Properties

Label 2151.2.a.f.1.1
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $7$
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] = [2151,2,Mod(1,2151)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2151, 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("2151.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.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: \(17.1758214748\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 10x^{5} + 8x^{4} + 22x^{3} - 5x^{2} - 7x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 717)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.19620\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.42067 q^{2} +3.85962 q^{4} -2.85962 q^{5} -2.49298 q^{7} -4.50152 q^{8} +O(q^{10})\) \(q-2.42067 q^{2} +3.85962 q^{4} -2.85962 q^{5} -2.49298 q^{7} -4.50152 q^{8} +6.92219 q^{10} +1.12006 q^{11} -0.971725 q^{13} +6.03467 q^{14} +3.17744 q^{16} +7.78181 q^{17} -4.63600 q^{19} -11.0371 q^{20} -2.71129 q^{22} -5.88790 q^{23} +3.17744 q^{25} +2.35222 q^{26} -9.62196 q^{28} +7.22556 q^{29} +3.96875 q^{31} +1.31153 q^{32} -18.8372 q^{34} +7.12898 q^{35} +6.21281 q^{37} +11.2222 q^{38} +12.8727 q^{40} -10.8562 q^{41} +12.3040 q^{43} +4.32300 q^{44} +14.2526 q^{46} +5.23663 q^{47} -0.785046 q^{49} -7.69151 q^{50} -3.75049 q^{52} +0.697008 q^{53} -3.20294 q^{55} +11.2222 q^{56} -17.4907 q^{58} -7.55604 q^{59} -2.56381 q^{61} -9.60703 q^{62} -9.52966 q^{64} +2.77876 q^{65} +6.16648 q^{67} +30.0348 q^{68} -17.2569 q^{70} +3.64042 q^{71} -13.1915 q^{73} -15.0391 q^{74} -17.8932 q^{76} -2.79229 q^{77} +12.4007 q^{79} -9.08627 q^{80} +26.2791 q^{82} -5.28871 q^{83} -22.2530 q^{85} -29.7840 q^{86} -5.04197 q^{88} +1.20236 q^{89} +2.42249 q^{91} -22.7251 q^{92} -12.6761 q^{94} +13.2572 q^{95} -3.49376 q^{97} +1.90033 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 10 q^{4} - 3 q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 10 q^{4} - 3 q^{5} + 3 q^{7} - 11 q^{11} - 5 q^{13} - 6 q^{14} - 4 q^{16} - 11 q^{17} - 8 q^{19} - 34 q^{20} - 19 q^{22} - 26 q^{23} - 4 q^{25} - 6 q^{26} - 2 q^{28} - 6 q^{29} - 2 q^{31} + 5 q^{32} - 40 q^{34} + 5 q^{35} + 12 q^{37} + 9 q^{38} - 5 q^{40} - 26 q^{41} + 10 q^{43} - 3 q^{44} + 6 q^{46} - 7 q^{47} + 2 q^{49} + 5 q^{50} - 22 q^{52} - 2 q^{53} - 8 q^{55} + 9 q^{56} - 6 q^{58} - 6 q^{59} - 8 q^{61} + 2 q^{62} - 18 q^{64} + 17 q^{65} + 24 q^{67} + 9 q^{68} - 3 q^{70} + 25 q^{71} - 16 q^{73} + 9 q^{74} - 32 q^{76} - 24 q^{77} + 19 q^{79} - 18 q^{80} + 39 q^{82} - 37 q^{83} - 20 q^{85} + q^{86} - 21 q^{88} - 29 q^{89} - 19 q^{91} - 52 q^{92} - 22 q^{94} + 24 q^{95} - 12 q^{97} - 23 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.42067 −1.71167 −0.855835 0.517250i \(-0.826956\pi\)
−0.855835 + 0.517250i \(0.826956\pi\)
\(3\) 0 0
\(4\) 3.85962 1.92981
\(5\) −2.85962 −1.27886 −0.639431 0.768849i \(-0.720830\pi\)
−0.639431 + 0.768849i \(0.720830\pi\)
\(6\) 0 0
\(7\) −2.49298 −0.942258 −0.471129 0.882064i \(-0.656153\pi\)
−0.471129 + 0.882064i \(0.656153\pi\)
\(8\) −4.50152 −1.59153
\(9\) 0 0
\(10\) 6.92219 2.18899
\(11\) 1.12006 0.337710 0.168855 0.985641i \(-0.445993\pi\)
0.168855 + 0.985641i \(0.445993\pi\)
\(12\) 0 0
\(13\) −0.971725 −0.269508 −0.134754 0.990879i \(-0.543024\pi\)
−0.134754 + 0.990879i \(0.543024\pi\)
\(14\) 6.03467 1.61283
\(15\) 0 0
\(16\) 3.17744 0.794359
\(17\) 7.78181 1.88737 0.943683 0.330851i \(-0.107336\pi\)
0.943683 + 0.330851i \(0.107336\pi\)
\(18\) 0 0
\(19\) −4.63600 −1.06357 −0.531786 0.846879i \(-0.678479\pi\)
−0.531786 + 0.846879i \(0.678479\pi\)
\(20\) −11.0371 −2.46796
\(21\) 0 0
\(22\) −2.71129 −0.578048
\(23\) −5.88790 −1.22771 −0.613856 0.789418i \(-0.710382\pi\)
−0.613856 + 0.789418i \(0.710382\pi\)
\(24\) 0 0
\(25\) 3.17744 0.635487
\(26\) 2.35222 0.461308
\(27\) 0 0
\(28\) −9.62196 −1.81838
\(29\) 7.22556 1.34175 0.670877 0.741569i \(-0.265918\pi\)
0.670877 + 0.741569i \(0.265918\pi\)
\(30\) 0 0
\(31\) 3.96875 0.712809 0.356405 0.934332i \(-0.384002\pi\)
0.356405 + 0.934332i \(0.384002\pi\)
\(32\) 1.31153 0.231849
\(33\) 0 0
\(34\) −18.8372 −3.23055
\(35\) 7.12898 1.20502
\(36\) 0 0
\(37\) 6.21281 1.02138 0.510690 0.859765i \(-0.329390\pi\)
0.510690 + 0.859765i \(0.329390\pi\)
\(38\) 11.2222 1.82048
\(39\) 0 0
\(40\) 12.8727 2.03534
\(41\) −10.8562 −1.69545 −0.847723 0.530439i \(-0.822027\pi\)
−0.847723 + 0.530439i \(0.822027\pi\)
\(42\) 0 0
\(43\) 12.3040 1.87635 0.938174 0.346165i \(-0.112516\pi\)
0.938174 + 0.346165i \(0.112516\pi\)
\(44\) 4.32300 0.651717
\(45\) 0 0
\(46\) 14.2526 2.10144
