Properties

Label 325.6.b.g.274.8
Level $325$
Weight $6$
Character 325.274
Analytic conductor $52.125$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,6,Mod(274,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.274");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.b (of order \(2\), degree \(1\), not 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: \(52.1247414392\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 326x^{10} + 38809x^{8} + 2034064x^{6} + 43897824x^{4} + 281822976x^{2} + 377913600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.8
Root \(2.75663i\) of defining polynomial
Character \(\chi\) \(=\) 325.274
Dual form 325.6.b.g.274.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.75663i q^{2} +23.9621i q^{3} +24.4010 q^{4} -66.0547 q^{6} -85.7300i q^{7} +155.477i q^{8} -331.182 q^{9} +431.046 q^{11} +584.699i q^{12} +169.000i q^{13} +236.326 q^{14} +352.239 q^{16} +438.894i q^{17} -912.946i q^{18} -1617.28 q^{19} +2054.27 q^{21} +1188.23i q^{22} +2173.44i q^{23} -3725.55 q^{24} -465.871 q^{26} -2113.02i q^{27} -2091.90i q^{28} -8289.72 q^{29} -2745.88 q^{31} +5946.25i q^{32} +10328.8i q^{33} -1209.87 q^{34} -8081.16 q^{36} +2137.03i q^{37} -4458.24i q^{38} -4049.59 q^{39} -19520.9 q^{41} +5662.87i q^{42} +8153.29i q^{43} +10517.9 q^{44} -5991.39 q^{46} +13235.5i q^{47} +8440.39i q^{48} +9457.36 q^{49} -10516.8 q^{51} +4123.77i q^{52} +1753.17i q^{53} +5824.83 q^{54} +13329.0 q^{56} -38753.4i q^{57} -22851.7i q^{58} +1976.46 q^{59} +45578.3 q^{61} -7569.39i q^{62} +28392.2i q^{63} -5119.96 q^{64} -28472.6 q^{66} +19457.7i q^{67} +10709.5i q^{68} -52080.3 q^{69} -64224.9 q^{71} -51491.1i q^{72} +1029.22i q^{73} -5891.00 q^{74} -39463.2 q^{76} -36953.6i q^{77} -11163.2i q^{78} +107661. q^{79} -29844.7 q^{81} -53812.1i q^{82} -46473.4i q^{83} +50126.2 q^{84} -22475.6 q^{86} -198639. i q^{87} +67017.6i q^{88} +3410.51 q^{89} +14488.4 q^{91} +53034.2i q^{92} -65797.1i q^{93} -36485.3 q^{94} -142485. q^{96} -133264. i q^{97} +26070.5i q^{98} -142755. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 268 q^{4} + 636 q^{6} - 1036 q^{9} - 340 q^{11} + 2880 q^{14} + 7012 q^{16} - 2436 q^{19} - 792 q^{21} - 25236 q^{24} - 16728 q^{29} + 5724 q^{31} + 42968 q^{34} - 4276 q^{36} - 12844 q^{39} + 4496 q^{41}+ \cdots + 64540 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.75663i 0.487308i 0.969862 + 0.243654i \(0.0783462\pi\)
−0.969862 + 0.243654i \(0.921654\pi\)
\(3\) 23.9621i 1.53717i 0.639748 + 0.768584i \(0.279039\pi\)
−0.639748 + 0.768584i \(0.720961\pi\)
\(4\) 24.4010 0.762531
\(5\) 0 0
\(6\) −66.0547 −0.749075
\(7\) − 85.7300i − 0.661284i −0.943756 0.330642i \(-0.892735\pi\)
0.943756 0.330642i \(-0.107265\pi\)
\(8\) 155.477i 0.858896i
\(9\) −331.182 −1.36289
\(10\) 0 0
\(11\) 431.046 1.07409 0.537046 0.843553i \(-0.319540\pi\)
0.537046 + 0.843553i \(0.319540\pi\)
\(12\) 584.699i 1.17214i
\(13\) 169.000i 0.277350i
\(14\) 236.326 0.322249
\(15\) 0 0
\(16\) 352.239 0.343984
\(17\) 438.894i 0.368331i 0.982895 + 0.184165i \(0.0589582\pi\)
−0.982895 + 0.184165i \(0.941042\pi\)
\(18\) − 912.946i − 0.664147i
\(19\) −1617.28 −1.02778 −0.513891 0.857856i \(-0.671796\pi\)
−0.513891 + 0.857856i \(0.671796\pi\)
\(20\) 0 0
\(21\) 2054.27 1.01650
\(22\) 1188.23i 0.523414i
\(23\) 2173.44i 0.856700i 0.903613 + 0.428350i \(0.140905\pi\)
−0.903613 + 0.428350i \(0.859095\pi\)
\(24\) −3725.55 −1.32027
\(25\) 0 0
\(26\) −465.871 −0.135155
\(27\) − 2113.02i − 0.557821i
\(28\) − 2091.90i − 0.504249i
\(29\) −8289.72 −1.83040 −0.915198 0.403005i \(-0.867966\pi\)
−0.915198 + 0.403005i \(0.867966\pi\)
\(30\) 0 0
\(31\) −2745.88 −0.513190 −0.256595 0.966519i \(-0.582601\pi\)
−0.256595 + 0.966519i \(0.582601\pi\)
\(32\) 5946.25i 1.02652i
\(33\) 10328.8i 1.65106i
\(34\) −1209.87 −0.179491
\(35\) 0 0
\(36\) −8081.16 −1.03924
\(37\) 2137.03i 0.256629i 0.991734 + 0.128315i \(0.0409567\pi\)
−0.991734 + 0.128315i \(0.959043\pi\)
\(38\) − 4458.24i − 0.500846i
\(39\) −4049.59 −0.426334
\(40\) 0 0
\(41\) −19520.9 −1.81360 −0.906799 0.421562i \(-0.861482\pi\)
−0.906799 + 0.421562i \(0.861482\pi\)
\(42\) 5662.87i 0.495351i
\(43\) 8153.29i 0.672452i 0.941781 + 0.336226i \(0.109151\pi\)
−0.941781 + 0.336226i \(0.890849\pi\)
\(44\) 10517.9 0.819029
\(45\) 0 0
\(46\) −5991.39 −0.417477
\(47\) 13235.5i 0.873966i 0.899470 + 0.436983i \(0.143953\pi\)
−0.899470 + 0.436983i \(0.856047\pi\)
\(48\) 8440.39i 0.528761i
\(49\) 9457.36 0.562704
\(50\) 0 0
\(51\) −10516.8 −0.566186
\(52\) 4123.77i 0.211488i
\(53\) 1753.17i 0.0857303i 0.999081 + 0.0428651i \(0.0136486\pi\)
−0.999081 + 0.0428651i \(0.986351\pi\)
\(54\) 5824.83 0.271831
\(55\) 0 0
\(56\) 13329.0 0.567974
\(57\) − 38753.4i − 1.57987i
\(58\) − 22851.7i − 0.891967i
\(59\) 1976.46 0.0739192 0.0369596 0.999317i \(-0.488233\pi\)
0.0369596 + 0.999317i \(0.488233\pi\)
\(60\) 0 0
\(61\) 45578.3 1.56832 0.784158 0.620561i \(-0.213095\pi\)
0.784158 + 0.620561i \(0.213095\pi\)
\(62\) − 7569.39i − 0.250082i
\(63\) 28392.2i 0.901256i
\(64\) −5119.96 −0.156249
\(65\) 0 0
\(66\) −28472.6 −0.804576
