Properties

Label 825.4.a.t.1.2
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $4$
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] = [825,4,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1540841.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 27x^{2} - 18x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.63835\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.63835 q^{2} -3.00000 q^{3} +5.23763 q^{4} +10.9151 q^{6} -20.8444 q^{7} +10.0505 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.63835 q^{2} -3.00000 q^{3} +5.23763 q^{4} +10.9151 q^{6} -20.8444 q^{7} +10.0505 q^{8} +9.00000 q^{9} -11.0000 q^{11} -15.7129 q^{12} -67.4988 q^{13} +75.8394 q^{14} -78.4683 q^{16} +57.8120 q^{17} -32.7452 q^{18} -7.98646 q^{19} +62.5333 q^{21} +40.0219 q^{22} -67.5185 q^{23} -30.1515 q^{24} +245.585 q^{26} -27.0000 q^{27} -109.175 q^{28} -56.1558 q^{29} -127.085 q^{31} +205.091 q^{32} +33.0000 q^{33} -210.341 q^{34} +47.1386 q^{36} +95.4222 q^{37} +29.0576 q^{38} +202.497 q^{39} -485.903 q^{41} -227.518 q^{42} +146.216 q^{43} -57.6139 q^{44} +245.656 q^{46} -164.296 q^{47} +235.405 q^{48} +91.4901 q^{49} -173.436 q^{51} -353.534 q^{52} -431.492 q^{53} +98.2356 q^{54} -209.497 q^{56} +23.9594 q^{57} +204.315 q^{58} -804.178 q^{59} -120.847 q^{61} +462.381 q^{62} -187.600 q^{63} -118.449 q^{64} -120.066 q^{66} +371.469 q^{67} +302.798 q^{68} +202.555 q^{69} +529.835 q^{71} +90.4545 q^{72} -1059.19 q^{73} -347.180 q^{74} -41.8301 q^{76} +229.289 q^{77} -736.754 q^{78} -168.663 q^{79} +81.0000 q^{81} +1767.89 q^{82} +144.130 q^{83} +327.526 q^{84} -531.986 q^{86} +168.468 q^{87} -110.555 q^{88} -1400.20 q^{89} +1406.97 q^{91} -353.637 q^{92} +381.256 q^{93} +597.767 q^{94} -615.274 q^{96} -29.6912 q^{97} -332.873 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 12 q^{3} + 26 q^{4} + 12 q^{6} - 34 q^{7} - 48 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 12 q^{3} + 26 q^{4} + 12 q^{6} - 34 q^{7} - 48 q^{8} + 36 q^{9} - 44 q^{11} - 78 q^{12} - 2 q^{13} - 52 q^{14} + 66 q^{16} - 74 q^{17} - 36 q^{18} + 136 q^{19} + 102 q^{21} + 44 q^{22} + 64 q^{23} + 144 q^{24} - 320 q^{26} - 108 q^{27} + 20 q^{28} + 52 q^{29} + 492 q^{31} - 208 q^{32} + 132 q^{33} + 244 q^{34} + 234 q^{36} + 4 q^{37} + 404 q^{38} + 6 q^{39} + 268 q^{41} + 156 q^{42} - 546 q^{43} - 286 q^{44} + 368 q^{46} + 276 q^{47} - 198 q^{48} - 496 q^{49} + 222 q^{51} + 1084 q^{52} + 184 q^{53} + 108 q^{54} - 852 q^{56} - 408 q^{57} + 444 q^{58} - 1032 q^{59} + 116 q^{61} + 1240 q^{62} - 306 q^{63} - 918 q^{64} - 132 q^{66} + 552 q^{67} + 720 q^{68} - 192 q^{69} - 920 q^{71} - 432 q^{72} - 926 q^{73} - 2856 q^{74} + 1572 q^{76} + 374 q^{77} + 960 q^{78} + 1152 q^{79} + 324 q^{81} + 1924 q^{82} + 134 q^{83} - 60 q^{84} + 236 q^{86} - 156 q^{87} + 528 q^{88} - 1064 q^{89} + 2780 q^{91} + 4896 q^{92} - 1476 q^{93} - 1432 q^{94} + 624 q^{96} + 1648 q^{97} + 188 q^{98} - 396 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.63835 −1.28635 −0.643176 0.765718i \(-0.722384\pi\)
−0.643176 + 0.765718i \(0.722384\pi\)
\(3\) −3.00000 −0.577350
\(4\) 5.23763 0.654703
\(5\) 0 0
\(6\) 10.9151 0.742676
\(7\) −20.8444 −1.12549 −0.562747 0.826629i \(-0.690255\pi\)
−0.562747 + 0.826629i \(0.690255\pi\)
\(8\) 10.0505 0.444173
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) −15.7129 −0.377993
\(13\) −67.4988 −1.44006 −0.720031 0.693942i \(-0.755872\pi\)
−0.720031 + 0.693942i \(0.755872\pi\)
\(14\) 75.8394 1.44778
\(15\) 0 0
\(16\) −78.4683 −1.22607
\(17\) 57.8120 0.824792 0.412396 0.911005i \(-0.364692\pi\)
0.412396 + 0.911005i \(0.364692\pi\)
\(18\) −32.7452 −0.428784
\(19\) −7.98646 −0.0964326 −0.0482163 0.998837i \(-0.515354\pi\)
−0.0482163 + 0.998837i \(0.515354\pi\)
\(20\) 0 0
\(21\) 62.5333 0.649804
\(22\) 40.0219 0.387850
\(23\) −67.5185 −0.612112 −0.306056 0.952013i \(-0.599010\pi\)
−0.306056 + 0.952013i \(0.599010\pi\)
\(24\) −30.1515 −0.256444
\(25\) 0 0
\(26\) 245.585 1.85243
\(27\) −27.0000 −0.192450
\(28\) −109.175 −0.736864
\(29\) −56.1558 −0.359582 −0.179791 0.983705i \(-0.557542\pi\)
−0.179791 + 0.983705i \(0.557542\pi\)
\(30\) 0 0
\(31\) −127.085 −0.736296 −0.368148 0.929767i \(-0.620008\pi\)
−0.368148 + 0.929767i \(0.620008\pi\)
\(32\) 205.091 1.13298
\(33\) 33.0000 0.174078
\(34\) −210.341 −1.06097
\(35\) 0 0
\(36\) 47.1386 0.218234
\(37\) 95.4222 0.423981 0.211991 0.977272i \(-0.432005\pi\)
0.211991 + 0.977272i \(0.432005\pi\)
\(38\) 29.0576 0.124046
\(39\) 202.497 0.831420
\(40\) 0 0
\(41\) −485.903 −1.85086 −0.925431 0.378917i \(-0.876297\pi\)
−0.925431 + 0.378917i \(0.876297\pi\)
\(42\) −227.518 −0.835877
\(43\) 146.216 0.518552 0.259276 0.965803i \(-0.416516\pi\)
0.259276 + 0.965803i \(0.416516\pi\)
\(44\) −57.6139 −0.197400
\(45\) 0 0
\(46\) 245.656 0.787392
\(47\) −164.296 −0.509894 −0.254947 0.966955i \(-0.582058\pi\)
−0.254947 + 0.966955i \(0.582058\pi\)
