Properties

Label 354.7.d.a.235.20
Level $354$
Weight $7$
Character 354.235
Analytic conductor $81.439$
Analytic rank $0$
Dimension $60$
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] = [354,7,Mod(235,354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, 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("354.235");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 354.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: \(81.4391456014\)
Analytic rank: \(0\)
Dimension: \(60\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 235.20
Character \(\chi\) \(=\) 354.235
Dual form 354.7.d.a.235.19

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.65685i q^{2} -15.5885 q^{3} -32.0000 q^{4} -96.2981 q^{5} -88.1816i q^{6} -484.887 q^{7} -181.019i q^{8} +243.000 q^{9} +O(q^{10})\) \(q+5.65685i q^{2} -15.5885 q^{3} -32.0000 q^{4} -96.2981 q^{5} -88.1816i q^{6} -484.887 q^{7} -181.019i q^{8} +243.000 q^{9} -544.744i q^{10} -1209.66i q^{11} +498.831 q^{12} +1553.53i q^{13} -2742.94i q^{14} +1501.14 q^{15} +1024.00 q^{16} +6745.19 q^{17} +1374.62i q^{18} +11357.4 q^{19} +3081.54 q^{20} +7558.64 q^{21} +6842.85 q^{22} +21259.9i q^{23} +2821.81i q^{24} -6351.68 q^{25} -8788.11 q^{26} -3788.00 q^{27} +15516.4 q^{28} -17805.3 q^{29} +8491.72i q^{30} -36428.3i q^{31} +5792.62i q^{32} +18856.7i q^{33} +38156.5i q^{34} +46693.7 q^{35} -7776.00 q^{36} -47212.4i q^{37} +64247.2i q^{38} -24217.2i q^{39} +17431.8i q^{40} +136149. q^{41} +42758.1i q^{42} -41734.3i q^{43} +38709.0i q^{44} -23400.4 q^{45} -120264. q^{46} +108720. i q^{47} -15962.6 q^{48} +117466. q^{49} -35930.5i q^{50} -105147. q^{51} -49713.0i q^{52} -142756. q^{53} -21428.1i q^{54} +116488. i q^{55} +87773.9i q^{56} -177044. q^{57} -100722. i q^{58} +(-81810.6 - 188381. i) q^{59} -48036.4 q^{60} +70678.1i q^{61} +206069. q^{62} -117828. q^{63} -32768.0 q^{64} -149602. i q^{65} -106669. q^{66} +221377. i q^{67} -215846. q^{68} -331410. i q^{69} +264139. i q^{70} +72810.0 q^{71} -43987.7i q^{72} +206404. i q^{73} +267074. q^{74} +99012.9 q^{75} -363437. q^{76} +586547. i q^{77} +136993. q^{78} -91271.7 q^{79} -98609.2 q^{80} +59049.0 q^{81} +770177. i q^{82} -588230. i q^{83} -241877. q^{84} -649549. q^{85} +236085. q^{86} +277557. q^{87} -218971. q^{88} +861372. i q^{89} -132373. i q^{90} -753288. i q^{91} -680318. i q^{92} +567860. i q^{93} -615012. q^{94} -1.09370e6 q^{95} -90298.0i q^{96} -686994. i q^{97} +664491. i q^{98} -293947. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q - 1920 q^{4} + 408 q^{7} + 14580 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 60 q - 1920 q^{4} + 408 q^{7} + 14580 q^{9} + 4536 q^{15} + 61440 q^{16} - 15840 q^{17} - 5616 q^{19} - 17472 q^{22} + 226260 q^{25} - 34048 q^{26} - 13056 q^{28} - 75392 q^{29} + 278000 q^{35} - 466560 q^{36} + 67376 q^{41} + 209856 q^{46} + 269100 q^{49} - 206064 q^{51} + 490000 q^{53} - 373248 q^{57} - 863472 q^{59} - 145152 q^{60} - 155072 q^{62} + 99144 q^{63} - 1966080 q^{64} - 404352 q^{66} + 506880 q^{68} - 2041856 q^{71} - 2146176 q^{74} + 808704 q^{75} + 179712 q^{76} + 228096 q^{78} + 670248 q^{79} + 3542940 q^{81} + 873408 q^{85} + 1832576 q^{86} - 2568024 q^{87} + 559104 q^{88} + 1049472 q^{94} - 245856 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/354\mathbb{Z}\right)^\times\).

\(n\) \(61\) \(119\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.65685i 0.707107i
\(3\) −15.5885 −0.577350
\(4\) −32.0000 −0.500000
\(5\) −96.2981 −0.770385 −0.385192 0.922836i \(-0.625865\pi\)
−0.385192 + 0.922836i \(0.625865\pi\)
\(6\) 88.1816i 0.408248i
\(7\) −484.887 −1.41366 −0.706832 0.707381i \(-0.749877\pi\)
−0.706832 + 0.707381i \(0.749877\pi\)
\(8\) 181.019i 0.353553i
\(9\) 243.000 0.333333
\(10\) 544.744i 0.544744i
\(11\) 1209.66i 0.908833i −0.890789 0.454416i \(-0.849848\pi\)
0.890789 0.454416i \(-0.150152\pi\)
\(12\) 498.831 0.288675
\(13\) 1553.53i 0.707115i 0.935413 + 0.353558i \(0.115028\pi\)
−0.935413 + 0.353558i \(0.884972\pi\)
\(14\) 2742.94i 0.999612i
\(15\) 1501.14 0.444782
\(16\) 1024.00 0.250000
\(17\) 6745.19 1.37293 0.686463 0.727164i \(-0.259162\pi\)
0.686463 + 0.727164i \(0.259162\pi\)
\(18\) 1374.62i 0.235702i
\(19\) 11357.4 1.65584 0.827920 0.560847i \(-0.189524\pi\)
0.827920 + 0.560847i \(0.189524\pi\)
\(20\) 3081.54 0.385192
\(21\) 7558.64 0.816180
\(22\) 6842.85 0.642642
\(23\) 21259.9i 1.74735i 0.486514 + 0.873673i \(0.338268\pi\)
−0.486514 + 0.873673i \(0.661732\pi\)
\(24\) 2821.81i 0.204124i
\(25\) −6351.68 −0.406507
\(26\) −8788.11 −0.500006
\(27\) −3788.00 −0.192450
\(28\) 15516.4 0.706832
\(29\) −17805.3 −0.730055 −0.365027 0.930997i \(-0.618940\pi\)
−0.365027 + 0.930997i \(0.618940\pi\)
\(30\) 8491.72i 0.314508i
\(31\) 36428.3i 1.22279i −0.791324 0.611397i \(-0.790608\pi\)
0.791324 0.611397i \(-0.209392\pi\)
\(32\) 5792.62i 0.176777i
\(33\) 18856.7i 0.524715i
\(34\) 38156.5i 0.970806i
\(35\) 46693.7 1.08907
\(36\) −7776.00 −0.166667
\(37\) 47212.4i 0.932075i −0.884765 0.466037i \(-0.845681\pi\)
0.884765 0.466037i \(-0.154319\pi\)
\(38\) 64247.2i 1.17086i
\(39\) 24217.2i 0.408253i
\(40\) 17431.8i 0.272372i
\(41\) 136149. 1.97544 0.987720 0.156236i \(-0.0499361\pi\)
0.987720 + 0.156236i \(0.0499361\pi\)
\(42\) 42758.1i 0.577126i
\(43\) 41734.3i 0.524913i −0.964944 0.262457i \(-0.915467\pi\)
0.964944 0.262457i \(-0.0845327\pi\)
\(44\) 38709.0i 0.454416i
\(45\) −23400.4 −0.256795
\(46\) −120264. −1.23556
\(47\) 108720.i 1.04717i 0.851975 + 0.523583i \(0.175405\pi\)
−0.851975 + 0.523583i \(0.824595\pi\)
\(48\) −15962.6 −0.144338
\(49\) 117466. 0.998449
