Properties

Label 2160.2.q.k.1441.2
Level $2160$
Weight $2$
Character 2160.1441
Analytic conductor $17.248$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2160,2,Mod(721,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.721");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2160.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.2476868366\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.954288.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1441.2
Root \(-1.62241 - 0.606458i\) of defining polynomial
Character \(\chi\) \(=\) 2160.1441
Dual form 2160.2.q.k.721.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +(0.714003 + 1.23669i) q^{7} +O(q^{10})\) \(q+(0.500000 - 0.866025i) q^{5} +(0.714003 + 1.23669i) q^{7} +(-1.33641 - 2.31473i) q^{11} +(-2.33641 + 4.04678i) q^{13} -2.67282 q^{17} -4.67282 q^{19} +(2.95882 - 5.12483i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-4.74482 - 8.21826i) q^{29} +(3.48040 - 6.02823i) q^{31} +1.42801 q^{35} -1.81681 q^{37} +(-0.735581 + 1.27406i) q^{41} +(0.235581 + 0.408039i) q^{43} +(-3.47842 - 6.02480i) q^{47} +(2.48040 - 4.29618i) q^{49} +1.14399 q^{53} -2.67282 q^{55} +(0.571993 - 0.990721i) q^{59} +(1.26442 + 2.19004i) q^{61} +(2.33641 + 4.04678i) q^{65} +(-3.29523 + 5.70751i) q^{67} -12.8745 q^{71} -1.71203 q^{73} +(1.90841 - 3.30545i) q^{77} +(-0.143987 - 0.249392i) q^{79} +(2.14201 + 3.71007i) q^{83} +(-1.33641 + 2.31473i) q^{85} +3.00000 q^{89} -6.67282 q^{91} +(-2.33641 + 4.04678i) q^{95} +(-3.91764 - 6.78555i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{5} + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{5} + 5 q^{7} + 2 q^{11} - 4 q^{13} + 4 q^{17} - 8 q^{19} - 3 q^{23} - 3 q^{25} - 7 q^{29} + 8 q^{31} + 10 q^{35} + 12 q^{37} - 13 q^{41} + 10 q^{43} - 13 q^{47} + 2 q^{49} + 4 q^{53} + 4 q^{55} + 2 q^{59} - q^{61} + 4 q^{65} + 11 q^{67} - 20 q^{71} - 16 q^{73} + 2 q^{79} + 15 q^{83} + 2 q^{85} + 18 q^{89} - 20 q^{91} - 4 q^{95} + 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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 0
\(4\) 0 0
\(5\) 0.500000 0.866025i 0.223607 0.387298i
\(6\) 0 0
\(7\) 0.714003 + 1.23669i 0.269868 + 0.467425i 0.968828 0.247736i \(-0.0796866\pi\)
−0.698960 + 0.715161i \(0.746353\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.33641 2.31473i −0.402943 0.697918i 0.591136 0.806572i \(-0.298679\pi\)
−0.994080 + 0.108653i \(0.965346\pi\)
\(12\) 0 0
\(13\) −2.33641 + 4.04678i −0.648004 + 1.12238i 0.335595 + 0.942006i \(0.391063\pi\)
−0.983599 + 0.180370i \(0.942271\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.67282 −0.648255 −0.324127 0.946013i \(-0.605071\pi\)
−0.324127 + 0.946013i \(0.605071\pi\)
\(18\) 0 0
\(19\) −4.67282 −1.07202 −0.536010 0.844212i \(-0.680069\pi\)
−0.536010 + 0.844212i \(0.680069\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.95882 5.12483i 0.616957 1.06860i −0.373081 0.927799i \(-0.621699\pi\)
0.990038 0.140802i \(-0.0449680\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.74482 8.21826i −0.881090 1.52609i −0.850130 0.526573i \(-0.823477\pi\)
−0.0309603 0.999521i \(-0.509857\pi\)
\(30\) 0 0
\(31\) 3.48040 6.02823i 0.625098 1.08270i −0.363424 0.931624i \(-0.618392\pi\)
0.988522 0.151078i \(-0.0482743\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.42801 0.241377
\(36\) 0 0
\(37\) −1.81681 −0.298682 −0.149341 0.988786i \(-0.547715\pi\)
−0.149341 + 0.988786i \(0.547715\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.735581 + 1.27406i −0.114879 + 0.198975i −0.917731 0.397202i \(-0.869981\pi\)
0.802853 + 0.596177i \(0.203315\pi\)
\(42\) 0 0
\(43\) 0.235581 + 0.408039i 0.0359258 + 0.0622254i 0.883429 0.468565i \(-0.155229\pi\)
−0.847503 + 0.530790i \(0.821895\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.47842 6.02480i −0.507380 0.878808i −0.999964 0.00854274i \(-0.997281\pi\)
0.492584 0.870265i \(-0.336053\pi\)
\(48\) 0 0
\(49\) 2.48040 4.29618i 0.354343 0.613739i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.14399 0.157139 0.0785693 0.996909i \(-0.474965\pi\)
0.0785693 + 0.996909i \(0.474965\pi\)
\(54\) 0 0
\(55\) −2.67282 −0.360403
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0.571993 0.990721i 0.0744672 0.128981i −0.826387 0.563102i \(-0.809608\pi\)
0.900854 + 0.434121i \(0.142941\pi\)
\(60\) 0 0
\(61\) 1.26442 + 2.19004i 0.161892 + 0.280406i 0.935547 0.353201i \(-0.114907\pi\)
−0.773655 + 0.633607i \(0.781574\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.33641 + 4.04678i 0.289796 + 0.501942i
\(66\) 0 0
\(67\) −3.29523 + 5.70751i −0.402577 + 0.697283i −0.994036 0.109051i \(-0.965219\pi\)
0.591459 + 0.806335i \(0.298552\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −12.8745 −1.52792 −0.763960 0.645263i \(-0.776748\pi\)
−0.763960 + 0.645263i \(0.776748\pi\)
\(72\) 0 0
