Properties

Label 2100.2.ce.c.1657.4
Level $2100$
Weight $2$
Character 2100.1657
Analytic conductor $16.769$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(157,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 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("2100.157");
 
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.ce (of order \(12\), degree \(4\), 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: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 1657.4
Character \(\chi\) \(=\) 2100.1657
Dual form 2100.2.ce.c.493.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.258819 - 0.965926i) q^{3} +(2.36577 + 1.18454i) q^{7} +(-0.866025 - 0.500000i) q^{9} +O(q^{10})\) \(q+(0.258819 - 0.965926i) q^{3} +(2.36577 + 1.18454i) q^{7} +(-0.866025 - 0.500000i) q^{9} +(-2.25649 - 3.90835i) q^{11} +(-4.37238 - 4.37238i) q^{13} +(-4.35920 - 1.16804i) q^{17} +(-2.51045 + 4.34823i) q^{19} +(1.75649 - 1.97857i) q^{21} +(1.44721 + 5.40108i) q^{23} +(-0.707107 + 0.707107i) q^{27} +(-3.92124 + 2.26393i) q^{31} +(-4.35920 + 1.16804i) q^{33} +(1.67303 - 0.448288i) q^{37} +(-5.35505 + 3.09174i) q^{39} +2.22509i q^{41} +(-7.85281 + 7.85281i) q^{43} +(-3.33357 - 12.4411i) q^{47} +(4.19371 + 5.60471i) q^{49} +(-2.25649 + 3.90835i) q^{51} +(-11.8489 - 3.17491i) q^{53} +(3.55031 + 3.55031i) q^{57} +(-1.11255 - 1.92699i) q^{59} +(6.34248 + 3.66183i) q^{61} +(-1.45654 - 2.20873i) q^{63} +(1.29950 - 4.84982i) q^{67} +5.59161 q^{69} +14.3670 q^{71} +(1.12353 - 4.19308i) q^{73} +(-0.708707 - 11.9192i) q^{77} +(-1.58312 - 0.914012i) q^{79} +(0.500000 + 0.866025i) q^{81} +(-11.2105 - 11.2105i) q^{83} +(-3.90835 + 6.76946i) q^{89} +(-5.16475 - 15.5233i) q^{91} +(1.17190 + 4.37357i) q^{93} +(-3.53553 + 3.53553i) q^{97} +4.51298i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 8 q^{11} - 4 q^{21} - 60 q^{31} - 8 q^{51} + 84 q^{61} + 112 q^{71} + 12 q^{81} - 136 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) 0.258819 0.965926i 0.149429 0.557678i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.36577 + 1.18454i 0.894176 + 0.447716i
\(8\) 0 0
\(9\) −0.866025 0.500000i −0.288675 0.166667i
\(10\) 0 0
\(11\) −2.25649 3.90835i −0.680357 1.17841i −0.974872 0.222766i \(-0.928492\pi\)
0.294515 0.955647i \(-0.404842\pi\)
\(12\) 0 0
\(13\) −4.37238 4.37238i −1.21268 1.21268i −0.970142 0.242537i \(-0.922020\pi\)
−0.242537 0.970142i \(-0.577980\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.35920 1.16804i −1.05726 0.283292i −0.312012 0.950078i \(-0.601003\pi\)
−0.745249 + 0.666786i \(0.767670\pi\)
\(18\) 0 0
\(19\) −2.51045 + 4.34823i −0.575937 + 0.997552i 0.420003 + 0.907523i \(0.362029\pi\)
−0.995939 + 0.0900286i \(0.971304\pi\)
\(20\) 0 0
\(21\) 1.75649 1.97857i 0.383297 0.431760i
\(22\) 0 0
\(23\) 1.44721 + 5.40108i 0.301765 + 1.12620i 0.935694 + 0.352812i \(0.114774\pi\)
−0.633929 + 0.773391i \(0.718559\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.707107 + 0.707107i −0.136083 + 0.136083i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −3.92124 + 2.26393i −0.704275 + 0.406613i −0.808938 0.587894i \(-0.799957\pi\)
0.104663 + 0.994508i \(0.466624\pi\)
\(32\) 0 0
\(33\) −4.35920 + 1.16804i −0.758839 + 0.203330i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.67303 0.448288i 0.275045 0.0736980i −0.118660 0.992935i \(-0.537860\pi\)
0.393705 + 0.919237i \(0.371193\pi\)
\(38\) 0 0
\(39\) −5.35505 + 3.09174i −0.857494 + 0.495074i
\(40\) 0 0
\(41\) 2.22509i 0.347501i 0.984790 + 0.173751i \(0.0555887\pi\)
−0.984790 + 0.173751i \(0.944411\pi\)
\(42\) 0 0
\(43\) −7.85281 + 7.85281i −1.19754 + 1.19754i −0.222642 + 0.974900i \(0.571468\pi\)
−0.974900 + 0.222642i \(0.928532\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.33357 12.4411i −0.486251 1.81471i −0.574359 0.818604i \(-0.694749\pi\)
0.0881077 0.996111i \(-0.471918\pi\)
\(48\) 0 0
\(49\) 4.19371 + 5.60471i 0.599101 + 0.800673i
\(50\) 0 0
\(51\) −2.25649 + 3.90835i −0.315972 + 0.547279i
\(52\) 0 0
\(53\) −11.8489 3.17491i −1.62757 0.436107i −0.674359 0.738404i \(-0.735580\pi\)
−0.953214 + 0.302297i \(0.902247\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.55031 + 3.55031i 0.470250 + 0.470250i
\(58\) 0 0
\(59\) −1.11255 1.92699i −0.144841 0.250873i 0.784472 0.620164i \(-0.212934\pi\)
−0.929314 + 0.369291i \(0.879601\pi\)
\(60\) 0 0
\(61\) 6.34248 + 3.66183i 0.812071 + 0.468849i 0.847674 0.530517i \(-0.178002\pi\)
−0.0356037 + 0.999366i \(0.511335\pi\)
\(62\) 0 0
\(63\) −1.45654 2.20873i −0.183507 0.278274i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.29950 4.84982i 0.158760 0.592499i −0.839994 0.542595i \(-0.817442\pi\)
0.998754 0.0499041i \(-0.0158916\pi\)
\(68\) 0 0
\(69\) 5.59161 0.673151
\(70\) 0 0
\(71\) 14.3670 1.70504 0.852522 0.522692i \(-0.175072\pi\)
0.852522 + 0.522692i \(0.175072\pi\)
\(72\) 0 0
\(73\) 1.12353 4.19308i 0.131500 0.490763i −0.868488 0.495710i \(-0.834908\pi\)
0.999988 + 0.00494658i \(0.00157455\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.708707 11.9192i −0.0807646 1.35831i
