Properties

Label 420.2.q.a.121.1
Level $420$
Weight $2$
Character 420.121
Analytic conductor $3.354$
Analytic rank $0$
Dimension $2$
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] = [420,2,Mod(121,420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(420, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("420.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.q (of order \(3\), degree \(2\), 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: \(3.35371688489\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 121.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 420.121
Dual form 420.2.q.a.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(1.00000 + 1.73205i) q^{11} +1.00000 q^{13} -1.00000 q^{15} +(2.00000 + 3.46410i) q^{17} +(0.500000 - 0.866025i) q^{19} +(2.00000 + 1.73205i) q^{21} +(-2.00000 + 3.46410i) q^{23} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} +(2.50000 + 4.33013i) q^{31} +(-1.00000 + 1.73205i) q^{33} +(-0.500000 + 2.59808i) q^{35} +(2.50000 - 4.33013i) q^{37} +(0.500000 + 0.866025i) q^{39} +2.00000 q^{41} -9.00000 q^{43} +(-0.500000 - 0.866025i) q^{45} +(1.00000 - 1.73205i) q^{47} +(5.50000 - 4.33013i) q^{49} +(-2.00000 + 3.46410i) q^{51} +(-6.00000 - 10.3923i) q^{53} -2.00000 q^{55} +1.00000 q^{57} +(4.00000 + 6.92820i) q^{59} +(7.00000 - 12.1244i) q^{61} +(-0.500000 + 2.59808i) q^{63} +(-0.500000 + 0.866025i) q^{65} +(-4.50000 - 7.79423i) q^{67} -4.00000 q^{69} +2.00000 q^{71} +(-0.500000 - 0.866025i) q^{73} +(0.500000 - 0.866025i) q^{75} +(4.00000 + 3.46410i) q^{77} +(1.50000 - 2.59808i) q^{79} +(-0.500000 - 0.866025i) q^{81} -18.0000 q^{83} -4.00000 q^{85} +(2.00000 - 3.46410i) q^{89} +(2.50000 - 0.866025i) q^{91} +(-2.50000 + 4.33013i) q^{93} +(0.500000 + 0.866025i) q^{95} +10.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} - q^{5} + 5 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} - q^{5} + 5 q^{7} - q^{9} + 2 q^{11} + 2 q^{13} - 2 q^{15} + 4 q^{17} + q^{19} + 4 q^{21} - 4 q^{23} - q^{25} - 2 q^{27} + 5 q^{31} - 2 q^{33} - q^{35} + 5 q^{37} + q^{39} + 4 q^{41} - 18 q^{43} - q^{45} + 2 q^{47} + 11 q^{49} - 4 q^{51} - 12 q^{53} - 4 q^{55} + 2 q^{57} + 8 q^{59} + 14 q^{61} - q^{63} - q^{65} - 9 q^{67} - 8 q^{69} + 4 q^{71} - q^{73} + q^{75} + 8 q^{77} + 3 q^{79} - q^{81} - 36 q^{83} - 8 q^{85} + 4 q^{89} + 5 q^{91} - 5 q^{93} + q^{95} + 20 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(241\) \(281\) \(337\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) 2.50000 0.866025i 0.944911 0.327327i
\(8\) 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
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 2.00000 + 3.46410i 0.485071 + 0.840168i 0.999853 0.0171533i \(-0.00546033\pi\)
−0.514782 + 0.857321i \(0.672127\pi\)
\(18\) 0 0
\(19\) 0.500000 0.866025i 0.114708 0.198680i −0.802955 0.596040i \(-0.796740\pi\)
0.917663 + 0.397360i \(0.130073\pi\)
\(20\) 0 0
\(21\) 2.00000 + 1.73205i 0.436436 + 0.377964i
\(22\) 0 0
\(23\) −2.00000 + 3.46410i −0.417029 + 0.722315i −0.995639 0.0932891i \(-0.970262\pi\)
0.578610 + 0.815604i \(0.303595\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 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
\(33\) −1.00000 + 1.73205i −0.174078 + 0.301511i
\(34\) 0 0
\(35\) −0.500000 + 2.59808i −0.0845154 + 0.439155i
\(36\) 0 0
\(37\) 2.50000 4.33013i 0.410997 0.711868i −0.584002 0.811752i \(-0.698514\pi\)
0.994999 + 0.0998840i \(0.0318472\pi\)
\(38\) 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
\(43\) −9.00000 −1.37249 −0.686244 0.727372i \(-0.740742\pi\)
−0.686244 + 0.727372i \(0.740742\pi\)
\(44\) 0 0
\(45\) −0.500000 0.866025i −0.0745356 0.129099i
\(46\) 0 0
\(47\) 1.00000 1.73205i 0.145865 0.252646i −0.783830 0.620975i \(-0.786737\pi\)
0.929695 + 0.368329i \(0.120070\pi\)
\(48\) 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
\(53\) −6.00000 10.3923i −0.824163 1.42749i −0.902557 0.430570i \(-0.858312\pi\)
0.0783936 0.996922i \(-0.475021\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 4.00000 + 6.92820i 0.520756 + 0.901975i 0.999709 + 0.0241347i \(0.00768307\pi\)
−0.478953 + 0.877841i \(0.658984\pi\)
\(60\) 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
\(63\) −0.500000 + 2.59808i −0.0629941 + 0.327327i
\(64\) 0 0
\(65\) −0.500000 + 0.866025i −0.0620174 + 0.107417i
