Properties

Label 275.6.b.b.199.5
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 199.5
Root \(3.64914 + 0.444721i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.6.b.b.199.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.18772i q^{2} +3.48600i q^{3} -35.0388 q^{4} -28.5424 q^{6} +145.071i q^{7} -24.8808i q^{8} +230.848 q^{9} +O(q^{10})\) \(q+8.18772i q^{2} +3.48600i q^{3} -35.0388 q^{4} -28.5424 q^{6} +145.071i q^{7} -24.8808i q^{8} +230.848 q^{9} +121.000 q^{11} -122.145i q^{12} -615.772i q^{13} -1187.80 q^{14} -917.524 q^{16} +1840.68i q^{17} +1890.12i q^{18} -366.633 q^{19} -505.718 q^{21} +990.714i q^{22} +4516.38i q^{23} +86.7344 q^{24} +5041.77 q^{26} +1651.83i q^{27} -5083.12i q^{28} +1717.00 q^{29} -2650.54 q^{31} -8308.62i q^{32} +421.806i q^{33} -15070.9 q^{34} -8088.63 q^{36} +9660.61i q^{37} -3001.89i q^{38} +2146.58 q^{39} -11154.8 q^{41} -4140.68i q^{42} -8368.48i q^{43} -4239.69 q^{44} -36978.9 q^{46} -2221.22i q^{47} -3198.49i q^{48} -4238.64 q^{49} -6416.60 q^{51} +21575.9i q^{52} -23707.9i q^{53} -13524.8 q^{54} +3609.48 q^{56} -1278.08i q^{57} +14058.3i q^{58} -19517.8 q^{59} +20937.3 q^{61} -21701.9i q^{62} +33489.4i q^{63} +38667.9 q^{64} -3453.63 q^{66} -51707.7i q^{67} -64495.1i q^{68} -15744.1 q^{69} -1398.38 q^{71} -5743.67i q^{72} -72466.6i q^{73} -79098.4 q^{74} +12846.4 q^{76} +17553.6i q^{77} +17575.6i q^{78} -64632.2 q^{79} +50337.7 q^{81} -91332.4i q^{82} +96790.3i q^{83} +17719.8 q^{84} +68518.8 q^{86} +5985.47i q^{87} -3010.57i q^{88} +47614.1 q^{89} +89330.7 q^{91} -158249. i q^{92} -9239.79i q^{93} +18186.7 q^{94} +28963.9 q^{96} -38399.6i q^{97} -34704.8i q^{98} +27932.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 168 q^{4} - 412 q^{6} + 14 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.18772i 1.44740i 0.690116 + 0.723699i \(0.257560\pi\)
−0.690116 + 0.723699i \(0.742440\pi\)
\(3\) 3.48600i 0.223627i 0.993729 + 0.111814i \(0.0356659\pi\)
−0.993729 + 0.111814i \(0.964334\pi\)
\(4\) −35.0388 −1.09496
\(5\) 0 0
\(6\) −28.5424 −0.323678
\(7\) 145.071i 1.11902i 0.828825 + 0.559508i \(0.189010\pi\)
−0.828825 + 0.559508i \(0.810990\pi\)
\(8\) − 24.8808i − 0.137448i
\(9\) 230.848 0.949991
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) − 122.145i − 0.244863i
\(13\) − 615.772i − 1.01056i −0.862956 0.505279i \(-0.831390\pi\)
0.862956 0.505279i \(-0.168610\pi\)
\(14\) −1187.80 −1.61966
\(15\) 0 0
\(16\) −917.524 −0.896020
\(17\) 1840.68i 1.54474i 0.635174 + 0.772369i \(0.280929\pi\)
−0.635174 + 0.772369i \(0.719071\pi\)
\(18\) 1890.12i 1.37502i
\(19\) −366.633 −0.232996 −0.116498 0.993191i \(-0.537167\pi\)
−0.116498 + 0.993191i \(0.537167\pi\)
\(20\) 0 0
\(21\) −505.718 −0.250242
\(22\) 990.714i 0.436407i
\(23\) 4516.38i 1.78021i 0.455757 + 0.890104i \(0.349369\pi\)
−0.455757 + 0.890104i \(0.650631\pi\)
\(24\) 86.7344 0.0307371
\(25\) 0 0
\(26\) 5041.77 1.46268
\(27\) 1651.83i 0.436071i
\(28\) − 5083.12i − 1.22528i
\(29\) 1717.00 0.379119 0.189560 0.981869i \(-0.439294\pi\)
0.189560 + 0.981869i \(0.439294\pi\)
\(30\) 0 0
\(31\) −2650.54 −0.495371 −0.247685 0.968841i \(-0.579670\pi\)
−0.247685 + 0.968841i \(0.579670\pi\)
\(32\) − 8308.62i − 1.43435i
\(33\) 421.806i 0.0674261i
\(34\) −15070.9 −2.23585
\(35\) 0 0
\(36\) −8088.63 −1.04020
\(37\) 9660.61i 1.16011i 0.814576 + 0.580057i \(0.196970\pi\)
−0.814576 + 0.580057i \(0.803030\pi\)
\(38\) − 3001.89i − 0.337238i
\(39\) 2146.58 0.225988
\(40\) 0 0
\(41\) −11154.8 −1.03634 −0.518170 0.855278i \(-0.673386\pi\)
−0.518170 + 0.855278i \(0.673386\pi\)
\(42\) − 4140.68i − 0.362200i
\(43\) − 8368.48i − 0.690201i −0.938566 0.345100i \(-0.887845\pi\)
0.938566 0.345100i \(-0.112155\pi\)
\(44\) −4239.69 −0.330144
\(45\) 0 0
\(46\) −36978.9 −2.57667
\(47\) − 2221.22i − 0.146672i −0.997307 0.0733360i \(-0.976635\pi\)
0.997307 0.0733360i \(-0.0233645\pi\)
\(48\) − 3198.49i − 0.200374i
\(49\) −4238.64 −0.252195
\(50\) 0 0
\(51\) −6416.60 −0.345445
\(52\) 21575.9i 1.10652i
\(53\) − 23707.9i − 1.15932i −0.814859 0.579659i \(-0.803186\pi\)
0.814859 0.579659i \(-0.196814\pi\)
\(54\) −13524.8 −0.631168
\(55\) 0 0
\(56\) 3609.48 0.153807
\(57\) − 1278.08i − 0.0521042i
\(58\) 14058.3i 0.548737i
\(59\) −19517.8 −0.729964 −0.364982 0.931015i \(-0.618925\pi\)
−0.364982 + 0.931015i \(0.618925\pi\)
\(60\) 0 0
\(61\) 20937.3 0.720436 0.360218 0.932868i \(-0.382702\pi\)
0.360218 + 0.932868i \(0.382702\pi\)
\(62\) − 21701.9i − 0.716999i
\(63\) 33489.4i 1.06305i
\(64\) 38667.9 1.18005
\(65\) 0 0
\(66\) −3453.63 −0.0975924
\(67\) − 51707.7i − 1.40724i −0.710577 0.703619i \(-0.751566\pi\)
0.710577 0.703619i \(-0.248434\pi\)
\(68\) − 64495.1i − 1.69143i
\(69\) −15744.1 −0.398103
\(70\) 0 0
\(71\) −1398.38 −0.0329216 −0.0164608 0.999865i \(-0.505240\pi\)
−0.0164608 + 0.999865i \(0.505240\pi\)
\(72\) − 5743.67i − 0.130574i
\(73\) − 72466.6i − 1.59159i −0.605567 0.795794i \(-0.707054\pi\)
0.605567 0.795794i \(-0.292946\pi\)
\(74\) −79098.4 −1.67915
\(75\) 0 0
\(76\) 12846.4 0.255122
\(77\) 17553.6i 0.337396i
\(78\) 17575.6i 0.327095i
