Properties

Label 325.2.a.k.1.2
Level $325$
Weight $2$
Character 325.1
Self dual yes
Analytic conductor $2.595$
Analytic rank $0$
Dimension $3$
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] = [325,2,Mod(1,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, 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("325.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.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: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 325.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.53919 q^{2} +3.17009 q^{3} +0.369102 q^{4} +4.87936 q^{6} -1.70928 q^{7} -2.51026 q^{8} +7.04945 q^{9} +O(q^{10})\) \(q+1.53919 q^{2} +3.17009 q^{3} +0.369102 q^{4} +4.87936 q^{6} -1.70928 q^{7} -2.51026 q^{8} +7.04945 q^{9} -2.53919 q^{11} +1.17009 q^{12} -1.00000 q^{13} -2.63090 q^{14} -4.60197 q^{16} +0.921622 q^{17} +10.8504 q^{18} -0.539189 q^{19} -5.41855 q^{21} -3.90829 q^{22} +2.82991 q^{23} -7.95774 q^{24} -1.53919 q^{26} +12.8371 q^{27} -0.630898 q^{28} -5.12783 q^{29} +0.879362 q^{31} -2.06278 q^{32} -8.04945 q^{33} +1.41855 q^{34} +2.60197 q^{36} -6.04945 q^{37} -0.829914 q^{38} -3.17009 q^{39} +1.26180 q^{41} -8.34017 q^{42} +6.43188 q^{43} -0.937221 q^{44} +4.35577 q^{46} +5.70928 q^{47} -14.5886 q^{48} -4.07838 q^{49} +2.92162 q^{51} -0.369102 q^{52} +8.49693 q^{53} +19.7587 q^{54} +4.29072 q^{56} -1.70928 q^{57} -7.89269 q^{58} -4.72261 q^{59} +8.04945 q^{61} +1.35350 q^{62} -12.0494 q^{63} +6.02893 q^{64} -12.3896 q^{66} +7.86603 q^{67} +0.340173 q^{68} +8.97107 q^{69} -14.4813 q^{71} -17.6959 q^{72} -1.95055 q^{73} -9.31124 q^{74} -0.199016 q^{76} +4.34017 q^{77} -4.87936 q^{78} +0.496928 q^{79} +19.5464 q^{81} +1.94214 q^{82} +8.63090 q^{83} -2.00000 q^{84} +9.89988 q^{86} -16.2557 q^{87} +6.37402 q^{88} +12.8371 q^{89} +1.70928 q^{91} +1.04453 q^{92} +2.78765 q^{93} +8.78765 q^{94} -6.53919 q^{96} +5.91548 q^{97} -6.27739 q^{98} -17.8999 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 4 q^{3} + 5 q^{4} + 2 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 4 q^{3} + 5 q^{4} + 2 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9} - 6 q^{11} - 2 q^{12} - 3 q^{13} - 4 q^{14} + 5 q^{16} + 6 q^{17} + 5 q^{18} - 2 q^{21} - 14 q^{22} + 14 q^{23} - 8 q^{24} - 3 q^{26} + 10 q^{27} + 2 q^{28} + 6 q^{29} - 10 q^{31} + 11 q^{32} - 6 q^{33} - 10 q^{34} - 11 q^{36} - 8 q^{38} - 4 q^{39} - 4 q^{41} - 14 q^{42} + 6 q^{43} - 20 q^{44} + 16 q^{46} + 10 q^{47} - 24 q^{48} - 9 q^{49} + 12 q^{51} - 5 q^{52} + 8 q^{53} + 34 q^{54} + 20 q^{56} + 2 q^{57} - 12 q^{58} - 8 q^{59} + 6 q^{61} - 6 q^{62} - 18 q^{63} + 33 q^{64} - 8 q^{66} + 10 q^{67} - 10 q^{68} + 12 q^{69} - 12 q^{71} - 45 q^{72} - 24 q^{73} - 2 q^{74} - 10 q^{76} + 2 q^{77} - 2 q^{78} - 16 q^{79} + 23 q^{81} - 24 q^{82} + 22 q^{83} - 6 q^{84} - 16 q^{86} - 6 q^{87} - 24 q^{88} + 10 q^{89} - 2 q^{91} + 32 q^{92} - 2 q^{93} + 16 q^{94} - 18 q^{96} - 14 q^{97} - 25 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.53919 1.08837 0.544185 0.838965i \(-0.316839\pi\)
0.544185 + 0.838965i \(0.316839\pi\)
\(3\) 3.17009 1.83025 0.915125 0.403170i \(-0.132092\pi\)
0.915125 + 0.403170i \(0.132092\pi\)
\(4\) 0.369102 0.184551
\(5\) 0 0
\(6\) 4.87936 1.99199
\(7\) −1.70928 −0.646045 −0.323023 0.946391i \(-0.604699\pi\)
−0.323023 + 0.946391i \(0.604699\pi\)
\(8\) −2.51026 −0.887511
\(9\) 7.04945 2.34982
\(10\) 0 0
\(11\) −2.53919 −0.765594 −0.382797 0.923832i \(-0.625039\pi\)
−0.382797 + 0.923832i \(0.625039\pi\)
\(12\) 1.17009 0.337775
\(13\) −1.00000 −0.277350
\(14\) −2.63090 −0.703137
\(15\) 0 0
\(16\) −4.60197 −1.15049
\(17\) 0.921622 0.223526 0.111763 0.993735i \(-0.464350\pi\)
0.111763 + 0.993735i \(0.464350\pi\)
\(18\) 10.8504 2.55747
\(19\) −0.539189 −0.123698 −0.0618492 0.998086i \(-0.519700\pi\)
−0.0618492 + 0.998086i \(0.519700\pi\)
\(20\) 0 0
\(21\) −5.41855 −1.18242
\(22\) −3.90829 −0.833250
\(23\) 2.82991 0.590078 0.295039 0.955485i \(-0.404667\pi\)
0.295039 + 0.955485i \(0.404667\pi\)
\(24\) −7.95774 −1.62437
\(25\) 0 0
\(26\) −1.53919 −0.301860
\(27\) 12.8371 2.47050
\(28\) −0.630898 −0.119228
\(29\) −5.12783 −0.952213 −0.476107 0.879388i \(-0.657952\pi\)
−0.476107 + 0.879388i \(0.657952\pi\)
\(30\) 0 0
\(31\) 0.879362 0.157938 0.0789690 0.996877i \(-0.474837\pi\)
0.0789690 + 0.996877i \(0.474837\pi\)
\(32\) −2.06278 −0.364651
\(33\) −8.04945 −1.40123
\(34\) 1.41855 0.243279
\(35\) 0 0
\(36\) 2.60197 0.433661
\(37\) −6.04945 −0.994523 −0.497262 0.867601i \(-0.665661\pi\)
−0.497262 + 0.867601i \(0.665661\pi\)
\(38\) −0.829914 −0.134630
\(39\) −3.17009 −0.507620
\(40\) 0 0
\(41\) 1.26180 0.197059 0.0985297 0.995134i \(-0.468586\pi\)
0.0985297 + 0.995134i \(0.468586\pi\)
\(42\) −8.34017 −1.28692
\(43\) 6.43188 0.980853 0.490426 0.871483i \(-0.336841\pi\)
0.490426 + 0.871483i \(0.336841\pi\)
\(44\) −0.937221 −0.141291
\(45\) 0 0
\(46\) 4.35577 0.642223
\(47\) 5.70928 0.832783 0.416392 0.909185i \(-0.363295\pi\)
0.416392 + 0.909185i \(0.363295\pi\)
\(48\) −14.5886 −2.10569
\(49\) −4.07838 −0.582625
\(50\) 0 0
\(51\) 2.92162 0.409109
\(52\) −0.369102 −0.0511853
