Properties

Label 2100.2.bc.c.1549.1
Level $2100$
Weight $2$
Character 2100.1549
Analytic conductor $16.769$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1549.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1549
Dual form 2100.2.bc.c.949.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 0.500000i) q^{3} +(-0.866025 - 2.50000i) q^{7} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\) \(q+(-0.866025 + 0.500000i) q^{3} +(-0.866025 - 2.50000i) q^{7} +(0.500000 - 0.866025i) q^{9} +(1.00000 + 1.73205i) q^{11} +1.00000i q^{13} +(3.46410 - 2.00000i) q^{17} +(-0.500000 + 0.866025i) q^{19} +(2.00000 + 1.73205i) q^{21} +(-3.46410 - 2.00000i) q^{23} +1.00000i q^{27} +(2.50000 + 4.33013i) q^{31} +(-1.73205 - 1.00000i) q^{33} +(-4.33013 - 2.50000i) q^{37} +(-0.500000 - 0.866025i) q^{39} +2.00000 q^{41} -9.00000i q^{43} +(-1.73205 - 1.00000i) q^{47} +(-5.50000 + 4.33013i) q^{49} +(-2.00000 + 3.46410i) q^{51} +(10.3923 - 6.00000i) q^{53} -1.00000i q^{57} +(-4.00000 - 6.92820i) q^{59} +(7.00000 - 12.1244i) q^{61} +(-2.59808 - 0.500000i) q^{63} +(-7.79423 + 4.50000i) q^{67} +4.00000 q^{69} +2.00000 q^{71} +(0.866025 - 0.500000i) q^{73} +(3.46410 - 4.00000i) q^{77} +(-1.50000 + 2.59808i) q^{79} +(-0.500000 - 0.866025i) q^{81} -18.0000i q^{83} +(-2.00000 + 3.46410i) q^{89} +(2.50000 - 0.866025i) q^{91} +(-4.33013 - 2.50000i) q^{93} -10.0000i q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} + 4 q^{11} - 2 q^{19} + 8 q^{21} + 10 q^{31} - 2 q^{39} + 8 q^{41} - 22 q^{49} - 8 q^{51} - 16 q^{59} + 28 q^{61} + 16 q^{69} + 8 q^{71} - 6 q^{79} - 2 q^{81} - 8 q^{89} + 10 q^{91} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(7\) −0.866025 2.50000i −0.327327 0.944911i
\(8\) 0 0
\(9\) 0.500000 0.866025i 0.166667 0.288675i
\(10\) 0 0
\(11\) 1.00000 + 1.73205i 0.301511 + 0.522233i 0.976478 0.215615i \(-0.0691756\pi\)
−0.674967 + 0.737848i \(0.735842\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i 0.990338 + 0.138675i \(0.0442844\pi\)
−0.990338 + 0.138675i \(0.955716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410 2.00000i 0.840168 0.485071i −0.0171533 0.999853i \(-0.505460\pi\)
0.857321 + 0.514782i \(0.172127\pi\)
\(18\) 0 0
\(19\) −0.500000 + 0.866025i −0.114708 + 0.198680i −0.917663 0.397360i \(-0.869927\pi\)
0.802955 + 0.596040i \(0.203260\pi\)
\(20\) 0 0
\(21\) 2.00000 + 1.73205i 0.436436 + 0.377964i
\(22\) 0 0
\(23\) −3.46410 2.00000i −0.722315 0.417029i 0.0932891 0.995639i \(-0.470262\pi\)
−0.815604 + 0.578610i \(0.803595\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2.50000 + 4.33013i 0.449013 + 0.777714i 0.998322 0.0579057i \(-0.0184423\pi\)
−0.549309 + 0.835619i \(0.685109\pi\)
\(32\) 0 0
\(33\) −1.73205 1.00000i −0.301511 0.174078i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.33013 2.50000i −0.711868 0.410997i 0.0998840 0.994999i \(-0.468153\pi\)
−0.811752 + 0.584002i \(0.801486\pi\)
\(38\) 0 0
\(39\) −0.500000 0.866025i −0.0800641 0.138675i
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 9.00000i 1.37249i −0.727372 0.686244i \(-0.759258\pi\)
0.727372 0.686244i \(-0.240742\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.73205 1.00000i −0.252646 0.145865i 0.368329 0.929695i \(-0.379930\pi\)
−0.620975 + 0.783830i \(0.713263\pi\)
\(48\) 0 0
\(49\) −5.50000 + 4.33013i −0.785714 + 0.618590i
\(50\) 0 0
\(51\) −2.00000 + 3.46410i −0.280056 + 0.485071i
\(52\) 0 0
\(53\) 10.3923 6.00000i 1.42749 0.824163i 0.430570 0.902557i \(-0.358312\pi\)
0.996922 + 0.0783936i \(0.0249791\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) 0 0
\(59\) −4.00000 6.92820i −0.520756 0.901975i −0.999709 0.0241347i \(-0.992317\pi\)
0.478953 0.877841i \(-0.341016\pi\)
\(60\) 0 0
\(61\) 7.00000 12.1244i 0.896258 1.55236i 0.0640184 0.997949i \(-0.479608\pi\)
0.832240 0.554416i \(-0.187058\pi\)
\(62\) 0 0
\(63\) −2.59808 0.500000i −0.327327 0.0629941i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.79423 + 4.50000i −0.952217 + 0.549762i −0.893769 0.448528i \(-0.851948\pi\)
−0.0584478 + 0.998290i \(0.518615\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 0.866025 0.500000i 0.101361 0.0585206i −0.448463 0.893801i \(-0.648028\pi\)
0.549823 + 0.835281i \(0.314695\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.46410 4.00000i 0.394771 0.455842i
\(78\) 0 0
\(79\) −1.50000 + 2.59808i −0.168763 + 0.292306i −0.937985 0.346675i \(-0.887311\pi\)
0.769222 + 0.638982i \(0.220644\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 18.0000i 1.97576i −0.155230 0.987878i \(-0.549612\pi\)
0.155230 0.987878i \(-0.450388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.00000 + 3.46410i −0.212000 + 0.367194i −0.952340 0.305038i \(-0.901331\pi\)
