Properties

Label 117.5.j.b.109.2
Level $117$
Weight $5$
Character 117.109
Analytic conductor $12.094$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,5,Mod(73,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.73");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 117.j (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.0942856808\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 5446 x^{16} - 1452 x^{15} + 106320 x^{13} + 8376897 x^{12} - 1643220 x^{11} + 1054152 x^{10} + \cdots + 2103506496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{10} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 109.2
Root \(-4.67933 + 4.67933i\) of defining polynomial
Character \(\chi\) \(=\) 117.109
Dual form 117.5.j.b.73.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-4.67933 + 4.67933i) q^{2} -27.7923i q^{4} +(-13.1418 + 13.1418i) q^{5} +(-30.2175 - 30.2175i) q^{7} +(55.1800 + 55.1800i) q^{8} -122.989i q^{10} +(-58.8012 - 58.8012i) q^{11} +(120.640 + 118.351i) q^{13} +282.795 q^{14} -71.7343 q^{16} -257.703i q^{17} +(336.208 - 336.208i) q^{19} +(365.240 + 365.240i) q^{20} +550.300 q^{22} +486.231i q^{23} +279.587i q^{25} +(-1118.32 + 10.7102i) q^{26} +(-839.813 + 839.813i) q^{28} +771.373 q^{29} +(-1304.18 + 1304.18i) q^{31} +(-547.211 + 547.211i) q^{32} +(1205.88 + 1205.88i) q^{34} +794.223 q^{35} +(1053.22 + 1053.22i) q^{37} +3146.45i q^{38} -1450.33 q^{40} +(1126.29 - 1126.29i) q^{41} -1930.35i q^{43} +(-1634.22 + 1634.22i) q^{44} +(-2275.24 - 2275.24i) q^{46} +(65.7645 + 65.7645i) q^{47} -574.806i q^{49} +(-1308.28 - 1308.28i) q^{50} +(3289.25 - 3352.86i) q^{52} +5040.23 q^{53} +1545.50 q^{55} -3334.80i q^{56} +(-3609.51 + 3609.51i) q^{58} +(206.279 + 206.279i) q^{59} +6998.83 q^{61} -12205.4i q^{62} -6268.91i q^{64} +(-3140.77 + 30.0794i) q^{65} +(-939.648 + 939.648i) q^{67} -7162.15 q^{68} +(-3716.43 + 3716.43i) q^{70} +(3049.34 - 3049.34i) q^{71} +(-840.175 - 840.175i) q^{73} -9856.69 q^{74} +(-9343.97 - 9343.97i) q^{76} +3553.65i q^{77} +4044.54 q^{79} +(942.716 - 942.716i) q^{80} +10540.5i q^{82} +(-2060.82 + 2060.82i) q^{83} +(3386.67 + 3386.67i) q^{85} +(9032.76 + 9032.76i) q^{86} -6489.30i q^{88} +(-6032.18 - 6032.18i) q^{89} +(-69.1630 - 7221.71i) q^{91} +13513.5 q^{92} -615.468 q^{94} +8836.73i q^{95} +(5575.10 - 5575.10i) q^{97} +(2689.71 + 2689.71i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 24 q^{5} - 24 q^{7} - 372 q^{11} - 224 q^{13} - 480 q^{14} - 2328 q^{16} - 840 q^{19} - 228 q^{20} + 3536 q^{22} + 828 q^{26} - 1984 q^{28} + 5064 q^{29} + 1712 q^{31} + 7260 q^{32} + 8040 q^{34}+ \cdots - 11544 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.67933 + 4.67933i −1.16983 + 1.16983i −0.187584 + 0.982249i \(0.560066\pi\)
−0.982249 + 0.187584i \(0.939934\pi\)
\(3\) 0 0
\(4\) 27.7923i 1.73702i
\(5\) −13.1418 + 13.1418i −0.525671 + 0.525671i −0.919279 0.393608i \(-0.871227\pi\)
0.393608 + 0.919279i \(0.371227\pi\)
\(6\) 0 0
\(7\) −30.2175 30.2175i −0.616684 0.616684i 0.327996 0.944679i \(-0.393627\pi\)
−0.944679 + 0.327996i \(0.893627\pi\)
\(8\) 55.1800 + 55.1800i 0.862187 + 0.862187i
\(9\) 0 0
\(10\) 122.989i 1.22989i
\(11\) −58.8012 58.8012i −0.485960 0.485960i 0.421069 0.907029i \(-0.361655\pi\)
−0.907029 + 0.421069i \(0.861655\pi\)
\(12\) 0 0
\(13\) 120.640 + 118.351i 0.713846 + 0.700303i
\(14\) 282.795 1.44283
\(15\) 0 0
\(16\) −71.7343 −0.280212
\(17\) 257.703i 0.891706i −0.895106 0.445853i \(-0.852900\pi\)
0.895106 0.445853i \(-0.147100\pi\)
\(18\) 0 0
\(19\) 336.208 336.208i 0.931323 0.931323i −0.0664659 0.997789i \(-0.521172\pi\)
0.997789 + 0.0664659i \(0.0211724\pi\)
\(20\) 365.240 + 365.240i 0.913100 + 0.913100i
\(21\) 0 0
\(22\) 550.300 1.13698
\(23\) 486.231i 0.919151i 0.888139 + 0.459576i \(0.151999\pi\)
−0.888139 + 0.459576i \(0.848001\pi\)
\(24\) 0 0
\(25\) 279.587i 0.447340i
\(26\) −1118.32 + 10.7102i −1.65432 + 0.0158436i
\(27\) 0 0
\(28\) −839.813 + 839.813i −1.07119 + 1.07119i
\(29\) 771.373 0.917209 0.458604 0.888641i \(-0.348350\pi\)
0.458604 + 0.888641i \(0.348350\pi\)
\(30\) 0 0
\(31\) −1304.18 + 1304.18i −1.35711 + 1.35711i −0.479654 + 0.877458i \(0.659238\pi\)
−0.877458 + 0.479654i \(0.840762\pi\)
\(32\) −547.211 + 547.211i −0.534386 + 0.534386i
\(33\) 0 0
\(34\) 1205.88 + 1205.88i 1.04315 + 1.04315i
\(35\) 794.223 0.648345
\(36\) 0 0
\(37\) 1053.22 + 1053.22i 0.769333 + 0.769333i 0.977989 0.208656i \(-0.0669091\pi\)
−0.208656 + 0.977989i \(0.566909\pi\)
\(38\) 3146.45i 2.17898i
\(39\) 0 0
\(40\) −1450.33 −0.906454
\(41\) 1126.29 1126.29i 0.670010 0.670010i −0.287708 0.957718i \(-0.592893\pi\)
0.957718 + 0.287708i \(0.0928932\pi\)
\(42\) 0 0
\(43\) 1930.35i 1.04400i −0.852946 0.521999i \(-0.825186\pi\)
0.852946 0.521999i \(-0.174814\pi\)
\(44\) −1634.22 + 1634.22i −0.844121 + 0.844121i
\(45\) 0 0
\(46\) −2275.24 2275.24i −1.07525 1.07525i
\(47\) 65.7645 + 65.7645i 0.0297712 + 0.0297712i 0.721836 0.692064i \(-0.243299\pi\)
−0.692064 + 0.721836i \(0.743299\pi\)
\(48\) 0 0
\(49\) 574.806i 0.239403i
\(50\) −1308.28 1308.28i −0.523313 0.523313i
\(51\) 0 0
\(52\) 3289.25 3352.86i 1.21644 1.23996i
\(53\) 5040.23 1.79432 0.897158 0.441710i \(-0.145628\pi\)
0.897158 + 0.441710i \(0.145628\pi\)
\(54\) 0 0
\(55\) 1545.50 0.510910
\(56\) 3334.80i 1.06339i
\(57\) 0 0
\(58\) −3609.51 + 3609.51i −1.07298 + 1.07298i
\(59\) 206.279 + 206.279i 0.0592586 + 0.0592586i 0.736115 0.676856i \(-0.236658\pi\)
−0.676856 + 0.736115i \(0.736658\pi\)
\(60\) 0 0
\(61\) 6998.83 1.88090 0.940450 0.339932i \(-0.110404\pi\)
0.940450 + 0.339932i \(0.110404\pi\)
\(62\) 12205.4i 3.17519i
