Properties

Label 1521.2.b.i.1351.2
Level $1521$
Weight $2$
Character 1521.1351
Analytic conductor $12.145$
Analytic rank $0$
Dimension $4$
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] = [1521,2,Mod(1351,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1521.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 117)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1521.1351
Dual form 1521.2.b.i.1351.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.73205i q^{8} +O(q^{10})\) \(q-1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.73205i q^{8} -3.46410i q^{11} +3.46410 q^{14} -5.00000 q^{16} +6.92820 q^{17} -2.00000i q^{19} -6.00000 q^{22} -6.92820 q^{23} +5.00000 q^{25} -2.00000i q^{28} +6.92820 q^{29} -2.00000i q^{31} +5.19615i q^{32} -12.0000i q^{34} +2.00000i q^{37} -3.46410 q^{38} -6.92820i q^{41} -8.00000 q^{43} +3.46410i q^{44} +12.0000i q^{46} -10.3923i q^{47} +3.00000 q^{49} -8.66025i q^{50} +3.46410 q^{56} -12.0000i q^{58} +3.46410i q^{59} -10.0000 q^{61} -3.46410 q^{62} -1.00000 q^{64} -14.0000i q^{67} -6.92820 q^{68} -3.46410i q^{71} -10.0000i q^{73} +3.46410 q^{74} +2.00000i q^{76} +6.92820 q^{77} -4.00000 q^{79} -12.0000 q^{82} -10.3923i q^{83} +13.8564i q^{86} -6.00000 q^{88} +6.92820i q^{89} +6.92820 q^{92} -18.0000 q^{94} +10.0000i q^{97} -5.19615i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 20 q^{16} - 24 q^{22} + 20 q^{25} - 32 q^{43} + 12 q^{49} - 40 q^{61} - 4 q^{64} - 16 q^{79} - 48 q^{82} - 24 q^{88} - 72 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.73205i − 1.22474i −0.790569 0.612372i \(-0.790215\pi\)
0.790569 0.612372i \(-0.209785\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 1.73205i − 0.612372i
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.46410i − 1.04447i −0.852803 0.522233i \(-0.825099\pi\)
0.852803 0.522233i \(-0.174901\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 6.92820 1.68034 0.840168 0.542326i \(-0.182456\pi\)
0.840168 + 0.542326i \(0.182456\pi\)
\(18\) 0 0
\(19\) − 2.00000i − 0.458831i −0.973329 0.229416i \(-0.926318\pi\)
0.973329 0.229416i \(-0.0736815\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) −6.92820 −1.44463 −0.722315 0.691564i \(-0.756922\pi\)
−0.722315 + 0.691564i \(0.756922\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) − 2.00000i − 0.377964i
\(29\) 6.92820 1.28654 0.643268 0.765641i \(-0.277578\pi\)
0.643268 + 0.765641i \(0.277578\pi\)
\(30\) 0 0
\(31\) − 2.00000i − 0.359211i −0.983739 0.179605i \(-0.942518\pi\)
0.983739 0.179605i \(-0.0574821\pi\)
\(32\) 5.19615i 0.918559i
\(33\) 0 0
\(34\) − 12.0000i − 2.05798i
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) −3.46410 −0.561951
\(39\) 0 0
\(40\) 0 0
\(41\) − 6.92820i − 1.08200i −0.841021 0.541002i \(-0.818045\pi\)
0.841021 0.541002i \(-0.181955\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 3.46410i 0.522233i
\(45\) 0 0
\(46\) 12.0000i 1.76930i
\(47\) − 10.3923i − 1.51587i −0.652328 0.757937i \(-0.726208\pi\)
0.652328 0.757937i \(-0.273792\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) − 8.66025i − 1.22474i
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.46410 0.462910
\(57\) 0 0
\(58\) − 12.0000i − 1.57568i
\(59\) 3.46410i 0.450988i 0.974245 + 0.225494i \(0.0723995\pi\)
−0.974245 + 0.225494i \(0.927600\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −3.46410 −0.439941
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 14.0000i − 1.71037i −0.518321 0.855186i \(-0.673443\pi\)
0.518321 0.855186i \(-0.326557\pi\)
\(68\) −6.92820 −0.840168
\(69\) 0 0
\(70\) 0 0
\(71\) − 3.46410i − 0.411113i −0.978645 0.205557i \(-0.934100\pi\)
0.978645 0.205557i \(-0.0659005\pi\)
\(72\) 0 0
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) 3.46410 0.402694
\(75\) 0 0
\(76\) 2.00000i 0.229416i
\(77\) 6.92820 0.789542
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −12.0000 −1.32518
\(83\) − 10.3923i − 1.14070i −0.821401 0.570352i \(-0.806807\pi\)