\(78\) 0 0
\(79\) −1.58312 0.914012i −0.178114 0.102834i 0.408292 0.912851i \(-0.366125\pi\)
−0.586406 + 0.810017i \(0.699458\pi\)
\(80\) 0 0
\(81\) 0.500000 + 0.866025i 0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) −11.2105 11.2105i −1.23051 1.23051i −0.963769 0.266738i \(-0.914054\pi\)
−0.266738 0.963769i \(-0.585946\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.90835 + 6.76946i −0.414284 + 0.717562i −0.995353 0.0962928i \(-0.969301\pi\)
0.581069 + 0.813855i \(0.302635\pi\)
\(90\) 0 0
\(91\) −5.16475 15.5233i −0.541413 1.62728i
\(92\) 0 0
\(93\) 1.17190 + 4.37357i 0.121520 + 0.453518i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −3.53553 + 3.53553i −0.358979 + 0.358979i −0.863437 0.504457i \(-0.831693\pi\)
0.504457 + 0.863437i \(0.331693\pi\)
\(98\) 0 0
\(99\) 4.51298i 0.453571i
\(100\) 0 0
\(101\) −6.08451 + 3.51290i −0.605432 + 0.349546i −0.771175 0.636623i \(-0.780331\pi\)
0.165744 + 0.986169i \(0.446998\pi\)
\(102\) 0 0
\(103\) 4.02982 1.07979i 0.397070 0.106395i −0.0547581 0.998500i \(-0.517439\pi\)
0.451828 + 0.892105i \(0.350772\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.78639 1.81841i 0.656065 0.175792i 0.0845956 0.996415i \(-0.473040\pi\)
0.571470 + 0.820623i \(0.306373\pi\)
\(108\) 0 0
\(109\) −11.7125 + 6.76224i −1.12186 + 0.647705i −0.941875 0.335965i \(-0.890938\pi\)
−0.179983 + 0.983670i \(0.557604\pi\)
\(110\) 0 0
\(111\) 1.73205i 0.164399i
\(112\) 0 0
\(113\) 7.34847 7.34847i 0.691286 0.691286i −0.271229 0.962515i \(-0.587430\pi\)
0.962515 + 0.271229i \(0.0874301\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.60040 + 5.97278i 0.147957 + 0.552184i
\(118\) 0 0
\(119\) −8.92925 7.92699i −0.818543 0.726666i
\(120\) 0 0
\(121\) −4.68348 + 8.11202i −0.425771 + 0.737456i
\(122\) 0 0
\(123\) 2.14928 + 0.575897i 0.193794 + 0.0519269i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −10.7059 10.7059i −0.949991 0.949991i 0.0488168 0.998808i \(-0.484455\pi\)
−0.998808 + 0.0488168i \(0.984455\pi\)
\(128\) 0 0
\(129\) 5.55278 + 9.61769i 0.488895 + 0.846790i
\(130\) 0 0
\(131\) 12.8540 + 7.42125i 1.12306 + 0.648397i 0.942180 0.335108i \(-0.108773\pi\)
0.180878 + 0.983506i \(0.442106\pi\)
\(132\) 0 0
\(133\) −11.0898 + 7.31315i −0.961608 + 0.634131i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.295421 + 1.10253i −0.0252395 + 0.0941953i −0.977397 0.211414i \(-0.932193\pi\)
0.952157 + 0.305609i \(0.0988600\pi\)
\(138\) 0 0
\(139\) −20.5766 −1.74529 −0.872644 0.488357i \(-0.837596\pi\)
−0.872644 + 0.488357i \(0.837596\pi\)
\(140\) 0 0
\(141\) −12.8799 −1.08469
\(142\) 0 0
\(143\) −7.22257 + 26.9550i −0.603982 + 2.25409i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.49915 2.60020i 0.536041 0.214461i
\(148\) 0 0
\(149\) −16.9212 9.76946i −1.38624 0.800346i −0.393350 0.919389i \(-0.628684\pi\)
−0.992889 + 0.119043i \(0.962017\pi\)
\(150\) 0 0
\(151\) 5.75649 + 9.97053i 0.468456 + 0.811390i 0.999350 0.0360482i \(-0.0114770\pi\)
−0.530894 + 0.847438i \(0.678144\pi\)
\(152\) 0 0
\(153\) 3.19116 + 3.19116i 0.257990 + 0.257990i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.75247 + 2.07727i 0.618715 + 0.165784i 0.554543 0.832155i \(-0.312893\pi\)
0.0641713 + 0.997939i \(0.479560\pi\)
\(158\) 0 0
\(159\) −6.13345 + 10.6234i −0.486414 + 0.842494i
\(160\) 0 0
\(161\) −2.97405 + 14.4920i −0.234388 + 1.14213i
\(162\) 0 0
\(163\) 1.11490 + 4.16087i 0.0873259 + 0.325905i 0.995744 0.0921571i \(-0.0293762\pi\)
−0.908419 + 0.418062i \(0.862710\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.119519 0.119519i 0.00924866 0.00924866i −0.702467 0.711716i \(-0.747918\pi\)
0.711716 + 0.702467i \(0.247918\pi\)
\(168\) 0 0
\(169\) 25.2354i 1.94118i
\(170\) 0 0
\(171\) 4.34823 2.51045i 0.332517 0.191979i
\(172\) 0 0
\(173\) −14.5140 + 3.88900i −1.10348 + 0.295675i −0.764179 0.645004i \(-0.776856\pi\)
−0.339296 + 0.940679i \(0.610189\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2.14928 + 0.575897i −0.161549 + 0.0432871i
\(178\) 0 0
\(179\) 17.0676 9.85398i 1.27569 0.736521i 0.299638 0.954053i \(-0.403134\pi\)
0.976053 + 0.217532i \(0.0698007\pi\)
\(180\) 0 0
\(181\) 9.32855i 0.693386i −0.937979 0.346693i \(-0.887305\pi\)
0.937979 0.346693i \(-0.112695\pi\)
\(182\) 0 0
\(183\) 5.17861 5.17861i 0.382814 0.382814i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 5.27136 + 19.6730i 0.385480 + 1.43863i
\(188\) 0 0
\(189\) −2.51045 + 0.835250i −0.182608 + 0.0607555i
\(190\) 0 0
\(191\) 7.95294 13.7749i 0.575455 0.996717i −0.420538 0.907275i \(-0.638158\pi\)
0.995992 0.0894414i \(-0.0285082\pi\)
\(192\) 0 0
\(193\) 4.37357 + 1.17190i 0.314817 + 0.0843549i 0.412768 0.910836i \(-0.364562\pi\)
−0.0979510 + 0.995191i \(0.531229\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −17.5959 17.5959i −1.25365 1.25365i −0.954071 0.299582i \(-0.903153\pi\)
−0.299582 0.954071i \(-0.596847\pi\)
\(198\) 0 0
\(199\) 4.59912 + 7.96592i 0.326023 + 0.564689i 0.981719 0.190336i \(-0.0609579\pi\)
