Properties

Label 2100.2.f.c.1049.2
Level $2100$
Weight $2$
Character 2100.1049
Analytic conductor $16.769$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(1049,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1049");
 
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.f (of order \(2\), degree \(1\), 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: \(4\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1049.2
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1049
Dual form 2100.2.f.c.1049.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 + 1.50000i) q^{3} +(-1.73205 + 2.00000i) q^{7} +(-1.50000 - 2.59808i) q^{9} +O(q^{10})\) \(q+(-0.866025 + 1.50000i) q^{3} +(-1.73205 + 2.00000i) q^{7} +(-1.50000 - 2.59808i) q^{9} -5.19615i q^{11} -1.73205 q^{13} +3.00000i q^{17} +3.46410i q^{19} +(-1.50000 - 4.33013i) q^{21} +5.19615 q^{27} +5.19615i q^{29} -10.3923i q^{31} +(7.79423 + 4.50000i) q^{33} -8.00000i q^{37} +(1.50000 - 2.59808i) q^{39} +6.00000 q^{41} -10.0000i q^{43} +3.00000i q^{47} +(-1.00000 - 6.92820i) q^{49} +(-4.50000 - 2.59808i) q^{51} +10.3923 q^{53} +(-5.19615 - 3.00000i) q^{57} -6.00000 q^{59} +6.92820i q^{61} +(7.79423 + 1.50000i) q^{63} +2.00000i q^{67} -10.3923i q^{71} +6.92820 q^{73} +(10.3923 + 9.00000i) q^{77} +13.0000 q^{79} +(-4.50000 + 7.79423i) q^{81} +12.0000i q^{83} +(-7.79423 - 4.50000i) q^{87} +(3.00000 - 3.46410i) q^{91} +(15.5885 + 9.00000i) q^{93} +1.73205 q^{97} +(-13.5000 + 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{9} - 6 q^{21} + 6 q^{39} + 24 q^{41} - 4 q^{49} - 18 q^{51} - 24 q^{59} + 52 q^{79} - 18 q^{81} + 12 q^{91} - 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.866025 + 1.50000i −0.500000 + 0.866025i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.73205 + 2.00000i −0.654654 + 0.755929i
\(8\) 0 0
\(9\) −1.50000 2.59808i −0.500000 0.866025i
\(10\) 0 0
\(11\) 5.19615i 1.56670i −0.621582 0.783349i \(-0.713510\pi\)
0.621582 0.783349i \(-0.286490\pi\)
\(12\) 0 0
\(13\) −1.73205 −0.480384 −0.240192 0.970725i \(-0.577210\pi\)
−0.240192 + 0.970725i \(0.577210\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 0 0
\(19\) 3.46410i 0.794719i 0.917663 + 0.397360i \(0.130073\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) −1.50000 4.33013i −0.327327 0.944911i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.19615 1.00000
\(28\) 0 0
\(29\) 5.19615i 0.964901i 0.875923 + 0.482451i \(0.160253\pi\)
−0.875923 + 0.482451i \(0.839747\pi\)
\(30\) 0 0
\(31\) 10.3923i 1.86651i −0.359211 0.933257i \(-0.616954\pi\)
0.359211 0.933257i \(-0.383046\pi\)
\(32\) 0 0
\(33\) 7.79423 + 4.50000i 1.35680 + 0.783349i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000i 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 1.50000 2.59808i 0.240192 0.416025i
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 10.0000i 1.52499i −0.646997 0.762493i \(-0.723975\pi\)
0.646997 0.762493i \(-0.276025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 0 0
\(49\) −1.00000 6.92820i −0.142857 0.989743i
\(50\) 0 0
\(51\) −4.50000 2.59808i −0.630126 0.363803i
\(52\) 0 0
\(53\) 10.3923 1.42749 0.713746 0.700404i \(-0.246997\pi\)
0.713746 + 0.700404i \(0.246997\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.19615 3.00000i −0.688247 0.397360i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 6.92820i 0.887066i 0.896258 + 0.443533i \(0.146275\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 7.79423 + 1.50000i 0.981981 + 0.188982i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.3923i 1.23334i −0.787222 0.616670i \(-0.788481\pi\)
0.787222 0.616670i \(-0.211519\pi\)
\(72\) 0 0
\(73\) 6.92820 0.810885 0.405442 0.914121i \(-0.367117\pi\)
0.405442 + 0.914121i \(0.367117\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3923 + 9.00000i 1.18431 + 1.02565i
\(78\) 0 0
\(79\) 13.0000 1.46261 0.731307 0.682048i \(-0.238911\pi\)
0.731307 + 0.682048i \(0.238911\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −7.79423 4.50000i −0.835629 0.482451i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 3.00000 3.46410i 0.314485 0.363137i
\(92\) 0 0
