Properties

Label 275.6.a.b.1.2
Level $275$
Weight $6$
Character 275.1
Self dual yes
Analytic conductor $44.106$
Analytic rank $0$
Dimension $3$
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] = [275,6,Mod(1,275)]
 
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([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("275.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.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: \(44.1055504486\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.54492.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 52x - 38 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(8.04796\) of defining polynomial
Character \(\chi\) \(=\) 275.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.20859 q^{2} -16.8394 q^{3} -27.1221 q^{4} +37.1913 q^{6} +225.525 q^{7} +130.577 q^{8} +40.5643 q^{9} +O(q^{10})\) \(q-2.20859 q^{2} -16.8394 q^{3} -27.1221 q^{4} +37.1913 q^{6} +225.525 q^{7} +130.577 q^{8} +40.5643 q^{9} +121.000 q^{11} +456.719 q^{12} -455.465 q^{13} -498.092 q^{14} +579.518 q^{16} -190.657 q^{17} -89.5900 q^{18} -135.393 q^{19} -3797.69 q^{21} -267.240 q^{22} -2796.65 q^{23} -2198.83 q^{24} +1005.94 q^{26} +3408.89 q^{27} -6116.71 q^{28} -2608.58 q^{29} -1056.76 q^{31} -5458.37 q^{32} -2037.56 q^{33} +421.082 q^{34} -1100.19 q^{36} -12536.8 q^{37} +299.028 q^{38} +7669.74 q^{39} +1130.09 q^{41} +8387.55 q^{42} +14671.0 q^{43} -3281.78 q^{44} +6176.65 q^{46} +16882.2 q^{47} -9758.71 q^{48} +34054.4 q^{49} +3210.54 q^{51} +12353.2 q^{52} -3313.02 q^{53} -7528.84 q^{54} +29448.3 q^{56} +2279.93 q^{57} +5761.29 q^{58} +11454.0 q^{59} -28227.5 q^{61} +2333.95 q^{62} +9148.26 q^{63} -6489.25 q^{64} +4500.15 q^{66} +51431.0 q^{67} +5171.01 q^{68} +47093.8 q^{69} -16218.0 q^{71} +5296.75 q^{72} +10168.8 q^{73} +27688.7 q^{74} +3672.15 q^{76} +27288.5 q^{77} -16939.3 q^{78} +60841.2 q^{79} -67260.7 q^{81} -2495.90 q^{82} -45770.6 q^{83} +103002. q^{84} -32402.3 q^{86} +43926.9 q^{87} +15799.8 q^{88} -82267.9 q^{89} -102719. q^{91} +75851.0 q^{92} +17795.1 q^{93} -37285.8 q^{94} +91915.5 q^{96} -53097.0 q^{97} -75212.2 q^{98} +4908.28 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 34 q^{3} + 84 q^{4} - 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 34 q^{3} + 84 q^{4} - 206 q^{6} - 84 q^{7} + 564 q^{8} - 7 q^{9} + 363 q^{11} - 992 q^{12} - 486 q^{13} - 1020 q^{14} + 1992 q^{16} - 1086 q^{17} + 3706 q^{18} + 1380 q^{19} - 908 q^{21} + 3066 q^{23} - 11748 q^{24} + 12132 q^{26} + 2990 q^{27} - 23712 q^{28} - 3426 q^{29} - 4098 q^{31} + 12408 q^{32} - 4114 q^{33} + 25320 q^{34} + 4756 q^{36} - 17724 q^{37} + 9240 q^{38} - 6560 q^{39} + 5994 q^{41} + 47828 q^{42} + 26208 q^{43} + 10164 q^{44} - 16806 q^{46} + 17232 q^{47} - 61064 q^{48} + 48531 q^{49} - 22724 q^{51} + 35304 q^{52} - 50586 q^{53} + 18814 q^{54} - 42312 q^{56} - 20160 q^{57} + 29172 q^{58} - 3738 q^{59} + 18486 q^{61} + 19974 q^{62} + 12496 q^{63} - 20352 q^{64} - 24926 q^{66} + 47754 q^{67} + 12600 q^{68} + 35042 q^{69} + 39282 q^{71} + 95040 q^{72} - 15426 q^{73} + 153294 q^{74} + 103920 q^{76} - 10164 q^{77} - 124984 q^{78} + 125148 q^{79} - 86917 q^{81} + 255372 q^{82} + 143928 q^{83} + 343616 q^{84} + 243060 q^{86} + 19368 q^{87} + 68244 q^{88} - 106824 q^{89} - 109632 q^{91} + 336528 q^{92} + 16622 q^{93} - 74928 q^{94} - 76456 q^{96} - 9684 q^{97} - 3480 q^{98} - 847 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.20859 −0.390428 −0.195214 0.980761i \(-0.562540\pi\)
−0.195214 + 0.980761i \(0.562540\pi\)
\(3\) −16.8394 −1.08025 −0.540123 0.841586i \(-0.681622\pi\)
−0.540123 + 0.841586i \(0.681622\pi\)
\(4\) −27.1221 −0.847566
\(5\) 0 0
\(6\) 37.1913 0.421758
\(7\) 225.525 1.73960 0.869799 0.493406i \(-0.164248\pi\)
0.869799 + 0.493406i \(0.164248\pi\)
\(8\) 130.577 0.721341
\(9\) 40.5643 0.166931
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 456.719 0.915580
\(13\) −455.465 −0.747474 −0.373737 0.927535i \(-0.621924\pi\)
−0.373737 + 0.927535i \(0.621924\pi\)
\(14\) −498.092 −0.679187
\(15\) 0 0
\(16\) 579.518 0.565935
\(17\) −190.657 −0.160003 −0.0800017 0.996795i \(-0.525493\pi\)
−0.0800017 + 0.996795i \(0.525493\pi\)
\(18\) −89.5900 −0.0651746
\(19\) −135.393 −0.0860424 −0.0430212 0.999074i \(-0.513698\pi\)
−0.0430212 + 0.999074i \(0.513698\pi\)
\(20\) 0 0
\(21\) −3797.69 −1.87919
\(22\) −267.240 −0.117718
\(23\) −2796.65 −1.10235 −0.551173 0.834391i \(-0.685820\pi\)
−0.551173 + 0.834391i \(0.685820\pi\)
\(24\) −2198.83 −0.779225
\(25\) 0 0
\(26\) 1005.94 0.291835
\(27\) 3408.89 0.899919
\(28\) −6116.71 −1.47443
\(29\) −2608.58 −0.575983 −0.287991 0.957633i \(-0.592987\pi\)
−0.287991 + 0.957633i \(0.592987\pi\)
\(30\) 0 0
\(31\) −1056.76 −0.197502 −0.0987510 0.995112i \(-0.531485\pi\)
−0.0987510 + 0.995112i \(0.531485\pi\)
\(32\) −5458.37 −0.942297
\(33\) −2037.56 −0.325706
\(34\) 421.082 0.0624697
\(35\) 0 0
\(36\) −1100.19 −0.141485
\(37\) −12536.8 −1.50550 −0.752752 0.658304i \(-0.771274\pi\)
−0.752752 + 0.658304i \(0.771274\pi\)
\(38\) 299.028 0.0335933
\(39\) 7669.74 0.807456
\(40\) 0 0
