Properties

Label 5225.2.a.r.1.2
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $15$
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] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.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: \(41.7218350561\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 5 x^{14} - 11 x^{13} + 87 x^{12} - 4 x^{11} - 545 x^{10} + 431 x^{9} + 1480 x^{8} - 1763 x^{7} + \cdots + 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.61792\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.61792 q^{2} -2.17802 q^{3} +4.85352 q^{4} +5.70188 q^{6} -4.94832 q^{7} -7.47030 q^{8} +1.74376 q^{9} +O(q^{10})\) \(q-2.61792 q^{2} -2.17802 q^{3} +4.85352 q^{4} +5.70188 q^{6} -4.94832 q^{7} -7.47030 q^{8} +1.74376 q^{9} +1.00000 q^{11} -10.5710 q^{12} +4.20557 q^{13} +12.9543 q^{14} +9.84962 q^{16} +6.38851 q^{17} -4.56502 q^{18} +1.00000 q^{19} +10.7775 q^{21} -2.61792 q^{22} -6.66227 q^{23} +16.2704 q^{24} -11.0099 q^{26} +2.73612 q^{27} -24.0168 q^{28} -6.65643 q^{29} -9.53855 q^{31} -10.8449 q^{32} -2.17802 q^{33} -16.7246 q^{34} +8.46336 q^{36} +3.69327 q^{37} -2.61792 q^{38} -9.15980 q^{39} -1.08046 q^{41} -28.2147 q^{42} -9.58018 q^{43} +4.85352 q^{44} +17.4413 q^{46} -2.90169 q^{47} -21.4526 q^{48} +17.4858 q^{49} -13.9143 q^{51} +20.4118 q^{52} +6.75151 q^{53} -7.16294 q^{54} +36.9654 q^{56} -2.17802 q^{57} +17.4260 q^{58} -0.613007 q^{59} +1.33665 q^{61} +24.9712 q^{62} -8.62866 q^{63} +8.69200 q^{64} +5.70188 q^{66} -0.386619 q^{67} +31.0067 q^{68} +14.5105 q^{69} +9.38483 q^{71} -13.0264 q^{72} +1.97456 q^{73} -9.66869 q^{74} +4.85352 q^{76} -4.94832 q^{77} +23.9797 q^{78} -6.61494 q^{79} -11.1906 q^{81} +2.82857 q^{82} +0.100761 q^{83} +52.3089 q^{84} +25.0802 q^{86} +14.4978 q^{87} -7.47030 q^{88} +2.48738 q^{89} -20.8105 q^{91} -32.3355 q^{92} +20.7751 q^{93} +7.59641 q^{94} +23.6205 q^{96} +17.3288 q^{97} -45.7766 q^{98} +1.74376 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 21 q^{7} - 9 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 5 q^{2} - 4 q^{3} + 17 q^{4} - q^{6} - 21 q^{7} - 9 q^{8} + 15 q^{9} + 15 q^{11} - 11 q^{12} - 13 q^{13} + 9 q^{14} + 21 q^{16} - 17 q^{17} - 22 q^{18} + 15 q^{19} + 6 q^{21} - 5 q^{22} - 26 q^{23} + q^{24} + 3 q^{26} - q^{27} - 46 q^{28} + 9 q^{29} + 14 q^{31} - 18 q^{32} - 4 q^{33} - 13 q^{34} + 12 q^{36} - 9 q^{37} - 5 q^{38} - 22 q^{39} + 4 q^{41} + 6 q^{42} - 28 q^{43} + 17 q^{44} + 27 q^{46} - 14 q^{47} + 4 q^{48} + 32 q^{49} - 40 q^{51} - 14 q^{52} - 3 q^{53} - 39 q^{54} + 34 q^{56} - 4 q^{57} - 26 q^{58} + q^{59} + 2 q^{61} + 3 q^{62} - 45 q^{63} + 5 q^{64} - q^{66} - 37 q^{67} - 26 q^{68} - 7 q^{69} - 7 q^{71} - 16 q^{72} - 42 q^{73} - 43 q^{74} + 17 q^{76} - 21 q^{77} + 64 q^{78} - 10 q^{79} + 31 q^{81} - 22 q^{82} - 14 q^{83} - 32 q^{84} + 37 q^{86} - 29 q^{87} - 9 q^{88} + 15 q^{89} - 22 q^{91} - 26 q^{92} + 18 q^{93} - 44 q^{94} + 71 q^{96} - 8 q^{97} + 10 q^{98} + 15 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.61792 −1.85115 −0.925575 0.378563i \(-0.876418\pi\)
−0.925575 + 0.378563i \(0.876418\pi\)
\(3\) −2.17802 −1.25748 −0.628739 0.777616i \(-0.716429\pi\)
−0.628739 + 0.777616i \(0.716429\pi\)
\(4\) 4.85352 2.42676
\(5\) 0 0
\(6\) 5.70188 2.32778
\(7\) −4.94832 −1.87029 −0.935144 0.354268i \(-0.884730\pi\)
−0.935144 + 0.354268i \(0.884730\pi\)
\(8\) −7.47030 −2.64115
\(9\) 1.74376 0.581253
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −10.5710 −3.05160
\(13\) 4.20557 1.16642 0.583208 0.812323i \(-0.301797\pi\)
0.583208 + 0.812323i \(0.301797\pi\)
\(14\) 12.9543 3.46218
\(15\) 0 0
\(16\) 9.84962 2.46240
\(17\) 6.38851 1.54944 0.774720 0.632304i \(-0.217891\pi\)
0.774720 + 0.632304i \(0.217891\pi\)
\(18\) −4.56502 −1.07599
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 10.7775 2.35185
\(22\) −2.61792 −0.558143
\(23\) −6.66227 −1.38918 −0.694590 0.719406i \(-0.744414\pi\)
−0.694590 + 0.719406i \(0.744414\pi\)
\(24\) 16.2704 3.32119
\(25\) 0 0
\(26\) −11.0099 −2.15921
\(27\) 2.73612 0.526566
\(28\) −24.0168 −4.53874
\(29\) −6.65643 −1.23607 −0.618034 0.786151i \(-0.712071\pi\)
−0.618034 + 0.786151i \(0.712071\pi\)
\(30\) 0 0
\(31\) −9.53855 −1.71317 −0.856587 0.516002i \(-0.827420\pi\)
−0.856587 + 0.516002i \(0.827420\pi\)
\(32\) −10.8449 −1.91713
\(33\) −2.17802 −0.379144
\(34\) −16.7246 −2.86825
\(35\) 0 0
\(36\) 8.46336 1.41056
\(37\) 3.69327 0.607170 0.303585 0.952804i \(-0.401816\pi\)
0.303585 + 0.952804i \(0.401816\pi\)
\(38\) −2.61792 −0.424683
\(39\) −9.15980 −1.46674
\(40\) 0 0
\(41\) −1.08046 −0.168740 −0.0843701 0.996434i \(-0.526888\pi\)
−0.0843701 + 0.996434i \(0.526888\pi\)
\(42\) −28.2147 −4.35362
\(43\) −9.58018 −1.46096 −0.730482 0.682932i \(-0.760704\pi\)
−0.730482 + 0.682932i \(0.760704\pi\)
\(44\) 4.85352 0.731696
\(45\) 0 0
\(46\) 17.4413 2.57158