\(79\) −64632.2 −1.16515 −0.582574 0.812777i \(-0.697955\pi\)
−0.582574 + 0.812777i \(0.697955\pi\)
\(80\) 0 0
\(81\) 50337.7 0.852474
\(82\) − 91332.4i − 1.50000i
\(83\) 96790.3i 1.54219i 0.636723 + 0.771093i \(0.280290\pi\)
−0.636723 + 0.771093i \(0.719710\pi\)
\(84\) 17719.8 0.274006
\(85\) 0 0
\(86\) 68518.8 0.998995
\(87\) 5985.47i 0.0847814i
\(88\) − 3010.57i − 0.0414422i
\(89\) 47614.1 0.637178 0.318589 0.947893i \(-0.396791\pi\)
0.318589 + 0.947893i \(0.396791\pi\)
\(90\) 0 0
\(91\) 89330.7 1.13083
\(92\) − 158249.i − 1.94926i
\(93\) − 9239.79i − 0.110778i
\(94\) 18186.7 0.212293
\(95\) 0 0
\(96\) 28963.9 0.320759
\(97\) − 38399.6i − 0.414378i −0.978301 0.207189i \(-0.933568\pi\)
0.978301 0.207189i \(-0.0664315\pi\)
\(98\) − 34704.8i − 0.365027i
\(99\) 27932.6 0.286433
\(100\) 0 0
\(101\) −41011.2 −0.400036 −0.200018 0.979792i \(-0.564100\pi\)
−0.200018 + 0.979792i \(0.564100\pi\)
\(102\) − 52537.3i − 0.499997i
\(103\) 49634.4i 0.460988i 0.973074 + 0.230494i \(0.0740343\pi\)
−0.973074 + 0.230494i \(0.925966\pi\)
\(104\) −15320.9 −0.138899
\(105\) 0 0
\(106\) 194113. 1.67800
\(107\) − 6791.34i − 0.0573450i −0.999589 0.0286725i \(-0.990872\pi\)
0.999589 0.0286725i \(-0.00912800\pi\)
\(108\) − 57878.3i − 0.477481i
\(109\) −96780.7 −0.780230 −0.390115 0.920766i \(-0.627565\pi\)
−0.390115 + 0.920766i \(0.627565\pi\)
\(110\) 0 0
\(111\) −33676.9 −0.259433
\(112\) − 133106.i − 1.00266i
\(113\) 212938.i 1.56876i 0.620281 + 0.784379i \(0.287018\pi\)
−0.620281 + 0.784379i \(0.712982\pi\)
\(114\) 10464.6 0.0754155
\(115\) 0 0
\(116\) −60161.7 −0.415121
\(117\) − 142149.i − 0.960021i
\(118\) − 159806.i − 1.05655i
\(119\) −267029. −1.72859
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 171428.i 1.04276i
\(123\) − 38885.6i − 0.231754i
\(124\) 92871.8 0.542412
\(125\) 0 0
\(126\) −274202. −1.53866
\(127\) − 90363.9i − 0.497148i −0.968613 0.248574i \(-0.920038\pi\)
0.968613 0.248574i \(-0.0799619\pi\)
\(128\) 50726.1i 0.273657i
\(129\) 29172.5 0.154348
\(130\) 0 0
\(131\) 65299.5 0.332454 0.166227 0.986088i \(-0.446842\pi\)
0.166227 + 0.986088i \(0.446842\pi\)
\(132\) − 14779.6i − 0.0738290i
\(133\) − 53187.9i − 0.260726i
\(134\) 423368. 2.03684
\(135\) 0 0
\(136\) 45797.4 0.212321
\(137\) 5322.74i 0.0242289i 0.999927 + 0.0121145i \(0.00385625\pi\)
−0.999927 + 0.0121145i \(0.996144\pi\)
\(138\) − 128908.i − 0.576214i
\(139\) −89967.1 −0.394954 −0.197477 0.980308i \(-0.563275\pi\)
−0.197477 + 0.980308i \(0.563275\pi\)
\(140\) 0 0
\(141\) 7743.18 0.0327998
\(142\) − 11449.6i − 0.0476506i
\(143\) − 74508.4i − 0.304695i
\(144\) −211808. −0.851211
\(145\) 0 0
\(146\) 593336. 2.30366
\(147\) − 14775.9i − 0.0563977i
\(148\) − 338496.i − 1.27028i
\(149\) −66489.8 −0.245352 −0.122676 0.992447i \(-0.539148\pi\)
−0.122676 + 0.992447i \(0.539148\pi\)
\(150\) 0 0
\(151\) −130866. −0.467074 −0.233537 0.972348i \(-0.575030\pi\)
−0.233537 + 0.972348i \(0.575030\pi\)
\(152\) 9122.12i 0.0320248i
\(153\) 424916.i 1.46749i
\(154\) −143724. −0.488346
\(155\) 0 0
\(156\) −75213.6 −0.247448
\(157\) − 163297.i − 0.528723i −0.964424 0.264362i \(-0.914839\pi\)
0.964424 0.264362i \(-0.0851612\pi\)
\(158\) − 529191.i − 1.68643i
\(159\) 82645.6 0.259255
\(160\) 0 0
\(161\) −655197. −1.99208
\(162\) 412151.i 1.23387i
\(163\) 535758.i 1.57943i 0.613477 + 0.789713i \(0.289771\pi\)
−0.613477 + 0.789713i \(0.710229\pi\)
\(164\) 390851. 1.13475
\(165\) 0 0
\(166\) −792492. −2.23216
\(167\) 553587.i 1.53601i 0.640443 + 0.768005i \(0.278751\pi\)
−0.640443 + 0.768005i \(0.721249\pi\)
\(168\) 12582.7i 0.0343953i
\(169\) −7881.54 −0.0212273
\(170\) 0 0
\(171\) −84636.5 −0.221344
\(172\) 293221.i 0.755744i
\(173\) − 266973.i − 0.678190i −0.940752 0.339095i \(-0.889879\pi\)
0.940752 0.339095i \(-0.110121\pi\)
\(174\) −49007.4 −0.122712
\(175\) 0 0
\(176\) −111020. −0.270160
\(177\) − 68039.2i − 0.163240i
\(178\) 389851.i 0.922250i
\(179\) −3030.33 −0.00706900 −0.00353450 0.999994i \(-0.501125\pi\)
−0.00353450 + 0.999994i \(0.501125\pi\)
\(180\) 0 0
\(181\) 761242. 1.72714 0.863568 0.504233i \(-0.168225\pi\)
0.863568 + 0.504233i \(0.168225\pi\)
\(182\) 731415.i 1.63676i
\(183\) 72987.3i 0.161109i
\(184\) 112371. 0.244686
\(185\) 0 0
\(186\) 75652.8 0.160340
\(187\) 222722.i 0.465756i
\(188\) 77828.9i 0.160600i
\(189\) −239634. −0.487970
\(190\) 0 0
\(191\) −430653. −0.854170 −0.427085 0.904212i \(-0.640459\pi\)
−0.427085 + 0.904212i \(0.640459\pi\)
\(192\) 134796.i 0.263891i
\(193\) 272285.i 0.526175i 0.964772 + 0.263088i \(0.0847409\pi\)
−0.964772 + 0.263088i \(0.915259\pi\)
\(194\) 314405. 0.599770
\(195\) 0 0
\(196\) 148517. 0.276144
\(197\) 574550.i 1.05478i 0.849623 + 0.527390i \(0.176829\pi\)
−0.849623 + 0.527390i \(0.823171\pi\)
\(198\) 228704.i 0.414583i
\(199\) −269926. −0.483183 −0.241592 0.970378i \(-0.577669\pi\)
−0.241592 + 0.970378i \(0.577669\pi\)
\(200\) 0 0
\(201\) 180253. 0.314697
\(202\) − 335788.i − 0.579011i
\(203\) 249088.i 0.424240i
\(204\) 224830. 0.378250
\(205\) 0 0
\(206\) −406393. −0.667234
\(207\) 1.04260e6i 1.69118i
