Properties

Label 1512.2.j.c.323.25
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 323.25
Character \(\chi\) \(=\) 1512.323
Dual form 1512.2.j.c.323.26

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.16953 - 0.795104i) q^{2} +(0.735620 - 1.85980i) q^{4} -2.09311 q^{5} +1.00000i q^{7} +(-0.618402 - 2.76000i) q^{8} +O(q^{10})\) \(q+(1.16953 - 0.795104i) q^{2} +(0.735620 - 1.85980i) q^{4} -2.09311 q^{5} +1.00000i q^{7} +(-0.618402 - 2.76000i) q^{8} +(-2.44797 + 1.66424i) q^{10} +0.961211i q^{11} -4.60210i q^{13} +(0.795104 + 1.16953i) q^{14} +(-2.91773 - 2.73622i) q^{16} -3.28066i q^{17} +3.66977 q^{19} +(-1.53974 + 3.89278i) q^{20} +(0.764262 + 1.12417i) q^{22} -2.40322 q^{23} -0.618872 q^{25} +(-3.65914 - 5.38231i) q^{26} +(1.85980 + 0.735620i) q^{28} -9.31771 q^{29} -1.84927i q^{31} +(-5.58795 - 0.880204i) q^{32} +(-2.60847 - 3.83684i) q^{34} -2.09311i q^{35} -2.29364i q^{37} +(4.29193 - 2.91785i) q^{38} +(1.29439 + 5.77699i) q^{40} +0.314406i q^{41} -8.64261 q^{43} +(1.78766 + 0.707086i) q^{44} +(-2.81064 + 1.91081i) q^{46} -2.34305 q^{47} -1.00000 q^{49} +(-0.723792 + 0.492067i) q^{50} +(-8.55899 - 3.38540i) q^{52} -7.73354 q^{53} -2.01192i q^{55} +(2.76000 - 0.618402i) q^{56} +(-10.8974 + 7.40855i) q^{58} -7.99530i q^{59} -10.6321i q^{61} +(-1.47036 - 2.16278i) q^{62} +(-7.23516 + 3.41357i) q^{64} +9.63272i q^{65} +4.12203 q^{67} +(-6.10138 - 2.41332i) q^{68} +(-1.66424 - 2.44797i) q^{70} +6.79440 q^{71} +16.0267 q^{73} +(-1.82368 - 2.68249i) q^{74} +(2.69956 - 6.82505i) q^{76} -0.961211 q^{77} -1.26231i q^{79} +(6.10713 + 5.72721i) q^{80} +(0.249985 + 0.367709i) q^{82} +2.99495i q^{83} +6.86680i q^{85} +(-10.1078 + 6.87177i) q^{86} +(2.65294 - 0.594415i) q^{88} +12.8990i q^{89} +4.60210 q^{91} +(-1.76785 + 4.46950i) q^{92} +(-2.74028 + 1.86297i) q^{94} -7.68126 q^{95} +4.98346 q^{97} +(-1.16953 + 0.795104i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32 q + 8 q^{4}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.16953 0.795104i 0.826986 0.562223i
\(3\) 0 0
\(4\) 0.735620 1.85980i 0.367810 0.929901i
\(5\) −2.09311 −0.936069 −0.468035 0.883710i \(-0.655038\pi\)
−0.468035 + 0.883710i \(0.655038\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) −0.618402 2.76000i −0.218638 0.975806i
\(9\) 0 0
\(10\) −2.44797 + 1.66424i −0.774116 + 0.526280i
\(11\) 0.961211i 0.289816i 0.989445 + 0.144908i \(0.0462886\pi\)
−0.989445 + 0.144908i \(0.953711\pi\)
\(12\) 0 0
\(13\) 4.60210i 1.27639i −0.769874 0.638196i \(-0.779681\pi\)
0.769874 0.638196i \(-0.220319\pi\)
\(14\) 0.795104 + 1.16953i 0.212500 + 0.312571i
\(15\) 0 0
\(16\) −2.91773 2.73622i −0.729431 0.684054i
\(17\) 3.28066i 0.795677i −0.917455 0.397839i \(-0.869760\pi\)
0.917455 0.397839i \(-0.130240\pi\)
\(18\) 0 0
\(19\) 3.66977 0.841904 0.420952 0.907083i \(-0.361696\pi\)
0.420952 + 0.907083i \(0.361696\pi\)
\(20\) −1.53974 + 3.89278i −0.344296 + 0.870452i
\(21\) 0 0
\(22\) 0.764262 + 1.12417i 0.162941 + 0.239674i
\(23\) −2.40322 −0.501105 −0.250553 0.968103i \(-0.580612\pi\)
−0.250553 + 0.968103i \(0.580612\pi\)
\(24\) 0 0
\(25\) −0.618872 −0.123774
\(26\) −3.65914 5.38231i −0.717617 1.05556i
\(27\) 0 0
\(28\) 1.85980 + 0.735620i 0.351470 + 0.139019i
\(29\) −9.31771 −1.73026 −0.865128 0.501552i \(-0.832763\pi\)
−0.865128 + 0.501552i \(0.832763\pi\)
\(30\) 0 0
\(31\) 1.84927i 0.332138i −0.986114 0.166069i \(-0.946892\pi\)
0.986114 0.166069i \(-0.0531075\pi\)
\(32\) −5.58795 0.880204i −0.987820 0.155600i
\(33\) 0 0
\(34\) −2.60847 3.83684i −0.447348 0.658013i
\(35\) 2.09311i 0.353801i
\(36\) 0 0
\(37\) 2.29364i 0.377072i −0.982066 0.188536i \(-0.939626\pi\)
0.982066 0.188536i \(-0.0603742\pi\)
\(38\) 4.29193 2.91785i 0.696242 0.473338i
\(39\) 0 0
\(40\) 1.29439 + 5.77699i 0.204660 + 0.913422i
\(41\) 0.314406i 0.0491020i 0.999699 + 0.0245510i \(0.00781561\pi\)
−0.999699 + 0.0245510i \(0.992184\pi\)
\(42\) 0 0
\(43\) −8.64261 −1.31799 −0.658993 0.752149i \(-0.729017\pi\)
−0.658993 + 0.752149i \(0.729017\pi\)
\(44\) 1.78766 + 0.707086i 0.269500 + 0.106597i
\(45\) 0 0
\(46\) −2.81064 + 1.91081i −0.414407 + 0.281733i
\(47\) −2.34305 −0.341769 −0.170884 0.985291i \(-0.554662\pi\)
−0.170884 + 0.985291i \(0.554662\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) −0.723792 + 0.492067i −0.102360 + 0.0695888i
\(51\) 0 0
\(52\) −8.55899 3.38540i −1.18692 0.469470i
\(53\) −7.73354 −1.06228 −0.531142 0.847283i \(-0.678237\pi\)
−0.531142 + 0.847283i \(0.678237\pi\)
\(54\) 0 0
\(55\) 2.01192i 0.271288i
\(56\) 2.76000 0.618402i 0.368820 0.0826374i
\(57\) 0 0
\(58\) −10.8974 + 7.40855i −1.43090 + 0.972790i
\(59\) 7.99530i 1.04090i −0.853892 0.520450i \(-0.825764\pi\)
0.853892 0.520450i \(-0.174236\pi\)
\(60\) 0 0
\(61\) 10.6321i 1.36130i −0.732610 0.680649i \(-0.761698\pi\)
0.732610 0.680649i \(-0.238302\pi\)
\(62\) −1.47036 2.16278i −0.186736 0.274674i
\(63\) 0 0
\(64\) −7.23516 + 3.41357i −0.904395 + 0.426697i
\(65\) 9.63272i 1.19479i
\(66\) 0 0
\(67\) 4.12203 0.503586 0.251793 0.967781i \(-0.418980\pi\)
0.251793 + 0.967781i \(0.418980\pi\)
\(68\) −6.10138 2.41332i −0.739901 0.292658i
\(69\) 0 0
\(70\) −1.66424 2.44797i −0.198915 0.292588i
\(71\) 6.79440 0.806347 0.403174 0.915124i \(-0.367907\pi\)
0.403174 + 0.915124i \(0.367907\pi\)
\(72\) 0 0
\(73\) 16.0267 1.87579 0.937894 0.346922i \(-0.112773\pi\)
0.937894 + 0.346922i \(0.112773\pi\)
\(74\) −1.82368 2.68249i −0.211999 0.311833i
\(75\) 0 0
\(76\) 2.69956 6.82505i 0.309661 0.782887i
