Properties

Label 256.2.e.b.65.3
Level $256$
Weight $2$
Character 256.65
Analytic conductor $2.044$
Analytic rank $0$
Dimension $8$
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] = [256,2,Mod(65,256)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(256, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("256.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 256.e (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.04417029174\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 65.3
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 256.65
Dual form 256.2.e.b.193.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.517638 + 0.517638i) q^{3} +(1.73205 - 1.73205i) q^{5} -3.86370i q^{7} -2.46410i q^{9} +O(q^{10})\) \(q+(0.517638 + 0.517638i) q^{3} +(1.73205 - 1.73205i) q^{5} -3.86370i q^{7} -2.46410i q^{9} +(-3.34607 + 3.34607i) q^{11} +(0.267949 + 0.267949i) q^{13} +1.79315 q^{15} +3.46410 q^{17} +(3.34607 + 3.34607i) q^{19} +(2.00000 - 2.00000i) q^{21} +1.79315i q^{23} -1.00000i q^{25} +(2.82843 - 2.82843i) q^{27} +(1.73205 + 1.73205i) q^{29} -5.65685 q^{31} -3.46410 q^{33} +(-6.69213 - 6.69213i) q^{35} +(-3.73205 + 3.73205i) q^{37} +0.277401i q^{39} +6.92820i q^{41} +(1.55291 - 1.55291i) q^{43} +(-4.26795 - 4.26795i) q^{45} +9.79796 q^{47} -7.92820 q^{49} +(1.79315 + 1.79315i) q^{51} +(-7.73205 + 7.73205i) q^{53} +11.5911i q^{55} +3.46410i q^{57} +(5.13922 - 5.13922i) q^{59} +(-3.73205 - 3.73205i) q^{61} -9.52056 q^{63} +0.928203 q^{65} +(-4.38134 - 4.38134i) q^{67} +(-0.928203 + 0.928203i) q^{69} +1.79315i q^{71} -2.53590i q^{73} +(0.517638 - 0.517638i) q^{75} +(12.9282 + 12.9282i) q^{77} -4.14110 q^{79} -4.46410 q^{81} +(-1.55291 - 1.55291i) q^{83} +(6.00000 - 6.00000i) q^{85} +1.79315i q^{87} +2.53590i q^{89} +(1.03528 - 1.03528i) q^{91} +(-2.92820 - 2.92820i) q^{93} +11.5911 q^{95} +3.46410 q^{97} +(8.24504 + 8.24504i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{13} + 16 q^{21} - 16 q^{37} - 48 q^{45} - 8 q^{49} - 48 q^{53} - 16 q^{61} - 48 q^{65} + 48 q^{69} + 48 q^{77} - 8 q^{81} + 48 q^{85} + 32 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.517638 + 0.517638i 0.298858 + 0.298858i 0.840567 0.541708i \(-0.182222\pi\)
−0.541708 + 0.840567i \(0.682222\pi\)
\(4\) 0 0
\(5\) 1.73205 1.73205i 0.774597 0.774597i −0.204310 0.978906i \(-0.565495\pi\)
0.978906 + 0.204310i \(0.0654949\pi\)
\(6\) 0 0
\(7\) 3.86370i 1.46034i −0.683264 0.730171i \(-0.739440\pi\)
0.683264 0.730171i \(-0.260560\pi\)
\(8\) 0 0
\(9\) 2.46410i 0.821367i
\(10\) 0 0
\(11\) −3.34607 + 3.34607i −1.00888 + 1.00888i −0.00891637 + 0.999960i \(0.502838\pi\)
−0.999960 + 0.00891637i \(0.997162\pi\)
\(12\) 0 0
\(13\) 0.267949 + 0.267949i 0.0743157 + 0.0743157i 0.743288 0.668972i \(-0.233265\pi\)
−0.668972 + 0.743288i \(0.733265\pi\)
\(14\) 0 0
\(15\) 1.79315 0.462990
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) 3.34607 + 3.34607i 0.767640 + 0.767640i 0.977691 0.210051i \(-0.0673629\pi\)
−0.210051 + 0.977691i \(0.567363\pi\)
\(20\) 0 0
\(21\) 2.00000 2.00000i 0.436436 0.436436i
\(22\) 0 0
\(23\) 1.79315i 0.373898i 0.982370 + 0.186949i \(0.0598599\pi\)
−0.982370 + 0.186949i \(0.940140\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) 0 0
\(27\) 2.82843 2.82843i 0.544331 0.544331i
\(28\) 0 0
\(29\) 1.73205 + 1.73205i 0.321634 + 0.321634i 0.849394 0.527760i \(-0.176968\pi\)
−0.527760 + 0.849394i \(0.676968\pi\)
\(30\) 0 0
\(31\) −5.65685 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) 0 0
\(33\) −3.46410 −0.603023
\(34\) 0 0
\(35\) −6.69213 6.69213i −1.13118 1.13118i
\(36\) 0 0
\(37\) −3.73205 + 3.73205i −0.613545 + 0.613545i −0.943868 0.330323i \(-0.892842\pi\)
0.330323 + 0.943868i \(0.392842\pi\)
\(38\) 0 0
\(39\) 0.277401i 0.0444198i
\(40\) 0 0
\(41\) 6.92820i 1.08200i 0.841021 + 0.541002i \(0.181955\pi\)
−0.841021 + 0.541002i \(0.818045\pi\)
\(42\) 0 0
\(43\) 1.55291 1.55291i 0.236817 0.236817i −0.578714 0.815531i \(-0.696445\pi\)
0.815531 + 0.578714i \(0.196445\pi\)
\(44\) 0 0
\(45\) −4.26795 4.26795i −0.636228 0.636228i
\(46\) 0 0
\(47\) 9.79796 1.42918 0.714590 0.699544i \(-0.246613\pi\)
0.714590 + 0.699544i \(0.246613\pi\)
\(48\) 0 0
\(49\) −7.92820 −1.13260
\(50\) 0 0
\(51\) 1.79315 + 1.79315i 0.251091 + 0.251091i
\(52\) 0 0
\(53\) −7.73205 + 7.73205i −1.06208 + 1.06208i −0.0641378 + 0.997941i \(0.520430\pi\)
−0.997941 + 0.0641378i \(0.979570\pi\)
\(54\) 0 0
\(55\) 11.5911i 1.56294i
\(56\) 0 0
\(57\) 3.46410i 0.458831i
\(58\) 0 0
\(59\) 5.13922 5.13922i 0.669069 0.669069i −0.288432 0.957501i \(-0.593134\pi\)
0.957501 + 0.288432i \(0.0931338\pi\)
\(60\) 0 0
\(61\) −3.73205 3.73205i −0.477840 0.477840i 0.426600 0.904440i \(-0.359711\pi\)
−0.904440 + 0.426600i \(0.859711\pi\)
\(62\) 0 0
\(63\) −9.52056 −1.19948
\(64\) 0 0
\(65\) 0.928203 0.115129
\(66\) 0 0
\(67\) −4.38134 4.38134i −0.535266 0.535266i 0.386869 0.922135i \(-0.373557\pi\)
−0.922135 + 0.386869i \(0.873557\pi\)
\(68\) 0 0
\(69\) −0.928203 + 0.928203i −0.111743 + 0.111743i
\(70\) 0 0
\(71\) 1.79315i 0.212808i 0.994323 + 0.106404i \(0.0339337\pi\)
−0.994323 + 0.106404i \(0.966066\pi\)