\(48\) 235.405 0.707870
\(49\) 91.4901 0.266735
\(50\) 0 0
\(51\) −173.436 −0.476194
\(52\) −353.534 −0.942813
\(53\) −431.492 −1.11830 −0.559151 0.829066i \(-0.688873\pi\)
−0.559151 + 0.829066i \(0.688873\pi\)
\(54\) 98.2356 0.247559
\(55\) 0 0
\(56\) −209.497 −0.499914
\(57\) 23.9594 0.0556754
\(58\) 204.315 0.462549
\(59\) −804.178 −1.77449 −0.887246 0.461296i \(-0.847385\pi\)
−0.887246 + 0.461296i \(0.847385\pi\)
\(60\) 0 0
\(61\) −120.847 −0.253653 −0.126826 0.991925i \(-0.540479\pi\)
−0.126826 + 0.991925i \(0.540479\pi\)
\(62\) 462.381 0.947137
\(63\) −187.600 −0.375164
\(64\) −118.449 −0.231346
\(65\) 0 0
\(66\) −120.066 −0.223925
\(67\) 371.469 0.677346 0.338673 0.940904i \(-0.390022\pi\)
0.338673 + 0.940904i \(0.390022\pi\)
\(68\) 302.798 0.539994
\(69\) 202.555 0.353403
\(70\) 0 0
\(71\) 529.835 0.885631 0.442816 0.896613i \(-0.353980\pi\)
0.442816 + 0.896613i \(0.353980\pi\)
\(72\) 90.4545 0.148058
\(73\) −1059.19 −1.69820 −0.849100 0.528232i \(-0.822855\pi\)
−0.849100 + 0.528232i \(0.822855\pi\)
\(74\) −347.180 −0.545389
\(75\) 0 0
\(76\) −41.8301 −0.0631348
\(77\) 229.289 0.339349
\(78\) −736.754 −1.06950
\(79\) −168.663 −0.240204 −0.120102 0.992762i \(-0.538322\pi\)
−0.120102 + 0.992762i \(0.538322\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 1767.89 2.38086
\(83\) 144.130 0.190606 0.0953032 0.995448i \(-0.469618\pi\)
0.0953032 + 0.995448i \(0.469618\pi\)
\(84\) 327.526 0.425429
\(85\) 0 0
\(86\) −531.986 −0.667041
\(87\) 168.468 0.207605
\(88\) −110.555 −0.133923
\(89\) −1400.20 −1.66765 −0.833823 0.552032i \(-0.813853\pi\)
−0.833823 + 0.552032i \(0.813853\pi\)
\(90\) 0 0
\(91\) 1406.97 1.62078
\(92\) −353.637 −0.400752
\(93\) 381.256 0.425101
\(94\) 597.767 0.655904
\(95\) 0 0
\(96\) −615.274 −0.654127
\(97\) −29.6912 −0.0310792 −0.0155396 0.999879i \(-0.504947\pi\)
−0.0155396 + 0.999879i \(0.504947\pi\)
\(98\) −332.873 −0.343115
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) −17.5982 −0.0173375 −0.00866874 0.999962i \(-0.502759\pi\)
−0.00866874 + 0.999962i \(0.502759\pi\)
\(102\) 631.022 0.612553
\(103\) −1423.03 −1.36131 −0.680655 0.732604i \(-0.738305\pi\)
−0.680655 + 0.732604i \(0.738305\pi\)
\(104\) −678.397 −0.639637
\(105\) 0 0
\(106\) 1569.92 1.43853
\(107\) 1335.15 1.20629 0.603147 0.797630i \(-0.293913\pi\)
0.603147 + 0.797630i \(0.293913\pi\)
\(108\) −141.416 −0.125998
\(109\) 1565.39 1.37557 0.687784 0.725916i \(-0.258584\pi\)
0.687784 + 0.725916i \(0.258584\pi\)
\(110\) 0 0
\(111\) −286.266 −0.244786
\(112\) 1635.63 1.37993
\(113\) −1176.83 −0.979709 −0.489854 0.871804i \(-0.662950\pi\)
−0.489854 + 0.871804i \(0.662950\pi\)
\(114\) −87.1728 −0.0716182
\(115\) 0 0
\(116\) −294.123 −0.235420
\(117\) −607.490 −0.480021
\(118\) 2925.89 2.28262
\(119\) −1205.06 −0.928298
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 439.683 0.326287
\(123\) 1457.71 1.06860
\(124\) −665.625 −0.482056
\(125\) 0 0
\(126\) 682.555 0.482594
\(127\) −311.155 −0.217406 −0.108703 0.994074i \(-0.534670\pi\)
−0.108703 + 0.994074i \(0.534670\pi\)
\(128\) −1209.77 −0.835388
\(129\) −438.648 −0.299386
\(130\) 0 0
\(131\) 582.818 0.388711 0.194355 0.980931i \(-0.437739\pi\)
0.194355 + 0.980931i \(0.437739\pi\)
\(132\) 172.842 0.113969
\(133\) 166.473 0.108534
\(134\) −1351.54 −0.871306
\(135\) 0 0
\(136\) 581.039 0.366351
\(137\) 2367.98 1.47672 0.738358 0.674410i \(-0.235602\pi\)
0.738358 + 0.674410i \(0.235602\pi\)
\(138\) −736.969 −0.454601
\(139\) 2573.48 1.57036 0.785179 0.619269i \(-0.212571\pi\)
0.785179 + 0.619269i \(0.212571\pi\)
\(140\) 0 0
\(141\) 492.888 0.294388
\(142\) −1927.73 −1.13923
\(143\) 742.487 0.434195
\(144\) −706.215 −0.408689
\(145\) 0 0
\(146\) 3853.70 2.18448
\(147\) −274.470 −0.153999
\(148\) 499.786 0.277582
\(149\) 1992.38 1.09545 0.547724 0.836659i \(-0.315494\pi\)
0.547724 + 0.836659i \(0.315494\pi\)
\(150\) 0 0
\(151\) 2961.20 1.59589 0.797944 0.602731i \(-0.205921\pi\)
0.797944 + 0.602731i \(0.205921\pi\)
\(152\) −80.2679 −0.0428328
\(153\) 520.308 0.274931
\(154\) −834.234 −0.436522
\(155\) 0 0
\(156\) 1060.60 0.544334
\(157\) −1302.56 −0.662139 −0.331070 0.943606i \(-0.607409\pi\)
−0.331070 + 0.943606i \(0.607409\pi\)
\(158\) 613.657 0.308987
\(159\) 1294.48 0.645652
\(160\) 0 0
\(161\) 1407.38 0.688928
\(162\) −294.707 −0.142928
\(163\) 563.163 0.270615 0.135308 0.990804i \(-0.456798\pi\)
0.135308 + 0.990804i \(0.456798\pi\)
\(164\) −2544.98 −1.21176
\(165\) 0 0
\(166\) −524.396 −0.245187
\(167\) −1128.96 −0.523124 −0.261562 0.965187i \(-0.584238\pi\)
−0.261562 + 0.965187i \(0.584238\pi\)
\(168\) 628.490 0.288626
\(169\) 2359.09 1.07378
\(170\) 0 0
\(171\) −71.8782 −0.0321442
\(172\) 765.825 0.339498
\(173\) −3091.97 −1.35883 −0.679416 0.733754i \(-0.737767\pi\)
−0.679416 + 0.733754i \(0.737767\pi\)
\(174\) −612.945 −0.267053
\(175\) 0 0
\(176\) 863.151 0.369673
\(177\) 2412.53 1.02450
