Properties

Label 60.7.g.a.41.2
Level $60$
Weight $7$
Character 60.41
Analytic conductor $13.803$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [60,7,Mod(41,60)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(60, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("60.41");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 60.g (of order \(2\), degree \(1\), 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: \(13.8032450172\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 202x^{6} + 620x^{5} + 12167x^{4} - 25372x^{3} - 177926x^{2} + 190716x + 977814 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{7}\cdot 5^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 41.2
Root \(-2.67163 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 60.41
Dual form 60.7.g.a.41.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-24.1542 + 12.0655i) q^{3} +55.9017i q^{5} -436.815 q^{7} +(437.848 - 582.864i) q^{9} +O(q^{10})\) \(q+(-24.1542 + 12.0655i) q^{3} +55.9017i q^{5} -436.815 q^{7} +(437.848 - 582.864i) q^{9} +1100.06i q^{11} +908.083 q^{13} +(-674.482 - 1350.26i) q^{15} -9747.84i q^{17} +7704.75 q^{19} +(10550.9 - 5270.39i) q^{21} -19320.0i q^{23} -3125.00 q^{25} +(-3543.30 + 19361.4i) q^{27} -8106.42i q^{29} -768.733 q^{31} +(-13272.8 - 26571.1i) q^{33} -24418.7i q^{35} +72219.1 q^{37} +(-21934.0 + 10956.5i) q^{39} +14041.3i q^{41} -121342. q^{43} +(32583.1 + 24476.4i) q^{45} +37066.0i q^{47} +73158.5 q^{49} +(117612. + 235451. i) q^{51} -173405. i q^{53} -61495.4 q^{55} +(-186102. + 92961.6i) q^{57} +138336. i q^{59} -278669. q^{61} +(-191258. + 254604. i) q^{63} +50763.4i q^{65} -96214.3 q^{67} +(233105. + 466658. i) q^{69} -338334. i q^{71} +37504.8 q^{73} +(75481.8 - 37704.7i) q^{75} -480524. i q^{77} +721768. q^{79} +(-148020. - 510411. i) q^{81} -310971. i q^{83} +544921. q^{85} +(97808.0 + 195804. i) q^{87} -1.11509e6i q^{89} -396665. q^{91} +(18568.1 - 9275.14i) q^{93} +430709. i q^{95} +306615. q^{97} +(641187. + 481660. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 20 q^{3} - 560 q^{7} + 1492 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 20 q^{3} - 560 q^{7} + 1492 q^{9} - 6440 q^{13} + 2000 q^{15} - 15272 q^{19} - 868 q^{21} - 25000 q^{25} - 18620 q^{27} + 35032 q^{31} - 111120 q^{33} + 99880 q^{37} + 39608 q^{39} - 161000 q^{43} - 5500 q^{45} + 202560 q^{49} + 429120 q^{51} - 33000 q^{55} - 27160 q^{57} - 135608 q^{61} + 377240 q^{63} + 404920 q^{67} - 254940 q^{69} - 356960 q^{73} + 62500 q^{75} + 707704 q^{79} - 1198112 q^{81} + 828000 q^{85} - 1528440 q^{87} - 2004112 q^{91} - 467920 q^{93} - 1326320 q^{97} + 2650080 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/60\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(41\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) −24.1542 + 12.0655i −0.894599 + 0.446870i
\(4\) 0 0
\(5\) 55.9017i 0.447214i
\(6\) 0 0
\(7\) −436.815 −1.27351 −0.636757 0.771065i \(-0.719724\pi\)
−0.636757 + 0.771065i \(0.719724\pi\)
\(8\) 0 0
\(9\) 437.848 582.864i 0.600614 0.799539i
\(10\) 0 0
\(11\) 1100.06i 0.826494i 0.910619 + 0.413247i \(0.135605\pi\)
−0.910619 + 0.413247i \(0.864395\pi\)
\(12\) 0 0
\(13\) 908.083 0.413329 0.206664 0.978412i \(-0.433739\pi\)
0.206664 + 0.978412i \(0.433739\pi\)
\(14\) 0 0
\(15\) −674.482 1350.26i −0.199846 0.400077i
\(16\) 0 0
\(17\) 9747.84i 1.98409i −0.125882 0.992045i \(-0.540176\pi\)
0.125882 0.992045i \(-0.459824\pi\)
\(18\) 0 0
\(19\) 7704.75 1.12331 0.561653 0.827373i \(-0.310166\pi\)
0.561653 + 0.827373i \(0.310166\pi\)
\(20\) 0 0
\(21\) 10550.9 5270.39i 1.13928 0.569095i
\(22\) 0 0
\(23\) 19320.0i 1.58790i −0.607982 0.793951i \(-0.708021\pi\)
0.607982 0.793951i \(-0.291979\pi\)
\(24\) 0 0
\(25\) −3125.00 −0.200000
\(26\) 0 0
\(27\) −3543.30 + 19361.4i −0.180018 + 0.983663i
\(28\) 0 0
\(29\) 8106.42i 0.332380i −0.986094 0.166190i \(-0.946853\pi\)
0.986094 0.166190i \(-0.0531465\pi\)
\(30\) 0 0
\(31\) −768.733 −0.0258042 −0.0129021 0.999917i \(-0.504107\pi\)
−0.0129021 + 0.999917i \(0.504107\pi\)
\(32\) 0 0
\(33\) −13272.8 26571.1i −0.369335 0.739380i
\(34\) 0 0
\(35\) 24418.7i 0.569533i
\(36\) 0 0
\(37\) 72219.1 1.42576 0.712881 0.701285i \(-0.247390\pi\)
0.712881 + 0.701285i \(0.247390\pi\)
\(38\) 0 0
\(39\) −21934.0 + 10956.5i −0.369763 + 0.184704i
\(40\) 0 0
\(41\) 14041.3i 0.203731i 0.994798 + 0.101865i \(0.0324811\pi\)
−0.994798 + 0.101865i \(0.967519\pi\)
\(42\) 0 0
\(43\) −121342. −1.52618 −0.763092 0.646289i \(-0.776320\pi\)
−0.763092 + 0.646289i \(0.776320\pi\)
\(44\) 0 0
\(45\) 32583.1 + 24476.4i 0.357565 + 0.268603i
\(46\) 0 0
\(47\) 37066.0i 0.357011i 0.983939 + 0.178506i \(0.0571263\pi\)
−0.983939 + 0.178506i \(0.942874\pi\)
\(48\) 0 0
\(49\) 73158.5 0.621837
\(50\) 0 0
\(51\) 117612. + 235451.i 0.886631 + 1.77496i
\(52\) 0 0
\(53\) 173405.i 1.16476i −0.812918 0.582378i \(-0.802122\pi\)
0.812918 0.582378i \(-0.197878\pi\)
\(54\) 0 0
\(55\) −61495.4 −0.369619
\(56\) 0 0