\(72\) 0 0
\(73\) 2.53590i 0.296804i −0.988927 0.148402i \(-0.952587\pi\)
0.988927 0.148402i \(-0.0474130\pi\)
\(74\) 0 0
\(75\) 0.517638 0.517638i 0.0597717 0.0597717i
\(76\) 0 0
\(77\) 12.9282 + 12.9282i 1.47331 + 1.47331i
\(78\) 0 0
\(79\) −4.14110 −0.465911 −0.232955 0.972487i \(-0.574840\pi\)
−0.232955 + 0.972487i \(0.574840\pi\)
\(80\) 0 0
\(81\) −4.46410 −0.496011
\(82\) 0 0
\(83\) −1.55291 1.55291i −0.170454 0.170454i 0.616725 0.787179i \(-0.288459\pi\)
−0.787179 + 0.616725i \(0.788459\pi\)
\(84\) 0 0
\(85\) 6.00000 6.00000i 0.650791 0.650791i
\(86\) 0 0
\(87\) 1.79315i 0.192246i
\(88\) 0 0
\(89\) 2.53590i 0.268805i 0.990927 + 0.134402i \(0.0429115\pi\)
−0.990927 + 0.134402i \(0.957089\pi\)
\(90\) 0 0
\(91\) 1.03528 1.03528i 0.108526 0.108526i
\(92\) 0 0
\(93\) −2.92820 2.92820i −0.303641 0.303641i
\(94\) 0 0
\(95\) 11.5911 1.18922
\(96\) 0 0
\(97\) 3.46410 0.351726 0.175863 0.984415i \(-0.443728\pi\)
0.175863 + 0.984415i \(0.443728\pi\)
\(98\) 0 0
\(99\) 8.24504 + 8.24504i 0.828658 + 0.828658i
\(100\) 0 0
\(101\) −0.803848 + 0.803848i −0.0799858 + 0.0799858i −0.745968 0.665982i \(-0.768013\pi\)
0.665982 + 0.745968i \(0.268013\pi\)
\(102\) 0 0
\(103\) 9.52056i 0.938088i −0.883175 0.469044i \(-0.844598\pi\)
0.883175 0.469044i \(-0.155402\pi\)
\(104\) 0 0
\(105\) 6.92820i 0.676123i
\(106\) 0 0
\(107\) −13.1440 + 13.1440i −1.27068 + 1.27068i −0.324949 + 0.945731i \(0.605347\pi\)
−0.945731 + 0.324949i \(0.894653\pi\)
\(108\) 0 0
\(109\) −13.1962 13.1962i −1.26396 1.26396i −0.949156 0.314806i \(-0.898060\pi\)
−0.314806 0.949156i \(-0.601940\pi\)
\(110\) 0 0
\(111\) −3.86370 −0.366726
\(112\) 0 0
\(113\) 12.9282 1.21618 0.608092 0.793867i \(-0.291935\pi\)
0.608092 + 0.793867i \(0.291935\pi\)
\(114\) 0 0
\(115\) 3.10583 + 3.10583i 0.289620 + 0.289620i
\(116\) 0 0
\(117\) 0.660254 0.660254i 0.0610405 0.0610405i
\(118\) 0 0
\(119\) 13.3843i 1.22693i
\(120\) 0 0
\(121\) 11.3923i 1.03566i
\(122\) 0 0
\(123\) −3.58630 + 3.58630i −0.323366 + 0.323366i
\(124\) 0 0
\(125\) 6.92820 + 6.92820i 0.619677 + 0.619677i
\(126\) 0 0
\(127\) 5.65685 0.501965 0.250982 0.967992i \(-0.419246\pi\)
0.250982 + 0.967992i \(0.419246\pi\)
\(128\) 0 0
\(129\) 1.60770 0.141550
\(130\) 0 0
\(131\) −0.240237 0.240237i −0.0209896 0.0209896i 0.696534 0.717524i \(-0.254724\pi\)
−0.717524 + 0.696534i \(0.754724\pi\)
\(132\) 0 0
\(133\) 12.9282 12.9282i 1.12102 1.12102i
\(134\) 0 0
\(135\) 9.79796i 0.843274i
\(136\) 0 0
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) 0 0
\(139\) 7.20977 7.20977i 0.611525 0.611525i −0.331819 0.943343i \(-0.607662\pi\)
0.943343 + 0.331819i \(0.107662\pi\)
\(140\) 0 0
\(141\) 5.07180 + 5.07180i 0.427122 + 0.427122i
\(142\) 0 0
\(143\) −1.79315 −0.149951
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) −4.10394 4.10394i −0.338487 0.338487i
\(148\) 0 0
\(149\) 13.7321 13.7321i 1.12497 1.12497i 0.133991 0.990983i \(-0.457221\pi\)
0.990983 0.133991i \(-0.0427793\pi\)
\(150\) 0 0
\(151\) 9.52056i 0.774772i −0.921918 0.387386i \(-0.873378\pi\)
0.921918 0.387386i \(-0.126622\pi\)
\(152\) 0 0
\(153\) 8.53590i 0.690086i
\(154\) 0 0
\(155\) −9.79796 + 9.79796i −0.786991 + 0.786991i
\(156\) 0 0
\(157\) 7.19615 + 7.19615i 0.574315 + 0.574315i 0.933331 0.359016i \(-0.116888\pi\)
−0.359016 + 0.933331i \(0.616888\pi\)
\(158\) 0 0
\(159\) −8.00481 −0.634823
\(160\) 0 0
\(161\) 6.92820 0.546019
\(162\) 0 0
\(163\) 11.8313 + 11.8313i 0.926703 + 0.926703i 0.997491 0.0707887i \(-0.0225516\pi\)
−0.0707887 + 0.997491i \(0.522552\pi\)
\(164\) 0 0
\(165\) −6.00000 + 6.00000i −0.467099 + 0.467099i
\(166\) 0 0
\(167\) 18.7637i 1.45198i 0.687705 + 0.725990i \(0.258618\pi\)
−0.687705 + 0.725990i \(0.741382\pi\)
\(168\) 0 0
\(169\) 12.8564i 0.988954i
\(170\) 0 0
\(171\) 8.24504 8.24504i 0.630514 0.630514i
\(172\) 0 0
\(173\) −7.73205 7.73205i −0.587857 0.587857i 0.349194 0.937051i \(-0.386456\pi\)
−0.937051 + 0.349194i \(0.886456\pi\)
\(174\) 0 0
\(175\) −3.86370 −0.292069
\(176\) 0 0
\(177\) 5.32051 0.399914
\(178\) 0 0
\(179\) −1.55291 1.55291i −0.116070 0.116070i 0.646686 0.762756i \(-0.276154\pi\)
−0.762756 + 0.646686i \(0.776154\pi\)
\(180\) 0 0
\(181\) −16.1244 + 16.1244i −1.19851 + 1.19851i −0.223902 + 0.974612i \(0.571880\pi\)
−0.974612 + 0.223902i \(0.928120\pi\)
\(182\) 0 0
\(183\) 3.86370i 0.285613i
\(184\) 0 0
\(185\) 12.9282i 0.950500i
\(186\) 0 0
\(187\) −11.5911 + 11.5911i −0.847626 + 0.847626i
\(188\) 0 0
\(189\) −10.9282 10.9282i −0.794910 0.794910i
\(190\) 0 0
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) 0 0
\(193\) −18.3923 −1.32391 −0.661954 0.749545i \(-0.730272\pi\)
−0.661954 + 0.749545i \(0.730272\pi\)
\(194\) 0 0
\(195\) 0.480473 + 0.480473i 0.0344074 + 0.0344074i
\(196\) 0 0
\(197\) 1.73205 1.73205i 0.123404 0.123404i −0.642708 0.766111i \(-0.722189\pi\)
0.766111 + 0.642708i \(0.222189\pi\)
\(198\) 0 0
\(199\) 15.7322i 1.11523i 0.830101 + 0.557614i \(0.188283\pi\)
−0.830101 + 0.557614i \(0.811717\pi\)
\(200\) 0 0
\(201\) 4.53590i 0.319938i
\(202\) 0 0
\(203\) 6.69213 6.69213i 0.469695 0.469695i
\(204\) 0 0
\(205\) 12.0000 + 12.0000i 0.838116 + 0.838116i
