Properties

Label 175.10.a.c.1.1
Level $175$
Weight $10$
Character 175.1
Self dual yes
Analytic conductor $90.131$
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] = [175,10,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, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
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: \(90.1312713287\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) 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)
 
\(f(q)\) \(=\) \(q+9.17157 q^{2} -65.7351 q^{3} -427.882 q^{4} -602.894 q^{6} -2401.00 q^{7} -8620.20 q^{8} -15361.9 q^{9} +O(q^{10})\) \(q+9.17157 q^{2} -65.7351 q^{3} -427.882 q^{4} -602.894 q^{6} -2401.00 q^{7} -8620.20 q^{8} -15361.9 q^{9} +35089.6 q^{11} +28126.9 q^{12} +77401.4 q^{13} -22020.9 q^{14} +140015. q^{16} +229907. q^{17} -140893. q^{18} +16433.6 q^{19} +157830. q^{21} +321827. q^{22} +2.57284e6 q^{23} +566649. q^{24} +709892. q^{26} +2.30368e6 q^{27} +1.02735e6 q^{28} -6.62817e6 q^{29} -8.17416e6 q^{31} +5.69770e6 q^{32} -2.30662e6 q^{33} +2.10861e6 q^{34} +6.57308e6 q^{36} -9.70272e6 q^{37} +150722. q^{38} -5.08798e6 q^{39} +2.98108e7 q^{41} +1.44755e6 q^{42} +1.95343e7 q^{43} -1.50142e7 q^{44} +2.35970e7 q^{46} -5.93794e6 q^{47} -9.20389e6 q^{48} +5.76480e6 q^{49} -1.51130e7 q^{51} -3.31187e7 q^{52} +2.74263e7 q^{53} +2.11284e7 q^{54} +2.06971e7 q^{56} -1.08026e6 q^{57} -6.07908e7 q^{58} +5.24915e7 q^{59} +2.23282e7 q^{61} -7.49699e7 q^{62} +3.68839e7 q^{63} -1.94308e7 q^{64} -2.11553e7 q^{66} -2.74351e8 q^{67} -9.83733e7 q^{68} -1.69126e8 q^{69} -3.63673e8 q^{71} +1.32423e8 q^{72} -2.09245e7 q^{73} -8.89892e7 q^{74} -7.03163e6 q^{76} -8.42501e7 q^{77} -4.66648e7 q^{78} -2.65896e8 q^{79} +1.50936e8 q^{81} +2.73412e8 q^{82} +9.43764e6 q^{83} -6.75326e7 q^{84} +1.79160e8 q^{86} +4.35704e8 q^{87} -3.02479e8 q^{88} -6.64876e8 q^{89} -1.85841e8 q^{91} -1.10087e9 q^{92} +5.37329e8 q^{93} -5.44603e7 q^{94} -3.74539e8 q^{96} +1.20731e9 q^{97} +5.28723e7 q^{98} -5.39042e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 24 q^{2} + 174 q^{3} - 720 q^{4} + 2952 q^{6} - 4802 q^{7} - 20544 q^{8} + 22428 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 24 q^{2} + 174 q^{3} - 720 q^{4} + 2952 q^{6} - 4802 q^{7} - 20544 q^{8} + 22428 q^{9} + 18566 q^{11} - 41904 q^{12} + 51090 q^{13} - 57624 q^{14} + 112768 q^{16} + 373910 q^{17} + 419472 q^{18} - 143276 q^{19} - 417774 q^{21} + 76808 q^{22} + 498908 q^{23} - 2291904 q^{24} + 319736 q^{26} + 6644538 q^{27} + 1728720 q^{28} - 11577554 q^{29} - 3953760 q^{31} + 11398656 q^{32} - 6267894 q^{33} + 4243944 q^{34} - 4466016 q^{36} + 3205412 q^{37} - 2217520 q^{38} - 11395746 q^{39} + 1058992 q^{41} - 7087752 q^{42} - 15948180 q^{43} - 10187376 q^{44} - 7156176 q^{46} - 65501290 q^{47} - 15735936 q^{48} + 11529602 q^{49} + 19409466 q^{51} - 25432656 q^{52} + 25114688 q^{53} + 85496472 q^{54} + 49326144 q^{56} - 39368244 q^{57} - 134182296 q^{58} - 116159208 q^{59} - 44688544 q^{61} - 12388000 q^{62} - 53849628 q^{63} + 79055872 q^{64} - 79894824 q^{66} - 118092496 q^{67} - 140439024 q^{68} - 666320892 q^{69} - 294165824 q^{71} - 318176640 q^{72} + 57419332 q^{73} + 102418064 q^{74} + 39622368 q^{76} - 44576966 q^{77} - 140199000 q^{78} - 692852854 q^{79} + 447773346 q^{81} - 152932128 q^{82} + 540679928 q^{83} + 100611504 q^{84} - 346989008 q^{86} - 750836190 q^{87} - 105455296 q^{88} - 779043704 q^{89} - 122667090 q^{91} - 495040608 q^{92} + 1549107360 q^{93} - 937691032 q^{94} + 992180736 q^{96} + 2673039406 q^{97} + 138355224 q^{98} - 1163466540 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.17157 0.405330 0.202665 0.979248i \(-0.435040\pi\)
0.202665 + 0.979248i \(0.435040\pi\)
\(3\) −65.7351 −0.468545 −0.234273 0.972171i \(-0.575271\pi\)
−0.234273 + 0.972171i \(0.575271\pi\)
\(4\) −427.882 −0.835708
\(5\) 0 0
\(6\) −602.894 −0.189915
\(7\) −2401.00 −0.377964
\(8\) −8620.20 −0.744067
\(9\) −15361.9 −0.780465
\(10\) 0 0
\(11\) 35089.6 0.722622 0.361311 0.932445i \(-0.382329\pi\)
0.361311 + 0.932445i \(0.382329\pi\)
\(12\) 28126.9 0.391567
\(13\) 77401.4 0.751629 0.375815 0.926695i \(-0.377363\pi\)
0.375815 + 0.926695i \(0.377363\pi\)
\(14\) −22020.9 −0.153200
\(15\) 0 0
\(16\) 140015. 0.534115
\(17\) 229907. 0.667626 0.333813 0.942639i \(-0.391665\pi\)
0.333813 + 0.942639i \(0.391665\pi\)
\(18\) −140893. −0.316346
\(19\) 16433.6 0.0289295 0.0144647 0.999895i \(-0.495396\pi\)
0.0144647 + 0.999895i \(0.495396\pi\)
\(20\) 0 0
\(21\) 157830. 0.177093
\(22\) 321827. 0.292900
\(23\) 2.57284e6 1.91707 0.958535 0.284975i \(-0.0919852\pi\)
0.958535 + 0.284975i \(0.0919852\pi\)
\(24\) 566649. 0.348629
\(25\) 0 0
\(26\) 709892. 0.304658
\(27\) 2.30368e6 0.834228
\(28\) 1.02735e6 0.315868
\(29\) −6.62817e6 −1.74022 −0.870108 0.492862i \(-0.835951\pi\)
−0.870108 + 0.492862i \(0.835951\pi\)
\(30\) 0 0
\(31\) −8.17416e6 −1.58970 −0.794851 0.606805i \(-0.792451\pi\)
−0.794851 + 0.606805i \(0.792451\pi\)
\(32\) 5.69770e6 0.960560
\(33\) −2.30662e6 −0.338581
\(34\) 2.10861e6 0.270609
\(35\) 0 0
\(36\) 6.57308e6 0.652241
\(37\) −9.70272e6 −0.851110 −0.425555 0.904933i \(-0.639921\pi\)
−0.425555 + 0.904933i \(0.639921\pi\)
\(38\) 150722. 0.0117260
\(39\) −5.08798e6 −0.352172
\(40\) 0 0
