Properties

Label 735.3.h.a.391.7
Level $735$
Weight $3$
Character 735.391
Analytic conductor $20.027$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [735,3,Mod(391,735)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(735, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("735.391");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 735.h (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(20.0272994305\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.523596960000.16
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 13x^{6} - 2x^{5} + 91x^{4} - 50x^{3} + 190x^{2} + 100x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 391.7
Root \(1.76021 + 3.04878i\) of defining polynomial
Character \(\chi\) \(=\) 735.391
Dual form 735.3.h.a.391.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.52043 q^{2} -1.73205i q^{3} +2.35256 q^{4} +2.23607i q^{5} -4.36551i q^{6} -4.15226 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+2.52043 q^{2} -1.73205i q^{3} +2.35256 q^{4} +2.23607i q^{5} -4.36551i q^{6} -4.15226 q^{8} -3.00000 q^{9} +5.63585i q^{10} -8.72072 q^{11} -4.07475i q^{12} -21.5286i q^{13} +3.87298 q^{15} -19.8757 q^{16} -21.6925i q^{17} -7.56128 q^{18} +3.13605i q^{19} +5.26047i q^{20} -21.9799 q^{22} +4.10842 q^{23} +7.19193i q^{24} -5.00000 q^{25} -54.2613i q^{26} +5.19615i q^{27} -50.8583 q^{29} +9.76157 q^{30} -39.1690i q^{31} -33.4862 q^{32} +15.1047i q^{33} -54.6743i q^{34} -7.05767 q^{36} +52.9812 q^{37} +7.90419i q^{38} -37.2886 q^{39} -9.28475i q^{40} -36.8122i q^{41} +17.6504 q^{43} -20.5160 q^{44} -6.70820i q^{45} +10.3550 q^{46} +4.03919i q^{47} +34.4257i q^{48} -12.6021 q^{50} -37.5724 q^{51} -50.6473i q^{52} +4.45186 q^{53} +13.0965i q^{54} -19.5001i q^{55} +5.43180 q^{57} -128.185 q^{58} +94.1118i q^{59} +9.11141 q^{60} +73.1827i q^{61} -98.7226i q^{62} -4.89677 q^{64} +48.1394 q^{65} +38.0704i q^{66} -100.532 q^{67} -51.0327i q^{68} -7.11598i q^{69} -56.6975 q^{71} +12.4568 q^{72} +74.8295i q^{73} +133.535 q^{74} +8.66025i q^{75} +7.37773i q^{76} -93.9833 q^{78} +28.9807 q^{79} -44.4434i q^{80} +9.00000 q^{81} -92.7824i q^{82} -21.1116i q^{83} +48.5058 q^{85} +44.4867 q^{86} +88.0892i q^{87} +36.2107 q^{88} -72.8692i q^{89} -16.9075i q^{90} +9.66528 q^{92} -67.8427 q^{93} +10.1805i q^{94} -7.01242 q^{95} +57.9998i q^{96} -73.7985i q^{97} +26.1622 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} + 12 q^{4} - 32 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} + 12 q^{4} - 32 q^{8} - 24 q^{9} - 40 q^{11} + 4 q^{16} + 12 q^{18} - 16 q^{22} - 124 q^{23} - 40 q^{25} - 100 q^{29} - 72 q^{32} - 36 q^{36} + 160 q^{37} + 24 q^{39} + 352 q^{43} + 36 q^{44} + 164 q^{46} + 20 q^{50} - 36 q^{51} + 152 q^{53} + 80 q^{58} + 120 q^{60} - 4 q^{64} + 120 q^{65} - 368 q^{67} + 164 q^{71} + 96 q^{72} + 280 q^{74} - 240 q^{78} + 412 q^{79} + 72 q^{81} - 60 q^{85} - 356 q^{86} - 248 q^{88} - 288 q^{92} - 252 q^{93} + 240 q^{95} + 120 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\) \(346\) \(442\) \(491\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.52043 1.26021 0.630107 0.776508i \(-0.283011\pi\)
0.630107 + 0.776508i \(0.283011\pi\)
\(3\) − 1.73205i − 0.577350i
\(4\) 2.35256 0.588139
\(5\) 2.23607i 0.447214i
\(6\) − 4.36551i − 0.727585i
\(7\) 0 0
\(8\) −4.15226 −0.519033
\(9\) −3.00000 −0.333333
\(10\) 5.63585i 0.563585i
\(11\) −8.72072 −0.792793 −0.396396 0.918079i \(-0.629739\pi\)
−0.396396 + 0.918079i \(0.629739\pi\)
\(12\) − 4.07475i − 0.339562i
\(13\) − 21.5286i − 1.65605i −0.560693 0.828024i \(-0.689465\pi\)
0.560693 0.828024i \(-0.310535\pi\)
\(14\) 0 0
\(15\) 3.87298 0.258199
\(16\) −19.8757 −1.24223
\(17\) − 21.6925i − 1.27603i −0.770025 0.638013i \(-0.779756\pi\)
0.770025 0.638013i \(-0.220244\pi\)
\(18\) −7.56128 −0.420071
\(19\) 3.13605i 0.165055i 0.996589 + 0.0825276i \(0.0262993\pi\)
−0.996589 + 0.0825276i \(0.973701\pi\)
\(20\) 5.26047i 0.263024i
\(21\) 0 0
\(22\) −21.9799 −0.999088
\(23\) 4.10842 0.178627 0.0893134 0.996004i \(-0.471533\pi\)
0.0893134 + 0.996004i \(0.471533\pi\)
\(24\) 7.19193i 0.299664i
\(25\) −5.00000 −0.200000
\(26\) − 54.2613i − 2.08697i
\(27\) 5.19615i 0.192450i
\(28\) 0 0
\(29\) −50.8583 −1.75373 −0.876867 0.480732i \(-0.840371\pi\)
−0.876867 + 0.480732i \(0.840371\pi\)
\(30\) 9.76157 0.325386
\(31\) − 39.1690i − 1.26352i −0.775166 0.631758i \(-0.782334\pi\)
0.775166 0.631758i \(-0.217666\pi\)
\(32\) −33.4862 −1.04644
\(33\) 15.1047i 0.457719i
\(34\) − 54.6743i − 1.60807i
\(35\) 0 0
\(36\) −7.05767 −0.196046
\(37\) 52.9812 1.43192 0.715962 0.698139i \(-0.245988\pi\)
0.715962 + 0.698139i \(0.245988\pi\)
\(38\) 7.90419i 0.208005i
\(39\) −37.2886 −0.956119
\(40\) − 9.28475i − 0.232119i
\(41\) − 36.8122i − 0.897857i −0.893568 0.448929i \(-0.851806\pi\)
0.893568 0.448929i \(-0.148194\pi\)
\(42\) 0 0
\(43\) 17.6504 0.410475 0.205238 0.978712i \(-0.434203\pi\)
0.205238 + 0.978712i \(0.434203\pi\)
\(44\) −20.5160 −0.466272
\(45\) − 6.70820i − 0.149071i
\(46\) 10.3550 0.225108
\(47\) 4.03919i 0.0859401i 0.999076 + 0.0429701i \(0.0136820\pi\)
−0.999076 + 0.0429701i \(0.986318\pi\)
\(48\) 34.4257i 0.717203i
\(49\) 0 0
\(50\) −12.6021 −0.252043
\(51\) −37.5724 −0.736715