0.821401 0.570352i \(-0.193193\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 13.8564i 1.49417i
\(87\) 0 0
\(88\) −6.00000 −0.639602
\(89\) 6.92820i 0.734388i 0.930144 + 0.367194i \(0.119682\pi\)
−0.930144 + 0.367194i \(0.880318\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.92820 0.722315
\(93\) 0 0
\(94\) −18.0000 −1.85656
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) − 5.19615i − 0.524891i
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) 13.8564 1.37876 0.689382 0.724398i \(-0.257882\pi\)
0.689382 + 0.724398i \(0.257882\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.92820 0.669775 0.334887 0.942258i \(-0.391302\pi\)
0.334887 + 0.942258i \(0.391302\pi\)
\(108\) 0 0
\(109\) 10.0000i 0.957826i 0.877862 + 0.478913i \(0.158969\pi\)
−0.877862 + 0.478913i \(0.841031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 10.0000i − 0.944911i
\(113\) −6.92820 −0.651751 −0.325875 0.945413i \(-0.605659\pi\)
−0.325875 + 0.945413i \(0.605659\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.92820 −0.643268
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 13.8564i 1.27021i
\(120\) 0 0
\(121\) −1.00000 −0.0909091
\(122\) 17.3205i 1.56813i
\(123\) 0 0
\(124\) 2.00000i 0.179605i
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 12.1244i 1.07165i
\(129\) 0 0
\(130\) 0 0
\(131\) −20.7846 −1.81596 −0.907980 0.419014i \(-0.862376\pi\)
−0.907980 + 0.419014i \(0.862376\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) −24.2487 −2.09477
\(135\) 0 0
\(136\) − 12.0000i − 1.02899i
\(137\) − 6.92820i − 0.591916i −0.955201 0.295958i \(-0.904361\pi\)
0.955201 0.295958i \(-0.0956389\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −6.00000 −0.503509
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −17.3205 −1.43346
\(147\) 0 0
\(148\) − 2.00000i − 0.164399i
\(149\) − 13.8564i − 1.13516i −0.823318 0.567581i \(-0.807880\pi\)
0.823318 0.567581i \(-0.192120\pi\)
\(150\) 0 0
\(151\) 14.0000i 1.13930i 0.821886 + 0.569652i \(0.192922\pi\)
−0.821886 + 0.569652i \(0.807078\pi\)
\(152\) −3.46410 −0.280976
\(153\) 0 0
\(154\) − 12.0000i − 0.966988i
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 6.92820i 0.551178i
\(159\) 0 0
\(160\) 0 0
\(161\) − 13.8564i − 1.09204i
\(162\) 0 0
\(163\) 14.0000i 1.09656i 0.836293 + 0.548282i \(0.184718\pi\)
−0.836293 + 0.548282i \(0.815282\pi\)
\(164\) 6.92820i 0.541002i
\(165\) 0 0
\(166\) −18.0000 −1.39707
\(167\) 17.3205i 1.34030i 0.742225 + 0.670151i \(0.233770\pi\)
−0.742225 + 0.670151i \(0.766230\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) −13.8564 −1.05348 −0.526742 0.850026i \(-0.676586\pi\)
−0.526742 + 0.850026i \(0.676586\pi\)
\(174\) 0 0
\(175\) 10.0000i 0.755929i
\(176\) 17.3205i 1.30558i
\(177\) 0 0
\(178\) 12.0000 0.899438
\(179\) 13.8564 1.03568 0.517838 0.855479i \(-0.326737\pi\)
0.517838 + 0.855479i \(0.326737\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 12.0000i 0.884652i
\(185\) 0 0
\(186\) 0 0
\(187\) − 24.0000i − 1.75505i
\(188\) 10.3923i 0.757937i
\(189\) 0 0
\(190\) 0 0
\(191\) 13.8564 1.00261 0.501307 0.865269i \(-0.332853\pi\)
0.501307 + 0.865269i \(0.332853\pi\)
\(192\) 0 0
\(193\) 26.0000i 1.87152i 0.352636 + 0.935760i \(0.385285\pi\)
−0.352636 + 0.935760i \(0.614715\pi\)
\(194\) 17.3205 1.24354
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 13.8564i 0.987228i 0.869681 + 0.493614i \(0.164324\pi\)
−0.869681 + 0.493614i \(0.835676\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) − 8.66025i − 0.612372i
\(201\) 0 0
\(202\) − 24.0000i − 1.68863i
\(203\) 13.8564i 0.972529i
\(204\) 0 0
\(205\) 0 0
\(206\) − 6.92820i − 0.482711i
\(207\) 0 0
\(208\) 0 0
\(209\) −6.92820 −0.479234
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) − 12.0000i − 0.820303i
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) 17.3205 1.17309
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) − 2.00000i − 0.133930i −0.997755 0.0669650i \(-0.978668\pi\)
0.997755 0.0669650i \(-0.0213316\pi\)
\(224\) −10.3923 −0.694365