\(50\) 35930.5i 0.287444i
\(51\) −105147. −0.792660
\(52\) 49713.0i 0.353558i
\(53\) −142756. −0.958888 −0.479444 0.877572i \(-0.659162\pi\)
−0.479444 + 0.877572i \(0.659162\pi\)
\(54\) 21428.1i 0.136083i
\(55\) 116488.i 0.700151i
\(56\) 87773.9i 0.499806i
\(57\) −177044. −0.955999
\(58\) 100722.i 0.516227i
\(59\) −81810.6 188381.i −0.398340 0.917238i
\(60\) −48036.4 −0.222391
\(61\) 70678.1i 0.311383i 0.987806 + 0.155692i \(0.0497606\pi\)
−0.987806 + 0.155692i \(0.950239\pi\)
\(62\) 206069. 0.864646
\(63\) −117828. −0.471222
\(64\) −32768.0 −0.125000
\(65\) 149602.i 0.544751i
\(66\) −106669. −0.371029
\(67\) 221377.i 0.736050i 0.929816 + 0.368025i \(0.119966\pi\)
−0.929816 + 0.368025i \(0.880034\pi\)
\(68\) −215846. −0.686463
\(69\) 331410.i 1.00883i
\(70\) 264139.i 0.770086i
\(71\) 72810.0 0.203430 0.101715 0.994814i \(-0.467567\pi\)
0.101715 + 0.994814i \(0.467567\pi\)
\(72\) 43987.7i 0.117851i
\(73\) 206404.i 0.530580i 0.964169 + 0.265290i \(0.0854676\pi\)
−0.964169 + 0.265290i \(0.914532\pi\)
\(74\) 267074. 0.659076
\(75\) 99012.9 0.234697
\(76\) −363437. −0.827920
\(77\) 586547.i 1.28479i
\(78\) 136993. 0.288679
\(79\) −91271.7 −0.185121 −0.0925604 0.995707i \(-0.529505\pi\)
−0.0925604 + 0.995707i \(0.529505\pi\)
\(80\) −98609.2 −0.192596
\(81\) 59049.0 0.111111
\(82\) 770177.i 1.39685i
\(83\) 588230.i 1.02876i −0.857563 0.514379i \(-0.828023\pi\)
0.857563 0.514379i \(-0.171977\pi\)
\(84\) −241877. −0.408090
\(85\) −649549. −1.05768
\(86\) 236085. 0.371170
\(87\) 277557. 0.421497
\(88\) −218971. −0.321321
\(89\) 861372.i 1.22186i 0.791686 + 0.610929i \(0.209204\pi\)
−0.791686 + 0.610929i \(0.790796\pi\)
\(90\) 132373.i 0.181581i
\(91\) 753288.i 0.999624i
\(92\) 680318.i 0.873673i
\(93\) 567860.i 0.705980i
\(94\) −615012. −0.740458
\(95\) −1.09370e6 −1.27563
\(96\) 90298.0i 0.102062i
\(97\) 686994.i 0.752727i −0.926472 0.376364i \(-0.877174\pi\)
0.926472 0.376364i \(-0.122826\pi\)
\(98\) 664491.i 0.706010i
\(99\) 293947.i 0.302944i
\(100\) 203254. 0.203254
\(101\) 1.58295e6i 1.53639i −0.640215 0.768196i \(-0.721155\pi\)
0.640215 0.768196i \(-0.278845\pi\)
\(102\) 594802.i 0.560495i
\(103\) 90265.4i 0.0826057i 0.999147 + 0.0413028i \(0.0131508\pi\)
−0.999147 + 0.0413028i \(0.986849\pi\)
\(104\) 281219. 0.250003
\(105\) −727883. −0.628772
\(106\) 807552.i 0.678036i
\(107\) −1.36331e6 −1.11286 −0.556432 0.830893i \(-0.687830\pi\)
−0.556432 + 0.830893i \(0.687830\pi\)
\(108\) 121216. 0.0962250
\(109\) 2.05270e6i 1.58506i 0.609832 + 0.792531i \(0.291237\pi\)
−0.609832 + 0.792531i \(0.708763\pi\)
\(110\) −658953. −0.495081
\(111\) 735968.i 0.538134i
\(112\) −496524. −0.353416
\(113\) 902711.i 0.625624i 0.949815 + 0.312812i \(0.101271\pi\)
−0.949815 + 0.312812i \(0.898729\pi\)
\(114\) 1.00151e6i 0.675993i
\(115\) 2.04729e6i 1.34613i
\(116\) 569770. 0.365027
\(117\) 377508.i 0.235705i
\(118\) 1.06565e6 462791.i 0.648585 0.281669i
\(119\) −3.27065e6 −1.94086
\(120\) 271735.i 0.157254i
\(121\) 308292. 0.174023
\(122\) −399816. −0.220181
\(123\) −2.12236e6 −1.14052
\(124\) 1.16570e6i 0.611397i
\(125\) 2.11631e6 1.08355
\(126\) 666533.i 0.333204i
\(127\) −889383. −0.434188 −0.217094 0.976151i \(-0.569658\pi\)
−0.217094 + 0.976151i \(0.569658\pi\)
\(128\) 185364.i 0.0883883i
\(129\) 650573.i 0.303059i
\(130\) 846278. 0.385197
\(131\) 2.93521e6i 1.30565i −0.757511 0.652823i \(-0.773585\pi\)
0.757511 0.652823i \(-0.226415\pi\)
\(132\) 603414.i 0.262357i
\(133\) −5.50706e6 −2.34080
\(134\) −1.25230e6 −0.520466
\(135\) 364777. 0.148261
\(136\) 1.22101e6i 0.485403i
\(137\) −1.03431e6 −0.402245 −0.201122 0.979566i \(-0.564459\pi\)
−0.201122 + 0.979566i \(0.564459\pi\)
\(138\) 1.87474e6 0.713351
\(139\) 2.44154e6 0.909116 0.454558 0.890717i \(-0.349797\pi\)
0.454558 + 0.890717i \(0.349797\pi\)
\(140\) −1.49420e6 −0.544533
\(141\) 1.69477e6i 0.604581i
\(142\) 411875.i 0.143847i
\(143\) 1.87924e6 0.642650
\(144\) 248832. 0.0833333
\(145\) 1.71462e6 0.562423
\(146\) −1.16760e6 −0.375176
\(147\) −1.83112e6 −0.576455
\(148\) 1.51080e6i 0.466037i
\(149\) 1.37696e6i 0.416259i 0.978101 + 0.208130i \(0.0667376\pi\)
−0.978101 + 0.208130i \(0.933262\pi\)
\(150\) 560101.i 0.165956i
\(151\) 260207.i 0.0755767i −0.999286 0.0377883i \(-0.987969\pi\)
0.999286 0.0377883i \(-0.0120313\pi\)
\(152\) 2.05591e6i 0.585428i
\(153\) 1.63908e6 0.457642
\(154\) −3.31801e6 −0.908480
\(155\) 3.50797e6i 0.942022i
\(156\) 774950.i 0.204127i
\(157\) 3.25031e6i 0.839896i 0.907548 + 0.419948i \(0.137952\pi\)
−0.907548 + 0.419948i \(0.862048\pi\)
\(158\) 516311.i 0.130900i
\(159\) 2.22535e6 0.553614
\(160\) 557818.i 0.136186i
\(161\) 1.03087e7i 2.47016i
\(162\) 334032.i 0.0785674i
\(163\) −4.35919e6 −1.00657 −0.503284 0.864121i \(-0.667875\pi\)
−0.503284 + 0.864121i \(0.667875\pi\)
\(164\) −4.35678e6 −0.987720
\(165\) 1.81586e6i 0.404232i
\(166\) 3.32753e6 0.727441
\(167\) −2.69927e6 −0.579558 −0.289779 0.957094i \(-0.593582\pi\)
−0.289779 + 0.957094i \(0.593582\pi\)
\(168\) 1.36826e6i 0.288563i
\(169\) 2.41335e6 0.499988
\(170\) 3.67440e6i 0.747894i
\(171\) 2.75985e6 0.551946
\(172\) 1.33550e6i 0.262457i
\(173\) 2.28386e6i 0.441094i 0.975376 + 0.220547i \(0.0707843\pi\)
−0.975376 + 0.220547i \(0.929216\pi\)
\(174\) 1.57010e6i 0.298044i
\(175\) 3.07985e6 0.574665
\(176\) 1.23869e6i 0.227208i
\(177\) 1.27530e6 + 2.93658e6i 0.229981 + 0.529568i
\(178\) −4.87265e6 −0.863984
\(179\) 6.94771e6i 1.21139i −0.795699 0.605693i \(-0.792896\pi\)
0.795699 0.605693i \(-0.207104\pi\)
\(180\) 748814. 0.128397
\(181\) −2.14818e6 −0.362273 −0.181136 0.983458i \(-0.557978\pi\)
