Properties

Label 1050.3.f.e.601.14
Level $1050$
Weight $3$
Character 1050.601
Analytic conductor $28.610$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1050,3,Mod(601,1050)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1050, 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("1050.601");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.f (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: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2744 x^{12} - 10640 x^{11} + 39126 x^{10} - 108488 x^{9} + 259514 x^{8} - 486436 x^{7} + 690168 x^{6} - 725188 x^{5} + \cdots + 33124 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 210)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 601.14
Root \(1.20711 + 3.39361i\) of defining polynomial
Character \(\chi\) \(=\) 1050.601
Dual form 1050.3.f.e.601.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.44949i q^{6} +(-1.57881 + 6.81963i) q^{7} +2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} +2.44949i q^{6} +(-1.57881 + 6.81963i) q^{7} +2.82843 q^{8} -3.00000 q^{9} -8.15965 q^{11} +3.46410i q^{12} -14.6064i q^{13} +(-2.23278 + 9.64441i) q^{14} +4.00000 q^{16} +5.81421i q^{17} -4.24264 q^{18} +33.7736i q^{19} +(-11.8119 - 2.73459i) q^{21} -11.5395 q^{22} -37.2576 q^{23} +4.89898i q^{24} -20.6566i q^{26} -5.19615i q^{27} +(-3.15763 + 13.6393i) q^{28} +9.25305 q^{29} +19.2558i q^{31} +5.65685 q^{32} -14.1329i q^{33} +8.22253i q^{34} -6.00000 q^{36} -63.4350 q^{37} +47.7631i q^{38} +25.2990 q^{39} -8.25880i q^{41} +(-16.7046 - 3.86729i) q^{42} +42.0893 q^{43} -16.3193 q^{44} -52.6901 q^{46} +23.3380i q^{47} +6.92820i q^{48} +(-44.0147 - 21.5339i) q^{49} -10.0705 q^{51} -29.2128i q^{52} -71.3497 q^{53} -7.34847i q^{54} +(-4.46556 + 19.2888i) q^{56} -58.4976 q^{57} +13.0858 q^{58} -42.9350i q^{59} +34.2864i q^{61} +27.2318i q^{62} +(4.73644 - 20.4589i) q^{63} +8.00000 q^{64} -19.9870i q^{66} -4.99889 q^{67} +11.6284i q^{68} -64.5320i q^{69} -38.8120 q^{71} -8.48528 q^{72} -124.629i q^{73} -89.7107 q^{74} +67.5472i q^{76} +(12.8826 - 55.6458i) q^{77} +35.7782 q^{78} +56.1842 q^{79} +9.00000 q^{81} -11.6797i q^{82} +90.3980i q^{83} +(-23.6239 - 5.46917i) q^{84} +59.5233 q^{86} +16.0267i q^{87} -23.0790 q^{88} +16.2289i q^{89} +(99.6102 + 23.0608i) q^{91} -74.5151 q^{92} -33.3520 q^{93} +33.0048i q^{94} +9.79796i q^{96} +82.7605i q^{97} +(-62.2462 - 30.4535i) q^{98} +24.4789 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} - 48 q^{9} + 96 q^{11} - 16 q^{14} + 64 q^{16} + 24 q^{21} - 64 q^{29} - 96 q^{36} + 144 q^{39} + 192 q^{44} - 176 q^{46} - 224 q^{49} - 48 q^{51} - 32 q^{56} + 128 q^{64} - 384 q^{71} - 224 q^{74} + 608 q^{79} + 144 q^{81} + 48 q^{84} + 416 q^{86} + 224 q^{91} - 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 0.707107
\(3\) 1.73205i 0.577350i
\(4\) 2.00000 0.500000
\(5\) 0 0
\(6\) 2.44949i 0.408248i
\(7\) −1.57881 + 6.81963i −0.225545 + 0.974233i
\(8\) 2.82843 0.353553
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) −8.15965 −0.741786 −0.370893 0.928676i \(-0.620948\pi\)
−0.370893 + 0.928676i \(0.620948\pi\)
\(12\) 3.46410i 0.288675i
\(13\) 14.6064i 1.12357i −0.827284 0.561785i \(-0.810115\pi\)
0.827284 0.561785i \(-0.189885\pi\)
\(14\) −2.23278 + 9.64441i −0.159484 + 0.688887i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 5.81421i 0.342012i 0.985270 + 0.171006i \(0.0547018\pi\)
−0.985270 + 0.171006i \(0.945298\pi\)
\(18\) −4.24264 −0.235702
\(19\) 33.7736i 1.77756i 0.458337 + 0.888778i \(0.348445\pi\)
−0.458337 + 0.888778i \(0.651555\pi\)
\(20\) 0 0
\(21\) −11.8119 2.73459i −0.562474 0.130218i
\(22\) −11.5395 −0.524522
\(23\) −37.2576 −1.61989 −0.809947 0.586503i \(-0.800504\pi\)
−0.809947 + 0.586503i \(0.800504\pi\)
\(24\) 4.89898i 0.204124i
\(25\) 0 0
\(26\) 20.6566i 0.794483i
\(27\) 5.19615i 0.192450i
\(28\) −3.15763 + 13.6393i −0.112772 + 0.487116i
\(29\) 9.25305 0.319071 0.159535 0.987192i \(-0.449000\pi\)
0.159535 + 0.987192i \(0.449000\pi\)
\(30\) 0 0
\(31\) 19.2558i 0.621154i 0.950548 + 0.310577i \(0.100522\pi\)
−0.950548 + 0.310577i \(0.899478\pi\)
\(32\) 5.65685 0.176777
\(33\) 14.1329i 0.428270i
\(34\) 8.22253i 0.241839i
\(35\) 0 0
\(36\) −6.00000 −0.166667
\(37\) −63.4350 −1.71446 −0.857230 0.514933i \(-0.827817\pi\)
−0.857230 + 0.514933i \(0.827817\pi\)
\(38\) 47.7631i 1.25692i
\(39\) 25.2990 0.648693
\(40\) 0 0
\(41\) 8.25880i 0.201434i −0.994915 0.100717i \(-0.967886\pi\)
0.994915 0.100717i \(-0.0321137\pi\)
\(42\) −16.7046 3.86729i −0.397729 0.0920783i
\(43\) 42.0893 0.978822 0.489411 0.872053i \(-0.337212\pi\)
0.489411 + 0.872053i \(0.337212\pi\)
\(44\) −16.3193 −0.370893
\(45\) 0 0
\(46\) −52.6901 −1.14544
\(47\) 23.3380i 0.496552i 0.968689 + 0.248276i \(0.0798640\pi\)
−0.968689 + 0.248276i \(0.920136\pi\)
\(48\) 6.92820i 0.144338i
\(49\) −44.0147 21.5339i −0.898259 0.439466i
\(50\) 0 0
\(51\) −10.0705 −0.197461
\(52\) 29.2128i 0.561785i
\(53\) −71.3497 −1.34622 −0.673110 0.739542i \(-0.735042\pi\)
−0.673110 + 0.739542i \(0.735042\pi\)
\(54\) 7.34847i 0.136083i
\(55\) 0 0
\(56\) −4.46556 + 19.2888i −0.0797422 + 0.344443i
\(57\) −58.4976 −1.02627
\(58\) 13.0858 0.225617
\(59\) 42.9350i 0.727712i −0.931455 0.363856i \(-0.881460\pi\)
0.931455 0.363856i \(-0.118540\pi\)
\(60\) 0 0
\(61\) 34.2864i 0.562072i 0.959697 + 0.281036i \(0.0906781\pi\)
