Properties

Label 2100.2.d.g.1301.1
Level $2100$
Weight $2$
Character 2100.1301
Analytic conductor $16.769$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(1301,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1301");
 
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.d (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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1301.1
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1301
Dual form 2100.2.d.g.1301.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.61803 - 0.618034i) q^{3} +(0.381966 - 2.61803i) q^{7} +(2.23607 + 2.00000i) q^{9} +O(q^{10})\) \(q+(-1.61803 - 0.618034i) q^{3} +(0.381966 - 2.61803i) q^{7} +(2.23607 + 2.00000i) q^{9} +0.763932i q^{11} -1.23607i q^{13} -4.47214 q^{17} +7.23607i q^{19} +(-2.23607 + 4.00000i) q^{21} +7.70820i q^{23} +(-2.38197 - 4.61803i) q^{27} -4.00000i q^{29} -3.23607i q^{31} +(0.472136 - 1.23607i) q^{33} -4.47214 q^{37} +(-0.763932 + 2.00000i) q^{39} -3.52786 q^{41} +7.23607 q^{43} +3.23607 q^{47} +(-6.70820 - 2.00000i) q^{49} +(7.23607 + 2.76393i) q^{51} +9.23607i q^{53} +(4.47214 - 11.7082i) q^{57} +8.94427 q^{59} -4.94427i q^{61} +(6.09017 - 5.09017i) q^{63} -9.70820 q^{67} +(4.76393 - 12.4721i) q^{69} +12.1803i q^{71} +6.76393i q^{73} +(2.00000 + 0.291796i) q^{77} +10.4721 q^{79} +(1.00000 + 8.94427i) q^{81} +7.23607 q^{83} +(-2.47214 + 6.47214i) q^{87} -12.4721 q^{89} +(-3.23607 - 0.472136i) q^{91} +(-2.00000 + 5.23607i) q^{93} +14.1803i q^{97} +(-1.52786 + 1.70820i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} + 6 q^{7} - 14 q^{27} - 16 q^{33} - 12 q^{39} - 32 q^{41} + 20 q^{43} + 4 q^{47} + 20 q^{51} + 2 q^{63} - 12 q^{67} + 28 q^{69} + 8 q^{77} + 24 q^{79} + 4 q^{81} + 20 q^{83} + 8 q^{87} - 32 q^{89} - 4 q^{91} - 8 q^{93} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(3\) −1.61803 0.618034i −0.934172 0.356822i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.381966 2.61803i 0.144370 0.989524i
\(8\) 0 0
\(9\) 2.23607 + 2.00000i 0.745356 + 0.666667i
\(10\) 0 0
\(11\) 0.763932i 0.230334i 0.993346 + 0.115167i \(0.0367403\pi\)
−0.993346 + 0.115167i \(0.963260\pi\)
\(12\) 0 0
\(13\) 1.23607i 0.342824i −0.985199 0.171412i \(-0.945167\pi\)
0.985199 0.171412i \(-0.0548329\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) 7.23607i 1.66007i 0.557713 + 0.830034i \(0.311679\pi\)
−0.557713 + 0.830034i \(0.688321\pi\)
\(20\) 0 0
\(21\) −2.23607 + 4.00000i −0.487950 + 0.872872i
\(22\) 0 0
\(23\) 7.70820i 1.60727i 0.595121 + 0.803636i \(0.297104\pi\)
−0.595121 + 0.803636i \(0.702896\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2.38197 4.61803i −0.458410 0.888741i
\(28\) 0 0
\(29\) 4.00000i 0.742781i −0.928477 0.371391i \(-0.878881\pi\)
0.928477 0.371391i \(-0.121119\pi\)
\(30\) 0 0
\(31\) 3.23607i 0.581215i −0.956842 0.290607i \(-0.906143\pi\)
0.956842 0.290607i \(-0.0938574\pi\)
\(32\) 0 0
\(33\) 0.472136 1.23607i 0.0821883 0.215172i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.47214 −0.735215 −0.367607 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) 0 0
\(39\) −0.763932 + 2.00000i −0.122327 + 0.320256i
\(40\) 0 0
\(41\) −3.52786 −0.550960 −0.275480 0.961307i \(-0.588837\pi\)
−0.275480 + 0.961307i \(0.588837\pi\)
\(42\) 0 0
\(43\) 7.23607 1.10349 0.551745 0.834013i \(-0.313962\pi\)
0.551745 + 0.834013i \(0.313962\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.23607 0.472029 0.236015 0.971750i \(-0.424159\pi\)
0.236015 + 0.971750i \(0.424159\pi\)
\(48\) 0 0
\(49\) −6.70820 2.00000i −0.958315 0.285714i
\(50\) 0 0
\(51\) 7.23607 + 2.76393i 1.01325 + 0.387028i
\(52\) 0 0
\(53\) 9.23607i 1.26867i 0.773058 + 0.634336i \(0.218726\pi\)
−0.773058 + 0.634336i \(0.781274\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 4.47214 11.7082i 0.592349 1.55079i
\(58\) 0 0
\(59\) 8.94427 1.16445 0.582223 0.813029i \(-0.302183\pi\)
0.582223 + 0.813029i \(0.302183\pi\)
\(60\) 0 0
\(61\) 4.94427i 0.633049i −0.948584 0.316525i \(-0.897484\pi\)
0.948584 0.316525i \(-0.102516\pi\)
\(62\) 0 0
\(63\) 6.09017 5.09017i 0.767289 0.641301i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −9.70820 −1.18605 −0.593023 0.805186i \(-0.702066\pi\)
−0.593023 + 0.805186i \(0.702066\pi\)
\(68\) 0 0
\(69\) 4.76393 12.4721i 0.573510 1.50147i
\(70\) 0 0
\(71\) 12.1803i 1.44554i 0.691088 + 0.722770i \(0.257131\pi\)
−0.691088 + 0.722770i \(0.742869\pi\)
\(72\) 0 0
\(73\) 6.76393i 0.791658i 0.918324 + 0.395829i \(0.129543\pi\)
−0.918324 + 0.395829i \(0.870457\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000 + 0.291796i 0.227921 + 0.0332532i
\(78\) 0 0
