Properties

Label 1300.2.bb.c.549.2
Level $1300$
Weight $2$
Character 1300.549
Analytic conductor $10.381$
Analytic rank $0$
Dimension $4$
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] = [1300,2,Mod(549,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1300.549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1300.bb (of order \(6\), degree \(2\), not 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: \(10.3805522628\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 549.2
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1300.549
Dual form 1300.2.bb.c.1049.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.866025 - 0.500000i) q^{3} +(-4.33013 - 2.50000i) q^{7} +(-1.00000 + 1.73205i) q^{9} +O(q^{10})\) \(q+(0.866025 - 0.500000i) q^{3} +(-4.33013 - 2.50000i) q^{7} +(-1.00000 + 1.73205i) q^{9} +(2.50000 + 4.33013i) q^{11} +(3.46410 + 1.00000i) q^{13} +(0.866025 + 0.500000i) q^{17} +(-1.50000 + 2.59808i) q^{19} -5.00000 q^{21} +(-2.59808 + 1.50000i) q^{23} +5.00000i q^{27} +(-0.500000 - 0.866025i) q^{29} +(4.33013 + 2.50000i) q^{33} +(6.06218 - 3.50000i) q^{37} +(3.50000 - 0.866025i) q^{39} +(2.50000 + 4.33013i) q^{41} +(4.33013 + 2.50000i) q^{43} +12.0000i q^{47} +(9.00000 + 15.5885i) q^{49} +1.00000 q^{51} -2.00000i q^{53} +3.00000i q^{57} +(-5.50000 + 9.52628i) q^{59} +(6.50000 - 11.2583i) q^{61} +(8.66025 - 5.00000i) q^{63} +(2.59808 - 1.50000i) q^{67} +(-1.50000 + 2.59808i) q^{69} +(-6.50000 + 11.2583i) q^{71} +2.00000i q^{73} -25.0000i q^{77} +4.00000 q^{79} +(-0.500000 - 0.866025i) q^{81} -12.0000i q^{83} +(-0.866025 - 0.500000i) q^{87} +(3.50000 + 6.06218i) q^{89} +(-12.5000 - 12.9904i) q^{91} +(-9.52628 - 5.50000i) q^{97} -10.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} + 10 q^{11} - 6 q^{19} - 20 q^{21} - 2 q^{29} + 14 q^{39} + 10 q^{41} + 36 q^{49} + 4 q^{51} - 22 q^{59} + 26 q^{61} - 6 q^{69} - 26 q^{71} + 16 q^{79} - 2 q^{81} + 14 q^{89} - 50 q^{91} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.866025 0.500000i 0.500000 0.288675i −0.228714 0.973494i \(-0.573452\pi\)
0.728714 + 0.684819i \(0.240119\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −4.33013 2.50000i −1.63663 0.944911i −0.981981 0.188982i \(-0.939481\pi\)
−0.654654 0.755929i \(-0.727186\pi\)
\(8\) 0 0
\(9\) −1.00000 + 1.73205i −0.333333 + 0.577350i
\(10\) 0 0
\(11\) 2.50000 + 4.33013i 0.753778 + 1.30558i 0.945979 + 0.324227i \(0.105104\pi\)
−0.192201 + 0.981356i \(0.561563\pi\)
\(12\) 0 0
\(13\) 3.46410 + 1.00000i 0.960769 + 0.277350i
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.866025 + 0.500000i 0.210042 + 0.121268i 0.601331 0.799000i \(-0.294637\pi\)
−0.391289 + 0.920268i \(0.627971\pi\)
\(18\) 0 0
\(19\) −1.50000 + 2.59808i −0.344124 + 0.596040i −0.985194 0.171442i \(-0.945157\pi\)
0.641071 + 0.767482i \(0.278491\pi\)
\(20\) 0 0
\(21\) −5.00000 −1.09109
\(22\) 0 0
\(23\) −2.59808 + 1.50000i −0.541736 + 0.312772i −0.745782 0.666190i \(-0.767924\pi\)
0.204046 + 0.978961i \(0.434591\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.00000i 0.962250i
\(28\) 0 0
\(29\) −0.500000 0.866025i −0.0928477 0.160817i 0.815861 0.578249i \(-0.196264\pi\)
−0.908708 + 0.417432i \(0.862930\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 4.33013 + 2.50000i 0.753778 + 0.435194i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.06218 3.50000i 0.996616 0.575396i 0.0893706 0.995998i \(-0.471514\pi\)
0.907245 + 0.420602i \(0.138181\pi\)
\(38\) 0 0
\(39\) 3.50000 0.866025i 0.560449 0.138675i
\(40\) 0 0
\(41\) 2.50000 + 4.33013i 0.390434 + 0.676252i 0.992507 0.122189i \(-0.0389915\pi\)
−0.602072 + 0.798441i \(0.705658\pi\)
\(42\) 0 0
\(43\) 4.33013 + 2.50000i 0.660338 + 0.381246i 0.792406 0.609994i \(-0.208828\pi\)
−0.132068 + 0.991241i \(0.542162\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.0000i 1.75038i 0.483779 + 0.875190i \(0.339264\pi\)
−0.483779 + 0.875190i \(0.660736\pi\)
\(48\) 0 0
\(49\) 9.00000 + 15.5885i 1.28571 + 2.22692i
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) 2.00000i 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.00000i 0.397360i
\(58\) 0 0
\(59\) −5.50000 + 9.52628i −0.716039 + 1.24022i 0.246518 + 0.969138i \(0.420713\pi\)
−0.962557 + 0.271078i \(0.912620\pi\)
\(60\) 0 0
\(61\) 6.50000 11.2583i 0.832240 1.44148i −0.0640184 0.997949i \(-0.520392\pi\)
0.896258 0.443533i \(-0.146275\pi\)
\(62\) 0 0
\(63\) 8.66025 5.00000i 1.09109 0.629941i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.59808 1.50000i 0.317406 0.183254i −0.332830 0.942987i \(-0.608004\pi\)
