Properties

Label 76.5.h.a.69.6
Level $76$
Weight $5$
Character 76.69
Analytic conductor $7.856$
Analytic rank $0$
Dimension $12$
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] = [76,5,Mod(65,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.65");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 76.h (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: \(7.85611719437\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 631 x^{10} - 3100 x^{9} + 142264 x^{8} - 550522 x^{7} + 14083117 x^{6} - 40335478 x^{5} + 638031136 x^{4} - 1209472584 x^{3} + \cdots + 90728724573 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 69.6
Root \(0.500000 + 15.2283i\) of defining polynomial
Character \(\chi\) \(=\) 76.69
Dual form 76.5.h.a.65.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(12.4381 - 7.18116i) q^{3} +(-3.11411 - 5.39380i) q^{5} +49.9415 q^{7} +(62.6382 - 108.493i) q^{9} +O(q^{10})\) \(q+(12.4381 - 7.18116i) q^{3} +(-3.11411 - 5.39380i) q^{5} +49.9415 q^{7} +(62.6382 - 108.493i) q^{9} +1.88575 q^{11} +(-69.0003 - 39.8373i) q^{13} +(-77.4675 - 44.7259i) q^{15} +(-119.788 - 207.478i) q^{17} +(72.4707 + 353.651i) q^{19} +(621.179 - 358.638i) q^{21} +(109.429 - 189.537i) q^{23} +(293.105 - 507.672i) q^{25} -635.911i q^{27} +(340.159 + 196.391i) q^{29} +580.549i q^{31} +(23.4553 - 13.5419i) q^{33} +(-155.523 - 269.374i) q^{35} +2478.77i q^{37} -1144.31 q^{39} +(-2840.91 + 1640.20i) q^{41} +(-162.917 - 282.181i) q^{43} -780.249 q^{45} +(-969.593 + 1679.39i) q^{47} +93.1537 q^{49} +(-2979.87 - 1720.43i) q^{51} +(2087.49 + 1205.21i) q^{53} +(-5.87245 - 10.1714i) q^{55} +(3441.03 + 3878.34i) q^{57} +(3046.44 - 1758.86i) q^{59} +(-1039.02 + 1799.64i) q^{61} +(3128.24 - 5418.28i) q^{63} +496.232i q^{65} +(-4389.01 - 2533.99i) q^{67} -3143.32i q^{69} +(2994.82 - 1729.06i) q^{71} +(4356.65 + 7545.94i) q^{73} -8419.33i q^{75} +94.1773 q^{77} +(3640.40 - 2101.79i) q^{79} +(507.109 + 878.339i) q^{81} -11775.2 q^{83} +(-746.064 + 1292.22i) q^{85} +5641.26 q^{87} +(-8212.16 - 4741.29i) q^{89} +(-3445.98 - 1989.54i) q^{91} +(4169.02 + 7220.95i) q^{93} +(1681.84 - 1492.20i) q^{95} +(5045.74 - 2913.16i) q^{97} +(118.120 - 204.590i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{3} + 9 q^{5} - 52 q^{7} + 136 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{3} + 9 q^{5} - 52 q^{7} + 136 q^{9} + 6 q^{11} - 93 q^{13} - 741 q^{15} - 483 q^{17} - 533 q^{19} + 972 q^{21} + 531 q^{23} - 217 q^{25} + 2025 q^{29} - 75 q^{33} - 1128 q^{35} - 2250 q^{39} - 1692 q^{41} - 63 q^{43} + 7976 q^{45} - 3471 q^{47} + 420 q^{49} + 6741 q^{51} - 3771 q^{53} - 2014 q^{55} + 7617 q^{57} - 9594 q^{59} + 1229 q^{61} + 1514 q^{63} + 7590 q^{67} + 963 q^{71} - 2838 q^{73} - 15408 q^{77} + 11073 q^{79} + 2086 q^{81} - 14202 q^{83} + 9455 q^{85} - 39510 q^{87} + 6525 q^{89} - 7686 q^{91} - 5316 q^{93} + 1521 q^{95} - 34110 q^{97} + 13220 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) 12.4381 7.18116i 1.38202 0.797907i 0.389617 0.920977i \(-0.372607\pi\)
0.992398 + 0.123070i \(0.0392740\pi\)
\(4\) 0 0
\(5\) −3.11411 5.39380i −0.124564 0.215752i 0.796998 0.603982i \(-0.206420\pi\)
−0.921563 + 0.388230i \(0.873087\pi\)
\(6\) 0 0
\(7\) 49.9415 1.01921 0.509607 0.860407i \(-0.329791\pi\)
0.509607 + 0.860407i \(0.329791\pi\)
\(8\) 0 0
\(9\) 62.6382 108.493i 0.773311 1.33941i
\(10\) 0 0
\(11\) 1.88575 0.0155847 0.00779237 0.999970i \(-0.497520\pi\)
0.00779237 + 0.999970i \(0.497520\pi\)
\(12\) 0 0
\(13\) −69.0003 39.8373i −0.408286 0.235724i 0.281767 0.959483i \(-0.409079\pi\)
−0.690053 + 0.723759i \(0.742413\pi\)
\(14\) 0 0
\(15\) −77.4675 44.7259i −0.344300 0.198782i
\(16\) 0 0
\(17\) −119.788 207.478i −0.414490 0.717918i 0.580885 0.813986i \(-0.302707\pi\)
−0.995375 + 0.0960680i \(0.969373\pi\)
\(18\) 0 0
\(19\) 72.4707 + 353.651i 0.200750 + 0.979643i
\(20\) 0 0
\(21\) 621.179 358.638i 1.40857 0.813238i
\(22\) 0 0
\(23\) 109.429 189.537i 0.206861 0.358294i −0.743863 0.668332i \(-0.767009\pi\)
0.950724 + 0.310038i \(0.100342\pi\)
\(24\) 0 0
\(25\) 293.105 507.672i 0.468967 0.812275i
\(26\) 0 0
\(27\) 635.911i 0.872306i
\(28\) 0 0
\(29\) 340.159 + 196.391i 0.404470 + 0.233521i 0.688411 0.725321i \(-0.258309\pi\)
−0.283941 + 0.958842i \(0.591642\pi\)
\(30\) 0 0
\(31\) 580.549i 0.604109i 0.953291 + 0.302055i \(0.0976725\pi\)
−0.953291 + 0.302055i \(0.902327\pi\)
\(32\) 0 0
\(33\) 23.4553 13.5419i 0.0215383 0.0124352i
\(34\) 0 0
\(35\) −155.523 269.374i −0.126958 0.219898i
\(36\) 0 0
\(37\) 2478.77i 1.81064i 0.424727 + 0.905321i \(0.360370\pi\)
−0.424727 + 0.905321i \(0.639630\pi\)
\(38\) 0 0
\(39\) −1144.31 −0.752343
\(40\) 0 0
\(41\) −2840.91 + 1640.20i −1.69001 + 0.975729i −0.735516 + 0.677507i \(0.763060\pi\)
−0.954497 + 0.298222i \(0.903606\pi\)
\(42\) 0 0
\(43\) −162.917 282.181i −0.0881111 0.152613i 0.818602 0.574362i \(-0.194750\pi\)
−0.906713 + 0.421749i \(0.861416\pi\)
\(44\) 0 0
\(45\) −780.249 −0.385308
\(46\) 0 0
\(47\) −969.593 + 1679.39i −0.438929 + 0.760247i −0.997607 0.0691376i \(-0.977975\pi\)
0.558678 + 0.829384i \(0.311309\pi\)
\(48\) 0 0
\(49\) 93.1537 0.0387979
\(50\) 0 0
\(51\) −2979.87 1720.43i −1.14566 0.661449i
\(52\) 0 0
\(53\) 2087.49 + 1205.21i 0.743142 + 0.429053i 0.823211 0.567736i \(-0.192181\pi\)
−0.0800686 + 0.996789i \(0.525514\pi\)