\(77\) −0.961211 −0.109540
\(78\) 0 0
\(79\) 1.26231i 0.142021i −0.997476 0.0710106i \(-0.977378\pi\)
0.997476 0.0710106i \(-0.0226224\pi\)
\(80\) 6.10713 + 5.72721i 0.682798 + 0.640322i
\(81\) 0 0
\(82\) 0.249985 + 0.367709i 0.0276063 + 0.0406066i
\(83\) 2.99495i 0.328738i 0.986399 + 0.164369i \(0.0525589\pi\)
−0.986399 + 0.164369i \(0.947441\pi\)
\(84\) 0 0
\(85\) 6.86680i 0.744809i
\(86\) −10.1078 + 6.87177i −1.08995 + 0.741002i
\(87\) 0 0
\(88\) 2.65294 0.594415i 0.282804 0.0633648i
\(89\) 12.8990i 1.36729i 0.729815 + 0.683645i \(0.239606\pi\)
−0.729815 + 0.683645i \(0.760394\pi\)
\(90\) 0 0
\(91\) 4.60210 0.482431
\(92\) −1.76785 + 4.46950i −0.184312 + 0.465978i
\(93\) 0 0
\(94\) −2.74028 + 1.86297i −0.282638 + 0.192150i
\(95\) −7.68126 −0.788080
\(96\) 0 0
\(97\) 4.98346 0.505994 0.252997 0.967467i \(-0.418584\pi\)
0.252997 + 0.967467i \(0.418584\pi\)
\(98\) −1.16953 + 0.795104i −0.118141 + 0.0803176i
\(99\) 0 0
\(100\) −0.455255 + 1.15098i −0.0455255 + 0.115098i
\(101\) 3.18764 0.317182 0.158591 0.987344i \(-0.449305\pi\)
0.158591 + 0.987344i \(0.449305\pi\)
\(102\) 0 0
\(103\) 0.535425i 0.0527570i 0.999652 + 0.0263785i \(0.00839750\pi\)
−0.999652 + 0.0263785i \(0.991602\pi\)
\(104\) −12.7018 + 2.84595i −1.24551 + 0.279068i
\(105\) 0 0
\(106\) −9.04464 + 6.14897i −0.878493 + 0.597240i
\(107\) 9.70673i 0.938385i −0.883096 0.469193i \(-0.844545\pi\)
0.883096 0.469193i \(-0.155455\pi\)
\(108\) 0 0
\(109\) 8.03576i 0.769686i −0.922982 0.384843i \(-0.874256\pi\)
0.922982 0.384843i \(-0.125744\pi\)
\(110\) −1.59969 2.35301i −0.152524 0.224351i
\(111\) 0 0
\(112\) 2.73622 2.91773i 0.258548 0.275699i
\(113\) 12.7809i 1.20233i 0.799126 + 0.601163i \(0.205296\pi\)
−0.799126 + 0.601163i \(0.794704\pi\)
\(114\) 0 0
\(115\) 5.03021 0.469069
\(116\) −6.85430 + 17.3291i −0.636406 + 1.60897i
\(117\) 0 0
\(118\) −6.35709 9.35078i −0.585218 0.860809i
\(119\) 3.28066 0.300738
\(120\) 0 0
\(121\) 10.0761 0.916007
\(122\) −8.45360 12.4346i −0.765353 1.12577i
\(123\) 0 0
\(124\) −3.43927 1.36036i −0.308856 0.122164i
\(125\) 11.7609 1.05193
\(126\) 0 0
\(127\) 16.6260i 1.47532i 0.675175 + 0.737658i \(0.264068\pi\)
−0.675175 + 0.737658i \(0.735932\pi\)
\(128\) −5.74762 + 9.74499i −0.508023 + 0.861344i
\(129\) 0 0
\(130\) 7.65901 + 11.2658i 0.671739 + 0.988075i
\(131\) 14.5603i 1.27214i −0.771632 0.636069i \(-0.780560\pi\)
0.771632 0.636069i \(-0.219440\pi\)
\(132\) 0 0
\(133\) 3.66977i 0.318210i
\(134\) 4.82086 3.27744i 0.416459 0.283128i
\(135\) 0 0
\(136\) −9.05461 + 2.02877i −0.776426 + 0.173965i
\(137\) 17.2672i 1.47524i −0.675216 0.737620i \(-0.735950\pi\)
0.675216 0.737620i \(-0.264050\pi\)
\(138\) 0 0
\(139\) 14.5580 1.23480 0.617398 0.786651i \(-0.288187\pi\)
0.617398 + 0.786651i \(0.288187\pi\)
\(140\) −3.89278 1.53974i −0.329000 0.130132i
\(141\) 0 0
\(142\) 7.94629 5.40225i 0.666837 0.453347i
\(143\) 4.42358 0.369919
\(144\) 0 0
\(145\) 19.5030 1.61964
\(146\) 18.7438 12.7429i 1.55125 1.05461i
\(147\) 0 0
\(148\) −4.26571 1.68725i −0.350640 0.138691i
\(149\) −20.1739 −1.65271 −0.826356 0.563148i \(-0.809590\pi\)
−0.826356 + 0.563148i \(0.809590\pi\)
\(150\) 0 0
\(151\) 11.9560i 0.972966i −0.873690 0.486483i \(-0.838280\pi\)
0.873690 0.486483i \(-0.161720\pi\)
\(152\) −2.26940 10.1286i −0.184072 0.821535i
\(153\) 0 0
\(154\) −1.12417 + 0.764262i −0.0905881 + 0.0615860i
\(155\) 3.87073i 0.310905i
\(156\) 0 0
\(157\) 22.1011i 1.76386i 0.471381 + 0.881930i \(0.343756\pi\)
−0.471381 + 0.881930i \(0.656244\pi\)
\(158\) −1.00367 1.47632i −0.0798476 0.117450i
\(159\) 0 0
\(160\) 11.6962 + 1.84237i 0.924668 + 0.145652i
\(161\) 2.40322i 0.189400i
\(162\) 0 0
\(163\) 11.2054 0.877674 0.438837 0.898567i \(-0.355391\pi\)
0.438837 + 0.898567i \(0.355391\pi\)
\(164\) 0.584733 + 0.231284i 0.0456600 + 0.0180602i
\(165\) 0 0
\(166\) 2.38130 + 3.50270i 0.184824 + 0.271862i
\(167\) 1.76628 0.136679 0.0683395 0.997662i \(-0.478230\pi\)
0.0683395 + 0.997662i \(0.478230\pi\)
\(168\) 0 0
\(169\) −8.17930 −0.629177
\(170\) 5.45982 + 8.03095i 0.418749 + 0.615946i
\(171\) 0 0
\(172\) −6.35768 + 16.0735i −0.484768 + 1.22560i
\(173\) 4.37240 0.332428 0.166214 0.986090i \(-0.446846\pi\)
0.166214 + 0.986090i \(0.446846\pi\)
\(174\) 0 0
\(175\) 0.618872i 0.0467823i
\(176\) 2.63008 2.80455i 0.198250 0.211401i
\(177\) 0 0
\(178\) 10.2560 + 15.0858i 0.768722 + 1.13073i
\(179\) 17.0068i 1.27115i −0.772041 0.635573i \(-0.780764\pi\)
0.772041 0.635573i \(-0.219236\pi\)
\(180\) 0 0
\(181\) 8.50624i 0.632264i −0.948715 0.316132i \(-0.897616\pi\)
0.948715 0.316132i \(-0.102384\pi\)
\(182\) 5.38231 3.65914i 0.398963 0.271234i
\(183\) 0 0
\(184\) 1.48615 + 6.63287i 0.109561 + 0.488981i
\(185\) 4.80085i 0.352965i
\(186\) 0 0
\(187\) 3.15341 0.230600
\(188\) −1.72359 + 4.35761i −0.125706 + 0.317811i
\(189\) 0 0
\(190\) −8.98349 + 6.10739i −0.651731 + 0.443077i
\(191\) −0.725194 −0.0524732 −0.0262366 0.999656i \(-0.508352\pi\)
−0.0262366 + 0.999656i \(0.508352\pi\)
\(192\) 0 0
\(193\) −15.7648 −1.13478 −0.567388 0.823450i \(-0.692046\pi\)
−0.567388 + 0.823450i \(0.692046\pi\)
\(194\) 5.82833 3.96237i 0.418450 0.284481i
\(195\) 0 0
\(196\) −0.735620 + 1.85980i −0.0525443 + 0.132843i
\(197\) 5.76032 0.410406 0.205203 0.978719i \(-0.434215\pi\)
0.205203 + 0.978719i \(0.434215\pi\)
\(198\) 0 0
\(199\) 11.7773i 0.834872i −0.908706 0.417436i \(-0.862929\pi\)
0.908706 0.417436i \(-0.137071\pi\)
\(200\) 0.382712 + 1.70808i 0.0270618 + 0.120780i
\(201\) 0 0