\(67\) 19457.7i 0.529546i 0.964311 + 0.264773i \(0.0852970\pi\)
−0.964311 + 0.264773i \(0.914703\pi\)
\(68\) 10709.5i 0.280863i
\(69\) −52080.3 −1.31689
\(70\) 0 0
\(71\) −64224.9 −1.51202 −0.756010 0.654561i \(-0.772854\pi\)
−0.756010 + 0.654561i \(0.772854\pi\)
\(72\) − 51491.1i − 1.17058i
\(73\) 1029.22i 0.0226048i 0.999936 + 0.0113024i \(0.00359775\pi\)
−0.999936 + 0.0113024i \(0.996402\pi\)
\(74\) −5891.00 −0.125058
\(75\) 0 0
\(76\) −39463.2 −0.783715
\(77\) − 36953.6i − 0.710280i
\(78\) − 11163.2i − 0.207756i
\(79\) 107661. 1.94085 0.970424 0.241407i \(-0.0776090\pi\)
0.970424 + 0.241407i \(0.0776090\pi\)
\(80\) 0 0
\(81\) −29844.7 −0.505423
\(82\) − 53812.1i − 0.883782i
\(83\) − 46473.4i − 0.740473i −0.928938 0.370237i \(-0.879277\pi\)
0.928938 0.370237i \(-0.120723\pi\)
\(84\) 50126.2 0.775116
\(85\) 0 0
\(86\) −22475.6 −0.327692
\(87\) − 198639.i − 2.81363i
\(88\) 67017.6i 0.922534i
\(89\) 3410.51 0.0456398 0.0228199 0.999740i \(-0.492736\pi\)
0.0228199 + 0.999740i \(0.492736\pi\)
\(90\) 0 0
\(91\) 14488.4 0.183407
\(92\) 53034.2i 0.653260i
\(93\) − 65797.1i − 0.788859i
\(94\) −36485.3 −0.425891
\(95\) 0 0
\(96\) −142485. −1.57794
\(97\) − 133264.i − 1.43808i −0.694967 0.719041i \(-0.744581\pi\)
0.694967 0.719041i \(-0.255419\pi\)
\(98\) 26070.5i 0.274210i
\(99\) −142755. −1.46387
\(100\) 0 0
\(101\) 112628. 1.09860 0.549302 0.835624i \(-0.314894\pi\)
0.549302 + 0.835624i \(0.314894\pi\)
\(102\) − 28991.0i − 0.275907i
\(103\) 102456.i 0.951581i 0.879559 + 0.475791i \(0.157838\pi\)
−0.879559 + 0.475791i \(0.842162\pi\)
\(104\) −26275.6 −0.238215
\(105\) 0 0
\(106\) −4832.84 −0.0417771
\(107\) 90417.9i 0.763475i 0.924271 + 0.381738i \(0.124674\pi\)
−0.924271 + 0.381738i \(0.875326\pi\)
\(108\) − 51559.8i − 0.425356i
\(109\) −164916. −1.32953 −0.664764 0.747053i \(-0.731468\pi\)
−0.664764 + 0.747053i \(0.731468\pi\)
\(110\) 0 0
\(111\) −51207.7 −0.394483
\(112\) − 30197.5i − 0.227471i
\(113\) − 36453.6i − 0.268562i −0.990943 0.134281i \(-0.957128\pi\)
0.990943 0.134281i \(-0.0428724\pi\)
\(114\) 106829. 0.769885
\(115\) 0 0
\(116\) −202277. −1.39573
\(117\) − 55969.7i − 0.377997i
\(118\) 5448.36i 0.0360214i
\(119\) 37626.4 0.243571
\(120\) 0 0
\(121\) 24749.6 0.153676
\(122\) 125643.i 0.764253i
\(123\) − 467763.i − 2.78781i
\(124\) −67002.3 −0.391323
\(125\) 0 0
\(126\) −78266.9 −0.439189
\(127\) − 178579.i − 0.982476i −0.871025 0.491238i \(-0.836544\pi\)
0.871025 0.491238i \(-0.163456\pi\)
\(128\) 176166.i 0.950381i
\(129\) −195370. −1.03367
\(130\) 0 0
\(131\) 43555.7 0.221752 0.110876 0.993834i \(-0.464634\pi\)
0.110876 + 0.993834i \(0.464634\pi\)
\(132\) 252032.i 1.25899i
\(133\) 138649.i 0.679655i
\(134\) −53637.6 −0.258052
\(135\) 0 0
\(136\) −68237.9 −0.316358
\(137\) 347518.i 1.58189i 0.611888 + 0.790944i \(0.290410\pi\)
−0.611888 + 0.790944i \(0.709590\pi\)
\(138\) − 143566.i − 0.641733i
\(139\) −54594.1 −0.239667 −0.119834 0.992794i \(-0.538236\pi\)
−0.119834 + 0.992794i \(0.538236\pi\)
\(140\) 0 0
\(141\) −317150. −1.34343
\(142\) − 177044.i − 0.736819i
\(143\) 72846.8i 0.297900i
\(144\) −116655. −0.468812
\(145\) 0 0
\(146\) −2837.18 −0.0110155
\(147\) 226618.i 0.864971i
\(148\) 52145.6i 0.195688i
\(149\) 249528. 0.920777 0.460388 0.887718i \(-0.347710\pi\)
0.460388 + 0.887718i \(0.347710\pi\)
\(150\) 0 0
\(151\) 398788. 1.42331 0.711655 0.702529i \(-0.247946\pi\)
0.711655 + 0.702529i \(0.247946\pi\)
\(152\) − 251449.i − 0.882757i
\(153\) − 145354.i − 0.501994i
\(154\) 101867. 0.346125
\(155\) 0 0
\(156\) −98814.1 −0.325093
\(157\) − 432099.i − 1.39905i −0.714606 0.699527i \(-0.753394\pi\)
0.714606 0.699527i \(-0.246606\pi\)
\(158\) 296782.i 0.945791i
\(159\) −42009.6 −0.131782
\(160\) 0 0
\(161\) 186329. 0.566522
\(162\) − 82271.0i − 0.246297i
\(163\) 56309.9i 0.166003i 0.996549 + 0.0830014i \(0.0264506\pi\)
−0.996549 + 0.0830014i \(0.973549\pi\)
\(164\) −476330. −1.38292
\(165\) 0 0
\(166\) 128110. 0.360839
\(167\) 512611.i 1.42232i 0.703031 + 0.711160i \(0.251830\pi\)
−0.703031 + 0.711160i \(0.748170\pi\)
\(168\) 319391.i 0.873072i
\(169\) −28561.0 −0.0769231
\(170\) 0 0
\(171\) 535613. 1.40075
\(172\) 198948.i 0.512766i
\(173\) − 363030.i − 0.922204i −0.887347 0.461102i \(-0.847454\pi\)
0.887347 0.461102i \(-0.152546\pi\)
\(174\) 547575. 1.37110
\(175\) 0 0
\(176\) 151831. 0.369471
\(177\) 47360.0i 0.113626i
\(178\) 9401.51i 0.0222406i
\(179\) 812489. 1.89533 0.947665 0.319265i \(-0.103436\pi\)
0.947665 + 0.319265i \(0.103436\pi\)
\(180\) 0 0
\(181\) −464974. −1.05495 −0.527475 0.849570i \(-0.676861\pi\)
−0.527475 + 0.849570i \(0.676861\pi\)
\(182\) 39939.1i 0.0893758i
\(183\) 1.09215e6i 2.41077i
\(184\) −337920. −0.735816
\(185\) 0 0
\(186\) 181378. 0.384418
\(187\) 189184.i 0.395621i
\(188\) 322959.i 0.666426i
\(189\) −181150. −0.368878
\(190\) 0 0
\(191\) −15537.6 −0.0308177 −0.0154088 0.999881i \(-0.504905\pi\)
−0.0154088 + 0.999881i \(0.504905\pi\)
\(192\) − 122685.i − 0.240181i
\(193\) − 319408.i − 0.617238i −0.951186 0.308619i \(-0.900133\pi\)
0.951186 0.308619i \(-0.0998667\pi\)
\(194\) 367360. 0.700789
\(195\) 0 0
\(196\) 230769. 0.429079
