Properties

Label 147.7.d.b.97.2
Level $147$
Weight $7$
Character 147.97
Analytic conductor $33.818$
Analytic rank $0$
Dimension $8$
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] = [147,7,Mod(97,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.97");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 147.d (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: \(33.8179502921\)
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: \( x^{8} - x^{7} + 212x^{6} + 473x^{5} + 39800x^{4} + 36821x^{3} + 985651x^{2} - 601290x + 21068100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 97.2
Root \(-6.30797 - 10.9257i\) of defining polynomial
Character \(\chi\) \(=\) 147.97
Dual form 147.7.d.b.97.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-11.6159 q^{2} +15.5885i q^{3} +70.9299 q^{4} -190.875i q^{5} -181.074i q^{6} -80.4975 q^{8} -243.000 q^{9} +O(q^{10})\) \(q-11.6159 q^{2} +15.5885i q^{3} +70.9299 q^{4} -190.875i q^{5} -181.074i q^{6} -80.4975 q^{8} -243.000 q^{9} +2217.19i q^{10} -2055.10 q^{11} +1105.69i q^{12} +3059.97i q^{13} +2975.44 q^{15} -3604.46 q^{16} -2850.24i q^{17} +2822.67 q^{18} -3949.13i q^{19} -13538.7i q^{20} +23872.0 q^{22} +660.543 q^{23} -1254.83i q^{24} -20808.1 q^{25} -35544.4i q^{26} -3788.00i q^{27} -9282.66 q^{29} -34562.5 q^{30} -2801.72i q^{31} +47021.0 q^{32} -32035.9i q^{33} +33108.2i q^{34} -17236.0 q^{36} -36932.5 q^{37} +45872.8i q^{38} -47700.1 q^{39} +15364.9i q^{40} -67941.3i q^{41} +12336.3 q^{43} -145768. q^{44} +46382.5i q^{45} -7672.83 q^{46} +147613. i q^{47} -56188.0i q^{48} +241705. q^{50} +44430.9 q^{51} +217043. i q^{52} +219635. q^{53} +44001.1i q^{54} +392267. i q^{55} +61560.8 q^{57} +107827. q^{58} -192518. i q^{59} +211048. q^{60} +332836. i q^{61} +32544.5i q^{62} -315508. q^{64} +584069. q^{65} +372127. i q^{66} +349184. q^{67} -202168. i q^{68} +10296.9i q^{69} +305650. q^{71} +19560.9 q^{72} +236498. i q^{73} +429005. q^{74} -324366. i q^{75} -280111. i q^{76} +554082. q^{78} -586437. q^{79} +688000. i q^{80} +59049.0 q^{81} +789201. i q^{82} +106377. i q^{83} -544039. q^{85} -143298. q^{86} -144702. i q^{87} +165431. q^{88} +13057.9i q^{89} -538776. i q^{90} +46852.3 q^{92} +43674.4 q^{93} -1.71466e6i q^{94} -753788. q^{95} +732985. i q^{96} +205209. i q^{97} +499390. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 10 q^{2} + 346 q^{4} + 3326 q^{8} - 1944 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 10 q^{2} + 346 q^{4} + 3326 q^{8} - 1944 q^{9} + 628 q^{11} + 5292 q^{15} + 25442 q^{16} - 2430 q^{18} + 86106 q^{22} - 7856 q^{23} + 34076 q^{25} - 8300 q^{29} - 61398 q^{30} + 372414 q^{32} - 84078 q^{36} - 129412 q^{37} - 58212 q^{39} + 45740 q^{43} - 185058 q^{44} + 223008 q^{46} + 967216 q^{50} - 99576 q^{51} + 1081948 q^{53} + 328212 q^{57} - 1079598 q^{58} + 292734 q^{60} + 2378626 q^{64} + 828408 q^{65} + 2317804 q^{67} + 1442344 q^{71} - 808218 q^{72} + 865880 q^{74} - 222588 q^{78} - 1222904 q^{79} + 472392 q^{81} - 275112 q^{85} - 1632448 q^{86} + 732882 q^{88} + 678720 q^{92} - 1611144 q^{93} + 1183584 q^{95} - 152604 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).

\(n\) \(50\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −11.6159 −1.45199 −0.725996 0.687699i \(-0.758621\pi\)
−0.725996 + 0.687699i \(0.758621\pi\)
\(3\) 15.5885i 0.577350i
\(4\) 70.9299 1.10828
\(5\) − 190.875i − 1.52700i −0.645810 0.763498i \(-0.723480\pi\)
0.645810 0.763498i \(-0.276520\pi\)
\(6\) − 181.074i − 0.838308i
\(7\) 0 0
\(8\) −80.4975 −0.157222
\(9\) −243.000 −0.333333
\(10\) 2217.19i 2.21719i
\(11\) −2055.10 −1.54403 −0.772015 0.635604i \(-0.780751\pi\)
−0.772015 + 0.635604i \(0.780751\pi\)
\(12\) 1105.69i 0.639866i
\(13\) 3059.97i 1.39279i 0.717657 + 0.696396i \(0.245214\pi\)
−0.717657 + 0.696396i \(0.754786\pi\)
\(14\) 0 0
\(15\) 2975.44 0.881612
\(16\) −3604.46 −0.879995
\(17\) − 2850.24i − 0.580143i −0.957005 0.290072i \(-0.906321\pi\)
0.957005 0.290072i \(-0.0936791\pi\)
\(18\) 2822.67 0.483997
\(19\) − 3949.13i − 0.575759i −0.957667 0.287879i \(-0.907050\pi\)
0.957667 0.287879i \(-0.0929502\pi\)
\(20\) − 13538.7i − 1.69234i
\(21\) 0 0
\(22\) 23872.0 2.24192
\(23\) 660.543 0.0542897 0.0271449 0.999632i \(-0.491358\pi\)
0.0271449 + 0.999632i \(0.491358\pi\)
\(24\) − 1254.83i − 0.0907720i
\(25\) −20808.1 −1.33172
\(26\) − 35544.4i − 2.02232i
\(27\) − 3788.00i − 0.192450i
\(28\) 0 0
\(29\) −9282.66 −0.380609 −0.190304 0.981725i \(-0.560947\pi\)
−0.190304 + 0.981725i \(0.560947\pi\)
\(30\) −34562.5 −1.28009
\(31\) − 2801.72i − 0.0940457i −0.998894 0.0470229i \(-0.985027\pi\)
0.998894 0.0470229i \(-0.0149734\pi\)
\(32\) 47021.0 1.43497
\(33\) − 32035.9i − 0.891446i
\(34\) 33108.2i 0.842363i
\(35\) 0 0
\(36\) −17236.0 −0.369427
\(37\) −36932.5 −0.729127 −0.364563 0.931179i \(-0.618782\pi\)
−0.364563 + 0.931179i \(0.618782\pi\)
\(38\) 45872.8i 0.835997i
\(39\) −47700.1 −0.804129
\(40\) 15364.9i 0.240077i
\(41\) − 67941.3i − 0.985785i −0.870090 0.492892i \(-0.835940\pi\)
0.870090 0.492892i \(-0.164060\pi\)
\(42\) 0 0
\(43\) 12336.3 0.155160 0.0775799 0.996986i \(-0.475281\pi\)
0.0775799 + 0.996986i \(0.475281\pi\)
\(44\) −145768. −1.71122
\(45\) 46382.5i 0.508999i
\(46\) −7672.83 −0.0788283
\(47\) 147613.i 1.42177i 0.703308 + 0.710886i \(0.251706\pi\)
−0.703308 + 0.710886i \(0.748294\pi\)
\(48\) − 56188.0i − 0.508066i
\(49\) 0 0
\(50\) 241705. 1.93364
\(51\) 44430.9 0.334946