\(41\) 1130.09 0.104991 0.0524954 0.998621i \(-0.483282\pi\)
0.0524954 + 0.998621i \(0.483282\pi\)
\(42\) 8387.55 0.733689
\(43\) 14671.0 1.21001 0.605005 0.796222i \(-0.293171\pi\)
0.605005 + 0.796222i \(0.293171\pi\)
\(44\) −3281.78 −0.255551
\(45\) 0 0
\(46\) 6176.65 0.430386
\(47\) 16882.2 1.11477 0.557383 0.830256i \(-0.311806\pi\)
0.557383 + 0.830256i \(0.311806\pi\)
\(48\) −9758.71 −0.611349
\(49\) 34054.4 2.02620
\(50\) 0 0
\(51\) 3210.54 0.172843
\(52\) 12353.2 0.633534
\(53\) −3313.02 −0.162007 −0.0810035 0.996714i \(-0.525813\pi\)
−0.0810035 + 0.996714i \(0.525813\pi\)
\(54\) −7528.84 −0.351353
\(55\) 0 0
\(56\) 29448.3 1.25484
\(57\) 2279.93 0.0929469
\(58\) 5761.29 0.224880
\(59\) 11454.0 0.428378 0.214189 0.976792i \(-0.431289\pi\)
0.214189 + 0.976792i \(0.431289\pi\)
\(60\) 0 0
\(61\) −28227.5 −0.971286 −0.485643 0.874157i \(-0.661415\pi\)
−0.485643 + 0.874157i \(0.661415\pi\)
\(62\) 2333.95 0.0771102
\(63\) 9148.26 0.290394
\(64\) −6489.25 −0.198036
\(65\) 0 0
\(66\) 4500.15 0.127165
\(67\) 51431.0 1.39971 0.699855 0.714285i \(-0.253248\pi\)
0.699855 + 0.714285i \(0.253248\pi\)
\(68\) 5171.01 0.135614
\(69\) 47093.8 1.19081
\(70\) 0 0
\(71\) −16218.0 −0.381814 −0.190907 0.981608i \(-0.561143\pi\)
−0.190907 + 0.981608i \(0.561143\pi\)
\(72\) 5296.75 0.120414
\(73\) 10168.8 0.223337 0.111669 0.993745i \(-0.464380\pi\)
0.111669 + 0.993745i \(0.464380\pi\)
\(74\) 27688.7 0.587791
\(75\) 0 0
\(76\) 3672.15 0.0729266
\(77\) 27288.5 0.524509
\(78\) −16939.3 −0.315253
\(79\) 60841.2 1.09681 0.548404 0.836214i \(-0.315236\pi\)
0.548404 + 0.836214i \(0.315236\pi\)
\(80\) 0 0
\(81\) −67260.7 −1.13907
\(82\) −2495.90 −0.0409913
\(83\) −45770.6 −0.729275 −0.364638 0.931150i \(-0.618807\pi\)
−0.364638 + 0.931150i \(0.618807\pi\)
\(84\) 103002. 1.59274
\(85\) 0 0
\(86\) −32402.3 −0.472421
\(87\) 43926.9 0.622203
\(88\) 15799.8 0.217492
\(89\) −82267.9 −1.10092 −0.550460 0.834862i \(-0.685548\pi\)
−0.550460 + 0.834862i \(0.685548\pi\)
\(90\) 0 0
\(91\) −102719. −1.30031
\(92\) 75851.0 0.934312
\(93\) 17795.1 0.213351
\(94\) −37285.8 −0.435235
\(95\) 0 0
\(96\) 91915.5 1.01791
\(97\) −53097.0 −0.572981 −0.286491 0.958083i \(-0.592489\pi\)
−0.286491 + 0.958083i \(0.592489\pi\)
\(98\) −75212.2 −0.791085
\(99\) 4908.28 0.0503317
\(100\) 0 0
\(101\) 186821. 1.82231 0.911153 0.412069i \(-0.135194\pi\)
0.911153 + 0.412069i \(0.135194\pi\)
\(102\) −7090.76 −0.0674827
\(103\) −34290.5 −0.318479 −0.159240 0.987240i \(-0.550904\pi\)
−0.159240 + 0.987240i \(0.550904\pi\)
\(104\) −59473.0 −0.539184
\(105\) 0 0
\(106\) 7317.10 0.0632520
\(107\) 224117. 1.89241 0.946206 0.323565i \(-0.104881\pi\)
0.946206 + 0.323565i \(0.104881\pi\)
\(108\) −92456.3 −0.762741
\(109\) 162229. 1.30786 0.653931 0.756554i \(-0.273118\pi\)
0.653931 + 0.756554i \(0.273118\pi\)
\(110\) 0 0
\(111\) 211112. 1.62632
\(112\) 130696. 0.984500
\(113\) −92225.0 −0.679442 −0.339721 0.940526i \(-0.610333\pi\)
−0.339721 + 0.940526i \(0.610333\pi\)
\(114\) −5035.44 −0.0362890
\(115\) 0 0
\(116\) 70750.3 0.488184
\(117\) −18475.6 −0.124777
\(118\) −25297.2 −0.167251
\(119\) −42997.8 −0.278342
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 62342.9 0.379217
\(123\) −19029.9 −0.113416
\(124\) 28661.5 0.167396
\(125\) 0 0
\(126\) −20204.8 −0.113378
\(127\) −138299. −0.760868 −0.380434 0.924808i \(-0.624225\pi\)
−0.380434 + 0.924808i \(0.624225\pi\)
\(128\) 189000. 1.01962
\(129\) −247051. −1.30711
\(130\) 0 0
\(131\) −54420.4 −0.277066 −0.138533 0.990358i \(-0.544239\pi\)
−0.138533 + 0.990358i \(0.544239\pi\)
\(132\) 55263.0 0.276058
\(133\) −30534.5 −0.149679
\(134\) −113590. −0.546485
\(135\) 0 0
\(136\) −24895.3 −0.115417
\(137\) −40555.1 −0.184605 −0.0923025 0.995731i \(-0.529423\pi\)
−0.0923025 + 0.995731i \(0.529423\pi\)
\(138\) −104011. −0.464923
\(139\) 140537. 0.616955 0.308477 0.951232i \(-0.400181\pi\)
0.308477 + 0.951232i \(0.400181\pi\)
\(140\) 0 0
\(141\) −284285. −1.20422
\(142\) 35818.9 0.149071
\(143\) −55111.2 −0.225372
\(144\) 23507.7 0.0944723
\(145\) 0 0
\(146\) −22458.7 −0.0871971
\(147\) −573454. −2.18880
\(148\) 340024. 1.27602
\(149\) 176073. 0.649722 0.324861 0.945762i \(-0.394683\pi\)
0.324861 + 0.945762i \(0.394683\pi\)
\(150\) 0 0
\(151\) 409241. 1.46062 0.730309 0.683117i \(-0.239376\pi\)
0.730309 + 0.683117i \(0.239376\pi\)
\(152\) −17679.2 −0.0620659
\(153\) −7733.85 −0.0267096
\(154\) −60269.1 −0.204783
\(155\) 0 0
\(156\) −208020. −0.684373
\(157\) −14294.5 −0.0462829 −0.0231414 0.999732i \(-0.507367\pi\)
−0.0231414 + 0.999732i \(0.507367\pi\)
\(158\) −134373. −0.428224
\(159\) 55789.1 0.175007
\(160\) 0 0
\(161\) −630713. −1.91764
\(162\) 148551. 0.444722
\(163\) 418474. 1.23367 0.616836 0.787091i \(-0.288414\pi\)
0.616836 + 0.787091i \(0.288414\pi\)
\(164\) −30650.3 −0.0889868
\(165\) 0 0
\(166\) 101089. 0.284729
\(167\) 139747. 0.387749 0.193875 0.981026i \(-0.437894\pi\)
0.193875 + 0.981026i \(0.437894\pi\)
\(168\) −495890. −1.35554
\(169\) −163845. −0.441282
\(170\) 0 0
\(171\) −5492.13 −0.0143632
\(172\) −397909. −1.02556
\(173\) 687104. 1.74545 0.872725 0.488213i \(-0.162351\pi\)
