Properties

Label 804.2.o.d.365.10
Level $804$
Weight $2$
Character 804.365
Analytic conductor $6.420$
Analytic rank $0$
Dimension $36$
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] = [804,2,Mod(365,804)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(804, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("804.365");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 804 = 2^{2} \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 804.o (of order \(6\), degree \(2\), 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: \(6.41997232251\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 365.10
Character \(\chi\) \(=\) 804.365
Dual form 804.2.o.d.641.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.355178 - 1.69524i) q^{3} +3.60368 q^{5} +(3.20029 + 1.84769i) q^{7} +(-2.74770 - 1.20423i) q^{9} +O(q^{10})\) \(q+(0.355178 - 1.69524i) q^{3} +3.60368 q^{5} +(3.20029 + 1.84769i) q^{7} +(-2.74770 - 1.20423i) q^{9} +(1.48078 - 2.56478i) q^{11} +(1.95258 - 1.12732i) q^{13} +(1.27995 - 6.10911i) q^{15} +(-4.00058 + 2.30974i) q^{17} +(2.85606 + 4.94684i) q^{19} +(4.26895 - 4.76900i) q^{21} +(-7.12280 + 4.11235i) q^{23} +7.98648 q^{25} +(-3.01738 + 4.23030i) q^{27} +(4.16985 + 2.40746i) q^{29} +(-6.73963 - 3.89113i) q^{31} +(-3.82199 - 3.42123i) q^{33} +(11.5328 + 6.65846i) q^{35} +(4.09202 + 7.08759i) q^{37} +(-1.21757 - 3.71049i) q^{39} +(-0.243870 + 0.422396i) q^{41} -11.0216i q^{43} +(-9.90181 - 4.33964i) q^{45} +(-7.37585 - 4.25845i) q^{47} +(3.32788 + 5.76406i) q^{49} +(2.49464 + 7.60232i) q^{51} -10.5924 q^{53} +(5.33624 - 9.24265i) q^{55} +(9.40051 - 3.08470i) q^{57} -11.3999i q^{59} +(-3.97066 + 2.29246i) q^{61} +(-6.56838 - 8.93075i) q^{63} +(7.03646 - 4.06250i) q^{65} +(2.48761 - 7.79819i) q^{67} +(4.44157 + 13.5355i) q^{69} +(-12.3588 - 7.13533i) q^{71} +(0.629763 + 1.09078i) q^{73} +(2.83662 - 13.5390i) q^{75} +(9.47783 - 5.47203i) q^{77} +(3.80258 + 2.19542i) q^{79} +(6.09967 + 6.61770i) q^{81} +(-2.80997 + 1.62234i) q^{83} +(-14.4168 + 8.32354i) q^{85} +(5.56227 - 6.21382i) q^{87} -2.12774i q^{89} +8.33174 q^{91} +(-8.99017 + 10.0433i) q^{93} +(10.2923 + 17.8268i) q^{95} +(8.97837 - 5.18366i) q^{97} +(-7.15731 + 5.26405i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 36 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 36 q + 4 q^{9} - 36 q^{13} + 18 q^{15} + 16 q^{21} + 76 q^{25} + 6 q^{31} + 4 q^{33} + 42 q^{37} - 21 q^{39} + 2 q^{49} + 18 q^{51} + 20 q^{55} + 18 q^{57} - 24 q^{61} - 12 q^{63} - 8 q^{67} + 3 q^{69} + 14 q^{73} + 72 q^{79} - 12 q^{81} - 18 q^{85} - 21 q^{87} - 68 q^{91} + 9 q^{93} - 48 q^{97} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/804\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\) \(403\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) 0.355178 1.69524i 0.205062 0.978749i
\(4\) 0 0
\(5\) 3.60368 1.61161 0.805806 0.592179i \(-0.201732\pi\)
0.805806 + 0.592179i \(0.201732\pi\)
\(6\) 0 0
\(7\) 3.20029 + 1.84769i 1.20959 + 0.698359i 0.962671 0.270676i \(-0.0872470\pi\)
0.246923 + 0.969035i \(0.420580\pi\)
\(8\) 0 0
\(9\) −2.74770 1.20423i −0.915899 0.401409i
\(10\) 0 0
\(11\) 1.48078 2.56478i 0.446471 0.773311i −0.551682 0.834055i \(-0.686014\pi\)
0.998153 + 0.0607433i \(0.0193471\pi\)
\(12\) 0 0
\(13\) 1.95258 1.12732i 0.541547 0.312663i −0.204158 0.978938i \(-0.565446\pi\)
0.745706 + 0.666275i \(0.232112\pi\)
\(14\) 0 0
\(15\) 1.27995 6.10911i 0.330481 1.57736i
\(16\) 0 0
\(17\) −4.00058 + 2.30974i −0.970283 + 0.560193i −0.899323 0.437286i \(-0.855940\pi\)
−0.0709605 + 0.997479i \(0.522606\pi\)
\(18\) 0 0
\(19\) 2.85606 + 4.94684i 0.655225 + 1.13488i 0.981837 + 0.189725i \(0.0607596\pi\)
−0.326612 + 0.945158i \(0.605907\pi\)
\(20\) 0 0
\(21\) 4.26895 4.76900i 0.931561 1.04068i
\(22\) 0 0
\(23\) −7.12280 + 4.11235i −1.48521 + 0.857484i −0.999858 0.0168370i \(-0.994640\pi\)
−0.485348 + 0.874321i \(0.661307\pi\)
\(24\) 0 0
\(25\) 7.98648 1.59730
\(26\) 0 0
\(27\) −3.01738 + 4.23030i −0.580695 + 0.814121i
\(28\) 0 0
\(29\) 4.16985 + 2.40746i 0.774321 + 0.447054i 0.834414 0.551138i \(-0.185806\pi\)
−0.0600929 + 0.998193i \(0.519140\pi\)
\(30\) 0 0
\(31\) −6.73963 3.89113i −1.21047 0.698867i −0.247611 0.968860i \(-0.579645\pi\)
−0.962862 + 0.269993i \(0.912979\pi\)
\(32\) 0 0
\(33\) −3.82199 3.42123i −0.665323 0.595560i
\(34\) 0 0
\(35\) 11.5328 + 6.65846i 1.94940 + 1.12549i
\(36\) 0 0
\(37\) 4.09202 + 7.08759i 0.672724 + 1.16519i 0.977129 + 0.212650i \(0.0682093\pi\)
−0.304404 + 0.952543i \(0.598457\pi\)
\(38\) 0 0
\(39\) −1.21757 3.71049i −0.194967 0.594154i
\(40\) 0 0
\(41\) −0.243870 + 0.422396i −0.0380862 + 0.0659672i −0.884440 0.466654i \(-0.845459\pi\)
0.846354 + 0.532621i \(0.178793\pi\)
\(42\) 0 0
\(43\) 11.0216i 1.68077i −0.541986 0.840387i \(-0.682328\pi\)
0.541986 0.840387i \(-0.317672\pi\)
\(44\) 0 0
\(45\) −9.90181 4.33964i −1.47607 0.646916i
\(46\) 0 0
\(47\) −7.37585 4.25845i −1.07588 0.621159i −0.146097 0.989270i \(-0.546671\pi\)
−0.929782 + 0.368111i \(0.880005\pi\)
\(48\) 0 0
\(49\) 3.32788 + 5.76406i 0.475412 + 0.823438i
\(50\) 0 0
\(51\) 2.49464 + 7.60232i 0.349320 + 1.06454i
\(52\) 0 0
\(53\) −10.5924 −1.45498 −0.727492 0.686116i \(-0.759314\pi\)
−0.727492 + 0.686116i \(0.759314\pi\)
\(54\) 0 0
\(55\) 5.33624 9.24265i 0.719539 1.24628i
\(56\) 0 0
\(57\) 9.40051 3.08470i 1.24513 0.408579i
\(58\) 0 0
