Properties

Label 275.6.b.b.199.4
Level $275$
Weight $6$
Character 275.199
Analytic conductor $44.106$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [275,6,Mod(199,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.199");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.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: \(44.1055504486\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.11877512256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 34x^{4} - 154x^{3} + 569x^{2} - 6512x + 17216 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.4
Root \(-3.52398 + 4.62828i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.6.b.b.199.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.20859i q^{2} -16.8394i q^{3} +27.1221 q^{4} +37.1913 q^{6} -225.525i q^{7} +130.577i q^{8} -40.5643 q^{9} +O(q^{10})\) \(q+2.20859i q^{2} -16.8394i q^{3} +27.1221 q^{4} +37.1913 q^{6} -225.525i q^{7} +130.577i q^{8} -40.5643 q^{9} +121.000 q^{11} -456.719i q^{12} -455.465i q^{13} +498.092 q^{14} +579.518 q^{16} +190.657i q^{17} -89.5900i q^{18} +135.393 q^{19} -3797.69 q^{21} +267.240i q^{22} -2796.65i q^{23} +2198.83 q^{24} +1005.94 q^{26} -3408.89i q^{27} -6116.71i q^{28} +2608.58 q^{29} -1056.76 q^{31} +5458.37i q^{32} -2037.56i q^{33} -421.082 q^{34} -1100.19 q^{36} +12536.8i q^{37} +299.028i q^{38} -7669.74 q^{39} +1130.09 q^{41} -8387.55i q^{42} +14671.0i q^{43} +3281.78 q^{44} +6176.65 q^{46} -16882.2i q^{47} -9758.71i q^{48} -34054.4 q^{49} +3210.54 q^{51} -12353.2i q^{52} -3313.02i q^{53} +7528.84 q^{54} +29448.3 q^{56} -2279.93i q^{57} +5761.29i q^{58} -11454.0 q^{59} -28227.5 q^{61} -2333.95i q^{62} +9148.26i q^{63} +6489.25 q^{64} +4500.15 q^{66} -51431.0i q^{67} +5171.01i q^{68} -47093.8 q^{69} -16218.0 q^{71} -5296.75i q^{72} +10168.8i q^{73} -27688.7 q^{74} +3672.15 q^{76} -27288.5i q^{77} -16939.3i q^{78} -60841.2 q^{79} -67260.7 q^{81} +2495.90i q^{82} -45770.6i q^{83} -103002. q^{84} -32402.3 q^{86} -43926.9i q^{87} +15799.8i q^{88} +82267.9 q^{89} -102719. q^{91} -75851.0i q^{92} +17795.1i q^{93} +37285.8 q^{94} +91915.5 q^{96} +53097.0i q^{97} -75212.2i q^{98} -4908.28 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 168 q^{4} - 412 q^{6} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 168 q^{4} - 412 q^{6} + 14 q^{9} + 726 q^{11} + 2040 q^{14} + 3984 q^{16} - 2760 q^{19} - 1816 q^{21} + 23496 q^{24} + 24264 q^{26} + 6852 q^{29} - 8196 q^{31} - 50640 q^{34} + 9512 q^{36} + 13120 q^{39} + 11988 q^{41} - 20328 q^{44} - 33612 q^{46} - 97062 q^{49} - 45448 q^{51} - 37628 q^{54} - 84624 q^{56} + 7476 q^{59} + 36972 q^{61} + 40704 q^{64} - 49852 q^{66} - 70084 q^{69} + 78564 q^{71} - 306588 q^{74} + 207840 q^{76} - 250296 q^{79} - 173834 q^{81} - 687232 q^{84} + 486120 q^{86} + 213648 q^{89} - 219264 q^{91} + 149856 q^{94} - 152912 q^{96} + 1694 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.20859i 0.390428i 0.980761 + 0.195214i \(0.0625401\pi\)
−0.980761 + 0.195214i \(0.937460\pi\)
\(3\) − 16.8394i − 1.08025i −0.841586 0.540123i \(-0.818378\pi\)
0.841586 0.540123i \(-0.181622\pi\)
\(4\) 27.1221 0.847566
\(5\) 0 0
\(6\) 37.1913 0.421758
\(7\) − 225.525i − 1.73960i −0.493406 0.869799i \(-0.664248\pi\)
0.493406 0.869799i \(-0.335752\pi\)
\(8\) 130.577i 0.721341i
\(9\) −40.5643 −0.166931
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) − 456.719i − 0.915580i
\(13\) − 455.465i − 0.747474i −0.927535 0.373737i \(-0.878076\pi\)
0.927535 0.373737i \(-0.121924\pi\)
\(14\) 498.092 0.679187
\(15\) 0 0
\(16\) 579.518 0.565935
\(17\) 190.657i 0.160003i 0.996795 + 0.0800017i \(0.0254926\pi\)
−0.996795 + 0.0800017i \(0.974507\pi\)
\(18\) − 89.5900i − 0.0651746i
\(19\) 135.393 0.0860424 0.0430212 0.999074i \(-0.486302\pi\)
0.0430212 + 0.999074i \(0.486302\pi\)
\(20\) 0 0
\(21\) −3797.69 −1.87919
\(22\) 267.240i 0.117718i
\(23\) − 2796.65i − 1.10235i −0.834391 0.551173i \(-0.814180\pi\)
0.834391 0.551173i \(-0.185820\pi\)
\(24\) 2198.83 0.779225
\(25\) 0 0
\(26\) 1005.94 0.291835
\(27\) − 3408.89i − 0.899919i
\(28\) − 6116.71i − 1.47443i
\(29\) 2608.58 0.575983 0.287991 0.957633i \(-0.407013\pi\)
0.287991 + 0.957633i \(0.407013\pi\)
\(30\) 0 0
\(31\) −1056.76 −0.197502 −0.0987510 0.995112i \(-0.531485\pi\)
−0.0987510 + 0.995112i \(0.531485\pi\)
\(32\) 5458.37i 0.942297i
\(33\) − 2037.56i − 0.325706i
\(34\) −421.082 −0.0624697
\(35\) 0 0
\(36\) −1100.19 −0.141485
\(37\) 12536.8i 1.50550i 0.658304 + 0.752752i \(0.271274\pi\)
−0.658304 + 0.752752i \(0.728726\pi\)
\(38\) 299.028i 0.0335933i
\(39\) −7669.74 −0.807456
\(40\) 0 0
\(41\) 1130.09 0.104991 0.0524954 0.998621i \(-0.483282\pi\)
0.0524954 + 0.998621i \(0.483282\pi\)
\(42\) − 8387.55i − 0.733689i
\(43\) 14671.0i 1.21001i 0.796222 + 0.605005i \(0.206829\pi\)
−0.796222 + 0.605005i \(0.793171\pi\)
\(44\) 3281.78 0.255551
\(45\) 0 0
\(46\) 6176.65 0.430386
\(47\) − 16882.2i − 1.11477i −0.830256 0.557383i \(-0.811806\pi\)
0.830256 0.557383i \(-0.188194\pi\)
\(48\) − 9758.71i − 0.611349i
\(49\) −34054.4 −2.02620
\(50\) 0 0
\(51\) 3210.54 0.172843
\(52\) − 12353.2i − 0.633534i
\(53\) − 3313.02i − 0.162007i −0.996714 0.0810035i \(-0.974187\pi\)
0.996714 0.0810035i \(-0.0258125\pi\)
