Properties

Label 4025.2.a.bb.1.6
Level $4025$
Weight $2$
Character 4025.1
Self dual yes
Analytic conductor $32.140$
Analytic rank $0$
Dimension $14$
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] = [4025,2,Mod(1,4025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4025, 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("4025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4025.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: \(32.1397868136\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} - 22 x^{12} + 18 x^{11} + 187 x^{10} - 118 x^{9} - 772 x^{8} + 346 x^{7} + 1581 x^{6} - 443 x^{5} - 1429 x^{4} + 193 x^{3} + 386 x^{2} - 3 x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-0.582976\) of defining polynomial
Character \(\chi\) \(=\) 4025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.582976 q^{2} -0.367598 q^{3} -1.66014 q^{4} +0.214301 q^{6} +1.00000 q^{7} +2.13377 q^{8} -2.86487 q^{9} +O(q^{10})\) \(q-0.582976 q^{2} -0.367598 q^{3} -1.66014 q^{4} +0.214301 q^{6} +1.00000 q^{7} +2.13377 q^{8} -2.86487 q^{9} +0.0267004 q^{11} +0.610264 q^{12} -3.10813 q^{13} -0.582976 q^{14} +2.07634 q^{16} -8.10612 q^{17} +1.67015 q^{18} -4.14248 q^{19} -0.367598 q^{21} -0.0155657 q^{22} -1.00000 q^{23} -0.784372 q^{24} +1.81197 q^{26} +2.15592 q^{27} -1.66014 q^{28} -1.48754 q^{29} +6.35245 q^{31} -5.47800 q^{32} -0.00981502 q^{33} +4.72568 q^{34} +4.75608 q^{36} +10.8928 q^{37} +2.41497 q^{38} +1.14254 q^{39} -8.32684 q^{41} +0.214301 q^{42} -3.73815 q^{43} -0.0443264 q^{44} +0.582976 q^{46} -11.9018 q^{47} -0.763258 q^{48} +1.00000 q^{49} +2.97980 q^{51} +5.15992 q^{52} +13.1021 q^{53} -1.25685 q^{54} +2.13377 q^{56} +1.52277 q^{57} +0.867201 q^{58} -1.06393 q^{59} +0.897659 q^{61} -3.70333 q^{62} -2.86487 q^{63} -0.959128 q^{64} +0.00572192 q^{66} +12.4481 q^{67} +13.4573 q^{68} +0.367598 q^{69} -11.6084 q^{71} -6.11299 q^{72} +3.27293 q^{73} -6.35025 q^{74} +6.87709 q^{76} +0.0267004 q^{77} -0.666075 q^{78} -4.17218 q^{79} +7.80210 q^{81} +4.85435 q^{82} -0.986115 q^{83} +0.610264 q^{84} +2.17925 q^{86} +0.546817 q^{87} +0.0569726 q^{88} -17.0415 q^{89} -3.10813 q^{91} +1.66014 q^{92} -2.33515 q^{93} +6.93848 q^{94} +2.01370 q^{96} -7.08771 q^{97} -0.582976 q^{98} -0.0764932 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + q^{2} + 6 q^{3} + 17 q^{4} - 4 q^{6} + 14 q^{7} + 9 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + q^{2} + 6 q^{3} + 17 q^{4} - 4 q^{6} + 14 q^{7} + 9 q^{8} + 18 q^{9} - 3 q^{11} + 11 q^{12} + 15 q^{13} + q^{14} + 23 q^{16} + 9 q^{17} + 17 q^{18} - 4 q^{19} + 6 q^{21} + 9 q^{22} - 14 q^{23} + 10 q^{24} - 5 q^{26} + 33 q^{27} + 17 q^{28} + 11 q^{29} - q^{31} + 24 q^{32} + 26 q^{33} - 6 q^{34} + 13 q^{36} + 18 q^{37} - 6 q^{38} + 6 q^{39} - 7 q^{41} - 4 q^{42} + 18 q^{43} - 16 q^{44} - q^{46} + 10 q^{47} + 40 q^{48} + 14 q^{49} + 28 q^{51} + 46 q^{52} + 5 q^{53} - 24 q^{54} + 9 q^{56} - 26 q^{57} + 2 q^{58} - 24 q^{59} - 6 q^{61} - 16 q^{62} + 18 q^{63} + 29 q^{64} + 27 q^{66} + 61 q^{67} + 35 q^{68} - 6 q^{69} + 11 q^{71} + 12 q^{72} + 28 q^{73} - 49 q^{74} - 27 q^{76} - 3 q^{77} + 38 q^{78} + 6 q^{79} + 26 q^{81} - 14 q^{82} + 16 q^{83} + 11 q^{84} + 46 q^{86} + 61 q^{87} + 58 q^{88} - 39 q^{89} + 15 q^{91} - 17 q^{92} + 21 q^{93} - 74 q^{94} + 41 q^{96} + 19 q^{97} + q^{98} - 9 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.582976 −0.412227 −0.206113 0.978528i \(-0.566082\pi\)
−0.206113 + 0.978528i \(0.566082\pi\)
\(3\) −0.367598 −0.212233 −0.106116 0.994354i \(-0.533842\pi\)
−0.106116 + 0.994354i \(0.533842\pi\)
\(4\) −1.66014 −0.830069
\(5\) 0 0
\(6\) 0.214301 0.0874880
\(7\) 1.00000 0.377964
\(8\) 2.13377 0.754403
\(9\) −2.86487 −0.954957
\(10\) 0 0
\(11\) 0.0267004 0.00805047 0.00402524 0.999992i \(-0.498719\pi\)
0.00402524 + 0.999992i \(0.498719\pi\)
\(12\) 0.610264 0.176168
\(13\) −3.10813 −0.862040 −0.431020 0.902342i \(-0.641846\pi\)
−0.431020 + 0.902342i \(0.641846\pi\)
\(14\) −0.582976 −0.155807
\(15\) 0 0
\(16\) 2.07634 0.519084
\(17\) −8.10612 −1.96602 −0.983012 0.183543i \(-0.941243\pi\)
−0.983012 + 0.183543i \(0.941243\pi\)
\(18\) 1.67015 0.393659
\(19\) −4.14248 −0.950349 −0.475175 0.879891i \(-0.657615\pi\)
−0.475175 + 0.879891i \(0.657615\pi\)
\(20\) 0 0
\(21\) −0.367598 −0.0802165
\(22\) −0.0155657 −0.00331862
\(23\) −1.00000 −0.208514
\(24\) −0.784372 −0.160109
\(25\) 0 0
\(26\) 1.81197 0.355356
\(27\) 2.15592 0.414906
\(28\) −1.66014 −0.313737
\(29\) −1.48754 −0.276229 −0.138115 0.990416i \(-0.544104\pi\)
−0.138115 + 0.990416i \(0.544104\pi\)
\(30\) 0 0
\(31\) 6.35245 1.14093 0.570467 0.821320i \(-0.306762\pi\)
0.570467 + 0.821320i \(0.306762\pi\)
\(32\) −5.47800 −0.968383
\(33\) −0.00981502 −0.00170858
\(34\) 4.72568 0.810447
\(35\) 0 0
\(36\) 4.75608 0.792681
\(37\) 10.8928 1.79077 0.895383 0.445296i \(-0.146902\pi\)
0.895383 + 0.445296i \(0.146902\pi\)
\(38\) 2.41497 0.391759
\(39\) 1.14254 0.182953
\(40\) 0 0
