Properties

Label 75.4.b.b.49.1
Level $75$
Weight $4$
Character 75.49
Analytic conductor $4.425$
Analytic rank $0$
Dimension $2$
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] = [75,4,Mod(49,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not 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: \(4.42514325043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.4.b.b.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} +3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} +24.0000i q^{7} -15.0000i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +3.00000i q^{3} +7.00000 q^{4} +3.00000 q^{6} +24.0000i q^{7} -15.0000i q^{8} -9.00000 q^{9} +52.0000 q^{11} +21.0000i q^{12} +22.0000i q^{13} +24.0000 q^{14} +41.0000 q^{16} +14.0000i q^{17} +9.00000i q^{18} +20.0000 q^{19} -72.0000 q^{21} -52.0000i q^{22} -168.000i q^{23} +45.0000 q^{24} +22.0000 q^{26} -27.0000i q^{27} +168.000i q^{28} -230.000 q^{29} -288.000 q^{31} -161.000i q^{32} +156.000i q^{33} +14.0000 q^{34} -63.0000 q^{36} +34.0000i q^{37} -20.0000i q^{38} -66.0000 q^{39} +122.000 q^{41} +72.0000i q^{42} -188.000i q^{43} +364.000 q^{44} -168.000 q^{46} -256.000i q^{47} +123.000i q^{48} -233.000 q^{49} -42.0000 q^{51} +154.000i q^{52} -338.000i q^{53} -27.0000 q^{54} +360.000 q^{56} +60.0000i q^{57} +230.000i q^{58} -100.000 q^{59} +742.000 q^{61} +288.000i q^{62} -216.000i q^{63} +167.000 q^{64} +156.000 q^{66} +84.0000i q^{67} +98.0000i q^{68} +504.000 q^{69} -328.000 q^{71} +135.000i q^{72} -38.0000i q^{73} +34.0000 q^{74} +140.000 q^{76} +1248.00i q^{77} +66.0000i q^{78} +240.000 q^{79} +81.0000 q^{81} -122.000i q^{82} +1212.00i q^{83} -504.000 q^{84} -188.000 q^{86} -690.000i q^{87} -780.000i q^{88} -330.000 q^{89} -528.000 q^{91} -1176.00i q^{92} -864.000i q^{93} -256.000 q^{94} +483.000 q^{96} -866.000i q^{97} +233.000i q^{98} -468.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} + 6 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 14 q^{4} + 6 q^{6} - 18 q^{9} + 104 q^{11} + 48 q^{14} + 82 q^{16} + 40 q^{19} - 144 q^{21} + 90 q^{24} + 44 q^{26} - 460 q^{29} - 576 q^{31} + 28 q^{34} - 126 q^{36} - 132 q^{39} + 244 q^{41} + 728 q^{44} - 336 q^{46} - 466 q^{49} - 84 q^{51} - 54 q^{54} + 720 q^{56} - 200 q^{59} + 1484 q^{61} + 334 q^{64} + 312 q^{66} + 1008 q^{69} - 656 q^{71} + 68 q^{74} + 280 q^{76} + 480 q^{79} + 162 q^{81} - 1008 q^{84} - 376 q^{86} - 660 q^{89} - 1056 q^{91} - 512 q^{94} + 966 q^{96} - 936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.353553i −0.984251 0.176777i \(-0.943433\pi\)
0.984251 0.176777i \(-0.0565670\pi\)
\(3\) 3.00000i 0.577350i
\(4\) 7.00000 0.875000
\(5\) 0 0
\(6\) 3.00000 0.204124
\(7\) 24.0000i 1.29588i 0.761692 + 0.647939i \(0.224369\pi\)
−0.761692 + 0.647939i \(0.775631\pi\)
\(8\) − 15.0000i − 0.662913i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 52.0000 1.42533 0.712663 0.701506i \(-0.247489\pi\)
0.712663 + 0.701506i \(0.247489\pi\)
\(12\) 21.0000i 0.505181i
\(13\) 22.0000i 0.469362i 0.972072 + 0.234681i \(0.0754045\pi\)
−0.972072 + 0.234681i \(0.924595\pi\)
\(14\) 24.0000 0.458162
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) 14.0000i 0.199735i 0.995001 + 0.0998676i \(0.0318419\pi\)
−0.995001 + 0.0998676i \(0.968158\pi\)
\(18\) 9.00000i 0.117851i
\(19\) 20.0000 0.241490 0.120745 0.992684i \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) 0 0
\(21\) −72.0000 −0.748176
\(22\) − 52.0000i − 0.503929i
\(23\) − 168.000i − 1.52306i −0.648129 0.761531i \(-0.724448\pi\)
0.648129 0.761531i \(-0.275552\pi\)
\(24\) 45.0000 0.382733
\(25\) 0 0
\(26\) 22.0000 0.165944
\(27\) − 27.0000i − 0.192450i
\(28\) 168.000i 1.13389i
\(29\) −230.000 −1.47276 −0.736378 0.676570i \(-0.763465\pi\)
−0.736378 + 0.676570i \(0.763465\pi\)
\(30\) 0 0
\(31\) −288.000 −1.66859 −0.834296 0.551317i \(-0.814125\pi\)
−0.834296 + 0.551317i \(0.814125\pi\)
\(32\) − 161.000i − 0.889408i
\(33\) 156.000i 0.822913i
\(34\) 14.0000 0.0706171
\(35\) 0 0
\(36\) −63.0000 −0.291667
\(37\) 34.0000i 0.151069i 0.997143 + 0.0755347i \(0.0240664\pi\)
−0.997143 + 0.0755347i \(0.975934\pi\)
\(38\) − 20.0000i − 0.0853797i
\(39\) −66.0000 −0.270986
\(40\) 0 0
\(41\) 122.000 0.464712 0.232356 0.972631i \(-0.425357\pi\)
0.232356 + 0.972631i \(0.425357\pi\)
\(42\) 72.0000i 0.264520i
\(43\) − 188.000i − 0.666738i −0.942796 0.333369i \(-0.891815\pi\)
0.942796 0.333369i \(-0.108185\pi\)
\(44\) 364.000 1.24716
\(45\) 0 0
\(46\) −168.000 −0.538484
\(47\) − 256.000i − 0.794499i −0.917711 0.397249i \(-0.869965\pi\)
0.917711 0.397249i \(-0.130035\pi\)
\(48\) 123.000i 0.369865i
\(49\) −233.000 −0.679300
\(50\) 0 0
\(51\) −42.0000 −0.115317
\(52\) 154.000i 0.410691i
\(53\) − 338.000i − 0.875998i −0.898976 0.437999i \(-0.855687\pi\)
0.898976 0.437999i \(-0.144313\pi\)
\(54\) −27.0000 −0.0680414
\(55\) 0 0
\(56\) 360.000 0.859054
\(57\) 60.0000i 0.139424i
\(58\) 230.000i 0.520698i
\(59\) −100.000 −0.220659 −0.110330 0.993895i \(-0.535191\pi\)
−0.110330 + 0.993895i \(0.535191\pi\)
\(60\) 0 0
\(61\) 742.000 1.55743 0.778716 0.627376i \(-0.215871\pi\)
0.778716 + 0.627376i \(0.215871\pi\)