\(93\) 15.5885 + 9.00000i 1.61645 + 0.933257i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.73205 0.175863 0.0879316 0.996127i \(-0.471974\pi\)
0.0879316 + 0.996127i \(0.471974\pi\)
\(98\) 0 0
\(99\) −13.5000 + 7.79423i −1.35680 + 0.783349i
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) 1.73205 0.170664 0.0853320 0.996353i \(-0.472805\pi\)
0.0853320 + 0.996353i \(0.472805\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −10.3923 −1.00466 −0.502331 0.864675i \(-0.667524\pi\)
−0.502331 + 0.864675i \(0.667524\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 12.0000 + 6.92820i 1.13899 + 0.657596i
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.59808 + 4.50000i 0.240192 + 0.416025i
\(118\) 0 0
\(119\) −6.00000 5.19615i −0.550019 0.476331i
\(120\) 0 0
\(121\) −16.0000 −1.45455
\(122\) 0 0
\(123\) −5.19615 + 9.00000i −0.468521 + 0.811503i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.00000i 0.177471i 0.996055 + 0.0887357i \(0.0282826\pi\)
−0.996055 + 0.0887357i \(0.971717\pi\)
\(128\) 0 0
\(129\) 15.0000 + 8.66025i 1.32068 + 0.762493i
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) −6.92820 6.00000i −0.600751 0.520266i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 20.7846 1.77575 0.887875 0.460086i \(-0.152181\pi\)
0.887875 + 0.460086i \(0.152181\pi\)
\(138\) 0 0
\(139\) 6.92820i 0.587643i −0.955860 0.293821i \(-0.905073\pi\)
0.955860 0.293821i \(-0.0949270\pi\)
\(140\) 0 0
\(141\) −4.50000 2.59808i −0.378968 0.218797i
\(142\) 0 0
\(143\) 9.00000i 0.752618i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 11.2583 + 4.50000i 0.928571 + 0.371154i
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −5.00000 −0.406894 −0.203447 0.979086i \(-0.565214\pi\)
−0.203447 + 0.979086i \(0.565214\pi\)
\(152\) 0 0
\(153\) 7.79423 4.50000i 0.630126 0.363803i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) −9.00000 + 15.5885i −0.713746 + 1.23625i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 8.00000i 0.626608i −0.949653 0.313304i \(-0.898564\pi\)
0.949653 0.313304i \(-0.101436\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.0000i 1.16073i −0.814355 0.580367i \(-0.802909\pi\)
0.814355 0.580367i \(-0.197091\pi\)
\(168\) 0 0
\(169\) −10.0000 −0.769231
\(170\) 0 0
\(171\) 9.00000 5.19615i 0.688247 0.397360i
\(172\) 0 0
\(173\) 9.00000i 0.684257i 0.939653 + 0.342129i \(0.111148\pi\)
−0.939653 + 0.342129i \(0.888852\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.19615 9.00000i 0.390567 0.676481i
\(178\) 0 0
\(179\) 10.3923i 0.776757i −0.921500 0.388379i \(-0.873035\pi\)
0.921500 0.388379i \(-0.126965\pi\)
\(180\) 0 0
\(181\) 17.3205i 1.28742i −0.765268 0.643712i \(-0.777394\pi\)
0.765268 0.643712i \(-0.222606\pi\)
\(182\) 0 0
\(183\) −10.3923 6.00000i −0.768221 0.443533i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 15.5885 1.13994
\(188\) 0 0
\(189\) −9.00000 + 10.3923i −0.654654 + 0.755929i
\(190\) 0 0
\(191\) 15.5885i 1.12794i 0.825795 + 0.563971i \(0.190727\pi\)
−0.825795 + 0.563971i \(0.809273\pi\)
\(192\) 0 0
\(193\) 16.0000i 1.15171i −0.817554 0.575853i \(-0.804670\pi\)
0.817554 0.575853i \(-0.195330\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.3923 −0.740421 −0.370211 0.928948i \(-0.620714\pi\)
−0.370211 + 0.928948i \(0.620714\pi\)
\(198\) 0 0
\(199\) 24.2487i 1.71895i −0.511182 0.859473i \(-0.670792\pi\)
0.511182 0.859473i \(-0.329208\pi\)
\(200\) 0 0
\(201\) −3.00000 1.73205i −0.211604 0.122169i
\(202\) 0 0
\(203\) −10.3923 9.00000i −0.729397 0.631676i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 18.0000 1.24509
\(210\) 0 0
\(211\) 25.0000 1.72107 0.860535 0.509390i \(-0.170129\pi\)
0.860535 + 0.509390i \(0.170129\pi\)
\(212\) 0 0
\(213\) 15.5885 + 9.00000i 1.06810 + 0.616670i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 20.7846 + 18.0000i 1.41095 + 1.22192i
\(218\) 0 0
\(219\) −6.00000 + 10.3923i −0.405442 + 0.702247i
\(220\) 0 0
\(221\) 5.19615i 0.349531i
\(222\) 0 0
\(223\) −5.19615 −0.347960 −0.173980 0.984749i \(-0.555663\pi\)
−0.173980 + 0.984749i \(0.555663\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 27.0000i 1.79205i −0.444001 0.896026i \(-0.646441\pi\)
