Properties

Label 1045.6.a.e.1.10
Level $1045$
Weight $6$
Character 1045.1
Self dual yes
Analytic conductor $167.601$
Analytic rank $1$
Dimension $38$
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] = [1045,6,Mod(1,1045)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1045, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1045.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1045.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: \(167.601091705\)
Analytic rank: \(1\)
Dimension: \(38\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Character \(\chi\) \(=\) 1045.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-7.34806 q^{2} +21.4317 q^{3} +21.9939 q^{4} +25.0000 q^{5} -157.482 q^{6} +84.2288 q^{7} +73.5250 q^{8} +216.320 q^{9} +O(q^{10})\) \(q-7.34806 q^{2} +21.4317 q^{3} +21.9939 q^{4} +25.0000 q^{5} -157.482 q^{6} +84.2288 q^{7} +73.5250 q^{8} +216.320 q^{9} -183.701 q^{10} +121.000 q^{11} +471.369 q^{12} -947.397 q^{13} -618.918 q^{14} +535.794 q^{15} -1244.07 q^{16} +938.005 q^{17} -1589.53 q^{18} -361.000 q^{19} +549.849 q^{20} +1805.17 q^{21} -889.115 q^{22} -752.093 q^{23} +1575.77 q^{24} +625.000 q^{25} +6961.53 q^{26} -571.803 q^{27} +1852.52 q^{28} +2571.32 q^{29} -3937.04 q^{30} -2474.20 q^{31} +6788.72 q^{32} +2593.24 q^{33} -6892.52 q^{34} +2105.72 q^{35} +4757.73 q^{36} -8564.08 q^{37} +2652.65 q^{38} -20304.4 q^{39} +1838.13 q^{40} +7460.09 q^{41} -13264.5 q^{42} -9669.23 q^{43} +2661.27 q^{44} +5408.00 q^{45} +5526.42 q^{46} +22597.7 q^{47} -26662.7 q^{48} -9712.51 q^{49} -4592.54 q^{50} +20103.1 q^{51} -20837.0 q^{52} -37071.2 q^{53} +4201.64 q^{54} +3025.00 q^{55} +6192.92 q^{56} -7736.86 q^{57} -18894.2 q^{58} -9732.55 q^{59} +11784.2 q^{60} -37427.8 q^{61} +18180.6 q^{62} +18220.4 q^{63} -10073.5 q^{64} -23684.9 q^{65} -19055.3 q^{66} +618.633 q^{67} +20630.4 q^{68} -16118.7 q^{69} -15472.9 q^{70} +47045.0 q^{71} +15904.9 q^{72} -63309.4 q^{73} +62929.3 q^{74} +13394.8 q^{75} -7939.82 q^{76} +10191.7 q^{77} +149198. q^{78} -20032.8 q^{79} -31101.8 q^{80} -64820.5 q^{81} -54817.2 q^{82} +87828.4 q^{83} +39702.8 q^{84} +23450.1 q^{85} +71050.0 q^{86} +55108.0 q^{87} +8896.53 q^{88} -10825.3 q^{89} -39738.3 q^{90} -79798.1 q^{91} -16541.5 q^{92} -53026.5 q^{93} -166049. q^{94} -9025.00 q^{95} +145494. q^{96} +23614.4 q^{97} +71368.1 q^{98} +26174.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 38 q - 24 q^{2} - 63 q^{3} + 594 q^{4} + 950 q^{5} - 67 q^{6} - 729 q^{7} - 1272 q^{8} + 3029 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 38 q - 24 q^{2} - 63 q^{3} + 594 q^{4} + 950 q^{5} - 67 q^{6} - 729 q^{7} - 1272 q^{8} + 3029 q^{9} - 600 q^{10} + 4598 q^{11} - 2008 q^{12} - 2663 q^{13} - 1565 q^{14} - 1575 q^{15} + 12390 q^{16} - 3311 q^{17} - 6383 q^{18} - 13718 q^{19} + 14850 q^{20} - 8179 q^{21} - 2904 q^{22} - 3412 q^{23} - 4100 q^{24} + 23750 q^{25} - 1399 q^{26} - 31596 q^{27} - 43653 q^{28} - 13633 q^{29} - 1675 q^{30} - 13789 q^{31} - 58603 q^{32} - 7623 q^{33} - 29149 q^{34} - 18225 q^{35} + 50641 q^{36} - 12103 q^{37} + 8664 q^{38} - 50960 q^{39} - 31800 q^{40} - 37885 q^{41} + 51100 q^{42} - 56119 q^{43} + 71874 q^{44} + 75725 q^{45} - 56291 q^{46} - 37532 q^{47} - 113895 q^{48} + 153501 q^{49} - 15000 q^{50} + 32882 q^{51} - 169554 q^{52} - 51511 q^{53} - 175060 q^{54} + 114950 q^{55} - 84247 q^{56} + 22743 q^{57} - 256962 q^{58} - 154267 q^{59} - 50200 q^{60} - 47165 q^{61} + 143002 q^{62} - 358780 q^{63} + 142292 q^{64} - 66575 q^{65} - 8107 q^{66} - 161712 q^{67} - 210188 q^{68} - 124602 q^{69} - 39125 q^{70} + 6118 q^{71} - 327878 q^{72} - 152182 q^{73} - 167349 q^{74} - 39375 q^{75} - 214434 q^{76} - 88209 q^{77} - 216594 q^{78} - 140433 q^{79} + 309750 q^{80} + 382874 q^{81} - 29842 q^{82} - 515287 q^{83} + 29222 q^{84} - 82775 q^{85} + 204974 q^{86} - 106764 q^{87} - 153912 q^{88} - 271610 q^{89} - 159575 q^{90} - 44332 q^{91} + 236348 q^{92} + 25202 q^{93} - 496224 q^{94} - 342950 q^{95} - 275218 q^{96} - 126390 q^{97} - 285506 q^{98} + 366509 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.34806 −1.29897 −0.649483 0.760376i \(-0.725014\pi\)
−0.649483 + 0.760376i \(0.725014\pi\)
\(3\) 21.4317 1.37485 0.687424 0.726257i \(-0.258742\pi\)
0.687424 + 0.726257i \(0.258742\pi\)
\(4\) 21.9939 0.687311
\(5\) 25.0000 0.447214
\(6\) −157.482 −1.78588
\(7\) 84.2288 0.649704 0.324852 0.945765i \(-0.394686\pi\)
0.324852 + 0.945765i \(0.394686\pi\)
\(8\) 73.5250 0.406172
\(9\) 216.320 0.890205
\(10\) −183.701 −0.580915
\(11\) 121.000 0.301511
\(12\) 471.369 0.944948
\(13\) −947.397 −1.55480 −0.777399 0.629008i \(-0.783461\pi\)
−0.777399 + 0.629008i \(0.783461\pi\)
\(14\) −618.918 −0.843943
\(15\) 535.794 0.614850
\(16\) −1244.07 −1.21491
\(17\) 938.005 0.787196 0.393598 0.919283i \(-0.371230\pi\)
0.393598 + 0.919283i \(0.371230\pi\)
\(18\) −1589.53 −1.15635
\(19\) −361.000 −0.229416
\(20\) 549.849 0.307375
\(21\) 1805.17 0.893243
\(22\) −889.115 −0.391653
\(23\) −752.093 −0.296450 −0.148225 0.988954i \(-0.547356\pi\)
−0.148225 + 0.988954i \(0.547356\pi\)
\(24\) 1575.77 0.558425
\(25\) 625.000 0.200000
\(26\) 6961.53 2.01963
\(27\) −571.803 −0.150951
\(28\) 1852.52 0.446548
\(29\) 2571.32 0.567756 0.283878 0.958860i \(-0.408379\pi\)
0.283878 + 0.958860i \(0.408379\pi\)
\(30\) −3937.04 −0.798669
\(31\) −2474.20 −0.462414 −0.231207 0.972905i \(-0.574267\pi\)
−0.231207 + 0.972905i \(0.574267\pi\)
\(32\) 6788.72 1.17196
\(33\) 2593.24 0.414532
\(34\) −6892.52 −1.02254
