Properties

Label 312.2.h.b.155.9
Level $312$
Weight $2$
Character 312.155
Analytic conductor $2.491$
Analytic rank $0$
Dimension $12$
CM discriminant -104
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [312,2,Mod(155,312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(312, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("312.155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.h (of order \(2\), degree \(1\), 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: \(2.49133254306\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{9} + 92x^{6} - 68x^{3} + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 155.9
Root \(0.803519 - 0.227114i\) of defining polynomial
Character \(\chi\) \(=\) 312.155
Dual form 312.2.h.b.155.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} +(0.111731 - 1.72844i) q^{3} -2.00000 q^{4} +4.04932i q^{5} +(2.44439 + 0.158012i) q^{6} -2.99062 q^{7} -2.82843i q^{8} +(-2.97503 - 0.386242i) q^{9} +O(q^{10})\) \(q+1.41421i q^{2} +(0.111731 - 1.72844i) q^{3} -2.00000 q^{4} +4.04932i q^{5} +(2.44439 + 0.158012i) q^{6} -2.99062 q^{7} -2.82843i q^{8} +(-2.97503 - 0.386242i) q^{9} -5.72660 q^{10} +(-0.223462 + 3.45689i) q^{12} -3.60555 q^{13} -4.22937i q^{14} +(6.99902 + 0.452435i) q^{15} +4.00000 q^{16} +6.14129i q^{17} +(0.546228 - 4.20733i) q^{18} -8.09864i q^{20} +(-0.334145 + 5.16911i) q^{21} +(-4.88878 - 0.316023i) q^{24} -11.3970 q^{25} -5.09902i q^{26} +(-1.00000 + 5.09902i) q^{27} +5.98123 q^{28} +(-0.639839 + 9.89811i) q^{30} -7.21110 q^{31} +5.65685i q^{32} -8.68510 q^{34} -12.1100i q^{35} +(5.95006 + 0.772483i) q^{36} +11.6757 q^{37} +(-0.402852 + 6.23199i) q^{39} +11.4532 q^{40} +(-7.31023 - 0.472552i) q^{42} +12.1236 q^{43} +(1.56402 - 12.0469i) q^{45} +4.99739i q^{47} +(0.446924 - 6.91377i) q^{48} +1.94378 q^{49} -16.1178i q^{50} +(10.6149 + 0.686173i) q^{51} +7.21110 q^{52} +(-7.21110 - 1.41421i) q^{54} +8.45874i q^{56} +(-13.9980 - 0.904869i) q^{60} -10.1980i q^{62} +(8.89718 + 1.15510i) q^{63} -8.00000 q^{64} -14.6000i q^{65} -12.2826i q^{68} +17.1261 q^{70} +3.10125i q^{71} +(-1.09246 + 8.41466i) q^{72} +16.5119i q^{74} +(-1.27340 + 19.6990i) q^{75} +(-8.81337 - 0.569719i) q^{78} +16.1973i q^{80} +(8.70163 + 2.29816i) q^{81} +(0.668289 - 10.3382i) q^{84} -24.8680 q^{85} +17.1453i q^{86} +(17.0368 + 2.21185i) q^{90} +10.7828 q^{91} +(-0.805704 + 12.4640i) q^{93} -7.06738 q^{94} +(9.77755 + 0.632046i) q^{96} +2.74893i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 24 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 24 q^{4} + 48 q^{16} - 60 q^{25} - 12 q^{27} + 24 q^{30} - 48 q^{42} + 84 q^{49} + 60 q^{51} - 96 q^{64} - 84 q^{75} + 96 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(157\) \(209\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) 0.111731 1.72844i 0.0645079 0.997917i
\(4\) −2.00000 −1.00000
\(5\) 4.04932i 1.81091i 0.424441 + 0.905455i \(0.360470\pi\)
−0.424441 + 0.905455i \(0.639530\pi\)
\(6\) 2.44439 + 0.158012i 0.997917 + 0.0645079i
\(7\) −2.99062 −1.13035 −0.565173 0.824972i \(-0.691191\pi\)
−0.565173 + 0.824972i \(0.691191\pi\)
\(8\) 2.82843i 1.00000i
\(9\) −2.97503 0.386242i −0.991677 0.128747i
\(10\) −5.72660 −1.81091
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −0.223462 + 3.45689i −0.0645079 + 0.997917i
\(13\) −3.60555 −1.00000
\(14\) 4.22937i 1.13035i
\(15\) 6.99902 + 0.452435i 1.80714 + 0.116818i
\(16\) 4.00000 1.00000
\(17\) 6.14129i 1.48948i 0.667354 + 0.744741i \(0.267427\pi\)
−0.667354 + 0.744741i \(0.732573\pi\)
\(18\) 0.546228 4.20733i 0.128747 0.991677i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 8.09864i 1.81091i
\(21\) −0.334145 + 5.16911i −0.0729163 + 1.12799i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −4.88878 0.316023i −0.997917 0.0645079i
\(25\) −11.3970 −2.27940
\(26\) 5.09902i 1.00000i
\(27\) −1.00000 + 5.09902i −0.192450 + 0.981307i
\(28\) 5.98123 1.13035
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) −0.639839 + 9.89811i −0.116818 + 1.80714i
\(31\) −7.21110 −1.29515 −0.647576 0.762001i \(-0.724217\pi\)
−0.647576 + 0.762001i \(0.724217\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 0 0
\(34\) −8.68510 −1.48948
\(35\) 12.1100i 2.04696i
\(36\) 5.95006 + 0.772483i 0.991677 + 0.128747i
\(37\) 11.6757 1.91948 0.959738 0.280898i \(-0.0906323\pi\)
