Properties

Label 1575.2.bc.a.899.2
Level $1575$
Weight $2$
Character 1575.899
Analytic conductor $12.576$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1575,2,Mod(899,1575)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1575, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1575.899");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.bc (of order \(6\), 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: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 899.2
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1575.899
Dual form 1575.2.bc.a.1349.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 - 1.22474i) q^{2} +(2.59808 + 0.500000i) q^{7} -2.82843 q^{8} +O(q^{10})\) \(q+(-0.707107 - 1.22474i) q^{2} +(2.59808 + 0.500000i) q^{7} -2.82843 q^{8} +(-1.22474 - 0.707107i) q^{11} +5.19615 q^{13} +(-1.22474 - 3.53553i) q^{14} +(2.00000 + 3.46410i) q^{16} +(4.24264 + 2.44949i) q^{17} +(-1.50000 + 0.866025i) q^{19} +2.00000i q^{22} +(2.82843 + 4.89898i) q^{23} +(-3.67423 - 6.36396i) q^{26} +2.82843i q^{29} +(1.50000 + 0.866025i) q^{31} -6.92820i q^{34} +(0.866025 - 0.500000i) q^{37} +(2.12132 + 1.22474i) q^{38} -7.34847 q^{41} -1.00000i q^{43} +(4.00000 - 6.92820i) q^{46} +(10.6066 - 6.12372i) q^{47} +(6.50000 + 2.59808i) q^{49} +(-1.41421 + 2.44949i) q^{53} +(-7.34847 - 1.41421i) q^{56} +(3.46410 - 2.00000i) q^{58} +(2.44949 - 4.24264i) q^{59} +(-3.00000 + 1.73205i) q^{61} -2.44949i q^{62} +8.00000 q^{64} +(9.52628 + 5.50000i) q^{67} -7.07107i q^{71} +(-0.866025 + 1.50000i) q^{73} +(-1.22474 - 0.707107i) q^{74} +(-2.82843 - 2.44949i) q^{77} +(2.50000 + 4.33013i) q^{79} +(5.19615 + 9.00000i) q^{82} +7.34847i q^{83} +(-1.22474 + 0.707107i) q^{86} +(3.46410 + 2.00000i) q^{88} +(-2.44949 - 4.24264i) q^{89} +(13.5000 + 2.59808i) q^{91} +(-15.0000 - 8.66025i) q^{94} -10.3923 q^{97} +(-1.41421 - 9.79796i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{16} - 12 q^{19} + 12 q^{31} + 32 q^{46} + 52 q^{49} - 24 q^{61} + 64 q^{64} + 20 q^{79} + 108 q^{91} - 120 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 1.22474i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.59808 + 0.500000i 0.981981 + 0.188982i
\(8\) −2.82843 −1.00000
\(9\) 0 0
\(10\) 0 0
\(11\) −1.22474 0.707107i −0.369274 0.213201i 0.303867 0.952714i \(-0.401722\pi\)
−0.673141 + 0.739514i \(0.735055\pi\)
\(12\) 0 0
\(13\) 5.19615 1.44115 0.720577 0.693375i \(-0.243877\pi\)
0.720577 + 0.693375i \(0.243877\pi\)
\(14\) −1.22474 3.53553i −0.327327 0.944911i
\(15\) 0 0
\(16\) 2.00000 + 3.46410i 0.500000 + 0.866025i
\(17\) 4.24264 + 2.44949i 1.02899 + 0.594089i 0.916696 0.399586i \(-0.130846\pi\)
0.112296 + 0.993675i \(0.464180\pi\)
\(18\) 0 0
\(19\) −1.50000 + 0.866025i −0.344124 + 0.198680i −0.662094 0.749421i \(-0.730332\pi\)
0.317970 + 0.948101i \(0.396999\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.00000i 0.426401i
\(23\) 2.82843 + 4.89898i 0.589768 + 1.02151i 0.994263 + 0.106967i \(0.0341141\pi\)
−0.404495 + 0.914540i \(0.632553\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.67423 6.36396i −0.720577 1.24808i
\(27\) 0 0
\(28\) 0 0
\(29\) 2.82843i 0.525226i 0.964901 + 0.262613i \(0.0845842\pi\)
−0.964901 + 0.262613i \(0.915416\pi\)
\(30\) 0 0
\(31\) 1.50000 + 0.866025i 0.269408 + 0.155543i 0.628619 0.777714i \(-0.283621\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 6.92820i 1.18818i
\(35\) 0 0
\(36\) 0 0
\(37\) 0.866025 0.500000i 0.142374 0.0821995i −0.427121 0.904194i \(-0.640472\pi\)
0.569495 + 0.821995i \(0.307139\pi\)
\(38\) 2.12132 + 1.22474i 0.344124 + 0.198680i
\(39\) 0 0
\(40\) 0 0
\(41\) −7.34847 −1.14764 −0.573819 0.818982i \(-0.694539\pi\)
−0.573819 + 0.818982i \(0.694539\pi\)
\(42\) 0 0
\(43\) 1.00000i 0.152499i −0.997089 0.0762493i \(-0.975706\pi\)
0.997089 0.0762493i \(-0.0242945\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.00000 6.92820i 0.589768 1.02151i
\(47\) 10.6066 6.12372i 1.54713 0.893237i 0.548773 0.835971i \(-0.315095\pi\)
0.998359 0.0572655i \(-0.0182382\pi\)
\(48\) 0 0
\(49\) 6.50000 + 2.59808i 0.928571 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.41421 + 2.44949i −0.194257 + 0.336463i −0.946657 0.322244i \(-0.895563\pi\)
0.752400 + 0.658707i \(0.228896\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −7.34847 1.41421i −0.981981 0.188982i
\(57\) 0 0
\(58\) 3.46410 2.00000i 0.454859 0.262613i
\(59\) 2.44949 4.24264i 0.318896 0.552345i −0.661362 0.750067i \(-0.730021\pi\)
0.980258 + 0.197722i \(0.0633545\pi\)
\(60\) 0 0
\(61\) −3.00000 + 1.73205i −0.384111 + 0.221766i −0.679605 0.733578i \(-0.737849\pi\)
0.295495 + 0.955344i \(0.404516\pi\)
\(62\) 2.44949i 0.311086i
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 9.52628 + 5.50000i 1.16382 + 0.671932i 0.952217 0.305424i \(-0.0987981\pi\)
0.211604 + 0.977356i \(0.432131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.07107i 0.839181i −0.907713 0.419591i \(-0.862174\pi\)
0.907713 0.419591i \(-0.137826\pi\)
\(72\) 0 0
\(73\) −0.866025 + 1.50000i −0.101361 + 0.175562i −0.912245 0.409644i \(-0.865653\pi\)
0.810885 + 0.585206i \(0.198986\pi\)
\(74\) −1.22474 0.707107i −0.142374 0.0821995i
\(75\) 0 0
\(76\) 0 0
\(77\) −2.82843 2.44949i −0.322329 0.279145i
\(78\) 0 0
\(79\) 2.50000 + 4.33013i 0.281272 + 0.487177i 0.971698 0.236225i \(-0.0759104\pi\)
