Properties

Label 1512.2.j.c.323.9
Level $1512$
Weight $2$
Character 1512.323
Analytic conductor $12.073$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1512,2,Mod(323,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1512.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.j (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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.9
Character \(\chi\) \(=\) 1512.323
Dual form 1512.2.j.c.323.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.985454 - 1.01434i) q^{2} +(-0.0577616 + 1.99917i) q^{4} +0.206991 q^{5} -1.00000i q^{7} +(2.08475 - 1.91150i) q^{8} +O(q^{10})\) \(q+(-0.985454 - 1.01434i) q^{2} +(-0.0577616 + 1.99917i) q^{4} +0.206991 q^{5} -1.00000i q^{7} +(2.08475 - 1.91150i) q^{8} +(-0.203980 - 0.209959i) q^{10} +6.42190i q^{11} -3.52336i q^{13} +(-1.01434 + 0.985454i) q^{14} +(-3.99333 - 0.230950i) q^{16} +3.90616i q^{17} -5.48691 q^{19} +(-0.0119562 + 0.413810i) q^{20} +(6.51397 - 6.32848i) q^{22} -0.880512 q^{23} -4.95715 q^{25} +(-3.57387 + 3.47211i) q^{26} +(1.99917 + 0.0577616i) q^{28} -3.20814 q^{29} -0.0631261i q^{31} +(3.70098 + 4.27817i) q^{32} +(3.96217 - 3.84934i) q^{34} -0.206991i q^{35} -9.91837i q^{37} +(5.40710 + 5.56558i) q^{38} +(0.431525 - 0.395663i) q^{40} -5.94221i q^{41} -1.13960 q^{43} +(-12.8384 - 0.370939i) q^{44} +(0.867704 + 0.893136i) q^{46} +6.81266 q^{47} -1.00000 q^{49} +(4.88505 + 5.02823i) q^{50} +(7.04378 + 0.203515i) q^{52} -12.6324 q^{53} +1.32928i q^{55} +(-1.91150 - 2.08475i) q^{56} +(3.16147 + 3.25414i) q^{58} -4.23338i q^{59} +12.3006i q^{61} +(-0.0640312 + 0.0622079i) q^{62} +(0.692368 - 7.96998i) q^{64} -0.729305i q^{65} -5.74666 q^{67} +(-7.80906 - 0.225626i) q^{68} +(-0.209959 + 0.203980i) q^{70} -10.1795 q^{71} +8.27618 q^{73} +(-10.0606 + 9.77409i) q^{74} +(0.316933 - 10.9692i) q^{76} +6.42190 q^{77} +4.86426i q^{79} +(-0.826584 - 0.0478047i) q^{80} +(-6.02741 + 5.85577i) q^{82} +7.51539i q^{83} +0.808542i q^{85} +(1.12302 + 1.15594i) q^{86} +(12.2754 + 13.3881i) q^{88} +4.44682i q^{89} -3.52336 q^{91} +(0.0508598 - 1.76029i) q^{92} +(-6.71356 - 6.91034i) q^{94} -1.13574 q^{95} -17.3929 q^{97} +(0.985454 + 1.01434i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 32 q + 8 q^{4} - 8 q^{10} - 8 q^{16} - 64 q^{19} + 24 q^{22} - 16 q^{25} - 8 q^{28} + 8 q^{34} - 24 q^{40} - 48 q^{43} - 8 q^{46} - 32 q^{49} - 24 q^{52} - 96 q^{58} - 40 q^{64} + 16 q^{67} - 16 q^{70} - 16 q^{73} + 16 q^{76} + 24 q^{82} + 72 q^{88} + 16 q^{91} - 56 q^{94} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.985454 1.01434i −0.696821 0.717245i
\(3\) 0 0
\(4\) −0.0577616 + 1.99917i −0.0288808 + 0.999583i
\(5\) 0.206991 0.0925694 0.0462847 0.998928i \(-0.485262\pi\)
0.0462847 + 0.998928i \(0.485262\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 2.08475 1.91150i 0.737071 0.675816i
\(9\) 0 0
\(10\) −0.203980 0.209959i −0.0645043 0.0663949i
\(11\) 6.42190i 1.93628i 0.250418 + 0.968138i \(0.419432\pi\)
−0.250418 + 0.968138i \(0.580568\pi\)
\(12\) 0 0
\(13\) 3.52336i 0.977204i −0.872507 0.488602i \(-0.837507\pi\)
0.872507 0.488602i \(-0.162493\pi\)
\(14\) −1.01434 + 0.985454i −0.271093 + 0.263374i
\(15\) 0 0
\(16\) −3.99333 0.230950i −0.998332 0.0577375i
\(17\) 3.90616i 0.947383i 0.880691 + 0.473692i \(0.157079\pi\)
−0.880691 + 0.473692i \(0.842921\pi\)
\(18\) 0 0
\(19\) −5.48691 −1.25878 −0.629392 0.777088i \(-0.716696\pi\)
−0.629392 + 0.777088i \(0.716696\pi\)
\(20\) −0.0119562 + 0.413810i −0.00267348 + 0.0925307i
\(21\) 0 0
\(22\) 6.51397 6.32848i 1.38878 1.34924i
\(23\) −0.880512 −0.183599 −0.0917997 0.995777i \(-0.529262\pi\)
−0.0917997 + 0.995777i \(0.529262\pi\)
\(24\) 0 0
\(25\) −4.95715 −0.991431
\(26\) −3.57387 + 3.47211i −0.700894 + 0.680936i
\(27\) 0 0
\(28\) 1.99917 + 0.0577616i 0.377807 + 0.0109159i
\(29\) −3.20814 −0.595737 −0.297868 0.954607i \(-0.596276\pi\)
−0.297868 + 0.954607i \(0.596276\pi\)
\(30\) 0 0
\(31\) 0.0631261i 0.0113378i −0.999984 0.00566890i \(-0.998196\pi\)
0.999984 0.00566890i \(-0.00180448\pi\)
\(32\) 3.70098 + 4.27817i 0.654247 + 0.756281i
\(33\) 0 0
\(34\) 3.96217 3.84934i 0.679506 0.660157i
\(35\) 0.206991i 0.0349879i
\(36\) 0 0
\(37\) 9.91837i 1.63057i −0.579060 0.815285i \(-0.696580\pi\)
0.579060 0.815285i \(-0.303420\pi\)
\(38\) 5.40710 + 5.56558i 0.877147 + 0.902856i
\(39\) 0 0
\(40\) 0.431525 0.395663i 0.0682302 0.0625598i
\(41\) 5.94221i 0.928017i −0.885831 0.464009i \(-0.846411\pi\)
0.885831 0.464009i \(-0.153589\pi\)
\(42\) 0 0
\(43\) −1.13960 −0.173788 −0.0868938 0.996218i \(-0.527694\pi\)
−0.0868938 + 0.996218i \(0.527694\pi\)
\(44\) −12.8384 0.370939i −1.93547 0.0559212i
\(45\) 0 0
\(46\) 0.867704 + 0.893136i 0.127936 + 0.131686i
\(47\) 6.81266 0.993729 0.496864 0.867828i \(-0.334485\pi\)
0.496864 + 0.867828i \(0.334485\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 4.88505 + 5.02823i 0.690850 + 0.711099i
\(51\) 0 0
\(52\) 7.04378 + 0.203515i 0.976796 + 0.0282224i
\(53\) −12.6324 −1.73520 −0.867598 0.497265i \(-0.834338\pi\)
−0.867598 + 0.497265i \(0.834338\pi\)
\(54\) 0 0
\(55\) 1.32928i 0.179240i
\(56\) −1.91150 2.08475i −0.255434 0.278586i
\(57\) 0 0
\(58\) 3.16147 + 3.25414i 0.415122 + 0.427289i
\(59\) 4.23338i 0.551139i −0.961281 0.275570i \(-0.911134\pi\)
0.961281 0.275570i \(-0.0888664\pi\)
\(60\) 0 0
\(61\) 12.3006i 1.57493i 0.616359 + 0.787465i \(0.288607\pi\)
−0.616359 + 0.787465i \(0.711393\pi\)
\(62\) −0.0640312 + 0.0622079i −0.00813197 + 0.00790041i
\(63\) 0 0
\(64\) 0.692368 7.96998i 0.0865460 0.996248i
\(65\) 0.729305i 0.0904591i
\(66\) 0 0
