Properties

Label 350.10.a.j.1.2
Level $350$
Weight $10$
Character 350.1
Self dual yes
Analytic conductor $180.263$
Analytic rank $0$
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] = [350,10,Mod(1,350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(350, 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("350.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 350.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: \(180.262542657\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2305}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 14)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(24.5052\) of defining polynomial
Character \(\chi\) \(=\) 350.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{2} +247.052 q^{3} +256.000 q^{4} +3952.83 q^{6} -2401.00 q^{7} +4096.00 q^{8} +41351.7 q^{9} +O(q^{10})\) \(q+16.0000 q^{2} +247.052 q^{3} +256.000 q^{4} +3952.83 q^{6} -2401.00 q^{7} +4096.00 q^{8} +41351.7 q^{9} -27940.9 q^{11} +63245.3 q^{12} -60943.3 q^{13} -38416.0 q^{14} +65536.0 q^{16} +358867. q^{17} +661628. q^{18} -391593. q^{19} -593172. q^{21} -447055. q^{22} +302894. q^{23} +1.01193e6 q^{24} -975093. q^{26} +5.35330e6 q^{27} -614656. q^{28} +6.73993e6 q^{29} +2.98262e6 q^{31} +1.04858e6 q^{32} -6.90287e6 q^{33} +5.74188e6 q^{34} +1.05860e7 q^{36} +3.49582e6 q^{37} -6.26549e6 q^{38} -1.50562e7 q^{39} +3.43724e7 q^{41} -9.49075e6 q^{42} +1.45085e7 q^{43} -7.15288e6 q^{44} +4.84631e6 q^{46} +2.76485e7 q^{47} +1.61908e7 q^{48} +5.76480e6 q^{49} +8.86589e7 q^{51} -1.56015e7 q^{52} +2.39217e7 q^{53} +8.56529e7 q^{54} -9.83450e6 q^{56} -9.67439e7 q^{57} +1.07839e8 q^{58} -1.20580e8 q^{59} +7.23140e7 q^{61} +4.77219e7 q^{62} -9.92855e7 q^{63} +1.67772e7 q^{64} -1.10446e8 q^{66} -8.70377e7 q^{67} +9.18701e7 q^{68} +7.48307e7 q^{69} +2.19622e8 q^{71} +1.69377e8 q^{72} -2.67792e8 q^{73} +5.59331e7 q^{74} -1.00248e8 q^{76} +6.70862e7 q^{77} -2.40899e8 q^{78} +2.85350e7 q^{79} +5.08619e8 q^{81} +5.49959e8 q^{82} +3.83237e8 q^{83} -1.51852e8 q^{84} +2.32136e8 q^{86} +1.66511e9 q^{87} -1.14446e8 q^{88} +7.21581e8 q^{89} +1.46325e8 q^{91} +7.75410e7 q^{92} +7.36861e8 q^{93} +4.42377e8 q^{94} +2.59053e8 q^{96} +6.73736e8 q^{97} +9.22368e7 q^{98} -1.15541e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 32 q^{2} + 14 q^{3} + 512 q^{4} + 224 q^{6} - 4802 q^{7} + 8192 q^{8} + 75982 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 32 q^{2} + 14 q^{3} + 512 q^{4} + 224 q^{6} - 4802 q^{7} + 8192 q^{8} + 75982 q^{9} + 44940 q^{11} + 3584 q^{12} - 100282 q^{13} - 76832 q^{14} + 131072 q^{16} + 870408 q^{17} + 1215712 q^{18} + 508774 q^{19} - 33614 q^{21} + 719040 q^{22} - 79800 q^{23} + 57344 q^{24} - 1604512 q^{26} + 1869812 q^{27} - 1229312 q^{28} + 2006328 q^{29} + 2188732 q^{31} + 2097152 q^{32} - 23887920 q^{33} + 13926528 q^{34} + 19451392 q^{36} + 20723576 q^{37} + 8140384 q^{38} - 5888224 q^{39} + 19016592 q^{41} - 537824 q^{42} - 4193716 q^{43} + 11504640 q^{44} - 1276800 q^{46} + 74542524 q^{47} + 917504 q^{48} + 11529602 q^{49} - 30556644 q^{51} - 25672192 q^{52} + 3239748 q^{53} + 29916992 q^{54} - 19668992 q^{56} - 306576332 q^{57} + 32101248 q^{58} - 133642362 q^{59} + 227801686 q^{61} + 35019712 q^{62} - 182432782 q^{63} + 33554432 q^{64} - 382206720 q^{66} - 332930272 q^{67} + 222824448 q^{68} + 164018400 q^{69} - 167985720 q^{71} + 311222272 q^{72} + 44684276 q^{73} + 331577216 q^{74} + 130246144 q^{76} - 107900940 q^{77} - 94211584 q^{78} + 269642776 q^{79} + 638826478 q^{81} + 304265472 q^{82} + 183105762 q^{83} - 8605184 q^{84} - 67099456 q^{86} + 2768288796 q^{87} + 184074240 q^{88} + 791657748 q^{89} + 240777082 q^{91} - 20428800 q^{92} + 921877624 q^{93} + 1192680384 q^{94} + 14680064 q^{96} + 4169480 q^{97} + 184473632 q^{98} + 1368480540 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 0.707107
\(3\) 247.052 1.76093 0.880467 0.474108i \(-0.157229\pi\)
0.880467 + 0.474108i \(0.157229\pi\)
\(4\) 256.000 0.500000
\(5\) 0 0
\(6\) 3952.83 1.24517
\(7\) −2401.00 −0.377964
\(8\) 4096.00 0.353553
\(9\) 41351.7 2.10089
\(10\) 0 0
\(11\) −27940.9 −0.575405 −0.287703 0.957720i \(-0.592891\pi\)
−0.287703 + 0.957720i \(0.592891\pi\)
\(12\) 63245.3 0.880467
\(13\) −60943.3 −0.591808 −0.295904 0.955218i \(-0.595621\pi\)
−0.295904 + 0.955218i \(0.595621\pi\)
\(14\) −38416.0 −0.267261
\(15\) 0 0
\(16\) 65536.0 0.250000
\(17\) 358867. 1.04211 0.521055 0.853523i \(-0.325538\pi\)
0.521055 + 0.853523i \(0.325538\pi\)
\(18\) 661628. 1.48555
\(19\) −391593. −0.689356 −0.344678 0.938721i \(-0.612012\pi\)
−0.344678 + 0.938721i \(0.612012\pi\)
\(20\) 0 0
\(21\) −593172. −0.665570
\(22\) −447055. −0.406873
\(23\) 302894. 0.225692 0.112846 0.993612i \(-0.464003\pi\)
0.112846 + 0.993612i \(0.464003\pi\)
\(24\) 1.01193e6 0.622584
\(25\) 0 0
\(26\) −975093. −0.418472
\(27\) 5.35330e6 1.93859
\(28\) −614656. −0.188982
\(29\) 6.73993e6 1.76956 0.884778 0.466013i \(-0.154310\pi\)
0.884778 + 0.466013i \(0.154310\pi\)
\(30\) 0 0
\(31\) 2.98262e6 0.580056 0.290028 0.957018i \(-0.406335\pi\)
0.290028 + 0.957018i \(0.406335\pi\)
\(32\) 1.04858e6 0.176777
\(33\) −6.90287e6 −1.01325
\(34\) 5.74188e6 0.736884
\(35\) 0 0
\(36\) 1.05860e7 1.05044
\(37\) 3.49582e6 0.306649 0.153324 0.988176i \(-0.451002\pi\)
0.153324 + 0.988176i \(0.451002\pi\)