\(54\) 7528.84 0.351353
\(55\) 0 0
\(56\) 29448.3 1.25484
\(57\) − 2279.93i − 0.0929469i
\(58\) 5761.29i 0.224880i
\(59\) −11454.0 −0.428378 −0.214189 0.976792i \(-0.568711\pi\)
−0.214189 + 0.976792i \(0.568711\pi\)
\(60\) 0 0
\(61\) −28227.5 −0.971286 −0.485643 0.874157i \(-0.661415\pi\)
−0.485643 + 0.874157i \(0.661415\pi\)
\(62\) − 2333.95i − 0.0771102i
\(63\) 9148.26i 0.290394i
\(64\) 6489.25 0.198036
\(65\) 0 0
\(66\) 4500.15 0.127165
\(67\) − 51431.0i − 1.39971i −0.714285 0.699855i \(-0.753248\pi\)
0.714285 0.699855i \(-0.246752\pi\)
\(68\) 5171.01i 0.135614i
\(69\) −47093.8 −1.19081
\(70\) 0 0
\(71\) −16218.0 −0.381814 −0.190907 0.981608i \(-0.561143\pi\)
−0.190907 + 0.981608i \(0.561143\pi\)
\(72\) − 5296.75i − 0.120414i
\(73\) 10168.8i 0.223337i 0.993745 + 0.111669i \(0.0356196\pi\)
−0.993745 + 0.111669i \(0.964380\pi\)
\(74\) −27688.7 −0.587791
\(75\) 0 0
\(76\) 3672.15 0.0729266
\(77\) − 27288.5i − 0.524509i
\(78\) − 16939.3i − 0.315253i
\(79\) −60841.2 −1.09681 −0.548404 0.836214i \(-0.684764\pi\)
−0.548404 + 0.836214i \(0.684764\pi\)
\(80\) 0 0
\(81\) −67260.7 −1.13907
\(82\) 2495.90i 0.0409913i
\(83\) − 45770.6i − 0.729275i −0.931150 0.364638i \(-0.881193\pi\)
0.931150 0.364638i \(-0.118807\pi\)
\(84\) −103002. −1.59274
\(85\) 0 0
\(86\) −32402.3 −0.472421
\(87\) − 43926.9i − 0.622203i
\(88\) 15799.8i 0.217492i
\(89\) 82267.9 1.10092 0.550460 0.834862i \(-0.314452\pi\)
0.550460 + 0.834862i \(0.314452\pi\)
\(90\) 0 0
\(91\) −102719. −1.30031
\(92\) − 75851.0i − 0.934312i
\(93\) 17795.1i 0.213351i
\(94\) 37285.8 0.435235
\(95\) 0 0
\(96\) 91915.5 1.01791
\(97\) 53097.0i 0.572981i 0.958083 + 0.286491i \(0.0924888\pi\)
−0.958083 + 0.286491i \(0.907511\pi\)
\(98\) − 75212.2i − 0.791085i
\(99\) −4908.28 −0.0503317
\(100\) 0 0
\(101\) 186821. 1.82231 0.911153 0.412069i \(-0.135194\pi\)
0.911153 + 0.412069i \(0.135194\pi\)
\(102\) 7090.76i 0.0674827i
\(103\) − 34290.5i − 0.318479i −0.987240 0.159240i \(-0.949096\pi\)
0.987240 0.159240i \(-0.0509042\pi\)
\(104\) 59473.0 0.539184
\(105\) 0 0
\(106\) 7317.10 0.0632520
\(107\) − 224117.i − 1.89241i −0.323565 0.946206i \(-0.604881\pi\)
0.323565 0.946206i \(-0.395119\pi\)
\(108\) − 92456.3i − 0.762741i
\(109\) −162229. −1.30786 −0.653931 0.756554i \(-0.726882\pi\)
−0.653931 + 0.756554i \(0.726882\pi\)
\(110\) 0 0
\(111\) 211112. 1.62632
\(112\) − 130696.i − 0.984500i
\(113\) − 92225.0i − 0.679442i −0.940526 0.339721i \(-0.889667\pi\)
0.940526 0.339721i \(-0.110333\pi\)
\(114\) 5035.44 0.0362890
\(115\) 0 0
\(116\) 70750.3 0.488184
\(117\) 18475.6i 0.124777i
\(118\) − 25297.2i − 0.167251i
\(119\) 42997.8 0.278342
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) − 62342.9i − 0.379217i
\(123\) − 19029.9i − 0.113416i
\(124\) −28661.5 −0.167396
\(125\) 0 0
\(126\) −20204.8 −0.113378
\(127\) 138299.i 0.760868i 0.924808 + 0.380434i \(0.124225\pi\)
−0.924808 + 0.380434i \(0.875775\pi\)
\(128\) 189000.i 1.01962i
\(129\) 247051. 1.30711
\(130\) 0 0
\(131\) −54420.4 −0.277066 −0.138533 0.990358i \(-0.544239\pi\)
−0.138533 + 0.990358i \(0.544239\pi\)
\(132\) − 55263.0i − 0.276058i
\(133\) − 30534.5i − 0.149679i
\(134\) 113590. 0.546485
\(135\) 0 0
\(136\) −24895.3 −0.115417
\(137\) 40555.1i 0.184605i 0.995731 + 0.0923025i \(0.0294227\pi\)
−0.995731 + 0.0923025i \(0.970577\pi\)
\(138\) − 104011.i − 0.464923i
\(139\) −140537. −0.616955 −0.308477 0.951232i \(-0.599819\pi\)
−0.308477 + 0.951232i \(0.599819\pi\)
\(140\) 0 0
\(141\) −284285. −1.20422
\(142\) − 35818.9i − 0.149071i
\(143\) − 55111.2i − 0.225372i
\(144\) −23507.7 −0.0944723
\(145\) 0 0
\(146\) −22458.7 −0.0871971
\(147\) 573454.i 2.18880i
\(148\) 340024.i 1.27602i
\(149\) −176073. −0.649722 −0.324861 0.945762i \(-0.605317\pi\)
−0.324861 + 0.945762i \(0.605317\pi\)
\(150\) 0 0
\(151\) 409241. 1.46062 0.730309 0.683117i \(-0.239376\pi\)
0.730309 + 0.683117i \(0.239376\pi\)
\(152\) 17679.2i 0.0620659i
\(153\) − 7733.85i − 0.0267096i
\(154\) 60269.1 0.204783
\(155\) 0 0
\(156\) −208020. −0.684373
\(157\) 14294.5i 0.0462829i 0.999732 + 0.0231414i \(0.00736681\pi\)
−0.999732 + 0.0231414i \(0.992633\pi\)
\(158\) − 134373.i − 0.428224i
\(159\) −55789.1 −0.175007
\(160\) 0 0
\(161\) −630713. −1.91764
\(162\) − 148551.i − 0.444722i
\(163\) 418474.i 1.23367i 0.787091 + 0.616836i \(0.211586\pi\)
−0.787091 + 0.616836i \(0.788414\pi\)
\(164\) 30650.3 0.0889868
\(165\) 0 0
\(166\) 101089. 0.284729
\(167\) − 139747.i − 0.387749i −0.981026 0.193875i \(-0.937894\pi\)
0.981026 0.193875i \(-0.0621055\pi\)
\(168\) − 495890.i − 1.35554i
\(169\) 163845. 0.441282
\(170\) 0 0
\(171\) −5492.13 −0.0143632
\(172\) 397909.i 1.02556i
\(173\) 687104.i 1.74545i 0.488213 + 0.872725i \(0.337649\pi\)
−0.488213 + 0.872725i \(0.662351\pi\)
\(174\) 97016.5 0.242925
\(175\) 0 0
\(176\) 70121.6 0.170636
\(177\) 192878.i 0.462754i
\(178\) 181696.i 0.429829i
\(179\) −35496.4 −0.0828042 −0.0414021 0.999143i \(-0.513182\pi\)
−0.0414021 + 0.999143i \(0.513182\pi\)
\(180\) 0 0
\(181\) 260469. 0.590963 0.295481 0.955349i \(-0.404520\pi\)
0.295481 + 0.955349i \(0.404520\pi\)
\(182\) − 226863.i − 0.507675i
\(183\) 475333.i 1.04923i
\(184\) 365177. 0.795167
\(185\) 0 0
\(186\) −39302.2 −0.0832980
