Properties

Label 2100.2.bc.h.949.2
Level $2100$
Weight $2$
Character 2100.949
Analytic conductor $16.769$
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] = [2100,2,Mod(949,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.949");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.bc (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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 949.2
Root \(-0.228425 - 1.39564i\) of defining polynomial
Character \(\chi\) \(=\) 2100.949
Dual form 2100.2.bc.h.1549.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{3} +(2.29129 + 1.32288i) q^{7} +(0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.866025 - 0.500000i) q^{3} +(2.29129 + 1.32288i) q^{7} +(0.500000 + 0.866025i) q^{9} +(2.82288 - 4.88936i) q^{11} -1.00000i q^{13} +(1.42526 + 0.822876i) q^{17} +(-2.32288 - 4.02334i) q^{19} +(-1.32288 - 2.29129i) q^{21} +(1.73205 - 1.00000i) q^{23} -1.00000i q^{27} -7.29150 q^{29} +(-2.00000 + 3.46410i) q^{31} +(-4.88936 + 2.82288i) q^{33} +(7.18065 - 4.14575i) q^{37} +(-0.500000 + 0.866025i) q^{39} -0.354249 q^{41} +7.29150i q^{43} +(2.54374 - 1.46863i) q^{47} +(3.50000 + 6.06218i) q^{49} +(-0.822876 - 1.42526i) q^{51} +(2.03884 + 1.17712i) q^{53} +4.64575i q^{57} +(7.46863 - 12.9360i) q^{59} +(-4.79150 - 8.29913i) q^{61} +2.64575i q^{63} +(6.87386 + 3.96863i) q^{67} -2.00000 q^{69} +3.29150 q^{71} +(-5.44860 - 3.14575i) q^{73} +(12.9360 - 7.46863i) q^{77} +(-3.32288 - 5.75539i) q^{79} +(-0.500000 + 0.866025i) q^{81} -3.64575i q^{83} +(6.31463 + 3.64575i) q^{87} +(-4.46863 - 7.73989i) q^{89} +(1.32288 - 2.29129i) q^{91} +(3.46410 - 2.00000i) q^{93} +14.2915i q^{97} +5.64575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} + 12 q^{11} - 8 q^{19} - 16 q^{29} - 16 q^{31} - 4 q^{39} - 24 q^{41} + 28 q^{49} + 4 q^{51} + 28 q^{59} + 4 q^{61} - 16 q^{69} - 16 q^{71} - 16 q^{79} - 4 q^{81} - 4 q^{89} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.29129 + 1.32288i 0.866025 + 0.500000i
\(8\) 0 0
\(9\) 0.500000 + 0.866025i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) 2.82288 4.88936i 0.851129 1.47420i −0.0290612 0.999578i \(-0.509252\pi\)
0.880190 0.474621i \(-0.157415\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i −0.990338 0.138675i \(-0.955716\pi\)
0.990338 0.138675i \(-0.0442844\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.42526 + 0.822876i 0.345677 + 0.199577i 0.662780 0.748815i \(-0.269377\pi\)
−0.317103 + 0.948391i \(0.602710\pi\)
\(18\) 0 0
\(19\) −2.32288 4.02334i −0.532904 0.923017i −0.999262 0.0384208i \(-0.987767\pi\)
0.466357 0.884596i \(-0.345566\pi\)
\(20\) 0 0
\(21\) −1.32288 2.29129i −0.288675 0.500000i
\(22\) 0 0
\(23\) 1.73205 1.00000i 0.361158 0.208514i −0.308431 0.951247i \(-0.599804\pi\)
0.669588 + 0.742732i \(0.266471\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −7.29150 −1.35400 −0.676999 0.735984i \(-0.736720\pi\)
−0.676999 + 0.735984i \(0.736720\pi\)
\(30\) 0 0
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) −4.88936 + 2.82288i −0.851129 + 0.491400i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.18065 4.14575i 1.18049 0.681557i 0.224364 0.974506i \(-0.427970\pi\)
0.956128 + 0.292948i \(0.0946363\pi\)
\(38\) 0 0
\(39\) −0.500000 + 0.866025i −0.0800641 + 0.138675i
\(40\) 0 0
\(41\) −0.354249 −0.0553244 −0.0276622 0.999617i \(-0.508806\pi\)
−0.0276622 + 0.999617i \(0.508806\pi\)
\(42\) 0 0
\(43\) 7.29150i 1.11194i 0.831201 + 0.555972i \(0.187654\pi\)
−0.831201 + 0.555972i \(0.812346\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.54374 1.46863i 0.371042 0.214221i −0.302872 0.953031i \(-0.597945\pi\)
0.673914 + 0.738810i \(0.264612\pi\)
\(48\) 0 0
\(49\) 3.50000 + 6.06218i 0.500000 + 0.866025i
\(50\) 0 0
\(51\) −0.822876 1.42526i −0.115226 0.199577i
\(52\) 0 0
\(53\) 2.03884 + 1.17712i 0.280056 + 0.161690i 0.633449 0.773785i \(-0.281639\pi\)
−0.353393 + 0.935475i \(0.614972\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.64575i 0.615345i
\(58\) 0 0
\(59\) 7.46863 12.9360i 0.972332 1.68413i 0.283861 0.958865i \(-0.408384\pi\)
0.688471 0.725264i \(-0.258282\pi\)
\(60\) 0 0
\(61\) −4.79150 8.29913i −0.613489 1.06259i −0.990648 0.136445i \(-0.956432\pi\)
0.377159 0.926149i \(-0.376901\pi\)
\(62\) 0 0
\(63\) 2.64575i 0.333333i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.87386 + 3.96863i 0.839776 + 0.484845i 0.857188 0.515003i \(-0.172209\pi\)
−0.0174120 + 0.999848i \(0.505543\pi\)
\(68\) 0 0
\(69\) −2.00000 −0.240772
\(70\) 0 0
\(71\) 3.29150 0.390629 0.195315 0.980741i \(-0.437427\pi\)
0.195315 + 0.980741i \(0.437427\pi\)
\(72\) 0 0
\(73\) −5.44860 3.14575i −0.637711 0.368182i 0.146022 0.989281i \(-0.453353\pi\)
−0.783732 + 0.621099i \(0.786686\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.9360 7.46863i 1.47420 0.851129i
\(78\) 0 0
