Properties

Label 315.8.a.c.1.1
Level $315$
Weight $8$
Character 315.1
Self dual yes
Analytic conductor $98.401$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,8,Mod(1,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 315.a (trivial)

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: yes
Analytic conductor: \(98.4012830275\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.31662\) of defining polynomial
Character \(\chi\) \(=\) 315.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-14.6332 q^{2} +86.1320 q^{4} -125.000 q^{5} -343.000 q^{7} +612.665 q^{8} +O(q^{10})\) \(q-14.6332 q^{2} +86.1320 q^{4} -125.000 q^{5} -343.000 q^{7} +612.665 q^{8} +1829.16 q^{10} +6473.63 q^{11} -11681.7 q^{13} +5019.20 q^{14} -19990.2 q^{16} -13460.5 q^{17} +34955.5 q^{19} -10766.5 q^{20} -94730.3 q^{22} -77831.4 q^{23} +15625.0 q^{25} +170941. q^{26} -29543.3 q^{28} +221135. q^{29} -23222.3 q^{31} +214100. q^{32} +196971. q^{34} +42875.0 q^{35} -422392. q^{37} -511512. q^{38} -76583.1 q^{40} -191818. q^{41} +310754. q^{43} +557587. q^{44} +1.13893e6 q^{46} +240747. q^{47} +117649. q^{49} -228645. q^{50} -1.00617e6 q^{52} +1.06654e6 q^{53} -809204. q^{55} -210144. q^{56} -3.23592e6 q^{58} -451838. q^{59} -831659. q^{61} +339818. q^{62} -574238. q^{64} +1.46021e6 q^{65} +2.26405e6 q^{67} -1.15938e6 q^{68} -627401. q^{70} +2.22036e6 q^{71} +4.99377e6 q^{73} +6.18096e6 q^{74} +3.01078e6 q^{76} -2.22046e6 q^{77} -2.72773e6 q^{79} +2.49877e6 q^{80} +2.80693e6 q^{82} +6.38392e6 q^{83} +1.68256e6 q^{85} -4.54734e6 q^{86} +3.96617e6 q^{88} +7.32978e6 q^{89} +4.00682e6 q^{91} -6.70378e6 q^{92} -3.52291e6 q^{94} -4.36943e6 q^{95} -2.38676e6 q^{97} -1.72159e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{2} - 40 q^{4} - 250 q^{5} - 686 q^{7} + 960 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{2} - 40 q^{4} - 250 q^{5} - 686 q^{7} + 960 q^{8} + 2000 q^{10} + 7906 q^{11} - 17818 q^{13} + 5488 q^{14} - 4320 q^{16} + 2398 q^{17} - 3612 q^{19} + 5000 q^{20} - 96688 q^{22} - 13844 q^{23} + 31250 q^{25} + 179328 q^{26} + 13720 q^{28} + 126898 q^{29} + 252768 q^{31} + 148224 q^{32} + 175296 q^{34} + 85750 q^{35} - 265860 q^{37} - 458800 q^{38} - 120000 q^{40} + 111920 q^{41} + 947572 q^{43} + 376920 q^{44} + 1051472 q^{46} - 271274 q^{47} + 235298 q^{49} - 250000 q^{50} - 232184 q^{52} + 1267792 q^{53} - 988250 q^{55} - 329280 q^{56} - 3107120 q^{58} + 1360120 q^{59} - 1813680 q^{61} - 37392 q^{62} - 2489984 q^{64} + 2227250 q^{65} - 2189312 q^{67} - 3159640 q^{68} - 686000 q^{70} + 1494928 q^{71} + 7169788 q^{73} + 5967024 q^{74} + 7875376 q^{76} - 2711758 q^{77} - 7942974 q^{79} + 540000 q^{80} + 2391792 q^{82} + 304712 q^{83} - 299750 q^{85} - 5417712 q^{86} + 4463680 q^{88} + 17943528 q^{89} + 6111574 q^{91} - 14774640 q^{92} - 2823104 q^{94} + 451500 q^{95} + 4258074 q^{97} - 1882384 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −14.6332 −1.29341 −0.646704 0.762741i \(-0.723853\pi\)
−0.646704 + 0.762741i \(0.723853\pi\)
\(3\) 0 0
\(4\) 86.1320 0.672906
\(5\) −125.000 −0.447214
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 612.665 0.423066
\(9\) 0 0
\(10\) 1829.16 0.578430
\(11\) 6473.63 1.46647 0.733236 0.679974i \(-0.238009\pi\)
0.733236 + 0.679974i \(0.238009\pi\)
\(12\) 0 0
\(13\) −11681.7 −1.47470 −0.737351 0.675510i \(-0.763924\pi\)
−0.737351 + 0.675510i \(0.763924\pi\)
\(14\) 5019.20 0.488863
\(15\) 0 0
\(16\) −19990.2 −1.22010
\(17\) −13460.5 −0.664491 −0.332246 0.943193i \(-0.607806\pi\)
−0.332246 + 0.943193i \(0.607806\pi\)
\(18\) 0 0
\(19\) 34955.5 1.16917 0.584585 0.811333i \(-0.301257\pi\)
0.584585 + 0.811333i \(0.301257\pi\)
\(20\) −10766.5 −0.300933
\(21\) 0 0
\(22\) −94730.3 −1.89675
\(23\) −77831.4 −1.33385 −0.666926 0.745124i \(-0.732390\pi\)
−0.666926 + 0.745124i \(0.732390\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) 170941. 1.90739
\(27\) 0 0
\(28\) −29543.3 −0.254335
\(29\) 221135. 1.68370 0.841848 0.539715i \(-0.181468\pi\)
0.841848 + 0.539715i \(0.181468\pi\)
\(30\) 0 0
\(31\) −23222.3 −0.140004 −0.0700018 0.997547i \(-0.522301\pi\)
−0.0700018 + 0.997547i \(0.522301\pi\)
\(32\) 214100. 1.15503
\(33\) 0 0
\(34\) 196971. 0.859459
\(35\) 42875.0 0.169031
\(36\) 0 0
\(37\) −422392. −1.37091 −0.685456 0.728114i \(-0.740397\pi\)
−0.685456 + 0.728114i \(0.740397\pi\)
\(38\) −511512. −1.51221
\(39\) 0 0
\(40\) −76583.1 −0.189201
\(41\) −191818. −0.434657 −0.217329 0.976099i \(-0.569734\pi\)
−0.217329 + 0.976099i \(0.569734\pi\)
\(42\) 0 0
\(43\) 310754. 0.596042 0.298021 0.954559i \(-0.403673\pi\)
0.298021 + 0.954559i \(0.403673\pi\)
\(44\) 557587. 0.986798
\(45\) 0 0
\(46\) 1.13893e6 1.72522
\(47\) 240747. 0.338235 0.169117 0.985596i \(-0.445908\pi\)
0.169117 + 0.985596i \(0.445908\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) −228645. −0.258682