\(187\) 23069.4i 0.0482429i
\(188\) − 457880.i − 0.944837i
\(189\) −768789. −1.56550
\(190\) 0 0
\(191\) 392051. 0.777605 0.388803 0.921321i \(-0.372889\pi\)
0.388803 + 0.921321i \(0.372889\pi\)
\(192\) − 109275.i − 0.213928i
\(193\) − 15776.8i − 0.0304878i −0.999884 0.0152439i \(-0.995148\pi\)
0.999884 0.0152439i \(-0.00485247\pi\)
\(194\) −117270. −0.223708
\(195\) 0 0
\(196\) −923627. −1.71734
\(197\) 545551.i 1.00154i 0.865580 + 0.500771i \(0.166950\pi\)
−0.865580 + 0.500771i \(0.833050\pi\)
\(198\) − 10840.4i − 0.0196509i
\(199\) 546514. 0.978293 0.489146 0.872202i \(-0.337308\pi\)
0.489146 + 0.872202i \(0.337308\pi\)
\(200\) 0 0
\(201\) −866065. −1.51203
\(202\) 412610.i 0.711478i
\(203\) − 588300.i − 1.00198i
\(204\) 87076.5 0.146496
\(205\) 0 0
\(206\) 75733.7 0.124343
\(207\) 113444.i 0.184016i
\(208\) − 263950.i − 0.423022i
\(209\) 16382.6 0.0259428
\(210\) 0 0
\(211\) 537150. 0.830596 0.415298 0.909686i \(-0.363677\pi\)
0.415298 + 0.909686i \(0.363677\pi\)
\(212\) − 89856.0i − 0.137312i
\(213\) 273101.i 0.412453i
\(214\) 494983. 0.738850
\(215\) 0 0
\(216\) 445121. 0.649148
\(217\) 238325.i 0.343574i
\(218\) − 358298.i − 0.510626i
\(219\) 171236. 0.241259
\(220\) 0 0
\(221\) 86837.3 0.119598
\(222\) 466259.i 0.634958i
\(223\) 189640.i 0.255368i 0.991815 + 0.127684i \(0.0407544\pi\)
−0.991815 + 0.127684i \(0.959246\pi\)
\(224\) 1.23100e6 1.63922
\(225\) 0 0
\(226\) 203687. 0.265273
\(227\) 363428.i 0.468116i 0.972223 + 0.234058i \(0.0752006\pi\)
−0.972223 + 0.234058i \(0.924799\pi\)
\(228\) − 61836.6i − 0.0787787i
\(229\) −504331. −0.635516 −0.317758 0.948172i \(-0.602930\pi\)
−0.317758 + 0.948172i \(0.602930\pi\)
\(230\) 0 0
\(231\) −459521. −0.566598
\(232\) 340620.i 0.415480i
\(233\) 1.20159e6i 1.45000i 0.688750 + 0.724999i \(0.258160\pi\)
−0.688750 + 0.724999i \(0.741840\pi\)
\(234\) −40805.1 −0.0487163
\(235\) 0 0
\(236\) −310657. −0.363079
\(237\) 1.02453e6i 1.18482i
\(238\) 94964.5i 0.108672i
\(239\) 185929. 0.210549 0.105275 0.994443i \(-0.466428\pi\)
0.105275 + 0.994443i \(0.466428\pi\)
\(240\) 0 0
\(241\) 174842. 0.193911 0.0969556 0.995289i \(-0.469090\pi\)
0.0969556 + 0.995289i \(0.469090\pi\)
\(242\) 32336.0i 0.0354934i
\(243\) 304267.i 0.330552i
\(244\) −765589. −0.823229
\(245\) 0 0
\(246\) 42029.3 0.0442807
\(247\) − 61666.8i − 0.0643145i
\(248\) − 137988.i − 0.142466i
\(249\) −770748. −0.787797
\(250\) 0 0
\(251\) 447906. 0.448748 0.224374 0.974503i \(-0.427966\pi\)
0.224374 + 0.974503i \(0.427966\pi\)
\(252\) 248120.i 0.246128i
\(253\) − 338394.i − 0.332370i
\(254\) −305446. −0.297064
\(255\) 0 0
\(256\) −209768. −0.200050
\(257\) − 1.14572e6i − 1.08204i −0.841009 0.541022i \(-0.818038\pi\)
0.841009 0.541022i \(-0.181962\pi\)
\(258\) 545634.i 0.510331i
\(259\) 2.82736e6 2.61897
\(260\) 0 0
\(261\) −105815. −0.0961496
\(262\) − 120192.i − 0.108174i
\(263\) − 443228.i − 0.395128i −0.980290 0.197564i \(-0.936697\pi\)
0.980290 0.197564i \(-0.0633030\pi\)
\(264\) 266058. 0.234945
\(265\) 0 0
\(266\) 67438.2 0.0584389
\(267\) − 1.38534e6i − 1.18926i
\(268\) − 1.39492e6i − 1.18635i
\(269\) 1.88722e6 1.59016 0.795082 0.606502i \(-0.207428\pi\)
0.795082 + 0.606502i \(0.207428\pi\)
\(270\) 0 0
\(271\) 2.24203e6 1.85446 0.927230 0.374491i \(-0.122183\pi\)
0.927230 + 0.374491i \(0.122183\pi\)
\(272\) 110489.i 0.0905516i
\(273\) 1.72972e6i 1.40465i
\(274\) −89569.6 −0.0720749
\(275\) 0 0
\(276\) −1.27728e6 −1.00929
\(277\) 1.27824e6i 1.00095i 0.865751 + 0.500474i \(0.166841\pi\)
−0.865751 + 0.500474i \(0.833159\pi\)
\(278\) − 310389.i − 0.240876i
\(279\) 42866.7 0.0329693
\(280\) 0 0
\(281\) 549325. 0.415015 0.207508 0.978233i \(-0.433465\pi\)
0.207508 + 0.978233i \(0.433465\pi\)
\(282\) − 627869.i − 0.470161i
\(283\) 135813.i 0.100803i 0.998729 + 0.0504016i \(0.0160501\pi\)
−0.998729 + 0.0504016i \(0.983950\pi\)
\(284\) −439867. −0.323612
\(285\) 0 0
\(286\) 121718. 0.0879914
\(287\) − 254862.i − 0.182642i
\(288\) − 221415.i − 0.157299i
\(289\) 1.38351e6 0.974399
\(290\) 0 0
\(291\) 894120. 0.618961
\(292\) 275799.i 0.189293i
\(293\) 1.76403e6i 1.20043i 0.799839 + 0.600215i \(0.204918\pi\)
−0.799839 + 0.600215i \(0.795082\pi\)
\(294\) −1.26653e6 −0.854567
\(295\) 0 0
\(296\) −1.63701e6 −1.08598
\(297\) − 412476.i − 0.271336i
\(298\) − 388874.i − 0.253669i
\(299\) −1.27377e6 −0.823976
\(300\) 0 0
\(301\) 3.30868e6 2.10493
\(302\) 903846.i 0.570266i
\(303\) − 3.14594e6i − 1.96854i
\(304\) 78462.7 0.0486944
\(305\) 0 0
\(306\) 17080.9 0.0104282
\(307\) − 1.93533e6i − 1.17195i −0.810329 0.585975i \(-0.800712\pi\)
0.810329 0.585975i \(-0.199288\pi\)
\(308\) − 740122.i − 0.444556i
\(309\) −577431. −0.344036
\(310\) 0 0
\(311\) 2.98327e6 1.74901 0.874504 0.485019i \(-0.161187\pi\)
0.874504 + 0.485019i \(0.161187\pi\)
\(312\) − 1.00149e6i − 0.582451i
\(313\) 10701.5i 0.00617426i 0.999995 + 0.00308713i \(0.000982666\pi\)
−0.999995 + 0.00308713i \(0.999017\pi\)
\(314\) −31570.8 −0.0180701
\(315\) 0 0
\(316\) −1.65014e6 −0.929617
\(317\) − 2.43658e6i − 1.36186i −0.732349 0.680929i \(-0.761576\pi\)
0.732349 0.680929i \(-0.238424\pi\)
\(318\) − 123215.i − 0.0683277i
\(319\) 315638. 0.173665
\(320\) 0 0
\(321\) −3.77399e6 −2.04427
