Properties

Label 1512.2.c.c.757.2
Level $1512$
Weight $2$
Character 1512.757
Analytic conductor $12.073$
Analytic rank $0$
Dimension $8$
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] = [1512,2,Mod(757,1512)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1512, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("1512.757");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1512 = 2^{3} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1512.c (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: \(12.0733807856\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 757.2
Root \(0.178197 - 1.40294i\) of defining polynomial
Character \(\chi\) \(=\) 1512.757
Dual form 1512.2.c.c.757.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.40294 + 0.178197i) q^{2} +(1.93649 - 0.500000i) q^{4} -0.356394i q^{5} -1.00000 q^{7} +(-2.62769 + 1.04655i) q^{8} +O(q^{10})\) \(q+(-1.40294 + 0.178197i) q^{2} +(1.93649 - 0.500000i) q^{4} -0.356394i q^{5} -1.00000 q^{7} +(-2.62769 + 1.04655i) q^{8} +(0.0635083 + 0.500000i) q^{10} +5.96816i q^{11} -2.87298i q^{13} +(1.40294 - 0.178197i) q^{14} +(3.50000 - 1.93649i) q^{16} +3.16228 q^{17} -0.127017i q^{19} +(-0.178197 - 0.690154i) q^{20} +(-1.06351 - 8.37298i) q^{22} -2.80588 q^{23} +4.87298 q^{25} +(0.511957 + 4.03063i) q^{26} +(-1.93649 + 0.500000i) q^{28} +2.44949i q^{29} -7.00000 q^{31} +(-4.56522 + 3.34047i) q^{32} +(-4.43649 + 0.563508i) q^{34} +0.356394i q^{35} +1.87298i q^{37} +(0.0226340 + 0.178197i) q^{38} +(0.372983 + 0.936492i) q^{40} -6.99208 q^{41} -1.12702i q^{43} +(2.98408 + 11.5573i) q^{44} +(3.93649 - 0.500000i) q^{46} -7.34847 q^{47} +1.00000 q^{49} +(-6.83651 + 0.868351i) q^{50} +(-1.43649 - 5.56351i) q^{52} +2.12702 q^{55} +(2.62769 - 1.04655i) q^{56} +(-0.436492 - 3.43649i) q^{58} +6.63568i q^{59} +13.7460i q^{61} +(9.82059 - 1.24738i) q^{62} +(5.80948 - 5.50000i) q^{64} -1.02391 q^{65} +8.61895i q^{67} +(6.12372 - 1.58114i) q^{68} +(-0.0635083 - 0.500000i) q^{70} +5.96816 q^{71} +0.872983 q^{73} +(-0.333760 - 2.62769i) q^{74} +(-0.0635083 - 0.245967i) q^{76} -5.96816i q^{77} +12.6190 q^{79} +(-0.690154 - 1.24738i) q^{80} +(9.80948 - 1.24597i) q^{82} +6.63568i q^{83} -1.12702i q^{85} +(0.200831 + 1.58114i) q^{86} +(-6.24597 - 15.6825i) q^{88} -6.99208 q^{89} +2.87298i q^{91} +(-5.43357 + 1.40294i) q^{92} +(10.3095 - 1.30948i) q^{94} -0.0452680 q^{95} -9.74597 q^{97} +(-1.40294 + 0.178197i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} + 16 q^{10} + 28 q^{16} - 24 q^{22} + 8 q^{25} - 56 q^{31} - 20 q^{34} - 28 q^{40} + 16 q^{46} + 8 q^{49} + 4 q^{52} + 48 q^{55} + 12 q^{58} - 16 q^{70} - 24 q^{73} - 16 q^{76} + 8 q^{79} + 32 q^{82} + 12 q^{88} + 36 q^{94} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times\).

\(n\) \(757\) \(785\) \(1081\) \(1135\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.40294 + 0.178197i −0.992030 + 0.126004i
\(3\) 0 0
\(4\) 1.93649 0.500000i 0.968246 0.250000i
\(5\) 0.356394i 0.159384i −0.996820 0.0796921i \(-0.974606\pi\)
0.996820 0.0796921i \(-0.0253937\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.62769 + 1.04655i −0.929028 + 0.370011i
\(9\) 0 0
\(10\) 0.0635083 + 0.500000i 0.0200831 + 0.158114i
\(11\) 5.96816i 1.79947i 0.436438 + 0.899734i \(0.356240\pi\)
−0.436438 + 0.899734i \(0.643760\pi\)
\(12\) 0 0
\(13\) 2.87298i 0.796822i −0.917207 0.398411i \(-0.869562\pi\)
0.917207 0.398411i \(-0.130438\pi\)
\(14\) 1.40294 0.178197i 0.374952 0.0476251i
\(15\) 0 0
\(16\) 3.50000 1.93649i 0.875000 0.484123i
\(17\) 3.16228 0.766965 0.383482 0.923548i \(-0.374725\pi\)
0.383482 + 0.923548i \(0.374725\pi\)
\(18\) 0 0
\(19\) 0.127017i 0.0291396i −0.999894 0.0145698i \(-0.995362\pi\)
0.999894 0.0145698i \(-0.00463788\pi\)
\(20\) −0.178197 0.690154i −0.0398461 0.154323i
\(21\) 0 0
\(22\) −1.06351 8.37298i −0.226741 1.78513i
\(23\) −2.80588 −0.585067 −0.292534 0.956255i \(-0.594498\pi\)
−0.292534 + 0.956255i \(0.594498\pi\)
\(24\) 0 0
\(25\) 4.87298 0.974597
\(26\) 0.511957 + 4.03063i 0.100403 + 0.790471i
\(27\) 0 0
\(28\) −1.93649 + 0.500000i −0.365963 + 0.0944911i
\(29\) 2.44949i 0.454859i 0.973795 + 0.227429i \(0.0730321\pi\)
−0.973795 + 0.227429i \(0.926968\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) −4.56522 + 3.34047i −0.807024 + 0.590518i
\(33\) 0 0
\(34\) −4.43649 + 0.563508i −0.760852 + 0.0966409i
\(35\) 0.356394i 0.0602416i
\(36\) 0 0
\(37\) 1.87298i 0.307917i 0.988077 + 0.153958i \(0.0492021\pi\)
−0.988077 + 0.153958i \(0.950798\pi\)
\(38\) 0.0226340 + 0.178197i 0.00367172 + 0.0289074i
\(39\) 0 0
\(40\) 0.372983 + 0.936492i 0.0589738 + 0.148072i
\(41\) −6.99208 −1.09198 −0.545989 0.837792i \(-0.683846\pi\)
−0.545989 + 0.837792i \(0.683846\pi\)
\(42\) 0 0
\(43\) 1.12702i 0.171868i −0.996301 0.0859342i \(-0.972613\pi\)
0.996301 0.0859342i \(-0.0273875\pi\)
\(44\) 2.98408 + 11.5573i 0.449867 + 1.74233i
\(45\) 0 0
\(46\) 3.93649 0.500000i 0.580404 0.0737210i
\(47\) −7.34847 −1.07188 −0.535942 0.844255i \(-0.680044\pi\)
−0.535942 + 0.844255i \(0.680044\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −6.83651 + 0.868351i −0.966829 + 0.122803i
\(51\) 0 0
\(52\) −1.43649 5.56351i −0.199206 0.771520i
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 2.12702 0.286807
\(56\) 2.62769 1.04655i 0.351139 0.139851i
\(57\) 0 0
\(58\) −0.436492 3.43649i −0.0573142 0.451233i
\(59\) 6.63568i 0.863892i 0.901899 + 0.431946i \(0.142173\pi\)
−0.901899 + 0.431946i \(0.857827\pi\)
\(60\) 0 0
\(61\) 13.7460i 1.75999i 0.474982 + 0.879995i \(0.342454\pi\)
−0.474982 + 0.879995i \(0.657546\pi\)