\(66\) 0 0
\(67\) −4.50000 7.79423i −0.549762 0.952217i −0.998290 0.0584478i \(-0.981385\pi\)
0.448528 0.893769i \(-0.351948\pi\)
\(68\) 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
\(73\) −0.500000 0.866025i −0.0585206 0.101361i 0.835281 0.549823i \(-0.185305\pi\)
−0.893801 + 0.448463i \(0.851972\pi\)
\(74\) 0 0
\(75\) 0.500000 0.866025i 0.0577350 0.100000i
\(76\) 0 0
\(77\) 4.00000 + 3.46410i 0.455842 + 0.394771i
\(78\) 0 0
\(79\) 1.50000 2.59808i 0.168763 0.292306i −0.769222 0.638982i \(-0.779356\pi\)
0.937985 + 0.346675i \(0.112689\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) −18.0000 −1.97576 −0.987878 0.155230i \(-0.950388\pi\)
−0.987878 + 0.155230i \(0.950388\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.00000 3.46410i 0.212000 0.367194i −0.740341 0.672232i \(-0.765336\pi\)
0.952340 + 0.305038i \(0.0986691\pi\)
\(90\) 0 0
\(91\) 2.50000 0.866025i 0.262071 0.0907841i
\(92\) 0 0
\(93\) −2.50000 + 4.33013i −0.259238 + 0.449013i
\(94\) 0 0
\(95\) 0.500000 + 0.866025i 0.0512989 + 0.0888523i
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 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
\(103\) 0.500000 0.866025i 0.0492665 0.0853320i −0.840341 0.542059i \(-0.817645\pi\)
0.889607 + 0.456727i \(0.150978\pi\)
\(104\) 0 0
\(105\) −2.50000 + 0.866025i −0.243975 + 0.0845154i
\(106\) 0 0
\(107\) −2.00000 + 3.46410i −0.193347 + 0.334887i −0.946357 0.323122i \(-0.895268\pi\)
0.753010 + 0.658009i \(0.228601\pi\)
\(108\) 0 0
\(109\) −6.50000 11.2583i −0.622587 1.07835i −0.989002 0.147901i \(-0.952748\pi\)
0.366415 0.930451i \(-0.380585\pi\)
\(110\) 0 0
\(111\) 5.00000 0.474579
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) −2.00000 3.46410i −0.186501 0.323029i
\(116\) 0 0
\(117\) −0.500000 + 0.866025i −0.0462250 + 0.0800641i
\(118\) 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
\(123\) 1.00000 + 1.73205i 0.0901670 + 0.156174i
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −9.00000 −0.798621 −0.399310 0.916816i \(-0.630750\pi\)
−0.399310 + 0.916816i \(0.630750\pi\)
\(128\) 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
\(133\) 0.500000 2.59808i 0.0433555 0.225282i
\(134\) 0 0
\(135\) 0.500000 0.866025i 0.0430331 0.0745356i
\(136\) 0 0
\(137\) −8.00000 13.8564i −0.683486 1.18383i −0.973910 0.226935i \(-0.927130\pi\)
0.290424 0.956898i \(-0.406204\pi\)
\(138\) 0 0
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) 2.00000 0.168430
\(142\) 0 0
\(143\) 1.00000 + 1.73205i 0.0836242 + 0.144841i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.50000 + 2.59808i 0.536111 + 0.214286i
\(148\) 0 0
\(149\) −6.00000 + 10.3923i −0.491539 + 0.851371i −0.999953 0.00974235i \(-0.996899\pi\)
0.508413 + 0.861113i \(0.330232\pi\)
\(150\) 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
\(153\) −4.00000 −0.323381
\(154\) 0 0
\(155\) −5.00000 −0.401610
\(156\) 0 0
\(157\) 11.0000 + 19.0526i 0.877896 + 1.52056i 0.853646 + 0.520854i \(0.174386\pi\)
0.0242497 + 0.999706i \(0.492280\pi\)
\(158\) 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
\(163\) −6.00000 + 10.3923i −0.469956 + 0.813988i −0.999410 0.0343508i \(-0.989064\pi\)
0.529454 + 0.848339i \(0.322397\pi\)
\(164\) 0 0
\(165\) −1.00000 1.73205i −0.0778499 0.134840i
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0.500000 + 0.866025i 0.0382360 + 0.0662266i
\(172\) 0 0
\(173\) 8.00000 13.8564i 0.608229 1.05348i −0.383304 0.923622i \(-0.625214\pi\)
0.991532 0.129861i \(-0.0414530\pi\)
\(174\) 0 0
\(175\) −2.00000 1.73205i −0.151186 0.130931i
\(176\) 0 0
\(177\) −4.00000 + 6.92820i −0.300658 + 0.520756i
\(178\) 0 0
\(179\) 13.0000 + 22.5167i 0.971666 + 1.68297i 0.690526 + 0.723307i \(0.257379\pi\)
0.281139 + 0.959667i \(0.409288\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 0 0
\(183\) 14.0000 1.03491
\(184\) 0 0
\(185\) 2.50000 + 4.33013i 0.183804 + 0.318357i
\(186\) 0 0
\(187\) −4.00000 + 6.92820i −0.292509 + 0.506640i
\(188\) 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
\(193\) 8.50000 + 14.7224i 0.611843 + 1.05974i 0.990930 + 0.134382i \(0.0429051\pi\)
−0.379086 + 0.925361i \(0.623762\pi\)
\(194\) 0 0
\(195\) −1.00000 −0.0716115
\(196\) 0 0
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) 8.00000 + 13.8564i 0.567105 + 0.982255i 0.996850 + 0.0793045i \(0.0252700\pi\)