\(53\) 8.49693 1.16714 0.583571 0.812062i \(-0.301655\pi\)
0.583571 + 0.812062i \(0.301655\pi\)
\(54\) 19.7587 2.68882
\(55\) 0 0
\(56\) 4.29072 0.573372
\(57\) −1.70928 −0.226399
\(58\) −7.89269 −1.03636
\(59\) −4.72261 −0.614831 −0.307415 0.951575i \(-0.599464\pi\)
−0.307415 + 0.951575i \(0.599464\pi\)
\(60\) 0 0
\(61\) 8.04945 1.03063 0.515313 0.857002i \(-0.327676\pi\)
0.515313 + 0.857002i \(0.327676\pi\)
\(62\) 1.35350 0.171895
\(63\) −12.0494 −1.51809
\(64\) 6.02893 0.753616
\(65\) 0 0
\(66\) −12.3896 −1.52506
\(67\) 7.86603 0.960989 0.480494 0.876998i \(-0.340457\pi\)
0.480494 + 0.876998i \(0.340457\pi\)
\(68\) 0.340173 0.0412520
\(69\) 8.97107 1.07999
\(70\) 0 0
\(71\) −14.4813 −1.71862 −0.859309 0.511457i \(-0.829106\pi\)
−0.859309 + 0.511457i \(0.829106\pi\)
\(72\) −17.6959 −2.08549
\(73\) −1.95055 −0.228295 −0.114147 0.993464i \(-0.536414\pi\)
−0.114147 + 0.993464i \(0.536414\pi\)
\(74\) −9.31124 −1.08241
\(75\) 0 0
\(76\) −0.199016 −0.0228287
\(77\) 4.34017 0.494609
\(78\) −4.87936 −0.552479
\(79\) 0.496928 0.0559088 0.0279544 0.999609i \(-0.491101\pi\)
0.0279544 + 0.999609i \(0.491101\pi\)
\(80\) 0 0
\(81\) 19.5464 2.17182
\(82\) 1.94214 0.214474
\(83\) 8.63090 0.947364 0.473682 0.880696i \(-0.342925\pi\)
0.473682 + 0.880696i \(0.342925\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 9.89988 1.06753
\(87\) −16.2557 −1.74279
\(88\) 6.37402 0.679473
\(89\) 12.8371 1.36073 0.680365 0.732873i \(-0.261821\pi\)
0.680365 + 0.732873i \(0.261821\pi\)
\(90\) 0 0
\(91\) 1.70928 0.179181
\(92\) 1.04453 0.108900
\(93\) 2.78765 0.289066
\(94\) 8.78765 0.906377
\(95\) 0 0
\(96\) −6.53919 −0.667403
\(97\) 5.91548 0.600626 0.300313 0.953841i \(-0.402909\pi\)
0.300313 + 0.953841i \(0.402909\pi\)
\(98\) −6.27739 −0.634113
\(99\) −17.8999 −1.79901
\(100\) 0 0
\(101\) −16.4391 −1.63575 −0.817874 0.575397i \(-0.804848\pi\)
−0.817874 + 0.575397i \(0.804848\pi\)
\(102\) 4.49693 0.445262
\(103\) −10.1906 −1.00411 −0.502055 0.864836i \(-0.667423\pi\)
−0.502055 + 0.864836i \(0.667423\pi\)
\(104\) 2.51026 0.246151
\(105\) 0 0
\(106\) 13.0784 1.27028
\(107\) 9.75154 0.942717 0.471358 0.881942i \(-0.343764\pi\)
0.471358 + 0.881942i \(0.343764\pi\)
\(108\) 4.73820 0.455934
\(109\) −16.8638 −1.61526 −0.807628 0.589693i \(-0.799249\pi\)
−0.807628 + 0.589693i \(0.799249\pi\)
\(110\) 0 0
\(111\) −19.1773 −1.82023
\(112\) 7.86603 0.743270
\(113\) −11.7587 −1.10617 −0.553084 0.833126i \(-0.686549\pi\)
−0.553084 + 0.833126i \(0.686549\pi\)
\(114\) −2.63090 −0.246406
\(115\) 0 0
\(116\) −1.89269 −0.175732
\(117\) −7.04945 −0.651722
\(118\) −7.26898 −0.669164
\(119\) −1.57531 −0.144408
\(120\) 0 0
\(121\) −4.55252 −0.413865
\(122\) 12.3896 1.12170
\(123\) 4.00000 0.360668
\(124\) 0.324575 0.0291477
\(125\) 0 0
\(126\) −18.5464 −1.65224
\(127\) −18.0072 −1.59788 −0.798940 0.601411i \(-0.794605\pi\)
−0.798940 + 0.601411i \(0.794605\pi\)
\(128\) 13.4052 1.18487
\(129\) 20.3896 1.79521
\(130\) 0 0
\(131\) 14.2557 1.24552 0.622761 0.782412i \(-0.286011\pi\)
0.622761 + 0.782412i \(0.286011\pi\)
\(132\) −2.97107 −0.258599
\(133\) 0.921622 0.0799148
\(134\) 12.1073 1.04591
\(135\) 0 0
\(136\) −2.31351 −0.198382
\(137\) 13.7854 1.17776 0.588882 0.808219i \(-0.299568\pi\)
0.588882 + 0.808219i \(0.299568\pi\)
\(138\) 13.8082 1.17543
\(139\) −6.65368 −0.564358 −0.282179 0.959362i \(-0.591057\pi\)
−0.282179 + 0.959362i \(0.591057\pi\)
\(140\) 0 0
\(141\) 18.0989 1.52420
\(142\) −22.2895 −1.87049
\(143\) 2.53919 0.212338
\(144\) −32.4413 −2.70344
\(145\) 0 0
\(146\) −3.00227 −0.248469
\(147\) −12.9288 −1.06635
\(148\) −2.23287 −0.183540
\(149\) −9.07838 −0.743730 −0.371865 0.928287i \(-0.621282\pi\)
−0.371865 + 0.928287i \(0.621282\pi\)
\(150\) 0 0
\(151\) 3.27739 0.266711 0.133355 0.991068i \(-0.457425\pi\)
0.133355 + 0.991068i \(0.457425\pi\)
\(152\) 1.35350 0.109784
\(153\) 6.49693 0.525246
\(154\) 6.68035 0.538318
\(155\) 0 0
\(156\) −1.17009 −0.0936819
\(157\) 12.8371 1.02451 0.512256 0.858833i \(-0.328810\pi\)
0.512256 + 0.858833i \(0.328810\pi\)
\(158\) 0.764867 0.0608495
\(159\) 26.9360 2.13616
\(160\) 0 0
\(161\) −4.83710 −0.381217
\(162\) 30.0856 2.36375
\(163\) 12.0494 0.943786 0.471893 0.881656i \(-0.343571\pi\)
0.471893 + 0.881656i \(0.343571\pi\)
\(164\) 0.465732 0.0363675
\(165\) 0 0
\(166\) 13.2846 1.03108
\(167\) 8.72979 0.675532 0.337766 0.941230i \(-0.390329\pi\)
0.337766 + 0.941230i \(0.390329\pi\)
\(168\) 13.6020 1.04941
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −3.80098 −0.290669
\(172\) 2.37402 0.181018
\(173\) 0.863763 0.0656707 0.0328354 0.999461i \(-0.489546\pi\)
0.0328354 + 0.999461i \(0.489546\pi\)
\(174\) −25.0205 −1.89680
\(175\) 0 0
\(176\) 11.6853 0.880810
\(177\) −14.9711 −1.12529
\(178\) 19.7587 1.48098
\(179\) 19.9155 1.48855 0.744276 0.667872i \(-0.232795\pi\)
0.744276 + 0.667872i \(0.232795\pi\)
\(180\) 0 0
\(181\) 14.3896 1.06957 0.534786 0.844987i \(-0.320392\pi\)
0.534786 + 0.844987i \(0.320392\pi\)
\(182\) 2.63090 0.195015
\(183\) 25.5174 1.88630
\(184\) −7.10382 −0.523700
\(185\) 0 0
\(186\) 4.29072 0.314611
\(187\) −2.34017 −0.171130