\(38\) −6.26549e6 −0.487449
\(39\) −1.50562e7 −1.04214
\(40\) 0 0
\(41\) 3.43724e7 1.89969 0.949845 0.312722i \(-0.101241\pi\)
0.949845 + 0.312722i \(0.101241\pi\)
\(42\) −9.49075e6 −0.470629
\(43\) 1.45085e7 0.647164 0.323582 0.946200i \(-0.395113\pi\)
0.323582 + 0.946200i \(0.395113\pi\)
\(44\) −7.15288e6 −0.287703
\(45\) 0 0
\(46\) 4.84631e6 0.159588
\(47\) 2.76485e7 0.826479 0.413239 0.910622i \(-0.364397\pi\)
0.413239 + 0.910622i \(0.364397\pi\)
\(48\) 1.61908e7 0.440233
\(49\) 5.76480e6 0.142857
\(50\) 0 0
\(51\) 8.86589e7 1.83509
\(52\) −1.56015e7 −0.295904
\(53\) 2.39217e7 0.416438 0.208219 0.978082i \(-0.433233\pi\)
0.208219 + 0.978082i \(0.433233\pi\)
\(54\) 8.56529e7 1.37079
\(55\) 0 0
\(56\) −9.83450e6 −0.133631
\(57\) −9.67439e7 −1.21391
\(58\) 1.07839e8 1.25127
\(59\) −1.20580e8 −1.29551 −0.647756 0.761848i \(-0.724293\pi\)
−0.647756 + 0.761848i \(0.724293\pi\)
\(60\) 0 0
\(61\) 7.23140e7 0.668710 0.334355 0.942447i \(-0.391482\pi\)
0.334355 + 0.942447i \(0.391482\pi\)
\(62\) 4.77219e7 0.410161
\(63\) −9.92855e7 −0.794060
\(64\) 1.67772e7 0.125000
\(65\) 0 0
\(66\) −1.10446e8 −0.716476
\(67\) −8.70377e7 −0.527680 −0.263840 0.964566i \(-0.584989\pi\)
−0.263840 + 0.964566i \(0.584989\pi\)
\(68\) 9.18701e7 0.521055
\(69\) 7.48307e7 0.397428
\(70\) 0 0
\(71\) 2.19622e8 1.02568 0.512842 0.858483i \(-0.328593\pi\)
0.512842 + 0.858483i \(0.328593\pi\)
\(72\) 1.69377e8 0.742775
\(73\) −2.67792e8 −1.10368 −0.551841 0.833949i \(-0.686074\pi\)
−0.551841 + 0.833949i \(0.686074\pi\)
\(74\) 5.59331e7 0.216833
\(75\) 0 0
\(76\) −1.00248e8 −0.344678
\(77\) 6.70862e7 0.217483
\(78\) −2.40899e8 −0.736901
\(79\) 2.85350e7 0.0824243 0.0412122 0.999150i \(-0.486878\pi\)
0.0412122 + 0.999150i \(0.486878\pi\)
\(80\) 0 0
\(81\) 5.08619e8 1.31283
\(82\) 5.49959e8 1.34328
\(83\) 3.83237e8 0.886372 0.443186 0.896430i \(-0.353848\pi\)
0.443186 + 0.896430i \(0.353848\pi\)
\(84\) −1.51852e8 −0.332785
\(85\) 0 0
\(86\) 2.32136e8 0.457614
\(87\) 1.66511e9 3.11607
\(88\) −1.14446e8 −0.203437
\(89\) 7.21581e8 1.21907 0.609537 0.792757i \(-0.291355\pi\)
0.609537 + 0.792757i \(0.291355\pi\)
\(90\) 0 0
\(91\) 1.46325e8 0.223683
\(92\) 7.75410e7 0.112846
\(93\) 7.36861e8 1.02144
\(94\) 4.42377e8 0.584409
\(95\) 0 0
\(96\) 2.59053e8 0.311292
\(97\) 6.73736e8 0.772710 0.386355 0.922350i \(-0.373734\pi\)
0.386355 + 0.922350i \(0.373734\pi\)
\(98\) 9.22368e7 0.101015
\(99\) −1.15541e9 −1.20886
\(100\) 0 0
\(101\) −1.04805e7 −0.0100216 −0.00501081 0.999987i \(-0.501595\pi\)
−0.00501081 + 0.999987i \(0.501595\pi\)
\(102\) 1.41854e9 1.29760
\(103\) −1.39782e9 −1.22373 −0.611864 0.790963i \(-0.709580\pi\)
−0.611864 + 0.790963i \(0.709580\pi\)
\(104\) −2.49624e8 −0.209236
\(105\) 0 0
\(106\) 3.82747e8 0.294466
\(107\) 4.84414e8 0.357264 0.178632 0.983916i \(-0.442833\pi\)
0.178632 + 0.983916i \(0.442833\pi\)
\(108\) 1.37045e9 0.969293
\(109\) −1.96298e8 −0.133198 −0.0665989 0.997780i \(-0.521215\pi\)
−0.0665989 + 0.997780i \(0.521215\pi\)
\(110\) 0 0
\(111\) 8.63649e8 0.539988
\(112\) −1.57352e8 −0.0944911
\(113\) 5.80849e8 0.335128 0.167564 0.985861i \(-0.446410\pi\)
0.167564 + 0.985861i \(0.446410\pi\)
\(114\) −1.54790e9 −0.858364
\(115\) 0 0
\(116\) 1.72542e9 0.884778
\(117\) −2.52011e9 −1.24332
\(118\) −1.92928e9 −0.916066
\(119\) −8.61641e8 −0.393881
\(120\) 0 0
\(121\) −1.57725e9 −0.668909
\(122\) 1.15702e9 0.472850
\(123\) 8.49178e9 3.34523
\(124\) 7.63550e8 0.290028
\(125\) 0 0
\(126\) −1.58857e9 −0.561485
\(127\) −2.59088e9 −0.883751 −0.441876 0.897076i \(-0.645687\pi\)
−0.441876 + 0.897076i \(0.645687\pi\)
\(128\) 2.68435e8 0.0883883
\(129\) 3.58436e9 1.13961
\(130\) 0 0
\(131\) 2.45598e9 0.728626 0.364313 0.931277i \(-0.381304\pi\)
0.364313 + 0.931277i \(0.381304\pi\)
\(132\) −1.76713e9 −0.506625
\(133\) 9.40215e8 0.260552
\(134\) −1.39260e9 −0.373126
\(135\) 0 0
\(136\) 1.46992e9 0.368442
\(137\) −6.08364e9 −1.47544 −0.737719 0.675108i \(-0.764097\pi\)
−0.737719 + 0.675108i \(0.764097\pi\)
\(138\) 1.19729e9 0.281024
\(139\) 2.25249e9 0.511796 0.255898 0.966704i \(-0.417629\pi\)
0.255898 + 0.966704i \(0.417629\pi\)
\(140\) 0 0
\(141\) 6.83063e9 1.45537
\(142\) 3.51395e9 0.725268
\(143\) 1.70281e9 0.340530
\(144\) 2.71003e9 0.525221
\(145\) 0 0
\(146\) −4.28466e9 −0.780421
\(147\) 1.42421e9 0.251562
\(148\) 8.94930e8 0.153324
\(149\) −3.13517e9 −0.521103 −0.260551 0.965460i \(-0.583904\pi\)
−0.260551 + 0.965460i \(0.583904\pi\)
\(150\) 0 0
\(151\) −6.20938e9 −0.971969 −0.485984 0.873968i \(-0.661539\pi\)
−0.485984 + 0.873968i \(0.661539\pi\)
\(152\) −1.60397e9 −0.243724
\(153\) 1.48398e10 2.18936
\(154\) 1.07338e9 0.153784
\(155\) 0 0
\(156\) −3.85438e9 −0.521068
\(157\) −1.33378e10 −1.75201 −0.876006 0.482300i \(-0.839802\pi\)
−0.876006 + 0.482300i \(0.839802\pi\)
\(158\) 4.56560e8 0.0582828
\(159\) 5.90990e9 0.733319
\(160\) 0 0
\(161\) −7.27249e8 −0.0853035
\(162\) 8.13790e9 0.928314
\(163\) −7.33621e9 −0.814006 −0.407003 0.913427i \(-0.633426\pi\)
−0.407003 + 0.913427i \(0.633426\pi\)
\(164\) 8.79934e9 0.949845
\(165\) 0 0
\(166\) 6.13179e9 0.626760
\(167\) 6.42205e9 0.638925 0.319462 0.947599i \(-0.396498\pi\)
0.319462 + 0.947599i \(0.396498\pi\)
\(168\) −2.42963e9 −0.235315
\(169\) −6.89041e9 −0.649763
\(170\) 0 0
\(171\) −1.61931e10 −1.44826