\(62\) 9.82059 1.24738i 1.24722 0.158417i
\(63\) 0 0
\(64\) 5.80948 5.50000i 0.726184 0.687500i
\(65\) −1.02391 −0.127001
\(66\) 0 0
\(67\) 8.61895i 1.05297i 0.850184 + 0.526486i \(0.176491\pi\)
−0.850184 + 0.526486i \(0.823509\pi\)
\(68\) 6.12372 1.58114i 0.742611 0.191741i
\(69\) 0 0
\(70\) −0.0635083 0.500000i −0.00759070 0.0597614i
\(71\) 5.96816 0.708290 0.354145 0.935190i \(-0.384772\pi\)
0.354145 + 0.935190i \(0.384772\pi\)
\(72\) 0 0
\(73\) 0.872983 0.102175 0.0510875 0.998694i \(-0.483731\pi\)
0.0510875 + 0.998694i \(0.483731\pi\)
\(74\) −0.333760 2.62769i −0.0387988 0.305462i
\(75\) 0 0
\(76\) −0.0635083 0.245967i −0.00728490 0.0282143i
\(77\) 5.96816i 0.680135i
\(78\) 0 0
\(79\) 12.6190 1.41974 0.709871 0.704331i \(-0.248753\pi\)
0.709871 + 0.704331i \(0.248753\pi\)
\(80\) −0.690154 1.24738i −0.0771616 0.139461i
\(81\) 0 0
\(82\) 9.80948 1.24597i 1.08328 0.137594i
\(83\) 6.63568i 0.728361i 0.931328 + 0.364180i \(0.118651\pi\)
−0.931328 + 0.364180i \(0.881349\pi\)
\(84\) 0 0
\(85\) 1.12702i 0.122242i
\(86\) 0.200831 + 1.58114i 0.0216562 + 0.170499i
\(87\) 0 0
\(88\) −6.24597 15.6825i −0.665822 1.67176i
\(89\) −6.99208 −0.741158 −0.370579 0.928801i \(-0.620841\pi\)
−0.370579 + 0.928801i \(0.620841\pi\)
\(90\) 0 0
\(91\) 2.87298i 0.301170i
\(92\) −5.43357 + 1.40294i −0.566489 + 0.146267i
\(93\) 0 0
\(94\) 10.3095 1.30948i 1.06334 0.135062i
\(95\) −0.0452680 −0.00464440
\(96\) 0 0
\(97\) −9.74597 −0.989553 −0.494776 0.869020i \(-0.664750\pi\)
−0.494776 + 0.869020i \(0.664750\pi\)
\(98\) −1.40294 + 0.178197i −0.141719 + 0.0180006i
\(99\) 0 0
\(100\) 9.43649 2.43649i 0.943649 0.243649i
\(101\) 11.5347i 1.14774i −0.818946 0.573871i \(-0.805441\pi\)
0.818946 0.573871i \(-0.194559\pi\)
\(102\) 0 0
\(103\) −10.7460 −1.05883 −0.529416 0.848363i \(-0.677589\pi\)
−0.529416 + 0.848363i \(0.677589\pi\)
\(104\) 3.00671 + 7.54930i 0.294833 + 0.740270i
\(105\) 0 0
\(106\) 0 0
\(107\) 14.3858i 1.39073i 0.718657 + 0.695364i \(0.244757\pi\)
−0.718657 + 0.695364i \(0.755243\pi\)
\(108\) 0 0
\(109\) 12.7460i 1.22084i 0.792077 + 0.610421i \(0.209000\pi\)
−0.792077 + 0.610421i \(0.791000\pi\)
\(110\) −2.98408 + 0.379028i −0.284521 + 0.0361389i
\(111\) 0 0
\(112\) −3.50000 + 1.93649i −0.330719 + 0.182981i
\(113\) 16.8353 1.58373 0.791866 0.610695i \(-0.209110\pi\)
0.791866 + 0.610695i \(0.209110\pi\)
\(114\) 0 0
\(115\) 1.00000i 0.0932505i
\(116\) 1.22474 + 4.74342i 0.113715 + 0.440415i
\(117\) 0 0
\(118\) −1.18246 9.30948i −0.108854 0.857007i
\(119\) −3.16228 −0.289886
\(120\) 0 0
\(121\) −24.6190 −2.23809
\(122\) −2.44949 19.2848i −0.221766 1.74596i
\(123\) 0 0
\(124\) −13.5554 + 3.50000i −1.21731 + 0.314309i
\(125\) 3.51867i 0.314720i
\(126\) 0 0
\(127\) 10.8730 0.964821 0.482411 0.875945i \(-0.339761\pi\)
0.482411 + 0.875945i \(0.339761\pi\)
\(128\) −7.17027 + 8.75141i −0.633769 + 0.773523i
\(129\) 0 0
\(130\) 1.43649 0.182458i 0.125989 0.0160027i
\(131\) 8.77405i 0.766592i 0.923626 + 0.383296i \(0.125211\pi\)
−0.923626 + 0.383296i \(0.874789\pi\)
\(132\) 0 0
\(133\) 0.127017i 0.0110137i
\(134\) −1.53587 12.0919i −0.132679 1.04458i
\(135\) 0 0
\(136\) −8.30948 + 3.30948i −0.712532 + 0.283785i
\(137\) −4.58785 −0.391967 −0.195983 0.980607i \(-0.562790\pi\)
−0.195983 + 0.980607i \(0.562790\pi\)
\(138\) 0 0
\(139\) 14.0000i 1.18746i 0.804663 + 0.593732i \(0.202346\pi\)
−0.804663 + 0.593732i \(0.797654\pi\)
\(140\) 0.178197 + 0.690154i 0.0150604 + 0.0583287i
\(141\) 0 0
\(142\) −8.37298 + 1.06351i −0.702645 + 0.0892476i
\(143\) 17.1464 1.43386
\(144\) 0 0
\(145\) 0.872983 0.0724973
\(146\) −1.22474 + 0.155563i −0.101361 + 0.0128745i
\(147\) 0 0
\(148\) 0.936492 + 3.62702i 0.0769791 + 0.298139i
\(149\) 9.79796i 0.802680i 0.915929 + 0.401340i \(0.131455\pi\)
−0.915929 + 0.401340i \(0.868545\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0.132929 + 0.333760i 0.0107820 + 0.0270715i
\(153\) 0 0
\(154\) 1.06351 + 8.37298i 0.0856999 + 0.674714i
\(155\) 2.49476i 0.200384i
\(156\) 0 0
\(157\) 7.12702i 0.568798i −0.958706 0.284399i \(-0.908206\pi\)
0.958706 0.284399i \(-0.0917940\pi\)
\(158\) −17.7037 + 2.24866i −1.40843 + 0.178894i
\(159\) 0 0
\(160\) 1.19052 + 1.62702i 0.0941193 + 0.128627i
\(161\) 2.80588 0.221135
\(162\) 0 0
\(163\) 22.8730i 1.79155i 0.444507 + 0.895775i \(0.353379\pi\)
−0.444507 + 0.895775i \(0.646621\pi\)
\(164\) −13.5401 + 3.49604i −1.05730 + 0.272995i
\(165\) 0 0
\(166\) −1.18246 9.30948i −0.0917766 0.722555i
\(167\) 1.42558 0.110314 0.0551572 0.998478i \(-0.482434\pi\)
0.0551572 + 0.998478i \(0.482434\pi\)
\(168\) 0 0
\(169\) 4.74597 0.365074
\(170\) 0.200831 + 1.58114i 0.0154030 + 0.121268i
\(171\) 0 0
\(172\) −0.563508 2.18246i −0.0429671 0.166411i
\(173\) 18.2156i 1.38491i −0.721462 0.692454i \(-0.756530\pi\)
0.721462 0.692454i \(-0.243470\pi\)
\(174\) 0 0
\(175\) −4.87298 −0.368363
\(176\) 11.5573 + 20.8886i 0.871164 + 1.57453i
\(177\) 0 0
\(178\) 9.80948 1.24597i 0.735251 0.0933892i
\(179\) 8.77405i 0.655803i 0.944712 + 0.327901i \(0.106341\pi\)
−0.944712 + 0.327901i \(0.893659\pi\)
\(180\) 0 0
\(181\) 21.7460i 1.61636i 0.588932 + 0.808182i \(0.299549\pi\)
−0.588932 + 0.808182i \(0.700451\pi\)
\(182\) −0.511957 4.03063i −0.0379488 0.298770i
\(183\) 0 0
\(184\) 7.37298 2.93649i 0.543544 0.216481i
\(185\) 0.667520 0.0490770
\(186\) 0 0
\(187\) 18.8730i 1.38013i
\(188\) −14.2302 + 3.67423i −1.03785 + 0.267971i
\(189\) 0 0
\(190\) 0.0635083 0.00806662i 0.00460738 0.000585214i
\(191\) −21.7795 −1.57591 −0.787956 0.615731i \(-0.788861\pi\)
−0.787956 + 0.615731i \(0.788861\pi\)
