Properties

Label 456.2.q.f.121.3
Level $456$
Weight $2$
Character 456.121
Analytic conductor $3.641$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [456,2,Mod(49,456)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(456, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("456.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 456.q (of order \(3\), 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: \(3.64117833217\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 121.3
Root \(0.939693 + 0.342020i\) of defining polynomial
Character \(\chi\) \(=\) 456.121
Dual form 456.2.q.f.49.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(1.87939 + 3.25519i) q^{5} +4.75877 q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{3} +(1.87939 + 3.25519i) q^{5} +4.75877 q^{7} +(-0.500000 + 0.866025i) q^{9} -4.36959 q^{11} +(-0.500000 + 0.866025i) q^{13} +(-1.87939 + 3.25519i) q^{15} +(-3.06418 - 5.30731i) q^{17} +(0.694593 - 4.30320i) q^{19} +(2.37939 + 4.12122i) q^{21} +(1.87939 - 3.25519i) q^{23} +(-4.56418 + 7.90539i) q^{25} -1.00000 q^{27} +(-2.69459 + 4.66717i) q^{29} +7.36959 q^{31} +(-2.18479 - 3.78417i) q^{33} +(8.94356 + 15.4907i) q^{35} -7.12836 q^{37} -1.00000 q^{39} +(1.75877 + 3.04628i) q^{41} +(0.379385 + 0.657115i) q^{43} -3.75877 q^{45} +(3.00000 - 5.19615i) q^{47} +15.6459 q^{49} +(3.06418 - 5.30731i) q^{51} +(-2.57398 + 4.45826i) q^{53} +(-8.21213 - 14.2238i) q^{55} +(4.07398 - 1.55007i) q^{57} +(-4.63816 - 8.03352i) q^{59} +(4.25877 - 7.37641i) q^{61} +(-2.37939 + 4.12122i) q^{63} -3.75877 q^{65} +(-0.684793 + 1.18610i) q^{67} +3.75877 q^{69} +(5.82295 + 10.0856i) q^{71} +(-2.25877 - 3.91231i) q^{73} -9.12836 q^{75} -20.7939 q^{77} +(-7.07398 - 12.2525i) q^{79} +(-0.500000 - 0.866025i) q^{81} +4.90673 q^{83} +(11.5175 - 19.9490i) q^{85} -5.38919 q^{87} +(-4.18479 + 7.24827i) q^{89} +(-2.37939 + 4.12122i) q^{91} +(3.68479 + 6.38225i) q^{93} +(15.3131 - 5.82634i) q^{95} +(5.82295 + 10.0856i) q^{97} +(2.18479 - 3.78417i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} + 6 q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} + 6 q^{7} - 3 q^{9} - 12 q^{11} - 3 q^{13} + 3 q^{21} - 9 q^{25} - 6 q^{27} - 12 q^{29} + 30 q^{31} - 6 q^{33} + 24 q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 9 q^{43} + 18 q^{47} + 12 q^{49} + 9 q^{57} + 6 q^{59} + 3 q^{61} - 3 q^{63} + 3 q^{67} - 6 q^{71} + 9 q^{73} - 18 q^{75} - 12 q^{77} - 27 q^{79} - 3 q^{81} - 24 q^{83} + 24 q^{85} - 24 q^{87} - 18 q^{89} - 3 q^{91} + 15 q^{93} + 48 q^{95} - 6 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/456\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(343\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) 1.87939 + 3.25519i 0.840487 + 1.45577i 0.889484 + 0.456967i \(0.151064\pi\)
−0.0489972 + 0.998799i \(0.515603\pi\)
\(6\) 0 0
\(7\) 4.75877 1.79865 0.899323 0.437285i \(-0.144060\pi\)
0.899323 + 0.437285i \(0.144060\pi\)
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −4.36959 −1.31748 −0.658740 0.752371i \(-0.728910\pi\)
−0.658740 + 0.752371i \(0.728910\pi\)
\(12\) 0 0
\(13\) −0.500000 + 0.866025i −0.138675 + 0.240192i −0.926995 0.375073i \(-0.877618\pi\)
0.788320 + 0.615265i \(0.210951\pi\)
\(14\) 0 0
\(15\) −1.87939 + 3.25519i −0.485255 + 0.840487i
\(16\) 0 0
\(17\) −3.06418 5.30731i −0.743172 1.28721i −0.951044 0.309056i \(-0.899987\pi\)
0.207872 0.978156i \(-0.433346\pi\)
\(18\) 0 0
\(19\) 0.694593 4.30320i 0.159350 0.987222i
\(20\) 0 0
\(21\) 2.37939 + 4.12122i 0.519224 + 0.899323i
\(22\) 0 0
\(23\) 1.87939 3.25519i 0.391879 0.678754i −0.600818 0.799385i \(-0.705159\pi\)
0.992697 + 0.120631i \(0.0384919\pi\)
\(24\) 0 0
\(25\) −4.56418 + 7.90539i −0.912836 + 1.58108i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −2.69459 + 4.66717i −0.500373 + 0.866672i 0.499627 + 0.866241i \(0.333471\pi\)
−1.00000 0.000431109i \(0.999863\pi\)
\(30\) 0 0
\(31\) 7.36959 1.32362 0.661808 0.749673i \(-0.269789\pi\)
0.661808 + 0.749673i \(0.269789\pi\)
\(32\) 0 0
\(33\) −2.18479 3.78417i −0.380324 0.658740i
\(34\) 0 0
\(35\) 8.94356 + 15.4907i 1.51174 + 2.61841i
\(36\) 0 0
\(37\) −7.12836 −1.17189 −0.585947 0.810349i \(-0.699277\pi\)
−0.585947 + 0.810349i \(0.699277\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 1.75877 + 3.04628i 0.274674 + 0.475749i 0.970053 0.242894i \(-0.0780968\pi\)
−0.695379 + 0.718643i \(0.744763\pi\)
\(42\) 0 0
\(43\) 0.379385 + 0.657115i 0.0578557 + 0.100209i 0.893503 0.449058i \(-0.148240\pi\)
−0.835647 + 0.549267i \(0.814907\pi\)
\(44\) 0 0
\(45\) −3.75877 −0.560324
\(46\) 0 0
\(47\) 3.00000 5.19615i 0.437595 0.757937i −0.559908 0.828554i \(-0.689164\pi\)
0.997503 + 0.0706177i \(0.0224970\pi\)
\(48\) 0 0
\(49\) 15.6459 2.23513
\(50\) 0 0
\(51\) 3.06418 5.30731i 0.429071 0.743172i
\(52\) 0 0
\(53\) −2.57398 + 4.45826i −0.353563 + 0.612389i −0.986871 0.161511i \(-0.948363\pi\)
0.633308 + 0.773900i \(0.281697\pi\)
\(54\) 0 0
\(55\) −8.21213 14.2238i −1.10732 1.91794i
\(56\) 0 0
\(57\) 4.07398 1.55007i 0.539612 0.205311i
\(58\) 0 0
\(59\) −4.63816 8.03352i −0.603836 1.04588i −0.992234 0.124384i \(-0.960305\pi\)
0.388398 0.921492i \(-0.373029\pi\)
\(60\) 0 0
\(61\) 4.25877 7.37641i 0.545280 0.944452i −0.453310 0.891353i \(-0.649757\pi\)
0.998589 0.0530990i \(-0.0169099\pi\)