\(52\) 217043.i 1.54360i
\(53\) 219635. 1.47528 0.737641 0.675194i \(-0.235940\pi\)
0.737641 + 0.675194i \(0.235940\pi\)
\(54\) 44001.1i 0.279436i
\(55\) 392267.i 2.35773i
\(56\) 0 0
\(57\) 61560.8 0.332414
\(58\) 107827. 0.552641
\(59\) − 192518.i − 0.937377i −0.883363 0.468689i \(-0.844727\pi\)
0.883363 0.468689i \(-0.155273\pi\)
\(60\) 211048. 0.977073
\(61\) 332836.i 1.46636i 0.680035 + 0.733180i \(0.261965\pi\)
−0.680035 + 0.733180i \(0.738035\pi\)
\(62\) 32544.5i 0.136554i
\(63\) 0 0
\(64\) −315508. −1.20357
\(65\) 584069. 2.12679
\(66\) 372127.i 1.29437i
\(67\) 349184. 1.16099 0.580496 0.814263i \(-0.302859\pi\)
0.580496 + 0.814263i \(0.302859\pi\)
\(68\) − 202168.i − 0.642961i
\(69\) 10296.9i 0.0313442i
\(70\) 0 0
\(71\) 305650. 0.853984 0.426992 0.904255i \(-0.359573\pi\)
0.426992 + 0.904255i \(0.359573\pi\)
\(72\) 19560.9 0.0524073
\(73\) 236498.i 0.607939i 0.952682 + 0.303969i \(0.0983120\pi\)
−0.952682 + 0.303969i \(0.901688\pi\)
\(74\) 429005. 1.05869
\(75\) − 324366.i − 0.768867i
\(76\) − 280111.i − 0.638102i
\(77\) 0 0
\(78\) 554082. 1.16759
\(79\) −586437. −1.18943 −0.594716 0.803936i \(-0.702736\pi\)
−0.594716 + 0.803936i \(0.702736\pi\)
\(80\) 688000.i 1.34375i
\(81\) 59049.0 0.111111
\(82\) 789201.i 1.43135i
\(83\) 106377.i 0.186044i 0.995664 + 0.0930218i \(0.0296526\pi\)
−0.995664 + 0.0930218i \(0.970347\pi\)
\(84\) 0 0
\(85\) −544039. −0.885876
\(86\) −143298. −0.225291
\(87\) − 144702.i − 0.219744i
\(88\) 165431. 0.242755
\(89\) 13057.9i 0.0185227i 0.999957 + 0.00926134i \(0.00294802\pi\)
−0.999957 + 0.00926134i \(0.997052\pi\)
\(90\) − 538776.i − 0.739062i
\(91\) 0 0
\(92\) 46852.3 0.0601682
\(93\) 43674.4 0.0542973
\(94\) − 1.71466e6i − 2.06440i
\(95\) −753788. −0.879181
\(96\) 732985.i 0.828479i
\(97\) 205209.i 0.224844i 0.993661 + 0.112422i \(0.0358609\pi\)
−0.993661 + 0.112422i \(0.964139\pi\)
\(98\) 0 0
\(99\) 499390. 0.514677
\(100\) −1.47592e6 −1.47592
\(101\) 1.40110e6i 1.35989i 0.733262 + 0.679946i \(0.237997\pi\)
−0.733262 + 0.679946i \(0.762003\pi\)
\(102\) −516106. −0.486339
\(103\) 517642.i 0.473716i 0.971544 + 0.236858i \(0.0761176\pi\)
−0.971544 + 0.236858i \(0.923882\pi\)
\(104\) − 246320.i − 0.218977i
\(105\) 0 0
\(106\) −2.55127e6 −2.14210
\(107\) 646744. 0.527936 0.263968 0.964531i \(-0.414969\pi\)
0.263968 + 0.964531i \(0.414969\pi\)
\(108\) − 268682.i − 0.213289i
\(109\) −687262. −0.530692 −0.265346 0.964153i \(-0.585486\pi\)
−0.265346 + 0.964153i \(0.585486\pi\)
\(110\) − 4.55655e6i − 3.42340i
\(111\) − 575720.i − 0.420962i
\(112\) 0 0
\(113\) 2.11668e6 1.46697 0.733483 0.679708i \(-0.237893\pi\)
0.733483 + 0.679708i \(0.237893\pi\)
\(114\) −715086. −0.482663
\(115\) − 126081.i − 0.0829002i
\(116\) −658419. −0.421821
\(117\) − 743572.i − 0.464264i
\(118\) 2.23627e6i 1.36106i
\(119\) 0 0
\(120\) −239516. −0.138609
\(121\) 2.45189e6 1.38403
\(122\) − 3.86620e6i − 2.12914i
\(123\) 1.05910e6 0.569143
\(124\) − 198725.i − 0.104229i
\(125\) 989318.i 0.506531i
\(126\) 0 0
\(127\) 3.29012e6 1.60620 0.803102 0.595841i \(-0.203181\pi\)
0.803102 + 0.595841i \(0.203181\pi\)
\(128\) 655570. 0.312600
\(129\) 192304.i 0.0895816i
\(130\) −6.78451e6 −3.08808
\(131\) 223347.i 0.0993497i 0.998765 + 0.0496748i \(0.0158185\pi\)
−0.998765 + 0.0496748i \(0.984181\pi\)
\(132\) − 2.27230e6i − 0.987972i
\(133\) 0 0
\(134\) −4.05609e6 −1.68575
\(135\) −723032. −0.293871
\(136\) 229438.i 0.0912111i
\(137\) −3.17050e6 −1.23301 −0.616504 0.787352i \(-0.711452\pi\)
−0.616504 + 0.787352i \(0.711452\pi\)
\(138\) − 119608.i − 0.0455115i
\(139\) 774740.i 0.288477i 0.989543 + 0.144239i \(0.0460733\pi\)
−0.989543 + 0.144239i \(0.953927\pi\)
\(140\) 0 0
\(141\) −2.30105e6 −0.820860
\(142\) −3.55042e6 −1.23998
\(143\) − 6.28855e6i − 2.15051i
\(144\) 875884. 0.293332
\(145\) 1.77182e6i 0.581188i
\(146\) − 2.74715e6i − 0.882722i
\(147\) 0 0
\(148\) −2.61962e6 −0.808077
\(149\) 3.50921e6 1.06084 0.530421 0.847734i \(-0.322034\pi\)
0.530421 + 0.847734i \(0.322034\pi\)
\(150\) 3.76781e6i 1.11639i
\(151\) 6.45201e6 1.87398 0.936989 0.349360i \(-0.113601\pi\)
0.936989 + 0.349360i \(0.113601\pi\)
\(152\) 317895.i 0.0905218i
\(153\) 692609.i 0.193381i
\(154\) 0 0
\(155\) −534776. −0.143607
\(156\) −3.38337e6 −0.891200
\(157\) 4.68297e6i 1.21010i 0.796187 + 0.605051i \(0.206847\pi\)
−0.796187 + 0.605051i \(0.793153\pi\)
\(158\) 6.81201e6 1.72705
\(159\) 3.42378e6i 0.851754i
\(160\) − 8.97512e6i − 2.19119i
\(161\) 0 0
\(162\) −685909. −0.161332
\(163\) 5.18233e6 1.19664 0.598318 0.801259i \(-0.295836\pi\)
0.598318 + 0.801259i \(0.295836\pi\)
\(164\) − 4.81907e6i − 1.09253i
\(165\) −6.11484e6 −1.36123
\(166\) − 1.23567e6i − 0.270134i
\(167\) 1.56630e6i 0.336298i 0.985762 + 0.168149i \(0.0537790\pi\)
−0.985762 + 0.168149i \(0.946221\pi\)
\(168\) 0 0
\(169\) −4.53658e6 −0.939871
\(170\) 6.31952e6 1.28629
\(171\) 959638.i 0.191920i
\(172\) 875012. 0.171961
\(173\) − 1.24462e6i − 0.240381i −0.992751 0.120190i \(-0.961649\pi\)
0.992751 0.120190i \(-0.0383505\pi\)
\(174\) 1.68085e6i 0.319067i
\(175\) 0 0
\(176\) 7.40754e6 1.35874
\(177\) 3.00105e6 0.541195
\(178\) − 151680.i − 0.0268948i
\(179\) 2.25600e6 0.393350 0.196675 0.980469i \(-0.436986\pi\)
0.196675 + 0.980469i \(0.436986\pi\)
\(180\) 3.28991e6i 0.564113i
\(181\) 7.97198e6i 1.34441i 0.740367 + 0.672203i \(0.234652\pi\)
−0.740367 + 0.672203i \(0.765348\pi\)
\(182\) 0 0
\(183\) −5.18839e6 −0.846603