\(41\) −8.32684 −1.30043 −0.650217 0.759748i \(-0.725322\pi\)
−0.650217 + 0.759748i \(0.725322\pi\)
\(42\) 0.214301 0.0330674
\(43\) −3.73815 −0.570063 −0.285031 0.958518i \(-0.592004\pi\)
−0.285031 + 0.958518i \(0.592004\pi\)
\(44\) −0.0443264 −0.00668245
\(45\) 0 0
\(46\) 0.582976 0.0859552
\(47\) −11.9018 −1.73606 −0.868030 0.496512i \(-0.834614\pi\)
−0.868030 + 0.496512i \(0.834614\pi\)
\(48\) −0.763258 −0.110167
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.97980 0.417255
\(52\) 5.15992 0.715553
\(53\) 13.1021 1.79971 0.899853 0.436194i \(-0.143674\pi\)
0.899853 + 0.436194i \(0.143674\pi\)
\(54\) −1.25685 −0.171035
\(55\) 0 0
\(56\) 2.13377 0.285138
\(57\) 1.52277 0.201695
\(58\) 0.867201 0.113869
\(59\) −1.06393 −0.138512 −0.0692560 0.997599i \(-0.522063\pi\)
−0.0692560 + 0.997599i \(0.522063\pi\)
\(60\) 0 0
\(61\) 0.897659 0.114933 0.0574667 0.998347i \(-0.481698\pi\)
0.0574667 + 0.998347i \(0.481698\pi\)
\(62\) −3.70333 −0.470323
\(63\) −2.86487 −0.360940
\(64\) −0.959128 −0.119891
\(65\) 0 0
\(66\) 0.00572192 0.000704320 0
\(67\) 12.4481 1.52078 0.760388 0.649469i \(-0.225009\pi\)
0.760388 + 0.649469i \(0.225009\pi\)
\(68\) 13.4573 1.63194
\(69\) 0.367598 0.0442536
\(70\) 0 0
\(71\) −11.6084 −1.37766 −0.688832 0.724921i \(-0.741876\pi\)
−0.688832 + 0.724921i \(0.741876\pi\)
\(72\) −6.11299 −0.720423
\(73\) 3.27293 0.383068 0.191534 0.981486i \(-0.438654\pi\)
0.191534 + 0.981486i \(0.438654\pi\)
\(74\) −6.35025 −0.738202
\(75\) 0 0
\(76\) 6.87709 0.788856
\(77\) 0.0267004 0.00304279
\(78\) −0.666075 −0.0754182
\(79\) −4.17218 −0.469407 −0.234703 0.972067i \(-0.575412\pi\)
−0.234703 + 0.972067i \(0.575412\pi\)
\(80\) 0 0
\(81\) 7.80210 0.866900
\(82\) 4.85435 0.536074
\(83\) −0.986115 −0.108240 −0.0541201 0.998534i \(-0.517235\pi\)
−0.0541201 + 0.998534i \(0.517235\pi\)
\(84\) 0.610264 0.0665853
\(85\) 0 0
\(86\) 2.17925 0.234995
\(87\) 0.546817 0.0586250
\(88\) 0.0569726 0.00607330
\(89\) −17.0415 −1.80639 −0.903196 0.429228i \(-0.858786\pi\)
−0.903196 + 0.429228i \(0.858786\pi\)
\(90\) 0 0
\(91\) −3.10813 −0.325820
\(92\) 1.66014 0.173081
\(93\) −2.33515 −0.242144
\(94\) 6.93848 0.715650
\(95\) 0 0
\(96\) 2.01370 0.205523
\(97\) −7.08771 −0.719648 −0.359824 0.933020i \(-0.617163\pi\)
−0.359824 + 0.933020i \(0.617163\pi\)
\(98\) −0.582976 −0.0588895
\(99\) −0.0764932 −0.00768786
\(100\) 0 0
\(101\) 6.26346 0.623237 0.311619 0.950207i \(-0.399129\pi\)
0.311619 + 0.950207i \(0.399129\pi\)
\(102\) −1.73715 −0.172004
\(103\) −0.647440 −0.0637941 −0.0318971 0.999491i \(-0.510155\pi\)
−0.0318971 + 0.999491i \(0.510155\pi\)
\(104\) −6.63204 −0.650325
\(105\) 0 0
\(106\) −7.63819 −0.741886
\(107\) 17.8099 1.72175 0.860875 0.508817i \(-0.169917\pi\)
0.860875 + 0.508817i \(0.169917\pi\)
\(108\) −3.57912 −0.344401
\(109\) −4.16066 −0.398519 −0.199259 0.979947i \(-0.563854\pi\)
−0.199259 + 0.979947i \(0.563854\pi\)
\(110\) 0 0
\(111\) −4.00418 −0.380060
\(112\) 2.07634 0.196195
\(113\) 2.35008 0.221077 0.110539 0.993872i \(-0.464742\pi\)
0.110539 + 0.993872i \(0.464742\pi\)
\(114\) −0.887737 −0.0831442
\(115\) 0 0
\(116\) 2.46952 0.229290
\(117\) 8.90439 0.823211
\(118\) 0.620247 0.0570984
\(119\) −8.10612 −0.743087
\(120\) 0 0
\(121\) −10.9993 −0.999935
\(122\) −0.523314 −0.0473786
\(123\) 3.06093 0.275995
\(124\) −10.5460 −0.947054
\(125\) 0 0
\(126\) 1.67015 0.148789
\(127\) −11.0139 −0.977323 −0.488662 0.872473i \(-0.662515\pi\)
−0.488662 + 0.872473i \(0.662515\pi\)
\(128\) 11.5152 1.01781
\(129\) 1.37414 0.120986
\(130\) 0 0
\(131\) 8.25785 0.721492 0.360746 0.932664i \(-0.382522\pi\)
0.360746 + 0.932664i \(0.382522\pi\)
\(132\) 0.0162943 0.00141824
\(133\) −4.14248 −0.359198
\(134\) −7.25694 −0.626904
\(135\) 0 0
\(136\) −17.2966 −1.48317
\(137\) 18.1430 1.55006 0.775031 0.631923i \(-0.217734\pi\)
0.775031 + 0.631923i \(0.217734\pi\)
\(138\) −0.214301 −0.0182425
\(139\) −10.9562 −0.929292 −0.464646 0.885497i \(-0.653818\pi\)
−0.464646 + 0.885497i \(0.653818\pi\)
\(140\) 0 0
\(141\) 4.37509 0.368449
\(142\) 6.76742 0.567909
\(143\) −0.0829883 −0.00693983
\(144\) −5.94844 −0.495703
\(145\) 0 0
\(146\) −1.90804 −0.157911
\(147\) −0.367598 −0.0303190
\(148\) −18.0836 −1.48646
\(149\) 11.2894 0.924864 0.462432 0.886655i \(-0.346977\pi\)
0.462432 + 0.886655i \(0.346977\pi\)
\(150\) 0 0
\(151\) −3.60271 −0.293185 −0.146592 0.989197i \(-0.546831\pi\)
−0.146592 + 0.989197i \(0.546831\pi\)
\(152\) −8.83911 −0.716946
\(153\) 23.2230 1.87747
\(154\) −0.0155657 −0.00125432
\(155\) 0 0
\(156\) −1.89678 −0.151864
\(157\) −7.96765 −0.635888 −0.317944 0.948110i \(-0.602992\pi\)
−0.317944 + 0.948110i \(0.602992\pi\)
\(158\) 2.43228 0.193502
\(159\) −4.81629 −0.381957
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) −4.54844 −0.357359
\(163\) 15.4157 1.20745 0.603724 0.797193i \(-0.293683\pi\)
0.603724 + 0.797193i \(0.293683\pi\)
\(164\) 13.8237 1.07945
\(165\) 0 0
\(166\) 0.574882 0.0446195
\(167\) 14.6991 1.13745 0.568726 0.822527i \(-0.307436\pi\)