\(67\) −5.74666 −0.702067 −0.351033 0.936363i \(-0.614170\pi\)
−0.351033 + 0.936363i \(0.614170\pi\)
\(68\) −7.80906 0.225626i −0.946988 0.0273612i
\(69\) 0 0
\(70\) −0.209959 + 0.203980i −0.0250949 + 0.0243803i
\(71\) −10.1795 −1.20809 −0.604043 0.796952i \(-0.706444\pi\)
−0.604043 + 0.796952i \(0.706444\pi\)
\(72\) 0 0
\(73\) 8.27618 0.968653 0.484327 0.874887i \(-0.339065\pi\)
0.484327 + 0.874887i \(0.339065\pi\)
\(74\) −10.0606 + 9.77409i −1.16952 + 1.13622i
\(75\) 0 0
\(76\) 0.316933 10.9692i 0.0363547 1.25826i
\(77\) 6.42190 0.731843
\(78\) 0 0
\(79\) 4.86426i 0.547272i 0.961833 + 0.273636i \(0.0882264\pi\)
−0.961833 + 0.273636i \(0.911774\pi\)
\(80\) −0.826584 0.0478047i −0.0924149 0.00534472i
\(81\) 0 0
\(82\) −6.02741 + 5.85577i −0.665616 + 0.646662i
\(83\) 7.51539i 0.824921i 0.910976 + 0.412460i \(0.135331\pi\)
−0.910976 + 0.412460i \(0.864669\pi\)
\(84\) 0 0
\(85\) 0.808542i 0.0876987i
\(86\) 1.12302 + 1.15594i 0.121099 + 0.124648i
\(87\) 0 0
\(88\) 12.2754 + 13.3881i 1.30857 + 1.42717i
\(89\) 4.44682i 0.471362i 0.971830 + 0.235681i \(0.0757320\pi\)
−0.971830 + 0.235681i \(0.924268\pi\)
\(90\) 0 0
\(91\) −3.52336 −0.369348
\(92\) 0.0508598 1.76029i 0.00530250 0.183523i
\(93\) 0 0
\(94\) −6.71356 6.91034i −0.692451 0.712747i
\(95\) −1.13574 −0.116525
\(96\) 0 0
\(97\) −17.3929 −1.76599 −0.882993 0.469387i \(-0.844475\pi\)
−0.882993 + 0.469387i \(0.844475\pi\)
\(98\) 0.985454 + 1.01434i 0.0995459 + 0.102464i
\(99\) 0 0
\(100\) 0.286333 9.91017i 0.0286333 0.991017i
\(101\) −19.0273 −1.89329 −0.946643 0.322284i \(-0.895550\pi\)
−0.946643 + 0.322284i \(0.895550\pi\)
\(102\) 0 0
\(103\) 10.7903i 1.06320i −0.846995 0.531601i \(-0.821591\pi\)
0.846995 0.531601i \(-0.178409\pi\)
\(104\) −6.73488 7.34532i −0.660410 0.720268i
\(105\) 0 0
\(106\) 12.4487 + 12.8135i 1.20912 + 1.24456i
\(107\) 4.97087i 0.480553i 0.970705 + 0.240276i \(0.0772380\pi\)
−0.970705 + 0.240276i \(0.922762\pi\)
\(108\) 0 0
\(109\) 2.84810i 0.272798i −0.990654 0.136399i \(-0.956447\pi\)
0.990654 0.136399i \(-0.0435530\pi\)
\(110\) 1.34834 1.30994i 0.128559 0.124898i
\(111\) 0 0
\(112\) −0.230950 + 3.99333i −0.0218227 + 0.377334i
\(113\) 2.88561i 0.271455i −0.990746 0.135728i \(-0.956663\pi\)
0.990746 0.135728i \(-0.0433372\pi\)
\(114\) 0 0
\(115\) −0.182258 −0.0169957
\(116\) 0.185307 6.41360i 0.0172054 0.595488i
\(117\) 0 0
\(118\) −4.29408 + 4.17180i −0.395302 + 0.384045i
\(119\) 3.90616 0.358077
\(120\) 0 0
\(121\) −30.2408 −2.74916
\(122\) 12.4770 12.1217i 1.12961 1.09744i
\(123\) 0 0
\(124\) 0.126200 + 0.00364627i 0.0113331 + 0.000327444i
\(125\) −2.06105 −0.184345
\(126\) 0 0
\(127\) 7.23082i 0.641631i 0.947142 + 0.320816i \(0.103957\pi\)
−0.947142 + 0.320816i \(0.896043\pi\)
\(128\) −8.76655 + 7.15175i −0.774861 + 0.632132i
\(129\) 0 0
\(130\) −0.739761 + 0.718696i −0.0648813 + 0.0630338i
\(131\) 12.8026i 1.11857i 0.828975 + 0.559286i \(0.188924\pi\)
−0.828975 + 0.559286i \(0.811076\pi\)
\(132\) 0 0
\(133\) 5.48691i 0.475776i
\(134\) 5.66307 + 5.82906i 0.489215 + 0.503554i
\(135\) 0 0
\(136\) 7.46661 + 8.14337i 0.640257 + 0.698288i
\(137\) 0.363873i 0.0310878i 0.999879 + 0.0155439i \(0.00494798\pi\)
−0.999879 + 0.0155439i \(0.995052\pi\)
\(138\) 0 0
\(139\) −9.32338 −0.790799 −0.395399 0.918509i \(-0.629394\pi\)
−0.395399 + 0.918509i \(0.629394\pi\)
\(140\) 0.413810 + 0.0119562i 0.0349733 + 0.00101048i
\(141\) 0 0
\(142\) 10.0314 + 10.3255i 0.841819 + 0.866493i
\(143\) 22.6266 1.89214
\(144\) 0 0
\(145\) −0.664057 −0.0551470
\(146\) −8.15579 8.39484i −0.674978 0.694762i
\(147\) 0 0
\(148\) 19.8285 + 0.572901i 1.62989 + 0.0470922i
\(149\) −4.82519 −0.395295 −0.197648 0.980273i \(-0.563330\pi\)
−0.197648 + 0.980273i \(0.563330\pi\)
\(150\) 0 0
\(151\) 12.7969i 1.04140i −0.853740 0.520700i \(-0.825671\pi\)
0.853740 0.520700i \(-0.174329\pi\)
\(152\) −11.4388 + 10.4882i −0.927813 + 0.850706i
\(153\) 0 0
\(154\) −6.32848 6.51397i −0.509964 0.524911i
\(155\) 0.0130666i 0.00104953i
\(156\) 0 0
\(157\) 15.5455i 1.24067i 0.784337 + 0.620335i \(0.213003\pi\)
−0.784337 + 0.620335i \(0.786997\pi\)
\(158\) 4.93400 4.79351i 0.392528 0.381351i
\(159\) 0 0
\(160\) 0.766071 + 0.885545i 0.0605632 + 0.0700085i
\(161\) 0.880512i 0.0693940i
\(162\) 0 0
\(163\) 18.4630 1.44613 0.723067 0.690778i \(-0.242732\pi\)
0.723067 + 0.690778i \(0.242732\pi\)
\(164\) 11.8795 + 0.343232i 0.927630 + 0.0268019i
\(165\) 0 0
\(166\) 7.62314 7.40607i 0.591670 0.574822i
\(167\) 12.5320 0.969754 0.484877 0.874582i \(-0.338864\pi\)
0.484877 + 0.874582i \(0.338864\pi\)
\(168\) 0 0
\(169\) 0.585951 0.0450731
\(170\) 0.820134 0.796781i 0.0629014 0.0611103i
\(171\) 0 0
\(172\) 0.0658252 2.27825i 0.00501912 0.173715i
\(173\) 5.49171 0.417527 0.208763 0.977966i \(-0.433056\pi\)
0.208763 + 0.977966i \(0.433056\pi\)
\(174\) 0 0
\(175\) 4.95715i 0.374726i
\(176\) 1.48314 25.6447i 0.111796 1.93305i
\(177\) 0 0
\(178\) 4.51058 4.38214i 0.338082 0.328455i
\(179\) 14.5830i 1.08998i 0.838441 + 0.544992i \(0.183467\pi\)
−0.838441 + 0.544992i \(0.816533\pi\)
\(180\) 0 0
\(181\) 17.2753i 1.28407i 0.766677 + 0.642033i \(0.221909\pi\)
−0.766677 + 0.642033i \(0.778091\pi\)
\(182\) 3.47211 + 3.57387i 0.257370 + 0.264913i
\(183\) 0 0
\(184\) −1.83565 + 1.68309i −0.135326 + 0.124079i
\(185\) 2.05302i 0.150941i
\(186\) 0 0
\(187\) −25.0850 −1.83440
\(188\) −0.393510 + 13.6196i −0.0286997 + 0.993314i
\(189\) 0 0
\(190\) 1.11922 + 1.15203i 0.0811969 + 0.0835768i
\(191\) −7.29828 −0.528085 −0.264042 0.964511i \(-0.585056\pi\)
−0.264042 + 0.964511i \(0.585056\pi\)
\(192\) 0 0
\(193\) 12.1106 0.871742 0.435871 0.900009i \(-0.356440\pi\)