−0.181136 + 0.983458i \(0.557978\pi\)
\(182\) 4.26124e6 0.706841
\(183\) 1.10176e6i 0.179777i
\(184\) 3.84846e6 0.617780
\(185\) 4.54646e6i 0.718056i
\(186\) −3.21230e6 −0.499203
\(187\) 8.15936e6i 1.24776i
\(188\) 3.47903e6i 0.523583i
\(189\) 1.83675e6 0.272060
\(190\) 6.18688e6i 0.902009i
\(191\) 5.87137e6i 0.842634i −0.906913 0.421317i \(-0.861568\pi\)
0.906913 0.421317i \(-0.138432\pi\)
\(192\) 510803. 0.0721688
\(193\) −1.38306e7 −1.92384 −0.961919 0.273333i \(-0.911874\pi\)
−0.961919 + 0.273333i \(0.911874\pi\)
\(194\) 3.88622e6 0.532259
\(195\) 2.33207e6i 0.314512i
\(196\) −3.75893e6 −0.499224
\(197\) −1.48020e7 −1.93608 −0.968039 0.250800i \(-0.919306\pi\)
−0.968039 + 0.250800i \(0.919306\pi\)
\(198\) 1.66281e6 0.214214
\(199\) 173100. 0.0219653 0.0109826 0.999940i \(-0.496504\pi\)
0.0109826 + 0.999940i \(0.496504\pi\)
\(200\) 1.14978e6i 0.143722i
\(201\) 3.45092e6i 0.424959i
\(202\) 8.95450e6 1.08639
\(203\) 8.63356e6 1.03205
\(204\) 3.36471e6 0.396330
\(205\) −1.31109e7 −1.52185
\(206\) −510618. −0.0584110
\(207\) 5.16617e6i 0.582448i
\(208\) 1.59082e6i 0.176779i
\(209\) 1.37386e7i 1.50488i
\(210\) 4.11753e6i 0.444609i
\(211\) 1.43898e7i 1.53182i 0.642947 + 0.765910i \(0.277711\pi\)
−0.642947 + 0.765910i \(0.722289\pi\)
\(212\) 4.56820e6 0.479444
\(213\) −1.13499e6 −0.117451
\(214\) 7.71202e6i 0.786914i
\(215\) 4.01893e6i 0.404385i
\(216\) 685700.i 0.0680414i
\(217\) 1.76636e7i 1.72862i
\(218\) −1.16118e7 −1.12081
\(219\) 3.21753e6i 0.306330i
\(220\) 3.72760e6i 0.350075i
\(221\) 1.04789e7i 0.970818i
\(222\) −4.16326e6 −0.380518
\(223\) 916397. 0.0826359 0.0413180 0.999146i \(-0.486844\pi\)
0.0413180 + 0.999146i \(0.486844\pi\)
\(224\) 2.80877e6i 0.249903i
\(225\) −1.54346e6 −0.135502
\(226\) −5.10651e6 −0.442383
\(227\) 2.09769e7i 1.79334i 0.442696 + 0.896672i \(0.354022\pi\)
−0.442696 + 0.896672i \(0.645978\pi\)
\(228\) 5.66542e6 0.478000
\(229\) 1.10845e7i 0.923016i 0.887136 + 0.461508i \(0.152691\pi\)
−0.887136 + 0.461508i \(0.847309\pi\)
\(230\) 1.15812e7 0.951856
\(231\) 9.14336e6i 0.741771i
\(232\) 3.22311e6i 0.258113i
\(233\) 4.82794e6i 0.381675i −0.981622 0.190838i \(-0.938880\pi\)
0.981622 0.190838i \(-0.0611204\pi\)
\(234\) −2.13551e6 −0.166669
\(235\) 1.04695e7i 0.806720i
\(236\) 2.61794e6 + 6.02821e6i 0.199170 + 0.458619i
\(237\) 1.42279e6 0.106880
\(238\) 1.85016e7i 1.37239i
\(239\) −1.02169e7 −0.748388 −0.374194 0.927350i \(-0.622081\pi\)
−0.374194 + 0.927350i \(0.622081\pi\)
\(240\) 1.53717e6 0.111195
\(241\) −1.83758e7 −1.31279 −0.656394 0.754419i \(-0.727919\pi\)
−0.656394 + 0.754419i \(0.727919\pi\)
\(242\) 1.74396e6i 0.123053i
\(243\) −920483. −0.0641500
\(244\) 2.26170e6i 0.155692i
\(245\) −1.13118e7 −0.769190
\(246\) 1.20059e7i 0.806470i
\(247\) 1.76441e7i 1.17087i
\(248\) −6.59422e6 −0.432323
\(249\) 9.16960e6i 0.593953i
\(250\) 1.19717e7i 0.766187i
\(251\) 5.81400e6 0.367666 0.183833 0.982957i \(-0.441149\pi\)
0.183833 + 0.982957i \(0.441149\pi\)
\(252\) 3.77048e6 0.235611
\(253\) 2.57172e7 1.58804
\(254\) 5.03111e6i 0.307017i
\(255\) 1.01255e7 0.610653
\(256\) 1.04858e6 0.0625000
\(257\) 2.44206e7 1.43866 0.719328 0.694670i \(-0.244450\pi\)
0.719328 + 0.694670i \(0.244450\pi\)
\(258\) −3.68020e6 −0.214295
\(259\) 2.28927e7i 1.31764i
\(260\) 4.78727e6i 0.272375i
\(261\) −4.32669e6 −0.243352
\(262\) 1.66041e7 0.923231
\(263\) 6.22939e6 0.342435 0.171217 0.985233i \(-0.445230\pi\)
0.171217 + 0.985233i \(0.445230\pi\)
\(264\) 3.41342e6 0.185515
\(265\) 1.37472e7 0.738713
\(266\) 3.11526e7i 1.65520i
\(267\) 1.34275e7i 0.705440i
\(268\) 7.08405e6i 0.368025i
\(269\) 4.34695e6i 0.223320i 0.993746 + 0.111660i \(0.0356168\pi\)
−0.993746 + 0.111660i \(0.964383\pi\)
\(270\) 2.06349e6i 0.104836i
\(271\) −5.24553e6 −0.263561 −0.131781 0.991279i \(-0.542069\pi\)
−0.131781 + 0.991279i \(0.542069\pi\)
\(272\) 6.90707e6 0.343232
\(273\) 1.17426e7i 0.577133i
\(274\) 5.85096e6i 0.284430i
\(275\) 7.68335e6i 0.369447i
\(276\) 1.06051e7i 0.504415i
\(277\) 1.01301e7 0.476623 0.238311 0.971189i \(-0.423406\pi\)
0.238311 + 0.971189i \(0.423406\pi\)
\(278\) 1.38114e7i 0.642842i
\(279\) 8.85206e6i 0.407598i
\(280\) 8.45246e6i 0.385043i
\(281\) 2.20004e7 0.991543 0.495771 0.868453i \(-0.334885\pi\)
0.495771 + 0.868453i \(0.334885\pi\)
\(282\) 9.58709e6 0.427503
\(283\) 2.54701e7i 1.12376i 0.827220 + 0.561878i \(0.189921\pi\)
−0.827220 + 0.561878i \(0.810079\pi\)
\(284\) −2.32992e6 −0.101715
\(285\) 1.70490e7 0.736487
\(286\) 1.06306e7i 0.454422i
\(287\) −6.60170e7 −2.79261
\(288\) 1.40761e6i 0.0589256i
\(289\) 2.13600e7 0.884928
\(290\) 9.69934e6i 0.397693i
\(291\) 1.07092e7i 0.434587i
\(292\) 6.60494e6i 0.265290i
\(293\) −4.78962e7 −1.90414 −0.952069 0.305884i \(-0.901048\pi\)
−0.952069 + 0.305884i \(0.901048\pi\)
\(294\) 1.03584e7i 0.407615i
\(295\) 7.87820e6 + 1.81408e7i 0.306875 + 0.706626i
\(296\) −8.54635e6 −0.329538
\(297\) 4.58217e6i 0.174905i
\(298\) −7.78928e6 −0.294340
\(299\) −3.30280e7 −1.23557
\(300\) −3.16841e6 −0.117349
\(301\) 2.02364e7i 0.742052i
\(302\) 1.47195e6 0.0534408
\(303\) 2.46757e7i 0.887036i
\(304\) 1.16300e7 0.413960
\(305\) 6.80617e6i 0.239885i
\(306\) 9.27204e6i 0.323602i
\(307\) 2.38135e7 0.823015 0.411507 0.911406i \(-0.365002\pi\)
0.411507 + 0.911406i \(0.365002\pi\)
\(308\) 1.87695e7i 0.642393i
\(309\) 1.40710e6i 0.0476924i
\(310\) −1.98441e7 −0.666110
\(311\) 2.44963e7 0.814366 0.407183 0.913347i \(-0.366511\pi\)
0.407183 + 0.913347i \(0.366511\pi\)
\(312\) −4.38378e6 −0.144339
\(313\) 1.58989e7i 0.518483i 0.965813 + 0.259242i \(0.0834726\pi\)
−0.965813 + 0.259242i \(0.916527\pi\)
\(314\) −1.83865e7 −0.593896