\(47\) 5.23663 0.763841 0.381920 0.924195i \(-0.375263\pi\)
0.381920 + 0.924195i \(0.375263\pi\)
\(48\) 0 0
\(49\) −0.785046 −0.112149
\(50\) −7.69151 −1.08774
\(51\) 0 0
\(52\) −3.75049 −0.520099
\(53\) 0.697008 0.0957414 0.0478707 0.998854i \(-0.484756\pi\)
0.0478707 + 0.998854i \(0.484756\pi\)
\(54\) 0 0
\(55\) −3.20294 −0.431885
\(56\) 11.2222 1.49963
\(57\) 0 0
\(58\) −17.4907 −2.29664
\(59\) −7.55604 −0.983713 −0.491857 0.870676i \(-0.663682\pi\)
−0.491857 + 0.870676i \(0.663682\pi\)
\(60\) 0 0
\(61\) −2.56381 −0.328262 −0.164131 0.986439i \(-0.552482\pi\)
−0.164131 + 0.986439i \(0.552482\pi\)
\(62\) −9.60703 −1.22009
\(63\) 0 0
\(64\) −9.52966 −1.19121
\(65\) 2.77876 0.344663
\(66\) 0 0
\(67\) 6.16648 0.753356 0.376678 0.926344i \(-0.377066\pi\)
0.376678 + 0.926344i \(0.377066\pi\)
\(68\) 30.0348 3.64226
\(69\) 0 0
\(70\) −17.2569 −2.06259
\(71\) 3.64042 0.432038 0.216019 0.976389i \(-0.430693\pi\)
0.216019 + 0.976389i \(0.430693\pi\)
\(72\) 0 0
\(73\) −13.1915 −1.54395 −0.771977 0.635650i \(-0.780732\pi\)
−0.771977 + 0.635650i \(0.780732\pi\)
\(74\) −15.0391 −1.74826
\(75\) 0 0
\(76\) −17.8932 −2.05249
\(77\) −2.79229 −0.318210
\(78\) 0 0
\(79\) 12.4007 1.39519 0.697595 0.716492i \(-0.254253\pi\)
0.697595 + 0.716492i \(0.254253\pi\)
\(80\) −9.08627 −1.01588
\(81\) 0 0
\(82\) 26.2791 2.90204
\(83\) −5.28871 −0.580512 −0.290256 0.956949i \(-0.593740\pi\)
−0.290256 + 0.956949i \(0.593740\pi\)
\(84\) 0 0
\(85\) −22.2530 −2.41368
\(86\) −29.7840 −3.21169
\(87\) 0 0
\(88\) −5.04197 −0.537476
\(89\) 1.20236 0.127450 0.0637249 0.997968i \(-0.479702\pi\)
0.0637249 + 0.997968i \(0.479702\pi\)
\(90\) 0 0
\(91\) 2.42249 0.253946
\(92\) −22.7251 −2.36925
\(93\) 0 0
\(94\) −12.6761 −1.30744
\(95\) 13.2572 1.36016
\(96\) 0 0
\(97\) −3.49376 −0.354737 −0.177369 0.984144i \(-0.556758\pi\)
−0.177369 + 0.984144i \(0.556758\pi\)
\(98\) 1.90033 0.191963
\(99\) 0 0
\(100\) 12.2637 1.22637
\(101\) −14.7746 −1.47013 −0.735064 0.677998i \(-0.762848\pi\)
−0.735064 + 0.677998i \(0.762848\pi\)
\(102\) 0 0
\(103\) 15.5456 1.53176 0.765879 0.642985i \(-0.222304\pi\)
0.765879 + 0.642985i \(0.222304\pi\)
\(104\) 4.37424 0.428930
\(105\) 0 0
\(106\) −1.68722 −0.163878
\(107\) 4.63008 0.447607 0.223803 0.974634i \(-0.428153\pi\)
0.223803 + 0.974634i \(0.428153\pi\)
\(108\) 0 0
\(109\) −4.46029 −0.427218 −0.213609 0.976919i \(-0.568522\pi\)
−0.213609 + 0.976919i \(0.568522\pi\)
\(110\) 7.75326 0.739244
\(111\) 0 0
\(112\) −7.92129 −0.748491
\(113\) −1.22312 −0.115062 −0.0575309 0.998344i \(-0.518323\pi\)
−0.0575309 + 0.998344i \(0.518323\pi\)
\(114\) 0 0
\(115\) 16.8372 1.57007
\(116\) 27.8879 2.58933
\(117\) 0 0
\(118\) 18.2907 1.68379
\(119\) −19.3999 −1.77839
\(120\) 0 0
\(121\) −9.74547 −0.885952
\(122\) 6.20613 0.561877
\(123\) 0 0
\(124\) 15.3179 1.37559
\(125\) 5.21184 0.466161
\(126\) 0 0
\(127\) 1.92155 0.170510 0.0852549 0.996359i \(-0.472830\pi\)
0.0852549 + 0.996359i \(0.472830\pi\)
\(128\) 20.4450 1.80710
\(129\) 0 0
\(130\) −6.72646 −0.589950
\(131\) −1.79786 −0.157080 −0.0785399 0.996911i \(-0.525026\pi\)
−0.0785399 + 0.996911i \(0.525026\pi\)
\(132\) 0 0
\(133\) 11.5575 1.00216
\(134\) −14.9270 −1.28950
\(135\) 0 0
\(136\) −35.0300 −3.00380
\(137\) 9.30901 0.795322 0.397661 0.917532i \(-0.369822\pi\)
0.397661 + 0.917532i \(0.369822\pi\)
\(138\) 0 0
\(139\) −21.9155 −1.85885 −0.929423 0.369016i \(-0.879695\pi\)
−0.929423 + 0.369016i \(0.879695\pi\)
\(140\) 27.5152 2.32546
\(141\) 0 0
\(142\) −8.81223 −0.739506
\(143\) −1.08839 −0.0910156
\(144\) 0 0
\(145\) −20.6624 −1.71592
\(146\) 31.9323 2.64274
\(147\) 0 0
\(148\) 23.9791 1.97107
\(149\) −5.26452 −0.431286 −0.215643 0.976472i \(-0.569185\pi\)
−0.215643 + 0.976472i \(0.569185\pi\)
\(150\) 0 0
\(151\) 11.9898 0.975715 0.487858 0.872923i \(-0.337778\pi\)
0.487858 + 0.872923i \(0.337778\pi\)
\(152\) 20.8691 1.69270
\(153\) 0 0
\(154\) 6.75919 0.544671
\(155\) −11.3491 −0.911584
\(156\) 0 0
\(157\) −12.0625 −0.962688 −0.481344 0.876532i \(-0.659851\pi\)
−0.481344 + 0.876532i \(0.659851\pi\)
\(158\) −30.0180 −2.38811
\(159\) 0 0
\(160\) −3.75049 −0.296502
\(161\) 14.6784 1.15682
\(162\) 0 0
\(163\) −11.5155 −0.901964 −0.450982 0.892533i \(-0.648926\pi\)
−0.450982 + 0.892533i \(0.648926\pi\)
\(164\) −41.9006 −3.27189
\(165\) 0 0
\(166\) 12.8022 0.993644
\(167\) −9.51806 −0.736530 −0.368265 0.929721i \(-0.620048\pi\)
−0.368265 + 0.929721i \(0.620048\pi\)
\(168\) 0 0
\(169\) −12.0558 −0.927365
\(170\) 53.8671 4.13142
\(171\) 0 0
\(172\) 47.4889 3.62100
\(173\) 18.3312 1.39370 0.696849 0.717218i \(-0.254585\pi\)
