Properties

Label 400.6.n.g.143.2
Level $400$
Weight $6$
Character 400.143
Analytic conductor $64.154$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,6,Mod(143,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.143");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.n (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 271 x^{18} + 109637 x^{16} + 25993614 x^{14} + 5522961902 x^{12} + 881545050522 x^{10} + 133816049059481 x^{8} + \cdots + 57\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{67}\cdot 3^{4}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 80)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 143.2
Root \(3.75557 - 3.81117i\) of defining polynomial
Character \(\chi\) \(=\) 400.143
Dual form 400.6.n.g.207.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-17.2921 + 17.2921i) q^{3} +(154.079 + 154.079i) q^{7} -355.037i q^{9} +O(q^{10})\) \(q+(-17.2921 + 17.2921i) q^{3} +(154.079 + 154.079i) q^{7} -355.037i q^{9} +127.489i q^{11} +(-335.067 - 335.067i) q^{13} +(1155.01 - 1155.01i) q^{17} -28.2166 q^{19} -5328.70 q^{21} +(-2783.04 + 2783.04i) q^{23} +(1937.35 + 1937.35i) q^{27} +3388.31i q^{29} -5384.41i q^{31} +(-2204.56 - 2204.56i) q^{33} +(-11534.0 + 11534.0i) q^{37} +11588.1 q^{39} -11147.8 q^{41} +(-1437.07 + 1437.07i) q^{43} +(-219.040 - 219.040i) q^{47} +30673.4i q^{49} +39945.1i q^{51} +(-22745.6 - 22745.6i) q^{53} +(487.926 - 487.926i) q^{57} +22196.6 q^{59} -1431.25 q^{61} +(54703.5 - 54703.5i) q^{63} +(-28948.2 - 28948.2i) q^{67} -96249.3i q^{69} -24188.3i q^{71} +(-28574.8 - 28574.8i) q^{73} +(-19643.4 + 19643.4i) q^{77} -23417.0 q^{79} +19271.9 q^{81} +(-18919.5 + 18919.5i) q^{83} +(-58591.1 - 58591.1i) q^{87} -8179.53i q^{89} -103253. i q^{91} +(93107.9 + 93107.9i) q^{93} +(76747.2 - 76747.2i) q^{97} +45263.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 804 q^{13} + 2236 q^{17} - 4520 q^{21} + 11096 q^{33} - 44260 q^{37} - 6760 q^{41} - 182452 q^{53} + 34288 q^{57} - 41080 q^{61} - 264372 q^{73} - 399304 q^{77} - 520220 q^{81} - 713496 q^{93} - 374772 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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 0
\(3\) −17.2921 + 17.2921i −1.10929 + 1.10929i −0.116048 + 0.993244i \(0.537023\pi\)
−0.993244 + 0.116048i \(0.962977\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 154.079 + 154.079i 1.18849 + 1.18849i 0.977484 + 0.211011i \(0.0676755\pi\)
0.211011 + 0.977484i \(0.432324\pi\)
\(8\) 0 0
\(9\) 355.037i 1.46106i
\(10\) 0 0
\(11\) 127.489i 0.317682i 0.987304 + 0.158841i \(0.0507757\pi\)
−0.987304 + 0.158841i \(0.949224\pi\)
\(12\) 0 0
\(13\) −335.067 335.067i −0.549887 0.549887i 0.376521 0.926408i \(-0.377120\pi\)
−0.926408 + 0.376521i \(0.877120\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1155.01 1155.01i 0.969310 0.969310i −0.0302331 0.999543i \(-0.509625\pi\)
0.999543 + 0.0302331i \(0.00962497\pi\)
\(18\) 0 0
\(19\) −28.2166 −0.0179317 −0.00896584 0.999960i \(-0.502854\pi\)
−0.00896584 + 0.999960i \(0.502854\pi\)
\(20\) 0 0
\(21\) −5328.70 −2.63677
\(22\) 0 0
\(23\) −2783.04 + 2783.04i −1.09698 + 1.09698i −0.102219 + 0.994762i \(0.532594\pi\)
−0.994762 + 0.102219i \(0.967406\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1937.35 + 1937.35i 0.511446 + 0.511446i
\(28\) 0 0
\(29\) 3388.31i 0.748149i 0.927399 + 0.374074i \(0.122040\pi\)
−0.927399 + 0.374074i \(0.877960\pi\)
\(30\) 0 0
\(31\) 5384.41i 1.00631i −0.864195 0.503157i \(-0.832172\pi\)
0.864195 0.503157i \(-0.167828\pi\)
\(32\) 0 0
\(33\) −2204.56 2204.56i −0.352402 0.352402i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −11534.0 + 11534.0i −1.38508 + 1.38508i −0.549757 + 0.835324i \(0.685280\pi\)
−0.835324 + 0.549757i \(0.814720\pi\)
\(38\) 0 0
\(39\) 11588.1 1.21997
\(40\) 0 0
\(41\) −11147.8 −1.03569 −0.517844 0.855475i \(-0.673265\pi\)
−0.517844 + 0.855475i \(0.673265\pi\)
\(42\) 0 0
\(43\) −1437.07 + 1437.07i −0.118524 + 0.118524i −0.763881 0.645357i \(-0.776709\pi\)
0.645357 + 0.763881i \(0.276709\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −219.040 219.040i −0.0144637 0.0144637i 0.699838 0.714302i \(-0.253256\pi\)
−0.714302 + 0.699838i \(0.753256\pi\)
\(48\) 0 0
\(49\) 30673.4i 1.82504i
\(50\) 0 0
\(51\) 39945.1i 2.15049i
\(52\) 0 0
\(53\) −22745.6 22745.6i −1.11226 1.11226i −0.992844 0.119421i \(-0.961896\pi\)
−0.119421 0.992844i \(-0.538104\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 487.926 487.926i 0.0198915 0.0198915i
\(58\) 0 0
\(59\) 22196.6 0.830149 0.415074 0.909787i \(-0.363756\pi\)
0.415074 + 0.909787i \(0.363756\pi\)
\(60\) 0 0
\(61\) −1431.25 −0.0492484 −0.0246242 0.999697i \(-0.507839\pi\)
−0.0246242 + 0.999697i \(0.507839\pi\)
\(62\) 0 0
\(63\) 54703.5 54703.5i 1.73646 1.73646i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −28948.2 28948.2i −0.787835 0.787835i 0.193304 0.981139i \(-0.438080\pi\)