0.444001 0.896026i \(-0.353559\pi\)
\(228\) 0 0
\(229\) 10.3923i 0.686743i −0.939200 0.343371i \(-0.888431\pi\)
0.939200 0.343371i \(-0.111569\pi\)
\(230\) 0 0
\(231\) −22.5000 + 7.79423i −1.48039 + 0.512823i
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −11.2583 + 19.5000i −0.731307 + 1.26666i
\(238\) 0 0
\(239\) 25.9808i 1.68056i 0.542156 + 0.840278i \(0.317608\pi\)
−0.542156 + 0.840278i \(0.682392\pi\)
\(240\) 0 0
\(241\) 6.92820i 0.446285i 0.974786 + 0.223142i \(0.0716315\pi\)
−0.974786 + 0.223142i \(0.928369\pi\)
\(242\) 0 0
\(243\) −7.79423 13.5000i −0.500000 0.866025i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.00000i 0.381771i
\(248\) 0 0
\(249\) −18.0000 10.3923i −1.14070 0.658586i
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) 0 0
\(259\) 16.0000 + 13.8564i 0.994192 + 0.860995i
\(260\) 0 0
\(261\) 13.5000 7.79423i 0.835629 0.482451i
\(262\) 0 0
\(263\) 31.1769 1.92245 0.961225 0.275764i \(-0.0889307\pi\)
0.961225 + 0.275764i \(0.0889307\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 12.0000 0.731653 0.365826 0.930683i \(-0.380786\pi\)
0.365826 + 0.930683i \(0.380786\pi\)
\(270\) 0 0
\(271\) 6.92820i 0.420858i −0.977609 0.210429i \(-0.932514\pi\)
0.977609 0.210429i \(-0.0674861\pi\)
\(272\) 0 0
\(273\) 2.59808 + 7.50000i 0.157243 + 0.453921i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 8.00000i 0.480673i 0.970690 + 0.240337i \(0.0772579\pi\)
−0.970690 + 0.240337i \(0.922742\pi\)
\(278\) 0 0
\(279\) −27.0000 + 15.5885i −1.61645 + 0.933257i
\(280\) 0 0
\(281\) 5.19615i 0.309976i 0.987916 + 0.154988i \(0.0495340\pi\)
−0.987916 + 0.154988i \(0.950466\pi\)
\(282\) 0 0
\(283\) −25.9808 −1.54440 −0.772198 0.635382i \(-0.780843\pi\)
−0.772198 + 0.635382i \(0.780843\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −10.3923 + 12.0000i −0.613438 + 0.708338i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −1.50000 + 2.59808i −0.0879316 + 0.152302i
\(292\) 0 0
\(293\) 3.00000i 0.175262i 0.996153 + 0.0876309i \(0.0279296\pi\)
−0.996153 + 0.0876309i \(0.972070\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 27.0000i 1.56670i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 20.0000 + 17.3205i 1.15278 + 0.998337i
\(302\) 0 0
\(303\) −15.5885 + 27.0000i −0.895533 + 1.55111i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −25.9808 −1.48280 −0.741400 0.671063i \(-0.765838\pi\)
−0.741400 + 0.671063i \(0.765838\pi\)
\(308\) 0 0
\(309\) −1.50000 + 2.59808i −0.0853320 + 0.147799i
\(310\) 0 0
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) −22.5167 −1.27272 −0.636358 0.771393i \(-0.719560\pi\)
−0.636358 + 0.771393i \(0.719560\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10.3923 −0.583690 −0.291845 0.956466i \(-0.594269\pi\)
−0.291845 + 0.956466i \(0.594269\pi\)
\(318\) 0 0
\(319\) 27.0000 1.51171
\(320\) 0 0
\(321\) 9.00000 15.5885i 0.502331 0.870063i
\(322\) 0 0
\(323\) −10.3923 −0.578243
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −9.52628 + 16.5000i −0.526804 + 0.912452i
\(328\) 0 0
\(329\) −6.00000 5.19615i −0.330791 0.286473i
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) −20.7846 + 12.0000i −1.13899 + 0.657596i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 20.0000i 1.08947i −0.838608 0.544735i \(-0.816630\pi\)
0.838608 0.544735i \(-0.183370\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −54.0000 −2.92426
\(342\) 0 0
\(343\) 15.5885 + 10.0000i 0.841698 + 0.539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 17.3205i 0.927146i −0.886059 0.463573i \(-0.846567\pi\)
0.886059 0.463573i \(-0.153433\pi\)
\(350\) 0 0
\(351\) −9.00000 −0.480384
\(352\) 0 0
\(353\) 21.0000i 1.11772i −0.829263 0.558859i \(-0.811239\pi\)
0.829263 0.558859i \(-0.188761\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 12.9904 4.50000i 0.687524 0.238165i
\(358\) 0 0
\(359\) 10.3923i 0.548485i −0.961661 0.274242i \(-0.911573\pi\)
0.961661 0.274242i \(-0.0884271\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) 13.8564 24.0000i 0.727273 1.25967i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −29.4449 −1.53701 −0.768505 0.639844i \(-0.778999\pi\)