\(41\) 2.98108e7 1.64758 0.823789 0.566896i \(-0.191856\pi\)
0.823789 + 0.566896i \(0.191856\pi\)
\(42\) 1.44755e6 0.0717813
\(43\) 1.95343e7 0.871343 0.435672 0.900106i \(-0.356511\pi\)
0.435672 + 0.900106i \(0.356511\pi\)
\(44\) −1.50142e7 −0.603900
\(45\) 0 0
\(46\) 2.35970e7 0.777046
\(47\) −5.93794e6 −0.177499 −0.0887494 0.996054i \(-0.528287\pi\)
−0.0887494 + 0.996054i \(0.528287\pi\)
\(48\) −9.20389e6 −0.250257
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) −1.51130e7 −0.312813
\(52\) −3.31187e7 −0.628142
\(53\) 2.74263e7 0.477448 0.238724 0.971088i \(-0.423271\pi\)
0.238724 + 0.971088i \(0.423271\pi\)
\(54\) 2.11284e7 0.338138
\(55\) 0 0
\(56\) 2.06971e7 0.281231
\(57\) −1.08026e6 −0.0135548
\(58\) −6.07908e7 −0.705362
\(59\) 5.24915e7 0.563969 0.281984 0.959419i \(-0.409007\pi\)
0.281984 + 0.959419i \(0.409007\pi\)
\(60\) 0 0
\(61\) 2.23282e7 0.206476 0.103238 0.994657i \(-0.467080\pi\)
0.103238 + 0.994657i \(0.467080\pi\)
\(62\) −7.49699e7 −0.644354
\(63\) 3.68839e7 0.294988
\(64\) −1.94308e7 −0.144771
\(65\) 0 0
\(66\) −2.11553e7 −0.137237
\(67\) −2.74351e8 −1.66330 −0.831649 0.555302i \(-0.812603\pi\)
−0.831649 + 0.555302i \(0.812603\pi\)
\(68\) −9.83733e7 −0.557940
\(69\) −1.69126e8 −0.898234
\(70\) 0 0
\(71\) −3.63673e8 −1.69843 −0.849216 0.528046i \(-0.822925\pi\)
−0.849216 + 0.528046i \(0.822925\pi\)
\(72\) 1.32423e8 0.580719
\(73\) −2.09245e7 −0.0862387 −0.0431193 0.999070i \(-0.513730\pi\)
−0.0431193 + 0.999070i \(0.513730\pi\)
\(74\) −8.89892e7 −0.344980
\(75\) 0 0
\(76\) −7.03163e6 −0.0241766
\(77\) −8.42501e7 −0.273125
\(78\) −4.66648e7 −0.142746
\(79\) −2.65896e8 −0.768051 −0.384025 0.923323i \(-0.625462\pi\)
−0.384025 + 0.923323i \(0.625462\pi\)
\(80\) 0 0
\(81\) 1.50936e8 0.389592
\(82\) 2.73412e8 0.667813
\(83\) 9.43764e6 0.0218279 0.0109140 0.999940i \(-0.496526\pi\)
0.0109140 + 0.999940i \(0.496526\pi\)
\(84\) −6.75326e7 −0.147998
\(85\) 0 0
\(86\) 1.79160e8 0.353182
\(87\) 4.35704e8 0.815369
\(88\) −3.02479e8 −0.537679
\(89\) −6.64876e8 −1.12327 −0.561637 0.827384i \(-0.689828\pi\)
−0.561637 + 0.827384i \(0.689828\pi\)
\(90\) 0 0
\(91\) −1.85841e8 −0.284089
\(92\) −1.10087e9 −1.60211
\(93\) 5.37329e8 0.744847
\(94\) −5.44603e7 −0.0719456
\(95\) 0 0
\(96\) −3.74539e8 −0.450066
\(97\) 1.20731e9 1.38467 0.692336 0.721575i \(-0.256582\pi\)
0.692336 + 0.721575i \(0.256582\pi\)
\(98\) 5.28723e7 0.0579043
\(99\) −5.39042e8 −0.563981
\(100\) 0 0
\(101\) 1.18204e9 1.13028 0.565139 0.824996i \(-0.308823\pi\)
0.565139 + 0.824996i \(0.308823\pi\)
\(102\) −1.38610e8 −0.126792
\(103\) −1.97811e9 −1.73174 −0.865870 0.500268i \(-0.833235\pi\)
−0.865870 + 0.500268i \(0.833235\pi\)
\(104\) −6.67215e8 −0.559263
\(105\) 0 0
\(106\) 2.51542e8 0.193524
\(107\) −1.67828e8 −0.123776 −0.0618881 0.998083i \(-0.519712\pi\)
−0.0618881 + 0.998083i \(0.519712\pi\)
\(108\) −9.85703e8 −0.697171
\(109\) −1.02540e9 −0.695784 −0.347892 0.937535i \(-0.613102\pi\)
−0.347892 + 0.937535i \(0.613102\pi\)
\(110\) 0 0
\(111\) 6.37809e8 0.398783
\(112\) −3.36176e8 −0.201876
\(113\) −1.27533e9 −0.735814 −0.367907 0.929863i \(-0.619926\pi\)
−0.367907 + 0.929863i \(0.619926\pi\)
\(114\) −9.90770e6 −0.00549415
\(115\) 0 0
\(116\) 2.83608e9 1.45431
\(117\) −1.18903e9 −0.586621
\(118\) 4.81430e8 0.228594
\(119\) −5.52008e8 −0.252339
\(120\) 0 0
\(121\) −1.12667e9 −0.477818
\(122\) 2.04785e8 0.0836909
\(123\) −1.95961e9 −0.771965
\(124\) 3.49758e9 1.32853
\(125\) 0 0
\(126\) 3.38284e8 0.119568
\(127\) 2.90339e9 0.990349 0.495174 0.868794i \(-0.335104\pi\)
0.495174 + 0.868794i \(0.335104\pi\)
\(128\) −3.09543e9 −1.01924
\(129\) −1.28409e9 −0.408264
\(130\) 0 0
\(131\) 2.05173e9 0.608694 0.304347 0.952561i \(-0.401562\pi\)
0.304347 + 0.952561i \(0.401562\pi\)
\(132\) 9.86960e8 0.282955
\(133\) −3.94570e7 −0.0109343
\(134\) −2.51623e9 −0.674185
\(135\) 0 0
\(136\) −1.98185e9 −0.496759
\(137\) −3.25539e9 −0.789514 −0.394757 0.918786i \(-0.629171\pi\)
−0.394757 + 0.918786i \(0.629171\pi\)
\(138\) −1.55115e9 −0.364081
\(139\) −8.26776e9 −1.87854 −0.939272 0.343173i \(-0.888498\pi\)
−0.939272 + 0.343173i \(0.888498\pi\)
\(140\) 0 0
\(141\) 3.90331e8 0.0831662
\(142\) −3.33545e9 −0.688425
\(143\) 2.71598e9 0.543143
\(144\) −2.15090e9 −0.416858
\(145\) 0 0
\(146\) −1.91910e8 −0.0349551
\(147\) −3.78950e8 −0.0669350
\(148\) 4.15162e9 0.711279
\(149\) 1.07127e9 0.178058 0.0890289 0.996029i \(-0.471624\pi\)
0.0890289 + 0.996029i \(0.471624\pi\)
\(150\) 0 0
\(151\) 1.97304e9 0.308844 0.154422 0.988005i \(-0.450649\pi\)
0.154422 + 0.988005i \(0.450649\pi\)
\(152\) −1.41661e8 −0.0215255
\(153\) −3.53182e9 −0.521059
\(154\) −7.72706e8 −0.110706
\(155\) 0 0
\(156\) 2.17706e9 0.294313
\(157\) 4.61623e9 0.606372 0.303186 0.952931i \(-0.401950\pi\)
0.303186 + 0.952931i \(0.401950\pi\)
\(158\) −2.43868e9 −0.311314
\(159\) −1.80287e9 −0.223706
\(160\) 0 0
\(161\) −6.17740e9 −0.724584
\(162\) 1.38432e9 0.157913
\(163\) −6.26525e9 −0.695175 −0.347588 0.937648i \(-0.612999\pi\)
−0.347588 + 0.937648i \(0.612999\pi\)
\(164\) −1.27555e10 −1.37689
\(165\) 0 0
\(166\) 8.65580e7 0.00884751
\(167\) 6.21672e9 0.618496 0.309248 0.950981i \(-0.399923\pi\)
0.309248 + 0.950981i \(0.399923\pi\)
\(168\) −1.36052e9 −0.131769
\(169\) −4.61353e9 −0.435054
\(170\) 0 0