0.959738 + 0.280898i \(0.0906323\pi\)
\(38\) 0 0
\(39\) −0.402852 + 6.23199i −0.0645079 + 0.997917i
\(40\) 11.4532 1.81091
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −7.31023 0.472552i −1.12799 0.0729163i
\(43\) 12.1236 1.84883 0.924415 0.381388i \(-0.124554\pi\)
0.924415 + 0.381388i \(0.124554\pi\)
\(44\) 0 0
\(45\) 1.56402 12.0469i 0.233150 1.79584i
\(46\) 0 0
\(47\) 4.99739i 0.728944i 0.931214 + 0.364472i \(0.118751\pi\)
−0.931214 + 0.364472i \(0.881249\pi\)
\(48\) 0.446924 6.91377i 0.0645079 0.997917i
\(49\) 1.94378 0.277683
\(50\) 16.1178i 2.27940i
\(51\) 10.6149 + 0.686173i 1.48638 + 0.0960834i
\(52\) 7.21110 1.00000
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −7.21110 1.41421i −0.981307 0.192450i
\(55\) 0 0
\(56\) 8.45874i 1.13035i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −13.9980 0.904869i −1.80714 0.116818i
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 10.1980i 1.29515i
\(63\) 8.89718 + 1.15510i 1.12094 + 0.145529i
\(64\) −8.00000 −1.00000
\(65\) 14.6000i 1.81091i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 12.2826i 1.48948i
\(69\) 0 0
\(70\) 17.1261 2.04696
\(71\) 3.10125i 0.368051i 0.982921 + 0.184025i \(0.0589129\pi\)
−0.982921 + 0.184025i \(0.941087\pi\)
\(72\) −1.09246 + 8.41466i −0.128747 + 0.991677i
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 16.5119i 1.91948i
\(75\) −1.27340 + 19.6990i −0.147039 + 2.27465i
\(76\) 0 0
\(77\) 0 0
\(78\) −8.81337 0.569719i −0.997917 0.0645079i
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 16.1973i 1.81091i
\(81\) 8.70163 + 2.29816i 0.966848 + 0.255351i
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0.668289 10.3382i 0.0729163 1.12799i
\(85\) −24.8680 −2.69732
\(86\) 17.1453i 1.84883i
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 17.0368 + 2.21185i 1.79584 + 0.233150i
\(91\) 10.7828 1.13035
\(92\) 0 0
\(93\) −0.805704 + 12.4640i −0.0835476 + 1.29245i
\(94\) −7.06738 −0.728944
\(95\) 0 0
\(96\) 9.77755 + 0.632046i 0.997917 + 0.0645079i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 2.74893i 0.277683i
\(99\) 0 0
\(100\) 22.7940 2.27940
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −0.970395 + 15.0117i −0.0960834 + 1.48638i
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 10.1980i 1.00000i
\(105\) −20.9314 1.35306i −2.04269 0.132045i
\(106\) 0 0
\(107\) 10.1980i 0.985882i −0.870063 0.492941i \(-0.835922\pi\)
0.870063 0.492941i \(-0.164078\pi\)
\(108\) 2.00000 10.1980i 0.192450 0.981307i
\(109\) 0.286752 0.0274659 0.0137329 0.999906i \(-0.495629\pi\)
0.0137329 + 0.999906i \(0.495629\pi\)
\(110\) 0 0
\(111\) 1.30454 20.1808i 0.123821 1.91548i
\(112\) −11.9625 −1.13035
\(113\) 20.3961i 1.91870i 0.282216 + 0.959351i \(0.408930\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 10.7266 + 1.39261i 0.991677 + 0.128747i
\(118\) 0 0
\(119\) 18.3662i 1.68363i
\(120\) 1.27968 19.7962i 0.116818 1.80714i
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 14.4222 1.29515
\(125\) 25.9035i 2.31688i
\(126\) −1.63356 + 12.5825i −0.145529 + 1.12094i
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 11.3137i 1.00000i
\(129\) 1.35458 20.9549i 0.119264 1.84498i
\(130\) 20.6476 1.81091
\(131\) 18.8294i 1.64513i 0.568669 + 0.822566i \(0.307458\pi\)
−0.568669 + 0.822566i \(0.692542\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −20.6476 4.04932i −1.77706 0.348510i
\(136\) 17.3702 1.48948
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) −23.5768 −1.99976 −0.999879 0.0155623i \(-0.995046\pi\)
−0.999879 + 0.0155623i \(0.995046\pi\)
\(140\) 24.2199i 2.04696i
\(141\) 8.63770 + 0.558363i 0.727426 + 0.0470227i
\(142\) −4.38583 −0.368051
\(143\) 0 0
\(144\) −11.9001 1.54497i −0.991677 0.128747i
\(145\) 0 0
\(146\) 0 0
\(147\) 0.217181 3.35972i 0.0179128 0.277105i
\(148\) −23.3514 −1.91948
\(149\) 11.3137i 0.926855i 0.886135 + 0.463428i \(0.153381\pi\)
−0.886135 + 0.463428i \(0.846619\pi\)
\(150\) −27.8587 1.80086i −2.27465 0.147039i
\(151\) −14.3796 −1.17019 −0.585097 0.810964i \(-0.698943\pi\)
−0.585097 + 0.810964i \(0.698943\pi\)
\(152\) 0 0
\(153\) 2.37202 18.2705i 0.191767 1.47709i
\(154\) 0 0
\(155\) 29.2001i 2.34541i
\(156\) 0.805704 12.4640i 0.0645079 0.997917i
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −22.9064 −1.81091
\(161\) 0 0
\(162\) −3.25009 + 12.3060i −0.255351 + 0.966848i
