Properties

Label 2925.2.c.s.2224.2
Level $2925$
Weight $2$
Character 2925.2224
Analytic conductor $23.356$
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] = [2925,2,Mod(2224,2925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2925, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2925.2224");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2925.c (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: \(23.3562425912\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 117)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2224.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2925.2224
Dual form 2925.2.c.s.2224.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.73205i q^{8} +O(q^{10})\) \(q-1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} -1.73205i q^{8} +3.46410 q^{11} -1.00000i q^{13} +3.46410 q^{14} -5.00000 q^{16} +6.92820i q^{17} -2.00000 q^{19} -6.00000i q^{22} +6.92820i q^{23} -1.73205 q^{26} -2.00000i q^{28} +6.92820 q^{29} +2.00000 q^{31} +5.19615i q^{32} +12.0000 q^{34} +2.00000i q^{37} +3.46410i q^{38} -6.92820 q^{41} -8.00000i q^{43} -3.46410 q^{44} +12.0000 q^{46} +10.3923i q^{47} +3.00000 q^{49} +1.00000i q^{52} +3.46410 q^{56} -12.0000i q^{58} +3.46410 q^{59} -10.0000 q^{61} -3.46410i q^{62} -1.00000 q^{64} +14.0000i q^{67} -6.92820i q^{68} -3.46410 q^{71} +10.0000i q^{73} +3.46410 q^{74} +2.00000 q^{76} +6.92820i q^{77} +4.00000 q^{79} +12.0000i q^{82} +10.3923i q^{83} -13.8564 q^{86} -6.00000i q^{88} +6.92820 q^{89} +2.00000 q^{91} -6.92820i q^{92} +18.0000 q^{94} -10.0000i q^{97} -5.19615i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 20 q^{16} - 8 q^{19} + 8 q^{31} + 48 q^{34} + 48 q^{46} + 12 q^{49} - 40 q^{61} - 4 q^{64} + 8 q^{76} + 16 q^{79} + 8 q^{91} + 72 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(2251\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 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
\(6\) 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
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) − 1.00000i − 0.277350i
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 6.92820i 1.68034i 0.542326 + 0.840168i \(0.317544\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 6.00000i − 1.27920i
\(23\) 6.92820i 1.44463i 0.691564 + 0.722315i \(0.256922\pi\)
−0.691564 + 0.722315i \(0.743078\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.73205 −0.339683
\(27\) 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
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 5.19615i 0.918559i
\(33\) 0 0
\(34\) 12.0000 2.05798
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 3.46410i 0.561951i
\(39\) 0 0
\(40\) 0 0
\(41\) −6.92820 −1.08200 −0.541002 0.841021i \(-0.681955\pi\)
−0.541002 + 0.841021i \(0.681955\pi\)
\(42\) 0 0
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) −3.46410 −0.522233
\(45\) 0 0
\(46\) 12.0000 1.76930
\(47\) 10.3923i 1.51587i 0.652328 + 0.757937i \(0.273792\pi\)
−0.652328 + 0.757937i \(0.726208\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 1.00000i 0.138675i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.46410 0.462910
\(57\) 0 0
\(58\) − 12.0000i − 1.57568i
\(59\) 3.46410 0.450988 0.225494 0.974245i \(-0.427600\pi\)
0.225494 + 0.974245i \(0.427600\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) − 3.46410i − 0.439941i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 14.0000i 1.71037i 0.518321 + 0.855186i \(0.326557\pi\)
−0.518321 + 0.855186i \(0.673443\pi\)
\(68\) − 6.92820i − 0.840168i
\(69\) 0 0
