Properties

Label 76.7.h.a.69.8
Level $76$
Weight $7$
Character 76.69
Analytic conductor $17.484$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [76,7,Mod(65,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(6))
 
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("76.65");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 76.h (of order \(6\), 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: \(17.4841103551\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 9460 x^{18} + 36670708 x^{16} + 75655761912 x^{14} + 90488614064544 x^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{8}\cdot 19^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 69.8
Root \(-21.3475i\) of defining polynomial
Character \(\chi\) \(=\) 76.69
Dual form 76.7.h.a.65.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(16.9875 - 9.80775i) q^{3} +(72.3330 + 125.284i) q^{5} +479.542 q^{7} +(-172.116 + 298.114i) q^{9} +O(q^{10})\) \(q+(16.9875 - 9.80775i) q^{3} +(72.3330 + 125.284i) q^{5} +479.542 q^{7} +(-172.116 + 298.114i) q^{9} -1675.76 q^{11} +(653.454 + 377.272i) q^{13} +(2457.52 + 1418.85i) q^{15} +(3910.26 + 6772.78i) q^{17} +(-6399.56 + 2468.11i) q^{19} +(8146.24 - 4703.23i) q^{21} +(10038.9 - 17387.9i) q^{23} +(-2651.63 + 4592.76i) q^{25} +21052.0i q^{27} +(39064.1 + 22553.7i) q^{29} -20987.2i q^{31} +(-28467.1 + 16435.5i) q^{33} +(34686.8 + 60079.2i) q^{35} +64541.3i q^{37} +14800.8 q^{39} +(38429.1 - 22187.0i) q^{41} +(-384.826 - 666.538i) q^{43} -49798.7 q^{45} +(24326.1 - 42134.0i) q^{47} +112312. q^{49} +(132851. + 76701.8i) q^{51} +(-154862. - 89409.9i) q^{53} +(-121213. - 209947. i) q^{55} +(-84506.0 + 104692. i) q^{57} +(-182560. + 105401. i) q^{59} +(200137. - 346647. i) q^{61} +(-82537.0 + 142958. i) q^{63} +109157. i q^{65} +(-516018. - 297923. i) q^{67} -393836. i q^{69} +(-20990.7 + 12119.0i) q^{71} +(-62525.1 - 108297. i) q^{73} +104026. i q^{75} -803599. q^{77} +(569365. - 328723. i) q^{79} +(80999.9 + 140296. i) q^{81} +992838. q^{83} +(-565682. + 979791. i) q^{85} +884804. q^{87} +(-462997. - 267312. i) q^{89} +(313359. + 180918. i) q^{91} +(-205838. - 356521. i) q^{93} +(-772115. - 623239. i) q^{95} +(-1.42085e6 + 820331. i) q^{97} +(288426. - 499568. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 30 q^{3} - 56 q^{5} + 464 q^{7} + 2200 q^{9} - 3644 q^{11} - 7140 q^{13} + 9168 q^{15} + 1132 q^{17} + 2110 q^{19} - 8748 q^{21} + 832 q^{23} - 27698 q^{25} - 10920 q^{29} - 30306 q^{33} + 4172 q^{35} + 81144 q^{39} + 109206 q^{41} + 110740 q^{43} - 785440 q^{45} + 107080 q^{47} + 136092 q^{49} + 199872 q^{51} + 254796 q^{53} + 354840 q^{55} + 212268 q^{57} - 610638 q^{59} + 47864 q^{61} - 254476 q^{63} - 839562 q^{67} + 366660 q^{71} + 854482 q^{73} + 763088 q^{77} + 1718592 q^{79} - 1054142 q^{81} + 439612 q^{83} - 400236 q^{85} - 1604736 q^{87} + 478032 q^{89} + 599856 q^{91} + 829380 q^{93} - 1055660 q^{95} - 191286 q^{97} - 2336728 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 16.9875 9.80775i 0.629167 0.363250i −0.151262 0.988494i \(-0.548334\pi\)
0.780430 + 0.625244i \(0.215000\pi\)
\(4\) 0 0
\(5\) 72.3330 + 125.284i 0.578664 + 1.00228i 0.995633 + 0.0933549i \(0.0297591\pi\)
−0.416969 + 0.908921i \(0.636908\pi\)
\(6\) 0 0
\(7\) 479.542 1.39808 0.699042 0.715081i \(-0.253610\pi\)
0.699042 + 0.715081i \(0.253610\pi\)
\(8\) 0 0
\(9\) −172.116 + 298.114i −0.236099 + 0.408935i
\(10\) 0 0
\(11\) −1675.76 −1.25903 −0.629513 0.776990i \(-0.716745\pi\)
−0.629513 + 0.776990i \(0.716745\pi\)
\(12\) 0 0
\(13\) 653.454 + 377.272i 0.297430 + 0.171721i 0.641288 0.767300i \(-0.278400\pi\)
−0.343858 + 0.939022i \(0.611734\pi\)
\(14\) 0 0
\(15\) 2457.52 + 1418.85i 0.728153 + 0.420399i
\(16\) 0 0
\(17\) 3910.26 + 6772.78i 0.795901 + 1.37854i 0.922265 + 0.386558i \(0.126336\pi\)
−0.126364 + 0.991984i \(0.540331\pi\)
\(18\) 0 0
\(19\) −6399.56 + 2468.11i −0.933016 + 0.359835i
\(20\) 0 0
\(21\) 8146.24 4703.23i 0.879628 0.507854i
\(22\) 0 0
\(23\) 10038.9 17387.9i 0.825092 1.42910i −0.0767564 0.997050i \(-0.524456\pi\)
0.901849 0.432052i \(-0.142210\pi\)
\(24\) 0 0
\(25\) −2651.63 + 4592.76i −0.169704 + 0.293937i
\(26\) 0 0
\(27\) 21052.0i 1.06955i
\(28\) 0 0
\(29\) 39064.1 + 22553.7i 1.60171 + 0.924749i 0.991145 + 0.132784i \(0.0423917\pi\)
0.610567 + 0.791965i \(0.290942\pi\)
\(30\) 0 0
\(31\) 20987.2i 0.704482i −0.935909 0.352241i \(-0.885420\pi\)
0.935909 0.352241i \(-0.114580\pi\)
\(32\) 0 0
\(33\) −28467.1 + 16435.5i −0.792138 + 0.457341i
\(34\) 0 0
\(35\) 34686.8 + 60079.2i 0.809021 + 1.40126i
\(36\) 0 0
\(37\) 64541.3i 1.27418i 0.770787 + 0.637092i \(0.219863\pi\)
−0.770787 + 0.637092i \(0.780137\pi\)
\(38\) 0 0
\(39\) 14800.8 0.249511
\(40\) 0 0
\(41\) 38429.1 22187.0i 0.557581 0.321920i −0.194593 0.980884i \(-0.562339\pi\)
0.752174 + 0.658964i \(0.229005\pi\)
\(42\) 0 0
\(43\) −384.826 666.538i −0.00484015 0.00838338i 0.863595 0.504186i \(-0.168207\pi\)
−0.868435 + 0.495802i \(0.834874\pi\)
\(44\) 0 0
\(45\) −49798.7 −0.546488
\(46\) 0 0
\(47\) 24326.1 42134.0i 0.234303 0.405825i −0.724767 0.688994i \(-0.758052\pi\)
0.959070 + 0.283169i \(0.0913857\pi\)
\(48\) 0 0
\(49\) 112312. 0.954636
\(50\) 0 0
\(51\) 132851. + 76701.8i 1.00151 + 0.578222i
\(52\) 0 0
\(53\) −154862. 89409.9i −1.04020 0.600562i −0.120313 0.992736i \(-0.538390\pi\)
−0.919891 + 0.392174i \(0.871723\pi\)
\(54\) 0 0
\(55\) −121213. 209947.i −0.728553 1.26189i
\(56\) 0 0