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.1421i 1.09435i −0.837018 0.547176i \(-0.815703\pi\)
0.837018 0.547176i \(-0.184297\pi\)
\(168\) 14.6205 + 0.945104i 1.12799 + 0.0729163i
\(169\) 13.0000 1.00000
\(170\) 35.1687i 2.69732i
\(171\) 0 0
\(172\) −24.2472 −1.84883
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 34.0840 2.57651
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 14.1945i 1.06095i 0.847702 + 0.530473i \(0.177986\pi\)
−0.847702 + 0.530473i \(0.822014\pi\)
\(180\) −3.12803 + 24.0937i −0.233150 + 1.79584i
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 15.2492i 1.13035i
\(183\) 0 0
\(184\) 0 0
\(185\) 47.2787i 3.47600i
\(186\) −17.6267 1.13944i −1.29245 0.0835476i
\(187\) 0 0
\(188\) 9.99478i 0.728944i
\(189\) 2.99062 15.2492i 0.217535 1.10922i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −0.893848 + 13.8275i −0.0645079 + 0.997917i
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) −25.2353 1.63128i −1.80714 0.116818i
\(196\) −3.88757 −0.277683
\(197\) 5.94546i 0.423596i 0.977313 + 0.211798i \(0.0679319\pi\)
−0.977313 + 0.211798i \(0.932068\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 32.2356i 2.27940i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) −21.2297 1.37235i −1.48638 0.0960834i
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −14.4222 −1.00000
\(209\) 0 0
\(210\) 1.91351 29.6014i 0.132045 2.04269i
\(211\) 13.4644 0.926925 0.463463 0.886117i \(-0.346607\pi\)
0.463463 + 0.886117i \(0.346607\pi\)
\(212\) 0 0
\(213\) 5.36034 + 0.346506i 0.367284 + 0.0237422i
\(214\) 14.4222 0.985882
\(215\) 49.0923i 3.34807i
\(216\) 14.4222 + 2.82843i 0.981307 + 0.192450i
\(217\) 21.5656 1.46397
\(218\) 0.405529i 0.0274659i
\(219\) 0 0
\(220\) 0 0
\(221\) 22.1427i 1.48948i
\(222\) 28.5400 + 1.84490i 1.91548 + 0.123821i
\(223\) 8.39834 0.562395 0.281197 0.959650i \(-0.409268\pi\)
0.281197 + 0.959650i \(0.409268\pi\)
\(224\) 16.9175i 1.13035i
\(225\) 33.9064 + 4.40199i 2.26043 + 0.293466i
\(226\) −28.8444 −1.91870
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −6.26798 −0.414200 −0.207100 0.978320i \(-0.566403\pi\)
−0.207100 + 0.978320i \(0.566403\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.96513i 0.652837i −0.945225 0.326419i \(-0.894158\pi\)
0.945225 0.326419i \(-0.105842\pi\)
\(234\) −1.96945 + 15.1697i −0.128747 + 0.991677i
\(235\) −20.2360 −1.32005
\(236\) 0 0
\(237\) 0 0
\(238\) 25.9738 1.68363
\(239\) 19.2985i 1.24832i −0.781297 0.624159i \(-0.785442\pi\)
0.781297 0.624159i \(-0.214558\pi\)
\(240\) 27.9961 + 1.80974i 1.80714 + 0.116818i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 15.5563i 1.00000i
\(243\) 4.94449 14.7835i 0.317189 0.948362i
\(244\) 0 0
\(245\) 7.87100i 0.502860i
\(246\) 0 0
\(247\) 0 0
\(248\) 20.3961i 1.29515i
\(249\) 0 0
\(250\) 36.6330 2.31688
\(251\) 10.1980i 0.643695i −0.946792 0.321847i \(-0.895696\pi\)
0.946792 0.321847i \(-0.104304\pi\)
\(252\) −17.7944 2.31020i −1.12094 0.145529i
\(253\) 0 0
\(254\) 0 0
\(255\) −2.77853 + 42.9830i −0.173998 + 2.69170i
\(256\) 16.0000 1.00000
\(257\) 10.7762i 0.672200i 0.941826 + 0.336100i \(0.109108\pi\)
−0.941826 + 0.336100i \(0.890892\pi\)
\(258\) 29.6348 + 1.91567i 1.84498 + 0.119264i
\(259\) −34.9176 −2.16967
\(260\) 29.2001i 1.81091i
\(261\) 0 0
\(262\) −26.6288 −1.64513
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 5.72660 29.2001i 0.348510 1.77706i
\(271\) −32.3233 −1.96350 −0.981749 0.190179i \(-0.939093\pi\)
−0.981749 + 0.190179i \(0.939093\pi\)
\(272\) 24.5652i 1.48948i
\(273\) 1.20478 18.6375i 0.0729163 1.12799i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 33.3426i 1.99976i
\(279\) 21.4533 + 2.78523i 1.28437 + 0.166747i
\(280\) −34.2521 −2.04696
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) −0.789645 + 12.2156i −0.0470227 + 0.727426i
\(283\) 14.0000 0.832214 0.416107 0.909316i \(-0.363394\pi\)
0.416107 + 0.909316i \(0.363394\pi\)
\(284\) 6.20250i 0.368051i
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 2.18491 16.8293i 0.128747 0.991677i
\(289\) −20.7154 −1.21856
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 30.2414i 1.76672i 0.468695 + 0.883360i \(0.344724\pi\)
−0.468695 + 0.883360i \(0.655276\pi\)
\(294\) 4.75136 + 0.307140i 0.277105 + 0.0179128i
\(295\) 0 0
\(296\) 33.0239i 1.91948i
\(297\) 0 0
\(298\) −16.0000 −0.926855