0.650236 + 0.759733i \(0.274670\pi\)
\(68\) 0 0
\(69\) −1.50000 + 2.59808i −0.180579 + 0.312772i
\(70\) 0 0
\(71\) −6.50000 + 11.2583i −0.771408 + 1.33612i 0.165383 + 0.986229i \(0.447114\pi\)
−0.936791 + 0.349889i \(0.886219\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 25.0000i 2.84901i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 12.0000i 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.866025 0.500000i −0.0928477 0.0536056i
\(88\) 0 0
\(89\) 3.50000 + 6.06218i 0.370999 + 0.642590i 0.989720 0.143022i \(-0.0456819\pi\)
−0.618720 + 0.785611i \(0.712349\pi\)
\(90\) 0 0
\(91\) −12.5000 12.9904i −1.31036 1.36176i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.52628 5.50000i −0.967247 0.558440i −0.0688512 0.997627i \(-0.521933\pi\)
−0.898396 + 0.439187i \(0.855267\pi\)
\(98\) 0 0
\(99\) −10.0000 −1.00504
\(100\) 0 0
\(101\) 6.50000 + 11.2583i 0.646774 + 1.12025i 0.983889 + 0.178782i \(0.0572157\pi\)
−0.337115 + 0.941464i \(0.609451\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.79423 + 4.50000i −0.753497 + 0.435031i −0.826956 0.562267i \(-0.809929\pi\)
0.0734594 + 0.997298i \(0.476596\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 3.50000 6.06218i 0.332205 0.575396i
\(112\) 0 0
\(113\) −0.866025 0.500000i −0.0814688 0.0470360i 0.458712 0.888585i \(-0.348311\pi\)
−0.540181 + 0.841549i \(0.681644\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −5.19615 + 5.00000i −0.480384 + 0.462250i
\(118\) 0 0
\(119\) −2.50000 4.33013i −0.229175 0.396942i
\(120\) 0 0
\(121\) −7.00000 + 12.1244i −0.636364 + 1.10221i
\(122\) 0 0
\(123\) 4.33013 + 2.50000i 0.390434 + 0.225417i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.06218 3.50000i 0.537931 0.310575i −0.206309 0.978487i \(-0.566145\pi\)
0.744240 + 0.667912i \(0.232812\pi\)
\(128\) 0 0
\(129\) 5.00000 0.440225
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) 12.9904 7.50000i 1.12641 0.650332i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.59808 1.50000i −0.221969 0.128154i 0.384893 0.922961i \(-0.374238\pi\)
−0.606861 + 0.794808i \(0.707572\pi\)
\(138\) 0 0
\(139\) 6.50000 11.2583i 0.551323 0.954919i −0.446857 0.894606i \(-0.647457\pi\)
0.998179 0.0603135i \(-0.0192101\pi\)
\(140\) 0 0
\(141\) 6.00000 + 10.3923i 0.505291 + 0.875190i
\(142\) 0 0
\(143\) 4.33013 + 17.5000i 0.362103 + 1.46342i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 15.5885 + 9.00000i 1.28571 + 0.742307i
\(148\) 0 0
\(149\) 5.50000 9.52628i 0.450578 0.780423i −0.547844 0.836580i \(-0.684551\pi\)
0.998422 + 0.0561570i \(0.0178847\pi\)
\(150\) 0 0
\(151\) −24.0000 −1.95309 −0.976546 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) 0 0
\(153\) −1.73205 + 1.00000i −0.140028 + 0.0808452i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000i 0.798087i 0.916932 + 0.399043i \(0.130658\pi\)
−0.916932 + 0.399043i \(0.869342\pi\)
\(158\) 0 0
\(159\) −1.00000 1.73205i −0.0793052 0.137361i
\(160\) 0 0
\(161\) 15.0000 1.18217
\(162\) 0 0
\(163\) 4.33013 + 2.50000i 0.339162 + 0.195815i 0.659901 0.751352i \(-0.270598\pi\)
−0.320740 + 0.947167i \(0.603931\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.2583 + 6.50000i −0.871196 + 0.502985i −0.867745 0.497009i \(-0.834432\pi\)
−0.00345033 + 0.999994i \(0.501098\pi\)
\(168\) 0 0
\(169\) 11.0000 + 6.92820i 0.846154 + 0.532939i
\(170\) 0 0
\(171\) −3.00000 5.19615i −0.229416 0.397360i
\(172\) 0 0
\(173\) −14.7224 8.50000i −1.11933 0.646243i −0.178097 0.984013i \(-0.556994\pi\)
−0.941229 + 0.337770i \(0.890327\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 11.0000i 0.826811i
\(178\) 0 0
\(179\) 5.50000 + 9.52628i 0.411089 + 0.712028i 0.995009 0.0997838i \(-0.0318151\pi\)
−0.583920 + 0.811811i \(0.698482\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 13.0000i 0.960988i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.00000i 0.365636i
\(188\) 0 0
\(189\) 12.5000 21.6506i 0.909241 1.57485i
\(190\) 0 0
\(191\) 7.50000 12.9904i 0.542681 0.939951i −0.456068 0.889945i \(-0.650743\pi\)
0.998749 0.0500060i \(-0.0159241\pi\)
\(192\) 0 0
\(193\) −19.9186 + 11.5000i −1.43377 + 0.827788i −0.997406 0.0719816i \(-0.977068\pi\)
−0.436365 + 0.899770i \(0.643734\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.3827 13.5000i 1.66595 0.961835i 0.696159 0.717888i \(-0.254891\pi\)
0.969788 0.243947i \(-0.0784425\pi\)
\(198\) 0 0
\(199\) 10.5000 18.1865i 0.744325 1.28921i −0.206184 0.978513i \(-0.566105\pi\)
0.950509 0.310696i \(-0.100562\pi\)
\(200\) 0 0