\(178\) 5094.41 2.14518
\(179\) 2879.27 1.20227 0.601136 0.799147i \(-0.294715\pi\)
0.601136 + 0.799147i \(0.294715\pi\)
\(180\) 0 0
\(181\) −702.137 −0.288339 −0.144170 0.989553i \(-0.546051\pi\)
−0.144170 + 0.989553i \(0.546051\pi\)
\(182\) −5119.07 −2.08489
\(183\) 362.540 0.146447
\(184\) −678.594 −0.271884
\(185\) 0 0
\(186\) −1387.14 −0.546830
\(187\) −635.932 −0.248684
\(188\) −860.521 −0.333829
\(189\) 562.799 0.216601
\(190\) 0 0
\(191\) 4294.28 1.62682 0.813412 0.581688i \(-0.197608\pi\)
0.813412 + 0.581688i \(0.197608\pi\)
\(192\) 355.348 0.133568
\(193\) 3888.11 1.45012 0.725059 0.688687i \(-0.241813\pi\)
0.725059 + 0.688687i \(0.241813\pi\)
\(194\) 108.027 0.0399788
\(195\) 0 0
\(196\) 479.191 0.174632
\(197\) 3062.26 1.10750 0.553748 0.832684i \(-0.313197\pi\)
0.553748 + 0.832684i \(0.313197\pi\)
\(198\) 360.197 0.129283
\(199\) 4874.64 1.73645 0.868226 0.496168i \(-0.165260\pi\)
0.868226 + 0.496168i \(0.165260\pi\)
\(200\) 0 0
\(201\) −1114.41 −0.391066
\(202\) 64.0285 0.0223021
\(203\) 1170.54 0.404707
\(204\) −908.393 −0.311766
\(205\) 0 0
\(206\) 5177.47 1.75112
\(207\) −607.666 −0.204037
\(208\) 5296.52 1.76561
\(209\) 87.8511 0.0290755
\(210\) 0 0
\(211\) −4324.02 −1.41080 −0.705398 0.708811i \(-0.749232\pi\)
−0.705398 + 0.708811i \(0.749232\pi\)
\(212\) −2259.99 −0.732156
\(213\) −1589.50 −0.511319
\(214\) −4857.74 −1.55172
\(215\) 0 0
\(216\) −271.363 −0.0854812
\(217\) 2649.02 0.828697
\(218\) −5695.43 −1.76947
\(219\) 3177.56 0.980456
\(220\) 0 0
\(221\) −3902.24 −1.18775
\(222\) 1041.54 0.314881
\(223\) 6205.69 1.86351 0.931757 0.363084i \(-0.118276\pi\)
0.931757 + 0.363084i \(0.118276\pi\)
\(224\) −4275.01 −1.27516
\(225\) 0 0
\(226\) 4281.73 1.26025
\(227\) −1535.81 −0.449055 −0.224528 0.974468i \(-0.572084\pi\)
−0.224528 + 0.974468i \(0.572084\pi\)
\(228\) 125.490 0.0364509
\(229\) 94.7362 0.0273378 0.0136689 0.999907i \(-0.495649\pi\)
0.0136689 + 0.999907i \(0.495649\pi\)
\(230\) 0 0
\(231\) −687.866 −0.195923
\(232\) −564.394 −0.159717
\(233\) −654.983 −0.184160 −0.0920801 0.995752i \(-0.529352\pi\)
−0.0920801 + 0.995752i \(0.529352\pi\)
\(234\) 2210.26 0.617476
\(235\) 0 0
\(236\) −4211.98 −1.16177
\(237\) 505.990 0.138682
\(238\) 4384.43 1.19412
\(239\) −5660.64 −1.53204 −0.766018 0.642819i \(-0.777765\pi\)
−0.766018 + 0.642819i \(0.777765\pi\)
\(240\) 0 0
\(241\) −4156.88 −1.11107 −0.555536 0.831493i \(-0.687487\pi\)
−0.555536 + 0.831493i \(0.687487\pi\)
\(242\) −440.241 −0.116941
\(243\) −243.000 −0.0641500
\(244\) −632.950 −0.166067
\(245\) 0 0
\(246\) −5303.66 −1.37459
\(247\) 539.077 0.138869
\(248\) −1277.27 −0.327043
\(249\) −432.390 −0.110047
\(250\) 0 0
\(251\) −7156.50 −1.79966 −0.899829 0.436243i \(-0.856309\pi\)
−0.899829 + 0.436243i \(0.856309\pi\)
\(252\) −982.578 −0.245621
\(253\) 742.703 0.184559
\(254\) 1132.09 0.279660
\(255\) 0 0
\(256\) 5349.17 1.30595
\(257\) 2188.36 0.531152 0.265576 0.964090i \(-0.414438\pi\)
0.265576 + 0.964090i \(0.414438\pi\)
\(258\) 1595.96 0.385116
\(259\) −1989.02 −0.477188
\(260\) 0 0
\(261\) −505.403 −0.119861
\(262\) −2120.50 −0.500019
\(263\) −757.388 −0.177576 −0.0887881 0.996051i \(-0.528299\pi\)
−0.0887881 + 0.996051i \(0.528299\pi\)
\(264\) 331.666 0.0773207
\(265\) 0 0
\(266\) −605.689 −0.139613
\(267\) 4200.59 0.962816
\(268\) 1945.62 0.443461
\(269\) −6782.64 −1.53734 −0.768671 0.639645i \(-0.779081\pi\)
−0.768671 + 0.639645i \(0.779081\pi\)
\(270\) 0 0
\(271\) 7040.87 1.57824 0.789119 0.614241i \(-0.210538\pi\)
0.789119 + 0.614241i \(0.210538\pi\)
\(272\) −4536.41 −1.01125
\(273\) −4220.92 −0.935758
\(274\) −8615.54 −1.89958
\(275\) 0 0
\(276\) 1060.91 0.231374
\(277\) 3211.46 0.696599 0.348300 0.937383i \(-0.386759\pi\)
0.348300 + 0.937383i \(0.386759\pi\)
\(278\) −9363.24 −2.02003
\(279\) −1143.77 −0.245432
\(280\) 0 0
\(281\) 3986.15 0.846241 0.423120 0.906073i \(-0.360935\pi\)
0.423120 + 0.906073i \(0.360935\pi\)
\(282\) −1793.30 −0.378686
\(283\) −7838.75 −1.64652 −0.823260 0.567664i \(-0.807847\pi\)
−0.823260 + 0.567664i \(0.807847\pi\)
\(284\) 2775.08 0.579826
\(285\) 0 0
\(286\) −2701.43 −0.558528
\(287\) 10128.4 2.08313
\(288\) 1845.82 0.377660
\(289\) −1570.77 −0.319718
\(290\) 0 0
\(291\) 89.0735 0.0179436
\(292\) −5547.63 −1.11182
\(293\) 5430.65 1.08280 0.541402 0.840764i \(-0.317894\pi\)
0.541402 + 0.840764i \(0.317894\pi\)
\(294\) 998.620 0.198098
\(295\) 0 0
\(296\) 959.040 0.188321
\(297\) 297.000 0.0580259
\(298\) −7248.97 −1.40913
\(299\) 4557.42 0.881480
\(300\) 0 0
\(301\) −3047.79 −0.583627
\(302\) −10773.9 −2.05288
\(303\) 52.7946 0.0100098
\(304\) 626.684 0.118233
\(305\) 0 0
\(306\) −1893.06 −0.353658
\(307\) −8045.25 −1.49566 −0.747828 0.663892i \(-0.768903\pi\)
−0.747828 + 0.663892i \(0.768903\pi\)
\(308\) 1200.93 0.222173
\(309\) 4269.08 0.785952
\(310\) 0 0
\(311\) −4712.06 −0.859152 −0.429576 0.903031i \(-0.641337\pi\)
