Properties

Label 2925.2.a.bj.1.1
Level $2925$
Weight $2$
Character 2925.1
Self dual yes
Analytic conductor $23.356$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2925,2,Mod(1,2925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2925, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2925.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2925.a (trivial)

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: yes
Analytic conductor: \(23.3562425912\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 2925.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.21432 q^{2} -0.525428 q^{4} -2.90321 q^{7} +3.06668 q^{8} +O(q^{10})\) \(q-1.21432 q^{2} -0.525428 q^{4} -2.90321 q^{7} +3.06668 q^{8} -0.214320 q^{11} +1.00000 q^{13} +3.52543 q^{14} -2.67307 q^{16} +6.42864 q^{17} +2.21432 q^{19} +0.260253 q^{22} +4.68889 q^{23} -1.21432 q^{26} +1.52543 q^{28} -8.70964 q^{29} -5.59210 q^{31} -2.88739 q^{32} -7.80642 q^{34} -2.28100 q^{37} -2.68889 q^{38} -3.05086 q^{41} -6.36196 q^{43} +0.112610 q^{44} -5.69381 q^{46} +1.09679 q^{47} +1.42864 q^{49} -0.525428 q^{52} -6.23506 q^{53} -8.90321 q^{56} +10.5763 q^{58} +9.26517 q^{59} -0.280996 q^{61} +6.79060 q^{62} +8.85236 q^{64} +7.76049 q^{67} -3.37778 q^{68} +6.08097 q^{71} +10.2810 q^{73} +2.76986 q^{74} -1.16346 q^{76} +0.622216 q^{77} -14.2351 q^{79} +3.70471 q^{82} +9.52543 q^{83} +7.72546 q^{86} -0.657249 q^{88} +5.61285 q^{89} -2.90321 q^{91} -2.46367 q^{92} -1.33185 q^{94} +18.0415 q^{97} -1.73483 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 5 q^{4} - 2 q^{7} + 9 q^{8} + 6 q^{11} + 3 q^{13} + 4 q^{14} + 5 q^{16} + 6 q^{17} + 14 q^{22} + 14 q^{23} + 3 q^{26} - 2 q^{28} - 6 q^{29} - 10 q^{31} + 11 q^{32} - 10 q^{34} - 8 q^{38} + 4 q^{41} - 6 q^{43} + 20 q^{44} + 16 q^{46} + 10 q^{47} - 9 q^{49} + 5 q^{52} + 8 q^{53} - 20 q^{56} + 12 q^{58} + 8 q^{59} + 6 q^{61} - 6 q^{62} + 33 q^{64} - 10 q^{67} - 10 q^{68} + 12 q^{71} + 24 q^{73} + 2 q^{74} - 10 q^{76} + 2 q^{77} - 16 q^{79} + 24 q^{82} + 22 q^{83} + 16 q^{86} + 24 q^{88} - 10 q^{89} - 2 q^{91} + 32 q^{92} + 16 q^{94} + 14 q^{97} - 25 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.21432 −0.858654 −0.429327 0.903149i \(-0.641249\pi\)
−0.429327 + 0.903149i \(0.641249\pi\)
\(3\) 0 0
\(4\) −0.525428 −0.262714
\(5\) 0 0
\(6\) 0 0
\(7\) −2.90321 −1.09731 −0.548655 0.836049i \(-0.684860\pi\)
−0.548655 + 0.836049i \(0.684860\pi\)
\(8\) 3.06668 1.08423
\(9\) 0 0
\(10\) 0 0
\(11\) −0.214320 −0.0646198 −0.0323099 0.999478i \(-0.510286\pi\)
−0.0323099 + 0.999478i \(0.510286\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 3.52543 0.942210
\(15\) 0 0
\(16\) −2.67307 −0.668268
\(17\) 6.42864 1.55917 0.779587 0.626294i \(-0.215429\pi\)
0.779587 + 0.626294i \(0.215429\pi\)
\(18\) 0 0
\(19\) 2.21432 0.508000 0.254000 0.967204i \(-0.418254\pi\)
0.254000 + 0.967204i \(0.418254\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.260253 0.0554861
\(23\) 4.68889 0.977702 0.488851 0.872367i \(-0.337416\pi\)
0.488851 + 0.872367i \(0.337416\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.21432 −0.238148
\(27\) 0 0
\(28\) 1.52543 0.288279
\(29\) −8.70964 −1.61734 −0.808669 0.588263i \(-0.799812\pi\)
−0.808669 + 0.588263i \(0.799812\pi\)
\(30\) 0 0
\(31\) −5.59210 −1.00437 −0.502186 0.864760i \(-0.667471\pi\)
−0.502186 + 0.864760i \(0.667471\pi\)
\(32\) −2.88739 −0.510423
\(33\) 0 0