\(172\) 3.71418e9 0.323582
\(173\) −1.91846e10 −1.62834 −0.814171 0.580626i \(-0.802808\pi\)
−0.814171 + 0.580626i \(0.802808\pi\)
\(174\) 2.66418e10 2.20339
\(175\) 0 0
\(176\) −1.83114e9 −0.143851
\(177\) −2.97896e10 −2.28131
\(178\) 1.15453e10 0.862016
\(179\) 1.53377e10 1.11666 0.558330 0.829619i \(-0.311442\pi\)
0.558330 + 0.829619i \(0.311442\pi\)
\(180\) 0 0
\(181\) −1.73475e10 −1.20139 −0.600694 0.799479i \(-0.705109\pi\)
−0.600694 + 0.799479i \(0.705109\pi\)
\(182\) 2.34120e9 0.158167
\(183\) 1.78653e10 1.17755
\(184\) 1.24066e9 0.0797941
\(185\) 0 0
\(186\) 1.17898e10 0.722267
\(187\) −1.00271e10 −0.599636
\(188\) 7.07802e9 0.413239
\(189\) −1.28533e10 −0.732717
\(190\) 0 0
\(191\) 2.70138e10 1.46871 0.734355 0.678765i \(-0.237485\pi\)
0.734355 + 0.678765i \(0.237485\pi\)
\(192\) 4.14485e9 0.220117
\(193\) −2.70232e9 −0.140194 −0.0700970 0.997540i \(-0.522331\pi\)
−0.0700970 + 0.997540i \(0.522331\pi\)
\(194\) 1.07798e10 0.546389
\(195\) 0 0
\(196\) 1.47579e9 0.0714286
\(197\) −2.39047e10 −1.13080 −0.565399 0.824818i \(-0.691278\pi\)
−0.565399 + 0.824818i \(0.691278\pi\)
\(198\) −1.84865e10 −0.854794
\(199\) −2.16111e9 −0.0976872 −0.0488436 0.998806i \(-0.515554\pi\)
−0.0488436 + 0.998806i \(0.515554\pi\)
\(200\) 0 0
\(201\) −2.15028e10 −0.929209
\(202\) −1.67689e8 −0.00708635
\(203\) −1.61826e10 −0.668829
\(204\) 2.26967e10 0.917544
\(205\) 0 0
\(206\) −2.23652e10 −0.865306
\(207\) 1.25252e10 0.474153
\(208\) −3.99398e9 −0.147952
\(209\) 1.09415e10 0.396659
\(210\) 0 0
\(211\) 3.61055e9 0.125401 0.0627007 0.998032i \(-0.480029\pi\)
0.0627007 + 0.998032i \(0.480029\pi\)
\(212\) 6.12395e9 0.208219
\(213\) 5.42581e10 1.80616
\(214\) 7.75062e9 0.252624
\(215\) 0 0
\(216\) 2.19271e10 0.685394
\(217\) −7.16126e9 −0.219240
\(218\) −3.14077e9 −0.0941851
\(219\) −6.61585e10 −1.94351
\(220\) 0 0
\(221\) −2.18706e10 −0.616730
\(222\) 1.38184e10 0.381829
\(223\) 7.13001e10 1.93072 0.965358 0.260929i \(-0.0840287\pi\)
0.965358 + 0.260929i \(0.0840287\pi\)
\(224\) −2.51763e9 −0.0668153
\(225\) 0 0
\(226\) 9.29359e9 0.236971
\(227\) −7.15361e10 −1.78817 −0.894086 0.447896i \(-0.852173\pi\)
−0.894086 + 0.447896i \(0.852173\pi\)
\(228\) −2.47664e10 −0.606955
\(229\) −3.56020e10 −0.855491 −0.427745 0.903899i \(-0.640692\pi\)
−0.427745 + 0.903899i \(0.640692\pi\)
\(230\) 0 0
\(231\) 1.65738e10 0.382973
\(232\) 2.76067e10 0.625633
\(233\) 3.80069e10 0.844814 0.422407 0.906406i \(-0.361185\pi\)
0.422407 + 0.906406i \(0.361185\pi\)
\(234\) −4.03218e10 −0.879161
\(235\) 0 0
\(236\) −3.08685e10 −0.647756
\(237\) 7.04962e9 0.145144
\(238\) −1.37863e10 −0.278516
\(239\) −8.67126e10 −1.71906 −0.859531 0.511083i \(-0.829244\pi\)
−0.859531 + 0.511083i \(0.829244\pi\)
\(240\) 0 0
\(241\) −8.18418e9 −0.156278 −0.0781391 0.996942i \(-0.524898\pi\)
−0.0781391 + 0.996942i \(0.524898\pi\)
\(242\) −2.52360e10 −0.472990
\(243\) 2.02863e10 0.373228
\(244\) 1.85124e10 0.334355
\(245\) 0 0
\(246\) 1.35868e11 2.36543
\(247\) 2.38650e10 0.407967
\(248\) 1.22168e10 0.205081
\(249\) 9.46795e10 1.56084
\(250\) 0 0
\(251\) 9.75467e10 1.55125 0.775624 0.631196i \(-0.217435\pi\)
0.775624 + 0.631196i \(0.217435\pi\)
\(252\) −2.54171e10 −0.397030
\(253\) −8.46315e9 −0.129864
\(254\) −4.14540e10 −0.624907
\(255\) 0 0
\(256\) 4.29497e9 0.0625000
\(257\) 4.43042e9 0.0633499 0.0316750 0.999498i \(-0.489916\pi\)
0.0316750 + 0.999498i \(0.489916\pi\)
\(258\) 5.73497e10 0.805828
\(259\) −8.39346e9 −0.115902
\(260\) 0 0
\(261\) 2.78708e11 3.71763
\(262\) 3.92957e10 0.515216
\(263\) 1.20620e11 1.55460 0.777301 0.629129i \(-0.216588\pi\)
0.777301 + 0.629129i \(0.216588\pi\)
\(264\) −2.82741e10 −0.358238
\(265\) 0 0
\(266\) 1.50434e10 0.184238
\(267\) 1.78268e11 2.14671
\(268\) −2.22816e10 −0.263840
\(269\) 7.59025e10 0.883835 0.441917 0.897056i \(-0.354298\pi\)
0.441917 + 0.897056i \(0.354298\pi\)
\(270\) 0 0
\(271\) 7.11397e10 0.801217 0.400608 0.916249i \(-0.368799\pi\)
0.400608 + 0.916249i \(0.368799\pi\)
\(272\) 2.35187e10 0.260528
\(273\) 3.61499e10 0.393890
\(274\) −9.73382e10 −1.04329
\(275\) 0 0
\(276\) 1.91567e10 0.198714
\(277\) −8.61542e10 −0.879261 −0.439630 0.898179i \(-0.644891\pi\)
−0.439630 + 0.898179i \(0.644891\pi\)
\(278\) 3.60399e10 0.361894
\(279\) 1.23336e11 1.21863
\(280\) 0 0
\(281\) −1.00179e11 −0.958511 −0.479256 0.877675i \(-0.659093\pi\)
−0.479256 + 0.877675i \(0.659093\pi\)
\(282\) 1.09290e11 1.02910
\(283\) 4.57444e10 0.423935 0.211967 0.977277i \(-0.432013\pi\)
0.211967 + 0.977277i \(0.432013\pi\)
\(284\) 5.62233e10 0.512842
\(285\) 0 0
\(286\) 2.72450e10 0.240791
\(287\) −8.25282e10 −0.718015
\(288\) 4.33604e10 0.371388
\(289\) 1.01980e10 0.0859950
\(290\) 0 0
\(291\) 1.66448e11 1.36069
\(292\) −6.85546e10 −0.551841
\(293\) 1.01615e10 0.0805476 0.0402738 0.999189i \(-0.487177\pi\)
0.0402738 + 0.999189i \(0.487177\pi\)
\(294\) 2.27873e10 0.177881
\(295\) 0 0
\(296\) 1.43189e10 0.108417
\(297\) −1.49576e11 −1.11547
\(298\) −5.01628e10 −0.368475
\(299\) −1.84594e10 −0.133566
\(300\) 0 0
\(301\) −3.48349e10 −0.244605
\(302\) −9.93501e10 −0.687286
\(303\) −2.58924e9 −0.0176474
\(304\) −2.56634e10 −0.172339
\(305\) 0 0
\(306\) 2.37437e11 1.54811
\(307\) 3.68957e10 0.237057 0.118529 0.992951i \(-0.462182\pi\)
0.118529 + 0.992951i \(0.462182\pi\)
