Properties

Label 2520.2.bi.e.1801.1
Level $2520$
Weight $2$
Character 2520.1801
Analytic conductor $20.122$
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] = [2520,2,Mod(361,2520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2520, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2520.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.bi (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: \(20.1223013094\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1801.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 2520.1801
Dual form 2520.2.bi.e.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{5} +(2.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q+(-0.500000 + 0.866025i) q^{5} +(2.00000 + 1.73205i) q^{7} +(-0.500000 - 0.866025i) q^{11} -3.00000 q^{13} +(-1.00000 - 1.73205i) q^{17} +(2.50000 - 4.33013i) q^{19} +(3.50000 - 6.06218i) q^{23} +(-0.500000 - 0.866025i) q^{25} +6.00000 q^{29} +(-2.00000 - 3.46410i) q^{31} +(-2.50000 + 0.866025i) q^{35} +(2.50000 - 4.33013i) q^{37} +5.00000 q^{41} +6.00000 q^{43} +(-4.50000 + 7.79423i) q^{47} +(1.00000 + 6.92820i) q^{49} +(5.50000 + 9.52628i) q^{53} +1.00000 q^{55} +(4.00000 + 6.92820i) q^{59} +(6.00000 - 10.3923i) q^{61} +(1.50000 - 2.59808i) q^{65} +(2.00000 + 3.46410i) q^{67} +4.00000 q^{71} +(-6.00000 - 10.3923i) q^{73} +(0.500000 - 2.59808i) q^{77} +(-7.00000 + 12.1244i) q^{79} +4.00000 q^{83} +2.00000 q^{85} +(3.00000 - 5.19615i) q^{89} +(-6.00000 - 5.19615i) q^{91} +(2.50000 + 4.33013i) q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + 4 q^{7} - q^{11} - 6 q^{13} - 2 q^{17} + 5 q^{19} + 7 q^{23} - q^{25} + 12 q^{29} - 4 q^{31} - 5 q^{35} + 5 q^{37} + 10 q^{41} + 12 q^{43} - 9 q^{47} + 2 q^{49} + 11 q^{53} + 2 q^{55} + 8 q^{59} + 12 q^{61} + 3 q^{65} + 4 q^{67} + 8 q^{71} - 12 q^{73} + q^{77} - 14 q^{79} + 8 q^{83} + 4 q^{85} + 6 q^{89} - 12 q^{91} + 5 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i
\(6\) 0 0
\(7\) 2.00000 + 1.73205i 0.755929 + 0.654654i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.500000 0.866025i −0.150756 0.261116i 0.780750 0.624844i \(-0.214837\pi\)
−0.931505 + 0.363727i \(0.881504\pi\)
\(12\) 0 0
\(13\) −3.00000 −0.832050 −0.416025 0.909353i \(-0.636577\pi\)
−0.416025 + 0.909353i \(0.636577\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 1.73205i −0.242536 0.420084i 0.718900 0.695113i \(-0.244646\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) 0 0
\(19\) 2.50000 4.33013i 0.573539 0.993399i −0.422659 0.906289i \(-0.638903\pi\)
0.996199 0.0871106i \(-0.0277634\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.50000 6.06218i 0.729800 1.26405i −0.227167 0.973856i \(-0.572946\pi\)
0.956967 0.290196i \(-0.0937204\pi\)
\(24\) 0 0
\(25\) −0.500000 0.866025i −0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −2.00000 3.46410i −0.359211 0.622171i 0.628619 0.777714i \(-0.283621\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.50000 + 0.866025i −0.422577 + 0.146385i
\(36\) 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 0
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.50000 + 7.79423i −0.656392 + 1.13691i 0.325150 + 0.945662i \(0.394585\pi\)
−0.981543 + 0.191243i \(0.938748\pi\)
\(48\) 0 0
\(49\) 1.00000 + 6.92820i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.50000 + 9.52628i 0.755483 + 1.30854i 0.945134 + 0.326683i \(0.105931\pi\)
−0.189651 + 0.981852i \(0.560736\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 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
\(61\) 6.00000 10.3923i 0.768221 1.33060i −0.170305 0.985391i \(-0.554475\pi\)
0.938527 0.345207i \(-0.112191\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.50000 2.59808i 0.186052 0.322252i
\(66\) 0 0
\(67\) 2.00000 + 3.46410i 0.244339 + 0.423207i 0.961946 0.273241i \(-0.0880957\pi\)
−0.717607 + 0.696449i \(0.754762\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −6.00000 10.3923i −0.702247 1.21633i −0.967676 0.252197i \(-0.918847\pi\)
0.265429 0.964130i \(-0.414486\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.500000 2.59808i 0.0569803 0.296078i
\(78\) 0 0
\(79\) −7.00000 + 12.1244i −0.787562 + 1.36410i 0.139895 + 0.990166i \(0.455323\pi\)
−0.927457 + 0.373930i \(0.878010\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 5.19615i 0.317999 0.550791i −0.662071 0.749441i \(-0.730322\pi\)