\(62\) 288.000i 0.589936i
\(63\) − 216.000i − 0.431959i
\(64\) 167.000 0.326172
\(65\) 0 0
\(66\) 156.000 0.290944
\(67\) 84.0000i 0.153168i 0.997063 + 0.0765838i \(0.0244013\pi\)
−0.997063 + 0.0765838i \(0.975599\pi\)
\(68\) 98.0000i 0.174768i
\(69\) 504.000 0.879340
\(70\) 0 0
\(71\) −328.000 −0.548260 −0.274130 0.961693i \(-0.588390\pi\)
−0.274130 + 0.961693i \(0.588390\pi\)
\(72\) 135.000i 0.220971i
\(73\) − 38.0000i − 0.0609255i −0.999536 0.0304628i \(-0.990302\pi\)
0.999536 0.0304628i \(-0.00969810\pi\)
\(74\) 34.0000 0.0534111
\(75\) 0 0
\(76\) 140.000 0.211304
\(77\) 1248.00i 1.84705i
\(78\) 66.0000i 0.0958081i
\(79\) 240.000 0.341799 0.170899 0.985288i \(-0.445333\pi\)
0.170899 + 0.985288i \(0.445333\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 122.000i − 0.164301i
\(83\) 1212.00i 1.60282i 0.598114 + 0.801411i \(0.295917\pi\)
−0.598114 + 0.801411i \(0.704083\pi\)
\(84\) −504.000 −0.654654
\(85\) 0 0
\(86\) −188.000 −0.235727
\(87\) − 690.000i − 0.850296i
\(88\) − 780.000i − 0.944867i
\(89\) −330.000 −0.393033 −0.196516 0.980501i \(-0.562963\pi\)
−0.196516 + 0.980501i \(0.562963\pi\)
\(90\) 0 0
\(91\) −528.000 −0.608236
\(92\) − 1176.00i − 1.33268i
\(93\) − 864.000i − 0.963362i
\(94\) −256.000 −0.280898
\(95\) 0 0
\(96\) 483.000 0.513500
\(97\) − 866.000i − 0.906484i −0.891387 0.453242i \(-0.850267\pi\)
0.891387 0.453242i \(-0.149733\pi\)
\(98\) 233.000i 0.240169i
\(99\) −468.000 −0.475109
\(100\) 0 0
\(101\) −1218.00 −1.19996 −0.599978 0.800017i \(-0.704824\pi\)
−0.599978 + 0.800017i \(0.704824\pi\)
\(102\) 42.0000i 0.0407708i
\(103\) − 88.0000i − 0.0841835i −0.999114 0.0420917i \(-0.986598\pi\)
0.999114 0.0420917i \(-0.0134022\pi\)
\(104\) 330.000 0.311146
\(105\) 0 0
\(106\) −338.000 −0.309712
\(107\) − 36.0000i − 0.0325257i −0.999868 0.0162629i \(-0.994823\pi\)
0.999868 0.0162629i \(-0.00517686\pi\)
\(108\) − 189.000i − 0.168394i
\(109\) 970.000 0.852378 0.426189 0.904634i \(-0.359856\pi\)
0.426189 + 0.904634i \(0.359856\pi\)
\(110\) 0 0
\(111\) −102.000 −0.0872199
\(112\) 984.000i 0.830172i
\(113\) 1042.00i 0.867461i 0.901043 + 0.433731i \(0.142803\pi\)
−0.901043 + 0.433731i \(0.857197\pi\)
\(114\) 60.0000 0.0492940
\(115\) 0 0
\(116\) −1610.00 −1.28866
\(117\) − 198.000i − 0.156454i
\(118\) 100.000i 0.0780148i
\(119\) −336.000 −0.258833
\(120\) 0 0
\(121\) 1373.00 1.03156
\(122\) − 742.000i − 0.550635i
\(123\) 366.000i 0.268302i
\(124\) −2016.00 −1.46002
\(125\) 0 0
\(126\) −216.000 −0.152721
\(127\) − 1936.00i − 1.35269i −0.736583 0.676347i \(-0.763562\pi\)
0.736583 0.676347i \(-0.236438\pi\)
\(128\) − 1455.00i − 1.00473i
\(129\) 564.000 0.384941
\(130\) 0 0
\(131\) 732.000 0.488207 0.244104 0.969749i \(-0.421506\pi\)
0.244104 + 0.969749i \(0.421506\pi\)
\(132\) 1092.00i 0.720048i
\(133\) 480.000i 0.312942i
\(134\) 84.0000 0.0541529
\(135\) 0 0
\(136\) 210.000 0.132407
\(137\) 2214.00i 1.38069i 0.723479 + 0.690346i \(0.242542\pi\)
−0.723479 + 0.690346i \(0.757458\pi\)
\(138\) − 504.000i − 0.310894i
\(139\) −20.0000 −0.0122042 −0.00610208 0.999981i \(-0.501942\pi\)
−0.00610208 + 0.999981i \(0.501942\pi\)
\(140\) 0 0
\(141\) 768.000 0.458704
\(142\) 328.000i 0.193839i
\(143\) 1144.00i 0.668994i
\(144\) −369.000 −0.213542
\(145\) 0 0
\(146\) −38.0000 −0.0215404
\(147\) − 699.000i − 0.392194i
\(148\) 238.000i 0.132186i
\(149\) 1330.00 0.731261 0.365630 0.930760i \(-0.380853\pi\)
0.365630 + 0.930760i \(0.380853\pi\)
\(150\) 0 0
\(151\) −1208.00 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) − 300.000i − 0.160087i
\(153\) − 126.000i − 0.0665784i
\(154\) 1248.00 0.653031
\(155\) 0 0
\(156\) −462.000 −0.237113
\(157\) 3514.00i 1.78629i 0.449768 + 0.893146i \(0.351507\pi\)
−0.449768 + 0.893146i \(0.648493\pi\)
\(158\) − 240.000i − 0.120844i
\(159\) 1014.00 0.505757
\(160\) 0 0
\(161\) 4032.00 1.97370
\(162\) − 81.0000i − 0.0392837i
\(163\) − 2068.00i − 0.993732i −0.867827 0.496866i \(-0.834484\pi\)
0.867827 0.496866i \(-0.165516\pi\)
\(164\) 854.000 0.406623
\(165\) 0 0
\(166\) 1212.00 0.566683
\(167\) 24.0000i 0.0111208i 0.999985 + 0.00556041i \(0.00176994\pi\)
−0.999985 + 0.00556041i \(0.998230\pi\)
\(168\) 1080.00i 0.495975i
\(169\) 1713.00 0.779700
\(170\) 0 0
\(171\) −180.000 −0.0804967
\(172\) − 1316.00i − 0.583396i
\(173\) − 618.000i − 0.271593i −0.990737 0.135797i \(-0.956641\pi\)
0.990737 0.135797i \(-0.0433594\pi\)
\(174\) −690.000 −0.300625
\(175\) 0 0
\(176\) 2132.00 0.913100
\(177\) − 300.000i − 0.127398i
\(178\) 330.000i 0.138958i
\(179\) −3340.00 −1.39466 −0.697328 0.716752i \(-0.745628\pi\)
−0.697328 + 0.716752i \(0.745628\pi\)
\(180\) 0 0
\(181\) −178.000 −0.0730974 −0.0365487 0.999332i \(-0.511636\pi\)
−0.0365487 + 0.999332i \(0.511636\pi\)
\(182\) 528.000i 0.215044i
\(183\) 2226.00i 0.899184i
\(184\) −2520.00 −1.00966
\(185\) 0 0
\(186\) −864.000 −0.340600
\(187\) 728.000i 0.284688i
\(188\) − 1792.00i − 0.695186i
\(189\) 648.000 0.249392
\(190\) 0 0
\(191\) −1888.00 −0.715240 −0.357620 0.933867i \(-0.616412\pi\)
−0.357620 + 0.933867i \(0.616412\pi\)
\(192\) 501.000i 0.188315i
\(193\) 1922.00i 0.716832i 0.933562 + 0.358416i \(0.116683\pi\)