\(79\) 10.4721 1.17821 0.589104 0.808057i \(-0.299481\pi\)
0.589104 + 0.808057i \(0.299481\pi\)
\(80\) 0 0
\(81\) 1.00000 + 8.94427i 0.111111 + 0.993808i
\(82\) 0 0
\(83\) 7.23607 0.794262 0.397131 0.917762i \(-0.370006\pi\)
0.397131 + 0.917762i \(0.370006\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −2.47214 + 6.47214i −0.265041 + 0.693886i
\(88\) 0 0
\(89\) −12.4721 −1.32204 −0.661022 0.750367i \(-0.729877\pi\)
−0.661022 + 0.750367i \(0.729877\pi\)
\(90\) 0 0
\(91\) −3.23607 0.472136i −0.339232 0.0494933i
\(92\) 0 0
\(93\) −2.00000 + 5.23607i −0.207390 + 0.542955i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.1803i 1.43980i 0.694080 + 0.719898i \(0.255811\pi\)
−0.694080 + 0.719898i \(0.744189\pi\)
\(98\) 0 0
\(99\) −1.52786 + 1.70820i −0.153556 + 0.171681i
\(100\) 0 0
\(101\) −6.94427 −0.690981 −0.345490 0.938422i \(-0.612287\pi\)
−0.345490 + 0.938422i \(0.612287\pi\)
\(102\) 0 0
\(103\) 10.7639i 1.06060i 0.847810 + 0.530301i \(0.177921\pi\)
−0.847810 + 0.530301i \(0.822079\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.291796i 0.0282090i −0.999901 0.0141045i \(-0.995510\pi\)
0.999901 0.0141045i \(-0.00448975\pi\)
\(108\) 0 0
\(109\) −2.94427 −0.282010 −0.141005 0.990009i \(-0.545033\pi\)
−0.141005 + 0.990009i \(0.545033\pi\)
\(110\) 0 0
\(111\) 7.23607 + 2.76393i 0.686817 + 0.262341i
\(112\) 0 0
\(113\) 17.2361i 1.62143i 0.585439 + 0.810716i \(0.300922\pi\)
−0.585439 + 0.810716i \(0.699078\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.47214 2.76393i 0.228549 0.255526i
\(118\) 0 0
\(119\) −1.70820 + 11.7082i −0.156591 + 1.07329i
\(120\) 0 0
\(121\) 10.4164 0.946946
\(122\) 0 0
\(123\) 5.70820 + 2.18034i 0.514691 + 0.196595i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 21.1246 1.87451 0.937253 0.348650i \(-0.113360\pi\)
0.937253 + 0.348650i \(0.113360\pi\)
\(128\) 0 0
\(129\) −11.7082 4.47214i −1.03085 0.393750i
\(130\) 0 0
\(131\) 15.4164 1.34694 0.673469 0.739216i \(-0.264804\pi\)
0.673469 + 0.739216i \(0.264804\pi\)
\(132\) 0 0
\(133\) 18.9443 + 2.76393i 1.64268 + 0.239663i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.23607i 0.105604i −0.998605 0.0528022i \(-0.983185\pi\)
0.998605 0.0528022i \(-0.0168153\pi\)
\(138\) 0 0
\(139\) 15.2361i 1.29231i 0.763208 + 0.646153i \(0.223623\pi\)
−0.763208 + 0.646153i \(0.776377\pi\)
\(140\) 0 0
\(141\) −5.23607 2.00000i −0.440956 0.168430i
\(142\) 0 0
\(143\) 0.944272 0.0789640
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 9.61803 + 7.38197i 0.793282 + 0.608854i
\(148\) 0 0
\(149\) 15.4164i 1.26296i −0.775392 0.631481i \(-0.782448\pi\)
0.775392 0.631481i \(-0.217552\pi\)
\(150\) 0 0
\(151\) 8.94427 0.727875 0.363937 0.931423i \(-0.381432\pi\)
0.363937 + 0.931423i \(0.381432\pi\)
\(152\) 0 0
\(153\) −10.0000 8.94427i −0.808452 0.723102i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.81966i 0.145225i −0.997360 0.0726123i \(-0.976866\pi\)
0.997360 0.0726123i \(-0.0231336\pi\)
\(158\) 0 0
\(159\) 5.70820 14.9443i 0.452690 1.18516i
\(160\) 0 0
\(161\) 20.1803 + 2.94427i 1.59043 + 0.232041i
\(162\) 0 0
\(163\) −18.6525 −1.46097 −0.730487 0.682926i \(-0.760707\pi\)
−0.730487 + 0.682926i \(0.760707\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.76393 0.678173 0.339087 0.940755i \(-0.389882\pi\)
0.339087 + 0.940755i \(0.389882\pi\)
\(168\) 0 0
\(169\) 11.4721 0.882472
\(170\) 0 0
\(171\) −14.4721 + 16.1803i −1.10671 + 1.23734i
\(172\) 0 0
\(173\) −2.00000 −0.152057 −0.0760286 0.997106i \(-0.524224\pi\)
−0.0760286 + 0.997106i \(0.524224\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −14.4721 5.52786i −1.08779 0.415500i
\(178\) 0 0
\(179\) 12.7639i 0.954021i 0.878898 + 0.477011i \(0.158280\pi\)
−0.878898 + 0.477011i \(0.841720\pi\)
\(180\) 0 0
\(181\) 6.47214i 0.481070i 0.970640 + 0.240535i \(0.0773229\pi\)
−0.970640 + 0.240535i \(0.922677\pi\)
\(182\) 0 0
\(183\) −3.05573 + 8.00000i −0.225886 + 0.591377i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.41641i 0.249832i
\(188\) 0 0
\(189\) −13.0000 + 4.47214i −0.945611 + 0.325300i
\(190\) 0 0
\(191\) 7.23607i 0.523584i 0.965124 + 0.261792i \(0.0843134\pi\)
−0.965124 + 0.261792i \(0.915687\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 25.2361i 1.79800i −0.437953 0.898998i \(-0.644296\pi\)
0.437953 0.898998i \(-0.355704\pi\)
\(198\) 0 0
\(199\) 8.76393i 0.621259i 0.950531 + 0.310629i \(0.100540\pi\)
−0.950531 + 0.310629i \(0.899460\pi\)
\(200\) 0 0
\(201\) 15.7082 + 6.00000i 1.10797 + 0.423207i
\(202\) 0 0
\(203\) −10.4721 1.52786i −0.735000 0.107235i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −15.4164 + 17.2361i −1.07151 + 1.19799i
\(208\) 0 0