\(201\) 1.50000 2.59808i 0.105802 0.183254i
\(202\) 0 0
\(203\) 5.00000i 0.350931i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 6.00000i 0.417029i
\(208\) 0 0
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) 2.50000 + 4.33013i 0.172107 + 0.298098i 0.939156 0.343490i \(-0.111609\pi\)
−0.767049 + 0.641588i \(0.778276\pi\)
\(212\) 0 0
\(213\) 13.0000i 0.890745i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1.00000 + 1.73205i 0.0675737 + 0.117041i
\(220\) 0 0
\(221\) 2.50000 + 2.59808i 0.168168 + 0.174766i
\(222\) 0 0
\(223\) −16.4545 + 9.50000i −1.10187 + 0.636167i −0.936713 0.350100i \(-0.886148\pi\)
−0.165161 + 0.986267i \(0.552814\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.7224 8.50000i −0.977162 0.564165i −0.0757500 0.997127i \(-0.524135\pi\)
−0.901412 + 0.432962i \(0.857468\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) −12.5000 21.6506i −0.822440 1.42451i
\(232\) 0 0
\(233\) 6.00000i 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 3.46410 2.00000i 0.225018 0.129914i
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −5.50000 + 9.52628i −0.354286 + 0.613642i −0.986996 0.160748i \(-0.948609\pi\)
0.632709 + 0.774389i \(0.281943\pi\)
\(242\) 0 0
\(243\) −13.8564 8.00000i −0.888889 0.513200i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −7.79423 + 7.50000i −0.495935 + 0.477214i
\(248\) 0 0
\(249\) −6.00000 10.3923i −0.380235 0.658586i
\(250\) 0 0
\(251\) 7.50000 12.9904i 0.473396 0.819946i −0.526140 0.850398i \(-0.676361\pi\)
0.999536 + 0.0304521i \(0.00969471\pi\)
\(252\) 0 0
\(253\) −12.9904 7.50000i −0.816698 0.471521i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.9904 7.50000i 0.810318 0.467837i −0.0367485 0.999325i \(-0.511700\pi\)
0.847066 + 0.531487i \(0.178367\pi\)
\(258\) 0 0
\(259\) −35.0000 −2.17479
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) −9.52628 + 5.50000i −0.587416 + 0.339145i −0.764075 0.645128i \(-0.776804\pi\)
0.176659 + 0.984272i \(0.443471\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.06218 + 3.50000i 0.370999 + 0.214197i
\(268\) 0 0
\(269\) −4.50000 + 7.79423i −0.274370 + 0.475223i −0.969976 0.243201i \(-0.921803\pi\)
0.695606 + 0.718423i \(0.255136\pi\)
\(270\) 0 0
\(271\) −3.50000 6.06218i −0.212610 0.368251i 0.739921 0.672694i \(-0.234863\pi\)
−0.952531 + 0.304443i \(0.901530\pi\)
\(272\) 0 0
\(273\) −17.3205 5.00000i −1.04828 0.302614i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 11.2583 + 6.50000i 0.676448 + 0.390547i 0.798515 0.601975i \(-0.205619\pi\)
−0.122068 + 0.992522i \(0.538953\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) −19.9186 + 11.5000i −1.18404 + 0.683604i −0.956945 0.290269i \(-0.906255\pi\)
−0.227092 + 0.973873i \(0.572922\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 25.0000i 1.47570i
\(288\) 0 0
\(289\) −8.00000 13.8564i −0.470588 0.815083i
\(290\) 0 0
\(291\) −11.0000 −0.644831
\(292\) 0 0
\(293\) 6.06218 + 3.50000i 0.354156 + 0.204472i 0.666514 0.745492i \(-0.267786\pi\)
−0.312358 + 0.949964i \(0.601119\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −21.6506 + 12.5000i −1.25630 + 0.725324i
\(298\) 0 0
\(299\) −10.5000 + 2.59808i −0.607231 + 0.150251i
\(300\) 0 0
\(301\) −12.5000 21.6506i −0.720488 1.24792i
\(302\) 0 0
\(303\) 11.2583 + 6.50000i 0.646774 + 0.373415i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0000i 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 6.00000i 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000i 1.01098i −0.862832 0.505490i \(-0.831312\pi\)
0.862832 0.505490i \(-0.168688\pi\)
\(318\) 0 0
\(319\) 2.50000 4.33013i 0.139973 0.242441i
\(320\) 0 0
\(321\) −4.50000 + 7.79423i −0.251166 + 0.435031i
\(322\) 0 0
\(323\) −2.59808 + 1.50000i −0.144561 + 0.0834622i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −15.5885 + 9.00000i −0.862044 + 0.497701i
\(328\) 0 0
\(329\) 30.0000 51.9615i 1.65395 2.86473i
\(330\) 0 0
\(331\) −0.500000 + 0.866025i −0.0274825 + 0.0476011i −0.879440 0.476011i \(-0.842082\pi\)
0.851957 + 0.523612i \(0.175416\pi\)
\(332\) 0 0
\(333\) 14.0000i 0.767195i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 0 0
\(339\) −1.00000 −0.0543125
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 55.0000i 2.96972i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 23.3827 + 13.5000i 1.25525 + 0.724718i 0.972147 0.234372i \(-0.0753034\pi\)
0.283101 + 0.959090i \(0.408637\pi\)
\(348\) 0 0
\(349\) 17.5000 + 30.3109i 0.936754 + 1.62250i 0.771477 + 0.636257i \(0.219518\pi\)
0.165277 + 0.986247i \(0.447148\pi\)