−0.429576 + 0.903031i \(0.641337\pi\)
\(312\) 2035.19 0.369295
\(313\) 1425.19 0.257369 0.128684 0.991686i \(-0.458925\pi\)
0.128684 + 0.991686i \(0.458925\pi\)
\(314\) 4739.19 0.851744
\(315\) 0 0
\(316\) −883.395 −0.157262
\(317\) 2031.54 0.359944 0.179972 0.983672i \(-0.442399\pi\)
0.179972 + 0.983672i \(0.442399\pi\)
\(318\) −4709.76 −0.830536
\(319\) 617.714 0.108418
\(320\) 0 0
\(321\) −4005.44 −0.696454
\(322\) −5120.56 −0.886204
\(323\) −461.713 −0.0795369
\(324\) 424.248 0.0727448
\(325\) 0 0
\(326\) −2048.99 −0.348107
\(327\) −4696.16 −0.794184
\(328\) −4883.57 −0.822103
\(329\) 3424.65 0.573882
\(330\) 0 0
\(331\) 316.790 0.0526054 0.0263027 0.999654i \(-0.491627\pi\)
0.0263027 + 0.999654i \(0.491627\pi\)
\(332\) 754.899 0.124791
\(333\) 858.799 0.141327
\(334\) 4107.57 0.672922
\(335\) 0 0
\(336\) −4906.88 −0.796703
\(337\) −4441.65 −0.717958 −0.358979 0.933346i \(-0.616875\pi\)
−0.358979 + 0.933346i \(0.616875\pi\)
\(338\) −8583.22 −1.38126
\(339\) 3530.50 0.565635
\(340\) 0 0
\(341\) 1397.94 0.222002
\(342\) 261.518 0.0413488
\(343\) 5242.58 0.825285
\(344\) 1469.54 0.230327
\(345\) 0 0
\(346\) 11249.7 1.74794
\(347\) −258.695 −0.0400215 −0.0200108 0.999800i \(-0.506370\pi\)
−0.0200108 + 0.999800i \(0.506370\pi\)
\(348\) 882.370 0.135920
\(349\) 9929.62 1.52298 0.761491 0.648176i \(-0.224468\pi\)
0.761491 + 0.648176i \(0.224468\pi\)
\(350\) 0 0
\(351\) 1822.47 0.277140
\(352\) −2256.01 −0.341607
\(353\) 1008.73 0.152095 0.0760474 0.997104i \(-0.475770\pi\)
0.0760474 + 0.997104i \(0.475770\pi\)
\(354\) −8777.66 −1.31787
\(355\) 0 0
\(356\) −7333.70 −1.09181
\(357\) 3615.17 0.535953
\(358\) −10475.8 −1.54655
\(359\) 5813.26 0.854630 0.427315 0.904103i \(-0.359460\pi\)
0.427315 + 0.904103i \(0.359460\pi\)
\(360\) 0 0
\(361\) −6795.22 −0.990701
\(362\) 2554.62 0.370906
\(363\) −363.000 −0.0524864
\(364\) 7369.21 1.06113
\(365\) 0 0
\(366\) −1319.05 −0.188382
\(367\) 10247.0 1.45747 0.728733 0.684798i \(-0.240110\pi\)
0.728733 + 0.684798i \(0.240110\pi\)
\(368\) 5298.06 0.750490
\(369\) −4373.13 −0.616954
\(370\) 0 0
\(371\) 8994.20 1.25864
\(372\) 1996.88 0.278315
\(373\) 2202.63 0.305759 0.152879 0.988245i \(-0.451145\pi\)
0.152879 + 0.988245i \(0.451145\pi\)
\(374\) 2313.75 0.319896
\(375\) 0 0
\(376\) −1651.26 −0.226481
\(377\) 3790.45 0.517821
\(378\) −2047.66 −0.278626
\(379\) −1851.13 −0.250887 −0.125444 0.992101i \(-0.540035\pi\)
−0.125444 + 0.992101i \(0.540035\pi\)
\(380\) 0 0
\(381\) 933.464 0.125519
\(382\) −15624.1 −2.09267
\(383\) −5880.65 −0.784562 −0.392281 0.919846i \(-0.628314\pi\)
−0.392281 + 0.919846i \(0.628314\pi\)
\(384\) 3629.31 0.482311
\(385\) 0 0
\(386\) −14146.3 −1.86536
\(387\) 1315.95 0.172851
\(388\) −155.511 −0.0203476
\(389\) −6963.66 −0.907639 −0.453820 0.891094i \(-0.649939\pi\)
−0.453820 + 0.891094i \(0.649939\pi\)
\(390\) 0 0
\(391\) −3903.38 −0.504865
\(392\) 919.520 0.118477
\(393\) −1748.46 −0.224422
\(394\) −11141.6 −1.42463
\(395\) 0 0
\(396\) −518.525 −0.0658002
\(397\) −3024.31 −0.382332 −0.191166 0.981558i \(-0.561227\pi\)
−0.191166 + 0.981558i \(0.561227\pi\)
\(398\) −17735.7 −2.23369
\(399\) −499.420 −0.0626623
\(400\) 0 0
\(401\) −13392.7 −1.66783 −0.833914 0.551895i \(-0.813905\pi\)
−0.833914 + 0.551895i \(0.813905\pi\)
\(402\) 4054.61 0.503049
\(403\) 8578.11 1.06031
\(404\) −92.1728 −0.0113509
\(405\) 0 0
\(406\) −4258.83 −0.520596
\(407\) −1049.64 −0.127835
\(408\) −1743.12 −0.211513
\(409\) 7637.16 0.923308 0.461654 0.887060i \(-0.347256\pi\)
0.461654 + 0.887060i \(0.347256\pi\)
\(410\) 0 0
\(411\) −7103.93 −0.852582
\(412\) −7453.28 −0.891254
\(413\) 16762.6 1.99718
\(414\) 2210.91 0.262464
\(415\) 0 0
\(416\) −13843.4 −1.63156
\(417\) −7720.44 −0.906647
\(418\) −319.633 −0.0374014
\(419\) 12523.9 1.46022 0.730112 0.683328i \(-0.239468\pi\)
0.730112 + 0.683328i \(0.239468\pi\)
\(420\) 0 0
\(421\) −11150.0 −1.29078 −0.645391 0.763852i \(-0.723306\pi\)
−0.645391 + 0.763852i \(0.723306\pi\)
\(422\) 15732.3 1.81478
\(423\) −1478.66 −0.169965
\(424\) −4336.71 −0.496720
\(425\) 0 0
\(426\) 5783.18 0.657737
\(427\) 2518.98 0.285485
\(428\) 6993.00 0.789765
\(429\) −2227.46 −0.250683
\(430\) 0 0
\(431\) −2093.77 −0.233998 −0.116999 0.993132i \(-0.537327\pi\)
−0.116999 + 0.993132i \(0.537327\pi\)
\(432\) 2118.64 0.235957
\(433\) 4391.47 0.487392 0.243696 0.969852i \(-0.421640\pi\)
0.243696 + 0.969852i \(0.421640\pi\)
\(434\) −9638.07 −1.06600
\(435\) 0 0
\(436\) 8198.91 0.900589
\(437\) 539.234 0.0590276
\(438\) −11561.1 −1.26121
\(439\) 15614.6 1.69759 0.848797 0.528719i \(-0.177328\pi\)
0.848797 + 0.528719i \(0.177328\pi\)
\(440\) 0 0
\(441\) 823.411 0.0889116
\(442\) 14197.7 1.52787
\(443\) −7909.57 −0.848296 −0.424148 0.905593i \(-0.639426\pi\)
−0.424148 + 0.905593i \(0.639426\pi\)
\(444\) −1499.36 −0.160262
\(445\) 0 0
\(446\) −22578.5 −2.39714
\(447\) −5977.13 −0.632458
\(448\) 2469.01 0.260379
