Properties

Label 2646.2.d.f.2645.9
Level $2646$
Weight $2$
Character 2646.2645
Analytic conductor $21.128$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2646,2,Mod(2645,2646)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2646, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2646.2645");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2646 = 2 \cdot 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2646.d (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: \(21.1284163748\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2645.9
Root \(0.130526 - 0.991445i\) of defining polynomial
Character \(\chi\) \(=\) 2646.2645
Dual form 2646.2.d.f.2645.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} -3.57981 q^{5} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} -3.57981 q^{5} -1.00000i q^{8} -3.57981i q^{10} -5.52607i q^{11} -6.91037i q^{13} +1.00000 q^{16} -2.44949 q^{17} +8.16176i q^{19} +3.57981 q^{20} +5.52607 q^{22} -3.08856i q^{23} +7.81504 q^{25} +6.91037 q^{26} +5.66096i q^{29} +0.400414i q^{31} +1.00000i q^{32} -2.44949i q^{34} +5.79174 q^{37} -8.16176 q^{38} +3.57981i q^{40} -5.49535 q^{41} -2.95367 q^{43} +5.52607i q^{44} +3.08856 q^{46} -0.0958559 q^{47} +7.81504i q^{50} +6.91037i q^{52} -5.17738i q^{53} +19.7823i q^{55} -5.66096 q^{58} -6.38401 q^{59} +7.06376i q^{61} -0.400414 q^{62} -1.00000 q^{64} +24.7378i q^{65} -1.45090 q^{67} +2.44949 q^{68} +8.77836i q^{71} -0.679758i q^{73} +5.79174i q^{74} -8.16176i q^{76} -6.43340 q^{79} -3.57981 q^{80} -5.49535i q^{82} -1.64156 q^{83} +8.76871 q^{85} -2.95367i q^{86} -5.52607 q^{88} -12.1921 q^{89} +3.08856i q^{92} -0.0958559i q^{94} -29.2176i q^{95} +11.3788i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} + 16 q^{16} + 16 q^{22} + 16 q^{37} - 32 q^{43} + 48 q^{46} - 32 q^{58} - 16 q^{64} - 32 q^{67} - 16 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2646\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −3.57981 −1.60094 −0.800470 0.599373i \(-0.795417\pi\)
−0.800470 + 0.599373i \(0.795417\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 1.00000i − 0.353553i
\(9\) 0 0
\(10\) − 3.57981i − 1.13204i
\(11\) − 5.52607i − 1.66617i −0.553144 0.833086i \(-0.686572\pi\)
0.553144 0.833086i \(-0.313428\pi\)
\(12\) 0 0
\(13\) − 6.91037i − 1.91659i −0.285776 0.958297i \(-0.592251\pi\)
0.285776 0.958297i \(-0.407749\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.44949 −0.594089 −0.297044 0.954864i \(-0.596001\pi\)
−0.297044 + 0.954864i \(0.596001\pi\)
\(18\) 0 0
\(19\) 8.16176i 1.87244i 0.351418 + 0.936219i \(0.385699\pi\)
−0.351418 + 0.936219i \(0.614301\pi\)
\(20\) 3.57981 0.800470
\(21\) 0 0
\(22\) 5.52607 1.17816
\(23\) − 3.08856i − 0.644009i −0.946738 0.322005i \(-0.895643\pi\)
0.946738 0.322005i \(-0.104357\pi\)
\(24\) 0 0
\(25\) 7.81504 1.56301
\(26\) 6.91037 1.35524
\(27\) 0 0
\(28\) 0 0
\(29\) 5.66096i 1.05121i 0.850728 + 0.525607i \(0.176162\pi\)
−0.850728 + 0.525607i \(0.823838\pi\)
\(30\) 0 0
\(31\) 0.400414i 0.0719164i 0.999353 + 0.0359582i \(0.0114483\pi\)
−0.999353 + 0.0359582i \(0.988552\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) − 2.44949i − 0.420084i
\(35\) 0 0
\(36\) 0 0
\(37\) 5.79174 0.952157 0.476078 0.879403i \(-0.342058\pi\)
0.476078 + 0.879403i \(0.342058\pi\)
\(38\) −8.16176 −1.32401
\(39\) 0 0
\(40\) 3.57981i 0.566018i
\(41\) −5.49535 −0.858229 −0.429115 0.903250i \(-0.641174\pi\)
−0.429115 + 0.903250i \(0.641174\pi\)
\(42\) 0 0
\(43\) −2.95367 −0.450430 −0.225215 0.974309i \(-0.572308\pi\)
−0.225215 + 0.974309i \(0.572308\pi\)
\(44\) 5.52607i 0.833086i
\(45\) 0 0
\(46\) 3.08856 0.455383
\(47\) −0.0958559 −0.0139820 −0.00699101 0.999976i \(-0.502225\pi\)
−0.00699101 + 0.999976i \(0.502225\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 7.81504i 1.10521i
\(51\) 0 0
\(52\) 6.91037i 0.958297i
\(53\) − 5.17738i − 0.711167i −0.934644 0.355584i \(-0.884282\pi\)
0.934644 0.355584i \(-0.115718\pi\)
\(54\) 0 0
\(55\) 19.7823i 2.66744i
\(56\) 0 0
\(57\) 0 0
\(58\) −5.66096 −0.743320
\(59\) −6.38401 −0.831127 −0.415563 0.909564i \(-0.636416\pi\)
−0.415563 + 0.909564i \(0.636416\pi\)
\(60\) 0 0
\(61\) 7.06376i 0.904422i 0.891911 + 0.452211i \(0.149365\pi\)
−0.891911 + 0.452211i \(0.850635\pi\)
\(62\) −0.400414 −0.0508526
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 24.7378i 3.06835i
\(66\) 0 0
\(67\) −1.45090 −0.177255 −0.0886276 0.996065i \(-0.528248\pi\)
−0.0886276 + 0.996065i \(0.528248\pi\)
\(68\) 2.44949 0.297044
\(69\) 0 0