−0.690426 + 0.723403i \(0.742577\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 5.19615 + 9.00000i 0.573819 + 0.993884i
\(83\) 7.34847i 0.806599i 0.915068 + 0.403300i \(0.132137\pi\)
−0.915068 + 0.403300i \(0.867863\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1.22474 + 0.707107i −0.132068 + 0.0762493i
\(87\) 0 0
\(88\) 3.46410 + 2.00000i 0.369274 + 0.213201i
\(89\) −2.44949 4.24264i −0.259645 0.449719i 0.706502 0.707712i \(-0.250272\pi\)
−0.966147 + 0.257993i \(0.916939\pi\)
\(90\) 0 0
\(91\) 13.5000 + 2.59808i 1.41518 + 0.272352i
\(92\) 0 0
\(93\) 0 0
\(94\) −15.0000 8.66025i −1.54713 0.893237i
\(95\) 0 0
\(96\) 0 0
\(97\) −10.3923 −1.05518 −0.527589 0.849500i \(-0.676904\pi\)
−0.527589 + 0.849500i \(0.676904\pi\)
\(98\) −1.41421 9.79796i −0.142857 0.989743i
\(99\) 0 0
\(100\) 0 0
\(101\) 8.57321 14.8492i 0.853067 1.47755i −0.0253604 0.999678i \(-0.508073\pi\)
0.878427 0.477876i \(-0.158593\pi\)
\(102\) 0 0
\(103\) −4.33013 7.50000i −0.426660 0.738997i 0.569914 0.821705i \(-0.306977\pi\)
−0.996574 + 0.0827075i \(0.973643\pi\)
\(104\) −14.6969 −1.44115
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) 1.41421 + 2.44949i 0.136717 + 0.236801i 0.926252 0.376905i \(-0.123012\pi\)
−0.789535 + 0.613706i \(0.789678\pi\)
\(108\) 0 0
\(109\) −0.500000 + 0.866025i −0.0478913 + 0.0829502i −0.888977 0.457951i \(-0.848583\pi\)
0.841086 + 0.540901i \(0.181917\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.46410 + 10.0000i 0.327327 + 0.944911i
\(113\) −1.41421 −0.133038 −0.0665190 0.997785i \(-0.521189\pi\)
−0.0665190 + 0.997785i \(0.521189\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −6.92820 −0.637793
\(119\) 9.79796 + 8.48528i 0.898177 + 0.777844i
\(120\) 0 0
\(121\) −4.50000 7.79423i −0.409091 0.708566i
\(122\) 4.24264 + 2.44949i 0.384111 + 0.221766i
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.0000i 0.976092i −0.872818 0.488046i \(-0.837710\pi\)
0.872818 0.488046i \(-0.162290\pi\)
\(128\) −5.65685 9.79796i −0.500000 0.866025i
\(129\) 0 0
\(130\) 0 0
\(131\) −1.22474 2.12132i −0.107006 0.185341i 0.807550 0.589799i \(-0.200793\pi\)
−0.914556 + 0.404459i \(0.867460\pi\)
\(132\) 0 0
\(133\) −4.33013 + 1.50000i −0.375470 + 0.130066i
\(134\) 15.5563i 1.34386i
\(135\) 0 0
\(136\) −12.0000 6.92820i −1.02899 0.594089i
\(137\) 5.65685 9.79796i 0.483298 0.837096i −0.516518 0.856276i \(-0.672772\pi\)
0.999816 + 0.0191800i \(0.00610555\pi\)
\(138\) 0 0
\(139\) 5.19615i 0.440732i 0.975417 + 0.220366i \(0.0707252\pi\)
−0.975417 + 0.220366i \(0.929275\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.66025 + 5.00000i −0.726752 + 0.419591i
\(143\) −6.36396 3.67423i −0.532181 0.307255i
\(144\) 0 0
\(145\) 0 0
\(146\) 2.44949 0.202721
\(147\) 0 0
\(148\) 0 0
\(149\) −4.89898 + 2.82843i −0.401340 + 0.231714i −0.687062 0.726599i \(-0.741100\pi\)
0.285722 + 0.958313i \(0.407767\pi\)
\(150\) 0 0
\(151\) 11.0000 19.0526i 0.895167 1.55048i 0.0615699 0.998103i \(-0.480389\pi\)
0.833597 0.552372i \(-0.186277\pi\)
\(152\) 4.24264 2.44949i 0.344124 0.198680i
\(153\) 0 0
\(154\) −1.00000 + 5.19615i −0.0805823 + 0.418718i
\(155\) 0 0
\(156\) 0 0
\(157\) 8.66025 15.0000i 0.691164 1.19713i −0.280293 0.959914i \(-0.590432\pi\)
0.971457 0.237216i \(-0.0762349\pi\)
\(158\) 3.53553 6.12372i 0.281272 0.487177i
\(159\) 0 0
\(160\) 0 0
\(161\) 4.89898 + 14.1421i 0.386094 + 1.11456i
\(162\) 0 0
\(163\) −8.66025 + 5.00000i −0.678323 + 0.391630i −0.799223 0.601035i \(-0.794755\pi\)
0.120900 + 0.992665i \(0.461422\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 9.00000 5.19615i 0.698535 0.403300i
\(167\) 7.34847i 0.568642i 0.958729 + 0.284321i \(0.0917681\pi\)
−0.958729 + 0.284321i \(0.908232\pi\)
\(168\) 0 0
\(169\) 14.0000 1.07692
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 8.48528 4.89898i 0.645124 0.372463i −0.141462 0.989944i \(-0.545180\pi\)
0.786586 + 0.617481i \(0.211847\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.65685i 0.426401i
\(177\) 0 0
\(178\) −3.46410 + 6.00000i −0.259645 + 0.449719i
\(179\) 8.57321 + 4.94975i 0.640792 + 0.369961i 0.784920 0.619598i \(-0.212704\pi\)
−0.144127 + 0.989559i \(0.546038\pi\)
\(180\) 0 0
\(181\) 15.5885i 1.15868i 0.815086 + 0.579340i \(0.196690\pi\)
−0.815086 + 0.579340i \(0.803310\pi\)
\(182\) −6.36396 18.3712i −0.471728 1.36176i
\(183\) 0 0
\(184\) −8.00000 13.8564i −0.589768 1.02151i
\(185\) 0 0
\(186\) 0 0
\(187\) −3.46410 6.00000i −0.253320 0.438763i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.22474 0.707107i 0.0886194 0.0511645i −0.455035 0.890473i \(-0.650373\pi\)
0.543655 + 0.839309i \(0.317040\pi\)
\(192\) 0 0
\(193\) −9.52628 5.50000i −0.685717 0.395899i 0.116289 0.993215i \(-0.462900\pi\)
−0.802005 + 0.597317i \(0.796234\pi\)
\(194\) 7.34847 + 12.7279i 0.527589 + 0.913812i
\(195\) 0 0
\(196\) 0 0
\(197\) −19.7990 −1.41062 −0.705310 0.708899i \(-0.749192\pi\)
−0.705310 + 0.708899i \(0.749192\pi\)
\(198\) 0 0
\(199\) 12.0000 + 6.92820i 0.850657 + 0.491127i 0.860873 0.508821i \(-0.169918\pi\)
−0.0102152 + 0.999948i \(0.503252\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −24.2487 −1.70613
\(203\) −1.41421 + 7.34847i −0.0992583 + 0.515761i
\(204\) 0 0
\(205\) 0 0
\(206\) −6.12372 + 10.6066i −0.426660 + 0.738997i
\(207\) 0 0
\(208\) 10.3923 + 18.0000i 0.720577 + 1.24808i
\(209\) 2.44949 0.169435
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 2.00000 3.46410i 0.136717 0.236801i