\(322\) − 1.39299e6i − 0.748699i
\(323\) 25813.6i 0.0137671i
\(324\) −1.82425e6 −0.965433
\(325\) 0 0
\(326\) −924239. −0.481660
\(327\) 2.73183e6i 1.41281i
\(328\) 147563.i 0.0757342i
\(329\) −3.80734e6 −1.93924
\(330\) 0 0
\(331\) 119576. 0.0599894 0.0299947 0.999550i \(-0.490451\pi\)
0.0299947 + 0.999550i \(0.490451\pi\)
\(332\) − 1.24140e6i − 0.618109i
\(333\) − 508546.i − 0.251316i
\(334\) 308644. 0.151388
\(335\) 0 0
\(336\) −2.20083e6 −1.06350
\(337\) − 2.02195e6i − 0.969830i −0.874561 0.484915i \(-0.838851\pi\)
0.874561 0.484915i \(-0.161149\pi\)
\(338\) 361867.i 0.172289i
\(339\) −1.55301e6 −0.733965
\(340\) 0 0
\(341\) −127868. −0.0595491
\(342\) − 12129.9i − 0.00560778i
\(343\) 3.88971e6i 1.78518i
\(344\) −1.91569e6 −0.872829
\(345\) 0 0
\(346\) −1.51753e6 −0.681471
\(347\) 3.01864e6i 1.34582i 0.739723 + 0.672912i \(0.234957\pi\)
−0.739723 + 0.672912i \(0.765043\pi\)
\(348\) − 1.19139e6i − 0.527358i
\(349\) −2.40399e6 −1.05650 −0.528250 0.849089i \(-0.677152\pi\)
−0.528250 + 0.849089i \(0.677152\pi\)
\(350\) 0 0
\(351\) −1.55263e6 −0.672666
\(352\) 660463.i 0.284113i
\(353\) − 3.62981e6i − 1.55041i −0.631709 0.775206i \(-0.717646\pi\)
0.631709 0.775206i \(-0.282354\pi\)
\(354\) −425989. −0.180672
\(355\) 0 0
\(356\) 2.23128e6 0.933102
\(357\) − 724055.i − 0.300678i
\(358\) − 78397.1i − 0.0323290i
\(359\) −939181. −0.384603 −0.192302 0.981336i \(-0.561595\pi\)
−0.192302 + 0.981336i \(0.561595\pi\)
\(360\) 0 0
\(361\) −2.45777e6 −0.992597
\(362\) 575270.i 0.230728i
\(363\) − 246545.i − 0.0982042i
\(364\) −2.78594e6 −1.10210
\(365\) 0 0
\(366\) −1.04982e6 −0.409647
\(367\) − 2.26697e6i − 0.878577i −0.898346 0.439288i \(-0.855231\pi\)
0.898346 0.439288i \(-0.144769\pi\)
\(368\) − 1.62071e6i − 0.623857i
\(369\) −45841.1 −0.0175263
\(370\) 0 0
\(371\) −747167. −0.281827
\(372\) 482642.i 0.180829i
\(373\) 4.55029e6i 1.69343i 0.532048 + 0.846714i \(0.321423\pi\)
−0.532048 + 0.846714i \(0.678577\pi\)
\(374\) −50951.0 −0.0188353
\(375\) 0 0
\(376\) 2.20442e6 0.804125
\(377\) − 1.18812e6i − 0.430532i
\(378\) − 1.69794e6i − 0.611213i
\(379\) −618788. −0.221281 −0.110641 0.993860i \(-0.535290\pi\)
−0.110641 + 0.993860i \(0.535290\pi\)
\(380\) 0 0
\(381\) 2.32886e6 0.821924
\(382\) 865881.i 0.303599i
\(383\) − 2.23829e6i − 0.779686i −0.920881 0.389843i \(-0.872529\pi\)
0.920881 0.389843i \(-0.127471\pi\)
\(384\) 3.18264e6 1.10144
\(385\) 0 0
\(386\) 34844.5 0.0119033
\(387\) − 595120.i − 0.201989i
\(388\) 1.44010e6i 0.485640i
\(389\) 4.60206e6 1.54198 0.770989 0.636848i \(-0.219762\pi\)
0.770989 + 0.636848i \(0.219762\pi\)
\(390\) 0 0
\(391\) 533199. 0.176379
\(392\) − 4.44671e6i − 1.46158i
\(393\) 916405.i 0.299299i
\(394\) −1.20490e6 −0.391030
\(395\) 0 0
\(396\) −133123. −0.0426595
\(397\) − 4.35532e6i − 1.38690i −0.720506 0.693448i \(-0.756091\pi\)
0.720506 0.693448i \(-0.243909\pi\)
\(398\) 1.20703e6i 0.381952i
\(399\) −514181. −0.161690
\(400\) 0 0
\(401\) 3.62515e6 1.12581 0.562905 0.826522i \(-0.309684\pi\)
0.562905 + 0.826522i \(0.309684\pi\)
\(402\) − 1.91278e6i − 0.590338i
\(403\) 481316.i 0.147628i
\(404\) 5.06697e6 1.54453
\(405\) 0 0
\(406\) 1.29931e6 0.391200
\(407\) 1.51695e6i 0.453927i
\(408\) 419221.i 0.124679i
\(409\) −4.13585e6 −1.22252 −0.611260 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(410\) 0 0
\(411\) 682922. 0.199419
\(412\) − 930032.i − 0.269932i
\(413\) 2.58316e6i 0.745206i
\(414\) −250552. −0.0718450
\(415\) 0 0
\(416\) 2.48609e6 0.704343
\(417\) 2.36655e6i 0.666463i
\(418\) 36182.4i 0.0101288i
\(419\) 2.46691e6 0.686464 0.343232 0.939251i \(-0.388478\pi\)
0.343232 + 0.939251i \(0.388478\pi\)
\(420\) 0 0
\(421\) 3.45258e6 0.949376 0.474688 0.880154i \(-0.342561\pi\)
0.474688 + 0.880154i \(0.342561\pi\)
\(422\) 1.18635e6i 0.324287i
\(423\) 684813.i 0.186089i
\(424\) 432602. 0.116862
\(425\) 0 0
\(426\) −603168. −0.161033
\(427\) 6.36599e6i 1.68965i
\(428\) − 6.07853e6i − 1.60394i
\(429\) −928038. −0.243457
\(430\) 0 0
\(431\) −3.65893e6 −0.948770 −0.474385 0.880318i \(-0.657329\pi\)
−0.474385 + 0.880318i \(0.657329\pi\)
\(432\) − 1.97551e6i − 0.509296i
\(433\) 1.59716e6i 0.409381i 0.978827 + 0.204690i \(0.0656188\pi\)
−0.978827 + 0.204690i \(0.934381\pi\)
\(434\) −526363. −0.134141
\(435\) 0 0
\(436\) −4.39999e6 −1.10850
\(437\) − 378647.i − 0.0948485i
\(438\) 378190.i 0.0941943i
\(439\) 1.58464e6 0.392437 0.196219 0.980560i \(-0.437134\pi\)
0.196219 + 0.980560i \(0.437134\pi\)
\(440\) 0 0
\(441\) 1.38139e6 0.338237
\(442\) 191788.i 0.0466945i
\(443\) − 2.29633e6i − 0.555936i −0.960590 0.277968i \(-0.910339\pi\)
0.960590 0.277968i \(-0.0896610\pi\)
\(444\) 5.72580e6 1.37841
\(445\) 0 0
\(446\) −418837. −0.0997029
\(447\) 2.96496e6i 0.701859i
\(448\) − 1.46349e6i − 0.344504i
\(449\) −3.31569e6 −0.776171 −0.388086 0.921623i \(-0.626864\pi\)
−0.388086 + 0.921623i \(0.626864\pi\)
\(450\) 0 0
\(451\) 136740. 0.0316559
\(452\) − 2.50134e6i − 0.575872i
\(453\) − 6.89136e6i − 1.57783i
\(454\) −802664. −0.182765
\(455\) 0 0
\(456\) 297706. 0.0670464
\(457\) 2.20892e6i 0.494754i 0.968919 + 0.247377i \(0.0795686\pi\)
−0.968919 + 0.247377i \(0.920431\pi\)
\(458\) − 1.11386e6i − 0.248123i
\(459\) 649927. 0.143990
\(460\) 0 0