\(59\) 11.3999i 1.48414i −0.670324 0.742068i \(-0.733845\pi\)
0.670324 0.742068i \(-0.266155\pi\)
\(60\) 0 0
\(61\) −3.97066 + 2.29246i −0.508391 + 0.293520i −0.732172 0.681120i \(-0.761493\pi\)
0.223781 + 0.974640i \(0.428160\pi\)
\(62\) 0 0
\(63\) −6.56838 8.93075i −0.827538 1.12517i
\(64\) 0 0
\(65\) 7.03646 4.06250i 0.872765 0.503891i
\(66\) 0 0
\(67\) 2.48761 7.79819i 0.303910 0.952701i
\(68\) 0 0
\(69\) 4.44157 + 13.5355i 0.534702 + 1.62948i
\(70\) 0 0
\(71\) −12.3588 7.13533i −1.46672 0.846808i −0.467408 0.884041i \(-0.654812\pi\)
−0.999307 + 0.0372332i \(0.988146\pi\)
\(72\) 0 0
\(73\) 0.629763 + 1.09078i 0.0737082 + 0.127666i 0.900524 0.434807i \(-0.143183\pi\)
−0.826816 + 0.562473i \(0.809850\pi\)
\(74\) 0 0
\(75\) 2.83662 13.5390i 0.327545 1.56335i
\(76\) 0 0
\(77\) 9.47783 5.47203i 1.08010 0.623595i
\(78\) 0 0
\(79\) 3.80258 + 2.19542i 0.427824 + 0.247004i 0.698419 0.715689i \(-0.253887\pi\)
−0.270595 + 0.962693i \(0.587220\pi\)
\(80\) 0 0
\(81\) 6.09967 + 6.61770i 0.677742 + 0.735300i
\(82\) 0 0
\(83\) −2.80997 + 1.62234i −0.308434 + 0.178075i −0.646226 0.763146i \(-0.723654\pi\)
0.337791 + 0.941221i \(0.390320\pi\)
\(84\) 0 0
\(85\) −14.4168 + 8.32354i −1.56372 + 0.902815i
\(86\) 0 0
\(87\) 5.56227 6.21382i 0.596338 0.666192i
\(88\) 0 0
\(89\) 2.12774i 0.225540i −0.993621 0.112770i \(-0.964028\pi\)
0.993621 0.112770i \(-0.0359723\pi\)
\(90\) 0 0
\(91\) 8.33174 0.873404
\(92\) 0 0
\(93\) −8.99017 + 10.0433i −0.932238 + 1.04144i
\(94\) 0 0
\(95\) 10.2923 + 17.8268i 1.05597 + 1.82899i
\(96\) 0 0
\(97\) 8.97837 5.18366i 0.911615 0.526321i 0.0306648 0.999530i \(-0.490238\pi\)
0.880951 + 0.473208i \(0.156904\pi\)
\(98\) 0 0
\(99\) −7.15731 + 5.26405i −0.719337 + 0.529057i
\(100\) 0 0
\(101\) 7.04180 12.1968i 0.700685 1.21362i −0.267541 0.963546i \(-0.586211\pi\)
0.968226 0.250076i \(-0.0804556\pi\)
\(102\) 0 0
\(103\) −3.26301 + 5.65170i −0.321514 + 0.556878i −0.980801 0.195013i \(-0.937525\pi\)
0.659287 + 0.751892i \(0.270858\pi\)
\(104\) 0 0
\(105\) 15.3839 17.1859i 1.50132 1.67718i
\(106\) 0 0
\(107\) 1.23281i 0.119180i 0.998223 + 0.0595899i \(0.0189793\pi\)
−0.998223 + 0.0595899i \(0.981021\pi\)
\(108\) 0 0
\(109\) 8.87642i 0.850207i 0.905145 + 0.425103i \(0.139762\pi\)
−0.905145 + 0.425103i \(0.860238\pi\)
\(110\) 0 0
\(111\) 13.4686 4.41961i 1.27838 0.419491i
\(112\) 0 0
\(113\) −0.367153 + 0.635928i −0.0345388 + 0.0598230i −0.882778 0.469790i \(-0.844330\pi\)
0.848239 + 0.529613i \(0.177663\pi\)
\(114\) 0 0
\(115\) −25.6683 + 14.8196i −2.39358 + 1.38193i
\(116\) 0 0
\(117\) −6.72264 + 0.746190i −0.621508 + 0.0689853i
\(118\) 0 0
\(119\) −17.0707 −1.56486
\(120\) 0 0
\(121\) 1.11459 + 1.93053i 0.101326 + 0.175503i
\(122\) 0 0
\(123\) 0.629446 + 0.563446i 0.0567553 + 0.0508042i
\(124\) 0 0
\(125\) 10.7623 0.962610
\(126\) 0 0
\(127\) −5.78672 + 10.0229i −0.513488 + 0.889388i 0.486389 + 0.873742i \(0.338314\pi\)
−0.999878 + 0.0156455i \(0.995020\pi\)
\(128\) 0 0
\(129\) −18.6842 3.91463i −1.64506 0.344664i
\(130\) 0 0
\(131\) 18.0364i 1.57585i 0.615770 + 0.787926i \(0.288845\pi\)
−0.615770 + 0.787926i \(0.711155\pi\)
\(132\) 0 0
\(133\) 21.1084i 1.83033i
\(134\) 0 0
\(135\) −10.8737 + 15.2446i −0.935856 + 1.31205i
\(136\) 0 0
\(137\) −2.18254 −0.186467 −0.0932334 0.995644i \(-0.529720\pi\)
−0.0932334 + 0.995644i \(0.529720\pi\)
\(138\) 0 0
\(139\) 1.48230i 0.125727i 0.998022 + 0.0628637i \(0.0200233\pi\)
−0.998022 + 0.0628637i \(0.979977\pi\)
\(140\) 0 0
\(141\) −9.83885 + 10.9914i −0.828581 + 0.925639i
\(142\) 0 0
\(143\) 6.67725i 0.558380i
\(144\) 0 0
\(145\) 15.0268 + 8.67571i 1.24791 + 0.720479i
\(146\) 0 0
\(147\) 10.9535 3.59430i 0.903428 0.296453i
\(148\) 0 0
\(149\) 2.54319i 0.208346i 0.994559 + 0.104173i \(0.0332196\pi\)
−0.994559 + 0.104173i \(0.966780\pi\)
\(150\) 0 0
\(151\) −4.61380 7.99133i −0.375466 0.650326i 0.614931 0.788581i \(-0.289184\pi\)
−0.990397 + 0.138255i \(0.955851\pi\)
\(152\) 0 0
\(153\) 13.7738 1.52885i 1.11355 0.123600i
\(154\) 0 0
\(155\) −24.2874 14.0224i −1.95081 1.12630i
\(156\) 0 0
\(157\) 1.35503 + 2.34698i 0.108143 + 0.187309i 0.915018 0.403413i \(-0.132176\pi\)
−0.806875 + 0.590722i \(0.798843\pi\)
\(158\) 0 0
\(159\) −3.76221 + 17.9568i −0.298363 + 1.42406i
\(160\) 0 0
\(161\) −30.3933 −2.39533
\(162\) 0 0
\(163\) 6.61168 11.4518i 0.517867 0.896972i −0.481918 0.876217i \(-0.660060\pi\)
0.999785 0.0207554i \(-0.00660713\pi\)
\(164\) 0 0
\(165\) −13.7732 12.3290i −1.07224 0.959813i
\(166\) 0 0
\(167\) 7.97167 + 4.60245i 0.616867 + 0.356148i 0.775648 0.631166i \(-0.217423\pi\)
−0.158781 + 0.987314i \(0.550757\pi\)
\(168\) 0 0
\(169\) −3.95829 + 6.85597i −0.304484 + 0.527382i
\(170\) 0 0
\(171\) −1.89047 17.0318i −0.144568 1.30245i
\(172\) 0 0
\(173\) 11.5022 6.64078i 0.874493 0.504889i 0.00565462 0.999984i \(-0.498200\pi\)
0.868839 + 0.495095i \(0.164867\pi\)
\(174\) 0 0
\(175\) 25.5590 + 14.7565i 1.93208 + 1.11549i
\(176\) 0 0
\(177\) −19.3255 4.04899i −1.45260 0.304340i
\(178\) 0 0
\(179\) 16.1945 1.21043 0.605217 0.796060i \(-0.293086\pi\)
0.605217 + 0.796060i \(0.293086\pi\)
\(180\) 0 0
\(181\) −0.831318 + 1.43989i −0.0617914 + 0.107026i −0.895266 0.445532i \(-0.853015\pi\)
0.833475 + 0.552557i \(0.186348\pi\)
\(182\) 0 0
\(183\) 2.47599 + 7.54547i 0.183030 + 0.557777i
\(184\) 0 0
\(185\) 14.7463 + 25.5414i 1.08417 + 1.87784i
\(186\) 0 0
\(187\) 13.6808i 1.00044i
\(188\) 0 0