\(63\) 0 0
\(64\) 6268.91i 1.53050i
\(65\) −3140.77 + 30.0794i −0.743377 + 0.00711939i
\(66\) 0 0
\(67\) −939.648 + 939.648i −0.209322 + 0.209322i −0.803979 0.594657i \(-0.797288\pi\)
0.594657 + 0.803979i \(0.297288\pi\)
\(68\) −7162.15 −1.54891
\(69\) 0 0
\(70\) −3716.43 + 3716.43i −0.758456 + 0.758456i
\(71\) 3049.34 3049.34i 0.604908 0.604908i −0.336703 0.941611i \(-0.609312\pi\)
0.941611 + 0.336703i \(0.109312\pi\)
\(72\) 0 0
\(73\) −840.175 840.175i −0.157661 0.157661i 0.623869 0.781529i \(-0.285560\pi\)
−0.781529 + 0.623869i \(0.785560\pi\)
\(74\) −9856.69 −1.79998
\(75\) 0 0
\(76\) −9343.97 9343.97i −1.61772 1.61772i
\(77\) 3553.65i 0.599367i
\(78\) 0 0
\(79\) 4044.54 0.648059 0.324030 0.946047i \(-0.394962\pi\)
0.324030 + 0.946047i \(0.394962\pi\)
\(80\) 942.716 942.716i 0.147299 0.147299i
\(81\) 0 0
\(82\) 10540.5i 1.56760i
\(83\) −2060.82 + 2060.82i −0.299147 + 0.299147i −0.840680 0.541533i \(-0.817844\pi\)
0.541533 + 0.840680i \(0.317844\pi\)
\(84\) 0 0
\(85\) 3386.67 + 3386.67i 0.468744 + 0.468744i
\(86\) 9032.76 + 9032.76i 1.22130 + 1.22130i
\(87\) 0 0
\(88\) 6489.30i 0.837977i
\(89\) −6032.18 6032.18i −0.761543 0.761543i 0.215059 0.976601i \(-0.431006\pi\)
−0.976601 + 0.215059i \(0.931006\pi\)
\(90\) 0 0
\(91\) −69.1630 7221.71i −0.00835202 0.872082i
\(92\) 13513.5 1.59658
\(93\) 0 0
\(94\) −615.468 −0.0696546
\(95\) 8836.73i 0.979139i
\(96\) 0 0
\(97\) 5575.10 5575.10i 0.592528 0.592528i −0.345785 0.938314i \(-0.612387\pi\)
0.938314 + 0.345785i \(0.112387\pi\)
\(98\) 2689.71 + 2689.71i 0.280061 + 0.280061i
\(99\) 0 0
\(100\) 7770.37 0.777037
\(101\) 14591.5i 1.43040i −0.698920 0.715200i \(-0.746335\pi\)
0.698920 0.715200i \(-0.253665\pi\)
\(102\) 0 0
\(103\) 13071.3i 1.23210i 0.787708 + 0.616048i \(0.211267\pi\)
−0.787708 + 0.616048i \(0.788733\pi\)
\(104\) 126.298 + 13187.5i 0.0116770 + 1.21926i
\(105\) 0 0
\(106\) −23584.9 + 23584.9i −2.09905 + 2.09905i
\(107\) 14255.0 1.24509 0.622545 0.782584i \(-0.286099\pi\)
0.622545 + 0.782584i \(0.286099\pi\)
\(108\) 0 0
\(109\) −4886.68 + 4886.68i −0.411302 + 0.411302i −0.882192 0.470890i \(-0.843933\pi\)
0.470890 + 0.882192i \(0.343933\pi\)
\(110\) −7231.92 + 7231.92i −0.597680 + 0.597680i
\(111\) 0 0
\(112\) 2167.63 + 2167.63i 0.172802 + 0.172802i
\(113\) 3433.45 0.268890 0.134445 0.990921i \(-0.457075\pi\)
0.134445 + 0.990921i \(0.457075\pi\)
\(114\) 0 0
\(115\) −6389.94 6389.94i −0.483171 0.483171i
\(116\) 21438.2i 1.59321i
\(117\) 0 0
\(118\) −1930.50 −0.138645
\(119\) −7787.14 + 7787.14i −0.549900 + 0.549900i
\(120\) 0 0
\(121\) 7725.84i 0.527685i
\(122\) −32749.8 + 32749.8i −2.20034 + 2.20034i
\(123\) 0 0
\(124\) 36246.3 + 36246.3i 2.35733 + 2.35733i
\(125\) −11887.9 11887.9i −0.760825 0.760825i
\(126\) 0 0
\(127\) 21948.2i 1.36079i −0.732846 0.680395i \(-0.761808\pi\)
0.732846 0.680395i \(-0.238192\pi\)
\(128\) 20578.9 + 20578.9i 1.25604 + 1.25604i
\(129\) 0 0
\(130\) 14555.9 14837.4i 0.861298 0.877955i
\(131\) 1658.15 0.0966231 0.0483116 0.998832i \(-0.484616\pi\)
0.0483116 + 0.998832i \(0.484616\pi\)
\(132\) 0 0
\(133\) −20318.7 −1.14866
\(134\) 8793.84i 0.489744i
\(135\) 0 0
\(136\) 14220.0 14220.0i 0.768817 0.768817i
\(137\) 21604.3 + 21604.3i 1.15106 + 1.15106i 0.986340 + 0.164724i \(0.0526732\pi\)
0.164724 + 0.986340i \(0.447327\pi\)
\(138\) 0 0
\(139\) −4658.17 −0.241094 −0.120547 0.992708i \(-0.538465\pi\)
−0.120547 + 0.992708i \(0.538465\pi\)
\(140\) 22073.3i 1.12619i
\(141\) 0 0
\(142\) 28537.8i 1.41528i
\(143\) −134.587 14053.0i −0.00658157 0.687220i
\(144\) 0 0
\(145\) −10137.2 + 10137.2i −0.482150 + 0.482150i
\(146\) 7862.91 0.368874
\(147\) 0 0
\(148\) 29271.3 29271.3i 1.33634 1.33634i
\(149\) −393.889 + 393.889i −0.0177420 + 0.0177420i −0.715922 0.698180i \(-0.753993\pi\)
0.698180 + 0.715922i \(0.253993\pi\)
\(150\) 0 0
\(151\) −7620.89 7620.89i −0.334235 0.334235i 0.519957 0.854192i \(-0.325948\pi\)
−0.854192 + 0.519957i \(0.825948\pi\)
\(152\) 37103.8 1.60595
\(153\) 0 0
\(154\) −16628.7 16628.7i −0.701159 0.701159i
\(155\) 34278.6i 1.42679i
\(156\) 0 0
\(157\) 25587.3 1.03807 0.519034 0.854754i \(-0.326292\pi\)
0.519034 + 0.854754i \(0.326292\pi\)
\(158\) −18925.7 + 18925.7i −0.758121 + 0.758121i
\(159\) 0 0
\(160\) 14382.7i 0.561822i
\(161\) 14692.7 14692.7i 0.566826 0.566826i
\(162\) 0 0
\(163\) −16600.8 16600.8i −0.624820 0.624820i 0.321940 0.946760i \(-0.395665\pi\)
−0.946760 + 0.321940i \(0.895665\pi\)
\(164\) −31302.1 31302.1i −1.16382 1.16382i
\(165\) 0 0
\(166\) 19286.6i 0.699904i
\(167\) 37894.4 + 37894.4i 1.35876 + 1.35876i 0.875451 + 0.483307i \(0.160565\pi\)
0.483307 + 0.875451i \(0.339435\pi\)
\(168\) 0 0
\(169\) 547.013 + 28555.8i 0.0191524 + 0.999817i
\(170\) −31694.7 −1.09670
\(171\) 0 0
\(172\) −53648.9 −1.81344
\(173\) 1712.90i 0.0572321i −0.999590 0.0286161i \(-0.990890\pi\)
0.999590 0.0286161i \(-0.00911002\pi\)
\(174\) 0 0
\(175\) 8448.43 8448.43i 0.275867 0.275867i
\(176\) 4218.06 + 4218.06i 0.136172 + 0.136172i
\(177\) 0 0
\(178\) 56453.1 1.78176
\(179\) 2700.43i 0.0842803i 0.999112 + 0.0421401i \(0.0134176\pi\)
−0.999112 + 0.0421401i \(0.986582\pi\)
\(180\) 0 0
\(181\) 31325.6i 0.956186i 0.878309 + 0.478093i \(0.158672\pi\)
−0.878309 + 0.478093i \(0.841328\pi\)
\(182\) 34116.4 + 33469.1i 1.02996 + 1.01042i
\(183\) 0 0
\(184\) −26830.2 + 26830.2i −0.792481 + 0.792481i
\(185\) −27682.3 −0.808832
\(186\) 0 0
\(187\) −15153.2 + 15153.2i −0.433333 + 0.433333i
\(188\) 1827.75 1827.75i 0.0517131 0.0517131i
\(189\) 0 0
\(190\) −41350.0 41350.0i −1.14543 1.14543i
\(191\) −52778.0 −1.44672 −0.723362 0.690469i \(-0.757404\pi\)
