Properties

Label 1305.2.a.h.1.1
Level $1305$
Weight $2$
Character 1305.1
Self dual yes
Analytic conductor $10.420$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(1,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, 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("1305.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.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: \(10.4204774638\)
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: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 1305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -4.23607 q^{7} +2.23607 q^{8} +O(q^{10})\) \(q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} -4.23607 q^{7} +2.23607 q^{8} +1.61803 q^{10} -0.236068 q^{11} +1.00000 q^{13} +6.85410 q^{14} -4.85410 q^{16} +7.47214 q^{17} +2.47214 q^{19} -0.618034 q^{20} +0.381966 q^{22} +4.47214 q^{23} +1.00000 q^{25} -1.61803 q^{26} -2.61803 q^{28} -1.00000 q^{29} -8.00000 q^{31} +3.38197 q^{32} -12.0902 q^{34} +4.23607 q^{35} -4.00000 q^{38} -2.23607 q^{40} -6.00000 q^{41} -6.00000 q^{43} -0.145898 q^{44} -7.23607 q^{46} +3.76393 q^{47} +10.9443 q^{49} -1.61803 q^{50} +0.618034 q^{52} +2.47214 q^{53} +0.236068 q^{55} -9.47214 q^{56} +1.61803 q^{58} -6.00000 q^{59} -8.47214 q^{61} +12.9443 q^{62} +4.23607 q^{64} -1.00000 q^{65} -1.29180 q^{67} +4.61803 q^{68} -6.85410 q^{70} -6.47214 q^{71} -6.00000 q^{73} +1.52786 q^{76} +1.00000 q^{77} -6.00000 q^{79} +4.85410 q^{80} +9.70820 q^{82} -2.47214 q^{83} -7.47214 q^{85} +9.70820 q^{86} -0.527864 q^{88} -9.94427 q^{89} -4.23607 q^{91} +2.76393 q^{92} -6.09017 q^{94} -2.47214 q^{95} +15.4164 q^{97} -17.7082 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - 2 q^{5} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} - q^{4} - 2 q^{5} - 4 q^{7} + q^{10} + 4 q^{11} + 2 q^{13} + 7 q^{14} - 3 q^{16} + 6 q^{17} - 4 q^{19} + q^{20} + 3 q^{22} + 2 q^{25} - q^{26} - 3 q^{28} - 2 q^{29} - 16 q^{31} + 9 q^{32} - 13 q^{34} + 4 q^{35} - 8 q^{38} - 12 q^{41} - 12 q^{43} - 7 q^{44} - 10 q^{46} + 12 q^{47} + 4 q^{49} - q^{50} - q^{52} - 4 q^{53} - 4 q^{55} - 10 q^{56} + q^{58} - 12 q^{59} - 8 q^{61} + 8 q^{62} + 4 q^{64} - 2 q^{65} - 16 q^{67} + 7 q^{68} - 7 q^{70} - 4 q^{71} - 12 q^{73} + 12 q^{76} + 2 q^{77} - 12 q^{79} + 3 q^{80} + 6 q^{82} + 4 q^{83} - 6 q^{85} + 6 q^{86} - 10 q^{88} - 2 q^{89} - 4 q^{91} + 10 q^{92} - q^{94} + 4 q^{95} + 4 q^{97} - 22 q^{98}+O(q^{100}) \) 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 \(-0.693856\pi\)
−0.572061 + 0.820211i \(0.693856\pi\)
\(3\) 0 0
\(4\) 0.618034 0.309017
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.23607 −1.60108 −0.800542 0.599277i \(-0.795455\pi\)
−0.800542 + 0.599277i \(0.795455\pi\)
\(8\) 2.23607 0.790569
\(9\) 0 0
\(10\) 1.61803 0.511667
\(11\) −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 \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 6.85410 1.83184
\(15\) 0 0
\(16\) −4.85410 −1.21353
\(17\) 7.47214 1.81226 0.906130 0.423000i \(-0.139023\pi\)
0.906130 + 0.423000i \(0.139023\pi\)