\(315\) 1.13466e7 0.363022
\(316\) 2.92070e6 0.0925604
\(317\) −3.23668e7 −1.01607 −0.508033 0.861338i \(-0.669627\pi\)
−0.508033 + 0.861338i \(0.669627\pi\)
\(318\) 1.25885e7i 0.391464i
\(319\) 2.15383e7i 0.663498i
\(320\) 3.15550e6 0.0962981
\(321\) 2.12518e7 0.642512
\(322\) 5.83147e7 1.74667
\(323\) 7.66078e7 2.27335
\(324\) −1.88957e6 −0.0555556
\(325\) 9.86754e6i 0.287448i
\(326\) 2.46593e7i 0.711752i
\(327\) 3.19984e7i 0.915136i
\(328\) 2.46456e7i 0.698423i
\(329\) 5.27168e7i 1.48034i
\(330\) 1.02721e7 0.285835
\(331\) −6.21208e7 −1.71298 −0.856492 0.516161i \(-0.827361\pi\)
−0.856492 + 0.516161i \(0.827361\pi\)
\(332\) 1.88234e7i 0.514379i
\(333\) 1.14726e7i 0.310692i
\(334\) 1.52694e7i 0.409809i
\(335\) 2.13182e7i 0.567042i
\(336\) 7.74005e6 0.204045
\(337\) 4.23741e7i 1.10716i 0.832796 + 0.553581i \(0.186739\pi\)
−0.832796 + 0.553581i \(0.813261\pi\)
\(338\) 1.36519e7i 0.353545i
\(339\) 1.40719e7i 0.361204i
\(340\) 2.07856e7 0.528841
\(341\) −4.40657e7 −1.11132
\(342\) 1.56121e7i 0.390285i
\(343\) 88499.0 0.00219309
\(344\) −7.55471e6 −0.185585
\(345\) 3.19141e7i 0.777187i
\(346\) −1.29195e7 −0.311901
\(347\) 2.48057e7i 0.593694i −0.954925 0.296847i \(-0.904065\pi\)
0.954925 0.296847i \(-0.0959352\pi\)
\(348\) −8.88183e6 −0.210749
\(349\) 9.69535e6i 0.228080i 0.993476 + 0.114040i \(0.0363792\pi\)
−0.993476 + 0.114040i \(0.963621\pi\)
\(350\) 1.74222e7i 0.406350i
\(351\) 5.88477e6i 0.136084i
\(352\) 7.00708e6 0.160660
\(353\) 5.91784e7i 1.34536i 0.739932 + 0.672681i \(0.234857\pi\)
−0.739932 + 0.672681i \(0.765143\pi\)
\(354\) −1.66118e7 + 7.21419e6i −0.374461 + 0.162621i
\(355\) −7.01146e6 −0.156720
\(356\) 2.75639e7i 0.610929i
\(357\) 5.09845e7 1.12056
\(358\) 3.93022e7 0.856579
\(359\) 5.54850e6 0.119920 0.0599601 0.998201i \(-0.480903\pi\)
0.0599601 + 0.998201i \(0.480903\pi\)
\(360\) 4.23593e6i 0.0907907i
\(361\) 8.19447e7 1.74180
\(362\) 1.21520e7i 0.256166i
\(363\) −4.80580e6 −0.100472
\(364\) 2.41052e7i 0.499812i
\(365\) 1.98764e7i 0.408750i
\(366\) 6.23251e6 0.127122
\(367\) 7.93803e6i 0.160589i 0.996771 + 0.0802943i \(0.0255860\pi\)
−0.996771 + 0.0802943i \(0.974414\pi\)
\(368\) 2.17702e7i 0.436836i
\(369\) 3.30843e7 0.658480
\(370\) −2.57187e7 −0.507742
\(371\) 6.92207e7 1.35555
\(372\) 1.81715e7i 0.352990i
\(373\) 6.18624e7 1.19207 0.596033 0.802960i \(-0.296743\pi\)
0.596033 + 0.802960i \(0.296743\pi\)
\(374\) 4.61563e7 0.882300
\(375\) −3.29900e7 −0.625589
\(376\) 1.96804e7 0.370229
\(377\) 2.76611e7i 0.516233i
\(378\) 1.03902e7i 0.192375i
\(379\) 7.05181e7 1.29534 0.647668 0.761922i \(-0.275744\pi\)
0.647668 + 0.761922i \(0.275744\pi\)
\(380\) 3.49983e7 0.637817
\(381\) 1.38641e7 0.250679
\(382\) 3.32135e7 0.595832
\(383\) −6.13632e7 −1.09222 −0.546112 0.837712i \(-0.683893\pi\)
−0.546112 + 0.837712i \(0.683893\pi\)
\(384\) 2.88954e6i 0.0510310i
\(385\) 5.64833e7i 0.989779i
\(386\) 7.82376e7i 1.36036i
\(387\) 1.01414e7i 0.174971i
\(388\) 2.19838e7i 0.376364i
\(389\) −7.29573e7 −1.23942 −0.619712 0.784829i \(-0.712751\pi\)
−0.619712 + 0.784829i \(0.712751\pi\)
\(390\) −1.31922e7 −0.222394
\(391\) 1.43402e8i 2.39898i
\(392\) 2.12637e7i 0.353005i
\(393\) 4.57554e7i 0.753815i
\(394\) 8.37330e7i 1.36901i
\(395\) 8.78929e6 0.142614
\(396\) 9.40629e6i 0.151472i
\(397\) 4.18309e7i 0.668537i −0.942478 0.334268i \(-0.891511\pi\)
0.942478 0.334268i \(-0.108489\pi\)
\(398\) 979200.i 0.0155318i
\(399\) 8.58465e7 1.35146
\(400\) −6.50412e6 −0.101627
\(401\) 3.32904e7i 0.516280i 0.966107 + 0.258140i \(0.0831096\pi\)
−0.966107 + 0.258140i \(0.916890\pi\)
\(402\) 1.95214e7 0.300491
\(403\) 5.65925e7 0.864656
\(404\) 5.06543e7i 0.768196i
\(405\) −5.68631e6 −0.0855983
\(406\) 4.88388e7i 0.729772i
\(407\) −5.71108e7 −0.847100
\(408\) 1.90337e7i 0.280247i
\(409\) 3.59064e7i 0.524809i −0.964958 0.262405i \(-0.915484\pi\)
0.964958 0.262405i \(-0.0845155\pi\)
\(410\) 7.41665e7i 1.07611i
\(411\) 1.61233e7 0.232236
\(412\) 2.88849e6i 0.0413028i
\(413\) 3.96689e7 + 9.13437e7i 0.563119 + 1.29667i
\(414\) −2.92243e7 −0.411853
\(415\) 5.66454e7i 0.792539i
\(416\) −8.99902e6 −0.125002
\(417\) −3.80598e7 −0.524879
\(418\) 7.77170e7 1.06411
\(419\) 3.52266e6i 0.0478882i 0.999713 + 0.0239441i \(0.00762237\pi\)
−0.999713 + 0.0239441i \(0.992378\pi\)
\(420\) 2.32922e7 0.314386
\(421\) 3.59243e7i 0.481439i −0.970595 0.240720i \(-0.922617\pi\)
0.970595 0.240720i \(-0.0773835\pi\)
\(422\) −8.14011e7 −1.08316
\(423\) 2.64189e7i 0.349055i
\(424\) 2.58417e7i 0.339018i
\(425\) −4.28433e7 −0.558105
\(426\) 6.42050e6i 0.0830501i
\(427\) 3.42709e7i 0.440192i
\(428\) 4.36258e7 0.556432
\(429\) −2.92945e7 −0.371034
\(430\) −2.27345e7 −0.285944
\(431\) 2.35726e6i 0.0294425i −0.999892 0.0147213i \(-0.995314\pi\)
0.999892 0.0147213i \(-0.00468609\pi\)
\(432\) −3.87891e6 −0.0481125
\(433\) 3.16666e7 0.390066 0.195033 0.980797i \(-0.437519\pi\)
0.195033 + 0.980797i \(0.437519\pi\)
\(434\) −9.99203e7 −1.22232
\(435\) −2.67282e7 −0.324715
\(436\) 6.56864e7i 0.792531i
\(437\) 2.41458e8i 2.89332i
\(438\) 1.82011e7 0.216608
\(439\) 1.11257e7 0.131502 0.0657511 0.997836i \(-0.479056\pi\)
0.0657511 + 0.997836i \(0.479056\pi\)
\(440\) 2.10865e7 0.247541
\(441\) 2.85444e7 0.332816
\(442\) −5.92774e7 −0.686472
\(443\) 1.42196e8i 1.63560i −0.575503 0.817800i \(-0.695194\pi\)
0.575503 0.817800i \(-0.304806\pi\)
\(444\) 2.35510e7i 0.269067i
\(445\) 8.29484e7i 0.941300i
\(446\) 5.18392e6i 0.0584324i
\(447\) 2.14647e7i 0.240327i
\(448\) 1.58888e7 0.176708
\(449\) −7.96695e7 −0.880142 −0.440071 0.897963i \(-0.645047\pi\)
−0.440071 + 0.897963i \(0.645047\pi\)