\(57\) −186102. + 92961.6i −1.00491 + 0.501972i
\(58\) 0 0
\(59\) 138336.i 0.673563i 0.941583 + 0.336782i \(0.109338\pi\)
−0.941583 + 0.336782i \(0.890662\pi\)
\(60\) 0 0
\(61\) −278669. −1.22772 −0.613861 0.789414i \(-0.710384\pi\)
−0.613861 + 0.789414i \(0.710384\pi\)
\(62\) 0 0
\(63\) −191258. + 254604.i −0.764890 + 1.01822i
\(64\) 0 0
\(65\) 50763.4i 0.184846i
\(66\) 0 0
\(67\) −96214.3 −0.319901 −0.159950 0.987125i \(-0.551133\pi\)
−0.159950 + 0.987125i \(0.551133\pi\)
\(68\) 0 0
\(69\) 233105. + 466658.i 0.709586 + 1.42053i
\(70\) 0 0
\(71\) 338334.i 0.945302i −0.881250 0.472651i \(-0.843297\pi\)
0.881250 0.472651i \(-0.156703\pi\)
\(72\) 0 0
\(73\) 37504.8 0.0964093 0.0482046 0.998837i \(-0.484650\pi\)
0.0482046 + 0.998837i \(0.484650\pi\)
\(74\) 0 0
\(75\) 75481.8 37704.7i 0.178920 0.0893740i
\(76\) 0 0
\(77\) 480524.i 1.05255i
\(78\) 0 0
\(79\) 721768. 1.46392 0.731958 0.681350i \(-0.238607\pi\)
0.731958 + 0.681350i \(0.238607\pi\)
\(80\) 0 0
\(81\) −148020. 510411.i −0.278526 0.960429i
\(82\) 0 0
\(83\) 310971.i 0.543858i −0.962317 0.271929i \(-0.912338\pi\)
0.962317 0.271929i \(-0.0876615\pi\)
\(84\) 0 0
\(85\) 544921. 0.887312
\(86\) 0 0
\(87\) 97808.0 + 195804.i 0.148531 + 0.297347i
\(88\) 0 0
\(89\) 1.11509e6i 1.58176i −0.611969 0.790882i \(-0.709622\pi\)
0.611969 0.790882i \(-0.290378\pi\)
\(90\) 0 0
\(91\) −396665. −0.526380
\(92\) 0 0
\(93\) 18568.1 9275.14i 0.0230844 0.0115311i
\(94\) 0 0
\(95\) 430709.i 0.502357i
\(96\) 0 0
\(97\) 306615. 0.335952 0.167976 0.985791i \(-0.446277\pi\)
0.167976 + 0.985791i \(0.446277\pi\)
\(98\) 0 0
\(99\) 641187. + 481660.i 0.660814 + 0.496404i
\(100\) 0 0
\(101\) 1.43599e6i 1.39376i −0.717189 0.696879i \(-0.754571\pi\)
0.717189 0.696879i \(-0.245429\pi\)
\(102\) 0 0
\(103\) −1.55067e6 −1.41908 −0.709540 0.704665i \(-0.751097\pi\)
−0.709540 + 0.704665i \(0.751097\pi\)
\(104\) 0 0
\(105\) 294624. + 589814.i 0.254507 + 0.509503i
\(106\) 0 0
\(107\) 1.02996e6i 0.840756i −0.907349 0.420378i \(-0.861897\pi\)
0.907349 0.420378i \(-0.138103\pi\)
\(108\) 0 0
\(109\) 906603. 0.700064 0.350032 0.936738i \(-0.386171\pi\)
0.350032 + 0.936738i \(0.386171\pi\)
\(110\) 0 0
\(111\) −1.74439e6 + 871359.i −1.27548 + 0.637131i
\(112\) 0 0
\(113\) 889378.i 0.616383i −0.951324 0.308192i \(-0.900276\pi\)
0.951324 0.308192i \(-0.0997238\pi\)
\(114\) 0 0
\(115\) 1.08002e6 0.710131
\(116\) 0 0
\(117\) 397602. 529289.i 0.248251 0.330473i
\(118\) 0 0
\(119\) 4.25800e6i 2.52677i
\(120\) 0 0
\(121\) 561422. 0.316908
\(122\) 0 0
\(123\) −169415. 339156.i −0.0910411 0.182257i
\(124\) 0 0
\(125\) 174693.i 0.0894427i
\(126\) 0 0
\(127\) −2.88160e6 −1.40677 −0.703383 0.710811i \(-0.748328\pi\)
−0.703383 + 0.710811i \(0.748328\pi\)
\(128\) 0 0
\(129\) 2.93092e6 1.46406e6i 1.36532 0.682007i
\(130\) 0 0
\(131\) 627919.i 0.279312i 0.990200 + 0.139656i \(0.0445997\pi\)
−0.990200 + 0.139656i \(0.955400\pi\)
\(132\) 0 0
\(133\) −3.36555e6 −1.43054
\(134\) 0 0
\(135\) −1.08234e6 198077.i −0.439908 0.0805066i
\(136\) 0 0
\(137\) 4.00040e6i 1.55576i 0.628415 + 0.777878i \(0.283704\pi\)
−0.628415 + 0.777878i \(0.716296\pi\)
\(138\) 0 0
\(139\) 4.05700e6 1.51064 0.755319 0.655357i \(-0.227482\pi\)
0.755319 + 0.655357i \(0.227482\pi\)
\(140\) 0 0
\(141\) −447219. 895298.i −0.159538 0.319382i
\(142\) 0 0
\(143\) 998949.i 0.341614i
\(144\) 0 0
\(145\) 453163. 0.148645
\(146\) 0 0
\(147\) −1.76708e6 + 882694.i −0.556295 + 0.277880i
\(148\) 0 0
\(149\) 3.19253e6i 0.965110i 0.875866 + 0.482555i \(0.160291\pi\)
−0.875866 + 0.482555i \(0.839709\pi\)
\(150\) 0 0
\(151\) −1.95081e6 −0.566610 −0.283305 0.959030i \(-0.591431\pi\)
−0.283305 + 0.959030i \(0.591431\pi\)
\(152\) 0 0
\(153\) −5.68166e6 4.26807e6i −1.58636 1.19167i
\(154\) 0 0
\(155\) 42973.5i 0.0115400i
\(156\) 0 0
\(157\) 3.91636e6 1.01201 0.506003 0.862532i \(-0.331122\pi\)
0.506003 + 0.862532i \(0.331122\pi\)
\(158\) 0 0
\(159\) 2.09222e6 + 4.18847e6i 0.520495 + 1.04199i
\(160\) 0 0
\(161\) 8.43927e6i 2.02221i
\(162\) 0 0
\(163\) −3.66984e6 −0.847392 −0.423696 0.905804i \(-0.639268\pi\)
−0.423696 + 0.905804i \(0.639268\pi\)
\(164\) 0 0
\(165\) 1.48537e6 741973.i 0.330661 0.165172i
\(166\) 0 0
\(167\) 3.16638e6i 0.679851i −0.940452 0.339925i \(-0.889598\pi\)
0.940452 0.339925i \(-0.110402\pi\)
\(168\) 0 0
\(169\) −4.00219e6 −0.829159
\(170\) 0 0
\(171\) 3.37351e6 4.49082e6i 0.674673 0.898126i
\(172\) 0 0
\(173\) 5.80835e6i 1.12180i −0.827884 0.560899i \(-0.810456\pi\)
0.827884 0.560899i \(-0.189544\pi\)
\(174\) 0 0
\(175\) 1.36505e6 0.254703
\(176\) 0 0
\(177\) −1.66909e6 3.34139e6i −0.300995 0.602569i
\(178\) 0 0
\(179\) 6.27743e6i 1.09452i −0.836963 0.547259i \(-0.815671\pi\)
0.836963 0.547259i \(-0.184329\pi\)
\(180\) 0 0
\(181\) 1.24869e6 0.210581 0.105291 0.994442i \(-0.466423\pi\)
0.105291 + 0.994442i \(0.466423\pi\)
\(182\) 0 0
\(183\) 6.73103e6 3.36228e6i 1.09832 0.548632i
\(184\) 0 0
\(185\) 4.03717e6i 0.637620i
\(186\) 0 0
\(187\) 1.07232e7 1.63984
\(188\) 0 0