\(299\) 0 0
\(300\) 2.54679 39.3981i 0.147039 2.27465i
\(301\) −36.2570 −2.08982
\(302\) 20.3358i 1.17019i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 25.8384 + 3.35454i 1.47709 + 0.191767i
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 41.2951 2.34541
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 17.6267 + 1.13944i 0.997917 + 0.0645079i
\(313\) 30.6442 1.73211 0.866055 0.499948i \(-0.166648\pi\)
0.866055 + 0.499948i \(0.166648\pi\)
\(314\) 0 0
\(315\) −4.67737 + 36.0275i −0.263540 + 2.02992i
\(316\) 0 0
\(317\) 28.2843i 1.58860i 0.607524 + 0.794301i \(0.292163\pi\)
−0.607524 + 0.794301i \(0.707837\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 32.3946i 1.81091i
\(321\) −17.6267 1.13944i −0.983828 0.0635972i
\(322\) 0 0
\(323\) 0 0
\(324\) −17.4033 4.59632i −0.966848 0.255351i
\(325\) 41.0924 2.27940
\(326\) 0 0
\(327\) 0.0320391 0.495635i 0.00177177 0.0274087i
\(328\) 0 0
\(329\) 14.9453i 0.823959i
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) −34.7356 4.50964i −1.90350 0.247127i
\(334\) 20.0000 1.09435
\(335\) 0 0
\(336\) −1.33658 + 20.6764i −0.0729163 + 1.12799i
\(337\) 27.9626 1.52322 0.761611 0.648035i \(-0.224409\pi\)
0.761611 + 0.648035i \(0.224409\pi\)
\(338\) 18.3848i 1.00000i
\(339\) 35.2535 + 2.27887i 1.91471 + 0.123771i
\(340\) 49.7361 2.69732
\(341\) 0 0
\(342\) 0 0
\(343\) 15.1212 0.816468
\(344\) 34.2907i 1.84883i
\(345\) 0 0
\(346\) 0 0
\(347\) 27.2881i 1.46490i −0.680819 0.732452i \(-0.738376\pi\)
0.680819 0.732452i \(-0.261624\pi\)
\(348\) 0 0
\(349\) 29.6194 1.58549 0.792745 0.609553i \(-0.208651\pi\)
0.792745 + 0.609553i \(0.208651\pi\)
\(350\) 48.2021i 2.57651i
\(351\) 3.60555 18.3848i 0.192450 0.981307i
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) −12.5580 −0.666507
\(356\) 0 0
\(357\) −31.7450 2.05208i −1.68012 0.108608i
\(358\) −20.0741 −1.06095
\(359\) 31.1127i 1.64207i −0.570881 0.821033i \(-0.693398\pi\)
0.570881 0.821033i \(-0.306602\pi\)
\(360\) −34.0737 4.42370i −1.79584 0.233150i
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) −1.22904 + 19.0129i −0.0645079 + 0.997917i
\(364\) −21.5656 −1.13035
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) −66.8622 −3.47600
\(371\) 0 0
\(372\) 1.61141 24.9280i 0.0835476 1.29245i
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −44.7726 2.89422i −2.31205 0.149457i
\(376\) 14.1348 0.728944
\(377\) 0 0
\(378\) 21.5656 + 4.22937i 1.10922 + 0.217535i
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 27.3972i 1.39993i 0.714177 + 0.699965i \(0.246801\pi\)
−0.714177 + 0.699965i \(0.753199\pi\)
\(384\) −19.5551 1.26409i −0.997917 0.0645079i
\(385\) 0 0
\(386\) 0 0
\(387\) −36.0681 4.68263i −1.83344 0.238032i
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 2.30697 35.6881i 0.116818 1.80714i
\(391\) 0 0
\(392\) 5.49785i 0.277683i
\(393\) 32.5455 + 2.10383i 1.64171 + 0.106124i
\(394\) −8.40815 −0.423596
\(395\) 0 0
\(396\) 0 0
\(397\) 36.0555 1.80957 0.904787 0.425864i \(-0.140030\pi\)
0.904787 + 0.425864i \(0.140030\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −45.5880 −2.27940
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 26.0000 1.29515
\(404\) 0 0
\(405\) −9.30599 + 35.2357i −0.462418 + 1.75088i
\(406\) 0 0
\(407\) 0 0
\(408\) 1.94079 30.0234i 0.0960834 1.48638i
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 20.3961i 1.00000i
\(417\) −2.63426 + 40.7512i −0.129000 + 1.99559i
\(418\) 0 0
\(419\) 34.9358i 1.70673i 0.521317 + 0.853363i \(0.325441\pi\)
−0.521317 + 0.853363i \(0.674559\pi\)
\(420\) 41.8628 + 2.70612i 2.04269 + 0.132045i
\(421\) 41.0084 1.99863 0.999313 0.0370678i \(-0.0118017\pi\)
0.999313 + 0.0370678i \(0.0118017\pi\)
\(422\) 19.0415i 0.926925i
\(423\) 1.93020 14.8674i 0.0938495 0.722877i
\(424\) 0 0
\(425\) 69.9922i 3.39512i
\(426\) −0.490033 + 7.58066i −0.0237422 + 0.367284i
\(427\) 0 0
\(428\) 20.3961i 0.985882i
\(429\) 0 0
\(430\) −69.4270 −3.34807
\(431\) 23.0908i 1.11224i −0.831100 0.556122i \(-0.812289\pi\)
0.831100 0.556122i \(-0.187711\pi\)
\(432\) −4.00000 + 20.3961i −0.192450 + 0.981307i
\(433\) −40.7566 −1.95864 −0.979319 0.202324i \(-0.935151\pi\)
−0.979319 + 0.202324i \(0.935151\pi\)
\(434\) 30.4984i 1.46397i
\(435\) 0 0
\(436\) −0.573504 −0.0274659
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −5.78282 0.750770i −0.275372 0.0357509i