0.696849 + 0.717218i \(0.254585\pi\)
\(174\) 0 0
\(175\) −7.92129 −0.598793
\(176\) 3.55892 0.268263
\(177\) 0 0
\(178\) −2.91051 −0.218152
\(179\) −9.47162 −0.707942 −0.353971 0.935256i \(-0.615169\pi\)
−0.353971 + 0.935256i \(0.615169\pi\)
\(180\) 0 0
\(181\) −5.52272 −0.410501 −0.205250 0.978710i \(-0.565801\pi\)
−0.205250 + 0.978710i \(0.565801\pi\)
\(182\) −5.86404 −0.434672
\(183\) 0 0
\(184\) 26.5045 1.95394
\(185\) −17.7663 −1.30620
\(186\) 0 0
\(187\) 8.71608 0.637383
\(188\) 20.2114 1.47407
\(189\) 0 0
\(190\) −32.0913 −2.32815
\(191\) 12.2366 0.885411 0.442706 0.896667i \(-0.354019\pi\)
0.442706 + 0.896667i \(0.354019\pi\)
\(192\) 0 0
\(193\) −3.57272 −0.257170 −0.128585 0.991698i \(-0.541044\pi\)
−0.128585 + 0.991698i \(0.541044\pi\)
\(194\) 8.45721 0.607193
\(195\) 0 0
\(196\) −3.02998 −0.216427
\(197\) −14.9639 −1.06614 −0.533068 0.846072i \(-0.678961\pi\)
−0.533068 + 0.846072i \(0.678961\pi\)
\(198\) 0 0
\(199\) −12.4882 −0.885262 −0.442631 0.896704i \(-0.645955\pi\)
−0.442631 + 0.896704i \(0.645955\pi\)
\(200\) −14.3033 −1.01140
\(201\) 0 0
\(202\) 35.7644 2.51637
\(203\) −18.0132 −1.26428
\(204\) 0 0
\(205\) 31.0445 2.16824
\(206\) −37.6308 −2.62186
\(207\) 0 0
\(208\) −3.08759 −0.214086
\(209\) −5.19259 −0.359179
\(210\) 0 0
\(211\) −8.95186 −0.616271 −0.308136 0.951342i \(-0.599705\pi\)
−0.308136 + 0.951342i \(0.599705\pi\)
\(212\) 2.69019 0.184763
\(213\) 0 0
\(214\) −11.2079 −0.766155
\(215\) −35.1849 −2.39959
\(216\) 0 0
\(217\) −9.89403 −0.671650
\(218\) 10.7969 0.731257
\(219\) 0 0
\(220\) −12.3622 −0.833456
\(221\) −7.56178 −0.508660
\(222\) 0 0
\(223\) −13.6518 −0.914190 −0.457095 0.889418i \(-0.651110\pi\)
−0.457095 + 0.889418i \(0.651110\pi\)
\(224\) −3.26963 −0.218461
\(225\) 0 0
\(226\) 2.96077 0.196948
\(227\) −2.30473 −0.152971 −0.0764853 0.997071i \(-0.524370\pi\)
−0.0764853 + 0.997071i \(0.524370\pi\)
\(228\) 0 0
\(229\) 21.1300 1.39631 0.698156 0.715945i \(-0.254004\pi\)
0.698156 + 0.715945i \(0.254004\pi\)
\(230\) −40.7571 −2.68745
\(231\) 0 0
\(232\) −32.5260 −2.13544
\(233\) −9.11468 −0.597122 −0.298561 0.954391i \(-0.596507\pi\)
−0.298561 + 0.954391i \(0.596507\pi\)
\(234\) 0 0
\(235\) −14.9748 −0.976847
\(236\) −29.1635 −1.89838
\(237\) 0 0
\(238\) 46.9607 3.04401
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) −14.7038 −0.947156 −0.473578 0.880752i \(-0.657038\pi\)
−0.473578 + 0.880752i \(0.657038\pi\)
\(242\) 23.5905 1.51646
\(243\) 0 0
\(244\) −9.89534 −0.633484
\(245\) 2.24494 0.143424
\(246\) 0 0
\(247\) 4.50492 0.286641
\(248\) −17.8654 −1.13446
\(249\) 0 0
\(250\) −12.6161 −0.797914
\(251\) −12.3028 −0.776546 −0.388273 0.921544i \(-0.626928\pi\)
−0.388273 + 0.921544i \(0.626928\pi\)
\(252\) 0 0
\(253\) −6.59479 −0.414611
\(254\) −4.65143 −0.291856
\(255\) 0 0
\(256\) −30.4313 −1.90196
\(257\) −30.2278 −1.88556 −0.942779 0.333418i \(-0.891798\pi\)
−0.942779 + 0.333418i \(0.891798\pi\)
\(258\) 0 0
\(259\) −15.4884 −0.962403
\(260\) 10.7250 0.665135
\(261\) 0 0
\(262\) 4.35202 0.268869
\(263\) −14.7836 −0.911595 −0.455797 0.890084i \(-0.650646\pi\)
−0.455797 + 0.890084i \(0.650646\pi\)
\(264\) 0 0
\(265\) −1.99318 −0.122440
\(266\) −27.9768 −1.71536
\(267\) 0 0
\(268\) 23.8003 1.45383
\(269\) 15.3623 0.936658 0.468329 0.883554i \(-0.344856\pi\)
0.468329 + 0.883554i \(0.344856\pi\)
\(270\) 0 0
\(271\) −8.71861 −0.529618 −0.264809 0.964301i \(-0.585309\pi\)
−0.264809 + 0.964301i \(0.585309\pi\)
\(272\) 24.7262 1.49925
\(273\) 0 0
\(274\) −22.5340 −1.36133
\(275\) 3.55892 0.214611
\(276\) 0 0
\(277\) −7.47755 −0.449282 −0.224641 0.974442i \(-0.572121\pi\)
−0.224641 + 0.974442i \(0.572121\pi\)
\(278\) 53.0500 3.18173
\(279\) 0 0
\(280\) −32.0913 −1.91782
\(281\) −19.3396 −1.15371 −0.576853 0.816848i \(-0.695720\pi\)
−0.576853 + 0.816848i \(0.695720\pi\)
\(282\) 0 0
\(283\) 8.53348 0.507263 0.253631 0.967301i \(-0.418375\pi\)
0.253631 + 0.967301i \(0.418375\pi\)
\(284\) 14.0506 0.833751
\(285\) 0 0
\(286\) 2.63463 0.155789
\(287\) 27.0642 1.59755
\(288\) 0 0
\(289\) 43.5566 2.56215
\(290\) 50.0167 2.93708
\(291\) 0 0
\(292\) −50.9144 −2.97954
\(293\) −16.6219 −0.971059 −0.485529 0.874220i \(-0.661373\pi\)
−0.485529 + 0.874220i \(0.661373\pi\)
\(294\) 0 0
\(295\) 21.6074 1.25803
\(296\) −27.9671 −1.62555
\(297\) 0 0
\(298\) 12.7436 0.738220
\(299\) 5.72141 0.330878
\(300\) 0 0
\(301\) −30.6737 −1.76800
\(302\) −29.0233 −1.67010
\(303\) 0 0
\(304\) −14.7306 −0.844858
\(305\) 7.33153 0.419802
\(306\) 0 0
\(307\) −16.5311 −0.943480 −0.471740 0.881738i \(-0.656374\pi\)