\(73\) −1.71203 −0.200378 −0.100189 0.994968i \(-0.531945\pi\)
−0.100189 + 0.994968i \(0.531945\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.90841 3.30545i 0.217483 0.376692i
\(78\) 0 0
\(79\) −0.143987 0.249392i −0.0161998 0.0280588i 0.857812 0.513964i \(-0.171823\pi\)
−0.874012 + 0.485905i \(0.838490\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.14201 + 3.71007i 0.235116 + 0.407233i 0.959306 0.282367i \(-0.0911196\pi\)
−0.724190 + 0.689600i \(0.757786\pi\)
\(84\) 0 0
\(85\) −1.33641 + 2.31473i −0.144954 + 0.251068i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −6.67282 −0.699502
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.33641 + 4.04678i −0.239711 + 0.415191i
\(96\) 0 0
\(97\) −3.91764 6.78555i −0.397776 0.688968i 0.595675 0.803225i \(-0.296885\pi\)
−0.993451 + 0.114257i \(0.963551\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.10083 3.63875i −0.209040 0.362069i 0.742372 0.669988i \(-0.233701\pi\)
−0.951413 + 0.307919i \(0.900367\pi\)
\(102\) 0 0
\(103\) −0.908405 + 1.57340i −0.0895078 + 0.155032i −0.907303 0.420477i \(-0.861863\pi\)
0.817795 + 0.575509i \(0.195196\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.9176 1.15212 0.576061 0.817407i \(-0.304589\pi\)
0.576061 + 0.817407i \(0.304589\pi\)
\(108\) 0 0
\(109\) −16.6521 −1.59498 −0.797491 0.603331i \(-0.793840\pi\)
−0.797491 + 0.603331i \(0.793840\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.0616 17.4272i 0.946518 1.63942i 0.193836 0.981034i \(-0.437907\pi\)
0.752682 0.658384i \(-0.228760\pi\)
\(114\) 0 0
\(115\) −2.95882 5.12483i −0.275911 0.477893i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.90841 3.30545i −0.174943 0.303011i
\(120\) 0 0
\(121\) 1.92801 3.33941i 0.175273 0.303582i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −2.18714 −0.194078 −0.0970388 0.995281i \(-0.530937\pi\)
−0.0970388 + 0.995281i \(0.530937\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.00000 5.19615i 0.262111 0.453990i −0.704692 0.709514i \(-0.748915\pi\)
0.966803 + 0.255524i \(0.0822479\pi\)
\(132\) 0 0
\(133\) −3.33641 5.77883i −0.289304 0.501089i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.10083 8.83490i −0.435793 0.754816i 0.561567 0.827431i \(-0.310199\pi\)
−0.997360 + 0.0726153i \(0.976865\pi\)
\(138\) 0 0
\(139\) 4.00000 6.92820i 0.339276 0.587643i −0.645021 0.764165i \(-0.723151\pi\)
0.984297 + 0.176522i \(0.0564848\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.4896 1.04444
\(144\) 0 0
\(145\) −9.48963 −0.788071
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.0381 + 17.3865i −0.822351 + 1.42435i 0.0815762 + 0.996667i \(0.474005\pi\)
−0.903927 + 0.427687i \(0.859329\pi\)
\(150\) 0 0
\(151\) 1.51960 + 2.63203i 0.123663 + 0.214191i 0.921210 0.389066i \(-0.127202\pi\)
−0.797546 + 0.603258i \(0.793869\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.48040 6.02823i −0.279552 0.484199i
\(156\) 0 0
\(157\) −0.100830 + 0.174643i −0.00804714 + 0.0139381i −0.870021 0.493015i \(-0.835895\pi\)
0.861974 + 0.506953i \(0.169228\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.45043 0.665987
\(162\) 0 0
\(163\) −17.8168 −1.39552 −0.697760 0.716331i \(-0.745820\pi\)
−0.697760 + 0.716331i \(0.745820\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.05042 + 12.2117i −0.545578 + 0.944968i 0.452993 + 0.891514i \(0.350356\pi\)
−0.998570 + 0.0534538i \(0.982977\pi\)
\(168\) 0 0
\(169\) −4.41764 7.65158i −0.339819 0.588583i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.18319 + 3.78140i 0.165985 + 0.287494i 0.937005 0.349317i \(-0.113586\pi\)
−0.771020 + 0.636811i \(0.780253\pi\)
\(174\) 0 0
\(175\) 0.714003 1.23669i 0.0539736 0.0934850i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −15.1625 −1.13330 −0.566648 0.823960i \(-0.691760\pi\)
−0.566648 + 0.823960i \(0.691760\pi\)
\(180\) 0 0
\(181\) 3.20166 0.237978 0.118989 0.992896i \(-0.462035\pi\)
0.118989 + 0.992896i \(0.462035\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.908405 + 1.57340i −0.0667873 + 0.115679i
\(186\) 0 0
\(187\) 3.57199 + 6.18687i 0.261210 + 0.452429i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.41877 + 2.45738i 0.102659 + 0.177810i 0.912779 0.408453i \(-0.133932\pi\)
−0.810121 + 0.586263i \(0.800598\pi\)
\(192\) 0 0
\(193\) 9.39409 16.2710i 0.676201 1.17121i −0.299915 0.953966i \(-0.596958\pi\)
0.976116 0.217249i \(-0.0697083\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −5.83528 −0.415747 −0.207873 0.978156i \(-0.566654\pi\)
−0.207873 + 0.978156i \(0.566654\pi\)
\(198\) 0 0
\(199\) −13.0761 −0.926943 −0.463472 0.886112i \(-0.653396\pi\)
−0.463472 + 0.886112i \(0.653396\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 6.77563 11.7357i 0.475556 0.823687i
\(204\) 0 0
\(205\) 0.735581 + 1.27406i 0.0513752 + 0.0889845i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.24482 + 10.8163i 0.431963 + 0.748182i