\(308\) 1.71741e10 0.108741
\(309\) −3.45335e11 −2.15490
\(310\) 0 0
\(311\) 1.88558e11 1.14294 0.571469 0.820624i \(-0.306374\pi\)
0.571469 + 0.820624i \(0.306374\pi\)
\(312\) −6.16701e10 −0.368450
\(313\) −1.31778e11 −0.776058 −0.388029 0.921647i \(-0.626844\pi\)
−0.388029 + 0.921647i \(0.626844\pi\)
\(314\) −2.13405e11 −1.23886
\(315\) 0 0
\(316\) 7.30495e9 0.0412122
\(317\) 1.50686e11 0.838118 0.419059 0.907959i \(-0.362360\pi\)
0.419059 + 0.907959i \(0.362360\pi\)
\(318\) 9.45584e10 0.518535
\(319\) −1.88320e11 −1.01821
\(320\) 0 0
\(321\) 1.19675e11 0.629118
\(322\) −1.16360e10 −0.0603187
\(323\) −1.40530e11 −0.718386
\(324\) 1.30206e11 0.656417
\(325\) 0 0
\(326\) −1.17379e11 −0.575589
\(327\) −4.84959e10 −0.234552
\(328\) 1.40789e11 0.671642
\(329\) −6.63841e10 −0.312380
\(330\) 0 0
\(331\) 3.38877e10 0.155173 0.0775865 0.996986i \(-0.475279\pi\)
0.0775865 + 0.996986i \(0.475279\pi\)
\(332\) 9.81087e10 0.443186
\(333\) 1.44558e11 0.644234
\(334\) 1.02753e11 0.451788
\(335\) 0 0
\(336\) −3.88741e10 −0.166393
\(337\) −1.98312e11 −0.837555 −0.418778 0.908089i \(-0.637541\pi\)
−0.418778 + 0.908089i \(0.637541\pi\)
\(338\) −1.10247e11 −0.459452
\(339\) 1.43500e11 0.590138
\(340\) 0 0
\(341\) −8.33371e10 −0.333767
\(342\) −2.59089e11 −1.02407
\(343\) −1.38413e10 −0.0539949
\(344\) 5.94268e10 0.228807
\(345\) 0 0
\(346\) −3.06954e11 −1.15141
\(347\) 1.71762e11 0.635981 0.317990 0.948094i \(-0.396992\pi\)
0.317990 + 0.948094i \(0.396992\pi\)
\(348\) 4.26269e11 1.55803
\(349\) −1.88189e11 −0.679014 −0.339507 0.940603i \(-0.610260\pi\)
−0.339507 + 0.940603i \(0.610260\pi\)
\(350\) 0 0
\(351\) −3.26248e11 −1.14727
\(352\) −2.92982e10 −0.101718
\(353\) −1.97995e11 −0.678686 −0.339343 0.940663i \(-0.610205\pi\)
−0.339343 + 0.940663i \(0.610205\pi\)
\(354\) −4.76633e11 −1.61313
\(355\) 0 0
\(356\) 1.84725e11 0.609537
\(357\) −2.12870e11 −0.693598
\(358\) 2.45403e11 0.789598
\(359\) 4.34669e11 1.38113 0.690563 0.723272i \(-0.257363\pi\)
0.690563 + 0.723272i \(0.257363\pi\)
\(360\) 0 0
\(361\) −1.69343e11 −0.524788
\(362\) −2.77560e11 −0.849510
\(363\) −3.89663e11 −1.17790
\(364\) 3.74592e10 0.111841
\(365\) 0 0
\(366\) 2.85845e11 0.832657
\(367\) −6.02066e10 −0.173240 −0.0866198 0.996241i \(-0.527607\pi\)
−0.0866198 + 0.996241i \(0.527607\pi\)
\(368\) 1.98505e10 0.0564230
\(369\) 1.42136e12 3.99103
\(370\) 0 0
\(371\) −5.74359e10 −0.157399
\(372\) 1.88637e11 0.510720
\(373\) 6.85521e11 1.83371 0.916855 0.399219i \(-0.130719\pi\)
0.916855 + 0.399219i \(0.130719\pi\)
\(374\) −1.60433e11 −0.424007
\(375\) 0 0
\(376\) 1.13248e11 0.292204
\(377\) −4.10754e11 −1.04724
\(378\) −2.05653e11 −0.518109
\(379\) 5.05522e11 1.25853 0.629266 0.777190i \(-0.283356\pi\)
0.629266 + 0.777190i \(0.283356\pi\)
\(380\) 0 0
\(381\) −6.40082e11 −1.55623
\(382\) 4.32221e11 1.03854
\(383\) −2.02054e11 −0.479814 −0.239907 0.970796i \(-0.577117\pi\)
−0.239907 + 0.970796i \(0.577117\pi\)
\(384\) 6.63175e10 0.155646
\(385\) 0 0
\(386\) −4.32371e10 −0.0991321
\(387\) 5.99952e11 1.35962
\(388\) 1.72476e11 0.386355
\(389\) 5.51954e11 1.22216 0.611082 0.791567i \(-0.290735\pi\)
0.611082 + 0.791567i \(0.290735\pi\)
\(390\) 0 0
\(391\) 1.08699e11 0.235196
\(392\) 2.36126e10 0.0505076
\(393\) 6.06756e11 1.28306
\(394\) −3.82475e11 −0.799595
\(395\) 0 0
\(396\) −2.95784e11 −0.604430
\(397\) −1.26816e11 −0.256222 −0.128111 0.991760i \(-0.540891\pi\)
−0.128111 + 0.991760i \(0.540891\pi\)
\(398\) −3.45777e10 −0.0690753
\(399\) 2.32282e11 0.458815
\(400\) 0 0
\(401\) 5.86957e11 1.13359 0.566795 0.823859i \(-0.308183\pi\)
0.566795 + 0.823859i \(0.308183\pi\)
\(402\) −3.44045e11 −0.657050
\(403\) −1.81771e11 −0.343282
\(404\) −2.68302e9 −0.00501081
\(405\) 0 0
\(406\) −2.58921e11 −0.472934
\(407\) −9.76764e10 −0.176447
\(408\) 3.63147e11 0.648801
\(409\) −9.08262e10 −0.160493 −0.0802465 0.996775i \(-0.525571\pi\)
−0.0802465 + 0.996775i \(0.525571\pi\)
\(410\) 0 0
\(411\) −1.50298e12 −2.59815
\(412\) −3.57843e11 −0.611864
\(413\) 2.89513e11 0.489658
\(414\) 2.00403e11 0.335277
\(415\) 0 0
\(416\) −6.39037e10 −0.104618
\(417\) 5.56483e11 0.901238
\(418\) 1.75064e11 0.280481
\(419\) −8.34669e11 −1.32297 −0.661487 0.749957i \(-0.730074\pi\)
−0.661487 + 0.749957i \(0.730074\pi\)
\(420\) 0 0
\(421\) −1.61360e10 −0.0250337 −0.0125169 0.999922i \(-0.503984\pi\)
−0.0125169 + 0.999922i \(0.503984\pi\)
\(422\) 5.77688e10 0.0886722
\(423\) 1.14331e12 1.73634
\(424\) 9.79832e10 0.147233
\(425\) 0 0
\(426\) 8.68130e11 1.27715
\(427\) −1.73626e11 −0.252749
\(428\) 1.24010e11 0.178632
\(429\) 4.20684e11 0.599650
\(430\) 0 0
\(431\) −4.81935e11 −0.672730 −0.336365 0.941732i \(-0.609198\pi\)
−0.336365 + 0.941732i \(0.609198\pi\)
\(432\) 3.50834e11 0.484646
\(433\) −6.68314e11 −0.913661 −0.456830 0.889554i \(-0.651015\pi\)
−0.456830 + 0.889554i \(0.651015\pi\)
\(434\) −1.14580e11 −0.155026
\(435\) 0 0
\(436\) −5.02523e10 −0.0665989
\(437\) −1.18611e11 −0.155582
\(438\) −1.05854e12 −1.37427
\(439\) −4.79819e11 −0.616577 −0.308288 0.951293i \(-0.599756\pi\)
−0.308288 + 0.951293i \(0.599756\pi\)
\(440\) 0 0
\(441\) 2.38384e11 0.300126
\(442\) −3.49929e11 −0.436094
\(443\) 6.53598e11 0.806295 0.403148 0.915135i \(-0.367916\pi\)
0.403148 + 0.915135i \(0.367916\pi\)
\(444\) 2.21094e11 0.269994
\(445\) 0 0
