Properties

Label 1792.2.m.e.449.6
Level $1792$
Weight $2$
Character 1792.449
Analytic conductor $14.309$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1792,2,Mod(449,1792)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1792, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1792.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1792 = 2^{8} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1792.m (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: \(14.3091920422\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 12 x^{14} - 48 x^{13} + 67 x^{12} - 24 x^{11} + 118 x^{10} - 176 x^{9} + 351 x^{8} - 180 x^{7} + 358 x^{6} - 336 x^{5} + 390 x^{4} - 344 x^{3} + 164 x^{2} - 40 x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 449.6
Root \(0.792206 - 1.03242i\) of defining polynomial
Character \(\chi\) \(=\) 1792.449
Dual form 1792.2.m.e.1345.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.18265 - 1.18265i) q^{3} +(-1.87820 - 1.87820i) q^{5} +1.00000i q^{7} +0.202696i q^{9} +O(q^{10})\) \(q+(1.18265 - 1.18265i) q^{3} +(-1.87820 - 1.87820i) q^{5} +1.00000i q^{7} +0.202696i q^{9} +(-0.584413 - 0.584413i) q^{11} +(3.94057 - 3.94057i) q^{13} -4.44250 q^{15} +1.74896 q^{17} +(4.19467 - 4.19467i) q^{19} +(1.18265 + 1.18265i) q^{21} -3.04150i q^{23} +2.05530i q^{25} +(3.78766 + 3.78766i) q^{27} +(-4.43316 + 4.43316i) q^{29} -7.90794 q^{31} -1.38231 q^{33} +(1.87820 - 1.87820i) q^{35} +(-5.87262 - 5.87262i) q^{37} -9.32061i q^{39} -1.38922i q^{41} +(-1.73902 - 1.73902i) q^{43} +(0.380704 - 0.380704i) q^{45} +1.80017 q^{47} -1.00000 q^{49} +(2.06840 - 2.06840i) q^{51} +(-9.73675 - 9.73675i) q^{53} +2.19529i q^{55} -9.92162i q^{57} +(4.74002 + 4.74002i) q^{59} +(3.10257 - 3.10257i) q^{61} -0.202696 q^{63} -14.8024 q^{65} +(4.81108 - 4.81108i) q^{67} +(-3.59702 - 3.59702i) q^{69} +1.11625i q^{71} -11.2521i q^{73} +(2.43070 + 2.43070i) q^{75} +(0.584413 - 0.584413i) q^{77} +7.61158 q^{79} +8.35083 q^{81} +(-11.1869 + 11.1869i) q^{83} +(-3.28490 - 3.28490i) q^{85} +10.4857i q^{87} -0.428825i q^{89} +(3.94057 + 3.94057i) q^{91} +(-9.35230 + 9.35230i) q^{93} -15.7569 q^{95} -19.2163 q^{97} +(0.118458 - 0.118458i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{3} - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{3} - 4 q^{5} + 8 q^{11} + 12 q^{13} - 8 q^{17} - 4 q^{19} - 4 q^{21} + 56 q^{27} - 8 q^{31} + 16 q^{33} + 4 q^{35} - 8 q^{37} + 24 q^{43} - 36 q^{45} - 40 q^{47} - 16 q^{49} - 24 q^{51} - 32 q^{53} + 4 q^{59} - 20 q^{61} + 24 q^{63} + 72 q^{65} - 32 q^{67} + 56 q^{69} + 28 q^{75} - 8 q^{77} - 40 q^{81} - 36 q^{83} + 12 q^{91} + 8 q^{93} - 80 q^{95} - 72 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times\).

\(n\) \(1023\) \(1025\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.18265 1.18265i 0.682801 0.682801i −0.277829 0.960630i \(-0.589615\pi\)
0.960630 + 0.277829i \(0.0896150\pi\)
\(4\) 0 0
\(5\) −1.87820 1.87820i −0.839958 0.839958i 0.148895 0.988853i \(-0.452428\pi\)
−0.988853 + 0.148895i \(0.952428\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0.202696i 0.0675652i
\(10\) 0 0
\(11\) −0.584413 0.584413i −0.176207 0.176207i 0.613493 0.789700i \(-0.289764\pi\)
−0.789700 + 0.613493i \(0.789764\pi\)
\(12\) 0 0
\(13\) 3.94057 3.94057i 1.09292 1.09292i 0.0977031 0.995216i \(-0.468850\pi\)
0.995216 0.0977031i \(-0.0311496\pi\)
\(14\) 0 0
\(15\) −4.44250 −1.14705
\(16\) 0 0
\(17\) 1.74896 0.424185 0.212093 0.977250i \(-0.431972\pi\)
0.212093 + 0.977250i \(0.431972\pi\)
\(18\) 0 0
\(19\) 4.19467 4.19467i 0.962323 0.962323i −0.0369927 0.999316i \(-0.511778\pi\)
0.999316 + 0.0369927i \(0.0117778\pi\)
\(20\) 0 0
\(21\) 1.18265 + 1.18265i 0.258075 + 0.258075i
\(22\) 0 0
\(23\) 3.04150i 0.634198i −0.948393 0.317099i \(-0.897291\pi\)
0.948393 0.317099i \(-0.102709\pi\)
\(24\) 0 0
\(25\) 2.05530i 0.411060i
\(26\) 0 0
\(27\) 3.78766 + 3.78766i 0.728935 + 0.728935i
\(28\) 0 0
\(29\) −4.43316 + 4.43316i −0.823217 + 0.823217i −0.986568 0.163351i \(-0.947770\pi\)
0.163351 + 0.986568i \(0.447770\pi\)
\(30\) 0 0
\(31\) −7.90794 −1.42031 −0.710154 0.704046i \(-0.751375\pi\)
−0.710154 + 0.704046i \(0.751375\pi\)
\(32\) 0 0
\(33\) −1.38231 −0.240629
\(34\) 0 0
\(35\) 1.87820 1.87820i 0.317474 0.317474i
\(36\) 0 0
\(37\) −5.87262 5.87262i −0.965453 0.965453i 0.0339701 0.999423i \(-0.489185\pi\)
−0.999423 + 0.0339701i \(0.989185\pi\)
\(38\) 0 0
\(39\) 9.32061i 1.49249i
\(40\) 0 0
\(41\) 1.38922i 0.216960i −0.994099 0.108480i \(-0.965402\pi\)
0.994099 0.108480i \(-0.0345983\pi\)
\(42\) 0 0
\(43\) −1.73902 1.73902i −0.265198 0.265198i 0.561964 0.827162i \(-0.310046\pi\)
−0.827162 + 0.561964i \(0.810046\pi\)
\(44\) 0 0
\(45\) 0.380704 0.380704i 0.0567520 0.0567520i
\(46\) 0 0
\(47\) 1.80017 0.262582 0.131291 0.991344i \(-0.458088\pi\)
0.131291 + 0.991344i \(0.458088\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 2.06840 2.06840i 0.289634 0.289634i
\(52\) 0 0
\(53\) −9.73675 9.73675i −1.33745 1.33745i −0.898527 0.438919i \(-0.855361\pi\)
−0.438919 0.898527i \(-0.644639\pi\)
\(54\) 0 0
\(55\) 2.19529i 0.296013i