0.435871 + 0.900009i \(0.356440\pi\)
\(194\) 17.1399 + 17.6423i 1.23058 + 1.26664i
\(195\) 0 0
\(196\) 0.0577616 1.99917i 0.00412583 0.142798i
\(197\) −17.3925 −1.23917 −0.619584 0.784931i \(-0.712699\pi\)
−0.619584 + 0.784931i \(0.712699\pi\)
\(198\) 0 0
\(199\) 3.07583i 0.218040i −0.994040 0.109020i \(-0.965229\pi\)
0.994040 0.109020i \(-0.0347712\pi\)
\(200\) −10.3344 + 9.47558i −0.730755 + 0.670025i
\(201\) 0 0
\(202\) 18.7505 + 19.3001i 1.31928 + 1.35795i
\(203\) 3.20814i 0.225167i
\(204\) 0 0
\(205\) 1.22999i 0.0859060i
\(206\) −10.9450 + 10.6334i −0.762576 + 0.740862i
\(207\) 0 0
\(208\) −0.813720 + 14.0699i −0.0564213 + 0.975573i
\(209\) 35.2364i 2.43735i
\(210\) 0 0
\(211\) 9.57154 0.658932 0.329466 0.944167i \(-0.393131\pi\)
0.329466 + 0.944167i \(0.393131\pi\)
\(212\) 0.729669 25.2543i 0.0501139 1.73447i
\(213\) 0 0
\(214\) 5.04214 4.89857i 0.344674 0.334859i
\(215\) −0.235888 −0.0160874
\(216\) 0 0
\(217\) −0.0631261 −0.00428528
\(218\) −2.88893 + 2.80667i −0.195663 + 0.190092i
\(219\) 0 0
\(220\) −2.65745 0.0767812i −0.179165 0.00517659i
\(221\) 13.7628 0.925786
\(222\) 0 0
\(223\) 26.5694i 1.77922i 0.456722 + 0.889610i \(0.349024\pi\)
−0.456722 + 0.889610i \(0.650976\pi\)
\(224\) 4.27817 3.70098i 0.285847 0.247282i
\(225\) 0 0
\(226\) −2.92698 + 2.84363i −0.194700 + 0.189156i
\(227\) 20.7345i 1.37620i 0.725616 + 0.688100i \(0.241555\pi\)
−0.725616 + 0.688100i \(0.758445\pi\)
\(228\) 0 0
\(229\) 19.2625i 1.27290i −0.771316 0.636452i \(-0.780401\pi\)
0.771316 0.636452i \(-0.219599\pi\)
\(230\) 0.179607 + 0.184871i 0.0118429 + 0.0121901i
\(231\) 0 0
\(232\) −6.68817 + 6.13235i −0.439100 + 0.402608i
\(233\) 25.0038i 1.63805i −0.573756 0.819027i \(-0.694514\pi\)
0.573756 0.819027i \(-0.305486\pi\)
\(234\) 0 0
\(235\) 1.41016 0.0919889
\(236\) 8.46323 + 0.244527i 0.550909 + 0.0159173i
\(237\) 0 0
\(238\) −3.84934 3.96217i −0.249516 0.256829i
\(239\) 15.0439 0.973107 0.486553 0.873651i \(-0.338254\pi\)
0.486553 + 0.873651i \(0.338254\pi\)
\(240\) 0 0
\(241\) −3.23274 −0.208239 −0.104120 0.994565i \(-0.533203\pi\)
−0.104120 + 0.994565i \(0.533203\pi\)
\(242\) 29.8009 + 30.6744i 1.91567 + 1.97182i
\(243\) 0 0
\(244\) −24.5909 0.710502i −1.57427 0.0454853i
\(245\) −0.206991 −0.0132242
\(246\) 0 0
\(247\) 19.3324i 1.23009i
\(248\) −0.120665 0.131602i −0.00766226 0.00835675i
\(249\) 0 0
\(250\) 2.03106 + 2.09060i 0.128456 + 0.132221i
\(251\) 10.4258i 0.658071i 0.944318 + 0.329036i \(0.106724\pi\)
−0.944318 + 0.329036i \(0.893276\pi\)
\(252\) 0 0
\(253\) 5.65456i 0.355499i
\(254\) 7.33449 7.12564i 0.460207 0.447102i
\(255\) 0 0
\(256\) 15.8933 + 1.84452i 0.993333 + 0.115282i
\(257\) 8.12452i 0.506794i −0.967362 0.253397i \(-0.918452\pi\)
0.967362 0.253397i \(-0.0815479\pi\)
\(258\) 0 0
\(259\) −9.91837 −0.616297
\(260\) 1.45800 + 0.0421258i 0.0904214 + 0.00261253i
\(261\) 0 0
\(262\) 12.9862 12.6164i 0.802290 0.779444i
\(263\) 14.7639 0.910382 0.455191 0.890394i \(-0.349571\pi\)
0.455191 + 0.890394i \(0.349571\pi\)
\(264\) 0 0
\(265\) −2.61480 −0.160626
\(266\) 5.56558 5.40710i 0.341248 0.331530i
\(267\) 0 0
\(268\) 0.331937 11.4885i 0.0202762 0.701774i
\(269\) 26.8961 1.63989 0.819943 0.572445i \(-0.194005\pi\)
0.819943 + 0.572445i \(0.194005\pi\)
\(270\) 0 0
\(271\) 17.5914i 1.06860i −0.845295 0.534301i \(-0.820575\pi\)
0.845295 0.534301i \(-0.179425\pi\)
\(272\) 0.902128 15.5986i 0.0546996 0.945803i
\(273\) 0 0
\(274\) 0.369090 0.358580i 0.0222976 0.0216626i
\(275\) 31.8343i 1.91968i
\(276\) 0 0
\(277\) 24.8383i 1.49239i −0.665729 0.746194i \(-0.731879\pi\)
0.665729 0.746194i \(-0.268121\pi\)
\(278\) 9.18776 + 9.45706i 0.551045 + 0.567197i
\(279\) 0 0
\(280\) −0.395663 0.431525i −0.0236454 0.0257886i
\(281\) 4.13370i 0.246596i −0.992370 0.123298i \(-0.960653\pi\)
0.992370 0.123298i \(-0.0393471\pi\)
\(282\) 0 0
\(283\) −23.9297 −1.42247 −0.711236 0.702954i \(-0.751864\pi\)
−0.711236 + 0.702954i \(0.751864\pi\)
\(284\) 0.587985 20.3505i 0.0348905 1.20758i
\(285\) 0 0
\(286\) −22.2975 22.9511i −1.31848 1.35712i
\(287\) −5.94221 −0.350758
\(288\) 0 0
\(289\) 1.74190 0.102465
\(290\) 0.654398 + 0.673578i 0.0384276 + 0.0395539i
\(291\) 0 0
\(292\) −0.478045 + 16.5454i −0.0279755 + 0.968249i
\(293\) −12.0225 −0.702360 −0.351180 0.936308i \(-0.614219\pi\)
−0.351180 + 0.936308i \(0.614219\pi\)
\(294\) 0 0
\(295\) 0.876273i 0.0510186i
\(296\) −18.9589 20.6773i −1.10196 1.20184i
\(297\) 0 0
\(298\) 4.75501 + 4.89438i 0.275450 + 0.283524i
\(299\) 3.10236i 0.179414i
\(300\) 0 0
\(301\) 1.13960i 0.0656855i
\(302\) −12.9804 + 12.6108i −0.746939 + 0.725670i
\(303\) 0 0
\(304\) 21.9110 + 1.26720i 1.25668 + 0.0726790i
\(305\) 2.54612i 0.145790i
\(306\) 0 0
\(307\) −5.82060 −0.332199 −0.166100 0.986109i \(-0.553117\pi\)
−0.166100 + 0.986109i \(0.553117\pi\)
\(308\) −0.370939 + 12.8384i −0.0211362 + 0.731538i
\(309\) 0 0
\(310\) −0.0132539 + 0.0128765i −0.000752772 + 0.000731336i
\(311\) 12.7884 0.725161 0.362580 0.931952i \(-0.381896\pi\)
0.362580 + 0.931952i \(0.381896\pi\)
\(312\) 0 0
\(313\) 1.74881 0.0988483 0.0494242 0.998778i \(-0.484261\pi\)
0.0494242 + 0.998778i \(0.484261\pi\)
\(314\) 15.7684 15.3194i 0.889864 0.864525i
\(315\) 0 0
\(316\) −9.72447 0.280968i −0.547044 0.0158057i
\(317\) −30.5938 −1.71832 −0.859159 0.511709i \(-0.829013\pi\)
−0.859159 + 0.511709i \(0.829013\pi\)
\(318\) 0 0
\(319\) 20.6024i 1.15351i
\(320\) 0.143314 1.64972i 0.00801151 0.0922220i
\(321\) 0 0
\(322\) 0.893136 0.867704i 0.0497725 0.0483552i
\(323\) 21.4328i 1.19255i
\(324\) 0 0
\(325\) 17.4658i 0.968830i
\(326\) −18.1944 18.7277i −1.00770 1.03723i