0.568726 + 0.822527i \(0.307436\pi\)
\(168\) −0.784372 −0.0605156
\(169\) −3.33954 −0.256888
\(170\) 0 0
\(171\) 11.8677 0.907543
\(172\) 6.20585 0.473192
\(173\) −6.31055 −0.479782 −0.239891 0.970800i \(-0.577112\pi\)
−0.239891 + 0.970800i \(0.577112\pi\)
\(174\) −0.318782 −0.0241668
\(175\) 0 0
\(176\) 0.0554390 0.00417888
\(177\) 0.391099 0.0293968
\(178\) 9.93478 0.744643
\(179\) 12.0397 0.899892 0.449946 0.893056i \(-0.351443\pi\)
0.449946 + 0.893056i \(0.351443\pi\)
\(180\) 0 0
\(181\) 17.4656 1.29821 0.649103 0.760700i \(-0.275144\pi\)
0.649103 + 0.760700i \(0.275144\pi\)
\(182\) 1.81197 0.134312
\(183\) −0.329978 −0.0243927
\(184\) −2.13377 −0.157304
\(185\) 0 0
\(186\) 1.36134 0.0998181
\(187\) −0.216437 −0.0158274
\(188\) 19.7587 1.44105
\(189\) 2.15592 0.156820
\(190\) 0 0
\(191\) −10.2513 −0.741756 −0.370878 0.928682i \(-0.620943\pi\)
−0.370878 + 0.928682i \(0.620943\pi\)
\(192\) 0.352574 0.0254448
\(193\) 12.8854 0.927514 0.463757 0.885963i \(-0.346501\pi\)
0.463757 + 0.885963i \(0.346501\pi\)
\(194\) 4.13197 0.296658
\(195\) 0 0
\(196\) −1.66014 −0.118581
\(197\) 18.5512 1.32172 0.660858 0.750511i \(-0.270192\pi\)
0.660858 + 0.750511i \(0.270192\pi\)
\(198\) 0.0445937 0.00316914
\(199\) 4.77181 0.338265 0.169132 0.985593i \(-0.445903\pi\)
0.169132 + 0.985593i \(0.445903\pi\)
\(200\) 0 0
\(201\) −4.57589 −0.322759
\(202\) −3.65145 −0.256915
\(203\) −1.48754 −0.104405
\(204\) −4.94687 −0.346350
\(205\) 0 0
\(206\) 0.377442 0.0262976
\(207\) 2.86487 0.199122
\(208\) −6.45352 −0.447471
\(209\) −0.110606 −0.00765076
\(210\) 0 0
\(211\) 1.82154 0.125400 0.0626999 0.998032i \(-0.480029\pi\)
0.0626999 + 0.998032i \(0.480029\pi\)
\(212\) −21.7512 −1.49388
\(213\) 4.26723 0.292386
\(214\) −10.3828 −0.709751
\(215\) 0 0
\(216\) 4.60024 0.313007
\(217\) 6.35245 0.431233
\(218\) 2.42556 0.164280
\(219\) −1.20312 −0.0812996
\(220\) 0 0
\(221\) 25.1949 1.69479
\(222\) 2.33434 0.156671
\(223\) 26.9390 1.80397 0.901983 0.431771i \(-0.142111\pi\)
0.901983 + 0.431771i \(0.142111\pi\)
\(224\) −5.47800 −0.366015
\(225\) 0 0
\(226\) −1.37004 −0.0911339
\(227\) −0.201134 −0.0133497 −0.00667486 0.999978i \(-0.502125\pi\)
−0.00667486 + 0.999978i \(0.502125\pi\)
\(228\) −2.52800 −0.167421
\(229\) 25.8753 1.70989 0.854943 0.518723i \(-0.173592\pi\)
0.854943 + 0.518723i \(0.173592\pi\)
\(230\) 0 0
\(231\) −0.00981502 −0.000645781 0
\(232\) −3.17408 −0.208388
\(233\) −8.30873 −0.544323 −0.272162 0.962252i \(-0.587739\pi\)
−0.272162 + 0.962252i \(0.587739\pi\)
\(234\) −5.19105 −0.339349
\(235\) 0 0
\(236\) 1.76627 0.114975
\(237\) 1.53369 0.0996236
\(238\) 4.72568 0.306320
\(239\) −18.5838 −1.20209 −0.601043 0.799217i \(-0.705248\pi\)
−0.601043 + 0.799217i \(0.705248\pi\)
\(240\) 0 0
\(241\) 24.9684 1.60836 0.804178 0.594389i \(-0.202606\pi\)
0.804178 + 0.594389i \(0.202606\pi\)
\(242\) 6.41232 0.412200
\(243\) −9.33579 −0.598891
\(244\) −1.49024 −0.0954027
\(245\) 0 0
\(246\) −1.78445 −0.113772
\(247\) 12.8753 0.819239
\(248\) 13.5547 0.860724
\(249\) 0.362494 0.0229721
\(250\) 0 0
\(251\) 3.55142 0.224163 0.112082 0.993699i \(-0.464248\pi\)
0.112082 + 0.993699i \(0.464248\pi\)
\(252\) 4.75608 0.299605
\(253\) −0.0267004 −0.00167864
\(254\) 6.42083 0.402879
\(255\) 0 0
\(256\) −4.79481 −0.299676
\(257\) −13.0752 −0.815607 −0.407803 0.913070i \(-0.633705\pi\)
−0.407803 + 0.913070i \(0.633705\pi\)
\(258\) −0.801090 −0.0498737
\(259\) 10.8928 0.676846
\(260\) 0 0
\(261\) 4.26161 0.263787
\(262\) −4.81413 −0.297418
\(263\) −14.6392 −0.902693 −0.451347 0.892349i \(-0.649056\pi\)
−0.451347 + 0.892349i \(0.649056\pi\)
\(264\) −0.0209430 −0.00128895
\(265\) 0 0
\(266\) 2.41497 0.148071
\(267\) 6.26442 0.383376
\(268\) −20.6655 −1.26235
\(269\) 4.50049 0.274400 0.137200 0.990543i \(-0.456190\pi\)
0.137200 + 0.990543i \(0.456190\pi\)
\(270\) 0 0
\(271\) −17.3892 −1.05632 −0.528161 0.849144i \(-0.677118\pi\)
−0.528161 + 0.849144i \(0.677118\pi\)
\(272\) −16.8310 −1.02053
\(273\) 1.14254 0.0691498
\(274\) −10.5770 −0.638977
\(275\) 0 0
\(276\) −0.610264 −0.0367336
\(277\) 9.94437 0.597499 0.298750 0.954332i \(-0.403430\pi\)
0.298750 + 0.954332i \(0.403430\pi\)
\(278\) 6.38720 0.383079
\(279\) −18.1990 −1.08954
\(280\) 0 0
\(281\) 4.03201 0.240530 0.120265 0.992742i \(-0.461626\pi\)
0.120265 + 0.992742i \(0.461626\pi\)
\(282\) −2.55057 −0.151884
\(283\) −18.5270 −1.10132 −0.550659 0.834730i \(-0.685623\pi\)
−0.550659 + 0.834730i \(0.685623\pi\)
\(284\) 19.2715 1.14356
\(285\) 0 0
\(286\) 0.0483802 0.00286078
\(287\) −8.32684 −0.491518
\(288\) 15.6938 0.924765
\(289\) 48.7092 2.86525
\(290\) 0 0
\(291\) 2.60543 0.152733
\(292\) −5.43352 −0.317973
\(293\) −9.00421 −0.526032 −0.263016 0.964792i \(-0.584717\pi\)
−0.263016 + 0.964792i \(0.584717\pi\)
\(294\) 0.214301 0.0124983
\(295\) 0 0
\(296\) 23.2428 1.35096
\(297\) 0.0575638 0.00334019
\(298\) −6.58145 −0.381253
\(299\) 3.10813 0.179748
\(300\) 0 0
\(301\) −3.73815 −0.215464
\(302\) 2.10030 0.120858