−0.655696 + 0.755025i \(0.727625\pi\)
\(200\) 0 0
\(201\) −4.34823 2.51045i −0.306700 0.177073i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.44721 5.40108i 0.100588 0.375401i
\(208\) 0 0
\(209\) 22.6592 1.56737
\(210\) 0 0
\(211\) 3.14602 0.216581 0.108291 0.994119i \(-0.465462\pi\)
0.108291 + 0.994119i \(0.465462\pi\)
\(212\) 0 0
\(213\) 3.71844 13.8774i 0.254783 0.950865i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −11.9585 + 0.711043i −0.811793 + 0.0482688i
\(218\) 0 0
\(219\) −3.75942 2.17050i −0.254038 0.146669i
\(220\) 0 0
\(221\) 13.9529 + 24.1672i 0.938576 + 1.62566i
\(222\) 0 0
\(223\) −6.46722 6.46722i −0.433077 0.433077i 0.456597 0.889674i \(-0.349068\pi\)
−0.889674 + 0.456597i \(0.849068\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.1505 4.05956i −1.00557 0.269443i −0.281795 0.959475i \(-0.590930\pi\)
−0.723779 + 0.690032i \(0.757596\pi\)
\(228\) 0 0
\(229\) 9.21461 15.9602i 0.608918 1.05468i −0.382501 0.923955i \(-0.624937\pi\)
0.991419 0.130722i \(-0.0417296\pi\)
\(230\) 0 0
\(231\) −11.6965 2.40035i −0.769570 0.157931i
\(232\) 0 0
\(233\) 0.666615 + 2.48784i 0.0436714 + 0.162984i 0.984318 0.176404i \(-0.0564466\pi\)
−0.940646 + 0.339388i \(0.889780\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.29261 + 1.29261i −0.0839640 + 0.0839640i
\(238\) 0 0
\(239\) 0.607095i 0.0392697i 0.999807 + 0.0196349i \(0.00625037\pi\)
−0.999807 + 0.0196349i \(0.993750\pi\)
\(240\) 0 0
\(241\) −7.50000 + 4.33013i −0.483117 + 0.278928i −0.721715 0.692191i \(-0.756646\pi\)
0.238597 + 0.971119i \(0.423312\pi\)
\(242\) 0 0
\(243\) 0.965926 0.258819i 0.0619642 0.0166032i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 29.9887 8.03545i 1.90814 0.511284i
\(248\) 0 0
\(249\) −13.7299 + 7.92699i −0.870100 + 0.502352i
\(250\) 0 0
\(251\) 28.2508i 1.78318i −0.452848 0.891588i \(-0.649592\pi\)
0.452848 0.891588i \(-0.350408\pi\)
\(252\) 0 0
\(253\) 17.8437 17.8437i 1.12182 1.12182i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.33861 + 4.99574i 0.0834999 + 0.311626i 0.995026 0.0996165i \(-0.0317616\pi\)
−0.911526 + 0.411243i \(0.865095\pi\)
\(258\) 0 0
\(259\) 4.48902 + 0.921238i 0.278934 + 0.0572429i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 15.4393 + 4.13694i 0.952027 + 0.255095i 0.701222 0.712943i \(-0.252638\pi\)
0.250805 + 0.968038i \(0.419305\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 5.52724 + 5.52724i 0.338262 + 0.338262i
\(268\) 0 0
\(269\) −8.75400 15.1624i −0.533741 0.924466i −0.999223 0.0394089i \(-0.987453\pi\)
0.465483 0.885057i \(-0.345881\pi\)
\(270\) 0 0
\(271\) −9.38140 5.41636i −0.569880 0.329020i 0.187222 0.982318i \(-0.440052\pi\)
−0.757101 + 0.653298i \(0.773385\pi\)
\(272\) 0 0
\(273\) −16.3311 + 0.971038i −0.988403 + 0.0587699i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.02719 + 11.2976i −0.181887 + 0.678810i 0.813389 + 0.581720i \(0.197620\pi\)
−0.995276 + 0.0970897i \(0.969047\pi\)
\(278\) 0 0
\(279\) 4.52786 0.271076
\(280\) 0 0
\(281\) 25.5389 1.52352 0.761762 0.647857i \(-0.224334\pi\)
0.761762 + 0.647857i \(0.224334\pi\)
\(282\) 0 0
\(283\) 4.41038 16.4598i 0.262170 0.978431i −0.701790 0.712384i \(-0.747616\pi\)
0.963960 0.266048i \(-0.0857178\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.63572 + 5.26406i −0.155582 + 0.310727i
\(288\) 0 0
\(289\) 2.91587 + 1.68348i 0.171522 + 0.0990280i
\(290\) 0 0
\(291\) 2.50000 + 4.33013i 0.146553 + 0.253837i
\(292\) 0 0
\(293\) −0.429281 0.429281i −0.0250789 0.0250789i 0.694456 0.719535i \(-0.255645\pi\)
−0.719535 + 0.694456i \(0.755645\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.35920 + 1.16804i 0.252946 + 0.0677768i
\(298\) 0 0
\(299\) 17.2878 29.9433i 0.999779 1.73167i
\(300\) 0 0
\(301\) −27.8799 + 9.27591i −1.60697 + 0.534655i
\(302\) 0 0
\(303\) 1.81841 + 6.78639i 0.104465 + 0.389868i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 11.2390 11.2390i 0.641445 0.641445i −0.309466 0.950911i \(-0.600150\pi\)
0.950911 + 0.309466i \(0.100150\pi\)
\(308\) 0 0
\(309\) 4.17198i 0.237335i
\(310\) 0 0
\(311\) −0.988499 + 0.570710i −0.0560526 + 0.0323620i −0.527764 0.849391i \(-0.676970\pi\)
0.471712 + 0.881753i \(0.343636\pi\)
\(312\) 0 0
\(313\) −4.34809 + 1.16507i −0.245769 + 0.0658535i −0.379600 0.925151i \(-0.623939\pi\)
0.133832 + 0.991004i \(0.457272\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.36107 + 2.50829i −0.525770 + 0.140880i −0.511934 0.859025i \(-0.671071\pi\)
−0.0138358 + 0.999904i \(0.504404\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 7.02579i 0.392141i
\(322\) 0 0
\(323\) 16.0225 16.0225i 0.891514 0.891514i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3.50039 + 13.0636i 0.193572 + 0.722421i
\(328\) 0 0
\(329\) 6.85054 33.3814i 0.377682 1.84038i
\(330\) 0 0
\(331\) −1.90251 + 3.29525i −0.104571 + 0.181123i −0.913563 0.406697i \(-0.866680\pi\)
0.808992 + 0.587820i \(0.200014\pi\)
\(332\) 0 0
\(333\) −1.67303 0.448288i −0.0916816 0.0245660i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 10.8616 + 10.8616i 0.591667 + 0.591667i 0.938082 0.346415i \(-0.112601\pi\)