\(206\) 0 0
\(207\) 4.41851 0.307107
\(208\) 0 0
\(209\) −22.3923 −1.54891
\(210\) 0 0
\(211\) −7.20977 7.20977i −0.496341 0.496341i 0.413956 0.910297i \(-0.364147\pi\)
−0.910297 + 0.413956i \(0.864147\pi\)
\(212\) 0 0
\(213\) −0.928203 + 0.928203i −0.0635994 + 0.0635994i
\(214\) 0 0
\(215\) 5.37945i 0.366876i
\(216\) 0 0
\(217\) 21.8564i 1.48371i
\(218\) 0 0
\(219\) 1.31268 1.31268i 0.0887025 0.0887025i
\(220\) 0 0
\(221\) 0.928203 + 0.928203i 0.0624377 + 0.0624377i
\(222\) 0 0
\(223\) −5.65685 −0.378811 −0.189405 0.981899i \(-0.560656\pi\)
−0.189405 + 0.981899i \(0.560656\pi\)
\(224\) 0 0
\(225\) −2.46410 −0.164273
\(226\) 0 0
\(227\) −14.9372 14.9372i −0.991415 0.991415i 0.00854833 0.999963i \(-0.497279\pi\)
−0.999963 + 0.00854833i \(0.997279\pi\)
\(228\) 0 0
\(229\) −9.19615 + 9.19615i −0.607699 + 0.607699i −0.942344 0.334645i \(-0.891383\pi\)
0.334645 + 0.942344i \(0.391383\pi\)
\(230\) 0 0
\(231\) 13.3843i 0.880620i
\(232\) 0 0
\(233\) 2.53590i 0.166132i −0.996544 0.0830661i \(-0.973529\pi\)
0.996544 0.0830661i \(-0.0264713\pi\)
\(234\) 0 0
\(235\) 16.9706 16.9706i 1.10704 1.10704i
\(236\) 0 0
\(237\) −2.14359 2.14359i −0.139241 0.139241i
\(238\) 0 0
\(239\) −26.7685 −1.73151 −0.865756 0.500467i \(-0.833162\pi\)
−0.865756 + 0.500467i \(0.833162\pi\)
\(240\) 0 0
\(241\) −2.39230 −0.154102 −0.0770510 0.997027i \(-0.524550\pi\)
−0.0770510 + 0.997027i \(0.524550\pi\)
\(242\) 0 0
\(243\) −10.7961 10.7961i −0.692568 0.692568i
\(244\) 0 0
\(245\) −13.7321 + 13.7321i −0.877309 + 0.877309i
\(246\) 0 0
\(247\) 1.79315i 0.114095i
\(248\) 0 0
\(249\) 1.60770i 0.101884i
\(250\) 0 0
\(251\) 14.9372 14.9372i 0.942826 0.942826i −0.0556256 0.998452i \(-0.517715\pi\)
0.998452 + 0.0556256i \(0.0177153\pi\)
\(252\) 0 0
\(253\) −6.00000 6.00000i −0.377217 0.377217i
\(254\) 0 0
\(255\) 6.21166 0.388989
\(256\) 0 0
\(257\) 12.9282 0.806439 0.403220 0.915103i \(-0.367891\pi\)
0.403220 + 0.915103i \(0.367891\pi\)
\(258\) 0 0
\(259\) 14.4195 + 14.4195i 0.895986 + 0.895986i
\(260\) 0 0
\(261\) 4.26795 4.26795i 0.264179 0.264179i
\(262\) 0 0
\(263\) 21.3891i 1.31891i 0.751746 + 0.659453i \(0.229212\pi\)
−0.751746 + 0.659453i \(0.770788\pi\)
\(264\) 0 0
\(265\) 26.7846i 1.64537i
\(266\) 0 0
\(267\) −1.31268 + 1.31268i −0.0803346 + 0.0803346i
\(268\) 0 0
\(269\) −5.19615 5.19615i −0.316815 0.316815i 0.530728 0.847543i \(-0.321919\pi\)
−0.847543 + 0.530728i \(0.821919\pi\)
\(270\) 0 0
\(271\) −1.51575 −0.0920752 −0.0460376 0.998940i \(-0.514659\pi\)
−0.0460376 + 0.998940i \(0.514659\pi\)
\(272\) 0 0
\(273\) 1.07180 0.0648681
\(274\) 0 0
\(275\) 3.34607 + 3.34607i 0.201775 + 0.201775i
\(276\) 0 0
\(277\) 3.19615 3.19615i 0.192038 0.192038i −0.604538 0.796576i \(-0.706642\pi\)
0.796576 + 0.604538i \(0.206642\pi\)
\(278\) 0 0
\(279\) 13.9391i 0.834510i
\(280\) 0 0
\(281\) 16.3923i 0.977883i −0.872317 0.488941i \(-0.837383\pi\)
0.872317 0.488941i \(-0.162617\pi\)
\(282\) 0 0
\(283\) −1.27551 + 1.27551i −0.0758214 + 0.0758214i −0.744000 0.668179i \(-0.767074\pi\)
0.668179 + 0.744000i \(0.267074\pi\)
\(284\) 0 0
\(285\) 6.00000 + 6.00000i 0.355409 + 0.355409i
\(286\) 0 0
\(287\) 26.7685 1.58010
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 1.79315 + 1.79315i 0.105116 + 0.105116i
\(292\) 0 0
\(293\) −12.1244 + 12.1244i −0.708312 + 0.708312i −0.966180 0.257868i \(-0.916980\pi\)
0.257868 + 0.966180i \(0.416980\pi\)
\(294\) 0 0
\(295\) 17.8028i 1.03652i
\(296\) 0 0
\(297\) 18.9282i 1.09833i
\(298\) 0 0
\(299\) −0.480473 + 0.480473i −0.0277865 + 0.0277865i
\(300\) 0 0
\(301\) −6.00000 6.00000i −0.345834 0.345834i
\(302\) 0 0
\(303\) −0.832204 −0.0478089
\(304\) 0 0
\(305\) −12.9282 −0.740267
\(306\) 0 0
\(307\) −21.9067 21.9067i −1.25028 1.25028i −0.955593 0.294688i \(-0.904784\pi\)
−0.294688 0.955593i \(-0.595216\pi\)
\(308\) 0 0
\(309\) 4.92820 4.92820i 0.280356 0.280356i
\(310\) 0 0
\(311\) 15.1774i 0.860632i −0.902678 0.430316i \(-0.858402\pi\)
0.902678 0.430316i \(-0.141598\pi\)
\(312\) 0 0
\(313\) 12.7846i 0.722629i −0.932444 0.361314i \(-0.882328\pi\)
0.932444 0.361314i \(-0.117672\pi\)
\(314\) 0 0
\(315\) −16.4901 + 16.4901i −0.929111 + 0.929111i
\(316\) 0 0
\(317\) 6.80385 + 6.80385i 0.382142 + 0.382142i 0.871873 0.489731i \(-0.162905\pi\)
−0.489731 + 0.871873i \(0.662905\pi\)
\(318\) 0 0
\(319\) −11.5911 −0.648978
\(320\) 0 0
\(321\) −13.6077 −0.759507
\(322\) 0 0
\(323\) 11.5911 + 11.5911i 0.644947 + 0.644947i
\(324\) 0 0
\(325\) 0.267949 0.267949i 0.0148631 0.0148631i
\(326\) 0 0
\(327\) 13.6617i 0.755492i
\(328\) 0 0
\(329\) 37.8564i 2.08709i
\(330\) 0 0
\(331\) −18.8009 + 18.8009i −1.03339 + 1.03339i −0.0339668 + 0.999423i \(0.510814\pi\)
−0.999423 + 0.0339668i \(0.989186\pi\)
\(332\) 0 0
\(333\) 9.19615 + 9.19615i 0.503946 + 0.503946i
\(334\) 0 0
\(335\) −15.1774 −0.829231
\(336\) 0 0
\(337\) 21.7128 1.18277 0.591386 0.806389i \(-0.298581\pi\)
0.591386 + 0.806389i \(0.298581\pi\)
\(338\) 0 0
\(339\) 6.69213 + 6.69213i 0.363467 + 0.363467i
\(340\) 0 0
\(341\) 18.9282 18.9282i 1.02502 1.02502i
\(342\) 0 0
\(343\) 3.58630i 0.193642i
\(344\) 0 0
\(345\) 3.21539i 0.173111i