\(79\) −3.32288 5.75539i −0.373853 0.647532i 0.616302 0.787510i \(-0.288630\pi\)
−0.990155 + 0.139978i \(0.955297\pi\)
\(80\) 0 0
\(81\) −0.500000 + 0.866025i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 3.64575i 0.400173i −0.979778 0.200087i \(-0.935878\pi\)
0.979778 0.200087i \(-0.0641224\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.31463 + 3.64575i 0.676999 + 0.390866i
\(88\) 0 0
\(89\) −4.46863 7.73989i −0.473674 0.820427i 0.525872 0.850564i \(-0.323739\pi\)
−0.999546 + 0.0301370i \(0.990406\pi\)
\(90\) 0 0
\(91\) 1.32288 2.29129i 0.138675 0.240192i
\(92\) 0 0
\(93\) 3.46410 2.00000i 0.359211 0.207390i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.2915i 1.45108i 0.688179 + 0.725541i \(0.258410\pi\)
−0.688179 + 0.725541i \(0.741590\pi\)
\(98\) 0 0
\(99\) 5.64575 0.567419
\(100\) 0 0
\(101\) 5.46863 9.47194i 0.544149 0.942493i −0.454511 0.890741i \(-0.650186\pi\)
0.998660 0.0517522i \(-0.0164806\pi\)
\(102\) 0 0
\(103\) −13.8021 + 7.96863i −1.35996 + 0.785172i −0.989618 0.143724i \(-0.954092\pi\)
−0.370340 + 0.928896i \(0.620759\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.85836 5.11438i 0.856370 0.494426i −0.00642483 0.999979i \(-0.502045\pi\)
0.862795 + 0.505554i \(0.168712\pi\)
\(108\) 0 0
\(109\) 4.79150 8.29913i 0.458943 0.794912i −0.539963 0.841689i \(-0.681562\pi\)
0.998905 + 0.0467769i \(0.0148950\pi\)
\(110\) 0 0
\(111\) −8.29150 −0.786995
\(112\) 0 0
\(113\) 12.5830i 1.18371i −0.806045 0.591855i \(-0.798396\pi\)
0.806045 0.591855i \(-0.201604\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.866025 0.500000i 0.0800641 0.0462250i
\(118\) 0 0
\(119\) 2.17712 + 3.77089i 0.199577 + 0.345677i
\(120\) 0 0
\(121\) −10.4373 18.0779i −0.948841 1.64344i
\(122\) 0 0
\(123\) 0.306788 + 0.177124i 0.0276622 + 0.0159708i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 6.64575i 0.589715i −0.955541 0.294858i \(-0.904728\pi\)
0.955541 0.294858i \(-0.0952722\pi\)
\(128\) 0 0
\(129\) 3.64575 6.31463i 0.320991 0.555972i
\(130\) 0 0
\(131\) 10.2915 + 17.8254i 0.899173 + 1.55741i 0.828554 + 0.559910i \(0.189164\pi\)
0.0706190 + 0.997503i \(0.477503\pi\)
\(132\) 0 0
\(133\) 12.2915i 1.06581i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.04668 + 4.64575i 0.687474 + 0.396913i 0.802665 0.596430i \(-0.203415\pi\)
−0.115191 + 0.993343i \(0.536748\pi\)
\(138\) 0 0
\(139\) −0.0627461 −0.00532205 −0.00266103 0.999996i \(-0.500847\pi\)
−0.00266103 + 0.999996i \(0.500847\pi\)
\(140\) 0 0
\(141\) −2.93725 −0.247361
\(142\) 0 0
\(143\) −4.88936 2.82288i −0.408869 0.236061i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 7.00000i 0.577350i
\(148\) 0 0
\(149\) −5.46863 9.47194i −0.448007 0.775972i 0.550249 0.835001i \(-0.314533\pi\)
−0.998256 + 0.0590292i \(0.981200\pi\)
\(150\) 0 0
\(151\) 8.32288 14.4156i 0.677306 1.17313i −0.298483 0.954415i \(-0.596481\pi\)
0.975789 0.218714i \(-0.0701860\pi\)
\(152\) 0 0
\(153\) 1.64575i 0.133051i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.7224 8.50000i −1.17498 0.678374i −0.220131 0.975470i \(-0.570648\pi\)
−0.954847 + 0.297097i \(0.903982\pi\)
\(158\) 0 0
\(159\) −1.17712 2.03884i −0.0933520 0.161690i
\(160\) 0 0
\(161\) 5.29150 0.417029
\(162\) 0 0
\(163\) −7.99234 + 4.61438i −0.626008 + 0.361426i −0.779204 0.626770i \(-0.784377\pi\)
0.153196 + 0.988196i \(0.451043\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.5203i 1.35576i −0.735173 0.677879i \(-0.762899\pi\)
0.735173 0.677879i \(-0.237101\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 2.32288 4.02334i 0.177635 0.307672i
\(172\) 0 0
\(173\) 0.504897 0.291503i 0.0383866 0.0221625i −0.480684 0.876894i \(-0.659612\pi\)
0.519071 + 0.854731i \(0.326278\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −12.9360 + 7.46863i −0.972332 + 0.561376i
\(178\) 0 0
\(179\) −6.35425 + 11.0059i −0.474939 + 0.822618i −0.999588 0.0287003i \(-0.990863\pi\)
0.524649 + 0.851319i \(0.324196\pi\)
\(180\) 0 0
\(181\) −6.58301 −0.489311 −0.244655 0.969610i \(-0.578675\pi\)
−0.244655 + 0.969610i \(0.578675\pi\)
\(182\) 0 0
\(183\) 9.58301i 0.708396i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 8.04668 4.64575i 0.588431 0.339731i
\(188\) 0 0
\(189\) 1.32288 2.29129i 0.0962250 0.166667i
\(190\) 0 0
\(191\) 11.7601 + 20.3691i 0.850933 + 1.47386i 0.880367 + 0.474293i \(0.157296\pi\)
−0.0294341 + 0.999567i \(0.509371\pi\)
\(192\) 0 0
\(193\) 13.8564 + 8.00000i 0.997406 + 0.575853i 0.907480 0.420096i \(-0.138004\pi\)
0.0899262 + 0.995948i \(0.471337\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 19.2915i 1.37446i −0.726439 0.687231i \(-0.758826\pi\)
0.726439 0.687231i \(-0.241174\pi\)
\(198\) 0 0
\(199\) −2.32288 + 4.02334i −0.164664 + 0.285207i −0.936536 0.350571i \(-0.885987\pi\)
0.771872 + 0.635778i \(0.219321\pi\)
\(200\) 0 0
\(201\) −3.96863 6.87386i −0.279925 0.484845i