\(189\) −17.4727 + 7.96299i −1.27095 + 0.579222i
\(190\) 0 0
\(191\) −7.65708 13.2625i −0.554047 0.959638i −0.997977 0.0635761i \(-0.979749\pi\)
0.443930 0.896061i \(-0.353584\pi\)
\(192\) 0 0
\(193\) 7.65307 0.550880 0.275440 0.961318i \(-0.411176\pi\)
0.275440 + 0.961318i \(0.411176\pi\)
\(194\) 0 0
\(195\) −4.38773 13.3714i −0.314212 0.957547i
\(196\) 0 0
\(197\) 1.55409 2.69175i 0.110724 0.191780i −0.805338 0.592815i \(-0.798016\pi\)
0.916062 + 0.401036i \(0.131350\pi\)
\(198\) 0 0
\(199\) 0.134799 + 0.233479i 0.00955567 + 0.0165509i 0.870764 0.491702i \(-0.163625\pi\)
−0.861208 + 0.508253i \(0.830292\pi\)
\(200\) 0 0
\(201\) −12.3363 6.98685i −0.870134 0.492815i
\(202\) 0 0
\(203\) 8.89646 + 15.4091i 0.624409 + 1.08151i
\(204\) 0 0
\(205\) −0.878830 + 1.52218i −0.0613802 + 0.106314i
\(206\) 0 0
\(207\) 24.5235 2.72202i 1.70450 0.189194i
\(208\) 0 0
\(209\) 16.9168 1.17016
\(210\) 0 0
\(211\) −13.5489 23.4673i −0.932743 1.61556i −0.778610 0.627508i \(-0.784075\pi\)
−0.154133 0.988050i \(-0.549258\pi\)
\(212\) 0 0
\(213\) −16.4857 + 18.4168i −1.12958 + 1.26190i
\(214\) 0 0
\(215\) 39.7182i 2.70876i
\(216\) 0 0
\(217\) −14.3792 24.9054i −0.976121 1.69069i
\(218\) 0 0
\(219\) 2.07282 0.680179i 0.140068 0.0459623i
\(220\) 0 0
\(221\) −5.20763 + 9.01987i −0.350303 + 0.606742i
\(222\) 0 0
\(223\) −8.51661 −0.570314 −0.285157 0.958481i \(-0.592046\pi\)
−0.285157 + 0.958481i \(0.592046\pi\)
\(224\) 0 0
\(225\) −21.9444 9.61754i −1.46296 0.641169i
\(226\) 0 0
\(227\) 14.5472 + 8.39882i 0.965530 + 0.557449i 0.897871 0.440259i \(-0.145114\pi\)
0.0676595 + 0.997708i \(0.478447\pi\)
\(228\) 0 0
\(229\) −4.13981 + 2.39012i −0.273566 + 0.157944i −0.630507 0.776183i \(-0.717153\pi\)
0.356941 + 0.934127i \(0.383820\pi\)
\(230\) 0 0
\(231\) −5.91009 18.0108i −0.388856 1.18502i
\(232\) 0 0
\(233\) −1.05550 + 1.82818i −0.0691480 + 0.119768i −0.898527 0.438919i \(-0.855361\pi\)
0.829378 + 0.558687i \(0.188695\pi\)
\(234\) 0 0
\(235\) −26.5802 15.3461i −1.73390 1.00107i
\(236\) 0 0
\(237\) 5.07237 5.66654i 0.329486 0.368081i
\(238\) 0 0
\(239\) −6.41223 + 11.1063i −0.414773 + 0.718407i −0.995405 0.0957589i \(-0.969472\pi\)
0.580632 + 0.814166i \(0.302806\pi\)
\(240\) 0 0
\(241\) 5.62450 0.362306 0.181153 0.983455i \(-0.442017\pi\)
0.181153 + 0.983455i \(0.442017\pi\)
\(242\) 0 0
\(243\) 13.3851 7.98996i 0.858654 0.512556i
\(244\) 0 0
\(245\) 11.9926 + 20.7718i 0.766180 + 1.32706i
\(246\) 0 0
\(247\) 11.1534 + 6.43939i 0.709671 + 0.409729i
\(248\) 0 0
\(249\) 1.75221 + 5.33980i 0.111042 + 0.338396i
\(250\) 0 0
\(251\) 10.2371 + 17.7313i 0.646163 + 1.11919i 0.984032 + 0.177994i \(0.0569607\pi\)
−0.337869 + 0.941193i \(0.609706\pi\)
\(252\) 0 0
\(253\) 24.3579i 1.53137i
\(254\) 0 0
\(255\) 8.98989 + 27.3963i 0.562969 + 1.71562i
\(256\) 0 0
\(257\) −16.3494 9.43933i −1.01985 0.588809i −0.105788 0.994389i \(-0.533736\pi\)
−0.914060 + 0.405580i \(0.867070\pi\)
\(258\) 0 0
\(259\) 30.2431i 1.87921i
\(260\) 0 0
\(261\) −8.55834 11.6364i −0.529748 0.720276i
\(262\) 0 0
\(263\) 18.8353i 1.16144i 0.814105 + 0.580718i \(0.197228\pi\)
−0.814105 + 0.580718i \(0.802772\pi\)
\(264\) 0 0
\(265\) −38.1717 −2.34487
\(266\) 0 0
\(267\) −3.60704 0.755727i −0.220747 0.0462498i
\(268\) 0 0
\(269\) 22.2511i 1.35667i −0.734752 0.678336i \(-0.762701\pi\)
0.734752 0.678336i \(-0.237299\pi\)
\(270\) 0 0
\(271\) 10.8294i 0.657838i 0.944358 + 0.328919i \(0.106684\pi\)
−0.944358 + 0.328919i \(0.893316\pi\)
\(272\) 0 0
\(273\) 2.95925 14.1243i 0.179102 0.854843i
\(274\) 0 0
\(275\) 11.8262 20.4836i 0.713147 1.23521i
\(276\) 0 0
\(277\) −5.46809 −0.328546 −0.164273 0.986415i \(-0.552528\pi\)
−0.164273 + 0.986415i \(0.552528\pi\)
\(278\) 0 0
\(279\) 13.8327 + 18.8077i 0.828139 + 1.12599i
\(280\) 0 0
\(281\) 7.71219 + 13.3579i 0.460071 + 0.796866i 0.998964 0.0455083i \(-0.0144907\pi\)
−0.538893 + 0.842374i \(0.681157\pi\)
\(282\) 0 0
\(283\) −14.7372 −0.876038 −0.438019 0.898966i \(-0.644320\pi\)
−0.438019 + 0.898966i \(0.644320\pi\)
\(284\) 0 0
\(285\) 33.8764 11.1163i 2.00666 0.658471i
\(286\) 0 0
\(287\) −1.56091 + 0.901192i −0.0921376 + 0.0531957i
\(288\) 0 0
\(289\) 2.16976 3.75813i 0.127633 0.221067i
\(290\) 0 0
\(291\) −5.59865 17.0616i −0.328199 1.00017i
\(292\) 0 0
\(293\) 7.13657i 0.416923i 0.978031 + 0.208461i \(0.0668456\pi\)
−0.978031 + 0.208461i \(0.933154\pi\)
\(294\) 0 0
\(295\) 41.0814i 2.39185i
\(296\) 0 0
\(297\) 6.38173 + 14.0031i 0.370305 + 0.812540i
\(298\) 0 0
\(299\) −9.27188 + 16.0594i −0.536206 + 0.928737i
\(300\) 0 0
\(301\) 20.3644 35.2722i 1.17378 2.03306i
\(302\) 0 0
\(303\) −18.1754 16.2696i −1.04415 0.934663i
\(304\) 0 0
\(305\) −14.3090 + 8.26130i −0.819330 + 0.473040i
\(306\) 0 0
\(307\) 0.0773160 + 0.133915i 0.00441266 + 0.00764295i 0.868223 0.496174i \(-0.165262\pi\)
−0.863811 + 0.503817i \(0.831929\pi\)
\(308\) 0 0
\(309\) 8.42205 + 7.53895i 0.479113 + 0.428876i
\(310\) 0 0
\(311\) 22.4333 1.27208 0.636039 0.771657i \(-0.280572\pi\)
0.636039 + 0.771657i \(0.280572\pi\)
\(312\) 0 0
\(313\) 6.54773i 0.370100i 0.982729 + 0.185050i \(0.0592447\pi\)
−0.982729 + 0.185050i \(0.940755\pi\)
\(314\) 0 0
\(315\) −23.6703 32.1835i −1.33367 1.81334i
\(316\) 0 0
\(317\) −13.9285 + 8.04165i −0.782305 + 0.451664i −0.837247 0.546826i \(-0.815836\pi\)
0.0549415 + 0.998490i \(0.482503\pi\)
\(318\) 0 0
\(319\) 12.3492 7.12983i 0.691424 0.399194i