−0.959697 + 0.281036i \(0.909322\pi\)
\(62\) 27.2318i 0.439222i
\(63\) 4.73644 20.4589i 0.0751816 0.324744i
\(64\) 8.00000 0.125000
\(65\) 0 0
\(66\) 19.9870i 0.302833i
\(67\) −4.99889 −0.0746102 −0.0373051 0.999304i \(-0.511877\pi\)
−0.0373051 + 0.999304i \(0.511877\pi\)
\(68\) 11.6284i 0.171006i
\(69\) 64.5320i 0.935246i
\(70\) 0 0
\(71\) −38.8120 −0.546648 −0.273324 0.961922i \(-0.588123\pi\)
−0.273324 + 0.961922i \(0.588123\pi\)
\(72\) −8.48528 −0.117851
\(73\) 124.629i 1.70725i −0.520886 0.853626i \(-0.674398\pi\)
0.520886 0.853626i \(-0.325602\pi\)
\(74\) −89.7107 −1.21231
\(75\) 0 0
\(76\) 67.5472i 0.888778i
\(77\) 12.8826 55.6458i 0.167306 0.722672i
\(78\) 35.7782 0.458695
\(79\) 56.1842 0.711192 0.355596 0.934640i \(-0.384278\pi\)
0.355596 + 0.934640i \(0.384278\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 11.6797i 0.142435i
\(83\) 90.3980i 1.08913i 0.838718 + 0.544566i \(0.183306\pi\)
−0.838718 + 0.544566i \(0.816694\pi\)
\(84\) −23.6239 5.46917i −0.281237 0.0651092i
\(85\) 0 0
\(86\) 59.5233 0.692131
\(87\) 16.0267i 0.184215i
\(88\) −23.0790 −0.262261
\(89\) 16.2289i 0.182347i 0.995835 + 0.0911736i \(0.0290618\pi\)
−0.995835 + 0.0911736i \(0.970938\pi\)
\(90\) 0 0
\(91\) 99.6102 + 23.0608i 1.09462 + 0.253415i
\(92\) −74.5151 −0.809947
\(93\) −33.3520 −0.358624
\(94\) 33.0048i 0.351115i
\(95\) 0 0
\(96\) 9.79796i 0.102062i
\(97\) 82.7605i 0.853201i 0.904440 + 0.426601i \(0.140289\pi\)
−0.904440 + 0.426601i \(0.859711\pi\)
\(98\) −62.2462 30.4535i −0.635165 0.310750i
\(99\) 24.4789 0.247262
\(100\) 0 0
\(101\) 87.2055i 0.863420i 0.902012 + 0.431710i \(0.142090\pi\)
−0.902012 + 0.431710i \(0.857910\pi\)
\(102\) −14.2418 −0.139626
\(103\) 187.567i 1.82104i 0.413462 + 0.910521i \(0.364319\pi\)
−0.413462 + 0.910521i \(0.635681\pi\)
\(104\) 41.3131i 0.397242i
\(105\) 0 0
\(106\) −100.904 −0.951922
\(107\) −157.858 −1.47531 −0.737653 0.675180i \(-0.764066\pi\)
−0.737653 + 0.675180i \(0.764066\pi\)
\(108\) 10.3923i 0.0962250i
\(109\) 112.073 1.02819 0.514097 0.857732i \(-0.328127\pi\)
0.514097 + 0.857732i \(0.328127\pi\)
\(110\) 0 0
\(111\) 109.873i 0.989844i
\(112\) −6.31526 + 27.2785i −0.0563862 + 0.243558i
\(113\) 141.987 1.25653 0.628263 0.778001i \(-0.283766\pi\)
0.628263 + 0.778001i \(0.283766\pi\)
\(114\) −82.7280 −0.725684
\(115\) 0 0
\(116\) 18.5061 0.159535
\(117\) 43.8192i 0.374523i
\(118\) 60.7192i 0.514570i
\(119\) −39.6507 9.17955i −0.333199 0.0771391i
\(120\) 0 0
\(121\) −54.4202 −0.449754
\(122\) 48.4883i 0.397445i
\(123\) 14.3047 0.116298
\(124\) 38.5116i 0.310577i
\(125\) 0 0
\(126\) 6.69834 28.9332i 0.0531614 0.229629i
\(127\) −202.414 −1.59381 −0.796905 0.604104i \(-0.793531\pi\)
−0.796905 + 0.604104i \(0.793531\pi\)
\(128\) 11.3137 0.0883883
\(129\) 72.9008i 0.565123i
\(130\) 0 0
\(131\) 141.260i 1.07832i −0.842203 0.539160i \(-0.818742\pi\)
0.842203 0.539160i \(-0.181258\pi\)
\(132\) 28.2658i 0.214135i
\(133\) −230.323 53.3222i −1.73175 0.400919i
\(134\) −7.06949 −0.0527574
\(135\) 0 0
\(136\) 16.4451i 0.120920i
\(137\) 50.6430 0.369657 0.184829 0.982771i \(-0.440827\pi\)
0.184829 + 0.982771i \(0.440827\pi\)
\(138\) 91.2620i 0.661319i
\(139\) 74.0256i 0.532559i 0.963896 + 0.266279i \(0.0857943\pi\)
−0.963896 + 0.266279i \(0.914206\pi\)
\(140\) 0 0
\(141\) −40.4225 −0.286685
\(142\) −54.8885 −0.386538
\(143\) 119.183i 0.833448i
\(144\) −12.0000 −0.0833333
\(145\) 0 0
\(146\) 176.253i 1.20721i
\(147\) 37.2977 76.2357i 0.253726 0.518610i
\(148\) −126.870 −0.857230
\(149\) 226.979 1.52335 0.761676 0.647958i \(-0.224377\pi\)
0.761676 + 0.647958i \(0.224377\pi\)
\(150\) 0 0
\(151\) 294.426 1.94984 0.974920 0.222558i \(-0.0714406\pi\)
0.974920 + 0.222558i \(0.0714406\pi\)
\(152\) 95.5261i 0.628461i
\(153\) 17.4426i 0.114004i
\(154\) 18.2187 78.6950i 0.118303 0.511006i
\(155\) 0 0
\(156\) 50.5981 0.324346
\(157\) 65.8596i 0.419488i −0.977756 0.209744i \(-0.932737\pi\)
0.977756 0.209744i \(-0.0672630\pi\)
\(158\) 79.4564 0.502889
\(159\) 123.581i 0.777241i
\(160\) 0 0
\(161\) 58.8227 254.083i 0.365359 1.57815i
\(162\) 12.7279 0.0785674
\(163\) 28.2075 0.173052 0.0865260 0.996250i \(-0.472423\pi\)
0.0865260 + 0.996250i \(0.472423\pi\)
\(164\) 16.5176i 0.100717i
\(165\) 0 0
\(166\) 127.842i 0.770133i
\(167\) 128.461i 0.769227i 0.923078 + 0.384614i \(0.125665\pi\)
−0.923078 + 0.384614i \(0.874335\pi\)
\(168\) −33.4092 7.73458i −0.198864 0.0460392i
\(169\) −44.3469 −0.262408
\(170\) 0 0
\(171\) 101.321i 0.592519i
\(172\) 84.1786 0.489411
\(173\) 112.446i 0.649976i −0.945718 0.324988i \(-0.894640\pi\)
0.945718 0.324988i \(-0.105360\pi\)
\(174\) 22.6652i 0.130260i
\(175\) 0 0
\(176\) −32.6386 −0.185446
\(177\) 74.3656 0.420144
\(178\) 22.9511i 0.128939i
\(179\) 68.9019 0.384927 0.192464 0.981304i \(-0.438352\pi\)
0.192464 + 0.981304i \(0.438352\pi\)
\(180\) 0 0
\(181\) 166.537i 0.920094i 0.887895 + 0.460047i \(0.152167\pi\)
−0.887895 + 0.460047i \(0.847833\pi\)
\(182\) 140.870 + 32.6129i 0.774012 + 0.179192i
\(183\) −59.3858 −0.324513
\(184\) −105.380 −0.572719
\(185\) 0 0
\(186\) −47.1668 −0.253585
\(187\) 47.4419i 0.253700i
\(188\) 46.6759i 0.248276i
\(189\) 35.4358 + 8.20376i 0.187491 + 0.0434061i
\(190\) 0 0
\(191\) −148.289 −0.776382 −0.388191 0.921579i \(-0.626900\pi\)
−0.388191 + 0.921579i \(0.626900\pi\)