\(52\) − 50.6473i − 0.973986i
\(53\) 4.45186 0.0839973 0.0419986 0.999118i \(-0.486627\pi\)
0.0419986 + 0.999118i \(0.486627\pi\)
\(54\) 13.0965i 0.242528i
\(55\) − 19.5001i − 0.354548i
\(56\) 0 0
\(57\) 5.43180 0.0952947
\(58\) −128.185 −2.21008
\(59\) 94.1118i 1.59512i 0.603243 + 0.797558i \(0.293875\pi\)
−0.603243 + 0.797558i \(0.706125\pi\)
\(60\) 9.11141 0.151857
\(61\) 73.1827i 1.19972i 0.800106 + 0.599858i \(0.204776\pi\)
−0.800106 + 0.599858i \(0.795224\pi\)
\(62\) − 98.7226i − 1.59230i
\(63\) 0 0
\(64\) −4.89677 −0.0765121
\(65\) 48.1394 0.740607
\(66\) 38.0704i 0.576824i
\(67\) −100.532 −1.50048 −0.750241 0.661165i \(-0.770062\pi\)
−0.750241 + 0.661165i \(0.770062\pi\)
\(68\) − 51.0327i − 0.750481i
\(69\) − 7.11598i − 0.103130i
\(70\) 0 0
\(71\) −56.6975 −0.798557 −0.399278 0.916830i \(-0.630739\pi\)
−0.399278 + 0.916830i \(0.630739\pi\)
\(72\) 12.4568 0.173011
\(73\) 74.8295i 1.02506i 0.858669 + 0.512531i \(0.171292\pi\)
−0.858669 + 0.512531i \(0.828708\pi\)
\(74\) 133.535 1.80453
\(75\) 8.66025i 0.115470i
\(76\) 7.37773i 0.0970754i
\(77\) 0 0
\(78\) −93.9833 −1.20491
\(79\) 28.9807 0.366844 0.183422 0.983034i \(-0.441283\pi\)
0.183422 + 0.983034i \(0.441283\pi\)
\(80\) − 44.4434i − 0.555543i
\(81\) 9.00000 0.111111
\(82\) − 92.7824i − 1.13149i
\(83\) − 21.1116i − 0.254357i −0.991880 0.127179i \(-0.959408\pi\)
0.991880 0.127179i \(-0.0405921\pi\)
\(84\) 0 0
\(85\) 48.5058 0.570657
\(86\) 44.4867 0.517287
\(87\) 88.0892i 1.01252i
\(88\) 36.2107 0.411486
\(89\) − 72.8692i − 0.818755i −0.912365 0.409378i \(-0.865746\pi\)
0.912365 0.409378i \(-0.134254\pi\)
\(90\) − 16.9075i − 0.187862i
\(91\) 0 0
\(92\) 9.66528 0.105057
\(93\) −67.8427 −0.729491
\(94\) 10.1805i 0.108303i
\(95\) −7.01242 −0.0738150
\(96\) 57.9998i 0.604165i
\(97\) − 73.7985i − 0.760809i −0.924820 0.380405i \(-0.875785\pi\)
0.924820 0.380405i \(-0.124215\pi\)
\(98\) 0 0
\(99\) 26.1622 0.264264
\(100\) −11.7628 −0.117628
\(101\) − 106.954i − 1.05895i −0.848327 0.529473i \(-0.822390\pi\)
0.848327 0.529473i \(-0.177610\pi\)
\(102\) −94.6986 −0.928418
\(103\) − 21.5392i − 0.209119i −0.994519 0.104559i \(-0.966657\pi\)
0.994519 0.104559i \(-0.0333432\pi\)
\(104\) 89.3925i 0.859543i
\(105\) 0 0
\(106\) 11.2206 0.105855
\(107\) 89.6367 0.837726 0.418863 0.908049i \(-0.362429\pi\)
0.418863 + 0.908049i \(0.362429\pi\)
\(108\) 12.2242i 0.113187i
\(109\) 27.3501 0.250919 0.125459 0.992099i \(-0.459960\pi\)
0.125459 + 0.992099i \(0.459960\pi\)
\(110\) − 49.1486i − 0.446806i
\(111\) − 91.7661i − 0.826722i
\(112\) 0 0
\(113\) −92.3372 −0.817144 −0.408572 0.912726i \(-0.633973\pi\)
−0.408572 + 0.912726i \(0.633973\pi\)
\(114\) 13.6905 0.120092
\(115\) 9.18670i 0.0798843i
\(116\) −119.647 −1.03144
\(117\) 64.5858i 0.552016i
\(118\) 237.202i 2.01019i
\(119\) 0 0
\(120\) −16.0817 −0.134014
\(121\) −44.9491 −0.371480
\(122\) 184.452i 1.51190i
\(123\) −63.7605 −0.518378
\(124\) − 92.1473i − 0.743123i
\(125\) − 11.1803i − 0.0894427i
\(126\) 0 0
\(127\) 191.591 1.50859 0.754297 0.656534i \(-0.227978\pi\)
0.754297 + 0.656534i \(0.227978\pi\)
\(128\) 121.603 0.950023
\(129\) − 30.5715i − 0.236988i
\(130\) 121.332 0.933323
\(131\) − 58.8123i − 0.448949i −0.974480 0.224474i \(-0.927933\pi\)
0.974480 0.224474i \(-0.0720665\pi\)
\(132\) 35.5347i 0.269202i
\(133\) 0 0
\(134\) −253.384 −1.89093
\(135\) −11.6190 −0.0860663
\(136\) 90.0728i 0.662300i
\(137\) 165.914 1.21105 0.605526 0.795825i \(-0.292963\pi\)
0.605526 + 0.795825i \(0.292963\pi\)
\(138\) − 17.9353i − 0.129966i
\(139\) 139.625i 1.00449i 0.864724 + 0.502247i \(0.167493\pi\)
−0.864724 + 0.502247i \(0.832507\pi\)
\(140\) 0 0
\(141\) 6.99607 0.0496176
\(142\) −142.902 −1.00635
\(143\) 187.745i 1.31290i
\(144\) 59.6271 0.414077
\(145\) − 113.723i − 0.784294i
\(146\) 188.602i 1.29180i
\(147\) 0 0
\(148\) 124.641 0.842171
\(149\) −14.3372 −0.0962230 −0.0481115 0.998842i \(-0.515320\pi\)
−0.0481115 + 0.998842i \(0.515320\pi\)
\(150\) 21.8275i 0.145517i
\(151\) 212.374 1.40645 0.703226 0.710966i \(-0.251742\pi\)
0.703226 + 0.710966i \(0.251742\pi\)
\(152\) − 13.0217i − 0.0856691i
\(153\) 65.0774i 0.425342i
\(154\) 0 0
\(155\) 87.5845 0.565062
\(156\) −87.7236 −0.562331
\(157\) − 242.918i − 1.54725i −0.633645 0.773624i \(-0.718442\pi\)
0.633645 0.773624i \(-0.281558\pi\)
\(158\) 73.0437 0.462302
\(159\) − 7.71084i − 0.0484959i
\(160\) − 74.8775i − 0.467984i
\(161\) 0 0
\(162\) 22.6838 0.140024
\(163\) 13.2346 0.0811936 0.0405968 0.999176i \(-0.487074\pi\)
0.0405968 + 0.999176i \(0.487074\pi\)
\(164\) − 86.6026i − 0.528065i
\(165\) −33.7752 −0.204698
\(166\) − 53.2104i − 0.320544i
\(167\) − 212.616i − 1.27315i −0.771216 0.636574i \(-0.780351\pi\)
0.771216 0.636574i \(-0.219649\pi\)
\(168\) 0 0
\(169\) −294.481 −1.74249
\(170\) 122.255 0.719149
\(171\) − 9.40815i − 0.0550184i
\(172\) 41.5236 0.241416
\(173\) 248.787i 1.43807i 0.694972 + 0.719037i \(0.255417\pi\)
−0.694972 + 0.719037i \(0.744583\pi\)
\(174\) 222.022i 1.27599i
\(175\) 0 0
\(176\) 173.330 0.984832
\(177\) 163.006 0.920940
\(178\) − 183.662i − 1.03181i
\(179\) 55.2703 0.308773 0.154386 0.988011i \(-0.450660\pi\)
0.154386 + 0.988011i \(0.450660\pi\)
\(180\) − 15.7814i − 0.0876746i