\(34\) −7.80642 −1.33879
\(35\) 0 0
\(36\) 0 0
\(37\) −2.28100 −0.374993 −0.187497 0.982265i \(-0.560037\pi\)
−0.187497 + 0.982265i \(0.560037\pi\)
\(38\) −2.68889 −0.436196
\(39\) 0 0
\(40\) 0 0
\(41\) −3.05086 −0.476464 −0.238232 0.971208i \(-0.576568\pi\)
−0.238232 + 0.971208i \(0.576568\pi\)
\(42\) 0 0
\(43\) −6.36196 −0.970190 −0.485095 0.874461i \(-0.661215\pi\)
−0.485095 + 0.874461i \(0.661215\pi\)
\(44\) 0.112610 0.0169765
\(45\) 0 0
\(46\) −5.69381 −0.839507
\(47\) 1.09679 0.159983 0.0799915 0.996796i \(-0.474511\pi\)
0.0799915 + 0.996796i \(0.474511\pi\)
\(48\) 0 0
\(49\) 1.42864 0.204091
\(50\) 0 0
\(51\) 0 0
\(52\) −0.525428 −0.0728637
\(53\) −6.23506 −0.856452 −0.428226 0.903672i \(-0.640861\pi\)
−0.428226 + 0.903672i \(0.640861\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −8.90321 −1.18974
\(57\) 0 0
\(58\) 10.5763 1.38873
\(59\) 9.26517 1.20622 0.603112 0.797657i \(-0.293927\pi\)
0.603112 + 0.797657i \(0.293927\pi\)
\(60\) 0 0
\(61\) −0.280996 −0.0359779 −0.0179889 0.999838i \(-0.505726\pi\)
−0.0179889 + 0.999838i \(0.505726\pi\)
\(62\) 6.79060 0.862407
\(63\) 0 0
\(64\) 8.85236 1.10654
\(65\) 0 0
\(66\) 0 0
\(67\) 7.76049 0.948095 0.474047 0.880499i \(-0.342793\pi\)
0.474047 + 0.880499i \(0.342793\pi\)
\(68\) −3.37778 −0.409617
\(69\) 0 0
\(70\) 0 0
\(71\) 6.08097 0.721678 0.360839 0.932628i \(-0.382490\pi\)
0.360839 + 0.932628i \(0.382490\pi\)
\(72\) 0 0
\(73\) 10.2810 1.20330 0.601650 0.798760i \(-0.294510\pi\)
0.601650 + 0.798760i \(0.294510\pi\)
\(74\) 2.76986 0.321990
\(75\) 0 0
\(76\) −1.16346 −0.133459
\(77\) 0.622216 0.0709081
\(78\) 0 0
\(79\) −14.2351 −1.60157 −0.800785 0.598952i \(-0.795584\pi\)
−0.800785 + 0.598952i \(0.795584\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 3.70471 0.409117
\(83\) 9.52543 1.04555 0.522776 0.852470i \(-0.324897\pi\)
0.522776 + 0.852470i \(0.324897\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.72546 0.833057
\(87\) 0 0
\(88\) −0.657249 −0.0700630
\(89\) 5.61285 0.594961 0.297480 0.954728i \(-0.403854\pi\)
0.297480 + 0.954728i \(0.403854\pi\)
\(90\) 0 0
\(91\) −2.90321 −0.304339
\(92\) −2.46367 −0.256856
\(93\) 0 0
\(94\) −1.33185 −0.137370
\(95\) 0 0
\(96\) 0 0
\(97\) 18.0415 1.83184 0.915918 0.401366i \(-0.131464\pi\)
0.915918 + 0.401366i \(0.131464\pi\)
\(98\) −1.73483 −0.175244
\(99\) 0 0
\(100\) 0 0
\(101\) −3.93978 −0.392022 −0.196011 0.980602i \(-0.562799\pi\)
−0.196011 + 0.980602i \(0.562799\pi\)
\(102\) 0 0
\(103\) −2.82225 −0.278084 −0.139042 0.990286i \(-0.544402\pi\)
−0.139042 + 0.990286i \(0.544402\pi\)
\(104\) 3.06668 0.300712
\(105\) 0 0
\(106\) 7.57136 0.735396
\(107\) 17.1175 1.65481 0.827407 0.561603i \(-0.189815\pi\)
0.827407 + 0.561603i \(0.189815\pi\)
\(108\) 0 0
\(109\) −16.7239 −1.60186 −0.800931 0.598757i \(-0.795662\pi\)
−0.800931 + 0.598757i \(0.795662\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 7.76049 0.733297
\(113\) 1.18421 0.111401 0.0557005 0.998448i \(-0.482261\pi\)
0.0557005 + 0.998448i \(0.482261\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4.57628 0.424897
\(117\) 0 0
\(118\) −11.2509 −1.03573
\(119\) −18.6637 −1.71090
\(120\) 0 0
\(121\) −10.9541 −0.995824
\(122\) 0.341219 0.0308925
\(123\) 0 0
\(124\) 2.93825 0.263862
\(125\) 0 0
\(126\) 0 0
\(127\) −2.30174 −0.204246 −0.102123 0.994772i \(-0.532564\pi\)
−0.102123 + 0.994772i \(0.532564\pi\)
\(128\) −4.97481 −0.439715
\(129\) 0 0