\(51\) 0 0
\(52\) −1.00617e6 −0.992336
\(53\) 1.06654e6 0.984040 0.492020 0.870584i \(-0.336259\pi\)
0.492020 + 0.870584i \(0.336259\pi\)
\(54\) 0 0
\(55\) −809204. −0.655826
\(56\) −210144. −0.159904
\(57\) 0 0
\(58\) −3.23592e6 −2.17771
\(59\) −451838. −0.286418 −0.143209 0.989692i \(-0.545742\pi\)
−0.143209 + 0.989692i \(0.545742\pi\)
\(60\) 0 0
\(61\) −831659. −0.469127 −0.234564 0.972101i \(-0.575366\pi\)
−0.234564 + 0.972101i \(0.575366\pi\)
\(62\) 339818. 0.181082
\(63\) 0 0
\(64\) −574238. −0.273818
\(65\) 1.46021e6 0.659507
\(66\) 0 0
\(67\) 2.26405e6 0.919654 0.459827 0.888009i \(-0.347912\pi\)
0.459827 + 0.888009i \(0.347912\pi\)
\(68\) −1.15938e6 −0.447140
\(69\) 0 0
\(70\) −627401. −0.218626
\(71\) 2.22036e6 0.736241 0.368120 0.929778i \(-0.380001\pi\)
0.368120 + 0.929778i \(0.380001\pi\)
\(72\) 0 0
\(73\) 4.99377e6 1.50244 0.751222 0.660049i \(-0.229465\pi\)
0.751222 + 0.660049i \(0.229465\pi\)
\(74\) 6.18096e6 1.77315
\(75\) 0 0
\(76\) 3.01078e6 0.786742
\(77\) −2.22046e6 −0.554274
\(78\) 0 0
\(79\) −2.72773e6 −0.622452 −0.311226 0.950336i \(-0.600740\pi\)
−0.311226 + 0.950336i \(0.600740\pi\)
\(80\) 2.49877e6 0.545647
\(81\) 0 0
\(82\) 2.80693e6 0.562189
\(83\) 6.38392e6 1.22550 0.612751 0.790276i \(-0.290063\pi\)
0.612751 + 0.790276i \(0.290063\pi\)
\(84\) 0 0
\(85\) 1.68256e6 0.297170
\(86\) −4.54734e6 −0.770926
\(87\) 0 0
\(88\) 3.96617e6 0.620414
\(89\) 7.32978e6 1.10211 0.551056 0.834468i \(-0.314225\pi\)
0.551056 + 0.834468i \(0.314225\pi\)
\(90\) 0 0
\(91\) 4.00682e6 0.557385
\(92\) −6.70378e6 −0.897557
\(93\) 0 0
\(94\) −3.52291e6 −0.437476
\(95\) −4.36943e6 −0.522869
\(96\) 0 0
\(97\) −2.38676e6 −0.265526 −0.132763 0.991148i \(-0.542385\pi\)
−0.132763 + 0.991148i \(0.542385\pi\)
\(98\) −1.72159e6 −0.184773
\(99\) 0 0
\(100\) 1.34581e6 0.134581
\(101\) 1.92113e7 1.85538 0.927688 0.373356i \(-0.121793\pi\)
0.927688 + 0.373356i \(0.121793\pi\)
\(102\) 0 0
\(103\) −4.45359e6 −0.401587 −0.200793 0.979634i \(-0.564352\pi\)
−0.200793 + 0.979634i \(0.564352\pi\)
\(104\) −7.15697e6 −0.623896
\(105\) 0 0
\(106\) −1.56070e7 −1.27277
\(107\) −7.61385e6 −0.600843 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(108\) 0 0
\(109\) −1.99698e7 −1.47700 −0.738502 0.674251i \(-0.764467\pi\)
−0.738502 + 0.674251i \(0.764467\pi\)
\(110\) 1.18413e7 0.848251
\(111\) 0 0
\(112\) 6.85663e6 0.461156
\(113\) −2.57944e7 −1.68171 −0.840855 0.541261i \(-0.817947\pi\)
−0.840855 + 0.541261i \(0.817947\pi\)
\(114\) 0 0
\(115\) 9.72893e6 0.596517
\(116\) 1.90468e7 1.13297
\(117\) 0 0
\(118\) 6.61185e6 0.370456
\(119\) 4.61695e6 0.251154
\(120\) 0 0
\(121\) 2.24208e7 1.15054
\(122\) 1.21699e7 0.606773
\(123\) 0 0
\(124\) −2.00018e6 −0.0942094
\(125\) −1.95312e6 −0.0894427
\(126\) 0 0
\(127\) −6.75687e6 −0.292707 −0.146353 0.989232i \(-0.546754\pi\)
−0.146353 + 0.989232i \(0.546754\pi\)
\(128\) −1.90018e7 −0.800868
\(129\) 0 0
\(130\) −2.13677e7 −0.853012
\(131\) 2.21063e6 0.0859144 0.0429572 0.999077i \(-0.486322\pi\)
0.0429572 + 0.999077i \(0.486322\pi\)
\(132\) 0 0
\(133\) −1.19897e7 −0.441905
\(134\) −3.31304e7 −1.18949
\(135\) 0 0
\(136\) −8.24677e6 −0.281124
\(137\) 7.10581e6 0.236097 0.118049 0.993008i \(-0.462336\pi\)
0.118049 + 0.993008i \(0.462336\pi\)
\(138\) 0 0
\(139\) 9.21848e6 0.291144 0.145572 0.989348i \(-0.453498\pi\)
0.145572 + 0.989348i \(0.453498\pi\)
\(140\) 3.69291e6 0.113742
\(141\) 0 0
\(142\) −3.24911e7 −0.952260
\(143\) −7.56230e7 −2.16261
\(144\) 0 0
\(145\) −2.76418e7 −0.752972
\(146\) −7.30751e7 −1.94328
\(147\) 0 0
\(148\) −3.63814e7 −0.922495
\(149\) −1.33298e7 −0.330119 −0.165059 0.986284i \(-0.552782\pi\)
−0.165059 + 0.986284i \(0.552782\pi\)
\(150\) 0 0
\(151\) −6.41939e7 −1.51731 −0.758656 0.651492i \(-0.774143\pi\)
−0.758656 + 0.651492i \(0.774143\pi\)
\(152\) 2.14160e7 0.494636
\(153\) 0 0
\(154\) 3.24925e7 0.716903
\(155\) 2.90279e6 0.0626116
\(156\) 0 0
\(157\) 7.36596e7 1.51908 0.759540 0.650461i \(-0.225424\pi\)
0.759540 + 0.650461i \(0.225424\pi\)
\(158\) 3.99155e7 0.805085
\(159\) 0 0
\(160\) −2.67625e7 −0.516544
\(161\) 2.66962e7 0.504149
\(162\) 0 0
\(163\) −3.50642e7 −0.634172 −0.317086 0.948397i \(-0.602704\pi\)
−0.317086 + 0.948397i \(0.602704\pi\)
\(164\) −1.65217e7 −0.292483
\(165\) 0 0
\(166\) −9.34175e7 −1.58508
\(167\) −2.56950e6 −0.0426915 −0.0213458 0.999772i \(-0.506795\pi\)
−0.0213458 + 0.999772i \(0.506795\pi\)
\(168\) 0 0
\(169\) 7.37136e7 1.17475
\(170\) −2.46213e7 −0.384362
\(171\) 0 0
\(172\) 2.67659e7 0.401081
\(173\) −8.03463e7 −1.17979 −0.589895 0.807480i \(-0.700831\pi\)
−0.589895 + 0.807480i \(0.700831\pi\)
\(174\) 0 0
\(175\) −5.35938e6 −0.0755929
\(176\) −1.29409e8 −1.78925
\(177\) 0 0
\(178\) −1.07259e8 −1.42548
\(179\) −9.99074e7 −1.30200 −0.651001 0.759077i \(-0.725651\pi\)