\(350\) 0 0
\(351\) −5.00000 + 17.3205i −0.266880 + 0.924500i
\(352\) 0 0
\(353\) 4.33013 2.50000i 0.230469 0.133062i −0.380319 0.924855i \(-0.624186\pi\)
0.610789 + 0.791794i \(0.290853\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −4.33013 2.50000i −0.229175 0.132314i
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) 5.00000 + 8.66025i 0.263158 + 0.455803i
\(362\) 0 0
\(363\) 14.0000i 0.734809i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 2.59808 1.50000i 0.135618 0.0782994i −0.430656 0.902516i \(-0.641718\pi\)
0.566274 + 0.824217i \(0.308384\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −5.00000 + 8.66025i −0.259587 + 0.449618i
\(372\) 0 0
\(373\) 16.4545 + 9.50000i 0.851981 + 0.491891i 0.861319 0.508065i \(-0.169639\pi\)
−0.00933789 + 0.999956i \(0.502972\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.866025 3.50000i −0.0446026 0.180259i
\(378\) 0 0
\(379\) −10.5000 18.1865i −0.539349 0.934179i −0.998939 0.0460485i \(-0.985337\pi\)
0.459590 0.888131i \(-0.347996\pi\)
\(380\) 0 0
\(381\) 3.50000 6.06218i 0.179310 0.310575i
\(382\) 0 0
\(383\) −2.59808 1.50000i −0.132755 0.0766464i 0.432151 0.901801i \(-0.357755\pi\)
−0.564907 + 0.825155i \(0.691088\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.66025 + 5.00000i −0.440225 + 0.254164i
\(388\) 0 0
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 0 0
\(393\) 3.46410 2.00000i 0.174741 0.100887i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 11.2583 + 6.50000i 0.565039 + 0.326226i 0.755166 0.655534i \(-0.227556\pi\)
−0.190126 + 0.981760i \(0.560890\pi\)
\(398\) 0 0
\(399\) 7.50000 12.9904i 0.375470 0.650332i
\(400\) 0 0
\(401\) −13.5000 23.3827i −0.674158 1.16768i −0.976714 0.214544i \(-0.931173\pi\)
0.302556 0.953131i \(-0.402160\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 30.3109 + 17.5000i 1.50245 + 0.867443i
\(408\) 0 0
\(409\) 9.50000 16.4545i 0.469745 0.813622i −0.529657 0.848212i \(-0.677679\pi\)
0.999402 + 0.0345902i \(0.0110126\pi\)
\(410\) 0 0
\(411\) −3.00000 −0.147979
\(412\) 0 0
\(413\) 47.6314 27.5000i 2.34379 1.35319i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 13.0000i 0.636613i
\(418\) 0 0
\(419\) −8.50000 14.7224i −0.415252 0.719238i 0.580203 0.814472i \(-0.302973\pi\)
−0.995455 + 0.0952342i \(0.969640\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) −20.7846 12.0000i −1.01058 0.583460i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −56.2917 + 32.5000i −2.72414 + 1.57279i
\(428\) 0 0
\(429\) 12.5000 + 12.9904i 0.603506 + 0.627182i
\(430\) 0 0
\(431\) 10.5000 + 18.1865i 0.505767 + 0.876014i 0.999978 + 0.00667224i \(0.00212386\pi\)
−0.494211 + 0.869342i \(0.664543\pi\)
\(432\) 0 0
\(433\) 6.06218 + 3.50000i 0.291330 + 0.168199i 0.638541 0.769588i \(-0.279538\pi\)
−0.347212 + 0.937787i \(0.612871\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9.00000i 0.430528i
\(438\) 0 0
\(439\) −14.5000 25.1147i −0.692047 1.19866i −0.971166 0.238404i \(-0.923376\pi\)
0.279119 0.960257i \(-0.409958\pi\)
\(440\) 0 0
\(441\) −36.0000 −1.71429
\(442\) 0 0
\(443\) 20.0000i 0.950229i 0.879924 + 0.475114i \(0.157593\pi\)
−0.879924 + 0.475114i \(0.842407\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 11.0000i 0.520282i
\(448\) 0 0
\(449\) −10.5000 + 18.1865i −0.495526 + 0.858276i −0.999987 0.00515887i \(-0.998358\pi\)
0.504461 + 0.863434i \(0.331691\pi\)
\(450\) 0 0
\(451\) −12.5000 + 21.6506i −0.588602 + 1.01949i
\(452\) 0 0
\(453\) −20.7846 + 12.0000i −0.976546 + 0.563809i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9.52628 5.50000i 0.445621 0.257279i −0.260358 0.965512i \(-0.583841\pi\)
0.705979 + 0.708233i \(0.250507\pi\)
\(458\) 0 0
\(459\) −2.50000 + 4.33013i −0.116690 + 0.202113i
\(460\) 0 0
\(461\) 16.5000 28.5788i 0.768482 1.33105i −0.169904 0.985461i \(-0.554346\pi\)
0.938386 0.345589i \(-0.112321\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.00000i 0.370196i −0.982720 0.185098i \(-0.940740\pi\)
0.982720 0.185098i \(-0.0592602\pi\)
\(468\) 0 0
\(469\) −15.0000 −0.692636
\(470\) 0 0
\(471\) 5.00000 + 8.66025i 0.230388 + 0.399043i
\(472\) 0 0
\(473\) 25.0000i 1.14950i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.46410 + 2.00000i 0.158610 + 0.0915737i
\(478\) 0 0
\(479\) 5.50000 + 9.52628i 0.251301 + 0.435267i 0.963884 0.266321i \(-0.0858081\pi\)
−0.712583 + 0.701588i \(0.752475\pi\)
\(480\) 0 0
\(481\) 24.5000 6.06218i 1.11710 0.276412i
\(482\) 0 0