\(442\) 31.3146 1.48948
\(443\) 39.5707i 1.88006i 0.341090 + 0.940031i \(0.389204\pi\)
−0.341090 + 0.940031i \(0.610796\pi\)
\(444\) −2.60908 + 40.3616i −0.123821 + 1.91548i
\(445\) 0 0
\(446\) 11.8771i 0.562395i
\(447\) 19.5551 + 1.26409i 0.924925 + 0.0597895i
\(448\) 23.9249 1.13035
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) −6.22535 + 47.9509i −0.293466 + 2.26043i
\(451\) 0 0
\(452\) 40.7922i 1.91870i
\(453\) −1.60664 + 24.8543i −0.0754868 + 1.16776i
\(454\) 0 0
\(455\) 43.6631i 2.04696i
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 8.86427i 0.414200i
\(459\) −31.3146 6.14129i −1.46164 0.286651i
\(460\) 0 0
\(461\) 18.3505i 0.854666i −0.904094 0.427333i \(-0.859453\pi\)
0.904094 0.427333i \(-0.140547\pi\)
\(462\) 0 0
\(463\) −7.21110 −0.335128 −0.167564 0.985861i \(-0.553590\pi\)
−0.167564 + 0.985861i \(0.553590\pi\)
\(464\) 0 0
\(465\) −50.4706 3.26255i −2.34052 0.151297i
\(466\) 14.0928 0.652837
\(467\) 10.1980i 0.471909i −0.971764 0.235954i \(-0.924178\pi\)
0.971764 0.235954i \(-0.0758216\pi\)
\(468\) −21.4533 2.78523i −0.991677 0.128747i
\(469\) 0 0
\(470\) 28.6181i 1.32005i
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 36.7325i 1.68363i
\(477\) 0 0
\(478\) 27.2922 1.24832
\(479\) 31.1894i 1.42508i 0.701630 + 0.712541i \(0.252456\pi\)
−0.701630 + 0.712541i \(0.747544\pi\)
\(480\) −2.55936 + 39.5924i −0.116818 + 1.80714i
\(481\) −42.0974 −1.91948
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000 1.00000
\(485\) 0 0
\(486\) 20.9070 + 6.99256i 0.948362 + 0.317189i
\(487\) 36.0555 1.63383 0.816916 0.576757i \(-0.195682\pi\)
0.816916 + 0.576757i \(0.195682\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −11.1313 −0.502860
\(491\) 43.3946i 1.95837i −0.202972 0.979185i \(-0.565060\pi\)
0.202972 0.979185i \(-0.434940\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −28.8444 −1.29515
\(497\) 9.27465i 0.416025i
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 51.8069i 2.31688i
\(501\) −24.4439 1.58012i −1.09207 0.0705944i
\(502\) 14.4222 0.643695
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 3.26712 25.1650i 0.145529 1.12094i
\(505\) 0 0
\(506\) 0 0
\(507\) 1.45250 22.4698i 0.0645079 0.997917i
\(508\) 0 0
\(509\) 39.5980i 1.75515i −0.479440 0.877575i \(-0.659160\pi\)
0.479440 0.877575i \(-0.340840\pi\)
\(510\) −60.7872 3.92944i −2.69170 0.173998i
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) 0 0
\(514\) −15.2398 −0.672200
\(515\) 0 0
\(516\) −2.70916 + 41.9099i −0.119264 + 1.84498i
\(517\) 0 0
\(518\) 49.3809i 2.16967i
\(519\) 0 0
\(520\) −41.2951 −1.81091
\(521\) 22.2477i 0.974690i 0.873210 + 0.487345i \(0.162035\pi\)
−0.873210 + 0.487345i \(0.837965\pi\)
\(522\) 0 0
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) 37.6588i 1.64513i
\(525\) 3.80824 58.9123i 0.166205 2.57114i
\(526\) 0 0
\(527\) 44.2855i 1.92911i
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 41.2951 1.78534
\(536\) 0 0
\(537\) 24.5344 + 1.58597i 1.05874 + 0.0684395i
\(538\) 0 0
\(539\) 0 0
\(540\) 41.2951 + 8.09864i 1.77706 + 0.348510i
\(541\) −11.1022 −0.477321 −0.238661 0.971103i \(-0.576708\pi\)
−0.238661 + 0.971103i \(0.576708\pi\)
\(542\) 45.7120i 1.96350i
\(543\) 0 0
\(544\) −34.7404 −1.48948
\(545\) 1.16115i 0.0497382i
\(546\) 26.3574 + 1.70381i 1.12799 + 0.0729163i
\(547\) −20.8953 −0.893416 −0.446708 0.894680i \(-0.647404\pi\)
−0.446708 + 0.894680i \(0.647404\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 81.7185 + 5.28250i 3.46876 + 0.224229i
\(556\) 47.1536 1.99976
\(557\) 46.4387i 1.96767i −0.179080 0.983834i \(-0.557312\pi\)
0.179080 0.983834i \(-0.442688\pi\)
\(558\) −3.93891 + 30.3395i −0.166747 + 1.28437i
\(559\) −43.7122 −1.84883
\(560\) 48.4398i 2.04696i
\(561\) 0 0
\(562\) 0 0
\(563\) 2.72298i 0.114760i 0.998352 + 0.0573799i \(0.0182746\pi\)
−0.998352 + 0.0573799i \(0.981725\pi\)
\(564\) −17.2754 1.11673i −0.727426 0.0470227i
\(565\) −82.5902 −3.47460
\(566\) 19.7990i 0.832214i
\(567\) −26.0232 6.87292i −1.09287 0.288636i
\(568\) 8.77166 0.368051
\(569\) 47.6239i 1.99650i 0.0591437 + 0.998249i \(0.481163\pi\)
−0.0591437 + 0.998249i \(0.518837\pi\)
\(570\) 0 0
\(571\) 46.4832 1.94526 0.972631 0.232356i \(-0.0746434\pi\)