\(54\) 0 0
\(55\) −5.87245 10.1714i −0.00194130 0.00336244i
\(56\) 0 0
\(57\) 3441.03 + 3878.34i 1.05910 + 1.19370i
\(58\) 0 0
\(59\) 3046.44 1758.86i 0.875163 0.505275i 0.00610218 0.999981i \(-0.498058\pi\)
0.869060 + 0.494706i \(0.164724\pi\)
\(60\) 0 0
\(61\) −1039.02 + 1799.64i −0.279232 + 0.483644i −0.971194 0.238290i \(-0.923413\pi\)
0.691962 + 0.721934i \(0.256747\pi\)
\(62\) 0 0
\(63\) 3128.24 5418.28i 0.788170 1.36515i
\(64\) 0 0
\(65\) 496.232i 0.117451i
\(66\) 0 0
\(67\) −4389.01 2533.99i −0.977725 0.564490i −0.0761425 0.997097i \(-0.524260\pi\)
−0.901583 + 0.432607i \(0.857594\pi\)
\(68\) 0 0
\(69\) 3143.32i 0.660223i
\(70\) 0 0
\(71\) 2994.82 1729.06i 0.594092 0.342999i −0.172622 0.984988i \(-0.555224\pi\)
0.766714 + 0.641989i \(0.221891\pi\)
\(72\) 0 0
\(73\) 4356.65 + 7545.94i 0.817536 + 1.41601i 0.907492 + 0.420069i \(0.137994\pi\)
−0.0899558 + 0.995946i \(0.528673\pi\)
\(74\) 0 0
\(75\) 8419.33i 1.49677i
\(76\) 0 0
\(77\) 94.1773 0.0158842
\(78\) 0 0
\(79\) 3640.40 2101.79i 0.583304 0.336771i −0.179141 0.983823i \(-0.557332\pi\)
0.762445 + 0.647053i \(0.223999\pi\)
\(80\) 0 0
\(81\) 507.109 + 878.339i 0.0772915 + 0.133873i
\(82\) 0 0
\(83\) −11775.2 −1.70927 −0.854635 0.519229i \(-0.826219\pi\)
−0.854635 + 0.519229i \(0.826219\pi\)
\(84\) 0 0
\(85\) −746.064 + 1292.22i −0.103261 + 0.178854i
\(86\) 0 0
\(87\) 5641.26 0.745311
\(88\) 0 0
\(89\) −8212.16 4741.29i −1.03676 0.598573i −0.117845 0.993032i \(-0.537599\pi\)
−0.918913 + 0.394460i \(0.870932\pi\)
\(90\) 0 0
\(91\) −3445.98 1989.54i −0.416131 0.240253i
\(92\) 0 0
\(93\) 4169.02 + 7220.95i 0.482023 + 0.834888i
\(94\) 0 0
\(95\) 1681.84 1492.20i 0.186354 0.165341i
\(96\) 0 0
\(97\) 5045.74 2913.16i 0.536267 0.309614i −0.207297 0.978278i \(-0.566467\pi\)
0.743565 + 0.668664i \(0.233133\pi\)
\(98\) 0 0
\(99\) 118.120 204.590i 0.0120518 0.0208744i
\(100\) 0 0
\(101\) 9167.14 15878.0i 0.898651 1.55651i 0.0694313 0.997587i \(-0.477882\pi\)
0.829220 0.558923i \(-0.188785\pi\)
\(102\) 0 0
\(103\) 15685.2i 1.47848i 0.673443 + 0.739239i \(0.264815\pi\)
−0.673443 + 0.739239i \(0.735185\pi\)
\(104\) 0 0
\(105\) −3868.84 2233.68i −0.350916 0.202601i
\(106\) 0 0
\(107\) 15464.3i 1.35071i −0.737494 0.675354i \(-0.763991\pi\)
0.737494 0.675354i \(-0.236009\pi\)
\(108\) 0 0
\(109\) 15286.3 8825.56i 1.28662 0.742830i 0.308569 0.951202i \(-0.400150\pi\)
0.978050 + 0.208372i \(0.0668166\pi\)
\(110\) 0 0
\(111\) 17800.4 + 30831.3i 1.44472 + 2.50234i
\(112\) 0 0
\(113\) 5351.37i 0.419090i −0.977799 0.209545i \(-0.932802\pi\)
0.977799 0.209545i \(-0.0671984\pi\)
\(114\) 0 0
\(115\) −1363.10 −0.103070
\(116\) 0 0
\(117\) −8644.10 + 4990.68i −0.631464 + 0.364576i
\(118\) 0 0
\(119\) −5982.37 10361.8i −0.422454 0.731712i
\(120\) 0 0
\(121\) −14637.4 −0.999757
\(122\) 0 0
\(123\) −23557.1 + 40802.1i −1.55708 + 2.69695i
\(124\) 0 0
\(125\) −7543.68 −0.482796
\(126\) 0 0
\(127\) −5218.76 3013.05i −0.323564 0.186810i 0.329416 0.944185i \(-0.393148\pi\)
−0.652980 + 0.757375i \(0.726481\pi\)
\(128\) 0 0
\(129\) −4052.78 2339.87i −0.243542 0.140609i
\(130\) 0 0
\(131\) −5324.73 9222.71i −0.310281 0.537423i 0.668142 0.744034i \(-0.267090\pi\)
−0.978423 + 0.206611i \(0.933757\pi\)
\(132\) 0 0
\(133\) 3619.30 + 17661.9i 0.204607 + 0.998466i
\(134\) 0 0
\(135\) −3429.98 + 1980.30i −0.188202 + 0.108658i
\(136\) 0 0
\(137\) −4402.76 + 7625.80i −0.234576 + 0.406298i −0.959149 0.282900i \(-0.908704\pi\)
0.724573 + 0.689198i \(0.242037\pi\)
\(138\) 0 0
\(139\) 3584.83 6209.10i 0.185540 0.321366i −0.758218 0.652001i \(-0.773930\pi\)
0.943759 + 0.330636i \(0.107263\pi\)
\(140\) 0 0
\(141\) 27851.2i 1.40090i
\(142\) 0 0
\(143\) −130.117 75.1234i −0.00636302 0.00367369i
\(144\) 0 0
\(145\) 2446.33i 0.116354i
\(146\) 0 0
\(147\) 1158.66 668.952i 0.0536192 0.0309571i
\(148\) 0 0
\(149\) −8324.08 14417.7i −0.374941 0.649418i 0.615377 0.788233i \(-0.289004\pi\)
−0.990318 + 0.138815i \(0.955671\pi\)
\(150\) 0 0
\(151\) 1780.79i 0.0781013i 0.999237 + 0.0390506i \(0.0124334\pi\)
−0.999237 + 0.0390506i \(0.987567\pi\)
\(152\) 0 0
\(153\) −30013.1 −1.28212
\(154\) 0 0
\(155\) 3131.37 1807.89i 0.130338 0.0752506i
\(156\) 0 0
\(157\) 4609.89 + 7984.57i 0.187022 + 0.323931i 0.944256 0.329212i \(-0.106783\pi\)
−0.757234 + 0.653143i \(0.773450\pi\)
\(158\) 0 0
\(159\) 34619.2 1.36938
\(160\) 0 0
\(161\) 5465.07 9465.78i 0.210836 0.365178i
\(162\) 0 0
\(163\) 29276.2 1.10189 0.550947 0.834540i \(-0.314267\pi\)
0.550947 + 0.834540i \(0.314267\pi\)
\(164\) 0 0
\(165\) −146.085 84.3420i −0.00536582 0.00309796i
\(166\) 0 0
\(167\) −37315.4 21544.1i −1.33800 0.772493i −0.351487 0.936193i \(-0.614324\pi\)
−0.986510 + 0.163700i \(0.947657\pi\)
\(168\) 0 0
\(169\) −11106.5 19237.0i −0.388869 0.673540i
\(170\) 0 0
\(171\) 42907.9 + 14289.5i 1.46739 + 0.488681i
\(172\) 0 0
\(173\) 38605.2 22288.7i 1.28989 0.744721i 0.311259 0.950325i \(-0.399249\pi\)
0.978635 + 0.205605i \(0.0659161\pi\)
\(174\) 0 0
\(175\) 14638.1 25353.9i 0.477978 0.827883i
\(176\) 0 0
\(177\) 25261.4 43754.0i 0.806325 1.39660i
\(178\) 0 0
\(179\) 18503.2i 0.577486i −0.957407 0.288743i \(-0.906763\pi\)
0.957407 0.288743i \(-0.0932373\pi\)
\(180\) 0 0
\(181\) 27630.5 + 15952.4i 0.843395 + 0.486934i 0.858417 0.512953i \(-0.171448\pi\)