0.980071 + 0.198650i \(0.0636557\pi\)
\(90\) 0 0
\(91\) −6.00000 5.19615i −0.628971 0.544705i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.50000 + 4.33013i 0.256495 + 0.444262i
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.00000 + 10.3923i 0.597022 + 1.03407i 0.993258 + 0.115924i \(0.0369830\pi\)
−0.396236 + 0.918149i \(0.629684\pi\)
\(102\) 0 0
\(103\) −10.0000 + 17.3205i −0.985329 + 1.70664i −0.344865 + 0.938652i \(0.612075\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.00000 10.3923i 0.580042 1.00466i −0.415432 0.909624i \(-0.636370\pi\)
0.995474 0.0950377i \(-0.0302972\pi\)
\(108\) 0 0
\(109\) 2.00000 + 3.46410i 0.191565 + 0.331801i 0.945769 0.324840i \(-0.105310\pi\)
−0.754204 + 0.656640i \(0.771977\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −20.0000 −1.88144 −0.940721 0.339182i \(-0.889850\pi\)
−0.940721 + 0.339182i \(0.889850\pi\)
\(114\) 0 0
\(115\) 3.50000 + 6.06218i 0.326377 + 0.565301i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.00000 5.19615i 0.0916698 0.476331i
\(120\) 0 0
\(121\) 5.00000 8.66025i 0.454545 0.787296i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 17.0000 1.50851 0.754253 0.656584i \(-0.227999\pi\)
0.754253 + 0.656584i \(0.227999\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.50000 6.06218i 0.305796 0.529655i −0.671642 0.740876i \(-0.734411\pi\)
0.977438 + 0.211221i \(0.0677440\pi\)
\(132\) 0 0
\(133\) 12.5000 4.33013i 1.08389 0.375470i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 10.3923i −0.512615 0.887875i −0.999893 0.0146279i \(-0.995344\pi\)
0.487278 0.873247i \(-0.337990\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.50000 + 2.59808i 0.125436 + 0.217262i
\(144\) 0 0
\(145\) −3.00000 + 5.19615i −0.249136 + 0.431517i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.00000 8.66025i 0.409616 0.709476i −0.585231 0.810867i \(-0.698996\pi\)
0.994847 + 0.101391i \(0.0323294\pi\)
\(150\) 0 0
\(151\) 5.00000 + 8.66025i 0.406894 + 0.704761i 0.994540 0.104357i \(-0.0332784\pi\)
−0.587646 + 0.809118i \(0.699945\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) −2.50000 4.33013i −0.199522 0.345582i 0.748852 0.662738i \(-0.230606\pi\)
−0.948373 + 0.317156i \(0.897272\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 17.5000 6.06218i 1.37919 0.477767i
\(162\) 0 0
\(163\) −2.00000 + 3.46410i −0.156652 + 0.271329i −0.933659 0.358162i \(-0.883403\pi\)
0.777007 + 0.629492i \(0.216737\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −5.00000 −0.386912 −0.193456 0.981109i \(-0.561970\pi\)
−0.193456 + 0.981109i \(0.561970\pi\)
\(168\) 0 0
\(169\) −4.00000 −0.307692
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.50000 16.4545i 0.722272 1.25101i −0.237816 0.971310i \(-0.576431\pi\)
0.960087 0.279701i \(-0.0902353\pi\)
\(174\) 0 0
\(175\) 0.500000 2.59808i 0.0377964 0.196396i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.50000 + 7.79423i 0.336346 + 0.582568i 0.983742 0.179585i \(-0.0574756\pi\)
−0.647397 + 0.762153i \(0.724142\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.50000 + 4.33013i 0.183804 + 0.318357i
\(186\) 0 0
\(187\) −1.00000 + 1.73205i −0.0731272 + 0.126660i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.00000 10.3923i 0.434145 0.751961i −0.563081 0.826402i \(-0.690384\pi\)
0.997225 + 0.0744412i \(0.0237173\pi\)
\(192\) 0 0
\(193\) 10.0000 + 17.3205i 0.719816 + 1.24676i 0.961073 + 0.276296i \(0.0891071\pi\)
−0.241257 + 0.970461i \(0.577560\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 27.0000 1.92367 0.961835 0.273629i \(-0.0882242\pi\)
0.961835 + 0.273629i \(0.0882242\pi\)
\(198\) 0 0
\(199\) 2.00000 + 3.46410i 0.141776 + 0.245564i 0.928166 0.372168i \(-0.121385\pi\)
−0.786389 + 0.617731i \(0.788052\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.0000 + 10.3923i 0.842235 + 0.729397i
\(204\) 0 0
\(205\) −2.50000 + 4.33013i −0.174608 + 0.302429i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −5.00000 −0.345857
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.00000 + 5.19615i −0.204598 + 0.354375i
\(216\) 0 0
\(217\) 2.00000 10.3923i 0.135769 0.705476i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.00000 + 5.19615i 0.201802 + 0.349531i
\(222\) 0 0
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.00000 6.92820i −0.265489 0.459841i 0.702202 0.711977i \(-0.252200\pi\)