\(215\) 0 0
\(216\) 0 0
\(217\) 3.46410 + 3.00000i 0.235159 + 0.203653i
\(218\) 1.41421 0.0957826
\(219\) 0 0
\(220\) 0 0
\(221\) 22.0454 + 12.7279i 1.48293 + 0.856173i
\(222\) 0 0
\(223\) −20.7846 −1.39184 −0.695920 0.718119i \(-0.745003\pi\)
−0.695920 + 0.718119i \(0.745003\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.00000 + 1.73205i 0.0665190 + 0.115214i
\(227\) 23.3345 + 13.4722i 1.54877 + 0.894181i 0.998236 + 0.0593658i \(0.0189078\pi\)
0.550530 + 0.834815i \(0.314425\pi\)
\(228\) 0 0
\(229\) −19.5000 + 11.2583i −1.28860 + 0.743971i −0.978404 0.206702i \(-0.933727\pi\)
−0.310192 + 0.950674i \(0.600393\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 8.00000i 0.525226i
\(233\) 4.94975 + 8.57321i 0.324269 + 0.561650i 0.981364 0.192158i \(-0.0615485\pi\)
−0.657095 + 0.753807i \(0.728215\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 3.46410 18.0000i 0.224544 1.16677i
\(239\) 26.8701i 1.73808i −0.494742 0.869040i \(-0.664738\pi\)
0.494742 0.869040i \(-0.335262\pi\)
\(240\) 0 0
\(241\) −12.0000 6.92820i −0.772988 0.446285i 0.0609515 0.998141i \(-0.480586\pi\)
−0.833939 + 0.551856i \(0.813920\pi\)
\(242\) −6.36396 + 11.0227i −0.409091 + 0.708566i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −7.79423 + 4.50000i −0.495935 + 0.286328i
\(248\) −4.24264 2.44949i −0.269408 0.155543i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 8.00000i 0.502956i
\(254\) −13.4722 + 7.77817i −0.845321 + 0.488046i
\(255\) 0 0
\(256\) 0 0
\(257\) −2.12132 + 1.22474i −0.132324 + 0.0763975i −0.564701 0.825296i \(-0.691008\pi\)
0.432377 + 0.901693i \(0.357675\pi\)
\(258\) 0 0
\(259\) 2.50000 0.866025i 0.155342 0.0538122i
\(260\) 0 0
\(261\) 0 0
\(262\) −1.73205 + 3.00000i −0.107006 + 0.185341i
\(263\) 7.07107 12.2474i 0.436021 0.755210i −0.561358 0.827573i \(-0.689721\pi\)
0.997378 + 0.0723633i \(0.0230541\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.89898 + 4.24264i 0.300376 + 0.260133i
\(267\) 0 0
\(268\) 0 0
\(269\) −8.57321 + 14.8492i −0.522718 + 0.905374i 0.476932 + 0.878940i \(0.341749\pi\)
−0.999651 + 0.0264343i \(0.991585\pi\)
\(270\) 0 0
\(271\) −12.0000 + 6.92820i −0.728948 + 0.420858i −0.818037 0.575165i \(-0.804938\pi\)
0.0890891 + 0.996024i \(0.471604\pi\)
\(272\) 19.5959i 1.18818i
\(273\) 0 0
\(274\) −16.0000 −0.966595
\(275\) 0 0
\(276\) 0 0
\(277\) 19.9186 + 11.5000i 1.19679 + 0.690968i 0.959839 0.280553i \(-0.0905179\pi\)
0.236953 + 0.971521i \(0.423851\pi\)
\(278\) 6.36396 3.67423i 0.381685 0.220366i
\(279\) 0 0
\(280\) 0 0
\(281\) 22.6274i 1.34984i 0.737892 + 0.674919i \(0.235822\pi\)
−0.737892 + 0.674919i \(0.764178\pi\)
\(282\) 0 0
\(283\) −0.866025 + 1.50000i −0.0514799 + 0.0891657i −0.890617 0.454754i \(-0.849727\pi\)
0.839137 + 0.543920i \(0.183060\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 10.3923i 0.614510i
\(287\) −19.0919 3.67423i −1.12696 0.216883i
\(288\) 0 0
\(289\) 3.50000 + 6.06218i 0.205882 + 0.356599i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.6969i 0.858604i 0.903161 + 0.429302i \(0.141240\pi\)
−0.903161 + 0.429302i \(0.858760\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −2.44949 + 1.41421i −0.142374 + 0.0821995i
\(297\) 0 0
\(298\) 6.92820 + 4.00000i 0.401340 + 0.231714i
\(299\) 14.6969 + 25.4558i 0.849946 + 1.47215i
\(300\) 0 0
\(301\) 0.500000 2.59808i 0.0288195 0.149751i
\(302\) −31.1127 −1.79033
\(303\) 0 0
\(304\) −6.00000 3.46410i −0.344124 0.198680i
\(305\) 0 0
\(306\) 0 0
\(307\) −15.5885 −0.889680 −0.444840 0.895610i \(-0.646740\pi\)
−0.444840 + 0.895610i \(0.646740\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 8.57321 14.8492i 0.486142 0.842023i −0.513731 0.857951i \(-0.671737\pi\)
0.999873 + 0.0159282i \(0.00507031\pi\)
\(312\) 0 0
\(313\) 6.06218 + 10.5000i 0.342655 + 0.593495i 0.984925 0.172983i \(-0.0553406\pi\)
−0.642270 + 0.766478i \(0.722007\pi\)
\(314\) −24.4949 −1.38233
\(315\) 0 0
\(316\) 0 0
\(317\) −7.07107 12.2474i −0.397151 0.687885i 0.596222 0.802819i \(-0.296668\pi\)
−0.993373 + 0.114934i \(0.963334\pi\)
\(318\) 0 0
\(319\) 2.00000 3.46410i 0.111979 0.193952i
\(320\) 0 0
\(321\) 0 0
\(322\) 13.8564 16.0000i 0.772187 0.891645i
\(323\) −8.48528 −0.472134
\(324\) 0 0
\(325\) 0 0
\(326\) 12.2474 + 7.07107i 0.678323 + 0.391630i
\(327\) 0 0
\(328\) 20.7846 1.14764
\(329\) 30.6186 10.6066i 1.68806 0.584761i
\(330\) 0 0
\(331\) 15.5000 + 26.8468i 0.851957 + 1.47563i 0.879440 + 0.476011i \(0.157918\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 9.00000 5.19615i 0.492458 0.284321i
\(335\) 0 0
\(336\) 0 0
\(337\) 23.0000i 1.25289i −0.779466 0.626445i \(-0.784509\pi\)
0.779466 0.626445i \(-0.215491\pi\)
\(338\) −9.89949 17.1464i −0.538462 0.932643i
\(339\) 0 0
\(340\) 0 0
\(341\) −1.22474 2.12132i −0.0663237 0.114876i
\(342\) 0 0
\(343\) 15.5885 + 10.0000i 0.841698 + 0.539949i
\(344\) 2.82843i 0.152499i
\(345\) 0 0
\(346\) −12.0000 6.92820i −0.645124 0.372463i
\(347\) −15.5563 + 26.9444i −0.835109 + 1.44645i 0.0588334 + 0.998268i \(0.481262\pi\)
−0.893942 + 0.448183i \(0.852071\pi\)
\(348\) 0 0
\(349\) 10.3923i 0.556287i 0.960539 + 0.278144i \(0.0897191\pi\)
−0.960539 + 0.278144i \(0.910281\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.8492 + 8.57321i 0.790345 + 0.456306i 0.840084 0.542456i \(-0.182506\pi\)
−0.0497387 + 0.998762i \(0.515839\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 14.0000i 0.739923i