\(303\) −2.30244 −0.132271
\(304\) −8.60118 −0.493311
\(305\) 0 0
\(306\) −13.5385 −0.773942
\(307\) 16.5249 0.943123 0.471562 0.881833i \(-0.343691\pi\)
0.471562 + 0.881833i \(0.343691\pi\)
\(308\) −0.0443264 −0.00252573
\(309\) 0.237998 0.0135392
\(310\) 0 0
\(311\) 9.67017 0.548345 0.274173 0.961680i \(-0.411596\pi\)
0.274173 + 0.961680i \(0.411596\pi\)
\(312\) 2.43793 0.138020
\(313\) 21.6189 1.22197 0.610986 0.791642i \(-0.290773\pi\)
0.610986 + 0.791642i \(0.290773\pi\)
\(314\) 4.64495 0.262130
\(315\) 0 0
\(316\) 6.92639 0.389640
\(317\) 2.10547 0.118255 0.0591276 0.998250i \(-0.481168\pi\)
0.0591276 + 0.998250i \(0.481168\pi\)
\(318\) 2.80778 0.157453
\(319\) −0.0397179 −0.00222378
\(320\) 0 0
\(321\) −6.54689 −0.365412
\(322\) 0.582976 0.0324880
\(323\) 33.5794 1.86841
\(324\) −12.9526 −0.719587
\(325\) 0 0
\(326\) −8.98697 −0.497742
\(327\) 1.52945 0.0845788
\(328\) −17.7676 −0.981052
\(329\) −11.9018 −0.656169
\(330\) 0 0
\(331\) 32.3019 1.77547 0.887737 0.460352i \(-0.152277\pi\)
0.887737 + 0.460352i \(0.152277\pi\)
\(332\) 1.63709 0.0898469
\(333\) −31.2065 −1.71011
\(334\) −8.56925 −0.468888
\(335\) 0 0
\(336\) −0.763258 −0.0416391
\(337\) 19.8860 1.08326 0.541629 0.840618i \(-0.317808\pi\)
0.541629 + 0.840618i \(0.317808\pi\)
\(338\) 1.94687 0.105896
\(339\) −0.863886 −0.0469198
\(340\) 0 0
\(341\) 0.169613 0.00918506
\(342\) −6.91857 −0.374113
\(343\) 1.00000 0.0539949
\(344\) −7.97637 −0.430057
\(345\) 0 0
\(346\) 3.67890 0.197779
\(347\) −3.71949 −0.199672 −0.0998362 0.995004i \(-0.531832\pi\)
−0.0998362 + 0.995004i \(0.531832\pi\)
\(348\) −0.907793 −0.0486628
\(349\) 14.6645 0.784975 0.392487 0.919757i \(-0.371615\pi\)
0.392487 + 0.919757i \(0.371615\pi\)
\(350\) 0 0
\(351\) −6.70086 −0.357666
\(352\) −0.146265 −0.00779595
\(353\) 26.0664 1.38738 0.693688 0.720275i \(-0.255984\pi\)
0.693688 + 0.720275i \(0.255984\pi\)
\(354\) −0.228002 −0.0121182
\(355\) 0 0
\(356\) 28.2912 1.49943
\(357\) 2.97980 0.157708
\(358\) −7.01888 −0.370959
\(359\) −16.6347 −0.877944 −0.438972 0.898501i \(-0.644657\pi\)
−0.438972 + 0.898501i \(0.644657\pi\)
\(360\) 0 0
\(361\) −1.83989 −0.0968361
\(362\) −10.1820 −0.535155
\(363\) 4.04332 0.212219
\(364\) 5.15992 0.270453
\(365\) 0 0
\(366\) 0.192369 0.0100553
\(367\) 31.7676 1.65825 0.829127 0.559061i \(-0.188838\pi\)
0.829127 + 0.559061i \(0.188838\pi\)
\(368\) −2.07634 −0.108237
\(369\) 23.8553 1.24186
\(370\) 0 0
\(371\) 13.1021 0.680225
\(372\) 3.87667 0.200996
\(373\) 0.115224 0.00596609 0.00298304 0.999996i \(-0.499050\pi\)
0.00298304 + 0.999996i \(0.499050\pi\)
\(374\) 0.126177 0.00652448
\(375\) 0 0
\(376\) −25.3958 −1.30969
\(377\) 4.62347 0.238121
\(378\) −1.25685 −0.0646453
\(379\) −37.6368 −1.93327 −0.966635 0.256157i \(-0.917544\pi\)
−0.966635 + 0.256157i \(0.917544\pi\)
\(380\) 0 0
\(381\) 4.04868 0.207420
\(382\) 5.97625 0.305772
\(383\) 34.0526 1.74000 0.870002 0.493048i \(-0.164117\pi\)
0.870002 + 0.493048i \(0.164117\pi\)
\(384\) −4.23295 −0.216012
\(385\) 0 0
\(386\) −7.51190 −0.382346
\(387\) 10.7093 0.544386
\(388\) 11.7666 0.597358
\(389\) 5.56952 0.282386 0.141193 0.989982i \(-0.454906\pi\)
0.141193 + 0.989982i \(0.454906\pi\)
\(390\) 0 0
\(391\) 8.10612 0.409944
\(392\) 2.13377 0.107772
\(393\) −3.03557 −0.153124
\(394\) −10.8149 −0.544847
\(395\) 0 0
\(396\) 0.126989 0.00638146
\(397\) −36.4612 −1.82994 −0.914968 0.403526i \(-0.867784\pi\)
−0.914968 + 0.403526i \(0.867784\pi\)
\(398\) −2.78185 −0.139442
\(399\) 1.52277 0.0762337
\(400\) 0 0
\(401\) −3.33616 −0.166600 −0.0833000 0.996525i \(-0.526546\pi\)
−0.0833000 + 0.996525i \(0.526546\pi\)
\(402\) 2.66764 0.133050
\(403\) −19.7442 −0.983531
\(404\) −10.3982 −0.517330
\(405\) 0 0
\(406\) 0.867201 0.0430385
\(407\) 0.290842 0.0144165
\(408\) 6.35821 0.314778
\(409\) −3.62760 −0.179373 −0.0896866 0.995970i \(-0.528587\pi\)
−0.0896866 + 0.995970i \(0.528587\pi\)
\(410\) 0 0
\(411\) −6.66934 −0.328974
\(412\) 1.07484 0.0529535
\(413\) −1.06393 −0.0523526
\(414\) −1.67015 −0.0820835
\(415\) 0 0
\(416\) 17.0263 0.834785
\(417\) 4.02748 0.197226
\(418\) 0.0644806 0.00315385
\(419\) −14.5798 −0.712268 −0.356134 0.934435i \(-0.615905\pi\)
−0.356134 + 0.934435i \(0.615905\pi\)
\(420\) 0 0
\(421\) −21.7980 −1.06237 −0.531184 0.847257i \(-0.678253\pi\)
−0.531184 + 0.847257i \(0.678253\pi\)
\(422\) −1.06191 −0.0516931
\(423\) 34.0972 1.65786
\(424\) 27.9568 1.35770
\(425\) 0 0
\(426\) −2.48769 −0.120529
\(427\) 0.897659 0.0434408
\(428\) −29.5669 −1.42917
\(429\) 0.0305063 0.00147286
\(430\) 0 0
\(431\) −34.6012 −1.66668 −0.833341 0.552759i \(-0.813575\pi\)
−0.833341 + 0.552759i \(0.813575\pi\)
\(432\) 4.47641 0.215371
\(433\) −10.5948 −0.509153 −0.254577 0.967053i \(-0.581936\pi\)
−0.254577 + 0.967053i \(0.581936\pi\)
\(434\) −3.70333 −0.177766
\(435\) 0 0
\(436\) 6.90727 0.330798
\(437\) 4.14248 0.198162
\(438\) 0.701393 0.0335138
\(439\) −37.4013 −1.78507 −0.892534 0.450979i \(-0.851075\pi\)