\(189\) 1.54777e6 8.45737e6i 0.229256 1.25271i
\(190\) 0 0
\(191\) 7.38849e6i 1.06037i −0.847883 0.530183i \(-0.822123\pi\)
0.847883 0.530183i \(-0.177877\pi\)
\(192\) 0 0
\(193\) −901327. −0.125375 −0.0626874 0.998033i \(-0.519967\pi\)
−0.0626874 + 0.998033i \(0.519967\pi\)
\(194\) 0 0
\(195\) −612486. 1.22615e6i −0.0826023 0.165363i
\(196\) 0 0
\(197\) 7.82439e6i 1.02341i 0.859160 + 0.511707i \(0.170987\pi\)
−0.859160 + 0.511707i \(0.829013\pi\)
\(198\) 0 0
\(199\) 4.89870e6 0.621615 0.310808 0.950473i \(-0.399401\pi\)
0.310808 + 0.950473i \(0.399401\pi\)
\(200\) 0 0
\(201\) 2.32398e6 1.16087e6i 0.286183 0.142954i
\(202\) 0 0
\(203\) 3.54101e6i 0.423291i
\(204\) 0 0
\(205\) −784933. −0.0911111
\(206\) 0 0
\(207\) −1.12609e7 8.45921e6i −1.26959 0.953716i
\(208\) 0 0
\(209\) 8.47571e6i 0.928405i
\(210\) 0 0
\(211\) −1.05764e7 −1.12587 −0.562935 0.826501i \(-0.690328\pi\)
−0.562935 + 0.826501i \(0.690328\pi\)
\(212\) 0 0
\(213\) 4.08217e6 + 8.17218e6i 0.422427 + 0.845666i
\(214\) 0 0
\(215\) 6.78325e6i 0.682531i
\(216\) 0 0
\(217\) 335794. 0.0328620
\(218\) 0 0
\(219\) −905898. + 452515.i −0.0862476 + 0.0430824i
\(220\) 0 0
\(221\) 8.85185e6i 0.820082i
\(222\) 0 0
\(223\) 9.73263e6 0.877639 0.438819 0.898575i \(-0.355397\pi\)
0.438819 + 0.898575i \(0.355397\pi\)
\(224\) 0 0
\(225\) −1.36827e6 + 1.82145e6i −0.120123 + 0.159908i
\(226\) 0 0
\(227\) 5.26593e6i 0.450192i −0.974337 0.225096i \(-0.927730\pi\)
0.974337 0.225096i \(-0.0722696\pi\)
\(228\) 0 0
\(229\) −1.98690e7 −1.65451 −0.827255 0.561827i \(-0.810099\pi\)
−0.827255 + 0.561827i \(0.810099\pi\)
\(230\) 0 0
\(231\) 5.79776e6 + 1.16067e7i 0.470354 + 0.941611i
\(232\) 0 0
\(233\) 2.42512e7i 1.91719i 0.284774 + 0.958595i \(0.408081\pi\)
−0.284774 + 0.958595i \(0.591919\pi\)
\(234\) 0 0
\(235\) −2.07205e6 −0.159660
\(236\) 0 0
\(237\) −1.74337e7 + 8.70849e6i −1.30962 + 0.654181i
\(238\) 0 0
\(239\) 9.38393e6i 0.687371i −0.939085 0.343685i \(-0.888325\pi\)
0.939085 0.343685i \(-0.111675\pi\)
\(240\) 0 0
\(241\) 239568. 0.0171151 0.00855753 0.999963i \(-0.497276\pi\)
0.00855753 + 0.999963i \(0.497276\pi\)
\(242\) 0 0
\(243\) 9.73366e6 + 1.05426e7i 0.678356 + 0.734734i
\(244\) 0 0
\(245\) 4.08968e6i 0.278094i
\(246\) 0 0
\(247\) 6.99656e6 0.464294
\(248\) 0 0
\(249\) 3.75202e6 + 7.51124e6i 0.243034 + 0.486534i
\(250\) 0 0
\(251\) 2.51087e6i 0.158783i 0.996844 + 0.0793913i \(0.0252977\pi\)
−0.996844 + 0.0793913i \(0.974702\pi\)
\(252\) 0 0
\(253\) 2.12532e7 1.31239
\(254\) 0 0
\(255\) −1.31621e7 + 6.57474e6i −0.793788 + 0.396513i
\(256\) 0 0
\(257\) 1.15026e7i 0.677634i −0.940852 0.338817i \(-0.889973\pi\)
0.940852 0.338817i \(-0.110027\pi\)
\(258\) 0 0
\(259\) −3.15464e7 −1.81573
\(260\) 0 0
\(261\) −4.72494e6 3.54938e6i −0.265751 0.199632i
\(262\) 0 0
\(263\) 8.74829e6i 0.480901i −0.970661 0.240451i \(-0.922705\pi\)
0.970661 0.240451i \(-0.0772953\pi\)
\(264\) 0 0
\(265\) 9.69366e6 0.520895
\(266\) 0 0
\(267\) 1.34542e7 + 2.69342e7i 0.706843 + 1.41504i
\(268\) 0 0
\(269\) 2.28060e7i 1.17163i −0.810444 0.585816i \(-0.800774\pi\)
0.810444 0.585816i \(-0.199226\pi\)
\(270\) 0 0
\(271\) 1.15949e7 0.582584 0.291292 0.956634i \(-0.405915\pi\)
0.291292 + 0.956634i \(0.405915\pi\)
\(272\) 0 0
\(273\) 9.58110e6 4.78596e6i 0.470899 0.235224i
\(274\) 0 0
\(275\) 3.43770e6i 0.165299i
\(276\) 0 0
\(277\) 1.44035e7 0.677688 0.338844 0.940843i \(-0.389964\pi\)
0.338844 + 0.940843i \(0.389964\pi\)
\(278\) 0 0
\(279\) −336588. + 448067.i −0.0154984 + 0.0206315i
\(280\) 0 0
\(281\) 3.74318e7i 1.68703i 0.537108 + 0.843514i \(0.319517\pi\)
−0.537108 + 0.843514i \(0.680483\pi\)
\(282\) 0 0
\(283\) −2.16540e7 −0.955384 −0.477692 0.878527i \(-0.658527\pi\)
−0.477692 + 0.878527i \(0.658527\pi\)
\(284\) 0 0
\(285\) −5.19671e6 1.04034e7i −0.224489 0.449408i
\(286\) 0 0
\(287\) 6.13346e6i 0.259454i
\(288\) 0 0
\(289\) −7.08827e7 −2.93661
\(290\) 0 0
\(291\) −7.40602e6 + 3.69946e6i −0.300543 + 0.150127i
\(292\) 0 0
\(293\) 1.13642e6i 0.0451790i −0.999745 0.0225895i \(-0.992809\pi\)
0.999745 0.0225895i \(-0.00719107\pi\)
\(294\) 0 0
\(295\) −7.73320e6 −0.301227
\(296\) 0 0
\(297\) −2.12988e7 3.89785e6i −0.812992 0.148784i
\(298\) 0 0
\(299\) 1.75442e7i 0.656325i
\(300\) 0 0
\(301\) 5.30042e7 1.94362
\(302\) 0 0
\(303\) 1.73259e7 + 3.46852e7i 0.622829 + 1.24685i
\(304\) 0 0
\(305\) 1.55781e7i 0.549054i
\(306\) 0 0
\(307\) 5.32583e7 1.84065 0.920326 0.391151i \(-0.127923\pi\)
0.920326 + 0.391151i \(0.127923\pi\)
\(308\) 0 0
\(309\) 3.74551e7 1.87096e7i 1.26951 0.634145i
\(310\) 0 0
\(311\) 5.42770e7i 1.80441i −0.431309 0.902204i \(-0.641948\pi\)
0.431309 0.902204i \(-0.358052\pi\)
\(312\) 0 0
\(313\) 5.24188e6 0.170944 0.0854721 0.996341i \(-0.472760\pi\)
0.0854721 + 0.996341i \(0.472760\pi\)
\(314\) 0 0
\(315\) −1.42328e7 1.06917e7i −0.455364 0.342069i
\(316\) 0 0
\(317\) 2.90245e6i 0.0911143i −0.998962 0.0455572i \(-0.985494\pi\)
0.998962 0.0455572i \(-0.0145063\pi\)
\(318\) 0 0
\(319\) 8.91758e6 0.274710
\(320\) 0 0