\(192\) 13.8564i 0.0721688i
\(193\) −35.1886 −0.182324 −0.0911622 0.995836i \(-0.529058\pi\)
−0.0911622 + 0.995836i \(0.529058\pi\)
\(194\) 117.041i 0.603304i
\(195\) 0 0
\(196\) −88.0294 43.0677i −0.449130 0.219733i
\(197\) −193.634 −0.982915 −0.491457 0.870902i \(-0.663536\pi\)
−0.491457 + 0.870902i \(0.663536\pi\)
\(198\) 34.6184 0.174841
\(199\) 181.838i 0.913759i 0.889529 + 0.456880i \(0.151033\pi\)
−0.889529 + 0.456880i \(0.848967\pi\)
\(200\) 0 0
\(201\) 8.65832i 0.0430762i
\(202\) 123.327i 0.610530i
\(203\) −14.6088 + 63.1023i −0.0719647 + 0.310849i
\(204\) −20.1410 −0.0987304
\(205\) 0 0
\(206\) 265.260i 1.28767i
\(207\) 111.773 0.539965
\(208\) 58.4256i 0.280892i
\(209\) 275.580i 1.31857i
\(210\) 0 0
\(211\) 175.914 0.833717 0.416859 0.908971i \(-0.363131\pi\)
0.416859 + 0.908971i \(0.363131\pi\)
\(212\) −142.699 −0.673110
\(213\) 67.2244i 0.315607i
\(214\) −223.244 −1.04320
\(215\) 0 0
\(216\) 14.6969i 0.0680414i
\(217\) −131.317 30.4013i −0.605149 0.140098i
\(218\) 158.495 0.727042
\(219\) 215.864 0.985683
\(220\) 0 0
\(221\) 84.9246 0.384274
\(222\) 155.384i 0.699926i
\(223\) 20.5675i 0.0922308i −0.998936 0.0461154i \(-0.985316\pi\)
0.998936 0.0461154i \(-0.0146842\pi\)
\(224\) −8.93112 + 38.5776i −0.0398711 + 0.172222i
\(225\) 0 0
\(226\) 200.801 0.888498
\(227\) 414.087i 1.82417i −0.410001 0.912085i \(-0.634472\pi\)
0.410001 0.912085i \(-0.365528\pi\)
\(228\) −116.995 −0.513136
\(229\) 195.520i 0.853799i 0.904299 + 0.426899i \(0.140394\pi\)
−0.904299 + 0.426899i \(0.859606\pi\)
\(230\) 0 0
\(231\) 96.3813 + 22.3133i 0.417235 + 0.0965942i
\(232\) 26.1716 0.112808
\(233\) −81.3006 −0.348930 −0.174465 0.984663i \(-0.555820\pi\)
−0.174465 + 0.984663i \(0.555820\pi\)
\(234\) 61.9697i 0.264828i
\(235\) 0 0
\(236\) 85.8700i 0.363856i
\(237\) 97.3138i 0.410607i
\(238\) −56.0746 12.9818i −0.235608 0.0545456i
\(239\) 249.265 1.04295 0.521475 0.853267i \(-0.325382\pi\)
0.521475 + 0.853267i \(0.325382\pi\)
\(240\) 0 0
\(241\) 262.343i 1.08856i 0.838904 + 0.544280i \(0.183197\pi\)
−0.838904 + 0.544280i \(0.816803\pi\)
\(242\) −76.9618 −0.318024
\(243\) 15.5885i 0.0641500i
\(244\) 68.5728i 0.281036i
\(245\) 0 0
\(246\) 20.2298 0.0822351
\(247\) 493.310 1.99721
\(248\) 54.4636i 0.219611i
\(249\) −156.574 −0.628811
\(250\) 0 0
\(251\) 141.917i 0.565406i 0.959208 + 0.282703i \(0.0912310\pi\)
−0.959208 + 0.282703i \(0.908769\pi\)
\(252\) 9.47288 40.9178i 0.0375908 0.162372i
\(253\) 304.008 1.20161
\(254\) −286.257 −1.12699
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 170.843i 0.664757i 0.943146 + 0.332379i \(0.107851\pi\)
−0.943146 + 0.332379i \(0.892149\pi\)
\(258\) 103.097i 0.399602i
\(259\) 100.152 432.604i 0.386688 1.67028i
\(260\) 0 0
\(261\) −27.7591 −0.106357
\(262\) 199.772i 0.762488i
\(263\) 383.417 1.45786 0.728930 0.684588i \(-0.240018\pi\)
0.728930 + 0.684588i \(0.240018\pi\)
\(264\) 39.9739i 0.151416i
\(265\) 0 0
\(266\) −325.726 75.4090i −1.22454 0.283492i
\(267\) −28.1093 −0.105278
\(268\) −9.99777 −0.0373051
\(269\) 154.512i 0.574393i −0.957872 0.287196i \(-0.907277\pi\)
0.957872 0.287196i \(-0.0927232\pi\)
\(270\) 0 0
\(271\) 320.328i 1.18202i 0.806664 + 0.591010i \(0.201271\pi\)
−0.806664 + 0.591010i \(0.798729\pi\)
\(272\) 23.2568i 0.0855030i
\(273\) −39.9425 + 172.530i −0.146309 + 0.631978i
\(274\) 71.6200 0.261387
\(275\) 0 0
\(276\) 129.064i 0.467623i
\(277\) 222.854 0.804526 0.402263 0.915524i \(-0.368224\pi\)
0.402263 + 0.915524i \(0.368224\pi\)
\(278\) 104.688i 0.376576i
\(279\) 57.7673i 0.207051i
\(280\) 0 0
\(281\) 218.973 0.779263 0.389631 0.920971i \(-0.372602\pi\)
0.389631 + 0.920971i \(0.372602\pi\)
\(282\) −57.1661 −0.202717
\(283\) 36.3957i 0.128607i −0.997930 0.0643034i \(-0.979517\pi\)
0.997930 0.0643034i \(-0.0204825\pi\)
\(284\) −77.6240 −0.273324
\(285\) 0 0
\(286\) 168.550i 0.589337i
\(287\) 56.3219 + 13.0391i 0.196244 + 0.0454324i
\(288\) −16.9706 −0.0589256
\(289\) 255.195 0.883028
\(290\) 0 0
\(291\) −143.345 −0.492596
\(292\) 249.259i 0.853626i
\(293\) 168.440i 0.574882i 0.957798 + 0.287441i \(0.0928045\pi\)
−0.957798 + 0.287441i \(0.907195\pi\)
\(294\) 52.7470 107.814i 0.179411 0.366713i
\(295\) 0 0
\(296\) −179.421 −0.606153
\(297\) 42.3988i 0.142757i
\(298\) 320.997 1.07717
\(299\) 544.199i 1.82006i
\(300\) 0 0
\(301\) −66.4512 + 287.034i −0.220768 + 0.953600i
\(302\) 416.381 1.37874
\(303\) −151.044 −0.498496
\(304\) 135.094i 0.444389i
\(305\) 0 0
\(306\) 24.6676i 0.0806130i
\(307\) 223.400i 0.727688i 0.931460 + 0.363844i \(0.118536\pi\)
−0.931460 + 0.363844i \(0.881464\pi\)
\(308\) 25.7651 111.292i 0.0836530 0.361336i
\(309\) −324.876 −1.05138
\(310\) 0 0
\(311\) 46.0396i 0.148037i −0.997257 0.0740186i \(-0.976418\pi\)
0.997257 0.0740186i \(-0.0235824\pi\)
\(312\) 71.5565 0.229348
\(313\) 80.0375i 0.255711i 0.991793 + 0.127855i \(0.0408094\pi\)
−0.991793 + 0.127855i \(0.959191\pi\)
\(314\) 93.1395i 0.296623i
\(315\) 0 0
\(316\) 112.368 0.355596
\(317\) 185.237 0.584343 0.292171 0.956366i \(-0.405622\pi\)
0.292171 + 0.956366i \(0.405622\pi\)
\(318\) 174.770i 0.549592i
\(319\) −75.5016 −0.236682
\(320\) 0 0
\(321\) 273.418i 0.851768i
\(322\) 83.1879 359.327i 0.258348 1.11592i
\(323\) −196.367 −0.607946
\(324\) 18.0000 0.0555556
\(325\) 0 0
\(326\) 39.8914 0.122366
\(327\) 194.116i 0.593628i