\(184\) −53172.1 −0.00853553
\(185\) 7.04947e6i 1.11337i
\(186\) −507319. −0.0788393
\(187\) 5.85755e6i 0.895759i
\(188\) 1.04701e7i 1.57572i
\(189\) 0 0
\(190\) 8.75595e6 1.27656
\(191\) 4.26174e6 0.611627 0.305814 0.952091i \(-0.401072\pi\)
0.305814 + 0.952091i \(0.401072\pi\)
\(192\) − 4.91828e6i − 0.694879i
\(193\) −145773. −0.0202771 −0.0101386 0.999949i \(-0.503227\pi\)
−0.0101386 + 0.999949i \(0.503227\pi\)
\(194\) − 2.38370e6i − 0.326472i
\(195\) 9.10474e6i 1.22790i
\(196\) 0 0
\(197\) −7.21313e6 −0.943463 −0.471732 0.881742i \(-0.656371\pi\)
−0.471732 + 0.881742i \(0.656371\pi\)
\(198\) −5.80089e6 −0.747306
\(199\) − 1.76480e6i − 0.223942i −0.993711 0.111971i \(-0.964284\pi\)
0.993711 0.111971i \(-0.0357164\pi\)
\(200\) 1.67500e6 0.209375
\(201\) 5.44323e6i 0.670299i
\(202\) − 1.62751e7i − 1.97455i
\(203\) 0 0
\(204\) 3.15148e6 0.371214
\(205\) −1.29683e7 −1.50529
\(206\) − 6.01290e6i − 0.687832i
\(207\) −160512. −0.0180966
\(208\) − 1.10295e7i − 1.22565i
\(209\) 8.11587e6i 0.888989i
\(210\) 0 0
\(211\) 1.37033e7 1.45874 0.729369 0.684121i \(-0.239814\pi\)
0.729369 + 0.684121i \(0.239814\pi\)
\(212\) 1.55787e7 1.63502
\(213\) 4.76462e6i 0.493048i
\(214\) −7.51254e6 −0.766558
\(215\) − 2.35468e6i − 0.236928i
\(216\) 304924.i 0.0302573i
\(217\) 0 0
\(218\) 7.98318e6 0.770560
\(219\) −3.68665e6 −0.350993
\(220\) 2.78235e7i 2.61302i
\(221\) 8.72165e6 0.808019
\(222\) 6.68753e6i 0.611233i
\(223\) − 2.15029e7i − 1.93902i −0.245055 0.969509i \(-0.578806\pi\)
0.245055 0.969509i \(-0.421194\pi\)
\(224\) 0 0
\(225\) 5.05636e6 0.443906
\(226\) −2.45872e7 −2.13002
\(227\) − 3.93130e6i − 0.336093i −0.985779 0.168046i \(-0.946254\pi\)
0.985779 0.168046i \(-0.0537458\pi\)
\(228\) 4.36650e6 0.368408
\(229\) 562326.i 0.0468254i 0.999726 + 0.0234127i \(0.00745317\pi\)
−0.999726 + 0.0234127i \(0.992547\pi\)
\(230\) 1.46455e6i 0.120370i
\(231\) 0 0
\(232\) 747232. 0.0598400
\(233\) −1.63331e7 −1.29122 −0.645612 0.763666i \(-0.723398\pi\)
−0.645612 + 0.763666i \(0.723398\pi\)
\(234\) 8.63728e6i 0.674108i
\(235\) 2.81755e7 2.17104
\(236\) − 1.36553e7i − 1.03888i
\(237\) − 9.14164e6i − 0.686719i
\(238\) 0 0
\(239\) −1.96132e7 −1.43666 −0.718331 0.695702i \(-0.755094\pi\)
−0.718331 + 0.695702i \(0.755094\pi\)
\(240\) −1.07249e7 −0.775814
\(241\) − 3.59901e6i − 0.257118i −0.991702 0.128559i \(-0.958965\pi\)
0.991702 0.128559i \(-0.0410351\pi\)
\(242\) −2.84810e7 −2.00960
\(243\) 920483.i 0.0641500i
\(244\) 2.36080e7i 1.62514i
\(245\) 0 0
\(246\) −1.23024e7 −0.826391
\(247\) 1.20842e7 0.801912
\(248\) 225531.i 0.0147860i
\(249\) −1.65826e6 −0.107412
\(250\) − 1.14918e7i − 0.735478i
\(251\) − 6.49855e6i − 0.410956i −0.978662 0.205478i \(-0.934125\pi\)
0.978662 0.205478i \(-0.0658749\pi\)
\(252\) 0 0
\(253\) −1.35749e6 −0.0838250
\(254\) −3.82178e7 −2.33220
\(255\) − 8.48073e6i − 0.511461i
\(256\) 1.25774e7 0.749673
\(257\) 2.85795e6i 0.168366i 0.996450 + 0.0841832i \(0.0268281\pi\)
−0.996450 + 0.0841832i \(0.973172\pi\)
\(258\) − 2.23379e6i − 0.130072i
\(259\) 0 0
\(260\) 4.14280e7 2.35708
\(261\) 2.25569e6 0.126870
\(262\) − 2.59439e6i − 0.144255i
\(263\) −4.99827e6 −0.274759 −0.137380 0.990518i \(-0.543868\pi\)
−0.137380 + 0.990518i \(0.543868\pi\)
\(264\) 2.57881e6i 0.140155i
\(265\) − 4.19228e7i − 2.25275i
\(266\) 0 0
\(267\) −203553. −0.0106941
\(268\) 2.47676e7 1.28670
\(269\) 2.04199e7i 1.04905i 0.851394 + 0.524527i \(0.175758\pi\)
−0.851394 + 0.524527i \(0.824242\pi\)
\(270\) 8.39869e6 0.426698
\(271\) 1.23420e7i 0.620124i 0.950716 + 0.310062i \(0.100350\pi\)
−0.950716 + 0.310062i \(0.899650\pi\)
\(272\) 1.02736e7i 0.510523i
\(273\) 0 0
\(274\) 3.68283e7 1.79032
\(275\) 4.27628e7 2.05621
\(276\) 730355.i 0.0347381i
\(277\) −2.08302e7 −0.980062 −0.490031 0.871705i \(-0.663015\pi\)
−0.490031 + 0.871705i \(0.663015\pi\)
\(278\) − 8.99933e6i − 0.418867i
\(279\) 680817.i 0.0313486i
\(280\) 0 0
\(281\) 2.10143e7 0.947100 0.473550 0.880767i \(-0.342972\pi\)
0.473550 + 0.880767i \(0.342972\pi\)
\(282\) 2.67289e7 1.19188
\(283\) − 1.04239e7i − 0.459908i −0.973201 0.229954i \(-0.926142\pi\)
0.973201 0.229954i \(-0.0738576\pi\)
\(284\) 2.16798e7 0.946454
\(285\) − 1.17504e7i − 0.507595i
\(286\) 7.30474e7i 3.12253i
\(287\) 0 0
\(288\) −1.14261e7 −0.478323
\(289\) 1.60137e7 0.663434
\(290\) − 2.05814e7i − 0.843880i
\(291\) −3.19890e6 −0.129814
\(292\) 1.67748e7i 0.673766i
\(293\) 2.85328e7i 1.13433i 0.823603 + 0.567167i \(0.191961\pi\)
−0.823603 + 0.567167i \(0.808039\pi\)
\(294\) 0 0
\(295\) −3.67467e7 −1.43137
\(296\) 2.97297e6 0.114635
\(297\) 7.78472e6i 0.297149i
\(298\) −4.07628e7 −1.54033
\(299\) 2.02124e6i 0.0756143i
\(300\) − 2.30072e7i − 0.852120i
\(301\) 0 0
\(302\) −7.49461e7 −2.72100
\(303\) −2.18410e7 −0.785134
\(304\) 1.42345e7i 0.506665i
\(305\) 6.35299e7 2.23912
\(306\) − 8.04530e6i − 0.280788i
\(307\) 1.02269e7i 0.353450i 0.984260 + 0.176725i \(0.0565503\pi\)
−0.984260 + 0.176725i \(0.943450\pi\)
\(308\) 0 0
\(309\) −8.06925e6 −0.273500
\(310\) 6.21192e6 0.208517
\(311\) 1.57422e7i 0.523339i 0.965158 + 0.261669i \(0.0842730\pi\)
−0.965158 + 0.261669i \(0.915727\pi\)
\(312\) 3.83974e6 0.126427
\(313\) 5.55274e7i 1.81082i 0.424543 + 0.905408i \(0.360435\pi\)
−0.424543 + 0.905408i \(0.639565\pi\)
\(314\) − 5.43970e7i − 1.75706i
\(315\) 0 0
\(316\) −4.15959e7 −1.31822
\(317\) −6.03079e6 −0.189320 −0.0946600 0.995510i \(-0.530176\pi\)
−0.0946600 + 0.995510i \(0.530176\pi\)
