Properties

Label 1800.2.k.r.901.2
Level $1800$
Weight $2$
Character 1800.901
Analytic conductor $14.373$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1800,2,Mod(901,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.901");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.k (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.3730723638\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.212336640000.29
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 2x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 901.2
Root \(-1.26979 + 0.622597i\) of defining polynomial
Character \(\chi\) \(=\) 1800.901
Dual form 1800.2.k.r.901.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.26979 + 0.622597i) q^{2} +(1.22474 - 1.58114i) q^{4} -3.44949 q^{7} +(-0.570759 + 2.77024i) q^{8} +O(q^{10})\) \(q+(-1.26979 + 0.622597i) q^{2} +(1.22474 - 1.58114i) q^{4} -3.44949 q^{7} +(-0.570759 + 2.77024i) q^{8} -5.54048i q^{11} +3.87298i q^{13} +(4.38014 - 2.14764i) q^{14} +(-1.00000 - 3.87298i) q^{16} +6.22069 q^{17} +7.03526i q^{19} +(3.44949 + 7.03526i) q^{22} -1.14152 q^{23} +(-2.41131 - 4.91788i) q^{26} +(-4.22474 + 5.45412i) q^{28} -3.05009i q^{29} -1.44949 q^{31} +(3.68110 + 4.29529i) q^{32} +(-7.89898 + 3.87298i) q^{34} -6.32456i q^{37} +(-4.38014 - 8.93332i) q^{38} +3.93765 q^{41} +0.710706i q^{43} +(-8.76027 - 6.78568i) q^{44} +(1.44949 - 0.710706i) q^{46} -10.1583 q^{47} +4.89898 q^{49} +(6.12372 + 4.74342i) q^{52} -8.03087i q^{53} +(1.96883 - 9.55592i) q^{56} +(1.89898 + 3.87298i) q^{58} -4.98078i q^{59} -10.1975i q^{61} +(1.84055 - 0.902449i) q^{62} +(-7.34847 - 3.16228i) q^{64} -5.61385i q^{67} +(7.61875 - 9.83577i) q^{68} -13.5829 q^{71} -6.89898 q^{73} +(3.93765 + 8.03087i) q^{74} +(11.1237 + 8.61640i) q^{76} +19.1118i q^{77} +2.89898 q^{79} +(-5.00000 + 2.45157i) q^{82} -6.10018i q^{83} +(-0.442484 - 0.902449i) q^{86} +(15.3485 + 3.16228i) q^{88} +7.87530 q^{89} -13.3598i q^{91} +(-1.39807 + 1.80490i) q^{92} +(12.8990 - 6.32456i) q^{94} -15.6969 q^{97} +(-6.22069 + 3.05009i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} - 8 q^{16} + 8 q^{22} - 24 q^{28} + 8 q^{31} - 24 q^{34} - 8 q^{46} - 24 q^{58} - 16 q^{73} + 40 q^{76} - 16 q^{79} - 40 q^{82} + 64 q^{88} + 64 q^{94} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.26979 + 0.622597i −0.897879 + 0.440243i
\(3\) 0 0
\(4\) 1.22474 1.58114i 0.612372 0.790569i
\(5\) 0 0
\(6\) 0 0
\(7\) −3.44949 −1.30378 −0.651892 0.758312i \(-0.726025\pi\)
−0.651892 + 0.758312i \(0.726025\pi\)
\(8\) −0.570759 + 2.77024i −0.201794 + 0.979428i
\(9\) 0 0
\(10\) 0 0
\(11\) 5.54048i 1.67052i −0.549857 0.835259i \(-0.685318\pi\)
0.549857 0.835259i \(-0.314682\pi\)
\(12\) 0 0
\(13\) 3.87298i 1.07417i 0.843527 + 0.537086i \(0.180475\pi\)
−0.843527 + 0.537086i \(0.819525\pi\)
\(14\) 4.38014 2.14764i 1.17064 0.573982i
\(15\) 0 0
\(16\) −1.00000 3.87298i −0.250000 0.968246i
\(17\) 6.22069 1.50874 0.754369 0.656451i \(-0.227943\pi\)
0.754369 + 0.656451i \(0.227943\pi\)
\(18\) 0 0
\(19\) 7.03526i 1.61400i 0.590552 + 0.807000i \(0.298910\pi\)
−0.590552 + 0.807000i \(0.701090\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 3.44949 + 7.03526i 0.735434 + 1.49992i
\(23\) −1.14152 −0.238023 −0.119011 0.992893i \(-0.537973\pi\)
−0.119011 + 0.992893i \(0.537973\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.41131 4.91788i −0.472897 0.964476i
\(27\) 0 0
\(28\) −4.22474 + 5.45412i −0.798402 + 1.03073i
\(29\) 3.05009i 0.566388i −0.959063 0.283194i \(-0.908606\pi\)
0.959063 0.283194i \(-0.0913940\pi\)
\(30\) 0 0
\(31\) −1.44949 −0.260336 −0.130168 0.991492i \(-0.541552\pi\)
−0.130168 + 0.991492i \(0.541552\pi\)
\(32\) 3.68110 + 4.29529i 0.650733 + 0.759307i
\(33\) 0 0
\(34\) −7.89898 + 3.87298i −1.35466 + 0.664211i
\(35\) 0 0
\(36\) 0 0
\(37\) 6.32456i 1.03975i −0.854242 0.519875i \(-0.825978\pi\)
0.854242 0.519875i \(-0.174022\pi\)
\(38\) −4.38014 8.93332i −0.710552 1.44918i
\(39\) 0 0
\(40\) 0 0
\(41\) 3.93765 0.614958 0.307479 0.951555i \(-0.400515\pi\)
0.307479 + 0.951555i \(0.400515\pi\)
\(42\) 0 0
\(43\) 0.710706i 0.108382i 0.998531 + 0.0541908i \(0.0172579\pi\)
−0.998531 + 0.0541908i \(0.982742\pi\)
\(44\) −8.76027 6.78568i −1.32066 1.02298i
\(45\) 0 0
\(46\) 1.44949 0.710706i 0.213716 0.104788i
\(47\) −10.1583 −1.48175 −0.740873 0.671645i \(-0.765588\pi\)
−0.740873 + 0.671645i \(0.765588\pi\)
\(48\) 0 0
\(49\) 4.89898 0.699854
\(50\) 0 0
\(51\) 0 0
\(52\) 6.12372 + 4.74342i 0.849208 + 0.657794i
\(53\) 8.03087i 1.10313i −0.834134 0.551563i \(-0.814032\pi\)
0.834134 0.551563i \(-0.185968\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.96883 9.55592i 0.263095 1.27696i
\(57\) 0 0
\(58\) 1.89898 + 3.87298i 0.249348 + 0.508548i
\(59\) 4.98078i 0.648442i −0.945981 0.324221i \(-0.894898\pi\)
0.945981 0.324221i \(-0.105102\pi\)
\(60\) 0 0
\(61\) 10.1975i 1.30566i −0.757504 0.652831i \(-0.773581\pi\)
0.757504 0.652831i \(-0.226419\pi\)
\(62\) 1.84055 0.902449i 0.233750 0.114611i
\(63\) 0 0
\(64\) −7.34847 3.16228i −0.918559 0.395285i
\(65\) 0 0
\(66\) 0 0
\(67\) 5.61385i 0.685841i −0.939364 0.342920i \(-0.888584\pi\)
0.939364 0.342920i \(-0.111416\pi\)
\(68\) 7.61875 9.83577i 0.923910 1.19276i
\(69\) 0 0
\(70\) 0 0
\(71\) −13.5829 −1.61199 −0.805996 0.591921i \(-0.798370\pi\)
−0.805996 + 0.591921i \(0.798370\pi\)
\(72\) 0 0