\(321\) 1.24270e7 + 2.48779e7i 0.375709 + 0.752140i
\(322\) 0 0
\(323\) 7.51046e7i 2.22874i
\(324\) 0 0
\(325\) −2.83776e6 −0.0826658
\(326\) 0 0
\(327\) −2.18983e7 + 1.09386e7i −0.626277 + 0.312838i
\(328\) 0 0
\(329\) 1.61910e7i 0.454659i
\(330\) 0 0
\(331\) −5.14619e6 −0.141906 −0.0709532 0.997480i \(-0.522604\pi\)
−0.0709532 + 0.997480i \(0.522604\pi\)
\(332\) 0 0
\(333\) 3.16210e7 4.20939e7i 0.856333 1.13995i
\(334\) 0 0
\(335\) 5.37854e6i 0.143064i
\(336\) 0 0
\(337\) −1.69798e7 −0.443652 −0.221826 0.975086i \(-0.571202\pi\)
−0.221826 + 0.975086i \(0.571202\pi\)
\(338\) 0 0
\(339\) 1.07308e7 + 2.14822e7i 0.275443 + 0.551416i
\(340\) 0 0
\(341\) 845655.i 0.0213270i
\(342\) 0 0
\(343\) 1.94341e7 0.481596
\(344\) 0 0
\(345\) −2.60870e7 + 1.30310e7i −0.635282 + 0.317336i
\(346\) 0 0
\(347\) 3.99529e7i 0.956225i 0.878299 + 0.478112i \(0.158679\pi\)
−0.878299 + 0.478112i \(0.841321\pi\)
\(348\) 0 0
\(349\) −5.61186e6 −0.132017 −0.0660086 0.997819i \(-0.521026\pi\)
−0.0660086 + 0.997819i \(0.521026\pi\)
\(350\) 0 0
\(351\) −3.21761e6 + 1.75818e7i −0.0744068 + 0.406576i
\(352\) 0 0
\(353\) 6.35649e7i 1.44508i 0.691327 + 0.722542i \(0.257027\pi\)
−0.691327 + 0.722542i \(0.742973\pi\)
\(354\) 0 0
\(355\) 1.89134e7 0.422752
\(356\) 0 0
\(357\) −5.13749e7 1.02849e8i −1.12914 2.26044i
\(358\) 0 0
\(359\) 2.08068e7i 0.449699i 0.974393 + 0.224850i \(0.0721891\pi\)
−0.974393 + 0.224850i \(0.927811\pi\)
\(360\) 0 0
\(361\) 1.23173e7 0.261814
\(362\) 0 0
\(363\) −1.35607e7 + 6.77383e6i −0.283505 + 0.141617i
\(364\) 0 0
\(365\) 2.09658e6i 0.0431155i
\(366\) 0 0
\(367\) 6.33583e7 1.28176 0.640878 0.767642i \(-0.278570\pi\)
0.640878 + 0.767642i \(0.278570\pi\)
\(368\) 0 0
\(369\) 8.18418e6 + 6.14796e6i 0.162891 + 0.122363i
\(370\) 0 0
\(371\) 7.57462e7i 1.48333i
\(372\) 0 0
\(373\) −7.53615e7 −1.45219 −0.726094 0.687595i \(-0.758666\pi\)
−0.726094 + 0.687595i \(0.758666\pi\)
\(374\) 0 0
\(375\) 2.10776e6 + 4.21956e6i 0.0399693 + 0.0800153i
\(376\) 0 0
\(377\) 7.36131e6i 0.137382i
\(378\) 0 0
\(379\) −1.07995e8 −1.98375 −0.991877 0.127204i \(-0.959400\pi\)
−0.991877 + 0.127204i \(0.959400\pi\)
\(380\) 0 0
\(381\) 6.96026e7 3.47679e7i 1.25849 0.628642i
\(382\) 0 0
\(383\) 8.99874e7i 1.60171i 0.598855 + 0.800857i \(0.295622\pi\)
−0.598855 + 0.800857i \(0.704378\pi\)
\(384\) 0 0
\(385\) 2.68621e7 0.470715
\(386\) 0 0
\(387\) −5.31295e7 + 7.07261e7i −0.916648 + 1.22024i
\(388\) 0 0
\(389\) 5.17288e7i 0.878788i 0.898295 + 0.439394i \(0.144807\pi\)
−0.898295 + 0.439394i \(0.855193\pi\)
\(390\) 0 0
\(391\) −1.88328e8 −3.15054
\(392\) 0 0
\(393\) −7.57615e6 1.51669e7i −0.124816 0.249872i
\(394\) 0 0
\(395\) 4.03481e7i 0.654683i
\(396\) 0 0
\(397\) 5.90854e7 0.944297 0.472149 0.881519i \(-0.343479\pi\)
0.472149 + 0.881519i \(0.343479\pi\)
\(398\) 0 0
\(399\) 8.12921e7 4.06071e7i 1.27976 0.639268i
\(400\) 0 0
\(401\) 1.40805e7i 0.218366i 0.994022 + 0.109183i \(0.0348235\pi\)
−0.994022 + 0.109183i \(0.965177\pi\)
\(402\) 0 0
\(403\) −698073. −0.0106656
\(404\) 0 0
\(405\) 2.85329e7 8.27457e6i 0.429517 0.124560i
\(406\) 0 0
\(407\) 7.94456e7i 1.17838i
\(408\) 0 0
\(409\) 1.05471e8 1.54157 0.770783 0.637097i \(-0.219865\pi\)
0.770783 + 0.637097i \(0.219865\pi\)
\(410\) 0 0
\(411\) −4.82668e7 9.66263e7i −0.695221 1.39178i
\(412\) 0 0
\(413\) 6.04272e7i 0.857792i
\(414\) 0 0
\(415\) 1.73838e7 0.243221
\(416\) 0 0
\(417\) −9.79934e7 + 4.89497e7i −1.35141 + 0.675059i
\(418\) 0 0
\(419\) 1.26383e7i 0.171809i 0.996303 + 0.0859046i \(0.0273780\pi\)
−0.996303 + 0.0859046i \(0.972622\pi\)
\(420\) 0 0
\(421\) 7.64436e6 0.102446 0.0512230 0.998687i \(-0.483688\pi\)
0.0512230 + 0.998687i \(0.483688\pi\)
\(422\) 0 0
\(423\) 2.16044e7 + 1.62293e7i 0.285444 + 0.214426i
\(424\) 0 0
\(425\) 3.04620e7i 0.396818i
\(426\) 0 0
\(427\) 1.21727e8 1.56352
\(428\) 0 0
\(429\) −1.20528e7 2.41288e7i −0.152657 0.305607i
\(430\) 0 0
\(431\) 1.11755e8i 1.39583i −0.716179 0.697917i \(-0.754111\pi\)
0.716179 0.697917i \(-0.245889\pi\)
\(432\) 0 0
\(433\) −5.93268e7 −0.730781 −0.365390 0.930854i \(-0.619065\pi\)
−0.365390 + 0.930854i \(0.619065\pi\)
\(434\) 0 0
\(435\) −1.09458e7 + 5.46763e6i −0.132978 + 0.0664250i
\(436\) 0 0
\(437\) 1.48856e8i 1.78370i
\(438\) 0 0
\(439\) −3.07400e6 −0.0363338 −0.0181669 0.999835i \(-0.505783\pi\)
−0.0181669 + 0.999835i \(0.505783\pi\)
\(440\) 0 0
\(441\) 3.20323e7 4.26415e7i 0.373484 0.497183i
\(442\) 0 0
\(443\) 7.68560e7i 0.884029i 0.897008 + 0.442014i \(0.145736\pi\)
−0.897008 + 0.442014i \(0.854264\pi\)
\(444\) 0 0
\(445\) 6.23357e7 0.707386
\(446\) 0 0
\(447\) −3.85195e7 7.71130e7i −0.431279 0.863386i
\(448\) 0 0
\(449\) 1.60168e8i 1.76945i 0.466117 + 0.884723i \(0.345653\pi\)
−0.466117 + 0.884723i \(0.654347\pi\)
\(450\) 0 0
\(451\) −1.54463e7 −0.168382
\(452\) 0 0
\(453\) 4.71202e7 2.35375e7i 0.506889 0.253201i
\(454\) 0 0
\(455\) 2.21742e7i 0.235404i
\(456\) 0 0
\(457\) −9.10136e7 −0.953581 −0.476790 0.879017i \(-0.658200\pi\)
−0.476790 + 0.879017i \(0.658200\pi\)