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) 10.0000i 1.17041i 0.810885 + 0.585206i \(0.198986\pi\)
−0.810885 + 0.585206i \(0.801014\pi\)
\(74\) 3.46410 0.402694
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 6.92820i 0.789542i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 12.0000i 1.32518i
\(83\) 10.3923i 1.14070i 0.821401 + 0.570352i \(0.193193\pi\)
−0.821401 + 0.570352i \(0.806807\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −13.8564 −1.49417
\(87\) 0 0
\(88\) − 6.00000i − 0.639602i
\(89\) 6.92820 0.734388 0.367194 0.930144i \(-0.380318\pi\)
0.367194 + 0.930144i \(0.380318\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) − 6.92820i − 0.722315i
\(93\) 0 0
\(94\) 18.0000 1.85656
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) − 5.19615i − 0.524891i
\(99\) 0 0
\(100\) 0 0
\(101\) 13.8564 1.37876 0.689382 0.724398i \(-0.257882\pi\)
0.689382 + 0.724398i \(0.257882\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) −1.73205 −0.169842
\(105\) 0 0
\(106\) 0 0
\(107\) − 6.92820i − 0.669775i −0.942258 0.334887i \(-0.891302\pi\)
0.942258 0.334887i \(-0.108698\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 10.0000i − 0.944911i
\(113\) − 6.92820i − 0.651751i −0.945413 0.325875i \(-0.894341\pi\)
0.945413 0.325875i \(-0.105659\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.92820 −0.643268
\(117\) 0 0
\(118\) − 6.00000i − 0.552345i
\(119\) −13.8564 −1.27021
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 17.3205i 1.56813i
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) − 16.0000i − 1.41977i −0.704317 0.709885i \(-0.748747\pi\)
0.704317 0.709885i \(-0.251253\pi\)
\(128\) 12.1244i 1.07165i
\(129\) 0 0
\(130\) 0 0
\(131\) 20.7846 1.81596 0.907980 0.419014i \(-0.137624\pi\)
0.907980 + 0.419014i \(0.137624\pi\)
\(132\) 0 0
\(133\) − 4.00000i − 0.346844i
\(134\) 24.2487 2.09477
\(135\) 0 0
\(136\) 12.0000 1.02899
\(137\) 6.92820i 0.591916i 0.955201 + 0.295958i \(0.0956389\pi\)
−0.955201 + 0.295958i \(0.904361\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.00000i 0.503509i
\(143\) − 3.46410i − 0.289683i
\(144\) 0 0
\(145\) 0 0
\(146\) 17.3205 1.43346
\(147\) 0 0
\(148\) − 2.00000i − 0.164399i
\(149\) 13.8564 1.13516 0.567581 0.823318i \(-0.307880\pi\)
0.567581 + 0.823318i \(0.307880\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) 3.46410i 0.280976i
\(153\) 0 0
\(154\) 12.0000 0.966988
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) − 6.92820i − 0.551178i
\(159\) 0 0
\(160\) 0 0
\(161\) −13.8564 −1.09204
\(162\) 0 0
\(163\) − 14.0000i − 1.09656i −0.836293 0.548282i \(-0.815282\pi\)
0.836293 0.548282i \(-0.184718\pi\)
\(164\) 6.92820 0.541002
\(165\) 0 0
\(166\) 18.0000 1.39707
\(167\) − 17.3205i − 1.34030i −0.742225 0.670151i \(-0.766230\pi\)
0.742225 0.670151i \(-0.233770\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000i 0.609994i
\(173\) 13.8564i 1.05348i 0.850026 + 0.526742i \(0.176586\pi\)
−0.850026 + 0.526742i \(0.823414\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −17.3205 −1.30558
\(177\) 0 0
\(178\) − 12.0000i − 0.899438i
\(179\) −13.8564 −1.03568 −0.517838 0.855479i \(-0.673263\pi\)
−0.517838 + 0.855479i \(0.673263\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) − 3.46410i − 0.256776i
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) 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 \(-0.667147\pi\)
−0.501307 + 0.865269i \(0.667147\pi\)
\(192\) 0 0
\(193\) − 26.0000i − 1.87152i −0.352636 0.935760i \(-0.614715\pi\)
0.352636 0.935760i \(-0.385285\pi\)