−0.892534 + 0.450979i \(0.851075\pi\)
\(440\) 0 0
\(441\) −2.86487 −0.136422
\(442\) −14.6880 −0.698637
\(443\) −1.77797 −0.0844739 −0.0422370 0.999108i \(-0.513448\pi\)
−0.0422370 + 0.999108i \(0.513448\pi\)
\(444\) 6.64749 0.315476
\(445\) 0 0
\(446\) −15.7048 −0.743643
\(447\) −4.14996 −0.196286
\(448\) −0.959128 −0.0453145
\(449\) 39.3938 1.85911 0.929554 0.368687i \(-0.120193\pi\)
0.929554 + 0.368687i \(0.120193\pi\)
\(450\) 0 0
\(451\) −0.222330 −0.0104691
\(452\) −3.90146 −0.183509
\(453\) 1.32435 0.0622234
\(454\) 0.117256 0.00550311
\(455\) 0 0
\(456\) 3.24924 0.152160
\(457\) −3.78655 −0.177127 −0.0885637 0.996071i \(-0.528228\pi\)
−0.0885637 + 0.996071i \(0.528228\pi\)
\(458\) −15.0847 −0.704860
\(459\) −17.4761 −0.815715
\(460\) 0 0
\(461\) −37.3471 −1.73943 −0.869715 0.493554i \(-0.835697\pi\)
−0.869715 + 0.493554i \(0.835697\pi\)
\(462\) 0.00572192 0.000266208 0
\(463\) 12.1071 0.562664 0.281332 0.959610i \(-0.409224\pi\)
0.281332 + 0.959610i \(0.409224\pi\)
\(464\) −3.08864 −0.143386
\(465\) 0 0
\(466\) 4.84379 0.224384
\(467\) 0.197125 0.00912185 0.00456093 0.999990i \(-0.498548\pi\)
0.00456093 + 0.999990i \(0.498548\pi\)
\(468\) −14.7825 −0.683322
\(469\) 12.4481 0.574799
\(470\) 0 0
\(471\) 2.92889 0.134956
\(472\) −2.27019 −0.104494
\(473\) −0.0998102 −0.00458928
\(474\) −0.894102 −0.0410675
\(475\) 0 0
\(476\) 13.4573 0.616814
\(477\) −37.5357 −1.71864
\(478\) 10.8339 0.495532
\(479\) 32.8344 1.50024 0.750121 0.661301i \(-0.229995\pi\)
0.750121 + 0.661301i \(0.229995\pi\)
\(480\) 0 0
\(481\) −33.8562 −1.54371
\(482\) −14.5560 −0.663007
\(483\) 0.367598 0.0167263
\(484\) 18.2603 0.830015
\(485\) 0 0
\(486\) 5.44254 0.246879
\(487\) −28.9171 −1.31036 −0.655179 0.755473i \(-0.727407\pi\)
−0.655179 + 0.755473i \(0.727407\pi\)
\(488\) 1.91540 0.0867062
\(489\) −5.66677 −0.256260
\(490\) 0 0
\(491\) −7.57281 −0.341756 −0.170878 0.985292i \(-0.554660\pi\)
−0.170878 + 0.985292i \(0.554660\pi\)
\(492\) −5.08157 −0.229095
\(493\) 12.0582 0.543074
\(494\) −7.50602 −0.337712
\(495\) 0 0
\(496\) 13.1898 0.592241
\(497\) −11.6084 −0.520708
\(498\) −0.211326 −0.00946972
\(499\) 29.5306 1.32197 0.660986 0.750398i \(-0.270138\pi\)
0.660986 + 0.750398i \(0.270138\pi\)
\(500\) 0 0
\(501\) −5.40337 −0.241405
\(502\) −2.07039 −0.0924061
\(503\) 14.8127 0.660468 0.330234 0.943899i \(-0.392872\pi\)
0.330234 + 0.943899i \(0.392872\pi\)
\(504\) −6.11299 −0.272294
\(505\) 0 0
\(506\) 0.0155657 0.000691980 0
\(507\) 1.22761 0.0545200
\(508\) 18.2846 0.811246
\(509\) 29.5150 1.30823 0.654115 0.756395i \(-0.273041\pi\)
0.654115 + 0.756395i \(0.273041\pi\)
\(510\) 0 0
\(511\) 3.27293 0.144786
\(512\) −20.2351 −0.894271
\(513\) −8.93083 −0.394306
\(514\) 7.62251 0.336215
\(515\) 0 0
\(516\) −2.28126 −0.100427
\(517\) −0.317784 −0.0139761
\(518\) −6.35025 −0.279014
\(519\) 2.31975 0.101826
\(520\) 0 0
\(521\) −6.76947 −0.296576 −0.148288 0.988944i \(-0.547376\pi\)
−0.148288 + 0.988944i \(0.547376\pi\)
\(522\) −2.48442 −0.108740
\(523\) 37.6825 1.64774 0.823870 0.566779i \(-0.191811\pi\)
0.823870 + 0.566779i \(0.191811\pi\)
\(524\) −13.7092 −0.598888
\(525\) 0 0
\(526\) 8.53432 0.372114
\(527\) −51.4938 −2.24310
\(528\) −0.0203793 −0.000886895 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 3.04803 0.132273
\(532\) 6.87709 0.298159
\(533\) 25.8809 1.12103
\(534\) −3.65201 −0.158038
\(535\) 0 0
\(536\) 26.5614 1.14728
\(537\) −4.42578 −0.190987
\(538\) −2.62368 −0.113115
\(539\) 0.0267004 0.00115007
\(540\) 0 0
\(541\) −29.7575 −1.27937 −0.639687 0.768636i \(-0.720936\pi\)
−0.639687 + 0.768636i \(0.720936\pi\)
\(542\) 10.1375 0.435444
\(543\) −6.42031 −0.275522
\(544\) 44.4054 1.90386
\(545\) 0 0
\(546\) −0.666075 −0.0285054
\(547\) −11.6592 −0.498510 −0.249255 0.968438i \(-0.580186\pi\)
−0.249255 + 0.968438i \(0.580186\pi\)
\(548\) −30.1199 −1.28666
\(549\) −2.57168 −0.109757
\(550\) 0 0
\(551\) 6.16210 0.262514
\(552\) 0.784372 0.0333851
\(553\) −4.17218 −0.177419
\(554\) −5.79733 −0.246305
\(555\) 0 0
\(556\) 18.1888 0.771377
\(557\) 13.9900 0.592777 0.296389 0.955067i \(-0.404218\pi\)
0.296389 + 0.955067i \(0.404218\pi\)
\(558\) 10.6096 0.449139
\(559\) 11.6187 0.491417
\(560\) 0 0
\(561\) 0.0795617 0.00335910
\(562\) −2.35057 −0.0991527
\(563\) −34.0803 −1.43631 −0.718157 0.695882i \(-0.755014\pi\)
−0.718157 + 0.695882i \(0.755014\pi\)
\(564\) −7.26325 −0.305838
\(565\) 0 0
\(566\) 10.8008 0.453992
\(567\) 7.80210 0.327658
\(568\) −24.7697 −1.03931
\(569\) −5.07887 −0.212917 −0.106459 0.994317i \(-0.533951\pi\)
−0.106459 + 0.994317i \(0.533951\pi\)
\(570\) 0 0
\(571\) 23.1865 0.970326 0.485163 0.874424i \(-0.338760\pi\)
0.485163 + 0.874424i \(0.338760\pi\)
\(572\) 0.137772 0.00576054
\(573\) 3.76835 0.157425
\(574\) 4.85435 0.202617
\(575\) 0 0
\(576\) 2.74778 0.114491
\(577\) 8.26129 0.343922 0.171961 0.985104i \(-0.444990\pi\)
0.171961 + 0.985104i \(0.444990\pi\)
\(578\) −28.3963 −1.18113
\(579\) −4.73666 −0.196849
\(580\) 0 0