\(70\) 0 0
\(71\) 8.77836i 1.04180i 0.853618 + 0.520900i \(0.174403\pi\)
−0.853618 + 0.520900i \(0.825597\pi\)
\(72\) 0 0
\(73\) − 0.679758i − 0.0795597i −0.999208 0.0397798i \(-0.987334\pi\)
0.999208 0.0397798i \(-0.0126657\pi\)
\(74\) 5.79174i 0.673277i
\(75\) 0 0
\(76\) − 8.16176i − 0.936219i
\(77\) 0 0
\(78\) 0 0
\(79\) −6.43340 −0.723815 −0.361907 0.932214i \(-0.617874\pi\)
−0.361907 + 0.932214i \(0.617874\pi\)
\(80\) −3.57981 −0.400235
\(81\) 0 0
\(82\) − 5.49535i − 0.606860i
\(83\) −1.64156 −0.180184 −0.0900921 0.995933i \(-0.528716\pi\)
−0.0900921 + 0.995933i \(0.528716\pi\)
\(84\) 0 0
\(85\) 8.76871 0.951100
\(86\) − 2.95367i − 0.318502i
\(87\) 0 0
\(88\) −5.52607 −0.589081
\(89\) −12.1921 −1.29236 −0.646182 0.763183i \(-0.723635\pi\)
−0.646182 + 0.763183i \(0.723635\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 3.08856i 0.322005i
\(93\) 0 0
\(94\) − 0.0958559i − 0.00988678i
\(95\) − 29.2176i − 2.99766i
\(96\) 0 0
\(97\) 11.3788i 1.15535i 0.816268 + 0.577673i \(0.196039\pi\)
−0.816268 + 0.577673i \(0.803961\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −7.81504 −0.781504
\(101\) 1.82679 0.181772 0.0908862 0.995861i \(-0.471030\pi\)
0.0908862 + 0.995861i \(0.471030\pi\)
\(102\) 0 0
\(103\) − 5.79030i − 0.570535i −0.958448 0.285268i \(-0.907917\pi\)
0.958448 0.285268i \(-0.0920825\pi\)
\(104\) −6.91037 −0.677618
\(105\) 0 0
\(106\) 5.17738 0.502871
\(107\) − 5.49867i − 0.531576i −0.964031 0.265788i \(-0.914368\pi\)
0.964031 0.265788i \(-0.0856322\pi\)
\(108\) 0 0
\(109\) 13.4705 1.29024 0.645118 0.764083i \(-0.276808\pi\)
0.645118 + 0.764083i \(0.276808\pi\)
\(110\) −19.7823 −1.88617
\(111\) 0 0
\(112\) 0 0
\(113\) − 11.8843i − 1.11798i −0.829174 0.558990i \(-0.811189\pi\)
0.829174 0.558990i \(-0.188811\pi\)
\(114\) 0 0
\(115\) 11.0565i 1.03102i
\(116\) − 5.66096i − 0.525607i
\(117\) 0 0
\(118\) − 6.38401i − 0.587695i
\(119\) 0 0
\(120\) 0 0
\(121\) −19.5374 −1.77613
\(122\) −7.06376 −0.639523
\(123\) 0 0
\(124\) − 0.400414i − 0.0359582i
\(125\) −10.0773 −0.901341
\(126\) 0 0
\(127\) 7.09410 0.629500 0.314750 0.949175i \(-0.398079\pi\)
0.314750 + 0.949175i \(0.398079\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 0 0
\(130\) −24.7378 −2.16965
\(131\) 16.0312 1.40066 0.700328 0.713821i \(-0.253037\pi\)
0.700328 + 0.713821i \(0.253037\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) − 1.45090i − 0.125338i
\(135\) 0 0
\(136\) 2.44949i 0.210042i
\(137\) 6.94172i 0.593071i 0.955022 + 0.296536i \(0.0958314\pi\)
−0.955022 + 0.296536i \(0.904169\pi\)
\(138\) 0 0
\(139\) − 0.273984i − 0.0232390i −0.999932 0.0116195i \(-0.996301\pi\)
0.999932 0.0116195i \(-0.00369868\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −8.77836 −0.736663
\(143\) −38.1872 −3.19337
\(144\) 0 0
\(145\) − 20.2652i − 1.68293i
\(146\) 0.679758 0.0562572
\(147\) 0 0
\(148\) −5.79174 −0.476078
\(149\) 19.1139i 1.56587i 0.622102 + 0.782937i \(0.286279\pi\)
−0.622102 + 0.782937i \(0.713721\pi\)
\(150\) 0 0
\(151\) −8.61463 −0.701048 −0.350524 0.936554i \(-0.613997\pi\)
−0.350524 + 0.936554i \(0.613997\pi\)
\(152\) 8.16176 0.662006
\(153\) 0 0
\(154\) 0 0
\(155\) − 1.43340i − 0.115134i
\(156\) 0 0
\(157\) 10.5807i 0.844430i 0.906496 + 0.422215i \(0.138747\pi\)
−0.906496 + 0.422215i \(0.861253\pi\)
\(158\) − 6.43340i − 0.511814i
\(159\) 0 0
\(160\) − 3.57981i − 0.283009i
\(161\) 0 0
\(162\) 0 0
\(163\) 9.25809 0.725150 0.362575 0.931955i \(-0.381898\pi\)
0.362575 + 0.931955i \(0.381898\pi\)
\(164\) 5.49535 0.429115
\(165\) 0 0
\(166\) − 1.64156i − 0.127409i
\(167\) −14.6539 −1.13395 −0.566976 0.823734i \(-0.691887\pi\)
−0.566976 + 0.823734i \(0.691887\pi\)
\(168\) 0 0
\(169\) −34.7533 −2.67333
\(170\) 8.76871i 0.672529i
\(171\) 0 0
\(172\) 2.95367 0.225215
\(173\) −7.93301 −0.603136 −0.301568 0.953445i \(-0.597510\pi\)
−0.301568 + 0.953445i \(0.597510\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 5.52607i − 0.416543i
\(177\) 0 0
\(178\) − 12.1921i − 0.913839i
\(179\) 16.7361i 1.25092i 0.780257 + 0.625458i \(0.215088\pi\)
−0.780257 + 0.625458i \(0.784912\pi\)
\(180\) 0 0
\(181\) 9.85077i 0.732202i 0.930575 + 0.366101i \(0.119308\pi\)
−0.930575 + 0.366101i \(0.880692\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −3.08856 −0.227692
\(185\) −20.7333 −1.52435
\(186\) 0 0
\(187\) 13.5360i 0.989854i
\(188\) 0.0958559 0.00699101
\(189\) 0 0
\(190\) 29.2176 2.11966
\(191\) 17.5738i 1.27160i 0.771854 + 0.635799i \(0.219329\pi\)
−0.771854 + 0.635799i \(0.780671\pi\)
\(192\) 0 0