\(192\) 0 0
\(193\) −9.49193 −0.683244 −0.341622 0.939837i \(-0.610976\pi\)
−0.341622 + 0.939837i \(0.610976\pi\)
\(194\) 13.6730 1.73670i 0.981666 0.124688i
\(195\) 0 0
\(196\) 1.93649 0.500000i 0.138321 0.0357143i
\(197\) 2.04783i 0.145902i 0.997336 + 0.0729508i \(0.0232416\pi\)
−0.997336 + 0.0729508i \(0.976758\pi\)
\(198\) 0 0
\(199\) 0.127017 0.00900397 0.00450199 0.999990i \(-0.498567\pi\)
0.00450199 + 0.999990i \(0.498567\pi\)
\(200\) −12.8047 + 5.09981i −0.905427 + 0.360611i
\(201\) 0 0
\(202\) 2.05544 + 16.1825i 0.144620 + 1.13859i
\(203\) 2.44949i 0.171920i
\(204\) 0 0
\(205\) 2.49193i 0.174044i
\(206\) 15.0760 1.91490i 1.05039 0.133417i
\(207\) 0 0
\(208\) −5.56351 10.0554i −0.385760 0.697219i
\(209\) 0.758056 0.0524358
\(210\) 0 0
\(211\) 20.0000i 1.37686i −0.725304 0.688428i \(-0.758301\pi\)
0.725304 0.688428i \(-0.241699\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −2.56351 20.1825i −0.175238 1.37964i
\(215\) −0.401662 −0.0273931
\(216\) 0 0
\(217\) 7.00000 0.475191
\(218\) −2.27129 17.8819i −0.153831 1.21111i
\(219\) 0 0
\(220\) 4.11895 1.06351i 0.277700 0.0717017i
\(221\) 9.08517i 0.611135i
\(222\) 0 0
\(223\) 7.61895 0.510203 0.255101 0.966914i \(-0.417891\pi\)
0.255101 + 0.966914i \(0.417891\pi\)
\(224\) 4.56522 3.34047i 0.305027 0.223195i
\(225\) 0 0
\(226\) −23.6190 + 3.00000i −1.57111 + 0.199557i
\(227\) 25.9205i 1.72040i −0.509955 0.860201i \(-0.670338\pi\)
0.509955 0.860201i \(-0.329662\pi\)
\(228\) 0 0
\(229\) 15.7460i 1.04052i 0.854007 + 0.520261i \(0.174165\pi\)
−0.854007 + 0.520261i \(0.825835\pi\)
\(230\) −0.178197 1.40294i −0.0117500 0.0925072i
\(231\) 0 0
\(232\) −2.56351 6.43649i −0.168303 0.422576i
\(233\) −5.61177 −0.367639 −0.183820 0.982960i \(-0.558846\pi\)
−0.183820 + 0.982960i \(0.558846\pi\)
\(234\) 0 0
\(235\) 2.61895i 0.170841i
\(236\) 3.31784 + 12.8499i 0.215973 + 0.836460i
\(237\) 0 0
\(238\) 4.43649 0.563508i 0.287575 0.0365268i
\(239\) −12.2474 −0.792222 −0.396111 0.918203i \(-0.629640\pi\)
−0.396111 + 0.918203i \(0.629640\pi\)
\(240\) 0 0
\(241\) 28.6190 1.84351 0.921754 0.387774i \(-0.126756\pi\)
0.921754 + 0.387774i \(0.126756\pi\)
\(242\) 34.5390 4.38702i 2.22025 0.282008i
\(243\) 0 0
\(244\) 6.87298 + 26.6190i 0.439998 + 1.70410i
\(245\) 0.356394i 0.0227692i
\(246\) 0 0
\(247\) −0.364917 −0.0232191
\(248\) 18.3938 7.32584i 1.16801 0.465191i
\(249\) 0 0
\(250\) 0.627017 + 4.93649i 0.0396560 + 0.312211i
\(251\) 16.8353i 1.06263i 0.847173 + 0.531317i \(0.178303\pi\)
−0.847173 + 0.531317i \(0.821697\pi\)
\(252\) 0 0
\(253\) 16.7460i 1.05281i
\(254\) −15.2542 + 1.93753i −0.957131 + 0.121572i
\(255\) 0 0
\(256\) 8.50000 13.5554i 0.531250 0.847215i
\(257\) 20.0428 1.25024 0.625119 0.780529i \(-0.285050\pi\)
0.625119 + 0.780529i \(0.285050\pi\)
\(258\) 0 0
\(259\) 1.87298i 0.116382i
\(260\) −1.98280 + 0.511957i −0.122968 + 0.0317502i
\(261\) 0 0
\(262\) −1.56351 12.3095i −0.0965939 0.760482i
\(263\) 2.80588 0.173018 0.0865091 0.996251i \(-0.472429\pi\)
0.0865091 + 0.996251i \(0.472429\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.0226340 0.178197i −0.00138778 0.0109260i
\(267\) 0 0
\(268\) 4.30948 + 16.6905i 0.263243 + 1.01954i
\(269\) 10.5560i 0.643612i −0.946806 0.321806i \(-0.895710\pi\)
0.946806 0.321806i \(-0.104290\pi\)
\(270\) 0 0
\(271\) −19.7460 −1.19948 −0.599741 0.800194i \(-0.704730\pi\)
−0.599741 + 0.800194i \(0.704730\pi\)
\(272\) 11.0680 6.12372i 0.671094 0.371305i
\(273\) 0 0
\(274\) 6.43649 0.817542i 0.388843 0.0493895i
\(275\) 29.0828i 1.75376i
\(276\) 0 0
\(277\) 5.87298i 0.352873i 0.984312 + 0.176437i \(0.0564571\pi\)
−0.984312 + 0.176437i \(0.943543\pi\)
\(278\) −2.49476 19.6412i −0.149626 1.17800i
\(279\) 0 0
\(280\) −0.372983 0.936492i −0.0222900 0.0559661i
\(281\) −22.0454 −1.31512 −0.657559 0.753403i \(-0.728411\pi\)
−0.657559 + 0.753403i \(0.728411\pi\)
\(282\) 0 0
\(283\) 11.4919i 0.683125i −0.939859 0.341562i \(-0.889044\pi\)
0.939859 0.341562i \(-0.110956\pi\)
\(284\) 11.5573 2.98408i 0.685799 0.177073i
\(285\) 0 0
\(286\) −24.0554 + 3.05544i −1.42243 + 0.180672i
\(287\) 6.99208 0.412729
\(288\) 0 0
\(289\) −7.00000 −0.411765
\(290\) −1.22474 + 0.155563i −0.0719195 + 0.00913497i
\(291\) 0 0
\(292\) 1.69052 0.436492i 0.0989305 0.0255437i
\(293\) 31.8434i 1.86031i −0.367168 0.930155i \(-0.619673\pi\)
0.367168 0.930155i \(-0.380327\pi\)
\(294\) 0 0
\(295\) 2.36492 0.137691
\(296\) −1.96017 4.92161i −0.113932 0.286063i
\(297\) 0 0
\(298\) −1.74597 13.7460i −0.101141 0.796282i
\(299\) 8.06126i 0.466195i
\(300\) 0 0
\(301\) 1.12702i 0.0649602i
\(302\) 16.8353 2.13836i 0.968763 0.123049i
\(303\) 0 0
\(304\) −0.245967 0.444558i −0.0141072 0.0254972i
\(305\) 4.89898 0.280515
\(306\) 0 0
\(307\) 29.6190i 1.69044i −0.534416 0.845221i \(-0.679469\pi\)
0.534416 0.845221i \(-0.320531\pi\)
\(308\) −2.98408 11.5573i −0.170034 0.658538i
\(309\) 0 0
\(310\) −0.444558 3.50000i −0.0252492 0.198787i
\(311\) −20.7104 −1.17438 −0.587189 0.809450i \(-0.699765\pi\)
−0.587189 + 0.809450i \(0.699765\pi\)
\(312\) 0 0
\(313\) 2.87298 0.162391 0.0811953 0.996698i \(-0.474126\pi\)
0.0811953 + 0.996698i \(0.474126\pi\)
\(314\) 1.27001 + 9.99879i 0.0716710 + 0.564264i
\(315\) 0 0
\(316\) 24.4365 6.30948i 1.37466 0.354936i
\(317\) 13.6730i 0.767954i −0.923343 0.383977i \(-0.874554\pi\)
0.923343 0.383977i \(-0.125446\pi\)
\(318\) 0 0
\(319\) −14.6190 −0.818504
\(320\) −1.96017 2.07046i −0.109577 0.115742i
\(321\) 0 0
\(322\) −3.93649 + 0.500000i −0.219372 + 0.0278639i
\(323\) 0.401662i 0.0223491i
\(324\) 0 0
\(325\) 14.0000i 0.776580i
