Properties

Label 1380.2.i.a.1241.1
Level $1380$
Weight $2$
Character 1380.1241
Analytic conductor $11.019$
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] = [1380,2,Mod(1241,1380)]
 
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, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1380.1241");
 
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.i (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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{14} - 4 x^{13} - 2 x^{12} + 12 x^{11} + 17 x^{10} + 16 x^{9} - 110 x^{8} + 48 x^{7} + 153 x^{6} + 324 x^{5} - 162 x^{4} - 972 x^{3} - 729 x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1241.1
Root \(1.68259 - 0.410981i\) of defining polynomial
Character \(\chi\) \(=\) 1380.1241
Dual form 1380.2.i.a.1241.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.68259 - 0.410981i) q^{3} -1.00000 q^{5} -4.25920i q^{7} +(2.66219 + 1.38302i) q^{9} +O(q^{10})\) \(q+(-1.68259 - 0.410981i) q^{3} -1.00000 q^{5} -4.25920i q^{7} +(2.66219 + 1.38302i) q^{9} -1.77042 q^{11} +4.14699 q^{13} +(1.68259 + 0.410981i) q^{15} +6.86944 q^{17} -1.46552i q^{19} +(-1.75045 + 7.16646i) q^{21} +(-1.73005 + 4.47291i) q^{23} +1.00000 q^{25} +(-3.91096 - 3.42116i) q^{27} -4.10332i q^{29} -5.05868 q^{31} +(2.97888 + 0.727608i) q^{33} +4.25920i q^{35} -2.88245i q^{37} +(-6.97767 - 1.70434i) q^{39} -4.56378i q^{41} -3.27999i q^{43} +(-2.66219 - 1.38302i) q^{45} +5.80230i q^{47} -11.1408 q^{49} +(-11.5584 - 2.82321i) q^{51} +0.831510 q^{53} +1.77042 q^{55} +(-0.602301 + 2.46586i) q^{57} -11.8329i q^{59} -12.0164i q^{61} +(5.89057 - 11.3388i) q^{63} -4.14699 q^{65} -10.3996i q^{67} +(4.74925 - 6.81503i) q^{69} +16.3046i q^{71} -1.69014 q^{73} +(-1.68259 - 0.410981i) q^{75} +7.54056i q^{77} -2.62553i q^{79} +(5.17450 + 7.36373i) q^{81} -5.22163 q^{83} -6.86944 q^{85} +(-1.68639 + 6.90419i) q^{87} -9.78113 q^{89} -17.6629i q^{91} +(8.51167 + 2.07902i) q^{93} +1.46552i q^{95} +9.26880i q^{97} +(-4.71318 - 2.44853i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{5} + 2 q^{9} - 4 q^{21} + 10 q^{23} + 16 q^{25} - 12 q^{27} + 4 q^{31} + 2 q^{33} - 14 q^{39} - 2 q^{45} + 8 q^{49} - 2 q^{51} + 4 q^{53} - 2 q^{57} + 30 q^{63} - 4 q^{73} + 10 q^{81} - 4 q^{83} - 10 q^{87} - 28 q^{89} - 10 q^{93} - 26 q^{99}+O(q^{100}) \) 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
\(3\) −1.68259 0.410981i −0.971441 0.237280i
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.25920i 1.60983i −0.593393 0.804913i \(-0.702212\pi\)
0.593393 0.804913i \(-0.297788\pi\)
\(8\) 0 0
\(9\) 2.66219 + 1.38302i 0.887396 + 0.461008i
\(10\) 0 0
\(11\) −1.77042 −0.533801 −0.266900 0.963724i \(-0.585999\pi\)
−0.266900 + 0.963724i \(0.585999\pi\)
\(12\) 0 0
\(13\) 4.14699 1.15017 0.575085 0.818094i \(-0.304969\pi\)
0.575085 + 0.818094i \(0.304969\pi\)
\(14\) 0 0
\(15\) 1.68259 + 0.410981i 0.434442 + 0.106115i
\(16\) 0 0
\(17\) 6.86944 1.66609 0.833043 0.553209i \(-0.186597\pi\)
0.833043 + 0.553209i \(0.186597\pi\)
\(18\) 0 0
\(19\) 1.46552i 0.336213i −0.985769 0.168106i \(-0.946235\pi\)
0.985769 0.168106i \(-0.0537652\pi\)
\(20\) 0 0
\(21\) −1.75045 + 7.16646i −0.381980 + 1.56385i
\(22\) 0 0
\(23\) −1.73005 + 4.47291i −0.360741 + 0.932666i
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.91096 3.42116i −0.752665 0.658403i
\(28\) 0 0
\(29\) 4.10332i 0.761968i −0.924582 0.380984i \(-0.875585\pi\)
0.924582 0.380984i \(-0.124415\pi\)
\(30\) 0 0
\(31\) −5.05868 −0.908566 −0.454283 0.890857i \(-0.650105\pi\)
−0.454283 + 0.890857i \(0.650105\pi\)
\(32\) 0 0
\(33\) 2.97888 + 0.727608i 0.518556 + 0.126660i
\(34\) 0 0
\(35\) 4.25920i 0.719936i
\(36\) 0 0
\(37\) 2.88245i 0.473873i −0.971525 0.236936i \(-0.923857\pi\)
0.971525 0.236936i \(-0.0761433\pi\)
\(38\) 0 0
\(39\) −6.97767 1.70434i −1.11732 0.272912i
\(40\) 0 0
\(41\) 4.56378i 0.712742i −0.934345 0.356371i \(-0.884014\pi\)
0.934345 0.356371i \(-0.115986\pi\)
\(42\) 0 0
\(43\) 3.27999i 0.500193i −0.968221 0.250097i \(-0.919538\pi\)
0.968221 0.250097i \(-0.0804624\pi\)
\(44\) 0 0
\(45\) −2.66219 1.38302i −0.396856 0.206169i
\(46\) 0 0
\(47\) 5.80230i 0.846353i 0.906047 + 0.423177i \(0.139085\pi\)
−0.906047 + 0.423177i \(0.860915\pi\)
\(48\) 0 0
\(49\) −11.1408 −1.59154
\(50\) 0 0
\(51\) −11.5584 2.82321i −1.61850 0.395329i
\(52\) 0 0
\(53\) 0.831510 0.114217 0.0571083 0.998368i \(-0.481812\pi\)
0.0571083 + 0.998368i \(0.481812\pi\)
\(54\) 0 0
\(55\) 1.77042 0.238723
\(56\) 0 0
\(57\) −0.602301 + 2.46586i −0.0797767 + 0.326611i
\(58\) 0 0
\(59\) 11.8329i 1.54052i −0.637732 0.770258i \(-0.720127\pi\)
0.637732 0.770258i \(-0.279873\pi\)
\(60\) 0 0
\(61\) 12.0164i 1.53855i −0.638919 0.769274i \(-0.720618\pi\)
0.638919 0.769274i \(-0.279382\pi\)
\(62\) 0 0