\(62\) 0 0
\(63\) −2.37939 + 4.12122i −0.299774 + 0.519224i
\(64\) 0 0
\(65\) −3.75877 −0.466218
\(66\) 0 0
\(67\) −0.684793 + 1.18610i −0.0836607 + 0.144905i −0.904820 0.425795i \(-0.859995\pi\)
0.821159 + 0.570699i \(0.193328\pi\)
\(68\) 0 0
\(69\) 3.75877 0.452503
\(70\) 0 0
\(71\) 5.82295 + 10.0856i 0.691057 + 1.19695i 0.971492 + 0.237073i \(0.0761879\pi\)
−0.280435 + 0.959873i \(0.590479\pi\)
\(72\) 0 0
\(73\) −2.25877 3.91231i −0.264369 0.457901i 0.703029 0.711161i \(-0.251830\pi\)
−0.967398 + 0.253260i \(0.918497\pi\)
\(74\) 0 0
\(75\) −9.12836 −1.05405
\(76\) 0 0
\(77\) −20.7939 −2.36968
\(78\) 0 0
\(79\) −7.07398 12.2525i −0.795885 1.37851i −0.922276 0.386532i \(-0.873673\pi\)
0.126391 0.991980i \(-0.459661\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 4.90673 0.538583 0.269292 0.963059i \(-0.413210\pi\)
0.269292 + 0.963059i \(0.413210\pi\)
\(84\) 0 0
\(85\) 11.5175 19.9490i 1.24925 2.16377i
\(86\) 0 0
\(87\) −5.38919 −0.577781
\(88\) 0 0
\(89\) −4.18479 + 7.24827i −0.443587 + 0.768315i −0.997953 0.0639579i \(-0.979628\pi\)
0.554365 + 0.832273i \(0.312961\pi\)
\(90\) 0 0
\(91\) −2.37939 + 4.12122i −0.249427 + 0.432021i
\(92\) 0 0
\(93\) 3.68479 + 6.38225i 0.382095 + 0.661808i
\(94\) 0 0
\(95\) 15.3131 5.82634i 1.57110 0.597770i
\(96\) 0 0
\(97\) 5.82295 + 10.0856i 0.591231 + 1.02404i 0.994067 + 0.108770i \(0.0346911\pi\)
−0.402836 + 0.915272i \(0.631976\pi\)
\(98\) 0 0
\(99\) 2.18479 3.78417i 0.219580 0.380324i
\(100\) 0 0
\(101\) 2.06418 3.57526i 0.205393 0.355752i −0.744865 0.667216i \(-0.767486\pi\)
0.950258 + 0.311464i \(0.100819\pi\)
\(102\) 0 0
\(103\) −6.75877 −0.665961 −0.332981 0.942934i \(-0.608054\pi\)
−0.332981 + 0.942934i \(0.608054\pi\)
\(104\) 0 0
\(105\) −8.94356 + 15.4907i −0.872802 + 1.51174i
\(106\) 0 0
\(107\) −4.12836 −0.399103 −0.199552 0.979887i \(-0.563949\pi\)
−0.199552 + 0.979887i \(0.563949\pi\)
\(108\) 0 0
\(109\) 0.0641778 + 0.111159i 0.00614712 + 0.0106471i 0.869083 0.494667i \(-0.164710\pi\)
−0.862935 + 0.505314i \(0.831377\pi\)
\(110\) 0 0
\(111\) −3.56418 6.17334i −0.338297 0.585947i
\(112\) 0 0
\(113\) −2.40879 −0.226600 −0.113300 0.993561i \(-0.536142\pi\)
−0.113300 + 0.993561i \(0.536142\pi\)
\(114\) 0 0
\(115\) 14.1284 1.31748
\(116\) 0 0
\(117\) −0.500000 0.866025i −0.0462250 0.0800641i
\(118\) 0 0
\(119\) −14.5817 25.2563i −1.33670 2.31524i
\(120\) 0 0
\(121\) 8.09327 0.735752
\(122\) 0 0
\(123\) −1.75877 + 3.04628i −0.158583 + 0.274674i
\(124\) 0 0
\(125\) −15.5175 −1.38793
\(126\) 0 0
\(127\) −0.241230 + 0.417822i −0.0214057 + 0.0370757i −0.876530 0.481348i \(-0.840147\pi\)
0.855124 + 0.518423i \(0.173481\pi\)
\(128\) 0 0
\(129\) −0.379385 + 0.657115i −0.0334030 + 0.0578557i
\(130\) 0 0
\(131\) 1.69459 + 2.93512i 0.148057 + 0.256443i 0.930509 0.366268i \(-0.119365\pi\)
−0.782452 + 0.622711i \(0.786031\pi\)
\(132\) 0 0
\(133\) 3.30541 20.4779i 0.286615 1.77566i
\(134\) 0 0
\(135\) −1.87939 3.25519i −0.161752 0.280162i
\(136\) 0 0
\(137\) 9.21213 15.9559i 0.787046 1.36320i −0.140724 0.990049i \(-0.544943\pi\)
0.927769 0.373154i \(-0.121724\pi\)
\(138\) 0 0
\(139\) −0.00980018 + 0.0169744i −0.000831240 + 0.00143975i −0.866441 0.499280i \(-0.833598\pi\)
0.865609 + 0.500720i \(0.166931\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 2.18479 3.78417i 0.182702 0.316448i
\(144\) 0 0
\(145\) −20.2567 −1.68223
\(146\) 0 0
\(147\) 7.82295 + 13.5497i 0.645226 + 1.11756i
\(148\) 0 0
\(149\) −3.12061 5.40506i −0.255651 0.442800i 0.709421 0.704785i \(-0.248956\pi\)
−0.965072 + 0.261985i \(0.915623\pi\)
\(150\) 0 0
\(151\) 0.739170 0.0601528 0.0300764 0.999548i \(-0.490425\pi\)
0.0300764 + 0.999548i \(0.490425\pi\)
\(152\) 0 0
\(153\) 6.12836 0.495448
\(154\) 0 0
\(155\) 13.8503 + 23.9894i 1.11248 + 1.92688i
\(156\) 0 0
\(157\) 3.32295 + 5.75552i 0.265200 + 0.459340i 0.967616 0.252427i \(-0.0812286\pi\)
−0.702416 + 0.711767i \(0.747895\pi\)
\(158\) 0 0
\(159\) −5.14796 −0.408259
\(160\) 0 0
\(161\) 8.94356 15.4907i 0.704852 1.22084i
\(162\) 0 0
\(163\) −11.3696 −0.890535 −0.445267 0.895398i \(-0.646891\pi\)
−0.445267 + 0.895398i \(0.646891\pi\)
\(164\) 0 0
\(165\) 8.21213 14.2238i 0.639314 1.10732i
\(166\) 0 0
\(167\) −1.49020 + 2.58110i −0.115315 + 0.199732i −0.917906 0.396799i \(-0.870121\pi\)
0.802591 + 0.596530i \(0.203454\pi\)
\(168\) 0 0
\(169\) 6.00000 + 10.3923i 0.461538 + 0.799408i
\(170\) 0 0
\(171\) 3.37939 + 2.75314i 0.258428 + 0.210538i
\(172\) 0 0
\(173\) 2.69459 + 4.66717i 0.204866 + 0.354838i 0.950090 0.311976i \(-0.100991\pi\)
−0.745224 + 0.666814i \(0.767657\pi\)
\(174\) 0 0
\(175\) −21.7199 + 37.6199i −1.64187 + 2.84380i
\(176\) 0 0
\(177\) 4.63816 8.03352i 0.348625 0.603836i
\(178\) 0 0
\(179\) 4.24123 0.317004 0.158502 0.987359i \(-0.449334\pi\)
0.158502 + 0.987359i \(0.449334\pi\)
\(180\) 0 0
\(181\) −6.30541 + 10.9213i −0.468677 + 0.811773i −0.999359 0.0357984i \(-0.988603\pi\)
0.530682 + 0.847571i \(0.321936\pi\)
\(182\) 0 0
\(183\) 8.51754 0.629635
\(184\) 0 0
\(185\) −13.3969 23.2042i −0.984962 1.70600i
\(186\) 0 0
\(187\) 13.3892 + 23.1907i 0.979114 + 1.69588i
\(188\) 0 0
\(189\) −4.75877 −0.346150
\(190\) 0 0
\(191\) −22.6263 −1.63718 −0.818591 0.574377i \(-0.805244\pi\)
−0.818591 + 0.574377i \(0.805244\pi\)