\(210\) 0 0
\(211\) 4.19243 7.26149i 0.288618 0.499902i −0.684862 0.728673i \(-0.740137\pi\)
0.973480 + 0.228771i \(0.0734708\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0.471163 0.0321330
\(216\) 0 0
\(217\) 9.94006 0.674776
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.24482 10.8163i 0.420072 0.727586i
\(222\) 0 0
\(223\) 4.58321 + 7.93834i 0.306914 + 0.531591i 0.977686 0.210073i \(-0.0673702\pi\)
−0.670772 + 0.741664i \(0.734037\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.33641 + 2.31473i 0.0887008 + 0.153634i 0.906962 0.421212i \(-0.138395\pi\)
−0.818261 + 0.574846i \(0.805062\pi\)
\(228\) 0 0
\(229\) −1.27365 + 2.20603i −0.0841654 + 0.145779i −0.905035 0.425336i \(-0.860156\pi\)
0.820870 + 0.571115i \(0.193489\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 6.22013 0.407494 0.203747 0.979024i \(-0.434688\pi\)
0.203747 + 0.979024i \(0.434688\pi\)
\(234\) 0 0
\(235\) −6.95684 −0.453814
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −4.06163 + 7.03494i −0.262725 + 0.455053i −0.966965 0.254909i \(-0.917954\pi\)
0.704240 + 0.709962i \(0.251288\pi\)
\(240\) 0 0
\(241\) 13.1821 + 22.8320i 0.849131 + 1.47074i 0.881985 + 0.471278i \(0.156207\pi\)
−0.0328536 + 0.999460i \(0.510460\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.48040 4.29618i −0.158467 0.274473i
\(246\) 0 0
\(247\) 10.9176 18.9099i 0.694673 1.20321i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −0.549569 −0.0346885 −0.0173443 0.999850i \(-0.505521\pi\)
−0.0173443 + 0.999850i \(0.505521\pi\)
\(252\) 0 0
\(253\) −15.8168 −0.994394
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −9.00000 + 15.5885i −0.561405 + 0.972381i 0.435970 + 0.899961i \(0.356405\pi\)
−0.997374 + 0.0724199i \(0.976928\pi\)
\(258\) 0 0
\(259\) −1.29721 2.24683i −0.0806046 0.139611i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −5.94761 10.3016i −0.366745 0.635221i 0.622309 0.782771i \(-0.286195\pi\)
−0.989055 + 0.147550i \(0.952861\pi\)
\(264\) 0 0
\(265\) 0.571993 0.990721i 0.0351373 0.0608595i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −28.5737 −1.74217 −0.871084 0.491134i \(-0.836583\pi\)
−0.871084 + 0.491134i \(0.836583\pi\)
\(270\) 0 0
\(271\) 23.3641 1.41927 0.709635 0.704570i \(-0.248860\pi\)
0.709635 + 0.704570i \(0.248860\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.33641 + 2.31473i −0.0805887 + 0.139584i
\(276\) 0 0
\(277\) 7.53807 + 13.0563i 0.452919 + 0.784479i 0.998566 0.0535366i \(-0.0170494\pi\)
−0.545647 + 0.838015i \(0.683716\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.32605 + 5.76088i 0.198415 + 0.343665i 0.948015 0.318226i \(-0.103087\pi\)
−0.749599 + 0.661892i \(0.769754\pi\)
\(282\) 0 0
\(283\) 13.4485 23.2934i 0.799428 1.38465i −0.120562 0.992706i \(-0.538470\pi\)
0.919989 0.391943i \(-0.128197\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.10083 −0.124008
\(288\) 0 0
\(289\) −9.85601 −0.579765
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 6.19243 10.7256i 0.361765 0.626596i −0.626486 0.779433i \(-0.715507\pi\)
0.988251 + 0.152837i \(0.0488408\pi\)
\(294\) 0 0
\(295\) −0.571993 0.990721i −0.0333027 0.0576820i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 13.8260 + 23.9474i 0.799581 + 1.38491i
\(300\) 0 0
\(301\) −0.336412 + 0.582682i −0.0193905 + 0.0335853i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.52884 0.144801
\(306\) 0 0
\(307\) 2.49359 0.142317 0.0711583 0.997465i \(-0.477330\pi\)
0.0711583 + 0.997465i \(0.477330\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.1101 20.9752i 0.686699 1.18940i −0.286201 0.958170i \(-0.592392\pi\)
0.972900 0.231228i \(-0.0742742\pi\)
\(312\) 0 0
\(313\) 17.5420 + 30.3837i 0.991534 + 1.71739i 0.608219 + 0.793770i \(0.291884\pi\)
0.383315 + 0.923618i \(0.374782\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.23558 9.06829i −0.294060 0.509326i 0.680706 0.732557i \(-0.261673\pi\)
−0.974766 + 0.223231i \(0.928340\pi\)
\(318\) 0 0
\(319\) −12.6821 + 21.9660i −0.710059 + 1.22986i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 12.4896 0.694942
\(324\) 0 0
\(325\) 4.67282 0.259202
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.96721 8.60346i 0.273851 0.474324i
\(330\) 0 0
\(331\) 8.38880 + 14.5298i 0.461090 + 0.798632i 0.999016 0.0443606i \(-0.0141251\pi\)
−0.537925 + 0.842993i \(0.680792\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.29523 + 5.70751i 0.180038 + 0.311835i
\(336\) 0 0
\(337\) 13.5905 23.5394i 0.740320 1.28227i −0.212030 0.977263i \(-0.568007\pi\)
0.952350 0.305008i \(-0.0986592\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −18.6050 −1.00752
\(342\) 0 0
\(343\) 17.0801 0.922239
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 11.7829 20.4086i 0.632539 1.09559i −0.354492 0.935059i \(-0.615346\pi\)