\(318\) − 3.97704e7i − 1.23674i
\(319\) 1.90768e7 0.587671
\(320\) 6.02224e7i 1.83784i
\(321\) 1.00817e7i 0.304804i
\(322\) 0 0
\(323\) −1.12560e7 −0.334022
\(324\) 4.18834e6 0.123142
\(325\) − 6.36720e7i − 1.85481i
\(326\) −6.01976e7 −1.73750
\(327\) − 1.07133e7i − 0.306395i
\(328\) 5.46910e6i 0.154987i
\(329\) 0 0
\(330\) 7.10295e7 1.97650
\(331\) 9.76554e6 0.269285 0.134643 0.990894i \(-0.457011\pi\)
0.134643 + 0.990894i \(0.457011\pi\)
\(332\) 7.54533e6i 0.206188i
\(333\) 8.97459e6 0.243042
\(334\) − 1.81940e7i − 0.488302i
\(335\) − 6.66502e7i − 1.77283i
\(336\) 0 0
\(337\) −5.70025e7 −1.48938 −0.744688 0.667412i \(-0.767402\pi\)
−0.744688 + 0.667412i \(0.767402\pi\)
\(338\) 5.26966e7 1.36469
\(339\) 3.29958e7i 0.846953i
\(340\) −3.85886e7 −0.981799
\(341\) 5.75782e6i 0.145209i
\(342\) − 1.11471e7i − 0.278666i
\(343\) 0 0
\(344\) −993041. −0.0243945
\(345\) 1.96541e6 0.0478625
\(346\) 1.44575e7i 0.349031i
\(347\) −3.67355e7 −0.879219 −0.439610 0.898189i \(-0.644883\pi\)
−0.439610 + 0.898189i \(0.644883\pi\)
\(348\) − 1.02637e7i − 0.243538i
\(349\) 6.58421e7i 1.54891i 0.632626 + 0.774457i \(0.281977\pi\)
−0.632626 + 0.774457i \(0.718023\pi\)
\(350\) 0 0
\(351\) 1.15911e7 0.268043
\(352\) −9.66331e7 −2.21563
\(353\) − 3.18706e7i − 0.724546i −0.932072 0.362273i \(-0.882001\pi\)
0.932072 0.362273i \(-0.117999\pi\)
\(354\) −3.48600e7 −0.785811
\(355\) − 5.83409e7i − 1.30403i
\(356\) 926197.i 0.0205283i
\(357\) 0 0
\(358\) −2.62055e7 −0.571141
\(359\) −3.82257e7 −0.826174 −0.413087 0.910691i \(-0.635550\pi\)
−0.413087 + 0.910691i \(0.635550\pi\)
\(360\) − 3.73368e6i − 0.0800257i
\(361\) 3.14503e7 0.668502
\(362\) − 9.26020e7i − 1.95207i
\(363\) 3.82212e7i 0.799070i
\(364\) 0 0
\(365\) 4.51415e7 0.928320
\(366\) 6.02681e7 1.22926
\(367\) − 9.73841e7i − 1.97011i −0.172249 0.985053i \(-0.555104\pi\)
0.172249 0.985053i \(-0.444896\pi\)
\(368\) −2.38090e6 −0.0477747
\(369\) 1.65097e7i 0.328595i
\(370\) − 8.18861e7i − 1.61661i
\(371\) 0 0
\(372\) 3.09782e6 0.0601766
\(373\) 2.10427e7 0.405484 0.202742 0.979232i \(-0.435015\pi\)
0.202742 + 0.979232i \(0.435015\pi\)
\(374\) − 6.80409e7i − 1.30063i
\(375\) −1.54219e7 −0.292446
\(376\) − 1.18824e7i − 0.223533i
\(377\) − 2.84046e7i − 0.530109i
\(378\) 0 0
\(379\) −4.11794e7 −0.756418 −0.378209 0.925720i \(-0.623460\pi\)
−0.378209 + 0.925720i \(0.623460\pi\)
\(380\) −5.34661e7 −0.974379
\(381\) 5.12879e7i 0.927343i
\(382\) −4.95041e7 −0.888078
\(383\) − 1.52188e7i − 0.270884i −0.990785 0.135442i \(-0.956755\pi\)
0.990785 0.135442i \(-0.0432454\pi\)
\(384\) 1.02193e7i 0.180480i
\(385\) 0 0
\(386\) 1.69329e6 0.0294422
\(387\) −2.99772e6 −0.0517199
\(388\) 1.45555e7i 0.249191i
\(389\) −6.74995e7 −1.14671 −0.573353 0.819309i \(-0.694357\pi\)
−0.573353 + 0.819309i \(0.694357\pi\)
\(390\) − 1.05760e8i − 1.78290i
\(391\) − 1.88271e6i − 0.0314958i
\(392\) 0 0
\(393\) −3.48164e6 −0.0573596
\(394\) 8.37872e7 1.36990
\(395\) 1.11936e8i 1.81626i
\(396\) 3.54217e7 0.570406
\(397\) 9.40155e6i 0.150255i 0.997174 + 0.0751273i \(0.0239363\pi\)
−0.997174 + 0.0751273i \(0.976064\pi\)
\(398\) 2.04998e7i 0.325162i
\(399\) 0 0
\(400\) 7.50019e7 1.17190
\(401\) −5.54972e7 −0.860673 −0.430336 0.902669i \(-0.641605\pi\)
−0.430336 + 0.902669i \(0.641605\pi\)
\(402\) − 6.32282e7i − 0.973269i
\(403\) 8.57315e6 0.130986
\(404\) 9.93798e7i 1.50714i
\(405\) − 1.12709e7i − 0.169666i
\(406\) 0 0
\(407\) 7.59001e7 1.12579
\(408\) −3.57658e6 −0.0526608
\(409\) − 1.63471e7i − 0.238931i −0.992838 0.119465i \(-0.961882\pi\)
0.992838 0.119465i \(-0.0381180\pi\)
\(410\) 1.50638e8 2.18567
\(411\) − 4.94232e7i − 0.711877i
\(412\) 3.67163e7i 0.525010i
\(413\) 0 0
\(414\) 1.86450e6 0.0262761
\(415\) 2.03047e7 0.284088
\(416\) 1.43883e8i 1.99861i
\(417\) −1.20770e7 −0.166552
\(418\) − 9.42734e7i − 1.29080i
\(419\) 3.75022e7i 0.509818i 0.966965 + 0.254909i \(0.0820455\pi\)
−0.966965 + 0.254909i \(0.917955\pi\)
\(420\) 0 0
\(421\) 9.17501e7 1.22959 0.614795 0.788687i \(-0.289239\pi\)
0.614795 + 0.788687i \(0.289239\pi\)
\(422\) −1.59176e8 −2.11807
\(423\) − 3.58699e7i − 0.473924i
\(424\) −1.76801e7 −0.231946
\(425\) 5.93081e7i 0.772587i
\(426\) − 5.53455e7i − 0.715902i
\(427\) 0 0
\(428\) 4.58735e7 0.585101
\(429\) 9.80288e7 1.24160
\(430\) 2.73518e7i 0.344018i
\(431\) −1.37807e8 −1.72123 −0.860616 0.509255i \(-0.829921\pi\)
−0.860616 + 0.509255i \(0.829921\pi\)
\(432\) 1.36537e7i 0.169355i
\(433\) − 2.00324e7i − 0.246757i −0.992360 0.123378i \(-0.960627\pi\)
0.992360 0.123378i \(-0.0393729\pi\)
\(434\) 0 0
\(435\) −2.76200e7 −0.335549
\(436\) −4.87474e7 −0.588155
\(437\) − 2.60857e6i − 0.0312578i
\(438\) 4.28238e7 0.509640
\(439\) 8.61025e7i 1.01771i 0.860854 + 0.508853i \(0.169930\pi\)
−0.860854 + 0.508853i \(0.830070\pi\)
\(440\) − 3.15765e7i − 0.370686i
\(441\) 0 0
\(442\) −1.01310e8 −1.17324
\(443\) −8.07874e6 −0.0929250 −0.0464625 0.998920i \(-0.514795\pi\)
−0.0464625 + 0.998920i \(0.514795\pi\)
\(444\) − 4.08358e7i − 0.466543i
\(445\) 2.49242e6 0.0282840
\(446\) 2.49776e8i 2.81544i
\(447\) 5.47032e7i 0.612477i
\(448\) 0 0
\(449\) −5.16486e7 −0.570584 −0.285292 0.958441i \(-0.592091\pi\)
−0.285292 + 0.958441i \(0.592091\pi\)
\(450\) −5.87344e7 −0.644547
\(451\) 1.39626e8i 1.52208i
\(452\) 1.50136e8 1.62581
\(453\) 1.00577e8i 1.08194i
\(454\) 4.56657e7i 0.488004i
\(455\) 0 0
\(456\) −4.95549e6 −0.0522628
\(457\) −3.99394e7 −0.418459 −0.209229 0.977867i \(-0.567096\pi\)