\(225\) 0 0
\(226\) 12.0000i 0.798228i
\(227\) − 3.46410i − 0.229920i −0.993370 0.114960i \(-0.963326\pi\)
0.993370 0.114960i \(-0.0366741\pi\)
\(228\) 0 0
\(229\) 14.0000i 0.925146i 0.886581 + 0.462573i \(0.153074\pi\)
−0.886581 + 0.462573i \(0.846926\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 12.0000i − 0.787839i
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) − 3.46410i − 0.225494i
\(237\) 0 0
\(238\) 24.0000 1.55569
\(239\) 10.3923i 0.672222i 0.941822 + 0.336111i \(0.109112\pi\)
−0.941822 + 0.336111i \(0.890888\pi\)
\(240\) 0 0
\(241\) − 10.0000i − 0.644157i −0.946713 0.322078i \(-0.895619\pi\)
0.946713 0.322078i \(-0.104381\pi\)
\(242\) 1.73205i 0.111340i
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) −3.46410 −0.219971
\(249\) 0 0
\(250\) 0 0
\(251\) 6.92820 0.437304 0.218652 0.975803i \(-0.429834\pi\)
0.218652 + 0.975803i \(0.429834\pi\)
\(252\) 0 0
\(253\) 24.0000i 1.50887i
\(254\) − 27.7128i − 1.73886i
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −27.7128 −1.72868 −0.864339 0.502910i \(-0.832263\pi\)
−0.864339 + 0.502910i \(0.832263\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 36.0000i 2.22409i
\(263\) 6.92820 0.427211 0.213606 0.976920i \(-0.431479\pi\)
0.213606 + 0.976920i \(0.431479\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) − 6.92820i − 0.424795i
\(267\) 0 0
\(268\) 14.0000i 0.855186i
\(269\) −6.92820 −0.422420 −0.211210 0.977441i \(-0.567740\pi\)
−0.211210 + 0.977441i \(0.567740\pi\)
\(270\) 0 0
\(271\) − 22.0000i − 1.33640i −0.743980 0.668202i \(-0.767064\pi\)
0.743980 0.668202i \(-0.232936\pi\)
\(272\) −34.6410 −2.10042
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) − 17.3205i − 1.04447i
\(276\) 0 0
\(277\) −2.00000 −0.120168 −0.0600842 0.998193i \(-0.519137\pi\)
−0.0600842 + 0.998193i \(0.519137\pi\)
\(278\) 6.92820i 0.415526i
\(279\) 0 0
\(280\) 0 0
\(281\) − 20.7846i − 1.23991i −0.784639 0.619953i \(-0.787152\pi\)
0.784639 0.619953i \(-0.212848\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 3.46410i 0.205557i
\(285\) 0 0
\(286\) 0 0
\(287\) 13.8564 0.817918
\(288\) 0 0
\(289\) 31.0000 1.82353
\(290\) 0 0
\(291\) 0 0
\(292\) 10.0000i 0.585206i
\(293\) 13.8564i 0.809500i 0.914427 + 0.404750i \(0.132641\pi\)
−0.914427 + 0.404750i \(0.867359\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.46410 0.201347
\(297\) 0 0
\(298\) −24.0000 −1.39028
\(299\) 0 0
\(300\) 0 0
\(301\) − 16.0000i − 0.922225i
\(302\) 24.2487 1.39536
\(303\) 0 0
\(304\) 10.0000i 0.573539i
\(305\) 0 0
\(306\) 0 0
\(307\) 26.0000i 1.48390i 0.670456 + 0.741949i \(0.266098\pi\)
−0.670456 + 0.741949i \(0.733902\pi\)
\(308\) −6.92820 −0.394771
\(309\) 0 0
\(310\) 0 0
\(311\) 20.7846 1.17859 0.589294 0.807919i \(-0.299406\pi\)
0.589294 + 0.807919i \(0.299406\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) − 3.46410i − 0.195491i
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) − 24.0000i − 1.34374i
\(320\) 0 0
\(321\) 0 0
\(322\) −24.0000 −1.33747
\(323\) − 13.8564i − 0.770991i
\(324\) 0 0
\(325\) 0 0
\(326\) 24.2487 1.34301
\(327\) 0 0
\(328\) −12.0000 −0.662589
\(329\) 20.7846 1.14589
\(330\) 0 0
\(331\) 10.0000i 0.549650i 0.961494 + 0.274825i \(0.0886199\pi\)
−0.961494 + 0.274825i \(0.911380\pi\)
\(332\) 10.3923i 0.570352i
\(333\) 0 0
\(334\) 30.0000 1.64153
\(335\) 0 0
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −6.92820 −0.375183
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 13.8564i 0.747087i
\(345\) 0 0
\(346\) 24.0000i 1.29025i
\(347\) −27.7128 −1.48770 −0.743851 0.668346i \(-0.767003\pi\)
−0.743851 + 0.668346i \(0.767003\pi\)
\(348\) 0 0
\(349\) 26.0000i 1.39175i 0.718164 + 0.695874i \(0.244983\pi\)
−0.718164 + 0.695874i \(0.755017\pi\)
\(350\) 17.3205 0.925820
\(351\) 0 0
\(352\) 18.0000 0.959403
\(353\) − 6.92820i − 0.368751i −0.982856 0.184376i \(-0.940974\pi\)
0.982856 0.184376i \(-0.0590263\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) − 6.92820i − 0.367194i
\(357\) 0 0