\(209\) −5.52786 −0.382370
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) 7.52786 19.7082i 0.515801 1.35038i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.47214 1.23607i −0.575126 0.0839098i
\(218\) 0 0
\(219\) 4.18034 10.9443i 0.282481 0.739545i
\(220\) 0 0
\(221\) 5.52786i 0.371844i
\(222\) 0 0
\(223\) 11.7082i 0.784039i 0.919957 + 0.392020i \(0.128223\pi\)
−0.919957 + 0.392020i \(0.871777\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −23.2361 −1.54223 −0.771116 0.636695i \(-0.780301\pi\)
−0.771116 + 0.636695i \(0.780301\pi\)
\(228\) 0 0
\(229\) 13.8885i 0.917781i 0.888493 + 0.458890i \(0.151753\pi\)
−0.888493 + 0.458890i \(0.848247\pi\)
\(230\) 0 0
\(231\) −3.05573 1.70820i −0.201052 0.112392i
\(232\) 0 0
\(233\) 1.81966i 0.119210i −0.998222 0.0596049i \(-0.981016\pi\)
0.998222 0.0596049i \(-0.0189841\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −16.9443 6.47214i −1.10065 0.420410i
\(238\) 0 0
\(239\) 0.180340i 0.0116652i 0.999983 + 0.00583261i \(0.00185659\pi\)
−0.999983 + 0.00583261i \(0.998143\pi\)
\(240\) 0 0
\(241\) 16.3607i 1.05388i 0.849901 + 0.526942i \(0.176662\pi\)
−0.849901 + 0.526942i \(0.823338\pi\)
\(242\) 0 0
\(243\) 3.90983 15.0902i 0.250816 0.968035i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.94427 0.569110
\(248\) 0 0
\(249\) −11.7082 4.47214i −0.741977 0.283410i
\(250\) 0 0
\(251\) −20.3607 −1.28515 −0.642577 0.766221i \(-0.722135\pi\)
−0.642577 + 0.766221i \(0.722135\pi\)
\(252\) 0 0
\(253\) −5.88854 −0.370210
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.4721 1.02750 0.513752 0.857939i \(-0.328255\pi\)
0.513752 + 0.857939i \(0.328255\pi\)
\(258\) 0 0
\(259\) −1.70820 + 11.7082i −0.106143 + 0.727512i
\(260\) 0 0
\(261\) 8.00000 8.94427i 0.495188 0.553637i
\(262\) 0 0
\(263\) 22.1803i 1.36770i −0.729623 0.683849i \(-0.760305\pi\)
0.729623 0.683849i \(-0.239695\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 20.1803 + 7.70820i 1.23502 + 0.471734i
\(268\) 0 0
\(269\) 19.5279 1.19063 0.595317 0.803491i \(-0.297026\pi\)
0.595317 + 0.803491i \(0.297026\pi\)
\(270\) 0 0
\(271\) 21.1246i 1.28323i −0.767027 0.641614i \(-0.778265\pi\)
0.767027 0.641614i \(-0.221735\pi\)
\(272\) 0 0
\(273\) 4.94427 + 2.76393i 0.299241 + 0.167281i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.41641 −0.0851037 −0.0425519 0.999094i \(-0.513549\pi\)
−0.0425519 + 0.999094i \(0.513549\pi\)
\(278\) 0 0
\(279\) 6.47214 7.23607i 0.387477 0.433212i
\(280\) 0 0
\(281\) 11.4164i 0.681046i −0.940236 0.340523i \(-0.889396\pi\)
0.940236 0.340523i \(-0.110604\pi\)
\(282\) 0 0
\(283\) 8.65248i 0.514336i 0.966367 + 0.257168i \(0.0827894\pi\)
−0.966367 + 0.257168i \(0.917211\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.34752 + 9.23607i −0.0795418 + 0.545188i
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 8.76393 22.9443i 0.513751 1.34502i
\(292\) 0 0
\(293\) 14.0000 0.817889 0.408944 0.912559i \(-0.365897\pi\)
0.408944 + 0.912559i \(0.365897\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 3.52786 1.81966i 0.204707 0.105587i
\(298\) 0 0
\(299\) 9.52786 0.551011
\(300\) 0 0
\(301\) 2.76393 18.9443i 0.159310 1.09193i
\(302\) 0 0
\(303\) 11.2361 + 4.29180i 0.645495 + 0.246557i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 18.1803i 1.03761i −0.854894 0.518803i \(-0.826378\pi\)
0.854894 0.518803i \(-0.173622\pi\)
\(308\) 0 0
\(309\) 6.65248 17.4164i 0.378446 0.990785i
\(310\) 0 0
\(311\) −32.3607 −1.83501 −0.917503 0.397729i \(-0.869798\pi\)
−0.917503 + 0.397729i \(0.869798\pi\)
\(312\) 0 0
\(313\) 4.29180i 0.242587i −0.992617 0.121293i \(-0.961296\pi\)
0.992617 0.121293i \(-0.0387042\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.18034i 0.347122i −0.984823 0.173561i \(-0.944473\pi\)
0.984823 0.173561i \(-0.0555275\pi\)
\(318\) 0 0
\(319\) 3.05573 0.171088
\(320\) 0 0
\(321\) −0.180340 + 0.472136i −0.0100656 + 0.0263521i
\(322\) 0 0
\(323\) 32.3607i 1.80060i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4.76393 + 1.81966i 0.263446 + 0.100627i
\(328\) 0 0
\(329\) 1.23607 8.47214i 0.0681466 0.467084i
\(330\) 0 0
\(331\) −20.9443 −1.15120 −0.575601 0.817731i \(-0.695232\pi\)
−0.575601 + 0.817731i \(0.695232\pi\)
\(332\) 0 0
\(333\) −10.0000 8.94427i −0.547997 0.490143i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) 10.6525 27.8885i 0.578563 1.51470i
\(340\) 0 0
\(341\) 2.47214 0.133874
\(342\) 0 0
\(343\) −7.79837 + 16.7984i −0.421073 + 0.907027i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 32.6525i 1.75288i 0.481514 + 0.876438i \(0.340087\pi\)
−0.481514 + 0.876438i \(0.659913\pi\)
\(348\) 0 0
\(349\) 8.94427i 0.478776i 0.970924 + 0.239388i \(0.0769468\pi\)