\(56\) 0 0
\(57\) 9.92162i 1.31415i
\(58\) 0 0
\(59\) 4.74002 + 4.74002i 0.617098 + 0.617098i 0.944786 0.327688i \(-0.106269\pi\)
−0.327688 + 0.944786i \(0.606269\pi\)
\(60\) 0 0
\(61\) 3.10257 3.10257i 0.397243 0.397243i −0.480016 0.877260i \(-0.659369\pi\)
0.877260 + 0.480016i \(0.159369\pi\)
\(62\) 0 0
\(63\) −0.202696 −0.0255372
\(64\) 0 0
\(65\) −14.8024 −1.83601
\(66\) 0 0
\(67\) 4.81108 4.81108i 0.587767 0.587767i −0.349259 0.937026i \(-0.613567\pi\)
0.937026 + 0.349259i \(0.113567\pi\)
\(68\) 0 0
\(69\) −3.59702 3.59702i −0.433031 0.433031i
\(70\) 0 0
\(71\) 1.11625i 0.132475i 0.997804 + 0.0662375i \(0.0210995\pi\)
−0.997804 + 0.0662375i \(0.978900\pi\)
\(72\) 0 0
\(73\) 11.2521i 1.31696i −0.752600 0.658478i \(-0.771201\pi\)
0.752600 0.658478i \(-0.228799\pi\)
\(74\) 0 0
\(75\) 2.43070 + 2.43070i 0.280673 + 0.280673i
\(76\) 0 0
\(77\) 0.584413 0.584413i 0.0666000 0.0666000i
\(78\) 0 0
\(79\) 7.61158 0.856370 0.428185 0.903691i \(-0.359153\pi\)
0.428185 + 0.903691i \(0.359153\pi\)
\(80\) 0 0
\(81\) 8.35083 0.927870
\(82\) 0 0
\(83\) −11.1869 + 11.1869i −1.22792 + 1.22792i −0.263176 + 0.964748i \(0.584770\pi\)
−0.964748 + 0.263176i \(0.915230\pi\)
\(84\) 0 0
\(85\) −3.28490 3.28490i −0.356298 0.356298i
\(86\) 0 0
\(87\) 10.4857i 1.12419i
\(88\) 0 0
\(89\) 0.428825i 0.0454554i −0.999742 0.0227277i \(-0.992765\pi\)
0.999742 0.0227277i \(-0.00723508\pi\)
\(90\) 0 0
\(91\) 3.94057 + 3.94057i 0.413084 + 0.413084i
\(92\) 0 0
\(93\) −9.35230 + 9.35230i −0.969788 + 0.969788i
\(94\) 0 0
\(95\) −15.7569 −1.61662
\(96\) 0 0
\(97\) −19.2163 −1.95111 −0.975557 0.219745i \(-0.929478\pi\)
−0.975557 + 0.219745i \(0.929478\pi\)
\(98\) 0 0
\(99\) 0.118458 0.118458i 0.0119055 0.0119055i
\(100\) 0 0
\(101\) 4.87547 + 4.87547i 0.485127 + 0.485127i 0.906765 0.421637i \(-0.138544\pi\)
−0.421637 + 0.906765i \(0.638544\pi\)
\(102\) 0 0
\(103\) 6.09849i 0.600902i −0.953797 0.300451i \(-0.902863\pi\)
0.953797 0.300451i \(-0.0971371\pi\)
\(104\) 0 0
\(105\) 4.44250i 0.433544i
\(106\) 0 0
\(107\) 5.19989 + 5.19989i 0.502693 + 0.502693i 0.912274 0.409581i \(-0.134325\pi\)
−0.409581 + 0.912274i \(0.634325\pi\)
\(108\) 0 0
\(109\) 7.70055 7.70055i 0.737579 0.737579i −0.234530 0.972109i \(-0.575355\pi\)
0.972109 + 0.234530i \(0.0753550\pi\)
\(110\) 0 0
\(111\) −13.8905 −1.31842
\(112\) 0 0
\(113\) −14.2646 −1.34190 −0.670952 0.741500i \(-0.734115\pi\)
−0.670952 + 0.741500i \(0.734115\pi\)
\(114\) 0 0
\(115\) −5.71257 + 5.71257i −0.532700 + 0.532700i
\(116\) 0 0
\(117\) 0.798737 + 0.798737i 0.0738433 + 0.0738433i
\(118\) 0 0
\(119\) 1.74896i 0.160327i
\(120\) 0 0
\(121\) 10.3169i 0.937902i
\(122\) 0 0
\(123\) −1.64296 1.64296i −0.148140 0.148140i
\(124\) 0 0
\(125\) −5.53074 + 5.53074i −0.494685 + 0.494685i
\(126\) 0 0
\(127\) 8.98310 0.797121 0.398561 0.917142i \(-0.369510\pi\)
0.398561 + 0.917142i \(0.369510\pi\)
\(128\) 0 0
\(129\) −4.11328 −0.362154
\(130\) 0 0
\(131\) 12.7547 12.7547i 1.11438 1.11438i 0.121828 0.992551i \(-0.461124\pi\)
0.992551 0.121828i \(-0.0388756\pi\)
\(132\) 0 0
\(133\) 4.19467 + 4.19467i 0.363724 + 0.363724i
\(134\) 0 0
\(135\) 14.2280i 1.22455i
\(136\) 0 0
\(137\) 11.7927i 1.00751i 0.863845 + 0.503757i \(0.168049\pi\)
−0.863845 + 0.503757i \(0.831951\pi\)
\(138\) 0 0
\(139\) 0.524016 + 0.524016i 0.0444465 + 0.0444465i 0.728981 0.684534i \(-0.239994\pi\)
−0.684534 + 0.728981i \(0.739994\pi\)
\(140\) 0 0
\(141\) 2.12896 2.12896i 0.179291 0.179291i
\(142\) 0 0
\(143\) −4.60584 −0.385160
\(144\) 0 0
\(145\) 16.6527 1.38294
\(146\) 0 0
\(147\) −1.18265 + 1.18265i −0.0975430 + 0.0975430i
\(148\) 0 0
\(149\) −1.98859 1.98859i −0.162912 0.162912i 0.620944 0.783855i \(-0.286750\pi\)
−0.783855 + 0.620944i \(0.786750\pi\)
\(150\) 0 0
\(151\) 15.1887i 1.23604i 0.786162 + 0.618020i \(0.212065\pi\)
−0.786162 + 0.618020i \(0.787935\pi\)
\(152\) 0 0
\(153\) 0.354507i 0.0286602i
\(154\) 0 0
\(155\) 14.8527 + 14.8527i 1.19300 + 1.19300i
\(156\) 0 0
\(157\) 7.91629 7.91629i 0.631789 0.631789i −0.316728 0.948516i \(-0.602584\pi\)
0.948516 + 0.316728i \(0.102584\pi\)
\(158\) 0 0
\(159\) −23.0303 −1.82642
\(160\) 0 0
\(161\) 3.04150 0.239704
\(162\) 0 0
\(163\) −6.32172 + 6.32172i −0.495155 + 0.495155i −0.909926 0.414771i \(-0.863862\pi\)
0.414771 + 0.909926i \(0.363862\pi\)
\(164\) 0 0
\(165\) 2.59626 + 2.59626i 0.202118 + 0.202118i
\(166\) 0 0
\(167\) 22.5263i 1.74314i 0.490271 + 0.871570i \(0.336898\pi\)
−0.490271 + 0.871570i \(0.663102\pi\)
\(168\) 0 0
\(169\) 18.0563i 1.38894i
\(170\) 0 0
\(171\) 0.850241 + 0.850241i 0.0650195 + 0.0650195i
\(172\) 0 0
\(173\) 0.0105911 0.0105911i 0.000805223 0.000805223i −0.706704 0.707509i \(-0.749819\pi\)
0.707509 + 0.706704i \(0.249819\pi\)
\(174\) 0 0
\(175\) −2.05530 −0.155366
\(176\) 0 0
\(177\) 11.2115 0.842710
\(178\) 0 0
\(179\) 14.3569 14.3569i 1.07309 1.07309i 0.0759779 0.997109i \(-0.475792\pi\)
0.997109 0.0759779i \(-0.0242079\pi\)
\(180\) 0 0
\(181\) −8.65321 8.65321i −0.643188 0.643188i 0.308150 0.951338i \(-0.400290\pi\)
−0.951338 + 0.308150i \(0.900290\pi\)
\(182\) 0 0
\(183\) 7.33848i 0.542476i
\(184\) 0 0