\(188\) 2.10731 0.153691
\(189\) −21.9421 −1.59606
\(190\) 0 0
\(191\) 1.47641 0.106829 0.0534146 0.998572i \(-0.482990\pi\)
0.0534146 + 0.998572i \(0.482990\pi\)
\(192\) 19.1122 1.37931
\(193\) −17.7321 −1.27638 −0.638191 0.769878i \(-0.720317\pi\)
−0.638191 + 0.769878i \(0.720317\pi\)
\(194\) 9.10504 0.653704
\(195\) 0 0
\(196\) −1.50534 −0.107524
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) −27.5513 −1.95799
\(199\) −5.39189 −0.382221 −0.191110 0.981569i \(-0.561209\pi\)
−0.191110 + 0.981569i \(0.561209\pi\)
\(200\) 0 0
\(201\) 24.9360 1.75885
\(202\) −25.3028 −1.78030
\(203\) 8.76487 0.615173
\(204\) 1.07838 0.0755015
\(205\) 0 0
\(206\) −15.6853 −1.09284
\(207\) 19.9493 1.38657
\(208\) 4.60197 0.319089
\(209\) 1.36910 0.0947028
\(210\) 0 0
\(211\) −22.7526 −1.56635 −0.783176 0.621800i \(-0.786402\pi\)
−0.783176 + 0.621800i \(0.786402\pi\)
\(212\) 3.13624 0.215398
\(213\) −45.9071 −3.14550
\(214\) 15.0095 1.02603
\(215\) 0 0
\(216\) −32.2245 −2.19260
\(217\) −1.50307 −0.102035
\(218\) −25.9565 −1.75800
\(219\) −6.18342 −0.417837
\(220\) 0 0
\(221\) −0.921622 −0.0619950
\(222\) −29.5174 −1.98108
\(223\) 8.76099 0.586679 0.293340 0.956008i \(-0.405233\pi\)
0.293340 + 0.956008i \(0.405233\pi\)
\(224\) 3.52586 0.235581
\(225\) 0 0
\(226\) −18.0989 −1.20392
\(227\) −17.2267 −1.14338 −0.571689 0.820470i \(-0.693712\pi\)
−0.571689 + 0.820470i \(0.693712\pi\)
\(228\) −0.630898 −0.0417822
\(229\) 3.07838 0.203425 0.101712 0.994814i \(-0.467568\pi\)
0.101712 + 0.994814i \(0.467568\pi\)
\(230\) 0 0
\(231\) 13.7587 0.905258
\(232\) 12.8722 0.845100
\(233\) 18.9360 1.24054 0.620269 0.784389i \(-0.287023\pi\)
0.620269 + 0.784389i \(0.287023\pi\)
\(234\) −10.8504 −0.709315
\(235\) 0 0
\(236\) −1.74313 −0.113468
\(237\) 1.57531 0.102327
\(238\) −2.42469 −0.157170
\(239\) 6.63809 0.429382 0.214691 0.976682i \(-0.431126\pi\)
0.214691 + 0.976682i \(0.431126\pi\)
\(240\) 0 0
\(241\) −9.47641 −0.610429 −0.305215 0.952284i \(-0.598728\pi\)
−0.305215 + 0.952284i \(0.598728\pi\)
\(242\) −7.00719 −0.450439
\(243\) 23.4524 1.50447
\(244\) 2.97107 0.190203
\(245\) 0 0
\(246\) 6.15676 0.392540
\(247\) 0.539189 0.0343078
\(248\) −2.20743 −0.140172
\(249\) 27.3607 1.73391
\(250\) 0 0
\(251\) 29.4596 1.85947 0.929736 0.368226i \(-0.120035\pi\)
0.929736 + 0.368226i \(0.120035\pi\)
\(252\) −4.44748 −0.280165
\(253\) −7.18568 −0.451760
\(254\) −27.7165 −1.73909
\(255\) 0 0
\(256\) 8.57531 0.535957
\(257\) −20.4657 −1.27662 −0.638309 0.769781i \(-0.720366\pi\)
−0.638309 + 0.769781i \(0.720366\pi\)
\(258\) 31.3835 1.95385
\(259\) 10.3402 0.642507
\(260\) 0 0
\(261\) −36.1483 −2.23753
\(262\) 21.9421 1.35559
\(263\) 9.14342 0.563808 0.281904 0.959443i \(-0.409034\pi\)
0.281904 + 0.959443i \(0.409034\pi\)
\(264\) 20.2062 1.24361
\(265\) 0 0
\(266\) 1.41855 0.0869769
\(267\) 40.6947 2.49048
\(268\) 2.90337 0.177352
\(269\) 11.3919 0.694576 0.347288 0.937759i \(-0.387103\pi\)
0.347288 + 0.937759i \(0.387103\pi\)
\(270\) 0 0
\(271\) −21.1350 −1.28386 −0.641930 0.766763i \(-0.721866\pi\)
−0.641930 + 0.766763i \(0.721866\pi\)
\(272\) −4.24128 −0.257165
\(273\) 5.41855 0.327946
\(274\) 21.2183 1.28185
\(275\) 0 0
\(276\) 3.31124 0.199313
\(277\) −13.0784 −0.785804 −0.392902 0.919580i \(-0.628529\pi\)
−0.392902 + 0.919580i \(0.628529\pi\)
\(278\) −10.2413 −0.614231
\(279\) 6.19902 0.371125
\(280\) 0 0
\(281\) −0.680346 −0.0405860 −0.0202930 0.999794i \(-0.506460\pi\)
−0.0202930 + 0.999794i \(0.506460\pi\)
\(282\) 27.8576 1.65890
\(283\) −19.2956 −1.14701 −0.573504 0.819203i \(-0.694416\pi\)
−0.573504 + 0.819203i \(0.694416\pi\)
\(284\) −5.34509 −0.317173
\(285\) 0 0
\(286\) 3.90829 0.231102
\(287\) −2.15676 −0.127309
\(288\) −14.5415 −0.856864
\(289\) −16.1506 −0.950036
\(290\) 0 0
\(291\) 18.7526 1.09930
\(292\) −0.719953 −0.0421321
\(293\) −9.46800 −0.553126 −0.276563 0.960996i \(-0.589196\pi\)
−0.276563 + 0.960996i \(0.589196\pi\)
\(294\) −19.8999 −1.16058
\(295\) 0 0
\(296\) 15.1857 0.882650
\(297\) −32.5958 −1.89140
\(298\) −13.9733 −0.809454
\(299\) −2.82991 −0.163658
\(300\) 0 0
\(301\) −10.9939 −0.633675
\(302\) 5.04453 0.290280
\(303\) −52.1133 −2.99383
\(304\) 2.48133 0.142314
\(305\) 0 0
\(306\) 10.0000 0.571662
\(307\) −0.264063 −0.0150709 −0.00753543 0.999972i \(-0.502399\pi\)
−0.00753543 + 0.999972i \(0.502399\pi\)
\(308\) 1.60197 0.0912806
\(309\) −32.3051 −1.83777
\(310\) 0 0
\(311\) 13.0472 0.739838 0.369919 0.929064i \(-0.379385\pi\)
0.369919 + 0.929064i \(0.379385\pi\)
\(312\) 7.95774 0.450518
\(313\) −33.7009 −1.90489 −0.952443 0.304718i \(-0.901438\pi\)
−0.952443 + 0.304718i \(0.901438\pi\)
\(314\) 19.7587 1.11505
\(315\) 0 0
\(316\) 0.183417 0.0103180
\(317\) −13.9506 −0.783541 −0.391771 0.920063i \(-0.628137\pi\)
−0.391771 + 0.920063i \(0.628137\pi\)
\(318\) 41.4596 2.32494
\(319\) 13.0205 0.729009
\(320\) 0 0
\(321\) 30.9132 1.72541
\(322\) −7.44521 −0.414905
\(323\) −0.496928 −0.0276498
\(324\) 7.21461 0.400812
\(325\) 0 0
\(326\) 18.5464 1.02719
\(327\) −53.4596 −2.95632
\(328\) −3.16743 −0.174892
\(329\) −9.75872 −0.538016
\(330\) 0 0