−0.981139 + 0.193304i \(0.938080\pi\)
\(68\) 0 0
\(69\) 96249.3i 2.43374i
\(70\) 0 0
\(71\) 24188.3i 0.569456i −0.958608 0.284728i \(-0.908097\pi\)
0.958608 0.284728i \(-0.0919032\pi\)
\(72\) 0 0
\(73\) −28574.8 28574.8i −0.627591 0.627591i 0.319871 0.947461i \(-0.396361\pi\)
−0.947461 + 0.319871i \(0.896361\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −19643.4 + 19643.4i −0.377563 + 0.377563i
\(78\) 0 0
\(79\) −23417.0 −0.422147 −0.211073 0.977470i \(-0.567696\pi\)
−0.211073 + 0.977470i \(0.567696\pi\)
\(80\) 0 0
\(81\) 19271.9 0.326371
\(82\) 0 0
\(83\) −18919.5 + 18919.5i −0.301450 + 0.301450i −0.841581 0.540131i \(-0.818375\pi\)
0.540131 + 0.841581i \(0.318375\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −58591.1 58591.1i −0.829915 0.829915i
\(88\) 0 0
\(89\) 8179.53i 0.109459i −0.998501 0.0547297i \(-0.982570\pi\)
0.998501 0.0547297i \(-0.0174297\pi\)
\(90\) 0 0
\(91\) 103253.i 1.30708i
\(92\) 0 0
\(93\) 93107.9 + 93107.9i 1.11630 + 1.11630i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 76747.2 76747.2i 0.828196 0.828196i −0.159071 0.987267i \(-0.550850\pi\)
0.987267 + 0.159071i \(0.0508499\pi\)
\(98\) 0 0
\(99\) 45263.4 0.464151
\(100\) 0 0
\(101\) −1379.60 −0.0134570 −0.00672851 0.999977i \(-0.502142\pi\)
−0.00672851 + 0.999977i \(0.502142\pi\)
\(102\) 0 0
\(103\) 127370. 127370.i 1.18297 1.18297i 0.203994 0.978972i \(-0.434608\pi\)
0.978972 0.203994i \(-0.0653924\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −17327.3 17327.3i −0.146309 0.146309i 0.630158 0.776467i \(-0.282990\pi\)
−0.776467 + 0.630158i \(0.782990\pi\)
\(108\) 0 0
\(109\) 105777.i 0.852759i −0.904544 0.426379i \(-0.859789\pi\)
0.904544 0.426379i \(-0.140211\pi\)
\(110\) 0 0
\(111\) 398895.i 3.07292i
\(112\) 0 0
\(113\) −63449.7 63449.7i −0.467449 0.467449i 0.433638 0.901087i \(-0.357230\pi\)
−0.901087 + 0.433638i \(0.857230\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −118961. + 118961.i −0.803416 + 0.803416i
\(118\) 0 0
\(119\) 355924. 2.30404
\(120\) 0 0
\(121\) 144797. 0.899078
\(122\) 0 0
\(123\) 192769. 192769.i 1.14888 1.14888i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 150685. + 150685.i 0.829009 + 0.829009i 0.987380 0.158370i \(-0.0506240\pi\)
−0.158370 + 0.987380i \(0.550624\pi\)
\(128\) 0 0
\(129\) 49700.1i 0.262956i
\(130\) 0 0
\(131\) 218406.i 1.11195i 0.831199 + 0.555976i \(0.187655\pi\)
−0.831199 + 0.555976i \(0.812345\pi\)
\(132\) 0 0
\(133\) −4347.58 4347.58i −0.0213117 0.0213117i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −48858.6 + 48858.6i −0.222403 + 0.222403i −0.809509 0.587107i \(-0.800267\pi\)
0.587107 + 0.809509i \(0.300267\pi\)
\(138\) 0 0
\(139\) −356608. −1.56550 −0.782752 0.622334i \(-0.786185\pi\)
−0.782752 + 0.622334i \(0.786185\pi\)
\(140\) 0 0
\(141\) 7575.36 0.0320889
\(142\) 0 0
\(143\) 42717.5 42717.5i 0.174689 0.174689i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −530409. 530409.i −2.02450 2.02450i
\(148\) 0 0
\(149\) 39079.0i 0.144204i −0.997397 0.0721022i \(-0.977029\pi\)
0.997397 0.0721022i \(-0.0229708\pi\)
\(150\) 0 0
\(151\) 355671.i 1.26942i 0.772750 + 0.634711i \(0.218881\pi\)
−0.772750 + 0.634711i \(0.781119\pi\)
\(152\) 0 0
\(153\) −410070. 410070.i −1.41622 1.41622i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 37305.0 37305.0i 0.120786 0.120786i −0.644130 0.764916i \(-0.722780\pi\)
0.764916 + 0.644130i \(0.222780\pi\)
\(158\) 0 0
\(159\) 786641. 2.46765
\(160\) 0 0
\(161\) −857612. −2.60751
\(162\) 0 0
\(163\) 180954. 180954.i 0.533458 0.533458i −0.388142 0.921600i \(-0.626883\pi\)
0.921600 + 0.388142i \(0.126883\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −174882. 174882.i −0.485236 0.485236i 0.421563 0.906799i \(-0.361482\pi\)
−0.906799 + 0.421563i \(0.861482\pi\)
\(168\) 0 0
\(169\) 146753.i 0.395249i
\(170\) 0 0
\(171\) 10017.9i 0.0261992i
\(172\) 0 0
\(173\) −376164. 376164.i −0.955568 0.955568i 0.0434856 0.999054i \(-0.486154\pi\)
−0.999054 + 0.0434856i \(0.986154\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −383826. + 383826.i −0.920877 + 0.920877i
\(178\) 0 0
\(179\) −384984. −0.898070 −0.449035 0.893514i \(-0.648232\pi\)
−0.449035 + 0.893514i \(0.648232\pi\)
\(180\) 0 0
\(181\) −332337. −0.754020 −0.377010 0.926209i \(-0.623048\pi\)
−0.377010 + 0.926209i \(0.623048\pi\)
\(182\) 0 0
\(183\) 24749.5 24749.5i 0.0546308 0.0546308i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 147251. + 147251.i 0.307932 + 0.307932i
\(188\) 0 0
\(189\) 597010.i 1.21570i
\(190\) 0 0
\(191\) 164553.i 0.326378i 0.986595 + 0.163189i \(0.0521781\pi\)
−0.986595 + 0.163189i \(0.947822\pi\)
\(192\) 0 0
\(193\) 58078.0 + 58078.0i 0.112232 + 0.112232i 0.760993 0.648760i \(-0.224712\pi\)