\(346\) 0 0
\(347\) 5.13922 5.13922i 0.275888 0.275888i −0.555577 0.831465i \(-0.687503\pi\)
0.831465 + 0.555577i \(0.187503\pi\)
\(348\) 0 0
\(349\) −4.12436 4.12436i −0.220772 0.220772i 0.588052 0.808823i \(-0.299895\pi\)
−0.808823 + 0.588052i \(0.799895\pi\)
\(350\) 0 0
\(351\) 1.51575 0.0809047
\(352\) 0 0
\(353\) −0.928203 −0.0494033 −0.0247016 0.999695i \(-0.507864\pi\)
−0.0247016 + 0.999695i \(0.507864\pi\)
\(354\) 0 0
\(355\) 3.10583 + 3.10583i 0.164840 + 0.164840i
\(356\) 0 0
\(357\) 6.92820 6.92820i 0.366679 0.366679i
\(358\) 0 0
\(359\) 17.8028i 0.939594i −0.882774 0.469797i \(-0.844327\pi\)
0.882774 0.469797i \(-0.155673\pi\)
\(360\) 0 0
\(361\) 3.39230i 0.178542i
\(362\) 0 0
\(363\) 5.89709 5.89709i 0.309517 0.309517i
\(364\) 0 0
\(365\) −4.39230 4.39230i −0.229904 0.229904i
\(366\) 0 0
\(367\) 32.4254 1.69259 0.846295 0.532714i \(-0.178828\pi\)
0.846295 + 0.532714i \(0.178828\pi\)
\(368\) 0 0
\(369\) 17.0718 0.888722
\(370\) 0 0
\(371\) 29.8744 + 29.8744i 1.55100 + 1.55100i
\(372\) 0 0
\(373\) 14.1244 14.1244i 0.731331 0.731331i −0.239552 0.970884i \(-0.577001\pi\)
0.970884 + 0.239552i \(0.0770006\pi\)
\(374\) 0 0
\(375\) 7.17260i 0.370392i
\(376\) 0 0
\(377\) 0.928203i 0.0478049i
\(378\) 0 0
\(379\) −2.03339 + 2.03339i −0.104448 + 0.104448i −0.757400 0.652952i \(-0.773531\pi\)
0.652952 + 0.757400i \(0.273531\pi\)
\(380\) 0 0
\(381\) 2.92820 + 2.92820i 0.150016 + 0.150016i
\(382\) 0 0
\(383\) −19.5959 −1.00130 −0.500652 0.865648i \(-0.666906\pi\)
−0.500652 + 0.865648i \(0.666906\pi\)
\(384\) 0 0
\(385\) 44.7846 2.28244
\(386\) 0 0
\(387\) −3.82654 3.82654i −0.194514 0.194514i
\(388\) 0 0
\(389\) 4.26795 4.26795i 0.216394 0.216394i −0.590583 0.806977i \(-0.701102\pi\)
0.806977 + 0.590583i \(0.201102\pi\)
\(390\) 0 0
\(391\) 6.21166i 0.314137i
\(392\) 0 0
\(393\) 0.248711i 0.0125458i
\(394\) 0 0
\(395\) −7.17260 + 7.17260i −0.360893 + 0.360893i
\(396\) 0 0
\(397\) 19.5885 + 19.5885i 0.983116 + 0.983116i 0.999860 0.0167433i \(-0.00532982\pi\)
−0.0167433 + 0.999860i \(0.505330\pi\)
\(398\) 0 0
\(399\) 13.3843 0.670051
\(400\) 0 0
\(401\) 13.6077 0.679536 0.339768 0.940509i \(-0.389651\pi\)
0.339768 + 0.940509i \(0.389651\pi\)
\(402\) 0 0
\(403\) −1.51575 1.51575i −0.0755049 0.0755049i
\(404\) 0 0
\(405\) −7.73205 + 7.73205i −0.384209 + 0.384209i
\(406\) 0 0
\(407\) 24.9754i 1.23798i
\(408\) 0 0
\(409\) 17.0718i 0.844146i −0.906562 0.422073i \(-0.861303\pi\)
0.906562 0.422073i \(-0.138697\pi\)
\(410\) 0 0
\(411\) −6.21166 + 6.21166i −0.306398 + 0.306398i
\(412\) 0 0
\(413\) −19.8564 19.8564i −0.977070 0.977070i
\(414\) 0 0
\(415\) −5.37945 −0.264067
\(416\) 0 0
\(417\) 7.46410 0.365519
\(418\) 0 0
\(419\) −10.0382 10.0382i −0.490398 0.490398i 0.418034 0.908432i \(-0.362719\pi\)
−0.908432 + 0.418034i \(0.862719\pi\)
\(420\) 0 0
\(421\) 18.5167 18.5167i 0.902447 0.902447i −0.0932005 0.995647i \(-0.529710\pi\)
0.995647 + 0.0932005i \(0.0297098\pi\)
\(422\) 0 0
\(423\) 24.1432i 1.17388i
\(424\) 0 0
\(425\) 3.46410i 0.168034i
\(426\) 0 0
\(427\) −14.4195 + 14.4195i −0.697810 + 0.697810i
\(428\) 0 0
\(429\) −0.928203 0.928203i −0.0448141 0.0448141i
\(430\) 0 0
\(431\) −26.7685 −1.28939 −0.644697 0.764438i \(-0.723017\pi\)
−0.644697 + 0.764438i \(0.723017\pi\)
\(432\) 0 0
\(433\) −34.3923 −1.65279 −0.826394 0.563092i \(-0.809612\pi\)
−0.826394 + 0.563092i \(0.809612\pi\)
\(434\) 0 0
\(435\) 3.10583 + 3.10583i 0.148913 + 0.148913i
\(436\) 0 0
\(437\) −6.00000 + 6.00000i −0.287019 + 0.287019i
\(438\) 0 0
\(439\) 12.1459i 0.579693i −0.957073 0.289846i \(-0.906396\pi\)
0.957073 0.289846i \(-0.0936042\pi\)
\(440\) 0 0
\(441\) 19.5359i 0.930281i
\(442\) 0 0
\(443\) 10.0382 10.0382i 0.476929 0.476929i −0.427219 0.904148i \(-0.640507\pi\)
0.904148 + 0.427219i \(0.140507\pi\)
\(444\) 0 0
\(445\) 4.39230 + 4.39230i 0.208215 + 0.208215i
\(446\) 0 0
\(447\) 14.2165 0.672416
\(448\) 0 0
\(449\) −24.2487 −1.14437 −0.572184 0.820125i \(-0.693904\pi\)
−0.572184 + 0.820125i \(0.693904\pi\)
\(450\) 0 0
\(451\) −23.1822 23.1822i −1.09161 1.09161i
\(452\) 0 0
\(453\) 4.92820 4.92820i 0.231547 0.231547i
\(454\) 0 0
\(455\) 3.58630i 0.168128i
\(456\) 0 0
\(457\) 20.7846i 0.972263i −0.873886 0.486132i \(-0.838408\pi\)
0.873886 0.486132i \(-0.161592\pi\)
\(458\) 0 0
\(459\) 9.79796 9.79796i 0.457330 0.457330i
\(460\) 0 0
\(461\) −24.1244 24.1244i −1.12358 1.12358i −0.991198 0.132385i \(-0.957737\pi\)
−0.132385 0.991198i \(-0.542263\pi\)
\(462\) 0 0
\(463\) 38.0822 1.76983 0.884916 0.465751i \(-0.154216\pi\)
0.884916 + 0.465751i \(0.154216\pi\)
\(464\) 0 0
\(465\) −10.1436 −0.470398
\(466\) 0 0
\(467\) 20.3166 + 20.3166i 0.940141 + 0.940141i 0.998307 0.0581655i \(-0.0185251\pi\)
−0.0581655 + 0.998307i \(0.518525\pi\)
\(468\) 0 0
\(469\) −16.9282 + 16.9282i −0.781672 + 0.781672i
\(470\) 0 0
\(471\) 7.45001i 0.343278i
\(472\) 0 0
\(473\) 10.3923i 0.477839i
\(474\) 0 0
\(475\) 3.34607 3.34607i 0.153528 0.153528i
\(476\) 0 0
\(477\) 19.0526 + 19.0526i 0.872357 + 0.872357i
\(478\) 0 0
\(479\) −36.5665 −1.67077 −0.835383 0.549669i \(-0.814754\pi\)
