Properties

Label 275.6.b.b.199.1
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 199.1
Root \(0.874840 - 6.07300i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.6.b.b.199.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-10.3963i q^{2} -20.6466i q^{3} -76.0833 q^{4} -214.649 q^{6} +164.454i q^{7} +458.304i q^{8} -183.283 q^{9} +O(q^{10})\) \(q-10.3963i q^{2} -20.6466i q^{3} -76.0833 q^{4} -214.649 q^{6} +164.454i q^{7} +458.304i q^{8} -183.283 q^{9} +121.000 q^{11} +1570.86i q^{12} +585.236i q^{13} +1709.71 q^{14} +2330.01 q^{16} -945.333i q^{17} +1905.47i q^{18} -1148.76 q^{19} +3395.41 q^{21} -1257.95i q^{22} +1346.27i q^{23} +9462.44 q^{24} +6084.30 q^{26} -1232.95i q^{27} -12512.2i q^{28} -899.585 q^{29} -390.700 q^{31} -9557.75i q^{32} -2498.24i q^{33} -9827.97 q^{34} +13944.8 q^{36} -4473.41i q^{37} +11942.9i q^{38} +12083.2 q^{39} +16018.7 q^{41} -35299.8i q^{42} +19905.5i q^{43} -9206.08 q^{44} +13996.2 q^{46} +1871.38i q^{47} -48106.8i q^{48} -10238.0 q^{49} -19517.9 q^{51} -44526.7i q^{52} -23565.1i q^{53} -12818.1 q^{54} -75369.7 q^{56} +23718.0i q^{57} +9352.37i q^{58} +34709.8 q^{59} +25776.2 q^{61} +4061.84i q^{62} -30141.6i q^{63} -24805.1 q^{64} -25972.5 q^{66} +55384.6i q^{67} +71924.1i q^{68} +27795.9 q^{69} +56898.4 q^{71} -83999.6i q^{72} +46871.8i q^{73} -46506.9 q^{74} +87401.5 q^{76} +19898.9i q^{77} -125620. i q^{78} +325.479 q^{79} -69994.1 q^{81} -166536. i q^{82} +92908.3i q^{83} -258334. q^{84} +206943. q^{86} +18573.4i q^{87} +55454.8i q^{88} -23058.0 q^{89} -96244.2 q^{91} -102428. i q^{92} +8066.65i q^{93} +19455.5 q^{94} -197335. q^{96} -5013.44i q^{97} +106437. i q^{98} -22177.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 168 q^{4} - 412 q^{6} + 14 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 10.3963i − 1.83783i −0.394460 0.918913i \(-0.629068\pi\)
0.394460 0.918913i \(-0.370932\pi\)
\(3\) − 20.6466i − 1.32448i −0.749291 0.662241i \(-0.769605\pi\)
0.749291 0.662241i \(-0.230395\pi\)
\(4\) −76.0833 −2.37760
\(5\) 0 0
\(6\) −214.649 −2.43417
\(7\) 164.454i 1.26852i 0.773119 + 0.634261i \(0.218696\pi\)
−0.773119 + 0.634261i \(0.781304\pi\)
\(8\) 458.304i 2.53180i
\(9\) −183.283 −0.754253
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 1570.86i 3.14909i
\(13\) 585.236i 0.960446i 0.877147 + 0.480223i \(0.159444\pi\)
−0.877147 + 0.480223i \(0.840556\pi\)
\(14\) 1709.71 2.33132
\(15\) 0 0
\(16\) 2330.01 2.27540
\(17\) − 945.333i − 0.793345i −0.917960 0.396673i \(-0.870165\pi\)
0.917960 0.396673i \(-0.129835\pi\)
\(18\) 1905.47i 1.38619i
\(19\) −1148.76 −0.730037 −0.365019 0.931000i \(-0.618937\pi\)
−0.365019 + 0.931000i \(0.618937\pi\)
\(20\) 0 0
\(21\) 3395.41 1.68014
\(22\) − 1257.95i − 0.554125i
\(23\) 1346.27i 0.530654i 0.964158 + 0.265327i \(0.0854799\pi\)
−0.964158 + 0.265327i \(0.914520\pi\)
\(24\) 9462.44 3.35332
\(25\) 0 0
\(26\) 6084.30 1.76513
\(27\) − 1232.95i − 0.325488i
\(28\) − 12512.2i − 3.01604i
\(29\) −899.585 −0.198631 −0.0993155 0.995056i \(-0.531665\pi\)
−0.0993155 + 0.995056i \(0.531665\pi\)
\(30\) 0 0
\(31\) −390.700 −0.0730196 −0.0365098 0.999333i \(-0.511624\pi\)
−0.0365098 + 0.999333i \(0.511624\pi\)
\(32\) − 9557.75i − 1.64999i
\(33\) − 2498.24i − 0.399346i
\(34\) −9827.97 −1.45803
\(35\) 0 0
\(36\) 13944.8 1.79332
\(37\) − 4473.41i − 0.537198i −0.963252 0.268599i \(-0.913439\pi\)
0.963252 0.268599i \(-0.0865606\pi\)
\(38\) 11942.9i 1.34168i
\(39\) 12083.2 1.27209
\(40\) 0 0
\(41\) 16018.7 1.48822 0.744111 0.668056i \(-0.232873\pi\)
0.744111 + 0.668056i \(0.232873\pi\)
\(42\) − 35299.8i − 3.08780i
\(43\) 19905.5i 1.64173i 0.571124 + 0.820864i \(0.306508\pi\)
−0.571124 + 0.820864i \(0.693492\pi\)
\(44\) −9206.08 −0.716875
\(45\) 0 0
\(46\) 13996.2 0.975250
\(47\) 1871.38i 0.123571i 0.998089 + 0.0617856i \(0.0196795\pi\)
−0.998089 + 0.0617856i \(0.980320\pi\)
\(48\) − 48106.8i − 3.01372i
\(49\) −10238.0 −0.609149
\(50\) 0 0
\(51\) −19517.9 −1.05077
\(52\) − 44526.7i − 2.28356i
\(53\) − 23565.1i − 1.15234i −0.817330 0.576169i \(-0.804547\pi\)
0.817330 0.576169i \(-0.195453\pi\)
\(54\) −12818.1 −0.598189
\(55\) 0 0
\(56\) −75369.7 −3.21164
\(57\) 23718.0i 0.966922i
\(58\) 9352.37i 0.365049i
\(59\) 34709.8 1.29814 0.649071 0.760727i \(-0.275158\pi\)
0.649071 + 0.760727i \(0.275158\pi\)
\(60\) 0 0
\(61\) 25776.2 0.886940 0.443470 0.896289i \(-0.353747\pi\)
0.443470 + 0.896289i \(0.353747\pi\)
\(62\) 4061.84i 0.134197i
\(63\) − 30141.6i − 0.956787i
\(64\) −24805.1 −0.756993
\(65\) 0 0
\(66\) −25972.5 −0.733929
\(67\) 55384.6i 1.50731i 0.657271 + 0.753655i \(0.271711\pi\)
−0.657271 + 0.753655i \(0.728289\pi\)
\(68\) 71924.1i 1.88626i
\(69\) 27795.9 0.702842
\(70\) 0 0
\(71\) 56898.4 1.33954 0.669768 0.742571i \(-0.266394\pi\)
0.669768 + 0.742571i \(0.266394\pi\)
\(72\) − 83999.6i − 1.90962i
\(73\) 46871.8i 1.02945i 0.857356 + 0.514724i \(0.172106\pi\)
−0.857356 + 0.514724i \(0.827894\pi\)
\(74\) −46506.9 −0.987276
\(75\) 0 0
\(76\) 87401.5 1.73574
\(77\) 19898.9i 0.382474i
\(78\) − 125620.i − 2.33789i
