Properties

Label 75.3.f.a.43.2
Level $75$
Weight $3$
Character 75.43
Analytic conductor $2.044$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,3,Mod(7,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.f (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: \(2.04360198270\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 43.2
Root \(1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 75.43
Dual form 75.3.f.a.7.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.22474 - 1.22474i) q^{2} +(1.22474 + 1.22474i) q^{3} +1.00000i q^{4} +3.00000 q^{6} +(7.34847 - 7.34847i) q^{7} +(6.12372 + 6.12372i) q^{8} +3.00000i q^{9} -18.0000 q^{11} +(-1.22474 + 1.22474i) q^{12} +(-7.34847 - 7.34847i) q^{13} -18.0000i q^{14} +11.0000 q^{16} +(-4.89898 + 4.89898i) q^{17} +(3.67423 + 3.67423i) q^{18} +10.0000i q^{19} +18.0000 q^{21} +(-22.0454 + 22.0454i) q^{22} +(-19.5959 - 19.5959i) q^{23} +15.0000i q^{24} -18.0000 q^{26} +(-3.67423 + 3.67423i) q^{27} +(7.34847 + 7.34847i) q^{28} +22.0000 q^{31} +(-11.0227 + 11.0227i) q^{32} +(-22.0454 - 22.0454i) q^{33} +12.0000i q^{34} -3.00000 q^{36} +(7.34847 - 7.34847i) q^{37} +(12.2474 + 12.2474i) q^{38} -18.0000i q^{39} -18.0000 q^{41} +(22.0454 - 22.0454i) q^{42} +(29.3939 + 29.3939i) q^{43} -18.0000i q^{44} -48.0000 q^{46} +(44.0908 - 44.0908i) q^{47} +(13.4722 + 13.4722i) q^{48} -59.0000i q^{49} -12.0000 q^{51} +(7.34847 - 7.34847i) q^{52} +(4.89898 + 4.89898i) q^{53} +9.00000i q^{54} +90.0000 q^{56} +(-12.2474 + 12.2474i) q^{57} +90.0000i q^{59} +2.00000 q^{61} +(26.9444 - 26.9444i) q^{62} +(22.0454 + 22.0454i) q^{63} +71.0000i q^{64} -54.0000 q^{66} +(44.0908 - 44.0908i) q^{67} +(-4.89898 - 4.89898i) q^{68} -48.0000i q^{69} +72.0000 q^{71} +(-18.3712 + 18.3712i) q^{72} +(-44.0908 - 44.0908i) q^{73} -18.0000i q^{74} -10.0000 q^{76} +(-132.272 + 132.272i) q^{77} +(-22.0454 - 22.0454i) q^{78} -70.0000i q^{79} -9.00000 q^{81} +(-22.0454 + 22.0454i) q^{82} +(53.8888 + 53.8888i) q^{83} +18.0000i q^{84} +72.0000 q^{86} +(-110.227 - 110.227i) q^{88} +90.0000i q^{89} -108.000 q^{91} +(19.5959 - 19.5959i) q^{92} +(26.9444 + 26.9444i) q^{93} -108.000i q^{94} -27.0000 q^{96} +(-102.879 + 102.879i) q^{97} +(-72.2599 - 72.2599i) q^{98} -54.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{6} - 72 q^{11} + 44 q^{16} + 72 q^{21} - 72 q^{26} + 88 q^{31} - 12 q^{36} - 72 q^{41} - 192 q^{46} - 48 q^{51} + 360 q^{56} + 8 q^{61} - 216 q^{66} + 288 q^{71} - 40 q^{76} - 36 q^{81} + 288 q^{86}+ \cdots - 108 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.22474 1.22474i 0.612372 0.612372i −0.331191 0.943564i \(-0.607451\pi\)
0.943564 + 0.331191i \(0.107451\pi\)
\(3\) 1.22474 + 1.22474i 0.408248 + 0.408248i
\(4\) 1.00000i 0.250000i
\(5\) 0 0
\(6\) 3.00000 0.500000
\(7\) 7.34847 7.34847i 1.04978 1.04978i 0.0510871 0.998694i \(-0.483731\pi\)
0.998694 0.0510871i \(-0.0162686\pi\)
\(8\) 6.12372 + 6.12372i 0.765466 + 0.765466i
\(9\) 3.00000i 0.333333i
\(10\) 0 0
\(11\) −18.0000 −1.63636 −0.818182 0.574960i \(-0.805018\pi\)
−0.818182 + 0.574960i \(0.805018\pi\)
\(12\) −1.22474 + 1.22474i −0.102062 + 0.102062i
\(13\) −7.34847 7.34847i −0.565267 0.565267i 0.365532 0.930799i \(-0.380887\pi\)
−0.930799 + 0.365532i \(0.880887\pi\)
\(14\) 18.0000i 1.28571i
\(15\) 0 0
\(16\) 11.0000 0.687500
\(17\) −4.89898 + 4.89898i −0.288175 + 0.288175i −0.836358 0.548183i \(-0.815320\pi\)
0.548183 + 0.836358i \(0.315320\pi\)
\(18\) 3.67423 + 3.67423i 0.204124 + 0.204124i
\(19\) 10.0000i 0.526316i 0.964753 + 0.263158i \(0.0847640\pi\)
−0.964753 + 0.263158i \(0.915236\pi\)
\(20\) 0 0
\(21\) 18.0000 0.857143
\(22\) −22.0454 + 22.0454i −1.00206 + 1.00206i
\(23\) −19.5959 19.5959i −0.851996 0.851996i 0.138382 0.990379i \(-0.455810\pi\)
−0.990379 + 0.138382i \(0.955810\pi\)
\(24\) 15.0000i 0.625000i
\(25\) 0 0
\(26\) −18.0000 −0.692308
\(27\) −3.67423 + 3.67423i −0.136083 + 0.136083i
\(28\) 7.34847 + 7.34847i 0.262445 + 0.262445i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 22.0000 0.709677 0.354839 0.934928i \(-0.384536\pi\)
0.354839 + 0.934928i \(0.384536\pi\)
\(32\) −11.0227 + 11.0227i −0.344459 + 0.344459i
\(33\) −22.0454 22.0454i −0.668043 0.668043i
\(34\) 12.0000i 0.352941i
\(35\) 0 0
\(36\) −3.00000 −0.0833333
\(37\) 7.34847 7.34847i 0.198607 0.198607i −0.600795 0.799403i \(-0.705149\pi\)
0.799403 + 0.600795i \(0.205149\pi\)
\(38\) 12.2474 + 12.2474i 0.322301 + 0.322301i
\(39\) 18.0000i 0.461538i
\(40\) 0 0
\(41\) −18.0000 −0.439024 −0.219512 0.975610i \(-0.570447\pi\)
−0.219512 + 0.975610i \(0.570447\pi\)
\(42\) 22.0454 22.0454i 0.524891 0.524891i
\(43\) 29.3939 + 29.3939i 0.683579 + 0.683579i 0.960805 0.277226i \(-0.0894151\pi\)
−0.277226 + 0.960805i \(0.589415\pi\)
\(44\) 18.0000i 0.409091i
\(45\) 0 0
\(46\) −48.0000 −1.04348
\(47\) 44.0908 44.0908i 0.938102 0.938102i −0.0600905 0.998193i \(-0.519139\pi\)
0.998193 + 0.0600905i \(0.0191389\pi\)
\(48\) 13.4722 + 13.4722i 0.280671 + 0.280671i
\(49\) 59.0000i 1.20408i
\(50\) 0 0
\(51\) −12.0000 −0.235294
\(52\) 7.34847 7.34847i 0.141317 0.141317i
\(53\) 4.89898 + 4.89898i 0.0924336 + 0.0924336i 0.751812 0.659378i \(-0.229180\pi\)
−0.659378 + 0.751812i \(0.729180\pi\)
\(54\) 9.00000i 0.166667i
\(55\) 0 0
\(56\) 90.0000 1.60714
\(57\) −12.2474 + 12.2474i −0.214868 + 0.214868i
\(58\) 0 0
\(59\) 90.0000i 1.52542i 0.646738 + 0.762712i \(0.276133\pi\)
−0.646738 + 0.762712i \(0.723867\pi\)
\(60\) 0 0
\(61\) 2.00000 0.0327869 0.0163934 0.999866i \(-0.494782\pi\)
0.0163934 + 0.999866i \(0.494782\pi\)
\(62\) 26.9444 26.9444i 0.434587 0.434587i
\(63\) 22.0454 + 22.0454i 0.349927 + 0.349927i
\(64\) 71.0000i 1.10938i
\(65\) 0 0
\(66\) −54.0000 −0.818182
\(67\) 44.0908 44.0908i 0.658072 0.658072i −0.296852 0.954924i \(-0.595937\pi\)
0.954924 + 0.296852i \(0.0959367\pi\)
\(68\) −4.89898 4.89898i −0.0720438 0.0720438i
\(69\) 48.0000i 0.695652i
\(70\) 0 0