\(130\) 0 0
\(131\) 13.4193 1.17245 0.586224 0.810149i \(-0.300614\pi\)
0.586224 + 0.810149i \(0.300614\pi\)
\(132\) 0 0
\(133\) −6.42864 −0.557434
\(134\) −9.42372 −0.814085
\(135\) 0 0
\(136\) 19.7146 1.69051
\(137\) 19.1526 1.63631 0.818157 0.574995i \(-0.194996\pi\)
0.818157 + 0.574995i \(0.194996\pi\)
\(138\) 0 0
\(139\) 19.0923 1.61939 0.809696 0.586850i \(-0.199632\pi\)
0.809696 + 0.586850i \(0.199632\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −7.38424 −0.619671
\(143\) −0.214320 −0.0179223
\(144\) 0 0
\(145\) 0 0
\(146\) −12.4844 −1.03322
\(147\) 0 0
\(148\) 1.19850 0.0985160
\(149\) 3.57136 0.292577 0.146289 0.989242i \(-0.453267\pi\)
0.146289 + 0.989242i \(0.453267\pi\)
\(150\) 0 0
\(151\) −1.26517 −0.102958 −0.0514792 0.998674i \(-0.516394\pi\)
−0.0514792 + 0.998674i \(0.516394\pi\)
\(152\) 6.79060 0.550791
\(153\) 0 0
\(154\) −0.755569 −0.0608855
\(155\) 0 0
\(156\) 0 0
\(157\) 5.61285 0.447954 0.223977 0.974594i \(-0.428096\pi\)
0.223977 + 0.974594i \(0.428096\pi\)
\(158\) 17.2859 1.37519
\(159\) 0 0
\(160\) 0 0
\(161\) −13.6128 −1.07284
\(162\) 0 0
\(163\) −3.71900 −0.291295 −0.145647 0.989337i \(-0.546527\pi\)
−0.145647 + 0.989337i \(0.546527\pi\)
\(164\) 1.60300 0.125174
\(165\) 0 0
\(166\) −11.5669 −0.897767
\(167\) −7.03657 −0.544506 −0.272253 0.962226i \(-0.587769\pi\)
−0.272253 + 0.962226i \(0.587769\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 3.34275 0.254882
\(173\) 0.723926 0.0550391 0.0275195 0.999621i \(-0.491239\pi\)
0.0275195 + 0.999621i \(0.491239\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.572892 0.0431833
\(177\) 0 0
\(178\) −6.81579 −0.510865
\(179\) 4.04149 0.302075 0.151037 0.988528i \(-0.451739\pi\)
0.151037 + 0.988528i \(0.451739\pi\)
\(180\) 0 0
\(181\) 2.34122 0.174021 0.0870107 0.996207i \(-0.472269\pi\)
0.0870107 + 0.996207i \(0.472269\pi\)
\(182\) 3.52543 0.261322
\(183\) 0 0
\(184\) 14.3793 1.06006
\(185\) 0 0
\(186\) 0 0
\(187\) −1.37778 −0.100754
\(188\) −0.576283 −0.0420297
\(189\) 0 0
\(190\) 0 0
\(191\) 2.10171 0.152074 0.0760372 0.997105i \(-0.475773\pi\)
0.0760372 + 0.997105i \(0.475773\pi\)
\(192\) 0 0
\(193\) −13.5210 −0.973262 −0.486631 0.873608i \(-0.661774\pi\)
−0.486631 + 0.873608i \(0.661774\pi\)
\(194\) −21.9081 −1.57291
\(195\) 0 0
\(196\) −0.750647 −0.0536176
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) 22.1432 1.56969 0.784845 0.619692i \(-0.212743\pi\)
0.784845 + 0.619692i \(0.212743\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 4.78415 0.336612
\(203\) 25.2859 1.77472
\(204\) 0 0
\(205\) 0 0
\(206\) 3.42711 0.238778
\(207\) 0 0
\(208\) −2.67307 −0.185344
\(209\) −0.474572 −0.0328269
\(210\) 0 0
\(211\) 19.6543 1.35306 0.676530 0.736415i \(-0.263483\pi\)
0.676530 + 0.736415i \(0.263483\pi\)
\(212\) 3.27607 0.225002
\(213\) 0 0
\(214\) −20.7862 −1.42091
\(215\) 0 0
\(216\) 0 0
\(217\) 16.2351 1.10211
\(218\) 20.3082 1.37544
\(219\) 0 0
\(220\) 0 0
\(221\) 6.42864 0.432437
\(222\) 0 0
\(223\) 19.6686 1.31711 0.658554 0.752533i \(-0.271168\pi\)
0.658554 + 0.752533i \(0.271168\pi\)
\(224\) 8.38271 0.560093
\(225\) 0 0
\(226\) −1.43801 −0.0956548
\(227\) 13.2716 0.880869 0.440434 0.897785i \(-0.354824\pi\)
0.440434 + 0.897785i \(0.354824\pi\)
\(228\) 0 0
\(229\) −2.42864 −0.160489 −0.0802445 0.996775i \(-0.525570\pi\)
−0.0802445 + 0.996775i \(0.525570\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −26.7096 −1.75357