\(171\) −2.52451e8 −0.0225785
\(172\) −8.35837e9 −0.728188
\(173\) −8.97209e9 −0.761528 −0.380764 0.924672i \(-0.624339\pi\)
−0.380764 + 0.924672i \(0.624339\pi\)
\(174\) 3.99609e9 0.330494
\(175\) 0 0
\(176\) 4.91306e9 0.385963
\(177\) −3.45053e9 −0.264245
\(178\) −6.09796e9 −0.455297
\(179\) 1.76242e10 1.28313 0.641565 0.767069i \(-0.278286\pi\)
0.641565 + 0.767069i \(0.278286\pi\)
\(180\) 0 0
\(181\) 1.62250e9 0.112365 0.0561824 0.998421i \(-0.482107\pi\)
0.0561824 + 0.998421i \(0.482107\pi\)
\(182\) −1.70445e9 −0.115150
\(183\) −1.46775e9 −0.0967433
\(184\) −2.21784e10 −1.42643
\(185\) 0 0
\(186\) 4.92815e9 0.301909
\(187\) 8.06735e9 0.482441
\(188\) 2.54074e9 0.148337
\(189\) −5.53113e9 −0.315309
\(190\) 0 0
\(191\) 1.66601e10 0.905788 0.452894 0.891564i \(-0.350392\pi\)
0.452894 + 0.891564i \(0.350392\pi\)
\(192\) 1.27728e9 0.0678316
\(193\) −2.41341e10 −1.25206 −0.626028 0.779801i \(-0.715320\pi\)
−0.626028 + 0.779801i \(0.715320\pi\)
\(194\) 1.10730e10 0.561249
\(195\) 0 0
\(196\) −2.46666e9 −0.119387
\(197\) 3.89843e9 0.184413 0.0922066 0.995740i \(-0.470608\pi\)
0.0922066 + 0.995740i \(0.470608\pi\)
\(198\) −4.94387e9 −0.228599
\(199\) −1.87489e10 −0.847493 −0.423747 0.905781i \(-0.639285\pi\)
−0.423747 + 0.905781i \(0.639285\pi\)
\(200\) 0 0
\(201\) 1.80345e10 0.779330
\(202\) 1.08411e10 0.458135
\(203\) 1.59142e10 0.657740
\(204\) 6.46658e9 0.261420
\(205\) 0 0
\(206\) −1.81424e10 −0.701927
\(207\) −3.95238e10 −1.49621
\(208\) 1.08373e10 0.401456
\(209\) 5.76647e8 0.0209051
\(210\) 0 0
\(211\) −2.20489e9 −0.0765801 −0.0382900 0.999267i \(-0.512191\pi\)
−0.0382900 + 0.999267i \(0.512191\pi\)
\(212\) −1.17352e10 −0.399007
\(213\) 2.39060e10 0.795792
\(214\) −1.53925e9 −0.0501702
\(215\) 0 0
\(216\) −1.98582e10 −0.620722
\(217\) 1.96262e10 0.600851
\(218\) −9.40454e9 −0.282022
\(219\) 1.37547e9 0.0404067
\(220\) 0 0
\(221\) 1.77952e10 0.501807
\(222\) 5.84971e9 0.161639
\(223\) 2.65324e10 0.718463 0.359231 0.933249i \(-0.383039\pi\)
0.359231 + 0.933249i \(0.383039\pi\)
\(224\) −1.36802e10 −0.363058
\(225\) 0 0
\(226\) −1.16967e10 −0.298248
\(227\) −7.78091e10 −1.94498 −0.972488 0.232955i \(-0.925161\pi\)
−0.972488 + 0.232955i \(0.925161\pi\)
\(228\) 4.62225e8 0.0113278
\(229\) 4.84637e10 1.16455 0.582274 0.812993i \(-0.302163\pi\)
0.582274 + 0.812993i \(0.302163\pi\)
\(230\) 0 0
\(231\) 5.53818e9 0.127972
\(232\) 5.71362e10 1.29484
\(233\) 2.38429e10 0.529978 0.264989 0.964251i \(-0.414632\pi\)
0.264989 + 0.964251i \(0.414632\pi\)
\(234\) −1.09053e10 −0.237775
\(235\) 0 0
\(236\) −2.24602e10 −0.471313
\(237\) 1.74787e10 0.359866
\(238\) −5.06278e9 −0.102280
\(239\) 6.25895e10 1.24083 0.620413 0.784275i \(-0.286965\pi\)
0.620413 + 0.784275i \(0.286965\pi\)
\(240\) 0 0
\(241\) −7.96605e10 −1.52113 −0.760565 0.649262i \(-0.775078\pi\)
−0.760565 + 0.649262i \(0.775078\pi\)
\(242\) −1.03333e10 −0.193674
\(243\) −5.52651e10 −1.01677
\(244\) −9.55384e9 −0.172554
\(245\) 0 0
\(246\) −1.79727e10 −0.312901
\(247\) 1.27198e9 0.0217442
\(248\) 7.04629e10 1.18285
\(249\) −6.20384e8 −0.0102274
\(250\) 0 0
\(251\) −5.44549e10 −0.865975 −0.432988 0.901400i \(-0.642541\pi\)
−0.432988 + 0.901400i \(0.642541\pi\)
\(252\) −1.57820e10 −0.246524
\(253\) 9.02799e10 1.38532
\(254\) 2.66286e10 0.401418
\(255\) 0 0
\(256\) −1.84414e10 −0.268358
\(257\) 5.35278e10 0.765385 0.382693 0.923876i \(-0.374997\pi\)
0.382693 + 0.923876i \(0.374997\pi\)
\(258\) −1.17771e10 −0.165482
\(259\) 2.32962e10 0.321689
\(260\) 0 0
\(261\) 1.01821e11 1.35818
\(262\) 1.88176e10 0.246722
\(263\) 5.81425e10 0.749364 0.374682 0.927153i \(-0.377752\pi\)
0.374682 + 0.927153i \(0.377752\pi\)
\(264\) 1.98835e10 0.251927
\(265\) 0 0
\(266\) −3.61883e8 −0.00443201
\(267\) 4.37057e10 0.526304
\(268\) 1.17390e11 1.39003
\(269\) 4.67380e10 0.544233 0.272116 0.962264i \(-0.412276\pi\)
0.272116 + 0.962264i \(0.412276\pi\)
\(270\) 0 0
\(271\) 2.68147e10 0.302003 0.151001 0.988534i \(-0.451750\pi\)
0.151001 + 0.988534i \(0.451750\pi\)
\(272\) 3.21905e10 0.356589
\(273\) 1.22163e10 0.133109
\(274\) −2.98570e10 −0.320014
\(275\) 0 0
\(276\) 7.23660e10 0.750661
\(277\) −1.12549e11 −1.14863 −0.574316 0.818633i \(-0.694732\pi\)
−0.574316 + 0.818633i \(0.694732\pi\)
\(278\) −7.58284e10 −0.761431
\(279\) 1.25571e11 1.24071
\(280\) 0 0
\(281\) −4.60761e10 −0.440857 −0.220428 0.975403i \(-0.570746\pi\)
−0.220428 + 0.975403i \(0.570746\pi\)
\(282\) 3.57995e9 0.0337098
\(283\) −7.94071e10 −0.735902 −0.367951 0.929845i \(-0.619941\pi\)
−0.367951 + 0.929845i \(0.619941\pi\)
\(284\) 1.55609e11 1.41939
\(285\) 0 0
\(286\) 2.49098e10 0.220152
\(287\) −7.15757e10 −0.622726
\(288\) −8.75275e10 −0.749684
\(289\) −6.57304e10 −0.554276
\(290\) 0 0
\(291\) −7.93628e10 −0.648782
\(292\) 8.95322e9 0.0720703
\(293\) 1.41265e11 1.11977 0.559887 0.828569i \(-0.310844\pi\)
0.559887 + 0.828569i \(0.310844\pi\)
\(294\) −3.47556e9 −0.0271308
\(295\) 0 0
\(296\) 8.36393e10 0.633283
\(297\) 8.08351e10 0.602832
\(298\) 9.82523e9 0.0721721
\(299\) 1.99142e11 1.44093
\(300\) 0 0
\(301\) −4.69018e10 −0.329337
\(302\) 1.80958e10 0.125184
\(303\) −7.77013e10 −0.529586
\(304\) 2.30094e9 0.0154517
\(305\) 0 0
\(306\) −3.23923e10 −0.211201
\(307\) 5.58349e10 0.358742 0.179371 0.983781i \(-0.442594\pi\)
