Properties

Label 175.6.a.c.1.2
Level $175$
Weight $6$
Character 175.1
Self dual yes
Analytic conductor $28.067$
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] = [175,6,Mod(1,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, 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("175.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 175.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: \(28.0671684673\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 175.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.725083 q^{2} -19.6495 q^{3} -31.4743 q^{4} +14.2475 q^{6} -49.0000 q^{7} +46.0241 q^{8} +143.103 q^{9} +O(q^{10})\) \(q-0.725083 q^{2} -19.6495 q^{3} -31.4743 q^{4} +14.2475 q^{6} -49.0000 q^{7} +46.0241 q^{8} +143.103 q^{9} +666.090 q^{11} +618.453 q^{12} +650.640 q^{13} +35.5291 q^{14} +973.805 q^{16} -1186.89 q^{17} -103.762 q^{18} -1565.05 q^{19} +962.826 q^{21} -482.970 q^{22} +1100.15 q^{23} -904.350 q^{24} -471.768 q^{26} +1962.93 q^{27} +1542.24 q^{28} +2396.72 q^{29} -2048.46 q^{31} -2178.86 q^{32} -13088.3 q^{33} +860.596 q^{34} -4504.06 q^{36} -1077.54 q^{37} +1134.79 q^{38} -12784.7 q^{39} +1098.21 q^{41} -698.128 q^{42} -16564.3 q^{43} -20964.7 q^{44} -797.702 q^{46} +8298.39 q^{47} -19134.8 q^{48} +2401.00 q^{49} +23321.9 q^{51} -20478.4 q^{52} -5519.18 q^{53} -1423.28 q^{54} -2255.18 q^{56} +30752.5 q^{57} -1737.82 q^{58} -14230.4 q^{59} -14234.7 q^{61} +1485.30 q^{62} -7012.05 q^{63} -29581.9 q^{64} +9490.12 q^{66} -19730.4 q^{67} +37356.6 q^{68} -21617.5 q^{69} +64562.7 q^{71} +6586.18 q^{72} -28567.0 q^{73} +781.309 q^{74} +49258.8 q^{76} -32638.4 q^{77} +9270.00 q^{78} -30633.4 q^{79} -73344.6 q^{81} -796.293 q^{82} +675.946 q^{83} -30304.2 q^{84} +12010.5 q^{86} -47094.4 q^{87} +30656.2 q^{88} +125971. q^{89} -31881.3 q^{91} -34626.5 q^{92} +40251.1 q^{93} -6017.02 q^{94} +42813.5 q^{96} +22906.8 q^{97} -1740.92 q^{98} +95319.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{2} + 6 q^{3} + 5 q^{4} - 198 q^{6} - 98 q^{7} + 9 q^{8} + 558 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{2} + 6 q^{3} + 5 q^{4} - 198 q^{6} - 98 q^{7} + 9 q^{8} + 558 q^{9} + 396 q^{11} + 1554 q^{12} + 350 q^{13} + 441 q^{14} + 113 q^{16} - 1800 q^{17} - 3537 q^{18} - 3266 q^{19} - 294 q^{21} + 1752 q^{22} - 2088 q^{23} - 1854 q^{24} + 2016 q^{26} + 6372 q^{27} - 245 q^{28} + 6696 q^{29} - 20 q^{31} + 6129 q^{32} - 20016 q^{33} + 5934 q^{34} + 10629 q^{36} - 6232 q^{37} + 15210 q^{38} - 20496 q^{39} - 6048 q^{41} + 9702 q^{42} + 3020 q^{43} - 30816 q^{44} + 25584 q^{46} - 11700 q^{47} - 41214 q^{48} + 4802 q^{49} + 7596 q^{51} - 31444 q^{52} - 9468 q^{53} - 37908 q^{54} - 441 q^{56} - 12876 q^{57} - 37314 q^{58} - 43938 q^{59} - 64754 q^{61} - 15300 q^{62} - 27342 q^{63} - 70783 q^{64} + 66816 q^{66} - 24784 q^{67} + 14994 q^{68} - 103392 q^{69} + 97416 q^{71} - 8775 q^{72} - 17452 q^{73} + 43434 q^{74} - 12782 q^{76} - 19404 q^{77} + 73080 q^{78} + 51256 q^{79} - 61074 q^{81} + 58338 q^{82} - 117558 q^{83} - 76146 q^{84} - 150048 q^{86} + 63180 q^{87} + 40656 q^{88} + 84276 q^{89} - 17150 q^{91} - 150912 q^{92} + 92280 q^{93} + 159468 q^{94} + 255906 q^{96} - 20776 q^{97} - 21609 q^{98} - 16740 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.725083 −0.128178 −0.0640889 0.997944i \(-0.520414\pi\)
−0.0640889 + 0.997944i \(0.520414\pi\)
\(3\) −19.6495 −1.26052 −0.630258 0.776386i \(-0.717051\pi\)
−0.630258 + 0.776386i \(0.717051\pi\)
\(4\) −31.4743 −0.983570
\(5\) 0 0
\(6\) 14.2475 0.161570
\(7\) −49.0000 −0.377964
\(8\) 46.0241 0.254250
\(9\) 143.103 0.588901
\(10\) 0 0
\(11\) 666.090 1.65978 0.829891 0.557926i \(-0.188403\pi\)
0.829891 + 0.557926i \(0.188403\pi\)
\(12\) 618.453 1.23981
\(13\) 650.640 1.06778 0.533890 0.845554i \(-0.320729\pi\)
0.533890 + 0.845554i \(0.320729\pi\)
\(14\) 35.5291 0.0484466
\(15\) 0 0
\(16\) 973.805 0.950981
\(17\) −1186.89 −0.996069 −0.498035 0.867157i \(-0.665945\pi\)
−0.498035 + 0.867157i \(0.665945\pi\)
\(18\) −103.762 −0.0754840
\(19\) −1565.05 −0.994591 −0.497296 0.867581i \(-0.665674\pi\)
−0.497296 + 0.867581i \(0.665674\pi\)
\(20\) 0 0
\(21\) 962.826 0.476430
\(22\) −482.970 −0.212747
\(23\) 1100.15 0.433644 0.216822 0.976211i \(-0.430431\pi\)
0.216822 + 0.976211i \(0.430431\pi\)
\(24\) −904.350 −0.320486
\(25\) 0 0
\(26\) −471.768 −0.136866
\(27\) 1962.93 0.518197
\(28\) 1542.24 0.371755
\(29\) 2396.72 0.529203 0.264602 0.964358i \(-0.414760\pi\)
0.264602 + 0.964358i \(0.414760\pi\)
\(30\) 0 0
\(31\) −2048.46 −0.382844 −0.191422 0.981508i \(-0.561310\pi\)
−0.191422 + 0.981508i \(0.561310\pi\)
\(32\) −2178.86 −0.376144
\(33\) −13088.3 −2.09218
\(34\) 860.596 0.127674
\(35\) 0 0
\(36\) −4504.06 −0.579226
\(37\) −1077.54 −0.129399 −0.0646995 0.997905i \(-0.520609\pi\)
−0.0646995 + 0.997905i \(0.520609\pi\)
\(38\) 1134.79 0.127484
\(39\) −12784.7 −1.34595
\(40\) 0 0
\(41\) 1098.21 0.102029 0.0510147 0.998698i \(-0.483754\pi\)
0.0510147 + 0.998698i \(0.483754\pi\)
\(42\) −698.128 −0.0610678
\(43\) −16564.3 −1.36616 −0.683081 0.730343i \(-0.739360\pi\)
−0.683081 + 0.730343i \(0.739360\pi\)
\(44\) −20964.7 −1.63251