\(331\) −18.4547 −1.01436 −0.507180 0.861840i \(-0.669312\pi\)
−0.507180 + 0.861840i \(0.669312\pi\)
\(332\) 3.18568 0.174837
\(333\) −42.6453 −2.33695
\(334\) 13.4368 0.735229
\(335\) 0 0
\(336\) 24.9360 1.36037
\(337\) −15.8576 −0.863820 −0.431910 0.901917i \(-0.642160\pi\)
−0.431910 + 0.901917i \(0.642160\pi\)
\(338\) 1.53919 0.0837208
\(339\) −37.2762 −2.02456
\(340\) 0 0
\(341\) −2.23287 −0.120916
\(342\) −5.85043 −0.316355
\(343\) 18.9360 1.02245
\(344\) −16.1457 −0.870517
\(345\) 0 0
\(346\) 1.32950 0.0714741
\(347\) 9.72487 0.522059 0.261029 0.965331i \(-0.415938\pi\)
0.261029 + 0.965331i \(0.415938\pi\)
\(348\) −6.00000 −0.321634
\(349\) −30.9093 −1.65454 −0.827269 0.561805i \(-0.810107\pi\)
−0.827269 + 0.561805i \(0.810107\pi\)
\(350\) 0 0
\(351\) −12.8371 −0.685194
\(352\) 5.23779 0.279175
\(353\) 5.95055 0.316716 0.158358 0.987382i \(-0.449380\pi\)
0.158358 + 0.987382i \(0.449380\pi\)
\(354\) −23.0433 −1.22474
\(355\) 0 0
\(356\) 4.73820 0.251124
\(357\) −4.99386 −0.264303
\(358\) 30.6537 1.62010
\(359\) 10.9783 0.579410 0.289705 0.957116i \(-0.406443\pi\)
0.289705 + 0.957116i \(0.406443\pi\)
\(360\) 0 0
\(361\) −18.7093 −0.984699
\(362\) 22.1483 1.16409
\(363\) −14.4319 −0.757477
\(364\) 0.630898 0.0330680
\(365\) 0 0
\(366\) 39.2762 2.05300
\(367\) 10.3740 0.541520 0.270760 0.962647i \(-0.412725\pi\)
0.270760 + 0.962647i \(0.412725\pi\)
\(368\) −13.0232 −0.678880
\(369\) 8.89496 0.463053
\(370\) 0 0
\(371\) −14.5236 −0.754027
\(372\) 1.02893 0.0533475
\(373\) 23.9877 1.24204 0.621018 0.783796i \(-0.286719\pi\)
0.621018 + 0.783796i \(0.286719\pi\)
\(374\) −3.60197 −0.186253
\(375\) 0 0
\(376\) −14.3318 −0.739104
\(377\) 5.12783 0.264096
\(378\) −33.7731 −1.73710
\(379\) 29.7575 1.52854 0.764270 0.644896i \(-0.223099\pi\)
0.764270 + 0.644896i \(0.223099\pi\)
\(380\) 0 0
\(381\) −57.0843 −2.92452
\(382\) 2.27247 0.116270
\(383\) 12.4163 0.634442 0.317221 0.948352i \(-0.397250\pi\)
0.317221 + 0.948352i \(0.397250\pi\)
\(384\) 42.4957 2.16860
\(385\) 0 0
\(386\) −27.2930 −1.38918
\(387\) 45.3412 2.30482
\(388\) 2.18342 0.110846
\(389\) −16.8371 −0.853675 −0.426837 0.904328i \(-0.640372\pi\)
−0.426837 + 0.904328i \(0.640372\pi\)
\(390\) 0 0
\(391\) 2.60811 0.131898
\(392\) 10.2378 0.517086
\(393\) 45.1917 2.27962
\(394\) 3.07838 0.155086
\(395\) 0 0
\(396\) −6.60689 −0.332009
\(397\) −3.89269 −0.195369 −0.0976843 0.995217i \(-0.531144\pi\)
−0.0976843 + 0.995217i \(0.531144\pi\)
\(398\) −8.29914 −0.415998
\(399\) 2.92162 0.146264
\(400\) 0 0
\(401\) −9.10504 −0.454684 −0.227342 0.973815i \(-0.573004\pi\)
−0.227342 + 0.973815i \(0.573004\pi\)
\(402\) 38.3812 1.91428
\(403\) −0.879362 −0.0438041
\(404\) −6.06770 −0.301879
\(405\) 0 0
\(406\) 13.4908 0.669536
\(407\) 15.3607 0.761401
\(408\) −7.33403 −0.363089
\(409\) −19.4186 −0.960186 −0.480093 0.877218i \(-0.659397\pi\)
−0.480093 + 0.877218i \(0.659397\pi\)
\(410\) 0 0
\(411\) 43.7009 2.15560
\(412\) −3.76138 −0.185310
\(413\) 8.07223 0.397209
\(414\) 30.7058 1.50911
\(415\) 0 0
\(416\) 2.06278 0.101136
\(417\) −21.0928 −1.03292
\(418\) 2.10731 0.103072
\(419\) −16.7792 −0.819720 −0.409860 0.912149i \(-0.634422\pi\)
−0.409860 + 0.912149i \(0.634422\pi\)
\(420\) 0 0
\(421\) −19.0205 −0.927003 −0.463502 0.886096i \(-0.653407\pi\)
−0.463502 + 0.886096i \(0.653407\pi\)
\(422\) −35.0205 −1.70477
\(423\) 40.2472 1.95689
\(424\) −21.3295 −1.03585
\(425\) 0 0
\(426\) −70.6596 −3.42347
\(427\) −13.7587 −0.665831
\(428\) 3.59932 0.173979
\(429\) 8.04945 0.388631
\(430\) 0 0
\(431\) 8.02997 0.386790 0.193395 0.981121i \(-0.438050\pi\)
0.193395 + 0.981121i \(0.438050\pi\)
\(432\) −59.0759 −2.84229
\(433\) 13.0472 0.627008 0.313504 0.949587i \(-0.398497\pi\)
0.313504 + 0.949587i \(0.398497\pi\)
\(434\) −2.31351 −0.111052
\(435\) 0 0
\(436\) −6.22446 −0.298097
\(437\) −1.52586 −0.0729917
\(438\) −9.51745 −0.454761
\(439\) −7.70086 −0.367542 −0.183771 0.982969i \(-0.558831\pi\)
−0.183771 + 0.982969i \(0.558831\pi\)
\(440\) 0 0
\(441\) −28.7503 −1.36906
\(442\) −1.41855 −0.0674736
\(443\) 6.39084 0.303638 0.151819 0.988408i \(-0.451487\pi\)
0.151819 + 0.988408i \(0.451487\pi\)
\(444\) −7.07838 −0.335925
\(445\) 0 0
\(446\) 13.4848 0.638525
\(447\) −28.7792 −1.36121
\(448\) −10.3051 −0.486870
\(449\) 31.6163 1.49207 0.746034 0.665908i \(-0.231956\pi\)
0.746034 + 0.665908i \(0.231956\pi\)
\(450\) 0 0
\(451\) −3.20394 −0.150867
\(452\) −4.34017 −0.204145
\(453\) 10.3896 0.488147
\(454\) −26.5152 −1.24442
\(455\) 0 0
\(456\) 4.29072 0.200932
\(457\) 35.6430 1.66731 0.833655 0.552286i \(-0.186244\pi\)
0.833655 + 0.552286i \(0.186244\pi\)
\(458\) 4.73820 0.221402
\(459\) 11.8310 0.552222
\(460\) 0 0
\(461\) −14.9795 −0.697664 −0.348832 0.937185i \(-0.613422\pi\)
−0.348832 + 0.937185i \(0.613422\pi\)
\(462\) 21.1773 0.985256
\(463\) 9.09663 0.422756 0.211378 0.977404i \(-0.432205\pi\)
0.211378 + 0.977404i \(0.432205\pi\)
\(464\) 23.5981 1.09551
\(465\) 0 0
\(466\) 29.1461 1.35017
\(467\) −1.87709 −0.0868616 −0.0434308 0.999056i \(-0.513829\pi\)
−0.0434308 + 0.999056i \(0.513829\pi\)
\(468\) −2.60197 −0.120276
\(469\) −13.4452 −0.620842
\(470\) 0 0