\(57\) −84506.0 + 104692.i −0.456313 + 0.565315i
\(58\) 0 0
\(59\) −182560. + 105401.i −0.888895 + 0.513204i −0.873581 0.486679i \(-0.838208\pi\)
−0.0153141 + 0.999883i \(0.504875\pi\)
\(60\) 0 0
\(61\) 200137. 346647.i 0.881734 1.52721i 0.0323229 0.999477i \(-0.489710\pi\)
0.849411 0.527731i \(-0.176957\pi\)
\(62\) 0 0
\(63\) −82537.0 + 142958.i −0.330086 + 0.571726i
\(64\) 0 0
\(65\) 109157.i 0.397476i
\(66\) 0 0
\(67\) −516018. 297923.i −1.71570 0.990558i −0.926395 0.376554i \(-0.877109\pi\)
−0.789303 0.614004i \(-0.789558\pi\)
\(68\) 0 0
\(69\) 393836.i 1.19886i
\(70\) 0 0
\(71\) −20990.7 + 12119.0i −0.0586479 + 0.0338604i −0.529037 0.848599i \(-0.677447\pi\)
0.470389 + 0.882459i \(0.344113\pi\)
\(72\) 0 0
\(73\) −62525.1 108297.i −0.160726 0.278385i 0.774403 0.632692i \(-0.218050\pi\)
−0.935129 + 0.354307i \(0.884717\pi\)
\(74\) 0 0
\(75\) 104026.i 0.246580i
\(76\) 0 0
\(77\) −803599. −1.76022
\(78\) 0 0
\(79\) 569365. 328723.i 1.15481 0.666729i 0.204754 0.978814i \(-0.434361\pi\)
0.950054 + 0.312085i \(0.101027\pi\)
\(80\) 0 0
\(81\) 80999.9 + 140296.i 0.152416 + 0.263992i
\(82\) 0 0
\(83\) 992838. 1.73638 0.868189 0.496234i \(-0.165284\pi\)
0.868189 + 0.496234i \(0.165284\pi\)
\(84\) 0 0
\(85\) −565682. + 979791.i −0.921119 + 1.59543i
\(86\) 0 0
\(87\) 884804. 1.34366
\(88\) 0 0
\(89\) −462997. 267312.i −0.656763 0.379182i 0.134280 0.990943i \(-0.457128\pi\)
−0.791042 + 0.611761i \(0.790461\pi\)
\(90\) 0 0
\(91\) 313359. + 180918.i 0.415832 + 0.240081i
\(92\) 0 0
\(93\) −205838. 356521.i −0.255903 0.443237i
\(94\) 0 0
\(95\) −772115. 623239.i −0.900557 0.726915i
\(96\) 0 0
\(97\) −1.42085e6 + 820331.i −1.55681 + 0.898822i −0.559246 + 0.829002i \(0.688909\pi\)
−0.997560 + 0.0698204i \(0.977757\pi\)
\(98\) 0 0
\(99\) 288426. 499568.i 0.297255 0.514860i
\(100\) 0 0
\(101\) 146191. 253210.i 0.141891 0.245763i −0.786318 0.617822i \(-0.788015\pi\)
0.928209 + 0.372060i \(0.121348\pi\)
\(102\) 0 0
\(103\) 637190.i 0.583119i −0.956553 0.291560i \(-0.905826\pi\)
0.956553 0.291560i \(-0.0941742\pi\)
\(104\) 0 0
\(105\) 1.17848e6 + 680398.i 1.01802 + 0.587753i
\(106\) 0 0
\(107\) 391710.i 0.319752i −0.987137 0.159876i \(-0.948891\pi\)
0.987137 0.159876i \(-0.0511095\pi\)
\(108\) 0 0
\(109\) −1.00949e6 + 582827.i −0.779509 + 0.450050i −0.836256 0.548339i \(-0.815260\pi\)
0.0567474 + 0.998389i \(0.481927\pi\)
\(110\) 0 0
\(111\) 633005. + 1.09640e6i 0.462848 + 0.801676i
\(112\) 0 0
\(113\) 65307.7i 0.0452615i −0.999744 0.0226308i \(-0.992796\pi\)
0.999744 0.0226308i \(-0.00720421\pi\)
\(114\) 0 0
\(115\) 2.90458e6 1.90981
\(116\) 0 0
\(117\) −224940. + 129869.i −0.140446 + 0.0810865i
\(118\) 0 0
\(119\) 1.87514e6 + 3.24783e6i 1.11274 + 1.92732i
\(120\) 0 0
\(121\) 1.03662e6 0.585145
\(122\) 0 0
\(123\) 435210. 753805.i 0.233875 0.405083i
\(124\) 0 0
\(125\) 1.49320e6 0.764521
\(126\) 0 0
\(127\) −116431. 67221.6i −0.0568406 0.0328169i 0.471310 0.881967i \(-0.343781\pi\)
−0.528151 + 0.849150i \(0.677114\pi\)
\(128\) 0 0
\(129\) −13074.5 7548.55i −0.00609053 0.00351637i
\(130\) 0 0
\(131\) 279525. + 484151.i 0.124339 + 0.215361i 0.921474 0.388439i \(-0.126986\pi\)
−0.797135 + 0.603800i \(0.793652\pi\)
\(132\) 0 0
\(133\) −3.06886e6 + 1.18356e6i −1.30443 + 0.503080i
\(134\) 0 0
\(135\) −2.63749e6 + 1.52275e6i −1.07199 + 0.618911i
\(136\) 0 0
\(137\) 922559. 1.59792e6i 0.358783 0.621431i −0.628975 0.777426i \(-0.716525\pi\)
0.987758 + 0.155995i \(0.0498584\pi\)
\(138\) 0 0
\(139\) 37752.2 65388.7i 0.0140572 0.0243477i −0.858911 0.512124i \(-0.828859\pi\)
0.872968 + 0.487777i \(0.162192\pi\)
\(140\) 0 0
\(141\) 954337.i 0.340443i
\(142\) 0 0
\(143\) −1.09503e6 632219.i −0.374472 0.216202i
\(144\) 0 0
\(145\) 6.52551e6i 2.14048i
\(146\) 0 0
\(147\) 1.90790e6 1.10153e6i 0.600626 0.346772i
\(148\) 0 0
\(149\) −582135. 1.00829e6i −0.175981 0.304807i 0.764520 0.644600i \(-0.222976\pi\)
−0.940500 + 0.339793i \(0.889643\pi\)
\(150\) 0 0
\(151\) 1.85186e6i 0.537869i −0.963158 0.268935i \(-0.913328\pi\)
0.963158 0.268935i \(-0.0866716\pi\)
\(152\) 0 0
\(153\) −2.69208e6 −0.751646
\(154\) 0 0
\(155\) 2.62937e6 1.51807e6i 0.706086 0.407659i
\(156\) 0 0
\(157\) −671911. 1.16378e6i −0.173625 0.300728i 0.766059 0.642770i \(-0.222215\pi\)
−0.939685 + 0.342042i \(0.888881\pi\)
\(158\) 0 0
\(159\) −3.50764e6 −0.872616
\(160\) 0 0
\(161\) 4.81408e6 8.33823e6i 1.15355 1.99800i
\(162\) 0 0
\(163\) −4.35438e6 −1.00546 −0.502728 0.864444i \(-0.667670\pi\)
−0.502728 + 0.864444i \(0.667670\pi\)
\(164\) 0 0
\(165\) −4.11822e6 2.37765e6i −0.916763 0.529294i
\(166\) 0 0
\(167\) 6.38284e6 + 3.68513e6i 1.37045 + 0.791232i 0.990985 0.133972i \(-0.0427733\pi\)
0.379469 + 0.925204i \(0.376107\pi\)
\(168\) 0 0
\(169\) −2.12874e6 3.68708e6i −0.441023 0.763875i
\(170\) 0 0
\(171\) 365689. 2.33260e6i 0.0731347 0.466500i
\(172\) 0 0
\(173\) −243263. + 140448.i −0.0469827 + 0.0271255i −0.523307 0.852144i \(-0.675302\pi\)
0.476325 + 0.879269i \(0.341969\pi\)
\(174\) 0 0
\(175\) −1.27157e6 + 2.20242e6i −0.237261 + 0.410948i
\(176\) 0 0
\(177\) −2.06750e6 + 3.58101e6i −0.372842 + 0.645782i
\(178\) 0 0
\(179\) 7.17348e6i 1.25075i −0.780324 0.625376i \(-0.784946\pi\)
0.780324 0.625376i \(-0.215054\pi\)
\(180\) 0 0
\(181\) 1.28844e6 + 743883.i 0.217285 + 0.125450i 0.604693 0.796459i \(-0.293296\pi\)
−0.387407 + 0.921909i \(0.626629\pi\)
\(182\) 0 0
\(183\) 7.85157e6i 1.28116i
\(184\) 0 0
\(185\) −8.08602e6 + 4.66847e6i −1.27708 + 0.737325i
\(186\) 0 0
