Properties

Label 9522.2.a.bs.1.1
Level $9522$
Weight $2$
Character 9522.1
Self dual yes
Analytic conductor $76.034$
Analytic rank $0$
Dimension $5$
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] = [9522,2,Mod(1,9522)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9522, 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("9522.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9522 = 2 \cdot 3^{2} \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9522.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: \(76.0335528047\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\Q(\zeta_{22})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 138)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.30972\) of defining polynomial
Character \(\chi\) \(=\) 9522.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.82306 q^{5} +2.89389 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.82306 q^{5} +2.89389 q^{7} -1.00000 q^{8} +3.82306 q^{10} +1.58065 q^{11} +4.70981 q^{13} -2.89389 q^{14} +1.00000 q^{16} +4.73780 q^{17} +2.14832 q^{19} -3.82306 q^{20} -1.58065 q^{22} +9.61578 q^{25} -4.70981 q^{26} +2.89389 q^{28} +0.588417 q^{29} +0.170752 q^{31} -1.00000 q^{32} -4.73780 q^{34} -11.0635 q^{35} +4.81881 q^{37} -2.14832 q^{38} +3.82306 q^{40} +10.3972 q^{41} +12.2681 q^{43} +1.58065 q^{44} -5.45520 q^{47} +1.37462 q^{49} -9.61578 q^{50} +4.70981 q^{52} +10.3078 q^{53} -6.04290 q^{55} -2.89389 q^{56} -0.588417 q^{58} +1.90272 q^{59} -12.6545 q^{61} -0.170752 q^{62} +1.00000 q^{64} -18.0059 q^{65} +5.87608 q^{67} +4.73780 q^{68} +11.0635 q^{70} +11.8216 q^{71} -13.2135 q^{73} -4.81881 q^{74} +2.14832 q^{76} +4.57422 q^{77} +11.6845 q^{79} -3.82306 q^{80} -10.3972 q^{82} +0.113248 q^{83} -18.1129 q^{85} -12.2681 q^{86} -1.58065 q^{88} +12.2386 q^{89} +13.6297 q^{91} +5.45520 q^{94} -8.21316 q^{95} +0.873537 q^{97} -1.37462 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} + 5 q^{4} + q^{5} + 11 q^{7} - 5 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{2} + 5 q^{4} + q^{5} + 11 q^{7} - 5 q^{8} - q^{10} - 11 q^{11} + 12 q^{13} - 11 q^{14} + 5 q^{16} + q^{17} + 15 q^{19} + q^{20} + 11 q^{22} + 6 q^{25} - 12 q^{26} + 11 q^{28} - q^{29} - 18 q^{31} - 5 q^{32} - q^{34} + 11 q^{35} + 10 q^{37} - 15 q^{38} - q^{40} + 16 q^{41} + 18 q^{43} - 11 q^{44} + 4 q^{47} + 20 q^{49} - 6 q^{50} + 12 q^{52} + q^{53} - 22 q^{55} - 11 q^{56} + q^{58} - 2 q^{59} - q^{61} + 18 q^{62} + 5 q^{64} - 24 q^{65} + 29 q^{67} + q^{68} - 11 q^{70} + 11 q^{71} - 8 q^{73} - 10 q^{74} + 15 q^{76} - 11 q^{77} + 40 q^{79} + q^{80} - 16 q^{82} - 8 q^{83} - 13 q^{85} - 18 q^{86} + 11 q^{88} - 2 q^{89} + 33 q^{91} - 4 q^{94} + 3 q^{95} + 17 q^{97} - 20 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.82306 −1.70972 −0.854862 0.518856i \(-0.826358\pi\)
−0.854862 + 0.518856i \(0.826358\pi\)
\(6\) 0 0
\(7\) 2.89389 1.09379 0.546895 0.837201i \(-0.315810\pi\)
0.546895 + 0.837201i \(0.315810\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.82306 1.20896
\(11\) 1.58065 0.476583 0.238291 0.971194i \(-0.423413\pi\)
0.238291 + 0.971194i \(0.423413\pi\)
\(12\) 0 0
\(13\) 4.70981 1.30627 0.653133 0.757243i \(-0.273454\pi\)
0.653133 + 0.757243i \(0.273454\pi\)
\(14\) −2.89389 −0.773426
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.73780 1.14909 0.574543 0.818475i \(-0.305180\pi\)
0.574543 + 0.818475i \(0.305180\pi\)
\(18\) 0 0
\(19\) 2.14832 0.492859 0.246430 0.969161i \(-0.420743\pi\)
0.246430 + 0.969161i \(0.420743\pi\)
\(20\) −3.82306 −0.854862
\(21\) 0 0
\(22\) −1.58065 −0.336995
\(23\) 0 0
\(24\) 0 0
\(25\) 9.61578 1.92316
\(26\) −4.70981 −0.923670
\(27\) 0 0
\(28\) 2.89389 0.546895
\(29\) 0.588417 0.109266 0.0546332 0.998506i \(-0.482601\pi\)
0.0546332 + 0.998506i \(0.482601\pi\)
\(30\) 0 0
\(31\) 0.170752 0.0306680 0.0153340 0.999882i \(-0.495119\pi\)
0.0153340 + 0.999882i \(0.495119\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −4.73780 −0.812526
\(35\) −11.0635 −1.87008
\(36\) 0 0
\(37\) 4.81881 0.792208 0.396104 0.918206i \(-0.370362\pi\)
0.396104 + 0.918206i \(0.370362\pi\)
\(38\) −2.14832 −0.348504
\(39\) 0 0
\(40\) 3.82306 0.604479
\(41\) 10.3972 1.62378 0.811889 0.583812i \(-0.198440\pi\)
0.811889 + 0.583812i \(0.198440\pi\)
\(42\) 0 0
\(43\) 12.2681 1.87087 0.935433 0.353505i \(-0.115010\pi\)
0.935433 + 0.353505i \(0.115010\pi\)
\(44\) 1.58065 0.238291
\(45\) 0 0
\(46\) 0 0
\(47\) −5.45520 −0.795723 −0.397862 0.917445i \(-0.630248\pi\)
−0.397862 + 0.917445i \(0.630248\pi\)
\(48\) 0 0
\(49\) 1.37462 0.196375
\(50\) −9.61578 −1.35988
\(51\) 0 0