0.179371 + 0.983781i \(0.442594\pi\)
\(308\) 3.60491e10 0.228253
\(309\) 1.30031e11 0.811399
\(310\) 0 0
\(311\) 5.26501e10 0.319137 0.159569 0.987187i \(-0.448990\pi\)
0.159569 + 0.987187i \(0.448990\pi\)
\(312\) 4.38594e10 0.262040
\(313\) 2.51256e11 1.47968 0.739838 0.672785i \(-0.234902\pi\)
0.739838 + 0.672785i \(0.234902\pi\)
\(314\) 4.23381e10 0.245781
\(315\) 0 0
\(316\) 1.13772e11 0.641866
\(317\) 1.16999e11 0.650749 0.325375 0.945585i \(-0.394510\pi\)
0.325375 + 0.945585i \(0.394510\pi\)
\(318\) −1.65351e10 −0.0906747
\(319\) −2.32580e11 −1.25752
\(320\) 0 0
\(321\) 1.10322e10 0.0579948
\(322\) −5.66564e10 −0.293696
\(323\) 3.77820e9 0.0193141
\(324\) −6.45828e10 −0.325585
\(325\) 0 0
\(326\) −5.74622e10 −0.281775
\(327\) 6.74048e10 0.326006
\(328\) −2.56975e11 −1.22591
\(329\) 1.42570e10 0.0670883
\(330\) 0 0
\(331\) −2.51419e11 −1.15126 −0.575629 0.817711i \(-0.695243\pi\)
−0.575629 + 0.817711i \(0.695243\pi\)
\(332\) −4.03820e9 −0.0182417
\(333\) 1.49052e11 0.664262
\(334\) 5.70171e10 0.250695
\(335\) 0 0
\(336\) 2.20985e10 0.0945882
\(337\) −6.11427e10 −0.258232 −0.129116 0.991630i \(-0.541214\pi\)
−0.129116 + 0.991630i \(0.541214\pi\)
\(338\) −4.23133e10 −0.176340
\(339\) 8.38336e10 0.344762
\(340\) 0 0
\(341\) −2.86828e11 −1.14875
\(342\) −2.31537e9 −0.00915173
\(343\) −1.38413e10 −0.0539949
\(344\) −1.68389e11 −0.648338
\(345\) 0 0
\(346\) −8.22882e10 −0.308670
\(347\) 1.68668e11 0.624524 0.312262 0.949996i \(-0.398913\pi\)
0.312262 + 0.949996i \(0.398913\pi\)
\(348\) −1.86430e11 −0.681410
\(349\) −3.31182e11 −1.19496 −0.597479 0.801885i \(-0.703831\pi\)
−0.597479 + 0.801885i \(0.703831\pi\)
\(350\) 0 0
\(351\) 1.78308e11 0.627030
\(352\) 1.99930e11 0.694122
\(353\) −3.78560e11 −1.29762 −0.648811 0.760949i \(-0.724734\pi\)
−0.648811 + 0.760949i \(0.724734\pi\)
\(354\) −3.16468e10 −0.107106
\(355\) 0 0
\(356\) 2.84489e11 0.938728
\(357\) 3.62863e10 0.118232
\(358\) 1.61641e11 0.520091
\(359\) −1.60137e11 −0.508822 −0.254411 0.967096i \(-0.581882\pi\)
−0.254411 + 0.967096i \(0.581882\pi\)
\(360\) 0 0
\(361\) −3.22418e11 −0.999163
\(362\) 1.48809e10 0.0455449
\(363\) 7.40617e10 0.223879
\(364\) 7.95179e10 0.237415
\(365\) 0 0
\(366\) −1.34615e10 −0.0392130
\(367\) −5.13837e11 −1.47852 −0.739261 0.673419i \(-0.764825\pi\)
−0.739261 + 0.673419i \(0.764825\pi\)
\(368\) 3.60236e11 1.02394
\(369\) −4.57950e11 −1.28588
\(370\) 0 0
\(371\) −6.58505e10 −0.180458
\(372\) −2.29914e11 −0.622474
\(373\) 6.70900e10 0.179460 0.0897301 0.995966i \(-0.471400\pi\)
0.0897301 + 0.995966i \(0.471400\pi\)
\(374\) 7.39903e10 0.195548
\(375\) 0 0
\(376\) 5.11862e10 0.132071
\(377\) −5.13030e11 −1.30800
\(378\) −5.07292e10 −0.127804
\(379\) 4.15471e11 1.03434 0.517171 0.855882i \(-0.326985\pi\)
0.517171 + 0.855882i \(0.326985\pi\)
\(380\) 0 0
\(381\) −1.90854e11 −0.464023
\(382\) 1.52799e11 0.367143
\(383\) 3.51976e11 0.835831 0.417915 0.908486i \(-0.362761\pi\)
0.417915 + 0.908486i \(0.362761\pi\)
\(384\) 2.03478e11 0.477560
\(385\) 0 0
\(386\) −2.21348e11 −0.507496
\(387\) −3.00084e11 −0.680053
\(388\) −5.16588e11 −1.15718
\(389\) −2.60061e11 −0.575840 −0.287920 0.957654i \(-0.592964\pi\)
−0.287920 + 0.957654i \(0.592964\pi\)
\(390\) 0 0
\(391\) 5.91516e11 1.27989
\(392\) −4.96937e10 −0.106295
\(393\) −1.34871e11 −0.285201
\(394\) 3.57548e10 0.0747482
\(395\) 0 0
\(396\) 2.30647e11 0.471323
\(397\) −7.34338e11 −1.48367 −0.741837 0.670580i \(-0.766045\pi\)
−0.741837 + 0.670580i \(0.766045\pi\)
\(398\) −1.71957e11 −0.343514
\(399\) 2.59371e9 0.00512322
\(400\) 0 0
\(401\) 8.08296e11 1.56106 0.780532 0.625116i \(-0.214948\pi\)
0.780532 + 0.625116i \(0.214948\pi\)
\(402\) 1.65405e11 0.315886
\(403\) −6.32691e11 −1.19487
\(404\) −5.05773e11 −0.944581
\(405\) 0 0
\(406\) 1.45959e11 0.266602
\(407\) −3.40464e11 −0.615030
\(408\) 1.30277e11 0.232754
\(409\) −9.11153e11 −1.61004 −0.805020 0.593248i \(-0.797845\pi\)
−0.805020 + 0.593248i \(0.797845\pi\)
\(410\) 0 0
\(411\) 2.13993e11 0.369923
\(412\) 8.46398e11 1.44723
\(413\) −1.26032e11 −0.213160
\(414\) −3.62495e11 −0.606458
\(415\) 0 0
\(416\) 4.41010e11 0.721985
\(417\) 5.43482e11 0.880183
\(418\) 5.28876e9 0.00847345
\(419\) 4.94109e11 0.783177 0.391589 0.920140i \(-0.371926\pi\)
0.391589 + 0.920140i \(0.371926\pi\)
\(420\) 0 0
\(421\) −1.15145e10 −0.0178639 −0.00893197 0.999960i \(-0.502843\pi\)
−0.00893197 + 0.999960i \(0.502843\pi\)
\(422\) −2.02223e10 −0.0310402
\(423\) 9.12181e10 0.138532
\(424\) −2.36420e11 −0.355253
\(425\) 0 0
\(426\) 2.19256e11 0.322558
\(427\) −5.36100e10 −0.0780406
\(428\) 7.18106e10 0.103441
\(429\) −1.78535e11 −0.254487
\(430\) 0 0
\(431\) −9.42534e11 −1.31568 −0.657839 0.753159i \(-0.728529\pi\)
−0.657839 + 0.753159i \(0.728529\pi\)
\(432\) 3.22549e11 0.445574
\(433\) −1.01849e12 −1.39239 −0.696196 0.717852i \(-0.745125\pi\)
−0.696196 + 0.717852i \(0.745125\pi\)
\(434\) 1.80003e11 0.243543
\(435\) 0 0
\(436\) 4.38751e11 0.581472
\(437\) 4.22810e10 0.0554598
\(438\) 1.26152e10 0.0163781
\(439\) −7.89357e11 −1.01434 −0.507169 0.861847i \(-0.669308\pi\)
−0.507169 + 0.861847i \(0.669308\pi\)
\(440\) 0 0
\(441\) −8.85583e10 −0.111495
\(442\) 1.63210e11 0.203397
\(443\) −1.06770e12 −1.31714 −0.658572 0.752518i \(-0.728839\pi\)