0.740341 + 0.672232i \(0.234664\pi\)
\(90\) 0 0
\(91\) 2.50000 0.866025i 0.262071 0.0907841i
\(92\) 0 0
\(93\) −4.33013 2.50000i −0.449013 0.259238i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i −0.861550 0.507673i \(-0.830506\pi\)
0.861550 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) −3.00000 5.19615i −0.298511 0.517036i 0.677284 0.735721i \(-0.263157\pi\)
−0.975796 + 0.218685i \(0.929823\pi\)
\(102\) 0 0
\(103\) 0.866025 + 0.500000i 0.0853320 + 0.0492665i 0.542059 0.840341i \(-0.317645\pi\)
−0.456727 + 0.889607i \(0.650978\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.46410 + 2.00000i 0.334887 + 0.193347i 0.658009 0.753010i \(-0.271399\pi\)
−0.323122 + 0.946357i \(0.604732\pi\)
\(108\) 0 0
\(109\) 6.50000 + 11.2583i 0.622587 + 1.07835i 0.989002 + 0.147901i \(0.0472517\pi\)
−0.366415 + 0.930451i \(0.619415\pi\)
\(110\) 0 0
\(111\) 5.00000 0.474579
\(112\) 0 0
\(113\) 14.0000i 1.31701i −0.752577 0.658505i \(-0.771189\pi\)
0.752577 0.658505i \(-0.228811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.866025 + 0.500000i 0.0800641 + 0.0462250i
\(118\) 0 0
\(119\) −8.00000 6.92820i −0.733359 0.635107i
\(120\) 0 0
\(121\) 3.50000 6.06218i 0.318182 0.551107i
\(122\) 0 0
\(123\) −1.73205 + 1.00000i −0.156174 + 0.0901670i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 9.00000i 0.798621i 0.916816 + 0.399310i \(0.130750\pi\)
−0.916816 + 0.399310i \(0.869250\pi\)
\(128\) 0 0
\(129\) 4.50000 + 7.79423i 0.396203 + 0.686244i
\(130\) 0 0
\(131\) 9.00000 15.5885i 0.786334 1.36197i −0.141865 0.989886i \(-0.545310\pi\)
0.928199 0.372084i \(-0.121357\pi\)
\(132\) 0 0
\(133\) 2.59808 + 0.500000i 0.225282 + 0.0433555i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.8564 + 8.00000i −1.18383 + 0.683486i −0.956898 0.290424i \(-0.906204\pi\)
−0.226935 + 0.973910i \(0.572870\pi\)
\(138\) 0 0
\(139\) 13.0000 1.10265 0.551323 0.834292i \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 0 0
\(143\) −1.73205 + 1.00000i −0.144841 + 0.0836242i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 2.59808 6.50000i 0.214286 0.536111i
\(148\) 0 0
\(149\) 6.00000 10.3923i 0.491539 0.851371i −0.508413 0.861113i \(-0.669768\pi\)
0.999953 + 0.00974235i \(0.00310113\pi\)
\(150\) 0 0
\(151\) 8.00000 + 13.8564i 0.651031 + 1.12762i 0.982873 + 0.184284i \(0.0589965\pi\)
−0.331842 + 0.943335i \(0.607670\pi\)
\(152\) 0 0
\(153\) 4.00000i 0.323381i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 19.0526 11.0000i 1.52056 0.877896i 0.520854 0.853646i \(-0.325614\pi\)
0.999706 0.0242497i \(-0.00771967\pi\)
\(158\) 0 0
\(159\) −6.00000 + 10.3923i −0.475831 + 0.824163i
\(160\) 0 0
\(161\) −2.00000 + 10.3923i −0.157622 + 0.819028i
\(162\) 0 0
\(163\) −10.3923 6.00000i −0.813988 0.469956i 0.0343508 0.999410i \(-0.489064\pi\)
−0.848339 + 0.529454i \(0.822397\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.00000i 0.154765i −0.997001 0.0773823i \(-0.975344\pi\)
0.997001 0.0773823i \(-0.0246562\pi\)
\(168\) 0 0
\(169\) 12.0000 0.923077
\(170\) 0 0
\(171\) 0.500000 + 0.866025i 0.0382360 + 0.0662266i
\(172\) 0 0
\(173\) 13.8564 + 8.00000i 1.05348 + 0.608229i 0.923622 0.383304i \(-0.125214\pi\)
0.129861 + 0.991532i \(0.458547\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.92820 + 4.00000i 0.520756 + 0.300658i
\(178\) 0 0
\(179\) −13.0000 22.5167i −0.971666 1.68297i −0.690526 0.723307i \(-0.742621\pi\)
−0.281139 0.959667i \(-0.590712\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 0 0
\(183\) 14.0000i 1.03491i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 6.92820 + 4.00000i 0.506640 + 0.292509i
\(188\) 0 0
\(189\) 2.50000 0.866025i 0.181848 0.0629941i
\(190\) 0 0
\(191\) 5.00000 8.66025i 0.361787 0.626634i −0.626468 0.779447i \(-0.715500\pi\)
0.988255 + 0.152813i \(0.0488333\pi\)
\(192\) 0 0
\(193\) −14.7224 + 8.50000i −1.05974 + 0.611843i −0.925361 0.379086i \(-0.876238\pi\)
−0.134382 + 0.990930i \(0.542905\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0000i 0.854965i −0.904024 0.427482i \(-0.859401\pi\)
0.904024 0.427482i \(-0.140599\pi\)
\(198\) 0 0
\(199\) −8.00000 13.8564i −0.567105 0.982255i −0.996850 0.0793045i \(-0.974730\pi\)
0.429745 0.902950i \(-0.358603\pi\)
\(200\) 0 0
\(201\) 4.50000 7.79423i 0.317406 0.549762i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.46410 + 2.00000i −0.240772 + 0.139010i
\(208\) 0 0
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) −28.0000 −1.92760 −0.963800 0.266627i \(-0.914091\pi\)
−0.963800 + 0.266627i \(0.914091\pi\)
\(212\) 0 0
\(213\) −1.73205 + 1.00000i −0.118678 + 0.0685189i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.66025 10.0000i 0.587896 0.678844i