\(63\) 5.89057 11.3388i 0.742142 1.42855i
\(64\) 0 0
\(65\) −4.14699 −0.514371
\(66\) 0 0
\(67\) 10.3996i 1.27051i −0.772303 0.635254i \(-0.780895\pi\)
0.772303 0.635254i \(-0.219105\pi\)
\(68\) 0 0
\(69\) 4.74925 6.81503i 0.571742 0.820433i
\(70\) 0 0
\(71\) 16.3046i 1.93501i 0.252861 + 0.967503i \(0.418629\pi\)
−0.252861 + 0.967503i \(0.581371\pi\)
\(72\) 0 0
\(73\) −1.69014 −0.197816 −0.0989080 0.995097i \(-0.531535\pi\)
−0.0989080 + 0.995097i \(0.531535\pi\)
\(74\) 0 0
\(75\) −1.68259 0.410981i −0.194288 0.0474560i
\(76\) 0 0
\(77\) 7.54056i 0.859326i
\(78\) 0 0
\(79\) 2.62553i 0.295395i −0.989033 0.147698i \(-0.952814\pi\)
0.989033 0.147698i \(-0.0471863\pi\)
\(80\) 0 0
\(81\) 5.17450 + 7.36373i 0.574944 + 0.818193i
\(82\) 0 0
\(83\) −5.22163 −0.573149 −0.286574 0.958058i \(-0.592517\pi\)
−0.286574 + 0.958058i \(0.592517\pi\)
\(84\) 0 0
\(85\) −6.86944 −0.745096
\(86\) 0 0
\(87\) −1.68639 + 6.90419i −0.180800 + 0.740207i
\(88\) 0 0
\(89\) −9.78113 −1.03680 −0.518399 0.855139i \(-0.673472\pi\)
−0.518399 + 0.855139i \(0.673472\pi\)
\(90\) 0 0
\(91\) 17.6629i 1.85157i
\(92\) 0 0
\(93\) 8.51167 + 2.07902i 0.882619 + 0.215585i
\(94\) 0 0
\(95\) 1.46552i 0.150359i
\(96\) 0 0
\(97\) 9.26880i 0.941104i 0.882372 + 0.470552i \(0.155945\pi\)
−0.882372 + 0.470552i \(0.844055\pi\)
\(98\) 0 0
\(99\) −4.71318 2.44853i −0.473693 0.246086i
\(100\) 0 0
\(101\) 10.1637i 1.01133i −0.862731 0.505663i \(-0.831248\pi\)
0.862731 0.505663i \(-0.168752\pi\)
\(102\) 0 0
\(103\) 9.36818i 0.923074i −0.887121 0.461537i \(-0.847298\pi\)
0.887121 0.461537i \(-0.152702\pi\)
\(104\) 0 0
\(105\) 1.75045 7.16646i 0.170827 0.699375i
\(106\) 0 0
\(107\) 1.58889 0.153604 0.0768019 0.997046i \(-0.475529\pi\)
0.0768019 + 0.997046i \(0.475529\pi\)
\(108\) 0 0
\(109\) 9.94643i 0.952695i −0.879257 0.476347i \(-0.841960\pi\)
0.879257 0.476347i \(-0.158040\pi\)
\(110\) 0 0
\(111\) −1.18464 + 4.84998i −0.112441 + 0.460339i
\(112\) 0 0
\(113\) −4.69423 −0.441596 −0.220798 0.975320i \(-0.570866\pi\)
−0.220798 + 0.975320i \(0.570866\pi\)
\(114\) 0 0
\(115\) 1.73005 4.47291i 0.161328 0.417101i
\(116\) 0 0
\(117\) 11.0401 + 5.73539i 1.02066 + 0.530237i
\(118\) 0 0
\(119\) 29.2583i 2.68211i
\(120\) 0 0
\(121\) −7.86562 −0.715057
\(122\) 0 0
\(123\) −1.87563 + 7.67894i −0.169120 + 0.692387i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 2.49918 0.221767 0.110883 0.993833i \(-0.464632\pi\)
0.110883 + 0.993833i \(0.464632\pi\)
\(128\) 0 0
\(129\) −1.34801 + 5.51886i −0.118686 + 0.485908i
\(130\) 0 0
\(131\) 6.14095i 0.536537i 0.963344 + 0.268269i \(0.0864515\pi\)
−0.963344 + 0.268269i \(0.913548\pi\)
\(132\) 0 0
\(133\) −6.24193 −0.541244
\(134\) 0 0
\(135\) 3.91096 + 3.42116i 0.336602 + 0.294447i
\(136\) 0 0
\(137\) 11.5065 0.983071 0.491535 0.870858i \(-0.336436\pi\)
0.491535 + 0.870858i \(0.336436\pi\)
\(138\) 0 0
\(139\) −13.8249 −1.17262 −0.586308 0.810088i \(-0.699419\pi\)
−0.586308 + 0.810088i \(0.699419\pi\)
\(140\) 0 0
\(141\) 2.38464 9.76287i 0.200823 0.822182i
\(142\) 0 0
\(143\) −7.34191 −0.613961
\(144\) 0 0
\(145\) 4.10332i 0.340762i
\(146\) 0 0
\(147\) 18.7453 + 4.57865i 1.54609 + 0.377640i
\(148\) 0 0
\(149\) −16.3047 −1.33573 −0.667867 0.744281i \(-0.732792\pi\)
−0.667867 + 0.744281i \(0.732792\pi\)
\(150\) 0 0
\(151\) 4.81062 0.391483 0.195741 0.980656i \(-0.437289\pi\)
0.195741 + 0.980656i \(0.437289\pi\)
\(152\) 0 0
\(153\) 18.2878 + 9.50060i 1.47848 + 0.768078i
\(154\) 0 0
\(155\) 5.05868 0.406323
\(156\) 0 0
\(157\) 12.9122i 1.03050i −0.857039 0.515252i \(-0.827698\pi\)
0.857039 0.515252i \(-0.172302\pi\)
\(158\) 0 0
\(159\) −1.39909 0.341735i −0.110955 0.0271014i
\(160\) 0 0
\(161\) 19.0510 + 7.36864i 1.50143 + 0.580730i
\(162\) 0 0
\(163\) −11.1862 −0.876174 −0.438087 0.898933i \(-0.644344\pi\)
−0.438087 + 0.898933i \(0.644344\pi\)
\(164\) 0 0
\(165\) −2.97888 0.727608i −0.231905 0.0566442i
\(166\) 0 0
\(167\) 0.273329i 0.0211508i −0.999944 0.0105754i \(-0.996634\pi\)
0.999944 0.0105754i \(-0.00336632\pi\)
\(168\) 0 0
\(169\) 4.19756 0.322889
\(170\) 0 0
\(171\) 2.02684 3.90149i 0.154997 0.298354i
\(172\) 0 0
\(173\) 12.8456i 0.976631i 0.872667 + 0.488315i \(0.162388\pi\)
−0.872667 + 0.488315i \(0.837612\pi\)
\(174\) 0 0
\(175\) 4.25920i 0.321965i
\(176\) 0 0
\(177\) −4.86311 + 19.9099i −0.365534 + 1.49652i
\(178\) 0 0
\(179\) 5.54592i 0.414522i −0.978286 0.207261i \(-0.933545\pi\)
0.978286 0.207261i \(-0.0664549\pi\)
\(180\) 0 0
\(181\) 9.79986i 0.728418i 0.931317 + 0.364209i \(0.118661\pi\)
−0.931317 + 0.364209i \(0.881339\pi\)
\(182\) 0 0
\(183\) −4.93853 + 20.2187i −0.365067 + 1.49461i
\(184\) 0 0
\(185\) 2.88245i 0.211922i
\(186\) 0 0
\(187\) −12.1618 −0.889358
\(188\) 0 0
\(189\) −14.5714 + 16.6576i −1.05991 + 1.21166i
\(190\) 0 0
