Properties

Label 2475.2.a.z.1.2
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2475,2,Mod(1,2475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2475.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.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: \(19.7629745003\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 825)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.363328\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.36333 q^{2} -0.141336 q^{4} +2.50466 q^{7} +2.91934 q^{8} +O(q^{10})\) \(q-1.36333 q^{2} -0.141336 q^{4} +2.50466 q^{7} +2.91934 q^{8} -1.00000 q^{11} +1.14134 q^{13} -3.41468 q^{14} -3.69735 q^{16} -7.64600 q^{17} +1.77801 q^{19} +1.36333 q^{22} -1.41468 q^{23} -1.55602 q^{26} -0.354000 q^{28} +0.726656 q^{29} +2.85866 q^{31} -0.797984 q^{32} +10.4240 q^{34} -8.42401 q^{37} -2.42401 q^{38} -0.636672 q^{41} -12.6974 q^{43} +0.141336 q^{44} +1.92867 q^{46} -6.14134 q^{47} -0.726656 q^{49} -0.161312 q^{52} +12.0187 q^{53} +7.31198 q^{56} -0.990671 q^{58} +3.41468 q^{59} +4.59465 q^{61} -3.89730 q^{62} +8.48262 q^{64} -9.32131 q^{67} +1.08066 q^{68} -5.85866 q^{71} +7.55602 q^{73} +11.4847 q^{74} -0.251297 q^{76} -2.50466 q^{77} +6.91934 q^{79} +0.867993 q^{82} +6.17997 q^{83} +17.3107 q^{86} -2.91934 q^{88} -3.45331 q^{89} +2.85866 q^{91} +0.199945 q^{92} +8.37266 q^{94} -19.4626 q^{97} +0.990671 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 8 q^{4} - 3 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 8 q^{4} - 3 q^{7} - 6 q^{8} - 3 q^{11} - 5 q^{13} - 6 q^{14} + 10 q^{16} - 4 q^{17} - q^{19} + 2 q^{22} + 8 q^{26} - 20 q^{28} - 2 q^{29} + 17 q^{31} - 34 q^{32} + 6 q^{34} + 18 q^{38} - 4 q^{41} - 17 q^{43} - 8 q^{44} - 30 q^{46} - 10 q^{47} + 2 q^{49} - 30 q^{52} - 6 q^{53} + 22 q^{56} - 24 q^{58} + 6 q^{59} - 3 q^{61} - 16 q^{62} + 34 q^{64} - 7 q^{67} + 18 q^{68} - 26 q^{71} + 10 q^{73} - 14 q^{74} - 24 q^{76} + 3 q^{77} + 6 q^{79} - 10 q^{82} + 6 q^{83} - 28 q^{86} + 6 q^{88} - 2 q^{89} + 17 q^{91} + 26 q^{92} + 2 q^{94} - 29 q^{97} + 24 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.36333 −0.964019 −0.482009 0.876166i \(-0.660093\pi\)
−0.482009 + 0.876166i \(0.660093\pi\)
\(3\) 0 0
\(4\) −0.141336 −0.0706681
\(5\) 0 0
\(6\) 0 0
\(7\) 2.50466 0.946674 0.473337 0.880881i \(-0.343049\pi\)
0.473337 + 0.880881i \(0.343049\pi\)
\(8\) 2.91934 1.03214
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.14134 0.316550 0.158275 0.987395i \(-0.449407\pi\)
0.158275 + 0.987395i \(0.449407\pi\)
\(14\) −3.41468 −0.912612
\(15\) 0 0
\(16\) −3.69735 −0.924338
\(17\) −7.64600 −1.85443 −0.927214 0.374533i \(-0.877803\pi\)
−0.927214 + 0.374533i \(0.877803\pi\)
\(18\) 0 0
\(19\) 1.77801 0.407903 0.203951 0.978981i \(-0.434622\pi\)
0.203951 + 0.978981i \(0.434622\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.36333 0.290663
\(23\) −1.41468 −0.294981 −0.147491 0.989063i \(-0.547120\pi\)
−0.147491 + 0.989063i \(0.547120\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.55602 −0.305160
\(27\) 0 0
\(28\) −0.354000 −0.0668996
\(29\) 0.726656 0.134937 0.0674684 0.997721i \(-0.478508\pi\)
0.0674684 + 0.997721i \(0.478508\pi\)
\(30\) 0 0
\(31\) 2.85866 0.513431 0.256716 0.966487i \(-0.417360\pi\)
0.256716 + 0.966487i \(0.417360\pi\)
\(32\) −0.797984 −0.141065
\(33\) 0 0
\(34\) 10.4240 1.78770
\(35\) 0 0
\(36\) 0 0
\(37\) −8.42401 −1.38490 −0.692449 0.721467i \(-0.743468\pi\)
−0.692449 + 0.721467i \(0.743468\pi\)
\(38\) −2.42401 −0.393226
\(39\) 0 0
\(40\) 0 0
\(41\) −0.636672 −0.0994314 −0.0497157 0.998763i \(-0.515832\pi\)
−0.0497157 + 0.998763i \(0.515832\pi\)
\(42\) 0 0
\(43\) −12.6974 −1.93633 −0.968164 0.250317i \(-0.919465\pi\)
−0.968164 + 0.250317i \(0.919465\pi\)
\(44\) 0.141336 0.0213072
\(45\) 0 0
\(46\) 1.92867 0.284367
\(47\) −6.14134 −0.895806 −0.447903 0.894082i \(-0.647829\pi\)
−0.447903 + 0.894082i \(0.647829\pi\)
\(48\) 0 0
\(49\) −0.726656 −0.103808
\(50\) 0 0
\(51\) 0 0
\(52\) −0.161312 −0.0223700
\(53\) 12.0187 1.65089 0.825445 0.564483i \(-0.190924\pi\)
0.825445 + 0.564483i \(0.190924\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 7.31198 0.977104
\(57\) 0 0
\(58\) −0.990671 −0.130082
\(59\) 3.41468 0.444553 0.222277 0.974984i \(-0.428651\pi\)
0.222277 + 0.974984i \(0.428651\pi\)
\(60\) 0 0
\(61\) 4.59465 0.588285 0.294142 0.955762i \(-0.404966\pi\)
0.294142 + 0.955762i \(0.404966\pi\)