\(192\) 0 0
\(193\) −12.3229 21.3440i −0.887025 1.53637i −0.843375 0.537326i \(-0.819435\pi\)
−0.0436505 0.999047i \(-0.513899\pi\)
\(194\) 0 0
\(195\) −1.87939 3.25519i −0.134586 0.233109i
\(196\) 0 0
\(197\) 22.6655 1.61485 0.807425 0.589970i \(-0.200861\pi\)
0.807425 + 0.589970i \(0.200861\pi\)
\(198\) 0 0
\(199\) −6.96110 + 12.0570i −0.493460 + 0.854697i −0.999972 0.00753584i \(-0.997601\pi\)
0.506512 + 0.862233i \(0.330935\pi\)
\(200\) 0 0
\(201\) −1.36959 −0.0966031
\(202\) 0 0
\(203\) −12.8229 + 22.2100i −0.899995 + 1.55884i
\(204\) 0 0
\(205\) −6.61081 + 11.4503i −0.461719 + 0.799721i
\(206\) 0 0
\(207\) 1.87939 + 3.25519i 0.130626 + 0.226251i
\(208\) 0 0
\(209\) −3.03508 + 18.8032i −0.209941 + 1.30064i
\(210\) 0 0
\(211\) 1.00980 + 1.74903i 0.0695175 + 0.120408i 0.898689 0.438586i \(-0.144521\pi\)
−0.829172 + 0.558994i \(0.811187\pi\)
\(212\) 0 0
\(213\) −5.82295 + 10.0856i −0.398982 + 0.691057i
\(214\) 0 0
\(215\) −1.42602 + 2.46994i −0.0972539 + 0.168449i
\(216\) 0 0
\(217\) 35.0702 2.38072
\(218\) 0 0
\(219\) 2.25877 3.91231i 0.152634 0.264369i
\(220\) 0 0
\(221\) 6.12836 0.412238
\(222\) 0 0
\(223\) −14.1827 24.5652i −0.949746 1.64501i −0.745958 0.665993i \(-0.768008\pi\)
−0.203789 0.979015i \(-0.565325\pi\)
\(224\) 0 0
\(225\) −4.56418 7.90539i −0.304279 0.527026i
\(226\) 0 0
\(227\) 20.1830 1.33960 0.669798 0.742544i \(-0.266381\pi\)
0.669798 + 0.742544i \(0.266381\pi\)
\(228\) 0 0
\(229\) 11.7784 0.778337 0.389168 0.921167i \(-0.372762\pi\)
0.389168 + 0.921167i \(0.372762\pi\)
\(230\) 0 0
\(231\) −10.3969 18.0080i −0.684068 1.18484i
\(232\) 0 0
\(233\) 0.0641778 + 0.111159i 0.00420443 + 0.00728228i 0.868120 0.496354i \(-0.165328\pi\)
−0.863916 + 0.503637i \(0.831995\pi\)
\(234\) 0 0
\(235\) 22.5526 1.47117
\(236\) 0 0
\(237\) 7.07398 12.2525i 0.459504 0.795885i
\(238\) 0 0
\(239\) 3.10876 0.201089 0.100544 0.994933i \(-0.467942\pi\)
0.100544 + 0.994933i \(0.467942\pi\)
\(240\) 0 0
\(241\) −5.64796 + 9.78255i −0.363817 + 0.630149i −0.988586 0.150661i \(-0.951860\pi\)
0.624769 + 0.780810i \(0.285193\pi\)
\(242\) 0 0
\(243\) 0.500000 0.866025i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) 29.4047 + 50.9304i 1.87860 + 3.25382i
\(246\) 0 0
\(247\) 3.37939 + 2.75314i 0.215025 + 0.175178i
\(248\) 0 0
\(249\) 2.45336 + 4.24935i 0.155476 + 0.269292i
\(250\) 0 0
\(251\) 0.758770 1.31423i 0.0478932 0.0829534i −0.841085 0.540903i \(-0.818083\pi\)
0.888978 + 0.457950i \(0.151416\pi\)
\(252\) 0 0
\(253\) −8.21213 + 14.2238i −0.516292 + 0.894245i
\(254\) 0 0
\(255\) 23.0351 1.44251
\(256\) 0 0
\(257\) 6.09152 10.5508i 0.379979 0.658142i −0.611080 0.791569i \(-0.709265\pi\)
0.991059 + 0.133427i \(0.0425980\pi\)
\(258\) 0 0
\(259\) −33.9222 −2.10782
\(260\) 0 0
\(261\) −2.69459 4.66717i −0.166791 0.288891i
\(262\) 0 0
\(263\) 9.69459 + 16.7915i 0.597794 + 1.03541i 0.993146 + 0.116880i \(0.0372893\pi\)
−0.395352 + 0.918530i \(0.629377\pi\)
\(264\) 0 0
\(265\) −19.3500 −1.18866
\(266\) 0 0
\(267\) −8.36959 −0.512210
\(268\) 0 0
\(269\) −11.7023 20.2690i −0.713504 1.23582i −0.963534 0.267587i \(-0.913774\pi\)
0.250030 0.968238i \(-0.419560\pi\)
\(270\) 0 0
\(271\) −0.482459 0.835644i −0.0293073 0.0507617i 0.851000 0.525166i \(-0.175997\pi\)
−0.880307 + 0.474404i \(0.842663\pi\)
\(272\) 0 0
\(273\) −4.75877 −0.288014
\(274\) 0 0
\(275\) 19.9436 34.5433i 1.20264 2.08304i
\(276\) 0 0
\(277\) 15.3892 0.924647 0.462323 0.886711i \(-0.347016\pi\)
0.462323 + 0.886711i \(0.347016\pi\)
\(278\) 0 0
\(279\) −3.68479 + 6.38225i −0.220603 + 0.382095i
\(280\) 0 0
\(281\) −4.24897 + 7.35943i −0.253472 + 0.439027i −0.964479 0.264158i \(-0.914906\pi\)
0.711007 + 0.703185i \(0.248239\pi\)
\(282\) 0 0
\(283\) 5.51754 + 9.55666i 0.327984 + 0.568085i 0.982112 0.188299i \(-0.0602975\pi\)
−0.654128 + 0.756384i \(0.726964\pi\)
\(284\) 0 0
\(285\) 12.7023 + 10.3484i 0.752421 + 0.612987i
\(286\) 0 0
\(287\) 8.36959 + 14.4965i 0.494041 + 0.855704i
\(288\) 0 0
\(289\) −10.2784 + 17.8027i −0.604610 + 1.04722i
\(290\) 0 0
\(291\) −5.82295 + 10.0856i −0.341347 + 0.591231i
\(292\) 0 0
\(293\) −1.03508 −0.0604701 −0.0302351 0.999543i \(-0.509626\pi\)
−0.0302351 + 0.999543i \(0.509626\pi\)
\(294\) 0 0
\(295\) 17.4338 30.1962i 1.01503 1.75809i
\(296\) 0 0
\(297\) 4.36959 0.253549
\(298\) 0 0
\(299\) 1.87939 + 3.25519i 0.108688 + 0.188253i
\(300\) 0 0
\(301\) 1.80541 + 3.12706i 0.104062 + 0.180241i
\(302\) 0 0
\(303\) 4.12836 0.237168
\(304\) 0 0
\(305\) 32.0155 1.83320
\(306\) 0 0
\(307\) −7.43376 12.8757i −0.424267 0.734852i 0.572084 0.820195i \(-0.306135\pi\)
−0.996352 + 0.0853423i \(0.972802\pi\)
\(308\) 0 0
\(309\) −3.37939 5.85327i −0.192247 0.332981i
\(310\) 0 0
\(311\) −16.0547 −0.910378 −0.455189 0.890395i \(-0.650428\pi\)
−0.455189 + 0.890395i \(0.650428\pi\)
\(312\) 0 0
\(313\) −17.5817 + 30.4524i −0.993777 + 1.72127i −0.400428 + 0.916328i \(0.631138\pi\)
−0.593350 + 0.804945i \(0.702195\pi\)
\(314\) 0 0
\(315\) −17.8871 −1.00783
\(316\) 0 0
\(317\) −9.26857 + 16.0536i −0.520575 + 0.901662i 0.479139 + 0.877739i \(0.340949\pi\)
−0.999714 + 0.0239230i \(0.992384\pi\)
\(318\) 0 0
\(319\) 11.7743 20.3936i 0.659232 1.14182i
\(320\) 0 0
\(321\) −2.06418 3.57526i −0.115211 0.199552i
\(322\) 0 0
\(323\) −24.9668 + 9.49935i −1.38919 + 0.528558i