\(208\) 564985.i 0.905480i
\(209\) −44362.6 −0.0702509
\(210\) 0 0
\(211\) −753372. −1.16494 −0.582470 0.812853i \(-0.697913\pi\)
−0.582470 + 0.812853i \(0.697913\pi\)
\(212\) 830695.i 1.26941i
\(213\) − 4874.77i − 0.00736216i
\(214\) 55605.6 0.0830011
\(215\) 0 0
\(216\) 41098.9 0.0599371
\(217\) − 384517.i − 0.554327i
\(218\) − 792414.i − 1.12930i
\(219\) 252619. 0.355922
\(220\) 0 0
\(221\) 1.13344e6 1.56105
\(222\) − 275737.i − 0.375503i
\(223\) 997692.i 1.34349i 0.740783 + 0.671745i \(0.234455\pi\)
−0.740783 + 0.671745i \(0.765545\pi\)
\(224\) 1.20534e6 1.60506
\(225\) 0 0
\(226\) −1.74347e6 −2.27062
\(227\) − 495214.i − 0.637864i −0.947778 0.318932i \(-0.896676\pi\)
0.947778 0.318932i \(-0.103324\pi\)
\(228\) 44782.5i 0.0570521i
\(229\) 221893. 0.279611 0.139806 0.990179i \(-0.455352\pi\)
0.139806 + 0.990179i \(0.455352\pi\)
\(230\) 0 0
\(231\) −61191.9 −0.0754508
\(232\) − 42720.4i − 0.0521093i
\(233\) − 619425.i − 0.747479i −0.927534 0.373739i \(-0.878075\pi\)
0.927534 0.373739i \(-0.121925\pi\)
\(234\) 1.16388e6 1.38953
\(235\) 0 0
\(236\) 683881. 0.799283
\(237\) − 225308.i − 0.260559i
\(238\) − 2.18636e6i − 2.50195i
\(239\) 295471. 0.334595 0.167298 0.985906i \(-0.446496\pi\)
0.167298 + 0.985906i \(0.446496\pi\)
\(240\) 0 0
\(241\) 693153. 0.768753 0.384376 0.923176i \(-0.374416\pi\)
0.384376 + 0.923176i \(0.374416\pi\)
\(242\) 119876.i 0.131582i
\(243\) 576873.i 0.626707i
\(244\) −733616. −0.788850
\(245\) 0 0
\(246\) 318385. 0.335440
\(247\) 225762.i 0.235456i
\(248\) 65947.5i 0.0680878i
\(249\) −337411. −0.344875
\(250\) 0 0
\(251\) 533816. 0.534820 0.267410 0.963583i \(-0.413832\pi\)
0.267410 + 0.963583i \(0.413832\pi\)
\(252\) − 1.17343e6i − 1.16400i
\(253\) 546482.i 0.536753i
\(254\) 739874. 0.719571
\(255\) 0 0
\(256\) 822041. 0.783960
\(257\) − 652296.i − 0.616044i −0.951379 0.308022i \(-0.900333\pi\)
0.951379 0.308022i \(-0.0996670\pi\)
\(258\) 238857.i 0.223402i
\(259\) −1.40148e6 −1.29818
\(260\) 0 0
\(261\) 396366. 0.360160
\(262\) 534654.i 0.481193i
\(263\) 622045.i 0.554540i 0.960792 + 0.277270i \(0.0894296\pi\)
−0.960792 + 0.277270i \(0.910570\pi\)
\(264\) 10494.9 0.00926759
\(265\) 0 0
\(266\) 435488. 0.377374
\(267\) 165983.i 0.142490i
\(268\) 1.81177e6i 1.54087i
\(269\) 482862. 0.406858 0.203429 0.979090i \(-0.434791\pi\)
0.203429 + 0.979090i \(0.434791\pi\)
\(270\) 0 0
\(271\) −1.10678e6 −0.915460 −0.457730 0.889091i \(-0.651337\pi\)
−0.457730 + 0.889091i \(0.651337\pi\)
\(272\) − 1.68887e6i − 1.38412i
\(273\) 311407.i 0.252884i
\(274\) −43581.1 −0.0350689
\(275\) 0 0
\(276\) 551655. 0.435908
\(277\) 639062.i 0.500430i 0.968190 + 0.250215i \(0.0805014\pi\)
−0.968190 + 0.250215i \(0.919499\pi\)
\(278\) − 736626.i − 0.571656i
\(279\) −611872. −0.470598
\(280\) 0 0
\(281\) −257984. −0.194907 −0.0974534 0.995240i \(-0.531070\pi\)
−0.0974534 + 0.995240i \(0.531070\pi\)
\(282\) 63399.0i 0.0474744i
\(283\) − 1.02991e6i − 0.764425i −0.924074 0.382213i \(-0.875162\pi\)
0.924074 0.382213i \(-0.124838\pi\)
\(284\) 48997.7 0.0360479
\(285\) 0 0
\(286\) 610054. 0.441015
\(287\) − 1.61824e6i − 1.15968i
\(288\) − 1.91803e6i − 1.36262i
\(289\) −1.96823e6 −1.38622
\(290\) 0 0
\(291\) 133861. 0.0926662
\(292\) 2.53914e6i 1.74273i
\(293\) − 877712.i − 0.597287i −0.954365 0.298644i \(-0.903466\pi\)
0.954365 0.298644i \(-0.0965342\pi\)
\(294\) 120981. 0.0816299
\(295\) 0 0
\(296\) 240363. 0.159455
\(297\) 199872.i 0.131480i
\(298\) − 544400.i − 0.355122i
\(299\) 2.78106e6 1.79900
\(300\) 0 0
\(301\) 1.21403e6 0.772345
\(302\) − 1.07150e6i − 0.676042i
\(303\) − 142965.i − 0.0894589i
\(304\) 336395. 0.208769
\(305\) 0 0
\(306\) −3.47909e6 −2.12404
\(307\) − 1.30925e6i − 0.792826i −0.918072 0.396413i \(-0.870255\pi\)
0.918072 0.396413i \(-0.129745\pi\)
\(308\) − 615057.i − 0.369436i
\(309\) −173026. −0.103089
\(310\) 0 0
\(311\) 3.35930e6 1.96946 0.984731 0.174083i \(-0.0556962\pi\)
0.984731 + 0.174083i \(0.0556962\pi\)
\(312\) − 53408.6i − 0.0310616i
\(313\) 3.00640e6i 1.73454i 0.497835 + 0.867272i \(0.334129\pi\)
−0.497835 + 0.867272i \(0.665871\pi\)
\(314\) 1.33703e6 0.765273
\(315\) 0 0
\(316\) 2.26464e6 1.27579
\(317\) 2.10147e6i 1.17456i 0.809385 + 0.587279i \(0.199801\pi\)
−0.809385 + 0.587279i \(0.800199\pi\)
\(318\) 676679.i 0.375245i
\(319\) 207757. 0.114309
\(320\) 0 0
\(321\) 23674.6 0.0128239
\(322\) − 5.36457e6i − 2.88333i
\(323\) − 674853.i − 0.359918i
\(324\) −1.76377e6 −0.933426
\(325\) 0 0
\(326\) −4.38663e6 −2.28606
\(327\) − 337378.i − 0.174481i
\(328\) 277540.i 0.142443i
\(329\) 322235. 0.164128
\(330\) 0 0
\(331\) 1.23338e6 0.618766 0.309383 0.950938i \(-0.399878\pi\)
0.309383 + 0.950938i \(0.399878\pi\)
\(332\) − 3.39142e6i − 1.68864i
\(333\) 2.23013e6i 1.10210i
\(334\) −4.53261e6 −2.22322
\(335\) 0 0
\(336\) 464009. 0.224222
\(337\) 679319.i 0.325836i 0.986640 + 0.162918i \(0.0520906\pi\)
−0.986640 + 0.162918i \(0.947909\pi\)
\(338\) − 64531.9i − 0.0307243i
\(339\) −742301. −0.350817
\(340\) 0 0
\(341\) −320715. −0.149360
\(342\) − 692980.i − 0.320373i