\(62\) −3.89730 −0.494957
\(63\) 0 0
\(64\) 8.48262 1.06033
\(65\) 0 0
\(66\) 0 0
\(67\) −9.32131 −1.13878 −0.569389 0.822068i \(-0.692820\pi\)
−0.569389 + 0.822068i \(0.692820\pi\)
\(68\) 1.08066 0.131049
\(69\) 0 0
\(70\) 0 0
\(71\) −5.85866 −0.695295 −0.347648 0.937625i \(-0.613019\pi\)
−0.347648 + 0.937625i \(0.613019\pi\)
\(72\) 0 0
\(73\) 7.55602 0.884365 0.442182 0.896925i \(-0.354204\pi\)
0.442182 + 0.896925i \(0.354204\pi\)
\(74\) 11.4847 1.33507
\(75\) 0 0
\(76\) −0.251297 −0.0288257
\(77\) −2.50466 −0.285433
\(78\) 0 0
\(79\) 6.91934 0.778487 0.389244 0.921135i \(-0.372736\pi\)
0.389244 + 0.921135i \(0.372736\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0.867993 0.0958537
\(83\) 6.17997 0.678340 0.339170 0.940725i \(-0.389854\pi\)
0.339170 + 0.940725i \(0.389854\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 17.3107 1.86666
\(87\) 0 0
\(88\) −2.91934 −0.311203
\(89\) −3.45331 −0.366050 −0.183025 0.983108i \(-0.558589\pi\)
−0.183025 + 0.983108i \(0.558589\pi\)
\(90\) 0 0
\(91\) 2.85866 0.299669
\(92\) 0.199945 0.0208457
\(93\) 0 0
\(94\) 8.37266 0.863574
\(95\) 0 0
\(96\) 0 0
\(97\) −19.4626 −1.97613 −0.988066 0.154032i \(-0.950774\pi\)
−0.988066 + 0.154032i \(0.950774\pi\)
\(98\) 0.990671 0.100073
\(99\) 0 0
\(100\) 0 0
\(101\) 8.37266 0.833111 0.416555 0.909110i \(-0.363237\pi\)
0.416555 + 0.909110i \(0.363237\pi\)
\(102\) 0 0
\(103\) 2.54669 0.250933 0.125466 0.992098i \(-0.459957\pi\)
0.125466 + 0.992098i \(0.459957\pi\)
\(104\) 3.33195 0.326725
\(105\) 0 0
\(106\) −16.3854 −1.59149
\(107\) 14.5653 1.40808 0.704042 0.710158i \(-0.251376\pi\)
0.704042 + 0.710158i \(0.251376\pi\)
\(108\) 0 0
\(109\) 2.41468 0.231284 0.115642 0.993291i \(-0.463107\pi\)
0.115642 + 0.993291i \(0.463107\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −9.26063 −0.875047
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.102703 −0.00953572
\(117\) 0 0
\(118\) −4.65533 −0.428558
\(119\) −19.1507 −1.75554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −6.26401 −0.567117
\(123\) 0 0
\(124\) −0.404032 −0.0362832
\(125\) 0 0
\(126\) 0 0
\(127\) −15.2020 −1.34896 −0.674480 0.738293i \(-0.735632\pi\)
−0.674480 + 0.738293i \(0.735632\pi\)
\(128\) −9.96862 −0.881110
\(129\) 0 0
\(130\) 0 0
\(131\) −18.7453 −1.63779 −0.818893 0.573946i \(-0.805412\pi\)
−0.818893 + 0.573946i \(0.805412\pi\)
\(132\) 0 0
\(133\) 4.45331 0.386151
\(134\) 12.7080 1.09780
\(135\) 0 0
\(136\) −22.3213 −1.91404
\(137\) −8.74531 −0.747163 −0.373581 0.927597i \(-0.621870\pi\)
−0.373581 + 0.927597i \(0.621870\pi\)
\(138\) 0 0
\(139\) −22.3013 −1.89157 −0.945787 0.324787i \(-0.894707\pi\)
−0.945787 + 0.324787i \(0.894707\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.98728 0.670278
\(143\) −1.14134 −0.0954433
\(144\) 0 0
\(145\) 0 0
\(146\) −10.3013 −0.852544
\(147\) 0 0
\(148\) 1.19062 0.0978681
\(149\) 15.9287 1.30493 0.652464 0.757820i \(-0.273735\pi\)
0.652464 + 0.757820i \(0.273735\pi\)
\(150\) 0 0
\(151\) −9.88665 −0.804564 −0.402282 0.915516i \(-0.631783\pi\)
−0.402282 + 0.915516i \(0.631783\pi\)
\(152\) 5.19062 0.421015
\(153\) 0 0
\(154\) 3.41468 0.275163
\(155\) 0 0
\(156\) 0 0
\(157\) −0.132007 −0.0105353 −0.00526767 0.999986i \(-0.501677\pi\)
−0.00526767 + 0.999986i \(0.501677\pi\)
\(158\) −9.43334 −0.750476
\(159\) 0 0
\(160\) 0 0
\(161\) −3.54330 −0.279251
\(162\) 0 0
\(163\) −4.31198 −0.337740 −0.168870 0.985638i \(-0.554012\pi\)
−0.168870 + 0.985638i \(0.554012\pi\)
\(164\) 0.0899847 0.00702663
\(165\) 0 0
\(166\) −8.42533 −0.653932
\(167\) 11.2920 0.873801 0.436901 0.899510i \(-0.356076\pi\)
0.436901 + 0.899510i \(0.356076\pi\)
\(168\) 0 0
\(169\) −11.6974 −0.899796
\(170\) 0 0
\(171\) 0 0
\(172\) 1.79459 0.136837
\(173\) −21.6460 −1.64571 −0.822857 0.568248i \(-0.807621\pi\)
−0.822857 + 0.568248i \(0.807621\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.69735 0.278698
\(177\) 0 0
\(178\) 4.70800 0.352879
\(179\) −16.1413 −1.20646 −0.603230 0.797567i \(-0.706120\pi\)
−0.603230 + 0.797567i \(0.706120\pi\)
\(180\) 0 0
\(181\) 20.7546 1.54268 0.771340 0.636423i \(-0.219587\pi\)
0.771340 + 0.636423i \(0.219587\pi\)
\(182\) −3.89730 −0.288887
\(183\) 0 0
\(184\) −4.12994 −0.304463
\(185\) 0 0
\(186\) 0 0
\(187\) 7.64600 0.559131
\(188\) 0.867993 0.0633049
\(189\) 0 0
\(190\) 0 0
\(191\) −3.31198 −0.239646 −0.119823 0.992795i \(-0.538233\pi\)
−0.119823 + 0.992795i \(0.538233\pi\)
\(192\) 0 0