\(324\) 0 0
\(325\) −4.56418 7.90539i −0.253175 0.438512i
\(326\) 0 0
\(327\) −0.0641778 + 0.111159i −0.00354904 + 0.00614712i
\(328\) 0 0
\(329\) 14.2763 24.7273i 0.787079 1.36326i
\(330\) 0 0
\(331\) −8.75877 −0.481426 −0.240713 0.970596i \(-0.577381\pi\)
−0.240713 + 0.970596i \(0.577381\pi\)
\(332\) 0 0
\(333\) 3.56418 6.17334i 0.195316 0.338297i
\(334\) 0 0
\(335\) −5.14796 −0.281263
\(336\) 0 0
\(337\) 10.6925 + 18.5200i 0.582459 + 1.00885i 0.995187 + 0.0979947i \(0.0312428\pi\)
−0.412728 + 0.910855i \(0.635424\pi\)
\(338\) 0 0
\(339\) −1.20439 2.08607i −0.0654136 0.113300i
\(340\) 0 0
\(341\) −32.2020 −1.74384
\(342\) 0 0
\(343\) 41.1438 2.22156
\(344\) 0 0
\(345\) 7.06418 + 12.2355i 0.380323 + 0.658738i
\(346\) 0 0
\(347\) −11.9632 20.7208i −0.642216 1.11235i −0.984937 0.172914i \(-0.944682\pi\)
0.342721 0.939437i \(-0.388652\pi\)
\(348\) 0 0
\(349\) −23.0702 −1.23492 −0.617459 0.786603i \(-0.711838\pi\)
−0.617459 + 0.786603i \(0.711838\pi\)
\(350\) 0 0
\(351\) 0.500000 0.866025i 0.0266880 0.0462250i
\(352\) 0 0
\(353\) −28.4979 −1.51679 −0.758396 0.651794i \(-0.774017\pi\)
−0.758396 + 0.651794i \(0.774017\pi\)
\(354\) 0 0
\(355\) −21.8871 + 37.9096i −1.16165 + 2.01203i
\(356\) 0 0
\(357\) 14.5817 25.2563i 0.771746 1.33670i
\(358\) 0 0
\(359\) −2.48246 4.29975i −0.131019 0.226932i 0.793051 0.609156i \(-0.208492\pi\)
−0.924070 + 0.382224i \(0.875158\pi\)
\(360\) 0 0
\(361\) −18.0351 5.97794i −0.949215 0.314629i
\(362\) 0 0
\(363\) 4.04664 + 7.00898i 0.212393 + 0.367876i
\(364\) 0 0
\(365\) 8.49020 14.7055i 0.444397 0.769719i
\(366\) 0 0
\(367\) −9.42396 + 16.3228i −0.491927 + 0.852042i −0.999957 0.00929712i \(-0.997041\pi\)
0.508030 + 0.861339i \(0.330374\pi\)
\(368\) 0 0
\(369\) −3.51754 −0.183116
\(370\) 0 0
\(371\) −12.2490 + 21.2158i −0.635935 + 1.10147i
\(372\) 0 0
\(373\) 35.9026 1.85897 0.929483 0.368864i \(-0.120253\pi\)
0.929483 + 0.368864i \(0.120253\pi\)
\(374\) 0 0
\(375\) −7.75877 13.4386i −0.400661 0.693966i
\(376\) 0 0
\(377\) −2.69459 4.66717i −0.138779 0.240372i
\(378\) 0 0
\(379\) −12.2763 −0.630592 −0.315296 0.948993i \(-0.602104\pi\)
−0.315296 + 0.948993i \(0.602104\pi\)
\(380\) 0 0
\(381\) −0.482459 −0.0247171
\(382\) 0 0
\(383\) −7.20439 12.4784i −0.368127 0.637615i 0.621145 0.783695i \(-0.286668\pi\)
−0.989273 + 0.146080i \(0.953334\pi\)
\(384\) 0 0
\(385\) −39.0797 67.6880i −1.99168 3.44970i
\(386\) 0 0
\(387\) −0.758770 −0.0385705
\(388\) 0 0
\(389\) −3.18479 + 5.51622i −0.161475 + 0.279684i −0.935398 0.353597i \(-0.884959\pi\)
0.773923 + 0.633280i \(0.218292\pi\)
\(390\) 0 0
\(391\) −23.0351 −1.16493
\(392\) 0 0
\(393\) −1.69459 + 2.93512i −0.0854809 + 0.148057i
\(394\) 0 0
\(395\) 26.5895 46.0543i 1.33786 2.31724i
\(396\) 0 0
\(397\) 5.27837 + 9.14241i 0.264914 + 0.458844i 0.967541 0.252714i \(-0.0813231\pi\)
−0.702627 + 0.711558i \(0.747990\pi\)
\(398\) 0 0
\(399\) 19.3871 7.37641i 0.970570 0.369282i
\(400\) 0 0
\(401\) −10.7219 18.5709i −0.535428 0.927388i −0.999143 0.0414036i \(-0.986817\pi\)
0.463715 0.885985i \(-0.346516\pi\)
\(402\) 0 0
\(403\) −3.68479 + 6.38225i −0.183553 + 0.317922i
\(404\) 0 0
\(405\) 1.87939 3.25519i 0.0933874 0.161752i
\(406\) 0 0
\(407\) 31.1480 1.54395
\(408\) 0 0
\(409\) −3.32501 + 5.75908i −0.164411 + 0.284768i −0.936446 0.350812i \(-0.885906\pi\)
0.772035 + 0.635580i \(0.219239\pi\)
\(410\) 0 0
\(411\) 18.4243 0.908802
\(412\) 0 0
\(413\) −22.0719 38.2297i −1.08609 1.88116i
\(414\) 0 0
\(415\) 9.22163 + 15.9723i 0.452672 + 0.784051i
\(416\) 0 0
\(417\) −0.0196004 −0.000959834
\(418\) 0 0
\(419\) 5.84793 0.285690 0.142845 0.989745i \(-0.454375\pi\)
0.142845 + 0.989745i \(0.454375\pi\)
\(420\) 0 0
\(421\) −0.714193 1.23702i −0.0348076 0.0602886i 0.848097 0.529841i \(-0.177749\pi\)
−0.882904 + 0.469553i \(0.844415\pi\)
\(422\) 0 0
\(423\) 3.00000 + 5.19615i 0.145865 + 0.252646i
\(424\) 0 0
\(425\) 55.9418 2.71358
\(426\) 0 0
\(427\) 20.2665 35.1026i 0.980765 1.69874i
\(428\) 0 0
\(429\) 4.36959 0.210966
\(430\) 0 0
\(431\) −0.389185 + 0.674089i −0.0187464 + 0.0324697i −0.875246 0.483677i \(-0.839301\pi\)
0.856500 + 0.516147i \(0.172634\pi\)
\(432\) 0 0
\(433\) −13.3425 + 23.1100i −0.641202 + 1.11059i 0.343963 + 0.938983i \(0.388231\pi\)
−0.985165 + 0.171611i \(0.945103\pi\)
\(434\) 0 0
\(435\) −10.1284 17.5428i −0.485617 0.841114i
\(436\) 0 0
\(437\) −12.7023 10.3484i −0.607635 0.495031i
\(438\) 0 0
\(439\) 14.2665 + 24.7103i 0.680903 + 1.17936i 0.974706 + 0.223493i \(0.0717460\pi\)
−0.293802 + 0.955866i \(0.594921\pi\)
\(440\) 0 0
\(441\) −7.82295 + 13.5497i −0.372521 + 0.645226i
\(442\) 0 0
\(443\) 8.82295 15.2818i 0.419191 0.726060i −0.576667 0.816979i \(-0.695647\pi\)
0.995858 + 0.0909191i \(0.0289805\pi\)
\(444\) 0 0
\(445\) −31.4593 −1.49132
\(446\) 0 0
\(447\) 3.12061 5.40506i 0.147600 0.255651i
\(448\) 0 0
\(449\) −1.38919 −0.0655597 −0.0327799 0.999463i \(-0.510436\pi\)
−0.0327799 + 0.999463i \(0.510436\pi\)
\(450\) 0 0
\(451\) −7.68510 13.3110i −0.361877 0.626790i
\(452\) 0 0
\(453\) 0.369585 + 0.640140i 0.0173646 + 0.0300764i
\(454\) 0 0
\(455\) −17.8871 −0.838561
\(456\) 0 0
\(457\) 24.6851 1.15472 0.577360 0.816490i \(-0.304083\pi\)
0.577360 + 0.816490i \(0.304083\pi\)
\(458\) 0 0
\(459\) 3.06418 + 5.30731i 0.143024 + 0.247724i