\(79\) 325.479 0.00586753 0.00293377 0.999996i \(-0.499066\pi\)
0.00293377 + 0.999996i \(0.499066\pi\)
\(80\) 0 0
\(81\) −69994.1 −1.18536
\(82\) − 166536.i − 2.73509i
\(83\) 92908.3i 1.48033i 0.672424 + 0.740166i \(0.265253\pi\)
−0.672424 + 0.740166i \(0.734747\pi\)
\(84\) −258334. −3.99470
\(85\) 0 0
\(86\) 206943. 3.01721
\(87\) 18573.4i 0.263083i
\(88\) 55454.8i 0.763365i
\(89\) −23058.0 −0.308565 −0.154283 0.988027i \(-0.549307\pi\)
−0.154283 + 0.988027i \(0.549307\pi\)
\(90\) 0 0
\(91\) −96244.2 −1.21835
\(92\) − 102428.i − 1.26169i
\(93\) 8066.65i 0.0967132i
\(94\) 19455.5 0.227103
\(95\) 0 0
\(96\) −197335. −2.18538
\(97\) − 5013.44i − 0.0541011i −0.999634 0.0270506i \(-0.991388\pi\)
0.999634 0.0270506i \(-0.00861151\pi\)
\(98\) 106437.i 1.11951i
\(99\) −22177.3 −0.227416
\(100\) 0 0
\(101\) 37928.6 0.369968 0.184984 0.982742i \(-0.440777\pi\)
0.184984 + 0.982742i \(0.440777\pi\)
\(102\) 202915.i 1.93114i
\(103\) − 180296.i − 1.67453i −0.546798 0.837265i \(-0.684153\pi\)
0.546798 0.837265i \(-0.315847\pi\)
\(104\) −268216. −2.43165
\(105\) 0 0
\(106\) −244990. −2.11780
\(107\) 92860.5i 0.784100i 0.919944 + 0.392050i \(0.128234\pi\)
−0.919944 + 0.392050i \(0.871766\pi\)
\(108\) 93806.6i 0.773881i
\(109\) −180736. −1.45707 −0.728533 0.685011i \(-0.759797\pi\)
−0.728533 + 0.685011i \(0.759797\pi\)
\(110\) 0 0
\(111\) −92360.8 −0.711509
\(112\) 383178.i 2.88639i
\(113\) 68275.4i 0.503000i 0.967857 + 0.251500i \(0.0809239\pi\)
−0.967857 + 0.251500i \(0.919076\pi\)
\(114\) 246580. 1.77703
\(115\) 0 0
\(116\) 68443.4 0.472266
\(117\) − 107264.i − 0.724419i
\(118\) − 360854.i − 2.38576i
\(119\) 155463. 1.00638
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) − 267977.i − 1.63004i
\(123\) − 330732.i − 1.97112i
\(124\) 29725.8 0.173612
\(125\) 0 0
\(126\) −313362. −1.75841
\(127\) 27233.1i 0.149826i 0.997190 + 0.0749130i \(0.0238679\pi\)
−0.997190 + 0.0749130i \(0.976132\pi\)
\(128\) − 47966.0i − 0.258767i
\(129\) 410981. 2.17444
\(130\) 0 0
\(131\) −11887.1 −0.0605199 −0.0302600 0.999542i \(-0.509634\pi\)
−0.0302600 + 0.999542i \(0.509634\pi\)
\(132\) 190075.i 0.949488i
\(133\) − 188918.i − 0.926069i
\(134\) 575796. 2.77017
\(135\) 0 0
\(136\) 433250. 2.00859
\(137\) 35302.2i 0.160694i 0.996767 + 0.0803471i \(0.0256029\pi\)
−0.996767 + 0.0803471i \(0.974397\pi\)
\(138\) − 288975.i − 1.29170i
\(139\) −26248.0 −0.115228 −0.0576141 0.998339i \(-0.518349\pi\)
−0.0576141 + 0.998339i \(0.518349\pi\)
\(140\) 0 0
\(141\) 38637.7 0.163668
\(142\) − 591533.i − 2.46183i
\(143\) 70813.6i 0.289585i
\(144\) −427052. −1.71623
\(145\) 0 0
\(146\) 487294. 1.89195
\(147\) 211379.i 0.806807i
\(148\) 340352.i 1.27724i
\(149\) 226321. 0.835139 0.417570 0.908645i \(-0.362882\pi\)
0.417570 + 0.908645i \(0.362882\pi\)
\(150\) 0 0
\(151\) −301067. −1.07453 −0.537267 0.843412i \(-0.680543\pi\)
−0.537267 + 0.843412i \(0.680543\pi\)
\(152\) − 526481.i − 1.84831i
\(153\) 173264.i 0.598383i
\(154\) 206875. 0.702920
\(155\) 0 0
\(156\) −919327. −3.02453
\(157\) 341482.i 1.10565i 0.833296 + 0.552827i \(0.186451\pi\)
−0.833296 + 0.552827i \(0.813549\pi\)
\(158\) − 3383.78i − 0.0107835i
\(159\) −486541. −1.52625
\(160\) 0 0
\(161\) −221398. −0.673147
\(162\) 727680.i 2.17848i
\(163\) − 604612.i − 1.78241i −0.453600 0.891205i \(-0.649861\pi\)
0.453600 0.891205i \(-0.350139\pi\)
\(164\) −1.21876e6 −3.53840
\(165\) 0 0
\(166\) 965904. 2.72059
\(167\) − 159824.i − 0.443455i −0.975109 0.221728i \(-0.928830\pi\)
0.975109 0.221728i \(-0.0711695\pi\)
\(168\) 1.55613e6i 4.25376i
\(169\) 28791.6 0.0775442
\(170\) 0 0
\(171\) 210549. 0.550633
\(172\) − 1.51447e6i − 3.90338i
\(173\) 499771.i 1.26957i 0.772690 + 0.634783i \(0.218911\pi\)
−0.772690 + 0.634783i \(0.781089\pi\)
\(174\) 193095. 0.483501
\(175\) 0 0
\(176\) 281931. 0.686058
\(177\) − 716641.i − 1.71937i
\(178\) 239719.i 0.567090i
\(179\) 626569. 1.46163 0.730813 0.682578i \(-0.239141\pi\)
0.730813 + 0.682578i \(0.239141\pi\)
\(180\) 0 0
\(181\) 393700. 0.893243 0.446621 0.894723i \(-0.352627\pi\)
0.446621 + 0.894723i \(0.352627\pi\)
\(182\) 1.00058e6i 2.23911i
\(183\) − 532192.i − 1.17474i
\(184\) −617000. −1.34351
\(185\) 0 0
\(186\) 83863.4 0.177742
\(187\) − 114385.i − 0.239203i
\(188\) − 142381.i − 0.293804i
\(189\) 202762. 0.412888
\(190\) 0 0
\(191\) 205468. 0.407531 0.203766 0.979020i \(-0.434682\pi\)
0.203766 + 0.979020i \(0.434682\pi\)
\(192\) 512143.i 1.00262i
\(193\) 349786.i 0.675941i 0.941157 + 0.337971i \(0.109740\pi\)
−0.941157 + 0.337971i \(0.890260\pi\)
\(194\) −52121.3 −0.0994285
\(195\) 0 0
\(196\) 778938. 1.44831
\(197\) 863902.i 1.58598i 0.609232 + 0.792992i \(0.291478\pi\)
−0.609232 + 0.792992i \(0.708522\pi\)
\(198\) 230562.i 0.417951i
\(199\) 610140. 1.09219 0.546093 0.837725i \(-0.316115\pi\)
0.546093 + 0.837725i \(0.316115\pi\)
\(200\) 0 0
\(201\) 1.14351e6 1.99640
\(202\) − 394318.i − 0.679936i
\(203\) − 147940.i − 0.251968i
\(204\) 1.48499e6 2.49832
\(205\) 0 0
\(206\) −1.87441e6 −3.07749
\(207\) − 246749.i − 0.400248i
\(208\) 1.36360e6i 2.18540i