−0.835383 + 0.549669i \(0.814754\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 3.58630 + 3.58630i 0.163182 + 0.163182i
\(484\) 0 0
\(485\) 6.00000 6.00000i 0.272446 0.272446i
\(486\) 0 0
\(487\) 27.0459i 1.22557i 0.790251 + 0.612784i \(0.209950\pi\)
−0.790251 + 0.612784i \(0.790050\pi\)
\(488\) 0 0
\(489\) 12.2487i 0.553906i
\(490\) 0 0
\(491\) −15.4176 + 15.4176i −0.695789 + 0.695789i −0.963499 0.267711i \(-0.913733\pi\)
0.267711 + 0.963499i \(0.413733\pi\)
\(492\) 0 0
\(493\) 6.00000 + 6.00000i 0.270226 + 0.270226i
\(494\) 0 0
\(495\) 28.5617 1.28375
\(496\) 0 0
\(497\) 6.92820 0.310772
\(498\) 0 0
\(499\) 4.10394 + 4.10394i 0.183718 + 0.183718i 0.792974 0.609256i \(-0.208532\pi\)
−0.609256 + 0.792974i \(0.708532\pi\)
\(500\) 0 0
\(501\) −9.71281 + 9.71281i −0.433936 + 0.433936i
\(502\) 0 0
\(503\) 40.9850i 1.82743i 0.406355 + 0.913715i \(0.366800\pi\)
−0.406355 + 0.913715i \(0.633200\pi\)
\(504\) 0 0
\(505\) 2.78461i 0.123914i
\(506\) 0 0
\(507\) 6.65497 6.65497i 0.295557 0.295557i
\(508\) 0 0
\(509\) −9.58846 9.58846i −0.425001 0.425001i 0.461920 0.886921i \(-0.347161\pi\)
−0.886921 + 0.461920i \(0.847161\pi\)
\(510\) 0 0
\(511\) −9.79796 −0.433436
\(512\) 0 0
\(513\) 18.9282 0.835701
\(514\) 0 0
\(515\) −16.4901 16.4901i −0.726640 0.726640i
\(516\) 0 0
\(517\) −32.7846 + 32.7846i −1.44187 + 1.44187i
\(518\) 0 0
\(519\) 8.00481i 0.351372i
\(520\) 0 0
\(521\) 12.0000i 0.525730i 0.964833 + 0.262865i \(0.0846673\pi\)
−0.964833 + 0.262865i \(0.915333\pi\)
\(522\) 0 0
\(523\) 13.6245 13.6245i 0.595758 0.595758i −0.343423 0.939181i \(-0.611586\pi\)
0.939181 + 0.343423i \(0.111586\pi\)
\(524\) 0 0
\(525\) −2.00000 2.00000i −0.0872872 0.0872872i
\(526\) 0 0
\(527\) −19.5959 −0.853612
\(528\) 0 0
\(529\) 19.7846 0.860200
\(530\) 0 0
\(531\) −12.6636 12.6636i −0.549551 0.549551i
\(532\) 0 0
\(533\) −1.85641 + 1.85641i −0.0804099 + 0.0804099i
\(534\) 0 0
\(535\) 45.5322i 1.96853i
\(536\) 0 0
\(537\) 1.60770i 0.0693772i
\(538\) 0 0
\(539\) 26.5283 26.5283i 1.14265 1.14265i
\(540\) 0 0
\(541\) 10.1244 + 10.1244i 0.435280 + 0.435280i 0.890420 0.455140i \(-0.150411\pi\)
−0.455140 + 0.890420i \(0.650411\pi\)
\(542\) 0 0
\(543\) −16.6932 −0.716372
\(544\) 0 0
\(545\) −45.7128 −1.95812
\(546\) 0 0
\(547\) −0.240237 0.240237i −0.0102718 0.0102718i 0.701952 0.712224i \(-0.252312\pi\)
−0.712224 + 0.701952i \(0.752312\pi\)
\(548\) 0 0
\(549\) −9.19615 + 9.19615i −0.392482 + 0.392482i
\(550\) 0 0
\(551\) 11.5911i 0.493798i
\(552\) 0 0
\(553\) 16.0000i 0.680389i
\(554\) 0 0
\(555\) −6.69213 + 6.69213i −0.284065 + 0.284065i
\(556\) 0 0
\(557\) 11.1962 + 11.1962i 0.474396 + 0.474396i 0.903334 0.428938i \(-0.141112\pi\)
−0.428938 + 0.903334i \(0.641112\pi\)
\(558\) 0 0
\(559\) 0.832204 0.0351985
\(560\) 0 0
\(561\) −12.0000 −0.506640
\(562\) 0 0
\(563\) 10.5187 + 10.5187i 0.443309 + 0.443309i 0.893123 0.449813i \(-0.148510\pi\)
−0.449813 + 0.893123i \(0.648510\pi\)
\(564\) 0 0
\(565\) 22.3923 22.3923i 0.942051 0.942051i
\(566\) 0 0
\(567\) 17.2480i 0.724346i
\(568\) 0 0
\(569\) 12.0000i 0.503066i −0.967849 0.251533i \(-0.919065\pi\)
0.967849 0.251533i \(-0.0809347\pi\)
\(570\) 0 0
\(571\) 21.3519 21.3519i 0.893549 0.893549i −0.101306 0.994855i \(-0.532302\pi\)
0.994855 + 0.101306i \(0.0323021\pi\)
\(572\) 0 0
\(573\) 8.78461 + 8.78461i 0.366982 + 0.366982i
\(574\) 0 0
\(575\) 1.79315 0.0747796
\(576\) 0 0
\(577\) −43.8564 −1.82577 −0.912883 0.408221i \(-0.866149\pi\)
−0.912883 + 0.408221i \(0.866149\pi\)
\(578\) 0 0
\(579\) −9.52056 9.52056i −0.395661 0.395661i
\(580\) 0 0
\(581\) −6.00000 + 6.00000i −0.248922 + 0.248922i
\(582\) 0 0
\(583\) 51.7439i 2.14301i
\(584\) 0 0
\(585\) 2.28719i 0.0945635i
\(586\) 0 0
\(587\) 6.45189 6.45189i 0.266298 0.266298i −0.561308 0.827607i \(-0.689702\pi\)
0.827607 + 0.561308i \(0.189702\pi\)
\(588\) 0 0
\(589\) −18.9282 18.9282i −0.779923 0.779923i
\(590\) 0 0
\(591\) 1.79315 0.0737604
\(592\) 0 0
\(593\) −19.8564 −0.815405 −0.407702 0.913115i \(-0.633670\pi\)
−0.407702 + 0.913115i \(0.633670\pi\)
\(594\) 0 0
\(595\) −23.1822 23.1822i −0.950378 0.950378i
\(596\) 0 0
\(597\) −8.14359 + 8.14359i −0.333295 + 0.333295i
\(598\) 0 0
\(599\) 4.41851i 0.180535i 0.995918 + 0.0902676i \(0.0287722\pi\)
−0.995918 + 0.0902676i \(0.971228\pi\)
\(600\) 0 0
\(601\) 40.3923i 1.64764i 0.566854 + 0.823818i \(0.308160\pi\)
−0.566854 + 0.823818i \(0.691840\pi\)
\(602\) 0 0
\(603\) −10.7961 + 10.7961i −0.439650 + 0.439650i
\(604\) 0 0
\(605\) −19.7321 19.7321i −0.802222 0.802222i
\(606\) 0 0
\(607\) 8.28221 0.336165 0.168082 0.985773i \(-0.446243\pi\)
0.168082 + 0.985773i \(0.446243\pi\)
\(608\) 0 0
\(609\) 6.92820 0.280745
\(610\) 0 0
\(611\) 2.62536 + 2.62536i 0.106211 + 0.106211i
\(612\) 0 0
\(613\) −15.0526 + 15.0526i −0.607967 + 0.607967i −0.942414 0.334447i \(-0.891450\pi\)
0.334447 + 0.942414i \(0.391450\pi\)
\(614\) 0 0
\(615\) 12.4233i 0.500956i
\(616\) 0 0
\(617\) 11.3205i 0.455746i −0.973691 0.227873i \(-0.926823\pi\)
0.973691 0.227873i \(-0.0731772\pi\)
\(618\) 0 0
\(619\) −18.8009 + 18.8009i −0.755671 + 0.755671i −0.975531 0.219860i \(-0.929440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(620\) 0 0