\(446\) 1.14080e12 1.36522
\(447\) −7.74551e11 −0.917627
\(448\) −4.02821e10 −0.0472456
\(449\) −1.88660e11 −0.219064 −0.109532 0.993983i \(-0.534935\pi\)
−0.109532 + 0.993983i \(0.534935\pi\)
\(450\) 0 0
\(451\) −9.60397e11 −1.09309
\(452\) 1.48697e11 0.167564
\(453\) −1.53404e12 −1.71157
\(454\) −1.14458e12 −1.26443
\(455\) 0 0
\(456\) −3.96263e11 −0.429182
\(457\) −1.31205e12 −1.40711 −0.703556 0.710639i \(-0.748406\pi\)
−0.703556 + 0.710639i \(0.748406\pi\)
\(458\) −5.69633e11 −0.604923
\(459\) 1.92113e12 2.02022
\(460\) 0 0
\(461\) 1.16594e12 1.20233 0.601165 0.799125i \(-0.294703\pi\)
0.601165 + 0.799125i \(0.294703\pi\)
\(462\) 2.65181e11 0.270803
\(463\) 3.87318e10 0.0391700 0.0195850 0.999808i \(-0.493766\pi\)
0.0195850 + 0.999808i \(0.493766\pi\)
\(464\) 4.41708e11 0.442389
\(465\) 0 0
\(466\) 6.08111e11 0.597374
\(467\) −1.17150e11 −0.113977 −0.0569883 0.998375i \(-0.518150\pi\)
−0.0569883 + 0.998375i \(0.518150\pi\)
\(468\) −6.45149e11 −0.621661
\(469\) 2.08977e11 0.199444
\(470\) 0 0
\(471\) −3.29514e12 −3.08518
\(472\) −4.93896e11 −0.458033
\(473\) −4.05381e11 −0.372382
\(474\) 1.12794e11 0.102632
\(475\) 0 0
\(476\) −2.20580e11 −0.196940
\(477\) 9.89203e11 0.874888
\(478\) −1.38740e12 −1.21556
\(479\) −1.94734e12 −1.69018 −0.845089 0.534626i \(-0.820452\pi\)
−0.845089 + 0.534626i \(0.820452\pi\)
\(480\) 0 0
\(481\) −2.13047e11 −0.181477
\(482\) −1.30947e11 −0.110505
\(483\) −1.79668e11 −0.150214
\(484\) −4.03776e11 −0.334454
\(485\) 0 0
\(486\) 3.24580e11 0.263912
\(487\) 1.95983e12 1.57884 0.789421 0.613852i \(-0.210381\pi\)
0.789421 + 0.613852i \(0.210381\pi\)
\(488\) 2.96198e11 0.236425
\(489\) −1.81243e12 −1.43341
\(490\) 0 0
\(491\) −2.93589e11 −0.227967 −0.113984 0.993483i \(-0.536361\pi\)
−0.113984 + 0.993483i \(0.536361\pi\)
\(492\) 2.17389e12 1.67261
\(493\) 2.41874e12 1.84407
\(494\) 3.81840e11 0.288476
\(495\) 0 0
\(496\) 1.95469e11 0.145014
\(497\) −5.27313e11 −0.387672
\(498\) 1.51487e12 1.10368
\(499\) 1.71390e12 1.23746 0.618732 0.785602i \(-0.287647\pi\)
0.618732 + 0.785602i \(0.287647\pi\)
\(500\) 0 0
\(501\) 1.58658e12 1.12510
\(502\) 1.56075e12 1.09690
\(503\) −1.61385e12 −1.12411 −0.562053 0.827101i \(-0.689988\pi\)
−0.562053 + 0.827101i \(0.689988\pi\)
\(504\) −4.06673e11 −0.280743
\(505\) 0 0
\(506\) −1.35410e11 −0.0918279
\(507\) −1.70229e12 −1.14419
\(508\) −6.63265e11 −0.441876
\(509\) 1.17088e12 0.773181 0.386590 0.922252i \(-0.373653\pi\)
0.386590 + 0.922252i \(0.373653\pi\)
\(510\) 0 0
\(511\) 6.42967e11 0.417153
\(512\) 6.87195e10 0.0441942
\(513\) −2.09632e12 −1.33638
\(514\) 7.08868e10 0.0447952
\(515\) 0 0
\(516\) 9.17595e11 0.569807
\(517\) −7.72526e11 −0.475560
\(518\) −1.34295e11 −0.0819553
\(519\) −4.73960e12 −2.86740
\(520\) 0 0
\(521\) 2.91339e12 1.73232 0.866161 0.499765i \(-0.166580\pi\)
0.866161 + 0.499765i \(0.166580\pi\)
\(522\) 4.45932e12 2.62876
\(523\) −2.53787e12 −1.48324 −0.741620 0.670820i \(-0.765942\pi\)
−0.741620 + 0.670820i \(0.765942\pi\)
\(524\) 6.28732e11 0.364313
\(525\) 0 0
\(526\) 1.92992e12 1.09927
\(527\) 1.07036e12 0.604482
\(528\) −4.52386e11 −0.253313
\(529\) −1.70941e12 −0.949063
\(530\) 0 0
\(531\) −4.98620e12 −2.72172
\(532\) 2.40695e11 0.130276
\(533\) −2.09477e12 −1.12425
\(534\) 2.85229e12 1.51795
\(535\) 0 0
\(536\) −3.56506e11 −0.186563
\(537\) 3.78920e12 1.96636
\(538\) 1.21444e12 0.624965
\(539\) −1.61074e11 −0.0822008
\(540\) 0 0
\(541\) 1.11804e12 0.561136 0.280568 0.959834i \(-0.409477\pi\)
0.280568 + 0.959834i \(0.409477\pi\)
\(542\) 1.13823e12 0.566546
\(543\) −4.28574e12 −2.11556
\(544\) 3.76300e11 0.184221
\(545\) 0 0
\(546\) 5.78398e11 0.278522
\(547\) −1.99438e12 −0.952500 −0.476250 0.879310i \(-0.658004\pi\)
−0.476250 + 0.879310i \(0.658004\pi\)
\(548\) −1.55741e12 −0.737719
\(549\) 2.99031e12 1.40488
\(550\) 0 0
\(551\) −2.63931e12 −1.21985
\(552\) 3.06506e11 0.140512
\(553\) −6.85125e10 −0.0311535
\(554\) −1.37847e12 −0.621731
\(555\) 0 0
\(556\) 5.76638e11 0.255898
\(557\) 4.69088e11 0.206493 0.103247 0.994656i \(-0.467077\pi\)
0.103247 + 0.994656i \(0.467077\pi\)
\(558\) 1.97338e12 0.861702
\(559\) −8.84197e11 −0.382997
\(560\) 0 0
\(561\) −2.47721e12 −1.05592
\(562\) −1.60286e12 −0.677770
\(563\) −1.74518e12 −0.732069 −0.366034 0.930601i \(-0.619285\pi\)
−0.366034 + 0.930601i \(0.619285\pi\)
\(564\) 1.74864e12 0.727687
\(565\) 0 0
\(566\) 7.31911e11 0.299767
\(567\) −1.22119e12 −0.496205
\(568\) 8.99572e11 0.362634
\(569\) −4.13341e12 −1.65311 −0.826557 0.562853i \(-0.809704\pi\)
−0.826557 + 0.562853i \(0.809704\pi\)
\(570\) 0 0
\(571\) 4.14262e12 1.63084 0.815422 0.578867i \(-0.196505\pi\)
0.815422 + 0.578867i \(0.196505\pi\)
\(572\) 4.35920e11 0.170265
\(573\) 6.67383e12 2.58630
\(574\) −1.32045e12 −0.507713
\(575\) 0 0
\(576\) 6.93767e11 0.262611
\(577\) −4.27130e12 −1.60424 −0.802120 0.597164i \(-0.796294\pi\)
−0.802120 + 0.597164i \(0.796294\pi\)
\(578\) 1.63167e11 0.0608076
\(579\) −6.67614e11 −0.246872
\(580\) 0 0
\(581\) −9.20152e11 −0.335017
\(582\) 2.66316e12 0.962154
\(583\) −6.68394e11 −0.239621
\(584\) −1.09687e12 −0.390211
\(585\) 0 0
\(586\) 1.62584e11 0.0569557
\(587\) −5.32282e12 −1.85042 −0.925210 0.379455i \(-0.876111\pi\)
−0.925210 + 0.379455i \(0.876111\pi\)
\(588\) 3.64597e11 0.125781
\(589\) −1.16797e12 −0.399865
\(590\) 0 0