\(202\) 3.72805 2.53450i 0.262305 0.178327i
\(203\) 9.31771i 0.653975i
\(204\) 0 0
\(205\) 0.658088i 0.0459629i
\(206\) 0.425718 + 0.626197i 0.0296612 + 0.0436292i
\(207\) 0 0
\(208\) −12.5923 + 13.4277i −0.873121 + 0.931040i
\(209\) 3.52742i 0.243997i
\(210\) 0 0
\(211\) 19.9971 1.37666 0.688328 0.725399i \(-0.258345\pi\)
0.688328 + 0.725399i \(0.258345\pi\)
\(212\) −5.68895 + 14.3829i −0.390719 + 0.987818i
\(213\) 0 0
\(214\) −7.71786 11.3524i −0.527582 0.776031i
\(215\) 18.0900 1.23373
\(216\) 0 0
\(217\) 1.84927 0.125537
\(218\) −6.38926 9.39810i −0.432735 0.636519i
\(219\) 0 0
\(220\) −3.74178 1.48001i −0.252271 0.0997824i
\(221\) −15.0979 −1.01560
\(222\) 0 0
\(223\) 6.84047i 0.458071i 0.973418 + 0.229036i \(0.0735573\pi\)
−0.973418 + 0.229036i \(0.926443\pi\)
\(224\) 0.880204 5.58795i 0.0588111 0.373361i
\(225\) 0 0
\(226\) 10.1621 + 14.9477i 0.675976 + 0.994307i
\(227\) 13.9061i 0.922981i −0.887145 0.461490i \(-0.847315\pi\)
0.887145 0.461490i \(-0.152685\pi\)
\(228\) 0 0
\(229\) 3.87934i 0.256354i 0.991751 + 0.128177i \(0.0409126\pi\)
−0.991751 + 0.128177i \(0.959087\pi\)
\(230\) 5.88300 3.99953i 0.387913 0.263722i
\(231\) 0 0
\(232\) 5.76209 + 25.7168i 0.378300 + 1.68839i
\(233\) 21.8307i 1.43017i 0.699035 + 0.715087i \(0.253613\pi\)
−0.699035 + 0.715087i \(0.746387\pi\)
\(234\) 0 0
\(235\) 4.90427 0.319919
\(236\) −14.8697 5.88151i −0.967933 0.382853i
\(237\) 0 0
\(238\) 3.83684 2.60847i 0.248706 0.169082i
\(239\) 27.8956 1.80441 0.902207 0.431303i \(-0.141946\pi\)
0.902207 + 0.431303i \(0.141946\pi\)
\(240\) 0 0
\(241\) −6.17606 −0.397835 −0.198918 0.980016i \(-0.563743\pi\)
−0.198918 + 0.980016i \(0.563743\pi\)
\(242\) 11.7843 8.01152i 0.757524 0.515000i
\(243\) 0 0
\(244\) −19.7736 7.82117i −1.26587 0.500699i
\(245\) 2.09311 0.133724
\(246\) 0 0
\(247\) 16.8887i 1.07460i
\(248\) −5.10397 + 1.14359i −0.324103 + 0.0726181i
\(249\) 0 0
\(250\) 13.7548 9.35117i 0.869931 0.591420i
\(251\) 18.6366i 1.17633i 0.808740 + 0.588167i \(0.200150\pi\)
−0.808740 + 0.588167i \(0.799850\pi\)
\(252\) 0 0
\(253\) 2.31000i 0.145228i
\(254\) 13.2194 + 19.4446i 0.829456 + 1.22006i
\(255\) 0 0
\(256\) 1.02624 + 15.9671i 0.0641402 + 0.997941i
\(257\) 8.25845i 0.515148i 0.966259 + 0.257574i \(0.0829231\pi\)
−0.966259 + 0.257574i \(0.917077\pi\)
\(258\) 0 0
\(259\) 2.29364 0.142520
\(260\) 17.9149 + 7.08602i 1.11104 + 0.439456i
\(261\) 0 0
\(262\) −11.5769 17.0287i −0.715225 1.05204i
\(263\) 9.34158 0.576027 0.288013 0.957626i \(-0.407005\pi\)
0.288013 + 0.957626i \(0.407005\pi\)
\(264\) 0 0
\(265\) 16.1872 0.994371
\(266\) 2.91785 + 4.29193i 0.178905 + 0.263155i
\(267\) 0 0
\(268\) 3.03225 7.66616i 0.185224 0.468286i
\(269\) −11.9972 −0.731484 −0.365742 0.930716i \(-0.619185\pi\)
−0.365742 + 0.930716i \(0.619185\pi\)
\(270\) 0 0
\(271\) 23.7030i 1.43986i −0.694049 0.719928i \(-0.744175\pi\)
0.694049 0.719928i \(-0.255825\pi\)
\(272\) −8.97660 + 9.57207i −0.544286 + 0.580392i
\(273\) 0 0
\(274\) −13.7292 20.1946i −0.829414 1.22000i
\(275\) 0.594866i 0.0358718i
\(276\) 0 0
\(277\) 13.7688i 0.827284i −0.910440 0.413642i \(-0.864256\pi\)
0.910440 0.413642i \(-0.135744\pi\)
\(278\) 17.0261 11.5751i 1.02116 0.694230i
\(279\) 0 0
\(280\) −5.77699 + 1.29439i −0.345241 + 0.0773544i
\(281\) 8.69078i 0.518448i 0.965817 + 0.259224i \(0.0834669\pi\)
−0.965817 + 0.259224i \(0.916533\pi\)
\(282\) 0 0
\(283\) −32.7518 −1.94689 −0.973447 0.228913i \(-0.926483\pi\)
−0.973447 + 0.228913i \(0.926483\pi\)
\(284\) 4.99810 12.6362i 0.296583 0.749823i
\(285\) 0 0
\(286\) 5.17353 3.51721i 0.305917 0.207977i
\(287\) −0.314406 −0.0185588
\(288\) 0 0
\(289\) 6.23727 0.366898
\(290\) 22.8095 15.5069i 1.33942 0.910599i
\(291\) 0 0
\(292\) 11.7896 29.8066i 0.689934 1.74430i
\(293\) −12.3629 −0.722249 −0.361124 0.932518i \(-0.617607\pi\)
−0.361124 + 0.932518i \(0.617607\pi\)
\(294\) 0 0
\(295\) 16.7351i 0.974354i
\(296\) −6.33044 + 1.41839i −0.367949 + 0.0824423i
\(297\) 0 0
\(298\) −23.5941 + 16.0404i −1.36677 + 0.929193i
\(299\) 11.0598i 0.639607i
\(300\) 0 0
\(301\) 8.64261i 0.498152i
\(302\) −9.50627 13.9830i −0.547024 0.804629i
\(303\) 0 0
\(304\) −10.7074 10.0413i −0.614111 0.575908i
\(305\) 22.2542i 1.27427i
\(306\) 0 0
\(307\) 15.2600 0.870935 0.435468 0.900204i \(-0.356583\pi\)
0.435468 + 0.900204i \(0.356583\pi\)
\(308\) −0.707086 + 1.78766i −0.0402900 + 0.101861i
\(309\) 0 0
\(310\) 3.07763 + 4.52695i 0.174798 + 0.257114i
\(311\) 5.12076 0.290372 0.145186 0.989404i \(-0.453622\pi\)
0.145186 + 0.989404i \(0.453622\pi\)
\(312\) 0 0
\(313\) −11.7058 −0.661649 −0.330825 0.943692i \(-0.607327\pi\)
−0.330825 + 0.943692i \(0.607327\pi\)
\(314\) 17.5727 + 25.8480i 0.991683 + 1.45869i
\(315\) 0 0
\(316\) −2.34765 0.928583i −0.132066 0.0522369i
\(317\) 18.5302 1.04076 0.520379 0.853935i \(-0.325791\pi\)
0.520379 + 0.853935i \(0.325791\pi\)
\(318\) 0 0
\(319\) 8.95628i 0.501456i
\(320\) 15.1440 7.14500i 0.846576 0.399418i
\(321\) 0 0
\(322\) −1.91081 2.81064i −0.106485 0.156631i
\(323\) 12.0393i 0.669883i
\(324\) 0 0
\(325\) 2.84811i 0.157985i
\(326\) 13.1051 8.90945i 0.725824 0.493449i
\(327\) 0 0
\(328\) 0.867760 0.194429i 0.0479140 0.0107356i
\(329\) 2.34305i 0.129176i
\(330\) 0 0
\(331\) −20.4688 −1.12507 −0.562534 0.826774i \(-0.690174\pi\)
−0.562534 + 0.826774i \(0.690174\pi\)
\(332\) 5.57001 + 2.20315i 0.305694 + 0.120913i
\(333\) 0 0
\(334\) 2.06573 1.40438i 0.113032 0.0768442i
\(335\) −8.62789 −0.471392
\(336\) 0 0
\(337\) −19.9419 −1.08631 −0.543153 0.839634i \(-0.682769\pi\)
