Properties

Label 1380.2.f.b.829.4
Level $1380$
Weight $2$
Character 1380.829
Analytic conductor $11.019$
Analytic rank $0$
Dimension $14$
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] = [1380,2,Mod(829,1380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1380, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1380.829");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.f (of order \(2\), degree \(1\), 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: \(11.0193554789\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 27x^{12} + 283x^{10} + 1441x^{8} + 3596x^{6} + 3740x^{4} + 772x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 829.4
Root \(-2.04690i\) of defining polynomial
Character \(\chi\) \(=\) 1380.829
Dual form 1380.2.f.b.829.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +(-0.181772 + 2.22867i) q^{5} +0.189781i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +(-0.181772 + 2.22867i) q^{5} +0.189781i q^{7} -1.00000 q^{9} +4.33196 q^{11} +4.67083i q^{13} +(2.22867 + 0.181772i) q^{15} -0.0397340i q^{17} -7.19716 q^{19} +0.189781 q^{21} +1.00000i q^{23} +(-4.93392 - 0.810219i) q^{25} +1.00000i q^{27} -6.49707 q^{29} +9.17655 q^{31} -4.33196i q^{33} +(-0.422959 - 0.0344969i) q^{35} +3.03881i q^{37} +4.67083 q^{39} -8.69330 q^{41} +7.83269i q^{43} +(0.181772 - 2.22867i) q^{45} +5.03127i q^{47} +6.96398 q^{49} -0.0397340 q^{51} +6.96306i q^{53} +(-0.787429 + 9.65450i) q^{55} +7.19716i q^{57} -4.64368 q^{59} +7.86428 q^{61} -0.189781i q^{63} +(-10.4097 - 0.849027i) q^{65} +2.52865i q^{67} +1.00000 q^{69} -1.70964 q^{71} +6.29127i q^{73} +(-0.810219 + 4.93392i) q^{75} +0.822124i q^{77} +9.82056 q^{79} +1.00000 q^{81} +0.417601i q^{83} +(0.0885538 + 0.00722253i) q^{85} +6.49707i q^{87} +3.92081 q^{89} -0.886435 q^{91} -9.17655i q^{93} +(1.30824 - 16.0401i) q^{95} +0.0584885i q^{97} -4.33196 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{9} + 2 q^{15} + 4 q^{19} - 2 q^{21} - 6 q^{25} - 30 q^{29} + 6 q^{31} - 14 q^{35} + 4 q^{39} + 46 q^{41} - 20 q^{49} + 2 q^{51} - 16 q^{55} - 10 q^{59} + 64 q^{61} - 36 q^{65} + 14 q^{69} + 42 q^{71} - 16 q^{75} - 32 q^{79} + 14 q^{81} - 42 q^{85} - 52 q^{89} + 28 q^{91} - 44 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(461\) \(691\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −0.181772 + 2.22867i −0.0812909 + 0.996690i
\(6\) 0 0
\(7\) 0.189781i 0.0717305i 0.999357 + 0.0358652i \(0.0114187\pi\)
−0.999357 + 0.0358652i \(0.988581\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.33196 1.30613 0.653067 0.757300i \(-0.273482\pi\)
0.653067 + 0.757300i \(0.273482\pi\)
\(12\) 0 0
\(13\) 4.67083i 1.29546i 0.761872 + 0.647728i \(0.224281\pi\)
−0.761872 + 0.647728i \(0.775719\pi\)
\(14\) 0 0
\(15\) 2.22867 + 0.181772i 0.575439 + 0.0469333i
\(16\) 0 0
\(17\) 0.0397340i 0.00963691i −0.999988 0.00481845i \(-0.998466\pi\)
0.999988 0.00481845i \(-0.00153377\pi\)
\(18\) 0 0
\(19\) −7.19716 −1.65114 −0.825571 0.564298i \(-0.809147\pi\)
−0.825571 + 0.564298i \(0.809147\pi\)
\(20\) 0 0
\(21\) 0.189781 0.0414136
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) −4.93392 0.810219i −0.986784 0.162044i
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −6.49707 −1.20648 −0.603238 0.797561i \(-0.706123\pi\)
−0.603238 + 0.797561i \(0.706123\pi\)
\(30\) 0 0
\(31\) 9.17655 1.64816 0.824079 0.566475i \(-0.191693\pi\)
0.824079 + 0.566475i \(0.191693\pi\)
\(32\) 0 0
\(33\) 4.33196i 0.754097i
\(34\) 0 0
\(35\) −0.422959 0.0344969i −0.0714931 0.00583104i
\(36\) 0 0
\(37\) 3.03881i 0.499577i 0.968300 + 0.249788i \(0.0803611\pi\)
−0.968300 + 0.249788i \(0.919639\pi\)
\(38\) 0 0
\(39\) 4.67083 0.747932
\(40\) 0 0
\(41\) −8.69330 −1.35767 −0.678833 0.734293i \(-0.737514\pi\)
−0.678833 + 0.734293i \(0.737514\pi\)
\(42\) 0 0
\(43\) 7.83269i 1.19447i 0.802065 + 0.597237i \(0.203735\pi\)
−0.802065 + 0.597237i \(0.796265\pi\)
\(44\) 0 0
\(45\) 0.181772 2.22867i 0.0270970 0.332230i
\(46\) 0 0
\(47\) 5.03127i 0.733886i 0.930243 + 0.366943i \(0.119596\pi\)
−0.930243 + 0.366943i \(0.880404\pi\)
\(48\) 0 0
\(49\) 6.96398 0.994855
\(50\) 0 0
\(51\) −0.0397340 −0.00556387
\(52\) 0 0
\(53\) 6.96306i 0.956449i 0.878238 + 0.478225i \(0.158720\pi\)
−0.878238 + 0.478225i \(0.841280\pi\)
\(54\) 0 0
\(55\) −0.787429 + 9.65450i −0.106177 + 1.30181i
\(56\) 0 0
\(57\) 7.19716i 0.953287i
\(58\) 0 0
\(59\) −4.64368 −0.604555 −0.302278 0.953220i \(-0.597747\pi\)
−0.302278 + 0.953220i \(0.597747\pi\)
\(60\) 0 0
\(61\) 7.86428 1.00692 0.503459 0.864019i \(-0.332061\pi\)
0.503459 + 0.864019i \(0.332061\pi\)
\(62\) 0 0