\(202\) 0 0
\(203\) −16.7069 9.64575i −1.17260 0.676999i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.73205 + 1.00000i 0.120386 + 0.0695048i
\(208\) 0 0
\(209\) −26.2288 −1.81428
\(210\) 0 0
\(211\) −17.9373 −1.23485 −0.617426 0.786629i \(-0.711824\pi\)
−0.617426 + 0.786629i \(0.711824\pi\)
\(212\) 0 0
\(213\) −2.85052 1.64575i −0.195315 0.112765i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −9.16515 + 5.29150i −0.622171 + 0.359211i
\(218\) 0 0
\(219\) 3.14575 + 5.44860i 0.212570 + 0.368182i
\(220\) 0 0
\(221\) 0.822876 1.42526i 0.0553526 0.0958735i
\(222\) 0 0
\(223\) 1.93725i 0.129728i 0.997894 + 0.0648641i \(0.0206614\pi\)
−0.997894 + 0.0648641i \(0.979339\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 16.7069 + 9.64575i 1.10888 + 0.640211i 0.938539 0.345174i \(-0.112180\pi\)
0.170339 + 0.985385i \(0.445514\pi\)
\(228\) 0 0
\(229\) −4.85425 8.40781i −0.320778 0.555603i 0.659871 0.751379i \(-0.270611\pi\)
−0.980649 + 0.195776i \(0.937278\pi\)
\(230\) 0 0
\(231\) −14.9373 −0.982799
\(232\) 0 0
\(233\) 1.93016 1.11438i 0.126449 0.0730053i −0.435441 0.900217i \(-0.643408\pi\)
0.561890 + 0.827212i \(0.310074\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 6.64575i 0.431688i
\(238\) 0 0
\(239\) 30.5830 1.97825 0.989125 0.147077i \(-0.0469864\pi\)
0.989125 + 0.147077i \(0.0469864\pi\)
\(240\) 0 0
\(241\) −5.14575 + 8.91270i −0.331467 + 0.574118i −0.982800 0.184675i \(-0.940877\pi\)
0.651333 + 0.758792i \(0.274210\pi\)
\(242\) 0 0
\(243\) 0.866025 0.500000i 0.0555556 0.0320750i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.02334 + 2.32288i −0.255999 + 0.147801i
\(248\) 0 0
\(249\) −1.82288 + 3.15731i −0.115520 + 0.200087i
\(250\) 0 0
\(251\) 18.8118 1.18739 0.593694 0.804691i \(-0.297669\pi\)
0.593694 + 0.804691i \(0.297669\pi\)
\(252\) 0 0
\(253\) 11.2915i 0.709891i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.6991 + 6.17712i −0.667391 + 0.385318i −0.795087 0.606495i \(-0.792575\pi\)
0.127696 + 0.991813i \(0.459242\pi\)
\(258\) 0 0
\(259\) 21.9373 1.36311
\(260\) 0 0
\(261\) −3.64575 6.31463i −0.225666 0.390866i
\(262\) 0 0
\(263\) 20.7846 + 12.0000i 1.28163 + 0.739952i 0.977147 0.212565i \(-0.0681817\pi\)
0.304487 + 0.952517i \(0.401515\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 8.93725i 0.546951i
\(268\) 0 0
\(269\) −3.11438 + 5.39426i −0.189887 + 0.328894i −0.945212 0.326456i \(-0.894145\pi\)
0.755325 + 0.655350i \(0.227479\pi\)
\(270\) 0 0
\(271\) 6.64575 + 11.5108i 0.403701 + 0.699230i 0.994169 0.107831i \(-0.0343904\pi\)
−0.590469 + 0.807061i \(0.701057\pi\)
\(272\) 0 0
\(273\) −2.29129 + 1.32288i −0.138675 + 0.0800641i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 21.6506 + 12.5000i 1.30086 + 0.751052i 0.980552 0.196261i \(-0.0628800\pi\)
0.320309 + 0.947313i \(0.396213\pi\)
\(278\) 0 0
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) 4.22876 0.252266 0.126133 0.992013i \(-0.459743\pi\)
0.126133 + 0.992013i \(0.459743\pi\)
\(282\) 0 0
\(283\) 26.4313 + 15.2601i 1.57118 + 0.907121i 0.996025 + 0.0890754i \(0.0283912\pi\)
0.575154 + 0.818045i \(0.304942\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.811686 0.468627i −0.0479123 0.0276622i
\(288\) 0 0
\(289\) −7.14575 12.3768i −0.420338 0.728047i
\(290\) 0 0
\(291\) 7.14575 12.3768i 0.418891 0.725541i
\(292\) 0 0
\(293\) 21.8745i 1.27792i −0.769239 0.638961i \(-0.779364\pi\)
0.769239 0.638961i \(-0.220636\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −4.88936 2.82288i −0.283710 0.163800i
\(298\) 0 0
\(299\) −1.00000 1.73205i −0.0578315 0.100167i
\(300\) 0 0
\(301\) −9.64575 + 16.7069i −0.555972 + 0.962972i
\(302\) 0 0
\(303\) −9.47194 + 5.46863i −0.544149 + 0.314164i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 29.8745i 1.70503i 0.522704 + 0.852514i \(0.324923\pi\)
−0.522704 + 0.852514i \(0.675077\pi\)
\(308\) 0 0
\(309\) 15.9373 0.906639
\(310\) 0 0
\(311\) 1.35425 2.34563i 0.0767924 0.133008i −0.825072 0.565028i \(-0.808865\pi\)
0.901864 + 0.432019i \(0.142199\pi\)
\(312\) 0 0
\(313\) −7.54178 + 4.35425i −0.426287 + 0.246117i −0.697763 0.716328i \(-0.745821\pi\)
0.271477 + 0.962445i \(0.412488\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.8972 + 6.29150i −0.612048 + 0.353366i −0.773767 0.633471i \(-0.781630\pi\)
0.161719 + 0.986837i \(0.448296\pi\)
\(318\) 0 0
\(319\) −20.5830 + 35.6508i −1.15243 + 1.99606i
\(320\) 0 0
\(321\) −10.2288 −0.570914
\(322\) 0 0
\(323\) 7.64575i 0.425421i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −8.29913 + 4.79150i −0.458943 + 0.264971i
\(328\) 0 0
\(329\) 7.77124 0.428443
\(330\) 0 0
\(331\) 4.32288 + 7.48744i 0.237607 + 0.411547i 0.960027 0.279907i \(-0.0903037\pi\)
−0.722420 + 0.691454i \(0.756970\pi\)
\(332\) 0 0
\(333\) 7.18065 + 4.14575i 0.393497 + 0.227186i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.70850i 0.474382i −0.971463 0.237191i \(-0.923773\pi\)