\(71\) 72.0000 1.01408 0.507042 0.861921i \(-0.330739\pi\)
0.507042 + 0.861921i \(0.330739\pi\)
\(72\) −18.3712 + 18.3712i −0.255155 + 0.255155i
\(73\) −44.0908 44.0908i −0.603984 0.603984i 0.337384 0.941367i \(-0.390458\pi\)
−0.941367 + 0.337384i \(0.890458\pi\)
\(74\) 18.0000i 0.243243i
\(75\) 0 0
\(76\) −10.0000 −0.131579
\(77\) −132.272 + 132.272i −1.71782 + 1.71782i
\(78\) −22.0454 22.0454i −0.282633 0.282633i
\(79\) 70.0000i 0.886076i −0.896503 0.443038i \(-0.853901\pi\)
0.896503 0.443038i \(-0.146099\pi\)
\(80\) 0 0
\(81\) −9.00000 −0.111111
\(82\) −22.0454 + 22.0454i −0.268846 + 0.268846i
\(83\) 53.8888 + 53.8888i 0.649262 + 0.649262i 0.952815 0.303552i \(-0.0981727\pi\)
−0.303552 + 0.952815i \(0.598173\pi\)
\(84\) 18.0000i 0.214286i
\(85\) 0 0
\(86\) 72.0000 0.837209
\(87\) 0 0
\(88\) −110.227 110.227i −1.25258 1.25258i
\(89\) 90.0000i 1.01124i 0.862757 + 0.505618i \(0.168735\pi\)
−0.862757 + 0.505618i \(0.831265\pi\)
\(90\) 0 0
\(91\) −108.000 −1.18681
\(92\) 19.5959 19.5959i 0.212999 0.212999i
\(93\) 26.9444 + 26.9444i 0.289725 + 0.289725i
\(94\) 108.000i 1.14894i
\(95\) 0 0
\(96\) −27.0000 −0.281250
\(97\) −102.879 + 102.879i −1.06060 + 1.06060i −0.0625628 + 0.998041i \(0.519927\pi\)
−0.998041 + 0.0625628i \(0.980073\pi\)
\(98\) −72.2599 72.2599i −0.737346 0.737346i
\(99\) 54.0000i 0.545455i
\(100\) 0 0
\(101\) −108.000 −1.06931 −0.534653 0.845071i \(-0.679558\pi\)
−0.534653 + 0.845071i \(0.679558\pi\)
\(102\) −14.6969 + 14.6969i −0.144088 + 0.144088i
\(103\) 66.1362 + 66.1362i 0.642099 + 0.642099i 0.951071 0.308972i \(-0.0999849\pi\)
−0.308972 + 0.951071i \(0.599985\pi\)
\(104\) 90.0000i 0.865385i
\(105\) 0 0
\(106\) 12.0000 0.113208
\(107\) 44.0908 44.0908i 0.412064 0.412064i −0.470393 0.882457i \(-0.655888\pi\)
0.882457 + 0.470393i \(0.155888\pi\)
\(108\) −3.67423 3.67423i −0.0340207 0.0340207i
\(109\) 170.000i 1.55963i 0.626008 + 0.779817i \(0.284688\pi\)
−0.626008 + 0.779817i \(0.715312\pi\)
\(110\) 0 0
\(111\) 18.0000 0.162162
\(112\) 80.8332 80.8332i 0.721725 0.721725i
\(113\) −44.0908 44.0908i −0.390184 0.390184i 0.484569 0.874753i \(-0.338976\pi\)
−0.874753 + 0.484569i \(0.838976\pi\)
\(114\) 30.0000i 0.263158i
\(115\) 0 0
\(116\) 0 0
\(117\) 22.0454 22.0454i 0.188422 0.188422i
\(118\) 110.227 + 110.227i 0.934127 + 0.934127i
\(119\) 72.0000i 0.605042i
\(120\) 0 0
\(121\) 203.000 1.67769
\(122\) 2.44949 2.44949i 0.0200778 0.0200778i
\(123\) −22.0454 22.0454i −0.179231 0.179231i
\(124\) 22.0000i 0.177419i
\(125\) 0 0
\(126\) 54.0000 0.428571
\(127\) 7.34847 7.34847i 0.0578620 0.0578620i −0.677584 0.735446i \(-0.736973\pi\)
0.735446 + 0.677584i \(0.236973\pi\)
\(128\) 42.8661 + 42.8661i 0.334891 + 0.334891i
\(129\) 72.0000i 0.558140i
\(130\) 0 0
\(131\) −18.0000 −0.137405 −0.0687023 0.997637i \(-0.521886\pi\)
−0.0687023 + 0.997637i \(0.521886\pi\)
\(132\) 22.0454 22.0454i 0.167011 0.167011i
\(133\) 73.4847 + 73.4847i 0.552516 + 0.552516i
\(134\) 108.000i 0.805970i
\(135\) 0 0
\(136\) −60.0000 −0.441176
\(137\) 142.070 142.070i 1.03701 1.03701i 0.0377220 0.999288i \(-0.487990\pi\)
0.999288 0.0377220i \(-0.0120101\pi\)
\(138\) −58.7878 58.7878i −0.425998 0.425998i
\(139\) 170.000i 1.22302i −0.791236 0.611511i \(-0.790562\pi\)
0.791236 0.611511i \(-0.209438\pi\)
\(140\) 0 0
\(141\) 108.000 0.765957
\(142\) 88.1816 88.1816i 0.620997 0.620997i
\(143\) 132.272 + 132.272i 0.924982 + 0.924982i
\(144\) 33.0000i 0.229167i
\(145\) 0 0
\(146\) −108.000 −0.739726
\(147\) 72.2599 72.2599i 0.491564 0.491564i
\(148\) 7.34847 + 7.34847i 0.0496518 + 0.0496518i
\(149\) 180.000i 1.20805i −0.796964 0.604027i \(-0.793562\pi\)
0.796964 0.604027i \(-0.206438\pi\)
\(150\) 0 0
\(151\) 22.0000 0.145695 0.0728477 0.997343i \(-0.476791\pi\)
0.0728477 + 0.997343i \(0.476791\pi\)
\(152\) −61.2372 + 61.2372i −0.402877 + 0.402877i
\(153\) −14.6969 14.6969i −0.0960584 0.0960584i
\(154\) 324.000i 2.10390i
\(155\) 0 0
\(156\) 18.0000 0.115385
\(157\) −139.621 + 139.621i −0.889305 + 0.889305i −0.994456 0.105151i \(-0.966467\pi\)
0.105151 + 0.994456i \(0.466467\pi\)
\(158\) −85.7321 85.7321i −0.542608 0.542608i
\(159\) 12.0000i 0.0754717i
\(160\) 0 0
\(161\) −288.000 −1.78882
\(162\) −11.0227 + 11.0227i −0.0680414 + 0.0680414i
\(163\) −117.576 117.576i −0.721322 0.721322i 0.247552 0.968875i \(-0.420374\pi\)
−0.968875 + 0.247552i \(0.920374\pi\)
\(164\) 18.0000i 0.109756i
\(165\) 0 0
\(166\) 132.000 0.795181
\(167\) 19.5959 19.5959i 0.117341 0.117341i −0.645998 0.763339i \(-0.723559\pi\)
0.763339 + 0.645998i \(0.223559\pi\)
\(168\) 110.227 + 110.227i 0.656113 + 0.656113i
\(169\) 61.0000i 0.360947i
\(170\) 0 0
\(171\) −30.0000 −0.175439
\(172\) −29.3939 + 29.3939i −0.170895 + 0.170895i
\(173\) −142.070 142.070i −0.821216 0.821216i 0.165066 0.986282i \(-0.447216\pi\)
−0.986282 + 0.165066i \(0.947216\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −198.000 −1.12500
\(177\) −110.227 + 110.227i −0.622752 + 0.622752i
\(178\) 110.227 + 110.227i 0.619253 + 0.619253i
\(179\) 90.0000i 0.502793i −0.967884 0.251397i \(-0.919110\pi\)
0.967884 0.251397i \(-0.0808899\pi\)
\(180\) 0 0
\(181\) −98.0000 −0.541436 −0.270718 0.962659i \(-0.587261\pi\)
−0.270718 + 0.962659i \(0.587261\pi\)
\(182\) −132.272 + 132.272i −0.726772 + 0.726772i
\(183\) 2.44949 + 2.44949i 0.0133852 + 0.0133852i
\(184\) 240.000i 1.30435i
\(185\) 0 0
\(186\) 66.0000 0.354839
\(187\) 88.1816 88.1816i 0.471560 0.471560i
\(188\) 44.0908 + 44.0908i 0.234526 + 0.234526i
\(189\) 54.0000i 0.285714i
\(190\) 0 0
\(191\) 252.000 1.31937 0.659686 0.751541i \(-0.270689\pi\)
0.659686 + 0.751541i \(0.270689\pi\)
\(192\) −86.9569 + 86.9569i −0.452900 + 0.452900i
\(193\) −264.545 264.545i −1.37070 1.37070i −0.859407 0.511292i \(-0.829167\pi\)
−0.511292 0.859407i \(-0.670833\pi\)
\(194\) 252.000i 1.29897i
\(195\) 0 0
\(196\) 59.0000 0.301020
\(197\) −127.373 + 127.373i −0.646566 + 0.646566i −0.952161 0.305596i \(-0.901144\pi\)
