Properties

Label 9025.2.a.bl.1.4
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9025,2,Mod(1,9025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("9025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.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: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1805)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.90211\) of defining polynomial
Character \(\chi\) \(=\) 9025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.90211 q^{2} +1.90211 q^{3} +1.61803 q^{4} +3.61803 q^{6} +4.23607 q^{7} -0.726543 q^{8} +0.618034 q^{9} +O(q^{10})\) \(q+1.90211 q^{2} +1.90211 q^{3} +1.61803 q^{4} +3.61803 q^{6} +4.23607 q^{7} -0.726543 q^{8} +0.618034 q^{9} -5.85410 q^{11} +3.07768 q^{12} -3.07768 q^{13} +8.05748 q^{14} -4.61803 q^{16} -5.23607 q^{17} +1.17557 q^{18} +8.05748 q^{21} -11.1352 q^{22} -4.09017 q^{23} -1.38197 q^{24} -5.85410 q^{26} -4.53077 q^{27} +6.85410 q^{28} -2.80017 q^{29} +1.90211 q^{31} -7.33094 q^{32} -11.1352 q^{33} -9.95959 q^{34} +1.00000 q^{36} +2.80017 q^{37} -5.85410 q^{39} -6.88191 q^{41} +15.3262 q^{42} -0.381966 q^{43} -9.47214 q^{44} -7.77997 q^{46} +1.47214 q^{47} -8.78402 q^{48} +10.9443 q^{49} -9.95959 q^{51} -4.97980 q^{52} -11.1352 q^{53} -8.61803 q^{54} -3.07768 q^{56} -5.32624 q^{58} +14.0413 q^{59} +3.94427 q^{61} +3.61803 q^{62} +2.61803 q^{63} -4.70820 q^{64} -21.1803 q^{66} -5.98385 q^{67} -8.47214 q^{68} -7.77997 q^{69} +0.171513 q^{71} -0.449028 q^{72} +1.00000 q^{73} +5.32624 q^{74} -24.7984 q^{77} -11.1352 q^{78} -5.25731 q^{79} -10.4721 q^{81} -13.0902 q^{82} +8.76393 q^{83} +13.0373 q^{84} -0.726543 q^{86} -5.32624 q^{87} +4.25325 q^{88} +7.77997 q^{89} -13.0373 q^{91} -6.61803 q^{92} +3.61803 q^{93} +2.80017 q^{94} -13.9443 q^{96} +2.62866 q^{97} +20.8172 q^{98} -3.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 10 q^{6} + 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} - 12 q^{17} + 6 q^{23} - 10 q^{24} - 10 q^{26} + 14 q^{28} + 4 q^{36} - 10 q^{39} + 30 q^{42} - 6 q^{43} - 20 q^{44} - 12 q^{47} + 8 q^{49} - 30 q^{54} + 10 q^{58} - 20 q^{61} + 10 q^{62} + 6 q^{63} + 8 q^{64} - 40 q^{66} - 16 q^{68} + 4 q^{73} - 10 q^{74} - 50 q^{77} - 24 q^{81} - 30 q^{82} + 44 q^{83} + 10 q^{87} - 22 q^{92} + 10 q^{93} - 20 q^{96} - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.90211 1.34500 0.672499 0.740098i \(-0.265221\pi\)
0.672499 + 0.740098i \(0.265221\pi\)
\(3\) 1.90211 1.09819 0.549093 0.835761i \(-0.314973\pi\)
0.549093 + 0.835761i \(0.314973\pi\)
\(4\) 1.61803 0.809017
\(5\) 0 0
\(6\) 3.61803 1.47706
\(7\) 4.23607 1.60108 0.800542 0.599277i \(-0.204545\pi\)
0.800542 + 0.599277i \(0.204545\pi\)
\(8\) −0.726543 −0.256872
\(9\) 0.618034 0.206011
\(10\) 0 0
\(11\) −5.85410 −1.76508 −0.882539 0.470239i \(-0.844168\pi\)
−0.882539 + 0.470239i \(0.844168\pi\)
\(12\) 3.07768 0.888451
\(13\) −3.07768 −0.853596 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(14\) 8.05748 2.15345
\(15\) 0 0
\(16\) −4.61803 −1.15451
\(17\) −5.23607 −1.26993 −0.634967 0.772540i \(-0.718986\pi\)
−0.634967 + 0.772540i \(0.718986\pi\)
\(18\) 1.17557 0.277085
\(19\) 0 0
\(20\) 0 0
\(21\) 8.05748 1.75829
\(22\) −11.1352 −2.37402
\(23\) −4.09017 −0.852859 −0.426430 0.904521i \(-0.640229\pi\)
−0.426430 + 0.904521i \(0.640229\pi\)
\(24\) −1.38197 −0.282093
\(25\) 0 0
\(26\) −5.85410 −1.14808
\(27\) −4.53077 −0.871947
\(28\) 6.85410 1.29530
\(29\) −2.80017 −0.519978 −0.259989 0.965612i \(-0.583719\pi\)
−0.259989 + 0.965612i \(0.583719\pi\)
\(30\) 0 0
\(31\) 1.90211 0.341630 0.170815 0.985303i \(-0.445360\pi\)
0.170815 + 0.985303i \(0.445360\pi\)
\(32\) −7.33094 −1.29594
\(33\) −11.1352 −1.93838
\(34\) −9.95959 −1.70806
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.80017 0.460345 0.230172 0.973150i \(-0.426071\pi\)
0.230172 + 0.973150i \(0.426071\pi\)
\(38\) 0 0
\(39\) −5.85410 −0.937407
\(40\) 0 0
\(41\) −6.88191 −1.07477 −0.537387 0.843336i \(-0.680588\pi\)
−0.537387 + 0.843336i \(0.680588\pi\)
\(42\) 15.3262 2.36489
\(43\) −0.381966 −0.0582493 −0.0291246 0.999576i \(-0.509272\pi\)
−0.0291246 + 0.999576i \(0.509272\pi\)
\(44\) −9.47214 −1.42798
\(45\) 0 0
\(46\) −7.77997 −1.14709
\(47\) 1.47214 0.214733 0.107367 0.994220i \(-0.465758\pi\)
0.107367 + 0.994220i \(0.465758\pi\)
\(48\) −8.78402 −1.26786
\(49\) 10.9443 1.56347
\(50\) 0 0
\(51\) −9.95959 −1.39462
\(52\) −4.97980 −0.690574
\(53\) −11.1352 −1.52953 −0.764766 0.644308i \(-0.777146\pi\)
−0.764766 + 0.644308i \(0.777146\pi\)
\(54\) −8.61803 −1.17277
\(55\) 0 0
\(56\) −3.07768 −0.411273
\(57\) 0 0
\(58\) −5.32624 −0.699369
\(59\) 14.0413 1.82803 0.914013 0.405685i \(-0.132967\pi\)
0.914013 + 0.405685i \(0.132967\pi\)