\(471\) 40.6947 1.87511
\(472\) 11.8550 0.545669
\(473\) −16.3318 −0.750935
\(474\) 2.42469 0.111370
\(475\) 0 0
\(476\) −0.581449 −0.0266507
\(477\) 59.8987 2.74257
\(478\) 10.2173 0.467327
\(479\) −15.7431 −0.719322 −0.359661 0.933083i \(-0.617108\pi\)
−0.359661 + 0.933083i \(0.617108\pi\)
\(480\) 0 0
\(481\) 6.04945 0.275831
\(482\) −14.5860 −0.664373
\(483\) −15.3340 −0.697723
\(484\) −1.68035 −0.0763794
\(485\) 0 0
\(486\) 36.0977 1.63742
\(487\) −4.94441 −0.224053 −0.112026 0.993705i \(-0.535734\pi\)
−0.112026 + 0.993705i \(0.535734\pi\)
\(488\) −20.2062 −0.914692
\(489\) 38.1978 1.72736
\(490\) 0 0
\(491\) 39.4863 1.78199 0.890995 0.454014i \(-0.150008\pi\)
0.890995 + 0.454014i \(0.150008\pi\)
\(492\) 1.47641 0.0665617
\(493\) −4.72592 −0.212845
\(494\) 0.829914 0.0373396
\(495\) 0 0
\(496\) −4.04680 −0.181706
\(497\) 24.7526 1.11030
\(498\) 42.1133 1.88714
\(499\) 1.67089 0.0747993 0.0373997 0.999300i \(-0.488093\pi\)
0.0373997 + 0.999300i \(0.488093\pi\)
\(500\) 0 0
\(501\) 27.6742 1.23639
\(502\) 45.3439 2.02380
\(503\) −9.08557 −0.405105 −0.202553 0.979271i \(-0.564924\pi\)
−0.202553 + 0.979271i \(0.564924\pi\)
\(504\) 30.2472 1.34732
\(505\) 0 0
\(506\) −11.0601 −0.491683
\(507\) 3.17009 0.140788
\(508\) −6.64650 −0.294891
\(509\) −19.5441 −0.866277 −0.433139 0.901327i \(-0.642594\pi\)
−0.433139 + 0.901327i \(0.642594\pi\)
\(510\) 0 0
\(511\) 3.33403 0.147489
\(512\) −13.6114 −0.601546
\(513\) −6.92162 −0.305597
\(514\) −31.5006 −1.38943
\(515\) 0 0
\(516\) 7.52586 0.331307
\(517\) −14.4969 −0.637574
\(518\) 15.9155 0.699286
\(519\) 2.73820 0.120194
\(520\) 0 0
\(521\) 6.50534 0.285004 0.142502 0.989795i \(-0.454485\pi\)
0.142502 + 0.989795i \(0.454485\pi\)
\(522\) −55.6391 −2.43526
\(523\) −36.5452 −1.59801 −0.799004 0.601326i \(-0.794639\pi\)
−0.799004 + 0.601326i \(0.794639\pi\)
\(524\) 5.26180 0.229863
\(525\) 0 0
\(526\) 14.0735 0.613632
\(527\) 0.810439 0.0353033
\(528\) 37.0433 1.61210
\(529\) −14.9916 −0.651808
\(530\) 0 0
\(531\) −33.2918 −1.44474
\(532\) 0.340173 0.0147484
\(533\) −1.26180 −0.0546544
\(534\) 62.6369 2.71056
\(535\) 0 0
\(536\) −19.7458 −0.852888
\(537\) 63.1338 2.72442
\(538\) 17.5343 0.755956
\(539\) 10.3558 0.446055
\(540\) 0 0
\(541\) 20.3402 0.874492 0.437246 0.899342i \(-0.355954\pi\)
0.437246 + 0.899342i \(0.355954\pi\)
\(542\) −32.5308 −1.39732
\(543\) 45.6163 1.95758
\(544\) −1.90110 −0.0815091
\(545\) 0 0
\(546\) 8.34017 0.356926
\(547\) 11.5948 0.495757 0.247879 0.968791i \(-0.420267\pi\)
0.247879 + 0.968791i \(0.420267\pi\)
\(548\) 5.08822 0.217358
\(549\) 56.7442 2.42178
\(550\) 0 0
\(551\) 2.76487 0.117787
\(552\) −22.5197 −0.958503
\(553\) −0.849388 −0.0361196
\(554\) −20.1301 −0.855246
\(555\) 0 0
\(556\) −2.45589 −0.104153
\(557\) −10.7298 −0.454636 −0.227318 0.973821i \(-0.572996\pi\)
−0.227318 + 0.973821i \(0.572996\pi\)
\(558\) 9.54146 0.403922
\(559\) −6.43188 −0.272040
\(560\) 0 0
\(561\) −7.41855 −0.313211
\(562\) −1.04718 −0.0441727
\(563\) −10.2485 −0.431921 −0.215961 0.976402i \(-0.569288\pi\)
−0.215961 + 0.976402i \(0.569288\pi\)
\(564\) 6.68035 0.281293
\(565\) 0 0
\(566\) −29.6996 −1.24837
\(567\) −33.4101 −1.40309
\(568\) 36.3519 1.52529
\(569\) −8.84551 −0.370823 −0.185412 0.982661i \(-0.559362\pi\)
−0.185412 + 0.982661i \(0.559362\pi\)
\(570\) 0 0
\(571\) 9.29299 0.388900 0.194450 0.980912i \(-0.437708\pi\)
0.194450 + 0.980912i \(0.437708\pi\)
\(572\) 0.937221 0.0391872
\(573\) 4.68035 0.195524
\(574\) −3.31965 −0.138560
\(575\) 0 0
\(576\) 42.5006 1.77086
\(577\) 19.5259 0.812872 0.406436 0.913679i \(-0.366771\pi\)
0.406436 + 0.913679i \(0.366771\pi\)
\(578\) −24.8588 −1.03399
\(579\) −56.2122 −2.33610
\(580\) 0 0
\(581\) −14.7526 −0.612040
\(582\) 28.8638 1.19644
\(583\) −21.5753 −0.893558
\(584\) 4.89639 0.202614
\(585\) 0 0
\(586\) −14.5730 −0.602007
\(587\) −22.5029 −0.928794 −0.464397 0.885627i \(-0.653729\pi\)
−0.464397 + 0.885627i \(0.653729\pi\)
\(588\) −4.77205 −0.196796
\(589\) −0.474142 −0.0195367
\(590\) 0 0
\(591\) 6.34017 0.260800
\(592\) 27.8394 1.14419
\(593\) −4.43907 −0.182291 −0.0911454 0.995838i \(-0.529053\pi\)
−0.0911454 + 0.995838i \(0.529053\pi\)
\(594\) −50.1711 −2.05855
\(595\) 0 0
\(596\) −3.35085 −0.137256
\(597\) −17.0928 −0.699560
\(598\) −4.35577 −0.178121
\(599\) 33.3607 1.36308 0.681540 0.731780i \(-0.261310\pi\)
0.681540 + 0.731780i \(0.261310\pi\)
\(600\) 0 0
\(601\) 13.3197 0.543320 0.271660 0.962393i \(-0.412427\pi\)
0.271660 + 0.962393i \(0.412427\pi\)
\(602\) −16.9216 −0.689674
\(603\) 55.4512 2.25815
\(604\) 1.20969 0.0492217
\(605\) 0 0
\(606\) −80.2122 −3.25840
\(607\) −14.1184 −0.573047 −0.286523 0.958073i \(-0.592500\pi\)
−0.286523 + 0.958073i \(0.592500\pi\)
\(608\) 1.11223 0.0451068
\(609\) 27.7854 1.12592
\(610\) 0 0
\(611\) −5.70928 −0.230973
\(612\) 2.39803 0.0969347
\(613\) 26.8104 1.08286 0.541432 0.840745i \(-0.317882\pi\)
0.541432 + 0.840745i \(0.317882\pi\)
\(614\) −0.406442 −0.0164027
\(615\) 0 0
\(616\) −10.8950 −0.438970
\(617\) −14.8950 −0.599649 −0.299824 0.953994i \(-0.596928\pi\)
−0.299824 + 0.953994i \(0.596928\pi\)
\(618\) −49.7237 −2.00018