−0.723362 + 0.690469i \(0.757404\pi\)
\(192\) 0 0
\(193\) 1405.79 + 1405.79i 0.0377403 + 0.0377403i 0.725725 0.687985i \(-0.241504\pi\)
−0.687985 + 0.725725i \(0.741504\pi\)
\(194\) 52175.5i 1.38632i
\(195\) 0 0
\(196\) −15975.2 −0.415847
\(197\) −10175.2 + 10175.2i −0.262188 + 0.262188i −0.825942 0.563755i \(-0.809356\pi\)
0.563755 + 0.825942i \(0.309356\pi\)
\(198\) 0 0
\(199\) 7439.54i 0.187862i −0.995579 0.0939312i \(-0.970057\pi\)
0.995579 0.0939312i \(-0.0299434\pi\)
\(200\) −15427.6 + 15427.6i −0.385691 + 0.385691i
\(201\) 0 0
\(202\) 68278.5 + 68278.5i 1.67333 + 1.67333i
\(203\) −23308.9 23308.9i −0.565627 0.565627i
\(204\) 0 0
\(205\) 29602.8i 0.704410i
\(206\) −61165.0 61165.0i −1.44135 1.44135i
\(207\) 0 0
\(208\) −8654.02 8489.84i −0.200028 0.196233i
\(209\) −39538.8 −0.905172
\(210\) 0 0
\(211\) −45876.3 −1.03044 −0.515221 0.857057i \(-0.672290\pi\)
−0.515221 + 0.857057i \(0.672290\pi\)
\(212\) 140080.i 3.11676i
\(213\) 0 0
\(214\) −66704.1 + 66704.1i −1.45655 + 1.45655i
\(215\) 25368.3 + 25368.3i 0.548800 + 0.548800i
\(216\) 0 0
\(217\) 78818.4 1.67382
\(218\) 45732.8i 0.962309i
\(219\) 0 0
\(220\) 42953.1i 0.887460i
\(221\) 30499.4 31089.3i 0.624464 0.636541i
\(222\) 0 0
\(223\) 26187.3 26187.3i 0.526601 0.526601i −0.392956 0.919557i \(-0.628548\pi\)
0.919557 + 0.392956i \(0.128548\pi\)
\(224\) 33070.7 0.659094
\(225\) 0 0
\(226\) −16066.3 + 16066.3i −0.314556 + 0.314556i
\(227\) 20418.0 20418.0i 0.396244 0.396244i −0.480662 0.876906i \(-0.659604\pi\)
0.876906 + 0.480662i \(0.159604\pi\)
\(228\) 0 0
\(229\) −52109.2 52109.2i −0.993672 0.993672i 0.00630803 0.999980i \(-0.497992\pi\)
−0.999980 + 0.00630803i \(0.997992\pi\)
\(230\) 59801.3 1.13046
\(231\) 0 0
\(232\) 42564.3 + 42564.3i 0.790806 + 0.790806i
\(233\) 45782.3i 0.843308i 0.906757 + 0.421654i \(0.138550\pi\)
−0.906757 + 0.421654i \(0.861450\pi\)
\(234\) 0 0
\(235\) −1728.53 −0.0312997
\(236\) 5732.97 5732.97i 0.102933 0.102933i
\(237\) 0 0
\(238\) 72877.2i 1.28658i
\(239\) 54954.6 54954.6i 0.962073 0.962073i −0.0372339 0.999307i \(-0.511855\pi\)
0.999307 + 0.0372339i \(0.0118546\pi\)
\(240\) 0 0
\(241\) 23474.0 + 23474.0i 0.404159 + 0.404159i 0.879696 0.475537i \(-0.157746\pi\)
−0.475537 + 0.879696i \(0.657746\pi\)
\(242\) 36151.8 + 36151.8i 0.617304 + 0.617304i
\(243\) 0 0
\(244\) 194513.i 3.26716i
\(245\) 7553.98 + 7553.98i 0.125847 + 0.125847i
\(246\) 0 0
\(247\) 80350.6 769.526i 1.31703 0.0126133i
\(248\) −143930. −2.34017
\(249\) 0 0
\(250\) 111255. 1.78008
\(251\) 51668.4i 0.820120i 0.912059 + 0.410060i \(0.134492\pi\)
−0.912059 + 0.410060i \(0.865508\pi\)
\(252\) 0 0
\(253\) 28591.0 28591.0i 0.446671 0.446671i
\(254\) 102703. + 102703.i 1.59190 + 1.59190i
\(255\) 0 0
\(256\) −92288.7 −1.40821
\(257\) 87980.2i 1.33204i 0.745932 + 0.666022i \(0.232004\pi\)
−0.745932 + 0.666022i \(0.767996\pi\)
\(258\) 0 0
\(259\) 63651.1i 0.948869i
\(260\) 835.976 + 87289.1i 0.0123665 + 1.29126i
\(261\) 0 0
\(262\) −7759.03 + 7759.03i −0.113033 + 0.113033i
\(263\) −113585. −1.64214 −0.821068 0.570831i \(-0.806621\pi\)
−0.821068 + 0.570831i \(0.806621\pi\)
\(264\) 0 0
\(265\) −66237.6 + 66237.6i −0.943220 + 0.943220i
\(266\) 95077.9 95077.9i 1.34374 1.34374i
\(267\) 0 0
\(268\) 26114.9 + 26114.9i 0.363596 + 0.363596i
\(269\) −5647.94 −0.0780523 −0.0390262 0.999238i \(-0.512426\pi\)
−0.0390262 + 0.999238i \(0.512426\pi\)
\(270\) 0 0
\(271\) −3722.39 3722.39i −0.0506854 0.0506854i 0.681310 0.731995i \(-0.261411\pi\)
−0.731995 + 0.681310i \(0.761411\pi\)
\(272\) 18486.1i 0.249867i
\(273\) 0 0
\(274\) −202187. −2.69310
\(275\) 16440.1 16440.1i 0.217389 0.217389i
\(276\) 0 0
\(277\) 31673.1i 0.412792i 0.978469 + 0.206396i \(0.0661736\pi\)
−0.978469 + 0.206396i \(0.933826\pi\)
\(278\) 21797.1 21797.1i 0.282039 0.282039i
\(279\) 0 0
\(280\) 43825.2 + 43825.2i 0.558995 + 0.558995i
\(281\) 10897.2 + 10897.2i 0.138007 + 0.138007i 0.772736 0.634728i \(-0.218888\pi\)
−0.634728 + 0.772736i \(0.718888\pi\)
\(282\) 0 0
\(283\) 30994.4i 0.387000i 0.981100 + 0.193500i \(0.0619839\pi\)
−0.981100 + 0.193500i \(0.938016\pi\)
\(284\) −84748.2 84748.2i −1.05074 1.05074i
\(285\) 0 0
\(286\) 66388.2 + 65128.7i 0.811632 + 0.796233i
\(287\) −68067.2 −0.826369
\(288\) 0 0
\(289\) 17110.2 0.204861
\(290\) 94870.7i 1.12807i
\(291\) 0 0
\(292\) −23350.4 + 23350.4i −0.273860 + 0.273860i
\(293\) 113536. + 113536.i 1.32251 + 1.32251i 0.911741 + 0.410765i \(0.134738\pi\)
0.410765 + 0.911741i \(0.365262\pi\)
\(294\) 0 0
\(295\) −5421.75 −0.0623011
\(296\) 116233.i 1.32662i
\(297\) 0 0
\(298\) 3686.28i 0.0415103i
\(299\) −57546.0 + 58658.9i −0.643684 + 0.656133i
\(300\) 0 0
\(301\) −58330.4 + 58330.4i −0.643817 + 0.643817i
\(302\) 71321.3 0.781998
\(303\) 0 0
\(304\) −24117.6 + 24117.6i −0.260968 + 0.260968i
\(305\) −91977.0 + 91977.0i −0.988735 + 0.988735i
\(306\) 0 0
\(307\) 75222.4 + 75222.4i 0.798124 + 0.798124i 0.982800 0.184675i \(-0.0591233\pi\)
−0.184675 + 0.982800i \(0.559123\pi\)
\(308\) 98764.0 1.04111
\(309\) 0 0
\(310\) 160401. + 160401.i 1.66910 + 1.66910i
\(311\) 150426.i 1.55525i −0.628726 0.777627i \(-0.716423\pi\)
0.628726 0.777627i \(-0.283577\pi\)
\(312\) 0 0
\(313\) 126604. 1.29229 0.646144 0.763216i \(-0.276381\pi\)
0.646144 + 0.763216i \(0.276381\pi\)
\(314\) −119732. + 119732.i −1.21437 + 1.21437i
\(315\) 0 0
\(316\) 112407.i 1.12569i
\(317\) 42350.8 42350.8i 0.421447 0.421447i −0.464255 0.885702i \(-0.653678\pi\)
0.885702 + 0.464255i \(0.153678\pi\)
\(318\) 0 0
\(319\) −45357.6 45357.6i −0.445727 0.445727i
\(320\) 82384.6 + 82384.6i 0.804537 + 0.804537i