\(191\) 18.3769 1.32971 0.664854 0.746973i \(-0.268494\pi\)
0.664854 + 0.746973i \(0.268494\pi\)
\(192\) 0 0
\(193\) −15.0944 −1.08652 −0.543260 0.839564i \(-0.682810\pi\)
−0.543260 + 0.839564i \(0.682810\pi\)
\(194\) 0 0
\(195\) 6.97767 + 1.70434i 0.499682 + 0.122050i
\(196\) 0 0
\(197\) 21.1615i 1.50769i 0.657050 + 0.753847i \(0.271804\pi\)
−0.657050 + 0.753847i \(0.728196\pi\)
\(198\) 0 0
\(199\) 3.16726i 0.224521i −0.993679 0.112261i \(-0.964191\pi\)
0.993679 0.112261i \(-0.0358092\pi\)
\(200\) 0 0
\(201\) −4.27402 + 17.4981i −0.301466 + 1.23422i
\(202\) 0 0
\(203\) −17.4769 −1.22663
\(204\) 0 0
\(205\) 4.56378i 0.318748i
\(206\) 0 0
\(207\) −10.7919 + 9.51502i −0.750086 + 0.661340i
\(208\) 0 0
\(209\) 2.59458i 0.179471i
\(210\) 0 0
\(211\) 5.16177 0.355351 0.177676 0.984089i \(-0.443142\pi\)
0.177676 + 0.984089i \(0.443142\pi\)
\(212\) 0 0
\(213\) 6.70091 27.4340i 0.459138 1.87974i
\(214\) 0 0
\(215\) 3.27999i 0.223693i
\(216\) 0 0
\(217\) 21.5459i 1.46263i
\(218\) 0 0
\(219\) 2.84381 + 0.694616i 0.192167 + 0.0469378i
\(220\) 0 0
\(221\) 28.4875 1.91628
\(222\) 0 0
\(223\) 29.1017 1.94879 0.974395 0.224842i \(-0.0721864\pi\)
0.974395 + 0.224842i \(0.0721864\pi\)
\(224\) 0 0
\(225\) 2.66219 + 1.38302i 0.177479 + 0.0922015i
\(226\) 0 0
\(227\) −5.42076 −0.359789 −0.179894 0.983686i \(-0.557576\pi\)
−0.179894 + 0.983686i \(0.557576\pi\)
\(228\) 0 0
\(229\) 15.5263i 1.02601i 0.858386 + 0.513004i \(0.171467\pi\)
−0.858386 + 0.513004i \(0.828533\pi\)
\(230\) 0 0
\(231\) 3.09903 12.6876i 0.203901 0.834785i
\(232\) 0 0
\(233\) 29.1310i 1.90844i −0.299113 0.954218i \(-0.596691\pi\)
0.299113 0.954218i \(-0.403309\pi\)
\(234\) 0 0
\(235\) 5.80230i 0.378501i
\(236\) 0 0
\(237\) −1.07904 + 4.41768i −0.0700915 + 0.286959i
\(238\) 0 0
\(239\) 25.1530i 1.62701i −0.581555 0.813507i \(-0.697556\pi\)
0.581555 0.813507i \(-0.302444\pi\)
\(240\) 0 0
\(241\) 23.5388i 1.51627i 0.652097 + 0.758135i \(0.273889\pi\)
−0.652097 + 0.758135i \(0.726111\pi\)
\(242\) 0 0
\(243\) −5.68018 14.5167i −0.364383 0.931249i
\(244\) 0 0
\(245\) 11.1408 0.711757
\(246\) 0 0
\(247\) 6.07750i 0.386702i
\(248\) 0 0
\(249\) 8.78584 + 2.14599i 0.556780 + 0.135997i
\(250\) 0 0
\(251\) 6.64673 0.419538 0.209769 0.977751i \(-0.432729\pi\)
0.209769 + 0.977751i \(0.432729\pi\)
\(252\) 0 0
\(253\) 3.06292 7.91891i 0.192564 0.497858i
\(254\) 0 0
\(255\) 11.5584 + 2.82321i 0.723817 + 0.176797i
\(256\) 0 0
\(257\) 6.11601i 0.381506i 0.981638 + 0.190753i \(0.0610930\pi\)
−0.981638 + 0.190753i \(0.938907\pi\)
\(258\) 0 0
\(259\) −12.2769 −0.762852
\(260\) 0 0
\(261\) 5.67499 10.9238i 0.351273 0.676167i
\(262\) 0 0
\(263\) −8.17715 −0.504225 −0.252112 0.967698i \(-0.581125\pi\)
−0.252112 + 0.967698i \(0.581125\pi\)
\(264\) 0 0
\(265\) −0.831510 −0.0510792
\(266\) 0 0
\(267\) 16.4576 + 4.01986i 1.00719 + 0.246012i
\(268\) 0 0
\(269\) 0.00864764i 0.000527256i 1.00000 0.000263628i \(8.39153e-5\pi\)
−1.00000 0.000263628i \(0.999916\pi\)
\(270\) 0 0
\(271\) 23.9068 1.45223 0.726117 0.687571i \(-0.241323\pi\)
0.726117 + 0.687571i \(0.241323\pi\)
\(272\) 0 0
\(273\) −7.25911 + 29.7193i −0.439341 + 1.79869i
\(274\) 0 0
\(275\) −1.77042 −0.106760
\(276\) 0 0
\(277\) 22.6195 1.35907 0.679537 0.733641i \(-0.262181\pi\)
0.679537 + 0.733641i \(0.262181\pi\)
\(278\) 0 0
\(279\) −13.4672 6.99627i −0.806258 0.418856i
\(280\) 0 0
\(281\) −32.6193 −1.94590 −0.972951 0.231013i \(-0.925796\pi\)
−0.972951 + 0.231013i \(0.925796\pi\)
\(282\) 0 0
\(283\) 12.2989i 0.731095i 0.930793 + 0.365548i \(0.119118\pi\)
−0.930793 + 0.365548i \(0.880882\pi\)
\(284\) 0 0
\(285\) 0.602301 2.46586i 0.0356772 0.146065i
\(286\) 0 0
\(287\) −19.4380 −1.14739
\(288\) 0 0
\(289\) 30.1893 1.77584
\(290\) 0 0
\(291\) 3.80930 15.5955i 0.223305 0.914227i
\(292\) 0 0
\(293\) 22.1635 1.29481 0.647403 0.762148i \(-0.275855\pi\)
0.647403 + 0.762148i \(0.275855\pi\)
\(294\) 0 0
\(295\) 11.8329i 0.688940i
\(296\) 0 0
\(297\) 6.92404 + 6.05689i 0.401773 + 0.351456i
\(298\) 0 0
\(299\) −7.17452 + 18.5491i −0.414913 + 1.07272i
\(300\) 0 0
\(301\) −13.9701 −0.805223
\(302\) 0 0
\(303\) −4.17709 + 17.1013i −0.239967 + 0.982443i
\(304\) 0 0
\(305\) 12.0164i 0.688059i
\(306\) 0 0
\(307\) 27.1627 1.55026 0.775129 0.631803i \(-0.217685\pi\)
0.775129 + 0.631803i \(0.217685\pi\)
\(308\) 0 0
\(309\) −3.85015 + 15.7628i −0.219027 + 0.896712i
\(310\) 0 0
\(311\) 13.8045i 0.782781i −0.920225 0.391391i \(-0.871994\pi\)
0.920225 0.391391i \(-0.128006\pi\)
\(312\) 0 0
\(313\) 32.9723i 1.86371i −0.362834 0.931854i \(-0.618191\pi\)
0.362834 0.931854i \(-0.381809\pi\)
\(314\) 0 0
\(315\) −5.89057 + 11.3388i −0.331896 + 0.638868i
\(316\) 0 0
\(317\) 10.9658i 0.615900i −0.951402 0.307950i \(-0.900357\pi\)
0.951402 0.307950i \(-0.0996430\pi\)
\(318\) 0 0