−0.651001 + 0.759077i \(0.725651\pi\)
\(180\) 0 0
\(181\) −1.07414e8 −1.34644 −0.673221 0.739442i \(-0.735090\pi\)
−0.673221 + 0.739442i \(0.735090\pi\)
\(182\) −5.86328e7 −0.720927
\(183\) 0 0
\(184\) −4.76846e7 −0.564307
\(185\) 5.27990e7 0.613090
\(186\) 0 0
\(187\) −8.71382e7 −0.974458
\(188\) 2.07360e7 0.227600
\(189\) 0 0
\(190\) 6.39390e7 0.676283
\(191\) 2.21085e7 0.229584 0.114792 0.993390i \(-0.463380\pi\)
0.114792 + 0.993390i \(0.463380\pi\)
\(192\) 0 0
\(193\) −1.49793e8 −1.49983 −0.749913 0.661537i \(-0.769905\pi\)
−0.749913 + 0.661537i \(0.769905\pi\)
\(194\) 3.49261e7 0.343434
\(195\) 0 0
\(196\) 1.01333e7 0.0961295
\(197\) 5.70107e7 0.531281 0.265641 0.964072i \(-0.414417\pi\)
0.265641 + 0.964072i \(0.414417\pi\)
\(198\) 0 0
\(199\) −6.84161e7 −0.615421 −0.307711 0.951480i \(-0.599563\pi\)
−0.307711 + 0.951480i \(0.599563\pi\)
\(200\) 9.57289e6 0.0846132
\(201\) 0 0
\(202\) −2.81124e8 −2.39976
\(203\) −7.58492e7 −0.636377
\(204\) 0 0
\(205\) 2.39773e7 0.194385
\(206\) 6.51704e7 0.519416
\(207\) 0 0
\(208\) 2.33519e8 1.79929
\(209\) 2.26289e8 1.71455
\(210\) 0 0
\(211\) −1.36201e8 −0.998141 −0.499071 0.866561i \(-0.666325\pi\)
−0.499071 + 0.866561i \(0.666325\pi\)
\(212\) 9.18635e7 0.662167
\(213\) 0 0
\(214\) 1.11415e8 0.777135
\(215\) −3.88442e7 −0.266558
\(216\) 0 0
\(217\) 7.96525e6 0.0529164
\(218\) 2.92224e8 1.91037
\(219\) 0 0
\(220\) −6.96984e7 −0.441310
\(221\) 1.57241e8 0.979927
\(222\) 0 0
\(223\) −1.47728e8 −0.892066 −0.446033 0.895017i \(-0.647164\pi\)
−0.446033 + 0.895017i \(0.647164\pi\)
\(224\) −7.34363e7 −0.436559
\(225\) 0 0
\(226\) 3.77456e8 2.17514
\(227\) −3.22427e8 −1.82954 −0.914768 0.403980i \(-0.867627\pi\)
−0.914768 + 0.403980i \(0.867627\pi\)
\(228\) 0 0
\(229\) 3.10033e8 1.70602 0.853010 0.521895i \(-0.174775\pi\)
0.853010 + 0.521895i \(0.174775\pi\)
\(230\) −1.42366e8 −0.771540
\(231\) 0 0
\(232\) 1.35481e8 0.712315
\(233\) −1.80410e8 −0.934361 −0.467181 0.884162i \(-0.654730\pi\)
−0.467181 + 0.884162i \(0.654730\pi\)
\(234\) 0 0
\(235\) −3.00934e7 −0.151263
\(236\) −3.89177e7 −0.192733
\(237\) 0 0
\(238\) −6.75609e7 −0.324845
\(239\) −3.66489e8 −1.73647 −0.868237 0.496150i \(-0.834747\pi\)
−0.868237 + 0.496150i \(0.834747\pi\)
\(240\) 0 0
\(241\) 2.19729e8 1.01118 0.505589 0.862775i \(-0.331275\pi\)
0.505589 + 0.862775i \(0.331275\pi\)
\(242\) −3.28089e8 −1.48812
\(243\) 0 0
\(244\) −7.16324e7 −0.315679
\(245\) −1.47061e7 −0.0638877
\(246\) 0 0
\(247\) −4.08339e8 −1.72418
\(248\) −1.42275e7 −0.0592308
\(249\) 0 0
\(250\) 2.85806e7 0.115686
\(251\) 1.29875e8 0.518404 0.259202 0.965823i \(-0.416540\pi\)
0.259202 + 0.965823i \(0.416540\pi\)
\(252\) 0 0
\(253\) −5.03852e8 −1.95606
\(254\) 9.88750e7 0.378589
\(255\) 0 0
\(256\) 3.51561e8 1.30967
\(257\) 2.94635e8 1.08273 0.541363 0.840789i \(-0.317909\pi\)
0.541363 + 0.840789i \(0.317909\pi\)
\(258\) 0 0
\(259\) 1.44880e8 0.518156
\(260\) 1.25771e8 0.443786
\(261\) 0 0
\(262\) −3.23486e7 −0.111122
\(263\) −3.13955e8 −1.06420 −0.532098 0.846683i \(-0.678596\pi\)
−0.532098 + 0.846683i \(0.678596\pi\)
\(264\) 0 0
\(265\) −1.33318e8 −0.440076
\(266\) 1.75449e8 0.571563
\(267\) 0 0
\(268\) 1.95007e8 0.618841
\(269\) 2.94745e8 0.923238 0.461619 0.887078i \(-0.347269\pi\)
0.461619 + 0.887078i \(0.347269\pi\)
\(270\) 0 0
\(271\) −8.47290e7 −0.258607 −0.129303 0.991605i \(-0.541274\pi\)
−0.129303 + 0.991605i \(0.541274\pi\)
\(272\) 2.69077e8 0.810748
\(273\) 0 0
\(274\) −1.03981e8 −0.305371
\(275\) 1.01151e8 0.293294
\(276\) 0 0
\(277\) 1.96909e8 0.556655 0.278327 0.960486i \(-0.410220\pi\)
0.278327 + 0.960486i \(0.410220\pi\)
\(278\) −1.34896e8 −0.376568
\(279\) 0 0
\(280\) 2.62680e7 0.0715112
\(281\) −3.32330e8 −0.893505 −0.446753 0.894658i \(-0.647420\pi\)
−0.446753 + 0.894658i \(0.647420\pi\)
\(282\) 0 0
\(283\) 4.40437e8 1.15513 0.577566 0.816344i \(-0.304003\pi\)
0.577566 + 0.816344i \(0.304003\pi\)
\(284\) 1.91244e8 0.495421
\(285\) 0 0
\(286\) 1.10661e9 2.79714
\(287\) 6.57937e7 0.164285
\(288\) 0 0
\(289\) −2.29154e8 −0.558451
\(290\) 4.04490e8 0.973900
\(291\) 0 0
\(292\) 4.30123e8 1.01100
\(293\) 3.05058e8 0.708510 0.354255 0.935149i \(-0.384735\pi\)
0.354255 + 0.935149i \(0.384735\pi\)
\(294\) 0 0
\(295\) 5.64797e7 0.128090
\(296\) −2.58785e8 −0.579986
\(297\) 0 0
\(298\) 1.95058e8 0.426979
\(299\) 9.09203e8 1.96703
\(300\) 0 0
\(301\) −1.06589e8 −0.225283
\(302\) 9.39366e8 1.96250
\(303\) 0 0
\(304\) −6.98766e8 −1.42651
\(305\) 1.03957e8 0.209800
\(306\) 0 0
\(307\) −2.41616e8 −0.476587 −0.238293 0.971193i \(-0.576588\pi\)
−0.238293 + 0.971193i \(0.576588\pi\)
\(308\) −1.91252e8 −0.372975
\(309\) 0 0
\(310\) −4.24772e7 −0.0809823
\(311\) 6.71768e7 0.126636 0.0633181 0.997993i \(-0.479832\pi\)
0.0633181 + 0.997993i \(0.479832\pi\)
\(312\) 0 0