\(197\) − 635878.i − 1.16737i −0.811980 0.583685i \(-0.801610\pi\)
0.811980 0.583685i \(-0.198390\pi\)
\(198\) − 393522.i − 0.713355i
\(199\) −57426.3 −0.102797 −0.0513983 0.998678i \(-0.516368\pi\)
−0.0513983 + 0.998678i \(0.516368\pi\)
\(200\) 0 0
\(201\) −466246. −0.814002
\(202\) 310473.i 0.535359i
\(203\) 710678.i 1.21041i
\(204\) −256621. −0.431734
\(205\) 0 0
\(206\) −282434. −0.463713
\(207\) − 719806.i − 1.16759i
\(208\) 59528.5i 0.0954039i
\(209\) −697121. −1.10393
\(210\) 0 0
\(211\) −407094. −0.629489 −0.314745 0.949176i \(-0.601919\pi\)
−0.314745 + 0.949176i \(0.601919\pi\)
\(212\) 42779.1i 0.0653720i
\(213\) − 1.53896e6i − 2.32423i
\(214\) −249249. −0.372048
\(215\) 0 0
\(216\) 328526. 0.479110
\(217\) 235405.i 0.339364i
\(218\) − 454614.i − 0.647890i
\(219\) −24662.3 −0.0347474
\(220\) 0 0
\(221\) −74173.2 −0.102157
\(222\) − 141161.i − 0.192235i
\(223\) 94338.9i 0.127037i 0.997981 + 0.0635183i \(0.0202321\pi\)
−0.997981 + 0.0635183i \(0.979768\pi\)
\(224\) 509772. 0.678822
\(225\) 0 0
\(226\) 100489. 0.130872
\(227\) − 251896.i − 0.324456i −0.986753 0.162228i \(-0.948132\pi\)
0.986753 0.162228i \(-0.0518680\pi\)
\(228\) − 945620.i − 1.20470i
\(229\) 428033. 0.539372 0.269686 0.962948i \(-0.413080\pi\)
0.269686 + 0.962948i \(0.413080\pi\)
\(230\) 0 0
\(231\) 885485. 1.09182
\(232\) − 1.28886e6i − 1.57212i
\(233\) 47973.8i 0.0578914i 0.999581 + 0.0289457i \(0.00921499\pi\)
−0.999581 + 0.0289457i \(0.990785\pi\)
\(234\) 154288. 0.184201
\(235\) 0 0
\(236\) 48227.5 0.0563657
\(237\) 2.57979e6i 2.98341i
\(238\) 103722.i 0.118694i
\(239\) 1.39917e6 1.58444 0.792220 0.610236i \(-0.208925\pi\)
0.792220 + 0.610236i \(0.208925\pi\)
\(240\) 0 0
\(241\) −576792. −0.639700 −0.319850 0.947468i \(-0.603633\pi\)
−0.319850 + 0.947468i \(0.603633\pi\)
\(242\) 68225.6i 0.0748875i
\(243\) − 1.22861e6i − 1.33474i
\(244\) 1.11216e6 1.19589
\(245\) 0 0
\(246\) 1.28945e6 1.35852
\(247\) − 273320.i − 0.285055i
\(248\) − 426921.i − 0.440776i
\(249\) 1.11360e6 1.13823
\(250\) 0 0
\(251\) 676974. 0.678247 0.339124 0.940742i \(-0.389870\pi\)
0.339124 + 0.940742i \(0.389870\pi\)
\(252\) 692798.i 0.687235i
\(253\) 936855.i 0.920176i
\(254\) 492278. 0.478769
\(255\) 0 0
\(256\) −649464. −0.619377
\(257\) 2.05124e6i 1.93724i 0.248545 + 0.968620i \(0.420047\pi\)
−0.248545 + 0.968620i \(0.579953\pi\)
\(258\) − 538562.i − 0.503717i
\(259\) 183208. 0.169705
\(260\) 0 0
\(261\) 2.74541e6 2.49462
\(262\) 120067.i 0.108061i
\(263\) 1.51723e6i 1.35258i 0.736636 + 0.676289i \(0.236413\pi\)
−0.736636 + 0.676289i \(0.763587\pi\)
\(264\) −1.60588e6 −1.41809
\(265\) 0 0
\(266\) −382205. −0.331201
\(267\) 81722.8i 0.0701561i
\(268\) 474786.i 0.403795i
\(269\) −180817. −0.152355 −0.0761777 0.997094i \(-0.524272\pi\)
−0.0761777 + 0.997094i \(0.524272\pi\)
\(270\) 0 0
\(271\) −1.58203e6 −1.30856 −0.654278 0.756254i \(-0.727028\pi\)
−0.654278 + 0.756254i \(0.727028\pi\)
\(272\) 154596.i 0.126700i
\(273\) 347172.i 0.281928i
\(274\) −957979. −0.770867
\(275\) 0 0
\(276\) −1.27081e6 −1.00417
\(277\) 1.53514e6i 1.20212i 0.799202 + 0.601062i \(0.205256\pi\)
−0.799202 + 0.601062i \(0.794744\pi\)
\(278\) − 150496.i − 0.116792i
\(279\) 909387. 0.699420
\(280\) 0 0
\(281\) 959201. 0.724676 0.362338 0.932047i \(-0.381979\pi\)
0.362338 + 0.932047i \(0.381979\pi\)
\(282\) − 874265.i − 0.654666i
\(283\) 2.63997e6i 1.95945i 0.200353 + 0.979724i \(0.435791\pi\)
−0.200353 + 0.979724i \(0.564209\pi\)
\(284\) −1.56715e6 −1.15296
\(285\) 0 0
\(286\) −200812. −0.145169
\(287\) 1.67353e6i 1.19930i
\(288\) − 1.96929e6i − 1.39903i
\(289\) 1.22723e6 0.864333
\(290\) 0 0
\(291\) 3.19329e6 2.21058
\(292\) 25114.0i 0.0172369i
\(293\) 286709.i 0.195106i 0.995230 + 0.0975532i \(0.0311016\pi\)
−0.995230 + 0.0975532i \(0.968898\pi\)
\(294\) −624703. −0.421507
\(295\) 0 0
\(296\) −332258. −0.220418
\(297\) − 910810.i − 0.599152i
\(298\) 687858.i 0.448702i
\(299\) −367312. −0.237606
\(300\) 0 0
\(301\) 698981. 0.444682
\(302\) 1.09931e6i 0.693591i
\(303\) 2.69879e6i 1.68874i
\(304\) −569669. −0.353540
\(305\) 0 0
\(306\) 400687. 0.244626
\(307\) − 1.89896e6i − 1.14993i −0.818179 0.574963i \(-0.805016\pi\)
0.818179 0.574963i \(-0.194984\pi\)
\(308\) − 901704.i − 0.541610i
\(309\) −2.45507e6 −1.46274
\(310\) 0 0
\(311\) 1.74416e6 1.02255 0.511277 0.859416i \(-0.329173\pi\)
0.511277 + 0.859416i \(0.329173\pi\)
\(312\) − 629618.i − 0.366176i
\(313\) 605507.i 0.349348i 0.984626 + 0.174674i \(0.0558871\pi\)
−0.984626 + 0.174674i \(0.944113\pi\)
\(314\) 1.19114e6 0.681770
\(315\) 0 0
\(316\) 2.62704e6 1.47996
\(317\) 2.19697e6i 1.22793i 0.789332 + 0.613967i \(0.210427\pi\)
−0.789332 + 0.613967i \(0.789573\pi\)
\(318\) − 115805.i − 0.0642184i
\(319\) −3.57325e6 −1.96601
\(320\) 0 0
\(321\) −2.16660e6 −1.17359
\(322\) 513642.i 0.276071i
\(323\) − 709814.i − 0.378563i
\(324\) −728241. −0.385401
\(325\) 0 0
\(326\) −155226. −0.0808945
\(327\) − 3.95174e6i − 2.04371i
\(328\) − 3.03505e6i − 1.55769i
\(329\) 1.13468e6 0.577940
\(330\) 0 0
\(331\) −3.68205e6 −1.84722 −0.923612 0.383329i \(-0.874777\pi\)