−0.648760 + 0.760993i \(0.724712\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −386719. + 386719.i −0.709954 + 0.709954i −0.966525 0.256571i \(-0.917407\pi\)
0.256571 + 0.966525i \(0.417407\pi\)
\(198\) 0 0
\(199\) −14055.2 −0.0251596 −0.0125798 0.999921i \(-0.504004\pi\)
−0.0125798 + 0.999921i \(0.504004\pi\)
\(200\) 0 0
\(201\) 1.00115e6 1.74788
\(202\) 0 0
\(203\) −522066. + 522066.i −0.889171 + 0.889171i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 988080. + 988080.i 1.60275 + 1.60275i
\(208\) 0 0
\(209\) 3597.32i 0.00569657i
\(210\) 0 0
\(211\) 438963.i 0.678768i −0.940648 0.339384i \(-0.889781\pi\)
0.940648 0.339384i \(-0.110219\pi\)
\(212\) 0 0
\(213\) 418268. + 418268.i 0.631692 + 0.631692i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 829622. 829622.i 1.19600 1.19600i
\(218\) 0 0
\(219\) 988240. 1.39236
\(220\) 0 0
\(221\) −774010. −1.06602
\(222\) 0 0
\(223\) 413890. 413890.i 0.557343 0.557343i −0.371207 0.928550i \(-0.621056\pi\)
0.928550 + 0.371207i \(0.121056\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −165742. 165742.i −0.213485 0.213485i 0.592261 0.805746i \(-0.298235\pi\)
−0.805746 + 0.592261i \(0.798235\pi\)
\(228\) 0 0
\(229\) 590717.i 0.744373i 0.928158 + 0.372187i \(0.121392\pi\)
−0.928158 + 0.372187i \(0.878608\pi\)
\(230\) 0 0
\(231\) 679352.i 0.837655i
\(232\) 0 0
\(233\) 849795. + 849795.i 1.02547 + 1.02547i 0.999667 + 0.0258063i \(0.00821531\pi\)
0.0258063 + 0.999667i \(0.491785\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 404930. 404930.i 0.468284 0.468284i
\(238\) 0 0
\(239\) −72025.5 −0.0815627 −0.0407814 0.999168i \(-0.512985\pi\)
−0.0407814 + 0.999168i \(0.512985\pi\)
\(240\) 0 0
\(241\) 358847. 0.397985 0.198992 0.980001i \(-0.436233\pi\)
0.198992 + 0.980001i \(0.436233\pi\)
\(242\) 0 0
\(243\) −804029. + 804029.i −0.873486 + 0.873486i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 9454.46 + 9454.46i 0.00986040 + 0.00986040i
\(248\) 0 0
\(249\) 654318.i 0.668791i
\(250\) 0 0
\(251\) 209249.i 0.209642i 0.994491 + 0.104821i \(0.0334270\pi\)
−0.994491 + 0.104821i \(0.966573\pi\)
\(252\) 0 0
\(253\) −354807. 354807.i −0.348491 0.348491i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 228218. 228218.i 0.215535 0.215535i −0.591079 0.806614i \(-0.701298\pi\)
0.806614 + 0.591079i \(0.201298\pi\)
\(258\) 0 0
\(259\) −3.55428e6 −3.29232
\(260\) 0 0
\(261\) 1.20297e6 1.09309
\(262\) 0 0
\(263\) −945242. + 945242.i −0.842663 + 0.842663i −0.989204 0.146542i \(-0.953186\pi\)
0.146542 + 0.989204i \(0.453186\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 141442. + 141442.i 0.121422 + 0.121422i
\(268\) 0 0
\(269\) 1.56559e6i 1.31916i −0.751635 0.659579i \(-0.770735\pi\)
0.751635 0.659579i \(-0.229265\pi\)
\(270\) 0 0
\(271\) 990821.i 0.819543i −0.912188 0.409772i \(-0.865608\pi\)
0.912188 0.409772i \(-0.134392\pi\)
\(272\) 0 0
\(273\) 1.78547e6 + 1.78547e6i 1.44993 + 1.44993i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 174283. 174283.i 0.136476 0.136476i −0.635569 0.772044i \(-0.719234\pi\)
0.772044 + 0.635569i \(0.219234\pi\)
\(278\) 0 0
\(279\) −1.91166e6 −1.47028
\(280\) 0 0
\(281\) 2.17766e6 1.64522 0.822609 0.568607i \(-0.192517\pi\)
0.822609 + 0.568607i \(0.192517\pi\)
\(282\) 0 0
\(283\) 502126. 502126.i 0.372689 0.372689i −0.495767 0.868456i \(-0.665113\pi\)
0.868456 + 0.495767i \(0.165113\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.71763e6 1.71763e6i −1.23091 1.23091i
\(288\) 0 0
\(289\) 1.24823e6i 0.879123i
\(290\) 0 0
\(291\) 2.65425e6i 1.83742i
\(292\) 0 0
\(293\) −45772.5 45772.5i −0.0311484 0.0311484i 0.691361 0.722509i \(-0.257011\pi\)
−0.722509 + 0.691361i \(0.757011\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −246992. + 246992.i −0.162477 + 0.162477i
\(298\) 0 0
\(299\) 1.86501e6 1.20643
\(300\) 0 0
\(301\) −442844. −0.281731
\(302\) 0 0
\(303\) 23856.2 23856.2i 0.0149277 0.0149277i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.39565e6 + 1.39565e6i 0.845144 + 0.845144i 0.989523 0.144379i \(-0.0461183\pi\)
−0.144379 + 0.989523i \(0.546118\pi\)
\(308\) 0 0
\(309\) 4.40498e6i 2.62451i
\(310\) 0 0
\(311\) 1.84115e6i 1.07942i 0.841852 + 0.539709i \(0.181466\pi\)
−0.841852 + 0.539709i \(0.818534\pi\)
\(312\) 0 0
\(313\) −2.22340e6 2.22340e6i −1.28279 1.28279i −0.939073 0.343718i \(-0.888314\pi\)
−0.343718 0.939073i \(-0.611686\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.35501e6 1.35501e6i 0.757344 0.757344i −0.218495 0.975838i \(-0.570115\pi\)
0.975838 + 0.218495i \(0.0701146\pi\)
\(318\) 0 0
\(319\) −431973. −0.237673
\(320\) 0 0
\(321\) 599254. 0.324600
\(322\) 0 0
\(323\) −32590.4 + 32590.4i −0.0173814 + 0.0173814i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.82912e6 + 1.82912e6i 0.945958 + 0.945958i
\(328\) 0 0
\(329\) 67498.9i 0.0343801i
\(330\) 0 0
\(331\) 2.40493e6i 1.20652i 0.797546 + 0.603258i \(0.206131\pi\)