−0.543153 + 0.839634i \(0.682769\pi\)
\(338\) −9.56597 + 6.50339i −0.520320 + 0.353738i
\(339\) 0 0
\(340\) 12.7709 + 5.05136i 0.692598 + 0.273948i
\(341\) 1.77754 0.0962590
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 5.34461 + 23.8536i 0.288162 + 1.28610i
\(345\) 0 0
\(346\) 5.11367 3.47651i 0.274913 0.186898i
\(347\) 10.1743i 0.546187i −0.961987 0.273094i \(-0.911953\pi\)
0.961987 0.273094i \(-0.0880469\pi\)
\(348\) 0 0
\(349\) 31.9386i 1.70963i 0.518930 + 0.854817i \(0.326331\pi\)
−0.518930 + 0.854817i \(0.673669\pi\)
\(350\) −0.492067 0.723792i −0.0263021 0.0386883i
\(351\) 0 0
\(352\) 0.846062 5.37120i 0.0450952 0.286286i
\(353\) 24.3932i 1.29832i −0.760653 0.649159i \(-0.775121\pi\)
0.760653 0.649159i \(-0.224879\pi\)
\(354\) 0 0
\(355\) −14.2215 −0.754797
\(356\) 23.9895 + 9.48875i 1.27144 + 0.502903i
\(357\) 0 0
\(358\) −13.5222 19.8900i −0.714668 1.05122i
\(359\) 8.59571 0.453664 0.226832 0.973934i \(-0.427163\pi\)
0.226832 + 0.973934i \(0.427163\pi\)
\(360\) 0 0
\(361\) −5.53276 −0.291198
\(362\) −6.76335 9.94834i −0.355474 0.522873i
\(363\) 0 0
\(364\) 3.38540 8.55899i 0.177443 0.448613i
\(365\) −33.5458 −1.75587
\(366\) 0 0
\(367\) 29.1285i 1.52050i 0.649632 + 0.760249i \(0.274923\pi\)
−0.649632 + 0.760249i \(0.725077\pi\)
\(368\) 7.01192 + 6.57572i 0.365522 + 0.342783i
\(369\) 0 0
\(370\) 3.81717 + 5.61476i 0.198445 + 0.291897i
\(371\) 7.73354i 0.401505i
\(372\) 0 0
\(373\) 4.56533i 0.236384i 0.992991 + 0.118192i \(0.0377098\pi\)
−0.992991 + 0.118192i \(0.962290\pi\)
\(374\) 3.68802 2.50728i 0.190703 0.129649i
\(375\) 0 0
\(376\) 1.44895 + 6.46680i 0.0747237 + 0.333500i
\(377\) 42.8810i 2.20848i
\(378\) 0 0
\(379\) 15.5673 0.799641 0.399820 0.916593i \(-0.369072\pi\)
0.399820 + 0.916593i \(0.369072\pi\)
\(380\) −5.65049 + 14.2856i −0.289864 + 0.732836i
\(381\) 0 0
\(382\) −0.848139 + 0.576604i −0.0433946 + 0.0295016i
\(383\) −3.08947 −0.157865 −0.0789323 0.996880i \(-0.525151\pi\)
−0.0789323 + 0.996880i \(0.525151\pi\)
\(384\) 0 0
\(385\) 2.01192 0.102537
\(386\) −18.4375 + 12.5347i −0.938444 + 0.637998i
\(387\) 0 0
\(388\) 3.66594 9.26825i 0.186110 0.470524i
\(389\) 30.6022 1.55160 0.775798 0.630982i \(-0.217348\pi\)
0.775798 + 0.630982i \(0.217348\pi\)
\(390\) 0 0
\(391\) 7.88413i 0.398718i
\(392\) 0.618402 + 2.76000i 0.0312340 + 0.139401i
\(393\) 0 0
\(394\) 6.73689 4.58005i 0.339400 0.230740i
\(395\) 2.64216i 0.132942i
\(396\) 0 0
\(397\) 0.403742i 0.0202632i −0.999949 0.0101316i \(-0.996775\pi\)
0.999949 0.0101316i \(-0.00322505\pi\)
\(398\) −9.36419 13.7740i −0.469385 0.690427i
\(399\) 0 0
\(400\) 1.80570 + 1.69337i 0.0902849 + 0.0846684i
\(401\) 30.3779i 1.51700i 0.651672 + 0.758501i \(0.274068\pi\)
−0.651672 + 0.758501i \(0.725932\pi\)
\(402\) 0 0
\(403\) −8.51051 −0.423939
\(404\) 2.34489 5.92838i 0.116663 0.294948i
\(405\) 0 0
\(406\) −7.40855 10.8974i −0.367680 0.540828i
\(407\) 2.20467 0.109281
\(408\) 0 0
\(409\) −17.8549 −0.882869 −0.441434 0.897293i \(-0.645530\pi\)
−0.441434 + 0.897293i \(0.645530\pi\)
\(410\) −0.523248 0.769656i −0.0258414 0.0380106i
\(411\) 0 0
\(412\) 0.995784 + 0.393869i 0.0490587 + 0.0194045i
\(413\) 7.99530 0.393423
\(414\) 0 0
\(415\) 6.26877i 0.307722i
\(416\) −4.05079 + 25.7163i −0.198606 + 1.26085i
\(417\) 0 0
\(418\) 2.80467 + 4.12544i 0.137181 + 0.201782i
\(419\) 31.1609i 1.52231i −0.648569 0.761156i \(-0.724632\pi\)
0.648569 0.761156i \(-0.275368\pi\)
\(420\) 0 0
\(421\) 7.19292i 0.350562i 0.984518 + 0.175281i \(0.0560833\pi\)
−0.984518 + 0.175281i \(0.943917\pi\)
\(422\) 23.3873 15.8998i 1.13848 0.773988i
\(423\) 0 0
\(424\) 4.78244 + 21.3445i 0.232256 + 1.03658i
\(425\) 2.03031i 0.0984844i
\(426\) 0 0
\(427\) 10.6321 0.514522
\(428\) −18.0526 7.14047i −0.872605 0.345148i
\(429\) 0 0
\(430\) 21.1568 14.3834i 1.02027 0.693629i
\(431\) 25.7598 1.24081 0.620404 0.784283i \(-0.286969\pi\)
0.620404 + 0.784283i \(0.286969\pi\)
\(432\) 0 0
\(433\) −3.66449 −0.176104 −0.0880521 0.996116i \(-0.528064\pi\)
−0.0880521 + 0.996116i \(0.528064\pi\)
\(434\) 2.16278 1.47036i 0.103817 0.0705796i
\(435\) 0 0
\(436\) −14.9449 5.91127i −0.715732 0.283098i
\(437\) −8.81926 −0.421882
\(438\) 0 0
\(439\) 39.4350i 1.88213i −0.338227 0.941064i \(-0.609827\pi\)
0.338227 0.941064i \(-0.390173\pi\)
\(440\) −5.55290 + 1.24418i −0.264724 + 0.0593138i
\(441\) 0 0
\(442\) −17.6575 + 12.0044i −0.839883 + 0.570992i
\(443\) 36.1437i 1.71724i 0.512612 + 0.858620i \(0.328678\pi\)
−0.512612 + 0.858620i \(0.671322\pi\)
\(444\) 0 0
\(445\) 26.9990i 1.27988i
\(446\) 5.43888 + 8.00016i 0.257538 + 0.378819i
\(447\) 0 0
\(448\) −3.41357 7.23516i −0.161276 0.341829i
\(449\) 24.9465i 1.17730i −0.808389 0.588649i \(-0.799660\pi\)
0.808389 0.588649i \(-0.200340\pi\)
\(450\) 0 0
\(451\) −0.302210 −0.0142305
\(452\) 23.7700 + 9.40190i 1.11804 + 0.442228i
\(453\) 0 0
\(454\) −11.0568 16.2637i −0.518921 0.763292i
\(455\) −9.63272 −0.451589
\(456\) 0 0
\(457\) 15.1856 0.710354 0.355177 0.934799i \(-0.384421\pi\)
0.355177 + 0.934799i \(0.384421\pi\)
\(458\) 3.08448 + 4.53702i 0.144128 + 0.212001i
\(459\) 0 0
\(460\) 3.70032 9.35518i 0.172528 0.436188i
\(461\) 9.29479 0.432902 0.216451 0.976294i \(-0.430552\pi\)
0.216451 + 0.976294i \(0.430552\pi\)
\(462\) 0 0
\(463\) 2.13438i 0.0991930i 0.998769 + 0.0495965i \(0.0157935\pi\)
−0.998769 + 0.0495965i \(0.984206\pi\)
\(464\) 27.1865 + 25.4953i 1.26210 + 1.18359i
\(465\) 0 0
\(466\) 17.3576 + 25.5317i 0.804077 + 1.18273i