\(35\) 2105.72 0.290556
\(36\) 4757.73 0.611848
\(37\) −8564.08 −1.02843 −0.514217 0.857660i \(-0.671917\pi\)
−0.514217 + 0.857660i \(0.671917\pi\)
\(38\) 2652.65 0.298003
\(39\) −20304.4 −2.13761
\(40\) 1838.13 0.181646
\(41\) 7460.09 0.693082 0.346541 0.938035i \(-0.387356\pi\)
0.346541 + 0.938035i \(0.387356\pi\)
\(42\) −13264.5 −1.16029
\(43\) −9669.23 −0.797482 −0.398741 0.917064i \(-0.630553\pi\)
−0.398741 + 0.917064i \(0.630553\pi\)
\(44\) 2661.27 0.207232
\(45\) 5408.00 0.398112
\(46\) 5526.42 0.385079
\(47\) 22597.7 1.49217 0.746087 0.665849i \(-0.231930\pi\)
0.746087 + 0.665849i \(0.231930\pi\)
\(48\) −26662.7 −1.67032
\(49\) −9712.51 −0.577885
\(50\) −4592.54 −0.259793
\(51\) 20103.1 1.08227
\(52\) −20837.0 −1.06863
\(53\) −37071.2 −1.81279 −0.906393 0.422435i \(-0.861175\pi\)
−0.906393 + 0.422435i \(0.861175\pi\)
\(54\) 4201.64 0.196080
\(55\) 3025.00 0.134840
\(56\) 6192.92 0.263892
\(57\) −7736.86 −0.315412
\(58\) −18894.2 −0.737496
\(59\) −9732.55 −0.363996 −0.181998 0.983299i \(-0.558256\pi\)
−0.181998 + 0.983299i \(0.558256\pi\)
\(60\) 11784.2 0.422593
\(61\) −37427.8 −1.28786 −0.643932 0.765082i \(-0.722698\pi\)
−0.643932 + 0.765082i \(0.722698\pi\)
\(62\) 18180.6 0.600660
\(63\) 18220.4 0.578370
\(64\) −10073.5 −0.307420
\(65\) −23684.9 −0.695327
\(66\) −19055.3 −0.538463
\(67\) 618.633 0.0168363 0.00841814 0.999965i \(-0.497320\pi\)
0.00841814 + 0.999965i \(0.497320\pi\)
\(68\) 20630.4 0.541048
\(69\) −16118.7 −0.407574
\(70\) −15472.9 −0.377423
\(71\) 47045.0 1.10756 0.553780 0.832663i \(-0.313185\pi\)
0.553780 + 0.832663i \(0.313185\pi\)
\(72\) 15904.9 0.361577
\(73\) −63309.4 −1.39047 −0.695234 0.718783i \(-0.744699\pi\)
−0.695234 + 0.718783i \(0.744699\pi\)
\(74\) 62929.3 1.33590
\(75\) 13394.8 0.274969
\(76\) −7939.82 −0.157680
\(77\) 10191.7 0.195893
\(78\) 149198. 2.77668
\(79\) −20032.8 −0.361139 −0.180570 0.983562i \(-0.557794\pi\)
−0.180570 + 0.983562i \(0.557794\pi\)
\(80\) −31101.8 −0.543326
\(81\) −64820.5 −1.09774
\(82\) −54817.2 −0.900289
\(83\) 87828.4 1.39939 0.699697 0.714440i \(-0.253318\pi\)
0.699697 + 0.714440i \(0.253318\pi\)
\(84\) 39702.8 0.613936
\(85\) 23450.1 0.352045
\(86\) 71050.0 1.03590
\(87\) 55108.0 0.780578
\(88\) 8896.53 0.122466
\(89\) −10825.3 −0.144866 −0.0724329 0.997373i \(-0.523076\pi\)
−0.0724329 + 0.997373i \(0.523076\pi\)
\(90\) −39738.3 −0.517133
\(91\) −79798.1 −1.01016
\(92\) −16541.5 −0.203754
\(93\) −53026.5 −0.635749
\(94\) −166049. −1.93828
\(95\) −9025.00 −0.102598
\(96\) 145494. 1.61127
\(97\) 23614.4 0.254829 0.127414 0.991850i \(-0.459332\pi\)
0.127414 + 0.991850i \(0.459332\pi\)
\(98\) 71368.1 0.750653
\(99\) 26174.7 0.268407
\(100\) 13746.2 0.137462
\(101\) −100508. −0.980386 −0.490193 0.871614i \(-0.663074\pi\)
−0.490193 + 0.871614i \(0.663074\pi\)
\(102\) −147719. −1.40584
\(103\) 76767.1 0.712987 0.356494 0.934298i \(-0.383972\pi\)
0.356494 + 0.934298i \(0.383972\pi\)
\(104\) −69657.4 −0.631516
\(105\) 45129.2 0.399471
\(106\) 272401. 2.35475
\(107\) −221627. −1.87138 −0.935691 0.352820i \(-0.885223\pi\)
−0.935691 + 0.352820i \(0.885223\pi\)
\(108\) −12576.2 −0.103750
\(109\) −139956. −1.12831 −0.564153 0.825670i \(-0.690797\pi\)
−0.564153 + 0.825670i \(0.690797\pi\)
\(110\) −22227.9 −0.175152
\(111\) −183543. −1.41394
\(112\) −104787. −0.789335
\(113\) 10011.3 0.0737558 0.0368779 0.999320i \(-0.488259\pi\)
0.0368779 + 0.999320i \(0.488259\pi\)
\(114\) 56850.9 0.409709
\(115\) −18802.3 −0.132577
\(116\) 56553.6 0.390225
\(117\) −204941. −1.38409
\(118\) 71515.3 0.472818
\(119\) 79007.0 0.511444
\(120\) 39394.3 0.249735
\(121\) 14641.0 0.0909091
\(122\) 275022. 1.67289
\(123\) 159883. 0.952882
\(124\) −54417.5 −0.317822
\(125\) 15625.0 0.0894427
\(126\) −133884. −0.751282
\(127\) −352689. −1.94036 −0.970181 0.242381i \(-0.922072\pi\)
−0.970181 + 0.242381i \(0.922072\pi\)
\(128\) −143218. −0.772631
\(129\) −207228. −1.09642
\(130\) 174038. 0.903205
\(131\) −219577. −1.11792 −0.558958 0.829196i \(-0.688799\pi\)
−0.558958 + 0.829196i \(0.688799\pi\)
\(132\) 57035.6 0.284912
\(133\) −30406.6 −0.149052
\(134\) −4545.75 −0.0218698
\(135\) −14295.1 −0.0675075
\(136\) 68966.9 0.319737
\(137\) 79257.9 0.360779 0.180389 0.983595i \(-0.442264\pi\)
0.180389 + 0.983595i \(0.442264\pi\)
\(138\) 118441. 0.529424
\(139\) −152220. −0.668244 −0.334122 0.942530i \(-0.608440\pi\)
−0.334122 + 0.942530i \(0.608440\pi\)
\(140\) 46313.1 0.199703
\(141\) 484308. 2.05151
\(142\) −345689. −1.43868
\(143\) −114635. −0.468789
\(144\) −269118. −1.08152
\(145\) 64283.1 0.253908
\(146\) 465201. 1.80617
\(147\) −208156. −0.794504
\(148\) −188358. −0.706854
\(149\) 224953. 0.830093 0.415046 0.909800i \(-0.363765\pi\)
0.415046 + 0.909800i \(0.363765\pi\)
\(150\) −98426.1 −0.357176
\(151\) 14688.3 0.0524241 0.0262120 0.999656i \(-0.491655\pi\)
0.0262120 + 0.999656i \(0.491655\pi\)
\(152\) −26542.5 −0.0931823
\(153\) 202909. 0.700766
\(154\) −74889.1 −0.254458
\(155\) −61855.0 −0.206798
\(156\) −446574. −1.46920
\(157\) 32994.6 0.106830 0.0534151 0.998572i \(-0.482989\pi\)
0.0534151 + 0.998572i \(0.482989\pi\)
\(158\) 147202. 0.469107
\(159\) −794500. −2.49230
\(160\) 169718. 0.524116
\(161\) −63347.9 −0.192605
\(162\) 476304. 1.42593
\(163\) 110502. 0.325762 0.162881 0.986646i \(-0.447921\pi\)
0.162881 + 0.986646i \(0.447921\pi\)
\(164\) 164077. 0.476363
\(165\) 64831.0 0.185384
\(166\) −645368. −1.81776
\(167\) 256704. 0.712264 0.356132 0.934436i \(-0.384095\pi\)