\(449\) −14958.7 −1.57226 −0.786128 0.618064i \(-0.787917\pi\)
−0.786128 + 0.618064i \(0.787917\pi\)
\(450\) 0 0
\(451\) 5344.93 0.558056
\(452\) −6163.81 −0.641419
\(453\) −8883.61 −0.921387
\(454\) 5587.84 0.577644
\(455\) 0 0
\(456\) 240.804 0.0247295
\(457\) −14584.6 −1.49286 −0.746430 0.665464i \(-0.768234\pi\)
−0.746430 + 0.665464i \(0.768234\pi\)
\(458\) −344.684 −0.0351660
\(459\) −1560.92 −0.158731
\(460\) 0 0
\(461\) −1802.35 −0.182091 −0.0910454 0.995847i \(-0.529021\pi\)
−0.0910454 + 0.995847i \(0.529021\pi\)
\(462\) 2502.70 0.252026
\(463\) −6753.79 −0.677916 −0.338958 0.940801i \(-0.610074\pi\)
−0.338958 + 0.940801i \(0.610074\pi\)
\(464\) 4406.45 0.440872
\(465\) 0 0
\(466\) 2383.06 0.236895
\(467\) 1744.55 0.172866 0.0864329 0.996258i \(-0.472453\pi\)
0.0864329 + 0.996258i \(0.472453\pi\)
\(468\) −3181.80 −0.314271
\(469\) −7743.06 −0.762349
\(470\) 0 0
\(471\) 3907.69 0.382286
\(472\) −8082.39 −0.788182
\(473\) −1608.38 −0.156349
\(474\) −1840.97 −0.178394
\(475\) 0 0
\(476\) −6311.64 −0.607760
\(477\) −3883.43 −0.372767
\(478\) 20595.4 1.97074
\(479\) 16930.6 1.61499 0.807495 0.589874i \(-0.200823\pi\)
0.807495 + 0.589874i \(0.200823\pi\)
\(480\) 0 0
\(481\) −6440.88 −0.610559
\(482\) 15124.2 1.42923
\(483\) −4222.15 −0.397753
\(484\) 633.753 0.0595185
\(485\) 0 0
\(486\) 884.120 0.0825196
\(487\) 3932.10 0.365874 0.182937 0.983125i \(-0.441440\pi\)
0.182937 + 0.983125i \(0.441440\pi\)
\(488\) −1214.57 −0.112666
\(489\) −1689.49 −0.156240
\(490\) 0 0
\(491\) 11477.9 1.05497 0.527484 0.849565i \(-0.323136\pi\)
0.527484 + 0.849565i \(0.323136\pi\)
\(492\) 7634.94 0.699613
\(493\) −3246.48 −0.296580
\(494\) −1961.35 −0.178635
\(495\) 0 0
\(496\) 9972.16 0.902749
\(497\) −11044.1 −0.996772
\(498\) 1573.19 0.141559
\(499\) 12806.6 1.14890 0.574451 0.818539i \(-0.305215\pi\)
0.574451 + 0.818539i \(0.305215\pi\)
\(500\) 0 0
\(501\) 3386.89 0.302026
\(502\) 26037.9 2.31500
\(503\) −1319.41 −0.116958 −0.0584789 0.998289i \(-0.518625\pi\)
−0.0584789 + 0.998289i \(0.518625\pi\)
\(504\) −1885.47 −0.166638
\(505\) 0 0
\(506\) −2702.22 −0.237408
\(507\) −7077.28 −0.619947
\(508\) −1629.71 −0.142336
\(509\) 1658.45 0.144419 0.0722097 0.997389i \(-0.476995\pi\)
0.0722097 + 0.997389i \(0.476995\pi\)
\(510\) 0 0
\(511\) 22078.2 1.91131
\(512\) −9784.02 −0.844524
\(513\) 215.635 0.0185585
\(514\) −7962.02 −0.683248
\(515\) 0 0
\(516\) −2297.48 −0.196009
\(517\) 1807.26 0.153739
\(518\) 7236.76 0.613832
\(519\) 9275.90 0.784522
\(520\) 0 0
\(521\) 2790.40 0.234644 0.117322 0.993094i \(-0.462569\pi\)
0.117322 + 0.993094i \(0.462569\pi\)
\(522\) 1838.83 0.154183
\(523\) 2440.70 0.204062 0.102031 0.994781i \(-0.467466\pi\)
0.102031 + 0.994781i \(0.467466\pi\)
\(524\) 3052.59 0.254490
\(525\) 0 0
\(526\) 2755.64 0.228426
\(527\) −7347.05 −0.607292
\(528\) −2589.45 −0.213431
\(529\) −7608.25 −0.625319
\(530\) 0 0
\(531\) −7237.60 −0.591498
\(532\) 871.925 0.0710578
\(533\) 32797.9 2.66536
\(534\) −15283.2 −1.23852
\(535\) 0 0
\(536\) 3733.45 0.300859
\(537\) −8637.81 −0.694132
\(538\) 24677.6 1.97756
\(539\) −1006.39 −0.0804236
\(540\) 0 0
\(541\) −7756.41 −0.616403 −0.308202 0.951321i \(-0.599727\pi\)
−0.308202 + 0.951321i \(0.599727\pi\)
\(542\) −25617.2 −2.03017
\(543\) 2106.41 0.166473
\(544\) 11856.7 0.934474
\(545\) 0 0
\(546\) 15357.2 1.20371
\(547\) 13056.6 1.02058 0.510290 0.860002i \(-0.329538\pi\)
0.510290 + 0.860002i \(0.329538\pi\)
\(548\) 12402.6 0.966810
\(549\) −1087.62 −0.0845510
\(550\) 0 0
\(551\) 448.487 0.0346754
\(552\) 2035.78 0.156972
\(553\) 3515.69 0.270348
\(554\) −11684.4 −0.896072
\(555\) 0 0
\(556\) 13478.9 1.02812
\(557\) 1837.48 0.139778 0.0698892 0.997555i \(-0.477735\pi\)
0.0698892 + 0.997555i \(0.477735\pi\)
\(558\) 4161.43 0.315712
\(559\) −9869.42 −0.746748
\(560\) 0 0
\(561\) 1907.80 0.143578
\(562\) −14503.0 −1.08856
\(563\) −11473.6 −0.858890 −0.429445 0.903093i \(-0.641291\pi\)
−0.429445 + 0.903093i \(0.641291\pi\)
\(564\) 2581.56 0.192736
\(565\) 0 0
\(566\) 28520.1 2.11801
\(567\) −1688.40 −0.125055
\(568\) 5325.10 0.393374
\(569\) 9698.61 0.714564 0.357282 0.933997i \(-0.383703\pi\)
0.357282 + 0.933997i \(0.383703\pi\)
\(570\) 0 0
\(571\) 14019.8 1.02752 0.513758 0.857935i \(-0.328253\pi\)
0.513758 + 0.857935i \(0.328253\pi\)
\(572\) 3888.87 0.284269
\(573\) −12882.8 −0.939247
\(574\) −36850.6 −2.67964
\(575\) 0 0
\(576\) −1066.04 −0.0771155
\(577\) −3912.25 −0.282269 −0.141134 0.989990i \(-0.545075\pi\)
−0.141134 + 0.989990i \(0.545075\pi\)
\(578\) 5715.03 0.411270
\(579\) −11664.3 −0.837226
\(580\) 0 0
\(581\) −3004.31 −0.214526
\(582\) −324.081 −0.0230818
\(583\) 4746.41 0.337181
\(584\) −10645.4 −0.754295
\(585\) 0 0
\(586\) −19758.6 −1.39287
\(587\) −11148.5 −0.783899 −0.391949 0.919987i \(-0.628199\pi\)
−0.391949 + 0.919987i \(0.628199\pi\)
\(588\) −1437.57 −0.100824
\(589\) 1014.96 0.0710030
\(590\) 0 0
