Properties

Label 6525.2.a.bb.1.2
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.a (trivial)

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: yes
Analytic conductor: \(52.1023873189\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1305)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 6525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.61803 q^{2} +0.618034 q^{4} +4.23607 q^{7} -2.23607 q^{8} +O(q^{10})\) \(q+1.61803 q^{2} +0.618034 q^{4} +4.23607 q^{7} -2.23607 q^{8} -0.236068 q^{11} -1.00000 q^{13} +6.85410 q^{14} -4.85410 q^{16} -7.47214 q^{17} +2.47214 q^{19} -0.381966 q^{22} -4.47214 q^{23} -1.61803 q^{26} +2.61803 q^{28} -1.00000 q^{29} -8.00000 q^{31} -3.38197 q^{32} -12.0902 q^{34} +4.00000 q^{38} -6.00000 q^{41} +6.00000 q^{43} -0.145898 q^{44} -7.23607 q^{46} -3.76393 q^{47} +10.9443 q^{49} -0.618034 q^{52} -2.47214 q^{53} -9.47214 q^{56} -1.61803 q^{58} -6.00000 q^{59} -8.47214 q^{61} -12.9443 q^{62} +4.23607 q^{64} +1.29180 q^{67} -4.61803 q^{68} -6.47214 q^{71} +6.00000 q^{73} +1.52786 q^{76} -1.00000 q^{77} -6.00000 q^{79} -9.70820 q^{82} +2.47214 q^{83} +9.70820 q^{86} +0.527864 q^{88} -9.94427 q^{89} -4.23607 q^{91} -2.76393 q^{92} -6.09017 q^{94} -15.4164 q^{97} +17.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} + 4 q^{7} + 4 q^{11} - 2 q^{13} + 7 q^{14} - 3 q^{16} - 6 q^{17} - 4 q^{19} - 3 q^{22} - q^{26} + 3 q^{28} - 2 q^{29} - 16 q^{31} - 9 q^{32} - 13 q^{34} + 8 q^{38} - 12 q^{41} + 12 q^{43} - 7 q^{44} - 10 q^{46} - 12 q^{47} + 4 q^{49} + q^{52} + 4 q^{53} - 10 q^{56} - q^{58} - 12 q^{59} - 8 q^{61} - 8 q^{62} + 4 q^{64} + 16 q^{67} - 7 q^{68} - 4 q^{71} + 12 q^{73} + 12 q^{76} - 2 q^{77} - 12 q^{79} - 6 q^{82} - 4 q^{83} + 6 q^{86} + 10 q^{88} - 2 q^{89} - 4 q^{91} - 10 q^{92} - q^{94} - 4 q^{97} + 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(3\) 0 0
\(4\) 0.618034 0.309017
\(5\) 0 0
\(6\) 0 0
\(7\) 4.23607 1.60108 0.800542 0.599277i \(-0.204545\pi\)
0.800542 + 0.599277i \(0.204545\pi\)
\(8\) −2.23607 −0.790569
\(9\) 0 0
\(10\) 0 0
\(11\) −0.236068 −0.0711772 −0.0355886 0.999367i \(-0.511331\pi\)
−0.0355886 + 0.999367i \(0.511331\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 6.85410 1.83184
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) −7.47214 −1.81226 −0.906130 0.423000i \(-0.860977\pi\)
−0.906130 + 0.423000i \(0.860977\pi\)
\(18\) 0 0
\(19\) 2.47214 0.567147 0.283573 0.958951i \(-0.408480\pi\)
0.283573 + 0.958951i \(0.408480\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.381966 −0.0814354
\(23\) −4.47214 −0.932505 −0.466252 0.884652i \(-0.654396\pi\)
−0.466252 + 0.884652i \(0.654396\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.61803 −0.317323
\(27\) 0 0
\(28\) 2.61803 0.494762
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) −3.38197 −0.597853
\(33\) 0 0
\(34\) −12.0902 −2.07345
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) −0.145898 −0.0219950
\(45\) 0 0
\(46\) −7.23607 −1.06690
\(47\) −3.76393 −0.549026 −0.274513 0.961583i \(-0.588517\pi\)
−0.274513 + 0.961583i \(0.588517\pi\)
\(48\) 0 0
\(49\) 10.9443 1.56347
\(50\) 0 0
\(51\) 0 0
\(52\) −0.618034 −0.0857059
\(53\) −2.47214 −0.339574 −0.169787 0.985481i \(-0.554308\pi\)
−0.169787 + 0.985481i \(0.554308\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −9.47214 −1.26577
\(57\) 0 0
\(58\) −1.61803 −0.212458
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −8.47214 −1.08475 −0.542373 0.840138i \(-0.682474\pi\)
−0.542373 + 0.840138i \(0.682474\pi\)
\(62\) −12.9443 −1.64392
\(63\) 0 0
\(64\) 4.23607 0.529508
\(65\) 0 0
\(66\) 0 0
\(67\) 1.29180 0.157818 0.0789090 0.996882i \(-0.474856\pi\)
0.0789090 + 0.996882i \(0.474856\pi\)
\(68\) −4.61803 −0.560019
\(69\) 0 0
\(70\) 0 0
\(71\) −6.47214 −0.768101 −0.384051 0.923312i \(-0.625471\pi\)
−0.384051 + 0.923312i \(0.625471\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 1.52786 0.175258
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) −6.00000 −0.675053 −0.337526 0.941316i \(-0.609590\pi\)
−0.337526 + 0.941316i \(0.609590\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −9.70820 −1.07209