−0.209229 + 0.977867i \(0.567096\pi\)
\(458\) − 6.53194e6i − 0.0679901i
\(459\) −1.07967e7 −0.111649
\(460\) − 8.94291e6i − 0.0918767i
\(461\) 1.40914e7i 0.143831i 0.997411 + 0.0719153i \(0.0229111\pi\)
−0.997411 + 0.0719153i \(0.977089\pi\)
\(462\) 0 0
\(463\) 1.25200e8 1.26142 0.630712 0.776017i \(-0.282763\pi\)
0.630712 + 0.776017i \(0.282763\pi\)
\(464\) 3.34590e7 0.334934
\(465\) − 8.33633e6i − 0.0829118i
\(466\) 1.89725e8 1.87485
\(467\) − 1.03527e8i − 1.01649i −0.861214 0.508243i \(-0.830295\pi\)
0.861214 0.508243i \(-0.169705\pi\)
\(468\) − 5.27415e7i − 0.514535i
\(469\) 0 0
\(470\) −3.27284e8 −3.15233
\(471\) −7.30002e7 −0.698653
\(472\) 1.54972e7i 0.147376i
\(473\) −2.53524e7 −0.239571
\(474\) 1.06189e8i 0.997111i
\(475\) 8.21738e7i 0.766747i
\(476\) 0 0
\(477\) −5.33714e7 −0.491760
\(478\) 2.27826e8 2.08602
\(479\) 7.79771e7i 0.709513i 0.934959 + 0.354757i \(0.115436\pi\)
−0.934959 + 0.354757i \(0.884564\pi\)
\(480\) 1.39908e8 1.26508
\(481\) − 1.13012e8i − 1.01552i
\(482\) 4.18058e7i 0.373333i
\(483\) 0 0
\(484\) 1.73913e8 1.53389
\(485\) 3.91692e7 0.343336
\(486\) − 1.06923e7i − 0.0931453i
\(487\) 1.42798e8 1.23634 0.618168 0.786046i \(-0.287875\pi\)
0.618168 + 0.786046i \(0.287875\pi\)
\(488\) − 2.67925e7i − 0.230544i
\(489\) 8.07845e7i 0.690878i
\(490\) 0 0
\(491\) 5.07818e7 0.429006 0.214503 0.976723i \(-0.431187\pi\)
0.214503 + 0.976723i \(0.431187\pi\)
\(492\) 7.51218e7 0.630770
\(493\) 2.64578e7i 0.220807i
\(494\) −1.40369e8 −1.16437
\(495\) − 9.53209e7i − 0.785909i
\(496\) 1.00987e7i 0.0827598i
\(497\) 0 0
\(498\) 1.92622e7 0.155962
\(499\) 1.26504e8 1.01813 0.509065 0.860728i \(-0.329991\pi\)
0.509065 + 0.860728i \(0.329991\pi\)
\(500\) 7.01722e7i 0.561378i
\(501\) −2.44161e7 −0.194162
\(502\) 7.54868e7i 0.596705i
\(503\) 7.36652e6i 0.0578840i 0.999581 + 0.0289420i \(0.00921381\pi\)
−0.999581 + 0.0289420i \(0.990786\pi\)
\(504\) 0 0
\(505\) 2.67434e8 2.07655
\(506\) 1.57685e7 0.121713
\(507\) − 7.07183e7i − 0.542635i
\(508\) 2.33368e8 1.78012
\(509\) − 2.29433e8i − 1.73981i −0.493217 0.869906i \(-0.664179\pi\)
0.493217 0.869906i \(-0.335821\pi\)
\(510\) 9.85116e7i 0.742637i
\(511\) 0 0
\(512\) −1.88055e8 −1.40112
\(513\) −1.49593e7 −0.110805
\(514\) − 3.31978e7i − 0.244466i
\(515\) 9.88047e7 0.723363
\(516\) 1.36401e7i 0.0992814i
\(517\) − 3.03359e8i − 2.19526i
\(518\) 0 0
\(519\) 1.94018e7 0.138784
\(520\) −4.70161e7 −0.334377
\(521\) 1.07368e8i 0.759210i 0.925149 + 0.379605i \(0.123940\pi\)
−0.925149 + 0.379605i \(0.876060\pi\)
\(522\) −2.62019e7 −0.184214
\(523\) 1.50523e8i 1.05220i 0.850423 + 0.526099i \(0.176346\pi\)
−0.850423 + 0.526099i \(0.823654\pi\)
\(524\) 1.58420e7i 0.110107i
\(525\) 0 0
\(526\) 5.80595e7 0.398948
\(527\) −7.98557e6 −0.0545600
\(528\) 1.15472e8i 0.784469i
\(529\) −1.47600e8 −0.997053
\(530\) 4.86973e8i 3.27097i
\(531\) 4.67818e7i 0.312459i
\(532\) 0 0
\(533\) 2.07898e8 1.37299
\(534\) 2.36445e6 0.0155277
\(535\) − 1.23447e8i − 0.806156i
\(536\) −2.81084e7 −0.182533
\(537\) 3.51675e7i 0.227101i
\(538\) − 2.37197e8i − 1.52322i
\(539\) 0 0
\(540\) −5.12846e7 −0.325691
\(541\) −6.05747e7 −0.382560 −0.191280 0.981535i \(-0.561264\pi\)
−0.191280 + 0.981535i \(0.561264\pi\)
\(542\) − 1.43364e8i − 0.900416i
\(543\) −1.24271e8 −0.776193
\(544\) − 1.34021e8i − 0.832487i
\(545\) 1.31181e8i 0.810365i
\(546\) 0 0
\(547\) −2.68955e8 −1.64330 −0.821651 0.569992i \(-0.806946\pi\)
−0.821651 + 0.569992i \(0.806946\pi\)
\(548\) −2.24883e8 −1.36652
\(549\) − 8.08791e7i − 0.488786i
\(550\) −4.96730e8 −2.98560
\(551\) 3.66584e7i 0.219139i
\(552\) − 828871.i − 0.00492799i
\(553\) 0 0
\(554\) 2.41962e8 1.42304
\(555\) −1.09890e8 −0.642807
\(556\) 5.49522e7i 0.319714i
\(557\) 2.01772e8 1.16760 0.583800 0.811897i \(-0.301565\pi\)
0.583800 + 0.811897i \(0.301565\pi\)
\(558\) − 7.90832e6i − 0.0455179i
\(559\) 3.77486e7i 0.216105i
\(560\) 0 0
\(561\) −9.13101e7 −0.517166
\(562\) −2.44101e8 −1.37518
\(563\) − 1.76315e8i − 0.988019i −0.869456 0.494010i \(-0.835531\pi\)
0.869456 0.494010i \(-0.164469\pi\)
\(564\) −1.63213e8 −0.909743
\(565\) − 4.04020e8i − 2.24005i
\(566\) 1.21083e8i 0.667783i
\(567\) 0 0
\(568\) −2.46041e7 −0.134265
\(569\) 2.00736e8 1.08966 0.544828 0.838548i \(-0.316595\pi\)
0.544828 + 0.838548i \(0.316595\pi\)
\(570\) 1.36492e8i 0.737024i
\(571\) −4.46408e7 −0.239786 −0.119893 0.992787i \(-0.538255\pi\)
−0.119893 + 0.992787i \(0.538255\pi\)
\(572\) − 4.46046e8i − 2.38337i
\(573\) 6.64340e7i 0.353123i
\(574\) 0 0
\(575\) −1.37446e7 −0.0722986
\(576\) 7.66683e7 0.401189
\(577\) 2.58790e8i 1.34716i 0.739114 + 0.673581i \(0.235245\pi\)
−0.739114 + 0.673581i \(0.764755\pi\)
\(578\) −1.86014e8 −0.963301
\(579\) − 2.27238e6i − 0.0117070i
\(580\) 1.25675e8i 0.644119i
\(581\) 0 0
\(582\) 3.71582e7 0.188489
\(583\) −4.51374e8 −2.27788
\(584\) − 1.90375e7i − 0.0955812i
\(585\) −1.41929e8 −0.708930
\(586\) − 3.31435e8i − 1.64704i
\(587\) − 3.41902e8i − 1.69039i −0.534457 0.845196i \(-0.679484\pi\)
0.534457 0.845196i \(-0.320516\pi\)
\(588\) 0 0
\(589\) −1.10643e7 −0.0541476
\(590\) 4.26847e8 2.07834
\(591\) − 1.12442e8i − 0.544709i
\(592\) 1.33122e8 0.641628
\(593\) 1.47561e8i 0.707630i 0.935315 + 0.353815i \(0.115116\pi\)
−0.935315 + 0.353815i \(0.884884\pi\)
\(594\) − 9.04268e7i − 0.431458i
\(595\) 0 0
\(596\) 2.48908e8 1.17571
\(597\) 2.75105e7 0.129293
\(598\) − 2.34786e7i − 0.109791i