−0.797546 + 0.603258i \(0.793869\pi\)
\(332\) 0 0
\(333\) 4.09499e6 + 4.09499e6i 2.02368 + 2.02368i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.72476e6 1.72476e6i 0.827284 0.827284i −0.159856 0.987140i \(-0.551103\pi\)
0.987140 + 0.159856i \(0.0511031\pi\)
\(338\) 0 0
\(339\) 2.19436e6 1.03707
\(340\) 0 0
\(341\) 686454. 0.319687
\(342\) 0 0
\(343\) −2.13652e6 + 2.13652e6i −0.980554 + 0.980554i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.06948e6 + 2.06948e6i 0.922653 + 0.922653i 0.997216 0.0745635i \(-0.0237563\pi\)
−0.0745635 + 0.997216i \(0.523756\pi\)
\(348\) 0 0
\(349\) 1.88338e6i 0.827702i 0.910345 + 0.413851i \(0.135817\pi\)
−0.910345 + 0.413851i \(0.864183\pi\)
\(350\) 0 0
\(351\) 1.29829e6i 0.562475i
\(352\) 0 0
\(353\) −1.77943e6 1.77943e6i −0.760054 0.760054i 0.216278 0.976332i \(-0.430608\pi\)
−0.976332 + 0.216278i \(0.930608\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −6.15469e6 + 6.15469e6i −2.55585 + 2.55585i
\(358\) 0 0
\(359\) −4.50238e6 −1.84377 −0.921884 0.387466i \(-0.873350\pi\)
−0.921884 + 0.387466i \(0.873350\pi\)
\(360\) 0 0
\(361\) −2.47530e6 −0.999678
\(362\) 0 0
\(363\) −2.50386e6 + 2.50386e6i −0.997340 + 0.997340i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −604369. 604369.i −0.234227 0.234227i 0.580227 0.814455i \(-0.302964\pi\)
−0.814455 + 0.580227i \(0.802964\pi\)
\(368\) 0 0
\(369\) 3.95787e6i 1.51320i
\(370\) 0 0
\(371\) 7.00922e6i 2.64384i
\(372\) 0 0
\(373\) −87601.5 87601.5i −0.0326016 0.0326016i 0.690618 0.723220i \(-0.257339\pi\)
−0.723220 + 0.690618i \(0.757339\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.13531e6 1.13531e6i 0.411397 0.411397i
\(378\) 0 0
\(379\) 2.48550e6 0.888824 0.444412 0.895823i \(-0.353413\pi\)
0.444412 + 0.895823i \(0.353413\pi\)
\(380\) 0 0
\(381\) −5.21132e6 −1.83923
\(382\) 0 0
\(383\) −601337. + 601337.i −0.209470 + 0.209470i −0.804042 0.594572i \(-0.797321\pi\)
0.594572 + 0.804042i \(0.297321\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 510213. + 510213.i 0.173171 + 0.173171i
\(388\) 0 0
\(389\) 3.73874e6i 1.25271i −0.779537 0.626356i \(-0.784546\pi\)
0.779537 0.626356i \(-0.215454\pi\)
\(390\) 0 0
\(391\) 6.42886e6i 2.12663i
\(392\) 0 0
\(393\) −3.77670e6 3.77670e6i −1.23348 1.23348i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 209001. 209001.i 0.0665538 0.0665538i −0.673046 0.739600i \(-0.735015\pi\)
0.739600 + 0.673046i \(0.235015\pi\)
\(398\) 0 0
\(399\) 150358. 0.0472818
\(400\) 0 0
\(401\) −3.88404e6 −1.20621 −0.603104 0.797662i \(-0.706070\pi\)
−0.603104 + 0.797662i \(0.706070\pi\)
\(402\) 0 0
\(403\) −1.80414e6 + 1.80414e6i −0.553359 + 0.553359i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.47046e6 1.47046e6i −0.440015 0.440015i
\(408\) 0 0
\(409\) 1.17358e6i 0.346899i −0.984843 0.173450i \(-0.944509\pi\)
0.984843 0.173450i \(-0.0554913\pi\)
\(410\) 0 0
\(411\) 1.68974e6i 0.493419i
\(412\) 0 0
\(413\) 3.42002e6 + 3.42002e6i 0.986627 + 0.986627i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 6.16652e6 6.16652e6i 1.73660 1.73660i
\(418\) 0 0
\(419\) −1.60340e6 −0.446177 −0.223089 0.974798i \(-0.571614\pi\)
−0.223089 + 0.974798i \(0.571614\pi\)
\(420\) 0 0
\(421\) −3.71020e6 −1.02022 −0.510108 0.860111i \(-0.670394\pi\)
−0.510108 + 0.860111i \(0.670394\pi\)
\(422\) 0 0
\(423\) −77767.4 + 77767.4i −0.0211323 + 0.0211323i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −220526. 220526.i −0.0585315 0.0585315i
\(428\) 0 0
\(429\) 1.47735e6i 0.387562i
\(430\) 0 0
\(431\) 2.29101e6i 0.594065i 0.954867 + 0.297033i \(0.0959971\pi\)
−0.954867 + 0.297033i \(0.904003\pi\)
\(432\) 0 0
\(433\) 972567. + 972567.i 0.249287 + 0.249287i 0.820678 0.571391i \(-0.193596\pi\)
−0.571391 + 0.820678i \(0.693596\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 78527.9 78527.9i 0.0196707 0.0196707i
\(438\) 0 0
\(439\) 3.49284e6 0.865002 0.432501 0.901633i \(-0.357631\pi\)
0.432501 + 0.901633i \(0.357631\pi\)
\(440\) 0 0
\(441\) 1.08902e7 2.66648
\(442\) 0 0
\(443\) 4.91464e6 4.91464e6i 1.18982 1.18982i 0.212707 0.977116i \(-0.431772\pi\)
0.977116 0.212707i \(-0.0682281\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 675761. + 675761.i 0.159965 + 0.159965i
\(448\) 0 0
\(449\) 3.79845e6i 0.889182i −0.895734 0.444591i \(-0.853349\pi\)
0.895734 0.444591i \(-0.146651\pi\)
\(450\) 0 0
\(451\) 1.42122e6i 0.329019i
\(452\) 0 0
\(453\) −6.15031e6 6.15031e6i −1.40816 1.40816i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.14446e6 + 4.14446e6i −0.928276 + 0.928276i −0.997595 0.0693189i \(-0.977917\pi\)
0.0693189 + 0.997595i \(0.477917\pi\)
\(458\) 0 0
\(459\) 4.47532e6 0.991499
\(460\) 0 0
\(461\) 1.33612e6 0.292815 0.146407 0.989224i \(-0.453229\pi\)
0.146407 + 0.989224i \(0.453229\pi\)
\(462\) 0 0
\(463\) 3.66263e6 3.66263e6i 0.794036 0.794036i −0.188112 0.982148i \(-0.560237\pi\)