−0.933562 + 0.358416i \(0.883317\pi\)
\(194\) −866.000 −0.320491
\(195\) 0 0
\(196\) −1631.00 −0.594388
\(197\) − 2526.00i − 0.913554i −0.889581 0.456777i \(-0.849004\pi\)
0.889581 0.456777i \(-0.150996\pi\)
\(198\) 468.000i 0.167976i
\(199\) 1160.00 0.413217 0.206609 0.978424i \(-0.433757\pi\)
0.206609 + 0.978424i \(0.433757\pi\)
\(200\) 0 0
\(201\) −252.000 −0.0884314
\(202\) 1218.00i 0.424248i
\(203\) − 5520.00i − 1.90851i
\(204\) −294.000 −0.100903
\(205\) 0 0
\(206\) −88.0000 −0.0297634
\(207\) 1512.00i 0.507687i
\(208\) 902.000i 0.300685i
\(209\) 1040.00 0.344202
\(210\) 0 0
\(211\) −4468.00 −1.45777 −0.728886 0.684635i \(-0.759961\pi\)
−0.728886 + 0.684635i \(0.759961\pi\)
\(212\) − 2366.00i − 0.766498i
\(213\) − 984.000i − 0.316538i
\(214\) −36.0000 −0.0114996
\(215\) 0 0
\(216\) −405.000 −0.127578
\(217\) − 6912.00i − 2.16229i
\(218\) − 970.000i − 0.301361i
\(219\) 114.000 0.0351754
\(220\) 0 0
\(221\) −308.000 −0.0937481
\(222\) 102.000i 0.0308369i
\(223\) 6032.00i 1.81136i 0.423965 + 0.905678i \(0.360638\pi\)
−0.423965 + 0.905678i \(0.639362\pi\)
\(224\) 3864.00 1.15256
\(225\) 0 0
\(226\) 1042.00 0.306694
\(227\) − 2636.00i − 0.770738i −0.922763 0.385369i \(-0.874074\pi\)
0.922763 0.385369i \(-0.125926\pi\)
\(228\) 420.000i 0.121996i
\(229\) −4830.00 −1.39378 −0.696889 0.717179i \(-0.745433\pi\)
−0.696889 + 0.717179i \(0.745433\pi\)
\(230\) 0 0
\(231\) −3744.00 −1.06639
\(232\) 3450.00i 0.976309i
\(233\) 2682.00i 0.754093i 0.926194 + 0.377046i \(0.123060\pi\)
−0.926194 + 0.377046i \(0.876940\pi\)
\(234\) −198.000 −0.0553148
\(235\) 0 0
\(236\) −700.000 −0.193077
\(237\) 720.000i 0.197338i
\(238\) 336.000i 0.0915111i
\(239\) −2320.00 −0.627901 −0.313950 0.949439i \(-0.601653\pi\)
−0.313950 + 0.949439i \(0.601653\pi\)
\(240\) 0 0
\(241\) 2002.00 0.535104 0.267552 0.963543i \(-0.413785\pi\)
0.267552 + 0.963543i \(0.413785\pi\)
\(242\) − 1373.00i − 0.364710i
\(243\) 243.000i 0.0641500i
\(244\) 5194.00 1.36275
\(245\) 0 0
\(246\) 366.000 0.0948590
\(247\) 440.000i 0.113346i
\(248\) 4320.00i 1.10613i
\(249\) −3636.00 −0.925390
\(250\) 0 0
\(251\) 132.000 0.0331943 0.0165971 0.999862i \(-0.494717\pi\)
0.0165971 + 0.999862i \(0.494717\pi\)
\(252\) − 1512.00i − 0.377964i
\(253\) − 8736.00i − 2.17086i
\(254\) −1936.00 −0.478250
\(255\) 0 0
\(256\) −119.000 −0.0290527
\(257\) 7614.00i 1.84805i 0.382335 + 0.924024i \(0.375120\pi\)
−0.382335 + 0.924024i \(0.624880\pi\)
\(258\) − 564.000i − 0.136097i
\(259\) −816.000 −0.195767
\(260\) 0 0
\(261\) 2070.00 0.490919
\(262\) − 732.000i − 0.172607i
\(263\) − 4888.00i − 1.14603i −0.819543 0.573017i \(-0.805773\pi\)
0.819543 0.573017i \(-0.194227\pi\)
\(264\) 2340.00 0.545519
\(265\) 0 0
\(266\) 480.000 0.110642
\(267\) − 990.000i − 0.226918i
\(268\) 588.000i 0.134022i
\(269\) −1270.00 −0.287856 −0.143928 0.989588i \(-0.545973\pi\)
−0.143928 + 0.989588i \(0.545973\pi\)
\(270\) 0 0
\(271\) 1072.00 0.240293 0.120146 0.992756i \(-0.461664\pi\)
0.120146 + 0.992756i \(0.461664\pi\)
\(272\) 574.000i 0.127955i
\(273\) − 1584.00i − 0.351165i
\(274\) 2214.00 0.488148
\(275\) 0 0
\(276\) 3528.00 0.769423
\(277\) 5394.00i 1.17001i 0.811028 + 0.585007i \(0.198908\pi\)
−0.811028 + 0.585007i \(0.801092\pi\)
\(278\) 20.0000i 0.00431482i
\(279\) 2592.00 0.556197
\(280\) 0 0
\(281\) 2442.00 0.518425 0.259213 0.965820i \(-0.416537\pi\)
0.259213 + 0.965820i \(0.416537\pi\)
\(282\) − 768.000i − 0.162176i
\(283\) 2772.00i 0.582255i 0.956684 + 0.291128i \(0.0940305\pi\)
−0.956684 + 0.291128i \(0.905970\pi\)
\(284\) −2296.00 −0.479727
\(285\) 0 0
\(286\) 1144.00 0.236525
\(287\) 2928.00i 0.602210i
\(288\) 1449.00i 0.296469i
\(289\) 4717.00 0.960106
\(290\) 0 0
\(291\) 2598.00 0.523359
\(292\) − 266.000i − 0.0533098i
\(293\) 4542.00i 0.905619i 0.891607 + 0.452810i \(0.149578\pi\)
−0.891607 + 0.452810i \(0.850422\pi\)
\(294\) −699.000 −0.138662
\(295\) 0 0
\(296\) 510.000 0.100146
\(297\) − 1404.00i − 0.274304i
\(298\) − 1330.00i − 0.258540i
\(299\) 3696.00 0.714867
\(300\) 0 0
\(301\) 4512.00 0.864011
\(302\) 1208.00i 0.230174i
\(303\) − 3654.00i − 0.692795i
\(304\) 820.000 0.154705
\(305\) 0 0
\(306\) −126.000 −0.0235390
\(307\) − 5116.00i − 0.951093i −0.879691 0.475546i \(-0.842250\pi\)
0.879691 0.475546i \(-0.157750\pi\)
\(308\) 8736.00i 1.61617i
\(309\) 264.000 0.0486034
\(310\) 0 0
\(311\) −2808.00 −0.511984 −0.255992 0.966679i \(-0.582402\pi\)
−0.255992 + 0.966679i \(0.582402\pi\)
\(312\) 990.000i 0.179640i
\(313\) − 7318.00i − 1.32153i −0.750594 0.660763i \(-0.770233\pi\)
0.750594 0.660763i \(-0.229767\pi\)
\(314\) 3514.00 0.631549
\(315\) 0 0
\(316\) 1680.00 0.299074
\(317\) − 2246.00i − 0.397943i −0.980005 0.198971i \(-0.936240\pi\)
0.980005 0.198971i \(-0.0637601\pi\)
\(318\) − 1014.00i − 0.178812i
\(319\) −11960.0 −2.09916
\(320\) 0 0
\(321\) 108.000 0.0187787
\(322\) − 4032.00i − 0.697809i
\(323\) 280.000i 0.0482341i
\(324\) 567.000 0.0972222
\(325\) 0 0
\(326\) −2068.00 −0.351337
\(327\) 2910.00i 0.492120i
\(328\) − 1830.00i − 0.308064i
\(329\) 6144.00 1.02957
\(330\) 0 0
\(331\) 1332.00 0.221188 0.110594 0.993866i \(-0.464725\pi\)