−0.967692 + 0.252136i \(0.918867\pi\)
\(228\) 0 0
\(229\) 14.0000 24.2487i 0.925146 1.60240i 0.133820 0.991006i \(-0.457276\pi\)
0.791326 0.611394i \(-0.209391\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 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
\(235\) −4.50000 7.79423i −0.293548 0.508439i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −11.5000 19.9186i −0.740780 1.28307i −0.952141 0.305661i \(-0.901123\pi\)
0.211360 0.977408i \(-0.432211\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.50000 2.59808i −0.415270 0.165985i
\(246\) 0 0
\(247\) −7.50000 + 12.9904i −0.477214 + 0.826558i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −29.0000 −1.83046 −0.915232 0.402928i \(-0.867993\pi\)
−0.915232 + 0.402928i \(0.867993\pi\)
\(252\) 0 0
\(253\) −7.00000 −0.440086
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.00000 10.3923i 0.374270 0.648254i −0.615948 0.787787i \(-0.711227\pi\)
0.990217 + 0.139533i \(0.0445601\pi\)
\(258\) 0 0
\(259\) 12.5000 4.33013i 0.776712 0.269061i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.00000 + 6.92820i 0.246651 + 0.427211i 0.962594 0.270947i \(-0.0873367\pi\)
−0.715944 + 0.698158i \(0.754003\pi\)
\(264\) 0 0
\(265\) −11.0000 −0.675725
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.00000 10.3923i −0.365826 0.633630i 0.623082 0.782157i \(-0.285880\pi\)
−0.988908 + 0.148527i \(0.952547\pi\)
\(270\) 0 0
\(271\) 4.00000 6.92820i 0.242983 0.420858i −0.718580 0.695444i \(-0.755208\pi\)
0.961563 + 0.274586i \(0.0885408\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.500000 + 0.866025i −0.0301511 + 0.0522233i
\(276\) 0 0
\(277\) −1.00000 1.73205i −0.0600842 0.104069i 0.834419 0.551131i \(-0.185804\pi\)
−0.894503 + 0.447062i \(0.852470\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) 11.0000 + 19.0526i 0.653882 + 1.13256i 0.982173 + 0.187980i \(0.0601941\pi\)
−0.328291 + 0.944577i \(0.606473\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 10.0000 + 8.66025i 0.590281 + 0.511199i
\(288\) 0 0
\(289\) 6.50000 11.2583i 0.382353 0.662255i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −21.0000 −1.22683 −0.613417 0.789760i \(-0.710205\pi\)
−0.613417 + 0.789760i \(0.710205\pi\)
\(294\) 0 0
\(295\) −8.00000 −0.465778
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −10.5000 + 18.1865i −0.607231 + 1.05175i
\(300\) 0 0
\(301\) 12.0000 + 10.3923i 0.691669 + 0.599002i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 6.00000 + 10.3923i 0.343559 + 0.595062i
\(306\) 0 0
\(307\) −6.00000 −0.342438 −0.171219 0.985233i \(-0.554771\pi\)
−0.171219 + 0.985233i \(0.554771\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2.00000 3.46410i −0.113410 0.196431i 0.803733 0.594990i \(-0.202844\pi\)
−0.917143 + 0.398559i \(0.869511\pi\)
\(312\) 0 0
\(313\) −8.00000 + 13.8564i −0.452187 + 0.783210i −0.998522 0.0543564i \(-0.982689\pi\)
0.546335 + 0.837567i \(0.316023\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.00000 1.73205i 0.0561656 0.0972817i −0.836576 0.547852i \(-0.815446\pi\)
0.892741 + 0.450570i \(0.148779\pi\)
\(318\) 0 0
\(319\) −3.00000 5.19615i −0.167968 0.290929i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −10.0000 −0.556415
\(324\) 0 0
\(325\) 1.50000 + 2.59808i 0.0832050 + 0.144115i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −22.5000 + 7.79423i −1.24047 + 0.429710i
\(330\) 0 0
\(331\) 13.5000 23.3827i 0.742027 1.28523i −0.209544 0.977799i \(-0.567198\pi\)
0.951571 0.307429i \(-0.0994688\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −26.0000 −1.41631 −0.708155 0.706057i \(-0.750472\pi\)
−0.708155 + 0.706057i \(0.750472\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.00000 + 3.46410i −0.108306 + 0.187592i
\(342\) 0 0
\(343\) −10.0000 + 15.5885i −0.539949 + 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.00000 + 6.92820i 0.214731 + 0.371925i 0.953189 0.302374i \(-0.0977791\pi\)
−0.738458 + 0.674299i \(0.764446\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.0000 + 17.3205i 0.532246 + 0.921878i 0.999291 + 0.0376440i \(0.0119853\pi\)
−0.467045 + 0.884234i \(0.654681\pi\)
\(354\) 0 0
\(355\) −2.00000 + 3.46410i −0.106149 + 0.183855i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 15.0000 25.9808i 0.791670 1.37121i −0.133263 0.991081i \(-0.542545\pi\)