\(193\) 19.8534 1.42908 0.714540 0.699594i \(-0.246636\pi\)
0.714540 + 0.699594i \(0.246636\pi\)
\(194\) −11.3788 −0.816953
\(195\) 0 0
\(196\) 0 0
\(197\) 21.5007i 1.53186i 0.642922 + 0.765932i \(0.277722\pi\)
−0.642922 + 0.765932i \(0.722278\pi\)
\(198\) 0 0
\(199\) − 13.4805i − 0.955610i −0.878466 0.477805i \(-0.841433\pi\)
0.878466 0.477805i \(-0.158567\pi\)
\(200\) − 7.81504i − 0.552607i
\(201\) 0 0
\(202\) 1.82679i 0.128533i
\(203\) 0 0
\(204\) 0 0
\(205\) 19.6723 1.37397
\(206\) 5.79030 0.403429
\(207\) 0 0
\(208\) − 6.91037i − 0.479148i
\(209\) 45.1025 3.11980
\(210\) 0 0
\(211\) −23.8575 −1.64242 −0.821210 0.570626i \(-0.806700\pi\)
−0.821210 + 0.570626i \(0.806700\pi\)
\(212\) 5.17738i 0.355584i
\(213\) 0 0
\(214\) 5.49867 0.375881
\(215\) 10.5736 0.721112
\(216\) 0 0
\(217\) 0 0
\(218\) 13.4705i 0.912334i
\(219\) 0 0
\(220\) − 19.7823i − 1.33372i
\(221\) 16.9269i 1.13863i
\(222\) 0 0
\(223\) − 7.75175i − 0.519095i −0.965730 0.259548i \(-0.916427\pi\)
0.965730 0.259548i \(-0.0835735\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 11.8843 0.790532
\(227\) 11.1867 0.742484 0.371242 0.928536i \(-0.378932\pi\)
0.371242 + 0.928536i \(0.378932\pi\)
\(228\) 0 0
\(229\) 16.0107i 1.05802i 0.848616 + 0.529010i \(0.177437\pi\)
−0.848616 + 0.529010i \(0.822563\pi\)
\(230\) −11.0565 −0.729041
\(231\) 0 0
\(232\) 5.66096 0.371660
\(233\) − 5.62801i − 0.368703i −0.982860 0.184352i \(-0.940981\pi\)
0.982860 0.184352i \(-0.0590186\pi\)
\(234\) 0 0
\(235\) 0.343146 0.0223844
\(236\) 6.38401 0.415563
\(237\) 0 0
\(238\) 0 0
\(239\) 25.9630i 1.67941i 0.543046 + 0.839703i \(0.317271\pi\)
−0.543046 + 0.839703i \(0.682729\pi\)
\(240\) 0 0
\(241\) − 14.5302i − 0.935970i −0.883736 0.467985i \(-0.844980\pi\)
0.883736 0.467985i \(-0.155020\pi\)
\(242\) − 19.5374i − 1.25591i
\(243\) 0 0
\(244\) − 7.06376i − 0.452211i
\(245\) 0 0
\(246\) 0 0
\(247\) 56.4008 3.58870
\(248\) 0.400414 0.0254263
\(249\) 0 0
\(250\) − 10.0773i − 0.637345i
\(251\) −21.5246 −1.35862 −0.679309 0.733852i \(-0.737721\pi\)
−0.679309 + 0.733852i \(0.737721\pi\)
\(252\) 0 0
\(253\) −17.0676 −1.07303
\(254\) 7.09410i 0.445124i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 9.85613 0.614809 0.307404 0.951579i \(-0.400540\pi\)
0.307404 + 0.951579i \(0.400540\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) − 24.7378i − 1.53417i
\(261\) 0 0
\(262\) 16.0312i 0.990413i
\(263\) 3.29041i 0.202896i 0.994841 + 0.101448i \(0.0323475\pi\)
−0.994841 + 0.101448i \(0.967653\pi\)
\(264\) 0 0
\(265\) 18.5340i 1.13854i
\(266\) 0 0
\(267\) 0 0
\(268\) 1.45090 0.0886276
\(269\) 18.9670 1.15644 0.578218 0.815883i \(-0.303748\pi\)
0.578218 + 0.815883i \(0.303748\pi\)
\(270\) 0 0
\(271\) 10.4806i 0.636650i 0.947982 + 0.318325i \(0.103120\pi\)
−0.947982 + 0.318325i \(0.896880\pi\)
\(272\) −2.44949 −0.148522
\(273\) 0 0
\(274\) −6.94172 −0.419365
\(275\) − 43.1864i − 2.60424i
\(276\) 0 0
\(277\) −6.37173 −0.382840 −0.191420 0.981508i \(-0.561309\pi\)
−0.191420 + 0.981508i \(0.561309\pi\)
\(278\) 0.273984 0.0164324
\(279\) 0 0
\(280\) 0 0
\(281\) 1.76520i 0.105303i 0.998613 + 0.0526516i \(0.0167673\pi\)
−0.998613 + 0.0526516i \(0.983233\pi\)
\(282\) 0 0
\(283\) 26.1243i 1.55293i 0.630160 + 0.776466i \(0.282989\pi\)
−0.630160 + 0.776466i \(0.717011\pi\)
\(284\) − 8.77836i − 0.520900i
\(285\) 0 0
\(286\) − 38.1872i − 2.25806i
\(287\) 0 0
\(288\) 0 0
\(289\) −11.0000 −0.647059
\(290\) 20.2652 1.19001
\(291\) 0 0
\(292\) 0.679758i 0.0397798i
\(293\) −12.8799 −0.752453 −0.376226 0.926528i \(-0.622779\pi\)
−0.376226 + 0.926528i \(0.622779\pi\)
\(294\) 0 0
\(295\) 22.8535 1.33058
\(296\) − 5.79174i − 0.336638i
\(297\) 0 0
\(298\) −19.1139 −1.10724
\(299\) −21.3431 −1.23430
\(300\) 0 0
\(301\) 0 0
\(302\) − 8.61463i − 0.495716i
\(303\) 0 0
\(304\) 8.16176i 0.468109i
\(305\) − 25.2869i − 1.44793i
\(306\) 0 0
\(307\) − 11.8601i − 0.676894i −0.940985 0.338447i \(-0.890098\pi\)
0.940985 0.338447i \(-0.109902\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.43340 0.0814119
\(311\) −9.41692 −0.533985 −0.266992 0.963699i \(-0.586030\pi\)
−0.266992 + 0.963699i \(0.586030\pi\)
\(312\) 0 0
\(313\) − 5.60415i − 0.316765i −0.987378 0.158383i \(-0.949372\pi\)
0.987378 0.158383i \(-0.0506279\pi\)
\(314\) −10.5807 −0.597102
\(315\) 0 0
\(316\) 6.43340 0.361907
\(317\) 25.8724i 1.45314i 0.687094 + 0.726568i \(0.258886\pi\)
−0.687094 + 0.726568i \(0.741114\pi\)
\(318\) 0 0
\(319\) 31.2828 1.75150
\(320\) 3.57981 0.200117
\(321\) 0 0
\(322\) 0 0