0.356132 + 0.934436i \(0.384095\pi\)
\(168\) 132725. 0.362811
\(169\) 526269. 1.41739
\(170\) −172313. −0.457294
\(171\) −78091.5 −0.204227
\(172\) −212664. −0.548118
\(173\) 565322. 1.43609 0.718043 0.695999i \(-0.245038\pi\)
0.718043 + 0.695999i \(0.245038\pi\)
\(174\) −404937. −1.01394
\(175\) 52643.0 0.129941
\(176\) −150533. −0.366311
\(177\) −208586. −0.500439
\(178\) 79545.1 0.188176
\(179\) 758958. 1.77046 0.885228 0.465157i \(-0.154002\pi\)
0.885228 + 0.465157i \(0.154002\pi\)
\(180\) 118943. 0.273627
\(181\) −558952. −1.26817 −0.634086 0.773263i \(-0.718623\pi\)
−0.634086 + 0.773263i \(0.718623\pi\)
\(182\) 586361. 1.31216
\(183\) −802144. −1.77062
\(184\) −55297.7 −0.120410
\(185\) −214102. −0.459929
\(186\) 389642. 0.825815
\(187\) 113499. 0.237349
\(188\) 497012. 1.02559
\(189\) −48162.2 −0.0980736
\(190\) 66316.2 0.133271
\(191\) 330907. 0.656330 0.328165 0.944620i \(-0.393570\pi\)
0.328165 + 0.944620i \(0.393570\pi\)
\(192\) −215894. −0.422656
\(193\) 504756. 0.975411 0.487706 0.873008i \(-0.337834\pi\)
0.487706 + 0.873008i \(0.337834\pi\)
\(194\) −173520. −0.331014
\(195\) −507610. −0.955968
\(196\) −213617. −0.397187
\(197\) 768297. 1.41047 0.705234 0.708974i \(-0.250842\pi\)
0.705234 + 0.708974i \(0.250842\pi\)
\(198\) −192333. −0.348651
\(199\) 390726. 0.699422 0.349711 0.936858i \(-0.386280\pi\)
0.349711 + 0.936858i \(0.386280\pi\)
\(200\) 45953.2 0.0812345
\(201\) 13258.4 0.0231473
\(202\) 738538. 1.27349
\(203\) 216580. 0.368873
\(204\) 442146. 0.743859
\(205\) 186502. 0.309956
\(206\) −564089. −0.926146
\(207\) −162693. −0.263902
\(208\) 1.17863e6 1.88895
\(209\) −43681.0 −0.0691714
\(210\) −331612. −0.518898
\(211\) −604166. −0.934222 −0.467111 0.884199i \(-0.654705\pi\)
−0.467111 + 0.884199i \(0.654705\pi\)
\(212\) −815341. −1.24595
\(213\) 1.00826e6 1.52273
\(214\) 1.62853e6 2.43086
\(215\) −241731. −0.356645
\(216\) −42041.8 −0.0613122
\(217\) −208399. −0.300432
\(218\) 1.02841e6 1.46563
\(219\) −1.35683e6 −1.91168
\(220\) 66531.7 0.0926770
\(221\) −888664. −1.22393
\(222\) 1.34869e6 1.83666
\(223\) 976246. 1.31461 0.657305 0.753624i \(-0.271696\pi\)
0.657305 + 0.753624i \(0.271696\pi\)
\(224\) 571805. 0.761427
\(225\) 135200. 0.178041
\(226\) −73563.9 −0.0958063
\(227\) −1.09207e6 −1.40665 −0.703326 0.710868i \(-0.748302\pi\)
−0.703326 + 0.710868i \(0.748302\pi\)
\(228\) −170164. −0.216786
\(229\) 57325.7 0.0722372 0.0361186 0.999348i \(-0.488501\pi\)
0.0361186 + 0.999348i \(0.488501\pi\)
\(230\) 138161. 0.172212
\(231\) 218426. 0.269323
\(232\) 189057. 0.230607
\(233\) −117833. −0.142193 −0.0710965 0.997469i \(-0.522650\pi\)
−0.0710965 + 0.997469i \(0.522650\pi\)
\(234\) 1.50592e6 1.79788
\(235\) 564942. 0.667320
\(236\) −214057. −0.250178
\(237\) −429338. −0.496511
\(238\) −580548. −0.664348
\(239\) 60104.5 0.0680631 0.0340316 0.999421i \(-0.489165\pi\)
0.0340316 + 0.999421i \(0.489165\pi\)
\(240\) −666566. −0.746991
\(241\) −836662. −0.927913 −0.463956 0.885858i \(-0.653571\pi\)
−0.463956 + 0.885858i \(0.653571\pi\)
\(242\) −107583. −0.118088
\(243\) −1.25027e6 −1.35827
\(244\) −823186. −0.885163
\(245\) −242813. −0.258438
\(246\) −1.17483e6 −1.23776
\(247\) 342010. 0.356695
\(248\) −181916. −0.187820
\(249\) 1.88232e6 1.92395
\(250\) −114813. −0.116183
\(251\) 277631. 0.278153 0.139077 0.990282i \(-0.455587\pi\)
0.139077 + 0.990282i \(0.455587\pi\)
\(252\) 400737. 0.397520
\(253\) −91003.2 −0.0893831
\(254\) 2.59158e6 2.52046
\(255\) 502577. 0.484008
\(256\) 1.37473e6 1.31104
\(257\) 1.15289e6 1.08882 0.544409 0.838820i \(-0.316754\pi\)
0.544409 + 0.838820i \(0.316754\pi\)
\(258\) 1.52273e6 1.42421
\(259\) −721342. −0.668177
\(260\) −520925. −0.477905
\(261\) 556229. 0.505419
\(262\) 1.61347e6 1.45213
\(263\) −80717.3 −0.0719577 −0.0359788 0.999353i \(-0.511455\pi\)
−0.0359788 + 0.999353i \(0.511455\pi\)
\(264\) 190668. 0.168371
\(265\) −926779. −0.810703
\(266\) 223429. 0.193614
\(267\) −232006. −0.199168
\(268\) 13606.2 0.0115718
\(269\) −1.72261e6 −1.45146 −0.725732 0.687977i \(-0.758499\pi\)
−0.725732 + 0.687977i \(0.758499\pi\)
\(270\) 105041. 0.0876899
\(271\) 513468. 0.424708 0.212354 0.977193i \(-0.431887\pi\)
0.212354 + 0.977193i \(0.431887\pi\)
\(272\) −1.16695e6 −0.956376
\(273\) −1.71021e6 −1.38881
\(274\) −582391. −0.468639
\(275\) 75625.0 0.0603023
\(276\) −354513. −0.280130
\(277\) −597435. −0.467833 −0.233917 0.972257i \(-0.575154\pi\)
−0.233917 + 0.972257i \(0.575154\pi\)
\(278\) 1.11852e6 0.868025
\(279\) −535219. −0.411643
\(280\) 154823. 0.118016
\(281\) −1.00075e6 −0.756068 −0.378034 0.925792i \(-0.623400\pi\)
−0.378034 + 0.925792i \(0.623400\pi\)
\(282\) −3.55872e6 −2.66484
\(283\) 178437. 0.132440 0.0662201 0.997805i \(-0.478906\pi\)
0.0662201 + 0.997805i \(0.478906\pi\)
\(284\) 1.03470e6 0.761238
\(285\) −193422. −0.141056
\(286\) 842345. 0.608941
\(287\) 628354. 0.450298
\(288\) 1.46853e6 1.04328
\(289\) −540003. −0.380322
\(290\) −472356. −0.329818
\(291\) 506099. 0.350351
\(292\) −1.39242e6 −0.955684
\(293\) −634516. −0.431791 −0.215896 0.976416i \(-0.569267\pi\)
−0.215896 + 0.976416i \(0.569267\pi\)
\(294\) 1.52954e6 1.03203
\(295\) −243314. −0.162784
\(296\) −629674. −0.417721
\(297\) −69188.1 −0.0455135
\(298\) −1.65297e6 −1.07826
\(299\) 712531. 0.460920
\(300\) 294605. 0.188990
\(301\) −814427. −0.518127
\(302\) −107931. −0.0680970
\(303\) −2.15406e6 −1.34788
\(304\) 449110. 0.278721
\(305\) −935696. −0.575951
\(306\) −1.49099e6 −0.910271