\(45\) 0 0
\(46\) −797.702 −0.0555835
\(47\) 8298.39 0.547960 0.273980 0.961735i \(-0.411660\pi\)
0.273980 + 0.961735i \(0.411660\pi\)
\(48\) −19134.8 −1.19873
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 23321.9 1.25556
\(52\) −20478.4 −1.05024
\(53\) −5519.18 −0.269889 −0.134944 0.990853i \(-0.543086\pi\)
−0.134944 + 0.990853i \(0.543086\pi\)
\(54\) −1423.28 −0.0664213
\(55\) 0 0
\(56\) −2255.18 −0.0960973
\(57\) 30752.5 1.25370
\(58\) −1737.82 −0.0678321
\(59\) −14230.4 −0.532216 −0.266108 0.963943i \(-0.585738\pi\)
−0.266108 + 0.963943i \(0.585738\pi\)
\(60\) 0 0
\(61\) −14234.7 −0.489807 −0.244904 0.969547i \(-0.578756\pi\)
−0.244904 + 0.969547i \(0.578756\pi\)
\(62\) 1485.30 0.0490721
\(63\) −7012.05 −0.222584
\(64\) −29581.9 −0.902768
\(65\) 0 0
\(66\) 9490.12 0.268171
\(67\) −19730.4 −0.536970 −0.268485 0.963284i \(-0.586523\pi\)
−0.268485 + 0.963284i \(0.586523\pi\)
\(68\) 37356.6 0.979704
\(69\) −21617.5 −0.546615
\(70\) 0 0
\(71\) 64562.7 1.51997 0.759986 0.649940i \(-0.225206\pi\)
0.759986 + 0.649940i \(0.225206\pi\)
\(72\) 6586.18 0.149728
\(73\) −28567.0 −0.627418 −0.313709 0.949519i \(-0.601572\pi\)
−0.313709 + 0.949519i \(0.601572\pi\)
\(74\) 781.309 0.0165861
\(75\) 0 0
\(76\) 49258.8 0.978251
\(77\) −32638.4 −0.627339
\(78\) 9270.00 0.172521
\(79\) −30633.4 −0.552239 −0.276119 0.961123i \(-0.589049\pi\)
−0.276119 + 0.961123i \(0.589049\pi\)
\(80\) 0 0
\(81\) −73344.6 −1.24210
\(82\) −796.293 −0.0130779
\(83\) 675.946 0.0107700 0.00538501 0.999986i \(-0.498286\pi\)
0.00538501 + 0.999986i \(0.498286\pi\)
\(84\) −30304.2 −0.468603
\(85\) 0 0
\(86\) 12010.5 0.175111
\(87\) −47094.4 −0.667069
\(88\) 30656.2 0.421999
\(89\) 125971. 1.68576 0.842882 0.538098i \(-0.180857\pi\)
0.842882 + 0.538098i \(0.180857\pi\)
\(90\) 0 0
\(91\) −31881.3 −0.403583
\(92\) −34626.5 −0.426520
\(93\) 40251.1 0.482582
\(94\) −6017.02 −0.0702363
\(95\) 0 0
\(96\) 42813.5 0.474136
\(97\) 22906.8 0.247192 0.123596 0.992333i \(-0.460557\pi\)
0.123596 + 0.992333i \(0.460557\pi\)
\(98\) −1740.92 −0.0183111
\(99\) 95319.4 0.977447
\(100\) 0 0
\(101\) −181474. −1.77015 −0.885077 0.465444i \(-0.845895\pi\)
−0.885077 + 0.465444i \(0.845895\pi\)
\(102\) −16910.3 −0.160935
\(103\) −64772.0 −0.601581 −0.300791 0.953690i \(-0.597251\pi\)
−0.300791 + 0.953690i \(0.597251\pi\)
\(104\) 29945.1 0.271483
\(105\) 0 0
\(106\) 4001.86 0.0345938
\(107\) 148170. 1.25112 0.625562 0.780175i \(-0.284870\pi\)
0.625562 + 0.780175i \(0.284870\pi\)
\(108\) −61781.7 −0.509683
\(109\) −111294. −0.897237 −0.448618 0.893723i \(-0.648084\pi\)
−0.448618 + 0.893723i \(0.648084\pi\)
\(110\) 0 0
\(111\) 21173.2 0.163110
\(112\) −47716.4 −0.359437
\(113\) 43175.5 0.318084 0.159042 0.987272i \(-0.449160\pi\)
0.159042 + 0.987272i \(0.449160\pi\)
\(114\) −22298.1 −0.160696
\(115\) 0 0
\(116\) −75435.0 −0.520509
\(117\) 93108.5 0.628817
\(118\) 10318.2 0.0682182
\(119\) 58157.8 0.376479
\(120\) 0 0
\(121\) 282625. 1.75488
\(122\) 10321.4 0.0627824
\(123\) −21579.3 −0.128610
\(124\) 64473.6 0.376554
\(125\) 0 0
\(126\) 5084.31 0.0285303
\(127\) 131449. 0.723182 0.361591 0.932337i \(-0.382234\pi\)
0.361591 + 0.932337i \(0.382234\pi\)
\(128\) 91172.8 0.491859
\(129\) 325480. 1.72207
\(130\) 0 0
\(131\) −349458. −1.77916 −0.889582 0.456775i \(-0.849005\pi\)
−0.889582 + 0.456775i \(0.849005\pi\)
\(132\) 411946. 2.05781
\(133\) 76687.5 0.375920
\(134\) 14306.2 0.0688276
\(135\) 0 0
\(136\) −54625.7 −0.253250
\(137\) −386434. −1.75903 −0.879516 0.475869i \(-0.842134\pi\)
−0.879516 + 0.475869i \(0.842134\pi\)
\(138\) 15674.4 0.0700639
\(139\) 17289.3 0.0758997 0.0379498 0.999280i \(-0.487917\pi\)
0.0379498 + 0.999280i \(0.487917\pi\)
\(140\) 0 0
\(141\) −163059. −0.690713
\(142\) −46813.3 −0.194827
\(143\) 433384. 1.77228
\(144\) 139354. 0.560034
\(145\) 0 0
\(146\) 20713.4 0.0804210
\(147\) −47178.5 −0.180074
\(148\) 33914.9 0.127273
\(149\) −112171. −0.413917 −0.206959 0.978350i \(-0.566357\pi\)
−0.206959 + 0.978350i \(0.566357\pi\)
\(150\) 0 0
\(151\) 30495.4 0.108841 0.0544205 0.998518i \(-0.482669\pi\)
0.0544205 + 0.998518i \(0.482669\pi\)
\(152\) −72030.1 −0.252874
\(153\) −169848. −0.586586
\(154\) 23665.5 0.0804108
\(155\) 0 0
\(156\) 402390. 1.32384
\(157\) −523509. −1.69502 −0.847510 0.530780i \(-0.821899\pi\)
−0.847510 + 0.530780i \(0.821899\pi\)
\(158\) 22211.7 0.0707847
\(159\) 108449. 0.340199
\(160\) 0 0
\(161\) −53907.5 −0.163902
\(162\) 53180.9 0.159209
\(163\) 439646. 1.29609 0.648043 0.761604i \(-0.275588\pi\)
0.648043 + 0.761604i \(0.275588\pi\)
\(164\) −34565.3 −0.100353
\(165\) 0 0
\(166\) −490.117 −0.00138048
\(167\) −279353. −0.775107 −0.387554 0.921847i \(-0.626680\pi\)
−0.387554 + 0.921847i \(0.626680\pi\)
\(168\) 44313.2 0.121132
\(169\) 52038.8 0.140156
\(170\) 0 0
\(171\) −223964. −0.585716
\(172\) 521349. 1.34372
\(173\) −99699.4 −0.253266 −0.126633 0.991950i \(-0.540417\pi\)
−0.126633 + 0.991950i \(0.540417\pi\)
\(174\) 34147.3 0.0855034
\(175\) 0 0
\(176\) 648641. 1.57842
\(177\) 279621. 0.670867
\(178\) −91339.7 −0.216077
\(179\) 329980. 0.769760 0.384880 0.922967i \(-0.374243\pi\)