0.971463 0.237191i \(-0.0762267\pi\)
\(338\) 0 0
\(339\) −6.29150 + 10.8972i −0.341708 + 0.591855i
\(340\) 0 0
\(341\) 11.2915 + 19.5575i 0.611469 + 1.05910i
\(342\) 0 0
\(343\) 18.5203i 1.00000i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −22.1012 12.7601i −1.18645 0.685000i −0.228955 0.973437i \(-0.573531\pi\)
−0.957499 + 0.288437i \(0.906864\pi\)
\(348\) 0 0
\(349\) −15.2915 −0.818535 −0.409268 0.912414i \(-0.634216\pi\)
−0.409268 + 0.912414i \(0.634216\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −0.920365 0.531373i −0.0489861 0.0282821i 0.475307 0.879820i \(-0.342337\pi\)
−0.524293 + 0.851538i \(0.675670\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4.35425i 0.230451i
\(358\) 0 0
\(359\) −4.53137 7.84857i −0.239157 0.414232i 0.721316 0.692606i \(-0.243538\pi\)
−0.960473 + 0.278375i \(0.910204\pi\)
\(360\) 0 0
\(361\) −1.29150 + 2.23695i −0.0679738 + 0.117734i
\(362\) 0 0
\(363\) 20.8745i 1.09563i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −17.2118 9.93725i −0.898450 0.518720i −0.0217531 0.999763i \(-0.506925\pi\)
−0.876697 + 0.481043i \(0.840258\pi\)
\(368\) 0 0
\(369\) −0.177124 0.306788i −0.00922073 0.0159708i
\(370\) 0 0
\(371\) 3.11438 + 5.39426i 0.161690 + 0.280056i
\(372\) 0 0
\(373\) −2.09318 + 1.20850i −0.108381 + 0.0625736i −0.553211 0.833041i \(-0.686597\pi\)
0.444830 + 0.895615i \(0.353264\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7.29150i 0.375531i
\(378\) 0 0
\(379\) 20.6458 1.06050 0.530251 0.847841i \(-0.322098\pi\)
0.530251 + 0.847841i \(0.322098\pi\)
\(380\) 0 0
\(381\) −3.32288 + 5.75539i −0.170236 + 0.294858i
\(382\) 0 0
\(383\) 1.11847 0.645751i 0.0571514 0.0329964i −0.471152 0.882052i \(-0.656162\pi\)
0.528303 + 0.849056i \(0.322828\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −6.31463 + 3.64575i −0.320991 + 0.185324i
\(388\) 0 0
\(389\) −18.7601 + 32.4935i −0.951176 + 1.64749i −0.208291 + 0.978067i \(0.566790\pi\)
−0.742885 + 0.669419i \(0.766543\pi\)
\(390\) 0 0
\(391\) 3.29150 0.166458
\(392\) 0 0
\(393\) 20.5830i 1.03828i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −28.1090 + 16.2288i −1.41075 + 0.814498i −0.995459 0.0951899i \(-0.969654\pi\)
−0.415293 + 0.909688i \(0.636321\pi\)
\(398\) 0 0
\(399\) −6.14575 + 10.6448i −0.307672 + 0.532904i
\(400\) 0 0
\(401\) 0.760130 + 1.31658i 0.0379591 + 0.0657470i 0.884381 0.466766i \(-0.154581\pi\)
−0.846422 + 0.532513i \(0.821248\pi\)
\(402\) 0 0
\(403\) 3.46410 + 2.00000i 0.172559 + 0.0996271i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 46.8118i 2.32037i
\(408\) 0 0
\(409\) −6.79150 + 11.7632i −0.335818 + 0.581654i −0.983642 0.180136i \(-0.942346\pi\)
0.647823 + 0.761790i \(0.275679\pi\)
\(410\) 0 0
\(411\) −4.64575 8.04668i −0.229158 0.396913i
\(412\) 0 0
\(413\) 34.2255 19.7601i 1.68413 0.972332i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.0543397 + 0.0313730i 0.00266103 + 0.00153634i
\(418\) 0 0
\(419\) −26.3542 −1.28749 −0.643745 0.765240i \(-0.722620\pi\)
−0.643745 + 0.765240i \(0.722620\pi\)
\(420\) 0 0
\(421\) −2.87451 −0.140095 −0.0700475 0.997544i \(-0.522315\pi\)
−0.0700475 + 0.997544i \(0.522315\pi\)
\(422\) 0 0
\(423\) 2.54374 + 1.46863i 0.123681 + 0.0714071i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 25.3542i 1.22698i
\(428\) 0 0
\(429\) 2.82288 + 4.88936i 0.136290 + 0.236061i
\(430\) 0 0
\(431\) −1.93725 + 3.35542i −0.0933142 + 0.161625i −0.908904 0.417006i \(-0.863079\pi\)
0.815590 + 0.578631i \(0.196413\pi\)
\(432\) 0 0
\(433\) 7.29150i 0.350407i −0.984532 0.175204i \(-0.943942\pi\)
0.984532 0.175204i \(-0.0560584\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8.04668 4.64575i −0.384925 0.222236i
\(438\) 0 0
\(439\) 3.90588 + 6.76518i 0.186418 + 0.322885i 0.944053 0.329793i \(-0.106979\pi\)
−0.757636 + 0.652678i \(0.773646\pi\)
\(440\) 0 0
\(441\) −3.50000 + 6.06218i −0.166667 + 0.288675i
\(442\) 0 0
\(443\) 14.9749 8.64575i 0.711478 0.410772i −0.100130 0.994974i \(-0.531926\pi\)
0.811608 + 0.584202i \(0.198593\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 10.9373i 0.517314i
\(448\) 0 0
\(449\) −26.1033 −1.23189 −0.615945 0.787789i \(-0.711225\pi\)
−0.615945 + 0.787789i \(0.711225\pi\)
\(450\) 0 0
\(451\) −1.00000 + 1.73205i −0.0470882 + 0.0815591i
\(452\) 0 0
\(453\) −14.4156 + 8.32288i −0.677306 + 0.391043i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −29.5886 + 17.0830i −1.38410 + 0.799109i −0.992642 0.121088i \(-0.961362\pi\)
−0.391456 + 0.920197i \(0.628028\pi\)
\(458\) 0 0
\(459\) 0.822876 1.42526i 0.0384085 0.0665256i
\(460\) 0 0
\(461\) 21.2915 0.991644 0.495822 0.868424i \(-0.334867\pi\)
0.495822 + 0.868424i \(0.334867\pi\)
\(462\) 0 0
\(463\) 18.0627i 0.839447i −0.907652 0.419723i \(-0.862127\pi\)