0.982148 + 0.188112i \(0.0602367\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.58287e6 4.58287e6i −0.972401 0.972401i 0.0272279 0.999629i \(-0.491332\pi\)
−0.999629 + 0.0272279i \(0.991332\pi\)
\(468\) 0 0
\(469\) 8.92061e6i 1.87267i
\(470\) 0 0
\(471\) 1.29017e6i 0.267975i
\(472\) 0 0
\(473\) −183211. 183211.i −0.0376530 0.0376530i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −8.07553e6 + 8.07553e6i −1.62508 + 1.62508i
\(478\) 0 0
\(479\) −7.00075e6 −1.39414 −0.697069 0.717004i \(-0.745513\pi\)
−0.697069 + 0.717004i \(0.745513\pi\)
\(480\) 0 0
\(481\) 7.72932e6 1.52328
\(482\) 0 0
\(483\) 1.48300e7 1.48300e7i 2.89249 2.89249i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −3.98892e6 3.98892e6i −0.762137 0.762137i 0.214571 0.976708i \(-0.431165\pi\)
−0.976708 + 0.214571i \(0.931165\pi\)
\(488\) 0 0
\(489\) 6.25818e6i 1.18352i
\(490\) 0 0
\(491\) 2.30379e6i 0.431260i −0.976475 0.215630i \(-0.930819\pi\)
0.976475 0.215630i \(-0.0691805\pi\)
\(492\) 0 0
\(493\) 3.91352e6 + 3.91352e6i 0.725188 + 0.725188i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.72690e6 3.72690e6i 0.676795 0.676795i
\(498\) 0 0
\(499\) 2.47982e6 0.445829 0.222914 0.974838i \(-0.428443\pi\)
0.222914 + 0.974838i \(0.428443\pi\)
\(500\) 0 0
\(501\) 6.04816e6 1.07654
\(502\) 0 0
\(503\) −4.08077e6 + 4.08077e6i −0.719154 + 0.719154i −0.968432 0.249278i \(-0.919807\pi\)
0.249278 + 0.968432i \(0.419807\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.53768e6 + 2.53768e6i 0.438446 + 0.438446i
\(508\) 0 0
\(509\) 2.80205e6i 0.479382i 0.970849 + 0.239691i \(0.0770461\pi\)
−0.970849 + 0.239691i \(0.922954\pi\)
\(510\) 0 0
\(511\) 8.80554e6i 1.49178i
\(512\) 0 0
\(513\) −54665.6 54665.6i −0.00917109 0.00917109i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 27925.3 27925.3i 0.00459485 0.00459485i
\(518\) 0 0
\(519\) 1.30094e7 2.12001
\(520\) 0 0
\(521\) 144765. 0.0233652 0.0116826 0.999932i \(-0.496281\pi\)
0.0116826 + 0.999932i \(0.496281\pi\)
\(522\) 0 0
\(523\) −3.79906e6 + 3.79906e6i −0.607327 + 0.607327i −0.942247 0.334920i \(-0.891291\pi\)
0.334920 + 0.942247i \(0.391291\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.21903e6 6.21903e6i −0.975430 0.975430i
\(528\) 0 0
\(529\) 9.05423e6i 1.40674i
\(530\) 0 0
\(531\) 7.88060e6i 1.21289i
\(532\) 0 0
\(533\) 3.73526e6 + 3.73526e6i 0.569511 + 0.569511i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 6.65720e6 6.65720e6i 0.996221 0.996221i
\(538\) 0 0
\(539\) −3.91053e6 −0.579781
\(540\) 0 0
\(541\) −1.17321e7 −1.72339 −0.861695 0.507426i \(-0.830597\pi\)
−0.861695 + 0.507426i \(0.830597\pi\)
\(542\) 0 0
\(543\) 5.74683e6 5.74683e6i 0.836428 0.836428i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −5.24186e6 5.24186e6i −0.749060 0.749060i 0.225243 0.974303i \(-0.427683\pi\)
−0.974303 + 0.225243i \(0.927683\pi\)
\(548\) 0 0
\(549\) 508148.i 0.0719547i
\(550\) 0 0
\(551\) 95606.6i 0.0134156i
\(552\) 0 0
\(553\) −3.60806e6 3.60806e6i −0.501719 0.501719i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −7.61782e6 + 7.61782e6i −1.04038 + 1.04038i −0.0412325 + 0.999150i \(0.513128\pi\)
−0.999150 + 0.0412325i \(0.986872\pi\)
\(558\) 0 0
\(559\) 963030. 0.130350
\(560\) 0 0
\(561\) −5.09258e6 −0.683173
\(562\) 0 0
\(563\) −86277.5 + 86277.5i −0.0114717 + 0.0114717i −0.712819 0.701348i \(-0.752582\pi\)
0.701348 + 0.712819i \(0.252582\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.96938e6 + 2.96938e6i 0.387890 + 0.387890i
\(568\) 0 0
\(569\) 5.02097e6i 0.650140i −0.945690 0.325070i \(-0.894612\pi\)
0.945690 0.325070i \(-0.105388\pi\)
\(570\) 0 0
\(571\) 8.08321e6i 1.03751i 0.854922 + 0.518756i \(0.173605\pi\)
−0.854922 + 0.518756i \(0.826395\pi\)
\(572\) 0 0
\(573\) −2.84547e6 2.84547e6i −0.362049 0.362049i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 3.25025e6 3.25025e6i 0.406422 0.406422i −0.474067 0.880489i \(-0.657214\pi\)
0.880489 + 0.474067i \(0.157214\pi\)
\(578\) 0 0
\(579\) −2.00859e6 −0.248997
\(580\) 0 0
\(581\) −5.83018e6 −0.716542
\(582\) 0 0
\(583\) 2.89982e6 2.89982e6i 0.353346 0.353346i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.79350e6 + 3.79350e6i 0.454407 + 0.454407i 0.896814 0.442407i \(-0.145875\pi\)
−0.442407 + 0.896814i \(0.645875\pi\)
\(588\) 0 0
\(589\) 151930.i 0.0180449i
\(590\) 0 0
\(591\) 1.33744e7i 1.57509i
\(592\) 0 0
\(593\) −1.01322e7 1.01322e7i −1.18323 1.18323i −0.978904 0.204321i \(-0.934501\pi\)
−0.204321 0.978904i \(-0.565499\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 243044. 243044.i 0.0279093 0.0279093i
\(598\) 0 0
\(599\) 1.84016e6 0.209550 0.104775 0.994496i \(-0.466588\pi\)
0.104775 + 0.994496i \(0.466588\pi\)
\(600\) 0 0
\(601\) 2.07660e6 0.234513 0.117257 0.993102i \(-0.462590\pi\)
0.117257 + 0.993102i \(0.462590\pi\)