\(47\) −2.90169 −0.423256 −0.211628 0.977350i \(-0.567876\pi\)
−0.211628 + 0.977350i \(0.567876\pi\)
\(48\) −21.4526 −3.09642
\(49\) 17.4858 2.49798
\(50\) 0 0
\(51\) −13.9143 −1.94839
\(52\) 20.4118 2.83061
\(53\) 6.75151 0.927391 0.463695 0.885995i \(-0.346523\pi\)
0.463695 + 0.885995i \(0.346523\pi\)
\(54\) −7.16294 −0.974753
\(55\) 0 0
\(56\) 36.9654 4.93971
\(57\) −2.17802 −0.288485
\(58\) 17.4260 2.28815
\(59\) −0.613007 −0.0798067 −0.0399034 0.999204i \(-0.512705\pi\)
−0.0399034 + 0.999204i \(0.512705\pi\)
\(60\) 0 0
\(61\) 1.33665 0.171140 0.0855701 0.996332i \(-0.472729\pi\)
0.0855701 + 0.996332i \(0.472729\pi\)
\(62\) 24.9712 3.17134
\(63\) −8.62866 −1.08711
\(64\) 8.69200 1.08650
\(65\) 0 0
\(66\) 5.70188 0.701853
\(67\) −0.386619 −0.0472330 −0.0236165 0.999721i \(-0.507518\pi\)
−0.0236165 + 0.999721i \(0.507518\pi\)
\(68\) 31.0067 3.76012
\(69\) 14.5105 1.74686
\(70\) 0 0
\(71\) 9.38483 1.11377 0.556887 0.830588i \(-0.311996\pi\)
0.556887 + 0.830588i \(0.311996\pi\)
\(72\) −13.0264 −1.53517
\(73\) 1.97456 0.231105 0.115552 0.993301i \(-0.463136\pi\)
0.115552 + 0.993301i \(0.463136\pi\)
\(74\) −9.66869 −1.12396
\(75\) 0 0
\(76\) 4.85352 0.556737
\(77\) −4.94832 −0.563913
\(78\) 23.9797 2.71516
\(79\) −6.61494 −0.744238 −0.372119 0.928185i \(-0.621369\pi\)
−0.372119 + 0.928185i \(0.621369\pi\)
\(80\) 0 0
\(81\) −11.1906 −1.24340
\(82\) 2.82857 0.312364
\(83\) 0.100761 0.0110600 0.00553000 0.999985i \(-0.498240\pi\)
0.00553000 + 0.999985i \(0.498240\pi\)
\(84\) 52.3089 5.70737
\(85\) 0 0
\(86\) 25.0802 2.70446
\(87\) 14.4978 1.55433
\(88\) −7.47030 −0.796336
\(89\) 2.48738 0.263661 0.131831 0.991272i \(-0.457914\pi\)
0.131831 + 0.991272i \(0.457914\pi\)
\(90\) 0 0
\(91\) −20.8105 −2.18153
\(92\) −32.3355 −3.37121
\(93\) 20.7751 2.15428
\(94\) 7.59641 0.783510
\(95\) 0 0
\(96\) 23.6205 2.41075
\(97\) 17.3288 1.75947 0.879737 0.475461i \(-0.157719\pi\)
0.879737 + 0.475461i \(0.157719\pi\)
\(98\) −45.7766 −4.62413
\(99\) 1.74376 0.175254
\(100\) 0 0
\(101\) −9.71209 −0.966389 −0.483194 0.875513i \(-0.660524\pi\)
−0.483194 + 0.875513i \(0.660524\pi\)
\(102\) 36.4265 3.60676
\(103\) −15.6212 −1.53920 −0.769602 0.638524i \(-0.779545\pi\)
−0.769602 + 0.638524i \(0.779545\pi\)
\(104\) −31.4168 −3.08068
\(105\) 0 0
\(106\) −17.6749 −1.71674
\(107\) 5.49210 0.530941 0.265471 0.964119i \(-0.414473\pi\)
0.265471 + 0.964119i \(0.414473\pi\)
\(108\) 13.2798 1.27785
\(109\) 13.5177 1.29476 0.647378 0.762169i \(-0.275865\pi\)
0.647378 + 0.762169i \(0.275865\pi\)
\(110\) 0 0
\(111\) −8.04400 −0.763503
\(112\) −48.7390 −4.60540
\(113\) 11.1897 1.05264 0.526318 0.850288i \(-0.323572\pi\)
0.526318 + 0.850288i \(0.323572\pi\)
\(114\) 5.70188 0.534030
\(115\) 0 0
\(116\) −32.3071 −2.99964
\(117\) 7.33349 0.677982
\(118\) 1.60481 0.147734
\(119\) −31.6124 −2.89790
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −3.49924 −0.316806
\(123\) 2.35327 0.212187
\(124\) −46.2955 −4.15746
\(125\) 0 0
\(126\) 22.5892 2.01240
\(127\) −6.65255 −0.590319 −0.295159 0.955448i \(-0.595373\pi\)
−0.295159 + 0.955448i \(0.595373\pi\)
\(128\) −1.06509 −0.0941412
\(129\) 20.8658 1.83713
\(130\) 0 0
\(131\) 18.4455 1.61159 0.805797 0.592192i \(-0.201737\pi\)
0.805797 + 0.592192i \(0.201737\pi\)
\(132\) −10.5710 −0.920092
\(133\) −4.94832 −0.429073
\(134\) 1.01214 0.0874355
\(135\) 0 0
\(136\) −47.7240 −4.09230
\(137\) 15.3705 1.31319 0.656597 0.754242i \(-0.271995\pi\)
0.656597 + 0.754242i \(0.271995\pi\)
\(138\) −37.9875 −3.23371
\(139\) −2.38661 −0.202430 −0.101215 0.994865i \(-0.532273\pi\)
−0.101215 + 0.994865i \(0.532273\pi\)
\(140\) 0 0
\(141\) 6.31994 0.532235
\(142\) −24.5688 −2.06176
\(143\) 4.20557 0.351687
\(144\) 17.1753 1.43128
\(145\) 0 0
\(146\) −5.16924 −0.427809
\(147\) −38.0844 −3.14115
\(148\) 17.9254 1.47346
\(149\) −10.6114 −0.869317 −0.434659 0.900595i \(-0.643131\pi\)
−0.434659 + 0.900595i \(0.643131\pi\)
\(150\) 0 0
\(151\) 14.8914 1.21185 0.605923 0.795523i \(-0.292804\pi\)
0.605923 + 0.795523i \(0.292804\pi\)
\(152\) −7.47030 −0.605921
\(153\) 11.1400 0.900616
\(154\) 12.9543 1.04389
\(155\) 0 0
\(156\) −44.4573 −3.55943
\(157\) −5.84649 −0.466601 −0.233300 0.972405i \(-0.574953\pi\)
−0.233300 + 0.972405i \(0.574953\pi\)
\(158\) 17.3174 1.37770
\(159\) −14.7049 −1.16617
\(160\) 0 0
\(161\) 32.9670 2.59817
\(162\) 29.2961 2.30172
\(163\) 3.65131 0.285993 0.142996 0.989723i \(-0.454326\pi\)
0.142996 + 0.989723i \(0.454326\pi\)
\(164\) −5.24405 −0.409492
\(165\) 0 0
\(166\) −0.263785 −0.0204737
\(167\) 8.34211 0.645532 0.322766 0.946479i \(-0.395387\pi\)
0.322766 + 0.946479i \(0.395387\pi\)
\(168\) −80.5112 −6.21158
\(169\) 4.68682 0.360524
\(170\) 0 0
\(171\) 1.74376 0.133348
\(172\) −46.4976 −3.54541
\(173\) −14.4715 −1.10025 −0.550124 0.835083i \(-0.685420\pi\)
−0.550124 + 0.835083i \(0.685420\pi\)