\(483\) 12.9904 7.50000i 0.591083 0.341262i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −14.7224 8.50000i −0.667137 0.385172i 0.127854 0.991793i \(-0.459191\pi\)
−0.794991 + 0.606621i \(0.792524\pi\)
\(488\) 0 0
\(489\) 5.00000 0.226108
\(490\) 0 0
\(491\) −7.50000 12.9904i −0.338470 0.586248i 0.645675 0.763612i \(-0.276576\pi\)
−0.984145 + 0.177365i \(0.943243\pi\)
\(492\) 0 0
\(493\) 1.00000i 0.0450377i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 56.2917 32.5000i 2.52503 1.45782i
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −6.50000 + 11.2583i −0.290399 + 0.502985i
\(502\) 0 0
\(503\) −9.52628 5.50000i −0.424756 0.245233i 0.272354 0.962197i \(-0.412198\pi\)
−0.697110 + 0.716964i \(0.745531\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 12.9904 + 0.500000i 0.576923 + 0.0222058i
\(508\) 0 0
\(509\) 1.50000 + 2.59808i 0.0664863 + 0.115158i 0.897352 0.441315i \(-0.145488\pi\)
−0.830866 + 0.556473i \(0.812154\pi\)
\(510\) 0 0
\(511\) 5.00000 8.66025i 0.221187 0.383107i
\(512\) 0 0
\(513\) −12.9904 7.50000i −0.573539 0.331133i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −51.9615 + 30.0000i −2.28527 + 1.31940i
\(518\) 0 0
\(519\) −17.0000 −0.746217
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) 18.1865 10.5000i 0.795242 0.459133i −0.0465630 0.998915i \(-0.514827\pi\)
0.841805 + 0.539782i \(0.181493\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 + 12.1244i −0.304348 + 0.527146i
\(530\) 0 0
\(531\) −11.0000 19.0526i −0.477359 0.826811i
\(532\) 0 0
\(533\) 4.33013 + 17.5000i 0.187559 + 0.758009i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 9.52628 + 5.50000i 0.411089 + 0.237343i
\(538\) 0 0
\(539\) −45.0000 + 77.9423i −1.93829 + 3.35721i
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 8.66025 5.00000i 0.371647 0.214571i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.0000i 0.855138i −0.903983 0.427569i \(-0.859370\pi\)
0.903983 0.427569i \(-0.140630\pi\)
\(548\) 0 0
\(549\) 13.0000 + 22.5167i 0.554826 + 0.960988i
\(550\) 0 0
\(551\) 3.00000 0.127804
\(552\) 0 0
\(553\) −17.3205 10.0000i −0.736543 0.425243i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −11.2583 + 6.50000i −0.477031 + 0.275414i −0.719178 0.694826i \(-0.755482\pi\)
0.242147 + 0.970239i \(0.422148\pi\)
\(558\) 0 0
\(559\) 12.5000 + 12.9904i 0.528694 + 0.549435i
\(560\) 0 0
\(561\) 2.50000 + 4.33013i 0.105550 + 0.182818i
\(562\) 0 0
\(563\) 7.79423 + 4.50000i 0.328488 + 0.189652i 0.655169 0.755482i \(-0.272597\pi\)
−0.326682 + 0.945134i \(0.605931\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 5.00000i 0.209980i
\(568\) 0 0
\(569\) 19.5000 + 33.7750i 0.817483 + 1.41592i 0.907532 + 0.419984i \(0.137964\pi\)
−0.0900490 + 0.995937i \(0.528702\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 15.0000i 0.626634i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 42.0000i 1.74848i −0.485491 0.874241i \(-0.661359\pi\)
0.485491 0.874241i \(-0.338641\pi\)
\(578\) 0 0
\(579\) −11.5000 + 19.9186i −0.477924 + 0.827788i
\(580\) 0 0
\(581\) −30.0000 + 51.9615i −1.24461 + 2.15573i
\(582\) 0 0
\(583\) 8.66025 5.00000i 0.358671 0.207079i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.59808 1.50000i 0.107234 0.0619116i −0.445424 0.895320i \(-0.646947\pi\)
0.552658 + 0.833408i \(0.313614\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 13.5000 23.3827i 0.555316 0.961835i
\(592\) 0 0
\(593\) 2.00000i 0.0821302i 0.999156 + 0.0410651i \(0.0130751\pi\)
−0.999156 + 0.0410651i \(0.986925\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 21.0000i 0.859473i
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 2.50000 + 4.33013i 0.101977 + 0.176630i 0.912499 0.409079i \(-0.134150\pi\)
−0.810522 + 0.585708i \(0.800816\pi\)
\(602\) 0 0
\(603\) 6.00000i 0.244339i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 26.8468 + 15.5000i 1.08968 + 0.629126i 0.933491 0.358602i \(-0.116746\pi\)
0.156187 + 0.987728i \(0.450080\pi\)
\(608\) 0 0
\(609\) 2.50000 + 4.33013i 0.101305 + 0.175466i
\(610\) 0 0
\(611\) −12.0000 + 41.5692i −0.485468 + 1.68171i
\(612\) 0 0
\(613\) 21.6506 12.5000i 0.874461 0.504870i 0.00563283 0.999984i \(-0.498207\pi\)
0.868828 + 0.495114i \(0.164874\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −23.3827 13.5000i −0.941351 0.543490i −0.0509678 0.998700i \(-0.516231\pi\)
−0.890384 + 0.455211i \(0.849564\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) −7.50000 12.9904i −0.300965 0.521286i
\(622\) 0 0
\(623\) 35.0000i 1.40225i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −12.9904 + 7.50000i −0.518786 + 0.299521i
\(628\) 0 0