\(83\) 2.47214 0.271352 0.135676 0.990753i \(-0.456679\pi\)
0.135676 + 0.990753i \(0.456679\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 9.70820 1.04686
\(87\) 0 0
\(88\) 0.527864 0.0562705
\(89\) −9.94427 −1.05409 −0.527045 0.849837i \(-0.676700\pi\)
−0.527045 + 0.849837i \(0.676700\pi\)
\(90\) 0 0
\(91\) −4.23607 −0.444061
\(92\) −2.76393 −0.288160
\(93\) 0 0
\(94\) −6.09017 −0.628153
\(95\) 0 0
\(96\) 0 0
\(97\) −15.4164 −1.56530 −0.782650 0.622463i \(-0.786132\pi\)
−0.782650 + 0.622463i \(0.786132\pi\)
\(98\) 17.7082 1.78880
\(99\) 0 0
\(100\) 0 0
\(101\) 6.52786 0.649547 0.324773 0.945792i \(-0.394712\pi\)
0.324773 + 0.945792i \(0.394712\pi\)
\(102\) 0 0
\(103\) 0.944272 0.0930419 0.0465209 0.998917i \(-0.485187\pi\)
0.0465209 + 0.998917i \(0.485187\pi\)
\(104\) 2.23607 0.219265
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) 4.47214 0.432338 0.216169 0.976356i \(-0.430644\pi\)
0.216169 + 0.976356i \(0.430644\pi\)
\(108\) 0 0
\(109\) 18.4164 1.76397 0.881986 0.471276i \(-0.156206\pi\)
0.881986 + 0.471276i \(0.156206\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −20.5623 −1.94296
\(113\) 1.47214 0.138487 0.0692435 0.997600i \(-0.477941\pi\)
0.0692435 + 0.997600i \(0.477941\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.618034 −0.0573830
\(117\) 0 0
\(118\) −9.70820 −0.893713
\(119\) −31.6525 −2.90158
\(120\) 0 0
\(121\) −10.9443 −0.994934
\(122\) −13.7082 −1.24108
\(123\) 0 0
\(124\) −4.94427 −0.444009
\(125\) 0 0
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) 13.6180 1.20368
\(129\) 0 0
\(130\) 0 0
\(131\) −4.23607 −0.370107 −0.185053 0.982728i \(-0.559246\pi\)
−0.185053 + 0.982728i \(0.559246\pi\)
\(132\) 0 0
\(133\) 10.4721 0.908049
\(134\) 2.09017 0.180563
\(135\) 0 0
\(136\) 16.7082 1.43272
\(137\) 6.94427 0.593289 0.296645 0.954988i \(-0.404132\pi\)
0.296645 + 0.954988i \(0.404132\pi\)
\(138\) 0 0
\(139\) 7.76393 0.658528 0.329264 0.944238i \(-0.393199\pi\)
0.329264 + 0.944238i \(0.393199\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −10.4721 −0.878802
\(143\) 0.236068 0.0197410
\(144\) 0 0
\(145\) 0 0
\(146\) 9.70820 0.803457
\(147\) 0 0
\(148\) 0 0
\(149\) −5.52786 −0.452860 −0.226430 0.974027i \(-0.572705\pi\)
−0.226430 + 0.974027i \(0.572705\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −5.52786 −0.448369
\(153\) 0 0
\(154\) −1.61803 −0.130385
\(155\) 0 0
\(156\) 0 0
\(157\) 15.4164 1.23036 0.615182 0.788385i \(-0.289083\pi\)
0.615182 + 0.788385i \(0.289083\pi\)
\(158\) −9.70820 −0.772343
\(159\) 0 0
\(160\) 0 0
\(161\) −18.9443 −1.49302
\(162\) 0 0
\(163\) 22.9443 1.79713 0.898567 0.438836i \(-0.144609\pi\)
0.898567 + 0.438836i \(0.144609\pi\)
\(164\) −3.70820 −0.289562
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) 13.4164 1.03819 0.519096 0.854716i \(-0.326269\pi\)
0.519096 + 0.854716i \(0.326269\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) 3.70820 0.282748
\(173\) −11.8885 −0.903869 −0.451935 0.892051i \(-0.649266\pi\)
−0.451935 + 0.892051i \(0.649266\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.14590 0.0863753
\(177\) 0 0
\(178\) −16.0902 −1.20601
\(179\) 14.9443 1.11699 0.558494 0.829509i \(-0.311380\pi\)
0.558494 + 0.829509i \(0.311380\pi\)
\(180\) 0 0
\(181\) −22.4164 −1.66620 −0.833099 0.553124i \(-0.813436\pi\)
−0.833099 + 0.553124i \(0.813436\pi\)
\(182\) −6.85410 −0.508060
\(183\) 0 0
\(184\) 10.0000 0.737210
\(185\) 0 0
\(186\) 0 0
\(187\) 1.76393 0.128991
\(188\) −2.32624 −0.169658
\(189\) 0 0
\(190\) 0 0
\(191\) 8.94427 0.647185 0.323592 0.946197i \(-0.395109\pi\)
0.323592 + 0.946197i \(0.395109\pi\)
\(192\) 0 0
\(193\) −20.0000 −1.43963 −0.719816 0.694165i \(-0.755774\pi\)
−0.719816 + 0.694165i \(0.755774\pi\)
\(194\) −24.9443 −1.79089
\(195\) 0 0
\(196\) 6.76393 0.483138
\(197\) 24.9443 1.77721 0.888603 0.458677i \(-0.151677\pi\)
0.888603 + 0.458677i \(0.151677\pi\)
\(198\) 0 0
\(199\) −20.1246 −1.42660 −0.713298 0.700861i \(-0.752799\pi\)
−0.713298 + 0.700861i \(0.752799\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 10.5623 0.743161
\(203\) −4.23607 −0.297314
\(204\) 0 0
\(205\) 0 0
\(206\) 1.52786 0.106451
\(207\) 0 0