\(174\) −37.9542 −2.87730
\(175\) 0 0
\(176\) 9.84962 0.742443
\(177\) 1.33514 0.100355
\(178\) −6.51176 −0.488077
\(179\) 23.2230 1.73577 0.867884 0.496766i \(-0.165479\pi\)
0.867884 + 0.496766i \(0.165479\pi\)
\(180\) 0 0
\(181\) 3.66090 0.272112 0.136056 0.990701i \(-0.456557\pi\)
0.136056 + 0.990701i \(0.456557\pi\)
\(182\) 54.4803 4.03834
\(183\) −2.91124 −0.215205
\(184\) 49.7691 3.66903
\(185\) 0 0
\(186\) −54.3877 −3.98790
\(187\) 6.38851 0.467174
\(188\) −14.0834 −1.02714
\(189\) −13.5392 −0.984830
\(190\) 0 0
\(191\) 23.3590 1.69020 0.845099 0.534610i \(-0.179541\pi\)
0.845099 + 0.534610i \(0.179541\pi\)
\(192\) −18.9313 −1.36625
\(193\) −25.1709 −1.81184 −0.905922 0.423445i \(-0.860821\pi\)
−0.905922 + 0.423445i \(0.860821\pi\)
\(194\) −45.3655 −3.25705
\(195\) 0 0
\(196\) 84.8678 6.06199
\(197\) −6.98835 −0.497899 −0.248950 0.968516i \(-0.580085\pi\)
−0.248950 + 0.968516i \(0.580085\pi\)
\(198\) −4.56502 −0.324422
\(199\) 11.5381 0.817911 0.408955 0.912554i \(-0.365893\pi\)
0.408955 + 0.912554i \(0.365893\pi\)
\(200\) 0 0
\(201\) 0.842063 0.0593946
\(202\) 25.4255 1.78893
\(203\) 32.9381 2.31180
\(204\) −67.5332 −4.72827
\(205\) 0 0
\(206\) 40.8951 2.84930
\(207\) −11.6174 −0.807464
\(208\) 41.4232 2.87219
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 13.7660 0.947687 0.473844 0.880609i \(-0.342866\pi\)
0.473844 + 0.880609i \(0.342866\pi\)
\(212\) 32.7686 2.25055
\(213\) −20.4403 −1.40055
\(214\) −14.3779 −0.982853
\(215\) 0 0
\(216\) −20.4396 −1.39074
\(217\) 47.1998 3.20413
\(218\) −35.3882 −2.39679
\(219\) −4.30062 −0.290609
\(220\) 0 0
\(221\) 26.8673 1.80729
\(222\) 21.0586 1.41336
\(223\) 18.7373 1.25474 0.627371 0.778720i \(-0.284131\pi\)
0.627371 + 0.778720i \(0.284131\pi\)
\(224\) 53.6642 3.58559
\(225\) 0 0
\(226\) −29.2937 −1.94859
\(227\) 21.7922 1.44640 0.723199 0.690640i \(-0.242671\pi\)
0.723199 + 0.690640i \(0.242671\pi\)
\(228\) −10.5710 −0.700085
\(229\) −15.5530 −1.02777 −0.513886 0.857859i \(-0.671795\pi\)
−0.513886 + 0.857859i \(0.671795\pi\)
\(230\) 0 0
\(231\) 10.7775 0.709109
\(232\) 49.7255 3.26464
\(233\) −27.0173 −1.76996 −0.884982 0.465625i \(-0.845830\pi\)
−0.884982 + 0.465625i \(0.845830\pi\)
\(234\) −19.1985 −1.25505
\(235\) 0 0
\(236\) −2.97524 −0.193672
\(237\) 14.4074 0.935864
\(238\) 82.7587 5.36445
\(239\) −26.5139 −1.71504 −0.857521 0.514449i \(-0.827997\pi\)
−0.857521 + 0.514449i \(0.827997\pi\)
\(240\) 0 0
\(241\) −16.1879 −1.04275 −0.521377 0.853327i \(-0.674581\pi\)
−0.521377 + 0.853327i \(0.674581\pi\)
\(242\) −2.61792 −0.168286
\(243\) 16.1649 1.03698
\(244\) 6.48745 0.415316
\(245\) 0 0
\(246\) −6.16068 −0.392790
\(247\) 4.20557 0.267594
\(248\) 71.2558 4.52475
\(249\) −0.219460 −0.0139077
\(250\) 0 0
\(251\) 2.48202 0.156664 0.0783319 0.996927i \(-0.475041\pi\)
0.0783319 + 0.996927i \(0.475041\pi\)
\(252\) −41.8794 −2.63815
\(253\) −6.66227 −0.418853
\(254\) 17.4159 1.09277
\(255\) 0 0
\(256\) −14.5957 −0.912230
\(257\) 1.75270 0.109330 0.0546651 0.998505i \(-0.482591\pi\)
0.0546651 + 0.998505i \(0.482591\pi\)
\(258\) −54.6250 −3.40081
\(259\) −18.2755 −1.13558
\(260\) 0 0
\(261\) −11.6072 −0.718468
\(262\) −48.2890 −2.98330
\(263\) 15.9328 0.982460 0.491230 0.871030i \(-0.336548\pi\)
0.491230 + 0.871030i \(0.336548\pi\)
\(264\) 16.2704 1.00138
\(265\) 0 0
\(266\) 12.9543 0.794280
\(267\) −5.41755 −0.331549
\(268\) −1.87646 −0.114623
\(269\) −5.40061 −0.329281 −0.164641 0.986354i \(-0.552646\pi\)
−0.164641 + 0.986354i \(0.552646\pi\)
\(270\) 0 0
\(271\) −0.761668 −0.0462680 −0.0231340 0.999732i \(-0.507364\pi\)
−0.0231340 + 0.999732i \(0.507364\pi\)
\(272\) 62.9243 3.81535
\(273\) 45.3256 2.74323
\(274\) −40.2389 −2.43092
\(275\) 0 0
\(276\) 70.4272 4.23922
\(277\) −26.9579 −1.61974 −0.809872 0.586606i \(-0.800464\pi\)
−0.809872 + 0.586606i \(0.800464\pi\)
\(278\) 6.24797 0.374729
\(279\) −16.6329 −0.995787
\(280\) 0 0
\(281\) 18.4983 1.10352 0.551759 0.834004i \(-0.313957\pi\)
0.551759 + 0.834004i \(0.313957\pi\)
\(282\) −16.5451 −0.985247
\(283\) −18.9028 −1.12365 −0.561827 0.827255i \(-0.689901\pi\)
−0.561827 + 0.827255i \(0.689901\pi\)
\(284\) 45.5495 2.70286
\(285\) 0 0
\(286\) −11.0099 −0.651026
\(287\) 5.34648 0.315593
\(288\) −18.9110 −1.11434
\(289\) 23.8130 1.40077
\(290\) 0 0
\(291\) −37.7424 −2.21250
\(292\) 9.58356 0.560835
\(293\) −13.8515 −0.809214 −0.404607 0.914491i \(-0.632592\pi\)
−0.404607 + 0.914491i \(0.632592\pi\)
\(294\) 99.7021 5.81475
\(295\) 0 0
\(296\) −27.5898 −1.60363
\(297\) 2.73612 0.158766
\(298\) 27.7797 1.60924
\(299\) −28.0186 −1.62036
\(300\) 0 0
\(301\) 47.4057 2.73242
\(302\) −38.9846 −2.24331
\(303\) 21.1531 1.21521
\(304\) 9.84962 0.564914
\(305\) 0 0
\(306\) −29.1637 −1.66718
\(307\) −0.233341 −0.0133175 −0.00665874 0.999978i \(-0.502120\pi\)
−0.00665874 + 0.999978i \(0.502120\pi\)