\(343\) 1.82331e6i 0.836805i
\(344\) −208214. −0.0948668
\(345\) 0 0
\(346\) 2.18590e6 0.981611
\(347\) 2.67540e6i 1.19279i 0.802690 + 0.596397i \(0.203402\pi\)
−0.802690 + 0.596397i \(0.796598\pi\)
\(348\) − 209724.i − 0.0928324i
\(349\) 2.37636e6 1.04435 0.522177 0.852837i \(-0.325120\pi\)
0.522177 + 0.852837i \(0.325120\pi\)
\(350\) 0 0
\(351\) 1.01715e6 0.440675
\(352\) − 1.00534e6i − 0.432472i
\(353\) − 638696.i − 0.272808i −0.990653 0.136404i \(-0.956445\pi\)
0.990653 0.136404i \(-0.0435545\pi\)
\(354\) 557086. 0.236273
\(355\) 0 0
\(356\) −1.66834e6 −0.697686
\(357\) − 930864.i − 0.386559i
\(358\) − 24811.5i − 0.0102317i
\(359\) −1.50842e6 −0.617712 −0.308856 0.951109i \(-0.599946\pi\)
−0.308856 + 0.951109i \(0.599946\pi\)
\(360\) 0 0
\(361\) −2.34168e6 −0.945713
\(362\) 6.23284e6i 2.49985i
\(363\) 51038.5i 0.0203297i
\(364\) −3.13004e6 −1.23822
\(365\) 0 0
\(366\) −597600. −0.233189
\(367\) 1.77368e6i 0.687403i 0.939079 + 0.343701i \(0.111681\pi\)
−0.939079 + 0.343701i \(0.888319\pi\)
\(368\) − 4.14389e6i − 1.59510i
\(369\) −2.57506e6 −0.984513
\(370\) 0 0
\(371\) 3.43933e6 1.29729
\(372\) 323751.i 0.121298i
\(373\) − 2.27176e6i − 0.845456i −0.906257 0.422728i \(-0.861073\pi\)
0.906257 0.422728i \(-0.138927\pi\)
\(374\) −1.82358e6 −0.674135
\(375\) 0 0
\(376\) −55265.7 −0.0201598
\(377\) − 1.05728e6i − 0.383122i
\(378\) − 1.96205e6i − 0.706287i
\(379\) 4.42409e6 1.58207 0.791035 0.611771i \(-0.209543\pi\)
0.791035 + 0.611771i \(0.209543\pi\)
\(380\) 0 0
\(381\) 315009. 0.111176
\(382\) − 3.52607e6i − 1.23632i
\(383\) − 2.37588e6i − 0.827615i −0.910364 0.413807i \(-0.864199\pi\)
0.910364 0.413807i \(-0.135801\pi\)
\(384\) −176831. −0.0611971
\(385\) 0 0
\(386\) −2.22939e6 −0.761586
\(387\) − 1.93185e6i − 0.655684i
\(388\) 1.34547e6i 0.453728i
\(389\) 2.65905e6 0.890949 0.445474 0.895295i \(-0.353035\pi\)
0.445474 + 0.895295i \(0.353035\pi\)
\(390\) 0 0
\(391\) −8.31319e6 −2.74996
\(392\) 105461.i 0.0346638i
\(393\) 227634.i 0.0743457i
\(394\) −4.70425e6 −1.52669
\(395\) 0 0
\(396\) −978724. −0.313633
\(397\) 2.15712e6i 0.686907i 0.939170 + 0.343453i \(0.111597\pi\)
−0.939170 + 0.343453i \(0.888403\pi\)
\(398\) − 2.21008e6i − 0.699359i
\(399\) 185413. 0.0583054
\(400\) 0 0
\(401\) −2.43031e6 −0.754744 −0.377372 0.926062i \(-0.623172\pi\)
−0.377372 + 0.926062i \(0.623172\pi\)
\(402\) 1.47586e6i 0.455492i
\(403\) 1.63213e6i 0.500601i
\(404\) 1.43698e6 0.438024
\(405\) 0 0
\(406\) −2.03946e6 −0.614045
\(407\) 1.16893e6i 0.349787i
\(408\) 159650.i 0.0474808i
\(409\) −6.12831e6 −1.81148 −0.905738 0.423839i \(-0.860682\pi\)
−0.905738 + 0.423839i \(0.860682\pi\)
\(410\) 0 0
\(411\) −18555.1 −0.00541824
\(412\) − 1.73913e6i − 0.504765i
\(413\) − 2.83147e6i − 0.816841i
\(414\) −8.53649e6 −2.44781
\(415\) 0 0
\(416\) −5.11621e6 −1.44949
\(417\) − 313625.i − 0.0883225i
\(418\) − 363229.i − 0.101681i
\(419\) 375626. 0.104525 0.0522626 0.998633i \(-0.483357\pi\)
0.0522626 + 0.998633i \(0.483357\pi\)
\(420\) 0 0
\(421\) 3.52333e6 0.968831 0.484416 0.874838i \(-0.339032\pi\)
0.484416 + 0.874838i \(0.339032\pi\)
\(422\) − 6.16840e6i − 1.68613i
\(423\) − 512764.i − 0.139337i
\(424\) −589870. −0.159346
\(425\) 0 0
\(426\) 39913.3 0.0106560
\(427\) 3.03739e6i 0.806178i
\(428\) 237960.i 0.0627906i
\(429\) 259736. 0.0681380
\(430\) 0 0
\(431\) 3.15287e6 0.817548 0.408774 0.912636i \(-0.365956\pi\)
0.408774 + 0.912636i \(0.365956\pi\)
\(432\) − 1.51560e6i − 0.390728i
\(433\) − 1.62168e6i − 0.415667i −0.978164 0.207833i \(-0.933359\pi\)
0.978164 0.207833i \(-0.0666412\pi\)
\(434\) 3.14832e6 0.802333
\(435\) 0 0
\(436\) 3.39108e6 0.854322
\(437\) − 1.65586e6i − 0.414781i
\(438\) 2.06837e6i 0.515161i
\(439\) 2.48145e6 0.614533 0.307266 0.951624i \(-0.400586\pi\)
0.307266 + 0.951624i \(0.400586\pi\)
\(440\) 0 0
\(441\) −978482. −0.239583
\(442\) 9.28026e6i 2.25946i
\(443\) − 3.75466e6i − 0.908994i −0.890748 0.454497i \(-0.849819\pi\)
0.890748 0.454497i \(-0.150181\pi\)
\(444\) 1.18000e6 0.284069
\(445\) 0 0
\(446\) −8.16882e6 −1.94456
\(447\) − 231783.i − 0.0548673i
\(448\) 5.60960e6i 1.32049i
\(449\) 4.80916e6 1.12578 0.562890 0.826532i \(-0.309689\pi\)
0.562890 + 0.826532i \(0.309689\pi\)
\(450\) 0 0
\(451\) −1.34973e6 −0.312468
\(452\) − 7.46108e6i − 1.71773i
\(453\) − 456200.i − 0.104450i
\(454\) 4.05467e6 0.923244
\(455\) 0 0
\(456\) −31799.7 −0.00716162
\(457\) 7.09951e6i 1.59015i 0.606512 + 0.795075i \(0.292568\pi\)
−0.606512 + 0.795075i \(0.707432\pi\)
\(458\) 1.81680e6i 0.404709i
\(459\) −3.04049e6 −0.673616
\(460\) 0 0
\(461\) 8.12745e6 1.78116 0.890578 0.454830i \(-0.150300\pi\)
0.890578 + 0.454830i \(0.150300\pi\)
\(462\) − 501022.i − 0.109207i
\(463\) 2.67361e6i 0.579623i 0.957084 + 0.289812i \(0.0935927\pi\)
−0.957084 + 0.289812i \(0.906407\pi\)
\(464\) −1.57539e6 −0.339699
\(465\) 0 0
\(466\) 5.07168e6 1.08190
\(467\) − 4.32733e6i − 0.918180i −0.888390 0.459090i \(-0.848175\pi\)
0.888390 0.459090i \(-0.151825\pi\)
\(468\) 4.98075e6i 1.05119i
\(469\) 7.50129e6 1.57472
\(470\) 0 0
\(471\) 569253. 0.118237
\(472\) 485618.i 0.100332i