−0.923612 + 0.383329i \(0.874777\pi\)
\(332\) − 1.13400e6i − 0.564633i
\(333\) − 707745.i − 0.349757i
\(334\) −1.41308e6 −0.693108
\(335\) 0 0
\(336\) 723595. 0.349661
\(337\) 3.46729e6i 1.66309i 0.555460 + 0.831543i \(0.312542\pi\)
−0.555460 + 0.831543i \(0.687458\pi\)
\(338\) − 78732.2i − 0.0374852i
\(339\) 873504. 0.412825
\(340\) 0 0
\(341\) −1.18360e6 −0.551214
\(342\) 1.47649e6i 0.682597i
\(343\) − 2.25164e6i − 1.03339i
\(344\) −1.26765e6 −0.577566
\(345\) 0 0
\(346\) 1.00074e6 0.449398
\(347\) − 3.91532e6i − 1.74559i −0.488083 0.872797i \(-0.662304\pi\)
0.488083 0.872797i \(-0.337696\pi\)
\(348\) − 4.84699e6i − 2.14548i
\(349\) 3.32051e6 1.45929 0.729644 0.683827i \(-0.239686\pi\)
0.729644 + 0.683827i \(0.239686\pi\)
\(350\) 0 0
\(351\) 357101. 0.154712
\(352\) 2.56311e6i 1.10258i
\(353\) 3.09575e6i 1.32230i 0.750255 + 0.661148i \(0.229931\pi\)
−0.750255 + 0.661148i \(0.770069\pi\)
\(354\) −130554. −0.0553710
\(355\) 0 0
\(356\) 83219.7 0.0348017
\(357\) 901608.i 0.374410i
\(358\) 2.23973e6i 0.923610i
\(359\) 2.71429e6 1.11153 0.555764 0.831340i \(-0.312426\pi\)
0.555764 + 0.831340i \(0.312426\pi\)
\(360\) 0 0
\(361\) 139488. 0.0563340
\(362\) − 1.28176e6i − 0.514086i
\(363\) 593053.i 0.236226i
\(364\) 353531. 0.139854
\(365\) 0 0
\(366\) −3.01066e6 −1.17479
\(367\) − 2.54964e6i − 0.988130i −0.869425 0.494065i \(-0.835511\pi\)
0.869425 0.494065i \(-0.164489\pi\)
\(368\) 765573.i 0.294691i
\(369\) 6.46498e6 2.47173
\(370\) 0 0
\(371\) 150299. 0.0566920
\(372\) − 1.60551e6i − 0.601530i
\(373\) 3.67946e6i 1.36934i 0.728852 + 0.684671i \(0.240054\pi\)
−0.728852 + 0.684671i \(0.759946\pi\)
\(374\) −521510. −0.192790
\(375\) 0 0
\(376\) −2.05781e6 −0.750646
\(377\) − 1.40096e6i − 0.507660i
\(378\) − 499362.i − 0.179757i
\(379\) 1.06822e6 0.382000 0.191000 0.981590i \(-0.438827\pi\)
0.191000 + 0.981590i \(0.438827\pi\)
\(380\) 0 0
\(381\) 4.27914e6 1.51023
\(382\) − 42831.4i − 0.0150177i
\(383\) − 601784.i − 0.209625i −0.994492 0.104813i \(-0.966576\pi\)
0.994492 0.104813i \(-0.0334243\pi\)
\(384\) −4.22131e6 −1.46090
\(385\) 0 0
\(386\) 880490. 0.300785
\(387\) − 2.70022e6i − 0.916478i
\(388\) − 3.25178e6i − 1.09658i
\(389\) −5.00396e6 −1.67664 −0.838319 0.545180i \(-0.816461\pi\)
−0.838319 + 0.545180i \(0.816461\pi\)
\(390\) 0 0
\(391\) −953913. −0.315549
\(392\) 1.47040e6i 0.483304i
\(393\) 1.04369e6i 0.340870i
\(394\) 1.75288e6 0.568869
\(395\) 0 0
\(396\) −3.48335e6 −1.11625
\(397\) − 3.47583e6i − 1.10683i −0.832904 0.553417i \(-0.813323\pi\)
0.832904 0.553417i \(-0.186677\pi\)
\(398\) − 158303.i − 0.0500936i
\(399\) −3.32233e6 −1.04474
\(400\) 0 0
\(401\) −5.15091e6 −1.59964 −0.799822 0.600238i \(-0.795073\pi\)
−0.799822 + 0.600238i \(0.795073\pi\)
\(402\) − 1.28527e6i − 0.396670i
\(403\) − 464054.i − 0.142333i
\(404\) 2.74822e6 0.837719
\(405\) 0 0
\(406\) −1.95908e6 −0.589843
\(407\) 921158.i 0.275644i
\(408\) − 1.63512e6i − 0.486295i
\(409\) 1.11646e6 0.330015 0.165008 0.986292i \(-0.447235\pi\)
0.165008 + 0.986292i \(0.447235\pi\)
\(410\) 0 0
\(411\) −8.32726e6 −2.43163
\(412\) 2.50004e6i 0.725610i
\(413\) − 169442.i − 0.0488816i
\(414\) 1.98424e6 0.568975
\(415\) 0 0
\(416\) −1.00492e6 −0.284706
\(417\) − 1.30819e6i − 0.368409i
\(418\) − 1.92171e6i − 0.537955i
\(419\) −1.79740e6 −0.500160 −0.250080 0.968225i \(-0.580457\pi\)
−0.250080 + 0.968225i \(0.580457\pi\)
\(420\) 0 0
\(421\) 5.31302e6 1.46095 0.730477 0.682937i \(-0.239298\pi\)
0.730477 + 0.682937i \(0.239298\pi\)
\(422\) − 1.12221e6i − 0.306755i
\(423\) − 4.38335e6i − 1.19112i
\(424\) −272577. −0.0736334
\(425\) 0 0
\(426\) 4.24235e6 1.13262
\(427\) − 3.90743e6i − 1.03710i
\(428\) 2.20629e6i 0.582173i
\(429\) −1.74556e6 −0.457922
\(430\) 0 0
\(431\) −4.63630e6 −1.20221 −0.601103 0.799172i \(-0.705272\pi\)
−0.601103 + 0.799172i \(0.705272\pi\)
\(432\) − 744290.i − 0.191881i
\(433\) − 3.68664e6i − 0.944955i −0.881343 0.472478i \(-0.843360\pi\)
0.881343 0.472478i \(-0.156640\pi\)
\(434\) −648924. −0.165375
\(435\) 0 0
\(436\) −4.02412e6 −1.01381
\(437\) − 3.51506e6i − 0.880501i
\(438\) − 67984.8i − 0.0169327i
\(439\) 1.67032e6 0.413656 0.206828 0.978377i \(-0.433686\pi\)
0.206828 + 0.978377i \(0.433686\pi\)
\(440\) 0 0
\(441\) −3.13211e6 −0.766903
\(442\) − 204468.i − 0.0497817i
\(443\) 3.59209e6i 0.869637i 0.900518 + 0.434819i \(0.143187\pi\)
−0.900518 + 0.434819i \(0.856813\pi\)
\(444\) −1.24952e6 −0.300805
\(445\) 0 0
\(446\) −260058. −0.0619060
\(447\) 5.97922e6i 1.41539i
\(448\) 438934.i 0.103325i
\(449\) −80455.2 −0.0188338 −0.00941690 0.999956i \(-0.502998\pi\)
−0.00941690 + 0.999956i \(0.502998\pi\)
\(450\) 0 0
\(451\) −8.41443e6 −1.94797
\(452\) − 889503.i − 0.204786i
\(453\) 9.55579e6i 2.18787i
\(454\) 694384. 0.158110
\(455\) 0 0
\(456\) 6.02525e6 1.35695
\(457\) − 2.87687e6i − 0.644362i −0.946678 0.322181i \(-0.895584\pi\)
0.946678 0.322181i \(-0.104416\pi\)
\(458\) 1.17993e6i 0.262840i
\(459\) 927394. 0.205463
\(460\) 0 0
\(461\) 81644.7 0.0178927 0.00894635 0.999960i \(-0.497152\pi\)
0.00894635 + 0.999960i \(0.497152\pi\)
\(462\) 2.44096e6i 0.532053i