\(467\) 41.6077i 1.92538i −0.270612 0.962688i \(-0.587226\pi\)
0.270612 0.962688i \(-0.412774\pi\)
\(468\) 0 0
\(469\) 4.12203i 0.190338i
\(470\) 5.73571 3.89940i 0.264569 0.179866i
\(471\) 0 0
\(472\) −22.0670 + 4.94431i −1.01572 + 0.227580i
\(473\) 8.30736i 0.381973i
\(474\) 0 0
\(475\) −2.27112 −0.104206
\(476\) 2.41332 6.10138i 0.110614 0.279656i
\(477\) 0 0
\(478\) 32.6248 22.1799i 1.49222 1.01448i
\(479\) 40.9442 1.87079 0.935394 0.353608i \(-0.115045\pi\)
0.935394 + 0.353608i \(0.115045\pi\)
\(480\) 0 0
\(481\) −10.5556 −0.481292
\(482\) −7.22312 + 4.91061i −0.329004 + 0.223672i
\(483\) 0 0
\(484\) 7.41217 18.7395i 0.336917 0.851795i
\(485\) −10.4310 −0.473645
\(486\) 0 0
\(487\) 14.2039i 0.643639i −0.946801 0.321820i \(-0.895706\pi\)
0.946801 0.321820i \(-0.104294\pi\)
\(488\) −29.3445 + 6.57490i −1.32836 + 0.297632i
\(489\) 0 0
\(490\) 2.44797 1.66424i 0.110588 0.0751828i
\(491\) 39.3013i 1.77364i 0.462113 + 0.886821i \(0.347091\pi\)
−0.462113 + 0.886821i \(0.652909\pi\)
\(492\) 0 0
\(493\) 30.5682i 1.37672i
\(494\) −13.4282 19.7519i −0.604165 0.888678i
\(495\) 0 0
\(496\) −5.06000 + 5.39566i −0.227201 + 0.242272i
\(497\) 6.79440i 0.304771i
\(498\) 0 0
\(499\) −32.3257 −1.44710 −0.723548 0.690274i \(-0.757490\pi\)
−0.723548 + 0.690274i \(0.757490\pi\)
\(500\) 8.65159 21.8730i 0.386911 0.978191i
\(501\) 0 0
\(502\) 14.8180 + 21.7962i 0.661362 + 0.972810i
\(503\) −19.7861 −0.882217 −0.441108 0.897454i \(-0.645415\pi\)
−0.441108 + 0.897454i \(0.645415\pi\)
\(504\) 0 0
\(505\) −6.67210 −0.296904
\(506\) −1.83669 2.70162i −0.0816507 0.120102i
\(507\) 0 0
\(508\) 30.9210 + 12.2304i 1.37190 + 0.542636i
\(509\) 33.7174 1.49450 0.747248 0.664545i \(-0.231375\pi\)
0.747248 + 0.664545i \(0.231375\pi\)
\(510\) 0 0
\(511\) 16.0267i 0.708981i
\(512\) 13.8957 + 17.8580i 0.614109 + 0.789222i
\(513\) 0 0
\(514\) 6.56632 + 9.65854i 0.289628 + 0.426020i
\(515\) 1.12070i 0.0493842i
\(516\) 0 0
\(517\) 2.25216i 0.0990500i
\(518\) 2.68249 1.82368i 0.117862 0.0801279i
\(519\) 0 0
\(520\) 26.5863 5.95689i 1.16588 0.261227i
\(521\) 28.4958i 1.24842i −0.781255 0.624212i \(-0.785420\pi\)
0.781255 0.624212i \(-0.214580\pi\)
\(522\) 0 0
\(523\) −4.42637 −0.193552 −0.0967758 0.995306i \(-0.530853\pi\)
−0.0967758 + 0.995306i \(0.530853\pi\)
\(524\) −27.0792 10.7108i −1.18296 0.467905i
\(525\) 0 0
\(526\) 10.9253 7.42753i 0.476366 0.323855i
\(527\) −6.06682 −0.264275
\(528\) 0 0
\(529\) −17.2246 −0.748894
\(530\) 18.9315 12.8705i 0.822330 0.559058i
\(531\) 0 0
\(532\) 6.82505 + 2.69956i 0.295903 + 0.117041i
\(533\) 1.44693 0.0626734
\(534\) 0 0
\(535\) 20.3173i 0.878394i
\(536\) −2.54907 11.3768i −0.110103 0.491403i
\(537\) 0 0
\(538\) −14.0312 + 9.53905i −0.604927 + 0.411258i
\(539\) 0.961211i 0.0414023i
\(540\) 0 0
\(541\) 33.7548i 1.45123i −0.688101 0.725615i \(-0.741555\pi\)
0.688101 0.725615i \(-0.258445\pi\)
\(542\) −18.8464 27.7215i −0.809520 1.19074i
\(543\) 0 0
\(544\) −2.88765 + 18.3322i −0.123807 + 0.785986i
\(545\) 16.8198i 0.720480i
\(546\) 0 0
\(547\) −14.6641 −0.626992 −0.313496 0.949589i \(-0.601500\pi\)
−0.313496 + 0.949589i \(0.601500\pi\)
\(548\) −32.1136 12.7021i −1.37183 0.542608i
\(549\) 0 0
\(550\) −0.472980 0.695716i −0.0201680 0.0296654i
\(551\) −34.1939 −1.45671
\(552\) 0 0
\(553\) 1.26231 0.0536790
\(554\) −10.9476 16.1030i −0.465118 0.684152i
\(555\) 0 0
\(556\) 10.7092 27.0750i 0.454170 1.14824i
\(557\) −15.4373 −0.654099 −0.327050 0.945007i \(-0.606054\pi\)
−0.327050 + 0.945007i \(0.606054\pi\)
\(558\) 0 0
\(559\) 39.7741i 1.68227i
\(560\) −5.72721 + 6.10713i −0.242019 + 0.258073i
\(561\) 0 0
\(562\) 6.91007 + 10.1642i 0.291484 + 0.428749i
\(563\) 26.9472i 1.13569i −0.823135 0.567845i \(-0.807777\pi\)
0.823135 0.567845i \(-0.192223\pi\)
\(564\) 0 0
\(565\) 26.7519i 1.12546i
\(566\) −38.3044 + 26.0411i −1.61005 + 1.09459i
\(567\) 0 0
\(568\) −4.20167 18.7525i −0.176298 0.786838i
\(569\) 3.63013i 0.152183i −0.997101 0.0760915i \(-0.975756\pi\)
0.997101 0.0760915i \(-0.0242441\pi\)
\(570\) 0 0
\(571\) −16.2293 −0.679174 −0.339587 0.940575i \(-0.610287\pi\)
−0.339587 + 0.940575i \(0.610287\pi\)
\(572\) 3.25408 8.22699i 0.136060 0.343988i
\(573\) 0 0
\(574\) −0.367709 + 0.249985i −0.0153479 + 0.0104342i
\(575\) 1.48728 0.0620240
\(576\) 0 0
\(577\) 42.1511 1.75477 0.877386 0.479786i \(-0.159286\pi\)
0.877386 + 0.479786i \(0.159286\pi\)
\(578\) 7.29470 4.95927i 0.303419 0.206279i
\(579\) 0 0
\(580\) 14.3468 36.2718i 0.595720 1.50610i
\(581\) −2.99495 −0.124251
\(582\) 0 0
\(583\) 7.43356i 0.307867i
\(584\) −9.91097 44.2337i −0.410119 1.83040i
\(585\) 0 0
\(586\) −14.4588 + 9.82979i −0.597289 + 0.406065i
\(587\) 22.7097i 0.937328i −0.883377 0.468664i \(-0.844736\pi\)
0.883377 0.468664i \(-0.155264\pi\)
\(588\) 0 0
\(589\) 6.78640i 0.279629i
\(590\) 13.3061 + 19.5722i 0.547804 + 0.805777i
\(591\) 0 0
\(592\) −6.27589 + 6.69221i −0.257938 + 0.275048i
\(593\) 22.8921i 0.940068i −0.882648 0.470034i \(-0.844242\pi\)
0.882648 0.470034i \(-0.155758\pi\)
\(594\) 0 0
\(595\) −6.86680 −0.281511
\(596\) −14.8404 + 37.5195i −0.607885 + 1.53686i
\(597\) 0 0
\(598\) 8.79371 + 12.9349i 0.359602 + 0.528945i
\(599\) 31.0216 1.26751 0.633755 0.773534i \(-0.281513\pi\)
0.633755 + 0.773534i \(0.281513\pi\)
\(600\) 0 0
\(601\) −36.3657 −1.48339 −0.741693 0.670739i \(-0.765977\pi\)
−0.741693 + 0.670739i \(0.765977\pi\)
\(602\) −6.87177 10.1078i −0.280072 0.411964i
\(603\) 0 0
\(604\) −22.2358 8.79508i −0.904762 0.357867i
\(605\) −21.0904 −0.857446