\(218\) 0 0
\(219\) −0.500000 + 0.866025i −0.0337869 + 0.0585206i
\(220\) 0 0
\(221\) 2.00000 + 3.46410i 0.134535 + 0.233021i
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −15.5885 + 9.00000i −1.03464 + 0.597351i −0.918311 0.395860i \(-0.870447\pi\)
−0.116331 + 0.993210i \(0.537113\pi\)
\(228\) 0 0
\(229\) 0.500000 0.866025i 0.0330409 0.0572286i −0.849032 0.528341i \(-0.822814\pi\)
0.882073 + 0.471113i \(0.156147\pi\)
\(230\) 0 0
\(231\) −1.00000 + 5.19615i −0.0657952 + 0.341882i
\(232\) 0 0
\(233\) 1.73205 + 1.00000i 0.113470 + 0.0655122i 0.555661 0.831409i \(-0.312465\pi\)
−0.442191 + 0.896921i \(0.645799\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 3.00000i 0.194871i
\(238\) 0 0
\(239\) −26.0000 −1.68180 −0.840900 0.541190i \(-0.817974\pi\)
−0.840900 + 0.541190i \(0.817974\pi\)
\(240\) 0 0
\(241\) 9.00000 + 15.5885i 0.579741 + 1.00414i 0.995509 + 0.0946700i \(0.0301796\pi\)
−0.415768 + 0.909471i \(0.636487\pi\)
\(242\) 0 0
\(243\) 0.866025 + 0.500000i 0.0555556 + 0.0320750i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.866025 0.500000i −0.0551039 0.0318142i
\(248\) 0 0
\(249\) 9.00000 + 15.5885i 0.570352 + 0.987878i
\(250\) 0 0
\(251\) 4.00000 0.252478 0.126239 0.992000i \(-0.459709\pi\)
0.126239 + 0.992000i \(0.459709\pi\)
\(252\) 0 0
\(253\) 8.00000i 0.502956i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 12.1244 + 7.00000i 0.756297 + 0.436648i 0.827964 0.560781i \(-0.189499\pi\)
−0.0716680 + 0.997429i \(0.522832\pi\)
\(258\) 0 0
\(259\) −2.50000 + 12.9904i −0.155342 + 0.807183i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 10.3923 6.00000i 0.640817 0.369976i −0.144112 0.989561i \(-0.546033\pi\)
0.784929 + 0.619586i \(0.212699\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4.00000i 0.244796i
\(268\) 0 0
\(269\) −9.00000 15.5885i −0.548740 0.950445i −0.998361 0.0572259i \(-0.981774\pi\)
0.449622 0.893219i \(-0.351559\pi\)
\(270\) 0 0
\(271\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(272\) 0 0
\(273\) −1.73205 + 2.00000i −0.104828 + 0.121046i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.33013 + 2.50000i −0.260172 + 0.150210i −0.624413 0.781094i \(-0.714662\pi\)
0.364241 + 0.931305i \(0.381328\pi\)
\(278\) 0 0
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) 0 0
\(283\) −21.6506 + 12.5000i −1.28700 + 0.743048i −0.978117 0.208053i \(-0.933287\pi\)
−0.308879 + 0.951101i \(0.599954\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.73205 5.00000i −0.102240 0.295141i
\(288\) 0 0
\(289\) −0.500000 + 0.866025i −0.0294118 + 0.0509427i
\(290\) 0 0
\(291\) 5.00000 + 8.66025i 0.293105 + 0.507673i
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.73205 + 1.00000i −0.100504 + 0.0580259i
\(298\) 0 0
\(299\) 2.00000 3.46410i 0.115663 0.200334i
\(300\) 0 0
\(301\) −22.5000 + 7.79423i −1.29688 + 0.449252i
\(302\) 0 0
\(303\) 5.19615 + 3.00000i 0.298511 + 0.172345i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 19.0000i 1.08439i 0.840254 + 0.542194i \(0.182406\pi\)
−0.840254 + 0.542194i \(0.817594\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) 0 0
\(311\) −13.0000 22.5167i −0.737162 1.27680i −0.953768 0.300544i \(-0.902832\pi\)
0.216606 0.976259i \(-0.430501\pi\)
\(312\) 0 0
\(313\) −16.4545 9.50000i −0.930062 0.536972i −0.0432311 0.999065i \(-0.513765\pi\)
−0.886831 + 0.462093i \(0.847098\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.46410 2.00000i −0.194563 0.112331i 0.399554 0.916710i \(-0.369165\pi\)
−0.594117 + 0.804379i \(0.702498\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 4.00000i 0.222566i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −11.2583 6.50000i −0.622587 0.359451i
\(328\) 0 0
\(329\) −1.00000 + 5.19615i −0.0551318 + 0.286473i
\(330\) 0 0
\(331\) −0.500000 + 0.866025i −0.0274825 + 0.0476011i −0.879440 0.476011i \(-0.842082\pi\)
0.851957 + 0.523612i \(0.175416\pi\)
\(332\) 0 0
\(333\) −4.33013 + 2.50000i −0.237289 + 0.136999i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 7.00000i 0.381314i 0.981657 + 0.190657i \(0.0610619\pi\)
−0.981657 + 0.190657i \(0.938938\pi\)
\(338\) 0 0
\(339\) 7.00000 + 12.1244i 0.380188 + 0.658505i
\(340\) 0 0
\(341\) −5.00000 + 8.66025i −0.270765 + 0.468979i
\(342\) 0 0
\(343\) 15.5885 + 10.0000i 0.841698 + 0.539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.7846 + 12.0000i −1.11578 + 0.644194i −0.940319 0.340293i \(-0.889474\pi\)
−0.175457 + 0.984487i \(0.556140\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) 25.9808 15.0000i 1.38282 0.798369i 0.390324 0.920677i \(-0.372363\pi\)