\(321\) 0 0
\(322\) 137504.i 1.32618i
\(323\) −86641.7 86641.7i −0.830466 0.830466i
\(324\) 0 0
\(325\) −33089.5 + 33729.4i −0.313273 + 0.319332i
\(326\) 155362. 1.46187
\(327\) 0 0
\(328\) 124297. 1.15535
\(329\) 3974.48i 0.0367188i
\(330\) 0 0
\(331\) 23593.9 23593.9i 0.215349 0.215349i −0.591186 0.806535i \(-0.701340\pi\)
0.806535 + 0.591186i \(0.201340\pi\)
\(332\) 57275.0 + 57275.0i 0.519624 + 0.519624i
\(333\) 0 0
\(334\) −354641. −3.17904
\(335\) 24697.3i 0.220069i
\(336\) 0 0
\(337\) 114863.i 1.01139i −0.862711 0.505697i \(-0.831235\pi\)
0.862711 0.505697i \(-0.168765\pi\)
\(338\) −136182. 131062.i −1.19202 1.14721i
\(339\) 0 0
\(340\) 94123.4 94123.4i 0.814216 0.814216i
\(341\) 153375. 1.31900
\(342\) 0 0
\(343\) −89921.4 + 89921.4i −0.764319 + 0.764319i
\(344\) 106517. 106517.i 0.900122 0.900122i
\(345\) 0 0
\(346\) 8015.22 + 8015.22i 0.0669520 + 0.0669520i
\(347\) −93409.6 −0.775769 −0.387885 0.921708i \(-0.626794\pi\)
−0.387885 + 0.921708i \(0.626794\pi\)
\(348\) 0 0
\(349\) −14479.3 14479.3i −0.118877 0.118877i 0.645166 0.764043i \(-0.276788\pi\)
−0.764043 + 0.645166i \(0.776788\pi\)
\(350\) 79066.0i 0.645437i
\(351\) 0 0
\(352\) 64353.3 0.519380
\(353\) −127078. + 127078.i −1.01981 + 1.01981i −0.0200140 + 0.999800i \(0.506371\pi\)
−0.999800 + 0.0200140i \(0.993629\pi\)
\(354\) 0 0
\(355\) 80147.6i 0.635966i
\(356\) −167648. + 167648.i −1.32281 + 1.32281i
\(357\) 0 0
\(358\) −12636.2 12636.2i −0.0985939 0.0985939i
\(359\) −40565.6 40565.6i −0.314752 0.314752i 0.531995 0.846747i \(-0.321442\pi\)
−0.846747 + 0.531995i \(0.821442\pi\)
\(360\) 0 0
\(361\) 95750.0i 0.734724i
\(362\) −146583. 146583.i −1.11858 1.11858i
\(363\) 0 0
\(364\) −200708. + 1922.20i −1.51482 + 0.0145076i
\(365\) 22082.8 0.165755
\(366\) 0 0
\(367\) −40747.7 −0.302532 −0.151266 0.988493i \(-0.548335\pi\)
−0.151266 + 0.988493i \(0.548335\pi\)
\(368\) 34879.4i 0.257557i
\(369\) 0 0
\(370\) 129534. 129534.i 0.946198 0.946198i
\(371\) −152303. 152303.i −1.10652 1.10652i
\(372\) 0 0
\(373\) −44353.1 −0.318791 −0.159396 0.987215i \(-0.550955\pi\)
−0.159396 + 0.987215i \(0.550955\pi\)
\(374\) 141814.i 1.01386i
\(375\) 0 0
\(376\) 7257.77i 0.0513367i
\(377\) 93058.4 + 91292.8i 0.654746 + 0.642324i
\(378\) 0 0
\(379\) −27066.7 + 27066.7i −0.188433 + 0.188433i −0.795018 0.606585i \(-0.792539\pi\)
0.606585 + 0.795018i \(0.292539\pi\)
\(380\) 245593. 1.70078
\(381\) 0 0
\(382\) 246966. 246966.i 1.69243 1.69243i
\(383\) 166419. 166419.i 1.13451 1.13451i 0.145087 0.989419i \(-0.453654\pi\)
0.989419 0.145087i \(-0.0463461\pi\)
\(384\) 0 0
\(385\) −46701.2 46701.2i −0.315070 0.315070i
\(386\) −13156.3 −0.0882997
\(387\) 0 0
\(388\) −154945. 154945.i −1.02923 1.02923i
\(389\) 16474.9i 0.108874i −0.998517 0.0544371i \(-0.982664\pi\)
0.998517 0.0544371i \(-0.0173364\pi\)
\(390\) 0 0
\(391\) 125303. 0.819613
\(392\) 31717.8 31717.8i 0.206410 0.206410i
\(393\) 0 0
\(394\) 95226.6i 0.613431i
\(395\) −53152.4 + 53152.4i −0.340666 + 0.340666i
\(396\) 0 0
\(397\) −155869. 155869.i −0.988957 0.988957i 0.0109822 0.999940i \(-0.496504\pi\)
−0.999940 + 0.0109822i \(0.996504\pi\)
\(398\) 34812.1 + 34812.1i 0.219768 + 0.219768i
\(399\) 0 0
\(400\) 20056.0i 0.125350i
\(401\) 142185. + 142185.i 0.884230 + 0.884230i 0.993961 0.109732i \(-0.0349991\pi\)
−0.109732 + 0.993961i \(0.534999\pi\)
\(402\) 0 0
\(403\) −311689. + 2985.07i −1.91916 + 0.0183800i
\(404\) −405531. −2.48463
\(405\) 0 0
\(406\) 218141. 1.32338
\(407\) 123861.i 0.747730i
\(408\) 0 0
\(409\) 9949.25 9949.25i 0.0594763 0.0594763i −0.676743 0.736219i \(-0.736609\pi\)
0.736219 + 0.676743i \(0.236609\pi\)
\(410\) −138521. 138521.i −0.824042 0.824042i
\(411\) 0 0
\(412\) 363282. 2.14017
\(413\) 12466.5i 0.0730876i
\(414\) 0 0
\(415\) 54165.8i 0.314506i
\(416\) −130779. + 1252.48i −0.755701 + 0.00723742i
\(417\) 0 0
\(418\) 185015. 185015.i 1.05890 1.05890i
\(419\) 17793.2 0.101351 0.0506753 0.998715i \(-0.483863\pi\)
0.0506753 + 0.998715i \(0.483863\pi\)
\(420\) 0 0
\(421\) −123242. + 123242.i −0.695334 + 0.695334i −0.963400 0.268066i \(-0.913615\pi\)
0.268066 + 0.963400i \(0.413615\pi\)
\(422\) 214671. 214671.i 1.20545 1.20545i
\(423\) 0 0
\(424\) 278120. + 278120.i 1.54704 + 1.54704i
\(425\) 72050.5 0.398896
\(426\) 0 0
\(427\) −211487. 211487.i −1.15992 1.15992i
\(428\) 396180.i 2.16274i
\(429\) 0 0
\(430\) −237413. −1.28401
\(431\) 81585.7 81585.7i 0.439197 0.439197i −0.452545 0.891742i \(-0.649484\pi\)
0.891742 + 0.452545i \(0.149484\pi\)
\(432\) 0 0
\(433\) 60799.1i 0.324281i −0.986768 0.162141i \(-0.948160\pi\)
0.986768 0.162141i \(-0.0518398\pi\)
\(434\) −368817. + 368817.i −1.95809 + 1.95809i
\(435\) 0 0
\(436\) 135812. + 135812.i 0.714439 + 0.714439i
\(437\) 163475. + 163475.i 0.856027 + 0.856027i
\(438\) 0 0
\(439\) 275044.i 1.42716i 0.700572 + 0.713581i \(0.252928\pi\)
−0.700572 + 0.713581i \(0.747072\pi\)
\(440\) 85280.9 + 85280.9i 0.440500 + 0.440500i
\(441\) 0 0
\(442\) 2760.06 + 288194.i 0.0141278 + 1.47516i
\(443\) 8716.50 0.0444155 0.0222077 0.999753i \(-0.492930\pi\)
0.0222077 + 0.999753i \(0.492930\pi\)
\(444\) 0 0
\(445\) 158547. 0.800642
\(446\) 245078.i 1.23207i
\(447\) 0 0
\(448\) −189431. + 189431.i −0.943832 + 0.943832i
\(449\) −246638. 246638.i −1.22340 1.22340i −0.966416 0.256982i \(-0.917272\pi\)
−0.256982 0.966416i \(-0.582728\pi\)
\(450\) 0 0
\(451\) −132454. −0.651197
\(452\) 95423.5i 0.467066i
\(453\) 0 0
\(454\) 191086.i 0.927078i
\(455\) 95815.0 + 93997.2i 0.462819 + 0.454038i
\(456\) 0 0
\(457\) 55246.3 55246.3i 0.264527 0.264527i −0.562363 0.826890i \(-0.690108\pi\)