\(323\) − 19.9922i − 1.11239i
\(324\) 0 0
\(325\) − 54.0048i − 2.99565i
\(326\) 9.25809i 0.512758i
\(327\) 0 0
\(328\) 5.49535i 0.303430i
\(329\) 0 0
\(330\) 0 0
\(331\) 11.0987 0.610041 0.305020 0.952346i \(-0.401337\pi\)
0.305020 + 0.952346i \(0.401337\pi\)
\(332\) 1.64156 0.0900921
\(333\) 0 0
\(334\) − 14.6539i − 0.801826i
\(335\) 5.19393 0.283775
\(336\) 0 0
\(337\) −16.4340 −0.895218 −0.447609 0.894229i \(-0.647724\pi\)
−0.447609 + 0.894229i \(0.647724\pi\)
\(338\) − 34.7533i − 1.89033i
\(339\) 0 0
\(340\) −8.76871 −0.475550
\(341\) 2.21271 0.119825
\(342\) 0 0
\(343\) 0 0
\(344\) 2.95367i 0.159251i
\(345\) 0 0
\(346\) − 7.93301i − 0.426481i
\(347\) 6.61282i 0.354995i 0.984121 + 0.177497i \(0.0568001\pi\)
−0.984121 + 0.177497i \(0.943200\pi\)
\(348\) 0 0
\(349\) − 17.9468i − 0.960669i −0.877085 0.480335i \(-0.840515\pi\)
0.877085 0.480335i \(-0.159485\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.52607 0.294540
\(353\) −3.64649 −0.194083 −0.0970414 0.995280i \(-0.530938\pi\)
−0.0970414 + 0.995280i \(0.530938\pi\)
\(354\) 0 0
\(355\) − 31.4248i − 1.66786i
\(356\) 12.1921 0.646182
\(357\) 0 0
\(358\) −16.7361 −0.884532
\(359\) − 3.05213i − 0.161085i −0.996751 0.0805427i \(-0.974335\pi\)
0.996751 0.0805427i \(-0.0256653\pi\)
\(360\) 0 0
\(361\) −47.6144 −2.50602
\(362\) −9.85077 −0.517745
\(363\) 0 0
\(364\) 0 0
\(365\) 2.43340i 0.127370i
\(366\) 0 0
\(367\) 22.7523i 1.18766i 0.804590 + 0.593831i \(0.202385\pi\)
−0.804590 + 0.593831i \(0.797615\pi\)
\(368\) − 3.08856i − 0.161002i
\(369\) 0 0
\(370\) − 20.7333i − 1.07788i
\(371\) 0 0
\(372\) 0 0
\(373\) −0.718273 −0.0371908 −0.0185954 0.999827i \(-0.505919\pi\)
−0.0185954 + 0.999827i \(0.505919\pi\)
\(374\) −13.5360 −0.699932
\(375\) 0 0
\(376\) 0.0958559i 0.00494339i
\(377\) 39.1193 2.01475
\(378\) 0 0
\(379\) 14.4935 0.744480 0.372240 0.928136i \(-0.378590\pi\)
0.372240 + 0.928136i \(0.378590\pi\)
\(380\) 29.2176i 1.49883i
\(381\) 0 0
\(382\) −17.5738 −0.899156
\(383\) 19.1105 0.976503 0.488251 0.872703i \(-0.337635\pi\)
0.488251 + 0.872703i \(0.337635\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 19.8534i 1.01051i
\(387\) 0 0
\(388\) − 11.3788i − 0.577673i
\(389\) − 1.90154i − 0.0964117i −0.998837 0.0482058i \(-0.984650\pi\)
0.998837 0.0482058i \(-0.0153503\pi\)
\(390\) 0 0
\(391\) 7.56539i 0.382598i
\(392\) 0 0
\(393\) 0 0
\(394\) −21.5007 −1.08319
\(395\) 23.0304 1.15878
\(396\) 0 0
\(397\) 15.5802i 0.781950i 0.920401 + 0.390975i \(0.127862\pi\)
−0.920401 + 0.390975i \(0.872138\pi\)
\(398\) 13.4805 0.675718
\(399\) 0 0
\(400\) 7.81504 0.390752
\(401\) − 34.2752i − 1.71162i −0.517287 0.855812i \(-0.673058\pi\)
0.517287 0.855812i \(-0.326942\pi\)
\(402\) 0 0
\(403\) 2.76701 0.137834
\(404\) −1.82679 −0.0908862
\(405\) 0 0
\(406\) 0 0
\(407\) − 32.0056i − 1.58646i
\(408\) 0 0
\(409\) − 4.14858i − 0.205134i −0.994726 0.102567i \(-0.967294\pi\)
0.994726 0.102567i \(-0.0327056\pi\)
\(410\) 19.6723i 0.971546i
\(411\) 0 0
\(412\) 5.79030i 0.285268i
\(413\) 0 0
\(414\) 0 0
\(415\) 5.87646 0.288464
\(416\) 6.91037 0.338809
\(417\) 0 0
\(418\) 45.1025i 2.20603i
\(419\) 16.8972 0.825481 0.412741 0.910849i \(-0.364572\pi\)
0.412741 + 0.910849i \(0.364572\pi\)
\(420\) 0 0
\(421\) 7.16773 0.349334 0.174667 0.984628i \(-0.444115\pi\)
0.174667 + 0.984628i \(0.444115\pi\)
\(422\) − 23.8575i − 1.16137i
\(423\) 0 0
\(424\) −5.17738 −0.251436
\(425\) −19.1429 −0.928565
\(426\) 0 0
\(427\) 0 0
\(428\) 5.49867i 0.265788i
\(429\) 0 0
\(430\) 10.5736i 0.509903i
\(431\) 19.6670i 0.947327i 0.880706 + 0.473663i \(0.157069\pi\)
−0.880706 + 0.473663i \(0.842931\pi\)
\(432\) 0 0
\(433\) 16.1213i 0.774740i 0.921924 + 0.387370i \(0.126616\pi\)
−0.921924 + 0.387370i \(0.873384\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −13.4705 −0.645118
\(437\) 25.2081 1.20587
\(438\) 0 0
\(439\) − 26.7007i − 1.27435i −0.770718 0.637177i \(-0.780102\pi\)
0.770718 0.637177i \(-0.219898\pi\)
\(440\) 19.7823 0.943083
\(441\) 0 0
\(442\) −16.9269 −0.805130
\(443\) 19.8643i 0.943779i 0.881658 + 0.471890i \(0.156428\pi\)
−0.881658 + 0.471890i \(0.843572\pi\)
\(444\) 0 0
\(445\) 43.6455 2.06900
\(446\) 7.75175 0.367056
\(447\) 0 0
\(448\) 0 0
\(449\) − 20.0132i − 0.944479i −0.881470 0.472240i \(-0.843446\pi\)
0.881470 0.472240i \(-0.156554\pi\)
\(450\) 0 0
\(451\) 30.3677i 1.42996i
\(452\) 11.8843i 0.558990i
\(453\) 0 0
\(454\) 11.1867i 0.525016i
\(455\) 0 0
\(456\) 0 0
\(457\) −23.9138 −1.11864 −0.559319 0.828952i \(-0.688937\pi\)
−0.559319 + 0.828952i \(0.688937\pi\)