\(619\) 45.3184 1.82150 0.910751 0.412956i \(-0.135504\pi\)
0.910751 + 0.412956i \(0.135504\pi\)
\(620\) 0 0
\(621\) 36.3279 1.45779
\(622\) 20.0821 0.805218
\(623\) −21.9421 −0.879093
\(624\) 14.5886 0.584013
\(625\) 0 0
\(626\) −51.8720 −2.07322
\(627\) 4.34017 0.173330
\(628\) 4.73820 0.189075
\(629\) −5.57531 −0.222302
\(630\) 0 0
\(631\) 37.8876 1.50828 0.754141 0.656713i \(-0.228054\pi\)
0.754141 + 0.656713i \(0.228054\pi\)
\(632\) −1.24742 −0.0496197
\(633\) −72.1276 −2.86682
\(634\) −21.4725 −0.852783
\(635\) 0 0
\(636\) 9.94214 0.394232
\(637\) 4.07838 0.161591
\(638\) 20.0410 0.793432
\(639\) −102.085 −4.03844
\(640\) 0 0
\(641\) −8.47027 −0.334555 −0.167278 0.985910i \(-0.553498\pi\)
−0.167278 + 0.985910i \(0.553498\pi\)
\(642\) 47.5813 1.87788
\(643\) −34.1750 −1.34773 −0.673865 0.738854i \(-0.735367\pi\)
−0.673865 + 0.738854i \(0.735367\pi\)
\(644\) −1.78539 −0.0703541
\(645\) 0 0
\(646\) −0.764867 −0.0300933
\(647\) 13.8238 0.543468 0.271734 0.962372i \(-0.412403\pi\)
0.271734 + 0.962372i \(0.412403\pi\)
\(648\) −49.0665 −1.92751
\(649\) 11.9916 0.470711
\(650\) 0 0
\(651\) −4.76487 −0.186750
\(652\) 4.44748 0.174177
\(653\) 42.8781 1.67795 0.838976 0.544169i \(-0.183155\pi\)
0.838976 + 0.544169i \(0.183155\pi\)
\(654\) −82.2844 −3.21757
\(655\) 0 0
\(656\) −5.80674 −0.226715
\(657\) −13.7503 −0.536451
\(658\) −15.0205 −0.585561
\(659\) −23.2495 −0.905672 −0.452836 0.891594i \(-0.649588\pi\)
−0.452836 + 0.891594i \(0.649588\pi\)
\(660\) 0 0
\(661\) 27.0661 1.05275 0.526374 0.850253i \(-0.323551\pi\)
0.526374 + 0.850253i \(0.323551\pi\)
\(662\) −28.4052 −1.10400
\(663\) −2.92162 −0.113466
\(664\) −21.6658 −0.840796
\(665\) 0 0
\(666\) −65.6391 −2.54346
\(667\) −14.5113 −0.561880
\(668\) 3.22219 0.124670
\(669\) 27.7731 1.07377
\(670\) 0 0
\(671\) −20.4391 −0.789042
\(672\) 11.1773 0.431173
\(673\) −16.1711 −0.623351 −0.311676 0.950189i \(-0.600890\pi\)
−0.311676 + 0.950189i \(0.600890\pi\)
\(674\) −24.4079 −0.940156
\(675\) 0 0
\(676\) 0.369102 0.0141962
\(677\) −43.1194 −1.65721 −0.828607 0.559831i \(-0.810866\pi\)
−0.828607 + 0.559831i \(0.810866\pi\)
\(678\) −57.3751 −2.20348
\(679\) −10.1112 −0.388032
\(680\) 0 0
\(681\) −54.6102 −2.09267
\(682\) −3.43680 −0.131602
\(683\) 17.7093 0.677627 0.338813 0.940854i \(-0.389974\pi\)
0.338813 + 0.940854i \(0.389974\pi\)
\(684\) −1.40295 −0.0536432
\(685\) 0 0
\(686\) 29.1461 1.11280
\(687\) 9.75872 0.372319
\(688\) −29.5993 −1.12846
\(689\) −8.49693 −0.323707
\(690\) 0 0
\(691\) −24.8794 −0.946456 −0.473228 0.880940i \(-0.656911\pi\)
−0.473228 + 0.880940i \(0.656911\pi\)
\(692\) 0.318817 0.0121196
\(693\) 30.5958 1.16224
\(694\) 14.9684 0.568193
\(695\) 0 0
\(696\) 40.8059 1.54674
\(697\) 1.16290 0.0440479
\(698\) −47.5753 −1.80075
\(699\) 60.0288 2.27050
\(700\) 0 0
\(701\) 33.0661 1.24889 0.624445 0.781069i \(-0.285325\pi\)
0.624445 + 0.781069i \(0.285325\pi\)
\(702\) −19.7587 −0.745745
\(703\) 3.26180 0.123021
\(704\) −15.3086 −0.576964
\(705\) 0 0
\(706\) 9.15902 0.344704
\(707\) 28.0989 1.05677
\(708\) −5.52586 −0.207674
\(709\) 2.18342 0.0820000 0.0410000 0.999159i \(-0.486946\pi\)
0.0410000 + 0.999159i \(0.486946\pi\)
\(710\) 0 0
\(711\) 3.50307 0.131375
\(712\) −32.2245 −1.20766
\(713\) 2.48852 0.0931957
\(714\) −7.68649 −0.287660
\(715\) 0 0
\(716\) 7.35085 0.274714
\(717\) 21.0433 0.785877
\(718\) 16.8976 0.630613
\(719\) 5.20847 0.194243 0.0971216 0.995273i \(-0.469036\pi\)
0.0971216 + 0.995273i \(0.469036\pi\)
\(720\) 0 0
\(721\) 17.4186 0.648701
\(722\) −28.7971 −1.07172
\(723\) −30.0410 −1.11724
\(724\) 5.31124 0.197391
\(725\) 0 0
\(726\) −22.2134 −0.824416
\(727\) 3.52464 0.130721 0.0653607 0.997862i \(-0.479180\pi\)
0.0653607 + 0.997862i \(0.479180\pi\)
\(728\) −4.29072 −0.159025
\(729\) 15.7070 0.581741
\(730\) 0 0
\(731\) 5.92777 0.219246
\(732\) 9.41855 0.348120
\(733\) 21.8310 0.806345 0.403172 0.915124i \(-0.367907\pi\)
0.403172 + 0.915124i \(0.367907\pi\)
\(734\) 15.9676 0.589374
\(735\) 0 0
\(736\) −5.83749 −0.215173
\(737\) −19.9733 −0.735727
\(738\) 13.6910 0.503974
\(739\) 50.3533 1.85228 0.926139 0.377184i \(-0.123107\pi\)
0.926139 + 0.377184i \(0.123107\pi\)
\(740\) 0 0
\(741\) 1.70928 0.0627918
\(742\) −22.3545 −0.820661
\(743\) −30.7877 −1.12949 −0.564745 0.825266i \(-0.691025\pi\)
−0.564745 + 0.825266i \(0.691025\pi\)
\(744\) −6.99773 −0.256549
\(745\) 0 0
\(746\) 36.9216 1.35180
\(747\) 60.8431 2.22613
\(748\) −0.863763 −0.0315823
\(749\) −16.6681 −0.609038
\(750\) 0 0
\(751\) −10.6225 −0.387620 −0.193810 0.981039i \(-0.562085\pi\)
−0.193810 + 0.981039i \(0.562085\pi\)
\(752\) −26.2739 −0.958111
\(753\) 93.3894 3.40330
\(754\) 7.89269 0.287435
\(755\) 0 0
\(756\) −8.09890 −0.294554
\(757\) 7.98562 0.290242 0.145121 0.989414i \(-0.453643\pi\)
0.145121 + 0.989414i \(0.453643\pi\)
\(758\) 45.8024 1.66362
\(759\) −22.7792 −0.826834
\(760\) 0 0
\(761\) −48.9360 −1.77393 −0.886964 0.461838i \(-0.847190\pi\)
−0.886964 + 0.461838i \(0.847190\pi\)
\(762\) −87.8636 −3.18296
\(763\) 28.8248 1.04353
\(764\) 0.544946 0.0197155
\(765\) 0 0
\(766\) 19.1110 0.690509
\(767\) 4.72261 0.170523