−0.429745 + 0.902950i \(0.641397\pi\)
\(200\) 0 0
\(201\) 4.50000 7.79423i 0.317406 0.549762i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1.00000 + 1.73205i −0.0698430 + 0.120972i
\(206\) 0 0
\(207\) −2.00000 3.46410i −0.139010 0.240772i
\(208\) 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
\(213\) 1.00000 + 1.73205i 0.0685189 + 0.118678i
\(214\) 0 0
\(215\) 4.50000 7.79423i 0.306897 0.531562i
\(216\) 0 0
\(217\) 10.0000 + 8.66025i 0.678844 + 0.587896i
\(218\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −9.00000 15.5885i −0.597351 1.03464i −0.993210 0.116331i \(-0.962887\pi\)
0.395860 0.918311i \(-0.370447\pi\)
\(228\) 0 0
\(229\) −0.500000 + 0.866025i −0.0330409 + 0.0572286i −0.882073 0.471113i \(-0.843853\pi\)
0.849032 + 0.528341i \(0.177186\pi\)
\(230\) 0 0
\(231\) −1.00000 + 5.19615i −0.0657952 + 0.341882i
\(232\) 0 0
\(233\) 1.00000 1.73205i 0.0655122 0.113470i −0.831409 0.555661i \(-0.812465\pi\)
0.896921 + 0.442191i \(0.145799\pi\)
\(234\) 0 0
\(235\) 1.00000 + 1.73205i 0.0652328 + 0.112987i
\(236\) 0 0
\(237\) 3.00000 0.194871
\(238\) 0 0
\(239\) 26.0000 1.68180 0.840900 0.541190i \(-0.182026\pi\)
0.840900 + 0.541190i \(0.182026\pi\)
\(240\) 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
\(243\) 0.500000 0.866025i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) 1.00000 + 6.92820i 0.0638877 + 0.442627i
\(246\) 0 0
\(247\) 0.500000 0.866025i 0.0318142 0.0551039i
\(248\) 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
\(253\) −8.00000 −0.502956
\(254\) 0 0
\(255\) −2.00000 3.46410i −0.125245 0.216930i
\(256\) 0 0
\(257\) −7.00000 + 12.1244i −0.436648 + 0.756297i −0.997429 0.0716680i \(-0.977168\pi\)
0.560781 + 0.827964i \(0.310501\pi\)
\(258\) 0 0
\(259\) 2.50000 12.9904i 0.155342 0.807183i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −6.00000 10.3923i −0.369976 0.640817i 0.619586 0.784929i \(-0.287301\pi\)
−0.989561 + 0.144112i \(0.953967\pi\)
\(264\) 0 0
\(265\) 12.0000 0.737154
\(266\) 0 0
\(267\) 4.00000 0.244796
\(268\) 0 0
\(269\) 9.00000 + 15.5885i 0.548740 + 0.950445i 0.998361 + 0.0572259i \(0.0182255\pi\)
−0.449622 + 0.893219i \(0.648441\pi\)
\(270\) 0 0
\(271\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(272\) 0 0
\(273\) 2.00000 + 1.73205i 0.121046 + 0.104828i
\(274\) 0 0
\(275\) 1.00000 1.73205i 0.0603023 0.104447i
\(276\) 0 0
\(277\) −2.50000 4.33013i −0.150210 0.260172i 0.781094 0.624413i \(-0.214662\pi\)
−0.931305 + 0.364241i \(0.881328\pi\)
\(278\) 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
\(283\) 12.5000 + 21.6506i 0.743048 + 1.28700i 0.951101 + 0.308879i \(0.0999539\pi\)
−0.208053 + 0.978117i \(0.566713\pi\)
\(284\) 0 0
\(285\) −0.500000 + 0.866025i −0.0296174 + 0.0512989i
\(286\) 0 0
\(287\) 5.00000 1.73205i 0.295141 0.102240i
\(288\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) −1.00000 1.73205i −0.0580259 0.100504i
\(298\) 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
\(303\) 3.00000 5.19615i 0.172345 0.298511i
\(304\) 0 0
\(305\) 7.00000 + 12.1244i 0.400819 + 0.694239i
\(306\) 0 0
\(307\) −19.0000 −1.08439 −0.542194 0.840254i \(-0.682406\pi\)
−0.542194 + 0.840254i \(0.682406\pi\)
\(308\) 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
\(313\) −9.50000 + 16.4545i −0.536972 + 0.930062i 0.462093 + 0.886831i \(0.347098\pi\)
−0.999065 + 0.0432311i \(0.986235\pi\)
\(314\) 0 0
\(315\) −2.00000 1.73205i −0.112687 0.0975900i
\(316\) 0 0
\(317\) 2.00000 3.46410i 0.112331 0.194563i −0.804379 0.594117i \(-0.797502\pi\)
0.916710 + 0.399554i \(0.130835\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) −0.500000 0.866025i −0.0277350 0.0480384i
\(326\) 0 0
\(327\) 6.50000 11.2583i 0.359451 0.622587i
\(328\) 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
\(333\) 2.50000 + 4.33013i 0.136999 + 0.237289i
\(334\) 0 0
\(335\) 9.00000 0.491723
\(336\) 0 0
\(337\) −7.00000 −0.381314 −0.190657 0.981657i \(-0.561062\pi\)
−0.190657 + 0.981657i \(0.561062\pi\)
\(338\) 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
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 2.00000 3.46410i 0.107676 0.186501i
\(346\) 0 0
\(347\) −12.0000 20.7846i −0.644194 1.11578i −0.984487 0.175457i \(-0.943860\pi\)
0.340293 0.940319i \(-0.389474\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −15.0000 25.9808i −0.798369 1.38282i −0.920677 0.390324i \(-0.872363\pi\)