0.872725 + 0.488213i \(0.162351\pi\)
\(174\) −97016.5 −0.242925
\(175\) 0 0
\(176\) 70121.6 0.170636
\(177\) −192878. −0.462754
\(178\) 181696. 0.429829
\(179\) 35496.4 0.0828042 0.0414021 0.999143i \(-0.486818\pi\)
0.0414021 + 0.999143i \(0.486818\pi\)
\(180\) 0 0
\(181\) 260469. 0.590963 0.295481 0.955349i \(-0.404520\pi\)
0.295481 + 0.955349i \(0.404520\pi\)
\(182\) 226863. 0.507675
\(183\) 475333. 1.04923
\(184\) −365177. −0.795167
\(185\) 0 0
\(186\) −39302.2 −0.0832980
\(187\) −23069.4 −0.0482429
\(188\) −457880. −0.944837
\(189\) 768789. 1.56550
\(190\) 0 0
\(191\) 392051. 0.777605 0.388803 0.921321i \(-0.372889\pi\)
0.388803 + 0.921321i \(0.372889\pi\)
\(192\) 109275. 0.213928
\(193\) −15776.8 −0.0304878 −0.0152439 0.999884i \(-0.504852\pi\)
−0.0152439 + 0.999884i \(0.504852\pi\)
\(194\) 117270. 0.223708
\(195\) 0 0
\(196\) −923627. −1.71734
\(197\) −545551. −1.00154 −0.500771 0.865580i \(-0.666950\pi\)
−0.500771 + 0.865580i \(0.666950\pi\)
\(198\) −10840.4 −0.0196509
\(199\) −546514. −0.978293 −0.489146 0.872202i \(-0.662692\pi\)
−0.489146 + 0.872202i \(0.662692\pi\)
\(200\) 0 0
\(201\) −866065. −1.51203
\(202\) −412610. −0.711478
\(203\) −588300. −1.00198
\(204\) −87076.5 −0.146496
\(205\) 0 0
\(206\) 75733.7 0.124343
\(207\) −113444. −0.184016
\(208\) −263950. −0.423022
\(209\) −16382.6 −0.0259428
\(210\) 0 0
\(211\) 537150. 0.830596 0.415298 0.909686i \(-0.363677\pi\)
0.415298 + 0.909686i \(0.363677\pi\)
\(212\) 89856.0 0.137312
\(213\) 273101. 0.412453
\(214\) −494983. −0.738850
\(215\) 0 0
\(216\) 445121. 0.649148
\(217\) −238325. −0.343574
\(218\) −358298. −0.510626
\(219\) −171236. −0.241259
\(220\) 0 0
\(221\) 86837.3 0.119598
\(222\) −466259. −0.634958
\(223\) 189640. 0.255368 0.127684 0.991815i \(-0.459246\pi\)
0.127684 + 0.991815i \(0.459246\pi\)
\(224\) −1.23100e6 −1.63922
\(225\) 0 0
\(226\) 203687. 0.265273
\(227\) −363428. −0.468116 −0.234058 0.972223i \(-0.575201\pi\)
−0.234058 + 0.972223i \(0.575201\pi\)
\(228\) −61836.6 −0.0787787
\(229\) 504331. 0.635516 0.317758 0.948172i \(-0.397070\pi\)
0.317758 + 0.948172i \(0.397070\pi\)
\(230\) 0 0
\(231\) −459521. −0.566598
\(232\) −340620. −0.415480
\(233\) 1.20159e6 1.45000 0.724999 0.688750i \(-0.241840\pi\)
0.724999 + 0.688750i \(0.241840\pi\)
\(234\) 40805.1 0.0487163
\(235\) 0 0
\(236\) −310657. −0.363079
\(237\) −1.02453e6 −1.18482
\(238\) 94964.5 0.108672
\(239\) −185929. −0.210549 −0.105275 0.994443i \(-0.533572\pi\)
−0.105275 + 0.994443i \(0.533572\pi\)
\(240\) 0 0
\(241\) 174842. 0.193911 0.0969556 0.995289i \(-0.469090\pi\)
0.0969556 + 0.995289i \(0.469090\pi\)
\(242\) −32336.0 −0.0354934
\(243\) 304267. 0.330552
\(244\) 765589. 0.823229
\(245\) 0 0
\(246\) 42029.3 0.0442807
\(247\) 61666.8 0.0643145
\(248\) −137988. −0.142466
\(249\) 770748. 0.787797
\(250\) 0 0
\(251\) 447906. 0.448748 0.224374 0.974503i \(-0.427966\pi\)
0.224374 + 0.974503i \(0.427966\pi\)
\(252\) −248120. −0.246128
\(253\) −338394. −0.332370
\(254\) 305446. 0.297064
\(255\) 0 0
\(256\) −209768. −0.200050
\(257\) 1.14572e6 1.08204 0.541022 0.841009i \(-0.318038\pi\)
0.541022 + 0.841009i \(0.318038\pi\)
\(258\) 545634. 0.510331
\(259\) −2.82736e6 −2.61897
\(260\) 0 0
\(261\) −105815. −0.0961496
\(262\) 120192. 0.108174
\(263\) −443228. −0.395128 −0.197564 0.980290i \(-0.563303\pi\)
−0.197564 + 0.980290i \(0.563303\pi\)
\(264\) −266058. −0.234945
\(265\) 0 0
\(266\) 67438.2 0.0584389
\(267\) 1.38534e6 1.18926
\(268\) −1.39492e6 −1.18635
\(269\) −1.88722e6 −1.59016 −0.795082 0.606502i \(-0.792572\pi\)
−0.795082 + 0.606502i \(0.792572\pi\)
\(270\) 0 0
\(271\) 2.24203e6 1.85446 0.927230 0.374491i \(-0.122183\pi\)
0.927230 + 0.374491i \(0.122183\pi\)
\(272\) −110489. −0.0905516
\(273\) 1.72972e6 1.40465
\(274\) 89569.6 0.0720749
\(275\) 0 0
\(276\) −1.27728e6 −1.00929
\(277\) −1.27824e6 −1.00095 −0.500474 0.865751i \(-0.666841\pi\)
−0.500474 + 0.865751i \(0.666841\pi\)
\(278\) −310389. −0.240876
\(279\) −42866.7 −0.0329693
\(280\) 0 0
\(281\) 549325. 0.415015 0.207508 0.978233i \(-0.433465\pi\)
0.207508 + 0.978233i \(0.433465\pi\)
\(282\) 627869. 0.470161
\(283\) 135813. 0.100803 0.0504016 0.998729i \(-0.483950\pi\)
0.0504016 + 0.998729i \(0.483950\pi\)
\(284\) 439867. 0.323612
\(285\) 0 0
\(286\) 121718. 0.0879914
\(287\) 254862. 0.182642
\(288\) −221415. −0.157299
\(289\) −1.38351e6 −0.974399
\(290\) 0 0
\(291\) 894120. 0.618961
\(292\) −275799. −0.189293
\(293\) 1.76403e6 1.20043 0.600215 0.799839i \(-0.295082\pi\)
0.600215 + 0.799839i \(0.295082\pi\)
\(294\) 1.26653e6 0.854567
\(295\) 0 0
\(296\) −1.63701e6 −1.08598
\(297\) 412476. 0.271336
\(298\) −388874. −0.253669
\(299\) 1.27377e6 0.823976
\(300\) 0 0
\(301\) 3.30868e6 2.10493
\(302\) −903846. −0.570266
\(303\) −3.14594e6 −1.96854
\(304\) −78462.7 −0.0486944
\(305\) 0 0
\(306\) 17080.9 0.0104282
\(307\) 1.93533e6 1.17195 0.585975 0.810329i \(-0.300712\pi\)
0.585975 + 0.810329i \(0.300712\pi\)
\(308\) −740122. −0.444556
\(309\) 577431. 0.344036
\(310\) 0 0
\(311\) 2.98327e6 1.74901 0.874504 0.485019i \(-0.161187\pi\)
0.874504 + 0.485019i \(0.161187\pi\)