\(181\) − 46.9001i − 0.259117i −0.991572 0.129558i \(-0.958644\pi\)
0.991572 0.129558i \(-0.0413559\pi\)
\(182\) 0 0
\(183\) 126.756 0.692656
\(184\) −17.0592 −0.0927132
\(185\) 118.470i 0.640376i
\(186\) −170.993 −0.919315
\(187\) 189.174i 1.01162i
\(188\) 9.50241i 0.0505447i
\(189\) 0 0
\(190\) −17.6743 −0.0930226
\(191\) −20.1123 −0.105300 −0.0526499 0.998613i \(-0.516767\pi\)
−0.0526499 + 0.998613i \(0.516767\pi\)
\(192\) 8.48146i 0.0441743i
\(193\) 28.7033 0.148722 0.0743609 0.997231i \(-0.476308\pi\)
0.0743609 + 0.997231i \(0.476308\pi\)
\(194\) − 186.004i − 0.958783i
\(195\) − 83.3800i − 0.427589i
\(196\) 0 0
\(197\) 224.436 1.13927 0.569636 0.821897i \(-0.307084\pi\)
0.569636 + 0.821897i \(0.307084\pi\)
\(198\) 65.9398 0.333029
\(199\) − 318.059i − 1.59829i −0.601140 0.799144i \(-0.705287\pi\)
0.601140 0.799144i \(-0.294713\pi\)
\(200\) 20.7613 0.103807
\(201\) 174.127i 0.866303i
\(202\) − 269.569i − 1.33450i
\(203\) 0 0
\(204\) −88.3913 −0.433290
\(205\) 82.3145 0.401534
\(206\) − 54.2881i − 0.263534i
\(207\) −12.3252 −0.0595423
\(208\) 427.896i 2.05719i
\(209\) − 27.3486i − 0.130855i
\(210\) 0 0
\(211\) 285.317 1.35221 0.676107 0.736804i \(-0.263666\pi\)
0.676107 + 0.736804i \(0.263666\pi\)
\(212\) 10.4732 0.0494021
\(213\) 98.2030i 0.461047i
\(214\) 225.923 1.05571
\(215\) 39.4676i 0.183570i
\(216\) − 21.5758i − 0.0998880i
\(217\) 0 0
\(218\) 68.9341 0.316211
\(219\) 129.608 0.591819
\(220\) − 45.8751i − 0.208523i
\(221\) −467.009 −2.11316
\(222\) − 231.290i − 1.04185i
\(223\) − 57.0977i − 0.256044i −0.991771 0.128022i \(-0.959137\pi\)
0.991771 0.128022i \(-0.0408627\pi\)
\(224\) 0 0
\(225\) 15.0000 0.0666667
\(226\) −232.729 −1.02978
\(227\) − 182.656i − 0.804651i −0.915497 0.402325i \(-0.868202\pi\)
0.915497 0.402325i \(-0.131798\pi\)
\(228\) 12.7786 0.0560465
\(229\) 16.7832i 0.0732890i 0.999328 + 0.0366445i \(0.0116669\pi\)
−0.999328 + 0.0366445i \(0.988333\pi\)
\(230\) 23.1544i 0.100671i
\(231\) 0 0
\(232\) 211.177 0.910246
\(233\) −266.407 −1.14338 −0.571688 0.820471i \(-0.693711\pi\)
−0.571688 + 0.820471i \(0.693711\pi\)
\(234\) 162.784i 0.695658i
\(235\) −9.03189 −0.0384336
\(236\) 221.403i 0.938150i
\(237\) − 50.1960i − 0.211797i
\(238\) 0 0
\(239\) −39.7012 −0.166114 −0.0830568 0.996545i \(-0.526468\pi\)
−0.0830568 + 0.996545i \(0.526468\pi\)
\(240\) −76.9783 −0.320743
\(241\) 83.3574i 0.345881i 0.984932 + 0.172941i \(0.0553269\pi\)
−0.984932 + 0.172941i \(0.944673\pi\)
\(242\) −113.291 −0.468144
\(243\) − 15.5885i − 0.0641500i
\(244\) 172.166i 0.705600i
\(245\) 0 0
\(246\) −160.704 −0.653267
\(247\) 67.5148 0.273339
\(248\) 162.640i 0.655807i
\(249\) −36.5664 −0.146853
\(250\) − 28.1792i − 0.112717i
\(251\) 111.464i 0.444079i 0.975038 + 0.222039i \(0.0712713\pi\)
−0.975038 + 0.222039i \(0.928729\pi\)
\(252\) 0 0
\(253\) −35.8283 −0.141614
\(254\) 482.892 1.90115
\(255\) − 84.0145i − 0.329469i
\(256\) 326.078 1.27374
\(257\) 222.907i 0.867343i 0.901071 + 0.433672i \(0.142782\pi\)
−0.901071 + 0.433672i \(0.857218\pi\)
\(258\) − 77.0531i − 0.298656i
\(259\) 0 0
\(260\) 113.251 0.435580
\(261\) 152.575 0.584578
\(262\) − 148.232i − 0.565772i
\(263\) −426.500 −1.62167 −0.810837 0.585272i \(-0.800988\pi\)
−0.810837 + 0.585272i \(0.800988\pi\)
\(264\) − 62.7188i − 0.237571i
\(265\) 9.95465i 0.0375647i
\(266\) 0 0
\(267\) −126.213 −0.472709
\(268\) −236.508 −0.882492
\(269\) − 59.4913i − 0.221157i −0.993867 0.110579i \(-0.964730\pi\)
0.993867 0.110579i \(-0.0352704\pi\)
\(270\) −29.2847 −0.108462
\(271\) 54.4810i 0.201037i 0.994935 + 0.100518i \(0.0320501\pi\)
−0.994935 + 0.100518i \(0.967950\pi\)
\(272\) 431.153i 1.58512i
\(273\) 0 0
\(274\) 418.175 1.52618
\(275\) 43.6036 0.158559
\(276\) − 16.7407i − 0.0606549i
\(277\) −472.378 −1.70534 −0.852669 0.522452i \(-0.825017\pi\)
−0.852669 + 0.522452i \(0.825017\pi\)
\(278\) 351.914i 1.26588i
\(279\) 117.507i 0.421172i
\(280\) 0 0
\(281\) −534.544 −1.90229 −0.951146 0.308743i \(-0.900092\pi\)
−0.951146 + 0.308743i \(0.900092\pi\)
\(282\) 17.6331 0.0625287
\(283\) − 447.275i − 1.58048i −0.612799 0.790239i \(-0.709957\pi\)
0.612799 0.790239i \(-0.290043\pi\)
\(284\) −133.384 −0.469662
\(285\) 12.1459i 0.0426171i
\(286\) 473.198i 1.65454i
\(287\) 0 0
\(288\) 100.459 0.348815
\(289\) −181.563 −0.628245
\(290\) − 286.630i − 0.988378i
\(291\) −127.823 −0.439254
\(292\) 176.040i 0.602878i
\(293\) 504.200i 1.72082i 0.509604 + 0.860409i \(0.329792\pi\)
−0.509604 + 0.860409i \(0.670208\pi\)
\(294\) 0 0
\(295\) −210.440 −0.713357
\(296\) −219.992 −0.743216
\(297\) − 45.3142i − 0.152573i
\(298\) −36.1359 −0.121262
\(299\) − 88.4485i − 0.295814i
\(300\) 20.3737i 0.0679124i
\(301\) 0 0
\(302\) 535.274 1.77243
\(303\) −185.249 −0.611383
\(304\) − 62.3312i − 0.205037i
\(305\) −163.641 −0.536529
\(306\) 164.023i 0.536022i
\(307\) − 398.792i − 1.29900i −0.760363 0.649499i \(-0.774979\pi\)
0.760363 0.649499i \(-0.225021\pi\)
\(308\) 0 0
\(309\) −37.3070 −0.120735
\(310\) 220.751 0.712098
\(311\) − 239.122i − 0.768880i −0.923150 0.384440i \(-0.874395\pi\)
0.923150 0.384440i \(-0.125605\pi\)
\(312\) 154.832 0.496257
\(313\) − 223.199i − 0.713095i −0.934277 0.356548i \(-0.883954\pi\)
0.934277 0.356548i \(-0.116046\pi\)
\(314\) − 612.257i − 1.94986i