\(591\) −5.90570e12 −1.99126
\(592\) 2.29102e11 0.0766622
\(593\) 2.69262e11 0.0894190 0.0447095 0.999000i \(-0.485764\pi\)
0.0447095 + 0.999000i \(0.485764\pi\)
\(594\) −2.39322e12 −0.788758
\(595\) 0 0
\(596\) −8.02605e11 −0.260551
\(597\) −5.33906e11 −0.172021
\(598\) −2.95350e11 −0.0944457
\(599\) 3.51807e12 1.11657 0.558283 0.829651i \(-0.311460\pi\)
0.558283 + 0.829651i \(0.311460\pi\)
\(600\) 0 0
\(601\) −1.97549e12 −0.617645 −0.308823 0.951120i \(-0.599935\pi\)
−0.308823 + 0.951120i \(0.599935\pi\)
\(602\) −5.57359e11 −0.172962
\(603\) −3.59916e12 −1.10860
\(604\) −1.58960e12 −0.485984
\(605\) 0 0
\(606\) −4.14279e10 −0.0124786
\(607\) 1.20562e12 0.360464 0.180232 0.983624i \(-0.442315\pi\)
0.180232 + 0.983624i \(0.442315\pi\)
\(608\) −4.10615e11 −0.121862
\(609\) −3.99794e12 −1.17776
\(610\) 0 0
\(611\) −1.68499e12 −0.489117
\(612\) 3.79899e12 1.09468
\(613\) 5.02026e11 0.143600 0.0717999 0.997419i \(-0.477126\pi\)
0.0717999 + 0.997419i \(0.477126\pi\)
\(614\) 5.90332e11 0.167625
\(615\) 0 0
\(616\) 2.74785e11 0.0768918
\(617\) −5.17852e12 −1.43854 −0.719271 0.694730i \(-0.755524\pi\)
−0.719271 + 0.694730i \(0.755524\pi\)
\(618\) −5.52536e12 −1.52375
\(619\) −4.69963e12 −1.28664 −0.643318 0.765599i \(-0.722443\pi\)
−0.643318 + 0.765599i \(0.722443\pi\)
\(620\) 0 0
\(621\) 1.62149e12 0.437523
\(622\) 3.01693e12 0.808179
\(623\) −1.73252e12 −0.460767
\(624\) −9.86722e11 −0.260534
\(625\) 0 0
\(626\) −2.10845e12 −0.548756
\(627\) 2.70312e12 0.698491
\(628\) −3.41449e12 −0.876006
\(629\) 1.25454e12 0.319562
\(630\) 0 0
\(631\) −5.53678e12 −1.39035 −0.695177 0.718839i \(-0.744674\pi\)
−0.695177 + 0.718839i \(0.744674\pi\)
\(632\) 1.16879e11 0.0291414
\(633\) 8.91994e11 0.220823
\(634\) 2.41097e12 0.592639
\(635\) 0 0
\(636\) 1.51293e12 0.366660
\(637\) −3.51326e11 −0.0845441
\(638\) −3.01312e12 −0.719985
\(639\) 9.08175e12 2.15484
\(640\) 0 0
\(641\) −2.85506e12 −0.667966 −0.333983 0.942579i \(-0.608393\pi\)
−0.333983 + 0.942579i \(0.608393\pi\)
\(642\) 1.91481e12 0.444854
\(643\) 7.48209e12 1.72613 0.863065 0.505093i \(-0.168542\pi\)
0.863065 + 0.505093i \(0.168542\pi\)
\(644\) −1.86176e11 −0.0426518
\(645\) 0 0
\(646\) −2.24848e12 −0.507976
\(647\) 3.83872e12 0.861225 0.430613 0.902537i \(-0.358297\pi\)
0.430613 + 0.902537i \(0.358297\pi\)
\(648\) 2.08330e12 0.464157
\(649\) 3.36912e12 0.745445
\(650\) 0 0
\(651\) −1.76920e12 −0.386068
\(652\) −1.87807e12 −0.407003
\(653\) 1.09261e11 0.0235157 0.0117578 0.999931i \(-0.496257\pi\)
0.0117578 + 0.999931i \(0.496257\pi\)
\(654\) −7.75934e11 −0.165854
\(655\) 0 0
\(656\) 2.25263e12 0.474922
\(657\) −1.10736e13 −2.31871
\(658\) −1.06215e12 −0.220886
\(659\) −2.10319e12 −0.434403 −0.217202 0.976127i \(-0.569693\pi\)
−0.217202 + 0.976127i \(0.569693\pi\)
\(660\) 0 0
\(661\) −1.76045e12 −0.358688 −0.179344 0.983786i \(-0.557397\pi\)
−0.179344 + 0.983786i \(0.557397\pi\)
\(662\) 5.42203e11 0.109724
\(663\) −5.40317e12 −1.08602
\(664\) 1.56974e12 0.313380
\(665\) 0 0
\(666\) 2.31293e12 0.455542
\(667\) 2.04149e12 0.399374
\(668\) 1.64404e12 0.319462
\(669\) 1.76148e13 3.39986
\(670\) 0 0
\(671\) −2.02052e12 −0.384779
\(672\) −6.21986e11 −0.117657
\(673\) −8.52389e12 −1.60166 −0.800830 0.598892i \(-0.795608\pi\)
−0.800830 + 0.598892i \(0.795608\pi\)
\(674\) −3.17299e12 −0.592241
\(675\) 0 0
\(676\) −1.76394e12 −0.324881
\(677\) 2.88497e12 0.527828 0.263914 0.964546i \(-0.414986\pi\)
0.263914 + 0.964546i \(0.414986\pi\)
\(678\) 2.29600e12 0.417290
\(679\) −1.61764e12 −0.292057
\(680\) 0 0
\(681\) −1.76732e13 −3.14885
\(682\) −1.33339e12 −0.236009
\(683\) −1.03086e13 −1.81262 −0.906309 0.422616i \(-0.861112\pi\)
−0.906309 + 0.422616i \(0.861112\pi\)
\(684\) −4.14542e12 −0.724130
\(685\) 0 0
\(686\) −2.21461e11 −0.0381802
\(687\) −8.79556e12 −1.50646
\(688\) 9.50829e11 0.161791
\(689\) −1.45787e12 −0.246451
\(690\) 0 0
\(691\) 7.85439e12 1.31057 0.655287 0.755380i \(-0.272548\pi\)
0.655287 + 0.755380i \(0.272548\pi\)
\(692\) −4.91126e12 −0.814171
\(693\) 2.77413e12 0.456906
\(694\) 2.74819e12 0.449706
\(695\) 0 0
\(696\) 6.82030e12 1.10170
\(697\) 1.23351e13 1.97969
\(698\) −3.01102e12 −0.480136
\(699\) 9.38969e12 1.48766
\(700\) 0 0
\(701\) 1.69579e12 0.265242 0.132621 0.991167i \(-0.457661\pi\)
0.132621 + 0.991167i \(0.457661\pi\)
\(702\) −5.21997e12 −0.811243
\(703\) −1.36894e12 −0.211390
\(704\) −4.68771e11 −0.0719257
\(705\) 0 0
\(706\) −3.16793e12 −0.479904
\(707\) 2.51638e10 0.00378781
\(708\) −7.62613e12 −1.14066
\(709\) −2.20703e12 −0.328020 −0.164010 0.986459i \(-0.552443\pi\)
−0.164010 + 0.986459i \(0.552443\pi\)
\(710\) 0 0
\(711\) 1.17997e12 0.173164
\(712\) 2.95560e12 0.431008
\(713\) 9.03417e11 0.130914
\(714\) −3.40592e12 −0.490448
\(715\) 0 0
\(716\) 3.92644e12 0.558330
\(717\) −2.14225e13 −3.02715
\(718\) 6.95470e12 0.976604
\(719\) −1.46869e12 −0.204952 −0.102476 0.994735i \(-0.532676\pi\)
−0.102476 + 0.994735i \(0.532676\pi\)
\(720\) 0 0
\(721\) 3.35617e12 0.462525
\(722\) −2.70948e12 −0.371081
\(723\) −2.02192e12 −0.275195
\(724\) −4.44096e12 −0.600694
\(725\) 0 0
\(726\) −6.23461e12 −0.832904
\(727\) 4.97511e11 0.0660538 0.0330269 0.999454i \(-0.489485\pi\)
0.0330269 + 0.999454i \(0.489485\pi\)
\(728\) 5.99347e11 0.0790837
\(729\) −4.99938e12 −0.655605
\(730\) 0 0