\(463\) − 2.50214e6i − 0.542450i −0.962516 0.271225i \(-0.912571\pi\)
0.962516 0.271225i \(-0.0874287\pi\)
\(464\) −2.91997e6 −0.629626
\(465\) 0 0
\(466\) −132246. −0.0282109
\(467\) − 1.38773e6i − 0.294452i −0.989103 0.147226i \(-0.952966\pi\)
0.989103 0.147226i \(-0.0470344\pi\)
\(468\) − 1.36572e6i − 0.288235i
\(469\) 1.66811e6 0.350180
\(470\) 0 0
\(471\) 1.03540e7 2.15058
\(472\) 307293.i 0.0634889i
\(473\) 3.51444e6i 0.722276i
\(474\) −7.11152e6 −1.45384
\(475\) 0 0
\(476\) 918122. 0.185730
\(477\) − 580618.i − 0.116841i
\(478\) 3.85700e6i 0.772111i
\(479\) 4.96374e6 0.988485 0.494243 0.869324i \(-0.335445\pi\)
0.494243 + 0.869324i \(0.335445\pi\)
\(480\) 0 0
\(481\) −361158. −0.0711761
\(482\) − 1.59000e6i − 0.311731i
\(483\) 4.46484e6i 0.870840i
\(484\) 603915. 0.117182
\(485\) 0 0
\(486\) 3.38682e6 0.650431
\(487\) − 8.57371e6i − 1.63812i −0.573706 0.819061i \(-0.694495\pi\)
0.573706 0.819061i \(-0.305505\pi\)
\(488\) 7.08637e6i 1.34702i
\(489\) −1.34930e6 −0.255174
\(490\) 0 0
\(491\) 2.55302e6 0.477915 0.238958 0.971030i \(-0.423194\pi\)
0.238958 + 0.971030i \(0.423194\pi\)
\(492\) − 1.14139e7i − 2.12579i
\(493\) − 3.63831e6i − 0.674191i
\(494\) 753442. 0.138910
\(495\) 0 0
\(496\) −967209. −0.176529
\(497\) 5.50600e6i 0.999874i
\(498\) 3.06979e6i 0.554670i
\(499\) −161861. −0.0290999 −0.0145500 0.999894i \(-0.504632\pi\)
−0.0145500 + 0.999894i \(0.504632\pi\)
\(500\) 0 0
\(501\) −1.22832e7 −2.18634
\(502\) 1.86617e6i 0.330515i
\(503\) 5.40474e6i 0.952477i 0.879316 + 0.476239i \(0.158000\pi\)
−0.879316 + 0.476239i \(0.842000\pi\)
\(504\) −4.41433e6 −0.774085
\(505\) 0 0
\(506\) −2.58256e6 −0.448409
\(507\) − 684381.i − 0.118244i
\(508\) − 4.35751e6i − 0.749169i
\(509\) −1.72999e6 −0.295971 −0.147985 0.988990i \(-0.547279\pi\)
−0.147985 + 0.988990i \(0.547279\pi\)
\(510\) 0 0
\(511\) 88235.1 0.0149482
\(512\) 3.84698e6i 0.648553i
\(513\) 3.41735e6i 0.573318i
\(514\) −5.65451e6 −0.944033
\(515\) 0 0
\(516\) −4.76721e6 −0.788207
\(517\) 5.70510e6i 0.938721i
\(518\) 505036.i 0.0826985i
\(519\) 8.69896e6 1.41758
\(520\) 0 0
\(521\) 8.15642e6 1.31645 0.658226 0.752820i \(-0.271307\pi\)
0.658226 + 0.752820i \(0.271307\pi\)
\(522\) 7.56807e6i 1.21565i
\(523\) − 7.57602e6i − 1.21112i −0.795800 0.605559i \(-0.792949\pi\)
0.795800 0.605559i \(-0.207051\pi\)
\(524\) 1.06280e6 0.169093
\(525\) 0 0
\(526\) −4.18245e6 −0.659123
\(527\) − 1.20515e6i − 0.189023i
\(528\) 3.63820e6i 0.567939i
\(529\) 1.71248e6 0.266064
\(530\) 0 0
\(531\) −654567. −0.100744
\(532\) 3.38318e6i 0.518258i
\(533\) − 3.29904e6i − 0.503002i
\(534\) −225280. −0.0341876
\(535\) 0 0
\(536\) −3.02521e6 −0.454825
\(537\) 1.94689e7i 2.91344i
\(538\) − 498445.i − 0.0742440i
\(539\) 4.07656e6 0.604396
\(540\) 0 0
\(541\) 1.29055e7 1.89575 0.947876 0.318640i \(-0.103226\pi\)
0.947876 + 0.318640i \(0.103226\pi\)
\(542\) − 4.36108e6i − 0.637670i
\(543\) − 1.11417e7i − 1.62164i
\(544\) −2.60978e6 −0.378099
\(545\) 0 0
\(546\) −957025. −0.137386
\(547\) 7.61965e6i 1.08885i 0.838811 + 0.544423i \(0.183252\pi\)
−0.838811 + 0.544423i \(0.816748\pi\)
\(548\) 8.47978e6i 1.20624i
\(549\) −1.50947e7 −2.13744
\(550\) 0 0
\(551\) 1.34068e7 1.88125
\(552\) − 8.09727e6i − 1.13107i
\(553\) − 9.22980e6i − 1.28345i
\(554\) −4.23182e6 −0.585805
\(555\) 0 0
\(556\) −1.33215e6 −0.182754
\(557\) 7.35984e6i 1.00515i 0.864534 + 0.502574i \(0.167614\pi\)
−0.864534 + 0.502574i \(0.832386\pi\)
\(558\) 2.50685e6i 0.340833i
\(559\) −1.37791e6 −0.186505
\(560\) 0 0
\(561\) −4.53324e6 −0.608137
\(562\) 2.64416e6i 0.353140i
\(563\) − 5.68526e6i − 0.755927i −0.925821 0.377963i \(-0.876625\pi\)
0.925821 0.377963i \(-0.123375\pi\)
\(564\) −7.73876e6 −1.02441
\(565\) 0 0
\(566\) −7.27744e6 −0.954855
\(567\) 2.55859e6i 0.334228i
\(568\) − 9.98547e6i − 1.29867i
\(569\) 1.43108e7 1.85304 0.926519 0.376249i \(-0.122786\pi\)
0.926519 + 0.376249i \(0.122786\pi\)
\(570\) 0 0
\(571\) −1.95049e6 −0.250354 −0.125177 0.992134i \(-0.539950\pi\)
−0.125177 + 0.992134i \(0.539950\pi\)
\(572\) 1.77753e6i 0.227158i
\(573\) − 372313.i − 0.0473720i
\(574\) −4.61331e6 −0.584430
\(575\) 0 0
\(576\) 1.69564e6 0.212950
\(577\) 987263.i 0.123451i 0.998093 + 0.0617253i \(0.0196603\pi\)
−0.998093 + 0.0617253i \(0.980340\pi\)
\(578\) 3.38302e6i 0.421196i
\(579\) 7.65368e6 0.948798
\(580\) 0 0
\(581\) −3.98417e6 −0.489663
\(582\) 8.80272e6i 1.07723i
\(583\) 755697.i 0.0920823i
\(584\) −160020. −0.0194152
\(585\) 0 0
\(586\) −790350. −0.0950770
\(587\) − 1.11813e6i − 0.133936i −0.997755 0.0669679i \(-0.978668\pi\)
0.997755 0.0669679i \(-0.0213325\pi\)
\(588\) 5.52971e6i 0.659567i
\(589\) 4.44086e6 0.527447
\(590\) 0 0
\(591\) 1.52370e7 1.79445
\(592\) 752746.i 0.0882763i
\(593\) − 905001.i − 0.105685i −0.998603 0.0528424i \(-0.983172\pi\)
0.998603 0.0528424i \(-0.0168281\pi\)
\(594\) 2.51077e6 0.291971
\(595\) 0 0
\(596\) 6.08874e6 0.702121
\(597\) − 1.37606e6i − 0.158016i
\(598\) − 1.01254e6i − 0.115787i
\(599\) 5.11859e6 0.582885 0.291443 0.956588i \(-0.405865\pi\)
0.291443 + 0.956588i \(0.405865\pi\)
\(600\) 0 0
\(601\) 8.84991e6 0.999431 0.499716 0.866190i \(-0.333438\pi\)