\(73\) −6.89898 −0.807464 −0.403732 0.914877i \(-0.632287\pi\)
−0.403732 + 0.914877i \(0.632287\pi\)
\(74\) 3.93765 + 8.03087i 0.457743 + 0.933570i
\(75\) 0 0
\(76\) 11.1237 + 8.61640i 1.27598 + 0.988369i
\(77\) 19.1118i 2.17800i
\(78\) 0 0
\(79\) 2.89898 0.326161 0.163080 0.986613i \(-0.447857\pi\)
0.163080 + 0.986613i \(0.447857\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −5.00000 + 2.45157i −0.552158 + 0.270731i
\(83\) 6.10018i 0.669582i −0.942292 0.334791i \(-0.891334\pi\)
0.942292 0.334791i \(-0.108666\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.442484 0.902449i −0.0477142 0.0973135i
\(87\) 0 0
\(88\) 15.3485 + 3.16228i 1.63615 + 0.337100i
\(89\) 7.87530 0.834781 0.417390 0.908727i \(-0.362945\pi\)
0.417390 + 0.908727i \(0.362945\pi\)
\(90\) 0 0
\(91\) 13.3598i 1.40049i
\(92\) −1.39807 + 1.80490i −0.145759 + 0.188174i
\(93\) 0 0
\(94\) 12.8990 6.32456i 1.33043 0.652328i
\(95\) 0 0
\(96\) 0 0
\(97\) −15.6969 −1.59378 −0.796891 0.604123i \(-0.793524\pi\)
−0.796891 + 0.604123i \(0.793524\pi\)
\(98\) −6.22069 + 3.05009i −0.628384 + 0.308106i
\(99\) 0 0
\(100\) 0 0
\(101\) 14.1311i 1.40609i −0.711144 0.703046i \(-0.751823\pi\)
0.711144 0.703046i \(-0.248177\pi\)
\(102\) 0 0
\(103\) −6.89898 −0.679777 −0.339888 0.940466i \(-0.610389\pi\)
−0.339888 + 0.940466i \(0.610389\pi\)
\(104\) −10.7291 2.21054i −1.05207 0.216761i
\(105\) 0 0
\(106\) 5.00000 + 10.1975i 0.485643 + 0.990473i
\(107\) 0.559702i 0.0541085i 0.999634 + 0.0270542i \(0.00861268\pi\)
−0.999634 + 0.0270542i \(0.991387\pi\)
\(108\) 0 0
\(109\) 16.5221i 1.58253i 0.611474 + 0.791265i \(0.290577\pi\)
−0.611474 + 0.791265i \(0.709423\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.44949 + 13.3598i 0.325946 + 1.26238i
\(113\) −12.4414 −1.17039 −0.585193 0.810894i \(-0.698981\pi\)
−0.585193 + 0.810894i \(0.698981\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.82262 3.73558i −0.447769 0.346840i
\(117\) 0 0
\(118\) 3.10102 + 6.32456i 0.285472 + 0.582223i
\(119\) −21.4582 −1.96707
\(120\) 0 0
\(121\) −19.6969 −1.79063
\(122\) 6.34896 + 12.9488i 0.574808 + 1.17233i
\(123\) 0 0
\(124\) −1.77526 + 2.29184i −0.159423 + 0.205814i
\(125\) 0 0
\(126\) 0 0
\(127\) −3.79796 −0.337014 −0.168507 0.985700i \(-0.553895\pi\)
−0.168507 + 0.985700i \(0.553895\pi\)
\(128\) 11.2999 0.559702i 0.998776 0.0494712i
\(129\) 0 0
\(130\) 0 0
\(131\) 10.5213i 0.919247i −0.888114 0.459623i \(-0.847984\pi\)
0.888114 0.459623i \(-0.152016\pi\)
\(132\) 0 0
\(133\) 24.2681i 2.10431i
\(134\) 3.49517 + 7.12842i 0.301937 + 0.615802i
\(135\) 0 0
\(136\) −3.55051 + 17.2328i −0.304454 + 1.47770i
\(137\) 10.1583 0.867885 0.433943 0.900940i \(-0.357122\pi\)
0.433943 + 0.900940i \(0.357122\pi\)
\(138\) 0 0
\(139\) 9.16738i 0.777567i −0.921329 0.388783i \(-0.872895\pi\)
0.921329 0.388783i \(-0.127105\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 17.2474 8.45667i 1.44737 0.709668i
\(143\) 21.4582 1.79442
\(144\) 0 0
\(145\) 0 0
\(146\) 8.76027 4.29529i 0.725005 0.355480i
\(147\) 0 0
\(148\) −10.0000 7.74597i −0.821995 0.636715i
\(149\) 21.0425i 1.72387i 0.507018 + 0.861935i \(0.330748\pi\)
−0.507018 + 0.861935i \(0.669252\pi\)
\(150\) 0 0
\(151\) −12.3485 −1.00490 −0.502452 0.864605i \(-0.667569\pi\)
−0.502452 + 0.864605i \(0.667569\pi\)
\(152\) −19.4894 4.01544i −1.58080 0.325695i
\(153\) 0 0
\(154\) −11.8990 24.2681i −0.958847 1.95558i
\(155\) 0 0
\(156\) 0 0
\(157\) 8.77613i 0.700411i −0.936673 0.350206i \(-0.886112\pi\)
0.936673 0.350206i \(-0.113888\pi\)
\(158\) −3.68110 + 1.80490i −0.292853 + 0.143590i
\(159\) 0 0
\(160\) 0 0
\(161\) 3.93765 0.310330
\(162\) 0 0
\(163\) 14.7812i 1.15776i −0.815414 0.578878i \(-0.803491\pi\)
0.815414 0.578878i \(-0.196509\pi\)
\(164\) 4.82262 6.22597i 0.376583 0.486167i
\(165\) 0 0
\(166\) 3.79796 + 7.74597i 0.294779 + 0.601204i
\(167\) 3.42455 0.265000 0.132500 0.991183i \(-0.457700\pi\)
0.132500 + 0.991183i \(0.457700\pi\)
\(168\) 0 0
\(169\) −2.00000 −0.153846
\(170\) 0 0
\(171\) 0 0
\(172\) 1.12372 + 0.870433i 0.0856832 + 0.0663699i
\(173\) 1.93069i 0.146787i −0.997303 0.0733937i \(-0.976617\pi\)
0.997303 0.0733937i \(-0.0233830\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −21.4582 + 5.54048i −1.61747 + 0.417630i
\(177\) 0 0
\(178\) −10.0000 + 4.90314i −0.749532 + 0.367506i
\(179\) 21.0425i 1.57279i −0.617723 0.786396i \(-0.711945\pi\)
0.617723 0.786396i \(-0.288055\pi\)
\(180\) 0 0
\(181\) 5.29439i 0.393529i 0.980451 + 0.196765i \(0.0630435\pi\)
−0.980451 + 0.196765i \(0.936957\pi\)
\(182\) 8.31779 + 16.9642i 0.616555 + 1.25747i
\(183\) 0 0
\(184\) 0.651531 3.16228i 0.0480315 0.233126i
\(185\) 0 0
\(186\) 0 0
\(187\) 34.4656i 2.52037i
\(188\) −12.4414 + 16.0617i −0.907380 + 1.17142i
\(189\) 0 0
\(190\) 0 0
\(191\) 13.5829 0.982823 0.491412 0.870927i \(-0.336481\pi\)
0.491412 + 0.870927i \(0.336481\pi\)
\(192\) 0 0
\(193\) 11.8990 0.856507 0.428254 0.903659i \(-0.359129\pi\)
0.428254 + 0.903659i \(0.359129\pi\)
\(194\) 19.9319 9.77287i 1.43102 0.701651i
\(195\) 0 0
\(196\) 6.00000 7.74597i 0.428571 0.553283i
\(197\) 23.2813i 1.65873i −0.558710 0.829363i \(-0.688704\pi\)
0.558710 0.829363i \(-0.311296\pi\)
\(198\) 0 0
\(199\) 21.2474 1.50619 0.753096 0.657911i \(-0.228560\pi\)
0.753096 + 0.657911i \(0.228560\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 8.79796 + 17.9435i 0.619022 + 1.26250i