\(327\) 0 0
\(328\) −11.3585 12.3880i −0.627169 0.684014i
\(329\) 6.81266i 0.375594i
\(330\) 0 0
\(331\) −20.7269 −1.13925 −0.569627 0.821904i \(-0.692912\pi\)
−0.569627 + 0.821904i \(0.692912\pi\)
\(332\) −15.0245 0.434101i −0.824577 0.0238244i
\(333\) 0 0
\(334\) −12.3497 12.7117i −0.675745 0.695551i
\(335\) −1.18951 −0.0649899
\(336\) 0 0
\(337\) 19.9716 1.08792 0.543961 0.839110i \(-0.316924\pi\)
0.543961 + 0.839110i \(0.316924\pi\)
\(338\) −0.577427 0.594352i −0.0314079 0.0323285i
\(339\) 0 0
\(340\) −1.61641 0.0467027i −0.0876621 0.00253281i
\(341\) 0.405390 0.0219531
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) −2.37578 + 2.17834i −0.128094 + 0.117448i
\(345\) 0 0
\(346\) −5.41182 5.57045i −0.290942 0.299469i
\(347\) 10.6443i 0.571414i −0.958317 0.285707i \(-0.907772\pi\)
0.958317 0.285707i \(-0.0922284\pi\)
\(348\) 0 0
\(349\) 21.1575i 1.13254i −0.824221 0.566268i \(-0.808387\pi\)
0.824221 0.566268i \(-0.191613\pi\)
\(350\) 5.02823 4.88505i 0.268770 0.261117i
\(351\) 0 0
\(352\) −27.4740 + 23.7673i −1.46437 + 1.26680i
\(353\) 7.85498i 0.418079i −0.977907 0.209039i \(-0.932966\pi\)
0.977907 0.209039i \(-0.0670337\pi\)
\(354\) 0 0
\(355\) −2.10707 −0.111832
\(356\) −8.88993 0.256855i −0.471165 0.0136133i
\(357\) 0 0
\(358\) 14.7921 14.3709i 0.781785 0.759523i
\(359\) −22.0102 −1.16165 −0.580826 0.814028i \(-0.697270\pi\)
−0.580826 + 0.814028i \(0.697270\pi\)
\(360\) 0 0
\(361\) 11.1062 0.584537
\(362\) 17.5230 17.0240i 0.920990 0.894764i
\(363\) 0 0
\(364\) 0.203515 7.04378i 0.0106671 0.369194i
\(365\) 1.71310 0.0896676
\(366\) 0 0
\(367\) 5.61196i 0.292942i 0.989215 + 0.146471i \(0.0467915\pi\)
−0.989215 + 0.146471i \(0.953209\pi\)
\(368\) 3.51617 + 0.203354i 0.183293 + 0.0106006i
\(369\) 0 0
\(370\) −2.08245 + 2.02315i −0.108262 + 0.105179i
\(371\) 12.6324i 0.655843i
\(372\) 0 0
\(373\) 21.5334i 1.11496i −0.830192 0.557478i \(-0.811769\pi\)
0.830192 0.557478i \(-0.188231\pi\)
\(374\) 24.7201 + 25.4446i 1.27825 + 1.31571i
\(375\) 0 0
\(376\) 14.2027 13.0224i 0.732448 0.671578i
\(377\) 11.3034i 0.582156i
\(378\) 0 0
\(379\) 36.2973 1.86447 0.932233 0.361858i \(-0.117857\pi\)
0.932233 + 0.361858i \(0.117857\pi\)
\(380\) 0.0656024 2.27054i 0.00336533 0.116476i
\(381\) 0 0
\(382\) 7.19211 + 7.40292i 0.367981 + 0.378766i
\(383\) −12.4362 −0.635462 −0.317731 0.948181i \(-0.602921\pi\)
−0.317731 + 0.948181i \(0.602921\pi\)
\(384\) 0 0
\(385\) 1.32928 0.0677463
\(386\) −11.9345 12.2843i −0.607448 0.625253i
\(387\) 0 0
\(388\) 1.00464 34.7714i 0.0510031 1.76525i
\(389\) −5.47648 −0.277669 −0.138834 0.990316i \(-0.544336\pi\)
−0.138834 + 0.990316i \(0.544336\pi\)
\(390\) 0 0
\(391\) 3.43942i 0.173939i
\(392\) −2.08475 + 1.91150i −0.105296 + 0.0965451i
\(393\) 0 0
\(394\) 17.1396 + 17.6419i 0.863478 + 0.888787i
\(395\) 1.00686i 0.0506606i
\(396\) 0 0
\(397\) 9.58947i 0.481282i −0.970614 0.240641i \(-0.922642\pi\)
0.970614 0.240641i \(-0.0773576\pi\)
\(398\) −3.11993 + 3.03108i −0.156388 + 0.151935i
\(399\) 0 0
\(400\) 19.7955 + 1.14486i 0.989777 + 0.0572428i
\(401\) 1.76702i 0.0882408i −0.999026 0.0441204i \(-0.985951\pi\)
0.999026 0.0441204i \(-0.0140485\pi\)
\(402\) 0 0
\(403\) −0.222416 −0.0110793
\(404\) 1.09905 38.0387i 0.0546796 1.89250i
\(405\) 0 0
\(406\) 3.25414 3.16147i 0.161500 0.156901i
\(407\) 63.6948 3.15723
\(408\) 0 0
\(409\) −5.87490 −0.290495 −0.145248 0.989395i \(-0.546398\pi\)
−0.145248 + 0.989395i \(0.546398\pi\)
\(410\) −1.24762 + 1.21209i −0.0616156 + 0.0598611i
\(411\) 0 0
\(412\) 21.5716 + 0.623266i 1.06276 + 0.0307061i
\(413\) −4.23338 −0.208311
\(414\) 0 0
\(415\) 1.55562i 0.0763624i
\(416\) 15.0735 13.0399i 0.739041 0.639332i
\(417\) 0 0
\(418\) −35.7416 + 34.7238i −1.74818 + 1.69840i
\(419\) 18.9459i 0.925567i −0.886471 0.462783i \(-0.846851\pi\)
0.886471 0.462783i \(-0.153149\pi\)
\(420\) 0 0
\(421\) 25.2598i 1.23109i 0.788103 + 0.615544i \(0.211064\pi\)
−0.788103 + 0.615544i \(0.788936\pi\)
\(422\) −9.43231 9.70878i −0.459158 0.472616i
\(423\) 0 0
\(424\) −26.3355 + 24.1468i −1.27896 + 1.17267i
\(425\) 19.3634i 0.939265i
\(426\) 0 0
\(427\) 12.3006 0.595268
\(428\) −9.93760 0.287126i −0.480352 0.0138787i
\(429\) 0 0
\(430\) 0.232456 + 0.239270i 0.0112100 + 0.0115386i
\(431\) 37.5774 1.81004 0.905020 0.425369i \(-0.139856\pi\)
0.905020 + 0.425369i \(0.139856\pi\)
\(432\) 0 0
\(433\) 21.1040 1.01419 0.507097 0.861889i \(-0.330719\pi\)
0.507097 + 0.861889i \(0.330719\pi\)
\(434\) 0.0622079 + 0.0640312i 0.00298607 + 0.00307360i
\(435\) 0 0
\(436\) 5.69382 + 0.164511i 0.272684 + 0.00787863i
\(437\) 4.83129 0.231112
\(438\) 0 0
\(439\) 2.88405i 0.137648i −0.997629 0.0688240i \(-0.978075\pi\)
0.997629 0.0688240i \(-0.0219247\pi\)
\(440\) 2.54091 + 2.77121i 0.121133 + 0.132112i
\(441\) 0 0
\(442\) −13.5626 13.9601i −0.645107 0.664016i
\(443\) 34.5978i 1.64379i 0.569640 + 0.821895i \(0.307083\pi\)
−0.569640 + 0.821895i \(0.692917\pi\)
\(444\) 0 0
\(445\) 0.920453i 0.0436337i
\(446\) 26.9504 26.1829i 1.27614 1.23980i
\(447\) 0 0
\(448\) −7.96998 0.692368i −0.376546 0.0327113i
\(449\) 14.8661i 0.701575i 0.936455 + 0.350787i \(0.114086\pi\)
−0.936455 + 0.350787i \(0.885914\pi\)
\(450\) 0 0
\(451\) 38.1603 1.79690
\(452\) 5.76881 + 0.166677i 0.271342 + 0.00783985i
\(453\) 0 0
\(454\) 21.0318 20.4329i 0.987072 0.958965i
\(455\) −0.729305 −0.0341903
\(456\) 0 0
\(457\) 11.0037 0.514730 0.257365 0.966314i \(-0.417146\pi\)
0.257365 + 0.966314i \(0.417146\pi\)
\(458\) −19.5387 + 18.9823i −0.912984 + 0.886987i
\(459\) 0 0
\(460\) 0.0105275 0.364365i 0.000490849 0.0169886i
\(461\) −32.9687 −1.53551 −0.767753 0.640746i \(-0.778625\pi\)