0.499716 + 0.866190i \(0.333438\pi\)
\(602\) 1.92683e6i 0.216697i
\(603\) − 6.44403e6i − 0.721712i
\(604\) 9.73082e6 1.08532
\(605\) 0 0
\(606\) −7.43957e6 −0.822937
\(607\) − 1.37995e7i − 1.52017i −0.649824 0.760085i \(-0.725157\pi\)
0.649824 0.760085i \(-0.274843\pi\)
\(608\) − 9.61674e6i − 1.05504i
\(609\) −1.70293e7 −1.86061
\(610\) 0 0
\(611\) −2.23679e6 −0.242395
\(612\) − 3.54678e6i − 0.382785i
\(613\) 1.55631e7i 1.67280i 0.548118 + 0.836401i \(0.315345\pi\)
−0.548118 + 0.836401i \(0.684655\pi\)
\(614\) 5.23473e6 0.560368
\(615\) 0 0
\(616\) 5.74542e6 0.610057
\(617\) − 1.57277e7i − 1.66323i −0.555353 0.831615i \(-0.687417\pi\)
0.555353 0.831615i \(-0.312583\pi\)
\(618\) − 6.76772e6i − 0.712806i
\(619\) 5.68261e6 0.596103 0.298051 0.954550i \(-0.403663\pi\)
0.298051 + 0.954550i \(0.403663\pi\)
\(620\) 0 0
\(621\) 4.59254e6 0.477885
\(622\) 4.80802e6i 0.498299i
\(623\) − 292383.i − 0.0301809i
\(624\) −1.42643e6 −0.146652
\(625\) 0 0
\(626\) −1.66916e6 −0.170240
\(627\) − 1.67045e7i − 1.69693i
\(628\) − 1.05436e7i − 1.06682i
\(629\) −937930. −0.0945244
\(630\) 0 0
\(631\) 7.92146e6 0.792012 0.396006 0.918248i \(-0.370396\pi\)
0.396006 + 0.918248i \(0.370396\pi\)
\(632\) 1.67388e7i 1.66699i
\(633\) − 9.75482e6i − 0.967631i
\(634\) −6.05622e6 −0.598382
\(635\) 0 0
\(636\) −1.02508e6 −0.100488
\(637\) 1.59829e6i 0.156066i
\(638\) − 9.85014e6i − 0.958055i
\(639\) 2.12701e7 2.06071
\(640\) 0 0
\(641\) 8.63330e6 0.829911 0.414956 0.909842i \(-0.363797\pi\)
0.414956 + 0.909842i \(0.363797\pi\)
\(642\) − 5.97252e6i − 0.571900i
\(643\) 5.41901e6i 0.516884i 0.966027 + 0.258442i \(0.0832090\pi\)
−0.966027 + 0.258442i \(0.916791\pi\)
\(644\) 4.54662e6 0.431990
\(645\) 0 0
\(646\) 1.95670e6 0.184477
\(647\) − 2.85189e6i − 0.267838i −0.990992 0.133919i \(-0.957244\pi\)
0.990992 0.133919i \(-0.0427562\pi\)
\(648\) − 4.64016e6i − 0.434106i
\(649\) 851944. 0.0793961
\(650\) 0 0
\(651\) −5.64079e6 −0.521660
\(652\) 1.37402e6i 0.126582i
\(653\) 1.31018e7i 1.20239i 0.799101 + 0.601197i \(0.205309\pi\)
−0.799101 + 0.601197i \(0.794691\pi\)
\(654\) 1.08935e7 0.995916
\(655\) 0 0
\(656\) −6.87605e6 −0.623849
\(657\) − 340859.i − 0.0308079i
\(658\) 3.12789e6i 0.281635i
\(659\) 987681. 0.0885937 0.0442969 0.999018i \(-0.485895\pi\)
0.0442969 + 0.999018i \(0.485895\pi\)
\(660\) 0 0
\(661\) 2.30367e6 0.205077 0.102538 0.994729i \(-0.467304\pi\)
0.102538 + 0.994729i \(0.467304\pi\)
\(662\) − 1.01500e7i − 0.900167i
\(663\) − 1.77734e6i − 0.157032i
\(664\) 7.22553e6 0.635989
\(665\) 0 0
\(666\) 1.95099e6 0.170439
\(667\) − 1.80172e7i − 1.56810i
\(668\) 1.25082e7i 1.08456i
\(669\) −2.26056e6 −0.195277
\(670\) 0 0
\(671\) 1.96464e7 1.68452
\(672\) 1.22152e7i 1.04346i
\(673\) − 3.89370e6i − 0.331379i −0.986178 0.165689i \(-0.947015\pi\)
0.986178 0.165689i \(-0.0529849\pi\)
\(674\) −9.55803e6 −0.810436
\(675\) 0 0
\(676\) −696916. −0.0586562
\(677\) − 2.83801e6i − 0.237981i −0.992895 0.118991i \(-0.962034\pi\)
0.992895 0.118991i \(-0.0379658\pi\)
\(678\) 2.40793e6i 0.201173i
\(679\) −1.14247e7 −0.950981
\(680\) 0 0
\(681\) 6.03595e6 0.498744
\(682\) − 3.26276e6i − 0.268611i
\(683\) − 6.98711e6i − 0.573121i −0.958062 0.286560i \(-0.907488\pi\)
0.958062 0.286560i \(-0.0925119\pi\)
\(684\) 1.30695e7 1.06812
\(685\) 0 0
\(686\) 6.20695e6 0.503580
\(687\) 1.02566e7i 0.829105i
\(688\) 2.87191e6i 0.231313i
\(689\) −296286. −0.0237773
\(690\) 0 0
\(691\) 3.58854e6 0.285905 0.142953 0.989730i \(-0.454340\pi\)
0.142953 + 0.989730i \(0.454340\pi\)
\(692\) − 8.85829e6i − 0.703209i
\(693\) 1.22384e7i 0.968033i
\(694\) 1.07931e7 0.850643
\(695\) 0 0
\(696\) 3.08838e7 2.41661
\(697\) − 8.56764e6i − 0.668004i
\(698\) 9.15342e6i 0.711123i
\(699\) −1.14955e6 −0.0889888
\(700\) 0 0
\(701\) −1.07723e7 −0.827971 −0.413986 0.910283i \(-0.635864\pi\)
−0.413986 + 0.910283i \(0.635864\pi\)
\(702\) 984396.i 0.0753923i
\(703\) − 3.45617e6i − 0.263759i
\(704\) −2.20694e6 −0.167826
\(705\) 0 0
\(706\) −8.53384e6 −0.644366
\(707\) − 9.65556e6i − 0.726489i
\(708\) 1.15563e6i 0.0866436i
\(709\) −1.68465e7 −1.25862 −0.629309 0.777155i \(-0.716662\pi\)
−0.629309 + 0.777155i \(0.716662\pi\)
\(710\) 0 0
\(711\) −3.56554e7 −2.64516
\(712\) 530254.i 0.0391998i
\(713\) − 5.96803e6i − 0.439650i
\(714\) −2.48540e6 −0.182453
\(715\) 0 0
\(716\) 1.98255e7 1.44525
\(717\) 3.35270e7i 2.43555i
\(718\) 7.48230e6i 0.541656i
\(719\) −7.10379e6 −0.512470 −0.256235 0.966615i \(-0.582482\pi\)
−0.256235 + 0.966615i \(0.582482\pi\)
\(720\) 0 0
\(721\) 8.78359e6 0.629265
\(722\) 384518.i 0.0274520i
\(723\) − 1.38211e7i − 0.983327i
\(724\) −1.13458e7 −0.804432
\(725\) 0 0
\(726\) −1.63483e6 −0.115115
\(727\) 2.50048e7i 1.75464i 0.479908 + 0.877319i \(0.340670\pi\)
−0.479908 + 0.877319i \(0.659330\pi\)
\(728\) 2.25260e6i 0.157528i
\(729\) 2.21877e7 1.54630
\(730\) 0 0
\(731\) −3.57843e6 −0.247685
\(732\) 2.66496e7i 1.83828i
\(733\) − 7.77802e6i − 0.534699i −0.963600 0.267349i \(-0.913852\pi\)
0.963600 0.267349i \(-0.0861477\pi\)
\(734\) 7.02842e6 0.481524
\(735\) 0 0
\(736\) −1.29238e7 −0.879422
\(737\) 8.38715e6i 0.568782i