\(203\) 10.5213i 0.738448i
\(204\) 0 0
\(205\) 0 0
\(206\) 8.76027 4.29529i 0.610357 0.299267i
\(207\) 0 0
\(208\) 15.0000 3.87298i 1.04006 0.268543i
\(209\) 38.9787 2.69622
\(210\) 0 0
\(211\) 2.13212i 0.146781i 0.997303 + 0.0733905i \(0.0233819\pi\)
−0.997303 + 0.0733905i \(0.976618\pi\)
\(212\) −12.6979 9.83577i −0.872097 0.675523i
\(213\) 0 0
\(214\) −0.348469 0.710706i −0.0238209 0.0485828i
\(215\) 0 0
\(216\) 0 0
\(217\) 5.00000 0.339422
\(218\) −10.2866 20.9796i −0.696697 1.42092i
\(219\) 0 0
\(220\) 0 0
\(221\) 24.0926i 1.62064i
\(222\) 0 0
\(223\) −0.348469 −0.0233352 −0.0116676 0.999932i \(-0.503714\pi\)
−0.0116676 + 0.999932i \(0.503714\pi\)
\(224\) −12.6979 14.8165i −0.848416 0.989972i
\(225\) 0 0
\(226\) 15.7980 7.74597i 1.05086 0.515254i
\(227\) 4.42108i 0.293437i −0.989178 0.146719i \(-0.953129\pi\)
0.989178 0.146719i \(-0.0468712\pi\)
\(228\) 0 0
\(229\) 5.29439i 0.349863i −0.984581 0.174932i \(-0.944030\pi\)
0.984581 0.174932i \(-0.0559705\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 8.44949 + 1.74087i 0.554736 + 0.114293i
\(233\) −24.2543 −1.58895 −0.794477 0.607294i \(-0.792255\pi\)
−0.794477 + 0.607294i \(0.792255\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.87530 6.10018i −0.512639 0.397088i
\(237\) 0 0
\(238\) 27.2474 13.3598i 1.76619 0.865988i
\(239\) −29.3335 −1.89743 −0.948713 0.316138i \(-0.897614\pi\)
−0.948713 + 0.316138i \(0.897614\pi\)
\(240\) 0 0
\(241\) 23.6969 1.52645 0.763227 0.646130i \(-0.223614\pi\)
0.763227 + 0.646130i \(0.223614\pi\)
\(242\) 25.0110 12.2633i 1.60777 0.788312i
\(243\) 0 0
\(244\) −16.1237 12.4894i −1.03222 0.799551i
\(245\) 0 0
\(246\) 0 0
\(247\) −27.2474 −1.73371
\(248\) 0.827309 4.01544i 0.0525342 0.254980i
\(249\) 0 0
\(250\) 0 0
\(251\) 6.10018i 0.385040i 0.981293 + 0.192520i \(0.0616661\pi\)
−0.981293 + 0.192520i \(0.938334\pi\)
\(252\) 0 0
\(253\) 6.32456i 0.397621i
\(254\) 4.82262 2.36460i 0.302598 0.148368i
\(255\) 0 0
\(256\) −14.0000 + 7.74597i −0.875000 + 0.484123i
\(257\) −2.28303 −0.142412 −0.0712059 0.997462i \(-0.522685\pi\)
−0.0712059 + 0.997462i \(0.522685\pi\)
\(258\) 0 0
\(259\) 21.8165i 1.35561i
\(260\) 0 0
\(261\) 0 0
\(262\) 6.55051 + 13.3598i 0.404692 + 0.825372i
\(263\) 9.01682 0.556001 0.278001 0.960581i \(-0.410328\pi\)
0.278001 + 0.960581i \(0.410328\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 15.1092 + 30.8154i 0.926406 + 1.88941i
\(267\) 0 0
\(268\) −8.87628 6.87553i −0.542205 0.419990i
\(269\) 4.16950i 0.254219i −0.991889 0.127109i \(-0.959430\pi\)
0.991889 0.127109i \(-0.0405699\pi\)
\(270\) 0 0
\(271\) 5.10102 0.309865 0.154932 0.987925i \(-0.450484\pi\)
0.154932 + 0.987925i \(0.450484\pi\)
\(272\) −6.22069 24.0926i −0.377185 1.46083i
\(273\) 0 0
\(274\) −12.8990 + 6.32456i −0.779256 + 0.382080i
\(275\) 0 0
\(276\) 0 0
\(277\) 24.2681i 1.45813i −0.684446 0.729063i \(-0.739956\pi\)
0.684446 0.729063i \(-0.260044\pi\)
\(278\) 5.70759 + 11.6407i 0.342318 + 0.698161i
\(279\) 0 0
\(280\) 0 0
\(281\) −3.93765 −0.234901 −0.117450 0.993079i \(-0.537472\pi\)
−0.117450 + 0.993079i \(0.537472\pi\)
\(282\) 0 0
\(283\) 0.710706i 0.0422471i 0.999777 + 0.0211235i \(0.00672433\pi\)
−0.999777 + 0.0211235i \(0.993276\pi\)
\(284\) −16.6356 + 21.4764i −0.987140 + 1.27439i
\(285\) 0 0
\(286\) −27.2474 + 13.3598i −1.61118 + 0.789983i
\(287\) −13.5829 −0.801773
\(288\) 0 0
\(289\) 21.6969 1.27629
\(290\) 0 0
\(291\) 0 0
\(292\) −8.44949 + 10.9082i −0.494469 + 0.638357i
\(293\) 16.0617i 0.938337i −0.883109 0.469169i \(-0.844554\pi\)
0.883109 0.469169i \(-0.155446\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 17.5205 + 3.60979i 1.01836 + 0.209815i
\(297\) 0 0
\(298\) −13.1010 26.7196i −0.758922 1.54783i
\(299\) 4.42108i 0.255677i
\(300\) 0 0
\(301\) 2.45157i 0.141306i
\(302\) 15.6800 7.68813i 0.902282 0.442402i
\(303\) 0 0
\(304\) 27.2474 7.03526i 1.56275 0.403500i
\(305\) 0 0
\(306\) 0 0
\(307\) 14.7812i 0.843609i 0.906687 + 0.421805i \(0.138603\pi\)
−0.906687 + 0.421805i \(0.861397\pi\)
\(308\) 30.2185 + 23.4071i 1.72186 + 1.33374i
\(309\) 0 0
\(310\) 0 0
\(311\) −15.7506 −0.893135 −0.446568 0.894750i \(-0.647354\pi\)
−0.446568 + 0.894750i \(0.647354\pi\)
\(312\) 0 0
\(313\) −22.5959 −1.27720 −0.638598 0.769540i \(-0.720485\pi\)
−0.638598 + 0.769540i \(0.720485\pi\)
\(314\) 5.46399 + 11.1439i 0.308351 + 0.628884i
\(315\) 0 0
\(316\) 3.55051 4.58369i 0.199732 0.257853i
\(317\) 21.0425i 1.18187i 0.806721 + 0.590933i \(0.201240\pi\)
−0.806721 + 0.590933i \(0.798760\pi\)
\(318\) 0 0
\(319\) −16.8990 −0.946161
\(320\) 0 0
\(321\) 0 0
\(322\) −5.00000 + 2.45157i −0.278639 + 0.136621i
\(323\) 43.7642i 2.43510i
\(324\) 0 0
\(325\) 0 0
\(326\) 9.20275 + 18.7691i 0.509693 + 1.03952i
\(327\) 0 0
\(328\) −2.24745 + 10.9082i −0.124095 + 0.602307i
\(329\) 35.0411 1.93188
\(330\) 0 0
\(331\) 12.6491i 0.695258i 0.937632 + 0.347629i \(0.113013\pi\)
−0.937632 + 0.347629i \(0.886987\pi\)
\(332\) −9.64524 7.47117i −0.529351 0.410034i
\(333\) 0 0
\(334\) −4.34847 + 2.13212i −0.237938 + 0.116664i
\(335\) 0 0
\(336\) 0 0
\(337\) −18.7980 −1.02399 −0.511995 0.858988i \(-0.671093\pi\)
−0.511995 + 0.858988i \(0.671093\pi\)
\(338\) 2.53958 1.24519i 0.138135 0.0677297i
\(339\) 0 0
\(340\) 0 0
\(341\) 8.03087i 0.434896i
\(342\) 0 0
\(343\) 7.24745 0.391325
\(344\) −1.96883 0.405641i −0.106152 0.0218707i
\(345\) 0 0