\(18\) 0 0
\(19\) 2.47214 0.567147 0.283573 0.958951i \(-0.408480\pi\)
0.283573 + 0.958951i \(0.408480\pi\)
\(20\) −0.618034 −0.138197
\(21\) 0 0
\(22\) 0.381966 0.0814354
\(23\) 4.47214 0.932505 0.466252 0.884652i \(-0.345604\pi\)
0.466252 + 0.884652i \(0.345604\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −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
\(35\) 4.23607 0.716026
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) −4.00000 −0.648886
\(39\) 0 0
\(40\) −2.23607 −0.353553
\(41\) −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 \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) −0.145898 −0.0219950
\(45\) 0 0
\(46\) −7.23607 −1.06690
\(47\) 3.76393 0.549026 0.274513 0.961583i \(-0.411483\pi\)
0.274513 + 0.961583i \(0.411483\pi\)
\(48\) 0 0
\(49\) 10.9443 1.56347
\(50\) −1.61803 −0.228825
\(51\) 0 0
\(52\) 0.618034 0.0857059
\(53\) 2.47214 0.339574 0.169787 0.985481i \(-0.445692\pi\)
0.169787 + 0.985481i \(0.445692\pi\)
\(54\) 0 0
\(55\) 0.236068 0.0318314
\(56\) −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
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −1.29180 −0.157818 −0.0789090 0.996882i \(-0.525144\pi\)
−0.0789090 + 0.996882i \(0.525144\pi\)
\(68\) 4.61803 0.560019
\(69\) 0 0
\(70\) −6.85410 −0.819222
\(71\) −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 \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 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 \(0.609590\pi\)
\(80\) 4.85410 0.542705
\(81\) 0 0
\(82\) 9.70820 1.07209
\(83\) −2.47214 −0.271352 −0.135676 0.990753i \(-0.543321\pi\)
−0.135676 + 0.990753i \(0.543321\pi\)
\(84\) 0 0
\(85\) −7.47214 −0.810467
\(86\) 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
\(95\) −2.47214 −0.253636
\(96\) 0 0
\(97\) 15.4164 1.56530 0.782650 0.622463i \(-0.213868\pi\)
0.782650 + 0.622463i \(0.213868\pi\)
\(98\) −17.7082 −1.78880
\(99\) 0 0
\(100\) 0.618034 0.0618034
\(101\) 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 \(-0.514813\pi\)
−0.0465209 + 0.998917i \(0.514813\pi\)
\(104\) 2.23607 0.219265
\(105\) 0 0
\(106\) −4.00000 −0.388514
\(107\) −4.47214 −0.432338 −0.216169 0.976356i \(-0.569356\pi\)
−0.216169 + 0.976356i \(0.569356\pi\)
\(108\) 0 0
\(109\) 18.4164 1.76397 0.881986 0.471276i \(-0.156206\pi\)
0.881986 + 0.471276i \(0.156206\pi\)
\(110\) −0.381966 −0.0364190
\(111\) 0 0
\(112\) 20.5623 1.94296
\(113\) −1.47214 −0.138487 −0.0692435 0.997600i \(-0.522059\pi\)
−0.0692435 + 0.997600i \(0.522059\pi\)
\(114\) 0 0
\(115\) −4.47214 −0.417029
\(116\) −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
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −13.6180 −1.20368
\(129\) 0 0
\(130\) 1.61803 0.141911
\(131\) −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 \(-0.595868\pi\)
−0.296645 + 0.954988i \(0.595868\pi\)
\(138\) 0 0
\(139\) 7.76393 0.658528 0.329264 0.944238i \(-0.393199\pi\)
0.329264 + 0.944238i \(0.393199\pi\)
\(140\) 2.61803 0.221264
\(141\) 0 0
\(142\) 10.4721 0.878802
\(143\) −0.236068 −0.0197410