−0.768505 + 0.639844i \(0.778999\pi\)
\(368\) 0 0
\(369\) −9.00000 15.5885i −0.468521 0.811503i
\(370\) 0 0
\(371\) −18.0000 + 20.7846i −0.934513 + 1.07908i
\(372\) 0 0
\(373\) 20.0000i 1.03556i −0.855514 0.517780i \(-0.826758\pi\)
0.855514 0.517780i \(-0.173242\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 9.00000i 0.463524i
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) −3.00000 1.73205i −0.153695 0.0887357i
\(382\) 0 0
\(383\) 24.0000i 1.22634i −0.789950 0.613171i \(-0.789894\pi\)
0.789950 0.613171i \(-0.210106\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −25.9808 + 15.0000i −1.32068 + 0.762493i
\(388\) 0 0
\(389\) 15.5885i 0.790366i −0.918602 0.395183i \(-0.870681\pi\)
0.918602 0.395183i \(-0.129319\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 5.19615 0.260787 0.130394 0.991462i \(-0.458376\pi\)
0.130394 + 0.991462i \(0.458376\pi\)
\(398\) 0 0
\(399\) 15.0000 5.19615i 0.750939 0.260133i
\(400\) 0 0
\(401\) 5.19615i 0.259483i −0.991548 0.129742i \(-0.958585\pi\)
0.991548 0.129742i \(-0.0414148\pi\)
\(402\) 0 0
\(403\) 18.0000i 0.896644i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −41.5692 −2.06051
\(408\) 0 0
\(409\) 3.46410i 0.171289i 0.996326 + 0.0856444i \(0.0272949\pi\)
−0.996326 + 0.0856444i \(0.972705\pi\)
\(410\) 0 0
\(411\) −18.0000 + 31.1769i −0.887875 + 1.53784i
\(412\) 0 0
\(413\) 10.3923 12.0000i 0.511372 0.590481i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 10.3923 + 6.00000i 0.508913 + 0.293821i
\(418\) 0 0
\(419\) 6.00000 0.293119 0.146560 0.989202i \(-0.453180\pi\)
0.146560 + 0.989202i \(0.453180\pi\)
\(420\) 0 0
\(421\) −13.0000 −0.633581 −0.316791 0.948495i \(-0.602605\pi\)
−0.316791 + 0.948495i \(0.602605\pi\)
\(422\) 0 0
\(423\) 7.79423 4.50000i 0.378968 0.218797i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −13.8564 12.0000i −0.670559 0.580721i
\(428\) 0 0
\(429\) −13.5000 7.79423i −0.651786 0.376309i
\(430\) 0 0
\(431\) 5.19615i 0.250290i 0.992138 + 0.125145i \(0.0399396\pi\)
−0.992138 + 0.125145i \(0.960060\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 20.7846i 0.991995i −0.868324 0.495998i \(-0.834802\pi\)
0.868324 0.495998i \(-0.165198\pi\)
\(440\) 0 0
\(441\) −16.5000 + 12.9904i −0.785714 + 0.618590i
\(442\) 0 0
\(443\) −31.1769 −1.48126 −0.740630 0.671913i \(-0.765473\pi\)
−0.740630 + 0.671913i \(0.765473\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 25.9808i 1.22611i −0.790041 0.613054i \(-0.789941\pi\)
0.790041 0.613054i \(-0.210059\pi\)
\(450\) 0 0
\(451\) 31.1769i 1.46806i
\(452\) 0 0
\(453\) 4.33013 7.50000i 0.203447 0.352381i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) 0 0
\(459\) 15.5885i 0.727607i
\(460\) 0 0
\(461\) −12.0000 −0.558896 −0.279448 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 16.0000i 0.743583i 0.928316 + 0.371792i \(0.121256\pi\)
−0.928316 + 0.371792i \(0.878744\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.00000i 0.416470i −0.978079 0.208235i \(-0.933228\pi\)
0.978079 0.208235i \(-0.0667719\pi\)
\(468\) 0 0
\(469\) −4.00000 3.46410i −0.184703 0.159957i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −51.9615 −2.38919
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −15.5885 27.0000i −0.713746 1.23625i
\(478\) 0 0
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) 13.8564i 0.631798i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8.00000i 0.362515i −0.983436 0.181257i \(-0.941983\pi\)
0.983436 0.181257i \(-0.0580167\pi\)
\(488\) 0 0
\(489\) 12.0000 + 6.92820i 0.542659 + 0.313304i
\(490\) 0 0
\(491\) 5.19615i 0.234499i 0.993102 + 0.117250i \(0.0374077\pi\)
−0.993102 + 0.117250i \(0.962592\pi\)
\(492\) 0 0
\(493\) −15.5885 −0.702069
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 20.7846 + 18.0000i 0.932317 + 0.807410i
\(498\) 0 0
\(499\) −13.0000 −0.581960 −0.290980 0.956729i \(-0.593981\pi\)
−0.290980 + 0.956729i \(0.593981\pi\)
\(500\) 0 0
\(501\) 22.5000 + 12.9904i 1.00523 + 0.580367i
\(502\) 0 0
\(503\) 3.00000i 0.133763i 0.997761 + 0.0668817i \(0.0213050\pi\)
−0.997761 + 0.0668817i \(0.978695\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 8.66025 15.0000i 0.384615 0.666173i