−0.658572 + 0.752518i \(0.728839\pi\)
\(444\) −2.72907e11 −0.333266
\(445\) 0 0
\(446\) 2.43344e11 0.291215
\(447\) −7.04200e10 −0.0834281
\(448\) 4.66533e10 0.0547182
\(449\) −3.31695e9 −0.00385150 −0.00192575 0.999998i \(-0.500613\pi\)
−0.00192575 + 0.999998i \(0.500613\pi\)
\(450\) 0 0
\(451\) 1.04605e12 1.19058
\(452\) 5.45689e11 0.614926
\(453\) −1.29698e11 −0.144707
\(454\) −7.13632e11 −0.788357
\(455\) 0 0
\(456\) 9.31207e9 0.0100857
\(457\) 7.03146e11 0.754089 0.377045 0.926195i \(-0.376940\pi\)
0.377045 + 0.926195i \(0.376940\pi\)
\(458\) 4.44489e11 0.472026
\(459\) 5.29633e11 0.556952
\(460\) 0 0
\(461\) −1.86192e12 −1.92003 −0.960015 0.279950i \(-0.909682\pi\)
−0.960015 + 0.279950i \(0.909682\pi\)
\(462\) 5.07938e10 0.0518707
\(463\) 1.06950e11 0.108160 0.0540799 0.998537i \(-0.482777\pi\)
0.0540799 + 0.998537i \(0.482777\pi\)
\(464\) −9.28043e11 −0.929474
\(465\) 0 0
\(466\) 2.18677e11 0.214816
\(467\) −4.13997e11 −0.402783 −0.201392 0.979511i \(-0.564546\pi\)
−0.201392 + 0.979511i \(0.564546\pi\)
\(468\) 5.08766e11 0.490243
\(469\) 6.58717e11 0.628667
\(470\) 0 0
\(471\) −3.03448e11 −0.284113
\(472\) −4.52487e11 −0.419631
\(473\) 6.85450e11 0.629652
\(474\) 1.60307e11 0.145865
\(475\) 0 0
\(476\) 2.36194e11 0.210881
\(477\) −4.21320e11 −0.372631
\(478\) 5.74044e11 0.502944
\(479\) −8.54131e10 −0.0741336 −0.0370668 0.999313i \(-0.511801\pi\)
−0.0370668 + 0.999313i \(0.511801\pi\)
\(480\) 0 0
\(481\) −7.51004e11 −0.639719
\(482\) −7.30612e11 −0.616560
\(483\) 4.06072e11 0.339501
\(484\) 4.82082e11 0.399316
\(485\) 0 0
\(486\) −5.06868e11 −0.412127
\(487\) −4.94789e11 −0.398602 −0.199301 0.979938i \(-0.563867\pi\)
−0.199301 + 0.979938i \(0.563867\pi\)
\(488\) −1.92474e11 −0.153632
\(489\) 4.11847e11 0.325721
\(490\) 0 0
\(491\) 1.11163e12 0.863168 0.431584 0.902073i \(-0.357955\pi\)
0.431584 + 0.902073i \(0.357955\pi\)
\(492\) 8.38484e11 0.645137
\(493\) −1.52387e12 −1.16181
\(494\) 1.16661e10 0.00881359
\(495\) 0 0
\(496\) −1.14450e12 −0.849083
\(497\) 8.73178e11 0.641947
\(498\) −5.68990e9 −0.00414546
\(499\) 8.18377e11 0.590882 0.295441 0.955361i \(-0.404533\pi\)
0.295441 + 0.955361i \(0.404533\pi\)
\(500\) 0 0
\(501\) −4.08656e11 −0.289793
\(502\) −4.99437e11 −0.351006
\(503\) −3.13384e11 −0.218284 −0.109142 0.994026i \(-0.534810\pi\)
−0.109142 + 0.994026i \(0.534810\pi\)
\(504\) −3.17947e11 −0.219491
\(505\) 0 0
\(506\) 8.28009e11 0.561510
\(507\) 3.03270e11 0.203842
\(508\) −1.24231e12 −0.827642
\(509\) −1.02554e12 −0.677211 −0.338606 0.940928i \(-0.609955\pi\)
−0.338606 + 0.940928i \(0.609955\pi\)
\(510\) 0 0
\(511\) 5.02397e10 0.0325951
\(512\) 1.41572e12 0.910467
\(513\) 3.78577e10 0.0241338
\(514\) 4.90934e11 0.310234
\(515\) 0 0
\(516\) 5.49438e11 0.341189
\(517\) −2.08360e11 −0.128265
\(518\) 2.13663e11 0.130390
\(519\) 5.89781e11 0.356810
\(520\) 0 0
\(521\) −3.18952e12 −1.89651 −0.948255 0.317510i \(-0.897153\pi\)
−0.948255 + 0.317510i \(0.897153\pi\)
\(522\) 9.33862e11 0.550510
\(523\) −9.43708e11 −0.551544 −0.275772 0.961223i \(-0.588933\pi\)
−0.275772 + 0.961223i \(0.588933\pi\)
\(524\) −8.77898e11 −0.508690
\(525\) 0 0
\(526\) 5.33258e11 0.303740
\(527\) −1.87930e12 −1.06133
\(528\) −3.22961e11 −0.180841
\(529\) 4.81837e12 2.67516
\(530\) 0 0
\(531\) −8.06370e11 −0.440158
\(532\) 1.68829e10 0.00913789
\(533\) 2.30740e12 1.23837
\(534\) 4.00850e11 0.213327
\(535\) 0 0
\(536\) 2.36496e12 1.23761
\(537\) −1.15853e12 −0.601204
\(538\) 4.28661e11 0.220594
\(539\) 2.02284e11 0.103232
\(540\) 0 0
\(541\) −8.78618e11 −0.440973 −0.220487 0.975390i \(-0.570765\pi\)
−0.220487 + 0.975390i \(0.570765\pi\)
\(542\) 2.45933e11 0.122411
\(543\) −1.06655e11 −0.0526480
\(544\) 1.30994e12 0.641295
\(545\) 0 0
\(546\) 1.12042e11 0.0539529
\(547\) −4.56216e11 −0.217885 −0.108943 0.994048i \(-0.534746\pi\)
−0.108943 + 0.994048i \(0.534746\pi\)
\(548\) 1.39292e12 0.659803
\(549\) −3.43004e11 −0.161147
\(550\) 0 0
\(551\) −1.08925e11 −0.0503435
\(552\) 1.45790e12 0.668347
\(553\) 6.38416e11 0.290296
\(554\) −1.03225e12 −0.465575
\(555\) 0 0
\(556\) 3.53763e12 1.56991
\(557\) −1.63980e12 −0.721842 −0.360921 0.932596i \(-0.617538\pi\)
−0.360921 + 0.932596i \(0.617538\pi\)
\(558\) 1.15168e12 0.502896
\(559\) 1.51198e12 0.654927
\(560\) 0 0
\(561\) −5.30308e11 −0.226045
\(562\) −4.22590e11 −0.178692
\(563\) −4.36151e11 −0.182957 −0.0914786 0.995807i \(-0.529159\pi\)
−0.0914786 + 0.995807i \(0.529159\pi\)
\(564\) −1.67016e11 −0.0695026
\(565\) 0 0
\(566\) −7.28288e11 −0.298283
\(567\) −3.62397e11 −0.147252
\(568\) 3.13493e12 1.26375
\(569\) 1.76284e12 0.705029 0.352514 0.935806i \(-0.385327\pi\)
0.352514 + 0.935806i \(0.385327\pi\)
\(570\) 0 0
\(571\) 2.37232e11 0.0933922 0.0466961 0.998909i \(-0.485131\pi\)
0.0466961 + 0.998909i \(0.485131\pi\)
\(572\) −1.16212e12 −0.453909
\(573\) −1.09515e12 −0.424403
\(574\) −6.56462e11 −0.252410
\(575\) 0 0
\(576\) 2.98494e11 0.112988
\(577\) 3.72080e12 1.39748 0.698739 0.715377i \(-0.253745\pi\)
0.698739 + 0.715377i \(0.253745\pi\)
\(578\) −6.02851e11 −0.224665
\(579\) 1.58646e12 0.586644
\(580\) 0 0
\(581\) −2.26598e10 −0.00825017
\(582\) −7.27882e11 −0.262971
\(583\) 9.62377e11 0.345014
\(584\) 1.80373e11 0.0641674
\(585\) 0 0
\(586\) 1.29562e12 0.453878