\(581\) −0.986115 −0.0409110
\(582\) −1.51890 −0.0629606
\(583\) 0.349830 0.0144885
\(584\) 6.98370 0.288987
\(585\) 0 0
\(586\) 5.24924 0.216844
\(587\) 31.4936 1.29988 0.649940 0.759985i \(-0.274794\pi\)
0.649940 + 0.759985i \(0.274794\pi\)
\(588\) 0.610264 0.0251669
\(589\) −26.3149 −1.08429
\(590\) 0 0
\(591\) −6.81938 −0.280512
\(592\) 22.6171 0.929559
\(593\) −29.1504 −1.19707 −0.598533 0.801098i \(-0.704249\pi\)
−0.598533 + 0.801098i \(0.704249\pi\)
\(594\) −0.0335584 −0.00137692
\(595\) 0 0
\(596\) −18.7420 −0.767701
\(597\) −1.75411 −0.0717909
\(598\) −1.81197 −0.0740968
\(599\) −22.2900 −0.910744 −0.455372 0.890301i \(-0.650494\pi\)
−0.455372 + 0.890301i \(0.650494\pi\)
\(600\) 0 0
\(601\) −8.35959 −0.340995 −0.170497 0.985358i \(-0.554537\pi\)
−0.170497 + 0.985358i \(0.554537\pi\)
\(602\) 2.17925 0.0888198
\(603\) −35.6622 −1.45228
\(604\) 5.98100 0.243364
\(605\) 0 0
\(606\) 1.34227 0.0545258
\(607\) 12.2796 0.498412 0.249206 0.968450i \(-0.419830\pi\)
0.249206 + 0.968450i \(0.419830\pi\)
\(608\) 22.6925 0.920303
\(609\) 0.546817 0.0221582
\(610\) 0 0
\(611\) 36.9924 1.49655
\(612\) −38.5534 −1.55843
\(613\) −5.40643 −0.218364 −0.109182 0.994022i \(-0.534823\pi\)
−0.109182 + 0.994022i \(0.534823\pi\)
\(614\) −9.63360 −0.388780
\(615\) 0 0
\(616\) 0.0569726 0.00229549
\(617\) 4.26177 0.171572 0.0857862 0.996314i \(-0.472660\pi\)
0.0857862 + 0.996314i \(0.472660\pi\)
\(618\) −0.138747 −0.00558122
\(619\) 22.7559 0.914636 0.457318 0.889303i \(-0.348810\pi\)
0.457318 + 0.889303i \(0.348810\pi\)
\(620\) 0 0
\(621\) −2.15592 −0.0865139
\(622\) −5.63748 −0.226042
\(623\) −17.0415 −0.682752
\(624\) 2.37230 0.0949681
\(625\) 0 0
\(626\) −12.6033 −0.503729
\(627\) 0.0406585 0.00162374
\(628\) 13.2274 0.527831
\(629\) −88.2984 −3.52069
\(630\) 0 0
\(631\) −34.6945 −1.38117 −0.690583 0.723253i \(-0.742646\pi\)
−0.690583 + 0.723253i \(0.742646\pi\)
\(632\) −8.90249 −0.354122
\(633\) −0.669594 −0.0266140
\(634\) −1.22744 −0.0487479
\(635\) 0 0
\(636\) 7.99571 0.317051
\(637\) −3.10813 −0.123149
\(638\) 0.0231546 0.000916700 0
\(639\) 33.2566 1.31561
\(640\) 0 0
\(641\) 25.9845 1.02633 0.513163 0.858291i \(-0.328474\pi\)
0.513163 + 0.858291i \(0.328474\pi\)
\(642\) 3.81668 0.150633
\(643\) −8.23620 −0.324804 −0.162402 0.986725i \(-0.551924\pi\)
−0.162402 + 0.986725i \(0.551924\pi\)
\(644\) 1.66014 0.0654186
\(645\) 0 0
\(646\) −19.5760 −0.770208
\(647\) −20.2308 −0.795354 −0.397677 0.917526i \(-0.630183\pi\)
−0.397677 + 0.917526i \(0.630183\pi\)
\(648\) 16.6479 0.653992
\(649\) −0.0284074 −0.00111509
\(650\) 0 0
\(651\) −2.33515 −0.0915217
\(652\) −25.5921 −1.00227
\(653\) −37.3573 −1.46190 −0.730952 0.682429i \(-0.760924\pi\)
−0.730952 + 0.682429i \(0.760924\pi\)
\(654\) −0.891633 −0.0348656
\(655\) 0 0
\(656\) −17.2893 −0.675035
\(657\) −9.37653 −0.365813
\(658\) 6.93848 0.270490
\(659\) 22.8235 0.889077 0.444538 0.895760i \(-0.353368\pi\)
0.444538 + 0.895760i \(0.353368\pi\)
\(660\) 0 0
\(661\) −5.10581 −0.198593 −0.0992964 0.995058i \(-0.531659\pi\)
−0.0992964 + 0.995058i \(0.531659\pi\)
\(662\) −18.8312 −0.731897
\(663\) −9.26159 −0.359690
\(664\) −2.10415 −0.0816567
\(665\) 0 0
\(666\) 18.1927 0.704951
\(667\) 1.48754 0.0575978
\(668\) −24.4026 −0.944165
\(669\) −9.90272 −0.382861
\(670\) 0 0
\(671\) 0.0239679 0.000925269 0
\(672\) 2.01370 0.0776803
\(673\) −17.4364 −0.672123 −0.336061 0.941840i \(-0.609095\pi\)
−0.336061 + 0.941840i \(0.609095\pi\)
\(674\) −11.5930 −0.446547
\(675\) 0 0
\(676\) 5.54410 0.213235
\(677\) 10.2077 0.392312 0.196156 0.980573i \(-0.437154\pi\)
0.196156 + 0.980573i \(0.437154\pi\)
\(678\) 0.503625 0.0193416
\(679\) −7.08771 −0.272002
\(680\) 0 0
\(681\) 0.0739364 0.00283325
\(682\) −0.0988804 −0.00378633
\(683\) 11.6087 0.444196 0.222098 0.975024i \(-0.428710\pi\)
0.222098 + 0.975024i \(0.428710\pi\)
\(684\) −19.7020 −0.753324
\(685\) 0 0
\(686\) −0.582976 −0.0222581
\(687\) −9.51170 −0.362894
\(688\) −7.76167 −0.295911
\(689\) −40.7229 −1.55142
\(690\) 0 0
\(691\) 3.04770 0.115940 0.0579700 0.998318i \(-0.481537\pi\)
0.0579700 + 0.998318i \(0.481537\pi\)
\(692\) 10.4764 0.398252
\(693\) −0.0764932 −0.00290574
\(694\) 2.16837 0.0823103
\(695\) 0 0
\(696\) 1.16678 0.0442269
\(697\) 67.4984 2.55668
\(698\) −8.54908 −0.323587
\(699\) 3.05427 0.115523
\(700\) 0 0
\(701\) −27.6080 −1.04274 −0.521370 0.853331i \(-0.674579\pi\)
−0.521370 + 0.853331i \(0.674579\pi\)
\(702\) 3.90645 0.147439
\(703\) −45.1232 −1.70185
\(704\) −0.0256091 −0.000965179 0
\(705\) 0 0
\(706\) −15.1961 −0.571914
\(707\) 6.26346 0.235562
\(708\) −0.649279 −0.0244014
\(709\) 10.7441 0.403504 0.201752 0.979437i \(-0.435337\pi\)
0.201752 + 0.979437i \(0.435337\pi\)
\(710\) 0 0
\(711\) 11.9528 0.448263
\(712\) −36.3627 −1.36275
\(713\) −6.35245 −0.237901
\(714\) −1.73715 −0.0650112
\(715\) 0 0
\(716\) −19.9876 −0.746972
\(717\) 6.83137 0.255122
\(718\) 9.69761 0.361912
\(719\) 18.6767 0.696522 0.348261 0.937398i \(-0.386772\pi\)