0.384880 + 0.922967i \(0.374243\pi\)
\(180\) 0 0
\(181\) −505810. −1.14760 −0.573800 0.818995i \(-0.694531\pi\)
−0.573800 + 0.818995i \(0.694531\pi\)
\(182\) 23116.6 0.0517304
\(183\) 279706. 0.617410
\(184\) 50633.5 0.110254
\(185\) 0 0
\(186\) −29185.4 −0.0618562
\(187\) −790578. −1.65326
\(188\) −261186. −0.538958
\(189\) −96183.4 −0.195860
\(190\) 0 0
\(191\) −63835.6 −0.126613 −0.0633067 0.997994i \(-0.520165\pi\)
−0.0633067 + 0.997994i \(0.520165\pi\)
\(192\) 581270. 1.13795
\(193\) −469355. −0.907001 −0.453501 0.891256i \(-0.649825\pi\)
−0.453501 + 0.891256i \(0.649825\pi\)
\(194\) −16609.3 −0.0316845
\(195\) 0 0
\(196\) −75569.7 −0.140510
\(197\) −268021. −0.492043 −0.246021 0.969264i \(-0.579123\pi\)
−0.246021 + 0.969264i \(0.579123\pi\)
\(198\) −69114.5 −0.125287
\(199\) −605167. −1.08328 −0.541642 0.840609i \(-0.682197\pi\)
−0.541642 + 0.840609i \(0.682197\pi\)
\(200\) 0 0
\(201\) 387693. 0.676859
\(202\) 131584. 0.226894
\(203\) −117439. −0.200020
\(204\) −734039. −1.23493
\(205\) 0 0
\(206\) 46965.1 0.0771094
\(207\) 157435. 0.255374
\(208\) 633596. 1.01544
\(209\) −1.04246e6 −1.65080
\(210\) 0 0
\(211\) 335389. 0.518612 0.259306 0.965795i \(-0.416506\pi\)
0.259306 + 0.965795i \(0.416506\pi\)
\(212\) 173712. 0.265455
\(213\) −1.26862e6 −1.91595
\(214\) −107435. −0.160366
\(215\) 0 0
\(216\) 90341.9 0.131751
\(217\) 100374. 0.144702
\(218\) 80697.7 0.115006
\(219\) 561327. 0.790871
\(220\) 0 0
\(221\) −772240. −1.06358
\(222\) −15352.3 −0.0209070
\(223\) −1.02526e6 −1.38061 −0.690305 0.723518i \(-0.742524\pi\)
−0.690305 + 0.723518i \(0.742524\pi\)
\(224\) 106764. 0.142169
\(225\) 0 0
\(226\) −31305.8 −0.0407713
\(227\) 504226. 0.649473 0.324736 0.945805i \(-0.394724\pi\)
0.324736 + 0.945805i \(0.394724\pi\)
\(228\) −967912. −1.23310
\(229\) −1.11939e6 −1.41057 −0.705283 0.708925i \(-0.749180\pi\)
−0.705283 + 0.708925i \(0.749180\pi\)
\(230\) 0 0
\(231\) 641328. 0.790770
\(232\) 110307. 0.134550
\(233\) −770703. −0.930031 −0.465015 0.885303i \(-0.653951\pi\)
−0.465015 + 0.885303i \(0.653951\pi\)
\(234\) −67511.3 −0.0806004
\(235\) 0 0
\(236\) 447892. 0.523472
\(237\) 601930. 0.696106
\(238\) −42169.2 −0.0482562
\(239\) −171646. −0.194374 −0.0971871 0.995266i \(-0.530985\pi\)
−0.0971871 + 0.995266i \(0.530985\pi\)
\(240\) 0 0
\(241\) −383779. −0.425637 −0.212818 0.977092i \(-0.568264\pi\)
−0.212818 + 0.977092i \(0.568264\pi\)
\(242\) −204926. −0.224936
\(243\) 964193. 1.04749
\(244\) 448028. 0.481760
\(245\) 0 0
\(246\) 15646.8 0.0164849
\(247\) −1.01828e6 −1.06201
\(248\) −94278.3 −0.0973380
\(249\) −13282.0 −0.0135758
\(250\) 0 0
\(251\) −1.57046e6 −1.57342 −0.786708 0.617325i \(-0.788216\pi\)
−0.786708 + 0.617325i \(0.788216\pi\)
\(252\) 220699. 0.218927
\(253\) 732801. 0.719755
\(254\) −95311.4 −0.0926959
\(255\) 0 0
\(256\) 880513. 0.839723
\(257\) −790656. −0.746715 −0.373357 0.927688i \(-0.621793\pi\)
−0.373357 + 0.927688i \(0.621793\pi\)
\(258\) −236000. −0.220731
\(259\) 52799.7 0.0489082
\(260\) 0 0
\(261\) 342978. 0.311648
\(262\) 253386. 0.228049
\(263\) −464416. −0.414017 −0.207008 0.978339i \(-0.566373\pi\)
−0.207008 + 0.978339i \(0.566373\pi\)
\(264\) −602379. −0.531936
\(265\) 0 0
\(266\) −55604.8 −0.0481846
\(267\) −2.47527e6 −2.12493
\(268\) 621001. 0.528147
\(269\) 1.99959e6 1.68484 0.842422 0.538818i \(-0.181129\pi\)
0.842422 + 0.538818i \(0.181129\pi\)
\(270\) 0 0
\(271\) 1.61296e6 1.33414 0.667070 0.744995i \(-0.267548\pi\)
0.667070 + 0.744995i \(0.267548\pi\)
\(272\) −1.15580e6 −0.947243
\(273\) 626452. 0.508723
\(274\) 280197. 0.225469
\(275\) 0 0
\(276\) 680393. 0.537635
\(277\) 2.08119e6 1.62972 0.814860 0.579658i \(-0.196814\pi\)
0.814860 + 0.579658i \(0.196814\pi\)
\(278\) −12536.2 −0.00972865
\(279\) −293140. −0.225457
\(280\) 0 0
\(281\) −982035. −0.741927 −0.370964 0.928647i \(-0.620973\pi\)
−0.370964 + 0.928647i \(0.620973\pi\)
\(282\) 118231. 0.0885340
\(283\) 1.39622e6 1.03630 0.518152 0.855289i \(-0.326620\pi\)
0.518152 + 0.855289i \(0.326620\pi\)
\(284\) −2.03206e6 −1.49500
\(285\) 0 0
\(286\) −314240. −0.227167
\(287\) −53812.3 −0.0385635
\(288\) −311801. −0.221512
\(289\) −11140.3 −0.00784609
\(290\) 0 0
\(291\) −450107. −0.311590
\(292\) 899124. 0.617110
\(293\) −2.56205e6 −1.74348 −0.871742 0.489965i \(-0.837010\pi\)
−0.871742 + 0.489965i \(0.837010\pi\)
\(294\) 34208.3 0.0230814
\(295\) 0 0
\(296\) −49593.0 −0.0328996
\(297\) 1.30749e6 0.860094
\(298\) 81333.0 0.0530550
\(299\) 715803. 0.463037
\(300\) 0 0
\(301\) 811651. 0.516361
\(302\) −22111.7 −0.0139510
\(303\) 3.56588e6 2.23131
\(304\) −1.52405e6 −0.945838
\(305\) 0 0
\(306\) 123154. 0.0751873
\(307\) 884855. 0.535829 0.267915 0.963443i \(-0.413666\pi\)
0.267915 + 0.963443i \(0.413666\pi\)
\(308\) 1.02727e6 0.617032
\(309\) 1.27274e6 0.758303
\(310\) 0 0
\(311\) 1.80120e6 1.05599 0.527997 0.849246i \(-0.322943\pi\)
0.527997 + 0.849246i \(0.322943\pi\)
\(312\) −588406. −0.342208
\(313\) −950366. −0.548315 −0.274158 0.961685i \(-0.588399\pi\)
−0.274158 + 0.961685i \(0.588399\pi\)
\(314\) 379587. 0.217264
\(315\) 0 0
\(316\) 964162. 0.543166
\(317\) −3.04277e6 −1.70068 −0.850338 0.526237i \(-0.823602\pi\)