\(587\) 6.46176e11 0.224636 0.112318 0.993672i \(-0.464172\pi\)
0.112318 + 0.993672i \(0.464172\pi\)
\(588\) 1.62146e11 0.0559381
\(589\) −1.34331e11 −0.0459892
\(590\) 0 0
\(591\) −2.56264e11 −0.0864059
\(592\) −1.35853e12 −0.454590
\(593\) 5.05774e12 1.67962 0.839808 0.542883i \(-0.182667\pi\)
0.839808 + 0.542883i \(0.182667\pi\)
\(594\) 7.41385e11 0.244346
\(595\) 0 0
\(596\) −4.58377e11 −0.148804
\(597\) 1.23246e12 0.397089
\(598\) 1.82644e12 0.584051
\(599\) −4.61588e11 −0.146499 −0.0732494 0.997314i \(-0.523337\pi\)
−0.0732494 + 0.997314i \(0.523337\pi\)
\(600\) 0 0
\(601\) −6.31800e12 −1.97535 −0.987677 0.156509i \(-0.949976\pi\)
−0.987677 + 0.156509i \(0.949976\pi\)
\(602\) −4.30163e11 −0.133490
\(603\) 4.21455e12 1.29815
\(604\) −8.44227e11 −0.258103
\(605\) 0 0
\(606\) −7.12643e11 −0.214657
\(607\) 1.45276e12 0.434356 0.217178 0.976132i \(-0.430315\pi\)
0.217178 + 0.976132i \(0.430315\pi\)
\(608\) 9.36335e10 0.0277885
\(609\) −1.04612e12 −0.308181
\(610\) 0 0
\(611\) −4.59605e11 −0.133413
\(612\) 1.51120e12 0.435453
\(613\) −1.20124e12 −0.343603 −0.171802 0.985132i \(-0.554959\pi\)
−0.171802 + 0.985132i \(0.554959\pi\)
\(614\) 5.12094e11 0.145409
\(615\) 0 0
\(616\) 7.26252e11 0.203224
\(617\) −3.13545e12 −0.870997 −0.435498 0.900189i \(-0.643428\pi\)
−0.435498 + 0.900189i \(0.643428\pi\)
\(618\) 1.19259e12 0.328884
\(619\) −5.17127e12 −1.41576 −0.707880 0.706333i \(-0.750348\pi\)
−0.707880 + 0.706333i \(0.750348\pi\)
\(620\) 0 0
\(621\) 5.92700e12 1.59927
\(622\) 4.82884e11 0.129356
\(623\) 1.59637e12 0.424557
\(624\) −7.12394e11 −0.188100
\(625\) 0 0
\(626\) 2.30441e12 0.599757
\(627\) −3.79059e10 −0.00979497
\(628\) −1.97520e12 −0.506749
\(629\) −2.23073e12 −0.568223
\(630\) 0 0
\(631\) −6.37331e12 −1.60042 −0.800208 0.599723i \(-0.795278\pi\)
−0.800208 + 0.599723i \(0.795278\pi\)
\(632\) 2.29208e12 0.571481
\(633\) 1.44939e11 0.0358812
\(634\) 1.07306e12 0.263768
\(635\) 0 0
\(636\) 7.71416e11 0.186953
\(637\) 4.46204e11 0.107376
\(638\) −2.13312e12 −0.509710
\(639\) 5.58670e12 1.32557
\(640\) 0 0
\(641\) −5.74174e12 −1.34333 −0.671665 0.740855i \(-0.734421\pi\)
−0.671665 + 0.740855i \(0.734421\pi\)
\(642\) 1.01182e11 0.0235070
\(643\) 5.85135e11 0.134992 0.0674958 0.997720i \(-0.478499\pi\)
0.0674958 + 0.997720i \(0.478499\pi\)
\(644\) 2.64320e12 0.605541
\(645\) 0 0
\(646\) 3.46520e10 0.00782857
\(647\) 1.80915e12 0.405887 0.202943 0.979190i \(-0.434949\pi\)
0.202943 + 0.979190i \(0.434949\pi\)
\(648\) −1.30110e12 −0.289883
\(649\) 1.84191e12 0.407536
\(650\) 0 0
\(651\) −1.29013e12 −0.281526
\(652\) 2.68079e12 0.580963
\(653\) −2.43900e12 −0.524932 −0.262466 0.964941i \(-0.584536\pi\)
−0.262466 + 0.964941i \(0.584536\pi\)
\(654\) 6.18208e11 0.132140
\(655\) 0 0
\(656\) 4.17396e12 0.879996
\(657\) 3.21440e11 0.0673063
\(658\) 1.30759e11 0.0271929
\(659\) 7.42836e12 1.53429 0.767147 0.641471i \(-0.221676\pi\)
0.767147 + 0.641471i \(0.221676\pi\)
\(660\) 0 0
\(661\) −6.34861e12 −1.29352 −0.646759 0.762695i \(-0.723876\pi\)
−0.646759 + 0.762695i \(0.723876\pi\)
\(662\) −2.30591e12 −0.466640
\(663\) −1.16977e12 −0.235119
\(664\) −8.13544e10 −0.0162414
\(665\) 0 0
\(666\) 1.36704e12 0.269245
\(667\) −1.70533e13 −3.33611
\(668\) −2.66002e12 −0.516882
\(669\) −1.74411e12 −0.336632
\(670\) 0 0
\(671\) 7.83487e11 0.149204
\(672\) 8.99267e11 0.170109
\(673\) −3.17186e12 −0.596000 −0.298000 0.954566i \(-0.596320\pi\)
−0.298000 + 0.954566i \(0.596320\pi\)
\(674\) −5.60774e11 −0.104669
\(675\) 0 0
\(676\) 1.97405e12 0.363578
\(677\) 2.32240e12 0.424901 0.212450 0.977172i \(-0.431856\pi\)
0.212450 + 0.977172i \(0.431856\pi\)
\(678\) 7.68886e11 0.139742
\(679\) −2.89876e12 −0.523357
\(680\) 0 0
\(681\) 5.11479e12 0.911309
\(682\) −2.63066e12 −0.465624
\(683\) −3.78639e12 −0.665782 −0.332891 0.942965i \(-0.608024\pi\)
−0.332891 + 0.942965i \(0.608024\pi\)
\(684\) 1.08019e11 0.0188690
\(685\) 0 0
\(686\) −1.26946e11 −0.0218858
\(687\) −3.18577e12 −0.545643
\(688\) 2.73509e12 0.465397
\(689\) 2.12283e12 0.358864
\(690\) 0 0
\(691\) −5.70532e12 −0.951982 −0.475991 0.879450i \(-0.657911\pi\)
−0.475991 + 0.879450i \(0.657911\pi\)
\(692\) 3.83900e12 0.636415
\(693\) 1.29424e12 0.213165
\(694\) 1.54695e12 0.253138
\(695\) 0 0
\(696\) −3.75585e12 −0.606690
\(697\) 6.85372e12 1.09997
\(698\) −3.03746e12 −0.484352
\(699\) −1.56732e12 −0.248318
\(700\) 0 0
\(701\) 6.25160e12 0.977823 0.488912 0.872333i \(-0.337394\pi\)
0.488912 + 0.872333i \(0.337394\pi\)
\(702\) 1.63536e12 0.254154
\(703\) −1.59450e11 −0.0246222
\(704\) −6.81818e11 −0.104614
\(705\) 0 0
\(706\) −3.47199e12 −0.525965
\(707\) −2.83807e12 −0.427205
\(708\) 1.47642e12 0.220831
\(709\) −8.01996e12 −1.19197 −0.595983 0.802997i \(-0.703238\pi\)
−0.595983 + 0.802997i \(0.703238\pi\)
\(710\) 0 0
\(711\) 4.08467e12 0.599437
\(712\) 5.73136e12 0.835791
\(713\) −2.10308e13 −3.04757
\(714\) 3.32802e11 0.0479230
\(715\) 0 0
\(716\) −7.54107e12 −1.07232
\(717\) −4.11433e12 −0.581383
\(718\) −1.46871e12 −0.206241
\(719\) 1.35002e12 0.188391 0.0941953 0.995554i \(-0.469972\pi\)
0.0941953 + 0.995554i \(0.469972\pi\)
\(720\) 0 0
\(721\) 4.74944e12 0.654537
\(722\) −2.95708e12 −0.404991
\(723\) 5.23649e12 0.712718
\(724\) −6.94238e11 −0.0939042
\(725\) 0 0
\(726\) 6.79262e11 0.0907450
