 12.6011 21.8258i 0.629269 1.08993i −0.358429 0.933557i \(-0.616688\pi\)
0.987699 0.156370i \(-0.0499791\pi\)
\(402\) 0 0
\(403\) −0.834588 + 3.11473i −0.0415738 + 0.155156i
\(404\) 0 0
\(405\) 66.0262 14.6356i 3.28087 0.727250i
\(406\) 0 0
\(407\) 15.2906 15.2906i 0.757928 0.757928i
\(408\) 0 0
\(409\) −11.7806 20.4046i −0.582513 1.00894i −0.995180 0.0980604i \(-0.968736\pi\)
0.412667 0.910882i \(-0.364597\pi\)
\(410\) 0 0
\(411\) −64.4147 37.1898i −3.17734 1.83444i
\(412\) 0 0
\(413\) 4.52024 + 4.11859i 0.222427 + 0.202663i
\(414\) 0 0
\(415\) −5.07662 9.75280i −0.249202 0.478746i
\(416\) 0 0
\(417\) −26.6312 7.13580i −1.30413 0.349441i
\(418\) 0 0
\(419\) 0.614382 0.0300145 0.0150073 0.999887i \(-0.495223\pi\)
0.0150073 + 0.999887i \(0.495223\pi\)
\(420\) 0 0
\(421\) 13.0643 0.636713 0.318356 0.947971i \(-0.396869\pi\)
0.318356 + 0.947971i \(0.396869\pi\)
\(422\) 0 0
\(423\) −31.0169 8.31095i −1.50809 0.404092i
\(424\) 0 0
\(425\) −12.5942 + 10.5665i −0.610909 + 0.512552i
\(426\) 0 0
\(427\) 4.32674 + 0.946388i 0.209385 + 0.0457989i
\(428\) 0 0
\(429\) 10.2867 + 5.93904i 0.496648 + 0.286740i
\(430\) 0 0
\(431\) 7.52522 + 13.0341i 0.362477 + 0.627828i 0.988368 0.152082i \(-0.0485979\pi\)
−0.625891 + 0.779911i \(0.715265\pi\)
\(432\) 0 0
\(433\) −18.5694 + 18.5694i −0.892387 + 0.892387i −0.994747 0.102361i \(-0.967360\pi\)
0.102361 + 0.994747i \(0.467360\pi\)
\(434\) 0 0
\(435\) −22.8256 14.5424i −1.09440 0.697256i
\(436\) 0 0
\(437\) 5.07564 18.9425i 0.242801 0.906144i
\(438\) 0 0
\(439\) 3.96987 6.87601i 0.189471 0.328174i −0.755603 0.655030i \(-0.772656\pi\)
0.945074 + 0.326856i \(0.105989\pi\)
\(440\) 0 0
\(441\) 55.3516 5.15819i 2.63579 0.245628i
\(442\) 0 0
\(443\) −9.70480 + 2.60039i −0.461089 + 0.123548i −0.481883 0.876235i \(-0.660047\pi\)
0.0207943 + 0.999784i \(0.493380\pi\)
\(444\) 0 0
\(445\) −8.66183 9.45329i −0.410610 0.448129i
\(446\) 0 0
\(447\) −20.1169 20.1169i −0.951496 0.951496i
\(448\) 0 0
\(449\) 15.6484i 0.738492i 0.929332 + 0.369246i \(0.120384\pi\)
−0.929332 + 0.369246i \(0.879616\pi\)
\(450\) 0 0
\(451\) −1.50737 + 0.870282i −0.0709794 + 0.0409800i
\(452\) 0 0
\(453\) 8.78726 + 32.7945i 0.412861 + 1.54082i
\(454\) 0 0
\(455\) 7.14940 + 7.10950i 0.335169 + 0.333298i
\(456\) 0 0
\(457\) −6.39543 23.8681i −0.299165 1.11650i −0.937853 0.347034i \(-0.887189\pi\)
0.638687 0.769467i \(-0.279478\pi\)
\(458\) 0 0
\(459\) −46.5443 + 26.8724i −2.17250 + 1.25430i
\(460\) 0 0
\(461\) 17.2862i 0.805099i −0.915398 0.402550i \(-0.868124\pi\)
0.915398 0.402550i \(-0.131876\pi\)
\(462\) 0 0
\(463\) 5.61564 + 5.61564i 0.260981 + 0.260981i 0.825453 0.564472i \(-0.190920\pi\)
−0.564472 + 0.825453i \(0.690920\pi\)
\(464\) 0 0
\(465\) −0.610849 + 13.9814i −0.0283274 + 0.648370i
\(466\) 0 0
\(467\) 40.3828 10.8205i 1.86869 0.500715i 0.868709 0.495322i \(-0.164950\pi\)
0.999985 0.00539309i \(-0.00171668\pi\)
\(468\) 0 0
\(469\) 33.5456 1.55967i 1.54899 0.0720189i
\(470\) 0 0
\(471\) −16.8308 + 29.1519i −0.775524 + 1.34325i
\(472\) 0 0
\(473\) 3.62606 13.5326i 0.166726 0.622231i