\(606\) 0 0
\(607\) 27.1019i 1.10003i 0.835154 + 0.550016i \(0.185378\pi\)
−0.835154 + 0.550016i \(0.814622\pi\)
\(608\) −20.5065 3.23015i −0.831649 0.131000i
\(609\) 0 0
\(610\) 17.6944 + 26.0270i 0.716424 + 1.05380i
\(611\) 10.7829i 0.436231i
\(612\) 0 0
\(613\) 21.7164i 0.877118i −0.898702 0.438559i \(-0.855489\pi\)
0.898702 0.438559i \(-0.144511\pi\)
\(614\) 17.8471 12.1333i 0.720251 0.489660i
\(615\) 0 0
\(616\) 0.594415 + 2.65294i 0.0239496 + 0.106890i
\(617\) 15.0335i 0.605224i 0.953114 + 0.302612i \(0.0978587\pi\)
−0.953114 + 0.302612i \(0.902141\pi\)
\(618\) 0 0
\(619\) −12.2249 −0.491359 −0.245680 0.969351i \(-0.579011\pi\)
−0.245680 + 0.969351i \(0.579011\pi\)
\(620\) 7.19879 + 2.84739i 0.289110 + 0.114354i
\(621\) 0 0
\(622\) 5.98891 4.07154i 0.240133 0.163254i
\(623\) −12.8990 −0.516787
\(624\) 0 0
\(625\) −21.5226 −0.860906
\(626\) −13.6903 + 9.30730i −0.547174 + 0.371995i
\(627\) 0 0
\(628\) 41.1037 + 16.2580i 1.64021 + 0.648765i
\(629\) −7.52465 −0.300028
\(630\) 0 0
\(631\) 0.290118i 0.0115494i 0.999983 + 0.00577472i \(0.00183816\pi\)
−0.999983 + 0.00577472i \(0.998162\pi\)
\(632\) −3.48398 + 0.780617i −0.138585 + 0.0310513i
\(633\) 0 0
\(634\) 21.6717 14.7334i 0.860692 0.585138i
\(635\) 34.8000i 1.38100i
\(636\) 0 0
\(637\) 4.60210i 0.182342i
\(638\) −7.12117 10.4747i −0.281930 0.414696i
\(639\) 0 0
\(640\) 12.0304 20.3974i 0.475544 0.806277i
\(641\) 32.2612i 1.27424i −0.770764 0.637120i \(-0.780125\pi\)
0.770764 0.637120i \(-0.219875\pi\)
\(642\) 0 0
\(643\) −25.2641 −0.996317 −0.498159 0.867086i \(-0.665990\pi\)
−0.498159 + 0.867086i \(0.665990\pi\)
\(644\) −4.46950 1.76785i −0.176123 0.0696632i
\(645\) 0 0
\(646\) −9.57248 14.0803i −0.376624 0.553984i
\(647\) 15.6735 0.616188 0.308094 0.951356i \(-0.400309\pi\)
0.308094 + 0.951356i \(0.400309\pi\)
\(648\) 0 0
\(649\) 7.68517 0.301669
\(650\) 2.26454 + 3.33096i 0.0888226 + 0.130651i
\(651\) 0 0
\(652\) 8.24291 20.8398i 0.322817 0.816150i
\(653\) 17.4580 0.683183 0.341591 0.939849i \(-0.389034\pi\)
0.341591 + 0.939849i \(0.389034\pi\)
\(654\) 0 0
\(655\) 30.4763i 1.19081i
\(656\) 0.860283 0.917351i 0.0335884 0.0358165i
\(657\) 0 0
\(658\) −1.86297 2.74028i −0.0726260 0.106827i
\(659\) 34.8692i 1.35831i 0.733995 + 0.679155i \(0.237653\pi\)
−0.733995 + 0.679155i \(0.762347\pi\)
\(660\) 0 0
\(661\) 13.7607i 0.535230i 0.963526 + 0.267615i \(0.0862355\pi\)
−0.963526 + 0.267615i \(0.913765\pi\)
\(662\) −23.9390 + 16.2748i −0.930415 + 0.632540i
\(663\) 0 0
\(664\) 8.26605 1.85208i 0.320785 0.0718748i
\(665\) 7.68126i 0.297866i
\(666\) 0 0
\(667\) 22.3925 0.867040
\(668\) 1.29931 3.28494i 0.0502720 0.127098i
\(669\) 0 0
\(670\) −10.0906 + 6.86006i −0.389834 + 0.265027i
\(671\) 10.2197 0.394526
\(672\) 0 0
\(673\) −8.50938 −0.328013 −0.164006 0.986459i \(-0.552442\pi\)
−0.164006 + 0.986459i \(0.552442\pi\)
\(674\) −23.3227 + 15.8559i −0.898359 + 0.610746i
\(675\) 0 0
\(676\) −6.01686 + 15.2119i −0.231418 + 0.585072i
\(677\) −3.14515 −0.120878 −0.0604390 0.998172i \(-0.519250\pi\)
−0.0604390 + 0.998172i \(0.519250\pi\)
\(678\) 0 0
\(679\) 4.98346i 0.191248i
\(680\) 18.9523 4.24644i 0.726789 0.162844i
\(681\) 0 0
\(682\) 2.07889 1.41333i 0.0796048 0.0541190i
\(683\) 9.12590i 0.349193i −0.984640 0.174596i \(-0.944138\pi\)
0.984640 0.174596i \(-0.0558621\pi\)
\(684\) 0 0
\(685\) 36.1423i 1.38093i
\(686\) −0.795104 1.16953i −0.0303572 0.0446530i
\(687\) 0 0
\(688\) 25.2168 + 23.6480i 0.961380 + 0.901573i
\(689\) 35.5905i 1.35589i
\(690\) 0 0
\(691\) −15.6298 −0.594586 −0.297293 0.954786i \(-0.596084\pi\)
−0.297293 + 0.954786i \(0.596084\pi\)
\(692\) 3.21643 8.13180i 0.122270 0.309125i
\(693\) 0 0
\(694\) −8.08965 11.8992i −0.307079 0.451689i
\(695\) −30.4716 −1.15585
\(696\) 0 0
\(697\) 1.03146 0.0390693
\(698\) 25.3945 + 37.3533i 0.961196 + 1.41384i
\(699\) 0 0
\(700\) −1.15098 0.455255i −0.0435029 0.0172070i
\(701\) −50.4853 −1.90680 −0.953402 0.301701i \(-0.902445\pi\)
−0.953402 + 0.301701i \(0.902445\pi\)
\(702\) 0 0
\(703\) 8.41714i 0.317458i
\(704\) −3.28116 6.95451i −0.123664 0.262108i
\(705\) 0 0
\(706\) −19.3951 28.5286i −0.729944 1.07369i
\(707\) 3.18764i 0.119884i
\(708\) 0 0
\(709\) 28.7706i 1.08050i 0.841504 + 0.540251i \(0.181671\pi\)
−0.841504 + 0.540251i \(0.818329\pi\)
\(710\) −16.6325 + 11.3075i −0.624206 + 0.424364i
\(711\) 0 0
\(712\) 35.6011 7.97676i 1.33421 0.298942i
\(713\) 4.44419i 0.166436i
\(714\) 0 0
\(715\) −9.25907 −0.346270
\(716\) −31.6292 12.5105i −1.18204 0.467541i
\(717\) 0 0
\(718\) 10.0530 6.83448i 0.375174 0.255061i
\(719\) 0.582170 0.0217113 0.0108556 0.999941i \(-0.496544\pi\)
0.0108556 + 0.999941i \(0.496544\pi\)
\(720\) 0 0
\(721\) −0.535425 −0.0199403
\(722\) −6.47076 + 4.39912i −0.240817 + 0.163718i
\(723\) 0 0
\(724\) −15.8199 6.25737i −0.587943 0.232553i
\(725\) 5.76647 0.214161
\(726\) 0 0
\(727\) 6.92907i 0.256985i 0.991710 + 0.128492i \(0.0410138\pi\)
−0.991710 + 0.128492i \(0.958986\pi\)
\(728\) −2.84595 12.7018i −0.105478 0.470759i
\(729\) 0 0
\(730\) −39.2330 + 26.6724i −1.45208 + 0.987189i
\(731\) 28.3535i 1.04869i
\(732\) 0 0
\(733\) 1.86770i 0.0689850i −0.999405 0.0344925i \(-0.989019\pi\)
0.999405 0.0344925i \(-0.0109815\pi\)
\(734\) 23.1602 + 34.0668i 0.854859 + 1.25743i
\(735\) 0 0
\(736\) 13.4291 + 2.11532i 0.495002 + 0.0779717i
\(737\) 3.96214i 0.145947i
\(738\) 0 0
\(739\) −38.7036 −1.42373 −0.711867 0.702314i \(-0.752150\pi\)
−0.711867 + 0.702314i \(0.752150\pi\)