\(727\) 1.47778e13 1.96203 0.981013 0.193941i \(-0.0621270\pi\)
0.981013 + 0.193941i \(0.0621270\pi\)
\(728\) 1.60198e12 0.211381
\(729\) 6.61984e11 0.0868108
\(730\) 0 0
\(731\) 4.49108e12 0.581731
\(732\) 6.28022e11 0.0808491
\(733\) 6.70116e12 0.857398 0.428699 0.903447i \(-0.358972\pi\)
0.428699 + 0.903447i \(0.358972\pi\)
\(734\) −4.71269e12 −0.599290
\(735\) 0 0
\(736\) 1.46593e13 1.84146
\(737\) −9.62686e12 −1.20193
\(738\) −4.20013e12 −0.521205
\(739\) −1.39054e13 −1.71508 −0.857540 0.514417i \(-0.828008\pi\)
−0.857540 + 0.514417i \(0.828008\pi\)
\(740\) 0 0
\(741\) −8.36137e10 −0.0101882
\(742\) −6.03953e11 −0.0731452
\(743\) 3.43953e12 0.414047 0.207024 0.978336i \(-0.433622\pi\)
0.207024 + 0.978336i \(0.433622\pi\)
\(744\) −4.63188e12 −0.554216
\(745\) 0 0
\(746\) 6.15321e11 0.0727407
\(747\) −1.44980e11 −0.0170359
\(748\) −3.45188e12 −0.403179
\(749\) 4.02955e11 0.0467830
\(750\) 0 0
\(751\) 1.00823e13 1.15660 0.578298 0.815826i \(-0.303717\pi\)
0.578298 + 0.815826i \(0.303717\pi\)
\(752\) −8.31401e11 −0.0948047
\(753\) 3.57960e12 0.405748
\(754\) −4.70529e12 −0.530170
\(755\) 0 0
\(756\) 2.36667e12 0.263506
\(757\) 1.02703e13 1.13672 0.568360 0.822780i \(-0.307578\pi\)
0.568360 + 0.822780i \(0.307578\pi\)
\(758\) 3.81052e12 0.419250
\(759\) −5.93456e12 −0.649083
\(760\) 0 0
\(761\) −3.20840e12 −0.346783 −0.173391 0.984853i \(-0.555473\pi\)
−0.173391 + 0.984853i \(0.555473\pi\)
\(762\) −1.75043e12 −0.188083
\(763\) 2.46199e12 0.262982
\(764\) −7.12855e12 −0.756974
\(765\) 0 0
\(766\) 3.22817e12 0.338787
\(767\) 4.06292e12 0.423896
\(768\) 1.21225e12 0.125738
\(769\) 1.52349e13 1.57098 0.785491 0.618872i \(-0.212410\pi\)
0.785491 + 0.618872i \(0.212410\pi\)
\(770\) 0 0
\(771\) −3.51865e12 −0.358618
\(772\) 1.03266e13 1.04635
\(773\) 8.54195e11 0.0860497 0.0430249 0.999074i \(-0.486301\pi\)
0.0430249 + 0.999074i \(0.486301\pi\)
\(774\) −2.75224e12 −0.275646
\(775\) 0 0
\(776\) −1.04073e13 −1.03029
\(777\) −1.53138e12 −0.150726
\(778\) −2.38517e12 −0.233405
\(779\) 4.89898e11 0.0476636
\(780\) 0 0
\(781\) −1.27611e13 −1.22732
\(782\) 5.42513e12 0.518776
\(783\) −1.52692e13 −1.45174
\(784\) 8.07158e11 0.0763021
\(785\) 0 0
\(786\) −1.23697e12 −0.115600
\(787\) −4.25016e12 −0.394929 −0.197465 0.980310i \(-0.563271\pi\)
−0.197465 + 0.980310i \(0.563271\pi\)
\(788\) −1.66807e12 −0.154116
\(789\) −3.82200e12 −0.351111
\(790\) 0 0
\(791\) 3.06206e12 0.278112
\(792\) 4.64665e12 0.419640
\(793\) 1.72823e12 0.155193
\(794\) −6.73503e12 −0.601378
\(795\) 0 0
\(796\) 8.02231e12 0.708256
\(797\) −2.02157e13 −1.77470 −0.887352 0.461093i \(-0.847458\pi\)
−0.887352 + 0.461093i \(0.847458\pi\)
\(798\) 2.37884e10 0.00207660
\(799\) −1.36518e12 −0.118503
\(800\) 0 0
\(801\) 1.02138e13 0.876676
\(802\) 7.41334e12 0.632746
\(803\) −7.34231e11 −0.0623179
\(804\) −7.71664e12 −0.651292
\(805\) 0 0
\(806\) −5.80278e12 −0.484315
\(807\) −3.07232e12 −0.254998
\(808\) −1.01894e13 −0.841002
\(809\) −6.65781e12 −0.546466 −0.273233 0.961948i \(-0.588093\pi\)
−0.273233 + 0.961948i \(0.588093\pi\)
\(810\) 0 0
\(811\) −9.35525e12 −0.759384 −0.379692 0.925113i \(-0.623970\pi\)
−0.379692 + 0.925113i \(0.623970\pi\)
\(812\) −6.80942e12 −0.549678
\(813\) −1.76267e12 −0.141502
\(814\) −3.12259e12 −0.249290
\(815\) 0 0
\(816\) −2.11604e12 −0.167078
\(817\) 3.21018e11 0.0252075
\(818\) −8.35671e12 −0.652597
\(819\) 2.85487e12 0.221722
\(820\) 0 0
\(821\) 1.61631e13 1.24160 0.620798 0.783971i \(-0.286809\pi\)
0.620798 + 0.783971i \(0.286809\pi\)
\(822\) 1.96265e12 0.149941
\(823\) −5.97042e12 −0.453634 −0.226817 0.973937i \(-0.572832\pi\)
−0.226817 + 0.973937i \(0.572832\pi\)
\(824\) 1.70517e13 1.28853
\(825\) 0 0
\(826\) −1.15591e12 −0.0864003
\(827\) −7.76424e12 −0.577197 −0.288599 0.957450i \(-0.593189\pi\)
−0.288599 + 0.957450i \(0.593189\pi\)
\(828\) 1.69115e13 1.25039
\(829\) −4.09585e12 −0.301195 −0.150598 0.988595i \(-0.548120\pi\)
−0.150598 + 0.988595i \(0.548120\pi\)
\(830\) 0 0
\(831\) 7.39839e12 0.538186
\(832\) −1.50397e12 −0.108814
\(833\) 1.32537e12 0.0953751
\(834\) 4.98458e12 0.356765
\(835\) 0 0
\(836\) −2.46737e11 −0.0174705
\(837\) −1.88306e13 −1.32617
\(838\) 4.53176e12 0.317445
\(839\) 4.46741e11 0.0311263 0.0155631 0.999879i \(-0.495046\pi\)
0.0155631 + 0.999879i \(0.495046\pi\)
\(840\) 0 0
\(841\) 2.94256e13 2.02835
\(842\) −1.05606e11 −0.00724079
\(843\) 3.02882e12 0.206561
\(844\) 9.43433e11 0.0639985
\(845\) 0 0
\(846\) 8.36613e11 0.0561511
\(847\) 2.70513e12 0.180598
\(848\) 3.84009e12 0.255012
\(849\) 5.21983e12 0.344803
\(850\) 0 0
\(851\) −2.49636e13 −1.63164
\(852\) −1.02290e13 −0.665049
\(853\) −2.72968e13 −1.76539 −0.882696 0.469945i \(-0.844274\pi\)
−0.882696 + 0.469945i \(0.844274\pi\)
\(854\) −4.91688e11 −0.0316322
\(855\) 0 0
\(856\) 1.44671e12 0.0920979
\(857\) −7.67571e12 −0.486077 −0.243038 0.970017i \(-0.578144\pi\)
−0.243038 + 0.970017i \(0.578144\pi\)
\(858\) −1.63745e12 −0.103151
\(859\) −1.67172e13 −1.04759 −0.523797 0.851843i \(-0.675485\pi\)
−0.523797 + 0.851843i \(0.675485\pi\)
\(860\) 0 0
\(861\) 4.70503e12 0.291775
\(862\) −8.64452e12 −0.533284
\(863\) 2.85615e13 1.75280 0.876401 0.481582i \(-0.159938\pi\)
0.876401 + 0.481582i \(0.159938\pi\)
\(864\) 1.31257e13 0.801327
\(865\) 0 0