\(328\) 23.3594i 0.0712177i
\(329\) −159.156 36.8463i −0.483757 0.111995i
\(330\) 0 0
\(331\) 61.4686 0.185706 0.0928529 0.995680i \(-0.470401\pi\)
0.0928529 + 0.995680i \(0.470401\pi\)
\(332\) 180.796i 0.544566i
\(333\) 190.305 0.571487
\(334\) 181.671i 0.543926i
\(335\) 0 0
\(336\) −47.2478 10.9383i −0.140618 0.0325546i
\(337\) 656.501 1.94807 0.974037 0.226387i \(-0.0726914\pi\)
0.974037 + 0.226387i \(0.0726914\pi\)
\(338\) −62.7160 −0.185550
\(339\) 245.929i 0.725456i
\(340\) 0 0
\(341\) 157.120i 0.460764i
\(342\) 143.289i 0.418974i
\(343\) 216.344 266.166i 0.630740 0.775994i
\(344\) 119.047 0.346066
\(345\) 0 0
\(346\) 159.022i 0.459603i
\(347\) 321.323 0.926002 0.463001 0.886358i \(-0.346773\pi\)
0.463001 + 0.886358i \(0.346773\pi\)
\(348\) 32.0535i 0.0921077i
\(349\) 185.135i 0.530474i −0.964183 0.265237i \(-0.914550\pi\)
0.964183 0.265237i \(-0.0854501\pi\)
\(350\) 0 0
\(351\) −75.8971 −0.216231
\(352\) −46.1579 −0.131130
\(353\) 597.338i 1.69218i −0.533043 0.846088i \(-0.678952\pi\)
0.533043 0.846088i \(-0.321048\pi\)
\(354\) 105.169 0.297087
\(355\) 0 0
\(356\) 32.4578i 0.0911736i
\(357\) 15.8995 68.6771i 0.0445363 0.192373i
\(358\) 97.4421 0.272185
\(359\) −239.446 −0.666982 −0.333491 0.942753i \(-0.608227\pi\)
−0.333491 + 0.942753i \(0.608227\pi\)
\(360\) 0 0
\(361\) −779.655 −2.15971
\(362\) 235.519i 0.650604i
\(363\) 94.2585i 0.259665i
\(364\) 199.220 + 46.1216i 0.547309 + 0.126708i
\(365\) 0 0
\(366\) −83.9842 −0.229465
\(367\) 398.743i 1.08649i 0.839573 + 0.543246i \(0.182805\pi\)
−0.839573 + 0.543246i \(0.817195\pi\)
\(368\) −149.030 −0.404973
\(369\) 24.7764i 0.0671447i
\(370\) 0 0
\(371\) 112.648 486.578i 0.303633 1.31153i
\(372\) −66.7040 −0.179312
\(373\) −34.7064 −0.0930466 −0.0465233 0.998917i \(-0.514814\pi\)
−0.0465233 + 0.998917i \(0.514814\pi\)
\(374\) 67.0929i 0.179393i
\(375\) 0 0
\(376\) 66.0097i 0.175558i
\(377\) 135.154i 0.358498i
\(378\) 50.1138 + 11.6019i 0.132576 + 0.0306928i
\(379\) 109.217 0.288172 0.144086 0.989565i \(-0.453976\pi\)
0.144086 + 0.989565i \(0.453976\pi\)
\(380\) 0 0
\(381\) 350.591i 0.920187i
\(382\) −209.712 −0.548985
\(383\) 336.420i 0.878381i −0.898394 0.439191i \(-0.855265\pi\)
0.898394 0.439191i \(-0.144735\pi\)
\(384\) 19.5959i 0.0510310i
\(385\) 0 0
\(386\) −49.7642 −0.128923
\(387\) −126.268 −0.326274
\(388\) 165.521i 0.426601i
\(389\) 311.173 0.799930 0.399965 0.916530i \(-0.369022\pi\)
0.399965 + 0.916530i \(0.369022\pi\)
\(390\) 0 0
\(391\) 216.623i 0.554023i
\(392\) −124.492 60.9069i −0.317583 0.155375i
\(393\) 244.669 0.622568
\(394\) −273.840 −0.695026
\(395\) 0 0
\(396\) 48.9579 0.123631
\(397\) 41.2679i 0.103949i −0.998648 0.0519747i \(-0.983448\pi\)
0.998648 0.0519747i \(-0.0165515\pi\)
\(398\) 257.158i 0.646125i
\(399\) 92.3568 398.932i 0.231471 0.999829i
\(400\) 0 0
\(401\) −495.642 −1.23601 −0.618007 0.786173i \(-0.712060\pi\)
−0.618007 + 0.786173i \(0.712060\pi\)
\(402\) 12.2447i 0.0304595i
\(403\) 281.258 0.697910
\(404\) 174.411i 0.431710i
\(405\) 0 0
\(406\) −20.6600 + 89.2402i −0.0508867 + 0.219803i
\(407\) 517.608 1.27176
\(408\) −28.4837 −0.0698129
\(409\) 359.920i 0.880001i 0.897998 + 0.440000i \(0.145022\pi\)
−0.897998 + 0.440000i \(0.854978\pi\)
\(410\) 0 0
\(411\) 87.7163i 0.213422i
\(412\) 375.135i 0.910521i
\(413\) 292.801 + 67.7863i 0.708960 + 0.164132i
\(414\) 158.070 0.381813
\(415\) 0 0
\(416\) 82.6263i 0.198621i
\(417\) −128.216 −0.307473
\(418\) 389.730i 0.932367i
\(419\) 482.910i 1.15253i −0.817263 0.576265i \(-0.804510\pi\)
0.817263 0.576265i \(-0.195490\pi\)
\(420\) 0 0
\(421\) −523.520 −1.24351 −0.621757 0.783210i \(-0.713581\pi\)
−0.621757 + 0.783210i \(0.713581\pi\)
\(422\) 248.780 0.589527
\(423\) 70.0139i 0.165517i
\(424\) −201.807 −0.475961
\(425\) 0 0
\(426\) 95.0696i 0.223168i
\(427\) −233.821 54.1319i −0.547589 0.126773i
\(428\) −315.715 −0.737653
\(429\) −206.431 −0.481191
\(430\) 0 0
\(431\) 517.027 1.19960 0.599799 0.800150i \(-0.295247\pi\)
0.599799 + 0.800150i \(0.295247\pi\)
\(432\) 20.7846i 0.0481125i
\(433\) 567.712i 1.31111i 0.755146 + 0.655557i \(0.227566\pi\)
−0.755146 + 0.655557i \(0.772434\pi\)
\(434\) −185.711 42.9939i −0.427905 0.0990644i
\(435\) 0 0
\(436\) 224.146 0.514097
\(437\) 1258.32i 2.87945i
\(438\) 305.279 0.696983
\(439\) 622.875i 1.41885i 0.704781 + 0.709425i \(0.251045\pi\)
−0.704781 + 0.709425i \(0.748955\pi\)
\(440\) 0 0
\(441\) 132.044 + 64.6016i 0.299420 + 0.146489i
\(442\) 120.102 0.271723
\(443\) −476.699 −1.07607 −0.538035 0.842923i \(-0.680833\pi\)
−0.538035 + 0.842923i \(0.680833\pi\)
\(444\) 219.745i 0.494922i
\(445\) 0 0
\(446\) 29.0868i 0.0652170i
\(447\) 393.140i 0.879507i
\(448\) −12.6305 + 54.5570i −0.0281931 + 0.121779i
\(449\) −861.931 −1.91967 −0.959834 0.280570i \(-0.909477\pi\)
−0.959834 + 0.280570i \(0.909477\pi\)
\(450\) 0 0
\(451\) 67.3889i 0.149421i
\(452\) 283.975 0.628263
\(453\) 509.960i 1.12574i
\(454\) 585.607i 1.28988i
\(455\) 0 0
\(456\) −165.456 −0.362842
\(457\) −27.5401 −0.0602629 −0.0301314 0.999546i \(-0.509593\pi\)
−0.0301314 + 0.999546i \(0.509593\pi\)
\(458\) 276.507i 0.603727i
\(459\) 30.2115 0.0658203
\(460\) 0 0
\(461\) 378.183i 0.820353i −0.912006 0.410176i \(-0.865467\pi\)
0.912006 0.410176i \(-0.134533\pi\)
\(462\) 136.304 + 31.5557i 0.295030 + 0.0683024i