\(194\) −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 \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 24.0000i − 1.68863i
\(203\) 13.8564i 0.972529i
\(204\) 0 0
\(205\) 0 0
\(206\) 6.92820 0.482711
\(207\) 0 0
\(208\) 5.00000i 0.346688i
\(209\) −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
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000i 0.271538i
\(218\) − 17.3205i − 1.17309i
\(219\) 0 0
\(220\) 0 0
\(221\) 6.92820 0.466041
\(222\) 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
\(226\) −12.0000 −0.798228
\(227\) − 3.46410i − 0.229920i −0.993370 0.114960i \(-0.963326\pi\)
0.993370 0.114960i \(-0.0366741\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 12.0000i − 0.787839i
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3.46410 −0.225494
\(237\) 0 0
\(238\) 24.0000i 1.55569i
\(239\) −10.3923 −0.672222 −0.336111 0.941822i \(-0.609112\pi\)
−0.336111 + 0.941822i \(0.609112\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) − 1.73205i − 0.111340i
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000i 0.127257i
\(248\) − 3.46410i − 0.219971i
\(249\) 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
\(254\) −27.7128 −1.73886
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) − 27.7128i − 1.72868i −0.502910 0.864339i \(-0.667737\pi\)
0.502910 0.864339i \(-0.332263\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) − 36.0000i − 2.22409i
\(263\) 6.92820i 0.427211i 0.976920 + 0.213606i \(0.0685208\pi\)
−0.976920 + 0.213606i \(0.931479\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.92820 −0.424795
\(267\) 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
\(271\) −22.0000 −1.33640 −0.668202 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) − 34.6410i − 2.10042i
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) − 6.92820i − 0.415526i
\(279\) 0 0
\(280\) 0 0
\(281\) 20.7846 1.23991 0.619953 0.784639i \(-0.287152\pi\)
0.619953 + 0.784639i \(0.287152\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 3.46410 0.205557
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) − 13.8564i − 0.817918i
\(288\) 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
\(298\) − 24.0000i − 1.39028i
\(299\) 6.92820 0.400668
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) − 24.2487i − 1.39536i
\(303\) 0 0
\(304\) 10.0000 0.573539
\(305\) 0 0
\(306\) 0 0
\(307\) 26.0000i 1.48390i 0.670456 + 0.741949i \(0.266098\pi\)
−0.670456 + 0.741949i \(0.733902\pi\)
\(308\) − 6.92820i − 0.394771i
\(309\) 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
\(313\) − 14.0000i − 0.791327i −0.918396 0.395663i \(-0.870515\pi\)
0.918396 0.395663i \(-0.129485\pi\)
\(314\) 3.46410 0.195491
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) 24.0000i 1.33747i
\(323\) − 13.8564i − 0.770991i
\(324\) 0 0
\(325\) 0 0
\(326\) −24.2487 −1.34301
\(327\) 0 0
\(328\) 12.0000i 0.662589i
\(329\) −20.7846 −1.14589
\(330\) 0 0
\(331\) −10.0000 −0.549650 −0.274825 0.961494i \(-0.588620\pi\)
−0.274825 + 0.961494i \(0.588620\pi\)
\(332\) − 10.3923i − 0.570352i
\(333\) 0 0
\(334\) −30.0000 −1.64153
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000i 0.108947i 0.998515 + 0.0544735i \(0.0173480\pi\)
−0.998515 + 0.0544735i \(0.982652\pi\)
\(338\) 1.73205i 0.0942111i
\(339\) 0 0
\(340\) 0 0
\(341\) 6.92820 0.375183
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) −13.8564 −0.747087
\(345\) 0 0
\(346\) 24.0000 1.29025
\(347\) 27.7128i 1.48770i 0.668346 + 0.743851i \(0.267003\pi\)
−0.668346 + 0.743851i \(0.732997\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 18.0000i 0.959403i