\(308\) −24.0168 −1.36848
\(309\) 34.0233 1.93552
\(310\) 0 0
\(311\) 7.70278 0.436784 0.218392 0.975861i \(-0.429919\pi\)
0.218392 + 0.975861i \(0.429919\pi\)
\(312\) 68.4264 3.87388
\(313\) −14.9273 −0.843742 −0.421871 0.906656i \(-0.638626\pi\)
−0.421871 + 0.906656i \(0.638626\pi\)
\(314\) 15.3057 0.863749
\(315\) 0 0
\(316\) −32.1057 −1.80609
\(317\) 3.39011 0.190408 0.0952038 0.995458i \(-0.469650\pi\)
0.0952038 + 0.995458i \(0.469650\pi\)
\(318\) 38.4963 2.15876
\(319\) −6.65643 −0.372689
\(320\) 0 0
\(321\) −11.9619 −0.667648
\(322\) −86.3051 −4.80960
\(323\) 6.38851 0.355466
\(324\) −54.3137 −3.01743
\(325\) 0 0
\(326\) −9.55885 −0.529416
\(327\) −29.4417 −1.62813
\(328\) 8.07139 0.445668
\(329\) 14.3585 0.791610
\(330\) 0 0
\(331\) −8.10646 −0.445571 −0.222786 0.974867i \(-0.571515\pi\)
−0.222786 + 0.974867i \(0.571515\pi\)
\(332\) 0.489047 0.0268399
\(333\) 6.44017 0.352919
\(334\) −21.8390 −1.19498
\(335\) 0 0
\(336\) 106.154 5.79120
\(337\) −13.4558 −0.732984 −0.366492 0.930421i \(-0.619441\pi\)
−0.366492 + 0.930421i \(0.619441\pi\)
\(338\) −12.2697 −0.667385
\(339\) −24.3713 −1.32367
\(340\) 0 0
\(341\) −9.53855 −0.516541
\(342\) −4.56502 −0.246848
\(343\) −51.8872 −2.80165
\(344\) 71.5667 3.85862
\(345\) 0 0
\(346\) 37.8853 2.03673
\(347\) 15.8117 0.848819 0.424409 0.905470i \(-0.360482\pi\)
0.424409 + 0.905470i \(0.360482\pi\)
\(348\) 70.3654 3.77198
\(349\) −7.24809 −0.387981 −0.193991 0.981003i \(-0.562143\pi\)
−0.193991 + 0.981003i \(0.562143\pi\)
\(350\) 0 0
\(351\) 11.5069 0.614194
\(352\) −10.8449 −0.578038
\(353\) 11.6864 0.622005 0.311003 0.950409i \(-0.399335\pi\)
0.311003 + 0.950409i \(0.399335\pi\)
\(354\) −3.49529 −0.185773
\(355\) 0 0
\(356\) 12.0725 0.639843
\(357\) 68.8522 3.64405
\(358\) −60.7960 −3.21317
\(359\) 27.1065 1.43063 0.715313 0.698804i \(-0.246284\pi\)
0.715313 + 0.698804i \(0.246284\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −9.58395 −0.503721
\(363\) −2.17802 −0.114316
\(364\) −101.004 −5.29405
\(365\) 0 0
\(366\) 7.62140 0.398377
\(367\) 12.9734 0.677207 0.338604 0.940929i \(-0.390045\pi\)
0.338604 + 0.940929i \(0.390045\pi\)
\(368\) −65.6208 −3.42072
\(369\) −1.88407 −0.0980806
\(370\) 0 0
\(371\) −33.4086 −1.73449
\(372\) 100.832 5.22792
\(373\) 25.2889 1.30941 0.654704 0.755885i \(-0.272793\pi\)
0.654704 + 0.755885i \(0.272793\pi\)
\(374\) −16.7246 −0.864809
\(375\) 0 0
\(376\) 21.6765 1.11788
\(377\) −27.9941 −1.44177
\(378\) 35.4445 1.82307
\(379\) −25.0016 −1.28425 −0.642123 0.766602i \(-0.721946\pi\)
−0.642123 + 0.766602i \(0.721946\pi\)
\(380\) 0 0
\(381\) 14.4894 0.742313
\(382\) −61.1521 −3.12881
\(383\) 2.66127 0.135985 0.0679923 0.997686i \(-0.478341\pi\)
0.0679923 + 0.997686i \(0.478341\pi\)
\(384\) 2.31978 0.118381
\(385\) 0 0
\(386\) 65.8956 3.35400
\(387\) −16.7055 −0.849189
\(388\) 84.1057 4.26982
\(389\) 6.77412 0.343462 0.171731 0.985144i \(-0.445064\pi\)
0.171731 + 0.985144i \(0.445064\pi\)
\(390\) 0 0
\(391\) −42.5620 −2.15245
\(392\) −130.624 −6.59753
\(393\) −40.1747 −2.02654
\(394\) 18.2949 0.921686
\(395\) 0 0
\(396\) 8.46336 0.425300
\(397\) −20.9430 −1.05110 −0.525550 0.850763i \(-0.676140\pi\)
−0.525550 + 0.850763i \(0.676140\pi\)
\(398\) −30.2057 −1.51408
\(399\) 10.7775 0.539551
\(400\) 0 0
\(401\) 15.8352 0.790774 0.395387 0.918515i \(-0.370611\pi\)
0.395387 + 0.918515i \(0.370611\pi\)
\(402\) −2.20446 −0.109948
\(403\) −40.1150 −1.99827
\(404\) −47.1378 −2.34519
\(405\) 0 0
\(406\) −86.2295 −4.27950
\(407\) 3.69327 0.183069
\(408\) 103.944 5.14598
\(409\) −1.16027 −0.0573715 −0.0286857 0.999588i \(-0.509132\pi\)
−0.0286857 + 0.999588i \(0.509132\pi\)
\(410\) 0 0
\(411\) −33.4773 −1.65131
\(412\) −75.8178 −3.73528
\(413\) 3.03335 0.149262
\(414\) 30.4134 1.49474
\(415\) 0 0
\(416\) −45.6092 −2.23617
\(417\) 5.19809 0.254551
\(418\) −2.61792 −0.128047
\(419\) 0.923724 0.0451269 0.0225634 0.999745i \(-0.492817\pi\)
0.0225634 + 0.999745i \(0.492817\pi\)
\(420\) 0 0
\(421\) 13.0507 0.636054 0.318027 0.948082i \(-0.396980\pi\)
0.318027 + 0.948082i \(0.396980\pi\)
\(422\) −36.0382 −1.75431
\(423\) −5.05985 −0.246018
\(424\) −50.4357 −2.44938
\(425\) 0 0
\(426\) 53.5112 2.59263
\(427\) −6.61415 −0.320081
\(428\) 26.6560 1.28847
\(429\) −9.15980 −0.442239
\(430\) 0 0
\(431\) 20.3251 0.979025 0.489513 0.871996i \(-0.337175\pi\)
0.489513 + 0.871996i \(0.337175\pi\)
\(432\) 26.9497 1.29662
\(433\) 20.5451 0.987333 0.493666 0.869651i \(-0.335656\pi\)
0.493666 + 0.869651i \(0.335656\pi\)
\(434\) −123.565 −5.93133
\(435\) 0 0
\(436\) 65.6082 3.14206
\(437\) −6.66227 −0.318700
\(438\) 11.2587 0.537961
\(439\) −7.67668 −0.366388 −0.183194 0.983077i \(-0.558644\pi\)
−0.183194 + 0.983077i \(0.558644\pi\)
\(440\) 0 0
\(441\) 30.4911 1.45196
\(442\) −70.3365 −3.34557
\(443\) −9.04317 −0.429654 −0.214827 0.976652i \(-0.568919\pi\)