\(359\) 24.4949 14.1421i 1.29279 0.746393i 0.313643 0.949541i \(-0.398450\pi\)
0.979148 + 0.203148i \(0.0651171\pi\)
\(360\) 0 0
\(361\) −8.00000 + 13.8564i −0.421053 + 0.729285i
\(362\) 19.0919 11.0227i 1.00345 0.579340i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0.866025 1.50000i 0.0452062 0.0782994i −0.842537 0.538639i \(-0.818939\pi\)
0.887743 + 0.460339i \(0.152272\pi\)
\(368\) −11.3137 + 19.5959i −0.589768 + 1.02151i
\(369\) 0 0
\(370\) 0 0
\(371\) −4.89898 + 5.65685i −0.254342 + 0.293689i
\(372\) 0 0
\(373\) 25.1147 14.5000i 1.30039 0.750782i 0.319921 0.947444i \(-0.396344\pi\)
0.980471 + 0.196663i \(0.0630104\pi\)
\(374\) −4.89898 + 8.48528i −0.253320 + 0.438763i
\(375\) 0 0
\(376\) −30.0000 + 17.3205i −1.54713 + 0.893237i
\(377\) 14.6969i 0.756931i
\(378\) 0 0
\(379\) 7.00000 0.359566 0.179783 0.983706i \(-0.442460\pi\)
0.179783 + 0.983706i \(0.442460\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.73205 1.00000i −0.0886194 0.0511645i
\(383\) −16.9706 + 9.79796i −0.867155 + 0.500652i −0.866402 0.499347i \(-0.833573\pi\)
−0.000753393 1.00000i \(0.500240\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 15.5563i 0.791797i
\(387\) 0 0
\(388\) 0 0
\(389\) 23.2702 + 13.4350i 1.17984 + 0.681183i 0.955978 0.293437i \(-0.0947991\pi\)
0.223865 + 0.974620i \(0.428132\pi\)
\(390\) 0 0
\(391\) 27.7128i 1.40150i
\(392\) −18.3848 7.34847i −0.928571 0.371154i
\(393\) 0 0
\(394\) 14.0000 + 24.2487i 0.705310 + 1.22163i
\(395\) 0 0
\(396\) 0 0
\(397\) −0.866025 1.50000i −0.0434646 0.0752828i 0.843475 0.537169i \(-0.180506\pi\)
−0.886939 + 0.461886i \(0.847173\pi\)
\(398\) 19.5959i 0.982255i
\(399\) 0 0
\(400\) 0 0
\(401\) −17.1464 + 9.89949i −0.856252 + 0.494357i −0.862755 0.505622i \(-0.831263\pi\)
0.00650355 + 0.999979i \(0.497930\pi\)
\(402\) 0 0
\(403\) 7.79423 + 4.50000i 0.388258 + 0.224161i
\(404\) 0 0
\(405\) 0 0
\(406\) 10.0000 3.46410i 0.496292 0.171920i
\(407\) −1.41421 −0.0701000
\(408\) 0 0
\(409\) −28.5000 16.4545i −1.40923 0.813622i −0.413920 0.910313i \(-0.635841\pi\)
−0.995314 + 0.0966915i \(0.969174\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.48528 9.79796i 0.417533 0.482126i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) −1.73205 3.00000i −0.0847174 0.146735i
\(419\) −36.7423 −1.79498 −0.897491 0.441034i \(-0.854612\pi\)
−0.897491 + 0.441034i \(0.854612\pi\)
\(420\) 0 0
\(421\) −1.00000 −0.0487370 −0.0243685 0.999703i \(-0.507758\pi\)
−0.0243685 + 0.999703i \(0.507758\pi\)
\(422\) 15.5563 + 26.9444i 0.757271 + 1.31163i
\(423\) 0 0
\(424\) 4.00000 6.92820i 0.194257 0.336463i
\(425\) 0 0
\(426\) 0 0
\(427\) −8.66025 + 3.00000i −0.419099 + 0.145180i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 13.4722 + 7.77817i 0.648933 + 0.374661i 0.788047 0.615615i \(-0.211092\pi\)
−0.139114 + 0.990276i \(0.544426\pi\)
\(432\) 0 0
\(433\) −15.5885 −0.749133 −0.374567 0.927200i \(-0.622209\pi\)
−0.374567 + 0.927200i \(0.622209\pi\)
\(434\) 1.22474 6.36396i 0.0587896 0.305480i
\(435\) 0 0
\(436\) 0 0
\(437\) −8.48528 4.89898i −0.405906 0.234350i
\(438\) 0 0
\(439\) −24.0000 + 13.8564i −1.14546 + 0.661330i −0.947776 0.318936i \(-0.896674\pi\)
−0.197681 + 0.980266i \(0.563341\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 36.0000i 1.71235i
\(443\) 19.7990 + 34.2929i 0.940678 + 1.62930i 0.764181 + 0.645002i \(0.223143\pi\)
0.176497 + 0.984301i \(0.443523\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 14.6969 + 25.4558i 0.695920 + 1.20537i
\(447\) 0 0
\(448\) 20.7846 + 4.00000i 0.981981 + 0.188982i
\(449\) 7.07107i 0.333704i 0.985982 + 0.166852i \(0.0533603\pi\)
−0.985982 + 0.166852i \(0.946640\pi\)
\(450\) 0 0
\(451\) 9.00000 + 5.19615i 0.423793 + 0.244677i
\(452\) 0 0
\(453\) 0 0
\(454\) 38.1051i 1.78836i
\(455\) 0 0
\(456\) 0 0
\(457\) −4.33013 + 2.50000i −0.202555 + 0.116945i −0.597847 0.801611i \(-0.703977\pi\)
0.395292 + 0.918556i \(0.370643\pi\)
\(458\) 27.5772 + 15.9217i 1.28860 + 0.743971i
\(459\) 0 0
\(460\) 0 0
\(461\) −14.6969 −0.684505 −0.342252 0.939608i \(-0.611190\pi\)
−0.342252 + 0.939608i \(0.611190\pi\)
\(462\) 0 0
\(463\) 13.0000i 0.604161i −0.953282 0.302081i \(-0.902319\pi\)
0.953282 0.302081i \(-0.0976812\pi\)
\(464\) −9.79796 + 5.65685i −0.454859 + 0.262613i
\(465\) 0 0
\(466\) 7.00000 12.1244i 0.324269 0.561650i
\(467\) 23.3345 13.4722i 1.07979 0.623419i 0.148952 0.988844i \(-0.452410\pi\)
0.930841 + 0.365426i \(0.119077\pi\)
\(468\) 0 0
\(469\) 22.0000 + 19.0526i 1.01587 + 0.879765i
\(470\) 0 0
\(471\) 0 0
\(472\) −6.92820 + 12.0000i −0.318896 + 0.552345i
\(473\) −0.707107 + 1.22474i −0.0325128 + 0.0563138i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −32.9090 + 19.0000i −1.50522 + 0.869040i
\(479\) 2.44949 4.24264i 0.111920 0.193851i −0.804624 0.593784i \(-0.797633\pi\)
0.916544 + 0.399933i \(0.130967\pi\)
\(480\) 0 0
\(481\) 4.50000 2.59808i 0.205182 0.118462i
\(482\) 19.5959i 0.892570i
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 14.7224 + 8.50000i 0.667137 + 0.385172i 0.794991 0.606621i \(-0.207476\pi\)
−0.127854 + 0.991793i \(0.540809\pi\)
\(488\) 8.48528 4.89898i 0.384111 0.221766i
\(489\) 0 0
\(490\) 0 0
\(491\) 11.3137i 0.510581i −0.966864 0.255290i \(-0.917829\pi\)
0.966864 0.255290i \(-0.0821710\pi\)
\(492\) 0 0
\(493\) −6.92820 + 12.0000i −0.312031 + 0.540453i
\(494\) 11.0227 + 6.36396i 0.495935 + 0.286328i