0.110594 + 0.993866i \(0.464725\pi\)
\(332\) 8484.00i 1.40247i
\(333\) − 306.000i − 0.0503564i
\(334\) 24.0000 0.00393180
\(335\) 0 0
\(336\) −2952.00 −0.479300
\(337\) 11534.0i 1.86438i 0.361966 + 0.932191i \(0.382106\pi\)
−0.361966 + 0.932191i \(0.617894\pi\)
\(338\) − 1713.00i − 0.275665i
\(339\) −3126.00 −0.500829
\(340\) 0 0
\(341\) −14976.0 −2.37829
\(342\) 180.000i 0.0284599i
\(343\) 2640.00i 0.415588i
\(344\) −2820.00 −0.441989
\(345\) 0 0
\(346\) −618.000 −0.0960228
\(347\) − 11956.0i − 1.84966i −0.380382 0.924830i \(-0.624207\pi\)
0.380382 0.924830i \(-0.375793\pi\)
\(348\) − 4830.00i − 0.744009i
\(349\) −4870.00 −0.746949 −0.373474 0.927640i \(-0.621834\pi\)
−0.373474 + 0.927640i \(0.621834\pi\)
\(350\) 0 0
\(351\) 594.000 0.0903287
\(352\) − 8372.00i − 1.26770i
\(353\) 10722.0i 1.61664i 0.588742 + 0.808321i \(0.299623\pi\)
−0.588742 + 0.808321i \(0.700377\pi\)
\(354\) −300.000 −0.0450419
\(355\) 0 0
\(356\) −2310.00 −0.343904
\(357\) − 1008.00i − 0.149437i
\(358\) 3340.00i 0.493085i
\(359\) −120.000 −0.0176417 −0.00882083 0.999961i \(-0.502808\pi\)
−0.00882083 + 0.999961i \(0.502808\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 178.000i 0.0258438i
\(363\) 4119.00i 0.595569i
\(364\) −3696.00 −0.532206
\(365\) 0 0
\(366\) 2226.00 0.317910
\(367\) − 3936.00i − 0.559830i −0.960025 0.279915i \(-0.909694\pi\)
0.960025 0.279915i \(-0.0903063\pi\)
\(368\) − 6888.00i − 0.975711i
\(369\) −1098.00 −0.154904
\(370\) 0 0
\(371\) 8112.00 1.13519
\(372\) − 6048.00i − 0.842941i
\(373\) 3022.00i 0.419499i 0.977755 + 0.209750i \(0.0672649\pi\)
−0.977755 + 0.209750i \(0.932735\pi\)
\(374\) 728.000 0.100652
\(375\) 0 0
\(376\) −3840.00 −0.526683
\(377\) − 5060.00i − 0.691255i
\(378\) − 648.000i − 0.0881733i
\(379\) 13340.0 1.80799 0.903997 0.427539i \(-0.140619\pi\)
0.903997 + 0.427539i \(0.140619\pi\)
\(380\) 0 0
\(381\) 5808.00 0.780979
\(382\) 1888.00i 0.252876i
\(383\) − 1008.00i − 0.134481i −0.997737 0.0672407i \(-0.978580\pi\)
0.997737 0.0672407i \(-0.0214195\pi\)
\(384\) 4365.00 0.580079
\(385\) 0 0
\(386\) 1922.00 0.253438
\(387\) 1692.00i 0.222246i
\(388\) − 6062.00i − 0.793174i
\(389\) −9630.00 −1.25517 −0.627584 0.778549i \(-0.715956\pi\)
−0.627584 + 0.778549i \(0.715956\pi\)
\(390\) 0 0
\(391\) 2352.00 0.304209
\(392\) 3495.00i 0.450317i
\(393\) 2196.00i 0.281867i
\(394\) −2526.00 −0.322990
\(395\) 0 0
\(396\) −3276.00 −0.415720
\(397\) − 7126.00i − 0.900866i −0.892810 0.450433i \(-0.851270\pi\)
0.892810 0.450433i \(-0.148730\pi\)
\(398\) − 1160.00i − 0.146094i
\(399\) −1440.00 −0.180677
\(400\) 0 0
\(401\) −8718.00 −1.08568 −0.542838 0.839837i \(-0.682650\pi\)
−0.542838 + 0.839837i \(0.682650\pi\)
\(402\) 252.000i 0.0312652i
\(403\) − 6336.00i − 0.783173i
\(404\) −8526.00 −1.04996
\(405\) 0 0
\(406\) −5520.00 −0.674761
\(407\) 1768.00i 0.215323i
\(408\) 630.000i 0.0764452i
\(409\) 10870.0 1.31415 0.657074 0.753826i \(-0.271794\pi\)
0.657074 + 0.753826i \(0.271794\pi\)
\(410\) 0 0
\(411\) −6642.00 −0.797143
\(412\) − 616.000i − 0.0736605i
\(413\) − 2400.00i − 0.285947i
\(414\) 1512.00 0.179495
\(415\) 0 0
\(416\) 3542.00 0.417454
\(417\) − 60.0000i − 0.00704607i
\(418\) − 1040.00i − 0.121694i
\(419\) 9700.00 1.13097 0.565484 0.824759i \(-0.308689\pi\)
0.565484 + 0.824759i \(0.308689\pi\)
\(420\) 0 0
\(421\) 862.000 0.0997893 0.0498947 0.998754i \(-0.484111\pi\)
0.0498947 + 0.998754i \(0.484111\pi\)
\(422\) 4468.00i 0.515400i
\(423\) 2304.00i 0.264833i
\(424\) −5070.00 −0.580710
\(425\) 0 0
\(426\) −984.000 −0.111913
\(427\) 17808.0i 2.01824i
\(428\) − 252.000i − 0.0284600i
\(429\) −3432.00 −0.386244
\(430\) 0 0
\(431\) 15792.0 1.76490 0.882452 0.470402i \(-0.155891\pi\)
0.882452 + 0.470402i \(0.155891\pi\)
\(432\) − 1107.00i − 0.123288i
\(433\) 11602.0i 1.28766i 0.765169 + 0.643830i \(0.222655\pi\)
−0.765169 + 0.643830i \(0.777345\pi\)
\(434\) −6912.00 −0.764485
\(435\) 0 0
\(436\) 6790.00 0.745830
\(437\) − 3360.00i − 0.367805i
\(438\) − 114.000i − 0.0124364i
\(439\) 440.000 0.0478361 0.0239181 0.999714i \(-0.492386\pi\)
0.0239181 + 0.999714i \(0.492386\pi\)
\(440\) 0 0
\(441\) 2097.00 0.226433
\(442\) 308.000i 0.0331449i
\(443\) − 10188.0i − 1.09266i −0.837571 0.546328i \(-0.816025\pi\)
0.837571 0.546328i \(-0.183975\pi\)
\(444\) −714.000 −0.0763174
\(445\) 0 0
\(446\) 6032.00 0.640411
\(447\) 3990.00i 0.422194i
\(448\) 4008.00i 0.422679i
\(449\) 13310.0 1.39897 0.699485 0.714647i \(-0.253413\pi\)
0.699485 + 0.714647i \(0.253413\pi\)
\(450\) 0 0
\(451\) 6344.00 0.662367
\(452\) 7294.00i 0.759029i
\(453\) − 3624.00i − 0.375873i
\(454\) −2636.00 −0.272497
\(455\) 0 0
\(456\) 900.000 0.0924262
\(457\) − 3226.00i − 0.330210i −0.986276 0.165105i \(-0.947204\pi\)
0.986276 0.165105i \(-0.0527963\pi\)
\(458\) 4830.00i 0.492775i
\(459\) 378.000 0.0384391
\(460\) 0 0
\(461\) 6582.00 0.664977 0.332488 0.943107i \(-0.392112\pi\)
0.332488 + 0.943107i \(0.392112\pi\)
\(462\) 3744.00i 0.377027i
\(463\) 15072.0i 1.51286i 0.654073 + 0.756431i \(0.273059\pi\)
−0.654073 + 0.756431i \(0.726941\pi\)
\(464\) −9430.00 −0.943484
\(465\) 0 0