\(208\) 4.85410 0.336571
\(209\) −0.583592 −0.0403679
\(210\) 0 0
\(211\) −0.944272 −0.0650064 −0.0325032 0.999472i \(-0.510348\pi\)
−0.0325032 + 0.999472i \(0.510348\pi\)
\(212\) −1.52786 −0.104934
\(213\) 0 0
\(214\) 7.23607 0.494647
\(215\) 0 0
\(216\) 0 0
\(217\) −33.8885 −2.30050
\(218\) 29.7984 2.01820
\(219\) 0 0
\(220\) 0 0
\(221\) 7.47214 0.502630
\(222\) 0 0
\(223\) −13.1803 −0.882621 −0.441310 0.897355i \(-0.645486\pi\)
−0.441310 + 0.897355i \(0.645486\pi\)
\(224\) −14.3262 −0.957212
\(225\) 0 0
\(226\) 2.38197 0.158446
\(227\) −13.5279 −0.897876 −0.448938 0.893563i \(-0.648198\pi\)
−0.448938 + 0.893563i \(0.648198\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.23607 0.146805
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3.70820 −0.241384
\(237\) 0 0
\(238\) −51.2148 −3.31976
\(239\) −20.9443 −1.35477 −0.677386 0.735628i \(-0.736887\pi\)
−0.677386 + 0.735628i \(0.736887\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) −17.7082 −1.13833
\(243\) 0 0
\(244\) −5.23607 −0.335205
\(245\) 0 0
\(246\) 0 0
\(247\) −2.47214 −0.157298
\(248\) 17.8885 1.13592
\(249\) 0 0
\(250\) 0 0
\(251\) 18.5967 1.17382 0.586908 0.809654i \(-0.300345\pi\)
0.586908 + 0.809654i \(0.300345\pi\)
\(252\) 0 0
\(253\) 1.05573 0.0663731
\(254\) 19.4164 1.21829
\(255\) 0 0
\(256\) 13.5623 0.847644
\(257\) −13.5279 −0.843845 −0.421922 0.906632i \(-0.638645\pi\)
−0.421922 + 0.906632i \(0.638645\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −6.85410 −0.423448
\(263\) 24.9443 1.53813 0.769065 0.639171i \(-0.220722\pi\)
0.769065 + 0.639171i \(0.220722\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 16.9443 1.03892
\(267\) 0 0
\(268\) 0.798374 0.0487684
\(269\) −9.47214 −0.577526 −0.288763 0.957401i \(-0.593244\pi\)
−0.288763 + 0.957401i \(0.593244\pi\)
\(270\) 0 0
\(271\) −3.52786 −0.214302 −0.107151 0.994243i \(-0.534173\pi\)
−0.107151 + 0.994243i \(0.534173\pi\)
\(272\) 36.2705 2.19922
\(273\) 0 0
\(274\) 11.2361 0.678796
\(275\) 0 0
\(276\) 0 0
\(277\) −19.9443 −1.19834 −0.599168 0.800624i \(-0.704502\pi\)
−0.599168 + 0.800624i \(0.704502\pi\)
\(278\) 12.5623 0.753437
\(279\) 0 0
\(280\) 0 0
\(281\) −22.4721 −1.34058 −0.670288 0.742101i \(-0.733829\pi\)
−0.670288 + 0.742101i \(0.733829\pi\)
\(282\) 0 0
\(283\) −4.94427 −0.293906 −0.146953 0.989143i \(-0.546947\pi\)
−0.146953 + 0.989143i \(0.546947\pi\)
\(284\) −4.00000 −0.237356
\(285\) 0 0
\(286\) 0.381966 0.0225861
\(287\) −25.4164 −1.50028
\(288\) 0 0
\(289\) 38.8328 2.28428
\(290\) 0 0
\(291\) 0 0
\(292\) 3.70820 0.217006
\(293\) 9.00000 0.525786 0.262893 0.964825i \(-0.415323\pi\)
0.262893 + 0.964825i \(0.415323\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) −8.94427 −0.518128
\(299\) 4.47214 0.258630
\(300\) 0 0
\(301\) 25.4164 1.46498
\(302\) −19.4164 −1.11729
\(303\) 0 0
\(304\) −12.0000 −0.688247
\(305\) 0 0
\(306\) 0 0
\(307\) 26.8328 1.53143 0.765715 0.643180i \(-0.222385\pi\)
0.765715 + 0.643180i \(0.222385\pi\)
\(308\) −0.618034 −0.0352158
\(309\) 0 0
\(310\) 0 0
\(311\) 20.1246 1.14116 0.570581 0.821241i \(-0.306718\pi\)
0.570581 + 0.821241i \(0.306718\pi\)
\(312\) 0 0
\(313\) −28.4164 −1.60619 −0.803095 0.595851i \(-0.796815\pi\)
−0.803095 + 0.595851i \(0.796815\pi\)
\(314\) 24.9443 1.40769
\(315\) 0 0
\(316\) −3.70820 −0.208603
\(317\) −20.8885 −1.17322 −0.586609 0.809870i \(-0.699537\pi\)
−0.586609 + 0.809870i \(0.699537\pi\)
\(318\) 0 0
\(319\) 0.236068 0.0132173
\(320\) 0 0
\(321\) 0 0
\(322\) −30.6525 −1.70820
\(323\) −18.4721 −1.02782
\(324\) 0 0
\(325\) 0 0
\(326\) 37.1246 2.05614
\(327\) 0 0
\(328\) 13.4164 0.740797
\(329\) −15.9443 −0.879036
\(330\) 0 0
\(331\) −9.88854 −0.543524 −0.271762 0.962365i \(-0.587606\pi\)
−0.271762 + 0.962365i \(0.587606\pi\)
\(332\) 1.52786 0.0838524
\(333\) 0 0
\(334\) 21.7082 1.18782
\(335\) 0 0
\(336\) 0 0
\(337\) −22.9443 −1.24985 −0.624927 0.780683i \(-0.714871\pi\)
−0.624927 + 0.780683i \(0.714871\pi\)
\(338\) −19.4164 −1.05611
\(339\) 0 0
\(340\) 0 0
\(341\) 1.88854 0.102270
\(342\) 0 0
\(343\) 16.7082 0.902158
\(344\) −13.4164 −0.723364
\(345\) 0 0
\(346\) −19.2361 −1.03414
\(347\) −29.8885 −1.60450 −0.802251 0.596987i \(-0.796364\pi\)