\(346\) 1.20204 + 2.45157i 0.0646221 + 0.131797i
\(347\) 16.0617i 0.862240i −0.902295 0.431120i \(-0.858119\pi\)
0.902295 0.431120i \(-0.141881\pi\)
\(348\) 0 0
\(349\) 12.6491i 0.677091i −0.940950 0.338546i \(-0.890065\pi\)
0.940950 0.338546i \(-0.109935\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 23.7980 20.3951i 1.26844 1.08706i
\(353\) 0.628417 0.0334473 0.0167236 0.999860i \(-0.494676\pi\)
0.0167236 + 0.999860i \(0.494676\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 9.64524 12.4519i 0.511197 0.659952i
\(357\) 0 0
\(358\) 13.1010 + 26.7196i 0.692410 + 1.41218i
\(359\) −21.4582 −1.13252 −0.566260 0.824227i \(-0.691610\pi\)
−0.566260 + 0.824227i \(0.691610\pi\)
\(360\) 0 0
\(361\) −30.4949 −1.60499
\(362\) −3.29628 6.72278i −0.173248 0.353342i
\(363\) 0 0
\(364\) −21.1237 16.3624i −1.10718 0.857621i
\(365\) 0 0
\(366\) 0 0
\(367\) 23.4495 1.22405 0.612027 0.790837i \(-0.290354\pi\)
0.612027 + 0.790837i \(0.290354\pi\)
\(368\) 1.14152 + 4.42108i 0.0595057 + 0.230465i
\(369\) 0 0
\(370\) 0 0
\(371\) 27.7024i 1.43824i
\(372\) 0 0
\(373\) 25.6895i 1.33015i 0.746776 + 0.665075i \(0.231601\pi\)
−0.746776 + 0.665075i \(0.768399\pi\)
\(374\) 21.4582 + 43.7642i 1.10958 + 2.26299i
\(375\) 0 0
\(376\) 5.79796 28.1410i 0.299007 1.45126i
\(377\) 11.8130 0.608398
\(378\) 0 0
\(379\) 13.3598i 0.686248i 0.939290 + 0.343124i \(0.111485\pi\)
−0.939290 + 0.343124i \(0.888515\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −17.2474 + 8.45667i −0.882456 + 0.432681i
\(383\) 12.4414 0.635724 0.317862 0.948137i \(-0.397035\pi\)
0.317862 + 0.948137i \(0.397035\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −15.1092 + 7.40827i −0.769040 + 0.377071i
\(387\) 0 0
\(388\) −19.2247 + 24.8190i −0.975989 + 1.26000i
\(389\) 35.1736i 1.78337i 0.452655 + 0.891686i \(0.350477\pi\)
−0.452655 + 0.891686i \(0.649523\pi\)
\(390\) 0 0
\(391\) −7.10102 −0.359114
\(392\) −2.79613 + 13.5714i −0.141226 + 0.685457i
\(393\) 0 0
\(394\) 14.4949 + 29.5625i 0.730242 + 1.48933i
\(395\) 0 0
\(396\) 0 0
\(397\) 2.45157i 0.123041i 0.998106 + 0.0615204i \(0.0195949\pi\)
−0.998106 + 0.0615204i \(0.980405\pi\)
\(398\) −26.9798 + 13.2286i −1.35238 + 0.663090i
\(399\) 0 0
\(400\) 0 0
\(401\) 31.1034 1.55323 0.776616 0.629975i \(-0.216935\pi\)
0.776616 + 0.629975i \(0.216935\pi\)
\(402\) 0 0
\(403\) 5.61385i 0.279646i
\(404\) −22.3432 17.3069i −1.11161 0.861052i
\(405\) 0 0
\(406\) −6.55051 13.3598i −0.325096 0.663037i
\(407\) −35.0411 −1.73692
\(408\) 0 0
\(409\) −7.89898 −0.390579 −0.195290 0.980746i \(-0.562565\pi\)
−0.195290 + 0.980746i \(0.562565\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8.44949 + 10.9082i −0.416276 + 0.537411i
\(413\) 17.1811i 0.845429i
\(414\) 0 0
\(415\) 0 0
\(416\) −16.6356 + 14.2568i −0.815626 + 0.698999i
\(417\) 0 0
\(418\) −49.4949 + 24.2681i −2.42087 + 1.18699i
\(419\) 23.8410i 1.16471i 0.812934 + 0.582355i \(0.197869\pi\)
−0.812934 + 0.582355i \(0.802131\pi\)
\(420\) 0 0
\(421\) 24.6593i 1.20182i −0.799316 0.600911i \(-0.794805\pi\)
0.799316 0.600911i \(-0.205195\pi\)
\(422\) −1.32745 2.70735i −0.0646193 0.131792i
\(423\) 0 0
\(424\) 22.2474 + 4.58369i 1.08043 + 0.222604i
\(425\) 0 0
\(426\) 0 0
\(427\) 35.1763i 1.70230i
\(428\) 0.884967 + 0.685493i 0.0427765 + 0.0331345i
\(429\) 0 0
\(430\) 0 0
\(431\) −21.4582 −1.03360 −0.516802 0.856105i \(-0.672878\pi\)
−0.516802 + 0.856105i \(0.672878\pi\)
\(432\) 0 0
\(433\) 12.5959 0.605321 0.302661 0.953098i \(-0.402125\pi\)
0.302661 + 0.953098i \(0.402125\pi\)
\(434\) −6.34896 + 3.11299i −0.304760 + 0.149428i
\(435\) 0 0
\(436\) 26.1237 + 20.2353i 1.25110 + 0.969098i
\(437\) 8.03087i 0.384169i
\(438\) 0 0
\(439\) −17.0454 −0.813533 −0.406766 0.913532i \(-0.633344\pi\)
−0.406766 + 0.913532i \(0.633344\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −15.0000 30.5926i −0.713477 1.45514i
\(443\) 7.21959i 0.343013i −0.985183 0.171507i \(-0.945137\pi\)
0.985183 0.171507i \(-0.0548635\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0.442484 0.216956i 0.0209522 0.0102732i
\(447\) 0 0
\(448\) 25.3485 + 10.9082i 1.19760 + 0.515366i
\(449\) −19.2905 −0.910374 −0.455187 0.890396i \(-0.650428\pi\)
−0.455187 + 0.890396i \(0.650428\pi\)
\(450\) 0 0
\(451\) 21.8165i 1.02730i
\(452\) −15.2375 + 19.6715i −0.716712 + 0.925271i
\(453\) 0 0
\(454\) 2.75255 + 5.61385i 0.129184 + 0.263471i
\(455\) 0 0
\(456\) 0 0
\(457\) 20.6969 0.968162 0.484081 0.875023i \(-0.339154\pi\)
0.484081 + 0.875023i \(0.339154\pi\)
\(458\) 3.29628 + 6.72278i 0.154025 + 0.314135i
\(459\) 0 0
\(460\) 0 0
\(461\) 25.2120i 1.17424i 0.809500 + 0.587120i \(0.199738\pi\)
−0.809500 + 0.587120i \(0.800262\pi\)
\(462\) 0 0
\(463\) 6.89898 0.320623 0.160311 0.987066i \(-0.448750\pi\)
0.160311 + 0.987066i \(0.448750\pi\)
\(464\) −11.8130 + 3.05009i −0.548403 + 0.141597i
\(465\) 0 0
\(466\) 30.7980 15.1007i 1.42669 0.699526i
\(467\) 32.6832i 1.51240i −0.654342 0.756199i \(-0.727054\pi\)
0.654342 0.756199i \(-0.272946\pi\)
\(468\) 0 0
\(469\) 19.3649i 0.894189i
\(470\) 0 0
\(471\) 0 0
\(472\) 13.7980 + 2.84282i 0.635103 + 0.130852i
\(473\) 3.93765 0.181053
\(474\) 0 0
\(475\) 0 0
\(476\) −26.2808 + 33.9284i −1.20458 + 1.55510i
\(477\) 0 0
\(478\) 37.2474 18.2630i 1.70366 0.835328i
\(479\) −7.87530 −0.359832 −0.179916 0.983682i \(-0.557583\pi\)
−0.179916 + 0.983682i \(0.557583\pi\)