\(60\) 0 0
\(61\) 3.94427 0.505012 0.252506 0.967595i \(-0.418745\pi\)
0.252506 + 0.967595i \(0.418745\pi\)
\(62\) 3.61803 0.459491
\(63\) 2.61803 0.329841
\(64\) −4.70820 −0.588525
\(65\) 0 0
\(66\) −21.1803 −2.60712
\(67\) −5.98385 −0.731044 −0.365522 0.930803i \(-0.619110\pi\)
−0.365522 + 0.930803i \(0.619110\pi\)
\(68\) −8.47214 −1.02740
\(69\) −7.77997 −0.936598
\(70\) 0 0
\(71\) 0.171513 0.0203549 0.0101774 0.999948i \(-0.496760\pi\)
0.0101774 + 0.999948i \(0.496760\pi\)
\(72\) −0.449028 −0.0529185
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 5.32624 0.619163
\(75\) 0 0
\(76\) 0 0
\(77\) −24.7984 −2.82604
\(78\) −11.1352 −1.26081
\(79\) −5.25731 −0.591494 −0.295747 0.955266i \(-0.595568\pi\)
−0.295747 + 0.955266i \(0.595568\pi\)
\(80\) 0 0
\(81\) −10.4721 −1.16357
\(82\) −13.0902 −1.44557
\(83\) 8.76393 0.961967 0.480983 0.876730i \(-0.340280\pi\)
0.480983 + 0.876730i \(0.340280\pi\)
\(84\) 13.0373 1.42248
\(85\) 0 0
\(86\) −0.726543 −0.0783451
\(87\) −5.32624 −0.571033
\(88\) 4.25325 0.453398
\(89\) 7.77997 0.824675 0.412337 0.911031i \(-0.364713\pi\)
0.412337 + 0.911031i \(0.364713\pi\)
\(90\) 0 0
\(91\) −13.0373 −1.36668
\(92\) −6.61803 −0.689978
\(93\) 3.61803 0.375173
\(94\) 2.80017 0.288815
\(95\) 0 0
\(96\) −13.9443 −1.42318
\(97\) 2.62866 0.266900 0.133450 0.991056i \(-0.457395\pi\)
0.133450 + 0.991056i \(0.457395\pi\)
\(98\) 20.8172 2.10286
\(99\) −3.61803 −0.363626
\(100\) 0 0
\(101\) 3.29180 0.327546 0.163773 0.986498i \(-0.447634\pi\)
0.163773 + 0.986498i \(0.447634\pi\)
\(102\) −18.9443 −1.87576
\(103\) 14.9394 1.47202 0.736011 0.676970i \(-0.236707\pi\)
0.736011 + 0.676970i \(0.236707\pi\)
\(104\) 2.23607 0.219265
\(105\) 0 0
\(106\) −21.1803 −2.05722
\(107\) 8.33499 0.805774 0.402887 0.915250i \(-0.368007\pi\)
0.402887 + 0.915250i \(0.368007\pi\)
\(108\) −7.33094 −0.705420
\(109\) −2.17963 −0.208770 −0.104385 0.994537i \(-0.533287\pi\)
−0.104385 + 0.994537i \(0.533287\pi\)
\(110\) 0 0
\(111\) 5.32624 0.505544
\(112\) −19.5623 −1.84846
\(113\) −18.4661 −1.73714 −0.868572 0.495562i \(-0.834962\pi\)
−0.868572 + 0.495562i \(0.834962\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.53077 −0.420671
\(117\) −1.90211 −0.175850
\(118\) 26.7082 2.45869
\(119\) −22.1803 −2.03327
\(120\) 0 0
\(121\) 23.2705 2.11550
\(122\) 7.50245 0.679240
\(123\) −13.0902 −1.18030
\(124\) 3.07768 0.276384
\(125\) 0 0
\(126\) 4.97980 0.443636
\(127\) −9.23305 −0.819301 −0.409650 0.912243i \(-0.634349\pi\)
−0.409650 + 0.912243i \(0.634349\pi\)
\(128\) 5.70634 0.504374
\(129\) −0.726543 −0.0639685
\(130\) 0 0
\(131\) −15.6180 −1.36455 −0.682277 0.731094i \(-0.739010\pi\)
−0.682277 + 0.731094i \(0.739010\pi\)
\(132\) −18.0171 −1.56818
\(133\) 0 0
\(134\) −11.3820 −0.983252
\(135\) 0 0
\(136\) 3.80423 0.326210
\(137\) −9.76393 −0.834189 −0.417095 0.908863i \(-0.636952\pi\)
−0.417095 + 0.908863i \(0.636952\pi\)
\(138\) −14.7984 −1.25972
\(139\) −22.0902 −1.87366 −0.936832 0.349780i \(-0.886256\pi\)
−0.936832 + 0.349780i \(0.886256\pi\)
\(140\) 0 0
\(141\) 2.80017 0.235817
\(142\) 0.326238 0.0273773
\(143\) 18.0171 1.50666
\(144\) −2.85410 −0.237842
\(145\) 0 0
\(146\) 1.90211 0.157420
\(147\) 20.8172 1.71698
\(148\) 4.53077 0.372427
\(149\) 7.09017 0.580849 0.290425 0.956898i \(-0.406203\pi\)
0.290425 + 0.956898i \(0.406203\pi\)
\(150\) 0 0
\(151\) 0.620541 0.0504989 0.0252495 0.999681i \(-0.491962\pi\)
0.0252495 + 0.999681i \(0.491962\pi\)
\(152\) 0 0
\(153\) −3.23607 −0.261621
\(154\) −47.1693 −3.80101
\(155\) 0 0
\(156\) −9.47214 −0.758378
\(157\) 16.2705 1.29853 0.649264 0.760563i \(-0.275077\pi\)
0.649264 + 0.760563i \(0.275077\pi\)
\(158\) −10.0000 −0.795557
\(159\) −21.1803 −1.67971
\(160\) 0 0
\(161\) −17.3262 −1.36550
\(162\) −19.9192 −1.56500
\(163\) 25.1803 1.97228 0.986138 0.165926i \(-0.0530613\pi\)
0.986138 + 0.165926i \(0.0530613\pi\)
\(164\) −11.1352 −0.869510
\(165\) 0 0
\(166\) 16.6700 1.29384
\(167\) −0.171513 −0.0132721 −0.00663605 0.999978i \(-0.502112\pi\)
−0.00663605 + 0.999978i \(0.502112\pi\)
\(168\) −5.85410 −0.451654
\(169\) −3.52786 −0.271374
\(170\) 0 0
\(171\) 0 0
\(172\) −0.618034 −0.0471246
\(173\) 13.2088 1.00425 0.502123 0.864796i \(-0.332553\pi\)
0.502123 + 0.864796i \(0.332553\pi\)
\(174\) −10.1311 −0.768037
\(175\) 0 0
\(176\) 27.0344 2.03780
\(177\) 26.7082 2.00751
\(178\) 14.7984 1.10919
\(179\) 10.1311 0.757234 0.378617 0.925553i \(-0.376400\pi\)
0.378617 + 0.925553i \(0.376400\pi\)
\(180\) 0 0
\(181\) −6.71040 −0.498780 −0.249390 0.968403i \(-0.580230\pi\)
−0.249390 + 0.968403i \(0.580230\pi\)
\(182\) −24.7984 −1.83818
\(183\) 7.50245 0.554597
\(184\) 2.97168 0.219075
\(185\) 0 0
\(186\) 6.88191 0.504606
\(187\) 30.6525 2.24153