−0.802251 + 0.596987i \(0.796364\pi\)
\(348\) 0 0
\(349\) −22.9443 −1.22818 −0.614089 0.789237i \(-0.710477\pi\)
−0.614089 + 0.789237i \(0.710477\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.798374 0.0425535
\(353\) 1.88854 0.100517 0.0502585 0.998736i \(-0.483995\pi\)
0.0502585 + 0.998736i \(0.483995\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.14590 −0.325732
\(357\) 0 0
\(358\) 24.1803 1.27797
\(359\) 35.7771 1.88824 0.944121 0.329598i \(-0.106913\pi\)
0.944121 + 0.329598i \(0.106913\pi\)
\(360\) 0 0
\(361\) −12.8885 −0.678344
\(362\) −36.2705 −1.90634
\(363\) 0 0
\(364\) −2.61803 −0.137222
\(365\) 0 0
\(366\) 0 0
\(367\) −1.41641 −0.0739359 −0.0369679 0.999316i \(-0.511770\pi\)
−0.0369679 + 0.999316i \(0.511770\pi\)
\(368\) 21.7082 1.13162
\(369\) 0 0
\(370\) 0 0
\(371\) −10.4721 −0.543686
\(372\) 0 0
\(373\) 28.8328 1.49291 0.746453 0.665438i \(-0.231755\pi\)
0.746453 + 0.665438i \(0.231755\pi\)
\(374\) 2.85410 0.147582
\(375\) 0 0
\(376\) 8.41641 0.434043
\(377\) 1.00000 0.0515026
\(378\) 0 0
\(379\) −2.58359 −0.132710 −0.0663551 0.997796i \(-0.521137\pi\)
−0.0663551 + 0.997796i \(0.521137\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 14.4721 0.740459
\(383\) 18.0000 0.919757 0.459879 0.887982i \(-0.347893\pi\)
0.459879 + 0.887982i \(0.347893\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −32.3607 −1.64712
\(387\) 0 0
\(388\) −9.52786 −0.483704
\(389\) −19.4721 −0.987276 −0.493638 0.869667i \(-0.664333\pi\)
−0.493638 + 0.869667i \(0.664333\pi\)
\(390\) 0 0
\(391\) 33.4164 1.68994
\(392\) −24.4721 −1.23603
\(393\) 0 0
\(394\) 40.3607 2.03334
\(395\) 0 0
\(396\) 0 0
\(397\) 7.88854 0.395915 0.197957 0.980211i \(-0.436569\pi\)
0.197957 + 0.980211i \(0.436569\pi\)
\(398\) −32.5623 −1.63220
\(399\) 0 0
\(400\) 0 0
\(401\) −10.3607 −0.517388 −0.258694 0.965959i \(-0.583292\pi\)
−0.258694 + 0.965959i \(0.583292\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) 4.03444 0.200721
\(405\) 0 0
\(406\) −6.85410 −0.340163
\(407\) 0 0
\(408\) 0 0
\(409\) 27.4164 1.35565 0.677827 0.735221i \(-0.262922\pi\)
0.677827 + 0.735221i \(0.262922\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.583592 0.0287515
\(413\) −25.4164 −1.25066
\(414\) 0 0
\(415\) 0 0
\(416\) 3.38197 0.165815
\(417\) 0 0
\(418\) −0.944272 −0.0461858
\(419\) −37.4164 −1.82791 −0.913956 0.405814i \(-0.866988\pi\)
−0.913956 + 0.405814i \(0.866988\pi\)
\(420\) 0 0
\(421\) −17.4164 −0.848824 −0.424412 0.905469i \(-0.639519\pi\)
−0.424412 + 0.905469i \(0.639519\pi\)
\(422\) −1.52786 −0.0743753
\(423\) 0 0
\(424\) 5.52786 0.268457
\(425\) 0 0
\(426\) 0 0
\(427\) −35.8885 −1.73677
\(428\) 2.76393 0.133600
\(429\) 0 0
\(430\) 0 0
\(431\) 12.4721 0.600762 0.300381 0.953819i \(-0.402886\pi\)
0.300381 + 0.953819i \(0.402886\pi\)
\(432\) 0 0
\(433\) 39.3050 1.88888 0.944438 0.328690i \(-0.106607\pi\)
0.944438 + 0.328690i \(0.106607\pi\)
\(434\) −54.8328 −2.63206
\(435\) 0 0
\(436\) 11.3820 0.545097
\(437\) −11.0557 −0.528867
\(438\) 0 0
\(439\) −6.23607 −0.297631 −0.148816 0.988865i \(-0.547546\pi\)
−0.148816 + 0.988865i \(0.547546\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 12.0902 0.575071
\(443\) −5.76393 −0.273853 −0.136926 0.990581i \(-0.543722\pi\)
−0.136926 + 0.990581i \(0.543722\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −21.3262 −1.00983
\(447\) 0 0
\(448\) 17.9443 0.847787
\(449\) 41.9443 1.97947 0.989736 0.142906i \(-0.0456446\pi\)
0.989736 + 0.142906i \(0.0456446\pi\)
\(450\) 0 0
\(451\) 1.41641 0.0666960
\(452\) 0.909830 0.0427948
\(453\) 0 0
\(454\) −21.8885 −1.02728
\(455\) 0 0
\(456\) 0 0
\(457\) 23.3607 1.09277 0.546383 0.837535i \(-0.316004\pi\)
0.546383 + 0.837535i \(0.316004\pi\)
\(458\) 6.47214 0.302423
\(459\) 0 0
\(460\) 0 0
\(461\) −41.7771 −1.94575 −0.972876 0.231325i \(-0.925694\pi\)
−0.972876 + 0.231325i \(0.925694\pi\)
\(462\) 0 0
\(463\) 7.76393 0.360821 0.180410 0.983591i \(-0.442257\pi\)
0.180410 + 0.983591i \(0.442257\pi\)
\(464\) 4.85410 0.225346
\(465\) 0 0
\(466\) −29.1246 −1.34917
\(467\) −24.0000 −1.11059 −0.555294 0.831654i \(-0.687394\pi\)
−0.555294 + 0.831654i \(0.687394\pi\)