\(52\) 4.70981 0.653133
\(53\) 10.3078 1.41588 0.707941 0.706272i \(-0.249624\pi\)
0.707941 + 0.706272i \(0.249624\pi\)
\(54\) 0 0
\(55\) −6.04290 −0.814825
\(56\) −2.89389 −0.386713
\(57\) 0 0
\(58\) −0.588417 −0.0772630
\(59\) 1.90272 0.247714 0.123857 0.992300i \(-0.460474\pi\)
0.123857 + 0.992300i \(0.460474\pi\)
\(60\) 0 0
\(61\) −12.6545 −1.62024 −0.810119 0.586265i \(-0.800598\pi\)
−0.810119 + 0.586265i \(0.800598\pi\)
\(62\) −0.170752 −0.0216856
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −18.0059 −2.23335
\(66\) 0 0
\(67\) 5.87608 0.717878 0.358939 0.933361i \(-0.383139\pi\)
0.358939 + 0.933361i \(0.383139\pi\)
\(68\) 4.73780 0.574543
\(69\) 0 0
\(70\) 11.0635 1.32234
\(71\) 11.8216 1.40297 0.701483 0.712686i \(-0.252522\pi\)
0.701483 + 0.712686i \(0.252522\pi\)
\(72\) 0 0
\(73\) −13.2135 −1.54653 −0.773264 0.634084i \(-0.781377\pi\)
−0.773264 + 0.634084i \(0.781377\pi\)
\(74\) −4.81881 −0.560176
\(75\) 0 0
\(76\) 2.14832 0.246430
\(77\) 4.57422 0.521281
\(78\) 0 0
\(79\) 11.6845 1.31461 0.657307 0.753623i \(-0.271696\pi\)
0.657307 + 0.753623i \(0.271696\pi\)
\(80\) −3.82306 −0.427431
\(81\) 0 0
\(82\) −10.3972 −1.14818
\(83\) 0.113248 0.0124306 0.00621528 0.999981i \(-0.498022\pi\)
0.00621528 + 0.999981i \(0.498022\pi\)
\(84\) 0 0
\(85\) −18.1129 −1.96462
\(86\) −12.2681 −1.32290
\(87\) 0 0
\(88\) −1.58065 −0.168497
\(89\) 12.2386 1.29729 0.648646 0.761090i \(-0.275336\pi\)
0.648646 + 0.761090i \(0.275336\pi\)
\(90\) 0 0
\(91\) 13.6297 1.42878
\(92\) 0 0
\(93\) 0 0
\(94\) 5.45520 0.562661
\(95\) −8.21316 −0.842653
\(96\) 0 0
\(97\) 0.873537 0.0886943 0.0443471 0.999016i \(-0.485879\pi\)
0.0443471 + 0.999016i \(0.485879\pi\)
\(98\) −1.37462 −0.138858
\(99\) 0 0
\(100\) 9.61578 0.961578
\(101\) 10.9103 1.08561 0.542807 0.839858i \(-0.317362\pi\)
0.542807 + 0.839858i \(0.317362\pi\)
\(102\) 0 0
\(103\) 3.69279 0.363862 0.181931 0.983311i \(-0.441765\pi\)
0.181931 + 0.983311i \(0.441765\pi\)
\(104\) −4.70981 −0.461835
\(105\) 0 0
\(106\) −10.3078 −1.00118
\(107\) −10.7665 −1.04083 −0.520416 0.853913i \(-0.674223\pi\)
−0.520416 + 0.853913i \(0.674223\pi\)
\(108\) 0 0
\(109\) −8.31357 −0.796296 −0.398148 0.917321i \(-0.630347\pi\)
−0.398148 + 0.917321i \(0.630347\pi\)
\(110\) 6.04290 0.576168
\(111\) 0 0
\(112\) 2.89389 0.273447
\(113\) −7.51278 −0.706743 −0.353372 0.935483i \(-0.614965\pi\)
−0.353372 + 0.935483i \(0.614965\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.588417 0.0546332
\(117\) 0 0
\(118\) −1.90272 −0.175160
\(119\) 13.7107 1.25686
\(120\) 0 0
\(121\) −8.50156 −0.772869
\(122\) 12.6545 1.14568
\(123\) 0 0
\(124\) 0.170752 0.0153340
\(125\) −17.6464 −1.57834
\(126\) 0 0
\(127\) −7.58758 −0.673289 −0.336645 0.941632i \(-0.609292\pi\)
−0.336645 + 0.941632i \(0.609292\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 18.0059 1.57922
\(131\) −17.8085 −1.55594 −0.777970 0.628302i \(-0.783750\pi\)
−0.777970 + 0.628302i \(0.783750\pi\)
\(132\) 0 0
\(133\) 6.21702 0.539084
\(134\) −5.87608 −0.507616
\(135\) 0 0
\(136\) −4.73780 −0.406263
\(137\) 0.762968 0.0651847 0.0325924 0.999469i \(-0.489624\pi\)
0.0325924 + 0.999469i \(0.489624\pi\)
\(138\) 0 0
\(139\) 9.22612 0.782549 0.391275 0.920274i \(-0.372034\pi\)
0.391275 + 0.920274i \(0.372034\pi\)
\(140\) −11.0635 −0.935039
\(141\) 0 0
\(142\) −11.8216 −0.992047
\(143\) 7.44454 0.622544
\(144\) 0 0
\(145\) −2.24955 −0.186815
\(146\) 13.2135 1.09356
\(147\) 0 0
\(148\) 4.81881 0.396104
\(149\) −18.4911 −1.51485 −0.757424 0.652923i \(-0.773542\pi\)
−0.757424 + 0.652923i \(0.773542\pi\)
\(150\) 0 0
\(151\) −0.991887 −0.0807186 −0.0403593 0.999185i \(-0.512850\pi\)
−0.0403593 + 0.999185i \(0.512850\pi\)
\(152\) −2.14832 −0.174252
\(153\) 0 0
\(154\) −4.57422 −0.368601
\(155\) −0.652796 −0.0524339
\(156\) 0 0
\(157\) 8.10972 0.647226 0.323613 0.946189i \(-0.395102\pi\)
0.323613 + 0.946189i \(0.395102\pi\)
\(158\) −11.6845 −0.929572
\(159\) 0 0
\(160\) 3.82306 0.302239
\(161\) 0 0
\(162\) 0 0
\(163\) −23.0759 −1.80745 −0.903724 0.428116i \(-0.859177\pi\)
−0.903724 + 0.428116i \(0.859177\pi\)
\(164\) 10.3972 0.811889
\(165\) 0 0
\(166\) −0.113248 −0.00878973
\(167\) 5.28482 0.408952 0.204476 0.978872i \(-0.434451\pi\)
0.204476 + 0.978872i \(0.434451\pi\)
\(168\) 0 0
\(169\) 9.18232 0.706332
\(170\) 18.1129 1.38919
\(171\) 0 0
\(172\) 12.2681 0.935433
\(173\) −14.3906 −1.09410 −0.547049 0.837101i \(-0.684249\pi\)
−0.547049 + 0.837101i \(0.684249\pi\)
\(174\) 0 0
\(175\) 27.8270 2.10353
\(176\) 1.58065 0.119146
\(177\) 0 0
\(178\) −12.2386 −0.917324
\(179\) −11.2001 −0.837132 −0.418566 0.908186i \(-0.637467\pi\)