\(480\) 0 0
\(481\) 24.4949 1.11687
\(482\) −30.0902 + 14.7537i −1.37057 + 0.672010i
\(483\) 0 0
\(484\) −24.1237 + 31.1436i −1.09653 + 1.41562i
\(485\) 0 0
\(486\) 0 0
\(487\) 1.04541 0.0473719 0.0236860 0.999719i \(-0.492460\pi\)
0.0236860 + 0.999719i \(0.492460\pi\)
\(488\) 28.2496 + 5.82033i 1.27880 + 0.263474i
\(489\) 0 0
\(490\) 0 0
\(491\) 1.11940i 0.0505180i 0.999681 + 0.0252590i \(0.00804105\pi\)
−0.999681 + 0.0252590i \(0.991959\pi\)
\(492\) 0 0
\(493\) 18.9737i 0.854531i
\(494\) 34.5986 16.9642i 1.55666 0.763255i
\(495\) 0 0
\(496\) 1.44949 + 5.61385i 0.0650840 + 0.252069i
\(497\) 46.8540 2.10169
\(498\) 0 0
\(499\) 19.6844i 0.881193i −0.897705 0.440597i \(-0.854767\pi\)
0.897705 0.440597i \(-0.145233\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −3.79796 7.74597i −0.169511 0.345719i
\(503\) 28.3073 1.26216 0.631080 0.775718i \(-0.282612\pi\)
0.631080 + 0.775718i \(0.282612\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −3.93765 8.03087i −0.175050 0.357016i
\(507\) 0 0
\(508\) −4.65153 + 6.00510i −0.206378 + 0.266433i
\(509\) 21.0425i 0.932693i 0.884602 + 0.466347i \(0.154430\pi\)
−0.884602 + 0.466347i \(0.845570\pi\)
\(510\) 0 0
\(511\) 23.7980 1.05276
\(512\) 12.9545 18.5521i 0.572512 0.819896i
\(513\) 0 0
\(514\) 2.89898 1.42141i 0.127869 0.0626958i
\(515\) 0 0
\(516\) 0 0
\(517\) 56.2821i 2.47528i
\(518\) −13.5829 27.7024i −0.596798 1.21717i
\(519\) 0 0
\(520\) 0 0
\(521\) 31.1034 1.36267 0.681333 0.731974i \(-0.261401\pi\)
0.681333 + 0.731974i \(0.261401\pi\)
\(522\) 0 0
\(523\) 33.7549i 1.47600i 0.674802 + 0.737999i \(0.264229\pi\)
−0.674802 + 0.737999i \(0.735771\pi\)
\(524\) −16.6356 12.8859i −0.726728 0.562921i
\(525\) 0 0
\(526\) −11.4495 + 5.61385i −0.499221 + 0.244775i
\(527\) −9.01682 −0.392779
\(528\) 0 0
\(529\) −21.6969 −0.943345
\(530\) 0 0
\(531\) 0 0
\(532\) −38.3712 29.7222i −1.66360 1.28862i
\(533\) 15.2505i 0.660571i
\(534\) 0 0
\(535\) 0 0
\(536\) 15.5517 + 3.20415i 0.671732 + 0.138398i
\(537\) 0 0
\(538\) 2.59592 + 5.29439i 0.111918 + 0.228258i
\(539\) 27.1427i 1.16912i
\(540\) 0 0
\(541\) 8.77613i 0.377315i 0.982043 + 0.188658i \(0.0604136\pi\)
−0.982043 + 0.188658i \(0.939586\pi\)
\(542\) −6.47724 + 3.17588i −0.278221 + 0.136416i
\(543\) 0 0
\(544\) 22.8990 + 26.7196i 0.981786 + 1.14559i
\(545\) 0 0
\(546\) 0 0
\(547\) 9.16738i 0.391969i −0.980607 0.195984i \(-0.937210\pi\)
0.980607 0.195984i \(-0.0627902\pi\)
\(548\) 12.4414 16.0617i 0.531469 0.686124i
\(549\) 0 0
\(550\) 0 0
\(551\) 21.4582 0.914150
\(552\) 0 0
\(553\) −10.0000 −0.425243
\(554\) 15.1092 + 30.8154i 0.641930 + 1.30922i
\(555\) 0 0
\(556\) −14.4949 11.2277i −0.614721 0.476161i
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −2.75255 −0.116421
\(560\) 0 0
\(561\) 0 0
\(562\) 5.00000 2.45157i 0.210912 0.103413i
\(563\) 43.7642i 1.84444i −0.386667 0.922220i \(-0.626374\pi\)
0.386667 0.922220i \(-0.373626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −0.442484 0.902449i −0.0185990 0.0379327i
\(567\) 0 0
\(568\) 7.75255 37.6279i 0.325290 1.57883i
\(569\) −3.93765 −0.165075 −0.0825375 0.996588i \(-0.526302\pi\)
−0.0825375 + 0.996588i \(0.526302\pi\)
\(570\) 0 0
\(571\) 8.45667i 0.353901i −0.984220 0.176950i \(-0.943377\pi\)
0.984220 0.176950i \(-0.0566232\pi\)
\(572\) 26.2808 33.9284i 1.09886 1.41862i
\(573\) 0 0
\(574\) 17.2474 8.45667i 0.719895 0.352975i
\(575\) 0 0
\(576\) 0 0
\(577\) 15.6969 0.653472 0.326736 0.945116i \(-0.394051\pi\)
0.326736 + 0.945116i \(0.394051\pi\)
\(578\) −27.5506 + 13.5085i −1.14595 + 0.561878i
\(579\) 0 0
\(580\) 0 0
\(581\) 21.0425i 0.872991i
\(582\) 0 0
\(583\) −44.4949 −1.84279
\(584\) 3.93765 19.1118i 0.162941 0.790853i
\(585\) 0 0
\(586\) 10.0000 + 20.3951i 0.413096 + 0.842513i
\(587\) 16.6214i 0.686040i −0.939328 0.343020i \(-0.888550\pi\)
0.939328 0.343020i \(-0.111450\pi\)
\(588\) 0 0
\(589\) 10.1975i 0.420182i
\(590\) 0 0
\(591\) 0 0
\(592\) −24.4949 + 6.32456i −1.00673 + 0.259938i
\(593\) −26.5374 −1.08976 −0.544879 0.838514i \(-0.683425\pi\)
−0.544879 + 0.838514i \(0.683425\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 33.2711 + 25.7717i 1.36284 + 1.05565i
\(597\) 0 0
\(598\) 2.75255 + 5.61385i 0.112560 + 0.229567i
\(599\) 19.2905 0.788187 0.394094 0.919070i \(-0.371059\pi\)
0.394094 + 0.919070i \(0.371059\pi\)
\(600\) 0 0
\(601\) −0.101021 −0.00412071 −0.00206036 0.999998i \(-0.500656\pi\)
−0.00206036 + 0.999998i \(0.500656\pi\)
\(602\) 1.52634 + 3.11299i 0.0622091 + 0.126876i
\(603\) 0 0
\(604\) −15.1237 + 19.5246i −0.615376 + 0.794447i
\(605\) 0 0
\(606\) 0 0
\(607\) 40.6969 1.65184 0.825919 0.563789i \(-0.190657\pi\)
0.825919 + 0.563789i \(0.190657\pi\)
\(608\) −30.2185 + 25.8975i −1.22552 + 1.05028i
\(609\) 0 0
\(610\) 0 0
\(611\) 39.3431i 1.59165i
\(612\) 0 0
\(613\) 31.6228i 1.27723i 0.769526 + 0.638616i \(0.220493\pi\)
−0.769526 + 0.638616i \(0.779507\pi\)
\(614\) −9.20275 18.7691i −0.371393 0.757459i
\(615\) 0 0
\(616\) −52.9444 10.9082i −2.13319 0.439506i
\(617\) 13.0698 0.526170 0.263085 0.964773i \(-0.415260\pi\)
0.263085 + 0.964773i \(0.415260\pi\)
\(618\) 0 0
\(619\) 37.2366i 1.49667i −0.663323 0.748333i \(-0.730854\pi\)
0.663323 0.748333i \(-0.269146\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 20.0000 9.80629i 0.801927 0.393196i
\(623\) −27.1658 −1.08837
\(624\) 0 0
\(625\) 0 0