\(458\) 0 0
\(459\) 1.88732e8 + 3.45395e7i 1.95168 + 0.357173i
\(460\) 0 0
\(461\) 5.62811e7i 0.574460i 0.957862 + 0.287230i \(0.0927344\pi\)
−0.957862 + 0.287230i \(0.907266\pi\)
\(462\) 0 0
\(463\) −7.87451e7 −0.793378 −0.396689 0.917953i \(-0.629841\pi\)
−0.396689 + 0.917953i \(0.629841\pi\)
\(464\) 0 0
\(465\) 518496. + 1.03799e6i 0.00515688 + 0.0103237i
\(466\) 0 0
\(467\) 3.51383e7i 0.345009i −0.985009 0.172504i \(-0.944814\pi\)
0.985009 0.172504i \(-0.0551859\pi\)
\(468\) 0 0
\(469\) 4.20279e7 0.407398
\(470\) 0 0
\(471\) −9.45963e7 + 4.72528e7i −0.905340 + 0.452236i
\(472\) 0 0
\(473\) 1.33484e8i 1.26138i
\(474\) 0 0
\(475\) −2.40773e7 −0.224661
\(476\) 0 0
\(477\) −1.01072e8 7.59252e7i −0.931269 0.699569i
\(478\) 0 0
\(479\) 1.04950e7i 0.0954942i −0.998859 0.0477471i \(-0.984796\pi\)
0.998859 0.0477471i \(-0.0152042\pi\)
\(480\) 0 0
\(481\) 6.55810e7 0.589309
\(482\) 0 0
\(483\) −1.01824e8 2.03843e8i −0.903667 1.80907i
\(484\) 0 0
\(485\) 1.71403e7i 0.150242i
\(486\) 0 0
\(487\) 8.61647e7 0.746007 0.373003 0.927830i \(-0.378328\pi\)
0.373003 + 0.927830i \(0.378328\pi\)
\(488\) 0 0
\(489\) 8.86420e7 4.42785e7i 0.758076 0.378674i
\(490\) 0 0
\(491\) 8.04932e7i 0.680009i −0.940424 0.340005i \(-0.889571\pi\)
0.940424 0.340005i \(-0.110429\pi\)
\(492\) 0 0
\(493\) −7.90201e7 −0.659472
\(494\) 0 0
\(495\) −2.69256e7 + 3.58435e7i −0.221999 + 0.295525i
\(496\) 0 0
\(497\) 1.47789e8i 1.20386i
\(498\) 0 0
\(499\) 2.12437e8 1.70974 0.854869 0.518845i \(-0.173638\pi\)
0.854869 + 0.518845i \(0.173638\pi\)
\(500\) 0 0
\(501\) 3.82039e7 + 7.64813e7i 0.303805 + 0.608194i
\(502\) 0 0
\(503\) 1.26961e8i 0.997625i −0.866710 0.498812i \(-0.833770\pi\)
0.866710 0.498812i \(-0.166230\pi\)
\(504\) 0 0
\(505\) 8.02743e7 0.623308
\(506\) 0 0
\(507\) 9.66696e7 4.82884e7i 0.741765 0.370527i
\(508\) 0 0
\(509\) 1.99418e8i 1.51221i −0.654452 0.756104i \(-0.727101\pi\)
0.654452 0.756104i \(-0.272899\pi\)
\(510\) 0 0
\(511\) −1.63827e7 −0.122779
\(512\) 0 0
\(513\) −2.73002e7 + 1.49175e8i −0.202216 + 1.10495i
\(514\) 0 0
\(515\) 8.66849e7i 0.634632i
\(516\) 0 0
\(517\) −4.07749e7 −0.295068
\(518\) 0 0
\(519\) 7.00807e7 + 1.40296e8i 0.501298 + 1.00356i
\(520\) 0 0
\(521\) 4.85935e7i 0.343609i 0.985131 + 0.171805i \(0.0549598\pi\)
−0.985131 + 0.171805i \(0.945040\pi\)
\(522\) 0 0
\(523\) −5.79820e7 −0.405311 −0.202655 0.979250i \(-0.564957\pi\)
−0.202655 + 0.979250i \(0.564957\pi\)
\(524\) 0 0
\(525\) −3.29716e7 + 1.64700e7i −0.227857 + 0.113819i
\(526\) 0 0
\(527\) 7.49348e6i 0.0511979i
\(528\) 0 0
\(529\) −2.25226e8 −1.52143
\(530\) 0 0
\(531\) 8.06309e7 + 6.05700e7i 0.538540 + 0.404552i
\(532\) 0 0
\(533\) 1.27507e7i 0.0842077i
\(534\) 0 0
\(535\) 5.75767e7 0.375998
\(536\) 0 0
\(537\) 7.57404e7 + 1.51626e8i 0.489108 + 0.979155i
\(538\) 0 0
\(539\) 8.04790e7i 0.513944i
\(540\) 0 0
\(541\) 4.20855e7 0.265791 0.132896 0.991130i \(-0.457572\pi\)
0.132896 + 0.991130i \(0.457572\pi\)
\(542\) 0 0
\(543\) −3.01611e7 + 1.50661e7i −0.188386 + 0.0941024i
\(544\) 0 0
\(545\) 5.06807e7i 0.313078i
\(546\) 0 0
\(547\) 7.51378e7 0.459088 0.229544 0.973298i \(-0.426276\pi\)
0.229544 + 0.973298i \(0.426276\pi\)
\(548\) 0 0
\(549\) −1.22015e8 + 1.62426e8i −0.737387 + 0.981611i
\(550\) 0 0
\(551\) 6.24580e7i 0.373364i
\(552\) 0 0
\(553\) −3.15279e8 −1.86432
\(554\) 0 0
\(555\) −4.87105e7 9.75145e7i −0.284933 0.570414i
\(556\) 0 0
\(557\) 1.38895e7i 0.0803750i 0.999192 + 0.0401875i \(0.0127955\pi\)
−0.999192 + 0.0401875i \(0.987204\pi\)
\(558\) 0 0
\(559\) −1.10189e8 −0.630816
\(560\) 0 0
\(561\) −2.59011e8 + 1.29381e8i −1.46700 + 0.732795i
\(562\) 0 0
\(563\) 2.69248e8i 1.50878i −0.656424 0.754392i \(-0.727932\pi\)
0.656424 0.754392i \(-0.272068\pi\)
\(564\) 0 0
\(565\) 4.97177e7 0.275655
\(566\) 0 0
\(567\) 6.46574e7 + 2.22955e8i 0.354706 + 1.22312i
\(568\) 0 0
\(569\) 1.49380e8i 0.810878i 0.914122 + 0.405439i \(0.132881\pi\)
−0.914122 + 0.405439i \(0.867119\pi\)
\(570\) 0 0
\(571\) −1.01093e7 −0.0543015 −0.0271507 0.999631i \(-0.508643\pi\)
−0.0271507 + 0.999631i \(0.508643\pi\)
\(572\) 0 0
\(573\) 8.91458e7 + 1.78463e8i 0.473846 + 0.948602i
\(574\) 0 0
\(575\) 6.03750e7i 0.317580i
\(576\) 0 0
\(577\) −1.19578e8 −0.622477 −0.311238 0.950332i \(-0.600744\pi\)
−0.311238 + 0.950332i \(0.600744\pi\)
\(578\) 0 0
\(579\) 2.17708e7 1.08750e7i 0.112160 0.0560263i
\(580\) 0 0
\(581\) 1.35837e8i 0.692610i
\(582\) 0 0
\(583\) 1.90757e8 0.962664
\(584\) 0 0
\(585\) 2.95882e7 + 2.22266e7i 0.147792 + 0.111021i
\(586\) 0 0
\(587\) 2.77627e7i 0.137261i 0.997642 + 0.0686305i \(0.0218630\pi\)
−0.997642 + 0.0686305i \(0.978137\pi\)
\(588\) 0 0
\(589\) −5.92289e6 −0.0289860
\(590\) 0 0
\(591\) −9.44051e7 1.88992e8i −0.457334 0.915546i
\(592\) 0 0
\(593\) 9.85398e7i 0.472550i 0.971686 + 0.236275i \(0.0759266\pi\)
−0.971686 + 0.236275i \(0.924073\pi\)
\(594\) 0 0
\(595\) −2.38030e8 −1.13000
\(596\) 0 0
\(597\) −1.18324e8 + 5.91052e7i −0.556096 + 0.277781i
\(598\) 0 0