\(353\) 6.92820i 0.368751i 0.982856 + 0.184376i \(0.0590263\pi\)
−0.982856 + 0.184376i \(0.940974\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.92820 −0.367194
\(357\) 0 0
\(358\) 24.0000i 1.26844i
\(359\) −10.3923 −0.548485 −0.274242 0.961661i \(-0.588427\pi\)
−0.274242 + 0.961661i \(0.588427\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 17.3205i 0.910346i
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) − 34.6410i − 1.80579i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 26.0000i − 1.34623i −0.739538 0.673114i \(-0.764956\pi\)
0.739538 0.673114i \(-0.235044\pi\)
\(374\) 41.5692 2.14949
\(375\) 0 0
\(376\) 18.0000 0.928279
\(377\) − 6.92820i − 0.356821i
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 24.0000i 1.22795i
\(383\) − 17.3205i − 0.885037i −0.896759 0.442518i \(-0.854085\pi\)
0.896759 0.442518i \(-0.145915\pi\)
\(384\) 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 \(-0.323350\pi\)
0.526911 + 0.849921i \(0.323350\pi\)
\(390\) 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
\(400\) 0 0
\(401\) 34.6410 1.72989 0.864945 0.501867i \(-0.167353\pi\)
0.864945 + 0.501867i \(0.167353\pi\)
\(402\) 0 0
\(403\) − 2.00000i − 0.0996271i
\(404\) −13.8564 −0.689382
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) 6.92820i 0.343418i
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 4.00000i − 0.197066i
\(413\) 6.92820i 0.340915i
\(414\) 0 0
\(415\) 0 0
\(416\) 5.19615 0.254762
\(417\) 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
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) − 13.8564i − 0.674519i
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 20.0000i − 0.967868i
\(428\) 6.92820i 0.334887i
\(429\) 0 0
\(430\) 0 0
\(431\) −38.1051 −1.83546 −0.917729 0.397206i \(-0.869980\pi\)
−0.917729 + 0.397206i \(0.869980\pi\)
\(432\) 0 0
\(433\) − 2.00000i − 0.0961139i −0.998845 0.0480569i \(-0.984697\pi\)
0.998845 0.0480569i \(-0.0153029\pi\)
\(434\) 6.92820 0.332564
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) − 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
\(442\) − 12.0000i − 0.570782i
\(443\) 20.7846i 0.987507i 0.869602 + 0.493753i \(0.164375\pi\)
−0.869602 + 0.493753i \(0.835625\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −3.46410 −0.164030
\(447\) 0 0
\(448\) − 2.00000i − 0.0944911i
\(449\) −6.92820 −0.326962 −0.163481 0.986546i \(-0.552272\pi\)
−0.163481 + 0.986546i \(0.552272\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 6.92820i 0.325875i
\(453\) 0 0
\(454\) −6.00000 −0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 24.2487i 1.13307i
\(459\) 0 0
\(460\) 0 0
\(461\) 27.7128 1.29071 0.645357 0.763881i \(-0.276709\pi\)
0.645357 + 0.763881i \(0.276709\pi\)
\(462\) 0 0
\(463\) − 2.00000i − 0.0929479i −0.998920 0.0464739i \(-0.985202\pi\)
0.998920 0.0464739i \(-0.0147984\pi\)
\(464\) −34.6410 −1.60817
\(465\) 0 0
\(466\) 0 0
\(467\) − 20.7846i − 0.961797i −0.876776 0.480899i \(-0.840311\pi\)
0.876776 0.480899i \(-0.159689\pi\)
\(468\) 0 0
\(469\) −28.0000 −1.29292
\(470\) 0 0
\(471\) 0 0
\(472\) − 6.00000i − 0.276172i
\(473\) − 27.7128i − 1.27424i
\(474\) 0 0
\(475\) 0 0
\(476\) 13.8564 0.635107
\(477\) 0 0
\(478\) 18.0000i 0.823301i
\(479\) −3.46410 −0.158279 −0.0791394 0.996864i \(-0.525217\pi\)
−0.0791394 + 0.996864i \(0.525217\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 17.3205i 0.788928i
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 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