−0.418566 + 0.908186i \(0.637467\pi\)
\(180\) 0 0
\(181\) 8.79467 0.653703 0.326852 0.945076i \(-0.394012\pi\)
0.326852 + 0.945076i \(0.394012\pi\)
\(182\) −13.6297 −1.01030
\(183\) 0 0
\(184\) 0 0
\(185\) −18.4226 −1.35446
\(186\) 0 0
\(187\) 7.48878 0.547634
\(188\) −5.45520 −0.397862
\(189\) 0 0
\(190\) 8.21316 0.595846
\(191\) 6.33200 0.458168 0.229084 0.973407i \(-0.426427\pi\)
0.229084 + 0.973407i \(0.426427\pi\)
\(192\) 0 0
\(193\) −3.07385 −0.221261 −0.110630 0.993862i \(-0.535287\pi\)
−0.110630 + 0.993862i \(0.535287\pi\)
\(194\) −0.873537 −0.0627163
\(195\) 0 0
\(196\) 1.37462 0.0981874
\(197\) 0.963415 0.0686405 0.0343202 0.999411i \(-0.489073\pi\)
0.0343202 + 0.999411i \(0.489073\pi\)
\(198\) 0 0
\(199\) 3.27328 0.232037 0.116018 0.993247i \(-0.462987\pi\)
0.116018 + 0.993247i \(0.462987\pi\)
\(200\) −9.61578 −0.679938
\(201\) 0 0
\(202\) −10.9103 −0.767645
\(203\) 1.70282 0.119514
\(204\) 0 0
\(205\) −39.7493 −2.77621
\(206\) −3.69279 −0.257289
\(207\) 0 0
\(208\) 4.70981 0.326567
\(209\) 3.39574 0.234888
\(210\) 0 0
\(211\) −24.4174 −1.68097 −0.840483 0.541838i \(-0.817728\pi\)
−0.840483 + 0.541838i \(0.817728\pi\)
\(212\) 10.3078 0.707941
\(213\) 0 0
\(214\) 10.7665 0.735979
\(215\) −46.9016 −3.19866
\(216\) 0 0
\(217\) 0.494139 0.0335444
\(218\) 8.31357 0.563066
\(219\) 0 0
\(220\) −6.04290 −0.407412
\(221\) 22.3141 1.50101
\(222\) 0 0
\(223\) 15.1991 1.01781 0.508904 0.860823i \(-0.330051\pi\)
0.508904 + 0.860823i \(0.330051\pi\)
\(224\) −2.89389 −0.193356
\(225\) 0 0
\(226\) 7.51278 0.499743
\(227\) −12.9325 −0.858358 −0.429179 0.903220i \(-0.641197\pi\)
−0.429179 + 0.903220i \(0.641197\pi\)
\(228\) 0 0
\(229\) −0.948502 −0.0626788 −0.0313394 0.999509i \(-0.509977\pi\)
−0.0313394 + 0.999509i \(0.509977\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.588417 −0.0386315
\(233\) −13.9482 −0.913778 −0.456889 0.889524i \(-0.651036\pi\)
−0.456889 + 0.889524i \(0.651036\pi\)
\(234\) 0 0
\(235\) 20.8556 1.36047
\(236\) 1.90272 0.123857
\(237\) 0 0
\(238\) −13.7107 −0.888732
\(239\) 2.44329 0.158043 0.0790217 0.996873i \(-0.474820\pi\)
0.0790217 + 0.996873i \(0.474820\pi\)
\(240\) 0 0
\(241\) 10.8636 0.699784 0.349892 0.936790i \(-0.386218\pi\)
0.349892 + 0.936790i \(0.386218\pi\)
\(242\) 8.50156 0.546501
\(243\) 0 0
\(244\) −12.6545 −0.810119
\(245\) −5.25527 −0.335747
\(246\) 0 0
\(247\) 10.1182 0.643805
\(248\) −0.170752 −0.0108428
\(249\) 0 0
\(250\) 17.6464 1.11606
\(251\) 14.3211 0.903940 0.451970 0.892033i \(-0.350721\pi\)
0.451970 + 0.892033i \(0.350721\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 7.58758 0.476087
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 4.24430 0.264752 0.132376 0.991200i \(-0.457739\pi\)
0.132376 + 0.991200i \(0.457739\pi\)
\(258\) 0 0
\(259\) 13.9451 0.866509
\(260\) −18.0059 −1.11668
\(261\) 0 0
\(262\) 17.8085 1.10022
\(263\) 7.10375 0.438036 0.219018 0.975721i \(-0.429715\pi\)
0.219018 + 0.975721i \(0.429715\pi\)
\(264\) 0 0
\(265\) −39.4072 −2.42077
\(266\) −6.21702 −0.381190
\(267\) 0 0
\(268\) 5.87608 0.358939
\(269\) 7.73235 0.471450 0.235725 0.971820i \(-0.424254\pi\)
0.235725 + 0.971820i \(0.424254\pi\)
\(270\) 0 0
\(271\) −5.08058 −0.308623 −0.154312 0.988022i \(-0.549316\pi\)
−0.154312 + 0.988022i \(0.549316\pi\)
\(272\) 4.73780 0.287271
\(273\) 0 0
\(274\) −0.762968 −0.0460926
\(275\) 15.1991 0.916543
\(276\) 0 0
\(277\) −15.3909 −0.924751 −0.462376 0.886684i \(-0.653003\pi\)
−0.462376 + 0.886684i \(0.653003\pi\)
\(278\) −9.22612 −0.553346
\(279\) 0 0
\(280\) 11.0635 0.661172
\(281\) 3.29958 0.196836 0.0984181 0.995145i \(-0.468622\pi\)
0.0984181 + 0.995145i \(0.468622\pi\)
\(282\) 0 0
\(283\) −25.8826 −1.53856 −0.769280 0.638911i \(-0.779385\pi\)
−0.769280 + 0.638911i \(0.779385\pi\)
\(284\) 11.8216 0.701483
\(285\) 0 0
\(286\) −7.44454 −0.440205
\(287\) 30.0885 1.77607
\(288\) 0 0
\(289\) 5.44675 0.320397
\(290\) 2.24955 0.132098
\(291\) 0 0
\(292\) −13.2135 −0.773264
\(293\) −6.52568 −0.381234 −0.190617 0.981664i \(-0.561049\pi\)
−0.190617 + 0.981664i \(0.561049\pi\)
\(294\) 0 0
\(295\) −7.27423 −0.423522
\(296\) −4.81881 −0.280088
\(297\) 0 0
\(298\) 18.4911 1.07116
\(299\) 0 0
\(300\) 0 0
\(301\) 35.5025 2.04633
\(302\) 0.991887 0.0570767
\(303\) 0 0
\(304\) 2.14832 0.123215
\(305\) 48.3788 2.77016
\(306\) 0 0
\(307\) 4.61726 0.263521 0.131761 0.991282i \(-0.457937\pi\)
0.131761 + 0.991282i \(0.457937\pi\)
\(308\) 4.57422 0.260641
\(309\) 0 0
\(310\) 0.652796 0.0370763
\(311\) 10.7400 0.609011 0.304506 0.952511i \(-0.401509\pi\)
0.304506 + 0.952511i \(0.401509\pi\)
\(312\) 0 0