\(495\) 0 0
\(496\) 6.92820i 0.311086i
\(497\) 3.53553 18.3712i 0.158590 0.824060i
\(498\) 0 0
\(499\) −12.5000 21.6506i −0.559577 0.969216i −0.997532 0.0702185i \(-0.977630\pi\)
0.437955 0.898997i \(-0.355703\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 22.0454i 0.982956i 0.870890 + 0.491478i \(0.163543\pi\)
−0.870890 + 0.491478i \(0.836457\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −9.79796 + 5.65685i −0.435572 + 0.251478i
\(507\) 0 0
\(508\) 0 0
\(509\) 1.22474 + 2.12132i 0.0542859 + 0.0940259i 0.891891 0.452250i \(-0.149378\pi\)
−0.837605 + 0.546276i \(0.816045\pi\)
\(510\) 0 0
\(511\) −3.00000 + 3.46410i −0.132712 + 0.153243i
\(512\) −22.6274 −1.00000
\(513\) 0 0
\(514\) 3.00000 + 1.73205i 0.132324 + 0.0763975i
\(515\) 0 0
\(516\) 0 0
\(517\) −17.3205 −0.761755
\(518\) −2.82843 2.44949i −0.124274 0.107624i
\(519\) 0 0
\(520\) 0 0
\(521\) −2.44949 + 4.24264i −0.107314 + 0.185873i −0.914681 0.404176i \(-0.867558\pi\)
0.807367 + 0.590049i \(0.200892\pi\)
\(522\) 0 0
\(523\) 0.866025 + 1.50000i 0.0378686 + 0.0655904i 0.884339 0.466846i \(-0.154610\pi\)
−0.846470 + 0.532437i \(0.821276\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −20.0000 −0.872041
\(527\) 4.24264 + 7.34847i 0.184812 + 0.320104i
\(528\) 0 0
\(529\) −4.50000 + 7.79423i −0.195652 + 0.338880i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −38.1838 −1.65392
\(534\) 0 0
\(535\) 0 0
\(536\) −26.9444 15.5563i −1.16382 0.671932i
\(537\) 0 0
\(538\) 24.2487 1.04544
\(539\) −6.12372 7.77817i −0.263767 0.335030i
\(540\) 0 0
\(541\) −8.50000 14.7224i −0.365444 0.632967i 0.623404 0.781900i \(-0.285749\pi\)
−0.988847 + 0.148933i \(0.952416\pi\)
\(542\) 16.9706 + 9.79796i 0.728948 + 0.420858i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 10.0000i 0.427569i 0.976881 + 0.213785i \(0.0685791\pi\)
−0.976881 + 0.213785i \(0.931421\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.44949 4.24264i −0.104352 0.180743i
\(552\) 0 0
\(553\) 4.33013 + 12.5000i 0.184136 + 0.531554i
\(554\) 32.5269i 1.38194i
\(555\) 0 0
\(556\) 0 0
\(557\) 7.77817 13.4722i 0.329572 0.570835i −0.652855 0.757483i \(-0.726429\pi\)
0.982427 + 0.186648i \(0.0597623\pi\)
\(558\) 0 0
\(559\) 5.19615i 0.219774i
\(560\) 0 0
\(561\) 0 0
\(562\) 27.7128 16.0000i 1.16899 0.674919i
\(563\) −23.3345 13.4722i −0.983433 0.567785i −0.0801281 0.996785i \(-0.525533\pi\)
−0.903305 + 0.428999i \(0.858866\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2.44949 0.102960
\(567\) 0 0
\(568\) 20.0000i 0.839181i
\(569\) −1.22474 + 0.707107i −0.0513440 + 0.0296435i −0.525452 0.850823i \(-0.676104\pi\)
0.474108 + 0.880467i \(0.342771\pi\)
\(570\) 0 0
\(571\) −5.50000 + 9.52628i −0.230168 + 0.398662i −0.957857 0.287244i \(-0.907261\pi\)
0.727690 + 0.685907i \(0.240594\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 9.00000 + 25.9808i 0.375653 + 1.08442i
\(575\) 0 0
\(576\) 0 0
\(577\) 0.866025 1.50000i 0.0360531 0.0624458i −0.847436 0.530898i \(-0.821855\pi\)
0.883489 + 0.468452i \(0.155188\pi\)
\(578\) 4.94975 8.57321i 0.205882 0.356599i
\(579\) 0 0
\(580\) 0 0
\(581\) −3.67423 + 19.0919i −0.152433 + 0.792065i
\(582\) 0 0
\(583\) 3.46410 2.00000i 0.143468 0.0828315i
\(584\) 2.44949 4.24264i 0.101361 0.175562i
\(585\) 0 0
\(586\) 18.0000 10.3923i 0.743573 0.429302i
\(587\) 14.6969i 0.606608i 0.952894 + 0.303304i \(0.0980897\pi\)
−0.952894 + 0.303304i \(0.901910\pi\)
\(588\) 0 0
\(589\) −3.00000 −0.123613
\(590\) 0 0
\(591\) 0 0
\(592\) 3.46410 + 2.00000i 0.142374 + 0.0821995i
\(593\) 14.8492 8.57321i 0.609785 0.352060i −0.163096 0.986610i \(-0.552148\pi\)
0.772881 + 0.634550i \(0.218815\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 20.7846 36.0000i 0.849946 1.47215i
\(599\) 4.89898 + 2.82843i 0.200167 + 0.115566i 0.596733 0.802440i \(-0.296465\pi\)
−0.396566 + 0.918006i \(0.629798\pi\)
\(600\) 0 0
\(601\) 25.9808i 1.05978i −0.848067 0.529889i \(-0.822234\pi\)
0.848067 0.529889i \(-0.177766\pi\)
\(602\) −3.53553 + 1.22474i −0.144098 + 0.0499169i
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 19.9186 + 34.5000i 0.808470 + 1.40031i 0.913923 + 0.405887i \(0.133038\pi\)
−0.105453 + 0.994424i \(0.533629\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 55.1135 31.8198i 2.22965 1.28729i
\(612\) 0 0
\(613\) −6.92820 4.00000i −0.279827 0.161558i 0.353518 0.935428i \(-0.384985\pi\)
−0.633345 + 0.773869i \(0.718319\pi\)
\(614\) 11.0227 + 19.0919i 0.444840 + 0.770486i
\(615\) 0 0
\(616\) 8.00000 + 6.92820i 0.322329 + 0.279145i
\(617\) −24.0416 −0.967880 −0.483940 0.875101i \(-0.660795\pi\)
−0.483940 + 0.875101i \(0.660795\pi\)
\(618\) 0 0
\(619\) 25.5000 + 14.7224i 1.02493 + 0.591744i 0.915529 0.402253i \(-0.131773\pi\)
0.109403 + 0.993997i \(0.465106\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −24.2487 −0.972285
\(623\) −4.24264 12.2474i −0.169978 0.490684i
\(624\) 0 0
\(625\) 0 0
\(626\) 8.57321 14.8492i 0.342655 0.593495i
\(627\) 0 0
\(628\) 0 0
\(629\) 4.89898 0.195335
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) −7.07107 12.2474i −0.281272 0.487177i
\(633\) 0 0
\(634\) −10.0000 + 17.3205i −0.397151 + 0.687885i
\(635\) 0 0
\(636\) 0 0
\(637\) 33.7750 + 13.5000i 1.33821 + 0.534889i
\(638\) −5.65685 −0.223957
\(639\) 0 0
\(640\) 0 0
\(641\) −12.2474 7.07107i −0.483745 0.279290i 0.238231 0.971209i \(-0.423433\pi\)