−0.471740 + 0.881738i \(0.656374\pi\)
\(308\) −10.7772 −0.614086
\(309\) 0 0
\(310\) 27.4725 1.56033
\(311\) −13.7367 −0.778939 −0.389470 0.921039i \(-0.627342\pi\)
−0.389470 + 0.921039i \(0.627342\pi\)
\(312\) 0 0
\(313\) 14.6071 0.825641 0.412821 0.910812i \(-0.364544\pi\)
0.412821 + 0.910812i \(0.364544\pi\)
\(314\) 29.1992 1.64780
\(315\) 0 0
\(316\) 47.8621 2.69245
\(317\) −8.04568 −0.451890 −0.225945 0.974140i \(-0.572547\pi\)
−0.225945 + 0.974140i \(0.572547\pi\)
\(318\) 0 0
\(319\) 8.09305 0.453124
\(320\) 27.2512 1.52339
\(321\) 0 0
\(322\) −35.5315 −1.98009
\(323\) −36.0765 −2.00735
\(324\) 0 0
\(325\) −3.08759 −0.171269
\(326\) 27.8752 1.54386
\(327\) 0 0
\(328\) 48.8692 2.69835
\(329\) −13.0548 −0.719735
\(330\) 0 0
\(331\) −22.2686 −1.22399 −0.611997 0.790860i \(-0.709634\pi\)
−0.611997 + 0.790860i \(0.709634\pi\)
\(332\) −20.4124 −1.12028
\(333\) 0 0
\(334\) 23.0400 1.26070
\(335\) −17.6338 −0.963438
\(336\) 0 0
\(337\) 31.6136 1.72210 0.861052 0.508517i \(-0.169806\pi\)
0.861052 + 0.508517i \(0.169806\pi\)
\(338\) 29.1829 1.58734
\(339\) 0 0
\(340\) −85.8883 −4.65795
\(341\) 4.44524 0.240723
\(342\) 0 0
\(343\) 19.4080 1.04793
\(344\) −55.3869 −2.98626
\(345\) 0 0
\(346\) −44.3738 −2.38555
\(347\) 10.8188 0.580784 0.290392 0.956908i \(-0.406214\pi\)
0.290392 + 0.956908i \(0.406214\pi\)
\(348\) 0 0
\(349\) −35.1766 −1.88296 −0.941479 0.337071i \(-0.890564\pi\)
−0.941479 + 0.337071i \(0.890564\pi\)
\(350\) 19.1748 1.02494
\(351\) 0 0
\(352\) 1.46899 0.0782977
\(353\) 17.1199 0.911203 0.455601 0.890184i \(-0.349424\pi\)
0.455601 + 0.890184i \(0.349424\pi\)
\(354\) 0 0
\(355\) −10.4102 −0.552517
\(356\) 4.64065 0.245954
\(357\) 0 0
\(358\) 22.9276 1.21176
\(359\) 11.2236 0.592361 0.296181 0.955132i \(-0.404287\pi\)
0.296181 + 0.955132i \(0.404287\pi\)
\(360\) 0 0
\(361\) 2.49251 0.131185
\(362\) 13.3687 0.702641
\(363\) 0 0
\(364\) 9.34990 0.490068
\(365\) 37.7228 1.97450
\(366\) 0 0
\(367\) 5.31570 0.277477 0.138739 0.990329i \(-0.455695\pi\)
0.138739 + 0.990329i \(0.455695\pi\)
\(368\) −18.7084 −0.975244
\(369\) 0 0
\(370\) 43.0062 2.23579
\(371\) −1.73763 −0.0902131
\(372\) 0 0
\(373\) 4.01475 0.207876 0.103938 0.994584i \(-0.466856\pi\)
0.103938 + 0.994584i \(0.466856\pi\)
\(374\) −21.0987 −1.09099
\(375\) 0 0
\(376\) −23.5728 −1.21567
\(377\) −7.02126 −0.361613
\(378\) 0 0
\(379\) 20.0669 1.03077 0.515383 0.856960i \(-0.327650\pi\)
0.515383 + 0.856960i \(0.327650\pi\)
\(380\) 51.1678 2.62485
\(381\) 0 0
\(382\) −29.6208 −1.51553
\(383\) −26.2363 −1.34061 −0.670306 0.742085i \(-0.733837\pi\)
−0.670306 + 0.742085i \(0.733837\pi\)
\(384\) 0 0
\(385\) 7.98488 0.406947
\(386\) 8.64837 0.440191
\(387\) 0 0
\(388\) −13.4846 −0.684576
\(389\) −13.1624 −0.667360 −0.333680 0.942686i \(-0.608291\pi\)
−0.333680 + 0.942686i \(0.608291\pi\)
\(390\) 0 0
\(391\) −45.8185 −2.31714
\(392\) 3.53390 0.178489
\(393\) 0 0
\(394\) 36.2227 1.82487
\(395\) −35.4614 −1.78426
\(396\) 0 0
\(397\) −29.5087 −1.48100 −0.740500 0.672056i \(-0.765411\pi\)
−0.740500 + 0.672056i \(0.765411\pi\)
\(398\) 30.2297 1.51528
\(399\) 0 0
\(400\) 10.0961 0.504805
\(401\) −27.1474 −1.35568 −0.677839 0.735211i \(-0.737083\pi\)
−0.677839 + 0.735211i \(0.737083\pi\)
\(402\) 0 0
\(403\) −3.85654 −0.192108
\(404\) −57.0244 −2.83707
\(405\) 0 0
\(406\) 43.6039 2.16403
\(407\) 6.95871 0.344931
\(408\) 0 0
\(409\) −14.8766 −0.735599 −0.367799 0.929905i \(-0.619889\pi\)
−0.367799 + 0.929905i \(0.619889\pi\)
\(410\) −75.1483 −3.71131
\(411\) 0 0
\(412\) 60.0003 2.95600
\(413\) 18.8371 0.926912
\(414\) 0 0
\(415\) 15.1237 0.742394
\(416\) −1.27445 −0.0624850
\(417\) 0 0
\(418\) 12.5695 0.614796
\(419\) 21.1669 1.03407 0.517035 0.855964i \(-0.327036\pi\)
0.517035 + 0.855964i \(0.327036\pi\)
\(420\) 0 0
\(421\) −8.12081 −0.395784 −0.197892 0.980224i \(-0.563410\pi\)
−0.197892 + 0.980224i \(0.563410\pi\)
\(422\) 21.6695 1.05485
\(423\) 0 0
\(424\) −3.13760 −0.152375
\(425\) 24.7262 1.19940
\(426\) 0 0
\(427\) 6.39153 0.309308
\(428\) 17.8704 0.863796
\(429\) 0 0
\(430\) 85.1708 4.10730
\(431\) 16.4514 0.792438 0.396219 0.918156i \(-0.370322\pi\)
0.396219 + 0.918156i \(0.370322\pi\)
\(432\) 0 0
\(433\) 15.7856 0.758606 0.379303 0.925273i \(-0.376164\pi\)
0.379303 + 0.925273i \(0.376164\pi\)
\(434\) 23.9501 1.14964
\(435\) 0 0
\(436\) −17.2150 −0.824451
\(437\) 27.2963 1.30576
\(438\) 0 0
\(439\) 0.441975 0.0210943 0.0105472 0.999944i \(-0.496643\pi\)
0.0105472 + 0.999944i \(0.496643\pi\)
\(440\) 14.4181 0.687357
\(441\) 0 0
\(442\) 18.3045 0.870658
\(443\) −34.6291 −1.64528 −0.822639 0.568564i \(-0.807499\pi\)