−0.0150220 + 0.999887i \(0.504782\pi\)
\(182\) 0 0
\(183\) 29845.6i 0.891205i
\(184\) 0 0
\(185\) 13370.0 7719.17i 0.390650 0.225542i
\(186\) 0 0
\(187\) −225.890 391.253i −0.00645972 0.0111886i
\(188\) 0 0
\(189\) 31758.4i 0.889067i
\(190\) 0 0
\(191\) −28246.4 −0.774276 −0.387138 0.922022i \(-0.626536\pi\)
−0.387138 + 0.922022i \(0.626536\pi\)
\(192\) 0 0
\(193\) −35466.8 + 20476.8i −0.952154 + 0.549727i −0.893750 0.448566i \(-0.851935\pi\)
−0.0584049 + 0.998293i \(0.518601\pi\)
\(194\) 0 0
\(195\) 3563.52 + 6172.20i 0.0937152 + 0.162319i
\(196\) 0 0
\(197\) −22359.6 −0.576144 −0.288072 0.957609i \(-0.593014\pi\)
−0.288072 + 0.957609i \(0.593014\pi\)
\(198\) 0 0
\(199\) −2166.66 + 3752.77i −0.0547123 + 0.0947645i −0.892084 0.451869i \(-0.850757\pi\)
0.837372 + 0.546633i \(0.184091\pi\)
\(200\) 0 0
\(201\) −72788.1 −1.80164
\(202\) 0 0
\(203\) 16988.1 + 9808.06i 0.412241 + 0.238008i
\(204\) 0 0
\(205\) 17693.8 + 10215.5i 0.421031 + 0.243082i
\(206\) 0 0
\(207\) −13708.9 23744.5i −0.319936 0.554145i
\(208\) 0 0
\(209\) 136.662 + 666.898i 0.00312863 + 0.0152675i
\(210\) 0 0
\(211\) −48243.9 + 27853.6i −1.08362 + 0.625629i −0.931871 0.362790i \(-0.881824\pi\)
−0.151750 + 0.988419i \(0.548491\pi\)
\(212\) 0 0
\(213\) 24833.3 43012.6i 0.547363 0.948060i
\(214\) 0 0
\(215\) −1014.69 + 1757.49i −0.0219510 + 0.0380203i
\(216\) 0 0
\(217\) 28993.5i 0.615717i
\(218\) 0 0
\(219\) 108377. + 62571.6i 2.25970 + 1.30464i
\(220\) 0 0
\(221\) 19088.1i 0.390821i
\(222\) 0 0
\(223\) −23671.3 + 13666.6i −0.476005 + 0.274822i −0.718750 0.695268i \(-0.755285\pi\)
0.242745 + 0.970090i \(0.421952\pi\)
\(224\) 0 0
\(225\) −36719.1 63599.3i −0.725315 1.25628i
\(226\) 0 0
\(227\) 42099.3i 0.817003i −0.912758 0.408501i \(-0.866051\pi\)
0.912758 0.408501i \(-0.133949\pi\)
\(228\) 0 0
\(229\) 12163.3 0.231942 0.115971 0.993253i \(-0.463002\pi\)
0.115971 + 0.993253i \(0.463002\pi\)
\(230\) 0 0
\(231\) 1171.39 676.303i 0.0219522 0.0126741i
\(232\) 0 0
\(233\) 12709.0 + 22012.6i 0.234098 + 0.405470i 0.959010 0.283372i \(-0.0914529\pi\)
−0.724912 + 0.688841i \(0.758120\pi\)
\(234\) 0 0
\(235\) 12077.7 0.218700
\(236\) 0 0
\(237\) 30186.5 52284.6i 0.537424 0.930845i
\(238\) 0 0
\(239\) −86525.4 −1.51477 −0.757387 0.652967i \(-0.773524\pi\)
−0.757387 + 0.652967i \(0.773524\pi\)
\(240\) 0 0
\(241\) 45400.9 + 26212.2i 0.781682 + 0.451305i 0.837026 0.547163i \(-0.184292\pi\)
−0.0553438 + 0.998467i \(0.517625\pi\)
\(242\) 0 0
\(243\) 57222.9 + 33037.7i 0.969076 + 0.559496i
\(244\) 0 0
\(245\) −290.091 502.452i −0.00483284 0.00837072i
\(246\) 0 0
\(247\) 9088.01 27289.1i 0.148962 0.447296i
\(248\) 0 0
\(249\) −146461. + 84559.4i −2.36224 + 1.36384i
\(250\) 0 0
\(251\) −19639.4 + 34016.4i −0.311732 + 0.539935i −0.978737 0.205118i \(-0.934242\pi\)
0.667006 + 0.745053i \(0.267576\pi\)
\(252\) 0 0
\(253\) 206.357 357.420i 0.00322387 0.00558391i
\(254\) 0 0
\(255\) 21430.4i 0.329572i
\(256\) 0 0
\(257\) 64968.9 + 37509.8i 0.983646 + 0.567908i 0.903369 0.428864i \(-0.141086\pi\)
0.0802773 + 0.996773i \(0.474419\pi\)
\(258\) 0 0
\(259\) 123793.i 1.84543i
\(260\) 0 0
\(261\) 42613.9 24603.1i 0.625562 0.361168i
\(262\) 0 0
\(263\) −14757.8 25561.3i −0.213359 0.369548i 0.739405 0.673261i \(-0.235107\pi\)
−0.952764 + 0.303713i \(0.901774\pi\)
\(264\) 0 0
\(265\) 15012.6i 0.213779i
\(266\) 0 0
\(267\) −136192. −1.91042
\(268\) 0 0
\(269\) 87976.7 50793.4i 1.21580 0.701944i 0.251785 0.967783i \(-0.418982\pi\)
0.964017 + 0.265839i \(0.0856490\pi\)
\(270\) 0 0
\(271\) 11915.4 + 20638.0i 0.162244 + 0.281015i 0.935673 0.352868i \(-0.114793\pi\)
−0.773429 + 0.633883i \(0.781460\pi\)
\(272\) 0 0
\(273\) −57148.7 −0.766799
\(274\) 0 0
\(275\) 552.723 957.344i 0.00730873 0.0126591i
\(276\) 0 0
\(277\) −109790. −1.43088 −0.715438 0.698677i \(-0.753773\pi\)
−0.715438 + 0.698677i \(0.753773\pi\)
\(278\) 0 0
\(279\) 62985.2 + 36364.5i 0.809152 + 0.467164i
\(280\) 0 0
\(281\) −52508.2 30315.6i −0.664989 0.383932i 0.129186 0.991620i \(-0.458764\pi\)
−0.794175 + 0.607689i \(0.792097\pi\)
\(282\) 0 0
\(283\) −8384.27 14522.0i −0.104687 0.181323i 0.808923 0.587914i \(-0.200051\pi\)
−0.913610 + 0.406591i \(0.866717\pi\)
\(284\) 0 0
\(285\) 10203.2 30637.8i 0.125617 0.377196i
\(286\) 0 0
\(287\) −141879. + 81914.1i −1.72249 + 0.994477i
\(288\) 0 0
\(289\) 13062.3 22624.6i 0.156396 0.270886i
\(290\) 0 0
\(291\) 41839.7 72468.6i 0.494087 0.855783i
\(292\) 0 0
\(293\) 127491.i 1.48506i 0.669813 + 0.742530i \(0.266374\pi\)
−0.669813 + 0.742530i \(0.733626\pi\)
\(294\) 0 0
\(295\) −18973.9 10954.6i −0.218028 0.125879i
\(296\) 0 0
\(297\) 1199.17i 0.0135947i
\(298\) 0 0
\(299\) −15101.3 + 8718.75i −0.168917 + 0.0975241i
\(300\) 0 0
\(301\) −8136.34 14092.6i −0.0898041 0.155545i
\(302\) 0 0
\(303\) 263323.i 2.86816i
\(304\) 0 0
\(305\) 12942.5 0.139130
\(306\) 0 0
\(307\) 94867.2 54771.6i 1.00656 0.581137i 0.0963769 0.995345i \(-0.469275\pi\)
0.910182 + 0.414208i \(0.135941\pi\)
\(308\) 0 0
\(309\) 112638. + 195094.i 1.17969 + 2.04328i
\(310\) 0 0
\(311\) 150216. 1.55309 0.776544 0.630063i \(-0.216971\pi\)
0.776544 + 0.630063i \(0.216971\pi\)
\(312\) 0 0
\(313\) 20012.3 34662.3i 0.204272 0.353809i −0.745629 0.666362i \(-0.767851\pi\)
0.949900 + 0.312553i \(0.101184\pi\)
\(314\) 0 0
\(315\) −38966.8 −0.392712
\(316\) 0 0