\(473\) − 1.01259e6i − 0.208103i
\(474\) 1.84476e6 0.377133
\(475\) 0 0
\(476\) 9.35637e6 1.89274
\(477\) − 5.47291e6i − 1.10134i
\(478\) 2.41923e6i 0.484293i
\(479\) −1.55878e6 −0.310417 −0.155208 0.987882i \(-0.549605\pi\)
−0.155208 + 0.987882i \(0.549605\pi\)
\(480\) 0 0
\(481\) 5.94873e6 1.17236
\(482\) 5.67535e6i 1.11269i
\(483\) − 2.28402e6i − 0.445483i
\(484\) −513003. −0.0995420
\(485\) 0 0
\(486\) −4.72328e6 −0.907095
\(487\) − 7.63818e6i − 1.45938i −0.683779 0.729689i \(-0.739665\pi\)
0.683779 0.729689i \(-0.260335\pi\)
\(488\) − 520935.i − 0.0990225i
\(489\) −1.86765e6 −0.353202
\(490\) 0 0
\(491\) 3.60872e6 0.675537 0.337768 0.941229i \(-0.390328\pi\)
0.337768 + 0.941229i \(0.390328\pi\)
\(492\) 1.36251e6i 0.253761i
\(493\) 3.16045e6i 0.585640i
\(494\) −1.84848e6 −0.340798
\(495\) 0 0
\(496\) 2.43194e6 0.443862
\(497\) − 202865.i − 0.0368397i
\(498\) − 2.76263e6i − 0.499171i
\(499\) 8.46131e6 1.52120 0.760599 0.649221i \(-0.224905\pi\)
0.760599 + 0.649221i \(0.224905\pi\)
\(500\) 0 0
\(501\) −1.92980e6 −0.343494
\(502\) 4.37074e6i 0.774098i
\(503\) 8.28353e6i 1.45981i 0.683550 + 0.729904i \(0.260435\pi\)
−0.683550 + 0.729904i \(0.739565\pi\)
\(504\) 833241. 0.146115
\(505\) 0 0
\(506\) −4.47444e6 −0.776896
\(507\) − 27475.1i − 0.00474700i
\(508\) 3.16624e6i 0.544358i
\(509\) 7.60138e6 1.30046 0.650232 0.759736i \(-0.274672\pi\)
0.650232 + 0.759736i \(0.274672\pi\)
\(510\) 0 0
\(511\) 1.05128e7 1.78101
\(512\) 8.35388e6i 1.40836i
\(513\) − 605618.i − 0.101603i
\(514\) 5.34082e6 0.891661
\(515\) 0 0
\(516\) −1.02217e6 −0.169005
\(517\) − 268768.i − 0.0442233i
\(518\) − 1.14749e7i − 1.87899i
\(519\) 930667. 0.151662
\(520\) 0 0
\(521\) 9.60432e6 1.55015 0.775073 0.631872i \(-0.217713\pi\)
0.775073 + 0.631872i \(0.217713\pi\)
\(522\) 3.24534e6i 0.521295i
\(523\) 9.97831e6i 1.59515i 0.603217 + 0.797577i \(0.293885\pi\)
−0.603217 + 0.797577i \(0.706115\pi\)
\(524\) −2.28802e6 −0.364025
\(525\) 0 0
\(526\) −5.09313e6 −0.802640
\(527\) − 4.87879e6i − 0.765218i
\(528\) − 387018.i − 0.0604151i
\(529\) −1.39613e7 −2.16914
\(530\) 0 0
\(531\) −4.50565e6 −0.693459
\(532\) 1.86364e6i 0.285485i
\(533\) 6.86881e6i 1.04728i
\(534\) −1.35902e6 −0.206240
\(535\) 0 0
\(536\) −1.28653e6 −0.193422
\(537\) − 10563.8i − 0.00158082i
\(538\) 3.95354e6i 0.588885i
\(539\) −512876. −0.0760397
\(540\) 0 0
\(541\) −4.34177e6 −0.637784 −0.318892 0.947791i \(-0.603311\pi\)
−0.318892 + 0.947791i \(0.603311\pi\)
\(542\) − 9.06203e6i − 1.32503i
\(543\) 2.65369e6i 0.386234i
\(544\) 1.52935e7 2.21569
\(545\) 0 0
\(546\) −2.54971e6 −0.366024
\(547\) 1.14668e7i 1.63860i 0.573363 + 0.819302i \(0.305639\pi\)
−0.573363 + 0.819302i \(0.694361\pi\)
\(548\) − 186503.i − 0.0265298i
\(549\) 4.83332e6 0.684407
\(550\) 0 0
\(551\) −629511. −0.0883332
\(552\) 391726.i 0.0547185i
\(553\) − 9.37627e6i − 1.30382i
\(554\) −5.23246e6 −0.724322
\(555\) 0 0
\(556\) 3.15234e6 0.432460
\(557\) − 1.57100e6i − 0.214555i −0.994229 0.107278i \(-0.965787\pi\)
0.994229 0.107278i \(-0.0342133\pi\)
\(558\) − 5.00983e6i − 0.681142i
\(559\) −5.15307e6 −0.697488
\(560\) 0 0
\(561\) −776409. −0.104156
\(562\) − 2.11230e6i − 0.282108i
\(563\) 850908.i 0.113139i 0.998399 + 0.0565694i \(0.0180162\pi\)
−0.998399 + 0.0565694i \(0.981984\pi\)
\(564\) −271312. −0.0359146
\(565\) 0 0
\(566\) 8.43265e6 1.10643
\(567\) 7.30255e6i 0.953931i
\(568\) 34792.9i 0.00452501i
\(569\) 1.19642e7 1.54919 0.774595 0.632458i \(-0.217954\pi\)
0.774595 + 0.632458i \(0.217954\pi\)
\(570\) 0 0
\(571\) −7.97842e6 −1.02406 −0.512032 0.858967i \(-0.671107\pi\)
−0.512032 + 0.858967i \(0.671107\pi\)
\(572\) 2.61068e6i 0.333629i
\(573\) − 1.50126e6i − 0.191016i
\(574\) 1.32497e7 1.67852
\(575\) 0 0
\(576\) 8.92640e6 1.12104
\(577\) 5.90743e6i 0.738685i 0.929293 + 0.369342i \(0.120417\pi\)
−0.929293 + 0.369342i \(0.879583\pi\)
\(578\) − 1.61153e7i − 2.00641i
\(579\) −949186. −0.117667
\(580\) 0 0
\(581\) −1.40415e7 −1.72573
\(582\) 1.09602e6i 0.134125i
\(583\) − 2.86865e6i − 0.349548i
\(584\) −1.80303e6 −0.218761
\(585\) 0 0
\(586\) 7.18646e6 0.864512
\(587\) 1.34766e6i 0.161430i 0.996737 + 0.0807151i \(0.0257204\pi\)
−0.996737 + 0.0807151i \(0.974280\pi\)
\(588\) 517730.i 0.0617533i
\(589\) 971777. 0.115419
\(590\) 0 0
\(591\) −2.00288e6 −0.235877
\(592\) − 8.86385e6i − 1.03948i
\(593\) − 1.05883e7i − 1.23649i −0.785987 0.618243i \(-0.787845\pi\)
0.785987 0.618243i \(-0.212155\pi\)
\(594\) −1.63650e6 −0.190304
\(595\) 0 0
\(596\) 2.32972e6 0.268651
\(597\) − 940962.i − 0.108053i
\(598\) 2.27705e7i 2.60388i
\(599\) −3.48377e6 −0.396718 −0.198359 0.980129i \(-0.563561\pi\)
−0.198359 + 0.980129i \(0.563561\pi\)
\(600\) 0 0
\(601\) −6.41433e6 −0.724378 −0.362189 0.932105i \(-0.617970\pi\)
−0.362189 + 0.932105i \(0.617970\pi\)
\(602\) 9.94010e6i 1.11789i
\(603\) − 1.19366e7i − 1.33686i
\(604\) 4.58540e6 0.511428
\(605\) 0 0
\(606\) 1.17056e6 0.129483
\(607\) − 700912.i − 0.0772132i −0.999254 0.0386066i \(-0.987708\pi\)
0.999254 0.0386066i \(-0.0122919\pi\)
\(608\) 3.04622e6i 0.334197i