−0.767753 + 0.640746i \(0.778625\pi\)
\(462\) 0 0
\(463\) 12.3348i 0.573245i 0.958044 + 0.286622i \(0.0925325\pi\)
−0.958044 + 0.286622i \(0.907468\pi\)
\(464\) 12.8112 + 0.740920i 0.594743 + 0.0343964i
\(465\) 0 0
\(466\) −25.3623 + 24.6401i −1.17489 + 1.14143i
\(467\) 33.5264i 1.55142i 0.631091 + 0.775709i \(0.282607\pi\)
−0.631091 + 0.775709i \(0.717393\pi\)
\(468\) 0 0
\(469\) 5.74666i 0.265356i
\(470\) −1.38965 1.43038i −0.0640998 0.0659785i
\(471\) 0 0
\(472\) −8.09209 8.82554i −0.372468 0.406228i
\(473\) 7.31840i 0.336501i
\(474\) 0 0
\(475\) 27.1995 1.24800
\(476\) −0.225626 + 7.80906i −0.0103416 + 0.357928i
\(477\) 0 0
\(478\) −14.8250 15.2596i −0.678081 0.697956i
\(479\) 9.65791 0.441281 0.220641 0.975355i \(-0.429185\pi\)
0.220641 + 0.975355i \(0.429185\pi\)
\(480\) 0 0
\(481\) −34.9460 −1.59340
\(482\) 3.18572 + 3.27909i 0.145106 + 0.149359i
\(483\) 0 0
\(484\) 1.74676 60.4563i 0.0793980 2.74802i
\(485\) −3.60019 −0.163476
\(486\) 0 0
\(487\) 25.1008i 1.13742i −0.822537 0.568712i \(-0.807442\pi\)
0.822537 0.568712i \(-0.192558\pi\)
\(488\) 23.5125 + 25.6437i 1.06436 + 1.16083i
\(489\) 0 0
\(490\) 0.203980 + 0.209959i 0.00921490 + 0.00948499i
\(491\) 10.5524i 0.476224i −0.971238 0.238112i \(-0.923471\pi\)
0.971238 0.238112i \(-0.0765285\pi\)
\(492\) 0 0
\(493\) 12.5315i 0.564391i
\(494\) 19.6095 19.0511i 0.882275 0.857151i
\(495\) 0 0
\(496\) −0.0145790 + 0.252083i −0.000654616 + 0.0113189i
\(497\) 10.1795i 0.456613i
\(498\) 0 0
\(499\) −11.6619 −0.522059 −0.261029 0.965331i \(-0.584062\pi\)
−0.261029 + 0.965331i \(0.584062\pi\)
\(500\) 0.119049 4.12037i 0.00532405 0.184269i
\(501\) 0 0
\(502\) 10.5753 10.2742i 0.471998 0.458558i
\(503\) −29.0202 −1.29394 −0.646972 0.762513i \(-0.723965\pi\)
−0.646972 + 0.762513i \(0.723965\pi\)
\(504\) 0 0
\(505\) −3.93849 −0.175260
\(506\) −5.73563 + 5.57230i −0.254980 + 0.247719i
\(507\) 0 0
\(508\) −14.4556 0.417664i −0.641364 0.0185308i
\(509\) 22.0894 0.979096 0.489548 0.871976i \(-0.337162\pi\)
0.489548 + 0.871976i \(0.337162\pi\)
\(510\) 0 0
\(511\) 8.27618i 0.366116i
\(512\) −13.7912 17.9389i −0.609489 0.792794i
\(513\) 0 0
\(514\) −8.24101 + 8.00634i −0.363495 + 0.353145i
\(515\) 2.23350i 0.0984199i
\(516\) 0 0
\(517\) 43.7502i 1.92413i
\(518\) 9.77409 + 10.0606i 0.429449 + 0.442036i
\(519\) 0 0
\(520\) −1.39406 1.52042i −0.0611337 0.0666747i
\(521\) 14.5460i 0.637273i 0.947877 + 0.318636i \(0.103225\pi\)
−0.947877 + 0.318636i \(0.896775\pi\)
\(522\) 0 0
\(523\) 31.6350 1.38330 0.691651 0.722232i \(-0.256884\pi\)
0.691651 + 0.722232i \(0.256884\pi\)
\(524\) −25.5946 0.739501i −1.11810 0.0323052i
\(525\) 0 0
\(526\) −14.5491 14.9756i −0.634373 0.652967i
\(527\) 0.246581 0.0107412
\(528\) 0 0
\(529\) −22.2247 −0.966291
\(530\) 2.57677 + 2.65229i 0.111928 + 0.115208i
\(531\) 0 0
\(532\) −10.9692 0.316933i −0.475577 0.0137408i
\(533\) −20.9365 −0.906862
\(534\) 0 0
\(535\) 1.02893i 0.0444844i
\(536\) −11.9804 + 10.9847i −0.517473 + 0.474468i
\(537\) 0 0
\(538\) −26.5049 27.2818i −1.14271 1.17620i
\(539\) 6.42190i 0.276611i
\(540\) 0 0
\(541\) 22.7160i 0.976639i 0.872665 + 0.488319i \(0.162390\pi\)
−0.872665 + 0.488319i \(0.837610\pi\)
\(542\) −17.8436 + 17.3355i −0.766449 + 0.744624i
\(543\) 0 0
\(544\) −16.7112 + 14.4566i −0.716488 + 0.619822i
\(545\) 0.589532i 0.0252528i
\(546\) 0 0
\(547\) −22.2601 −0.951774 −0.475887 0.879506i \(-0.657873\pi\)
−0.475887 + 0.879506i \(0.657873\pi\)
\(548\) −0.727443 0.0210179i −0.0310748 0.000897840i
\(549\) 0 0
\(550\) −32.2908 + 31.3713i −1.37688 + 1.33768i
\(551\) 17.6028 0.749904
\(552\) 0 0
\(553\) 4.86426 0.206849
\(554\) −25.1944 + 24.4770i −1.07041 + 1.03993i
\(555\) 0 0
\(556\) 0.538533 18.6390i 0.0228389 0.790469i
\(557\) −2.99670 −0.126974 −0.0634872 0.997983i \(-0.520222\pi\)
−0.0634872 + 0.997983i \(0.520222\pi\)
\(558\) 0 0
\(559\) 4.01522i 0.169826i
\(560\) −0.0478047 + 0.826584i −0.00202012 + 0.0349296i
\(561\) 0 0
\(562\) −4.19297 + 4.07357i −0.176870 + 0.171833i
\(563\) 13.4913i 0.568589i 0.958737 + 0.284295i \(0.0917594\pi\)
−0.958737 + 0.284295i \(0.908241\pi\)
\(564\) 0 0
\(565\) 0.597296i 0.0251284i
\(566\) 23.5816 + 24.2728i 0.991208 + 1.02026i
\(567\) 0 0
\(568\) −21.2217 + 19.4581i −0.890444 + 0.816443i
\(569\) 40.7445i 1.70810i −0.520191 0.854050i \(-0.674139\pi\)
0.520191 0.854050i \(-0.325861\pi\)
\(570\) 0 0
\(571\) 18.0110 0.753739 0.376869 0.926266i \(-0.377001\pi\)
0.376869 + 0.926266i \(0.377001\pi\)
\(572\) −1.30695 + 45.2344i −0.0546464 + 1.89135i
\(573\) 0 0
\(574\) 5.85577 + 6.02741i 0.244415 + 0.251579i
\(575\) 4.36483 0.182026
\(576\) 0 0
\(577\) 29.3495 1.22184 0.610919 0.791693i \(-0.290800\pi\)
0.610919 + 0.791693i \(0.290800\pi\)
\(578\) −1.71656 1.76688i −0.0713997 0.0734924i
\(579\) 0 0
\(580\) 0.0383570 1.32756i 0.00159269 0.0551240i
\(581\) 7.51539 0.311791
\(582\) 0 0
\(583\) 81.1242i 3.35982i
\(584\) 17.2538 15.8199i 0.713966 0.654631i
\(585\) 0 0
\(586\) 11.8476 + 12.1948i 0.489419 + 0.503764i
\(587\) 11.1732i 0.461168i 0.973052 + 0.230584i \(0.0740637\pi\)
−0.973052 + 0.230584i \(0.925936\pi\)
\(588\) 0 0
\(589\) 0.346368i 0.0142718i
\(590\) −0.888837 + 0.863527i −0.0365928 + 0.0355508i
\(591\) 0 0
\(592\) −2.29065 + 39.6073i −0.0941450 + 1.62785i
\(593\) 29.9792i 1.23110i −0.788099 0.615549i \(-0.788934\pi\)
0.788099 0.615549i \(-0.211066\pi\)
\(594\) 0 0
\(595\) 0.808542 0.0331470
\(596\) 0.278711 9.64636i 0.0114164 0.395130i
\(597\) 0 0
\(598\) 3.14684 3.05723i 0.128684 0.125019i
\(599\) 6.38388 0.260838 0.130419 0.991459i \(-0.458368\pi\)
0.130419 + 0.991459i \(0.458368\pi\)