\(629\) 7.00000 0.279108
\(630\) 0 0
\(631\) 13.5000 23.3827i 0.537427 0.930850i −0.461615 0.887080i \(-0.652730\pi\)
0.999042 0.0437697i \(-0.0139368\pi\)
\(632\) 0 0
\(633\) 4.33013 + 2.50000i 0.172107 + 0.0993661i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 15.5885 + 63.0000i 0.617637 + 2.49615i
\(638\) 0 0
\(639\) −13.0000 22.5167i −0.514272 0.890745i
\(640\) 0 0
\(641\) −13.5000 + 23.3827i −0.533218 + 0.923561i 0.466029 + 0.884769i \(0.345684\pi\)
−0.999247 + 0.0387913i \(0.987649\pi\)
\(642\) 0 0
\(643\) 4.33013 + 2.50000i 0.170764 + 0.0985904i 0.582946 0.812511i \(-0.301900\pi\)
−0.412182 + 0.911101i \(0.635233\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −7.79423 + 4.50000i −0.306423 + 0.176913i −0.645325 0.763908i \(-0.723278\pi\)
0.338902 + 0.940822i \(0.389945\pi\)
\(648\) 0 0
\(649\) −55.0000 −2.15894
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −33.7750 + 19.5000i −1.32172 + 0.763094i −0.984003 0.178154i \(-0.942987\pi\)
−0.337715 + 0.941248i \(0.609654\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.46410 2.00000i −0.135147 0.0780274i
\(658\) 0 0
\(659\) 8.50000 14.7224i 0.331113 0.573505i −0.651617 0.758548i \(-0.725909\pi\)
0.982730 + 0.185043i \(0.0592425\pi\)
\(660\) 0 0
\(661\) −1.50000 2.59808i −0.0583432 0.101053i 0.835379 0.549675i \(-0.185248\pi\)
−0.893722 + 0.448622i \(0.851915\pi\)
\(662\) 0 0
\(663\) 3.46410 + 1.00000i 0.134535 + 0.0388368i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 2.59808 + 1.50000i 0.100598 + 0.0580802i
\(668\) 0 0
\(669\) −9.50000 + 16.4545i −0.367291 + 0.636167i
\(670\) 0 0
\(671\) 65.0000 2.50930
\(672\) 0 0
\(673\) −9.52628 + 5.50000i −0.367211 + 0.212009i −0.672239 0.740334i \(-0.734667\pi\)
0.305028 + 0.952343i \(0.401334\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 42.0000i 1.61419i 0.590421 + 0.807096i \(0.298962\pi\)
−0.590421 + 0.807096i \(0.701038\pi\)
\(678\) 0 0
\(679\) 27.5000 + 47.6314i 1.05535 + 1.82793i
\(680\) 0 0
\(681\) −17.0000 −0.651441
\(682\) 0 0
\(683\) 42.4352 + 24.5000i 1.62374 + 0.937466i 0.985908 + 0.167291i \(0.0535019\pi\)
0.637832 + 0.770176i \(0.279831\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −8.66025 + 5.00000i −0.330409 + 0.190762i
\(688\) 0 0
\(689\) 2.00000 6.92820i 0.0761939 0.263944i
\(690\) 0 0
\(691\) 2.50000 + 4.33013i 0.0951045 + 0.164726i 0.909652 0.415371i \(-0.136348\pi\)
−0.814548 + 0.580097i \(0.803015\pi\)
\(692\) 0 0
\(693\) 43.3013 + 25.0000i 1.64488 + 0.949671i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 5.00000i 0.189389i
\(698\) 0 0
\(699\) −3.00000 5.19615i −0.113470 0.196537i
\(700\) 0 0
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) 21.0000i 0.792030i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 65.0000i 2.44458i
\(708\) 0 0
\(709\) 11.5000 19.9186i 0.431892 0.748058i −0.565145 0.824992i \(-0.691180\pi\)
0.997036 + 0.0769337i \(0.0245130\pi\)
\(710\) 0 0
\(711\) −4.00000 + 6.92820i −0.150012 + 0.259828i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.92820 4.00000i 0.258738 0.149383i
\(718\) 0 0
\(719\) 16.5000 28.5788i 0.615346 1.06581i −0.374978 0.927034i \(-0.622350\pi\)
0.990324 0.138777i \(-0.0443171\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 11.0000i 0.409094i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 8.00000i 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) 0 0
\(729\) −13.0000 −0.481481
\(730\) 0 0
\(731\) 2.50000 + 4.33013i 0.0924658 + 0.160156i
\(732\) 0 0
\(733\) 34.0000i 1.25582i 0.778287 + 0.627909i \(0.216089\pi\)
−0.778287 + 0.627909i \(0.783911\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.9904 + 7.50000i 0.478507 + 0.276266i
\(738\) 0 0
\(739\) 7.50000 + 12.9904i 0.275892 + 0.477859i 0.970360 0.241665i \(-0.0776935\pi\)
−0.694468 + 0.719524i \(0.744360\pi\)
\(740\) 0 0
\(741\) −3.00000 + 10.3923i −0.110208 + 0.381771i
\(742\) 0 0
\(743\) 7.79423 4.50000i 0.285943 0.165089i −0.350168 0.936687i \(-0.613876\pi\)
0.636111 + 0.771598i \(0.280542\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 20.7846 + 12.0000i 0.760469 + 0.439057i
\(748\) 0 0
\(749\) 45.0000 1.64426
\(750\) 0 0
\(751\) 12.5000 + 21.6506i 0.456131 + 0.790043i 0.998752 0.0499348i \(-0.0159013\pi\)
−0.542621 + 0.839978i \(0.682568\pi\)
\(752\) 0 0
\(753\) 15.0000i 0.546630i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −7.79423 + 4.50000i −0.283286 + 0.163555i −0.634910 0.772586i \(-0.718963\pi\)
0.351624 + 0.936141i \(0.385630\pi\)
\(758\) 0 0
\(759\) −15.0000 −0.544466
\(760\) 0 0