\(313\) −6.75084e8 −1.24438 −0.622190 0.782867i \(-0.713757\pi\)
−0.622190 + 0.782867i \(0.713757\pi\)
\(314\) −1.07788e9 −1.96479
\(315\) 0 0
\(316\) −2.34944e8 −0.418852
\(317\) 4.65149e8 0.820135 0.410067 0.912055i \(-0.365505\pi\)
0.410067 + 0.912055i \(0.365505\pi\)
\(318\) 0 0
\(319\) 1.43154e9 2.46909
\(320\) 7.17797e7 0.122455
\(321\) 0 0
\(322\) −3.90652e8 −0.652070
\(323\) −4.70517e8 −0.776903
\(324\) 0 0
\(325\) −1.82527e8 −0.294940
\(326\) 5.13103e8 0.820244
\(327\) 0 0
\(328\) −1.17520e8 −0.183889
\(329\) −8.25762e7 −0.127841
\(330\) 0 0
\(331\) 2.69779e8 0.408893 0.204447 0.978878i \(-0.434460\pi\)
0.204447 + 0.978878i \(0.434460\pi\)
\(332\) 5.49860e8 0.824648
\(333\) 0 0
\(334\) 3.76002e7 0.0552176
\(335\) −2.83006e8 −0.411282
\(336\) 0 0
\(337\) −6.29093e8 −0.895385 −0.447693 0.894187i \(-0.647754\pi\)
−0.447693 + 0.894187i \(0.647754\pi\)
\(338\) −1.07867e9 −1.51943
\(339\) 0 0
\(340\) 1.44922e8 0.199967
\(341\) −1.50333e8 −0.205312
\(342\) 0 0
\(343\) −4.03536e7 −0.0539949
\(344\) 1.90388e8 0.252165
\(345\) 0 0
\(346\) 1.17573e9 1.52595
\(347\) 7.61715e8 0.978676 0.489338 0.872094i \(-0.337238\pi\)
0.489338 + 0.872094i \(0.337238\pi\)
\(348\) 0 0
\(349\) −3.31639e8 −0.417616 −0.208808 0.977957i \(-0.566958\pi\)
−0.208808 + 0.977957i \(0.566958\pi\)
\(350\) 7.84251e7 0.0977725
\(351\) 0 0
\(352\) 1.38601e9 1.69381
\(353\) −7.57419e8 −0.916484 −0.458242 0.888828i \(-0.651521\pi\)
−0.458242 + 0.888828i \(0.651521\pi\)
\(354\) 0 0
\(355\) −2.77545e8 −0.329257
\(356\) 6.31329e8 0.741619
\(357\) 0 0
\(358\) 1.46197e9 1.68402
\(359\) −1.46796e9 −1.67449 −0.837246 0.546827i \(-0.815836\pi\)
−0.837246 + 0.546827i \(0.815836\pi\)
\(360\) 0 0
\(361\) 3.28013e8 0.366958
\(362\) 1.57182e9 1.74150
\(363\) 0 0
\(364\) 3.45116e8 0.375068
\(365\) −6.24221e8 −0.671914
\(366\) 0 0
\(367\) 1.64615e9 1.73835 0.869177 0.494502i \(-0.164649\pi\)
0.869177 + 0.494502i \(0.164649\pi\)
\(368\) 1.55586e9 1.62744
\(369\) 0 0
\(370\) −7.72620e8 −0.792976
\(371\) −3.65824e8 −0.371932
\(372\) 0 0
\(373\) −1.16387e9 −1.16124 −0.580622 0.814173i \(-0.697191\pi\)
−0.580622 + 0.814173i \(0.697191\pi\)
\(374\) 1.27512e9 1.26037
\(375\) 0 0
\(376\) 1.47497e8 0.143096
\(377\) −2.58323e9 −2.48295
\(378\) 0 0
\(379\) 4.07762e8 0.384742 0.192371 0.981322i \(-0.438382\pi\)
0.192371 + 0.981322i \(0.438382\pi\)
\(380\) −3.76348e8 −0.351841
\(381\) 0 0
\(382\) −3.23519e8 −0.296946
\(383\) −7.10345e8 −0.646061 −0.323031 0.946389i \(-0.604702\pi\)
−0.323031 + 0.946389i \(0.604702\pi\)
\(384\) 0 0
\(385\) 2.77557e8 0.247879
\(386\) 2.19196e9 1.93989
\(387\) 0 0
\(388\) −2.05576e8 −0.178674
\(389\) −1.95091e8 −0.168040 −0.0840202 0.996464i \(-0.526776\pi\)
−0.0840202 + 0.996464i \(0.526776\pi\)
\(390\) 0 0
\(391\) 1.04765e9 0.886333
\(392\) 7.20794e7 0.0604380
\(393\) 0 0
\(394\) −8.34252e8 −0.687164
\(395\) 3.40966e8 0.278369
\(396\) 0 0
\(397\) 1.58231e9 1.26919 0.634593 0.772846i \(-0.281168\pi\)
0.634593 + 0.772846i \(0.281168\pi\)
\(398\) 1.00115e9 0.795991
\(399\) 0 0
\(400\) −3.12346e8 −0.244021
\(401\) 2.13647e9 1.65460 0.827298 0.561763i \(-0.189877\pi\)
0.827298 + 0.561763i \(0.189877\pi\)
\(402\) 0 0
\(403\) 2.71276e8 0.206464
\(404\) 1.65471e9 1.24849
\(405\) 0 0
\(406\) 1.10992e9 0.823096
\(407\) −2.73441e9 −2.01040
\(408\) 0 0
\(409\) −4.05635e8 −0.293159 −0.146580 0.989199i \(-0.546826\pi\)
−0.146580 + 0.989199i \(0.546826\pi\)
\(410\) −3.50866e8 −0.251419
\(411\) 0 0
\(412\) −3.83596e8 −0.270230
\(413\) 1.54980e8 0.108256
\(414\) 0 0
\(415\) −7.97990e8 −0.548061
\(416\) −2.50105e9 −1.70332
\(417\) 0 0
\(418\) −3.31134e9 −2.21762
\(419\) −1.07475e9 −0.713771 −0.356886 0.934148i \(-0.616161\pi\)
−0.356886 + 0.934148i \(0.616161\pi\)
\(420\) 0 0
\(421\) −8.32900e8 −0.544009 −0.272004 0.962296i \(-0.587686\pi\)
−0.272004 + 0.962296i \(0.587686\pi\)
\(422\) 1.99306e9 1.29100
\(423\) 0 0
\(424\) 6.53434e8 0.416314
\(425\) −2.10320e8 −0.132898
\(426\) 0 0
\(427\) 2.85259e8 0.177313
\(428\) −6.55796e8 −0.404311
\(429\) 0 0
\(430\) 5.68418e8 0.344769
\(431\) −8.26292e6 −0.00497122 −0.00248561 0.999997i \(-0.500791\pi\)
−0.00248561 + 0.999997i \(0.500791\pi\)
\(432\) 0 0
\(433\) −3.10619e9 −1.83874 −0.919369 0.393396i \(-0.871300\pi\)
−0.919369 + 0.393396i \(0.871300\pi\)
\(434\) −1.16558e8 −0.0684426
\(435\) 0 0
\(436\) −1.72004e9 −0.993885
\(437\) −2.72063e9 −1.55950
\(438\) 0 0
\(439\) −1.28295e9 −0.723742 −0.361871 0.932228i \(-0.617862\pi\)
−0.361871 + 0.932228i \(0.617862\pi\)
\(440\) −4.95771e8 −0.277458
\(441\) 0 0
\(442\) −2.30095e9 −1.26745
\(443\) 1.73263e9 0.946878 0.473439 0.880827i \(-0.343012\pi\)
0.473439 + 0.880827i \(0.343012\pi\)
\(444\) 0 0
\(445\) −9.16223e8 −0.492880
\(446\) 2.16175e9 1.15381
\(447\) 0 0
\(448\) 1.96964e8 0.103493
\(449\) −1.86903e9 −0.974439 −0.487219 0.873280i \(-0.661989\pi\)