\(209\) −139000. −0.220115
\(210\) 0 0
\(211\) 166602. 0.257616 0.128808 0.991670i \(-0.458885\pi\)
0.128808 + 0.991670i \(0.458885\pi\)
\(212\) 1.79291e6i 2.73981i
\(213\) − 1.17476e6i − 1.77419i
\(214\) 965407. 1.44104
\(215\) 0 0
\(216\) 565064. 0.824068
\(217\) − 64252.0i − 0.0926270i
\(218\) 1.87899e6i 2.67783i
\(219\) 967746. 1.36349
\(220\) 0 0
\(221\) 553243. 0.761965
\(222\) 960212.i 1.30763i
\(223\) 1.05575e6i 1.42167i 0.703358 + 0.710836i \(0.251683\pi\)
−0.703358 + 0.710836i \(0.748317\pi\)
\(224\) 1.57181e6 2.09305
\(225\) 0 0
\(226\) 709812. 0.924427
\(227\) 526562.i 0.678242i 0.940743 + 0.339121i \(0.110130\pi\)
−0.940743 + 0.339121i \(0.889870\pi\)
\(228\) − 1.80455e6i − 2.29896i
\(229\) −1.11694e6 −1.40748 −0.703740 0.710458i \(-0.748488\pi\)
−0.703740 + 0.710458i \(0.748488\pi\)
\(230\) 0 0
\(231\) 410845. 0.506580
\(232\) − 412283.i − 0.502893i
\(233\) 29262.0i 0.0353113i 0.999844 + 0.0176557i \(0.00562026\pi\)
−0.999844 + 0.0176557i \(0.994380\pi\)
\(234\) −1.11515e6 −1.33136
\(235\) 0 0
\(236\) −2.64084e6 −3.08647
\(237\) − 6720.05i − 0.00777144i
\(238\) − 1.61625e6i − 1.84954i
\(239\) −822476. −0.931384 −0.465692 0.884947i \(-0.654194\pi\)
−0.465692 + 0.884947i \(0.654194\pi\)
\(240\) 0 0
\(241\) 762439. 0.845595 0.422797 0.906224i \(-0.361048\pi\)
0.422797 + 0.906224i \(0.361048\pi\)
\(242\) − 152212.i − 0.167075i
\(243\) 1.14554e6i 1.24449i
\(244\) −1.96114e6 −2.10879
\(245\) 0 0
\(246\) −3.43840e6 −3.62258
\(247\) − 672296.i − 0.701161i
\(248\) − 179060.i − 0.184871i
\(249\) 1.91824e6 1.96067
\(250\) 0 0
\(251\) 561364. 0.562419 0.281209 0.959646i \(-0.409264\pi\)
0.281209 + 0.959646i \(0.409264\pi\)
\(252\) 2.29327e6i 2.27486i
\(253\) 162898.i 0.159998i
\(254\) 283123. 0.275354
\(255\) 0 0
\(256\) −1.29243e6 −1.23256
\(257\) − 764965.i − 0.722451i −0.932478 0.361226i \(-0.882358\pi\)
0.932478 0.361226i \(-0.117642\pi\)
\(258\) − 4.27269e6i − 3.99624i
\(259\) 735668. 0.681447
\(260\) 0 0
\(261\) 164879. 0.149818
\(262\) 123582.i 0.111225i
\(263\) 763627.i 0.680756i 0.940289 + 0.340378i \(0.110555\pi\)
−0.940289 + 0.340378i \(0.889445\pi\)
\(264\) 1.14495e6 1.01106
\(265\) 0 0
\(266\) −1.96405e6 −1.70195
\(267\) 476071.i 0.408689i
\(268\) − 4.21385e6i − 3.58378i
\(269\) −800885. −0.674823 −0.337411 0.941357i \(-0.609551\pi\)
−0.337411 + 0.941357i \(0.609551\pi\)
\(270\) 0 0
\(271\) 98139.7 0.0811749 0.0405874 0.999176i \(-0.487077\pi\)
0.0405874 + 0.999176i \(0.487077\pi\)
\(272\) − 2.20263e6i − 1.80518i
\(273\) 1.98712e6i 1.61368i
\(274\) 367013. 0.295328
\(275\) 0 0
\(276\) −2.11480e6 −1.67108
\(277\) − 620993.i − 0.486281i −0.969991 0.243140i \(-0.921822\pi\)
0.969991 0.243140i \(-0.0781776\pi\)
\(278\) 272882.i 0.211769i
\(279\) 71608.9 0.0550753
\(280\) 0 0
\(281\) 1.31191e6 0.991149 0.495575 0.868565i \(-0.334957\pi\)
0.495575 + 0.868565i \(0.334957\pi\)
\(282\) − 401690.i − 0.300793i
\(283\) 26897.8i 0.0199641i 0.999950 + 0.00998205i \(0.00317744\pi\)
−0.999950 + 0.00998205i \(0.996823\pi\)
\(284\) −4.32902e6 −3.18488
\(285\) 0 0
\(286\) 736200. 0.532207
\(287\) 2.63433e6i 1.88784i
\(288\) 1.75178e6i 1.24451i
\(289\) 526203. 0.370603
\(290\) 0 0
\(291\) −103511. −0.0716560
\(292\) − 3.56617e6i − 2.44762i
\(293\) − 638546.i − 0.434534i −0.976112 0.217267i \(-0.930286\pi\)
0.976112 0.217267i \(-0.0697142\pi\)
\(294\) 2.19757e6 1.48277
\(295\) 0 0
\(296\) 2.05018e6 1.36008
\(297\) − 149186.i − 0.0981382i
\(298\) − 2.35290e6i − 1.53484i
\(299\) −787884. −0.509665
\(300\) 0 0
\(301\) −3.27352e6 −2.08257
\(302\) 3.12998e6i 1.97481i
\(303\) − 783099.i − 0.490016i
\(304\) −2.67662e6 −1.66113
\(305\) 0 0
\(306\) 1.80131e6 1.09972
\(307\) 550428.i 0.333315i 0.986015 + 0.166658i \(0.0532974\pi\)
−0.986015 + 0.166658i \(0.946703\pi\)
\(308\) − 1.51397e6i − 0.909371i
\(309\) −3.72250e6 −2.21788
\(310\) 0 0
\(311\) 186775. 0.109501 0.0547504 0.998500i \(-0.482564\pi\)
0.0547504 + 0.998500i \(0.482564\pi\)
\(312\) 5.53776e6i 3.22068i
\(313\) 934239.i 0.539010i 0.962999 + 0.269505i \(0.0868601\pi\)
−0.962999 + 0.269505i \(0.913140\pi\)
\(314\) 3.55016e6 2.03200
\(315\) 0 0
\(316\) −24763.5 −0.0139507
\(317\) − 1.88280e6i − 1.05234i −0.850379 0.526170i \(-0.823628\pi\)
0.850379 0.526170i \(-0.176372\pi\)
\(318\) 5.05823e6i 2.80499i
\(319\) −108850. −0.0598895
\(320\) 0 0
\(321\) 1.91726e6 1.03853
\(322\) 2.30173e6i 1.23713i
\(323\) 1.08596e6i 0.579172i
\(324\) 5.32538e6 2.81831
\(325\) 0 0
\(326\) −6.28574e6 −3.27576
\(327\) 3.73160e6i 1.92986i
\(328\) 7.34144e6i 3.76788i
\(329\) −307755. −0.156753
\(330\) 0 0
\(331\) 197056. 0.0988596 0.0494298 0.998778i \(-0.484260\pi\)
0.0494298 + 0.998778i \(0.484260\pi\)
\(332\) − 7.06877e6i − 3.51964i
\(333\) 819901.i 0.405183i
\(334\) −1.66158e6 −0.814993
\(335\) 0 0
\(336\) 7.91133e6 3.82298
\(337\) 387484.i 0.185857i 0.995673 + 0.0929285i \(0.0296228\pi\)
−0.995673 + 0.0929285i \(0.970377\pi\)
\(338\) − 299327.i − 0.142513i
\(339\) 1.40966e6 0.666215
\(340\) 0 0
\(341\) −47274.7 −0.0220162
\(342\) − 2.18893e6i − 1.01197i
\(343\) 1.08030e6i 0.495803i