\(621\) 5.07180 + 5.07180i 0.203524 + 0.203524i
\(622\) 0 0
\(623\) 9.79796 0.392547
\(624\) 0 0
\(625\) 29.0000 1.16000
\(626\) 0 0
\(627\) −11.5911 11.5911i −0.462904 0.462904i
\(628\) 0 0
\(629\) −12.9282 + 12.9282i −0.515481 + 0.515481i
\(630\) 0 0
\(631\) 1.23835i 0.0492979i −0.999696 0.0246489i \(-0.992153\pi\)
0.999696 0.0246489i \(-0.00784679\pi\)
\(632\) 0 0
\(633\) 7.46410i 0.296671i
\(634\) 0 0
\(635\) 9.79796 9.79796i 0.388820 0.388820i
\(636\) 0 0
\(637\) −2.12436 2.12436i −0.0841700 0.0841700i
\(638\) 0 0
\(639\) 4.41851 0.174793
\(640\) 0 0
\(641\) 13.6077 0.537472 0.268736 0.963214i \(-0.413394\pi\)
0.268736 + 0.963214i \(0.413394\pi\)
\(642\) 0 0
\(643\) 6.93237 + 6.93237i 0.273386 + 0.273386i 0.830462 0.557076i \(-0.188077\pi\)
−0.557076 + 0.830462i \(0.688077\pi\)
\(644\) 0 0
\(645\) 2.78461 2.78461i 0.109644 0.109644i
\(646\) 0 0
\(647\) 1.79315i 0.0704960i 0.999379 + 0.0352480i \(0.0112221\pi\)
−0.999379 + 0.0352480i \(0.988778\pi\)
\(648\) 0 0
\(649\) 34.3923i 1.35002i
\(650\) 0 0
\(651\) −11.3137 + 11.3137i −0.443419 + 0.443419i
\(652\) 0 0
\(653\) −15.3397 15.3397i −0.600291 0.600291i 0.340099 0.940390i \(-0.389539\pi\)
−0.940390 + 0.340099i \(0.889539\pi\)
\(654\) 0 0
\(655\) −0.832204 −0.0325169
\(656\) 0 0
\(657\) −6.24871 −0.243785
\(658\) 0 0
\(659\) 18.0430 + 18.0430i 0.702856 + 0.702856i 0.965023 0.262167i \(-0.0844371\pi\)
−0.262167 + 0.965023i \(0.584437\pi\)
\(660\) 0 0
\(661\) −18.2679 + 18.2679i −0.710541 + 0.710541i −0.966648 0.256107i \(-0.917560\pi\)
0.256107 + 0.966648i \(0.417560\pi\)
\(662\) 0 0
\(663\) 0.960947i 0.0373201i
\(664\) 0 0
\(665\) 44.7846i 1.73667i
\(666\) 0 0
\(667\) −3.10583 + 3.10583i −0.120258 + 0.120258i
\(668\) 0 0
\(669\) −2.92820 2.92820i −0.113211 0.113211i
\(670\) 0 0
\(671\) 24.9754 0.964163
\(672\) 0 0
\(673\) 41.3205 1.59279 0.796394 0.604778i \(-0.206738\pi\)
0.796394 + 0.604778i \(0.206738\pi\)
\(674\) 0 0
\(675\) −2.82843 2.82843i −0.108866 0.108866i
\(676\) 0 0
\(677\) 28.2679 28.2679i 1.08643 1.08643i 0.0905320 0.995894i \(-0.471143\pi\)
0.995894 0.0905320i \(-0.0288567\pi\)
\(678\) 0 0
\(679\) 13.3843i 0.513641i
\(680\) 0 0
\(681\) 15.4641i 0.592586i
\(682\) 0 0
\(683\) 8.72552 8.72552i 0.333873 0.333873i −0.520182 0.854055i \(-0.674136\pi\)
0.854055 + 0.520182i \(0.174136\pi\)
\(684\) 0 0
\(685\) 20.7846 + 20.7846i 0.794139 + 0.794139i
\(686\) 0 0
\(687\) −9.52056 −0.363232
\(688\) 0 0
\(689\) −4.14359 −0.157858
\(690\) 0 0
\(691\) 27.8410 + 27.8410i 1.05912 + 1.05912i 0.998139 + 0.0609812i \(0.0194230\pi\)
0.0609812 + 0.998139i \(0.480577\pi\)
\(692\) 0 0
\(693\) 31.8564 31.8564i 1.21012 1.21012i
\(694\) 0 0
\(695\) 24.9754i 0.947370i
\(696\) 0 0
\(697\) 24.0000i 0.909065i
\(698\) 0 0
\(699\) 1.31268 1.31268i 0.0496500 0.0496500i
\(700\) 0 0
\(701\) 9.33975 + 9.33975i 0.352757 + 0.352757i 0.861135 0.508377i \(-0.169754\pi\)
−0.508377 + 0.861135i \(0.669754\pi\)
\(702\) 0 0
\(703\) −24.9754 −0.941964
\(704\) 0 0
\(705\) 17.5692 0.661695
\(706\) 0 0
\(707\) 3.10583 + 3.10583i 0.116807 + 0.116807i
\(708\) 0 0
\(709\) 29.0526 29.0526i 1.09109 1.09109i 0.0956796 0.995412i \(-0.469498\pi\)
0.995412 0.0956796i \(-0.0305024\pi\)
\(710\) 0 0
\(711\) 10.2041i 0.382684i
\(712\) 0 0
\(713\) 10.1436i 0.379881i
\(714\) 0 0
\(715\) −3.10583 + 3.10583i −0.116151 + 0.116151i
\(716\) 0 0
\(717\) −13.8564 13.8564i −0.517477 0.517477i
\(718\) 0 0
\(719\) −12.4233 −0.463311 −0.231656 0.972798i \(-0.574414\pi\)
−0.231656 + 0.972798i \(0.574414\pi\)
\(720\) 0 0
\(721\) −36.7846 −1.36993
\(722\) 0 0
\(723\) −1.23835 1.23835i −0.0460547 0.0460547i
\(724\) 0 0
\(725\) 1.73205 1.73205i 0.0643268 0.0643268i
\(726\) 0 0
\(727\) 23.4596i 0.870069i −0.900414 0.435035i \(-0.856736\pi\)
0.900414 0.435035i \(-0.143264\pi\)
\(728\) 0 0
\(729\) 2.21539i 0.0820515i
\(730\) 0 0
\(731\) 5.37945 5.37945i 0.198966 0.198966i
\(732\) 0 0
\(733\) 14.8038 + 14.8038i 0.546793 + 0.546793i 0.925512 0.378719i \(-0.123635\pi\)
−0.378719 + 0.925512i \(0.623635\pi\)
\(734\) 0 0
\(735\) −14.2165 −0.524382
\(736\) 0 0
\(737\) 29.3205 1.08003
\(738\) 0 0
\(739\) −2.10772 2.10772i −0.0775336 0.0775336i 0.667276 0.744810i \(-0.267460\pi\)
−0.744810 + 0.667276i \(0.767460\pi\)
\(740\) 0 0
\(741\) −0.928203 + 0.928203i −0.0340984 + 0.0340984i
\(742\) 0 0
\(743\) 15.1774i 0.556805i −0.960464 0.278403i \(-0.910195\pi\)
0.960464 0.278403i \(-0.0898050\pi\)
\(744\) 0 0
\(745\) 47.5692i 1.74280i
\(746\) 0 0
\(747\) −3.82654 + 3.82654i −0.140006 + 0.140006i
\(748\) 0 0
\(749\) 50.7846 + 50.7846i 1.85563 + 1.85563i
\(750\) 0 0
\(751\) 4.14110 0.151111 0.0755555 0.997142i \(-0.475927\pi\)
0.0755555 + 0.997142i \(0.475927\pi\)
\(752\) 0 0
\(753\) 15.4641 0.563543
\(754\) 0 0
\(755\) −16.4901 16.4901i −0.600136 0.600136i
\(756\) 0 0
\(757\) 21.7321 21.7321i 0.789865 0.789865i −0.191607 0.981472i \(-0.561370\pi\)
0.981472 + 0.191607i \(0.0613699\pi\)
\(758\) 0 0
\(759\) 6.21166i 0.225469i
\(760\) 0 0
\(761\) 34.6410i 1.25574i −0.778320 0.627868i \(-0.783928\pi\)
0.778320 0.627868i \(-0.216072\pi\)
\(762\) 0 0
\(763\) −50.9860 + 50.9860i −1.84582 + 1.84582i