\(313\) 26.6899 1.50860 0.754301 0.656529i \(-0.227976\pi\)
0.754301 + 0.656529i \(0.227976\pi\)
\(314\) −8.10972 −0.457658
\(315\) 0 0
\(316\) 11.6845 0.657307
\(317\) −19.1814 −1.07733 −0.538667 0.842518i \(-0.681072\pi\)
−0.538667 + 0.842518i \(0.681072\pi\)
\(318\) 0 0
\(319\) 0.930080 0.0520745
\(320\) −3.82306 −0.213715
\(321\) 0 0
\(322\) 0 0
\(323\) 10.1783 0.566337
\(324\) 0 0
\(325\) 45.2885 2.51215
\(326\) 23.0759 1.27806
\(327\) 0 0
\(328\) −10.3972 −0.574092
\(329\) −15.7868 −0.870353
\(330\) 0 0
\(331\) 8.74066 0.480430 0.240215 0.970720i \(-0.422782\pi\)
0.240215 + 0.970720i \(0.422782\pi\)
\(332\) 0.113248 0.00621528
\(333\) 0 0
\(334\) −5.28482 −0.289173
\(335\) −22.4646 −1.22737
\(336\) 0 0
\(337\) −3.25394 −0.177253 −0.0886267 0.996065i \(-0.528248\pi\)
−0.0886267 + 0.996065i \(0.528248\pi\)
\(338\) −9.18232 −0.499452
\(339\) 0 0
\(340\) −18.1129 −0.982309
\(341\) 0.269899 0.0146159
\(342\) 0 0
\(343\) −16.2792 −0.878997
\(344\) −12.2681 −0.661451
\(345\) 0 0
\(346\) 14.3906 0.773644
\(347\) 5.21179 0.279783 0.139892 0.990167i \(-0.455325\pi\)
0.139892 + 0.990167i \(0.455325\pi\)
\(348\) 0 0
\(349\) 15.4731 0.828256 0.414128 0.910219i \(-0.364087\pi\)
0.414128 + 0.910219i \(0.364087\pi\)
\(350\) −27.8270 −1.48742
\(351\) 0 0
\(352\) −1.58065 −0.0842487
\(353\) 1.61663 0.0860448 0.0430224 0.999074i \(-0.486301\pi\)
0.0430224 + 0.999074i \(0.486301\pi\)
\(354\) 0 0
\(355\) −45.1947 −2.39868
\(356\) 12.2386 0.648646
\(357\) 0 0
\(358\) 11.2001 0.591942
\(359\) −13.5579 −0.715561 −0.357780 0.933806i \(-0.616466\pi\)
−0.357780 + 0.933806i \(0.616466\pi\)
\(360\) 0 0
\(361\) −14.3847 −0.757090
\(362\) −8.79467 −0.462238
\(363\) 0 0
\(364\) 13.6297 0.714390
\(365\) 50.5161 2.64414
\(366\) 0 0
\(367\) 16.2735 0.849467 0.424734 0.905318i \(-0.360368\pi\)
0.424734 + 0.905318i \(0.360368\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 18.4226 0.957746
\(371\) 29.8296 1.54868
\(372\) 0 0
\(373\) 6.97489 0.361146 0.180573 0.983562i \(-0.442205\pi\)
0.180573 + 0.983562i \(0.442205\pi\)
\(374\) −7.48878 −0.387236
\(375\) 0 0
\(376\) 5.45520 0.281331
\(377\) 2.77133 0.142731
\(378\) 0 0
\(379\) 16.5116 0.848143 0.424072 0.905629i \(-0.360600\pi\)
0.424072 + 0.905629i \(0.360600\pi\)
\(380\) −8.21316 −0.421326
\(381\) 0 0
\(382\) −6.33200 −0.323973
\(383\) 22.3879 1.14397 0.571984 0.820265i \(-0.306174\pi\)
0.571984 + 0.820265i \(0.306174\pi\)
\(384\) 0 0
\(385\) −17.4875 −0.891247
\(386\) 3.07385 0.156455
\(387\) 0 0
\(388\) 0.873537 0.0443471
\(389\) 7.54227 0.382408 0.191204 0.981550i \(-0.438761\pi\)
0.191204 + 0.981550i \(0.438761\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.37462 −0.0694290
\(393\) 0 0
\(394\) −0.963415 −0.0485361
\(395\) −44.6707 −2.24763
\(396\) 0 0
\(397\) 28.7582 1.44333 0.721665 0.692242i \(-0.243377\pi\)
0.721665 + 0.692242i \(0.243377\pi\)
\(398\) −3.27328 −0.164075
\(399\) 0 0
\(400\) 9.61578 0.480789
\(401\) −6.81566 −0.340358 −0.170179 0.985413i \(-0.554435\pi\)
−0.170179 + 0.985413i \(0.554435\pi\)
\(402\) 0 0
\(403\) 0.804211 0.0400606
\(404\) 10.9103 0.542807
\(405\) 0 0
\(406\) −1.70282 −0.0845094
\(407\) 7.61684 0.377553
\(408\) 0 0
\(409\) −27.7947 −1.37436 −0.687179 0.726488i \(-0.741151\pi\)
−0.687179 + 0.726488i \(0.741151\pi\)
\(410\) 39.7493 1.96308
\(411\) 0 0
\(412\) 3.69279 0.181931
\(413\) 5.50628 0.270947
\(414\) 0 0
\(415\) −0.432953 −0.0212528
\(416\) −4.70981 −0.230917
\(417\) 0 0
\(418\) −3.39574 −0.166091
\(419\) 15.1417 0.739718 0.369859 0.929088i \(-0.379406\pi\)
0.369859 + 0.929088i \(0.379406\pi\)
\(420\) 0 0
\(421\) 38.1445 1.85905 0.929524 0.368762i \(-0.120218\pi\)
0.929524 + 0.368762i \(0.120218\pi\)
\(422\) 24.4174 1.18862
\(423\) 0 0
\(424\) −10.3078 −0.500590
\(425\) 45.5576 2.20987
\(426\) 0 0
\(427\) −36.6207 −1.77220
\(428\) −10.7665 −0.520416
\(429\) 0 0
\(430\) 46.9016 2.26180
\(431\) −10.2138 −0.491981 −0.245991 0.969272i \(-0.579113\pi\)
−0.245991 + 0.969272i \(0.579113\pi\)
\(432\) 0 0
\(433\) −9.82271 −0.472049 −0.236025 0.971747i \(-0.575845\pi\)
−0.236025 + 0.971747i \(0.575845\pi\)
\(434\) −0.494139 −0.0237194
\(435\) 0 0
\(436\) −8.31357 −0.398148
\(437\) 0 0
\(438\) 0 0
\(439\) −7.79601 −0.372083 −0.186042 0.982542i \(-0.559566\pi\)
−0.186042 + 0.982542i \(0.559566\pi\)
\(440\) 6.04290 0.288084
\(441\) 0 0
\(442\) −22.3141 −1.06138
\(443\) 28.2208 1.34081 0.670406 0.741994i \(-0.266120\pi\)
0.670406 + 0.741994i \(0.266120\pi\)
\(444\) 0 0
\(445\) −46.7890 −2.21801
\(446\) −15.1991 −0.719699
\(447\) 0 0
\(448\) 2.89389 0.136724
\(449\) 12.7574 0.602061 0.301030 0.953615i \(-0.402669\pi\)