\(187\) −6.55268e6 1.13496e7i −1.00206 1.73562i
\(188\) 0 0
\(189\) 1.00953e7i 1.49532i
\(190\) 0 0
\(191\) −5.64832e6 −0.810623 −0.405312 0.914179i \(-0.632837\pi\)
−0.405312 + 0.914179i \(0.632837\pi\)
\(192\) 0 0
\(193\) 458436. 264678.i 0.0637686 0.0368168i −0.467777 0.883847i \(-0.654945\pi\)
0.531545 + 0.847030i \(0.321612\pi\)
\(194\) 0 0
\(195\) 1.07058e6 + 1.85431e6i 0.144383 + 0.250079i
\(196\) 0 0
\(197\) 870815. 0.113901 0.0569505 0.998377i \(-0.481862\pi\)
0.0569505 + 0.998377i \(0.481862\pi\)
\(198\) 0 0
\(199\) −750692. + 1.30024e6i −0.0952582 + 0.164992i −0.909716 0.415230i \(-0.863701\pi\)
0.814458 + 0.580222i \(0.197034\pi\)
\(200\) 0 0
\(201\) −1.16878e7 −1.43928
\(202\) 0 0
\(203\) 1.87329e7 + 1.08155e7i 2.23933 + 1.29288i
\(204\) 0 0
\(205\) 5.55938e6 + 3.20971e6i 0.645305 + 0.372567i
\(206\) 0 0
\(207\) 3.45571e6 + 5.98547e6i 0.389607 + 0.674819i
\(208\) 0 0
\(209\) 1.07241e7 4.13597e6i 1.17469 0.453042i
\(210\) 0 0
\(211\) −3.22064e6 + 1.85944e6i −0.342843 + 0.197941i −0.661529 0.749920i \(-0.730092\pi\)
0.318686 + 0.947861i \(0.396759\pi\)
\(212\) 0 0
\(213\) −237720. + 411744.i −0.0245996 + 0.0426077i
\(214\) 0 0
\(215\) 55671.2 96425.4i 0.00560164 0.00970233i
\(216\) 0 0
\(217\) 1.00643e7i 0.984925i
\(218\) 0 0
\(219\) −2.12429e6 1.22646e6i −0.202247 0.116767i
\(220\) 0 0
\(221\) 5.90093e6i 0.546694i
\(222\) 0 0
\(223\) 2.07682e6 1.19905e6i 0.187277 0.108124i −0.403430 0.915010i \(-0.632182\pi\)
0.590707 + 0.806886i \(0.298849\pi\)
\(224\) 0 0
\(225\) −912777. 1.58098e6i −0.0801340 0.138796i
\(226\) 0 0
\(227\) 1.02844e7i 0.879231i 0.898186 + 0.439615i \(0.144885\pi\)
−0.898186 + 0.439615i \(0.855115\pi\)
\(228\) 0 0
\(229\) 2.02771e7 1.68849 0.844245 0.535957i \(-0.180049\pi\)
0.844245 + 0.535957i \(0.180049\pi\)
\(230\) 0 0
\(231\) −1.36512e7 + 7.88150e6i −1.10747 + 0.639401i
\(232\) 0 0
\(233\) −6.84113e6 1.18492e7i −0.540829 0.936744i −0.998857 0.0478058i \(-0.984777\pi\)
0.458027 0.888938i \(-0.348556\pi\)
\(234\) 0 0
\(235\) 7.03832e6 0.542332
\(236\) 0 0
\(237\) 6.44807e6 1.11684e7i 0.484378 0.838968i
\(238\) 0 0
\(239\) 8.65777e6 0.634180 0.317090 0.948396i \(-0.397294\pi\)
0.317090 + 0.948396i \(0.397294\pi\)
\(240\) 0 0
\(241\) −292332. 168778.i −0.0208846 0.0120577i 0.489521 0.871991i \(-0.337172\pi\)
−0.510406 + 0.859934i \(0.670505\pi\)
\(242\) 0 0
\(243\) −1.05388e7 6.08459e6i −0.734469 0.424046i
\(244\) 0 0
\(245\) 8.12386e6 + 1.40709e7i 0.552414 + 0.956809i
\(246\) 0 0
\(247\) −5.11297e6 801576.i −0.339299 0.0531929i
\(248\) 0 0
\(249\) 1.68659e7 9.73750e6i 1.09247 0.630739i
\(250\) 0 0
\(251\) 1.69434e6 2.93468e6i 0.107147 0.185584i −0.807467 0.589913i \(-0.799162\pi\)
0.914613 + 0.404330i \(0.132495\pi\)
\(252\) 0 0
\(253\) −1.68228e7 + 2.91380e7i −1.03881 + 1.79928i
\(254\) 0 0
\(255\) 2.21923e7i 1.33839i
\(256\) 0 0
\(257\) 1.86463e7 + 1.07654e7i 1.09848 + 0.634209i 0.935822 0.352474i \(-0.114659\pi\)
0.162660 + 0.986682i \(0.447993\pi\)
\(258\) 0 0
\(259\) 3.09503e7i 1.78142i
\(260\) 0 0
\(261\) −1.34471e7 + 7.76371e6i −0.756325 + 0.436664i
\(262\) 0 0
\(263\) −5.11051e6 8.85167e6i −0.280930 0.486584i 0.690684 0.723156i \(-0.257309\pi\)
−0.971614 + 0.236572i \(0.923976\pi\)
\(264\) 0 0
\(265\) 2.58691e7i 1.39009i
\(266\) 0 0
\(267\) −1.04869e7 −0.550952
\(268\) 0 0
\(269\) 2.12047e7 1.22425e7i 1.08937 0.628948i 0.155962 0.987763i \(-0.450152\pi\)
0.933409 + 0.358815i \(0.116819\pi\)
\(270\) 0 0
\(271\) −4.65278e6 8.05885e6i −0.233778 0.404916i 0.725139 0.688603i \(-0.241776\pi\)
−0.958917 + 0.283687i \(0.908442\pi\)
\(272\) 0 0
\(273\) 7.09759e6 0.348838
\(274\) 0 0
\(275\) 4.44350e6 7.69638e6i 0.213662 0.370074i
\(276\) 0 0
\(277\) 2.23186e7 1.05009 0.525047 0.851073i \(-0.324048\pi\)
0.525047 + 0.851073i \(0.324048\pi\)
\(278\) 0 0
\(279\) 6.25659e6 + 3.61224e6i 0.288088 + 0.166328i
\(280\) 0 0
\(281\) −1.23215e7 7.11380e6i −0.555320 0.320614i 0.195945 0.980615i \(-0.437223\pi\)
−0.751265 + 0.660001i \(0.770556\pi\)
\(282\) 0 0
\(283\) −8.32695e6 1.44227e7i −0.367390 0.636337i 0.621767 0.783202i \(-0.286415\pi\)
−0.989157 + 0.146865i \(0.953082\pi\)
\(284\) 0 0
\(285\) −1.92289e7 3.01458e6i −0.830653 0.130224i
\(286\) 0 0
\(287\) 1.84284e7 1.06396e7i 0.779545 0.450071i
\(288\) 0 0
\(289\) −1.85115e7 + 3.20629e7i −0.766918 + 1.32834i
\(290\) 0 0
\(291\) −1.60912e7 + 2.78708e7i −0.652994 + 1.13102i
\(292\) 0 0
\(293\) 6.25894e6i 0.248827i −0.992230 0.124414i \(-0.960295\pi\)
0.992230 0.124414i \(-0.0397050\pi\)
\(294\) 0 0
\(295\) −2.64103e7 1.52480e7i −1.02874 0.593945i
\(296\) 0 0
\(297\) 3.52781e7i 1.34659i
\(298\) 0 0
\(299\) 1.31199e7 7.57479e6i 0.490815 0.283372i
\(300\) 0 0
\(301\) −184540. 319633.i −0.00676693 0.0117207i
\(302\) 0 0
\(303\) 5.73520e6i 0.206168i
\(304\) 0 0
\(305\) 5.79060e7 2.04091
\(306\) 0 0
\(307\) −2.54271e7 + 1.46803e7i −0.878783 + 0.507366i −0.870257 0.492598i \(-0.836047\pi\)
−0.00852607 + 0.999964i \(0.502714\pi\)
\(308\) 0 0
\(309\) −6.24940e6 1.08243e7i −0.211818 0.366880i
\(310\) 0 0
\(311\) 3.57357e7 1.18801 0.594006 0.804460i \(-0.297545\pi\)
0.594006 + 0.804460i \(0.297545\pi\)
\(312\) 0 0
\(313\) −1.78953e7 + 3.09956e7i −0.583587 + 1.01080i 0.411463 + 0.911427i \(0.365018\pi\)
−0.995050 + 0.0993761i \(0.968315\pi\)
\(314\) 0 0
\(315\) −2.38806e7 −0.764036
\(316\) 0 0
\(317\) −2.12209e7 1.22519e7i −0.666171 0.384614i 0.128453 0.991716i \(-0.458999\pi\)
−0.794624 + 0.607102i \(0.792332\pi\)
\(318\) 0 0