\(63\) 0.189781i 0.0239102i
\(64\) 0 0
\(65\) −10.4097 0.849027i −1.29117 0.105309i
\(66\) 0 0
\(67\) 2.52865i 0.308924i 0.987999 + 0.154462i \(0.0493645\pi\)
−0.987999 + 0.154462i \(0.950636\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −1.70964 −0.202897 −0.101448 0.994841i \(-0.532348\pi\)
−0.101448 + 0.994841i \(0.532348\pi\)
\(72\) 0 0
\(73\) 6.29127i 0.736337i 0.929759 + 0.368169i \(0.120015\pi\)
−0.929759 + 0.368169i \(0.879985\pi\)
\(74\) 0 0
\(75\) −0.810219 + 4.93392i −0.0935560 + 0.569720i
\(76\) 0 0
\(77\) 0.822124i 0.0936897i
\(78\) 0 0
\(79\) 9.82056 1.10490 0.552450 0.833546i \(-0.313693\pi\)
0.552450 + 0.833546i \(0.313693\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.417601i 0.0458377i 0.999737 + 0.0229188i \(0.00729593\pi\)
−0.999737 + 0.0229188i \(0.992704\pi\)
\(84\) 0 0
\(85\) 0.0885538 + 0.00722253i 0.00960501 + 0.000783393i
\(86\) 0 0
\(87\) 6.49707i 0.696559i
\(88\) 0 0
\(89\) 3.92081 0.415605 0.207803 0.978171i \(-0.433369\pi\)
0.207803 + 0.978171i \(0.433369\pi\)
\(90\) 0 0
\(91\) −0.886435 −0.0929237
\(92\) 0 0
\(93\) 9.17655i 0.951564i
\(94\) 0 0
\(95\) 1.30824 16.0401i 0.134223 1.64568i
\(96\) 0 0
\(97\) 0.0584885i 0.00593861i 0.999996 + 0.00296930i \(0.000945160\pi\)
−0.999996 + 0.00296930i \(0.999055\pi\)
\(98\) 0 0
\(99\) −4.33196 −0.435378
\(100\) 0 0
\(101\) −6.61569 −0.658286 −0.329143 0.944280i \(-0.606760\pi\)
−0.329143 + 0.944280i \(0.606760\pi\)
\(102\) 0 0
\(103\) 16.9236i 1.66753i −0.552118 0.833766i \(-0.686180\pi\)
0.552118 0.833766i \(-0.313820\pi\)
\(104\) 0 0
\(105\) −0.0344969 + 0.422959i −0.00336655 + 0.0412766i
\(106\) 0 0
\(107\) 17.5775i 1.69928i 0.527364 + 0.849639i \(0.323180\pi\)
−0.527364 + 0.849639i \(0.676820\pi\)
\(108\) 0 0
\(109\) 0.243838 0.0233554 0.0116777 0.999932i \(-0.496283\pi\)
0.0116777 + 0.999932i \(0.496283\pi\)
\(110\) 0 0
\(111\) 3.03881 0.288431
\(112\) 0 0
\(113\) 1.22452i 0.115194i −0.998340 0.0575968i \(-0.981656\pi\)
0.998340 0.0575968i \(-0.0183438\pi\)
\(114\) 0 0
\(115\) −2.22867 0.181772i −0.207824 0.0169503i
\(116\) 0 0
\(117\) 4.67083i 0.431819i
\(118\) 0 0
\(119\) 0.00754075 0.000691260
\(120\) 0 0
\(121\) 7.76587 0.705988
\(122\) 0 0
\(123\) 8.69330i 0.783849i
\(124\) 0 0
\(125\) 2.70256 10.8488i 0.241724 0.970345i
\(126\) 0 0
\(127\) 20.6764i 1.83474i 0.398041 + 0.917368i \(0.369690\pi\)
−0.398041 + 0.917368i \(0.630310\pi\)
\(128\) 0 0
\(129\) 7.83269 0.689630
\(130\) 0 0
\(131\) 18.3253 1.60109 0.800546 0.599272i \(-0.204543\pi\)
0.800546 + 0.599272i \(0.204543\pi\)
\(132\) 0 0
\(133\) 1.36588i 0.118437i
\(134\) 0 0
\(135\) −2.22867 0.181772i −0.191813 0.0156444i
\(136\) 0 0
\(137\) 3.46899i 0.296376i −0.988959 0.148188i \(-0.952656\pi\)
0.988959 0.148188i \(-0.0473440\pi\)
\(138\) 0 0
\(139\) −6.40878 −0.543585 −0.271793 0.962356i \(-0.587617\pi\)
−0.271793 + 0.962356i \(0.587617\pi\)
\(140\) 0 0
\(141\) 5.03127 0.423709
\(142\) 0 0
\(143\) 20.2339i 1.69204i
\(144\) 0 0
\(145\) 1.18099 14.4798i 0.0980755 1.20248i
\(146\) 0 0
\(147\) 6.96398i 0.574380i
\(148\) 0 0
\(149\) −13.9989 −1.14684 −0.573418 0.819263i \(-0.694383\pi\)
−0.573418 + 0.819263i \(0.694383\pi\)
\(150\) 0 0
\(151\) 14.5758 1.18616 0.593081 0.805143i \(-0.297911\pi\)
0.593081 + 0.805143i \(0.297911\pi\)
\(152\) 0 0
\(153\) 0.0397340i 0.00321230i
\(154\) 0 0
\(155\) −1.66804 + 20.4515i −0.133980 + 1.64270i
\(156\) 0 0
\(157\) 18.9563i 1.51287i −0.654066 0.756437i \(-0.726938\pi\)
0.654066 0.756437i \(-0.273062\pi\)
\(158\) 0 0
\(159\) 6.96306 0.552206
\(160\) 0 0
\(161\) −0.189781 −0.0149568
\(162\) 0 0
\(163\) 13.2785i 1.04005i −0.854150 0.520026i \(-0.825922\pi\)
0.854150 0.520026i \(-0.174078\pi\)
\(164\) 0 0
\(165\) 9.65450 + 0.787429i 0.751601 + 0.0613013i
\(166\) 0 0
\(167\) 13.9737i 1.08132i 0.841242 + 0.540659i \(0.181825\pi\)
−0.841242 + 0.540659i \(0.818175\pi\)
\(168\) 0 0
\(169\) −8.81667 −0.678206
\(170\) 0 0
\(171\) 7.19716 0.550381
\(172\) 0 0
\(173\) 12.7917i 0.972534i −0.873810 0.486267i \(-0.838358\pi\)
0.873810 0.486267i \(-0.161642\pi\)
\(174\) 0 0
\(175\) 0.153764 0.936364i 0.0116235 0.0707825i
\(176\) 0 0
\(177\) 4.64368i 0.349040i
\(178\) 0 0
\(179\) −0.0511629 −0.00382410 −0.00191205 0.999998i \(-0.500609\pi\)
−0.00191205 + 0.999998i \(0.500609\pi\)
\(180\) 0 0
\(181\) 21.7448 1.61628 0.808141 0.588989i \(-0.200474\pi\)
0.808141 + 0.588989i \(0.200474\pi\)
\(182\) 0 0
\(183\) 7.86428i 0.581344i
\(184\) 0 0
\(185\) −6.77249 0.552370i −0.497924 0.0406111i
\(186\) 0 0
\(187\) 0.172126i 0.0125871i
\(188\) 0 0
\(189\) −0.189781 −0.0138045
\(190\) 0 0