0.907652 0.419723i \(-0.137873\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −28.8313 + 16.6458i −1.33415 + 0.770274i −0.985933 0.167139i \(-0.946547\pi\)
−0.348220 + 0.937413i \(0.613214\pi\)
\(468\) 0 0
\(469\) 10.5000 + 18.1865i 0.484845 + 0.839776i
\(470\) 0 0
\(471\) 8.50000 + 14.7224i 0.391659 + 0.678374i
\(472\) 0 0
\(473\) 35.6508 + 20.5830i 1.63923 + 0.946408i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 2.35425i 0.107794i
\(478\) 0 0
\(479\) 10.1771 17.6273i 0.465005 0.805412i −0.534197 0.845360i \(-0.679386\pi\)
0.999202 + 0.0399482i \(0.0127193\pi\)
\(480\) 0 0
\(481\) −4.14575 7.18065i −0.189030 0.327410i
\(482\) 0 0
\(483\) −4.58258 2.64575i −0.208514 0.120386i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −10.7885 6.22876i −0.488875 0.282252i 0.235233 0.971939i \(-0.424415\pi\)
−0.724107 + 0.689687i \(0.757748\pi\)
\(488\) 0 0
\(489\) 9.22876 0.417339
\(490\) 0 0
\(491\) 18.4575 0.832976 0.416488 0.909141i \(-0.363261\pi\)
0.416488 + 0.909141i \(0.363261\pi\)
\(492\) 0 0
\(493\) −10.3923 6.00000i −0.468046 0.270226i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.54178 + 4.35425i 0.338295 + 0.195315i
\(498\) 0 0
\(499\) 13.9059 + 24.0857i 0.622513 + 1.07822i 0.989016 + 0.147807i \(0.0472214\pi\)
−0.366504 + 0.930417i \(0.619445\pi\)
\(500\) 0 0
\(501\) −8.76013 + 15.1730i −0.391374 + 0.677879i
\(502\) 0 0
\(503\) 27.2915i 1.21687i 0.793604 + 0.608434i \(0.208202\pi\)
−0.793604 + 0.608434i \(0.791798\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −10.3923 6.00000i −0.461538 0.266469i
\(508\) 0 0
\(509\) 2.93725 + 5.08747i 0.130191 + 0.225498i 0.923750 0.382995i \(-0.125107\pi\)
−0.793559 + 0.608494i \(0.791774\pi\)
\(510\) 0 0
\(511\) −8.32288 14.4156i −0.368182 0.637711i
\(512\) 0 0
\(513\) −4.02334 + 2.32288i −0.177635 + 0.102557i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 16.5830i 0.729320i
\(518\) 0 0
\(519\) −0.583005 −0.0255911
\(520\) 0 0
\(521\) −7.93725 + 13.7477i −0.347737 + 0.602299i −0.985847 0.167647i \(-0.946383\pi\)
0.638110 + 0.769945i \(0.279717\pi\)
\(522\) 0 0
\(523\) −36.2644 + 20.9373i −1.58573 + 0.915522i −0.591732 + 0.806135i \(0.701556\pi\)
−0.993999 + 0.109387i \(0.965111\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.70105 + 3.29150i −0.248342 + 0.143380i
\(528\) 0 0
\(529\) −9.50000 + 16.4545i −0.413043 + 0.715412i
\(530\) 0 0
\(531\) 14.9373 0.648222
\(532\) 0 0
\(533\) 0.354249i 0.0153442i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 11.0059 6.35425i 0.474939 0.274206i
\(538\) 0 0
\(539\) 39.5203 1.70226
\(540\) 0 0
\(541\) 2.43725 + 4.22145i 0.104786 + 0.181494i 0.913651 0.406500i \(-0.133251\pi\)
−0.808865 + 0.587995i \(0.799918\pi\)
\(542\) 0 0
\(543\) 5.70105 + 3.29150i 0.244655 + 0.141252i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 31.8745i 1.36286i 0.731885 + 0.681428i \(0.238641\pi\)
−0.731885 + 0.681428i \(0.761359\pi\)
\(548\) 0 0
\(549\) 4.79150 8.29913i 0.204496 0.354198i
\(550\) 0 0
\(551\) 16.9373 + 29.3362i 0.721551 + 1.24976i
\(552\) 0 0
\(553\) 17.5830i 0.747705i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 23.6351 + 13.6458i 1.00145 + 0.578189i 0.908678 0.417498i \(-0.137093\pi\)
0.0927750 + 0.995687i \(0.470426\pi\)
\(558\) 0 0
\(559\) 7.29150 0.308398
\(560\) 0 0
\(561\) −9.29150 −0.392288
\(562\) 0 0
\(563\) −27.2973 15.7601i −1.15045 0.664210i −0.201450 0.979499i \(-0.564565\pi\)
−0.948996 + 0.315289i \(0.897899\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.29129 + 1.32288i −0.0962250 + 0.0555556i
\(568\) 0 0
\(569\) −5.46863 9.47194i −0.229257 0.397084i 0.728331 0.685225i \(-0.240296\pi\)
−0.957588 + 0.288141i \(0.906963\pi\)
\(570\) 0 0
\(571\) 4.96863 8.60591i 0.207931 0.360146i −0.743132 0.669145i \(-0.766661\pi\)
0.951062 + 0.308999i \(0.0999939\pi\)
\(572\) 0 0
\(573\) 23.5203i 0.982573i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −14.3613 8.29150i −0.597869 0.345180i 0.170334 0.985386i \(-0.445515\pi\)
−0.768203 + 0.640207i \(0.778849\pi\)
\(578\) 0 0
\(579\) −8.00000 13.8564i −0.332469 0.575853i
\(580\) 0 0
\(581\) 4.82288 8.35347i 0.200087 0.346560i
\(582\) 0 0
\(583\) 11.5108 6.64575i 0.476728 0.275239i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.00000i 0.247647i 0.992304 + 0.123823i \(0.0395156\pi\)
−0.992304 + 0.123823i \(0.960484\pi\)
\(588\) 0 0
\(589\) 18.5830 0.765699
\(590\) 0 0
\(591\) −9.64575 + 16.7069i −0.396773 + 0.687231i
\(592\) 0 0
\(593\) 17.8254 10.2915i 0.732002 0.422621i −0.0871523 0.996195i \(-0.527777\pi\)
0.819154 + 0.573574i \(0.194443\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4.02334 2.32288i 0.164664 0.0950690i
\(598\) 0 0
\(599\) −8.53137 + 14.7768i −0.348582 + 0.603763i −0.985998 0.166758i \(-0.946670\pi\)
0.637415 + 0.770520i \(0.280004\pi\)
\(600\) 0 0
\(601\) 3.70850 0.151273 0.0756364 0.997135i \(-0.475901\pi\)