0.301030 + 0.953615i \(0.402669\pi\)
\(450\) 0 0
\(451\) 16.4344 0.773864
\(452\) −7.51278 −0.353372
\(453\) 0 0
\(454\) 12.9325 0.606950
\(455\) −52.1071 −2.44282
\(456\) 0 0
\(457\) 9.37038 0.438328 0.219164 0.975688i \(-0.429667\pi\)
0.219164 + 0.975688i \(0.429667\pi\)
\(458\) 0.948502 0.0443206
\(459\) 0 0
\(460\) 0 0
\(461\) 11.6569 0.542916 0.271458 0.962450i \(-0.412494\pi\)
0.271458 + 0.962450i \(0.412494\pi\)
\(462\) 0 0
\(463\) 13.6279 0.633341 0.316670 0.948536i \(-0.397435\pi\)
0.316670 + 0.948536i \(0.397435\pi\)
\(464\) 0.588417 0.0273166
\(465\) 0 0
\(466\) 13.9482 0.646139
\(467\) −6.57700 −0.304347 −0.152174 0.988354i \(-0.548627\pi\)
−0.152174 + 0.988354i \(0.548627\pi\)
\(468\) 0 0
\(469\) 17.0048 0.785207
\(470\) −20.8556 −0.961995
\(471\) 0 0
\(472\) −1.90272 −0.0875800
\(473\) 19.3915 0.891622
\(474\) 0 0
\(475\) 20.6578 0.947845
\(476\) 13.7107 0.628428
\(477\) 0 0
\(478\) −2.44329 −0.111754
\(479\) −9.24047 −0.422208 −0.211104 0.977464i \(-0.567706\pi\)
−0.211104 + 0.977464i \(0.567706\pi\)
\(480\) 0 0
\(481\) 22.6957 1.03483
\(482\) −10.8636 −0.494822
\(483\) 0 0
\(484\) −8.50156 −0.386434
\(485\) −3.33958 −0.151643
\(486\) 0 0
\(487\) −38.0246 −1.72306 −0.861529 0.507708i \(-0.830493\pi\)
−0.861529 + 0.507708i \(0.830493\pi\)
\(488\) 12.6545 0.572841
\(489\) 0 0
\(490\) 5.25527 0.237409
\(491\) 4.28664 0.193453 0.0967267 0.995311i \(-0.469163\pi\)
0.0967267 + 0.995311i \(0.469163\pi\)
\(492\) 0 0
\(493\) 2.78780 0.125556
\(494\) −10.1182 −0.455239
\(495\) 0 0
\(496\) 0.170752 0.00766701
\(497\) 34.2105 1.53455
\(498\) 0 0
\(499\) −30.2794 −1.35549 −0.677746 0.735297i \(-0.737043\pi\)
−0.677746 + 0.735297i \(0.737043\pi\)
\(500\) −17.6464 −0.789170
\(501\) 0 0
\(502\) −14.3211 −0.639182
\(503\) −11.6641 −0.520077 −0.260038 0.965598i \(-0.583735\pi\)
−0.260038 + 0.965598i \(0.583735\pi\)
\(504\) 0 0
\(505\) −41.7107 −1.85610
\(506\) 0 0
\(507\) 0 0
\(508\) −7.58758 −0.336645
\(509\) 15.3035 0.678317 0.339159 0.940729i \(-0.389858\pi\)
0.339159 + 0.940729i \(0.389858\pi\)
\(510\) 0 0
\(511\) −38.2386 −1.69158
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −4.24430 −0.187208
\(515\) −14.1178 −0.622103
\(516\) 0 0
\(517\) −8.62274 −0.379228
\(518\) −13.9451 −0.612714
\(519\) 0 0
\(520\) 18.0059 0.789610
\(521\) −1.93070 −0.0845857 −0.0422928 0.999105i \(-0.513466\pi\)
−0.0422928 + 0.999105i \(0.513466\pi\)
\(522\) 0 0
\(523\) −18.2743 −0.799081 −0.399540 0.916716i \(-0.630830\pi\)
−0.399540 + 0.916716i \(0.630830\pi\)
\(524\) −17.8085 −0.777970
\(525\) 0 0
\(526\) −7.10375 −0.309738
\(527\) 0.808991 0.0352402
\(528\) 0 0
\(529\) 0 0
\(530\) 39.4072 1.71174
\(531\) 0 0
\(532\) 6.21702 0.269542
\(533\) 48.9691 2.12109
\(534\) 0 0
\(535\) 41.1608 1.77954
\(536\) −5.87608 −0.253808
\(537\) 0 0
\(538\) −7.73235 −0.333365
\(539\) 2.17279 0.0935888
\(540\) 0 0
\(541\) 44.4499 1.91105 0.955525 0.294910i \(-0.0952898\pi\)
0.955525 + 0.294910i \(0.0952898\pi\)
\(542\) 5.08058 0.218230
\(543\) 0 0
\(544\) −4.73780 −0.203131
\(545\) 31.7833 1.36145
\(546\) 0 0
\(547\) 19.1076 0.816981 0.408490 0.912763i \(-0.366055\pi\)
0.408490 + 0.912763i \(0.366055\pi\)
\(548\) 0.762968 0.0325924
\(549\) 0 0
\(550\) −15.1991 −0.648093
\(551\) 1.26411 0.0538529
\(552\) 0 0
\(553\) 33.8138 1.43791
\(554\) 15.3909 0.653898
\(555\) 0 0
\(556\) 9.22612 0.391275
\(557\) 44.0806 1.86776 0.933878 0.357591i \(-0.116402\pi\)
0.933878 + 0.357591i \(0.116402\pi\)
\(558\) 0 0
\(559\) 57.7804 2.44385
\(560\) −11.0635 −0.467519
\(561\) 0 0
\(562\) −3.29958 −0.139184
\(563\) −39.4399 −1.66219 −0.831096 0.556129i \(-0.812286\pi\)
−0.831096 + 0.556129i \(0.812286\pi\)
\(564\) 0 0
\(565\) 28.7218 1.20834
\(566\) 25.8826 1.08793
\(567\) 0 0
\(568\) −11.8216 −0.496024
\(569\) 38.3267 1.60674 0.803370 0.595480i \(-0.203038\pi\)
0.803370 + 0.595480i \(0.203038\pi\)
\(570\) 0 0
\(571\) −16.9708 −0.710204 −0.355102 0.934828i \(-0.615554\pi\)
−0.355102 + 0.934828i \(0.615554\pi\)
\(572\) 7.44454 0.311272
\(573\) 0 0
\(574\) −30.0885 −1.25587
\(575\) 0 0
\(576\) 0 0
\(577\) −29.1794 −1.21475 −0.607377 0.794414i \(-0.707778\pi\)
−0.607377 + 0.794414i \(0.707778\pi\)
\(578\) −5.44675 −0.226555
\(579\) 0 0
\(580\) −2.24955 −0.0934077
\(581\) 0.327727 0.0135964
\(582\) 0 0
\(583\) 16.2929 0.674785
\(584\) 13.2135 0.546780
\(585\) 0 0
\(586\) 6.52568 0.269573
\(587\) 15.1816 0.626613 0.313306 0.949652i \(-0.398563\pi\)
0.313306 + 0.949652i \(0.398563\pi\)
\(588\) 0 0
\(589\) 0.366831 0.0151150
\(590\) 7.27423 0.299475
\(591\) 0 0
\(592\) 4.81881 0.198052