0.122308 0.992492i \(-0.460970\pi\)
\(354\) 0 0
\(355\) −1.00000 + 1.73205i −0.0530745 + 0.0919277i
\(356\) 0 0
\(357\) −2.00000 + 10.3923i −0.105851 + 0.550019i
\(358\) 0 0
\(359\) −8.00000 + 13.8564i −0.422224 + 0.731313i −0.996157 0.0875892i \(-0.972084\pi\)
0.573933 + 0.818902i \(0.305417\pi\)
\(360\) 0 0
\(361\) 9.00000 + 15.5885i 0.473684 + 0.820445i
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 1.00000 0.0523424
\(366\) 0 0
\(367\) −0.500000 0.866025i −0.0260998 0.0452062i 0.852680 0.522433i \(-0.174975\pi\)
−0.878780 + 0.477227i \(0.841642\pi\)
\(368\) 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
\(373\) 12.5000 21.6506i 0.647225 1.12103i −0.336557 0.941663i \(-0.609263\pi\)
0.983783 0.179364i \(-0.0574041\pi\)
\(374\) 0 0
\(375\) 0.500000 + 0.866025i 0.0258199 + 0.0447214i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −11.0000 −0.565032 −0.282516 0.959263i \(-0.591169\pi\)
−0.282516 + 0.959263i \(0.591169\pi\)
\(380\) 0 0
\(381\) −4.50000 7.79423i −0.230542 0.399310i
\(382\) 0 0
\(383\) 18.0000 31.1769i 0.919757 1.59307i 0.119974 0.992777i \(-0.461719\pi\)
0.799783 0.600289i \(-0.204948\pi\)
\(384\) 0 0
\(385\) −5.00000 + 1.73205i −0.254824 + 0.0882735i
\(386\) 0 0
\(387\) 4.50000 7.79423i 0.228748 0.396203i
\(388\) 0 0
\(389\) −15.0000 25.9808i −0.760530 1.31728i −0.942578 0.333987i \(-0.891606\pi\)
0.182047 0.983290i \(-0.441728\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 18.0000 0.907980
\(394\) 0 0
\(395\) 1.50000 + 2.59808i 0.0754732 + 0.130723i
\(396\) 0 0
\(397\) −15.5000 + 26.8468i −0.777923 + 1.34740i 0.155214 + 0.987881i \(0.450393\pi\)
−0.933137 + 0.359521i \(0.882940\pi\)
\(398\) 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
\(403\) 2.50000 + 4.33013i 0.124534 + 0.215699i
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 10.0000 0.495682
\(408\) 0 0
\(409\) −6.50000 11.2583i −0.321404 0.556689i 0.659374 0.751815i \(-0.270822\pi\)
−0.980778 + 0.195127i \(0.937488\pi\)
\(410\) 0 0
\(411\) 8.00000 13.8564i 0.394611 0.683486i
\(412\) 0 0
\(413\) 16.0000 + 13.8564i 0.787309 + 0.681829i
\(414\) 0 0
\(415\) 9.00000 15.5885i 0.441793 0.765207i
\(416\) 0 0
\(417\) −6.50000 11.2583i −0.318306 0.551323i
\(418\) 0 0
\(419\) −6.00000 −0.293119 −0.146560 0.989202i \(-0.546820\pi\)
−0.146560 + 0.989202i \(0.546820\pi\)
\(420\) 0 0
\(421\) −35.0000 −1.70580 −0.852898 0.522078i \(-0.825157\pi\)
−0.852898 + 0.522078i \(0.825157\pi\)
\(422\) 0 0
\(423\) 1.00000 + 1.73205i 0.0486217 + 0.0842152i
\(424\) 0 0
\(425\) 2.00000 3.46410i 0.0970143 0.168034i
\(426\) 0 0
\(427\) 7.00000 36.3731i 0.338754 1.76022i
\(428\) 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
\(433\) 27.0000 1.29754 0.648769 0.760986i \(-0.275284\pi\)
0.648769 + 0.760986i \(0.275284\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.00000 + 3.46410i 0.0956730 + 0.165710i
\(438\) 0 0
\(439\) −4.00000 + 6.92820i −0.190910 + 0.330665i −0.945552 0.325471i \(-0.894477\pi\)
0.754642 + 0.656136i \(0.227810\pi\)
\(440\) 0 0
\(441\) 1.00000 + 6.92820i 0.0476190 + 0.329914i
\(442\) 0 0
\(443\) 2.00000 3.46410i 0.0950229 0.164584i −0.814595 0.580030i \(-0.803041\pi\)
0.909618 + 0.415445i \(0.136374\pi\)
\(444\) 0 0
\(445\) 2.00000 + 3.46410i 0.0948091 + 0.164214i
\(446\) 0 0
\(447\) −12.0000 −0.567581
\(448\) 0 0
\(449\) 22.0000 1.03824 0.519122 0.854700i \(-0.326259\pi\)
0.519122 + 0.854700i \(0.326259\pi\)
\(450\) 0 0
\(451\) 2.00000 + 3.46410i 0.0941763 + 0.163118i
\(452\) 0 0
\(453\) −8.00000 + 13.8564i −0.375873 + 0.651031i
\(454\) 0 0
\(455\) −0.500000 + 2.59808i −0.0234404 + 0.121800i
\(456\) 0 0
\(457\) −0.500000 + 0.866025i −0.0233890 + 0.0405110i −0.877483 0.479608i \(-0.840779\pi\)
0.854094 + 0.520119i \(0.174112\pi\)
\(458\) 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
\(463\) 41.0000 1.90543 0.952716 0.303863i \(-0.0982765\pi\)
0.952716 + 0.303863i \(0.0982765\pi\)
\(464\) 0 0
\(465\) −2.50000 4.33013i −0.115935 0.200805i
\(466\) 0 0
\(467\) −17.0000 + 29.4449i −0.786666 + 1.36255i 0.141332 + 0.989962i \(0.454861\pi\)
−0.927999 + 0.372584i \(0.878472\pi\)
\(468\) 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
\(473\) −9.00000 15.5885i −0.413820 0.716758i
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0 0