−0.850338 + 0.526237i \(0.823602\pi\)
\(318\) −78634.6 −0.0436060
\(319\) 1.59643e6 0.878362
\(320\) 0 0
\(321\) −2.91146e6 −1.57706
\(322\) 39087.4 0.0210086
\(323\) 1.85755e6 0.990682
\(324\) 2.30847e6 1.22169
\(325\) 0 0
\(326\) −318780. −0.166129
\(327\) 2.18688e6 1.13098
\(328\) 50544.1 0.0259409
\(329\) −406621. −0.207110
\(330\) 0 0
\(331\) −2.19616e6 −1.10178 −0.550889 0.834579i \(-0.685711\pi\)
−0.550889 + 0.834579i \(0.685711\pi\)
\(332\) −21274.9 −0.0105931
\(333\) −154200. −0.0762032
\(334\) 202554. 0.0993515
\(335\) 0 0
\(336\) 937604. 0.453076
\(337\) 2.41491e6 1.15832 0.579158 0.815216i \(-0.303382\pi\)
0.579158 + 0.815216i \(0.303382\pi\)
\(338\) −37732.5 −0.0179648
\(339\) −848377. −0.400950
\(340\) 0 0
\(341\) −1.36446e6 −0.635438
\(342\) 162392. 0.0750757
\(343\) −117649. −0.0539949
\(344\) −762357. −0.347346
\(345\) 0 0
\(346\) 72290.3 0.0324631
\(347\) 1.08833e6 0.485219 0.242609 0.970124i \(-0.421997\pi\)
0.242609 + 0.970124i \(0.421997\pi\)
\(348\) 1.48226e6 0.656110
\(349\) 2.79267e6 1.22731 0.613657 0.789573i \(-0.289698\pi\)
0.613657 + 0.789573i \(0.289698\pi\)
\(350\) 0 0
\(351\) 1.27716e6 0.553321
\(352\) −1.45132e6 −0.624317
\(353\) −2.53134e6 −1.08122 −0.540610 0.841273i \(-0.681807\pi\)
−0.540610 + 0.841273i \(0.681807\pi\)
\(354\) −202748. −0.0859902
\(355\) 0 0
\(356\) −3.96485e6 −1.65807
\(357\) −1.14277e6 −0.474558
\(358\) −239263. −0.0986661
\(359\) 1.09028e6 0.446480 0.223240 0.974763i \(-0.428337\pi\)
0.223240 + 0.974763i \(0.428337\pi\)
\(360\) 0 0
\(361\) −26712.8 −0.0107883
\(362\) 366754. 0.147097
\(363\) −5.55343e6 −2.21205
\(364\) 1.00344e6 0.396953
\(365\) 0 0
\(366\) −202810. −0.0791382
\(367\) −188070. −0.0728879 −0.0364439 0.999336i \(-0.511603\pi\)
−0.0364439 + 0.999336i \(0.511603\pi\)
\(368\) 1.07133e6 0.412387
\(369\) 157157. 0.0600853
\(370\) 0 0
\(371\) 270440. 0.102008
\(372\) −1.26687e6 −0.474653
\(373\) 1.79371e6 0.667545 0.333772 0.942654i \(-0.391678\pi\)
0.333772 + 0.942654i \(0.391678\pi\)
\(374\) 573234. 0.211911
\(375\) 0 0
\(376\) 381926. 0.139319
\(377\) 1.55940e6 0.565073
\(378\) 69740.9 0.0251049
\(379\) 3.58806e6 1.28310 0.641551 0.767080i \(-0.278291\pi\)
0.641551 + 0.767080i \(0.278291\pi\)
\(380\) 0 0
\(381\) −2.58291e6 −0.911583
\(382\) 46286.1 0.0162290
\(383\) 3.42457e6 1.19291 0.596457 0.802645i \(-0.296575\pi\)
0.596457 + 0.802645i \(0.296575\pi\)
\(384\) −1.79150e6 −0.619996
\(385\) 0 0
\(386\) 340321. 0.116257
\(387\) −2.37040e6 −0.804534
\(388\) −720974. −0.243131
\(389\) 8625.27 0.00289001 0.00144500 0.999999i \(-0.499540\pi\)
0.00144500 + 0.999999i \(0.499540\pi\)
\(390\) 0 0
\(391\) −1.30576e6 −0.431940
\(392\) 110504. 0.0363214
\(393\) 6.86667e6 2.24267
\(394\) 194337. 0.0630690
\(395\) 0 0
\(396\) −3.00011e6 −0.961388
\(397\) −1.25709e6 −0.400306 −0.200153 0.979765i \(-0.564144\pi\)
−0.200153 + 0.979765i \(0.564144\pi\)
\(398\) 438796. 0.138853
\(399\) −1.50687e6 −0.473853
\(400\) 0 0
\(401\) 1.42670e6 0.443070 0.221535 0.975152i \(-0.428893\pi\)
0.221535 + 0.975152i \(0.428893\pi\)
\(402\) −281110. −0.0867583
\(403\) −1.33281e6 −0.408794
\(404\) 5.71176e6 1.74107
\(405\) 0 0
\(406\) 85153.2 0.0256381
\(407\) −717741. −0.214774
\(408\) 1.07337e6 0.319226
\(409\) −3.06529e6 −0.906073 −0.453036 0.891492i \(-0.649659\pi\)
−0.453036 + 0.891492i \(0.649659\pi\)
\(410\) 0 0
\(411\) 7.59324e6 2.21729
\(412\) 2.03865e6 0.591698
\(413\) 697291. 0.201159
\(414\) −114154. −0.0327332
\(415\) 0 0
\(416\) −1.41765e6 −0.401640
\(417\) −339726. −0.0956728
\(418\) 755873. 0.211596
\(419\) −248240. −0.0690776 −0.0345388 0.999403i \(-0.510996\pi\)
−0.0345388 + 0.999403i \(0.510996\pi\)
\(420\) 0 0
\(421\) 5.96280e6 1.63963 0.819814 0.572630i \(-0.194077\pi\)
0.819814 + 0.572630i \(0.194077\pi\)
\(422\) −243185. −0.0664746
\(423\) 1.18752e6 0.322695
\(424\) −254015. −0.0686192
\(425\) 0 0
\(426\) 919857. 0.245582
\(427\) 697503. 0.185130
\(428\) −4.66353e6 −1.23057
\(429\) −8.51579e6 −2.23399
\(430\) 0 0
\(431\) −4.93538e6 −1.27976 −0.639879 0.768476i \(-0.721015\pi\)
−0.639879 + 0.768476i \(0.721015\pi\)
\(432\) 1.91151e6 0.492795
\(433\) −4.15513e6 −1.06504 −0.532519 0.846418i \(-0.678755\pi\)
−0.532519 + 0.846418i \(0.678755\pi\)
\(434\) −72779.7 −0.0185475
\(435\) 0 0
\(436\) 3.50291e6 0.882496
\(437\) −1.72180e6 −0.431299
\(438\) −407008. −0.101372
\(439\) −227955. −0.0564531 −0.0282265 0.999602i \(-0.508986\pi\)
−0.0282265 + 0.999602i \(0.508986\pi\)
\(440\) 0 0
\(441\) 343590. 0.0841287
\(442\) 559938. 0.136328
\(443\) 1.98462e6 0.480472 0.240236 0.970715i \(-0.422775\pi\)
0.240236 + 0.970715i \(0.422775\pi\)
\(444\) −666411. −0.160430
\(445\) 0 0
\(446\) 743397. 0.176963
\(447\) 2.20410e6 0.521749
\(448\) 1.44951e6 0.341214
\(449\) 2.61077e6 0.611157 0.305579 0.952167i \(-0.401150\pi\)
0.305579 + 0.952167i \(0.401150\pi\)
\(450\) 0 0
\(451\) 731506. 0.169347
\(452\) −1.35892e6 −0.312858
\(453\) −599220. −0.137196
\(454\) −365606. −0.0832480
\(455\) 0 0
\(456\) 1.41536e6 0.318752
\(457\) 4.09917e6 0.918132 0.459066 0.888402i \(-0.348184\pi\)
0.459066 + 0.888402i \(0.348184\pi\)
\(458\) 811652. 0.180803