\(599\) 6.51155e7i 0.302973i −0.988459 0.151486i \(-0.951594\pi\)
0.988459 0.151486i \(-0.0484060\pi\)
\(600\) 0 0
\(601\) 2.53264e8 1.16668 0.583339 0.812229i \(-0.301746\pi\)
0.583339 + 0.812229i \(0.301746\pi\)
\(602\) 0 0
\(603\) −4.21272e7 + 5.60799e7i −0.192137 + 0.255773i
\(604\) 0 0
\(605\) 3.13844e7i 0.141726i
\(606\) 0 0
\(607\) −5.70547e7 −0.255109 −0.127554 0.991832i \(-0.540713\pi\)
−0.127554 + 0.991832i \(0.540713\pi\)
\(608\) 0 0
\(609\) −4.27240e7 8.55301e7i −0.189156 0.378675i
\(610\) 0 0
\(611\) 3.36590e7i 0.147563i
\(612\) 0 0
\(613\) 3.26889e8 1.41912 0.709560 0.704645i \(-0.248894\pi\)
0.709560 + 0.704645i \(0.248894\pi\)
\(614\) 0 0
\(615\) 1.89594e7 9.47061e6i 0.0815079 0.0407148i
\(616\) 0 0
\(617\) 4.01353e8i 1.70872i −0.519680 0.854361i \(-0.673949\pi\)
0.519680 0.854361i \(-0.326051\pi\)
\(618\) 0 0
\(619\) 1.37298e8 0.578884 0.289442 0.957195i \(-0.406530\pi\)
0.289442 + 0.957195i \(0.406530\pi\)
\(620\) 0 0
\(621\) 3.74063e8 + 6.84566e7i 1.56196 + 0.285851i
\(622\) 0 0
\(623\) 4.87090e8i 2.01440i
\(624\) 0 0
\(625\) 9.76562e6 0.0400000
\(626\) 0 0
\(627\) −1.02264e8 2.04724e8i −0.414876 0.830550i
\(628\) 0 0
\(629\) 7.03980e8i 2.82884i
\(630\) 0 0
\(631\) −1.95608e8 −0.778572 −0.389286 0.921117i \(-0.627278\pi\)
−0.389286 + 0.921117i \(0.627278\pi\)
\(632\) 0 0
\(633\) 2.55463e8 1.27609e8i 1.00720 0.503118i
\(634\) 0 0
\(635\) 1.61086e8i 0.629125i
\(636\) 0 0
\(637\) 6.64340e7 0.257023
\(638\) 0 0
\(639\) −1.97203e8 1.48139e8i −0.755806 0.567762i
\(640\) 0 0
\(641\) 1.86584e8i 0.708437i −0.935163 0.354219i \(-0.884747\pi\)
0.935163 0.354219i \(-0.115253\pi\)
\(642\) 0 0
\(643\) 3.74191e8 1.40754 0.703770 0.710427i \(-0.251498\pi\)
0.703770 + 0.710427i \(0.251498\pi\)
\(644\) 0 0
\(645\) 8.18432e7 + 1.63844e8i 0.305003 + 0.610591i
\(646\) 0 0
\(647\) 3.66258e8i 1.35230i −0.736762 0.676152i \(-0.763646\pi\)
0.736762 0.676152i \(-0.236354\pi\)
\(648\) 0 0
\(649\) −1.52178e8 −0.556696
\(650\) 0 0
\(651\) −8.11083e6 + 4.05152e6i −0.0293983 + 0.0146850i
\(652\) 0 0
\(653\) 1.50492e8i 0.540472i −0.962794 0.270236i \(-0.912898\pi\)
0.962794 0.270236i \(-0.0871017\pi\)
\(654\) 0 0
\(655\) −3.51017e7 −0.124912
\(656\) 0 0
\(657\) 1.64214e7 2.18602e7i 0.0579048 0.0770830i
\(658\) 0 0
\(659\) 4.31848e8i 1.50895i 0.656329 + 0.754475i \(0.272109\pi\)
−0.656329 + 0.754475i \(0.727891\pi\)
\(660\) 0 0
\(661\) −1.04838e8 −0.363007 −0.181504 0.983390i \(-0.558096\pi\)
−0.181504 + 0.983390i \(0.558096\pi\)
\(662\) 0 0
\(663\) 1.06802e8 + 2.13809e8i 0.366470 + 0.733644i
\(664\) 0 0
\(665\) 1.88140e8i 0.639759i
\(666\) 0 0
\(667\) −1.56616e8 −0.527787
\(668\) 0 0
\(669\) −2.35084e8 + 1.17429e8i −0.785135 + 0.392191i
\(670\) 0 0
\(671\) 3.06554e8i 1.01470i
\(672\) 0 0
\(673\) 2.07160e8 0.679611 0.339805 0.940496i \(-0.389639\pi\)
0.339805 + 0.940496i \(0.389639\pi\)
\(674\) 0 0
\(675\) 1.10728e7 6.05045e7i 0.0360037 0.196733i
\(676\) 0 0
\(677\) 4.59856e8i 1.48203i 0.671491 + 0.741013i \(0.265654\pi\)
−0.671491 + 0.741013i \(0.734346\pi\)
\(678\) 0 0
\(679\) −1.33934e8 −0.427840
\(680\) 0 0
\(681\) 6.35361e7 + 1.27194e8i 0.201177 + 0.402741i
\(682\) 0 0
\(683\) 4.04044e8i 1.26814i 0.773277 + 0.634069i \(0.218616\pi\)
−0.773277 + 0.634069i \(0.781384\pi\)
\(684\) 0 0
\(685\) −2.23629e8 −0.695755
\(686\) 0 0
\(687\) 4.79919e8 2.39729e8i 1.48012 0.739351i
\(688\) 0 0
\(689\) 1.57467e8i 0.481428i
\(690\) 0 0
\(691\) −2.98324e8 −0.904179 −0.452090 0.891973i \(-0.649321\pi\)
−0.452090 + 0.891973i \(0.649321\pi\)
\(692\) 0 0
\(693\) −2.80080e8 2.10396e8i −0.841556 0.632177i
\(694\) 0 0
\(695\) 2.26793e8i 0.675578i
\(696\) 0 0
\(697\) 1.36872e8 0.404220
\(698\) 0 0
\(699\) −2.92602e8 5.85767e8i −0.856735 1.71512i
\(700\) 0 0
\(701\) 4.10378e8i 1.19132i −0.803235 0.595662i \(-0.796890\pi\)
0.803235 0.595662i \(-0.203110\pi\)
\(702\) 0 0
\(703\) 5.56430e8 1.60157
\(704\) 0 0
\(705\) 5.00487e7 2.50003e7i 0.142832 0.0713474i
\(706\) 0 0
\(707\) 6.27262e8i 1.77497i
\(708\) 0 0
\(709\) −4.38412e8 −1.23011 −0.615054 0.788485i \(-0.710866\pi\)
−0.615054 + 0.788485i \(0.710866\pi\)
\(710\) 0 0
\(711\) 3.16024e8 4.20693e8i 0.879249 1.17046i
\(712\) 0 0
\(713\) 1.48519e7i 0.0409745i
\(714\) 0 0
\(715\) −5.58430e7 −0.152774
\(716\) 0 0
\(717\) 1.13222e8 + 2.26661e8i 0.307166 + 0.614921i
\(718\) 0 0
\(719\) 1.35325e8i 0.364076i −0.983291 0.182038i \(-0.941731\pi\)
0.983291 0.182038i \(-0.0582694\pi\)
\(720\) 0 0
\(721\) 6.77355e8 1.80722
\(722\) 0 0
\(723\) −5.78657e6 + 2.89051e6i −0.0153111 + 0.00764821i
\(724\) 0 0
\(725\) 2.53326e7i 0.0664761i
\(726\) 0 0
\(727\) 1.04885e7 0.0272966 0.0136483 0.999907i \(-0.495655\pi\)
0.0136483 + 0.999907i \(0.495655\pi\)
\(728\) 0 0
\(729\) −3.62311e8 1.37207e8i −0.935187 0.354155i
\(730\) 0 0
\(731\) 1.18283e9i 3.02809i
\(732\) 0 0
\(733\) −2.59655e7 −0.0659303 −0.0329652 0.999457i \(-0.510495\pi\)
−0.0329652 + 0.999457i \(0.510495\pi\)
\(734\) 0 0
\(735\) −4.93441e7 9.87829e7i −0.124272 0.248783i
\(736\) 0 0