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 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
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −15.4164 −1.23036 −0.615182 0.788385i \(-0.710917\pi\)
−0.615182 + 0.788385i \(0.710917\pi\)
\(158\) 9.70820 0.772343
\(159\) 0 0
\(160\) −3.38197 −0.267368
\(161\) −18.9443 −1.49302
\(162\) 0 0
\(163\) −22.9443 −1.79713 −0.898567 0.438836i \(-0.855391\pi\)
−0.898567 + 0.438836i \(0.855391\pi\)
\(164\) −3.70820 −0.289562
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −13.4164 −1.03819 −0.519096 0.854716i \(-0.673731\pi\)
−0.519096 + 0.854716i \(0.673731\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 12.0902 0.927274
\(171\) 0 0
\(172\) −3.70820 −0.282748
\(173\) 11.8885 0.903869 0.451935 0.892051i \(-0.350734\pi\)
0.451935 + 0.892051i \(0.350734\pi\)
\(174\) 0 0
\(175\) −4.23607 −0.320217
\(176\) 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
\(190\) 4.00000 0.290191
\(191\) 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 \(-0.244226\pi\)
0.719816 + 0.694165i \(0.244226\pi\)
\(194\) −24.9443 −1.79089
\(195\) 0 0
\(196\) 6.76393 0.483138
\(197\) −24.9443 −1.77721 −0.888603 0.458677i \(-0.848323\pi\)
−0.888603 + 0.458677i \(0.848323\pi\)
\(198\) 0 0
\(199\) −20.1246 −1.42660 −0.713298 0.700861i \(-0.752799\pi\)
−0.713298 + 0.700861i \(0.752799\pi\)
\(200\) 2.23607 0.158114
\(201\) 0 0
\(202\) −10.5623 −0.743161
\(203\) 4.23607 0.297314
\(204\) 0 0
\(205\) 6.00000 0.419058
\(206\) 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
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 33.8885 2.30050
\(218\) −29.7984 −2.01820
\(219\) 0 0
\(220\) 0.145898 0.00983644
\(221\) 7.47214 0.502630
\(222\) 0 0
\(223\) 13.1803 0.882621 0.441310 0.897355i \(-0.354514\pi\)
0.441310 + 0.897355i \(0.354514\pi\)
\(224\) −14.3262 −0.957212
\(225\) 0 0
\(226\) 2.38197 0.158446
\(227\) 13.5279 0.897876 0.448938 0.893563i \(-0.351802\pi\)
0.448938 + 0.893563i \(0.351802\pi\)
\(228\) 0 0
\(229\) 4.00000 0.264327 0.132164 0.991228i \(-0.457808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 7.23607 0.477132
\(231\) 0 0
\(232\) −2.23607 −0.146805
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) −3.76393 −0.245532
\(236\) −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
\(245\) −10.9443 −0.699204
\(246\) 0 0
\(247\) 2.47214 0.157298
\(248\) −17.8885 −1.13592
\(249\) 0 0
\(250\) 1.61803 0.102333
\(251\) 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 \(-0.361355\pi\)
0.421922 + 0.906632i \(0.361355\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −0.618034 −0.0383288
\(261\) 0 0
\(262\) 6.85410 0.423448
\(263\) −24.9443 −1.53813 −0.769065 0.639171i \(-0.779278\pi\)
−0.769065 + 0.639171i \(0.779278\pi\)
\(264\) 0 0
\(265\) −2.47214 −0.151862
\(266\) 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
\(275\) −0.236068 −0.0142354
\(276\) 0 0
\(277\) 19.9443 1.19834 0.599168 0.800624i \(-0.295498\pi\)
0.599168 + 0.800624i \(0.295498\pi\)
\(278\) −12.5623 −0.753437
\(279\) 0 0