\(185\) 22.0600i 1.62188i
\(186\) 0 0
\(187\) −1.02211 1.02211i −0.0747444 0.0747444i
\(188\) 0 0
\(189\) −3.78766 + 3.78766i −0.275511 + 0.275511i
\(190\) 0 0
\(191\) 17.2085 1.24517 0.622583 0.782554i \(-0.286083\pi\)
0.622583 + 0.782554i \(0.286083\pi\)
\(192\) 0 0
\(193\) 7.00982 0.504578 0.252289 0.967652i \(-0.418817\pi\)
0.252289 + 0.967652i \(0.418817\pi\)
\(194\) 0 0
\(195\) −17.5060 + 17.5060i −1.25363 + 1.25363i
\(196\) 0 0
\(197\) 15.4175 + 15.4175i 1.09845 + 1.09845i 0.994592 + 0.103860i \(0.0331194\pi\)
0.103860 + 0.994592i \(0.466881\pi\)
\(198\) 0 0
\(199\) 15.3483i 1.08801i 0.839082 + 0.544005i \(0.183093\pi\)
−0.839082 + 0.544005i \(0.816907\pi\)
\(200\) 0 0
\(201\) 11.3796i 0.802656i
\(202\) 0 0
\(203\) −4.43316 4.43316i −0.311147 0.311147i
\(204\) 0 0
\(205\) −2.60924 + 2.60924i −0.182237 + 0.182237i
\(206\) 0 0
\(207\) 0.616500 0.0428497
\(208\) 0 0
\(209\) −4.90283 −0.339136
\(210\) 0 0
\(211\) 4.38104 4.38104i 0.301603 0.301603i −0.540038 0.841641i \(-0.681590\pi\)
0.841641 + 0.540038i \(0.181590\pi\)
\(212\) 0 0
\(213\) 1.32013 + 1.32013i 0.0904541 + 0.0904541i
\(214\) 0 0
\(215\) 6.53246i 0.445510i
\(216\) 0 0
\(217\) 7.90794i 0.536826i
\(218\) 0 0
\(219\) −13.3072 13.3072i −0.899219 0.899219i
\(220\) 0 0
\(221\) 6.89191 6.89191i 0.463600 0.463600i
\(222\) 0 0
\(223\) −0.528935 −0.0354201 −0.0177101 0.999843i \(-0.505638\pi\)
−0.0177101 + 0.999843i \(0.505638\pi\)
\(224\) 0 0
\(225\) −0.416601 −0.0277734
\(226\) 0 0
\(227\) 17.8735 17.8735i 1.18631 1.18631i 0.208224 0.978081i \(-0.433232\pi\)
0.978081 0.208224i \(-0.0667684\pi\)
\(228\) 0 0
\(229\) 8.73248 + 8.73248i 0.577059 + 0.577059i 0.934092 0.357033i \(-0.116212\pi\)
−0.357033 + 0.934092i \(0.616212\pi\)
\(230\) 0 0
\(231\) 1.38231i 0.0909491i
\(232\) 0 0
\(233\) 26.9485i 1.76545i −0.469885 0.882727i \(-0.655705\pi\)
0.469885 0.882727i \(-0.344295\pi\)
\(234\) 0 0
\(235\) −3.38109 3.38109i −0.220558 0.220558i
\(236\) 0 0
\(237\) 9.00181 9.00181i 0.584730 0.584730i
\(238\) 0 0
\(239\) 19.8050 1.28108 0.640539 0.767926i \(-0.278711\pi\)
0.640539 + 0.767926i \(0.278711\pi\)
\(240\) 0 0
\(241\) 3.73993 0.240910 0.120455 0.992719i \(-0.461565\pi\)
0.120455 + 0.992719i \(0.461565\pi\)
\(242\) 0 0
\(243\) −1.48689 + 1.48689i −0.0953842 + 0.0953842i
\(244\) 0 0
\(245\) 1.87820 + 1.87820i 0.119994 + 0.119994i
\(246\) 0 0
\(247\) 33.0588i 2.10348i
\(248\) 0 0
\(249\) 26.4603i 1.67686i
\(250\) 0 0
\(251\) −10.1476 10.1476i −0.640510 0.640510i 0.310171 0.950681i \(-0.399614\pi\)
−0.950681 + 0.310171i \(0.899614\pi\)
\(252\) 0 0
\(253\) −1.77749 + 1.77749i −0.111750 + 0.111750i
\(254\) 0 0
\(255\) −7.76976 −0.486561
\(256\) 0 0
\(257\) 12.6111 0.786657 0.393328 0.919398i \(-0.371324\pi\)
0.393328 + 0.919398i \(0.371324\pi\)
\(258\) 0 0
\(259\) 5.87262 5.87262i 0.364907 0.364907i
\(260\) 0 0
\(261\) −0.898582 0.898582i −0.0556208 0.0556208i
\(262\) 0 0
\(263\) 7.01176i 0.432364i −0.976353 0.216182i \(-0.930640\pi\)
0.976353 0.216182i \(-0.0693604\pi\)
\(264\) 0 0
\(265\) 36.5752i 2.24680i
\(266\) 0 0
\(267\) −0.507149 0.507149i −0.0310370 0.0310370i
\(268\) 0 0
\(269\) 3.92307 3.92307i 0.239194 0.239194i −0.577323 0.816516i \(-0.695902\pi\)
0.816516 + 0.577323i \(0.195902\pi\)
\(270\) 0 0
\(271\) 23.5746 1.43205 0.716026 0.698073i \(-0.245959\pi\)
0.716026 + 0.698073i \(0.245959\pi\)
\(272\) 0 0
\(273\) 9.32061 0.564109
\(274\) 0 0
\(275\) 1.20114 1.20114i 0.0724318 0.0724318i
\(276\) 0 0
\(277\) −7.89677 7.89677i −0.474471 0.474471i 0.428887 0.903358i \(-0.358906\pi\)
−0.903358 + 0.428887i \(0.858906\pi\)
\(278\) 0 0
\(279\) 1.60290i 0.0959634i
\(280\) 0 0
\(281\) 33.1753i 1.97907i 0.144288 + 0.989536i \(0.453911\pi\)
−0.144288 + 0.989536i \(0.546089\pi\)
\(282\) 0 0
\(283\) 2.06245 + 2.06245i 0.122600 + 0.122600i 0.765745 0.643145i \(-0.222371\pi\)
−0.643145 + 0.765745i \(0.722371\pi\)
\(284\) 0 0
\(285\) −18.6348 + 18.6348i −1.10383 + 1.10383i
\(286\) 0 0
\(287\) 1.38922 0.0820030
\(288\) 0 0
\(289\) −13.9411 −0.820067
\(290\) 0 0
\(291\) −22.7260 + 22.7260i −1.33222 + 1.33222i
\(292\) 0 0
\(293\) −5.32453 5.32453i −0.311062 0.311062i 0.534259 0.845321i \(-0.320591\pi\)
−0.845321 + 0.534259i \(0.820591\pi\)
\(294\) 0 0
\(295\) 17.8054i 1.03667i
\(296\) 0 0
\(297\) 4.42711i 0.256887i
\(298\) 0 0
\(299\) −11.9853 11.9853i −0.693126 0.693126i
\(300\) 0 0
\(301\) 1.73902 1.73902i 0.100235 0.100235i
\(302\) 0 0
\(303\) 11.5319 0.662491
\(304\) 0 0
\(305\) −11.6545 −0.667336
\(306\) 0 0
\(307\) −9.92446 + 9.92446i −0.566419 + 0.566419i −0.931123 0.364705i \(-0.881170\pi\)
0.364705 + 0.931123i \(0.381170\pi\)
\(308\) 0 0
\(309\) −7.21235 7.21235i −0.410296 0.410296i
\(310\) 0 0
\(311\) 9.78126i 0.554644i −0.960777 0.277322i \(-0.910553\pi\)
0.960777 0.277322i \(-0.0894469\pi\)
\(312\) 0 0
\(313\) 5.68720i 0.321460i 0.986998 + 0.160730i \(0.0513848\pi\)
−0.986998 + 0.160730i \(0.948615\pi\)
\(314\) 0 0
\(315\) 0.380704 + 0.380704i 0.0214502 + 0.0214502i
\(316\) 0 0
\(317\) −21.8662 + 21.8662i −1.22813 + 1.22813i −0.263457 + 0.964671i \(0.584863\pi\)
−0.964671 + 0.263457i \(0.915137\pi\)