0.924932 0.380131i \(-0.124121\pi\)
\(360\) 0 0
\(361\) −3.00000 5.19615i −0.157895 0.273482i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.0000 0.628109
\(366\) 0 0
\(367\) −9.50000 16.4545i −0.495896 0.858917i 0.504093 0.863649i \(-0.331827\pi\)
−0.999989 + 0.00473247i \(0.998494\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.50000 + 28.5788i −0.285546 + 1.48374i
\(372\) 0 0
\(373\) −7.00000 + 12.1244i −0.362446 + 0.627775i −0.988363 0.152115i \(-0.951392\pi\)
0.625917 + 0.779890i \(0.284725\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −18.0000 −0.927047
\(378\) 0 0
\(379\) −21.0000 −1.07870 −0.539349 0.842082i \(-0.681330\pi\)
−0.539349 + 0.842082i \(0.681330\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10.5000 + 18.1865i −0.536525 + 0.929288i 0.462563 + 0.886586i \(0.346930\pi\)
−0.999088 + 0.0427020i \(0.986403\pi\)
\(384\) 0 0
\(385\) 2.00000 + 1.73205i 0.101929 + 0.0882735i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −8.00000 13.8564i −0.405616 0.702548i 0.588777 0.808296i \(-0.299610\pi\)
−0.994393 + 0.105748i \(0.966276\pi\)
\(390\) 0 0
\(391\) −14.0000 −0.708010
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −7.00000 12.1244i −0.352208 0.610043i
\(396\) 0 0
\(397\) −9.00000 + 15.5885i −0.451697 + 0.782362i −0.998492 0.0549046i \(-0.982515\pi\)
0.546795 + 0.837267i \(0.315848\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.50000 11.2583i 0.324595 0.562214i −0.656836 0.754034i \(-0.728105\pi\)
0.981430 + 0.191820i \(0.0614388\pi\)
\(402\) 0 0
\(403\) 6.00000 + 10.3923i 0.298881 + 0.517678i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.00000 −0.247841
\(408\) 0 0
\(409\) 3.00000 + 5.19615i 0.148340 + 0.256933i 0.930614 0.366002i \(-0.119274\pi\)
−0.782274 + 0.622935i \(0.785940\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4.00000 + 20.7846i −0.196827 + 1.02274i
\(414\) 0 0
\(415\) −2.00000 + 3.46410i −0.0981761 + 0.170046i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.00000 0.244266 0.122133 0.992514i \(-0.461027\pi\)
0.122133 + 0.992514i \(0.461027\pi\)
\(420\) 0 0
\(421\) 30.0000 1.46211 0.731055 0.682318i \(-0.239028\pi\)
0.731055 + 0.682318i \(0.239028\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.00000 + 1.73205i −0.0485071 + 0.0840168i
\(426\) 0 0
\(427\) 30.0000 10.3923i 1.45180 0.502919i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −8.00000 13.8564i −0.385346 0.667440i 0.606471 0.795106i \(-0.292585\pi\)
−0.991817 + 0.127666i \(0.959251\pi\)
\(432\) 0 0
\(433\) 24.0000 1.15337 0.576683 0.816968i \(-0.304347\pi\)
0.576683 + 0.816968i \(0.304347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −17.5000 30.3109i −0.837139 1.44997i
\(438\) 0 0
\(439\) 5.00000 8.66025i 0.238637 0.413331i −0.721686 0.692220i \(-0.756633\pi\)
0.960323 + 0.278889i \(0.0899661\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.0000 + 27.7128i −0.760183 + 1.31668i 0.182573 + 0.983192i \(0.441557\pi\)
−0.942756 + 0.333483i \(0.891776\pi\)
\(444\) 0 0
\(445\) 3.00000 + 5.19615i 0.142214 + 0.246321i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −13.0000 −0.613508 −0.306754 0.951789i \(-0.599243\pi\)
−0.306754 + 0.951789i \(0.599243\pi\)
\(450\) 0 0
\(451\) −2.50000 4.33013i −0.117720 0.203898i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 7.50000 2.59808i 0.351605 0.121800i
\(456\) 0 0
\(457\) 11.0000 19.0526i 0.514558 0.891241i −0.485299 0.874348i \(-0.661289\pi\)
0.999857 0.0168929i \(-0.00537742\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 9.00000 0.418265 0.209133 0.977887i \(-0.432936\pi\)
0.209133 + 0.977887i \(0.432936\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.00000 + 8.66025i −0.231372 + 0.400749i −0.958212 0.286058i \(-0.907655\pi\)
0.726840 + 0.686807i \(0.240988\pi\)
\(468\) 0 0
\(469\) −2.00000 + 10.3923i −0.0923514 + 0.479872i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.00000 5.19615i −0.137940 0.238919i
\(474\) 0 0
\(475\) −5.00000 −0.229416
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.00000 13.8564i −0.365529 0.633115i 0.623332 0.781958i \(-0.285779\pi\)
−0.988861 + 0.148842i \(0.952445\pi\)
\(480\) 0 0
\(481\) −7.50000 + 12.9904i −0.341971 + 0.592310i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.00000 + 5.19615i −0.136223 + 0.235945i
\(486\) 0 0