\(450\) 8.73112e6i 0.0958147i
\(451\) 1.64694e8i 1.79534i
\(452\) 2.88868e7i 0.312812i
\(453\) 4.05622e6i 0.0436342i
\(454\) −1.18663e8 −1.26809
\(455\) 7.25402e7i 0.770095i
\(456\) 3.20484e7i 0.337997i
\(457\) 1.52943e8i 1.60244i −0.598373 0.801218i \(-0.704186\pi\)
0.598373 0.801218i \(-0.295814\pi\)
\(458\) −6.27033e7 −0.652671
\(459\) −2.55507e7 −0.264220
\(460\) 6.55134e7i 0.673064i
\(461\) −2.69808e7 −0.275392 −0.137696 0.990475i \(-0.543970\pi\)
−0.137696 + 0.990475i \(0.543970\pi\)
\(462\) 5.17227e7 0.524511
\(463\) 1.28669e7i 0.129638i 0.997897 + 0.0648188i \(0.0206469\pi\)
−0.997897 + 0.0648188i \(0.979353\pi\)
\(464\) −1.82326e7 −0.182514
\(465\) 5.46839e7i 0.543876i
\(466\) 2.73110e7 0.269885
\(467\) 1.81660e8i 1.78365i −0.452385 0.891823i \(-0.649427\pi\)
0.452385 0.891823i \(-0.350573\pi\)
\(468\) 1.20803e7i 0.117853i
\(469\) 1.07343e8i 1.04053i
\(470\) 5.92245e7 0.570437
\(471\) 5.06673e7i 0.484914i
\(472\) −3.41007e7 + 1.48093e7i −0.324293 + 0.140834i
\(473\) −5.04842e7 −0.477059
\(474\) 8.04849e6i 0.0755752i
\(475\) −7.21385e7 −0.673111
\(476\) 1.04661e8 0.970429
\(477\) −3.46898e7 −0.319629
\(478\) 5.77957e7i 0.529190i
\(479\) 7.95713e7 0.724019 0.362009 0.932174i \(-0.382091\pi\)
0.362009 + 0.932174i \(0.382091\pi\)
\(480\) 8.69552e6i 0.0786271i
\(481\) 7.33460e7 0.659084
\(482\) 1.03949e8i 0.928281i
\(483\) 1.60696e8i 1.42615i
\(484\) −9.86535e6 −0.0870115
\(485\) 6.61562e7i 0.579890i
\(486\) 5.20704e6i 0.0453609i
\(487\) −1.77168e8 −1.53391 −0.766953 0.641704i \(-0.778228\pi\)
−0.766953 + 0.641704i \(0.778228\pi\)
\(488\) 1.27941e7 0.110091
\(489\) 6.79531e7 0.581143
\(490\) 6.39892e7i 0.543899i
\(491\) 6.28678e7 0.531109 0.265555 0.964096i \(-0.414445\pi\)
0.265555 + 0.964096i \(0.414445\pi\)
\(492\) 6.79154e7 0.570260
\(493\) −1.20100e8 −1.00231
\(494\) −9.98101e7 −0.827930
\(495\) 2.83065e7i 0.233384i
\(496\) 3.73025e7i 0.305698i
\(497\) −3.53046e7 −0.287582
\(498\) −5.18711e7 −0.419988
\(499\) −2.45486e8 −1.97572 −0.987860 0.155344i \(-0.950351\pi\)
−0.987860 + 0.155344i \(0.950351\pi\)
\(500\) −6.77220e7 −0.541776
\(501\) 4.20775e7 0.334608
\(502\) 3.28889e7i 0.259979i
\(503\) 1.55411e8i 1.22118i −0.791949 0.610588i \(-0.790933\pi\)
0.791949 0.610588i \(-0.209067\pi\)
\(504\) 2.13291e7i 0.166602i
\(505\) 1.52435e8i 1.18361i
\(506\) 1.45479e8i 1.12292i
\(507\) −3.76203e7 −0.288668
\(508\) 2.84603e7 0.217094
\(509\) 2.69695e7i 0.204512i 0.994758 + 0.102256i \(0.0326061\pi\)
−0.994758 + 0.102256i \(0.967394\pi\)
\(510\) 5.72783e7i 0.431797i
\(511\) 1.00083e8i 0.750062i
\(512\) 5.93164e6i 0.0441942i
\(513\) −4.30218e7 −0.318666
\(514\) 1.38144e8i 1.01728i
\(515\) 8.69239e6i 0.0636381i
\(516\) 2.08183e7i 0.151529i
\(517\) 1.31514e8 0.951698
\(518\) −1.29501e8 −0.931713
\(519\) 3.56019e7i 0.254666i
\(520\) −2.70809e7 −0.192599
\(521\) −1.73264e8 −1.22517 −0.612583 0.790406i \(-0.709869\pi\)
−0.612583 + 0.790406i \(0.709869\pi\)
\(522\) 2.44755e7i 0.172076i
\(523\) 5.70724e7 0.398952 0.199476 0.979903i \(-0.436076\pi\)
0.199476 + 0.979903i \(0.436076\pi\)
\(524\) 9.39267e7i 0.652823i
\(525\) −4.80101e7 −0.331783
\(526\) 3.52387e7i 0.242138i
\(527\) 2.45715e8i 1.67881i
\(528\) 1.93092e7i 0.131179i
\(529\) −3.03950e8 −2.05322
\(530\) 7.77657e7i 0.522349i
\(531\) −1.98800e7 4.57767e7i −0.132780 0.305746i
\(532\) 1.76226e8 1.17040
\(533\) 2.11512e8i 1.39686i
\(534\) 7.59572e7 0.498821
\(535\) 1.31284e8 0.857333
\(536\) 4.00735e7 0.260233
\(537\) 1.08304e8i 0.699394i
\(538\) −2.45901e7 −0.157911
\(539\) 1.42094e8i 0.907423i
\(540\) −1.16729e7 −0.0741303
\(541\) 3.08310e8i 1.94713i 0.228406 + 0.973566i \(0.426649\pi\)
−0.228406 + 0.973566i \(0.573351\pi\)
\(542\) 2.96732e7i 0.186366i
\(543\) 3.34869e7 0.209158
\(544\) 3.90723e7i 0.242701i
\(545\) 1.97671e8i 1.22111i
\(546\) −6.64262e7 −0.408095
\(547\) 1.01408e8 0.619597 0.309799 0.950802i \(-0.399738\pi\)
0.309799 + 0.950802i \(0.399738\pi\)
\(548\) 3.30980e7 0.201122
\(549\) 1.71748e7i 0.103794i
\(550\) −4.34636e7 −0.261239
\(551\) −2.02222e8 −1.20885
\(552\) −5.99916e7 −0.356675
\(553\) 4.42565e7 0.261699
\(554\) 5.73046e7i 0.337023i
\(555\) 7.08723e7i 0.414570i
\(556\) −7.81293e7 −0.454558
\(557\) −8.87719e7 −0.513701 −0.256850 0.966451i \(-0.582685\pi\)
−0.256850 + 0.966451i \(0.582685\pi\)
\(558\) 5.00748e7 0.288215
\(559\) 6.48356e7 0.371174
\(560\) 4.78143e7 0.272266
\(561\) 1.27192e8i 0.720395i
\(562\) 1.24453e8i 0.701127i
\(563\) 1.40331e8i 0.786370i 0.919459 + 0.393185i \(0.128627\pi\)
−0.919459 + 0.393185i \(0.871373\pi\)
\(564\) 5.42328e7i 0.302291i
\(565\) 8.69294e7i 0.481971i
\(566\) −1.44081e8 −0.794616
\(567\) −2.86321e7 −0.157074
\(568\) 1.31800e7i 0.0719235i
\(569\) 1.25934e7i 0.0683606i −0.999416 0.0341803i \(-0.989118\pi\)
0.999416 0.0341803i \(-0.0108820\pi\)
\(570\) 9.64439e7i 0.520775i
\(571\) 1.67521e8i 0.899830i 0.893071 + 0.449915i \(0.148546\pi\)
−0.893071 + 0.449915i \(0.851454\pi\)
\(572\) −6.01357e7 −0.321325
\(573\) 9.15256e7i 0.486495i
\(574\) 3.73449e8i 1.97467i
\(575\) 1.35036e8i 0.710309i
\(576\) −7.96262e6 −0.0416667
\(577\) 6.14313e7 0.319788 0.159894 0.987134i \(-0.448885\pi\)
0.159894 + 0.987134i \(0.448885\pi\)
\(578\) 1.20830e8i 0.625738i
\(579\) 2.15598e8 1.11073
\(580\) −5.48677e7 −0.281212
\(581\) 2.85225e8i 1.45432i
\(582\) −6.05802e7 −0.307300
\(583\) 1.72686e8i 0.871469i
\(584\) 3.73632e7 0.187588
\(585\) 3.63533e7i 0.181584i
\(586\) 2.70942e8i 1.34643i
\(587\) 2.18923e8i 1.08237i −0.840902 0.541187i \(-0.817975\pi\)
0.840902 0.541187i \(-0.182025\pi\)
\(588\) 5.85959e7 0.288227
\(589\) 4.13730e8i 2.02475i