\(233\) −16.1748 −1.05965 −0.529825 0.848107i \(-0.677742\pi\)
−0.529825 + 0.848107i \(0.677742\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.86818 −0.316891
\(237\) 0 0
\(238\) 22.6637 1.46907
\(239\) 12.7763 0.826431 0.413215 0.910633i \(-0.364406\pi\)
0.413215 + 0.910633i \(0.364406\pi\)
\(240\) 0 0
\(241\) −5.89829 −0.379942 −0.189971 0.981790i \(-0.560839\pi\)
−0.189971 + 0.981790i \(0.560839\pi\)
\(242\) 13.3017 0.855068
\(243\) 0 0
\(244\) 0.147643 0.00945189
\(245\) 0 0
\(246\) 0 0
\(247\) 2.21432 0.140894
\(248\) −17.1492 −1.08897
\(249\) 0 0
\(250\) 0 0
\(251\) 2.07313 0.130855 0.0654274 0.997857i \(-0.479159\pi\)
0.0654274 + 0.997857i \(0.479159\pi\)
\(252\) 0 0
\(253\) −1.00492 −0.0631789
\(254\) 2.79505 0.175377
\(255\) 0 0
\(256\) −11.6637 −0.728981
\(257\) −18.3970 −1.14757 −0.573787 0.819005i \(-0.694526\pi\)
−0.573787 + 0.819005i \(0.694526\pi\)
\(258\) 0 0
\(259\) 6.62222 0.411484
\(260\) 0 0
\(261\) 0 0
\(262\) −16.2953 −1.00673
\(263\) −11.0257 −0.679872 −0.339936 0.940449i \(-0.610405\pi\)
−0.339936 + 0.940449i \(0.610405\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 7.80642 0.478643
\(267\) 0 0
\(268\) −4.07758 −0.249078
\(269\) 16.1432 0.984268 0.492134 0.870519i \(-0.336217\pi\)
0.492134 + 0.870519i \(0.336217\pi\)
\(270\) 0 0
\(271\) 13.0114 0.790385 0.395192 0.918598i \(-0.370678\pi\)
0.395192 + 0.918598i \(0.370678\pi\)
\(272\) −17.1842 −1.04195
\(273\) 0 0
\(274\) −23.2573 −1.40503
\(275\) 0 0
\(276\) 0 0
\(277\) 7.57136 0.454919 0.227459 0.973788i \(-0.426958\pi\)
0.227459 + 0.973788i \(0.426958\pi\)
\(278\) −23.1842 −1.39050
\(279\) 0 0
\(280\) 0 0
\(281\) −6.75557 −0.403003 −0.201502 0.979488i \(-0.564582\pi\)
−0.201502 + 0.979488i \(0.564582\pi\)
\(282\) 0 0
\(283\) 19.0859 1.13454 0.567269 0.823532i \(-0.308000\pi\)
0.567269 + 0.823532i \(0.308000\pi\)
\(284\) −3.19511 −0.189595
\(285\) 0 0
\(286\) 0.260253 0.0153891
\(287\) 8.85728 0.522829
\(288\) 0 0
\(289\) 24.3274 1.43102
\(290\) 0 0
\(291\) 0 0
\(292\) −5.40192 −0.316123
\(293\) 8.08742 0.472472 0.236236 0.971696i \(-0.424086\pi\)
0.236236 + 0.971696i \(0.424086\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −6.99508 −0.406581
\(297\) 0 0
\(298\) −4.33677 −0.251223
\(299\) 4.68889 0.271166
\(300\) 0 0
\(301\) 18.4701 1.06460
\(302\) 1.53633 0.0884057
\(303\) 0 0
\(304\) −5.91903 −0.339480
\(305\) 0 0
\(306\) 0 0
\(307\) −13.4336 −0.766694 −0.383347 0.923604i \(-0.625229\pi\)
−0.383347 + 0.923604i \(0.625229\pi\)
\(308\) −0.326929 −0.0186285
\(309\) 0 0
\(310\) 0 0
\(311\) −20.2034 −1.14563 −0.572815 0.819684i \(-0.694149\pi\)
−0.572815 + 0.819684i \(0.694149\pi\)
\(312\) 0 0
\(313\) 15.1111 0.854129 0.427064 0.904221i \(-0.359548\pi\)
0.427064 + 0.904221i \(0.359548\pi\)
\(314\) −6.81579 −0.384637
\(315\) 0 0
\(316\) 7.47949 0.420754
\(317\) −22.2810 −1.25143 −0.625713 0.780054i \(-0.715192\pi\)
−0.625713 + 0.780054i \(0.715192\pi\)
\(318\) 0 0
\(319\) 1.86665 0.104512
\(320\) 0 0
\(321\) 0 0
\(322\) 16.5303 0.921200
\(323\) 14.2351 0.792060
\(324\) 0 0
\(325\) 0 0
\(326\) 4.51606 0.250121
\(327\) 0 0
\(328\) −9.35599 −0.516598
\(329\) −3.18421 −0.175551
\(330\) 0 0
\(331\) 8.25581 0.453780 0.226890 0.973920i \(-0.427144\pi\)
0.226890 + 0.973920i \(0.427144\pi\)
\(332\) −5.00492 −0.274681
\(333\) 0 0
\(334\) 8.54464 0.467542
\(335\) 0 0
\(336\) 0 0
\(337\) −13.7462 −0.748803 −0.374402 0.927267i \(-0.622152\pi\)