\(458\) −16.0107 −0.748133
\(459\) 0 0
\(460\) − 11.0565i − 0.515510i
\(461\) 18.6642 0.869279 0.434640 0.900604i \(-0.356876\pi\)
0.434640 + 0.900604i \(0.356876\pi\)
\(462\) 0 0
\(463\) −7.64166 −0.355138 −0.177569 0.984108i \(-0.556823\pi\)
−0.177569 + 0.984108i \(0.556823\pi\)
\(464\) 5.66096i 0.262803i
\(465\) 0 0
\(466\) 5.62801 0.260713
\(467\) 34.7012 1.60578 0.802890 0.596127i \(-0.203294\pi\)
0.802890 + 0.596127i \(0.203294\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0.343146i 0.0158281i
\(471\) 0 0
\(472\) 6.38401i 0.293848i
\(473\) 16.3222i 0.750494i
\(474\) 0 0
\(475\) 63.7845i 2.92663i
\(476\) 0 0
\(477\) 0 0
\(478\) −25.9630 −1.18752
\(479\) 26.2363 1.19877 0.599383 0.800462i \(-0.295413\pi\)
0.599383 + 0.800462i \(0.295413\pi\)
\(480\) 0 0
\(481\) − 40.0231i − 1.82490i
\(482\) 14.5302 0.661831
\(483\) 0 0
\(484\) 19.5374 0.888064
\(485\) − 40.7341i − 1.84964i
\(486\) 0 0
\(487\) −7.73022 −0.350290 −0.175145 0.984543i \(-0.556039\pi\)
−0.175145 + 0.984543i \(0.556039\pi\)
\(488\) 7.06376 0.319762
\(489\) 0 0
\(490\) 0 0
\(491\) − 33.0173i − 1.49005i −0.667038 0.745024i \(-0.732438\pi\)
0.667038 0.745024i \(-0.267562\pi\)
\(492\) 0 0
\(493\) − 13.8665i − 0.624514i
\(494\) 56.4008i 2.53759i
\(495\) 0 0
\(496\) 0.400414i 0.0179791i
\(497\) 0 0
\(498\) 0 0
\(499\) −7.61078 −0.340705 −0.170353 0.985383i \(-0.554491\pi\)
−0.170353 + 0.985383i \(0.554491\pi\)
\(500\) 10.0773 0.450671
\(501\) 0 0
\(502\) − 21.5246i − 0.960688i
\(503\) −39.9386 −1.78078 −0.890388 0.455203i \(-0.849567\pi\)
−0.890388 + 0.455203i \(0.849567\pi\)
\(504\) 0 0
\(505\) −6.53956 −0.291007
\(506\) − 17.0676i − 0.758747i
\(507\) 0 0
\(508\) −7.09410 −0.314750
\(509\) 25.5868 1.13411 0.567057 0.823678i \(-0.308082\pi\)
0.567057 + 0.823678i \(0.308082\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 9.85613i 0.434735i
\(515\) 20.7282i 0.913393i
\(516\) 0 0
\(517\) 0.529706i 0.0232964i
\(518\) 0 0
\(519\) 0 0
\(520\) 24.7378 1.08483
\(521\) 14.8838 0.652070 0.326035 0.945358i \(-0.394287\pi\)
0.326035 + 0.945358i \(0.394287\pi\)
\(522\) 0 0
\(523\) 7.63492i 0.333852i 0.985969 + 0.166926i \(0.0533841\pi\)
−0.985969 + 0.166926i \(0.946616\pi\)
\(524\) −16.0312 −0.700328
\(525\) 0 0
\(526\) −3.29041 −0.143469
\(527\) − 0.980809i − 0.0427247i
\(528\) 0 0
\(529\) 13.4608 0.585252
\(530\) −18.5340 −0.805066
\(531\) 0 0
\(532\) 0 0
\(533\) 37.9749i 1.64488i
\(534\) 0 0
\(535\) 19.6842i 0.851022i
\(536\) 1.45090i 0.0626692i
\(537\) 0 0
\(538\) 18.9670i 0.817723i
\(539\) 0 0
\(540\) 0 0
\(541\) −36.0600 −1.55034 −0.775170 0.631752i \(-0.782336\pi\)
−0.775170 + 0.631752i \(0.782336\pi\)
\(542\) −10.4806 −0.450179
\(543\) 0 0
\(544\) − 2.44949i − 0.105021i
\(545\) −48.2217 −2.06559
\(546\) 0 0
\(547\) −34.3548 −1.46890 −0.734452 0.678661i \(-0.762561\pi\)
−0.734452 + 0.678661i \(0.762561\pi\)
\(548\) − 6.94172i − 0.296536i
\(549\) 0 0
\(550\) 43.1864 1.84148
\(551\) −46.2034 −1.96833
\(552\) 0 0
\(553\) 0 0
\(554\) − 6.37173i − 0.270709i
\(555\) 0 0
\(556\) 0.273984i 0.0116195i
\(557\) − 36.6100i − 1.55122i −0.631214 0.775608i \(-0.717443\pi\)
0.631214 0.775608i \(-0.282557\pi\)
\(558\) 0 0
\(559\) 20.4110i 0.863292i
\(560\) 0 0
\(561\) 0 0
\(562\) −1.76520 −0.0744606
\(563\) 33.0644 1.39350 0.696749 0.717315i \(-0.254629\pi\)
0.696749 + 0.717315i \(0.254629\pi\)
\(564\) 0 0
\(565\) 42.5435i 1.78982i
\(566\) −26.1243 −1.09809
\(567\) 0 0
\(568\) 8.77836 0.368332
\(569\) − 44.0195i − 1.84539i −0.385528 0.922696i \(-0.625981\pi\)
0.385528 0.922696i \(-0.374019\pi\)
\(570\) 0 0
\(571\) −24.7407 −1.03537 −0.517684 0.855572i \(-0.673206\pi\)
−0.517684 + 0.855572i \(0.673206\pi\)
\(572\) 38.1872 1.59669
\(573\) 0 0
\(574\) 0 0
\(575\) − 24.1372i − 1.00659i
\(576\) 0 0
\(577\) 20.6883i 0.861264i 0.902528 + 0.430632i \(0.141709\pi\)
−0.902528 + 0.430632i \(0.858291\pi\)
\(578\) − 11.0000i − 0.457540i
\(579\) 0 0
\(580\) 20.2652i 0.841465i
\(581\) 0 0
\(582\) 0 0
\(583\) −28.6105 −1.18493
\(584\) −0.679758 −0.0281286
\(585\) 0 0
\(586\) − 12.8799i − 0.532065i
\(587\) 4.24396 0.175167 0.0875835 0.996157i \(-0.472086\pi\)
0.0875835 + 0.996157i \(0.472086\pi\)
\(588\) 0 0
\(589\) −3.26808 −0.134659
\(590\) 22.8535i 0.940865i
\(591\) 0 0
\(592\) 5.79174 0.238039
\(593\) −14.8951 −0.611669 −0.305835 0.952085i \(-0.598935\pi\)
−0.305835 + 0.952085i \(0.598935\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 19.1139i − 0.782937i
\(597\) 0 0
\(598\) − 21.3431i − 0.872784i