\(474\) 0 0
\(475\) −21.4339 + 9.99327i −0.983455 + 0.458523i
\(476\) 0 0
\(477\) −14.0107 + 14.0107i −0.641506 + 0.641506i
\(478\) 0 0
\(479\) 18.2812 + 31.6639i 0.835287 + 1.44676i 0.893796 + 0.448473i \(0.148032\pi\)
−0.0585091 + 0.998287i \(0.518635\pi\)
\(480\) 0 0
\(481\) 15.1475 + 8.74544i 0.690669 + 0.398758i
\(482\) 0 0
\(483\) −34.5757 + 11.0092i −1.57325 + 0.500938i
\(484\) 0 0
\(485\) −2.88114 + 1.49972i −0.130826 + 0.0680989i
\(486\) 0 0
\(487\) −3.00674 0.805655i −0.136249 0.0365077i 0.190050 0.981774i \(-0.439135\pi\)
−0.326299 + 0.945267i \(0.605802\pi\)
\(488\) 0 0
\(489\) 9.58102 0.433269
\(490\) 0 0
\(491\) 32.3286 1.45897 0.729484 0.683998i \(-0.239760\pi\)
0.729484 + 0.683998i \(0.239760\pi\)
\(492\) 0 0
\(493\) 11.6210 + 3.11384i 0.523384 + 0.140240i
\(494\) 0 0
\(495\) 35.6843 + 11.2524i 1.60389 + 0.505756i
\(496\) 0 0
\(497\) −27.2057 + 8.66257i −1.22034 + 0.388569i
\(498\) 0 0
\(499\) 19.3372 + 11.1643i 0.865651 + 0.499784i 0.865901 0.500216i \(-0.166746\pi\)
−0.000249428 1.00000i \(0.500079\pi\)
\(500\) 0 0
\(501\) 19.6293 + 33.9989i 0.876972 + 1.51896i
\(502\) 0 0
\(503\) −20.3631 + 20.3631i −0.907947 + 0.907947i −0.996106 0.0881596i \(-0.971901\pi\)
0.0881596 + 0.996106i \(0.471901\pi\)
\(504\) 0 0
\(505\) −0.625093 2.82000i −0.0278163 0.125489i
\(506\) 0 0
\(507\) 8.64298 32.2560i 0.383848 1.43254i
\(508\) 0 0
\(509\) −16.5423 + 28.6521i −0.733224 + 1.26998i 0.222274 + 0.974984i \(0.428652\pi\)
−0.955498 + 0.294997i \(0.904681\pi\)
\(510\) 0 0
\(511\) −34.5116 + 1.60459i −1.52670 + 0.0709827i
\(512\) 0 0
\(513\) −74.6790 + 20.0102i −3.29716 + 0.883471i
\(514\) 0 0
\(515\) 24.4225 + 1.06703i 1.07619 + 0.0470188i
\(516\) 0 0
\(517\) −6.02417 6.02417i −0.264943 0.264943i
\(518\) 0 0
\(519\) 0.834889i 0.0366475i
\(520\) 0 0
\(521\) −20.9365 + 12.0877i −0.917246 + 0.529572i −0.882755 0.469833i \(-0.844314\pi\)
−0.0344903 + 0.999405i \(0.510981\pi\)
\(522\) 0 0
\(523\) 6.17690 + 23.0525i 0.270097 + 1.00801i 0.959056 + 0.283215i \(0.0914011\pi\)
−0.688960 + 0.724800i \(0.741932\pi\)
\(524\) 0 0
\(525\) 36.9703 + 23.4090i 1.61352 + 1.02165i
\(526\) 0 0
\(527\) −1.61012 6.00906i −0.0701381 0.261759i
\(528\) 0 0
\(529\) −5.03074 + 2.90450i −0.218728 + 0.126283i
\(530\) 0 0
\(531\) 18.3556i 0.796566i
\(532\) 0 0
\(533\) −0.995512 0.995512i −0.0431204 0.0431204i
\(534\) 0 0
\(535\) 6.64790 + 0.290448i 0.287414 + 0.0125572i
\(536\) 0 0
\(537\) 69.4503 18.6092i 2.99700 0.803045i
\(538\) 0 0
\(539\) 13.4022 + 6.15754i 0.577275 + 0.265224i
\(540\) 0 0
\(541\) 0.226204 0.391797i 0.00972527 0.0168447i −0.861122 0.508399i \(-0.830238\pi\)
0.870847 + 0.491554i \(0.163571\pi\)
\(542\) 0 0
\(543\) 2.62017 9.77860i 0.112442 0.419640i
\(544\) 0 0
\(545\) −7.30279 32.9453i −0.312817 1.41122i
\(546\) 0 0
\(547\) −24.0665 + 24.0665i −1.02901 + 1.02901i −0.0294424 + 0.999566i \(0.509373\pi\)
−0.999566 + 0.0294424i \(0.990627\pi\)
\(548\) 0 0
\(549\) 6.64720 + 11.5133i 0.283696 + 0.491375i
\(550\) 0 0
\(551\) 14.9882 + 8.65343i 0.638518 + 0.368649i
\(552\) 0 0
\(553\) −24.7498 5.41353i −1.05247 0.230207i