\(319\) 7.26459i 0.406739i
\(320\) 0 0
\(321\) −2.67344 0.653004i −0.149217 0.0364471i
\(322\) 0 0
\(323\) 10.0673i 0.560159i
\(324\) 0 0
\(325\) 4.14699 0.230034
\(326\) 0 0
\(327\) −4.08780 + 16.7357i −0.226056 + 0.925487i
\(328\) 0 0
\(329\) 24.7132 1.36248
\(330\) 0 0
\(331\) −18.0444 −0.991811 −0.495905 0.868377i \(-0.665164\pi\)
−0.495905 + 0.868377i \(0.665164\pi\)
\(332\) 0 0
\(333\) 3.98650 7.67364i 0.218459 0.420513i
\(334\) 0 0
\(335\) 10.3996i 0.568188i
\(336\) 0 0
\(337\) 0.830721i 0.0452523i −0.999744 0.0226261i \(-0.992797\pi\)
0.999744 0.0226261i \(-0.00720274\pi\)
\(338\) 0 0
\(339\) 7.89845 + 1.92924i 0.428985 + 0.104782i
\(340\) 0 0
\(341\) 8.95598 0.484993
\(342\) 0 0
\(343\) 17.6363i 0.952272i
\(344\) 0 0
\(345\) −4.74925 + 6.81503i −0.255691 + 0.366909i
\(346\) 0 0
\(347\) 6.54648i 0.351434i 0.984441 + 0.175717i \(0.0562243\pi\)
−0.984441 + 0.175717i \(0.943776\pi\)
\(348\) 0 0
\(349\) 17.4584 0.934529 0.467264 0.884118i \(-0.345240\pi\)
0.467264 + 0.884118i \(0.345240\pi\)
\(350\) 0 0
\(351\) −16.2187 14.1875i −0.865693 0.757275i
\(352\) 0 0
\(353\) 11.3823i 0.605819i −0.953019 0.302909i \(-0.902042\pi\)
0.953019 0.302909i \(-0.0979579\pi\)
\(354\) 0 0
\(355\) 16.3046i 0.865361i
\(356\) 0 0
\(357\) −12.0246 + 49.2296i −0.636411 + 2.60551i
\(358\) 0 0
\(359\) 30.3119 1.59980 0.799900 0.600134i \(-0.204886\pi\)
0.799900 + 0.600134i \(0.204886\pi\)
\(360\) 0 0
\(361\) 16.8523 0.886961
\(362\) 0 0
\(363\) 13.2346 + 3.23262i 0.694636 + 0.169669i
\(364\) 0 0
\(365\) 1.69014 0.0884660
\(366\) 0 0
\(367\) 11.1215i 0.580540i 0.956945 + 0.290270i \(0.0937451\pi\)
−0.956945 + 0.290270i \(0.906255\pi\)
\(368\) 0 0
\(369\) 6.31180 12.1496i 0.328579 0.632485i
\(370\) 0 0
\(371\) 3.54156i 0.183869i
\(372\) 0 0
\(373\) 7.32753i 0.379405i 0.981842 + 0.189703i \(0.0607524\pi\)
−0.981842 + 0.189703i \(0.939248\pi\)
\(374\) 0 0
\(375\) 1.68259 + 0.410981i 0.0868883 + 0.0212230i
\(376\) 0 0
\(377\) 17.0164i 0.876392i
\(378\) 0 0
\(379\) 1.54185i 0.0791997i 0.999216 + 0.0395998i \(0.0126083\pi\)
−0.999216 + 0.0395998i \(0.987392\pi\)
\(380\) 0 0
\(381\) −4.20509 1.02712i −0.215433 0.0526208i
\(382\) 0 0
\(383\) 33.1055 1.69161 0.845805 0.533492i \(-0.179121\pi\)
0.845805 + 0.533492i \(0.179121\pi\)
\(384\) 0 0
\(385\) 7.54056i 0.384302i
\(386\) 0 0
\(387\) 4.53629 8.73194i 0.230593 0.443869i
\(388\) 0 0
\(389\) −27.5503 −1.39686 −0.698428 0.715680i \(-0.746117\pi\)
−0.698428 + 0.715680i \(0.746117\pi\)
\(390\) 0 0
\(391\) −11.8845 + 30.7264i −0.601025 + 1.55390i
\(392\) 0 0
\(393\) 2.52382 10.3327i 0.127310 0.521215i
\(394\) 0 0
\(395\) 2.62553i 0.132105i
\(396\) 0 0
\(397\) 13.4334 0.674205 0.337102 0.941468i \(-0.390553\pi\)
0.337102 + 0.941468i \(0.390553\pi\)
\(398\) 0 0
\(399\) 10.5026 + 2.56532i 0.525787 + 0.128427i
\(400\) 0 0
\(401\) −17.8416 −0.890965 −0.445483 0.895291i \(-0.646968\pi\)
−0.445483 + 0.895291i \(0.646968\pi\)
\(402\) 0 0
\(403\) −20.9783 −1.04501
\(404\) 0 0
\(405\) −5.17450 7.36373i −0.257123 0.365907i
\(406\) 0 0
\(407\) 5.10315i 0.252954i
\(408\) 0 0
\(409\) −7.54963 −0.373305 −0.186653 0.982426i \(-0.559764\pi\)
−0.186653 + 0.982426i \(0.559764\pi\)
\(410\) 0 0
\(411\) −19.3607 4.72898i −0.954995 0.233263i
\(412\) 0 0
\(413\) −50.3988 −2.47996
\(414\) 0 0
\(415\) 5.22163 0.256320
\(416\) 0 0
\(417\) 23.2616 + 5.68179i 1.13913 + 0.278238i
\(418\) 0 0
\(419\) −6.14328 −0.300119 −0.150059 0.988677i \(-0.547946\pi\)
−0.150059 + 0.988677i \(0.547946\pi\)
\(420\) 0 0
\(421\) 38.8307i 1.89249i −0.323449 0.946246i \(-0.604842\pi\)
0.323449 0.946246i \(-0.395158\pi\)
\(422\) 0 0
\(423\) −8.02472 + 15.4468i −0.390175 + 0.751051i
\(424\) 0 0
\(425\) 6.86944 0.333217
\(426\) 0 0
\(427\) −51.1804 −2.47679
\(428\) 0 0
\(429\) 12.3534 + 3.01739i 0.596427 + 0.145681i
\(430\) 0 0
\(431\) −25.7768 −1.24163 −0.620813 0.783959i \(-0.713197\pi\)
−0.620813 + 0.783959i \(0.713197\pi\)
\(432\) 0 0
\(433\) 32.6236i 1.56779i 0.620894 + 0.783894i \(0.286770\pi\)
−0.620894 + 0.783894i \(0.713230\pi\)
\(434\) 0 0
\(435\) 1.68639 6.90419i 0.0808561 0.331031i
\(436\) 0 0
\(437\) 6.55513 + 2.53543i 0.313574 + 0.121286i
\(438\) 0 0
\(439\) −19.8107 −0.945513 −0.472757 0.881193i \(-0.656741\pi\)
−0.472757 + 0.881193i \(0.656741\pi\)
\(440\) 0 0
\(441\) −29.6588 15.4079i −1.41232 0.733711i
\(442\) 0 0
\(443\) 1.51886i 0.0721631i 0.999349 + 0.0360816i \(0.0114876\pi\)
−0.999349 + 0.0360816i \(0.988512\pi\)
\(444\) 0 0
\(445\) 9.78113 0.463670
\(446\) 0 0
\(447\) 27.4341 + 6.70093i 1.29759 + 0.316943i
\(448\) 0 0
\(449\) 12.0706i 0.569646i −0.958580 0.284823i \(-0.908065\pi\)
0.958580 0.284823i \(-0.0919348\pi\)
\(450\) 0 0
\(451\) 8.07979i 0.380462i
\(452\) 0 0
\(453\) −8.09428 1.97707i −0.380302 0.0928911i
\(454\) 0 0
\(455\) 17.6629i 0.828048i