\(591\) −9186.77 −0.639413
\(592\) −7487.61 −0.519829
\(593\) 17155.0 1.18798 0.593989 0.804473i \(-0.297552\pi\)
0.593989 + 0.804473i \(0.297552\pi\)
\(594\) −1080.59 −0.0746418
\(595\) 0 0
\(596\) 10435.3 0.717194
\(597\) −14623.9 −1.00254
\(598\) −16581.5 −1.13389
\(599\) −2891.03 −0.197203 −0.0986014 0.995127i \(-0.531437\pi\)
−0.0986014 + 0.995127i \(0.531437\pi\)
\(600\) 0 0
\(601\) 11439.4 0.776412 0.388206 0.921573i \(-0.373095\pi\)
0.388206 + 0.921573i \(0.373095\pi\)
\(602\) 11088.9 0.750750
\(603\) 3343.22 0.225782
\(604\) 15509.7 1.04483
\(605\) 0 0
\(606\) −192.085 −0.0128761
\(607\) 7768.78 0.519481 0.259741 0.965678i \(-0.416363\pi\)
0.259741 + 0.965678i \(0.416363\pi\)
\(608\) −1637.96 −0.109256
\(609\) −3511.61 −0.233658
\(610\) 0 0
\(611\) 11089.8 0.734279
\(612\) 2725.18 0.179998
\(613\) 6469.74 0.426281 0.213140 0.977022i \(-0.431631\pi\)
0.213140 + 0.977022i \(0.431631\pi\)
\(614\) 29271.5 1.92394
\(615\) 0 0
\(616\) 2304.46 0.150730
\(617\) 18323.7 1.19560 0.597799 0.801646i \(-0.296042\pi\)
0.597799 + 0.801646i \(0.296042\pi\)
\(618\) −15532.4 −1.01101
\(619\) −16697.1 −1.08419 −0.542096 0.840317i \(-0.682369\pi\)
−0.542096 + 0.840317i \(0.682369\pi\)
\(620\) 0 0
\(621\) 1823.00 0.117801
\(622\) 17144.1 1.10517
\(623\) 29186.3 1.87692
\(624\) −15889.6 −1.01938
\(625\) 0 0
\(626\) −5185.35 −0.331067
\(627\) −263.553 −0.0167868
\(628\) −6822.34 −0.433505
\(629\) 5516.54 0.349696
\(630\) 0 0
\(631\) 20458.8 1.29073 0.645367 0.763873i \(-0.276705\pi\)
0.645367 + 0.763873i \(0.276705\pi\)
\(632\) −1695.15 −0.106692
\(633\) 12972.1 0.814524
\(634\) −7391.45 −0.463016
\(635\) 0 0
\(636\) 6779.98 0.422710
\(637\) −6175.47 −0.384115
\(638\) −2247.46 −0.139464
\(639\) 4768.51 0.295210
\(640\) 0 0
\(641\) 4676.71 0.288173 0.144087 0.989565i \(-0.453976\pi\)
0.144087 + 0.989565i \(0.453976\pi\)
\(642\) 14573.2 0.895886
\(643\) 2321.97 0.142410 0.0712049 0.997462i \(-0.477316\pi\)
0.0712049 + 0.997462i \(0.477316\pi\)
\(644\) 7371.35 0.451043
\(645\) 0 0
\(646\) 1679.88 0.102312
\(647\) −18149.3 −1.10282 −0.551408 0.834236i \(-0.685909\pi\)
−0.551408 + 0.834236i \(0.685909\pi\)
\(648\) 814.090 0.0493526
\(649\) 8845.96 0.535030
\(650\) 0 0
\(651\) −7947.06 −0.478448
\(652\) 2949.64 0.177173
\(653\) −23089.9 −1.38374 −0.691868 0.722024i \(-0.743212\pi\)
−0.691868 + 0.722024i \(0.743212\pi\)
\(654\) 17086.3 1.02160
\(655\) 0 0
\(656\) 38128.0 2.26928
\(657\) −9532.69 −0.566067
\(658\) −12460.1 −0.738215
\(659\) 415.639 0.0245690 0.0122845 0.999925i \(-0.496090\pi\)
0.0122845 + 0.999925i \(0.496090\pi\)
\(660\) 0 0
\(661\) 12044.0 0.708710 0.354355 0.935111i \(-0.384700\pi\)
0.354355 + 0.935111i \(0.384700\pi\)
\(662\) −1152.60 −0.0676691
\(663\) 11706.7 0.685749
\(664\) 1448.58 0.0846622
\(665\) 0 0
\(666\) −3124.62 −0.181796
\(667\) 3791.56 0.220105
\(668\) −5913.09 −0.342491
\(669\) −18617.1 −1.07590
\(670\) 0 0
\(671\) 1329.31 0.0764792
\(672\) 12825.0 0.736215
\(673\) −8699.88 −0.498300 −0.249150 0.968465i \(-0.580151\pi\)
−0.249150 + 0.968465i \(0.580151\pi\)
\(674\) 16160.3 0.923547
\(675\) 0 0
\(676\) 12356.0 0.703007
\(677\) 29389.8 1.66845 0.834225 0.551424i \(-0.185915\pi\)
0.834225 + 0.551424i \(0.185915\pi\)
\(678\) −12845.2 −0.727606
\(679\) 618.895 0.0349794
\(680\) 0 0
\(681\) 4607.44 0.259262
\(682\) −5086.20 −0.285573
\(683\) −12261.0 −0.686905 −0.343452 0.939170i \(-0.611596\pi\)
−0.343452 + 0.939170i \(0.611596\pi\)
\(684\) −376.471 −0.0210449
\(685\) 0 0
\(686\) −19074.4 −1.06161
\(687\) −284.209 −0.0157835
\(688\) −11473.3 −0.635780
\(689\) 29125.2 1.61042
\(690\) 0 0
\(691\) −22711.4 −1.25034 −0.625168 0.780490i \(-0.714970\pi\)
−0.625168 + 0.780490i \(0.714970\pi\)
\(692\) −16194.6 −0.889631
\(693\) 2063.60 0.113116
\(694\) 941.224 0.0514818
\(695\) 0 0
\(696\) 1693.18 0.0922125
\(697\) −28091.0 −1.52658
\(698\) −36127.5 −1.95909
\(699\) 1964.95 0.106325
\(700\) 0 0
\(701\) −21070.9 −1.13529 −0.567643 0.823275i \(-0.692145\pi\)
−0.567643 + 0.823275i \(0.692145\pi\)
\(702\) −6630.79 −0.356500
\(703\) −762.086 −0.0408856
\(704\) 1302.94 0.0697536
\(705\) 0 0
\(706\) −3670.13 −0.195648
\(707\) 366.824 0.0195132
\(708\) 12636.0 0.670746
\(709\) 8521.30 0.451374 0.225687 0.974200i \(-0.427537\pi\)
0.225687 + 0.974200i \(0.427537\pi\)
\(710\) 0 0
\(711\) −1517.97 −0.0800679
\(712\) −14072.7 −0.740724
\(713\) 8580.61 0.450696
\(714\) −13153.3 −0.689425
\(715\) 0 0
\(716\) 15080.5 0.787132
\(717\) 16981.9 0.884521
\(718\) −21150.7 −1.09936
\(719\) 8217.19 0.426216 0.213108 0.977029i \(-0.431641\pi\)
0.213108 + 0.977029i \(0.431641\pi\)
\(720\) 0 0
\(721\) 29662.1 1.53214
\(722\) 24723.4 1.27439
\(723\) 12470.6 0.641478
\(724\) −3677.53 −0.188777
\(725\) 0 0
\(726\) 1320.72 0.0675160
\(727\) 20870.8 1.06473 0.532363 0.846516i \(-0.321304\pi\)
0.532363 + 0.846516i \(0.321304\pi\)
\(728\) 14140.8 0.719907