\(508\) 0 0
\(509\) −24.0000 −1.06378 −0.531891 0.846813i \(-0.678518\pi\)
−0.531891 + 0.846813i \(0.678518\pi\)
\(510\) 0 0
\(511\) −12.0000 + 13.8564i −0.530849 + 0.612971i
\(512\) 0 0
\(513\) 18.0000i 0.794719i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 15.5885 0.685580
\(518\) 0 0
\(519\) −13.5000 7.79423i −0.592584 0.342129i
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) −17.3205 −0.757373 −0.378686 0.925525i \(-0.623624\pi\)
−0.378686 + 0.925525i \(0.623624\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 31.1769 1.35809
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 9.00000 + 15.5885i 0.390567 + 0.676481i
\(532\) 0 0
\(533\) −10.3923 −0.450141
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.5885 + 9.00000i 0.672692 + 0.388379i
\(538\) 0 0
\(539\) −36.0000 + 5.19615i −1.55063 + 0.223814i
\(540\) 0 0
\(541\) 17.0000 0.730887 0.365444 0.930834i \(-0.380917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 0 0
\(543\) 25.9808 + 15.0000i 1.11494 + 0.643712i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 32.0000i 1.36822i 0.729378 + 0.684111i \(0.239809\pi\)
−0.729378 + 0.684111i \(0.760191\pi\)
\(548\) 0 0
\(549\) 18.0000 10.3923i 0.768221 0.443533i
\(550\) 0 0
\(551\) −18.0000 −0.766826
\(552\) 0 0
\(553\) −22.5167 + 26.0000i −0.957506 + 1.10563i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −20.7846 −0.880672 −0.440336 0.897833i \(-0.645141\pi\)
−0.440336 + 0.897833i \(0.645141\pi\)
\(558\) 0 0
\(559\) 17.3205i 0.732579i
\(560\) 0 0
\(561\) −13.5000 + 23.3827i −0.569970 + 0.987218i
\(562\) 0 0
\(563\) 36.0000i 1.51722i −0.651546 0.758610i \(-0.725879\pi\)
0.651546 0.758610i \(-0.274121\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −7.79423 22.5000i −0.327327 0.944911i
\(568\) 0 0
\(569\) 20.7846i 0.871336i 0.900107 + 0.435668i \(0.143488\pi\)
−0.900107 + 0.435668i \(0.856512\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) −23.3827 13.5000i −0.976826 0.563971i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.73205 −0.0721062 −0.0360531 0.999350i \(-0.511479\pi\)
−0.0360531 + 0.999350i \(0.511479\pi\)
\(578\) 0 0
\(579\) 24.0000 + 13.8564i 0.997406 + 0.575853i
\(580\) 0 0
\(581\) −24.0000 20.7846i −0.995688 0.862291i
\(582\) 0 0
\(583\) 54.0000i 2.23645i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 36.0000i 1.48588i −0.669359 0.742940i \(-0.733431\pi\)
0.669359 0.742940i \(-0.266569\pi\)
\(588\) 0 0
\(589\) 36.0000 1.48335
\(590\) 0 0
\(591\) 9.00000 15.5885i 0.370211 0.641223i
\(592\) 0 0
\(593\) 21.0000i 0.862367i 0.902264 + 0.431183i \(0.141904\pi\)
−0.902264 + 0.431183i \(0.858096\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 36.3731 + 21.0000i 1.48865 + 0.859473i
\(598\) 0 0
\(599\) 25.9808i 1.06155i 0.847514 + 0.530773i \(0.178098\pi\)
−0.847514 + 0.530773i \(0.821902\pi\)
\(600\) 0 0
\(601\) 31.1769i 1.27173i 0.771799 + 0.635866i \(0.219357\pi\)
−0.771799 + 0.635866i \(0.780643\pi\)
\(602\) 0 0
\(603\) 5.19615 3.00000i 0.211604 0.122169i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −8.66025 −0.351509 −0.175754 0.984434i \(-0.556236\pi\)
−0.175754 + 0.984434i \(0.556236\pi\)
\(608\) 0 0
\(609\) 22.5000 7.79423i 0.911746 0.315838i
\(610\) 0 0
\(611\) 5.19615i 0.210214i
\(612\) 0 0
\(613\) 26.0000i 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 41.5692 1.67351 0.836757 0.547575i \(-0.184449\pi\)
0.836757 + 0.547575i \(0.184449\pi\)
\(618\) 0 0
\(619\) 6.92820i 0.278468i 0.990260 + 0.139234i \(0.0444640\pi\)
−0.990260 + 0.139234i \(0.955536\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −15.5885 + 27.0000i −0.622543 + 1.07828i
\(628\) 0 0
\(629\) 24.0000 0.956943
\(630\) 0 0
\(631\) 17.0000 0.676759 0.338380 0.941010i \(-0.390121\pi\)
0.338380 + 0.941010i \(0.390121\pi\)
\(632\) 0 0
\(633\) −21.6506 + 37.5000i −0.860535 + 1.49049i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.73205 + 12.0000i 0.0686264 + 0.475457i
\(638\) 0 0
\(639\) −27.0000 + 15.5885i −1.06810 + 0.616670i
\(640\) 0 0
\(641\) 20.7846i 0.820943i 0.911873 + 0.410471i \(0.134636\pi\)
−0.911873 + 0.410471i \(0.865364\pi\)