0.992492 + 0.122308i \(0.0390296\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 10.3923 + 2.00000i 0.550019 + 0.105851i
\(358\) 0 0
\(359\) 8.00000 13.8564i 0.422224 0.731313i −0.573933 0.818902i \(-0.694583\pi\)
0.996157 + 0.0875892i \(0.0279163\pi\)
\(360\) 0 0
\(361\) 9.00000 + 15.5885i 0.473684 + 0.820445i
\(362\) 0 0
\(363\) 7.00000i 0.367405i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.866025 + 0.500000i −0.0452062 + 0.0260998i −0.522433 0.852680i \(-0.674975\pi\)
0.477227 + 0.878780i \(0.341642\pi\)
\(368\) 0 0
\(369\) 1.00000 1.73205i 0.0520579 0.0901670i
\(370\) 0 0
\(371\) −24.0000 20.7846i −1.24602 1.07908i
\(372\) 0 0
\(373\) 21.6506 + 12.5000i 1.12103 + 0.647225i 0.941663 0.336557i \(-0.109263\pi\)
0.179364 + 0.983783i \(0.442596\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 11.0000 0.565032 0.282516 0.959263i \(-0.408831\pi\)
0.282516 + 0.959263i \(0.408831\pi\)
\(380\) 0 0
\(381\) −4.50000 7.79423i −0.230542 0.399310i
\(382\) 0 0
\(383\) 31.1769 + 18.0000i 1.59307 + 0.919757i 0.992777 + 0.119974i \(0.0382810\pi\)
0.600289 + 0.799783i \(0.295052\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −7.79423 4.50000i −0.396203 0.228748i
\(388\) 0 0
\(389\) 15.0000 + 25.9808i 0.760530 + 1.31728i 0.942578 + 0.333987i \(0.108394\pi\)
−0.182047 + 0.983290i \(0.558272\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 18.0000i 0.907980i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 26.8468 + 15.5000i 1.34740 + 0.777923i 0.987881 0.155214i \(-0.0496068\pi\)
0.359521 + 0.933137i \(0.382940\pi\)
\(398\) 0 0
\(399\) −2.50000 + 0.866025i −0.125157 + 0.0433555i
\(400\) 0 0
\(401\) −6.00000 + 10.3923i −0.299626 + 0.518967i −0.976050 0.217545i \(-0.930195\pi\)
0.676425 + 0.736512i \(0.263528\pi\)
\(402\) 0 0
\(403\) −4.33013 + 2.50000i −0.215699 + 0.124534i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.0000i 0.495682i
\(408\) 0 0
\(409\) 6.50000 + 11.2583i 0.321404 + 0.556689i 0.980778 0.195127i \(-0.0625118\pi\)
−0.659374 + 0.751815i \(0.729178\pi\)
\(410\) 0 0
\(411\) 8.00000 13.8564i 0.394611 0.683486i
\(412\) 0 0
\(413\) −13.8564 + 16.0000i −0.681829 + 0.787309i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −11.2583 + 6.50000i −0.551323 + 0.318306i
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −35.0000 −1.70580 −0.852898 0.522078i \(-0.825157\pi\)
−0.852898 + 0.522078i \(0.825157\pi\)
\(422\) 0 0
\(423\) −1.73205 + 1.00000i −0.0842152 + 0.0486217i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −36.3731 7.00000i −1.76022 0.338754i
\(428\) 0 0
\(429\) 1.00000 1.73205i 0.0482805 0.0836242i
\(430\) 0 0
\(431\) 15.0000 + 25.9808i 0.722525 + 1.25145i 0.959985 + 0.280052i \(0.0903517\pi\)
−0.237460 + 0.971397i \(0.576315\pi\)
\(432\) 0 0
\(433\) 27.0000i 1.29754i 0.760986 + 0.648769i \(0.224716\pi\)
−0.760986 + 0.648769i \(0.775284\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.46410 2.00000i 0.165710 0.0956730i
\(438\) 0 0
\(439\) 4.00000 6.92820i 0.190910 0.330665i −0.754642 0.656136i \(-0.772190\pi\)
0.945552 + 0.325471i \(0.105523\pi\)
\(440\) 0 0
\(441\) 1.00000 + 6.92820i 0.0476190 + 0.329914i
\(442\) 0 0
\(443\) 3.46410 + 2.00000i 0.164584 + 0.0950229i 0.580030 0.814595i \(-0.303041\pi\)
−0.415445 + 0.909618i \(0.636374\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 12.0000i 0.567581i
\(448\) 0 0
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) 2.00000 + 3.46410i 0.0941763 + 0.163118i
\(452\) 0 0
\(453\) −13.8564 8.00000i −0.651031 0.375873i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.866025 + 0.500000i 0.0405110 + 0.0233890i 0.520119 0.854094i \(-0.325888\pi\)
−0.479608 + 0.877483i \(0.659221\pi\)
\(458\) 0 0
\(459\) 2.00000 + 3.46410i 0.0933520 + 0.161690i
\(460\) 0 0
\(461\) −16.0000 −0.745194 −0.372597 0.927993i \(-0.621533\pi\)
−0.372597 + 0.927993i \(0.621533\pi\)
\(462\) 0 0
\(463\) 41.0000i 1.90543i 0.303863 + 0.952716i \(0.401724\pi\)
−0.303863 + 0.952716i \(0.598276\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 29.4449 + 17.0000i 1.36255 + 0.786666i 0.989962 0.141332i \(-0.0451386\pi\)
0.372584 + 0.927999i \(0.378472\pi\)
\(468\) 0 0
\(469\) 18.0000 + 15.5885i 0.831163 + 0.719808i
\(470\) 0 0
\(471\) −11.0000 + 19.0526i −0.506853 + 0.877896i
\(472\) 0 0
\(473\) 15.5885 9.00000i 0.716758 0.413820i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000i 0.549442i
\(478\) 0 0
\(479\) 6.00000 + 10.3923i 0.274147 + 0.474837i 0.969920 0.243426i \(-0.0782712\pi\)
−0.695773 + 0.718262i \(0.744938\pi\)
\(480\) 0 0
\(481\) 2.50000 4.33013i 0.113990 0.197437i
\(482\) 0 0