\(599\) −6.64923e7 −0.309379 −0.154690 0.987963i \(-0.549438\pi\)
−0.154690 + 0.987963i \(0.549438\pi\)
\(600\) 2.61107e7i 0.120883i
\(601\) − 3.71302e7i − 0.171042i −0.996336 0.0855212i \(-0.972744\pi\)
0.996336 0.0855212i \(-0.0272555\pi\)
\(602\) 0 0
\(603\) −8.48516e7 −0.386998
\(604\) 4.57641e8 2.07689
\(605\) − 4.68004e8i − 2.11341i
\(606\) 2.53703e8 1.14001
\(607\) 1.27170e8i 0.568616i 0.958733 + 0.284308i \(0.0917638\pi\)
−0.958733 + 0.284308i \(0.908236\pi\)
\(608\) − 1.85692e8i − 0.826195i
\(609\) 0 0
\(610\) −7.37959e8 −3.25119
\(611\) −4.51689e8 −1.98023
\(612\) 4.91267e7i 0.214320i
\(613\) −1.42995e8 −0.620783 −0.310391 0.950609i \(-0.600460\pi\)
−0.310391 + 0.950609i \(0.600460\pi\)
\(614\) − 1.18795e8i − 0.513206i
\(615\) − 2.02155e8i − 0.869079i
\(616\) 0 0
\(617\) −2.96115e8 −1.26068 −0.630339 0.776320i \(-0.717084\pi\)
−0.630339 + 0.776320i \(0.717084\pi\)
\(618\) 9.37318e7 0.397120
\(619\) − 3.24866e8i − 1.36972i −0.728673 0.684862i \(-0.759863\pi\)
0.728673 0.684862i \(-0.240137\pi\)
\(620\) −3.79316e7 −0.159157
\(621\) − 2.50213e6i − 0.0104481i
\(622\) − 1.82860e8i − 0.759884i
\(623\) 0 0
\(624\) 1.71933e8 0.707630
\(625\) −1.36291e8 −0.558247
\(626\) − 6.45002e8i − 2.62929i
\(627\) −1.26514e8 −0.513258
\(628\) 3.32162e8i 1.34113i
\(629\) 1.05267e8i 0.422998i
\(630\) 0 0
\(631\) −1.40735e8 −0.560163 −0.280082 0.959976i \(-0.590362\pi\)
−0.280082 + 0.959976i \(0.590362\pi\)
\(632\) 4.72067e7 0.187005
\(633\) 2.13613e8i 0.842202i
\(634\) 7.00533e7 0.274891
\(635\) − 6.28000e8i − 2.45267i
\(636\) 2.42848e8i 0.943982i
\(637\) 0 0
\(638\) −2.21595e8 −0.853294
\(639\) −7.42731e7 −0.284661
\(640\) − 1.25132e8i − 0.477339i
\(641\) −2.16677e8 −0.822696 −0.411348 0.911478i \(-0.634942\pi\)
−0.411348 + 0.911478i \(0.634942\pi\)
\(642\) − 1.17109e8i − 0.442573i
\(643\) 2.61645e8i 0.984192i 0.870541 + 0.492096i \(0.163769\pi\)
−0.870541 + 0.492096i \(0.836231\pi\)
\(644\) 0 0
\(645\) 3.67059e7 0.136791
\(646\) 1.30749e8 0.484998
\(647\) 1.07319e8i 0.396245i 0.980177 + 0.198122i \(0.0634843\pi\)
−0.980177 + 0.198122i \(0.936516\pi\)
\(648\) −4.75330e6 −0.0174691
\(649\) 3.95644e8i 1.44734i
\(650\) 7.39610e8i 2.69316i
\(651\) 0 0
\(652\) 3.67582e8 1.32621
\(653\) 4.43953e8 1.59440 0.797200 0.603715i \(-0.206313\pi\)
0.797200 + 0.603715i \(0.206313\pi\)
\(654\) 1.24446e8i 0.444883i
\(655\) 4.26313e7 0.151707
\(656\) 2.44892e8i 0.867486i
\(657\) − 5.74691e7i − 0.202646i
\(658\) 0 0
\(659\) −1.87037e7 −0.0653538 −0.0326769 0.999466i \(-0.510403\pi\)
−0.0326769 + 0.999466i \(0.510403\pi\)
\(660\) −4.33725e8 −1.50863
\(661\) 2.14286e7i 0.0741976i 0.999312 + 0.0370988i \(0.0118116\pi\)
−0.999312 + 0.0370988i \(0.988188\pi\)
\(662\) −1.13436e8 −0.391000
\(663\) 1.35957e8i 0.466510i
\(664\) − 8.56311e6i − 0.0292501i
\(665\) 0 0
\(666\) −1.04248e8 −0.352895
\(667\) −6.13160e6 −0.0206631
\(668\) 1.11097e8i 0.372712i
\(669\) 3.35197e8 1.11949
\(670\) 7.74205e8i 2.57414i
\(671\) − 6.84012e8i − 2.26410i
\(672\) 0 0
\(673\) 2.29593e8 0.753205 0.376603 0.926375i \(-0.377092\pi\)
0.376603 + 0.926375i \(0.377092\pi\)
\(674\) 6.62138e8 2.16256
\(675\) 7.88209e7i 0.256289i
\(676\) −3.21779e8 −1.04164
\(677\) 2.26448e7i 0.0729798i 0.999334 + 0.0364899i \(0.0116177\pi\)
−0.999334 + 0.0364899i \(0.988382\pi\)
\(678\) − 3.83277e8i − 1.22977i
\(679\) 0 0
\(680\) 4.37938e7 0.139279
\(681\) 6.12829e7 0.194043
\(682\) − 6.68824e7i − 0.210843i
\(683\) 2.24621e8 0.704998 0.352499 0.935812i \(-0.385332\pi\)
0.352499 + 0.935812i \(0.385332\pi\)
\(684\) 6.80671e7i 0.212701i
\(685\) 6.05167e8i 1.88280i
\(686\) 0 0
\(687\) −8.76579e6 −0.0270346
\(688\) −4.44657e7 −0.136540
\(689\) 6.72077e8i 2.05476i
\(690\) −2.28300e7 −0.0694959
\(691\) 2.53201e8i 0.767417i 0.923454 + 0.383709i \(0.125353\pi\)
−0.923454 + 0.383709i \(0.874647\pi\)
\(692\) − 8.82811e7i − 0.266409i
\(693\) 0 0
\(694\) 4.26717e8 1.27662
\(695\) 1.47878e8 0.440504
\(696\) 1.16482e7i 0.0345486i
\(697\) −1.93649e8 −0.571896
\(698\) − 7.64817e8i − 2.24901i
\(699\) − 2.54608e8i − 0.745489i
\(700\) 0 0
\(701\) 3.18180e8 0.923674 0.461837 0.886965i \(-0.347191\pi\)
0.461837 + 0.886965i \(0.347191\pi\)
\(702\) −1.34642e8 −0.389196
\(703\) 1.45851e8i 0.419801i
\(704\) 6.48401e8 1.85834
\(705\) 4.39212e8i 1.25345i
\(706\) 3.70207e8i 1.05204i
\(707\) 0 0
\(708\) 2.12864e8 0.599796
\(709\) 2.70283e8 0.758369 0.379185 0.925321i \(-0.376205\pi\)
0.379185 + 0.925321i \(0.376205\pi\)
\(710\) 6.77684e8i 1.89344i
\(711\) 1.42504e8 0.396478
\(712\) − 1.05113e6i − 0.00291217i
\(713\) − 1.85065e6i − 0.00510572i
\(714\) 0 0
\(715\) −1.20032e9 −3.28383
\(716\) 1.60018e8 0.435942
\(717\) − 3.05739e8i − 0.829457i
\(718\) 4.44027e8 1.19960
\(719\) 1.05505e8i 0.283849i 0.989877 + 0.141924i \(0.0453290\pi\)
−0.989877 + 0.141924i \(0.954671\pi\)
\(720\) − 1.67184e8i − 0.447916i
\(721\) 0 0
\(722\) −3.65324e8 −0.970660
\(723\) 5.61030e7 0.148447
\(724\) 5.65452e8i 1.48998i
\(725\) 1.93154e8 0.506863
\(726\) − 4.43975e8i − 1.16024i
\(727\) − 3.47133e8i − 0.903425i −0.892164 0.451713i \(-0.850813\pi\)
0.892164 0.451713i \(-0.149187\pi\)
\(728\) 0 0
\(729\) −1.43489e7 −0.0370370
\(730\) −5.24361e8 −1.34791
\(731\) − 3.51614e7i − 0.0900149i
\(732\) −3.68012e8 −0.938273
\(733\) − 4.67191e8i − 1.18627i −0.805104 0.593134i \(-0.797890\pi\)
0.805104 0.593134i \(-0.202110\pi\)
\(734\) 1.13121e9i 2.86058i
\(735\) 0 0
\(736\) 3.10594e7 0.0779040
\(737\) −7.17609e8 −1.79261