\(312\) 1.00149e6 0.582451
\(313\) 10701.5 0.00617426 0.00308713 0.999995i \(-0.499017\pi\)
0.00308713 + 0.999995i \(0.499017\pi\)
\(314\) 31570.8 0.0180701
\(315\) 0 0
\(316\) −1.65014e6 −0.929617
\(317\) 2.43658e6 1.36186 0.680929 0.732349i \(-0.261576\pi\)
0.680929 + 0.732349i \(0.261576\pi\)
\(318\) −123215. −0.0683277
\(319\) −315638. −0.173665
\(320\) 0 0
\(321\) −3.77399e6 −2.04427
\(322\) 1.39299e6 0.748699
\(323\) 25813.6 0.0137671
\(324\) 1.82425e6 0.965433
\(325\) 0 0
\(326\) −924239. −0.481660
\(327\) −2.73183e6 −1.41281
\(328\) 147563. 0.0757342
\(329\) 3.80734e6 1.93924
\(330\) 0 0
\(331\) 119576. 0.0599894 0.0299947 0.999550i \(-0.490451\pi\)
0.0299947 + 0.999550i \(0.490451\pi\)
\(332\) 1.24140e6 0.618109
\(333\) −508546. −0.251316
\(334\) −308644. −0.151388
\(335\) 0 0
\(336\) −2.20083e6 −1.06350
\(337\) 2.02195e6 0.969830 0.484915 0.874561i \(-0.338851\pi\)
0.484915 + 0.874561i \(0.338851\pi\)
\(338\) 361867. 0.172289
\(339\) 1.55301e6 0.733965
\(340\) 0 0
\(341\) −127868. −0.0595491
\(342\) 12129.9 0.00560778
\(343\) 3.88971e6 1.78518
\(344\) 1.91569e6 0.872829
\(345\) 0 0
\(346\) −1.51753e6 −0.681471
\(347\) −3.01864e6 −1.34582 −0.672912 0.739723i \(-0.734957\pi\)
−0.672912 + 0.739723i \(0.734957\pi\)
\(348\) −1.19139e6 −0.527358
\(349\) 2.40399e6 1.05650 0.528250 0.849089i \(-0.322848\pi\)
0.528250 + 0.849089i \(0.322848\pi\)
\(350\) 0 0
\(351\) −1.55263e6 −0.672666
\(352\) −660463. −0.284113
\(353\) −3.62981e6 −1.55041 −0.775206 0.631709i \(-0.782354\pi\)
−0.775206 + 0.631709i \(0.782354\pi\)
\(354\) 425989. 0.180672
\(355\) 0 0
\(356\) 2.23128e6 0.933102
\(357\) 724055. 0.300678
\(358\) −78397.1 −0.0323290
\(359\) 939181. 0.384603 0.192302 0.981336i \(-0.438405\pi\)
0.192302 + 0.981336i \(0.438405\pi\)
\(360\) 0 0
\(361\) −2.45777e6 −0.992597
\(362\) −575270. −0.230728
\(363\) −246545. −0.0982042
\(364\) 2.78594e6 1.10210
\(365\) 0 0
\(366\) −1.04982e6 −0.409647
\(367\) 2.26697e6 0.878577 0.439288 0.898346i \(-0.355231\pi\)
0.439288 + 0.898346i \(0.355231\pi\)
\(368\) −1.62071e6 −0.623857
\(369\) 45841.1 0.0175263
\(370\) 0 0
\(371\) −747167. −0.281827
\(372\) −482642. −0.180829
\(373\) 4.55029e6 1.69343 0.846714 0.532048i \(-0.178577\pi\)
0.846714 + 0.532048i \(0.178577\pi\)
\(374\) 50951.0 0.0188353
\(375\) 0 0
\(376\) 2.20442e6 0.804125
\(377\) 1.18812e6 0.430532
\(378\) −1.69794e6 −0.611213
\(379\) 618788. 0.221281 0.110641 0.993860i \(-0.464710\pi\)
0.110641 + 0.993860i \(0.464710\pi\)
\(380\) 0 0
\(381\) 2.32886e6 0.821924
\(382\) −865881. −0.303599
\(383\) −2.23829e6 −0.779686 −0.389843 0.920881i \(-0.627471\pi\)
−0.389843 + 0.920881i \(0.627471\pi\)
\(384\) −3.18264e6 −1.10144
\(385\) 0 0
\(386\) 34844.5 0.0119033
\(387\) 595120. 0.201989
\(388\) 1.44010e6 0.485640
\(389\) −4.60206e6 −1.54198 −0.770989 0.636848i \(-0.780238\pi\)
−0.770989 + 0.636848i \(0.780238\pi\)
\(390\) 0 0
\(391\) 533199. 0.176379
\(392\) 4.44671e6 1.46158
\(393\) 916405. 0.299299
\(394\) 1.20490e6 0.391030
\(395\) 0 0
\(396\) −133123. −0.0426595
\(397\) 4.35532e6 1.38690 0.693448 0.720506i \(-0.256091\pi\)
0.693448 + 0.720506i \(0.256091\pi\)
\(398\) 1.20703e6 0.381952
\(399\) 514181. 0.161690
\(400\) 0 0
\(401\) 3.62515e6 1.12581 0.562905 0.826522i \(-0.309684\pi\)
0.562905 + 0.826522i \(0.309684\pi\)
\(402\) 1.91278e6 0.590338
\(403\) 481316. 0.147628
\(404\) −5.06697e6 −1.54453
\(405\) 0 0
\(406\) 1.29931e6 0.391200
\(407\) −1.51695e6 −0.453927
\(408\) 419221. 0.124679
\(409\) 4.13585e6 1.22252 0.611260 0.791430i \(-0.290663\pi\)
0.611260 + 0.791430i \(0.290663\pi\)
\(410\) 0 0
\(411\) 682922. 0.199419
\(412\) 930032. 0.269932
\(413\) 2.58316e6 0.745206
\(414\) 250552. 0.0718450
\(415\) 0 0
\(416\) 2.48609e6 0.704343
\(417\) −2.36655e6 −0.666463
\(418\) 36182.4 0.0101288
\(419\) −2.46691e6 −0.686464 −0.343232 0.939251i \(-0.611522\pi\)
−0.343232 + 0.939251i \(0.611522\pi\)
\(420\) 0 0
\(421\) 3.45258e6 0.949376 0.474688 0.880154i \(-0.342561\pi\)
0.474688 + 0.880154i \(0.342561\pi\)
\(422\) −1.18635e6 −0.324287
\(423\) 684813. 0.186089
\(424\) −432602. −0.116862
\(425\) 0 0
\(426\) −603168. −0.161033
\(427\) −6.36599e6 −1.68965
\(428\) −6.07853e6 −1.60394
\(429\) 928038. 0.243457
\(430\) 0 0
\(431\) −3.65893e6 −0.948770 −0.474385 0.880318i \(-0.657329\pi\)
−0.474385 + 0.880318i \(0.657329\pi\)
\(432\) 1.97551e6 0.509296
\(433\) 1.59716e6 0.409381 0.204690 0.978827i \(-0.434381\pi\)
0.204690 + 0.978827i \(0.434381\pi\)
\(434\) 526363. 0.134141
\(435\) 0 0
\(436\) −4.39999e6 −1.10850
\(437\) 378647. 0.0948485
\(438\) 378190. 0.0941943
\(439\) −1.58464e6 −0.392437 −0.196219 0.980560i \(-0.562866\pi\)
−0.196219 + 0.980560i \(0.562866\pi\)
\(440\) 0 0
\(441\) 1.38139e6 0.338237
\(442\) −191788. −0.0466945
\(443\) −2.29633e6 −0.555936 −0.277968 0.960590i \(-0.589661\pi\)
−0.277968 + 0.960590i \(0.589661\pi\)
\(444\) −5.72580e6 −1.37841
\(445\) 0 0
\(446\) −418837. −0.0997029
\(447\) −2.96496e6 −0.701859
\(448\) −1.46349e6 −0.344504
\(449\) 3.31569e6 0.776171 0.388086 0.921623i \(-0.373136\pi\)
0.388086 + 0.921623i \(0.373136\pi\)
\(450\) 0 0
\(451\) 136740. 0.0316559