−0.374402 + 0.927267i \(0.622152\pi\)
\(338\) −1.21432 −0.0660503
\(339\) 0 0
\(340\) 0 0
\(341\) 1.19850 0.0649023
\(342\) 0 0
\(343\) 16.1748 0.873359
\(344\) −19.5101 −1.05191
\(345\) 0 0
\(346\) −0.879077 −0.0472595
\(347\) −1.21924 −0.0654523 −0.0327262 0.999464i \(-0.510419\pi\)
−0.0327262 + 0.999464i \(0.510419\pi\)
\(348\) 0 0
\(349\) 22.5116 1.20502 0.602510 0.798112i \(-0.294168\pi\)
0.602510 + 0.798112i \(0.294168\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.618825 0.0329835
\(353\) 14.2810 0.760101 0.380050 0.924966i \(-0.375907\pi\)
0.380050 + 0.924966i \(0.375907\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −2.94914 −0.156304
\(357\) 0 0
\(358\) −4.90766 −0.259378
\(359\) 12.1541 0.641469 0.320734 0.947169i \(-0.396070\pi\)
0.320734 + 0.947169i \(0.396070\pi\)
\(360\) 0 0
\(361\) −14.0968 −0.741936
\(362\) −2.84299 −0.149424
\(363\) 0 0
\(364\) 1.52543 0.0799541
\(365\) 0 0
\(366\) 0 0
\(367\) −4.65725 −0.243106 −0.121553 0.992585i \(-0.538788\pi\)
−0.121553 + 0.992585i \(0.538788\pi\)
\(368\) −12.5337 −0.653366
\(369\) 0 0
\(370\) 0 0
\(371\) 18.1017 0.939794
\(372\) 0 0
\(373\) 34.9403 1.80914 0.904569 0.426328i \(-0.140193\pi\)
0.904569 + 0.426328i \(0.140193\pi\)
\(374\) 1.67307 0.0865124
\(375\) 0 0
\(376\) 3.36349 0.173459
\(377\) −8.70964 −0.448569
\(378\) 0 0
\(379\) −17.4717 −0.897459 −0.448729 0.893668i \(-0.648123\pi\)
−0.448729 + 0.893668i \(0.648123\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2.55215 −0.130579
\(383\) 18.6780 0.954401 0.477200 0.878794i \(-0.341652\pi\)
0.477200 + 0.878794i \(0.341652\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 16.4188 0.835695
\(387\) 0 0
\(388\) −9.47949 −0.481248
\(389\) −1.61285 −0.0817746 −0.0408873 0.999164i \(-0.513018\pi\)
−0.0408873 + 0.999164i \(0.513018\pi\)
\(390\) 0 0
\(391\) 30.1432 1.52441
\(392\) 4.38118 0.221283
\(393\) 0 0
\(394\) −2.42864 −0.122353
\(395\) 0 0
\(396\) 0 0
\(397\) 6.57628 0.330054 0.165027 0.986289i \(-0.447229\pi\)
0.165027 + 0.986289i \(0.447229\pi\)
\(398\) −26.8889 −1.34782
\(399\) 0 0
\(400\) 0 0
\(401\) 21.9081 1.09404 0.547020 0.837120i \(-0.315762\pi\)
0.547020 + 0.837120i \(0.315762\pi\)
\(402\) 0 0
\(403\) −5.59210 −0.278563
\(404\) 2.07007 0.102990
\(405\) 0 0
\(406\) −30.7052 −1.52387
\(407\) 0.488863 0.0242320
\(408\) 0 0
\(409\) −10.1936 −0.504040 −0.252020 0.967722i \(-0.581095\pi\)
−0.252020 + 0.967722i \(0.581095\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.48289 0.0730565
\(413\) −26.8988 −1.32360
\(414\) 0 0
\(415\) 0 0
\(416\) −2.88739 −0.141566
\(417\) 0 0
\(418\) 0.576283 0.0281869
\(419\) −7.31756 −0.357486 −0.178743 0.983896i \(-0.557203\pi\)
−0.178743 + 0.983896i \(0.557203\pi\)
\(420\) 0 0
\(421\) −7.86665 −0.383397 −0.191698 0.981454i \(-0.561400\pi\)
−0.191698 + 0.981454i \(0.561400\pi\)
\(422\) −23.8666 −1.16181
\(423\) 0 0
\(424\) −19.1209 −0.928594
\(425\) 0 0
\(426\) 0 0
\(427\) 0.815792 0.0394789
\(428\) −8.99402 −0.434743
\(429\) 0 0
\(430\) 0 0
\(431\) 38.9195 1.87469 0.937343 0.348407i \(-0.113277\pi\)
0.937343 + 0.348407i \(0.113277\pi\)
\(432\) 0 0
\(433\) −20.2034 −0.970914 −0.485457 0.874260i \(-0.661347\pi\)
−0.485457 + 0.874260i \(0.661347\pi\)
\(434\) −19.7146 −0.946329
\(435\) 0 0
\(436\) 8.78721 0.420831
\(437\) 10.3827 0.496672
\(438\) 0 0
\(439\) 10.8889 0.519700 0.259850 0.965649i \(-0.416327\pi\)