\(193\) −2.13201 −0.153465 −0.0767326 0.997052i \(-0.524449\pi\)
−0.0767326 + 0.997052i \(0.524449\pi\)
\(194\) 26.5340 1.90503
\(195\) 0 0
\(196\) 0.102703 0.00733591
\(197\) −17.9287 −1.27737 −0.638683 0.769470i \(-0.720520\pi\)
−0.638683 + 0.769470i \(0.720520\pi\)
\(198\) 0 0
\(199\) 15.1600 1.07466 0.537332 0.843371i \(-0.319432\pi\)
0.537332 + 0.843371i \(0.319432\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −11.4147 −0.803134
\(203\) 1.82003 0.127741
\(204\) 0 0
\(205\) 0 0
\(206\) −3.47197 −0.241904
\(207\) 0 0
\(208\) −4.21992 −0.292599
\(209\) −1.77801 −0.122987
\(210\) 0 0
\(211\) −13.3213 −0.917076 −0.458538 0.888675i \(-0.651627\pi\)
−0.458538 + 0.888675i \(0.651627\pi\)
\(212\) −1.69867 −0.116665
\(213\) 0 0
\(214\) −19.8573 −1.35742
\(215\) 0 0
\(216\) 0 0
\(217\) 7.15999 0.486052
\(218\) −3.29200 −0.222962
\(219\) 0 0
\(220\) 0 0
\(221\) −8.72666 −0.587018
\(222\) 0 0
\(223\) −12.2534 −0.820546 −0.410273 0.911963i \(-0.634567\pi\)
−0.410273 + 0.911963i \(0.634567\pi\)
\(224\) −1.99868 −0.133543
\(225\) 0 0
\(226\) 8.17997 0.544123
\(227\) −8.74531 −0.580447 −0.290223 0.956959i \(-0.593730\pi\)
−0.290223 + 0.956959i \(0.593730\pi\)
\(228\) 0 0
\(229\) −11.4626 −0.757473 −0.378736 0.925505i \(-0.623641\pi\)
−0.378736 + 0.925505i \(0.623641\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.12136 0.139274
\(233\) −7.48469 −0.490338 −0.245169 0.969480i \(-0.578843\pi\)
−0.245169 + 0.969480i \(0.578843\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.482618 −0.0314157
\(237\) 0 0
\(238\) 26.1086 1.69237
\(239\) −12.9066 −0.834860 −0.417430 0.908709i \(-0.637069\pi\)
−0.417430 + 0.908709i \(0.637069\pi\)
\(240\) 0 0
\(241\) −16.8773 −1.08716 −0.543582 0.839356i \(-0.682932\pi\)
−0.543582 + 0.839356i \(0.682932\pi\)
\(242\) −1.36333 −0.0876381
\(243\) 0 0
\(244\) −0.649390 −0.0415729
\(245\) 0 0
\(246\) 0 0
\(247\) 2.02930 0.129122
\(248\) 8.34542 0.529935
\(249\) 0 0
\(250\) 0 0
\(251\) −2.28267 −0.144081 −0.0720405 0.997402i \(-0.522951\pi\)
−0.0720405 + 0.997402i \(0.522951\pi\)
\(252\) 0 0
\(253\) 1.41468 0.0889401
\(254\) 20.7253 1.30042
\(255\) 0 0
\(256\) −3.37473 −0.210920
\(257\) 26.8667 1.67590 0.837949 0.545749i \(-0.183755\pi\)
0.837949 + 0.545749i \(0.183755\pi\)
\(258\) 0 0
\(259\) −21.0993 −1.31105
\(260\) 0 0
\(261\) 0 0
\(262\) 25.5560 1.57886
\(263\) −12.0187 −0.741102 −0.370551 0.928812i \(-0.620831\pi\)
−0.370551 + 0.928812i \(0.620831\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.07133 −0.372257
\(267\) 0 0
\(268\) 1.31744 0.0804753
\(269\) 14.3854 0.877092 0.438546 0.898709i \(-0.355494\pi\)
0.438546 + 0.898709i \(0.355494\pi\)
\(270\) 0 0
\(271\) 28.6553 1.74069 0.870344 0.492445i \(-0.163897\pi\)
0.870344 + 0.492445i \(0.163897\pi\)
\(272\) 28.2700 1.71412
\(273\) 0 0
\(274\) 11.9227 0.720279
\(275\) 0 0
\(276\) 0 0
\(277\) −29.1600 −1.75205 −0.876027 0.482262i \(-0.839815\pi\)
−0.876027 + 0.482262i \(0.839815\pi\)
\(278\) 30.4040 1.82351
\(279\) 0 0
\(280\) 0 0
\(281\) 5.36333 0.319949 0.159975 0.987121i \(-0.448859\pi\)
0.159975 + 0.987121i \(0.448859\pi\)
\(282\) 0 0
\(283\) −10.0420 −0.596936 −0.298468 0.954420i \(-0.596476\pi\)
−0.298468 + 0.954420i \(0.596476\pi\)
\(284\) 0.828041 0.0491352
\(285\) 0 0
\(286\) 1.55602 0.0920091
\(287\) −1.59465 −0.0941292
\(288\) 0 0
\(289\) 41.4613 2.43890
\(290\) 0 0
\(291\) 0 0
\(292\) −1.06794 −0.0624963
\(293\) −20.3540 −1.18909 −0.594547 0.804061i \(-0.702668\pi\)
−0.594547 + 0.804061i \(0.702668\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −24.5926 −1.42941
\(297\) 0 0
\(298\) −21.7160 −1.25797
\(299\) −1.61462 −0.0933762
\(300\) 0 0
\(301\) −31.8026 −1.83307
\(302\) 13.4787 0.775615
\(303\) 0 0
\(304\) −6.57392 −0.377040
\(305\) 0 0
\(306\) 0 0
\(307\) 25.6226 1.46236 0.731181 0.682184i \(-0.238970\pi\)
0.731181 + 0.682184i \(0.238970\pi\)
\(308\) 0.354000 0.0201710
\(309\) 0 0
\(310\) 0 0
\(311\) −11.9414 −0.677134 −0.338567 0.940942i \(-0.609942\pi\)
−0.338567 + 0.940942i \(0.609942\pi\)
\(312\) 0 0
\(313\) 17.1986 0.972124 0.486062 0.873924i \(-0.338433\pi\)
0.486062 + 0.873924i \(0.338433\pi\)
\(314\) 0.179969 0.0101563
\(315\) 0 0
\(316\) −0.977953 −0.0550142
\(317\) 22.7453 1.27750 0.638752 0.769413i \(-0.279451\pi\)
0.638752 + 0.769413i \(0.279451\pi\)
\(318\) 0 0
\(319\) −0.726656 −0.0406850
\(320\) 0 0
\(321\) 0 0
\(322\) 4.83068 0.269203
\(323\) −13.5946 −0.756427