\(188\) 2.38197 0.173723
\(189\) −19.1926 −1.39606
\(190\) 0 0
\(191\) 13.0000 0.940647 0.470323 0.882494i \(-0.344137\pi\)
0.470323 + 0.882494i \(0.344137\pi\)
\(192\) −8.95554 −0.646310
\(193\) −1.45309 −0.104595 −0.0522977 0.998632i \(-0.516654\pi\)
−0.0522977 + 0.998632i \(0.516654\pi\)
\(194\) 5.00000 0.358979
\(195\) 0 0
\(196\) 17.7082 1.26487
\(197\) −14.8885 −1.06076 −0.530382 0.847759i \(-0.677952\pi\)
−0.530382 + 0.847759i \(0.677952\pi\)
\(198\) −6.88191 −0.489076
\(199\) 13.4164 0.951064 0.475532 0.879698i \(-0.342256\pi\)
0.475532 + 0.879698i \(0.342256\pi\)
\(200\) 0 0
\(201\) −11.3820 −0.802822
\(202\) 6.26137 0.440548
\(203\) −11.8617 −0.832529
\(204\) −16.1150 −1.12827
\(205\) 0 0
\(206\) 28.4164 1.97986
\(207\) −2.52786 −0.175699
\(208\) 14.2128 0.985484
\(209\) 0 0
\(210\) 0 0
\(211\) −1.73060 −0.119139 −0.0595697 0.998224i \(-0.518973\pi\)
−0.0595697 + 0.998224i \(0.518973\pi\)
\(212\) −18.0171 −1.23742
\(213\) 0.326238 0.0223535
\(214\) 15.8541 1.08376
\(215\) 0 0
\(216\) 3.29180 0.223978
\(217\) 8.05748 0.546977
\(218\) −4.14590 −0.280796
\(219\) 1.90211 0.128533
\(220\) 0 0
\(221\) 16.1150 1.08401
\(222\) 10.1311 0.679955
\(223\) 14.8334 0.993317 0.496659 0.867946i \(-0.334560\pi\)
0.496659 + 0.867946i \(0.334560\pi\)
\(224\) −31.0543 −2.07491
\(225\) 0 0
\(226\) −35.1246 −2.33645
\(227\) −17.3965 −1.15465 −0.577324 0.816515i \(-0.695903\pi\)
−0.577324 + 0.816515i \(0.695903\pi\)
\(228\) 0 0
\(229\) −18.1459 −1.19911 −0.599557 0.800332i \(-0.704657\pi\)
−0.599557 + 0.800332i \(0.704657\pi\)
\(230\) 0 0
\(231\) −47.1693 −3.10351
\(232\) 2.03444 0.133568
\(233\) −1.76393 −0.115559 −0.0577795 0.998329i \(-0.518402\pi\)
−0.0577795 + 0.998329i \(0.518402\pi\)
\(234\) −3.61803 −0.236518
\(235\) 0 0
\(236\) 22.7194 1.47890
\(237\) −10.0000 −0.649570
\(238\) −42.1895 −2.73474
\(239\) −23.2148 −1.50164 −0.750820 0.660507i \(-0.770341\pi\)
−0.750820 + 0.660507i \(0.770341\pi\)
\(240\) 0 0
\(241\) −6.32688 −0.407550 −0.203775 0.979018i \(-0.565321\pi\)
−0.203775 + 0.979018i \(0.565321\pi\)
\(242\) 44.2631 2.84534
\(243\) −6.32688 −0.405870
\(244\) 6.38197 0.408564
\(245\) 0 0
\(246\) −24.8990 −1.58750
\(247\) 0 0
\(248\) −1.38197 −0.0877549
\(249\) 16.6700 1.05642
\(250\) 0 0
\(251\) −3.00000 −0.189358 −0.0946792 0.995508i \(-0.530183\pi\)
−0.0946792 + 0.995508i \(0.530183\pi\)
\(252\) 4.23607 0.266847
\(253\) 23.9443 1.50536
\(254\) −17.5623 −1.10196
\(255\) 0 0
\(256\) 20.2705 1.26691
\(257\) 3.24920 0.202679 0.101340 0.994852i \(-0.467687\pi\)
0.101340 + 0.994852i \(0.467687\pi\)
\(258\) −1.38197 −0.0860374
\(259\) 11.8617 0.737051
\(260\) 0 0
\(261\) −1.73060 −0.107121
\(262\) −29.7073 −1.83532
\(263\) 6.00000 0.369976 0.184988 0.982741i \(-0.440775\pi\)
0.184988 + 0.982741i \(0.440775\pi\)
\(264\) 8.09017 0.497916
\(265\) 0 0
\(266\) 0 0
\(267\) 14.7984 0.905646
\(268\) −9.68208 −0.591427
\(269\) 19.6417 1.19757 0.598787 0.800908i \(-0.295650\pi\)
0.598787 + 0.800908i \(0.295650\pi\)
\(270\) 0 0
\(271\) 8.61803 0.523508 0.261754 0.965135i \(-0.415699\pi\)
0.261754 + 0.965135i \(0.415699\pi\)
\(272\) 24.1803 1.46615
\(273\) −24.7984 −1.50087
\(274\) −18.5721 −1.12198
\(275\) 0 0
\(276\) −12.5882 −0.757724
\(277\) −24.8328 −1.49206 −0.746030 0.665913i \(-0.768042\pi\)
−0.746030 + 0.665913i \(0.768042\pi\)
\(278\) −42.0180 −2.52007
\(279\) 1.17557 0.0703796
\(280\) 0 0
\(281\) −25.0705 −1.49558 −0.747790 0.663935i \(-0.768885\pi\)
−0.747790 + 0.663935i \(0.768885\pi\)
\(282\) 5.32624 0.317173
\(283\) 28.2705 1.68051 0.840254 0.542193i \(-0.182406\pi\)
0.840254 + 0.542193i \(0.182406\pi\)
\(284\) 0.277515 0.0164675
\(285\) 0 0
\(286\) 34.2705 2.02646
\(287\) −29.1522 −1.72080
\(288\) −4.53077 −0.266978
\(289\) 10.4164 0.612730
\(290\) 0 0
\(291\) 5.00000 0.293105
\(292\) 1.61803 0.0946883
\(293\) −13.8293 −0.807918 −0.403959 0.914777i \(-0.632366\pi\)
−0.403959 + 0.914777i \(0.632366\pi\)
\(294\) 39.5967 2.30933
\(295\) 0 0
\(296\) −2.03444 −0.118250
\(297\) 26.5236 1.53905
\(298\) 13.4863 0.781241
\(299\) 12.5882 0.727997
\(300\) 0 0
\(301\) −1.61803 −0.0932619
\(302\) 1.18034 0.0679209
\(303\) 6.26137 0.359706
\(304\) 0 0
\(305\) 0 0
\(306\) −6.15537 −0.351879
\(307\) −18.3601 −1.04787 −0.523933 0.851759i \(-0.675536\pi\)
−0.523933 + 0.851759i \(0.675536\pi\)
\(308\) −40.1246 −2.28631
\(309\) 28.4164 1.61655
\(310\) 0 0
\(311\) −2.76393 −0.156728 −0.0783641 0.996925i \(-0.524970\pi\)
−0.0783641 + 0.996925i \(0.524970\pi\)
\(312\) 4.25325 0.240793
\(313\) −3.27051 −0.184860 −0.0924301 0.995719i \(-0.529463\pi\)
−0.0924301 + 0.995719i \(0.529463\pi\)
\(314\) 30.9483 1.74652
\(315\) 0 0
\(316\) −8.50651 −0.478528
\(317\) 16.4580 0.924373 0.462186 0.886783i \(-0.347065\pi\)