0.987031 0.160530i \(-0.0513204\pi\)
\(348\) 0 0
\(349\) 5.35601 + 9.27689i 0.286701 + 0.496580i 0.973020 0.230720i \(-0.0741081\pi\)
−0.686319 + 0.727300i \(0.740775\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 13.6336 + 23.6141i 0.725644 + 1.25685i 0.958708 + 0.284392i \(0.0917916\pi\)
−0.233064 + 0.972461i \(0.574875\pi\)
\(354\) 0 0
\(355\) −6.43724 + 11.1496i −0.341653 + 0.591761i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.6807 −0.563707 −0.281854 0.959457i \(-0.590949\pi\)
−0.281854 + 0.959457i \(0.590949\pi\)
\(360\) 0 0
\(361\) 2.83528 0.149225
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −0.856013 + 1.48266i −0.0448058 + 0.0776059i
\(366\) 0 0
\(367\) −4.23558 7.33624i −0.221096 0.382949i 0.734045 0.679100i \(-0.237630\pi\)
−0.955141 + 0.296152i \(0.904297\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.816810 + 1.41476i 0.0424067 + 0.0734505i
\(372\) 0 0
\(373\) −5.06163 + 8.76700i −0.262081 + 0.453938i −0.966795 0.255554i \(-0.917742\pi\)
0.704714 + 0.709492i \(0.251075\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 44.3434 2.28380
\(378\) 0 0
\(379\) −11.9216 −0.612371 −0.306186 0.951972i \(-0.599053\pi\)
−0.306186 + 0.951972i \(0.599053\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.90841 + 8.50161i −0.250808 + 0.434412i −0.963748 0.266813i \(-0.914030\pi\)
0.712941 + 0.701224i \(0.247363\pi\)
\(384\) 0 0
\(385\) −1.90841 3.30545i −0.0972613 0.168462i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 4.61007 + 7.98487i 0.233740 + 0.404849i 0.958906 0.283725i \(-0.0915703\pi\)
−0.725166 + 0.688574i \(0.758237\pi\)
\(390\) 0 0
\(391\) −7.90841 + 13.6978i −0.399945 + 0.692725i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.287973 −0.0144895
\(396\) 0 0
\(397\) −22.9793 −1.15330 −0.576648 0.816993i \(-0.695640\pi\)
−0.576648 + 0.816993i \(0.695640\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5.53279 + 9.58307i −0.276294 + 0.478556i −0.970461 0.241259i \(-0.922440\pi\)
0.694167 + 0.719814i \(0.255773\pi\)
\(402\) 0 0
\(403\) 16.2633 + 28.1688i 0.810132 + 1.40319i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.42801 + 4.20543i 0.120352 + 0.208455i
\(408\) 0 0
\(409\) 8.81681 15.2712i 0.435963 0.755110i −0.561411 0.827537i \(-0.689741\pi\)
0.997374 + 0.0724270i \(0.0230744\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.63362 0.0803852
\(414\) 0 0
\(415\) 4.28402 0.210294
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −18.5173 + 32.0730i −0.904631 + 1.56687i −0.0832199 + 0.996531i \(0.526520\pi\)
−0.821411 + 0.570336i \(0.806813\pi\)
\(420\) 0 0
\(421\) −2.52884 4.38007i −0.123248 0.213472i 0.797799 0.602924i \(-0.205998\pi\)
−0.921047 + 0.389452i \(0.872664\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.33641 + 2.31473i 0.0648255 + 0.112281i
\(426\) 0 0
\(427\) −1.80560 + 3.12739i −0.0873790 + 0.151345i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −5.23030 −0.251935 −0.125967 0.992034i \(-0.540203\pi\)
−0.125967 + 0.992034i \(0.540203\pi\)
\(432\) 0 0
\(433\) 34.3434 1.65044 0.825219 0.564813i \(-0.191052\pi\)
0.825219 + 0.564813i \(0.191052\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −13.8260 + 23.9474i −0.661389 + 1.14556i
\(438\) 0 0
\(439\) −9.77365 16.9285i −0.466471 0.807952i 0.532796 0.846244i \(-0.321141\pi\)
−0.999267 + 0.0382924i \(0.987808\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.25208 9.09686i −0.249534 0.432205i 0.713863 0.700286i \(-0.246944\pi\)
−0.963396 + 0.268081i \(0.913611\pi\)
\(444\) 0 0
\(445\) 1.50000 2.59808i 0.0711068 0.123161i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −22.8560 −1.07864 −0.539321 0.842100i \(-0.681319\pi\)
−0.539321 + 0.842100i \(0.681319\pi\)
\(450\) 0 0
\(451\) 3.93216 0.185158
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3.33641 + 5.77883i −0.156413 + 0.270916i
\(456\) 0 0
\(457\) 7.01319 + 12.1472i 0.328063 + 0.568222i 0.982127 0.188217i \(-0.0602708\pi\)
−0.654064 + 0.756439i \(0.726937\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −12.0513 20.8734i −0.561283 0.972171i −0.997385 0.0722736i \(-0.976975\pi\)
0.436102 0.899897i \(-0.356359\pi\)
\(462\) 0 0
\(463\) −16.9700 + 29.3930i −0.788664 + 1.36601i 0.138121 + 0.990415i \(0.455894\pi\)
−0.926785 + 0.375591i \(0.877440\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −27.3720 −1.26663 −0.633313 0.773896i \(-0.718305\pi\)
−0.633313 + 0.773896i \(0.718305\pi\)
\(468\) 0 0
\(469\) −9.41123 −0.434570
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.629668 1.09062i 0.0289521 0.0501466i
\(474\) 0 0
\(475\) 2.33641 + 4.04678i 0.107202 + 0.185679i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −2.61515 4.52957i −0.119489 0.206961i 0.800076 0.599898i \(-0.204792\pi\)
−0.919565 + 0.392937i \(0.871459\pi\)
\(480\) 0 0