−0.822639 + 0.568564i \(0.807499\pi\)
\(444\) 0 0
\(445\) −3.43829 −0.162991
\(446\) 33.0464 1.56479
\(447\) 0 0
\(448\) 23.7573 1.12242
\(449\) −27.3140 −1.28903 −0.644513 0.764593i \(-0.722940\pi\)
−0.644513 + 0.764593i \(0.722940\pi\)
\(450\) 0 0
\(451\) −12.1595 −0.572570
\(452\) −4.72079 −0.222047
\(453\) 0 0
\(454\) 5.57899 0.261835
\(455\) −6.92741 −0.324762
\(456\) 0 0
\(457\) −22.4901 −1.05204 −0.526021 0.850472i \(-0.676317\pi\)
−0.526021 + 0.850472i \(0.676317\pi\)
\(458\) −51.1488 −2.39003
\(459\) 0 0
\(460\) 64.9851 3.02994
\(461\) 34.2275 1.59413 0.797067 0.603891i \(-0.206384\pi\)
0.797067 + 0.603891i \(0.206384\pi\)
\(462\) 0 0
\(463\) −20.5949 −0.957125 −0.478563 0.878053i \(-0.658842\pi\)
−0.478563 + 0.878053i \(0.658842\pi\)
\(464\) 22.9588 1.06583
\(465\) 0 0
\(466\) 22.0636 1.02208
\(467\) 18.8946 0.874338 0.437169 0.899379i \(-0.355981\pi\)
0.437169 + 0.899379i \(0.355981\pi\)
\(468\) 0 0
\(469\) −15.3729 −0.709856
\(470\) 36.2489 1.67204
\(471\) 0 0
\(472\) 34.0137 1.56561
\(473\) 13.7812 0.633662
\(474\) 0 0
\(475\) −14.7306 −0.675886
\(476\) −74.8763 −3.43195
\(477\) 0 0
\(478\) −2.42067 −0.110719
\(479\) 33.5961 1.53504 0.767522 0.641023i \(-0.221490\pi\)
0.767522 + 0.641023i \(0.221490\pi\)
\(480\) 0 0
\(481\) −6.03714 −0.275270
\(482\) 35.5930 1.62122
\(483\) 0 0
\(484\) −37.6138 −1.70972
\(485\) 9.99082 0.453660
\(486\) 0 0
\(487\) 21.7272 0.984553 0.492276 0.870439i \(-0.336165\pi\)
0.492276 + 0.870439i \(0.336165\pi\)
\(488\) 11.5411 0.522439
\(489\) 0 0
\(490\) −5.43424 −0.245494
\(491\) 19.6300 0.885891 0.442946 0.896548i \(-0.353933\pi\)
0.442946 + 0.896548i \(0.353933\pi\)
\(492\) 0 0
\(493\) 56.2279 2.53238
\(494\) −10.9049 −0.490635
\(495\) 0 0
\(496\) 12.6105 0.566227
\(497\) −9.07549 −0.407091
\(498\) 0 0
\(499\) 27.2445 1.21963 0.609816 0.792543i \(-0.291243\pi\)
0.609816 + 0.792543i \(0.291243\pi\)
\(500\) 20.1157 0.899603
\(501\) 0 0
\(502\) 29.7810 1.32919
\(503\) −11.8818 −0.529782 −0.264891 0.964278i \(-0.585336\pi\)
−0.264891 + 0.964278i \(0.585336\pi\)
\(504\) 0 0
\(505\) 42.2498 1.88009
\(506\) 15.9638 0.709677
\(507\) 0 0
\(508\) 7.41645 0.329052
\(509\) 33.3751 1.47933 0.739663 0.672977i \(-0.234985\pi\)
0.739663 + 0.672977i \(0.234985\pi\)
\(510\) 0 0
\(511\) 32.8863 1.45480
\(512\) 32.7739 1.44842
\(513\) 0 0
\(514\) 73.1714 3.22745
\(515\) −44.4546 −1.95891
\(516\) 0 0
\(517\) 5.86533 0.257957
\(518\) 37.4923 1.64732
\(519\) 0 0
\(520\) −12.5087 −0.548542
\(521\) −23.8798 −1.04619 −0.523097 0.852273i \(-0.675223\pi\)
−0.523097 + 0.852273i \(0.675223\pi\)
\(522\) 0 0
\(523\) 43.3692 1.89640 0.948202 0.317669i \(-0.102900\pi\)
0.948202 + 0.317669i \(0.102900\pi\)
\(524\) −6.93906 −0.303134
\(525\) 0 0
\(526\) 35.7861 1.56035
\(527\) 30.8841 1.34533
\(528\) 0 0
\(529\) 11.6673 0.507275
\(530\) 4.82482 0.209577
\(531\) 0 0
\(532\) 44.6074 1.93398
\(533\) 10.5492 0.456936
\(534\) 0 0
\(535\) −13.2403 −0.572427
\(536\) −27.7586 −1.19899
\(537\) 0 0
\(538\) −37.1871 −1.60325
\(539\) −0.879298 −0.0378740
\(540\) 0 0
\(541\) 44.3614 1.90724 0.953622 0.301008i \(-0.0973230\pi\)
0.953622 + 0.301008i \(0.0973230\pi\)
\(542\) 21.1048 0.906530
\(543\) 0 0
\(544\) 10.2061 0.437583
\(545\) 12.7547 0.546353
\(546\) 0 0
\(547\) −17.4733 −0.747103 −0.373551 0.927610i \(-0.621860\pi\)
−0.373551 + 0.927610i \(0.621860\pi\)
\(548\) 35.9292 1.53482
\(549\) 0 0
\(550\) −8.61494 −0.367342
\(551\) −33.4977 −1.42705
\(552\) 0 0
\(553\) −30.9148 −1.31463
\(554\) 18.1006 0.769022
\(555\) 0 0
\(556\) −84.5854 −3.58722
\(557\) 26.1559 1.10826 0.554130 0.832430i \(-0.313051\pi\)
0.554130 + 0.832430i \(0.313051\pi\)
\(558\) 0 0
\(559\) −11.9561 −0.505691
\(560\) 22.6519 0.957217
\(561\) 0 0
\(562\) 46.8148 1.97476
\(563\) 39.7284 1.67435 0.837176 0.546933i \(-0.184205\pi\)
0.837176 + 0.546933i \(0.184205\pi\)
\(564\) 0 0
\(565\) 3.49767 0.147148
\(566\) −20.6567 −0.868266
\(567\) 0 0
\(568\) −16.3874 −0.687601
\(569\) −2.15075 −0.0901640 −0.0450820 0.998983i \(-0.514355\pi\)
−0.0450820 + 0.998983i \(0.514355\pi\)
\(570\) 0 0
\(571\) −17.3630 −0.726619 −0.363310 0.931669i \(-0.618353\pi\)
−0.363310 + 0.931669i \(0.618353\pi\)
\(572\) −4.20077 −0.175643
\(573\) 0 0
\(574\) −65.5133 −2.73447
\(575\) −18.7084 −0.780195
\(576\) 0 0
\(577\) −22.1707 −0.922977 −0.461489 0.887146i \(-0.652684\pi\)
−0.461489 + 0.887146i \(0.652684\pi\)
\(578\) −105.436 −4.38555
\(579\) 0 0
\(580\) −79.7490 −3.31139
\(581\) 13.1847 0.546992
\(582\) 0 0
\(583\) 0.780690 0.0323329
\(584\) 59.3820 2.45725