\(479\) −6.00000 10.3923i −0.274147 0.474837i 0.695773 0.718262i \(-0.255062\pi\)
−0.969920 + 0.243426i \(0.921729\pi\)
\(480\) 0 0
\(481\) 2.50000 4.33013i 0.113990 0.197437i
\(482\) 0 0
\(483\) −10.0000 + 3.46410i −0.455016 + 0.157622i
\(484\) 0 0
\(485\) −5.00000 + 8.66025i −0.227038 + 0.393242i
\(486\) 0 0
\(487\) −16.5000 28.5788i −0.747686 1.29503i −0.948929 0.315489i \(-0.897831\pi\)
0.201243 0.979541i \(-0.435502\pi\)
\(488\) 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
\(495\) 1.00000 1.73205i 0.0449467 0.0778499i
\(496\) 0 0
\(497\) 5.00000 1.73205i 0.224281 0.0776931i
\(498\) 0 0
\(499\) 2.50000 4.33013i 0.111915 0.193843i −0.804627 0.593780i \(-0.797635\pi\)
0.916542 + 0.399937i \(0.130968\pi\)
\(500\) 0 0
\(501\) 1.00000 + 1.73205i 0.0446767 + 0.0773823i
\(502\) 0 0
\(503\) 10.0000 0.445878 0.222939 0.974832i \(-0.428435\pi\)
0.222939 + 0.974832i \(0.428435\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) −6.00000 10.3923i −0.266469 0.461538i
\(508\) 0 0
\(509\) −13.0000 + 22.5167i −0.576215 + 0.998033i 0.419694 + 0.907666i \(0.362138\pi\)
−0.995908 + 0.0903676i \(0.971196\pi\)
\(510\) 0 0
\(511\) −2.00000 1.73205i −0.0884748 0.0766214i
\(512\) 0 0
\(513\) −0.500000 + 0.866025i −0.0220755 + 0.0382360i
\(514\) 0 0
\(515\) 0.500000 + 0.866025i 0.0220326 + 0.0381616i
\(516\) 0 0
\(517\) 4.00000 0.175920
\(518\) 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
\(523\) −14.5000 + 25.1147i −0.634041 + 1.09819i 0.352677 + 0.935745i \(0.385272\pi\)
−0.986718 + 0.162446i \(0.948062\pi\)
\(524\) 0 0
\(525\) 0.500000 2.59808i 0.0218218 0.113389i
\(526\) 0 0
\(527\) −10.0000 + 17.3205i −0.435607 + 0.754493i
\(528\) 0 0
\(529\) 3.50000 + 6.06218i 0.152174 + 0.263573i
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) −2.00000 3.46410i −0.0864675 0.149766i
\(536\) 0 0
\(537\) −13.0000 + 22.5167i −0.560991 + 0.971666i
\(538\) 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
\(543\) 8.50000 + 14.7224i 0.364770 + 0.631800i
\(544\) 0 0
\(545\) 13.0000 0.556859
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) 7.00000 + 12.1244i 0.298753 + 0.517455i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 1.50000 7.79423i 0.0637865 0.331444i
\(554\) 0 0
\(555\) −2.50000 + 4.33013i −0.106119 + 0.183804i
\(556\) 0 0
\(557\) 1.00000 + 1.73205i 0.0423714 + 0.0733893i 0.886433 0.462856i \(-0.153175\pi\)
−0.844062 + 0.536246i \(0.819842\pi\)
\(558\) 0 0
\(559\) −9.00000 −0.380659
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) 15.0000 + 25.9808i 0.632175 + 1.09496i 0.987106 + 0.160066i \(0.0511708\pi\)
−0.354932 + 0.934892i \(0.615496\pi\)
\(564\) 0 0
\(565\) 7.00000 12.1244i 0.294492 0.510075i
\(566\) 0 0
\(567\) −2.00000 1.73205i −0.0839921 0.0727393i
\(568\) 0 0
\(569\) 3.00000 5.19615i 0.125767 0.217834i −0.796266 0.604947i \(-0.793194\pi\)
0.922032 + 0.387113i \(0.126528\pi\)
\(570\) 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
\(573\) 10.0000 0.417756
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 6.50000 + 11.2583i 0.270599 + 0.468690i 0.969015 0.247001i \(-0.0794451\pi\)
−0.698417 + 0.715691i \(0.746112\pi\)
\(578\) 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
\(583\) 12.0000 20.7846i 0.496989 0.860811i
\(584\) 0 0
\(585\) −0.500000 0.866025i −0.0206725 0.0358057i
\(586\) 0 0
\(587\) −36.0000 −1.48588 −0.742940 0.669359i \(-0.766569\pi\)
−0.742940 + 0.669359i \(0.766569\pi\)
\(588\) 0 0
\(589\) 5.00000 0.206021
\(590\) 0 0
\(591\) 6.00000 + 10.3923i 0.246807 + 0.427482i
\(592\) 0 0
\(593\) −3.00000 + 5.19615i −0.123195 + 0.213380i −0.921026 0.389501i \(-0.872647\pi\)
0.797831 + 0.602881i \(0.205981\pi\)
\(594\) 0 0
\(595\) −10.0000 + 3.46410i −0.409960 + 0.142014i
\(596\) 0 0
\(597\) −8.00000 + 13.8564i −0.327418 + 0.567105i
\(598\) 0 0
\(599\) 14.0000 + 24.2487i 0.572024 + 0.990775i 0.996358 + 0.0852695i \(0.0271751\pi\)
−0.424333 + 0.905506i \(0.639492\pi\)
\(600\) 0 0
\(601\) 23.0000 0.938190 0.469095 0.883148i \(-0.344580\pi\)
0.469095 + 0.883148i \(0.344580\pi\)
\(602\) 0 0
\(603\) 9.00000 0.366508
\(604\) 0 0
\(605\) 3.50000 + 6.06218i 0.142295 + 0.246463i
\(606\) 0 0
\(607\) −4.50000 + 7.79423i −0.182649 + 0.316358i −0.942782 0.333410i \(-0.891801\pi\)
0.760133 + 0.649768i \(0.225134\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.00000 1.73205i 0.0404557 0.0700713i