\(280\) 9.47214 0.566068
\(281\) −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 \(-0.453053\pi\)
0.146953 + 0.989143i \(0.453053\pi\)
\(284\) −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
\(290\) −1.61803 −0.0950142
\(291\) 0 0
\(292\) −3.70820 −0.217006
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 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
\(305\) 8.47214 0.485113
\(306\) 0 0
\(307\) −26.8328 −1.53143 −0.765715 0.643180i \(-0.777615\pi\)
−0.765715 + 0.643180i \(0.777615\pi\)
\(308\) 0.618034 0.0352158
\(309\) 0 0
\(310\) −12.9443 −0.735185
\(311\) 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 \(-0.203185\pi\)
0.803095 + 0.595851i \(0.203185\pi\)
\(314\) 24.9443 1.40769
\(315\) 0 0
\(316\) −3.70820 −0.208603
\(317\) 20.8885 1.17322 0.586609 0.809870i \(-0.300463\pi\)
0.586609 + 0.809870i \(0.300463\pi\)
\(318\) 0 0
\(319\) 0.236068 0.0132173
\(320\) −4.23607 −0.236803
\(321\) 0 0
\(322\) 30.6525 1.70820
\(323\) 18.4721 1.02782
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 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
\(335\) 1.29180 0.0705784
\(336\) 0 0
\(337\) 22.9443 1.24985 0.624927 0.780683i \(-0.285129\pi\)
0.624927 + 0.780683i \(0.285129\pi\)
\(338\) 19.4164 1.05611
\(339\) 0 0
\(340\) −4.61803 −0.250448
\(341\) 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 \(-0.203636\pi\)
0.802251 + 0.596987i \(0.203636\pi\)
\(348\) 0 0
\(349\) −22.9443 −1.22818 −0.614089 0.789237i \(-0.710477\pi\)
−0.614089 + 0.789237i \(0.710477\pi\)
\(350\) 6.85410 0.366367
\(351\) 0 0
\(352\) −0.798374 −0.0425535
\(353\) −1.88854 −0.100517 −0.0502585 0.998736i \(-0.516005\pi\)
−0.0502585 + 0.998736i \(0.516005\pi\)
\(354\) 0 0
\(355\) 6.47214 0.343505
\(356\) −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
\(365\) 6.00000 0.314054
\(366\) 0 0
\(367\) 1.41641 0.0739359 0.0369679 0.999316i \(-0.488230\pi\)
0.0369679 + 0.999316i \(0.488230\pi\)
\(368\) −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 \(-0.768245\pi\)
−0.746453 + 0.665438i \(0.768245\pi\)
\(374\) 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 \(0.521137\pi\)
\(380\) −1.52786 −0.0783778
\(381\) 0 0
\(382\) −14.4721 −0.740459
\(383\) −18.0000 −0.919757 −0.459879 0.887982i \(-0.652107\pi\)
−0.459879 + 0.887982i \(0.652107\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) −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
\(395\) 6.00000 0.301893
\(396\) 0 0
\(397\) −7.88854 −0.395915 −0.197957 0.980211i \(-0.563431\pi\)
−0.197957 + 0.980211i \(0.563431\pi\)
\(398\) 32.5623 1.63220
\(399\) 0 0
\(400\) −4.85410 −0.242705
\(401\) −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 \(0.262922\pi\)
\(410\) −9.70820 −0.479454
\(411\) 0 0
\(412\) −0.583592 −0.0287515
\(413\) 25.4164 1.25066
\(414\) 0 0
\(415\) 2.47214 0.121352
\(416\) 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
\(425\) 7.47214 0.362452
\(426\) 0 0
\(427\) 35.8885 1.73677
\(428\) −2.76393 −0.133600
\(429\) 0 0
\(430\) −9.70820 −0.468171