\(468\) 0 0
\(469\) 5.47214 0.252680
\(470\) 0 0
\(471\) 0 0
\(472\) 13.4164 0.617540
\(473\) −1.41641 −0.0651265
\(474\) 0 0
\(475\) 0 0
\(476\) −19.5623 −0.896637
\(477\) 0 0
\(478\) −33.8885 −1.55003
\(479\) −33.8885 −1.54841 −0.774204 0.632937i \(-0.781849\pi\)
−0.774204 + 0.632937i \(0.781849\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −8.09017 −0.368497
\(483\) 0 0
\(484\) −6.76393 −0.307451
\(485\) 0 0
\(486\) 0 0
\(487\) 7.05573 0.319726 0.159863 0.987139i \(-0.448895\pi\)
0.159863 + 0.987139i \(0.448895\pi\)
\(488\) 18.9443 0.857567
\(489\) 0 0
\(490\) 0 0
\(491\) 14.8328 0.669396 0.334698 0.942326i \(-0.391366\pi\)
0.334698 + 0.942326i \(0.391366\pi\)
\(492\) 0 0
\(493\) 7.47214 0.336528
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 38.8328 1.74364
\(497\) −27.4164 −1.22979
\(498\) 0 0
\(499\) −26.1246 −1.16950 −0.584749 0.811214i \(-0.698807\pi\)
−0.584749 + 0.811214i \(0.698807\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 30.0902 1.34299
\(503\) 14.1246 0.629785 0.314893 0.949127i \(-0.398031\pi\)
0.314893 + 0.949127i \(0.398031\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.70820 0.0759389
\(507\) 0 0
\(508\) 7.41641 0.329050
\(509\) 38.3607 1.70031 0.850154 0.526535i \(-0.176509\pi\)
0.850154 + 0.526535i \(0.176509\pi\)
\(510\) 0 0
\(511\) 25.4164 1.12436
\(512\) −5.29180 −0.233867
\(513\) 0 0
\(514\) −21.8885 −0.965462
\(515\) 0 0
\(516\) 0 0
\(517\) 0.888544 0.0390781
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −12.4721 −0.546414 −0.273207 0.961955i \(-0.588084\pi\)
−0.273207 + 0.961955i \(0.588084\pi\)
\(522\) 0 0
\(523\) 33.5410 1.46665 0.733323 0.679880i \(-0.237968\pi\)
0.733323 + 0.679880i \(0.237968\pi\)
\(524\) −2.61803 −0.114369
\(525\) 0 0
\(526\) 40.3607 1.75981
\(527\) 59.7771 2.60393
\(528\) 0 0
\(529\) −3.00000 −0.130435
\(530\) 0 0
\(531\) 0 0
\(532\) 6.47214 0.280603
\(533\) 6.00000 0.259889
\(534\) 0 0
\(535\) 0 0
\(536\) −2.88854 −0.124766
\(537\) 0 0
\(538\) −15.3262 −0.660761
\(539\) −2.58359 −0.111283
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) −5.70820 −0.245188
\(543\) 0 0
\(544\) 25.2705 1.08346
\(545\) 0 0
\(546\) 0 0
\(547\) 30.7082 1.31299 0.656494 0.754331i \(-0.272039\pi\)
0.656494 + 0.754331i \(0.272039\pi\)
\(548\) 4.29180 0.183336
\(549\) 0 0
\(550\) 0 0
\(551\) −2.47214 −0.105317
\(552\) 0 0
\(553\) −25.4164 −1.08082
\(554\) −32.2705 −1.37104
\(555\) 0 0
\(556\) 4.79837 0.203496
\(557\) −31.4164 −1.33116 −0.665578 0.746328i \(-0.731815\pi\)
−0.665578 + 0.746328i \(0.731815\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) 0 0
\(561\) 0 0
\(562\) −36.3607 −1.53378
\(563\) 19.1803 0.808355 0.404177 0.914681i \(-0.367558\pi\)
0.404177 + 0.914681i \(0.367558\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −8.00000 −0.336265
\(567\) 0 0
\(568\) 14.4721 0.607237
\(569\) 31.9443 1.33917 0.669587 0.742734i \(-0.266471\pi\)
0.669587 + 0.742734i \(0.266471\pi\)
\(570\) 0 0
\(571\) 30.8328 1.29031 0.645157 0.764050i \(-0.276792\pi\)
0.645157 + 0.764050i \(0.276792\pi\)
\(572\) 0.145898 0.00610030
\(573\) 0 0
\(574\) −41.1246 −1.71651
\(575\) 0 0
\(576\) 0 0
\(577\) 29.0557 1.20961 0.604803 0.796375i \(-0.293252\pi\)
0.604803 + 0.796375i \(0.293252\pi\)
\(578\) 62.8328 2.61350
\(579\) 0 0
\(580\) 0 0
\(581\) 10.4721 0.434457
\(582\) 0 0
\(583\) 0.583592 0.0241699
\(584\) −13.4164 −0.555175
\(585\) 0 0
\(586\) 14.5623 0.601563
\(587\) −6.94427 −0.286621 −0.143310 0.989678i \(-0.545775\pi\)
−0.143310 + 0.989678i \(0.545775\pi\)
\(588\) 0 0
\(589\) −19.7771 −0.814901
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −12.4721 −0.512169 −0.256085 0.966654i \(-0.582433\pi\)
−0.256085 + 0.966654i \(0.582433\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3.41641 −0.139942
\(597\) 0 0
\(598\) 7.23607 0.295905
\(599\) 29.0689 1.18772 0.593861 0.804568i \(-0.297603\pi\)
0.593861 + 0.804568i \(0.297603\pi\)
\(600\) 0 0
\(601\) 34.8328 1.42086 0.710430 0.703768i \(-0.248500\pi\)
0.710430 + 0.703768i \(0.248500\pi\)
\(602\) 41.1246 1.67611
\(603\) 0 0
\(604\) −7.41641 −0.301769
\(605\) 0 0
\(606\) 0 0