\(191\) −20.3680 −1.47378 −0.736890 0.676013i \(-0.763706\pi\)
−0.736890 + 0.676013i \(0.763706\pi\)
\(192\) 0 0
\(193\) 15.8592i 1.14157i 0.821099 + 0.570786i \(0.193361\pi\)
−0.821099 + 0.570786i \(0.806639\pi\)
\(194\) 0 0
\(195\) −0.849027 + 10.4097i −0.0608001 + 0.745456i
\(196\) 0 0
\(197\) 6.16978i 0.439579i −0.975547 0.219789i \(-0.929463\pi\)
0.975547 0.219789i \(-0.0705370\pi\)
\(198\) 0 0
\(199\) 4.81534 0.341350 0.170675 0.985327i \(-0.445405\pi\)
0.170675 + 0.985327i \(0.445405\pi\)
\(200\) 0 0
\(201\) 2.52865 0.178358
\(202\) 0 0
\(203\) 1.23302i 0.0865411i
\(204\) 0 0
\(205\) 1.58020 19.3745i 0.110366 1.35317i
\(206\) 0 0
\(207\) 1.00000i 0.0695048i
\(208\) 0 0
\(209\) −31.1778 −2.15661
\(210\) 0 0
\(211\) −23.9874 −1.65136 −0.825679 0.564140i \(-0.809208\pi\)
−0.825679 + 0.564140i \(0.809208\pi\)
\(212\) 0 0
\(213\) 1.70964i 0.117143i
\(214\) 0 0
\(215\) −17.4565 1.42376i −1.19052 0.0970999i
\(216\) 0 0
\(217\) 1.74154i 0.118223i
\(218\) 0 0
\(219\) 6.29127 0.425125
\(220\) 0 0
\(221\) 0.185591 0.0124842
\(222\) 0 0
\(223\) 11.4215i 0.764840i −0.923989 0.382420i \(-0.875091\pi\)
0.923989 0.382420i \(-0.124909\pi\)
\(224\) 0 0
\(225\) 4.93392 + 0.810219i 0.328928 + 0.0540146i
\(226\) 0 0
\(227\) 15.7360i 1.04443i −0.852813 0.522217i \(-0.825105\pi\)
0.852813 0.522217i \(-0.174895\pi\)
\(228\) 0 0
\(229\) −3.46335 −0.228865 −0.114432 0.993431i \(-0.536505\pi\)
−0.114432 + 0.993431i \(0.536505\pi\)
\(230\) 0 0
\(231\) 0.822124 0.0540918
\(232\) 0 0
\(233\) 0.319746i 0.0209473i −0.999945 0.0104736i \(-0.996666\pi\)
0.999945 0.0104736i \(-0.00333392\pi\)
\(234\) 0 0
\(235\) −11.2130 0.914544i −0.731457 0.0596583i
\(236\) 0 0
\(237\) 9.82056i 0.637914i
\(238\) 0 0
\(239\) 2.93697 0.189977 0.0949885 0.995478i \(-0.469719\pi\)
0.0949885 + 0.995478i \(0.469719\pi\)
\(240\) 0 0
\(241\) −18.6764 −1.20305 −0.601527 0.798853i \(-0.705441\pi\)
−0.601527 + 0.798853i \(0.705441\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) −1.26586 + 15.5204i −0.0808727 + 0.991562i
\(246\) 0 0
\(247\) 33.6167i 2.13898i
\(248\) 0 0
\(249\) 0.417601 0.0264644
\(250\) 0 0
\(251\) −3.76059 −0.237366 −0.118683 0.992932i \(-0.537867\pi\)
−0.118683 + 0.992932i \(0.537867\pi\)
\(252\) 0 0
\(253\) 4.33196i 0.272348i
\(254\) 0 0
\(255\) 0.00722253 0.0885538i 0.000452292 0.00554546i
\(256\) 0 0
\(257\) 26.1810i 1.63312i −0.577259 0.816561i \(-0.695878\pi\)
0.577259 0.816561i \(-0.304122\pi\)
\(258\) 0 0
\(259\) −0.576708 −0.0358349
\(260\) 0 0
\(261\) 6.49707 0.402158
\(262\) 0 0
\(263\) 22.6844i 1.39878i −0.714739 0.699391i \(-0.753455\pi\)
0.714739 0.699391i \(-0.246545\pi\)
\(264\) 0 0
\(265\) −15.5183 1.26569i −0.953284 0.0777507i
\(266\) 0 0
\(267\) 3.92081i 0.239950i
\(268\) 0 0
\(269\) −0.570220 −0.0347669 −0.0173835 0.999849i \(-0.505534\pi\)
−0.0173835 + 0.999849i \(0.505534\pi\)
\(270\) 0 0
\(271\) 21.9506 1.33341 0.666703 0.745323i \(-0.267705\pi\)
0.666703 + 0.745323i \(0.267705\pi\)
\(272\) 0 0
\(273\) 0.886435i 0.0536495i
\(274\) 0 0
\(275\) −21.3735 3.50984i −1.28887 0.211651i
\(276\) 0 0
\(277\) 12.5514i 0.754141i 0.926185 + 0.377070i \(0.123069\pi\)
−0.926185 + 0.377070i \(0.876931\pi\)
\(278\) 0 0
\(279\) −9.17655 −0.549386
\(280\) 0 0
\(281\) 21.7956 1.30022 0.650108 0.759842i \(-0.274724\pi\)
0.650108 + 0.759842i \(0.274724\pi\)
\(282\) 0 0
\(283\) 24.0765i 1.43120i 0.698510 + 0.715601i \(0.253847\pi\)
−0.698510 + 0.715601i \(0.746153\pi\)
\(284\) 0 0
\(285\) −16.0401 1.30824i −0.950132 0.0774936i
\(286\) 0 0
\(287\) 1.64982i 0.0973860i
\(288\) 0 0
\(289\) 16.9984 0.999907
\(290\) 0 0
\(291\) 0.0584885 0.00342866
\(292\) 0 0
\(293\) 27.2212i 1.59028i −0.606427 0.795139i \(-0.707398\pi\)
0.606427 0.795139i \(-0.292602\pi\)
\(294\) 0 0
\(295\) 0.844091 10.3492i 0.0491449 0.602555i
\(296\) 0 0
\(297\) 4.33196i 0.251366i
\(298\) 0 0
\(299\) −4.67083 −0.270121
\(300\) 0 0
\(301\) −1.48650 −0.0856802
\(302\) 0 0
\(303\) 6.61569i 0.380061i
\(304\) 0 0
\(305\) −1.42951 + 17.5269i −0.0818533 + 1.00358i
\(306\) 0 0
\(307\) 6.48076i 0.369877i 0.982750 + 0.184938i \(0.0592085\pi\)
−0.982750 + 0.184938i \(0.940791\pi\)
\(308\) 0 0
\(309\) −16.9236 −0.962750
\(310\) 0 0
\(311\) 4.11457 0.233316 0.116658 0.993172i \(-0.462782\pi\)
0.116658 + 0.993172i \(0.462782\pi\)
\(312\) 0 0
\(313\) 0.760034i 0.0429597i −0.999769 0.0214798i \(-0.993162\pi\)
0.999769 0.0214798i \(-0.00683777\pi\)
\(314\) 0 0
\(315\) 0.422959 + 0.0344969i 0.0238310 + 0.00194368i
\(316\) 0 0
\(317\) 7.48188i 0.420224i 0.977677 + 0.210112i \(0.0673829\pi\)
−0.977677 + 0.210112i \(0.932617\pi\)
\(318\) 0 0
\(319\) −28.1450 −1.57582