−0.721976 + 0.691918i \(0.756766\pi\)
\(642\) 0 0
\(643\) −25.9808 −1.02458 −0.512291 0.858812i \(-0.671203\pi\)
−0.512291 + 0.858812i \(0.671203\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 + 10.3923i 0.236067 + 0.408880i
\(647\) −14.8492 8.57321i −0.583784 0.337048i 0.178852 0.983876i \(-0.442762\pi\)
−0.762636 + 0.646828i \(0.776095\pi\)
\(648\) 0 0
\(649\) −6.00000 + 3.46410i −0.235521 + 0.135978i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3.53553 6.12372i −0.138356 0.239640i 0.788518 0.615011i \(-0.210849\pi\)
−0.926875 + 0.375371i \(0.877515\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −14.6969 25.4558i −0.573819 0.993884i
\(657\) 0 0
\(658\) −34.6410 30.0000i −1.35045 1.16952i
\(659\) 22.6274i 0.881439i −0.897645 0.440720i \(-0.854723\pi\)
0.897645 0.440720i \(-0.145277\pi\)
\(660\) 0 0
\(661\) −25.5000 14.7224i −0.991835 0.572636i −0.0860127 0.996294i \(-0.527413\pi\)
−0.905822 + 0.423658i \(0.860746\pi\)
\(662\) 21.9203 37.9671i 0.851957 1.47563i
\(663\) 0 0
\(664\) 20.7846i 0.806599i
\(665\) 0 0
\(666\) 0 0
\(667\) −13.8564 + 8.00000i −0.536522 + 0.309761i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.89898 0.189123
\(672\) 0 0
\(673\) 35.0000i 1.34915i 0.738206 + 0.674575i \(0.235673\pi\)
−0.738206 + 0.674575i \(0.764327\pi\)
\(674\) −28.1691 + 16.2635i −1.08503 + 0.626445i
\(675\) 0 0
\(676\) 0 0
\(677\) −8.48528 + 4.89898i −0.326116 + 0.188283i −0.654115 0.756395i \(-0.726959\pi\)
0.327999 + 0.944678i \(0.393626\pi\)
\(678\) 0 0
\(679\) −27.0000 5.19615i −1.03616 0.199410i
\(680\) 0 0
\(681\) 0 0
\(682\) −1.73205 + 3.00000i −0.0663237 + 0.114876i
\(683\) 24.0416 41.6413i 0.919927 1.59336i 0.120405 0.992725i \(-0.461581\pi\)
0.799522 0.600636i \(-0.205086\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.22474 26.1630i 0.0467610 0.998906i
\(687\) 0 0
\(688\) 3.46410 2.00000i 0.132068 0.0762493i
\(689\) −7.34847 + 12.7279i −0.279954 + 0.484895i
\(690\) 0 0
\(691\) 37.5000 21.6506i 1.42657 0.823629i 0.429719 0.902963i \(-0.358613\pi\)
0.996848 + 0.0793336i \(0.0252792\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 44.0000 1.67022
\(695\) 0 0
\(696\) 0 0
\(697\) −31.1769 18.0000i −1.18091 0.681799i
\(698\) 12.7279 7.34847i 0.481759 0.278144i
\(699\) 0 0
\(700\) 0 0
\(701\) 5.65685i 0.213656i 0.994277 + 0.106828i \(0.0340695\pi\)
−0.994277 + 0.106828i \(0.965931\pi\)
\(702\) 0 0
\(703\) −0.866025 + 1.50000i −0.0326628 + 0.0565736i
\(704\) −9.79796 5.65685i −0.369274 0.213201i
\(705\) 0 0
\(706\) 24.2487i 0.912612i
\(707\) 29.6985 34.2929i 1.11693 1.28972i
\(708\) 0 0
\(709\) −20.0000 34.6410i −0.751116 1.30097i −0.947282 0.320400i \(-0.896183\pi\)
0.196167 0.980571i \(-0.437151\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.92820 + 12.0000i 0.259645 + 0.449719i
\(713\) 9.79796i 0.366936i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) −34.6410 20.0000i −1.29279 0.746393i
\(719\) −13.4722 23.3345i −0.502428 0.870231i −0.999996 0.00280593i \(-0.999107\pi\)
0.497568 0.867425i \(-0.334226\pi\)
\(720\) 0 0
\(721\) −7.50000 21.6506i −0.279315 0.806312i
\(722\) 22.6274 0.842105
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −25.9808 −0.963573 −0.481787 0.876289i \(-0.660012\pi\)
−0.481787 + 0.876289i \(0.660012\pi\)
\(728\) −38.1838 7.34847i −1.41518 0.272352i
\(729\) 0 0
\(730\) 0 0
\(731\) 2.44949 4.24264i 0.0905977 0.156920i
\(732\) 0 0
\(733\) −19.9186 34.5000i −0.735710 1.27429i −0.954411 0.298495i \(-0.903515\pi\)
0.218702 0.975792i \(-0.429818\pi\)
\(734\) −2.44949 −0.0904123
\(735\) 0 0
\(736\) 0 0
\(737\) −7.77817 13.4722i −0.286513 0.496255i
\(738\) 0 0
\(739\) −0.500000 + 0.866025i −0.0183928 + 0.0318573i −0.875075 0.483987i \(-0.839188\pi\)
0.856683 + 0.515844i \(0.172522\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 10.3923 + 2.00000i 0.381514 + 0.0734223i
\(743\) 24.0416 0.882002 0.441001 0.897507i \(-0.354624\pi\)
0.441001 + 0.897507i \(0.354624\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −35.5176 20.5061i −1.30039 0.750782i
\(747\) 0 0
\(748\) 0 0
\(749\) 2.44949 + 7.07107i 0.0895024 + 0.258371i
\(750\) 0 0
\(751\) −14.5000 25.1147i −0.529113 0.916450i −0.999424 0.0339490i \(-0.989192\pi\)
0.470311 0.882501i \(-0.344142\pi\)
\(752\) 42.4264 + 24.4949i 1.54713 + 0.893237i
\(753\) 0 0
\(754\) 18.0000 10.3923i 0.655521 0.378465i
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000i 0.0726912i −0.999339 0.0363456i \(-0.988428\pi\)
0.999339 0.0363456i \(-0.0115717\pi\)
\(758\) −4.94975 8.57321i −0.179783 0.311393i
\(759\) 0 0
\(760\) 0 0
\(761\) −12.2474 21.2132i −0.443970 0.768978i 0.554010 0.832510i \(-0.313097\pi\)
−0.997980 + 0.0635319i \(0.979764\pi\)
\(762\) 0 0
\(763\) −1.73205 + 2.00000i −0.0627044 + 0.0724049i
\(764\) 0 0
\(765\) 0 0
\(766\) 24.0000 + 13.8564i 0.867155 + 0.500652i
\(767\) 12.7279 22.0454i 0.459579 0.796014i
\(768\) 0 0
\(769\) 25.9808i 0.936890i −0.883493 0.468445i \(-0.844814\pi\)
0.883493 0.468445i \(-0.155186\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −23.3345 13.4722i −0.839284 0.484561i 0.0177365 0.999843i \(-0.494354\pi\)
−0.857021 + 0.515282i \(0.827687\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 29.3939 1.05518
\(777\) 0 0
\(778\) 38.0000i 1.36237i
\(779\) 11.0227 6.36396i 0.394929 0.228013i
\(780\) 0 0
\(781\) −5.00000 + 8.66025i −0.178914 + 0.309888i