\(456\) 0 0
\(457\) 36.0870i 1.68808i −0.536281 0.844039i \(-0.680171\pi\)
0.536281 0.844039i \(-0.319829\pi\)
\(458\) 0 0
\(459\) −26.8661 23.5015i −1.25400 1.09696i
\(460\) 0 0
\(461\) 39.9476i 1.86054i 0.366872 + 0.930271i \(0.380429\pi\)
−0.366872 + 0.930271i \(0.619571\pi\)
\(462\) 0 0
\(463\) 4.60520 0.214022 0.107011 0.994258i \(-0.465872\pi\)
0.107011 + 0.994258i \(0.465872\pi\)
\(464\) 0 0
\(465\) −8.51167 2.07902i −0.394719 0.0964125i
\(466\) 0 0
\(467\) 30.4087 1.40715 0.703573 0.710623i \(-0.251587\pi\)
0.703573 + 0.710623i \(0.251587\pi\)
\(468\) 0 0
\(469\) −44.2938 −2.04530
\(470\) 0 0
\(471\) −5.30666 + 21.7258i −0.244518 + 1.00107i
\(472\) 0 0
\(473\) 5.80694i 0.267003i
\(474\) 0 0
\(475\) 1.46552i 0.0672426i
\(476\) 0 0
\(477\) 2.21364 + 1.15000i 0.101355 + 0.0526547i
\(478\) 0 0
\(479\) −31.0474 −1.41859 −0.709297 0.704910i \(-0.750987\pi\)
−0.709297 + 0.704910i \(0.750987\pi\)
\(480\) 0 0
\(481\) 11.9535i 0.545034i
\(482\) 0 0
\(483\) −29.0266 20.2280i −1.32075 0.920405i
\(484\) 0 0
\(485\) 9.26880i 0.420875i
\(486\) 0 0
\(487\) −11.7747 −0.533563 −0.266782 0.963757i \(-0.585960\pi\)
−0.266782 + 0.963757i \(0.585960\pi\)
\(488\) 0 0
\(489\) 18.8218 + 4.59733i 0.851151 + 0.207899i
\(490\) 0 0
\(491\) 23.6929i 1.06924i 0.845091 + 0.534622i \(0.179546\pi\)
−0.845091 + 0.534622i \(0.820454\pi\)
\(492\) 0 0
\(493\) 28.1875i 1.26950i
\(494\) 0 0
\(495\) 4.71318 + 2.44853i 0.211842 + 0.110053i
\(496\) 0 0
\(497\) 69.4447 3.11502
\(498\) 0 0
\(499\) 36.7997 1.64738 0.823689 0.567041i \(-0.191912\pi\)
0.823689 + 0.567041i \(0.191912\pi\)
\(500\) 0 0
\(501\) −0.112333 + 0.459899i −0.00501867 + 0.0205468i
\(502\) 0 0
\(503\) 39.7986 1.77453 0.887266 0.461259i \(-0.152602\pi\)
0.887266 + 0.461259i \(0.152602\pi\)
\(504\) 0 0
\(505\) 10.1637i 0.452278i
\(506\) 0 0
\(507\) −7.06276 1.72512i −0.313668 0.0766153i
\(508\) 0 0
\(509\) 10.3012i 0.456593i 0.973592 + 0.228297i \(0.0733156\pi\)
−0.973592 + 0.228297i \(0.926684\pi\)
\(510\) 0 0
\(511\) 7.19864i 0.318449i
\(512\) 0 0
\(513\) −5.01378 + 5.73159i −0.221364 + 0.253056i
\(514\) 0 0
\(515\) 9.36818i 0.412811i
\(516\) 0 0
\(517\) 10.2725i 0.451784i
\(518\) 0 0
\(519\) 5.27929 21.6138i 0.231735 0.948739i
\(520\) 0 0
\(521\) −24.8553 −1.08893 −0.544465 0.838784i \(-0.683267\pi\)
−0.544465 + 0.838784i \(0.683267\pi\)
\(522\) 0 0
\(523\) 12.6351i 0.552496i −0.961086 0.276248i \(-0.910909\pi\)
0.961086 0.276248i \(-0.0890912\pi\)
\(524\) 0 0
\(525\) −1.75045 + 7.16646i −0.0763959 + 0.312770i
\(526\) 0 0
\(527\) −34.7503 −1.51375
\(528\) 0 0
\(529\) −17.0138 15.4767i −0.739732 0.672902i
\(530\) 0 0
\(531\) 16.3652 31.5015i 0.710190 1.36705i
\(532\) 0 0
\(533\) 18.9260i 0.819774i
\(534\) 0 0
\(535\) −1.58889 −0.0686937
\(536\) 0 0
\(537\) −2.27927 + 9.33149i −0.0983578 + 0.402683i
\(538\) 0 0
\(539\) 19.7238 0.849564
\(540\) 0 0
\(541\) −21.7627 −0.935653 −0.467826 0.883820i \(-0.654963\pi\)
−0.467826 + 0.883820i \(0.654963\pi\)
\(542\) 0 0
\(543\) 4.02756 16.4891i 0.172839 0.707615i
\(544\) 0 0
\(545\) 9.94643i 0.426058i
\(546\) 0 0
\(547\) 0.652614 0.0279038 0.0139519 0.999903i \(-0.495559\pi\)
0.0139519 + 0.999903i \(0.495559\pi\)
\(548\) 0 0
\(549\) 16.6190 31.9900i 0.709282 1.36530i
\(550\) 0 0
\(551\) −6.01349 −0.256183
\(552\) 0 0
\(553\) −11.1827 −0.475535
\(554\) 0 0
\(555\) 1.18464 4.84998i 0.0502850 0.205870i
\(556\) 0 0
\(557\) −43.7961 −1.85570 −0.927850 0.372953i \(-0.878345\pi\)
−0.927850 + 0.372953i \(0.878345\pi\)
\(558\) 0 0
\(559\) 13.6021i 0.575307i
\(560\) 0 0
\(561\) 20.4632 + 4.99827i 0.863959 + 0.211027i
\(562\) 0 0
\(563\) 4.04662 0.170545 0.0852723 0.996358i \(-0.472824\pi\)
0.0852723 + 0.996358i \(0.472824\pi\)
\(564\) 0 0
\(565\) 4.69423 0.197488
\(566\) 0 0
\(567\) 31.3636 22.0392i 1.31715 0.925560i
\(568\) 0 0
\(569\) −17.6778 −0.741092 −0.370546 0.928814i \(-0.620829\pi\)
−0.370546 + 0.928814i \(0.620829\pi\)
\(570\) 0 0
\(571\) 9.47431i 0.396488i −0.980153 0.198244i \(-0.936476\pi\)
0.980153 0.198244i \(-0.0635238\pi\)
\(572\) 0 0
\(573\) −30.9208 7.55257i −1.29173 0.315513i
\(574\) 0 0
\(575\) −1.73005 + 4.47291i −0.0721482 + 0.186533i
\(576\) 0 0
\(577\) 27.6920 1.15284 0.576418 0.817155i \(-0.304450\pi\)
0.576418 + 0.817155i \(0.304450\pi\)
\(578\) 0 0
\(579\) 25.3977 + 6.20353i 1.05549 + 0.257810i
\(580\) 0 0
\(581\) 22.2400i 0.922669i
\(582\) 0 0
\(583\) −1.47212 −0.0609689
\(584\) 0 0
\(585\) −11.0401 5.73539i −0.456451 0.237129i
\(586\) 0 0
\(587\) 33.9922i 1.40301i −0.712665 0.701504i \(-0.752512\pi\)
0.712665 0.701504i \(-0.247488\pi\)
\(588\) 0 0
\(589\) 7.41359i 0.305472i
\(590\) 0 0
\(591\) 8.69698 35.6060i 0.357746 1.46464i
\(592\) 0 0
\(593\) 45.4678i 1.86714i −0.358398 0.933569i \(-0.616677\pi\)
0.358398 0.933569i \(-0.383323\pi\)