−0.214827 + 0.976652i \(0.568919\pi\)
\(444\) −39.0417 −1.85284
\(445\) 0 0
\(446\) −49.0528 −2.32272
\(447\) 23.1117 1.09315
\(448\) −43.0107 −2.03207
\(449\) −9.97433 −0.470718 −0.235359 0.971909i \(-0.575627\pi\)
−0.235359 + 0.971909i \(0.575627\pi\)
\(450\) 0 0
\(451\) −1.08046 −0.0508771
\(452\) 54.3093 2.55449
\(453\) −32.4338 −1.52387
\(454\) −57.0502 −2.67750
\(455\) 0 0
\(456\) 16.2704 0.761933
\(457\) −18.7284 −0.876077 −0.438038 0.898956i \(-0.644327\pi\)
−0.438038 + 0.898956i \(0.644327\pi\)
\(458\) 40.7166 1.90256
\(459\) 17.4797 0.815883
\(460\) 0 0
\(461\) −26.5036 −1.23440 −0.617198 0.786808i \(-0.711732\pi\)
−0.617198 + 0.786808i \(0.711732\pi\)
\(462\) −28.2147 −1.31267
\(463\) −18.1944 −0.845564 −0.422782 0.906231i \(-0.638946\pi\)
−0.422782 + 0.906231i \(0.638946\pi\)
\(464\) −65.5633 −3.04370
\(465\) 0 0
\(466\) 70.7293 3.27647
\(467\) −13.5114 −0.625234 −0.312617 0.949879i \(-0.601206\pi\)
−0.312617 + 0.949879i \(0.601206\pi\)
\(468\) 35.5933 1.64530
\(469\) 1.91311 0.0883394
\(470\) 0 0
\(471\) 12.7338 0.586741
\(472\) 4.57934 0.210781
\(473\) −9.58018 −0.440497
\(474\) −37.7176 −1.73243
\(475\) 0 0
\(476\) −153.431 −7.03251
\(477\) 11.7730 0.539048
\(478\) 69.4114 3.17480
\(479\) −15.0739 −0.688743 −0.344371 0.938833i \(-0.611908\pi\)
−0.344371 + 0.938833i \(0.611908\pi\)
\(480\) 0 0
\(481\) 15.5323 0.708212
\(482\) 42.3786 1.93029
\(483\) −71.8027 −3.26714
\(484\) 4.85352 0.220615
\(485\) 0 0
\(486\) −42.3185 −1.91961
\(487\) 41.3382 1.87321 0.936607 0.350381i \(-0.113948\pi\)
0.936607 + 0.350381i \(0.113948\pi\)
\(488\) −9.98515 −0.452007
\(489\) −7.95262 −0.359630
\(490\) 0 0
\(491\) −29.4695 −1.32994 −0.664970 0.746870i \(-0.731556\pi\)
−0.664970 + 0.746870i \(0.731556\pi\)
\(492\) 11.4216 0.514927
\(493\) −42.5246 −1.91521
\(494\) −11.0099 −0.495357
\(495\) 0 0
\(496\) −93.9511 −4.21853
\(497\) −46.4391 −2.08308
\(498\) 0.574529 0.0257453
\(499\) 0.211110 0.00945058 0.00472529 0.999989i \(-0.498496\pi\)
0.00472529 + 0.999989i \(0.498496\pi\)
\(500\) 0 0
\(501\) −18.1693 −0.811743
\(502\) −6.49774 −0.290008
\(503\) 7.94133 0.354086 0.177043 0.984203i \(-0.443347\pi\)
0.177043 + 0.984203i \(0.443347\pi\)
\(504\) 64.4587 2.87122
\(505\) 0 0
\(506\) 17.4413 0.775361
\(507\) −10.2080 −0.453352
\(508\) −32.2883 −1.43256
\(509\) 7.40748 0.328331 0.164165 0.986433i \(-0.447507\pi\)
0.164165 + 0.986433i \(0.447507\pi\)
\(510\) 0 0
\(511\) −9.77074 −0.432232
\(512\) 40.3405 1.78282
\(513\) 2.73612 0.120803
\(514\) −4.58842 −0.202387
\(515\) 0 0
\(516\) 101.273 4.45827
\(517\) −2.90169 −0.127616
\(518\) 47.8438 2.10213
\(519\) 31.5192 1.38354
\(520\) 0 0
\(521\) 14.9680 0.655760 0.327880 0.944719i \(-0.393666\pi\)
0.327880 + 0.944719i \(0.393666\pi\)
\(522\) 30.3868 1.32999
\(523\) −6.78937 −0.296878 −0.148439 0.988922i \(-0.547425\pi\)
−0.148439 + 0.988922i \(0.547425\pi\)
\(524\) 89.5257 3.91095
\(525\) 0 0
\(526\) −41.7109 −1.81868
\(527\) −60.9371 −2.65446
\(528\) −21.4526 −0.933606
\(529\) 21.3859 0.929820
\(530\) 0 0
\(531\) −1.06894 −0.0463879
\(532\) −24.0168 −1.04126
\(533\) −4.54397 −0.196821
\(534\) 14.1827 0.613746
\(535\) 0 0
\(536\) 2.88816 0.124749
\(537\) −50.5801 −2.18269
\(538\) 14.1384 0.609549
\(539\) 17.4858 0.753168
\(540\) 0 0
\(541\) −21.8957 −0.941368 −0.470684 0.882302i \(-0.655993\pi\)
−0.470684 + 0.882302i \(0.655993\pi\)
\(542\) 1.99399 0.0856491
\(543\) −7.97350 −0.342176
\(544\) −69.2830 −2.97048
\(545\) 0 0
\(546\) −118.659 −5.07813
\(547\) 16.7053 0.714266 0.357133 0.934054i \(-0.383754\pi\)
0.357133 + 0.934054i \(0.383754\pi\)
\(548\) 74.6012 3.18681
\(549\) 2.33079 0.0994757
\(550\) 0 0
\(551\) −6.65643 −0.283573
\(552\) −108.398 −4.61373
\(553\) 32.7328 1.39194
\(554\) 70.5738 2.99839
\(555\) 0 0
\(556\) −11.5835 −0.491249
\(557\) 23.8953 1.01248 0.506238 0.862394i \(-0.331036\pi\)
0.506238 + 0.862394i \(0.331036\pi\)
\(558\) 43.5437 1.84335
\(559\) −40.2901 −1.70409
\(560\) 0 0
\(561\) −13.9143 −0.587461
\(562\) −48.4272 −2.04278
\(563\) 17.6090 0.742129 0.371065 0.928607i \(-0.378993\pi\)
0.371065 + 0.928607i \(0.378993\pi\)
\(564\) 30.6739 1.29161
\(565\) 0 0
\(566\) 49.4860 2.08005
\(567\) 55.3745 2.32551
\(568\) −70.1075 −2.94164
\(569\) 38.1265 1.59835 0.799174 0.601100i \(-0.205271\pi\)
0.799174 + 0.601100i \(0.205271\pi\)
\(570\) 0 0
\(571\) −6.37976 −0.266985 −0.133492 0.991050i \(-0.542619\pi\)
−0.133492 + 0.991050i \(0.542619\pi\)
\(572\) 20.4118 0.853461
\(573\) −50.8763 −2.12539
\(574\) −13.9967 −0.584210
\(575\) 0 0
\(576\) 15.1567 0.631531
\(577\) 19.3120 0.803969 0.401985 0.915646i \(-0.368321\pi\)
0.401985 + 0.915646i \(0.368321\pi\)
\(578\) −62.3406 −2.59303
\(579\) 54.8227 2.27835
\(580\) 0 0
\(581\) −0.498599 −0.0206854
\(582\) 98.8068 4.09567
\(583\) 6.75151 0.279619
\(584\) −14.7505 −0.610381
\(585\) 0 0