0.259850 + 0.965649i \(0.416327\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −7.80642 −0.371314
\(443\) 28.6287 1.36019 0.680095 0.733124i \(-0.261939\pi\)
0.680095 + 0.733124i \(0.261939\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −23.8840 −1.13094
\(447\) 0 0
\(448\) −25.7003 −1.21422
\(449\) 10.9304 0.515838 0.257919 0.966167i \(-0.416963\pi\)
0.257919 + 0.966167i \(0.416963\pi\)
\(450\) 0 0
\(451\) 0.653858 0.0307890
\(452\) −0.622216 −0.0292666
\(453\) 0 0
\(454\) −16.1160 −0.756361
\(455\) 0 0
\(456\) 0 0
\(457\) −11.4064 −0.533567 −0.266784 0.963756i \(-0.585961\pi\)
−0.266784 + 0.963756i \(0.585961\pi\)
\(458\) 2.94914 0.137804
\(459\) 0 0
\(460\) 0 0
\(461\) 26.1334 1.21715 0.608576 0.793496i \(-0.291741\pi\)
0.608576 + 0.793496i \(0.291741\pi\)
\(462\) 0 0
\(463\) −7.92242 −0.368186 −0.184093 0.982909i \(-0.558935\pi\)
−0.184093 + 0.982909i \(0.558935\pi\)
\(464\) 23.2815 1.08082
\(465\) 0 0
\(466\) 19.6414 0.909872
\(467\) −10.8923 −0.504036 −0.252018 0.967723i \(-0.581094\pi\)
−0.252018 + 0.967723i \(0.581094\pi\)
\(468\) 0 0
\(469\) −22.5303 −1.04035
\(470\) 0 0
\(471\) 0 0
\(472\) 28.4133 1.30783
\(473\) 1.36349 0.0626935
\(474\) 0 0
\(475\) 0 0
\(476\) 9.80642 0.449477
\(477\) 0 0
\(478\) −15.5145 −0.709618
\(479\) 9.13182 0.417244 0.208622 0.977996i \(-0.433102\pi\)
0.208622 + 0.977996i \(0.433102\pi\)
\(480\) 0 0
\(481\) −2.28100 −0.104004
\(482\) 7.16241 0.326239
\(483\) 0 0
\(484\) 5.75557 0.261617
\(485\) 0 0
\(486\) 0 0
\(487\) −16.1891 −0.733600 −0.366800 0.930300i \(-0.619547\pi\)
−0.366800 + 0.930300i \(0.619547\pi\)
\(488\) −0.861725 −0.0390084
\(489\) 0 0
\(490\) 0 0
\(491\) −26.2636 −1.18526 −0.592631 0.805474i \(-0.701911\pi\)
−0.592631 + 0.805474i \(0.701911\pi\)
\(492\) 0 0
\(493\) −55.9911 −2.52171
\(494\) −2.68889 −0.120979
\(495\) 0 0
\(496\) 14.9481 0.671189
\(497\) −17.6543 −0.791905
\(498\) 0 0
\(499\) 30.0306 1.34435 0.672177 0.740391i \(-0.265359\pi\)
0.672177 + 0.740391i \(0.265359\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −2.51744 −0.112359
\(503\) 16.7304 0.745971 0.372985 0.927837i \(-0.378334\pi\)
0.372985 + 0.927837i \(0.378334\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.22030 0.0542488
\(507\) 0 0
\(508\) 1.20940 0.0536583
\(509\) 11.9684 0.530488 0.265244 0.964181i \(-0.414547\pi\)
0.265244 + 0.964181i \(0.414547\pi\)
\(510\) 0 0
\(511\) −29.8479 −1.32039
\(512\) 24.1131 1.06566
\(513\) 0 0
\(514\) 22.3398 0.985368
\(515\) 0 0
\(516\) 0 0
\(517\) −0.235063 −0.0103381
\(518\) −8.04149 −0.353323
\(519\) 0 0
\(520\) 0 0
\(521\) −5.75065 −0.251940 −0.125970 0.992034i \(-0.540204\pi\)
−0.125970 + 0.992034i \(0.540204\pi\)
\(522\) 0 0
\(523\) −20.8035 −0.909674 −0.454837 0.890575i \(-0.650302\pi\)
−0.454837 + 0.890575i \(0.650302\pi\)
\(524\) −7.05086 −0.308018
\(525\) 0 0
\(526\) 13.3887 0.583774
\(527\) −35.9496 −1.56599
\(528\) 0 0
\(529\) −1.01429 −0.0440996
\(530\) 0 0
\(531\) 0 0
\(532\) 3.37778 0.146446
\(533\) −3.05086 −0.132147
\(534\) 0 0
\(535\) 0 0
\(536\) 23.7989 1.02796
\(537\) 0 0
\(538\) −19.6030 −0.845145
\(539\) −0.306186 −0.0131883
\(540\) 0 0
\(541\) 16.6222 0.714645 0.357322 0.933981i \(-0.383690\pi\)
0.357322 + 0.933981i \(0.383690\pi\)
\(542\) −15.8000 −0.678667
\(543\) 0 0
\(544\) −18.5620 −0.795839
\(545\) 0 0
\(546\) 0 0
\(547\) −29.9748 −1.28163 −0.640815 0.767695i \(-0.721404\pi\)
−0.640815 + 0.767695i \(0.721404\pi\)