\(481\) 4.24482 7.35224i 0.193547 0.335233i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.83528 −0.355782
\(486\) 0 0
\(487\) 24.0185 1.08838 0.544190 0.838962i \(-0.316837\pi\)
0.544190 + 0.838962i \(0.316837\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7.38880 + 12.7978i −0.333452 + 0.577556i −0.983186 0.182606i \(-0.941547\pi\)
0.649734 + 0.760161i \(0.274880\pi\)
\(492\) 0 0
\(493\) 12.6821 + 21.9660i 0.571171 + 0.989298i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.19243 15.9217i −0.412337 0.714188i
\(498\) 0 0
\(499\) 12.4280 21.5259i 0.556354 0.963633i −0.441443 0.897289i \(-0.645533\pi\)
0.997797 0.0663440i \(-0.0211335\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 38.9154 1.73515 0.867576 0.497305i \(-0.165677\pi\)
0.867576 + 0.497305i \(0.165677\pi\)
\(504\) 0 0
\(505\) −4.20166 −0.186971
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.01037 1.75001i 0.0447837 0.0775676i −0.842765 0.538282i \(-0.819073\pi\)
0.887548 + 0.460715i \(0.152407\pi\)
\(510\) 0 0
\(511\) −1.22239 2.11725i −0.0540755 0.0936615i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0.908405 + 1.57340i 0.0400291 + 0.0693325i
\(516\) 0 0
\(517\) −9.29721 + 16.1032i −0.408891 + 0.708220i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 23.0290 1.00892 0.504460 0.863435i \(-0.331692\pi\)
0.504460 + 0.863435i \(0.331692\pi\)
\(522\) 0 0
\(523\) 41.1170 1.79792 0.898961 0.438028i \(-0.144323\pi\)
0.898961 + 0.438028i \(0.144323\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9.30249 + 16.1124i −0.405223 + 0.701867i
\(528\) 0 0
\(529\) −6.00924 10.4083i −0.261271 0.452535i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −3.43724 5.95348i −0.148883 0.257874i
\(534\) 0 0
\(535\) 5.95882 10.3210i 0.257622 0.446215i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −13.2593 −0.571120
\(540\) 0 0
\(541\) 5.20957 0.223977 0.111988 0.993710i \(-0.464278\pi\)
0.111988 + 0.993710i \(0.464278\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −8.32605 + 14.4211i −0.356649 + 0.617734i
\(546\) 0 0
\(547\) 20.0204 + 34.6764i 0.856013 + 1.48266i 0.875702 + 0.482851i \(0.160399\pi\)
−0.0196900 + 0.999806i \(0.506268\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 22.1717 + 38.4025i 0.944546 + 1.63600i
\(552\) 0 0
\(553\) 0.205614 0.356133i 0.00874359 0.0151443i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.4033 0.610288 0.305144 0.952306i \(-0.401295\pi\)
0.305144 + 0.952306i \(0.401295\pi\)
\(558\) 0 0
\(559\) −2.20166 −0.0931203
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −14.6840 + 25.4335i −0.618858 + 1.07189i 0.370836 + 0.928698i \(0.379071\pi\)
−0.989694 + 0.143196i \(0.954262\pi\)
\(564\) 0 0
\(565\) −10.0616 17.4272i −0.423296 0.733170i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.4033 + 40.5357i 0.981118 + 1.69935i 0.658056 + 0.752969i \(0.271379\pi\)
0.323062 + 0.946378i \(0.395288\pi\)
\(570\) 0 0
\(571\) 10.0000 17.3205i 0.418487 0.724841i −0.577301 0.816532i \(-0.695894\pi\)
0.995788 + 0.0916910i \(0.0292272\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.91764 −0.246783
\(576\) 0 0
\(577\) 28.2386 1.17559 0.587794 0.809010i \(-0.299996\pi\)
0.587794 + 0.809010i \(0.299996\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −3.05880 + 5.29801i −0.126901 + 0.219798i
\(582\) 0 0
\(583\) −1.52884 2.64802i −0.0633180 0.109670i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −9.04118 15.6598i −0.373169 0.646348i 0.616882 0.787056i \(-0.288396\pi\)
−0.990051 + 0.140707i \(0.955062\pi\)
\(588\) 0 0
\(589\) −16.2633 + 28.1688i −0.670117 + 1.16068i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7.73840 0.317778 0.158889 0.987296i \(-0.449209\pi\)
0.158889 + 0.987296i \(0.449209\pi\)
\(594\) 0 0
\(595\) −3.81681 −0.156474
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 13.9608 24.1808i 0.570423 0.988001i −0.426100 0.904676i \(-0.640113\pi\)
0.996522 0.0833249i \(-0.0265539\pi\)
\(600\) 0 0
\(601\) −19.2201 33.2902i −0.784006 1.35794i −0.929591 0.368592i \(-0.879840\pi\)
0.145586 0.989346i \(-0.453493\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.92801 3.33941i −0.0783846 0.135766i
\(606\) 0 0
\(607\) −0.319917 + 0.554113i −0.0129850 + 0.0224907i −0.872445 0.488712i \(-0.837467\pi\)
0.859460 + 0.511203i \(0.170800\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 32.5081 1.31514
\(612\) 0 0
\(613\) 42.7467 1.72652 0.863262 0.504757i \(-0.168418\pi\)
0.863262 + 0.504757i \(0.168418\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10.5513 18.2753i 0.424778 0.735737i −0.571622 0.820517i \(-0.693686\pi\)
0.996400 + 0.0847805i \(0.0270189\pi\)
\(618\) 0 0
\(619\) −6.82605 11.8231i −0.274362 0.475209i 0.695612 0.718418i \(-0.255133\pi\)
−0.969974 + 0.243209i \(0.921800\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.14201 + 3.71007i 0.0858178 + 0.148641i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.85601 0.193622