\(460\) 0 0
\(461\) −0.795607 1.37803i −0.0370551 0.0641813i 0.846903 0.531747i \(-0.178464\pi\)
−0.883958 + 0.467566i \(0.845131\pi\)
\(462\) 0 0
\(463\) −21.7547 −1.01102 −0.505512 0.862819i \(-0.668696\pi\)
−0.505512 + 0.862819i \(0.668696\pi\)
\(464\) 0 0
\(465\) −13.8503 + 23.9894i −0.642292 + 1.11248i
\(466\) 0 0
\(467\) 15.5175 0.718066 0.359033 0.933325i \(-0.383107\pi\)
0.359033 + 0.933325i \(0.383107\pi\)
\(468\) 0 0
\(469\) −3.25877 + 5.64436i −0.150476 + 0.260632i
\(470\) 0 0
\(471\) −3.32295 + 5.75552i −0.153113 + 0.265200i
\(472\) 0 0
\(473\) −1.65776 2.87132i −0.0762237 0.132023i
\(474\) 0 0
\(475\) 30.8482 + 25.1316i 1.41541 + 1.15312i
\(476\) 0 0
\(477\) −2.57398 4.45826i −0.117854 0.204130i
\(478\) 0 0
\(479\) 5.82295 10.0856i 0.266057 0.460825i −0.701783 0.712391i \(-0.747612\pi\)
0.967840 + 0.251566i \(0.0809456\pi\)
\(480\) 0 0
\(481\) 3.56418 6.17334i 0.162513 0.281480i
\(482\) 0 0
\(483\) 17.8871 0.813892
\(484\) 0 0
\(485\) −21.8871 + 37.9096i −0.993843 + 1.72139i
\(486\) 0 0
\(487\) 38.5526 1.74699 0.873493 0.486837i \(-0.161849\pi\)
0.873493 + 0.486837i \(0.161849\pi\)
\(488\) 0 0
\(489\) −5.68479 9.84635i −0.257075 0.445267i
\(490\) 0 0
\(491\) −3.82295 6.62154i −0.172527 0.298826i 0.766776 0.641915i \(-0.221860\pi\)
−0.939303 + 0.343089i \(0.888527\pi\)
\(492\) 0 0
\(493\) 33.0268 1.48745
\(494\) 0 0
\(495\) 16.4243 0.738216
\(496\) 0 0
\(497\) 27.7101 + 47.9953i 1.24297 + 2.15288i
\(498\) 0 0
\(499\) −15.3152 26.5267i −0.685603 1.18750i −0.973247 0.229762i \(-0.926205\pi\)
0.287644 0.957737i \(-0.407128\pi\)
\(500\) 0 0
\(501\) −2.98040 −0.133154
\(502\) 0 0
\(503\) 2.69459 4.66717i 0.120146 0.208099i −0.799679 0.600428i \(-0.794997\pi\)
0.919825 + 0.392329i \(0.128330\pi\)
\(504\) 0 0
\(505\) 15.5175 0.690522
\(506\) 0 0
\(507\) −6.00000 + 10.3923i −0.266469 + 0.461538i
\(508\) 0 0
\(509\) 1.12836 1.95437i 0.0500135 0.0866259i −0.839935 0.542687i \(-0.817407\pi\)
0.889948 + 0.456061i \(0.150740\pi\)
\(510\) 0 0
\(511\) −10.7490 18.6178i −0.475506 0.823601i
\(512\) 0 0
\(513\) −0.694593 + 4.30320i −0.0306670 + 0.189991i
\(514\) 0 0
\(515\) −12.7023 22.0011i −0.559732 0.969484i
\(516\) 0 0
\(517\) −13.1088 + 22.7050i −0.576522 + 0.998566i
\(518\) 0 0
\(519\) −2.69459 + 4.66717i −0.118279 + 0.204866i
\(520\) 0 0
\(521\) 37.2763 1.63310 0.816552 0.577271i \(-0.195882\pi\)
0.816552 + 0.577271i \(0.195882\pi\)
\(522\) 0 0
\(523\) 0.186852 0.323637i 0.00817046 0.0141517i −0.861911 0.507059i \(-0.830733\pi\)
0.870082 + 0.492907i \(0.164066\pi\)
\(524\) 0 0
\(525\) −43.4397 −1.89587
\(526\) 0 0
\(527\) −22.5817 39.1127i −0.983675 1.70378i
\(528\) 0 0
\(529\) 4.43582 + 7.68307i 0.192862 + 0.334046i
\(530\) 0 0
\(531\) 9.27631 0.402558
\(532\) 0 0
\(533\) −3.51754 −0.152362
\(534\) 0 0
\(535\) −7.75877 13.4386i −0.335441 0.581001i
\(536\) 0 0
\(537\) 2.12061 + 3.67301i 0.0915113 + 0.158502i
\(538\) 0 0
\(539\) −68.3661 −2.94474
\(540\) 0 0
\(541\) 9.54458 16.5317i 0.410353 0.710753i −0.584575 0.811340i \(-0.698739\pi\)
0.994928 + 0.100587i \(0.0320720\pi\)
\(542\) 0 0
\(543\) −12.6108 −0.541182
\(544\) 0 0
\(545\) −0.241230 + 0.417822i −0.0103331 + 0.0178975i
\(546\) 0 0
\(547\) −21.8773 + 37.8926i −0.935407 + 1.62017i −0.161500 + 0.986873i \(0.551633\pi\)
−0.773907 + 0.633300i \(0.781700\pi\)
\(548\) 0 0
\(549\) 4.25877 + 7.37641i 0.181760 + 0.314817i
\(550\) 0 0
\(551\) 18.2121 + 14.8372i 0.775863 + 0.632084i
\(552\) 0 0
\(553\) −33.6634 58.3068i −1.43151 2.47946i
\(554\) 0 0
\(555\) 13.3969 23.2042i 0.568668 0.984962i
\(556\) 0 0
\(557\) −15.9513 + 27.6285i −0.675878 + 1.17066i 0.300333 + 0.953834i \(0.402902\pi\)
−0.976211 + 0.216821i \(0.930431\pi\)
\(558\) 0 0
\(559\) −0.758770 −0.0320926
\(560\) 0 0
\(561\) −13.3892 + 23.1907i −0.565292 + 0.979114i
\(562\) 0 0
\(563\) −3.87164 −0.163170 −0.0815852 0.996666i \(-0.525998\pi\)
−0.0815852 + 0.996666i \(0.525998\pi\)
\(564\) 0 0
\(565\) −4.52704 7.84106i −0.190454 0.329876i
\(566\) 0 0
\(567\) −2.37939 4.12122i −0.0999248 0.173075i
\(568\) 0 0
\(569\) −32.9959 −1.38326 −0.691630 0.722252i \(-0.743107\pi\)
−0.691630 + 0.722252i \(0.743107\pi\)
\(570\) 0 0
\(571\) −24.1789 −1.01186 −0.505928 0.862576i \(-0.668850\pi\)
−0.505928 + 0.862576i \(0.668850\pi\)
\(572\) 0 0
\(573\) −11.3131 19.5949i −0.472614 0.818591i
\(574\) 0 0
\(575\) 17.1557 + 29.7145i 0.715442 + 1.23918i
\(576\) 0 0
\(577\) −3.16344 −0.131696 −0.0658478 0.997830i \(-0.520975\pi\)
−0.0658478 + 0.997830i \(0.520975\pi\)
\(578\) 0 0
\(579\) 12.3229 21.3440i 0.512124 0.887025i
\(580\) 0 0
\(581\) 23.3500 0.968721
\(582\) 0 0
\(583\) 11.2472 19.4807i 0.465812 0.806810i
\(584\) 0 0
\(585\) 1.87939 3.25519i 0.0777030 0.134586i
\(586\) 0 0
\(587\) 3.00774 + 5.20956i 0.124143 + 0.215022i 0.921398 0.388621i \(-0.127049\pi\)
−0.797255 + 0.603643i \(0.793715\pi\)
\(588\) 0 0
\(589\) 5.11886 31.7128i 0.210919 1.30670i
\(590\) 0 0
\(591\) 11.3327 + 19.6289i 0.466167 + 0.807425i
\(592\) 0 0
\(593\) −11.4611 + 19.8512i −0.470651 + 0.815192i −0.999437 0.0335639i \(-0.989314\pi\)
0.528785 + 0.848756i \(0.322648\pi\)
\(594\) 0 0
\(595\) 54.8093 94.9326i 2.24696 3.89186i
\(596\) 0 0
\(597\) −13.9222 −0.569798
\(598\) 0 0
\(599\) −17.2199 + 29.8257i −0.703585 + 1.21864i 0.263615 + 0.964628i \(0.415085\pi\)