−0.970924 + 0.239388i \(0.923053\pi\)
\(350\) 0 0
\(351\) −5.70820 + 2.94427i −0.304681 + 0.157154i
\(352\) 0 0
\(353\) −25.4164 −1.35278 −0.676389 0.736544i \(-0.736456\pi\)
−0.676389 + 0.736544i \(0.736456\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 10.0000 17.8885i 0.529256 0.946762i
\(358\) 0 0
\(359\) 34.0689i 1.79809i 0.437859 + 0.899043i \(0.355737\pi\)
−0.437859 + 0.899043i \(0.644263\pi\)
\(360\) 0 0
\(361\) −33.3607 −1.75583
\(362\) 0 0
\(363\) −16.8541 6.43769i −0.884611 0.337891i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 24.6525i 1.28685i 0.765510 + 0.643424i \(0.222487\pi\)
−0.765510 + 0.643424i \(0.777513\pi\)
\(368\) 0 0
\(369\) −7.88854 7.05573i −0.410661 0.367307i
\(370\) 0 0
\(371\) 24.1803 + 3.52786i 1.25538 + 0.183158i
\(372\) 0 0
\(373\) −7.52786 −0.389778 −0.194889 0.980825i \(-0.562435\pi\)
−0.194889 + 0.980825i \(0.562435\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.94427 −0.254643
\(378\) 0 0
\(379\) 1.52786 0.0784811 0.0392406 0.999230i \(-0.487506\pi\)
0.0392406 + 0.999230i \(0.487506\pi\)
\(380\) 0 0
\(381\) −34.1803 13.0557i −1.75111 0.668865i
\(382\) 0 0
\(383\) 15.2361 0.778527 0.389263 0.921127i \(-0.372730\pi\)
0.389263 + 0.921127i \(0.372730\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 16.1803 + 14.4721i 0.822493 + 0.735660i
\(388\) 0 0
\(389\) 2.47214i 0.125342i 0.998034 + 0.0626711i \(0.0199619\pi\)
−0.998034 + 0.0626711i \(0.980038\pi\)
\(390\) 0 0
\(391\) 34.4721i 1.74333i
\(392\) 0 0
\(393\) −24.9443 9.52786i −1.25827 0.480617i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.70820i 0.186109i −0.995661 0.0930547i \(-0.970337\pi\)
0.995661 0.0930547i \(-0.0296631\pi\)
\(398\) 0 0
\(399\) −28.9443 16.1803i −1.44903 0.810030i
\(400\) 0 0
\(401\) 28.9443i 1.44541i −0.691158 0.722704i \(-0.742899\pi\)
0.691158 0.722704i \(-0.257101\pi\)
\(402\) 0 0
\(403\) −4.00000 −0.199254
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3.41641i 0.169345i
\(408\) 0 0
\(409\) 12.0000i 0.593362i 0.954977 + 0.296681i \(0.0958798\pi\)
−0.954977 + 0.296681i \(0.904120\pi\)
\(410\) 0 0
\(411\) −0.763932 + 2.00000i −0.0376820 + 0.0986527i
\(412\) 0 0
\(413\) 3.41641 23.4164i 0.168110 1.15225i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 9.41641 24.6525i 0.461123 1.20724i
\(418\) 0 0
\(419\) −28.0000 −1.36789 −0.683945 0.729534i \(-0.739737\pi\)
−0.683945 + 0.729534i \(0.739737\pi\)
\(420\) 0 0
\(421\) 32.4721 1.58260 0.791298 0.611431i \(-0.209406\pi\)
0.791298 + 0.611431i \(0.209406\pi\)
\(422\) 0 0
\(423\) 7.23607 + 6.47214i 0.351830 + 0.314686i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −12.9443 1.88854i −0.626417 0.0913930i
\(428\) 0 0
\(429\) −1.52786 0.583592i −0.0737660 0.0281761i
\(430\) 0 0
\(431\) 0.763932i 0.0367973i −0.999831 0.0183987i \(-0.994143\pi\)
0.999831 0.0183987i \(-0.00585680\pi\)
\(432\) 0 0
\(433\) 32.6525i 1.56918i 0.620016 + 0.784589i \(0.287126\pi\)
−0.620016 + 0.784589i \(0.712874\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −55.7771 −2.66818
\(438\) 0 0
\(439\) 4.18034i 0.199517i −0.995012 0.0997584i \(-0.968193\pi\)
0.995012 0.0997584i \(-0.0318070\pi\)
\(440\) 0 0
\(441\) −11.0000 17.8885i −0.523810 0.851835i
\(442\) 0 0
\(443\) 24.0689i 1.14355i −0.820411 0.571774i \(-0.806256\pi\)
0.820411 0.571774i \(-0.193744\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −9.52786 + 24.9443i −0.450653 + 1.17982i
\(448\) 0 0
\(449\) 29.8885i 1.41053i −0.708945 0.705264i \(-0.750829\pi\)
0.708945 0.705264i \(-0.249171\pi\)
\(450\) 0 0
\(451\) 2.69505i 0.126905i
\(452\) 0 0
\(453\) −14.4721 5.52786i −0.679960 0.259722i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9.05573 0.423609 0.211805 0.977312i \(-0.432066\pi\)
0.211805 + 0.977312i \(0.432066\pi\)
\(458\) 0 0
\(459\) 10.6525 + 20.6525i 0.497215 + 0.963975i
\(460\) 0 0
\(461\) −24.4721 −1.13978 −0.569891 0.821721i \(-0.693014\pi\)
−0.569891 + 0.821721i \(0.693014\pi\)
\(462\) 0 0
\(463\) 20.7639 0.964982 0.482491 0.875901i \(-0.339732\pi\)
0.482491 + 0.875901i \(0.339732\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 36.5410 1.69092 0.845458 0.534041i \(-0.179327\pi\)
0.845458 + 0.534041i \(0.179327\pi\)
\(468\) 0 0
\(469\) −3.70820 + 25.4164i −0.171229 + 1.17362i
\(470\) 0 0
\(471\) −1.12461 + 2.94427i −0.0518194 + 0.135665i
\(472\) 0 0
\(473\) 5.52786i 0.254171i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −18.4721 + 20.6525i −0.845781 + 0.945612i
\(478\) 0 0
\(479\) 12.9443 0.591439 0.295719 0.955275i \(-0.404441\pi\)
0.295719 + 0.955275i \(0.404441\pi\)
\(480\) 0 0
\(481\) 5.52786i 0.252049i