\(768\) 27.1845 0.980935
\(769\) −7.99547 −0.288324 −0.144162 0.989554i \(-0.546049\pi\)
−0.144162 + 0.989554i \(0.546049\pi\)
\(770\) 0 0
\(771\) −64.8781 −2.33653
\(772\) −6.54495 −0.235558
\(773\) −26.6141 −0.957242 −0.478621 0.878022i \(-0.658863\pi\)
−0.478621 + 0.878022i \(0.658863\pi\)
\(774\) 69.7887 2.50850
\(775\) 0 0
\(776\) −14.8494 −0.533062
\(777\) 32.7792 1.17595
\(778\) −25.9155 −0.929115
\(779\) −0.680346 −0.0243759
\(780\) 0 0
\(781\) 36.7708 1.31576
\(782\) 4.01438 0.143554
\(783\) −65.8264 −2.35244
\(784\) 18.7686 0.670306
\(785\) 0 0
\(786\) 69.5585 2.48107
\(787\) −9.25792 −0.330009 −0.165005 0.986293i \(-0.552764\pi\)
−0.165005 + 0.986293i \(0.552764\pi\)
\(788\) 0.738205 0.0262975
\(789\) 28.9854 1.03191
\(790\) 0 0
\(791\) 20.0989 0.714634
\(792\) 44.9333 1.59664
\(793\) −8.04945 −0.285844
\(794\) −5.99159 −0.212634
\(795\) 0 0
\(796\) −1.99016 −0.0705393
\(797\) −15.9421 −0.564700 −0.282350 0.959312i \(-0.591114\pi\)
−0.282350 + 0.959312i \(0.591114\pi\)
\(798\) 4.49693 0.159190
\(799\) 5.26180 0.186149
\(800\) 0 0
\(801\) 90.4945 3.19747
\(802\) −14.0144 −0.494865
\(803\) 4.95282 0.174781
\(804\) 9.20394 0.324598
\(805\) 0 0
\(806\) −1.35350 −0.0476751
\(807\) 36.1133 1.27125
\(808\) 41.2663 1.45174
\(809\) 17.9239 0.630170 0.315085 0.949063i \(-0.397967\pi\)
0.315085 + 0.949063i \(0.397967\pi\)
\(810\) 0 0
\(811\) 7.43415 0.261048 0.130524 0.991445i \(-0.458334\pi\)
0.130524 + 0.991445i \(0.458334\pi\)
\(812\) 3.23513 0.113531
\(813\) −66.9998 −2.34979
\(814\) 23.6430 0.828687
\(815\) 0 0
\(816\) −13.4452 −0.470677
\(817\) −3.46800 −0.121330
\(818\) −29.8888 −1.04504
\(819\) 12.0494 0.421042
\(820\) 0 0
\(821\) 20.4801 0.714761 0.357380 0.933959i \(-0.383670\pi\)
0.357380 + 0.933959i \(0.383670\pi\)
\(822\) 67.2639 2.34610
\(823\) 3.75154 0.130770 0.0653852 0.997860i \(-0.479172\pi\)
0.0653852 + 0.997860i \(0.479172\pi\)
\(824\) 25.5811 0.891159
\(825\) 0 0
\(826\) 12.4247 0.432310
\(827\) 48.1483 1.67428 0.837141 0.546987i \(-0.184225\pi\)
0.837141 + 0.546987i \(0.184225\pi\)
\(828\) 7.36334 0.255894
\(829\) 36.5608 1.26981 0.634904 0.772591i \(-0.281040\pi\)
0.634904 + 0.772591i \(0.281040\pi\)
\(830\) 0 0
\(831\) −41.4596 −1.43822
\(832\) −6.02893 −0.209016
\(833\) −3.75872 −0.130232
\(834\) −32.4657 −1.12420
\(835\) 0 0
\(836\) 0.505339 0.0174775
\(837\) 11.2885 0.390186
\(838\) −25.8264 −0.892159
\(839\) 45.2294 1.56149 0.780746 0.624849i \(-0.214839\pi\)
0.780746 + 0.624849i \(0.214839\pi\)
\(840\) 0 0
\(841\) −2.70540 −0.0932896
\(842\) −29.2762 −1.00892
\(843\) −2.15676 −0.0742826
\(844\) −8.39803 −0.289072
\(845\) 0 0
\(846\) 61.9481 2.12982
\(847\) 7.78151 0.267376
\(848\) −39.1026 −1.34279
\(849\) −61.1689 −2.09931
\(850\) 0 0
\(851\) −17.1194 −0.586846
\(852\) −16.9444 −0.580506
\(853\) 37.2534 1.27553 0.637766 0.770230i \(-0.279859\pi\)
0.637766 + 0.770230i \(0.279859\pi\)
\(854\) −21.1773 −0.724671
\(855\) 0 0
\(856\) −24.4789 −0.836671
\(857\) −10.8371 −0.370188 −0.185094 0.982721i \(-0.559259\pi\)
−0.185094 + 0.982721i \(0.559259\pi\)
\(858\) 12.3896 0.422975
\(859\) 14.6081 0.498422 0.249211 0.968449i \(-0.419829\pi\)
0.249211 + 0.968449i \(0.419829\pi\)
\(860\) 0 0
\(861\) −6.83710 −0.233008
\(862\) 12.3596 0.420971
\(863\) 10.3440 0.352116 0.176058 0.984380i \(-0.443665\pi\)
0.176058 + 0.984380i \(0.443665\pi\)
\(864\) −26.4801 −0.900872
\(865\) 0 0
\(866\) 20.0821 0.682417
\(867\) −51.1988 −1.73880
\(868\) −0.554787 −0.0188307
\(869\) −1.26180 −0.0428035
\(870\) 0 0
\(871\) −7.86603 −0.266530
\(872\) 42.3324 1.43356
\(873\) 41.7009 1.41136
\(874\) −2.34858 −0.0794420
\(875\) 0 0
\(876\) −2.28231 −0.0771122
\(877\) −38.0677 −1.28545 −0.642727 0.766095i \(-0.722197\pi\)
−0.642727 + 0.766095i \(0.722197\pi\)
\(878\) −11.8531 −0.400022
\(879\) −30.0144 −1.01236
\(880\) 0 0
\(881\) 12.0494 0.405956 0.202978 0.979183i \(-0.434938\pi\)
0.202978 + 0.979183i \(0.434938\pi\)
\(882\) −44.2522 −1.49005
\(883\) 0.320699 0.0107924 0.00539619 0.999985i \(-0.498282\pi\)
0.00539619 + 0.999985i \(0.498282\pi\)
\(884\) −0.340173 −0.0114413
\(885\) 0 0
\(886\) 9.83672 0.330471
\(887\) −3.62144 −0.121596 −0.0607981 0.998150i \(-0.519365\pi\)
−0.0607981 + 0.998150i \(0.519365\pi\)
\(888\) 48.1399 1.61547
\(889\) 30.7792 1.03230
\(890\) 0 0
\(891\) −49.6319 −1.66273
\(892\) 3.23370 0.108272
\(893\) −3.07838 −0.103014
\(894\) −44.2967 −1.48150
\(895\) 0 0
\(896\) −22.9132 −0.765477
\(897\) −8.97107 −0.299535
\(898\) 48.6635 1.62392
\(899\) −4.50921 −0.150391
\(900\) 0 0
\(901\) 7.83096 0.260887
\(902\) −4.93146 −0.164200
\(903\) −34.8515 −1.15978
\(904\) 29.5174 0.981736
\(905\) 0 0
\(906\) 15.9916 0.531285
\(907\) −10.9333 −0.363036 −0.181518 0.983388i \(-0.558101\pi\)
−0.181518 + 0.983388i \(0.558101\pi\)
\(908\) −6.35842 −0.211012
\(909\) −115.886 −3.84371
\(910\) 0 0
\(911\) −37.5897 −1.24540 −0.622701 0.782460i \(-0.713965\pi\)
−0.622701 + 0.782460i \(0.713965\pi\)
\(912\) 7.86603 0.260470
\(913\) −21.9155 −0.725297
\(914\) 54.8613 1.81465
\(915\) 0 0
\(916\) 1.13624 0.0375423
\(917\) −24.3668 −0.804664
\(918\) 18.2101 0.601022