\(320\) 0 0
\(321\) 17.5775 0.981079
\(322\) 0 0
\(323\) 0.285972i 0.0159119i
\(324\) 0 0
\(325\) 3.78440 23.0455i 0.209921 1.27833i
\(326\) 0 0
\(327\) 0.243838i 0.0134843i
\(328\) 0 0
\(329\) −0.954839 −0.0526420
\(330\) 0 0
\(331\) 28.5981 1.57189 0.785946 0.618295i \(-0.212176\pi\)
0.785946 + 0.618295i \(0.212176\pi\)
\(332\) 0 0
\(333\) 3.03881i 0.166526i
\(334\) 0 0
\(335\) −5.63553 0.459639i −0.307902 0.0251127i
\(336\) 0 0
\(337\) 22.9098i 1.24798i −0.781434 0.623988i \(-0.785512\pi\)
0.781434 0.623988i \(-0.214488\pi\)
\(338\) 0 0
\(339\) −1.22452 −0.0665071
\(340\) 0 0
\(341\) 39.7525 2.15272
\(342\) 0 0
\(343\) 2.65010i 0.143092i
\(344\) 0 0
\(345\) −0.181772 + 2.22867i −0.00978628 + 0.119987i
\(346\) 0 0
\(347\) 10.7314i 0.576090i 0.957617 + 0.288045i \(0.0930053\pi\)
−0.957617 + 0.288045i \(0.906995\pi\)
\(348\) 0 0
\(349\) 9.67523 0.517903 0.258952 0.965890i \(-0.416623\pi\)
0.258952 + 0.965890i \(0.416623\pi\)
\(350\) 0 0
\(351\) −4.67083 −0.249311
\(352\) 0 0
\(353\) 29.4875i 1.56946i −0.619838 0.784730i \(-0.712802\pi\)
0.619838 0.784730i \(-0.287198\pi\)
\(354\) 0 0
\(355\) 0.310765 3.81022i 0.0164937 0.202225i
\(356\) 0 0
\(357\) 0.00754075i 0.000399099i
\(358\) 0 0
\(359\) −4.53528 −0.239363 −0.119681 0.992812i \(-0.538187\pi\)
−0.119681 + 0.992812i \(0.538187\pi\)
\(360\) 0 0
\(361\) 32.7991 1.72627
\(362\) 0 0
\(363\) 7.76587i 0.407602i
\(364\) 0 0
\(365\) −14.0211 1.14358i −0.733900 0.0598576i
\(366\) 0 0
\(367\) 8.15285i 0.425576i −0.977098 0.212788i \(-0.931746\pi\)
0.977098 0.212788i \(-0.0682543\pi\)
\(368\) 0 0
\(369\) 8.69330 0.452555
\(370\) 0 0
\(371\) −1.32146 −0.0686066
\(372\) 0 0
\(373\) 28.9482i 1.49888i 0.662070 + 0.749442i \(0.269678\pi\)
−0.662070 + 0.749442i \(0.730322\pi\)
\(374\) 0 0
\(375\) −10.8488 2.70256i −0.560229 0.139559i
\(376\) 0 0
\(377\) 30.3467i 1.56294i
\(378\) 0 0
\(379\) −2.46596 −0.126668 −0.0633338 0.997992i \(-0.520173\pi\)
−0.0633338 + 0.997992i \(0.520173\pi\)
\(380\) 0 0
\(381\) 20.6764 1.05928
\(382\) 0 0
\(383\) 19.0579i 0.973813i −0.873454 0.486906i \(-0.838125\pi\)
0.873454 0.486906i \(-0.161875\pi\)
\(384\) 0 0
\(385\) −1.83224 0.149439i −0.0933796 0.00761612i
\(386\) 0 0
\(387\) 7.83269i 0.398158i
\(388\) 0 0
\(389\) −37.4975 −1.90120 −0.950599 0.310421i \(-0.899530\pi\)
−0.950599 + 0.310421i \(0.899530\pi\)
\(390\) 0 0
\(391\) 0.0397340 0.00200943
\(392\) 0 0
\(393\) 18.3253i 0.924391i
\(394\) 0 0
\(395\) −1.78510 + 21.8868i −0.0898183 + 1.10124i
\(396\) 0 0
\(397\) 0.171284i 0.00859652i −0.999991 0.00429826i \(-0.998632\pi\)
0.999991 0.00429826i \(-0.00136818\pi\)
\(398\) 0 0
\(399\) −1.36588 −0.0683798
\(400\) 0 0
\(401\) −22.7490 −1.13603 −0.568015 0.823018i \(-0.692289\pi\)
−0.568015 + 0.823018i \(0.692289\pi\)
\(402\) 0 0
\(403\) 42.8621i 2.13512i
\(404\) 0 0
\(405\) −0.181772 + 2.22867i −0.00903233 + 0.110743i
\(406\) 0 0
\(407\) 13.1640i 0.652515i
\(408\) 0 0
\(409\) 11.0110 0.544460 0.272230 0.962232i \(-0.412239\pi\)
0.272230 + 0.962232i \(0.412239\pi\)
\(410\) 0 0
\(411\) −3.46899 −0.171113
\(412\) 0 0
\(413\) 0.881282i 0.0433651i
\(414\) 0 0
\(415\) −0.930694 0.0759082i −0.0456860 0.00372619i
\(416\) 0 0
\(417\) 6.40878i 0.313839i
\(418\) 0 0
\(419\) 30.9587 1.51243 0.756215 0.654323i \(-0.227046\pi\)
0.756215 + 0.654323i \(0.227046\pi\)
\(420\) 0 0
\(421\) 18.0710 0.880727 0.440364 0.897819i \(-0.354850\pi\)
0.440364 + 0.897819i \(0.354850\pi\)
\(422\) 0 0
\(423\) 5.03127i 0.244629i
\(424\) 0 0
\(425\) −0.0321932 + 0.196044i −0.00156160 + 0.00950954i
\(426\) 0 0
\(427\) 1.49249i 0.0722267i
\(428\) 0 0
\(429\) 20.2339 0.976900
\(430\) 0 0
\(431\) 27.0949 1.30512 0.652558 0.757738i \(-0.273696\pi\)
0.652558 + 0.757738i \(0.273696\pi\)
\(432\) 0 0
\(433\) 35.7515i 1.71811i −0.511887 0.859053i \(-0.671053\pi\)
0.511887 0.859053i \(-0.328947\pi\)
\(434\) 0 0
\(435\) −14.4798 1.18099i −0.694254 0.0566239i
\(436\) 0 0
\(437\) 7.19716i 0.344287i
\(438\) 0 0
\(439\) 27.8294 1.32822 0.664112 0.747633i \(-0.268810\pi\)
0.664112 + 0.747633i \(0.268810\pi\)
\(440\) 0 0
\(441\) −6.96398 −0.331618
\(442\) 0 0
\(443\) 8.72908i 0.414731i 0.978264 + 0.207366i \(0.0664889\pi\)
−0.978264 + 0.207366i \(0.933511\pi\)
\(444\) 0 0
\(445\) −0.712694 + 8.73819i −0.0337849 + 0.414230i
\(446\) 0 0
\(447\) 13.9989i 0.662126i
\(448\) 0 0
\(449\) −30.6145 −1.44479 −0.722394 0.691481i \(-0.756958\pi\)
−0.722394 + 0.691481i \(0.756958\pi\)
\(450\) 0 0
\(451\) −37.6590 −1.77329
\(452\) 0 0
\(453\) 14.5758i 0.684831i
\(454\) 0 0
\(455\) 0.161129 1.97557i 0.00755385 0.0926161i
\(456\) 0 0