\(482\) 0 0
\(483\) −30.8328 17.2361i −1.40294 0.784268i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3.23607 0.146640 0.0733201 0.997308i \(-0.476641\pi\)
0.0733201 + 0.997308i \(0.476641\pi\)
\(488\) 0 0
\(489\) 30.1803 + 11.5279i 1.36480 + 0.521308i
\(490\) 0 0
\(491\) 41.1246i 1.85593i −0.372670 0.927964i \(-0.621558\pi\)
0.372670 0.927964i \(-0.378442\pi\)
\(492\) 0 0
\(493\) 17.8885i 0.805659i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 31.8885 + 4.65248i 1.43040 + 0.208692i
\(498\) 0 0
\(499\) −16.3607 −0.732405 −0.366202 0.930535i \(-0.619342\pi\)
−0.366202 + 0.930535i \(0.619342\pi\)
\(500\) 0 0
\(501\) −14.1803 5.41641i −0.633531 0.241987i
\(502\) 0 0
\(503\) −18.6525 −0.831673 −0.415836 0.909439i \(-0.636511\pi\)
−0.415836 + 0.909439i \(0.636511\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −18.5623 7.09017i −0.824381 0.314886i
\(508\) 0 0
\(509\) −22.9443 −1.01699 −0.508493 0.861066i \(-0.669797\pi\)
−0.508493 + 0.861066i \(0.669797\pi\)
\(510\) 0 0
\(511\) 17.7082 + 2.58359i 0.783365 + 0.114291i
\(512\) 0 0
\(513\) 33.4164 17.2361i 1.47537 0.760991i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2.47214i 0.108724i
\(518\) 0 0
\(519\) 3.23607 + 1.23607i 0.142048 + 0.0542574i
\(520\) 0 0
\(521\) −7.88854 −0.345603 −0.172802 0.984957i \(-0.555282\pi\)
−0.172802 + 0.984957i \(0.555282\pi\)
\(522\) 0 0
\(523\) 8.29180i 0.362575i −0.983430 0.181287i \(-0.941974\pi\)
0.983430 0.181287i \(-0.0580264\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 14.4721i 0.630416i
\(528\) 0 0
\(529\) −36.4164 −1.58332
\(530\) 0 0
\(531\) 20.0000 + 17.8885i 0.867926 + 0.776297i
\(532\) 0 0
\(533\) 4.36068i 0.188882i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 7.88854 20.6525i 0.340416 0.891220i
\(538\) 0 0
\(539\) 1.52786 5.12461i 0.0658098 0.220733i
\(540\) 0 0
\(541\) −43.8885 −1.88692 −0.943458 0.331492i \(-0.892448\pi\)
−0.943458 + 0.331492i \(0.892448\pi\)
\(542\) 0 0
\(543\) 4.00000 10.4721i 0.171656 0.449402i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −17.7082 −0.757148 −0.378574 0.925571i \(-0.623585\pi\)
−0.378574 + 0.925571i \(0.623585\pi\)
\(548\) 0 0
\(549\) 9.88854 11.0557i 0.422033 0.471847i
\(550\) 0 0
\(551\) 28.9443 1.23307
\(552\) 0 0
\(553\) 4.00000 27.4164i 0.170097 1.16586i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 14.1803i 0.600840i −0.953807 0.300420i \(-0.902873\pi\)
0.953807 0.300420i \(-0.0971269\pi\)
\(558\) 0 0
\(559\) 8.94427i 0.378302i
\(560\) 0 0
\(561\) −2.11146 + 5.52786i −0.0891457 + 0.233387i
\(562\) 0 0
\(563\) −18.0689 −0.761513 −0.380756 0.924675i \(-0.624336\pi\)
−0.380756 + 0.924675i \(0.624336\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 23.7984 + 0.798374i 0.999438 + 0.0335286i
\(568\) 0 0
\(569\) 21.8885i 0.917615i −0.888536 0.458808i \(-0.848277\pi\)
0.888536 0.458808i \(-0.151723\pi\)
\(570\) 0 0
\(571\) 6.83282 0.285944 0.142972 0.989727i \(-0.454334\pi\)
0.142972 + 0.989727i \(0.454334\pi\)
\(572\) 0 0
\(573\) 4.47214 11.7082i 0.186826 0.489117i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 32.0689i 1.33505i 0.744590 + 0.667523i \(0.232645\pi\)
−0.744590 + 0.667523i \(0.767355\pi\)
\(578\) 0 0
\(579\) −9.70820 3.70820i −0.403459 0.154108i
\(580\) 0 0
\(581\) 2.76393 18.9443i 0.114667 0.785941i
\(582\) 0 0
\(583\) −7.05573 −0.292218
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.29180 0.259690 0.129845 0.991534i \(-0.458552\pi\)
0.129845 + 0.991534i \(0.458552\pi\)
\(588\) 0 0
\(589\) 23.4164 0.964856
\(590\) 0 0
\(591\) −15.5967 + 40.8328i −0.641564 + 1.67964i
\(592\) 0 0
\(593\) 19.5279 0.801913 0.400957 0.916097i \(-0.368678\pi\)
0.400957 + 0.916097i \(0.368678\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.41641 14.1803i 0.221679 0.580363i
\(598\) 0 0
\(599\) 30.6525i 1.25243i −0.779652 0.626213i \(-0.784604\pi\)
0.779652 0.626213i \(-0.215396\pi\)
\(600\) 0 0
\(601\) 45.3050i 1.84803i 0.382359 + 0.924014i \(0.375112\pi\)
−0.382359 + 0.924014i \(0.624888\pi\)
\(602\) 0 0
\(603\) −21.7082 19.4164i −0.884026 0.790697i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 40.0689i 1.62635i −0.582022 0.813173i \(-0.697738\pi\)
0.582022 0.813173i \(-0.302262\pi\)
\(608\) 0 0
\(609\) 16.0000 + 8.94427i 0.648353 + 0.362440i
\(610\) 0 0
\(611\) 4.00000i 0.161823i
\(612\) 0 0
\(613\) −22.3607 −0.903139 −0.451570 0.892236i \(-0.649136\pi\)
−0.451570 + 0.892236i \(0.649136\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.5410i 1.06850i 0.845326 + 0.534251i \(0.179406\pi\)
−0.845326 + 0.534251i \(0.820594\pi\)
\(618\) 0 0
\(619\) 5.70820i 0.229432i −0.993398 0.114716i \(-0.963404\pi\)