\(626\) 28.6921 14.0682i 1.14677 0.562277i
\(627\) 0 0
\(628\) −13.8763 10.7485i −0.553724 0.428913i
\(629\) 39.3431i 1.56871i
\(630\) 0 0
\(631\) −18.5505 −0.738484 −0.369242 0.929333i \(-0.620383\pi\)
−0.369242 + 0.929333i \(0.620383\pi\)
\(632\) −1.65462 + 8.03087i −0.0658171 + 0.319451i
\(633\) 0 0
\(634\) −13.1010 26.7196i −0.520308 1.06117i
\(635\) 0 0
\(636\) 0 0
\(637\) 18.9737i 0.751764i
\(638\) 21.4582 10.5213i 0.849538 0.416541i
\(639\) 0 0
\(640\) 0 0
\(641\) −35.0411 −1.38404 −0.692020 0.721879i \(-0.743279\pi\)
−0.692020 + 0.721879i \(0.743279\pi\)
\(642\) 0 0
\(643\) 34.4656i 1.35919i 0.733587 + 0.679595i \(0.237845\pi\)
−0.733587 + 0.679595i \(0.762155\pi\)
\(644\) 4.82262 6.22597i 0.190038 0.245338i
\(645\) 0 0
\(646\) −27.2474 55.5714i −1.07204 2.18643i
\(647\) −24.8827 −0.978242 −0.489121 0.872216i \(-0.662682\pi\)
−0.489121 + 0.872216i \(0.662682\pi\)
\(648\) 0 0
\(649\) −27.5959 −1.08323
\(650\) 0 0
\(651\) 0 0
\(652\) −23.3712 18.1032i −0.915286 0.708977i
\(653\) 6.10018i 0.238719i −0.992851 0.119359i \(-0.961916\pi\)
0.992851 0.119359i \(-0.0380840\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.93765 15.2505i −0.153739 0.595430i
\(657\) 0 0
\(658\) −44.4949 + 21.8165i −1.73459 + 0.850495i
\(659\) 22.7216i 0.885109i −0.896742 0.442555i \(-0.854072\pi\)
0.896742 0.442555i \(-0.145928\pi\)
\(660\) 0 0
\(661\) 37.3084i 1.45113i 0.688154 + 0.725565i \(0.258421\pi\)
−0.688154 + 0.725565i \(0.741579\pi\)
\(662\) −7.87530 16.0617i −0.306082 0.624257i
\(663\) 0 0
\(664\) 16.8990 + 3.48173i 0.655808 + 0.135117i
\(665\) 0 0
\(666\) 0 0
\(667\) 3.48173i 0.134813i
\(668\) 4.19420 5.41469i 0.162279 0.209501i
\(669\) 0 0
\(670\) 0 0
\(671\) −56.4993 −2.18113
\(672\) 0 0
\(673\) 26.8990 1.03688 0.518440 0.855114i \(-0.326513\pi\)
0.518440 + 0.855114i \(0.326513\pi\)
\(674\) 23.8695 11.7036i 0.919419 0.450804i
\(675\) 0 0
\(676\) −2.44949 + 3.16228i −0.0942111 + 0.121626i
\(677\) 35.1736i 1.35183i 0.736979 + 0.675915i \(0.236251\pi\)
−0.736979 + 0.675915i \(0.763749\pi\)
\(678\) 0 0
\(679\) 54.1464 2.07795
\(680\) 0 0
\(681\) 0 0
\(682\) −5.00000 10.1975i −0.191460 0.390484i
\(683\) 33.2429i 1.27200i 0.771687 + 0.636002i \(0.219413\pi\)
−0.771687 + 0.636002i \(0.780587\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −9.20275 + 4.51224i −0.351363 + 0.172278i
\(687\) 0 0
\(688\) 2.75255 0.710706i 0.104940 0.0270954i
\(689\) 31.1034 1.18495
\(690\) 0 0
\(691\) 37.9473i 1.44358i 0.692110 + 0.721792i \(0.256681\pi\)
−0.692110 + 0.721792i \(0.743319\pi\)
\(692\) −3.05268 2.36460i −0.116046 0.0898886i
\(693\) 0 0
\(694\) 10.0000 + 20.3951i 0.379595 + 0.774187i
\(695\) 0 0
\(696\) 0 0
\(697\) 24.4949 0.927810
\(698\) 7.87530 + 16.0617i 0.298085 + 0.607946i
\(699\) 0 0
\(700\) 0 0
\(701\) 41.2738i 1.55889i −0.626472 0.779444i \(-0.715502\pi\)
0.626472 0.779444i \(-0.284498\pi\)
\(702\) 0 0
\(703\) 44.4949 1.67816
\(704\) −17.5205 + 40.7141i −0.660330 + 1.53447i
\(705\) 0 0
\(706\) −0.797959 + 0.391251i −0.0300316 + 0.0147249i
\(707\) 48.7449i 1.83324i
\(708\) 0 0
\(709\) 1.03016i 0.0386885i −0.999813 0.0193442i \(-0.993842\pi\)
0.999813 0.0193442i \(-0.00615785\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −4.49490 + 21.8165i −0.168453 + 0.817607i
\(713\) 1.65462 0.0619659
\(714\) 0 0
\(715\) 0 0
\(716\) −33.2711 25.7717i −1.24340 0.963134i
\(717\) 0 0
\(718\) 27.2474 13.3598i 1.01687 0.498584i
\(719\) 2.16772 0.0808422 0.0404211 0.999183i \(-0.487130\pi\)
0.0404211 + 0.999183i \(0.487130\pi\)
\(720\) 0 0
\(721\) 23.7980 0.886282
\(722\) 38.7222 18.9860i 1.44109 0.706587i
\(723\) 0 0
\(724\) 8.37117 + 6.48428i 0.311112 + 0.240986i
\(725\) 0 0
\(726\) 0 0
\(727\) −43.4495 −1.61145 −0.805726 0.592288i \(-0.798225\pi\)
−0.805726 + 0.592288i \(0.798225\pi\)
\(728\) 37.0099 + 7.62523i 1.37168 + 0.282610i
\(729\) 0 0
\(730\) 0 0
\(731\) 4.42108i 0.163519i
\(732\) 0 0
\(733\) 30.9839i 1.14442i −0.820109 0.572208i \(-0.806087\pi\)
0.820109 0.572208i \(-0.193913\pi\)
\(734\) −29.7760 + 14.5996i −1.09905 + 0.538881i
\(735\) 0 0
\(736\) −4.20204 4.90314i −0.154889 0.180732i
\(737\) −31.1034 −1.14571
\(738\) 0 0
\(739\) 30.9839i 1.13976i −0.821728 0.569880i \(-0.806990\pi\)
0.821728 0.569880i \(-0.193010\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −17.2474 35.1763i −0.633174 1.29136i
\(743\) −6.84910 −0.251269 −0.125635 0.992077i \(-0.540097\pi\)
−0.125635 + 0.992077i \(0.540097\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −15.9942 32.6203i −0.585589 1.19431i
\(747\) 0 0
\(748\) −54.4949 42.2116i −1.99253 1.54341i
\(749\) 1.93069i 0.0705458i
\(750\) 0 0
\(751\) 35.7980 1.30629 0.653143 0.757235i \(-0.273450\pi\)
0.653143 + 0.757235i \(0.273450\pi\)
\(752\) 10.1583 + 39.3431i 0.370436 + 1.43469i
\(753\) 0 0
\(754\) −15.0000 + 7.35472i −0.546268 + 0.267843i
\(755\) 0 0
\(756\) 0 0
\(757\) 12.2579i 0.445519i 0.974873 + 0.222760i \(0.0715065\pi\)
−0.974873 + 0.222760i \(0.928493\pi\)
\(758\) −8.31779 16.9642i −0.302116 0.616167i
\(759\) 0 0
\(760\) 0 0
\(761\) 15.7506 0.570959 0.285480 0.958385i \(-0.407847\pi\)
0.285480 + 0.958385i \(0.407847\pi\)
\(762\) 0 0
\(763\) 56.9928i 2.06328i
\(764\) 16.6356 21.4764i 0.601854 0.776990i
\(765\) 0 0
\(766\) −15.7980 + 7.74597i −0.570803 + 0.279873i
\(767\) 19.2905 0.696539
\(768\) 0 0
\(769\) 5.20204 0.187590 0.0937952 0.995592i \(-0.470100\pi\)