\(318\) 0 0
\(319\) 5.18159 0.290113
\(320\) 0 0
\(321\) 12.2993 0.686478
\(322\) 0 0
\(323\) 7.33631 7.33631i 0.408203 0.408203i
\(324\) 0 0
\(325\) 8.09907 + 8.09907i 0.449256 + 0.449256i
\(326\) 0 0
\(327\) 18.2141i 1.00724i
\(328\) 0 0
\(329\) 1.80017i 0.0992466i
\(330\) 0 0
\(331\) 8.75943 + 8.75943i 0.481462 + 0.481462i 0.905598 0.424137i \(-0.139422\pi\)
−0.424137 + 0.905598i \(0.639422\pi\)
\(332\) 0 0
\(333\) 1.19035 1.19035i 0.0652310 0.0652310i
\(334\) 0 0
\(335\) −18.0724 −0.987400
\(336\) 0 0
\(337\) 19.1758 1.04457 0.522286 0.852771i \(-0.325079\pi\)
0.522286 + 0.852771i \(0.325079\pi\)
\(338\) 0 0
\(339\) −16.8700 + 16.8700i −0.916254 + 0.916254i
\(340\) 0 0
\(341\) 4.62150 + 4.62150i 0.250268 + 0.250268i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 13.5119i 0.727456i
\(346\) 0 0
\(347\) −6.37041 6.37041i −0.341982 0.341982i 0.515130 0.857112i \(-0.327744\pi\)
−0.857112 + 0.515130i \(0.827744\pi\)
\(348\) 0 0
\(349\) 14.0831 14.0831i 0.753850 0.753850i −0.221346 0.975195i \(-0.571045\pi\)
0.975195 + 0.221346i \(0.0710450\pi\)
\(350\) 0 0
\(351\) 29.8511 1.59333
\(352\) 0 0
\(353\) 12.8957 0.686371 0.343185 0.939268i \(-0.388494\pi\)
0.343185 + 0.939268i \(0.388494\pi\)
\(354\) 0 0
\(355\) 2.09655 2.09655i 0.111274 0.111274i
\(356\) 0 0
\(357\) 2.06840 + 2.06840i 0.109471 + 0.109471i
\(358\) 0 0
\(359\) 33.5832i 1.77245i 0.463252 + 0.886227i \(0.346682\pi\)
−0.463252 + 0.886227i \(0.653318\pi\)
\(360\) 0 0
\(361\) 16.1905i 0.852130i
\(362\) 0 0
\(363\) −12.2013 12.2013i −0.640401 0.640401i
\(364\) 0 0
\(365\) −21.1337 + 21.1337i −1.10619 + 1.10619i
\(366\) 0 0
\(367\) 7.74580 0.404328 0.202164 0.979352i \(-0.435203\pi\)
0.202164 + 0.979352i \(0.435203\pi\)
\(368\) 0 0
\(369\) 0.281589 0.0146589
\(370\) 0 0
\(371\) 9.73675 9.73675i 0.505507 0.505507i
\(372\) 0 0
\(373\) 1.84606 + 1.84606i 0.0955855 + 0.0955855i 0.753283 0.657697i \(-0.228469\pi\)
−0.657697 + 0.753283i \(0.728469\pi\)
\(374\) 0 0
\(375\) 13.0818i 0.675543i
\(376\) 0 0
\(377\) 34.9384i 1.79942i
\(378\) 0 0
\(379\) −24.4450 24.4450i −1.25566 1.25566i −0.953146 0.302511i \(-0.902175\pi\)
−0.302511 0.953146i \(-0.597825\pi\)
\(380\) 0 0
\(381\) 10.6238 10.6238i 0.544275 0.544275i
\(382\) 0 0
\(383\) 22.2480 1.13682 0.568411 0.822745i \(-0.307559\pi\)
0.568411 + 0.822745i \(0.307559\pi\)
\(384\) 0 0
\(385\) −2.19529 −0.111882
\(386\) 0 0
\(387\) 0.352491 0.352491i 0.0179181 0.0179181i
\(388\) 0 0
\(389\) 6.93666 + 6.93666i 0.351703 + 0.351703i 0.860743 0.509040i \(-0.170000\pi\)
−0.509040 + 0.860743i \(0.670000\pi\)
\(390\) 0 0
\(391\) 5.31947i 0.269017i
\(392\) 0 0
\(393\) 30.1685i 1.52180i
\(394\) 0 0
\(395\) −14.2961 14.2961i −0.719315 0.719315i
\(396\) 0 0
\(397\) 16.4042 16.4042i 0.823304 0.823304i −0.163277 0.986580i \(-0.552206\pi\)
0.986580 + 0.163277i \(0.0522063\pi\)
\(398\) 0 0
\(399\) 9.92162 0.496702
\(400\) 0 0
\(401\) −22.8150 −1.13932 −0.569662 0.821879i \(-0.692926\pi\)
−0.569662 + 0.821879i \(0.692926\pi\)
\(402\) 0 0
\(403\) −31.1618 + 31.1618i −1.55228 + 1.55228i
\(404\) 0 0
\(405\) −15.6846 15.6846i −0.779372 0.779372i
\(406\) 0 0
\(407\) 6.86407i 0.340239i
\(408\) 0 0
\(409\) 13.9196i 0.688281i 0.938918 + 0.344140i \(0.111830\pi\)
−0.938918 + 0.344140i \(0.888170\pi\)
\(410\) 0 0
\(411\) 13.9465 + 13.9465i 0.687932 + 0.687932i
\(412\) 0 0
\(413\) −4.74002 + 4.74002i −0.233241 + 0.233241i
\(414\) 0 0
\(415\) 42.0226 2.06281
\(416\) 0 0
\(417\) 1.23945 0.0606962
\(418\) 0 0
\(419\) −25.1837 + 25.1837i −1.23030 + 1.23030i −0.266455 + 0.963847i \(0.585852\pi\)
−0.963847 + 0.266455i \(0.914148\pi\)
\(420\) 0 0
\(421\) 16.0774 + 16.0774i 0.783566 + 0.783566i 0.980431 0.196865i \(-0.0630761\pi\)
−0.196865 + 0.980431i \(0.563076\pi\)
\(422\) 0 0
\(423\) 0.364887i 0.0177414i
\(424\) 0 0
\(425\) 3.59464i 0.174366i
\(426\) 0 0
\(427\) 3.10257 + 3.10257i 0.150144 + 0.150144i
\(428\) 0 0
\(429\) −5.44708 + 5.44708i −0.262988 + 0.262988i
\(430\) 0 0
\(431\) −0.695976 −0.0335240 −0.0167620 0.999860i \(-0.505336\pi\)
−0.0167620 + 0.999860i \(0.505336\pi\)
\(432\) 0 0
\(433\) 26.4982 1.27342 0.636711 0.771103i \(-0.280295\pi\)
0.636711 + 0.771103i \(0.280295\pi\)
\(434\) 0 0
\(435\) 19.6943 19.6943i 0.944270 0.944270i
\(436\) 0 0
\(437\) −12.7581 12.7581i −0.610303 0.610303i
\(438\) 0 0
\(439\) 5.34131i 0.254927i 0.991843 + 0.127463i \(0.0406835\pi\)
−0.991843 + 0.127463i \(0.959316\pi\)
\(440\) 0 0
\(441\) 0.202696i 0.00965217i
\(442\) 0 0
\(443\) −18.0337 18.0337i −0.856805 0.856805i 0.134155 0.990960i \(-0.457168\pi\)
−0.990960 + 0.134155i \(0.957168\pi\)
\(444\) 0 0
\(445\) −0.805422 + 0.805422i −0.0381807 + 0.0381807i
\(446\) 0 0
\(447\) −4.70360 −0.222473
\(448\) 0 0
\(449\) 5.57561 0.263129 0.131565 0.991308i \(-0.458000\pi\)
0.131565 + 0.991308i \(0.458000\pi\)
\(450\) 0 0
\(451\) −0.811878 + 0.811878i −0.0382298 + 0.0382298i
\(452\) 0 0
\(453\) 17.9629 + 17.9629i 0.843970 + 0.843970i
\(454\) 0 0
\(455\) 14.8024i 0.693948i
\(456\) 0 0
\(457\) 11.8678i 0.555151i 0.960704 + 0.277576i \(0.0895309\pi\)
−0.960704 + 0.277576i \(0.910469\pi\)