0.993398 0.114716i \(-0.0365958\pi\)
\(620\) 0 0
\(621\) 35.5967 18.3607i 1.42845 0.736789i
\(622\) 0 0
\(623\) −4.76393 + 32.6525i −0.190863 + 1.30819i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 8.94427 + 3.41641i 0.357200 + 0.136438i
\(628\) 0 0
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) 8.94427 0.356066 0.178033 0.984025i \(-0.443027\pi\)
0.178033 + 0.984025i \(0.443027\pi\)
\(632\) 0 0
\(633\) 12.9443 + 4.94427i 0.514489 + 0.196517i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −2.47214 + 8.29180i −0.0979496 + 0.328533i
\(638\) 0 0
\(639\) −24.3607 + 27.2361i −0.963694 + 1.07744i
\(640\) 0 0
\(641\) 12.5836i 0.497022i −0.968629 0.248511i \(-0.920059\pi\)
0.968629 0.248511i \(-0.0799412\pi\)
\(642\) 0 0
\(643\) 24.2918i 0.957975i −0.877822 0.478987i \(-0.841004\pi\)
0.877822 0.478987i \(-0.158996\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.12461 −0.358726 −0.179363 0.983783i \(-0.557404\pi\)
−0.179363 + 0.983783i \(0.557404\pi\)
\(648\) 0 0
\(649\) 6.83282i 0.268211i
\(650\) 0 0
\(651\) 12.9443 + 7.23607i 0.507326 + 0.283604i
\(652\) 0 0
\(653\) 21.5967i 0.845146i −0.906329 0.422573i \(-0.861127\pi\)
0.906329 0.422573i \(-0.138873\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −13.5279 + 15.1246i −0.527772 + 0.590067i
\(658\) 0 0
\(659\) 36.7639i 1.43212i 0.698039 + 0.716060i \(0.254056\pi\)
−0.698039 + 0.716060i \(0.745944\pi\)
\(660\) 0 0
\(661\) 22.8328i 0.888094i 0.896004 + 0.444047i \(0.146458\pi\)
−0.896004 + 0.444047i \(0.853542\pi\)
\(662\) 0 0
\(663\) 3.41641 8.94427i 0.132682 0.347367i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 30.8328 1.19385
\(668\) 0 0
\(669\) 7.23607 18.9443i 0.279763 0.732428i
\(670\) 0 0
\(671\) 3.77709 0.145813
\(672\) 0 0
\(673\) −14.9443 −0.576059 −0.288030 0.957621i \(-0.593000\pi\)
−0.288030 + 0.957621i \(0.593000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) 0 0
\(679\) 37.1246 + 5.41641i 1.42471 + 0.207863i
\(680\) 0 0
\(681\) 37.5967 + 14.3607i 1.44071 + 0.550302i
\(682\) 0 0
\(683\) 20.2918i 0.776444i −0.921566 0.388222i \(-0.873089\pi\)
0.921566 0.388222i \(-0.126911\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 8.58359 22.4721i 0.327484 0.857365i
\(688\) 0 0
\(689\) 11.4164 0.434931
\(690\) 0 0
\(691\) 8.18034i 0.311195i 0.987821 + 0.155597i \(0.0497302\pi\)
−0.987821 + 0.155597i \(0.950270\pi\)
\(692\) 0 0
\(693\) 3.88854 + 4.65248i 0.147714 + 0.176733i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 15.7771 0.597600
\(698\) 0 0
\(699\) −1.12461 + 2.94427i −0.0425367 + 0.111363i
\(700\) 0 0
\(701\) 3.05573i 0.115413i 0.998334 + 0.0577066i \(0.0183788\pi\)
−0.998334 + 0.0577066i \(0.981621\pi\)
\(702\) 0 0
\(703\) 32.3607i 1.22051i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.65248 + 18.1803i −0.0997566 + 0.683742i
\(708\) 0 0
\(709\) −22.9443 −0.861690 −0.430845 0.902426i \(-0.641784\pi\)
−0.430845 + 0.902426i \(0.641784\pi\)
\(710\) 0 0
\(711\) 23.4164 + 20.9443i 0.878184 + 0.785472i
\(712\) 0 0
\(713\) 24.9443 0.934170
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0.111456 0.291796i 0.00416241 0.0108973i
\(718\) 0 0
\(719\) −6.11146 −0.227919 −0.113959 0.993485i \(-0.536353\pi\)
−0.113959 + 0.993485i \(0.536353\pi\)
\(720\) 0 0
\(721\) 28.1803 + 4.11146i 1.04949 + 0.153119i
\(722\) 0 0
\(723\) 10.1115 26.4721i 0.376049 0.984509i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.70820i 0.137530i 0.997633 + 0.0687648i \(0.0219058\pi\)
−0.997633 + 0.0687648i \(0.978094\pi\)
\(728\) 0 0
\(729\) −15.6525 + 22.0000i −0.579721 + 0.814815i
\(730\) 0 0
\(731\) −32.3607 −1.19690
\(732\) 0 0
\(733\) 48.6525i 1.79702i 0.438953 + 0.898510i \(0.355350\pi\)
−0.438953 + 0.898510i \(0.644650\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.41641i 0.273187i
\(738\) 0 0
\(739\) 40.3607 1.48469 0.742346 0.670017i \(-0.233713\pi\)
0.742346 + 0.670017i \(0.233713\pi\)
\(740\) 0 0
\(741\) −14.4721 5.52786i −0.531647 0.203071i
\(742\) 0 0
\(743\) 24.6525i 0.904412i 0.891914 + 0.452206i \(0.149363\pi\)
−0.891914 + 0.452206i \(0.850637\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 16.1803 + 14.4721i 0.592008 + 0.529508i
\(748\) 0 0
\(749\) −0.763932 0.111456i −0.0279135 0.00407252i
\(750\) 0 0
\(751\) −44.0000 −1.60558 −0.802791 0.596260i \(-0.796653\pi\)
−0.802791 + 0.596260i \(0.796653\pi\)
\(752\) 0 0
\(753\) 32.9443 + 12.5836i 1.20056 + 0.458572i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −35.3050 −1.28318 −0.641590 0.767048i \(-0.721725\pi\)
−0.641590 + 0.767048i \(0.721725\pi\)
\(758\) 0 0
\(759\) 9.52786 + 3.63932i 0.345840 + 0.132099i
\(760\) 0 0