\(493\) 48.0000i 2.16181i
\(494\) 3.46410 0.155857
\(495\) 0 0
\(496\) −10.0000 −0.449013
\(497\) − 6.92820i − 0.310772i
\(498\) 0 0
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 12.0000i − 0.535586i
\(503\) − 6.92820i − 0.308913i −0.988000 0.154457i \(-0.950637\pi\)
0.988000 0.154457i \(-0.0493627\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 41.5692 1.84798
\(507\) 0 0
\(508\) 16.0000i 0.709885i
\(509\) −13.8564 −0.614174 −0.307087 0.951681i \(-0.599354\pi\)
−0.307087 + 0.951681i \(0.599354\pi\)
\(510\) 0 0
\(511\) −20.0000 −0.884748
\(512\) − 8.66025i − 0.382733i
\(513\) 0 0
\(514\) −48.0000 −2.11719
\(515\) 0 0
\(516\) 0 0
\(517\) 36.0000i 1.58328i
\(518\) 6.92820i 0.304408i
\(519\) 0 0
\(520\) 0 0
\(521\) −20.7846 −0.910590 −0.455295 0.890341i \(-0.650466\pi\)
−0.455295 + 0.890341i \(0.650466\pi\)
\(522\) 0 0
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) −20.7846 −0.907980
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 13.8564i 0.603595i
\(528\) 0 0
\(529\) −25.0000 −1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000i 0.173422i
\(533\) 6.92820i 0.300094i
\(534\) 0 0
\(535\) 0 0
\(536\) 24.2487 1.04738
\(537\) 0 0
\(538\) 12.0000i 0.517357i
\(539\) 10.3923 0.447628
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 38.1051i 1.63675i
\(543\) 0 0
\(544\) −36.0000 −1.54349
\(545\) 0 0
\(546\) 0 0
\(547\) − 4.00000i − 0.171028i −0.996337 0.0855138i \(-0.972747\pi\)
0.996337 0.0855138i \(-0.0272532\pi\)
\(548\) − 6.92820i − 0.295958i
\(549\) 0 0
\(550\) 0 0
\(551\) −13.8564 −0.590303
\(552\) 0 0
\(553\) 8.00000i 0.340195i
\(554\) 3.46410 0.147176
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) − 27.7128i − 1.17423i −0.809504 0.587115i \(-0.800264\pi\)
0.809504 0.587115i \(-0.199736\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) − 36.0000i − 1.51857i
\(563\) 13.8564i 0.583978i 0.956422 + 0.291989i \(0.0943171\pi\)
−0.956422 + 0.291989i \(0.905683\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 6.92820 0.291214
\(567\) 0 0
\(568\) 6.00000i 0.251754i
\(569\) −13.8564 −0.580891 −0.290445 0.956892i \(-0.593803\pi\)
−0.290445 + 0.956892i \(0.593803\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 3.46410i 0.144841i
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) 0 0
\(576\) 0 0
\(577\) 26.0000i 1.08239i 0.840896 + 0.541197i \(0.182029\pi\)
−0.840896 + 0.541197i \(0.817971\pi\)
\(578\) 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
\(596\) −13.8564 −0.567581
\(597\) 0 0
\(598\) − 12.0000i − 0.490716i
\(599\) 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 \(0.743908\pi\)
\(602\) − 27.7128i − 1.12949i
\(603\) 0 0
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) 0 0
\(607\) − 40.0000i − 1.62355i −0.583970 0.811775i \(-0.698502\pi\)
0.583970 0.811775i \(-0.301498\pi\)
\(608\) − 10.3923i − 0.421464i
\(609\) 0 0
\(610\) 0 0
\(611\) 10.3923 0.420428
\(612\) 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
\(616\) 12.0000 0.483494
\(617\) 6.92820i 0.278919i 0.990228 + 0.139459i \(0.0445365\pi\)
−0.990228 + 0.139459i \(0.955464\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 36.0000i − 1.44347i
\(623\) 13.8564i 0.555145i
\(624\) 0 0
\(625\) 0 0
\(626\) −24.2487 −0.969173
\(627\) 0 0
\(628\) − 2.00000i − 0.0798087i
\(629\) −13.8564 −0.552491
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) − 6.92820i − 0.275589i
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.00000i − 0.118864i
\(638\) − 41.5692i − 1.64574i
\(639\) 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 \(-0.0890266\pi\)