\(548\) −10.0633 −0.429882
\(549\) 0 0
\(550\) 0 0
\(551\) −19.2859 −0.821608
\(552\) 0 0
\(553\) 41.3274 1.75742
\(554\) −9.19405 −0.390618
\(555\) 0 0
\(556\) −10.0316 −0.425436
\(557\) 5.03657 0.213406 0.106703 0.994291i \(-0.465971\pi\)
0.106703 + 0.994291i \(0.465971\pi\)
\(558\) 0 0
\(559\) −6.36196 −0.269082
\(560\) 0 0
\(561\) 0 0
\(562\) 8.20342 0.346040
\(563\) −2.88247 −0.121482 −0.0607408 0.998154i \(-0.519346\pi\)
−0.0607408 + 0.998154i \(0.519346\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −23.1764 −0.974176
\(567\) 0 0
\(568\) 18.6484 0.782468
\(569\) 4.37286 0.183320 0.0916600 0.995790i \(-0.470783\pi\)
0.0916600 + 0.995790i \(0.470783\pi\)
\(570\) 0 0
\(571\) −1.58120 −0.0661714 −0.0330857 0.999453i \(-0.510533\pi\)
−0.0330857 + 0.999453i \(0.510533\pi\)
\(572\) 0.112610 0.00470844
\(573\) 0 0
\(574\) −10.7556 −0.448929
\(575\) 0 0
\(576\) 0 0
\(577\) −7.61729 −0.317112 −0.158556 0.987350i \(-0.550684\pi\)
−0.158556 + 0.987350i \(0.550684\pi\)
\(578\) −29.5412 −1.22875
\(579\) 0 0
\(580\) 0 0
\(581\) −27.6543 −1.14730
\(582\) 0 0
\(583\) 1.33630 0.0553438
\(584\) 31.5285 1.30466
\(585\) 0 0
\(586\) −9.82071 −0.405690
\(587\) 46.8243 1.93264 0.966322 0.257336i \(-0.0828449\pi\)
0.966322 + 0.257336i \(0.0828449\pi\)
\(588\) 0 0
\(589\) −12.3827 −0.510221
\(590\) 0 0
\(591\) 0 0
\(592\) 6.09726 0.250596
\(593\) 15.9398 0.654568 0.327284 0.944926i \(-0.393867\pi\)
0.327284 + 0.944926i \(0.393867\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.87649 −0.0768641
\(597\) 0 0
\(598\) −5.69381 −0.232837
\(599\) −18.4889 −0.755434 −0.377717 0.925921i \(-0.623291\pi\)
−0.377717 + 0.925921i \(0.623291\pi\)
\(600\) 0 0
\(601\) 20.7556 0.846637 0.423319 0.905981i \(-0.360865\pi\)
0.423319 + 0.905981i \(0.360865\pi\)
\(602\) −22.4286 −0.914123
\(603\) 0 0
\(604\) 0.664758 0.0270486
\(605\) 0 0
\(606\) 0 0
\(607\) 36.0765 1.46430 0.732150 0.681143i \(-0.238517\pi\)
0.732150 + 0.681143i \(0.238517\pi\)
\(608\) −6.39361 −0.259295
\(609\) 0 0
\(610\) 0 0
\(611\) 1.09679 0.0443713
\(612\) 0 0
\(613\) 9.94962 0.401861 0.200931 0.979605i \(-0.435603\pi\)
0.200931 + 0.979605i \(0.435603\pi\)
\(614\) 16.3126 0.658325
\(615\) 0 0
\(616\) 1.90813 0.0768809
\(617\) −2.09187 −0.0842154 −0.0421077 0.999113i \(-0.513407\pi\)
−0.0421077 + 0.999113i \(0.513407\pi\)
\(618\) 0 0
\(619\) 18.4681 0.742296 0.371148 0.928574i \(-0.378964\pi\)
0.371148 + 0.928574i \(0.378964\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 24.5334 0.983700
\(623\) −16.2953 −0.652857
\(624\) 0 0
\(625\) 0 0
\(626\) −18.3497 −0.733401
\(627\) 0 0
\(628\) −2.94914 −0.117684
\(629\) −14.6637 −0.584680
\(630\) 0 0
\(631\) −38.6657 −1.53926 −0.769629 0.638492i \(-0.779559\pi\)
−0.769629 + 0.638492i \(0.779559\pi\)
\(632\) −43.6543 −1.73648
\(633\) 0 0
\(634\) 27.0563 1.07454
\(635\) 0 0
\(636\) 0 0
\(637\) 1.42864 0.0566048
\(638\) −2.26671 −0.0897398
\(639\) 0 0
\(640\) 0 0
\(641\) −24.5718 −0.970529 −0.485265 0.874367i \(-0.661277\pi\)
−0.485265 + 0.874367i \(0.661277\pi\)
\(642\) 0 0
\(643\) 27.4938 1.08425 0.542125 0.840298i \(-0.317620\pi\)
0.542125 + 0.840298i \(0.317620\pi\)
\(644\) 7.15257 0.281851
\(645\) 0 0
\(646\) −17.2859 −0.680105
\(647\) −13.7812 −0.541796 −0.270898 0.962608i \(-0.587321\pi\)
−0.270898 + 0.962608i \(0.587321\pi\)
\(648\) 0 0
\(649\) −1.98571 −0.0779459
\(650\) 0 0
\(651\) 0 0
\(652\) 1.95407 0.0765272