\(590\) −1.02620e8 + 4.45658e7i −0.499660 + 0.216993i
\(591\) 2.30741e8 1.11780
\(592\) 4.83455e7i 0.233019i
\(593\) −1.32061e8 −0.633301 −0.316651 0.948542i \(-0.602558\pi\)
−0.316651 + 0.948542i \(0.602558\pi\)
\(594\) −2.59207e7 −0.123676
\(595\) 3.14958e8 1.49521
\(596\) 4.40628e7i 0.208130i
\(597\) −2.69836e6 −0.0126817
\(598\) 1.86835e8i 0.873683i
\(599\) −3.65742e8 −1.70174 −0.850872 0.525373i \(-0.823926\pi\)
−0.850872 + 0.525373i \(0.823926\pi\)
\(600\) 1.79232e7i 0.0829780i
\(601\) 1.30989e8i 0.603408i 0.953402 + 0.301704i \(0.0975554\pi\)
−0.953402 + 0.301704i \(0.902445\pi\)
\(602\) −1.14474e8 −0.524710
\(603\) 5.37945e7i 0.245350i
\(604\) 8.32662e6i 0.0377883i
\(605\) −2.96880e7 −0.134065
\(606\) −1.39587e8 −0.627230
\(607\) 2.79266e8 1.24868 0.624342 0.781151i \(-0.285367\pi\)
0.624342 + 0.781151i \(0.285367\pi\)
\(608\) 6.57891e7i 0.292714i
\(609\) −1.34584e8 −0.595856
\(610\) 3.85015e7 0.169624
\(611\) −1.68900e8 −0.740467
\(612\) −5.24506e7 −0.228821
\(613\) 3.16953e8i 1.37599i 0.725717 + 0.687993i \(0.241508\pi\)
−0.725717 + 0.687993i \(0.758492\pi\)
\(614\) 1.34709e8i 0.581959i
\(615\) 2.04379e8 0.878640
\(616\) 1.06176e8 0.454240
\(617\) −4.07741e8 −1.73592 −0.867959 0.496636i \(-0.834568\pi\)
−0.867959 + 0.496636i \(0.834568\pi\)
\(618\) 7.95975e6 0.0337236
\(619\) −2.85584e8 −1.20410 −0.602049 0.798459i \(-0.705649\pi\)
−0.602049 + 0.798459i \(0.705649\pi\)
\(620\) 1.12255e8i 0.471011i
\(621\) 8.05326e7i 0.336277i
\(622\) 1.38572e8i 0.575844i
\(623\) 4.17668e8i 1.72730i
\(624\) 2.47984e7i 0.102063i
\(625\) −1.04552e8 −0.428244
\(626\) −8.99379e7 −0.366623
\(627\) 2.14163e8i 0.868843i
\(628\) 1.04010e8i 0.419948i
\(629\) 3.18456e8i 1.27967i
\(630\) 6.41859e7i 0.256695i
\(631\) 1.30499e8 0.519421 0.259710 0.965687i \(-0.416373\pi\)
0.259710 + 0.965687i \(0.416373\pi\)
\(632\) 1.65220e7i 0.0654501i
\(633\) 2.24315e8i 0.884397i
\(634\) 1.83094e8i 0.718467i
\(635\) 8.56459e7 0.334492
\(636\) −7.12113e7 −0.276807
\(637\) 1.82488e8i 0.706018i
\(638\) −1.21839e8 −0.469164
\(639\) 1.76928e7 0.0678101
\(640\) 1.78502e7i 0.0680930i
\(641\) 1.85260e8 0.703409 0.351704 0.936111i \(-0.385602\pi\)
0.351704 + 0.936111i \(0.385602\pi\)
\(642\) 1.20219e8i 0.454325i
\(643\) 9.95522e6 0.0374471 0.0187235 0.999825i \(-0.494040\pi\)
0.0187235 + 0.999825i \(0.494040\pi\)
\(644\) 3.29878e8i 1.23508i
\(645\) 6.26490e7i 0.233472i
\(646\) 4.33359e8i 1.60750i
\(647\) 1.38325e8 0.510724 0.255362 0.966845i \(-0.417805\pi\)
0.255362 + 0.966845i \(0.417805\pi\)
\(648\) 1.06890e7i 0.0392837i
\(649\) −2.27877e8 + 9.89627e7i −0.833616 + 0.362024i
\(650\) 5.58192e7 0.203256
\(651\) 2.75348e8i 0.998020i
\(652\) 1.39494e8 0.503284
\(653\) −3.95730e8 −1.42121 −0.710606 0.703590i \(-0.751579\pi\)
−0.710606 + 0.703590i \(0.751579\pi\)
\(654\) 1.81011e8 0.647099
\(655\) 2.82655e8i 1.00585i
\(656\) 1.39417e8 0.493860
\(657\) 5.01563e7i 0.176860i
\(658\) 2.98211e8 1.04676
\(659\) 2.30747e8i 0.806269i −0.915141 0.403134i \(-0.867921\pi\)
0.915141 0.403134i \(-0.132079\pi\)
\(660\) 5.81076e7i 0.202116i
\(661\) −5.18516e8 −1.79539 −0.897693 0.440622i \(-0.854758\pi\)
−0.897693 + 0.440622i \(0.854758\pi\)
\(662\) 3.51408e8i 1.21126i
\(663\) 1.63349e8i 0.560502i
\(664\) −1.06481e8 −0.363721
\(665\) 5.30319e8 1.80332
\(666\) 6.48989e7 0.219692
\(667\) 3.78540e8i 1.27566i
\(668\) 8.63766e7 0.289779
\(669\) −1.42852e7 −0.0477099
\(670\) 1.20594e8 0.400959
\(671\) 8.54962e7 0.282995
\(672\) 4.37843e7i 0.144282i
\(673\) 5.69561e8i 1.86851i 0.356609 + 0.934254i \(0.383933\pi\)
−0.356609 + 0.934254i \(0.616067\pi\)
\(674\) −2.39704e8 −0.782881
\(675\) 2.40601e7 0.0782324
\(676\) −7.72271e7 −0.249994
\(677\) −2.95557e8 −0.952523 −0.476261 0.879304i \(-0.658008\pi\)
−0.476261 + 0.879304i \(0.658008\pi\)
\(678\) 7.96025e7 0.255410
\(679\) 3.33114e8i 1.06410i
\(680\) 1.17581e8i 0.373947i
\(681\) 3.26997e8i 1.03539i
\(682\) 2.49273e8i 0.785818i
\(683\) 6.28756e7i 0.197342i −0.995120 0.0986711i \(-0.968541\pi\)
0.995120 0.0986711i \(-0.0314592\pi\)
\(684\) −8.83151e7 −0.275973
\(685\) 9.96023e7 0.309883
\(686\) 500626.i 0.00155075i
\(687\) 1.72790e8i 0.532903i
\(688\) 4.27359e7i 0.131228i
\(689\) 2.21777e8i 0.678045i
\(690\) −1.80534e8 −0.549554
\(691\) 5.64077e8i 1.70964i −0.518927 0.854818i \(-0.673668\pi\)
0.518927 0.854818i \(-0.326332\pi\)
\(692\) 7.30836e7i 0.220547i
\(693\) 1.42531e8i 0.428262i
\(694\) 1.40322e8 0.419805
\(695\) −2.35116e8 −0.700369
\(696\) 5.02432e7i 0.149022i
\(697\) 9.18352e8 2.71213
\(698\) −5.48452e7 −0.161277
\(699\) 7.52602e7i 0.220360i
\(700\) −9.85551e7 −0.287333
\(701\) 2.43173e8i 0.705930i −0.935636 0.352965i \(-0.885173\pi\)
0.935636 0.352965i \(-0.114827\pi\)
\(702\) 3.32893e7 0.0962262
\(703\) 5.36210e8i 1.54337i
\(704\) 3.96380e7i 0.113604i
\(705\) 1.63204e8i 0.465760i
\(706\) −3.34764e8 −0.951315
\(707\) 7.67550e8i 2.17194i
\(708\) −4.08096e7 9.39704e7i −0.114991 0.264784i
\(709\) 2.32305e6 0.00651808 0.00325904 0.999995i \(-0.498963\pi\)
0.00325904 + 0.999995i \(0.498963\pi\)
\(710\) 3.96628e7i 0.110818i
\(711\) −2.21790e7 −0.0617069
\(712\) 1.55925e8 0.431992
\(713\) 7.74463e8 2.13664
\(714\) 2.88412e8i 0.792352i
\(715\) −1.80967e8 −0.495087
\(716\) 2.22327e8i 0.605693i
\(717\) 1.59266e8 0.432082
\(718\) 3.13870e7i 0.0847963i
\(719\) 2.75575e8i 0.741401i −0.928752 0.370701i \(-0.879118\pi\)
0.928752 0.370701i \(-0.120882\pi\)
\(720\) −2.39620e7 −0.0641987
\(721\) 4.37685e7i 0.116777i
\(722\) 4.63549e8i 1.23164i
\(723\) 2.86450e8 0.757938
\(724\) 6.87419e7 0.181136
\(725\) 1.13094e8 0.296773
\(726\) 2.71857e7i 0.0710446i