\(431\) 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 \(-0.893393\pi\)
−0.944438 + 0.328690i \(0.893393\pi\)
\(434\) −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 \(0.547546\pi\)
\(440\) 0.527864 0.0251649
\(441\) 0 0
\(442\) −12.0902 −0.575071
\(443\) 5.76393 0.273853 0.136926 0.990581i \(-0.456278\pi\)
0.136926 + 0.990581i \(0.456278\pi\)
\(444\) 0 0
\(445\) 9.94427 0.471404
\(446\) −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
\(455\) 4.23607 0.198590
\(456\) 0 0
\(457\) −23.3607 −1.09277 −0.546383 0.837535i \(-0.683996\pi\)
−0.546383 + 0.837535i \(0.683996\pi\)
\(458\) −6.47214 −0.302423
\(459\) 0 0
\(460\) −2.76393 −0.128869
\(461\) −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 \(-0.557743\pi\)
−0.180410 + 0.983591i \(0.557743\pi\)
\(464\) 4.85410 0.225346
\(465\) 0 0
\(466\) −29.1246 −1.34917
\(467\) 24.0000 1.11059 0.555294 0.831654i \(-0.312606\pi\)
0.555294 + 0.831654i \(0.312606\pi\)
\(468\) 0 0
\(469\) 5.47214 0.252680
\(470\) 6.09017 0.280919
\(471\) 0 0
\(472\) −13.4164 −0.617540
\(473\) 1.41641 0.0651265
\(474\) 0 0
\(475\) 2.47214 0.113429
\(476\) −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
\(485\) −15.4164 −0.700023
\(486\) 0 0
\(487\) −7.05573 −0.319726 −0.159863 0.987139i \(-0.551105\pi\)
−0.159863 + 0.987139i \(0.551105\pi\)
\(488\) −18.9443 −0.857567
\(489\) 0 0
\(490\) 17.7082 0.799975
\(491\) 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 \(0.698807\pi\)
\(500\) −0.618034 −0.0276393
\(501\) 0 0
\(502\) −30.0902 −1.34299
\(503\) −14.1246 −0.629785 −0.314893 0.949127i \(-0.601969\pi\)
−0.314893 + 0.949127i \(0.601969\pi\)
\(504\) 0 0
\(505\) −6.52786 −0.290486
\(506\) 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
\(515\) 0.944272 0.0416096
\(516\) 0 0
\(517\) −0.888544 −0.0390781
\(518\) 0 0
\(519\) 0 0
\(520\) −2.23607 −0.0980581
\(521\) −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 \(-0.762032\pi\)
−0.733323 + 0.679880i \(0.762032\pi\)
\(524\) −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
\(530\) 4.00000 0.173749
\(531\) 0 0
\(532\) −6.47214 −0.280603
\(533\) −6.00000 −0.259889
\(534\) 0 0
\(535\) 4.47214 0.193347
\(536\) −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
\(545\) −18.4164 −0.788872
\(546\) 0 0
\(547\) −30.7082 −1.31299 −0.656494 0.754331i \(-0.727961\pi\)
−0.656494 + 0.754331i \(0.727961\pi\)
\(548\) −4.29180 −0.183336
\(549\) 0 0
\(550\) 0.381966 0.0162871
\(551\) −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 \(-0.268185\pi\)
0.665578 + 0.746328i \(0.268185\pi\)
\(558\) 0 0
\(559\) −6.00000 −0.253773
\(560\) −20.5623 −0.868916
\(561\) 0 0
\(562\) 36.3607 1.53378
\(563\) −19.1803 −0.808355 −0.404177 0.914681i \(-0.632442\pi\)
−0.404177 + 0.914681i \(0.632442\pi\)
\(564\) 0 0
\(565\) 1.47214 0.0619332
\(566\) −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
\(575\) 4.47214 0.186501
\(576\) 0 0
\(577\) −29.0557 −1.20961 −0.604803 0.796375i \(-0.706748\pi\)
−0.604803 + 0.796375i \(0.706748\pi\)
\(578\) −62.8328 −2.61350
\(579\) 0 0
\(580\) 0.618034 0.0256625