\(761\) −9.50000 + 16.4545i −0.344375 + 0.596475i −0.985240 0.171179i \(-0.945242\pi\)
0.640865 + 0.767653i \(0.278576\pi\)
\(762\) 0 0
\(763\) 77.9423 + 45.0000i 2.82170 + 1.62911i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −28.5788 + 27.5000i −1.03192 + 0.992967i
\(768\) 0 0
\(769\) 9.50000 + 16.4545i 0.342579 + 0.593364i 0.984911 0.173063i \(-0.0553663\pi\)
−0.642332 + 0.766426i \(0.722033\pi\)
\(770\) 0 0
\(771\) 7.50000 12.9904i 0.270106 0.467837i
\(772\) 0 0
\(773\) −32.0429 18.5000i −1.15250 0.665399i −0.203008 0.979177i \(-0.565072\pi\)
−0.949496 + 0.313778i \(0.898405\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −30.3109 + 17.5000i −1.08740 + 0.627809i
\(778\) 0 0
\(779\) −15.0000 −0.537431
\(780\) 0 0
\(781\) −65.0000 −2.32588
\(782\) 0 0
\(783\) 4.33013 2.50000i 0.154746 0.0893427i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 9.52628 + 5.50000i 0.339575 + 0.196054i 0.660084 0.751192i \(-0.270521\pi\)
−0.320509 + 0.947245i \(0.603854\pi\)
\(788\) 0 0
\(789\) −5.50000 + 9.52628i −0.195805 + 0.339145i
\(790\) 0 0
\(791\) 2.50000 + 4.33013i 0.0888898 + 0.153962i
\(792\) 0 0
\(793\) 33.7750 32.5000i 1.19939 1.15411i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −19.9186 11.5000i −0.705552 0.407351i 0.103860 0.994592i \(-0.466881\pi\)
−0.809412 + 0.587241i \(0.800214\pi\)
\(798\) 0 0
\(799\) −6.00000 + 10.3923i −0.212265 + 0.367653i
\(800\) 0 0
\(801\) −14.0000 −0.494666
\(802\) 0 0
\(803\) −8.66025 + 5.00000i −0.305614 + 0.176446i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 9.00000i 0.316815i
\(808\) 0 0
\(809\) −18.5000 32.0429i −0.650425 1.12657i −0.983020 0.183500i \(-0.941257\pi\)
0.332594 0.943070i \(-0.392076\pi\)
\(810\) 0 0
\(811\) −8.00000 −0.280918 −0.140459 0.990086i \(-0.544858\pi\)
−0.140459 + 0.990086i \(0.544858\pi\)
\(812\) 0 0
\(813\) −6.06218 3.50000i −0.212610 0.122750i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −12.9904 + 7.50000i −0.454476 + 0.262392i
\(818\) 0 0
\(819\) 35.0000 8.66025i 1.22300 0.302614i
\(820\) 0 0
\(821\) 12.5000 + 21.6506i 0.436253 + 0.755612i 0.997397 0.0721058i \(-0.0229719\pi\)
−0.561144 + 0.827718i \(0.689639\pi\)
\(822\) 0 0
\(823\) −26.8468 15.5000i −0.935820 0.540296i −0.0471726 0.998887i \(-0.515021\pi\)
−0.888648 + 0.458591i \(0.848354\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) −12.5000 21.6506i −0.434143 0.751958i 0.563082 0.826401i \(-0.309615\pi\)
−0.997225 + 0.0744432i \(0.976282\pi\)
\(830\) 0 0
\(831\) 13.0000 0.450965
\(832\) 0 0
\(833\) 18.0000i 0.623663i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −21.5000 + 37.2391i −0.742262 + 1.28564i 0.209200 + 0.977873i \(0.432914\pi\)
−0.951463 + 0.307763i \(0.900419\pi\)
\(840\) 0 0
\(841\) 14.0000 24.2487i 0.482759 0.836162i
\(842\) 0 0
\(843\) 25.9808 15.0000i 0.894825 0.516627i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 60.6218 35.0000i 2.08299 1.20261i
\(848\) 0 0
\(849\) −11.5000 + 19.9186i −0.394679 + 0.683604i
\(850\) 0 0
\(851\) −10.5000 + 18.1865i −0.359935 + 0.623426i
\(852\) 0 0
\(853\) 34.0000i 1.16414i −0.813139 0.582069i \(-0.802243\pi\)
0.813139 0.582069i \(-0.197757\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 26.0000i 0.888143i −0.895991 0.444072i \(-0.853534\pi\)
0.895991 0.444072i \(-0.146466\pi\)
\(858\) 0 0
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) 0 0
\(861\) −12.5000 21.6506i −0.425999 0.737852i
\(862\) 0 0
\(863\) 24.0000i 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −13.8564 8.00000i −0.470588 0.271694i
\(868\) 0 0
\(869\) 10.0000 + 17.3205i 0.339227 + 0.587558i
\(870\) 0 0
\(871\) 10.5000 2.59808i 0.355779 0.0880325i
\(872\) 0 0
\(873\) 19.0526 11.0000i 0.644831 0.372294i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −26.8468 15.5000i −0.906552 0.523398i −0.0272316 0.999629i \(-0.508669\pi\)
−0.879320 + 0.476231i \(0.842002\pi\)
\(878\) 0 0
\(879\) 7.00000 0.236104
\(880\) 0 0
\(881\) 22.5000 + 38.9711i 0.758044 + 1.31297i 0.943847 + 0.330384i \(0.107178\pi\)
−0.185802 + 0.982587i \(0.559488\pi\)
\(882\) 0 0
\(883\) 20.0000i 0.673054i −0.941674 0.336527i \(-0.890748\pi\)
0.941674 0.336527i \(-0.109252\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 26.8468 15.5000i 0.901427 0.520439i 0.0237640 0.999718i \(-0.492435\pi\)
0.877663 + 0.479279i \(0.159102\pi\)
\(888\) 0 0
\(889\) −35.0000 −1.17386
\(890\) 0 0
\(891\) 2.50000 4.33013i 0.0837532 0.145065i
\(892\) 0 0