\(315\) 0 0
\(316\) 68.1786 0.215755
\(317\) −286.014 −0.902252 −0.451126 0.892460i \(-0.648978\pi\)
−0.451126 + 0.892460i \(0.648978\pi\)
\(318\) − 19.4346i − 0.0611152i
\(319\) 443.521 1.39035
\(320\) − 10.9495i − 0.0342172i
\(321\) − 155.255i − 0.483662i
\(322\) 0 0
\(323\) 68.0286 0.210615
\(324\) 21.1730 0.0653488
\(325\) 107.643i 0.331209i
\(326\) 33.3568 0.102321
\(327\) − 47.3718i − 0.144868i
\(328\) 152.854i 0.466018i
\(329\) 0 0
\(330\) −85.1279 −0.257963
\(331\) 539.024 1.62847 0.814236 0.580534i \(-0.197156\pi\)
0.814236 + 0.580534i \(0.197156\pi\)
\(332\) − 49.6663i − 0.149597i
\(333\) −158.944 −0.477308
\(334\) − 535.883i − 1.60444i
\(335\) − 224.797i − 0.671036i
\(336\) 0 0
\(337\) 68.2484 0.202518 0.101259 0.994860i \(-0.467713\pi\)
0.101259 + 0.994860i \(0.467713\pi\)
\(338\) −742.218 −2.19591
\(339\) 159.933i 0.471778i
\(340\) 114.113 0.335625
\(341\) 341.582i 1.00171i
\(342\) − 23.7126i − 0.0693350i
\(343\) 0 0
\(344\) −73.2893 −0.213050
\(345\) 15.9118 0.0461212
\(346\) 627.049i 1.81228i
\(347\) 381.895 1.10056 0.550281 0.834980i \(-0.314521\pi\)
0.550281 + 0.834980i \(0.314521\pi\)
\(348\) 207.235i 0.595502i
\(349\) − 301.869i − 0.864953i −0.901645 0.432477i \(-0.857640\pi\)
0.901645 0.432477i \(-0.142360\pi\)
\(350\) 0 0
\(351\) 111.866 0.318706
\(352\) 292.024 0.829613
\(353\) 128.045i 0.362735i 0.983415 + 0.181367i \(0.0580523\pi\)
−0.983415 + 0.181367i \(0.941948\pi\)
\(354\) 410.846 1.16058
\(355\) − 126.780i − 0.357125i
\(356\) − 171.429i − 0.481542i
\(357\) 0 0
\(358\) 139.305 0.389120
\(359\) −524.226 −1.46024 −0.730119 0.683320i \(-0.760535\pi\)
−0.730119 + 0.683320i \(0.760535\pi\)
\(360\) 27.8542i 0.0773729i
\(361\) 351.165 0.972757
\(362\) − 118.208i − 0.326542i
\(363\) 77.8541i 0.214474i
\(364\) 0 0
\(365\) −167.324 −0.458421
\(366\) 319.480 0.872895
\(367\) 35.5881i 0.0969704i 0.998824 + 0.0484852i \(0.0154394\pi\)
−0.998824 + 0.0484852i \(0.984561\pi\)
\(368\) −81.6576 −0.221896
\(369\) 110.436i 0.299286i
\(370\) 298.594i 0.807011i
\(371\) 0 0
\(372\) −159.604 −0.429042
\(373\) −267.091 −0.716063 −0.358031 0.933710i \(-0.616552\pi\)
−0.358031 + 0.933710i \(0.616552\pi\)
\(374\) 476.799i 1.27486i
\(375\) −19.3649 −0.0516398
\(376\) − 16.7718i − 0.0446058i
\(377\) 1094.91i 2.90427i
\(378\) 0 0
\(379\) −125.687 −0.331627 −0.165813 0.986157i \(-0.553025\pi\)
−0.165813 + 0.986157i \(0.553025\pi\)
\(380\) −16.4971 −0.0434135
\(381\) − 331.846i − 0.870987i
\(382\) −50.6915 −0.132700
\(383\) 356.520i 0.930861i 0.885084 + 0.465430i \(0.154100\pi\)
−0.885084 + 0.465430i \(0.845900\pi\)
\(384\) − 210.622i − 0.548496i
\(385\) 0 0
\(386\) 72.3446 0.187421
\(387\) −52.9513 −0.136825
\(388\) − 173.615i − 0.447462i
\(389\) 446.632 1.14816 0.574078 0.818801i \(-0.305361\pi\)
0.574078 + 0.818801i \(0.305361\pi\)
\(390\) − 210.153i − 0.538854i
\(391\) − 89.1216i − 0.227933i
\(392\) 0 0
\(393\) −101.866 −0.259201
\(394\) 565.676 1.43573
\(395\) 64.8027i 0.164058i
\(396\) 61.5479 0.155424
\(397\) − 606.320i − 1.52725i −0.645657 0.763627i \(-0.723417\pi\)
0.645657 0.763627i \(-0.276583\pi\)
\(398\) − 801.645i − 2.01418i
\(399\) 0 0
\(400\) 99.3785 0.248446
\(401\) −728.804 −1.81747 −0.908734 0.417377i \(-0.862950\pi\)
−0.908734 + 0.417377i \(0.862950\pi\)
\(402\) 438.874i 1.09173i
\(403\) −843.254 −2.09244
\(404\) − 251.614i − 0.622808i
\(405\) 20.1246i 0.0496904i
\(406\) 0 0
\(407\) −462.034 −1.13522
\(408\) 156.011 0.382379
\(409\) − 530.657i − 1.29745i −0.761023 0.648725i \(-0.775302\pi\)
0.761023 0.648725i \(-0.224698\pi\)
\(410\) 207.468 0.506019
\(411\) − 287.372i − 0.699201i
\(412\) − 50.6722i − 0.122991i
\(413\) 0 0
\(414\) −31.0649 −0.0750360
\(415\) 47.2071 0.113752
\(416\) 720.912i 1.73296i
\(417\) 241.837 0.579945
\(418\) − 68.9302i − 0.164905i
\(419\) 282.637i 0.674552i 0.941406 + 0.337276i \(0.109506\pi\)
−0.941406 + 0.337276i \(0.890494\pi\)
\(420\) 0 0
\(421\) 440.590 1.04653 0.523267 0.852169i \(-0.324713\pi\)
0.523267 + 0.852169i \(0.324713\pi\)
\(422\) 719.121 1.70408
\(423\) − 12.1176i − 0.0286467i
\(424\) −18.4853 −0.0435974
\(425\) 108.462i 0.255205i
\(426\) 247.514i 0.581018i
\(427\) 0 0
\(428\) 210.875 0.492700
\(429\) 325.184 0.758004
\(430\) 99.4752i 0.231338i
\(431\) −127.435 −0.295672 −0.147836 0.989012i \(-0.547231\pi\)
−0.147836 + 0.989012i \(0.547231\pi\)
\(432\) − 103.277i − 0.239068i
\(433\) − 433.284i − 1.00066i −0.865836 0.500328i \(-0.833213\pi\)
0.865836 0.500328i \(-0.166787\pi\)
\(434\) 0 0
\(435\) −196.973 −0.452812
\(436\) 64.3427 0.147575
\(437\) 12.8842i 0.0294833i
\(438\) 326.669 0.745819
\(439\) 63.2288i 0.144029i 0.997404 + 0.0720146i \(0.0229428\pi\)
−0.997404 + 0.0720146i \(0.977057\pi\)
\(440\) 80.9697i 0.184022i
\(441\) 0 0
\(442\) −1177.06 −2.66303
\(443\) 438.379 0.989570 0.494785 0.869016i \(-0.335247\pi\)
0.494785 + 0.869016i \(0.335247\pi\)
\(444\) − 215.885i − 0.486227i
\(445\) 162.941 0.366159
\(446\) − 143.911i − 0.322670i
\(447\) 24.8328i 0.0555544i
\(448\) 0 0
\(449\) 214.986 0.478810 0.239405 0.970920i \(-0.423048\pi\)
0.239405 + 0.970920i \(0.423048\pi\)
\(450\) 37.8064 0.0840143
\(451\) 321.028i 0.711815i
\(452\) −217.228 −0.480594
\(453\) − 367.843i − 0.812016i
\(454\) − 460.371i − 1.01403i