\(609\) −868320. −0.0948717
\(610\) 0 0
\(611\) −1.36776e6 −0.148221
\(612\) − 1.48885e7i − 1.60684i
\(613\) 1.17591e7i 1.26393i 0.774996 + 0.631966i \(0.217752\pi\)
−0.774996 + 0.631966i \(0.782248\pi\)
\(614\) 1.07198e7 1.14753
\(615\) 0 0
\(616\) 436747. 0.0463744
\(617\) − 1.00683e6i − 0.106474i −0.998582 0.0532371i \(-0.983046\pi\)
0.998582 0.0532371i \(-0.0169539\pi\)
\(618\) − 1.41669e6i − 0.149212i
\(619\) −1.27458e7 −1.33703 −0.668513 0.743700i \(-0.733069\pi\)
−0.668513 + 0.743700i \(0.733069\pi\)
\(620\) 0 0
\(621\) −7.46031e6 −0.776297
\(622\) 2.75050e7i 2.85060i
\(623\) 6.90744e6i 0.713012i
\(624\) −1.96954e6 −0.202490
\(625\) 0 0
\(626\) −2.46155e7 −2.51058
\(627\) − 154648.i − 0.0157100i
\(628\) 5.72172e6i 0.578932i
\(629\) −1.77821e7 −1.79207
\(630\) 0 0
\(631\) 1.41284e7 1.41260 0.706299 0.707913i \(-0.250363\pi\)
0.706299 + 0.707913i \(0.250363\pi\)
\(632\) 1.60810e6i 0.160148i
\(633\) − 2.62626e6i − 0.260512i
\(634\) −1.72062e7 −1.70005
\(635\) 0 0
\(636\) −2.89580e6 −0.283874
\(637\) 2.61004e6i 0.254858i
\(638\) 1.70106e6i 0.165450i
\(639\) −322814. −0.0312752
\(640\) 0 0
\(641\) −4.36680e6 −0.419777 −0.209888 0.977725i \(-0.567310\pi\)
−0.209888 + 0.977725i \(0.567310\pi\)
\(642\) 193841.i 0.0185613i
\(643\) − 7.81597e6i − 0.745513i −0.927929 0.372757i \(-0.878413\pi\)
0.927929 0.372757i \(-0.121587\pi\)
\(644\) 2.29573e7 2.18125
\(645\) 0 0
\(646\) 5.52551e6 0.520944
\(647\) 2.01624e7i 1.89357i 0.321863 + 0.946786i \(0.395691\pi\)
−0.321863 + 0.946786i \(0.604309\pi\)
\(648\) − 1.25244e6i − 0.117171i
\(649\) −2.36166e6 −0.220092
\(650\) 0 0
\(651\) 1.34043e6 0.123963
\(652\) − 1.87723e7i − 1.72941i
\(653\) − 324619.i − 0.0297914i −0.999889 0.0148957i \(-0.995258\pi\)
0.999889 0.0148957i \(-0.00474163\pi\)
\(654\) 2.76235e6 0.252543
\(655\) 0 0
\(656\) 1.02348e7 0.928581
\(657\) − 1.67288e7i − 1.51199i
\(658\) 2.63837e6i 0.237559i
\(659\) 1.07107e7 0.960740 0.480370 0.877066i \(-0.340502\pi\)
0.480370 + 0.877066i \(0.340502\pi\)
\(660\) 0 0
\(661\) −1.11064e7 −0.988712 −0.494356 0.869260i \(-0.664596\pi\)
−0.494356 + 0.869260i \(0.664596\pi\)
\(662\) 1.00986e7i 0.895601i
\(663\) 3.95116e6i 0.349093i
\(664\) 2.40822e6 0.211971
\(665\) 0 0
\(666\) −1.82597e7 −1.59517
\(667\) 7.75464e6i 0.674912i
\(668\) − 1.93970e7i − 1.68187i
\(669\) −3.47795e6 −0.300441
\(670\) 0 0
\(671\) 2.53341e6 0.217219
\(672\) 4.20182e6i 0.358934i
\(673\) 1.38137e7i 1.17564i 0.808993 + 0.587818i \(0.200013\pi\)
−0.808993 + 0.587818i \(0.799987\pi\)
\(674\) −5.56208e6 −0.471615
\(675\) 0 0
\(676\) 276160. 0.0232431
\(677\) 2.29090e6i 0.192103i 0.995376 + 0.0960514i \(0.0306213\pi\)
−0.995376 + 0.0960514i \(0.969379\pi\)
\(678\) − 6.07775e6i − 0.507772i
\(679\) 5.57067e6 0.463695
\(680\) 0 0
\(681\) 1.72632e6 0.142644
\(682\) − 2.62593e6i − 0.216183i
\(683\) − 4.40512e6i − 0.361332i −0.983545 0.180666i \(-0.942175\pi\)
0.983545 0.180666i \(-0.0578253\pi\)
\(684\) 2.96556e6 0.242363
\(685\) 0 0
\(686\) −1.49287e7 −1.21119
\(687\) 773519.i 0.0625287i
\(688\) 7.67828e6i 0.618434i
\(689\) −1.45986e7 −1.17156
\(690\) 0 0
\(691\) −5.86199e6 −0.467035 −0.233518 0.972353i \(-0.575024\pi\)
−0.233518 + 0.972353i \(0.575024\pi\)
\(692\) 9.35440e6i 0.742592i
\(693\) 4.05221e6i 0.320523i
\(694\) −2.19054e7 −1.72645
\(695\) 0 0
\(696\) 148923. 0.0116530
\(697\) − 2.05324e7i − 1.60087i
\(698\) 1.94569e7i 1.51160i
\(699\) 2.15932e6 0.167157
\(700\) 0 0
\(701\) −8.02106e6 −0.616505 −0.308253 0.951305i \(-0.599744\pi\)
−0.308253 + 0.951305i \(0.599744\pi\)
\(702\) 8.32816e6i 0.637832i
\(703\) − 3.54190e6i − 0.270301i
\(704\) 4.67881e6 0.355799
\(705\) 0 0
\(706\) 5.22946e6 0.394862
\(707\) − 5.94954e6i − 0.447646i
\(708\) 2.38401e6i 0.178741i
\(709\) −2.17891e7 −1.62788 −0.813941 0.580948i \(-0.802682\pi\)
−0.813941 + 0.580948i \(0.802682\pi\)
\(710\) 0 0
\(711\) −1.49202e7 −1.10688
\(712\) − 1.18468e6i − 0.0875789i
\(713\) − 1.19709e7i − 0.881863i
\(714\) 7.62165e6 0.559505
\(715\) 0 0
\(716\) 106179. 0.00774029
\(717\) 1.03001e6i 0.0748246i
\(718\) − 1.23505e7i − 0.894076i
\(719\) −1.03483e7 −0.746531 −0.373266 0.927724i \(-0.621762\pi\)
−0.373266 + 0.927724i \(0.621762\pi\)
\(720\) 0 0
\(721\) −7.20052e6 −0.515853
\(722\) − 1.91730e7i − 1.36882i
\(723\) 2.41633e6i 0.171914i
\(724\) −2.66730e7 −1.89115
\(725\) 0 0
\(726\) −417889. −0.0294252
\(727\) − 2.03348e7i − 1.42693i −0.700690 0.713466i \(-0.747124\pi\)
0.700690 0.713466i \(-0.252876\pi\)
\(728\) − 2.22262e6i − 0.155430i
\(729\) 1.02211e7 0.712325
\(730\) 0 0
\(731\) 1.54037e7 1.06618
\(732\) − 2.55739e6i − 0.176408i
\(733\) − 4.78280e6i − 0.328793i −0.986394 0.164396i \(-0.947432\pi\)
0.986394 0.164396i \(-0.0525676\pi\)
\(734\) −1.45224e7 −0.994946
\(735\) 0 0
\(736\) 3.75249e7 2.55344
\(737\) − 6.25663e6i − 0.424298i
\(738\) − 2.10839e7i − 1.42498i
\(739\) 1.08737e7 0.732429 0.366215 0.930530i \(-0.380654\pi\)
0.366215 + 0.930530i \(0.380654\pi\)
\(740\) 0 0
\(741\) −787008. −0.0526543
\(742\) 2.81603e7i 1.87770i
\(743\) 1.01036e7i 0.671434i 0.941963 + 0.335717i \(0.108979\pi\)