\(452\) 2.50134e6 0.575872
\(453\) −6.89136e6 −1.57783
\(454\) 802664. 0.182765
\(455\) 0 0
\(456\) 297706. 0.0670464
\(457\) −2.20892e6 −0.494754 −0.247377 0.968919i \(-0.579569\pi\)
−0.247377 + 0.968919i \(0.579569\pi\)
\(458\) −1.11386e6 −0.248123
\(459\) −649927. −0.143990
\(460\) 0 0
\(461\) −1.86064e6 −0.407764 −0.203882 0.978995i \(-0.565356\pi\)
−0.203882 + 0.978995i \(0.565356\pi\)
\(462\) 1.01489e6 0.221216
\(463\) −1.20592e6 −0.261437 −0.130718 0.991420i \(-0.541728\pi\)
−0.130718 + 0.991420i \(0.541728\pi\)
\(464\) −1.51172e6 −0.325969
\(465\) 0 0
\(466\) −2.65383e6 −0.566119
\(467\) −2.29388e6 −0.486719 −0.243360 0.969936i \(-0.578250\pi\)
−0.243360 + 0.969936i \(0.578250\pi\)
\(468\) 501098. 0.105757
\(469\) 1.15990e7 2.43493
\(470\) 0 0
\(471\) 240711. 0.0499969
\(472\) 1.49563e6 0.309007
\(473\) 1.77519e6 0.364832
\(474\) 2.26276e6 0.462587
\(475\) 0 0
\(476\) 1.16619e6 0.235913
\(477\) −134390. −0.0270441
\(478\) 410642. 0.0822041
\(479\) −7.90892e6 −1.57499 −0.787496 0.616320i \(-0.788623\pi\)
−0.787496 + 0.616320i \(0.788623\pi\)
\(480\) 0 0
\(481\) 5.71007e6 1.12533
\(482\) −386154. −0.0757083
\(483\) 1.06208e7 2.07152
\(484\) −397095. −0.0770515
\(485\) 0 0
\(486\) −672002. −0.129056
\(487\) −3.48410e6 −0.665684 −0.332842 0.942983i \(-0.608008\pi\)
−0.332842 + 0.942983i \(0.608008\pi\)
\(488\) −3.68585e6 −0.700628
\(489\) −7.04684e6 −1.33267
\(490\) 0 0
\(491\) −8.98096e6 −1.68120 −0.840599 0.541658i \(-0.817797\pi\)
−0.840599 + 0.541658i \(0.817797\pi\)
\(492\) 516132. 0.0961276
\(493\) 497343. 0.0921592
\(494\) −136197. −0.0251101
\(495\) 0 0
\(496\) −612410. −0.111773
\(497\) −3.65756e6 −0.664203
\(498\) −1.70227e6 −0.307577
\(499\) 6.67736e6 1.20048 0.600238 0.799821i \(-0.295072\pi\)
0.600238 + 0.799821i \(0.295072\pi\)
\(500\) 0 0
\(501\) −2.35325e6 −0.418865
\(502\) −989242. −0.175204
\(503\) −7.58428e6 −1.33658 −0.668289 0.743902i \(-0.732973\pi\)
−0.668289 + 0.743902i \(0.732973\pi\)
\(504\) 1.19455e6 0.209473
\(505\) 0 0
\(506\) 747375. 0.129766
\(507\) 2.75905e6 0.476693
\(508\) 3.75096e6 0.644886
\(509\) −6.02580e6 −1.03091 −0.515454 0.856917i \(-0.672377\pi\)
−0.515454 + 0.856917i \(0.672377\pi\)
\(510\) 0 0
\(511\) 2.29331e6 0.388518
\(512\) −5.58471e6 −0.941511
\(513\) −461540. −0.0774312
\(514\) −2.53042e6 −0.422460
\(515\) 0 0
\(516\) 6.70054e6 1.10786
\(517\) 2.04274e6 0.336114
\(518\) 6.24448e6 1.02252
\(519\) −1.15704e7 −1.88551
\(520\) 0 0
\(521\) −4.58541e6 −0.740088 −0.370044 0.929014i \(-0.620657\pi\)
−0.370044 + 0.929014i \(0.620657\pi\)
\(522\) 233703. 0.0375394
\(523\) 4.88145e6 0.780359 0.390179 0.920739i \(-0.372413\pi\)
0.390179 + 0.920739i \(0.372413\pi\)
\(524\) 1.47600e6 0.234832
\(525\) 0 0
\(526\) 978910. 0.154269
\(527\) 201478. 0.0316010
\(528\) −1.18080e6 −0.184329
\(529\) 1.38489e6 0.215168
\(530\) 0 0
\(531\) 464624. 0.0715098
\(532\) 828160. 0.126863
\(533\) −514714. −0.0784780
\(534\) −3.05965e6 −0.464321
\(535\) 0 0
\(536\) 6.71568e6 1.00967
\(537\) −597738. −0.0894489
\(538\) 4.16810e6 0.620844
\(539\) 4.12058e6 0.610923
\(540\) 0 0
\(541\) 6.21940e6 0.913598 0.456799 0.889570i \(-0.348996\pi\)
0.456799 + 0.889570i \(0.348996\pi\)
\(542\) −4.95172e6 −0.724033
\(543\) −4.38614e6 −0.638385
\(544\) 1.04067e6 0.150771
\(545\) 0 0
\(546\) −3.82023e6 −0.548414
\(547\) 9.49047e6 1.35619 0.678093 0.734976i \(-0.262807\pi\)
0.678093 + 0.734976i \(0.262807\pi\)
\(548\) 1.09994e6 0.156465
\(549\) −1.14503e6 −0.162138
\(550\) 0 0
\(551\) 353184. 0.0495589
\(552\) 6.14935e6 0.858976
\(553\) 1.37212e7 1.90800
\(554\) 2.82310e6 0.390798
\(555\) 0 0
\(556\) −3.81166e6 −0.522910
\(557\) 2.92907e6 0.400029 0.200014 0.979793i \(-0.435901\pi\)
0.200014 + 0.979793i \(0.435901\pi\)
\(558\) 94675.0 0.0128721
\(559\) −6.68213e6 −0.904451
\(560\) 0 0
\(561\) 388475. 0.0521141
\(562\) −1.21324e6 −0.162033
\(563\) 455079. 0.0605084 0.0302542 0.999542i \(-0.490368\pi\)
0.0302542 + 0.999542i \(0.490368\pi\)
\(564\) 7.71041e6 1.02066
\(565\) 0 0
\(566\) −299955. −0.0393563
\(567\) −1.51689e7 −1.98152
\(568\) −2.11769e6 −0.275418
\(569\) −6.27664e6 −0.812730 −0.406365 0.913711i \(-0.633204\pi\)
−0.406365 + 0.913711i \(0.633204\pi\)
\(570\) 0 0
\(571\) −621794. −0.0798098 −0.0399049 0.999203i \(-0.512706\pi\)
−0.0399049 + 0.999203i \(0.512706\pi\)
\(572\) 1.49473e6 0.191018
\(573\) −6.60189e6 −0.840005
\(574\) −562886. −0.0713085
\(575\) 0 0
\(576\) −263232. −0.0330585
\(577\) −1.28776e7 −1.61026 −0.805130 0.593098i \(-0.797905\pi\)
−0.805130 + 0.593098i \(0.797905\pi\)
\(578\) 3.05560e6 0.380432
\(579\) 265671. 0.0329343
\(580\) 0 0
\(581\) −1.03224e7 −1.26865
\(582\) −1.97475e6 −0.241659
\(583\) −400875. −0.0488470
\(584\) 1.32780e6 0.161102
\(585\) 0 0
\(586\) −3.89602e6 −0.468681
\(587\) −1.08775e7 −1.30296 −0.651482 0.758664i \(-0.725852\pi\)
−0.651482 + 0.758664i \(0.725852\pi\)
\(588\) 1.55533e7 1.85515
\(589\) 143078. 0.0169935
\(590\) 0 0
\(591\) 9.18673e6 1.08191
\(592\) −7.26529e6 −0.852018
\(593\) 7.50449e6 0.876364 0.438182 0.898886i \(-0.355622\pi\)
0.438182 + 0.898886i \(0.355622\pi\)
\(594\) −910990. −0.105937