0.305596 + 0.952161i \(0.401144\pi\)
\(198\) −66.1362 66.1362i −0.334021 0.334021i
\(199\) 290.000i 1.45729i 0.684893 + 0.728643i \(0.259849\pi\)
−0.684893 + 0.728643i \(0.740151\pi\)
\(200\) 0 0
\(201\) 108.000 0.537313
\(202\) −132.272 + 132.272i −0.654814 + 0.654814i
\(203\) 0 0
\(204\) 12.0000i 0.0588235i
\(205\) 0 0
\(206\) 162.000 0.786408
\(207\) 58.7878 58.7878i 0.283999 0.283999i
\(208\) −80.8332 80.8332i −0.388621 0.388621i
\(209\) 180.000i 0.861244i
\(210\) 0 0
\(211\) 122.000 0.578199 0.289100 0.957299i \(-0.406644\pi\)
0.289100 + 0.957299i \(0.406644\pi\)
\(212\) −4.89898 + 4.89898i −0.0231084 + 0.0231084i
\(213\) 88.1816 + 88.1816i 0.413998 + 0.413998i
\(214\) 108.000i 0.504673i
\(215\) 0 0
\(216\) −45.0000 −0.208333
\(217\) 161.666 161.666i 0.745006 0.745006i
\(218\) 208.207 + 208.207i 0.955076 + 0.955076i
\(219\) 108.000i 0.493151i
\(220\) 0 0
\(221\) 72.0000 0.325792
\(222\) 22.0454 22.0454i 0.0993036 0.0993036i
\(223\) −80.8332 80.8332i −0.362481 0.362481i 0.502245 0.864725i \(-0.332508\pi\)
−0.864725 + 0.502245i \(0.832508\pi\)
\(224\) 162.000i 0.723214i
\(225\) 0 0
\(226\) −108.000 −0.477876
\(227\) −53.8888 + 53.8888i −0.237395 + 0.237395i −0.815771 0.578375i \(-0.803687\pi\)
0.578375 + 0.815771i \(0.303687\pi\)
\(228\) −12.2474 12.2474i −0.0537169 0.0537169i
\(229\) 50.0000i 0.218341i −0.994023 0.109170i \(-0.965181\pi\)
0.994023 0.109170i \(-0.0348194\pi\)
\(230\) 0 0
\(231\) −324.000 −1.40260
\(232\) 0 0
\(233\) −44.0908 44.0908i −0.189231 0.189231i 0.606133 0.795364i \(-0.292720\pi\)
−0.795364 + 0.606133i \(0.792720\pi\)
\(234\) 54.0000i 0.230769i
\(235\) 0 0
\(236\) −90.0000 −0.381356
\(237\) 85.7321 85.7321i 0.361739 0.361739i
\(238\) 88.1816 + 88.1816i 0.370511 + 0.370511i
\(239\) 180.000i 0.753138i 0.926389 + 0.376569i \(0.122896\pi\)
−0.926389 + 0.376569i \(0.877104\pi\)
\(240\) 0 0
\(241\) −178.000 −0.738589 −0.369295 0.929312i \(-0.620401\pi\)
−0.369295 + 0.929312i \(0.620401\pi\)
\(242\) 248.623 248.623i 1.02737 1.02737i
\(243\) −11.0227 11.0227i −0.0453609 0.0453609i
\(244\) 2.00000i 0.00819672i
\(245\) 0 0
\(246\) −54.0000 −0.219512
\(247\) 73.4847 73.4847i 0.297509 0.297509i
\(248\) 134.722 + 134.722i 0.543234 + 0.543234i
\(249\) 132.000i 0.530120i
\(250\) 0 0
\(251\) 342.000 1.36255 0.681275 0.732028i \(-0.261426\pi\)
0.681275 + 0.732028i \(0.261426\pi\)
\(252\) −22.0454 + 22.0454i −0.0874818 + 0.0874818i
\(253\) 352.727 + 352.727i 1.39418 + 1.39418i
\(254\) 18.0000i 0.0708661i
\(255\) 0 0
\(256\) −179.000 −0.699219
\(257\) 289.040 289.040i 1.12467 1.12467i 0.133638 0.991030i \(-0.457334\pi\)
0.991030 0.133638i \(-0.0426660\pi\)
\(258\) 88.1816 + 88.1816i 0.341789 + 0.341789i
\(259\) 108.000i 0.416988i
\(260\) 0 0
\(261\) 0 0
\(262\) −22.0454 + 22.0454i −0.0841428 + 0.0841428i
\(263\) −264.545 264.545i −1.00587 1.00587i −0.999983 0.00589147i \(-0.998125\pi\)
−0.00589147 0.999983i \(-0.501875\pi\)
\(264\) 270.000i 1.02273i
\(265\) 0 0
\(266\) 180.000 0.676692
\(267\) −110.227 + 110.227i −0.412835 + 0.412835i
\(268\) 44.0908 + 44.0908i 0.164518 + 0.164518i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −478.000 −1.76384 −0.881919 0.471401i \(-0.843748\pi\)
−0.881919 + 0.471401i \(0.843748\pi\)
\(272\) −53.8888 + 53.8888i −0.198120 + 0.198120i
\(273\) −132.272 132.272i −0.484514 0.484514i
\(274\) 348.000i 1.27007i
\(275\) 0 0
\(276\) 48.0000 0.173913
\(277\) 80.8332 80.8332i 0.291816 0.291816i −0.545981 0.837798i \(-0.683843\pi\)
0.837798 + 0.545981i \(0.183843\pi\)
\(278\) −208.207 208.207i −0.748945 0.748945i
\(279\) 66.0000i 0.236559i
\(280\) 0 0
\(281\) 162.000 0.576512 0.288256 0.957553i \(-0.406925\pi\)
0.288256 + 0.957553i \(0.406925\pi\)
\(282\) 132.272 132.272i 0.469051 0.469051i
\(283\) 249.848 + 249.848i 0.882855 + 0.882855i 0.993824 0.110969i \(-0.0353954\pi\)
−0.110969 + 0.993824i \(0.535395\pi\)
\(284\) 72.0000i 0.253521i
\(285\) 0 0
\(286\) 324.000 1.13287
\(287\) −132.272 + 132.272i −0.460880 + 0.460880i
\(288\) −33.0681 33.0681i −0.114820 0.114820i
\(289\) 241.000i 0.833910i
\(290\) 0 0
\(291\) −252.000 −0.865979
\(292\) 44.0908 44.0908i 0.150996 0.150996i
\(293\) 372.322 + 372.322i 1.27073 + 1.27073i 0.945708 + 0.325017i \(0.105370\pi\)
0.325017 + 0.945708i \(0.394630\pi\)
\(294\) 177.000i 0.602041i
\(295\) 0 0
\(296\) 90.0000 0.304054
\(297\) 66.1362 66.1362i 0.222681 0.222681i
\(298\) −220.454 220.454i −0.739779 0.739779i
\(299\) 288.000i 0.963211i
\(300\) 0 0
\(301\) 432.000 1.43522
\(302\) 26.9444 26.9444i 0.0892198 0.0892198i
\(303\) −132.272 132.272i −0.436543 0.436543i
\(304\) 110.000i 0.361842i
\(305\) 0 0
\(306\) −36.0000 −0.117647
\(307\) −396.817 + 396.817i −1.29256 + 1.29256i −0.359369 + 0.933195i \(0.617008\pi\)
−0.933195 + 0.359369i \(0.882992\pi\)
\(308\) −132.272 132.272i −0.429456 0.429456i
\(309\) 162.000i 0.524272i
\(310\) 0 0
\(311\) 252.000 0.810289 0.405145 0.914253i \(-0.367221\pi\)
0.405145 + 0.914253i \(0.367221\pi\)
\(312\) 110.227 110.227i 0.353292 0.353292i
\(313\) −117.576 117.576i −0.375641 0.375641i 0.493886 0.869527i \(-0.335576\pi\)
−0.869527 + 0.493886i \(0.835576\pi\)
\(314\) 342.000i 1.08917i
\(315\) 0 0
\(316\) 70.0000 0.221519
\(317\) −274.343 + 274.343i −0.865435 + 0.865435i −0.991963 0.126528i \(-0.959617\pi\)
0.126528 + 0.991963i \(0.459617\pi\)
\(318\) 14.6969 + 14.6969i 0.0462168 + 0.0462168i
\(319\) 0 0
\(320\) 0 0
\(321\) 108.000 0.336449
\(322\) −352.727 + 352.727i −1.09542 + 1.09542i
\(323\) −48.9898 48.9898i −0.151671 0.151671i
\(324\) 9.00000i 0.0277778i
\(325\) 0 0
\(326\) −288.000 −0.883436
\(327\) −208.207 + 208.207i −0.636718 + 0.636718i
\(328\) −110.227 110.227i −0.336058 0.336058i
\(329\) 648.000i 1.96960i
\(330\) 0 0
\(331\) −418.000 −1.26284 −0.631420 0.775441i \(-0.717528\pi\)
−0.631420 + 0.775441i \(0.717528\pi\)
\(332\) −53.8888 + 53.8888i −0.162316 + 0.162316i