\(324\) 0 0
\(325\) 0 0
\(326\) 5.87864 0.325588
\(327\) 0 0
\(328\) −1.85866 −0.102628
\(329\) −15.3820 −0.848036
\(330\) 0 0
\(331\) 25.2733 1.38915 0.694574 0.719421i \(-0.255593\pi\)
0.694574 + 0.719421i \(0.255593\pi\)
\(332\) −0.873453 −0.0479370
\(333\) 0 0
\(334\) −15.3947 −0.842361
\(335\) 0 0
\(336\) 0 0
\(337\) −17.6880 −0.963528 −0.481764 0.876301i \(-0.660004\pi\)
−0.481764 + 0.876301i \(0.660004\pi\)
\(338\) 15.9473 0.867420
\(339\) 0 0
\(340\) 0 0
\(341\) −2.85866 −0.154805
\(342\) 0 0
\(343\) −19.3527 −1.04495
\(344\) −37.0679 −1.99857
\(345\) 0 0
\(346\) 29.5106 1.58650
\(347\) −20.1027 −1.07917 −0.539585 0.841931i \(-0.681419\pi\)
−0.539585 + 0.841931i \(0.681419\pi\)
\(348\) 0 0
\(349\) −10.9907 −0.588317 −0.294159 0.955757i \(-0.595039\pi\)
−0.294159 + 0.955757i \(0.595039\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.797984 0.0425327
\(353\) −16.7267 −0.890270 −0.445135 0.895463i \(-0.646844\pi\)
−0.445135 + 0.895463i \(0.646844\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0.488078 0.0258681
\(357\) 0 0
\(358\) 22.0059 1.16305
\(359\) 13.5933 0.717429 0.358714 0.933447i \(-0.383215\pi\)
0.358714 + 0.933447i \(0.383215\pi\)
\(360\) 0 0
\(361\) −15.8387 −0.833615
\(362\) −28.2954 −1.48717
\(363\) 0 0
\(364\) −0.404032 −0.0211771
\(365\) 0 0
\(366\) 0 0
\(367\) 28.9987 1.51372 0.756859 0.653578i \(-0.226733\pi\)
0.756859 + 0.653578i \(0.226733\pi\)
\(368\) 5.23057 0.272662
\(369\) 0 0
\(370\) 0 0
\(371\) 30.1027 1.56285
\(372\) 0 0
\(373\) −17.8867 −0.926136 −0.463068 0.886323i \(-0.653251\pi\)
−0.463068 + 0.886323i \(0.653251\pi\)
\(374\) −10.4240 −0.539013
\(375\) 0 0
\(376\) −17.9287 −0.924601
\(377\) 0.829359 0.0427142
\(378\) 0 0
\(379\) 23.7839 1.22170 0.610850 0.791747i \(-0.290828\pi\)
0.610850 + 0.791747i \(0.290828\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 4.51531 0.231023
\(383\) 31.7360 1.62163 0.810817 0.585300i \(-0.199023\pi\)
0.810817 + 0.585300i \(0.199023\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.90663 0.147943
\(387\) 0 0
\(388\) 2.75077 0.139649
\(389\) −25.7546 −1.30581 −0.652906 0.757439i \(-0.726450\pi\)
−0.652906 + 0.757439i \(0.726450\pi\)
\(390\) 0 0
\(391\) 10.8166 0.547021
\(392\) −2.12136 −0.107145
\(393\) 0 0
\(394\) 24.4427 1.23140
\(395\) 0 0
\(396\) 0 0
\(397\) −3.03863 −0.152505 −0.0762523 0.997089i \(-0.524295\pi\)
−0.0762523 + 0.997089i \(0.524295\pi\)
\(398\) −20.6680 −1.03600
\(399\) 0 0
\(400\) 0 0
\(401\) 18.1800 0.907864 0.453932 0.891036i \(-0.350021\pi\)
0.453932 + 0.891036i \(0.350021\pi\)
\(402\) 0 0
\(403\) 3.26270 0.162526
\(404\) −1.18336 −0.0588743
\(405\) 0 0
\(406\) −2.48130 −0.123145
\(407\) 8.42401 0.417563
\(408\) 0 0
\(409\) −20.5946 −1.01834 −0.509170 0.860666i \(-0.670048\pi\)
−0.509170 + 0.860666i \(0.670048\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.359939 −0.0177329
\(413\) 8.55263 0.420847
\(414\) 0 0
\(415\) 0 0
\(416\) −0.910768 −0.0446541
\(417\) 0 0
\(418\) 2.42401 0.118562
\(419\) −32.7253 −1.59874 −0.799369 0.600841i \(-0.794833\pi\)
−0.799369 + 0.600841i \(0.794833\pi\)
\(420\) 0 0
\(421\) 32.5640 1.58707 0.793537 0.608522i \(-0.208237\pi\)
0.793537 + 0.608522i \(0.208237\pi\)
\(422\) 18.1613 0.884079
\(423\) 0 0
\(424\) 35.0866 1.70396
\(425\) 0 0
\(426\) 0 0
\(427\) 11.5081 0.556914
\(428\) −2.05861 −0.0995066
\(429\) 0 0
\(430\) 0 0
\(431\) −1.82003 −0.0876678 −0.0438339 0.999039i \(-0.513957\pi\)
−0.0438339 + 0.999039i \(0.513957\pi\)
\(432\) 0 0
\(433\) −29.8280 −1.43344 −0.716722 0.697359i \(-0.754358\pi\)
−0.716722 + 0.697359i \(0.754358\pi\)
\(434\) −9.76142 −0.468563
\(435\) 0 0
\(436\) −0.341281 −0.0163444
\(437\) −2.51531 −0.120324
\(438\) 0 0
\(439\) 18.2220 0.869688 0.434844 0.900506i \(-0.356803\pi\)
0.434844 + 0.900506i \(0.356803\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 11.8973 0.565897
\(443\) 22.2627 1.05773 0.528866 0.848705i \(-0.322617\pi\)
0.528866 + 0.848705i \(0.322617\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 16.7054 0.791022
\(447\) 0 0
\(448\) 21.2461 1.00378
\(449\) −25.8760 −1.22116 −0.610582 0.791953i \(-0.709064\pi\)
−0.610582 + 0.791953i \(0.709064\pi\)
\(450\) 0 0
\(451\) 0.636672 0.0299797
\(452\) 0.848017 0.0398874
\(453\) 0 0
\(454\) 11.9227 0.559562
\(455\) 0 0
\(456\) 0 0
\(457\) −10.4626 −0.489422 −0.244711 0.969596i \(-0.578693\pi\)
−0.244711 + 0.969596i \(0.578693\pi\)