\(458\) 0 0
\(459\) 6.62446 + 6.62446i 0.309203 + 0.309203i
\(460\) 0 0
\(461\) 25.1973 25.1973i 1.17356 1.17356i 0.192200 0.981356i \(-0.438438\pi\)
0.981356 0.192200i \(-0.0615624\pi\)
\(462\) 0 0
\(463\) −31.1785 −1.44899 −0.724494 0.689281i \(-0.757926\pi\)
−0.724494 + 0.689281i \(0.757926\pi\)
\(464\) 0 0
\(465\) 35.1310 1.62916
\(466\) 0 0
\(467\) 4.86260 4.86260i 0.225014 0.225014i −0.585592 0.810606i \(-0.699138\pi\)
0.810606 + 0.585592i \(0.199138\pi\)
\(468\) 0 0
\(469\) 4.81108 + 4.81108i 0.222155 + 0.222155i
\(470\) 0 0
\(471\) 18.7243i 0.862772i
\(472\) 0 0
\(473\) 2.03261i 0.0934594i
\(474\) 0 0
\(475\) 8.62131 + 8.62131i 0.395573 + 0.395573i
\(476\) 0 0
\(477\) 1.97360 1.97360i 0.0903648 0.0903648i
\(478\) 0 0
\(479\) −13.2313 −0.604555 −0.302277 0.953220i \(-0.597747\pi\)
−0.302277 + 0.953220i \(0.597747\pi\)
\(480\) 0 0
\(481\) −46.2830 −2.11032
\(482\) 0 0
\(483\) 3.59702 3.59702i 0.163670 0.163670i
\(484\) 0 0
\(485\) 36.0920 + 36.0920i 1.63886 + 1.63886i
\(486\) 0 0
\(487\) 27.6451i 1.25272i 0.779534 + 0.626360i \(0.215456\pi\)
−0.779534 + 0.626360i \(0.784544\pi\)
\(488\) 0 0
\(489\) 14.9527i 0.676185i
\(490\) 0 0
\(491\) −5.28148 5.28148i −0.238350 0.238350i 0.577817 0.816167i \(-0.303905\pi\)
−0.816167 + 0.577817i \(0.803905\pi\)
\(492\) 0 0
\(493\) −7.75342 + 7.75342i −0.349196 + 0.349196i
\(494\) 0 0
\(495\) −0.444976 −0.0200002
\(496\) 0 0
\(497\) −1.11625 −0.0500709
\(498\) 0 0
\(499\) 28.8375 28.8375i 1.29094 1.29094i 0.356741 0.934203i \(-0.383888\pi\)
0.934203 0.356741i \(-0.116112\pi\)
\(500\) 0 0
\(501\) 26.6407 + 26.6407i 1.19022 + 1.19022i
\(502\) 0 0
\(503\) 16.8595i 0.751729i 0.926675 + 0.375864i \(0.122654\pi\)
−0.926675 + 0.375864i \(0.877346\pi\)
\(504\) 0 0
\(505\) 18.3142i 0.814973i
\(506\) 0 0
\(507\) −21.3542 21.3542i −0.948372 0.948372i
\(508\) 0 0
\(509\) −21.3899 + 21.3899i −0.948093 + 0.948093i −0.998718 0.0506250i \(-0.983879\pi\)
0.0506250 + 0.998718i \(0.483879\pi\)
\(510\) 0 0
\(511\) 11.2521 0.497762
\(512\) 0 0
\(513\) 31.7759 1.40294
\(514\) 0 0
\(515\) −11.4542 + 11.4542i −0.504732 + 0.504732i
\(516\) 0 0
\(517\) −1.05204 1.05204i −0.0462688 0.0462688i
\(518\) 0 0
\(519\) 0.0250509i 0.00109961i
\(520\) 0 0
\(521\) 29.9153i 1.31061i 0.755363 + 0.655307i \(0.227461\pi\)
−0.755363 + 0.655307i \(0.772539\pi\)
\(522\) 0 0
\(523\) 12.6212 + 12.6212i 0.551885 + 0.551885i 0.926985 0.375099i \(-0.122391\pi\)
−0.375099 + 0.926985i \(0.622391\pi\)
\(524\) 0 0
\(525\) −2.43070 + 2.43070i −0.106084 + 0.106084i
\(526\) 0 0
\(527\) −13.8307 −0.602473
\(528\) 0 0
\(529\) 13.7492 0.597793
\(530\) 0 0
\(531\) −0.960781 + 0.960781i −0.0416943 + 0.0416943i
\(532\) 0 0
\(533\) −5.47432 5.47432i −0.237119 0.237119i
\(534\) 0 0
\(535\) 19.5329i 0.844482i
\(536\) 0 0
\(537\) 33.9583i 1.46541i
\(538\) 0 0
\(539\) 0.584413 + 0.584413i 0.0251724 + 0.0251724i
\(540\) 0 0
\(541\) 6.98421 6.98421i 0.300275 0.300275i −0.540847 0.841121i \(-0.681896\pi\)
0.841121 + 0.540847i \(0.181896\pi\)
\(542\) 0 0
\(543\) −20.4674 −0.878339
\(544\) 0 0
\(545\) −28.9264 −1.23907
\(546\) 0 0
\(547\) −19.0691 + 19.0691i −0.815337 + 0.815337i −0.985428 0.170091i \(-0.945594\pi\)
0.170091 + 0.985428i \(0.445594\pi\)
\(548\) 0 0
\(549\) 0.628877 + 0.628877i 0.0268398 + 0.0268398i
\(550\) 0 0
\(551\) 37.1912i 1.58440i
\(552\) 0 0
\(553\) 7.61158i 0.323677i
\(554\) 0 0
\(555\) 26.0891 + 26.0891i 1.10742 + 1.10742i
\(556\) 0 0
\(557\) 10.7024 10.7024i 0.453474 0.453474i −0.443032 0.896506i \(-0.646097\pi\)
0.896506 + 0.443032i \(0.146097\pi\)
\(558\) 0 0
\(559\) −13.7055 −0.579679
\(560\) 0 0
\(561\) −2.41760 −0.102071
\(562\) 0 0
\(563\) −6.53316 + 6.53316i −0.275340 + 0.275340i −0.831246 0.555905i \(-0.812372\pi\)
0.555905 + 0.831246i \(0.312372\pi\)
\(564\) 0 0
\(565\) 26.7919 + 26.7919i 1.12714 + 1.12714i
\(566\) 0 0
\(567\) 8.35083i 0.350702i
\(568\) 0 0
\(569\) 36.7286i 1.53974i −0.638199 0.769871i \(-0.720320\pi\)
0.638199 0.769871i \(-0.279680\pi\)
\(570\) 0 0
\(571\) 23.7908 + 23.7908i 0.995613 + 0.995613i 0.999990 0.00437721i \(-0.00139331\pi\)
−0.00437721 + 0.999990i \(0.501393\pi\)
\(572\) 0 0
\(573\) 20.3516 20.3516i 0.850200 0.850200i
\(574\) 0 0
\(575\) 6.25121 0.260694
\(576\) 0 0
\(577\) −5.87217 −0.244462 −0.122231 0.992502i \(-0.539005\pi\)
−0.122231 + 0.992502i \(0.539005\pi\)
\(578\) 0 0
\(579\) 8.29014 8.29014i 0.344527 0.344527i
\(580\) 0 0
\(581\) −11.1869 11.1869i −0.464112 0.464112i
\(582\) 0 0
\(583\) 11.3806i 0.471335i
\(584\) 0 0
\(585\) 3.00038i 0.124051i
\(586\) 0 0
\(587\) 11.0010 + 11.0010i 0.454060 + 0.454060i 0.896700 0.442639i \(-0.145958\pi\)
−0.442639 + 0.896700i \(0.645958\pi\)
\(588\) 0 0
\(589\) −33.1712 + 33.1712i −1.36679 + 1.36679i
\(590\) 0 0
\(591\) 36.4669 1.50005
\(592\) 0 0
\(593\) 26.8167 1.10123 0.550616 0.834759i \(-0.314393\pi\)
0.550616 + 0.834759i \(0.314393\pi\)
\(594\) 0 0
\(595\) 3.28490 3.28490i 0.134668 0.134668i
\(596\) 0 0
\(597\) 18.1516 + 18.1516i 0.742894 + 0.742894i
\(598\) 0 0
\(599\) 44.9307i 1.83582i 0.396791 + 0.917909i \(0.370124\pi\)
−0.396791 + 0.917909i \(0.629876\pi\)