\(919\) 33.6742 1.11081 0.555405 0.831580i \(-0.312563\pi\)
0.555405 + 0.831580i \(0.312563\pi\)
\(920\) 0 0
\(921\) −0.837101 −0.0275834
\(922\) −23.0563 −0.759317
\(923\) 14.4813 0.476659
\(924\) 5.07838 0.167066
\(925\) 0 0
\(926\) 14.0014 0.460116
\(927\) −71.8381 −2.35947
\(928\) 10.5776 0.347226
\(929\) −43.2039 −1.41748 −0.708738 0.705472i \(-0.750735\pi\)
−0.708738 + 0.705472i \(0.750735\pi\)
\(930\) 0 0
\(931\) 2.19902 0.0720698
\(932\) 6.98932 0.228943
\(933\) 41.3607 1.35409
\(934\) −2.88920 −0.0945376
\(935\) 0 0
\(936\) 17.6959 0.578410
\(937\) 27.5630 0.900445 0.450222 0.892916i \(-0.351345\pi\)
0.450222 + 0.892916i \(0.351345\pi\)
\(938\) −20.6947 −0.675707
\(939\) −106.835 −3.48642
\(940\) 0 0
\(941\) 58.1666 1.89618 0.948088 0.318007i \(-0.103013\pi\)
0.948088 + 0.318007i \(0.103013\pi\)
\(942\) 62.6369 2.04082
\(943\) 3.57077 0.116280
\(944\) 21.7333 0.707358
\(945\) 0 0
\(946\) −25.1377 −0.817296
\(947\) 48.5152 1.57653 0.788266 0.615335i \(-0.210979\pi\)
0.788266 + 0.615335i \(0.210979\pi\)
\(948\) 0.581449 0.0188846
\(949\) 1.95055 0.0633176
\(950\) 0 0
\(951\) −44.2245 −1.43408
\(952\) 3.95443 0.128164
\(953\) −23.0349 −0.746173 −0.373087 0.927796i \(-0.621701\pi\)
−0.373087 + 0.927796i \(0.621701\pi\)
\(954\) 92.1953 2.98493
\(955\) 0 0
\(956\) 2.45013 0.0792430
\(957\) 41.2762 1.33427
\(958\) −24.2316 −0.782889
\(959\) −23.5630 −0.760890
\(960\) 0 0
\(961\) −30.2267 −0.975056
\(962\) 9.31124 0.300207
\(963\) 68.7429 2.21521
\(964\) −3.49777 −0.112655
\(965\) 0 0
\(966\) −23.6020 −0.759381
\(967\) 54.9998 1.76868 0.884338 0.466848i \(-0.154611\pi\)
0.884338 + 0.466848i \(0.154611\pi\)
\(968\) 11.4280 0.367310
\(969\) −1.57531 −0.0506061
\(970\) 0 0
\(971\) 9.70540 0.311461 0.155731 0.987800i \(-0.450227\pi\)
0.155731 + 0.987800i \(0.450227\pi\)
\(972\) 8.65634 0.277652
\(973\) 11.3730 0.364601
\(974\) −7.61038 −0.243852
\(975\) 0 0
\(976\) −37.0433 −1.18573
\(977\) 32.2062 1.03037 0.515184 0.857080i \(-0.327724\pi\)
0.515184 + 0.857080i \(0.327724\pi\)
\(978\) 58.7936 1.88001
\(979\) −32.5958 −1.04177
\(980\) 0 0
\(981\) −118.880 −3.79555
\(982\) 60.7768 1.93947
\(983\) 44.0782 1.40588 0.702938 0.711251i \(-0.251871\pi\)
0.702938 + 0.711251i \(0.251871\pi\)
\(984\) −10.0410 −0.320097
\(985\) 0 0
\(986\) −7.27408 −0.231654
\(987\) −30.9360 −0.984704
\(988\) 0.199016 0.00633154
\(989\) 18.2017 0.578779
\(990\) 0 0
\(991\) 12.0677 0.383343 0.191672 0.981459i \(-0.438609\pi\)
0.191672 + 0.981459i \(0.438609\pi\)
\(992\) −1.81393 −0.0575923
\(993\) −58.5029 −1.85653
\(994\) 38.0989 1.20842
\(995\) 0 0
\(996\) 10.0989 0.319996
\(997\) 45.7587 1.44919 0.724597 0.689173i \(-0.242026\pi\)
0.724597 + 0.689173i \(0.242026\pi\)
\(998\) 2.57182 0.0814094
\(999\) −77.6574 −2.45697
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 325.2.a.k.1.2 3
3.2 odd 2 2925.2.a.bf.1.2 3
4.3 odd 2 5200.2.a.cb.1.1 3
5.2 odd 4 65.2.b.a.14.5 yes 6
5.3 odd 4 65.2.b.a.14.2 6
5.4 even 2 325.2.a.j.1.2 3
13.12 even 2 4225.2.a.ba.1.2 3
15.2 even 4 585.2.c.b.469.2 6
15.8 even 4 585.2.c.b.469.5 6
15.14 odd 2 2925.2.a.bj.1.2 3
20.3 even 4 1040.2.d.c.209.1 6
20.7 even 4 1040.2.d.c.209.6 6
20.19 odd 2 5200.2.a.cj.1.3 3
65.2 even 12 845.2.l.d.654.4 12
65.3 odd 12 845.2.n.f.529.5 12
65.7 even 12 845.2.l.e.699.3 12
65.8 even 4 845.2.d.b.844.4 6
65.12 odd 4 845.2.b.c.339.2 6
65.17 odd 12 845.2.n.g.484.2 12
65.18 even 4 845.2.d.a.844.4 6
65.22 odd 12 845.2.n.f.484.5 12
65.23 odd 12 845.2.n.g.529.2 12
65.28 even 12 845.2.l.e.654.3 12
65.32 even 12 845.2.l.d.699.3 12
65.33 even 12 845.2.l.d.699.4 12
65.37 even 12 845.2.l.e.654.4 12
65.38 odd 4 845.2.b.c.339.5 6
65.42 odd 12 845.2.n.f.529.2 12
65.43 odd 12 845.2.n.g.484.5 12
65.47 even 4 845.2.d.a.844.3 6
65.48 odd 12 845.2.n.f.484.2 12
65.57 even 4 845.2.d.b.844.3 6
65.58 even 12 845.2.l.e.699.4 12
65.62 odd 12 845.2.n.g.529.5 12
65.63 even 12 845.2.l.d.654.3 12
65.64 even 2 4225.2.a.bh.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.b.a.14.2 6 5.3 odd 4
65.2.b.a.14.5 yes 6 5.2 odd 4
325.2.a.j.1.2 3 5.4 even 2
325.2.a.k.1.2 3 1.1 even 1 trivial
585.2.c.b.469.2 6 15.2 even 4
585.2.c.b.469.5 6 15.8 even 4
845.2.b.c.339.2 6 65.12 odd 4
845.2.b.c.339.5 6 65.38 odd 4
845.2.d.a.844.3 6 65.47 even 4
845.2.d.a.844.4 6 65.18 even 4
845.2.d.b.844.3 6 65.57 even 4
845.2.d.b.844.4 6 65.8 even 4
845.2.l.d.654.3 12 65.63 even 12
845.2.l.d.654.4 12 65.2 even 12
845.2.l.d.699.3 12 65.32 even 12
845.2.l.d.699.4 12 65.33 even 12
845.2.l.e.654.3 12 65.28 even 12
845.2.l.e.654.4 12 65.37 even 12
845.2.l.e.699.3 12 65.7 even 12
845.2.l.e.699.4 12 65.58 even 12
845.2.n.f.484.2 12 65.48 odd 12
845.2.n.f.484.5 12 65.22 odd 12
845.2.n.f.529.2 12 65.42 odd 12
845.2.n.f.529.5 12 65.3 odd 12
845.2.n.g.484.2 12 65.17 odd 12
845.2.n.g.484.5 12 65.43 odd 12
845.2.n.g.529.2 12 65.23 odd 12
845.2.n.g.529.5 12 65.62 odd 12
1040.2.d.c.209.1 6 20.3 even 4
1040.2.d.c.209.6 6 20.7 even 4
2925.2.a.bf.1.2 3 3.2 odd 2
2925.2.a.bj.1.2 3 15.14 odd 2
4225.2.a.ba.1.2 3 13.12 even 2
4225.2.a.bh.1.2 3 65.64 even 2
5200.2.a.cb.1.1 3 4.3 odd 2
5200.2.a.cj.1.3 3 20.19 odd 2