\(595\) 0 0
\(596\) −4.77548e6 −0.550682
\(597\) 9.20295e6 1.05680
\(598\) −2.81325e6 −0.321703
\(599\) −7.69438e6 −0.876207 −0.438104 0.898925i \(-0.644350\pi\)
−0.438104 + 0.898925i \(0.644350\pi\)
\(600\) 0 0
\(601\) 3.14770e6 0.355473 0.177737 0.984078i \(-0.443123\pi\)
0.177737 + 0.984078i \(0.443123\pi\)
\(602\) −7.30751e6 −0.821823
\(603\) 2.08626e6 0.233655
\(604\) −1.10995e7 −1.23797
\(605\) 0 0
\(606\) 6.94810e6 0.768572
\(607\) 4.57397e6 0.503874 0.251937 0.967744i \(-0.418932\pi\)
0.251937 + 0.967744i \(0.418932\pi\)
\(608\) 739025. 0.0810775
\(609\) 9.90660e6 1.08238
\(610\) 0 0
\(611\) −7.68923e6 −0.833258
\(612\) 209758. 0.0226381
\(613\) 1.56075e7 1.67758 0.838790 0.544455i \(-0.183264\pi\)
0.838790 + 0.544455i \(0.183264\pi\)
\(614\) −4.27436e6 −0.457562
\(615\) 0 0
\(616\) 3.56324e6 0.378349
\(617\) 1.18602e7 1.25424 0.627119 0.778924i \(-0.284234\pi\)
0.627119 + 0.778924i \(0.284234\pi\)
\(618\) −1.27531e6 −0.134321
\(619\) 7.91821e6 0.830616 0.415308 0.909681i \(-0.363674\pi\)
0.415308 + 0.909681i \(0.363674\pi\)
\(620\) 0 0
\(621\) −9.53346e6 −0.992023
\(622\) −6.58883e6 −0.682861
\(623\) −1.85534e7 −1.91516
\(624\) 4.44475e6 0.456968
\(625\) 0 0
\(626\) −23635.3 −0.00241060
\(627\) 275872. 0.0280246
\(628\) 387698. 0.0392278
\(629\) 2.39022e6 0.240886
\(630\) 0 0
\(631\) −1.11561e7 −1.11542 −0.557709 0.830037i \(-0.688319\pi\)
−0.557709 + 0.830037i \(0.688319\pi\)
\(632\) 7.94444e6 0.791172
\(633\) −9.04527e6 −0.897248
\(634\) −5.38140e6 −0.531707
\(635\) 0 0
\(636\) −1.51312e6 −0.148330
\(637\) −1.55106e7 −1.51453
\(638\) 697116. 0.0678037
\(639\) −657872. −0.0637367
\(640\) 0 0
\(641\) 7.17389e6 0.689620 0.344810 0.938672i \(-0.387943\pi\)
0.344810 + 0.938672i \(0.387943\pi\)
\(642\) 8.33521e6 0.798139
\(643\) −7.14025e6 −0.681061 −0.340531 0.940233i \(-0.610607\pi\)
−0.340531 + 0.940233i \(0.610607\pi\)
\(644\) 1.71063e7 1.62533
\(645\) 0 0
\(646\) −57011.6 −0.00537505
\(647\) −1.56897e7 −1.47351 −0.736756 0.676159i \(-0.763643\pi\)
−0.736756 + 0.676159i \(0.763643\pi\)
\(648\) −8.78267e6 −0.821654
\(649\) 1.38594e6 0.129161
\(650\) 0 0
\(651\) 4.01324e6 0.371145
\(652\) −1.13499e7 −1.04562
\(653\) 5.04236e6 0.462755 0.231378 0.972864i \(-0.425677\pi\)
0.231378 + 0.972864i \(0.425677\pi\)
\(654\) 6.03350e6 0.551601
\(655\) 0 0
\(656\) 654904. 0.0594180
\(657\) 412489. 0.0372820
\(658\) −8.40887e6 −0.757134
\(659\) −9.10902e6 −0.817068 −0.408534 0.912743i \(-0.633960\pi\)
−0.408534 + 0.912743i \(0.633960\pi\)
\(660\) 0 0
\(661\) 1.31308e7 1.16893 0.584464 0.811420i \(-0.301305\pi\)
0.584464 + 0.811420i \(0.301305\pi\)
\(662\) −264095. −0.0234215
\(663\) −1.46229e6 −0.129196
\(664\) −5.97657e6 −0.526056
\(665\) 0 0
\(666\) 1.12317e6 0.0981207
\(667\) 7.29528e6 0.634933
\(668\) −3.79023e6 −0.328643
\(669\) −3.19341e6 −0.275861
\(670\) 0 0
\(671\) −3.41552e6 −0.292854
\(672\) 2.07292e7 1.77076
\(673\) −1.55171e7 −1.32061 −0.660303 0.750999i \(-0.729572\pi\)
−0.660303 + 0.750999i \(0.729572\pi\)
\(674\) −4.46566e6 −0.378648
\(675\) 0 0
\(676\) 4.44382e6 0.374016
\(677\) 1.40356e7 1.17695 0.588476 0.808515i \(-0.299728\pi\)
0.588476 + 0.808515i \(0.299728\pi\)
\(678\) −3.42997e6 −0.286560
\(679\) −1.19747e7 −0.996758
\(680\) 0 0
\(681\) 6.11990e6 0.505681
\(682\) 282408. 0.0232496
\(683\) −5.34969e6 −0.438810 −0.219405 0.975634i \(-0.570412\pi\)
−0.219405 + 0.975634i \(0.570412\pi\)
\(684\) 148958. 0.0121737
\(685\) 0 0
\(686\) −8.59079e6 −0.696984
\(687\) −8.49261e6 −0.686514
\(688\) 8.50211e6 0.684787
\(689\) 1.50896e6 0.121096
\(690\) 0 0
\(691\) 1.31390e7 1.04681 0.523404 0.852084i \(-0.324662\pi\)
0.523404 + 0.852084i \(0.324662\pi\)
\(692\) −1.86357e7 −1.47938
\(693\) 1.10694e6 0.0875569
\(694\) 6.66695e6 0.525447
\(695\) 0 0
\(696\) 5.73582e6 0.448820
\(697\) −215458. −0.0167989
\(698\) −5.30944e6 −0.412487
\(699\) −2.02341e7 −1.56636
\(700\) 0 0
\(701\) −2.49888e7 −1.92066 −0.960330 0.278865i \(-0.910042\pi\)
−0.960330 + 0.278865i \(0.910042\pi\)
\(702\) 3.42912e6 0.262627
\(703\) 1.69740e6 0.129537
\(704\) −785200. −0.0597102
\(705\) 0 0
\(706\) 8.01676e6 0.605323
\(707\) 4.21327e7 3.17008
\(708\) 5.23127e6 0.392215
\(709\) −8.86200e6 −0.662089 −0.331044 0.943615i \(-0.607401\pi\)
−0.331044 + 0.943615i \(0.607401\pi\)
\(710\) 0 0
\(711\) 2.46798e6 0.183092
\(712\) −1.07423e7 −0.794138
\(713\) 2.95538e6 0.217716
\(714\) −1.59914e6 −0.117393
\(715\) 0 0
\(716\) −962739. −0.0701820
\(717\) 3.13093e6 0.227445
\(718\) −2.07427e6 −0.150160
\(719\) 2.58635e7 1.86580 0.932901 0.360132i \(-0.117268\pi\)
0.932901 + 0.360132i \(0.117268\pi\)
\(720\) 0 0
\(721\) −7.73336e6 −0.554026
\(722\) 5.42820e6 0.387537
\(723\) −2.94423e6 −0.209472
\(724\) −7.06448e6 −0.500880
\(725\) 0 0
\(726\) 544518. 0.0383416
\(727\) 1.71871e7 1.20605 0.603026 0.797721i \(-0.293961\pi\)
0.603026 + 0.797721i \(0.293961\pi\)
\(728\) −1.34126e7 −0.937963
\(729\) 1.12207e7 0.781988
\(730\) 0 0
\(731\) −2.79712e6 −0.193606
\(732\) −1.28920e7 −0.889290
\(733\) −1.85650e7 −1.27625 −0.638125 0.769932i \(-0.720290\pi\)
−0.638125 + 0.769932i \(0.720290\pi\)
\(734\) −5.00680e6 −0.343021