\(320\) 0 0
\(321\) 2.08991 + 0.437866i 0.116647 + 0.0244393i
\(322\) 0 0
\(323\) −22.8518 13.1935i −1.27151 0.734105i
\(324\) 0 0
\(325\) 15.5942 9.00333i 0.865012 0.499415i
\(326\) 0 0
\(327\) 15.0477 + 3.15271i 0.832139 + 0.174345i
\(328\) 0 0
\(329\) −15.7366 27.2565i −0.867585 1.50270i
\(330\) 0 0
\(331\) 21.7622 + 12.5644i 1.19616 + 0.690602i 0.959696 0.281039i \(-0.0906790\pi\)
0.236461 + 0.971641i \(0.424012\pi\)
\(332\) 0 0
\(333\) −2.70857 24.4023i −0.148429 1.33724i
\(334\) 0 0
\(335\) 8.96454 28.1022i 0.489785 1.53538i
\(336\) 0 0
\(337\) 19.8300 11.4488i 1.08021 0.623659i 0.149256 0.988799i \(-0.452312\pi\)
0.930953 + 0.365140i \(0.118979\pi\)
\(338\) 0 0
\(339\) 0.947647 + 0.848281i 0.0514691 + 0.0460723i
\(340\) 0 0
\(341\) −19.9598 + 11.5238i −1.08088 + 0.624048i
\(342\) 0 0
\(343\) 1.27207i 0.0686853i
\(344\) 0 0
\(345\) 16.0060 + 48.7775i 0.861732 + 2.62609i
\(346\) 0 0
\(347\) −12.4805 + 21.6168i −0.669986 + 1.16045i 0.307922 + 0.951412i \(0.400367\pi\)
−0.977908 + 0.209038i \(0.932967\pi\)
\(348\) 0 0
\(349\) 20.2064 1.08162 0.540812 0.841144i \(-0.318117\pi\)
0.540812 + 0.841144i \(0.318117\pi\)
\(350\) 0 0
\(351\) −1.12276 + 11.6615i −0.0599287 + 0.622447i
\(352\) 0 0
\(353\) 17.5145 + 30.3359i 0.932200 + 1.61462i 0.779552 + 0.626338i \(0.215447\pi\)
0.152648 + 0.988281i \(0.451220\pi\)
\(354\) 0 0
\(355\) −44.5370 25.7134i −2.36378 1.36473i
\(356\) 0 0
\(357\) −6.06313 + 28.9389i −0.320895 + 1.53161i
\(358\) 0 0
\(359\) 13.5980i 0.717674i −0.933400 0.358837i \(-0.883173\pi\)
0.933400 0.358837i \(-0.116827\pi\)
\(360\) 0 0
\(361\) −6.81416 + 11.8025i −0.358640 + 0.621183i
\(362\) 0 0
\(363\) 3.66859 1.20382i 0.192551 0.0631842i
\(364\) 0 0
\(365\) 2.26946 + 3.93082i 0.118789 + 0.205749i
\(366\) 0 0
\(367\) 4.52728 + 2.61383i 0.236322 + 0.136441i 0.613485 0.789706i \(-0.289767\pi\)
−0.377163 + 0.926147i \(0.623100\pi\)
\(368\) 0 0
\(369\) 1.17874 0.866941i 0.0613629 0.0451311i
\(370\) 0 0
\(371\) −33.8989 19.5715i −1.75994 1.01610i
\(372\) 0 0
\(373\) −1.06973 0.617607i −0.0553884 0.0319785i 0.472050 0.881572i \(-0.343514\pi\)
−0.527438 + 0.849593i \(0.676847\pi\)
\(374\) 0 0
\(375\) 3.82254 18.2447i 0.197395 0.942153i
\(376\) 0 0
\(377\) 10.8559 0.559109
\(378\) 0 0
\(379\) −0.461731 + 0.266580i −0.0237175 + 0.0136933i −0.511812 0.859098i \(-0.671026\pi\)
0.488094 + 0.872791i \(0.337692\pi\)
\(380\) 0 0
\(381\) 14.9359 + 13.3698i 0.765190 + 0.684956i
\(382\) 0 0
\(383\) −1.76314 3.05384i −0.0900920 0.156044i 0.817458 0.575989i \(-0.195383\pi\)
−0.907550 + 0.419945i \(0.862049\pi\)
\(384\) 0 0
\(385\) 34.1550 19.7194i 1.74070 1.00499i
\(386\) 0 0
\(387\) −13.2725 + 30.2840i −0.674678 + 1.53942i
\(388\) 0 0
\(389\) −6.31060 + 3.64343i −0.319960 + 0.184729i −0.651375 0.758756i \(-0.725807\pi\)
0.331415 + 0.943485i \(0.392474\pi\)
\(390\) 0 0
\(391\) 18.9969 32.9036i 0.960714 1.66400i
\(392\) 0 0
\(393\) 30.5761 + 6.40615i 1.54236 + 0.323148i
\(394\) 0 0
\(395\) 13.7033 + 7.91159i 0.689487 + 0.398075i
\(396\) 0 0
\(397\) 14.1142 0.708371 0.354185 0.935175i \(-0.384758\pi\)
0.354185 + 0.935175i \(0.384758\pi\)
\(398\) 0 0
\(399\) 35.7839 + 7.49725i 1.79143 + 0.375332i
\(400\) 0 0
\(401\) 2.05940 0.102842 0.0514208 0.998677i \(-0.483625\pi\)
0.0514208 + 0.998677i \(0.483625\pi\)
\(402\) 0 0
\(403\) −17.5462 −0.874038
\(404\) 0 0
\(405\) 21.9812 + 23.8481i 1.09226 + 1.18502i
\(406\) 0 0
\(407\) 24.2375 1.20141
\(408\) 0 0
\(409\) −22.0958 12.7570i −1.09257 0.630794i −0.158309 0.987390i \(-0.550604\pi\)
−0.934259 + 0.356595i \(0.883938\pi\)
\(410\) 0 0
\(411\) −0.775190 + 3.69993i −0.0382373 + 0.182504i
\(412\) 0 0
\(413\) 21.0634 36.4828i 1.03646 1.79520i
\(414\) 0 0
\(415\) −10.1262 + 5.84638i −0.497077 + 0.286987i
\(416\) 0 0
\(417\) 2.51287 + 0.526482i 0.123056 + 0.0257820i
\(418\) 0 0
\(419\) 13.2976 7.67735i 0.649628 0.375063i −0.138686 0.990336i \(-0.544288\pi\)
0.788314 + 0.615273i \(0.210954\pi\)
\(420\) 0 0
\(421\) 15.1923 + 26.3139i 0.740428 + 1.28246i 0.952300 + 0.305162i \(0.0987106\pi\)
−0.211872 + 0.977297i \(0.567956\pi\)
\(422\) 0 0
\(423\) 15.1385 + 20.5831i 0.736057 + 1.00079i
\(424\) 0 0
\(425\) −31.9505 + 18.4467i −1.54983 + 0.894794i
\(426\) 0 0
\(427\) −16.9430 −0.819930
\(428\) 0 0
\(429\) −11.3196 2.37161i −0.546514 0.114503i
\(430\) 0 0
\(431\) 19.8202 + 11.4432i 0.954707 + 0.551201i 0.894540 0.446988i \(-0.147503\pi\)
0.0601674 + 0.998188i \(0.480837\pi\)
\(432\) 0 0
\(433\) 22.6525 + 13.0784i 1.08861 + 0.628508i 0.933206 0.359343i \(-0.116999\pi\)
0.155403 + 0.987851i \(0.450332\pi\)
\(434\) 0 0
\(435\) 20.0446 22.3926i 0.961066 1.07364i
\(436\) 0 0
\(437\) −40.6863 23.4902i −1.94629 1.12369i
\(438\) 0 0
\(439\) −7.95944 13.7861i −0.379883 0.657977i 0.611162 0.791506i \(-0.290702\pi\)
−0.991045 + 0.133529i \(0.957369\pi\)
\(440\) 0 0
\(441\) −2.20277 19.8454i −0.104894 0.945020i
\(442\) 0 0
\(443\) 7.66008 13.2677i 0.363942 0.630365i −0.624664 0.780894i \(-0.714764\pi\)
0.988606 + 0.150528i \(0.0480974\pi\)
\(444\) 0 0
\(445\) 7.66769i 0.363483i
\(446\) 0 0
\(447\) 4.31132 + 0.903285i 0.203918 + 0.0427239i
\(448\) 0 0
\(449\) 6.40241 + 3.69643i 0.302149 + 0.174446i 0.643408 0.765524i \(-0.277520\pi\)
−0.341259 + 0.939969i \(0.610853\pi\)
\(450\) 0 0
\(451\) 0.722236 + 1.25095i 0.0340088 + 0.0589049i
\(452\) 0 0
\(453\) −15.1860 + 4.98316i −0.713499 + 0.234129i