\(487\) 4.00000 + 6.92820i 0.181257 + 0.313947i 0.942309 0.334744i \(-0.108650\pi\)
−0.761052 + 0.648691i \(0.775317\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) 0 0
\(493\) −6.00000 10.3923i −0.270226 0.468046i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 + 6.92820i 0.358849 + 0.310772i
\(498\) 0 0
\(499\) −14.0000 + 24.2487i −0.626726 + 1.08552i 0.361478 + 0.932381i \(0.382272\pi\)
−0.988204 + 0.153141i \(0.951061\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −9.00000 + 15.5885i −0.398918 + 0.690946i −0.993593 0.113020i \(-0.963948\pi\)
0.594675 + 0.803966i \(0.297281\pi\)
\(510\) 0 0
\(511\) 6.00000 31.1769i 0.265424 1.37919i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −10.0000 17.3205i −0.440653 0.763233i
\(516\) 0 0
\(517\) 9.00000 0.395820
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −7.50000 12.9904i −0.328581 0.569119i 0.653650 0.756797i \(-0.273237\pi\)
−0.982231 + 0.187678i \(0.939904\pi\)
\(522\) 0 0
\(523\) −14.0000 + 24.2487i −0.612177 + 1.06032i 0.378695 + 0.925521i \(0.376373\pi\)
−0.990873 + 0.134801i \(0.956961\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4.00000 + 6.92820i −0.174243 + 0.301797i
\(528\) 0 0
\(529\) −13.0000 22.5167i −0.565217 0.978985i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −15.0000 −0.649722
\(534\) 0 0
\(535\) 6.00000 + 10.3923i 0.259403 + 0.449299i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.50000 4.33013i 0.236902 0.186512i
\(540\) 0 0
\(541\) 20.0000 34.6410i 0.859867 1.48933i −0.0121878 0.999926i \(-0.503880\pi\)
0.872055 0.489408i \(-0.162787\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.00000 −0.171341
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 15.0000 25.9808i 0.639021 1.10682i
\(552\) 0 0
\(553\) −35.0000 + 12.1244i −1.48835 + 0.515580i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10.5000 + 18.1865i 0.444899 + 0.770588i 0.998045 0.0624962i \(-0.0199061\pi\)
−0.553146 + 0.833084i \(0.686573\pi\)
\(558\) 0 0
\(559\) −18.0000 −0.761319
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.00000 + 5.19615i 0.126435 + 0.218992i 0.922293 0.386492i \(-0.126313\pi\)
−0.795858 + 0.605483i \(0.792980\pi\)
\(564\) 0 0
\(565\) 10.0000 17.3205i 0.420703 0.728679i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5.50000 + 9.52628i −0.230572 + 0.399362i −0.957977 0.286846i \(-0.907393\pi\)
0.727405 + 0.686209i \(0.240726\pi\)
\(570\) 0 0
\(571\) −14.0000 24.2487i −0.585882 1.01478i −0.994765 0.102190i \(-0.967415\pi\)
0.408883 0.912587i \(-0.365918\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −7.00000 −0.291920
\(576\) 0 0
\(577\) 2.00000 + 3.46410i 0.0832611 + 0.144212i 0.904649 0.426158i \(-0.140133\pi\)
−0.821388 + 0.570370i \(0.806800\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.00000 + 6.92820i 0.331896 + 0.287430i
\(582\) 0 0
\(583\) 5.50000 9.52628i 0.227787 0.394538i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) −20.0000 −0.824086
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.00000 + 10.3923i −0.246390 + 0.426761i −0.962522 0.271205i \(-0.912578\pi\)
0.716131 + 0.697966i \(0.245911\pi\)
\(594\) 0 0
\(595\) 4.00000 + 3.46410i 0.163984 + 0.142014i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 17.0000 + 29.4449i 0.694601 + 1.20308i 0.970315 + 0.241845i \(0.0777525\pi\)
−0.275714 + 0.961240i \(0.588914\pi\)
\(600\) 0 0
\(601\) −14.0000 −0.571072 −0.285536 0.958368i \(-0.592172\pi\)
−0.285536 + 0.958368i \(0.592172\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.00000 + 8.66025i 0.203279 + 0.352089i
\(606\) 0 0
\(607\) −16.5000 + 28.5788i −0.669714 + 1.15998i 0.308270 + 0.951299i \(0.400250\pi\)
−0.977984 + 0.208680i \(0.933083\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 13.5000 23.3827i 0.546152 0.945962i
\(612\) 0 0
\(613\) 20.5000 + 35.5070i 0.827987 + 1.43412i 0.899615 + 0.436684i \(0.143847\pi\)
−0.0716275 + 0.997431i \(0.522819\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −42.0000 −1.69086 −0.845428 0.534089i \(-0.820655\pi\)
−0.845428 + 0.534089i \(0.820655\pi\)
\(618\) 0 0
\(619\) −17.5000 30.3109i −0.703384 1.21830i −0.967271 0.253744i \(-0.918338\pi\)
0.263887 0.964554i \(-0.414995\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 15.0000 5.19615i 0.600962 0.208179i
\(624\) 0 0