\(612\) 0 0
\(613\) −1.00000 1.73205i −0.0403896 0.0699569i 0.845124 0.534570i \(-0.179527\pi\)
−0.885514 + 0.464614i \(0.846193\pi\)
\(614\) 0 0
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) 34.0000 1.36879 0.684394 0.729112i \(-0.260067\pi\)
0.684394 + 0.729112i \(0.260067\pi\)
\(618\) 0 0
\(619\) 5.50000 + 9.52628i 0.221064 + 0.382893i 0.955131 0.296183i \(-0.0957138\pi\)
−0.734068 + 0.679076i \(0.762380\pi\)
\(620\) 0 0
\(621\) 2.00000 3.46410i 0.0802572 0.139010i
\(622\) 0 0
\(623\) 2.00000 10.3923i 0.0801283 0.416359i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 1.00000 + 1.73205i 0.0399362 + 0.0691714i
\(628\) 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
\(633\) −14.0000 24.2487i −0.556450 0.963800i
\(634\) 0 0
\(635\) 4.50000 7.79423i 0.178577 0.309305i
\(636\) 0 0
\(637\) 5.50000 4.33013i 0.217918 0.171566i
\(638\) 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
\(643\) 7.00000 0.276053 0.138027 0.990429i \(-0.455924\pi\)
0.138027 + 0.990429i \(0.455924\pi\)
\(644\) 0 0
\(645\) 9.00000 0.354375
\(646\) 0 0
\(647\) 7.00000 + 12.1244i 0.275198 + 0.476658i 0.970185 0.242365i \(-0.0779231\pi\)
−0.694987 + 0.719023i \(0.744590\pi\)
\(648\) 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
\(653\) −7.00000 + 12.1244i −0.273931 + 0.474463i −0.969865 0.243643i \(-0.921657\pi\)
0.695934 + 0.718106i \(0.254991\pi\)
\(654\) 0 0
\(655\) 9.00000 + 15.5885i 0.351659 + 0.609091i
\(656\) 0 0
\(657\) 1.00000 0.0390137
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 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
\(663\) −2.00000 + 3.46410i −0.0776736 + 0.134535i
\(664\) 0 0
\(665\) 2.00000 + 1.73205i 0.0775567 + 0.0671660i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 28.0000 1.08093
\(672\) 0 0
\(673\) 11.0000 0.424019 0.212009 0.977268i \(-0.431999\pi\)
0.212009 + 0.977268i \(0.431999\pi\)
\(674\) 0 0
\(675\) 0.500000 + 0.866025i 0.0192450 + 0.0333333i
\(676\) 0 0
\(677\) −12.0000 + 20.7846i −0.461197 + 0.798817i −0.999021 0.0442400i \(-0.985913\pi\)
0.537823 + 0.843057i \(0.319247\pi\)
\(678\) 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
\(683\) 8.00000 + 13.8564i 0.306111 + 0.530201i 0.977508 0.210898i \(-0.0676386\pi\)
−0.671397 + 0.741098i \(0.734305\pi\)
\(684\) 0 0
\(685\) 16.0000 0.611329
\(686\) 0 0
\(687\) −1.00000 −0.0381524
\(688\) 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
\(693\) −5.00000 + 1.73205i −0.189934 + 0.0657952i
\(694\) 0 0
\(695\) 6.50000 11.2583i 0.246559 0.427053i
\(696\) 0 0
\(697\) 4.00000 + 6.92820i 0.151511 + 0.262424i
\(698\) 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
\(703\) −2.50000 4.33013i −0.0942893 0.163314i
\(704\) 0 0
\(705\) −1.00000 + 1.73205i −0.0376622 + 0.0652328i
\(706\) 0 0
\(707\) −12.0000 10.3923i −0.451306 0.390843i
\(708\) 0 0
\(709\) 13.0000 22.5167i 0.488225 0.845631i −0.511683 0.859174i \(-0.670978\pi\)
0.999908 + 0.0135434i \(0.00431112\pi\)
\(710\) 0 0
\(711\) 1.50000 + 2.59808i 0.0562544 + 0.0974355i
\(712\) 0 0
\(713\) −20.0000 −0.749006
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) 0 0
\(717\) 13.0000 + 22.5167i 0.485494 + 0.840900i
\(718\) 0 0
\(719\) −21.0000 + 36.3731i −0.783168 + 1.35649i 0.146920 + 0.989148i \(0.453064\pi\)
−0.930087 + 0.367338i \(0.880269\pi\)
\(720\) 0 0
\(721\) 0.500000 2.59808i 0.0186210 0.0967574i
\(722\) 0 0
\(723\) −9.00000 + 15.5885i −0.334714 + 0.579741i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −3.00000 −0.111264 −0.0556319 0.998451i \(-0.517717\pi\)
−0.0556319 + 0.998451i \(0.517717\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −18.0000 31.1769i −0.665754 1.15312i
\(732\) 0 0
\(733\) −4.50000 + 7.79423i −0.166211 + 0.287886i −0.937085 0.349102i \(-0.886487\pi\)
0.770873 + 0.636988i \(0.219820\pi\)
\(734\) 0 0
\(735\) −5.50000 + 4.33013i −0.202871 + 0.159719i
\(736\) 0 0
\(737\) 9.00000 15.5885i 0.331519 0.574208i
\(738\) 0 0
\(739\) −7.50000 12.9904i −0.275892 0.477859i 0.694468 0.719524i \(-0.255640\pi\)
−0.970360 + 0.241665i \(0.922307\pi\)
\(740\) 0 0
\(741\) 1.00000 0.0367359
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) −6.00000 10.3923i −0.219823 0.380745i
\(746\) 0 0
\(747\) 9.00000 15.5885i 0.329293 0.570352i
\(748\) 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