0.0756364 + 0.997135i \(0.475901\pi\)
\(602\) 0 0
\(603\) 7.93725i 0.323230i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −9.83307 + 5.67712i −0.399112 + 0.230427i −0.686101 0.727507i \(-0.740679\pi\)
0.286989 + 0.957934i \(0.407346\pi\)
\(608\) 0 0
\(609\) 9.64575 + 16.7069i 0.390866 + 0.676999i
\(610\) 0 0
\(611\) −1.46863 2.54374i −0.0594143 0.102909i
\(612\) 0 0
\(613\) 21.3982 + 12.3542i 0.864265 + 0.498983i 0.865438 0.501016i \(-0.167040\pi\)
−0.00117347 + 0.999999i \(0.500374\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 10.1255i 0.407637i −0.979009 0.203818i \(-0.934665\pi\)
0.979009 0.203818i \(-0.0653352\pi\)
\(618\) 0 0
\(619\) 17.6458 30.5633i 0.709243 1.22844i −0.255896 0.966704i \(-0.582370\pi\)
0.965138 0.261740i \(-0.0842963\pi\)
\(620\) 0 0
\(621\) −1.00000 1.73205i −0.0401286 0.0695048i
\(622\) 0 0
\(623\) 23.6458i 0.947347i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 22.7148 + 13.1144i 0.907141 + 0.523738i
\(628\) 0 0
\(629\) 13.6458 0.544092
\(630\) 0 0
\(631\) −6.64575 −0.264563 −0.132282 0.991212i \(-0.542230\pi\)
−0.132282 + 0.991212i \(0.542230\pi\)
\(632\) 0 0
\(633\) 15.5341 + 8.96863i 0.617426 + 0.356471i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.06218 3.50000i 0.240192 0.138675i
\(638\) 0 0
\(639\) 1.64575 + 2.85052i 0.0651049 + 0.112765i
\(640\) 0 0
\(641\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(642\) 0 0
\(643\) 32.6458i 1.28742i 0.765268 + 0.643711i \(0.222606\pi\)
−0.765268 + 0.643711i \(0.777394\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −24.8430 14.3431i −0.976681 0.563887i −0.0754143 0.997152i \(-0.524028\pi\)
−0.901266 + 0.433265i \(0.857361\pi\)
\(648\) 0 0
\(649\) −42.1660 73.0337i −1.65516 2.86682i
\(650\) 0 0
\(651\) 10.5830 0.414781
\(652\) 0 0
\(653\) 8.85836 5.11438i 0.346655 0.200141i −0.316556 0.948574i \(-0.602527\pi\)
0.663211 + 0.748433i \(0.269193\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.29150i 0.245455i
\(658\) 0 0
\(659\) 18.1033 0.705203 0.352602 0.935774i \(-0.385297\pi\)
0.352602 + 0.935774i \(0.385297\pi\)
\(660\) 0 0
\(661\) −6.14575 + 10.6448i −0.239042 + 0.414033i −0.960440 0.278488i \(-0.910167\pi\)
0.721398 + 0.692521i \(0.243500\pi\)
\(662\) 0 0
\(663\) −1.42526 + 0.822876i −0.0553526 + 0.0319578i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.6293 + 7.29150i −0.489007 + 0.282328i
\(668\) 0 0
\(669\) 0.968627 1.67771i 0.0374493 0.0648641i
\(670\) 0 0
\(671\) −54.1033 −2.08863
\(672\) 0 0
\(673\) 8.29150i 0.319614i 0.987148 + 0.159807i \(0.0510872\pi\)
−0.987148 + 0.159807i \(0.948913\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −42.6877 + 24.6458i −1.64062 + 0.947213i −0.660008 + 0.751258i \(0.729447\pi\)
−0.980613 + 0.195955i \(0.937219\pi\)
\(678\) 0 0
\(679\) −18.9059 + 32.7459i −0.725541 + 1.25667i
\(680\) 0 0
\(681\) −9.64575 16.7069i −0.369626 0.640211i
\(682\) 0 0
\(683\) −32.9984 19.0516i −1.26265 0.728990i −0.289062 0.957310i \(-0.593343\pi\)
−0.973586 + 0.228320i \(0.926677\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 9.70850i 0.370402i
\(688\) 0 0
\(689\) 1.17712 2.03884i 0.0448449 0.0776736i
\(690\) 0 0
\(691\) 5.61438 + 9.72439i 0.213581 + 0.369933i 0.952833 0.303496i \(-0.0981539\pi\)
−0.739252 + 0.673429i \(0.764821\pi\)
\(692\) 0 0
\(693\) 12.9360 + 7.46863i 0.491400 + 0.283710i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −0.504897 0.291503i −0.0191244 0.0110414i
\(698\) 0 0
\(699\) −2.22876 −0.0842993
\(700\) 0 0
\(701\) −40.9373 −1.54618 −0.773089 0.634297i \(-0.781290\pi\)
−0.773089 + 0.634297i \(0.781290\pi\)
\(702\) 0 0
\(703\) −33.3595 19.2601i −1.25818 0.726410i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 25.0604 14.4686i 0.942493 0.544149i
\(708\) 0 0
\(709\) 18.0830 + 31.3207i 0.679122 + 1.17627i 0.975246 + 0.221124i \(0.0709724\pi\)
−0.296124 + 0.955149i \(0.595694\pi\)
\(710\) 0 0
\(711\) 3.32288 5.75539i 0.124618 0.215844i
\(712\) 0 0
\(713\) 8.00000i 0.299602i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −26.4857 15.2915i −0.989125 0.571072i
\(718\) 0 0
\(719\) −12.1144 20.9827i −0.451790 0.782523i 0.546707 0.837324i \(-0.315881\pi\)
−0.998497 + 0.0548005i \(0.982548\pi\)
\(720\) 0 0
\(721\) −42.1660 −1.57034
\(722\) 0 0
\(723\) 8.91270 5.14575i 0.331467 0.191373i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 31.1033i 1.15356i −0.816901 0.576778i \(-0.804310\pi\)
0.816901 0.576778i \(-0.195690\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −6.00000 + 10.3923i −0.221918 + 0.384373i
\(732\) 0 0
\(733\) 36.4082 21.0203i 1.34477 0.776401i 0.357263 0.934004i \(-0.383710\pi\)
0.987503 + 0.157603i \(0.0503765\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 38.8081 22.4059i 1.42952 0.825331i
\(738\) 0 0
\(739\) −6.73987 + 11.6738i −0.247930 + 0.429428i −0.962951 0.269675i \(-0.913084\pi\)