\(893\) −31.1769 18.0000i −1.04330 0.602347i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −7.79423 + 7.50000i −0.260242 + 0.250418i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 1.00000 1.73205i 0.0333148 0.0577030i
\(902\) 0 0
\(903\) −21.6506 12.5000i −0.720488 0.415974i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −14.7224 + 8.50000i −0.488850 + 0.282238i −0.724097 0.689698i \(-0.757743\pi\)
0.235247 + 0.971936i \(0.424410\pi\)
\(908\) 0 0
\(909\) −26.0000 −0.862366
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) 51.9615 30.0000i 1.71968 0.992855i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −17.3205 10.0000i −0.571974 0.330229i
\(918\) 0 0
\(919\) 24.5000 42.4352i 0.808180 1.39981i −0.105942 0.994372i \(-0.533786\pi\)
0.914123 0.405437i \(-0.132881\pi\)
\(920\) 0 0
\(921\) −6.00000 10.3923i −0.197707 0.342438i
\(922\) 0 0
\(923\) −33.7750 + 32.5000i −1.11172 + 1.06975i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −22.5000 + 38.9711i −0.738201 + 1.27860i 0.215104 + 0.976591i \(0.430991\pi\)
−0.953305 + 0.302010i \(0.902342\pi\)
\(930\) 0 0
\(931\) −54.0000 −1.76978
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 18.0000i 0.588034i −0.955800 0.294017i \(-0.905008\pi\)
0.955800 0.294017i \(-0.0949923\pi\)
\(938\) 0 0
\(939\) −3.00000 5.19615i −0.0979013 0.169570i
\(940\) 0 0
\(941\) 14.0000 0.456387 0.228193 0.973616i \(-0.426718\pi\)
0.228193 + 0.973616i \(0.426718\pi\)
\(942\) 0 0
\(943\) −12.9904 7.50000i −0.423025 0.244234i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 26.8468 15.5000i 0.872403 0.503682i 0.00425721 0.999991i \(-0.498645\pi\)
0.868146 + 0.496309i \(0.165312\pi\)
\(948\) 0 0
\(949\) −2.00000 + 6.92820i −0.0649227 + 0.224899i
\(950\) 0 0
\(951\) −9.00000 15.5885i −0.291845 0.505490i
\(952\) 0 0
\(953\) 2.59808 + 1.50000i 0.0841599 + 0.0485898i 0.541489 0.840708i \(-0.317861\pi\)
−0.457329 + 0.889297i \(0.651194\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 5.00000i 0.161627i
\(958\) 0 0
\(959\) 7.50000 + 12.9904i 0.242188 + 0.419481i
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 18.0000i 0.580042i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 16.0000i 0.514525i 0.966342 + 0.257263i \(0.0828206\pi\)
−0.966342 + 0.257263i \(0.917179\pi\)
\(968\) 0 0
\(969\) −1.50000 + 2.59808i −0.0481869 + 0.0834622i
\(970\) 0 0
\(971\) 3.50000 6.06218i 0.112320 0.194545i −0.804385 0.594108i \(-0.797505\pi\)
0.916705 + 0.399564i \(0.130838\pi\)
\(972\) 0 0
\(973\) −56.2917 + 32.5000i −1.80463 + 1.04190i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −38.9711 + 22.5000i −1.24680 + 0.719839i −0.970469 0.241225i \(-0.922451\pi\)
−0.276328 + 0.961063i \(0.589118\pi\)
\(978\) 0 0
\(979\) −17.5000 + 30.3109i −0.559302 + 0.968740i
\(980\) 0 0
\(981\) 18.0000 31.1769i 0.574696 0.995402i
\(982\) 0 0
\(983\) 16.0000i 0.510321i 0.966899 + 0.255160i \(0.0821283\pi\)
−0.966899 + 0.255160i \(0.917872\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 60.0000i 1.90982i
\(988\) 0 0
\(989\) −15.0000 −0.476972
\(990\) 0 0
\(991\) −1.50000 2.59808i −0.0476491 0.0825306i 0.841217 0.540697i \(-0.181840\pi\)
−0.888866 + 0.458167i \(0.848506\pi\)
\(992\) 0 0
\(993\) 1.00000i 0.0317340i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −6.06218 3.50000i −0.191991 0.110846i 0.400923 0.916112i \(-0.368689\pi\)
−0.592914 + 0.805266i \(0.702023\pi\)
\(998\) 0 0
\(999\) 17.5000 + 30.3109i 0.553675 + 0.958994i
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 1300.2.bb.c.549.2 4
5.2 odd 4 1300.2.i.c.601.1 2
5.3 odd 4 260.2.i.c.81.1 yes 2
5.4 even 2 inner 1300.2.bb.c.549.1 4
13.9 even 3 inner 1300.2.bb.c.1049.1 4
15.8 even 4 2340.2.q.c.2161.1 2
20.3 even 4 1040.2.q.f.81.1 2
65.3 odd 12 3380.2.a.d.1.1 1
65.9 even 6 inner 1300.2.bb.c.1049.2 4
65.22 odd 12 1300.2.i.c.1101.1 2
65.23 odd 12 3380.2.a.e.1.1 1
65.28 even 12 3380.2.f.c.3041.2 2
65.48 odd 12 260.2.i.c.61.1 2
65.63 even 12 3380.2.f.c.3041.1 2
195.113 even 12 2340.2.q.c.1621.1 2
260.243 even 12 1040.2.q.f.321.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.i.c.61.1 2 65.48 odd 12
260.2.i.c.81.1 yes 2 5.3 odd 4
1040.2.q.f.81.1 2 20.3 even 4
1040.2.q.f.321.1 2 260.243 even 12
1300.2.i.c.601.1 2 5.2 odd 4
1300.2.i.c.1101.1 2 65.22 odd 12
1300.2.bb.c.549.1 4 5.4 even 2 inner
1300.2.bb.c.549.2 4 1.1 even 1 trivial
1300.2.bb.c.1049.1 4 13.9 even 3 inner
1300.2.bb.c.1049.2 4 65.9 even 6 inner
2340.2.q.c.1621.1 2 195.113 even 12
2340.2.q.c.2161.1 2 15.8 even 4
3380.2.a.d.1.1 1 65.3 odd 12
3380.2.a.e.1.1 1 65.23 odd 12
3380.2.f.c.3041.1 2 65.63 even 12
3380.2.f.c.3041.2 2 65.28 even 12