\(454\) 0 0
\(455\) 30.0249 1.40759
\(456\) 0 0
\(457\) −2.60191 + 4.50665i −0.121712 + 0.210812i −0.920443 0.390877i \(-0.872172\pi\)
0.798731 + 0.601689i \(0.205505\pi\)
\(458\) 0 0
\(459\) 2.30040 23.8930i 0.107373 1.11523i
\(460\) 0 0
\(461\) 33.6857i 1.56890i −0.620192 0.784450i \(-0.712945\pi\)
0.620192 0.784450i \(-0.287055\pi\)
\(462\) 0 0
\(463\) −13.4453 + 7.76263i −0.624854 + 0.360760i −0.778757 0.627326i \(-0.784149\pi\)
0.153902 + 0.988086i \(0.450816\pi\)
\(464\) 0 0
\(465\) −32.3977 + 36.1927i −1.50241 + 1.67839i
\(466\) 0 0
\(467\) −11.3144 + 6.53235i −0.523567 + 0.302281i −0.738393 0.674371i \(-0.764415\pi\)
0.214826 + 0.976652i \(0.431082\pi\)
\(468\) 0 0
\(469\) 22.3697 20.3601i 1.03294 0.940143i
\(470\) 0 0
\(471\) 4.45997 1.46350i 0.205505 0.0674347i
\(472\) 0 0
\(473\) −28.2680 16.3205i −1.29976 0.750418i
\(474\) 0 0
\(475\) 22.8099 + 39.5078i 1.04659 + 1.81274i
\(476\) 0 0
\(477\) 29.1048 + 12.7557i 1.33262 + 0.584044i
\(478\) 0 0
\(479\) 0.387072 0.223476i 0.0176858 0.0102109i −0.491131 0.871086i \(-0.663416\pi\)
0.508817 + 0.860875i \(0.330083\pi\)
\(480\) 0 0
\(481\) 15.9800 + 9.22604i 0.728624 + 0.420671i
\(482\) 0 0
\(483\) −10.7950 + 51.5240i −0.491192 + 2.34443i
\(484\) 0 0
\(485\) 32.3551 18.6802i 1.46917 0.848226i
\(486\) 0 0
\(487\) −11.0165 + 6.36037i −0.499204 + 0.288216i −0.728385 0.685168i \(-0.759729\pi\)
0.229181 + 0.973384i \(0.426395\pi\)
\(488\) 0 0
\(489\) −17.0652 15.2758i −0.771715 0.690797i
\(490\) 0 0
\(491\) 44.0337i 1.98721i −0.112892 0.993607i \(-0.536011\pi\)
0.112892 0.993607i \(-0.463989\pi\)
\(492\) 0 0
\(493\) −22.2424 −1.00175
\(494\) 0 0
\(495\) −25.7926 + 18.9699i −1.15929 + 0.852635i
\(496\) 0 0
\(497\) −26.3677 45.6702i −1.18275 2.04859i
\(498\) 0 0
\(499\) 3.43529 1.98336i 0.153785 0.0887876i −0.421133 0.906999i \(-0.638367\pi\)
0.574918 + 0.818211i \(0.305034\pi\)
\(500\) 0 0
\(501\) 10.6336 11.8792i 0.475076 0.530725i
\(502\) 0 0
\(503\) −8.23418 + 14.2620i −0.367144 + 0.635912i −0.989118 0.147126i \(-0.952998\pi\)
0.621974 + 0.783038i \(0.286331\pi\)
\(504\) 0 0
\(505\) 25.3764 43.9531i 1.12923 1.95589i
\(506\) 0 0
\(507\) 10.2166 + 9.14536i 0.453736 + 0.406160i
\(508\) 0 0
\(509\) 9.04756i 0.401026i 0.979691 + 0.200513i \(0.0642609\pi\)
−0.979691 + 0.200513i \(0.935739\pi\)
\(510\) 0 0
\(511\) 4.65442i 0.205899i
\(512\) 0 0
\(513\) −29.5444 2.84451i −1.30442 0.125588i
\(514\) 0 0
\(515\) −11.7588 + 20.3669i −0.518156 + 0.897472i
\(516\) 0 0
\(517\) −21.8440 + 12.6116i −0.960698 + 0.554660i
\(518\) 0 0
\(519\) −7.17241 21.8576i −0.314834 0.959443i
\(520\) 0 0
\(521\) 18.9377 0.829678 0.414839 0.909895i \(-0.363838\pi\)
0.414839 + 0.909895i \(0.363838\pi\)
\(522\) 0 0
\(523\) −11.4271 19.7922i −0.499670 0.865454i 0.500330 0.865835i \(-0.333212\pi\)
−1.00000 0.000380701i \(0.999879\pi\)
\(524\) 0 0
\(525\) 34.0939 38.0875i 1.48798 1.66228i
\(526\) 0 0
\(527\) 35.9499 1.56600
\(528\) 0 0
\(529\) 22.3228 38.6643i 0.970558 1.68106i
\(530\) 0 0
\(531\) −13.7280 + 31.3234i −0.595746 + 1.35932i
\(532\) 0 0
\(533\) 1.09968i 0.0476325i
\(534\) 0 0
\(535\) 4.44263i 0.192072i
\(536\) 0 0
\(537\) 5.75194 27.4536i 0.248214 1.18471i
\(538\) 0 0
\(539\) 19.7114 0.849031
\(540\) 0 0
\(541\) 42.8342i 1.84159i −0.390051 0.920793i \(-0.627543\pi\)
0.390051 0.920793i \(-0.372457\pi\)
\(542\) 0 0
\(543\) 2.14569 + 1.92070i 0.0920803 + 0.0824252i
\(544\) 0 0
\(545\) 31.9877i 1.37020i
\(546\) 0 0
\(547\) 27.8728 + 16.0924i 1.19175 + 0.688060i 0.958704 0.284405i \(-0.0917959\pi\)
0.233051 + 0.972465i \(0.425129\pi\)
\(548\) 0 0
\(549\) 13.6708 1.51741i 0.583457 0.0647617i
\(550\) 0 0
\(551\) 27.5034i 1.17169i
\(552\) 0 0
\(553\) 8.11290 + 14.0520i 0.344996 + 0.597550i
\(554\) 0 0
\(555\) 48.5364 15.9268i 2.06026 0.676057i
\(556\) 0 0
\(557\) 13.7497 + 7.93838i 0.582593 + 0.336360i 0.762163 0.647385i \(-0.224137\pi\)
−0.179570 + 0.983745i \(0.557471\pi\)
\(558\) 0 0
\(559\) −12.4249 21.5205i −0.525515 0.910219i
\(560\) 0 0
\(561\) 23.1923 + 4.85913i 0.979181 + 0.205153i
\(562\) 0 0
\(563\) −21.9295 −0.924217 −0.462108 0.886823i \(-0.652907\pi\)
−0.462108 + 0.886823i \(0.652907\pi\)
\(564\) 0 0
\(565\) −1.32310 + 2.29168i −0.0556633 + 0.0964116i
\(566\) 0 0
\(567\) 7.29326 + 32.4488i 0.306288 + 1.36272i
\(568\) 0 0
\(569\) 10.0417 + 5.79755i 0.420968 + 0.243046i 0.695491 0.718534i \(-0.255187\pi\)
−0.274523 + 0.961580i \(0.588520\pi\)
\(570\) 0 0
\(571\) 18.0411 31.2481i 0.754998 1.30769i −0.190378 0.981711i \(-0.560971\pi\)
0.945376 0.325983i \(-0.105695\pi\)
\(572\) 0 0
\(573\) −25.2027 + 8.27008i −1.05286 + 0.345487i
\(574\) 0 0
\(575\) −56.8861 + 32.8432i −2.37231 + 1.36966i
\(576\) 0 0
\(577\) −31.7861 18.3517i −1.32327 0.763993i −0.339025 0.940778i \(-0.610097\pi\)
−0.984250 + 0.176785i \(0.943430\pi\)
\(578\) 0 0
\(579\) 2.71821 12.9738i 0.112965 0.539173i
\(580\) 0 0
\(581\) −11.9903 −0.497440
\(582\) 0 0
\(583\) −15.6851 + 27.1673i −0.649609 + 1.12516i
\(584\) 0 0
\(585\) −24.2262 + 2.68903i −1.00163 + 0.111178i
\(586\) 0 0
\(587\) −3.02789 5.24447i −0.124975 0.216462i 0.796748 0.604311i \(-0.206552\pi\)
−0.921723 + 0.387849i \(0.873218\pi\)
\(588\) 0 0
\(589\) 44.4532i 1.83166i
\(590\) 0 0
\(591\) −4.01120 3.59061i −0.164999 0.147698i
\(592\) 0 0
\(593\) 0.587963 + 1.01838i 0.0241447 + 0.0418199i 0.877845 0.478944i \(-0.158980\pi\)
−0.853701 + 0.520764i \(0.825647\pi\)