\(307\) −1.20427e6 −0.729255 −0.364628 0.931153i \(-0.618804\pi\)
−0.364628 + 0.931153i \(0.618804\pi\)
\(308\) 224155. 0.134639
\(309\) 1.64525e6 0.980249
\(310\) 454514. 0.268623
\(311\) −773440. −0.453446 −0.226723 0.973959i \(-0.572801\pi\)
−0.226723 + 0.973959i \(0.572801\pi\)
\(312\) −1.49288e6 −0.868238
\(313\) −1.71303e6 −0.988333 −0.494166 0.869367i \(-0.664527\pi\)
−0.494166 + 0.869367i \(0.664527\pi\)
\(314\) −242447. −0.138769
\(315\) 455509. 0.258655
\(316\) −440601. −0.248215
\(317\) −2.30187e6 −1.28657 −0.643285 0.765627i \(-0.722429\pi\)
−0.643285 + 0.765627i \(0.722429\pi\)
\(318\) 5.83803e6 3.23742
\(319\) 311130. 0.171185
\(320\) −251839. −0.137483
\(321\) −4.74985e6 −2.57286
\(322\) 465484. 0.250187
\(323\) −338620. −0.180595
\(324\) −1.42566e6 −0.754489
\(325\) −592123. −0.310959
\(326\) −811975. −0.423154
\(327\) −2.99951e6 −1.55125
\(328\) 548504. 0.281511
\(329\) 1.90338e6 0.969471
\(330\) −476382. −0.240808
\(331\) −1.01111e6 −0.507259 −0.253630 0.967301i \(-0.581624\pi\)
−0.253630 + 0.967301i \(0.581624\pi\)
\(332\) 1.93169e6 0.961819
\(333\) −1.85258e6 −0.915517
\(334\) −1.88628e6 −0.925207
\(335\) 15465.8 0.00752942
\(336\) −2.24576e6 −1.08521
\(337\) 2.52958e6 1.21331 0.606657 0.794964i \(-0.292510\pi\)
0.606657 + 0.794964i \(0.292510\pi\)
\(338\) −3.86705e6 −1.84115
\(339\) 214561. 0.101403
\(340\) 515761. 0.241964
\(341\) −299378. −0.139423
\(342\) 573821. 0.265284
\(343\) −2.23371e6 −1.02516
\(344\) −710930. −0.323915
\(345\) −402967. −0.182273
\(346\) −4.15401e6 −1.86543
\(347\) −1.80606e6 −0.805207 −0.402603 0.915375i \(-0.631895\pi\)
−0.402603 + 0.915375i \(0.631895\pi\)
\(348\) 1.21204e6 0.536500
\(349\) −2.24094e6 −0.984843 −0.492421 0.870357i \(-0.663888\pi\)
−0.492421 + 0.870357i \(0.663888\pi\)
\(350\) −386824. −0.168789
\(351\) 541724. 0.234699
\(352\) 821435. 0.353359
\(353\) 2.21799e6 0.947375 0.473687 0.880693i \(-0.342923\pi\)
0.473687 + 0.880693i \(0.342923\pi\)
\(354\) 1.53270e6 0.650053
\(355\) 1.17612e6 0.495316
\(356\) −238092. −0.0995679
\(357\) 1.69326e6 0.703158
\(358\) −5.57687e6 −2.29976
\(359\) 3.63641e6 1.48914 0.744572 0.667542i \(-0.232654\pi\)
0.744572 + 0.667542i \(0.232654\pi\)
\(360\) 397623. 0.161702
\(361\) 130321. 0.0526316
\(362\) 4.10721e6 1.64731
\(363\) 313782. 0.124986
\(364\) −1.75508e6 −0.694292
\(365\) −1.58274e6 −0.621836
\(366\) 5.89420e6 2.29997
\(367\) −1.69594e6 −0.657271 −0.328636 0.944457i \(-0.606589\pi\)
−0.328636 + 0.944457i \(0.606589\pi\)
\(368\) 935658. 0.360162
\(369\) 1.61377e6 0.616985
\(370\) 1.57323e6 0.597432
\(371\) −3.12246e6 −1.17777
\(372\) −1.16626e6 −0.436957
\(373\) −4.78535e6 −1.78091 −0.890454 0.455072i \(-0.849613\pi\)
−0.890454 + 0.455072i \(0.849613\pi\)
\(374\) −833994. −0.308308
\(375\) 334871. 0.122970
\(376\) 1.66150e6 0.606080
\(377\) −2.43607e6 −0.882746
\(378\) 353899. 0.127394
\(379\) −223946. −0.0800840 −0.0400420 0.999198i \(-0.512749\pi\)
−0.0400420 + 0.999198i \(0.512749\pi\)
\(380\) −198495. −0.0705166
\(381\) −7.55875e6 −2.66770
\(382\) −2.43152e6 −0.852550
\(383\) 873082. 0.304129 0.152065 0.988371i \(-0.451408\pi\)
0.152065 + 0.988371i \(0.451408\pi\)
\(384\) −3.06941e6 −1.06225
\(385\) 254792. 0.0876060
\(386\) −3.70897e6 −1.26703
\(387\) −2.09165e6 −0.709922
\(388\) 519375. 0.175147
\(389\) −1.29218e6 −0.432960 −0.216480 0.976287i \(-0.569458\pi\)
−0.216480 + 0.976287i \(0.569458\pi\)
\(390\) 3.72994e6 1.24177
\(391\) −705467. −0.233364
\(392\) −714113. −0.234721
\(393\) −4.70592e6 −1.53696
\(394\) −5.64549e6 −1.83215
\(395\) −500821. −0.161506
\(396\) 575685. 0.184479
\(397\) 1.90173e6 0.605582 0.302791 0.953057i \(-0.402082\pi\)
0.302791 + 0.953057i \(0.402082\pi\)
\(398\) −2.87108e6 −0.908526
\(399\) −651666. −0.204924
\(400\) −777545. −0.242983
\(401\) −3.27319e6 −1.01651 −0.508253 0.861208i \(-0.669709\pi\)
−0.508253 + 0.861208i \(0.669709\pi\)
\(402\) −97423.5 −0.0300676
\(403\) 2.34405e6 0.718960
\(404\) −2.21057e6 −0.673830
\(405\) −1.62051e6 −0.490924
\(406\) −1.59144e6 −0.479154
\(407\) −1.03625e6 −0.310084
\(408\) 1.47808e6 0.439590
\(409\) −2.46121e6 −0.727511 −0.363755 0.931494i \(-0.618506\pi\)
−0.363755 + 0.931494i \(0.618506\pi\)
\(410\) −1.37043e6 −0.402622
\(411\) 1.69863e6 0.496016
\(412\) 1.68841e6 0.490044
\(413\) −819761. −0.236490
\(414\) 1.19547e6 0.342799
\(415\) 2.19571e6 0.625828
\(416\) −6.43161e6 −1.82216
\(417\) −3.26234e6 −0.918733
\(418\) 320970. 0.0898513
\(419\) 2.05556e6 0.572000 0.286000 0.958230i \(-0.407674\pi\)
0.286000 + 0.958230i \(0.407674\pi\)
\(420\) 992570. 0.274560
\(421\) 820076. 0.225501 0.112751 0.993623i \(-0.464034\pi\)
0.112751 + 0.993623i \(0.464034\pi\)
\(422\) 4.43945e6 1.21352
\(423\) 4.88833e6 1.32834
\(424\) −2.72566e6 −0.736304
\(425\) 586253. 0.157439
\(426\) −7.40872e6 −1.97797
\(427\) −3.15250e6 −0.836730
\(428\) −4.87444e6 −1.28622
\(429\) −2.45683e6 −0.644513
\(430\) 1.77625e6 0.463269
\(431\) −4.59815e6 −1.19231 −0.596156 0.802869i \(-0.703306\pi\)
−0.596156 + 0.802869i \(0.703306\pi\)
\(432\) 711364. 0.183393
\(433\) 3.67289e6 0.941431 0.470715 0.882285i \(-0.343996\pi\)
0.470715 + 0.882285i \(0.343996\pi\)
\(434\) 1.53133e6 0.390251
\(435\) 1.37770e6 0.349085
\(436\) −3.07819e6 −0.775497
\(437\) 271506. 0.0680104
\(438\) 9.97008e6 2.48321
\(439\) 1.01093e6 0.250356 0.125178 0.992134i \(-0.460050\pi\)
0.125178 + 0.992134i \(0.460050\pi\)
\(440\) 222413. 0.0547683
\(441\) −2.10101e6 −0.514436
\(442\) 6.52995e6 1.58984
\(443\) 67001.0 0.0162208 0.00811040 0.999967i \(-0.497418\pi\)