\(458\) 15.6273 0.730218
\(459\) 0 0
\(460\) 0 0
\(461\) −11.8387 −0.551383 −0.275691 0.961246i \(-0.588907\pi\)
−0.275691 + 0.961246i \(0.588907\pi\)
\(462\) 0 0
\(463\) 27.7546 1.28987 0.644934 0.764238i \(-0.276885\pi\)
0.644934 + 0.764238i \(0.276885\pi\)
\(464\) −2.68670 −0.124727
\(465\) 0 0
\(466\) 10.2041 0.472695
\(467\) −20.8294 −0.963868 −0.481934 0.876208i \(-0.660065\pi\)
−0.481934 + 0.876208i \(0.660065\pi\)
\(468\) 0 0
\(469\) −23.3467 −1.07805
\(470\) 0 0
\(471\) 0 0
\(472\) 9.96862 0.458843
\(473\) 12.6974 0.583825
\(474\) 0 0
\(475\) 0 0
\(476\) 2.70668 0.124061
\(477\) 0 0
\(478\) 17.5960 0.804821
\(479\) −2.64939 −0.121054 −0.0605269 0.998167i \(-0.519278\pi\)
−0.0605269 + 0.998167i \(0.519278\pi\)
\(480\) 0 0
\(481\) −9.61462 −0.438389
\(482\) 23.0093 1.04805
\(483\) 0 0
\(484\) −0.141336 −0.00642437
\(485\) 0 0
\(486\) 0 0
\(487\) 40.5360 1.83686 0.918432 0.395580i \(-0.129456\pi\)
0.918432 + 0.395580i \(0.129456\pi\)
\(488\) 13.4134 0.607194
\(489\) 0 0
\(490\) 0 0
\(491\) 16.1214 0.727547 0.363773 0.931487i \(-0.381488\pi\)
0.363773 + 0.931487i \(0.381488\pi\)
\(492\) 0 0
\(493\) −5.55602 −0.250230
\(494\) −2.76661 −0.124476
\(495\) 0 0
\(496\) −10.5695 −0.474584
\(497\) −14.6740 −0.658218
\(498\) 0 0
\(499\) −28.0666 −1.25643 −0.628217 0.778038i \(-0.716215\pi\)
−0.628217 + 0.778038i \(0.716215\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 3.11203 0.138897
\(503\) −0.906626 −0.0404245 −0.0202122 0.999796i \(-0.506434\pi\)
−0.0202122 + 0.999796i \(0.506434\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.92867 −0.0857400
\(507\) 0 0
\(508\) 2.14859 0.0953284
\(509\) 14.3013 0.633895 0.316948 0.948443i \(-0.397342\pi\)
0.316948 + 0.948443i \(0.397342\pi\)
\(510\) 0 0
\(511\) 18.9253 0.837205
\(512\) 24.5381 1.08444
\(513\) 0 0
\(514\) −36.6281 −1.61560
\(515\) 0 0
\(516\) 0 0
\(517\) 6.14134 0.270096
\(518\) 28.7653 1.26387
\(519\) 0 0
\(520\) 0 0
\(521\) −20.1214 −0.881533 −0.440766 0.897622i \(-0.645293\pi\)
−0.440766 + 0.897622i \(0.645293\pi\)
\(522\) 0 0
\(523\) −12.8459 −0.561714 −0.280857 0.959750i \(-0.590619\pi\)
−0.280857 + 0.959750i \(0.590619\pi\)
\(524\) 2.64939 0.115739
\(525\) 0 0
\(526\) 16.3854 0.714436
\(527\) −21.8573 −0.952121
\(528\) 0 0
\(529\) −20.9987 −0.912986
\(530\) 0 0
\(531\) 0 0
\(532\) −0.629414 −0.0272886
\(533\) −0.726656 −0.0314750
\(534\) 0 0
\(535\) 0 0
\(536\) −27.2121 −1.17538
\(537\) 0 0
\(538\) −19.6120 −0.845533
\(539\) 0.726656 0.0312993
\(540\) 0 0
\(541\) 17.6040 0.756854 0.378427 0.925631i \(-0.376465\pi\)
0.378427 + 0.925631i \(0.376465\pi\)
\(542\) −39.0666 −1.67805
\(543\) 0 0
\(544\) 6.10138 0.261595
\(545\) 0 0
\(546\) 0 0
\(547\) 19.9873 0.854594 0.427297 0.904111i \(-0.359466\pi\)
0.427297 + 0.904111i \(0.359466\pi\)
\(548\) 1.23603 0.0528005
\(549\) 0 0
\(550\) 0 0
\(551\) 1.29200 0.0550411
\(552\) 0 0
\(553\) 17.3306 0.736974
\(554\) 39.7546 1.68901
\(555\) 0 0
\(556\) 3.15198 0.133674
\(557\) 41.5933 1.76237 0.881183 0.472775i \(-0.156748\pi\)
0.881183 + 0.472775i \(0.156748\pi\)
\(558\) 0 0
\(559\) −14.4919 −0.612944
\(560\) 0 0
\(561\) 0 0
\(562\) −7.31198 −0.308437
\(563\) −2.28267 −0.0962032 −0.0481016 0.998842i \(-0.515317\pi\)
−0.0481016 + 0.998842i \(0.515317\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 13.6906 0.575458
\(567\) 0 0
\(568\) −17.1035 −0.717645
\(569\) 2.37266 0.0994670 0.0497335 0.998763i \(-0.484163\pi\)
0.0497335 + 0.998763i \(0.484163\pi\)
\(570\) 0 0
\(571\) −31.9053 −1.33520 −0.667598 0.744522i \(-0.732677\pi\)
−0.667598 + 0.744522i \(0.732677\pi\)
\(572\) 0.161312 0.00674479
\(573\) 0 0
\(574\) 2.17403 0.0907423
\(575\) 0 0
\(576\) 0 0
\(577\) 3.28267 0.136659 0.0683297 0.997663i \(-0.478233\pi\)
0.0683297 + 0.997663i \(0.478233\pi\)
\(578\) −56.5254 −2.35115
\(579\) 0 0
\(580\) 0 0
\(581\) 15.4787 0.642167
\(582\) 0 0
\(583\) −12.0187 −0.497762
\(584\) 22.0586 0.912792
\(585\) 0 0
\(586\) 27.7492 1.14631
\(587\) −26.3400 −1.08717 −0.543583 0.839355i \(-0.682933\pi\)
−0.543583 + 0.839355i \(0.682933\pi\)
\(588\) 0 0
\(589\) 5.08273 0.209430
\(590\) 0 0
\(591\) 0 0
\(592\) 31.1465 1.28011
\(593\) 32.9694 1.35389 0.676945 0.736034i \(-0.263303\pi\)
0.676945 + 0.736034i \(0.263303\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2.25130 −0.0922167
\(597\) 0 0
\(598\) 2.20126 0.0900164
\(599\) −24.5454 −1.00290 −0.501448 0.865188i \(-0.667199\pi\)