\(753\) 2.00000 + 3.46410i 0.0728841 + 0.126239i
\(754\) 0 0
\(755\) −16.0000 −0.582300
\(756\) 0 0
\(757\) −22.0000 −0.799604 −0.399802 0.916602i \(-0.630921\pi\)
−0.399802 + 0.916602i \(0.630921\pi\)
\(758\) 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
\(763\) −26.0000 22.5167i −0.941263 0.815158i
\(764\) 0 0
\(765\) 2.00000 3.46410i 0.0723102 0.125245i
\(766\) 0 0
\(767\) 4.00000 + 6.92820i 0.144432 + 0.250163i
\(768\) 0 0
\(769\) 31.0000 1.11789 0.558944 0.829205i \(-0.311207\pi\)
0.558944 + 0.829205i \(0.311207\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 0 0
\(773\) 7.00000 + 12.1244i 0.251773 + 0.436083i 0.964014 0.265852i \(-0.0856532\pi\)
−0.712241 + 0.701935i \(0.752320\pi\)
\(774\) 0 0
\(775\) 2.50000 4.33013i 0.0898027 0.155543i
\(776\) 0 0
\(777\) 12.5000 4.33013i 0.448435 0.155342i
\(778\) 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
\(785\) −22.0000 −0.785214
\(786\) 0 0
\(787\) 8.00000 + 13.8564i 0.285169 + 0.493928i 0.972650 0.232275i \(-0.0746169\pi\)
−0.687481 + 0.726202i \(0.741284\pi\)
\(788\) 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
\(793\) 7.00000 12.1244i 0.248577 0.430548i
\(794\) 0 0
\(795\) 6.00000 + 10.3923i 0.212798 + 0.368577i
\(796\) 0 0
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) 0 0
\(801\) 2.00000 + 3.46410i 0.0706665 + 0.122398i
\(802\) 0 0
\(803\) 1.00000 1.73205i 0.0352892 0.0611227i
\(804\) 0 0
\(805\) −8.00000 6.92820i −0.281963 0.244187i
\(806\) 0 0
\(807\) −9.00000 + 15.5885i −0.316815 + 0.548740i
\(808\) 0 0
\(809\) −23.0000 39.8372i −0.808637 1.40060i −0.913808 0.406146i \(-0.866872\pi\)
0.105171 0.994454i \(-0.466461\pi\)
\(810\) 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
\(815\) −6.00000 10.3923i −0.210171 0.364027i
\(816\) 0 0
\(817\) −4.50000 + 7.79423i −0.157435 + 0.272686i
\(818\) 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
\(823\) −8.00000 13.8564i −0.278862 0.483004i 0.692240 0.721668i \(-0.256624\pi\)
−0.971102 + 0.238664i \(0.923291\pi\)
\(824\) 0 0
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) 6.00000 0.208640 0.104320 0.994544i \(-0.466733\pi\)
0.104320 + 0.994544i \(0.466733\pi\)
\(828\) 0 0
\(829\) −2.50000 4.33013i −0.0868286 0.150392i 0.819340 0.573307i \(-0.194340\pi\)
−0.906169 + 0.422916i \(0.861007\pi\)
\(830\) 0 0
\(831\) 2.50000 4.33013i 0.0867240 0.150210i
\(832\) 0 0
\(833\) 26.0000 + 10.3923i 0.900847 + 0.360072i
\(834\) 0 0
\(835\) −1.00000 + 1.73205i −0.0346064 + 0.0599401i
\(836\) 0 0
\(837\) −2.50000 4.33013i −0.0864126 0.149671i
\(838\) 0 0
\(839\) −48.0000 −1.65714 −0.828572 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) −10.0000 17.3205i −0.344418 0.596550i
\(844\) 0 0
\(845\) 6.00000 10.3923i 0.206406 0.357506i
\(846\) 0 0
\(847\) 3.50000 18.1865i 0.120261 0.624897i
\(848\) 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
\(853\) −29.0000 −0.992941 −0.496471 0.868054i \(-0.665371\pi\)
−0.496471 + 0.868054i \(0.665371\pi\)
\(854\) 0 0
\(855\) −1.00000 −0.0341993
\(856\) 0 0
\(857\) −22.0000 38.1051i −0.751506 1.30165i −0.947093 0.320960i \(-0.895995\pi\)
0.195587 0.980686i \(-0.437339\pi\)
\(858\) 0 0
\(859\) −4.00000 + 6.92820i −0.136478 + 0.236387i −0.926161 0.377128i \(-0.876912\pi\)
0.789683 + 0.613515i \(0.210245\pi\)
\(860\) 0 0
\(861\) 4.00000 + 3.46410i 0.136320 + 0.118056i
\(862\) 0 0
\(863\) −15.0000 + 25.9808i −0.510606 + 0.884395i 0.489319 + 0.872105i \(0.337246\pi\)
−0.999924 + 0.0122903i \(0.996088\pi\)
\(864\) 0 0
\(865\) 8.00000 + 13.8564i 0.272008 + 0.471132i
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 6.00000 0.203536
\(870\) 0 0
\(871\) −4.50000 7.79423i −0.152477 0.264097i
\(872\) 0 0
\(873\) −5.00000 + 8.66025i −0.169224 + 0.293105i
\(874\) 0 0
\(875\) 2.50000 0.866025i 0.0845154 0.0292770i
\(876\) 0 0
\(877\) 11.0000 19.0526i 0.371444 0.643359i −0.618344 0.785907i \(-0.712196\pi\)
0.989788 + 0.142548i \(0.0455296\pi\)
\(878\) 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
\(883\) 1.00000 0.0336527 0.0168263 0.999858i \(-0.494644\pi\)
0.0168263 + 0.999858i \(0.494644\pi\)
\(884\) 0 0
\(885\) −4.00000 6.92820i −0.134459 0.232889i
\(886\) 0 0
\(887\) −11.0000 + 19.0526i −0.369344 + 0.639722i −0.989463 0.144785i \(-0.953751\pi\)
0.620119 + 0.784508i \(0.287084\pi\)
\(888\) 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