\(455\) 0 0
\(456\) −22.5543 −0.0494611
\(457\) −241.199 −0.527789 −0.263894 0.964552i \(-0.585007\pi\)
−0.263894 + 0.964552i \(0.585007\pi\)
\(458\) 42.3008i 0.0923598i
\(459\) 112.717 0.245572
\(460\) 21.6122i 0.0469831i
\(461\) − 343.383i − 0.744865i −0.928059 0.372432i \(-0.878524\pi\)
0.928059 0.372432i \(-0.121476\pi\)
\(462\) 0 0
\(463\) 74.7714 0.161493 0.0807467 0.996735i \(-0.474270\pi\)
0.0807467 + 0.996735i \(0.474270\pi\)
\(464\) 1010.84 2.17854
\(465\) − 151.701i − 0.326238i
\(466\) −671.459 −1.44090
\(467\) 356.190i 0.762720i 0.924427 + 0.381360i \(0.124544\pi\)
−0.924427 + 0.381360i \(0.875456\pi\)
\(468\) 151.942i 0.324662i
\(469\) 0 0
\(470\) −22.7642 −0.0484345
\(471\) −420.746 −0.893304
\(472\) − 390.777i − 0.827918i
\(473\) −153.925 −0.325422
\(474\) − 126.515i − 0.266910i
\(475\) − 15.6803i − 0.0330111i
\(476\) 0 0
\(477\) −13.3556 −0.0279991
\(478\) −100.064 −0.209339
\(479\) 373.751i 0.780274i 0.920757 + 0.390137i \(0.127572\pi\)
−0.920757 + 0.390137i \(0.872428\pi\)
\(480\) −129.692 −0.270191
\(481\) − 1140.61i − 2.37133i
\(482\) 210.096i 0.435884i
\(483\) 0 0
\(484\) −105.745 −0.218482
\(485\) 165.019 0.340244
\(486\) − 39.2896i − 0.0808428i
\(487\) 777.563 1.59664 0.798319 0.602235i \(-0.205723\pi\)
0.798319 + 0.602235i \(0.205723\pi\)
\(488\) − 303.874i − 0.622692i
\(489\) − 22.9229i − 0.0468772i
\(490\) 0 0
\(491\) 458.794 0.934407 0.467203 0.884150i \(-0.345262\pi\)
0.467203 + 0.884150i \(0.345262\pi\)
\(492\) −150.000 −0.304878
\(493\) 1103.24i 2.23781i
\(494\) 170.166 0.344466
\(495\) 58.5004i 0.118183i
\(496\) 778.511i 1.56958i
\(497\) 0 0
\(498\) −92.1631 −0.185066
\(499\) 635.543 1.27363 0.636817 0.771015i \(-0.280251\pi\)
0.636817 + 0.771015i \(0.280251\pi\)
\(500\) − 26.3024i − 0.0526047i
\(501\) −368.261 −0.735052
\(502\) 280.936i 0.559634i
\(503\) 10.6561i 0.0211852i 0.999944 + 0.0105926i \(0.00337179\pi\)
−0.999944 + 0.0105926i \(0.996628\pi\)
\(504\) 0 0
\(505\) 239.156 0.473575
\(506\) −90.3027 −0.178464
\(507\) 510.056i 1.00603i
\(508\) 450.729 0.887262
\(509\) − 815.315i − 1.60180i −0.598799 0.800899i \(-0.704355\pi\)
0.598799 0.800899i \(-0.295645\pi\)
\(510\) − 211.753i − 0.415201i
\(511\) 0 0
\(512\) 335.445 0.655167
\(513\) −16.2954 −0.0317649
\(514\) 561.822i 1.09304i
\(515\) 48.1632 0.0935207
\(516\) − 71.9211i − 0.139382i
\(517\) − 35.2246i − 0.0681327i
\(518\) 0 0
\(519\) 430.911 0.830273
\(520\) −199.888 −0.384399
\(521\) − 443.325i − 0.850911i −0.904979 0.425456i \(-0.860114\pi\)
0.904979 0.425456i \(-0.139886\pi\)
\(522\) 384.554 0.736694
\(523\) 634.102i 1.21243i 0.795300 + 0.606216i \(0.207313\pi\)
−0.795300 + 0.606216i \(0.792687\pi\)
\(524\) − 138.359i − 0.264044i
\(525\) 0 0
\(526\) −1074.96 −2.04366
\(527\) −849.672 −1.61228
\(528\) − 300.217i − 0.568593i
\(529\) −512.121 −0.968092
\(530\) 25.0900i 0.0473396i
\(531\) − 282.335i − 0.531705i
\(532\) 0 0
\(533\) −792.515 −1.48689
\(534\) −318.111 −0.595714
\(535\) 200.434i 0.374643i
\(536\) 417.437 0.778799
\(537\) − 95.7310i − 0.178270i
\(538\) − 149.944i − 0.278706i
\(539\) 0 0
\(540\) −27.3342 −0.0506189
\(541\) −175.150 −0.323752 −0.161876 0.986811i \(-0.551755\pi\)
−0.161876 + 0.986811i \(0.551755\pi\)
\(542\) 137.315i 0.253349i
\(543\) −81.2333 −0.149601
\(544\) 726.398i 1.33529i
\(545\) 61.1568i 0.112214i
\(546\) 0 0
\(547\) −773.543 −1.41416 −0.707078 0.707136i \(-0.749987\pi\)
−0.707078 + 0.707136i \(0.749987\pi\)
\(548\) 390.322 0.712267
\(549\) − 219.548i − 0.399905i
\(550\) 109.900 0.199818
\(551\) − 159.494i − 0.289463i
\(552\) 29.5474i 0.0535280i
\(553\) 0 0
\(554\) −1190.60 −2.14909
\(555\) 205.195 0.369721
\(556\) 328.475i 0.590782i
\(557\) −756.528 −1.35822 −0.679110 0.734037i \(-0.737634\pi\)
−0.679110 + 0.734037i \(0.737634\pi\)
\(558\) 296.168i 0.530767i
\(559\) − 379.989i − 0.679766i
\(560\) 0 0
\(561\) 327.659 0.584062
\(562\) −1347.28 −2.39729
\(563\) − 520.983i − 0.925370i −0.886523 0.462685i \(-0.846886\pi\)
0.886523 0.462685i \(-0.153114\pi\)
\(564\) 16.4587 0.0291820
\(565\) − 206.472i − 0.365438i
\(566\) − 1127.32i − 1.99174i
\(567\) 0 0
\(568\) 235.423 0.414477
\(569\) −183.066 −0.321733 −0.160867 0.986976i \(-0.551429\pi\)
−0.160867 + 0.986976i \(0.551429\pi\)
\(570\) 30.6128i 0.0537066i
\(571\) 997.600 1.74711 0.873555 0.486725i \(-0.161808\pi\)
0.873555 + 0.486725i \(0.161808\pi\)
\(572\) 441.680i 0.772169i
\(573\) 34.8355i 0.0607949i
\(574\) 0 0
\(575\) −20.5421 −0.0357254
\(576\) 14.6903 0.0255040
\(577\) 322.317i 0.558608i 0.960203 + 0.279304i \(0.0901038\pi\)
−0.960203 + 0.279304i \(0.909896\pi\)
\(578\) −457.616 −0.791723
\(579\) − 49.7156i − 0.0858645i
\(580\) − 267.539i − 0.461274i
\(581\) 0 0
\(582\) −322.168 −0.553553
\(583\) −38.8234 −0.0665924
\(584\) − 310.712i − 0.532041i
\(585\) −144.418 −0.246869
\(586\) 1270.80i 2.16860i
\(587\) 406.391i 0.692318i 0.938176 + 0.346159i \(0.112514\pi\)
−0.938176 + 0.346159i \(0.887486\pi\)
\(588\) 0 0
\(589\) 122.836 0.208550
\(590\) −530.400 −0.898983
\(591\) − 388.735i − 0.657759i
\(592\) −1053.04 −1.77878
\(593\) − 385.309i − 0.649763i −0.945755 0.324881i \(-0.894676\pi\)
0.945755 0.324881i \(-0.105324\pi\)
\(594\) − 114.211i − 0.192275i
\(595\) 0 0
\(596\) −33.7291 −0.0565925