\(358\) − 24.0000i − 1.26844i
\(359\) − 10.3923i − 0.548485i −0.961661 0.274242i \(-0.911573\pi\)
0.961661 0.274242i \(-0.0884271\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) − 17.3205i − 0.910346i
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) 34.6410 1.80579
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) −41.5692 −2.14949
\(375\) 0 0
\(376\) −18.0000 −0.928279
\(377\) 0 0
\(378\) 0 0
\(379\) − 2.00000i − 0.102733i −0.998680 0.0513665i \(-0.983642\pi\)
0.998680 0.0513665i \(-0.0163577\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 24.0000i − 1.22795i
\(383\) 17.3205i 0.885037i 0.896759 + 0.442518i \(0.145915\pi\)
−0.896759 + 0.442518i \(0.854085\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 45.0333 2.29214
\(387\) 0 0
\(388\) − 10.0000i − 0.507673i
\(389\) −20.7846 −1.05382 −0.526911 0.849921i \(-0.676650\pi\)
−0.526911 + 0.849921i \(0.676650\pi\)
\(390\) 0 0
\(391\) −48.0000 −2.42746
\(392\) − 5.19615i − 0.262445i
\(393\) 0 0
\(394\) 24.0000 1.20910
\(395\) 0 0
\(396\) 0 0
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) − 27.7128i − 1.38912i
\(399\) 0 0
\(400\) −25.0000 −1.25000
\(401\) − 34.6410i − 1.72989i −0.501867 0.864945i \(-0.667353\pi\)
0.501867 0.864945i \(-0.332647\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −13.8564 −0.689382
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) 6.92820 0.343418
\(408\) 0 0
\(409\) 10.0000i 0.494468i 0.968956 + 0.247234i \(0.0795217\pi\)
−0.968956 + 0.247234i \(0.920478\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) −6.92820 −0.340915
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 12.0000i 0.586939i
\(419\) 6.92820 0.338465 0.169232 0.985576i \(-0.445871\pi\)
0.169232 + 0.985576i \(0.445871\pi\)
\(420\) 0 0
\(421\) 10.0000i 0.487370i 0.969854 + 0.243685i \(0.0783563\pi\)
−0.969854 + 0.243685i \(0.921644\pi\)
\(422\) − 13.8564i − 0.674519i
\(423\) 0 0
\(424\) 0 0
\(425\) 34.6410 1.68034
\(426\) 0 0
\(427\) − 20.0000i − 0.967868i
\(428\) −6.92820 −0.334887
\(429\) 0 0
\(430\) 0 0
\(431\) − 38.1051i − 1.83546i −0.397206 0.917729i \(-0.630020\pi\)
0.397206 0.917729i \(-0.369980\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) − 6.92820i − 0.332564i
\(435\) 0 0
\(436\) − 10.0000i − 0.478913i
\(437\) 13.8564i 0.662842i
\(438\) 0 0
\(439\) 28.0000 1.33637 0.668184 0.743996i \(-0.267072\pi\)
0.668184 + 0.743996i \(0.267072\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.7846 0.987507 0.493753 0.869602i \(-0.335625\pi\)
0.493753 + 0.869602i \(0.335625\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −3.46410 −0.164030
\(447\) 0 0
\(448\) − 2.00000i − 0.0944911i
\(449\) − 6.92820i − 0.326962i −0.986546 0.163481i \(-0.947728\pi\)
0.986546 0.163481i \(-0.0522723\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 6.92820 0.325875
\(453\) 0 0
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) 24.2487 1.13307
\(459\) 0 0
\(460\) 0 0
\(461\) 27.7128i 1.29071i 0.763881 + 0.645357i \(0.223291\pi\)
−0.763881 + 0.645357i \(0.776709\pi\)
\(462\) 0 0
\(463\) 2.00000i 0.0929479i 0.998920 + 0.0464739i \(0.0147984\pi\)
−0.998920 + 0.0464739i \(0.985202\pi\)
\(464\) −34.6410 −1.60817
\(465\) 0 0
\(466\) 0 0
\(467\) −20.7846 −0.961797 −0.480899 0.876776i \(-0.659689\pi\)
−0.480899 + 0.876776i \(0.659689\pi\)
\(468\) 0 0
\(469\) 28.0000 1.29292
\(470\) 0 0
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) 27.7128i 1.27424i
\(474\) 0 0
\(475\) − 10.0000i − 0.458831i
\(476\) − 13.8564i − 0.635107i
\(477\) 0 0
\(478\) 18.0000 0.823301
\(479\) − 3.46410i − 0.158279i −0.996864 0.0791394i \(-0.974783\pi\)
0.996864 0.0791394i \(-0.0252172\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −17.3205 −0.788928
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) 17.3205i 0.784063i
\(489\) 0 0
\(490\) 0 0
\(491\) −6.92820 −0.312665 −0.156333 0.987704i \(-0.549967\pi\)
−0.156333 + 0.987704i \(0.549967\pi\)
\(492\) 0 0
\(493\) 48.0000 2.16181
\(494\) 0 0
\(495\) 0 0