\(740\) 8.92863 + 3.53160i 0.328223 + 0.129824i
\(741\) 0 0
\(742\) −6.14897 9.04464i −0.225736 0.332039i
\(743\) 23.3088 0.855118 0.427559 0.903987i \(-0.359374\pi\)
0.427559 + 0.903987i \(0.359374\pi\)
\(744\) 0 0
\(745\) 42.2263 1.54705
\(746\) 3.62991 + 5.33931i 0.132900 + 0.195486i
\(747\) 0 0
\(748\) 2.31971 5.86471i 0.0848170 0.214435i
\(749\) 9.70673 0.354676
\(750\) 0 0
\(751\) 26.6345i 0.971908i 0.873984 + 0.485954i \(0.161528\pi\)
−0.873984 + 0.485954i \(0.838472\pi\)
\(752\) 6.83637 + 6.41109i 0.249297 + 0.233788i
\(753\) 0 0
\(754\) 34.0948 + 50.1508i 1.24166 + 1.82638i
\(755\) 25.0253i 0.910764i
\(756\) 0 0
\(757\) 26.4285i 0.960562i 0.877115 + 0.480281i \(0.159465\pi\)
−0.877115 + 0.480281i \(0.840535\pi\)
\(758\) 18.2065 12.3777i 0.661292 0.449577i
\(759\) 0 0
\(760\) 4.75010 + 21.2002i 0.172304 + 0.769013i
\(761\) 13.2617i 0.480736i −0.970682 0.240368i \(-0.922732\pi\)
0.970682 0.240368i \(-0.0772681\pi\)
\(762\) 0 0
\(763\) 8.03576 0.290914
\(764\) −0.533467 + 1.34872i −0.0193002 + 0.0487949i
\(765\) 0 0
\(766\) −3.61324 + 2.45645i −0.130552 + 0.0887552i
\(767\) −36.7951 −1.32860
\(768\) 0 0
\(769\) −4.34986 −0.156860 −0.0784300 0.996920i \(-0.524991\pi\)
−0.0784300 + 0.996920i \(0.524991\pi\)
\(770\) 2.35301 1.59969i 0.0847967 0.0576488i
\(771\) 0 0
\(772\) −11.5969 + 29.3194i −0.417382 + 1.05523i
\(773\) −34.8861 −1.25477 −0.627383 0.778711i \(-0.715874\pi\)
−0.627383 + 0.778711i \(0.715874\pi\)
\(774\) 0 0
\(775\) 1.14446i 0.0411102i
\(776\) −3.08178 13.7543i −0.110630 0.493752i
\(777\) 0 0
\(778\) 35.7904 24.3320i 1.28315 0.872343i
\(779\) 1.15380i 0.0413391i
\(780\) 0 0
\(781\) 6.53085i 0.233692i
\(782\) 6.26870 + 9.22076i 0.224168 + 0.329734i
\(783\) 0 0
\(784\) 2.91773 + 2.73622i 0.104204 + 0.0977220i
\(785\) 46.2601i 1.65109i
\(786\) 0 0
\(787\) 12.9317 0.460964 0.230482 0.973077i \(-0.425970\pi\)
0.230482 + 0.973077i \(0.425970\pi\)
\(788\) 4.23741 10.7131i 0.150951 0.381637i
\(789\) 0 0
\(790\) 2.10079 + 3.09010i 0.0747429 + 0.109941i
\(791\) −12.7809 −0.454437
\(792\) 0 0
\(793\) −48.9299 −1.73755
\(794\) −0.321017 0.472190i −0.0113925 0.0167574i
\(795\) 0 0
\(796\) −21.9035 8.66364i −0.776349 0.307075i
\(797\) −11.1981 −0.396656 −0.198328 0.980136i \(-0.563551\pi\)
−0.198328 + 0.980136i \(0.563551\pi\)
\(798\) 0 0
\(799\) 7.68675i 0.271938i
\(800\) 3.45823 + 0.544734i 0.122267 + 0.0192592i
\(801\) 0 0
\(802\) 24.1536 + 35.5280i 0.852894 + 1.25454i
\(803\) 15.4051i 0.543633i
\(804\) 0 0
\(805\) 5.03021i 0.177291i
\(806\) −9.95334 + 6.76674i −0.350591 + 0.238348i
\(807\) 0 0
\(808\) −1.97124 8.79788i −0.0693481 0.309508i
\(809\) 23.6434i 0.831258i 0.909534 + 0.415629i \(0.136439\pi\)
−0.909534 + 0.415629i \(0.863561\pi\)
\(810\) 0 0
\(811\) 34.8213 1.22274 0.611371 0.791344i \(-0.290618\pi\)
0.611371 + 0.791344i \(0.290618\pi\)
\(812\) −17.3291 6.85430i −0.608132 0.240539i
\(813\) 0 0
\(814\) 2.57844 1.75294i 0.0903742 0.0614406i
\(815\) −23.4542 −0.821564
\(816\) 0 0
\(817\) −31.7164 −1.10962
\(818\) −20.8819 + 14.1965i −0.730120 + 0.496369i
\(819\) 0 0
\(820\) −1.22391 0.484103i −0.0427409 0.0169056i
\(821\) 0.361884 0.0126298 0.00631491 0.999980i \(-0.497990\pi\)
0.00631491 + 0.999980i \(0.497990\pi\)
\(822\) 0 0
\(823\) 49.2033i 1.71512i −0.514386 0.857559i \(-0.671980\pi\)
0.514386 0.857559i \(-0.328020\pi\)
\(824\) 1.47777 0.331108i 0.0514806 0.0115347i
\(825\) 0 0
\(826\) 9.35078 6.35709i 0.325355 0.221192i
\(827\) 16.2920i 0.566529i −0.959042 0.283264i \(-0.908583\pi\)
0.959042 0.283264i \(-0.0914174\pi\)
\(828\) 0 0
\(829\) 6.40526i 0.222464i 0.993794 + 0.111232i \(0.0354796\pi\)
−0.993794 + 0.111232i \(0.964520\pi\)
\(830\) −4.98432 7.33154i −0.173008 0.254482i
\(831\) 0 0
\(832\) 15.7096 + 33.2969i 0.544632 + 1.15436i
\(833\) 3.28066i 0.113668i
\(834\) 0 0
\(835\) −3.69703 −0.127941
\(836\) 6.56031 + 2.59485i 0.226893 + 0.0897446i
\(837\) 0 0
\(838\) −24.7762 36.4438i −0.855879 1.25893i
\(839\) −23.3877 −0.807432 −0.403716 0.914884i \(-0.632282\pi\)
−0.403716 + 0.914884i \(0.632282\pi\)
\(840\) 0 0
\(841\) 57.8197 1.99378
\(842\) 5.71912 + 8.41237i 0.197094 + 0.289909i
\(843\) 0 0
\(844\) 14.7103 37.1906i 0.506348 1.28015i
\(845\) 17.1202 0.588953
\(846\) 0 0
\(847\) 10.0761i 0.346218i
\(848\) 22.5643 + 21.1606i 0.774863 + 0.726659i
\(849\) 0 0
\(850\) 1.61431 + 2.37452i 0.0553702 + 0.0814452i
\(851\) 5.51211i 0.188953i
\(852\) 0 0
\(853\) 22.3667i 0.765821i −0.923786 0.382910i \(-0.874922\pi\)
0.923786 0.382910i \(-0.125078\pi\)
\(854\) 12.4346 8.45360i 0.425502 0.289276i
\(855\) 0 0
\(856\) −26.7906 + 6.00266i −0.915682 + 0.205167i
\(857\) 34.7703i 1.18773i −0.804564 0.593866i \(-0.797601\pi\)
0.804564 0.593866i \(-0.202399\pi\)
\(858\) 0 0
\(859\) 16.1287 0.550305 0.275152 0.961401i \(-0.411272\pi\)
0.275152 + 0.961401i \(0.411272\pi\)
\(860\) 13.3073 33.6437i 0.453777 1.14724i
\(861\) 0 0
\(862\) 30.1270 20.4817i 1.02613 0.697611i
\(863\) −34.3951 −1.17082 −0.585412 0.810736i \(-0.699067\pi\)
−0.585412 + 0.810736i \(0.699067\pi\)
\(864\) 0 0
\(865\) −9.15194 −0.311175
\(866\) −4.28575 + 2.91365i −0.145636 + 0.0990099i
\(867\) 0 0
\(868\) 1.36036 3.43927i 0.0461736 0.116737i
\(869\) 1.21335 0.0411600
\(870\) 0 0
\(871\) 18.9700i 0.642774i
\(872\) −22.1787 + 4.96933i −0.751064 + 0.168283i
\(873\) 0 0
\(874\) −10.3144 + 7.01222i −0.348891 + 0.237192i
\(875\) 11.7609i 0.397592i
\(876\) 0 0
\(877\) 14.0555i 0.474620i −0.971434 0.237310i \(-0.923734\pi\)
0.971434 0.237310i \(-0.0762658\pi\)