\(457\) 27.4809i 1.28550i −0.766074 0.642752i \(-0.777793\pi\)
0.766074 0.642752i \(-0.222207\pi\)
\(458\) 0 0
\(459\) 0.0397340 0.00185462
\(460\) 0 0
\(461\) 25.3722 1.18170 0.590850 0.806782i \(-0.298793\pi\)
0.590850 + 0.806782i \(0.298793\pi\)
\(462\) 0 0
\(463\) 32.2448i 1.49854i 0.662264 + 0.749271i \(0.269596\pi\)
−0.662264 + 0.749271i \(0.730404\pi\)
\(464\) 0 0
\(465\) 20.4515 + 1.66804i 0.948415 + 0.0773536i
\(466\) 0 0
\(467\) 0.845923i 0.0391446i 0.999808 + 0.0195723i \(0.00623046\pi\)
−0.999808 + 0.0195723i \(0.993770\pi\)
\(468\) 0 0
\(469\) −0.479891 −0.0221593
\(470\) 0 0
\(471\) −18.9563 −0.873459
\(472\) 0 0
\(473\) 33.9309i 1.56014i
\(474\) 0 0
\(475\) 35.5102 + 5.83128i 1.62932 + 0.267557i
\(476\) 0 0
\(477\) 6.96306i 0.318816i
\(478\) 0 0
\(479\) 8.01308 0.366127 0.183063 0.983101i \(-0.441399\pi\)
0.183063 + 0.983101i \(0.441399\pi\)
\(480\) 0 0
\(481\) −14.1938 −0.647180
\(482\) 0 0
\(483\) 0.189781i 0.00863534i
\(484\) 0 0
\(485\) −0.130351 0.0106316i −0.00591895 0.000482755i
\(486\) 0 0
\(487\) 17.2251i 0.780544i −0.920700 0.390272i \(-0.872381\pi\)
0.920700 0.390272i \(-0.127619\pi\)
\(488\) 0 0
\(489\) −13.2785 −0.600474
\(490\) 0 0
\(491\) −9.41331 −0.424817 −0.212408 0.977181i \(-0.568131\pi\)
−0.212408 + 0.977181i \(0.568131\pi\)
\(492\) 0 0
\(493\) 0.258154i 0.0116267i
\(494\) 0 0
\(495\) 0.787429 9.65450i 0.0353923 0.433937i
\(496\) 0 0
\(497\) 0.324457i 0.0145539i
\(498\) 0 0
\(499\) 9.17655 0.410799 0.205399 0.978678i \(-0.434151\pi\)
0.205399 + 0.978678i \(0.434151\pi\)
\(500\) 0 0
\(501\) 13.9737 0.624300
\(502\) 0 0
\(503\) 5.02474i 0.224042i −0.993706 0.112021i \(-0.964268\pi\)
0.993706 0.112021i \(-0.0357324\pi\)
\(504\) 0 0
\(505\) 1.20255 14.7442i 0.0535127 0.656107i
\(506\) 0 0
\(507\) 8.81667i 0.391562i
\(508\) 0 0
\(509\) 14.3326 0.635282 0.317641 0.948211i \(-0.397109\pi\)
0.317641 + 0.948211i \(0.397109\pi\)
\(510\) 0 0
\(511\) −1.19396 −0.0528178
\(512\) 0 0
\(513\) 7.19716i 0.317762i
\(514\) 0 0
\(515\) 37.7171 + 3.07624i 1.66201 + 0.135555i
\(516\) 0 0
\(517\) 21.7952i 0.958554i
\(518\) 0 0
\(519\) −12.7917 −0.561493
\(520\) 0 0
\(521\) −25.4727 −1.11598 −0.557991 0.829847i \(-0.688427\pi\)
−0.557991 + 0.829847i \(0.688427\pi\)
\(522\) 0 0
\(523\) 15.7464i 0.688543i 0.938870 + 0.344271i \(0.111874\pi\)
−0.938870 + 0.344271i \(0.888126\pi\)
\(524\) 0 0
\(525\) −0.936364 0.153764i −0.0408663 0.00671082i
\(526\) 0 0
\(527\) 0.364621i 0.0158831i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 4.64368 0.201518
\(532\) 0 0
\(533\) 40.6050i 1.75880i
\(534\) 0 0
\(535\) −39.1743 3.19509i −1.69365 0.138136i
\(536\) 0 0
\(537\) 0.0511629i 0.00220784i
\(538\) 0 0
\(539\) 30.1677 1.29941
\(540\) 0 0
\(541\) −5.58372 −0.240063 −0.120031 0.992770i \(-0.538300\pi\)
−0.120031 + 0.992770i \(0.538300\pi\)
\(542\) 0 0
\(543\) 21.7448i 0.933161i
\(544\) 0 0
\(545\) −0.0443229 + 0.543434i −0.00189859 + 0.0232781i
\(546\) 0 0
\(547\) 13.9544i 0.596648i −0.954465 0.298324i \(-0.903572\pi\)
0.954465 0.298324i \(-0.0964276\pi\)
\(548\) 0 0
\(549\) −7.86428 −0.335639
\(550\) 0 0
\(551\) 46.7605 1.99206
\(552\) 0 0
\(553\) 1.86376i 0.0792550i
\(554\) 0 0
\(555\) −0.552370 + 6.77249i −0.0234468 + 0.287476i
\(556\) 0 0
\(557\) 30.6343i 1.29802i 0.760781 + 0.649009i \(0.224816\pi\)
−0.760781 + 0.649009i \(0.775184\pi\)
\(558\) 0 0
\(559\) −36.5852 −1.54739
\(560\) 0 0
\(561\) −0.172126 −0.00726716
\(562\) 0 0
\(563\) 40.9163i 1.72442i −0.506553 0.862209i \(-0.669080\pi\)
0.506553 0.862209i \(-0.330920\pi\)
\(564\) 0 0
\(565\) 2.72906 + 0.222584i 0.114812 + 0.00936420i
\(566\) 0 0
\(567\) 0.189781i 0.00797005i
\(568\) 0 0
\(569\) −17.2131 −0.721612 −0.360806 0.932641i \(-0.617498\pi\)
−0.360806 + 0.932641i \(0.617498\pi\)
\(570\) 0 0
\(571\) −14.7070 −0.615469 −0.307734 0.951472i \(-0.599571\pi\)
−0.307734 + 0.951472i \(0.599571\pi\)
\(572\) 0 0
\(573\) 20.3680i 0.850887i
\(574\) 0 0
\(575\) 0.810219 4.93392i 0.0337885 0.205759i
\(576\) 0 0
\(577\) 15.7617i 0.656168i 0.944648 + 0.328084i \(0.106403\pi\)
−0.944648 + 0.328084i \(0.893597\pi\)
\(578\) 0 0
\(579\) 15.8592 0.659086
\(580\) 0 0
\(581\) −0.0792528 −0.00328796
\(582\) 0 0
\(583\) 30.1637i 1.24925i
\(584\) 0 0
\(585\) 10.4097 + 0.849027i 0.430389 + 0.0351029i
\(586\) 0 0
\(587\) 40.4757i 1.67061i 0.549787 + 0.835305i \(0.314709\pi\)
−0.549787 + 0.835305i \(0.685291\pi\)
\(588\) 0 0
\(589\) −66.0451 −2.72134
\(590\) 0 0
\(591\) −6.16978 −0.253791
\(592\) 0 0
\(593\) 35.8541i 1.47235i 0.676791 + 0.736175i \(0.263370\pi\)
−0.676791 + 0.736175i \(0.736630\pi\)
\(594\) 0 0
\(595\) −0.00137070 + 0.0168058i −5.61932e−5 + 0.000688972i