0.972631 + 0.232356i \(0.0746434\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 23.8003 + 3.08993i 0.991677 + 0.128747i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 29.2961i 1.21856i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −5.63914 + 43.4356i −0.233150 + 1.79584i
\(586\) −42.7678 −1.76672
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) −0.434362 + 6.71944i −0.0179128 + 0.277105i
\(589\) 0 0
\(590\) 0 0
\(591\) 10.2764 + 0.664292i 0.422714 + 0.0273253i
\(592\) 46.7028 1.91948
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 74.3708 3.04890
\(596\) 22.6274i 0.926855i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 55.7173 + 3.60171i 2.27465 + 0.147039i
\(601\) −39.4158 −1.60781 −0.803903 0.594761i \(-0.797247\pi\)
−0.803903 + 0.594761i \(0.797247\pi\)
\(602\) 51.2751i 2.08982i
\(603\) 0 0
\(604\) 28.7592 1.17019
\(605\) 44.5425i 1.81091i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 18.0183i 0.728944i
\(612\) −4.74404 + 36.5411i −0.191767 + 1.47709i
\(613\) 36.0555 1.45627 0.728134 0.685435i \(-0.240388\pi\)
0.728134 + 0.685435i \(0.240388\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 58.4001i 2.34541i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −1.61141 + 24.9280i −0.0645079 + 0.997917i
\(625\) 47.9064 1.91626
\(626\) 43.3374i 1.73211i
\(627\) 0 0
\(628\) 0 0
\(629\) 71.7039i 2.85902i
\(630\) −50.9506 6.61480i −2.02992 0.263540i
\(631\) 44.2857 1.76299 0.881494 0.472196i \(-0.156539\pi\)
0.881494 + 0.472196i \(0.156539\pi\)
\(632\) 0 0
\(633\) 1.50439 23.2724i 0.0597940 0.924994i
\(634\) −40.0000 −1.58860
\(635\) 0 0
\(636\) 0 0
\(637\) −7.00841 −0.277683
\(638\) 0 0
\(639\) 1.19783 9.22632i 0.0473855 0.364988i
\(640\) 45.8128 1.81091
\(641\) 40.7922i 1.61119i −0.592464 0.805597i \(-0.701845\pi\)
0.592464 0.805597i \(-0.298155\pi\)
\(642\) 1.61141 24.9280i 0.0635972 0.983828i
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 84.8532 + 5.48513i 3.34109 + 0.215977i
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 6.50018 24.6119i 0.255351 0.966848i
\(649\) 0 0
\(650\) 58.1135i 2.27940i
\(651\) 2.40955 37.2750i 0.0944378 1.46092i
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0.700933 + 0.0453101i 0.0274087 + 0.00177177i
\(655\) −76.2463 −2.97919
\(656\) 0 0
\(657\) 0 0
\(658\) 21.1358 0.823959
\(659\) 50.9902i 1.98630i 0.116863 + 0.993148i \(0.462716\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) −7.21110 −0.280479 −0.140240 0.990118i \(-0.544787\pi\)
−0.140240 + 0.990118i \(0.544787\pi\)
\(662\) 0 0
\(663\) −38.2725 2.47403i −1.48638 0.0960834i
\(664\) 0 0
\(665\) 0 0
\(666\) 6.37760 49.1236i 0.247127 1.90350i
\(667\) 0 0
\(668\) 28.2843i 1.09435i
\(669\) 0.938356 14.5161i 0.0362789 0.561223i
\(670\) 0 0
\(671\) 0 0
\(672\) −29.2409 1.89021i −1.12799 0.0729163i
\(673\) 26.6219 1.02620 0.513099 0.858330i \(-0.328498\pi\)
0.513099 + 0.858330i \(0.328498\pi\)
\(674\) 39.5451i 1.52322i
\(675\) 11.3970 58.1135i 0.438670 2.23679i
\(676\) −26.0000 −1.00000
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) −3.22282 + 49.8559i −0.123771 + 1.91471i
\(679\) 0 0
\(680\) 70.3375i 2.69732i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 21.3846i 0.816468i
\(687\) −0.700328 + 10.8339i −0.0267192 + 0.413337i
\(688\) 48.4944 1.84883
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 38.5913 1.46490
\(695\) 95.4700i 3.62138i
\(696\) 0 0
\(697\) 0 0
\(698\) 41.8882i 1.58549i
\(699\) −17.2242 1.11341i −0.651478 0.0421132i
\(700\) −68.1680 −2.57651
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 26.0000 + 5.09902i 0.981307 + 0.192450i
\(703\) 0 0
\(704\) 0 0
\(705\) −2.26099 + 34.9768i −0.0851539 + 1.31730i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −50.4777 −1.89573 −0.947865 0.318671i \(-0.896763\pi\)
−0.947865 + 0.318671i \(0.896763\pi\)
\(710\) 17.7596i 0.666507i
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 2.90208 44.8942i 0.108608 1.68012i
\(715\) 0 0
\(716\) 28.3890i 1.06095i
\(717\) −33.3564 2.15624i −1.24572 0.0805264i
\(718\) 44.0000 1.64207
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 6.25606 48.1874i 0.233150 1.79584i
\(721\) 0 0
\(722\) 26.8701i 1.00000i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) −26.8883 1.73813i −0.997917 0.0645079i
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 30.4984i 1.13035i
\(729\) −25.0000 10.1980i −0.925926 0.377705i
\(730\) 0 0
\(731\) 74.4545i 2.75380i