\(893\) −1.00000 1.73205i −0.0334637 0.0579609i
\(894\) 0 0
\(895\) −26.0000 −0.869084
\(896\) 0 0
\(897\) −4.00000 −0.133556
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 24.0000 41.5692i 0.799556 1.38487i
\(902\) 0 0
\(903\) −18.0000 15.5885i −0.599002 0.518751i
\(904\) 0 0
\(905\) −8.50000 + 14.7224i −0.282550 + 0.489390i
\(906\) 0 0
\(907\) −4.50000 7.79423i −0.149420 0.258803i 0.781593 0.623788i \(-0.214407\pi\)
−0.931013 + 0.364985i \(0.881074\pi\)
\(908\) 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
\(913\) −18.0000 31.1769i −0.595713 1.03181i
\(914\) 0 0
\(915\) −7.00000 + 12.1244i −0.231413 + 0.400819i
\(916\) 0 0
\(917\) 9.00000 46.7654i 0.297206 1.54433i
\(918\) 0 0
\(919\) 5.50000 9.52628i 0.181428 0.314243i −0.760939 0.648824i \(-0.775261\pi\)
0.942367 + 0.334581i \(0.108595\pi\)
\(920\) 0 0
\(921\) −9.50000 16.4545i −0.313036 0.542194i
\(922\) 0 0
\(923\) 2.00000 0.0658308
\(924\) 0 0
\(925\) −5.00000 −0.164399
\(926\) 0 0
\(927\) 0.500000 + 0.866025i 0.0164222 + 0.0284440i
\(928\) 0 0
\(929\) 7.00000 12.1244i 0.229663 0.397787i −0.728046 0.685529i \(-0.759571\pi\)
0.957708 + 0.287742i \(0.0929044\pi\)
\(930\) 0 0
\(931\) −1.00000 6.92820i −0.0327737 0.227063i
\(932\) 0 0
\(933\) 13.0000 22.5167i 0.425601 0.737162i
\(934\) 0 0
\(935\) −4.00000 6.92820i −0.130814 0.226576i
\(936\) 0 0
\(937\) −21.0000 −0.686040 −0.343020 0.939328i \(-0.611450\pi\)
−0.343020 + 0.939328i \(0.611450\pi\)
\(938\) 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
\(943\) −4.00000 + 6.92820i −0.130258 + 0.225613i
\(944\) 0 0
\(945\) 0.500000 2.59808i 0.0162650 0.0845154i
\(946\) 0 0
\(947\) 9.00000 15.5885i 0.292461 0.506557i −0.681930 0.731417i \(-0.738859\pi\)
0.974391 + 0.224860i \(0.0721926\pi\)
\(948\) 0 0
\(949\) −0.500000 0.866025i −0.0162307 0.0281124i
\(950\) 0 0
\(951\) 4.00000 0.129709
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) 5.00000 + 8.66025i 0.161796 + 0.280239i
\(956\) 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
\(963\) −2.00000 3.46410i −0.0644491 0.111629i
\(964\) 0 0
\(965\) −17.0000 −0.547249
\(966\) 0 0
\(967\) 37.0000 1.18984 0.594920 0.803785i \(-0.297184\pi\)
0.594920 + 0.803785i \(0.297184\pi\)
\(968\) 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
\(973\) −32.5000 + 11.2583i −1.04190 + 0.360925i
\(974\) 0 0
\(975\) 0.500000 0.866025i 0.0160128 0.0277350i
\(976\) 0 0
\(977\) −9.00000 15.5885i −0.287936 0.498719i 0.685381 0.728184i \(-0.259636\pi\)
−0.973317 + 0.229465i \(0.926302\pi\)
\(978\) 0 0
\(979\) 8.00000 0.255681
\(980\) 0 0
\(981\) 13.0000 0.415058
\(982\) 0 0
\(983\) 8.00000 + 13.8564i 0.255160 + 0.441951i 0.964939 0.262474i \(-0.0845384\pi\)
−0.709779 + 0.704425i \(0.751205\pi\)
\(984\) 0 0
\(985\) −6.00000 + 10.3923i −0.191176 + 0.331126i
\(986\) 0 0
\(987\) 5.00000 1.73205i 0.159152 0.0551318i
\(988\) 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
\(993\) −1.00000 −0.0317340
\(994\) 0 0
\(995\) −16.0000 −0.507234
\(996\) 0 0
\(997\) −5.50000 9.52628i −0.174187 0.301700i 0.765693 0.643206i \(-0.222396\pi\)
−0.939880 + 0.341506i \(0.889063\pi\)
\(998\) 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 420.2.q.a.121.1 2
3.2 odd 2 1260.2.s.d.541.1 2
4.3 odd 2 1680.2.bg.a.961.1 2
5.2 odd 4 2100.2.bc.c.1549.2 4
5.3 odd 4 2100.2.bc.c.1549.1 4
5.4 even 2 2100.2.q.a.1801.1 2
7.2 even 3 2940.2.a.d.1.1 1
7.3 odd 6 2940.2.q.h.361.1 2
7.4 even 3 inner 420.2.q.a.361.1 yes 2
7.5 odd 6 2940.2.a.h.1.1 1
7.6 odd 2 2940.2.q.h.961.1 2
21.2 odd 6 8820.2.a.j.1.1 1
21.5 even 6 8820.2.a.y.1.1 1
21.11 odd 6 1260.2.s.d.361.1 2
28.11 odd 6 1680.2.bg.a.1201.1 2
35.4 even 6 2100.2.q.a.1201.1 2
35.18 odd 12 2100.2.bc.c.949.2 4
35.32 odd 12 2100.2.bc.c.949.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.q.a.121.1 2 1.1 even 1 trivial
420.2.q.a.361.1 yes 2 7.4 even 3 inner
1260.2.s.d.361.1 2 21.11 odd 6
1260.2.s.d.541.1 2 3.2 odd 2
1680.2.bg.a.961.1 2 4.3 odd 2
1680.2.bg.a.1201.1 2 28.11 odd 6
2100.2.q.a.1201.1 2 35.4 even 6
2100.2.q.a.1801.1 2 5.4 even 2
2100.2.bc.c.949.1 4 35.32 odd 12
2100.2.bc.c.949.2 4 35.18 odd 12
2100.2.bc.c.1549.1 4 5.3 odd 4
2100.2.bc.c.1549.2 4 5.2 odd 4
2940.2.a.d.1.1 1 7.2 even 3
2940.2.a.h.1.1 1 7.5 odd 6
2940.2.q.h.361.1 2 7.3 odd 6
2940.2.q.h.961.1 2 7.6 odd 2
8820.2.a.j.1.1 1 21.2 odd 6
8820.2.a.y.1.1 1 21.5 even 6