−0.501448 + 0.865188i \(0.667199\pi\)
\(600\) 0 0
\(601\) −9.70668 −0.395944 −0.197972 0.980208i \(-0.563435\pi\)
−0.197972 + 0.980208i \(0.563435\pi\)
\(602\) 43.3574 1.76712
\(603\) 0 0
\(604\) 1.39734 0.0568570
\(605\) 0 0
\(606\) 0 0
\(607\) 1.73599 0.0704615 0.0352307 0.999379i \(-0.488783\pi\)
0.0352307 + 0.999379i \(0.488783\pi\)
\(608\) −1.41882 −0.0575408
\(609\) 0 0
\(610\) 0 0
\(611\) −7.00933 −0.283567
\(612\) 0 0
\(613\) 12.0187 0.485429 0.242715 0.970098i \(-0.421962\pi\)
0.242715 + 0.970098i \(0.421962\pi\)
\(614\) −34.9321 −1.40974
\(615\) 0 0
\(616\) −7.31198 −0.294608
\(617\) −42.1587 −1.69724 −0.848622 0.528999i \(-0.822567\pi\)
−0.848622 + 0.528999i \(0.822567\pi\)
\(618\) 0 0
\(619\) 9.03863 0.363293 0.181647 0.983364i \(-0.441857\pi\)
0.181647 + 0.983364i \(0.441857\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 16.2800 0.652770
\(623\) −8.64939 −0.346530
\(624\) 0 0
\(625\) 0 0
\(626\) −23.4474 −0.937146
\(627\) 0 0
\(628\) 0.0186574 0.000744512 0
\(629\) 64.4100 2.56819
\(630\) 0 0
\(631\) 10.9614 0.436365 0.218183 0.975908i \(-0.429987\pi\)
0.218183 + 0.975908i \(0.429987\pi\)
\(632\) 20.1999 0.803511
\(633\) 0 0
\(634\) −31.0093 −1.23154
\(635\) 0 0
\(636\) 0 0
\(637\) −0.829359 −0.0328604
\(638\) 0.990671 0.0392211
\(639\) 0 0
\(640\) 0 0
\(641\) 44.1587 1.74416 0.872081 0.489361i \(-0.162770\pi\)
0.872081 + 0.489361i \(0.162770\pi\)
\(642\) 0 0
\(643\) 22.5840 0.890626 0.445313 0.895375i \(-0.353092\pi\)
0.445313 + 0.895375i \(0.353092\pi\)
\(644\) 0.500796 0.0197341
\(645\) 0 0
\(646\) 18.5340 0.729209
\(647\) 13.3561 0.525081 0.262541 0.964921i \(-0.415440\pi\)
0.262541 + 0.964921i \(0.415440\pi\)
\(648\) 0 0
\(649\) −3.41468 −0.134038
\(650\) 0 0
\(651\) 0 0
\(652\) 0.609438 0.0238674
\(653\) −37.5933 −1.47114 −0.735570 0.677448i \(-0.763086\pi\)
−0.735570 + 0.677448i \(0.763086\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.35400 0.0919082
\(657\) 0 0
\(658\) 20.9707 0.817523
\(659\) −7.00933 −0.273045 −0.136522 0.990637i \(-0.543593\pi\)
−0.136522 + 0.990637i \(0.543593\pi\)
\(660\) 0 0
\(661\) 26.6226 1.03550 0.517750 0.855532i \(-0.326770\pi\)
0.517750 + 0.855532i \(0.326770\pi\)
\(662\) −34.4559 −1.33917
\(663\) 0 0
\(664\) 18.0415 0.700144
\(665\) 0 0
\(666\) 0 0
\(667\) −1.02799 −0.0398038
\(668\) −1.59597 −0.0617498
\(669\) 0 0
\(670\) 0 0
\(671\) −4.59465 −0.177374
\(672\) 0 0
\(673\) 35.6960 1.37598 0.687990 0.725720i \(-0.258493\pi\)
0.687990 + 0.725720i \(0.258493\pi\)
\(674\) 24.1146 0.928859
\(675\) 0 0
\(676\) 1.65326 0.0635869
\(677\) −33.2920 −1.27952 −0.639758 0.768577i \(-0.720965\pi\)
−0.639758 + 0.768577i \(0.720965\pi\)
\(678\) 0 0
\(679\) −48.7474 −1.87075
\(680\) 0 0
\(681\) 0 0
\(682\) 3.89730 0.149235
\(683\) 36.1787 1.38434 0.692169 0.721736i \(-0.256655\pi\)
0.692169 + 0.721736i \(0.256655\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 26.3841 1.00735
\(687\) 0 0
\(688\) 46.9466 1.78982
\(689\) 13.7173 0.522589
\(690\) 0 0
\(691\) 35.3293 1.34399 0.671995 0.740555i \(-0.265438\pi\)
0.671995 + 0.740555i \(0.265438\pi\)
\(692\) 3.05936 0.116299
\(693\) 0 0
\(694\) 27.4066 1.04034
\(695\) 0 0
\(696\) 0 0
\(697\) 4.86799 0.184388
\(698\) 14.9839 0.567149
\(699\) 0 0
\(700\) 0 0
\(701\) 49.4006 1.86584 0.932918 0.360088i \(-0.117253\pi\)
0.932918 + 0.360088i \(0.117253\pi\)
\(702\) 0 0
\(703\) −14.9780 −0.564904
\(704\) −8.48262 −0.319701
\(705\) 0 0
\(706\) 22.8039 0.858237
\(707\) 20.9707 0.788684
\(708\) 0 0
\(709\) −11.1027 −0.416971 −0.208485 0.978025i \(-0.566853\pi\)
−0.208485 + 0.978025i \(0.566853\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −10.0814 −0.377817
\(713\) −4.04409 −0.151452
\(714\) 0 0
\(715\) 0 0
\(716\) 2.28135 0.0852582
\(717\) 0 0
\(718\) −18.5322 −0.691614
\(719\) 43.9787 1.64013 0.820064 0.572271i \(-0.193938\pi\)
0.820064 + 0.572271i \(0.193938\pi\)
\(720\) 0 0
\(721\) 6.37860 0.237551
\(722\) 21.5933 0.803621
\(723\) 0 0
\(724\) −2.93338 −0.109018
\(725\) 0 0
\(726\) 0 0
\(727\) −44.1880 −1.63884 −0.819421 0.573193i \(-0.805705\pi\)
−0.819421 + 0.573193i \(0.805705\pi\)
\(728\) 8.34542 0.309302
\(729\) 0 0
\(730\) 0 0
\(731\) 97.0840 3.59078
\(732\) 0 0
\(733\) −47.3879 −1.75031 −0.875156 0.483840i \(-0.839242\pi\)
−0.875156 + 0.483840i \(0.839242\pi\)
\(734\) −39.5347 −1.45925
\(735\) 0 0
\(736\) 1.12889 0.0416115
\(737\) 9.32131 0.343355
\(738\) 0 0