\(496\) 10.0000i 0.449013i
\(497\) 6.92820 0.310772
\(498\) 0 0
\(499\) − 14.0000i − 0.626726i −0.949633 0.313363i \(-0.898544\pi\)
0.949633 0.313363i \(-0.101456\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 12.0000i − 0.535586i
\(503\) −6.92820 −0.308913 −0.154457 0.988000i \(-0.549363\pi\)
−0.154457 + 0.988000i \(0.549363\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 41.5692 1.84798
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) 13.8564i 0.614174i 0.951681 + 0.307087i \(0.0993543\pi\)
−0.951681 + 0.307087i \(0.900646\pi\)
\(510\) 0 0
\(511\) 20.0000 0.884748
\(512\) − 8.66025i − 0.382733i
\(513\) 0 0
\(514\) 48.0000i 2.11719i
\(515\) 0 0
\(516\) 0 0
\(517\) −36.0000 −1.58328
\(518\) 6.92820i 0.304408i
\(519\) 0 0
\(520\) 0 0
\(521\) 20.7846 0.910590 0.455295 0.890341i \(-0.349534\pi\)
0.455295 + 0.890341i \(0.349534\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) 20.7846 0.907980
\(525\) 0 0
\(526\) − 12.0000i − 0.523225i
\(527\) − 13.8564i − 0.603595i
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −24.2487 −1.04738
\(537\) 0 0
\(538\) 12.0000i 0.517357i
\(539\) − 10.3923i − 0.447628i
\(540\) 0 0
\(541\) − 10.0000i − 0.429934i −0.976621 0.214967i \(-0.931036\pi\)
0.976621 0.214967i \(-0.0689643\pi\)
\(542\) −38.1051 −1.63675
\(543\) 0 0
\(544\) 36.0000i 1.54349i
\(545\) 0 0
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 6.92820i 0.295958i
\(549\) 0 0
\(550\) −30.0000 −1.27920
\(551\) − 13.8564i − 0.590303i
\(552\) 0 0
\(553\) − 8.00000i − 0.340195i
\(554\) 3.46410i 0.147176i
\(555\) 0 0
\(556\) 4.00000 0.169638
\(557\) 27.7128i 1.17423i 0.809504 + 0.587115i \(0.199736\pi\)
−0.809504 + 0.587115i \(0.800264\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −36.0000 −1.51857
\(563\) −13.8564 −0.583978 −0.291989 0.956422i \(-0.594317\pi\)
−0.291989 + 0.956422i \(0.594317\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) − 6.92820i − 0.291214i
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 13.8564 0.580891 0.290445 0.956892i \(-0.406197\pi\)
0.290445 + 0.956892i \(0.406197\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 24.0000i − 1.00174i
\(575\) −34.6410 −1.44463
\(576\) 0 0
\(577\) − 26.0000i − 1.08239i −0.840896 0.541197i \(-0.817971\pi\)
0.840896 0.541197i \(-0.182029\pi\)
\(578\) − 53.6936i − 2.23336i
\(579\) 0 0
\(580\) 0 0
\(581\) 20.7846 0.862291
\(582\) 0 0
\(583\) 0 0
\(584\) −17.3205 −0.716728
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 45.0333i 1.85872i 0.369170 + 0.929362i \(0.379642\pi\)
−0.369170 + 0.929362i \(0.620358\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) − 10.0000i − 0.410997i
\(593\) 20.7846i 0.853522i 0.904365 + 0.426761i \(0.140345\pi\)
−0.904365 + 0.426761i \(0.859655\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 13.8564i 0.567581i
\(597\) 0 0
\(598\) 0 0
\(599\) 20.7846 0.849236 0.424618 0.905373i \(-0.360408\pi\)
0.424618 + 0.905373i \(0.360408\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) −27.7128 −1.12949
\(603\) 0 0
\(604\) − 14.0000i − 0.569652i
\(605\) 0 0
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) 10.3923 0.421464
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 45.0333 1.81740
\(615\) 0 0
\(616\) − 12.0000i − 0.483494i
\(617\) 6.92820i 0.278919i 0.990228 + 0.139459i \(0.0445365\pi\)
−0.990228 + 0.139459i \(0.955464\pi\)
\(618\) 0 0
\(619\) 26.0000i 1.04503i 0.852631 + 0.522514i \(0.175006\pi\)
−0.852631 + 0.522514i \(0.824994\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 36.0000i − 1.44347i
\(623\) −13.8564 −0.555145
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) − 24.2487i − 0.969173i
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) 13.8564i 0.552491i
\(630\) 0 0
\(631\) 38.0000i 1.51276i 0.654135 + 0.756378i \(0.273033\pi\)
−0.654135 + 0.756378i \(0.726967\pi\)
\(632\) 6.92820i 0.275589i
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −41.5692 −1.64574
\(639\) 0 0
\(640\) 0 0