\(317\) 143247. + 82703.8i 1.42550 + 0.823013i 0.996761 0.0804148i \(-0.0256245\pi\)
0.428739 + 0.903428i \(0.358958\pi\)
\(318\) 0 0
\(319\) 641.456 + 370.345i 0.00630356 + 0.00363936i
\(320\) 0 0
\(321\) −111051. 192347.i −1.07774 1.86670i
\(322\) 0 0
\(323\) 64693.8 57399.1i 0.620094 0.550174i
\(324\) 0 0
\(325\) −40448.6 + 23353.0i −0.382945 + 0.221094i
\(326\) 0 0
\(327\) 126756. 219547.i 1.18542 2.05320i
\(328\) 0 0
\(329\) −48423.0 + 83871.0i −0.447362 + 0.774854i
\(330\) 0 0
\(331\) 188621.i 1.72161i −0.508935 0.860805i \(-0.669961\pi\)
0.508935 0.860805i \(-0.330039\pi\)
\(332\) 0 0
\(333\) 268928. + 155266.i 2.42520 + 1.40019i
\(334\) 0 0
\(335\) 31564.6i 0.281262i
\(336\) 0 0
\(337\) 31324.9 18085.5i 0.275823 0.159246i −0.355708 0.934597i \(-0.615760\pi\)
0.631531 + 0.775351i \(0.282427\pi\)
\(338\) 0 0
\(339\) −38429.0 66561.0i −0.334395 0.579189i
\(340\) 0 0
\(341\) 1094.77i 0.00941488i
\(342\) 0 0
\(343\) −115257. −0.979671
\(344\) 0 0
\(345\) −16954.4 + 9788.65i −0.142444 + 0.0822403i
\(346\) 0 0
\(347\) −90478.1 156713.i −0.751423 1.30150i −0.947133 0.320841i \(-0.896035\pi\)
0.195710 0.980662i \(-0.437299\pi\)
\(348\) 0 0
\(349\) 128784. 1.05733 0.528666 0.848830i \(-0.322692\pi\)
0.528666 + 0.848830i \(0.322692\pi\)
\(350\) 0 0
\(351\) −25333.0 + 43878.1i −0.205623 + 0.356150i
\(352\) 0 0
\(353\) 71514.4 0.573910 0.286955 0.957944i \(-0.407357\pi\)
0.286955 + 0.957944i \(0.407357\pi\)
\(354\) 0 0
\(355\) −18652.4 10769.0i −0.148006 0.0854510i
\(356\) 0 0
\(357\) −148819. 85920.8i −1.16768 0.674158i
\(358\) 0 0
\(359\) 45696.9 + 79149.3i 0.354566 + 0.614127i 0.987044 0.160452i \(-0.0512951\pi\)
−0.632477 + 0.774579i \(0.717962\pi\)
\(360\) 0 0
\(361\) −119817. + 51258.7i −0.919399 + 0.393326i
\(362\) 0 0
\(363\) −182063. + 105114.i −1.38168 + 0.797713i
\(364\) 0 0
\(365\) 27134.2 46997.8i 0.203672 0.352770i
\(366\) 0 0
\(367\) 103142. 178648.i 0.765781 1.32637i −0.174052 0.984736i \(-0.555686\pi\)
0.939833 0.341634i \(-0.110980\pi\)
\(368\) 0 0
\(369\) 410957.i 3.01817i
\(370\) 0 0
\(371\) 104252. + 60190.0i 0.757421 + 0.437297i
\(372\) 0 0
\(373\) 63012.8i 0.452909i −0.974022 0.226454i \(-0.927287\pi\)
0.974022 0.226454i \(-0.0727134\pi\)
\(374\) 0 0
\(375\) −93829.4 + 54172.4i −0.667231 + 0.385226i
\(376\) 0 0
\(377\) −15647.4 27102.1i −0.110093 0.190686i
\(378\) 0 0
\(379\) 74224.9i 0.516739i 0.966046 + 0.258370i \(0.0831852\pi\)
−0.966046 + 0.258370i \(0.916815\pi\)
\(380\) 0 0
\(381\) −86548.9 −0.596227
\(382\) 0 0
\(383\) −191250. + 110418.i −1.30378 + 0.752738i −0.981050 0.193753i \(-0.937934\pi\)
−0.322731 + 0.946491i \(0.604601\pi\)
\(384\) 0 0
\(385\) −293.279 507.974i −0.00197861 0.00342704i
\(386\) 0 0
\(387\) −40819.4 −0.272549
\(388\) 0 0
\(389\) −119430. + 206858.i −0.789248 + 1.36702i 0.137181 + 0.990546i \(0.456196\pi\)
−0.926428 + 0.376471i \(0.877137\pi\)
\(390\) 0 0
\(391\) −52433.2 −0.342967
\(392\) 0 0
\(393\) −132460. 76475.6i −0.857627 0.495151i
\(394\) 0 0
\(395\) −22673.2 13090.4i −0.145318 0.0838994i
\(396\) 0 0
\(397\) −121860. 211069.i −0.773182 1.33919i −0.935810 0.352504i \(-0.885330\pi\)
0.162628 0.986687i \(-0.448003\pi\)
\(398\) 0 0
\(399\) 171850. + 193690.i 1.07945 + 1.21664i
\(400\) 0 0
\(401\) 181404. 104733.i 1.12812 0.651323i 0.184662 0.982802i \(-0.440881\pi\)
0.943463 + 0.331479i \(0.107548\pi\)
\(402\) 0 0
\(403\) 23127.5 40058.1i 0.142403 0.246649i
\(404\) 0 0
\(405\) 3158.39 5470.49i 0.0192555 0.0333516i
\(406\) 0 0
\(407\) 4674.35i 0.0282184i
\(408\) 0 0
\(409\) 102139. + 58969.8i 0.610581 + 0.352519i 0.773193 0.634171i \(-0.218658\pi\)
−0.162612 + 0.986690i \(0.551992\pi\)
\(410\) 0 0
\(411\) 126468.i 0.748679i
\(412\) 0 0
\(413\) 152144. 87840.3i 0.891978 0.514984i
\(414\) 0 0
\(415\) 36669.2 + 63512.9i 0.212914 + 0.368779i
\(416\) 0 0
\(417\) 102973.i 0.592176i
\(418\) 0 0
\(419\) −59069.1 −0.336459 −0.168230 0.985748i \(-0.553805\pi\)
−0.168230 + 0.985748i \(0.553805\pi\)
\(420\) 0 0
\(421\) 45326.1 26169.0i 0.255731 0.147647i −0.366654 0.930357i \(-0.619497\pi\)
0.622386 + 0.782711i \(0.286164\pi\)
\(422\) 0 0
\(423\) 121467. + 210387.i 0.678857 + 1.17581i
\(424\) 0 0
\(425\) −140441. −0.777529
\(426\) 0 0
\(427\) −51890.4 + 89876.8i −0.284597 + 0.492937i
\(428\) 0 0
\(429\) −2157.89 −0.0117251
\(430\) 0 0
\(431\) 203718. + 117616.i 1.09667 + 0.633160i 0.935343 0.353742i \(-0.115091\pi\)
0.161322 + 0.986902i \(0.448424\pi\)
\(432\) 0 0
\(433\) −127748. 73755.1i −0.681360 0.393384i 0.119007 0.992893i \(-0.462029\pi\)
−0.800367 + 0.599510i \(0.795362\pi\)
\(434\) 0 0
\(435\) −17567.5 30427.8i −0.0928393 0.160802i
\(436\) 0 0
\(437\) 74960.5 + 24963.9i 0.392527 + 0.130722i
\(438\) 0 0
\(439\) −7033.58 + 4060.84i −0.0364962 + 0.0210711i −0.518137 0.855298i \(-0.673374\pi\)
0.481641 + 0.876369i \(0.340041\pi\)
\(440\) 0 0
\(441\) 5834.98 10106.5i 0.0300028 0.0519664i
\(442\) 0 0
\(443\) 102446. 177441.i 0.522018 0.904162i −0.477654 0.878548i \(-0.658513\pi\)
0.999672 0.0256140i \(-0.00815408\pi\)
\(444\) 0 0
\(445\) 59059.7i 0.298244i
\(446\) 0 0
\(447\) −207072. 119553.i −1.03635 0.598337i
\(448\) 0 0
\(449\) 114834.i 0.569611i 0.958585 + 0.284806i \(0.0919291\pi\)
−0.958585 + 0.284806i \(0.908071\pi\)
\(450\) 0 0
\(451\) −5357.26 + 3093.01i −0.0263384 + 0.0152065i