0.00811040 + 0.999967i \(0.497418\pi\)
\(444\) −4.03684e6 −0.971816
\(445\) −270633. −0.0647860
\(446\) −7.17351e6 −1.70763
\(447\) 4.82114e6 1.14125
\(448\) −848483. −0.199732
\(449\) −1.86511e6 −0.436606 −0.218303 0.975881i \(-0.570052\pi\)
−0.218303 + 0.975881i \(0.570052\pi\)
\(450\) −993457. −0.231269
\(451\) 902671. 0.208972
\(452\) 220189. 0.0506932
\(453\) 314797. 0.0720751
\(454\) 8.02460e6 1.82719
\(455\) −1.99495e6 −0.451756
\(456\) −568853. −0.128111
\(457\) −5.27822e6 −1.18222 −0.591109 0.806592i \(-0.701310\pi\)
−0.591109 + 0.806592i \(0.701310\pi\)
\(458\) −421233. −0.0938336
\(459\) −536354. −0.118828
\(460\) −413537. −0.0911213
\(461\) 2.01902e6 0.442475 0.221238 0.975220i \(-0.428990\pi\)
0.221238 + 0.975220i \(0.428990\pi\)
\(462\) −1.60500e6 −0.349841
\(463\) 6.33610e6 1.37363 0.686814 0.726833i \(-0.259009\pi\)
0.686814 + 0.726833i \(0.259009\pi\)
\(464\) −3.19891e6 −0.689775
\(465\) −1.32566e6 −0.284315
\(466\) 865846. 0.184704
\(467\) −8.47933e6 −1.79916 −0.899579 0.436759i \(-0.856126\pi\)
−0.899579 + 0.436759i \(0.856126\pi\)
\(468\) −4.50746e6 −0.951299
\(469\) 52106.7 0.0109386
\(470\) −4.15123e6 −0.866826
\(471\) 707133. 0.146875
\(472\) −715586. −0.147845
\(473\) −1.16998e6 −0.240450
\(474\) 3.15480e6 0.644951
\(475\) −225625. −0.0458831
\(476\) 1.73768e6 0.351521
\(477\) −8.01923e6 −1.61375
\(478\) −441651. −0.0884117
\(479\) 5.40868e6 1.07709 0.538546 0.842596i \(-0.318974\pi\)
0.538546 + 0.842596i \(0.318974\pi\)
\(480\) 3.63735e6 0.720580
\(481\) 8.11358e6 1.59901
\(482\) 6.14784e6 1.20533
\(483\) −1.35766e6 −0.264802
\(484\) 322013. 0.0624828
\(485\) 590361. 0.113963
\(486\) 9.18704e6 1.76435
\(487\) −2.56793e6 −0.490637 −0.245318 0.969443i \(-0.578892\pi\)
−0.245318 + 0.969443i \(0.578892\pi\)
\(488\) −2.75188e6 −0.523095
\(489\) 2.36825e6 0.447874
\(490\) 1.78420e6 0.335702
\(491\) 1.13750e6 0.212935 0.106467 0.994316i \(-0.466046\pi\)
0.106467 + 0.994316i \(0.466046\pi\)
\(492\) 3.51645e6 0.654926
\(493\) 2.41192e6 0.446935
\(494\) −2.51311e6 −0.463334
\(495\) 654367. 0.120035
\(496\) 3.07809e6 0.561793
\(497\) 3.96254e6 0.719586
\(498\) −1.38314e7 −2.49915
\(499\) 6.62654e6 1.19134 0.595670 0.803230i \(-0.296887\pi\)
0.595670 + 0.803230i \(0.296887\pi\)
\(500\) 343655. 0.0614750
\(501\) 5.50161e6 0.979255
\(502\) −2.04005e6 −0.361311
\(503\) −8.10433e6 −1.42823 −0.714113 0.700030i \(-0.753170\pi\)
−0.714113 + 0.700030i \(0.753170\pi\)
\(504\) 1.33965e6 0.234918
\(505\) −2.51270e6 −0.438442
\(506\) 668697. 0.116106
\(507\) 1.12789e7 1.94870
\(508\) −7.75703e6 −1.33363
\(509\) 1.09648e7 1.87588 0.937939 0.346802i \(-0.112732\pi\)
0.937939 + 0.346802i \(0.112732\pi\)
\(510\) −3.69297e6 −0.628709
\(511\) −5.33247e6 −0.903393
\(512\) −5.51860e6 −0.930366
\(513\) 206421. 0.0346306
\(514\) −8.47150e6 −1.41434
\(515\) 1.91918e6 0.318858
\(516\) −4.55777e6 −0.753578
\(517\) 2.73432e6 0.449907
\(518\) 5.30046e6 0.867939
\(519\) 1.21158e7 1.97440
\(520\) −1.74144e6 −0.282422
\(521\) −4.35822e6 −0.703420 −0.351710 0.936109i \(-0.614400\pi\)
−0.351710 + 0.936109i \(0.614400\pi\)
\(522\) −4.08720e6 −0.656522
\(523\) 2.19655e6 0.351145 0.175573 0.984466i \(-0.443822\pi\)
0.175573 + 0.984466i \(0.443822\pi\)
\(524\) −4.82937e6 −0.768356
\(525\) 1.12823e6 0.178649
\(526\) 593115. 0.0934705
\(527\) −2.32081e6 −0.364010
\(528\) −3.22618e6 −0.503621
\(529\) −5.87070e6 −0.912117
\(530\) 6.81003e6 1.05307
\(531\) −2.10534e6 −0.324031
\(532\) −668761. −0.102445
\(533\) −7.06767e6 −1.07760
\(534\) 1.70479e6 0.258713
\(535\) −5.54067e6 −0.836908
\(536\) 45485.0 0.00683843
\(537\) 1.62658e7 2.43411
\(538\) 1.26578e7 1.88540
\(539\) −1.17521e6 −0.174239
\(540\) −314405. −0.0463986
\(541\) 5.41278e6 0.795109 0.397555 0.917578i \(-0.369859\pi\)
0.397555 + 0.917578i \(0.369859\pi\)
\(542\) −3.77299e6 −0.551681
\(543\) −1.19793e7 −1.74354
\(544\) 6.36785e6 0.922562
\(545\) −3.49891e6 −0.504593
\(546\) 1.25667e7 1.80402
\(547\) −1.20209e7 −1.71778 −0.858889 0.512162i \(-0.828845\pi\)
−0.858889 + 0.512162i \(0.828845\pi\)
\(548\) 1.74319e6 0.247967
\(549\) −8.09638e6 −1.14646
\(550\) −555697. −0.0783306
\(551\) −928248. −0.130252
\(552\) −1.18513e6 −0.165545
\(553\) −1.68734e6 −0.234633
\(554\) 4.38998e6 0.607699
\(555\) −4.58858e6 −0.632333
\(556\) −3.34792e6 −0.459291
\(557\) 1.29791e7 1.77258 0.886291 0.463128i \(-0.153273\pi\)
0.886291 + 0.463128i \(0.153273\pi\)
\(558\) 3.93282e6 0.534710
\(559\) 9.16060e6 1.23992
\(560\) −2.61967e6 −0.353001
\(561\) 2.43247e6 0.326318
\(562\) 7.35359e6 0.982106
\(563\) −1.83258e6 −0.243665 −0.121832 0.992551i \(-0.538877\pi\)
−0.121832 + 0.992551i \(0.538877\pi\)
\(564\) 1.06518e7 1.41003
\(565\) 250284. 0.0329846
\(566\) −1.31117e6 −0.172035
\(567\) −5.45975e6 −0.713206
\(568\) 3.45898e6 0.449860
\(569\) −5.51647e6 −0.714300 −0.357150 0.934047i \(-0.616251\pi\)
−0.357150 + 0.934047i \(0.616251\pi\)
\(570\) 1.42127e6 0.183227
\(571\) −2.25143e6 −0.288980 −0.144490 0.989506i \(-0.546154\pi\)
−0.144490 + 0.989506i \(0.546154\pi\)
\(572\) −2.52128e6 −0.322204
\(573\) 7.09191e6 0.902354
\(574\) −4.61718e6 −0.584921
\(575\) −470058. −0.0592901
\(576\) −2.17911e6 −0.273667
\(577\) −1.38710e6 −0.173447 −0.0867236 0.996232i \(-0.527640\pi\)
−0.0867236 + 0.996232i \(0.527640\pi\)
\(578\) 3.96798e6 0.494026
\(579\) 1.08178e7 1.34104
\(580\) 1.41384e6 0.174514
\(581\) 7.39768e6 0.909191
\(582\) −3.71884e6 −0.455093
\(583\) −4.48561e6 −0.546576
\(584\) −4.65483e6 −0.564770