−0.346415 + 0.938082i \(0.612601\pi\)
\(338\) 0 0
\(339\) −5.19615 9.00000i −0.282216 0.488813i
\(340\) 0 0
\(341\) 17.6965 + 10.2171i 0.958317 + 0.553284i
\(342\) 0 0
\(343\) 3.28230 + 18.2271i 0.177227 + 0.984170i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.47033 12.9514i 0.186297 0.695270i −0.808052 0.589111i \(-0.799478\pi\)
0.994349 0.106159i \(-0.0338552\pi\)
\(348\) 0 0
\(349\) 35.2766 1.88831 0.944156 0.329497i \(-0.106879\pi\)
0.944156 + 0.329497i \(0.106879\pi\)
\(350\) 0 0
\(351\) 6.18348 0.330050
\(352\) 0 0
\(353\) −1.15461 + 4.30906i −0.0614537 + 0.229348i −0.989821 0.142314i \(-0.954546\pi\)
0.928368 + 0.371663i \(0.121212\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −9.96794 + 6.57334i −0.527559 + 0.347898i
\(358\) 0 0
\(359\) 10.0129 + 5.78097i 0.528462 + 0.305108i 0.740390 0.672178i \(-0.234641\pi\)
−0.211928 + 0.977285i \(0.567974\pi\)
\(360\) 0 0
\(361\) −3.10471 5.37752i −0.163406 0.283028i
\(362\) 0 0
\(363\) 6.62344 + 6.62344i 0.347640 + 0.347640i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2.72055 0.728970i −0.142012 0.0380519i 0.187113 0.982338i \(-0.440087\pi\)
−0.329125 + 0.944286i \(0.606754\pi\)
\(368\) 0 0
\(369\) 1.11255 1.92699i 0.0579169 0.100315i
\(370\) 0 0
\(371\) −24.2709 21.5467i −1.26008 1.11865i
\(372\) 0 0
\(373\) 6.10306 + 22.7769i 0.316004 + 1.17934i 0.923051 + 0.384678i \(0.125688\pi\)
−0.607046 + 0.794667i \(0.707646\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 12.6445i 0.649507i 0.945799 + 0.324753i \(0.105281\pi\)
−0.945799 + 0.324753i \(0.894719\pi\)
\(380\) 0 0
\(381\) −13.1119 + 7.57018i −0.671745 + 0.387832i
\(382\) 0 0
\(383\) −8.08186 + 2.16553i −0.412964 + 0.110653i −0.459318 0.888272i \(-0.651906\pi\)
0.0463547 + 0.998925i \(0.485240\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.7271 2.87433i 0.545291 0.146110i
\(388\) 0 0
\(389\) 4.77182 2.75501i 0.241941 0.139685i −0.374128 0.927377i \(-0.622058\pi\)
0.616069 + 0.787693i \(0.288724\pi\)
\(390\) 0 0
\(391\) 25.2348i 1.27618i
\(392\) 0 0
\(393\) 10.4952 10.4952i 0.529414 0.529414i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.49612 + 20.5118i 0.275842 + 1.02946i 0.955275 + 0.295720i \(0.0955594\pi\)
−0.679432 + 0.733738i \(0.737774\pi\)
\(398\) 0 0
\(399\) 4.19371 + 12.6047i 0.209948 + 0.631025i
\(400\) 0 0
\(401\) −7.26799 + 12.5885i −0.362946 + 0.628641i −0.988444 0.151585i \(-0.951562\pi\)
0.625498 + 0.780226i \(0.284896\pi\)
\(402\) 0 0
\(403\) 27.0439 + 7.24639i 1.34715 + 0.360968i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.52724 5.52724i −0.273975 0.273975i
\(408\) 0 0
\(409\) −9.21461 15.9602i −0.455633 0.789179i 0.543091 0.839674i \(-0.317254\pi\)
−0.998724 + 0.0504942i \(0.983920\pi\)
\(410\) 0 0
\(411\) 0.988499 + 0.570710i 0.0487591 + 0.0281511i
\(412\) 0 0
\(413\) −0.349423 5.87667i −0.0171940 0.289172i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5.32563 + 19.8755i −0.260797 + 0.973308i
\(418\) 0 0
\(419\) −9.74904 −0.476272 −0.238136 0.971232i \(-0.576536\pi\)
−0.238136 + 0.971232i \(0.576536\pi\)
\(420\) 0 0
\(421\) 10.6330 0.518223 0.259112 0.965847i \(-0.416570\pi\)
0.259112 + 0.965847i \(0.416570\pi\)
\(422\) 0 0
\(423\) −3.33357 + 12.4411i −0.162084 + 0.604905i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 10.6672 + 16.1760i 0.516223 + 0.782811i
\(428\) 0 0
\(429\) 24.1672 + 13.9529i 1.16680 + 0.673654i
\(430\) 0 0
\(431\) −16.2354 28.1205i −0.782031 1.35452i −0.930757 0.365639i \(-0.880851\pi\)
0.148726 0.988878i \(-0.452483\pi\)
\(432\) 0 0
\(433\) −23.2782 23.2782i −1.11868 1.11868i −0.991936 0.126743i \(-0.959548\pi\)
−0.126743 0.991936i \(-0.540452\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −27.1183 7.26632i −1.29724 0.347595i
\(438\) 0 0
\(439\) −6.99947 + 12.1234i −0.334067 + 0.578620i −0.983305 0.181965i \(-0.941754\pi\)
0.649239 + 0.760585i \(0.275088\pi\)
\(440\) 0 0
\(441\) −0.829500 6.95068i −0.0395000 0.330985i
\(442\) 0 0
\(443\) −0.204704 0.763964i −0.00972576 0.0362970i 0.960892 0.276922i \(-0.0893143\pi\)
−0.970618 + 0.240625i \(0.922648\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −13.8161 + 13.8161i −0.653480 + 0.653480i
\(448\) 0 0
\(449\) 15.9059i 0.750645i −0.926894 0.375322i \(-0.877532\pi\)
0.926894 0.375322i \(-0.122468\pi\)
\(450\) 0 0
\(451\) 8.69645 5.02090i 0.409500 0.236425i
\(452\) 0 0
\(453\) 11.1207 2.97978i 0.522495 0.140002i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −35.4090 + 9.48783i −1.65637 + 0.443822i −0.961385 0.275207i \(-0.911254\pi\)
−0.694980 + 0.719029i \(0.744587\pi\)
\(458\) 0 0
\(459\) 3.90835 2.25649i 0.182426 0.105324i
\(460\) 0 0
\(461\) 14.1415i 0.658634i 0.944219 + 0.329317i \(0.106819\pi\)
−0.944219 + 0.329317i \(0.893181\pi\)
\(462\) 0 0
\(463\) 16.4119 16.4119i 0.762728 0.762728i −0.214087 0.976815i \(-0.568678\pi\)
0.976815 + 0.214087i \(0.0686776\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.22509 23.2324i −0.288063 1.07507i −0.946572 0.322492i \(-0.895479\pi\)