\(738\) 1.78216e7i 1.20450i
\(739\) −1.00573e7 −0.677439 −0.338720 0.940887i \(-0.609994\pi\)
−0.338720 + 0.940887i \(0.609994\pi\)
\(740\) 0 0
\(741\) 6.54932e6 0.438178
\(742\) 414320.i 0.0276265i
\(743\) − 3.40838e6i − 0.226504i −0.993566 0.113252i \(-0.963873\pi\)
0.993566 0.113252i \(-0.0361268\pi\)
\(744\) 1.02299e7 0.677548
\(745\) 0 0
\(746\) −1.01429e7 −0.667292
\(747\) 1.53912e7i 1.00918i
\(748\) 4.61627e6i 0.301673i
\(749\) 7.75153e6 0.504874
\(750\) 0 0
\(751\) 1.57084e7 1.01633 0.508163 0.861261i \(-0.330325\pi\)
0.508163 + 0.861261i \(0.330325\pi\)
\(752\) 4.66205e6i 0.300630i
\(753\) 1.62217e7i 1.04258i
\(754\) 3.86194e6 0.247387
\(755\) 0 0
\(756\) −4.42023e6 −0.281281
\(757\) 2.52846e7i 1.60367i 0.597544 + 0.801836i \(0.296143\pi\)
−0.597544 + 0.801836i \(0.703857\pi\)
\(758\) 2.94469e6i 0.186152i
\(759\) −2.24490e7 −1.41447
\(760\) 0 0
\(761\) 1.34808e7 0.843825 0.421913 0.906636i \(-0.361359\pi\)
0.421913 + 0.906636i \(0.361359\pi\)
\(762\) 1.17960e7i 0.735949i
\(763\) 1.41383e7i 0.879195i
\(764\) −379132. −0.0234994
\(765\) 0 0
\(766\) 1.65890e6 0.102152
\(767\) 334021.i 0.0205015i
\(768\) − 1.55625e7i − 0.952087i
\(769\) −8.09020e6 −0.493337 −0.246668 0.969100i \(-0.579336\pi\)
−0.246668 + 0.969100i \(0.579336\pi\)
\(770\) 0 0
\(771\) −4.91520e7 −2.97787
\(772\) − 7.79387e6i − 0.470663i
\(773\) 2.75295e6i 0.165710i 0.996562 + 0.0828551i \(0.0264039\pi\)
−0.996562 + 0.0828551i \(0.973596\pi\)
\(774\) 7.44351e6 0.446607
\(775\) 0 0
\(776\) 2.07195e7 1.23516
\(777\) 4.39004e6i 0.260865i
\(778\) − 1.37941e7i − 0.817040i
\(779\) 3.15708e7 1.86398
\(780\) 0 0
\(781\) −2.76839e7 −1.62405
\(782\) − 2.62959e6i − 0.153770i
\(783\) 1.75164e7i 1.02103i
\(784\) 3.33126e6 0.193561
\(785\) 0 0
\(786\) −2.87706e6 −0.166109
\(787\) − 2.06536e7i − 1.18867i −0.804219 0.594333i \(-0.797416\pi\)
0.804219 0.594333i \(-0.202584\pi\)
\(788\) − 1.55161e7i − 0.890156i
\(789\) −3.63560e7 −2.07914
\(790\) 0 0
\(791\) −3.12516e6 −0.177595
\(792\) − 2.21950e7i − 1.25731i
\(793\) 7.70274e6i 0.434973i
\(794\) 9.58160e6 0.539370
\(795\) 0 0
\(796\) −1.40126e6 −0.0783855
\(797\) − 5.27093e6i − 0.293928i −0.989142 0.146964i \(-0.953050\pi\)
0.989142 0.146964i \(-0.0469502\pi\)
\(798\) − 9.15843e6i − 0.509113i
\(799\) −5.80897e6 −0.321909
\(800\) 0 0
\(801\) −1.12950e6 −0.0622020
\(802\) − 1.41992e7i − 0.779519i
\(803\) 443641.i 0.0242797i
\(804\) −1.13769e7 −0.620701
\(805\) 0 0
\(806\) 1.27923e6 0.0693602
\(807\) − 4.33275e6i − 0.234196i
\(808\) 1.75110e7i 0.943586i
\(809\) 2.62432e7 1.40976 0.704882 0.709325i \(-0.251000\pi\)
0.704882 + 0.709325i \(0.251000\pi\)
\(810\) 0 0
\(811\) 1.50454e7 0.803251 0.401626 0.915804i \(-0.368445\pi\)
0.401626 + 0.915804i \(0.368445\pi\)
\(812\) 1.73412e7i 0.922975i
\(813\) − 3.79088e7i − 2.01147i
\(814\) −2.53929e6 −0.134323
\(815\) 0 0
\(816\) −3.70444e6 −0.194759
\(817\) − 1.31861e7i − 0.691134i
\(818\) 3.07766e6i 0.160819i
\(819\) −4.79829e6 −0.249963
\(820\) 0 0
\(821\) 1.82770e6 0.0946338 0.0473169 0.998880i \(-0.484933\pi\)
0.0473169 + 0.998880i \(0.484933\pi\)
\(822\) − 2.29552e7i − 1.18495i
\(823\) 2.45074e7i 1.26124i 0.776093 + 0.630619i \(0.217199\pi\)
−0.776093 + 0.630619i \(0.782801\pi\)
\(824\) −1.59296e7 −0.817309
\(825\) 0 0
\(826\) 467088. 0.0238204
\(827\) 7.03772e6i 0.357823i 0.983865 + 0.178911i \(0.0572576\pi\)
−0.983865 + 0.178911i \(0.942742\pi\)
\(828\) − 1.75640e7i − 0.890321i
\(829\) −3.44967e7 −1.74338 −0.871689 0.490059i \(-0.836975\pi\)
−0.871689 + 0.490059i \(0.836975\pi\)
\(830\) 0 0
\(831\) −3.67852e7 −1.84787
\(832\) − 865273.i − 0.0433356i
\(833\) 4.15078e6i 0.207261i
\(834\) 3.60619e6 0.179529
\(835\) 0 0
\(836\) −1.70104e7 −0.841782
\(837\) 5.80212e6i 0.286268i
\(838\) − 4.95476e6i − 0.243732i
\(839\) −2.79128e6 −0.136899 −0.0684493 0.997655i \(-0.521805\pi\)
−0.0684493 + 0.997655i \(0.521805\pi\)
\(840\) 0 0
\(841\) 4.82083e7 2.35035
\(842\) 1.46460e7i 0.711935i
\(843\) 2.29845e7i 1.11395i
\(844\) −9.93349e6 −0.480005
\(845\) 0 0
\(846\) 1.20833e7 0.580442
\(847\) − 2.12179e6i − 0.101623i
\(848\) 617535.i 0.0294898i
\(849\) −6.32593e7 −3.01200
\(850\) 0 0
\(851\) −4.64471e6 −0.219854
\(852\) − 3.75522e7i − 1.77230i
\(853\) − 2.48047e7i − 1.16724i −0.812026 0.583621i \(-0.801635\pi\)
0.812026 0.583621i \(-0.198365\pi\)
\(854\) 1.07713e7 0.505388
\(855\) 0 0
\(856\) −1.40579e7 −0.655746
\(857\) 1.83304e7i 0.852550i 0.904594 + 0.426275i \(0.140174\pi\)
−0.904594 + 0.426275i \(0.859826\pi\)
\(858\) − 4.81187e6i − 0.223149i
\(859\) −1.67092e7 −0.772632 −0.386316 0.922366i \(-0.626253\pi\)
−0.386316 + 0.922366i \(0.626253\pi\)
\(860\) 0 0
\(861\) −4.01013e7 −1.84353
\(862\) − 1.27806e7i − 0.585844i
\(863\) 8.29285e6i 0.379033i 0.981878 + 0.189516i \(0.0606920\pi\)
−0.981878 + 0.189516i \(0.939308\pi\)
\(864\) 1.25646e7 0.572615
\(865\) 0 0
\(866\) 1.01627e7 0.460484
\(867\) 2.94070e7i 1.32863i
\(868\) 5.74411e6i 0.258775i
\(869\) 4.64069e7 2.08465
\(870\) 0 0
\(871\) −3.28834e6 −0.146870
\(872\) − 2.56407e7i − 1.14193i
\(873\) 4.41347e7i 1.95995i
\(874\) 9.68974e6 0.429075
\(875\) 0 0
\(876\) −601784. −0.0264960