0.462186 + 0.886783i \(0.347065\pi\)
\(318\) −40.2874 −2.25921
\(319\) 16.3925 0.917802
\(320\) 0 0
\(321\) 15.8541 0.884890
\(322\) −32.9565 −1.83659
\(323\) 0 0
\(324\) −16.9443 −0.941348
\(325\) 0 0
\(326\) 47.8959 2.65271
\(327\) −4.14590 −0.229269
\(328\) 5.00000 0.276079
\(329\) 6.23607 0.343806
\(330\) 0 0
\(331\) −19.9192 −1.09486 −0.547429 0.836852i \(-0.684393\pi\)
−0.547429 + 0.836852i \(0.684393\pi\)
\(332\) 14.1803 0.778247
\(333\) 1.73060 0.0948363
\(334\) −0.326238 −0.0178509
\(335\) 0 0
\(336\) −37.2097 −2.02996
\(337\) −28.4257 −1.54845 −0.774223 0.632913i \(-0.781859\pi\)
−0.774223 + 0.632913i \(0.781859\pi\)
\(338\) −6.71040 −0.364997
\(339\) −35.1246 −1.90771
\(340\) 0 0
\(341\) −11.1352 −0.603003
\(342\) 0 0
\(343\) 16.7082 0.902158
\(344\) 0.277515 0.0149626
\(345\) 0 0
\(346\) 25.1246 1.35071
\(347\) −16.4721 −0.884271 −0.442135 0.896948i \(-0.645779\pi\)
−0.442135 + 0.896948i \(0.645779\pi\)
\(348\) −8.61803 −0.461975
\(349\) −2.56231 −0.137157 −0.0685785 0.997646i \(-0.521846\pi\)
−0.0685785 + 0.997646i \(0.521846\pi\)
\(350\) 0 0
\(351\) 13.9443 0.744290
\(352\) 42.9161 2.28743
\(353\) −3.67376 −0.195535 −0.0977673 0.995209i \(-0.531170\pi\)
−0.0977673 + 0.995209i \(0.531170\pi\)
\(354\) 50.8020 2.70010
\(355\) 0 0
\(356\) 12.5882 0.667176
\(357\) −42.1895 −2.23291
\(358\) 19.2705 1.01848
\(359\) 7.38197 0.389605 0.194803 0.980842i \(-0.437593\pi\)
0.194803 + 0.980842i \(0.437593\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) −12.7639 −0.670857
\(363\) 44.2631 2.32321
\(364\) −21.0948 −1.10567
\(365\) 0 0
\(366\) 14.2705 0.745931
\(367\) −31.0689 −1.62178 −0.810891 0.585197i \(-0.801017\pi\)
−0.810891 + 0.585197i \(0.801017\pi\)
\(368\) 18.8885 0.984633
\(369\) −4.25325 −0.221416
\(370\) 0 0
\(371\) −47.1693 −2.44891
\(372\) 5.85410 0.303521
\(373\) −5.42882 −0.281094 −0.140547 0.990074i \(-0.544886\pi\)
−0.140547 + 0.990074i \(0.544886\pi\)
\(374\) 58.3045 3.01485
\(375\) 0 0
\(376\) −1.06957 −0.0551588
\(377\) 8.61803 0.443851
\(378\) −36.5066 −1.87770
\(379\) −12.8658 −0.660870 −0.330435 0.943829i \(-0.607195\pi\)
−0.330435 + 0.943829i \(0.607195\pi\)
\(380\) 0 0
\(381\) −17.5623 −0.899744
\(382\) 24.7275 1.26517
\(383\) 4.25325 0.217331 0.108666 0.994078i \(-0.465342\pi\)
0.108666 + 0.994078i \(0.465342\pi\)
\(384\) 10.8541 0.553896
\(385\) 0 0
\(386\) −2.76393 −0.140680
\(387\) −0.236068 −0.0120000
\(388\) 4.25325 0.215926
\(389\) −20.8541 −1.05734 −0.528672 0.848826i \(-0.677310\pi\)
−0.528672 + 0.848826i \(0.677310\pi\)
\(390\) 0 0
\(391\) 21.4164 1.08307
\(392\) −7.95148 −0.401610
\(393\) −29.7073 −1.49853
\(394\) −28.3197 −1.42673
\(395\) 0 0
\(396\) −5.85410 −0.294180
\(397\) 23.8328 1.19613 0.598067 0.801446i \(-0.295936\pi\)
0.598067 + 0.801446i \(0.295936\pi\)
\(398\) 25.5195 1.27918
\(399\) 0 0
\(400\) 0 0
\(401\) 9.06154 0.452512 0.226256 0.974068i \(-0.427351\pi\)
0.226256 + 0.974068i \(0.427351\pi\)
\(402\) −21.6498 −1.07979
\(403\) −5.85410 −0.291614
\(404\) 5.32624 0.264990
\(405\) 0 0
\(406\) −22.5623 −1.11975
\(407\) −16.3925 −0.812545
\(408\) 7.23607 0.358239
\(409\) 21.7153 1.07375 0.536876 0.843661i \(-0.319604\pi\)
0.536876 + 0.843661i \(0.319604\pi\)
\(410\) 0 0
\(411\) −18.5721 −0.916094
\(412\) 24.1724 1.19089
\(413\) 59.4800 2.92682
\(414\) −4.80828 −0.236314
\(415\) 0 0
\(416\) 22.5623 1.10621
\(417\) −42.0180 −2.05763
\(418\) 0 0
\(419\) 26.1803 1.27899 0.639497 0.768794i \(-0.279143\pi\)
0.639497 + 0.768794i \(0.279143\pi\)
\(420\) 0 0
\(421\) −11.5842 −0.564579 −0.282289 0.959329i \(-0.591094\pi\)
−0.282289 + 0.959329i \(0.591094\pi\)
\(422\) −3.29180 −0.160242
\(423\) 0.909830 0.0442375
\(424\) 8.09017 0.392893
\(425\) 0 0
\(426\) 0.620541 0.0300653
\(427\) 16.7082 0.808567
\(428\) 13.4863 0.651885
\(429\) 34.2705 1.65460
\(430\) 0 0
\(431\) 6.88191 0.331490 0.165745 0.986169i \(-0.446997\pi\)
0.165745 + 0.986169i \(0.446997\pi\)
\(432\) 20.9232 1.00667
\(433\) 6.04937 0.290714 0.145357 0.989379i \(-0.453567\pi\)
0.145357 + 0.989379i \(0.453567\pi\)
\(434\) 15.3262 0.735683
\(435\) 0 0
\(436\) −3.52671 −0.168899
\(437\) 0 0
\(438\) 3.61803 0.172876
\(439\) 5.64083 0.269222 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(440\) 0 0
\(441\) 6.76393 0.322092
\(442\) 30.6525 1.45799
\(443\) −34.7771 −1.65231 −0.826155 0.563443i \(-0.809476\pi\)
−0.826155 + 0.563443i \(0.809476\pi\)
\(444\) 8.61803 0.408994
\(445\) 0 0
\(446\) 28.2148 1.33601
\(447\) 13.4863 0.637880
\(448\) −19.9443 −0.942278
\(449\) 5.64083 0.266207 0.133104 0.991102i \(-0.457506\pi\)
0.133104 + 0.991102i \(0.457506\pi\)
\(450\) 0 0
\(451\) 40.2874 1.89706
\(452\) −29.8788 −1.40538
\(453\) 1.18034 0.0554572
\(454\) −33.0902 −1.55300