\(554\) 0 0
\(555\) 72.3960 + 22.8287i 3.07304 + 0.969024i
\(556\) 0 0
\(557\) 43.9643 + 11.7802i 1.86283 + 0.499143i 0.999980 0.00632392i \(-0.00201298\pi\)
0.862846 + 0.505467i \(0.168680\pi\)
\(558\) 0 0
\(559\) 11.3321 0.479297
\(560\) 0 0
\(561\) −22.9157 −0.967502
\(562\) 0 0
\(563\) 22.3526 + 5.98936i 0.942050 + 0.252422i 0.696986 0.717085i \(-0.254524\pi\)
0.245065 + 0.969507i \(0.421191\pi\)
\(564\) 0 0
\(565\) 11.8700 6.17871i 0.499376 0.259940i
\(566\) 0 0
\(567\) 59.1492 + 53.8934i 2.48403 + 2.26331i
\(568\) 0 0
\(569\) 6.46781 + 3.73419i 0.271145 + 0.156545i 0.629408 0.777075i \(-0.283298\pi\)
−0.358263 + 0.933621i \(0.616631\pi\)
\(570\) 0 0
\(571\) −0.421350 0.729799i −0.0176329 0.0305412i 0.857074 0.515193i \(-0.172280\pi\)
−0.874707 + 0.484652i \(0.838946\pi\)
\(572\) 0 0
\(573\) −49.3517 + 49.3517i −2.06170 + 2.06170i
\(574\) 0 0
\(575\) −19.4804 7.09156i −0.812387 0.295739i
\(576\) 0 0
\(577\) −9.36558 + 34.9528i −0.389894 + 1.45511i 0.440411 + 0.897796i \(0.354833\pi\)
−0.830305 + 0.557309i \(0.811834\pi\)
\(578\) 0 0
\(579\) 34.4486 59.6668i 1.43164 2.47967i
\(580\) 0 0
\(581\) 5.97443 11.5564i 0.247861 0.479441i
\(582\) 0 0
\(583\) −5.07781 + 1.36059i −0.210301 + 0.0563500i
\(584\) 0 0
\(585\) −1.32100 + 30.2356i −0.0546167 + 1.25009i
\(586\) 0 0
\(587\) 0.777106 + 0.777106i 0.0320746 + 0.0320746i 0.722962 0.690888i \(-0.242780\pi\)
−0.690888 + 0.722962i \(0.742780\pi\)
\(588\) 0 0
\(589\) 8.94914i 0.368743i
\(590\) 0 0
\(591\) 37.6655 21.7462i 1.54935 0.894519i
\(592\) 0 0
\(593\) −5.14887 19.2159i −0.211439 0.789101i −0.987390 0.158307i \(-0.949396\pi\)
0.775951 0.630793i \(-0.217270\pi\)
\(594\) 0 0
\(595\) −18.8030 4.98190i −0.770848 0.204238i
\(596\) 0 0
\(597\) 6.46934 + 24.1439i 0.264772 + 0.988144i
\(598\) 0 0
\(599\) −19.4628 + 11.2369i −0.795229 + 0.459126i −0.841800 0.539789i \(-0.818504\pi\)
0.0465712 + 0.998915i \(0.485171\pi\)
\(600\) 0 0
\(601\) 30.0046i 1.22391i 0.790891 + 0.611957i \(0.209618\pi\)
−0.790891 + 0.611957i \(0.790382\pi\)
\(602\) 0 0
\(603\) 71.2771 + 71.2771i 2.90263 + 2.90263i
\(604\) 0 0
\(605\) −9.91041 10.8159i −0.402915 0.439731i
\(606\) 0 0
\(607\) 4.14580 1.11086i 0.168273 0.0450886i −0.173699 0.984799i \(-0.555572\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(608\) 0 0
\(609\) −1.48728 31.9886i −0.0602677 1.29624i
\(610\) 0 0
\(611\) 3.44551 5.96780i 0.139390 0.241431i
\(612\) 0 0
\(613\) −6.74007 + 25.1543i −0.272229 + 1.01597i 0.685447 + 0.728123i \(0.259607\pi\)
−0.957676 + 0.287850i \(0.907060\pi\)
\(614\) 0 0
\(615\) −5.15312 3.28311i −0.207794 0.132388i
\(616\) 0 0
\(617\) 18.5489 18.5489i 0.746750 0.746750i −0.227117 0.973867i \(-0.572930\pi\)
0.973867 + 0.227117i \(0.0729300\pi\)
\(618\) 0 0
\(619\) −14.9263 25.8532i −0.599940 1.03913i −0.992829 0.119541i \(-0.961858\pi\)
0.392889 0.919586i \(-0.371476\pi\)
\(620\) 0 0
\(621\) −58.6938 33.8869i −2.35530 1.35983i
\(622\) 0 0
\(623\) 3.24164 14.8203i 0.129874 0.593762i
\(624\) 0 0
\(625\) 8.54776 + 23.4933i 0.341910 + 0.939733i
\(626\) 0 0