\(466\) 2682.00 0.266612
\(467\) − 476.000i − 0.0471663i −0.999722 0.0235831i \(-0.992493\pi\)
0.999722 0.0235831i \(-0.00750744\pi\)
\(468\) − 1386.00i − 0.136897i
\(469\) −2016.00 −0.198487
\(470\) 0 0
\(471\) −10542.0 −1.03132
\(472\) 1500.00i 0.146278i
\(473\) − 9776.00i − 0.950319i
\(474\) 720.000 0.0697694
\(475\) 0 0
\(476\) −2352.00 −0.226478
\(477\) 3042.00i 0.291999i
\(478\) 2320.00i 0.221997i
\(479\) 19680.0 1.87725 0.938624 0.344941i \(-0.112101\pi\)
0.938624 + 0.344941i \(0.112101\pi\)
\(480\) 0 0
\(481\) −748.000 −0.0709062
\(482\) − 2002.00i − 0.189188i
\(483\) 12096.0i 1.13952i
\(484\) 9611.00 0.902611
\(485\) 0 0
\(486\) 243.000 0.0226805
\(487\) 5944.00i 0.553077i 0.961003 + 0.276538i \(0.0891873\pi\)
−0.961003 + 0.276538i \(0.910813\pi\)
\(488\) − 11130.0i − 1.03244i
\(489\) 6204.00 0.573731
\(490\) 0 0
\(491\) 10772.0 0.990089 0.495044 0.868868i \(-0.335152\pi\)
0.495044 + 0.868868i \(0.335152\pi\)
\(492\) 2562.00i 0.234764i
\(493\) − 3220.00i − 0.294161i
\(494\) 440.000 0.0400740
\(495\) 0 0
\(496\) −11808.0 −1.06894
\(497\) − 7872.00i − 0.710478i
\(498\) 3636.00i 0.327175i
\(499\) −8140.00 −0.730253 −0.365127 0.930958i \(-0.618974\pi\)
−0.365127 + 0.930958i \(0.618974\pi\)
\(500\) 0 0
\(501\) −72.0000 −0.00642060
\(502\) − 132.000i − 0.0117360i
\(503\) − 13768.0i − 1.22045i −0.792229 0.610223i \(-0.791080\pi\)
0.792229 0.610223i \(-0.208920\pi\)
\(504\) −3240.00 −0.286351
\(505\) 0 0
\(506\) −8736.00 −0.767515
\(507\) 5139.00i 0.450160i
\(508\) − 13552.0i − 1.18361i
\(509\) −22150.0 −1.92884 −0.964422 0.264368i \(-0.914837\pi\)
−0.964422 + 0.264368i \(0.914837\pi\)
\(510\) 0 0
\(511\) 912.000 0.0789521
\(512\) − 11521.0i − 0.994455i
\(513\) − 540.000i − 0.0464748i
\(514\) 7614.00 0.653384
\(515\) 0 0
\(516\) 3948.00 0.336824
\(517\) − 13312.0i − 1.13242i
\(518\) 816.000i 0.0692143i
\(519\) 1854.00 0.156805
\(520\) 0 0
\(521\) 1562.00 0.131348 0.0656741 0.997841i \(-0.479080\pi\)
0.0656741 + 0.997841i \(0.479080\pi\)
\(522\) − 2070.00i − 0.173566i
\(523\) − 668.000i − 0.0558501i −0.999610 0.0279250i \(-0.991110\pi\)
0.999610 0.0279250i \(-0.00888997\pi\)
\(524\) 5124.00 0.427181
\(525\) 0 0
\(526\) −4888.00 −0.405184
\(527\) − 4032.00i − 0.333276i
\(528\) 6396.00i 0.527178i
\(529\) −16057.0 −1.31972
\(530\) 0 0
\(531\) 900.000 0.0735531
\(532\) 3360.00i 0.273824i
\(533\) 2684.00i 0.218118i
\(534\) −990.000 −0.0802275
\(535\) 0 0
\(536\) 1260.00 0.101537
\(537\) − 10020.0i − 0.805205i
\(538\) 1270.00i 0.101772i
\(539\) −12116.0 −0.968225
\(540\) 0 0
\(541\) −6138.00 −0.487788 −0.243894 0.969802i \(-0.578425\pi\)
−0.243894 + 0.969802i \(0.578425\pi\)
\(542\) − 1072.00i − 0.0849564i
\(543\) − 534.000i − 0.0422028i
\(544\) 2254.00 0.177646
\(545\) 0 0
\(546\) −1584.00 −0.124156
\(547\) 10484.0i 0.819494i 0.912199 + 0.409747i \(0.134383\pi\)
−0.912199 + 0.409747i \(0.865617\pi\)
\(548\) 15498.0i 1.20811i
\(549\) −6678.00 −0.519144
\(550\) 0 0
\(551\) −4600.00 −0.355656
\(552\) − 7560.00i − 0.582926i
\(553\) 5760.00i 0.442930i
\(554\) 5394.00 0.413663
\(555\) 0 0
\(556\) −140.000 −0.0106786
\(557\) − 3606.00i − 0.274311i −0.990550 0.137155i \(-0.956204\pi\)
0.990550 0.137155i \(-0.0437960\pi\)
\(558\) − 2592.00i − 0.196645i
\(559\) 4136.00 0.312941
\(560\) 0 0
\(561\) −2184.00 −0.164365
\(562\) − 2442.00i − 0.183291i
\(563\) 12252.0i 0.917159i 0.888654 + 0.458579i \(0.151641\pi\)
−0.888654 + 0.458579i \(0.848359\pi\)
\(564\) 5376.00 0.401366
\(565\) 0 0
\(566\) 2772.00 0.205858
\(567\) 1944.00i 0.143986i
\(568\) 4920.00i 0.363448i
\(569\) 14550.0 1.07200 0.536000 0.844218i \(-0.319935\pi\)
0.536000 + 0.844218i \(0.319935\pi\)
\(570\) 0 0
\(571\) −25468.0 −1.86655 −0.933277 0.359157i \(-0.883064\pi\)
−0.933277 + 0.359157i \(0.883064\pi\)
\(572\) 8008.00i 0.585369i
\(573\) − 5664.00i − 0.412944i
\(574\) 2928.00 0.212914
\(575\) 0 0
\(576\) −1503.00 −0.108724
\(577\) − 12866.0i − 0.928282i −0.885761 0.464141i \(-0.846363\pi\)
0.885761 0.464141i \(-0.153637\pi\)
\(578\) − 4717.00i − 0.339449i
\(579\) −5766.00 −0.413863
\(580\) 0 0
\(581\) −29088.0 −2.07706
\(582\) − 2598.00i − 0.185035i
\(583\) − 17576.0i − 1.24858i
\(584\) −570.000 −0.0403883
\(585\) 0 0
\(586\) 4542.00 0.320185
\(587\) 14844.0i 1.04374i 0.853024 + 0.521872i \(0.174766\pi\)
−0.853024 + 0.521872i \(0.825234\pi\)
\(588\) − 4893.00i − 0.343170i
\(589\) −5760.00 −0.402948
\(590\) 0 0
\(591\) 7578.00 0.527440
\(592\) 1394.00i 0.0967788i
\(593\) 20402.0i 1.41283i 0.707797 + 0.706416i \(0.249689\pi\)
−0.707797 + 0.706416i \(0.750311\pi\)
\(594\) −1404.00 −0.0969812
\(595\) 0 0
\(596\) 9310.00 0.639853
\(597\) 3480.00i 0.238571i
\(598\) − 3696.00i − 0.252744i
\(599\) −10760.0 −0.733959 −0.366980 0.930229i \(-0.619608\pi\)
−0.366980 + 0.930229i \(0.619608\pi\)
\(600\) 0 0
\(601\) 14282.0 0.969343 0.484671 0.874696i \(-0.338939\pi\)
0.484671 + 0.874696i \(0.338939\pi\)
\(602\) − 4512.00i − 0.305474i
\(603\) − 756.000i − 0.0510559i
\(604\) −8456.00 −0.569652
\(605\) 0 0
\(606\) −3654.00 −0.244940
\(607\) − 11056.0i − 0.739290i −0.929173 0.369645i \(-0.879479\pi\)
0.929173 0.369645i \(-0.120521\pi\)