\(644\) 13.8564 0.546019
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) 13.8564i 0.544752i 0.962191 + 0.272376i \(0.0878094\pi\)
−0.962191 + 0.272376i \(0.912191\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 14.0000i 0.548282i
\(653\) − 34.6410i − 1.35561i −0.735243 0.677804i \(-0.762932\pi\)
0.735243 0.677804i \(-0.237068\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 34.6410 1.35250
\(657\) 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
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 17.3205i 0.673181i
\(663\) 0 0
\(664\) 18.0000 0.698535
\(665\) 0 0
\(666\) 0 0
\(667\) 48.0000i 1.85857i
\(668\) 17.3205i 0.670151i
\(669\) 0 0
\(670\) 0 0
\(671\) −34.6410 −1.33730
\(672\) 0 0
\(673\) 10.0000i 0.385472i 0.981251 + 0.192736i \(0.0617360\pi\)
−0.981251 + 0.192736i \(0.938264\pi\)
\(674\) 3.46410 0.133432
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 41.5692i 1.59763i 0.601574 + 0.798817i \(0.294541\pi\)
−0.601574 + 0.798817i \(0.705459\pi\)
\(678\) 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
\(688\) 40.0000i 1.52499i
\(689\) 0 0
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) − 13.8564i − 0.526742i
\(693\) 0 0
\(694\) 48.0000 1.82206
\(695\) 0 0
\(696\) 0 0
\(697\) − 48.0000i − 1.81813i
\(698\) 45.0333i 1.70454i
\(699\) 0 0
\(700\) 0 0
\(701\) 20.7846 0.785024 0.392512 0.919747i \(-0.371606\pi\)
0.392512 + 0.919747i \(0.371606\pi\)
\(702\) 0 0
\(703\) − 4.00000i − 0.150863i
\(704\) −3.46410 −0.130558
\(705\) 0 0
\(706\) 12.0000 0.451626
\(707\) 27.7128i 1.04225i
\(708\) 0 0
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 12.0000i − 0.449719i
\(713\) 13.8564i 0.518927i
\(714\) 0 0
\(715\) 0 0
\(716\) 13.8564 0.517838
\(717\) 0 0
\(718\) 18.0000i 0.671754i
\(719\) 34.6410 1.29189 0.645946 0.763383i \(-0.276463\pi\)
0.645946 + 0.763383i \(0.276463\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 25.9808i 0.966904i
\(723\) 0 0
\(724\) 10.0000 0.371647
\(725\) 0 0
\(726\) 0 0
\(727\) 32.0000i 1.18681i 0.804902 + 0.593407i \(0.202218\pi\)
−0.804902 + 0.593407i \(0.797782\pi\)
\(728\) − 3.46410i − 0.128388i
\(729\) 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 \(0.783911\pi\)
\(734\) 13.8564 0.511449
\(735\) 0 0
\(736\) −36.0000 −1.32698
\(737\) 48.4974i 1.78643i
\(738\) 0 0
\(739\) 22.0000 0.809283 0.404642 0.914475i \(-0.367396\pi\)
0.404642 + 0.914475i \(0.367396\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 24.2487i − 0.889599i −0.895630 0.444799i \(-0.853275\pi\)
0.895630 0.444799i \(-0.146725\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −45.0333 −1.64879
\(747\) 0 0
\(748\) − 24.0000i − 0.877527i
\(749\) 13.8564 0.506302
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) − 51.9615i − 1.89484i
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 3.46410i 0.125822i
\(759\) 0 0
\(760\) 0 0
\(761\) −34.6410 −1.25574 −0.627868 0.778320i \(-0.716072\pi\)
−0.627868 + 0.778320i \(0.716072\pi\)
\(762\) 0 0
\(763\) 20.0000i 0.724049i
\(764\) 13.8564 0.501307
\(765\) 0 0
\(766\) −30.0000 −1.08394
\(767\) − 3.46410i − 0.125081i
\(768\) 0 0
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 26.0000i 0.935760i
\(773\) 13.8564i 0.498380i 0.968455 + 0.249190i \(0.0801644\pi\)
−0.968455 + 0.249190i \(0.919836\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −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
\(791\) 13.8564 0.492677
\(792\) 0 0
\(793\) 10.0000i 0.355110i
\(794\) 3.46410 0.122936
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) − 34.6410i − 1.22705i −0.789676 0.613524i \(-0.789751\pi\)