\(627\) −31.8416 8.53194i −1.27163 0.340733i
\(628\) 0 0
\(629\) −33.7441 −1.34547
\(630\) 0 0
\(631\) −0.755612 −0.0300804 −0.0150402 0.999887i \(-0.504788\pi\)
−0.0150402 + 0.999887i \(0.504788\pi\)
\(632\) 0 0
\(633\) −4.24108 1.13639i −0.168568 0.0451676i
\(634\) 0 0
\(635\) −6.61589 12.7099i −0.262544 0.504378i
\(636\) 0 0
\(637\) −2.00514 + 11.7602i −0.0794464 + 0.465955i
\(638\) 0 0
\(639\) −74.2199 42.8509i −2.93609 1.69515i
\(640\) 0 0
\(641\) −10.1050 17.5023i −0.399122 0.691299i 0.594496 0.804099i \(-0.297352\pi\)
−0.993618 + 0.112799i \(0.964018\pi\)
\(642\) 0 0
\(643\) −0.933128 + 0.933128i −0.0367990 + 0.0367990i −0.725267 0.688468i \(-0.758284\pi\)
0.688468 + 0.725267i \(0.258284\pi\)
\(644\) 0 0
\(645\) 48.0155 10.6433i 1.89061 0.419080i
\(646\) 0 0
\(647\) 7.85058 29.2988i 0.308638 1.15185i −0.621130 0.783708i \(-0.713326\pi\)
0.929768 0.368146i \(-0.120007\pi\)
\(648\) 0 0
\(649\) −2.43499 + 4.21752i −0.0955816 + 0.165552i
\(650\) 0 0
\(651\) −13.9403 + 8.93646i −0.546363 + 0.350247i
\(652\) 0 0
\(653\) 40.0331 10.7268i 1.56662 0.419774i 0.631867 0.775077i \(-0.282289\pi\)
0.934751 + 0.355303i \(0.115622\pi\)
\(654\) 0 0
\(655\) 4.58669 4.20268i 0.179217 0.164212i
\(656\) 0 0
\(657\) −73.3297 73.3297i −2.86086 2.86086i
\(658\) 0 0
\(659\) 7.54235i 0.293808i −0.989151 0.146904i \(-0.953069\pi\)
0.989151 0.146904i \(-0.0469309\pi\)
\(660\) 0 0
\(661\) −40.6248 + 23.4548i −1.58012 + 0.912284i −0.585282 + 0.810830i \(0.699016\pi\)
−0.994840 + 0.101455i \(0.967650\pi\)
\(662\) 0 0
\(663\) −4.79735 17.9039i −0.186314 0.695332i
\(664\) 0 0
\(665\) −24.2721 13.9231i −0.941233 0.539915i
\(666\) 0 0
\(667\) 3.92664 + 14.6544i 0.152040 + 0.567422i
\(668\) 0 0
\(669\) −29.3700 + 16.9568i −1.13551 + 0.655588i
\(670\) 0 0
\(671\) 3.52717i 0.136165i
\(672\) 0 0
\(673\) 29.6391 + 29.6391i 1.14250 + 1.14250i 0.987991 + 0.154511i \(0.0493803\pi\)
0.154511 + 0.987991i \(0.450620\pi\)
\(674\) 0 0
\(675\) 14.1875 + 80.4890i 0.546078 + 3.09802i
\(676\) 0 0
\(677\) 38.0992 10.2087i 1.46427 0.392351i 0.563309 0.826246i \(-0.309528\pi\)
0.900963 + 0.433896i \(0.142861\pi\)
\(678\) 0 0
\(679\) −3.41397 1.76495i −0.131016 0.0677325i
\(680\) 0 0
\(681\) 49.6385 85.9764i 1.90215 3.29462i
\(682\) 0 0
\(683\) −1.10186 + 4.11219i −0.0421615 + 0.157349i −0.983797 0.179285i \(-0.942622\pi\)
0.941636 + 0.336633i \(0.109288\pi\)
\(684\) 0 0
\(685\) 27.0169 42.4052i 1.03226 1.62022i
\(686\) 0 0
\(687\) −48.4093 + 48.4093i −1.84693 + 1.84693i
\(688\) 0 0
\(689\) −2.12605 3.68243i −0.0809961 0.140289i
\(690\) 0 0
\(691\) −0.538345 0.310814i −0.0204796 0.0118239i 0.489725 0.871877i \(-0.337097\pi\)
−0.510205 + 0.860053i \(0.670430\pi\)
\(692\) 0 0
\(693\) 13.4321 + 42.1848i 0.510242 + 1.60247i
\(694\) 0 0
\(695\) 5.60496 17.7748i 0.212608 0.674238i
\(696\) 0 0
\(697\) 2.62357 + 0.702983i 0.0993747 + 0.0266274i
\(698\) 0 0
\(699\) −13.1042 −0.495645
\(700\) 0 0
\(701\) −30.9626 −1.16944 −0.584721 0.811235i \(-0.698796\pi\)
−0.584721 + 0.811235i \(0.698796\pi\)
\(702\) 0 0
\(703\) −46.8879 12.5636i −1.76841 0.473844i