−0.487219 + 0.873280i \(0.661989\pi\)
\(450\) 0 0
\(451\) −1.24176e9 −0.637413
\(452\) −2.22172e9 −1.13163
\(453\) 0 0
\(454\) 4.71816e9 2.36634
\(455\) −5.00853e8 −0.249270
\(456\) 0 0
\(457\) 1.76868e9 0.866849 0.433425 0.901190i \(-0.357305\pi\)
0.433425 + 0.901190i \(0.357305\pi\)
\(458\) −4.53679e9 −2.20658
\(459\) 0 0
\(460\) 8.37972e8 0.401400
\(461\) 2.55825e8 0.121616 0.0608078 0.998149i \(-0.480632\pi\)
0.0608078 + 0.998149i \(0.480632\pi\)
\(462\) 0 0
\(463\) −4.19121e9 −1.96249 −0.981243 0.192777i \(-0.938250\pi\)
−0.981243 + 0.192777i \(0.938250\pi\)
\(464\) −4.42052e9 −2.05428
\(465\) 0 0
\(466\) 2.63998e9 1.20851
\(467\) 2.94239e9 1.33688 0.668438 0.743768i \(-0.266963\pi\)
0.668438 + 0.743768i \(0.266963\pi\)
\(468\) 0 0
\(469\) −7.76569e8 −0.347596
\(470\) 4.40364e8 0.195645
\(471\) 0 0
\(472\) −2.76825e8 −0.121174
\(473\) 2.01171e9 0.874079
\(474\) 0 0
\(475\) 5.46179e8 0.233834
\(476\) 3.97667e8 0.169003
\(477\) 0 0
\(478\) 5.36292e9 2.24597
\(479\) −4.57933e9 −1.90383 −0.951914 0.306366i \(-0.900887\pi\)
−0.951914 + 0.306366i \(0.900887\pi\)
\(480\) 0 0
\(481\) 4.93425e9 2.02169
\(482\) −3.21535e9 −1.30787
\(483\) 0 0
\(484\) 1.93115e9 0.774206
\(485\) 2.98345e8 0.118747
\(486\) 0 0
\(487\) −1.40131e9 −0.549771 −0.274886 0.961477i \(-0.588640\pi\)
−0.274886 + 0.961477i \(0.588640\pi\)
\(488\) −5.09528e8 −0.198472
\(489\) 0 0
\(490\) 2.15198e8 0.0826329
\(491\) 4.79712e9 1.82892 0.914462 0.404672i \(-0.132614\pi\)
0.914462 + 0.404672i \(0.132614\pi\)
\(492\) 0 0
\(493\) −2.97658e9 −1.11880
\(494\) 5.97533e9 2.23007
\(495\) 0 0
\(496\) 4.64218e8 0.170819
\(497\) −7.61585e8 −0.278273
\(498\) 0 0
\(499\) −2.01049e9 −0.724351 −0.362176 0.932110i \(-0.617966\pi\)
−0.362176 + 0.932110i \(0.617966\pi\)
\(500\) −1.68227e8 −0.0601866
\(501\) 0 0
\(502\) −1.90050e9 −0.670509
\(503\) −1.68618e9 −0.590766 −0.295383 0.955379i \(-0.595447\pi\)
−0.295383 + 0.955379i \(0.595447\pi\)
\(504\) 0 0
\(505\) −2.40141e9 −0.829749
\(506\) 7.37300e9 2.52998
\(507\) 0 0
\(508\) −5.81983e8 −0.196964
\(509\) −1.63477e9 −0.549470 −0.274735 0.961520i \(-0.588590\pi\)
−0.274735 + 0.961520i \(0.588590\pi\)
\(510\) 0 0
\(511\) −1.71286e9 −0.567871
\(512\) −2.71225e9 −0.893068
\(513\) 0 0
\(514\) −4.31147e9 −1.40041
\(515\) 5.56698e8 0.179595
\(516\) 0 0
\(517\) 1.55851e9 0.496012
\(518\) −2.12007e9 −0.670187
\(519\) 0 0
\(520\) 8.94621e8 0.279015
\(521\) −3.28595e9 −1.01796 −0.508978 0.860779i \(-0.669977\pi\)
−0.508978 + 0.860779i \(0.669977\pi\)
\(522\) 0 0
\(523\) 1.29734e9 0.396549 0.198274 0.980147i \(-0.436466\pi\)
0.198274 + 0.980147i \(0.436466\pi\)
\(524\) 1.90406e8 0.0578123
\(525\) 0 0
\(526\) 4.59418e9 1.37644
\(527\) 3.12583e8 0.0930313
\(528\) 0 0
\(529\) 2.65291e9 0.779161
\(530\) 1.95087e9 0.569198
\(531\) 0 0
\(532\) −1.03270e9 −0.297360
\(533\) 2.24076e9 0.640990
\(534\) 0 0
\(535\) 9.51731e8 0.268705
\(536\) 1.38710e9 0.389074
\(537\) 0 0
\(538\) −4.31308e9 −1.19412
\(539\) 7.61617e8 0.209496
\(540\) 0 0
\(541\) 1.23726e9 0.335948 0.167974 0.985791i \(-0.446278\pi\)
0.167974 + 0.985791i \(0.446278\pi\)
\(542\) 1.23986e9 0.334484
\(543\) 0 0
\(544\) −2.88189e9 −0.767505
\(545\) 2.49623e9 0.660536
\(546\) 0 0
\(547\) −4.27288e9 −1.11626 −0.558130 0.829754i \(-0.688481\pi\)
−0.558130 + 0.829754i \(0.688481\pi\)
\(548\) 6.12037e8 0.158871
\(549\) 0 0
\(550\) −1.48016e9 −0.379350
\(551\) 7.72986e9 1.96853
\(552\) 0 0
\(553\) 9.35610e8 0.235265
\(554\) −2.88141e9 −0.719982
\(555\) 0 0
\(556\) 7.94006e8 0.195912
\(557\) 2.41921e9 0.593171 0.296586 0.955006i \(-0.404152\pi\)
0.296586 + 0.955006i \(0.404152\pi\)
\(558\) 0 0
\(559\) −3.63013e9 −0.878985
\(560\) −8.57079e8 −0.206235
\(561\) 0 0
\(562\) 4.86306e9 1.15567
\(563\) 3.03839e9 0.717570 0.358785 0.933420i \(-0.383191\pi\)
0.358785 + 0.933420i \(0.383191\pi\)
\(564\) 0 0
\(565\) 3.22430e9 0.752084
\(566\) −6.44503e9 −1.49406
\(567\) 0 0
\(568\) 1.36034e9 0.311478
\(569\) −2.85539e9 −0.649788 −0.324894 0.945750i \(-0.605329\pi\)
−0.324894 + 0.945750i \(0.605329\pi\)
\(570\) 0 0
\(571\) 1.87867e9 0.422304 0.211152 0.977453i \(-0.432279\pi\)
0.211152 + 0.977453i \(0.432279\pi\)
\(572\) −6.51356e9 −1.45523
\(573\) 0 0
\(574\) −9.62776e8 −0.212488
\(575\) −1.21612e9 −0.266770
\(576\) 0 0
\(577\) 2.19182e9 0.474996 0.237498 0.971388i \(-0.423673\pi\)
0.237498 + 0.971388i \(0.423673\pi\)
\(578\) 3.35327e9 0.722306
\(579\) 0 0
\(580\) −2.38085e9 −0.506679
\(581\) −2.18969e9 −0.463196
\(582\) 0 0
\(583\) 6.90441e9 1.44307
\(584\) 3.05951e9 0.635633
\(585\) 0 0
\(586\) −4.46399e9 −0.916392
\(587\) 4.70415e9 0.959949 0.479974 0.877283i \(-0.340646\pi\)
0.479974 + 0.877283i \(0.340646\pi\)
\(588\) 0 0
\(589\) −8.11747e8 −0.163688
\(590\) −8.26482e8 −0.165673
\(591\) 0 0
\(592\) 8.44368e9 1.67265