\(333\) 22.0454 + 22.0454i 0.0662024 + 0.0662024i
\(334\) 48.0000i 0.143713i
\(335\) 0 0
\(336\) 198.000 0.589286
\(337\) 191.060 191.060i 0.566944 0.566944i −0.364327 0.931271i \(-0.618701\pi\)
0.931271 + 0.364327i \(0.118701\pi\)
\(338\) −74.7094 74.7094i −0.221034 0.221034i
\(339\) 108.000i 0.318584i
\(340\) 0 0
\(341\) −396.000 −1.16129
\(342\) −36.7423 + 36.7423i −0.107434 + 0.107434i
\(343\) −73.4847 73.4847i −0.214241 0.214241i
\(344\) 360.000i 1.04651i
\(345\) 0 0
\(346\) −348.000 −1.00578
\(347\) 44.0908 44.0908i 0.127063 0.127063i −0.640716 0.767778i \(-0.721362\pi\)
0.767778 + 0.640716i \(0.221362\pi\)
\(348\) 0 0
\(349\) 70.0000i 0.200573i 0.994959 + 0.100287i \(0.0319759\pi\)
−0.994959 + 0.100287i \(0.968024\pi\)
\(350\) 0 0
\(351\) 54.0000 0.153846
\(352\) 198.409 198.409i 0.563661 0.563661i
\(353\) −44.0908 44.0908i −0.124903 0.124903i 0.641892 0.766795i \(-0.278150\pi\)
−0.766795 + 0.641892i \(0.778150\pi\)
\(354\) 270.000i 0.762712i
\(355\) 0 0
\(356\) −90.0000 −0.252809
\(357\) −88.1816 + 88.1816i −0.247007 + 0.247007i
\(358\) −110.227 110.227i −0.307897 0.307897i
\(359\) 540.000i 1.50418i −0.659061 0.752089i \(-0.729046\pi\)
0.659061 0.752089i \(-0.270954\pi\)
\(360\) 0 0
\(361\) 261.000 0.722992
\(362\) −120.025 + 120.025i −0.331561 + 0.331561i
\(363\) 248.623 + 248.623i 0.684912 + 0.684912i
\(364\) 108.000i 0.296703i
\(365\) 0 0
\(366\) 6.00000 0.0163934
\(367\) −66.1362 + 66.1362i −0.180208 + 0.180208i −0.791446 0.611239i \(-0.790672\pi\)
0.611239 + 0.791446i \(0.290672\pi\)
\(368\) −215.555 215.555i −0.585748 0.585748i
\(369\) 54.0000i 0.146341i
\(370\) 0 0
\(371\) 72.0000 0.194070
\(372\) −26.9444 + 26.9444i −0.0724311 + 0.0724311i
\(373\) 213.106 + 213.106i 0.571329 + 0.571329i 0.932500 0.361171i \(-0.117623\pi\)
−0.361171 + 0.932500i \(0.617623\pi\)
\(374\) 216.000i 0.577540i
\(375\) 0 0
\(376\) 540.000 1.43617
\(377\) 0 0
\(378\) 66.1362 + 66.1362i 0.174964 + 0.174964i
\(379\) 170.000i 0.448549i −0.974526 0.224274i \(-0.927999\pi\)
0.974526 0.224274i \(-0.0720012\pi\)
\(380\) 0 0
\(381\) 18.0000 0.0472441
\(382\) 308.636 308.636i 0.807947 0.807947i
\(383\) 151.868 + 151.868i 0.396523 + 0.396523i 0.877005 0.480482i \(-0.159538\pi\)
−0.480482 + 0.877005i \(0.659538\pi\)
\(384\) 105.000i 0.273438i
\(385\) 0 0
\(386\) −648.000 −1.67876
\(387\) −88.1816 + 88.1816i −0.227860 + 0.227860i
\(388\) −102.879 102.879i −0.265151 0.265151i
\(389\) 360.000i 0.925450i −0.886502 0.462725i \(-0.846872\pi\)
0.886502 0.462725i \(-0.153128\pi\)
\(390\) 0 0
\(391\) 192.000 0.491049
\(392\) 361.300 361.300i 0.921683 0.921683i
\(393\) −22.0454 22.0454i −0.0560952 0.0560952i
\(394\) 312.000i 0.791878i
\(395\) 0 0
\(396\) 54.0000 0.136364
\(397\) 301.287 301.287i 0.758910 0.758910i −0.217214 0.976124i \(-0.569697\pi\)
0.976124 + 0.217214i \(0.0696970\pi\)
\(398\) 355.176 + 355.176i 0.892402 + 0.892402i
\(399\) 180.000i 0.451128i
\(400\) 0 0
\(401\) −558.000 −1.39152 −0.695761 0.718274i \(-0.744933\pi\)
−0.695761 + 0.718274i \(0.744933\pi\)
\(402\) 132.272 132.272i 0.329036 0.329036i
\(403\) −161.666 161.666i −0.401157 0.401157i
\(404\) 108.000i 0.267327i
\(405\) 0 0
\(406\) 0 0
\(407\) −132.272 + 132.272i −0.324994 + 0.324994i
\(408\) −73.4847 73.4847i −0.180110 0.180110i
\(409\) 670.000i 1.63814i 0.573692 + 0.819071i \(0.305511\pi\)
−0.573692 + 0.819071i \(0.694489\pi\)
\(410\) 0 0
\(411\) 348.000 0.846715
\(412\) −66.1362 + 66.1362i −0.160525 + 0.160525i
\(413\) 661.362 + 661.362i 1.60136 + 1.60136i
\(414\) 144.000i 0.347826i
\(415\) 0 0
\(416\) 162.000 0.389423
\(417\) 208.207 208.207i 0.499296 0.499296i
\(418\) −220.454 220.454i −0.527402 0.527402i
\(419\) 630.000i 1.50358i 0.659403 + 0.751790i \(0.270809\pi\)
−0.659403 + 0.751790i \(0.729191\pi\)
\(420\) 0 0
\(421\) 142.000 0.337292 0.168646 0.985677i \(-0.446061\pi\)
0.168646 + 0.985677i \(0.446061\pi\)
\(422\) 149.419 149.419i 0.354073 0.354073i
\(423\) 132.272 + 132.272i 0.312701 + 0.312701i
\(424\) 60.0000i 0.141509i
\(425\) 0 0
\(426\) 216.000 0.507042
\(427\) 14.6969 14.6969i 0.0344191 0.0344191i
\(428\) 44.0908 + 44.0908i 0.103016 + 0.103016i
\(429\) 324.000i 0.755245i
\(430\) 0 0
\(431\) 612.000 1.41995 0.709977 0.704225i \(-0.248705\pi\)
0.709977 + 0.704225i \(0.248705\pi\)
\(432\) −40.4166 + 40.4166i −0.0935569 + 0.0935569i
\(433\) 102.879 + 102.879i 0.237595 + 0.237595i 0.815853 0.578259i \(-0.196268\pi\)
−0.578259 + 0.815853i \(0.696268\pi\)
\(434\) 396.000i 0.912442i
\(435\) 0 0
\(436\) −170.000 −0.389908
\(437\) 195.959 195.959i 0.448419 0.448419i
\(438\) −132.272 132.272i −0.301992 0.301992i
\(439\) 430.000i 0.979499i −0.871863 0.489749i \(-0.837088\pi\)
0.871863 0.489749i \(-0.162912\pi\)
\(440\) 0 0
\(441\) 177.000 0.401361
\(442\) 88.1816 88.1816i 0.199506 0.199506i
\(443\) −582.979 582.979i −1.31598 1.31598i −0.916930 0.399049i \(-0.869340\pi\)
−0.399049 0.916930i \(-0.630660\pi\)
\(444\) 18.0000i 0.0405405i
\(445\) 0 0
\(446\) −198.000 −0.443946
\(447\) 220.454 220.454i 0.493186 0.493186i
\(448\) 521.741 + 521.741i 1.16460 + 1.16460i
\(449\) 90.0000i 0.200445i 0.994965 + 0.100223i \(0.0319555\pi\)
−0.994965 + 0.100223i \(0.968044\pi\)
\(450\) 0 0
\(451\) 324.000 0.718404
\(452\) 44.0908 44.0908i 0.0975461 0.0975461i
\(453\) 26.9444 + 26.9444i 0.0594799 + 0.0594799i
\(454\) 132.000i 0.290749i
\(455\) 0 0
\(456\) −150.000 −0.328947
\(457\) 191.060 191.060i 0.418075 0.418075i −0.466465 0.884540i \(-0.654473\pi\)
0.884540 + 0.466465i \(0.154473\pi\)
\(458\) −61.2372 61.2372i −0.133706 0.133706i
\(459\) 36.0000i 0.0784314i
\(460\) 0 0
\(461\) −828.000 −1.79610 −0.898048 0.439898i \(-0.855015\pi\)
−0.898048 + 0.439898i \(0.855015\pi\)
\(462\) −396.817 + 396.817i −0.858912 + 0.858912i
\(463\) −7.34847 7.34847i −0.0158714 0.0158714i 0.699127 0.714998i \(-0.253572\pi\)
−0.714998 + 0.699127i \(0.753572\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −108.000 −0.231760