\(630\) 0 0
\(631\) −33.2593 −1.32403 −0.662017 0.749489i \(-0.730299\pi\)
−0.662017 + 0.749489i \(0.730299\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.09357 + 1.89412i −0.0433971 + 0.0751659i
\(636\) 0 0
\(637\) 11.5905 + 20.0753i 0.459231 + 0.795411i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 13.1429 + 22.7641i 0.519112 + 0.899128i 0.999753 + 0.0222106i \(0.00707044\pi\)
−0.480642 + 0.876917i \(0.659596\pi\)
\(642\) 0 0
\(643\) 10.2913 17.8250i 0.405848 0.702950i −0.588571 0.808445i \(-0.700309\pi\)
0.994420 + 0.105495i \(0.0336427\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 23.2527 0.914159 0.457079 0.889426i \(-0.348896\pi\)
0.457079 + 0.889426i \(0.348896\pi\)
\(648\) 0 0
\(649\) −3.05767 −0.120024
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.37957 16.2459i 0.367051 0.635751i −0.622052 0.782976i \(-0.713701\pi\)
0.989103 + 0.147225i \(0.0470342\pi\)
\(654\) 0 0
\(655\) −3.00000 5.19615i −0.117220 0.203030i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0.140034 + 0.242545i 0.00545494 + 0.00944823i 0.868740 0.495268i \(-0.164930\pi\)
−0.863285 + 0.504717i \(0.831597\pi\)
\(660\) 0 0
\(661\) 19.8930 34.4556i 0.773746 1.34017i −0.161750 0.986832i \(-0.551714\pi\)
0.935496 0.353336i \(-0.114953\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.67282 −0.258761
\(666\) 0 0
\(667\) −56.1562 −2.17438
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3.37957 5.85358i 0.130467 0.225975i
\(672\) 0 0
\(673\) −16.7644 29.0368i −0.646221 1.11929i −0.984018 0.178068i \(-0.943015\pi\)
0.337797 0.941219i \(-0.390318\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.7437 + 23.8048i 0.528213 + 0.914891i 0.999459 + 0.0328897i \(0.0104710\pi\)
−0.471246 + 0.882002i \(0.656196\pi\)
\(678\) 0 0
\(679\) 5.59442 9.68981i 0.214694 0.371861i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −34.5865 −1.32342 −0.661708 0.749762i \(-0.730168\pi\)
−0.661708 + 0.749762i \(0.730168\pi\)
\(684\) 0 0
\(685\) −10.2017 −0.389785
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.67282 + 4.62947i −0.101826 + 0.176369i
\(690\) 0 0
\(691\) 20.3641 + 35.2717i 0.774688 + 1.34180i 0.934970 + 0.354727i \(0.115426\pi\)
−0.160282 + 0.987071i \(0.551240\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 6.92820i −0.151729 0.262802i
\(696\) 0 0
\(697\) 1.96608 3.40535i 0.0744706 0.128987i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 19.4712 0.735416 0.367708 0.929941i \(-0.380143\pi\)
0.367708 + 0.929941i \(0.380143\pi\)
\(702\) 0 0
\(703\) 8.48963 0.320193
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.00000 5.19615i 0.112827 0.195421i
\(708\) 0 0
\(709\) −7.54316 13.0651i −0.283289 0.490671i 0.688904 0.724853i \(-0.258092\pi\)
−0.972193 + 0.234182i \(0.924759\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −20.5957 35.6729i −0.771317 1.33596i
\(714\) 0 0
\(715\) 6.24482 10.8163i 0.233543 0.404508i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −3.43196 −0.127990 −0.0639952 0.997950i \(-0.520384\pi\)
−0.0639952 + 0.997950i \(0.520384\pi\)
\(720\) 0 0
\(721\) −2.59442 −0.0966211
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −4.74482 + 8.21826i −0.176218 + 0.305219i
\(726\) 0 0
\(727\) −17.8857 30.9789i −0.663344 1.14895i −0.979732 0.200315i \(-0.935803\pi\)
0.316388 0.948630i \(-0.397530\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −0.629668 1.09062i −0.0232891 0.0403379i
\(732\) 0 0
\(733\) 11.0000 19.0526i 0.406294 0.703722i −0.588177 0.808732i \(-0.700154\pi\)
0.994471 + 0.105010i \(0.0334875\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17.6151 0.648862
\(738\) 0 0
\(739\) −6.08631 −0.223889 −0.111944 0.993714i \(-0.535708\pi\)
−0.111944 + 0.993714i \(0.535708\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.7509 22.0853i 0.467787 0.810231i −0.531536 0.847036i \(-0.678385\pi\)
0.999322 + 0.0368054i \(0.0117182\pi\)
\(744\) 0 0
\(745\) 10.0381 + 17.3865i 0.367767 + 0.636990i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 8.50924 + 14.7384i 0.310921 + 0.538530i
\(750\) 0 0
\(751\) −9.19638 + 15.9286i −0.335581 + 0.581243i −0.983596 0.180384i \(-0.942266\pi\)
0.648016 + 0.761627i \(0.275599\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.03920 0.110608
\(756\) 0 0
\(757\) −41.8986 −1.52283 −0.761415 0.648264i \(-0.775495\pi\)
−0.761415 + 0.648264i \(0.775495\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.98568 + 6.90340i −0.144481 + 0.250248i −0.929179 0.369630i \(-0.879485\pi\)
0.784698 + 0.619878i \(0.212818\pi\)
\(762\) 0 0
\(763\) −11.8896 20.5935i −0.430434 0.745534i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.67282 + 4.62947i 0.0965101 + 0.167160i
\(768\) 0 0