\(319\) −6.54623e7 3.77946e7i −2.01660 1.16428i
\(320\) 0 0
\(321\) −3.84180e6 6.65419e6i −0.116150 0.201178i
\(322\) 0 0
\(323\) −4.17399e7 3.36918e7i −1.23864 0.999808i
\(324\) 0 0
\(325\) −3.46544e6 + 2.00077e6i −0.100950 + 0.0582838i
\(326\) 0 0
\(327\) −1.14324e7 + 1.98016e7i −0.326961 + 0.566313i
\(328\) 0 0
\(329\) 1.16654e7 2.02051e7i 0.327576 0.567378i
\(330\) 0 0
\(331\) 5.39323e6i 0.148718i −0.997232 0.0743592i \(-0.976309\pi\)
0.997232 0.0743592i \(-0.0236911\pi\)
\(332\) 0 0
\(333\) −1.92407e7 1.11086e7i −0.521059 0.300834i
\(334\) 0 0
\(335\) 8.61988e7i 2.29280i
\(336\) 0 0
\(337\) 2.11633e6 1.22186e6i 0.0552960 0.0319251i −0.472097 0.881547i \(-0.656503\pi\)
0.527393 + 0.849621i \(0.323170\pi\)
\(338\) 0 0
\(339\) −640522. 1.10942e6i −0.0164412 0.0284771i
\(340\) 0 0
\(341\) 3.51696e7i 0.886961i
\(342\) 0 0
\(343\) −2.55933e6 −0.0634225
\(344\) 0 0
\(345\) 4.93415e7 2.84873e7i 1.20159 0.693737i
\(346\) 0 0
\(347\) 2.44727e7 + 4.23879e7i 0.585724 + 1.01450i 0.994785 + 0.101996i \(0.0325230\pi\)
−0.409061 + 0.912507i \(0.634144\pi\)
\(348\) 0 0
\(349\) −3.31844e7 −0.780652 −0.390326 0.920677i \(-0.627638\pi\)
−0.390326 + 0.920677i \(0.627638\pi\)
\(350\) 0 0
\(351\) −7.94233e6 + 1.37565e7i −0.183665 + 0.318117i
\(352\) 0 0
\(353\) 2.56483e7 0.583089 0.291544 0.956557i \(-0.405831\pi\)
0.291544 + 0.956557i \(0.405831\pi\)
\(354\) 0 0
\(355\) −3.03665e6 1.75321e6i −0.0678749 0.0391876i
\(356\) 0 0
\(357\) 6.37079e7 + 3.67818e7i 1.40019 + 0.808403i
\(358\) 0 0
\(359\) 3.70037e7 + 6.40922e7i 0.799763 + 1.38523i 0.919770 + 0.392457i \(0.128375\pi\)
−0.120007 + 0.992773i \(0.538292\pi\)
\(360\) 0 0
\(361\) 3.48627e7 3.15896e7i 0.741037 0.671464i
\(362\) 0 0
\(363\) 1.76096e7 1.01669e7i 0.368154 0.212554i
\(364\) 0 0
\(365\) 9.04525e6 1.56668e7i 0.186013 0.322183i
\(366\) 0 0
\(367\) −2.76966e7 + 4.79719e7i −0.560310 + 0.970485i 0.437159 + 0.899384i \(0.355985\pi\)
−0.997469 + 0.0711009i \(0.977349\pi\)
\(368\) 0 0
\(369\) 1.52750e7i 0.304020i
\(370\) 0 0
\(371\) −7.42631e7 4.28758e7i −1.45429 0.839635i
\(372\) 0 0
\(373\) 7.16019e7i 1.37974i 0.723932 + 0.689871i \(0.242333\pi\)
−0.723932 + 0.689871i \(0.757667\pi\)
\(374\) 0 0
\(375\) 2.53658e7 1.46450e7i 0.481012 0.277712i
\(376\) 0 0
\(377\) 1.70178e7 + 2.94756e7i 0.317598 + 0.550097i
\(378\) 0 0
\(379\) 9.27549e7i 1.70380i 0.523702 + 0.851901i \(0.324551\pi\)
−0.523702 + 0.851901i \(0.675449\pi\)
\(380\) 0 0
\(381\) −2.63717e6 −0.0476830
\(382\) 0 0
\(383\) −1.75745e7 + 1.01467e7i −0.312815 + 0.180604i −0.648185 0.761482i \(-0.724472\pi\)
0.335370 + 0.942086i \(0.391139\pi\)
\(384\) 0 0
\(385\) −5.81268e7 1.00679e8i −1.01858 1.76423i
\(386\) 0 0
\(387\) 264939. 0.00457102
\(388\) 0 0
\(389\) −2.31042e7 + 4.00176e7i −0.392502 + 0.679833i −0.992779 0.119959i \(-0.961724\pi\)
0.600277 + 0.799792i \(0.295057\pi\)
\(390\) 0 0
\(391\) 1.57019e8 2.62677
\(392\) 0 0
\(393\) 9.49687e6 + 5.48302e6i 0.156460 + 0.0903321i
\(394\) 0 0
\(395\) 8.23678e7 + 4.75551e7i 1.33649 + 0.771624i
\(396\) 0 0
\(397\) −1.30268e7 2.25631e7i −0.208193 0.360602i 0.742952 0.669345i \(-0.233425\pi\)
−0.951145 + 0.308743i \(0.900092\pi\)
\(398\) 0 0
\(399\) −4.05242e7 + 5.02044e7i −0.637963 + 0.790357i
\(400\) 0 0
\(401\) −1.66330e7 + 9.60308e6i −0.257951 + 0.148928i −0.623400 0.781903i \(-0.714249\pi\)
0.365448 + 0.930832i \(0.380916\pi\)
\(402\) 0 0
\(403\) 7.91790e6 1.37142e7i 0.120975 0.209534i
\(404\) 0 0
\(405\) −1.17179e7 + 2.02961e7i −0.176395 + 0.305525i
\(406\) 0 0
\(407\) 1.08156e8i 1.60423i
\(408\) 0 0
\(409\) −7.57833e7 4.37535e7i −1.10765 0.639504i −0.169433 0.985542i \(-0.554193\pi\)
−0.938220 + 0.346038i \(0.887527\pi\)
\(410\) 0 0
\(411\) 3.61929e7i 0.521312i
\(412\) 0 0
\(413\) −8.75454e7 + 5.05444e7i −1.24275 + 0.717501i
\(414\) 0 0
\(415\) 7.18150e7 + 1.24387e8i 1.00478 + 1.74033i
\(416\) 0 0
\(417\) 1.48105e6i 0.0204250i
\(418\) 0 0
\(419\) 5.07404e7 0.689782 0.344891 0.938643i \(-0.387916\pi\)
0.344891 + 0.938643i \(0.387916\pi\)
\(420\) 0 0
\(421\) 4.66586e7 2.69383e7i 0.625295 0.361014i −0.153633 0.988128i \(-0.549097\pi\)
0.778928 + 0.627114i \(0.215764\pi\)
\(422\) 0 0
\(423\) 8.37382e6 + 1.45039e7i 0.110638 + 0.191630i
\(424\) 0 0
\(425\) −4.14743e7 −0.540272
\(426\) 0 0
\(427\) 9.59742e7 1.66232e8i 1.23274 2.13516i
\(428\) 0 0
\(429\) −2.48026e7 −0.314141
\(430\) 0 0
\(431\) −2.19013e7 1.26447e7i −0.273551 0.157935i 0.356949 0.934124i \(-0.383817\pi\)
−0.630500 + 0.776189i \(0.717150\pi\)
\(432\) 0 0
\(433\) 7.16881e7 + 4.13891e7i 0.883046 + 0.509827i 0.871661 0.490108i \(-0.163043\pi\)
0.0113843 + 0.999935i \(0.496376\pi\)
\(434\) 0 0
\(435\) 6.40005e7 + 1.10852e8i 0.777528 + 1.34672i
\(436\) 0 0
\(437\) −2.13293e7 + 1.36052e8i −0.255583 + 1.63027i
\(438\) 0 0
\(439\) 1.04607e8 6.03947e7i 1.23642 0.713847i 0.268058 0.963403i \(-0.413618\pi\)
0.968360 + 0.249556i \(0.0802847\pi\)
\(440\) 0 0
\(441\) −1.93307e7 + 3.34818e7i −0.225389 + 0.390384i
\(442\) 0 0
\(443\) 3.44943e7 5.97459e7i 0.396768 0.687222i −0.596557 0.802571i \(-0.703465\pi\)
0.993325 + 0.115348i \(0.0367984\pi\)
\(444\) 0 0
\(445\) 7.73418e7i 0.877676i
\(446\) 0 0
\(447\) −1.97780e7 1.14189e7i −0.221442 0.127850i
\(448\) 0 0
\(449\) 7.48148e7i 0.826511i −0.910615 0.413256i \(-0.864392\pi\)
0.910615 0.413256i \(-0.135608\pi\)
\(450\) 0 0
\(451\) −6.43980e7 + 3.71802e7i −0.702009 + 0.405305i
\(452\) 0 0
\(453\) −1.81626e7 3.14585e7i −0.195381 0.338410i
\(454\) 0 0