\(641\) −34.6410 −1.36824 −0.684119 0.729370i \(-0.739813\pi\)
−0.684119 + 0.729370i \(0.739813\pi\)
\(642\) 0 0
\(643\) − 14.0000i − 0.552106i −0.961142 0.276053i \(-0.910973\pi\)
0.961142 0.276053i \(-0.0890266\pi\)
\(644\) 13.8564i 0.546019i
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 13.8564 0.544752 0.272376 0.962191i \(-0.412191\pi\)
0.272376 + 0.962191i \(0.412191\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) − 14.0000i − 0.548282i
\(653\) −34.6410 −1.35561 −0.677804 0.735243i \(-0.737068\pi\)
−0.677804 + 0.735243i \(0.737068\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 34.6410i 1.35250i
\(657\) 0 0
\(658\) − 36.0000i − 1.40343i
\(659\) 13.8564 0.539769 0.269884 0.962893i \(-0.413014\pi\)
0.269884 + 0.962893i \(0.413014\pi\)
\(660\) 0 0
\(661\) 26.0000i 1.01128i 0.862744 + 0.505641i \(0.168744\pi\)
−0.862744 + 0.505641i \(0.831256\pi\)
\(662\) 17.3205 0.673181
\(663\) 0 0
\(664\) −18.0000 −0.698535
\(665\) 0 0
\(666\) 0 0
\(667\) −48.0000 −1.85857
\(668\) − 17.3205i − 0.670151i
\(669\) 0 0
\(670\) 0 0
\(671\) 34.6410i 1.33730i
\(672\) 0 0
\(673\) 10.0000 0.385472 0.192736 0.981251i \(-0.438264\pi\)
0.192736 + 0.981251i \(0.438264\pi\)
\(674\) 3.46410i 0.133432i
\(675\) 0 0
\(676\) 0 0
\(677\) −41.5692 −1.59763 −0.798817 0.601574i \(-0.794541\pi\)
−0.798817 + 0.601574i \(0.794541\pi\)
\(678\) 0 0
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) 0 0
\(682\) 12.0000i 0.459504i
\(683\) − 17.3205i − 0.662751i −0.943499 0.331375i \(-0.892487\pi\)
0.943499 0.331375i \(-0.107513\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 34.6410 1.32260
\(687\) 0 0
\(688\) 40.0000 1.52499
\(689\) 0 0
\(690\) 0 0
\(691\) 10.0000i 0.380418i 0.981744 + 0.190209i \(0.0609166\pi\)
−0.981744 + 0.190209i \(0.939083\pi\)
\(692\) 13.8564 0.526742
\(693\) 0 0
\(694\) 48.0000i 1.82206i
\(695\) 0 0
\(696\) 0 0
\(697\) − 48.0000i − 1.81813i
\(698\) 45.0333 1.70454
\(699\) 0 0
\(700\) − 10.0000i − 0.377964i
\(701\) 20.7846 0.785024 0.392512 0.919747i \(-0.371606\pi\)
0.392512 + 0.919747i \(0.371606\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 3.46410i 0.130558i
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 27.7128i 1.04225i
\(708\) 0 0
\(709\) 38.0000i 1.42712i 0.700594 + 0.713560i \(0.252918\pi\)
−0.700594 + 0.713560i \(0.747082\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 12.0000 0.449719
\(713\) 13.8564i 0.518927i
\(714\) 0 0
\(715\) 0 0
\(716\) −13.8564 −0.517838
\(717\) 0 0
\(718\) −18.0000 −0.671754
\(719\) −34.6410 −1.29189 −0.645946 0.763383i \(-0.723537\pi\)
−0.645946 + 0.763383i \(0.723537\pi\)
\(720\) 0 0
\(721\) 8.00000i 0.297936i
\(722\) − 25.9808i − 0.966904i
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 34.6410 1.28654
\(726\) 0 0
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −55.4256 −2.04999
\(732\) 0 0
\(733\) 34.0000i 1.25582i 0.778287 + 0.627909i \(0.216089\pi\)
−0.778287 + 0.627909i \(0.783911\pi\)
\(734\) − 13.8564i − 0.511449i
\(735\) 0 0
\(736\) − 36.0000i − 1.32698i
\(737\) −48.4974 −1.78643
\(738\) 0 0
\(739\) − 22.0000i − 0.809283i −0.914475 0.404642i \(-0.867396\pi\)
0.914475 0.404642i \(-0.132604\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.2487i 0.889599i 0.895630 + 0.444799i \(0.146725\pi\)
−0.895630 + 0.444799i \(0.853275\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) − 45.0333i − 1.64879i
\(747\) 0 0
\(748\) 24.0000i 0.877527i
\(749\) 13.8564i 0.506302i
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 51.9615i 1.89484i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −3.46410 −0.125822
\(759\) 0 0
\(760\) 0 0
\(761\) 34.6410i 1.25574i 0.778320 + 0.627868i \(0.216072\pi\)
−0.778320 + 0.627868i \(0.783928\pi\)
\(762\) 0 0
\(763\) −20.0000 −0.724049
\(764\) −13.8564 −0.501307
\(765\) 0 0
\(766\) 30.0000 1.08394
\(767\) 0 0
\(768\) 0 0
\(769\) − 50.0000i − 1.80305i −0.432731 0.901523i \(-0.642450\pi\)
0.432731 0.901523i \(-0.357550\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 26.0000i − 0.935760i