\(467\) −347.828 + 347.828i −0.744813 + 0.744813i −0.973500 0.228687i \(-0.926557\pi\)
0.228687 + 0.973500i \(0.426557\pi\)
\(468\) 22.0454 + 22.0454i 0.0471056 + 0.0471056i
\(469\) 648.000i 1.38166i
\(470\) 0 0
\(471\) −342.000 −0.726115
\(472\) −551.135 + 551.135i −1.16766 + 1.16766i
\(473\) −529.090 529.090i −1.11858 1.11858i
\(474\) 210.000i 0.443038i
\(475\) 0 0
\(476\) −72.0000 −0.151261
\(477\) −14.6969 + 14.6969i −0.0308112 + 0.0308112i
\(478\) 220.454 + 220.454i 0.461201 + 0.461201i
\(479\) 360.000i 0.751566i −0.926708 0.375783i \(-0.877374\pi\)
0.926708 0.375783i \(-0.122626\pi\)
\(480\) 0 0
\(481\) −108.000 −0.224532
\(482\) −218.005 + 218.005i −0.452292 + 0.452292i
\(483\) −352.727 352.727i −0.730283 0.730283i
\(484\) 203.000i 0.419421i
\(485\) 0 0
\(486\) −27.0000 −0.0555556
\(487\) −580.529 + 580.529i −1.19205 + 1.19205i −0.215561 + 0.976490i \(0.569158\pi\)
−0.976490 + 0.215561i \(0.930842\pi\)
\(488\) 12.2474 + 12.2474i 0.0250972 + 0.0250972i
\(489\) 288.000i 0.588957i
\(490\) 0 0
\(491\) −18.0000 −0.0366599 −0.0183299 0.999832i \(-0.505835\pi\)
−0.0183299 + 0.999832i \(0.505835\pi\)
\(492\) 22.0454 22.0454i 0.0448077 0.0448077i
\(493\) 0 0
\(494\) 180.000i 0.364372i
\(495\) 0 0
\(496\) 242.000 0.487903
\(497\) 529.090 529.090i 1.06457 1.06457i
\(498\) 161.666 + 161.666i 0.324631 + 0.324631i
\(499\) 590.000i 1.18236i 0.806538 + 0.591182i \(0.201339\pi\)
−0.806538 + 0.591182i \(0.798661\pi\)
\(500\) 0 0
\(501\) 48.0000 0.0958084
\(502\) 418.863 418.863i 0.834388 0.834388i
\(503\) −93.0806 93.0806i −0.185051 0.185051i 0.608502 0.793553i \(-0.291771\pi\)
−0.793553 + 0.608502i \(0.791771\pi\)
\(504\) 270.000i 0.535714i
\(505\) 0 0
\(506\) 864.000 1.70751
\(507\) 74.7094 74.7094i 0.147356 0.147356i
\(508\) 7.34847 + 7.34847i 0.0144655 + 0.0144655i
\(509\) 540.000i 1.06090i 0.847715 + 0.530452i \(0.177978\pi\)
−0.847715 + 0.530452i \(0.822022\pi\)
\(510\) 0 0
\(511\) −648.000 −1.26810
\(512\) −390.694 + 390.694i −0.763073 + 0.763073i
\(513\) −36.7423 36.7423i −0.0716225 0.0716225i
\(514\) 708.000i 1.37743i
\(515\) 0 0
\(516\) −72.0000 −0.139535
\(517\) −793.635 + 793.635i −1.53508 + 1.53508i
\(518\) −132.272 132.272i −0.255352 0.255352i
\(519\) 348.000i 0.670520i
\(520\) 0 0
\(521\) 342.000 0.656430 0.328215 0.944603i \(-0.393553\pi\)
0.328215 + 0.944603i \(0.393553\pi\)
\(522\) 0 0
\(523\) −264.545 264.545i −0.505822 0.505822i 0.407419 0.913241i \(-0.366429\pi\)
−0.913241 + 0.407419i \(0.866429\pi\)
\(524\) 18.0000i 0.0343511i
\(525\) 0 0
\(526\) −648.000 −1.23194
\(527\) −107.778 + 107.778i −0.204511 + 0.204511i
\(528\) −242.499 242.499i −0.459279 0.459279i
\(529\) 239.000i 0.451796i
\(530\) 0 0
\(531\) −270.000 −0.508475
\(532\) −73.4847 + 73.4847i −0.138129 + 0.138129i
\(533\) 132.272 + 132.272i 0.248166 + 0.248166i
\(534\) 270.000i 0.505618i
\(535\) 0 0
\(536\) 540.000 1.00746
\(537\) 110.227 110.227i 0.205265 0.205265i
\(538\) 0 0
\(539\) 1062.00i 1.97032i
\(540\) 0 0
\(541\) 2.00000 0.00369686 0.00184843 0.999998i \(-0.499412\pi\)
0.00184843 + 0.999998i \(0.499412\pi\)
\(542\) −585.428 + 585.428i −1.08013 + 1.08013i
\(543\) −120.025 120.025i −0.221041 0.221041i
\(544\) 108.000i 0.198529i
\(545\) 0 0
\(546\) −324.000 −0.593407
\(547\) 264.545 264.545i 0.483629 0.483629i −0.422660 0.906288i \(-0.638903\pi\)
0.906288 + 0.422660i \(0.138903\pi\)
\(548\) 142.070 + 142.070i 0.259253 + 0.259253i
\(549\) 6.00000i 0.0109290i
\(550\) 0 0
\(551\) 0 0
\(552\) 293.939 293.939i 0.532498 0.532498i
\(553\) −514.393 514.393i −0.930186 0.930186i
\(554\) 198.000i 0.357401i
\(555\) 0 0
\(556\) 170.000 0.305755
\(557\) 484.999 484.999i 0.870734 0.870734i −0.121818 0.992552i \(-0.538872\pi\)
0.992552 + 0.121818i \(0.0388725\pi\)
\(558\) 80.8332 + 80.8332i 0.144862 + 0.144862i
\(559\) 432.000i 0.772809i
\(560\) 0 0
\(561\) 216.000 0.385027
\(562\) 198.409 198.409i 0.353040 0.353040i
\(563\) 396.817 + 396.817i 0.704827 + 0.704827i 0.965443 0.260616i \(-0.0839256\pi\)
−0.260616 + 0.965443i \(0.583926\pi\)
\(564\) 108.000i 0.191489i
\(565\) 0 0
\(566\) 612.000 1.08127
\(567\) −66.1362 + 66.1362i −0.116642 + 0.116642i
\(568\) 440.908 + 440.908i 0.776247 + 0.776247i
\(569\) 630.000i 1.10721i 0.832781 + 0.553603i \(0.186747\pi\)
−0.832781 + 0.553603i \(0.813253\pi\)
\(570\) 0 0
\(571\) 302.000 0.528897 0.264448 0.964400i \(-0.414810\pi\)
0.264448 + 0.964400i \(0.414810\pi\)
\(572\) −132.272 + 132.272i −0.231246 + 0.231246i
\(573\) 308.636 + 308.636i 0.538631 + 0.538631i
\(574\) 324.000i 0.564460i
\(575\) 0 0
\(576\) −213.000 −0.369792
\(577\) 484.999 484.999i 0.840553 0.840553i −0.148378 0.988931i \(-0.547405\pi\)
0.988931 + 0.148378i \(0.0474052\pi\)
\(578\) 295.164 + 295.164i 0.510664 + 0.510664i
\(579\) 648.000i 1.11917i
\(580\) 0 0
\(581\) 792.000 1.36317
\(582\) −308.636 + 308.636i −0.530302 + 0.530302i
\(583\) −88.1816 88.1816i −0.151255 0.151255i
\(584\) 540.000i 0.924658i
\(585\) 0 0
\(586\) 912.000 1.55631
\(587\) 582.979 582.979i 0.993149 0.993149i −0.00682753 0.999977i \(-0.502173\pi\)
0.999977 + 0.00682753i \(0.00217329\pi\)
\(588\) 72.2599 + 72.2599i 0.122891 + 0.122891i
\(589\) 220.000i 0.373514i
\(590\) 0 0
\(591\) −312.000 −0.527919
\(592\) 80.8332 80.8332i 0.136543 0.136543i
\(593\) −533.989 533.989i −0.900487 0.900487i 0.0949912 0.995478i \(-0.469718\pi\)
−0.995478 + 0.0949912i \(0.969718\pi\)
\(594\) 162.000i 0.272727i
\(595\) 0 0
\(596\) 180.000 0.302013
\(597\) −355.176 + 355.176i −0.594935 + 0.594935i
\(598\) 352.727 + 352.727i 0.589844 + 0.589844i
\(599\) 540.000i 0.901503i −0.892650 0.450751i \(-0.851156\pi\)
0.892650 0.450751i \(-0.148844\pi\)
\(600\) 0 0
\(601\) −758.000 −1.26123 −0.630616 0.776095i \(-0.717198\pi\)
−0.630616 + 0.776095i \(0.717198\pi\)
\(602\) 529.090 529.090i 0.878887 0.878887i
\(603\) 132.272 + 132.272i 0.219357 + 0.219357i