−0.941963 + 0.335717i \(0.891021\pi\)
\(744\) −229893. −0.0152263
\(745\) 0 0
\(746\) 1.86006e7 1.22371
\(747\) 2.23438e7i 1.46506i
\(748\) − 7.80390e6i − 0.509986i
\(749\) 985227. 0.0641700
\(750\) 0 0
\(751\) 9.91947e6 0.641784 0.320892 0.947116i \(-0.396017\pi\)
0.320892 + 0.947116i \(0.396017\pi\)
\(752\) 2.03802e6i 0.131421i
\(753\) 1.86088e6i 0.119600i
\(754\) 8.65673e6 0.554530
\(755\) 0 0
\(756\) 8.39647e6 0.534309
\(757\) − 1.33506e7i − 0.846759i −0.905952 0.423380i \(-0.860844\pi\)
0.905952 0.423380i \(-0.139156\pi\)
\(758\) 3.62232e7i 2.28988i
\(759\) −1.90504e6 −0.120033
\(760\) 0 0
\(761\) 1.28109e7 0.801898 0.400949 0.916100i \(-0.368680\pi\)
0.400949 + 0.916100i \(0.368680\pi\)
\(762\) 2.57920e6i 0.160916i
\(763\) − 1.40401e7i − 0.873089i
\(764\) 1.50896e7 0.935284
\(765\) 0 0
\(766\) 1.94531e7 1.19789
\(767\) 1.20185e7i 0.737671i
\(768\) 2.86564e6i 0.175315i
\(769\) −1.90629e7 −1.16245 −0.581224 0.813743i \(-0.697426\pi\)
−0.581224 + 0.813743i \(0.697426\pi\)
\(770\) 0 0
\(771\) 2.27390e6 0.137764
\(772\) − 9.54054e6i − 0.576142i
\(773\) 2.34434e6i 0.141115i 0.997508 + 0.0705574i \(0.0224778\pi\)
−0.997508 + 0.0705574i \(0.977522\pi\)
\(774\) 1.58174e7 0.949037
\(775\) 0 0
\(776\) −955410. −0.0569555
\(777\) − 4.88555e6i − 0.290309i
\(778\) 2.17716e7i 1.28956i
\(779\) 4.08972e6 0.241463
\(780\) 0 0
\(781\) −169204. −0.00992623
\(782\) − 6.80661e7i − 3.98028i
\(783\) 2.83620e6i 0.165323i
\(784\) 3.88906e6 0.225972
\(785\) 0 0
\(786\) −1.86381e6 −0.107608
\(787\) − 1.52283e7i − 0.876425i −0.898871 0.438212i \(-0.855612\pi\)
0.898871 0.438212i \(-0.144388\pi\)
\(788\) − 2.01315e7i − 1.15494i
\(789\) −2.16845e6 −0.124010
\(790\) 0 0
\(791\) −3.08911e7 −1.75547
\(792\) − 694984.i − 0.0393697i
\(793\) − 1.28926e7i − 0.728042i
\(794\) −1.76619e7 −0.994228
\(795\) 0 0
\(796\) 9.45788e6 0.529068
\(797\) − 2.70618e7i − 1.50907i −0.656257 0.754537i \(-0.727861\pi\)
0.656257 0.754537i \(-0.272139\pi\)
\(798\) 1.51811e6i 0.0843911i
\(799\) 4.08855e6 0.226570
\(800\) 0 0
\(801\) 1.09916e7 0.605313
\(802\) − 1.98987e7i − 1.09242i
\(803\) − 8.76846e6i − 0.479882i
\(804\) −6.31585e6 −0.344581
\(805\) 0 0
\(806\) −1.33634e7 −0.724569
\(807\) 1.68326e6i 0.0909844i
\(808\) 1.02039e6i 0.0549842i
\(809\) 2.25663e6 0.121224 0.0606121 0.998161i \(-0.480695\pi\)
0.0606121 + 0.998161i \(0.480695\pi\)
\(810\) 0 0
\(811\) 4.49121e6 0.239779 0.119890 0.992787i \(-0.461746\pi\)
0.119890 + 0.992787i \(0.461746\pi\)
\(812\) − 8.72773e6i − 0.464527i
\(813\) − 3.85825e6i − 0.204722i
\(814\) −9.57091e6 −0.506282
\(815\) 0 0
\(816\) 5.88739e6 0.309526
\(817\) 3.06816e6i 0.160814i
\(818\) − 5.01769e7i − 2.62193i
\(819\) 2.06218e7 1.07428
\(820\) 0 0
\(821\) 592581. 0.0306825 0.0153412 0.999882i \(-0.495117\pi\)
0.0153412 + 0.999882i \(0.495117\pi\)
\(822\) − 151924.i − 0.00784236i
\(823\) 1.14748e7i 0.590533i 0.955415 + 0.295266i \(0.0954084\pi\)
−0.955415 + 0.295266i \(0.904592\pi\)
\(824\) 1.23494e6 0.0633620
\(825\) 0 0
\(826\) 2.31833e7 1.18229
\(827\) − 8.47060e6i − 0.430676i −0.976540 0.215338i \(-0.930915\pi\)
0.976540 0.215338i \(-0.0690853\pi\)
\(828\) − 3.65313e7i − 1.85178i
\(829\) 1.58876e7 0.802919 0.401460 0.915877i \(-0.368503\pi\)
0.401460 + 0.915877i \(0.368503\pi\)
\(830\) 0 0
\(831\) −2.22777e6 −0.111910
\(832\) − 2.38106e7i − 1.19251i
\(833\) − 7.80197e6i − 0.389576i
\(834\) 2.56788e6 0.127838
\(835\) 0 0
\(836\) 1.55441e6 0.0769221
\(837\) − 4.37825e6i − 0.216017i
\(838\) 3.07552e6i 0.151290i
\(839\) −2.66963e7 −1.30932 −0.654659 0.755924i \(-0.727188\pi\)
−0.654659 + 0.755924i \(0.727188\pi\)
\(840\) 0 0
\(841\) −1.75631e7 −0.856268
\(842\) 2.88480e7i 1.40228i
\(843\) − 899333.i − 0.0435864i
\(844\) 2.63972e7 1.27556
\(845\) 0 0
\(846\) 4.19837e6 0.201676
\(847\) 2.12399e6i 0.101729i
\(848\) 2.17525e7i 1.03877i
\(849\) 3.59028e6 0.170946
\(850\) 0 0
\(851\) −4.36310e7 −2.06524
\(852\) 170806.i 0.00806128i
\(853\) − 3.58773e6i − 0.168829i −0.996431 0.0844146i \(-0.973098\pi\)
0.996431 0.0844146i \(-0.0269020\pi\)
\(854\) −2.48693e7 −1.16686
\(855\) 0 0
\(856\) −168974. −0.00788197
\(857\) − 6.00941e6i − 0.279499i −0.990187 0.139749i \(-0.955370\pi\)
0.990187 0.139749i \(-0.0446297\pi\)
\(858\) 2.12665e6i 0.0986228i
\(859\) 1.74629e7 0.807484 0.403742 0.914873i \(-0.367709\pi\)
0.403742 + 0.914873i \(0.367709\pi\)
\(860\) 0 0
\(861\) 5.64118e6 0.259336
\(862\) 2.58149e7i 1.18332i
\(863\) − 2.34431e6i − 0.107149i −0.998564 0.0535746i \(-0.982939\pi\)
0.998564 0.0535746i \(-0.0170615\pi\)
\(864\) 1.37245e7 0.625476
\(865\) 0 0
\(866\) 1.32779e7 0.601635
\(867\) − 6.86126e6i − 0.309996i
\(868\) 1.34730e7i 0.606968i
\(869\) −7.82050e6 −0.351306
\(870\) 0 0
\(871\) −3.18401e7 −1.42210
\(872\) 2.40798e6i 0.107241i
\(873\) − 8.86445e6i − 0.393655i
\(874\) 1.35577e7 0.600354
\(875\) 0 0
\(876\) −8.85145e6 −0.389721
\(877\) 1.98979e7i 0.873591i 0.899561 + 0.436796i \(0.143887\pi\)
−0.899561 + 0.436796i \(0.856113\pi\)
\(878\) 2.03175e7i 0.889474i
\(879\) 3.05971e6 0.133570
\(880\) 0 0