\(600\) 0 0
\(601\) −23.5445 −0.960402 −0.480201 0.877159i \(-0.659436\pi\)
−0.480201 + 0.877159i \(0.659436\pi\)
\(602\) 1.15594 1.12302i 0.0471126 0.0457710i
\(603\) 0 0
\(604\) 25.5832 + 0.739172i 1.04097 + 0.0300765i
\(605\) −6.25958 −0.254488
\(606\) 0 0
\(607\) 33.9599i 1.37839i 0.724577 + 0.689194i \(0.242035\pi\)
−0.724577 + 0.689194i \(0.757965\pi\)
\(608\) −20.3069 23.4740i −0.823555 0.951995i
\(609\) 0 0
\(610\) 2.58262 2.50908i 0.104567 0.101590i
\(611\) 24.0034i 0.971076i
\(612\) 0 0
\(613\) 20.6042i 0.832197i −0.909320 0.416098i \(-0.863397\pi\)
0.909320 0.416098i \(-0.136603\pi\)
\(614\) 5.73593 + 5.90405i 0.231483 + 0.238268i
\(615\) 0 0
\(616\) 13.3881 12.2754i 0.539420 0.494591i
\(617\) 37.4489i 1.50764i 0.657083 + 0.753818i \(0.271790\pi\)
−0.657083 + 0.753818i \(0.728210\pi\)
\(618\) 0 0
\(619\) 14.2766 0.573825 0.286913 0.957957i \(-0.407371\pi\)
0.286913 + 0.957957i \(0.407371\pi\)
\(620\) 0.0261222 0.000754746i 0.00104909 3.03113e-5i
\(621\) 0 0
\(622\) −12.6023 12.9717i −0.505307 0.520118i
\(623\) 4.44682 0.178158
\(624\) 0 0
\(625\) 24.3592 0.974366
\(626\) −1.72337 1.77388i −0.0688796 0.0708985i
\(627\) 0 0
\(628\) −31.0781 0.897936i −1.24015 0.0358315i
\(629\) 38.7427 1.54477
\(630\) 0 0
\(631\) 7.00448i 0.278844i 0.990233 + 0.139422i \(0.0445244\pi\)
−0.990233 + 0.139422i \(0.955476\pi\)
\(632\) 9.29802 + 10.1408i 0.369855 + 0.403378i
\(633\) 0 0
\(634\) 30.1488 + 31.0324i 1.19736 + 1.23245i
\(635\) 1.49672i 0.0593954i
\(636\) 0 0
\(637\) 3.52336i 0.139601i
\(638\) −20.8977 + 20.3027i −0.827350 + 0.803790i
\(639\) 0 0
\(640\) −1.81460 + 1.48035i −0.0717284 + 0.0585160i
\(641\) 9.23092i 0.364599i 0.983243 + 0.182300i \(0.0583541\pi\)
−0.983243 + 0.182300i \(0.941646\pi\)
\(642\) 0 0
\(643\) 1.48603 0.0586031 0.0293016 0.999571i \(-0.490672\pi\)
0.0293016 + 0.999571i \(0.490672\pi\)
\(644\) −1.76029 0.0508598i −0.0693651 0.00200416i
\(645\) 0 0
\(646\) −21.7401 + 21.1210i −0.855351 + 0.830995i
\(647\) −16.2654 −0.639457 −0.319729 0.947509i \(-0.603592\pi\)
−0.319729 + 0.947509i \(0.603592\pi\)
\(648\) 0 0
\(649\) 27.1863 1.06716
\(650\) 17.7162 17.2118i 0.694888 0.675101i
\(651\) 0 0
\(652\) −1.06645 + 36.9106i −0.0417655 + 1.44553i
\(653\) −6.92310 −0.270922 −0.135461 0.990783i \(-0.543252\pi\)
−0.135461 + 0.990783i \(0.543252\pi\)
\(654\) 0 0
\(655\) 2.65003i 0.103545i
\(656\) −1.37235 + 23.7292i −0.0535814 + 0.926469i
\(657\) 0 0
\(658\) −6.91034 + 6.71356i −0.269393 + 0.261722i
\(659\) 11.3057i 0.440407i −0.975454 0.220203i \(-0.929328\pi\)
0.975454 0.220203i \(-0.0706721\pi\)
\(660\) 0 0
\(661\) 18.9877i 0.738536i 0.929323 + 0.369268i \(0.120392\pi\)
−0.929323 + 0.369268i \(0.879608\pi\)
\(662\) 20.4254 + 21.0241i 0.793856 + 0.817124i
\(663\) 0 0
\(664\) 14.3656 + 15.6677i 0.557495 + 0.608025i
\(665\) 1.13574i 0.0440422i
\(666\) 0 0
\(667\) 2.82481 0.109377
\(668\) −0.723867 + 25.0535i −0.0280073 + 0.969349i
\(669\) 0 0
\(670\) 1.17221 + 1.20656i 0.0452863 + 0.0466137i
\(671\) −78.9932 −3.04950
\(672\) 0 0
\(673\) 17.6164 0.679061 0.339531 0.940595i \(-0.389732\pi\)
0.339531 + 0.940595i \(0.389732\pi\)
\(674\) −19.6811 20.2580i −0.758088 0.780307i
\(675\) 0 0
\(676\) −0.0338455 + 1.17141i −0.00130175 + 0.0450543i
\(677\) 7.42579 0.285396 0.142698 0.989766i \(-0.454422\pi\)
0.142698 + 0.989766i \(0.454422\pi\)
\(678\) 0 0
\(679\) 17.3929i 0.667480i
\(680\) 1.54552 + 1.68561i 0.0592681 + 0.0646401i
\(681\) 0 0
\(682\) −0.399493 0.411202i −0.0152974 0.0157457i
\(683\) 22.8832i 0.875600i −0.899072 0.437800i \(-0.855758\pi\)
0.899072 0.437800i \(-0.144242\pi\)
\(684\) 0 0
\(685\) 0.0753186i 0.00287778i
\(686\) 1.01434 0.985454i 0.0387276 0.0376248i
\(687\) 0 0
\(688\) 4.55080 + 0.263191i 0.173498 + 0.0100341i
\(689\) 44.5086i 1.69564i
\(690\) 0 0
\(691\) 33.6439 1.27987 0.639936 0.768428i \(-0.278961\pi\)
0.639936 + 0.768428i \(0.278961\pi\)
\(692\) −0.317210 + 10.9788i −0.0120585 + 0.417353i
\(693\) 0 0
\(694\) −10.7969 + 10.4894i −0.409844 + 0.398173i
\(695\) −1.92986 −0.0732038
\(696\) 0 0
\(697\) 23.2112 0.879188
\(698\) −21.4609 + 20.8498i −0.812306 + 0.789175i
\(699\) 0 0
\(700\) −9.91017 0.286333i −0.374569 0.0108224i
\(701\) −35.4474 −1.33883 −0.669414 0.742890i \(-0.733455\pi\)
−0.669414 + 0.742890i \(0.733455\pi\)
\(702\) 0 0
\(703\) 54.4212i 2.05253i
\(704\) 51.1824 + 4.44632i 1.92901 + 0.167577i
\(705\) 0 0
\(706\) −7.96760 + 7.74072i −0.299865 + 0.291326i
\(707\) 19.0273i 0.715595i
\(708\) 0 0
\(709\) 8.71118i 0.327155i 0.986530 + 0.163578i \(0.0523034\pi\)
−0.986530 + 0.163578i \(0.947697\pi\)
\(710\) 2.07642 + 2.13728i 0.0779267 + 0.0802107i
\(711\) 0 0
\(712\) 8.50008 + 9.27051i 0.318554 + 0.347427i
\(713\) 0.0555833i 0.00208161i
\(714\) 0 0
\(715\) 4.68352 0.175154
\(716\) −29.1538 0.842337i −1.08953 0.0314796i
\(717\) 0 0
\(718\) 21.6900 + 22.3257i 0.809463 + 0.833189i
\(719\) 0.821108 0.0306222 0.0153111 0.999883i \(-0.495126\pi\)
0.0153111 + 0.999883i \(0.495126\pi\)
\(720\) 0 0
\(721\) −10.7903 −0.401853
\(722\) −10.9446 11.2654i −0.407318 0.419256i
\(723\) 0 0
\(724\) −34.5363 0.997851i −1.28353 0.0370848i
\(725\) 15.9032 0.590632
\(726\) 0 0
\(727\) 28.0509i 1.04035i 0.854059 + 0.520176i \(0.174134\pi\)
−0.854059 + 0.520176i \(0.825866\pi\)
\(728\) −7.34532 + 6.73488i −0.272236 + 0.249611i
\(729\) 0 0
\(730\) −1.68818 1.73766i −0.0624823 0.0643136i
\(731\) 4.45147i 0.164643i
\(732\) 0 0
\(733\) 21.9677i 0.811395i −0.914007 0.405697i \(-0.867029\pi\)
0.914007 0.405697i \(-0.132971\pi\)
\(734\) 5.69242 5.53032i 0.210111 0.204128i
\(735\) 0 0
\(736\) −3.25875 3.76698i −0.120119 0.138853i