\(773\) − 13.8564i − 0.498380i −0.968455 0.249190i \(-0.919836\pi\)
0.968455 0.249190i \(-0.0801644\pi\)
\(774\) 0 0
\(775\) − 10.0000i − 0.359211i
\(776\) 17.3205 0.621770
\(777\) 0 0
\(778\) 36.0000i 1.29066i
\(779\) −13.8564 −0.496457
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 83.1384i 2.97302i
\(783\) 0 0
\(784\) −15.0000 −0.535714
\(785\) 0 0
\(786\) 0 0
\(787\) − 22.0000i − 0.784215i −0.919919 0.392108i \(-0.871746\pi\)
0.919919 0.392108i \(-0.128254\pi\)
\(788\) − 13.8564i − 0.493614i
\(789\) 0 0
\(790\) 0 0
\(791\) − 13.8564i − 0.492677i
\(792\) 0 0
\(793\) 0 0
\(794\) 3.46410 0.122936
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) −34.6410 −1.22705 −0.613524 0.789676i \(-0.710249\pi\)
−0.613524 + 0.789676i \(0.710249\pi\)
\(798\) 0 0
\(799\) − 72.0000i − 2.54718i
\(800\) 25.9808i 0.918559i
\(801\) 0 0
\(802\) −60.0000 −2.11867
\(803\) −34.6410 −1.22245
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) − 24.0000i − 0.844317i
\(809\) 6.92820 0.243583 0.121791 0.992556i \(-0.461136\pi\)
0.121791 + 0.992556i \(0.461136\pi\)
\(810\) 0 0
\(811\) − 26.0000i − 0.912983i −0.889728 0.456492i \(-0.849106\pi\)
0.889728 0.456492i \(-0.150894\pi\)
\(812\) − 13.8564i − 0.486265i
\(813\) 0 0
\(814\) − 12.0000i − 0.420600i
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000i 0.559769i
\(818\) 17.3205 0.605597
\(819\) 0 0
\(820\) 0 0
\(821\) − 27.7128i − 0.967184i −0.875294 0.483592i \(-0.839332\pi\)
0.875294 0.483592i \(-0.160668\pi\)
\(822\) 0 0
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) − 6.92820i − 0.241355i
\(825\) 0 0
\(826\) 12.0000i 0.417533i
\(827\) 10.3923i 0.361376i 0.983540 + 0.180688i \(0.0578324\pi\)
−0.983540 + 0.180688i \(0.942168\pi\)
\(828\) 0 0
\(829\) −2.00000 −0.0694629 −0.0347314 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 20.7846 0.720144
\(834\) 0 0
\(835\) 0 0
\(836\) 6.92820 0.239617
\(837\) 0 0
\(838\) − 12.0000i − 0.414533i
\(839\) 24.2487i 0.837158i 0.908180 + 0.418579i \(0.137472\pi\)
−0.908180 + 0.418579i \(0.862528\pi\)
\(840\) 0 0
\(841\) 19.0000 0.655172
\(842\) 17.3205 0.596904
\(843\) 0 0
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.00000i − 0.0687208i
\(848\) 0 0
\(849\) 0 0
\(850\) − 60.0000i − 2.05798i
\(851\) − 13.8564i − 0.474991i
\(852\) 0 0
\(853\) − 46.0000i − 1.57501i −0.616308 0.787505i \(-0.711372\pi\)
0.616308 0.787505i \(-0.288628\pi\)
\(854\) −34.6410 −1.18539
\(855\) 0 0
\(856\) − 12.0000i − 0.410152i
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −66.0000 −2.24797
\(863\) 31.1769i 1.06127i 0.847599 + 0.530637i \(0.178047\pi\)
−0.847599 + 0.530637i \(0.821953\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 3.46410i 0.117715i
\(867\) 0 0
\(868\) −4.00000 −0.135769
\(869\) 13.8564i 0.470046i
\(870\) 0 0
\(871\) 0 0
\(872\) 17.3205 0.586546
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) − 48.4974i − 1.63671i
\(879\) 0 0
\(880\) 0 0
\(881\) 13.8564 0.466834 0.233417 0.972377i \(-0.425009\pi\)
0.233417 + 0.972377i \(0.425009\pi\)
\(882\) 0 0
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) − 36.0000i − 1.20944i
\(887\) 27.7128 0.930505 0.465253 0.885178i \(-0.345963\pi\)
0.465253 + 0.885178i \(0.345963\pi\)
\(888\) 0 0
\(889\) 32.0000i 1.07325i
\(890\) 0 0
\(891\) 0 0
\(892\) 2.00000i 0.0669650i
\(893\) −20.7846 −0.695530
\(894\) 0 0
\(895\) 0 0
\(896\) −24.2487 −0.810093
\(897\) 0 0
\(898\) −12.0000 −0.400445
\(899\) − 13.8564i − 0.462137i
\(900\) 0 0
\(901\) 0 0
\(902\) 41.5692i 1.38410i
\(903\) 0 0
\(904\) 12.0000i 0.399114i
\(905\) 0 0
\(906\) 0 0
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) 3.46410i 0.114960i
\(909\) 0 0
\(910\) 0 0
\(911\) −41.5692 −1.37725 −0.688625 0.725118i \(-0.741785\pi\)
−0.688625 + 0.725118i \(0.741785\pi\)
\(912\) 0 0
\(913\) −36.0000 −1.19143
\(914\) 17.3205 0.572911
\(915\) 0 0
\(916\) − 14.0000i − 0.462573i
\(917\) − 41.5692i − 1.37274i
\(918\) 0 0
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 48.0000 1.58080