0.658509 0.752573i \(-0.271187\pi\)
\(468\) 0 0
\(469\) 8.81915 9.93421i 0.407230 0.458719i
\(470\) 0 0
\(471\) 4.01298 6.95068i 0.184908 0.320270i
\(472\) 0 0
\(473\) 48.4113 + 12.9718i 2.22596 + 0.596443i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 8.67400 + 8.67400i 0.397155 + 0.397155i
\(478\) 0 0
\(479\) 0.146380 + 0.253538i 0.00668829 + 0.0115845i 0.869350 0.494197i \(-0.164538\pi\)
−0.862662 + 0.505781i \(0.831204\pi\)
\(480\) 0 0
\(481\) −9.27521 5.35505i −0.422913 0.244169i
\(482\) 0 0
\(483\) 13.2284 + 6.62351i 0.601915 + 0.301380i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.08118 + 4.03501i −0.0489928 + 0.182844i −0.986086 0.166235i \(-0.946839\pi\)
0.937093 + 0.349079i \(0.113506\pi\)
\(488\) 0 0
\(489\) 4.30765 0.194799
\(490\) 0 0
\(491\) 5.28910 0.238694 0.119347 0.992853i \(-0.461920\pi\)
0.119347 + 0.992853i \(0.461920\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 33.9889 + 17.0183i 1.52461 + 0.763375i
\(498\) 0 0
\(499\) −25.4675 14.7037i −1.14008 0.658227i −0.193632 0.981074i \(-0.562027\pi\)
−0.946451 + 0.322847i \(0.895360\pi\)
\(500\) 0 0
\(501\) −0.0845127 0.146380i −0.00377575 0.00653979i
\(502\) 0 0
\(503\) −25.4558 25.4558i −1.13502 1.13502i −0.989330 0.145690i \(-0.953460\pi\)
−0.145690 0.989330i \(-0.546540\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 24.3755 + 6.53140i 1.08255 + 0.290070i
\(508\) 0 0
\(509\) −5.76686 + 9.98850i −0.255612 + 0.442732i −0.965061 0.262024i \(-0.915610\pi\)
0.709450 + 0.704756i \(0.248944\pi\)
\(510\) 0 0
\(511\) 7.62492 8.58899i 0.337306 0.379954i
\(512\) 0 0
\(513\) −1.29950 4.84982i −0.0573745 0.214125i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −41.1019 + 41.1019i −1.80766 + 1.80766i
\(518\) 0 0
\(519\) 15.0260i 0.659566i
\(520\) 0 0
\(521\) 0.684951 0.395457i 0.0300083 0.0173253i −0.484921 0.874558i \(-0.661152\pi\)
0.514929 + 0.857233i \(0.327818\pi\)
\(522\) 0 0
\(523\) 35.9888 9.64318i 1.57368 0.421667i 0.636719 0.771096i \(-0.280291\pi\)
0.936963 + 0.349429i \(0.113624\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 19.7378 5.28874i 0.859793 0.230381i
\(528\) 0 0
\(529\) −7.15865 + 4.13305i −0.311246 + 0.179698i
\(530\) 0 0
\(531\) 2.22509i 0.0965609i
\(532\) 0 0
\(533\) 9.72895 9.72895i 0.421408 0.421408i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −5.10079 19.0364i −0.220115 0.821482i
\(538\) 0 0
\(539\) 12.4421 29.0375i 0.535921 1.25073i
\(540\) 0 0
\(541\) −1.05429 + 1.82608i −0.0453273 + 0.0785091i −0.887799 0.460232i \(-0.847766\pi\)
0.842472 + 0.538741i \(0.181100\pi\)
\(542\) 0 0
\(543\) −9.01069 2.41441i −0.386686 0.103612i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 20.7426 + 20.7426i 0.886890 + 0.886890i 0.994223 0.107333i \(-0.0342311\pi\)
−0.107333 + 0.994223i \(0.534231\pi\)
\(548\) 0 0
\(549\) −3.66183 6.34248i −0.156283 0.270690i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −2.66259 4.03761i −0.113225 0.171697i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.98515 + 11.1407i −0.126485 + 0.472048i −0.999888 0.0149497i \(-0.995241\pi\)
0.873403 + 0.486997i \(0.161908\pi\)
\(558\) 0 0
\(559\) 68.6709 2.90447
\(560\) 0 0
\(561\) 20.3670 0.859893
\(562\) 0 0
\(563\) 1.62353 6.05909i 0.0684236 0.255360i −0.923238 0.384228i \(-0.874468\pi\)
0.991662 + 0.128868i \(0.0411343\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.157038 + 2.64109i 0.00659496 + 0.110915i
\(568\) 0 0
\(569\) −26.1721 15.1105i −1.09719 0.633464i −0.161709 0.986838i \(-0.551701\pi\)
−0.935482 + 0.353375i \(0.885034\pi\)
\(570\) 0 0
\(571\) 18.7037 + 32.3957i 0.782725 + 1.35572i 0.930349 + 0.366675i \(0.119504\pi\)
−0.147624 + 0.989043i \(0.547163\pi\)
\(572\) 0 0
\(573\) −11.2472 11.2472i −0.469857 0.469857i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 17.8711 + 4.78853i 0.743982 + 0.199349i 0.610847 0.791749i \(-0.290829\pi\)
0.133135 + 0.991098i \(0.457496\pi\)
\(578\) 0 0
\(579\) 2.26393 3.92124i 0.0940856 0.162961i
\(580\) 0 0
\(581\) −13.2420 39.8006i −0.549372 1.65121i
\(582\) 0 0
\(583\) 14.3283 + 53.4738i 0.593416 + 2.21466i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.17461 5.17461i 0.213579 0.213579i −0.592207 0.805786i \(-0.701743\pi\)
0.805786 + 0.592207i \(0.201743\pi\)
\(588\) 0 0
\(589\) 22.7339i 0.936734i
\(590\) 0 0
\(591\) −21.5504 + 12.4421i −0.886466 + 0.511802i
\(592\) 0 0
\(593\) 40.0827 10.7401i 1.64600 0.441045i 0.687512 0.726173i \(-0.258703\pi\)
0.958489 + 0.285128i \(0.0920363\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.88483 2.38068i 0.363632 0.0974348i
\(598\) 0 0
\(599\) 36.2716 20.9414i 1.48202 0.855644i 0.482227 0.876046i \(-0.339828\pi\)
0.999792 + 0.0204023i \(0.00649469\pi\)
\(600\) 0 0
\(601\) 3.49682i 0.142638i −0.997454 0.0713191i \(-0.977279\pi\)
0.997454 0.0713191i \(-0.0227209\pi\)
\(602\) 0 0
\(603\) −3.55031 + 3.55031i −0.144580 + 0.144580i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −4.54613 16.9664i −0.184522 0.688644i −0.994732 0.102505i \(-0.967314\pi\)