\(599\) − 21.4812i − 0.877697i −0.898561 0.438849i \(-0.855386\pi\)
0.898561 0.438849i \(-0.144614\pi\)
\(600\) 0 0
\(601\) − 37.5607i − 1.53213i −0.642761 0.766067i \(-0.722211\pi\)
0.642761 0.766067i \(-0.277789\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 8.61463 0.350524
\(605\) 69.9402 2.84347
\(606\) 0 0
\(607\) 41.7681i 1.69532i 0.530543 + 0.847658i \(0.321988\pi\)
−0.530543 + 0.847658i \(0.678012\pi\)
\(608\) −8.16176 −0.331003
\(609\) 0 0
\(610\) 25.2869 1.02384
\(611\) 0.662400i 0.0267978i
\(612\) 0 0
\(613\) 31.5940 1.27607 0.638035 0.770007i \(-0.279747\pi\)
0.638035 + 0.770007i \(0.279747\pi\)
\(614\) 11.8601 0.478636
\(615\) 0 0
\(616\) 0 0
\(617\) − 48.9718i − 1.97153i −0.168127 0.985765i \(-0.553772\pi\)
0.168127 0.985765i \(-0.446228\pi\)
\(618\) 0 0
\(619\) − 6.67646i − 0.268350i −0.990958 0.134175i \(-0.957162\pi\)
0.990958 0.134175i \(-0.0428384\pi\)
\(620\) 1.43340i 0.0575669i
\(621\) 0 0
\(622\) − 9.41692i − 0.377584i
\(623\) 0 0
\(624\) 0 0
\(625\) −3.00036 −0.120015
\(626\) 5.60415 0.223987
\(627\) 0 0
\(628\) − 10.5807i − 0.422215i
\(629\) −14.1868 −0.565666
\(630\) 0 0
\(631\) −0.503852 −0.0200580 −0.0100290 0.999950i \(-0.503192\pi\)
−0.0100290 + 0.999950i \(0.503192\pi\)
\(632\) 6.43340i 0.255907i
\(633\) 0 0
\(634\) −25.8724 −1.02752
\(635\) −25.3955 −1.00779
\(636\) 0 0
\(637\) 0 0
\(638\) 31.2828i 1.23850i
\(639\) 0 0
\(640\) 3.57981i 0.141504i
\(641\) 20.8631i 0.824042i 0.911174 + 0.412021i \(0.135177\pi\)
−0.911174 + 0.412021i \(0.864823\pi\)
\(642\) 0 0
\(643\) − 33.6322i − 1.32632i −0.748476 0.663162i \(-0.769214\pi\)
0.748476 0.663162i \(-0.230786\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 19.9922 0.786581
\(647\) 17.6878 0.695378 0.347689 0.937610i \(-0.386967\pi\)
0.347689 + 0.937610i \(0.386967\pi\)
\(648\) 0 0
\(649\) 35.2784i 1.38480i
\(650\) 54.0048 2.11824
\(651\) 0 0
\(652\) −9.25809 −0.362575
\(653\) 9.80139i 0.383558i 0.981438 + 0.191779i \(0.0614257\pi\)
−0.981438 + 0.191779i \(0.938574\pi\)
\(654\) 0 0
\(655\) −57.3888 −2.24237
\(656\) −5.49535 −0.214557
\(657\) 0 0
\(658\) 0 0
\(659\) 28.9869i 1.12917i 0.825376 + 0.564584i \(0.190963\pi\)
−0.825376 + 0.564584i \(0.809037\pi\)
\(660\) 0 0
\(661\) − 7.41931i − 0.288578i −0.989536 0.144289i \(-0.953911\pi\)
0.989536 0.144289i \(-0.0460894\pi\)
\(662\) 11.0987i 0.431364i
\(663\) 0 0
\(664\) 1.64156i 0.0637047i
\(665\) 0 0
\(666\) 0 0
\(667\) 17.4842 0.676991
\(668\) 14.6539 0.566976
\(669\) 0 0
\(670\) 5.19393i 0.200659i
\(671\) 39.0348 1.50692
\(672\) 0 0
\(673\) 27.5802 1.06314 0.531570 0.847014i \(-0.321602\pi\)
0.531570 + 0.847014i \(0.321602\pi\)
\(674\) − 16.4340i − 0.633015i
\(675\) 0 0
\(676\) 34.7533 1.33666
\(677\) 21.7087 0.834334 0.417167 0.908830i \(-0.363023\pi\)
0.417167 + 0.908830i \(0.363023\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) − 8.76871i − 0.336265i
\(681\) 0 0
\(682\) 2.21271i 0.0847291i
\(683\) 22.0113i 0.842241i 0.907005 + 0.421120i \(0.138363\pi\)
−0.907005 + 0.421120i \(0.861637\pi\)
\(684\) 0 0
\(685\) − 24.8500i − 0.949472i
\(686\) 0 0
\(687\) 0 0
\(688\) −2.95367 −0.112608
\(689\) −35.7776 −1.36302
\(690\) 0 0
\(691\) 19.8589i 0.755470i 0.925914 + 0.377735i \(0.123297\pi\)
−0.925914 + 0.377735i \(0.876703\pi\)
\(692\) 7.93301 0.301568
\(693\) 0 0
\(694\) −6.61282 −0.251019
\(695\) 0.980809i 0.0372042i
\(696\) 0 0
\(697\) 13.4608 0.509864
\(698\) 17.9468 0.679296
\(699\) 0 0
\(700\) 0 0
\(701\) − 11.8055i − 0.445887i −0.974831 0.222944i \(-0.928433\pi\)
0.974831 0.222944i \(-0.0715666\pi\)
\(702\) 0 0
\(703\) 47.2709i 1.78285i
\(704\) 5.52607i 0.208271i
\(705\) 0 0
\(706\) − 3.64649i − 0.137237i
\(707\) 0 0
\(708\) 0 0
\(709\) −9.33992 −0.350768 −0.175384 0.984500i \(-0.556117\pi\)
−0.175384 + 0.984500i \(0.556117\pi\)
\(710\) 31.4248 1.17935
\(711\) 0 0
\(712\) 12.1921i 0.456920i
\(713\) 1.23670 0.0463148
\(714\) 0 0
\(715\) 136.703 5.11240
\(716\) − 16.7361i − 0.625458i
\(717\) 0 0
\(718\) 3.05213 0.113905
\(719\) −38.6032 −1.43966 −0.719828 0.694152i \(-0.755780\pi\)
−0.719828 + 0.694152i \(0.755780\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 47.6144i − 1.77202i
\(723\) 0 0
\(724\) − 9.85077i − 0.366101i
\(725\) 44.2406i 1.64305i
\(726\) 0 0
\(727\) 14.9640i 0.554984i 0.960728 + 0.277492i \(0.0895033\pi\)
−0.960728 + 0.277492i \(0.910497\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −2.43340 −0.0900643
\(731\) 7.23498 0.267595
\(732\) 0 0
\(733\) − 28.6378i − 1.05776i −0.848696 0.528881i \(-0.822612\pi\)