0.826890 + 0.562363i \(0.190108\pi\)
\(458\) 487672. 2.32486
\(459\) 0 0
\(460\) −177591. + 177591.i −0.839277 + 0.839277i
\(461\) 31473.0 31473.0i 0.148093 0.148093i −0.629172 0.777266i \(-0.716606\pi\)
0.777266 + 0.629172i \(0.216606\pi\)
\(462\) 0 0
\(463\) 134703. + 134703.i 0.628371 + 0.628371i 0.947658 0.319287i \(-0.103443\pi\)
−0.319287 + 0.947658i \(0.603443\pi\)
\(464\) −55333.9 −0.257013
\(465\) 0 0
\(466\) −214231. 214231.i −0.986529 0.986529i
\(467\) 106801.i 0.489714i −0.969559 0.244857i \(-0.921259\pi\)
0.969559 0.244857i \(-0.0787410\pi\)
\(468\) 0 0
\(469\) 56787.6 0.258171
\(470\) 8088.34 8088.34i 0.0366154 0.0366154i
\(471\) 0 0
\(472\) 22765.0i 0.102184i
\(473\) −113507. + 113507.i −0.507342 + 0.507342i
\(474\) 0 0
\(475\) 93999.4 + 93999.4i 0.416618 + 0.416618i
\(476\) 216422. + 216422.i 0.955186 + 0.955186i
\(477\) 0 0
\(478\) 514301.i 2.25093i
\(479\) −69279.4 69279.4i −0.301949 0.301949i 0.539827 0.841776i \(-0.318490\pi\)
−0.841776 + 0.539827i \(0.818490\pi\)
\(480\) 0 0
\(481\) 2410.64 + 251709.i 0.0104194 + 1.08795i
\(482\) −219685. −0.945597
\(483\) 0 0
\(484\) −214719. −0.916599
\(485\) 146533.i 0.622950i
\(486\) 0 0
\(487\) 334191. 334191.i 1.40908 1.40908i 0.644372 0.764713i \(-0.277119\pi\)
0.764713 0.644372i \(-0.222881\pi\)
\(488\) 386195. + 386195.i 1.62169 + 1.62169i
\(489\) 0 0
\(490\) −70695.1 −0.294440
\(491\) 204363.i 0.847696i 0.905733 + 0.423848i \(0.139321\pi\)
−0.905733 + 0.423848i \(0.860679\pi\)
\(492\) 0 0
\(493\) 198785.i 0.817880i
\(494\) −372386. + 379588.i −1.52595 + 1.55546i
\(495\) 0 0
\(496\) 93554.7 93554.7i 0.380279 0.380279i
\(497\) −184287. −0.746074
\(498\) 0 0
\(499\) 263314. 263314.i 1.05748 1.05748i 0.0592381 0.998244i \(-0.481133\pi\)
0.998244 0.0592381i \(-0.0188671\pi\)
\(500\) −330391. + 330391.i −1.32157 + 1.32157i
\(501\) 0 0
\(502\) −241773. 241773.i −0.959403 0.959403i
\(503\) −120971. −0.478131 −0.239065 0.971003i \(-0.576841\pi\)
−0.239065 + 0.971003i \(0.576841\pi\)
\(504\) 0 0
\(505\) 191758. + 191758.i 0.751920 + 0.751920i
\(506\) 267573.i 1.04506i
\(507\) 0 0
\(508\) −609990. −2.36371
\(509\) 50199.3 50199.3i 0.193759 0.193759i −0.603559 0.797318i \(-0.706251\pi\)
0.797318 + 0.603559i \(0.206251\pi\)
\(510\) 0 0
\(511\) 50775.9i 0.194454i
\(512\) 102587. 102587.i 0.391337 0.391337i
\(513\) 0 0
\(514\) −411688. 411688.i −1.55827 1.55827i
\(515\) −171780. 171780.i −0.647677 0.647677i
\(516\) 0 0
\(517\) 7734.07i 0.0289352i
\(518\) 297845. + 297845.i 1.11002 + 1.11002i
\(519\) 0 0
\(520\) −174967. 171648.i −0.647068 0.634792i
\(521\) 307242. 1.13189 0.565946 0.824443i \(-0.308511\pi\)
0.565946 + 0.824443i \(0.308511\pi\)
\(522\) 0 0
\(523\) 271249. 0.991664 0.495832 0.868418i \(-0.334863\pi\)
0.495832 + 0.868418i \(0.334863\pi\)
\(524\) 46083.7i 0.167836i
\(525\) 0 0
\(526\) 531501. 531501.i 1.92102 1.92102i
\(527\) 336092. + 336092.i 1.21014 + 1.21014i
\(528\) 0 0
\(529\) 43420.3 0.155161
\(530\) 619895.i 2.20682i
\(531\) 0 0
\(532\) 564703.i 1.99525i
\(533\) 269173. 2577.89i 0.947494 0.00907424i
\(534\) 0 0
\(535\) −187337. + 187337.i −0.654508 + 0.654508i
\(536\) −103699. −0.360950
\(537\) 0 0
\(538\) 26428.6 26428.6i 0.0913081 0.0913081i
\(539\) −33799.3 + 33799.3i −0.116340 + 0.116340i
\(540\) 0 0
\(541\) 184394. + 184394.i 0.630017 + 0.630017i 0.948072 0.318055i \(-0.103030\pi\)
−0.318055 + 0.948072i \(0.603030\pi\)
\(542\) 34836.6 0.118587
\(543\) 0 0
\(544\) 141018. + 141018.i 0.476515 + 0.476515i
\(545\) 128439.i 0.432419i
\(546\) 0 0
\(547\) 373290. 1.24759 0.623795 0.781588i \(-0.285590\pi\)
0.623795 + 0.781588i \(0.285590\pi\)
\(548\) 600433. 600433.i 1.99942 1.99942i
\(549\) 0 0
\(550\) 153857.i 0.508619i
\(551\) 259341. 259341.i 0.854217 0.854217i
\(552\) 0 0
\(553\) −122216. 122216.i −0.399647 0.399647i
\(554\) −148209. 148209.i −0.482898 0.482898i
\(555\) 0 0
\(556\) 129461.i 0.418784i
\(557\) −200107. 200107.i −0.644987 0.644987i 0.306790 0.951777i \(-0.400745\pi\)
−0.951777 + 0.306790i \(0.900745\pi\)
\(558\) 0 0
\(559\) 228460. 232878.i 0.731115 0.745254i
\(560\) −56973.0 −0.181674
\(561\) 0 0
\(562\) −101983. −0.322891
\(563\) 488676.i 1.54172i −0.637007 0.770858i \(-0.719828\pi\)
0.637007 0.770858i \(-0.280172\pi\)
\(564\) 0 0
\(565\) −45121.7 + 45121.7i −0.141348 + 0.141348i
\(566\) −145033. 145033.i −0.452725 0.452725i
\(567\) 0 0
\(568\) 336525. 1.04309
\(569\) 562947.i 1.73877i 0.494132 + 0.869387i \(0.335486\pi\)
−0.494132 + 0.869387i \(0.664514\pi\)
\(570\) 0 0
\(571\) 183359.i 0.562380i −0.959652 0.281190i \(-0.909271\pi\)
0.959652 0.281190i \(-0.0907291\pi\)
\(572\) −390564. + 3740.47i −1.19371 + 0.0114323i
\(573\) 0 0
\(574\) 318509. 318509.i 0.966713 0.966713i
\(575\) −135944. −0.411173
\(576\) 0 0
\(577\) 196407. 196407.i 0.589938 0.589938i −0.347677 0.937614i \(-0.613029\pi\)
0.937614 + 0.347677i \(0.113029\pi\)
\(578\) −80064.2 + 80064.2i −0.239653 + 0.239653i
\(579\) 0 0
\(580\) 281736. + 281736.i 0.837503 + 0.837503i
\(581\) 124546. 0.368958
\(582\) 0 0
\(583\) −296372. 296372.i −0.871966 0.871966i
\(584\) 92721.6i 0.271866i
\(585\) 0 0
\(586\) −1.06254e6 −3.09422
\(587\) 127057. 127057.i 0.368742 0.368742i −0.498276 0.867018i \(-0.666034\pi\)
0.867018 + 0.498276i \(0.166034\pi\)
\(588\) 0 0
\(589\) 876953.i 2.52782i
\(590\) 25370.2 25370.2i 0.0728818 0.0728818i
\(591\) 0 0
\(592\) −75551.7 75551.7i −0.215576 0.215576i
\(593\) −168617. 168617.i −0.479503 0.479503i 0.425470 0.904973i \(-0.360109\pi\)
−0.904973 + 0.425470i \(0.860109\pi\)
\(594\) 0 0
\(595\) 204674.i 0.578133i