\(594\) 0 0
\(595\) −61.5171 −2.52196
\(596\) 0 0
\(597\) 0.443682 0.145591i 0.0181587 0.00595864i
\(598\) 0 0
\(599\) −6.14540 + 10.6441i −0.251094 + 0.434908i −0.963827 0.266527i \(-0.914124\pi\)
0.712733 + 0.701435i \(0.247457\pi\)
\(600\) 0 0
\(601\) −4.36040 7.55243i −0.177864 0.308070i 0.763284 0.646063i \(-0.223586\pi\)
−0.941149 + 0.337992i \(0.890252\pi\)
\(602\) 0 0
\(603\) −16.2260 + 18.4314i −0.660773 + 0.750585i
\(604\) 0 0
\(605\) 4.01663 + 6.95700i 0.163299 + 0.282842i
\(606\) 0 0
\(607\) −1.97906 + 3.42782i −0.0803274 + 0.139131i −0.903391 0.428819i \(-0.858930\pi\)
0.823063 + 0.567950i \(0.192263\pi\)
\(608\) 0 0
\(609\) 29.2820 9.60868i 1.18657 0.389363i
\(610\) 0 0
\(611\) −19.2026 −0.776853
\(612\) 0 0
\(613\) 3.85267 + 6.67301i 0.155608 + 0.269520i 0.933280 0.359149i \(-0.116933\pi\)
−0.777672 + 0.628670i \(0.783600\pi\)
\(614\) 0 0
\(615\) 2.26832 + 2.03048i 0.0914675 + 0.0818767i
\(616\) 0 0
\(617\) 28.9608i 1.16592i −0.812502 0.582958i \(-0.801895\pi\)
0.812502 0.582958i \(-0.198105\pi\)
\(618\) 0 0
\(619\) −6.14678 10.6465i −0.247060 0.427920i 0.715649 0.698460i \(-0.246131\pi\)
−0.962709 + 0.270540i \(0.912798\pi\)
\(620\) 0 0
\(621\) 4.09573 42.5401i 0.164356 1.70707i
\(622\) 0 0
\(623\) 3.93140 6.80938i 0.157508 0.272812i
\(624\) 0 0
\(625\) −1.14854 −0.0459418
\(626\) 0 0
\(627\) 6.00847 28.6780i 0.239955 1.14529i
\(628\) 0 0
\(629\) −32.7409 18.9030i −1.30547 0.753711i
\(630\) 0 0
\(631\) −2.05504 + 1.18648i −0.0818097 + 0.0472328i −0.540347 0.841442i \(-0.681707\pi\)
0.458537 + 0.888675i \(0.348374\pi\)
\(632\) 0 0
\(633\) −44.5951 + 14.6335i −1.77250 + 0.581631i
\(634\) 0 0
\(635\) −20.8535 + 36.1192i −0.827544 + 1.43335i
\(636\) 0 0
\(637\) 12.9959 + 7.50319i 0.514916 + 0.297287i
\(638\) 0 0
\(639\) 25.3656 + 34.4885i 1.00345 + 1.36434i
\(640\) 0 0
\(641\) 14.2442 24.6717i 0.562613 0.974473i −0.434655 0.900597i \(-0.643130\pi\)
0.997267 0.0738763i \(-0.0235370\pi\)
\(642\) 0 0
\(643\) −2.32535 −0.0917029 −0.0458514 0.998948i \(-0.514600\pi\)
−0.0458514 + 0.998948i \(0.514600\pi\)
\(644\) 0 0
\(645\) −67.3320 14.1070i −2.65119 0.555464i
\(646\) 0 0
\(647\) 19.9762 + 34.5999i 0.785347 + 1.36026i 0.928791 + 0.370603i \(0.120849\pi\)
−0.143444 + 0.989658i \(0.545818\pi\)
\(648\) 0 0
\(649\) −29.2382 16.8807i −1.14770 0.662624i
\(650\) 0 0
\(651\) −47.3279 + 15.5303i −1.85493 + 0.608680i
\(652\) 0 0
\(653\) 4.75318 + 8.23275i 0.186006 + 0.322172i 0.943915 0.330188i \(-0.107112\pi\)
−0.757909 + 0.652360i \(0.773779\pi\)
\(654\) 0 0
\(655\) 64.9975i 2.53966i
\(656\) 0 0
\(657\) −0.416849 3.75551i −0.0162628 0.146517i
\(658\) 0 0
\(659\) −40.1336 23.1711i −1.56338 0.902620i −0.996911 0.0785391i \(-0.974974\pi\)
−0.566472 0.824081i \(-0.691692\pi\)
\(660\) 0 0
\(661\) 29.0198i 1.12874i 0.825523 + 0.564369i \(0.190881\pi\)
−0.825523 + 0.564369i \(0.809119\pi\)
\(662\) 0 0
\(663\) 13.4412 + 12.0319i 0.522015 + 0.467279i
\(664\) 0 0
\(665\) 76.0678i 2.94978i
\(666\) 0 0
\(667\) −39.6013 −1.53337
\(668\) 0 0
\(669\) −3.02492 + 14.4377i −0.116950 + 0.558195i
\(670\) 0 0
\(671\) 13.5785i 0.524193i
\(672\) 0 0
\(673\) 18.6836i 0.720199i 0.932914 + 0.360100i \(0.117257\pi\)
−0.932914 + 0.360100i \(0.882743\pi\)
\(674\) 0 0
\(675\) −24.0982 + 33.7852i −0.927542 + 1.30039i
\(676\) 0 0
\(677\) −9.74938 + 16.8864i −0.374699 + 0.648998i −0.990282 0.139074i \(-0.955587\pi\)
0.615583 + 0.788072i \(0.288921\pi\)
\(678\) 0 0
\(679\) 38.3111 1.47025
\(680\) 0 0
\(681\) 19.4049 21.6779i 0.743596 0.830700i
\(682\) 0 0
\(683\) 7.63754 + 13.2286i 0.292242 + 0.506179i 0.974340 0.225083i \(-0.0722652\pi\)
−0.682097 + 0.731262i \(0.738932\pi\)
\(684\) 0 0
\(685\) −7.86516 −0.300512
\(686\) 0 0
\(687\) 2.58146 + 7.86690i 0.0984890 + 0.300141i
\(688\) 0 0
\(689\) −20.6826 + 11.9411i −0.787943 + 0.454919i
\(690\) 0 0
\(691\) 12.1115 20.9777i 0.460743 0.798030i −0.538255 0.842782i \(-0.680916\pi\)
0.998998 + 0.0447519i \(0.0142497\pi\)
\(692\) 0 0
\(693\) −32.6318 + 3.62201i −1.23958 + 0.137589i
\(694\) 0 0
\(695\) 5.34174i 0.202624i
\(696\) 0 0
\(697\) 2.25311i 0.0853424i
\(698\) 0 0
\(699\) 2.72431 + 2.43866i 0.103043 + 0.0922384i
\(700\) 0 0
\(701\) 4.11912 7.13453i 0.155577 0.269467i −0.777692 0.628646i \(-0.783610\pi\)
0.933269 + 0.359178i \(0.116943\pi\)
\(702\) 0 0
\(703\) −23.3741 + 40.4852i −0.881572 + 1.52693i
\(704\) 0 0
\(705\) −35.4560 + 39.6093i −1.33535 + 1.49177i
\(706\) 0 0
\(707\) 45.0715 26.0221i 1.69509 0.978660i
\(708\) 0 0
\(709\) 13.5151 + 23.4089i 0.507571 + 0.879139i 0.999962 + 0.00876447i \(0.00278985\pi\)
−0.492391 + 0.870374i \(0.663877\pi\)
\(710\) 0 0
\(711\) −7.80456 10.6115i −0.292694 0.397963i
\(712\) 0 0
\(713\) 64.0067 2.39707
\(714\) 0 0
\(715\) 24.0626i 0.899892i
\(716\) 0 0
\(717\) 16.5504 + 14.8150i 0.618086 + 0.553277i
\(718\) 0 0
\(719\) 37.1549 21.4514i 1.38564 0.800002i 0.392823 0.919614i \(-0.371498\pi\)
0.992821 + 0.119612i \(0.0381650\pi\)
\(720\) 0 0
\(721\) −20.8851 + 12.0580i −0.777802 + 0.449064i
\(722\) 0 0
\(723\) 1.99770 9.53490i 0.0742953 0.354607i
\(724\) 0 0
\(725\) 33.3024 + 19.2271i 1.23682 + 0.714078i
\(726\) 0 0
\(727\) 21.5056 12.4163i 0.797599 0.460494i −0.0450320 0.998986i \(-0.514339\pi\)
0.842631 + 0.538492i \(0.181006\pi\)
\(728\) 0 0
\(729\) −8.79084 25.5288i −0.325587 0.945512i
\(730\) 0 0
\(731\) 25.4569 + 44.0927i 0.941559 + 1.63083i
\(732\) 0 0