0.789676 0.613524i \(-0.210249\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 0 0
\(801\) 0 0
\(802\) − 60.0000i − 2.11867i
\(803\) 34.6410i 1.22245i
\(804\) 0 0
\(805\) 0 0
\(806\) −3.46410 −0.122018
\(807\) 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
\(811\) 26.0000 0.912983 0.456492 0.889728i \(-0.349106\pi\)
0.456492 + 0.889728i \(0.349106\pi\)
\(812\) − 13.8564i − 0.486265i
\(813\) 0 0
\(814\) 12.0000 0.420600
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000i 0.559769i
\(818\) − 17.3205i − 0.605597i
\(819\) 0 0
\(820\) 0 0
\(821\) −27.7128 −0.967184 −0.483592 0.875294i \(-0.660668\pi\)
−0.483592 + 0.875294i \(0.660668\pi\)
\(822\) 0 0
\(823\) 4.00000i 0.139431i 0.997567 + 0.0697156i \(0.0222092\pi\)
−0.997567 + 0.0697156i \(0.977791\pi\)
\(824\) 6.92820 0.241355
\(825\) 0 0
\(826\) 12.0000 0.417533
\(827\) − 10.3923i − 0.361376i −0.983540 0.180688i \(-0.942168\pi\)
0.983540 0.180688i \(-0.0578324\pi\)
\(828\) 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
\(832\) 1.00000i 0.0346688i
\(833\) 20.7846i 0.720144i
\(834\) 0 0
\(835\) 0 0
\(836\) 6.92820 0.239617
\(837\) 0 0
\(838\) − 12.0000i − 0.414533i
\(839\) 24.2487 0.837158 0.418579 0.908180i \(-0.362528\pi\)
0.418579 + 0.908180i \(0.362528\pi\)
\(840\) 0 0
\(841\) 19.0000 0.655172
\(842\) 17.3205i 0.596904i
\(843\) 0 0
\(844\) −8.00000 −0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −13.8564 −0.474991
\(852\) 0 0
\(853\) 46.0000i 1.57501i 0.616308 + 0.787505i \(0.288628\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) −34.6410 −1.18539
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 40.0000 1.36478 0.682391 0.730987i \(-0.260940\pi\)
0.682391 + 0.730987i \(0.260940\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 66.0000i 2.24797i
\(863\) − 31.1769i − 1.06127i −0.847599 0.530637i \(-0.821953\pi\)
0.847599 0.530637i \(-0.178047\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −3.46410 −0.117715
\(867\) 0 0
\(868\) − 4.00000i − 0.135769i
\(869\) 13.8564 0.470046
\(870\) 0 0
\(871\) 14.0000 0.474372
\(872\) − 17.3205i − 0.586546i
\(873\) 0 0
\(874\) −24.0000 −0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) − 22.0000i − 0.742887i −0.928456 0.371444i \(-0.878863\pi\)
0.928456 0.371444i \(-0.121137\pi\)
\(878\) − 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
\(883\) − 44.0000i − 1.48072i −0.672212 0.740359i \(-0.734656\pi\)
0.672212 0.740359i \(-0.265344\pi\)
\(884\) −6.92820 −0.233021
\(885\) 0 0
\(886\) 36.0000 1.20944
\(887\) − 27.7128i − 0.930505i −0.885178 0.465253i \(-0.845963\pi\)
0.885178 0.465253i \(-0.154037\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) 2.00000i 0.0669650i
\(893\) − 20.7846i − 0.695530i
\(894\) 0 0
\(895\) 0 0
\(896\) −24.2487 −0.810093
\(897\) 0 0
\(898\) 12.0000i 0.400445i
\(899\) 13.8564 0.462137
\(900\) 0 0
\(901\) 0 0
\(902\) 41.5692i 1.38410i
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) 0 0
\(907\) 44.0000i 1.46100i 0.682915 + 0.730498i \(0.260712\pi\)
−0.682915 + 0.730498i \(0.739288\pi\)
\(908\) 3.46410i 0.114960i
\(909\) 0 0
\(910\) 0 0
\(911\) 41.5692 1.37725 0.688625 0.725118i \(-0.258215\pi\)
0.688625 + 0.725118i \(0.258215\pi\)
\(912\) 0 0
\(913\) 36.0000i 1.19143i
\(914\) −17.3205 −0.572911
\(915\) 0 0
\(916\) 14.0000 0.462573
\(917\) 41.5692i 1.37274i
\(918\) 0 0
\(919\) 4.00000 0.131948 0.0659739 0.997821i \(-0.478985\pi\)
0.0659739 + 0.997821i \(0.478985\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 48.0000i − 1.58080i