\(596\) 10947.1 + 10947.1i 0.0308181 + 0.0308181i
\(597\) 0 0
\(598\) −5207.65 543761.i −0.0145626 1.52057i
\(599\) 22624.2 0.0630550 0.0315275 0.999503i \(-0.489963\pi\)
0.0315275 + 0.999503i \(0.489963\pi\)
\(600\) 0 0
\(601\) 621958. 1.72192 0.860958 0.508675i \(-0.169865\pi\)
0.860958 + 0.508675i \(0.169865\pi\)
\(602\) 545895.i 1.50632i
\(603\) 0 0
\(604\) −211802. + 211802.i −0.580572 + 0.580572i
\(605\) 101531. + 101531.i 0.277389 + 0.277389i
\(606\) 0 0
\(607\) −600769. −1.63054 −0.815268 0.579084i \(-0.803410\pi\)
−0.815268 + 0.579084i \(0.803410\pi\)
\(608\) 367953.i 0.995371i
\(609\) 0 0
\(610\) 860782.i 2.31331i
\(611\) 150.525 + 15717.1i 0.000403204 + 0.0421009i
\(612\) 0 0
\(613\) −230276. + 230276.i −0.612812 + 0.612812i −0.943678 0.330866i \(-0.892659\pi\)
0.330866 + 0.943678i \(0.392659\pi\)
\(614\) −703981. −1.86734
\(615\) 0 0
\(616\) −196090. + 196090.i −0.516767 + 0.516767i
\(617\) −34766.2 + 34766.2i −0.0913244 + 0.0913244i −0.751293 0.659969i \(-0.770570\pi\)
0.659969 + 0.751293i \(0.270570\pi\)
\(618\) 0 0
\(619\) −420362. 420362.i −1.09709 1.09709i −0.994750 0.102339i \(-0.967367\pi\)
−0.102339 0.994750i \(-0.532633\pi\)
\(620\) −952680. −2.47836
\(621\) 0 0
\(622\) 703892. + 703892.i 1.81939 + 1.81939i
\(623\) 364555.i 0.939262i
\(624\) 0 0
\(625\) 137714. 0.352547
\(626\) −592423. + 592423.i −1.51176 + 1.51176i
\(627\) 0 0
\(628\) 711130.i 1.80314i
\(629\) 271417. 271417.i 0.686018 0.686018i
\(630\) 0 0
\(631\) 93922.3 + 93922.3i 0.235890 + 0.235890i 0.815146 0.579256i \(-0.196657\pi\)
−0.579256 + 0.815146i \(0.696657\pi\)
\(632\) 223177. + 223177.i 0.558748 + 0.558748i
\(633\) 0 0
\(634\) 396347.i 0.986045i
\(635\) 288438. + 288438.i 0.715327 + 0.715327i
\(636\) 0 0
\(637\) 68029.0 69344.6i 0.167654 0.170897i
\(638\) 424487. 1.04285
\(639\) 0 0
\(640\) −540888. −1.32053
\(641\) 123299.i 0.300084i 0.988680 + 0.150042i \(0.0479409\pi\)
−0.988680 + 0.150042i \(0.952059\pi\)
\(642\) 0 0
\(643\) −280732. + 280732.i −0.678999 + 0.678999i −0.959774 0.280774i \(-0.909409\pi\)
0.280774 + 0.959774i \(0.409409\pi\)
\(644\) −408343. 408343.i −0.984586 0.984586i
\(645\) 0 0
\(646\) 810850. 1.94301
\(647\) 45435.6i 0.108539i 0.998526 + 0.0542697i \(0.0172831\pi\)
−0.998526 + 0.0542697i \(0.982717\pi\)
\(648\) 0 0
\(649\) 24258.9i 0.0575946i
\(650\) −2994.45 312668.i −0.00708746 0.740042i
\(651\) 0 0
\(652\) −461375. + 461375.i −1.08532 + 1.08532i
\(653\) 378723. 0.888168 0.444084 0.895985i \(-0.353529\pi\)
0.444084 + 0.895985i \(0.353529\pi\)
\(654\) 0 0
\(655\) −21791.0 + 21791.0i −0.0507920 + 0.0507920i
\(656\) −80793.4 + 80793.4i −0.187745 + 0.187745i
\(657\) 0 0
\(658\) 18597.9 + 18597.9i 0.0429548 + 0.0429548i
\(659\) −289116. −0.665735 −0.332867 0.942974i \(-0.608016\pi\)
−0.332867 + 0.942974i \(0.608016\pi\)
\(660\) 0 0
\(661\) −168774. 168774.i −0.386281 0.386281i 0.487078 0.873359i \(-0.338063\pi\)
−0.873359 + 0.487078i \(0.838063\pi\)
\(662\) 220807.i 0.503845i
\(663\) 0 0
\(664\) −227433. −0.515842
\(665\) 267024. 267024.i 0.603819 0.603819i
\(666\) 0 0
\(667\) 375065.i 0.843054i
\(668\) 1.05317e6 1.05317e6i 2.36019 2.36019i
\(669\) 0 0
\(670\) 115567. + 115567.i 0.257444 + 0.257444i
\(671\) −411539. 411539.i −0.914043 0.914043i
\(672\) 0 0
\(673\) 163719.i 0.361467i 0.983532 + 0.180734i \(0.0578472\pi\)
−0.983532 + 0.180734i \(0.942153\pi\)
\(674\) 537482. + 537482.i 1.18316 + 1.18316i
\(675\) 0 0
\(676\) 793630. 15202.7i 1.73670 0.0332681i
\(677\) −462025. −1.00806 −0.504032 0.863685i \(-0.668151\pi\)
−0.504032 + 0.863685i \(0.668151\pi\)
\(678\) 0 0
\(679\) −336931. −0.730805
\(680\) 373753.i 0.808290i
\(681\) 0 0
\(682\) −717693. + 717693.i −1.54301 + 1.54301i
\(683\) 213724. + 213724.i 0.458155 + 0.458155i 0.898049 0.439895i \(-0.144984\pi\)
−0.439895 + 0.898049i \(0.644984\pi\)
\(684\) 0 0
\(685\) −567838. −1.21016
\(686\) 841544.i 1.78825i
\(687\) 0 0
\(688\) 138473.i 0.292541i
\(689\) 608054. + 596517.i 1.28087 + 1.25656i
\(690\) 0 0
\(691\) −344817. + 344817.i −0.722159 + 0.722159i −0.969045 0.246886i \(-0.920593\pi\)
0.246886 + 0.969045i \(0.420593\pi\)
\(692\) −47605.4 −0.0994132
\(693\) 0 0
\(694\) 437094. 437094.i 0.907520 0.907520i
\(695\) 61216.6 61216.6i 0.126736 0.126736i
\(696\) 0 0
\(697\) −290248. 290248.i −0.597452 0.597452i
\(698\) 135507. 0.278132
\(699\) 0 0
\(700\) −234801. 234801.i −0.479186 0.479186i
\(701\) 83587.3i 0.170100i 0.996377 + 0.0850500i \(0.0271050\pi\)
−0.996377 + 0.0850500i \(0.972895\pi\)
\(702\) 0 0
\(703\) 708198. 1.43299
\(704\) −368619. + 368619.i −0.743760 + 0.743760i
\(705\) 0 0
\(706\) 1.18928e6i 2.38602i
\(707\) −440919. + 440919.i −0.882104 + 0.882104i
\(708\) 0 0
\(709\) 507737. + 507737.i 1.01006 + 1.01006i 0.999949 + 0.0101092i \(0.00321793\pi\)
0.0101092 + 0.999949i \(0.496782\pi\)
\(710\) −375037. 375037.i −0.743973 0.743973i
\(711\) 0 0
\(712\) 665711.i 1.31318i
\(713\) −634135. 634135.i −1.24739 1.24739i
\(714\) 0 0
\(715\) 186450. + 182912.i 0.364711 + 0.357792i
\(716\) 75051.0 0.146396
\(717\) 0 0
\(718\) 379640. 0.736415
\(719\) 50596.5i 0.0978729i −0.998802 0.0489364i \(-0.984417\pi\)
0.998802 0.0489364i \(-0.0155832\pi\)
\(720\) 0 0
\(721\) 394982. 394982.i 0.759814 0.759814i
\(722\) 448046. + 448046.i 0.859505 + 0.859505i
\(723\) 0 0
\(724\) 870610. 1.66091
\(725\) 215666.i 0.410304i
\(726\) 0 0
\(727\) 891410.i 1.68659i −0.537453 0.843294i \(-0.680613\pi\)
0.537453 0.843294i \(-0.319387\pi\)
\(728\) 394678. 402310.i 0.744697 0.759099i
\(729\) 0 0
\(730\) −103333. + 103333.i −0.193906 + 0.193906i
\(731\) −497458. −0.930939
\(732\) 0 0