\(642\) 0 0
\(643\) −15.5885 −0.614749 −0.307374 0.951589i \(-0.599450\pi\)
−0.307374 + 0.951589i \(0.599450\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 36.0000i 1.41531i −0.706560 0.707653i \(-0.749754\pi\)
0.706560 0.707653i \(-0.250246\pi\)
\(648\) 0 0
\(649\) 31.1769i 1.22380i
\(650\) 0 0
\(651\) −45.0000 + 15.5885i −1.76369 + 0.610960i
\(652\) 0 0
\(653\) 10.3923 0.406682 0.203341 0.979108i \(-0.434820\pi\)
0.203341 + 0.979108i \(0.434820\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −10.3923 18.0000i −0.405442 0.702247i
\(658\) 0 0
\(659\) 15.5885i 0.607240i 0.952793 + 0.303620i \(0.0981953\pi\)
−0.952793 + 0.303620i \(0.901805\pi\)
\(660\) 0 0
\(661\) 48.4974i 1.88633i −0.332323 0.943166i \(-0.607832\pi\)
0.332323 0.943166i \(-0.392168\pi\)
\(662\) 0 0
\(663\) 7.79423 + 4.50000i 0.302703 + 0.174766i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 4.50000 7.79423i 0.173980 0.301342i
\(670\) 0 0
\(671\) 36.0000 1.38976
\(672\) 0 0
\(673\) 32.0000i 1.23351i 0.787155 + 0.616755i \(0.211553\pi\)
−0.787155 + 0.616755i \(0.788447\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 27.0000i 1.03769i 0.854867 + 0.518847i \(0.173639\pi\)
−0.854867 + 0.518847i \(0.826361\pi\)
\(678\) 0 0
\(679\) −3.00000 + 3.46410i −0.115129 + 0.132940i
\(680\) 0 0
\(681\) 40.5000 + 23.3827i 1.55196 + 0.896026i
\(682\) 0 0
\(683\) −20.7846 −0.795301 −0.397650 0.917537i \(-0.630174\pi\)
−0.397650 + 0.917537i \(0.630174\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 15.5885 + 9.00000i 0.594737 + 0.343371i
\(688\) 0 0
\(689\) −18.0000 −0.685745
\(690\) 0 0
\(691\) 41.5692i 1.58137i 0.612225 + 0.790684i \(0.290275\pi\)
−0.612225 + 0.790684i \(0.709725\pi\)
\(692\) 0 0
\(693\) 7.79423 40.5000i 0.296078 1.53847i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 18.0000i 0.681799i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 36.3731i 1.37379i 0.726756 + 0.686896i \(0.241027\pi\)
−0.726756 + 0.686896i \(0.758973\pi\)
\(702\) 0 0
\(703\) 27.7128 1.04521
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −31.1769 + 36.0000i −1.17253 + 1.35392i
\(708\) 0 0
\(709\) −13.0000 −0.488225 −0.244113 0.969747i \(-0.578497\pi\)
−0.244113 + 0.969747i \(0.578497\pi\)
\(710\) 0 0
\(711\) −19.5000 33.7750i −0.731307 1.26666i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −38.9711 22.5000i −1.45540 0.840278i
\(718\) 0 0
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) 0 0
\(721\) −3.00000 + 3.46410i −0.111726 + 0.129010i
\(722\) 0 0
\(723\) −10.3923 6.00000i −0.386494 0.223142i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 51.9615 1.92715 0.963573 0.267445i \(-0.0861794\pi\)
0.963573 + 0.267445i \(0.0861794\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 30.0000 1.10959
\(732\) 0 0
\(733\) 36.3731 1.34347 0.671735 0.740792i \(-0.265549\pi\)
0.671735 + 0.740792i \(0.265549\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10.3923 0.382805
\(738\) 0 0
\(739\) −11.0000 −0.404642 −0.202321 0.979319i \(-0.564848\pi\)
−0.202321 + 0.979319i \(0.564848\pi\)
\(740\) 0 0
\(741\) 9.00000 + 5.19615i 0.330623 + 0.190885i
\(742\) 0 0
\(743\) 10.3923 0.381257 0.190628 0.981662i \(-0.438947\pi\)
0.190628 + 0.981662i \(0.438947\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 31.1769 18.0000i 1.14070 0.658586i
\(748\) 0 0
\(749\) 18.0000 20.7846i 0.657706 0.759453i
\(750\) 0 0
\(751\) −11.0000 −0.401396 −0.200698 0.979653i \(-0.564321\pi\)
−0.200698 + 0.979653i \(0.564321\pi\)
\(752\) 0 0
\(753\) 20.7846 36.0000i 0.757433 1.31191i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 8.00000i 0.290765i −0.989376 0.145382i \(-0.953559\pi\)
0.989376 0.145382i \(-0.0464413\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −12.0000 −0.435000 −0.217500 0.976060i \(-0.569790\pi\)
−0.217500 + 0.976060i \(0.569790\pi\)
\(762\) 0 0
\(763\) −19.0526 + 22.0000i −0.689749 + 0.796453i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 10.3923 0.375244
\(768\) 0 0
\(769\) 17.3205i 0.624593i −0.949985 0.312297i \(-0.898902\pi\)
0.949985 0.312297i \(-0.101098\pi\)
\(770\) 0 0
\(771\) −27.0000 15.5885i −0.972381 0.561405i