\(483\) −3.46410 10.0000i −0.157622 0.455016i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −28.5788 + 16.5000i −1.29503 + 0.747686i −0.979541 0.201243i \(-0.935502\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.73205 5.00000i −0.0776931 0.224281i
\(498\) 0 0
\(499\) −2.50000 + 4.33013i −0.111915 + 0.193843i −0.916542 0.399937i \(-0.869032\pi\)
0.804627 + 0.593780i \(0.202365\pi\)
\(500\) 0 0
\(501\) 1.00000 + 1.73205i 0.0446767 + 0.0773823i
\(502\) 0 0
\(503\) 10.0000i 0.445878i 0.974832 + 0.222939i \(0.0715651\pi\)
−0.974832 + 0.222939i \(0.928435\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −10.3923 + 6.00000i −0.461538 + 0.266469i
\(508\) 0 0
\(509\) 13.0000 22.5167i 0.576215 0.998033i −0.419694 0.907666i \(-0.637862\pi\)
0.995908 0.0903676i \(-0.0288042\pi\)
\(510\) 0 0
\(511\) −2.00000 1.73205i −0.0884748 0.0766214i
\(512\) 0 0
\(513\) −0.866025 0.500000i −0.0382360 0.0220755i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 4.00000i 0.175920i
\(518\) 0 0
\(519\) −16.0000 −0.702322
\(520\) 0 0
\(521\) −18.0000 31.1769i −0.788594 1.36589i −0.926828 0.375486i \(-0.877476\pi\)
0.138234 0.990400i \(-0.455857\pi\)
\(522\) 0 0
\(523\) −25.1147 14.5000i −1.09819 0.634041i −0.162446 0.986718i \(-0.551938\pi\)
−0.935745 + 0.352677i \(0.885272\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 17.3205 + 10.0000i 0.754493 + 0.435607i
\(528\) 0 0
\(529\) −3.50000 6.06218i −0.152174 0.263573i
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 2.00000i 0.0866296i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 22.5167 + 13.0000i 0.971666 + 0.560991i
\(538\) 0 0
\(539\) −13.0000 5.19615i −0.559950 0.223814i
\(540\) 0 0
\(541\) 15.5000 26.8468i 0.666397 1.15423i −0.312507 0.949915i \(-0.601169\pi\)
0.978905 0.204318i \(-0.0654977\pi\)
\(542\) 0 0
\(543\) −14.7224 + 8.50000i −0.631800 + 0.364770i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000i 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) 0 0
\(549\) −7.00000 12.1244i −0.298753 0.517455i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 7.79423 + 1.50000i 0.331444 + 0.0637865i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.73205 1.00000i 0.0733893 0.0423714i −0.462856 0.886433i \(-0.653175\pi\)
0.536246 + 0.844062i \(0.319842\pi\)
\(558\) 0 0
\(559\) 9.00000 0.380659
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) −25.9808 + 15.0000i −1.09496 + 0.632175i −0.934892 0.354932i \(-0.884504\pi\)
−0.160066 + 0.987106i \(0.551171\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.73205 + 2.00000i −0.0727393 + 0.0839921i
\(568\) 0 0
\(569\) −3.00000 + 5.19615i −0.125767 + 0.217834i −0.922032 0.387113i \(-0.873472\pi\)
0.796266 + 0.604947i \(0.206806\pi\)
\(570\) 0 0
\(571\) 2.50000 + 4.33013i 0.104622 + 0.181210i 0.913584 0.406651i \(-0.133303\pi\)
−0.808962 + 0.587861i \(0.799970\pi\)
\(572\) 0 0
\(573\) 10.0000i 0.417756i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 11.2583 6.50000i 0.468690 0.270599i −0.247001 0.969015i \(-0.579445\pi\)
0.715691 + 0.698417i \(0.246112\pi\)
\(578\) 0 0
\(579\) 8.50000 14.7224i 0.353248 0.611843i
\(580\) 0 0
\(581\) −45.0000 + 15.5885i −1.86691 + 0.646718i
\(582\) 0 0
\(583\) 20.7846 + 12.0000i 0.860811 + 0.496989i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.0000i 1.48588i 0.669359 + 0.742940i \(0.266569\pi\)
−0.669359 + 0.742940i \(0.733431\pi\)
\(588\) 0 0
\(589\) −5.00000 −0.206021
\(590\) 0 0
\(591\) 6.00000 + 10.3923i 0.246807 + 0.427482i
\(592\) 0 0
\(593\) −5.19615 3.00000i −0.213380 0.123195i 0.389501 0.921026i \(-0.372647\pi\)
−0.602881 + 0.797831i \(0.705981\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13.8564 + 8.00000i 0.567105 + 0.327418i
\(598\) 0 0
\(599\) −14.0000 24.2487i −0.572024 0.990775i −0.996358 0.0852695i \(-0.972825\pi\)
0.424333 0.905506i \(-0.360508\pi\)
\(600\) 0 0
\(601\) 23.0000 0.938190 0.469095 0.883148i \(-0.344580\pi\)
0.469095 + 0.883148i \(0.344580\pi\)
\(602\) 0 0
\(603\) 9.00000i 0.366508i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 7.79423 + 4.50000i 0.316358 + 0.182649i 0.649768 0.760133i \(-0.274866\pi\)
−0.333410 + 0.942782i \(0.608199\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.00000 1.73205i 0.0404557 0.0700713i
\(612\) 0 0
\(613\) 1.73205 1.00000i 0.0699569 0.0403896i −0.464614 0.885514i \(-0.653807\pi\)
0.534570 + 0.845124i \(0.320473\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 34.0000i 1.36879i −0.729112 0.684394i \(-0.760067\pi\)
0.729112 0.684394i \(-0.239933\pi\)
\(618\) 0 0
\(619\) −5.50000 9.52628i −0.221064 0.382893i 0.734068 0.679076i \(-0.237620\pi\)
−0.955131 + 0.296183i \(0.904286\pi\)
\(620\) 0 0
\(621\) 2.00000 3.46410i 0.0802572 0.139010i