\(455\) 0 0
\(456\) 0 0
\(457\) −5.23607 −0.244933 −0.122466 0.992473i \(-0.539080\pi\)
−0.122466 + 0.992473i \(0.539080\pi\)
\(458\) −34.5155 −1.61281
\(459\) 23.7234 1.10731
\(460\) 0 0
\(461\) −8.00000 −0.372597 −0.186299 0.982493i \(-0.559649\pi\)
−0.186299 + 0.982493i \(0.559649\pi\)
\(462\) −89.7214 −4.17422
\(463\) −0.145898 −0.00678046 −0.00339023 0.999994i \(-0.501079\pi\)
−0.00339023 + 0.999994i \(0.501079\pi\)
\(464\) 12.9313 0.600319
\(465\) 0 0
\(466\) −3.35520 −0.155427
\(467\) 9.18034 0.424815 0.212408 0.977181i \(-0.431870\pi\)
0.212408 + 0.977181i \(0.431870\pi\)
\(468\) −3.07768 −0.142266
\(469\) −25.3480 −1.17046
\(470\) 0 0
\(471\) 30.9483 1.42602
\(472\) −10.2016 −0.469568
\(473\) 2.23607 0.102815
\(474\) −19.0211 −0.873669
\(475\) 0 0
\(476\) −35.8885 −1.64495
\(477\) −6.88191 −0.315101
\(478\) −44.1571 −2.01970
\(479\) −11.4377 −0.522602 −0.261301 0.965257i \(-0.584151\pi\)
−0.261301 + 0.965257i \(0.584151\pi\)
\(480\) 0 0
\(481\) −8.61803 −0.392949
\(482\) −12.0344 −0.548154
\(483\) −32.9565 −1.49957
\(484\) 37.6525 1.71148
\(485\) 0 0
\(486\) −12.0344 −0.545893
\(487\) 21.0292 0.952926 0.476463 0.879195i \(-0.341919\pi\)
0.476463 + 0.879195i \(0.341919\pi\)
\(488\) −2.86568 −0.129723
\(489\) 47.8959 2.16593
\(490\) 0 0
\(491\) 37.2705 1.68199 0.840997 0.541039i \(-0.181969\pi\)
0.840997 + 0.541039i \(0.181969\pi\)
\(492\) −21.1803 −0.954883
\(493\) 14.6619 0.660338
\(494\) 0 0
\(495\) 0 0
\(496\) −8.78402 −0.394414
\(497\) 0.726543 0.0325899
\(498\) 31.7082 1.42088
\(499\) −29.5279 −1.32185 −0.660924 0.750453i \(-0.729836\pi\)
−0.660924 + 0.750453i \(0.729836\pi\)
\(500\) 0 0
\(501\) −0.326238 −0.0145752
\(502\) −5.70634 −0.254686
\(503\) −31.6525 −1.41131 −0.705657 0.708554i \(-0.749348\pi\)
−0.705657 + 0.708554i \(0.749348\pi\)
\(504\) −1.90211 −0.0847268
\(505\) 0 0
\(506\) 45.5447 2.02471
\(507\) −6.71040 −0.298019
\(508\) −14.9394 −0.662828
\(509\) 30.1563 1.33665 0.668327 0.743868i \(-0.267011\pi\)
0.668327 + 0.743868i \(0.267011\pi\)
\(510\) 0 0
\(511\) 4.23607 0.187393
\(512\) 27.1441 1.19961
\(513\) 0 0
\(514\) 6.18034 0.272603
\(515\) 0 0
\(516\) −1.17557 −0.0517516
\(517\) −8.61803 −0.379021
\(518\) 22.5623 0.991331
\(519\) 25.1246 1.10285
\(520\) 0 0
\(521\) −32.6789 −1.43169 −0.715845 0.698259i \(-0.753958\pi\)
−0.715845 + 0.698259i \(0.753958\pi\)
\(522\) −3.29180 −0.144078
\(523\) 6.53888 0.285925 0.142963 0.989728i \(-0.454337\pi\)
0.142963 + 0.989728i \(0.454337\pi\)
\(524\) −25.2705 −1.10395
\(525\) 0 0
\(526\) 11.4127 0.497616
\(527\) −9.95959 −0.433847
\(528\) 51.4226 2.23788
\(529\) −6.27051 −0.272631
\(530\) 0 0
\(531\) 8.67802 0.376594
\(532\) 0 0
\(533\) 21.1803 0.917422
\(534\) 28.1482 1.21809
\(535\) 0 0
\(536\) 4.34752 0.187784
\(537\) 19.2705 0.831584
\(538\) 37.3607 1.61073
\(539\) −64.0689 −2.75964
\(540\) 0 0
\(541\) −24.5967 −1.05750 −0.528748 0.848779i \(-0.677338\pi\)
−0.528748 + 0.848779i \(0.677338\pi\)
\(542\) 16.3925 0.704117
\(543\) −12.7639 −0.547753
\(544\) 38.3853 1.64576
\(545\) 0 0
\(546\) −47.1693 −2.01866
\(547\) 21.2663 0.909280 0.454640 0.890675i \(-0.349768\pi\)
0.454640 + 0.890675i \(0.349768\pi\)
\(548\) −15.7984 −0.674873
\(549\) 2.43769 0.104038
\(550\) 0 0
\(551\) 0 0
\(552\) 5.65248 0.240585
\(553\) −22.2703 −0.947031
\(554\) −47.2348 −2.00682
\(555\) 0 0
\(556\) −35.7426 −1.51583
\(557\) 31.0689 1.31643 0.658215 0.752830i \(-0.271312\pi\)
0.658215 + 0.752830i \(0.271312\pi\)
\(558\) 2.23607 0.0946603
\(559\) 1.17557 0.0497213
\(560\) 0 0
\(561\) 58.3045 2.46162
\(562\) −47.6869 −2.01155
\(563\) 4.35926 0.183721 0.0918604 0.995772i \(-0.470719\pi\)
0.0918604 + 0.995772i \(0.470719\pi\)
\(564\) 4.53077 0.190780
\(565\) 0 0
\(566\) 53.7737 2.26028
\(567\) −44.3607 −1.86297
\(568\) −0.124612 −0.00522859
\(569\) −25.7970 −1.08147 −0.540734 0.841194i \(-0.681853\pi\)
−0.540734 + 0.841194i \(0.681853\pi\)
\(570\) 0 0
\(571\) −25.8541 −1.08196 −0.540980 0.841035i \(-0.681947\pi\)
−0.540980 + 0.841035i \(0.681947\pi\)
\(572\) 29.1522 1.21892
\(573\) 24.7275 1.03300
\(574\) −55.4508 −2.31447
\(575\) 0 0
\(576\) −2.90983 −0.121243
\(577\) −8.52786 −0.355020 −0.177510 0.984119i \(-0.556804\pi\)
−0.177510 + 0.984119i \(0.556804\pi\)
\(578\) 19.8132 0.824120
\(579\) −2.76393 −0.114865
\(580\) 0 0
\(581\) 37.1246 1.54019
\(582\) 9.51057 0.394226
\(583\) 65.1864 2.69974
\(584\) −0.726543 −0.0300645
\(585\) 0 0
\(586\) −26.3050 −1.08665
\(587\) −36.4164 −1.50307 −0.751533 0.659695i \(-0.770685\pi\)
−0.751533 + 0.659695i \(0.770685\pi\)
\(588\) 33.6830 1.38906
\(589\) 0 0
\(590\) 0 0
\(591\) −28.3197 −1.16492
\(592\) −12.9313 −0.531472
\(593\) −20.7082 −0.850384 −0.425192 0.905103i \(-0.639793\pi\)