\(459\) −2.32979e6 −0.516160
\(460\) 0 0
\(461\) −2.62378e6 −0.575009 −0.287505 0.957779i \(-0.592826\pi\)
−0.287505 + 0.957779i \(0.592826\pi\)
\(462\) −465016. −0.101359
\(463\) 4.28563e6 0.929100 0.464550 0.885547i \(-0.346216\pi\)
0.464550 + 0.885547i \(0.346216\pi\)
\(464\) 2.33394e6 0.503262
\(465\) 0 0
\(466\) 558824. 0.119209
\(467\) −990118. −0.210085 −0.105042 0.994468i \(-0.533498\pi\)
−0.105042 + 0.994468i \(0.533498\pi\)
\(468\) −2.93052e6 −0.618486
\(469\) 966792. 0.202955
\(470\) 0 0
\(471\) 1.02867e7 2.13660
\(472\) −654942. −0.135316
\(473\) −1.10333e7 −2.26753
\(474\) −436449. −0.0892253
\(475\) 0 0
\(476\) −1.83047e6 −0.370293
\(477\) −789812. −0.158938
\(478\) 124457. 0.0249144
\(479\) 2.57976e6 0.513737 0.256868 0.966446i \(-0.417309\pi\)
0.256868 + 0.966446i \(0.417309\pi\)
\(480\) 0 0
\(481\) −701093. −0.138170
\(482\) 278272. 0.0545571
\(483\) 1.05926e6 0.206601
\(484\) −8.89540e6 −1.72604
\(485\) 0 0
\(486\) −699120. −0.134264
\(487\) −3.21474e6 −0.614219 −0.307109 0.951674i \(-0.599362\pi\)
−0.307109 + 0.951674i \(0.599362\pi\)
\(488\) −655141. −0.124533
\(489\) −8.63882e6 −1.63374
\(490\) 0 0
\(491\) −7.86108e6 −1.47156 −0.735781 0.677220i \(-0.763185\pi\)
−0.735781 + 0.677220i \(0.763185\pi\)
\(492\) 679192. 0.126497
\(493\) −2.84465e6 −0.527123
\(494\) 738341. 0.136125
\(495\) 0 0
\(496\) −1.99480e6 −0.364078
\(497\) −3.16357e6 −0.574495
\(498\) 9630.55 0.00174011
\(499\) −1.35382e6 −0.243395 −0.121697 0.992567i \(-0.538834\pi\)
−0.121697 + 0.992567i \(0.538834\pi\)
\(500\) 0 0
\(501\) 5.48914e6 0.977035
\(502\) 1.13872e6 0.201677
\(503\) −3.85775e6 −0.679851 −0.339926 0.940452i \(-0.610402\pi\)
−0.339926 + 0.940452i \(0.610402\pi\)
\(504\) −322723. −0.0565918
\(505\) 0 0
\(506\) −531341. −0.0922565
\(507\) −1.02254e6 −0.176669
\(508\) −4.13726e6 −0.711301
\(509\) −1.06060e7 −1.81451 −0.907253 0.420585i \(-0.861825\pi\)
−0.907253 + 0.420585i \(0.861825\pi\)
\(510\) 0 0
\(511\) 1.39978e6 0.237142
\(512\) −3.55598e6 −0.599493
\(513\) −3.07208e6 −0.515394
\(514\) 573291. 0.0957122
\(515\) 0 0
\(516\) −1.02443e7 −1.69378
\(517\) 5.52747e6 0.909495
\(518\) −38284.1 −0.00626895
\(519\) 1.95904e6 0.319246
\(520\) 0 0
\(521\) −9.17989e6 −1.48164 −0.740821 0.671703i \(-0.765563\pi\)
−0.740821 + 0.671703i \(0.765563\pi\)
\(522\) −248687. −0.0399464
\(523\) −9.05585e6 −1.44769 −0.723844 0.689964i \(-0.757627\pi\)
−0.723844 + 0.689964i \(0.757627\pi\)
\(524\) 1.09989e7 1.74993
\(525\) 0 0
\(526\) 336740. 0.0530677
\(527\) 2.43130e6 0.381339
\(528\) −1.27455e7 −1.98963
\(529\) −5.22601e6 −0.811953
\(530\) 0 0
\(531\) −2.03642e6 −0.313422
\(532\) −2.41368e6 −0.369744
\(533\) 714539. 0.108945
\(534\) 1.79478e6 0.272369
\(535\) 0 0
\(536\) −908075. −0.136524
\(537\) −6.48394e6 −0.970295
\(538\) −1.44987e6 −0.215960
\(539\) 1.59928e6 0.237112
\(540\) 0 0
\(541\) 1.21783e7 1.78894 0.894468 0.447132i \(-0.147555\pi\)
0.894468 + 0.447132i \(0.147555\pi\)
\(542\) −1.16953e6 −0.171007
\(543\) 9.93891e6 1.44657
\(544\) 2.58608e6 0.374666
\(545\) 0 0
\(546\) −454230. −0.0652070
\(547\) 9.00451e6 1.28674 0.643372 0.765554i \(-0.277535\pi\)
0.643372 + 0.765554i \(0.277535\pi\)
\(548\) 1.21627e7 1.73013
\(549\) −2.03703e6 −0.288448
\(550\) 0 0
\(551\) −3.75099e6 −0.526341
\(552\) −994924. −0.138977
\(553\) 1.50103e6 0.208727
\(554\) −1.50904e6 −0.208894
\(555\) 0 0
\(556\) −544167. −0.0746527
\(557\) −1.54461e6 −0.210950 −0.105475 0.994422i \(-0.533636\pi\)
−0.105475 + 0.994422i \(0.533636\pi\)
\(558\) 212551. 0.0288986
\(559\) −1.07774e7 −1.45876
\(560\) 0 0
\(561\) 1.55345e7 2.08396
\(562\) 712057. 0.0950986
\(563\) 1.18748e7 1.57890 0.789449 0.613816i \(-0.210366\pi\)
0.789449 + 0.613816i \(0.210366\pi\)
\(564\) 5.13217e6 0.679365
\(565\) 0 0
\(566\) −1.01237e6 −0.132831
\(567\) 3.59388e6 0.469468
\(568\) 2.97144e6 0.386452
\(569\) −1.54308e6 −0.199806 −0.0999031 0.994997i \(-0.531853\pi\)
−0.0999031 + 0.994997i \(0.531853\pi\)
\(570\) 0 0
\(571\) −7.01812e6 −0.900804 −0.450402 0.892826i \(-0.648719\pi\)
−0.450402 + 0.892826i \(0.648719\pi\)
\(572\) −1.36404e7 −1.74317
\(573\) 1.25434e6 0.159598
\(574\) 39018.4 0.00494298
\(575\) 0 0
\(576\) −4.23326e6 −0.531641
\(577\) 5.37543e6 0.672162 0.336081 0.941833i \(-0.390898\pi\)
0.336081 + 0.941833i \(0.390898\pi\)
\(578\) 8077.66 0.00100569
\(579\) 9.22259e6 1.14329
\(580\) 0 0
\(581\) −33121.4 −0.00407069
\(582\) 326365. 0.0399389
\(583\) −3.67627e6 −0.447957
\(584\) −1.31477e6 −0.159521
\(585\) 0 0
\(586\) 1.85770e6 0.223476
\(587\) 2.06682e6 0.247575 0.123788 0.992309i \(-0.460496\pi\)
0.123788 + 0.992309i \(0.460496\pi\)
\(588\) 1.48491e6 0.177115
\(589\) 3.20594e6 0.380774
\(590\) 0 0
\(591\) 5.26648e6 0.620228
\(592\) −1.04932e6 −0.123056
\(593\) 5.46947e6 0.638717 0.319358 0.947634i \(-0.396533\pi\)
0.319358 + 0.947634i \(0.396533\pi\)
\(594\) −948035. −0.110245
\(595\) 0 0
\(596\) 3.53049e6 0.407117
\(597\) 1.18912e7 1.36550
\(598\) −519016. −0.0593510
\(599\) 1.09943e7 1.25199 0.625996 0.779826i \(-0.284693\pi\)
0.625996 + 0.779826i \(0.284693\pi\)
\(600\) 0 0
\(601\) 1.58788e7 1.79322 0.896608 0.442826i \(-0.146024\pi\)