0.848696 0.528881i \(-0.177388\pi\)
\(734\) −22.7523 −0.839804
\(735\) 0 0
\(736\) 3.08856 0.113846
\(737\) 8.01775i 0.295338i
\(738\) 0 0
\(739\) −33.8925 −1.24676 −0.623378 0.781921i \(-0.714240\pi\)
−0.623378 + 0.781921i \(0.714240\pi\)
\(740\) 20.7333 0.762173
\(741\) 0 0
\(742\) 0 0
\(743\) 29.8058i 1.09347i 0.837307 + 0.546734i \(0.184129\pi\)
−0.837307 + 0.546734i \(0.815871\pi\)
\(744\) 0 0
\(745\) − 68.4242i − 2.50687i
\(746\) − 0.718273i − 0.0262979i
\(747\) 0 0
\(748\) − 13.5360i − 0.494927i
\(749\) 0 0
\(750\) 0 0
\(751\) −20.0036 −0.729942 −0.364971 0.931019i \(-0.618921\pi\)
−0.364971 + 0.931019i \(0.618921\pi\)
\(752\) −0.0958559 −0.00349550
\(753\) 0 0
\(754\) 39.1193i 1.42464i
\(755\) 30.8387 1.12234
\(756\) 0 0
\(757\) −27.6330 −1.00434 −0.502169 0.864770i \(-0.667464\pi\)
−0.502169 + 0.864770i \(0.667464\pi\)
\(758\) 14.4935i 0.526427i
\(759\) 0 0
\(760\) −29.2176 −1.05983
\(761\) −26.3712 −0.955954 −0.477977 0.878372i \(-0.658630\pi\)
−0.477977 + 0.878372i \(0.658630\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) − 17.5738i − 0.635799i
\(765\) 0 0
\(766\) 19.1105i 0.690492i
\(767\) 44.1159i 1.59293i
\(768\) 0 0
\(769\) − 46.4263i − 1.67418i −0.547068 0.837088i \(-0.684256\pi\)
0.547068 0.837088i \(-0.315744\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −19.8534 −0.714540
\(773\) −9.18820 −0.330477 −0.165238 0.986254i \(-0.552839\pi\)
−0.165238 + 0.986254i \(0.552839\pi\)
\(774\) 0 0
\(775\) 3.12925i 0.112406i
\(776\) 11.3788 0.408477
\(777\) 0 0
\(778\) 1.90154 0.0681733
\(779\) − 44.8517i − 1.60698i
\(780\) 0 0
\(781\) 48.5098 1.73582
\(782\) −7.56539 −0.270538
\(783\) 0 0
\(784\) 0 0
\(785\) − 37.8768i − 1.35188i
\(786\) 0 0
\(787\) − 16.9582i − 0.604495i −0.953230 0.302247i \(-0.902263\pi\)
0.953230 0.302247i \(-0.0977369\pi\)
\(788\) − 21.5007i − 0.765932i
\(789\) 0 0
\(790\) 23.0304i 0.819384i
\(791\) 0 0
\(792\) 0 0
\(793\) 48.8133 1.73341
\(794\) −15.5802 −0.552922
\(795\) 0 0
\(796\) 13.4805i 0.477805i
\(797\) −28.7507 −1.01840 −0.509200 0.860648i \(-0.670059\pi\)
−0.509200 + 0.860648i \(0.670059\pi\)
\(798\) 0 0
\(799\) 0.234798 0.00830655
\(800\) 7.81504i 0.276303i
\(801\) 0 0
\(802\) 34.2752 1.21030
\(803\) −3.75639 −0.132560
\(804\) 0 0
\(805\) 0 0
\(806\) 2.76701i 0.0974637i
\(807\) 0 0
\(808\) − 1.82679i − 0.0642663i
\(809\) 20.9726i 0.737358i 0.929557 + 0.368679i \(0.120190\pi\)
−0.929557 + 0.368679i \(0.879810\pi\)
\(810\) 0 0
\(811\) − 35.0116i − 1.22942i −0.788752 0.614711i \(-0.789272\pi\)
0.788752 0.614711i \(-0.210728\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 32.0056 1.12179
\(815\) −33.1422 −1.16092
\(816\) 0 0
\(817\) − 24.1071i − 0.843402i
\(818\) 4.14858 0.145052
\(819\) 0 0
\(820\) −19.6723 −0.686987
\(821\) − 39.6709i − 1.38452i −0.721647 0.692261i \(-0.756615\pi\)
0.721647 0.692261i \(-0.243385\pi\)
\(822\) 0 0
\(823\) −3.17619 −0.110715 −0.0553575 0.998467i \(-0.517630\pi\)
−0.0553575 + 0.998467i \(0.517630\pi\)
\(824\) −5.79030 −0.201715
\(825\) 0 0
\(826\) 0 0
\(827\) − 25.7069i − 0.893918i −0.894554 0.446959i \(-0.852507\pi\)
0.894554 0.446959i \(-0.147493\pi\)
\(828\) 0 0
\(829\) − 0.943062i − 0.0327539i −0.999866 0.0163769i \(-0.994787\pi\)
0.999866 0.0163769i \(-0.00521318\pi\)
\(830\) 5.87646i 0.203975i
\(831\) 0 0
\(832\) 6.91037i 0.239574i
\(833\) 0 0
\(834\) 0 0
\(835\) 52.4582 1.81539
\(836\) −45.1025 −1.55990
\(837\) 0 0
\(838\) 16.8972i 0.583704i
\(839\) 9.97589 0.344406 0.172203 0.985061i \(-0.444911\pi\)
0.172203 + 0.985061i \(0.444911\pi\)
\(840\) 0 0
\(841\) −3.04644 −0.105050
\(842\) 7.16773i 0.247016i
\(843\) 0 0
\(844\) 23.8575 0.821210
\(845\) 124.410 4.27984
\(846\) 0 0
\(847\) 0 0
\(848\) − 5.17738i − 0.177792i
\(849\) 0 0
\(850\) − 19.1429i − 0.656595i
\(851\) − 17.8881i − 0.613198i
\(852\) 0 0
\(853\) − 0.267352i − 0.00915394i −0.999990 0.00457697i \(-0.998543\pi\)
0.999990 0.00457697i \(-0.00145690\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −5.49867 −0.187941
\(857\) −36.3969 −1.24330 −0.621648 0.783297i \(-0.713536\pi\)
−0.621648 + 0.783297i \(0.713536\pi\)
\(858\) 0 0
\(859\) 8.12650i 0.277273i 0.990343 + 0.138636i \(0.0442719\pi\)
−0.990343 + 0.138636i \(0.955728\pi\)
\(860\) −10.5736 −0.360556
\(861\) 0 0
\(862\) −19.6670 −0.669861
\(863\) 36.7419i 1.25071i 0.780340 + 0.625355i \(0.215046\pi\)
−0.780340 + 0.625355i \(0.784954\pi\)
\(864\) 0 0
\(865\) 28.3987 0.965584
\(866\) −16.1213 −0.547824
\(867\) 0 0
\(868\) 0 0
\(869\) 35.5514i 1.20600i
\(870\) 0 0
\(871\) 10.0262i 0.339726i