\(782\) 33.9411 19.5959i 1.21373 0.700749i
\(783\) 0 0
\(784\) 4.00000 + 27.7128i 0.142857 + 0.989743i
\(785\) 0 0
\(786\) 0 0
\(787\) −22.5167 + 39.0000i −0.802632 + 1.39020i 0.115246 + 0.993337i \(0.463234\pi\)
−0.917878 + 0.396863i \(0.870099\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.67423 0.707107i −0.130641 0.0251418i
\(792\) 0 0
\(793\) −15.5885 + 9.00000i −0.553562 + 0.319599i
\(794\) −1.22474 + 2.12132i −0.0434646 + 0.0752828i
\(795\) 0 0
\(796\) 0 0
\(797\) 29.3939i 1.04118i 0.853805 + 0.520592i \(0.174289\pi\)
−0.853805 + 0.520592i \(0.825711\pi\)
\(798\) 0 0
\(799\) 60.0000 2.12265
\(800\) 0 0
\(801\) 0 0
\(802\) 24.2487 + 14.0000i 0.856252 + 0.494357i
\(803\) 2.12132 1.22474i 0.0748598 0.0432203i
\(804\) 0 0
\(805\) 0 0
\(806\) 12.7279i 0.448322i
\(807\) 0 0
\(808\) −24.2487 + 42.0000i −0.853067 + 1.47755i
\(809\) −35.5176 20.5061i −1.24873 0.720956i −0.277876 0.960617i \(-0.589630\pi\)
−0.970857 + 0.239661i \(0.922964\pi\)
\(810\) 0 0
\(811\) 31.1769i 1.09477i −0.836881 0.547385i \(-0.815623\pi\)
0.836881 0.547385i \(-0.184377\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1.00000 + 1.73205i 0.0350500 + 0.0607083i
\(815\) 0 0
\(816\) 0 0
\(817\) 0.866025 + 1.50000i 0.0302984 + 0.0524784i
\(818\) 46.5403i 1.62724i
\(819\) 0 0
\(820\) 0 0
\(821\) −20.8207 + 12.0208i −0.726646 + 0.419529i −0.817194 0.576363i \(-0.804472\pi\)
0.0905478 + 0.995892i \(0.471138\pi\)
\(822\) 0 0
\(823\) 29.4449 + 17.0000i 1.02638 + 0.592583i 0.915947 0.401300i \(-0.131442\pi\)
0.110437 + 0.993883i \(0.464775\pi\)
\(824\) 12.2474 + 21.2132i 0.426660 + 0.738997i
\(825\) 0 0
\(826\) −18.0000 3.46410i −0.626300 0.120532i
\(827\) −7.07107 −0.245885 −0.122943 0.992414i \(-0.539233\pi\)
−0.122943 + 0.992414i \(0.539233\pi\)
\(828\) 0 0
\(829\) −1.50000 0.866025i −0.0520972 0.0300783i 0.473725 0.880673i \(-0.342909\pi\)
−0.525822 + 0.850594i \(0.676242\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 41.5692 1.44115
\(833\) 21.2132 + 26.9444i 0.734994 + 0.933568i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 25.9808 + 45.0000i 0.897491 + 1.55450i
\(839\) 14.6969 0.507395 0.253697 0.967284i \(-0.418353\pi\)
0.253697 + 0.967284i \(0.418353\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) 0.707107 + 1.22474i 0.0243685 + 0.0422075i
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −7.79423 22.5000i −0.267813 0.773109i
\(848\) −11.3137 −0.388514
\(849\) 0 0
\(850\) 0 0
\(851\) 4.89898 + 2.82843i 0.167935 + 0.0969572i
\(852\) 0 0
\(853\) −36.3731 −1.24539 −0.622695 0.782465i \(-0.713962\pi\)
−0.622695 + 0.782465i \(0.713962\pi\)
\(854\) 9.79796 + 8.48528i 0.335279 + 0.290360i
\(855\) 0 0
\(856\) −4.00000 6.92820i −0.136717 0.236801i
\(857\) −33.9411 19.5959i −1.15941 0.669384i −0.208245 0.978077i \(-0.566775\pi\)
−0.951162 + 0.308693i \(0.900108\pi\)
\(858\) 0 0
\(859\) −33.0000 + 19.0526i −1.12595 + 0.650065i −0.942912 0.333042i \(-0.891925\pi\)
−0.183033 + 0.983107i \(0.558592\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 22.0000i 0.749323i
\(863\) −20.5061 35.5176i −0.698036 1.20903i −0.969147 0.246485i \(-0.920724\pi\)
0.271111 0.962548i \(-0.412609\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 11.0227 + 19.0919i 0.374567 + 0.648769i
\(867\) 0 0
\(868\) 0 0
\(869\) 7.07107i 0.239870i
\(870\) 0 0
\(871\) 49.5000 + 28.5788i 1.67724 + 0.968357i
\(872\) 1.41421 2.44949i 0.0478913 0.0829502i
\(873\) 0 0
\(874\) 13.8564i 0.468700i
\(875\) 0 0
\(876\) 0 0
\(877\) −17.3205 + 10.0000i −0.584872 + 0.337676i −0.763067 0.646319i \(-0.776307\pi\)
0.178195 + 0.983995i \(0.442974\pi\)
\(878\) 33.9411 + 19.5959i 1.14546 + 0.661330i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 11.0000i 0.370179i 0.982722 + 0.185090i \(0.0592576\pi\)
−0.982722 + 0.185090i \(0.940742\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 28.0000 48.4974i 0.940678 1.62930i
\(887\) 36.0624 20.8207i 1.21086 0.699089i 0.247912 0.968783i \(-0.420256\pi\)
0.962946 + 0.269693i \(0.0869222\pi\)
\(888\) 0 0
\(889\) 5.50000 28.5788i 0.184464 0.958503i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −10.6066 + 18.3712i −0.354936 + 0.614768i
\(894\) 0 0
\(895\) 0 0
\(896\) −9.79796 28.2843i −0.327327 0.944911i
\(897\) 0 0
\(898\) 8.66025 5.00000i 0.288996 0.166852i
\(899\) −2.44949 + 4.24264i −0.0816951 + 0.141500i
\(900\) 0 0
\(901\) −12.0000 + 6.92820i −0.399778 + 0.230812i
\(902\) 14.6969i 0.489355i
\(903\) 0 0
\(904\) 4.00000 0.133038
\(905\) 0 0
\(906\) 0 0
\(907\) 4.33013 + 2.50000i 0.143780 + 0.0830111i 0.570164 0.821531i \(-0.306880\pi\)
−0.426385 + 0.904542i \(0.640213\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 53.7401i 1.78049i −0.455483 0.890245i \(-0.650533\pi\)
0.455483 0.890245i \(-0.349467\pi\)
\(912\) 0 0
\(913\) 5.19615 9.00000i 0.171968 0.297857i
\(914\) 6.12372 + 3.53553i 0.202555 + 0.116945i
\(915\) 0 0
\(916\) 0 0
\(917\) −2.12132 6.12372i −0.0700522 0.202223i
\(918\) 0 0
\(919\) 8.50000 + 14.7224i 0.280389 + 0.485648i 0.971481 0.237119i \(-0.0762032\pi\)
−0.691091 + 0.722767i \(0.742870\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 10.3923 + 18.0000i 0.342252 + 0.592798i
\(923\) 36.7423i 1.20939i
\(924\) 0 0
\(925\) 0 0
\(926\) −15.9217 + 9.19239i −0.523219 + 0.302081i
\(927\) 0 0
\(928\) 0 0
\(929\) −28.1691 48.7904i −0.924199 1.60076i −0.792844 0.609425i \(-0.791400\pi\)
−0.131355 0.991335i \(-0.541933\pi\)