\(593\) 31.2609 1.28373 0.641865 0.766817i \(-0.278161\pi\)
0.641865 + 0.766817i \(0.278161\pi\)
\(594\) 0 0
\(595\) −52.4168 −2.14888
\(596\) −18.4911 −0.757424
\(597\) 0 0
\(598\) 0 0
\(599\) 30.8576 1.26081 0.630405 0.776267i \(-0.282889\pi\)
0.630405 + 0.776267i \(0.282889\pi\)
\(600\) 0 0
\(601\) 6.62774 0.270351 0.135176 0.990822i \(-0.456840\pi\)
0.135176 + 0.990822i \(0.456840\pi\)
\(602\) −35.5025 −1.44698
\(603\) 0 0
\(604\) −0.991887 −0.0403593
\(605\) 32.5020 1.32139
\(606\) 0 0
\(607\) −23.5777 −0.956991 −0.478495 0.878090i \(-0.658818\pi\)
−0.478495 + 0.878090i \(0.658818\pi\)
\(608\) −2.14832 −0.0871260
\(609\) 0 0
\(610\) −48.3788 −1.95880
\(611\) −25.6930 −1.03943
\(612\) 0 0
\(613\) −32.5703 −1.31550 −0.657750 0.753236i \(-0.728492\pi\)
−0.657750 + 0.753236i \(0.728492\pi\)
\(614\) −4.61726 −0.186338
\(615\) 0 0
\(616\) −4.57422 −0.184301
\(617\) 6.00021 0.241559 0.120780 0.992679i \(-0.461461\pi\)
0.120780 + 0.992679i \(0.461461\pi\)
\(618\) 0 0
\(619\) −12.8869 −0.517968 −0.258984 0.965882i \(-0.583388\pi\)
−0.258984 + 0.965882i \(0.583388\pi\)
\(620\) −0.652796 −0.0262169
\(621\) 0 0
\(622\) −10.7400 −0.430636
\(623\) 35.4173 1.41896
\(624\) 0 0
\(625\) 19.3843 0.775371
\(626\) −26.6899 −1.06674
\(627\) 0 0
\(628\) 8.10972 0.323613
\(629\) 22.8306 0.910315
\(630\) 0 0
\(631\) 21.4418 0.853585 0.426792 0.904350i \(-0.359644\pi\)
0.426792 + 0.904350i \(0.359644\pi\)
\(632\) −11.6845 −0.464786
\(633\) 0 0
\(634\) 19.1814 0.761791
\(635\) 29.0078 1.15114
\(636\) 0 0
\(637\) 6.47422 0.256518
\(638\) −0.930080 −0.0368222
\(639\) 0 0
\(640\) 3.82306 0.151120
\(641\) −30.7587 −1.21489 −0.607447 0.794360i \(-0.707806\pi\)
−0.607447 + 0.794360i \(0.707806\pi\)
\(642\) 0 0
\(643\) 28.7980 1.13568 0.567842 0.823138i \(-0.307778\pi\)
0.567842 + 0.823138i \(0.307778\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −10.1783 −0.400461
\(647\) 20.8594 0.820069 0.410035 0.912070i \(-0.365517\pi\)
0.410035 + 0.912070i \(0.365517\pi\)
\(648\) 0 0
\(649\) 3.00753 0.118056
\(650\) −45.2885 −1.77636
\(651\) 0 0
\(652\) −23.0759 −0.903724
\(653\) −26.5838 −1.04031 −0.520153 0.854073i \(-0.674125\pi\)
−0.520153 + 0.854073i \(0.674125\pi\)
\(654\) 0 0
\(655\) 68.0831 2.66023
\(656\) 10.3972 0.405944
\(657\) 0 0
\(658\) 15.7868 0.615433
\(659\) 27.4167 1.06800 0.534002 0.845483i \(-0.320687\pi\)
0.534002 + 0.845483i \(0.320687\pi\)
\(660\) 0 0
\(661\) 8.13934 0.316583 0.158292 0.987392i \(-0.449401\pi\)
0.158292 + 0.987392i \(0.449401\pi\)
\(662\) −8.74066 −0.339715
\(663\) 0 0
\(664\) −0.113248 −0.00439486
\(665\) −23.7680 −0.921685
\(666\) 0 0
\(667\) 0 0
\(668\) 5.28482 0.204476
\(669\) 0 0
\(670\) 22.4646 0.867884
\(671\) −20.0022 −0.772178
\(672\) 0 0
\(673\) 31.0127 1.19545 0.597726 0.801700i \(-0.296071\pi\)
0.597726 + 0.801700i \(0.296071\pi\)
\(674\) 3.25394 0.125337
\(675\) 0 0
\(676\) 9.18232 0.353166
\(677\) 20.4045 0.784208 0.392104 0.919921i \(-0.371747\pi\)
0.392104 + 0.919921i \(0.371747\pi\)
\(678\) 0 0
\(679\) 2.52792 0.0970128
\(680\) 18.1129 0.694597
\(681\) 0 0
\(682\) −0.269899 −0.0103350
\(683\) −20.7857 −0.795342 −0.397671 0.917528i \(-0.630181\pi\)
−0.397671 + 0.917528i \(0.630181\pi\)
\(684\) 0 0
\(685\) −2.91687 −0.111448
\(686\) 16.2792 0.621544
\(687\) 0 0
\(688\) 12.2681 0.467716
\(689\) 48.5477 1.84952
\(690\) 0 0
\(691\) −13.6695 −0.520012 −0.260006 0.965607i \(-0.583725\pi\)
−0.260006 + 0.965607i \(0.583725\pi\)
\(692\) −14.3906 −0.547049
\(693\) 0 0
\(694\) −5.21179 −0.197837
\(695\) −35.2720 −1.33794
\(696\) 0 0
\(697\) 49.2601 1.86586
\(698\) −15.4731 −0.585665
\(699\) 0 0
\(700\) 27.8270 1.05176
\(701\) 2.20983 0.0834641 0.0417320 0.999129i \(-0.486712\pi\)
0.0417320 + 0.999129i \(0.486712\pi\)
\(702\) 0 0
\(703\) 10.3524 0.390447
\(704\) 1.58065 0.0595728
\(705\) 0 0
\(706\) −1.61663 −0.0608428
\(707\) 31.5732 1.18743
\(708\) 0 0
\(709\) −18.7677 −0.704837 −0.352418 0.935843i \(-0.614641\pi\)
−0.352418 + 0.935843i \(0.614641\pi\)
\(710\) 45.1947 1.69613
\(711\) 0 0
\(712\) −12.2386 −0.458662
\(713\) 0 0
\(714\) 0 0
\(715\) −28.4609 −1.06438
\(716\) −11.2001 −0.418566
\(717\) 0 0
\(718\) 13.5579 0.505978
\(719\) −20.6700 −0.770862 −0.385431 0.922737i \(-0.625947\pi\)
−0.385431 + 0.922737i \(0.625947\pi\)
\(720\) 0 0
\(721\) 10.6865 0.397988
\(722\) 14.3847 0.535343
\(723\) 0 0
\(724\) 8.79467 0.326852
\(725\) 5.65809 0.210136
\(726\) 0 0
\(727\) −31.5454 −1.16996 −0.584978 0.811049i \(-0.698897\pi\)
−0.584978 + 0.811049i \(0.698897\pi\)
\(728\) −13.6297 −0.505150
\(729\) 0 0
\(730\) −50.5161 −1.86969
\(731\) 58.1237 2.14978
\(732\) 0 0