−0.967200 + 0.254017i \(0.918248\pi\)
\(600\) 0 0
\(601\) 24.1634 0.985647 0.492824 0.870129i \(-0.335965\pi\)
0.492824 + 0.870129i \(0.335965\pi\)
\(602\) 0 0
\(603\) −0.684793 1.18610i −0.0278869 0.0483015i
\(604\) 0 0
\(605\) 15.2104 + 26.3451i 0.618390 + 1.07108i
\(606\) 0 0
\(607\) −6.84793 −0.277949 −0.138974 0.990296i \(-0.544381\pi\)
−0.138974 + 0.990296i \(0.544381\pi\)
\(608\) 0 0
\(609\) −25.6459 −1.03922
\(610\) 0 0
\(611\) 3.00000 + 5.19615i 0.121367 + 0.210214i
\(612\) 0 0
\(613\) −8.47296 14.6756i −0.342220 0.592742i 0.642625 0.766181i \(-0.277845\pi\)
−0.984845 + 0.173439i \(0.944512\pi\)
\(614\) 0 0
\(615\) −13.2216 −0.533148
\(616\) 0 0
\(617\) −21.1361 + 36.6088i −0.850907 + 1.47381i 0.0294834 + 0.999565i \(0.490614\pi\)
−0.880391 + 0.474249i \(0.842720\pi\)
\(618\) 0 0
\(619\) 4.59121 0.184536 0.0922682 0.995734i \(-0.470588\pi\)
0.0922682 + 0.995734i \(0.470588\pi\)
\(620\) 0 0
\(621\) −1.87939 + 3.25519i −0.0754171 + 0.130626i
\(622\) 0 0
\(623\) −19.9145 + 34.4929i −0.797856 + 1.38193i
\(624\) 0 0
\(625\) −6.34255 10.9856i −0.253702 0.439425i
\(626\) 0 0
\(627\) −17.8016 + 6.77314i −0.710927 + 0.270493i
\(628\) 0 0
\(629\) 21.8425 + 37.8324i 0.870919 + 1.50848i
\(630\) 0 0
\(631\) 18.7841 32.5349i 0.747781 1.29520i −0.201103 0.979570i \(-0.564452\pi\)
0.948884 0.315625i \(-0.102214\pi\)
\(632\) 0 0
\(633\) −1.00980 + 1.74903i −0.0401360 + 0.0695175i
\(634\) 0 0
\(635\) −1.81345 −0.0719647
\(636\) 0 0
\(637\) −7.82295 + 13.5497i −0.309956 + 0.536860i
\(638\) 0 0
\(639\) −11.6459 −0.460705
\(640\) 0 0
\(641\) 2.75877 + 4.77833i 0.108965 + 0.188733i 0.915351 0.402656i \(-0.131913\pi\)
−0.806386 + 0.591389i \(0.798580\pi\)
\(642\) 0 0
\(643\) 17.5915 + 30.4694i 0.693742 + 1.20160i 0.970603 + 0.240686i \(0.0773724\pi\)
−0.276861 + 0.960910i \(0.589294\pi\)
\(644\) 0 0
\(645\) −2.85204 −0.112299
\(646\) 0 0
\(647\) −3.57573 −0.140577 −0.0702883 0.997527i \(-0.522392\pi\)
−0.0702883 + 0.997527i \(0.522392\pi\)
\(648\) 0 0
\(649\) 20.2668 + 35.1032i 0.795542 + 1.37792i
\(650\) 0 0
\(651\) 17.5351 + 30.3717i 0.687254 + 1.19036i
\(652\) 0 0
\(653\) 32.6418 1.27737 0.638686 0.769468i \(-0.279478\pi\)
0.638686 + 0.769468i \(0.279478\pi\)
\(654\) 0 0
\(655\) −6.36959 + 11.0324i −0.248880 + 0.431073i
\(656\) 0 0
\(657\) 4.51754 0.176246
\(658\) 0 0
\(659\) −24.4807 + 42.4018i −0.953633 + 1.65174i −0.216167 + 0.976356i \(0.569355\pi\)
−0.737466 + 0.675384i \(0.763978\pi\)
\(660\) 0 0
\(661\) −17.7392 + 30.7251i −0.689974 + 1.19507i 0.281872 + 0.959452i \(0.409045\pi\)
−0.971846 + 0.235618i \(0.924289\pi\)
\(662\) 0 0
\(663\) 3.06418 + 5.30731i 0.119003 + 0.206119i
\(664\) 0 0
\(665\) 72.8718 27.7262i 2.82585 1.07518i
\(666\) 0 0
\(667\) 10.1284 + 17.5428i 0.392171 + 0.679261i
\(668\) 0 0
\(669\) 14.1827 24.5652i 0.548336 0.949746i
\(670\) 0 0
\(671\) −18.6091 + 32.2318i −0.718395 + 1.24430i
\(672\) 0 0
\(673\) 2.68510 0.103503 0.0517514 0.998660i \(-0.483520\pi\)
0.0517514 + 0.998660i \(0.483520\pi\)
\(674\) 0 0
\(675\) 4.56418 7.90539i 0.175675 0.304279i
\(676\) 0 0
\(677\) 18.3351 0.704676 0.352338 0.935873i \(-0.385387\pi\)
0.352338 + 0.935873i \(0.385387\pi\)
\(678\) 0 0
\(679\) 27.7101 + 47.9953i 1.06342 + 1.84189i
\(680\) 0 0
\(681\) 10.0915 + 17.4790i 0.386708 + 0.669798i
\(682\) 0 0
\(683\) 4.07367 0.155875 0.0779374 0.996958i \(-0.475167\pi\)
0.0779374 + 0.996958i \(0.475167\pi\)
\(684\) 0 0
\(685\) 69.2526 2.64601
\(686\) 0 0
\(687\) 5.88919 + 10.2004i 0.224686 + 0.389168i
\(688\) 0 0
\(689\) −2.57398 4.45826i −0.0980608 0.169846i
\(690\) 0 0
\(691\) 43.5485 1.65666 0.828332 0.560238i \(-0.189290\pi\)
0.828332 + 0.560238i \(0.189290\pi\)
\(692\) 0 0
\(693\) 10.3969 18.0080i 0.394947 0.684068i
\(694\) 0 0
\(695\) −0.0736733 −0.00279459
\(696\) 0 0
\(697\) 10.7784 18.6687i 0.408260 0.707127i
\(698\) 0 0
\(699\) −0.0641778 + 0.111159i −0.00242743 + 0.00420443i
\(700\) 0 0
\(701\) −15.6536 27.1129i −0.591230 1.02404i −0.994067 0.108768i \(-0.965309\pi\)
0.402837 0.915272i \(-0.368024\pi\)
\(702\) 0 0
\(703\) −4.95130 + 30.6747i −0.186742 + 1.15692i
\(704\) 0 0
\(705\) 11.2763 + 19.5311i 0.424690 + 0.735585i
\(706\) 0 0
\(707\) 9.82295 17.0138i 0.369430 0.639872i
\(708\) 0 0
\(709\) −11.9047 + 20.6195i −0.447089 + 0.774381i −0.998195 0.0600541i \(-0.980873\pi\)
0.551106 + 0.834435i \(0.314206\pi\)
\(710\) 0 0
\(711\) 14.1480 0.530590
\(712\) 0 0
\(713\) 13.8503 23.9894i 0.518697 0.898410i
\(714\) 0 0
\(715\) 16.4243 0.614233
\(716\) 0 0
\(717\) 1.55438 + 2.69226i 0.0580493 + 0.100544i
\(718\) 0 0
\(719\) −15.8949 27.5307i −0.592779 1.02672i −0.993856 0.110678i \(-0.964698\pi\)
0.401078 0.916044i \(-0.368636\pi\)
\(720\) 0 0
\(721\) −32.1634 −1.19783
\(722\) 0 0
\(723\) −11.2959 −0.420099
\(724\) 0 0
\(725\) −24.5972 42.6036i −0.913517 1.58226i
\(726\) 0 0
\(727\) 3.22193 + 5.58055i 0.119495 + 0.206971i 0.919568 0.392932i \(-0.128539\pi\)
−0.800073 + 0.599903i \(0.795206\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.32501 4.02703i 0.0859935 0.148945i
\(732\) 0 0
\(733\) −33.9728 −1.25481 −0.627406 0.778692i \(-0.715884\pi\)
−0.627406 + 0.778692i \(0.715884\pi\)
\(734\) 0 0
\(735\) −29.4047 + 50.9304i −1.08461 + 1.87860i
\(736\) 0 0
\(737\) 2.99226 5.18274i 0.110221 0.190909i