\(452\) 0 0
\(453\) 12788.1 + 22149.7i 0.0623175 + 0.107937i
\(454\) 0 0
\(455\) 24782.6i 0.119708i
\(456\) 0 0
\(457\) −230282. −1.10262 −0.551312 0.834299i \(-0.685873\pi\)
−0.551312 + 0.834299i \(0.685873\pi\)
\(458\) 0 0
\(459\) −131938. + 76174.3i −0.626244 + 0.361562i
\(460\) 0 0
\(461\) −52622.7 91145.2i −0.247612 0.428876i 0.715251 0.698868i \(-0.246312\pi\)
−0.962863 + 0.269992i \(0.912979\pi\)
\(462\) 0 0
\(463\) 103267. 0.481724 0.240862 0.970559i \(-0.422570\pi\)
0.240862 + 0.970559i \(0.422570\pi\)
\(464\) 0 0
\(465\) 25965.6 44973.7i 0.120086 0.207995i
\(466\) 0 0
\(467\) −203377. −0.932542 −0.466271 0.884642i \(-0.654403\pi\)
−0.466271 + 0.884642i \(0.654403\pi\)
\(468\) 0 0
\(469\) −219194. 126552.i −0.996511 0.575336i
\(470\) 0 0
\(471\) 114677. + 66208.8i 0.516933 + 0.298452i
\(472\) 0 0
\(473\) −307.222 532.124i −0.00137319 0.00237843i
\(474\) 0 0
\(475\) 200780. + 66865.4i 0.889885 + 0.296356i
\(476\) 0 0
\(477\) 261513. 150984.i 1.14936 0.663583i
\(478\) 0 0
\(479\) −221111. + 382976.i −0.963697 + 1.66917i −0.250619 + 0.968086i \(0.580634\pi\)
−0.713077 + 0.701086i \(0.752699\pi\)
\(480\) 0 0
\(481\) 98747.6 171036.i 0.426812 0.739260i
\(482\) 0 0
\(483\) 156982.i 0.672909i
\(484\) 0 0
\(485\) −31426.0 18143.8i −0.133600 0.0771339i
\(486\) 0 0
\(487\) 212627.i 0.896519i −0.893903 0.448260i \(-0.852044\pi\)
0.893903 0.448260i \(-0.147956\pi\)
\(488\) 0 0
\(489\) 364141. 210237.i 1.52283 0.879208i
\(490\) 0 0
\(491\) 216804. + 375515.i 0.899298 + 1.55763i 0.828393 + 0.560147i \(0.189255\pi\)
0.0709052 + 0.997483i \(0.477411\pi\)
\(492\) 0 0
\(493\) 94100.8i 0.387168i
\(494\) 0 0
\(495\) −1471.36 −0.00600493
\(496\) 0 0
\(497\) 149566. 86351.8i 0.605507 0.349590i
\(498\) 0 0
\(499\) −80809.6 139966.i −0.324535 0.562111i 0.656883 0.753992i \(-0.271875\pi\)
−0.981418 + 0.191881i \(0.938541\pi\)
\(500\) 0 0
\(501\) −618846. −2.46551
\(502\) 0 0
\(503\) 27970.9 48446.9i 0.110553 0.191483i −0.805440 0.592677i \(-0.798071\pi\)
0.915993 + 0.401194i \(0.131405\pi\)
\(504\) 0 0
\(505\) −114190. −0.447760
\(506\) 0 0
\(507\) −276288. 159515.i −1.07484 0.620562i
\(508\) 0 0
\(509\) 180101. + 103981.i 0.695152 + 0.401346i 0.805539 0.592542i \(-0.201876\pi\)
−0.110387 + 0.993889i \(0.535209\pi\)
\(510\) 0 0
\(511\) 217578. + 376856.i 0.833245 + 1.44322i
\(512\) 0 0
\(513\) 224891. 46085.0i 0.854549 0.175115i
\(514\) 0 0
\(515\) 84602.7 48845.4i 0.318985 0.184166i
\(516\) 0 0
\(517\) −1828.41 + 3166.91i −0.00684059 + 0.0118482i
\(518\) 0 0
\(519\) 320118. 554461.i 1.18844 2.05843i
\(520\) 0 0
\(521\) 207601.i 0.764810i 0.923995 + 0.382405i \(0.124904\pi\)
−0.923995 + 0.382405i \(0.875096\pi\)
\(522\) 0 0
\(523\) −35996.4 20782.5i −0.131600 0.0759792i 0.432755 0.901512i \(-0.357542\pi\)
−0.564355 + 0.825532i \(0.690875\pi\)
\(524\) 0 0
\(525\) 420474.i 1.52553i
\(526\) 0 0
\(527\) 120451. 69542.6i 0.433701 0.250397i
\(528\) 0 0
\(529\) 115971. + 200868.i 0.414417 + 0.717792i
\(530\) 0 0
\(531\) 440688.i 1.56294i
\(532\) 0 0
\(533\) 261365. 0.920011
\(534\) 0 0
\(535\) −83411.1 + 48157.4i −0.291418 + 0.168250i
\(536\) 0 0
\(537\) −132875. 230146.i −0.460780 0.798094i
\(538\) 0 0
\(539\) 175.665 0.000604654
\(540\) 0 0
\(541\) −51917.0 + 89922.8i −0.177384 + 0.307238i −0.940984 0.338452i \(-0.890097\pi\)
0.763600 + 0.645690i \(0.223430\pi\)
\(542\) 0 0
\(543\) 458229. 1.55411
\(544\) 0 0
\(545\) −95206.6 54967.6i −0.320534 0.185060i
\(546\) 0 0
\(547\) 142979. + 82548.7i 0.477855 + 0.275890i 0.719522 0.694469i \(-0.244361\pi\)
−0.241667 + 0.970359i \(0.577694\pi\)
\(548\) 0 0
\(549\) 130165. + 225452.i 0.431867 + 0.748015i
\(550\) 0 0
\(551\) −44802.3 + 134530.i −0.147570 + 0.443115i
\(552\) 0 0
\(553\) 181807. 104966.i 0.594512 0.343242i
\(554\) 0 0
\(555\) 110865. 192024.i 0.359923 0.623404i
\(556\) 0 0
\(557\) −30248.0 + 52391.2i −0.0974960 + 0.168868i −0.910648 0.413184i \(-0.864417\pi\)
0.813152 + 0.582052i \(0.197750\pi\)
\(558\) 0 0
\(559\) 25960.8i 0.0830796i
\(560\) 0 0
\(561\) −5619.30 3244.30i −0.0178549 0.0103085i
\(562\) 0 0
\(563\) 248939.i 0.785374i 0.919672 + 0.392687i \(0.128454\pi\)
−0.919672 + 0.392687i \(0.871546\pi\)
\(564\) 0 0
\(565\) −28864.2 + 16664.8i −0.0904196 + 0.0522038i
\(566\) 0 0
\(567\) 25325.8 + 43865.6i 0.0787766 + 0.136445i
\(568\) 0 0
\(569\) 87739.1i 0.270999i 0.990777 + 0.135500i \(0.0432640\pi\)
−0.990777 + 0.135500i \(0.956736\pi\)
\(570\) 0 0
\(571\) 130764. 0.401067 0.200533 0.979687i \(-0.435732\pi\)
0.200533 + 0.979687i \(0.435732\pi\)
\(572\) 0 0
\(573\) −351332. + 202842.i −1.07006 + 0.617800i
\(574\) 0 0
\(575\) −64148.5 111108.i −0.194022 0.336056i
\(576\) 0 0
\(577\) −572106. −1.71840 −0.859201 0.511639i \(-0.829039\pi\)
−0.859201 + 0.511639i \(0.829039\pi\)
\(578\) 0 0
\(579\) −294094. + 509386.i −0.877261 + 1.51946i
\(580\) 0 0
\(581\) −588069. −1.74211
\(582\) 0 0
\(583\) 3936.48 + 2272.73i 0.0115817 + 0.00668668i
\(584\) 0 0
\(585\) 53837.4 + 31083.0i 0.157316 + 0.0908264i
\(586\) 0 0
\(587\) −17337.7 30029.7i −0.0503169 0.0871515i 0.839770 0.542942i \(-0.182690\pi\)
−0.890087 + 0.455791i \(0.849356\pi\)
\(588\) 0 0
\(589\) −205312. + 42072.8i −0.591811 + 0.121275i
\(590\) 0 0
\(591\) −278112. + 160568.i −0.796240 + 0.459710i
\(592\) 0 0