\(761\) 31.3050 1.13480 0.567402 0.823441i \(-0.307949\pi\)
0.567402 + 0.823441i \(0.307949\pi\)
\(762\) 0 0
\(763\) −1.12461 + 7.70820i −0.0407137 + 0.279056i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11.0557i 0.399199i
\(768\) 0 0
\(769\) 31.4164i 1.13290i −0.824095 0.566452i \(-0.808316\pi\)
0.824095 0.566452i \(-0.191684\pi\)
\(770\) 0 0
\(771\) −26.6525 10.1803i −0.959865 0.366636i
\(772\) 0 0
\(773\) −18.0000 −0.647415 −0.323708 0.946157i \(-0.604929\pi\)
−0.323708 + 0.946157i \(0.604929\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 10.0000 17.8885i 0.358748 0.641748i
\(778\) 0 0
\(779\) 25.5279i 0.914631i
\(780\) 0 0
\(781\) −9.30495 −0.332957
\(782\) 0 0
\(783\) −18.4721 + 9.52786i −0.660140 + 0.340498i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 16.2918i 0.580740i −0.956914 0.290370i \(-0.906222\pi\)
0.956914 0.290370i \(-0.0937784\pi\)
\(788\) 0 0
\(789\) −13.7082 + 35.8885i −0.488025 + 1.27767i
\(790\) 0 0
\(791\) 45.1246 + 6.58359i 1.60445 + 0.234086i
\(792\) 0 0
\(793\) −6.11146 −0.217024
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 28.8328 1.02131 0.510655 0.859785i \(-0.329403\pi\)
0.510655 + 0.859785i \(0.329403\pi\)
\(798\) 0 0
\(799\) −14.4721 −0.511987
\(800\) 0 0
\(801\) −27.8885 24.9443i −0.985393 0.881363i
\(802\) 0 0
\(803\) −5.16718 −0.182346
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −31.5967 12.0689i −1.11226 0.424845i
\(808\) 0 0
\(809\) 12.3607i 0.434578i 0.976107 + 0.217289i \(0.0697215\pi\)
−0.976107 + 0.217289i \(0.930279\pi\)
\(810\) 0 0
\(811\) 1.70820i 0.0599832i −0.999550 0.0299916i \(-0.990452\pi\)
0.999550 0.0299916i \(-0.00954805\pi\)
\(812\) 0 0
\(813\) −13.0557 + 34.1803i −0.457884 + 1.19876i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 52.3607i 1.83187i
\(818\) 0 0
\(819\) −6.29180 7.52786i −0.219853 0.263045i
\(820\) 0 0
\(821\) 46.8328i 1.63448i 0.576300 + 0.817238i \(0.304496\pi\)
−0.576300 + 0.817238i \(0.695504\pi\)
\(822\) 0 0
\(823\) 11.5967 0.404237 0.202119 0.979361i \(-0.435217\pi\)
0.202119 + 0.979361i \(0.435217\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 39.7082i 1.38079i 0.723433 + 0.690395i \(0.242563\pi\)
−0.723433 + 0.690395i \(0.757437\pi\)
\(828\) 0 0
\(829\) 20.3607i 0.707156i −0.935405 0.353578i \(-0.884965\pi\)
0.935405 0.353578i \(-0.115035\pi\)
\(830\) 0 0
\(831\) 2.29180 + 0.875388i 0.0795015 + 0.0303669i
\(832\) 0 0
\(833\) 30.0000 + 8.94427i 1.03944 + 0.309901i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −14.9443 + 7.70820i −0.516550 + 0.266435i
\(838\) 0 0
\(839\) −12.9443 −0.446886 −0.223443 0.974717i \(-0.571730\pi\)
−0.223443 + 0.974717i \(0.571730\pi\)
\(840\) 0 0
\(841\) 13.0000 0.448276
\(842\) 0 0
\(843\) −7.05573 + 18.4721i −0.243012 + 0.636214i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3.97871 27.2705i 0.136710 0.937026i
\(848\) 0 0
\(849\) 5.34752 14.0000i 0.183527 0.480479i
\(850\) 0 0
\(851\) 34.4721i 1.18169i
\(852\) 0 0
\(853\) 19.1246i 0.654814i −0.944883 0.327407i \(-0.893825\pi\)
0.944883 0.327407i \(-0.106175\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −28.4721 −0.972590 −0.486295 0.873795i \(-0.661652\pi\)
−0.486295 + 0.873795i \(0.661652\pi\)
\(858\) 0 0
\(859\) 13.3475i 0.455412i 0.973730 + 0.227706i \(0.0731225\pi\)
−0.973730 + 0.227706i \(0.926878\pi\)
\(860\) 0 0
\(861\) 7.88854 14.1115i 0.268841 0.480917i
\(862\) 0 0
\(863\) 52.6525i 1.79231i 0.443740 + 0.896156i \(0.353651\pi\)
−0.443740 + 0.896156i \(0.646349\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −4.85410 1.85410i −0.164854 0.0629686i
\(868\) 0 0
\(869\) 8.00000i 0.271381i
\(870\) 0 0
\(871\) 12.0000i 0.406604i
\(872\) 0 0
\(873\) −28.3607 + 31.7082i −0.959864 + 1.07316i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 32.4721 1.09651 0.548253 0.836312i \(-0.315293\pi\)
0.548253 + 0.836312i \(0.315293\pi\)
\(878\) 0 0
\(879\) −22.6525 8.65248i −0.764049 0.291841i
\(880\) 0 0
\(881\) −16.8328 −0.567112 −0.283556 0.958956i \(-0.591514\pi\)
−0.283556 + 0.958956i \(0.591514\pi\)
\(882\) 0 0
\(883\) 35.5967 1.19793 0.598963 0.800777i \(-0.295580\pi\)
0.598963 + 0.800777i \(0.295580\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.1803 0.543283 0.271641 0.962399i \(-0.412434\pi\)
0.271641 + 0.962399i \(0.412434\pi\)
\(888\) 0 0
\(889\) 8.06888 55.3050i 0.270622 1.85487i
\(890\) 0 0
\(891\) −6.83282 + 0.763932i −0.228908 + 0.0255927i
\(892\) 0 0
\(893\) 23.4164i 0.783600i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −15.4164 5.88854i −0.514739 0.196613i
\(898\) 0 0
\(899\) −12.9443 −0.431716
\(900\) 0 0
\(901\) 41.3050i 1.37607i
\(902\) 0 0