\(738\) − 1.91776e8i − 0.477117i
\(739\) 3.97629e8 0.985245 0.492623 0.870243i \(-0.336038\pi\)
0.492623 + 0.870243i \(0.336038\pi\)
\(740\) 5.00018e8i 1.23393i
\(741\) 1.88374e8i 0.462984i
\(742\) 0 0
\(743\) 6.88913e8 1.67957 0.839784 0.542920i \(-0.182681\pi\)
0.839784 + 0.542920i \(0.182681\pi\)
\(744\) −3.51568e6 −0.00853672
\(745\) − 6.69819e8i − 1.61990i
\(746\) −2.44430e8 −0.588760
\(747\) − 2.58497e7i − 0.0620145i
\(748\) 4.15475e8i 0.992751i
\(749\) 0 0
\(750\) 1.79140e8 0.424629
\(751\) 3.66564e8 0.865425 0.432712 0.901532i \(-0.357557\pi\)
0.432712 + 0.901532i \(0.357557\pi\)
\(752\) − 5.32064e8i − 1.25115i
\(753\) 1.01302e8 0.237266
\(754\) 3.29946e8i 0.769714i
\(755\) − 1.23152e9i − 2.86156i
\(756\) 0 0
\(757\) −2.07241e8 −0.477735 −0.238868 0.971052i \(-0.576776\pi\)
−0.238868 + 0.971052i \(0.576776\pi\)
\(758\) 4.78337e8 1.09831
\(759\) − 2.11611e7i − 0.0483964i
\(760\) 6.06781e7 0.138226
\(761\) − 6.76679e8i − 1.53542i −0.640795 0.767712i \(-0.721395\pi\)
0.640795 0.767712i \(-0.278605\pi\)
\(762\) − 5.95757e8i − 1.34649i
\(763\) 0 0
\(764\) 3.02285e8 0.677854
\(765\) 1.32201e8 0.295292
\(766\) 1.76780e8i 0.393321i
\(767\) 5.89097e8 1.30557
\(768\) 1.96063e8i 0.432824i
\(769\) 5.58023e8i 1.22708i 0.789663 + 0.613540i \(0.210255\pi\)
−0.789663 + 0.613540i \(0.789745\pi\)
\(770\) 0 0
\(771\) −4.45510e7 −0.0972063
\(772\) −1.03397e7 −0.0224727
\(773\) − 8.28716e7i − 0.179419i −0.995968 0.0897093i \(-0.971406\pi\)
0.995968 0.0897093i \(-0.0285938\pi\)
\(774\) 3.48213e7 0.0750969
\(775\) 5.82983e7i 0.125242i
\(776\) − 1.65188e7i − 0.0353504i
\(777\) 0 0
\(778\) 7.84070e8 1.66501
\(779\) −2.68309e8 −0.567574
\(780\) 6.45799e8i 1.36086i
\(781\) −6.28143e8 −1.31858
\(782\) 2.18694e7i 0.0457317i
\(783\) 3.51627e7i 0.0732482i
\(784\) 0 0
\(785\) 8.93859e8 1.84782
\(786\) 4.04425e7 0.0832856
\(787\) 5.30510e8i 1.08835i 0.838971 + 0.544176i \(0.183158\pi\)
−0.838971 + 0.544176i \(0.816842\pi\)
\(788\) −5.11627e8 −1.04562
\(789\) − 7.79152e7i − 0.158632i
\(790\) − 1.30024e9i − 2.63719i
\(791\) 0 0
\(792\) −4.01997e7 −0.0809184
\(793\) −1.01847e9 −2.04233
\(794\) − 1.09208e8i − 0.218169i
\(795\) 6.53512e8 1.30063
\(796\) − 1.25177e8i − 0.248191i
\(797\) 4.96880e8i 0.981469i 0.871309 + 0.490735i \(0.163272\pi\)
−0.871309 + 0.490735i \(0.836728\pi\)
\(798\) 0 0
\(799\) 4.20732e8 0.824831
\(800\) −9.78417e8 −1.91097
\(801\) − 3.17307e6i − 0.00617422i
\(802\) 6.44652e8 1.24969
\(803\) − 4.86029e8i − 0.938675i
\(804\) 3.86088e8i 0.742879i
\(805\) 0 0
\(806\) −9.95852e7 −0.190191
\(807\) −3.18315e8 −0.605672
\(808\) − 1.12785e8i − 0.213805i
\(809\) −4.20304e8 −0.793812 −0.396906 0.917859i \(-0.629916\pi\)
−0.396906 + 0.917859i \(0.629916\pi\)
\(810\) 1.30923e8i 0.246354i
\(811\) − 4.27809e8i − 0.802025i −0.916073 0.401012i \(-0.868658\pi\)
0.916073 0.401012i \(-0.131342\pi\)
\(812\) 0 0
\(813\) −1.92393e8 −0.358029
\(814\) −8.81650e8 −1.63464
\(815\) − 9.89174e8i − 1.82726i
\(816\) −1.60149e8 −0.294751
\(817\) − 4.87176e7i − 0.0893346i
\(818\) 1.89887e8i 0.346925i
\(819\) 0 0
\(820\) −9.19837e8 −1.66828
\(821\) 9.52803e8 1.72176 0.860882 0.508805i \(-0.169912\pi\)
0.860882 + 0.508805i \(0.169912\pi\)
\(822\) 5.74096e8i 1.03364i
\(823\) −7.41229e8 −1.32970 −0.664848 0.746978i \(-0.731504\pi\)
−0.664848 + 0.746978i \(0.731504\pi\)
\(824\) − 4.16689e7i − 0.0744785i
\(825\) 6.66606e8i 1.18715i
\(826\) 0 0
\(827\) −7.61072e8 −1.34558 −0.672790 0.739834i \(-0.734904\pi\)
−0.672790 + 0.739834i \(0.734904\pi\)
\(828\) −1.13851e7 −0.0200561
\(829\) 8.15761e8i 1.43186i 0.698174 + 0.715928i \(0.253996\pi\)
−0.698174 + 0.715928i \(0.746004\pi\)
\(830\) −2.35858e8 −0.412493
\(831\) − 3.24710e8i − 0.565839i
\(832\) − 9.65442e8i − 1.67632i
\(833\) 0 0
\(834\) 1.40286e8 0.241833
\(835\) 2.98966e8 0.513526
\(836\) 5.75658e8i 0.985248i
\(837\) −1.06129e7 −0.0180991
\(838\) − 4.35623e8i − 0.740251i
\(839\) 1.07654e9i 1.82283i 0.411491 + 0.911414i \(0.365008\pi\)
−0.411491 + 0.911414i \(0.634992\pi\)
\(840\) 0 0
\(841\) −5.08655e8 −0.855137
\(842\) −1.06576e9 −1.78535
\(843\) 3.27581e8i 0.546809i
\(844\) 9.71972e8 1.61669
\(845\) 8.65917e8i 1.43518i
\(846\) 4.16662e8i 0.688133i
\(847\) 0 0
\(848\) −7.91667e8 −1.29824
\(849\) 1.62493e8 0.265528
\(850\) − 6.88919e8i − 1.12179i
\(851\) −2.43955e7 −0.0395841
\(852\) 3.37954e8i 0.546435i
\(853\) 9.74565e8i 1.57023i 0.619349 + 0.785116i \(0.287397\pi\)
−0.619349 + 0.785116i \(0.712603\pi\)
\(854\) 0 0
\(855\) 1.83170e8 0.293060
\(856\) −5.20613e7 −0.0830030
\(857\) 9.60657e8i 1.52625i 0.646251 + 0.763125i \(0.276336\pi\)
−0.646251 + 0.763125i \(0.723664\pi\)
\(858\) −1.13870e9 −1.80279
\(859\) 4.88078e8i 0.770034i 0.922910 + 0.385017i \(0.125804\pi\)
−0.922910 + 0.385017i \(0.874196\pi\)
\(860\) − 1.67018e8i − 0.262583i
\(861\) 0 0
\(862\) 1.60076e9 2.49921
\(863\) 3.22747e8 0.502146 0.251073 0.967968i \(-0.419217\pi\)
0.251073 + 0.967968i \(0.419217\pi\)
\(864\) − 1.78115e8i − 0.276160i
\(865\) −2.37567e8 −0.367061
\(866\) 2.32695e8i 0.358289i
\(867\) 2.49629e8i 0.383034i
\(868\) 0 0
\(869\) 1.20519e9 1.83652
\(870\) 3.20832e8 0.487214
\(871\) 1.06849e9i 1.61702i
\(872\) 5.53229e7 0.0834363
\(873\) − 4.98659e7i − 0.0749481i
\(874\) 3.03010e7i 0.0453860i
\(875\) 0 0
\(876\) −2.61494e8 −0.388999
\(877\) 9.37896e8 1.39045 0.695226 0.718791i \(-0.255304\pi\)
0.695226 + 0.718791i \(0.255304\pi\)
\(878\) − 1.00016e9i − 1.47770i