\(581\) 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 \(-0.454225\pi\)
0.143310 + 0.989678i \(0.454225\pi\)
\(588\) 0 0
\(589\) −19.7771 −0.814901
\(590\) −9.70820 −0.399680
\(591\) 0 0
\(592\) 0 0
\(593\) 12.4721 0.512169 0.256085 0.966654i \(-0.417567\pi\)
0.256085 + 0.966654i \(0.417567\pi\)
\(594\) 0 0
\(595\) 31.6525 1.29762
\(596\) −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
\(605\) 10.9443 0.444948
\(606\) 0 0
\(607\) −23.4164 −0.950443 −0.475221 0.879866i \(-0.657632\pi\)
−0.475221 + 0.879866i \(0.657632\pi\)
\(608\) 8.36068 0.339070
\(609\) 0 0
\(610\) −13.7082 −0.555029
\(611\) 3.76393 0.152272
\(612\) 0 0
\(613\) −23.0000 −0.928961 −0.464481 0.885583i \(-0.653759\pi\)
−0.464481 + 0.885583i \(0.653759\pi\)
\(614\) 43.4164 1.75214
\(615\) 0 0
\(616\) 2.23607 0.0900937
\(617\) −26.9443 −1.08474 −0.542368 0.840141i \(-0.682472\pi\)
−0.542368 + 0.840141i \(0.682472\pi\)
\(618\) 0 0
\(619\) 35.3050 1.41903 0.709513 0.704692i \(-0.248915\pi\)
0.709513 + 0.704692i \(0.248915\pi\)
\(620\) 4.94427 0.198567
\(621\) 0 0
\(622\) −32.5623 −1.30563
\(623\) 42.1246 1.68769
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −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
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) 10.9443 0.433628
\(638\) −0.381966 −0.0151222
\(639\) 0 0
\(640\) 13.6180 0.538300
\(641\) 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 \(-0.145034\pi\)
0.897981 + 0.440034i \(0.145034\pi\)
\(644\) −11.7082 −0.461368
\(645\) 0 0
\(646\) −29.8885 −1.17595
\(647\) −46.9443 −1.84557 −0.922785 0.385316i \(-0.874093\pi\)
−0.922785 + 0.385316i \(0.874093\pi\)
\(648\) 0 0
\(649\) 1.41641 0.0555989
\(650\) −1.61803 −0.0634645
\(651\) 0 0
\(652\) −14.1803 −0.555345
\(653\) 16.8885 0.660900 0.330450 0.943824i \(-0.392800\pi\)
0.330450 + 0.943824i \(0.392800\pi\)
\(654\) 0 0
\(655\) 4.23607 0.165517
\(656\) 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
\(665\) 10.4721 0.406092
\(666\) 0 0
\(667\) −4.47214 −0.173162
\(668\) −8.29180 −0.320819
\(669\) 0 0
\(670\) −2.09017 −0.0807503
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) 28.4164 1.09537 0.547686 0.836684i \(-0.315509\pi\)
0.547686 + 0.836684i \(0.315509\pi\)
\(674\) −37.1246 −1.42999
\(675\) 0 0
\(676\) −7.41641 −0.285246
\(677\) 17.9443 0.689654 0.344827 0.938666i \(-0.387938\pi\)
0.344827 + 0.938666i \(0.387938\pi\)
\(678\) 0 0
\(679\) −65.3050 −2.50617
\(680\) −16.7082 −0.640730
\(681\) 0 0
\(682\) −3.05573 −0.117010
\(683\) 19.0557 0.729147 0.364574 0.931175i \(-0.381215\pi\)
0.364574 + 0.931175i \(0.381215\pi\)
\(684\) 0 0
\(685\) 6.94427 0.265327
\(686\) 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
\(695\) −7.76393 −0.294503
\(696\) 0 0
\(697\) −44.8328 −1.69816
\(698\) 37.1246 1.40519
\(699\) 0 0
\(700\) −2.61803 −0.0989524
\(701\) −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 \(0.415324\pi\)
\(710\) −10.4721 −0.393012
\(711\) 0 0
\(712\) −22.2361 −0.833332
\(713\) −35.7771 −1.33986
\(714\) 0 0
\(715\) 0.236068 0.00882844