\(586\) 36.2622 1.49798
\(587\) 10.2671 0.423769 0.211884 0.977295i \(-0.432040\pi\)
0.211884 + 0.977295i \(0.432040\pi\)
\(588\) −184.844 −7.62282
\(589\) −9.53855 −0.393029
\(590\) 0 0
\(591\) 15.2207 0.626097
\(592\) 36.3773 1.49510
\(593\) −28.6849 −1.17795 −0.588975 0.808151i \(-0.700468\pi\)
−0.588975 + 0.808151i \(0.700468\pi\)
\(594\) −7.16294 −0.293899
\(595\) 0 0
\(596\) −51.5025 −2.10962
\(597\) −25.1301 −1.02851
\(598\) 73.3507 2.99953
\(599\) −21.3984 −0.874315 −0.437158 0.899385i \(-0.644015\pi\)
−0.437158 + 0.899385i \(0.644015\pi\)
\(600\) 0 0
\(601\) −36.6616 −1.49546 −0.747729 0.664004i \(-0.768856\pi\)
−0.747729 + 0.664004i \(0.768856\pi\)
\(602\) −124.105 −5.05812
\(603\) −0.674170 −0.0274543
\(604\) 72.2758 2.94086
\(605\) 0 0
\(606\) −55.3772 −2.24954
\(607\) −47.9867 −1.94772 −0.973860 0.227150i \(-0.927059\pi\)
−0.973860 + 0.227150i \(0.927059\pi\)
\(608\) −10.8449 −0.439821
\(609\) −71.7398 −2.90704
\(610\) 0 0
\(611\) −12.2033 −0.493692
\(612\) 54.0682 2.18558
\(613\) −25.0991 −1.01374 −0.506872 0.862021i \(-0.669198\pi\)
−0.506872 + 0.862021i \(0.669198\pi\)
\(614\) 0.610869 0.0246527
\(615\) 0 0
\(616\) 36.9654 1.48938
\(617\) 7.05585 0.284058 0.142029 0.989863i \(-0.454637\pi\)
0.142029 + 0.989863i \(0.454637\pi\)
\(618\) −89.0703 −3.58293
\(619\) 15.8586 0.637410 0.318705 0.947854i \(-0.396752\pi\)
0.318705 + 0.947854i \(0.396752\pi\)
\(620\) 0 0
\(621\) −18.2288 −0.731495
\(622\) −20.1653 −0.808554
\(623\) −12.3083 −0.493123
\(624\) −90.2205 −3.61171
\(625\) 0 0
\(626\) 39.0786 1.56189
\(627\) −2.17802 −0.0869816
\(628\) −28.3761 −1.13233
\(629\) 23.5945 0.940773
\(630\) 0 0
\(631\) 3.11637 0.124061 0.0620303 0.998074i \(-0.480242\pi\)
0.0620303 + 0.998074i \(0.480242\pi\)
\(632\) 49.4155 1.96564
\(633\) −29.9825 −1.19170
\(634\) −8.87505 −0.352473
\(635\) 0 0
\(636\) −71.3705 −2.83002
\(637\) 73.5379 2.91368
\(638\) 17.4260 0.689903
\(639\) 16.3649 0.647384
\(640\) 0 0
\(641\) −23.2203 −0.917147 −0.458573 0.888657i \(-0.651639\pi\)
−0.458573 + 0.888657i \(0.651639\pi\)
\(642\) 31.3153 1.23592
\(643\) −17.3300 −0.683428 −0.341714 0.939804i \(-0.611007\pi\)
−0.341714 + 0.939804i \(0.611007\pi\)
\(644\) 160.006 6.30512
\(645\) 0 0
\(646\) −16.7246 −0.658021
\(647\) 33.8000 1.32881 0.664407 0.747371i \(-0.268684\pi\)
0.664407 + 0.747371i \(0.268684\pi\)
\(648\) 83.5970 3.28400
\(649\) −0.613007 −0.0240626
\(650\) 0 0
\(651\) −102.802 −4.02912
\(652\) 17.7217 0.694036
\(653\) −23.8491 −0.933289 −0.466644 0.884445i \(-0.654537\pi\)
−0.466644 + 0.884445i \(0.654537\pi\)
\(654\) 77.0761 3.01391
\(655\) 0 0
\(656\) −10.6422 −0.415506
\(657\) 3.44315 0.134330
\(658\) −37.5894 −1.46539
\(659\) −34.8028 −1.35572 −0.677862 0.735189i \(-0.737093\pi\)
−0.677862 + 0.735189i \(0.737093\pi\)
\(660\) 0 0
\(661\) −41.3984 −1.61021 −0.805105 0.593132i \(-0.797891\pi\)
−0.805105 + 0.593132i \(0.797891\pi\)
\(662\) 21.2221 0.824820
\(663\) −58.5175 −2.27263
\(664\) −0.752717 −0.0292111
\(665\) 0 0
\(666\) −16.8599 −0.653306
\(667\) 44.3469 1.71712
\(668\) 40.4886 1.56655
\(669\) −40.8102 −1.57781
\(670\) 0 0
\(671\) 1.33665 0.0516007
\(672\) −116.882 −4.50880
\(673\) −49.3984 −1.90417 −0.952085 0.305835i \(-0.901065\pi\)
−0.952085 + 0.305835i \(0.901065\pi\)
\(674\) 35.2262 1.35686
\(675\) 0 0
\(676\) 22.7476 0.874906
\(677\) −36.6932 −1.41023 −0.705117 0.709091i \(-0.749105\pi\)
−0.705117 + 0.709091i \(0.749105\pi\)
\(678\) 63.8022 2.45031
\(679\) −85.7484 −3.29072
\(680\) 0 0
\(681\) −47.4637 −1.81881
\(682\) 24.9712 0.956196
\(683\) 26.3703 1.00903 0.504516 0.863403i \(-0.331671\pi\)
0.504516 + 0.863403i \(0.331671\pi\)
\(684\) 8.46336 0.323605
\(685\) 0 0
\(686\) 135.837 5.18627
\(687\) 33.8747 1.29240
\(688\) −94.3611 −3.59748
\(689\) 28.3939 1.08172
\(690\) 0 0
\(691\) 13.3427 0.507581 0.253791 0.967259i \(-0.418323\pi\)
0.253791 + 0.967259i \(0.418323\pi\)
\(692\) −70.2378 −2.67004
\(693\) −8.62866 −0.327776
\(694\) −41.3939 −1.57129
\(695\) 0 0
\(696\) −108.303 −4.10521
\(697\) −6.90255 −0.261453
\(698\) 18.9749 0.718212
\(699\) 58.8442 2.22569
\(700\) 0 0
\(701\) −7.00399 −0.264537 −0.132268 0.991214i \(-0.542226\pi\)
−0.132268 + 0.991214i \(0.542226\pi\)
\(702\) −30.1243 −1.13697
\(703\) 3.69327 0.139294
\(704\) 8.69200 0.327592
\(705\) 0 0
\(706\) −30.5941 −1.15143
\(707\) 48.0585 1.80743
\(708\) 6.48013 0.243538
\(709\) −30.5543 −1.14749 −0.573745 0.819034i \(-0.694510\pi\)
−0.573745 + 0.819034i \(0.694510\pi\)
\(710\) 0 0
\(711\) −11.5348 −0.432591
\(712\) −18.5814 −0.696369
\(713\) 63.5484 2.37991
\(714\) −180.250 −6.74568
\(715\) 0 0
\(716\) 112.713 4.21229
\(717\) 57.7478 2.15663
\(718\) −70.9627 −2.64831
\(719\) −29.1224 −1.08608 −0.543042 0.839706i \(-0.682727\pi\)
−0.543042 + 0.839706i \(0.682727\pi\)
\(720\) 0 0
\(721\) 77.2987 2.87875
\(722\) −2.61792 −0.0974290