\(602\) 0 0
\(603\) −1.02777e7 + 1.02777e7i −1.15107 + 1.15107i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.47671e6 + 7.47671e6i 0.823643 + 0.823643i 0.986628 0.162986i \(-0.0521125\pi\)
−0.162986 + 0.986628i \(0.552112\pi\)
\(608\) 0 0
\(609\) 1.80553e7i 1.97270i
\(610\) 0 0
\(611\) 146786.i 0.0159068i
\(612\) 0 0
\(613\) −421185. 421185.i −0.0452712 0.0452712i 0.684109 0.729380i \(-0.260191\pi\)
−0.729380 + 0.684109i \(0.760191\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −4.79146e6 + 4.79146e6i −0.506705 + 0.506705i −0.913514 0.406808i \(-0.866642\pi\)
0.406808 + 0.913514i \(0.366642\pi\)
\(618\) 0 0
\(619\) 8.81805e6 0.925009 0.462505 0.886617i \(-0.346951\pi\)
0.462505 + 0.886617i \(0.346951\pi\)
\(620\) 0 0
\(621\) −1.07835e7 −1.12209
\(622\) 0 0
\(623\) 1.26029e6 1.26029e6i 0.130092 0.130092i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 62205.4 + 62205.4i 0.00631915 + 0.00631915i
\(628\) 0 0
\(629\) 2.66437e7i 2.68515i
\(630\) 0 0
\(631\) 4.57807e6i 0.457730i 0.973458 + 0.228865i \(0.0735014\pi\)
−0.973458 + 0.228865i \(0.926499\pi\)
\(632\) 0 0
\(633\) 7.59061e6 + 7.59061e6i 0.752952 + 0.752952i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.02777e7 1.02777e7i 1.00356 1.00356i
\(638\) 0 0
\(639\) −8.58774e6 −0.832007
\(640\) 0 0
\(641\) −1.62758e7 −1.56458 −0.782288 0.622917i \(-0.785947\pi\)
−0.782288 + 0.622917i \(0.785947\pi\)
\(642\) 0 0
\(643\) −2.99416e6 + 2.99416e6i −0.285593 + 0.285593i −0.835335 0.549742i \(-0.814726\pi\)
0.549742 + 0.835335i \(0.314726\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.92387e6 + 4.92387e6i 0.462430 + 0.462430i 0.899451 0.437021i \(-0.143967\pi\)
−0.437021 + 0.899451i \(0.643967\pi\)
\(648\) 0 0
\(649\) 2.82983e6i 0.263723i
\(650\) 0 0
\(651\) 2.86919e7i 2.65342i
\(652\) 0 0
\(653\) −1.05307e7 1.05307e7i −0.966441 0.966441i 0.0330138 0.999455i \(-0.489489\pi\)
−0.999455 + 0.0330138i \(0.989489\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.01451e7 + 1.01451e7i −0.916945 + 0.916945i
\(658\) 0 0
\(659\) 1.29023e7 1.15732 0.578659 0.815570i \(-0.303576\pi\)
0.578659 + 0.815570i \(0.303576\pi\)
\(660\) 0 0
\(661\) 6.60437e6 0.587933 0.293967 0.955816i \(-0.405025\pi\)
0.293967 + 0.955816i \(0.405025\pi\)
\(662\) 0 0
\(663\) 1.33843e7 1.33843e7i 1.18253 1.18253i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −9.42978e6 9.42978e6i −0.820705 0.820705i
\(668\) 0 0
\(669\) 1.43141e7i 1.23651i
\(670\) 0 0
\(671\) 182470.i 0.0156453i
\(672\) 0 0
\(673\) −1.01077e7 1.01077e7i −0.860230 0.860230i 0.131135 0.991365i \(-0.458138\pi\)
−0.991365 + 0.131135i \(0.958138\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5.24850e6 + 5.24850e6i −0.440112 + 0.440112i −0.892050 0.451938i \(-0.850733\pi\)
0.451938 + 0.892050i \(0.350733\pi\)
\(678\) 0 0
\(679\) 2.36502e7 1.96861
\(680\) 0 0
\(681\) 5.73207e6 0.473635
\(682\) 0 0
\(683\) −1.03798e7 + 1.03798e7i −0.851407 + 0.851407i −0.990306 0.138900i \(-0.955643\pi\)
0.138900 + 0.990306i \(0.455643\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.02148e7 1.02148e7i −0.825727 0.825727i
\(688\) 0 0
\(689\) 1.52426e7i 1.22324i
\(690\) 0 0
\(691\) 6.06155e6i 0.482935i −0.970409 0.241468i \(-0.922371\pi\)
0.970409 0.241468i \(-0.0776288\pi\)
\(692\) 0 0
\(693\) 6.97412e6 + 6.97412e6i 0.551641 + 0.551641i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1.28758e7 + 1.28758e7i −1.00390 + 1.00390i
\(698\) 0 0
\(699\) −2.93896e7 −2.27510
\(700\) 0 0
\(701\) 2.87503e6 0.220977 0.110489 0.993877i \(-0.464758\pi\)
0.110489 + 0.993877i \(0.464758\pi\)
\(702\) 0 0
\(703\) 325450. 325450.i 0.0248368 0.0248368i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −212566. 212566.i −0.0159936 0.0159936i
\(708\) 0 0
\(709\) 528547.i 0.0394883i 0.999805 + 0.0197441i \(0.00628516\pi\)
−0.999805 + 0.0197441i \(0.993715\pi\)
\(710\) 0 0
\(711\) 8.31390e6i 0.616780i
\(712\) 0 0
\(713\) 1.49850e7 + 1.49850e7i 1.10391 + 1.10391i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.24548e6 1.24548e6i 0.0904769 0.0904769i
\(718\) 0 0
\(719\) −1.66055e7 −1.19793 −0.598963 0.800777i \(-0.704420\pi\)
−0.598963 + 0.800777i \(0.704420\pi\)
\(720\) 0 0
\(721\) 3.92498e7 2.81190
\(722\) 0 0
\(723\) −6.20523e6 + 6.20523e6i −0.441481 + 0.441481i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.54075e7 + 1.54075e7i 1.08118 + 1.08118i 0.996400 + 0.0847783i \(0.0270182\pi\)
0.0847783 + 0.996400i \(0.472982\pi\)
\(728\) 0 0
\(729\) 2.31237e7i 1.61153i
\(730\) 0 0
\(731\) 3.31966e6i 0.229773i
\(732\) 0 0
\(733\) 1.69113e7 + 1.69113e7i 1.16256 + 1.16256i 0.983913 + 0.178650i \(0.0571731\pi\)
0.178650 + 0.983913i \(0.442827\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.69059e6 3.69059e6i 0.250281 0.250281i
\(738\) 0 0
\(739\) −2.76180e7 −1.86029 −0.930147 0.367187i \(-0.880321\pi\)