\(596\) 0 0
\(597\) 4.81534i 0.197079i
\(598\) 0 0
\(599\) 21.0263 0.859112 0.429556 0.903040i \(-0.358670\pi\)
0.429556 + 0.903040i \(0.358670\pi\)
\(600\) 0 0
\(601\) 6.16524 0.251485 0.125743 0.992063i \(-0.459869\pi\)
0.125743 + 0.992063i \(0.459869\pi\)
\(602\) 0 0
\(603\) 2.52865i 0.102975i
\(604\) 0 0
\(605\) −1.41162 + 17.3075i −0.0573904 + 0.703651i
\(606\) 0 0
\(607\) 27.5809i 1.11947i −0.828671 0.559736i \(-0.810903\pi\)
0.828671 0.559736i \(-0.189097\pi\)
\(608\) 0 0
\(609\) −1.23302 −0.0499645
\(610\) 0 0
\(611\) −23.5002 −0.950716
\(612\) 0 0
\(613\) 21.7384i 0.878005i 0.898486 + 0.439003i \(0.144668\pi\)
−0.898486 + 0.439003i \(0.855332\pi\)
\(614\) 0 0
\(615\) −19.3745 1.58020i −0.781255 0.0637198i
\(616\) 0 0
\(617\) 25.2023i 1.01461i 0.861767 + 0.507304i \(0.169358\pi\)
−0.861767 + 0.507304i \(0.830642\pi\)
\(618\) 0 0
\(619\) −6.45053 −0.259269 −0.129634 0.991562i \(-0.541380\pi\)
−0.129634 + 0.991562i \(0.541380\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 0.744096i 0.0298116i
\(624\) 0 0
\(625\) 23.6871 + 7.99511i 0.947484 + 0.319804i
\(626\) 0 0
\(627\) 31.1778i 1.24512i
\(628\) 0 0
\(629\) 0.120744 0.00481438
\(630\) 0 0
\(631\) −24.0614 −0.957871 −0.478935 0.877850i \(-0.658977\pi\)
−0.478935 + 0.877850i \(0.658977\pi\)
\(632\) 0 0
\(633\) 23.9874i 0.953412i
\(634\) 0 0
\(635\) −46.0809 3.75839i −1.82866 0.149147i
\(636\) 0 0
\(637\) 32.5276i 1.28879i
\(638\) 0 0
\(639\) 1.70964 0.0676323
\(640\) 0 0
\(641\) 0.0204978 0.000809615 0.000404807 1.00000i \(-0.499871\pi\)
0.000404807 1.00000i \(0.499871\pi\)
\(642\) 0 0
\(643\) 46.4037i 1.82998i 0.403471 + 0.914992i \(0.367804\pi\)
−0.403471 + 0.914992i \(0.632196\pi\)
\(644\) 0 0
\(645\) −1.42376 + 17.4565i −0.0560607 + 0.687348i
\(646\) 0 0
\(647\) 16.8386i 0.661992i −0.943632 0.330996i \(-0.892615\pi\)
0.943632 0.330996i \(-0.107385\pi\)
\(648\) 0 0
\(649\) −20.1162 −0.789631
\(650\) 0 0
\(651\) 1.74154 0.0682562
\(652\) 0 0
\(653\) 40.9087i 1.60088i 0.599413 + 0.800440i \(0.295401\pi\)
−0.599413 + 0.800440i \(0.704599\pi\)
\(654\) 0 0
\(655\) −3.33103 + 40.8411i −0.130154 + 1.59579i
\(656\) 0 0
\(657\) 6.29127i 0.245446i
\(658\) 0 0
\(659\) 38.3970 1.49573 0.747867 0.663849i \(-0.231078\pi\)
0.747867 + 0.663849i \(0.231078\pi\)
\(660\) 0 0
\(661\) −10.8022 −0.420155 −0.210078 0.977685i \(-0.567372\pi\)
−0.210078 + 0.977685i \(0.567372\pi\)
\(662\) 0 0
\(663\) 0.185591i 0.00720775i
\(664\) 0 0
\(665\) 3.04410 + 0.248280i 0.118045 + 0.00962787i
\(666\) 0 0
\(667\) 6.49707i 0.251568i
\(668\) 0 0
\(669\) −11.4215 −0.441581
\(670\) 0 0
\(671\) 34.0677 1.31517
\(672\) 0 0
\(673\) 11.3951i 0.439247i −0.975585 0.219624i \(-0.929517\pi\)
0.975585 0.219624i \(-0.0704829\pi\)
\(674\) 0 0
\(675\) 0.810219 4.93392i 0.0311853 0.189907i
\(676\) 0 0
\(677\) 3.87400i 0.148890i 0.997225 + 0.0744449i \(0.0237185\pi\)
−0.997225 + 0.0744449i \(0.976282\pi\)
\(678\) 0 0
\(679\) −0.0111000 −0.000425979
\(680\) 0 0
\(681\) −15.7360 −0.603004
\(682\) 0 0
\(683\) 27.9517i 1.06954i −0.844997 0.534770i \(-0.820398\pi\)
0.844997 0.534770i \(-0.179602\pi\)
\(684\) 0 0
\(685\) 7.73122 + 0.630565i 0.295395 + 0.0240927i
\(686\) 0 0
\(687\) 3.46335i 0.132135i
\(688\) 0 0
\(689\) −32.5233 −1.23904
\(690\) 0 0
\(691\) 35.1448 1.33697 0.668485 0.743726i \(-0.266943\pi\)
0.668485 + 0.743726i \(0.266943\pi\)
\(692\) 0 0
\(693\) 0.822124i 0.0312299i
\(694\) 0 0
\(695\) 1.16494 14.2830i 0.0441886 0.541786i
\(696\) 0 0
\(697\) 0.345420i 0.0130837i
\(698\) 0 0
\(699\) −0.319746 −0.0120939
\(700\) 0 0
\(701\) −7.89656 −0.298249 −0.149124 0.988818i \(-0.547645\pi\)
−0.149124 + 0.988818i \(0.547645\pi\)
\(702\) 0 0
\(703\) 21.8708i 0.824872i
\(704\) 0 0
\(705\) −0.914544 + 11.2130i −0.0344437 + 0.422307i
\(706\) 0 0
\(707\) 1.25553i 0.0472191i
\(708\) 0 0
\(709\) 16.3686 0.614734 0.307367 0.951591i \(-0.400552\pi\)
0.307367 + 0.951591i \(0.400552\pi\)
\(710\) 0 0
\(711\) −9.82056 −0.368300
\(712\) 0 0
\(713\) 9.17655i 0.343665i
\(714\) 0 0
\(715\) −45.0945 3.67795i −1.68644 0.137547i
\(716\) 0 0
\(717\) 2.93697i 0.109683i
\(718\) 0 0
\(719\) −12.0861 −0.450737 −0.225368 0.974274i \(-0.572359\pi\)
−0.225368 + 0.974274i \(0.572359\pi\)
\(720\) 0 0
\(721\) 3.21178 0.119613
\(722\) 0 0
\(723\) 18.6764i 0.694583i
\(724\) 0 0
\(725\) 32.0560 + 5.26405i 1.19053 + 0.195502i
\(726\) 0 0
\(727\) 7.89415i 0.292778i −0.989227 0.146389i \(-0.953235\pi\)
0.989227 0.146389i \(-0.0467651\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0.311224 0.0115110
\(732\) 0 0
\(733\) 17.1444i 0.633242i −0.948552 0.316621i \(-0.897452\pi\)