\(455\) 5.23454e7i 0.555705i
\(456\) 0 0
\(457\) 5.28041e7 0.553247 0.276623 0.960978i \(-0.410785\pi\)
0.276623 + 0.960978i \(0.410785\pi\)
\(458\) 0 0
\(459\) −1.42580e8 + 8.23188e7i −1.47442 + 0.851258i
\(460\) 0 0
\(461\) 6.25780e7 + 1.08388e8i 0.638733 + 1.10632i 0.985711 + 0.168444i \(0.0538743\pi\)
−0.346979 + 0.937873i \(0.612792\pi\)
\(462\) 0 0
\(463\) −1.70244e8 −1.71526 −0.857630 0.514267i \(-0.828064\pi\)
−0.857630 + 0.514267i \(0.828064\pi\)
\(464\) 0 0
\(465\) 2.97777e7 5.15765e7i 0.296164 0.512971i
\(466\) 0 0
\(467\) 3.00664e7 0.295210 0.147605 0.989046i \(-0.452844\pi\)
0.147605 + 0.989046i \(0.452844\pi\)
\(468\) 0 0
\(469\) −2.47453e8 1.42867e8i −2.39869 1.38488i
\(470\) 0 0
\(471\) −2.28282e7 1.31799e7i −0.218479 0.126139i
\(472\) 0 0
\(473\) 644877. + 1.11696e6i 0.00609387 + 0.0105549i
\(474\) 0 0
\(475\) 5.63382e6 3.59361e7i 0.0525681 0.335313i
\(476\) 0 0
\(477\) 5.33086e7 3.07778e7i 0.491182 0.283584i
\(478\) 0 0
\(479\) 4.25052e6 7.36212e6i 0.0386755 0.0669879i −0.846040 0.533120i \(-0.821019\pi\)
0.884715 + 0.466132i \(0.154353\pi\)
\(480\) 0 0
\(481\) −2.43496e7 + 4.21748e7i −0.218805 + 0.378981i
\(482\) 0 0
\(483\) 1.88861e8i 1.67610i
\(484\) 0 0
\(485\) −2.05549e8 1.18674e8i −1.80174 1.04023i
\(486\) 0 0
\(487\) 5.86160e7i 0.507492i −0.967271 0.253746i \(-0.918337\pi\)
0.967271 0.253746i \(-0.0816627\pi\)
\(488\) 0 0
\(489\) −7.39701e7 + 4.27066e7i −0.632601 + 0.365232i
\(490\) 0 0
\(491\) 1.73857e7 + 3.01129e7i 0.146875 + 0.254395i 0.930071 0.367380i \(-0.119745\pi\)
−0.783196 + 0.621775i \(0.786412\pi\)
\(492\) 0 0
\(493\) 3.52764e8i 2.94404i
\(494\) 0 0
\(495\) 8.34508e7 0.688042
\(496\) 0 0
\(497\) −1.00660e7 + 5.81158e6i −0.0819947 + 0.0473396i
\(498\) 0 0
\(499\) −1.06825e7 1.85026e7i −0.0859749 0.148913i 0.819831 0.572605i \(-0.194067\pi\)
−0.905806 + 0.423692i \(0.860734\pi\)
\(500\) 0 0
\(501\) 1.44571e8 1.14966
\(502\) 0 0
\(503\) 4.00789e7 6.94186e7i 0.314928 0.545472i −0.664494 0.747294i \(-0.731353\pi\)
0.979422 + 0.201822i \(0.0646862\pi\)
\(504\) 0 0
\(505\) 4.22976e7 0.328429
\(506\) 0 0
\(507\) −7.23239e7 4.17562e7i −0.554955 0.320404i
\(508\) 0 0
\(509\) 7.04620e7 + 4.06812e7i 0.534320 + 0.308490i 0.742774 0.669542i \(-0.233510\pi\)
−0.208454 + 0.978032i \(0.566843\pi\)
\(510\) 0 0
\(511\) −2.99834e7 5.19328e7i −0.224708 0.389206i
\(512\) 0 0
\(513\) −5.19586e7 1.34723e8i −0.384862 0.997909i
\(514\) 0 0
\(515\) 7.98300e7 4.60899e7i 0.584446 0.337430i
\(516\) 0 0
\(517\) −4.07648e7 + 7.06066e7i −0.294994 + 0.510945i
\(518\) 0 0
\(519\) −2.75496e6 + 4.77172e6i −0.0197066 + 0.0341329i
\(520\) 0 0
\(521\) 1.41254e8i 0.998821i −0.866366 0.499410i \(-0.833550\pi\)
0.866366 0.499410i \(-0.166450\pi\)
\(522\) 0 0
\(523\) 4.14406e7 + 2.39258e7i 0.289682 + 0.167248i 0.637798 0.770203i \(-0.279845\pi\)
−0.348117 + 0.937451i \(0.613179\pi\)
\(524\) 0 0
\(525\) 4.98849e7i 0.344740i
\(526\) 0 0
\(527\) 1.42142e8 8.20656e7i 0.971158 0.560699i
\(528\) 0 0
\(529\) −1.27541e8 2.20908e8i −0.861555 1.49226i
\(530\) 0 0
\(531\) 7.25650e7i 0.484667i
\(532\) 0 0
\(533\) 3.34822e7 0.221122
\(534\) 0 0
\(535\) 4.90752e7 2.83336e7i 0.320480 0.185029i
\(536\) 0 0
\(537\) −7.03557e7 1.21860e8i −0.454335 0.786932i
\(538\) 0 0
\(539\) −1.88208e8 −1.20191
\(540\) 0 0
\(541\) −1.33126e8 + 2.30580e8i −0.840755 + 1.45623i 0.0485022 + 0.998823i \(0.484555\pi\)
−0.889257 + 0.457407i \(0.848778\pi\)
\(542\) 0 0
\(543\) 2.91833e7 0.182278
\(544\) 0 0
\(545\) −1.46038e8 8.43153e7i −0.902148 0.520855i
\(546\) 0 0
\(547\) 1.90348e8 + 1.09898e8i 1.16302 + 0.671470i 0.952026 0.306017i \(-0.0989965\pi\)
0.210994 + 0.977487i \(0.432330\pi\)
\(548\) 0 0
\(549\) 6.88936e7 + 1.19327e8i 0.416353 + 0.721145i
\(550\) 0 0
\(551\) −3.05658e8 4.79190e7i −1.82718 0.286453i
\(552\) 0 0
\(553\) 2.73035e8 1.57637e8i 1.61452 0.932142i
\(554\) 0 0
\(555\) −9.15743e7 + 1.58611e8i −0.535667 + 0.927802i
\(556\) 0 0
\(557\) −5.05811e7 + 8.76091e7i −0.292700 + 0.506972i −0.974447 0.224616i \(-0.927887\pi\)
0.681747 + 0.731588i \(0.261220\pi\)
\(558\) 0 0
\(559\) 580736.i 0.00332463i
\(560\) 0 0
\(561\) −2.22627e8 1.28534e8i −1.26093 0.727997i
\(562\) 0 0
\(563\) 1.75466e8i 0.983260i 0.870804 + 0.491630i \(0.163599\pi\)
−0.870804 + 0.491630i \(0.836401\pi\)
\(564\) 0 0
\(565\) 8.18204e6 4.72390e6i 0.0453645 0.0261912i
\(566\) 0 0
\(567\) 3.88429e7 + 6.72779e7i 0.213090 + 0.369082i
\(568\) 0 0
\(569\) 1.29570e8i 0.703342i 0.936124 + 0.351671i \(0.114386\pi\)
−0.936124 + 0.351671i \(0.885614\pi\)
\(570\) 0 0
\(571\) 1.78253e8 0.957475 0.478737 0.877958i \(-0.341095\pi\)
0.478737 + 0.877958i \(0.341095\pi\)
\(572\) 0 0
\(573\) −9.59509e7 + 5.53973e7i −0.510018 + 0.294459i
\(574\) 0 0
\(575\) 5.32389e7 + 9.22125e7i 0.280044 + 0.485050i
\(576\) 0 0
\(577\) 3.66437e8 1.90753 0.953766 0.300550i \(-0.0971704\pi\)
0.953766 + 0.300550i \(0.0971704\pi\)
\(578\) 0 0
\(579\) 5.19180e6 8.99246e6i 0.0267474 0.0463279i
\(580\) 0 0
\(581\) 4.76108e8 2.42760
\(582\) 0 0
\(583\) 2.59513e8 + 1.49830e8i 1.30964 + 0.756123i
\(584\) 0 0
\(585\) −3.25412e7 1.87877e7i −0.162542 0.0938437i
\(586\) 0 0
\(587\) 3.95703e7 + 6.85377e7i 0.195639 + 0.338856i 0.947110 0.320910i \(-0.103989\pi\)
−0.751471 + 0.659766i \(0.770655\pi\)
\(588\) 0 0
\(589\) 5.17988e7 + 1.34309e8i 0.253498 + 0.657293i
\(590\) 0 0
\(591\) 1.47930e7 8.54074e6i 0.0716628 0.0413745i
\(592\) 0 0
\(593\) −1.76464e8 + 3.05645e8i −0.846239 + 1.46573i 0.0383017 + 0.999266i \(0.487805\pi\)