\(923\) 3.46410i 0.114022i
\(924\) 0 0
\(925\) 0 0
\(926\) −3.46410 −0.113837
\(927\) 0 0
\(928\) 36.0000i 1.18176i
\(929\) −48.4974 −1.59115 −0.795574 0.605856i \(-0.792831\pi\)
−0.795574 + 0.605856i \(0.792831\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) 26.0000i 0.849383i 0.905338 + 0.424691i \(0.139617\pi\)
−0.905338 + 0.424691i \(0.860383\pi\)
\(938\) 48.4974i 1.58350i
\(939\) 0 0
\(940\) 0 0
\(941\) −41.5692 −1.35512 −0.677559 0.735469i \(-0.736962\pi\)
−0.677559 + 0.735469i \(0.736962\pi\)
\(942\) 0 0
\(943\) − 48.0000i − 1.56310i
\(944\) −17.3205 −0.563735
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) − 17.3205i − 0.562841i −0.959585 0.281420i \(-0.909194\pi\)
0.959585 0.281420i \(-0.0908056\pi\)
\(948\) 0 0
\(949\) 10.0000 0.324614
\(950\) 0 0
\(951\) 0 0
\(952\) 24.0000i 0.777844i
\(953\) − 27.7128i − 0.897706i −0.893606 0.448853i \(-0.851833\pi\)
0.893606 0.448853i \(-0.148167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 10.3923 0.336111
\(957\) 0 0
\(958\) 6.00000i 0.193851i
\(959\) −13.8564 −0.447447
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) − 3.46410i − 0.111687i
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) 2.00000i 0.0643157i 0.999483 + 0.0321578i \(0.0102379\pi\)
−0.999483 + 0.0321578i \(0.989762\pi\)
\(968\) − 1.73205i − 0.0556702i
\(969\) 0 0
\(970\) 0 0
\(971\) −55.4256 −1.77869 −0.889346 0.457234i \(-0.848840\pi\)
−0.889346 + 0.457234i \(0.848840\pi\)
\(972\) 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
\(986\) 83.1384 2.64767
\(987\) 0 0
\(988\) − 2.00000i − 0.0636285i
\(989\) 55.4256 1.76243
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 10.3923i 0.329956i
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) 0 0
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) 24.2487i 0.767580i
\(999\) 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 2925.2.c.s.2224.2 4
3.2 odd 2 inner 2925.2.c.s.2224.4 4
5.2 odd 4 2925.2.a.y.1.2 2
5.3 odd 4 117.2.a.b.1.1 2
5.4 even 2 inner 2925.2.c.s.2224.3 4
15.2 even 4 2925.2.a.y.1.1 2
15.8 even 4 117.2.a.b.1.2 yes 2
15.14 odd 2 inner 2925.2.c.s.2224.1 4
20.3 even 4 1872.2.a.v.1.1 2
35.13 even 4 5733.2.a.t.1.1 2
40.3 even 4 7488.2.a.cj.1.2 2
40.13 odd 4 7488.2.a.cq.1.1 2
45.13 odd 12 1053.2.e.i.703.2 4
45.23 even 12 1053.2.e.i.703.1 4
45.38 even 12 1053.2.e.i.352.1 4
45.43 odd 12 1053.2.e.i.352.2 4
60.23 odd 4 1872.2.a.v.1.2 2
65.8 even 4 1521.2.b.i.1351.1 4
65.18 even 4 1521.2.b.i.1351.4 4
65.38 odd 4 1521.2.a.j.1.2 2
105.83 odd 4 5733.2.a.t.1.2 2
120.53 even 4 7488.2.a.cq.1.2 2
120.83 odd 4 7488.2.a.cj.1.1 2
195.8 odd 4 1521.2.b.i.1351.3 4
195.38 even 4 1521.2.a.j.1.1 2
195.83 odd 4 1521.2.b.i.1351.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.a.b.1.1 2 5.3 odd 4
117.2.a.b.1.2 yes 2 15.8 even 4
1053.2.e.i.352.1 4 45.38 even 12
1053.2.e.i.352.2 4 45.43 odd 12
1053.2.e.i.703.1 4 45.23 even 12
1053.2.e.i.703.2 4 45.13 odd 12
1521.2.a.j.1.1 2 195.38 even 4
1521.2.a.j.1.2 2 65.38 odd 4
1521.2.b.i.1351.1 4 65.8 even 4
1521.2.b.i.1351.2 4 195.83 odd 4
1521.2.b.i.1351.3 4 195.8 odd 4
1521.2.b.i.1351.4 4 65.18 even 4
1872.2.a.v.1.1 2 20.3 even 4
1872.2.a.v.1.2 2 60.23 odd 4
2925.2.a.y.1.1 2 15.2 even 4
2925.2.a.y.1.2 2 5.2 odd 4
2925.2.c.s.2224.1 4 15.14 odd 2 inner
2925.2.c.s.2224.2 4 1.1 even 1 trivial
2925.2.c.s.2224.3 4 5.4 even 2 inner
2925.2.c.s.2224.4 4 3.2 odd 2 inner
5733.2.a.t.1.1 2 35.13 even 4
5733.2.a.t.1.2 2 105.83 odd 4
7488.2.a.cj.1.1 2 120.83 odd 4
7488.2.a.cj.1.2 2 40.3 even 4
7488.2.a.cq.1.1 2 40.13 odd 4
7488.2.a.cq.1.2 2 120.53 even 4