\(727\) −4.51085e8 −1.17396 −0.586982 0.809600i \(-0.699684\pi\)
−0.586982 + 0.809600i \(0.699684\pi\)
\(728\) −1.36360e8 −0.353421
\(729\) 1.43489e7 0.0370370
\(730\) 1.12438e8 0.289030
\(731\) 2.81506e8i 0.720668i
\(732\) 3.52564e7i 0.0898886i
\(733\) 5.04739e8 1.28161 0.640803 0.767705i \(-0.278601\pi\)
0.640803 + 0.767705i \(0.278601\pi\)
\(734\) −4.49043e7 −0.113553
\(735\) 1.76333e8 0.444092
\(736\) −1.23151e8 −0.308890
\(737\) 2.67790e8 0.668947
\(738\) 1.87153e8i 0.465616i
\(739\) 3.61892e8i 0.896697i 0.893859 + 0.448349i \(0.147988\pi\)
−0.893859 + 0.448349i \(0.852012\pi\)
\(740\) 1.45487e8i 0.359028i
\(741\) 2.75044e8i 0.676002i
\(742\) 3.91572e8i 0.958516i
\(743\) 1.82011e8 0.443742 0.221871 0.975076i \(-0.428784\pi\)
0.221871 + 0.975076i \(0.428784\pi\)
\(744\) 1.02794e8 0.249602
\(745\) 1.32599e8i 0.320680i
\(746\) 3.49947e8i 0.842918i
\(747\) 1.42940e8i 0.342919i
\(748\) 2.61100e8i 0.623880i
\(749\) 6.61050e8 1.57322
\(750\) 1.86620e8i 0.442358i
\(751\) 2.25377e8i 0.532096i 0.963960 + 0.266048i \(0.0857179\pi\)
−0.963960 + 0.266048i \(0.914282\pi\)
\(752\) 1.11329e8i 0.261791i
\(753\) −9.06312e7 −0.212272
\(754\) 1.56475e8 0.365032
\(755\) 2.50574e7i 0.0582231i
\(756\) −5.87760e7 −0.136030
\(757\) −1.78181e8 −0.410746 −0.205373 0.978684i \(-0.565841\pi\)
−0.205373 + 0.978684i \(0.565841\pi\)
\(758\) 3.98910e8i 0.915941i
\(759\) −4.00892e8 −0.916858
\(760\) 1.97980e8i 0.451004i
\(761\) 1.51836e7 0.0344525 0.0172262 0.999852i \(-0.494516\pi\)
0.0172262 + 0.999852i \(0.494516\pi\)
\(762\) 7.84273e7i 0.177257i
\(763\) 9.95328e8i 2.24075i
\(764\) 1.87884e8i 0.421317i
\(765\) −1.57840e8 −0.352561
\(766\) 3.47123e8i 0.772319i
\(767\) 2.92657e8 1.27095e8i 0.648593 0.281672i
\(768\) −1.63457e7 −0.0360844
\(769\) 1.40680e7i 0.0309353i −0.999880 0.0154676i \(-0.995076\pi\)
0.999880 0.0154676i \(-0.00492370\pi\)
\(770\) 3.19518e8 0.699879
\(771\) −3.80680e8 −0.830609
\(772\) 4.42579e8 0.961919
\(773\) 2.32040e8i 0.502370i 0.967939 + 0.251185i \(0.0808203\pi\)
−0.967939 + 0.251185i \(0.919180\pi\)
\(774\) 5.73686e7 0.123723
\(775\) 2.31381e8i 0.497075i
\(776\) −1.24359e8 −0.266129
\(777\) 3.56861e8i 0.760741i
\(778\) 4.12709e8i 0.876405i
\(779\) 1.54630e9 3.27101
\(780\) 7.46262e7i 0.157256i
\(781\) 8.80750e7i 0.184884i
\(782\) −8.11206e8 −1.69633
\(783\) 6.74464e7 0.140499
\(784\) 1.20286e8 0.249612
\(785\) 3.12998e8i 0.647043i
\(786\) −2.58832e8 −0.533027
\(787\) −4.47870e7 −0.0918814 −0.0459407 0.998944i \(-0.514629\pi\)
−0.0459407 + 0.998944i \(0.514629\pi\)
\(788\) 4.73665e8 0.968039
\(789\) −9.71065e7 −0.197705
\(790\) 4.97198e7i 0.100843i
\(791\) 4.37713e8i 0.884423i
\(792\) −5.32100e7 −0.107107
\(793\) −1.09801e8 −0.220184
\(794\) 2.36631e8 0.472727
\(795\) −2.14297e8 −0.426496
\(796\) −5.53919e6 −0.0109826
\(797\) 5.69778e8i 1.12546i −0.826640 0.562731i \(-0.809751\pi\)
0.826640 0.562731i \(-0.190249\pi\)
\(798\) 4.85621e8i 0.955628i
\(799\) 7.33336e8i 1.43768i
\(800\) 3.67928e7i 0.0718610i
\(801\) 2.09313e8i 0.407286i
\(802\) −1.88319e8 −0.365065
\(803\) 2.49678e8 0.482208
\(804\) 1.10429e8i 0.212479i
\(805\) 9.92706e8i 1.90297i
\(806\) 3.20135e8i 0.611404i
\(807\) 6.77623e7i 0.128934i
\(808\) −2.86544e8 −0.543197
\(809\) 7.98785e8i 1.50864i 0.656509 + 0.754318i \(0.272032\pi\)
−0.656509 + 0.754318i \(0.727968\pi\)
\(810\) 3.21666e7i 0.0605271i
\(811\) 2.80075e8i 0.525064i 0.964923 + 0.262532i \(0.0845575\pi\)
−0.964923 + 0.262532i \(0.915442\pi\)
\(812\) −2.76274e8 −0.516026
\(813\) 8.17697e7 0.152167
\(814\) 3.23067e8i 0.598990i
\(815\) 4.19782e8 0.775445
\(816\) −1.07671e8 −0.198165
\(817\) 4.73993e8i 0.869172i
\(818\) 2.03117e8 0.371096
\(819\) 1.83049e8i 0.333208i
\(820\) 4.19549e8 0.760924
\(821\) 4.15041e8i 0.750001i −0.927025 0.375001i \(-0.877642\pi\)
0.927025 0.375001i \(-0.122358\pi\)
\(822\) 9.12074e7i 0.164216i
\(823\) 5.73292e8i 1.02843i 0.857660 + 0.514217i \(0.171917\pi\)
−0.857660 + 0.514217i \(0.828083\pi\)
\(824\) 1.63398e7 0.0292055
\(825\) 1.19772e8i 0.213300i
\(826\) −5.16718e8 + 2.24401e8i −0.916882 + 0.398185i
\(827\) −2.00658e8 −0.354765 −0.177382 0.984142i \(-0.556763\pi\)
−0.177382 + 0.984142i \(0.556763\pi\)
\(828\) 1.65317e8i 0.291224i
\(829\) −4.37927e8 −0.768667 −0.384334 0.923194i \(-0.625569\pi\)
−0.384334 + 0.923194i \(0.625569\pi\)
\(830\) −3.20435e8 −0.560410
\(831\) −1.57913e8 −0.275178
\(832\) 5.09062e7i 0.0883894i
\(833\) 7.92334e8 1.37080
\(834\) 2.15299e8i 0.371145i
\(835\) 2.59935e8 0.446483
\(836\) 4.39634e8i 0.752440i
\(837\) 1.37990e8i 0.235327i
\(838\) −1.99272e7 −0.0338621
\(839\) 5.51306e8i 0.933485i −0.884393 0.466742i \(-0.845428\pi\)
0.884393 0.466742i \(-0.154572\pi\)
\(840\) 1.31761e8i 0.222305i
\(841\) −2.77794e8 −0.467020
\(842\) 2.03218e8 0.340429
\(843\) −3.42952e8 −0.572467
\(844\) 4.60474e8i 0.765910i
\(845\) −2.32401e8 −0.385183
\(846\) −1.49448e8 −0.246819
\(847\) −1.49487e8 −0.246010
\(848\) −1.46183e8 −0.239722
\(849\) 3.97040e8i 0.648801i
\(850\) 2.42358e8i 0.394640i
\(851\) 1.00373e9 1.62866
\(852\) 3.63198e7 0.0587253
\(853\) −3.16637e8 −0.510169 −0.255085 0.966919i \(-0.582103\pi\)
−0.255085 + 0.966919i \(0.582103\pi\)
\(854\) 1.93865e8 0.311263
\(855\) −2.65768e8 −0.425211
\(856\) 2.46785e8i 0.393457i
\(857\) 8.21327e8i 1.30489i −0.757837 0.652444i \(-0.773744\pi\)
0.757837 0.652444i \(-0.226256\pi\)
\(858\) 1.65715e8i 0.262361i
\(859\) 8.41438e7i 0.132752i −0.997795 0.0663762i \(-0.978856\pi\)
0.997795 0.0663762i \(-0.0211437\pi\)
\(860\) 1.28606e8i 0.202193i
\(861\) 1.02910e9 1.61231
\(862\) 1.33347e7 0.0208190
\(863\) 4.15889e8i 0.647061i −0.946218 0.323531i \(-0.895130\pi\)