\(732\) 0 0
\(733\) 5.12098 0.189148 0.0945738 0.995518i \(-0.469851\pi\)
0.0945738 + 0.995518i \(0.469851\pi\)
\(734\) 0 0
\(735\) 13.6046 + 0.879435i 0.501812 + 0.0324385i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 94.5574i 3.47600i
\(741\) 0 0
\(742\) 0 0
\(743\) 53.5892i 1.96600i 0.183611 + 0.982999i \(0.441221\pi\)
−0.183611 + 0.982999i \(0.558779\pi\)
\(744\) 35.2535 + 2.27887i 1.29245 + 0.0835476i
\(745\) −45.8128 −1.67845
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 30.4984i 1.11439i
\(750\) 4.09304 63.3181i 0.149457 2.31205i
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 19.9896i 0.728944i
\(753\) −17.6267 1.13944i −0.642354 0.0415234i
\(754\) 0 0
\(755\) 58.2275i 2.11912i
\(756\) −5.98123 + 30.4984i −0.217535 + 1.10922i
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −0.857565 −0.0310459
\(764\) 0 0
\(765\) 73.9832 + 9.60507i 2.67487 + 0.347272i
\(766\) −38.7454 −1.39993
\(767\) 0 0
\(768\) 1.78770 27.6551i 0.0645079 0.997917i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 18.6260 + 1.20403i 0.670800 + 0.0433622i
\(772\) 0 0
\(773\) 32.1375i 1.15591i 0.816070 + 0.577953i \(0.196148\pi\)
−0.816070 + 0.577953i \(0.803852\pi\)
\(774\) 6.62225 51.0080i 0.238032 1.83344i
\(775\) 82.1849 2.95217
\(776\) 0 0
\(777\) −3.90138 + 60.3530i −0.139961 + 2.16515i
\(778\) 0 0
\(779\) 0 0
\(780\) 50.4706 + 3.26255i 1.80714 + 0.116818i
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 7.77513 0.277683
\(785\) 0 0
\(786\) −2.97526 + 46.0264i −0.106124 + 1.64171i
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 11.8909i 0.423596i
\(789\) 0 0
\(790\) 0 0
\(791\) 60.9968i 2.16880i
\(792\) 0 0
\(793\) 0 0
\(794\) 50.9902i 1.80957i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) −30.6904 −1.08575
\(800\) 64.4711i 2.27940i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 36.7696i 1.29515i
\(807\) 0 0
\(808\) 0 0
\(809\) 56.0827i 1.97176i −0.167449 0.985881i \(-0.553553\pi\)
0.167449 0.985881i \(-0.446447\pi\)
\(810\) −49.8308 13.1607i −1.75088 0.462418i
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) −3.61151 + 55.8689i −0.126661 + 1.95941i
\(814\) 0 0
\(815\) 0 0
\(816\) 42.4595 + 2.74469i 1.48638 + 0.0960834i
\(817\) 0 0
\(818\) 0 0
\(819\) −32.0792 4.16477i −1.12094 0.145529i
\(820\) 0 0
\(821\) 7.84160i 0.273674i 0.990594 + 0.136837i \(0.0436936\pi\)
−0.990594 + 0.136837i \(0.956306\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 28.8444 1.00000
\(833\) 11.9373i 0.413604i
\(834\) −57.6308 3.72541i −1.99559 0.129000i
\(835\) 57.2660 1.98177
\(836\) 0 0
\(837\) 7.21110 36.7696i 0.249252 1.27094i
\(838\) −49.4067 −1.70673
\(839\) 53.7401i 1.85531i 0.373432 + 0.927657i \(0.378181\pi\)
−0.373432 + 0.927657i \(0.621819\pi\)
\(840\) −3.82703 + 59.2029i −0.132045 + 2.04269i
\(841\) −29.0000 −1.00000
\(842\) 57.9946i 1.99863i
\(843\) 0 0
\(844\) −26.9287 −0.926925
\(845\) 52.6412i 1.81091i
\(846\) 21.0257 + 2.72971i 0.722877 + 0.0938495i
\(847\) 32.8968 1.13035
\(848\) 0 0
\(849\) 1.56423 24.1982i 0.0536844 0.830480i
\(850\) 98.9839 3.39512
\(851\) 0 0
\(852\) −10.7207 0.693012i −0.367284 0.0237422i
\(853\) 52.3973 1.79405 0.897025 0.441980i \(-0.145724\pi\)
0.897025 + 0.441980i \(0.145724\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −28.8444 −0.985882
\(857\) 40.7922i 1.39343i −0.717346 0.696717i \(-0.754643\pi\)
0.717346 0.696717i \(-0.245357\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) 98.1846i 3.34807i
\(861\) 0 0
\(862\) 32.6553 1.11224
\(863\) 55.4854i 1.88874i 0.328880 + 0.944372i \(0.393329\pi\)
−0.328880 + 0.944372i \(0.606671\pi\)
\(864\) −28.8444 5.65685i −0.981307 0.192450i
\(865\) 0 0
\(866\) 57.6385i 1.95864i
\(867\) −2.31456 + 35.8055i −0.0786065 + 1.21602i
\(868\) −43.1313 −1.46397
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0.811057i 0.0274659i
\(873\) 0 0
\(874\) 0 0
\(875\) 77.4673i 2.61887i
\(876\) 0 0
\(877\) −40.4349 −1.36539 −0.682694 0.730704i \(-0.739192\pi\)
−0.682694 + 0.730704i \(0.739192\pi\)
\(878\) 0 0
\(879\) 52.2705 + 3.37890i 1.76304 + 0.113967i
\(880\) 0 0
\(881\) 39.9762i 1.34683i −0.739263 0.673417i \(-0.764826\pi\)
0.739263 0.673417i \(-0.235174\pi\)
\(882\) 1.06175 8.17814i 0.0357509 0.275372i
\(883\) −59.2772 −1.99484 −0.997418 0.0718139i \(-0.977121\pi\)
−0.997418 + 0.0718139i \(0.977121\pi\)