\(593\) −200472. + 347228.i −0.570092 + 0.987429i 0.426463 + 0.904505i \(0.359759\pi\)
−0.996556 + 0.0829242i \(0.973574\pi\)
\(594\) 0 0
\(595\) −37259.6 + 64535.5i −0.105246 + 0.182291i
\(596\) 0 0
\(597\) 62236.6i 0.174621i
\(598\) 0 0
\(599\) −461443. 266414.i −1.28607 0.742512i −0.308118 0.951348i \(-0.599699\pi\)
−0.977951 + 0.208836i \(0.933032\pi\)
\(600\) 0 0
\(601\) 169049.i 0.468019i 0.972234 + 0.234009i \(0.0751846\pi\)
−0.972234 + 0.234009i \(0.924815\pi\)
\(602\) 0 0
\(603\) −549839. + 317450.i −1.51217 + 0.873052i
\(604\) 0 0
\(605\) 45582.6 + 78951.4i 0.124534 + 0.215700i
\(606\) 0 0
\(607\) 527599.i 1.43195i 0.698127 + 0.715974i \(0.254017\pi\)
−0.698127 + 0.715974i \(0.745983\pi\)
\(608\) 0 0
\(609\) 281733. 0.759632
\(610\) 0 0
\(611\) 133804. 77252.0i 0.358417 0.206932i
\(612\) 0 0
\(613\) 144166. + 249704.i 0.383657 + 0.664513i 0.991582 0.129481i \(-0.0413312\pi\)
−0.607925 + 0.793995i \(0.707998\pi\)
\(614\) 0 0
\(615\) 293438. 0.775829
\(616\) 0 0
\(617\) 215208. 372751.i 0.565311 0.979147i −0.431710 0.902013i \(-0.642090\pi\)
0.997021 0.0771347i \(-0.0245772\pi\)
\(618\) 0 0
\(619\) 490418. 1.27993 0.639963 0.768405i \(-0.278950\pi\)
0.639963 + 0.768405i \(0.278950\pi\)
\(620\) 0 0
\(621\) −120529. 69587.4i −0.312542 0.180446i
\(622\) 0 0
\(623\) −410128. 236787.i −1.05668 0.610074i
\(624\) 0 0
\(625\) −159699. 276606.i −0.408828 0.708111i
\(626\) 0 0
\(627\) 6488.92 + 7313.58i 0.0165058 + 0.0186035i
\(628\) 0 0
\(629\) 514291. 296926.i 1.29989 0.750493i
\(630\) 0 0
\(631\) 65005.3 112593.i 0.163264 0.282781i −0.772774 0.634682i \(-0.781131\pi\)
0.936037 + 0.351901i \(0.114464\pi\)
\(632\) 0 0
\(633\) −400043. + 692894.i −0.998387 + 1.72926i
\(634\) 0 0
\(635\) 37531.9i 0.0930794i
\(636\) 0 0
\(637\) −6427.63 3710.99i −0.0158406 0.00914558i
\(638\) 0 0
\(639\) 433220.i 1.06098i
\(640\) 0 0
\(641\) 395283. 228217.i 0.962037 0.555432i 0.0652376 0.997870i \(-0.479219\pi\)
0.896799 + 0.442437i \(0.145886\pi\)
\(642\) 0 0
\(643\) 194677. + 337190.i 0.470860 + 0.815554i 0.999445 0.0333268i \(-0.0106102\pi\)
−0.528584 + 0.848881i \(0.677277\pi\)
\(644\) 0 0
\(645\) 29146.5i 0.0700595i
\(646\) 0 0
\(647\) −656698. −1.56876 −0.784381 0.620279i \(-0.787019\pi\)
−0.784381 + 0.620279i \(0.787019\pi\)
\(648\) 0 0
\(649\) 5744.83 3316.78i 0.0136392 0.00787458i
\(650\) 0 0
\(651\) 208207. + 360625.i 0.491285 + 0.850930i
\(652\) 0 0
\(653\) 280986. 0.658959 0.329480 0.944163i \(-0.393127\pi\)
0.329480 + 0.944163i \(0.393127\pi\)
\(654\) 0 0
\(655\) −33163.6 + 57441.1i −0.0773000 + 0.133888i
\(656\) 0 0
\(657\) 1.09157e6 2.52884
\(658\) 0 0
\(659\) −331713. 191514.i −0.763820 0.440992i 0.0668456 0.997763i \(-0.478707\pi\)
−0.830666 + 0.556772i \(0.812040\pi\)
\(660\) 0 0
\(661\) −314250. 181433.i −0.719239 0.415253i 0.0952338 0.995455i \(-0.469640\pi\)
−0.814472 + 0.580202i \(0.802973\pi\)
\(662\) 0 0
\(663\) 137075. + 237420.i 0.311839 + 0.540120i
\(664\) 0 0
\(665\) 83993.6 74522.8i 0.189934 0.168518i
\(666\) 0 0
\(667\) 74446.8 42981.9i 0.167338 0.0966126i
\(668\) 0 0
\(669\) −196284. + 339974.i −0.438564 + 0.759616i
\(670\) 0 0
\(671\) −1959.34 + 3393.68i −0.00435176 + 0.00753747i
\(672\) 0 0
\(673\) 191020.i 0.421744i −0.977514 0.210872i \(-0.932370\pi\)
0.977514 0.210872i \(-0.0676304\pi\)
\(674\) 0 0
\(675\) −322834. 186389.i −0.708553 0.409083i
\(676\) 0 0
\(677\) 336677.i 0.734575i −0.930107 0.367287i \(-0.880287\pi\)
0.930107 0.367287i \(-0.119713\pi\)
\(678\) 0 0
\(679\) 251992. 145488.i 0.546571 0.315563i
\(680\) 0 0
\(681\) −302322. 523637.i −0.651892 1.12911i
\(682\) 0 0
\(683\) 313252.i 0.671509i 0.941950 + 0.335755i \(0.108991\pi\)
−0.941950 + 0.335755i \(0.891009\pi\)
\(684\) 0 0
\(685\) 54842.7 0.116879
\(686\) 0 0
\(687\) 151288. 87346.4i 0.320547 0.185068i
\(688\) 0 0
\(689\) −96024.8 166320.i −0.202276 0.350353i
\(690\) 0 0
\(691\) 803808. 1.68344 0.841718 0.539918i \(-0.181545\pi\)
0.841718 + 0.539918i \(0.181545\pi\)
\(692\) 0 0
\(693\) 5899.10 10217.5i 0.0122834 0.0212755i
\(694\) 0 0
\(695\) −44654.2 −0.0924470
\(696\) 0 0
\(697\) 680612. + 392952.i 1.40099 + 0.808860i
\(698\) 0 0
\(699\) 316152. + 182530.i 0.647055 + 0.373577i
\(700\) 0 0
\(701\) −247655. 428950.i −0.503977 0.872913i −0.999989 0.00459793i \(-0.998536\pi\)
0.496013 0.868315i \(-0.334797\pi\)
\(702\) 0 0
\(703\) −876619. + 179638.i −1.77378 + 0.363486i
\(704\) 0 0
\(705\) 150224. 86731.9i 0.302246 0.174502i
\(706\) 0 0
\(707\) 457821. 792969.i 0.915918 1.58642i
\(708\) 0 0
\(709\) −7754.66 + 13431.5i −0.0154266 + 0.0267197i −0.873636 0.486581i \(-0.838244\pi\)
0.858209 + 0.513300i \(0.171577\pi\)
\(710\) 0 0
\(711\) 526608.i 1.04171i
\(712\) 0 0
\(713\) 110036. + 63529.1i 0.216448 + 0.124967i
\(714\) 0 0
\(715\) 935.770i 0.00183045i
\(716\) 0 0
\(717\) −1.07621e6 + 621353.i −2.09344 + 1.20865i
\(718\) 0 0
\(719\) −377951. 654630.i −0.731102 1.26631i −0.956413 0.292018i \(-0.905673\pi\)
0.225311 0.974287i \(-0.427660\pi\)
\(720\) 0 0
\(721\) 783342.i 1.50689i
\(722\) 0 0
\(723\) 752937. 1.44040
\(724\) 0 0
\(725\) 199404. 115126.i 0.379366 0.219027i
\(726\) 0 0
\(727\) −326273. 565122.i −0.617323 1.06923i −0.989972 0.141262i \(-0.954884\pi\)
0.372649 0.927972i \(-0.378449\pi\)
\(728\) 0 0
\(729\) 866844. 1.63112
\(730\) 0 0
\(731\) −39031.0 + 67603.7i −0.0730424 + 0.126513i