\(923\) 0 0
\(924\) 0 0
\(925\) 10.0000i 0.328798i
\(926\) 3.46410 0.113837
\(927\) 0 0
\(928\) 36.0000i 1.18176i
\(929\) 48.4974i 1.59115i 0.605856 + 0.795574i \(0.292831\pi\)
−0.605856 + 0.795574i \(0.707169\pi\)
\(930\) 0 0
\(931\) − 6.00000i − 0.196642i
\(932\) 0 0
\(933\) 0 0
\(934\) 36.0000i 1.17796i
\(935\) 0 0
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) − 48.4974i − 1.58350i
\(939\) 0 0
\(940\) 0 0
\(941\) − 41.5692i − 1.35512i −0.735469 0.677559i \(-0.763038\pi\)
0.735469 0.677559i \(-0.236962\pi\)
\(942\) 0 0
\(943\) 48.0000i 1.56310i
\(944\) − 17.3205i − 0.563735i
\(945\) 0 0
\(946\) 48.0000 1.56061
\(947\) 17.3205i 0.562841i 0.959585 + 0.281420i \(0.0908056\pi\)
−0.959585 + 0.281420i \(0.909194\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −17.3205 −0.561951
\(951\) 0 0
\(952\) 24.0000 0.777844
\(953\) 27.7128 0.897706 0.448853 0.893606i \(-0.351833\pi\)
0.448853 + 0.893606i \(0.351833\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) − 10.3923i − 0.336111i
\(957\) 0 0
\(958\) −6.00000 −0.193851
\(959\) 13.8564 0.447447
\(960\) 0 0
\(961\) 27.0000 0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 10.0000i 0.322078i
\(965\) 0 0
\(966\) 0 0
\(967\) − 2.00000i − 0.0643157i −0.999483 0.0321578i \(-0.989762\pi\)
0.999483 0.0321578i \(-0.0102379\pi\)
\(968\) 1.73205i 0.0556702i
\(969\) 0 0
\(970\) 0 0
\(971\) 55.4256 1.77869 0.889346 0.457234i \(-0.151160\pi\)
0.889346 + 0.457234i \(0.151160\pi\)
\(972\) 0 0
\(973\) − 8.00000i − 0.256468i
\(974\) −3.46410 −0.110997
\(975\) 0 0
\(976\) 50.0000 1.60046
\(977\) − 6.92820i − 0.221653i −0.993840 0.110826i \(-0.964650\pi\)
0.993840 0.110826i \(-0.0353498\pi\)
\(978\) 0 0
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) 12.0000i 0.382935i
\(983\) 10.3923i 0.331463i 0.986171 + 0.165732i \(0.0529985\pi\)
−0.986171 + 0.165732i \(0.947001\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) − 83.1384i − 2.64767i
\(987\) 0 0
\(988\) 0 0
\(989\) 55.4256 1.76243
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 10.3923 0.329956
\(993\) 0 0
\(994\) − 12.0000i − 0.380617i
\(995\) 0 0
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) −24.2487 −0.767580
\(999\) 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 1521.2.b.i.1351.2 4
3.2 odd 2 inner 1521.2.b.i.1351.4 4
13.5 odd 4 1521.2.a.j.1.1 2
13.8 odd 4 117.2.a.b.1.2 yes 2
13.12 even 2 inner 1521.2.b.i.1351.3 4
39.5 even 4 1521.2.a.j.1.2 2
39.8 even 4 117.2.a.b.1.1 2
39.38 odd 2 inner 1521.2.b.i.1351.1 4
52.47 even 4 1872.2.a.v.1.2 2
65.8 even 4 2925.2.c.s.2224.1 4
65.34 odd 4 2925.2.a.y.1.1 2
65.47 even 4 2925.2.c.s.2224.4 4
91.34 even 4 5733.2.a.t.1.2 2
104.21 odd 4 7488.2.a.cq.1.2 2
104.99 even 4 7488.2.a.cj.1.1 2
117.34 odd 12 1053.2.e.i.352.1 4
117.47 even 12 1053.2.e.i.352.2 4
117.86 even 12 1053.2.e.i.703.2 4
117.112 odd 12 1053.2.e.i.703.1 4
156.47 odd 4 1872.2.a.v.1.1 2
195.8 odd 4 2925.2.c.s.2224.3 4
195.47 odd 4 2925.2.c.s.2224.2 4
195.164 even 4 2925.2.a.y.1.2 2
273.125 odd 4 5733.2.a.t.1.1 2
312.125 even 4 7488.2.a.cq.1.1 2
312.203 odd 4 7488.2.a.cj.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.a.b.1.1 2 39.8 even 4
117.2.a.b.1.2 yes 2 13.8 odd 4
1053.2.e.i.352.1 4 117.34 odd 12
1053.2.e.i.352.2 4 117.47 even 12
1053.2.e.i.703.1 4 117.112 odd 12
1053.2.e.i.703.2 4 117.86 even 12
1521.2.a.j.1.1 2 13.5 odd 4
1521.2.a.j.1.2 2 39.5 even 4
1521.2.b.i.1351.1 4 39.38 odd 2 inner
1521.2.b.i.1351.2 4 1.1 even 1 trivial
1521.2.b.i.1351.3 4 13.12 even 2 inner
1521.2.b.i.1351.4 4 3.2 odd 2 inner
1872.2.a.v.1.1 2 156.47 odd 4
1872.2.a.v.1.2 2 52.47 even 4
2925.2.a.y.1.1 2 65.34 odd 4
2925.2.a.y.1.2 2 195.164 even 4
2925.2.c.s.2224.1 4 65.8 even 4
2925.2.c.s.2224.2 4 195.47 odd 4
2925.2.c.s.2224.3 4 195.8 odd 4
2925.2.c.s.2224.4 4 65.47 even 4
5733.2.a.t.1.1 2 273.125 odd 4
5733.2.a.t.1.2 2 91.34 even 4
7488.2.a.cj.1.1 2 104.99 even 4
7488.2.a.cj.1.2 2 312.203 odd 4
7488.2.a.cq.1.1 2 312.125 even 4
7488.2.a.cq.1.2 2 104.21 odd 4