\(593\) −3.66996e9 −0.722719 −0.361359 0.932427i \(-0.617687\pi\)
−0.361359 + 0.932427i \(0.617687\pi\)
\(594\) 0 0
\(595\) −5.77118e8 −0.112320
\(596\) −1.14812e9 −0.222139
\(597\) 0 0
\(598\) −1.33046e10 −2.54418
\(599\) 5.46935e9 1.03978 0.519890 0.854233i \(-0.325973\pi\)
0.519890 + 0.854233i \(0.325973\pi\)
\(600\) 0 0
\(601\) 2.74417e9 0.515645 0.257822 0.966192i \(-0.416995\pi\)
0.257822 + 0.966192i \(0.416995\pi\)
\(602\) 1.55974e9 0.291383
\(603\) 0 0
\(604\) −5.52915e9 −1.02101
\(605\) −2.80260e9 −0.514537
\(606\) 0 0
\(607\) −2.26388e9 −0.410859 −0.205430 0.978672i \(-0.565859\pi\)
−0.205430 + 0.978672i \(0.565859\pi\)
\(608\) 7.48397e9 1.35042
\(609\) 0 0
\(610\) −1.52123e9 −0.271357
\(611\) −2.81233e9 −0.498795
\(612\) 0 0
\(613\) −9.92731e9 −1.74068 −0.870342 0.492448i \(-0.836102\pi\)
−0.870342 + 0.492448i \(0.836102\pi\)
\(614\) 3.53563e9 0.616422
\(615\) 0 0
\(616\) −1.36040e9 −0.234495
\(617\) −2.27156e9 −0.389338 −0.194669 0.980869i \(-0.562363\pi\)
−0.194669 + 0.980869i \(0.562363\pi\)
\(618\) 0 0
\(619\) 6.90552e8 0.117025 0.0585126 0.998287i \(-0.481364\pi\)
0.0585126 + 0.998287i \(0.481364\pi\)
\(620\) 2.50023e8 0.0421317
\(621\) 0 0
\(622\) −9.83015e8 −0.163792
\(623\) −2.51412e9 −0.416560
\(624\) 0 0
\(625\) 2.44141e8 0.0400000
\(626\) 9.87867e9 1.60949
\(627\) 0 0
\(628\) 6.34445e9 1.02220
\(629\) 5.68560e9 0.910959
\(630\) 0 0
\(631\) −1.00992e10 −1.60023 −0.800116 0.599846i \(-0.795229\pi\)
−0.800116 + 0.599846i \(0.795229\pi\)
\(632\) −1.67118e9 −0.263338
\(633\) 0 0
\(634\) −6.80665e9 −1.06077
\(635\) 8.44609e8 0.130902
\(636\) 0 0
\(637\) −1.37434e9 −0.210672
\(638\) −2.09482e10 −3.19355
\(639\) 0 0
\(640\) 2.37523e9 0.358159
\(641\) −8.50418e8 −0.127535 −0.0637675 0.997965i \(-0.520312\pi\)
−0.0637675 + 0.997965i \(0.520312\pi\)
\(642\) 0 0
\(643\) 1.56624e9 0.232339 0.116169 0.993229i \(-0.462938\pi\)
0.116169 + 0.993229i \(0.462938\pi\)
\(644\) 2.29940e9 0.339245
\(645\) 0 0
\(646\) 6.88520e9 1.00485
\(647\) −1.06053e10 −1.53942 −0.769712 0.638391i \(-0.779600\pi\)
−0.769712 + 0.638391i \(0.779600\pi\)
\(648\) 0 0
\(649\) −2.92503e9 −0.420024
\(650\) 2.67096e9 0.381478
\(651\) 0 0
\(652\) −3.02015e9 −0.426739
\(653\) 1.83644e9 0.258096 0.129048 0.991638i \(-0.458808\pi\)
0.129048 + 0.991638i \(0.458808\pi\)
\(654\) 0 0
\(655\) −2.76328e8 −0.0384221
\(656\) 3.83448e9 0.530327
\(657\) 0 0
\(658\) 1.20836e9 0.165350
\(659\) 5.03583e9 0.685444 0.342722 0.939437i \(-0.388651\pi\)
0.342722 + 0.939437i \(0.388651\pi\)
\(660\) 0 0
\(661\) −1.27453e10 −1.71651 −0.858254 0.513225i \(-0.828451\pi\)
−0.858254 + 0.513225i \(0.828451\pi\)
\(662\) −3.94774e9 −0.528866
\(663\) 0 0
\(664\) 3.91121e9 0.518469
\(665\) 1.49872e9 0.197626
\(666\) 0 0
\(667\) −1.72112e10 −2.24580
\(668\) −2.21316e8 −0.0287274
\(669\) 0 0
\(670\) 4.14130e9 0.531955
\(671\) −5.38386e9 −0.687962
\(672\) 0 0
\(673\) 1.02193e10 1.29231 0.646155 0.763206i \(-0.276376\pi\)
0.646155 + 0.763206i \(0.276376\pi\)
\(674\) 9.20567e9 1.15810
\(675\) 0 0
\(676\) 6.34910e9 0.790494
\(677\) −1.04119e10 −1.28965 −0.644823 0.764332i \(-0.723069\pi\)
−0.644823 + 0.764332i \(0.723069\pi\)
\(678\) 0 0
\(679\) 8.18659e8 0.100360
\(680\) 1.03085e9 0.125722
\(681\) 0 0
\(682\) 2.19986e9 0.265552
\(683\) −6.56705e9 −0.788675 −0.394338 0.918966i \(-0.629026\pi\)
−0.394338 + 0.918966i \(0.629026\pi\)
\(684\) 0 0
\(685\) −8.88226e8 −0.105586
\(686\) 5.90504e8 0.0698375
\(687\) 0 0
\(688\) −6.21203e9 −0.727233
\(689\) −1.24590e10 −1.45117
\(690\) 0 0
\(691\) 4.44242e9 0.512208 0.256104 0.966649i \(-0.417561\pi\)
0.256104 + 0.966649i \(0.417561\pi\)
\(692\) −6.92039e9 −0.793888
\(693\) 0 0
\(694\) −1.11464e10 −1.26583
\(695\) −1.15231e9 −0.130203
\(696\) 0 0
\(697\) 2.58197e9 0.288826
\(698\) 4.85296e9 0.540148
\(699\) 0 0
\(700\) −4.61614e8 −0.0508669
\(701\) 7.92343e9 0.868761 0.434380 0.900729i \(-0.356967\pi\)
0.434380 + 0.900729i \(0.356967\pi\)
\(702\) 0 0
\(703\) −1.47649e10 −1.60283
\(704\) −3.71741e9 −0.401546
\(705\) 0 0
\(706\) 1.10835e10 1.18539
\(707\) −6.58948e9 −0.701266
\(708\) 0 0
\(709\) 9.27216e9 0.977055 0.488528 0.872548i \(-0.337534\pi\)
0.488528 + 0.872548i \(0.337534\pi\)
\(710\) 4.06139e9 0.425864
\(711\) 0 0
\(712\) 4.49070e9 0.466266
\(713\) 1.80743e9 0.186744
\(714\) 0 0
\(715\) 9.45288e9 0.967148
\(716\) −8.60522e9 −0.876126
\(717\) 0 0
\(718\) 2.14810e10 2.16580
\(719\) 5.48244e9 0.550076 0.275038 0.961433i \(-0.411310\pi\)
0.275038 + 0.961433i \(0.411310\pi\)
\(720\) 0 0
\(721\) 1.52758e9 0.151786
\(722\) −4.79990e9 −0.474626
\(723\) 0 0
\(724\) −9.25181e9 −0.906029
\(725\) 3.45523e9 0.336739
\(726\) 0 0
\(727\) −9.22691e9 −0.890606 −0.445303 0.895380i \(-0.646904\pi\)
−0.445303 + 0.895380i \(0.646904\pi\)
\(728\) 2.45484e9 0.235811
\(729\) 0 0
\(730\) 9.13438e9 0.869059
\(731\) −4.18290e9 −0.396065