\(739\) −13.0407 −0.479710 −0.239855 0.970809i \(-0.577100\pi\)
−0.239855 + 0.970809i \(0.577100\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −41.0399 −1.50662
\(743\) 14.1800 0.520213 0.260106 0.965580i \(-0.416242\pi\)
0.260106 + 0.965580i \(0.416242\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 24.3854 0.892812
\(747\) 0 0
\(748\) −1.08066 −0.0395127
\(749\) 36.4813 1.33300
\(750\) 0 0
\(751\) 19.1120 0.697408 0.348704 0.937233i \(-0.386622\pi\)
0.348704 + 0.937233i \(0.386622\pi\)
\(752\) 22.7067 0.828027
\(753\) 0 0
\(754\) −1.13069 −0.0411773
\(755\) 0 0
\(756\) 0 0
\(757\) 24.7347 0.898997 0.449498 0.893281i \(-0.351603\pi\)
0.449498 + 0.893281i \(0.351603\pi\)
\(758\) −32.4253 −1.17774
\(759\) 0 0
\(760\) 0 0
\(761\) 9.82003 0.355976 0.177988 0.984033i \(-0.443041\pi\)
0.177988 + 0.984033i \(0.443041\pi\)
\(762\) 0 0
\(763\) 6.04796 0.218951
\(764\) 0.468102 0.0169353
\(765\) 0 0
\(766\) −43.2666 −1.56328
\(767\) 3.89730 0.140723
\(768\) 0 0
\(769\) −15.3026 −0.551828 −0.275914 0.961182i \(-0.588980\pi\)
−0.275914 + 0.961182i \(0.588980\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.301330 0.0108451
\(773\) −5.36927 −0.193119 −0.0965596 0.995327i \(-0.530784\pi\)
−0.0965596 + 0.995327i \(0.530784\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −56.8181 −2.03965
\(777\) 0 0
\(778\) 35.1120 1.25883
\(779\) −1.13201 −0.0405584
\(780\) 0 0
\(781\) 5.85866 0.209639
\(782\) −14.7466 −0.527338
\(783\) 0 0
\(784\) 2.68670 0.0959537
\(785\) 0 0
\(786\) 0 0
\(787\) 15.4767 0.551684 0.275842 0.961203i \(-0.411043\pi\)
0.275842 + 0.961203i \(0.411043\pi\)
\(788\) 2.53397 0.0902689
\(789\) 0 0
\(790\) 0 0
\(791\) −15.0280 −0.534334
\(792\) 0 0
\(793\) 5.24404 0.186221
\(794\) 4.14265 0.147017
\(795\) 0 0
\(796\) −2.14265 −0.0759444
\(797\) 43.6774 1.54713 0.773566 0.633716i \(-0.218471\pi\)
0.773566 + 0.633716i \(0.218471\pi\)
\(798\) 0 0
\(799\) 46.9567 1.66121
\(800\) 0 0
\(801\) 0 0
\(802\) −24.7853 −0.875198
\(803\) −7.55602 −0.266646
\(804\) 0 0
\(805\) 0 0
\(806\) −4.44813 −0.156679
\(807\) 0 0
\(808\) 24.4427 0.859890
\(809\) 26.9966 0.949150 0.474575 0.880215i \(-0.342602\pi\)
0.474575 + 0.880215i \(0.342602\pi\)
\(810\) 0 0
\(811\) −14.2220 −0.499402 −0.249701 0.968323i \(-0.580332\pi\)
−0.249701 + 0.968323i \(0.580332\pi\)
\(812\) −0.257236 −0.00902722
\(813\) 0 0
\(814\) −11.4847 −0.402538
\(815\) 0 0
\(816\) 0 0
\(817\) −22.5760 −0.789834
\(818\) 28.0773 0.981699
\(819\) 0 0
\(820\) 0 0
\(821\) −23.7801 −0.829930 −0.414965 0.909837i \(-0.636206\pi\)
−0.414965 + 0.909837i \(0.636206\pi\)
\(822\) 0 0
\(823\) −1.84934 −0.0644638 −0.0322319 0.999480i \(-0.510262\pi\)
−0.0322319 + 0.999480i \(0.510262\pi\)
\(824\) 7.43466 0.258998
\(825\) 0 0
\(826\) −11.6600 −0.405705
\(827\) −15.6987 −0.545896 −0.272948 0.962029i \(-0.587999\pi\)
−0.272948 + 0.962029i \(0.587999\pi\)
\(828\) 0 0
\(829\) −4.34128 −0.150779 −0.0753895 0.997154i \(-0.524020\pi\)
−0.0753895 + 0.997154i \(0.524020\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 9.68152 0.335646
\(833\) 5.55602 0.192505
\(834\) 0 0
\(835\) 0 0
\(836\) 0.251297 0.00869128
\(837\) 0 0
\(838\) 44.6154 1.54121
\(839\) −28.3013 −0.977070 −0.488535 0.872544i \(-0.662469\pi\)
−0.488535 + 0.872544i \(0.662469\pi\)
\(840\) 0 0
\(841\) −28.4720 −0.981792
\(842\) −44.3955 −1.52997
\(843\) 0 0
\(844\) 1.88278 0.0648080
\(845\) 0 0
\(846\) 0 0
\(847\) 2.50466 0.0860613
\(848\) −44.4372 −1.52598
\(849\) 0 0
\(850\) 0 0
\(851\) 11.9173 0.408519
\(852\) 0 0
\(853\) −35.7653 −1.22458 −0.612290 0.790633i \(-0.709752\pi\)
−0.612290 + 0.790633i \(0.709752\pi\)
\(854\) −15.6893 −0.536875
\(855\) 0 0
\(856\) 42.5213 1.45335
\(857\) 24.7513 0.845487 0.422743 0.906249i \(-0.361067\pi\)
0.422743 + 0.906249i \(0.361067\pi\)
\(858\) 0 0
\(859\) −10.8039 −0.368625 −0.184313 0.982868i \(-0.559006\pi\)
−0.184313 + 0.982868i \(0.559006\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2.48130 0.0845134
\(863\) 16.6027 0.565161 0.282581 0.959244i \(-0.408810\pi\)
0.282581 + 0.959244i \(0.408810\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 40.6654 1.38187
\(867\) 0 0
\(868\) −1.01197 −0.0343484
\(869\) −6.91934 −0.234723
\(870\) 0 0
\(871\) −10.6387 −0.360480
\(872\) 7.04928 0.238719
\(873\) 0 0
\(874\) 3.42920 0.115994
\(875\) 0 0
\(876\) 0 0
\(877\) −11.1600 −0.376846 −0.188423 0.982088i \(-0.560338\pi\)
−0.188423 + 0.982088i \(0.560338\pi\)