\(732\) 0 0
\(733\) −811947. −1.51119 −0.755597 0.655037i \(-0.772653\pi\)
−0.755597 + 0.655037i \(0.772653\pi\)
\(734\) 0 0
\(735\) −7216.38 4166.38i −0.0133581 0.00771230i
\(736\) 0 0
\(737\) −8276.58 4778.49i −0.0152376 0.00879742i
\(738\) 0 0
\(739\) −389853. 675246.i −0.713859 1.23644i −0.963398 0.268075i \(-0.913612\pi\)
0.249539 0.968365i \(-0.419721\pi\)
\(740\) 0 0
\(741\) −82929.2 404688.i −0.151033 0.737027i
\(742\) 0 0
\(743\) −497760. + 287382.i −0.901660 + 0.520574i −0.877738 0.479140i \(-0.840948\pi\)
−0.0239215 + 0.999714i \(0.507615\pi\)
\(744\) 0 0
\(745\) −51844.2 + 89796.8i −0.0934088 + 0.161789i
\(746\) 0 0
\(747\) −737575. + 1.27752e6i −1.32180 + 2.28942i
\(748\) 0 0
\(749\) 772308.i 1.37666i
\(750\) 0 0
\(751\) 168301. + 97168.5i 0.298405 + 0.172284i 0.641726 0.766934i \(-0.278219\pi\)
−0.343321 + 0.939218i \(0.611552\pi\)
\(752\) 0 0
\(753\) 564135.i 0.994931i
\(754\) 0 0
\(755\) 9605.21 5545.57i 0.0168505 0.00972864i
\(756\) 0 0
\(757\) −144106. 249600.i −0.251473 0.435564i 0.712458 0.701714i \(-0.247582\pi\)
−0.963932 + 0.266150i \(0.914248\pi\)
\(758\) 0 0
\(759\) 5927.53i 0.0102894i
\(760\) 0 0
\(761\) 660814. 1.14106 0.570532 0.821275i \(-0.306737\pi\)
0.570532 + 0.821275i \(0.306737\pi\)
\(762\) 0 0
\(763\) 763422. 440762.i 1.31134 0.757103i
\(764\) 0 0
\(765\) 93464.2 + 161885.i 0.159706 + 0.276620i
\(766\) 0 0
\(767\) −280274. −0.476422
\(768\) 0 0
\(769\) 367592. 636688.i 0.621603 1.07665i −0.367584 0.929990i \(-0.619815\pi\)
0.989187 0.146658i \(-0.0468516\pi\)
\(770\) 0 0
\(771\) 1.07746e6 1.81255
\(772\) 0 0
\(773\) 503015. + 290416.i 0.841824 + 0.486028i 0.857884 0.513843i \(-0.171779\pi\)
−0.0160595 + 0.999871i \(0.505112\pi\)
\(774\) 0 0
\(775\) 294729. + 170162.i 0.490703 + 0.283308i
\(776\) 0 0
\(777\) 888981. + 1.53976e6i 1.47248 + 2.55042i
\(778\) 0 0
\(779\) −785942. 885824.i −1.29514 1.45973i
\(780\) 0 0
\(781\) 5647.49 3260.58i 0.00925877 0.00534555i
\(782\) 0 0
\(783\) 124887. 216311.i 0.203702 0.352822i
\(784\) 0 0
\(785\) 28711.5 49729.7i 0.0465925 0.0807006i
\(786\) 0 0
\(787\) 209530.i 0.338296i −0.985591 0.169148i \(-0.945898\pi\)
0.985591 0.169148i \(-0.0541015\pi\)
\(788\) 0 0
\(789\) −367119. 211956.i −0.589730 0.340481i
\(790\) 0 0
\(791\) 267255.i 0.427143i
\(792\) 0 0
\(793\) 143386. 82783.8i 0.228013 0.131643i
\(794\) 0 0
\(795\) −107808. 186729.i −0.170576 0.295446i
\(796\) 0 0
\(797\) 384377.i 0.605118i 0.953131 + 0.302559i \(0.0978410\pi\)
−0.953131 + 0.302559i \(0.902159\pi\)
\(798\) 0 0
\(799\) 464581. 0.727726
\(800\) 0 0
\(801\) −1.02879e6 + 593972.i −1.60347 + 0.925765i
\(802\) 0 0
\(803\) 8215.57 + 14229.8i 0.0127411 + 0.0220682i
\(804\) 0 0
\(805\) −68075.3 −0.105050
\(806\) 0 0
\(807\) 729511. 1.26355e6i 1.12017 1.94019i
\(808\) 0 0
\(809\) −133009. −0.203228 −0.101614 0.994824i \(-0.532401\pi\)
−0.101614 + 0.994824i \(0.532401\pi\)
\(810\) 0 0
\(811\) −71351.1 41194.6i −0.108482 0.0626323i 0.444777 0.895641i \(-0.353283\pi\)
−0.553259 + 0.833009i \(0.686616\pi\)
\(812\) 0 0
\(813\) 296410. + 171132.i 0.448448 + 0.258911i
\(814\) 0 0
\(815\) −91169.4 157910.i −0.137257 0.237736i
\(816\) 0 0
\(817\) 87986.9 78065.8i 0.131818 0.116954i
\(818\) 0 0
\(819\) −431700. + 249242.i −0.643597 + 0.371581i
\(820\) 0 0
\(821\) −268164. + 464473.i −0.397845 + 0.689087i −0.993460 0.114183i \(-0.963575\pi\)
0.595615 + 0.803270i \(0.296908\pi\)
\(822\) 0 0
\(823\) −236126. + 408982.i −0.348613 + 0.603816i −0.986003 0.166725i \(-0.946681\pi\)
0.637390 + 0.770541i \(0.280014\pi\)
\(824\) 0 0
\(825\) 15876.8i 0.0233268i
\(826\) 0 0
\(827\) −189650. 109494.i −0.277294 0.160096i 0.354903 0.934903i \(-0.384514\pi\)
−0.632198 + 0.774807i \(0.717847\pi\)
\(828\) 0 0
\(829\) 1.15812e6i 1.68517i 0.538561 + 0.842586i \(0.318968\pi\)
−0.538561 + 0.842586i \(0.681032\pi\)
\(830\) 0 0
\(831\) −1.36558e6 + 788417.i −1.97749 + 1.14171i
\(832\) 0 0
\(833\) −11158.7 19327.4i −0.0160813 0.0278537i
\(834\) 0 0
\(835\) 268362.i 0.384901i
\(836\) 0 0
\(837\) 369178. 0.526969
\(838\) 0 0
\(839\) 576430. 332802.i 0.818885 0.472783i −0.0311469 0.999515i \(-0.509916\pi\)
0.850032 + 0.526731i \(0.176583\pi\)
\(840\) 0 0
\(841\) −276502. 478915.i −0.390936 0.677121i
\(842\) 0 0
\(843\) −870806. −1.22537
\(844\) 0 0
\(845\) −69173.6 + 119812.i −0.0968784 + 0.167798i
\(846\) 0 0
\(847\) −731016. −1.01897
\(848\) 0 0
\(849\) −208569. 120418.i −0.289358 0.167061i
\(850\) 0 0
\(851\) 469819. + 271250.i 0.648742 + 0.374551i
\(852\) 0 0
\(853\) 441621. + 764910.i 0.606949 + 1.05127i 0.991740 + 0.128262i \(0.0409399\pi\)
−0.384792 + 0.923003i \(0.625727\pi\)
\(854\) 0 0
\(855\) −56545.2 275936.i −0.0773506 0.377464i
\(856\) 0 0
\(857\) −680285. + 392763.i −0.926252 + 0.534772i −0.885624 0.464403i \(-0.846269\pi\)
−0.0406278 + 0.999174i \(0.512936\pi\)
\(858\) 0 0
\(859\) −727091. + 1.25936e6i −0.985377 + 1.70672i −0.345128 + 0.938556i \(0.612164\pi\)
−0.640249 + 0.768167i \(0.721169\pi\)
\(860\) 0 0
\(861\) −1.17648e6 + 2.03772e6i −1.58700 + 2.74877i
\(862\) 0 0
\(863\) 899567.i 1.20785i −0.797042 0.603924i \(-0.793603\pi\)
0.797042 0.603924i \(-0.206397\pi\)
\(864\) 0 0
\(865\) −240442. 138819.i −0.321350 0.185531i
\(866\) 0 0
\(867\) 375211.i 0.499158i
\(868\) 0 0
\(869\) 6864.90 3963.45i 0.00909064 0.00524848i