\(881\) 2.32718e7 1.01016 0.505081 0.863072i \(-0.331463\pi\)
0.505081 + 0.863072i \(0.331463\pi\)
\(882\) − 8.01154e6i − 0.346772i
\(883\) 2.71777e7i 1.17304i 0.809936 + 0.586518i \(0.199502\pi\)
−0.809936 + 0.586518i \(0.800498\pi\)
\(884\) −3.97142e7 −1.70929
\(885\) 0 0
\(886\) 3.07421e7 1.31568
\(887\) − 1.39671e7i − 0.596069i −0.954555 0.298034i \(-0.903669\pi\)
0.954555 0.298034i \(-0.0963310\pi\)
\(888\) 837907.i 0.0356585i
\(889\) 1.31092e7 0.556316
\(890\) 0 0
\(891\) 6.09086e6 0.257030
\(892\) − 3.49579e7i − 1.47107i
\(893\) 814374.i 0.0341740i
\(894\) 1.89778e6 0.0794149
\(895\) 0 0
\(896\) −7.35889e6 −0.306226
\(897\) 9.69477e6i 0.402306i
\(898\) 3.93761e7i 1.62945i
\(899\) −4.55099e6 −0.187805
\(900\) 0 0
\(901\) 4.36385e7 1.79084
\(902\) − 1.10512e7i − 0.452266i
\(903\) 4.23209e6i 0.172717i
\(904\) 5.29805e6 0.215623
\(905\) 0 0
\(906\) 3.73524e6 0.151181
\(907\) 1.77875e7i 0.717954i 0.933346 + 0.358977i \(0.116874\pi\)
−0.933346 + 0.358977i \(0.883126\pi\)
\(908\) 1.73517e7i 0.698437i
\(909\) −9.46734e6 −0.380031
\(910\) 0 0
\(911\) 30398.8 0.00121356 0.000606780 1.00000i \(-0.499807\pi\)
0.000606780 1.00000i \(0.499807\pi\)
\(912\) 1.17267e6i 0.0466864i
\(913\) 1.17116e7i 0.464987i
\(914\) −5.81288e7 −2.30158
\(915\) 0 0
\(916\) −7.77486e6 −0.306164
\(917\) 9.47308e6i 0.372021i
\(918\) − 2.48947e7i − 0.974990i
\(919\) 4.10055e6 0.160160 0.0800798 0.996788i \(-0.474482\pi\)
0.0800798 + 0.996788i \(0.474482\pi\)
\(920\) 0 0
\(921\) 4.56406e6 0.177297
\(922\) 6.65453e7i 2.57804i
\(923\) 861085.i 0.0332692i
\(924\) 2.14409e6 0.0826158
\(925\) 0 0
\(926\) −2.18908e7 −0.838946
\(927\) 1.14580e7i 0.437935i
\(928\) − 1.42659e7i − 0.543788i
\(929\) 1.46532e7 0.557048 0.278524 0.960429i \(-0.410155\pi\)
0.278524 + 0.960429i \(0.410155\pi\)
\(930\) 0 0
\(931\) 1.55403e6 0.0587604
\(932\) 2.17039e7i 0.818461i
\(933\) 1.17105e7i 0.440425i
\(934\) 3.54310e7 1.32897
\(935\) 0 0
\(936\) −3.53679e6 −0.131953
\(937\) − 3.97538e7i − 1.47921i −0.673042 0.739604i \(-0.735013\pi\)
0.673042 0.739604i \(-0.264987\pi\)
\(938\) 6.14185e7i 2.27925i
\(939\) −1.04803e7 −0.387891
\(940\) 0 0
\(941\) 5.32850e6 0.196169 0.0980847 0.995178i \(-0.468728\pi\)
0.0980847 + 0.995178i \(0.468728\pi\)
\(942\) 4.66088e6i 0.171136i
\(943\) − 5.03793e7i − 1.84490i
\(944\) 1.79081e7 0.654062
\(945\) 0 0
\(946\) 8.29077e6 0.301208
\(947\) 3.11430e7i 1.12846i 0.825618 + 0.564230i \(0.190827\pi\)
−0.825618 + 0.564230i \(0.809173\pi\)
\(948\) 7.89452e6i 0.285302i
\(949\) −4.46229e7 −1.60839
\(950\) 0 0
\(951\) −7.32571e6 −0.262663
\(952\) 6.64389e6i 0.237591i
\(953\) − 4.87227e7i − 1.73780i −0.494990 0.868899i \(-0.664828\pi\)
0.494990 0.868899i \(-0.335172\pi\)
\(954\) 4.48106e7 1.59408
\(955\) 0 0
\(956\) −1.03529e7 −0.366369
\(957\) 724242.i 0.0255625i
\(958\) − 1.27628e7i − 0.449297i
\(959\) −772177. −0.0271125
\(960\) 0 0
\(961\) −2.16038e7 −0.754608
\(962\) 4.87065e7i 1.69687i
\(963\) − 1.56776e6i − 0.0544773i
\(964\) −2.42873e7 −0.841755
\(965\) 0 0
\(966\) 1.87009e7 0.644792
\(967\) 4.85436e7i 1.66942i 0.550688 + 0.834711i \(0.314365\pi\)
−0.550688 + 0.834711i \(0.685635\pi\)
\(968\) − 364279.i − 0.0124953i
\(969\) 2.35254e6 0.0804873
\(970\) 0 0
\(971\) −3.15035e7 −1.07229 −0.536143 0.844127i \(-0.680119\pi\)
−0.536143 + 0.844127i \(0.680119\pi\)
\(972\) − 2.02129e7i − 0.686221i
\(973\) − 1.30516e7i − 0.441960i
\(974\) 6.25393e7 2.11230
\(975\) 0 0
\(976\) −1.92104e7 −0.645525
\(977\) − 5.35354e7i − 1.79434i −0.441684 0.897170i \(-0.645619\pi\)
0.441684 0.897170i \(-0.354381\pi\)
\(978\) − 1.52918e7i − 0.511225i
\(979\) 5.76131e6 0.192116
\(980\) 0 0
\(981\) −2.23416e7 −0.741211
\(982\) 2.95472e7i 0.977771i
\(983\) 5.40925e7i 1.78547i 0.450580 + 0.892736i \(0.351217\pi\)
−0.450580 + 0.892736i \(0.648783\pi\)
\(984\) −967505. −0.0318541
\(985\) 0 0
\(986\) −2.58769e7 −0.847655
\(987\) 1.12331e6i 0.0367035i
\(988\) − 7.91044e6i − 0.257815i
\(989\) 3.77952e7 1.22870
\(990\) 0 0
\(991\) 2.31007e7 0.747208 0.373604 0.927588i \(-0.378122\pi\)
0.373604 + 0.927588i \(0.378122\pi\)
\(992\) 2.20223e7i 0.710533i
\(993\) 4.29956e6i 0.138373i
\(994\) 1.66100e6 0.0533218
\(995\) 0 0
\(996\) 1.18225e7 0.377625
\(997\) − 4.54061e7i − 1.44669i −0.690486 0.723346i \(-0.742603\pi\)
0.690486 0.723346i \(-0.257397\pi\)
\(998\) 6.92788e7i 2.20178i
\(999\) −1.59577e7 −0.505891
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.199.5 6
5.2 odd 4 275.6.a.b.1.1 3
5.3 odd 4 11.6.a.b.1.3 3
5.4 even 2 inner 275.6.b.b.199.2 6
15.8 even 4 99.6.a.g.1.1 3
20.3 even 4 176.6.a.i.1.3 3
35.13 even 4 539.6.a.e.1.3 3
40.3 even 4 704.6.a.t.1.1 3
40.13 odd 4 704.6.a.q.1.3 3
55.43 even 4 121.6.a.d.1.1 3
165.98 odd 4 1089.6.a.r.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.3 3 5.3 odd 4
99.6.a.g.1.1 3 15.8 even 4
121.6.a.d.1.1 3 55.43 even 4
176.6.a.i.1.3 3 20.3 even 4
275.6.a.b.1.1 3 5.2 odd 4
275.6.b.b.199.2 6 5.4 even 2 inner
275.6.b.b.199.5 6 1.1 even 1 trivial
539.6.a.e.1.3 3 35.13 even 4
704.6.a.q.1.3 3 40.13 odd 4
704.6.a.t.1.1 3 40.3 even 4
1089.6.a.r.1.3 3 165.98 odd 4