0.715021 + 0.699103i \(0.246417\pi\)
\(740\) 0 0
\(741\) 4.64575 0.170666
\(742\) 0 0
\(743\) 32.5830i 1.19535i 0.801737 + 0.597677i \(0.203910\pi\)
−0.801737 + 0.597677i \(0.796090\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3.15731 1.82288i 0.115520 0.0666955i
\(748\) 0 0
\(749\) 27.0627 0.988851
\(750\) 0 0
\(751\) 19.3229 + 33.4682i 0.705102 + 1.22127i 0.966655 + 0.256083i \(0.0824320\pi\)
−0.261553 + 0.965189i \(0.584235\pi\)
\(752\) 0 0
\(753\) −16.2915 9.40588i −0.593694 0.342769i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 11.0000i 0.399802i 0.979816 + 0.199901i \(0.0640620\pi\)
−0.979816 + 0.199901i \(0.935938\pi\)
\(758\) 0 0
\(759\) −5.64575 + 9.77873i −0.204928 + 0.354945i
\(760\) 0 0
\(761\) 24.8229 + 42.9945i 0.899829 + 1.55855i 0.827712 + 0.561153i \(0.189642\pi\)
0.0721161 + 0.997396i \(0.477025\pi\)
\(762\) 0 0
\(763\) 21.9574 12.6771i 0.794912 0.458943i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.9360 7.46863i −0.467093 0.269676i
\(768\) 0 0
\(769\) −4.00000 −0.144244 −0.0721218 0.997396i \(-0.522977\pi\)
−0.0721218 + 0.997396i \(0.522977\pi\)
\(770\) 0 0
\(771\) 12.3542 0.444927
\(772\) 0 0
\(773\) −4.69126 2.70850i −0.168733 0.0974179i 0.413255 0.910615i \(-0.364392\pi\)
−0.581988 + 0.813197i \(0.697725\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −18.9982 10.9686i −0.681557 0.393497i
\(778\) 0 0
\(779\) 0.822876 + 1.42526i 0.0294826 + 0.0510653i
\(780\) 0 0
\(781\) 9.29150 16.0934i 0.332476 0.575866i
\(782\) 0 0
\(783\) 7.29150i 0.260577i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 38.9519 + 22.4889i 1.38849 + 0.801642i 0.993145 0.116892i \(-0.0372933\pi\)
0.395340 + 0.918535i \(0.370627\pi\)
\(788\) 0 0
\(789\) −12.0000 20.7846i −0.427211 0.739952i
\(790\) 0 0
\(791\) 16.6458 28.8313i 0.591855 1.02512i
\(792\) 0 0
\(793\) −8.29913 + 4.79150i −0.294711 + 0.170151i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 27.8745i 0.987366i 0.869642 + 0.493683i \(0.164350\pi\)
−0.869642 + 0.493683i \(0.835650\pi\)
\(798\) 0 0
\(799\) 4.83399 0.171014
\(800\) 0 0
\(801\) 4.46863 7.73989i 0.157891 0.273476i
\(802\) 0 0
\(803\) −30.7614 + 17.7601i −1.08555 + 0.626741i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 5.39426 3.11438i 0.189887 0.109631i
\(808\) 0 0
\(809\) −22.6458 + 39.2236i −0.796182 + 1.37903i 0.125904 + 0.992042i \(0.459817\pi\)
−0.922086 + 0.386986i \(0.873516\pi\)
\(810\) 0 0
\(811\) 7.93725 0.278715 0.139357 0.990242i \(-0.455496\pi\)
0.139357 + 0.990242i \(0.455496\pi\)
\(812\) 0 0
\(813\) 13.2915i 0.466153i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 29.3362 16.9373i 1.02634 0.592560i
\(818\) 0 0
\(819\) 2.64575 0.0924500
\(820\) 0 0
\(821\) −25.6974 44.5092i −0.896845 1.55338i −0.831505 0.555518i \(-0.812520\pi\)
−0.0653402 0.997863i \(-0.520813\pi\)
\(822\) 0 0
\(823\) 17.7711 + 10.2601i 0.619460 + 0.357646i 0.776659 0.629921i \(-0.216913\pi\)
−0.157198 + 0.987567i \(0.550246\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 21.4170i 0.744742i −0.928084 0.372371i \(-0.878545\pi\)
0.928084 0.372371i \(-0.121455\pi\)
\(828\) 0 0
\(829\) 22.6660 39.2587i 0.787223 1.36351i −0.140439 0.990089i \(-0.544851\pi\)
0.927662 0.373421i \(-0.121815\pi\)
\(830\) 0 0
\(831\) −12.5000 21.6506i −0.433620 0.751052i
\(832\) 0 0
\(833\) 11.5203i 0.399153i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.46410 + 2.00000i 0.119737 + 0.0691301i
\(838\) 0 0
\(839\) 23.1660 0.799780 0.399890 0.916563i \(-0.369048\pi\)
0.399890 + 0.916563i \(0.369048\pi\)
\(840\) 0 0
\(841\) 24.1660 0.833311
\(842\) 0 0
\(843\) −3.66221 2.11438i −0.126133 0.0728231i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 55.2288i 1.89768i
\(848\) 0 0
\(849\) −15.2601 26.4313i −0.523726 0.907121i
\(850\) 0 0
\(851\) 8.29150 14.3613i 0.284229 0.492299i
\(852\) 0 0
\(853\) 2.00000i 0.0684787i 0.999414 + 0.0342393i \(0.0109009\pi\)
−0.999414 + 0.0342393i \(0.989099\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13.1342 + 7.58301i 0.448654 + 0.259031i 0.707262 0.706952i \(-0.249930\pi\)
−0.258608 + 0.965982i \(0.583264\pi\)
\(858\) 0 0
\(859\) 0.583005 + 1.00979i 0.0198919 + 0.0344538i 0.875800 0.482674i \(-0.160334\pi\)
−0.855908 + 0.517128i \(0.827001\pi\)
\(860\) 0 0
\(861\) 0.468627 + 0.811686i 0.0159708 + 0.0276622i
\(862\) 0 0
\(863\) −45.1228 + 26.0516i −1.53600 + 0.886808i −0.536929 + 0.843627i \(0.680416\pi\)
−0.999067 + 0.0431804i \(0.986251\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 14.2915i 0.485365i
\(868\) 0 0
\(869\) −37.5203 −1.27279
\(870\) 0 0
\(871\) 3.96863 6.87386i 0.134472 0.232912i
\(872\) 0 0
\(873\) −12.3768 + 7.14575i −0.418891 + 0.241847i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 16.2371 9.37451i 0.548289 0.316555i −0.200143 0.979767i \(-0.564141\pi\)
0.748431 + 0.663212i \(0.230807\pi\)