0.948552 0.316621i \(-0.102548\pi\)
\(734\) 0 0
\(735\) 15.5204 + 1.26586i 0.572479 + 0.0466919i
\(736\) 0 0
\(737\) 10.9540i 0.403497i
\(738\) 0 0
\(739\) 43.1409 1.58696 0.793482 0.608593i \(-0.208266\pi\)
0.793482 + 0.608593i \(0.208266\pi\)
\(740\) 0 0
\(741\) −33.6167 −1.23494
\(742\) 0 0
\(743\) 41.0320i 1.50532i −0.658410 0.752660i \(-0.728771\pi\)
0.658410 0.752660i \(-0.271229\pi\)
\(744\) 0 0
\(745\) 2.54461 31.1989i 0.0932273 1.14304i
\(746\) 0 0
\(747\) 0.417601i 0.0152792i
\(748\) 0 0
\(749\) −3.33587 −0.121890
\(750\) 0 0
\(751\) 32.4190 1.18299 0.591494 0.806309i \(-0.298538\pi\)
0.591494 + 0.806309i \(0.298538\pi\)
\(752\) 0 0
\(753\) 3.76059i 0.137043i
\(754\) 0 0
\(755\) −2.64947 + 32.4846i −0.0964242 + 1.18224i
\(756\) 0 0
\(757\) 31.1373i 1.13170i −0.824507 0.565852i \(-0.808547\pi\)
0.824507 0.565852i \(-0.191453\pi\)
\(758\) 0 0
\(759\) 4.33196 0.157240
\(760\) 0 0
\(761\) −43.4836 −1.57628 −0.788139 0.615497i \(-0.788955\pi\)
−0.788139 + 0.615497i \(0.788955\pi\)
\(762\) 0 0
\(763\) 0.0462758i 0.00167530i
\(764\) 0 0
\(765\) −0.0885538 0.00722253i −0.00320167 0.000261131i
\(766\) 0 0
\(767\) 21.6898i 0.783175i
\(768\) 0 0
\(769\) 4.35659 0.157103 0.0785514 0.996910i \(-0.474971\pi\)
0.0785514 + 0.996910i \(0.474971\pi\)
\(770\) 0 0
\(771\) −26.1810 −0.942884
\(772\) 0 0
\(773\) 52.3629i 1.88336i 0.336508 + 0.941681i \(0.390754\pi\)
−0.336508 + 0.941681i \(0.609246\pi\)
\(774\) 0 0
\(775\) −45.2764 7.43502i −1.62637 0.267074i
\(776\) 0 0
\(777\) 0.576708i 0.0206893i
\(778\) 0 0
\(779\) 62.5671 2.24170
\(780\) 0 0
\(781\) −7.40609 −0.265011
\(782\) 0 0
\(783\) 6.49707i 0.232186i
\(784\) 0 0
\(785\) 42.2472 + 3.44572i 1.50787 + 0.122983i
\(786\) 0 0
\(787\) 15.0229i 0.535507i 0.963487 + 0.267754i \(0.0862813\pi\)
−0.963487 + 0.267754i \(0.913719\pi\)
\(788\) 0 0
\(789\) −22.6844 −0.807587
\(790\) 0 0
\(791\) 0.232392 0.00826289
\(792\) 0 0
\(793\) 36.7327i 1.30442i
\(794\) 0 0
\(795\) −1.26569 + 15.5183i −0.0448894 + 0.550379i
\(796\) 0 0
\(797\) 31.3217i 1.10947i −0.832026 0.554737i \(-0.812819\pi\)
0.832026 0.554737i \(-0.187181\pi\)
\(798\) 0 0
\(799\) 0.199912 0.00707239
\(800\) 0 0
\(801\) −3.92081 −0.138535
\(802\) 0 0
\(803\) 27.2535i 0.961756i
\(804\) 0 0
\(805\) 0.0344969 0.422959i 0.00121586 0.0149073i
\(806\) 0 0
\(807\) 0.570220i 0.0200727i
\(808\) 0 0
\(809\) 22.0261 0.774397 0.387198 0.921996i \(-0.373443\pi\)
0.387198 + 0.921996i \(0.373443\pi\)
\(810\) 0 0
\(811\) 22.2449 0.781125 0.390563 0.920576i \(-0.372281\pi\)
0.390563 + 0.920576i \(0.372281\pi\)
\(812\) 0 0
\(813\) 21.9506i 0.769843i
\(814\) 0 0
\(815\) 29.5933 + 2.41366i 1.03661 + 0.0845468i
\(816\) 0 0
\(817\) 56.3731i 1.97225i
\(818\) 0 0
\(819\) 0.886435 0.0309746
\(820\) 0 0
\(821\) 55.8300 1.94848 0.974240 0.225513i \(-0.0724058\pi\)
0.974240 + 0.225513i \(0.0724058\pi\)
\(822\) 0 0
\(823\) 8.57848i 0.299027i −0.988760 0.149514i \(-0.952229\pi\)
0.988760 0.149514i \(-0.0477708\pi\)
\(824\) 0 0
\(825\) −3.50984 + 21.3735i −0.122197 + 0.744131i
\(826\) 0 0
\(827\) 12.4646i 0.433437i 0.976234 + 0.216718i \(0.0695354\pi\)
−0.976234 + 0.216718i \(0.930465\pi\)
\(828\) 0 0
\(829\) 5.83650 0.202710 0.101355 0.994850i \(-0.467682\pi\)
0.101355 + 0.994850i \(0.467682\pi\)
\(830\) 0 0
\(831\) 12.5514 0.435403
\(832\) 0 0
\(833\) 0.276707i 0.00958732i
\(834\) 0 0
\(835\) −31.1428 2.54003i −1.07774 0.0879014i
\(836\) 0 0
\(837\) 9.17655i 0.317188i
\(838\) 0 0
\(839\) 21.9925 0.759265 0.379633 0.925137i \(-0.376050\pi\)
0.379633 + 0.925137i \(0.376050\pi\)
\(840\) 0 0
\(841\) 13.2119 0.455583
\(842\) 0 0
\(843\) 21.7956i 0.750680i
\(844\) 0 0
\(845\) 1.60262 19.6494i 0.0551320 0.675961i
\(846\) 0 0
\(847\) 1.47381i 0.0506409i
\(848\) 0 0
\(849\) 24.0765 0.826304
\(850\) 0 0
\(851\) −3.03881 −0.104169
\(852\) 0 0
\(853\) 30.7842i 1.05403i 0.849856 + 0.527015i \(0.176689\pi\)
−0.849856 + 0.527015i \(0.823311\pi\)
\(854\) 0 0
\(855\) −1.30824 + 16.0401i −0.0447410 + 0.548559i
\(856\) 0 0
\(857\) 15.0207i 0.513098i 0.966531 + 0.256549i \(0.0825855\pi\)
−0.966531 + 0.256549i \(0.917414\pi\)
\(858\) 0 0
\(859\) −0.352932 −0.0120419 −0.00602094 0.999982i \(-0.501917\pi\)
−0.00602094 + 0.999982i \(0.501917\pi\)
\(860\) 0 0
\(861\) −1.64982 −0.0562259
\(862\) 0 0
\(863\) 8.55180i 0.291107i −0.989350 0.145553i \(-0.953504\pi\)
0.989350 0.145553i \(-0.0464962\pi\)
\(864\) 0 0
\(865\) 28.5084 + 2.32517i 0.969315 + 0.0790582i
\(866\) 0 0
\(867\) 16.9984i 0.577297i
\(868\) 0 0
\(869\) 42.5423 1.44315
\(870\) 0 0
\(871\) −11.8109 −0.400198
\(872\) 0 0
\(873\) 0.0584885i 0.00197954i