\(585\) 0 0
\(586\) 40.2359 1.66213
\(587\) −2.43336 −0.100436 −0.0502178 0.998738i \(-0.515992\pi\)
−0.0502178 + 0.998738i \(0.515992\pi\)
\(588\) 0 0
\(589\) −18.3991 −0.758124
\(590\) −52.3044 −2.15334
\(591\) 0 0
\(592\) 19.7408 0.811342
\(593\) 15.5820 0.639874 0.319937 0.947439i \(-0.396338\pi\)
0.319937 + 0.947439i \(0.396338\pi\)
\(594\) 0 0
\(595\) 55.4764 2.27431
\(596\) −20.3191 −0.832301
\(597\) 0 0
\(598\) −13.8496 −0.566354
\(599\) −13.6123 −0.556181 −0.278091 0.960555i \(-0.589702\pi\)
−0.278091 + 0.960555i \(0.589702\pi\)
\(600\) 0 0
\(601\) −4.30181 −0.175475 −0.0877373 0.996144i \(-0.527964\pi\)
−0.0877373 + 0.996144i \(0.527964\pi\)
\(602\) 74.2508 3.02624
\(603\) 0 0
\(604\) 46.2761 1.88295
\(605\) 27.8684 1.13301
\(606\) 0 0
\(607\) 11.1853 0.453997 0.226999 0.973895i \(-0.427109\pi\)
0.226999 + 0.973895i \(0.427109\pi\)
\(608\) −6.08027 −0.246588
\(609\) 0 0
\(610\) −17.7472 −0.718562
\(611\) −5.08856 −0.205861
\(612\) 0 0
\(613\) 40.7783 1.64702 0.823511 0.567301i \(-0.192012\pi\)
0.823511 + 0.567301i \(0.192012\pi\)
\(614\) 40.0163 1.61492
\(615\) 0 0
\(616\) 12.5695 0.506441
\(617\) −42.0334 −1.69220 −0.846100 0.533024i \(-0.821056\pi\)
−0.846100 + 0.533024i \(0.821056\pi\)
\(618\) 0 0
\(619\) −11.0986 −0.446090 −0.223045 0.974808i \(-0.571600\pi\)
−0.223045 + 0.974808i \(0.571600\pi\)
\(620\) −43.8034 −1.75919
\(621\) 0 0
\(622\) 33.2521 1.33329
\(623\) −2.99746 −0.120091
\(624\) 0 0
\(625\) −30.7911 −1.23164
\(626\) −35.3589 −1.41322
\(627\) 0 0
\(628\) −46.5565 −1.85781
\(629\) 48.3469 1.92772
\(630\) 0 0
\(631\) −26.2393 −1.04457 −0.522285 0.852771i \(-0.674920\pi\)
−0.522285 + 0.852771i \(0.674920\pi\)
\(632\) −55.8222 −2.22049
\(633\) 0 0
\(634\) 19.4759 0.773486
\(635\) −5.49490 −0.218059
\(636\) 0 0
\(637\) 0.762849 0.0302252
\(638\) −19.5906 −0.775598
\(639\) 0 0
\(640\) −58.4651 −2.31104
\(641\) −12.1081 −0.478240 −0.239120 0.970990i \(-0.576859\pi\)
−0.239120 + 0.970990i \(0.576859\pi\)
\(642\) 0 0
\(643\) 26.6604 1.05138 0.525692 0.850675i \(-0.323806\pi\)
0.525692 + 0.850675i \(0.323806\pi\)
\(644\) 56.6531 2.23245
\(645\) 0 0
\(646\) 87.3291 3.43592
\(647\) 29.7009 1.16766 0.583832 0.811875i \(-0.301553\pi\)
0.583832 + 0.811875i \(0.301553\pi\)
\(648\) 0 0
\(649\) −8.46321 −0.332210
\(650\) 7.47403 0.293156
\(651\) 0 0
\(652\) −44.4455 −1.74062
\(653\) 9.28628 0.363400 0.181700 0.983354i \(-0.441840\pi\)
0.181700 + 0.983354i \(0.441840\pi\)
\(654\) 0 0
\(655\) 5.14120 0.200883
\(656\) −34.4947 −1.34679
\(657\) 0 0
\(658\) 31.6013 1.23195
\(659\) −16.1495 −0.629094 −0.314547 0.949242i \(-0.601853\pi\)
−0.314547 + 0.949242i \(0.601853\pi\)
\(660\) 0 0
\(661\) −31.2682 −1.21619 −0.608097 0.793863i \(-0.708067\pi\)
−0.608097 + 0.793863i \(0.708067\pi\)
\(662\) 53.9049 2.09507
\(663\) 0 0
\(664\) 23.8073 0.923901
\(665\) −33.0500 −1.28162
\(666\) 0 0
\(667\) −42.5434 −1.64729
\(668\) −36.7361 −1.42136
\(669\) 0 0
\(670\) 42.6856 1.64909
\(671\) −2.87162 −0.110858
\(672\) 0 0
\(673\) −18.6955 −0.720657 −0.360329 0.932825i \(-0.617335\pi\)
−0.360329 + 0.932825i \(0.617335\pi\)
\(674\) −76.5260 −2.94767
\(675\) 0 0
\(676\) −46.5306 −1.78964
\(677\) −24.3890 −0.937346 −0.468673 0.883372i \(-0.655268\pi\)
−0.468673 + 0.883372i \(0.655268\pi\)
\(678\) 0 0
\(679\) 8.70987 0.334254
\(680\) 100.173 3.84144
\(681\) 0 0
\(682\) −10.7604 −0.412038
\(683\) −44.2443 −1.69296 −0.846481 0.532419i \(-0.821283\pi\)
−0.846481 + 0.532419i \(0.821283\pi\)
\(684\) 0 0
\(685\) −26.6202 −1.01711
\(686\) −46.9802 −1.79371
\(687\) 0 0
\(688\) 39.0953 1.49049
\(689\) −0.677300 −0.0258031
\(690\) 0 0
\(691\) 3.11643 0.118555 0.0592773 0.998242i \(-0.481120\pi\)
0.0592773 + 0.998242i \(0.481120\pi\)
\(692\) 70.7516 2.68957
\(693\) 0 0
\(694\) −26.1887 −0.994110
\(695\) 62.6699 2.37721
\(696\) 0 0
\(697\) −84.4805 −3.19993
\(698\) 85.1507 3.22300
\(699\) 0 0
\(700\) −30.5732 −1.15556
\(701\) 26.4851 1.00033 0.500164 0.865931i \(-0.333273\pi\)
0.500164 + 0.865931i \(0.333273\pi\)
\(702\) 0 0
\(703\) −28.8026 −1.08631
\(704\) −10.6738 −0.402283
\(705\) 0 0
\(706\) −41.4417 −1.55968
\(707\) 36.8328 1.38524
\(708\) 0 0
\(709\) 19.0210 0.714350 0.357175 0.934038i \(-0.383740\pi\)
0.357175 + 0.934038i \(0.383740\pi\)
\(710\) 25.1996 0.945726
\(711\) 0 0
\(712\) −5.41244 −0.202840
\(713\) −23.3676 −0.875124
\(714\) 0 0
\(715\) 3.11238 0.116396
\(716\) −36.5569 −1.36619
\(717\) 0 0
\(718\) −27.1687 −1.01393
\(719\) 31.3712 1.16995 0.584974 0.811052i \(-0.301105\pi\)
0.584974 + 0.811052i \(0.301105\pi\)
\(720\) 0 0
\(721\) −38.7550 −1.44331