\(704\) 0 0
\(705\) 8.99400 28.5224i 0.338734 1.07422i
\(706\) 0 0
\(707\) 2.30181 2.52628i 0.0865683 0.0950107i
\(708\) 0 0
\(709\) −26.5831 15.3478i −0.998350 0.576398i −0.0905903 0.995888i \(-0.528875\pi\)
−0.907760 + 0.419491i \(0.862209\pi\)
\(710\) 0 0
\(711\) −38.0233 65.8583i −1.42599 2.46988i
\(712\) 0 0
\(713\) 5.54719 5.54719i 0.207744 0.207744i
\(714\) 0 0
\(715\) −4.31446 + 6.77192i −0.161352 + 0.253255i
\(716\) 0 0
\(717\) −12.2073 + 45.5583i −0.455891 + 1.70141i
\(718\) 0 0
\(719\) 4.00357 6.93439i 0.149308 0.258609i −0.781664 0.623700i \(-0.785629\pi\)
0.930972 + 0.365091i \(0.118962\pi\)
\(720\) 0 0
\(721\) 15.6101 + 24.3508i 0.581352 + 0.906871i
\(722\) 0 0
\(723\) 15.5585 4.16889i 0.578628 0.155043i
\(724\) 0 0
\(725\) 10.4948 14.9862i 0.389765 0.556573i
\(726\) 0 0
\(727\) −2.24335 2.24335i −0.0832011 0.0832011i 0.664281 0.747483i \(-0.268738\pi\)
−0.747483 + 0.664281i \(0.768738\pi\)
\(728\) 0 0
\(729\) 77.9825i 2.88824i
\(730\) 0 0
\(731\) −18.9334 + 10.9312i −0.700276 + 0.404304i
\(732\) 0 0
\(733\) −9.69369 36.1773i −0.358045 1.33624i −0.876610 0.481202i \(-0.840200\pi\)
0.518565 0.855038i \(-0.326466\pi\)
\(734\) 0 0
\(735\) 2.54936 + 51.7127i 0.0940347 + 1.90745i
\(736\) 0 0
\(737\) 6.92179 + 25.8325i 0.254967 + 0.951552i
\(738\) 0 0
\(739\) 8.55973 4.94196i 0.314875 0.181793i −0.334231 0.942491i \(-0.608477\pi\)
0.649106 + 0.760698i \(0.275143\pi\)
\(740\) 0 0
\(741\) 26.6639i 0.979522i
\(742\) 0 0
\(743\) −23.1188 23.1188i −0.848147 0.848147i 0.141755 0.989902i \(-0.454725\pi\)
−0.989902 + 0.141755i \(0.954725\pi\)
\(744\) 0 0
\(745\) 14.1795 12.9924i 0.519498 0.476005i
\(746\) 0 0
\(747\) 37.7191 10.1068i 1.38007 0.369789i
\(748\) 0 0
\(749\) 4.24914 + 6.62838i 0.155260 + 0.242196i
\(750\) 0 0
\(751\) 19.0861 33.0582i 0.696463 1.20631i −0.273222 0.961951i \(-0.588089\pi\)
0.969685 0.244358i \(-0.0785773\pi\)
\(752\) 0 0
\(753\) 14.1362 52.7569i 0.515150 1.92257i
\(754\) 0 0
\(755\) −22.4071 + 4.96685i −0.815478 + 0.180762i
\(756\) 0 0
\(757\) −24.5560 + 24.5560i −0.892504 + 0.892504i −0.994758 0.102255i \(-0.967394\pi\)
0.102255 + 0.994758i \(0.467394\pi\)
\(758\) 0 0
\(759\) −14.4487 25.0259i −0.524454 0.908381i
\(760\) 0 0
\(761\) 11.7157 + 6.76405i 0.424693 + 0.245197i 0.697083 0.716990i \(-0.254481\pi\)
−0.272390 + 0.962187i \(0.587814\pi\)
\(762\) 0 0
\(763\) 26.8914 29.5139i 0.973533 1.06847i
\(764\) 0 0
\(765\) −26.9588 51.7911i −0.974699 1.87251i
\(766\) 0 0
\(767\) −3.80489 1.01952i −0.137387 0.0368126i
\(768\) 0 0
\(769\) 39.3025 1.41728 0.708641 0.705569i \(-0.249308\pi\)
0.708641 + 0.705569i \(0.249308\pi\)
\(770\) 0 0
\(771\) 32.4426 1.16839
\(772\) 0 0
\(773\) −27.5742 7.38848i −0.991774 0.265745i −0.273778 0.961793i \(-0.588274\pi\)
−0.717995 + 0.696048i \(0.754940\pi\)
\(774\) 0 0
\(775\) −9.42431 0.825076i −0.338531 0.0296376i
\(776\) 0 0
\(777\) 27.2509 + 85.5842i 0.977619 + 3.07031i
\(778\) 0 0
\(779\) 3.38374 + 1.95360i 0.121235 + 0.0699952i
\(780\) 0 0
\(781\) −11.3689 19.6915i −0.406810 0.704616i
\(782\) 0 0
\(783\) 42.2932 42.2932i 1.51143 1.51143i