\(733\) −41.1568 23.7619i −1.52016 0.877665i −0.999718 0.0237679i \(-0.992434\pi\)
−0.520442 0.853897i \(-0.674233\pi\)
\(734\) 0 0
\(735\) 39.4728 12.9527i 1.45598 0.477767i
\(736\) 0 0
\(737\) −16.3171 17.9276i −0.601047 0.660371i
\(738\) 0 0
\(739\) −6.28091 + 3.62629i −0.231047 + 0.133395i −0.611055 0.791588i \(-0.709255\pi\)
0.380008 + 0.924983i \(0.375921\pi\)
\(740\) 0 0
\(741\) 14.8778 16.6205i 0.546548 0.610570i
\(742\) 0 0
\(743\) 20.7112 11.9576i 0.759819 0.438681i −0.0694121 0.997588i \(-0.522112\pi\)
0.829231 + 0.558907i \(0.188779\pi\)
\(744\) 0 0
\(745\) 9.16482i 0.335773i
\(746\) 0 0
\(747\) 9.67461 1.07385i 0.353975 0.0392900i
\(748\) 0 0
\(749\) −2.27784 + 3.94533i −0.0832304 + 0.144159i
\(750\) 0 0
\(751\) 26.7088 0.974618 0.487309 0.873230i \(-0.337979\pi\)
0.487309 + 0.873230i \(0.337979\pi\)
\(752\) 0 0
\(753\) 33.6948 11.0567i 1.22791 0.402928i
\(754\) 0 0
\(755\) −16.6266 28.7982i −0.605105 1.04807i
\(756\) 0 0
\(757\) −38.1680 22.0363i −1.38724 0.800924i −0.394237 0.919009i \(-0.628991\pi\)
−0.993003 + 0.118085i \(0.962324\pi\)
\(758\) 0 0
\(759\) 41.2926 + 8.65140i 1.49883 + 0.314026i
\(760\) 0 0
\(761\) 3.93893i 0.142786i −0.997448 0.0713931i \(-0.977256\pi\)
0.997448 0.0713931i \(-0.0227445\pi\)
\(762\) 0 0
\(763\) −16.4008 + 28.4071i −0.593750 + 1.02841i
\(764\) 0 0
\(765\) 49.6364 5.50947i 1.79461 0.199195i
\(766\) 0 0
\(767\) −12.8513 22.2591i −0.464034 0.803730i
\(768\) 0 0
\(769\) −13.6091 7.85722i −0.490757 0.283339i 0.234131 0.972205i \(-0.424775\pi\)
−0.724888 + 0.688866i \(0.758109\pi\)
\(770\) 0 0
\(771\) −21.8089 + 24.3636i −0.785429 + 0.877432i
\(772\) 0 0
\(773\) −5.69280 3.28674i −0.204756 0.118216i 0.394116 0.919061i \(-0.371051\pi\)
−0.598872 + 0.800845i \(0.704384\pi\)
\(774\) 0 0
\(775\) −53.8259 31.0764i −1.93348 1.11630i
\(776\) 0 0
\(777\) 51.2694 + 10.7417i 1.83928 + 0.385356i
\(778\) 0 0
\(779\) −2.78603 −0.0998200
\(780\) 0 0
\(781\) −36.6012 + 21.1317i −1.30969 + 0.756151i
\(782\) 0 0
\(783\) −22.7663 + 10.3755i −0.813601 + 0.370789i
\(784\) 0 0
\(785\) 4.88308 + 8.45774i 0.174285 + 0.301870i
\(786\) 0 0
\(787\) −16.0667 + 9.27613i −0.572717 + 0.330658i −0.758234 0.651983i \(-0.773937\pi\)
0.185517 + 0.982641i \(0.440604\pi\)
\(788\) 0 0
\(789\) 31.9304 + 6.68990i 1.13675 + 0.238167i
\(790\) 0 0
\(791\) −2.34999 + 1.35677i −0.0835560 + 0.0482411i
\(792\) 0 0
\(793\) −5.16869 + 8.95243i −0.183545 + 0.317910i
\(794\) 0 0
\(795\) −13.5578 + 64.7104i −0.480845 + 2.29504i
\(796\) 0 0
\(797\) 9.71007 + 5.60611i 0.343948 + 0.198579i 0.662017 0.749489i \(-0.269701\pi\)
−0.318068 + 0.948068i \(0.603034\pi\)
\(798\) 0 0
\(799\) 39.3436 1.39188
\(800\) 0 0
\(801\) −2.56228 + 5.84639i −0.0905338 + 0.206572i
\(802\) 0 0
\(803\) 3.73016 0.131634
\(804\) 0 0
\(805\) −109.528 −3.86034
\(806\) 0 0
\(807\) −37.7210 7.90310i −1.32784 0.278202i
\(808\) 0 0
\(809\) 5.69575 0.200252 0.100126 0.994975i \(-0.468075\pi\)
0.100126 + 0.994975i \(0.468075\pi\)
\(810\) 0 0
\(811\) −3.58570 2.07020i −0.125911 0.0726947i 0.435722 0.900081i \(-0.356493\pi\)
−0.561633 + 0.827387i \(0.689827\pi\)
\(812\) 0 0
\(813\) 18.3584 + 3.84636i 0.643858 + 0.134898i
\(814\) 0 0
\(815\) 23.8264 41.2685i 0.834601 1.44557i
\(816\) 0 0
\(817\) 54.5220 31.4783i 1.90748 1.10129i
\(818\) 0 0
\(819\) −22.8931 10.0333i −0.799949 0.350592i
\(820\) 0 0
\(821\) −20.9010 + 12.0672i −0.729450 + 0.421148i −0.818221 0.574904i \(-0.805039\pi\)
0.0887707 + 0.996052i \(0.471706\pi\)
\(822\) 0 0
\(823\) 20.4375 + 35.3987i 0.712406 + 1.23392i 0.963952 + 0.266077i \(0.0857277\pi\)
−0.251546 + 0.967845i \(0.580939\pi\)
\(824\) 0 0
\(825\) −30.5242 27.3236i −1.06272 0.951286i
\(826\) 0 0
\(827\) 37.1522 21.4499i 1.29191 0.745884i 0.312917 0.949780i \(-0.398694\pi\)
0.978993 + 0.203896i \(0.0653604\pi\)
\(828\) 0 0
\(829\) 18.5255 0.643419 0.321709 0.946838i \(-0.395743\pi\)
0.321709 + 0.946838i \(0.395743\pi\)
\(830\) 0 0
\(831\) −1.94215 + 9.26974i −0.0673723 + 0.321564i
\(832\) 0 0
\(833\) −26.6269 15.3731i −0.922568 0.532645i
\(834\) 0 0
\(835\) 28.7273 + 16.5857i 0.994150 + 0.573973i
\(836\) 0 0
\(837\) 36.7966 16.7696i 1.27188 0.579643i
\(838\) 0 0
\(839\) 28.7508 + 16.5993i 0.992586 + 0.573070i 0.906046 0.423178i \(-0.139086\pi\)
0.0865399 + 0.996248i \(0.472419\pi\)
\(840\) 0 0
\(841\) −2.90826 5.03725i −0.100285 0.173698i
\(842\) 0 0
\(843\) 25.3841 8.32960i 0.874275 0.286887i
\(844\) 0 0
\(845\) −14.2644 + 24.7067i −0.490711 + 0.849936i
\(846\) 0 0
\(847\) 8.23766i 0.283049i
\(848\) 0 0
\(849\) −5.23435 + 24.9832i −0.179642 + 0.857422i
\(850\) 0 0
\(851\) −58.2933 33.6556i −1.99827 1.15370i
\(852\) 0 0
\(853\) 16.2252 + 28.1029i 0.555541 + 0.962225i 0.997861 + 0.0653678i \(0.0208221\pi\)
−0.442320 + 0.896857i \(0.645845\pi\)
\(854\) 0 0
\(855\) −6.81263 61.3770i −0.232987 2.09905i
\(856\) 0 0
\(857\) 47.9928 1.63940 0.819701 0.572791i \(-0.194139\pi\)
0.819701 + 0.572791i \(0.194139\pi\)
\(858\) 0 0
\(859\) −3.44272 + 5.96297i −0.117464 + 0.203454i −0.918762 0.394812i \(-0.870810\pi\)
0.801298 + 0.598265i \(0.204143\pi\)
\(860\) 0 0
\(861\) 0.973337 + 2.96621i 0.0331713 + 0.101088i
\(862\) 0 0
\(863\) 0.688741i 0.0234450i −0.999931 0.0117225i \(-0.996269\pi\)
0.999931 0.0117225i \(-0.00373148\pi\)
\(864\) 0 0
\(865\) 41.4501 23.9312i 1.40934 0.813686i
\(866\) 0 0
\(867\) −5.60029 5.01307i −0.190196 0.170253i
\(868\) 0 0