−0.884541 + 0.466463i \(0.845528\pi\)
\(594\) 0 0
\(595\) −2.71269e8 + 4.69851e8i −1.28780 + 2.23054i
\(596\) 0 0
\(597\) 2.94504e7i 0.138410i
\(598\) 0 0
\(599\) −2.62481e8 1.51544e8i −1.22129 0.705110i −0.256095 0.966652i \(-0.582436\pi\)
−0.965192 + 0.261541i \(0.915769\pi\)
\(600\) 0 0
\(601\) 3.56371e8i 1.64164i −0.571184 0.820822i \(-0.693516\pi\)
0.571184 0.820822i \(-0.306484\pi\)
\(602\) 0 0
\(603\) 1.77630e8 1.02555e8i 0.810149 0.467740i
\(604\) 0 0
\(605\) 7.49819e7 + 1.29872e8i 0.338603 + 0.586477i
\(606\) 0 0
\(607\) 7.93413e7i 0.354759i 0.984143 + 0.177379i \(0.0567620\pi\)
−0.984143 + 0.177379i \(0.943238\pi\)
\(608\) 0 0
\(609\) 4.24301e8 1.87855
\(610\) 0 0
\(611\) 3.17920e7 1.83551e7i 0.139378 0.0804699i
\(612\) 0 0
\(613\) −1.83104e8 3.17145e8i −0.794907 1.37682i −0.922898 0.385044i \(-0.874186\pi\)
0.127991 0.991775i \(-0.459147\pi\)
\(614\) 0 0
\(615\) 1.25920e8 0.541340
\(616\) 0 0
\(617\) 1.57432e8 2.72680e8i 0.670250 1.16091i −0.307583 0.951521i \(-0.599520\pi\)
0.977833 0.209386i \(-0.0671464\pi\)
\(618\) 0 0
\(619\) −1.06434e8 −0.448754 −0.224377 0.974502i \(-0.572035\pi\)
−0.224377 + 0.974502i \(0.572035\pi\)
\(620\) 0 0
\(621\) 3.66049e8 + 2.11339e8i 1.52850 + 0.882479i
\(622\) 0 0
\(623\) −2.22027e8 1.28187e8i −0.918209 0.530128i
\(624\) 0 0
\(625\) 1.49440e8 + 2.58837e8i 0.612105 + 1.06020i
\(626\) 0 0
\(627\) 1.41612e8 1.75439e8i 0.574510 0.711745i
\(628\) 0 0
\(629\) −4.37124e8 + 2.52373e8i −1.75652 + 1.01413i
\(630\) 0 0
\(631\) 2.96465e7 5.13493e7i 0.118001 0.204384i −0.800974 0.598699i \(-0.795685\pi\)
0.918975 + 0.394315i \(0.129018\pi\)
\(632\) 0 0
\(633\) −3.64738e7 + 6.31745e7i −0.143804 + 0.249075i
\(634\) 0 0
\(635\) 1.94494e7i 0.0759599i
\(636\) 0 0
\(637\) 7.33908e7 + 4.23722e7i 0.283938 + 0.163932i
\(638\) 0 0
\(639\) 8.34351e6i 0.0319776i
\(640\) 0 0
\(641\) −4.29264e8 + 2.47836e8i −1.62986 + 0.941000i −0.645727 + 0.763569i \(0.723445\pi\)
−0.984133 + 0.177431i \(0.943221\pi\)
\(642\) 0 0
\(643\) 3.39583e6 + 5.88176e6i 0.0127736 + 0.0221245i 0.872342 0.488897i \(-0.162601\pi\)
−0.859568 + 0.511021i \(0.829267\pi\)
\(644\) 0 0
\(645\) 2.18404e6i 0.00813918i
\(646\) 0 0
\(647\) 1.56052e7 0.0576179 0.0288090 0.999585i \(-0.490829\pi\)
0.0288090 + 0.999585i \(0.490829\pi\)
\(648\) 0 0
\(649\) 3.05928e8 1.76628e8i 1.11914 0.646137i
\(650\) 0 0
\(651\) −9.87078e7 1.70967e8i −0.357774 0.619683i
\(652\) 0 0
\(653\) 1.38685e8 0.498069 0.249034 0.968495i \(-0.419887\pi\)
0.249034 + 0.968495i \(0.419887\pi\)
\(654\) 0 0
\(655\) −4.04378e7 + 7.00403e7i −0.143901 + 0.249244i
\(656\) 0 0
\(657\) 4.30463e7 0.151789
\(658\) 0 0
\(659\) −3.26135e8 1.88294e8i −1.13957 0.657932i −0.193247 0.981150i \(-0.561902\pi\)
−0.946324 + 0.323218i \(0.895235\pi\)
\(660\) 0 0
\(661\) −4.70005e8 2.71358e8i −1.62741 0.939588i −0.984861 0.173348i \(-0.944541\pi\)
−0.642554 0.766240i \(-0.722125\pi\)
\(662\) 0 0
\(663\) 5.78749e7 + 1.00242e8i 0.198586 + 0.343962i
\(664\) 0 0
\(665\) −3.70262e8 2.98870e8i −1.25905 1.01629i
\(666\) 0 0
\(667\) 7.84322e8 4.52829e8i 2.64312 1.52601i
\(668\) 0 0
\(669\) 2.35200e7 4.07379e7i 0.0785524 0.136057i
\(670\) 0 0
\(671\) −3.35382e8 + 5.80899e8i −1.11013 + 1.92279i
\(672\) 0 0
\(673\) 9.85863e7i 0.323423i −0.986838 0.161712i \(-0.948299\pi\)
0.986838 0.161712i \(-0.0517014\pi\)
\(674\) 0 0
\(675\) −9.66867e7 5.58221e7i −0.314380 0.181508i
\(676\) 0 0
\(677\) 2.46094e8i 0.793112i −0.918010 0.396556i \(-0.870205\pi\)
0.918010 0.396556i \(-0.129795\pi\)
\(678\) 0 0
\(679\) −6.81360e8 + 3.93383e8i −2.17654 + 1.25663i
\(680\) 0 0
\(681\) 1.00867e8 + 1.74707e8i 0.319380 + 0.553183i
\(682\) 0 0
\(683\) 1.80543e8i 0.566654i 0.959023 + 0.283327i \(0.0914382\pi\)
−0.959023 + 0.283327i \(0.908562\pi\)
\(684\) 0 0
\(685\) 2.66926e8 0.830460
\(686\) 0 0
\(687\) 3.44457e8 1.98872e8i 1.06234 0.613344i
\(688\) 0 0
\(689\) −6.74637e7 1.16851e8i −0.206259 0.357251i
\(690\) 0 0
\(691\) −1.85003e8 −0.560720 −0.280360 0.959895i \(-0.590454\pi\)
−0.280360 + 0.959895i \(0.590454\pi\)
\(692\) 0 0
\(693\) 1.38312e8 2.39564e8i 0.415587 0.719817i
\(694\) 0 0
\(695\) 1.09229e7 0.0325375
\(696\) 0 0
\(697\) 3.00536e8 + 1.73514e8i 0.887560 + 0.512433i
\(698\) 0 0
\(699\) −2.32428e8 1.34192e8i −0.680544 0.392912i
\(700\) 0 0
\(701\) 1.09718e8 + 1.90037e8i 0.318510 + 0.551675i 0.980177 0.198122i \(-0.0634842\pi\)
−0.661667 + 0.749797i \(0.730151\pi\)
\(702\) 0 0
\(703\) −1.59295e8 4.13036e8i −0.458497 1.18883i
\(704\) 0 0
\(705\) 1.19564e8 6.90300e7i 0.341218 0.197002i
\(706\) 0 0
\(707\) 7.01046e7 1.21425e8i 0.198376 0.343597i
\(708\) 0 0
\(709\) −2.15984e8 + 3.74096e8i −0.606016 + 1.04965i 0.385875 + 0.922551i \(0.373900\pi\)
−0.991890 + 0.127098i \(0.959434\pi\)
\(710\) 0 0
\(711\) 2.26314e8i 0.629656i
\(712\) 0 0
\(713\) −3.64924e8 2.10689e8i −1.00678 0.581263i
\(714\) 0 0
\(715\) 1.82921e8i 0.500433i
\(716\) 0 0
\(717\) 1.47074e8 8.49132e7i 0.399005 0.230366i
\(718\) 0 0
\(719\) −2.60987e8 4.52043e8i −0.702153 1.21617i −0.967709 0.252070i \(-0.918889\pi\)
0.265556 0.964096i \(-0.414445\pi\)
\(720\) 0 0
\(721\) 3.05560e8i 0.815249i
\(722\) 0 0
\(723\) −6.62133e6 −0.0175198
\(724\) 0 0
\(725\) −2.07167e8 + 1.19608e8i −0.543635 + 0.313868i
\(726\) 0 0
\(727\) −2.32657e6 4.02973e6i −0.00605498 0.0104875i 0.862982 0.505235i \(-0.168594\pi\)
−0.869037 + 0.494747i \(0.835261\pi\)
\(728\) 0 0
\(729\) −3.56803e8 −0.920970
\(730\) 0 0