0.0937952 + 0.995592i \(0.470100\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.5732 18.8139i 0.524501 0.677128i
\(773\) 9.15028i 0.329113i −0.986368 0.164556i \(-0.947381\pi\)
0.986368 0.164556i \(-0.0526192\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.95916 43.4843i 0.321615 1.56100i
\(777\) 0 0
\(778\) −21.8990 44.6631i −0.785116 1.60125i
\(779\) 27.7024i 0.992542i
\(780\) 0 0
\(781\) 75.2558i 2.69286i
\(782\) 9.01682 4.42108i 0.322441 0.158097i
\(783\) 0 0
\(784\) −4.89898 18.9737i −0.174964 0.677631i
\(785\) 0 0
\(786\) 0 0
\(787\) 10.5170i 0.374890i 0.982275 + 0.187445i \(0.0600207\pi\)
−0.982275 + 0.187445i \(0.939979\pi\)
\(788\) −36.8110 28.5137i −1.31134 1.01576i
\(789\) 0 0
\(790\) 0 0
\(791\) 42.9164 1.52593
\(792\) 0 0
\(793\) 39.4949 1.40250
\(794\) −1.52634 3.11299i −0.0541679 0.110476i
\(795\) 0 0
\(796\) 26.0227 33.5952i 0.922350 1.19075i
\(797\) 27.1427i 0.961444i −0.876873 0.480722i \(-0.840375\pi\)
0.876873 0.480722i \(-0.159625\pi\)
\(798\) 0 0
\(799\) −63.1918 −2.23557
\(800\) 0 0
\(801\) 0 0
\(802\) −39.4949 + 19.3649i −1.39461 + 0.683799i
\(803\) 38.2237i 1.34888i
\(804\) 0 0
\(805\) 0 0
\(806\) 3.49517 + 7.12842i 0.123112 + 0.251088i
\(807\) 0 0
\(808\) 39.1464 + 8.06542i 1.37717 + 0.283741i
\(809\) 3.93765 0.138440 0.0692202 0.997601i \(-0.477949\pi\)
0.0692202 + 0.997601i \(0.477949\pi\)
\(810\) 0 0
\(811\) 39.4405i 1.38494i 0.721444 + 0.692472i \(0.243479\pi\)
−0.721444 + 0.692472i \(0.756521\pi\)
\(812\) 16.6356 + 12.8859i 0.583794 + 0.452205i
\(813\) 0 0
\(814\) 44.4949 21.8165i 1.55955 0.764668i
\(815\) 0 0
\(816\) 0 0
\(817\) −5.00000 −0.174928
\(818\) 10.0301 4.91788i 0.350693 0.171950i
\(819\) 0 0
\(820\) 0 0
\(821\) 26.0233i 0.908220i −0.890946 0.454110i \(-0.849957\pi\)
0.890946 0.454110i \(-0.150043\pi\)
\(822\) 0 0
\(823\) 34.8434 1.21456 0.607282 0.794487i \(-0.292260\pi\)
0.607282 + 0.794487i \(0.292260\pi\)
\(824\) 3.93765 19.1118i 0.137175 0.665792i
\(825\) 0 0
\(826\) −10.6969 21.8165i −0.372194 0.759093i
\(827\) 11.6407i 0.404786i 0.979304 + 0.202393i \(0.0648718\pi\)
−0.979304 + 0.202393i \(0.935128\pi\)
\(828\) 0 0
\(829\) 37.3084i 1.29578i −0.761736 0.647888i \(-0.775653\pi\)
0.761736 0.647888i \(-0.224347\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 12.2474 28.4605i 0.424604 0.986690i
\(833\) 30.4750 1.05590
\(834\) 0 0
\(835\) 0 0
\(836\) 47.7390 61.6308i 1.65109 2.13155i
\(837\) 0 0
\(838\) −14.8434 30.2732i −0.512756 1.04577i
\(839\) −5.70759 −0.197048 −0.0985239 0.995135i \(-0.531412\pi\)
−0.0985239 + 0.995135i \(0.531412\pi\)
\(840\) 0 0
\(841\) 19.6969 0.679205
\(842\) 15.3528 + 31.3122i 0.529093 + 1.07909i
\(843\) 0 0
\(844\) 3.37117 + 2.61130i 0.116041 + 0.0898846i
\(845\) 0 0
\(846\) 0 0
\(847\) 67.9444 2.33460
\(848\) −31.1034 + 8.03087i −1.06810 + 0.275781i
\(849\) 0 0
\(850\) 0 0
\(851\) 7.21959i 0.247484i
\(852\) 0 0
\(853\) 5.29439i 0.181277i 0.995884 + 0.0906383i \(0.0288907\pi\)
−0.995884 + 0.0906383i \(0.971109\pi\)
\(854\) −21.9007 44.6666i −0.749426 1.52846i
\(855\) 0 0
\(856\) −1.55051 0.319455i −0.0529953 0.0109187i
\(857\) −18.0336 −0.616017 −0.308009 0.951384i \(-0.599663\pi\)
−0.308009 + 0.951384i \(0.599663\pi\)
\(858\) 0 0
\(859\) 43.6330i 1.48874i 0.667769 + 0.744369i \(0.267250\pi\)
−0.667769 + 0.744369i \(0.732750\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 27.2474 13.3598i 0.928052 0.455037i
\(863\) −12.4414 −0.423509 −0.211755 0.977323i \(-0.567918\pi\)
−0.211755 + 0.977323i \(0.567918\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −15.9942 + 7.84219i −0.543505 + 0.266488i
\(867\) 0 0
\(868\) 6.12372 7.90569i 0.207853 0.268337i
\(869\) 16.0617i 0.544857i
\(870\) 0 0
\(871\) 21.7423 0.736711
\(872\) −45.7702 9.43013i −1.54997 0.319344i
\(873\) 0 0
\(874\) 5.00000 + 10.1975i 0.169128 + 0.344937i
\(875\) 0 0
\(876\) 0 0
\(877\) 20.7863i 0.701904i −0.936393 0.350952i \(-0.885858\pi\)
0.936393 0.350952i \(-0.114142\pi\)
\(878\) 21.6441 10.6124i 0.730454 0.358152i
\(879\) 0 0
\(880\) 0 0
\(881\) −19.2905 −0.649913 −0.324956 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(882\) 0 0
\(883\) 30.9121i 1.04027i 0.854083 + 0.520137i \(0.174119\pi\)
−0.854083 + 0.520137i \(0.825881\pi\)
\(884\) 38.0938 + 29.5073i 1.28123 + 0.992438i
\(885\) 0 0
\(886\) 4.49490 + 9.16738i 0.151009 + 0.307984i
\(887\) 53.0747 1.78207 0.891037 0.453930i \(-0.149978\pi\)
0.891037 + 0.453930i \(0.149978\pi\)
\(888\) 0 0
\(889\) 13.1010 0.439394
\(890\) 0 0
\(891\) 0 0
\(892\) −0.426786 + 0.550978i −0.0142898 + 0.0184481i
\(893\) 71.4666i 2.39154i
\(894\) 0 0
\(895\) 0 0
\(896\) −38.9787 + 1.93069i −1.30219 + 0.0644997i
\(897\) 0 0
\(898\) 24.4949 12.0102i 0.817405 0.400786i
\(899\) 4.42108i 0.147451i
\(900\) 0 0
\(901\) 49.9575i 1.66433i
\(902\) 13.5829 + 27.7024i 0.452261 + 0.922389i
\(903\) 0 0
\(904\) 7.10102 34.4656i 0.236176 1.14631i
\(905\) 0 0
\(906\) 0 0
\(907\) 56.2821i 1.86882i 0.356204 + 0.934408i \(0.384071\pi\)
−0.356204 + 0.934408i \(0.615929\pi\)
\(908\) −6.99034 5.41469i −0.231982 0.179693i
\(909\) 0 0
\(910\) 0 0
\(911\) 23.6259 0.782761 0.391381 0.920229i \(-0.371998\pi\)
0.391381 + 0.920229i \(0.371998\pi\)
\(912\) 0 0
\(913\) −33.7980 −1.11855
\(914\) −26.2808 + 12.8859i −0.869292 + 0.426226i
\(915\) 0 0
\(916\) −8.37117 6.48428i −0.276591 0.214247i