\(735\) 0 0
\(736\) 1.52651e7 1.03874
\(737\) 6.22315e6 0.422028
\(738\) −101244. −0.00684274
\(739\) 5.94724e6 0.400594 0.200297 0.979735i \(-0.435809\pi\)
0.200297 + 0.979735i \(0.435809\pi\)
\(740\) 0 0
\(741\) −1.03843e6 −0.0694755
\(742\) 1.65019e6 0.110033
\(743\) 2.72654e7 1.81193 0.905963 0.423357i \(-0.139148\pi\)
0.905963 + 0.423357i \(0.139148\pi\)
\(744\) 2.32363e6 0.153899
\(745\) 0 0
\(746\) −1.00497e7 −0.661161
\(747\) −1.85665e6 −0.121739
\(748\) 625692. 0.0408890
\(749\) 5.05440e7 3.29204
\(750\) 0 0
\(751\) 1.30069e7 0.841541 0.420770 0.907167i \(-0.361760\pi\)
0.420770 + 0.907167i \(0.361760\pi\)
\(752\) 9.78351e6 0.630885
\(753\) −7.54245e6 −0.484758
\(754\) −2.62407e6 −0.168092
\(755\) 0 0
\(756\) −2.08512e7 −1.32686
\(757\) 9.22009e6 0.584784 0.292392 0.956299i \(-0.405549\pi\)
0.292392 + 0.956299i \(0.405549\pi\)
\(758\) −1.36665e6 −0.0863942
\(759\) 5.69835e6 0.359041
\(760\) 0 0
\(761\) 328083. 0.0205363 0.0102682 0.999947i \(-0.496731\pi\)
0.0102682 + 0.999947i \(0.496731\pi\)
\(762\) −5.14351e6 −0.320902
\(763\) 3.65866e7 2.27516
\(764\) −1.06333e7 −0.659072
\(765\) 0 0
\(766\) 4.94347e6 0.304411
\(767\) −5.21690e6 −0.320202
\(768\) 3.53235e6 0.216103
\(769\) 2.19214e6 0.133676 0.0668380 0.997764i \(-0.478709\pi\)
0.0668380 + 0.997764i \(0.478709\pi\)
\(770\) 0 0
\(771\) −1.92932e7 −1.16887
\(772\) 427900. 0.0258404
\(773\) −2.18539e7 −1.31547 −0.657735 0.753249i \(-0.728485\pi\)
−0.657735 + 0.753249i \(0.728485\pi\)
\(774\) −1.31438e6 −0.0788619
\(775\) 0 0
\(776\) −6.93323e6 −0.413315
\(777\) 4.76109e7 2.82914
\(778\) 1.01641e7 0.602031
\(779\) −153006. −0.00903367
\(780\) 0 0
\(781\) −1.96238e6 −0.115121
\(782\) −1.17762e6 −0.0688633
\(783\) −8.89237e6 −0.518338
\(784\) 1.97351e7 1.14670
\(785\) 0 0
\(786\) −2.02396e6 −0.116855
\(787\) −2.61010e7 −1.50217 −0.751087 0.660203i \(-0.770470\pi\)
−0.751087 + 0.660203i \(0.770470\pi\)
\(788\) 1.47965e7 0.848874
\(789\) 7.46368e6 0.426835
\(790\) 0 0
\(791\) −2.07990e7 −1.18196
\(792\) 640907. 0.0363063
\(793\) 1.28566e7 0.726011
\(794\) −9.61913e6 −0.541483
\(795\) 0 0
\(796\) 1.48226e7 0.829168
\(797\) −1.39846e7 −0.779840 −0.389920 0.920849i \(-0.627497\pi\)
−0.389920 + 0.920849i \(0.627497\pi\)
\(798\) −1.13562e6 −0.0631284
\(799\) −3.21869e6 −0.178366
\(800\) 0 0
\(801\) −3.33714e6 −0.183778
\(802\) −8.00647e6 −0.439547
\(803\) 1.23042e6 0.0673388
\(804\) 2.34895e7 1.28155
\(805\) 0 0
\(806\) −1.06303e6 −0.0576379
\(807\) 3.17796e7 1.71777
\(808\) 2.43944e7 1.31450
\(809\) 2.70989e7 1.45573 0.727865 0.685721i \(-0.240513\pi\)
0.727865 + 0.685721i \(0.240513\pi\)
\(810\) 0 0
\(811\) 1.99644e7 1.06587 0.532936 0.846156i \(-0.321089\pi\)
0.532936 + 0.846156i \(0.321089\pi\)
\(812\) 1.59559e7 0.849244
\(813\) −3.77543e7 −2.00327
\(814\) 3.35033e6 0.177226
\(815\) 0 0
\(816\) 1.86056e6 0.0978180
\(817\) −1.98635e6 −0.104112
\(818\) −9.13439e6 −0.477306
\(819\) −4.16671e6 −0.217062
\(820\) 0 0
\(821\) 3.18829e7 1.65082 0.825410 0.564533i \(-0.190944\pi\)
0.825410 + 0.564533i \(0.190944\pi\)
\(822\) −1.50829e6 −0.0778586
\(823\) 1.34203e7 0.690655 0.345328 0.938482i \(-0.387768\pi\)
0.345328 + 0.938482i \(0.387768\pi\)
\(824\) −4.47754e6 −0.229732
\(825\) 0 0
\(826\) −5.70515e6 −0.290949
\(827\) −1.19386e7 −0.607002 −0.303501 0.952831i \(-0.598156\pi\)
−0.303501 + 0.952831i \(0.598156\pi\)
\(828\) 3.07684e6 0.155966
\(829\) 2.59274e7 1.31031 0.655153 0.755497i \(-0.272604\pi\)
0.655153 + 0.755497i \(0.272604\pi\)
\(830\) 0 0
\(831\) 2.15247e7 1.08127
\(832\) 2.95563e6 0.148027
\(833\) −6.49269e6 −0.324199
\(834\) 5.22675e6 0.260206
\(835\) 0 0
\(836\) 444330. 0.0219882
\(837\) −3.60237e6 −0.177736
\(838\) 5.44839e6 0.268015
\(839\) −3.21482e7 −1.57671 −0.788354 0.615222i \(-0.789067\pi\)
−0.788354 + 0.615222i \(0.789067\pi\)
\(840\) 0 0
\(841\) −1.37064e7 −0.668244
\(842\) −7.62533e6 −0.370662
\(843\) −9.25029e6 −0.448318
\(844\) −1.45687e7 −0.703985
\(845\) 0 0
\(846\) −1.51247e6 −0.0726544
\(847\) 3.30191e6 0.158145
\(848\) −1.91995e6 −0.0916855
\(849\) −2.28700e6 −0.108892
\(850\) 0 0
\(851\) 3.50610e7 1.65959
\(852\) −7.40708e6 −0.349581
\(853\) 5.64308e6 0.265548 0.132774 0.991146i \(-0.457612\pi\)
0.132774 + 0.991146i \(0.457612\pi\)
\(854\) 1.40599e7 0.659685
\(855\) 0 0
\(856\) 2.92645e7 1.36507
\(857\) 1.77067e7 0.823543 0.411772 0.911287i \(-0.364910\pi\)
0.411772 + 0.911287i \(0.364910\pi\)
\(858\) −2.04966e6 −0.0950524
\(859\) 1.57119e7 0.726515 0.363258 0.931689i \(-0.381664\pi\)
0.363258 + 0.931689i \(0.381664\pi\)
\(860\) 0 0
\(861\) −4.29172e6 −0.197298
\(862\) 8.08108e6 0.370426
\(863\) −245263. −0.0112100 −0.00560500 0.999984i \(-0.501784\pi\)
−0.00560500 + 0.999984i \(0.501784\pi\)
\(864\) −1.86070e7 −0.847991
\(865\) 0 0
\(866\) −3.52747e6 −0.159834
\(867\) 2.32974e7 1.05259
\(868\) 6.46388e6 0.291202
\(869\) 7.36179e6 0.330700
\(870\) 0 0
\(871\) −2.34250e7 −1.04625
\(872\) 2.11833e7 0.943415
\(873\) −2.15384e6 −0.0956486
\(874\) −836276. −0.0370315
\(875\) 0 0
\(876\) 4.64428e6 0.204483
\(877\) −1.04352e7 −0.458142 −0.229071 0.973410i \(-0.573569\pi\)