\(903\) −16.1803 + 28.9443i −0.538448 + 0.963205i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 27.2361 0.904359 0.452179 0.891927i \(-0.350647\pi\)
0.452179 + 0.891927i \(0.350647\pi\)
\(908\) 0 0
\(909\) −15.5279 13.8885i −0.515027 0.460654i
\(910\) 0 0
\(911\) 46.0689i 1.52633i 0.646204 + 0.763165i \(0.276356\pi\)
−0.646204 + 0.763165i \(0.723644\pi\)
\(912\) 0 0
\(913\) 5.52786i 0.182946i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.88854 40.3607i 0.194457 1.33283i
\(918\) 0 0
\(919\) −34.4721 −1.13713 −0.568565 0.822638i \(-0.692501\pi\)
−0.568565 + 0.822638i \(0.692501\pi\)
\(920\) 0 0
\(921\) −11.2361 + 29.4164i −0.370241 + 0.969304i
\(922\) 0 0
\(923\) 15.0557 0.495565
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −21.5279 + 24.0689i −0.707068 + 0.790526i
\(928\) 0 0
\(929\) −19.8885 −0.652522 −0.326261 0.945280i \(-0.605789\pi\)
−0.326261 + 0.945280i \(0.605789\pi\)
\(930\) 0 0
\(931\) 14.4721 48.5410i 0.474305 1.59087i
\(932\) 0 0
\(933\) 52.3607 + 20.0000i 1.71421 + 0.654771i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 14.7639i 0.482317i −0.970486 0.241158i \(-0.922473\pi\)
0.970486 0.241158i \(-0.0775273\pi\)
\(938\) 0 0
\(939\) −2.65248 + 6.94427i −0.0865603 + 0.226618i
\(940\) 0 0
\(941\) −44.2492 −1.44248 −0.721242 0.692683i \(-0.756428\pi\)
−0.721242 + 0.692683i \(0.756428\pi\)
\(942\) 0 0
\(943\) 27.1935i 0.885542i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 25.2361i 0.820062i −0.912072 0.410031i \(-0.865518\pi\)
0.912072 0.410031i \(-0.134482\pi\)
\(948\) 0 0
\(949\) 8.36068 0.271399
\(950\) 0 0
\(951\) −3.81966 + 10.0000i −0.123861 + 0.324272i
\(952\) 0 0
\(953\) 3.70820i 0.120121i −0.998195 0.0600603i \(-0.980871\pi\)
0.998195 0.0600603i \(-0.0191293\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −4.94427 1.88854i −0.159826 0.0610480i
\(958\) 0 0
\(959\) −3.23607 0.472136i −0.104498 0.0152461i
\(960\) 0 0
\(961\) 20.5279 0.662189
\(962\) 0 0
\(963\) 0.583592 0.652476i 0.0188060 0.0210257i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −44.1803 −1.42074 −0.710372 0.703826i \(-0.751473\pi\)
−0.710372 + 0.703826i \(0.751473\pi\)
\(968\) 0 0
\(969\) −20.0000 + 52.3607i −0.642493 + 1.68207i
\(970\) 0 0
\(971\) −0.583592 −0.0187284 −0.00936418 0.999956i \(-0.502981\pi\)
−0.00936418 + 0.999956i \(0.502981\pi\)
\(972\) 0 0
\(973\) 39.8885 + 5.81966i 1.27877 + 0.186570i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 49.9574i 1.59828i −0.601145 0.799140i \(-0.705289\pi\)
0.601145 0.799140i \(-0.294711\pi\)
\(978\) 0 0
\(979\) 9.52786i 0.304512i
\(980\) 0 0
\(981\) −6.58359 5.88854i −0.210198 0.188007i
\(982\) 0 0
\(983\) 40.5410 1.29306 0.646529 0.762890i \(-0.276220\pi\)
0.646529 + 0.762890i \(0.276220\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −7.23607 + 12.9443i −0.230327 + 0.412021i
\(988\) 0 0
\(989\) 55.7771i 1.77361i
\(990\) 0 0
\(991\) 39.7771 1.26356 0.631780 0.775147i \(-0.282324\pi\)
0.631780 + 0.775147i \(0.282324\pi\)
\(992\) 0 0
\(993\) 33.8885 + 12.9443i 1.07542 + 0.410774i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 16.0689i 0.508907i 0.967085 + 0.254453i \(0.0818955\pi\)
−0.967085 + 0.254453i \(0.918104\pi\)
\(998\) 0 0
\(999\) 10.6525 + 20.6525i 0.337029 + 0.653415i
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.d.g.1301.1 4
3.2 odd 2 2100.2.d.h.1301.3 4
5.2 odd 4 2100.2.f.h.1049.2 4
5.3 odd 4 2100.2.f.b.1049.3 4
5.4 even 2 420.2.d.d.41.4 yes 4
7.6 odd 2 2100.2.d.h.1301.4 4
15.2 even 4 2100.2.f.g.1049.1 4
15.8 even 4 2100.2.f.a.1049.4 4
15.14 odd 2 420.2.d.c.41.2 yes 4
20.19 odd 2 1680.2.f.f.881.1 4
21.20 even 2 inner 2100.2.d.g.1301.2 4
35.13 even 4 2100.2.f.g.1049.2 4
35.27 even 4 2100.2.f.a.1049.3 4
35.34 odd 2 420.2.d.c.41.1 4
60.59 even 2 1680.2.f.j.881.3 4
105.62 odd 4 2100.2.f.b.1049.4 4
105.83 odd 4 2100.2.f.h.1049.1 4
105.104 even 2 420.2.d.d.41.3 yes 4
140.139 even 2 1680.2.f.j.881.4 4
420.419 odd 2 1680.2.f.f.881.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.d.c.41.1 4 35.34 odd 2
420.2.d.c.41.2 yes 4 15.14 odd 2
420.2.d.d.41.3 yes 4 105.104 even 2
420.2.d.d.41.4 yes 4 5.4 even 2
1680.2.f.f.881.1 4 20.19 odd 2
1680.2.f.f.881.2 4 420.419 odd 2
1680.2.f.j.881.3 4 60.59 even 2
1680.2.f.j.881.4 4 140.139 even 2
2100.2.d.g.1301.1 4 1.1 even 1 trivial
2100.2.d.g.1301.2 4 21.20 even 2 inner
2100.2.d.h.1301.3 4 3.2 odd 2
2100.2.d.h.1301.4 4 7.6 odd 2
2100.2.f.a.1049.3 4 35.27 even 4
2100.2.f.a.1049.4 4 15.8 even 4
2100.2.f.b.1049.3 4 5.3 odd 4
2100.2.f.b.1049.4 4 105.62 odd 4
2100.2.f.g.1049.1 4 15.2 even 4
2100.2.f.g.1049.2 4 35.13 even 4
2100.2.f.h.1049.1 4 105.83 odd 4
2100.2.f.h.1049.2 4 5.2 odd 4