\(597\) −550.895 −0.922772
\(598\) − 222.928i − 0.372789i
\(599\) −896.544 −1.49673 −0.748367 0.663285i \(-0.769162\pi\)
−0.748367 + 0.663285i \(0.769162\pi\)
\(600\) − 35.9597i − 0.0599328i
\(601\) 599.296i 0.997166i 0.866842 + 0.498583i \(0.166146\pi\)
−0.866842 + 0.498583i \(0.833854\pi\)
\(602\) 0 0
\(603\) 301.597 0.500161
\(604\) 499.622 0.827189
\(605\) − 100.509i − 0.166131i
\(606\) −466.907 −0.770474
\(607\) 492.970i 0.812142i 0.913841 + 0.406071i \(0.133101\pi\)
−0.913841 + 0.406071i \(0.866899\pi\)
\(608\) − 105.014i − 0.172721i
\(609\) 0 0
\(610\) −412.446 −0.676142
\(611\) 86.9581 0.142321
\(612\) 153.098i 0.250160i
\(613\) −140.964 −0.229958 −0.114979 0.993368i \(-0.536680\pi\)
−0.114979 + 0.993368i \(0.536680\pi\)
\(614\) − 1005.13i − 1.63701i
\(615\) − 142.573i − 0.231826i
\(616\) 0 0
\(617\) 61.9853 0.100462 0.0502312 0.998738i \(-0.484004\pi\)
0.0502312 + 0.998738i \(0.484004\pi\)
\(618\) −94.0297 −0.152152
\(619\) − 634.457i − 1.02497i −0.858696 0.512485i \(-0.828725\pi\)
0.858696 0.512485i \(-0.171275\pi\)
\(620\) 206.048 0.332335
\(621\) 21.3480i 0.0343767i
\(622\) − 602.689i − 0.968953i
\(623\) 0 0
\(624\) 741.138 1.18772
\(625\) 25.0000 0.0400000
\(626\) − 562.557i − 0.898653i
\(627\) −47.3692 −0.0755489
\(628\) − 571.478i − 0.909997i
\(629\) − 1149.29i − 1.82717i
\(630\) 0 0
\(631\) 93.3216 0.147895 0.0739474 0.997262i \(-0.476440\pi\)
0.0739474 + 0.997262i \(0.476440\pi\)
\(632\) −120.335 −0.190404
\(633\) − 494.184i − 0.780701i
\(634\) −720.878 −1.13703
\(635\) 428.411i 0.674663i
\(636\) − 18.1402i − 0.0285223i
\(637\) 0 0
\(638\) 1117.86 1.75214
\(639\) 170.093 0.266186
\(640\) 271.912i 0.424863i
\(641\) 307.921 0.480377 0.240188 0.970726i \(-0.422791\pi\)
0.240188 + 0.970726i \(0.422791\pi\)
\(642\) − 391.310i − 0.609517i
\(643\) − 296.519i − 0.461150i −0.973055 0.230575i \(-0.925939\pi\)
0.973055 0.230575i \(-0.0740607\pi\)
\(644\) 0 0
\(645\) 68.3599 0.105984
\(646\) 171.461 0.265420
\(647\) 235.244i 0.363592i 0.983336 + 0.181796i \(0.0581911\pi\)
−0.983336 + 0.181796i \(0.941809\pi\)
\(648\) −37.3704 −0.0576703
\(649\) − 820.723i − 1.26460i
\(650\) 271.307i 0.417395i
\(651\) 0 0
\(652\) 31.1350 0.0477531
\(653\) 297.646 0.455813 0.227906 0.973683i \(-0.426812\pi\)
0.227906 + 0.973683i \(0.426812\pi\)
\(654\) − 119.397i − 0.182565i
\(655\) 131.508 0.200776
\(656\) 731.668i 1.11535i
\(657\) − 224.488i − 0.341687i
\(658\) 0 0
\(659\) −127.740 −0.193839 −0.0969197 0.995292i \(-0.530899\pi\)
−0.0969197 + 0.995292i \(0.530899\pi\)
\(660\) −79.4580 −0.120391
\(661\) − 951.023i − 1.43876i −0.694614 0.719382i \(-0.744425\pi\)
0.694614 0.719382i \(-0.255575\pi\)
\(662\) 1358.57 2.05222
\(663\) 808.882i 1.22003i
\(664\) 87.6611i 0.132020i
\(665\) 0 0
\(666\) −400.606 −0.601510
\(667\) −208.947 −0.313264
\(668\) − 500.190i − 0.748788i
\(669\) −98.8962 −0.147827
\(670\) − 566.585i − 0.845649i
\(671\) − 638.206i − 0.951126i
\(672\) 0 0
\(673\) −1003.39 −1.49092 −0.745460 0.666550i \(-0.767770\pi\)
−0.745460 + 0.666550i \(0.767770\pi\)
\(674\) 172.015 0.255215
\(675\) − 25.9808i − 0.0384900i
\(676\) −692.783 −1.02483
\(677\) − 471.815i − 0.696920i −0.937324 0.348460i \(-0.886705\pi\)
0.937324 0.348460i \(-0.113295\pi\)
\(678\) 403.099i 0.594541i
\(679\) 0 0
\(680\) −201.409 −0.296190
\(681\) −316.369 −0.464565
\(682\) 860.932i 1.26236i
\(683\) 417.228 0.610876 0.305438 0.952212i \(-0.401197\pi\)
0.305438 + 0.952212i \(0.401197\pi\)
\(684\) − 22.1332i − 0.0323585i
\(685\) 370.995i 0.541599i
\(686\) 0 0
\(687\) 29.0693 0.0423134
\(688\) −350.815 −0.509905
\(689\) − 95.8423i − 0.139103i
\(690\) 40.1046 0.0581226
\(691\) − 185.799i − 0.268884i −0.990921 0.134442i \(-0.957076\pi\)
0.990921 0.134442i \(-0.0429242\pi\)
\(692\) 585.285i 0.845787i
\(693\) 0 0
\(694\) 962.538 1.38694
\(695\) −312.210 −0.449223
\(696\) − 365.770i − 0.525531i
\(697\) −798.546 −1.14569
\(698\) − 760.838i − 1.09003i
\(699\) 461.430i 0.660129i
\(700\) 0 0
\(701\) 1034.80 1.47618 0.738089 0.674704i \(-0.235729\pi\)
0.738089 + 0.674704i \(0.235729\pi\)
\(702\) 281.950 0.401638
\(703\) 166.152i 0.236347i
\(704\) 42.7034 0.0606582
\(705\) 15.6437i 0.0221896i
\(706\) 322.729i 0.457123i
\(707\) 0 0
\(708\) 383.482 0.541641
\(709\) −216.642 −0.305560 −0.152780 0.988260i \(-0.548823\pi\)
−0.152780 + 0.988260i \(0.548823\pi\)
\(710\) − 319.539i − 0.450054i
\(711\) −86.9420 −0.122281
\(712\) 302.572i 0.424961i
\(713\) − 160.923i − 0.225698i
\(714\) 0 0
\(715\) −419.811 −0.587148
\(716\) 130.027 0.181601
\(717\) 68.7644i 0.0959057i
\(718\) −1321.27 −1.84021
\(719\) − 0.375796i 0 0.000522665i −1.00000 0.000261332i \(-0.999917\pi\)
1.00000 0.000261332i \(-8.31847e-5\pi\)
\(720\) 133.330i 0.185181i
\(721\) 0 0
\(722\) 885.086 1.22588
\(723\) 144.379 0.199695
\(724\) − 110.335i − 0.152397i
\(725\) 254.292 0.350747
\(726\) 196.226i 0.270283i
\(727\) 174.857i 0.240518i 0.992743 + 0.120259i \(0.0383726\pi\)
−0.992743 + 0.120259i \(0.961627\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) −421.727 −0.577709
\(731\) − 382.881i − 0.523778i
\(732\) 298.201 0.407378
\(733\) 852.412i 1.16291i 0.813579 + 0.581454i \(0.197516\pi\)
−0.813579 + 0.581454i \(0.802484\pi\)