\(604\) 22.0000i 0.0364238i
\(605\) 0 0
\(606\) −324.000 −0.534653
\(607\) 668.711 668.711i 1.10167 1.10167i 0.107455 0.994210i \(-0.465730\pi\)
0.994210 0.107455i \(-0.0342702\pi\)
\(608\) −110.227 110.227i −0.181294 0.181294i
\(609\) 0 0
\(610\) 0 0
\(611\) −648.000 −1.06056
\(612\) 14.6969 14.6969i 0.0240146 0.0240146i
\(613\) 66.1362 + 66.1362i 0.107889 + 0.107889i 0.758991 0.651101i \(-0.225693\pi\)
−0.651101 + 0.758991i \(0.725693\pi\)
\(614\) 972.000i 1.58306i
\(615\) 0 0
\(616\) −1620.00 −2.62987
\(617\) −592.777 + 592.777i −0.960740 + 0.960740i −0.999258 0.0385180i \(-0.987736\pi\)
0.0385180 + 0.999258i \(0.487736\pi\)
\(618\) 198.409 + 198.409i 0.321050 + 0.321050i
\(619\) 1030.00i 1.66397i −0.554795 0.831987i \(-0.687203\pi\)
0.554795 0.831987i \(-0.312797\pi\)
\(620\) 0 0
\(621\) 144.000 0.231884
\(622\) 308.636 308.636i 0.496199 0.496199i
\(623\) 661.362 + 661.362i 1.06158 + 1.06158i
\(624\) 198.000i 0.317308i
\(625\) 0 0
\(626\) −288.000 −0.460064
\(627\) 220.454 220.454i 0.351601 0.351601i
\(628\) −139.621 139.621i −0.222326 0.222326i
\(629\) 72.0000i 0.114467i
\(630\) 0 0
\(631\) 242.000 0.383518 0.191759 0.981442i \(-0.438581\pi\)
0.191759 + 0.981442i \(0.438581\pi\)
\(632\) 428.661 428.661i 0.678261 0.678261i
\(633\) 149.419 + 149.419i 0.236049 + 0.236049i
\(634\) 672.000i 1.05994i
\(635\) 0 0
\(636\) −12.0000 −0.0188679
\(637\) −433.560 + 433.560i −0.680627 + 0.680627i
\(638\) 0 0
\(639\) 216.000i 0.338028i
\(640\) 0 0
\(641\) 162.000 0.252730 0.126365 0.991984i \(-0.459669\pi\)
0.126365 + 0.991984i \(0.459669\pi\)
\(642\) 132.272 132.272i 0.206032 0.206032i
\(643\) −264.545 264.545i −0.411423 0.411423i 0.470811 0.882234i \(-0.343961\pi\)
−0.882234 + 0.470811i \(0.843961\pi\)
\(644\) 288.000i 0.447205i
\(645\) 0 0
\(646\) −120.000 −0.185759
\(647\) 93.0806 93.0806i 0.143865 0.143865i −0.631506 0.775371i \(-0.717563\pi\)
0.775371 + 0.631506i \(0.217563\pi\)
\(648\) −55.1135 55.1135i −0.0850517 0.0850517i
\(649\) 1620.00i 2.49615i
\(650\) 0 0
\(651\) 396.000 0.608295
\(652\) 117.576 117.576i 0.180331 0.180331i
\(653\) −68.5857 68.5857i −0.105032 0.105032i 0.652638 0.757670i \(-0.273662\pi\)
−0.757670 + 0.652638i \(0.773662\pi\)
\(654\) 510.000i 0.779817i
\(655\) 0 0
\(656\) −198.000 −0.301829
\(657\) 132.272 132.272i 0.201328 0.201328i
\(658\) −793.635 793.635i −1.20613 1.20613i
\(659\) 630.000i 0.955994i 0.878361 + 0.477997i \(0.158637\pi\)
−0.878361 + 0.477997i \(0.841363\pi\)
\(660\) 0 0
\(661\) 622.000 0.940998 0.470499 0.882400i \(-0.344074\pi\)
0.470499 + 0.882400i \(0.344074\pi\)
\(662\) −511.943 + 511.943i −0.773328 + 0.773328i
\(663\) 88.1816 + 88.1816i 0.133004 + 0.133004i
\(664\) 660.000i 0.993976i
\(665\) 0 0
\(666\) 54.0000 0.0810811
\(667\) 0 0
\(668\) 19.5959 + 19.5959i 0.0293352 + 0.0293352i
\(669\) 198.000i 0.295964i
\(670\) 0 0
\(671\) −36.0000 −0.0536513
\(672\) −198.409 + 198.409i −0.295251 + 0.295251i
\(673\) 102.879 + 102.879i 0.152866 + 0.152866i 0.779397 0.626531i \(-0.215526\pi\)
−0.626531 + 0.779397i \(0.715526\pi\)
\(674\) 468.000i 0.694362i
\(675\) 0 0
\(676\) 61.0000 0.0902367
\(677\) −176.363 + 176.363i −0.260507 + 0.260507i −0.825260 0.564753i \(-0.808971\pi\)
0.564753 + 0.825260i \(0.308971\pi\)
\(678\) −132.272 132.272i −0.195092 0.195092i
\(679\) 1512.00i 2.22680i
\(680\) 0 0
\(681\) −132.000 −0.193833
\(682\) −484.999 + 484.999i −0.711142 + 0.711142i
\(683\) 445.807 + 445.807i 0.652719 + 0.652719i 0.953647 0.300928i \(-0.0972963\pi\)
−0.300928 + 0.953647i \(0.597296\pi\)
\(684\) 30.0000i 0.0438596i
\(685\) 0 0
\(686\) −180.000 −0.262391
\(687\) 61.2372 61.2372i 0.0891372 0.0891372i
\(688\) 323.333 + 323.333i 0.469960 + 0.469960i
\(689\) 72.0000i 0.104499i
\(690\) 0 0
\(691\) 682.000 0.986975 0.493488 0.869753i \(-0.335722\pi\)
0.493488 + 0.869753i \(0.335722\pi\)
\(692\) 142.070 142.070i 0.205304 0.205304i
\(693\) −396.817 396.817i −0.572608 0.572608i
\(694\) 108.000i 0.155620i
\(695\) 0 0
\(696\) 0 0
\(697\) 88.1816 88.1816i 0.126516 0.126516i
\(698\) 85.7321 + 85.7321i 0.122825 + 0.122825i
\(699\) 108.000i 0.154506i
\(700\) 0 0
\(701\) −468.000 −0.667618 −0.333809 0.942641i \(-0.608334\pi\)
−0.333809 + 0.942641i \(0.608334\pi\)
\(702\) 66.1362 66.1362i 0.0942111 0.0942111i
\(703\) 73.4847 + 73.4847i 0.104530 + 0.104530i
\(704\) 1278.00i 1.81534i
\(705\) 0 0
\(706\) −108.000 −0.152975
\(707\) −793.635 + 793.635i −1.12254 + 1.12254i
\(708\) −110.227 110.227i −0.155688 0.155688i
\(709\) 310.000i 0.437236i 0.975811 + 0.218618i \(0.0701548\pi\)
−0.975811 + 0.218618i \(0.929845\pi\)
\(710\) 0 0
\(711\) 210.000 0.295359
\(712\) −551.135 + 551.135i −0.774066 + 0.774066i
\(713\) −431.110 431.110i −0.604643 0.604643i
\(714\) 216.000i 0.302521i
\(715\) 0 0
\(716\) 90.0000 0.125698
\(717\) −220.454 + 220.454i −0.307467 + 0.307467i
\(718\) −661.362 661.362i −0.921117 0.921117i
\(719\) 180.000i 0.250348i 0.992135 + 0.125174i \(0.0399489\pi\)
−0.992135 + 0.125174i \(0.960051\pi\)
\(720\) 0 0
\(721\) 972.000 1.34813
\(722\) 319.658 319.658i 0.442740 0.442740i
\(723\) −218.005 218.005i −0.301528 0.301528i
\(724\) 98.0000i 0.135359i
\(725\) 0 0
\(726\) 609.000 0.838843
\(727\) −507.044 + 507.044i −0.697448 + 0.697448i −0.963859 0.266412i \(-0.914162\pi\)
0.266412 + 0.963859i \(0.414162\pi\)
\(728\) −661.362 661.362i −0.908465 0.908465i
\(729\) 27.0000i 0.0370370i
\(730\) 0 0
\(731\) −288.000 −0.393981
\(732\) −2.44949 + 2.44949i −0.00334630 + 0.00334630i
\(733\) 800.983 + 800.983i 1.09275 + 1.09275i 0.995234 + 0.0975121i \(0.0310885\pi\)
0.0975121 + 0.995234i \(0.468912\pi\)
\(734\) 162.000i 0.220708i
\(735\) 0 0
\(736\) 432.000 0.586957
\(737\) −793.635 + 793.635i −1.07684 + 1.07684i
\(738\) −66.1362 66.1362i −0.0896155 0.0896155i
\(739\) 350.000i 0.473613i −0.971557 0.236806i \(-0.923899\pi\)
0.971557 0.236806i \(-0.0761007\pi\)
\(740\) 0 0
\(741\) 180.000 0.242915