−0.930147 + 0.367187i \(0.880321\pi\)
\(740\) 0 0
\(741\) −326976. −0.0218761
\(742\) 0 0
\(743\) −1.71922e7 + 1.71922e7i −1.14251 + 1.14251i −0.154521 + 0.987990i \(0.549383\pi\)
−0.987990 + 0.154521i \(0.950617\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 6.71712e6 + 6.71712e6i 0.440435 + 0.440435i
\(748\) 0 0
\(749\) 5.33954e6i 0.347776i
\(750\) 0 0
\(751\) 1.11896e7i 0.723961i −0.932186 0.361981i \(-0.882101\pi\)
0.932186 0.361981i \(-0.117899\pi\)
\(752\) 0 0
\(753\) −3.61836e6 3.61836e6i −0.232554 0.232554i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.17247e6 1.17247e6i 0.0743637 0.0743637i −0.668947 0.743310i \(-0.733255\pi\)
0.743310 + 0.668947i \(0.233255\pi\)
\(758\) 0 0
\(759\) 1.22708e7 0.773156
\(760\) 0 0
\(761\) −9.79138e6 −0.612890 −0.306445 0.951888i \(-0.599140\pi\)
−0.306445 + 0.951888i \(0.599140\pi\)
\(762\) 0 0
\(763\) 1.62980e7 1.62980e7i 1.01350 1.01350i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −7.43734e6 7.43734e6i −0.456488 0.456488i
\(768\) 0 0
\(769\) 1.10059e7i 0.671138i 0.942016 + 0.335569i \(0.108929\pi\)
−0.942016 + 0.335569i \(0.891071\pi\)
\(770\) 0 0
\(771\) 7.89277e6i 0.478182i
\(772\) 0 0
\(773\) −8.90959e6 8.90959e6i −0.536302 0.536302i 0.386139 0.922441i \(-0.373809\pi\)
−0.922441 + 0.386139i \(0.873809\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 6.14612e7 6.14612e7i 3.65215 3.65215i
\(778\) 0 0
\(779\) 314553. 0.0185716
\(780\) 0 0
\(781\) 3.08375e6 0.180906
\(782\) 0 0
\(783\) −6.56435e6 + 6.56435e6i −0.382638 + 0.382638i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 4.88094e6 + 4.88094e6i 0.280910 + 0.280910i 0.833472 0.552562i \(-0.186350\pi\)
−0.552562 + 0.833472i \(0.686350\pi\)
\(788\) 0 0
\(789\) 3.26905e7i 1.86952i
\(790\) 0 0
\(791\) 1.95525e7i 1.11112i
\(792\) 0 0
\(793\) 479566. + 479566.i 0.0270811 + 0.0270811i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −7.86412e6 + 7.86412e6i −0.438535 + 0.438535i −0.891519 0.452984i \(-0.850360\pi\)
0.452984 + 0.891519i \(0.350360\pi\)
\(798\) 0 0
\(799\) −505987. −0.0280396
\(800\) 0 0
\(801\) −2.90403e6 −0.159926
\(802\) 0 0
\(803\) 3.64299e6 3.64299e6i 0.199374 0.199374i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 2.70724e7 + 2.70724e7i 1.46333 + 1.46333i
\(808\) 0 0
\(809\) 3.59398e6i 0.193065i 0.995330 + 0.0965327i \(0.0307752\pi\)
−0.995330 + 0.0965327i \(0.969225\pi\)
\(810\) 0 0
\(811\) 1.96828e7i 1.05084i 0.850844 + 0.525418i \(0.176091\pi\)
−0.850844 + 0.525418i \(0.823909\pi\)
\(812\) 0 0
\(813\) 1.71334e7 + 1.71334e7i 0.909113 + 0.909113i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 40549.3 40549.3i 0.00212534 0.00212534i
\(818\) 0 0
\(819\) −3.66587e7 −1.90971
\(820\) 0 0
\(821\) −6.87305e6 −0.355870 −0.177935 0.984042i \(-0.556942\pi\)
−0.177935 + 0.984042i \(0.556942\pi\)
\(822\) 0 0
\(823\) −2.46199e7 + 2.46199e7i −1.26703 + 1.26703i −0.319415 + 0.947615i \(0.603486\pi\)
−0.947615 + 0.319415i \(0.896514\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.64914e7 2.64914e7i −1.34692 1.34692i −0.888988 0.457931i \(-0.848591\pi\)
−0.457931 0.888988i \(-0.651409\pi\)
\(828\) 0 0
\(829\) 4.71839e6i 0.238456i −0.992867 0.119228i \(-0.961958\pi\)
0.992867 0.119228i \(-0.0380419\pi\)
\(830\) 0 0
\(831\) 6.02745e6i 0.302782i
\(832\) 0 0
\(833\) 3.54280e7 + 3.54280e7i 1.76903 + 1.76903i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.04315e7 1.04315e7i 0.514675 0.514675i
\(838\) 0 0
\(839\) −2.18851e6 −0.107336 −0.0536678 0.998559i \(-0.517091\pi\)
−0.0536678 + 0.998559i \(0.517091\pi\)
\(840\) 0 0
\(841\) 9.03051e6 0.440273
\(842\) 0 0
\(843\) −3.76564e7 + 3.76564e7i −1.82503 + 1.82503i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.23102e7 + 2.23102e7i 1.06855 + 1.06855i
\(848\) 0 0
\(849\) 1.73657e7i 0.826842i
\(850\) 0 0
\(851\) 6.41990e7i 3.03882i
\(852\) 0 0
\(853\) 1.82806e7 + 1.82806e7i 0.860238 + 0.860238i 0.991366 0.131127i \(-0.0418596\pi\)
−0.131127 + 0.991366i \(0.541860\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.39122e7 + 2.39122e7i −1.11216 + 1.11216i −0.119303 + 0.992858i \(0.538066\pi\)
−0.992858 + 0.119303i \(0.961934\pi\)
\(858\) 0 0
\(859\) −1.15768e7 −0.535309 −0.267654 0.963515i \(-0.586249\pi\)
−0.267654 + 0.963515i \(0.586249\pi\)
\(860\) 0 0
\(861\) 5.94032e7 2.73088
\(862\) 0 0
\(863\) −3.08022e6 + 3.08022e6i −0.140784 + 0.140784i −0.773987 0.633202i \(-0.781740\pi\)
0.633202 + 0.773987i \(0.281740\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.15846e7 + 2.15846e7i 0.975204 + 0.975204i
\(868\) 0 0
\(869\) 2.98542e6i 0.134108i
\(870\) 0 0
\(871\) 1.93992e7i 0.866440i
\(872\) 0 0
\(873\) −2.72481e7 2.72481e7i −1.21004 1.21004i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.24010e7 1.24010e7i 0.544449 0.544449i −0.380381 0.924830i \(-0.624207\pi\)
0.924830 + 0.380381i \(0.124207\pi\)