\(625\) −0.500000 + 0.866025i −0.0200000 + 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −10.0000 −0.398726
\(630\) 0 0
\(631\) 40.0000 1.59237 0.796187 0.605050i \(-0.206847\pi\)
0.796187 + 0.605050i \(0.206847\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.50000 + 14.7224i −0.337312 + 0.584242i
\(636\) 0 0
\(637\) −3.00000 20.7846i −0.118864 0.823516i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.5000 + 30.3109i 0.691208 + 1.19721i 0.971442 + 0.237276i \(0.0762547\pi\)
−0.280234 + 0.959932i \(0.590412\pi\)
\(642\) 0 0
\(643\) −34.0000 −1.34083 −0.670415 0.741987i \(-0.733884\pi\)
−0.670415 + 0.741987i \(0.733884\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 11.5000 + 19.9186i 0.452112 + 0.783080i 0.998517 0.0544405i \(-0.0173375\pi\)
−0.546405 + 0.837521i \(0.684004\pi\)
\(648\) 0 0
\(649\) 4.00000 6.92820i 0.157014 0.271956i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22.5000 38.9711i 0.880493 1.52506i 0.0296993 0.999559i \(-0.490545\pi\)
0.850794 0.525500i \(-0.176122\pi\)
\(654\) 0 0
\(655\) 3.50000 + 6.06218i 0.136756 + 0.236869i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 16.0000 0.623272 0.311636 0.950202i \(-0.399123\pi\)
0.311636 + 0.950202i \(0.399123\pi\)
\(660\) 0 0
\(661\) 18.0000 + 31.1769i 0.700119 + 1.21264i 0.968424 + 0.249308i \(0.0802030\pi\)
−0.268306 + 0.963334i \(0.586464\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.50000 + 12.9904i −0.0969458 + 0.503745i
\(666\) 0 0
\(667\) 21.0000 36.3731i 0.813123 1.40837i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −12.0000 −0.463255
\(672\) 0 0
\(673\) −42.0000 −1.61898 −0.809491 0.587133i \(-0.800257\pi\)
−0.809491 + 0.587133i \(0.800257\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 10.5000 18.1865i 0.403548 0.698965i −0.590603 0.806962i \(-0.701110\pi\)
0.994151 + 0.107997i \(0.0344436\pi\)
\(678\) 0 0
\(679\) 12.0000 + 10.3923i 0.460518 + 0.398820i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −18.0000 31.1769i −0.688751 1.19295i −0.972242 0.233977i \(-0.924826\pi\)
0.283491 0.958975i \(-0.408507\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −16.5000 28.5788i −0.628600 1.08877i
\(690\) 0 0
\(691\) −4.00000 + 6.92820i −0.152167 + 0.263561i −0.932024 0.362397i \(-0.881959\pi\)
0.779857 + 0.625958i \(0.215292\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.00000 + 3.46410i −0.0758643 + 0.131401i
\(696\) 0 0
\(697\) −5.00000 8.66025i −0.189389 0.328031i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) −12.5000 21.6506i −0.471446 0.816569i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.00000 + 31.1769i −0.225653 + 1.17253i
\(708\) 0 0
\(709\) 9.00000 15.5885i 0.338002 0.585437i −0.646055 0.763291i \(-0.723582\pi\)
0.984057 + 0.177854i \(0.0569156\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −28.0000 −1.04861
\(714\) 0 0
\(715\) −3.00000 −0.112194
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −18.0000 + 31.1769i −0.671287 + 1.16270i 0.306253 + 0.951950i \(0.400925\pi\)
−0.977539 + 0.210752i \(0.932409\pi\)
\(720\) 0 0
\(721\) −50.0000 + 17.3205i −1.86210 + 0.645049i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.00000 5.19615i −0.111417 0.192980i
\(726\) 0 0
\(727\) 41.0000 1.52061 0.760303 0.649569i \(-0.225051\pi\)
0.760303 + 0.649569i \(0.225051\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −6.00000 10.3923i −0.221918 0.384373i
\(732\) 0 0
\(733\) −15.5000 + 26.8468i −0.572506 + 0.991609i 0.423802 + 0.905755i \(0.360695\pi\)
−0.996308 + 0.0858539i \(0.972638\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 2.00000 3.46410i 0.0736709 0.127602i
\(738\) 0 0
\(739\) 20.5000 + 35.5070i 0.754105 + 1.30615i 0.945818 + 0.324697i \(0.105262\pi\)
−0.191714 + 0.981451i \(0.561404\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.00000 0.110059 0.0550297 0.998485i \(-0.482475\pi\)
0.0550297 + 0.998485i \(0.482475\pi\)
\(744\) 0 0
\(745\) 5.00000 + 8.66025i 0.183186 + 0.317287i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 30.0000 10.3923i 1.09618 0.379727i
\(750\) 0 0
\(751\) −9.00000 + 15.5885i −0.328415 + 0.568831i −0.982197 0.187851i \(-0.939848\pi\)
0.653783 + 0.756682i \(0.273181\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.50000 + 2.59808i −0.0543750 + 0.0941802i −0.891932 0.452170i \(-0.850650\pi\)
0.837557 + 0.546350i \(0.183983\pi\)