\(461\) −1.86064e6 −0.407764 −0.203882 0.978995i \(-0.565356\pi\)
−0.203882 + 0.978995i \(0.565356\pi\)
\(462\) − 1.01489e6i − 0.221216i
\(463\) − 1.20592e6i − 0.261437i −0.991420 0.130718i \(-0.958272\pi\)
0.991420 0.130718i \(-0.0417284\pi\)
\(464\) 1.51172e6 0.325969
\(465\) 0 0
\(466\) −2.65383e6 −0.566119
\(467\) 2.29388e6i 0.486719i 0.969936 + 0.243360i \(0.0782495\pi\)
−0.969936 + 0.243360i \(0.921750\pi\)
\(468\) 501098.i 0.105757i
\(469\) −1.15990e7 −2.43493
\(470\) 0 0
\(471\) 240711. 0.0499969
\(472\) − 1.49563e6i − 0.309007i
\(473\) 1.77519e6i 0.364832i
\(474\) −2.26276e6 −0.462587
\(475\) 0 0
\(476\) 1.16619e6 0.235913
\(477\) 134390.i 0.0270441i
\(478\) 410642.i 0.0822041i
\(479\) 7.90892e6 1.57499 0.787496 0.616320i \(-0.211377\pi\)
0.787496 + 0.616320i \(0.211377\pi\)
\(480\) 0 0
\(481\) 5.71007e6 1.12533
\(482\) 386154.i 0.0757083i
\(483\) 1.06208e7i 2.07152i
\(484\) 397095. 0.0770515
\(485\) 0 0
\(486\) −672002. −0.129056
\(487\) 3.48410e6i 0.665684i 0.942983 + 0.332842i \(0.108008\pi\)
−0.942983 + 0.332842i \(0.891992\pi\)
\(488\) − 3.68585e6i − 0.700628i
\(489\) 7.04684e6 1.33267
\(490\) 0 0
\(491\) −8.98096e6 −1.68120 −0.840599 0.541658i \(-0.817797\pi\)
−0.840599 + 0.541658i \(0.817797\pi\)
\(492\) − 516132.i − 0.0961276i
\(493\) 497343.i 0.0921592i
\(494\) 136197. 0.0251101
\(495\) 0 0
\(496\) −612410. −0.111773
\(497\) 3.65756e6i 0.664203i
\(498\) − 1.70227e6i − 0.307577i
\(499\) −6.67736e6 −1.20048 −0.600238 0.799821i \(-0.704928\pi\)
−0.600238 + 0.799821i \(0.704928\pi\)
\(500\) 0 0
\(501\) −2.35325e6 −0.418865
\(502\) 989242.i 0.175204i
\(503\) − 7.58428e6i − 1.33658i −0.743902 0.668289i \(-0.767027\pi\)
0.743902 0.668289i \(-0.232973\pi\)
\(504\) −1.19455e6 −0.209473
\(505\) 0 0
\(506\) 747375. 0.129766
\(507\) − 2.75905e6i − 0.476693i
\(508\) 3.75096e6i 0.644886i
\(509\) 6.02580e6 1.03091 0.515454 0.856917i \(-0.327623\pi\)
0.515454 + 0.856917i \(0.327623\pi\)
\(510\) 0 0
\(511\) 2.29331e6 0.388518
\(512\) 5.58471e6i 0.941511i
\(513\) − 461540.i − 0.0774312i
\(514\) 2.53042e6 0.422460
\(515\) 0 0
\(516\) 6.70054e6 1.10786
\(517\) − 2.04274e6i − 0.336114i
\(518\) 6.24448e6i 1.02252i
\(519\) 1.15704e7 1.88551
\(520\) 0 0
\(521\) −4.58541e6 −0.740088 −0.370044 0.929014i \(-0.620657\pi\)
−0.370044 + 0.929014i \(0.620657\pi\)
\(522\) − 233703.i − 0.0375394i
\(523\) 4.88145e6i 0.780359i 0.920739 + 0.390179i \(0.127587\pi\)
−0.920739 + 0.390179i \(0.872413\pi\)
\(524\) −1.47600e6 −0.234832
\(525\) 0 0
\(526\) 978910. 0.154269
\(527\) − 201478.i − 0.0316010i
\(528\) − 1.18080e6i − 0.184329i
\(529\) −1.38489e6 −0.215168
\(530\) 0 0
\(531\) 464624. 0.0715098
\(532\) − 828160.i − 0.126863i
\(533\) − 514714.i − 0.0784780i
\(534\) 3.05965e6 0.464321
\(535\) 0 0
\(536\) 6.71568e6 1.00967
\(537\) 597738.i 0.0894489i
\(538\) 4.16810e6i 0.620844i
\(539\) −4.12058e6 −0.610923
\(540\) 0 0
\(541\) 6.21940e6 0.913598 0.456799 0.889570i \(-0.348996\pi\)
0.456799 + 0.889570i \(0.348996\pi\)
\(542\) 4.95172e6i 0.724033i
\(543\) − 4.38614e6i − 0.638385i
\(544\) −1.04067e6 −0.150771
\(545\) 0 0
\(546\) −3.82023e6 −0.548414
\(547\) − 9.49047e6i − 1.35619i −0.734976 0.678093i \(-0.762807\pi\)
0.734976 0.678093i \(-0.237193\pi\)
\(548\) 1.09994e6i 0.156465i
\(549\) 1.14503e6 0.162138
\(550\) 0 0
\(551\) 353184. 0.0495589
\(552\) − 6.14935e6i − 0.858976i
\(553\) 1.37212e7i 1.90800i
\(554\) −2.82310e6 −0.390798
\(555\) 0 0
\(556\) −3.81166e6 −0.522910
\(557\) − 2.92907e6i − 0.400029i −0.979793 0.200014i \(-0.935901\pi\)
0.979793 0.200014i \(-0.0640989\pi\)
\(558\) 94675.0i 0.0128721i
\(559\) 6.68213e6 0.904451
\(560\) 0 0
\(561\) 388475. 0.0521141
\(562\) 1.21324e6i 0.162033i
\(563\) 455079.i 0.0605084i 0.999542 + 0.0302542i \(0.00963168\pi\)
−0.999542 + 0.0302542i \(0.990368\pi\)
\(564\) −7.71041e6 −1.02066
\(565\) 0 0
\(566\) −299955. −0.0393563
\(567\) 1.51689e7i 1.98152i
\(568\) − 2.11769e6i − 0.275418i
\(569\) 6.27664e6 0.812730 0.406365 0.913711i \(-0.366796\pi\)
0.406365 + 0.913711i \(0.366796\pi\)
\(570\) 0 0
\(571\) −621794. −0.0798098 −0.0399049 0.999203i \(-0.512706\pi\)
−0.0399049 + 0.999203i \(0.512706\pi\)
\(572\) − 1.49473e6i − 0.191018i
\(573\) − 6.60189e6i − 0.840005i
\(574\) 562886. 0.0713085
\(575\) 0 0
\(576\) −263232. −0.0330585
\(577\) 1.28776e7i 1.61026i 0.593098 + 0.805130i \(0.297905\pi\)
−0.593098 + 0.805130i \(0.702095\pi\)
\(578\) 3.05560e6i 0.380432i
\(579\) −265671. −0.0329343
\(580\) 0 0
\(581\) −1.03224e7 −1.26865
\(582\) 1.97475e6i 0.241659i
\(583\) − 400875.i − 0.0488470i
\(584\) −1.32780e6 −0.161102
\(585\) 0 0
\(586\) −3.89602e6 −0.468681
\(587\) 1.08775e7i 1.30296i 0.758664 + 0.651482i \(0.225852\pi\)
−0.758664 + 0.651482i \(0.774148\pi\)
\(588\) 1.55533e7i 1.85515i
\(589\) −143078. −0.0169935
\(590\) 0 0
\(591\) 9.18673e6 1.08191
\(592\) 7.26529e6i 0.852018i
\(593\) 7.50449e6i 0.876364i 0.898886 + 0.438182i \(0.144378\pi\)
−0.898886 + 0.438182i \(0.855622\pi\)
\(594\) 910990. 0.105937
\(595\) 0 0
\(596\) −4.77548e6 −0.550682
\(597\) − 9.20295e6i − 1.05680i
\(598\) − 2.81325e6i − 0.321703i
\(599\) 7.69438e6 0.876207 0.438104 0.898925i \(-0.355650\pi\)
0.438104 + 0.898925i \(0.355650\pi\)
\(600\) 0 0