\(607\) 23.4164 0.950443 0.475221 0.879866i \(-0.342368\pi\)
0.475221 + 0.879866i \(0.342368\pi\)
\(608\) −8.36068 −0.339070
\(609\) 0 0
\(610\) 0 0
\(611\) 3.76393 0.152272
\(612\) 0 0
\(613\) 23.0000 0.928961 0.464481 0.885583i \(-0.346241\pi\)
0.464481 + 0.885583i \(0.346241\pi\)
\(614\) 43.4164 1.75214
\(615\) 0 0
\(616\) 2.23607 0.0900937
\(617\) 26.9443 1.08474 0.542368 0.840141i \(-0.317528\pi\)
0.542368 + 0.840141i \(0.317528\pi\)
\(618\) 0 0
\(619\) 35.3050 1.41903 0.709513 0.704692i \(-0.248915\pi\)
0.709513 + 0.704692i \(0.248915\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 32.5623 1.30563
\(623\) −42.1246 −1.68769
\(624\) 0 0
\(625\) 0 0
\(626\) −45.9787 −1.83768
\(627\) 0 0
\(628\) 9.52786 0.380203
\(629\) 0 0
\(630\) 0 0
\(631\) −48.1246 −1.91581 −0.957905 0.287084i \(-0.907314\pi\)
−0.957905 + 0.287084i \(0.907314\pi\)
\(632\) 13.4164 0.533676
\(633\) 0 0
\(634\) −33.7984 −1.34230
\(635\) 0 0
\(636\) 0 0
\(637\) −10.9443 −0.433628
\(638\) 0.381966 0.0151222
\(639\) 0 0
\(640\) 0 0
\(641\) 3.94427 0.155789 0.0778947 0.996962i \(-0.475180\pi\)
0.0778947 + 0.996962i \(0.475180\pi\)
\(642\) 0 0
\(643\) −45.5410 −1.79596 −0.897981 0.440034i \(-0.854966\pi\)
−0.897981 + 0.440034i \(0.854966\pi\)
\(644\) −11.7082 −0.461368
\(645\) 0 0
\(646\) −29.8885 −1.17595
\(647\) 46.9443 1.84557 0.922785 0.385316i \(-0.125907\pi\)
0.922785 + 0.385316i \(0.125907\pi\)
\(648\) 0 0
\(649\) 1.41641 0.0555989
\(650\) 0 0
\(651\) 0 0
\(652\) 14.1803 0.555345
\(653\) −16.8885 −0.660900 −0.330450 0.943824i \(-0.607200\pi\)
−0.330450 + 0.943824i \(0.607200\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 29.1246 1.13713
\(657\) 0 0
\(658\) −25.7984 −1.00573
\(659\) −36.4853 −1.42127 −0.710633 0.703563i \(-0.751591\pi\)
−0.710633 + 0.703563i \(0.751591\pi\)
\(660\) 0 0
\(661\) 0.416408 0.0161964 0.00809819 0.999967i \(-0.497422\pi\)
0.00809819 + 0.999967i \(0.497422\pi\)
\(662\) −16.0000 −0.621858
\(663\) 0 0
\(664\) −5.52786 −0.214523
\(665\) 0 0
\(666\) 0 0
\(667\) 4.47214 0.173162
\(668\) 8.29180 0.320819
\(669\) 0 0
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) −28.4164 −1.09537 −0.547686 0.836684i \(-0.684491\pi\)
−0.547686 + 0.836684i \(0.684491\pi\)
\(674\) −37.1246 −1.42999
\(675\) 0 0
\(676\) −7.41641 −0.285246
\(677\) −17.9443 −0.689654 −0.344827 0.938666i \(-0.612062\pi\)
−0.344827 + 0.938666i \(0.612062\pi\)
\(678\) 0 0
\(679\) −65.3050 −2.50617
\(680\) 0 0
\(681\) 0 0
\(682\) 3.05573 0.117010
\(683\) −19.0557 −0.729147 −0.364574 0.931175i \(-0.618785\pi\)
−0.364574 + 0.931175i \(0.618785\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 27.0344 1.03218
\(687\) 0 0
\(688\) −29.1246 −1.11037
\(689\) 2.47214 0.0941809
\(690\) 0 0
\(691\) −28.7082 −1.09211 −0.546056 0.837749i \(-0.683871\pi\)
−0.546056 + 0.837749i \(0.683871\pi\)
\(692\) −7.34752 −0.279311
\(693\) 0 0
\(694\) −48.3607 −1.83575
\(695\) 0 0
\(696\) 0 0
\(697\) 44.8328 1.69816
\(698\) −37.1246 −1.40519
\(699\) 0 0
\(700\) 0 0
\(701\) −34.4721 −1.30199 −0.650997 0.759080i \(-0.725649\pi\)
−0.650997 + 0.759080i \(0.725649\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 3.05573 0.115004
\(707\) 27.6525 1.03998
\(708\) 0 0
\(709\) 14.0000 0.525781 0.262891 0.964826i \(-0.415324\pi\)
0.262891 + 0.964826i \(0.415324\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 22.2361 0.833332
\(713\) 35.7771 1.33986
\(714\) 0 0
\(715\) 0 0
\(716\) 9.23607 0.345168
\(717\) 0 0
\(718\) 57.8885 2.16038
\(719\) 23.8885 0.890892 0.445446 0.895309i \(-0.353045\pi\)
0.445446 + 0.895309i \(0.353045\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) −20.8541 −0.776109
\(723\) 0 0
\(724\) −13.8541 −0.514884
\(725\) 0 0
\(726\) 0 0
\(727\) 48.2492 1.78946 0.894732 0.446603i \(-0.147366\pi\)
0.894732 + 0.446603i \(0.147366\pi\)
\(728\) 9.47214 0.351061
\(729\) 0 0
\(730\) 0 0
\(731\) −44.8328 −1.65820
\(732\) 0 0
\(733\) 3.52786 0.130305 0.0651523 0.997875i \(-0.479247\pi\)
0.0651523 + 0.997875i \(0.479247\pi\)
\(734\) −2.29180 −0.0845917
\(735\) 0 0
\(736\) 15.1246 0.557501
\(737\) −0.304952 −0.0112330
\(738\) 0 0
\(739\) 43.4164 1.59710 0.798549 0.601930i \(-0.205601\pi\)