\(585\) −5.12352e6 −0.618983
\(586\) 4.66246e6 0.560882
\(587\) −159778. −0.0191391 −0.00956955 0.999954i \(-0.503046\pi\)
−0.00956955 + 0.999954i \(0.503046\pi\)
\(588\) −4.57818e6 −0.546071
\(589\) 893187. 0.106085
\(590\) 1.78788e6 0.211451
\(591\) 1.64659e7 1.93918
\(592\) 1.06543e7 1.24946
\(593\) −1.18442e7 −1.38314 −0.691572 0.722308i \(-0.743081\pi\)
−0.691572 + 0.722308i \(0.743081\pi\)
\(594\) 508398. 0.0591205
\(595\) 1.97518e6 0.228725
\(596\) 4.94761e6 0.570532
\(597\) 8.37394e6 0.961599
\(598\) −5.23572e6 −0.598719
\(599\) −1.87540e6 −0.213564 −0.106782 0.994282i \(-0.534055\pi\)
−0.106782 + 0.994282i \(0.534055\pi\)
\(600\) 984856. 0.111685
\(601\) −1.59381e7 −1.79991 −0.899954 0.435985i \(-0.856400\pi\)
−0.899954 + 0.435985i \(0.856400\pi\)
\(602\) 5.98446e6 0.673029
\(603\) 133823. 0.0149877
\(604\) 323055. 0.0360316
\(605\) 366025. 0.0406558
\(606\) 1.58282e7 1.75085
\(607\) 1.35588e7 1.49365 0.746825 0.665021i \(-0.231577\pi\)
0.746825 + 0.665021i \(0.231577\pi\)
\(608\) −2.45073e6 −0.268866
\(609\) 4.64168e6 0.507145
\(610\) 6.87555e6 0.748140
\(611\) −2.14090e7 −2.32003
\(612\) 4.46277e6 0.481644
\(613\) −7.25709e6 −0.780030 −0.390015 0.920809i \(-0.627530\pi\)
−0.390015 + 0.920809i \(0.627530\pi\)
\(614\) 8.84908e6 0.947277
\(615\) 3.99707e6 0.426142
\(616\) 749344. 0.0795663
\(617\) −1.82352e7 −1.92840 −0.964199 0.265181i \(-0.914568\pi\)
−0.964199 + 0.265181i \(0.914568\pi\)
\(618\) −1.20894e7 −1.27331
\(619\) 4.92013e6 0.516119 0.258060 0.966129i \(-0.416917\pi\)
0.258060 + 0.966129i \(0.416917\pi\)
\(620\) −1.36044e6 −0.142134
\(621\) 430049. 0.0447496
\(622\) 5.68328e6 0.589010
\(623\) −911804. −0.0941199
\(624\) 2.52601e7 2.59701
\(625\) 390625. 0.0400000
\(626\) 1.25874e7 1.28381
\(627\) −936160. −0.0951002
\(628\) 725682. 0.0734256
\(629\) −8.03315e6 −0.809579
\(630\) −3.34710e6 −0.335983
\(631\) −8.54707e6 −0.854562 −0.427281 0.904119i \(-0.640529\pi\)
−0.427281 + 0.904119i \(0.640529\pi\)
\(632\) −1.47291e6 −0.146685
\(633\) −1.29483e7 −1.28441
\(634\) 1.69143e7 1.67121
\(635\) −8.81723e6 −0.867756
\(636\) −1.74742e7 −1.71299
\(637\) 9.20161e6 0.898494
\(638\) −2.28620e6 −0.222363
\(639\) 1.01768e7 0.985956
\(640\) −3.58045e6 −0.345531
\(641\) 4.12709e6 0.396734 0.198367 0.980128i \(-0.436436\pi\)
0.198367 + 0.980128i \(0.436436\pi\)
\(642\) 3.49021e7 3.34206
\(643\) −1.02660e7 −0.979201 −0.489601 0.871947i \(-0.662857\pi\)
−0.489601 + 0.871947i \(0.662857\pi\)
\(644\) −1.39327e6 −0.132379
\(645\) −5.18071e6 −0.490332
\(646\) 2.48820e6 0.234587
\(647\) −1.07012e7 −1.00502 −0.502508 0.864573i \(-0.667589\pi\)
−0.502508 + 0.864573i \(0.667589\pi\)
\(648\) −4.76593e6 −0.445872
\(649\) −1.17764e6 −0.109749
\(650\) 4.35096e6 0.403926
\(651\) −4.46635e6 −0.413048
\(652\) 2.43037e6 0.223900
\(653\) −2.06722e7 −1.89716 −0.948579 0.316540i \(-0.897479\pi\)
−0.948579 + 0.316540i \(0.897479\pi\)
\(654\) 2.20406e7 2.01502
\(655\) −5.48943e6 −0.499947
\(656\) −9.28090e6 −0.842035
\(657\) −1.36951e7 −1.23780
\(658\) −1.39861e7 −1.25931
\(659\) 7.97936e6 0.715739 0.357870 0.933772i \(-0.383503\pi\)
0.357870 + 0.933772i \(0.383503\pi\)
\(660\) 1.42589e6 0.127417
\(661\) 7.50235e6 0.667872 0.333936 0.942596i \(-0.391623\pi\)
0.333936 + 0.942596i \(0.391623\pi\)
\(662\) 7.42972e6 0.658912
\(663\) −1.90456e7 −1.68272
\(664\) 6.45759e6 0.568395
\(665\) −760165. −0.0666582
\(666\) 1.36129e7 1.18922
\(667\) −1.93387e6 −0.168312
\(668\) 5.64593e6 0.489547
\(669\) 2.09227e7 1.80739
\(670\) −113644. −0.00978045
\(671\) −4.52877e6 −0.388306
\(672\) 1.22548e7 1.04685
\(673\) −9.47243e6 −0.806164 −0.403082 0.915164i \(-0.632061\pi\)
−0.403082 + 0.915164i \(0.632061\pi\)
\(674\) −1.85875e7 −1.57605
\(675\) −357377. −0.0301903
\(676\) 1.15747e7 0.974191
\(677\) −9.79457e6 −0.821323 −0.410661 0.911788i \(-0.634702\pi\)
−0.410661 + 0.911788i \(0.634702\pi\)
\(678\) −1.57660e6 −0.131719
\(679\) 1.98902e6 0.165563
\(680\) 1.72417e6 0.142991
\(681\) −2.34050e7 −1.93393
\(682\) 2.19985e6 0.181106
\(683\) 5.50489e6 0.451540 0.225770 0.974181i \(-0.427510\pi\)
0.225770 + 0.974181i \(0.427510\pi\)
\(684\) −1.71754e6 −0.140367
\(685\) 1.98145e6 0.161345
\(686\) 1.64134e7 1.33164
\(687\) 1.22859e6 0.0993151
\(688\) 1.20292e7 0.968872
\(689\) 3.51211e7 2.81852
\(690\) 2.96102e6 0.236766
\(691\) −1.22205e6 −0.0973630 −0.0486815 0.998814i \(-0.515502\pi\)
−0.0486815 + 0.998814i \(0.515502\pi\)
\(692\) 1.24337e7 0.987037
\(693\) 2.20466e6 0.174385
\(694\) 1.32710e7 1.04594
\(695\) −3.80550e6 −0.298848
\(696\) 4.05182e6 0.317049
\(697\) 6.99760e6 0.545591
\(698\) 1.64666e7 1.27928
\(699\) −2.52537e6 −0.195494
\(700\) 1.15783e6 0.0893097
\(701\) −1.24814e7 −0.959331 −0.479665 0.877451i \(-0.659242\pi\)
−0.479665 + 0.877451i \(0.659242\pi\)
\(702\) −3.98062e6 −0.304865
\(703\) 3.09163e6 0.235939
\(704\) −1.21890e6 −0.0926907
\(705\) 1.21077e7 0.917463
\(706\) −1.62979e7 −1.23061
\(707\) −8.46566e6 −0.636960
\(708\) −4.58762e6 −0.343957
\(709\) −2.21328e7 −1.65356 −0.826782 0.562522i \(-0.809831\pi\)
−0.826782 + 0.562522i \(0.809831\pi\)
\(710\) −8.64223e6 −0.643398
\(711\) −4.33350e6 −0.321488
\(712\) −795933. −0.0588405
\(713\) 1.86083e6 0.137083
\(714\) −1.24422e7 −0.913377
\(715\) −2.86588e6 −0.209649
\(716\) 1.66925e7 1.21685
\(717\) 1.28814e6 0.0935764
\(718\) −2.67205e7 −1.93435
\(719\) 2.39803e7 1.72995 0.864973 0.501819i \(-0.167336\pi\)
0.864973 + 0.501819i \(0.167336\pi\)
\(720\) −6.72794e6 −0.483672
\(721\) 6.46600e6 0.463231