0.810211 0.586138i \(-0.199353\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −39.8214 + 68.9726i −1.61100 + 2.79033i
\(612\) 0 0
\(613\) −35.6025 9.53966i −1.43797 0.385303i −0.546149 0.837688i \(-0.683907\pi\)
−0.891823 + 0.452385i \(0.850573\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −11.3023 11.3023i −0.455015 0.455015i 0.442000 0.897015i \(-0.354269\pi\)
−0.897015 + 0.442000i \(0.854269\pi\)
\(618\) 0 0
\(619\) −22.7956 39.4832i −0.916233 1.58696i −0.805086 0.593159i \(-0.797881\pi\)
−0.111148 0.993804i \(-0.535453\pi\)
\(620\) 0 0
\(621\) −4.84248 2.79580i −0.194322 0.112192i
\(622\) 0 0
\(623\) −17.2650 + 11.3854i −0.691707 + 0.456145i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 5.86463 21.8871i 0.234211 0.874087i
\(628\) 0 0
\(629\) −7.81670 −0.311672
\(630\) 0 0
\(631\) 33.9029 1.34965 0.674827 0.737976i \(-0.264218\pi\)
0.674827 + 0.737976i \(0.264218\pi\)
\(632\) 0 0
\(633\) 0.814251 3.03883i 0.0323636 0.120782i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.16946 42.8424i 0.244443 1.69748i
\(638\) 0 0
\(639\) −12.4421 7.18348i −0.492204 0.284174i
\(640\) 0 0
\(641\) 20.0634 + 34.7508i 0.792457 + 1.37258i 0.924442 + 0.381324i \(0.124532\pi\)
−0.131985 + 0.991252i \(0.542135\pi\)
\(642\) 0 0
\(643\) 18.5960 + 18.5960i 0.733357 + 0.733357i 0.971283 0.237927i \(-0.0764678\pi\)
−0.237927 + 0.971283i \(0.576468\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.2049 + 2.73439i 0.401196 + 0.107500i 0.453774 0.891117i \(-0.350077\pi\)
−0.0525785 + 0.998617i \(0.516744\pi\)
\(648\) 0 0
\(649\) −5.02090 + 8.69645i −0.197088 + 0.341366i
\(650\) 0 0
\(651\) −2.40826 + 11.7350i −0.0943872 + 0.459932i
\(652\) 0 0
\(653\) 11.1534 + 41.6249i 0.436465 + 1.62891i 0.737536 + 0.675308i \(0.235989\pi\)
−0.301071 + 0.953602i \(0.597344\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.06955 + 3.06955i −0.119755 + 0.119755i
\(658\) 0 0
\(659\) 10.4611i 0.407506i 0.979022 + 0.203753i \(0.0653139\pi\)
−0.979022 + 0.203753i \(0.934686\pi\)
\(660\) 0 0
\(661\) −36.7584 + 21.2225i −1.42974 + 0.825458i −0.997099 0.0761106i \(-0.975750\pi\)
−0.432636 + 0.901569i \(0.642416\pi\)
\(662\) 0 0
\(663\) 26.9550 7.22257i 1.04685 0.280502i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −7.92069 + 4.57301i −0.306232 + 0.176803i
\(670\) 0 0
\(671\) 33.0515i 1.27594i
\(672\) 0 0
\(673\) 23.0223 23.0223i 0.887445 0.887445i −0.106832 0.994277i \(-0.534071\pi\)
0.994277 + 0.106832i \(0.0340708\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.39579 + 5.20915i 0.0536444 + 0.200204i 0.987547 0.157324i \(-0.0502866\pi\)
−0.933903 + 0.357527i \(0.883620\pi\)
\(678\) 0 0
\(679\) −12.5522 + 4.17625i −0.481711 + 0.160270i
\(680\) 0 0
\(681\) −7.84248 + 13.5836i −0.300524 + 0.520523i
\(682\) 0 0
\(683\) 36.9010 + 9.88759i 1.41198 + 0.378338i 0.882630 0.470068i \(-0.155771\pi\)
0.529346 + 0.848406i \(0.322437\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −13.0314 13.0314i −0.497180 0.497180i
\(688\) 0 0
\(689\) 37.9260 + 65.6898i 1.44487 + 2.50258i
\(690\) 0 0
\(691\) −4.58451 2.64687i −0.174403 0.100692i 0.410257 0.911970i \(-0.365439\pi\)
−0.584660 + 0.811278i \(0.698772\pi\)
\(692\) 0 0
\(693\) −5.34582 + 10.6766i −0.203071 + 0.405572i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.59901 9.69963i 0.0984445 0.367400i
\(698\) 0 0
\(699\) 2.57560 0.0974182
\(700\) 0 0
\(701\) 17.8310 0.673467 0.336733 0.941600i \(-0.390678\pi\)
0.336733 + 0.941600i \(0.390678\pi\)
\(702\) 0 0
\(703\) −2.25081 + 8.40013i −0.0848908 + 0.316817i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −18.5557 + 1.10331i −0.697860 + 0.0414944i
\(708\) 0 0
\(709\) −6.63289 3.82950i −0.249103 0.143820i 0.370250 0.928932i \(-0.379272\pi\)
−0.619354 + 0.785112i \(0.712605\pi\)
\(710\) 0 0
\(711\) 0.914012 + 1.58312i 0.0342781 + 0.0593715i
\(712\) 0 0
\(713\) −17.9025 17.9025i −0.670455 0.670455i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0.586409 + 0.157128i 0.0218998 + 0.00586804i
\(718\) 0 0
\(719\) −6.30870 + 10.9270i −0.235275 + 0.407508i −0.959353 0.282210i \(-0.908932\pi\)
0.724078 + 0.689718i \(0.242266\pi\)
\(720\) 0 0
\(721\) 10.8127 + 2.21898i 0.402685 + 0.0826390i
\(722\) 0 0
\(723\) 2.24144 + 8.36516i 0.0833600 + 0.311104i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 13.4799 13.4799i 0.499940 0.499940i −0.411479 0.911419i \(-0.634988\pi\)
0.911419 + 0.411479i \(0.134988\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 43.4044 25.0595i 1.60537 0.926861i
\(732\) 0 0
\(733\) 8.67111 2.32342i 0.320275 0.0858174i −0.0951000 0.995468i \(-0.530317\pi\)
0.415375 + 0.909650i \(0.363650\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −21.8871 + 5.86463i −0.806222 + 0.216026i
\(738\) 0 0
\(739\) −12.5911 + 7.26946i −0.463170 + 0.267412i −0.713376 0.700781i \(-0.752835\pi\)
0.250206 + 0.968193i \(0.419502\pi\)
\(740\) 0 0
\(741\) 31.0466i 1.14053i
\(742\) 0 0
\(743\) −24.4486 + 24.4486i −0.896933 + 0.896933i −0.995164 0.0982305i \(-0.968682\pi\)