\(463\) −724.994 −1.56586 −0.782931 0.622108i \(-0.786276\pi\)
−0.782931 + 0.622108i \(0.786276\pi\)
\(464\) 37.0122 0.0797676
\(465\) 0 0
\(466\) −114.976 −0.246731
\(467\) 371.451i 0.795398i −0.917516 0.397699i \(-0.869809\pi\)
0.917516 0.397699i \(-0.130191\pi\)
\(468\) 87.6384i 0.187262i
\(469\) 7.89231 34.0905i 0.0168280 0.0726877i
\(470\) 0 0
\(471\) 114.072 0.242191
\(472\) 121.438i 0.257285i
\(473\) −343.434 −0.726076
\(474\) 137.623i 0.290343i
\(475\) 0 0
\(476\) −79.3015 18.3591i −0.166600 0.0385695i
\(477\) 214.049 0.448740
\(478\) 352.514 0.737477
\(479\) 860.650i 1.79677i −0.439214 0.898383i \(-0.644743\pi\)
0.439214 0.898383i \(-0.355257\pi\)
\(480\) 0 0
\(481\) 926.558i 1.92632i
\(482\) 371.009i 0.769728i
\(483\) 440.084 + 101.884i 0.911147 + 0.210940i
\(484\) −108.840 −0.224877
\(485\) 0 0
\(486\) 22.0454i 0.0453609i
\(487\) −887.173 −1.82171 −0.910855 0.412726i \(-0.864577\pi\)
−0.910855 + 0.412726i \(0.864577\pi\)
\(488\) 96.9766i 0.198723i
\(489\) 48.8568i 0.0999116i
\(490\) 0 0
\(491\) −50.1823 −0.102204 −0.0511021 0.998693i \(-0.516273\pi\)
−0.0511021 + 0.998693i \(0.516273\pi\)
\(492\) 28.6093 0.0581490
\(493\) 53.7991i 0.109126i
\(494\) 697.646 1.41224
\(495\) 0 0
\(496\) 77.0231i 0.155289i
\(497\) 61.2769 264.683i 0.123294 0.532562i
\(498\) −221.429 −0.444636
\(499\) −160.443 −0.321529 −0.160765 0.986993i \(-0.551396\pi\)
−0.160765 + 0.986993i \(0.551396\pi\)
\(500\) 0 0
\(501\) −222.501 −0.444113
\(502\) 200.701i 0.399802i
\(503\) 742.042i 1.47523i −0.675220 0.737616i \(-0.735951\pi\)
0.675220 0.737616i \(-0.264049\pi\)
\(504\) 13.3967 57.8665i 0.0265807 0.114814i
\(505\) 0 0
\(506\) 429.933 0.849670
\(507\) 76.8111i 0.151501i
\(508\) −404.828 −0.796905
\(509\) 978.447i 1.92229i 0.276038 + 0.961147i \(0.410979\pi\)
−0.276038 + 0.961147i \(0.589021\pi\)
\(510\) 0 0
\(511\) 849.927 + 196.767i 1.66326 + 0.385062i
\(512\) 22.6274 0.0441942
\(513\) 175.493 0.342091
\(514\) 241.608i 0.470054i
\(515\) 0 0
\(516\) 145.802i 0.282561i
\(517\) 190.429i 0.368335i
\(518\) 141.637 611.794i 0.273430 1.18107i
\(519\) 194.762 0.375264
\(520\) 0 0
\(521\) 575.650i 1.10489i 0.833548 + 0.552447i \(0.186306\pi\)
−0.833548 + 0.552447i \(0.813694\pi\)
\(522\) −39.2573 −0.0752056
\(523\) 339.090i 0.648355i 0.945996 + 0.324178i \(0.105088\pi\)
−0.945996 + 0.324178i \(0.894912\pi\)
\(524\) 282.520i 0.539160i
\(525\) 0 0
\(526\) 542.234 1.03086
\(527\) −111.957 −0.212442
\(528\) 56.5317i 0.107068i
\(529\) 859.125 1.62406
\(530\) 0 0
\(531\) 128.805i 0.242571i
\(532\) −460.647 106.644i −0.865877 0.200459i
\(533\) −120.631 −0.226325
\(534\) −39.7525 −0.0744429
\(535\) 0 0
\(536\) −14.1390 −0.0263787
\(537\) 119.342i 0.222238i
\(538\) 218.512i 0.406157i
\(539\) 359.144 + 175.709i 0.666316 + 0.325990i
\(540\) 0 0
\(541\) 35.8638 0.0662916 0.0331458 0.999451i \(-0.489447\pi\)
0.0331458 + 0.999451i \(0.489447\pi\)
\(542\) 453.012i 0.835815i
\(543\) −288.450 −0.531216
\(544\) 32.8901i 0.0604598i
\(545\) 0 0
\(546\) −56.4872 + 243.994i −0.103456 + 0.446876i
\(547\) 700.845 1.28125 0.640626 0.767853i \(-0.278675\pi\)
0.640626 + 0.767853i \(0.278675\pi\)
\(548\) 101.286 0.184829
\(549\) 102.859i 0.187357i
\(550\) 0 0
\(551\) 312.508i 0.567166i
\(552\) 182.524i 0.330659i
\(553\) −88.7044 + 383.155i −0.160406 + 0.692867i
\(554\) 315.163 0.568886
\(555\) 0 0
\(556\) 148.051i 0.266279i
\(557\) −639.349 −1.14784 −0.573922 0.818910i \(-0.694579\pi\)
−0.573922 + 0.818910i \(0.694579\pi\)
\(558\) 81.6954i 0.146407i
\(559\) 614.773i 1.09977i
\(560\) 0 0
\(561\) 82.1717 0.146474
\(562\) 309.674 0.551022
\(563\) 227.519i 0.404120i 0.979373 + 0.202060i \(0.0647636\pi\)
−0.979373 + 0.202060i \(0.935236\pi\)
\(564\) −80.8450 −0.143342
\(565\) 0 0
\(566\) 51.4713i 0.0909387i
\(567\) −14.2093 + 61.3767i −0.0250605 + 0.108248i
\(568\) −109.777 −0.193269
\(569\) −613.901 −1.07891 −0.539456 0.842014i \(-0.681370\pi\)
−0.539456 + 0.842014i \(0.681370\pi\)
\(570\) 0 0
\(571\) 366.674 0.642161 0.321081 0.947052i \(-0.395954\pi\)
0.321081 + 0.947052i \(0.395954\pi\)
\(572\) 238.366i 0.416724i
\(573\) 256.844i 0.448244i
\(574\) 79.6513 + 18.4401i 0.138765 + 0.0321256i
\(575\) 0 0
\(576\) −24.0000 −0.0416667
\(577\) 62.1951i 0.107790i −0.998547 0.0538952i \(-0.982836\pi\)
0.998547 0.0538952i \(-0.0171637\pi\)
\(578\) 360.900 0.624395
\(579\) 60.9485i 0.105265i
\(580\) 0 0
\(581\) −616.481 142.722i −1.06107 0.245648i
\(582\) −202.721 −0.348318
\(583\) 582.188 0.998607
\(584\) 352.505i 0.603605i
\(585\) 0 0
\(586\) 238.211i 0.406503i
\(587\) 176.872i 0.301315i −0.988586 0.150658i \(-0.951861\pi\)
0.988586 0.150658i \(-0.0481391\pi\)
\(588\) 74.5955 152.471i 0.126863 0.259305i
\(589\) −650.337 −1.10414
\(590\) 0 0
\(591\) 335.384i 0.567486i
\(592\) −253.740 −0.428615
\(593\) 154.878i 0.261176i −0.991437 0.130588i \(-0.958313\pi\)
0.991437 0.130588i \(-0.0416866\pi\)
\(594\) 59.9609i 0.100944i
\(595\) 0 0
\(596\) 453.959 0.761676
\(597\) −314.953 −0.527559
\(598\) 769.613i 1.28698i
\(599\) −900.825 −1.50388 −0.751940 0.659231i \(-0.770882\pi\)
−0.751940 + 0.659231i \(0.770882\pi\)
\(600\) 0 0
\(601\) 682.387i 1.13542i −0.823229 0.567710i \(-0.807830\pi\)
0.823229 0.567710i \(-0.192170\pi\)
\(602\) −93.9762 + 405.927i −0.156107 + 0.674297i
\(603\) 14.9967 0.0248701
\(604\) 588.851 0.974920
\(605\) 0 0