\(870\) 0 0
\(871\) 201895. + 349693.i 0.266127 + 0.460946i
\(872\) 0 0
\(873\) 729900.i 0.957712i
\(874\) 0 0
\(875\) −376743. −0.492072
\(876\) 0 0
\(877\) 279525. 161384.i 0.363431 0.209827i −0.307154 0.951660i \(-0.599377\pi\)
0.670585 + 0.741833i \(0.266043\pi\)
\(878\) 0 0
\(879\) 915533. + 1.58575e6i 1.18494 + 2.05238i
\(880\) 0 0
\(881\) 547085. 0.704860 0.352430 0.935838i \(-0.385355\pi\)
0.352430 + 0.935838i \(0.385355\pi\)
\(882\) 0 0
\(883\) 524253. 908034.i 0.672388 1.16461i −0.304837 0.952404i \(-0.598602\pi\)
0.977225 0.212205i \(-0.0680646\pi\)
\(884\) 0 0
\(885\) −314667. −0.401758
\(886\) 0 0
\(887\) 318039. + 183620.i 0.404234 + 0.233385i 0.688309 0.725417i \(-0.258353\pi\)
−0.284075 + 0.958802i \(0.591687\pi\)
\(888\) 0 0
\(889\) −260633. 150476.i −0.329781 0.190399i
\(890\) 0 0
\(891\) 956.283 + 1656.33i 0.00120457 + 0.00208637i
\(892\) 0 0
\(893\) −664183. 221191.i −0.832885 0.277374i
\(894\) 0 0
\(895\) −99802.7 + 57621.1i −0.124594 + 0.0719342i
\(896\) 0 0
\(897\) −125222. + 216890.i −0.155630 + 0.269560i
\(898\) 0 0
\(899\) −114015. + 197479.i −0.141072 + 0.244344i
\(900\) 0 0
\(901\) 577477.i 0.711353i
\(902\) 0 0
\(903\) −202402. 116857.i −0.248221 0.143311i
\(904\) 0 0
\(905\) 198711.i 0.242619i
\(906\) 0 0
\(907\) 132002. 76211.4i 0.160460 0.0926415i −0.417620 0.908622i \(-0.637136\pi\)
0.578080 + 0.815980i \(0.303802\pi\)
\(908\) 0 0
\(909\) −1.14843e6 1.98913e6i −1.38987 2.40733i
\(910\) 0 0
\(911\) 826053.i 0.995339i 0.867367 + 0.497669i \(0.165811\pi\)
−0.867367 + 0.497669i \(0.834189\pi\)
\(912\) 0 0
\(913\) −22205.0 −0.0266385
\(914\) 0 0
\(915\) 160981. 92942.5i 0.192279 0.111013i
\(916\) 0 0
\(917\) −265925. 460596.i −0.316243 0.547749i
\(918\) 0 0
\(919\) −139504. −0.165180 −0.0825898 0.996584i \(-0.526319\pi\)
−0.0825898 + 0.996584i \(0.526319\pi\)
\(920\) 0 0
\(921\) 786648. 1.36251e6i 0.927387 1.60628i
\(922\) 0 0
\(923\) −275524. −0.323412
\(924\) 0 0
\(925\) 1.25840e6 + 726539.i 1.47074 + 0.849132i
\(926\) 0 0
\(927\) 1.70172e6 + 982491.i 1.98029 + 1.14332i
\(928\) 0 0
\(929\) 193588. + 335304.i 0.224309 + 0.388515i 0.956112 0.293002i \(-0.0946541\pi\)
−0.731803 + 0.681516i \(0.761321\pi\)
\(930\) 0 0
\(931\) 6750.91 + 32943.9i 0.00778867 + 0.0380080i
\(932\) 0 0
\(933\) 1.86841e6 1.07873e6i 2.14639 1.23922i
\(934\) 0 0
\(935\) −1406.89 + 2436.81i −0.00160930 + 0.00278739i
\(936\) 0 0
\(937\) 31843.4 55154.4i 0.0362694 0.0628205i −0.847321 0.531081i \(-0.821786\pi\)
0.883590 + 0.468261i \(0.155119\pi\)
\(938\) 0 0
\(939\) 574846.i 0.651959i
\(940\) 0 0
\(941\) −783860. 452562.i −0.885237 0.511092i −0.0128552 0.999917i \(-0.504092\pi\)
−0.872381 + 0.488826i \(0.837425\pi\)
\(942\) 0 0
\(943\) 717945.i 0.807361i
\(944\) 0 0
\(945\) −171298. + 98899.1i −0.191818 + 0.110746i
\(946\) 0 0
\(947\) 86501.2 + 149824.i 0.0964544 + 0.167064i 0.910215 0.414137i \(-0.135916\pi\)
−0.813760 + 0.581201i \(0.802583\pi\)
\(948\) 0 0
\(949\) 694230.i 0.770851i
\(950\) 0 0
\(951\) 2.37564e6 2.62675
\(952\) 0 0
\(953\) 796366. 459782.i 0.876853 0.506251i 0.00723358 0.999974i \(-0.497697\pi\)
0.869620 + 0.493722i \(0.164364\pi\)
\(954\) 0 0
\(955\) 87962.4 + 152355.i 0.0964473 + 0.167052i
\(956\) 0 0
\(957\) 10638.0 0.0116155
\(958\) 0 0
\(959\) −219880. + 380844.i −0.239083 + 0.414104i
\(960\) 0 0
\(961\) 586484. 0.635052
\(962\) 0 0
\(963\) −1.67776e6 968653.i −1.80916 1.04452i
\(964\) 0 0
\(965\) 220895. + 127534.i 0.237209 + 0.136953i
\(966\) 0 0
\(967\) −51509.8 89217.7i −0.0550855 0.0954109i 0.837168 0.546946i \(-0.184210\pi\)
−0.892253 + 0.451535i \(0.850876\pi\)
\(968\) 0 0
\(969\) 392478. 1.17851e6i 0.417992 1.25513i
\(970\) 0 0
\(971\) −157988. + 91214.6i −0.167566 + 0.0967445i −0.581438 0.813591i \(-0.697510\pi\)
0.413871 + 0.910335i \(0.364176\pi\)
\(972\) 0 0
\(973\) 179032. 310092.i 0.189105 0.327540i
\(974\) 0 0
\(975\) −335404. + 580936.i −0.352824 + 0.611110i
\(976\) 0 0
\(977\) 436847.i 0.457657i 0.973467 + 0.228828i \(0.0734895\pi\)
−0.973467 + 0.228828i \(0.926511\pi\)
\(978\) 0 0
\(979\) −15486.1 8940.91i −0.0161576 0.00932859i
\(980\) 0 0
\(981\) 2.21127e6i 2.29775i
\(982\) 0 0
\(983\) 577269. 333286.i 0.597408 0.344914i −0.170613 0.985338i \(-0.554575\pi\)
0.768021 + 0.640424i \(0.221241\pi\)
\(984\) 0 0
\(985\) 69630.3 + 120603.i 0.0717671 + 0.124304i
\(986\) 0 0
\(987\) 1.39093e6i 1.42781i
\(988\) 0 0
\(989\) −71311.8 −0.0729070
\(990\) 0 0
\(991\) −706100. + 407667.i −0.718984 + 0.415105i −0.814378 0.580334i \(-0.802922\pi\)
0.0953949 + 0.995440i \(0.469589\pi\)
\(992\) 0 0
\(993\) −1.35452e6 2.34610e6i −1.37368 2.37929i
\(994\) 0 0
\(995\) 26988.9 0.0272608
\(996\) 0 0
\(997\) −720639. + 1.24818e6i −0.724983 + 1.25571i 0.233999 + 0.972237i \(0.424819\pi\)
−0.958981 + 0.283470i \(0.908514\pi\)
\(998\) 0 0
\(999\) 1.57628e6 1.57944
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 76.5.h.a.69.6 yes 12
3.2 odd 2 684.5.y.c.145.4 12
4.3 odd 2 304.5.r.b.145.1 12
19.8 odd 6 inner 76.5.h.a.65.6 12
57.8 even 6 684.5.y.c.217.4 12
76.27 even 6 304.5.r.b.65.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.5.h.a.65.6 12 19.8 odd 6 inner
76.5.h.a.69.6 yes 12 1.1 even 1 trivial
304.5.r.b.65.1 12 76.27 even 6
304.5.r.b.145.1 12 4.3 odd 2
684.5.y.c.145.4 12 3.2 odd 2
684.5.y.c.217.4 12 57.8 even 6