\(594\) 0 0
\(595\) 29.2583i 1.19947i
\(596\) 0 0
\(597\) −1.30169 + 5.32919i −0.0532745 + 0.218109i
\(598\) 0 0
\(599\) 23.7247i 0.969364i −0.874691 0.484682i \(-0.838935\pi\)
0.874691 0.484682i \(-0.161065\pi\)
\(600\) 0 0
\(601\) 34.9900 1.42727 0.713636 0.700517i \(-0.247047\pi\)
0.713636 + 0.700517i \(0.247047\pi\)
\(602\) 0 0
\(603\) 14.3828 27.6856i 0.585714 1.12744i
\(604\) 0 0
\(605\) 7.86562 0.319783
\(606\) 0 0
\(607\) 26.2120 1.06391 0.531955 0.846773i \(-0.321457\pi\)
0.531955 + 0.846773i \(0.321457\pi\)
\(608\) 0 0
\(609\) 29.4063 + 7.18266i 1.19160 + 0.291056i
\(610\) 0 0
\(611\) 24.0621i 0.973449i
\(612\) 0 0
\(613\) 27.1765i 1.09765i 0.835938 + 0.548824i \(0.184924\pi\)
−0.835938 + 0.548824i \(0.815076\pi\)
\(614\) 0 0
\(615\) 1.87563 7.67894i 0.0756326 0.309645i
\(616\) 0 0
\(617\) 0.364713 0.0146828 0.00734140 0.999973i \(-0.497663\pi\)
0.00734140 + 0.999973i \(0.497663\pi\)
\(618\) 0 0
\(619\) 0.854713i 0.0343538i −0.999852 0.0171769i \(-0.994532\pi\)
0.999852 0.0171769i \(-0.00546785\pi\)
\(620\) 0 0
\(621\) 22.0687 11.5746i 0.885588 0.464472i
\(622\) 0 0
\(623\) 41.6598i 1.66906i
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.06632 4.36560i 0.0425848 0.174345i
\(628\) 0 0
\(629\) 19.8009i 0.789512i
\(630\) 0 0
\(631\) 43.6340i 1.73704i 0.495654 + 0.868520i \(0.334928\pi\)
−0.495654 + 0.868520i \(0.665072\pi\)
\(632\) 0 0
\(633\) −8.68512 2.12139i −0.345203 0.0843178i
\(634\) 0 0
\(635\) −2.49918 −0.0991770
\(636\) 0 0
\(637\) −46.2007 −1.83054
\(638\) 0 0
\(639\) −22.5497 + 43.4060i −0.892052 + 1.71712i
\(640\) 0 0
\(641\) −10.9445 −0.432282 −0.216141 0.976362i \(-0.569347\pi\)
−0.216141 + 0.976362i \(0.569347\pi\)
\(642\) 0 0
\(643\) 27.3836i 1.07990i 0.841696 + 0.539952i \(0.181558\pi\)
−0.841696 + 0.539952i \(0.818442\pi\)
\(644\) 0 0
\(645\) 1.34801 5.51886i 0.0530780 0.217305i
\(646\) 0 0
\(647\) 11.8734i 0.466791i 0.972382 + 0.233395i \(0.0749836\pi\)
−0.972382 + 0.233395i \(0.925016\pi\)
\(648\) 0 0
\(649\) 20.9492i 0.822329i
\(650\) 0 0
\(651\) 8.85498 36.2529i 0.347054 1.42086i
\(652\) 0 0
\(653\) 15.0075i 0.587289i −0.955915 0.293644i \(-0.905132\pi\)
0.955915 0.293644i \(-0.0948681\pi\)
\(654\) 0 0
\(655\) 6.14095i 0.239947i
\(656\) 0 0
\(657\) −4.49947 2.33750i −0.175541 0.0911946i
\(658\) 0 0
\(659\) 25.1920 0.981342 0.490671 0.871345i \(-0.336752\pi\)
0.490671 + 0.871345i \(0.336752\pi\)
\(660\) 0 0
\(661\) 11.3817i 0.442698i 0.975195 + 0.221349i \(0.0710459\pi\)
−0.975195 + 0.221349i \(0.928954\pi\)
\(662\) 0 0
\(663\) −47.9327 11.7079i −1.86155 0.454695i
\(664\) 0 0
\(665\) 6.24193 0.242052
\(666\) 0 0
\(667\) 18.3538 + 7.09897i 0.710661 + 0.274873i
\(668\) 0 0
\(669\) −48.9660 11.9602i −1.89314 0.462409i
\(670\) 0 0
\(671\) 21.2741i 0.821278i
\(672\) 0 0
\(673\) 7.04251 0.271469 0.135734 0.990745i \(-0.456661\pi\)
0.135734 + 0.990745i \(0.456661\pi\)
\(674\) 0 0
\(675\) −3.91096 3.42116i −0.150533 0.131681i
\(676\) 0 0
\(677\) 4.06545 0.156248 0.0781239 0.996944i \(-0.475107\pi\)
0.0781239 + 0.996944i \(0.475107\pi\)
\(678\) 0 0
\(679\) 39.4776 1.51501
\(680\) 0 0
\(681\) 9.12090 + 2.22783i 0.349514 + 0.0853707i
\(682\) 0 0
\(683\) 45.4211i 1.73799i −0.494819 0.868996i \(-0.664766\pi\)
0.494819 0.868996i \(-0.335234\pi\)
\(684\) 0 0
\(685\) −11.5065 −0.439643
\(686\) 0 0
\(687\) 6.38102 26.1243i 0.243451 0.996706i
\(688\) 0 0
\(689\) 3.44827 0.131369
\(690\) 0 0
\(691\) −15.9715 −0.607586 −0.303793 0.952738i \(-0.598253\pi\)
−0.303793 + 0.952738i \(0.598253\pi\)
\(692\) 0 0
\(693\) −10.4288 + 20.0744i −0.396156 + 0.762563i
\(694\) 0 0
\(695\) 13.8249 0.524410
\(696\) 0 0
\(697\) 31.3506i 1.18749i
\(698\) 0 0
\(699\) −11.9723 + 49.0154i −0.452834 + 1.85393i
\(700\) 0 0
\(701\) 19.3116 0.729391 0.364695 0.931127i \(-0.381173\pi\)
0.364695 + 0.931127i \(0.381173\pi\)
\(702\) 0 0
\(703\) −4.22429 −0.159322
\(704\) 0 0
\(705\) −2.38464 + 9.76287i −0.0898107 + 0.367691i
\(706\) 0 0
\(707\) −43.2892 −1.62806
\(708\) 0 0
\(709\) 35.6708i 1.33964i −0.742521 0.669822i \(-0.766370\pi\)
0.742521 0.669822i \(-0.233630\pi\)
\(710\) 0 0
\(711\) 3.63117 6.98966i 0.136180 0.262133i
\(712\) 0 0
\(713\) 8.75179 22.6270i 0.327757 0.847389i
\(714\) 0 0
\(715\) 7.34191 0.274572
\(716\) 0 0
\(717\) −10.3374 + 42.3221i −0.386058 + 1.58055i
\(718\) 0 0
\(719\) 8.77126i 0.327113i 0.986534 + 0.163556i \(0.0522966\pi\)
−0.986534 + 0.163556i \(0.947703\pi\)
\(720\) 0 0
\(721\) −39.9009 −1.48599
\(722\) 0 0
\(723\) 9.67403 39.6061i 0.359781 1.47297i
\(724\) 0 0
\(725\) 4.10332i 0.152394i
\(726\) 0 0
\(727\) 26.4591i 0.981315i 0.871353 + 0.490657i \(0.163243\pi\)
−0.871353 + 0.490657i \(0.836757\pi\)
\(728\) 0 0
\(729\) 3.59127 + 26.7601i 0.133010 + 0.991115i
\(730\) 0 0
\(731\) 22.5317i 0.833364i