\(608\) − 3220.00i − 0.214783i
\(609\) 16560.0 1.10188
\(610\) 0 0
\(611\) 5632.00 0.372907
\(612\) − 882.000i − 0.0582561i
\(613\) − 16418.0i − 1.08176i −0.841101 0.540878i \(-0.818092\pi\)
0.841101 0.540878i \(-0.181908\pi\)
\(614\) −5116.00 −0.336262
\(615\) 0 0
\(616\) 18720.0 1.22443
\(617\) 10374.0i 0.676891i 0.940986 + 0.338445i \(0.109901\pi\)
−0.940986 + 0.338445i \(0.890099\pi\)
\(618\) − 264.000i − 0.0171839i
\(619\) 5260.00 0.341546 0.170773 0.985310i \(-0.445373\pi\)
0.170773 + 0.985310i \(0.445373\pi\)
\(620\) 0 0
\(621\) −4536.00 −0.293113
\(622\) 2808.00i 0.181014i
\(623\) − 7920.00i − 0.509323i
\(624\) −2706.00 −0.173600
\(625\) 0 0
\(626\) −7318.00 −0.467230
\(627\) 3120.00i 0.198725i
\(628\) 24598.0i 1.56300i
\(629\) −476.000 −0.0301739
\(630\) 0 0
\(631\) 21352.0 1.34708 0.673542 0.739149i \(-0.264772\pi\)
0.673542 + 0.739149i \(0.264772\pi\)
\(632\) − 3600.00i − 0.226583i
\(633\) − 13404.0i − 0.841645i
\(634\) −2246.00 −0.140694
\(635\) 0 0
\(636\) 7098.00 0.442538
\(637\) − 5126.00i − 0.318838i
\(638\) 11960.0i 0.742164i
\(639\) 2952.00 0.182753
\(640\) 0 0
\(641\) −29118.0 −1.79422 −0.897108 0.441812i \(-0.854336\pi\)
−0.897108 + 0.441812i \(0.854336\pi\)
\(642\) − 108.000i − 0.00663928i
\(643\) 5772.00i 0.354005i 0.984210 + 0.177003i \(0.0566401\pi\)
−0.984210 + 0.177003i \(0.943360\pi\)
\(644\) 28224.0 1.72699
\(645\) 0 0
\(646\) 280.000 0.0170533
\(647\) 14264.0i 0.866732i 0.901218 + 0.433366i \(0.142674\pi\)
−0.901218 + 0.433366i \(0.857326\pi\)
\(648\) − 1215.00i − 0.0736570i
\(649\) −5200.00 −0.314511
\(650\) 0 0
\(651\) 20736.0 1.24840
\(652\) − 14476.0i − 0.869515i
\(653\) 6902.00i 0.413623i 0.978381 + 0.206812i \(0.0663088\pi\)
−0.978381 + 0.206812i \(0.933691\pi\)
\(654\) 2910.00 0.173991
\(655\) 0 0
\(656\) 5002.00 0.297706
\(657\) 342.000i 0.0203085i
\(658\) − 6144.00i − 0.364009i
\(659\) −20140.0 −1.19051 −0.595253 0.803539i \(-0.702948\pi\)
−0.595253 + 0.803539i \(0.702948\pi\)
\(660\) 0 0
\(661\) −3218.00 −0.189358 −0.0946790 0.995508i \(-0.530182\pi\)
−0.0946790 + 0.995508i \(0.530182\pi\)
\(662\) − 1332.00i − 0.0782019i
\(663\) − 924.000i − 0.0541255i
\(664\) 18180.0 1.06253
\(665\) 0 0
\(666\) −306.000 −0.0178037
\(667\) 38640.0i 2.24310i
\(668\) 168.000i 0.00973071i
\(669\) −18096.0 −1.04579
\(670\) 0 0
\(671\) 38584.0 2.21985
\(672\) 11592.0i 0.665433i
\(673\) − 7518.00i − 0.430606i −0.976547 0.215303i \(-0.930926\pi\)
0.976547 0.215303i \(-0.0690739\pi\)
\(674\) 11534.0 0.659159
\(675\) 0 0
\(676\) 11991.0 0.682237
\(677\) 18114.0i 1.02833i 0.857692 + 0.514164i \(0.171898\pi\)
−0.857692 + 0.514164i \(0.828102\pi\)
\(678\) 3126.00i 0.177070i
\(679\) 20784.0 1.17469
\(680\) 0 0
\(681\) 7908.00 0.444986
\(682\) 14976.0i 0.840851i
\(683\) − 23868.0i − 1.33716i −0.743638 0.668582i \(-0.766901\pi\)
0.743638 0.668582i \(-0.233099\pi\)
\(684\) −1260.00 −0.0704347
\(685\) 0 0
\(686\) 2640.00 0.146932
\(687\) − 14490.0i − 0.804699i
\(688\) − 7708.00i − 0.427129i
\(689\) 7436.00 0.411160
\(690\) 0 0
\(691\) 172.000 0.00946916 0.00473458 0.999989i \(-0.498493\pi\)
0.00473458 + 0.999989i \(0.498493\pi\)
\(692\) − 4326.00i − 0.237644i
\(693\) − 11232.0i − 0.615683i
\(694\) −11956.0 −0.653953
\(695\) 0 0
\(696\) −10350.0 −0.563672
\(697\) 1708.00i 0.0928194i
\(698\) 4870.00i 0.264086i
\(699\) −8046.00 −0.435376
\(700\) 0 0
\(701\) −22138.0 −1.19278 −0.596391 0.802694i \(-0.703399\pi\)
−0.596391 + 0.802694i \(0.703399\pi\)
\(702\) − 594.000i − 0.0319360i
\(703\) 680.000i 0.0364818i
\(704\) 8684.00 0.464901
\(705\) 0 0
\(706\) 10722.0 0.571569
\(707\) − 29232.0i − 1.55500i
\(708\) − 2100.00i − 0.111473i
\(709\) −3070.00 −0.162618 −0.0813091 0.996689i \(-0.525910\pi\)
−0.0813091 + 0.996689i \(0.525910\pi\)
\(710\) 0 0
\(711\) −2160.00 −0.113933
\(712\) 4950.00i 0.260546i
\(713\) 48384.0i 2.54137i
\(714\) −1008.00 −0.0528340
\(715\) 0 0
\(716\) −23380.0 −1.22032
\(717\) − 6960.00i − 0.362519i
\(718\) 120.000i 0.00623727i
\(719\) −15600.0 −0.809154 −0.404577 0.914504i \(-0.632581\pi\)
−0.404577 + 0.914504i \(0.632581\pi\)
\(720\) 0 0
\(721\) 2112.00 0.109092
\(722\) 6459.00i 0.332935i
\(723\) 6006.00i 0.308943i
\(724\) −1246.00 −0.0639603
\(725\) 0 0
\(726\) 4119.00 0.210565
\(727\) − 20696.0i − 1.05581i −0.849304 0.527904i \(-0.822978\pi\)
0.849304 0.527904i \(-0.177022\pi\)
\(728\) 7920.00i 0.403207i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 2632.00 0.133171
\(732\) 15582.0i 0.786786i
\(733\) − 30778.0i − 1.55090i −0.631408 0.775451i \(-0.717522\pi\)
0.631408 0.775451i \(-0.282478\pi\)
\(734\) −3936.00 −0.197930
\(735\) 0 0
\(736\) −27048.0 −1.35462
\(737\) 4368.00i 0.218314i
\(738\) 1098.00i 0.0547669i
\(739\) −11740.0 −0.584388 −0.292194 0.956359i \(-0.594385\pi\)
−0.292194 + 0.956359i \(0.594385\pi\)
\(740\) 0 0
\(741\) −1320.00 −0.0654405
\(742\) − 8112.00i − 0.401349i
\(743\) 2632.00i 0.129958i 0.997887 + 0.0649789i \(0.0206980\pi\)
−0.997887 + 0.0649789i \(0.979302\pi\)
\(744\) −12960.0 −0.638625
\(745\) 0 0
\(746\) 3022.00 0.148315
\(747\) − 10908.0i − 0.534274i
\(748\) 5096.00i 0.249102i
\(749\) 864.000 0.0421494
\(750\) 0 0