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 8453.04 0.427698
\(732\) 1898.85 0.0958790
\(733\) 33776.4 1.70199 0.850997 0.525171i \(-0.175998\pi\)
0.850997 + 0.525171i \(0.175998\pi\)
\(734\) −37282.3 −1.87482
\(735\) 0 0
\(736\) −13847.5 −0.693512
\(737\) −4086.16 −0.204228
\(738\) 15911.0 0.793620
\(739\) −14342.3 −0.713922 −0.356961 0.934119i \(-0.616187\pi\)
−0.356961 + 0.934119i \(0.616187\pi\)
\(740\) 0 0
\(741\) −1617.23 −0.0801761
\(742\) −32724.1 −1.61906
\(743\) 19225.5 0.949279 0.474640 0.880180i \(-0.342578\pi\)
0.474640 + 0.880180i \(0.342578\pi\)
\(744\) 3831.81 0.188819
\(745\) 0 0
\(746\) −8013.96 −0.393313
\(747\) 1297.17 0.0635354
\(748\) −3330.77 −0.162814
\(749\) −27830.4 −1.35768
\(750\) 0 0
\(751\) −19629.4 −0.953776 −0.476888 0.878964i \(-0.658235\pi\)
−0.476888 + 0.878964i \(0.658235\pi\)
\(752\) 12892.0 0.625164
\(753\) 21469.5 1.03903
\(754\) −13791.0 −0.666100
\(755\) 0 0
\(756\) 2947.73 0.141810
\(757\) 26801.5 1.28681 0.643406 0.765525i \(-0.277521\pi\)
0.643406 + 0.765525i \(0.277521\pi\)
\(758\) 6735.07 0.322729
\(759\) −2228.11 −0.106555
\(760\) 0 0
\(761\) 6490.31 0.309164 0.154582 0.987980i \(-0.450597\pi\)
0.154582 + 0.987980i \(0.450597\pi\)
\(762\) −3396.27 −0.161462
\(763\) −32629.6 −1.54819
\(764\) 22491.8 1.06509
\(765\) 0 0
\(766\) 21395.9 1.00922
\(767\) 54281.1 2.55538
\(768\) −16047.5 −0.753991
\(769\) 14356.1 0.673204 0.336602 0.941647i \(-0.390722\pi\)
0.336602 + 0.941647i \(0.390722\pi\)
\(770\) 0 0
\(771\) −6565.07 −0.306661
\(772\) 20364.5 0.949397
\(773\) −19127.6 −0.890004 −0.445002 0.895529i \(-0.646797\pi\)
−0.445002 + 0.895529i \(0.646797\pi\)
\(774\) −4787.88 −0.222347
\(775\) 0 0
\(776\) −298.411 −0.0138045
\(777\) 5967.06 0.275505
\(778\) 25336.3 1.16754
\(779\) 3880.65 0.178483
\(780\) 0 0
\(781\) −5828.18 −0.267028
\(782\) 14201.9 0.649435
\(783\) 1516.21 0.0692016
\(784\) −7179.07 −0.327035
\(785\) 0 0
\(786\) 6361.50 0.288686
\(787\) −23509.7 −1.06484 −0.532420 0.846480i \(-0.678717\pi\)
−0.532420 + 0.846480i \(0.678717\pi\)
\(788\) 16039.0 0.725082
\(789\) 2272.16 0.102524
\(790\) 0 0
\(791\) 24530.4 1.10266
\(792\) −994.999 −0.0446411
\(793\) 8157.01 0.365276
\(794\) 11003.5 0.491814
\(795\) 0 0
\(796\) 25531.5 1.13686
\(797\) −43145.1 −1.91754 −0.958769 0.284187i \(-0.908276\pi\)
−0.958769 + 0.284187i \(0.908276\pi\)
\(798\) 1817.07 0.0806058
\(799\) −9498.27 −0.420557
\(800\) 0 0
\(801\) −12601.8 −0.555882
\(802\) 48727.3 2.14541
\(803\) 11651.1 0.512027
\(804\) −5836.85 −0.256032
\(805\) 0 0
\(806\) −31210.2 −1.36394
\(807\) 20347.9 0.887584
\(808\) −176.871 −0.00770085
\(809\) −23470.4 −1.02000 −0.509998 0.860176i \(-0.670354\pi\)
−0.509998 + 0.860176i \(0.670354\pi\)
\(810\) 0 0
\(811\) −4906.52 −0.212443 −0.106221 0.994343i \(-0.533875\pi\)
−0.106221 + 0.994343i \(0.533875\pi\)
\(812\) 6130.83 0.264963
\(813\) −21122.6 −0.911196
\(814\) 3818.98 0.164441
\(815\) 0 0
\(816\) 13609.2 0.583846
\(817\) −1167.75 −0.0500054
\(818\) −27786.7 −1.18770
\(819\) 12662.8 0.540260
\(820\) 0 0
\(821\) 45229.6 1.92268 0.961342 0.275356i \(-0.0887958\pi\)
0.961342 + 0.275356i \(0.0887958\pi\)
\(822\) 25846.6 1.09672
\(823\) −24192.1 −1.02465 −0.512323 0.858793i \(-0.671215\pi\)
−0.512323 + 0.858793i \(0.671215\pi\)
\(824\) −14302.1 −0.604657
\(825\) 0 0
\(826\) −60988.4 −2.56908
\(827\) −2031.70 −0.0854282 −0.0427141 0.999087i \(-0.513600\pi\)
−0.0427141 + 0.999087i \(0.513600\pi\)
\(828\) −3182.73 −0.133584
\(829\) 17010.6 0.712671 0.356335 0.934358i \(-0.384026\pi\)
0.356335 + 0.934358i \(0.384026\pi\)
\(830\) 0 0
\(831\) −9634.38 −0.402182
\(832\) 7995.20 0.333153
\(833\) 5289.22 0.220001
\(834\) 28089.7 1.16627
\(835\) 0 0
\(836\) 460.131 0.0190358
\(837\) 3431.30 0.141700
\(838\) −45566.5 −1.87836
\(839\) 3184.10 0.131022 0.0655109 0.997852i \(-0.479132\pi\)
0.0655109 + 0.997852i \(0.479132\pi\)
\(840\) 0 0
\(841\) −21235.5 −0.870701
\(842\) 40567.8 1.66040
\(843\) −11958.4 −0.488577
\(844\) −22647.6 −0.923653
\(845\) 0 0
\(846\) 5379.90 0.218635
\(847\) −2522.18 −0.102318
\(848\) 33858.4 1.37111
\(849\) 23516.2 0.950619
\(850\) 0 0
\(851\) −6442.76 −0.259524
\(852\) −8325.23 −0.334762
\(853\) −20566.8 −0.825548 −0.412774 0.910833i \(-0.635440\pi\)
−0.412774 + 0.910833i \(0.635440\pi\)
\(854\) −9164.94 −0.367234
\(855\) 0 0
\(856\) 13418.9 0.535804
\(857\) −48125.8 −1.91826 −0.959129 0.282971i \(-0.908680\pi\)
−0.959129 + 0.282971i \(0.908680\pi\)
\(858\) 8104.30 0.322466
\(859\) −22013.7 −0.874387 −0.437194 0.899367i \(-0.644028\pi\)
−0.437194 + 0.899367i \(0.644028\pi\)
\(860\) 0 0
\(861\) −30385.1 −1.20270
\(862\) 7617.86 0.301004
\(863\) 23417.0 0.923667 0.461834 0.886967i \(-0.347192\pi\)
0.461834 + 0.886967i \(0.347192\pi\)
\(864\) −5537.47 −0.218042
\(865\) 0 0
\(866\) −15977.7 −0.626958
\(867\) 4712.32 0.184589
\(868\) 13874.6 0.542550
\(869\) 1855.30 0.0724241
\(870\) 0 0
\(871\) −25073.7 −0.975421