\(601\) 3.14770e6 0.355473 0.177737 0.984078i \(-0.443123\pi\)
0.177737 + 0.984078i \(0.443123\pi\)
\(602\) 7.30751e6i 0.821823i
\(603\) 2.08626e6i 0.233655i
\(604\) 1.10995e7 1.23797
\(605\) 0 0
\(606\) 6.94810e6 0.768572
\(607\) − 4.57397e6i − 0.503874i −0.967744 0.251937i \(-0.918932\pi\)
0.967744 0.251937i \(-0.0810676\pi\)
\(608\) 739025.i 0.0810775i
\(609\) −9.90660e6 −1.08238
\(610\) 0 0
\(611\) −7.68923e6 −0.833258
\(612\) − 209758.i − 0.0226381i
\(613\) 1.56075e7i 1.67758i 0.544455 + 0.838790i \(0.316736\pi\)
−0.544455 + 0.838790i \(0.683264\pi\)
\(614\) 4.27436e6 0.457562
\(615\) 0 0
\(616\) 3.56324e6 0.378349
\(617\) − 1.18602e7i − 1.25424i −0.778924 0.627119i \(-0.784234\pi\)
0.778924 0.627119i \(-0.215766\pi\)
\(618\) − 1.27531e6i − 0.134321i
\(619\) −7.91821e6 −0.830616 −0.415308 0.909681i \(-0.636326\pi\)
−0.415308 + 0.909681i \(0.636326\pi\)
\(620\) 0 0
\(621\) −9.53346e6 −0.992023
\(622\) 6.58883e6i 0.682861i
\(623\) − 1.85534e7i − 1.91516i
\(624\) −4.44475e6 −0.456968
\(625\) 0 0
\(626\) −23635.3 −0.00241060
\(627\) − 275872.i − 0.0280246i
\(628\) 387698.i 0.0392278i
\(629\) −2.39022e6 −0.240886
\(630\) 0 0
\(631\) −1.11561e7 −1.11542 −0.557709 0.830037i \(-0.688319\pi\)
−0.557709 + 0.830037i \(0.688319\pi\)
\(632\) − 7.94444e6i − 0.791172i
\(633\) − 9.04527e6i − 0.897248i
\(634\) 5.38140e6 0.531707
\(635\) 0 0
\(636\) −1.51312e6 −0.148330
\(637\) 1.55106e7i 1.51453i
\(638\) 697116.i 0.0678037i
\(639\) 657872. 0.0637367
\(640\) 0 0
\(641\) 7.17389e6 0.689620 0.344810 0.938672i \(-0.387943\pi\)
0.344810 + 0.938672i \(0.387943\pi\)
\(642\) − 8.33521e6i − 0.798139i
\(643\) − 7.14025e6i − 0.681061i −0.940233 0.340531i \(-0.889393\pi\)
0.940233 0.340531i \(-0.110607\pi\)
\(644\) −1.71063e7 −1.62533
\(645\) 0 0
\(646\) −57011.6 −0.00537505
\(647\) 1.56897e7i 1.47351i 0.676159 + 0.736756i \(0.263643\pi\)
−0.676159 + 0.736756i \(0.736357\pi\)
\(648\) − 8.78267e6i − 0.821654i
\(649\) −1.38594e6 −0.129161
\(650\) 0 0
\(651\) 4.01324e6 0.371145
\(652\) 1.13499e7i 1.04562i
\(653\) 5.04236e6i 0.462755i 0.972864 + 0.231378i \(0.0743233\pi\)
−0.972864 + 0.231378i \(0.925677\pi\)
\(654\) −6.03350e6 −0.551601
\(655\) 0 0
\(656\) 654904. 0.0594180
\(657\) − 412489.i − 0.0372820i
\(658\) − 8.40887e6i − 0.757134i
\(659\) 9.10902e6 0.817068 0.408534 0.912743i \(-0.366040\pi\)
0.408534 + 0.912743i \(0.366040\pi\)
\(660\) 0 0
\(661\) 1.31308e7 1.16893 0.584464 0.811420i \(-0.301305\pi\)
0.584464 + 0.811420i \(0.301305\pi\)
\(662\) 264095.i 0.0234215i
\(663\) − 1.46229e6i − 0.129196i
\(664\) 5.97657e6 0.526056
\(665\) 0 0
\(666\) 1.12317e6 0.0981207
\(667\) − 7.29528e6i − 0.634933i
\(668\) − 3.79023e6i − 0.328643i
\(669\) 3.19341e6 0.275861
\(670\) 0 0
\(671\) −3.41552e6 −0.292854
\(672\) − 2.07292e7i − 1.77076i
\(673\) − 1.55171e7i − 1.32061i −0.750999 0.660303i \(-0.770428\pi\)
0.750999 0.660303i \(-0.229572\pi\)
\(674\) 4.46566e6 0.378648
\(675\) 0 0
\(676\) 4.44382e6 0.374016
\(677\) − 1.40356e7i − 1.17695i −0.808515 0.588476i \(-0.799728\pi\)
0.808515 0.588476i \(-0.200272\pi\)
\(678\) − 3.42997e6i − 0.286560i
\(679\) 1.19747e7 0.996758
\(680\) 0 0
\(681\) 6.11990e6 0.505681
\(682\) − 282408.i − 0.0232496i
\(683\) − 5.34969e6i − 0.438810i −0.975634 0.219405i \(-0.929588\pi\)
0.975634 0.219405i \(-0.0704117\pi\)
\(684\) −148958. −0.0121737
\(685\) 0 0
\(686\) −8.59079e6 −0.696984
\(687\) 8.49261e6i 0.686514i
\(688\) 8.50211e6i 0.684787i
\(689\) −1.50896e6 −0.121096
\(690\) 0 0
\(691\) 1.31390e7 1.04681 0.523404 0.852084i \(-0.324662\pi\)
0.523404 + 0.852084i \(0.324662\pi\)
\(692\) 1.86357e7i 1.47938i
\(693\) 1.10694e6i 0.0875569i
\(694\) −6.66695e6 −0.525447
\(695\) 0 0
\(696\) 5.73582e6 0.448820
\(697\) 215458.i 0.0167989i
\(698\) − 5.30944e6i − 0.412487i
\(699\) 2.02341e7 1.56636
\(700\) 0 0
\(701\) −2.49888e7 −1.92066 −0.960330 0.278865i \(-0.910042\pi\)
−0.960330 + 0.278865i \(0.910042\pi\)
\(702\) − 3.42912e6i − 0.262627i
\(703\) 1.69740e6i 0.129537i
\(704\) 785200. 0.0597102
\(705\) 0 0
\(706\) 8.01676e6 0.605323
\(707\) − 4.21327e7i − 3.17008i
\(708\) 5.23127e6i 0.392215i
\(709\) 8.86200e6 0.662089 0.331044 0.943615i \(-0.392599\pi\)
0.331044 + 0.943615i \(0.392599\pi\)
\(710\) 0 0
\(711\) 2.46798e6 0.183092
\(712\) 1.07423e7i 0.794138i
\(713\) 2.95538e6i 0.217716i
\(714\) 1.59914e6 0.117393
\(715\) 0 0
\(716\) −962739. −0.0701820
\(717\) − 3.13093e6i − 0.227445i
\(718\) − 2.07427e6i − 0.150160i
\(719\) −2.58635e7 −1.86580 −0.932901 0.360132i \(-0.882732\pi\)
−0.932901 + 0.360132i \(0.882732\pi\)
\(720\) 0 0
\(721\) −7.73336e6 −0.554026
\(722\) − 5.42820e6i − 0.387537i
\(723\) − 2.94423e6i − 0.209472i
\(724\) 7.06448e6 0.500880
\(725\) 0 0
\(726\) 544518. 0.0383416
\(727\) − 1.71871e7i − 1.20605i −0.797721 0.603026i \(-0.793961\pi\)
0.797721 0.603026i \(-0.206039\pi\)
\(728\) − 1.34126e7i − 0.937963i
\(729\) −1.12207e7 −0.781988
\(730\) 0 0
\(731\) −2.79712e6 −0.193606
\(732\) 1.28920e7i 0.889290i
\(733\) − 1.85650e7i − 1.27625i −0.769932 0.638125i \(-0.779710\pi\)
0.769932 0.638125i \(-0.220290\pi\)
\(734\) 5.00680e6 0.343021
\(735\) 0 0
\(736\) 1.52651e7 1.03874
\(737\) − 6.22315e6i − 0.422028i
\(738\) − 101244.i − 0.00684274i