\(772\) 0 0
\(773\) 15.0000i 0.539513i 0.962929 + 0.269756i \(0.0869431\pi\)
−0.962929 + 0.269756i \(0.913057\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −34.6410 + 12.0000i −1.24274 + 0.430498i
\(778\) 0 0
\(779\) 20.7846i 0.744686i
\(780\) 0 0
\(781\) −54.0000 −1.93227
\(782\) 0 0
\(783\) 27.0000i 0.964901i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −50.2295 −1.79049 −0.895244 0.445577i \(-0.852999\pi\)
−0.895244 + 0.445577i \(0.852999\pi\)
\(788\) 0 0
\(789\) −27.0000 + 46.7654i −0.961225 + 1.66489i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9.00000i 0.318796i −0.987214 0.159398i \(-0.949045\pi\)
0.987214 0.159398i \(-0.0509554\pi\)
\(798\) 0 0
\(799\) −9.00000 −0.318397
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 36.0000i 1.27041i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −10.3923 + 18.0000i −0.365826 + 0.633630i
\(808\) 0 0
\(809\) 36.3731i 1.27881i 0.768871 + 0.639404i \(0.220819\pi\)
−0.768871 + 0.639404i \(0.779181\pi\)
\(810\) 0 0
\(811\) 20.7846i 0.729846i −0.931038 0.364923i \(-0.881095\pi\)
0.931038 0.364923i \(-0.118905\pi\)
\(812\) 0 0
\(813\) 10.3923 + 6.00000i 0.364474 + 0.210429i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 34.6410 1.21194
\(818\) 0 0
\(819\) −13.5000 2.59808i −0.471728 0.0907841i
\(820\) 0 0
\(821\) 36.3731i 1.26943i −0.772747 0.634714i \(-0.781118\pi\)
0.772747 0.634714i \(-0.218882\pi\)
\(822\) 0 0
\(823\) 2.00000i 0.0697156i −0.999392 0.0348578i \(-0.988902\pi\)
0.999392 0.0348578i \(-0.0110978\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 41.5692 1.44550 0.722752 0.691108i \(-0.242877\pi\)
0.722752 + 0.691108i \(0.242877\pi\)
\(828\) 0 0
\(829\) 45.0333i 1.56407i −0.623233 0.782036i \(-0.714181\pi\)
0.623233 0.782036i \(-0.285819\pi\)
\(830\) 0 0
\(831\) −12.0000 6.92820i −0.416275 0.240337i
\(832\) 0 0
\(833\) 20.7846 3.00000i 0.720144 0.103944i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 54.0000i 1.86651i
\(838\) 0 0
\(839\) −12.0000 −0.414286 −0.207143 0.978311i \(-0.566417\pi\)
−0.207143 + 0.978311i \(0.566417\pi\)
\(840\) 0 0
\(841\) 2.00000 0.0689655
\(842\) 0 0
\(843\) −7.79423 4.50000i −0.268447 0.154988i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 27.7128 32.0000i 0.952224 1.09953i
\(848\) 0 0
\(849\) 22.5000 38.9711i 0.772198 1.33749i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 13.8564 0.474434 0.237217 0.971457i \(-0.423765\pi\)
0.237217 + 0.971457i \(0.423765\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.00000i 0.204956i −0.994735 0.102478i \(-0.967323\pi\)
0.994735 0.102478i \(-0.0326771\pi\)
\(858\) 0 0
\(859\) 6.92820i 0.236387i 0.992991 + 0.118194i \(0.0377103\pi\)
−0.992991 + 0.118194i \(0.962290\pi\)
\(860\) 0 0
\(861\) −9.00000 25.9808i −0.306719 0.885422i
\(862\) 0 0
\(863\) 31.1769 1.06127 0.530637 0.847599i \(-0.321953\pi\)
0.530637 + 0.847599i \(0.321953\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −6.92820 + 12.0000i −0.235294 + 0.407541i
\(868\) 0 0
\(869\) 67.5500i 2.29148i
\(870\) 0 0
\(871\) 3.46410i 0.117377i
\(872\) 0 0
\(873\) −2.59808 4.50000i −0.0879316 0.152302i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.00000i 0.0675352i 0.999430 + 0.0337676i \(0.0107506\pi\)
−0.999430 + 0.0337676i \(0.989249\pi\)
\(878\) 0 0
\(879\) −4.50000 2.59808i −0.151781 0.0876309i
\(880\) 0 0
\(881\) 36.0000 1.21287 0.606435 0.795133i \(-0.292599\pi\)
0.606435 + 0.795133i \(0.292599\pi\)
\(882\) 0 0
\(883\) 44.0000i 1.48072i 0.672212 + 0.740359i \(0.265344\pi\)
−0.672212 + 0.740359i \(0.734656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 48.0000i 1.61168i 0.592132 + 0.805841i \(0.298286\pi\)
−0.592132 + 0.805841i \(0.701714\pi\)
\(888\) 0 0
\(889\) −4.00000 3.46410i −0.134156 0.116182i
\(890\) 0 0
\(891\) 40.5000 + 23.3827i 1.35680 + 0.783349i
\(892\) 0 0
\(893\) −10.3923 −0.347765
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 54.0000 1.80100
\(900\) 0 0
\(901\) 31.1769i 1.03865i
\(902\) 0 0
\(903\) −43.3013 + 15.0000i −1.44098 + 0.499169i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 32.0000i 1.06254i −0.847202 0.531271i \(-0.821714\pi\)