\(734\) 89.6973i 0.122203i
\(735\) 0 0
\(736\) −137.575 −0.186923
\(737\) 876.714 1.18957
\(738\) 278.347i 0.377164i
\(739\) −1168.25 −1.58086 −0.790428 0.612555i \(-0.790142\pi\)
−0.790428 + 0.612555i \(0.790142\pi\)
\(740\) 278.706i 0.376630i
\(741\) − 116.939i − 0.157813i
\(742\) 0 0
\(743\) −558.877 −0.752190 −0.376095 0.926581i \(-0.622733\pi\)
−0.376095 + 0.926581i \(0.622733\pi\)
\(744\) 281.701 0.378630
\(745\) − 32.0590i − 0.0430322i
\(746\) −673.184 −0.902392
\(747\) 63.3349i 0.0847857i
\(748\) 445.042i 0.594976i
\(749\) 0 0
\(750\) −48.8079 −0.0650772
\(751\) 1261.31 1.67950 0.839752 0.542970i \(-0.182700\pi\)
0.839752 + 0.542970i \(0.182700\pi\)
\(752\) − 80.2817i − 0.106758i
\(753\) 193.061 0.256389
\(754\) 2759.64i 3.66000i
\(755\) 474.883i 0.628985i
\(756\) 0 0
\(757\) −1269.13 −1.67652 −0.838262 0.545268i \(-0.816428\pi\)
−0.838262 + 0.545268i \(0.816428\pi\)
\(758\) −316.784 −0.417921
\(759\) 62.0565i 0.0817609i
\(760\) 29.1174 0.0383124
\(761\) 182.117i 0.239313i 0.992815 + 0.119656i \(0.0381793\pi\)
−0.992815 + 0.119656i \(0.961821\pi\)
\(762\) − 836.394i − 1.09763i
\(763\) 0 0
\(764\) −47.3152 −0.0619309
\(765\) −145.517 −0.190219
\(766\) 898.582i 1.17308i
\(767\) 2026.10 2.64159
\(768\) − 564.784i − 0.735396i
\(769\) 810.237i 1.05362i 0.849982 + 0.526812i \(0.176613\pi\)
−0.849982 + 0.526812i \(0.823387\pi\)
\(770\) 0 0
\(771\) 386.087 0.500761
\(772\) 67.5261 0.0874690
\(773\) 245.365i 0.317419i 0.987325 + 0.158709i \(0.0507333\pi\)
−0.987325 + 0.158709i \(0.949267\pi\)
\(774\) −133.460 −0.172429
\(775\) 195.845i 0.252703i
\(776\) 306.431i 0.394885i
\(777\) 0 0
\(778\) 1125.70 1.44692
\(779\) 115.445 0.148196
\(780\) − 196.156i − 0.251482i
\(781\) 494.443 0.633090
\(782\) − 224.625i − 0.287244i
\(783\) − 264.268i − 0.337506i
\(784\) 0 0
\(785\) 543.181 0.691950
\(786\) −256.746 −0.326648
\(787\) − 430.726i − 0.547302i −0.961829 0.273651i \(-0.911769\pi\)
0.961829 0.273651i \(-0.0882312\pi\)
\(788\) 527.999 0.670050
\(789\) 738.720i 0.936274i
\(790\) 163.331i 0.206748i
\(791\) 0 0
\(792\) −108.632 −0.137162
\(793\) 1575.52 1.98679
\(794\) − 1528.19i − 1.92467i
\(795\) 17.2420 0.0216880
\(796\) − 748.252i − 0.940015i
\(797\) 1137.61i 1.42737i 0.700468 + 0.713684i \(0.252975\pi\)
−0.700468 + 0.713684i \(0.747025\pi\)
\(798\) 0 0
\(799\) 87.6199 0.109662
\(800\) 167.431 0.209289
\(801\) 218.608i 0.272918i
\(802\) −1836.90 −2.29040
\(803\) − 652.567i − 0.812661i
\(804\) 409.643i 0.509507i
\(805\) 0 0
\(806\) −2125.36 −2.63692
\(807\) −103.042 −0.127685
\(808\) 444.100i 0.549628i
\(809\) 910.672 1.12568 0.562838 0.826567i \(-0.309710\pi\)
0.562838 + 0.826567i \(0.309710\pi\)
\(810\) 50.7226i 0.0626205i
\(811\) − 546.361i − 0.673688i −0.941560 0.336844i \(-0.890640\pi\)
0.941560 0.336844i \(-0.109360\pi\)
\(812\) 0 0
\(813\) 94.3638 0.116069
\(814\) −1164.52 −1.43062
\(815\) 29.5934i 0.0363109i
\(816\) 746.779 0.915170
\(817\) 55.3527i 0.0677511i
\(818\) − 1337.48i − 1.63507i
\(819\) 0 0
\(820\) 193.649 0.236158
\(821\) 644.300 0.784775 0.392388 0.919800i \(-0.371649\pi\)
0.392388 + 0.919800i \(0.371649\pi\)
\(822\) − 724.300i − 0.881143i
\(823\) 769.321 0.934777 0.467388 0.884052i \(-0.345195\pi\)
0.467388 + 0.884052i \(0.345195\pi\)
\(824\) 89.4365i 0.108539i
\(825\) − 75.5236i − 0.0915438i
\(826\) 0 0
\(827\) 715.404 0.865060 0.432530 0.901620i \(-0.357621\pi\)
0.432530 + 0.901620i \(0.357621\pi\)
\(828\) −28.9958 −0.0350191
\(829\) 78.8630i 0.0951303i 0.998868 + 0.0475651i \(0.0151462\pi\)
−0.998868 + 0.0475651i \(0.984854\pi\)
\(830\) 118.982 0.143352
\(831\) 818.184i 0.984577i
\(832\) 105.421i 0.126708i
\(833\) 0 0
\(834\) 609.533 0.730855
\(835\) 475.423 0.569369
\(836\) − 64.3391i − 0.0769607i
\(837\) 203.528 0.243164
\(838\) 712.367i 0.850080i
\(839\) 165.698i 0.197494i 0.995113 + 0.0987471i \(0.0314835\pi\)
−0.995113 + 0.0987471i \(0.968517\pi\)
\(840\) 0 0
\(841\) 1745.57 2.07559
\(842\) 1110.48 1.31886
\(843\) 925.857i 1.09829i
\(844\) 671.224 0.795289
\(845\) − 658.480i − 0.779266i
\(846\) − 30.5414i − 0.0361010i
\(847\) 0 0
\(848\) −88.4838 −0.104344
\(849\) −774.703 −0.912489
\(850\) 273.371i 0.321613i
\(851\) 217.669 0.255780
\(852\) 231.028i 0.271160i
\(853\) 1066.17i 1.24991i 0.780661 + 0.624955i \(0.214883\pi\)
−0.780661 + 0.624955i \(0.785117\pi\)
\(854\) 0 0
\(855\) 21.0373 0.0246050
\(856\) −372.195 −0.434808
\(857\) 222.650i 0.259802i 0.991527 + 0.129901i \(0.0414659\pi\)
−0.991527 + 0.129901i \(0.958534\pi\)
\(858\) 819.602 0.955247
\(859\) − 1516.66i − 1.76561i −0.469740 0.882805i \(-0.655652\pi\)
0.469740 0.882805i \(-0.344348\pi\)
\(860\) 92.8497i 0.107965i
\(861\) 0 0
\(862\) −321.190 −0.372610
\(863\) 138.036 0.159949 0.0799745 0.996797i \(-0.474516\pi\)
0.0799745 + 0.996797i \(0.474516\pi\)
\(864\) − 173.999i − 0.201388i
\(865\) −556.304 −0.643126
\(866\) − 1092.06i − 1.26104i
\(867\) 314.476i 0.362717i
\(868\) 0 0
\(869\) −252.732 −0.290831
\(870\) −496.457 −0.570640
\(871\) 2164.32i 2.48487i
\(872\) −113.565 −0.130235
\(873\) 221.396i 0.253603i
\(874\) 32.4737i 0.0371552i
\(875\) 0 0
\(876\) 304.911 0.348072
\(877\) −617.621 −0.704243 −0.352121 0.935954i \(-0.614540\pi\)
−0.352121 + 0.935954i \(0.614540\pi\)
\(878\) 159.364i 0.181508i