−0.425192 + 0.905103i \(0.639793\pi\)
\(594\) 50.4508 2.07002
\(595\) 0 0
\(596\) 11.4721 0.469917
\(597\) 25.5195 1.04444
\(598\) 23.9443 0.979154
\(599\) −30.0503 −1.22782 −0.613911 0.789375i \(-0.710405\pi\)
−0.613911 + 0.789375i \(0.710405\pi\)
\(600\) 0 0
\(601\) 10.5146 0.428900 0.214450 0.976735i \(-0.431204\pi\)
0.214450 + 0.976735i \(0.431204\pi\)
\(602\) −3.07768 −0.125437
\(603\) −3.69822 −0.150603
\(604\) 1.00406 0.0408545
\(605\) 0 0
\(606\) 11.9098 0.483804
\(607\) 16.2210 0.658389 0.329194 0.944262i \(-0.393223\pi\)
0.329194 + 0.944262i \(0.393223\pi\)
\(608\) 0 0
\(609\) −22.5623 −0.914271
\(610\) 0 0
\(611\) −4.53077 −0.183295
\(612\) −5.23607 −0.211656
\(613\) −26.5279 −1.07145 −0.535725 0.844392i \(-0.679962\pi\)
−0.535725 + 0.844392i \(0.679962\pi\)
\(614\) −34.9230 −1.40938
\(615\) 0 0
\(616\) 18.0171 0.725929
\(617\) −21.6180 −0.870309 −0.435155 0.900356i \(-0.643306\pi\)
−0.435155 + 0.900356i \(0.643306\pi\)
\(618\) 54.0512 2.17426
\(619\) 39.5410 1.58929 0.794644 0.607076i \(-0.207658\pi\)
0.794644 + 0.607076i \(0.207658\pi\)
\(620\) 0 0
\(621\) 18.5316 0.743648
\(622\) −5.25731 −0.210799
\(623\) 32.9565 1.32037
\(624\) 27.0344 1.08224
\(625\) 0 0
\(626\) −6.22088 −0.248636
\(627\) 0 0
\(628\) 26.3262 1.05053
\(629\) −14.6619 −0.584607
\(630\) 0 0
\(631\) −41.1803 −1.63936 −0.819682 0.572819i \(-0.805850\pi\)
−0.819682 + 0.572819i \(0.805850\pi\)
\(632\) 3.81966 0.151938
\(633\) −3.29180 −0.130837
\(634\) 31.3050 1.24328
\(635\) 0 0
\(636\) −34.2705 −1.35891
\(637\) −33.6830 −1.33457
\(638\) 31.1803 1.23444
\(639\) 0.106001 0.00419334
\(640\) 0 0
\(641\) −36.6952 −1.44937 −0.724686 0.689079i \(-0.758015\pi\)
−0.724686 + 0.689079i \(0.758015\pi\)
\(642\) 30.1563 1.19017
\(643\) 38.4721 1.51719 0.758596 0.651561i \(-0.225885\pi\)
0.758596 + 0.651561i \(0.225885\pi\)
\(644\) −28.0344 −1.10471
\(645\) 0 0
\(646\) 0 0
\(647\) −8.52786 −0.335265 −0.167632 0.985850i \(-0.553612\pi\)
−0.167632 + 0.985850i \(0.553612\pi\)
\(648\) 7.60845 0.298888
\(649\) −82.1994 −3.22661
\(650\) 0 0
\(651\) 15.3262 0.600683
\(652\) 40.7426 1.59561
\(653\) 13.7984 0.539972 0.269986 0.962864i \(-0.412981\pi\)
0.269986 + 0.962864i \(0.412981\pi\)
\(654\) −7.88597 −0.308366
\(655\) 0 0
\(656\) 31.7809 1.24084
\(657\) 0.618034 0.0241118
\(658\) 11.8617 0.462417
\(659\) 7.26543 0.283021 0.141510 0.989937i \(-0.454804\pi\)
0.141510 + 0.989937i \(0.454804\pi\)
\(660\) 0 0
\(661\) −29.3238 −1.14056 −0.570281 0.821450i \(-0.693166\pi\)
−0.570281 + 0.821450i \(0.693166\pi\)
\(662\) −37.8885 −1.47258
\(663\) 30.6525 1.19044
\(664\) −6.36737 −0.247102
\(665\) 0 0
\(666\) 3.29180 0.127555
\(667\) 11.4532 0.443468
\(668\) −0.277515 −0.0107374
\(669\) 28.2148 1.09085
\(670\) 0 0
\(671\) −23.0902 −0.891386
\(672\) −59.0689 −2.27863
\(673\) −24.2380 −0.934304 −0.467152 0.884177i \(-0.654720\pi\)
−0.467152 + 0.884177i \(0.654720\pi\)
\(674\) −54.0689 −2.08266
\(675\) 0 0
\(676\) −5.70820 −0.219546
\(677\) −33.5770 −1.29047 −0.645235 0.763985i \(-0.723240\pi\)
−0.645235 + 0.763985i \(0.723240\pi\)
\(678\) −66.8110 −2.56586
\(679\) 11.1352 0.427328
\(680\) 0 0
\(681\) −33.0902 −1.26802
\(682\) −21.1803 −0.811037
\(683\) −28.0422 −1.07300 −0.536502 0.843899i \(-0.680255\pi\)
−0.536502 + 0.843899i \(0.680255\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 31.7809 1.21340
\(687\) −34.5155 −1.31685
\(688\) 1.76393 0.0672493
\(689\) 34.2705 1.30560
\(690\) 0 0
\(691\) −12.2361 −0.465482 −0.232741 0.972539i \(-0.574769\pi\)
−0.232741 + 0.972539i \(0.574769\pi\)
\(692\) 21.3723 0.812452
\(693\) −15.3262 −0.582196
\(694\) −31.3319 −1.18934
\(695\) 0 0
\(696\) 3.86974 0.146682
\(697\) 36.0341 1.36489
\(698\) −4.87380 −0.184476
\(699\) −3.35520 −0.126905
\(700\) 0 0
\(701\) −31.9098 −1.20522 −0.602609 0.798037i \(-0.705872\pi\)
−0.602609 + 0.798037i \(0.705872\pi\)
\(702\) 26.5236 1.00107
\(703\) 0 0
\(704\) 27.5623 1.03879
\(705\) 0 0
\(706\) −6.98791 −0.262993
\(707\) 13.9443 0.524428
\(708\) 43.2148 1.62411
\(709\) 36.1803 1.35878 0.679391 0.733777i \(-0.262244\pi\)
0.679391 + 0.733777i \(0.262244\pi\)
\(710\) 0 0
\(711\) −3.24920 −0.121854
\(712\) −5.65248 −0.211835
\(713\) −7.77997 −0.291362
\(714\) −80.2492 −3.00325
\(715\) 0 0
\(716\) 16.3925 0.612616
\(717\) −44.1571 −1.64908
\(718\) 14.0413 0.524018
\(719\) −30.9098 −1.15274 −0.576371 0.817188i \(-0.695532\pi\)
−0.576371 + 0.817188i \(0.695532\pi\)
\(720\) 0 0
\(721\) 63.2843 2.35683
\(722\) 0 0
\(723\) −12.0344 −0.447566
\(724\) −10.8576 −0.403521
\(725\) 0 0
\(726\) 84.1935 3.12471
\(727\) −34.3607 −1.27437 −0.637184 0.770712i \(-0.719901\pi\)
−0.637184 + 0.770712i \(0.719901\pi\)
\(728\) 9.47214 0.351061
\(729\) 19.3820 0.717851
\(730\) 0 0