\(600\) 0 0
\(601\) 40.2092i 1.64017i 0.572244 + 0.820083i \(0.306073\pi\)
−0.572244 + 0.820083i \(0.693927\pi\)
\(602\) 0 0
\(603\) 0.975185 + 0.975185i 0.0397126 + 0.0397126i
\(604\) 0 0
\(605\) −19.3773 + 19.3773i −0.787799 + 0.787799i
\(606\) 0 0
\(607\) 20.7271 0.841286 0.420643 0.907226i \(-0.361805\pi\)
0.420643 + 0.907226i \(0.361805\pi\)
\(608\) 0 0
\(609\) −10.4857 −0.424903
\(610\) 0 0
\(611\) 7.09371 7.09371i 0.286981 0.286981i
\(612\) 0 0
\(613\) 5.43581 + 5.43581i 0.219550 + 0.219550i 0.808309 0.588759i \(-0.200383\pi\)
−0.588759 + 0.808309i \(0.700383\pi\)
\(614\) 0 0
\(615\) 6.17161i 0.248863i
\(616\) 0 0
\(617\) 24.8297i 0.999607i −0.866139 0.499803i \(-0.833406\pi\)
0.866139 0.499803i \(-0.166594\pi\)
\(618\) 0 0
\(619\) 13.2178 + 13.2178i 0.531267 + 0.531267i 0.920949 0.389683i \(-0.127415\pi\)
−0.389683 + 0.920949i \(0.627415\pi\)
\(620\) 0 0
\(621\) 11.5202 11.5202i 0.462289 0.462289i
\(622\) 0 0
\(623\) 0.428825 0.0171805
\(624\) 0 0
\(625\) 31.0522 1.24209
\(626\) 0 0
\(627\) −5.79832 + 5.79832i −0.231563 + 0.231563i
\(628\) 0 0
\(629\) −10.2710 10.2710i −0.409531 0.409531i
\(630\) 0 0
\(631\) 30.4138i 1.21075i −0.795939 0.605377i \(-0.793022\pi\)
0.795939 0.605377i \(-0.206978\pi\)
\(632\) 0 0
\(633\) 10.3624i 0.411870i
\(634\) 0 0
\(635\) −16.8721 16.8721i −0.669549 0.669549i
\(636\) 0 0
\(637\) −3.94057 + 3.94057i −0.156131 + 0.156131i
\(638\) 0 0
\(639\) −0.226260 −0.00895071
\(640\) 0 0
\(641\) −41.7486 −1.64897 −0.824485 0.565884i \(-0.808535\pi\)
−0.824485 + 0.565884i \(0.808535\pi\)
\(642\) 0 0
\(643\) −15.4137 + 15.4137i −0.607857 + 0.607857i −0.942386 0.334529i \(-0.891423\pi\)
0.334529 + 0.942386i \(0.391423\pi\)
\(644\) 0 0
\(645\) 7.72559 + 7.72559i 0.304195 + 0.304195i
\(646\) 0 0
\(647\) 8.85923i 0.348292i 0.984720 + 0.174146i \(0.0557165\pi\)
−0.984720 + 0.174146i \(0.944283\pi\)
\(648\) 0 0
\(649\) 5.54026i 0.217474i
\(650\) 0 0
\(651\) −9.35230 9.35230i −0.366545 0.366545i
\(652\) 0 0
\(653\) −2.42659 + 2.42659i −0.0949598 + 0.0949598i −0.752991 0.658031i \(-0.771390\pi\)
0.658031 + 0.752991i \(0.271390\pi\)
\(654\) 0 0
\(655\) −47.9117 −1.87206
\(656\) 0 0
\(657\) 2.28075 0.0889804
\(658\) 0 0
\(659\) 23.3045 23.3045i 0.907815 0.907815i −0.0882806 0.996096i \(-0.528137\pi\)
0.996096 + 0.0882806i \(0.0281372\pi\)
\(660\) 0 0
\(661\) −23.2433 23.2433i −0.904059 0.904059i 0.0917252 0.995784i \(-0.470762\pi\)
−0.995784 + 0.0917252i \(0.970762\pi\)
\(662\) 0 0
\(663\) 16.3014i 0.633093i
\(664\) 0 0
\(665\) 15.7569i 0.611026i
\(666\) 0 0
\(667\) 13.4835 + 13.4835i 0.522082 + 0.522082i
\(668\) 0 0
\(669\) −0.625543 + 0.625543i −0.0241849 + 0.0241849i
\(670\) 0 0
\(671\) −3.62636 −0.139994
\(672\) 0 0
\(673\) 46.7430 1.80181 0.900906 0.434014i \(-0.142903\pi\)
0.900906 + 0.434014i \(0.142903\pi\)
\(674\) 0 0
\(675\) −7.78478 + 7.78478i −0.299636 + 0.299636i
\(676\) 0 0
\(677\) −30.0928 30.0928i −1.15656 1.15656i −0.985210 0.171351i \(-0.945187\pi\)
−0.171351 0.985210i \(-0.554813\pi\)
\(678\) 0 0
\(679\) 19.2163i 0.737452i
\(680\) 0 0
\(681\) 42.2760i 1.62002i
\(682\) 0 0
\(683\) −25.4691 25.4691i −0.974548 0.974548i 0.0251358 0.999684i \(-0.491998\pi\)
−0.999684 + 0.0251358i \(0.991998\pi\)
\(684\) 0 0
\(685\) 22.1490 22.1490i 0.846270 0.846270i
\(686\) 0 0
\(687\) 20.6549 0.788033
\(688\) 0 0
\(689\) −76.7368 −2.92344
\(690\) 0 0
\(691\) −15.5437 + 15.5437i −0.591311 + 0.591311i −0.937986 0.346674i \(-0.887311\pi\)
0.346674 + 0.937986i \(0.387311\pi\)
\(692\) 0 0
\(693\) 0.118458 + 0.118458i 0.00449984 + 0.00449984i
\(694\) 0 0
\(695\) 1.96842i 0.0746664i
\(696\) 0 0
\(697\) 2.42969i 0.0920311i
\(698\) 0 0
\(699\) −31.8705 31.8705i −1.20545 1.20545i
\(700\) 0 0
\(701\) 5.59915 5.59915i 0.211477 0.211477i −0.593418 0.804895i \(-0.702222\pi\)
0.804895 + 0.593418i \(0.202222\pi\)
\(702\) 0 0
\(703\) −49.2674 −1.85815
\(704\) 0 0
\(705\) −7.99726 −0.301194
\(706\) 0 0
\(707\) −4.87547 + 4.87547i −0.183361 + 0.183361i
\(708\) 0 0
\(709\) 0.685858 + 0.685858i 0.0257579 + 0.0257579i 0.719868 0.694111i \(-0.244202\pi\)
−0.694111 + 0.719868i \(0.744202\pi\)
\(710\) 0 0
\(711\) 1.54283i 0.0578608i
\(712\) 0 0
\(713\) 24.0520i 0.900756i
\(714\) 0 0
\(715\) 8.65072 + 8.65072i 0.323518 + 0.323518i
\(716\) 0 0
\(717\) 23.4223 23.4223i 0.874721 0.874721i
\(718\) 0 0
\(719\) 10.0440 0.374579 0.187289 0.982305i \(-0.440030\pi\)
0.187289 + 0.982305i \(0.440030\pi\)
\(720\) 0 0
\(721\) 6.09849 0.227119
\(722\) 0 0
\(723\) 4.42302 4.42302i 0.164494 0.164494i
\(724\) 0 0
\(725\) −9.11148 9.11148i −0.338392 0.338392i
\(726\) 0 0
\(727\) 5.53235i 0.205184i 0.994724 + 0.102592i \(0.0327135\pi\)
−0.994724 + 0.102592i \(0.967286\pi\)
\(728\) 0 0
\(729\) 28.5694i 1.05813i
\(730\) 0 0
\(731\) −3.04147 3.04147i −0.112493 0.112493i
\(732\) 0 0
\(733\) −3.98509 + 3.98509i −0.147193 + 0.147193i −0.776863 0.629670i \(-0.783190\pi\)
0.629670 + 0.776863i \(0.283190\pi\)
\(734\) 0 0
\(735\) 4.44250 0.163864
\(736\) 0 0
\(737\) −5.62331 −0.207137
\(738\) 0 0
\(739\) 2.19669 2.19669i 0.0808065 0.0808065i −0.665548 0.746355i \(-0.731802\pi\)