0.0982305 + 0.995164i \(0.468682\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 4.10331 + 15.3138i 0.150132 + 0.560301i
\(748\) 0 0
\(749\) 18.2090 + 3.73686i 0.665343 + 0.136542i
\(750\) 0 0
\(751\) −21.4644 + 37.1775i −0.783249 + 1.35663i 0.146791 + 0.989168i \(0.453105\pi\)
−0.930040 + 0.367459i \(0.880228\pi\)
\(752\) 0 0
\(753\) −27.2882 7.31185i −0.994437 0.266459i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 6.43514 + 6.43514i 0.233889 + 0.233889i 0.814314 0.580425i \(-0.197113\pi\)
−0.580425 + 0.814314i \(0.697113\pi\)
\(758\) 0 0
\(759\) −12.6174 21.8540i −0.457983 0.793249i
\(760\) 0 0
\(761\) 12.2969 + 7.09961i 0.445762 + 0.257361i 0.706039 0.708173i \(-0.250480\pi\)
−0.260277 + 0.965534i \(0.583814\pi\)
\(762\) 0 0
\(763\) −35.7193 + 2.12385i −1.29313 + 0.0768886i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −3.56104 + 13.2900i −0.128582 + 0.479874i
\(768\) 0 0
\(769\) −12.7272 −0.458955 −0.229478 0.973314i \(-0.573702\pi\)
−0.229478 + 0.973314i \(0.573702\pi\)
\(770\) 0 0
\(771\) 5.17198 0.186264
\(772\) 0 0
\(773\) −6.21166 + 23.1822i −0.223418 + 0.833806i 0.759614 + 0.650374i \(0.225388\pi\)
−0.983032 + 0.183433i \(0.941279\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.05169 4.09763i 0.0736040 0.147002i
\(778\) 0 0
\(779\) −9.67521 5.58599i −0.346651 0.200139i
\(780\) 0 0
\(781\) −32.4189 56.1511i −1.16004 2.00924i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 4.00190 + 1.07231i 0.142652 + 0.0382236i 0.329439 0.944177i \(-0.393141\pi\)
−0.186786 + 0.982401i \(0.559807\pi\)
\(788\) 0 0
\(789\) 7.99196 13.8425i 0.284521 0.492805i
\(790\) 0 0
\(791\) 26.0894 8.68017i 0.927631 0.308631i
\(792\) 0 0
\(793\) −11.7208 43.7426i −0.416218 1.55335i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −13.2306 + 13.2306i −0.468652 + 0.468652i −0.901478 0.432826i \(-0.857517\pi\)
0.432826 + 0.901478i \(0.357517\pi\)
\(798\) 0 0
\(799\) 58.1268i 2.05638i
\(800\) 0 0
\(801\) 6.76946 3.90835i 0.239187 0.138095i
\(802\) 0 0
\(803\) −18.9233 + 5.07048i −0.667788 + 0.178933i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −16.9114 + 4.53140i −0.595310 + 0.159513i
\(808\) 0 0
\(809\) −8.68017 + 5.01150i −0.305179 + 0.176195i −0.644767 0.764379i \(-0.723046\pi\)
0.339588 + 0.940574i \(0.389712\pi\)
\(810\) 0 0
\(811\) 31.1243i 1.09292i −0.837485 0.546461i \(-0.815975\pi\)
0.837485 0.546461i \(-0.184025\pi\)
\(812\) 0 0
\(813\) −7.65988 + 7.65988i −0.268644 + 0.268644i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −14.4317 53.8599i −0.504902 1.88432i
\(818\) 0 0
\(819\) −3.28885 + 16.0260i −0.114922 + 0.559992i
\(820\) 0 0
\(821\) −6.74351 + 11.6801i −0.235350 + 0.407638i −0.959374 0.282136i \(-0.908957\pi\)
0.724024 + 0.689775i \(0.242290\pi\)
\(822\) 0 0
\(823\) −36.7944 9.85902i −1.28257 0.343664i −0.447737 0.894165i \(-0.647770\pi\)
−0.834834 + 0.550502i \(0.814436\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 28.6503 + 28.6503i 0.996270 + 0.996270i 0.999993 0.00372312i \(-0.00118511\pi\)
−0.00372312 + 0.999993i \(0.501185\pi\)
\(828\) 0 0
\(829\) 23.8430 + 41.2973i 0.828102 + 1.43431i 0.899525 + 0.436869i \(0.143913\pi\)
−0.0714231 + 0.997446i \(0.522754\pi\)
\(830\) 0 0
\(831\) 10.1292 + 5.84809i 0.351378 + 0.202868i
\(832\) 0 0
\(833\) −11.7347 29.3305i −0.406582 1.01624i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.17190 4.37357i 0.0405066 0.151173i
\(838\) 0 0
\(839\) −28.8941 −0.997534 −0.498767 0.866736i \(-0.666214\pi\)
−0.498767 + 0.866736i \(0.666214\pi\)
\(840\) 0 0
\(841\) 29.0000 1.00000
\(842\) 0 0
\(843\) 6.60996 24.6687i 0.227659 0.849636i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −20.6891 + 13.6434i −0.710885 + 0.468791i
\(848\) 0 0
\(849\) −14.7574 8.52020i −0.506473 0.292413i
\(850\) 0 0
\(851\) 4.84248 + 8.38741i 0.165998 + 0.287517i
\(852\) 0 0
\(853\) 9.80855 + 9.80855i 0.335838 + 0.335838i 0.854798 0.518960i \(-0.173681\pi\)
−0.518960 + 0.854798i \(0.673681\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −10.6280 2.84777i −0.363047 0.0972781i 0.0726843 0.997355i \(-0.476843\pi\)
−0.435731 + 0.900077i \(0.643510\pi\)
\(858\) 0 0
\(859\) −3.76197 + 6.51593i −0.128357 + 0.222321i −0.923040 0.384704i \(-0.874304\pi\)
0.794683 + 0.607024i \(0.207637\pi\)
\(860\) 0 0
\(861\) 4.40251 + 3.90835i 0.150037 + 0.133196i
\(862\) 0 0
\(863\) −4.21271 15.7221i −0.143402 0.535185i −0.999821 0.0189036i \(-0.993982\pi\)
0.856419 0.516282i \(-0.172684\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.38080 2.38080i 0.0808560 0.0808560i
\(868\) 0 0
\(869\) 8.24983i 0.279856i
\(870\) 0 0
\(871\) −26.8872 + 15.5233i −0.911036 + 0.525987i
\(872\) 0 0
\(873\) 4.82963 1.29410i 0.163458 0.0437985i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 26.3233 7.05332i 0.888876 0.238174i 0.214643 0.976693i \(-0.431141\pi\)
0.674233 + 0.738519i \(0.264474\pi\)
\(878\) 0 0
\(879\) −0.525760 + 0.303548i −0.0177334 + 0.0102384i
\(880\) 0 0
\(881\) 15.9839i 0.538512i −0.963069 0.269256i \(-0.913222\pi\)