\(731\) 2.00000 0.0739727
\(732\) 12.1392 0.448679
\(733\) 11.0000 0.406294 0.203147 0.979148i \(-0.434883\pi\)
0.203147 + 0.979148i \(0.434883\pi\)
\(734\) −59.0965 −2.18129
\(735\) 0 0
\(736\) 29.9848 1.10525
\(737\) 35.0301 1.29035
\(738\) −8.09017 −0.297803
\(739\) −27.7639 −1.02131 −0.510656 0.859785i \(-0.670598\pi\)
−0.510656 + 0.859785i \(0.670598\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −89.7214 −3.29377
\(743\) 5.98385 0.219526 0.109763 0.993958i \(-0.464991\pi\)
0.109763 + 0.993958i \(0.464991\pi\)
\(744\) −2.62866 −0.0963712
\(745\) 0 0
\(746\) −10.3262 −0.378070
\(747\) 5.41641 0.198176
\(748\) 49.5967 1.81344
\(749\) 35.3076 1.29011
\(750\) 0 0
\(751\) 50.4590 1.84127 0.920637 0.390419i \(-0.127670\pi\)
0.920637 + 0.390419i \(0.127670\pi\)
\(752\) −6.79837 −0.247911
\(753\) −5.70634 −0.207951
\(754\) 16.3925 0.596979
\(755\) 0 0
\(756\) −31.0543 −1.12944
\(757\) −2.97871 −0.108263 −0.0541316 0.998534i \(-0.517239\pi\)
−0.0541316 + 0.998534i \(0.517239\pi\)
\(758\) −24.4721 −0.888868
\(759\) 45.5447 1.65317
\(760\) 0 0
\(761\) −18.8197 −0.682212 −0.341106 0.940025i \(-0.610802\pi\)
−0.341106 + 0.940025i \(0.610802\pi\)
\(762\) −33.4055 −1.21015
\(763\) −9.23305 −0.334259
\(764\) 21.0344 0.760999
\(765\) 0 0
\(766\) 8.09017 0.292310
\(767\) −43.2148 −1.56040
\(768\) 38.5568 1.39130
\(769\) −13.2705 −0.478547 −0.239273 0.970952i \(-0.576909\pi\)
−0.239273 + 0.970952i \(0.576909\pi\)
\(770\) 0 0
\(771\) 6.18034 0.222580
\(772\) −2.35114 −0.0846194
\(773\) 32.5074 1.16921 0.584606 0.811318i \(-0.301249\pi\)
0.584606 + 0.811318i \(0.301249\pi\)
\(774\) −0.449028 −0.0161400
\(775\) 0 0
\(776\) −1.90983 −0.0685589
\(777\) 22.5623 0.809418
\(778\) −39.6669 −1.42213
\(779\) 0 0
\(780\) 0 0
\(781\) −1.00406 −0.0359280
\(782\) 40.7364 1.45673
\(783\) 12.6869 0.453393
\(784\) −50.5410 −1.80504
\(785\) 0 0
\(786\) −56.5066 −2.01552
\(787\) −20.1312 −0.717599 −0.358800 0.933415i \(-0.616814\pi\)
−0.358800 + 0.933415i \(0.616814\pi\)
\(788\) −24.0902 −0.858177
\(789\) 11.4127 0.406302
\(790\) 0 0
\(791\) −78.2237 −2.78131
\(792\) 2.62866 0.0934052
\(793\) −12.1392 −0.431076
\(794\) 45.3327 1.60880
\(795\) 0 0
\(796\) 21.7082 0.769427
\(797\) 23.7234 0.840326 0.420163 0.907449i \(-0.361973\pi\)
0.420163 + 0.907449i \(0.361973\pi\)
\(798\) 0 0
\(799\) −7.70820 −0.272697
\(800\) 0 0
\(801\) 4.80828 0.169892
\(802\) 17.2361 0.608627
\(803\) −5.85410 −0.206587
\(804\) −18.4164 −0.649497
\(805\) 0 0
\(806\) −11.1352 −0.392219
\(807\) 37.3607 1.31516
\(808\) −2.39163 −0.0841372
\(809\) −53.2148 −1.87093 −0.935466 0.353417i \(-0.885020\pi\)
−0.935466 + 0.353417i \(0.885020\pi\)
\(810\) 0 0
\(811\) −13.1433 −0.461523 −0.230761 0.973010i \(-0.574122\pi\)
−0.230761 + 0.973010i \(0.574122\pi\)
\(812\) −19.1926 −0.673530
\(813\) 16.3925 0.574909
\(814\) −31.1803 −1.09287
\(815\) 0 0
\(816\) 45.9937 1.61010
\(817\) 0 0
\(818\) 41.3050 1.44419
\(819\) −8.05748 −0.281551
\(820\) 0 0
\(821\) −27.4508 −0.958041 −0.479021 0.877804i \(-0.659008\pi\)
−0.479021 + 0.877804i \(0.659008\pi\)
\(822\) −35.3262 −1.23214
\(823\) 16.7639 0.584354 0.292177 0.956364i \(-0.405620\pi\)
0.292177 + 0.956364i \(0.405620\pi\)
\(824\) −10.8541 −0.378121
\(825\) 0 0
\(826\) 113.138 3.93657
\(827\) 9.23305 0.321065 0.160532 0.987031i \(-0.448679\pi\)
0.160532 + 0.987031i \(0.448679\pi\)
\(828\) −4.09017 −0.142143
\(829\) 25.9686 0.901925 0.450963 0.892543i \(-0.351081\pi\)
0.450963 + 0.892543i \(0.351081\pi\)
\(830\) 0 0
\(831\) −47.2348 −1.63856
\(832\) 14.4904 0.502363
\(833\) −57.3050 −1.98550
\(834\) −79.9230 −2.76751
\(835\) 0 0
\(836\) 0 0
\(837\) −8.61803 −0.297883
\(838\) 49.7980 1.72024
\(839\) −4.08174 −0.140917 −0.0704587 0.997515i \(-0.522446\pi\)
−0.0704587 + 0.997515i \(0.522446\pi\)
\(840\) 0 0
\(841\) −21.1591 −0.729623
\(842\) −22.0344 −0.759357
\(843\) −47.6869 −1.64242
\(844\) −2.80017 −0.0963858
\(845\) 0 0
\(846\) 1.73060 0.0594992
\(847\) 98.5755 3.38709
\(848\) 51.4226 1.76586
\(849\) 53.7737 1.84551
\(850\) 0 0
\(851\) −11.4532 −0.392610
\(852\) 0.527864 0.0180843
\(853\) 15.4721 0.529756 0.264878 0.964282i \(-0.414668\pi\)
0.264878 + 0.964282i \(0.414668\pi\)
\(854\) 31.7809 1.08752
\(855\) 0 0
\(856\) −6.05573 −0.206981
\(857\) −6.22088 −0.212501 −0.106251 0.994339i \(-0.533885\pi\)
−0.106251 + 0.994339i \(0.533885\pi\)
\(858\) 65.1864 2.22543
\(859\) −22.7639 −0.776695 −0.388348 0.921513i \(-0.626954\pi\)
−0.388348 + 0.921513i \(0.626954\pi\)
\(860\) 0 0
\(861\) −55.4508 −1.88976
\(862\) 13.0902 0.445853
\(863\) −36.3772 −1.23829 −0.619147 0.785275i \(-0.712521\pi\)
−0.619147 + 0.785275i \(0.712521\pi\)
\(864\) 33.2148 1.12999
\(865\) 0 0
\(866\) 11.5066 0.391009