\(737\) 1.05842e8i 0.264396i
\(738\) 0 0
\(739\) 1.95717e8 0.484948 0.242474 0.970158i \(-0.422041\pi\)
0.242474 + 0.970158i \(0.422041\pi\)
\(740\) 0 0
\(741\) −1.68996e8 + 8.44169e7i −0.415357 + 0.207479i
\(742\) 0 0
\(743\) 1.21923e8i 0.297247i −0.988894 0.148624i \(-0.952516\pi\)
0.988894 0.148624i \(-0.0474843\pi\)
\(744\) 0 0
\(745\) −1.78468e8 −0.431610
\(746\) 0 0
\(747\) −1.81254e8 1.36158e8i −0.434835 0.326648i
\(748\) 0 0
\(749\) 4.49903e8i 1.07071i
\(750\) 0 0
\(751\) −3.28855e8 −0.776399 −0.388199 0.921575i \(-0.626903\pi\)
−0.388199 + 0.921575i \(0.626903\pi\)
\(752\) 0 0
\(753\) −3.02949e7 6.06480e7i −0.0709553 0.142047i
\(754\) 0 0
\(755\) 1.09054e8i 0.253396i
\(756\) 0 0
\(757\) −2.45441e8 −0.565797 −0.282898 0.959150i \(-0.591296\pi\)
−0.282898 + 0.959150i \(0.591296\pi\)
\(758\) 0 0
\(759\) −5.13354e8 + 2.56431e8i −1.17406 + 0.586468i
\(760\) 0 0
\(761\) 5.19490e8i 1.17875i 0.807858 + 0.589377i \(0.200627\pi\)
−0.807858 + 0.589377i \(0.799373\pi\)
\(762\) 0 0
\(763\) −3.96018e8 −0.891541
\(764\) 0 0
\(765\) 2.38592e8 3.17615e8i 0.532932 0.709441i
\(766\) 0 0
\(767\) 1.25620e8i 0.278403i
\(768\) 0 0
\(769\) 4.01913e8 0.883797 0.441899 0.897065i \(-0.354305\pi\)
0.441899 + 0.897065i \(0.354305\pi\)
\(770\) 0 0
\(771\) 1.38784e8 + 2.77835e8i 0.302814 + 0.606211i
\(772\) 0 0
\(773\) 3.45715e8i 0.748478i 0.927332 + 0.374239i \(0.122096\pi\)
−0.927332 + 0.374239i \(0.877904\pi\)
\(774\) 0 0
\(775\) 2.40229e6 0.00516084
\(776\) 0 0
\(777\) 7.61977e8 3.80623e8i 1.62435 0.811394i
\(778\) 0 0
\(779\) 1.08185e8i 0.228852i
\(780\) 0 0
\(781\) 3.72189e8 0.781286
\(782\) 0 0
\(783\) 1.56952e8 + 2.87235e7i 0.326950 + 0.0598345i
\(784\) 0 0
\(785\) 2.18931e8i 0.452583i
\(786\) 0 0
\(787\) −7.70491e8 −1.58068 −0.790339 0.612670i \(-0.790095\pi\)
−0.790339 + 0.612670i \(0.790095\pi\)
\(788\) 0 0
\(789\) 1.05553e8 + 2.11308e8i 0.214901 + 0.430214i
\(790\) 0 0
\(791\) 3.88494e8i 0.784973i
\(792\) 0 0
\(793\) −2.53055e8 −0.507453
\(794\) 0 0
\(795\) −2.34142e8 + 1.16959e8i −0.465992 + 0.232772i
\(796\) 0 0
\(797\) 2.84076e8i 0.561125i −0.959836 0.280563i \(-0.909479\pi\)
0.959836 0.280563i \(-0.0905210\pi\)
\(798\) 0 0
\(799\) 3.61313e8 0.708343
\(800\) 0 0
\(801\) −6.49948e8 4.88241e8i −1.26468 0.950029i
\(802\) 0 0
\(803\) 4.12577e7i 0.0796817i
\(804\) 0 0
\(805\) −4.71769e8 −0.904362
\(806\) 0 0
\(807\) 2.75165e8 + 5.50859e8i 0.523568 + 1.04814i
\(808\) 0 0
\(809\) 6.17256e8i 1.16579i −0.812548 0.582894i \(-0.801920\pi\)
0.812548 0.582894i \(-0.198080\pi\)
\(810\) 0 0
\(811\) 5.32210e8 0.997748 0.498874 0.866675i \(-0.333747\pi\)
0.498874 + 0.866675i \(0.333747\pi\)
\(812\) 0 0
\(813\) −2.80065e8 + 1.39898e8i −0.521179 + 0.260339i
\(814\) 0 0
\(815\) 2.05150e8i 0.378965i
\(816\) 0 0
\(817\) −9.34913e8 −1.71437
\(818\) 0 0
\(819\) −1.73679e8 + 2.31202e8i −0.316151 + 0.420861i
\(820\) 0 0
\(821\) 3.60333e8i 0.651140i 0.945518 + 0.325570i \(0.105556\pi\)
−0.945518 + 0.325570i \(0.894444\pi\)
\(822\) 0 0
\(823\) 9.46142e7 0.169729 0.0848646 0.996392i \(-0.472954\pi\)
0.0848646 + 0.996392i \(0.472954\pi\)
\(824\) 0 0
\(825\) 4.14775e7 + 8.30347e7i 0.0738671 + 0.147876i
\(826\) 0 0
\(827\) 7.55768e8i 1.33620i 0.744070 + 0.668101i \(0.232893\pi\)
−0.744070 + 0.668101i \(0.767107\pi\)
\(828\) 0 0
\(829\) 4.78503e8 0.839887 0.419944 0.907550i \(-0.362050\pi\)
0.419944 + 0.907550i \(0.362050\pi\)
\(830\) 0 0
\(831\) −3.47906e8 + 1.73786e8i −0.606259 + 0.302839i
\(832\) 0 0
\(833\) 7.13137e8i 1.23378i
\(834\) 0 0
\(835\) 1.77006e8 0.304039
\(836\) 0 0
\(837\) 2.72385e6 1.48838e7i 0.00464523 0.0253826i
\(838\) 0 0
\(839\) 1.45796e8i 0.246866i 0.992353 + 0.123433i \(0.0393903\pi\)
−0.992353 + 0.123433i \(0.960610\pi\)
\(840\) 0 0
\(841\) 5.29109e8 0.889523
\(842\) 0 0
\(843\) −4.51634e8 9.04135e8i −0.753882 1.50921i
\(844\) 0 0
\(845\) 2.23729e8i 0.370811i
\(846\) 0 0
\(847\) −2.45238e8 −0.403587
\(848\) 0 0
\(849\) 5.23033e8 2.61266e8i 0.854685 0.426933i
\(850\) 0 0
\(851\) 1.39527e9i 2.26397i
\(852\) 0 0
\(853\) 7.64922e8 1.23245 0.616226 0.787569i \(-0.288661\pi\)
0.616226 + 0.787569i \(0.288661\pi\)
\(854\) 0 0
\(855\) 2.51045e8 + 1.88585e8i 0.401654 + 0.301723i
\(856\) 0 0
\(857\) 4.79513e8i 0.761830i −0.924610 0.380915i \(-0.875609\pi\)
0.924610 0.380915i \(-0.124391\pi\)
\(858\) 0 0
\(859\) −2.90859e8 −0.458884 −0.229442 0.973322i \(-0.573690\pi\)
−0.229442 + 0.973322i \(0.573690\pi\)
\(860\) 0 0
\(861\) 7.40032e7 + 1.48149e8i 0.115942 + 0.232107i
\(862\) 0 0
\(863\) 3.78195e8i 0.588415i −0.955742 0.294207i \(-0.904944\pi\)
0.955742 0.294207i \(-0.0950556\pi\)
\(864\) 0 0
\(865\) 3.24697e8 0.501683
\(866\) 0 0
\(867\) 1.71211e9 8.55235e8i 2.62709 1.31229i
\(868\) 0 0
\(869\) 7.93990e8i 1.20992i
\(870\) 0 0
\(871\) −8.73706e7 −0.132224
\(872\) 0 0
\(873\) 1.34250e8 1.78715e8i 0.201778 0.268607i
\(874\) 0 0
\(875\) 7.63085e7i 0.113907i
\(876\) 0 0
\(877\) 3.32209e8 0.492508 0.246254 0.969205i \(-0.420800\pi\)
0.246254 + 0.969205i \(0.420800\pi\)