\(606\) −213.609 −0.352490
\(607\) 367.814i 0.605954i 0.952998 + 0.302977i \(0.0979806\pi\)
−0.952998 + 0.302977i \(0.902019\pi\)
\(608\) 191.052i 0.314231i
\(609\) −109.296 25.3032i −0.179469 0.0415488i
\(610\) 0 0
\(611\) 340.883 0.557911
\(612\) 34.8852i 0.0570020i
\(613\) 117.271 0.191306 0.0956532 0.995415i \(-0.469506\pi\)
0.0956532 + 0.995415i \(0.469506\pi\)
\(614\) 315.936i 0.514553i
\(615\) 0 0
\(616\) 36.4374 157.390i 0.0591516 0.255503i
\(617\) −1070.18 −1.73449 −0.867245 0.497881i \(-0.834112\pi\)
−0.867245 + 0.497881i \(0.834112\pi\)
\(618\) −459.444 −0.743438
\(619\) 532.192i 0.859761i 0.902886 + 0.429880i \(0.141444\pi\)
−0.902886 + 0.429880i \(0.858556\pi\)
\(620\) 0 0
\(621\) 193.596i 0.311749i
\(622\) 65.1098i 0.104678i
\(623\) −110.675 25.6224i −0.177649 0.0411275i
\(624\) 101.196 0.162173
\(625\) 0 0
\(626\) 113.190i 0.180815i
\(627\) 477.319 0.761275
\(628\) 131.719i 0.209744i
\(629\) 368.825i 0.586366i
\(630\) 0 0
\(631\) 472.933 0.749498 0.374749 0.927126i \(-0.377729\pi\)
0.374749 + 0.927126i \(0.377729\pi\)
\(632\) 158.913 0.251444
\(633\) 304.693i 0.481347i
\(634\) 261.964 0.413193
\(635\) 0 0
\(636\) 247.163i 0.388620i
\(637\) −314.532 + 642.896i −0.493771 + 1.00926i
\(638\) −106.775 −0.167359
\(639\) 116.436 0.182216
\(640\) 0 0
\(641\) −486.990 −0.759734 −0.379867 0.925041i \(-0.624030\pi\)
−0.379867 + 0.925041i \(0.624030\pi\)
\(642\) 386.671i 0.602291i
\(643\) 111.425i 0.173289i 0.996239 + 0.0866447i \(0.0276145\pi\)
−0.996239 + 0.0866447i \(0.972386\pi\)
\(644\) 117.645 508.165i 0.182679 0.789077i
\(645\) 0 0
\(646\) −277.704 −0.429883
\(647\) 737.696i 1.14018i 0.821583 + 0.570090i \(0.193092\pi\)
−0.821583 + 0.570090i \(0.806908\pi\)
\(648\) 25.4558 0.0392837
\(649\) 350.334i 0.539806i
\(650\) 0 0
\(651\) 52.6566 227.448i 0.0808857 0.349383i
\(652\) 56.4149 0.0865260
\(653\) 193.734 0.296682 0.148341 0.988936i \(-0.452607\pi\)
0.148341 + 0.988936i \(0.452607\pi\)
\(654\) 274.522i 0.419758i
\(655\) 0 0
\(656\) 33.0352i 0.0503585i
\(657\) 373.888i 0.569084i
\(658\) −225.081 52.1085i −0.342068 0.0791923i
\(659\) 823.339 1.24938 0.624688 0.780874i \(-0.285226\pi\)
0.624688 + 0.780874i \(0.285226\pi\)
\(660\) 0 0
\(661\) 657.833i 0.995208i −0.867404 0.497604i \(-0.834213\pi\)
0.867404 0.497604i \(-0.165787\pi\)
\(662\) 86.9298 0.131314
\(663\) 147.094i 0.221861i
\(664\) 255.684i 0.385066i
\(665\) 0 0
\(666\) 269.132 0.404102
\(667\) −344.746 −0.516860
\(668\) 256.922i 0.384614i
\(669\) 35.6239 0.0532495
\(670\) 0 0
\(671\) 279.765i 0.416937i
\(672\) −66.8184 15.4692i −0.0994322 0.0230196i
\(673\) −727.300 −1.08068 −0.540342 0.841446i \(-0.681705\pi\)
−0.540342 + 0.841446i \(0.681705\pi\)
\(674\) 928.433 1.37750
\(675\) 0 0
\(676\) −88.6938 −0.131204
\(677\) 385.964i 0.570110i −0.958511 0.285055i \(-0.907988\pi\)
0.958511 0.285055i \(-0.0920118\pi\)
\(678\) 347.797i 0.512975i
\(679\) −564.396 130.663i −0.831217 0.192435i
\(680\) 0 0
\(681\) 717.219 1.05319
\(682\) 222.202i 0.325809i
\(683\) 236.488 0.346249 0.173124 0.984900i \(-0.444614\pi\)
0.173124 + 0.984900i \(0.444614\pi\)
\(684\) 202.641i 0.296259i
\(685\) 0 0
\(686\) 305.957 376.415i 0.446001 0.548711i
\(687\) −338.650 −0.492941
\(688\) 168.357 0.244705
\(689\) 1042.16i 1.51257i
\(690\) 0 0
\(691\) 337.426i 0.488316i −0.969735 0.244158i \(-0.921488\pi\)
0.969735 0.244158i \(-0.0785116\pi\)
\(692\) 224.892i 0.324988i
\(693\) −38.6477 + 166.937i −0.0557687 + 0.240891i
\(694\) 454.419 0.654782
\(695\) 0 0
\(696\) 45.3305i 0.0651300i
\(697\) 48.0184 0.0688929
\(698\) 261.821i 0.375101i
\(699\) 140.817i 0.201455i
\(700\) 0 0
\(701\) 89.2192 0.127274 0.0636371 0.997973i \(-0.479730\pi\)
0.0636371 + 0.997973i \(0.479730\pi\)
\(702\) −107.335 −0.152898
\(703\) 2142.43i 3.04755i
\(704\) −65.2772 −0.0927232
\(705\) 0 0
\(706\) 844.764i 1.19655i
\(707\) −594.709 137.681i −0.841172 0.194740i
\(708\) 148.731 0.210072
\(709\) −569.646 −0.803450 −0.401725 0.915760i \(-0.631589\pi\)
−0.401725 + 0.915760i \(0.631589\pi\)
\(710\) 0 0
\(711\) −168.552 −0.237064
\(712\) 45.9023i 0.0644695i
\(713\) 717.423i 1.00620i
\(714\) 22.4852 97.1241i 0.0314919 0.136028i
\(715\) 0 0
\(716\) 137.804 0.192464
\(717\) 431.740i 0.602147i
\(718\) −338.628 −0.471627
\(719\) 1261.33i 1.75428i 0.480233 + 0.877141i \(0.340552\pi\)
−0.480233 + 0.877141i \(0.659448\pi\)
\(720\) 0 0
\(721\) −1279.14 296.134i −1.77412 0.410727i
\(722\) −1102.60 −1.52714
\(723\) −454.391 −0.628481
\(724\) 333.074i 0.460047i
\(725\) 0 0
\(726\) 133.302i 0.183611i
\(727\) 307.746i 0.423310i −0.977344 0.211655i \(-0.932115\pi\)
0.977344 0.211655i \(-0.0678852\pi\)
\(728\) 281.740 + 65.2258i 0.387006 + 0.0895958i
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) 244.716i 0.334769i
\(732\) −118.772 −0.162256
\(733\) 633.490i 0.864243i 0.901815 + 0.432121i \(0.142235\pi\)
−0.901815 + 0.432121i \(0.857765\pi\)
\(734\) 563.908i 0.768266i
\(735\) 0 0
\(736\) −210.761 −0.286359
\(737\) 40.7891 0.0553448
\(738\) 35.0391i 0.0474785i
\(739\) −749.557 −1.01428 −0.507142 0.861862i \(-0.669298\pi\)
−0.507142 + 0.861862i \(0.669298\pi\)
\(740\) 0 0
\(741\) 854.439i 1.15309i
\(742\) 159.308 688.126i 0.214701 0.927393i
\(743\) −145.699 −0.196095 −0.0980476 0.995182i \(-0.531260\pi\)
−0.0980476 + 0.995182i \(0.531260\pi\)
\(744\) −94.3337 −0.126793