\(784\) 0 0
\(785\) −19.1911 12.2269i −0.684961 0.436397i
\(786\) 0 0
\(787\) −4.15609 + 15.5107i −0.148149 + 0.552898i 0.851446 + 0.524442i \(0.175726\pi\)
−0.999595 + 0.0284567i \(0.990941\pi\)
\(788\) 0 0
\(789\) 20.5240 35.5487i 0.730675 1.26557i
\(790\) 0 0
\(791\) 14.0652 + 7.27142i 0.500101 + 0.258542i
\(792\) 0 0
\(793\) −2.75576 + 0.738404i −0.0978600 + 0.0262215i
\(794\) 0 0
\(795\) −12.4670 13.6061i −0.442158 0.482559i
\(796\) 0 0
\(797\) 11.4141 + 11.4141i 0.404310 + 0.404310i 0.879749 0.475439i \(-0.157711\pi\)
−0.475439 + 0.879749i \(0.657711\pi\)
\(798\) 0 0
\(799\) 13.2945i 0.470324i
\(800\) 0 0
\(801\) 39.4362 22.7685i 1.39341 0.804486i
\(802\) 0 0
\(803\) −7.12113 26.5764i −0.251299 0.937861i
\(804\) 0 0
\(805\) −6.41491 23.6756i −0.226096 0.834455i
\(806\) 0 0
\(807\) −3.19385 11.9196i −0.112429 0.419591i
\(808\) 0 0
\(809\) 15.1396 8.74085i 0.532280 0.307312i −0.209665 0.977773i \(-0.567237\pi\)
0.741944 + 0.670461i \(0.233904\pi\)
\(810\) 0 0
\(811\) 51.1883i 1.79747i −0.438497 0.898733i \(-0.644489\pi\)
0.438497 0.898733i \(-0.355511\pi\)
\(812\) 0 0
\(813\) −3.87674 3.87674i −0.135963 0.135963i
\(814\) 0 0
\(815\) −0.282700 + 6.47056i −0.00990256 + 0.226654i
\(816\) 0 0
\(817\) −30.3780 + 8.13976i −1.06279 + 0.284774i
\(818\) 0 0
\(819\) −30.1468 + 19.3257i −1.05341 + 0.675294i
\(820\) 0 0
\(821\) 0.0218989 0.0379301i 0.000764278 0.00132377i −0.865643 0.500662i \(-0.833090\pi\)
0.866407 + 0.499338i \(0.166423\pi\)
\(822\) 0 0
\(823\) −1.97965 + 7.38815i −0.0690062 + 0.257535i −0.991807 0.127742i \(-0.959227\pi\)
0.922801 + 0.385276i \(0.125894\pi\)
\(824\) 0 0
\(825\) −11.9206 + 32.7457i −0.415023 + 1.14006i
\(826\) 0 0
\(827\) 25.9919 25.9919i 0.903826 0.903826i −0.0919384 0.995765i \(-0.529306\pi\)
0.995765 + 0.0919384i \(0.0293063\pi\)
\(828\) 0 0
\(829\) 10.5010 + 18.1883i 0.364715 + 0.631704i 0.988730 0.149708i \(-0.0478332\pi\)
−0.624016 + 0.781412i \(0.714500\pi\)
\(830\) 0 0
\(831\) 43.6657 + 25.2104i 1.51475 + 0.874539i
\(832\) 0 0
\(833\) −7.99399 21.5828i −0.276975 0.747799i
\(834\) 0 0
\(835\) −23.5404 + 12.2535i −0.814650 + 0.424050i
\(836\) 0 0
\(837\) −29.8739 8.00469i −1.03259 0.276683i
\(838\) 0 0
\(839\) −43.5751 −1.50438 −0.752189 0.658947i \(-0.771002\pi\)
−0.752189 + 0.658947i \(0.771002\pi\)
\(840\) 0 0
\(841\) 15.6110 0.538309
\(842\) 0 0
\(843\) 7.21205 + 1.93246i 0.248396 + 0.0665576i
\(844\) 0 0
\(845\) 21.5291 + 6.78880i 0.740625 + 0.233542i
\(846\) 0 0
\(847\) 3.70891 16.9566i 0.127440 0.582635i
\(848\) 0 0
\(849\) −17.4117 10.0526i −0.597567 0.345005i
\(850\) 0 0
\(851\) −21.2762 36.8514i −0.729338 1.26325i
\(852\) 0 0
\(853\) 17.6117 17.6117i 0.603014 0.603014i −0.338098 0.941111i \(-0.609783\pi\)
0.941111 + 0.338098i \(0.109783\pi\)
\(854\) 0 0
\(855\) −18.1768 82.0016i −0.621634 2.80440i
\(856\) 0 0
\(857\) 9.04624 33.7610i 0.309014 1.15325i −0.620422 0.784268i \(-0.713038\pi\)
0.929435 0.368986i \(-0.120295\pi\)
\(858\) 0 0
\(859\) −17.2794 + 29.9288i −0.589565 + 1.02116i 0.404724 + 0.914439i \(0.367367\pi\)