0.847202 0.531271i \(-0.178286\pi\)
\(908\) 0 0
\(909\) −27.0000 46.7654i −0.895533 1.55111i
\(910\) 0 0
\(911\) 31.1769i 1.03294i −0.856306 0.516469i \(-0.827246\pi\)
0.856306 0.516469i \(-0.172754\pi\)
\(912\) 0 0
\(913\) 62.3538 2.06361
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 11.0000 0.362857 0.181428 0.983404i \(-0.441928\pi\)
0.181428 + 0.983404i \(0.441928\pi\)
\(920\) 0 0
\(921\) 22.5000 38.9711i 0.741400 1.28414i
\(922\) 0 0
\(923\) 18.0000i 0.592477i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −2.59808 4.50000i −0.0853320 0.147799i
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 24.0000 3.46410i 0.786568 0.113531i
\(932\) 0 0
\(933\) −25.9808 + 45.0000i −0.850572 + 1.47323i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −22.5167 −0.735587 −0.367794 0.929907i \(-0.619887\pi\)
−0.367794 + 0.929907i \(0.619887\pi\)
\(938\) 0 0
\(939\) 19.5000 33.7750i 0.636358 1.10221i
\(940\) 0 0
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −10.3923 −0.337705 −0.168852 0.985641i \(-0.554006\pi\)
−0.168852 + 0.985641i \(0.554006\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 9.00000 15.5885i 0.291845 0.505490i
\(952\) 0 0
\(953\) 51.9615 1.68320 0.841599 0.540102i \(-0.181614\pi\)
0.841599 + 0.540102i \(0.181614\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −23.3827 + 40.5000i −0.755855 + 1.30918i
\(958\) 0 0
\(959\) −36.0000 + 41.5692i −1.16250 + 1.34234i
\(960\) 0 0
\(961\) −77.0000 −2.48387
\(962\) 0 0
\(963\) 15.5885 + 27.0000i 0.502331 + 0.870063i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 2.00000i 0.0643157i 0.999483 + 0.0321578i \(0.0102379\pi\)
−0.999483 + 0.0321578i \(0.989762\pi\)
\(968\) 0 0
\(969\) 9.00000 15.5885i 0.289122 0.500773i
\(970\) 0 0
\(971\) 6.00000 0.192549 0.0962746 0.995355i \(-0.469307\pi\)
0.0962746 + 0.995355i \(0.469307\pi\)
\(972\) 0 0
\(973\) 13.8564 + 12.0000i 0.444216 + 0.384702i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −51.9615 −1.66240 −0.831198 0.555976i \(-0.812345\pi\)
−0.831198 + 0.555976i \(0.812345\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −16.5000 28.5788i −0.526804 0.912452i
\(982\) 0 0
\(983\) 57.0000i 1.81802i 0.416777 + 0.909009i \(0.363160\pi\)
−0.416777 + 0.909009i \(0.636840\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 12.9904 4.50000i 0.413488 0.143237i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 28.0000 0.889449 0.444725 0.895667i \(-0.353302\pi\)
0.444725 + 0.895667i \(0.353302\pi\)
\(992\) 0 0
\(993\) −3.46410 + 6.00000i −0.109930 + 0.190404i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 15.5885 0.493691 0.246846 0.969055i \(-0.420606\pi\)
0.246846 + 0.969055i \(0.420606\pi\)
\(998\) 0 0
\(999\) 41.5692i 1.31519i
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.f.c.1049.2 4
3.2 odd 2 2100.2.f.d.1049.1 4
5.2 odd 4 2100.2.d.e.1301.2 2
5.3 odd 4 420.2.d.a.41.1 2
5.4 even 2 inner 2100.2.f.c.1049.3 4
7.6 odd 2 2100.2.f.d.1049.3 4
15.2 even 4 2100.2.d.a.1301.2 2
15.8 even 4 420.2.d.b.41.1 yes 2
15.14 odd 2 2100.2.f.d.1049.4 4
20.3 even 4 1680.2.f.d.881.2 2
21.20 even 2 inner 2100.2.f.c.1049.4 4
35.13 even 4 420.2.d.b.41.2 yes 2
35.27 even 4 2100.2.d.a.1301.1 2
35.34 odd 2 2100.2.f.d.1049.2 4
60.23 odd 4 1680.2.f.a.881.2 2
105.62 odd 4 2100.2.d.e.1301.1 2
105.83 odd 4 420.2.d.a.41.2 yes 2
105.104 even 2 inner 2100.2.f.c.1049.1 4
140.83 odd 4 1680.2.f.a.881.1 2
420.83 even 4 1680.2.f.d.881.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
420.2.d.a.41.1 2 5.3 odd 4
420.2.d.a.41.2 yes 2 105.83 odd 4
420.2.d.b.41.1 yes 2 15.8 even 4
420.2.d.b.41.2 yes 2 35.13 even 4
1680.2.f.a.881.1 2 140.83 odd 4
1680.2.f.a.881.2 2 60.23 odd 4
1680.2.f.d.881.1 2 420.83 even 4
1680.2.f.d.881.2 2 20.3 even 4
2100.2.d.a.1301.1 2 35.27 even 4
2100.2.d.a.1301.2 2 15.2 even 4
2100.2.d.e.1301.1 2 105.62 odd 4
2100.2.d.e.1301.2 2 5.2 odd 4
2100.2.f.c.1049.1 4 105.104 even 2 inner
2100.2.f.c.1049.2 4 1.1 even 1 trivial
2100.2.f.c.1049.3 4 5.4 even 2 inner
2100.2.f.c.1049.4 4 21.20 even 2 inner
2100.2.f.d.1049.1 4 3.2 odd 2
2100.2.f.d.1049.2 4 35.34 odd 2
2100.2.f.d.1049.3 4 7.6 odd 2
2100.2.f.d.1049.4 4 15.14 odd 2