\(344\) −9.12276e6 −4.15652
\(345\) 0 0
\(346\) 5.19577e6 2.33324
\(347\) − 2.94793e6i − 1.31430i −0.753761 0.657148i \(-0.771762\pi\)
0.753761 0.657148i \(-0.228238\pi\)
\(348\) − 1.41313e6i − 0.625508i
\(349\) 924908. 0.406476 0.203238 0.979129i \(-0.434854\pi\)
0.203238 + 0.979129i \(0.434854\pi\)
\(350\) 0 0
\(351\) 721564. 0.312613
\(352\) − 1.15649e6i − 0.497490i
\(353\) 4.46816e6i 1.90850i 0.299012 + 0.954249i \(0.403343\pi\)
−0.299012 + 0.954249i \(0.596657\pi\)
\(354\) −7.45043e6 −3.15990
\(355\) 0 0
\(356\) 1.75433e6 0.733647
\(357\) − 3.20979e6i − 1.33293i
\(358\) − 6.51401e6i − 2.68621i
\(359\) 995937. 0.407846 0.203923 0.978987i \(-0.434631\pi\)
0.203923 + 0.978987i \(0.434631\pi\)
\(360\) 0 0
\(361\) −1.15645e6 −0.467045
\(362\) − 4.09303e6i − 1.64162i
\(363\) − 302287.i − 0.120407i
\(364\) 7.32258e6 2.89675
\(365\) 0 0
\(366\) −5.53283e6 −2.15896
\(367\) − 1.21088e6i − 0.469284i −0.972082 0.234642i \(-0.924608\pi\)
0.972082 0.234642i \(-0.0753919\pi\)
\(368\) 3.13681e6i 1.20745i
\(369\) −2.93596e6 −1.12250
\(370\) 0 0
\(371\) 3.87537e6 1.46177
\(372\) − 613737.i − 0.229946i
\(373\) − 1.82235e6i − 0.678203i −0.940750 0.339102i \(-0.889877\pi\)
0.940750 0.339102i \(-0.110123\pi\)
\(374\) −1.18918e6 −0.439613
\(375\) 0 0
\(376\) −857662. −0.312857
\(377\) − 526470.i − 0.190774i
\(378\) − 2.10798e6i − 0.758817i
\(379\) 419357. 0.149964 0.0749819 0.997185i \(-0.476110\pi\)
0.0749819 + 0.997185i \(0.476110\pi\)
\(380\) 0 0
\(381\) 562271. 0.198442
\(382\) − 2.13611e6i − 0.748972i
\(383\) − 2.95656e6i − 1.02989i −0.857224 0.514943i \(-0.827813\pi\)
0.857224 0.514943i \(-0.172187\pi\)
\(384\) −990336. −0.342732
\(385\) 0 0
\(386\) 3.63648e6 1.24226
\(387\) − 3.64834e6i − 1.23828i
\(388\) 381439.i 0.128631i
\(389\) −2.35429e6 −0.788834 −0.394417 0.918932i \(-0.629053\pi\)
−0.394417 + 0.918932i \(0.629053\pi\)
\(390\) 0 0
\(391\) 1.27267e6 0.420992
\(392\) − 4.69210e6i − 1.54224i
\(393\) 245429.i 0.0801576i
\(394\) 8.98139e6 2.91476
\(395\) 0 0
\(396\) 1.68732e6 0.540705
\(397\) 3.94809e6i 1.25722i 0.777722 + 0.628609i \(0.216375\pi\)
−0.777722 + 0.628609i \(0.783625\pi\)
\(398\) − 6.34320e6i − 2.00725i
\(399\) −3.90051e6 −1.22656
\(400\) 0 0
\(401\) −5.76535e6 −1.79046 −0.895230 0.445604i \(-0.852989\pi\)
−0.895230 + 0.445604i \(0.852989\pi\)
\(402\) − 1.18883e7i − 3.66904i
\(403\) − 228652.i − 0.0701314i
\(404\) −2.88574e6 −0.879637
\(405\) 0 0
\(406\) −1.53803e6 −0.463073
\(407\) − 541282.i − 0.161971i
\(408\) − 8.94515e6i − 2.66034i
\(409\) −2.39693e6 −0.708512 −0.354256 0.935148i \(-0.615266\pi\)
−0.354256 + 0.935148i \(0.615266\pi\)
\(410\) 0 0
\(411\) 728872. 0.212837
\(412\) 1.37175e7i 3.98137i
\(413\) 5.70815e6i 1.64672i
\(414\) −2.56527e6 −0.735585
\(415\) 0 0
\(416\) 5.59354e6 1.58472
\(417\) 541932.i 0.152618i
\(418\) 1.44509e6i 0.404532i
\(419\) 1.41668e6 0.394220 0.197110 0.980381i \(-0.436844\pi\)
0.197110 + 0.980381i \(0.436844\pi\)
\(420\) 0 0
\(421\) −4.80538e6 −1.32136 −0.660682 0.750666i \(-0.729733\pi\)
−0.660682 + 0.750666i \(0.729733\pi\)
\(422\) − 1.73204e6i − 0.473454i
\(423\) − 342993.i − 0.0932040i
\(424\) 1.08000e7 2.91749
\(425\) 0 0
\(426\) −1.22132e7 −3.26065
\(427\) 4.23899e6i 1.12510i
\(428\) − 7.06514e6i − 1.86428i
\(429\) 1.46206e6 0.383551
\(430\) 0 0
\(431\) −2.73465e6 −0.709103 −0.354551 0.935037i \(-0.615366\pi\)
−0.354551 + 0.935037i \(0.615366\pi\)
\(432\) − 2.87277e6i − 0.740613i
\(433\) − 2.71922e6i − 0.696986i −0.937311 0.348493i \(-0.886693\pi\)
0.937311 0.348493i \(-0.113307\pi\)
\(434\) −667984. −0.170232
\(435\) 0 0
\(436\) 1.37510e7 3.46433
\(437\) − 1.54654e6i − 0.387397i
\(438\) − 1.00610e7i − 2.50585i
\(439\) 4.17101e6 1.03295 0.516476 0.856301i \(-0.327243\pi\)
0.516476 + 0.856301i \(0.327243\pi\)
\(440\) 0 0
\(441\) 1.87645e6 0.459452
\(442\) − 5.75169e6i − 1.40036i
\(443\) − 6.86870e6i − 1.66290i −0.555603 0.831448i \(-0.687513\pi\)
0.555603 0.831448i \(-0.312487\pi\)
\(444\) 7.02712e6 1.69169
\(445\) 0 0
\(446\) 1.09759e7 2.61278
\(447\) − 4.67276e6i − 1.10613i
\(448\) − 4.07929e6i − 0.960262i
\(449\) 693812. 0.162415 0.0812075 0.996697i \(-0.474122\pi\)
0.0812075 + 0.996697i \(0.474122\pi\)
\(450\) 0 0
\(451\) 1.93826e6 0.448716
\(452\) − 5.19462e6i − 1.19594i
\(453\) 6.21601e6i 1.42320i
\(454\) 5.47431e6 1.24649
\(455\) 0 0
\(456\) −1.08701e7 −2.44805
\(457\) 8.12461e6i 1.81975i 0.414880 + 0.909876i \(0.363824\pi\)
−0.414880 + 0.909876i \(0.636176\pi\)
\(458\) 1.16121e7i 2.58670i
\(459\) −1.16554e6 −0.258224
\(460\) 0 0
\(461\) −4.48975e6 −0.983944 −0.491972 0.870611i \(-0.663724\pi\)
−0.491972 + 0.870611i \(0.663724\pi\)
\(462\) − 4.27127e6i − 0.931006i
\(463\) 9.04494e6i 1.96089i 0.196793 + 0.980445i \(0.436947\pi\)
−0.196793 + 0.980445i \(0.563053\pi\)
\(464\) −2.09604e6 −0.451965
\(465\) 0 0
\(466\) 304217. 0.0648961
\(467\) 7.17275e6i 1.52192i 0.648796 + 0.760962i \(0.275273\pi\)
−0.648796 + 0.760962i \(0.724727\pi\)
\(468\) 8.16101e6i 1.72238i
\(469\) −9.10820e6 −1.91206
\(470\) 0 0
\(471\) 7.05046e6 1.46442
\(472\) 1.59077e7i 3.28663i