\(872\) 15732.9 0.610991
\(873\) −267.221 −0.0103597
\(874\) −1961.92 −0.0759303
\(875\) 0 0
\(876\) 16642.9 0.641908
\(877\) 29017.9 1.11729 0.558645 0.829407i \(-0.311321\pi\)
0.558645 + 0.829407i \(0.311321\pi\)
\(878\) −56811.4 −2.18370
\(879\) −16291.9 −0.625158
\(880\) 0 0
\(881\) −33241.1 −1.27119 −0.635597 0.772021i \(-0.719246\pi\)
−0.635597 + 0.772021i \(0.719246\pi\)
\(882\) −2995.86 −0.114372
\(883\) 5618.39 0.214127 0.107063 0.994252i \(-0.465855\pi\)
0.107063 + 0.994252i \(0.465855\pi\)
\(884\) −20438.5 −0.777625
\(885\) 0 0
\(886\) 28777.8 1.09121
\(887\) 14334.2 0.542612 0.271306 0.962493i \(-0.412545\pi\)
0.271306 + 0.962493i \(0.412545\pi\)
\(888\) −2877.12 −0.108727
\(889\) 6485.84 0.244688
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) 32503.1 1.22005
\(893\) 1312.14 0.0491704
\(894\) 21746.9 0.813564
\(895\) 0 0
\(896\) 25217.0 0.940223
\(897\) −13672.3 −0.508922
\(898\) 54424.9 2.02247
\(899\) 7136.58 0.264759
\(900\) 0 0
\(901\) −24945.4 −0.922367
\(902\) −19446.8 −0.717856
\(903\) 9143.37 0.336957
\(904\) −11827.7 −0.435161
\(905\) 0 0
\(906\) 32321.7 1.18523
\(907\) 36659.3 1.34207 0.671033 0.741428i \(-0.265851\pi\)
0.671033 + 0.741428i \(0.265851\pi\)
\(908\) −8044.02 −0.293998
\(909\) −158.384 −0.00577916
\(910\) 0 0
\(911\) −9050.97 −0.329168 −0.164584 0.986363i \(-0.552628\pi\)
−0.164584 + 0.986363i \(0.552628\pi\)
\(912\) −1880.05 −0.0682618
\(913\) −1585.43 −0.0574700
\(914\) 53063.8 1.92035
\(915\) 0 0
\(916\) 496.193 0.0178981
\(917\) −12148.5 −0.437491
\(918\) 5679.19 0.204184
\(919\) −46444.5 −1.66710 −0.833549 0.552445i \(-0.813695\pi\)
−0.833549 + 0.552445i \(0.813695\pi\)
\(920\) 0 0
\(921\) 24135.7 0.863518
\(922\) 6557.59 0.234233
\(923\) −35763.2 −1.27536
\(924\) −3602.79 −0.128272
\(925\) 0 0
\(926\) 24572.7 0.872039
\(927\) −12807.2 −0.453770
\(928\) −11517.1 −0.407400
\(929\) −18695.8 −0.660267 −0.330133 0.943934i \(-0.607094\pi\)
−0.330133 + 0.943934i \(0.607094\pi\)
\(930\) 0 0
\(931\) −730.682 −0.0257219
\(932\) −3430.55 −0.120570
\(933\) 14136.2 0.496032
\(934\) −6347.30 −0.222366
\(935\) 0 0
\(936\) −6105.57 −0.213212
\(937\) 13323.6 0.464530 0.232265 0.972653i \(-0.425386\pi\)
0.232265 + 0.972653i \(0.425386\pi\)
\(938\) 28172.0 0.980649
\(939\) −4275.57 −0.148592
\(940\) 0 0
\(941\) −9500.50 −0.329126 −0.164563 0.986367i \(-0.552621\pi\)
−0.164563 + 0.986367i \(0.552621\pi\)
\(942\) −14217.6 −0.491755
\(943\) 32807.4 1.13293
\(944\) 63102.5 2.17565
\(945\) 0 0
\(946\) 5851.85 0.201120
\(947\) −15179.4 −0.520869 −0.260435 0.965491i \(-0.583866\pi\)
−0.260435 + 0.965491i \(0.583866\pi\)
\(948\) 2650.18 0.0907953
\(949\) 71494.0 2.44551
\(950\) 0 0
\(951\) −6094.61 −0.207814
\(952\) −12111.4 −0.412325
\(953\) −5051.55 −0.171706 −0.0858530 0.996308i \(-0.527362\pi\)
−0.0858530 + 0.996308i \(0.527362\pi\)
\(954\) 14129.3 0.479510
\(955\) 0 0
\(956\) −29648.3 −1.00303
\(957\) −1853.14 −0.0625952
\(958\) −61599.6 −2.07745
\(959\) −49359.1 −1.66203
\(960\) 0 0
\(961\) −13640.3 −0.457867
\(962\) 23434.2 0.785395
\(963\) 12016.3 0.402098
\(964\) −21772.2 −0.727422
\(965\) 0 0
\(966\) 15361.7 0.511650
\(967\) −15341.1 −0.510172 −0.255086 0.966918i \(-0.582104\pi\)
−0.255086 + 0.966918i \(0.582104\pi\)
\(968\) 1216.11 0.0403794
\(969\) 1385.14 0.0459206
\(970\) 0 0
\(971\) −5397.65 −0.178392 −0.0891961 0.996014i \(-0.528430\pi\)
−0.0891961 + 0.996014i \(0.528430\pi\)
\(972\) −1272.74 −0.0419992
\(973\) −53642.7 −1.76743
\(974\) −14306.4 −0.470643
\(975\) 0 0
\(976\) 9482.63 0.310995
\(977\) −24180.4 −0.791812 −0.395906 0.918291i \(-0.629569\pi\)
−0.395906 + 0.918291i \(0.629569\pi\)
\(978\) 6146.96 0.200980
\(979\) 15402.2 0.502814
\(980\) 0 0
\(981\) 14088.5 0.458523
\(982\) −41760.5 −1.35706
\(983\) −15159.6 −0.491877 −0.245938 0.969285i \(-0.579096\pi\)
−0.245938 + 0.969285i \(0.579096\pi\)
\(984\) 14650.7 0.474642
\(985\) 0 0
\(986\) 11811.9 0.381507
\(987\) −10274.0 −0.331331
\(988\) 2823.48 0.0909180
\(989\) −9872.29 −0.317412
\(990\) 0 0
\(991\) −3654.97 −0.117158 −0.0585792 0.998283i \(-0.518657\pi\)
−0.0585792 + 0.998283i \(0.518657\pi\)
\(992\) −26064.1 −0.834210
\(993\) −950.371 −0.0303717
\(994\) 40182.4 1.28220
\(995\) 0 0
\(996\) −2264.70 −0.0720479
\(997\) −46219.9 −1.46820 −0.734102 0.679040i \(-0.762396\pi\)
−0.734102 + 0.679040i \(0.762396\pi\)
\(998\) −46595.0 −1.47789
\(999\) −2576.40 −0.0815952
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 825.4.a.t.1.2 4
3.2 odd 2 2475.4.a.be.1.3 4
5.2 odd 4 825.4.c.p.199.3 8
5.3 odd 4 825.4.c.p.199.6 8
5.4 even 2 165.4.a.h.1.3 4
15.14 odd 2 495.4.a.m.1.2 4
55.54 odd 2 1815.4.a.t.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.h.1.3 4 5.4 even 2
495.4.a.m.1.2 4 15.14 odd 2
825.4.a.t.1.2 4 1.1 even 1 trivial
825.4.c.p.199.3 8 5.2 odd 4
825.4.c.p.199.6 8 5.3 odd 4
1815.4.a.t.1.2 4 55.54 odd 2
2475.4.a.be.1.3 4 3.2 odd 2