\(764\) 0 0
\(765\) −14.7846 14.7846i −0.534539 0.534539i
\(766\) 0 0
\(767\) 2.75410 0.0994447
\(768\) 0 0
\(769\) −2.39230 −0.0862687 −0.0431344 0.999069i \(-0.513734\pi\)
−0.0431344 + 0.999069i \(0.513734\pi\)
\(770\) 0 0
\(771\) 6.69213 + 6.69213i 0.241011 + 0.241011i
\(772\) 0 0
\(773\) −14.6603 + 14.6603i −0.527293 + 0.527293i −0.919764 0.392471i \(-0.871620\pi\)
0.392471 + 0.919764i \(0.371620\pi\)
\(774\) 0 0
\(775\) 5.65685i 0.203200i
\(776\) 0 0
\(777\) 14.9282i 0.535546i
\(778\) 0 0
\(779\) −23.1822 + 23.1822i −0.830589 + 0.830589i
\(780\) 0 0
\(781\) −6.00000 6.00000i −0.214697 0.214697i
\(782\) 0 0
\(783\) 9.79796 0.350150
\(784\) 0 0
\(785\) 24.9282 0.889726
\(786\) 0 0
\(787\) 25.2156 + 25.2156i 0.898839 + 0.898839i 0.995334 0.0964942i \(-0.0307629\pi\)
−0.0964942 + 0.995334i \(0.530763\pi\)
\(788\) 0 0
\(789\) −11.0718 + 11.0718i −0.394166 + 0.394166i
\(790\) 0 0
\(791\) 49.9507i 1.77604i
\(792\) 0 0
\(793\) 2.00000i 0.0710221i
\(794\) 0 0
\(795\) −13.8647 + 13.8647i −0.491731 + 0.491731i
\(796\) 0 0
\(797\) −33.5885 33.5885i −1.18976 1.18976i −0.977132 0.212632i \(-0.931797\pi\)
−0.212632 0.977132i \(-0.568203\pi\)
\(798\) 0 0
\(799\) 33.9411 1.20075
\(800\) 0 0
\(801\) 6.24871 0.220787
\(802\) 0 0
\(803\) 8.48528 + 8.48528i 0.299439 + 0.299439i
\(804\) 0 0
\(805\) 12.0000 12.0000i 0.422944 0.422944i
\(806\) 0 0
\(807\) 5.37945i 0.189366i
\(808\) 0 0
\(809\) 17.0718i 0.600212i 0.953906 + 0.300106i \(0.0970221\pi\)
−0.953906 + 0.300106i \(0.902978\pi\)
\(810\) 0 0
\(811\) 33.2204 33.2204i 1.16653 1.16653i 0.183508 0.983018i \(-0.441255\pi\)
0.983018 0.183508i \(-0.0587455\pi\)
\(812\) 0 0
\(813\) −0.784610 0.784610i −0.0275175 0.0275175i
\(814\) 0 0
\(815\) 40.9850 1.43564
\(816\) 0 0
\(817\) 10.3923 0.363581
\(818\) 0 0
\(819\) −2.55103 2.55103i −0.0891401 0.0891401i
\(820\) 0 0
\(821\) 19.9808 19.9808i 0.697333 0.697333i −0.266501 0.963835i \(-0.585868\pi\)
0.963835 + 0.266501i \(0.0858676\pi\)
\(822\) 0 0
\(823\) 15.7322i 0.548391i 0.961674 + 0.274195i \(0.0884115\pi\)
−0.961674 + 0.274195i \(0.911589\pi\)
\(824\) 0 0
\(825\) 3.46410i 0.120605i
\(826\) 0 0
\(827\) −16.7303 + 16.7303i −0.581770 + 0.581770i −0.935390 0.353619i \(-0.884951\pi\)
0.353619 + 0.935390i \(0.384951\pi\)
\(828\) 0 0
\(829\) 2.12436 + 2.12436i 0.0737819 + 0.0737819i 0.743035 0.669253i \(-0.233386\pi\)
−0.669253 + 0.743035i \(0.733386\pi\)
\(830\) 0 0
\(831\) 3.30890 0.114784
\(832\) 0 0
\(833\) −27.4641 −0.951575
\(834\) 0 0
\(835\) 32.4997 + 32.4997i 1.12470 + 1.12470i
\(836\) 0 0
\(837\) −16.0000 + 16.0000i −0.553041 + 0.553041i
\(838\) 0 0
\(839\) 38.3596i 1.32432i 0.749362 + 0.662161i \(0.230360\pi\)
−0.749362 + 0.662161i \(0.769640\pi\)
\(840\) 0 0
\(841\) 23.0000i 0.793103i
\(842\) 0 0
\(843\) 8.48528 8.48528i 0.292249 0.292249i
\(844\) 0 0
\(845\) −22.2679 22.2679i −0.766041 0.766041i
\(846\) 0 0
\(847\) −44.0165 −1.51242
\(848\) 0 0
\(849\) −1.32051 −0.0453197
\(850\) 0 0
\(851\) −6.69213 6.69213i −0.229403 0.229403i
\(852\) 0 0
\(853\) −9.87564 + 9.87564i −0.338136 + 0.338136i −0.855665 0.517530i \(-0.826852\pi\)
0.517530 + 0.855665i \(0.326852\pi\)
\(854\) 0 0
\(855\) 28.5617i 0.976789i
\(856\) 0 0
\(857\) 25.8564i 0.883238i 0.897203 + 0.441619i \(0.145596\pi\)
−0.897203 + 0.441619i \(0.854404\pi\)
\(858\) 0 0
\(859\) −14.4567 + 14.4567i −0.493256 + 0.493256i −0.909331 0.416074i \(-0.863406\pi\)
0.416074 + 0.909331i \(0.363406\pi\)
\(860\) 0 0
\(861\) 13.8564 + 13.8564i 0.472225 + 0.472225i
\(862\) 0 0
\(863\) 2.62536 0.0893681 0.0446841 0.999001i \(-0.485772\pi\)
0.0446841 + 0.999001i \(0.485772\pi\)
\(864\) 0 0
\(865\) −26.7846 −0.910704
\(866\) 0 0
\(867\) −2.58819 2.58819i −0.0878996 0.0878996i
\(868\) 0 0
\(869\) 13.8564 13.8564i 0.470046 0.470046i
\(870\) 0 0
\(871\) 2.34795i 0.0795574i
\(872\) 0 0
\(873\) 8.53590i 0.288896i
\(874\) 0 0
\(875\) 26.7685 26.7685i 0.904941 0.904941i
\(876\) 0 0
\(877\) −16.1244 16.1244i −0.544481 0.544481i 0.380358 0.924839i \(-0.375801\pi\)
−0.924839 + 0.380358i \(0.875801\pi\)
\(878\) 0 0
\(879\) −12.5521 −0.423370
\(880\) 0 0
\(881\) 31.8564 1.07327 0.536635 0.843815i \(-0.319695\pi\)
0.536635 + 0.843815i \(0.319695\pi\)
\(882\) 0 0
\(883\) −40.3930 40.3930i −1.35933 1.35933i −0.874739 0.484594i \(-0.838967\pi\)
−0.484594 0.874739i \(-0.661033\pi\)
\(884\) 0 0
\(885\) 9.21539 9.21539i 0.309772 0.309772i
\(886\) 0 0
\(887\) 15.1774i 0.509608i −0.966993 0.254804i \(-0.917989\pi\)
0.966993 0.254804i \(-0.0820109\pi\)
\(888\) 0 0
\(889\) 21.8564i 0.733040i
\(890\) 0 0
\(891\) 14.9372 14.9372i 0.500414 0.500414i
\(892\) 0 0
\(893\) 32.7846 + 32.7846i 1.09710 + 1.09710i
\(894\) 0 0
\(895\) −5.37945 −0.179815
\(896\) 0 0
\(897\) −0.497423 −0.0166085
\(898\) 0 0
\(899\) −9.79796 9.79796i −0.326780 0.326780i
\(900\) 0 0
\(901\) −26.7846 + 26.7846i −0.892325 + 0.892325i
\(902\) 0 0
\(903\) 6.21166i 0.206711i
\(904\) 0 0
\(905\) 55.8564i 1.85673i
\(906\) 0 0
\(907\) 21.1488 21.1488i 0.702235 0.702235i −0.262655 0.964890i \(-0.584598\pi\)
0.964890 + 0.262655i \(0.0845979\pi\)
\(908\) 0 0
\(909\) 1.98076 + 1.98076i 0.0656977 + 0.0656977i