\(733\) −439182. + 439182.i −0.817404 + 0.817404i −0.985731 0.168327i \(-0.946164\pi\)
0.168327 + 0.985731i \(0.446164\pi\)
\(734\) 190672. 190672.i 0.353912 0.353912i
\(735\) 0 0
\(736\) −266071. 266071.i −0.491182 0.491182i
\(737\) 110505. 0.203445
\(738\) 0 0
\(739\) 83855.7 + 83855.7i 0.153548 + 0.153548i 0.779701 0.626153i \(-0.215371\pi\)
−0.626153 + 0.779701i \(0.715371\pi\)
\(740\) 769353.i 1.40495i
\(741\) 0 0
\(742\) 1.42535e6 2.58890
\(743\) 153273. 153273.i 0.277643 0.277643i −0.554524 0.832167i \(-0.687100\pi\)
0.832167 + 0.554524i \(0.187100\pi\)
\(744\) 0 0
\(745\) 10352.8i 0.0186529i
\(746\) 207543. 207543.i 0.372933 0.372933i
\(747\) 0 0
\(748\) 421143. + 421143.i 0.752708 + 0.752708i
\(749\) −430752. 430752.i −0.767827 0.767827i
\(750\) 0 0
\(751\) 208551.i 0.369771i −0.982760 0.184886i \(-0.940808\pi\)
0.982760 0.184886i \(-0.0591915\pi\)
\(752\) −4717.57 4717.57i −0.00834225 0.00834225i
\(753\) 0 0
\(754\) −862640. + 8261.59i −1.51735 + 0.0145319i
\(755\) 200304. 0.351395
\(756\) 0 0
\(757\) −762290. −1.33024 −0.665118 0.746739i \(-0.731619\pi\)
−0.665118 + 0.746739i \(0.731619\pi\)
\(758\) 253308.i 0.440870i
\(759\) 0 0
\(760\) −487610. + 487610.i −0.844201 + 0.844201i
\(761\) −790313. 790313.i −1.36468 1.36468i −0.867850 0.496826i \(-0.834499\pi\)
−0.496826 0.867850i \(-0.665501\pi\)
\(762\) 0 0
\(763\) 295326. 0.507287
\(764\) 1.46682e6i 2.51299i
\(765\) 0 0
\(766\) 1.55746e6i 2.65436i
\(767\) 472.140 + 49298.9i 0.000802565 + 0.0838005i
\(768\) 0 0
\(769\) 170652. 170652.i 0.288574 0.288574i −0.547942 0.836516i \(-0.684589\pi\)
0.836516 + 0.547942i \(0.184589\pi\)
\(770\) 437061. 0.737158
\(771\) 0 0
\(772\) 39070.1 39070.1i 0.0655555 0.0655555i
\(773\) −238774. + 238774.i −0.399602 + 0.399602i −0.878093 0.478490i \(-0.841184\pi\)
0.478490 + 0.878093i \(0.341184\pi\)
\(774\) 0 0
\(775\) −364634. 364634.i −0.607090 0.607090i
\(776\) 615268. 1.02174
\(777\) 0 0
\(778\) 77091.7 + 77091.7i 0.127365 + 0.127365i
\(779\) 757333.i 1.24799i
\(780\) 0 0
\(781\) −358610. −0.587923
\(782\) −586335. + 586335.i −0.958810 + 0.958810i
\(783\) 0 0
\(784\) 41233.3i 0.0670836i
\(785\) −336263. + 336263.i −0.545682 + 0.545682i
\(786\) 0 0
\(787\) −460701. 460701.i −0.743823 0.743823i 0.229488 0.973311i \(-0.426295\pi\)
−0.973311 + 0.229488i \(0.926295\pi\)
\(788\) 282793. + 282793.i 0.455424 + 0.455424i
\(789\) 0 0
\(790\) 497435.i 0.797044i
\(791\) −103750. 103750.i −0.165820 0.165820i
\(792\) 0 0
\(793\) 844339. + 828320.i 1.34267 + 1.31720i
\(794\) 1.45872e6 2.31383
\(795\) 0 0
\(796\) −206762. −0.326320
\(797\) 940096.i 1.47998i 0.672618 + 0.739989i \(0.265170\pi\)
−0.672618 + 0.739989i \(0.734830\pi\)
\(798\) 0 0
\(799\) 16947.7 16947.7i 0.0265471 0.0265471i
\(800\) −152993. 152993.i −0.239052 0.239052i
\(801\) 0 0
\(802\) −1.33066e6 −2.06880
\(803\) 98806.5i 0.153234i
\(804\) 0 0
\(805\) 386176.i 0.595928i
\(806\) 1.44453e6 1.47246e6i 2.22359 2.26660i
\(807\) 0 0
\(808\) 805159. 805159.i 1.23327 1.23327i
\(809\) −612686. −0.936140 −0.468070 0.883691i \(-0.655050\pi\)
−0.468070 + 0.883691i \(0.655050\pi\)
\(810\) 0 0
\(811\) 431732. 431732.i 0.656406 0.656406i −0.298121 0.954528i \(-0.596360\pi\)
0.954528 + 0.298121i \(0.0963601\pi\)
\(812\) −647809. + 647809.i −0.982505 + 0.982505i
\(813\) 0 0
\(814\) 579585. + 579585.i 0.874719 + 0.874719i
\(815\) 436329. 0.656900
\(816\) 0 0
\(817\) −648999. 648999.i −0.972300 0.972300i
\(818\) 93111.7i 0.139155i
\(819\) 0 0
\(820\) 822730. 1.22357
\(821\) 193284. 193284.i 0.286754 0.286754i −0.549041 0.835795i \(-0.685007\pi\)
0.835795 + 0.549041i \(0.185007\pi\)
\(822\) 0 0
\(823\) 547967.i 0.809011i 0.914536 + 0.404506i \(0.132556\pi\)
−0.914536 + 0.404506i \(0.867444\pi\)
\(824\) −721275. + 721275.i −1.06230 + 1.06230i
\(825\) 0 0
\(826\) 58334.8 + 58334.8i 0.0855003 + 0.0855003i
\(827\) 859953. + 859953.i 1.25737 + 1.25737i 0.952343 + 0.305029i \(0.0986660\pi\)
0.305029 + 0.952343i \(0.401334\pi\)
\(828\) 0 0
\(829\) 726172.i 1.05665i −0.849043 0.528324i \(-0.822821\pi\)
0.849043 0.528324i \(-0.177179\pi\)
\(830\) 253460. + 253460.i 0.367919 + 0.367919i
\(831\) 0 0
\(832\) 741933. 756282.i 1.07181 1.09254i
\(833\) −148129. −0.213477
\(834\) 0 0
\(835\) −995999. −1.42852
\(836\) 1.09887e6i 1.57230i
\(837\) 0 0
\(838\) −83260.3 + 83260.3i −0.118563 + 0.118563i
\(839\) 507389. + 507389.i 0.720804 + 0.720804i 0.968769 0.247965i \(-0.0797618\pi\)
−0.247965 + 0.968769i \(0.579762\pi\)
\(840\) 0 0
\(841\) −112265. −0.158728
\(842\) 1.15338e6i 1.62685i
\(843\) 0 0
\(844\) 1.27501e6i 1.78990i
\(845\) −382462. 368085.i −0.535642 0.515507i
\(846\) 0 0
\(847\) −233456. + 233456.i −0.325415 + 0.325415i
\(848\) −361557. −0.502789
\(849\) 0 0
\(850\) −337148. + 337148.i −0.466641 + 0.466641i
\(851\) −512107. + 512107.i −0.707133 + 0.707133i
\(852\) 0 0
\(853\) 192677. + 192677.i 0.264808 + 0.264808i 0.827004 0.562196i \(-0.190043\pi\)
−0.562196 + 0.827004i \(0.690043\pi\)
\(854\) 1.97924e6 2.71383
\(855\) 0 0
\(856\) 786593. + 786593.i 1.07350 + 1.07350i
\(857\) 96833.2i 0.131845i −0.997825 0.0659223i \(-0.979001\pi\)
0.997825 0.0659223i \(-0.0209990\pi\)
\(858\) 0 0
\(859\) 248211. 0.336383 0.168192 0.985754i \(-0.446207\pi\)
0.168192 + 0.985754i \(0.446207\pi\)
\(860\) 705042. 705042.i 0.953275 0.953275i
\(861\) 0 0
\(862\) 763533.i 1.02757i
\(863\) −1.02273e6 + 1.02273e6i −1.37321 + 1.37321i −0.517571 + 0.855640i \(0.673164\pi\)
−0.855640 + 0.517571i \(0.826836\pi\)
\(864\) 0 0
\(865\) 22510.5 + 22510.5i 0.0300853 + 0.0300853i
\(866\) 284499. + 284499.i 0.379355 + 0.379355i
\(867\) 0 0