0.348261 + 0.937398i \(0.386772\pi\)
\(720\) 0 0
\(721\) −0.647440 −0.0241119
\(722\) 1.07261 0.0399184
\(723\) −9.17834 −0.341346
\(724\) −28.9953 −1.07760
\(725\) 0 0
\(726\) −2.35716 −0.0874824
\(727\) 34.5497 1.28138 0.640688 0.767801i \(-0.278649\pi\)
0.640688 + 0.767801i \(0.278649\pi\)
\(728\) −6.63204 −0.245800
\(729\) −19.9745 −0.739796
\(730\) 0 0
\(731\) 30.3019 1.12076
\(732\) 0.547809 0.0202476
\(733\) −26.7884 −0.989454 −0.494727 0.869049i \(-0.664732\pi\)
−0.494727 + 0.869049i \(0.664732\pi\)
\(734\) −18.5197 −0.683576
\(735\) 0 0
\(736\) 5.47800 0.201922
\(737\) 0.332369 0.0122430
\(738\) −13.9071 −0.511927
\(739\) 30.0904 1.10689 0.553447 0.832885i \(-0.313312\pi\)
0.553447 + 0.832885i \(0.313312\pi\)
\(740\) 0 0
\(741\) −4.73296 −0.173869
\(742\) −7.63819 −0.280407
\(743\) 32.4616 1.19090 0.595450 0.803392i \(-0.296974\pi\)
0.595450 + 0.803392i \(0.296974\pi\)
\(744\) −4.98268 −0.182674
\(745\) 0 0
\(746\) −0.0671730 −0.00245938
\(747\) 2.82509 0.103365
\(748\) 0.359315 0.0131379
\(749\) 17.8099 0.650760
\(750\) 0 0
\(751\) 36.3064 1.32484 0.662420 0.749133i \(-0.269529\pi\)
0.662420 + 0.749133i \(0.269529\pi\)
\(752\) −24.7122 −0.901161
\(753\) −1.30549 −0.0475749
\(754\) −2.69537 −0.0981597
\(755\) 0 0
\(756\) −3.57912 −0.130171
\(757\) −12.4321 −0.451853 −0.225926 0.974144i \(-0.572541\pi\)
−0.225926 + 0.974144i \(0.572541\pi\)
\(758\) 21.9413 0.796945
\(759\) 0.00981502 0.000356263 0
\(760\) 0 0
\(761\) −34.8319 −1.26266 −0.631329 0.775515i \(-0.717490\pi\)
−0.631329 + 0.775515i \(0.717490\pi\)
\(762\) −2.36028 −0.0855041
\(763\) −4.16066 −0.150626
\(764\) 17.0185 0.615709
\(765\) 0 0
\(766\) −19.8518 −0.717276
\(767\) 3.30684 0.119403
\(768\) 1.76256 0.0636010
\(769\) 41.4728 1.49555 0.747774 0.663953i \(-0.231123\pi\)
0.747774 + 0.663953i \(0.231123\pi\)
\(770\) 0 0
\(771\) 4.80641 0.173099
\(772\) −21.3916 −0.769901
\(773\) −29.0392 −1.04447 −0.522234 0.852802i \(-0.674901\pi\)
−0.522234 + 0.852802i \(0.674901\pi\)
\(774\) −6.24328 −0.224410
\(775\) 0 0
\(776\) −15.1236 −0.542905
\(777\) −4.00418 −0.143649
\(778\) −3.24690 −0.116407
\(779\) 34.4938 1.23587
\(780\) 0 0
\(781\) −0.309949 −0.0110908
\(782\) −4.72568 −0.168990
\(783\) −3.20701 −0.114609
\(784\) 2.07634 0.0741549
\(785\) 0 0
\(786\) 1.76967 0.0631219
\(787\) 5.23260 0.186522 0.0932610 0.995642i \(-0.470271\pi\)
0.0932610 + 0.995642i \(0.470271\pi\)
\(788\) −30.7975 −1.09712
\(789\) 5.38135 0.191581
\(790\) 0 0
\(791\) 2.35008 0.0835593
\(792\) −0.163219 −0.00579974
\(793\) −2.79004 −0.0990772
\(794\) 21.2560 0.754348
\(795\) 0 0
\(796\) −7.92187 −0.280783
\(797\) −36.0192 −1.27587 −0.637933 0.770092i \(-0.720210\pi\)
−0.637933 + 0.770092i \(0.720210\pi\)
\(798\) −0.887737 −0.0314256
\(799\) 96.4776 3.41313
\(800\) 0 0
\(801\) 48.8216 1.72503
\(802\) 1.94490 0.0686770
\(803\) 0.0873886 0.00308388
\(804\) 7.59662 0.267912
\(805\) 0 0
\(806\) 11.5104 0.405437
\(807\) −1.65437 −0.0582367
\(808\) 13.3648 0.470172
\(809\) −1.76325 −0.0619925 −0.0309962 0.999520i \(-0.509868\pi\)
−0.0309962 + 0.999520i \(0.509868\pi\)
\(810\) 0 0
\(811\) −20.9994 −0.737388 −0.368694 0.929551i \(-0.620195\pi\)
−0.368694 + 0.929551i \(0.620195\pi\)
\(812\) 2.46952 0.0866633
\(813\) 6.39226 0.224186
\(814\) −0.169554 −0.00594287
\(815\) 0 0
\(816\) 6.18706 0.216590
\(817\) 15.4852 0.541759
\(818\) 2.11480 0.0739424
\(819\) 8.90439 0.311145
\(820\) 0 0
\(821\) 13.7805 0.480942 0.240471 0.970656i \(-0.422698\pi\)
0.240471 + 0.970656i \(0.422698\pi\)
\(822\) 3.88807 0.135612
\(823\) 27.1880 0.947715 0.473857 0.880602i \(-0.342861\pi\)
0.473857 + 0.880602i \(0.342861\pi\)
\(824\) −1.38149 −0.0481265
\(825\) 0 0
\(826\) 0.620247 0.0215812
\(827\) 34.4957 1.19953 0.599766 0.800175i \(-0.295260\pi\)
0.599766 + 0.800175i \(0.295260\pi\)
\(828\) −4.75608 −0.165285
\(829\) 33.2015 1.15314 0.576568 0.817049i \(-0.304392\pi\)
0.576568 + 0.817049i \(0.304392\pi\)
\(830\) 0 0
\(831\) −3.65553 −0.126809
\(832\) 2.98109 0.103351
\(833\) −8.10612 −0.280860
\(834\) −2.34792 −0.0813019
\(835\) 0 0
\(836\) 0.183621 0.00635066
\(837\) 13.6954 0.473381
\(838\) 8.49965 0.293616
\(839\) 25.4737 0.879451 0.439726 0.898132i \(-0.355076\pi\)
0.439726 + 0.898132i \(0.355076\pi\)
\(840\) 0 0
\(841\) −26.7872 −0.923697
\(842\) 12.7077 0.437936
\(843\) −1.48216 −0.0510483
\(844\) −3.02401 −0.104091
\(845\) 0 0
\(846\) −19.8779 −0.683415
\(847\) −10.9993 −0.377940
\(848\) 27.2043 0.934199
\(849\) 6.81050 0.233736
\(850\) 0 0
\(851\) −10.8928 −0.373401
\(852\) −7.08419 −0.242700
\(853\) −33.0155 −1.13043 −0.565215 0.824944i \(-0.691207\pi\)
−0.565215 + 0.824944i \(0.691207\pi\)
\(854\) −0.523314 −0.0179074
\(855\) 0 0
\(856\) 38.0023 1.29889
\(857\) 32.6486 1.11525 0.557627 0.830091i \(-0.311712\pi\)
0.557627 + 0.830091i \(0.311712\pi\)
\(858\) −0.0177845 −0.000607152 0
\(859\) −34.8450 −1.18890 −0.594448 0.804134i \(-0.702630\pi\)
−0.594448 + 0.804134i \(0.702630\pi\)
\(860\) 0 0