0.963069 0.269256i \(-0.0867777\pi\)
\(882\) 0 0
\(883\) 37.9349 37.9349i 1.27661 1.27661i 0.334059 0.942552i \(-0.391581\pi\)
0.942552 0.334059i \(-0.108419\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.35779 + 34.9237i 0.314204 + 1.17262i 0.924729 + 0.380627i \(0.124292\pi\)
−0.610525 + 0.791997i \(0.709041\pi\)
\(888\) 0 0
\(889\) −12.6460 38.0091i −0.424133 1.27478i
\(890\) 0 0
\(891\) 2.25649 3.90835i 0.0755952 0.130935i
\(892\) 0 0
\(893\) 62.4653 + 16.7375i 2.09032 + 0.560100i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −24.4486 24.4486i −0.816316 0.816316i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 47.9433 + 27.6801i 1.59722 + 0.922158i
\(902\) 0 0
\(903\) 1.74399 + 29.3307i 0.0580363 + 0.976065i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 7.04499 26.2923i 0.233925 0.873021i −0.744705 0.667394i \(-0.767410\pi\)
0.978630 0.205627i \(-0.0659234\pi\)
\(908\) 0 0
\(909\) 7.02579 0.233031
\(910\) 0 0
\(911\) −13.1009 −0.434051 −0.217025 0.976166i \(-0.569635\pi\)
−0.217025 + 0.976166i \(0.569635\pi\)
\(912\) 0 0
\(913\) −18.5181 + 69.1107i −0.612861 + 2.28723i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 21.6187 + 32.7831i 0.713913 + 1.08259i
\(918\) 0 0
\(919\) 7.08965 + 4.09321i 0.233866 + 0.135023i 0.612354 0.790583i \(-0.290223\pi\)
−0.378488 + 0.925606i \(0.623556\pi\)
\(920\) 0 0
\(921\) −7.94719 13.7649i −0.261869 0.453570i
\(922\) 0 0
\(923\) −62.8178 62.8178i −2.06767 2.06767i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −4.02982 1.07979i −0.132357 0.0354648i
\(928\) 0 0
\(929\) −14.8136 + 25.6579i −0.486019 + 0.841810i −0.999871 0.0160692i \(-0.994885\pi\)
0.513852 + 0.857879i \(0.328218\pi\)
\(930\) 0 0
\(931\) −34.8987 + 4.16483i −1.14376 + 0.136497i
\(932\) 0 0
\(933\) 0.295421 + 1.10253i 0.00967166 + 0.0360951i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.01202 + 2.01202i −0.0657298 + 0.0657298i −0.739208 0.673478i \(-0.764800\pi\)
0.673478 + 0.739208i \(0.264800\pi\)
\(938\) 0 0
\(939\) 4.50147i 0.146900i
\(940\) 0 0
\(941\) −32.2239 + 18.6045i −1.05047 + 0.606488i −0.922780 0.385326i \(-0.874089\pi\)
−0.127688 + 0.991814i \(0.540756\pi\)
\(942\) 0 0
\(943\) −12.0179 + 3.22019i −0.391357 + 0.104864i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 24.4618 6.55451i 0.794901 0.212993i 0.161557 0.986863i \(-0.448348\pi\)
0.633344 + 0.773870i \(0.281682\pi\)
\(948\) 0 0
\(949\) −23.2463 + 13.4212i −0.754606 + 0.435672i
\(950\) 0 0
\(951\) 9.69129i 0.314262i
\(952\) 0 0
\(953\) −23.4345 + 23.4345i −0.759118 + 0.759118i −0.976162 0.217044i \(-0.930359\pi\)
0.217044 + 0.976162i \(0.430359\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.00489 + 2.25838i −0.0647413 + 0.0729270i
\(960\) 0 0
\(961\) −5.24926 + 9.09199i −0.169331 + 0.293290i
\(962\) 0 0
\(963\) −6.78639 1.81841i −0.218688 0.0585974i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −35.2925 35.2925i −1.13493 1.13493i −0.989346 0.145584i \(-0.953494\pi\)
−0.145584 0.989346i \(-0.546506\pi\)
\(968\) 0 0
\(969\) −11.3296 19.6234i −0.363959 0.630396i
\(970\) 0 0
\(971\) −51.5438 29.7588i −1.65412 0.955006i −0.975353 0.220651i \(-0.929182\pi\)
−0.678765 0.734355i \(-0.737485\pi\)
\(972\) 0 0
\(973\) −48.6795 24.3740i −1.56059 0.781393i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −5.45522 + 20.3592i −0.174528 + 0.651348i 0.822103 + 0.569338i \(0.192800\pi\)
−0.996632 + 0.0820096i \(0.973866\pi\)
\(978\) 0 0
\(979\) 35.2766 1.12744
\(980\) 0 0
\(981\) 13.5245 0.431803
\(982\) 0 0
\(983\) −9.45773 + 35.2967i −0.301655 + 1.12579i 0.634132 + 0.773225i \(0.281358\pi\)
−0.935787 + 0.352566i \(0.885309\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −30.4709 15.2569i −0.969900 0.485631i
\(988\) 0 0
\(989\) −53.7784 31.0490i −1.71005 0.987299i
\(990\) 0 0
\(991\) 14.0903 + 24.4051i 0.447592 + 0.775252i 0.998229 0.0594929i \(-0.0189484\pi\)
−0.550637 + 0.834745i \(0.685615\pi\)
\(992\) 0 0
\(993\) 2.69056 + 2.69056i 0.0853823 + 0.0853823i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 25.0890 + 6.72258i 0.794577 + 0.212906i 0.633202 0.773987i \(-0.281740\pi\)
0.161375 + 0.986893i \(0.448407\pi\)
\(998\) 0 0
\(999\) −0.866025 + 1.50000i −0.0273998 + 0.0474579i
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.ce.c.1657.4 yes 24
5.2 odd 4 inner 2100.2.ce.c.1993.3 yes 24
5.3 odd 4 inner 2100.2.ce.c.1993.4 yes 24
5.4 even 2 inner 2100.2.ce.c.1657.2 yes 24
7.3 odd 6 inner 2100.2.ce.c.157.4 yes 24
35.3 even 12 inner 2100.2.ce.c.493.4 yes 24
35.17 even 12 inner 2100.2.ce.c.493.2 yes 24
35.24 odd 6 inner 2100.2.ce.c.157.3 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.ce.c.157.3 24 35.24 odd 6 inner
2100.2.ce.c.157.4 yes 24 7.3 odd 6 inner
2100.2.ce.c.493.2 yes 24 35.17 even 12 inner
2100.2.ce.c.493.4 yes 24 35.3 even 12 inner
2100.2.ce.c.1657.2 yes 24 5.4 even 2 inner
2100.2.ce.c.1657.4 yes 24 1.1 even 1 trivial
2100.2.ce.c.1993.3 yes 24 5.2 odd 4 inner
2100.2.ce.c.1993.4 yes 24 5.3 odd 4 inner