\(326\) −4.07590 32.0895i −0.225743 1.77727i
\(327\) 0 0
\(328\) 18.3730 7.31754i 1.01448 0.404044i
\(329\) 7.34847 0.405134
\(330\) 0 0
\(331\) 10.0000i 0.549650i 0.961494 + 0.274825i \(0.0886199\pi\)
−0.961494 + 0.274825i \(0.911380\pi\)
\(332\) 3.31784 + 12.8499i 0.182090 + 0.705232i
\(333\) 0 0
\(334\) −2.00000 + 0.254033i −0.109435 + 0.0139001i
\(335\) 3.07174 0.167827
\(336\) 0 0
\(337\) −3.25403 −0.177258 −0.0886292 0.996065i \(-0.528249\pi\)
−0.0886292 + 0.996065i \(0.528249\pi\)
\(338\) −6.65832 + 0.845717i −0.362165 + 0.0460009i
\(339\) 0 0
\(340\) −0.563508 2.18246i −0.0305605 0.118360i
\(341\) 41.7771i 2.26236i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 1.17948 + 2.96145i 0.0635931 + 0.159671i
\(345\) 0 0
\(346\) 3.24597 + 25.5554i 0.174504 + 1.37387i
\(347\) 36.4765i 1.95816i −0.203475 0.979080i \(-0.565223\pi\)
0.203475 0.979080i \(-0.434777\pi\)
\(348\) 0 0
\(349\) 23.4919i 1.25749i −0.777610 0.628747i \(-0.783568\pi\)
0.777610 0.628747i \(-0.216432\pi\)
\(350\) 6.83651 0.868351i 0.365427 0.0464153i
\(351\) 0 0
\(352\) −19.9365 27.2460i −1.06262 1.45221i
\(353\) 13.3166 0.708773 0.354386 0.935099i \(-0.384690\pi\)
0.354386 + 0.935099i \(0.384690\pi\)
\(354\) 0 0
\(355\) 2.12702i 0.112890i
\(356\) −13.5401 + 3.49604i −0.717624 + 0.185290i
\(357\) 0 0
\(358\) −1.56351 12.3095i −0.0826340 0.650576i
\(359\) 29.1733 1.53971 0.769854 0.638221i \(-0.220329\pi\)
0.769854 + 0.638221i \(0.220329\pi\)
\(360\) 0 0
\(361\) 18.9839 0.999151
\(362\) −3.87507 30.5083i −0.203669 1.60348i
\(363\) 0 0
\(364\) 1.43649 + 5.56351i 0.0752926 + 0.291607i
\(365\) 0.311126i 0.0162851i
\(366\) 0 0
\(367\) 21.8730 1.14176 0.570880 0.821033i \(-0.306602\pi\)
0.570880 + 0.821033i \(0.306602\pi\)
\(368\) −9.82059 + 5.43357i −0.511934 + 0.283244i
\(369\) 0 0
\(370\) −0.936492 + 0.118950i −0.0486859 + 0.00618392i
\(371\) 0 0
\(372\) 0 0
\(373\) 28.2379i 1.46210i 0.682322 + 0.731052i \(0.260970\pi\)
−0.682322 + 0.731052i \(0.739030\pi\)
\(374\) −3.36311 26.4777i −0.173902 1.36913i
\(375\) 0 0
\(376\) 19.3095 7.69052i 0.995810 0.396609i
\(377\) 7.03734 0.362442
\(378\) 0 0
\(379\) 8.87298i 0.455775i 0.973688 + 0.227887i \(0.0731818\pi\)
−0.973688 + 0.227887i \(0.926818\pi\)
\(380\) −0.0876610 + 0.0226340i −0.00449692 + 0.00116110i
\(381\) 0 0
\(382\) 30.5554 3.88105i 1.56335 0.198572i
\(383\) 17.4576 0.892039 0.446020 0.895023i \(-0.352841\pi\)
0.446020 + 0.895023i \(0.352841\pi\)
\(384\) 0 0
\(385\) −2.12702 −0.108403
\(386\) 13.3166 1.69143i 0.677799 0.0860917i
\(387\) 0 0
\(388\) −18.8730 + 4.87298i −0.958131 + 0.247388i
\(389\) 15.4097i 0.781304i −0.920538 0.390652i \(-0.872250\pi\)
0.920538 0.390652i \(-0.127750\pi\)
\(390\) 0 0
\(391\) −8.87298 −0.448726
\(392\) −2.62769 + 1.04655i −0.132718 + 0.0528587i
\(393\) 0 0
\(394\) −0.364917 2.87298i −0.0183842 0.144739i
\(395\) 4.49732i 0.226285i
\(396\) 0 0
\(397\) 10.0000i 0.501886i −0.968002 0.250943i \(-0.919259\pi\)
0.968002 0.250943i \(-0.0807406\pi\)
\(398\) −0.178197 + 0.0226340i −0.00893221 + 0.00113454i
\(399\) 0 0
\(400\) 17.0554 9.43649i 0.852772 0.471825i
\(401\) 16.5242 0.825178 0.412589 0.910917i \(-0.364625\pi\)
0.412589 + 0.910917i \(0.364625\pi\)
\(402\) 0 0
\(403\) 20.1109i 1.00179i
\(404\) −5.76733 22.3368i −0.286935 1.11130i
\(405\) 0 0
\(406\) 0.436492 + 3.43649i 0.0216627 + 0.170550i
\(407\) −11.1783 −0.554086
\(408\) 0 0
\(409\) −12.3649 −0.611406 −0.305703 0.952127i \(-0.598891\pi\)
−0.305703 + 0.952127i \(0.598891\pi\)
\(410\) −0.444055 3.49604i −0.0219303 0.172657i
\(411\) 0 0
\(412\) −20.8095 + 5.37298i −1.02521 + 0.264708i
\(413\) 6.63568i 0.326521i
\(414\) 0 0
\(415\) 2.36492 0.116089
\(416\) 9.59713 + 13.1158i 0.470538 + 0.643055i
\(417\) 0 0
\(418\) −1.06351 + 0.135083i −0.0520179 + 0.00660714i
\(419\) 28.7716i 1.40559i 0.711394 + 0.702793i \(0.248064\pi\)
−0.711394 + 0.702793i \(0.751936\pi\)
\(420\) 0 0
\(421\) 19.1109i 0.931407i −0.884941 0.465704i \(-0.845801\pi\)
0.884941 0.465704i \(-0.154199\pi\)
\(422\) 3.56394 + 28.0588i 0.173490 + 1.36588i
\(423\) 0 0
\(424\) 0 0
\(425\) 15.4097 0.747482
\(426\) 0 0
\(427\) 13.7460i 0.665214i
\(428\) 7.19291 + 27.8580i 0.347682 + 1.34657i
\(429\) 0 0
\(430\) 0.563508 0.0715749i 0.0271748 0.00345165i
\(431\) 23.1146 1.11339 0.556695 0.830717i \(-0.312069\pi\)
0.556695 + 0.830717i \(0.312069\pi\)
\(432\) 0 0
\(433\) −4.87298 −0.234181 −0.117090 0.993121i \(-0.537357\pi\)
−0.117090 + 0.993121i \(0.537357\pi\)
\(434\) −9.82059 + 1.24738i −0.471404 + 0.0598761i
\(435\) 0 0
\(436\) 6.37298 + 24.6825i 0.305211 + 1.18208i
\(437\) 0.356394i 0.0170486i
\(438\) 0 0
\(439\) 25.7460 1.22879 0.614394 0.788999i \(-0.289401\pi\)
0.614394 + 0.788999i \(0.289401\pi\)
\(440\) −5.58913 + 2.22602i −0.266451 + 0.106122i
\(441\) 0 0
\(442\) 1.61895 + 12.7460i 0.0770056 + 0.606264i
\(443\) 27.7024i 1.31618i 0.752938 + 0.658091i \(0.228636\pi\)
−0.752938 + 0.658091i \(0.771364\pi\)
\(444\) 0 0
\(445\) 2.49193i 0.118129i
\(446\) −10.6889 + 1.35767i −0.506136 + 0.0642877i
\(447\) 0 0
\(448\) −5.80948 + 5.50000i −0.274472 + 0.259851i
\(449\) −30.9100 −1.45873 −0.729366 0.684123i \(-0.760185\pi\)
−0.729366 + 0.684123i \(0.760185\pi\)
\(450\) 0 0
\(451\) 41.7298i 1.96498i
\(452\) 32.6014 8.41765i 1.53344 0.395933i
\(453\) 0 0
\(454\) 4.61895 + 36.3649i 0.216778 + 1.70669i
\(455\) 1.02391 0.0480018
\(456\) 0 0
\(457\) 9.25403 0.432885 0.216443 0.976295i \(-0.430555\pi\)
0.216443 + 0.976295i \(0.430555\pi\)
\(458\) −2.80588 22.0907i −0.131110 1.03223i
\(459\) 0 0
\(460\) 0.500000 + 1.93649i 0.0233126 + 0.0902894i