\(861\) 3.06093 0.104316
\(862\) 20.1717 0.687051
\(863\) −26.8030 −0.912385 −0.456192 0.889881i \(-0.650787\pi\)
−0.456192 + 0.889881i \(0.650787\pi\)
\(864\) −11.8101 −0.401788
\(865\) 0 0
\(866\) 6.17651 0.209886
\(867\) −17.9054 −0.608100
\(868\) −10.5460 −0.357953
\(869\) −0.111399 −0.00377895
\(870\) 0 0
\(871\) −38.6902 −1.31097
\(872\) −8.87790 −0.300644
\(873\) 20.3054 0.687233
\(874\) −2.41497 −0.0816874
\(875\) 0 0
\(876\) 1.99735 0.0674843
\(877\) 51.2702 1.73127 0.865635 0.500675i \(-0.166915\pi\)
0.865635 + 0.500675i \(0.166915\pi\)
\(878\) 21.8041 0.735853
\(879\) 3.30993 0.111641
\(880\) 0 0
\(881\) −22.0726 −0.743646 −0.371823 0.928304i \(-0.621267\pi\)
−0.371823 + 0.928304i \(0.621267\pi\)
\(882\) 1.67015 0.0562370
\(883\) 23.5847 0.793687 0.396844 0.917886i \(-0.370105\pi\)
0.396844 + 0.917886i \(0.370105\pi\)
\(884\) −41.8270 −1.40679
\(885\) 0 0
\(886\) 1.03651 0.0348224
\(887\) −14.6114 −0.490602 −0.245301 0.969447i \(-0.578887\pi\)
−0.245301 + 0.969447i \(0.578887\pi\)
\(888\) −8.54401 −0.286718
\(889\) −11.0139 −0.369393
\(890\) 0 0
\(891\) 0.208319 0.00697896
\(892\) −44.7224 −1.49742
\(893\) 49.3030 1.64986
\(894\) 2.41933 0.0809145
\(895\) 0 0
\(896\) 11.5152 0.384694
\(897\) −1.14254 −0.0381484
\(898\) −22.9656 −0.766373
\(899\) −9.44953 −0.315160
\(900\) 0 0
\(901\) −106.207 −3.53826
\(902\) 0.129613 0.00431565
\(903\) 1.37414 0.0457285
\(904\) 5.01454 0.166781
\(905\) 0 0
\(906\) −0.772065 −0.0256501
\(907\) −10.0900 −0.335034 −0.167517 0.985869i \(-0.553575\pi\)
−0.167517 + 0.985869i \(0.553575\pi\)
\(908\) 0.333910 0.0110812
\(909\) −17.9440 −0.595165
\(910\) 0 0
\(911\) −45.5530 −1.50924 −0.754619 0.656163i \(-0.772179\pi\)
−0.754619 + 0.656163i \(0.772179\pi\)
\(912\) 3.16178 0.104697
\(913\) −0.0263297 −0.000871385 0
\(914\) 2.20747 0.0730166
\(915\) 0 0
\(916\) −42.9565 −1.41932
\(917\) 8.25785 0.272698
\(918\) 10.1882 0.336260
\(919\) 41.8047 1.37901 0.689504 0.724282i \(-0.257829\pi\)
0.689504 + 0.724282i \(0.257829\pi\)
\(920\) 0 0
\(921\) −6.07451 −0.200162
\(922\) 21.7725 0.717039
\(923\) 36.0804 1.18760
\(924\) 0.0162943 0.000536043 0
\(925\) 0 0
\(926\) −7.05815 −0.231945
\(927\) 1.85483 0.0609207
\(928\) 8.14876 0.267496
\(929\) 8.45666 0.277454 0.138727 0.990331i \(-0.455699\pi\)
0.138727 + 0.990331i \(0.455699\pi\)
\(930\) 0 0
\(931\) −4.14248 −0.135764
\(932\) 13.7936 0.451826
\(933\) −3.55474 −0.116377
\(934\) −0.114919 −0.00376027
\(935\) 0 0
\(936\) 19.0000 0.621033
\(937\) 26.1767 0.855155 0.427578 0.903979i \(-0.359367\pi\)
0.427578 + 0.903979i \(0.359367\pi\)
\(938\) −7.25694 −0.236947
\(939\) −7.94706 −0.259343
\(940\) 0 0
\(941\) −38.0124 −1.23917 −0.619585 0.784929i \(-0.712699\pi\)
−0.619585 + 0.784929i \(0.712699\pi\)
\(942\) −1.70748 −0.0556326
\(943\) 8.32684 0.271159
\(944\) −2.20908 −0.0718995
\(945\) 0 0
\(946\) 0.0581870 0.00189182
\(947\) −16.9759 −0.551642 −0.275821 0.961209i \(-0.588950\pi\)
−0.275821 + 0.961209i \(0.588950\pi\)
\(948\) −2.54613 −0.0826945
\(949\) −10.1727 −0.330219
\(950\) 0 0
\(951\) −0.773969 −0.0250977
\(952\) −17.2966 −0.560587
\(953\) −3.41279 −0.110551 −0.0552756 0.998471i \(-0.517604\pi\)
−0.0552756 + 0.998471i \(0.517604\pi\)
\(954\) 21.8824 0.708470
\(955\) 0 0
\(956\) 30.8517 0.997815
\(957\) 0.0146002 0.000471959 0
\(958\) −19.1417 −0.618440
\(959\) 18.1430 0.585869
\(960\) 0 0
\(961\) 9.35365 0.301731
\(962\) 19.7374 0.636359
\(963\) −51.0231 −1.64420
\(964\) −41.4510 −1.33505
\(965\) 0 0
\(966\) −0.214301 −0.00689502
\(967\) −43.4582 −1.39752 −0.698761 0.715355i \(-0.746265\pi\)
−0.698761 + 0.715355i \(0.746265\pi\)
\(968\) −23.4700 −0.754354
\(969\) −12.3437 −0.396538
\(970\) 0 0
\(971\) −56.8918 −1.82575 −0.912873 0.408244i \(-0.866141\pi\)
−0.912873 + 0.408244i \(0.866141\pi\)
\(972\) 15.4987 0.497121
\(973\) −10.9562 −0.351239
\(974\) 16.8580 0.540165
\(975\) 0 0
\(976\) 1.86384 0.0596602
\(977\) 49.9260 1.59727 0.798637 0.601813i \(-0.205555\pi\)
0.798637 + 0.601813i \(0.205555\pi\)
\(978\) 3.30359 0.105637
\(979\) −0.455014 −0.0145423
\(980\) 0 0
\(981\) 11.9197 0.380568
\(982\) 4.41477 0.140881
\(983\) −16.2972 −0.519798 −0.259899 0.965636i \(-0.583689\pi\)
−0.259899 + 0.965636i \(0.583689\pi\)
\(984\) 6.53134 0.208212
\(985\) 0 0
\(986\) −7.02964 −0.223869
\(987\) 4.37509 0.139261
\(988\) −21.3749 −0.680025
\(989\) 3.73815 0.118866
\(990\) 0 0
\(991\) 46.1841 1.46709 0.733544 0.679642i \(-0.237865\pi\)
0.733544 + 0.679642i \(0.237865\pi\)
\(992\) −34.7988 −1.10486
\(993\) −11.8741 −0.376814
\(994\) 6.76742 0.214650
\(995\) 0 0
\(996\) −0.601791 −0.0190685
\(997\) −33.5035 −1.06107 −0.530533 0.847664i \(-0.678008\pi\)
−0.530533 + 0.847664i \(0.678008\pi\)
\(998\) −17.2157 −0.544952
\(999\) 23.4840 0.743000
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 4025.2.a.bb.1.6 yes 14
5.4 even 2 4025.2.a.ba.1.9 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4025.2.a.ba.1.9 14 5.4 even 2
4025.2.a.bb.1.6 yes 14 1.1 even 1 trivial