\(878\) −24.8426 −0.838396
\(879\) 0 0
\(880\) 0 0
\(881\) −54.5254 −1.83701 −0.918504 0.395413i \(-0.870602\pi\)
−0.918504 + 0.395413i \(0.870602\pi\)
\(882\) 0 0
\(883\) 12.7746 0.429900 0.214950 0.976625i \(-0.431041\pi\)
0.214950 + 0.976625i \(0.431041\pi\)
\(884\) 1.23339 0.0414835
\(885\) 0 0
\(886\) −30.3514 −1.01967
\(887\) 47.1053 1.58164 0.790820 0.612049i \(-0.209655\pi\)
0.790820 + 0.612049i \(0.209655\pi\)
\(888\) 0 0
\(889\) −38.0759 −1.27703
\(890\) 0 0
\(891\) 0 0
\(892\) 1.73184 0.0579864
\(893\) −10.9193 −0.365402
\(894\) 0 0
\(895\) 0 0
\(896\) −24.9681 −0.834124
\(897\) 0 0
\(898\) 35.2775 1.17722
\(899\) 2.07727 0.0692807
\(900\) 0 0
\(901\) −91.8947 −3.06146
\(902\) −0.867993 −0.0289010
\(903\) 0 0
\(904\) −17.5161 −0.582576
\(905\) 0 0
\(906\) 0 0
\(907\) −24.7826 −0.822894 −0.411447 0.911434i \(-0.634976\pi\)
−0.411447 + 0.911434i \(0.634976\pi\)
\(908\) 1.23603 0.0410191
\(909\) 0 0
\(910\) 0 0
\(911\) 6.08273 0.201530 0.100765 0.994910i \(-0.467871\pi\)
0.100765 + 0.994910i \(0.467871\pi\)
\(912\) 0 0
\(913\) −6.17997 −0.204527
\(914\) 14.2640 0.471812
\(915\) 0 0
\(916\) 1.62009 0.0535291
\(917\) −46.9507 −1.55045
\(918\) 0 0
\(919\) 17.9953 0.593610 0.296805 0.954938i \(-0.404079\pi\)
0.296805 + 0.954938i \(0.404079\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 16.1400 0.531543
\(923\) −6.68670 −0.220096
\(924\) 0 0
\(925\) 0 0
\(926\) −37.8387 −1.24346
\(927\) 0 0
\(928\) −0.579860 −0.0190348
\(929\) 16.7453 0.549396 0.274698 0.961531i \(-0.411422\pi\)
0.274698 + 0.961531i \(0.411422\pi\)
\(930\) 0 0
\(931\) −1.29200 −0.0423436
\(932\) 1.05786 0.0346513
\(933\) 0 0
\(934\) 28.3973 0.929187
\(935\) 0 0
\(936\) 0 0
\(937\) −31.5853 −1.03185 −0.515924 0.856635i \(-0.672551\pi\)
−0.515924 + 0.856635i \(0.672551\pi\)
\(938\) 31.8293 1.03926
\(939\) 0 0
\(940\) 0 0
\(941\) 14.8421 0.483838 0.241919 0.970296i \(-0.422223\pi\)
0.241919 + 0.970296i \(0.422223\pi\)
\(942\) 0 0
\(943\) 0.900687 0.0293304
\(944\) −12.6253 −0.410918
\(945\) 0 0
\(946\) −17.3107 −0.562818
\(947\) −2.68802 −0.0873490 −0.0436745 0.999046i \(-0.513906\pi\)
−0.0436745 + 0.999046i \(0.513906\pi\)
\(948\) 0 0
\(949\) 8.62395 0.279945
\(950\) 0 0
\(951\) 0 0
\(952\) −55.9074 −1.81197
\(953\) −14.2700 −0.462249 −0.231125 0.972924i \(-0.574240\pi\)
−0.231125 + 0.972924i \(0.574240\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.82417 0.0589980
\(957\) 0 0
\(958\) 3.61199 0.116698
\(959\) −21.9041 −0.707320
\(960\) 0 0
\(961\) −22.8280 −0.736388
\(962\) 13.1079 0.422615
\(963\) 0 0
\(964\) 2.38538 0.0768278
\(965\) 0 0
\(966\) 0 0
\(967\) 10.0586 0.323463 0.161732 0.986835i \(-0.448292\pi\)
0.161732 + 0.986835i \(0.448292\pi\)
\(968\) 2.91934 0.0938313
\(969\) 0 0
\(970\) 0 0
\(971\) 13.4520 0.431695 0.215848 0.976427i \(-0.430749\pi\)
0.215848 + 0.976427i \(0.430749\pi\)
\(972\) 0 0
\(973\) −55.8573 −1.79070
\(974\) −55.2639 −1.77077
\(975\) 0 0
\(976\) −16.9880 −0.543774
\(977\) 32.8294 1.05030 0.525152 0.851008i \(-0.324008\pi\)
0.525152 + 0.851008i \(0.324008\pi\)
\(978\) 0 0
\(979\) 3.45331 0.110368
\(980\) 0 0
\(981\) 0 0
\(982\) −21.9787 −0.701369
\(983\) −43.7746 −1.39619 −0.698097 0.716003i \(-0.745969\pi\)
−0.698097 + 0.716003i \(0.745969\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 7.57467 0.241227
\(987\) 0 0
\(988\) −0.286814 −0.00912477
\(989\) 17.9627 0.571180
\(990\) 0 0
\(991\) 42.1507 1.33896 0.669480 0.742830i \(-0.266517\pi\)
0.669480 + 0.742830i \(0.266517\pi\)
\(992\) −2.28117 −0.0724271
\(993\) 0 0
\(994\) 20.0055 0.634535
\(995\) 0 0
\(996\) 0 0
\(997\) −17.4347 −0.552161 −0.276081 0.961135i \(-0.589036\pi\)
−0.276081 + 0.961135i \(0.589036\pi\)
\(998\) 38.2640 1.21123
\(999\) 0 0
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 2475.2.a.z.1.2 3
3.2 odd 2 825.2.a.m.1.2 yes 3
5.2 odd 4 2475.2.c.q.199.3 6
5.3 odd 4 2475.2.c.q.199.4 6
5.4 even 2 2475.2.a.bd.1.2 3
15.2 even 4 825.2.c.f.199.4 6
15.8 even 4 825.2.c.f.199.3 6
15.14 odd 2 825.2.a.i.1.2 3
33.32 even 2 9075.2.a.cd.1.2 3
165.164 even 2 9075.2.a.cj.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.2 3 15.14 odd 2
825.2.a.m.1.2 yes 3 3.2 odd 2
825.2.c.f.199.3 6 15.8 even 4
825.2.c.f.199.4 6 15.2 even 4
2475.2.a.z.1.2 3 1.1 even 1 trivial
2475.2.a.bd.1.2 3 5.4 even 2
2475.2.c.q.199.3 6 5.2 odd 4
2475.2.c.q.199.4 6 5.3 odd 4
9075.2.a.cd.1.2 3 33.32 even 2
9075.2.a.cj.1.2 3 165.164 even 2