\(878\) 0 0
\(879\) −10.9373 + 18.9439i −0.368904 + 0.638961i
\(880\) 0 0
\(881\) 55.2915 1.86282 0.931409 0.363974i \(-0.118580\pi\)
0.931409 + 0.363974i \(0.118580\pi\)
\(882\) 0 0
\(883\) 28.6458i 0.964006i −0.876169 0.482003i \(-0.839909\pi\)
0.876169 0.482003i \(-0.160091\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −26.5943 + 15.3542i −0.892951 + 0.515545i −0.874906 0.484292i \(-0.839077\pi\)
−0.0180441 + 0.999837i \(0.505744\pi\)
\(888\) 0 0
\(889\) 8.79150 15.2273i 0.294858 0.510708i
\(890\) 0 0
\(891\) 2.82288 + 4.88936i 0.0945699 + 0.163800i
\(892\) 0 0
\(893\) −11.8176 6.82288i −0.395460 0.228319i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.00000i 0.0667781i
\(898\) 0 0
\(899\) 14.5830 25.2585i 0.486370 0.842418i
\(900\) 0 0
\(901\) 1.93725 + 3.35542i 0.0645393 + 0.111785i
\(902\) 0 0
\(903\) 16.7069 9.64575i 0.555972 0.320991i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 51.0762 + 29.4889i 1.69596 + 0.979162i 0.949519 + 0.313710i \(0.101572\pi\)
0.746440 + 0.665453i \(0.231761\pi\)
\(908\) 0 0
\(909\) 10.9373 0.362766
\(910\) 0 0
\(911\) −52.3320 −1.73384 −0.866919 0.498450i \(-0.833903\pi\)
−0.866919 + 0.498450i \(0.833903\pi\)
\(912\) 0 0
\(913\) −17.8254 10.2915i −0.589935 0.340599i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 54.4575i 1.79835i
\(918\) 0 0
\(919\) −28.4575 49.2899i −0.938727 1.62592i −0.767850 0.640630i \(-0.778673\pi\)
−0.170877 0.985292i \(-0.554660\pi\)
\(920\) 0 0
\(921\) 14.9373 25.8721i 0.492199 0.852514i
\(922\) 0 0
\(923\) 3.29150i 0.108341i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −13.8021 7.96863i −0.453319 0.261724i
\(928\) 0 0
\(929\) 3.82288 + 6.62141i 0.125424 + 0.217242i 0.921899 0.387431i \(-0.126637\pi\)
−0.796474 + 0.604672i \(0.793304\pi\)
\(930\) 0 0
\(931\) 16.2601 28.1634i 0.532904 0.923017i
\(932\) 0 0
\(933\) −2.34563 + 1.35425i −0.0767924 + 0.0443361i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 8.70850 0.284191
\(940\) 0 0
\(941\) 24.8745 43.0839i 0.810886 1.40450i −0.101359 0.994850i \(-0.532319\pi\)
0.912245 0.409645i \(-0.134348\pi\)
\(942\) 0 0
\(943\) −0.613577 + 0.354249i −0.0199808 + 0.0115359i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 32.2954 18.6458i 1.04946 0.605906i 0.126961 0.991908i \(-0.459478\pi\)
0.922498 + 0.386002i \(0.126144\pi\)
\(948\) 0 0
\(949\) −3.14575 + 5.44860i −0.102115 + 0.176869i
\(950\) 0 0
\(951\) 12.5830 0.408032
\(952\) 0 0
\(953\) 10.4797i 0.339472i −0.985490 0.169736i \(-0.945708\pi\)
0.985490 0.169736i \(-0.0542915\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 35.6508 20.5830i 1.15243 0.665354i
\(958\) 0 0
\(959\) 12.2915 + 21.2895i 0.396913 + 0.687474i
\(960\) 0 0
\(961\) 7.50000 + 12.9904i 0.241935 + 0.419045i
\(962\) 0 0
\(963\) 8.85836 + 5.11438i 0.285457 + 0.164809i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 14.0627i 0.452227i 0.974101 + 0.226114i \(0.0726021\pi\)
−0.974101 + 0.226114i \(0.927398\pi\)
\(968\) 0 0
\(969\) −3.82288 + 6.62141i −0.122808 + 0.212710i
\(970\) 0 0
\(971\) −29.2804 50.7151i −0.939652 1.62753i −0.766120 0.642698i \(-0.777815\pi\)
−0.173532 0.984828i \(-0.555518\pi\)
\(972\) 0 0
\(973\) −0.143769 0.0830052i −0.00460903 0.00266103i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 47.3597 + 27.3431i 1.51517 + 0.874784i 0.999842 + 0.0177921i \(0.00566371\pi\)
0.515329 + 0.856992i \(0.327670\pi\)
\(978\) 0 0
\(979\) −50.4575 −1.61263
\(980\) 0 0
\(981\) 9.58301 0.305962
\(982\) 0 0
\(983\) −0.306788 0.177124i −0.00978503 0.00564939i 0.495100 0.868836i \(-0.335132\pi\)
−0.504885 + 0.863187i \(0.668465\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −6.73009 3.88562i −0.214221 0.123681i
\(988\) 0 0
\(989\) 7.29150 + 12.6293i 0.231856 + 0.401587i
\(990\) 0 0
\(991\) −3.93725 + 6.81952i −0.125071 + 0.216629i −0.921761 0.387759i \(-0.873249\pi\)
0.796690 + 0.604389i \(0.206583\pi\)
\(992\) 0 0
\(993\) 8.64575i 0.274365i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 12.7730 + 7.37451i 0.404526 + 0.233553i 0.688435 0.725298i \(-0.258298\pi\)
−0.283909 + 0.958851i \(0.591631\pi\)
\(998\) 0 0
\(999\) −4.14575 7.18065i −0.131166 0.227186i
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 2100.2.bc.h.949.2 8
5.2 odd 4 2100.2.q.g.1201.1 4
5.3 odd 4 2100.2.q.i.1201.2 yes 4
5.4 even 2 inner 2100.2.bc.h.949.3 8
7.2 even 3 inner 2100.2.bc.h.1549.3 8
35.2 odd 12 2100.2.q.g.1801.1 yes 4
35.9 even 6 inner 2100.2.bc.h.1549.2 8
35.23 odd 12 2100.2.q.i.1801.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.q.g.1201.1 4 5.2 odd 4
2100.2.q.g.1801.1 yes 4 35.2 odd 12
2100.2.q.i.1201.2 yes 4 5.3 odd 4
2100.2.q.i.1801.2 yes 4 35.23 odd 12
2100.2.bc.h.949.2 8 1.1 even 1 trivial
2100.2.bc.h.949.3 8 5.4 even 2 inner
2100.2.bc.h.1549.2 8 35.9 even 6 inner
2100.2.bc.h.1549.3 8 7.2 even 3 inner