\(737\) 36.9045i 1.35939i
\(738\) 0 0
\(739\) −18.6044 −0.684374 −0.342187 0.939632i \(-0.611168\pi\)
−0.342187 + 0.939632i \(0.611168\pi\)
\(740\) 4.10432 + 0.118586i 0.150878 + 0.00435929i
\(741\) 0 0
\(742\) 12.8135 12.4487i 0.470400 0.457005i
\(743\) −29.0849 −1.06702 −0.533512 0.845793i \(-0.679128\pi\)
−0.533512 + 0.845793i \(0.679128\pi\)
\(744\) 0 0
\(745\) −0.998774 −0.0365922
\(746\) −21.8421 + 21.2201i −0.799696 + 0.776925i
\(747\) 0 0
\(748\) 1.44895 50.1490i 0.0529788 1.83363i
\(749\) 4.97087 0.181632
\(750\) 0 0
\(751\) 14.4693i 0.527991i −0.964524 0.263995i \(-0.914960\pi\)
0.964524 0.263995i \(-0.0850403\pi\)
\(752\) −27.2052 1.57338i −0.992071 0.0573754i
\(753\) 0 0
\(754\) 11.4655 11.1390i 0.417549 0.405659i
\(755\) 2.64886i 0.0964018i
\(756\) 0 0
\(757\) 13.6988i 0.497891i −0.968517 0.248946i \(-0.919916\pi\)
0.968517 0.248946i \(-0.0800840\pi\)
\(758\) −35.7693 36.8177i −1.29920 1.33728i
\(759\) 0 0
\(760\) −2.36774 + 2.17097i −0.0858870 + 0.0787493i
\(761\) 46.5579i 1.68772i 0.536562 + 0.843861i \(0.319723\pi\)
−0.536562 + 0.843861i \(0.680277\pi\)
\(762\) 0 0
\(763\) −2.84810 −0.103108
\(764\) 0.421560 14.5905i 0.0152515 0.527864i
\(765\) 0 0
\(766\) 12.2553 + 12.6145i 0.442804 + 0.455782i
\(767\) −14.9157 −0.538575
\(768\) 0 0
\(769\) 50.6427 1.82622 0.913111 0.407711i \(-0.133673\pi\)
0.913111 + 0.407711i \(0.133673\pi\)
\(770\) −1.30994 1.34834i −0.0472070 0.0485907i
\(771\) 0 0
\(772\) −0.699530 + 24.2112i −0.0251766 + 0.871379i
\(773\) 6.66928 0.239877 0.119939 0.992781i \(-0.461730\pi\)
0.119939 + 0.992781i \(0.461730\pi\)
\(774\) 0 0
\(775\) 0.312926i 0.0112406i
\(776\) −36.2599 + 33.2465i −1.30166 + 1.19348i
\(777\) 0 0
\(778\) 5.39682 + 5.55500i 0.193485 + 0.199156i
\(779\) 32.6044i 1.16817i
\(780\) 0 0
\(781\) 65.3718i 2.33919i
\(782\) −3.48873 + 3.38939i −0.124757 + 0.121204i
\(783\) 0 0
\(784\) 3.99333 + 0.230950i 0.142619 + 0.00824822i
\(785\) 3.21779i 0.114848i
\(786\) 0 0
\(787\) −0.759732 −0.0270815 −0.0135408 0.999908i \(-0.504310\pi\)
−0.0135408 + 0.999908i \(0.504310\pi\)
\(788\) 1.00462 34.7706i 0.0357882 1.23865i
\(789\) 0 0
\(790\) 1.02130 0.992214i 0.0363361 0.0353014i
\(791\) −2.88561 −0.102600
\(792\) 0 0
\(793\) 43.3394 1.53903
\(794\) −9.72696 + 9.44998i −0.345197 + 0.335367i
\(795\) 0 0
\(796\) 6.14909 + 0.177665i 0.217949 + 0.00629716i
\(797\) 26.1758 0.927194 0.463597 0.886046i \(-0.346559\pi\)
0.463597 + 0.886046i \(0.346559\pi\)
\(798\) 0 0
\(799\) 26.6114i 0.941442i
\(800\) −18.3463 21.2076i −0.648640 0.749801i
\(801\) 0 0
\(802\) −1.79236 + 1.74132i −0.0632903 + 0.0614880i
\(803\) 53.1488i 1.87558i
\(804\) 0 0
\(805\) 0.182258i 0.00642376i
\(806\) 0.219181 + 0.225605i 0.00772031 + 0.00794659i
\(807\) 0 0
\(808\) −39.6672 + 36.3706i −1.39549 + 1.27951i
\(809\) 14.6876i 0.516388i 0.966093 + 0.258194i \(0.0831273\pi\)
−0.966093 + 0.258194i \(0.916873\pi\)
\(810\) 0 0
\(811\) −4.13912 −0.145344 −0.0726721 0.997356i \(-0.523153\pi\)
−0.0726721 + 0.997356i \(0.523153\pi\)
\(812\) −6.41360 0.185307i −0.225073 0.00650301i
\(813\) 0 0
\(814\) −62.7682 64.6080i −2.20003 2.26451i
\(815\) 3.82168 0.133868
\(816\) 0 0
\(817\) 6.25289 0.218761
\(818\) 5.78944 + 5.95913i 0.202423 + 0.208356i
\(819\) 0 0
\(820\) 2.45895 + 0.0710460i 0.0858701 + 0.00248103i
\(821\) 30.7760 1.07409 0.537045 0.843554i \(-0.319540\pi\)
0.537045 + 0.843554i \(0.319540\pi\)
\(822\) 0 0
\(823\) 28.3750i 0.989090i −0.869152 0.494545i \(-0.835335\pi\)
0.869152 0.494545i \(-0.164665\pi\)
\(824\) −20.6257 22.4951i −0.718529 0.783655i
\(825\) 0 0
\(826\) 4.17180 + 4.29408i 0.145155 + 0.149410i
\(827\) 26.2522i 0.912880i −0.889754 0.456440i \(-0.849124\pi\)
0.889754 0.456440i \(-0.150876\pi\)
\(828\) 0 0
\(829\) 8.40833i 0.292033i −0.989282 0.146017i \(-0.953355\pi\)
0.989282 0.146017i \(-0.0466453\pi\)
\(830\) 1.57792 1.53299i 0.0547706 0.0532109i
\(831\) 0 0
\(832\) −28.0811 2.43946i −0.973537 0.0845731i
\(833\) 3.90616i 0.135340i
\(834\) 0 0
\(835\) 2.59401 0.0897695
\(836\) 70.4434 + 2.03531i 2.43634 + 0.0703927i
\(837\) 0 0
\(838\) −19.2175 + 18.6703i −0.663858 + 0.644954i
\(839\) −10.3099 −0.355936 −0.177968 0.984036i \(-0.556952\pi\)
−0.177968 + 0.984036i \(0.556952\pi\)
\(840\) 0 0
\(841\) −18.7078 −0.645098
\(842\) 25.6220 24.8924i 0.882991 0.857848i
\(843\) 0 0
\(844\) −0.552868 + 19.1351i −0.0190305 + 0.658657i
\(845\) 0.121287 0.00417239
\(846\) 0 0
\(847\) 30.2408i 1.03909i
\(848\) 50.4454 + 2.91746i 1.73230 + 0.100186i
\(849\) 0 0
\(850\) −19.6411 + 19.0818i −0.673683 + 0.654500i
\(851\) 8.73324i 0.299372i
\(852\) 0 0
\(853\) 42.6992i 1.46199i 0.682382 + 0.730996i \(0.260944\pi\)
−0.682382 + 0.730996i \(0.739056\pi\)
\(854\) −12.1217 12.4770i −0.414795 0.426953i
\(855\) 0 0
\(856\) 9.50180 + 10.3630i 0.324765 + 0.354201i
\(857\) 5.06990i 0.173185i 0.996244 + 0.0865923i \(0.0275977\pi\)
−0.996244 + 0.0865923i \(0.972402\pi\)
\(858\) 0 0
\(859\) 10.9220 0.372654 0.186327 0.982488i \(-0.440342\pi\)
0.186327 + 0.982488i \(0.440342\pi\)
\(860\) 0.0136252 0.471578i 0.000464617 0.0160807i
\(861\) 0 0
\(862\) −37.0308 38.1162i −1.26127 1.29824i
\(863\) −38.1030 −1.29704 −0.648520 0.761197i \(-0.724612\pi\)
−0.648520 + 0.761197i \(0.724612\pi\)
\(864\) 0 0
\(865\) 1.13674 0.0386502
\(866\) −20.7970 21.4066i −0.706711 0.727425i
\(867\) 0 0
\(868\) 0.00364627 0.126200i 0.000123762 0.00428349i
\(869\) −31.2378 −1.05967
\(870\) 0 0
\(871\) 20.2476i 0.686062i
\(872\) −5.44413 5.93757i −0.184361 0.201072i
\(873\) 0 0
\(874\) −4.76101 4.90056i −0.161044 0.165764i
\(875\) 2.06105i 0.0696760i
\(876\) 0 0