\(738\) 0 0
\(739\) −11.4632 19.8548i −0.421679 0.730370i 0.574425 0.818557i \(-0.305226\pi\)
−0.996104 + 0.0881876i \(0.971893\pi\)
\(740\) 0 0
\(741\) −0.694593 + 4.30320i −0.0255165 + 0.158082i
\(742\) 0 0
\(743\) 3.98276 + 6.89835i 0.146113 + 0.253076i 0.929788 0.368096i \(-0.119990\pi\)
−0.783674 + 0.621172i \(0.786657\pi\)
\(744\) 0 0
\(745\) 11.7297 20.3164i 0.429742 0.744335i
\(746\) 0 0
\(747\) −2.45336 + 4.24935i −0.0897639 + 0.155476i
\(748\) 0 0
\(749\) −19.6459 −0.717845
\(750\) 0 0
\(751\) 11.3794 19.7097i 0.415240 0.719216i −0.580214 0.814464i \(-0.697031\pi\)
0.995454 + 0.0952479i \(0.0303644\pi\)
\(752\) 0 0
\(753\) 1.51754 0.0553023
\(754\) 0 0
\(755\) 1.38919 + 2.40614i 0.0505576 + 0.0875684i
\(756\) 0 0
\(757\) −2.38713 4.13462i −0.0867616 0.150275i 0.819379 0.573252i \(-0.194318\pi\)
−0.906140 + 0.422977i \(0.860985\pi\)
\(758\) 0 0
\(759\) −16.4243 −0.596163
\(760\) 0 0
\(761\) 28.2222 1.02306 0.511528 0.859267i \(-0.329080\pi\)
0.511528 + 0.859267i \(0.329080\pi\)
\(762\) 0 0
\(763\) 0.305407 + 0.528981i 0.0110565 + 0.0191504i
\(764\) 0 0
\(765\) 11.5175 + 19.9490i 0.416418 + 0.721256i
\(766\) 0 0
\(767\) 9.27631 0.334948
\(768\) 0 0
\(769\) −12.1108 + 20.9765i −0.436727 + 0.756434i −0.997435 0.0715802i \(-0.977196\pi\)
0.560708 + 0.828014i \(0.310529\pi\)
\(770\) 0 0
\(771\) 12.1830 0.438761
\(772\) 0 0
\(773\) −11.9804 + 20.7507i −0.430905 + 0.746349i −0.996951 0.0780239i \(-0.975139\pi\)
0.566046 + 0.824373i \(0.308472\pi\)
\(774\) 0 0
\(775\) −33.6361 + 58.2594i −1.20824 + 2.09274i
\(776\) 0 0
\(777\) −16.9611 29.3775i −0.608476 1.05391i
\(778\) 0 0
\(779\) 14.3304 5.45242i 0.513439 0.195353i
\(780\) 0 0
\(781\) −25.4439 44.0701i −0.910453 1.57695i
\(782\) 0 0
\(783\) 2.69459 4.66717i 0.0962969 0.166791i
\(784\) 0 0
\(785\) −12.4902 + 21.6337i −0.445794 + 0.772138i
\(786\) 0 0
\(787\) −28.8871 −1.02971 −0.514857 0.857276i \(-0.672155\pi\)
−0.514857 + 0.857276i \(0.672155\pi\)
\(788\) 0 0
\(789\) −9.69459 + 16.7915i −0.345137 + 0.597794i
\(790\) 0 0
\(791\) −11.4629 −0.407572
\(792\) 0 0
\(793\) 4.25877 + 7.37641i 0.151233 + 0.261944i
\(794\) 0 0
\(795\) −9.67499 16.7576i −0.343137 0.594330i
\(796\) 0 0
\(797\) −7.61493 −0.269735 −0.134867 0.990864i \(-0.543061\pi\)
−0.134867 + 0.990864i \(0.543061\pi\)
\(798\) 0 0
\(799\) −36.7701 −1.30083
\(800\) 0 0
\(801\) −4.18479 7.24827i −0.147862 0.256105i
\(802\) 0 0
\(803\) 9.86989 + 17.0952i 0.348301 + 0.603275i
\(804\) 0 0
\(805\) 67.2336 2.36967
\(806\) 0 0
\(807\) 11.7023 20.2690i 0.411942 0.713504i
\(808\) 0 0
\(809\) 22.5681 0.793452 0.396726 0.917937i \(-0.370146\pi\)
0.396726 + 0.917937i \(0.370146\pi\)
\(810\) 0 0
\(811\) 12.3696 21.4247i 0.434355 0.752325i −0.562888 0.826533i \(-0.690310\pi\)
0.997243 + 0.0742086i \(0.0236431\pi\)
\(812\) 0 0
\(813\) 0.482459 0.835644i 0.0169206 0.0293073i
\(814\) 0 0
\(815\) −21.3678 37.0102i −0.748482 1.29641i
\(816\) 0 0
\(817\) 3.09121 1.17614i 0.108148 0.0411481i
\(818\) 0 0
\(819\) −2.37939 4.12122i −0.0831424 0.144007i
\(820\) 0 0
\(821\) 9.30541 16.1174i 0.324761 0.562502i −0.656703 0.754149i \(-0.728049\pi\)
0.981464 + 0.191647i \(0.0613828\pi\)
\(822\) 0 0
\(823\) 10.9804 19.0186i 0.382753 0.662947i −0.608702 0.793399i \(-0.708309\pi\)
0.991455 + 0.130452i \(0.0416428\pi\)
\(824\) 0 0
\(825\) 39.8871 1.38869
\(826\) 0 0
\(827\) 12.9513 22.4323i 0.450361 0.780048i −0.548047 0.836447i \(-0.684629\pi\)
0.998408 + 0.0563993i \(0.0179620\pi\)
\(828\) 0 0
\(829\) −43.0310 −1.49453 −0.747264 0.664528i \(-0.768633\pi\)
−0.747264 + 0.664528i \(0.768633\pi\)
\(830\) 0 0
\(831\) 7.69459 + 13.3274i 0.266922 + 0.462323i
\(832\) 0 0
\(833\) −47.9418 83.0376i −1.66109 2.87708i
\(834\) 0 0
\(835\) −11.2026 −0.387683
\(836\) 0 0
\(837\) −7.36959 −0.254730
\(838\) 0 0
\(839\) −4.16519 7.21432i −0.143798 0.249066i 0.785126 0.619337i \(-0.212598\pi\)
−0.928924 + 0.370270i \(0.879265\pi\)
\(840\) 0 0
\(841\) −0.0216598 0.0375158i −0.000746888 0.00129365i
\(842\) 0 0
\(843\) −8.49794 −0.292685
\(844\) 0 0
\(845\) −22.5526 + 39.0623i −0.775834 + 1.34378i
\(846\) 0 0
\(847\) 38.5140 1.32336
\(848\) 0 0
\(849\) −5.51754 + 9.55666i −0.189362 + 0.327984i
\(850\) 0 0
\(851\) −13.3969 + 23.2042i −0.459241 + 0.795428i
\(852\) 0 0
\(853\) 21.6925 + 37.5726i 0.742738 + 1.28646i 0.951244 + 0.308439i \(0.0998066\pi\)
−0.208506 + 0.978021i \(0.566860\pi\)
\(854\) 0 0
\(855\) −2.61081 + 16.1747i −0.0892880 + 0.553165i
\(856\) 0 0
\(857\) −17.2567 29.8895i −0.589478 1.02101i −0.994301 0.106611i \(-0.966000\pi\)
0.404823 0.914395i \(-0.367333\pi\)
\(858\) 0 0
\(859\) 9.08946 15.7434i 0.310128 0.537158i −0.668262 0.743926i \(-0.732961\pi\)
0.978390 + 0.206768i \(0.0662946\pi\)
\(860\) 0 0
\(861\) −8.36959 + 14.4965i −0.285235 + 0.494041i
\(862\) 0 0
\(863\) 44.3269 1.50890 0.754452 0.656355i \(-0.227903\pi\)
0.754452 + 0.656355i \(0.227903\pi\)
\(864\) 0 0
\(865\) −10.1284 + 17.5428i −0.344374 + 0.596474i
\(866\) 0 0
\(867\) −20.5567 −0.698144
\(868\) 0 0
\(869\) 30.9103 + 53.5383i 1.04856 + 1.81616i
\(870\) 0 0
\(871\) −0.684793 1.18610i −0.0232033 0.0401893i
\(872\) 0 0
\(873\) −11.6459 −0.394154
\(874\) 0 0
\(875\) −73.8444 −2.49640
\(876\) 0 0
\(877\) −5.95336 10.3115i −0.201031 0.348196i 0.747830 0.663890i \(-0.231096\pi\)