\(716\) 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
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) −48.2492 −1.78946 −0.894732 0.446603i \(-0.852634\pi\)
−0.894732 + 0.446603i \(0.852634\pi\)
\(728\) −9.47214 −0.351061
\(729\) 0 0
\(730\) −9.70820 −0.359317
\(731\) −44.8328 −1.65820
\(732\) 0 0
\(733\) −3.52786 −0.130305 −0.0651523 0.997875i \(-0.520753\pi\)
−0.0651523 + 0.997875i \(0.520753\pi\)
\(734\) −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 \(-0.239800\pi\)
0.729399 + 0.684089i \(0.239800\pi\)
\(744\) 0 0
\(745\) 5.52786 0.202525
\(746\) 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
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) −28.8328 −1.04795 −0.523973 0.851735i \(-0.675551\pi\)
−0.523973 + 0.851735i \(0.675551\pi\)
\(758\) 4.18034 0.151837
\(759\) 0 0
\(760\) −5.52786 −0.200517
\(761\) −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 \(0.601004\pi\)
\(770\) 1.61803 0.0583099
\(771\) 0 0
\(772\) 12.3607 0.444871
\(773\) −20.8328 −0.749304 −0.374652 0.927165i \(-0.622238\pi\)
−0.374652 + 0.927165i \(0.622238\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 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
\(785\) 15.4164 0.550235
\(786\) 0 0
\(787\) 27.7771 0.990146 0.495073 0.868851i \(-0.335141\pi\)
0.495073 + 0.868851i \(0.335141\pi\)
\(788\) −15.4164 −0.549187
\(789\) 0 0
\(790\) −9.70820 −0.345402
\(791\) 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 \(-0.773158\pi\)
−0.756634 + 0.653839i \(0.773158\pi\)
\(798\) 0 0
\(799\) 28.1246 0.994977
\(800\) 3.38197 0.119571
\(801\) 0 0
\(802\) 16.7639 0.591955
\(803\) 1.41641 0.0499839
\(804\) 0 0
\(805\) 18.9443 0.667698
\(806\) 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
\(815\) 22.9443 0.803703
\(816\) 0 0
\(817\) −14.8328 −0.518935
\(818\) −44.3607 −1.55103
\(819\) 0 0
\(820\) 3.70820 0.129496
\(821\) 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 \(-0.511098\pi\)
−0.0348578 + 0.999392i \(0.511098\pi\)
\(824\) −2.11146 −0.0735561
\(825\) 0 0
\(826\) −41.1246 −1.43091
\(827\) −6.11146 −0.212516 −0.106258 0.994339i \(-0.533887\pi\)
−0.106258 + 0.994339i \(0.533887\pi\)
\(828\) 0 0
\(829\) 15.8885 0.551832 0.275916 0.961182i \(-0.411019\pi\)
0.275916 + 0.961182i \(0.411019\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) 4.23607 0.146859
\(833\) 81.7771 2.83341
\(834\) 0 0
\(835\) 13.4164 0.464294
\(836\) −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
\(845\) 12.0000 0.412813
\(846\) 0 0
\(847\) 46.3607 1.59297
\(848\) −12.0000 −0.412082
\(849\) 0 0
\(850\) −12.0902 −0.414689
\(851\) 0 0
\(852\) 0 0
\(853\) 51.3050 1.75665 0.878324 0.478066i \(-0.158662\pi\)
0.878324 + 0.478066i \(0.158662\pi\)
\(854\) −58.0689 −1.98708
\(855\) 0 0
\(856\) −10.0000 −0.341793
\(857\) 48.7214 1.66429 0.832145 0.554558i \(-0.187113\pi\)
0.832145 + 0.554558i \(0.187113\pi\)
\(858\) 0 0
\(859\) 36.8328 1.25672 0.628360 0.777923i \(-0.283727\pi\)
0.628360 + 0.777923i \(0.283727\pi\)
\(860\) 3.70820 0.126449
\(861\) 0 0