\(622\) 0 0
\(623\) 10.3923 + 2.00000i 0.416359 + 0.0801283i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.73205 1.00000i 0.0691714 0.0399362i
\(628\) 0 0
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 24.2487 14.0000i 0.963800 0.556450i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −4.33013 5.50000i −0.171566 0.217918i
\(638\) 0 0
\(639\) 1.00000 1.73205i 0.0395594 0.0685189i
\(640\) 0 0
\(641\) 18.0000 + 31.1769i 0.710957 + 1.23141i 0.964498 + 0.264089i \(0.0850714\pi\)
−0.253541 + 0.967325i \(0.581595\pi\)
\(642\) 0 0
\(643\) 7.00000i 0.276053i 0.990429 + 0.138027i \(0.0440759\pi\)
−0.990429 + 0.138027i \(0.955924\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.1244 7.00000i 0.476658 0.275198i −0.242365 0.970185i \(-0.577923\pi\)
0.719023 + 0.694987i \(0.244590\pi\)
\(648\) 0 0
\(649\) 8.00000 13.8564i 0.314027 0.543912i
\(650\) 0 0
\(651\) −2.50000 + 12.9904i −0.0979827 + 0.509133i
\(652\) 0 0
\(653\) −12.1244 7.00000i −0.474463 0.273931i 0.243643 0.969865i \(-0.421657\pi\)
−0.718106 + 0.695934i \(0.754991\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.00000i 0.0390137i
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 0 0
\(661\) −17.5000 30.3109i −0.680671 1.17896i −0.974776 0.223184i \(-0.928355\pi\)
0.294105 0.955773i \(-0.404978\pi\)
\(662\) 0 0
\(663\) −3.46410 2.00000i −0.134535 0.0776736i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 28.0000 1.08093
\(672\) 0 0
\(673\) 11.0000i 0.424019i 0.977268 + 0.212009i \(0.0680008\pi\)
−0.977268 + 0.212009i \(0.931999\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 20.7846 + 12.0000i 0.798817 + 0.461197i 0.843057 0.537823i \(-0.180753\pi\)
−0.0442400 + 0.999021i \(0.514087\pi\)
\(678\) 0 0
\(679\) −25.0000 + 8.66025i −0.959412 + 0.332350i
\(680\) 0 0
\(681\) 9.00000 15.5885i 0.344881 0.597351i
\(682\) 0 0
\(683\) −13.8564 + 8.00000i −0.530201 + 0.306111i −0.741098 0.671397i \(-0.765695\pi\)
0.210898 + 0.977508i \(0.432361\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 1.00000i 0.0381524i
\(688\) 0 0
\(689\) 6.00000 + 10.3923i 0.228582 + 0.395915i
\(690\) 0 0
\(691\) 5.50000 9.52628i 0.209230 0.362397i −0.742242 0.670132i \(-0.766238\pi\)
0.951472 + 0.307735i \(0.0995710\pi\)
\(692\) 0 0
\(693\) −1.73205 5.00000i −0.0657952 0.189934i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 6.92820 4.00000i 0.262424 0.151511i
\(698\) 0 0
\(699\) −2.00000 −0.0756469
\(700\) 0 0
\(701\) 36.0000 1.35970 0.679851 0.733351i \(-0.262045\pi\)
0.679851 + 0.733351i \(0.262045\pi\)
\(702\) 0 0
\(703\) 4.33013 2.50000i 0.163314 0.0942893i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10.3923 + 12.0000i −0.390843 + 0.451306i
\(708\) 0 0
\(709\) −13.0000 + 22.5167i −0.488225 + 0.845631i −0.999908 0.0135434i \(-0.995689\pi\)
0.511683 + 0.859174i \(0.329022\pi\)
\(710\) 0 0
\(711\) 1.50000 + 2.59808i 0.0562544 + 0.0974355i
\(712\) 0 0
\(713\) 20.0000i 0.749006i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 22.5167 13.0000i 0.840900 0.485494i
\(718\) 0 0
\(719\) 21.0000 36.3731i 0.783168 1.35649i −0.146920 0.989148i \(-0.546936\pi\)
0.930087 0.367338i \(-0.119731\pi\)
\(720\) 0 0
\(721\) 0.500000 2.59808i 0.0186210 0.0967574i
\(722\) 0 0
\(723\) −15.5885 9.00000i −0.579741 0.334714i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.00000i 0.111264i 0.998451 + 0.0556319i \(0.0177173\pi\)
−0.998451 + 0.0556319i \(0.982283\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −18.0000 31.1769i −0.665754 1.15312i
\(732\) 0 0
\(733\) −7.79423 4.50000i −0.287886 0.166211i 0.349102 0.937085i \(-0.386487\pi\)
−0.636988 + 0.770873i \(0.719820\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −15.5885 9.00000i −0.574208 0.331519i
\(738\) 0 0
\(739\) 7.50000 + 12.9904i 0.275892 + 0.477859i 0.970360 0.241665i \(-0.0776935\pi\)
−0.694468 + 0.719524i \(0.744360\pi\)
\(740\) 0 0
\(741\) 1.00000 0.0367359
\(742\) 0 0
\(743\) 6.00000i 0.220119i −0.993925 0.110059i \(-0.964896\pi\)
0.993925 0.110059i \(-0.0351041\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −15.5885 9.00000i −0.570352 0.329293i
\(748\) 0 0
\(749\) 2.00000 10.3923i 0.0730784 0.379727i
\(750\) 0 0
\(751\) −15.5000 + 26.8468i −0.565603 + 0.979653i 0.431390 + 0.902165i \(0.358023\pi\)
−0.996993 + 0.0774878i \(0.975310\pi\)
\(752\) 0 0
\(753\) −3.46410 + 2.00000i −0.126239 + 0.0728841i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22.0000i 0.799604i 0.916602 + 0.399802i \(0.130921\pi\)
−0.916602 + 0.399802i \(0.869079\pi\)
\(758\) 0 0
\(759\) 4.00000 + 6.92820i 0.145191 + 0.251478i
\(760\) 0 0
\(761\) −10.0000 + 17.3205i −0.362500 + 0.627868i −0.988372 0.152058i \(-0.951410\pi\)