−0.994289 + 0.106718i \(0.965966\pi\)
\(860\) 0 0
\(861\) −0.335770 7.22177i −0.0114430 0.246117i
\(862\) 0 0
\(863\) −42.6135 + 11.4183i −1.45058 + 0.388682i −0.896226 0.443598i \(-0.853702\pi\)
−0.554355 + 0.832280i \(0.687035\pi\)
\(864\) 0 0
\(865\) 0.563844 + 0.0246345i 0.0191713 + 0.000837597i
\(866\) 0 0
\(867\) −14.4768 14.4768i −0.491658 0.491658i
\(868\) 0 0
\(869\) 20.1761i 0.684428i
\(870\) 0 0
\(871\) −18.7337 + 10.8159i −0.634769 + 0.366484i
\(872\) 0 0
\(873\) −2.98573 11.1429i −0.101052 0.377130i
\(874\) 0 0
\(875\) −16.9002 + 24.2772i −0.571330 + 0.820720i
\(876\) 0 0
\(877\) −0.217309 0.811006i −0.00733799 0.0273857i 0.962160 0.272486i \(-0.0878458\pi\)
−0.969498 + 0.245100i \(0.921179\pi\)
\(878\) 0 0
\(879\) 71.2083 41.1121i 2.40180 1.38668i
\(880\) 0 0
\(881\) 39.6633i 1.33629i 0.744031 + 0.668145i \(0.232911\pi\)
−0.744031 + 0.668145i \(0.767089\pi\)
\(882\) 0 0
\(883\) 28.9598 + 28.9598i 0.974576 + 0.974576i 0.999685 0.0251084i \(-0.00799308\pi\)
−0.0251084 + 0.999685i \(0.507993\pi\)
\(884\) 0 0
\(885\) −17.0794 0.746202i −0.574117 0.0250833i
\(886\) 0 0
\(887\) 5.40217 1.44751i 0.181387 0.0486025i −0.166982 0.985960i \(-0.553402\pi\)
0.348369 + 0.937357i \(0.386736\pi\)
\(888\) 0 0
\(889\) 7.78592 15.0604i 0.261131 0.505110i
\(890\) 0 0
\(891\) −31.8628 + 55.1880i −1.06744 + 1.84887i
\(892\) 0 0
\(893\) −4.94977 + 18.4728i −0.165638 + 0.618168i
\(894\) 0 0
\(895\) 10.5185 + 47.4525i 0.351595 + 1.58616i
\(896\) 0 0
\(897\) 16.5278 16.5278i 0.551847 0.551847i
\(898\) 0 0
\(899\) 3.46164 + 5.99574i 0.115452 + 0.199969i
\(900\) 0 0
\(901\) 7.10430 + 4.10167i 0.236679 + 0.136646i
\(902\) 0 0
\(903\) 43.0145 + 39.1923i 1.43143 + 1.30424i
\(904\) 0 0
\(905\) 6.52668 + 2.05807i 0.216954 + 0.0684124i
\(906\) 0 0
\(907\) 20.2095 + 5.41512i 0.671046 + 0.179806i 0.578226 0.815877i \(-0.303745\pi\)
0.0928200 + 0.995683i \(0.470412\pi\)
\(908\) 0 0
\(909\) 10.2586 0.340258
\(910\) 0 0
\(911\) 13.0212 0.431413 0.215707 0.976458i \(-0.430795\pi\)
0.215707 + 0.976458i \(0.430795\pi\)
\(912\) 0 0
\(913\) 10.0074 + 2.68146i 0.331195 + 0.0887435i
\(914\) 0 0
\(915\) −10.9830 + 5.71698i −0.363087 + 0.188997i
\(916\) 0 0
\(917\) 7.19072 + 1.57283i 0.237459 + 0.0519394i
\(918\) 0 0
\(919\) 4.38681 + 2.53273i 0.144708 + 0.0835470i 0.570606 0.821224i \(-0.306708\pi\)
−0.425898 + 0.904771i \(0.640042\pi\)
\(920\) 0 0
\(921\) −49.3807 85.5299i −1.62715 2.81831i
\(922\) 0 0
\(923\) 13.0048 13.0048i 0.428059 0.428059i
\(924\) 0 0
\(925\) −17.5535 + 48.2192i −0.577157 + 1.58544i
\(926\) 0 0
\(927\) −22.4712 + 83.8635i −0.738050 + 2.75444i
\(928\) 0 0
\(929\) −24.7411 + 42.8528i −0.811728 + 1.40595i 0.0999258 + 0.994995i \(0.468139\pi\)
−0.911654 + 0.410959i \(0.865194\pi\)
\(930\) 0 0
\(931\) −3.07207 32.9658i −0.100683 1.08041i
\(932\) 0 0
\(933\) 60.2919 16.1552i 1.97387 0.528897i
\(934\) 0 0
\(935\) 0.676157 15.4762i 0.0221127 0.506125i
\(936\) 0 0
\(937\) 22.1742 + 22.1742i 0.724399 + 0.724399i 0.969498 0.245099i \(-0.0788204\pi\)
−0.245099 + 0.969498i \(0.578820\pi\)
\(938\) 0 0
\(939\) 85.8324i 2.80103i