\(731\) 5.20663e12 0.674417
\(732\) 4.57352e12 0.588777
\(733\) 5.22437e12 0.668445 0.334223 0.942494i \(-0.391526\pi\)
0.334223 + 0.942494i \(0.391526\pi\)
\(734\) −9.63306e11 −0.122499
\(735\) 0 0
\(736\) 3.17608e11 0.0398971
\(737\) 2.43191e12 0.303630
\(738\) 2.27417e13 2.82208
\(739\) 7.35495e11 0.0907152 0.0453576 0.998971i \(-0.485557\pi\)
0.0453576 + 0.998971i \(0.485557\pi\)
\(740\) 0 0
\(741\) 5.89590e12 0.718403
\(742\) −9.18975e11 −0.111298
\(743\) 1.12076e12 0.134916 0.0674580 0.997722i \(-0.478511\pi\)
0.0674580 + 0.997722i \(0.478511\pi\)
\(744\) 3.01818e12 0.361133
\(745\) 0 0
\(746\) 1.09683e13 1.29663
\(747\) 1.58475e13 1.86217
\(748\) −2.56694e12 −0.299818
\(749\) −1.16308e12 −0.135033
\(750\) 0 0
\(751\) −1.62183e12 −0.186049 −0.0930244 0.995664i \(-0.529653\pi\)
−0.0930244 + 0.995664i \(0.529653\pi\)
\(752\) 1.81197e12 0.206620
\(753\) 2.40991e13 2.73164
\(754\) −6.57206e12 −0.740509
\(755\) 0 0
\(756\) −3.29044e12 −0.366358
\(757\) 1.55150e12 0.171720 0.0858600 0.996307i \(-0.472636\pi\)
0.0858600 + 0.996307i \(0.472636\pi\)
\(758\) 8.08836e12 0.889916
\(759\) −2.09084e12 −0.228682
\(760\) 0 0
\(761\) 1.62863e13 1.76032 0.880159 0.474679i \(-0.157436\pi\)
0.880159 + 0.474679i \(0.157436\pi\)
\(762\) −1.02413e13 −1.10042
\(763\) 4.71312e11 0.0503440
\(764\) 6.91554e12 0.734355
\(765\) 0 0
\(766\) −3.23286e12 −0.339280
\(767\) 7.34856e12 0.766695
\(768\) 1.06108e12 0.110058
\(769\) −1.32915e13 −1.37059 −0.685293 0.728267i \(-0.740326\pi\)
−0.685293 + 0.728267i \(0.740326\pi\)
\(770\) 0 0
\(771\) 1.09455e12 0.111555
\(772\) −6.91794e11 −0.0700970
\(773\) 7.58593e12 0.764190 0.382095 0.924123i \(-0.375203\pi\)
0.382095 + 0.924123i \(0.375203\pi\)
\(774\) 9.59923e12 0.961395
\(775\) 0 0
\(776\) 2.75962e12 0.273194
\(777\) −2.07362e12 −0.204096
\(778\) 8.83126e12 0.864200
\(779\) −1.34600e13 −1.30956
\(780\) 0 0
\(781\) −6.13645e12 −0.590184
\(782\) 1.73918e12 0.166309
\(783\) 3.60809e13 3.43044
\(784\) 3.77802e11 0.0357143
\(785\) 0 0
\(786\) 9.70809e12 0.907262
\(787\) −3.30985e12 −0.307554 −0.153777 0.988106i \(-0.549144\pi\)
−0.153777 + 0.988106i \(0.549144\pi\)
\(788\) −6.11960e12 −0.565399
\(789\) 2.97995e13 2.73755
\(790\) 0 0
\(791\) −1.39462e12 −0.126666
\(792\) −4.73254e12 −0.427397
\(793\) −4.40706e12 −0.395748
\(794\) −2.02906e12 −0.181176
\(795\) 0 0
\(796\) −5.53244e11 −0.0488436
\(797\) 3.26446e12 0.286582 0.143291 0.989681i \(-0.454232\pi\)
0.143291 + 0.989681i \(0.454232\pi\)
\(798\) 3.71651e12 0.324431
\(799\) 9.92216e12 0.861283
\(800\) 0 0
\(801\) 2.98386e13 2.56114
\(802\) 9.39131e12 0.801570
\(803\) 7.48235e12 0.635064
\(804\) −5.50473e12 −0.464605
\(805\) 0 0
\(806\) −2.90833e12 −0.242737
\(807\) 1.87519e13 1.55637
\(808\) −4.29283e10 −0.00354318
\(809\) 3.46768e12 0.284624 0.142312 0.989822i \(-0.454546\pi\)
0.142312 + 0.989822i \(0.454546\pi\)
\(810\) 0 0
\(811\) 1.02915e13 0.835381 0.417690 0.908589i \(-0.362840\pi\)
0.417690 + 0.908589i \(0.362840\pi\)
\(812\) −4.14274e12 −0.334415
\(813\) 1.75752e13 1.41089
\(814\) −1.56282e12 −0.124767
\(815\) 0 0
\(816\) 5.81035e12 0.458772
\(817\) −5.68143e12 −0.446127
\(818\) −1.45322e12 −0.113486
\(819\) 6.05079e12 0.469931
\(820\) 0 0
\(821\) 6.23700e12 0.479106 0.239553 0.970883i \(-0.422999\pi\)
0.239553 + 0.970883i \(0.422999\pi\)
\(822\) −2.40476e13 −1.83717
\(823\) −2.46689e13 −1.87435 −0.937175 0.348860i \(-0.886569\pi\)
−0.937175 + 0.348860i \(0.886569\pi\)
\(824\) −5.72548e12 −0.432653
\(825\) 0 0
\(826\) 4.63221e12 0.346240
\(827\) 1.45027e13 1.07814 0.539068 0.842262i \(-0.318776\pi\)
0.539068 + 0.842262i \(0.318776\pi\)
\(828\) 3.20645e12 0.237076
\(829\) −1.07681e13 −0.791854 −0.395927 0.918282i \(-0.629577\pi\)
−0.395927 + 0.918282i \(0.629577\pi\)
\(830\) 0 0
\(831\) −2.12846e13 −1.54832
\(832\) −1.02246e12 −0.0739761
\(833\) 2.06880e12 0.148873
\(834\) 8.90373e12 0.637272
\(835\) 0 0
\(836\) 2.80102e12 0.198330
\(837\) 1.59668e13 1.12449
\(838\) −1.33547e13 −0.935484
\(839\) −2.36129e13 −1.64521 −0.822603 0.568616i \(-0.807479\pi\)
−0.822603 + 0.568616i \(0.807479\pi\)
\(840\) 0 0
\(841\) 3.09195e13 2.13133
\(842\) −2.58176e11 −0.0177015
\(843\) −2.47494e13 −1.68787
\(844\) 9.24301e11 0.0627007
\(845\) 0 0
\(846\) 1.82930e13 1.22778
\(847\) 3.78698e12 0.252824
\(848\) 1.56773e12 0.104109
\(849\) 1.13013e13 0.746521
\(850\) 0 0
\(851\) 1.05886e12 0.0692081
\(852\) 1.38901e13 0.903080
\(853\) 1.67503e13 1.08331 0.541655 0.840601i \(-0.317798\pi\)
0.541655 + 0.840601i \(0.317798\pi\)
\(854\) −2.77801e12 −0.178720
\(855\) 0 0
\(856\) 1.98416e12 0.126312
\(857\) 1.16182e13 0.735742 0.367871 0.929877i \(-0.380087\pi\)
0.367871 + 0.929877i \(0.380087\pi\)
\(858\) 6.73094e12 0.424017
\(859\) 1.58818e13 0.995246 0.497623 0.867393i \(-0.334206\pi\)
0.497623 + 0.867393i \(0.334206\pi\)
\(860\) 0 0
\(861\) −2.03888e13 −1.26438
\(862\) −7.71096e12 −0.475692
\(863\) −1.35971e13 −0.834446 −0.417223 0.908804i \(-0.636997\pi\)
−0.417223 + 0.908804i \(0.636997\pi\)
\(864\) 5.61335e12 0.342697
\(865\) 0 0
\(866\) −1.06930e13 −0.646056
\(867\) 2.51943e12 0.151431
\(868\) −1.83328e12 −0.109620
\(869\) −7.97294e11 −0.0474274
\(870\) 0 0
\(871\) 5.30437e12 0.312285
\(872\) −8.04037e11 −0.0470925
\(873\) 2.78601e13 1.62338
\(874\) −1.89778e12 −0.110013
\(875\) 0 0