\(722\) −957606. −0.0683666
\(723\) −1.79311e7 −1.27574
\(724\) −1.22936e7 −0.871628
\(725\) 1.60708e6 0.113551
\(726\) −2.30569e6 −0.162353
\(727\) 2.14734e7 1.50683 0.753417 0.657543i \(-0.228404\pi\)
0.753417 + 0.657543i \(0.228404\pi\)
\(728\) −5.86716e6 −0.410298
\(729\) −1.10440e7 −0.769679
\(730\) 1.16300e7 0.807744
\(731\) −9.06979e6 −0.627774
\(732\) −1.76423e7 −1.21696
\(733\) −718866. −0.0494183 −0.0247092 0.999695i \(-0.507866\pi\)
−0.0247092 + 0.999695i \(0.507866\pi\)
\(734\) 1.24618e7 0.853772
\(735\) −5.20390e6 −0.355313
\(736\) −5.10574e6 −0.347428
\(737\) 74854.6 0.00507633
\(738\) −1.18580e7 −0.801442
\(739\) 2.57636e7 1.73538 0.867692 0.497101i \(-0.165602\pi\)
0.867692 + 0.497101i \(0.165602\pi\)
\(740\) −4.70895e6 −0.316115
\(741\) 7.32988e6 0.490401
\(742\) 2.29440e7 1.52989
\(743\) 1.72902e7 1.14902 0.574511 0.818497i \(-0.305192\pi\)
0.574511 + 0.818497i \(0.305192\pi\)
\(744\) −3.89877e6 −0.258223
\(745\) 5.62383e6 0.371229
\(746\) 3.51630e7 2.31334
\(747\) 1.89990e7 1.24575
\(748\) 2.49628e6 0.163132
\(749\) −1.86673e7 −1.21584
\(750\) −2.46065e6 −0.159734
\(751\) 2.42401e7 1.56832 0.784161 0.620558i \(-0.213094\pi\)
0.784161 + 0.620558i \(0.213094\pi\)
\(752\) −2.81132e7 −1.81286
\(753\) 5.95012e6 0.382418
\(754\) 1.79004e7 1.14666
\(755\) 367209. 0.0234448
\(756\) −1.05928e6 −0.0674071
\(757\) −2.34776e6 −0.148906 −0.0744532 0.997225i \(-0.523721\pi\)
−0.0744532 + 0.997225i \(0.523721\pi\)
\(758\) 1.64557e6 0.104026
\(759\) −1.95036e6 −0.122888
\(760\) −663563. −0.0416724
\(761\) 1.11190e7 0.695993 0.347996 0.937496i \(-0.386862\pi\)
0.347996 + 0.937496i \(0.386862\pi\)
\(762\) 5.55421e7 3.46525
\(763\) −1.17884e7 −0.733064
\(764\) 7.27795e6 0.451103
\(765\) 5.07273e6 0.313392
\(766\) −6.41545e6 −0.395053
\(767\) 9.22059e6 0.565940
\(768\) 2.94628e7 1.80248
\(769\) 1.74490e7 1.06403 0.532017 0.846734i \(-0.321434\pi\)
0.532017 + 0.846734i \(0.321434\pi\)
\(770\) −1.87223e6 −0.113797
\(771\) 2.47085e7 1.49696
\(772\) 1.11016e7 0.670411
\(773\) 5.91190e6 0.355860 0.177930 0.984043i \(-0.443060\pi\)
0.177930 + 0.984043i \(0.443060\pi\)
\(774\) 1.53695e7 0.922164
\(775\) −1.54638e6 −0.0924828
\(776\) 1.73625e6 0.103504
\(777\) −1.54596e7 −0.918641
\(778\) 9.49498e6 0.562400
\(779\) −2.69309e6 −0.159004
\(780\) −1.11643e7 −0.657047
\(781\) 5.69244e6 0.333942
\(782\) 5.18381e6 0.303132
\(783\) −1.47029e6 −0.0857035
\(784\) 1.20831e7 0.702081
\(785\) 824866. 0.0477759
\(786\) 3.45794e7 1.99646
\(787\) −8.19379e6 −0.471572 −0.235786 0.971805i \(-0.575766\pi\)
−0.235786 + 0.971805i \(0.575766\pi\)
\(788\) 1.68979e7 0.969430
\(789\) −1.72991e6 −0.0989308
\(790\) 3.68006e6 0.209791
\(791\) 843243. 0.0479194
\(792\) 1.92450e6 0.109019
\(793\) 3.54590e7 2.00237
\(794\) −1.39740e7 −0.786630
\(795\) −1.98625e7 −1.11459
\(796\) 8.59361e6 0.480721
\(797\) 2.73550e6 0.152543 0.0762714 0.997087i \(-0.475698\pi\)
0.0762714 + 0.997087i \(0.475698\pi\)
\(798\) 4.78848e6 0.266189
\(799\) 2.11967e7 1.17463
\(800\) 4.24295e6 0.234392
\(801\) −2.34173e6 −0.128960
\(802\) 2.40516e7 1.32041
\(803\) −7.66044e6 −0.419242
\(804\) 291604. 0.0159094
\(805\) −1.58370e6 −0.0861355
\(806\) −1.72242e7 −0.933904
\(807\) −3.69186e7 −1.99554
\(808\) −7.38985e6 −0.398206
\(809\) 3.16480e7 1.70010 0.850051 0.526701i \(-0.176571\pi\)
0.850051 + 0.526701i \(0.176571\pi\)
\(810\) 1.19076e7 0.637694
\(811\) 1.23445e7 0.659056 0.329528 0.944146i \(-0.393110\pi\)
0.329528 + 0.944146i \(0.393110\pi\)
\(812\) 4.76344e6 0.253531
\(813\) 1.10045e7 0.583908
\(814\) 7.61445e6 0.402789
\(815\) 2.76255e6 0.145685
\(816\) −2.50097e7 −1.31487
\(817\) 3.49059e6 0.182955
\(818\) 1.80851e7 0.945011
\(819\) −1.72619e7 −0.899247
\(820\) 4.10192e6 0.213036
\(821\) 1.32448e7 0.685786 0.342893 0.939374i \(-0.388593\pi\)
0.342893 + 0.939374i \(0.388593\pi\)
\(822\) −1.24817e7 −0.644307
\(823\) −1.49357e7 −0.768648 −0.384324 0.923198i \(-0.625565\pi\)
−0.384324 + 0.923198i \(0.625565\pi\)
\(824\) 5.64430e6 0.289596
\(825\) 1.62078e6 0.0829064
\(826\) 6.02365e6 0.307192
\(827\) 1.00695e7 0.511970 0.255985 0.966681i \(-0.417600\pi\)
0.255985 + 0.966681i \(0.417600\pi\)
\(828\) −3.57825e6 −0.181382
\(829\) −1.10311e7 −0.557483 −0.278741 0.960366i \(-0.589917\pi\)
−0.278741 + 0.960366i \(0.589917\pi\)
\(830\) −1.61342e7 −0.812929
\(831\) −1.28041e7 −0.643199
\(832\) 9.54365e6 0.477976
\(833\) −9.11039e6 −0.454909
\(834\) 2.39719e7 1.19340
\(835\) 6.41760e6 0.318534
\(836\) −960718. −0.0475423
\(837\) 1.41476e6 0.0698020
\(838\) −1.51044e7 −0.743008
\(839\) 1.60476e7 0.787053 0.393526 0.919313i \(-0.371255\pi\)
0.393526 + 0.919313i \(0.371255\pi\)
\(840\) 3.31813e6 0.162254
\(841\) −1.38994e7 −0.677653
\(842\) −6.02597e6 −0.292918
\(843\) −2.14479e7 −1.03948
\(844\) −1.32880e7 −0.642101
\(845\) 1.31567e7 0.633878
\(846\) −3.59197e7 −1.72547
\(847\) 1.23319e6 0.0590640
\(848\) 4.61192e7 2.20238
\(849\) 3.82422e6 0.182085
\(850\) −4.30782e6 −0.204508
\(851\) 6.44098e6 0.304879
\(852\) 2.21755e7 1.04659
\(853\) −4.31366e6 −0.202989 −0.101495 0.994836i \(-0.532362\pi\)
−0.101495 + 0.994836i \(0.532362\pi\)
\(854\) 2.31648e7 1.08688
\(855\) −1.95229e6 −0.0913331
\(856\) −1.62951e7 −0.760104
\(857\) 2.80413e6 0.130421 0.0652103 0.997872i \(-0.479228\pi\)
0.0652103 + 0.997872i \(0.479228\pi\)
\(858\) 1.80529e7 0.837200
\(859\) 4.08154e7 1.88730 0.943649 0.330947i \(-0.107368\pi\)
0.943649 + 0.330947i \(0.107368\pi\)
\(860\) −5.31661e6 −0.245126
\(861\) 1.34667e7 0.619091
\(862\) 3.37875e7 1.54877