−0.948861 + 0.315695i \(0.897762\pi\)
\(878\) 0 0
\(879\) −0.517541 0.896407i −0.0174562 0.0302351i
\(880\) 0 0
\(881\) 28.9804 0.976374 0.488187 0.872739i \(-0.337658\pi\)
0.488187 + 0.872739i \(0.337658\pi\)
\(882\) 0 0
\(883\) −2.23143 + 3.86495i −0.0750936 + 0.130066i −0.901127 0.433555i \(-0.857259\pi\)
0.826033 + 0.563621i \(0.190592\pi\)
\(884\) 0 0
\(885\) 34.8675 1.17206
\(886\) 0 0
\(887\) 14.1557 24.5184i 0.475302 0.823247i −0.524298 0.851535i \(-0.675672\pi\)
0.999600 + 0.0282880i \(0.00900555\pi\)
\(888\) 0 0
\(889\) −1.14796 + 1.98832i −0.0385012 + 0.0666860i
\(890\) 0 0
\(891\) 2.18479 + 3.78417i 0.0731933 + 0.126775i
\(892\) 0 0
\(893\) −20.2763 16.5188i −0.678521 0.552781i
\(894\) 0 0
\(895\) 7.97090 + 13.8060i 0.266438 + 0.461484i
\(896\) 0 0
\(897\) −1.87939 + 3.25519i −0.0627508 + 0.108688i
\(898\) 0 0
\(899\) −19.8580 + 34.3951i −0.662302 + 1.14714i
\(900\) 0 0
\(901\) 31.5485 1.05103
\(902\) 0 0
\(903\) −1.80541 + 3.12706i −0.0600802 + 0.104062i
\(904\) 0 0
\(905\) −47.4012 −1.57567
\(906\) 0 0
\(907\) −10.2567 17.7651i −0.340569 0.589882i 0.643970 0.765051i \(-0.277286\pi\)
−0.984538 + 0.175169i \(0.943953\pi\)
\(908\) 0 0
\(909\) 2.06418 + 3.57526i 0.0684645 + 0.118584i
\(910\) 0 0
\(911\) 21.9608 0.727594 0.363797 0.931478i \(-0.381480\pi\)
0.363797 + 0.931478i \(0.381480\pi\)
\(912\) 0 0
\(913\) −21.4404 −0.709572
\(914\) 0 0
\(915\) 16.0077 + 27.7262i 0.529200 + 0.916601i
\(916\) 0 0
\(917\) 8.06418 + 13.9676i 0.266303 + 0.461250i
\(918\) 0 0
\(919\) −22.0898 −0.728674 −0.364337 0.931267i \(-0.618704\pi\)
−0.364337 + 0.931267i \(0.618704\pi\)
\(920\) 0 0
\(921\) 7.43376 12.8757i 0.244951 0.424267i
\(922\) 0 0
\(923\) −11.6459 −0.383329
\(924\) 0 0
\(925\) 32.5351 56.3524i 1.06975 1.85286i
\(926\) 0 0
\(927\) 3.37939 5.85327i 0.110994 0.192247i
\(928\) 0 0
\(929\) −26.2645 45.4914i −0.861709 1.49252i −0.870278 0.492560i \(-0.836061\pi\)
0.00856978 0.999963i \(-0.497272\pi\)
\(930\) 0 0
\(931\) 10.8675 67.3274i 0.356169 2.20657i
\(932\) 0 0
\(933\) −8.02734 13.9038i −0.262803 0.455189i
\(934\) 0 0
\(935\) −50.3269 + 87.1687i −1.64586 + 2.85072i
\(936\) 0 0
\(937\) 12.9688 22.4627i 0.423674 0.733824i −0.572622 0.819820i \(-0.694074\pi\)
0.996296 + 0.0859953i \(0.0274070\pi\)
\(938\) 0 0
\(939\) −35.1634 −1.14752
\(940\) 0 0
\(941\) −8.90673 + 15.4269i −0.290351 + 0.502903i −0.973893 0.227009i \(-0.927105\pi\)
0.683542 + 0.729911i \(0.260439\pi\)
\(942\) 0 0
\(943\) 13.2216 0.430555
\(944\) 0 0
\(945\) −8.94356 15.4907i −0.290934 0.503913i
\(946\) 0 0
\(947\) 17.0446 + 29.5221i 0.553874 + 0.959339i 0.997990 + 0.0633691i \(0.0201845\pi\)
−0.444116 + 0.895969i \(0.646482\pi\)
\(948\) 0 0
\(949\) 4.51754 0.146646
\(950\) 0 0
\(951\) −18.5371 −0.601108
\(952\) 0 0
\(953\) 6.34224 + 10.9851i 0.205445 + 0.355842i 0.950275 0.311413i \(-0.100802\pi\)
−0.744829 + 0.667255i \(0.767469\pi\)
\(954\) 0 0
\(955\) −42.5235 73.6529i −1.37603 2.38335i
\(956\) 0 0
\(957\) 23.5485 0.761215
\(958\) 0 0
\(959\) 43.8384 75.9304i 1.41562 2.45192i
\(960\) 0 0
\(961\) 23.3108 0.751961
\(962\) 0 0
\(963\) 2.06418 3.57526i 0.0665172 0.115211i
\(964\) 0 0
\(965\) 46.3191 80.2271i 1.49107 2.58260i
\(966\) 0 0
\(967\) 23.6266 + 40.9225i 0.759780 + 1.31598i 0.942962 + 0.332899i \(0.108027\pi\)
−0.183182 + 0.983079i \(0.558640\pi\)
\(968\) 0 0
\(969\) −20.7101 16.8722i −0.665303 0.542013i
\(970\) 0 0
\(971\) 17.6810 + 30.6244i 0.567410 + 0.982782i 0.996821 + 0.0796733i \(0.0253877\pi\)
−0.429411 + 0.903109i \(0.641279\pi\)
\(972\) 0 0
\(973\) −0.0466368 + 0.0807773i −0.00149511 + 0.00258960i
\(974\) 0 0
\(975\) 4.56418 7.90539i 0.146171 0.253175i
\(976\) 0 0
\(977\) 45.0660 1.44179 0.720895 0.693044i \(-0.243731\pi\)
0.720895 + 0.693044i \(0.243731\pi\)
\(978\) 0 0
\(979\) 18.2858 31.6719i 0.584417 1.01224i
\(980\) 0 0
\(981\) −0.128356 −0.00409808
\(982\) 0 0
\(983\) 13.0915 + 22.6752i 0.417555 + 0.723226i 0.995693 0.0927130i \(-0.0295539\pi\)
−0.578138 + 0.815939i \(0.696221\pi\)
\(984\) 0 0
\(985\) 42.5972 + 73.7805i 1.35726 + 2.35084i
\(986\) 0 0
\(987\) 28.5526 0.908840
\(988\) 0 0
\(989\) 2.85204 0.0906897
\(990\) 0 0
\(991\) 10.6070 + 18.3719i 0.336942 + 0.583601i 0.983856 0.178962i \(-0.0572739\pi\)
−0.646914 + 0.762563i \(0.723941\pi\)
\(992\) 0 0
\(993\) −4.37939 7.58532i −0.138976 0.240713i
\(994\) 0 0
\(995\) −52.3304 −1.65898
\(996\) 0 0
\(997\) 27.1364 47.0016i 0.859418 1.48856i −0.0130661 0.999915i \(-0.504159\pi\)
0.872485 0.488642i \(-0.162507\pi\)
\(998\) 0 0
\(999\) 7.12836 0.225531
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 456.2.q.f.121.3 yes 6
3.2 odd 2 1368.2.s.j.577.1 6
4.3 odd 2 912.2.q.k.577.3 6
12.11 even 2 2736.2.s.x.577.1 6
19.7 even 3 8664.2.a.x.1.1 3
19.11 even 3 inner 456.2.q.f.49.3 6
19.12 odd 6 8664.2.a.z.1.1 3
57.11 odd 6 1368.2.s.j.505.1 6
76.11 odd 6 912.2.q.k.49.3 6
228.11 even 6 2736.2.s.x.1873.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.2.q.f.49.3 6 19.11 even 3 inner
456.2.q.f.121.3 yes 6 1.1 even 1 trivial
912.2.q.k.49.3 6 76.11 odd 6
912.2.q.k.577.3 6 4.3 odd 2
1368.2.s.j.505.1 6 57.11 odd 6
1368.2.s.j.577.1 6 3.2 odd 2
2736.2.s.x.577.1 6 12.11 even 2
2736.2.s.x.1873.1 6 228.11 even 6
8664.2.a.x.1.1 3 19.7 even 3
8664.2.a.z.1.1 3 19.12 odd 6