\(732\) 0 0
\(733\) 40.6758i 1.50240i 0.660077 + 0.751198i \(0.270523\pi\)
−0.660077 + 0.751198i \(0.729477\pi\)
\(734\) 0 0
\(735\) −18.7453 4.57865i −0.691430 0.168886i
\(736\) 0 0
\(737\) 18.4115i 0.678198i
\(738\) 0 0
\(739\) −4.06596 −0.149569 −0.0747844 0.997200i \(-0.523827\pi\)
−0.0747844 + 0.997200i \(0.523827\pi\)
\(740\) 0 0
\(741\) −2.49774 + 10.2259i −0.0917567 + 0.375658i
\(742\) 0 0
\(743\) 37.8267 1.38773 0.693863 0.720107i \(-0.255907\pi\)
0.693863 + 0.720107i \(0.255907\pi\)
\(744\) 0 0
\(745\) 16.3047 0.597358
\(746\) 0 0
\(747\) −13.9010 7.22164i −0.508610 0.264226i
\(748\) 0 0
\(749\) 6.76739i 0.247275i
\(750\) 0 0
\(751\) 37.9249i 1.38390i −0.721946 0.691950i \(-0.756752\pi\)
0.721946 0.691950i \(-0.243248\pi\)
\(752\) 0 0
\(753\) −11.1837 2.73168i −0.407556 0.0995480i
\(754\) 0 0
\(755\) −4.81062 −0.175076
\(756\) 0 0
\(757\) 14.6543i 0.532618i 0.963888 + 0.266309i \(0.0858041\pi\)
−0.963888 + 0.266309i \(0.914196\pi\)
\(758\) 0 0
\(759\) −8.40815 + 12.0654i −0.305196 + 0.437948i
\(760\) 0 0
\(761\) 7.11993i 0.258097i 0.991638 + 0.129049i \(0.0411923\pi\)
−0.991638 + 0.129049i \(0.958808\pi\)
\(762\) 0 0
\(763\) −42.3638 −1.53367
\(764\) 0 0
\(765\) −18.2878 9.50060i −0.661195 0.343495i
\(766\) 0 0
\(767\) 49.0711i 1.77185i
\(768\) 0 0
\(769\) 32.0781i 1.15676i −0.815766 0.578382i \(-0.803684\pi\)
0.815766 0.578382i \(-0.196316\pi\)
\(770\) 0 0
\(771\) 2.51356 10.2907i 0.0905238 0.370611i
\(772\) 0 0
\(773\) 8.51413 0.306232 0.153116 0.988208i \(-0.451069\pi\)
0.153116 + 0.988208i \(0.451069\pi\)
\(774\) 0 0
\(775\) −5.05868 −0.181713
\(776\) 0 0
\(777\) 20.6570 + 5.04559i 0.741066 + 0.181010i
\(778\) 0 0
\(779\) −6.68830 −0.239633
\(780\) 0 0
\(781\) 28.8660i 1.03291i
\(782\) 0 0
\(783\) −14.0381 + 16.0479i −0.501682 + 0.573507i
\(784\) 0 0
\(785\) 12.9122i 0.460855i
\(786\) 0 0
\(787\) 11.9810i 0.427078i 0.976935 + 0.213539i \(0.0684990\pi\)
−0.976935 + 0.213539i \(0.931501\pi\)
\(788\) 0 0
\(789\) 13.7588 + 3.36066i 0.489825 + 0.119643i
\(790\) 0 0
\(791\) 19.9937i 0.710893i
\(792\) 0 0
\(793\) 49.8321i 1.76959i
\(794\) 0 0
\(795\) 1.39909 + 0.341735i 0.0496205 + 0.0121201i
\(796\) 0 0
\(797\) −15.5987 −0.552533 −0.276266 0.961081i \(-0.589097\pi\)
−0.276266 + 0.961081i \(0.589097\pi\)
\(798\) 0 0
\(799\) 39.8586i 1.41010i
\(800\) 0 0
\(801\) −26.0392 13.5275i −0.920051 0.477972i
\(802\) 0 0
\(803\) 2.99225 0.105594
\(804\) 0 0
\(805\) −19.0510 7.36864i −0.671460 0.259710i
\(806\) 0 0
\(807\) 0.00355402 0.0145504i 0.000125107 0.000512198i
\(808\) 0 0
\(809\) 12.8819i 0.452904i 0.974022 + 0.226452i \(0.0727127\pi\)
−0.974022 + 0.226452i \(0.927287\pi\)
\(810\) 0 0
\(811\) 18.9804 0.666491 0.333245 0.942840i \(-0.391856\pi\)
0.333245 + 0.942840i \(0.391856\pi\)
\(812\) 0 0
\(813\) −40.2252 9.82525i −1.41076 0.344587i
\(814\) 0 0
\(815\) 11.1862 0.391837
\(816\) 0 0
\(817\) −4.80688 −0.168171
\(818\) 0 0
\(819\) 24.4281 47.0219i 0.853589 1.64308i
\(820\) 0 0
\(821\) 10.5143i 0.366952i −0.983024 0.183476i \(-0.941265\pi\)
0.983024 0.183476i \(-0.0587350\pi\)
\(822\) 0 0
\(823\) 20.7205 0.722270 0.361135 0.932514i \(-0.382389\pi\)
0.361135 + 0.932514i \(0.382389\pi\)
\(824\) 0 0
\(825\) 2.97888 + 0.727608i 0.103711 + 0.0253321i
\(826\) 0 0
\(827\) −45.9796 −1.59887 −0.799434 0.600754i \(-0.794867\pi\)
−0.799434 + 0.600754i \(0.794867\pi\)
\(828\) 0 0
\(829\) 46.1501 1.60286 0.801429 0.598089i \(-0.204073\pi\)
0.801429 + 0.598089i \(0.204073\pi\)
\(830\) 0 0
\(831\) −38.0593 9.29619i −1.32026 0.322481i
\(832\) 0 0
\(833\) −76.5309 −2.65164
\(834\) 0 0
\(835\) 0.273329i 0.00945894i
\(836\) 0 0
\(837\) 19.7843 + 17.3066i 0.683846 + 0.598203i
\(838\) 0 0
\(839\) −53.3483 −1.84179 −0.920894 0.389812i \(-0.872540\pi\)
−0.920894 + 0.389812i \(0.872540\pi\)
\(840\) 0 0
\(841\) 12.1628 0.419405
\(842\) 0 0
\(843\) 54.8847 + 13.4059i 1.89033 + 0.461724i
\(844\) 0 0
\(845\) −4.19756 −0.144401
\(846\) 0 0
\(847\) 33.5012i 1.15112i
\(848\) 0 0
\(849\) 5.05463 20.6940i 0.173474 0.710216i
\(850\) 0 0
\(851\) 12.8930 + 4.98680i 0.441965 + 0.170945i
\(852\) 0 0
\(853\) −20.9563 −0.717530 −0.358765 0.933428i \(-0.616802\pi\)
−0.358765 + 0.933428i \(0.616802\pi\)
\(854\) 0 0
\(855\) −2.02684 + 3.90149i −0.0693166 + 0.133428i
\(856\) 0 0
\(857\) 44.3899i 1.51633i 0.652063 + 0.758165i \(0.273904\pi\)
−0.652063 + 0.758165i \(0.726096\pi\)
\(858\) 0 0
\(859\) −13.2571 −0.452326 −0.226163 0.974089i \(-0.572618\pi\)
−0.226163 + 0.974089i \(0.572618\pi\)
\(860\) 0 0
\(861\) 32.7061 + 7.98866i 1.11462 + 0.272253i
\(862\) 0 0
\(863\) 34.1455i 1.16233i −0.813787 0.581164i \(-0.802598\pi\)
0.813787 0.581164i \(-0.197402\pi\)
\(864\) 0 0
\(865\) 12.8456i 0.436762i
\(866\) 0 0
\(867\) −50.7960 12.4072i −1.72512 0.421372i
\(868\) 0 0