\(722\) −6.03353 −0.224545
\(723\) 0 0
\(724\) −21.3156 −0.792189
\(725\) 22.9588 0.852667
\(726\) 0 0
\(727\) −43.3958 −1.60946 −0.804730 0.593641i \(-0.797690\pi\)
−0.804730 + 0.593641i \(0.797690\pi\)
\(728\) −10.9049 −0.404162
\(729\) 0 0
\(730\) −91.3144 −3.37970
\(731\) 95.7476 3.54135
\(732\) 0 0
\(733\) −16.8576 −0.622649 −0.311325 0.950304i \(-0.600773\pi\)
−0.311325 + 0.950304i \(0.600773\pi\)
\(734\) −12.8675 −0.474949
\(735\) 0 0
\(736\) −7.72217 −0.284643
\(737\) 6.90682 0.254416
\(738\) 0 0
\(739\) 50.0747 1.84203 0.921013 0.389531i \(-0.127363\pi\)
0.921013 + 0.389531i \(0.127363\pi\)
\(740\) −68.5711 −2.52073
\(741\) 0 0
\(742\) 4.20621 0.154415
\(743\) 43.8762 1.60966 0.804831 0.593504i \(-0.202256\pi\)
0.804831 + 0.593504i \(0.202256\pi\)
\(744\) 0 0
\(745\) 15.0545 0.551556
\(746\) −9.71837 −0.355815
\(747\) 0 0
\(748\) 33.6408 1.23003
\(749\) −11.5427 −0.421761
\(750\) 0 0
\(751\) −24.6107 −0.898059 −0.449029 0.893517i \(-0.648230\pi\)
−0.449029 + 0.893517i \(0.648230\pi\)
\(752\) 16.6391 0.606764
\(753\) 0 0
\(754\) 16.9961 0.618962
\(755\) −34.2863 −1.24781
\(756\) 0 0
\(757\) −47.2113 −1.71593 −0.857963 0.513712i \(-0.828270\pi\)
−0.857963 + 0.513712i \(0.828270\pi\)
\(758\) −48.5752 −1.76433
\(759\) 0 0
\(760\) −59.6776 −2.16474
\(761\) −42.8822 −1.55448 −0.777239 0.629206i \(-0.783380\pi\)
−0.777239 + 0.629206i \(0.783380\pi\)
\(762\) 0 0
\(763\) 11.1194 0.402550
\(764\) 47.2288 1.70868
\(765\) 0 0
\(766\) 63.5093 2.29469
\(767\) 7.34239 0.265118
\(768\) 0 0
\(769\) −35.8695 −1.29349 −0.646744 0.762707i \(-0.723870\pi\)
−0.646744 + 0.762707i \(0.723870\pi\)
\(770\) −19.3287 −0.696559
\(771\) 0 0
\(772\) −13.7894 −0.496290
\(773\) 5.98233 0.215170 0.107585 0.994196i \(-0.465688\pi\)
0.107585 + 0.994196i \(0.465688\pi\)
\(774\) 0 0
\(775\) 12.6105 0.452981
\(776\) 15.7272 0.564574
\(777\) 0 0
\(778\) 31.8618 1.14230
\(779\) 50.3291 1.80323
\(780\) 0 0
\(781\) 4.07748 0.145904
\(782\) 110.911 3.96618
\(783\) 0 0
\(784\) −2.49443 −0.0890870
\(785\) 34.4940 1.23115
\(786\) 0 0
\(787\) 35.5595 1.26756 0.633780 0.773513i \(-0.281502\pi\)
0.633780 + 0.773513i \(0.281502\pi\)
\(788\) −57.7552 −2.05744
\(789\) 0 0
\(790\) 85.8402 3.05406
\(791\) 3.04922 0.108418
\(792\) 0 0
\(793\) 2.49132 0.0884693
\(794\) 71.4307 2.53498
\(795\) 0 0
\(796\) −48.1996 −1.70839
\(797\) 13.1917 0.467273 0.233636 0.972324i \(-0.424937\pi\)
0.233636 + 0.972324i \(0.424937\pi\)
\(798\) 0 0
\(799\) 40.7505 1.44165
\(800\) 4.16731 0.147337
\(801\) 0 0
\(802\) 65.7148 2.32047
\(803\) −14.7753 −0.521409
\(804\) 0 0
\(805\) −41.9747 −1.47941
\(806\) 9.33538 0.328825
\(807\) 0 0
\(808\) 66.5082 2.33975
\(809\) 27.1698 0.955238 0.477619 0.878567i \(-0.341500\pi\)
0.477619 + 0.878567i \(0.341500\pi\)
\(810\) 0 0
\(811\) −10.2045 −0.358329 −0.179165 0.983819i \(-0.557339\pi\)
−0.179165 + 0.983819i \(0.557339\pi\)
\(812\) −69.5241 −2.43982
\(813\) 0 0
\(814\) −16.8447 −0.590407
\(815\) 32.9300 1.15349
\(816\) 0 0
\(817\) −57.0415 −1.99563
\(818\) 36.0112 1.25910
\(819\) 0 0
\(820\) 119.820 4.18430
\(821\) −33.9031 −1.18323 −0.591613 0.806222i \(-0.701509\pi\)
−0.591613 + 0.806222i \(0.701509\pi\)
\(822\) 0 0
\(823\) 8.71662 0.303842 0.151921 0.988393i \(-0.451454\pi\)
0.151921 + 0.988393i \(0.451454\pi\)
\(824\) −69.9790 −2.43783
\(825\) 0 0
\(826\) −45.5983 −1.58657
\(827\) 4.52090 0.157207 0.0786036 0.996906i \(-0.474954\pi\)
0.0786036 + 0.996906i \(0.474954\pi\)
\(828\) 0 0
\(829\) −14.1324 −0.490839 −0.245419 0.969417i \(-0.578926\pi\)
−0.245419 + 0.969417i \(0.578926\pi\)
\(830\) −36.6095 −1.27073
\(831\) 0 0
\(832\) 9.26020 0.321040
\(833\) −6.10908 −0.211667
\(834\) 0 0
\(835\) 27.2181 0.941920
\(836\) −20.0414 −0.693148
\(837\) 0 0
\(838\) −51.2379 −1.76999
\(839\) −27.3228 −0.943286 −0.471643 0.881789i \(-0.656339\pi\)
−0.471643 + 0.881789i \(0.656339\pi\)
\(840\) 0 0
\(841\) 23.2087 0.800302
\(842\) 19.6578 0.677452
\(843\) 0 0
\(844\) −34.5508 −1.18929
\(845\) 34.4749 1.18597
\(846\) 0 0
\(847\) 24.2953 0.834795
\(848\) 2.21470 0.0760531
\(849\) 0 0
\(850\) −59.8539 −2.05297
\(851\) −36.5804 −1.25396
\(852\) 0 0
\(853\) −28.6150 −0.979758 −0.489879 0.871791i \(-0.662959\pi\)
−0.489879 + 0.871791i \(0.662959\pi\)
\(854\) −15.4718 −0.529433
\(855\) 0 0
\(856\) −20.8424 −0.712379
\(857\) 25.9084 0.885015 0.442507 0.896765i \(-0.354089\pi\)
0.442507 + 0.896765i \(0.354089\pi\)
\(858\) 0 0
\(859\) 24.3172 0.829693 0.414847 0.909891i \(-0.363835\pi\)
0.414847 + 0.909891i \(0.363835\pi\)
\(860\) −135.800 −4.63075
\(861\) 0 0
\(862\) −39.8234 −1.35639