0.798549 + 0.601930i \(0.205601\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −16.9443 −0.622044
\(743\) −39.7639 −1.45880 −0.729399 0.684089i \(-0.760200\pi\)
−0.729399 + 0.684089i \(0.760200\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 46.6525 1.70807
\(747\) 0 0
\(748\) 1.09017 0.0398606
\(749\) 18.9443 0.692209
\(750\) 0 0
\(751\) −10.1115 −0.368972 −0.184486 0.982835i \(-0.559062\pi\)
−0.184486 + 0.982835i \(0.559062\pi\)
\(752\) 18.2705 0.666257
\(753\) 0 0
\(754\) 1.61803 0.0589253
\(755\) 0 0
\(756\) 0 0
\(757\) 28.8328 1.04795 0.523973 0.851735i \(-0.324449\pi\)
0.523973 + 0.851735i \(0.324449\pi\)
\(758\) −4.18034 −0.151837
\(759\) 0 0
\(760\) 0 0
\(761\) −44.4721 −1.61211 −0.806057 0.591838i \(-0.798402\pi\)
−0.806057 + 0.591838i \(0.798402\pi\)
\(762\) 0 0
\(763\) 78.0132 2.82427
\(764\) 5.52786 0.199991
\(765\) 0 0
\(766\) 29.1246 1.05231
\(767\) 6.00000 0.216647
\(768\) 0 0
\(769\) −17.3050 −0.624033 −0.312016 0.950077i \(-0.601004\pi\)
−0.312016 + 0.950077i \(0.601004\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −12.3607 −0.444871
\(773\) 20.8328 0.749304 0.374652 0.927165i \(-0.377762\pi\)
0.374652 + 0.927165i \(0.377762\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 34.4721 1.23748
\(777\) 0 0
\(778\) −31.5066 −1.12957
\(779\) −14.8328 −0.531441
\(780\) 0 0
\(781\) 1.52786 0.0546713
\(782\) 54.0689 1.93350
\(783\) 0 0
\(784\) −53.1246 −1.89731
\(785\) 0 0
\(786\) 0 0
\(787\) −27.7771 −0.990146 −0.495073 0.868851i \(-0.664859\pi\)
−0.495073 + 0.868851i \(0.664859\pi\)
\(788\) 15.4164 0.549187
\(789\) 0 0
\(790\) 0 0
\(791\) 6.23607 0.221729
\(792\) 0 0
\(793\) 8.47214 0.300854
\(794\) 12.7639 0.452975
\(795\) 0 0
\(796\) −12.4377 −0.440842
\(797\) 42.7214 1.51327 0.756634 0.653839i \(-0.226842\pi\)
0.756634 + 0.653839i \(0.226842\pi\)
\(798\) 0 0
\(799\) 28.1246 0.994977
\(800\) 0 0
\(801\) 0 0
\(802\) −16.7639 −0.591955
\(803\) −1.41641 −0.0499839
\(804\) 0 0
\(805\) 0 0
\(806\) 12.9443 0.455943
\(807\) 0 0
\(808\) −14.5967 −0.513512
\(809\) −19.9443 −0.701203 −0.350602 0.936525i \(-0.614023\pi\)
−0.350602 + 0.936525i \(0.614023\pi\)
\(810\) 0 0
\(811\) 25.1803 0.884201 0.442101 0.896965i \(-0.354233\pi\)
0.442101 + 0.896965i \(0.354233\pi\)
\(812\) −2.61803 −0.0918750
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 14.8328 0.518935
\(818\) 44.3607 1.55103
\(819\) 0 0
\(820\) 0 0
\(821\) 31.0557 1.08385 0.541926 0.840426i \(-0.317695\pi\)
0.541926 + 0.840426i \(0.317695\pi\)
\(822\) 0 0
\(823\) 2.00000 0.0697156 0.0348578 0.999392i \(-0.488902\pi\)
0.0348578 + 0.999392i \(0.488902\pi\)
\(824\) −2.11146 −0.0735561
\(825\) 0 0
\(826\) −41.1246 −1.43091
\(827\) 6.11146 0.212516 0.106258 0.994339i \(-0.466113\pi\)
0.106258 + 0.994339i \(0.466113\pi\)
\(828\) 0 0
\(829\) 15.8885 0.551832 0.275916 0.961182i \(-0.411019\pi\)
0.275916 + 0.961182i \(0.411019\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.23607 −0.146859
\(833\) −81.7771 −2.83341
\(834\) 0 0
\(835\) 0 0
\(836\) −0.360680 −0.0124744
\(837\) 0 0
\(838\) −60.5410 −2.09135
\(839\) 5.29180 0.182693 0.0913465 0.995819i \(-0.470883\pi\)
0.0913465 + 0.995819i \(0.470883\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −28.1803 −0.971159
\(843\) 0 0
\(844\) −0.583592 −0.0200881
\(845\) 0 0
\(846\) 0 0
\(847\) −46.3607 −1.59297
\(848\) 12.0000 0.412082
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −51.3050 −1.75665 −0.878324 0.478066i \(-0.841338\pi\)
−0.878324 + 0.478066i \(0.841338\pi\)
\(854\) −58.0689 −1.98708
\(855\) 0 0
\(856\) −10.0000 −0.341793
\(857\) −48.7214 −1.66429 −0.832145 0.554558i \(-0.812887\pi\)
−0.832145 + 0.554558i \(0.812887\pi\)
\(858\) 0 0
\(859\) 36.8328 1.25672 0.628360 0.777923i \(-0.283727\pi\)
0.628360 + 0.777923i \(0.283727\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 20.1803 0.687345
\(863\) 45.5279 1.54979 0.774893 0.632092i \(-0.217804\pi\)
0.774893 + 0.632092i \(0.217804\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 63.5967 2.16111
\(867\) 0 0
\(868\) −20.9443 −0.710895
\(869\) 1.41641 0.0480483
\(870\) 0 0
\(871\) −1.29180 −0.0437708
\(872\) −41.1803 −1.39454
\(873\) 0 0