0.746355 + 0.665548i \(0.231802\pi\)
\(740\) 0 0
\(741\) −39.0969 39.0969i −1.43626 1.43626i
\(742\) 0 0
\(743\) 22.5404i 0.826927i −0.910521 0.413464i \(-0.864319\pi\)
0.910521 0.413464i \(-0.135681\pi\)
\(744\) 0 0
\(745\) 7.46996i 0.273678i
\(746\) 0 0
\(747\) −2.26754 2.26754i −0.0829649 0.0829649i
\(748\) 0 0
\(749\) −5.19989 + 5.19989i −0.190000 + 0.190000i
\(750\) 0 0
\(751\) −14.8752 −0.542805 −0.271403 0.962466i \(-0.587487\pi\)
−0.271403 + 0.962466i \(0.587487\pi\)
\(752\) 0 0
\(753\) −24.0020 −0.874682
\(754\) 0 0
\(755\) 28.5275 28.5275i 1.03822 1.03822i
\(756\) 0 0
\(757\) −0.982488 0.982488i −0.0357091 0.0357091i 0.689027 0.724736i \(-0.258038\pi\)
−0.724736 + 0.689027i \(0.758038\pi\)
\(758\) 0 0
\(759\) 4.20429i 0.152606i
\(760\) 0 0
\(761\) 19.9186i 0.722049i −0.932556 0.361024i \(-0.882427\pi\)
0.932556 0.361024i \(-0.117573\pi\)
\(762\) 0 0
\(763\) 7.70055 + 7.70055i 0.278779 + 0.278779i
\(764\) 0 0
\(765\) 0.665836 0.665836i 0.0240733 0.0240733i
\(766\) 0 0
\(767\) 37.3568 1.34888
\(768\) 0 0
\(769\) −9.78665 −0.352915 −0.176458 0.984308i \(-0.556464\pi\)
−0.176458 + 0.984308i \(0.556464\pi\)
\(770\) 0 0
\(771\) 14.9144 14.9144i 0.537130 0.537130i
\(772\) 0 0
\(773\) −19.6983 19.6983i −0.708499 0.708499i 0.257720 0.966220i \(-0.417029\pi\)
−0.966220 + 0.257720i \(0.917029\pi\)
\(774\) 0 0
\(775\) 16.2532i 0.583832i
\(776\) 0 0
\(777\) 13.8905i 0.498318i
\(778\) 0 0
\(779\) −5.82731 5.82731i −0.208785 0.208785i
\(780\) 0 0
\(781\) 0.652354 0.652354i 0.0233430 0.0233430i
\(782\) 0 0
\(783\) −33.5825 −1.20014
\(784\) 0 0
\(785\) −29.7368 −1.06135
\(786\) 0 0
\(787\) −30.9238 + 30.9238i −1.10231 + 1.10231i −0.108184 + 0.994131i \(0.534504\pi\)
−0.994131 + 0.108184i \(0.965496\pi\)
\(788\) 0 0
\(789\) −8.29244 8.29244i −0.295219 0.295219i
\(790\) 0 0
\(791\) 14.2646i 0.507192i
\(792\) 0 0
\(793\) 24.4518i 0.868309i
\(794\) 0 0
\(795\) 43.2555 + 43.2555i 1.53412 + 1.53412i
\(796\) 0 0
\(797\) −32.5005 + 32.5005i −1.15123 + 1.15123i −0.164919 + 0.986307i \(0.552736\pi\)
−0.986307 + 0.164919i \(0.947264\pi\)
\(798\) 0 0
\(799\) 3.14843 0.111383
\(800\) 0 0
\(801\) 0.0869210 0.00307120
\(802\) 0 0
\(803\) −6.57585 + 6.57585i −0.232057 + 0.232057i
\(804\) 0 0
\(805\) −5.71257 5.71257i −0.201342 0.201342i
\(806\) 0 0
\(807\) 9.27920i 0.326643i
\(808\) 0 0
\(809\) 20.8417i 0.732756i 0.930466 + 0.366378i \(0.119402\pi\)
−0.930466 + 0.366378i \(0.880598\pi\)
\(810\) 0 0
\(811\) −1.93354 1.93354i −0.0678959 0.0678959i 0.672343 0.740239i \(-0.265288\pi\)
−0.740239 + 0.672343i \(0.765288\pi\)
\(812\) 0 0
\(813\) 27.8804 27.8804i 0.977807 0.977807i
\(814\) 0 0
\(815\) 23.7469 0.831819
\(816\) 0 0
\(817\) −14.5892 −0.510411
\(818\) 0 0
\(819\) −0.798737 + 0.798737i −0.0279101 + 0.0279101i
\(820\) 0 0
\(821\) 6.76535 + 6.76535i 0.236112 + 0.236112i 0.815238 0.579126i \(-0.196606\pi\)
−0.579126 + 0.815238i \(0.696606\pi\)
\(822\) 0 0
\(823\) 35.9651i 1.25366i −0.779154 0.626832i \(-0.784351\pi\)
0.779154 0.626832i \(-0.215649\pi\)
\(824\) 0 0
\(825\) 2.84106i 0.0989130i
\(826\) 0 0
\(827\) 23.1871 + 23.1871i 0.806296 + 0.806296i 0.984071 0.177775i \(-0.0568900\pi\)
−0.177775 + 0.984071i \(0.556890\pi\)
\(828\) 0 0
\(829\) −18.4289 + 18.4289i −0.640063 + 0.640063i −0.950571 0.310508i \(-0.899501\pi\)
0.310508 + 0.950571i \(0.399501\pi\)
\(830\) 0 0
\(831\) −18.6782 −0.647938
\(832\) 0 0
\(833\) −1.74896 −0.0605979
\(834\) 0 0
\(835\) 42.3091 42.3091i 1.46417 1.46417i
\(836\) 0 0
\(837\) −29.9526 29.9526i −1.03531 1.03531i
\(838\) 0 0
\(839\) 34.0165i 1.17438i 0.809449 + 0.587191i \(0.199766\pi\)
−0.809449 + 0.587191i \(0.800234\pi\)
\(840\) 0 0
\(841\) 10.3058i 0.355371i
\(842\) 0 0
\(843\) 39.2346 + 39.2346i 1.35131 + 1.35131i
\(844\) 0 0
\(845\) −33.9133 + 33.9133i −1.16665 + 1.16665i
\(846\) 0 0
\(847\) 10.3169 0.354494
\(848\) 0 0
\(849\) 4.87829 0.167423
\(850\) 0 0
\(851\) −17.8616 + 17.8616i −0.612288 + 0.612288i
\(852\) 0 0
\(853\) 11.0005 + 11.0005i 0.376649 + 0.376649i 0.869892 0.493243i \(-0.164188\pi\)
−0.493243 + 0.869892i \(0.664188\pi\)
\(854\) 0 0
\(855\) 3.19385i 0.109227i
\(856\) 0 0
\(857\) 0.996249i 0.0340312i −0.999855 0.0170156i \(-0.994584\pi\)
0.999855 0.0170156i \(-0.00541650\pi\)
\(858\) 0 0
\(859\) 23.5504 + 23.5504i 0.803530 + 0.803530i 0.983645 0.180116i \(-0.0576472\pi\)
−0.180116 + 0.983645i \(0.557647\pi\)
\(860\) 0 0
\(861\) 1.64296 1.64296i 0.0559918 0.0559918i
\(862\) 0 0
\(863\) −45.1680 −1.53754 −0.768768 0.639528i \(-0.779130\pi\)
−0.768768 + 0.639528i \(0.779130\pi\)
\(864\) 0 0
\(865\) −0.0397843 −0.00135271
\(866\) 0 0
\(867\) −16.4874 + 16.4874i −0.559943 + 0.559943i
\(868\) 0 0
\(869\) −4.44831 4.44831i −0.150898 0.150898i
\(870\) 0 0
\(871\) 37.9168i 1.28476i
\(872\) 0 0
\(873\) 3.89505i 0.131827i
\(874\) 0 0
\(875\) −5.53074 5.53074i −0.186973 0.186973i
\(876\) 0 0
\(877\) 4.57165 4.57165i 0.154374 0.154374i −0.625695 0.780068i \(-0.715184\pi\)
0.780068 + 0.625695i \(0.215184\pi\)
\(878\) 0 0
\(879\) −12.5941 −0.424788
\(880\) 0 0
\(881\) 29.5811 0.996613 0.498306 0.867001i \(-0.333955\pi\)