\(473\) 2.40856e6i 0.495000i
\(474\) −69863.7 −0.0142826
\(475\) 0 0
\(476\) −1.18282e7 −2.39276
\(477\) 4.31910e6i 0.869155i
\(478\) 8.55072e6i 1.71172i
\(479\) 1.51089e6 0.300881 0.150440 0.988619i \(-0.451931\pi\)
0.150440 + 0.988619i \(0.451931\pi\)
\(480\) 0 0
\(481\) 2.61800e6 0.515949
\(482\) − 7.92655e6i − 1.55406i
\(483\) 4.57113e6i 0.891571i
\(484\) −1.11394e6 −0.216146
\(485\) 0 0
\(486\) 1.19093e7 2.28716
\(487\) 1.63265e6i 0.311940i 0.987762 + 0.155970i \(0.0498504\pi\)
−0.987762 + 0.155970i \(0.950150\pi\)
\(488\) 1.18133e7i 2.24555i
\(489\) −1.24832e7 −2.36077
\(490\) 0 0
\(491\) 1.24459e6 0.232982 0.116491 0.993192i \(-0.462835\pi\)
0.116491 + 0.993192i \(0.462835\pi\)
\(492\) 2.51632e7i 4.68655i
\(493\) 850407.i 0.157583i
\(494\) −6.98940e6 −1.28861
\(495\) 0 0
\(496\) −910334. −0.166149
\(497\) 9.35714e6i 1.69923i
\(498\) − 1.99427e7i − 3.60338i
\(499\) 7.66211e6 1.37752 0.688759 0.724991i \(-0.258156\pi\)
0.688759 + 0.724991i \(0.258156\pi\)
\(500\) 0 0
\(501\) −3.29982e6 −0.587348
\(502\) − 5.83611e6i − 1.03363i
\(503\) 1.07030e7i 1.88619i 0.332518 + 0.943097i \(0.392102\pi\)
−0.332518 + 0.943097i \(0.607898\pi\)
\(504\) 1.38140e7 2.42239
\(505\) 0 0
\(506\) 1.69354e6 0.294049
\(507\) − 594450.i − 0.102706i
\(508\) − 2.07198e6i − 0.356227i
\(509\) −6.27494e6 −1.07353 −0.536766 0.843731i \(-0.680354\pi\)
−0.536766 + 0.843731i \(0.680354\pi\)
\(510\) 0 0
\(511\) −7.70824e6 −1.30588
\(512\) 1.19016e7i 2.00647i
\(513\) 1.41636e6i 0.237618i
\(514\) −7.95281e6 −1.32774
\(515\) 0 0
\(516\) −3.12688e7 −5.16996
\(517\) 226437.i 0.0372581i
\(518\) − 7.64823e6i − 1.25238i
\(519\) 1.03186e7 1.68152
\(520\) 0 0
\(521\) 3.60326e6 0.581570 0.290785 0.956788i \(-0.406084\pi\)
0.290785 + 0.956788i \(0.406084\pi\)
\(522\) − 1.71413e6i − 0.275340i
\(523\) − 3.56925e6i − 0.570589i −0.958440 0.285294i \(-0.907909\pi\)
0.958440 0.285294i \(-0.0920913\pi\)
\(524\) 904412. 0.143892
\(525\) 0 0
\(526\) 7.93890e6 1.25111
\(527\) 369342.i 0.0579298i
\(528\) − 5.82092e6i − 0.908672i
\(529\) 4.62391e6 0.718406
\(530\) 0 0
\(531\) −6.36174e6 −0.979128
\(532\) 1.43735e7i 2.20183i
\(533\) 9.37473e6i 1.42936i
\(534\) 4.94938e6 0.751100
\(535\) 0 0
\(536\) −2.53830e7 −3.81620
\(537\) − 1.29365e7i − 1.93590i
\(538\) 8.32625e6i 1.24021i
\(539\) −1.23879e6 −0.183665
\(540\) 0 0
\(541\) 1.16842e7 1.71635 0.858176 0.513355i \(-0.171598\pi\)
0.858176 + 0.513355i \(0.171598\pi\)
\(542\) − 1.02029e6i − 0.149185i
\(543\) − 8.12859e6i − 1.18308i
\(544\) −9.03525e6 −1.30901
\(545\) 0 0
\(546\) 2.06587e7 2.96566
\(547\) − 4.05598e6i − 0.579600i −0.957087 0.289800i \(-0.906411\pi\)
0.957087 0.289800i \(-0.0935887\pi\)
\(548\) − 2.68591e6i − 0.382067i
\(549\) −4.72435e6 −0.668977
\(550\) 0 0
\(551\) 1.03341e6 0.145008
\(552\) 1.27390e7i 1.77945i
\(553\) 53526.2i 0.00744309i
\(554\) −6.45603e6 −0.893699
\(555\) 0 0
\(556\) 1.99703e6 0.273967
\(557\) − 284385.i − 0.0388390i −0.999811 0.0194195i \(-0.993818\pi\)
0.999811 0.0194195i \(-0.00618181\pi\)
\(558\) − 744469.i − 0.101219i
\(559\) −1.16494e7 −1.57679
\(560\) 0 0
\(561\) −2.36167e6 −0.316820
\(562\) − 1.36391e7i − 1.82156i
\(563\) − 1.20582e6i − 0.160329i −0.996782 0.0801646i \(-0.974455\pi\)
0.996782 0.0801646i \(-0.0255446\pi\)
\(564\) −2.93969e6 −0.389138
\(565\) 0 0
\(566\) 279637. 0.0366906
\(567\) − 1.15108e7i − 1.50365i
\(568\) 2.60768e7i 3.39143i
\(569\) −3.94580e6 −0.510922 −0.255461 0.966819i \(-0.582227\pi\)
−0.255461 + 0.966819i \(0.582227\pi\)
\(570\) 0 0
\(571\) 5.88346e6 0.755166 0.377583 0.925976i \(-0.376755\pi\)
0.377583 + 0.925976i \(0.376755\pi\)
\(572\) − 5.38773e6i − 0.688519i
\(573\) − 4.24222e6i − 0.539768i
\(574\) 2.73874e7 3.46953
\(575\) 0 0
\(576\) 4.54637e6 0.570964
\(577\) 3.61005e6i 0.451413i 0.974195 + 0.225706i \(0.0724690\pi\)
−0.974195 + 0.225706i \(0.927531\pi\)
\(578\) − 5.47057e6i − 0.681104i
\(579\) 7.22190e6 0.895272
\(580\) 0 0
\(581\) −1.52791e7 −1.87783
\(582\) 1.07613e6i 0.131691i
\(583\) − 2.85138e6i − 0.347443i
\(584\) −2.14816e7 −2.60636
\(585\) 0 0
\(586\) −6.63853e6 −0.798597
\(587\) 5.01469e6i 0.600688i 0.953831 + 0.300344i \(0.0971015\pi\)
−0.953831 + 0.300344i \(0.902899\pi\)
\(588\) − 1.60825e7i − 1.91827i
\(589\) 448821. 0.0533070
\(590\) 0 0
\(591\) 1.78367e7 2.10061
\(592\) − 1.04231e7i − 1.22234i
\(593\) − 1.64451e7i − 1.92044i −0.279244 0.960220i \(-0.590084\pi\)
0.279244 0.960220i \(-0.409916\pi\)
\(594\) −1.55099e6 −0.180361
\(595\) 0 0
\(596\) −1.72192e7 −1.98563
\(597\) − 1.25973e7i − 1.44658i
\(598\) 8.19109e6i 0.936675i
\(599\) −6.45089e6 −0.734603 −0.367302 0.930102i \(-0.619718\pi\)
−0.367302 + 0.930102i \(0.619718\pi\)
\(600\) 0 0
\(601\) −8.32443e6 −0.940087 −0.470044 0.882643i \(-0.655762\pi\)
−0.470044 + 0.882643i \(0.655762\pi\)
\(602\) 3.40326e7i 3.82740i
\(603\) − 1.01511e7i − 1.13689i
\(604\) 2.29062e7 2.55482
\(605\) 0 0
\(606\) −8.14134e6 −0.900563
\(607\) 1.47290e7i 1.62256i 0.584659 + 0.811279i \(0.301228\pi\)
−0.584659 + 0.811279i \(0.698772\pi\)
\(608\) 1.09796e7i 1.20455i
\(609\) −3.05446e6 −0.333727