\(879\) −4.44782e8 −0.654908
\(880\) − 1.41391e9i − 2.07479i
\(881\) − 9.47337e8i − 1.38540i −0.721224 0.692702i \(-0.756420\pi\)
0.721224 0.692702i \(-0.243580\pi\)
\(882\) 0 0
\(883\) 1.52054e8 0.220860 0.110430 0.993884i \(-0.464777\pi\)
0.110430 + 0.993884i \(0.464777\pi\)
\(884\) 6.18626e8 0.895511
\(885\) − 5.72824e8i − 0.826403i
\(886\) 9.38422e7 0.134926
\(887\) 1.18390e9i 1.69646i 0.529632 + 0.848228i \(0.322330\pi\)
−0.529632 + 0.848228i \(0.677670\pi\)
\(888\) 4.63441e7i 0.0661843i
\(889\) 0 0
\(890\) −2.89518e7 −0.0410682
\(891\) −1.21352e8 −0.171559
\(892\) − 1.52520e9i − 2.14898i
\(893\) 5.82941e8 0.818597
\(894\) − 6.35429e8i − 0.889312i
\(895\) − 4.30612e8i − 0.600644i
\(896\) 0 0
\(897\) −3.15080e7 −0.0436560
\(898\) 5.99947e8 0.828484
\(899\) 2.60074e7i 0.0357946i
\(900\) 3.58647e8 0.491972
\(901\) − 6.26014e8i − 0.855874i
\(902\) − 1.62189e9i − 2.21005i
\(903\) 0 0
\(904\) −1.70388e8 −0.230639
\(905\) 1.52165e9 2.05290
\(906\) − 1.16829e9i − 1.57097i
\(907\) 3.39773e8 0.455372 0.227686 0.973735i \(-0.426884\pi\)
0.227686 + 0.973735i \(0.426884\pi\)
\(908\) − 2.78847e8i − 0.372485i
\(909\) − 3.40467e8i − 0.453297i
\(910\) 0 0
\(911\) −9.44286e8 −1.24896 −0.624480 0.781041i \(-0.714689\pi\)
−0.624480 + 0.781041i \(0.714689\pi\)
\(912\) −2.21894e8 −0.292523
\(913\) − 2.18616e8i − 0.287257i
\(914\) 4.63933e8 0.607599
\(915\) 9.90332e8i 1.29276i
\(916\) 3.98857e7i 0.0518956i
\(917\) 0 0
\(918\) 1.25414e8 0.162113
\(919\) −3.74554e8 −0.482579 −0.241290 0.970453i \(-0.577570\pi\)
−0.241290 + 0.970453i \(0.577570\pi\)
\(920\) 1.01492e7i 0.0130337i
\(921\) −1.59421e8 −0.204064
\(922\) − 1.63685e8i − 0.208841i
\(923\) 9.35280e8i 1.18942i
\(924\) 0 0
\(925\) 7.68494e8 0.970991
\(926\) −1.45431e9 −1.83158
\(927\) − 1.25787e8i − 0.157905i
\(928\) −4.36480e8 −0.546161
\(929\) 9.11640e8i 1.13704i 0.822669 + 0.568521i \(0.192484\pi\)
−0.822669 + 0.568521i \(0.807516\pi\)
\(930\) 9.68343e7i 0.120387i
\(931\) 0 0
\(932\) −1.15851e9 −1.43104
\(933\) −2.45396e8 −0.302150
\(934\) 1.20256e9i 1.47593i
\(935\) 1.11806e9 1.36782
\(936\) 5.98557e7i 0.0729924i
\(937\) 3.76597e8i 0.457781i 0.973452 + 0.228890i \(0.0735097\pi\)
−0.973452 + 0.228890i \(0.926490\pi\)
\(938\) 0 0
\(939\) −8.65586e8 −1.04547
\(940\) 1.99848e9 2.40612
\(941\) − 2.51723e8i − 0.302103i −0.988526 0.151051i \(-0.951734\pi\)
0.988526 0.151051i \(-0.0482659\pi\)
\(942\) 8.47966e8 1.01444
\(943\) − 4.48781e7i − 0.0535180i
\(944\) 6.93922e8i 0.824888i
\(945\) 0 0
\(946\) 2.94491e8 0.347856
\(947\) −2.65866e8 −0.313050 −0.156525 0.987674i \(-0.550029\pi\)
−0.156525 + 0.987674i \(0.550029\pi\)
\(948\) − 6.48416e8i − 0.761077i
\(949\) −7.23677e8 −0.846732
\(950\) − 9.54525e8i − 1.11331i
\(951\) − 9.40107e7i − 0.109304i
\(952\) 0 0
\(953\) −5.38039e8 −0.621635 −0.310817 0.950470i \(-0.600603\pi\)
−0.310817 + 0.950470i \(0.600603\pi\)
\(954\) 6.19959e8 0.714032
\(955\) − 8.13458e8i − 0.933953i
\(956\) −1.39116e9 −1.59222
\(957\) 2.97379e8i 0.339292i
\(958\) − 9.05777e8i − 1.03021i
\(959\) 0 0
\(960\) −9.38774e8 −1.06108
\(961\) 8.79654e8 0.991155
\(962\) 1.31274e9i 1.47453i
\(963\) −1.57159e8 −0.175979
\(964\) − 2.55277e8i − 0.284958i
\(965\) 2.78244e7i 0.0309631i
\(966\) 0 0
\(967\) −1.16138e9 −1.28439 −0.642194 0.766542i \(-0.721976\pi\)
−0.642194 + 0.766542i \(0.721976\pi\)
\(968\) −1.97371e8 −0.217600
\(969\) − 1.75463e8i − 0.192848i
\(970\) −4.54987e8 −0.498522
\(971\) 1.50850e9i 1.64773i 0.566784 + 0.823866i \(0.308187\pi\)
−0.566784 + 0.823866i \(0.691813\pi\)
\(972\) 6.52898e7i 0.0710962i
\(973\) 0 0
\(974\) −1.65874e9 −1.79515
\(975\) 9.92548e8 1.07087
\(976\) − 1.19969e9i − 1.29039i
\(977\) −1.20754e8 −0.129485 −0.0647425 0.997902i \(-0.520623\pi\)
−0.0647425 + 0.997902i \(0.520623\pi\)
\(978\) − 9.38387e8i − 1.00315i
\(979\) − 2.68354e7i − 0.0285996i
\(980\) 0 0
\(981\) 1.67005e8 0.176897
\(982\) −5.89878e8 −0.622914
\(983\) 2.15159e7i 0.0226516i 0.999936 + 0.0113258i \(0.00360520\pi\)
−0.999936 + 0.0113258i \(0.996395\pi\)
\(984\) −8.52549e7 −0.0894817
\(985\) 1.37680e9i 1.44067i
\(986\) − 3.07333e8i − 0.320611i
\(987\) 0 0
\(988\) 8.57131e8 0.888743
\(989\) 8.14865e6 0.00842359
\(990\) 1.10724e9i 1.14113i
\(991\) −6.15881e7 −0.0632813 −0.0316407 0.999499i \(-0.510073\pi\)
−0.0316407 + 0.999499i \(0.510073\pi\)
\(992\) − 1.31740e8i − 0.134953i
\(993\) 1.52230e8i 0.155472i
\(994\) 0 0
\(995\) −3.36855e8 −0.341959
\(996\) −1.17620e8 −0.119043
\(997\) − 7.76327e8i − 0.783356i −0.920102 0.391678i \(-0.871895\pi\)
0.920102 0.391678i \(-0.128105\pi\)
\(998\) −1.46947e9 −1.47832
\(999\) 1.39900e8i 0.140321i
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 147.7.d.b.97.2 8
3.2 odd 2 441.7.d.c.244.8 8
7.2 even 3 21.7.f.a.10.4 8
7.3 odd 6 21.7.f.a.19.4 yes 8
7.4 even 3 147.7.f.d.19.4 8
7.5 odd 6 147.7.f.d.31.4 8
7.6 odd 2 inner 147.7.d.b.97.1 8
21.2 odd 6 63.7.m.d.10.1 8
21.17 even 6 63.7.m.d.19.1 8
21.20 even 2 441.7.d.c.244.7 8
28.3 even 6 336.7.bh.d.145.1 8
28.23 odd 6 336.7.bh.d.241.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.7.f.a.10.4 8 7.2 even 3
21.7.f.a.19.4 yes 8 7.3 odd 6
63.7.m.d.10.1 8 21.2 odd 6
63.7.m.d.19.1 8 21.17 even 6
147.7.d.b.97.1 8 7.6 odd 2 inner
147.7.d.b.97.2 8 1.1 even 1 trivial
147.7.f.d.19.4 8 7.4 even 3
147.7.f.d.31.4 8 7.5 odd 6
336.7.bh.d.145.1 8 28.3 even 6
336.7.bh.d.241.1 8 28.23 odd 6
441.7.d.c.244.7 8 21.20 even 2
441.7.d.c.244.8 8 3.2 odd 2