\(739\) −5.94724e6 −0.400594 −0.200297 0.979735i \(-0.564191\pi\)
−0.200297 + 0.979735i \(0.564191\pi\)
\(740\) 0 0
\(741\) −1.03843e6 −0.0694755
\(742\) − 1.65019e6i − 0.110033i
\(743\) 2.72654e7i 1.81193i 0.423357 + 0.905963i \(0.360852\pi\)
−0.423357 + 0.905963i \(0.639148\pi\)
\(744\) −2.32363e6 −0.153899
\(745\) 0 0
\(746\) −1.00497e7 −0.661161
\(747\) 1.85665e6i 0.121739i
\(748\) 625692.i 0.0408890i
\(749\) −5.05440e7 −3.29204
\(750\) 0 0
\(751\) 1.30069e7 0.841541 0.420770 0.907167i \(-0.361760\pi\)
0.420770 + 0.907167i \(0.361760\pi\)
\(752\) − 9.78351e6i − 0.630885i
\(753\) − 7.54245e6i − 0.484758i
\(754\) 2.62407e6 0.168092
\(755\) 0 0
\(756\) −2.08512e7 −1.32686
\(757\) − 9.22009e6i − 0.584784i −0.956299 0.292392i \(-0.905549\pi\)
0.956299 0.292392i \(-0.0944512\pi\)
\(758\) − 1.36665e6i − 0.0863942i
\(759\) −5.69835e6 −0.359041
\(760\) 0 0
\(761\) 328083. 0.0205363 0.0102682 0.999947i \(-0.496731\pi\)
0.0102682 + 0.999947i \(0.496731\pi\)
\(762\) 5.14351e6i 0.320902i
\(763\) 3.65866e7i 2.27516i
\(764\) 1.06333e7 0.659072
\(765\) 0 0
\(766\) 4.94347e6 0.304411
\(767\) 5.21690e6i 0.320202i
\(768\) 3.53235e6i 0.216103i
\(769\) −2.19214e6 −0.133676 −0.0668380 0.997764i \(-0.521291\pi\)
−0.0668380 + 0.997764i \(0.521291\pi\)
\(770\) 0 0
\(771\) −1.92932e7 −1.16887
\(772\) − 427900.i − 0.0258404i
\(773\) − 2.18539e7i − 1.31547i −0.753249 0.657735i \(-0.771515\pi\)
0.753249 0.657735i \(-0.228485\pi\)
\(774\) 1.31438e6 0.0788619
\(775\) 0 0
\(776\) −6.93323e6 −0.413315
\(777\) − 4.76109e7i − 2.82914i
\(778\) 1.01641e7i 0.602031i
\(779\) 153006. 0.00903367
\(780\) 0 0
\(781\) −1.96238e6 −0.115121
\(782\) 1.17762e6i 0.0688633i
\(783\) − 8.89237e6i − 0.518338i
\(784\) −1.97351e7 −1.14670
\(785\) 0 0
\(786\) −2.02396e6 −0.116855
\(787\) 2.61010e7i 1.50217i 0.660203 + 0.751087i \(0.270470\pi\)
−0.660203 + 0.751087i \(0.729530\pi\)
\(788\) 1.47965e7i 0.848874i
\(789\) −7.46368e6 −0.426835
\(790\) 0 0
\(791\) −2.07990e7 −1.18196
\(792\) − 640907.i − 0.0363063i
\(793\) 1.28566e7i 0.726011i
\(794\) 9.61913e6 0.541483
\(795\) 0 0
\(796\) 1.48226e7 0.829168
\(797\) 1.39846e7i 0.779840i 0.920849 + 0.389920i \(0.127497\pi\)
−0.920849 + 0.389920i \(0.872503\pi\)
\(798\) − 1.13562e6i − 0.0631284i
\(799\) 3.21869e6 0.178366
\(800\) 0 0
\(801\) −3.33714e6 −0.183778
\(802\) 8.00647e6i 0.439547i
\(803\) 1.23042e6i 0.0673388i
\(804\) −2.34895e7 −1.28155
\(805\) 0 0
\(806\) −1.06303e6 −0.0576379
\(807\) − 3.17796e7i − 1.71777i
\(808\) 2.43944e7i 1.31450i
\(809\) −2.70989e7 −1.45573 −0.727865 0.685721i \(-0.759487\pi\)
−0.727865 + 0.685721i \(0.759487\pi\)
\(810\) 0 0
\(811\) 1.99644e7 1.06587 0.532936 0.846156i \(-0.321089\pi\)
0.532936 + 0.846156i \(0.321089\pi\)
\(812\) − 1.59559e7i − 0.849244i
\(813\) − 3.77543e7i − 2.00327i
\(814\) −3.35033e6 −0.177226
\(815\) 0 0
\(816\) 1.86056e6 0.0978180
\(817\) 1.98635e6i 0.104112i
\(818\) − 9.13439e6i − 0.477306i
\(819\) 4.16671e6 0.217062
\(820\) 0 0
\(821\) 3.18829e7 1.65082 0.825410 0.564533i \(-0.190944\pi\)
0.825410 + 0.564533i \(0.190944\pi\)
\(822\) 1.50829e6i 0.0778586i
\(823\) 1.34203e7i 0.690655i 0.938482 + 0.345328i \(0.112232\pi\)
−0.938482 + 0.345328i \(0.887768\pi\)
\(824\) 4.47754e6 0.229732
\(825\) 0 0
\(826\) −5.70515e6 −0.290949
\(827\) 1.19386e7i 0.607002i 0.952831 + 0.303501i \(0.0981556\pi\)
−0.952831 + 0.303501i \(0.901844\pi\)
\(828\) 3.07684e6i 0.155966i
\(829\) −2.59274e7 −1.31031 −0.655153 0.755497i \(-0.727396\pi\)
−0.655153 + 0.755497i \(0.727396\pi\)
\(830\) 0 0
\(831\) 2.15247e7 1.08127
\(832\) − 2.95563e6i − 0.148027i
\(833\) − 6.49269e6i − 0.324199i
\(834\) −5.22675e6 −0.260206
\(835\) 0 0
\(836\) 444330. 0.0219882
\(837\) 3.60237e6i 0.177736i
\(838\) 5.44839e6i 0.268015i
\(839\) 3.21482e7 1.57671 0.788354 0.615222i \(-0.210933\pi\)
0.788354 + 0.615222i \(0.210933\pi\)
\(840\) 0 0
\(841\) −1.37064e7 −0.668244
\(842\) 7.62533e6i 0.370662i
\(843\) − 9.25029e6i − 0.448318i
\(844\) 1.45687e7 0.703985
\(845\) 0 0
\(846\) −1.51247e6 −0.0726544
\(847\) − 3.30191e6i − 0.158145i
\(848\) − 1.91995e6i − 0.0916855i
\(849\) 2.28700e6 0.108892
\(850\) 0 0
\(851\) 3.50610e7 1.65959
\(852\) 7.40708e6i 0.349581i
\(853\) 5.64308e6i 0.265548i 0.991146 + 0.132774i \(0.0423885\pi\)
−0.991146 + 0.132774i \(0.957612\pi\)
\(854\) −1.40599e7 −0.659685
\(855\) 0 0
\(856\) 2.92645e7 1.36507
\(857\) − 1.77067e7i − 0.823543i −0.911287 0.411772i \(-0.864910\pi\)
0.911287 0.411772i \(-0.135090\pi\)
\(858\) − 2.04966e6i − 0.0950524i
\(859\) −1.57119e7 −0.726515 −0.363258 0.931689i \(-0.618336\pi\)
−0.363258 + 0.931689i \(0.618336\pi\)
\(860\) 0 0
\(861\) −4.29172e6 −0.197298
\(862\) − 8.08108e6i − 0.370426i
\(863\) − 245263.i − 0.0112100i −0.999984 0.00560500i \(-0.998216\pi\)
0.999984 0.00560500i \(-0.00178414\pi\)
\(864\) 1.86070e7 0.847991
\(865\) 0 0
\(866\) −3.52747e6 −0.159834
\(867\) − 2.32974e7i − 1.05259i
\(868\) 6.46388e6i 0.291202i
\(869\) −7.36179e6 −0.330700
\(870\) 0 0
\(871\) −2.34250e7 −1.04625
\(872\) − 2.11833e7i − 0.943415i
\(873\) − 2.15384e6i − 0.0956486i
\(874\) 836276. 0.0370315
\(875\) 0 0
\(876\) 4.64428e6 0.204483
\(877\) 1.04352e7i 0.458142i 0.973410 + 0.229071i \(0.0735688\pi\)
−0.973410 + 0.229071i \(0.926431\pi\)