\(723\) 35.2575 1.31124
\(724\) 17.7682 0.660351
\(725\) 0 0
\(726\) 5.70188 0.211617
\(727\) −20.4576 −0.758731 −0.379366 0.925247i \(-0.623858\pi\)
−0.379366 + 0.925247i \(0.623858\pi\)
\(728\) 155.460 5.76175
\(729\) −1.63574 −0.0605829
\(730\) 0 0
\(731\) −61.2030 −2.26368
\(732\) −14.1298 −0.522251
\(733\) 43.2636 1.59798 0.798989 0.601346i \(-0.205369\pi\)
0.798989 + 0.601346i \(0.205369\pi\)
\(734\) −33.9634 −1.25361
\(735\) 0 0
\(736\) 72.2520 2.66324
\(737\) −0.386619 −0.0142413
\(738\) 4.93234 0.181562
\(739\) −31.1858 −1.14719 −0.573595 0.819139i \(-0.694452\pi\)
−0.573595 + 0.819139i \(0.694452\pi\)
\(740\) 0 0
\(741\) −9.15980 −0.336494
\(742\) 87.4611 3.21080
\(743\) 45.0080 1.65118 0.825592 0.564268i \(-0.190842\pi\)
0.825592 + 0.564268i \(0.190842\pi\)
\(744\) −155.196 −5.68977
\(745\) 0 0
\(746\) −66.2043 −2.42391
\(747\) 0.175703 0.00642865
\(748\) 31.0067 1.13372
\(749\) −27.1767 −0.993013
\(750\) 0 0
\(751\) −29.7716 −1.08638 −0.543191 0.839609i \(-0.682784\pi\)
−0.543191 + 0.839609i \(0.682784\pi\)
\(752\) −28.5806 −1.04223
\(753\) −5.40588 −0.197001
\(754\) 73.2863 2.66893
\(755\) 0 0
\(756\) −65.7126 −2.38995
\(757\) 32.1474 1.16842 0.584209 0.811603i \(-0.301405\pi\)
0.584209 + 0.811603i \(0.301405\pi\)
\(758\) 65.4522 2.37733
\(759\) 14.5105 0.526699
\(760\) 0 0
\(761\) −2.83297 −0.102695 −0.0513475 0.998681i \(-0.516352\pi\)
−0.0513475 + 0.998681i \(0.516352\pi\)
\(762\) −37.9321 −1.37413
\(763\) −66.8896 −2.42157
\(764\) 113.373 4.10170
\(765\) 0 0
\(766\) −6.96700 −0.251728
\(767\) −2.57804 −0.0930878
\(768\) 31.7896 1.14711
\(769\) −0.197238 −0.00711259 −0.00355630 0.999994i \(-0.501132\pi\)
−0.00355630 + 0.999994i \(0.501132\pi\)
\(770\) 0 0
\(771\) −3.81740 −0.137480
\(772\) −122.168 −4.39691
\(773\) −39.5489 −1.42247 −0.711237 0.702952i \(-0.751865\pi\)
−0.711237 + 0.702952i \(0.751865\pi\)
\(774\) 43.7337 1.57198
\(775\) 0 0
\(776\) −129.451 −4.64703
\(777\) 39.8043 1.42797
\(778\) −17.7341 −0.635799
\(779\) −1.08046 −0.0387116
\(780\) 0 0
\(781\) 9.38483 0.335816
\(782\) 111.424 3.98451
\(783\) −18.2128 −0.650871
\(784\) 172.229 6.15103
\(785\) 0 0
\(786\) 105.174 3.75144
\(787\) −20.2522 −0.721912 −0.360956 0.932583i \(-0.617550\pi\)
−0.360956 + 0.932583i \(0.617550\pi\)
\(788\) −33.9181 −1.20828
\(789\) −34.7019 −1.23542
\(790\) 0 0
\(791\) −55.3700 −1.96873
\(792\) −13.0264 −0.462872
\(793\) 5.62136 0.199620
\(794\) 54.8272 1.94574
\(795\) 0 0
\(796\) 56.0002 1.98487
\(797\) −24.5195 −0.868524 −0.434262 0.900787i \(-0.642991\pi\)
−0.434262 + 0.900787i \(0.642991\pi\)
\(798\) −28.2147 −0.998790
\(799\) −18.5375 −0.655809
\(800\) 0 0
\(801\) 4.33738 0.153254
\(802\) −41.4554 −1.46384
\(803\) 1.97456 0.0696806
\(804\) 4.08697 0.144136
\(805\) 0 0
\(806\) 105.018 3.69910
\(807\) 11.7626 0.414064
\(808\) 72.5522 2.55238
\(809\) 18.8544 0.662885 0.331442 0.943475i \(-0.392465\pi\)
0.331442 + 0.943475i \(0.392465\pi\)
\(810\) 0 0
\(811\) 30.8152 1.08207 0.541034 0.841001i \(-0.318033\pi\)
0.541034 + 0.841001i \(0.318033\pi\)
\(812\) 159.866 5.61019
\(813\) 1.65893 0.0581811
\(814\) −9.66869 −0.338888
\(815\) 0 0
\(816\) −137.050 −4.79772
\(817\) −9.58018 −0.335168
\(818\) 3.03749 0.106203
\(819\) −36.2884 −1.26802
\(820\) 0 0
\(821\) −3.91462 −0.136621 −0.0683106 0.997664i \(-0.521761\pi\)
−0.0683106 + 0.997664i \(0.521761\pi\)
\(822\) 87.6410 3.05683
\(823\) −30.0384 −1.04707 −0.523537 0.852003i \(-0.675388\pi\)
−0.523537 + 0.852003i \(0.675388\pi\)
\(824\) 116.695 4.06526
\(825\) 0 0
\(826\) −7.94108 −0.276306
\(827\) 19.0520 0.662504 0.331252 0.943542i \(-0.392529\pi\)
0.331252 + 0.943542i \(0.392529\pi\)
\(828\) −56.3852 −1.95952
\(829\) −2.78078 −0.0965806 −0.0482903 0.998833i \(-0.515377\pi\)
−0.0482903 + 0.998833i \(0.515377\pi\)
\(830\) 0 0
\(831\) 58.7148 2.03679
\(832\) 36.5548 1.26731
\(833\) 111.708 3.87047
\(834\) −13.6082 −0.471213
\(835\) 0 0
\(836\) 4.85352 0.167862
\(837\) −26.0986 −0.902099
\(838\) −2.41824 −0.0835367
\(839\) −12.1680 −0.420087 −0.210044 0.977692i \(-0.567361\pi\)
−0.210044 + 0.977692i \(0.567361\pi\)
\(840\) 0 0
\(841\) 15.3081 0.527864
\(842\) −34.1658 −1.17743
\(843\) −40.2897 −1.38765
\(844\) 66.8133 2.29981
\(845\) 0 0
\(846\) 13.2463 0.455417
\(847\) −4.94832 −0.170026
\(848\) 66.4997 2.28361
\(849\) 41.1706 1.41297
\(850\) 0 0
\(851\) −24.6056 −0.843468
\(852\) −99.2075 −3.39879
\(853\) −30.8061 −1.05478 −0.527390 0.849623i \(-0.676829\pi\)
−0.527390 + 0.849623i \(0.676829\pi\)
\(854\) 17.3153 0.592519
\(855\) 0 0
\(856\) −41.0276 −1.40229
\(857\) 5.90481 0.201705 0.100852 0.994901i \(-0.467843\pi\)
0.100852 + 0.994901i \(0.467843\pi\)
\(858\) 23.9797 0.818652
\(859\) 28.0485 0.957003 0.478502 0.878087i \(-0.341180\pi\)
0.478502 + 0.878087i \(0.341180\pi\)
\(860\) 0 0
\(861\) −11.6447 −0.396851
\(862\) −53.2095 −1.81232