\(869\) 11.2616 6.50187i 0.382023 0.220561i
\(870\) 0 0
\(871\) −3.93381 18.0309i −0.133292 0.610954i
\(872\) 0 0
\(873\) −30.9121 + 3.43114i −1.04622 + 0.116127i
\(874\) 0 0
\(875\) 34.4424 + 19.8854i 1.16437 + 0.672248i
\(876\) 0 0
\(877\) 2.02107 + 3.50059i 0.0682466 + 0.118207i 0.898130 0.439731i \(-0.144926\pi\)
−0.829883 + 0.557938i \(0.811593\pi\)
\(878\) 0 0
\(879\) 12.0982 + 2.53475i 0.408063 + 0.0854951i
\(880\) 0 0
\(881\) 25.1225 14.5045i 0.846399 0.488668i −0.0130355 0.999915i \(-0.504149\pi\)
0.859434 + 0.511247i \(0.170816\pi\)
\(882\) 0 0
\(883\) −18.3892 10.6170i −0.618847 0.357292i 0.157573 0.987507i \(-0.449633\pi\)
−0.776420 + 0.630216i \(0.782966\pi\)
\(884\) 0 0
\(885\) −69.6430 14.5912i −2.34102 0.490479i
\(886\) 0 0
\(887\) −29.7696 + 17.1875i −0.999566 + 0.577099i −0.908120 0.418711i \(-0.862482\pi\)
−0.0914458 + 0.995810i \(0.529149\pi\)
\(888\) 0 0
\(889\) −37.0383 + 21.3841i −1.24222 + 0.717199i
\(890\) 0 0
\(891\) 26.0052 5.84499i 0.871208 0.195815i
\(892\) 0 0
\(893\) 48.6496i 1.62800i
\(894\) 0 0
\(895\) 58.3598 1.95075
\(896\) 0 0
\(897\) 23.9313 + 21.4220i 0.799044 + 0.715260i
\(898\) 0 0
\(899\) −18.7355 32.4508i −0.624863 1.08229i
\(900\) 0 0
\(901\) 42.3759 24.4658i 1.41175 0.815073i
\(902\) 0 0
\(903\) −52.5619 47.0505i −1.74915 1.56574i
\(904\) 0 0
\(905\) −2.99580 + 5.18888i −0.0995838 + 0.172484i
\(906\) 0 0
\(907\) 11.4099 19.7625i 0.378858 0.656202i −0.612038 0.790828i \(-0.709650\pi\)
0.990896 + 0.134627i \(0.0429835\pi\)
\(908\) 0 0
\(909\) −34.0364 + 25.0331i −1.12892 + 0.830294i
\(910\) 0 0
\(911\) 33.8372i 1.12108i 0.828129 + 0.560538i \(0.189406\pi\)
−0.828129 + 0.560538i \(0.810594\pi\)
\(912\) 0 0
\(913\) 9.60929i 0.318021i
\(914\) 0 0
\(915\) 8.92266 + 27.1914i 0.294974 + 0.898921i
\(916\) 0 0
\(917\) −33.3257 + 57.7218i −1.10051 + 1.90614i
\(918\) 0 0
\(919\) 1.39111 0.803156i 0.0458884 0.0264937i −0.476880 0.878968i \(-0.658232\pi\)
0.522769 + 0.852475i \(0.324899\pi\)
\(920\) 0 0
\(921\) 0.254480 0.0835056i 0.00838540 0.00275160i
\(922\) 0 0
\(923\) −32.1752 −1.05906
\(924\) 0 0
\(925\) 32.6808 + 56.6049i 1.07454 + 1.86116i
\(926\) 0 0
\(927\) 15.7717 11.5997i 0.518010 0.380986i
\(928\) 0 0
\(929\) −33.5542 −1.10088 −0.550439 0.834875i \(-0.685540\pi\)
−0.550439 + 0.834875i \(0.685540\pi\)
\(930\) 0 0
\(931\) −19.0093 + 32.9250i −0.623004 + 1.07907i
\(932\) 0 0
\(933\) 7.96783 38.0299i 0.260855 1.24504i
\(934\) 0 0
\(935\) 49.3013i 1.61232i
\(936\) 0 0
\(937\) 30.6902i 1.00261i −0.865272 0.501303i \(-0.832854\pi\)
0.865272 0.501303i \(-0.167146\pi\)
\(938\) 0 0
\(939\) 11.1000 + 2.32561i 0.362235 + 0.0758935i
\(940\) 0 0
\(941\) −27.0344 −0.881296 −0.440648 0.897680i \(-0.645251\pi\)
−0.440648 + 0.897680i \(0.645251\pi\)
\(942\) 0 0
\(943\) 4.01152i 0.130633i
\(944\) 0 0
\(945\) −62.9661 + 28.6960i −2.04829 + 0.933482i
\(946\) 0 0
\(947\) 19.3183i 0.627760i 0.949463 + 0.313880i \(0.101629\pi\)
−0.949463 + 0.313880i \(0.898371\pi\)
\(948\) 0 0
\(949\) 2.45932 + 1.41989i 0.0798330 + 0.0460916i
\(950\) 0 0
\(951\) 8.68543 + 26.4685i 0.281644 + 0.858299i
\(952\) 0 0
\(953\) 24.5725i 0.795983i −0.917389 0.397991i \(-0.869707\pi\)
0.917389 0.397991i \(-0.130293\pi\)
\(954\) 0 0
\(955\) −27.5936 47.7936i −0.892909 1.54656i
\(956\) 0 0
\(957\) −7.70062 23.4673i −0.248926 0.758591i
\(958\) 0 0
\(959\) −6.98474 4.03264i −0.225549 0.130221i
\(960\) 0 0
\(961\) 14.7817 + 25.6027i 0.476830 + 0.825894i
\(962\) 0 0
\(963\) 1.48458 3.38738i 0.0478399 0.109157i
\(964\) 0 0
\(965\) 27.5792 0.887806
\(966\) 0 0
\(967\) −22.5787 + 39.1075i −0.726083 + 1.25761i 0.232444 + 0.972610i \(0.425328\pi\)
−0.958527 + 0.285003i \(0.908006\pi\)
\(968\) 0 0
\(969\) −30.4826 + 34.0533i −0.979243 + 1.09395i
\(970\) 0 0
\(971\) 31.2612 + 18.0487i 1.00322 + 0.579209i 0.909199 0.416361i \(-0.136695\pi\)
0.0940204 + 0.995570i \(0.470028\pi\)
\(972\) 0 0
\(973\) −2.73883 + 4.74380i −0.0878029 + 0.152079i
\(974\) 0 0
\(975\) −9.72410 29.6338i −0.311420 0.949040i
\(976\) 0 0
\(977\) −16.5345 + 9.54621i −0.528986 + 0.305410i −0.740604 0.671942i \(-0.765460\pi\)
0.211617 + 0.977353i \(0.432127\pi\)
\(978\) 0 0
\(979\) −5.45719 3.15071i −0.174413 0.100697i
\(980\) 0 0
\(981\) 10.6892 24.3897i 0.341281 0.778703i
\(982\) 0 0
\(983\) −37.2412 −1.18781 −0.593906 0.804535i \(-0.702415\pi\)
−0.593906 + 0.804535i \(0.702415\pi\)
\(984\) 0 0
\(985\) 5.60042 9.70021i 0.178444 0.309074i
\(986\) 0 0
\(987\) −51.7957 + 16.9964i −1.64868 + 0.541000i
\(988\) 0 0
\(989\) 45.3246 + 78.5045i 1.44124 + 2.49630i
\(990\) 0 0
\(991\) 25.4788i 0.809360i 0.914458 + 0.404680i \(0.132617\pi\)
−0.914458 + 0.404680i \(0.867383\pi\)
\(992\) 0 0
\(993\) 29.0292 32.4296i 0.921213 1.02912i
\(994\) 0 0
\(995\) 0.485773 + 0.841384i 0.0154000 + 0.0266737i
\(996\) 0 0
\(997\) −38.6275 −1.22335 −0.611673 0.791111i \(-0.709503\pi\)
−0.611673 + 0.791111i \(0.709503\pi\)
\(998\) 0 0
\(999\) −42.3298 4.07548i −1.33926 0.128942i
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 804.2.o.d.365.10 yes 36
3.2 odd 2 inner 804.2.o.d.365.9 36
67.38 odd 6 inner 804.2.o.d.641.9 yes 36
201.38 even 6 inner 804.2.o.d.641.10 yes 36
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
804.2.o.d.365.9 36 3.2 odd 2 inner
804.2.o.d.365.10 yes 36 1.1 even 1 trivial
804.2.o.d.641.9 yes 36 67.38 odd 6 inner
804.2.o.d.641.10 yes 36 201.38 even 6 inner