\(879\) 873.300 0.993515
\(880\) 387.579i 0.440430i
\(881\) − 425.629i − 0.483120i −0.970386 0.241560i \(-0.922341\pi\)
0.970386 0.241560i \(-0.0776591\pi\)
\(882\) 0 0
\(883\) 295.270 0.334394 0.167197 0.985923i \(-0.446528\pi\)
0.167197 + 0.985923i \(0.446528\pi\)
\(884\) −1098.66 −1.24283
\(885\) 364.494i 0.411857i
\(886\) 1104.90 1.24707
\(887\) 1689.77i 1.90504i 0.304481 + 0.952519i \(0.401517\pi\)
−0.304481 + 0.952519i \(0.598483\pi\)
\(888\) 381.037i 0.429096i
\(889\) 0 0
\(890\) 410.680 0.461438
\(891\) −78.4865 −0.0880881
\(892\) − 134.326i − 0.150589i
\(893\) −12.6671 −0.0141849
\(894\) 62.5893i 0.0700104i
\(895\) 123.588i 0.138087i
\(896\) 0 0
\(897\) −153.197 −0.170788
\(898\) 541.856 0.603403
\(899\) 1992.07i 2.21587i
\(900\) 35.2883 0.0392093
\(901\) − 96.5717i − 0.107183i
\(902\) 809.129i 0.897039i
\(903\) 0 0
\(904\) 383.409 0.424124
\(905\) 104.872 0.115880
\(906\) − 927.122i − 1.02331i
\(907\) 442.075 0.487404 0.243702 0.969850i \(-0.421638\pi\)
0.243702 + 0.969850i \(0.421638\pi\)
\(908\) − 429.708i − 0.473246i
\(909\) 320.861i 0.352982i
\(910\) 0 0
\(911\) 998.378 1.09591 0.547957 0.836507i \(-0.315406\pi\)
0.547957 + 0.836507i \(0.315406\pi\)
\(912\) −107.961 −0.118378
\(913\) 184.109i 0.201653i
\(914\) −607.926 −0.665127
\(915\) 283.435i 0.309765i
\(916\) 39.4834i 0.0431041i
\(917\) 0 0
\(918\) 284.096 0.309473
\(919\) 309.594 0.336881 0.168441 0.985712i \(-0.446127\pi\)
0.168441 + 0.985712i \(0.446127\pi\)
\(920\) − 38.1456i − 0.0414626i
\(921\) −690.728 −0.749976
\(922\) − 865.471i − 0.938689i
\(923\) 1220.62i 1.32245i
\(924\) 0 0
\(925\) −264.906 −0.286385
\(926\) 188.456 0.203516
\(927\) 64.6177i 0.0697062i
\(928\) 1703.05 1.83519
\(929\) 511.843i 0.550961i 0.961307 + 0.275480i \(0.0888369\pi\)
−0.961307 + 0.275480i \(0.911163\pi\)
\(930\) − 382.351i − 0.411130i
\(931\) 0 0
\(932\) −626.737 −0.672464
\(933\) −414.171 −0.443913
\(934\) 897.752i 0.961190i
\(935\) −423.006 −0.452412
\(936\) − 268.177i − 0.286514i
\(937\) 592.935i 0.632801i 0.948626 + 0.316401i \(0.102474\pi\)
−0.948626 + 0.316401i \(0.897526\pi\)
\(938\) 0 0
\(939\) −386.592 −0.411706
\(940\) −21.2480 −0.0226043
\(941\) − 634.459i − 0.674240i −0.941462 0.337120i \(-0.890547\pi\)
0.941462 0.337120i \(-0.109453\pi\)
\(942\) −1060.46 −1.12575
\(943\) − 151.240i − 0.160381i
\(944\) − 1870.54i − 1.98150i
\(945\) 0 0
\(946\) −387.956 −0.410101
\(947\) 659.609 0.696525 0.348263 0.937397i \(-0.386772\pi\)
0.348263 + 0.937397i \(0.386772\pi\)
\(948\) − 118.089i − 0.124566i
\(949\) 1610.97 1.69755
\(950\) − 39.5209i − 0.0416010i
\(951\) 495.391i 0.520916i
\(952\) 0 0
\(953\) −615.571 −0.645930 −0.322965 0.946411i \(-0.604680\pi\)
−0.322965 + 0.946411i \(0.604680\pi\)
\(954\) −33.6617 −0.0352849
\(955\) − 44.9724i − 0.0470915i
\(956\) −93.3992 −0.0976979
\(957\) − 768.201i − 0.802718i
\(958\) 942.013i 0.983312i
\(959\) 0 0
\(960\) −18.9651 −0.0197553
\(961\) −573.211 −0.596473
\(962\) − 2874.83i − 2.98839i
\(963\) −268.910 −0.279242
\(964\) 196.103i 0.203426i
\(965\) 64.1825i 0.0665104i
\(966\) 0 0
\(967\) 386.702 0.399899 0.199949 0.979806i \(-0.435922\pi\)
0.199949 + 0.979806i \(0.435922\pi\)
\(968\) 186.640 0.192810
\(969\) − 117.829i − 0.121599i
\(970\) 415.917 0.428781
\(971\) − 562.498i − 0.579298i −0.957133 0.289649i \(-0.906461\pi\)
0.957133 0.289649i \(-0.0935386\pi\)
\(972\) − 36.6727i − 0.0377291i
\(973\) 0 0
\(974\) 1959.79 2.01211
\(975\) 186.443 0.191224
\(976\) − 1454.56i − 1.49033i
\(977\) 712.083 0.728846 0.364423 0.931234i \(-0.381266\pi\)
0.364423 + 0.931234i \(0.381266\pi\)
\(978\) − 57.7756i − 0.0590753i
\(979\) 635.472i 0.649103i
\(980\) 0 0
\(981\) −82.0504 −0.0836396
\(982\) 1156.36 1.17755
\(983\) 1228.81i 1.25006i 0.780599 + 0.625032i \(0.214914\pi\)
−0.780599 + 0.625032i \(0.785086\pi\)
\(984\) 264.751 0.269055
\(985\) 501.855i 0.509498i
\(986\) 2780.64i 2.82012i
\(987\) 0 0
\(988\) 158.832 0.160761
\(989\) 72.5153 0.0733219
\(990\) 147.446i 0.148935i
\(991\) −1242.50 −1.25378 −0.626892 0.779107i \(-0.715673\pi\)
−0.626892 + 0.779107i \(0.715673\pi\)
\(992\) 1311.62i 1.32220i
\(993\) − 933.617i − 0.940199i
\(994\) 0 0
\(995\) 711.202 0.714776
\(996\) −86.0246 −0.0863701
\(997\) − 801.766i − 0.804178i −0.915601 0.402089i \(-0.868284\pi\)
0.915601 0.402089i \(-0.131716\pi\)
\(998\) 1601.84 1.60505
\(999\) 275.298i 0.275574i
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 735.3.h.a.391.7 8
7.4 even 3 105.3.n.a.61.1 yes 8
7.5 odd 6 105.3.n.a.31.1 8
7.6 odd 2 inner 735.3.h.a.391.8 8
21.5 even 6 315.3.w.a.136.4 8
21.11 odd 6 315.3.w.a.271.4 8
35.4 even 6 525.3.o.l.376.4 8
35.12 even 12 525.3.s.h.199.7 16
35.18 odd 12 525.3.s.h.124.7 16
35.19 odd 6 525.3.o.l.451.4 8
35.32 odd 12 525.3.s.h.124.2 16
35.33 even 12 525.3.s.h.199.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.3.n.a.31.1 8 7.5 odd 6
105.3.n.a.61.1 yes 8 7.4 even 3
315.3.w.a.136.4 8 21.5 even 6
315.3.w.a.271.4 8 21.11 odd 6
525.3.o.l.376.4 8 35.4 even 6
525.3.o.l.451.4 8 35.19 odd 6
525.3.s.h.124.2 16 35.32 odd 12
525.3.s.h.124.7 16 35.18 odd 12
525.3.s.h.199.2 16 35.33 even 12
525.3.s.h.199.7 16 35.12 even 12
735.3.h.a.391.7 8 1.1 even 1 trivial
735.3.h.a.391.8 8 7.6 odd 2 inner