\(863\) 2.39288e7 1.09369 0.546844 0.837235i \(-0.315829\pi\)
0.546844 + 0.837235i \(0.315829\pi\)
\(864\) −3.88181e6 −0.176909
\(865\) 1.41330e7 0.642237
\(866\) −2.69886e7 −1.22289
\(867\) −1.15732e7 −0.522885
\(868\) −4.58352e6 −0.206490
\(869\) −2.42397e6 −0.108888
\(870\) −1.01234e7 −0.453450
\(871\) −586092. −0.0261770
\(872\) −1.02903e7 −0.458286
\(873\) 5.10827e6 0.226850
\(874\) −1.99504e6 −0.0883431
\(875\) 1.31607e6 0.0581113
\(876\) −2.98421e7 −1.31392
\(877\) −2.09756e7 −0.920907 −0.460454 0.887684i \(-0.652313\pi\)
−0.460454 + 0.887684i \(0.652313\pi\)
\(878\) −7.42834e6 −0.325204
\(879\) −1.35988e7 −0.593647
\(880\) −3.76332e6 −0.163819
\(881\) −966060. −0.0419338 −0.0209669 0.999780i \(-0.506674\pi\)
−0.0209669 + 0.999780i \(0.506674\pi\)
\(882\) 1.54383e7 0.668235
\(883\) −1.30372e7 −0.562709 −0.281355 0.959604i \(-0.590784\pi\)
−0.281355 + 0.959604i \(0.590784\pi\)
\(884\) −1.95452e7 −0.841221
\(885\) −5.21464e6 −0.223803
\(886\) −492328. −0.0210703
\(887\) −9.46650e6 −0.403999 −0.202000 0.979386i \(-0.564744\pi\)
−0.202000 + 0.979386i \(0.564744\pi\)
\(888\) −1.34950e7 −0.574303
\(889\) −2.97066e7 −1.26066
\(890\) 1.98863e6 0.0841548
\(891\) −7.84327e6 −0.330981
\(892\) 2.14715e7 0.903546
\(893\) −8.15776e6 −0.342328
\(894\) −3.54260e7 −1.48244
\(895\) 1.89740e7 0.791772
\(896\) −1.20631e7 −0.501981
\(897\) 1.52708e7 0.633695
\(898\) 1.37050e7 0.567136
\(899\) −6.36198e6 −0.262538
\(900\) 2.97358e6 0.122370
\(901\) −3.47729e7 −1.42702
\(902\) −6.63288e6 −0.271447
\(903\) −1.74546e7 −0.712345
\(904\) 736085. 0.0299576
\(905\) −1.39738e7 −0.567143
\(906\) −2.31315e6 −0.0936230
\(907\) −4.28420e7 −1.72923 −0.864613 0.502438i \(-0.832436\pi\)
−0.864613 + 0.502438i \(0.832436\pi\)
\(908\) −2.40190e7 −0.966807
\(909\) −2.17419e7 −0.872745
\(910\) 1.46590e7 0.586816
\(911\) 4.32180e7 1.72532 0.862658 0.505788i \(-0.168798\pi\)
0.862658 + 0.505788i \(0.168798\pi\)
\(912\) 9.62522e6 0.383198
\(913\) 1.06272e7 0.421933
\(914\) 3.87847e7 1.53566
\(915\) −2.00536e7 −0.791844
\(916\) 1.26082e6 0.0496494
\(917\) −1.84947e7 −0.726314
\(918\) 3.94116e6 0.154354
\(919\) −7.50162e6 −0.292999 −0.146500 0.989211i \(-0.546801\pi\)
−0.146500 + 0.989211i \(0.546801\pi\)
\(920\) −1.38244e6 −0.0538489
\(921\) −2.58097e7 −1.00261
\(922\) −1.48359e7 −0.574760
\(923\) −4.45703e7 −1.72203
\(924\) 4.80404e6 0.185109
\(925\) −5.35255e6 −0.205687
\(926\) −4.65580e7 −1.78430
\(927\) 1.66062e7 0.634705
\(928\) 1.74560e7 0.665387
\(929\) −4.62847e7 −1.75954 −0.879768 0.475404i \(-0.842302\pi\)
−0.879768 + 0.475404i \(0.842302\pi\)
\(930\) 9.74104e6 0.369316
\(931\) 3.50622e6 0.132576
\(932\) −2.59162e6 −0.0977308
\(933\) −1.65762e7 −0.623419
\(934\) 6.23066e7 2.33704
\(935\) 2.83747e6 0.106145
\(936\) −1.50683e7 −0.562178
\(937\) 1.19512e7 0.444696 0.222348 0.974967i \(-0.428628\pi\)
0.222348 + 0.974967i \(0.428628\pi\)
\(938\) −382883. −0.0142089
\(939\) −3.67132e7 −1.35881
\(940\) 1.24253e7 0.458656
\(941\) 1.48031e6 0.0544976 0.0272488 0.999629i \(-0.491325\pi\)
0.0272488 + 0.999629i \(0.491325\pi\)
\(942\) −5.19605e6 −0.190786
\(943\) −5.61068e6 −0.205464
\(944\) 1.21080e7 0.442224
\(945\) −1.20406e6 −0.0438599
\(946\) 8.59706e6 0.312336
\(947\) 1.06634e7 0.386385 0.193193 0.981161i \(-0.438116\pi\)
0.193193 + 0.981161i \(0.438116\pi\)
\(948\) −9.44285e6 −0.341257
\(949\) 5.99792e7 2.16190
\(950\) 1.65791e6 0.0596006
\(951\) −4.93332e7 −1.76884
\(952\) 5.80899e6 0.207734
\(953\) −7.45014e6 −0.265725 −0.132862 0.991134i \(-0.542417\pi\)
−0.132862 + 0.991134i \(0.542417\pi\)
\(954\) 5.89258e7 2.09621
\(955\) 8.27267e6 0.293520
\(956\) 1.32193e6 0.0467805
\(957\) 6.66807e6 0.235353
\(958\) −3.97433e7 −1.39910
\(959\) 6.67579e6 0.234399
\(960\) −5.39734e6 −0.189018
\(961\) −2.25075e7 −0.786173
\(962\) −5.96191e7 −2.07705
\(963\) −4.79422e7 −1.66591
\(964\) −1.84015e7 −0.637765
\(965\) 1.26189e7 0.436217
\(966\) 9.97613e6 0.343969
\(967\) 7.81001e6 0.268587 0.134294 0.990942i \(-0.457123\pi\)
0.134294 + 0.990942i \(0.457123\pi\)
\(968\) 1.07648e6 0.0369248
\(969\) −7.25722e6 −0.248291
\(970\) −4.33801e6 −0.148034
\(971\) 5.69979e7 1.94004 0.970021 0.243021i \(-0.0781385\pi\)
0.970021 + 0.243021i \(0.0781385\pi\)
\(972\) −2.74983e7 −0.933556
\(973\) −1.28213e7 −0.434160
\(974\) 1.88693e7 0.637320
\(975\) −1.26902e7 −0.427522
\(976\) 4.65630e7 1.56465
\(977\) −392600. −0.0131587 −0.00657936 0.999978i \(-0.502094\pi\)
−0.00657936 + 0.999978i \(0.502094\pi\)
\(978\) −1.74020e7 −0.581772
\(979\) −1.30986e6 −0.0436787
\(980\) −5.34041e6 −0.177627
\(981\) −3.02754e7 −1.00442
\(982\) −8.35840e6 −0.276595
\(983\) 2.60437e7 0.859645 0.429822 0.902914i \(-0.358576\pi\)
0.429822 + 0.902914i \(0.358576\pi\)
\(984\) 1.17554e7 0.387034
\(985\) 1.92074e7 0.630781
\(986\) −1.77229e7 −0.580554
\(987\) 4.07927e7 1.33287
\(988\) 7.52216e6 0.245160
\(989\) 7.27216e6 0.236414
\(990\) −4.80833e6 −0.155922
\(991\) 1.64529e7 0.532179 0.266090 0.963948i \(-0.414268\pi\)
0.266090 + 0.963948i \(0.414268\pi\)
\(992\) −1.67967e7 −0.541930
\(993\) −2.16699e7 −0.697404
\(994\) −2.91170e7 −0.934717
\(995\) 9.76815e6 0.312791
\(996\) 4.13996e7 1.32235
\(997\) 3.71006e6 0.118207 0.0591034 0.998252i \(-0.481176\pi\)
0.0591034 + 0.998252i \(0.481176\pi\)
\(998\) −4.86922e7 −1.54751
\(999\) 4.89696e6 0.155243
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 1045.6.a.e.1.10 38
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.6.a.e.1.10 38 1.1 even 1 trivial