\(731\) 3.00954e6 5.21268e6i 0.00770456 0.0133447i
\(732\) 0 0
\(733\) −7.07531e7 −0.179653 −0.0898263 0.995957i \(-0.528631\pi\)
−0.0898263 + 0.995957i \(0.528631\pi\)
\(734\) 0 0
\(735\) 2.76009e8 + 1.59354e8i 0.695121 + 0.401328i
\(736\) 0 0
\(737\) 8.64724e8 + 4.99249e8i 2.16011 + 1.24714i
\(738\) 0 0
\(739\) −1.88312e8 3.26166e8i −0.466600 0.808175i 0.532672 0.846322i \(-0.321188\pi\)
−0.999272 + 0.0381465i \(0.987855\pi\)
\(740\) 0 0
\(741\) −9.47183e7 + 3.65299e7i −0.232798 + 0.0897830i
\(742\) 0 0
\(743\) −2.78200e8 + 1.60619e8i −0.678252 + 0.391589i −0.799196 0.601070i \(-0.794741\pi\)
0.120944 + 0.992659i \(0.461408\pi\)
\(744\) 0 0
\(745\) 8.42151e7 1.45865e8i 0.203667 0.352762i
\(746\) 0 0
\(747\) −1.70883e8 + 2.95979e8i −0.409957 + 0.710066i
\(748\) 0 0
\(749\) 1.87842e8i 0.447040i
\(750\) 0 0
\(751\) 7.54561e7 + 4.35646e7i 0.178145 + 0.102852i 0.586421 0.810006i \(-0.300536\pi\)
−0.408276 + 0.912859i \(0.633870\pi\)
\(752\) 0 0
\(753\) 6.64706e7i 0.155684i
\(754\) 0 0
\(755\) 2.32009e8 1.33950e8i 0.539093 0.311246i
\(756\) 0 0
\(757\) −3.40754e8 5.90202e8i −0.785512 1.36055i −0.928693 0.370850i \(-0.879066\pi\)
0.143181 0.989697i \(-0.454267\pi\)
\(758\) 0 0
\(759\) 6.59976e8i 1.50939i
\(760\) 0 0
\(761\) −7.29867e7 −0.165611 −0.0828057 0.996566i \(-0.526388\pi\)
−0.0828057 + 0.996566i \(0.526388\pi\)
\(762\) 0 0
\(763\) −4.84092e8 + 2.79490e8i −1.08982 + 0.629207i
\(764\) 0 0
\(765\) −1.94726e8 3.37276e8i −0.434951 0.753356i
\(766\) 0 0
\(767\) −1.59060e8 −0.352512
\(768\) 0 0
\(769\) 9.82080e7 1.70101e8i 0.215957 0.374049i −0.737611 0.675226i \(-0.764046\pi\)
0.953568 + 0.301177i \(0.0973795\pi\)
\(770\) 0 0
\(771\) 4.22339e8 0.921505
\(772\) 0 0
\(773\) 1.72911e8 + 9.98303e7i 0.374356 + 0.216134i 0.675360 0.737488i \(-0.263988\pi\)
−0.301004 + 0.953623i \(0.597322\pi\)
\(774\) 0 0
\(775\) 9.63893e7 + 5.56504e7i 0.207073 + 0.119554i
\(776\) 0 0
\(777\) 3.03553e8 + 5.25769e8i 0.647099 + 1.12081i
\(778\) 0 0
\(779\) −1.91169e8 + 2.36834e8i −0.404394 + 0.500994i
\(780\) 0 0
\(781\) 3.51755e7 2.03086e7i 0.0738392 0.0426311i
\(782\) 0 0
\(783\) −4.74800e8 + 8.22378e8i −0.989067 + 1.71311i
\(784\) 0 0
\(785\) 9.72027e7 1.68360e8i 0.200941 0.348041i
\(786\) 0 0
\(787\) 7.06372e8i 1.44914i −0.689203 0.724568i \(-0.742039\pi\)
0.689203 0.724568i \(-0.257961\pi\)
\(788\) 0 0
\(789\) −1.73630e8 1.00245e8i −0.353503 0.204095i
\(790\) 0 0
\(791\) 3.13178e7i 0.0632794i
\(792\) 0 0
\(793\) 2.61561e8 1.51012e8i 0.524509 0.302825i
\(794\) 0 0
\(795\) −2.53718e8 4.39452e8i −0.504952 0.874602i
\(796\) 0 0
\(797\) 2.45124e8i 0.484185i 0.970253 + 0.242092i \(0.0778337\pi\)
−0.970253 + 0.242092i \(0.922166\pi\)
\(798\) 0 0
\(799\) 3.80486e8 0.745930
\(800\) 0 0
\(801\) 1.59379e8 9.20173e7i 0.310122 0.179049i
\(802\) 0 0
\(803\) 1.04777e8 + 1.81479e8i 0.202358 + 0.350494i
\(804\) 0 0
\(805\) 1.39287e9 2.67007
\(806\) 0 0
\(807\) 2.40144e8 4.15941e8i 0.456931 0.791428i
\(808\) 0 0
\(809\) −3.06400e8 −0.578686 −0.289343 0.957225i \(-0.593437\pi\)
−0.289343 + 0.957225i \(0.593437\pi\)
\(810\) 0 0
\(811\) −5.16295e8 2.98083e8i −0.967911 0.558823i −0.0693120 0.997595i \(-0.522080\pi\)
−0.898599 + 0.438772i \(0.855414\pi\)
\(812\) 0 0
\(813\) −1.58078e8 9.12666e7i −0.294172 0.169840i
\(814\) 0 0
\(815\) −3.14965e8 5.45536e8i −0.581822 1.00774i
\(816\) 0 0
\(817\) 4.10780e6 + 3.31575e6i 0.00753257 + 0.00608017i
\(818\) 0 0
\(819\) −1.07868e8 + 6.22778e7i −0.196355 + 0.113366i
\(820\) 0 0
\(821\) 2.07421e8 3.59264e8i 0.374821 0.649209i −0.615479 0.788153i \(-0.711038\pi\)
0.990300 + 0.138944i \(0.0443709\pi\)
\(822\) 0 0
\(823\) −2.93877e8 + 5.09009e8i −0.527188 + 0.913116i 0.472310 + 0.881432i \(0.343420\pi\)
−0.999498 + 0.0316837i \(0.989913\pi\)
\(824\) 0 0
\(825\) 1.74323e8i 0.310451i
\(826\) 0 0
\(827\) 2.14159e8 + 1.23645e8i 0.378635 + 0.218605i 0.677224 0.735777i \(-0.263183\pi\)
−0.298589 + 0.954382i \(0.596516\pi\)
\(828\) 0 0
\(829\) 1.10394e9i 1.93768i 0.247694 + 0.968838i \(0.420327\pi\)
−0.247694 + 0.968838i \(0.579673\pi\)
\(830\) 0 0
\(831\) 3.79138e8 2.18895e8i 0.660684 0.381446i
\(832\) 0 0
\(833\) 4.39169e8 + 7.60664e8i 0.759796 + 1.31601i
\(834\) 0 0
\(835\) 1.06623e9i 1.83143i
\(836\) 0 0
\(837\) 4.41823e8 0.753480
\(838\) 0 0
\(839\) 4.45140e8 2.57002e8i 0.753721 0.435161i −0.0733159 0.997309i \(-0.523358\pi\)
0.827037 + 0.562148i \(0.190025\pi\)
\(840\) 0 0
\(841\) 7.19927e8 + 1.24695e9i 1.21032 + 2.09634i
\(842\) 0 0
\(843\) −2.79082e8 −0.465853
\(844\) 0 0
\(845\) 3.07956e8 5.33395e8i 0.510409 0.884054i
\(846\) 0 0
\(847\) 4.97103e8 0.818082
\(848\) 0 0
\(849\) −2.82909e8 1.63337e8i −0.462299 0.266908i
\(850\) 0 0
\(851\) 1.12224e9 + 6.47923e8i 1.82094 + 1.05132i
\(852\) 0 0
\(853\) 5.98430e7 + 1.03651e8i 0.0964198 + 0.167004i 0.910200 0.414168i \(-0.135928\pi\)
−0.813780 + 0.581172i \(0.802594\pi\)
\(854\) 0 0
\(855\) 3.18690e8 1.22909e8i 0.509882 0.196646i
\(856\) 0 0
\(857\) −6.76218e8 + 3.90415e8i −1.07435 + 0.620274i −0.929366 0.369160i \(-0.879645\pi\)
−0.144981 + 0.989434i \(0.546312\pi\)
\(858\) 0 0
\(859\) −4.99149e8 + 8.64552e8i −0.787501 + 1.36399i 0.139993 + 0.990152i \(0.455292\pi\)
−0.927494 + 0.373838i \(0.878041\pi\)
\(860\) 0 0
\(861\) 2.08702e8 3.61482e8i 0.326976 0.566339i
\(862\) 0 0
\(863\) 5.85694e8i 0.911251i −0.890171 0.455626i \(-0.849416\pi\)
0.890171 0.455626i \(-0.150584\pi\)
\(864\) 0 0
\(865\) −3.51919e7 2.03180e7i −0.0543744 0.0313931i
\(866\) 0 0