\(751\) −20528.0 −0.997440 −0.498720 0.866763i \(-0.666196\pi\)
−0.498720 + 0.866763i \(0.666196\pi\)
\(752\) − 10496.0i − 0.508976i
\(753\) 396.000i 0.0191647i
\(754\) −5060.00 −0.244396
\(755\) 0 0
\(756\) 4536.00 0.218218
\(757\) − 21646.0i − 1.03928i −0.854384 0.519642i \(-0.826066\pi\)
0.854384 0.519642i \(-0.173934\pi\)
\(758\) − 13340.0i − 0.639222i
\(759\) 26208.0 1.25335
\(760\) 0 0
\(761\) 18282.0 0.870857 0.435428 0.900223i \(-0.356597\pi\)
0.435428 + 0.900223i \(0.356597\pi\)
\(762\) − 5808.00i − 0.276118i
\(763\) 23280.0i 1.10458i
\(764\) −13216.0 −0.625835
\(765\) 0 0
\(766\) −1008.00 −0.0475464
\(767\) − 2200.00i − 0.103569i
\(768\) − 357.000i − 0.0167736i
\(769\) 24190.0 1.13435 0.567174 0.823598i \(-0.308037\pi\)
0.567174 + 0.823598i \(0.308037\pi\)
\(770\) 0 0
\(771\) −22842.0 −1.06697
\(772\) 13454.0i 0.627228i
\(773\) − 25698.0i − 1.19572i −0.801600 0.597861i \(-0.796018\pi\)
0.801600 0.597861i \(-0.203982\pi\)
\(774\) 1692.00 0.0785758
\(775\) 0 0
\(776\) −12990.0 −0.600920
\(777\) − 2448.00i − 0.113026i
\(778\) 9630.00i 0.443769i
\(779\) 2440.00 0.112223
\(780\) 0 0
\(781\) −17056.0 −0.781449
\(782\) − 2352.00i − 0.107554i
\(783\) 6210.00i 0.283432i
\(784\) −9553.00 −0.435177
\(785\) 0 0
\(786\) 2196.00 0.0996549
\(787\) − 33436.0i − 1.51444i −0.653160 0.757220i \(-0.726557\pi\)
0.653160 0.757220i \(-0.273443\pi\)
\(788\) − 17682.0i − 0.799359i
\(789\) 14664.0 0.661663
\(790\) 0 0
\(791\) −25008.0 −1.12412
\(792\) 7020.00i 0.314956i
\(793\) 16324.0i 0.730999i
\(794\) −7126.00 −0.318504
\(795\) 0 0
\(796\) 8120.00 0.361565
\(797\) 37594.0i 1.67083i 0.549623 + 0.835413i \(0.314771\pi\)
−0.549623 + 0.835413i \(0.685229\pi\)
\(798\) 1440.00i 0.0638790i
\(799\) 3584.00 0.158689
\(800\) 0 0
\(801\) 2970.00 0.131011
\(802\) 8718.00i 0.383844i
\(803\) − 1976.00i − 0.0868388i
\(804\) −1764.00 −0.0773775
\(805\) 0 0
\(806\) −6336.00 −0.276893
\(807\) − 3810.00i − 0.166194i
\(808\) 18270.0i 0.795466i
\(809\) −4730.00 −0.205560 −0.102780 0.994704i \(-0.532774\pi\)
−0.102780 + 0.994704i \(0.532774\pi\)
\(810\) 0 0
\(811\) −8748.00 −0.378772 −0.189386 0.981903i \(-0.560650\pi\)
−0.189386 + 0.981903i \(0.560650\pi\)
\(812\) − 38640.0i − 1.66995i
\(813\) 3216.00i 0.138733i
\(814\) 1768.00 0.0761282
\(815\) 0 0
\(816\) −1722.00 −0.0738751
\(817\) − 3760.00i − 0.161011i
\(818\) − 10870.0i − 0.464622i
\(819\) 4752.00 0.202745
\(820\) 0 0
\(821\) 44142.0 1.87645 0.938226 0.346024i \(-0.112468\pi\)
0.938226 + 0.346024i \(0.112468\pi\)
\(822\) 6642.00i 0.281833i
\(823\) 3992.00i 0.169079i 0.996420 + 0.0845397i \(0.0269420\pi\)
−0.996420 + 0.0845397i \(0.973058\pi\)
\(824\) −1320.00 −0.0558063
\(825\) 0 0
\(826\) −2400.00 −0.101098
\(827\) 14444.0i 0.607336i 0.952778 + 0.303668i \(0.0982114\pi\)
−0.952778 + 0.303668i \(0.901789\pi\)
\(828\) 10584.0i 0.444226i
\(829\) −42150.0 −1.76590 −0.882949 0.469468i \(-0.844446\pi\)
−0.882949 + 0.469468i \(0.844446\pi\)
\(830\) 0 0
\(831\) −16182.0 −0.675508
\(832\) 3674.00i 0.153093i
\(833\) − 3262.00i − 0.135680i
\(834\) −60.0000 −0.00249116
\(835\) 0 0
\(836\) 7280.00 0.301177
\(837\) 7776.00i 0.321121i
\(838\) − 9700.00i − 0.399858i
\(839\) −13400.0 −0.551394 −0.275697 0.961245i \(-0.588909\pi\)
−0.275697 + 0.961245i \(0.588909\pi\)
\(840\) 0 0
\(841\) 28511.0 1.16901
\(842\) − 862.000i − 0.0352809i
\(843\) 7326.00i 0.299313i
\(844\) −31276.0 −1.27555
\(845\) 0 0
\(846\) 2304.00 0.0936326
\(847\) 32952.0i 1.33677i
\(848\) − 13858.0i − 0.561186i
\(849\) −8316.00 −0.336165
\(850\) 0 0
\(851\) 5712.00 0.230088
\(852\) − 6888.00i − 0.276971i
\(853\) − 8658.00i − 0.347531i −0.984787 0.173766i \(-0.944406\pi\)
0.984787 0.173766i \(-0.0555935\pi\)
\(854\) 17808.0 0.713556
\(855\) 0 0
\(856\) −540.000 −0.0215617
\(857\) − 42826.0i − 1.70701i −0.521084 0.853505i \(-0.674472\pi\)
0.521084 0.853505i \(-0.325528\pi\)
\(858\) 3432.00i 0.136558i
\(859\) 35900.0 1.42595 0.712976 0.701189i \(-0.247347\pi\)
0.712976 + 0.701189i \(0.247347\pi\)
\(860\) 0 0
\(861\) −8784.00 −0.347686
\(862\) − 15792.0i − 0.623988i
\(863\) − 3088.00i − 0.121804i −0.998144 0.0609019i \(-0.980602\pi\)
0.998144 0.0609019i \(-0.0193977\pi\)
\(864\) −4347.00 −0.171167
\(865\) 0 0
\(866\) 11602.0 0.455256
\(867\) 14151.0i 0.554317i
\(868\) − 48384.0i − 1.89200i
\(869\) 12480.0 0.487175
\(870\) 0 0
\(871\) −1848.00 −0.0718910
\(872\) − 14550.0i − 0.565052i
\(873\) 7794.00i 0.302161i
\(874\) −3360.00 −0.130039
\(875\) 0 0
\(876\) 798.000 0.0307784
\(877\) 35274.0i 1.35817i 0.734058 + 0.679087i \(0.237624\pi\)
−0.734058 + 0.679087i \(0.762376\pi\)
\(878\) − 440.000i − 0.0169126i
\(879\) −13626.0 −0.522860
\(880\) 0 0
\(881\) 25042.0 0.957646 0.478823 0.877911i \(-0.341064\pi\)
0.478823 + 0.877911i \(0.341064\pi\)
\(882\) − 2097.00i − 0.0800563i
\(883\) 12572.0i 0.479141i 0.970879 + 0.239570i \(0.0770066\pi\)
−0.970879 + 0.239570i \(0.922993\pi\)
\(884\) −2156.00 −0.0820296
\(885\) 0 0
\(886\) −10188.0 −0.386312
\(887\) 21864.0i 0.827645i 0.910358 + 0.413823i \(0.135807\pi\)
−0.910358 + 0.413823i \(0.864193\pi\)
\(888\) 1530.00i 0.0578192i
\(889\) 46464.0 1.75293
\(890\) 0 0