\(878\) 0 0
\(879\) 1.37115e7 + 2.74493e7i 0.0201891 + 0.0404171i
\(880\) 0 0
\(881\) 1.22274e9i 1.78815i −0.447914 0.894077i \(-0.647833\pi\)
0.447914 0.894077i \(-0.352167\pi\)
\(882\) 0 0
\(883\) −8.56383e8 −1.24390 −0.621951 0.783056i \(-0.713660\pi\)
−0.621951 + 0.783056i \(0.713660\pi\)
\(884\) 0 0
\(885\) 1.86789e8 9.33049e7i 0.269477 0.134609i
\(886\) 0 0
\(887\) 5.70473e8i 0.817456i −0.912656 0.408728i \(-0.865973\pi\)
0.912656 0.408728i \(-0.134027\pi\)
\(888\) 0 0
\(889\) 1.25873e9 1.79154
\(890\) 0 0
\(891\) 5.61485e8 1.62831e8i 0.793788 0.230200i
\(892\) 0 0
\(893\) 2.85584e8i 0.401033i
\(894\) 0 0
\(895\) 3.50919e8 0.489484
\(896\) 0 0
\(897\) 2.11679e8 + 4.23765e8i 0.293292 + 0.587148i
\(898\) 0 0
\(899\) 6.23167e6i 0.00857680i
\(900\) 0 0
\(901\) −1.69033e9 −2.31098
\(902\) 0 0
\(903\) −1.28027e9 + 6.39522e8i −1.73876 + 0.868545i
\(904\) 0 0
\(905\) 6.98040e7i 0.0941747i
\(906\) 0 0
\(907\) −2.31544e8 −0.310321 −0.155160 0.987889i \(-0.549589\pi\)
−0.155160 + 0.987889i \(0.549589\pi\)
\(908\) 0 0
\(909\) −8.36987e8 6.28745e8i −1.11436 0.837111i
\(910\) 0 0
\(911\) 4.23991e8i 0.560792i −0.959884 0.280396i \(-0.909534\pi\)
0.959884 0.280396i \(-0.0904658\pi\)
\(912\) 0 0
\(913\) 3.42087e8 0.449495
\(914\) 0 0
\(915\) 1.87957e8 + 3.76276e8i 0.245356 + 0.491183i
\(916\) 0 0
\(917\) 2.74285e8i 0.355708i
\(918\) 0 0
\(919\) 1.08546e9 1.39852 0.699259 0.714869i \(-0.253514\pi\)
0.699259 + 0.714869i \(0.253514\pi\)
\(920\) 0 0
\(921\) −1.28641e9 + 6.42587e8i −1.64665 + 0.822533i
\(922\) 0 0
\(923\) 3.07236e8i 0.390721i
\(924\) 0 0
\(925\) −2.25685e8 −0.285152
\(926\) 0 0
\(927\) −6.78956e8 + 9.03828e8i −0.852319 + 1.13461i
\(928\) 0 0
\(929\) 8.68457e7i 0.108318i 0.998532 + 0.0541591i \(0.0172478\pi\)
−0.998532 + 0.0541591i \(0.982752\pi\)
\(930\) 0 0
\(931\) 5.63668e8 0.698513
\(932\) 0 0
\(933\) 6.54879e8 + 1.31102e9i 0.806336 + 1.61422i
\(934\) 0 0
\(935\) 5.99447e8i 0.733358i
\(936\) 0 0
\(937\) −3.79010e8 −0.460715 −0.230358 0.973106i \(-0.573990\pi\)
−0.230358 + 0.973106i \(0.573990\pi\)
\(938\) 0 0
\(939\) −1.26613e8 + 6.32459e7i −0.152926 + 0.0763898i
\(940\) 0 0
\(941\) 5.55151e6i 0.00666258i 0.999994 + 0.00333129i \(0.00106038\pi\)
−0.999994 + 0.00333129i \(0.998940\pi\)
\(942\) 0 0
\(943\) 2.71278e8 0.323504
\(944\) 0 0
\(945\) 4.72782e8 + 8.65228e7i 0.560228 + 0.102526i
\(946\) 0 0
\(947\) 5.18303e8i 0.610286i −0.952306 0.305143i \(-0.901296\pi\)
0.952306 0.305143i \(-0.0987043\pi\)
\(948\) 0 0
\(949\) 3.40575e7 0.0398487
\(950\) 0 0
\(951\) 3.50195e7 + 7.01062e7i 0.0407163 + 0.0815108i
\(952\) 0 0
\(953\) 3.23349e8i 0.373588i −0.982399 0.186794i \(-0.940190\pi\)
0.982399 0.186794i \(-0.0598097\pi\)
\(954\) 0 0
\(955\) 4.13029e8 0.474210
\(956\) 0 0
\(957\) −2.15397e8 + 1.07595e8i −0.245755 + 0.122760i
\(958\) 0 0
\(959\) 1.74743e9i 1.98128i
\(960\) 0 0
\(961\) −8.86913e8 −0.999334
\(962\) 0 0
\(963\) −6.00328e8 4.50967e8i −0.672218 0.504970i
\(964\) 0 0
\(965\) 5.03857e7i 0.0560694i
\(966\) 0 0
\(967\) −1.56777e9 −1.73381 −0.866907 0.498470i \(-0.833895\pi\)
−0.866907 + 0.498470i \(0.833895\pi\)
\(968\) 0 0
\(969\) 9.06175e8 + 1.81409e9i 0.995957 + 1.99383i
\(970\) 0 0
\(971\) 1.19138e9i 1.30135i 0.759358 + 0.650673i \(0.225513\pi\)
−0.759358 + 0.650673i \(0.774487\pi\)
\(972\) 0 0
\(973\) −1.77216e9 −1.92382
\(974\) 0 0
\(975\) 6.85437e7 3.42390e7i 0.0739527 0.0369409i
\(976\) 0 0
\(977\) 2.83913e8i 0.304440i −0.988347 0.152220i \(-0.951358\pi\)
0.988347 0.152220i \(-0.0486422\pi\)
\(978\) 0 0
\(979\) 1.22667e9 1.30732
\(980\) 0 0
\(981\) 3.96954e8 5.28427e8i 0.420468 0.559729i
\(982\) 0 0
\(983\) 2.30425e8i 0.242587i −0.992617 0.121294i \(-0.961296\pi\)
0.992617 0.121294i \(-0.0387043\pi\)
\(984\) 0 0
\(985\) −4.37397e8 −0.457685
\(986\) 0 0
\(987\) 1.95352e8 + 3.91080e8i 0.203173 + 0.406737i
\(988\) 0 0
\(989\) 2.34433e9i 2.42343i
\(990\) 0 0
\(991\) −1.46001e8 −0.150015 −0.0750076 0.997183i \(-0.523898\pi\)
−0.0750076 + 0.997183i \(0.523898\pi\)
\(992\) 0 0
\(993\) 1.24302e8 6.20914e7i 0.126949 0.0634138i
\(994\) 0 0
\(995\) 2.73846e8i 0.277995i
\(996\) 0 0
\(997\) 1.78931e8 0.180551 0.0902756 0.995917i \(-0.471225\pi\)
0.0902756 + 0.995917i \(0.471225\pi\)
\(998\) 0 0
\(999\) −2.55894e8 + 1.39827e9i −0.256663 + 1.40247i
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 60.7.g.a.41.2 yes 8
3.2 odd 2 inner 60.7.g.a.41.1 8
4.3 odd 2 240.7.l.c.161.7 8
5.2 odd 4 300.7.b.e.149.12 16
5.3 odd 4 300.7.b.e.149.5 16
5.4 even 2 300.7.g.h.101.7 8
12.11 even 2 240.7.l.c.161.8 8
15.2 even 4 300.7.b.e.149.6 16
15.8 even 4 300.7.b.e.149.11 16
15.14 odd 2 300.7.g.h.101.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.7.g.a.41.1 8 3.2 odd 2 inner
60.7.g.a.41.2 yes 8 1.1 even 1 trivial
240.7.l.c.161.7 8 4.3 odd 2
240.7.l.c.161.8 8 12.11 even 2
300.7.b.e.149.5 16 5.3 odd 4
300.7.b.e.149.6 16 15.2 even 4
300.7.b.e.149.11 16 15.8 even 4
300.7.b.e.149.12 16 5.2 odd 4
300.7.g.h.101.7 8 5.4 even 2
300.7.g.h.101.8 8 15.14 odd 2