\(940\) 0 0
\(941\) 26.5134 15.3075i 0.864313 0.499011i −0.00114123 0.999999i \(-0.500363\pi\)
0.865454 + 0.500988i \(0.167030\pi\)
\(942\) 0 0
\(943\) 0.886481 + 3.30839i 0.0288678 + 0.107736i
\(944\) 0 0
\(945\) −68.1885 + 68.5712i −2.21817 + 2.23062i
\(946\) 0 0
\(947\) −5.48860 20.4837i −0.178355 0.665632i −0.995956 0.0898453i \(-0.971363\pi\)
0.817600 0.575786i \(-0.195304\pi\)
\(948\) 0 0
\(949\) 19.2732 11.1274i 0.625636 0.361211i
\(950\) 0 0
\(951\) 4.73414i 0.153515i
\(952\) 0 0
\(953\) −30.4466 30.4466i −0.986262 0.986262i 0.0136447 0.999907i \(-0.495657\pi\)
−0.999907 + 0.0136447i \(0.995657\pi\)
\(954\) 0 0
\(955\) −31.8736 34.7860i −1.03140 1.12565i
\(956\) 0 0
\(957\) 24.6335 6.60053i 0.796288 0.213365i
\(958\) 0 0
\(959\) 59.4283 2.76306i 1.91904 0.0892240i
\(960\) 0 0
\(961\) −13.7100 + 23.7465i −0.442259 + 0.766015i
\(962\) 0 0
\(963\) −6.11673 + 22.8280i −0.197109 + 0.735620i
\(964\) 0 0
\(965\) 39.2796 + 25.0255i 1.26446 + 0.805599i
\(966\) 0 0
\(967\) −40.2212 + 40.2212i −1.29343 + 1.29343i −0.360774 + 0.932653i \(0.617487\pi\)
−0.932653 + 0.360774i \(0.882513\pi\)
\(968\) 0 0
\(969\) 25.7205 + 44.5493i 0.826263 + 1.43113i
\(970\) 0 0
\(971\) 11.4574 + 6.61493i 0.367685 + 0.212283i 0.672447 0.740145i \(-0.265243\pi\)
−0.304761 + 0.952429i \(0.598577\pi\)
\(972\) 0 0
\(973\) 21.0128 6.69070i 0.673641 0.214494i
\(974\) 0 0
\(975\) −28.0796 2.45830i −0.899268 0.0787288i
\(976\) 0 0
\(977\) 28.2330 + 7.56500i 0.903253 + 0.242026i 0.680413 0.732829i \(-0.261800\pi\)
0.222840 + 0.974855i \(0.428467\pi\)
\(978\) 0 0
\(979\) 12.0815 0.386128
\(980\) 0 0
\(981\) 119.849 3.82648
\(982\) 0 0
\(983\) −13.3912 3.58817i −0.427114 0.114445i 0.0388569 0.999245i \(-0.487628\pi\)
−0.465971 + 0.884800i \(0.654295\pi\)
\(984\) 0 0
\(985\) 13.5750 + 26.0791i 0.432534 + 0.830950i
\(986\) 0 0
\(987\) 33.7183 10.7362i 1.07326 0.341738i
\(988\) 0 0
\(989\) −23.8755 13.7845i −0.759197 0.438323i
\(990\) 0 0
\(991\) −2.37408 4.11203i −0.0754152 0.130623i 0.825852 0.563888i \(-0.190695\pi\)
−0.901267 + 0.433265i \(0.857362\pi\)
\(992\) 0 0
\(993\) −38.4461 + 38.4461i −1.22005 + 1.22005i
\(994\) 0 0
\(995\) −16.4965 + 3.65668i −0.522974 + 0.115925i
\(996\) 0 0
\(997\) 1.26721 4.72928i 0.0401329 0.149778i −0.942952 0.332929i \(-0.891963\pi\)
0.983085 + 0.183151i \(0.0586297\pi\)
\(998\) 0 0
\(999\) −83.8791 + 145.283i −2.65382 + 4.59655i
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 560.2.ci.e.33.1 48
4.3 odd 2 280.2.bo.a.33.12 yes 48
5.2 odd 4 inner 560.2.ci.e.257.1 48
7.3 odd 6 inner 560.2.ci.e.353.1 48
20.7 even 4 280.2.bo.a.257.12 yes 48
28.3 even 6 280.2.bo.a.73.12 yes 48
35.17 even 12 inner 560.2.ci.e.17.1 48
140.87 odd 12 280.2.bo.a.17.12 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.bo.a.17.12 48 140.87 odd 12
280.2.bo.a.33.12 yes 48 4.3 odd 2
280.2.bo.a.73.12 yes 48 28.3 even 6
280.2.bo.a.257.12 yes 48 20.7 even 4
560.2.ci.e.17.1 48 35.17 even 12 inner
560.2.ci.e.33.1 48 1.1 even 1 trivial
560.2.ci.e.257.1 48 5.2 odd 4 inner
560.2.ci.e.353.1 48 7.3 odd 6 inner