\(461\) 13.9389i 0.649198i 0.945852 + 0.324599i \(0.105229\pi\)
−0.945852 + 0.324599i \(0.894771\pi\)
\(462\) 0 0
\(463\) 0.508067 0.0236119 0.0118059 0.999930i \(-0.496242\pi\)
0.0118059 + 0.999930i \(0.496242\pi\)
\(464\) 4.74342 + 8.57321i 0.220208 + 0.398001i
\(465\) 0 0
\(466\) 7.87298 1.00000i 0.364709 0.0463241i
\(467\) 0.401662i 0.0185867i 0.999957 + 0.00929335i \(0.00295821\pi\)
−0.999957 + 0.00929335i \(0.997042\pi\)
\(468\) 0 0
\(469\) 8.61895i 0.397986i
\(470\) −0.466689 3.67423i −0.0215268 0.169480i
\(471\) 0 0
\(472\) −6.94456 17.4365i −0.319649 0.802580i
\(473\) 6.72622 0.309272
\(474\) 0 0
\(475\) 0.618950i 0.0283994i
\(476\) −6.12372 + 1.58114i −0.280680 + 0.0724714i
\(477\) 0 0
\(478\) 17.1825 2.18246i 0.785907 0.0998233i
\(479\) −0.712788 −0.0325681 −0.0162841 0.999867i \(-0.505184\pi\)
−0.0162841 + 0.999867i \(0.505184\pi\)
\(480\) 0 0
\(481\) 5.38105 0.245355
\(482\) −40.1507 + 5.09981i −1.82882 + 0.232290i
\(483\) 0 0
\(484\) −47.6744 + 12.3095i −2.16702 + 0.559522i
\(485\) 3.47340i 0.157719i
\(486\) 0 0
\(487\) 26.1109 1.18320 0.591599 0.806233i \(-0.298497\pi\)
0.591599 + 0.806233i \(0.298497\pi\)
\(488\) −14.3858 36.1201i −0.651215 1.63508i
\(489\) 0 0
\(490\) 0.0635083 + 0.500000i 0.00286901 + 0.0225877i
\(491\) 13.4072i 0.605057i −0.953140 0.302528i \(-0.902169\pi\)
0.953140 0.302528i \(-0.0978307\pi\)
\(492\) 0 0
\(493\) 7.74597i 0.348861i
\(494\) 0.511957 0.0650271i 0.0230340 0.00292571i
\(495\) 0 0
\(496\) −24.5000 + 13.5554i −1.10008 + 0.608657i
\(497\) −5.96816 −0.267709
\(498\) 0 0
\(499\) 34.3649i 1.53838i −0.639017 0.769192i \(-0.720659\pi\)
0.639017 0.769192i \(-0.279341\pi\)
\(500\) −1.75934 6.81388i −0.0786799 0.304726i
\(501\) 0 0
\(502\) −3.00000 23.6190i −0.133897 1.05417i
\(503\) 7.34847 0.327652 0.163826 0.986489i \(-0.447616\pi\)
0.163826 + 0.986489i \(0.447616\pi\)
\(504\) 0 0
\(505\) −4.11088 −0.182932
\(506\) 2.98408 + 23.4936i 0.132659 + 1.04442i
\(507\) 0 0
\(508\) 21.0554 5.43649i 0.934184 0.241205i
\(509\) 0.311126i 0.0137904i −0.999976 0.00689521i \(-0.997805\pi\)
0.999976 0.00689521i \(-0.00219483\pi\)
\(510\) 0 0
\(511\) −0.872983 −0.0386185
\(512\) −9.50947 + 20.5322i −0.420263 + 0.907402i
\(513\) 0 0
\(514\) −28.1190 + 3.57157i −1.24027 + 0.157535i
\(515\) 3.82980i 0.168761i
\(516\) 0 0
\(517\) 43.8569i 1.92882i
\(518\) 0.333760 + 2.62769i 0.0146646 + 0.115454i
\(519\) 0 0
\(520\) 2.69052 1.07157i 0.117987 0.0469917i
\(521\) −29.8408 −1.30735 −0.653675 0.756776i \(-0.726774\pi\)
−0.653675 + 0.756776i \(0.726774\pi\)
\(522\) 0 0
\(523\) 34.7460i 1.51934i 0.650312 + 0.759668i \(0.274638\pi\)
−0.650312 + 0.759668i \(0.725362\pi\)
\(524\) 4.38702 + 16.9909i 0.191648 + 0.742249i
\(525\) 0 0
\(526\) −3.93649 + 0.500000i −0.171639 + 0.0218010i
\(527\) −22.1359 −0.964257
\(528\) 0 0
\(529\) −15.1270 −0.657696
\(530\) 0 0
\(531\) 0 0
\(532\) 0.0635083 + 0.245967i 0.00275344 + 0.0106640i
\(533\) 20.0881i 0.870113i
\(534\) 0 0
\(535\) 5.12702 0.221660
\(536\) −9.02014 22.6479i −0.389611 0.978240i
\(537\) 0 0
\(538\) 1.88105 + 14.8095i 0.0810978 + 0.638482i
\(539\) 5.96816i 0.257067i
\(540\) 0 0
\(541\) 30.8569i 1.32664i −0.748336 0.663320i \(-0.769147\pi\)
0.748336 0.663320i \(-0.230853\pi\)
\(542\) 27.7024 3.51867i 1.18992 0.151140i
\(543\) 0 0
\(544\) −14.4365 + 10.5635i −0.618959 + 0.452907i
\(545\) 4.54259 0.194583
\(546\) 0 0
\(547\) 39.2379i 1.67769i −0.544369 0.838846i \(-0.683231\pi\)
0.544369 0.838846i \(-0.316769\pi\)
\(548\) −8.88434 + 2.29393i −0.379520 + 0.0979917i
\(549\) 0 0
\(550\) −5.18246 40.8014i −0.220981 1.73978i
\(551\) 0.311126 0.0132544
\(552\) 0 0
\(553\) −12.6190 −0.536612
\(554\) −1.04655 8.23945i −0.0444636 0.350061i
\(555\) 0 0
\(556\) 7.00000 + 27.1109i 0.296866 + 1.14976i
\(557\) 9.17571i 0.388787i −0.980924 0.194394i \(-0.937726\pi\)
0.980924 0.194394i \(-0.0622739\pi\)
\(558\) 0 0
\(559\) −3.23790 −0.136949
\(560\) 0.690154 + 1.24738i 0.0291643 + 0.0527114i
\(561\) 0 0
\(562\) 30.9284 3.92843i 1.30464 0.165711i
\(563\) 7.34847i 0.309701i −0.987938 0.154851i \(-0.950510\pi\)
0.987938 0.154851i \(-0.0494896\pi\)
\(564\) 0 0
\(565\) 6.00000i 0.252422i
\(566\) 2.04783 + 16.1225i 0.0860766 + 0.677680i
\(567\) 0 0
\(568\) −15.6825 + 6.24597i −0.658021 + 0.262075i
\(569\) −5.52123 −0.231462 −0.115731 0.993281i \(-0.536921\pi\)
−0.115731 + 0.993281i \(0.536921\pi\)
\(570\) 0 0
\(571\) 32.9839i 1.38033i 0.723651 + 0.690166i \(0.242462\pi\)
−0.723651 + 0.690166i \(0.757538\pi\)
\(572\) 33.2039 8.57321i 1.38833 0.358464i
\(573\) 0 0
\(574\) −9.80948 + 1.24597i −0.409440 + 0.0520056i
\(575\) −13.6730 −0.570205
\(576\) 0 0
\(577\) −22.2540 −0.926448 −0.463224 0.886241i \(-0.653307\pi\)
−0.463224 + 0.886241i \(0.653307\pi\)
\(578\) 9.82059 1.24738i 0.408483 0.0518841i
\(579\) 0 0
\(580\) 1.69052 0.436492i 0.0701952 0.0181243i
\(581\) 6.63568i 0.275294i
\(582\) 0 0
\(583\) 0 0
\(584\) −2.29393 + 0.913619i −0.0949234 + 0.0378058i
\(585\) 0 0
\(586\) 5.67439 + 44.6744i 0.234407 + 1.84548i
\(587\) 37.8568i 1.56252i 0.624208 + 0.781259i \(0.285422\pi\)
−0.624208 + 0.781259i \(0.714578\pi\)
\(588\) 0 0
\(589\) 0.889117i 0.0366354i
\(590\) −3.31784 + 0.421421i −0.136593 + 0.0173496i
\(591\) 0 0
\(592\) 3.62702 + 6.55544i 0.149069 + 0.269427i
\(593\) −22.8035 −0.936426 −0.468213 0.883616i \(-0.655102\pi\)
−0.468213 + 0.883616i \(0.655102\pi\)
\(594\) 0 0
\(595\) 1.12702i 0.0462032i
\(596\) 4.89898 + 18.9737i 0.200670 + 0.777192i
\(597\) 0 0
\(598\) −1.43649 11.3095i −0.0587425 0.462479i
\(599\) −41.3755 −1.69056 −0.845278 0.534327i \(-0.820565\pi\)