\(910\) 0 0
\(911\) −24.1432 −0.799899 −0.399949 0.916537i \(-0.630972\pi\)
−0.399949 + 0.916537i \(0.630972\pi\)
\(912\) 0 0
\(913\) 10.3923 0.343935
\(914\) 0 0
\(915\) −6.69213 6.69213i −0.221235 0.221235i
\(916\) 0 0
\(917\) −0.928203 + 0.928203i −0.0306520 + 0.0306520i
\(918\) 0 0
\(919\) 18.3576i 0.605560i 0.953060 + 0.302780i \(0.0979148\pi\)
−0.953060 + 0.302780i \(0.902085\pi\)
\(920\) 0 0
\(921\) 22.6795i 0.747315i
\(922\) 0 0
\(923\) −0.480473 + 0.480473i −0.0158150 + 0.0158150i
\(924\) 0 0
\(925\) 3.73205 + 3.73205i 0.122709 + 0.122709i
\(926\) 0 0
\(927\) −23.4596 −0.770515
\(928\) 0 0
\(929\) 3.46410 0.113653 0.0568267 0.998384i \(-0.481902\pi\)
0.0568267 + 0.998384i \(0.481902\pi\)
\(930\) 0 0
\(931\) −26.5283 26.5283i −0.869429 0.869429i
\(932\) 0 0
\(933\) 7.85641 7.85641i 0.257207 0.257207i
\(934\) 0 0
\(935\) 40.1528i 1.31314i
\(936\) 0 0
\(937\) 9.17691i 0.299797i 0.988701 + 0.149898i \(0.0478946\pi\)
−0.988701 + 0.149898i \(0.952105\pi\)
\(938\) 0 0
\(939\) 6.61780 6.61780i 0.215964 0.215964i
\(940\) 0 0
\(941\) 18.8038 + 18.8038i 0.612988 + 0.612988i 0.943723 0.330736i \(-0.107297\pi\)
−0.330736 + 0.943723i \(0.607297\pi\)
\(942\) 0 0
\(943\) −12.4233 −0.404559
\(944\) 0 0
\(945\) −37.8564 −1.23147
\(946\) 0 0
\(947\) −13.6245 13.6245i −0.442737 0.442737i 0.450194 0.892931i \(-0.351355\pi\)
−0.892931 + 0.450194i \(0.851355\pi\)
\(948\) 0 0
\(949\) 0.679492 0.679492i 0.0220572 0.0220572i
\(950\) 0 0
\(951\) 7.04386i 0.228413i
\(952\) 0 0
\(953\) 53.5692i 1.73528i 0.497195 + 0.867639i \(0.334363\pi\)
−0.497195 + 0.867639i \(0.665637\pi\)
\(954\) 0 0
\(955\) 29.3939 29.3939i 0.951164 0.951164i
\(956\) 0 0
\(957\) −6.00000 6.00000i −0.193952 0.193952i
\(958\) 0 0
\(959\) 46.3644 1.49719
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 32.3882 + 32.3882i 1.04370 + 1.04370i
\(964\) 0 0
\(965\) −31.8564 + 31.8564i −1.02549 + 1.02549i
\(966\) 0 0
\(967\) 27.0459i 0.869738i 0.900494 + 0.434869i \(0.143205\pi\)
−0.900494 + 0.434869i \(0.856795\pi\)
\(968\) 0 0
\(969\) 12.0000i 0.385496i
\(970\) 0 0
\(971\) −27.8410 + 27.8410i −0.893459 + 0.893459i −0.994847 0.101388i \(-0.967672\pi\)
0.101388 + 0.994847i \(0.467672\pi\)
\(972\) 0 0
\(973\) −27.8564 27.8564i −0.893035 0.893035i
\(974\) 0 0
\(975\) 0.277401 0.00888396
\(976\) 0 0
\(977\) −44.5359 −1.42483 −0.712415 0.701759i \(-0.752399\pi\)
−0.712415 + 0.701759i \(0.752399\pi\)
\(978\) 0 0
\(979\) −8.48528 8.48528i −0.271191 0.271191i
\(980\) 0 0
\(981\) −32.5167 + 32.5167i −1.03818 + 1.03818i
\(982\) 0 0
\(983\) 52.7048i 1.68102i 0.541793 + 0.840512i \(0.317745\pi\)
−0.541793 + 0.840512i \(0.682255\pi\)
\(984\) 0 0
\(985\) 6.00000i 0.191176i
\(986\) 0 0
\(987\) 19.5959 19.5959i 0.623745 0.623745i
\(988\) 0 0
\(989\) 2.78461 + 2.78461i 0.0885454 + 0.0885454i
\(990\) 0 0
\(991\) 61.8193 1.96375 0.981877 0.189521i \(-0.0606936\pi\)
0.981877 + 0.189521i \(0.0606936\pi\)
\(992\) 0 0
\(993\) −19.4641 −0.617675
\(994\) 0 0
\(995\) 27.2490 + 27.2490i 0.863851 + 0.863851i
\(996\) 0 0
\(997\) 15.1962 15.1962i 0.481267 0.481267i −0.424269 0.905536i \(-0.639469\pi\)
0.905536 + 0.424269i \(0.139469\pi\)
\(998\) 0 0
\(999\) 21.1117i 0.667944i
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 256.2.e.b.65.3 yes 8
3.2 odd 2 2304.2.k.k.577.1 8
4.3 odd 2 inner 256.2.e.b.65.2 yes 8
8.3 odd 2 256.2.e.a.65.3 yes 8
8.5 even 2 256.2.e.a.65.2 8
12.11 even 2 2304.2.k.k.577.2 8
16.3 odd 4 256.2.e.a.193.3 yes 8
16.5 even 4 inner 256.2.e.b.193.3 yes 8
16.11 odd 4 inner 256.2.e.b.193.2 yes 8
16.13 even 4 256.2.e.a.193.2 yes 8
24.5 odd 2 2304.2.k.f.577.3 8
24.11 even 2 2304.2.k.f.577.4 8
32.3 odd 8 1024.2.b.h.513.4 8
32.5 even 8 1024.2.a.j.1.2 4
32.11 odd 8 1024.2.a.j.1.1 4
32.13 even 8 1024.2.b.h.513.3 8
32.19 odd 8 1024.2.b.h.513.5 8
32.21 even 8 1024.2.a.g.1.3 4
32.27 odd 8 1024.2.a.g.1.4 4
32.29 even 8 1024.2.b.h.513.6 8
48.5 odd 4 2304.2.k.k.1729.2 8
48.11 even 4 2304.2.k.k.1729.1 8
48.29 odd 4 2304.2.k.f.1729.4 8
48.35 even 4 2304.2.k.f.1729.3 8
96.5 odd 8 9216.2.a.bb.1.2 4
96.11 even 8 9216.2.a.bb.1.3 4
96.53 odd 8 9216.2.a.bk.1.4 4
96.59 even 8 9216.2.a.bk.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
256.2.e.a.65.2 8 8.5 even 2
256.2.e.a.65.3 yes 8 8.3 odd 2
256.2.e.a.193.2 yes 8 16.13 even 4
256.2.e.a.193.3 yes 8 16.3 odd 4
256.2.e.b.65.2 yes 8 4.3 odd 2 inner
256.2.e.b.65.3 yes 8 1.1 even 1 trivial
256.2.e.b.193.2 yes 8 16.11 odd 4 inner
256.2.e.b.193.3 yes 8 16.5 even 4 inner
1024.2.a.g.1.3 4 32.21 even 8
1024.2.a.g.1.4 4 32.27 odd 8
1024.2.a.j.1.1 4 32.11 odd 8
1024.2.a.j.1.2 4 32.5 even 8
1024.2.b.h.513.3 8 32.13 even 8
1024.2.b.h.513.4 8 32.3 odd 8
1024.2.b.h.513.5 8 32.19 odd 8
1024.2.b.h.513.6 8 32.29 even 8
2304.2.k.f.577.3 8 24.5 odd 2
2304.2.k.f.577.4 8 24.11 even 2
2304.2.k.f.1729.3 8 48.35 even 4
2304.2.k.f.1729.4 8 48.29 odd 4
2304.2.k.k.577.1 8 3.2 odd 2
2304.2.k.k.577.2 8 12.11 even 2
2304.2.k.k.1729.1 8 48.11 even 4
2304.2.k.k.1729.2 8 48.5 odd 4
9216.2.a.bb.1.2 4 96.5 odd 8
9216.2.a.bb.1.3 4 96.11 even 8
9216.2.a.bk.1.1 4 96.59 even 8
9216.2.a.bk.1.4 4 96.53 odd 8