\(878\) 3.49983e6i 0.153218i
\(879\) 2.97051e7 1.29676
\(880\) 0 0
\(881\) −1.10430e7 −0.479344 −0.239672 0.970854i \(-0.577040\pi\)
−0.239672 + 0.970854i \(0.577040\pi\)
\(882\) 3.05093e6i 0.132057i
\(883\) − 5.41498e6i − 0.233720i −0.993148 0.116860i \(-0.962717\pi\)
0.993148 0.116860i \(-0.0372828\pi\)
\(884\) 2.35521e6 0.101368
\(885\) 0 0
\(886\) 5.07166e6 0.217053
\(887\) − 1.52663e7i − 0.651515i −0.945453 0.325757i \(-0.894381\pi\)
0.945453 0.325757i \(-0.105619\pi\)
\(888\) 2.75663e7i 1.17313i
\(889\) 3.11898e7 1.32360
\(890\) 0 0
\(891\) −8.13854e6 −0.343441
\(892\) 5.14343e6i 0.216442i
\(893\) − 2.28573e6i − 0.0959170i
\(894\) −6.54838e6 −0.274025
\(895\) 0 0
\(896\) 4.26242e7 1.77372
\(897\) 2.14496e7i 0.890096i
\(898\) − 7.32300e6i − 0.303039i
\(899\) −2.75664e6 −0.113758
\(900\) 0 0
\(901\) 631648. 0.0259217
\(902\) 302004.i 0.0123594i
\(903\) − 5.57160e7i − 2.27384i
\(904\) 1.20424e7 0.490109
\(905\) 0 0
\(906\) 1.52202e7 0.616027
\(907\) 2.02437e7i 0.817094i 0.912737 + 0.408547i \(0.133964\pi\)
−0.912737 + 0.408547i \(0.866036\pi\)
\(908\) 9.85694e6i 0.396759i
\(909\) −7.57825e6 −0.304200
\(910\) 0 0
\(911\) −1.17158e7 −0.467708 −0.233854 0.972272i \(-0.575134\pi\)
−0.233854 + 0.972272i \(0.575134\pi\)
\(912\) − 1.32126e6i − 0.0526019i
\(913\) − 5.53824e6i − 0.219885i
\(914\) −4.87860e6 −0.193166
\(915\) 0 0
\(916\) −1.36785e7 −0.538642
\(917\) 1.22731e7i 0.481984i
\(918\) 1.43542e6i 0.0562177i
\(919\) 1.95296e7 0.762788 0.381394 0.924413i \(-0.375444\pi\)
0.381394 + 0.924413i \(0.375444\pi\)
\(920\) 0 0
\(921\) −3.25898e7 −1.26599
\(922\) − 4.10939e6i − 0.159202i
\(923\) 7.38673e6i 0.285396i
\(924\) −1.24632e7 −0.480230
\(925\) 0 0
\(926\) 2.66339e6 0.102072
\(927\) 1.39097e6i 0.0531641i
\(928\) 1.42386e7i 0.542747i
\(929\) 4.29425e7 1.63248 0.816240 0.577713i \(-0.196055\pi\)
0.816240 + 0.577713i \(0.196055\pi\)
\(930\) 0 0
\(931\) −4.61073e6 −0.174339
\(932\) 3.25898e7i 1.22897i
\(933\) − 5.02364e7i − 1.88936i
\(934\) −5.06625e6 −0.190029
\(935\) 0 0
\(936\) −2.41248e6 −0.0900067
\(937\) 2.27191e7i 0.845361i 0.906279 + 0.422681i \(0.138911\pi\)
−0.906279 + 0.422681i \(0.861089\pi\)
\(938\) − 2.56174e7i − 0.950665i
\(939\) 180207. 0.00666972
\(940\) 0 0
\(941\) 3.98095e6 0.146559 0.0732795 0.997311i \(-0.476653\pi\)
0.0732795 + 0.997311i \(0.476653\pi\)
\(942\) 531632.i 0.0195202i
\(943\) − 3.16045e6i − 0.115736i
\(944\) −6.63780e6 −0.242434
\(945\) 0 0
\(946\) −3.92067e6 −0.142440
\(947\) − 2.43639e7i − 0.882818i −0.897306 0.441409i \(-0.854479\pi\)
0.897306 0.441409i \(-0.145521\pi\)
\(948\) 2.77874e7i 1.00422i
\(949\) 4.63152e6 0.166939
\(950\) 0 0
\(951\) −4.10304e7 −1.47114
\(952\) 5.61450e6i 0.200779i
\(953\) − 1.39017e7i − 0.495833i −0.968781 0.247916i \(-0.920254\pi\)
0.968781 0.247916i \(-0.0797458\pi\)
\(954\) −296813. −0.0105587
\(955\) 0 0
\(956\) 5.04280e6 0.178454
\(957\) − 5.31515e6i − 0.187601i
\(958\) 1.74676e7i 0.614920i
\(959\) 9.14617e6 0.321139
\(960\) 0 0
\(961\) −2.75124e7 −0.960993
\(962\) 1.26112e7i 0.439358i
\(963\) 9.09116e6i 0.315903i
\(964\) 4.74208e6 0.164353
\(965\) 0 0
\(966\) −2.34570e7 −0.808780
\(967\) 5.16682e7i 1.77688i 0.458998 + 0.888438i \(0.348209\pi\)
−0.458998 + 0.888438i \(0.651791\pi\)
\(968\) 1.91177e6i 0.0655764i
\(969\) 434684. 0.0148718
\(970\) 0 0
\(971\) −1.45794e7 −0.496240 −0.248120 0.968729i \(-0.579813\pi\)
−0.248120 + 0.968729i \(0.579813\pi\)
\(972\) 8.25237e6i 0.280164i
\(973\) 3.16945e7i 1.07325i
\(974\) −7.69495e6 −0.259901
\(975\) 0 0
\(976\) −1.63583e7 −0.549685
\(977\) − 3.09921e6i − 0.103876i −0.998650 0.0519379i \(-0.983460\pi\)
0.998650 0.0519379i \(-0.0165398\pi\)
\(978\) 1.55636e7i 0.520311i
\(979\) 9.95442e6 0.331940
\(980\) 0 0
\(981\) 6.58071e6 0.218323
\(982\) − 1.98353e7i − 0.656386i
\(983\) − 1.53445e7i − 0.506489i −0.967402 0.253245i \(-0.918502\pi\)
0.967402 0.253245i \(-0.0814977\pi\)
\(984\) 2.48486e6 0.0818116
\(985\) 0 0
\(986\) −1.09843e6 −0.0359815
\(987\) 6.41133e7i 2.09486i
\(988\) − 1.67253e6i − 0.0545108i
\(989\) 4.10296e7 1.33385
\(990\) 0 0
\(991\) 1.57747e7 0.510244 0.255122 0.966909i \(-0.417884\pi\)
0.255122 + 0.966909i \(0.417884\pi\)
\(992\) − 5.76818e6i − 0.186106i
\(993\) − 2.01359e6i − 0.0648033i
\(994\) −8.07806e6 −0.259323
\(995\) 0 0
\(996\) −2.09043e7 −0.667710
\(997\) 1.85577e6i 0.0591270i 0.999563 + 0.0295635i \(0.00941173\pi\)
−0.999563 + 0.0295635i \(0.990588\pi\)
\(998\) − 1.47476e7i − 0.468699i
\(999\) 4.27365e7 1.35483
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 275.6.b.b.199.4 6
5.2 odd 4 275.6.a.b.1.2 3
5.3 odd 4 11.6.a.b.1.2 3
5.4 even 2 inner 275.6.b.b.199.3 6
15.8 even 4 99.6.a.g.1.2 3
20.3 even 4 176.6.a.i.1.2 3
35.13 even 4 539.6.a.e.1.2 3
40.3 even 4 704.6.a.t.1.2 3
40.13 odd 4 704.6.a.q.1.2 3
55.43 even 4 121.6.a.d.1.2 3
165.98 odd 4 1089.6.a.r.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.2 3 5.3 odd 4
99.6.a.g.1.2 3 15.8 even 4
121.6.a.d.1.2 3 55.43 even 4
176.6.a.i.1.2 3 20.3 even 4
275.6.a.b.1.2 3 5.2 odd 4
275.6.b.b.199.3 6 5.4 even 2 inner
275.6.b.b.199.4 6 1.1 even 1 trivial
539.6.a.e.1.2 3 35.13 even 4
704.6.a.q.1.2 3 40.13 odd 4
704.6.a.t.1.2 3 40.3 even 4
1089.6.a.r.1.2 3 165.98 odd 4