−0.845278 + 0.534327i \(0.820565\pi\)
\(600\) 0 0
\(601\) 32.1109 1.30983 0.654915 0.755702i \(-0.272704\pi\)
0.654915 + 0.755702i \(0.272704\pi\)
\(602\) −0.200831 1.58114i −0.00818526 0.0644424i
\(603\) 0 0
\(604\) −23.2379 + 6.00000i −0.945537 + 0.244137i
\(605\) 8.77405i 0.356716i
\(606\) 0 0
\(607\) 15.2379 0.618487 0.309244 0.950983i \(-0.399924\pi\)
0.309244 + 0.950983i \(0.399924\pi\)
\(608\) 0.424296 + 0.579859i 0.0172075 + 0.0235164i
\(609\) 0 0
\(610\) −6.87298 + 0.872983i −0.278279 + 0.0353461i
\(611\) 21.1120i 0.854101i
\(612\) 0 0
\(613\) 40.2379i 1.62519i 0.582826 + 0.812597i \(0.301947\pi\)
−0.582826 + 0.812597i \(0.698053\pi\)
\(614\) 5.27801 + 41.5537i 0.213003 + 1.67697i
\(615\) 0 0
\(616\) 6.24597 + 15.6825i 0.251657 + 0.631864i
\(617\) 15.7209 0.632898 0.316449 0.948610i \(-0.397509\pi\)
0.316449 + 0.948610i \(0.397509\pi\)
\(618\) 0 0
\(619\) 0.491933i 0.0197725i −0.999951 0.00988624i \(-0.996853\pi\)
0.999951 0.00988624i \(-0.00314694\pi\)
\(620\) 1.24738 + 4.83108i 0.0500959 + 0.194021i
\(621\) 0 0
\(622\) 29.0554 3.69052i 1.16502 0.147977i
\(623\) 6.99208 0.280132
\(624\) 0 0
\(625\) 23.1109 0.924435
\(626\) −4.03063 + 0.511957i −0.161096 + 0.0204619i
\(627\) 0 0
\(628\) −3.56351 13.8014i −0.142199 0.550736i
\(629\) 5.92289i 0.236161i
\(630\) 0 0
\(631\) −8.87298 −0.353228 −0.176614 0.984280i \(-0.556514\pi\)
−0.176614 + 0.984280i \(0.556514\pi\)
\(632\) −33.1586 + 13.2063i −1.31898 + 0.525320i
\(633\) 0 0
\(634\) 2.43649 + 19.1825i 0.0967654 + 0.761833i
\(635\) 3.87507i 0.153777i
\(636\) 0 0
\(637\) 2.87298i 0.113832i
\(638\) 20.5095 2.60505i 0.811980 0.103135i
\(639\) 0 0
\(640\) 3.11895 + 2.55544i 0.123287 + 0.101013i
\(641\) 18.1703 0.717685 0.358843 0.933398i \(-0.383171\pi\)
0.358843 + 0.933398i \(0.383171\pi\)
\(642\) 0 0
\(643\) 8.38105i 0.330516i −0.986250 0.165258i \(-0.947154\pi\)
0.986250 0.165258i \(-0.0528457\pi\)
\(644\) 5.43357 1.40294i 0.214113 0.0552837i
\(645\) 0 0
\(646\) 0.0715749 + 0.563508i 0.00281608 + 0.0221709i
\(647\) 21.4232 0.842231 0.421116 0.907007i \(-0.361639\pi\)
0.421116 + 0.907007i \(0.361639\pi\)
\(648\) 0 0
\(649\) −39.6028 −1.55455
\(650\) 2.49476 + 19.6412i 0.0978524 + 0.770391i
\(651\) 0 0
\(652\) 11.4365 + 44.2933i 0.447888 + 1.73466i
\(653\) 0.311126i 0.0121753i 0.999981 + 0.00608765i \(0.00193777\pi\)
−0.999981 + 0.00608765i \(0.998062\pi\)
\(654\) 0 0
\(655\) 3.12702 0.122183
\(656\) −24.4723 + 13.5401i −0.955481 + 0.528652i
\(657\) 0 0
\(658\) −10.3095 + 1.30948i −0.401905 + 0.0510487i
\(659\) 35.3620i 1.37751i −0.724994 0.688755i \(-0.758158\pi\)
0.724994 0.688755i \(-0.241842\pi\)
\(660\) 0 0
\(661\) 45.8569i 1.78362i −0.452405 0.891812i \(-0.649434\pi\)
0.452405 0.891812i \(-0.350566\pi\)
\(662\) −1.78197 14.0294i −0.0692582 0.545269i
\(663\) 0 0
\(664\) −6.94456 17.4365i −0.269501 0.676667i
\(665\) 0.0452680 0.00175542
\(666\) 0 0
\(667\) 6.87298i 0.266123i
\(668\) 2.76062 0.712788i 0.106811 0.0275786i
\(669\) 0 0
\(670\) −4.30948 + 0.547375i −0.166490 + 0.0211469i
\(671\) −82.0381 −3.16705
\(672\) 0 0
\(673\) 29.4919 1.13683 0.568415 0.822742i \(-0.307557\pi\)
0.568415 + 0.822742i \(0.307557\pi\)
\(674\) 4.56522 0.579859i 0.175846 0.0223353i
\(675\) 0 0
\(676\) 9.19052 2.37298i 0.353482 0.0912686i
\(677\) 17.9045i 0.688125i −0.938947 0.344063i \(-0.888197\pi\)
0.938947 0.344063i \(-0.111803\pi\)
\(678\) 0 0
\(679\) 9.74597 0.374016
\(680\) 1.17948 + 2.96145i 0.0452309 + 0.113566i
\(681\) 0 0
\(682\) 7.44456 + 58.6109i 0.285067 + 2.24433i
\(683\) 9.44157i 0.361271i −0.983550 0.180636i \(-0.942184\pi\)
0.983550 0.180636i \(-0.0578155\pi\)
\(684\) 0 0
\(685\) 1.63508i 0.0624733i
\(686\) 1.40294 0.178197i 0.0535646 0.00680359i
\(687\) 0 0
\(688\) −2.18246 3.94456i −0.0832054 0.150385i
\(689\) 0 0
\(690\) 0 0
\(691\) 22.0000i 0.836919i −0.908235 0.418460i \(-0.862570\pi\)
0.908235 0.418460i \(-0.137430\pi\)
\(692\) −9.10781 35.2744i −0.346227 1.34093i
\(693\) 0 0
\(694\) 6.50000 + 51.1744i 0.246737 + 1.94255i
\(695\) 4.98952 0.189263
\(696\) 0 0
\(697\) −22.1109 −0.837509
\(698\) 4.18619 + 32.9578i 0.158450 + 1.24747i
\(699\) 0 0
\(700\) −9.43649 + 2.43649i −0.356666 + 0.0920907i
\(701\) 48.7692i 1.84199i 0.389577 + 0.920994i \(0.372621\pi\)
−0.389577 + 0.920994i \(0.627379\pi\)
\(702\) 0 0
\(703\) 0.237900 0.00897257
\(704\) 32.8249 + 34.6719i 1.23713 + 1.30675i
\(705\) 0 0
\(706\) −18.6825 + 2.37298i −0.703124 + 0.0893084i
\(707\) 11.5347i 0.433806i
\(708\) 0 0
\(709\) 21.2540i 0.798212i 0.916905 + 0.399106i \(0.130679\pi\)
−0.916905 + 0.399106i \(0.869321\pi\)
\(710\) 0.379028 + 2.98408i 0.0142247 + 0.111991i
\(711\) 0 0
\(712\) 18.3730 7.31754i 0.688557 0.274236i
\(713\) 19.6412 0.735568
\(714\) 0 0
\(715\) 6.11088i 0.228534i
\(716\) 4.38702 + 16.9909i 0.163951 + 0.634978i
\(717\) 0 0
\(718\) −40.9284 + 5.19859i −1.52744 + 0.194010i
\(719\) 36.8329 1.37363 0.686817 0.726830i \(-0.259007\pi\)
0.686817 + 0.726830i \(0.259007\pi\)
\(720\) 0 0
\(721\) 10.7460 0.400201
\(722\) −26.6333 + 3.38287i −0.991187 + 0.125897i
\(723\) 0 0
\(724\) 10.8730 + 42.1109i 0.404091 + 1.56504i
\(725\) 11.9363i 0.443304i
\(726\) 0 0
\(727\) 3.49193 0.129509 0.0647543 0.997901i \(-0.479374\pi\)
0.0647543 + 0.997901i \(0.479374\pi\)
\(728\) −3.00671 7.54930i −0.111436 0.279796i
\(729\) 0 0
\(730\) 0.0554417 + 0.436492i 0.00205199 + 0.0161553i
\(731\) 3.56394i 0.131817i
\(732\) 0 0
\(733\) 8.36492i 0.308965i 0.987996 + 0.154483i \(0.0493711\pi\)
−0.987996 + 0.154483i \(0.950629\pi\)
\(734\) −30.6865 + 3.89770i −1.13266 + 0.143867i
\(735\) 0 0