\(869\) 4.64829i 0.157682i
\(870\) 0 0
\(871\) 43.1269i 1.46130i
\(872\) 0 0
\(873\) −12.8190 + 24.6753i −0.433856 + 0.835132i
\(874\) 0 0
\(875\) 4.25920i 0.143987i
\(876\) 0 0
\(877\) −6.23583 −0.210569 −0.105284 0.994442i \(-0.533575\pi\)
−0.105284 + 0.994442i \(0.533575\pi\)
\(878\) 0 0
\(879\) −37.2920 9.10879i −1.25783 0.307232i
\(880\) 0 0
\(881\) 32.4198 1.09225 0.546125 0.837704i \(-0.316102\pi\)
0.546125 + 0.837704i \(0.316102\pi\)
\(882\) 0 0
\(883\) 47.9132 1.61241 0.806204 0.591637i \(-0.201518\pi\)
0.806204 + 0.591637i \(0.201518\pi\)
\(884\) 0 0
\(885\) 4.86311 19.9099i 0.163472 0.669265i
\(886\) 0 0
\(887\) 6.06146i 0.203524i 0.994809 + 0.101762i \(0.0324480\pi\)
−0.994809 + 0.101762i \(0.967552\pi\)
\(888\) 0 0
\(889\) 10.6445i 0.357005i
\(890\) 0 0
\(891\) −9.16102 13.0369i −0.306906 0.436752i
\(892\) 0 0
\(893\) 8.50338 0.284555
\(894\) 0 0
\(895\) 5.54592i 0.185380i
\(896\) 0 0
\(897\) 19.6951 28.2619i 0.657600 0.943637i
\(898\) 0 0
\(899\) 20.7574i 0.692298i
\(900\) 0 0
\(901\) 5.71201 0.190295
\(902\) 0 0
\(903\) 23.5059 + 5.74145i 0.782227 + 0.191064i
\(904\) 0 0
\(905\) 9.79986i 0.325758i
\(906\) 0 0
\(907\) 26.5492i 0.881552i −0.897617 0.440776i \(-0.854703\pi\)
0.897617 0.440776i \(-0.145297\pi\)
\(908\) 0 0
\(909\) 14.0566 27.0577i 0.466229 0.897446i
\(910\) 0 0
\(911\) 33.4165 1.10714 0.553569 0.832804i \(-0.313266\pi\)
0.553569 + 0.832804i \(0.313266\pi\)
\(912\) 0 0
\(913\) 9.24447 0.305947
\(914\) 0 0
\(915\) 4.93853 20.2187i 0.163263 0.668409i
\(916\) 0 0
\(917\) 26.1555 0.863732
\(918\) 0 0
\(919\) 43.3201i 1.42900i 0.699637 + 0.714498i \(0.253345\pi\)
−0.699637 + 0.714498i \(0.746655\pi\)
\(920\) 0 0
\(921\) −45.7036 11.1634i −1.50599 0.367846i
\(922\) 0 0
\(923\) 67.6153i 2.22558i
\(924\) 0 0
\(925\) 2.88245i 0.0947745i
\(926\) 0 0
\(927\) 12.9564 24.9399i 0.425544 0.819132i
\(928\) 0 0
\(929\) 18.3726i 0.602785i 0.953500 + 0.301393i \(0.0974515\pi\)
−0.953500 + 0.301393i \(0.902549\pi\)
\(930\) 0 0
\(931\) 16.3270i 0.535095i
\(932\) 0 0
\(933\) −5.67339 + 23.2273i −0.185739 + 0.760426i
\(934\) 0 0
\(935\) 12.1618 0.397733
\(936\) 0 0
\(937\) 36.2126i 1.18301i 0.806300 + 0.591507i \(0.201467\pi\)
−0.806300 + 0.591507i \(0.798533\pi\)
\(938\) 0 0
\(939\) −13.5510 + 55.4788i −0.442221 + 1.81048i
\(940\) 0 0
\(941\) 16.1480 0.526409 0.263204 0.964740i \(-0.415221\pi\)
0.263204 + 0.964740i \(0.415221\pi\)
\(942\) 0 0
\(943\) 20.4134 + 7.89558i 0.664750 + 0.257115i
\(944\) 0 0
\(945\) 14.5714 16.6576i 0.474008 0.541871i
\(946\) 0 0
\(947\) 32.7857i 1.06539i −0.846306 0.532697i \(-0.821179\pi\)
0.846306 0.532697i \(-0.178821\pi\)
\(948\) 0 0
\(949\) −7.00900 −0.227522
\(950\) 0 0
\(951\) −4.50674 + 18.4509i −0.146141 + 0.598311i
\(952\) 0 0
\(953\) 3.84790 0.124646 0.0623228 0.998056i \(-0.480149\pi\)
0.0623228 + 0.998056i \(0.480149\pi\)
\(954\) 0 0
\(955\) −18.3769 −0.594663
\(956\) 0 0
\(957\) 2.98561 12.2233i 0.0965111 0.395123i
\(958\) 0 0
\(959\) 49.0087i 1.58257i
\(960\) 0 0
\(961\) −5.40973 −0.174507
\(962\) 0 0
\(963\) 4.22992 + 2.19747i 0.136307 + 0.0708125i
\(964\) 0 0
\(965\) 15.0944 0.485907
\(966\) 0 0
\(967\) 25.0017 0.804001 0.402001 0.915639i \(-0.368315\pi\)
0.402001 + 0.915639i \(0.368315\pi\)
\(968\) 0 0
\(969\) −4.13747 + 16.9391i −0.132915 + 0.544162i
\(970\) 0 0
\(971\) −26.2788 −0.843326 −0.421663 0.906753i \(-0.638553\pi\)
−0.421663 + 0.906753i \(0.638553\pi\)
\(972\) 0 0
\(973\) 58.8831i 1.88771i
\(974\) 0 0
\(975\) −6.97767 1.70434i −0.223464 0.0545825i
\(976\) 0 0
\(977\) −21.4585 −0.686517 −0.343258 0.939241i \(-0.611531\pi\)
−0.343258 + 0.939241i \(0.611531\pi\)
\(978\) 0 0
\(979\) 17.3167 0.553444
\(980\) 0 0
\(981\) 13.7561 26.4793i 0.439199 0.845418i
\(982\) 0 0
\(983\) −14.7143 −0.469313 −0.234656 0.972078i \(-0.575397\pi\)
−0.234656 + 0.972078i \(0.575397\pi\)
\(984\) 0 0
\(985\) 21.1615i 0.674262i
\(986\) 0 0
\(987\) −41.5820 10.1566i −1.32357 0.323290i
\(988\) 0 0
\(989\) 14.6711 + 5.67455i 0.466513 + 0.180440i
\(990\) 0 0
\(991\) 38.7963 1.23240 0.616202 0.787588i \(-0.288670\pi\)
0.616202 + 0.787588i \(0.288670\pi\)
\(992\) 0 0
\(993\) 30.3613 + 7.41592i 0.963486 + 0.235337i
\(994\) 0 0
\(995\) 3.16726i 0.100409i
\(996\) 0 0
\(997\) 11.4649 0.363097 0.181549 0.983382i \(-0.441889\pi\)
0.181549 + 0.983382i \(0.441889\pi\)
\(998\) 0 0
\(999\) −9.86135 + 11.2732i −0.311999 + 0.356667i
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.i.a.1241.1 16
3.2 odd 2 1380.2.i.b.1241.2 yes 16
23.22 odd 2 1380.2.i.b.1241.1 yes 16
69.68 even 2 inner 1380.2.i.a.1241.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.i.a.1241.1 16 1.1 even 1 trivial
1380.2.i.a.1241.2 yes 16 69.68 even 2 inner
1380.2.i.b.1241.1 yes 16 23.22 odd 2
1380.2.i.b.1241.2 yes 16 3.2 odd 2