\(872\) − 13.4705i − 0.456167i
\(873\) 0 0
\(874\) 25.2081i 0.852676i
\(875\) 0 0
\(876\) 0 0
\(877\) −55.2141 −1.86445 −0.932224 0.361883i \(-0.882134\pi\)
−0.932224 + 0.361883i \(0.882134\pi\)
\(878\) 26.7007 0.901104
\(879\) 0 0
\(880\) 19.7823i 0.666860i
\(881\) 22.0434 0.742660 0.371330 0.928501i \(-0.378902\pi\)
0.371330 + 0.928501i \(0.378902\pi\)
\(882\) 0 0
\(883\) −16.7683 −0.564300 −0.282150 0.959370i \(-0.591048\pi\)
−0.282150 + 0.959370i \(0.591048\pi\)
\(884\) − 16.9269i − 0.569313i
\(885\) 0 0
\(886\) −19.8643 −0.667353
\(887\) 15.9974 0.537139 0.268569 0.963260i \(-0.413449\pi\)
0.268569 + 0.963260i \(0.413449\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 43.6455i 1.46300i
\(891\) 0 0
\(892\) 7.75175i 0.259548i
\(893\) − 0.782353i − 0.0261804i
\(894\) 0 0
\(895\) − 59.9122i − 2.00264i
\(896\) 0 0
\(897\) 0 0
\(898\) 20.0132 0.667848
\(899\) −2.26672 −0.0755995
\(900\) 0 0
\(901\) 12.6819i 0.422496i
\(902\) −30.3677 −1.01113
\(903\) 0 0
\(904\) −11.8843 −0.395266
\(905\) − 35.2639i − 1.17221i
\(906\) 0 0
\(907\) 20.5899 0.683676 0.341838 0.939759i \(-0.388951\pi\)
0.341838 + 0.939759i \(0.388951\pi\)
\(908\) −11.1867 −0.371242
\(909\) 0 0
\(910\) 0 0
\(911\) 23.5024i 0.778670i 0.921096 + 0.389335i \(0.127295\pi\)
−0.921096 + 0.389335i \(0.872705\pi\)
\(912\) 0 0
\(913\) 9.07135i 0.300218i
\(914\) − 23.9138i − 0.790997i
\(915\) 0 0
\(916\) − 16.0107i − 0.529010i
\(917\) 0 0
\(918\) 0 0
\(919\) 26.8861 0.886891 0.443445 0.896301i \(-0.353756\pi\)
0.443445 + 0.896301i \(0.353756\pi\)
\(920\) 11.0565 0.364521
\(921\) 0 0
\(922\) 18.6642i 0.614673i
\(923\) 60.6617 1.99671
\(924\) 0 0
\(925\) 45.2627 1.48823
\(926\) − 7.64166i − 0.251121i
\(927\) 0 0
\(928\) −5.66096 −0.185830
\(929\) −10.2186 −0.335262 −0.167631 0.985850i \(-0.553612\pi\)
−0.167631 + 0.985850i \(0.553612\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 5.62801i 0.184352i
\(933\) 0 0
\(934\) 34.7012i 1.13546i
\(935\) − 48.4565i − 1.58470i
\(936\) 0 0
\(937\) 42.9366i 1.40268i 0.712829 + 0.701338i \(0.247414\pi\)
−0.712829 + 0.701338i \(0.752586\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −0.343146 −0.0111922
\(941\) 9.15254 0.298364 0.149182 0.988810i \(-0.452336\pi\)
0.149182 + 0.988810i \(0.452336\pi\)
\(942\) 0 0
\(943\) 16.9727i 0.552708i
\(944\) −6.38401 −0.207782
\(945\) 0 0
\(946\) −16.3222 −0.530680
\(947\) 40.6714i 1.32164i 0.750543 + 0.660821i \(0.229792\pi\)
−0.750543 + 0.660821i \(0.770208\pi\)
\(948\) 0 0
\(949\) −4.69738 −0.152483
\(950\) −63.7845 −2.06944
\(951\) 0 0
\(952\) 0 0
\(953\) 7.49289i 0.242719i 0.992609 + 0.121359i \(0.0387253\pi\)
−0.992609 + 0.121359i \(0.961275\pi\)
\(954\) 0 0
\(955\) − 62.9110i − 2.03575i
\(956\) − 25.9630i − 0.839703i
\(957\) 0 0
\(958\) 26.2363i 0.847656i
\(959\) 0 0
\(960\) 0 0
\(961\) 30.8397 0.994828
\(962\) 40.0231 1.29040
\(963\) 0 0
\(964\) 14.5302i 0.467985i
\(965\) −71.0715 −2.28787
\(966\) 0 0
\(967\) 9.71864 0.312530 0.156265 0.987715i \(-0.450055\pi\)
0.156265 + 0.987715i \(0.450055\pi\)
\(968\) 19.5374i 0.627956i
\(969\) 0 0
\(970\) 40.7341 1.30789
\(971\) −5.15136 −0.165315 −0.0826576 0.996578i \(-0.526341\pi\)
−0.0826576 + 0.996578i \(0.526341\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) − 7.73022i − 0.247692i
\(975\) 0 0
\(976\) 7.06376i 0.226106i
\(977\) − 50.1137i − 1.60328i −0.597808 0.801639i \(-0.703961\pi\)
0.597808 0.801639i \(-0.296039\pi\)
\(978\) 0 0
\(979\) 67.3746i 2.15330i
\(980\) 0 0
\(981\) 0 0
\(982\) 33.0173 1.05362
\(983\) 28.5480 0.910538 0.455269 0.890354i \(-0.349543\pi\)
0.455269 + 0.890354i \(0.349543\pi\)
\(984\) 0 0
\(985\) − 76.9685i − 2.45242i
\(986\) 13.8665 0.441598
\(987\) 0 0
\(988\) −56.4008 −1.79435
\(989\) 9.12258i 0.290081i
\(990\) 0 0
\(991\) −62.0232 −1.97023 −0.985116 0.171888i \(-0.945013\pi\)
−0.985116 + 0.171888i \(0.945013\pi\)
\(992\) −0.400414 −0.0127131
\(993\) 0 0
\(994\) 0 0
\(995\) 48.2578i 1.52987i
\(996\) 0 0
\(997\) − 7.05415i − 0.223407i −0.993742 0.111704i \(-0.964369\pi\)
0.993742 0.111704i \(-0.0356307\pi\)
\(998\) − 7.61078i − 0.240915i
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2646.2.d.f.2645.9 yes 16
3.2 odd 2 inner 2646.2.d.f.2645.8 yes 16
7.6 odd 2 inner 2646.2.d.f.2645.16 yes 16
21.20 even 2 inner 2646.2.d.f.2645.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2646.2.d.f.2645.1 16 21.20 even 2 inner
2646.2.d.f.2645.8 yes 16 3.2 odd 2 inner
2646.2.d.f.2645.9 yes 16 1.1 even 1 trivial
2646.2.d.f.2645.16 yes 16 7.6 odd 2 inner