\(736\) 12.8095 9.37298i 0.472164 0.345493i
\(737\) −51.4393 −1.89479
\(738\) 0 0
\(739\) 0.254033i 0.00934477i −0.999989 0.00467238i \(-0.998513\pi\)
0.999989 0.00467238i \(-0.00148727\pi\)
\(740\) 1.29265 0.333760i 0.0475186 0.0122693i
\(741\) 0 0
\(742\) 0 0
\(743\) 38.3037 1.40523 0.702614 0.711571i \(-0.252016\pi\)
0.702614 + 0.711571i \(0.252016\pi\)
\(744\) 0 0
\(745\) 3.49193 0.127935
\(746\) −5.03191 39.6161i −0.184231 1.45045i
\(747\) 0 0
\(748\) 9.43649 + 36.5474i 0.345032 + 1.33630i
\(749\) 14.3858i 0.525646i
\(750\) 0 0
\(751\) −51.3488 −1.87374 −0.936872 0.349673i \(-0.886293\pi\)
−0.936872 + 0.349673i \(0.886293\pi\)
\(752\) −25.7196 + 14.2302i −0.937899 + 0.518924i
\(753\) 0 0
\(754\) −9.87298 + 1.25403i −0.359553 + 0.0456692i
\(755\) 4.27673i 0.155646i
\(756\) 0 0
\(757\) 25.7460i 0.935753i −0.883794 0.467877i \(-0.845019\pi\)
0.883794 0.467877i \(-0.154981\pi\)
\(758\) −1.58114 12.4483i −0.0574295 0.452142i
\(759\) 0 0
\(760\) 0.118950 0.0473751i 0.00431477 0.00171848i
\(761\) 34.6945 1.25768 0.628838 0.777537i \(-0.283531\pi\)
0.628838 + 0.777537i \(0.283531\pi\)
\(762\) 0 0
\(763\) 12.7460i 0.461435i
\(764\) −42.1759 + 10.8898i −1.52587 + 0.393978i
\(765\) 0 0
\(766\) −24.4919 + 3.11088i −0.884930 + 0.112401i
\(767\) 19.0642 0.688368
\(768\) 0 0
\(769\) 11.6351 0.419572 0.209786 0.977747i \(-0.432723\pi\)
0.209786 + 0.977747i \(0.432723\pi\)
\(770\) 2.98408 0.379028i 0.107539 0.0136592i
\(771\) 0 0
\(772\) −18.3810 + 4.74597i −0.661548 + 0.170811i
\(773\) 22.0001i 0.791290i 0.918404 + 0.395645i \(0.129479\pi\)
−0.918404 + 0.395645i \(0.870521\pi\)
\(774\) 0 0
\(775\) −34.1109 −1.22530
\(776\) 25.6093 10.1996i 0.919322 0.366145i
\(777\) 0 0
\(778\) 2.74597 + 21.6190i 0.0984477 + 0.775077i
\(779\) 0.888110i 0.0318198i
\(780\) 0 0
\(781\) 35.6190i 1.27455i
\(782\) 12.4483 1.58114i 0.445150 0.0565414i
\(783\) 0 0
\(784\) 3.50000 1.93649i 0.125000 0.0691604i
\(785\) −2.54003 −0.0906574
\(786\) 0 0
\(787\) 4.25403i 0.151640i 0.997122 + 0.0758200i \(0.0241574\pi\)
−0.997122 + 0.0758200i \(0.975843\pi\)
\(788\) 1.02391 + 3.96560i 0.0364754 + 0.141269i
\(789\) 0 0
\(790\) 0.801408 + 6.30948i 0.0285128 + 0.224481i
\(791\) −16.8353 −0.598594
\(792\) 0 0
\(793\) 39.4919 1.40240
\(794\) 1.78197 + 14.0294i 0.0632397 + 0.497885i
\(795\) 0 0
\(796\) 0.245967 0.0635083i 0.00871806 0.00225099i
\(797\) 31.2664i 1.10751i 0.832679 + 0.553756i \(0.186806\pi\)
−0.832679 + 0.553756i \(0.813194\pi\)
\(798\) 0 0
\(799\) −23.2379 −0.822098
\(800\) −22.2462 + 16.2781i −0.786523 + 0.575517i
\(801\) 0 0
\(802\) −23.1825 + 2.94456i −0.818601 + 0.103976i
\(803\) 5.21011i 0.183861i
\(804\) 0 0
\(805\) 1.00000i 0.0352454i
\(806\) −3.58370 28.2144i −0.126230 0.993810i
\(807\) 0 0
\(808\) 12.0716 + 30.3095i 0.424677 + 1.06628i
\(809\) 4.58785 0.161300 0.0806502 0.996742i \(-0.474300\pi\)
0.0806502 + 0.996742i \(0.474300\pi\)
\(810\) 0 0
\(811\) 41.9839i 1.47425i −0.675755 0.737126i \(-0.736182\pi\)
0.675755 0.737126i \(-0.263818\pi\)
\(812\) −1.22474 4.74342i −0.0429801 0.166461i
\(813\) 0 0
\(814\) 15.6825 1.99193i 0.549670 0.0698172i
\(815\) 8.15179 0.285545
\(816\) 0 0
\(817\) −0.143150 −0.00500818
\(818\) 17.3473 2.20339i 0.606533 0.0770398i
\(819\) 0 0
\(820\) 1.24597 + 4.82561i 0.0435110 + 0.168518i
\(821\) 22.1359i 0.772550i 0.922384 + 0.386275i \(0.126238\pi\)
−0.922384 + 0.386275i \(0.873762\pi\)
\(822\) 0 0
\(823\) −37.4919 −1.30689 −0.653443 0.756975i \(-0.726676\pi\)
−0.653443 + 0.756975i \(0.726676\pi\)
\(824\) 28.2370 11.2462i 0.983684 0.391779i
\(825\) 0 0
\(826\) 1.18246 + 9.30948i 0.0411430 + 0.323918i
\(827\) 30.1519i 1.04849i −0.851569 0.524243i \(-0.824348\pi\)
0.851569 0.524243i \(-0.175652\pi\)
\(828\) 0 0
\(829\) 2.36492i 0.0821370i −0.999156 0.0410685i \(-0.986924\pi\)
0.999156 0.0410685i \(-0.0130762\pi\)
\(830\) −3.31784 + 0.421421i −0.115164 + 0.0146277i
\(831\) 0 0
\(832\) −15.8014 16.6905i −0.547815 0.578640i
\(833\) 3.16228 0.109566
\(834\) 0 0
\(835\) 0.508067i 0.0175824i
\(836\) 1.46797 0.379028i 0.0507708 0.0131090i
\(837\) 0 0
\(838\) −5.12702 40.3649i −0.177110 1.39438i
\(839\) 37.5457 1.29622 0.648110 0.761547i \(-0.275560\pi\)
0.648110 + 0.761547i \(0.275560\pi\)
\(840\) 0 0
\(841\) 23.0000 0.793103
\(842\) 3.40550 + 26.8115i 0.117361 + 0.923984i
\(843\) 0 0
\(844\) −10.0000 38.7298i −0.344214 1.33314i
\(845\) 1.69143i 0.0581871i
\(846\) 0 0
\(847\) 24.6190 0.845917
\(848\) 0 0
\(849\) 0 0
\(850\) −21.6190 + 2.74597i −0.741524 + 0.0941859i
\(851\) 5.25537i 0.180152i
\(852\) 0 0
\(853\) 5.23790i 0.179342i −0.995971 0.0896711i \(-0.971418\pi\)
0.995971 0.0896711i \(-0.0285816\pi\)
\(854\) 2.44949 + 19.2848i 0.0838198 + 0.659912i
\(855\) 0 0
\(856\) −15.0554 37.8014i −0.514584 1.29203i
\(857\) 47.6095 1.62631 0.813155 0.582048i \(-0.197748\pi\)
0.813155 + 0.582048i \(0.197748\pi\)
\(858\) 0 0
\(859\) 31.8730i 1.08749i 0.839250 + 0.543746i \(0.182995\pi\)
−0.839250 + 0.543746i \(0.817005\pi\)
\(860\) −0.777815 + 0.200831i −0.0265233 + 0.00684828i
\(861\) 0 0
\(862\) −32.4284 + 4.11895i −1.10452 + 0.140292i
\(863\) −26.3221 −0.896016 −0.448008 0.894030i \(-0.647866\pi\)
−0.448008 + 0.894030i \(0.647866\pi\)
\(864\) 0 0
\(865\) −6.49193 −0.220732
\(866\) 6.83651 0.868351i 0.232314 0.0295078i
\(867\) 0 0
\(868\) 13.5554 3.50000i 0.460102 0.118798i
\(869\) 75.3119i 2.55478i
\(870\) 0 0
\(871\) 24.7621 0.839032
\(872\) −13.3393 33.4924i −0.451724 1.13420i
\(873\) 0 0
\(874\) −0.0635083 0.500000i −0.00214820 0.0169128i
\(875\) 3.51867i 0.118953i
\(876\) 0 0