\(930\) 0 0
\(931\) −12.0000 + 1.73205i −0.393284 + 0.0567657i
\(932\) 0 0
\(933\) 0 0
\(934\) −33.0000 19.0526i −1.07979 0.623419i
\(935\) 0 0
\(936\) 0 0
\(937\) 46.7654 1.52776 0.763879 0.645359i \(-0.223292\pi\)
0.763879 + 0.645359i \(0.223292\pi\)
\(938\) 7.77817 40.4166i 0.253966 1.31965i
\(939\) 0 0
\(940\) 0 0
\(941\) −24.4949 + 42.4264i −0.798511 + 1.38306i 0.122075 + 0.992521i \(0.461045\pi\)
−0.920586 + 0.390540i \(0.872288\pi\)
\(942\) 0 0
\(943\) −20.7846 36.0000i −0.676840 1.17232i
\(944\) 19.5959 0.637793
\(945\) 0 0
\(946\) 2.00000 0.0650256
\(947\) 24.7487 + 42.8661i 0.804226 + 1.39296i 0.916812 + 0.399318i \(0.130753\pi\)
−0.112586 + 0.993642i \(0.535914\pi\)
\(948\) 0 0
\(949\) −4.50000 + 7.79423i −0.146076 + 0.253011i
\(950\) 0 0
\(951\) 0 0
\(952\) −27.7128 24.0000i −0.898177 0.777844i
\(953\) −5.65685 −0.183243 −0.0916217 0.995794i \(-0.529205\pi\)
−0.0916217 + 0.995794i \(0.529205\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) −6.92820 −0.223840
\(959\) 19.5959 22.6274i 0.632785 0.730677i
\(960\) 0 0
\(961\) −14.0000 24.2487i −0.451613 0.782216i
\(962\) −6.36396 3.67423i −0.205182 0.118462i
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 13.0000i 0.418052i 0.977910 + 0.209026i \(0.0670293\pi\)
−0.977910 + 0.209026i \(0.932971\pi\)
\(968\) 12.7279 + 22.0454i 0.409091 + 0.708566i
\(969\) 0 0
\(970\) 0 0
\(971\) −19.5959 33.9411i −0.628863 1.08922i −0.987780 0.155853i \(-0.950187\pi\)
0.358917 0.933369i \(-0.383146\pi\)
\(972\) 0 0
\(973\) −2.59808 + 13.5000i −0.0832905 + 0.432790i
\(974\) 24.0416i 0.770344i
\(975\) 0 0
\(976\) −12.0000 6.92820i −0.384111 0.221766i
\(977\) 3.53553 6.12372i 0.113112 0.195915i −0.803912 0.594749i \(-0.797252\pi\)
0.917023 + 0.398833i \(0.130585\pi\)
\(978\) 0 0
\(979\) 6.92820i 0.221426i
\(980\) 0 0
\(981\) 0 0
\(982\) −13.8564 + 8.00000i −0.442176 + 0.255290i
\(983\) −4.24264 2.44949i −0.135319 0.0781266i 0.430812 0.902442i \(-0.358227\pi\)
−0.566131 + 0.824315i \(0.691561\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 19.5959 0.624061
\(987\) 0 0
\(988\) 0 0
\(989\) 4.89898 2.82843i 0.155778 0.0899388i
\(990\) 0 0
\(991\) −17.5000 + 30.3109i −0.555906 + 0.962857i 0.441927 + 0.897051i \(0.354295\pi\)
−0.997832 + 0.0658059i \(0.979038\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −25.0000 + 8.66025i −0.792952 + 0.274687i
\(995\) 0 0
\(996\) 0 0
\(997\) −14.7224 + 25.5000i −0.466264 + 0.807593i −0.999258 0.0385262i \(-0.987734\pi\)
0.532993 + 0.846119i \(0.321067\pi\)
\(998\) −17.6777 + 30.6186i −0.559577 + 0.969216i
\(999\) 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 1575.2.bc.a.899.2 8
3.2 odd 2 inner 1575.2.bc.a.899.4 8
5.2 odd 4 63.2.p.a.17.2 yes 4
5.3 odd 4 1575.2.bk.c.1151.1 4
5.4 even 2 inner 1575.2.bc.a.899.3 8
7.5 odd 6 inner 1575.2.bc.a.1349.1 8
15.2 even 4 63.2.p.a.17.1 4
15.8 even 4 1575.2.bk.c.1151.2 4
15.14 odd 2 inner 1575.2.bc.a.899.1 8
20.7 even 4 1008.2.bt.b.17.1 4
21.5 even 6 inner 1575.2.bc.a.1349.3 8
35.2 odd 12 441.2.p.a.215.1 4
35.12 even 12 63.2.p.a.26.1 yes 4
35.17 even 12 441.2.c.a.440.3 4
35.19 odd 6 inner 1575.2.bc.a.1349.4 8
35.27 even 4 441.2.p.a.80.2 4
35.32 odd 12 441.2.c.a.440.4 4
35.33 even 12 1575.2.bk.c.26.2 4
45.2 even 12 567.2.s.d.458.2 4
45.7 odd 12 567.2.s.d.458.1 4
45.22 odd 12 567.2.i.d.269.2 4
45.32 even 12 567.2.i.d.269.1 4
60.47 odd 4 1008.2.bt.b.17.2 4
105.2 even 12 441.2.p.a.215.2 4
105.17 odd 12 441.2.c.a.440.2 4
105.32 even 12 441.2.c.a.440.1 4
105.47 odd 12 63.2.p.a.26.2 yes 4
105.62 odd 4 441.2.p.a.80.1 4
105.68 odd 12 1575.2.bk.c.26.1 4
105.89 even 6 inner 1575.2.bc.a.1349.2 8
140.47 odd 12 1008.2.bt.b.593.2 4
140.67 even 12 7056.2.k.b.881.3 4
140.87 odd 12 7056.2.k.b.881.1 4
315.47 odd 12 567.2.i.d.215.1 4
315.187 even 12 567.2.i.d.215.2 4
315.257 odd 12 567.2.s.d.26.1 4
315.292 even 12 567.2.s.d.26.2 4
420.47 even 12 1008.2.bt.b.593.1 4
420.227 even 12 7056.2.k.b.881.4 4
420.347 odd 12 7056.2.k.b.881.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.p.a.17.1 4 15.2 even 4
63.2.p.a.17.2 yes 4 5.2 odd 4
63.2.p.a.26.1 yes 4 35.12 even 12
63.2.p.a.26.2 yes 4 105.47 odd 12
441.2.c.a.440.1 4 105.32 even 12
441.2.c.a.440.2 4 105.17 odd 12
441.2.c.a.440.3 4 35.17 even 12
441.2.c.a.440.4 4 35.32 odd 12
441.2.p.a.80.1 4 105.62 odd 4
441.2.p.a.80.2 4 35.27 even 4
441.2.p.a.215.1 4 35.2 odd 12
441.2.p.a.215.2 4 105.2 even 12
567.2.i.d.215.1 4 315.47 odd 12
567.2.i.d.215.2 4 315.187 even 12
567.2.i.d.269.1 4 45.32 even 12
567.2.i.d.269.2 4 45.22 odd 12
567.2.s.d.26.1 4 315.257 odd 12
567.2.s.d.26.2 4 315.292 even 12
567.2.s.d.458.1 4 45.7 odd 12
567.2.s.d.458.2 4 45.2 even 12
1008.2.bt.b.17.1 4 20.7 even 4
1008.2.bt.b.17.2 4 60.47 odd 4
1008.2.bt.b.593.1 4 420.47 even 12
1008.2.bt.b.593.2 4 140.47 odd 12
1575.2.bc.a.899.1 8 15.14 odd 2 inner
1575.2.bc.a.899.2 8 1.1 even 1 trivial
1575.2.bc.a.899.3 8 5.4 even 2 inner
1575.2.bc.a.899.4 8 3.2 odd 2 inner
1575.2.bc.a.1349.1 8 7.5 odd 6 inner
1575.2.bc.a.1349.2 8 105.89 even 6 inner
1575.2.bc.a.1349.3 8 21.5 even 6 inner
1575.2.bc.a.1349.4 8 35.19 odd 6 inner
1575.2.bk.c.26.1 4 105.68 odd 12
1575.2.bk.c.26.2 4 35.33 even 12
1575.2.bk.c.1151.1 4 5.3 odd 4
1575.2.bk.c.1151.2 4 15.8 even 4
7056.2.k.b.881.1 4 140.87 odd 12
7056.2.k.b.881.2 4 420.347 odd 12
7056.2.k.b.881.3 4 140.67 even 12
7056.2.k.b.881.4 4 420.227 even 12