0.896608 + 0.442826i \(0.146024\pi\)
\(602\) −588514. −0.0661859
\(603\) −2.82348e6 −0.316222
\(604\) −959820. −0.107053
\(605\) 0 0
\(606\) −2.58555e6 −0.286004
\(607\) −5.33262e6 −0.587447 −0.293724 0.955890i \(-0.594895\pi\)
−0.293724 + 0.955890i \(0.594895\pi\)
\(608\) 3.41003e6 0.374110
\(609\) 2.30762e6 0.252128
\(610\) 0 0
\(611\) 5.39926e6 0.585102
\(612\) 5.34584e6 0.576949
\(613\) 8.91838e6 0.958594 0.479297 0.877653i \(-0.340892\pi\)
0.479297 + 0.877653i \(0.340892\pi\)
\(614\) −641593. −0.0686814
\(615\) 0 0
\(616\) −1.50215e6 −0.159501
\(617\) −9.63586e6 −1.01901 −0.509504 0.860468i \(-0.670171\pi\)
−0.509504 + 0.860468i \(0.670171\pi\)
\(618\) −922841. −0.0971976
\(619\) 1.21747e7 1.27712 0.638560 0.769572i \(-0.279530\pi\)
0.638560 + 0.769572i \(0.279530\pi\)
\(620\) 0 0
\(621\) 2.15952e6 0.224713
\(622\) −1.30602e6 −0.135355
\(623\) −6.17260e6 −0.637159
\(624\) −1.24498e7 −1.27998
\(625\) 0 0
\(626\) 689094. 0.0702818
\(627\) 2.04839e7 2.08087
\(628\) 1.64770e7 1.66717
\(629\) 1.27893e6 0.128890
\(630\) 0 0
\(631\) −1.30854e7 −1.30832 −0.654161 0.756356i \(-0.726978\pi\)
−0.654161 + 0.756356i \(0.726978\pi\)
\(632\) −1.40987e6 −0.140407
\(633\) −6.59023e6 −0.653719
\(634\) 2.20626e6 0.217989
\(635\) 0 0
\(636\) −3.41336e6 −0.334610
\(637\) 1.56219e6 0.152540
\(638\) −1.15754e6 −0.112586
\(639\) 9.23911e6 0.895113
\(640\) 0 0
\(641\) −441107. −0.0424032 −0.0212016 0.999775i \(-0.506749\pi\)
−0.0212016 + 0.999775i \(0.506749\pi\)
\(642\) 2.11105e6 0.202144
\(643\) 4.18888e6 0.399550 0.199775 0.979842i \(-0.435979\pi\)
0.199775 + 0.979842i \(0.435979\pi\)
\(644\) 1.69670e6 0.161209
\(645\) 0 0
\(646\) −1.34688e6 −0.126983
\(647\) −1.87822e7 −1.76395 −0.881973 0.471300i \(-0.843785\pi\)
−0.881973 + 0.471300i \(0.843785\pi\)
\(648\) −3.37562e6 −0.315803
\(649\) −9.47874e6 −0.883362
\(650\) 0 0
\(651\) −1.97231e6 −0.182399
\(652\) −1.38375e7 −1.27479
\(653\) 1.51733e6 0.139251 0.0696254 0.997573i \(-0.477820\pi\)
0.0696254 + 0.997573i \(0.477820\pi\)
\(654\) −1.58567e6 −0.144967
\(655\) 0 0
\(656\) 1.06944e6 0.0970281
\(657\) −4.08802e6 −0.369487
\(658\) 294834. 0.0265468
\(659\) 1.84809e7 1.65772 0.828859 0.559458i \(-0.188991\pi\)
0.828859 + 0.559458i \(0.188991\pi\)
\(660\) 0 0
\(661\) −1.03952e7 −0.925403 −0.462702 0.886514i \(-0.653120\pi\)
−0.462702 + 0.886514i \(0.653120\pi\)
\(662\) 1.59240e6 0.141223
\(663\) 1.51741e7 1.34066
\(664\) 31109.8 0.00273828
\(665\) 0 0
\(666\) 111808. 0.00976756
\(667\) 2.63676e6 0.229486
\(668\) 8.79242e6 0.762372
\(669\) 2.01458e7 1.74028
\(670\) 0 0
\(671\) −9.48162e6 −0.812973
\(672\) −2.09786e6 −0.179207
\(673\) 1.10398e7 0.939556 0.469778 0.882785i \(-0.344334\pi\)
0.469778 + 0.882785i \(0.344334\pi\)
\(674\) −1.75101e6 −0.148470
\(675\) 0 0
\(676\) −1.63788e6 −0.137853
\(677\) 8.23485e6 0.690532 0.345266 0.938505i \(-0.387789\pi\)
0.345266 + 0.938505i \(0.387789\pi\)
\(678\) 615144. 0.0513928
\(679\) −1.12243e6 −0.0934298
\(680\) 0 0
\(681\) −9.90780e6 −0.818671
\(682\) 989343. 0.0814490
\(683\) −1.98437e7 −1.62768 −0.813842 0.581086i \(-0.802628\pi\)
−0.813842 + 0.581086i \(0.802628\pi\)
\(684\) 7.04909e6 0.576093
\(685\) 0 0
\(686\) 85305.3 0.00692095
\(687\) 2.19955e7 1.77804
\(688\) −1.61304e7 −1.29919
\(689\) −3.59100e6 −0.288182
\(690\) 0 0
\(691\) −2.37019e6 −0.188837 −0.0944185 0.995533i \(-0.530099\pi\)
−0.0944185 + 0.995533i \(0.530099\pi\)
\(692\) 3.13796e6 0.249105
\(693\) −4.67065e6 −0.369440
\(694\) −789131. −0.0621943
\(695\) 0 0
\(696\) −2.16748e6 −0.169602
\(697\) −1.30346e6 −0.101628
\(698\) −2.02491e6 −0.157314
\(699\) 1.51439e7 1.17232
\(700\) 0 0
\(701\) −1.56833e7 −1.20543 −0.602714 0.797957i \(-0.705914\pi\)
−0.602714 + 0.797957i \(0.705914\pi\)
\(702\) −926045. −0.0709234
\(703\) 1.68641e6 0.128699
\(704\) −1.97042e7 −1.49840
\(705\) 0 0
\(706\) 1.83543e6 0.138588
\(707\) 8.89223e6 0.669056
\(708\) −8.80085e6 −0.659844
\(709\) 3.68544e6 0.275343 0.137671 0.990478i \(-0.456038\pi\)
0.137671 + 0.990478i \(0.456038\pi\)
\(710\) 0 0
\(711\) −4.38373e6 −0.325214
\(712\) 5.79772e6 0.428605
\(713\) −2.25361e6 −0.166018
\(714\) 828604. 0.0608277
\(715\) 0 0
\(716\) −1.03859e7 −0.757113
\(717\) 3.37276e6 0.245012
\(718\) −790544. −0.0572289
\(719\) −1.56427e7 −1.12847 −0.564233 0.825615i \(-0.690828\pi\)
−0.564233 + 0.825615i \(0.690828\pi\)
\(720\) 0 0
\(721\) 3.17383e6 0.227376
\(722\) 19369.0 0.00138282
\(723\) 7.54107e6 0.536522
\(724\) 1.59200e7 1.12875
\(725\) 0 0
\(726\) 4.02670e6 0.283536
\(727\) −1.85908e7 −1.30456 −0.652279 0.757979i \(-0.726187\pi\)
−0.652279 + 0.757979i \(0.726187\pi\)
\(728\) −1.46731e6 −0.102611
\(729\) −1.12318e6 −0.0782767
\(730\) 0 0
\(731\) 1.96601e7 1.36079
\(732\) −8.80353e6 −0.607266
\(733\) −2.49466e7 −1.71495 −0.857476 0.514524i \(-0.827969\pi\)
−0.857476 + 0.514524i \(0.827969\pi\)
\(734\) 136367. 0.00934260
\(735\) 0 0
\(736\) −2.39708e6 −0.163113
\(737\) −1.31422e7 −0.891253
\(738\) −113952. −0.00770159
\(739\) −2.42944e7 −1.63642 −0.818211 0.574918i \(-0.805034\pi\)
−0.818211 + 0.574918i \(0.805034\pi\)
\(740\) 0 0
\(741\) 2.00088e7 1.33868
\(742\) −196091. −0.0130752