\(610\) 0 0
\(611\) −1.09520e6 −0.118683
\(612\) − 1.31825e7i − 1.42272i
\(613\) 1.14564e7i 1.23139i 0.787983 + 0.615697i \(0.211126\pi\)
−0.787983 + 0.615697i \(0.788874\pi\)
\(614\) 5.72243e6 0.612575
\(615\) 0 0
\(616\) −9.11974e6 −0.968346
\(617\) − 413797.i − 0.0437597i −0.999761 0.0218799i \(-0.993035\pi\)
0.999761 0.0218799i \(-0.00696513\pi\)
\(618\) 3.87003e7i 4.07609i
\(619\) 1.25898e7 1.32067 0.660333 0.750973i \(-0.270415\pi\)
0.660333 + 0.750973i \(0.270415\pi\)
\(620\) 0 0
\(621\) 1.65987e6 0.172721
\(622\) − 1.94177e6i − 0.201243i
\(623\) − 3.79198e6i − 0.391422i
\(624\) 2.81538e7 2.89452
\(625\) 0 0
\(626\) 9.71264e6 0.990607
\(627\) 2.86988e6i 0.291538i
\(628\) − 2.59811e7i − 2.62881i
\(629\) −4.22886e6 −0.426183
\(630\) 0 0
\(631\) −1.55648e7 −1.55621 −0.778107 0.628132i \(-0.783820\pi\)
−0.778107 + 0.628132i \(0.783820\pi\)
\(632\) 149168.i 0.0148554i
\(633\) − 3.43976e6i − 0.341208i
\(634\) −1.95742e7 −1.93402
\(635\) 0 0
\(636\) 3.70176e7 3.62882
\(637\) − 5.99163e6i − 0.585054i
\(638\) 1.13164e6i 0.110067i
\(639\) −1.04285e7 −1.01035
\(640\) 0 0
\(641\) −1.23075e7 −1.18311 −0.591556 0.806264i \(-0.701486\pi\)
−0.591556 + 0.806264i \(0.701486\pi\)
\(642\) − 1.99324e7i − 1.90863i
\(643\) 1.44400e6i 0.137733i 0.997626 + 0.0688667i \(0.0219383\pi\)
−0.997626 + 0.0688667i \(0.978062\pi\)
\(644\) 1.68447e7 1.60048
\(645\) 0 0
\(646\) 1.12900e7 1.06442
\(647\) − 7.35610e6i − 0.690855i −0.938445 0.345427i \(-0.887734\pi\)
0.938445 0.345427i \(-0.112266\pi\)
\(648\) − 3.20786e7i − 3.00108i
\(649\) 4.19989e6 0.391405
\(650\) 0 0
\(651\) −1.32659e6 −0.122683
\(652\) 4.60009e7i 4.23787i
\(653\) − 3.83734e6i − 0.352166i −0.984375 0.176083i \(-0.943657\pi\)
0.984375 0.176083i \(-0.0563427\pi\)
\(654\) 3.87948e7 3.54674
\(655\) 0 0
\(656\) 3.73237e7 3.38630
\(657\) − 8.59083e6i − 0.776465i
\(658\) 3.19952e6i 0.288085i
\(659\) 1.98049e7 1.77648 0.888239 0.459382i \(-0.151929\pi\)
0.888239 + 0.459382i \(0.151929\pi\)
\(660\) 0 0
\(661\) 1.75724e7 1.56433 0.782164 0.623073i \(-0.214116\pi\)
0.782164 + 0.623073i \(0.214116\pi\)
\(662\) − 2.04865e6i − 0.181687i
\(663\) − 1.14226e7i − 1.00921i
\(664\) −4.25803e7 −3.74790
\(665\) 0 0
\(666\) 8.52395e6 0.744656
\(667\) − 1.21108e6i − 0.105404i
\(668\) 1.21599e7i 1.05436i
\(669\) 2.17977e7 1.88298
\(670\) 0 0
\(671\) 3.11892e6 0.267423
\(672\) − 3.24525e7i − 2.77220i
\(673\) − 4.88569e6i − 0.415804i −0.978150 0.207902i \(-0.933337\pi\)
0.978150 0.207902i \(-0.0666634\pi\)
\(674\) 4.02840e6 0.341573
\(675\) 0 0
\(676\) −2.19056e6 −0.184369
\(677\) − 1.56799e7i − 1.31483i −0.753528 0.657416i \(-0.771649\pi\)
0.753528 0.657416i \(-0.228351\pi\)
\(678\) − 1.46552e7i − 1.22439i
\(679\) 824478. 0.0686285
\(680\) 0 0
\(681\) 1.08717e7 0.898320
\(682\) 491483.i 0.0404620i
\(683\) 1.94703e6i 0.159705i 0.996807 + 0.0798527i \(0.0254450\pi\)
−0.996807 + 0.0798527i \(0.974555\pi\)
\(684\) −1.60192e7 −1.30919
\(685\) 0 0
\(686\) 1.12311e7 0.911200
\(687\) 2.30611e7i 1.86418i
\(688\) 4.63799e7i 3.73558i
\(689\) 1.37912e7 1.10676
\(690\) 0 0
\(691\) −5.39805e6 −0.430073 −0.215036 0.976606i \(-0.568987\pi\)
−0.215036 + 0.976606i \(0.568987\pi\)
\(692\) − 3.80242e7i − 3.01853i
\(693\) − 3.64714e6i − 0.288482i
\(694\) −3.06476e7 −2.41545
\(695\) 0 0
\(696\) −8.51227e6 −0.666073
\(697\) − 1.51430e7i − 1.18067i
\(698\) − 9.61563e6i − 0.747032i
\(699\) 604162. 0.0467692
\(700\) 0 0
\(701\) 5.48228e6 0.421373 0.210686 0.977554i \(-0.432430\pi\)
0.210686 + 0.977554i \(0.432430\pi\)
\(702\) − 7.50161e6i − 0.574528i
\(703\) 5.13887e6i 0.392174i
\(704\) −3.00142e6 −0.228242
\(705\) 0 0
\(706\) 4.64524e7 3.50749
\(707\) 6.23750e6i 0.469312i
\(708\) 5.45244e7i 4.08797i
\(709\) 1.30530e7 0.975200 0.487600 0.873067i \(-0.337872\pi\)
0.487600 + 0.873067i \(0.337872\pi\)
\(710\) 0 0
\(711\) −59654.9 −0.00442560
\(712\) − 1.05676e7i − 0.781225i
\(713\) − 525987.i − 0.0387482i
\(714\) −3.33700e7 −2.44969
\(715\) 0 0
\(716\) −4.76714e7 −3.47517
\(717\) 1.69814e7i 1.23360i
\(718\) − 1.03541e7i − 0.749550i
\(719\) 2.17045e7 1.56577 0.782885 0.622167i \(-0.213748\pi\)
0.782885 + 0.622167i \(0.213748\pi\)
\(720\) 0 0
\(721\) 2.96503e7 2.12418
\(722\) 1.20228e7i 0.858348i
\(723\) − 1.57418e7i − 1.11998i
\(724\) −2.99540e7 −2.12378
\(725\) 0 0
\(726\) −3.14267e6 −0.221288
\(727\) − 1.07681e7i − 0.755619i −0.925883 0.377809i \(-0.876677\pi\)
0.925883 0.377809i \(-0.123323\pi\)
\(728\) − 4.41091e7i − 3.08461i
\(729\) 6.64290e6 0.462955
\(730\) 0 0
\(731\) 1.88173e7 1.30246
\(732\) 4.04909e7i 2.79306i
\(733\) − 7.35742e6i − 0.505785i −0.967494 0.252892i \(-0.918618\pi\)
0.967494 0.252892i \(-0.0813819\pi\)
\(734\) −1.25887e7 −0.862463
\(735\) 0 0
\(736\) 1.28673e7 0.875573
\(737\) 6.70154e6i 0.454471i
\(738\) 3.05232e7i 2.06295i
\(739\) 6.81140e6 0.458802 0.229401 0.973332i \(-0.426323\pi\)
0.229401 + 0.973332i \(0.426323\pi\)
\(740\) 0 0
\(741\) −1.38806e7 −0.928676
\(742\) − 4.02895e7i − 2.68647i
\(743\) 1.01182e7i 0.672405i 0.941790 + 0.336203i \(0.109143\pi\)