\(876\) −1.69366e13 −0.971755
\(877\) −3.40277e12 −0.194238 −0.0971189 0.995273i \(-0.530963\pi\)
−0.0971189 + 0.995273i \(0.530963\pi\)
\(878\) −7.67711e12 −0.435986
\(879\) 2.51041e12 0.141839
\(880\) 0 0
\(881\) 9.30779e12 0.520541 0.260270 0.965536i \(-0.416188\pi\)
0.260270 + 0.965536i \(0.416188\pi\)
\(882\) 3.81415e12 0.212221
\(883\) −1.00154e13 −0.554430 −0.277215 0.960808i \(-0.589411\pi\)
−0.277215 + 0.960808i \(0.589411\pi\)
\(884\) −5.59887e12 −0.308365
\(885\) 0 0
\(886\) 1.04576e13 0.570137
\(887\) 2.58440e13 1.40186 0.700928 0.713232i \(-0.252769\pi\)
0.700928 + 0.713232i \(0.252769\pi\)
\(888\) 3.53751e12 0.190915
\(889\) 6.22070e12 0.334027
\(890\) 0 0
\(891\) −1.42113e13 −0.755412
\(892\) 1.82528e13 0.965358
\(893\) −1.08270e13 −0.569739
\(894\) −1.23928e13 −0.648860
\(895\) 0 0
\(896\) −6.44514e11 −0.0334077
\(897\) −4.56043e12 −0.235201
\(898\) −3.01855e12 −0.154901
\(899\) 2.01026e13 1.02644
\(900\) 0 0
\(901\) 8.58471e12 0.433974
\(902\) −1.53664e13 −0.772932
\(903\) −8.60604e12 −0.430733
\(904\) 2.37916e12 0.118486
\(905\) 0 0
\(906\) −2.45447e13 −1.21026
\(907\) −3.82049e13 −1.87451 −0.937253 0.348650i \(-0.886640\pi\)
−0.937253 + 0.348650i \(0.886640\pi\)
\(908\) −1.83133e13 −0.894086
\(909\) −4.33389e11 −0.0210543
\(910\) 0 0
\(911\) 9.85478e12 0.474039 0.237020 0.971505i \(-0.423829\pi\)
0.237020 + 0.971505i \(0.423829\pi\)
\(912\) −6.34021e12 −0.303478
\(913\) −1.07080e13 −0.510023
\(914\) −2.09929e13 −0.994979
\(915\) 0 0
\(916\) −9.11412e12 −0.427745
\(917\) −5.89682e12 −0.275395
\(918\) 3.07380e13 1.42851
\(919\) 2.60834e13 1.20627 0.603135 0.797639i \(-0.293918\pi\)
0.603135 + 0.797639i \(0.293918\pi\)
\(920\) 0 0
\(921\) 9.11517e12 0.417442
\(922\) 1.86551e13 0.850176
\(923\) −1.33845e13 −0.607008
\(924\) 4.24289e12 0.191486
\(925\) 0 0
\(926\) 6.19709e11 0.0276974
\(927\) −5.78024e13 −2.57091
\(928\) 7.06733e12 0.312816
\(929\) −3.60592e13 −1.58835 −0.794174 0.607690i \(-0.792096\pi\)
−0.794174 + 0.607690i \(0.792096\pi\)
\(930\) 0 0
\(931\) −2.25746e12 −0.0984795
\(932\) 9.72977e12 0.422407
\(933\) 4.65836e13 2.01264
\(934\) −1.87440e12 −0.0805936
\(935\) 0 0
\(936\) −1.03224e13 −0.439581
\(937\) −2.75358e13 −1.16700 −0.583499 0.812114i \(-0.698317\pi\)
−0.583499 + 0.812114i \(0.698317\pi\)
\(938\) 3.34364e12 0.141028
\(939\) −3.25561e13 −1.36659
\(940\) 0 0
\(941\) −7.60157e12 −0.316046 −0.158023 0.987435i \(-0.550512\pi\)
−0.158023 + 0.987435i \(0.550512\pi\)
\(942\) −5.27223e13 −2.18155
\(943\) 1.04112e13 0.428745
\(944\) −7.90234e12 −0.323878
\(945\) 0 0
\(946\) −6.48610e12 −0.263314
\(947\) 2.35269e13 0.950582 0.475291 0.879829i \(-0.342343\pi\)
0.475291 + 0.879829i \(0.342343\pi\)
\(948\) 1.80470e12 0.0725719
\(949\) 1.63201e13 0.653168
\(950\) 0 0
\(951\) 3.72272e13 1.47587
\(952\) −3.52928e12 −0.139258
\(953\) −3.76056e12 −0.147684 −0.0738422 0.997270i \(-0.523526\pi\)
−0.0738422 + 0.997270i \(0.523526\pi\)
\(954\) 1.58272e13 0.618639
\(955\) 0 0
\(956\) −2.21984e13 −0.859531
\(957\) −4.65248e13 −1.79300
\(958\) −3.11575e13 −1.19514
\(959\) 1.46068e13 0.557663
\(960\) 0 0
\(961\) −1.75436e13 −0.663535
\(962\) −3.40875e12 −0.128324
\(963\) 2.00313e13 0.750571
\(964\) −2.09515e12 −0.0781391
\(965\) 0 0
\(966\) −2.87470e12 −0.106217
\(967\) −3.59678e13 −1.32280 −0.661400 0.750033i \(-0.730037\pi\)
−0.661400 + 0.750033i \(0.730037\pi\)
\(968\) −6.46042e12 −0.236495
\(969\) −3.47182e13 −1.26503
\(970\) 0 0
\(971\) −1.20049e13 −0.433383 −0.216692 0.976240i \(-0.569527\pi\)
−0.216692 + 0.976240i \(0.569527\pi\)
\(972\) 5.19328e12 0.186614
\(973\) −5.40823e12 −0.193441
\(974\) 3.13573e13 1.11641
\(975\) 0 0
\(976\) 4.73917e12 0.167178
\(977\) 4.59954e12 0.161506 0.0807530 0.996734i \(-0.474268\pi\)
0.0807530 + 0.996734i \(0.474268\pi\)
\(978\) −2.89988e13 −1.01357
\(979\) −2.01617e13 −0.701462
\(980\) 0 0
\(981\) −8.11727e12 −0.279833
\(982\) −4.69742e12 −0.161197
\(983\) 3.13830e12 0.107202 0.0536011 0.998562i \(-0.482930\pi\)
0.0536011 + 0.998562i \(0.482930\pi\)
\(984\) 3.47823e13 1.18272
\(985\) 0 0
\(986\) 3.86999e13 1.30396
\(987\) −1.64003e13 −0.550080
\(988\) 6.10944e12 0.203984
\(989\) 4.39454e12 0.146060
\(990\) 0 0
\(991\) 1.38736e13 0.456939 0.228470 0.973551i \(-0.426628\pi\)
0.228470 + 0.973551i \(0.426628\pi\)
\(992\) 3.12750e12 0.102540
\(993\) 8.37203e12 0.273249
\(994\) −8.43700e12 −0.274126
\(995\) 0 0
\(996\) 2.42380e13 0.780421
\(997\) 1.65453e13 0.530331 0.265166 0.964203i \(-0.414573\pi\)
0.265166 + 0.964203i \(0.414573\pi\)
\(998\) 2.74224e13 0.875019
\(999\) 1.87142e13 0.594465
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 350.10.a.j.1.2 2
5.2 odd 4 350.10.c.j.99.3 4
5.3 odd 4 350.10.c.j.99.2 4
5.4 even 2 14.10.a.c.1.1 2
15.14 odd 2 126.10.a.o.1.2 2
20.19 odd 2 112.10.a.c.1.2 2
35.4 even 6 98.10.c.j.79.2 4
35.9 even 6 98.10.c.j.67.2 4
35.19 odd 6 98.10.c.h.67.1 4
35.24 odd 6 98.10.c.h.79.1 4
35.34 odd 2 98.10.a.e.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.10.a.c.1.1 2 5.4 even 2
98.10.a.e.1.2 2 35.34 odd 2
98.10.c.h.67.1 4 35.19 odd 6
98.10.c.h.79.1 4 35.24 odd 6
98.10.c.j.67.2 4 35.9 even 6
98.10.c.j.79.2 4 35.4 even 6
112.10.a.c.1.2 2 20.19 odd 2
126.10.a.o.1.2 2 15.14 odd 2
350.10.a.j.1.2 2 1.1 even 1 trivial
350.10.c.j.99.2 4 5.3 odd 4
350.10.c.j.99.3 4 5.2 odd 4