\(743\) −4.16541e6 −0.276812 −0.138406 0.990376i \(-0.544198\pi\)
−0.138406 + 0.990376i \(0.544198\pi\)
\(744\) 1.85252e6 0.122696
\(745\) 0 0
\(746\) −1.30059e6 −0.0855644
\(747\) 96729.9 0.00634248
\(748\) 2.48828e7 1.62610
\(749\) −7.26032e6 −0.472880
\(750\) 0 0
\(751\) −2.36434e7 −1.52972 −0.764858 0.644199i \(-0.777191\pi\)
−0.764858 + 0.644199i \(0.777191\pi\)
\(752\) 8.08101e6 0.521100
\(753\) 3.08588e7 1.98332
\(754\) −1.13070e6 −0.0724298
\(755\) 0 0
\(756\) 3.02730e6 0.192642
\(757\) −1.50108e7 −0.952062 −0.476031 0.879429i \(-0.657925\pi\)
−0.476031 + 0.879429i \(0.657925\pi\)
\(758\) −2.60164e6 −0.164465
\(759\) −1.43992e7 −0.907262
\(760\) 0 0
\(761\) 2.92191e6 0.182897 0.0914483 0.995810i \(-0.470850\pi\)
0.0914483 + 0.995810i \(0.470850\pi\)
\(762\) 1.87282e6 0.116845
\(763\) 5.45343e6 0.339124
\(764\) 2.00918e6 0.124533
\(765\) 0 0
\(766\) −2.48310e6 −0.152905
\(767\) −9.25887e6 −0.568290
\(768\) −1.73016e7 −1.05848
\(769\) 1.42847e7 0.871073 0.435536 0.900171i \(-0.356559\pi\)
0.435536 + 0.900171i \(0.356559\pi\)
\(770\) 0 0
\(771\) 1.55360e7 0.941246
\(772\) 1.47726e7 0.892100
\(773\) 1.09012e7 0.656186 0.328093 0.944645i \(-0.393594\pi\)
0.328093 + 0.944645i \(0.393594\pi\)
\(774\) 1.71874e6 0.103123
\(775\) 0 0
\(776\) 1.05426e6 0.0628485
\(777\) −1.03749e6 −0.0616496
\(778\) −6254.04 −0.000370434 0
\(779\) −1.71875e6 −0.101478
\(780\) 0 0
\(781\) 4.30045e7 2.52282
\(782\) 946787. 0.0553650
\(783\) 4.70459e6 0.274231
\(784\) 2.33811e6 0.135854
\(785\) 0 0
\(786\) −4.97890e6 −0.287460
\(787\) 2.56449e7 1.47593 0.737963 0.674841i \(-0.235788\pi\)
0.737963 + 0.674841i \(0.235788\pi\)
\(788\) 8.43576e6 0.483959
\(789\) 9.12555e6 0.521875
\(790\) 0 0
\(791\) −2.11560e6 −0.120224
\(792\) 4.38699e6 0.248516
\(793\) −9.26169e6 −0.523007
\(794\) 911498. 0.0513103
\(795\) 0 0
\(796\) 1.90472e7 1.06549
\(797\) −7.73086e6 −0.431104 −0.215552 0.976492i \(-0.569155\pi\)
−0.215552 + 0.976492i \(0.569155\pi\)
\(798\) 1.09261e6 0.0607375
\(799\) −9.84931e6 −0.545807
\(800\) 0 0
\(801\) 1.80269e7 0.992748
\(802\) −1.03448e6 −0.0567917
\(803\) −1.90282e7 −1.04138
\(804\) −1.22024e7 −0.665738
\(805\) 0 0
\(806\) 966395. 0.0523983
\(807\) −3.92909e7 −2.12377
\(808\) −8.35218e6 −0.450061
\(809\) −1.87811e7 −1.00890 −0.504452 0.863440i \(-0.668305\pi\)
−0.504452 + 0.863440i \(0.668305\pi\)
\(810\) 0 0
\(811\) −9.00729e6 −0.480886 −0.240443 0.970663i \(-0.577293\pi\)
−0.240443 + 0.970663i \(0.577293\pi\)
\(812\) 3.69632e6 0.196734
\(813\) −3.16939e7 −1.68170
\(814\) 520422. 0.0275293
\(815\) 0 0
\(816\) 2.27110e7 1.19402
\(817\) 2.59240e7 1.35877
\(818\) 2.22259e6 0.116138
\(819\) −4.56231e6 −0.237671
\(820\) 0 0
\(821\) −9.27965e6 −0.480478 −0.240239 0.970714i \(-0.577226\pi\)
−0.240239 + 0.970714i \(0.577226\pi\)
\(822\) −5.50572e6 −0.284207
\(823\) 1.08308e7 0.557393 0.278697 0.960379i \(-0.410098\pi\)
0.278697 + 0.960379i \(0.410098\pi\)
\(824\) −2.98107e6 −0.152952
\(825\) 0 0
\(826\) −505593. −0.0257841
\(827\) 2.05230e7 1.04346 0.521731 0.853110i \(-0.325286\pi\)
0.521731 + 0.853110i \(0.325286\pi\)
\(828\) −4.95515e6 −0.251178
\(829\) −1.42216e7 −0.718724 −0.359362 0.933198i \(-0.617006\pi\)
−0.359362 + 0.933198i \(0.617006\pi\)
\(830\) 0 0
\(831\) −4.08944e7 −2.05429
\(832\) −1.92472e7 −0.963958
\(833\) −2.84973e6 −0.142296
\(834\) 246329. 0.0122631
\(835\) 0 0
\(836\) 3.28108e7 1.62368
\(837\) −4.02097e6 −0.198389
\(838\) 179995. 0.00885421
\(839\) −9.29934e6 −0.456087 −0.228043 0.973651i \(-0.573233\pi\)
−0.228043 + 0.973651i \(0.573233\pi\)
\(840\) 0 0
\(841\) −1.47669e7 −0.719944
\(842\) −4.32353e6 −0.210164
\(843\) 1.92965e7 0.935211
\(844\) −1.05561e7 −0.510092
\(845\) 0 0
\(846\) −861053. −0.0413623
\(847\) −1.38486e7 −0.663281
\(848\) −5.37461e6 −0.256659
\(849\) −2.74350e7 −1.30628
\(850\) 0 0
\(851\) −1.18546e6 −0.0561131
\(852\) 3.99290e7 1.88447
\(853\) 3.07436e6 0.144671 0.0723357 0.997380i \(-0.476955\pi\)
0.0723357 + 0.997380i \(0.476955\pi\)
\(854\) −505747. −0.0237295
\(855\) 0 0
\(856\) 6.81938e6 0.318098
\(857\) −3.45835e7 −1.60848 −0.804242 0.594302i \(-0.797428\pi\)
−0.804242 + 0.594302i \(0.797428\pi\)
\(858\) 6.17465e6 0.286348
\(859\) 1.63022e7 0.753814 0.376907 0.926251i \(-0.376988\pi\)
0.376907 + 0.926251i \(0.376988\pi\)
\(860\) 0 0
\(861\) 1.05738e6 0.0486099
\(862\) 3.57856e6 0.164036
\(863\) −2.56962e7 −1.17447 −0.587235 0.809416i \(-0.699784\pi\)
−0.587235 + 0.809416i \(0.699784\pi\)
\(864\) −4.27694e6 −0.194917
\(865\) 0 0
\(866\) 3.01281e6 0.136514
\(867\) 218902. 0.00989012
\(868\) −3.15921e6 −0.142324
\(869\) −2.04046e7 −0.916596
\(870\) 0 0
\(871\) −1.28374e7 −0.573366
\(872\) −5.12222e6 −0.228122
\(873\) 3.27803e6 0.145572
\(874\) 1.24844e6 0.0552829
\(875\) 0 0
\(876\) −1.76673e7 −0.777877
\(877\) −3.30060e7 −1.44908 −0.724542 0.689230i \(-0.757949\pi\)
−0.724542 + 0.689230i \(0.757949\pi\)
\(878\) 165286. 0.00723603
\(879\) 5.03429e7 2.19769
\(880\) 0 0
\(881\) 2.26705e6 0.0984060 0.0492030 0.998789i \(-0.484332\pi\)
0.0492030 + 0.998789i \(0.484332\pi\)
\(882\) −249131. −0.0107834
\(883\) −1.97779e7 −0.853649 −0.426825 0.904334i \(-0.640368\pi\)