\(877\) 1.44569e7i 0.634711i 0.948307 + 0.317356i \(0.102795\pi\)
−0.948307 + 0.317356i \(0.897205\pi\)
\(878\) 4.60447e6i 0.201578i
\(879\) −6.87014e6 −0.299912
\(880\) 0 0
\(881\) −3.42081e7 −1.48487 −0.742437 0.669916i \(-0.766330\pi\)
−0.742437 + 0.669916i \(0.766330\pi\)
\(882\) − 8.63407e6i − 0.373718i
\(883\) − 1.41341e7i − 0.610051i −0.952344 0.305026i \(-0.901335\pi\)
0.952344 0.305026i \(-0.0986650\pi\)
\(884\) −1.80990e6 −0.0778975
\(885\) 0 0
\(886\) −9.90207e6 −0.423781
\(887\) − 2.15650e7i − 0.920325i −0.887835 0.460163i \(-0.847791\pi\)
0.887835 0.460163i \(-0.152209\pi\)
\(888\) − 7.96160e6i − 0.338819i
\(889\) −1.53096e7 −0.649696
\(890\) 0 0
\(891\) −1.28645e7 −0.542872
\(892\) 2.30196e6i 0.0968693i
\(893\) − 2.14054e7i − 0.898246i
\(894\) −1.64825e7 −0.689731
\(895\) 0 0
\(896\) 1.51027e7 0.628471
\(897\) − 8.80157e6i − 0.365241i
\(898\) − 221785.i − 0.00917787i
\(899\) 2.27626e7 0.939340
\(900\) 0 0
\(901\) −769456. −0.0315771
\(902\) − 2.31955e7i − 0.949264i
\(903\) 1.67491e7i 0.683551i
\(904\) 5.66768e6 0.230666
\(905\) 0 0
\(906\) −2.63418e7 −1.06617
\(907\) − 1.53494e7i − 0.619547i −0.950810 0.309773i \(-0.899747\pi\)
0.950810 0.309773i \(-0.100253\pi\)
\(908\) − 6.14650e6i − 0.247408i
\(909\) −3.73002e7 −1.49727
\(910\) 0 0
\(911\) −8.00925e6 −0.319739 −0.159870 0.987138i \(-0.551107\pi\)
−0.159870 + 0.987138i \(0.551107\pi\)
\(912\) − 1.36505e7i − 0.543451i
\(913\) − 2.00322e7i − 0.795337i
\(914\) 7.93047e6 0.314003
\(915\) 0 0
\(916\) 1.04444e7 0.411287
\(917\) − 3.73403e6i − 0.146641i
\(918\) 2.55648e6i 0.100124i
\(919\) 1.00787e7 0.393655 0.196828 0.980438i \(-0.436936\pi\)
0.196828 + 0.980438i \(0.436936\pi\)
\(920\) 0 0
\(921\) 4.55031e7 1.76763
\(922\) 225064.i 0.00871926i
\(923\) − 1.08540e7i − 0.419359i
\(924\) 2.16067e7 0.832547
\(925\) 0 0
\(926\) 6.89748e6 0.264340
\(927\) − 3.39317e7i − 1.29690i
\(928\) − 4.92927e7i − 1.87894i
\(929\) 1.20808e7 0.459257 0.229629 0.973278i \(-0.426249\pi\)
0.229629 + 0.973278i \(0.426249\pi\)
\(930\) 0 0
\(931\) −1.52952e7 −0.578336
\(932\) 1.17061e6i 0.0441440i
\(933\) 4.17938e7i 1.57184i
\(934\) 3.82547e6 0.143489
\(935\) 0 0
\(936\) 8.70199e6 0.324660
\(937\) − 1.85069e7i − 0.688628i −0.938854 0.344314i \(-0.888111\pi\)
0.938854 0.344314i \(-0.111889\pi\)
\(938\) 4.59835e6i 0.170646i
\(939\) −1.45092e7 −0.537007
\(940\) 0 0
\(941\) −1.31161e7 −0.482871 −0.241436 0.970417i \(-0.577618\pi\)
−0.241436 + 0.970417i \(0.577618\pi\)
\(942\) 2.85422e7i 1.04800i
\(943\) − 4.24277e7i − 1.55371i
\(944\) 696186. 0.0254270
\(945\) 0 0
\(946\) −9.68802e6 −0.351971
\(947\) 2.46835e7i 0.894399i 0.894434 + 0.447199i \(0.147579\pi\)
−0.894434 + 0.447199i \(0.852421\pi\)
\(948\) 6.29494e7i 2.27494i
\(949\) −173938. −0.00626945
\(950\) 0 0
\(951\) −5.26439e7 −1.88754
\(952\) 5.85003e6i 0.209202i
\(953\) 3.21978e7i 1.14840i 0.818715 + 0.574201i \(0.194687\pi\)
−0.818715 + 0.574201i \(0.805313\pi\)
\(954\) 1.60055e6 0.0569375
\(955\) 0 0
\(956\) 3.41411e7 1.20818
\(957\) − 8.56226e7i − 3.02210i
\(958\) 1.36832e7i 0.481697i
\(959\) 2.97927e7 1.04608
\(960\) 0 0
\(961\) −2.10893e7 −0.736636
\(962\) − 995579.i − 0.0346847i
\(963\) − 2.99448e7i − 1.04053i
\(964\) −1.40743e7 −0.487791
\(965\) 0 0
\(966\) −1.23079e7 −0.424368
\(967\) − 2.28108e7i − 0.784465i −0.919866 0.392232i \(-0.871703\pi\)
0.919866 0.392232i \(-0.128297\pi\)
\(968\) 3.84799e6i 0.131991i
\(969\) 1.70086e7 0.581916
\(970\) 0 0
\(971\) 4.02182e7 1.36891 0.684455 0.729055i \(-0.260040\pi\)
0.684455 + 0.729055i \(0.260040\pi\)
\(972\) − 2.99792e7i − 1.01778i
\(973\) 4.68035e6i 0.158488i
\(974\) 2.36346e7 0.798271
\(975\) 0 0
\(976\) 1.60545e7 0.539475
\(977\) − 3.17409e7i − 1.06386i −0.846790 0.531928i \(-0.821468\pi\)
0.846790 0.531928i \(-0.178532\pi\)
\(978\) − 3.71953e6i − 0.124349i
\(979\) 1.47008e6 0.0490214
\(980\) 0 0
\(981\) 5.46173e7 1.81200
\(982\) 7.03774e6i 0.232892i
\(983\) 4.52276e6i 0.149286i 0.997210 + 0.0746431i \(0.0237818\pi\)
−0.997210 + 0.0746431i \(0.976218\pi\)
\(984\) 7.27262e7 2.39444
\(985\) 0 0
\(986\) 1.00295e7 0.328539
\(987\) 2.71892e7i 0.888391i
\(988\) − 6.66928e6i − 0.217363i
\(989\) −1.77207e7 −0.576090
\(990\) 0 0
\(991\) −2.04730e7 −0.662213 −0.331107 0.943593i \(-0.607422\pi\)
−0.331107 + 0.943593i \(0.607422\pi\)
\(992\) − 1.63277e7i − 0.526801i
\(993\) − 8.82296e7i − 2.83949i
\(994\) −1.51780e7 −0.487247
\(995\) 0 0
\(996\) 2.71729e7 0.867937
\(997\) 3.49736e7i 1.11430i 0.830412 + 0.557150i \(0.188105\pi\)
−0.830412 + 0.557150i \(0.811895\pi\)
\(998\) − 446192.i − 0.0141806i
\(999\) 4.51559e6 0.143153
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 325.6.b.g.274.8 12
5.2 odd 4 65.6.a.d.1.3 6
5.3 odd 4 325.6.a.g.1.4 6
5.4 even 2 inner 325.6.b.g.274.5 12
15.2 even 4 585.6.a.m.1.4 6
20.7 even 4 1040.6.a.q.1.2 6
65.12 odd 4 845.6.a.h.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.6.a.d.1.3 6 5.2 odd 4
325.6.a.g.1.4 6 5.3 odd 4
325.6.b.g.274.5 12 5.4 even 2 inner
325.6.b.g.274.8 12 1.1 even 1 trivial
585.6.a.m.1.4 6 15.2 even 4
845.6.a.h.1.4 6 65.12 odd 4
1040.6.a.q.1.2 6 20.7 even 4