\(769\) −3.01432 + 5.22095i −0.108699 + 0.188272i −0.915244 0.402901i \(-0.868002\pi\)
0.806544 + 0.591174i \(0.201335\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −44.4033 −1.59708 −0.798538 0.601944i \(-0.794393\pi\)
−0.798538 + 0.601944i \(0.794393\pi\)
\(774\) 0 0
\(775\) −6.96080 −0.250039
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 3.43724 5.95348i 0.123152 0.213305i
\(780\) 0 0
\(781\) 17.2056 + 29.8010i 0.615665 + 1.06636i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.100830 + 0.174643i 0.00359879 + 0.00623329i
\(786\) 0 0
\(787\) 8.41877 14.5817i 0.300097 0.519783i −0.676061 0.736846i \(-0.736314\pi\)
0.976158 + 0.217063i \(0.0696477\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 28.7361 1.02174
\(792\) 0 0
\(793\) −11.8168 −0.419627
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −16.6860 + 28.9010i −0.591049 + 1.02373i 0.403043 + 0.915181i \(0.367953\pi\)
−0.994091 + 0.108545i \(0.965381\pi\)
\(798\) 0 0
\(799\) 9.29721 + 16.1032i 0.328912 + 0.569692i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.28797 + 3.96289i 0.0807408 + 0.139847i
\(804\) 0 0
\(805\) 4.22522 7.31829i 0.148919 0.257936i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −29.1809 −1.02595 −0.512973 0.858404i \(-0.671456\pi\)
−0.512973 + 0.858404i \(0.671456\pi\)
\(810\) 0 0
\(811\) 15.5552 0.546217 0.273109 0.961983i \(-0.411948\pi\)
0.273109 + 0.961983i \(0.411948\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −8.90841 + 15.4298i −0.312048 + 0.540483i
\(816\) 0 0
\(817\) −1.10083 1.90669i −0.0385132 0.0667068i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −11.8588 20.5401i −0.413876 0.716855i 0.581434 0.813594i \(-0.302492\pi\)
−0.995310 + 0.0967393i \(0.969159\pi\)
\(822\) 0 0
\(823\) 9.68404 16.7732i 0.337564 0.584678i −0.646410 0.762990i \(-0.723730\pi\)
0.983974 + 0.178312i \(0.0570636\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 52.2241 1.81601 0.908005 0.418960i \(-0.137605\pi\)
0.908005 + 0.418960i \(0.137605\pi\)
\(828\) 0 0
\(829\) −26.6442 −0.925391 −0.462695 0.886517i \(-0.653118\pi\)
−0.462695 + 0.886517i \(0.653118\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.62967 + 11.4829i −0.229704 + 0.397860i
\(834\) 0 0
\(835\) 7.05042 + 12.2117i 0.243990 + 0.422603i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 12.5196 + 21.6846i 0.432225 + 0.748635i 0.997065 0.0765655i \(-0.0243954\pi\)
−0.564840 + 0.825201i \(0.691062\pi\)
\(840\) 0 0
\(841\) −30.5266 + 52.8736i −1.05264 + 1.82323i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −8.83528 −0.303943
\(846\) 0 0
\(847\) 5.50641 0.189203
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −5.37562 + 9.31084i −0.184274 + 0.319171i
\(852\) 0 0
\(853\) 5.43724 + 9.41758i 0.186168 + 0.322452i 0.943969 0.330033i \(-0.107060\pi\)
−0.757802 + 0.652485i \(0.773727\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −9.29721 16.1032i −0.317587 0.550076i 0.662397 0.749153i \(-0.269539\pi\)
−0.979984 + 0.199077i \(0.936206\pi\)
\(858\) 0 0
\(859\) −2.33246 + 4.03994i −0.0795825 + 0.137841i −0.903070 0.429494i \(-0.858692\pi\)
0.823487 + 0.567335i \(0.192025\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −28.0594 −0.955152 −0.477576 0.878590i \(-0.658484\pi\)
−0.477576 + 0.878590i \(0.658484\pi\)
\(864\) 0 0
\(865\) 4.36638 0.148461
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −0.384851 + 0.666581i −0.0130552 + 0.0226122i
\(870\) 0 0
\(871\) −15.3980 26.6702i −0.521743 0.903685i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.714003 1.23669i −0.0241377 0.0418078i
\(876\) 0 0
\(877\) −17.3342 + 30.0236i −0.585333 + 1.01383i 0.409501 + 0.912310i \(0.365703\pi\)
−0.994834 + 0.101516i \(0.967631\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 5.29854 0.178512 0.0892561 0.996009i \(-0.471551\pi\)
0.0892561 + 0.996009i \(0.471551\pi\)
\(882\) 0 0
\(883\) 10.3025 0.346706 0.173353 0.984860i \(-0.444540\pi\)
0.173353 + 0.984860i \(0.444540\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −20.7252 + 35.8971i −0.695885 + 1.20531i 0.273997 + 0.961731i \(0.411654\pi\)
−0.969882 + 0.243577i \(0.921679\pi\)
\(888\) 0 0
\(889\) −1.56163 2.70482i −0.0523753 0.0907167i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 16.2541 + 28.1528i 0.543921 + 0.942099i
\(894\) 0 0
\(895\) −7.58123 + 13.1311i −0.253413 + 0.438923i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −66.0554 −2.20307
\(900\) 0 0
\(901\) −3.05767 −0.101866
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.60083 2.77272i 0.0532134 0.0921683i
\(906\) 0 0
\(907\) −8.39606 14.5424i −0.278787 0.482873i 0.692297 0.721613i \(-0.256599\pi\)
−0.971083 + 0.238740i \(0.923266\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −18.6768 32.3491i −0.618789 1.07177i −0.989707 0.143109i \(-0.954290\pi\)
0.370918 0.928666i \(-0.379043\pi\)
\(912\) 0 0