\(884\) 44.2855i 1.48948i
\(885\) 0 0
\(886\) −55.9614 −1.88006
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −57.0799 3.68979i −1.91548 0.123821i
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −16.7967 −0.562395
\(893\) 0 0
\(894\) −1.78770 + 27.6551i −0.0597895 + 0.924925i
\(895\) −57.4781 −1.92128
\(896\) 33.8350i 1.13035i
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −67.8128 8.80398i −2.26043 0.293466i
\(901\) 0 0
\(902\) 0 0
\(903\) −4.05103 + 62.6682i −0.134810 + 2.08547i
\(904\) 57.6888 1.91870
\(905\) 0 0
\(906\) −35.1493 2.27214i −1.16776 0.0754868i
\(907\) −57.9364 −1.92375 −0.961874 0.273495i \(-0.911820\pi\)
−0.961874 + 0.273495i \(0.911820\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) −61.7489 −2.04696
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 12.5360 0.414200
\(917\) 56.3115i 1.85957i
\(918\) 8.68510 44.2855i 0.286651 1.46164i
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 25.9515 0.854666
\(923\) 11.1817i 0.368051i
\(924\) 0 0
\(925\) −133.068 −4.37525
\(926\) 10.1980i 0.335128i
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 4.61395 71.3763i 0.151297 2.34052i
\(931\) 0 0
\(932\) 19.9303i 0.652837i
\(933\) 0 0
\(934\) 14.4222 0.471909
\(935\) 0 0
\(936\) 3.93891 30.3395i 0.128747 0.991677i
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 3.42391 52.9667i 0.111735 1.72850i
\(940\) 40.4720 1.32005
\(941\) 48.3348i 1.57567i −0.615887 0.787835i \(-0.711202\pi\)
0.615887 0.787835i \(-0.288798\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 61.7489 + 12.1100i 2.00869 + 0.393937i
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 48.8878 + 3.16023i 1.58529 + 0.102478i
\(952\) −51.9476 −1.68363
\(953\) 52.2588i 1.69283i 0.532524 + 0.846415i \(0.321244\pi\)
−0.532524 + 0.846415i \(0.678756\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 38.5971i 1.24832i
\(957\) 0 0
\(958\) −44.1085 −1.42508
\(959\) 0 0
\(960\) −55.9922 3.61948i −1.80714 0.116818i
\(961\) 21.0000 0.677419
\(962\) 59.5347i 1.91948i
\(963\) −3.93891 + 30.3395i −0.126929 + 0.977677i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −61.6559 −1.98272 −0.991360 0.131166i \(-0.958128\pi\)
−0.991360 + 0.131166i \(0.958128\pi\)
\(968\) 31.1127i 1.00000i
\(969\) 0 0
\(970\) 0 0
\(971\) 60.3120i 1.93551i 0.251902 + 0.967753i \(0.418944\pi\)
−0.251902 + 0.967753i \(0.581056\pi\)
\(972\) −9.88897 + 29.5670i −0.317189 + 0.948362i
\(973\) 70.5091 2.26042
\(974\) 50.9902i 1.63383i
\(975\) 4.59130 71.0259i 0.147039 2.27465i
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 15.7420i 0.502860i
\(981\) −0.853097 0.110756i −0.0272373 0.00353615i
\(982\) 61.3692 1.95837
\(983\) 25.5010i 0.813357i 0.913571 + 0.406678i \(0.133313\pi\)
−0.913571 + 0.406678i \(0.866687\pi\)
\(984\) 0 0
\(985\) −24.0751 −0.767095
\(986\) 0 0
\(987\) −25.8321 1.66985i −0.822243 0.0531519i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 40.7922i 1.29515i
\(993\) 0 0
\(994\) 13.1163 0.416025
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) −11.6757 + 59.5347i −0.369403 + 1.88359i
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 312.2.h.b.155.9 yes 12
3.2 odd 2 inner 312.2.h.b.155.4 yes 12
4.3 odd 2 1248.2.h.b.623.8 12
8.3 odd 2 inner 312.2.h.b.155.3 12
8.5 even 2 1248.2.h.b.623.7 12
12.11 even 2 1248.2.h.b.623.5 12
13.12 even 2 inner 312.2.h.b.155.3 12
24.5 odd 2 1248.2.h.b.623.6 12
24.11 even 2 inner 312.2.h.b.155.10 yes 12
39.38 odd 2 inner 312.2.h.b.155.10 yes 12
52.51 odd 2 1248.2.h.b.623.7 12
104.51 odd 2 CM 312.2.h.b.155.9 yes 12
104.77 even 2 1248.2.h.b.623.8 12
156.155 even 2 1248.2.h.b.623.6 12
312.77 odd 2 1248.2.h.b.623.5 12
312.155 even 2 inner 312.2.h.b.155.4 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.2.h.b.155.3 12 8.3 odd 2 inner
312.2.h.b.155.3 12 13.12 even 2 inner
312.2.h.b.155.4 yes 12 3.2 odd 2 inner
312.2.h.b.155.4 yes 12 312.155 even 2 inner
312.2.h.b.155.9 yes 12 1.1 even 1 trivial
312.2.h.b.155.9 yes 12 104.51 odd 2 CM
312.2.h.b.155.10 yes 12 24.11 even 2 inner
312.2.h.b.155.10 yes 12 39.38 odd 2 inner
1248.2.h.b.623.5 12 12.11 even 2
1248.2.h.b.623.5 12 312.77 odd 2
1248.2.h.b.623.6 12 24.5 odd 2
1248.2.h.b.623.6 12 156.155 even 2
1248.2.h.b.623.7 12 8.5 even 2
1248.2.h.b.623.7 12 52.51 odd 2
1248.2.h.b.623.8 12 4.3 odd 2
1248.2.h.b.623.8 12 104.77 even 2