−0.941790 + 0.336203i \(0.890857\pi\)
\(744\) −3.69698e6 −0.244858
\(745\) 0 0
\(746\) −1.89457e7 −1.24642
\(747\) − 1.70286e7i − 1.11655i
\(748\) 8.70281e6i 0.568729i
\(749\) −1.52712e7 −0.994649
\(750\) 0 0
\(751\) −1.91140e7 −1.23667 −0.618333 0.785916i \(-0.712192\pi\)
−0.618333 + 0.785916i \(0.712192\pi\)
\(752\) 4.36033e6i 0.281174i
\(753\) − 1.15903e7i − 0.744914i
\(754\) −5.47334e6 −0.350610
\(755\) 0 0
\(756\) −1.54268e7 −0.981685
\(757\) 1.48895e7i 0.944366i 0.881501 + 0.472183i \(0.156534\pi\)
−0.881501 + 0.472183i \(0.843466\pi\)
\(758\) − 4.35977e6i − 0.275607i
\(759\) 3.36330e6 0.211915
\(760\) 0 0
\(761\) −2.14078e7 −1.34002 −0.670009 0.742353i \(-0.733710\pi\)
−0.670009 + 0.742353i \(0.733710\pi\)
\(762\) − 5.84554e6i − 0.364702i
\(763\) − 2.97227e7i − 1.84832i
\(764\) −1.56327e7 −0.968948
\(765\) 0 0
\(766\) −3.07373e7 −1.89275
\(767\) 2.03134e7i 1.24680i
\(768\) 2.66844e7i 1.63251i
\(769\) −1.45027e7 −0.884369 −0.442184 0.896924i \(-0.645796\pi\)
−0.442184 + 0.896924i \(0.645796\pi\)
\(770\) 0 0
\(771\) −1.57939e7 −0.956874
\(772\) − 2.66129e7i − 1.60712i
\(773\) − 3.18546e7i − 1.91745i −0.284342 0.958723i \(-0.591775\pi\)
0.284342 0.958723i \(-0.408225\pi\)
\(774\) −3.79293e7 −2.27574
\(775\) 0 0
\(776\) 2.29768e6 0.136973
\(777\) − 1.51891e7i − 0.902565i
\(778\) 2.44759e7i 1.44974i
\(779\) −1.84016e7 −1.08646
\(780\) 0 0
\(781\) 6.88470e6 0.403885
\(782\) − 1.32311e7i − 0.773710i
\(783\) 1.10914e6i 0.0646519i
\(784\) −2.38545e7 −1.38606
\(785\) 0 0
\(786\) 2.55156e6 0.147316
\(787\) − 1.68239e7i − 0.968253i −0.874998 0.484126i \(-0.839138\pi\)
0.874998 0.484126i \(-0.160862\pi\)
\(788\) − 6.57285e7i − 3.77084i
\(789\) 1.57663e7 0.901650
\(790\) 0 0
\(791\) −1.12281e7 −0.638067
\(792\) − 1.01639e7i − 0.575771i
\(793\) 1.50852e7i 0.851858i
\(794\) 4.10455e7 2.31055
\(795\) 0 0
\(796\) −4.64215e7 −2.59679
\(797\) − 2.00376e7i − 1.11738i −0.829377 0.558690i \(-0.811304\pi\)
0.829377 0.558690i \(-0.188696\pi\)
\(798\) 4.05509e7i 2.25421i
\(799\) 1.76908e6 0.0980347
\(800\) 0 0
\(801\) 4.22616e6 0.232736
\(802\) 5.99384e7i 3.29055i
\(803\) 5.67149e6i 0.310391i
\(804\) −8.70018e7 −4.74666
\(805\) 0 0
\(806\) −2.37714e6 −0.128889
\(807\) 1.65356e7i 0.893790i
\(808\) 1.73829e7i 0.936683i
\(809\) 1.59711e7 0.857954 0.428977 0.903315i \(-0.358874\pi\)
0.428977 + 0.903315i \(0.358874\pi\)
\(810\) 0 0
\(811\) 1.37309e7 0.733074 0.366537 0.930403i \(-0.380543\pi\)
0.366537 + 0.930403i \(0.380543\pi\)
\(812\) 1.12558e7i 0.599080i
\(813\) − 2.02625e6i − 0.107515i
\(814\) −5.62734e6 −0.297675
\(815\) 0 0
\(816\) −4.54769e7 −2.39092
\(817\) − 2.28666e7i − 1.19852i
\(818\) 2.49192e7i 1.30212i
\(819\) 1.76400e7 0.918942
\(820\) 0 0
\(821\) 2.26402e7 1.17226 0.586128 0.810218i \(-0.300652\pi\)
0.586128 + 0.810218i \(0.300652\pi\)
\(822\) − 7.57758e6i − 0.391157i
\(823\) − 1.16510e7i − 0.599603i −0.954002 0.299802i \(-0.903079\pi\)
0.954002 0.299802i \(-0.0969205\pi\)
\(824\) 8.26304e7 4.23957
\(825\) 0 0
\(826\) 5.93438e7 3.02639
\(827\) 5.48883e6i 0.279072i 0.990217 + 0.139536i \(0.0445611\pi\)
−0.990217 + 0.139536i \(0.955439\pi\)
\(828\) 1.87734e7i 0.951630i
\(829\) −1.81680e7 −0.918164 −0.459082 0.888394i \(-0.651822\pi\)
−0.459082 + 0.888394i \(0.651822\pi\)
\(830\) 0 0
\(831\) −1.28214e7 −0.644070
\(832\) − 1.45169e7i − 0.727050i
\(833\) 9.67828e6i 0.483265i
\(834\) 5.63410e6 0.280485
\(835\) 0 0
\(836\) 1.05756e7 0.523345
\(837\) 481712.i 0.0237670i
\(838\) − 1.47283e7i − 0.724507i
\(839\) 1.83237e7 0.898689 0.449344 0.893359i \(-0.351658\pi\)
0.449344 + 0.893359i \(0.351658\pi\)
\(840\) 0 0
\(841\) −1.97019e7 −0.960546
\(842\) 4.99582e7i 2.42844i
\(843\) − 2.70866e7i − 1.31276i
\(844\) −1.26756e7 −0.612509
\(845\) 0 0
\(846\) −3.56586e6 −0.171293
\(847\) 2.40776e6i 0.115320i
\(848\) − 5.49069e7i − 2.62203i
\(849\) 555348. 0.0264421
\(850\) 0 0
\(851\) 6.02240e6 0.285066
\(852\) 8.93797e7i 4.21832i
\(853\) − 3.66987e7i − 1.72694i −0.504398 0.863471i \(-0.668285\pi\)
0.504398 0.863471i \(-0.331715\pi\)
\(854\) 4.40698e7 2.06774
\(855\) 0 0
\(856\) −4.25584e7 −1.98518
\(857\) 3.80021e7i 1.76749i 0.467972 + 0.883743i \(0.344985\pi\)
−0.467972 + 0.883743i \(0.655015\pi\)
\(858\) − 1.52001e7i − 0.704899i
\(859\) 7.40664e6 0.342482 0.171241 0.985229i \(-0.445222\pi\)
0.171241 + 0.985229i \(0.445222\pi\)
\(860\) 0 0
\(861\) 5.43901e7 2.50042
\(862\) 2.84303e7i 1.30321i
\(863\) 9.79342e6i 0.447618i 0.974633 + 0.223809i \(0.0718491\pi\)
−0.974633 + 0.223809i \(0.928151\pi\)
\(864\) −1.17842e7 −0.537050
\(865\) 0 0
\(866\) −2.82698e7 −1.28094
\(867\) − 1.08643e7i − 0.490857i
\(868\) 4.88851e6i 0.220230i
\(869\) 39383.0 0.00176913
\(870\) 0 0
\(871\) −3.24131e7 −1.44769
\(872\) − 8.28322e7i − 3.68899i
\(873\) 918880.i 0.0408059i
\(874\) −1.60783e7 −0.711969
\(875\) 0 0
\(876\) −7.36293e7 −3.24183
\(877\) − 3.57898e7i − 1.57130i −0.618669 0.785652i \(-0.712328\pi\)
0.618669 0.785652i \(-0.287672\pi\)
\(878\) − 4.33632e7i − 1.89839i
\(879\) −1.31838e7 −0.575532
\(880\) 0 0
\(881\) 7.14038e6 0.309943 0.154971 0.987919i \(-0.450471\pi\)