−0.229071 + 0.973410i \(0.573569\pi\)
\(878\) 3.49983e6 0.153218
\(879\) −2.97051e7 −1.29676
\(880\) 0 0
\(881\) −1.10430e7 −0.479344 −0.239672 0.970854i \(-0.577040\pi\)
−0.239672 + 0.970854i \(0.577040\pi\)
\(882\) −3.05093e6 −0.132057
\(883\) −5.41498e6 −0.233720 −0.116860 0.993148i \(-0.537283\pi\)
−0.116860 + 0.993148i \(0.537283\pi\)
\(884\) −2.35521e6 −0.101368
\(885\) 0 0
\(886\) 5.07166e6 0.217053
\(887\) 1.52663e7 0.651515 0.325757 0.945453i \(-0.394381\pi\)
0.325757 + 0.945453i \(0.394381\pi\)
\(888\) 2.75663e7 1.17313
\(889\) −3.11898e7 −1.32360
\(890\) 0 0
\(891\) −8.13854e6 −0.343441
\(892\) −5.14343e6 −0.216442
\(893\) −2.28573e6 −0.0959170
\(894\) 6.54838e6 0.274025
\(895\) 0 0
\(896\) 4.26242e7 1.77372
\(897\) −2.14496e7 −0.890096
\(898\) −7.32300e6 −0.303039
\(899\) 2.75664e6 0.113758
\(900\) 0 0
\(901\) 631648. 0.0259217
\(902\) −302004. −0.0123594
\(903\) −5.57160e7 −2.27384
\(904\) −1.20424e7 −0.490109
\(905\) 0 0
\(906\) 1.52202e7 0.616027
\(907\) −2.02437e7 −0.817094 −0.408547 0.912737i \(-0.633964\pi\)
−0.408547 + 0.912737i \(0.633964\pi\)
\(908\) 9.85694e6 0.396759
\(909\) 7.57825e6 0.304200
\(910\) 0 0
\(911\) −1.17158e7 −0.467708 −0.233854 0.972272i \(-0.575134\pi\)
−0.233854 + 0.972272i \(0.575134\pi\)
\(912\) 1.32126e6 0.0526019
\(913\) −5.53824e6 −0.219885
\(914\) 4.87860e6 0.193166
\(915\) 0 0
\(916\) −1.36785e7 −0.538642
\(917\) −1.22731e7 −0.481984
\(918\) 1.43542e6 0.0562177
\(919\) −1.95296e7 −0.762788 −0.381394 0.924413i \(-0.624556\pi\)
−0.381394 + 0.924413i \(0.624556\pi\)
\(920\) 0 0
\(921\) −3.25898e7 −1.26599
\(922\) 4.10939e6 0.159202
\(923\) 7.38673e6 0.285396
\(924\) 1.24632e7 0.480230
\(925\) 0 0
\(926\) 2.66339e6 0.102072
\(927\) −1.39097e6 −0.0531641
\(928\) 1.42386e7 0.542747
\(929\) −4.29425e7 −1.63248 −0.816240 0.577713i \(-0.803945\pi\)
−0.816240 + 0.577713i \(0.803945\pi\)
\(930\) 0 0
\(931\) −4.61073e6 −0.174339
\(932\) −3.25898e7 −1.22897
\(933\) −5.02364e7 −1.88936
\(934\) 5.06625e6 0.190029
\(935\) 0 0
\(936\) −2.41248e6 −0.0900067
\(937\) −2.27191e7 −0.845361 −0.422681 0.906279i \(-0.638911\pi\)
−0.422681 + 0.906279i \(0.638911\pi\)
\(938\) −2.56174e7 −0.950665
\(939\) −180207. −0.00666972
\(940\) 0 0
\(941\) 3.98095e6 0.146559 0.0732795 0.997311i \(-0.476653\pi\)
0.0732795 + 0.997311i \(0.476653\pi\)
\(942\) −531632. −0.0195202
\(943\) −3.16045e6 −0.115736
\(944\) 6.63780e6 0.242434
\(945\) 0 0
\(946\) −3.92067e6 −0.142440
\(947\) 2.43639e7 0.882818 0.441409 0.897306i \(-0.354479\pi\)
0.441409 + 0.897306i \(0.354479\pi\)
\(948\) 2.77874e7 1.00422
\(949\) −4.63152e6 −0.166939
\(950\) 0 0
\(951\) −4.10304e7 −1.47114
\(952\) −5.61450e6 −0.200779
\(953\) −1.39017e7 −0.495833 −0.247916 0.968781i \(-0.579746\pi\)
−0.247916 + 0.968781i \(0.579746\pi\)
\(954\) 296813. 0.0105587
\(955\) 0 0
\(956\) 5.04280e6 0.178454
\(957\) 5.31515e6 0.187601
\(958\) 1.74676e7 0.614920
\(959\) −9.14617e6 −0.321139
\(960\) 0 0
\(961\) −2.75124e7 −0.960993
\(962\) −1.26112e7 −0.439358
\(963\) 9.09116e6 0.315903
\(964\) −4.74208e6 −0.164353
\(965\) 0 0
\(966\) −2.34570e7 −0.808780
\(967\) −5.16682e7 −1.77688 −0.888438 0.458998i \(-0.848209\pi\)
−0.888438 + 0.458998i \(0.848209\pi\)
\(968\) 1.91177e6 0.0655764
\(969\) −434684. −0.0148718
\(970\) 0 0
\(971\) −1.45794e7 −0.496240 −0.248120 0.968729i \(-0.579813\pi\)
−0.248120 + 0.968729i \(0.579813\pi\)
\(972\) −8.25237e6 −0.280164
\(973\) 3.16945e7 1.07325
\(974\) 7.69495e6 0.259901
\(975\) 0 0
\(976\) −1.63583e7 −0.549685
\(977\) 3.09921e6 0.103876 0.0519379 0.998650i \(-0.483460\pi\)
0.0519379 + 0.998650i \(0.483460\pi\)
\(978\) 1.55636e7 0.520311
\(979\) −9.95442e6 −0.331940
\(980\) 0 0
\(981\) 6.58071e6 0.218323
\(982\) 1.98353e7 0.656386
\(983\) −1.53445e7 −0.506489 −0.253245 0.967402i \(-0.581498\pi\)
−0.253245 + 0.967402i \(0.581498\pi\)
\(984\) −2.48486e6 −0.0818116
\(985\) 0 0
\(986\) −1.09843e6 −0.0359815
\(987\) −6.41133e7 −2.09486
\(988\) −1.67253e6 −0.0545108
\(989\) −4.10296e7 −1.33385
\(990\) 0 0
\(991\) 1.57747e7 0.510244 0.255122 0.966909i \(-0.417884\pi\)
0.255122 + 0.966909i \(0.417884\pi\)
\(992\) 5.76818e6 0.186106
\(993\) −2.01359e6 −0.0648033
\(994\) 8.07806e6 0.259323
\(995\) 0 0
\(996\) −2.09043e7 −0.667710
\(997\) −1.85577e6 −0.0591270 −0.0295635 0.999563i \(-0.509412\pi\)
−0.0295635 + 0.999563i \(0.509412\pi\)
\(998\) −1.47476e7 −0.468699
\(999\) −4.27365e7 −1.35483
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 275.6.a.b.1.2 3
5.2 odd 4 275.6.b.b.199.3 6
5.3 odd 4 275.6.b.b.199.4 6
5.4 even 2 11.6.a.b.1.2 3
15.14 odd 2 99.6.a.g.1.2 3
20.19 odd 2 176.6.a.i.1.2 3
35.34 odd 2 539.6.a.e.1.2 3
40.19 odd 2 704.6.a.t.1.2 3
40.29 even 2 704.6.a.q.1.2 3
55.54 odd 2 121.6.a.d.1.2 3
165.164 even 2 1089.6.a.r.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.6.a.b.1.2 3 5.4 even 2
99.6.a.g.1.2 3 15.14 odd 2
121.6.a.d.1.2 3 55.54 odd 2
176.6.a.i.1.2 3 20.19 odd 2
275.6.a.b.1.2 3 1.1 even 1 trivial
275.6.b.b.199.3 6 5.2 odd 4
275.6.b.b.199.4 6 5.3 odd 4
539.6.a.e.1.2 3 35.34 odd 2
704.6.a.q.1.2 3 40.29 even 2
704.6.a.t.1.2 3 40.19 odd 2
1089.6.a.r.1.2 3 165.164 even 2