\(745\) 0 0
\(746\) −49.0822 −0.0657939
\(747\) 271.194i 0.363044i
\(748\) 94.8837i 0.126850i
\(749\) 249.228 1076.53i 0.332748 1.43729i
\(750\) 0 0
\(751\) 156.811 0.208803 0.104402 0.994535i \(-0.466707\pi\)
0.104402 + 0.994535i \(0.466707\pi\)
\(752\) 93.3518i 0.124138i
\(753\) −245.807 −0.326437
\(754\) 191.136i 0.253496i
\(755\) 0 0
\(756\) 70.8717 + 16.4075i 0.0937456 + 0.0217031i
\(757\) −296.377 −0.391515 −0.195758 0.980652i \(-0.562717\pi\)
−0.195758 + 0.980652i \(0.562717\pi\)
\(758\) 154.457 0.203769
\(759\) 526.558i 0.693752i
\(760\) 0 0
\(761\) 312.747i 0.410968i 0.978660 + 0.205484i \(0.0658769\pi\)
−0.978660 + 0.205484i \(0.934123\pi\)
\(762\) 495.811i 0.650670i
\(763\) −176.942 + 764.297i −0.231904 + 1.00170i
\(764\) −296.578 −0.388191
\(765\) 0 0
\(766\) 475.770i 0.621109i
\(767\) −627.125 −0.817634
\(768\) 27.7128i 0.0360844i
\(769\) 91.1460i 0.118525i −0.998242 0.0592627i \(-0.981125\pi\)
0.998242 0.0592627i \(-0.0188750\pi\)
\(770\) 0 0
\(771\) −295.908 −0.383798
\(772\) −70.3772 −0.0911622
\(773\) 417.539i 0.540154i −0.962839 0.270077i \(-0.912951\pi\)
0.962839 0.270077i \(-0.0870492\pi\)
\(774\) −178.570 −0.230710
\(775\) 0 0
\(776\) 234.082i 0.301652i
\(777\) 749.291 + 173.469i 0.964339 + 0.223254i
\(778\) 440.065 0.565636
\(779\) 278.929 0.358061
\(780\) 0 0
\(781\) 316.692 0.405496
\(782\) 306.351i 0.391754i
\(783\) 48.0802i 0.0614052i
\(784\) −176.059 86.1354i −0.224565 0.109867i
\(785\) 0 0
\(786\) 346.015 0.440222
\(787\) 318.111i 0.404207i 0.979364 + 0.202104i \(0.0647778\pi\)
−0.979364 + 0.202104i \(0.935222\pi\)
\(788\) −387.268 −0.491457
\(789\) 664.098i 0.841696i
\(790\) 0 0
\(791\) −224.172 + 968.302i −0.283403 + 1.22415i
\(792\) 69.2369 0.0874203
\(793\) 500.801 0.631527
\(794\) 58.3616i 0.0735033i
\(795\) 0 0
\(796\) 363.676i 0.456880i
\(797\) 1188.06i 1.49067i 0.666690 + 0.745335i \(0.267711\pi\)
−0.666690 + 0.745335i \(0.732289\pi\)
\(798\) 130.612 564.175i 0.163674 0.706986i
\(799\) −135.692 −0.169827
\(800\) 0 0
\(801\) 48.6867i 0.0607824i
\(802\) −700.943 −0.873994
\(803\) 1016.93i 1.26642i
\(804\) 17.3166i 0.0215381i
\(805\) 0 0
\(806\) 397.758 0.493497
\(807\) 267.622 0.331626
\(808\) 246.654i 0.305265i
\(809\) 223.374 0.276111 0.138055 0.990425i \(-0.455915\pi\)
0.138055 + 0.990425i \(0.455915\pi\)
\(810\) 0 0
\(811\) 366.185i 0.451523i −0.974183 0.225761i \(-0.927513\pi\)
0.974183 0.225761i \(-0.0724870\pi\)
\(812\) −29.2177 + 126.205i −0.0359824 + 0.155424i
\(813\) −554.824 −0.682440
\(814\) 732.008 0.899272
\(815\) 0 0
\(816\) −40.2820 −0.0493652
\(817\) 1421.51i 1.73991i
\(818\) 509.004i 0.622255i
\(819\) −298.831 69.1824i −0.364873 0.0844718i
\(820\) 0 0
\(821\) −1547.39 −1.88476 −0.942381 0.334543i \(-0.891418\pi\)
−0.942381 + 0.334543i \(0.891418\pi\)
\(822\) 124.050i 0.150912i
\(823\) −1284.25 −1.56045 −0.780225 0.625499i \(-0.784895\pi\)
−0.780225 + 0.625499i \(0.784895\pi\)
\(824\) 530.521i 0.643836i
\(825\) 0 0
\(826\) 414.083 + 95.8644i 0.501311 + 0.116059i
\(827\) −807.319 −0.976202 −0.488101 0.872787i \(-0.662310\pi\)
−0.488101 + 0.872787i \(0.662310\pi\)
\(828\) 223.545 0.269982
\(829\) 623.615i 0.752250i −0.926569 0.376125i \(-0.877256\pi\)
0.926569 0.376125i \(-0.122744\pi\)
\(830\) 0 0
\(831\) 385.994i 0.464493i
\(832\) 116.851i 0.140446i
\(833\) 125.202 255.911i 0.150303 0.307216i
\(834\) −181.325 −0.217416
\(835\) 0 0
\(836\) 551.161i 0.659283i
\(837\) 100.056 0.119541
\(838\) 682.938i 0.814962i
\(839\) 1146.31i 1.36628i −0.730286 0.683142i \(-0.760613\pi\)
0.730286 0.683142i \(-0.239387\pi\)
\(840\) 0 0
\(841\) −755.381 −0.898194
\(842\) −740.368 −0.879297
\(843\) 379.272i 0.449908i
\(844\) 351.829 0.416859
\(845\) 0 0
\(846\) 99.0145i 0.117038i
\(847\) 85.9194 371.126i 0.101440 0.438165i
\(848\) −285.399 −0.336555
\(849\) 63.0392 0.0742511
\(850\) 0 0
\(851\) 2363.43 2.77724
\(852\) 134.449i 0.157804i
\(853\) 948.920i 1.11245i 0.831032 + 0.556225i \(0.187751\pi\)
−0.831032 + 0.556225i \(0.812249\pi\)
\(854\) −330.672 76.5540i −0.387204 0.0896417i
\(855\) 0 0
\(856\) −446.489 −0.521599
\(857\) 868.842i 1.01382i −0.862000 0.506909i \(-0.830788\pi\)
0.862000 0.506909i \(-0.169212\pi\)
\(858\) −291.938 −0.340254
\(859\) 119.981i 0.139675i −0.997558 0.0698377i \(-0.977752\pi\)
0.997558 0.0698377i \(-0.0222482\pi\)
\(860\) 0 0
\(861\) −22.5844 + 97.5525i −0.0262304 + 0.113301i
\(862\) 731.187 0.848245
\(863\) 788.430 0.913592 0.456796 0.889571i \(-0.348997\pi\)
0.456796 + 0.889571i \(0.348997\pi\)
\(864\) 29.3939i 0.0340207i
\(865\) 0 0
\(866\) 802.866i 0.927097i
\(867\) 442.011i 0.509816i
\(868\) −262.635 60.8026i −0.302574 0.0700491i
\(869\) −458.443 −0.527552
\(870\) 0 0
\(871\) 73.0157i 0.0838298i
\(872\) 316.990 0.363521
\(873\) 248.282i 0.284400i
\(874\) 1779.53i 2.03608i
\(875\) 0 0
\(876\) 431.729 0.492841
\(877\) −234.663 −0.267574 −0.133787 0.991010i \(-0.542714\pi\)
−0.133787 + 0.991010i \(0.542714\pi\)
\(878\) 880.879i 1.00328i
\(879\) −291.747 −0.331908
\(880\) 0 0
\(881\) 750.816i 0.852232i 0.904669 + 0.426116i \(0.140118\pi\)
−0.904669 + 0.426116i \(0.859882\pi\)
\(882\) 186.739 + 91.3604i 0.211722 + 0.103583i
\(883\) 1293.79 1.46522 0.732610 0.680649i \(-0.238302\pi\)
0.732610 + 0.680649i \(0.238302\pi\)
\(884\) 169.849 0.192137
\(885\) 0 0
\(886\) −674.154 −0.760896