\(891\) 4212.00 0.158370
\(892\) 42224.0i 1.58494i
\(893\) − 5120.00i − 0.191864i
\(894\) 3990.00 0.149268
\(895\) 0 0
\(896\) 34920.0 1.30200
\(897\) 11088.0i 0.412729i
\(898\) − 13310.0i − 0.494611i
\(899\) 66240.0 2.45743
\(900\) 0 0
\(901\) 4732.00 0.174968
\(902\) − 6344.00i − 0.234182i
\(903\) 13536.0i 0.498837i
\(904\) 15630.0 0.575051
\(905\) 0 0
\(906\) −3624.00 −0.132891
\(907\) − 31236.0i − 1.14352i −0.820420 0.571761i \(-0.806260\pi\)
0.820420 0.571761i \(-0.193740\pi\)
\(908\) − 18452.0i − 0.674396i
\(909\) 10962.0 0.399985
\(910\) 0 0
\(911\) 8272.00 0.300838 0.150419 0.988622i \(-0.451938\pi\)
0.150419 + 0.988622i \(0.451938\pi\)
\(912\) 2460.00i 0.0893188i
\(913\) 63024.0i 2.28455i
\(914\) −3226.00 −0.116747
\(915\) 0 0
\(916\) −33810.0 −1.21956
\(917\) 17568.0i 0.632657i
\(918\) − 378.000i − 0.0135903i
\(919\) −20200.0 −0.725067 −0.362533 0.931971i \(-0.618088\pi\)
−0.362533 + 0.931971i \(0.618088\pi\)
\(920\) 0 0
\(921\) 15348.0 0.549114
\(922\) − 6582.00i − 0.235105i
\(923\) − 7216.00i − 0.257332i
\(924\) −26208.0 −0.933095
\(925\) 0 0
\(926\) 15072.0 0.534878
\(927\) 792.000i 0.0280612i
\(928\) 37030.0i 1.30988i
\(929\) −31010.0 −1.09516 −0.547581 0.836753i \(-0.684451\pi\)
−0.547581 + 0.836753i \(0.684451\pi\)
\(930\) 0 0
\(931\) −4660.00 −0.164044
\(932\) 18774.0i 0.659831i
\(933\) − 8424.00i − 0.295594i
\(934\) −476.000 −0.0166758
\(935\) 0 0
\(936\) −2970.00 −0.103715
\(937\) 39174.0i 1.36580i 0.730510 + 0.682902i \(0.239283\pi\)
−0.730510 + 0.682902i \(0.760717\pi\)
\(938\) 2016.00i 0.0701756i
\(939\) 21954.0 0.762984
\(940\) 0 0
\(941\) −4138.00 −0.143353 −0.0716764 0.997428i \(-0.522835\pi\)
−0.0716764 + 0.997428i \(0.522835\pi\)
\(942\) 10542.0i 0.364625i
\(943\) − 20496.0i − 0.707785i
\(944\) −4100.00 −0.141360
\(945\) 0 0
\(946\) −9776.00 −0.335989
\(947\) − 23676.0i − 0.812425i −0.913779 0.406213i \(-0.866849\pi\)
0.913779 0.406213i \(-0.133151\pi\)
\(948\) 5040.00i 0.172670i
\(949\) 836.000 0.0285961
\(950\) 0 0
\(951\) 6738.00 0.229752
\(952\) 5040.00i 0.171583i
\(953\) 18922.0i 0.643173i 0.946880 + 0.321586i \(0.104216\pi\)
−0.946880 + 0.321586i \(0.895784\pi\)
\(954\) 3042.00 0.103237
\(955\) 0 0
\(956\) −16240.0 −0.549413
\(957\) − 35880.0i − 1.21195i
\(958\) − 19680.0i − 0.663708i
\(959\) −53136.0 −1.78921
\(960\) 0 0
\(961\) 53153.0 1.78420
\(962\) 748.000i 0.0250691i
\(963\) 324.000i 0.0108419i
\(964\) 14014.0 0.468216
\(965\) 0 0
\(966\) 12096.0 0.402880
\(967\) − 39656.0i − 1.31877i −0.751805 0.659385i \(-0.770817\pi\)
0.751805 0.659385i \(-0.229183\pi\)
\(968\) − 20595.0i − 0.683831i
\(969\) −840.000 −0.0278480
\(970\) 0 0
\(971\) −33228.0 −1.09818 −0.549092 0.835762i \(-0.685026\pi\)
−0.549092 + 0.835762i \(0.685026\pi\)
\(972\) 1701.00i 0.0561313i
\(973\) − 480.000i − 0.0158151i
\(974\) 5944.00 0.195542
\(975\) 0 0
\(976\) 30422.0 0.997730
\(977\) 974.000i 0.0318946i 0.999873 + 0.0159473i \(0.00507640\pi\)
−0.999873 + 0.0159473i \(0.994924\pi\)
\(978\) − 6204.00i − 0.202845i
\(979\) −17160.0 −0.560200
\(980\) 0 0
\(981\) −8730.00 −0.284126
\(982\) − 10772.0i − 0.350049i
\(983\) − 13608.0i − 0.441534i −0.975327 0.220767i \(-0.929144\pi\)
0.975327 0.220767i \(-0.0708560\pi\)
\(984\) 5490.00 0.177861
\(985\) 0 0
\(986\) −3220.00 −0.104002
\(987\) 18432.0i 0.594425i
\(988\) 3080.00i 0.0991780i
\(989\) −31584.0 −1.01548
\(990\) 0 0
\(991\) 13472.0 0.431839 0.215919 0.976411i \(-0.430725\pi\)
0.215919 + 0.976411i \(0.430725\pi\)
\(992\) 46368.0i 1.48406i
\(993\) 3996.00i 0.127703i
\(994\) −7872.00 −0.251192
\(995\) 0 0
\(996\) −25452.0 −0.809716
\(997\) 3234.00i 0.102730i 0.998680 + 0.0513650i \(0.0163572\pi\)
−0.998680 + 0.0513650i \(0.983643\pi\)
\(998\) 8140.00i 0.258184i
\(999\) 918.000 0.0290733
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 75.4.b.b.49.1 2
3.2 odd 2 225.4.b.e.199.2 2
4.3 odd 2 1200.4.f.b.49.1 2
5.2 odd 4 15.4.a.a.1.1 1
5.3 odd 4 75.4.a.b.1.1 1
5.4 even 2 inner 75.4.b.b.49.2 2
15.2 even 4 45.4.a.c.1.1 1
15.8 even 4 225.4.a.f.1.1 1
15.14 odd 2 225.4.b.e.199.1 2
20.3 even 4 1200.4.a.t.1.1 1
20.7 even 4 240.4.a.e.1.1 1
20.19 odd 2 1200.4.f.b.49.2 2
35.27 even 4 735.4.a.e.1.1 1
40.27 even 4 960.4.a.ba.1.1 1
40.37 odd 4 960.4.a.b.1.1 1
45.2 even 12 405.4.e.i.271.1 2
45.7 odd 12 405.4.e.g.271.1 2
45.22 odd 12 405.4.e.g.136.1 2
45.32 even 12 405.4.e.i.136.1 2
55.32 even 4 1815.4.a.e.1.1 1
60.47 odd 4 720.4.a.n.1.1 1
105.62 odd 4 2205.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 5.2 odd 4
45.4.a.c.1.1 1 15.2 even 4
75.4.a.b.1.1 1 5.3 odd 4
75.4.b.b.49.1 2 1.1 even 1 trivial
75.4.b.b.49.2 2 5.4 even 2 inner
225.4.a.f.1.1 1 15.8 even 4
225.4.b.e.199.1 2 15.14 odd 2
225.4.b.e.199.2 2 3.2 odd 2
240.4.a.e.1.1 1 20.7 even 4
405.4.e.g.136.1 2 45.22 odd 12
405.4.e.g.271.1 2 45.7 odd 12
405.4.e.i.136.1 2 45.32 even 12
405.4.e.i.271.1 2 45.2 even 12
720.4.a.n.1.1 1 60.47 odd 4
735.4.a.e.1.1 1 35.27 even 4
960.4.a.b.1.1 1 40.37 odd 4
960.4.a.ba.1.1 1 40.27 even 4
1200.4.a.t.1.1 1 20.3 even 4
1200.4.f.b.49.1 2 4.3 odd 2
1200.4.f.b.49.2 2 20.19 odd 2
1815.4.a.e.1.1 1 55.32 even 4
2205.4.a.l.1.1 1 105.62 odd 4