\(867\) 7.26226e8i 1.11433i
\(868\) 0 0
\(869\) −9.54121e8 + 5.50862e8i −1.45393 + 0.839428i
\(870\) 0 0
\(871\) −2.24796e8 3.89359e8i −0.340200 0.589244i
\(872\) 0 0
\(873\) 5.64769e8i 0.848844i
\(874\) 0 0
\(875\) 7.16055e8 1.06886
\(876\) 0 0
\(877\) −1.05820e9 + 6.10949e8i −1.56880 + 0.905746i −0.572489 + 0.819912i \(0.694022\pi\)
−0.996309 + 0.0858337i \(0.972645\pi\)
\(878\) 0 0
\(879\) −6.13861e7 1.06324e8i −0.0903865 0.156554i
\(880\) 0 0
\(881\) −3.76883e8 −0.551162 −0.275581 0.961278i \(-0.588870\pi\)
−0.275581 + 0.961278i \(0.588870\pi\)
\(882\) 0 0
\(883\) −1.66348e8 + 2.88123e8i −0.241621 + 0.418500i −0.961176 0.275935i \(-0.911012\pi\)
0.719555 + 0.694435i \(0.244346\pi\)
\(884\) 0 0
\(885\) −5.98194e8 −0.863002
\(886\) 0 0
\(887\) −5.36249e8 3.09604e8i −0.768415 0.443645i 0.0638937 0.997957i \(-0.479648\pi\)
−0.832309 + 0.554312i \(0.812981\pi\)
\(888\) 0 0
\(889\) −5.58337e7 3.22356e7i −0.0794678 0.0458808i
\(890\) 0 0
\(891\) −1.35737e8 2.35103e8i −0.191895 0.332372i
\(892\) 0 0
\(893\) −5.16848e7 + 3.29678e8i −0.0725785 + 0.462952i
\(894\) 0 0
\(895\) 8.98726e8 5.18880e8i 1.25360 0.723765i
\(896\) 0 0
\(897\) 1.48583e8 2.57354e8i 0.205870 0.356577i
\(898\) 0 0
\(899\) 4.73340e8 8.19848e8i 0.651469 1.12838i
\(900\) 0 0
\(901\) 1.39846e9i 1.91195i
\(902\) 0 0
\(903\) −6.26976e6 3.61985e6i −0.00851506 0.00491617i
\(904\) 0 0
\(905\) 2.15229e8i 0.290373i
\(906\) 0 0
\(907\) −3.44639e8 + 1.98977e8i −0.461894 + 0.266675i −0.712840 0.701326i \(-0.752592\pi\)
0.250946 + 0.968001i \(0.419258\pi\)
\(908\) 0 0
\(909\) 5.03235e7 + 8.71629e7i 0.0670007 + 0.116049i
\(910\) 0 0
\(911\) 2.15131e8i 0.284543i −0.989828 0.142271i \(-0.954559\pi\)
0.989828 0.142271i \(-0.0454406\pi\)
\(912\) 0 0
\(913\) −1.66376e9 −2.18614
\(914\) 0 0
\(915\) 9.83680e8 5.67928e8i 1.28408 0.741361i
\(916\) 0 0
\(917\) 1.34044e8 + 2.32171e8i 0.173836 + 0.301093i
\(918\) 0 0
\(919\) 8.50647e8 1.09598 0.547990 0.836485i \(-0.315393\pi\)
0.547990 + 0.836485i \(0.315393\pi\)
\(920\) 0 0
\(921\) −2.87962e8 + 4.98765e8i −0.368601 + 0.638436i
\(922\) 0 0
\(923\) −1.82887e7 −0.0232582
\(924\) 0 0
\(925\) −2.96423e8 1.71140e8i −0.374530 0.216235i
\(926\) 0 0
\(927\) 1.89955e8 + 1.09671e8i 0.238458 + 0.137674i
\(928\) 0 0
\(929\) 5.39242e8 + 9.33995e8i 0.672569 + 1.16492i 0.977173 + 0.212444i \(0.0681423\pi\)
−0.304605 + 0.952479i \(0.598524\pi\)
\(930\) 0 0
\(931\) −7.18747e8 + 2.77198e8i −0.890691 + 0.343512i
\(932\) 0 0
\(933\) 6.07061e8 3.50487e8i 0.747459 0.431546i
\(934\) 0 0
\(935\) 9.47950e8 1.64190e9i 1.15971 2.00868i
\(936\) 0 0
\(937\) 6.24661e8 1.08194e9i 0.759321 1.31518i −0.183877 0.982949i \(-0.558865\pi\)
0.943197 0.332233i \(-0.107802\pi\)
\(938\) 0 0
\(939\) 7.02050e8i 0.847952i
\(940\) 0 0
\(941\) −7.01517e8 4.05021e8i −0.841917 0.486081i 0.0159981 0.999872i \(-0.494907\pi\)
−0.857916 + 0.513791i \(0.828241\pi\)
\(942\) 0 0
\(943\) 8.90933e8i 1.06245i
\(944\) 0 0
\(945\) −1.26479e9 + 7.30225e8i −1.49872 + 0.865289i
\(946\) 0 0
\(947\) 3.63881e8 + 6.30261e8i 0.428460 + 0.742114i 0.996737 0.0807237i \(-0.0257231\pi\)
−0.568277 + 0.822837i \(0.692390\pi\)
\(948\) 0 0
\(949\) 9.43559e7i 0.110400i
\(950\) 0 0
\(951\) −4.80654e8 −0.558844
\(952\) 0 0
\(953\) 3.12803e7 1.80597e7i 0.0361403 0.0208656i −0.481821 0.876270i \(-0.660024\pi\)
0.517961 + 0.855404i \(0.326691\pi\)
\(954\) 0 0
\(955\) −4.08560e8 7.07646e8i −0.469079 0.812468i
\(956\) 0 0
\(957\) −1.48272e9 −1.69170
\(958\) 0 0
\(959\) 4.42406e8 7.66270e8i 0.501609 0.868812i
\(960\) 0 0
\(961\) 4.47040e8 0.503705
\(962\) 0 0
\(963\) 1.16774e8 + 6.74196e7i 0.130758 + 0.0754932i
\(964\) 0 0
\(965\) 6.63202e7 + 3.82900e7i 0.0738012 + 0.0426092i
\(966\) 0 0
\(967\) 1.78062e8 + 3.08412e8i 0.196920 + 0.341076i 0.947528 0.319672i \(-0.103573\pi\)
−0.750608 + 0.660748i \(0.770239\pi\)
\(968\) 0 0
\(969\) −1.03950e9 1.62965e8i −1.14249 0.179112i
\(970\) 0 0
\(971\) −1.15308e9 + 6.65730e8i −1.25951 + 0.727177i −0.972979 0.230895i \(-0.925835\pi\)
−0.286529 + 0.958072i \(0.592501\pi\)
\(972\) 0 0
\(973\) 1.81038e7 3.13566e7i 0.0196531 0.0340401i
\(974\) 0 0
\(975\) −3.92462e7 + 6.79763e7i −0.0423432 + 0.0733405i
\(976\) 0 0
\(977\) 1.13278e9i 1.21468i −0.794442 0.607339i \(-0.792237\pi\)
0.794442 0.607339i \(-0.207763\pi\)
\(978\) 0 0
\(979\) 7.75874e8 + 4.47951e8i 0.826881 + 0.477400i
\(980\) 0 0
\(981\) 4.01256e8i 0.425025i
\(982\) 0 0
\(983\) 1.03981e9 6.00334e8i 1.09469 0.632022i 0.159872 0.987138i \(-0.448892\pi\)
0.934822 + 0.355116i \(0.115559\pi\)
\(984\) 0 0
\(985\) 6.29887e7 + 1.09100e8i 0.0659104 + 0.114160i
\(986\) 0 0
\(987\) 4.57645e8i 0.475967i
\(988\) 0 0
\(989\) −1.54529e7 −0.0159743
\(990\) 0 0
\(991\) −1.08820e9 + 6.28275e8i −1.11812 + 0.645548i −0.940921 0.338626i \(-0.890038\pi\)
−0.177201 + 0.984175i \(0.556704\pi\)
\(992\) 0 0
\(993\) −5.28954e7 9.16176e7i −0.0540220 0.0935688i
\(994\) 0 0
\(995\) −2.17199e8 −0.220490
\(996\) 0 0
\(997\) 1.66824e8 2.88947e8i 0.168334 0.291564i −0.769500 0.638647i \(-0.779495\pi\)
0.937834 + 0.347083i \(0.112828\pi\)
\(998\) 0 0
\(999\) −1.35872e9 −1.36281
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 76.7.h.a.69.8 yes 20
3.2 odd 2 684.7.y.c.145.2 20
19.8 odd 6 inner 76.7.h.a.65.8 20
57.8 even 6 684.7.y.c.217.2 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.7.h.a.65.8 20 19.8 odd 6 inner
76.7.h.a.69.8 yes 20 1.1 even 1 trivial
684.7.y.c.145.2 20 3.2 odd 2
684.7.y.c.217.2 20 57.8 even 6