0.625872 + 0.779926i \(0.284743\pi\)
\(762\) 0 0
\(763\) 22.5167 26.0000i 0.815158 0.941263i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 6.92820 4.00000i 0.250163 0.144432i
\(768\) 0 0
\(769\) −31.0000 −1.11789 −0.558944 0.829205i \(-0.688793\pi\)
−0.558944 + 0.829205i \(0.688793\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 0 0
\(773\) −12.1244 + 7.00000i −0.436083 + 0.251773i −0.701935 0.712241i \(-0.747680\pi\)
0.265852 + 0.964014i \(0.414347\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −4.33013 12.5000i −0.155342 0.448435i
\(778\) 0 0
\(779\) −1.00000 + 1.73205i −0.0358287 + 0.0620572i
\(780\) 0 0
\(781\) 2.00000 + 3.46410i 0.0715656 + 0.123955i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 13.8564 8.00000i 0.493928 0.285169i −0.232275 0.972650i \(-0.574617\pi\)
0.726202 + 0.687481i \(0.241284\pi\)
\(788\) 0 0
\(789\) −6.00000 + 10.3923i −0.213606 + 0.369976i
\(790\) 0 0
\(791\) −35.0000 + 12.1244i −1.24446 + 0.431092i
\(792\) 0 0
\(793\) 12.1244 + 7.00000i 0.430548 + 0.248577i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000i 0.425062i −0.977154 0.212531i \(-0.931829\pi\)
0.977154 0.212531i \(-0.0681706\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 2.00000 + 3.46410i 0.0706665 + 0.122398i
\(802\) 0 0
\(803\) 1.73205 + 1.00000i 0.0611227 + 0.0352892i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 15.5885 + 9.00000i 0.548740 + 0.316815i
\(808\) 0 0
\(809\) 23.0000 + 39.8372i 0.808637 + 1.40060i 0.913808 + 0.406146i \(0.133128\pi\)
−0.105171 + 0.994454i \(0.533539\pi\)
\(810\) 0 0
\(811\) 48.0000 1.68551 0.842754 0.538299i \(-0.180933\pi\)
0.842754 + 0.538299i \(0.180933\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7.79423 + 4.50000i 0.272686 + 0.157435i
\(818\) 0 0
\(819\) 0.500000 2.59808i 0.0174714 0.0907841i
\(820\) 0 0
\(821\) 25.0000 43.3013i 0.872506 1.51122i 0.0131101 0.999914i \(-0.495827\pi\)
0.859396 0.511311i \(-0.170840\pi\)
\(822\) 0 0
\(823\) 13.8564 8.00000i 0.483004 0.278862i −0.238664 0.971102i \(-0.576709\pi\)
0.721668 + 0.692240i \(0.243376\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 6.00000i 0.208640i −0.994544 0.104320i \(-0.966733\pi\)
0.994544 0.104320i \(-0.0332667\pi\)
\(828\) 0 0
\(829\) 2.50000 + 4.33013i 0.0868286 + 0.150392i 0.906169 0.422916i \(-0.138993\pi\)
−0.819340 + 0.573307i \(0.805660\pi\)
\(830\) 0 0
\(831\) 2.50000 4.33013i 0.0867240 0.150210i
\(832\) 0 0
\(833\) −10.3923 + 26.0000i −0.360072 + 0.900847i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.33013 + 2.50000i −0.149671 + 0.0864126i
\(838\) 0 0
\(839\) 48.0000 1.65714 0.828572 0.559883i \(-0.189154\pi\)
0.828572 + 0.559883i \(0.189154\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 17.3205 10.0000i 0.596550 0.344418i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −18.1865 3.50000i −0.624897 0.120261i
\(848\) 0 0
\(849\) 12.5000 21.6506i 0.428999 0.743048i
\(850\) 0 0
\(851\) 10.0000 + 17.3205i 0.342796 + 0.593739i
\(852\) 0 0
\(853\) 29.0000i 0.992941i −0.868054 0.496471i \(-0.834629\pi\)
0.868054 0.496471i \(-0.165371\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −38.1051 + 22.0000i −1.30165 + 0.751506i −0.980686 0.195587i \(-0.937339\pi\)
−0.320960 + 0.947093i \(0.604005\pi\)
\(858\) 0 0
\(859\) 4.00000 6.92820i 0.136478 0.236387i −0.789683 0.613515i \(-0.789755\pi\)
0.926161 + 0.377128i \(0.123088\pi\)
\(860\) 0 0
\(861\) 4.00000 + 3.46410i 0.136320 + 0.118056i
\(862\) 0 0
\(863\) −25.9808 15.0000i −0.884395 0.510606i −0.0122903 0.999924i \(-0.503912\pi\)
−0.872105 + 0.489319i \(0.837246\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.00000i 0.0339618i
\(868\) 0 0
\(869\) −6.00000 −0.203536
\(870\) 0 0
\(871\) −4.50000 7.79423i −0.152477 0.264097i
\(872\) 0 0
\(873\) −8.66025 5.00000i −0.293105 0.169224i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −19.0526 11.0000i −0.643359 0.371444i 0.142548 0.989788i \(-0.454470\pi\)
−0.785907 + 0.618344i \(0.787804\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 24.0000 0.808581 0.404290 0.914631i \(-0.367519\pi\)
0.404290 + 0.914631i \(0.367519\pi\)
\(882\) 0 0
\(883\) 1.00000i 0.0336527i 0.999858 + 0.0168263i \(0.00535624\pi\)
−0.999858 + 0.0168263i \(0.994644\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 19.0526 + 11.0000i 0.639722 + 0.369344i 0.784508 0.620119i \(-0.212916\pi\)
−0.144785 + 0.989463i \(0.546249\pi\)
\(888\) 0 0
\(889\) 22.5000 7.79423i 0.754626 0.261410i
\(890\) 0 0
\(891\) 1.00000 1.73205i 0.0335013 0.0580259i
\(892\) 0 0
\(893\) 1.73205 1.00000i 0.0579609 0.0334637i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.00000i 0.133556i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 24.0000 41.5692i 0.799556 1.38487i