\(878\) −31.3549 46.1206i −1.05818 1.55649i
\(879\) 0 0
\(880\) −5.50506 + 5.87024i −0.185575 + 0.197886i
\(881\) 41.1209i 1.38540i 0.721226 + 0.692700i \(0.243579\pi\)
−0.721226 + 0.692700i \(0.756421\pi\)
\(882\) 0 0
\(883\) −18.8276 −0.633598 −0.316799 0.948493i \(-0.602608\pi\)
−0.316799 + 0.948493i \(0.602608\pi\)
\(884\) −11.1063 + 28.0791i −0.373547 + 0.944404i
\(885\) 0 0
\(886\) 28.7380 + 42.2713i 0.965473 + 1.42013i
\(887\) −10.8728 −0.365071 −0.182536 0.983199i \(-0.558430\pi\)
−0.182536 + 0.983199i \(0.558430\pi\)
\(888\) 0 0
\(889\) −16.6260 −0.557617
\(890\) −21.4670 31.5763i −0.719577 1.05844i
\(891\) 0 0
\(892\) 12.7219 + 5.03199i 0.425961 + 0.168483i
\(893\) −8.59846 −0.287736
\(894\) 0 0
\(895\) 35.5971i 1.18988i
\(896\) −9.74499 5.74762i −0.325557 0.192014i
\(897\) 0 0
\(898\) −19.8350 29.1758i −0.661904 0.973608i
\(899\) 17.2310i 0.574684i
\(900\) 0 0
\(901\) 25.3711i 0.845235i
\(902\) −0.353446 + 0.240289i −0.0117684 + 0.00800074i
\(903\) 0 0
\(904\) 35.2753 7.90374i 1.17324 0.262874i
\(905\) 17.8045i 0.591843i
\(906\) 0 0
\(907\) 15.4589 0.513304 0.256652 0.966504i \(-0.417381\pi\)
0.256652 + 0.966504i \(0.417381\pi\)
\(908\) −25.8626 10.2296i −0.858281 0.339482i
\(909\) 0 0
\(910\) −11.2658 + 7.65901i −0.373457 + 0.253894i
\(911\) −23.9492 −0.793473 −0.396737 0.917933i \(-0.629857\pi\)
−0.396737 + 0.917933i \(0.629857\pi\)
\(912\) 0 0
\(913\) −2.87878 −0.0952736
\(914\) 17.7601 12.0741i 0.587452 0.399377i
\(915\) 0 0
\(916\) 7.21480 + 2.85372i 0.238384 + 0.0942896i
\(917\) 14.5603 0.480823
\(918\) 0 0
\(919\) 24.4853i 0.807694i −0.914826 0.403847i \(-0.867673\pi\)
0.914826 0.403847i \(-0.132327\pi\)
\(920\) −3.11069 13.8833i −0.102556 0.457720i
\(921\) 0 0
\(922\) 10.8706 7.39032i 0.358003 0.243387i
\(923\) 31.2685i 1.02922i
\(924\) 0 0
\(925\) 1.41947i 0.0466719i
\(926\) 1.69705 + 2.49623i 0.0557686 + 0.0820312i
\(927\) 0 0
\(928\) 52.0669 + 8.20149i 1.70918 + 0.269227i
\(929\) 35.7122i 1.17168i −0.810428 0.585839i \(-0.800765\pi\)
0.810428 0.585839i \(-0.199235\pi\)
\(930\) 0 0
\(931\) −3.66977 −0.120272
\(932\) 40.6007 + 16.0591i 1.32992 + 0.526033i
\(933\) 0 0
\(934\) −33.0825 48.6617i −1.08249 1.59226i
\(935\) −6.60044 −0.215857
\(936\) 0 0
\(937\) 0.263959 0.00862315 0.00431157 0.999991i \(-0.498628\pi\)
0.00431157 + 0.999991i \(0.498628\pi\)
\(938\) 3.27744 + 4.82086i 0.107012 + 0.157407i
\(939\) 0 0
\(940\) 3.60768 9.12097i 0.117670 0.297493i
\(941\) 40.9124 1.33371 0.666853 0.745189i \(-0.267641\pi\)
0.666853 + 0.745189i \(0.267641\pi\)
\(942\) 0 0
\(943\) 0.755586i 0.0246053i
\(944\) −21.8769 + 23.3281i −0.712031 + 0.759265i
\(945\) 0 0
\(946\) −6.60522 9.71575i −0.214754 0.315886i
\(947\) 43.3236i 1.40783i 0.710286 + 0.703913i \(0.248566\pi\)
−0.710286 + 0.703913i \(0.751434\pi\)
\(948\) 0 0
\(949\) 73.7566i 2.39424i
\(950\) −2.65615 + 1.80578i −0.0861770 + 0.0585871i
\(951\) 0 0
\(952\) −2.02877 9.05461i −0.0657527 0.293462i
\(953\) 52.0756i 1.68690i −0.537211 0.843448i \(-0.680522\pi\)
0.537211 0.843448i \(-0.319478\pi\)
\(954\) 0 0
\(955\) 1.51791 0.0491185
\(956\) 20.5205 51.8802i 0.663682 1.67793i
\(957\) 0 0
\(958\) 47.8856 32.5549i 1.54711 1.05180i
\(959\) 17.2672 0.557588
\(960\) 0 0
\(961\) 27.5802 0.889684
\(962\) −12.3451 + 8.39276i −0.398021 + 0.270593i
\(963\) 0 0
\(964\) −4.54324 + 11.4863i −0.146328 + 0.369947i
\(965\) 32.9976 1.06223
\(966\) 0 0
\(967\) 17.3116i 0.556705i −0.960479 0.278352i \(-0.910212\pi\)
0.960479 0.278352i \(-0.0897883\pi\)
\(968\) −6.23106 27.8099i −0.200274 0.893845i
\(969\) 0 0
\(970\) −12.1994 + 8.29369i −0.391698 + 0.266294i
\(971\) 8.86074i 0.284355i −0.989841 0.142177i \(-0.954590\pi\)
0.989841 0.142177i \(-0.0454103\pi\)
\(972\) 0 0
\(973\) 14.5580i 0.466709i
\(974\) −11.2936 16.6119i −0.361869 0.532280i
\(975\) 0 0
\(976\) −29.0917 + 31.0215i −0.931201 + 0.992973i
\(977\) 8.48509i 0.271462i 0.990746 + 0.135731i \(0.0433383\pi\)
−0.990746 + 0.135731i \(0.956662\pi\)
\(978\) 0 0
\(979\) −12.3986 −0.396262
\(980\) 1.53974 3.89278i 0.0491851 0.124350i
\(981\) 0 0
\(982\) 31.2486 + 45.9642i 0.997183 + 1.46678i
\(983\) −43.6181 −1.39120 −0.695600 0.718429i \(-0.744862\pi\)
−0.695600 + 0.718429i \(0.744862\pi\)
\(984\) 0 0
\(985\) −12.0570 −0.384168
\(986\) 24.3049 + 35.7506i 0.774026 + 1.13853i
\(987\) 0 0
\(988\) −31.4095 12.4236i −0.999271 0.395249i
\(989\) 20.7700 0.660449
\(990\) 0 0
\(991\) 51.0937i 1.62304i −0.584322 0.811522i \(-0.698639\pi\)
0.584322 0.811522i \(-0.301361\pi\)
\(992\) −1.62773 + 10.3336i −0.0516806 + 0.328093i
\(993\) 0 0
\(994\) 5.40225 + 7.94629i 0.171349 + 0.252041i
\(995\) 24.6513i 0.781498i
\(996\) 0 0
\(997\) 1.59455i 0.0504998i 0.999681 + 0.0252499i \(0.00803815\pi\)
−0.999681 + 0.0252499i \(0.991962\pi\)
\(998\) −37.8060 + 25.7023i −1.19673 + 0.813591i
\(999\) 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.323.25 yes 32
3.2 odd 2 inner 1512.2.j.c.323.8 yes 32
4.3 odd 2 6048.2.j.c.5615.8 32
8.3 odd 2 inner 1512.2.j.c.323.7 32
8.5 even 2 6048.2.j.c.5615.26 32
12.11 even 2 6048.2.j.c.5615.25 32
24.5 odd 2 6048.2.j.c.5615.7 32
24.11 even 2 inner 1512.2.j.c.323.26 yes 32
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.j.c.323.7 32 8.3 odd 2 inner
1512.2.j.c.323.8 yes 32 3.2 odd 2 inner
1512.2.j.c.323.25 yes 32 1.1 even 1 trivial
1512.2.j.c.323.26 yes 32 24.11 even 2 inner
6048.2.j.c.5615.7 32 24.5 odd 2
6048.2.j.c.5615.8 32 4.3 odd 2
6048.2.j.c.5615.25 32 12.11 even 2
6048.2.j.c.5615.26 32 8.5 even 2