\(742\) 88.1816 88.1816i 0.118843 0.118843i
\(743\) −925.907 925.907i −1.24617 1.24617i −0.957396 0.288778i \(-0.906751\pi\)
−0.288778 0.957396i \(-0.593249\pi\)
\(744\) 330.000i 0.443548i
\(745\) 0 0
\(746\) 522.000 0.699732
\(747\) −161.666 + 161.666i −0.216421 + 0.216421i
\(748\) 88.1816 + 88.1816i 0.117890 + 0.117890i
\(749\) 648.000i 0.865154i
\(750\) 0 0
\(751\) −338.000 −0.450067 −0.225033 0.974351i \(-0.572249\pi\)
−0.225033 + 0.974351i \(0.572249\pi\)
\(752\) 484.999 484.999i 0.644945 0.644945i
\(753\) 418.863 + 418.863i 0.556259 + 0.556259i
\(754\) 0 0
\(755\) 0 0
\(756\) −54.0000 −0.0714286
\(757\) −286.590 + 286.590i −0.378587 + 0.378587i −0.870592 0.492005i \(-0.836264\pi\)
0.492005 + 0.870592i \(0.336264\pi\)
\(758\) −208.207 208.207i −0.274679 0.274679i
\(759\) 864.000i 1.13834i
\(760\) 0 0
\(761\) −1278.00 −1.67937 −0.839685 0.543074i \(-0.817260\pi\)
−0.839685 + 0.543074i \(0.817260\pi\)
\(762\) 22.0454 22.0454i 0.0289310 0.0289310i
\(763\) 1249.24 + 1249.24i 1.63727 + 1.63727i
\(764\) 252.000i 0.329843i
\(765\) 0 0
\(766\) 372.000 0.485640
\(767\) 661.362 661.362i 0.862271 0.862271i
\(768\) −219.229 219.229i −0.285455 0.285455i
\(769\) 590.000i 0.767230i 0.923493 + 0.383615i \(0.125321\pi\)
−0.923493 + 0.383615i \(0.874679\pi\)
\(770\) 0 0
\(771\) 708.000 0.918288
\(772\) 264.545 264.545i 0.342675 0.342675i
\(773\) 127.373 + 127.373i 0.164778 + 0.164778i 0.784680 0.619902i \(-0.212827\pi\)
−0.619902 + 0.784680i \(0.712827\pi\)
\(774\) 216.000i 0.279070i
\(775\) 0 0
\(776\) −1260.00 −1.62371
\(777\) 132.272 132.272i 0.170235 0.170235i
\(778\) −440.908 440.908i −0.566720 0.566720i
\(779\) 180.000i 0.231065i
\(780\) 0 0
\(781\) −1296.00 −1.65941
\(782\) 235.151 235.151i 0.300705 0.300705i
\(783\) 0 0
\(784\) 649.000i 0.827806i
\(785\) 0 0
\(786\) −54.0000 −0.0687023
\(787\) 264.545 264.545i 0.336143 0.336143i −0.518770 0.854914i \(-0.673610\pi\)
0.854914 + 0.518770i \(0.173610\pi\)
\(788\) −127.373 127.373i −0.161641 0.161641i
\(789\) 648.000i 0.821293i
\(790\) 0 0
\(791\) −648.000 −0.819216
\(792\) 330.681 330.681i 0.417527 0.417527i
\(793\) −14.6969 14.6969i −0.0185333 0.0185333i
\(794\) 738.000i 0.929471i
\(795\) 0 0
\(796\) −290.000 −0.364322
\(797\) −200.858 + 200.858i −0.252018 + 0.252018i −0.821797 0.569780i \(-0.807029\pi\)
0.569780 + 0.821797i \(0.307029\pi\)
\(798\) 220.454 + 220.454i 0.276258 + 0.276258i
\(799\) 432.000i 0.540676i
\(800\) 0 0
\(801\) −270.000 −0.337079
\(802\) −683.408 + 683.408i −0.852129 + 0.852129i
\(803\) 793.635 + 793.635i 0.988337 + 0.988337i
\(804\) 108.000i 0.134328i
\(805\) 0 0
\(806\) −396.000 −0.491315
\(807\) 0 0
\(808\) −661.362 661.362i −0.818518 0.818518i
\(809\) 630.000i 0.778739i −0.921082 0.389370i \(-0.872693\pi\)
0.921082 0.389370i \(-0.127307\pi\)
\(810\) 0 0
\(811\) −218.000 −0.268804 −0.134402 0.990927i \(-0.542911\pi\)
−0.134402 + 0.990927i \(0.542911\pi\)
\(812\) 0 0
\(813\) −585.428 585.428i −0.720084 0.720084i
\(814\) 324.000i 0.398034i
\(815\) 0 0
\(816\) −132.000 −0.161765
\(817\) −293.939 + 293.939i −0.359778 + 0.359778i
\(818\) 820.579 + 820.579i 1.00315 + 1.00315i
\(819\) 324.000i 0.395604i
\(820\) 0 0
\(821\) 432.000 0.526188 0.263094 0.964770i \(-0.415257\pi\)
0.263094 + 0.964770i \(0.415257\pi\)
\(822\) 426.211 426.211i 0.518505 0.518505i
\(823\) 360.075 + 360.075i 0.437515 + 0.437515i 0.891175 0.453660i \(-0.149882\pi\)
−0.453660 + 0.891175i \(0.649882\pi\)
\(824\) 810.000i 0.983010i
\(825\) 0 0
\(826\) 1620.00 1.96126
\(827\) 876.917 876.917i 1.06036 1.06036i 0.0623022 0.998057i \(-0.480156\pi\)
0.998057 0.0623022i \(-0.0198443\pi\)
\(828\) 58.7878 + 58.7878i 0.0709997 + 0.0709997i
\(829\) 70.0000i 0.0844391i 0.999108 + 0.0422195i \(0.0134429\pi\)
−0.999108 + 0.0422195i \(0.986557\pi\)
\(830\) 0 0
\(831\) 198.000 0.238267
\(832\) 521.741 521.741i 0.627093 0.627093i
\(833\) 289.040 + 289.040i 0.346987 + 0.346987i
\(834\) 510.000i 0.611511i
\(835\) 0 0
\(836\) 180.000 0.215311
\(837\) −80.8332 + 80.8332i −0.0965749 + 0.0965749i
\(838\) 771.589 + 771.589i 0.920751 + 0.920751i
\(839\) 540.000i 0.643623i −0.946804 0.321812i \(-0.895708\pi\)
0.946804 0.321812i \(-0.104292\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 173.914 173.914i 0.206548 0.206548i
\(843\) 198.409 + 198.409i 0.235360 + 0.235360i
\(844\) 122.000i 0.144550i
\(845\) 0 0
\(846\) 324.000 0.382979
\(847\) 1491.74 1491.74i 1.76120 1.76120i
\(848\) 53.8888 + 53.8888i 0.0635481 + 0.0635481i
\(849\) 612.000i 0.720848i
\(850\) 0 0
\(851\) −288.000 −0.338425
\(852\) −88.1816 + 88.1816i −0.103500 + 0.103500i
\(853\) −815.680 815.680i −0.956249 0.956249i 0.0428336 0.999082i \(-0.486361\pi\)
−0.999082 + 0.0428336i \(0.986361\pi\)
\(854\) 36.0000i 0.0421546i
\(855\) 0 0
\(856\) 540.000 0.630841
\(857\) −298.838 + 298.838i −0.348702 + 0.348702i −0.859626 0.510924i \(-0.829303\pi\)
0.510924 + 0.859626i \(0.329303\pi\)
\(858\) 396.817 + 396.817i 0.462491 + 0.462491i
\(859\) 310.000i 0.360885i −0.983586 0.180442i \(-0.942247\pi\)
0.983586 0.180442i \(-0.0577529\pi\)
\(860\) 0 0
\(861\) −324.000 −0.376307
\(862\) 749.544 749.544i 0.869540 0.869540i
\(863\) 225.353 + 225.353i 0.261128 + 0.261128i 0.825512 0.564385i \(-0.190886\pi\)
−0.564385 + 0.825512i \(0.690886\pi\)
\(864\) 81.0000i 0.0937500i
\(865\) 0 0
\(866\) 252.000 0.290993
\(867\) −295.164 + 295.164i −0.340442 + 0.340442i
\(868\) 161.666 + 161.666i 0.186252 + 0.186252i
\(869\) 1260.00i 1.44994i
\(870\) 0 0
\(871\) −648.000 −0.743972
\(872\) −1041.03 + 1041.03i −1.19385 + 1.19385i
\(873\) −308.636 308.636i −0.353535 0.353535i
\(874\) 480.000i 0.549199i
\(875\) 0 0
\(876\) 108.000 0.123288
\(877\) −213.106 + 213.106i −0.242994 + 0.242994i −0.818088 0.575094i \(-0.804966\pi\)
0.575094 + 0.818088i \(0.304966\pi\)
\(878\) −526.640 526.640i −0.599818 0.599818i
\(879\) 912.000i 1.03754i
\(880\) 0 0