\(762\) 0 0
\(763\) −2.00000 + 10.3923i −0.0724049 + 0.376227i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.0000 20.7846i −0.433295 0.750489i
\(768\) 0 0
\(769\) −41.0000 −1.47850 −0.739249 0.673432i \(-0.764819\pi\)
−0.739249 + 0.673432i \(0.764819\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 7.50000 + 12.9904i 0.269756 + 0.467232i 0.968799 0.247849i \(-0.0797235\pi\)
−0.699043 + 0.715080i \(0.746390\pi\)
\(774\) 0 0
\(775\) −2.00000 + 3.46410i −0.0718421 + 0.124434i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.5000 21.6506i 0.447859 0.775715i
\(780\) 0 0
\(781\) −2.00000 3.46410i −0.0715656 0.123955i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5.00000 0.178458
\(786\) 0 0
\(787\) 9.00000 + 15.5885i 0.320815 + 0.555668i 0.980656 0.195737i \(-0.0627098\pi\)
−0.659841 + 0.751405i \(0.729376\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −40.0000 34.6410i −1.42224 1.23169i
\(792\) 0 0
\(793\) −18.0000 + 31.1769i −0.639199 + 1.10712i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −2.00000 −0.0708436 −0.0354218 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.00000 + 10.3923i −0.211735 + 0.366736i
\(804\) 0 0
\(805\) −3.50000 + 18.1865i −0.123359 + 0.640991i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 27.5000 + 47.6314i 0.966849 + 1.67463i 0.704564 + 0.709640i \(0.251142\pi\)
0.262284 + 0.964991i \(0.415524\pi\)
\(810\) 0 0
\(811\) −5.00000 −0.175574 −0.0877869 0.996139i \(-0.527979\pi\)
−0.0877869 + 0.996139i \(0.527979\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.00000 3.46410i −0.0700569 0.121342i
\(816\) 0 0
\(817\) 15.0000 25.9808i 0.524784 0.908952i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 15.0000 25.9808i 0.523504 0.906735i −0.476122 0.879379i \(-0.657958\pi\)
0.999626 0.0273557i \(-0.00870868\pi\)
\(822\) 0 0
\(823\) −2.00000 3.46410i −0.0697156 0.120751i 0.829060 0.559159i \(-0.188876\pi\)
−0.898776 + 0.438408i \(0.855543\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 14.0000 0.486828 0.243414 0.969923i \(-0.421733\pi\)
0.243414 + 0.969923i \(0.421733\pi\)
\(828\) 0 0
\(829\) −19.0000 32.9090i −0.659897 1.14298i −0.980642 0.195810i \(-0.937266\pi\)
0.320745 0.947166i \(-0.396067\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 11.0000 8.66025i 0.381127 0.300060i
\(834\) 0 0
\(835\) 2.50000 4.33013i 0.0865161 0.149850i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 14.0000 0.483334 0.241667 0.970359i \(-0.422306\pi\)
0.241667 + 0.970359i \(0.422306\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.00000 3.46410i 0.0688021 0.119169i
\(846\) 0 0
\(847\) 25.0000 8.66025i 0.859010 0.297570i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −17.5000 30.3109i −0.599892 1.03904i
\(852\) 0 0
\(853\) 23.0000 0.787505 0.393753 0.919216i \(-0.371177\pi\)
0.393753 + 0.919216i \(0.371177\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −21.0000 36.3731i −0.717346 1.24248i −0.962048 0.272882i \(-0.912023\pi\)
0.244701 0.969599i \(-0.421310\pi\)
\(858\) 0 0
\(859\) −20.0000 + 34.6410i −0.682391 + 1.18194i 0.291858 + 0.956462i \(0.405727\pi\)
−0.974249 + 0.225475i \(0.927607\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.50000 + 11.2583i −0.221263 + 0.383238i −0.955192 0.295988i \(-0.904351\pi\)
0.733929 + 0.679226i \(0.237684\pi\)
\(864\) 0 0
\(865\) 9.50000 + 16.4545i 0.323010 + 0.559469i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 14.0000 0.474917
\(870\) 0 0
\(871\) −6.00000 10.3923i −0.203302 0.352130i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.00000 + 1.73205i 0.0676123 + 0.0585540i
\(876\) 0 0
\(877\) −4.50000 + 7.79423i −0.151954 + 0.263192i −0.931946 0.362598i \(-0.881890\pi\)
0.779992 + 0.625790i \(0.215223\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 7.00000 0.235836 0.117918 0.993023i \(-0.462378\pi\)
0.117918 + 0.993023i \(0.462378\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(888\) 0 0
\(889\) 34.0000 + 29.4449i 1.14032 + 0.987549i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 22.5000 + 38.9711i 0.752934 + 1.30412i
\(894\) 0 0
\(895\) −9.00000 −0.300837
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −12.0000 20.7846i −0.400222 0.693206i
\(900\) 0 0
\(901\) 11.0000 19.0526i 0.366463 0.634733i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.00000 + 1.73205i −0.0332411 + 0.0575753i