\(733\) −24.4270 −0.902232 −0.451116 0.892465i \(-0.648974\pi\)
−0.451116 + 0.892465i \(0.648974\pi\)
\(734\) −16.2735 −0.600664
\(735\) 0 0
\(736\) 0 0
\(737\) 9.28801 0.342128
\(738\) 0 0
\(739\) −42.1528 −1.55062 −0.775308 0.631583i \(-0.782405\pi\)
−0.775308 + 0.631583i \(0.782405\pi\)
\(740\) −18.4226 −0.677229
\(741\) 0 0
\(742\) −29.8296 −1.09508
\(743\) −2.98913 −0.109661 −0.0548303 0.998496i \(-0.517462\pi\)
−0.0548303 + 0.998496i \(0.517462\pi\)
\(744\) 0 0
\(745\) 70.6925 2.58997
\(746\) −6.97489 −0.255369
\(747\) 0 0
\(748\) 7.48878 0.273817
\(749\) −31.1570 −1.13845
\(750\) 0 0
\(751\) 39.2254 1.43135 0.715677 0.698431i \(-0.246118\pi\)
0.715677 + 0.698431i \(0.246118\pi\)
\(752\) −5.45520 −0.198931
\(753\) 0 0
\(754\) −2.77133 −0.100926
\(755\) 3.79204 0.138007
\(756\) 0 0
\(757\) −47.4506 −1.72462 −0.862311 0.506379i \(-0.830984\pi\)
−0.862311 + 0.506379i \(0.830984\pi\)
\(758\) −16.5116 −0.599728
\(759\) 0 0
\(760\) 8.21316 0.297923
\(761\) 30.7455 1.11452 0.557261 0.830337i \(-0.311852\pi\)
0.557261 + 0.830337i \(0.311852\pi\)
\(762\) 0 0
\(763\) −24.0586 −0.870980
\(764\) 6.33200 0.229084
\(765\) 0 0
\(766\) −22.3879 −0.808907
\(767\) 8.96147 0.323580
\(768\) 0 0
\(769\) −32.4747 −1.17107 −0.585533 0.810648i \(-0.699115\pi\)
−0.585533 + 0.810648i \(0.699115\pi\)
\(770\) 17.4875 0.630206
\(771\) 0 0
\(772\) −3.07385 −0.110630
\(773\) −12.4335 −0.447202 −0.223601 0.974681i \(-0.571781\pi\)
−0.223601 + 0.974681i \(0.571781\pi\)
\(774\) 0 0
\(775\) 1.64192 0.0589794
\(776\) −0.873537 −0.0313582
\(777\) 0 0
\(778\) −7.54227 −0.270403
\(779\) 22.3366 0.800293
\(780\) 0 0
\(781\) 18.6858 0.668629
\(782\) 0 0
\(783\) 0 0
\(784\) 1.37462 0.0490937
\(785\) −31.0039 −1.10658
\(786\) 0 0
\(787\) −4.10539 −0.146341 −0.0731707 0.997319i \(-0.523312\pi\)
−0.0731707 + 0.997319i \(0.523312\pi\)
\(788\) 0.963415 0.0343202
\(789\) 0 0
\(790\) 44.6707 1.58931
\(791\) −21.7412 −0.773028
\(792\) 0 0
\(793\) −59.6001 −2.11646
\(794\) −28.7582 −1.02059
\(795\) 0 0
\(796\) 3.27328 0.116018
\(797\) 38.6182 1.36793 0.683963 0.729517i \(-0.260255\pi\)
0.683963 + 0.729517i \(0.260255\pi\)
\(798\) 0 0
\(799\) −25.8457 −0.914354
\(800\) −9.61578 −0.339969
\(801\) 0 0
\(802\) 6.81566 0.240669
\(803\) −20.8859 −0.737048
\(804\) 0 0
\(805\) 0 0
\(806\) −0.804211 −0.0283271
\(807\) 0 0
\(808\) −10.9103 −0.383822
\(809\) −36.8332 −1.29499 −0.647494 0.762071i \(-0.724183\pi\)
−0.647494 + 0.762071i \(0.724183\pi\)
\(810\) 0 0
\(811\) −44.7013 −1.56968 −0.784838 0.619701i \(-0.787254\pi\)
−0.784838 + 0.619701i \(0.787254\pi\)
\(812\) 1.70282 0.0597572
\(813\) 0 0
\(814\) −7.61684 −0.266970
\(815\) 88.2207 3.09024
\(816\) 0 0
\(817\) 26.3558 0.922073
\(818\) 27.7947 0.971818
\(819\) 0 0
\(820\) −39.7493 −1.38811
\(821\) −37.2008 −1.29832 −0.649159 0.760653i \(-0.724879\pi\)
−0.649159 + 0.760653i \(0.724879\pi\)
\(822\) 0 0
\(823\) 41.0413 1.43061 0.715304 0.698813i \(-0.246288\pi\)
0.715304 + 0.698813i \(0.246288\pi\)
\(824\) −3.69279 −0.128645
\(825\) 0 0
\(826\) −5.50628 −0.191588
\(827\) −6.42848 −0.223540 −0.111770 0.993734i \(-0.535652\pi\)
−0.111770 + 0.993734i \(0.535652\pi\)
\(828\) 0 0
\(829\) 15.8721 0.551262 0.275631 0.961264i \(-0.411113\pi\)
0.275631 + 0.961264i \(0.411113\pi\)
\(830\) 0.432953 0.0150280
\(831\) 0 0
\(832\) 4.70981 0.163283
\(833\) 6.51269 0.225651
\(834\) 0 0
\(835\) −20.2042 −0.699195
\(836\) 3.39574 0.117444
\(837\) 0 0
\(838\) −15.1417 −0.523060
\(839\) −10.4265 −0.359963 −0.179981 0.983670i \(-0.557604\pi\)
−0.179981 + 0.983670i \(0.557604\pi\)
\(840\) 0 0
\(841\) −28.6538 −0.988061
\(842\) −38.1445 −1.31455
\(843\) 0 0
\(844\) −24.4174 −0.840483
\(845\) −35.1045 −1.20763
\(846\) 0 0
\(847\) −24.6026 −0.845356
\(848\) 10.3078 0.353971
\(849\) 0 0
\(850\) −45.5576 −1.56261
\(851\) 0 0
\(852\) 0 0
\(853\) 26.2935 0.900271 0.450135 0.892960i \(-0.351376\pi\)
0.450135 + 0.892960i \(0.351376\pi\)
\(854\) 36.6207 1.25313
\(855\) 0 0
\(856\) 10.7665 0.367990
\(857\) −32.4336 −1.10791 −0.553955 0.832547i \(-0.686882\pi\)
−0.553955 + 0.832547i \(0.686882\pi\)
\(858\) 0 0
\(859\) −55.1643 −1.88218 −0.941091 0.338154i \(-0.890198\pi\)
−0.941091 + 0.338154i \(0.890198\pi\)
\(860\) −46.9016 −1.59933
\(861\) 0 0
\(862\) 10.2138 0.347883
\(863\) −18.2143 −0.620023 −0.310012 0.950733i \(-0.600333\pi\)
−0.310012 + 0.950733i \(0.600333\pi\)
\(864\) 0 0
\(865\) 55.0161 1.87060
\(866\) 9.82271 0.333789
\(867\) 0 0
\(868\) 0.494139 0.0167722
\(869\) 18.4691 0.626522
\(870\) 0 0
\(871\) 27.6752 0.937740
\(872\) 8.31357 0.281533
\(873\) 0 0
\(874\) 0 0
\(875\) −51.0668 −1.72637
\(876\) 0 0