\(887\) 728.248i 0.821024i 0.911855 + 0.410512i \(0.134650\pi\)
−0.911855 + 0.410512i \(0.865350\pi\)
\(888\) 310.767i 0.349963i
\(889\) 319.574 1380.39i 0.359476 1.55274i
\(890\) 0 0
\(891\) −73.4368 −0.0824207
\(892\) 41.1349i 0.0461154i
\(893\) −788.206 −0.882650
\(894\) 555.984i 0.621906i
\(895\) 0 0
\(896\) −17.8622 + 77.1553i −0.0199355 + 0.0861108i
\(897\) −942.580 −1.05081
\(898\) −1218.95 −1.35741
\(899\) 178.175i 0.198192i
\(900\) 0 0
\(901\) 414.842i 0.460424i
\(902\) 95.3022i 0.105657i
\(903\) −497.157 115.097i −0.550561 0.127461i
\(904\) 401.601 0.444249
\(905\) 0 0
\(906\) 721.193i 0.796018i
\(907\) −913.713 −1.00740 −0.503700 0.863878i \(-0.668028\pi\)
−0.503700 + 0.863878i \(0.668028\pi\)
\(908\) 828.173i 0.912085i
\(909\) 261.616i 0.287807i
\(910\) 0 0
\(911\) −1331.32 −1.46138 −0.730690 0.682709i \(-0.760802\pi\)
−0.730690 + 0.682709i \(0.760802\pi\)
\(912\) −233.990 −0.256568
\(913\) 737.615i 0.807903i
\(914\) −38.9476 −0.0426123
\(915\) 0 0
\(916\) 391.040i 0.426899i
\(917\) 963.341 + 223.023i 1.05053 + 0.243210i
\(918\) 42.7255 0.0465420
\(919\) 111.140 0.120936 0.0604678 0.998170i \(-0.480741\pi\)
0.0604678 + 0.998170i \(0.480741\pi\)
\(920\) 0 0
\(921\) −386.941 −0.420131
\(922\) 534.831i 0.580077i
\(923\) 566.904i 0.614197i
\(924\) 192.763 + 44.6265i 0.208617 + 0.0482971i
\(925\) 0 0
\(926\) −1025.30 −1.10723
\(927\) 562.702i 0.607014i
\(928\) 52.3431 0.0564042
\(929\) 571.383i 0.615051i −0.951540 0.307526i \(-0.900499\pi\)
0.951540 0.307526i \(-0.0995010\pi\)
\(930\) 0 0
\(931\) 727.275 1486.53i 0.781176 1.59671i
\(932\) −162.601 −0.174465
\(933\) 79.7428 0.0854693
\(934\) 525.311i 0.562431i
\(935\) 0 0
\(936\) 123.939i 0.132414i
\(937\) 1657.10i 1.76852i −0.466997 0.884259i \(-0.654664\pi\)
0.466997 0.884259i \(-0.345336\pi\)
\(938\) 11.1614 48.2113i 0.0118992 0.0513980i
\(939\) −138.629 −0.147635
\(940\) 0 0
\(941\) 519.083i 0.551629i 0.961211 + 0.275815i \(0.0889476\pi\)
−0.961211 + 0.275815i \(0.911052\pi\)
\(942\) 161.322 0.171255
\(943\) 307.703i 0.326302i
\(944\) 171.740i 0.181928i
\(945\) 0 0
\(946\) −485.689 −0.513413
\(947\) −558.258 −0.589502 −0.294751 0.955574i \(-0.595237\pi\)
−0.294751 + 0.955574i \(0.595237\pi\)
\(948\) 194.628i 0.205303i
\(949\) −1820.39 −1.91822
\(950\) 0 0
\(951\) 320.839i 0.337370i
\(952\) −112.149 25.9637i −0.117804 0.0272728i
\(953\) 291.413 0.305784 0.152892 0.988243i \(-0.451141\pi\)
0.152892 + 0.988243i \(0.451141\pi\)
\(954\) 302.711 0.317307
\(955\) 0 0
\(956\) 498.530 0.521475
\(957\) 130.773i 0.136648i
\(958\) 1217.14i 1.27050i
\(959\) −79.9559 + 345.367i −0.0833742 + 0.360132i
\(960\) 0 0
\(961\) 590.215 0.614167
\(962\) 1310.35i 1.36211i
\(963\) 473.573 0.491768
\(964\) 524.686i 0.544280i
\(965\) 0 0
\(966\) 622.373 + 144.086i 0.644278 + 0.149157i
\(967\) 323.558 0.334600 0.167300 0.985906i \(-0.446495\pi\)
0.167300 + 0.985906i \(0.446495\pi\)
\(968\) −153.924 −0.159012
\(969\) 340.117i 0.350998i
\(970\) 0 0
\(971\) 219.659i 0.226219i −0.993583 0.113110i \(-0.963919\pi\)
0.993583 0.113110i \(-0.0360812\pi\)
\(972\) 31.1769i 0.0320750i
\(973\) −504.827 116.873i −0.518836 0.120116i
\(974\) −1254.65 −1.28814
\(975\) 0 0
\(976\) 137.146i 0.140518i
\(977\) 1302.29 1.33295 0.666475 0.745527i \(-0.267802\pi\)
0.666475 + 0.745527i \(0.267802\pi\)
\(978\) 69.0939i 0.0706482i
\(979\) 132.422i 0.135263i
\(980\) 0 0
\(981\) −336.219 −0.342731
\(982\) −70.9685 −0.0722693
\(983\) 649.905i 0.661144i 0.943781 + 0.330572i \(0.107242\pi\)
−0.943781 + 0.330572i \(0.892758\pi\)
\(984\) 40.4597 0.0411176
\(985\) 0 0
\(986\) 76.0834i 0.0771637i
\(987\) 63.8196 275.667i 0.0646602 0.279297i
\(988\) 986.621 0.998604
\(989\) −1568.15 −1.58559
\(990\) 0 0
\(991\) −575.568 −0.580795 −0.290397 0.956906i \(-0.593788\pi\)
−0.290397 + 0.956906i \(0.593788\pi\)
\(992\) 108.927i 0.109806i
\(993\) 106.467i 0.107217i
\(994\) 86.6587 374.319i 0.0871818 0.376578i
\(995\) 0 0
\(996\) −313.148 −0.314405
\(997\) 705.512i 0.707635i 0.935315 + 0.353817i \(0.115117\pi\)
−0.935315 + 0.353817i \(0.884883\pi\)
\(998\) −226.901 −0.227355
\(999\) 329.618i 0.329948i
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 1050.3.f.e.601.14 16
5.2 odd 4 210.3.h.a.139.13 yes 16
5.3 odd 4 210.3.h.a.139.4 16
5.4 even 2 inner 1050.3.f.e.601.3 16
7.6 odd 2 inner 1050.3.f.e.601.10 16
15.2 even 4 630.3.h.e.559.7 16
15.8 even 4 630.3.h.e.559.10 16
20.3 even 4 1680.3.bd.a.769.15 16
20.7 even 4 1680.3.bd.a.769.1 16
35.13 even 4 210.3.h.a.139.5 yes 16
35.27 even 4 210.3.h.a.139.12 yes 16
35.34 odd 2 inner 1050.3.f.e.601.7 16
105.62 odd 4 630.3.h.e.559.2 16
105.83 odd 4 630.3.h.e.559.15 16
140.27 odd 4 1680.3.bd.a.769.16 16
140.83 odd 4 1680.3.bd.a.769.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
210.3.h.a.139.4 16 5.3 odd 4
210.3.h.a.139.5 yes 16 35.13 even 4
210.3.h.a.139.12 yes 16 35.27 even 4
210.3.h.a.139.13 yes 16 5.2 odd 4
630.3.h.e.559.2 16 105.62 odd 4
630.3.h.e.559.7 16 15.2 even 4
630.3.h.e.559.10 16 15.8 even 4
630.3.h.e.559.15 16 105.83 odd 4
1050.3.f.e.601.3 16 5.4 even 2 inner
1050.3.f.e.601.7 16 35.34 odd 2 inner
1050.3.f.e.601.10 16 7.6 odd 2 inner
1050.3.f.e.601.14 16 1.1 even 1 trivial
1680.3.bd.a.769.1 16 20.7 even 4
1680.3.bd.a.769.2 16 140.83 odd 4
1680.3.bd.a.769.15 16 20.3 even 4
1680.3.bd.a.769.16 16 140.27 odd 4