\(906\) 0 0
\(907\) −11.0000 19.0526i −0.365249 0.632630i 0.623567 0.781770i \(-0.285683\pi\)
−0.988816 + 0.149140i \(0.952349\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −42.0000 −1.39152 −0.695761 0.718273i \(-0.744933\pi\)
−0.695761 + 0.718273i \(0.744933\pi\)
\(912\) 0 0
\(913\) −2.00000 3.46410i −0.0661903 0.114645i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 17.5000 6.06218i 0.577901 0.200191i
\(918\) 0 0
\(919\) 9.00000 15.5885i 0.296883 0.514216i −0.678538 0.734565i \(-0.737386\pi\)
0.975421 + 0.220349i \(0.0707197\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −5.00000 −0.164399
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −15.5000 + 26.8468i −0.508539 + 0.880815i 0.491413 + 0.870927i \(0.336481\pi\)
−0.999951 + 0.00988764i \(0.996853\pi\)
\(930\) 0 0
\(931\) 32.5000 + 12.9904i 1.06514 + 0.425743i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.00000 1.73205i −0.0327035 0.0566441i
\(936\) 0 0
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 12.0000 + 20.7846i 0.391189 + 0.677559i 0.992607 0.121376i \(-0.0387306\pi\)
−0.601418 + 0.798935i \(0.705397\pi\)
\(942\) 0 0
\(943\) 17.5000 30.3109i 0.569878 0.987058i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7.00000 12.1244i 0.227469 0.393989i −0.729588 0.683887i \(-0.760288\pi\)
0.957057 + 0.289898i \(0.0936215\pi\)
\(948\) 0 0
\(949\) 18.0000 + 31.1769i 0.584305 + 1.01205i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 20.0000 0.647864 0.323932 0.946080i \(-0.394995\pi\)
0.323932 + 0.946080i \(0.394995\pi\)
\(954\) 0 0
\(955\) 6.00000 + 10.3923i 0.194155 + 0.336287i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 6.00000 31.1769i 0.193750 1.00676i
\(960\) 0 0
\(961\) 7.50000 12.9904i 0.241935 0.419045i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −20.0000 −0.643823
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3.50000 6.06218i 0.112320 0.194545i −0.804385 0.594108i \(-0.797505\pi\)
0.916705 + 0.399564i \(0.130838\pi\)
\(972\) 0 0
\(973\) 8.00000 + 6.92820i 0.256468 + 0.222108i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.00000 + 15.5885i 0.287936 + 0.498719i 0.973317 0.229465i \(-0.0736978\pi\)
−0.685381 + 0.728184i \(0.740364\pi\)
\(978\) 0 0
\(979\) −6.00000 −0.191761
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.50000 2.59808i −0.0478426 0.0828658i 0.841112 0.540860i \(-0.181901\pi\)
−0.888955 + 0.457995i \(0.848568\pi\)
\(984\) 0 0
\(985\) −13.5000 + 23.3827i −0.430146 + 0.745034i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 21.0000 36.3731i 0.667761 1.15660i
\(990\) 0 0
\(991\) −22.0000 38.1051i −0.698853 1.21045i −0.968864 0.247592i \(-0.920361\pi\)
0.270011 0.962857i \(-0.412973\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.00000 −0.126809
\(996\) 0 0
\(997\) −23.0000 39.8372i −0.728417 1.26166i −0.957552 0.288261i \(-0.906923\pi\)
0.229135 0.973395i \(-0.426410\pi\)
\(998\) 0 0
\(999\) 0 0
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 2520.2.bi.e.1801.1 2
3.2 odd 2 280.2.q.c.121.1 yes 2
7.4 even 3 inner 2520.2.bi.e.361.1 2
12.11 even 2 560.2.q.c.401.1 2
15.2 even 4 1400.2.bh.e.849.2 4
15.8 even 4 1400.2.bh.e.849.1 4
15.14 odd 2 1400.2.q.a.401.1 2
21.2 odd 6 1960.2.a.a.1.1 1
21.5 even 6 1960.2.a.m.1.1 1
21.11 odd 6 280.2.q.c.81.1 2
21.17 even 6 1960.2.q.c.361.1 2
21.20 even 2 1960.2.q.c.961.1 2
84.11 even 6 560.2.q.c.81.1 2
84.23 even 6 3920.2.a.bf.1.1 1
84.47 odd 6 3920.2.a.i.1.1 1
105.32 even 12 1400.2.bh.e.249.1 4
105.44 odd 6 9800.2.a.bi.1.1 1
105.53 even 12 1400.2.bh.e.249.2 4
105.74 odd 6 1400.2.q.a.1201.1 2
105.89 even 6 9800.2.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.q.c.81.1 2 21.11 odd 6
280.2.q.c.121.1 yes 2 3.2 odd 2
560.2.q.c.81.1 2 84.11 even 6
560.2.q.c.401.1 2 12.11 even 2
1400.2.q.a.401.1 2 15.14 odd 2
1400.2.q.a.1201.1 2 105.74 odd 6
1400.2.bh.e.249.1 4 105.32 even 12
1400.2.bh.e.249.2 4 105.53 even 12
1400.2.bh.e.849.1 4 15.8 even 4
1400.2.bh.e.849.2 4 15.2 even 4
1960.2.a.a.1.1 1 21.2 odd 6
1960.2.a.m.1.1 1 21.5 even 6
1960.2.q.c.361.1 2 21.17 even 6
1960.2.q.c.961.1 2 21.20 even 2
2520.2.bi.e.361.1 2 7.4 even 3 inner
2520.2.bi.e.1801.1 2 1.1 even 1 trivial
3920.2.a.i.1.1 1 84.47 odd 6
3920.2.a.bf.1.1 1 84.23 even 6
9800.2.a.g.1.1 1 105.89 even 6
9800.2.a.bi.1.1 1 105.44 odd 6