\(902\) 0 0
\(903\) 15.5885 18.0000i 0.518751 0.599002i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −7.79423 + 4.50000i −0.258803 + 0.149420i −0.623788 0.781593i \(-0.714407\pi\)
0.364985 + 0.931013i \(0.381074\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 31.1769 18.0000i 1.03181 0.595713i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −46.7654 9.00000i −1.54433 0.297206i
\(918\) 0 0
\(919\) −5.50000 + 9.52628i −0.181428 + 0.314243i −0.942367 0.334581i \(-0.891405\pi\)
0.760939 + 0.648824i \(0.224739\pi\)
\(920\) 0 0
\(921\) −9.50000 16.4545i −0.313036 0.542194i
\(922\) 0 0
\(923\) 2.00000i 0.0658308i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0.866025 0.500000i 0.0284440 0.0164222i
\(928\) 0 0
\(929\) −7.00000 + 12.1244i −0.229663 + 0.397787i −0.957708 0.287742i \(-0.907096\pi\)
0.728046 + 0.685529i \(0.240429\pi\)
\(930\) 0 0
\(931\) −1.00000 6.92820i −0.0327737 0.227063i
\(932\) 0 0
\(933\) 22.5167 + 13.0000i 0.737162 + 0.425601i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 21.0000i 0.686040i 0.939328 + 0.343020i \(0.111450\pi\)
−0.939328 + 0.343020i \(0.888550\pi\)
\(938\) 0 0
\(939\) 19.0000 0.620042
\(940\) 0 0
\(941\) 24.0000 + 41.5692i 0.782378 + 1.35512i 0.930553 + 0.366157i \(0.119327\pi\)
−0.148176 + 0.988961i \(0.547340\pi\)
\(942\) 0 0
\(943\) −6.92820 4.00000i −0.225613 0.130258i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −15.5885 9.00000i −0.506557 0.292461i 0.224860 0.974391i \(-0.427807\pi\)
−0.731417 + 0.681930i \(0.761141\pi\)
\(948\) 0 0
\(949\) 0.500000 + 0.866025i 0.0162307 + 0.0281124i
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) 36.0000i 1.16615i −0.812417 0.583077i \(-0.801849\pi\)
0.812417 0.583077i \(-0.198151\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 32.0000 + 27.7128i 1.03333 + 0.894893i
\(960\) 0 0
\(961\) 3.00000 5.19615i 0.0967742 0.167618i
\(962\) 0 0
\(963\) 3.46410 2.00000i 0.111629 0.0644491i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 37.0000i 1.18984i −0.803785 0.594920i \(-0.797184\pi\)
0.803785 0.594920i \(-0.202816\pi\)
\(968\) 0 0
\(969\) −2.00000 3.46410i −0.0642493 0.111283i
\(970\) 0 0
\(971\) 18.0000 31.1769i 0.577647 1.00051i −0.418101 0.908401i \(-0.637304\pi\)
0.995748 0.0921142i \(-0.0293625\pi\)
\(972\) 0 0
\(973\) −11.2583 32.5000i −0.360925 1.04190i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −15.5885 + 9.00000i −0.498719 + 0.287936i −0.728184 0.685381i \(-0.759636\pi\)
0.229465 + 0.973317i \(0.426302\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) 13.0000 0.415058
\(982\) 0 0
\(983\) −13.8564 + 8.00000i −0.441951 + 0.255160i −0.704425 0.709779i \(-0.748795\pi\)
0.262474 + 0.964939i \(0.415462\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −1.73205 5.00000i −0.0551318 0.159152i
\(988\) 0 0
\(989\) −18.0000 + 31.1769i −0.572367 + 0.991368i
\(990\) 0 0
\(991\) 26.5000 + 45.8993i 0.841800 + 1.45804i 0.888371 + 0.459126i \(0.151837\pi\)
−0.0465710 + 0.998915i \(0.514829\pi\)
\(992\) 0 0
\(993\) 1.00000i 0.0317340i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −9.52628 + 5.50000i −0.301700 + 0.174187i −0.643206 0.765693i \(-0.722396\pi\)
0.341506 + 0.939880i \(0.389063\pi\)
\(998\) 0 0
\(999\) 2.50000 4.33013i 0.0790965 0.136999i
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.c.1549.1 4
5.2 odd 4 420.2.q.a.121.1 2
5.3 odd 4 2100.2.q.a.1801.1 2
5.4 even 2 inner 2100.2.bc.c.1549.2 4
7.4 even 3 inner 2100.2.bc.c.949.2 4
15.2 even 4 1260.2.s.d.541.1 2
20.7 even 4 1680.2.bg.a.961.1 2
35.2 odd 12 2940.2.a.d.1.1 1
35.4 even 6 inner 2100.2.bc.c.949.1 4
35.12 even 12 2940.2.a.h.1.1 1
35.17 even 12 2940.2.q.h.361.1 2
35.18 odd 12 2100.2.q.a.1201.1 2
35.27 even 4 2940.2.q.h.961.1 2
35.32 odd 12 420.2.q.a.361.1 yes 2
105.2 even 12 8820.2.a.j.1.1 1
105.32 even 12 1260.2.s.d.361.1 2
105.47 odd 12 8820.2.a.y.1.1 1
140.67 even 12 1680.2.bg.a.1201.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.q.a.121.1 2 5.2 odd 4
420.2.q.a.361.1 yes 2 35.32 odd 12
1260.2.s.d.361.1 2 105.32 even 12
1260.2.s.d.541.1 2 15.2 even 4
1680.2.bg.a.961.1 2 20.7 even 4
1680.2.bg.a.1201.1 2 140.67 even 12
2100.2.q.a.1201.1 2 35.18 odd 12
2100.2.q.a.1801.1 2 5.3 odd 4
2100.2.bc.c.949.1 4 35.4 even 6 inner
2100.2.bc.c.949.2 4 7.4 even 3 inner
2100.2.bc.c.1549.1 4 1.1 even 1 trivial
2100.2.bc.c.1549.2 4 5.4 even 2 inner
2940.2.a.d.1.1 1 35.2 odd 12
2940.2.a.h.1.1 1 35.12 even 12
2940.2.q.h.361.1 2 35.17 even 12
2940.2.q.h.961.1 2 35.27 even 4
8820.2.a.j.1.1 1 105.2 even 12
8820.2.a.y.1.1 1 105.47 odd 12