\(867\) 19.8132 0.672891
\(868\) 13.0373 0.442514
\(869\) 30.7768 1.04403
\(870\) 0 0
\(871\) 18.4164 0.624016
\(872\) 1.58359 0.0536272
\(873\) 1.62460 0.0549843
\(874\) 0 0
\(875\) 0 0
\(876\) 3.07768 0.103985
\(877\) 11.9677 0.404121 0.202060 0.979373i \(-0.435236\pi\)
0.202060 + 0.979373i \(0.435236\pi\)
\(878\) 10.7295 0.362103
\(879\) −26.3050 −0.887244
\(880\) 0 0
\(881\) 10.3262 0.347900 0.173950 0.984755i \(-0.444347\pi\)
0.173950 + 0.984755i \(0.444347\pi\)
\(882\) 12.8658 0.433213
\(883\) 5.78522 0.194688 0.0973440 0.995251i \(-0.468965\pi\)
0.0973440 + 0.995251i \(0.468965\pi\)
\(884\) 26.0746 0.876982
\(885\) 0 0
\(886\) −66.1500 −2.22235
\(887\) 14.3188 0.480780 0.240390 0.970676i \(-0.422725\pi\)
0.240390 + 0.970676i \(0.422725\pi\)
\(888\) −3.86974 −0.129860
\(889\) −39.1118 −1.31177
\(890\) 0 0
\(891\) 61.3050 2.05379
\(892\) 24.0009 0.803610
\(893\) 0 0
\(894\) 25.6525 0.857947
\(895\) 0 0
\(896\) 24.1724 0.807545
\(897\) 23.9443 0.799476
\(898\) 10.7295 0.358048
\(899\) −5.32624 −0.177640
\(900\) 0 0
\(901\) 58.3045 1.94240
\(902\) 76.6312 2.55154
\(903\) −3.07768 −0.102419
\(904\) 13.4164 0.446223
\(905\) 0 0
\(906\) 2.24514 0.0745898
\(907\) 51.9371 1.72454 0.862272 0.506446i \(-0.169041\pi\)
0.862272 + 0.506446i \(0.169041\pi\)
\(908\) −28.1482 −0.934130
\(909\) 2.03444 0.0674782
\(910\) 0 0
\(911\) 34.3035 1.13653 0.568264 0.822847i \(-0.307615\pi\)
0.568264 + 0.822847i \(0.307615\pi\)
\(912\) 0 0
\(913\) −51.3050 −1.69795
\(914\) −9.95959 −0.329434
\(915\) 0 0
\(916\) −29.3607 −0.970104
\(917\) −66.1591 −2.18476
\(918\) 45.1246 1.48933
\(919\) 7.36068 0.242806 0.121403 0.992603i \(-0.461261\pi\)
0.121403 + 0.992603i \(0.461261\pi\)
\(920\) 0 0
\(921\) −34.9230 −1.15075
\(922\) −15.2169 −0.501142
\(923\) −0.527864 −0.0173749
\(924\) −76.3215 −2.51079
\(925\) 0 0
\(926\) −0.277515 −0.00911969
\(927\) 9.23305 0.303253
\(928\) 20.5279 0.673860
\(929\) 37.5623 1.23238 0.616190 0.787598i \(-0.288675\pi\)
0.616190 + 0.787598i \(0.288675\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.85410 −0.0934892
\(933\) −5.25731 −0.172117
\(934\) 17.4620 0.571376
\(935\) 0 0
\(936\) 1.38197 0.0451710
\(937\) −23.2016 −0.757964 −0.378982 0.925404i \(-0.623726\pi\)
−0.378982 + 0.925404i \(0.623726\pi\)
\(938\) −48.2148 −1.57427
\(939\) −6.22088 −0.203011
\(940\) 0 0
\(941\) −50.7615 −1.65478 −0.827389 0.561629i \(-0.810175\pi\)
−0.827389 + 0.561629i \(0.810175\pi\)
\(942\) 58.8673 1.91800
\(943\) 28.1482 0.916631
\(944\) −64.8434 −2.11047
\(945\) 0 0
\(946\) 4.25325 0.138285
\(947\) −1.41641 −0.0460271 −0.0230135 0.999735i \(-0.507326\pi\)
−0.0230135 + 0.999735i \(0.507326\pi\)
\(948\) −16.1803 −0.525513
\(949\) −3.07768 −0.0999058
\(950\) 0 0
\(951\) 31.3050 1.01513
\(952\) 16.1150 0.522289
\(953\) −42.2300 −1.36796 −0.683982 0.729499i \(-0.739753\pi\)
−0.683982 + 0.729499i \(0.739753\pi\)
\(954\) −13.0902 −0.423810
\(955\) 0 0
\(956\) −37.5623 −1.21485
\(957\) 31.1803 1.00792
\(958\) −21.7558 −0.702898
\(959\) −41.3607 −1.33561
\(960\) 0 0
\(961\) −27.3820 −0.883289
\(962\) −16.3925 −0.528515
\(963\) 5.15131 0.165999
\(964\) −10.2371 −0.329715
\(965\) 0 0
\(966\) −62.6869 −2.01692
\(967\) 29.0557 0.934369 0.467185 0.884160i \(-0.345268\pi\)
0.467185 + 0.884160i \(0.345268\pi\)
\(968\) −16.9070 −0.543412
\(969\) 0 0
\(970\) 0 0
\(971\) −27.4216 −0.880002 −0.440001 0.897997i \(-0.645022\pi\)
−0.440001 + 0.897997i \(0.645022\pi\)
\(972\) −10.2371 −0.328355
\(973\) −93.5755 −2.99989
\(974\) 40.0000 1.28168
\(975\) 0 0
\(976\) −18.2148 −0.583041
\(977\) 36.7607 1.17608 0.588039 0.808832i \(-0.299900\pi\)
0.588039 + 0.808832i \(0.299900\pi\)
\(978\) 91.1033 2.91316
\(979\) −45.5447 −1.45562
\(980\) 0 0
\(981\) −1.34708 −0.0430091
\(982\) 70.8927 2.26228
\(983\) 19.6417 0.626472 0.313236 0.949675i \(-0.398587\pi\)
0.313236 + 0.949675i \(0.398587\pi\)
\(984\) 9.51057 0.303186
\(985\) 0 0
\(986\) 27.8885 0.888152
\(987\) 11.8617 0.377562
\(988\) 0 0
\(989\) 1.56231 0.0496784
\(990\) 0 0
\(991\) 57.6839 1.83239 0.916195 0.400732i \(-0.131244\pi\)
0.916195 + 0.400732i \(0.131244\pi\)
\(992\) −13.9443 −0.442731
\(993\) −37.8885 −1.20236
\(994\) 1.38197 0.0438333
\(995\) 0 0
\(996\) 26.9726 0.854660
\(997\) 36.9443 1.17004 0.585018 0.811020i \(-0.301087\pi\)
0.585018 + 0.811020i \(0.301087\pi\)
\(998\) −56.1653 −1.77788
\(999\) −12.6869 −0.401396
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 9025.2.a.bl.1.4 4
5.4 even 2 1805.2.a.l.1.1 4
19.18 odd 2 inner 9025.2.a.bl.1.1 4
95.94 odd 2 1805.2.a.l.1.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.l.1.1 4 5.4 even 2
1805.2.a.l.1.4 yes 4 95.94 odd 2
9025.2.a.bl.1.1 4 19.18 odd 2 inner
9025.2.a.bl.1.4 4 1.1 even 1 trivial