\(732\) 0 0
\(733\) 4.96865e9 0.465988 0.232994 0.972478i \(-0.425148\pi\)
0.232994 + 0.972478i \(0.425148\pi\)
\(734\) −2.40885e10 −2.24840
\(735\) 0 0
\(736\) −1.66637e10 −1.54063
\(737\) 1.46566e10 1.34865
\(738\) 0 0
\(739\) −1.96084e9 −0.178726 −0.0893628 0.995999i \(-0.528483\pi\)
−0.0893628 + 0.995999i \(0.528483\pi\)
\(740\) 4.54768e9 0.412552
\(741\) 0 0
\(742\) 5.35320e9 0.481060
\(743\) −9.56947e9 −0.855908 −0.427954 0.903801i \(-0.640765\pi\)
−0.427954 + 0.903801i \(0.640765\pi\)
\(744\) 0 0
\(745\) 1.66622e9 0.147634
\(746\) 1.70312e10 1.50196
\(747\) 0 0
\(748\) −7.50539e9 −0.655719
\(749\) 2.61155e9 0.227097
\(750\) 0 0
\(751\) 8.11719e9 0.699304 0.349652 0.936880i \(-0.386300\pi\)
0.349652 + 0.936880i \(0.386300\pi\)
\(752\) −4.81257e9 −0.412681
\(753\) 0 0
\(754\) 3.78010e10 3.21147
\(755\) 8.02424e9 0.678562
\(756\) 0 0
\(757\) −9.11117e9 −0.763376 −0.381688 0.924291i \(-0.624657\pi\)
−0.381688 + 0.924291i \(0.624657\pi\)
\(758\) −5.96689e9 −0.497629
\(759\) 0 0
\(760\) −2.67700e9 −0.221208
\(761\) −1.71359e10 −1.40948 −0.704742 0.709464i \(-0.748937\pi\)
−0.704742 + 0.709464i \(0.748937\pi\)
\(762\) 0 0
\(763\) 6.84965e9 0.558255
\(764\) 1.90425e9 0.154489
\(765\) 0 0
\(766\) 1.03947e10 0.835621
\(767\) 5.27823e9 0.422381
\(768\) 0 0
\(769\) −1.82316e10 −1.44572 −0.722858 0.690997i \(-0.757172\pi\)
−0.722858 + 0.690997i \(0.757172\pi\)
\(770\) −4.06156e9 −0.320609
\(771\) 0 0
\(772\) −1.29020e10 −1.00924
\(773\) 1.45965e10 1.13663 0.568316 0.822810i \(-0.307595\pi\)
0.568316 + 0.822810i \(0.307595\pi\)
\(774\) 0 0
\(775\) −3.62849e8 −0.0280007
\(776\) −1.46228e9 −0.112335
\(777\) 0 0
\(778\) 2.85481e9 0.217345
\(779\) −6.70510e9 −0.508188
\(780\) 0 0
\(781\) 1.43738e10 1.07968
\(782\) −1.53305e10 −1.14639
\(783\) 0 0
\(784\) −2.35182e9 −0.174300
\(785\) −9.20745e9 −0.679353
\(786\) 0 0
\(787\) 1.42358e10 1.04104 0.520522 0.853848i \(-0.325737\pi\)
0.520522 + 0.853848i \(0.325737\pi\)
\(788\) 4.91045e9 0.357503
\(789\) 0 0
\(790\) −4.98944e9 −0.360045
\(791\) 8.84748e9 0.635627
\(792\) 0 0
\(793\) 9.71519e9 0.691823
\(794\) −2.31544e10 −1.64158
\(795\) 0 0
\(796\) −5.89281e9 −0.414121
\(797\) −1.03325e10 −0.722935 −0.361467 0.932385i \(-0.617724\pi\)
−0.361467 + 0.932385i \(0.617724\pi\)
\(798\) 0 0
\(799\) −3.24057e9 −0.224754
\(800\) 3.34531e9 0.231005
\(801\) 0 0
\(802\) −3.12635e10 −2.14007
\(803\) 3.23278e10 2.20329
\(804\) 0 0
\(805\) −3.33702e9 −0.225462
\(806\) −3.96965e9 −0.267042
\(807\) 0 0
\(808\) 1.17701e10 0.784947
\(809\) 1.58064e10 1.04957 0.524787 0.851233i \(-0.324145\pi\)
0.524787 + 0.851233i \(0.324145\pi\)
\(810\) 0 0
\(811\) 2.87547e9 0.189294 0.0946469 0.995511i \(-0.469828\pi\)
0.0946469 + 0.995511i \(0.469828\pi\)
\(812\) −6.53304e9 −0.428222
\(813\) 0 0
\(814\) 4.00133e10 2.60027
\(815\) 4.38303e9 0.283611
\(816\) 0 0
\(817\) 1.08626e10 0.696875
\(818\) 5.93575e9 0.379175
\(819\) 0 0
\(820\) 2.06521e9 0.130803
\(821\) 1.42014e10 0.895633 0.447817 0.894125i \(-0.352202\pi\)
0.447817 + 0.894125i \(0.352202\pi\)
\(822\) 0 0
\(823\) −2.79354e10 −1.74685 −0.873424 0.486961i \(-0.838105\pi\)
−0.873424 + 0.486961i \(0.838105\pi\)
\(824\) −2.72856e9 −0.169898
\(825\) 0 0
\(826\) −2.26787e9 −0.140019
\(827\) 1.48560e8 0.00913341 0.00456670 0.999990i \(-0.498546\pi\)
0.00456670 + 0.999990i \(0.498546\pi\)
\(828\) 0 0
\(829\) 3.98861e9 0.243154 0.121577 0.992582i \(-0.461205\pi\)
0.121577 + 0.992582i \(0.461205\pi\)
\(830\) 1.16772e10 0.708868
\(831\) 0 0
\(832\) 6.70807e9 0.403800
\(833\) −1.58361e9 −0.0949273
\(834\) 0 0
\(835\) 3.21188e8 0.0190922
\(836\) 1.94907e10 1.15373
\(837\) 0 0
\(838\) 1.57271e10 0.923198
\(839\) −2.43693e9 −0.142455 −0.0712273 0.997460i \(-0.522692\pi\)
−0.0712273 + 0.997460i \(0.522692\pi\)
\(840\) 0 0
\(841\) 3.16506e10 1.83483
\(842\) 1.21880e10 0.703625
\(843\) 0 0
\(844\) −1.17313e10 −0.671655
\(845\) −9.21419e9 −0.525362
\(846\) 0 0
\(847\) −7.69033e9 −0.434863
\(848\) −2.13204e10 −1.20063
\(849\) 0 0
\(850\) 3.07767e9 0.171892
\(851\) 3.28754e10 1.82859
\(852\) 0 0
\(853\) 1.54310e9 0.0851282 0.0425641 0.999094i \(-0.486447\pi\)
0.0425641 + 0.999094i \(0.486447\pi\)
\(854\) −4.17427e9 −0.229339
\(855\) 0 0
\(856\) −4.66474e9 −0.254196
\(857\) 1.29972e10 0.705369 0.352684 0.935742i \(-0.385269\pi\)
0.352684 + 0.935742i \(0.385269\pi\)
\(858\) 0 0
\(859\) −1.98316e10 −1.06754 −0.533768 0.845631i \(-0.679224\pi\)
−0.533768 + 0.845631i \(0.679224\pi\)
\(860\) −3.34573e9 −0.179369
\(861\) 0 0
\(862\) 1.20913e8 0.00642982
\(863\) −4.94264e9 −0.261771 −0.130886 0.991397i \(-0.541782\pi\)
−0.130886 + 0.991397i \(0.541782\pi\)
\(864\) 0 0
\(865\) 1.00433e10 0.527618
\(866\) 4.54536e10 2.37824
\(867\) 0 0
\(868\) 6.86063e8 0.0356078
\(869\) −1.76583e10 −0.912809
\(870\) 0 0
\(871\) −2.64480e10 −1.35621