\(877\) 26.1674i 0.883610i 0.897111 + 0.441805i \(0.145662\pi\)
−0.897111 + 0.441805i \(0.854338\pi\)
\(878\) −2.92540 + 2.84209i −0.0987273 + 0.0959160i
\(879\) 0 0
\(880\) 0.306997 5.30824i 0.0103489 0.178941i
\(881\) 29.2903i 0.986817i 0.869798 + 0.493408i \(0.164249\pi\)
−0.869798 + 0.493408i \(0.835751\pi\)
\(882\) 0 0
\(883\) −49.0887 −1.65197 −0.825983 0.563696i \(-0.809379\pi\)
−0.825983 + 0.563696i \(0.809379\pi\)
\(884\) −0.794962 + 27.5141i −0.0267375 + 0.925400i
\(885\) 0 0
\(886\) 35.0938 34.0945i 1.17900 1.14543i
\(887\) −20.0617 −0.673606 −0.336803 0.941575i \(-0.609346\pi\)
−0.336803 + 0.941575i \(0.609346\pi\)
\(888\) 0 0
\(889\) 7.23082 0.242514
\(890\) 0.933651 0.907064i 0.0312960 0.0304049i
\(891\) 0 0
\(892\) −53.1167 1.53469i −1.77848 0.0513853i
\(893\) −37.3805 −1.25089
\(894\) 0 0
\(895\) 3.01855i 0.100899i
\(896\) 7.15175 + 8.76655i 0.238923 + 0.292870i
\(897\) 0 0
\(898\) 15.0792 14.6499i 0.503201 0.488872i
\(899\) 0.202518i 0.00675434i
\(900\) 0 0
\(901\) 49.3443i 1.64390i
\(902\) −37.6052 38.7074i −1.25212 1.28882i
\(903\) 0 0
\(904\) −5.51583 6.01577i −0.183454 0.200082i
\(905\) 3.57585i 0.118865i
\(906\) 0 0
\(907\) 17.0142 0.564948 0.282474 0.959275i \(-0.408845\pi\)
0.282474 + 0.959275i \(0.408845\pi\)
\(908\) −41.4518 1.19766i −1.37563 0.0397457i
\(909\) 0 0
\(910\) 0.718696 + 0.739761i 0.0238245 + 0.0245228i
\(911\) −8.54462 −0.283096 −0.141548 0.989931i \(-0.545208\pi\)
−0.141548 + 0.989931i \(0.545208\pi\)
\(912\) 0 0
\(913\) −48.2631 −1.59727
\(914\) −10.8436 11.1614i −0.358675 0.369187i
\(915\) 0 0
\(916\) 38.5090 + 1.11264i 1.27237 + 0.0367625i
\(917\) 12.8026 0.422780
\(918\) 0 0
\(919\) 6.49841i 0.214363i −0.994239 0.107181i \(-0.965817\pi\)
0.994239 0.107181i \(-0.0341826\pi\)
\(920\) −0.379963 + 0.348386i −0.0125270 + 0.0114859i
\(921\) 0 0
\(922\) 32.4891 + 33.4414i 1.06997 + 1.10133i
\(923\) 35.8660i 1.18055i
\(924\) 0 0
\(925\) 49.1669i 1.61660i
\(926\) 12.5116 12.1553i 0.411157 0.399449i
\(927\) 0 0
\(928\) −11.8733 13.7250i −0.389759 0.450544i
\(929\) 16.9216i 0.555181i −0.960699 0.277591i \(-0.910464\pi\)
0.960699 0.277591i \(-0.0895358\pi\)
\(930\) 0 0
\(931\) 5.48691 0.179826
\(932\) 49.9867 + 1.44426i 1.63737 + 0.0473083i
\(933\) 0 0
\(934\) 34.0071 33.0387i 1.11275 1.08106i
\(935\) −5.19237 −0.169809
\(936\) 0 0
\(937\) 14.4249 0.471240 0.235620 0.971845i \(-0.424288\pi\)
0.235620 + 0.971845i \(0.424288\pi\)
\(938\) 5.82906 5.66307i 0.190325 0.184906i
\(939\) 0 0
\(940\) −0.0814532 + 2.81915i −0.00265671 + 0.0919505i
\(941\) −58.5364 −1.90823 −0.954116 0.299436i \(-0.903201\pi\)
−0.954116 + 0.299436i \(0.903201\pi\)
\(942\) 0 0
\(943\) 5.23218i 0.170383i
\(944\) −0.977699 + 16.9053i −0.0318214 + 0.550220i
\(945\) 0 0
\(946\) −7.42333 + 7.21195i −0.241353 + 0.234481i
\(947\) 15.3006i 0.497201i −0.968606 0.248601i \(-0.920029\pi\)
0.968606 0.248601i \(-0.0799707\pi\)
\(948\) 0 0
\(949\) 29.1599i 0.946571i
\(950\) −26.8038 27.5894i −0.869631 0.895120i
\(951\) 0 0
\(952\) 8.14337 7.46661i 0.263928 0.241994i
\(953\) 14.9133i 0.483089i 0.970390 + 0.241544i \(0.0776539\pi\)
−0.970390 + 0.241544i \(0.922346\pi\)
\(954\) 0 0
\(955\) −1.51068 −0.0488845
\(956\) −0.868958 + 30.0752i −0.0281041 + 0.972701i
\(957\) 0 0
\(958\) −9.51743 9.79638i −0.307494 0.316507i
\(959\) 0.363873 0.0117501
\(960\) 0 0
\(961\) 30.9960 0.999871
\(962\) 34.4376 + 35.4470i 1.11031 + 1.14286i
\(963\) 0 0
\(964\) 0.186729 6.46279i 0.00601412 0.208153i
\(965\) 2.50680 0.0806966
\(966\) 0 0
\(967\) 27.1896i 0.874358i −0.899375 0.437179i \(-0.855978\pi\)
0.899375 0.437179i \(-0.144022\pi\)
\(968\) −63.0445 + 57.8051i −2.02633 + 1.85793i
\(969\) 0 0
\(970\) 3.54782 + 3.65181i 0.113914 + 0.117252i
\(971\) 35.0892i 1.12607i −0.826434 0.563034i \(-0.809634\pi\)
0.826434 0.563034i \(-0.190366\pi\)
\(972\) 0 0
\(973\) 9.32338i 0.298894i
\(974\) −25.4606 + 24.7356i −0.815811 + 0.792581i
\(975\) 0 0
\(976\) 2.84082 49.1203i 0.0909326 1.57230i
\(977\) 32.0438i 1.02517i 0.858636 + 0.512586i \(0.171312\pi\)
−0.858636 + 0.512586i \(0.828688\pi\)
\(978\) 0 0
\(979\) −28.5570 −0.912687
\(980\) 0.0119562 0.413810i 0.000381925 0.0132187i
\(981\) 0 0
\(982\) −10.7037 + 10.3989i −0.341569 + 0.331843i
\(983\) 43.5719 1.38973 0.694864 0.719142i \(-0.255465\pi\)
0.694864 + 0.719142i \(0.255465\pi\)
\(984\) 0 0
\(985\) −3.60011 −0.114709
\(986\) −12.7112 + 12.3492i −0.404807 + 0.393280i
\(987\) 0 0
\(988\) −38.6486 1.11667i −1.22958 0.0355259i
\(989\) 1.00343 0.0319073
\(990\) 0 0
\(991\) 15.2929i 0.485794i −0.970052 0.242897i \(-0.921902\pi\)
0.970052 0.242897i \(-0.0780978\pi\)
\(992\) 0.270065 0.233628i 0.00857456 0.00741771i
\(993\) 0 0
\(994\) 10.3255 10.0314i 0.327504 0.318178i
\(995\) 0.636670i 0.0201838i
\(996\) 0 0
\(997\) 29.8050i 0.943933i 0.881616 + 0.471967i \(0.156456\pi\)
−0.881616 + 0.471967i \(0.843544\pi\)
\(998\) 11.4923 + 11.8291i 0.363782 + 0.374444i
\(999\) 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 1512.2.j.c.323.9 32
3.2 odd 2 inner 1512.2.j.c.323.24 yes 32
4.3 odd 2 6048.2.j.c.5615.20 32
8.3 odd 2 inner 1512.2.j.c.323.23 yes 32
8.5 even 2 6048.2.j.c.5615.14 32
12.11 even 2 6048.2.j.c.5615.13 32
24.5 odd 2 6048.2.j.c.5615.19 32
24.11 even 2 inner 1512.2.j.c.323.10 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.c.323.9 32 1.1 even 1 trivial
1512.2.j.c.323.10 yes 32 24.11 even 2 inner
1512.2.j.c.323.23 yes 32 8.3 odd 2 inner
1512.2.j.c.323.24 yes 32 3.2 odd 2 inner
6048.2.j.c.5615.13 32 12.11 even 2
6048.2.j.c.5615.14 32 8.5 even 2
6048.2.j.c.5615.19 32 24.5 odd 2
6048.2.j.c.5615.20 32 4.3 odd 2