\(863\) 38.4808 1.30990 0.654952 0.755671i \(-0.272689\pi\)
0.654952 + 0.755671i \(0.272689\pi\)
\(864\) 0 0
\(865\) −52.4204 −1.78235
\(866\) −38.2116 −1.29848
\(867\) 0 0
\(868\) −38.1872 −1.29616
\(869\) 13.8895 0.471171
\(870\) 0 0
\(871\) −5.99212 −0.203035
\(872\) 20.0781 0.679930
\(873\) 0 0
\(874\) −66.0752 −2.23503
\(875\) −12.9930 −0.439244
\(876\) 0 0
\(877\) 6.81200 0.230025 0.115013 0.993364i \(-0.463309\pi\)
0.115013 + 0.993364i \(0.463309\pi\)
\(878\) −1.06987 −0.0361065
\(879\) 0 0
\(880\) −10.1772 −0.343072
\(881\) −51.1946 −1.72479 −0.862395 0.506235i \(-0.831037\pi\)
−0.862395 + 0.506235i \(0.831037\pi\)
\(882\) 0 0
\(883\) 23.9388 0.805606 0.402803 0.915287i \(-0.368036\pi\)
0.402803 + 0.915287i \(0.368036\pi\)
\(884\) −29.1856 −0.981618
\(885\) 0 0
\(886\) 83.8255 2.81617
\(887\) −22.9111 −0.769278 −0.384639 0.923067i \(-0.625674\pi\)
−0.384639 + 0.923067i \(0.625674\pi\)
\(888\) 0 0
\(889\) −4.79038 −0.160664
\(890\) 8.32295 0.278986
\(891\) 0 0
\(892\) −52.6907 −1.76421
\(893\) −24.2770 −0.812400
\(894\) 0 0
\(895\) 27.0853 0.905360
\(896\) −50.9691 −1.70276
\(897\) 0 0
\(898\) 66.1180 2.20639
\(899\) 28.6765 0.956414
\(900\) 0 0
\(901\) 5.42398 0.180699
\(902\) 29.4342 0.980050
\(903\) 0 0
\(904\) 5.50592 0.183124
\(905\) 15.7929 0.524974
\(906\) 0 0
\(907\) 30.2708 1.00512 0.502562 0.864541i \(-0.332391\pi\)
0.502562 + 0.864541i \(0.332391\pi\)
\(908\) −8.89540 −0.295204
\(909\) 0 0
\(910\) 16.7689 0.555885
\(911\) −49.9174 −1.65384 −0.826919 0.562322i \(-0.809908\pi\)
−0.826919 + 0.562322i \(0.809908\pi\)
\(912\) 0 0
\(913\) −5.92367 −0.196045
\(914\) 54.4409 1.80075
\(915\) 0 0
\(916\) 81.5540 2.69462
\(917\) 4.48203 0.148010
\(918\) 0 0
\(919\) −15.2909 −0.504400 −0.252200 0.967675i \(-0.581154\pi\)
−0.252200 + 0.967675i \(0.581154\pi\)
\(920\) −75.7928 −2.49882
\(921\) 0 0
\(922\) −82.8533 −2.72863
\(923\) −3.53748 −0.116438
\(924\) 0 0
\(925\) 19.7408 0.649074
\(926\) 49.8533 1.63828
\(927\) 0 0
\(928\) 9.47657 0.311084
\(929\) −18.9994 −0.623349 −0.311674 0.950189i \(-0.600890\pi\)
−0.311674 + 0.950189i \(0.600890\pi\)
\(930\) 0 0
\(931\) 3.63948 0.119279
\(932\) −35.1792 −1.15233
\(933\) 0 0
\(934\) −45.7375 −1.49658
\(935\) −24.9247 −0.815125
\(936\) 0 0
\(937\) −15.8277 −0.517068 −0.258534 0.966002i \(-0.583239\pi\)
−0.258534 + 0.966002i \(0.583239\pi\)
\(938\) 37.2127 1.21504
\(939\) 0 0
\(940\) −57.7970 −1.88513
\(941\) 48.8260 1.59168 0.795840 0.605506i \(-0.207029\pi\)
0.795840 + 0.605506i \(0.207029\pi\)
\(942\) 0 0
\(943\) 63.9199 2.08152
\(944\) −24.0089 −0.781422
\(945\) 0 0
\(946\) −33.3598 −1.08462
\(947\) −44.8868 −1.45862 −0.729312 0.684181i \(-0.760160\pi\)
−0.729312 + 0.684181i \(0.760160\pi\)
\(948\) 0 0
\(949\) 12.8186 0.416108
\(950\) 35.6579 1.15689
\(951\) 0 0
\(952\) 87.3291 2.83035
\(953\) −37.1889 −1.20467 −0.602334 0.798244i \(-0.705762\pi\)
−0.602334 + 0.798244i \(0.705762\pi\)
\(954\) 0 0
\(955\) −34.9921 −1.13232
\(956\) 3.85962 0.124829
\(957\) 0 0
\(958\) −81.3249 −2.62749
\(959\) −23.2072 −0.749399
\(960\) 0 0
\(961\) −15.2490 −0.491903
\(962\) 14.6139 0.471171
\(963\) 0 0
\(964\) −56.7511 −1.82783
\(965\) 10.2166 0.328885
\(966\) 0 0
\(967\) 6.40043 0.205824 0.102912 0.994690i \(-0.467184\pi\)
0.102912 + 0.994690i \(0.467184\pi\)
\(968\) 43.8694 1.41002
\(969\) 0 0
\(970\) −24.1844 −0.776515
\(971\) −1.23491 −0.0396301 −0.0198150 0.999804i \(-0.506308\pi\)
−0.0198150 + 0.999804i \(0.506308\pi\)
\(972\) 0 0
\(973\) 54.6348 1.75151
\(974\) −52.5943 −1.68523
\(975\) 0 0
\(976\) −8.14635 −0.260758
\(977\) 5.05273 0.161651 0.0808256 0.996728i \(-0.474244\pi\)
0.0808256 + 0.996728i \(0.474244\pi\)
\(978\) 0 0
\(979\) 1.34671 0.0430411
\(980\) 8.66460 0.276781
\(981\) 0 0
\(982\) −47.5178 −1.51635
\(983\) 60.0002 1.91371 0.956855 0.290566i \(-0.0938436\pi\)
0.956855 + 0.290566i \(0.0938436\pi\)
\(984\) 0 0
\(985\) 42.7912 1.36344
\(986\) −136.109 −4.33460
\(987\) 0 0
\(988\) 17.3873 0.553163
\(989\) −72.4449 −2.30361
\(990\) 0 0
\(991\) 5.43534 0.172659 0.0863296 0.996267i \(-0.472486\pi\)
0.0863296 + 0.996267i \(0.472486\pi\)
\(992\) 5.20515 0.165264
\(993\) 0 0
\(994\) 21.9687 0.696805
\(995\) 35.7114 1.13213
\(996\) 0 0
\(997\) −33.7441 −1.06869 −0.534343 0.845268i \(-0.679441\pi\)
−0.534343 + 0.845268i \(0.679441\pi\)
\(998\) −65.9498 −2.08760
\(999\) 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 2151.2.a.f.1.1 7
3.2 odd 2 717.2.a.e.1.7 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
717.2.a.e.1.7 7 3.2 odd 2
2151.2.a.f.1.1 7 1.1 even 1 trivial