\(868\) 2.19054e6i 2.90745i
\(869\) −237824. 237824.i −0.314931 0.314931i
\(870\) 0 0
\(871\) −224567. + 2150.70i −0.296013 + 0.00283494i
\(872\) −539294. −0.709239
\(873\) 0 0
\(874\) −1.52990e6 −2.00282
\(875\) 718444.i 0.938376i
\(876\) 0 0
\(877\) 263404. 263404.i 0.342470 0.342470i −0.514825 0.857295i \(-0.672143\pi\)
0.857295 + 0.514825i \(0.172143\pi\)
\(878\) −1.28702e6 1.28702e6i −1.66954 1.66954i
\(879\) 0 0
\(880\) −110866. −0.143163
\(881\) 580702.i 0.748171i −0.927394 0.374086i \(-0.877957\pi\)
0.927394 0.374086i \(-0.122043\pi\)
\(882\) 0 0
\(883\) 1.28367e6i 1.64638i 0.567763 + 0.823192i \(0.307809\pi\)
−0.567763 + 0.823192i \(0.692191\pi\)
\(884\) −864042. 847649.i −1.10568 1.08470i
\(885\) 0 0
\(886\) −40787.4 + 40787.4i −0.0519587 + 0.0519587i
\(887\) −166700. −0.211879 −0.105940 0.994373i \(-0.533785\pi\)
−0.105940 + 0.994373i \(0.533785\pi\)
\(888\) 0 0
\(889\) −663219. + 663219.i −0.839176 + 0.839176i
\(890\) −741894. + 741894.i −0.936617 + 0.936617i
\(891\) 0 0
\(892\) −727805. 727805.i −0.914714 0.914714i
\(893\) 44221.1 0.0554532
\(894\) 0 0
\(895\) −35488.4 35488.4i −0.0443037 0.0443037i
\(896\) 1.24369e6i 1.54916i
\(897\) 0 0
\(898\) 2.30820e6 2.86234
\(899\) −1.00601e6 + 1.00601e6i −1.24475 + 1.24475i
\(900\) 0 0
\(901\) 1.29888e6i 1.60000i
\(902\) 619796. 619796.i 0.761791 0.761791i
\(903\) 0 0
\(904\) 189458. + 189458.i 0.231833 + 0.231833i
\(905\) −411674. 411674.i −0.502639 0.502639i
\(906\) 0 0
\(907\) 1.06525e6i 1.29490i 0.762108 + 0.647450i \(0.224164\pi\)
−0.762108 + 0.647450i \(0.775836\pi\)
\(908\) −567464. 567464.i −0.688282 0.688282i
\(909\) 0 0
\(910\) −888194. + 8506.32i −1.07257 + 0.0102721i
\(911\) 794671. 0.957526 0.478763 0.877944i \(-0.341085\pi\)
0.478763 + 0.877944i \(0.341085\pi\)
\(912\) 0 0
\(913\) 242358. 0.290747
\(914\) 517031.i 0.618905i
\(915\) 0 0
\(916\) −1.44823e6 + 1.44823e6i −1.72603 + 1.72603i
\(917\) −50105.1 50105.1i −0.0595859 0.0595859i
\(918\) 0 0
\(919\) 514196. 0.608833 0.304416 0.952539i \(-0.401539\pi\)
0.304416 + 0.952539i \(0.401539\pi\)
\(920\) 705193.i 0.833168i
\(921\) 0 0
\(922\) 294545.i 0.346489i
\(923\) 728766. 6979.46i 0.855431 0.00819254i
\(924\) 0 0
\(925\) −294466. + 294466.i −0.344153 + 0.344153i
\(926\) −1.26064e6 −1.47018
\(927\) 0 0
\(928\) −422104. + 422104.i −0.490143 + 0.490143i
\(929\) 746377. 746377.i 0.864823 0.864823i −0.127071 0.991894i \(-0.540558\pi\)
0.991894 + 0.127071i \(0.0405576\pi\)
\(930\) 0 0
\(931\) −193254. 193254.i −0.222961 0.222961i
\(932\) 1.27240e6 1.46484
\(933\) 0 0
\(934\) 499758. + 499758.i 0.572884 + 0.572884i
\(935\) 398281.i 0.455582i
\(936\) 0 0
\(937\) 379998. 0.432815 0.216408 0.976303i \(-0.430566\pi\)
0.216408 + 0.976303i \(0.430566\pi\)
\(938\) −265728. + 265728.i −0.302017 + 0.302017i
\(939\) 0 0
\(940\) 48039.7i 0.0543681i
\(941\) 906449. 906449.i 1.02368 1.02368i 0.0239676 0.999713i \(-0.492370\pi\)
0.999713 0.0239676i \(-0.00762984\pi\)
\(942\) 0 0
\(943\) 547636. + 547636.i 0.615841 + 0.615841i
\(944\) −14797.3 14797.3i −0.0166050 0.0166050i
\(945\) 0 0
\(946\) 1.06227e6i 1.18701i
\(947\) −290385. 290385.i −0.323798 0.323798i 0.526424 0.850222i \(-0.323532\pi\)
−0.850222 + 0.526424i \(0.823532\pi\)
\(948\) 0 0
\(949\) −1923.03 200794.i −0.00213527 0.222956i
\(950\) −879709. −0.974747
\(951\) 0 0
\(952\) −859388. −0.948234
\(953\) 1.13841e6i 1.25347i 0.779234 + 0.626733i \(0.215608\pi\)
−0.779234 + 0.626733i \(0.784392\pi\)
\(954\) 0 0
\(955\) 693596. 693596.i 0.760501 0.760501i
\(956\) −1.52731e6 1.52731e6i −1.67114 1.67114i
\(957\) 0 0
\(958\) 648362. 0.706459
\(959\) 1.30566e6i 1.41968i
\(960\) 0 0
\(961\) 2.47827e6i 2.68350i
\(962\) −1.18911e6 1.16655e6i −1.28491 1.26053i
\(963\) 0 0
\(964\) 652395. 652395.i 0.702032 0.702032i
\(965\) −36949.1 −0.0396780
\(966\) 0 0
\(967\) 19218.1 19218.1i 0.0205521 0.0205521i −0.696756 0.717308i \(-0.745374\pi\)
0.717308 + 0.696756i \(0.245374\pi\)
\(968\) 426312. 426312.i 0.454964 0.454964i
\(969\) 0 0
\(970\) −685678. 685678.i −0.728747 0.728747i
\(971\) −416812. −0.442080 −0.221040 0.975265i \(-0.570945\pi\)
−0.221040 + 0.975265i \(0.570945\pi\)
\(972\) 0 0
\(973\) 140758. + 140758.i 0.148678 + 0.148678i
\(974\) 3.12758e6i 3.29679i
\(975\) 0 0
\(976\) −502056. −0.527051
\(977\) −394416. + 394416.i −0.413205 + 0.413205i −0.882853 0.469649i \(-0.844381\pi\)
0.469649 + 0.882853i \(0.344381\pi\)
\(978\) 0 0
\(979\) 709399.i 0.740159i
\(980\) 209942. 209942.i 0.218599 0.218599i
\(981\) 0 0
\(982\) −956284. 956284.i −0.991663 0.991663i
\(983\) 159479. + 159479.i 0.165043 + 0.165043i 0.784796 0.619754i \(-0.212767\pi\)
−0.619754 + 0.784796i \(0.712767\pi\)
\(984\) 0 0
\(985\) 267441.i 0.275649i
\(986\) 930181. + 930181.i 0.956783 + 0.956783i
\(987\) 0 0
\(988\) −21386.9 2.23313e6i −0.0219095 2.28770i
\(989\) 938598. 0.959593
\(990\) 0 0
\(991\) 869851. 0.885722 0.442861 0.896590i \(-0.353964\pi\)
0.442861 + 0.896590i \(0.353964\pi\)
\(992\) 1.42733e6i 1.45044i
\(993\) 0 0
\(994\) 862340. 862340.i 0.872782 0.872782i
\(995\) 97768.8 + 97768.8i 0.0987539 + 0.0987539i
\(996\) 0 0
\(997\) 798123. 0.802934 0.401467 0.915874i \(-0.368501\pi\)
0.401467 + 0.915874i \(0.368501\pi\)
\(998\) 2.46427e6i 2.47415i
\(999\) 0 0
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 117.5.j.b.109.2 20
3.2 odd 2 39.5.g.a.31.9 20
13.8 odd 4 inner 117.5.j.b.73.2 20
39.8 even 4 39.5.g.a.34.9 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.5.g.a.31.9 20 3.2 odd 2
39.5.g.a.34.9 yes 20 39.8 even 4
117.5.j.b.73.2 20 13.8 odd 4 inner
117.5.j.b.109.2 20 1.1 even 1 trivial