\(877\) −28.7970 −0.972407 −0.486203 0.873846i \(-0.661619\pi\)
−0.486203 + 0.873846i \(0.661619\pi\)
\(878\) 7.79601 0.263103
\(879\) 0 0
\(880\) −6.04290 −0.203706
\(881\) 21.0631 0.709633 0.354816 0.934936i \(-0.384543\pi\)
0.354816 + 0.934936i \(0.384543\pi\)
\(882\) 0 0
\(883\) −14.1593 −0.476498 −0.238249 0.971204i \(-0.576573\pi\)
−0.238249 + 0.971204i \(0.576573\pi\)
\(884\) 22.3141 0.750506
\(885\) 0 0
\(886\) −28.2208 −0.948098
\(887\) −57.5302 −1.93167 −0.965837 0.259150i \(-0.916558\pi\)
−0.965837 + 0.259150i \(0.916558\pi\)
\(888\) 0 0
\(889\) −21.9577 −0.736436
\(890\) 46.7890 1.56837
\(891\) 0 0
\(892\) 15.1991 0.508904
\(893\) −11.7195 −0.392179
\(894\) 0 0
\(895\) 42.8185 1.43127
\(896\) −2.89389 −0.0966782
\(897\) 0 0
\(898\) −12.7574 −0.425721
\(899\) 0.100474 0.00335098
\(900\) 0 0
\(901\) 48.8362 1.62697
\(902\) −16.4344 −0.547205
\(903\) 0 0
\(904\) 7.51278 0.249871
\(905\) −33.6226 −1.11765
\(906\) 0 0
\(907\) 45.1699 1.49984 0.749921 0.661528i \(-0.230092\pi\)
0.749921 + 0.661528i \(0.230092\pi\)
\(908\) −12.9325 −0.429179
\(909\) 0 0
\(910\) 52.1071 1.72733
\(911\) −24.4377 −0.809656 −0.404828 0.914393i \(-0.632669\pi\)
−0.404828 + 0.914393i \(0.632669\pi\)
\(912\) 0 0
\(913\) 0.179005 0.00592419
\(914\) −9.37038 −0.309945
\(915\) 0 0
\(916\) −0.948502 −0.0313394
\(917\) −51.5360 −1.70187
\(918\) 0 0
\(919\) −1.40194 −0.0462457 −0.0231229 0.999733i \(-0.507361\pi\)
−0.0231229 + 0.999733i \(0.507361\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −11.6569 −0.383899
\(923\) 55.6775 1.83265
\(924\) 0 0
\(925\) 46.3366 1.52354
\(926\) −13.6279 −0.447840
\(927\) 0 0
\(928\) −0.588417 −0.0193157
\(929\) 8.73879 0.286710 0.143355 0.989671i \(-0.454211\pi\)
0.143355 + 0.989671i \(0.454211\pi\)
\(930\) 0 0
\(931\) 2.95313 0.0967851
\(932\) −13.9482 −0.456889
\(933\) 0 0
\(934\) 6.57700 0.215206
\(935\) −28.6301 −0.936303
\(936\) 0 0
\(937\) 7.95252 0.259798 0.129899 0.991527i \(-0.458535\pi\)
0.129899 + 0.991527i \(0.458535\pi\)
\(938\) −17.0048 −0.555225
\(939\) 0 0
\(940\) 20.8556 0.680233
\(941\) −20.7252 −0.675622 −0.337811 0.941214i \(-0.609686\pi\)
−0.337811 + 0.941214i \(0.609686\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 1.90272 0.0619284
\(945\) 0 0
\(946\) −19.3915 −0.630472
\(947\) −54.9184 −1.78461 −0.892304 0.451435i \(-0.850912\pi\)
−0.892304 + 0.451435i \(0.850912\pi\)
\(948\) 0 0
\(949\) −62.2333 −2.02018
\(950\) −20.6578 −0.670227
\(951\) 0 0
\(952\) −13.7107 −0.444366
\(953\) −8.68947 −0.281480 −0.140740 0.990047i \(-0.544948\pi\)
−0.140740 + 0.990047i \(0.544948\pi\)
\(954\) 0 0
\(955\) −24.2076 −0.783340
\(956\) 2.44329 0.0790217
\(957\) 0 0
\(958\) 9.24047 0.298546
\(959\) 2.20795 0.0712984
\(960\) 0 0
\(961\) −30.9708 −0.999059
\(962\) −22.6957 −0.731739
\(963\) 0 0
\(964\) 10.8636 0.349892
\(965\) 11.7515 0.378295
\(966\) 0 0
\(967\) 39.3485 1.26536 0.632681 0.774412i \(-0.281954\pi\)
0.632681 + 0.774412i \(0.281954\pi\)
\(968\) 8.50156 0.273250
\(969\) 0 0
\(970\) 3.33958 0.107228
\(971\) 16.1185 0.517268 0.258634 0.965975i \(-0.416728\pi\)
0.258634 + 0.965975i \(0.416728\pi\)
\(972\) 0 0
\(973\) 26.6994 0.855944
\(974\) 38.0246 1.21839
\(975\) 0 0
\(976\) −12.6545 −0.405060
\(977\) 35.5575 1.13758 0.568792 0.822482i \(-0.307411\pi\)
0.568792 + 0.822482i \(0.307411\pi\)
\(978\) 0 0
\(979\) 19.3449 0.618267
\(980\) −5.25527 −0.167873
\(981\) 0 0
\(982\) −4.28664 −0.136792
\(983\) 54.6156 1.74197 0.870984 0.491311i \(-0.163482\pi\)
0.870984 + 0.491311i \(0.163482\pi\)
\(984\) 0 0
\(985\) −3.68319 −0.117356
\(986\) −2.78780 −0.0887818
\(987\) 0 0
\(988\) 10.1182 0.321903
\(989\) 0 0
\(990\) 0 0
\(991\) 26.8734 0.853663 0.426832 0.904331i \(-0.359630\pi\)
0.426832 + 0.904331i \(0.359630\pi\)
\(992\) −0.170752 −0.00542139
\(993\) 0 0
\(994\) −34.2105 −1.08509
\(995\) −12.5140 −0.396719
\(996\) 0 0
\(997\) −26.9729 −0.854239 −0.427119 0.904195i \(-0.640472\pi\)
−0.427119 + 0.904195i \(0.640472\pi\)
\(998\) 30.2794 0.958477
\(999\) 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 9522.2.a.bs.1.1 5
3.2 odd 2 3174.2.a.ba.1.5 5
23.3 even 11 414.2.i.e.55.1 10
23.8 even 11 414.2.i.e.271.1 10
23.22 odd 2 9522.2.a.br.1.5 5
69.8 odd 22 138.2.e.b.133.1 yes 10
69.26 odd 22 138.2.e.b.55.1 10
69.68 even 2 3174.2.a.bb.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
138.2.e.b.55.1 10 69.26 odd 22
138.2.e.b.133.1 yes 10 69.8 odd 22
414.2.i.e.55.1 10 23.3 even 11
414.2.i.e.271.1 10 23.8 even 11
3174.2.a.ba.1.5 5 3.2 odd 2
3174.2.a.bb.1.1 5 69.68 even 2
9522.2.a.br.1.5 5 23.22 odd 2
9522.2.a.bs.1.1 5 1.1 even 1 trivial