\(872\) −1.22348e10 −0.624870
\(873\) 0 0
\(874\) 3.98117e10 2.01707
\(875\) 6.69922e8 0.0338062
\(876\) 0 0
\(877\) −7.37011e9 −0.368957 −0.184478 0.982837i \(-0.559060\pi\)
−0.184478 + 0.982837i \(0.559060\pi\)
\(878\) 1.87737e10 0.936094
\(879\) 0 0
\(880\) 1.61761e10 0.800176
\(881\) −9.74385e9 −0.480081 −0.240041 0.970763i \(-0.577161\pi\)
−0.240041 + 0.970763i \(0.577161\pi\)
\(882\) 0 0
\(883\) −2.79540e10 −1.36641 −0.683206 0.730226i \(-0.739415\pi\)
−0.683206 + 0.730226i \(0.739415\pi\)
\(884\) 1.35435e10 0.659399
\(885\) 0 0
\(886\) −2.53541e10 −1.22470
\(887\) 2.08176e10 1.00161 0.500804 0.865561i \(-0.333038\pi\)
0.500804 + 0.865561i \(0.333038\pi\)
\(888\) 0 0
\(889\) 2.31761e9 0.110633
\(890\) 1.34073e10 0.637495
\(891\) 0 0
\(892\) −1.27241e10 −0.600276
\(893\) 8.41542e9 0.395454
\(894\) 0 0
\(895\) 1.24884e10 0.582273
\(896\) 6.51763e9 0.302700
\(897\) 0 0
\(898\) 2.73500e10 1.26035
\(899\) −5.13526e9 −0.235724
\(900\) 0 0
\(901\) −1.43562e10 −0.653886
\(902\) 1.81710e10 0.824435
\(903\) 0 0
\(904\) −1.58033e10 −0.711474
\(905\) 1.34268e10 0.602147
\(906\) 0 0
\(907\) 3.61602e10 1.60918 0.804591 0.593830i \(-0.202385\pi\)
0.804591 + 0.593830i \(0.202385\pi\)
\(908\) −2.77713e10 −1.23111
\(909\) 0 0
\(910\) 7.32910e9 0.322408
\(911\) −1.79811e10 −0.787954 −0.393977 0.919120i \(-0.628901\pi\)
−0.393977 + 0.919120i \(0.628901\pi\)
\(912\) 0 0
\(913\) 4.13272e10 1.79717
\(914\) −2.58816e10 −1.12119
\(915\) 0 0
\(916\) 2.67038e10 1.14799
\(917\) −7.58245e8 −0.0324726
\(918\) 0 0
\(919\) −1.01373e10 −0.430841 −0.215421 0.976521i \(-0.569112\pi\)
−0.215421 + 0.976521i \(0.569112\pi\)
\(920\) 5.96057e9 0.252366
\(921\) 0 0
\(922\) −3.74355e9 −0.157299
\(923\) −2.59376e10 −1.08574
\(924\) 0 0
\(925\) −6.59987e9 −0.274182
\(926\) 6.13311e10 2.53830
\(927\) 0 0
\(928\) 4.73449e10 1.94471
\(929\) 2.39605e10 0.980485 0.490242 0.871586i \(-0.336908\pi\)
0.490242 + 0.871586i \(0.336908\pi\)
\(930\) 0 0
\(931\) 4.11248e9 0.167024
\(932\) −1.55391e10 −0.628738
\(933\) 0 0
\(934\) −4.30567e10 −1.72913
\(935\) 1.08923e10 0.435791
\(936\) 0 0
\(937\) 1.16627e10 0.463138 0.231569 0.972818i \(-0.425614\pi\)
0.231569 + 0.972818i \(0.425614\pi\)
\(938\) 1.13637e10 0.449584
\(939\) 0 0
\(940\) −2.59200e9 −0.101786
\(941\) −3.03134e10 −1.18596 −0.592982 0.805216i \(-0.702049\pi\)
−0.592982 + 0.805216i \(0.702049\pi\)
\(942\) 0 0
\(943\) 1.49295e10 0.579768
\(944\) 9.03231e9 0.349460
\(945\) 0 0
\(946\) −2.94378e10 −1.13054
\(947\) −2.84339e10 −1.08796 −0.543979 0.839099i \(-0.683083\pi\)
−0.543979 + 0.839099i \(0.683083\pi\)
\(948\) 0 0
\(949\) −5.83357e10 −2.21566
\(950\) −7.99238e9 −0.302443
\(951\) 0 0
\(952\) 2.82864e9 0.106255
\(953\) 4.09127e10 1.53120 0.765602 0.643315i \(-0.222441\pi\)
0.765602 + 0.643315i \(0.222441\pi\)
\(954\) 0 0
\(955\) −2.76356e9 −0.102673
\(956\) −3.15664e10 −1.16848
\(957\) 0 0
\(958\) 6.70105e10 2.46243
\(959\) −2.43729e9 −0.0892365
\(960\) 0 0
\(961\) −2.69733e10 −0.980399
\(962\) −7.22042e10 −2.61487
\(963\) 0 0
\(964\) 1.89257e10 0.680428
\(965\) 1.87241e10 0.670742
\(966\) 0 0
\(967\) −4.11324e10 −1.46282 −0.731411 0.681937i \(-0.761138\pi\)
−0.731411 + 0.681937i \(0.761138\pi\)
\(968\) 1.37364e10 0.486754
\(969\) 0 0
\(970\) −4.36576e9 −0.153588
\(971\) −2.85539e10 −1.00092 −0.500459 0.865760i \(-0.666835\pi\)
−0.500459 + 0.865760i \(0.666835\pi\)
\(972\) 0 0
\(973\) −3.16194e9 −0.110042
\(974\) 2.05057e10 0.711079
\(975\) 0 0
\(976\) 1.66250e10 0.572384
\(977\) −3.64395e10 −1.25009 −0.625045 0.780589i \(-0.714919\pi\)
−0.625045 + 0.780589i \(0.714919\pi\)
\(978\) 0 0
\(979\) 4.74503e10 1.61622
\(980\) −1.26667e9 −0.0429904
\(981\) 0 0
\(982\) −7.01975e10 −2.36555
\(983\) 3.71636e10 1.24790 0.623951 0.781463i \(-0.285526\pi\)
0.623951 + 0.781463i \(0.285526\pi\)
\(984\) 0 0
\(985\) −7.12634e9 −0.237596
\(986\) 4.35570e10 1.44707
\(987\) 0 0
\(988\) −3.51711e10 −1.16021
\(989\) −2.41864e10 −0.795032
\(990\) 0 0
\(991\) 1.80657e10 0.589655 0.294827 0.955551i \(-0.404738\pi\)
0.294827 + 0.955551i \(0.404738\pi\)
\(992\) −4.97190e9 −0.161708
\(993\) 0 0
\(994\) 1.11445e10 0.359921
\(995\) 8.55201e9 0.275225
\(996\) 0 0
\(997\) −2.62408e10 −0.838580 −0.419290 0.907852i \(-0.637721\pi\)
−0.419290 + 0.907852i \(0.637721\pi\)
\(998\) 2.94199e10 0.936882
\(999\) 0 0
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 315.8.a.c.1.1 2
3.2 odd 2 35.8.a.a.1.2 2
12.11 even 2 560.8.a.i.1.2 2
15.2 even 4 175.8.b.c.99.4 4
15.8 even 4 175.8.b.c.99.1 4
15.14 odd 2 175.8.a.b.1.1 2
21.20 even 2 245.8.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.8.a.a.1.2 2 3.2 odd 2
175.8.a.b.1.1 2 15.14 odd 2
175.8.b.c.99.1 4 15.8 even 4
175.8.b.c.99.4 4 15.2 even 4
245.8.a.b.1.2 2 21.20 even 2
315.8.a.c.1.1 2 1.1 even 1 trivial
560.8.a.i.1.2 2 12.11 even 2