\(866\) −9.34116e12 −0.564378
\(867\) 4.32079e12 0.259703
\(868\) −8.39769e12 −0.502135
\(869\) −9.33018e12 −0.555010
\(870\) 0 0
\(871\) −2.12351e13 −1.25018
\(872\) 8.83916e12 0.517710
\(873\) −1.85466e13 −1.08069
\(874\) 3.87783e11 0.0224795
\(875\) 0 0
\(876\) −5.88540e11 −0.0337682
\(877\) 8.40714e12 0.479899 0.239950 0.970785i \(-0.422869\pi\)
0.239950 + 0.970785i \(0.422869\pi\)
\(878\) −7.23964e12 −0.411142
\(879\) −9.28607e12 −0.524665
\(880\) 0 0
\(881\) −1.99694e13 −1.11680 −0.558398 0.829573i \(-0.688584\pi\)
−0.558398 + 0.829573i \(0.688584\pi\)
\(882\) −8.12219e11 −0.0451923
\(883\) 2.36498e13 1.30919 0.654597 0.755978i \(-0.272838\pi\)
0.654597 + 0.755978i \(0.272838\pi\)
\(884\) −7.61423e12 −0.419364
\(885\) 0 0
\(886\) −9.79250e12 −0.533878
\(887\) 3.83859e12 0.208217 0.104108 0.994566i \(-0.466801\pi\)
0.104108 + 0.994566i \(0.466801\pi\)
\(888\) −5.49804e12 −0.296722
\(889\) −6.97103e12 −0.374317
\(890\) 0 0
\(891\) 5.29627e12 0.281527
\(892\) −1.13527e13 −0.600425
\(893\) −9.75816e10 −0.00513495
\(894\) −6.45862e11 −0.0338159
\(895\) 0 0
\(896\) 7.43213e12 0.385237
\(897\) −1.30906e13 −0.675139
\(898\) −3.04216e10 −0.00156113
\(899\) 5.41798e13 2.76642
\(900\) 0 0
\(901\) 6.30551e12 0.318756
\(902\) 9.59390e12 0.482576
\(903\) 3.08309e12 0.154309
\(904\) 1.09936e13 0.547495
\(905\) 0 0
\(906\) −1.18953e12 −0.0586542
\(907\) 2.31944e13 1.13802 0.569009 0.822331i \(-0.307327\pi\)
0.569009 + 0.822331i \(0.307327\pi\)
\(908\) 3.32931e13 1.62543
\(909\) −1.81583e13 −0.882142
\(910\) 0 0
\(911\) 1.70743e13 0.821317 0.410659 0.911789i \(-0.365299\pi\)
0.410659 + 0.911789i \(0.365299\pi\)
\(912\) −1.51253e11 −0.00723980
\(913\) 3.31163e11 0.0157733
\(914\) 6.44896e12 0.305655
\(915\) 0 0
\(916\) −2.07368e13 −0.973221
\(917\) −4.92620e12 −0.230065
\(918\) 4.85757e12 0.225750
\(919\) 1.49265e13 0.690301 0.345151 0.938547i \(-0.387828\pi\)
0.345151 + 0.938547i \(0.387828\pi\)
\(920\) 0 0
\(921\) −3.67031e12 −0.168087
\(922\) −1.70768e13 −0.778246
\(923\) −2.81488e13 −1.27659
\(924\) −2.36969e12 −0.106947
\(925\) 0 0
\(926\) 9.80898e11 0.0438404
\(927\) 3.03875e13 1.35156
\(928\) −3.77653e13 −1.67158
\(929\) 2.92103e12 0.128666 0.0643332 0.997928i \(-0.479508\pi\)
0.0643332 + 0.997928i \(0.479508\pi\)
\(930\) 0 0
\(931\) 9.47362e10 0.00413278
\(932\) −1.02020e13 −0.442906
\(933\) −3.46096e12 −0.149530
\(934\) −3.79701e12 −0.163260
\(935\) 0 0
\(936\) 1.02497e13 0.436485
\(937\) 3.53996e13 1.50027 0.750135 0.661284i \(-0.229988\pi\)
0.750135 + 0.661284i \(0.229988\pi\)
\(938\) 6.04147e12 0.254818
\(939\) −1.65163e13 −0.693295
\(940\) 0 0
\(941\) −4.67286e13 −1.94280 −0.971402 0.237439i \(-0.923692\pi\)
−0.971402 + 0.237439i \(0.923692\pi\)
\(942\) −2.78310e12 −0.115159
\(943\) 7.66985e13 3.15852
\(944\) 7.34960e12 0.301224
\(945\) 0 0
\(946\) 6.28665e12 0.255217
\(947\) 5.22799e12 0.211232 0.105616 0.994407i \(-0.466319\pi\)
0.105616 + 0.994407i \(0.466319\pi\)
\(948\) −7.47882e12 −0.300743
\(949\) −1.61958e12 −0.0648195
\(950\) 0 0
\(951\) −7.69091e12 −0.304905
\(952\) 4.75842e12 0.187757
\(953\) 2.46017e13 0.966156 0.483078 0.875577i \(-0.339519\pi\)
0.483078 + 0.875577i \(0.339519\pi\)
\(954\) −3.86417e12 −0.151039
\(955\) 0 0
\(956\) −2.67809e13 −1.03697
\(957\) 1.52886e13 0.589204
\(958\) −7.83373e11 −0.0300486
\(959\) 7.81618e12 0.298408
\(960\) 0 0
\(961\) 4.03773e13 1.52715
\(962\) −6.88788e12 −0.259297
\(963\) 2.57816e12 0.0966031
\(964\) 3.40853e13 1.27122
\(965\) 0 0
\(966\) 3.72431e12 0.137610
\(967\) −1.24062e13 −0.456267 −0.228134 0.973630i \(-0.573262\pi\)
−0.228134 + 0.973630i \(0.573262\pi\)
\(968\) 9.71212e12 0.355529
\(969\) −2.48360e11 −0.00904951
\(970\) 0 0
\(971\) 5.30324e12 0.191450 0.0957248 0.995408i \(-0.469483\pi\)
0.0957248 + 0.995408i \(0.469483\pi\)
\(972\) 2.36469e13 0.849722
\(973\) 1.98509e13 0.710023
\(974\) −4.53799e12 −0.161565
\(975\) 0 0
\(976\) 3.12628e12 0.110282
\(977\) 1.45131e13 0.509606 0.254803 0.966993i \(-0.417989\pi\)
0.254803 + 0.966993i \(0.417989\pi\)
\(978\) 3.77728e12 0.132024
\(979\) −2.33302e13 −0.811702
\(980\) 0 0
\(981\) 1.57521e13 0.543035
\(982\) 1.01954e13 0.349868
\(983\) 3.61534e13 1.23498 0.617488 0.786580i \(-0.288150\pi\)
0.617488 + 0.786580i \(0.288150\pi\)
\(984\) 1.68923e13 0.574394
\(985\) 0 0
\(986\) −1.39763e13 −0.470917
\(987\) −9.37185e11 −0.0314339
\(988\) −5.44258e11 −0.0181718
\(989\) 5.02586e13 1.67043
\(990\) 0 0
\(991\) 4.94162e13 1.62756 0.813782 0.581170i \(-0.197405\pi\)
0.813782 + 0.581170i \(0.197405\pi\)
\(992\) −4.65739e13 −1.52700
\(993\) 1.65271e13 0.539417
\(994\) 8.00841e12 0.260200
\(995\) 0 0
\(996\) 2.65451e11 0.00854708
\(997\) 3.03522e13 0.972886 0.486443 0.873712i \(-0.338294\pi\)
0.486443 + 0.873712i \(0.338294\pi\)
\(998\) 7.50580e12 0.239502
\(999\) −2.23519e13 −0.710020
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 175.10.a.c.1.1 2
5.2 odd 4 175.10.b.c.99.3 4
5.3 odd 4 175.10.b.c.99.2 4
5.4 even 2 35.10.a.b.1.2 2
15.14 odd 2 315.10.a.b.1.1 2
35.34 odd 2 245.10.a.c.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.10.a.b.1.2 2 5.4 even 2
175.10.a.c.1.1 2 1.1 even 1 trivial
175.10.b.c.99.2 4 5.3 odd 4
175.10.b.c.99.3 4 5.2 odd 4
245.10.a.c.1.2 2 35.34 odd 2
315.10.a.b.1.1 2 15.14 odd 2