\(881\) 1062.00 1.20545 0.602724 0.797950i \(-0.294082\pi\)
0.602724 + 0.797950i \(0.294082\pi\)
\(882\) 216.780 216.780i 0.245782 0.245782i
\(883\) −117.576 117.576i −0.133155 0.133155i 0.637388 0.770543i \(-0.280015\pi\)
−0.770543 + 0.637388i \(0.780015\pi\)
\(884\) 72.0000i 0.0814480i
\(885\) 0 0
\(886\) −1428.00 −1.61174
\(887\) 509.494 509.494i 0.574401 0.574401i −0.358954 0.933355i \(-0.616867\pi\)
0.933355 + 0.358954i \(0.116867\pi\)
\(888\) 110.227 + 110.227i 0.124130 + 0.124130i
\(889\) 108.000i 0.121485i
\(890\) 0 0
\(891\) 162.000 0.181818
\(892\) 80.8332 80.8332i 0.0906201 0.0906201i
\(893\) 440.908 + 440.908i 0.493738 + 0.493738i
\(894\) 540.000i 0.604027i
\(895\) 0 0
\(896\) 630.000 0.703125
\(897\) −352.727 + 352.727i −0.393229 + 0.393229i
\(898\) 110.227 + 110.227i 0.122747 + 0.122747i
\(899\) 0 0
\(900\) 0 0
\(901\) −48.0000 −0.0532741
\(902\) 396.817 396.817i 0.439931 0.439931i
\(903\) 529.090 + 529.090i 0.585924 + 0.585924i
\(904\) 540.000i 0.597345i
\(905\) 0 0
\(906\) 66.0000 0.0728477
\(907\) 411.514 411.514i 0.453709 0.453709i −0.442874 0.896584i \(-0.646041\pi\)
0.896584 + 0.442874i \(0.146041\pi\)
\(908\) −53.8888 53.8888i −0.0593489 0.0593489i
\(909\) 324.000i 0.356436i
\(910\) 0 0
\(911\) 792.000 0.869374 0.434687 0.900582i \(-0.356859\pi\)
0.434687 + 0.900582i \(0.356859\pi\)
\(912\) −134.722 + 134.722i −0.147721 + 0.147721i
\(913\) −969.998 969.998i −1.06243 1.06243i
\(914\) 468.000i 0.512035i
\(915\) 0 0
\(916\) 50.0000 0.0545852
\(917\) −132.272 + 132.272i −0.144245 + 0.144245i
\(918\) −44.0908 44.0908i −0.0480292 0.0480292i
\(919\) 430.000i 0.467900i 0.972249 + 0.233950i \(0.0751652\pi\)
−0.972249 + 0.233950i \(0.924835\pi\)
\(920\) 0 0
\(921\) −972.000 −1.05537
\(922\) −1014.09 + 1014.09i −1.09988 + 1.09988i
\(923\) −529.090 529.090i −0.573228 0.573228i
\(924\) 324.000i 0.350649i
\(925\) 0 0
\(926\) −18.0000 −0.0194384
\(927\) −198.409 + 198.409i −0.214033 + 0.214033i
\(928\) 0 0
\(929\) 1530.00i 1.64693i 0.567365 + 0.823466i \(0.307963\pi\)
−0.567365 + 0.823466i \(0.692037\pi\)
\(930\) 0 0
\(931\) 590.000 0.633727
\(932\) 44.0908 44.0908i 0.0473077 0.0473077i
\(933\) 308.636 + 308.636i 0.330799 + 0.330799i
\(934\) 852.000i 0.912206i
\(935\) 0 0
\(936\) 270.000 0.288462
\(937\) −323.333 + 323.333i −0.345072 + 0.345072i −0.858270 0.513198i \(-0.828461\pi\)
0.513198 + 0.858270i \(0.328461\pi\)
\(938\) −793.635 793.635i −0.846092 0.846092i
\(939\) 288.000i 0.306709i
\(940\) 0 0
\(941\) 1152.00 1.22423 0.612115 0.790769i \(-0.290319\pi\)
0.612115 + 0.790769i \(0.290319\pi\)
\(942\) −418.863 + 418.863i −0.444653 + 0.444653i
\(943\) 352.727 + 352.727i 0.374047 + 0.374047i
\(944\) 990.000i 1.04873i
\(945\) 0 0
\(946\) −1296.00 −1.36998
\(947\) −739.746 + 739.746i −0.781147 + 0.781147i −0.980024 0.198878i \(-0.936270\pi\)
0.198878 + 0.980024i \(0.436270\pi\)
\(948\) 85.7321 + 85.7321i 0.0904347 + 0.0904347i
\(949\) 648.000i 0.682824i
\(950\) 0 0
\(951\) −672.000 −0.706625
\(952\) −440.908 + 440.908i −0.463139 + 0.463139i
\(953\) −44.0908 44.0908i −0.0462653 0.0462653i 0.683596 0.729861i \(-0.260415\pi\)
−0.729861 + 0.683596i \(0.760415\pi\)
\(954\) 36.0000i 0.0377358i
\(955\) 0 0
\(956\) −180.000 −0.188285
\(957\) 0 0
\(958\) −440.908 440.908i −0.460238 0.460238i
\(959\) 2088.00i 2.17727i
\(960\) 0 0
\(961\) −477.000 −0.496358
\(962\) −132.272 + 132.272i −0.137497 + 0.137497i
\(963\) 132.272 + 132.272i 0.137355 + 0.137355i
\(964\) 178.000i 0.184647i
\(965\) 0 0
\(966\) −864.000 −0.894410
\(967\) −727.498 + 727.498i −0.752325 + 0.752325i −0.974913 0.222588i \(-0.928550\pi\)
0.222588 + 0.974913i \(0.428550\pi\)
\(968\) 1243.12 + 1243.12i 1.28421 + 1.28421i
\(969\) 120.000i 0.123839i
\(970\) 0 0
\(971\) −1278.00 −1.31617 −0.658084 0.752944i \(-0.728633\pi\)
−0.658084 + 0.752944i \(0.728633\pi\)
\(972\) 11.0227 11.0227i 0.0113402 0.0113402i
\(973\) −1249.24 1249.24i −1.28391 1.28391i
\(974\) 1422.00i 1.45996i
\(975\) 0 0
\(976\) 22.0000 0.0225410
\(977\) −396.817 + 396.817i −0.406159 + 0.406159i −0.880397 0.474238i \(-0.842724\pi\)
0.474238 + 0.880397i \(0.342724\pi\)
\(978\) −352.727 352.727i −0.360661 0.360661i
\(979\) 1620.00i 1.65475i
\(980\) 0 0
\(981\) −510.000 −0.519878
\(982\) −22.0454 + 22.0454i −0.0224495 + 0.0224495i
\(983\) −827.928 827.928i −0.842246 0.842246i 0.146905 0.989151i \(-0.453069\pi\)
−0.989151 + 0.146905i \(0.953069\pi\)
\(984\) 270.000i 0.274390i
\(985\) 0 0
\(986\) 0 0
\(987\) 793.635 793.635i 0.804088 0.804088i
\(988\) 73.4847 + 73.4847i 0.0743772 + 0.0743772i
\(989\) 1152.00i 1.16481i
\(990\) 0 0
\(991\) −118.000 −0.119072 −0.0595358 0.998226i \(-0.518962\pi\)
−0.0595358 + 0.998226i \(0.518962\pi\)
\(992\) −242.499 + 242.499i −0.244455 + 0.244455i
\(993\) −511.943 511.943i −0.515552 0.515552i
\(994\) 1296.00i 1.30382i
\(995\) 0 0
\(996\) −132.000 −0.132530
\(997\) 815.680 815.680i 0.818134 0.818134i −0.167703 0.985838i \(-0.553635\pi\)
0.985838 + 0.167703i \(0.0536350\pi\)
\(998\) 722.599 + 722.599i 0.724048 + 0.724048i
\(999\) 54.0000i 0.0540541i
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 75.3.f.a.43.2 yes 4
3.2 odd 2 225.3.g.f.118.1 4
4.3 odd 2 1200.3.bg.j.193.1 4
5.2 odd 4 inner 75.3.f.a.7.2 yes 4
5.3 odd 4 inner 75.3.f.a.7.1 4
5.4 even 2 inner 75.3.f.a.43.1 yes 4
15.2 even 4 225.3.g.f.82.1 4
15.8 even 4 225.3.g.f.82.2 4
15.14 odd 2 225.3.g.f.118.2 4
20.3 even 4 1200.3.bg.j.1057.2 4
20.7 even 4 1200.3.bg.j.1057.1 4
20.19 odd 2 1200.3.bg.j.193.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.3.f.a.7.1 4 5.3 odd 4 inner
75.3.f.a.7.2 yes 4 5.2 odd 4 inner
75.3.f.a.43.1 yes 4 5.4 even 2 inner
75.3.f.a.43.2 yes 4 1.1 even 1 trivial
225.3.g.f.82.1 4 15.2 even 4
225.3.g.f.82.2 4 15.8 even 4
225.3.g.f.118.1 4 3.2 odd 2
225.3.g.f.118.2 4 15.14 odd 2
1200.3.bg.j.193.1 4 4.3 odd 2
1200.3.bg.j.193.2 4 20.19 odd 2
1200.3.bg.j.1057.1 4 20.7 even 4
1200.3.bg.j.1057.2 4 20.3 even 4