0.946218 0.323531i \(-0.104870\pi\)
\(864\) 2.19424e7i 0.0340207i
\(865\) 2.19932e8i 0.339812i
\(866\) 1.79134e8i 0.275818i
\(867\) −3.32969e8 −0.510913
\(868\) 5.65235e8i 0.864310i
\(869\) 1.10407e8i 0.168244i
\(870\) 1.51198e8i 0.229608i
\(871\) −3.43916e8 −0.520473
\(872\) 3.71579e8 0.560404
\(873\) 1.66939e8i 0.250909i
\(874\) −1.36589e9 −2.04589
\(875\) −1.02617e9 −1.53178
\(876\) 1.02961e8i 0.153165i
\(877\) 6.99610e8 1.03719 0.518593 0.855021i \(-0.326456\pi\)
0.518593 + 0.855021i \(0.326456\pi\)
\(878\) 6.29363e7i 0.0929861i
\(879\) 7.46628e8 1.09935
\(880\) 1.19283e8i 0.175038i
\(881\) 8.40662e8i 1.22940i 0.788760 + 0.614701i \(0.210723\pi\)
−0.788760 + 0.614701i \(0.789277\pi\)
\(882\) 1.61471e8i 0.235337i
\(883\) −8.39866e8 −1.21991 −0.609955 0.792436i \(-0.708812\pi\)
−0.609955 + 0.792436i \(0.708812\pi\)
\(884\) 3.35324e8i 0.485409i
\(885\) −1.22809e8 2.82787e8i −0.177174 0.407971i
\(886\) 8.04384e8 1.15654
\(887\) 6.76545e8i 0.969451i 0.874666 + 0.484725i \(0.161080\pi\)
−0.874666 + 0.484725i \(0.838920\pi\)
\(888\) 1.33224e8 0.190259
\(889\) 4.31250e8 0.613796
\(890\) 4.69227e8 0.665600
\(891\) 7.14290e7i 0.100981i
\(892\) −2.93247e7 −0.0413180
\(893\) 1.23477e9i 1.73394i
\(894\) 1.21423e8 0.169937
\(895\) 6.69051e8i 0.933233i
\(896\) 8.98805e7i 0.124952i
\(897\) 5.14856e8 0.713359
\(898\) 4.50678e8i 0.622354i
\(899\) 6.48616e8i 0.892707i
\(900\) 4.93906e7 0.0677512
\(901\) −9.62919e8 −1.31648
\(902\) 9.31649e8 1.26950
\(903\) 3.15455e8i 0.428424i
\(904\) 1.63408e8 0.221192
\(905\) 2.06866e8 0.279090
\(906\) −2.29455e7 −0.0308540
\(907\) −1.01769e9 −1.36393 −0.681967 0.731383i \(-0.738875\pi\)
−0.681967 + 0.731383i \(0.738875\pi\)
\(908\) 6.71261e8i 0.896672i
\(909\) 3.84656e8i 0.512131i
\(910\) −4.10349e8 −0.544540
\(911\) 8.60976e8 1.13877 0.569385 0.822071i \(-0.307181\pi\)
0.569385 + 0.822071i \(0.307181\pi\)
\(912\) −1.81293e8 −0.239000
\(913\) −7.11556e8 −0.934968
\(914\) 8.65175e8 1.13309
\(915\) 1.06098e8i 0.138498i
\(916\) 3.54704e8i 0.461508i
\(917\) 1.42324e9i 1.84574i
\(918\) 1.44537e8i 0.186832i
\(919\) 8.40861e8i 1.08337i 0.840581 + 0.541686i \(0.182214\pi\)
−0.840581 + 0.541686i \(0.817786\pi\)
\(920\) −3.70600e8 −0.475928
\(921\) −3.71215e8 −0.475168
\(922\) 1.52626e8i 0.194732i
\(923\) 1.13113e8i 0.143849i
\(924\) 2.92588e8i 0.370886i
\(925\) 2.99878e8i 0.378895i
\(926\) −7.27861e7 −0.0916676
\(927\) 2.19345e7i 0.0275352i
\(928\) 1.03139e8i 0.129057i
\(929\) 6.55950e8i 0.818132i −0.912505 0.409066i \(-0.865855\pi\)
0.912505 0.409066i \(-0.134145\pi\)
\(930\) 3.09339e8 0.384579
\(931\) 1.33411e9 1.65327
\(932\) 1.54494e8i 0.190838i
\(933\) −3.81860e8 −0.470175
\(934\) 1.02762e9 1.26123
\(935\) 7.85731e8i 0.961256i
\(936\) 6.83363e7 0.0833344
\(937\) 5.07264e8i 0.616616i 0.951287 + 0.308308i \(0.0997628\pi\)
−0.951287 + 0.308308i \(0.900237\pi\)
\(938\) 6.07222e8 0.735765
\(939\) 2.47840e8i 0.299346i
\(940\) 3.35024e8i 0.403360i
\(941\) 1.31730e9i 1.58094i 0.612499 + 0.790471i \(0.290164\pi\)
−0.612499 + 0.790471i \(0.709836\pi\)
\(942\) 2.86617e8 0.342886
\(943\) 2.89453e9i 3.45177i
\(944\) −8.37740e7 1.92903e8i −0.0995849 0.229310i
\(945\) −1.76876e8 −0.209591
\(946\) 2.85582e8i 0.337331i
\(947\) 2.50024e8 0.294396 0.147198 0.989107i \(-0.452974\pi\)
0.147198 + 0.989107i \(0.452974\pi\)
\(948\) −4.55291e7 −0.0534398
\(949\) −3.20656e8 −0.375181
\(950\) 4.08077e8i 0.475961i
\(951\) 5.04548e8 0.586626
\(952\) 5.92052e8i 0.686197i
\(953\) −9.44099e8 −1.09078 −0.545392 0.838181i \(-0.683619\pi\)
−0.545392 + 0.838181i \(0.683619\pi\)
\(954\) 1.96235e8i 0.226012i
\(955\) 5.65401e8i 0.649153i
\(956\) 3.26942e8 0.374194
\(957\) 3.35749e8i 0.383071i
\(958\) 4.50123e8i 0.511958i
\(959\) 5.01525e8 0.568639
\(960\) −4.91893e7 −0.0555977
\(961\) −4.39514e8 −0.495225
\(962\) 4.14907e8i 0.466043i
\(963\) −3.31283e8 −0.370955
\(964\) 5.88024e8 0.656394
\(965\) 1.33186e9 1.48210
\(966\) −9.09036e8 −1.00844
\(967\) 1.02380e9i 1.13223i −0.824325 0.566117i \(-0.808445\pi\)
0.824325 0.566117i \(-0.191555\pi\)
\(968\) 5.58069e7i 0.0615264i
\(969\) −1.19420e9 −1.31252
\(970\) −3.74236e8 −0.410044
\(971\) −7.02238e8 −0.767055 −0.383528 0.923529i \(-0.625291\pi\)
−0.383528 + 0.923529i \(0.625291\pi\)
\(972\) 2.94555e7 0.0320750
\(973\) −1.18387e9 −1.28519
\(974\) 1.00221e9i 1.08463i
\(975\) 1.53820e8i 0.165958i
\(976\) 7.23744e7i 0.0778458i
\(977\) 1.30511e9i 1.39946i −0.714405 0.699732i \(-0.753303\pi\)
0.714405 0.699732i \(-0.246697\pi\)
\(978\) 3.84401e8i 0.410930i
\(979\) 1.04196e9 1.11046
\(980\) 3.61978e8 0.384595
\(981\) 4.98806e8i 0.528354i
\(982\) 3.55634e8i 0.375551i
\(983\) 7.66861e8i 0.807339i 0.914905 + 0.403670i \(0.132265\pi\)
−0.914905 + 0.403670i \(0.867735\pi\)
\(984\) 3.84188e8i 0.403235i
\(985\) 1.42541e9 1.49152
\(986\) 6.79389e8i 0.708741i
\(987\) 8.21774e8i 0.854675i
\(988\) 5.64611e8i 0.585435i
\(989\) 8.87269e8 0.917205
\(990\) −1.60126e8 −0.165027
\(991\) 1.81812e9i 1.86811i −0.357133 0.934054i \(-0.616246\pi\)
0.357133 0.934054i \(-0.383754\pi\)
\(992\) 2.11015e8 0.216161
\(993\) 9.68368e8 0.988991
\(994\) 1.99713e8i 0.203351i
\(995\) −1.66692e7 −0.0169217
\(996\) 2.93427e8i 0.296977i
\(997\) −7.87647e8 −0.794778 −0.397389 0.917650i \(-0.630084\pi\)
−0.397389 + 0.917650i \(0.630084\pi\)
\(998\) 1.38868e9i 1.39705i
\(999\) 1.78840e8i 0.179378i
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 354.7.d.a.235.20 yes 60
59.58 odd 2 inner 354.7.d.a.235.19 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
354.7.d.a.235.19 60 59.58 odd 2 inner
354.7.d.a.235.20 yes 60 1.1 even 1 trivial