\(863\) 24.0412 0.818374 0.409187 0.912451i \(-0.365812\pi\)
0.409187 + 0.912451i \(0.365812\pi\)
\(864\) −29.6730 −1.00950
\(865\) 0 0
\(866\) −53.7854 −1.82770
\(867\) −51.8652 −1.76143
\(868\) 229.085 7.77565
\(869\) −6.61494 −0.224396
\(870\) 0 0
\(871\) −1.62595 −0.0550933
\(872\) −100.981 −3.41964
\(873\) 30.2172 1.02270
\(874\) 17.4413 0.589961
\(875\) 0 0
\(876\) −20.8731 −0.705238
\(877\) 46.1837 1.55951 0.779757 0.626083i \(-0.215343\pi\)
0.779757 + 0.626083i \(0.215343\pi\)
\(878\) 20.0970 0.678240
\(879\) 30.1688 1.01757
\(880\) 0 0
\(881\) −16.5583 −0.557863 −0.278931 0.960311i \(-0.589980\pi\)
−0.278931 + 0.960311i \(0.589980\pi\)
\(882\) −79.8232 −2.68779
\(883\) −39.6210 −1.33335 −0.666676 0.745347i \(-0.732284\pi\)
−0.666676 + 0.745347i \(0.732284\pi\)
\(884\) 130.401 4.38586
\(885\) 0 0
\(886\) 23.6743 0.795354
\(887\) −2.31838 −0.0778436 −0.0389218 0.999242i \(-0.512392\pi\)
−0.0389218 + 0.999242i \(0.512392\pi\)
\(888\) 60.0911 2.01652
\(889\) 32.9189 1.10407
\(890\) 0 0
\(891\) −11.1906 −0.374899
\(892\) 90.9419 3.04496
\(893\) −2.90169 −0.0971015
\(894\) −60.5048 −2.02358
\(895\) 0 0
\(896\) 5.27038 0.176071
\(897\) 61.0251 2.03757
\(898\) 26.1120 0.871370
\(899\) 63.4927 2.11760
\(900\) 0 0
\(901\) 43.1320 1.43694
\(902\) 2.82857 0.0941811
\(903\) −103.251 −3.43596
\(904\) −83.5901 −2.78017
\(905\) 0 0
\(906\) 84.9091 2.82092
\(907\) −3.05923 −0.101580 −0.0507900 0.998709i \(-0.516174\pi\)
−0.0507900 + 0.998709i \(0.516174\pi\)
\(908\) 105.769 3.51006
\(909\) −16.9355 −0.561716
\(910\) 0 0
\(911\) −51.4953 −1.70611 −0.853057 0.521817i \(-0.825254\pi\)
−0.853057 + 0.521817i \(0.825254\pi\)
\(912\) −21.4526 −0.710368
\(913\) 0.100761 0.00333471
\(914\) 49.0295 1.62175
\(915\) 0 0
\(916\) −75.4868 −2.49416
\(917\) −91.2743 −3.01414
\(918\) −45.7605 −1.51032
\(919\) 47.9995 1.58336 0.791678 0.610938i \(-0.209208\pi\)
0.791678 + 0.610938i \(0.209208\pi\)
\(920\) 0 0
\(921\) 0.508221 0.0167464
\(922\) 69.3844 2.28505
\(923\) 39.4686 1.29912
\(924\) 52.3089 1.72084
\(925\) 0 0
\(926\) 47.6315 1.56527
\(927\) −27.2396 −0.894666
\(928\) 72.1886 2.36971
\(929\) 27.5397 0.903548 0.451774 0.892132i \(-0.350791\pi\)
0.451774 + 0.892132i \(0.350791\pi\)
\(930\) 0 0
\(931\) 17.4858 0.573075
\(932\) −131.129 −4.29528
\(933\) −16.7768 −0.549247
\(934\) 35.3719 1.15740
\(935\) 0 0
\(936\) −54.7834 −1.79065
\(937\) −7.50039 −0.245027 −0.122513 0.992467i \(-0.539095\pi\)
−0.122513 + 0.992467i \(0.539095\pi\)
\(938\) −5.00838 −0.163530
\(939\) 32.5119 1.06099
\(940\) 0 0
\(941\) −42.0396 −1.37045 −0.685226 0.728330i \(-0.740297\pi\)
−0.685226 + 0.728330i \(0.740297\pi\)
\(942\) −33.3360 −1.08615
\(943\) 7.19835 0.234410
\(944\) −6.03789 −0.196516
\(945\) 0 0
\(946\) 25.0802 0.815426
\(947\) −20.1743 −0.655576 −0.327788 0.944751i \(-0.606303\pi\)
−0.327788 + 0.944751i \(0.606303\pi\)
\(948\) 69.9268 2.27112
\(949\) 8.30414 0.269564
\(950\) 0 0
\(951\) −7.38372 −0.239434
\(952\) 236.154 7.65378
\(953\) 1.49342 0.0483766 0.0241883 0.999707i \(-0.492300\pi\)
0.0241883 + 0.999707i \(0.492300\pi\)
\(954\) −30.8208 −0.997860
\(955\) 0 0
\(956\) −128.686 −4.16200
\(957\) 14.4978 0.468648
\(958\) 39.4622 1.27497
\(959\) −76.0583 −2.45605
\(960\) 0 0
\(961\) 59.9839 1.93497
\(962\) −40.6624 −1.31101
\(963\) 9.57689 0.308611
\(964\) −78.5682 −2.53051
\(965\) 0 0
\(966\) 187.974 6.04797
\(967\) −37.4587 −1.20459 −0.602295 0.798274i \(-0.705747\pi\)
−0.602295 + 0.798274i \(0.705747\pi\)
\(968\) −7.47030 −0.240104
\(969\) −13.9143 −0.446991
\(970\) 0 0
\(971\) −8.24513 −0.264599 −0.132299 0.991210i \(-0.542236\pi\)
−0.132299 + 0.991210i \(0.542236\pi\)
\(972\) 78.4568 2.51650
\(973\) 11.8097 0.378602
\(974\) −108.220 −3.46760
\(975\) 0 0
\(976\) 13.1655 0.421416
\(977\) −29.8558 −0.955171 −0.477586 0.878585i \(-0.658488\pi\)
−0.477586 + 0.878585i \(0.658488\pi\)
\(978\) 20.8193 0.665729
\(979\) 2.48738 0.0794969
\(980\) 0 0
\(981\) 23.5715 0.752581
\(982\) 77.1489 2.46192
\(983\) −25.9753 −0.828484 −0.414242 0.910167i \(-0.635953\pi\)
−0.414242 + 0.910167i \(0.635953\pi\)
\(984\) −17.5796 −0.560418
\(985\) 0 0
\(986\) 111.326 3.54535
\(987\) −31.2731 −0.995432
\(988\) 20.4118 0.649386
\(989\) 63.8257 2.02954
\(990\) 0 0
\(991\) −33.3837 −1.06047 −0.530235 0.847851i \(-0.677896\pi\)
−0.530235 + 0.847851i \(0.677896\pi\)
\(992\) 103.445 3.28438
\(993\) 17.6560 0.560297
\(994\) 121.574 3.85609
\(995\) 0 0
\(996\) −1.06515 −0.0337507
\(997\) −11.8321 −0.374725 −0.187363 0.982291i \(-0.559994\pi\)
−0.187363 + 0.982291i \(0.559994\pi\)
\(998\) −0.552670 −0.0174945
\(999\) 10.1052 0.319715
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 5225.2.a.r.1.2 15
5.4 even 2 5225.2.a.y.1.14 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5225.2.a.r.1.2 15 1.1 even 1 trivial
5225.2.a.y.1.14 yes 15 5.4 even 2