\(874\) 0 0
\(875\) 2.05889 + 0.512894i 0.0696033 + 0.0173390i
\(876\) 0 0
\(877\) 46.9633i 1.58584i 0.609327 + 0.792919i \(0.291440\pi\)
−0.609327 + 0.792919i \(0.708560\pi\)
\(878\) 0 0
\(879\) −27.2212 −0.918147
\(880\) 0 0
\(881\) −47.9481 −1.61541 −0.807706 0.589585i \(-0.799291\pi\)
−0.807706 + 0.589585i \(0.799291\pi\)
\(882\) 0 0
\(883\) 49.6286i 1.67014i 0.550147 + 0.835068i \(0.314572\pi\)
−0.550147 + 0.835068i \(0.685428\pi\)
\(884\) 0 0
\(885\) −10.3492 0.844091i −0.347885 0.0283738i
\(886\) 0 0
\(887\) 39.1853i 1.31572i 0.753142 + 0.657858i \(0.228537\pi\)
−0.753142 + 0.657858i \(0.771463\pi\)
\(888\) 0 0
\(889\) −3.92399 −0.131606
\(890\) 0 0
\(891\) 4.33196 0.145126
\(892\) 0 0
\(893\) 36.2108i 1.21175i
\(894\) 0 0
\(895\) 0.00929999 0.114025i 0.000310865 0.00381144i
\(896\) 0 0
\(897\) 4.67083i 0.155955i
\(898\) 0 0
\(899\) −59.6207 −1.98846
\(900\) 0 0
\(901\) 0.276670 0.00921721
\(902\) 0 0
\(903\) 1.48650i 0.0494675i
\(904\) 0 0
\(905\) −3.95261 + 48.4620i −0.131389 + 1.61093i
\(906\) 0 0
\(907\) 39.3323i 1.30601i −0.757355 0.653004i \(-0.773509\pi\)
0.757355 0.653004i \(-0.226491\pi\)
\(908\) 0 0
\(909\) 6.61569 0.219429
\(910\) 0 0
\(911\) −17.8676 −0.591980 −0.295990 0.955191i \(-0.595649\pi\)
−0.295990 + 0.955191i \(0.595649\pi\)
\(912\) 0 0
\(913\) 1.80903i 0.0598702i
\(914\) 0 0
\(915\) 17.5269 + 1.42951i 0.579420 + 0.0472580i
\(916\) 0 0
\(917\) 3.47780i 0.114847i
\(918\) 0 0
\(919\) 20.0858 0.662570 0.331285 0.943531i \(-0.392518\pi\)
0.331285 + 0.943531i \(0.392518\pi\)
\(920\) 0 0
\(921\) 6.48076 0.213548
\(922\) 0 0
\(923\) 7.98544i 0.262844i
\(924\) 0 0
\(925\) 2.46210 14.9932i 0.0809533 0.492974i
\(926\) 0 0
\(927\) 16.9236i 0.555844i
\(928\) 0 0
\(929\) 56.6085 1.85727 0.928633 0.371001i \(-0.120985\pi\)
0.928633 + 0.371001i \(0.120985\pi\)
\(930\) 0 0
\(931\) −50.1209 −1.64265
\(932\) 0 0
\(933\) 4.11457i 0.134705i
\(934\) 0 0
\(935\) 0.383612 + 0.0312877i 0.0125454 + 0.00102322i
\(936\) 0 0
\(937\) 48.1869i 1.57420i 0.616826 + 0.787099i \(0.288418\pi\)
−0.616826 + 0.787099i \(0.711582\pi\)
\(938\) 0 0
\(939\) −0.760034 −0.0248028
\(940\) 0 0
\(941\) −10.7084 −0.349083 −0.174542 0.984650i \(-0.555844\pi\)
−0.174542 + 0.984650i \(0.555844\pi\)
\(942\) 0 0
\(943\) 8.69330i 0.283093i
\(944\) 0 0
\(945\) 0.0344969 0.422959i 0.00112218 0.0137589i
\(946\) 0 0
\(947\) 17.5646i 0.570774i 0.958412 + 0.285387i \(0.0921221\pi\)
−0.958412 + 0.285387i \(0.907878\pi\)
\(948\) 0 0
\(949\) −29.3855 −0.953893
\(950\) 0 0
\(951\) 7.48188 0.242617
\(952\) 0 0
\(953\) 36.3058i 1.17606i 0.808839 + 0.588031i \(0.200097\pi\)
−0.808839 + 0.588031i \(0.799903\pi\)
\(954\) 0 0
\(955\) 3.70234 45.3936i 0.119805 1.46890i
\(956\) 0 0
\(957\) 28.1450i 0.909800i
\(958\) 0 0
\(959\) 0.658348 0.0212592
\(960\) 0 0
\(961\) 53.2091 1.71642
\(962\) 0 0
\(963\) 17.5775i 0.566426i
\(964\) 0 0
\(965\) −35.3449 2.88276i −1.13779 0.0927994i
\(966\) 0 0
\(967\) 30.4808i 0.980195i −0.871668 0.490098i \(-0.836961\pi\)
0.871668 0.490098i \(-0.163039\pi\)
\(968\) 0 0
\(969\) 0.285972 0.00918674
\(970\) 0 0
\(971\) 43.2929 1.38934 0.694668 0.719330i \(-0.255551\pi\)
0.694668 + 0.719330i \(0.255551\pi\)
\(972\) 0 0
\(973\) 1.21626i 0.0389916i
\(974\) 0 0
\(975\) −23.0455 3.78440i −0.738047 0.121198i
\(976\) 0 0
\(977\) 6.01624i 0.192477i −0.995358 0.0962383i \(-0.969319\pi\)
0.995358 0.0962383i \(-0.0306811\pi\)
\(978\) 0 0
\(979\) 16.9848 0.542836
\(980\) 0 0
\(981\) −0.243838 −0.00778515
\(982\) 0 0
\(983\) 36.0552i 1.14998i 0.818159 + 0.574992i \(0.194995\pi\)
−0.818159 + 0.574992i \(0.805005\pi\)
\(984\) 0 0
\(985\) 13.7504 + 1.12149i 0.438124 + 0.0357338i
\(986\) 0 0
\(987\) 0.954839i 0.0303929i
\(988\) 0 0
\(989\) −7.83269 −0.249065
\(990\) 0 0
\(991\) −51.5584 −1.63781 −0.818904 0.573931i \(-0.805418\pi\)
−0.818904 + 0.573931i \(0.805418\pi\)
\(992\) 0 0
\(993\) 28.5981i 0.907532i
\(994\) 0 0
\(995\) −0.875293 + 10.7318i −0.0277487 + 0.340220i
\(996\) 0 0
\(997\) 18.9389i 0.599799i 0.953971 + 0.299900i \(0.0969532\pi\)
−0.953971 + 0.299900i \(0.903047\pi\)
\(998\) 0 0
\(999\) −3.03881 −0.0961436
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 1380.2.f.b.829.4 14
3.2 odd 2 4140.2.f.c.829.7 14
5.2 odd 4 6900.2.a.bc.1.4 7
5.3 odd 4 6900.2.a.bd.1.4 7
5.4 even 2 inner 1380.2.f.b.829.11 yes 14
15.14 odd 2 4140.2.f.c.829.8 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.f.b.829.4 14 1.1 even 1 trivial
1380.2.f.b.829.11 yes 14 5.4 even 2 inner
4140.2.f.c.829.7 14 3.2 odd 2
4140.2.f.c.829.8 14 15.14 odd 2
6900.2.a.bc.1.4 7 5.2 odd 4
6900.2.a.bd.1.4 7 5.3 odd 4