0.498306 + 0.867001i \(0.333955\pi\)
\(882\) 0 0
\(883\) 20.9519 20.9519i 0.705087 0.705087i −0.260411 0.965498i \(-0.583858\pi\)
0.965498 + 0.260411i \(0.0838581\pi\)
\(884\) 0 0
\(885\) −21.0575 21.0575i −0.707842 0.707842i
\(886\) 0 0
\(887\) 33.7805i 1.13424i −0.823635 0.567120i \(-0.808058\pi\)
0.823635 0.567120i \(-0.191942\pi\)
\(888\) 0 0
\(889\) 8.98310i 0.301284i
\(890\) 0 0
\(891\) −4.88033 4.88033i −0.163497 0.163497i
\(892\) 0 0
\(893\) 7.55112 7.55112i 0.252688 0.252688i
\(894\) 0 0
\(895\) −53.9305 −1.80270
\(896\) 0 0
\(897\) −28.3487 −0.946535
\(898\) 0 0
\(899\) 35.0571 35.0571i 1.16922 1.16922i
\(900\) 0 0
\(901\) −17.0292 17.0292i −0.567325 0.567325i
\(902\) 0 0
\(903\) 4.11328i 0.136881i
\(904\) 0 0
\(905\) 32.5050i 1.08050i
\(906\) 0 0
\(907\) −37.9368 37.9368i −1.25967 1.25967i −0.951251 0.308418i \(-0.900200\pi\)
−0.308418 0.951251i \(-0.599800\pi\)
\(908\) 0 0
\(909\) −0.988236 + 0.988236i −0.0327777 + 0.0327777i
\(910\) 0 0
\(911\) 16.3562 0.541906 0.270953 0.962593i \(-0.412661\pi\)
0.270953 + 0.962593i \(0.412661\pi\)
\(912\) 0 0
\(913\) 13.0756 0.432738
\(914\) 0 0
\(915\) −13.7832 + 13.7832i −0.455658 + 0.455658i
\(916\) 0 0
\(917\) 12.7547 + 12.7547i 0.421196 + 0.421196i
\(918\) 0 0
\(919\) 6.74372i 0.222455i −0.993795 0.111227i \(-0.964522\pi\)
0.993795 0.111227i \(-0.0354782\pi\)
\(920\) 0 0
\(921\) 23.4742i 0.773503i
\(922\) 0 0
\(923\) 4.39869 + 4.39869i 0.144785 + 0.144785i
\(924\) 0 0
\(925\) 12.0700 12.0700i 0.396859 0.396859i
\(926\) 0 0
\(927\) 1.23614 0.0406000
\(928\) 0 0
\(929\) −3.61776 −0.118695 −0.0593475 0.998237i \(-0.518902\pi\)
−0.0593475 + 0.998237i \(0.518902\pi\)
\(930\) 0 0
\(931\) −4.19467 + 4.19467i −0.137475 + 0.137475i
\(932\) 0 0
\(933\) −11.5678 11.5678i −0.378712 0.378712i
\(934\) 0 0
\(935\) 3.83948i 0.125564i
\(936\) 0 0
\(937\) 2.79455i 0.0912940i 0.998958 + 0.0456470i \(0.0145349\pi\)
−0.998958 + 0.0456470i \(0.985465\pi\)
\(938\) 0 0
\(939\) 6.72595 + 6.72595i 0.219493 + 0.219493i
\(940\) 0 0
\(941\) 17.3862 17.3862i 0.566775 0.566775i −0.364448 0.931224i \(-0.618742\pi\)
0.931224 + 0.364448i \(0.118742\pi\)
\(942\) 0 0
\(943\) −4.22532 −0.137595
\(944\) 0 0
\(945\) 14.2280 0.462836
\(946\) 0 0
\(947\) 29.8903 29.8903i 0.971304 0.971304i −0.0282955 0.999600i \(-0.509008\pi\)
0.999600 + 0.0282955i \(0.00900794\pi\)
\(948\) 0 0
\(949\) −44.3396 44.3396i −1.43933 1.43933i
\(950\) 0 0
\(951\) 51.7199i 1.67713i
\(952\) 0 0
\(953\) 26.9944i 0.874434i −0.899356 0.437217i \(-0.855964\pi\)
0.899356 0.437217i \(-0.144036\pi\)
\(954\) 0 0
\(955\) −32.3211 32.3211i −1.04589 1.04589i
\(956\) 0 0
\(957\) 6.12799 6.12799i 0.198090 0.198090i
\(958\) 0 0
\(959\) −11.7927 −0.380805
\(960\) 0 0
\(961\) 31.5355 1.01727
\(962\) 0 0
\(963\) −1.05400 + 1.05400i −0.0339645 + 0.0339645i
\(964\) 0 0
\(965\) −13.1659 13.1659i −0.423825 0.423825i
\(966\) 0 0
\(967\) 3.64431i 0.117193i 0.998282 + 0.0585966i \(0.0186626\pi\)
−0.998282 + 0.0585966i \(0.981337\pi\)
\(968\) 0 0
\(969\) 17.3525i 0.557443i
\(970\) 0 0
\(971\) 10.3490 + 10.3490i 0.332114 + 0.332114i 0.853389 0.521275i \(-0.174543\pi\)
−0.521275 + 0.853389i \(0.674543\pi\)
\(972\) 0 0
\(973\) −0.524016 + 0.524016i −0.0167992 + 0.0167992i
\(974\) 0 0
\(975\) 19.1567 0.613504
\(976\) 0 0
\(977\) −11.7791 −0.376846 −0.188423 0.982088i \(-0.560338\pi\)
−0.188423 + 0.982088i \(0.560338\pi\)
\(978\) 0 0
\(979\) −0.250611 + 0.250611i −0.00800956 + 0.00800956i
\(980\) 0 0
\(981\) 1.56087 + 1.56087i 0.0498347 + 0.0498347i
\(982\) 0 0
\(983\) 3.39640i 0.108328i −0.998532 0.0541642i \(-0.982751\pi\)
0.998532 0.0541642i \(-0.0172494\pi\)
\(984\) 0 0
\(985\) 57.9145i 1.84531i
\(986\) 0 0
\(987\) 2.12896 + 2.12896i 0.0677657 + 0.0677657i
\(988\) 0 0
\(989\) −5.28923 + 5.28923i −0.168188 + 0.168188i
\(990\) 0 0
\(991\) 24.0326 0.763421 0.381710 0.924282i \(-0.375335\pi\)
0.381710 + 0.924282i \(0.375335\pi\)
\(992\) 0 0
\(993\) 20.7186 0.657485
\(994\) 0 0
\(995\) 28.8272 28.8272i 0.913883 0.913883i
\(996\) 0 0
\(997\) 1.71006 + 1.71006i 0.0541583 + 0.0541583i 0.733667 0.679509i \(-0.237807\pi\)
−0.679509 + 0.733667i \(0.737807\pi\)
\(998\) 0 0
\(999\) 44.4869i 1.40750i
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 1792.2.m.e.449.6 16
4.3 odd 2 1792.2.m.g.449.3 yes 16
8.3 odd 2 1792.2.m.f.449.6 yes 16
8.5 even 2 1792.2.m.h.449.3 yes 16
16.3 odd 4 1792.2.m.g.1345.3 yes 16
16.5 even 4 1792.2.m.h.1345.3 yes 16
16.11 odd 4 1792.2.m.f.1345.6 yes 16
16.13 even 4 inner 1792.2.m.e.1345.6 yes 16
32.3 odd 8 7168.2.a.ba.1.3 8
32.13 even 8 7168.2.a.bf.1.3 8
32.19 odd 8 7168.2.a.be.1.6 8
32.29 even 8 7168.2.a.bb.1.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1792.2.m.e.449.6 16 1.1 even 1 trivial
1792.2.m.e.1345.6 yes 16 16.13 even 4 inner
1792.2.m.f.449.6 yes 16 8.3 odd 2
1792.2.m.f.1345.6 yes 16 16.11 odd 4
1792.2.m.g.449.3 yes 16 4.3 odd 2
1792.2.m.g.1345.3 yes 16 16.3 odd 4
1792.2.m.h.449.3 yes 16 8.5 even 2
1792.2.m.h.1345.3 yes 16 16.5 even 4
7168.2.a.ba.1.3 8 32.3 odd 8
7168.2.a.bb.1.6 8 32.29 even 8
7168.2.a.be.1.6 8 32.19 odd 8
7168.2.a.bf.1.3 8 32.13 even 8