\(917\) 36.2930i 1.19850i
\(918\) 0 0
\(919\) −34.6413 −1.14271 −0.571356 0.820702i \(-0.693582\pi\)
−0.571356 + 0.820702i \(0.693582\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −15.6969 32.0140i −0.516951 1.05433i
\(923\) 52.6063i 1.73156i
\(924\) 0 0
\(925\) 0 0
\(926\) −8.76027 + 4.29529i −0.287880 + 0.141152i
\(927\) 0 0
\(928\) 13.1010 11.2277i 0.430062 0.368567i
\(929\) −38.9787 −1.27885 −0.639425 0.768853i \(-0.720828\pi\)
−0.639425 + 0.768853i \(0.720828\pi\)
\(930\) 0 0
\(931\) 34.4656i 1.12956i
\(932\) −29.7054 + 38.3495i −0.973032 + 1.25618i
\(933\) 0 0
\(934\) 20.3485 + 41.5009i 0.665822 + 1.35795i
\(935\) 0 0
\(936\) 0 0
\(937\) −1.20204 −0.0392690 −0.0196345 0.999807i \(-0.506250\pi\)
−0.0196345 + 0.999807i \(0.506250\pi\)
\(938\) −12.0565 24.5894i −0.393660 0.802873i
\(939\) 0 0
\(940\) 0 0
\(941\) 40.4625i 1.31904i −0.751687 0.659520i \(-0.770760\pi\)
0.751687 0.659520i \(-0.229240\pi\)
\(942\) 0 0
\(943\) −4.49490 −0.146374
\(944\) −19.2905 + 4.98078i −0.627852 + 0.162111i
\(945\) 0 0
\(946\) −5.00000 + 2.45157i −0.162564 + 0.0797075i
\(947\) 4.42108i 0.143666i −0.997417 0.0718329i \(-0.977115\pi\)
0.997417 0.0718329i \(-0.0228848\pi\)
\(948\) 0 0
\(949\) 26.7196i 0.867356i
\(950\) 0 0
\(951\) 0 0
\(952\) 12.2474 59.4444i 0.396942 1.92660i
\(953\) 4.56607 0.147909 0.0739547 0.997262i \(-0.476438\pi\)
0.0739547 + 0.997262i \(0.476438\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −35.9261 + 46.3803i −1.16193 + 1.50005i
\(957\) 0 0
\(958\) 10.0000 4.90314i 0.323085 0.158413i
\(959\) −35.0411 −1.13154
\(960\) 0 0
\(961\) −28.8990 −0.932225
\(962\) −31.1034 + 15.2505i −1.00281 + 0.491695i
\(963\) 0 0
\(964\) 29.0227 37.4681i 0.934758 1.20677i
\(965\) 0 0
\(966\) 0 0
\(967\) 8.29286 0.266680 0.133340 0.991070i \(-0.457430\pi\)
0.133340 + 0.991070i \(0.457430\pi\)
\(968\) 11.2422 54.5653i 0.361338 1.75379i
\(969\) 0 0
\(970\) 0 0
\(971\) 45.4433i 1.45834i −0.684331 0.729172i \(-0.739906\pi\)
0.684331 0.729172i \(-0.260094\pi\)
\(972\) 0 0
\(973\) 31.6228i 1.01378i
\(974\) −1.32745 + 0.650868i −0.0425343 + 0.0208552i
\(975\) 0 0
\(976\) −39.4949 + 10.1975i −1.26420 + 0.326415i
\(977\) −33.7842 −1.08085 −0.540427 0.841391i \(-0.681737\pi\)
−0.540427 + 0.841391i \(0.681737\pi\)
\(978\) 0 0
\(979\) 43.6330i 1.39452i
\(980\) 0 0
\(981\) 0 0
\(982\) −0.696938 1.42141i −0.0222402 0.0453591i
\(983\) 22.5997 0.720819 0.360409 0.932794i \(-0.382637\pi\)
0.360409 + 0.932794i \(0.382637\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 11.8130 + 24.0926i 0.376201 + 0.767265i
\(987\) 0 0
\(988\) −33.3712 + 43.0820i −1.06168 + 1.37062i
\(989\) 0.811283i 0.0257973i
\(990\) 0 0
\(991\) 4.75255 0.150970 0.0754849 0.997147i \(-0.475950\pi\)
0.0754849 + 0.997147i \(0.475950\pi\)
\(992\) −5.33572 6.22597i −0.169409 0.197675i
\(993\) 0 0
\(994\) −59.4949 + 29.1712i −1.88706 + 0.925254i
\(995\) 0 0
\(996\) 0 0
\(997\) 12.6491i 0.400601i −0.979734 0.200301i \(-0.935808\pi\)
0.979734 0.200301i \(-0.0641919\pi\)
\(998\) 12.2554 + 24.9951i 0.387939 + 0.791205i
\(999\) 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 1800.2.k.r.901.2 yes 8
3.2 odd 2 inner 1800.2.k.r.901.7 yes 8
4.3 odd 2 7200.2.k.t.3601.8 8
5.2 odd 4 1800.2.d.u.1549.5 16
5.3 odd 4 1800.2.d.u.1549.12 16
5.4 even 2 1800.2.k.s.901.7 yes 8
8.3 odd 2 7200.2.k.t.3601.5 8
8.5 even 2 inner 1800.2.k.r.901.1 8
12.11 even 2 7200.2.k.t.3601.6 8
15.2 even 4 1800.2.d.u.1549.11 16
15.8 even 4 1800.2.d.u.1549.6 16
15.14 odd 2 1800.2.k.s.901.2 yes 8
20.3 even 4 7200.2.d.u.2449.3 16
20.7 even 4 7200.2.d.u.2449.16 16
20.19 odd 2 7200.2.k.q.3601.3 8
24.5 odd 2 inner 1800.2.k.r.901.8 yes 8
24.11 even 2 7200.2.k.t.3601.7 8
40.3 even 4 7200.2.d.u.2449.2 16
40.13 odd 4 1800.2.d.u.1549.8 16
40.19 odd 2 7200.2.k.q.3601.2 8
40.27 even 4 7200.2.d.u.2449.13 16
40.29 even 2 1800.2.k.s.901.8 yes 8
40.37 odd 4 1800.2.d.u.1549.9 16
60.23 odd 4 7200.2.d.u.2449.1 16
60.47 odd 4 7200.2.d.u.2449.14 16
60.59 even 2 7200.2.k.q.3601.1 8
120.29 odd 2 1800.2.k.s.901.1 yes 8
120.53 even 4 1800.2.d.u.1549.10 16
120.59 even 2 7200.2.k.q.3601.4 8
120.77 even 4 1800.2.d.u.1549.7 16
120.83 odd 4 7200.2.d.u.2449.4 16
120.107 odd 4 7200.2.d.u.2449.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1800.2.d.u.1549.5 16 5.2 odd 4
1800.2.d.u.1549.6 16 15.8 even 4
1800.2.d.u.1549.7 16 120.77 even 4
1800.2.d.u.1549.8 16 40.13 odd 4
1800.2.d.u.1549.9 16 40.37 odd 4
1800.2.d.u.1549.10 16 120.53 even 4
1800.2.d.u.1549.11 16 15.2 even 4
1800.2.d.u.1549.12 16 5.3 odd 4
1800.2.k.r.901.1 8 8.5 even 2 inner
1800.2.k.r.901.2 yes 8 1.1 even 1 trivial
1800.2.k.r.901.7 yes 8 3.2 odd 2 inner
1800.2.k.r.901.8 yes 8 24.5 odd 2 inner
1800.2.k.s.901.1 yes 8 120.29 odd 2
1800.2.k.s.901.2 yes 8 15.14 odd 2
1800.2.k.s.901.7 yes 8 5.4 even 2
1800.2.k.s.901.8 yes 8 40.29 even 2
7200.2.d.u.2449.1 16 60.23 odd 4
7200.2.d.u.2449.2 16 40.3 even 4
7200.2.d.u.2449.3 16 20.3 even 4
7200.2.d.u.2449.4 16 120.83 odd 4
7200.2.d.u.2449.13 16 40.27 even 4
7200.2.d.u.2449.14 16 60.47 odd 4
7200.2.d.u.2449.15 16 120.107 odd 4
7200.2.d.u.2449.16 16 20.7 even 4
7200.2.k.q.3601.1 8 60.59 even 2
7200.2.k.q.3601.2 8 40.19 odd 2
7200.2.k.q.3601.3 8 20.19 odd 2
7200.2.k.q.3601.4 8 120.59 even 2
7200.2.k.t.3601.5 8 8.3 odd 2
7200.2.k.t.3601.6 8 12.11 even 2
7200.2.k.t.3601.7 8 24.11 even 2
7200.2.k.t.3601.8 8 4.3 odd 2