\(862\) −20.1803 −0.687345
\(863\) −45.5279 −1.54979 −0.774893 0.632092i \(-0.782196\pi\)
−0.774893 + 0.632092i \(0.782196\pi\)
\(864\) 0 0
\(865\) −11.8885 −0.404223
\(866\) 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
\(875\) 4.23607 0.143205
\(876\) 0 0
\(877\) −19.8885 −0.671588 −0.335794 0.941935i \(-0.609005\pi\)
−0.335794 + 0.941935i \(0.609005\pi\)
\(878\) 10.0902 0.340527
\(879\) 0 0
\(880\) −1.14590 −0.0386282
\(881\) −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 \(-0.580293\pi\)
−0.249582 + 0.968354i \(0.580293\pi\)
\(884\) 4.61803 0.155321
\(885\) 0 0
\(886\) −9.32624 −0.313321
\(887\) 5.18034 0.173939 0.0869694 0.996211i \(-0.472282\pi\)
0.0869694 + 0.996211i \(0.472282\pi\)
\(888\) 0 0
\(889\) 50.8328 1.70488
\(890\) −16.0902 −0.539344
\(891\) 0 0
\(892\) 8.14590 0.272745
\(893\) 9.30495 0.311378
\(894\) 0 0
\(895\) −14.9443 −0.499532
\(896\) 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
\(905\) 22.4164 0.745147
\(906\) 0 0
\(907\) −24.8328 −0.824560 −0.412280 0.911057i \(-0.635267\pi\)
−0.412280 + 0.911057i \(0.635267\pi\)
\(908\) 8.36068 0.277459
\(909\) 0 0
\(910\) −6.85410 −0.227211
\(911\) −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 \(0.384383\pi\)
\(920\) −10.0000 −0.329690
\(921\) 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
\(935\) 1.76393 0.0576867
\(936\) 0 0
\(937\) 45.4721 1.48551 0.742755 0.669563i \(-0.233519\pi\)
0.742755 + 0.669563i \(0.233519\pi\)
\(938\) −8.85410 −0.289097
\(939\) 0 0
\(940\) −2.32624 −0.0758735
\(941\) 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 \(-0.884525\pi\)
−0.934916 + 0.354869i \(0.884525\pi\)
\(948\) 0 0
\(949\) −6.00000 −0.194768
\(950\) −4.00000 −0.129777
\(951\) 0 0
\(952\) −70.7771 −2.29390
\(953\) 1.63932 0.0531028 0.0265514 0.999647i \(-0.491547\pi\)
0.0265514 + 0.999647i \(0.491547\pi\)
\(954\) 0 0
\(955\) −8.94427 −0.289430
\(956\) −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
\(965\) −20.0000 −0.643823
\(966\) 0 0
\(967\) 11.4164 0.367127 0.183563 0.983008i \(-0.441237\pi\)
0.183563 + 0.983008i \(0.441237\pi\)
\(968\) −24.4721 −0.786564
\(969\) 0 0
\(970\) 24.9443 0.800912
\(971\) 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 \(-0.786277\pi\)
−0.782933 + 0.622106i \(0.786277\pi\)
\(978\) 0 0
\(979\) 2.34752 0.0750272
\(980\) −6.76393 −0.216066
\(981\) 0 0
\(982\) −24.0000 −0.765871
\(983\) 0.944272 0.0301176 0.0150588 0.999887i \(-0.495206\pi\)
0.0150588 + 0.999887i \(0.495206\pi\)
\(984\) 0 0
\(985\) 24.9443 0.794791
\(986\) 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
\(995\) 20.1246 0.637993
\(996\) 0 0
\(997\) 12.9443 0.409949 0.204975 0.978767i \(-0.434289\pi\)
0.204975 + 0.978767i \(0.434289\pi\)
\(998\) 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 1305.2.a.h.1.1 2
3.2 odd 2 1305.2.a.l.1.2 yes 2
5.4 even 2 6525.2.a.bb.1.2 2
15.14 odd 2 6525.2.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.a.h.1.1 2 1.1 even 1 trivial
1305.2.a.l.1.2 yes 2 3.2 odd 2
6525.2.a.r.1.1 2 15.14 odd 2
6525.2.a.bb.1.2 2 5.4 even 2