0.154971 + 0.987919i \(0.450471\pi\)
\(882\) − 1.95082e7i − 0.844393i
\(883\) 2.77023e7i 1.19568i 0.801617 + 0.597838i \(0.203973\pi\)
−0.801617 + 0.597838i \(0.796027\pi\)
\(884\) −4.20926e7 −1.81165
\(885\) 0 0
\(886\) −7.14091e7 −3.05611
\(887\) 387870.i 0.0165530i 0.999966 + 0.00827651i \(0.00263453\pi\)
−0.999966 + 0.00827651i \(0.997365\pi\)
\(888\) − 4.23293e7i − 1.80140i
\(889\) −4.47857e6 −0.190058
\(890\) 0 0
\(891\) −8.46928e6 −0.357398
\(892\) − 8.03250e7i − 3.38017i
\(893\) − 2.14977e6i − 0.0902117i
\(894\) −4.85795e7 −2.03287
\(895\) 0 0
\(896\) 7.88818e6 0.328251
\(897\) 1.62672e7i 0.675042i
\(898\) − 7.21309e6i − 0.298491i
\(899\) 351468. 0.0145040
\(900\) 0 0
\(901\) −2.22769e7 −0.914203
\(902\) − 2.01508e7i − 0.824662i
\(903\) 6.75873e7i 2.75833i
\(904\) −3.12909e7 −1.27349
\(905\) 0 0
\(906\) 6.46236e7 2.61560
\(907\) 5.05051e6i 0.203853i 0.994792 + 0.101926i \(0.0325006\pi\)
−0.994792 + 0.101926i \(0.967499\pi\)
\(908\) − 4.00626e7i − 1.61259i
\(909\) −6.95169e6 −0.279049
\(910\) 0 0
\(911\) −1.72283e7 −0.687774 −0.343887 0.939011i \(-0.611744\pi\)
−0.343887 + 0.939011i \(0.611744\pi\)
\(912\) 5.52631e7i 2.20013i
\(913\) 1.12419e7i 0.446337i
\(914\) 8.44660e7 3.34439
\(915\) 0 0
\(916\) 8.49807e7 3.34643
\(917\) − 1.95488e6i − 0.0767709i
\(918\) 1.21174e7i 0.474571i
\(919\) −3.89056e7 −1.51958 −0.759789 0.650170i \(-0.774698\pi\)
−0.759789 + 0.650170i \(0.774698\pi\)
\(920\) 0 0
\(921\) 1.13645e7 0.441470
\(922\) 4.66769e7i 1.80832i
\(923\) 3.32990e7i 1.28655i
\(924\) −3.12584e7 −1.20445
\(925\) 0 0
\(926\) 9.40340e7 3.60377
\(927\) 3.30453e7i 1.26302i
\(928\) 8.59801e6i 0.327739i
\(929\) −3.18602e7 −1.21118 −0.605590 0.795777i \(-0.707063\pi\)
−0.605590 + 0.795777i \(0.707063\pi\)
\(930\) 0 0
\(931\) 1.17610e7 0.444701
\(932\) − 2.22635e6i − 0.0839564i
\(933\) − 3.85627e6i − 0.145032i
\(934\) 7.45701e7 2.79703
\(935\) 0 0
\(936\) 4.91596e7 1.83408
\(937\) − 4.09748e7i − 1.52464i −0.647198 0.762322i \(-0.724059\pi\)
0.647198 0.762322i \(-0.275941\pi\)
\(938\) 9.46917e7i 3.51402i
\(939\) 1.92889e7 0.713909
\(940\) 0 0
\(941\) −1.76369e7 −0.649304 −0.324652 0.945833i \(-0.605247\pi\)
−0.324652 + 0.945833i \(0.605247\pi\)
\(942\) − 7.32988e7i − 2.69135i
\(943\) 2.15655e7i 0.789732i
\(944\) 8.08741e7 2.95379
\(945\) 0 0
\(946\) 2.50402e7 0.909723
\(947\) 4.07232e7i 1.47559i 0.675022 + 0.737797i \(0.264134\pi\)
−0.675022 + 0.737797i \(0.735866\pi\)
\(948\) 511284.i 0.0184774i
\(949\) −2.74311e7 −0.988730
\(950\) 0 0
\(951\) −3.88735e7 −1.39381
\(952\) 7.12495e7i 2.54794i
\(953\) 1.02210e7i 0.364553i 0.983247 + 0.182276i \(0.0583465\pi\)
−0.983247 + 0.182276i \(0.941653\pi\)
\(954\) 4.49027e7 1.59736
\(955\) 0 0
\(956\) 6.25767e7 2.21446
\(957\) 2.24738e6i 0.0793226i
\(958\) − 1.57077e7i − 0.552967i
\(959\) −5.80557e6 −0.203844
\(960\) 0 0
\(961\) −2.84765e7 −0.994668
\(962\) − 2.72175e7i − 0.948225i
\(963\) − 1.70198e7i − 0.591410i
\(964\) −5.80089e7 −2.01049
\(965\) 0 0
\(966\) 4.75229e7 1.63855
\(967\) − 7.46133e6i − 0.256596i −0.991736 0.128298i \(-0.959049\pi\)
0.991736 0.128298i \(-0.0409514\pi\)
\(968\) 6.71003e6i 0.230163i
\(969\) 2.24214e7 0.767103
\(970\) 0 0
\(971\) 4.34819e7 1.47999 0.739997 0.672610i \(-0.234827\pi\)
0.739997 + 0.672610i \(0.234827\pi\)
\(972\) − 8.71562e7i − 2.95892i
\(973\) − 4.31657e6i − 0.146170i
\(974\) 1.69736e7 0.573292
\(975\) 0 0
\(976\) 6.00587e7 2.01814
\(977\) − 1.52214e7i − 0.510173i −0.966918 0.255087i \(-0.917896\pi\)
0.966918 0.255087i \(-0.0821040\pi\)
\(978\) 1.29779e8i 4.33869i
\(979\) −2.79002e6 −0.0930360
\(980\) 0 0
\(981\) 3.31260e7 1.09900
\(982\) − 1.29392e7i − 0.428181i
\(983\) − 2.90192e7i − 0.957858i −0.877854 0.478929i \(-0.841025\pi\)
0.877854 0.478929i \(-0.158975\pi\)
\(984\) 1.51576e8 4.99049
\(985\) 0 0
\(986\) 8.84110e6 0.289610
\(987\) 6.35411e6i 0.207616i
\(988\) 5.11505e7i 1.66708i
\(989\) −2.67981e7 −0.871190
\(990\) 0 0
\(991\) 3.46744e7 1.12156 0.560782 0.827963i \(-0.310500\pi\)
0.560782 + 0.827963i \(0.310500\pi\)
\(992\) 3.73422e6i 0.120481i
\(993\) − 4.06853e6i − 0.130938i
\(994\) 9.72798e7 3.12289
\(995\) 0 0
\(996\) −1.45946e8 −4.66171
\(997\) 1.05268e7i 0.335397i 0.985838 + 0.167699i \(0.0536336\pi\)
−0.985838 + 0.167699i \(0.946366\pi\)
\(998\) − 7.96577e7i − 2.53164i
\(999\) −5.51546e6 −0.174851
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.199.1 6
5.2 odd 4 275.6.a.b.1.3 3
5.3 odd 4 11.6.a.b.1.1 3
5.4 even 2 inner 275.6.b.b.199.6 6
15.8 even 4 99.6.a.g.1.3 3
20.3 even 4 176.6.a.i.1.1 3
35.13 even 4 539.6.a.e.1.1 3
40.3 even 4 704.6.a.t.1.3 3
40.13 odd 4 704.6.a.q.1.1 3
55.43 even 4 121.6.a.d.1.3 3
165.98 odd 4 1089.6.a.r.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.1 3 5.3 odd 4
99.6.a.g.1.3 3 15.8 even 4
121.6.a.d.1.3 3 55.43 even 4
176.6.a.i.1.1 3 20.3 even 4
275.6.a.b.1.3 3 5.2 odd 4
275.6.b.b.199.1 6 1.1 even 1 trivial
275.6.b.b.199.6 6 5.4 even 2 inner
539.6.a.e.1.1 3 35.13 even 4
704.6.a.q.1.1 3 40.13 odd 4
704.6.a.t.1.3 3 40.3 even 4
1089.6.a.r.1.1 3 165.98 odd 4