\(913\) 5.72522 9.91636i 0.189477 0.328184i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.56804 0.282942
\(918\) 0 0
\(919\) −37.1316 −1.22486 −0.612429 0.790526i \(-0.709807\pi\)
−0.612429 + 0.790526i \(0.709807\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 30.0801 52.1003i 0.990098 1.71490i
\(924\) 0 0
\(925\) 0.908405 + 1.57340i 0.0298682 + 0.0517332i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −23.9977 41.5653i −0.787340 1.36371i −0.927591 0.373597i \(-0.878124\pi\)
0.140251 0.990116i \(-0.455209\pi\)
\(930\) 0 0
\(931\) −11.5905 + 20.0753i −0.379862 + 0.657941i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 7.14399 0.233633
\(936\) 0 0
\(937\) 22.0079 0.718967 0.359483 0.933151i \(-0.382953\pi\)
0.359483 + 0.933151i \(0.382953\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 20.2921 35.1470i 0.661504 1.14576i −0.318716 0.947850i \(-0.603252\pi\)
0.980220 0.197909i \(-0.0634150\pi\)
\(942\) 0 0
\(943\) 4.35291 + 7.53946i 0.141750 + 0.245518i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −15.1160 26.1817i −0.491204 0.850790i 0.508745 0.860917i \(-0.330110\pi\)
−0.999949 + 0.0101273i \(0.996776\pi\)
\(948\) 0 0
\(949\) 4.00000 6.92820i 0.129845 0.224899i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2.50811 −0.0812455 −0.0406227 0.999175i \(-0.512934\pi\)
−0.0406227 + 0.999175i \(0.512934\pi\)
\(954\) 0 0
\(955\) 2.83754 0.0918207
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 7.28402 12.6163i 0.235213 0.407401i
\(960\) 0 0
\(961\) −8.72635 15.1145i −0.281495 0.487564i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −9.39409 16.2710i −0.302406 0.523783i
\(966\) 0 0
\(967\) −10.8765 + 18.8386i −0.349763 + 0.605808i −0.986207 0.165515i \(-0.947071\pi\)
0.636444 + 0.771323i \(0.280405\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 22.7512 0.730122 0.365061 0.930984i \(-0.381048\pi\)
0.365061 + 0.930984i \(0.381048\pi\)
\(972\) 0 0
\(973\) 11.4241 0.366238
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 19.6389 34.0156i 0.628304 1.08825i −0.359588 0.933111i \(-0.617083\pi\)
0.987892 0.155143i \(-0.0495840\pi\)
\(978\) 0 0
\(979\) −4.00924 6.94420i −0.128136 0.221938i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −16.8949 29.2629i −0.538865 0.933341i −0.998966 0.0454743i \(-0.985520\pi\)
0.460101 0.887867i \(-0.347813\pi\)
\(984\) 0 0
\(985\) −2.91764 + 5.05350i −0.0929638 + 0.161018i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.78817 0.0886587
\(990\) 0 0
\(991\) 23.7983 0.755979 0.377990 0.925810i \(-0.376616\pi\)
0.377990 + 0.925810i \(0.376616\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.53807 + 11.3243i −0.207271 + 0.359004i
\(996\) 0 0
\(997\) −25.8392 44.7549i −0.818337 1.41740i −0.906907 0.421331i \(-0.861563\pi\)
0.0885702 0.996070i \(-0.471770\pi\)
\(998\) 0 0
\(999\) 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 2160.2.q.k.1441.2 6
3.2 odd 2 720.2.q.i.481.2 6
4.3 odd 2 135.2.e.b.91.2 6
9.2 odd 6 720.2.q.i.241.2 6
9.4 even 3 6480.2.a.bs.1.2 3
9.5 odd 6 6480.2.a.bv.1.2 3
9.7 even 3 inner 2160.2.q.k.721.2 6
12.11 even 2 45.2.e.b.31.2 yes 6
20.3 even 4 675.2.k.b.199.4 12
20.7 even 4 675.2.k.b.199.3 12
20.19 odd 2 675.2.e.b.226.2 6
36.7 odd 6 135.2.e.b.46.2 6
36.11 even 6 45.2.e.b.16.2 6
36.23 even 6 405.2.a.j.1.2 3
36.31 odd 6 405.2.a.i.1.2 3
60.23 odd 4 225.2.k.b.49.3 12
60.47 odd 4 225.2.k.b.49.4 12
60.59 even 2 225.2.e.b.76.2 6
180.7 even 12 675.2.k.b.424.4 12
180.23 odd 12 2025.2.b.l.649.3 6
180.43 even 12 675.2.k.b.424.3 12
180.47 odd 12 225.2.k.b.124.3 12
180.59 even 6 2025.2.a.n.1.2 3
180.67 even 12 2025.2.b.m.649.3 6
180.79 odd 6 675.2.e.b.451.2 6
180.83 odd 12 225.2.k.b.124.4 12
180.103 even 12 2025.2.b.m.649.4 6
180.119 even 6 225.2.e.b.151.2 6
180.139 odd 6 2025.2.a.o.1.2 3
180.167 odd 12 2025.2.b.l.649.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.2.e.b.16.2 6 36.11 even 6
45.2.e.b.31.2 yes 6 12.11 even 2
135.2.e.b.46.2 6 36.7 odd 6
135.2.e.b.91.2 6 4.3 odd 2
225.2.e.b.76.2 6 60.59 even 2
225.2.e.b.151.2 6 180.119 even 6
225.2.k.b.49.3 12 60.23 odd 4
225.2.k.b.49.4 12 60.47 odd 4
225.2.k.b.124.3 12 180.47 odd 12
225.2.k.b.124.4 12 180.83 odd 12
405.2.a.i.1.2 3 36.31 odd 6
405.2.a.j.1.2 3 36.23 even 6
675.2.e.b.226.2 6 20.19 odd 2
675.2.e.b.451.2 6 180.79 odd 6
675.2.k.b.199.3 12 20.7 even 4
675.2.k.b.199.4 12 20.3 even 4
675.2.k.b.424.3 12 180.43 even 12
675.2.k.b.424.4 12 180.7 even 12
720.2.q.i.241.2 6 9.2 odd 6
720.2.q.i.481.2 6 3.2 odd 2
2025.2.a.n.1.2 3 180.59 even 6
2025.2.a.o.1.2 3 180.139 odd 6
2025.2.b.l.649.3 6 180.23 odd 12
2025.2.b.l.649.4 6 180.167 odd 12
2025.2.b.m.649.3 6 180.67 even 12
2025.2.b.m.649.4 6 180.103 even 12
2160.2.q.k.721.2 6 9.7 even 3 inner
2160.2.q.k.1441.2 6 1.1 even 1 trivial
6480.2.a.bs.1.2 3 9.4 even 3
6480.2.a.bv.1.2 3 9.5 odd 6