\(874\) −17.8885 −0.605089
\(875\) 0 0
\(876\) 0 0
\(877\) 19.8885 0.671588 0.335794 0.941935i \(-0.390995\pi\)
0.335794 + 0.941935i \(0.390995\pi\)
\(878\) −10.0902 −0.340527
\(879\) 0 0
\(880\) 0 0
\(881\) −15.1115 −0.509118 −0.254559 0.967057i \(-0.581930\pi\)
−0.254559 + 0.967057i \(0.581930\pi\)
\(882\) 0 0
\(883\) 14.8328 0.499164 0.249582 0.968354i \(-0.419707\pi\)
0.249582 + 0.968354i \(0.419707\pi\)
\(884\) 4.61803 0.155321
\(885\) 0 0
\(886\) −9.32624 −0.313321
\(887\) −5.18034 −0.173939 −0.0869694 0.996211i \(-0.527718\pi\)
−0.0869694 + 0.996211i \(0.527718\pi\)
\(888\) 0 0
\(889\) 50.8328 1.70488
\(890\) 0 0
\(891\) 0 0
\(892\) −8.14590 −0.272745
\(893\) −9.30495 −0.311378
\(894\) 0 0
\(895\) 0 0
\(896\) 57.6869 1.92718
\(897\) 0 0
\(898\) 67.8673 2.26476
\(899\) 8.00000 0.266815
\(900\) 0 0
\(901\) 18.4721 0.615396
\(902\) 2.29180 0.0763085
\(903\) 0 0
\(904\) −3.29180 −0.109484
\(905\) 0 0
\(906\) 0 0
\(907\) 24.8328 0.824560 0.412280 0.911057i \(-0.364733\pi\)
0.412280 + 0.911057i \(0.364733\pi\)
\(908\) −8.36068 −0.277459
\(909\) 0 0
\(910\) 0 0
\(911\) −21.7639 −0.721071 −0.360536 0.932745i \(-0.617406\pi\)
−0.360536 + 0.932745i \(0.617406\pi\)
\(912\) 0 0
\(913\) −0.583592 −0.0193141
\(914\) 37.7984 1.25026
\(915\) 0 0
\(916\) 2.47214 0.0816817
\(917\) −17.9443 −0.592572
\(918\) 0 0
\(919\) 21.5410 0.710573 0.355286 0.934758i \(-0.384383\pi\)
0.355286 + 0.934758i \(0.384383\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −67.5967 −2.22618
\(923\) 6.47214 0.213033
\(924\) 0 0
\(925\) 0 0
\(926\) 12.5623 0.412823
\(927\) 0 0
\(928\) 3.38197 0.111018
\(929\) 9.05573 0.297109 0.148554 0.988904i \(-0.452538\pi\)
0.148554 + 0.988904i \(0.452538\pi\)
\(930\) 0 0
\(931\) 27.0557 0.886716
\(932\) −11.1246 −0.364399
\(933\) 0 0
\(934\) −38.8328 −1.27065
\(935\) 0 0
\(936\) 0 0
\(937\) −45.4721 −1.48551 −0.742755 0.669563i \(-0.766481\pi\)
−0.742755 + 0.669563i \(0.766481\pi\)
\(938\) 8.85410 0.289097
\(939\) 0 0
\(940\) 0 0
\(941\) 13.4164 0.437362 0.218681 0.975796i \(-0.429825\pi\)
0.218681 + 0.975796i \(0.429825\pi\)
\(942\) 0 0
\(943\) 26.8328 0.873797
\(944\) 29.1246 0.947925
\(945\) 0 0
\(946\) −2.29180 −0.0745127
\(947\) 57.5410 1.86983 0.934916 0.354869i \(-0.115475\pi\)
0.934916 + 0.354869i \(0.115475\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) 0 0
\(952\) 70.7771 2.29390
\(953\) −1.63932 −0.0531028 −0.0265514 0.999647i \(-0.508453\pi\)
−0.0265514 + 0.999647i \(0.508453\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −12.9443 −0.418648
\(957\) 0 0
\(958\) −54.8328 −1.77157
\(959\) 29.4164 0.949905
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) −3.09017 −0.0995277
\(965\) 0 0
\(966\) 0 0
\(967\) −11.4164 −0.367127 −0.183563 0.983008i \(-0.558763\pi\)
−0.183563 + 0.983008i \(0.558763\pi\)
\(968\) 24.4721 0.786564
\(969\) 0 0
\(970\) 0 0
\(971\) 10.1115 0.324492 0.162246 0.986750i \(-0.448126\pi\)
0.162246 + 0.986750i \(0.448126\pi\)
\(972\) 0 0
\(973\) 32.8885 1.05436
\(974\) 11.4164 0.365805
\(975\) 0 0
\(976\) 41.1246 1.31637
\(977\) 48.9443 1.56587 0.782933 0.622106i \(-0.213723\pi\)
0.782933 + 0.622106i \(0.213723\pi\)
\(978\) 0 0
\(979\) 2.34752 0.0750272
\(980\) 0 0
\(981\) 0 0
\(982\) 24.0000 0.765871
\(983\) −0.944272 −0.0301176 −0.0150588 0.999887i \(-0.504794\pi\)
−0.0150588 + 0.999887i \(0.504794\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 12.0902 0.385029
\(987\) 0 0
\(988\) −1.52786 −0.0486078
\(989\) −26.8328 −0.853234
\(990\) 0 0
\(991\) −2.70820 −0.0860289 −0.0430145 0.999074i \(-0.513696\pi\)
−0.0430145 + 0.999074i \(0.513696\pi\)
\(992\) 27.0557 0.859020
\(993\) 0 0
\(994\) −44.3607 −1.40704
\(995\) 0 0
\(996\) 0 0
\(997\) −12.9443 −0.409949 −0.204975 0.978767i \(-0.565711\pi\)
−0.204975 + 0.978767i \(0.565711\pi\)
\(998\) −42.2705 −1.33805
\(999\) 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 6525.2.a.bb.1.2 2
3.2 odd 2 6525.2.a.r.1.1 2
5.4 even 2 1305.2.a.h.1.1 2
15.14 odd 2 1305.2.a.l.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.h.1.1 2 5.4 even 2
1305.2.a.l.1.2 yes 2 15.14 odd 2
6525.2.a.r.1.1 2 3.2 odd 2
6525.2.a.bb.1.2 2 1.1 even 1 trivial