\(877\) 13.4919i 0.455590i −0.973709 0.227795i \(-0.926848\pi\)
0.973709 0.227795i \(-0.0731516\pi\)
\(878\) −36.1201 + 4.58785i −1.21899 + 0.154833i
\(879\) 0 0
\(880\) 7.44456 4.11895i 0.250956 0.138850i
\(881\) −43.4233 −1.46297 −0.731484 0.681859i \(-0.761172\pi\)
−0.731484 + 0.681859i \(0.761172\pi\)
\(882\) 0 0
\(883\) 15.2379i 0.512796i 0.966571 + 0.256398i \(0.0825358\pi\)
−0.966571 + 0.256398i \(0.917464\pi\)
\(884\) −4.54259 17.5934i −0.152784 0.591729i
\(885\) 0 0
\(886\) −4.93649 38.8649i −0.165845 1.30569i
\(887\) −48.0564 −1.61358 −0.806788 0.590841i \(-0.798796\pi\)
−0.806788 + 0.590841i \(0.798796\pi\)
\(888\) 0 0
\(889\) −10.8730 −0.364668
\(890\) −0.444055 3.49604i −0.0148848 0.117187i
\(891\) 0 0
\(892\) 14.7540 3.80948i 0.494002 0.127551i
\(893\) 0.933378i 0.0312343i
\(894\) 0 0
\(895\) 3.12702 0.104525
\(896\) 7.17027 8.75141i 0.239542 0.292364i
\(897\) 0 0
\(898\) 43.3649 5.50807i 1.44711 0.183807i
\(899\) 17.1464i 0.571865i
\(900\) 0 0
\(901\) 0 0
\(902\) 7.43613 + 58.5445i 0.247596 + 1.94932i
\(903\) 0 0
\(904\) −44.2379 + 17.6190i −1.47133 + 0.585998i
\(905\) 7.75013 0.257623
\(906\) 0 0
\(907\) 8.00000i 0.265636i 0.991140 + 0.132818i \(0.0424025\pi\)
−0.991140 + 0.132818i \(0.957597\pi\)
\(908\) −12.9602 50.1948i −0.430101 1.66577i
\(909\) 0 0
\(910\) −1.43649 + 0.182458i −0.0476192 + 0.00604844i
\(911\) 33.3595 1.10525 0.552624 0.833430i \(-0.313626\pi\)
0.552624 + 0.833430i \(0.313626\pi\)
\(912\) 0 0
\(913\) −39.6028 −1.31066
\(914\) −12.9829 + 1.64904i −0.429435 + 0.0545454i
\(915\) 0 0
\(916\) 7.87298 + 30.4919i 0.260131 + 1.00748i
\(917\) 8.77405i 0.289744i
\(918\) 0 0
\(919\) −14.5081 −0.478577 −0.239288 0.970949i \(-0.576914\pi\)
−0.239288 + 0.970949i \(0.576914\pi\)
\(920\) −1.04655 2.62769i −0.0345037 0.0866323i
\(921\) 0 0
\(922\) −2.48387 19.5554i −0.0818018 0.644024i
\(923\) 17.1464i 0.564382i
\(924\) 0 0
\(925\) 9.12702i 0.300094i
\(926\) −0.712788 + 0.0905359i −0.0234237 + 0.00297520i
\(927\) 0 0
\(928\) −8.18246 11.1825i −0.268602 0.367082i
\(929\) −23.4710 −0.770058 −0.385029 0.922904i \(-0.625809\pi\)
−0.385029 + 0.922904i \(0.625809\pi\)
\(930\) 0 0
\(931\) 0.127017i 0.00416280i
\(932\) −10.8671 + 2.80588i −0.355965 + 0.0919098i
\(933\) 0 0
\(934\) −0.0715749 0.563508i −0.00234200 0.0184386i
\(935\) 6.72622 0.219971
\(936\) 0 0
\(937\) 41.4919 1.35548 0.677741 0.735301i \(-0.262959\pi\)
0.677741 + 0.735301i \(0.262959\pi\)
\(938\) 1.53587 + 12.0919i 0.0501480 + 0.394814i
\(939\) 0 0
\(940\) 1.30948 + 5.07157i 0.0427104 + 0.165417i
\(941\) 11.4894i 0.374544i −0.982308 0.187272i \(-0.940036\pi\)
0.982308 0.187272i \(-0.0599645\pi\)
\(942\) 0 0
\(943\) 19.6190 0.638881
\(944\) 12.8499 + 23.2249i 0.418230 + 0.755906i
\(945\) 0 0
\(946\) −9.43649 + 1.19859i −0.306807 + 0.0389696i
\(947\) 2.00256i 0.0650745i 0.999471 + 0.0325372i \(0.0103587\pi\)
−0.999471 + 0.0325372i \(0.989641\pi\)
\(948\) 0 0
\(949\) 2.50807i 0.0814153i
\(950\) 0.110295 + 0.868351i 0.00357844 + 0.0281730i
\(951\) 0 0
\(952\) 8.30948 3.30948i 0.269312 0.107261i
\(953\) 0.311126 0.0100784 0.00503918 0.999987i \(-0.498396\pi\)
0.00503918 + 0.999987i \(0.498396\pi\)
\(954\) 0 0
\(955\) 7.76210i 0.251176i
\(956\) −23.7171 + 6.12372i −0.767065 + 0.198055i
\(957\) 0 0
\(958\) 1.00000 0.127017i 0.0323085 0.00410372i
\(959\) 4.58785 0.148150
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) −7.54930 + 0.958887i −0.243399 + 0.0309158i
\(963\) 0 0
\(964\) 55.4204 14.3095i 1.78497 0.460877i
\(965\) 3.38287i 0.108898i
\(966\) 0 0
\(967\) 25.2379 0.811596 0.405798 0.913963i \(-0.366994\pi\)
0.405798 + 0.913963i \(0.366994\pi\)
\(968\) 64.6909 25.7649i 2.07924 0.828116i
\(969\) 0 0
\(970\) −0.618950 4.87298i −0.0198733 0.156462i
\(971\) 13.7636i 0.441694i 0.975309 + 0.220847i \(0.0708821\pi\)
−0.975309 + 0.220847i \(0.929118\pi\)
\(972\) 0 0
\(973\) 14.0000i 0.448819i
\(974\) −36.6321 + 4.65288i −1.17377 + 0.149088i
\(975\) 0 0
\(976\) 26.6190 + 48.1109i 0.852052 + 1.53999i
\(977\) −23.1599 −0.740949 −0.370475 0.928843i \(-0.620805\pi\)
−0.370475 + 0.928843i \(0.620805\pi\)
\(978\) 0 0
\(979\) 41.7298i 1.33369i
\(980\) −0.178197 0.690154i −0.00569229 0.0220462i
\(981\) 0 0
\(982\) 2.38912 + 18.8095i 0.0762398 + 0.600234i
\(983\) −53.6682 −1.71175 −0.855875 0.517183i \(-0.826981\pi\)
−0.855875 + 0.517183i \(0.826981\pi\)
\(984\) 0 0
\(985\) 0.729833 0.0232544
\(986\) −1.38031 10.8671i −0.0439580 0.346080i
\(987\) 0 0
\(988\) −0.706658 + 0.182458i −0.0224818 + 0.00580477i
\(989\) 3.16228i 0.100555i
\(990\) 0 0
\(991\) 17.3810 0.552127 0.276064 0.961139i \(-0.410970\pi\)
0.276064 + 0.961139i \(0.410970\pi\)
\(992\) 31.9565 23.3833i 1.01462 0.742421i
\(993\) 0 0
\(994\) 8.37298 1.06351i 0.265575 0.0337324i
\(995\) 0.0452680i 0.00143509i
\(996\) 0 0
\(997\) 18.0000i 0.570066i −0.958518 0.285033i \(-0.907995\pi\)
0.958518 0.285033i \(-0.0920045\pi\)
\(998\) 6.12372 + 48.2120i 0.193843 + 1.52612i
\(999\) 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 1512.2.c.c.757.2 yes 8
3.2 odd 2 inner 1512.2.c.c.757.7 yes 8
4.3 odd 2 6048.2.c.c.3025.3 8
8.3 odd 2 6048.2.c.c.3025.6 8
8.5 even 2 inner 1512.2.c.c.757.1 8
12.11 even 2 6048.2.c.c.3025.5 8
24.5 odd 2 inner 1512.2.c.c.757.8 yes 8
24.11 even 2 6048.2.c.c.3025.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.c.757.1 8 8.5 even 2 inner
1512.2.c.c.757.2 yes 8 1.1 even 1 trivial
1512.2.c.c.757.7 yes 8 3.2 odd 2 inner
1512.2.c.c.757.8 yes 8 24.5 odd 2 inner
6048.2.c.c.3025.3 8 4.3 odd 2
6048.2.c.c.3025.4 8 24.11 even 2
6048.2.c.c.3025.5 8 12.11 even 2
6048.2.c.c.3025.6 8 8.3 odd 2