\(878\) 0 0
\(879\) 1.58301e6 0.0691053
\(880\) 0 0
\(881\) 8.61119e6 0.373786 0.186893 0.982380i \(-0.440158\pi\)
0.186893 + 0.982380i \(0.440158\pi\)
\(882\) 0 0
\(883\) 2.44433e7 2.44433e7i 1.05502 1.05502i 0.0566197 0.998396i \(-0.481968\pi\)
0.998396 0.0566197i \(-0.0180323\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.94644e7 1.94644e7i −0.830678 0.830678i 0.156932 0.987609i \(-0.449840\pi\)
−0.987609 + 0.156932i \(0.949840\pi\)
\(888\) 0 0
\(889\) 4.64345e7i 1.97055i
\(890\) 0 0
\(891\) 2.45696e6i 0.103682i
\(892\) 0 0
\(893\) 6180.58 + 6180.58i 0.000259359 + 0.000259359i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −3.22500e7 + 3.22500e7i −1.33828 + 1.33828i
\(898\) 0 0
\(899\) 1.82440e7 0.752873
\(900\) 0 0
\(901\) −5.25427e7 −2.15626
\(902\) 0 0
\(903\) 7.65772e6 7.65772e6i 0.312522 0.312522i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 2.02098e7 + 2.02098e7i 0.815724 + 0.815724i 0.985485 0.169761i \(-0.0542997\pi\)
−0.169761 + 0.985485i \(0.554300\pi\)
\(908\) 0 0
\(909\) 489807.i 0.0196614i
\(910\) 0 0
\(911\) 2.45272e7i 0.979155i 0.871960 + 0.489577i \(0.162849\pi\)
−0.871960 + 0.489577i \(0.837151\pi\)
\(912\) 0 0
\(913\) −2.41204e6 2.41204e6i −0.0957650 0.0957650i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.36516e7 + 3.36516e7i −1.32155 + 1.32155i
\(918\) 0 0
\(919\) −1.34572e7 −0.525613 −0.262806 0.964849i \(-0.584648\pi\)
−0.262806 + 0.964849i \(0.584648\pi\)
\(920\) 0 0
\(921\) −4.82676e7 −1.87502
\(922\) 0 0
\(923\) −8.10471e6 + 8.10471e6i −0.313136 + 0.313136i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −4.52208e7 4.52208e7i −1.72838 1.72838i
\(928\) 0 0
\(929\) 1.09539e7i 0.416417i 0.978084 + 0.208209i \(0.0667633\pi\)
−0.978084 + 0.208209i \(0.933237\pi\)
\(930\) 0 0
\(931\) 865500.i 0.0327260i
\(932\) 0 0
\(933\) −3.18375e7 3.18375e7i −1.19739 1.19739i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.46236e7 2.46236e7i 0.916224 0.916224i −0.0805282 0.996752i \(-0.525661\pi\)
0.996752 + 0.0805282i \(0.0256607\pi\)
\(938\) 0 0
\(939\) 7.68946e7 2.84598
\(940\) 0 0
\(941\) 2.79774e7 1.02999 0.514995 0.857193i \(-0.327794\pi\)
0.514995 + 0.857193i \(0.327794\pi\)
\(942\) 0 0
\(943\) 3.10247e7 3.10247e7i 1.13613 1.13613i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.34277e7 1.34277e7i −0.486550 0.486550i 0.420666 0.907216i \(-0.361796\pi\)
−0.907216 + 0.420666i \(0.861796\pi\)
\(948\) 0 0
\(949\) 1.91490e7i 0.690208i
\(950\) 0 0
\(951\) 4.68619e7i 1.68023i
\(952\) 0 0
\(953\) 2.47107e7 + 2.47107e7i 0.881359 + 0.881359i 0.993673 0.112313i \(-0.0358261\pi\)
−0.112313 + 0.993673i \(0.535826\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 7.46974e6 7.46974e6i 0.263649 0.263649i
\(958\) 0 0
\(959\) −1.50561e7 −0.528648
\(960\) 0 0
\(961\) −362678. −0.0126681
\(962\) 0 0
\(963\) −6.15184e6 + 6.15184e6i −0.213766 + 0.213766i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 2.49726e7 + 2.49726e7i 0.858811 + 0.858811i 0.991198 0.132387i \(-0.0422641\pi\)
−0.132387 + 0.991198i \(0.542264\pi\)
\(968\) 0 0
\(969\) 1.12712e6i 0.0385620i
\(970\) 0 0
\(971\) 1.77496e7i 0.604143i 0.953285 + 0.302071i \(0.0976781\pi\)
−0.953285 + 0.302071i \(0.902322\pi\)
\(972\) 0 0
\(973\) −5.49457e7 5.49457e7i −1.86059 1.86059i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.06328e7 2.06328e7i 0.691548 0.691548i −0.271024 0.962572i \(-0.587362\pi\)
0.962572 + 0.271024i \(0.0873624\pi\)
\(978\) 0 0
\(979\) 1.04280e6 0.0347733
\(980\) 0 0
\(981\) −3.75548e7 −1.24593
\(982\) 0 0
\(983\) −1.82547e7 + 1.82547e7i −0.602546 + 0.602546i −0.940987 0.338442i \(-0.890100\pi\)
0.338442 + 0.940987i \(0.390100\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 1.16720e6 + 1.16720e6i 0.0381375 + 0.0381375i
\(988\) 0 0
\(989\) 7.99884e6i 0.260038i
\(990\) 0 0
\(991\) 3.83326e7i 1.23989i 0.784644 + 0.619947i \(0.212846\pi\)
−0.784644 + 0.619947i \(0.787154\pi\)
\(992\) 0 0
\(993\) −4.15865e7 4.15865e7i −1.33838 1.33838i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.22874e7 + 1.22874e7i −0.391491 + 0.391491i −0.875219 0.483727i \(-0.839283\pi\)
0.483727 + 0.875219i \(0.339283\pi\)
\(998\) 0 0
\(999\) −4.46909e7 −1.41679
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 400.6.n.g.143.2 20
4.3 odd 2 inner 400.6.n.g.143.9 20
5.2 odd 4 inner 400.6.n.g.207.9 20
5.3 odd 4 80.6.n.d.47.2 20
5.4 even 2 80.6.n.d.63.9 yes 20
20.3 even 4 80.6.n.d.47.9 yes 20
20.7 even 4 inner 400.6.n.g.207.2 20
20.19 odd 2 80.6.n.d.63.2 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
80.6.n.d.47.2 20 5.3 odd 4
80.6.n.d.47.9 yes 20 20.3 even 4
80.6.n.d.63.2 yes 20 20.19 odd 2
80.6.n.d.63.9 yes 20 5.4 even 2
400.6.n.g.143.2 20 1.1 even 1 trivial
400.6.n.g.143.9 20 4.3 odd 2 inner
400.6.n.g.207.2 20 20.7 even 4 inner
400.6.n.g.207.9 20 5.2 odd 4 inner