−0.426825 + 0.904334i \(0.640368\pi\)
\(884\) 2.43057e7 1.04611
\(885\) 0 0
\(886\) −1.43901e6 −0.0615858
\(887\) 36468.3 0.00155635 0.000778173 1.00000i \(-0.499752\pi\)
0.000778173 1.00000i \(0.499752\pi\)
\(888\) 974478. 0.0414705
\(889\) −6.44100e6 −0.273337
\(890\) 0 0
\(891\) −4.88541e7 −2.06161
\(892\) 3.22692e7 1.35793
\(893\) −1.29874e7 −0.544997
\(894\) −1.59815e6 −0.0668767
\(895\) 0 0
\(896\) −4.46747e6 −0.185905
\(897\) −1.40652e7 −0.583665
\(898\) −1.89303e6 −0.0783367
\(899\) −4.90958e6 −0.202602
\(900\) 0 0
\(901\) 6.55068e6 0.268828
\(902\) −530403. −0.0217065
\(903\) −1.59485e7 −0.650881
\(904\) 1.98711e6 0.0808727
\(905\) 0 0
\(906\) 434484. 0.0175854
\(907\) −1.62298e7 −0.655079 −0.327540 0.944837i \(-0.606219\pi\)
−0.327540 + 0.944837i \(0.606219\pi\)
\(908\) −1.58702e7 −0.638802
\(909\) −2.59695e7 −1.04245
\(910\) 0 0
\(911\) −2.61699e7 −1.04474 −0.522368 0.852720i \(-0.674951\pi\)
−0.522368 + 0.852720i \(0.674951\pi\)
\(912\) 2.99469e7 1.19224
\(913\) 450241. 0.0178759
\(914\) −2.97224e6 −0.117684
\(915\) 0 0
\(916\) 3.52320e7 1.38739
\(917\) 1.71234e7 0.672461
\(918\) 1.68929e6 0.0661602
\(919\) 4.05973e6 0.158565 0.0792826 0.996852i \(-0.474737\pi\)
0.0792826 + 0.996852i \(0.474737\pi\)
\(920\) 0 0
\(921\) −1.73870e7 −0.675421
\(922\) 1.90246e6 0.0737034
\(923\) 4.20070e7 1.62300
\(924\) −2.01853e7 −0.777778
\(925\) 0 0
\(926\) −3.10744e6 −0.119090
\(927\) −9.26907e6 −0.354272
\(928\) −5.22212e6 −0.199057
\(929\) 3.69518e6 0.140474 0.0702370 0.997530i \(-0.477624\pi\)
0.0702370 + 0.997530i \(0.477624\pi\)
\(930\) 0 0
\(931\) −3.75769e6 −0.142084
\(932\) 2.42573e7 0.914751
\(933\) −3.53927e7 −1.33110
\(934\) 717918. 0.0269282
\(935\) 0 0
\(936\) 4.28523e6 0.159877
\(937\) 1.91384e7 0.712126 0.356063 0.934462i \(-0.384119\pi\)
0.356063 + 0.934462i \(0.384119\pi\)
\(938\) −701004. −0.0260144
\(939\) 1.86742e7 0.691160
\(940\) 0 0
\(941\) 1.20954e7 0.445294 0.222647 0.974899i \(-0.428530\pi\)
0.222647 + 0.974899i \(0.428530\pi\)
\(942\) −7.45870e6 −0.273865
\(943\) 1.20820e6 0.0442445
\(944\) −1.38577e7 −0.506127
\(945\) 0 0
\(946\) 8.00007e6 0.290647
\(947\) 1.95969e7 0.710087 0.355044 0.934850i \(-0.384466\pi\)
0.355044 + 0.934850i \(0.384466\pi\)
\(948\) −1.89453e7 −0.684669
\(949\) −1.85868e7 −0.669945
\(950\) 0 0
\(951\) 5.97890e7 2.14373
\(952\) 2.67666e6 0.0957196
\(953\) −4.23265e7 −1.50966 −0.754832 0.655918i \(-0.772282\pi\)
−0.754832 + 0.655918i \(0.772282\pi\)
\(954\) 572679. 0.0203723
\(955\) 0 0
\(956\) 5.40243e6 0.191181
\(957\) −3.13691e7 −1.10719
\(958\) −1.87054e6 −0.0658496
\(959\) 1.89353e7 0.664852
\(960\) 0 0
\(961\) −2.44330e7 −0.853430
\(962\) 508351. 0.0177103
\(963\) 2.12035e7 0.736788
\(964\) 1.20792e7 0.418644
\(965\) 0 0
\(966\) −768048. −0.0264817
\(967\) −1.39211e6 −0.0478749 −0.0239375 0.999713i \(-0.507620\pi\)
−0.0239375 + 0.999713i \(0.507620\pi\)
\(968\) 1.30075e7 0.446176
\(969\) −3.64999e7 −1.24877
\(970\) 0 0
\(971\) 4.41877e7 1.50402 0.752009 0.659153i \(-0.229085\pi\)
0.752009 + 0.659153i \(0.229085\pi\)
\(972\) −3.03473e7 −1.03028
\(973\) −847175. −0.0286874
\(974\) 2.33095e6 0.0787292
\(975\) 0 0
\(976\) −1.38619e7 −0.465798
\(977\) 5.60457e7 1.87848 0.939239 0.343264i \(-0.111533\pi\)
0.939239 + 0.343264i \(0.111533\pi\)
\(978\) 6.26386e6 0.209409
\(979\) 8.39082e7 2.79800
\(980\) 0 0
\(981\) −1.59266e7 −0.528384
\(982\) 5.69993e6 0.188621
\(983\) 5.00560e7 1.65224 0.826118 0.563497i \(-0.190544\pi\)
0.826118 + 0.563497i \(0.190544\pi\)
\(984\) −993166. −0.0326990
\(985\) 0 0
\(986\) 2.06261e6 0.0675654
\(987\) 7.98990e6 0.261065
\(988\) 3.20497e7 1.04456
\(989\) −1.82233e7 −0.592428
\(990\) 0 0
\(991\) 1.59116e7 0.514670 0.257335 0.966322i \(-0.417156\pi\)
0.257335 + 0.966322i \(0.417156\pi\)
\(992\) 4.46330e6 0.144005
\(993\) 4.31534e7 1.38881
\(994\) 2.29385e6 0.0736375
\(995\) 0 0
\(996\) 418041. 0.0133528
\(997\) −4.25995e7 −1.35727 −0.678635 0.734476i \(-0.737428\pi\)
−0.678635 + 0.734476i \(0.737428\pi\)
\(998\) 981634. 0.0311978
\(999\) −2.11514e6 −0.0670542
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 175.6.a.c.1.2 2
5.2 odd 4 175.6.b.c.99.2 4
5.3 odd 4 175.6.b.c.99.3 4
5.4 even 2 7.6.a.b.1.1 2
15.14 odd 2 63.6.a.f.1.2 2
20.19 odd 2 112.6.a.h.1.1 2
35.4 even 6 49.6.c.e.30.2 4
35.9 even 6 49.6.c.e.18.2 4
35.19 odd 6 49.6.c.d.18.2 4
35.24 odd 6 49.6.c.d.30.2 4
35.34 odd 2 49.6.a.f.1.1 2
40.19 odd 2 448.6.a.u.1.2 2
40.29 even 2 448.6.a.w.1.1 2
55.54 odd 2 847.6.a.c.1.2 2
60.59 even 2 1008.6.a.bq.1.2 2
105.104 even 2 441.6.a.l.1.2 2
140.139 even 2 784.6.a.v.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.6.a.b.1.1 2 5.4 even 2
49.6.a.f.1.1 2 35.34 odd 2
49.6.c.d.18.2 4 35.19 odd 6
49.6.c.d.30.2 4 35.24 odd 6
49.6.c.e.18.2 4 35.9 even 6
49.6.c.e.30.2 4 35.4 even 6
63.6.a.f.1.2 2 15.14 odd 2
112.6.a.h.1.1 2 20.19 odd 2
175.6.a.c.1.2 2 1.1 even 1 trivial
175.6.b.c.99.2 4 5.2 odd 4
175.6.b.c.99.3 4 5.3 odd 4
441.6.a.l.1.2 2 105.104 even 2
448.6.a.u.1.2 2 40.19 odd 2
448.6.a.w.1.1 2 40.29 even 2
784.6.a.v.1.2 2 140.139 even 2
847.6.a.c.1.2 2 55.54 odd 2
1008.6.a.bq.1.2 2 60.59 even 2