\(653\) 2.12045 0.0829795 0.0414897 0.999139i \(-0.486790\pi\)
0.0414897 + 0.999139i \(0.486790\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 8.15515 0.318405
\(657\) 0 0
\(658\) 3.86665 0.150738
\(659\) −33.8894 −1.32014 −0.660072 0.751203i \(-0.729474\pi\)
−0.660072 + 0.751203i \(0.729474\pi\)
\(660\) 0 0
\(661\) −37.3689 −1.45348 −0.726741 0.686912i \(-0.758966\pi\)
−0.726741 + 0.686912i \(0.758966\pi\)
\(662\) −10.0252 −0.389640
\(663\) 0 0
\(664\) 29.2114 1.13362
\(665\) 0 0
\(666\) 0 0
\(667\) −40.8385 −1.58127
\(668\) 3.69721 0.143049
\(669\) 0 0
\(670\) 0 0
\(671\) 0.0602231 0.00232489
\(672\) 0 0
\(673\) −35.4608 −1.36691 −0.683456 0.729992i \(-0.739524\pi\)
−0.683456 + 0.729992i \(0.739524\pi\)
\(674\) 16.6923 0.642963
\(675\) 0 0
\(676\) −0.525428 −0.0202088
\(677\) −15.3047 −0.588206 −0.294103 0.955774i \(-0.595021\pi\)
−0.294103 + 0.955774i \(0.595021\pi\)
\(678\) 0 0
\(679\) −52.3783 −2.01009
\(680\) 0 0
\(681\) 0 0
\(682\) −1.45536 −0.0557286
\(683\) 13.0968 0.501135 0.250567 0.968099i \(-0.419383\pi\)
0.250567 + 0.968099i \(0.419383\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −19.6414 −0.749913
\(687\) 0 0
\(688\) 17.0060 0.648347
\(689\) −6.23506 −0.237537
\(690\) 0 0
\(691\) −18.4079 −0.700269 −0.350135 0.936699i \(-0.613864\pi\)
−0.350135 + 0.936699i \(0.613864\pi\)
\(692\) −0.380371 −0.0144595
\(693\) 0 0
\(694\) 1.48055 0.0562009
\(695\) 0 0
\(696\) 0 0
\(697\) −19.6128 −0.742890
\(698\) −27.3363 −1.03469
\(699\) 0 0
\(700\) 0 0
\(701\) 31.3689 1.18479 0.592393 0.805649i \(-0.298183\pi\)
0.592393 + 0.805649i \(0.298183\pi\)
\(702\) 0 0
\(703\) −5.05086 −0.190497
\(704\) −1.89723 −0.0715047
\(705\) 0 0
\(706\) −17.3417 −0.652663
\(707\) 11.4380 0.430171
\(708\) 0 0
\(709\) 9.47949 0.356010 0.178005 0.984030i \(-0.443036\pi\)
0.178005 + 0.984030i \(0.443036\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 17.2128 0.645077
\(713\) −26.2208 −0.981976
\(714\) 0 0
\(715\) 0 0
\(716\) −2.12351 −0.0793592
\(717\) 0 0
\(718\) −14.7590 −0.550799
\(719\) 29.6227 1.10474 0.552370 0.833599i \(-0.313724\pi\)
0.552370 + 0.833599i \(0.313724\pi\)
\(720\) 0 0
\(721\) 8.19358 0.305145
\(722\) 17.1180 0.637066
\(723\) 0 0
\(724\) −1.23014 −0.0457178
\(725\) 0 0
\(726\) 0 0
\(727\) 42.6702 1.58255 0.791274 0.611461i \(-0.209418\pi\)
0.791274 + 0.611461i \(0.209418\pi\)
\(728\) −8.90321 −0.329975
\(729\) 0 0
\(730\) 0 0
\(731\) −40.8988 −1.51270
\(732\) 0 0
\(733\) 26.0830 0.963397 0.481698 0.876337i \(-0.340020\pi\)
0.481698 + 0.876337i \(0.340020\pi\)
\(734\) 5.65539 0.208744
\(735\) 0 0
\(736\) −13.5387 −0.499042
\(737\) −1.66323 −0.0612657
\(738\) 0 0
\(739\) −28.2687 −1.03988 −0.519941 0.854202i \(-0.674046\pi\)
−0.519941 + 0.854202i \(0.674046\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −21.9813 −0.806958
\(743\) −20.6681 −0.758241 −0.379120 0.925347i \(-0.623773\pi\)
−0.379120 + 0.925347i \(0.623773\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −42.4286 −1.55342
\(747\) 0 0
\(748\) 0.723926 0.0264694
\(749\) −49.6958 −1.81585
\(750\) 0 0
\(751\) 2.46028 0.0897770 0.0448885 0.998992i \(-0.485707\pi\)
0.0448885 + 0.998992i \(0.485707\pi\)
\(752\) −2.93179 −0.106911
\(753\) 0 0
\(754\) 10.5763 0.385165
\(755\) 0 0
\(756\) 0 0
\(757\) −48.6035 −1.76652 −0.883262 0.468880i \(-0.844658\pi\)
−0.883262 + 0.468880i \(0.844658\pi\)
\(758\) 21.2162 0.770606
\(759\) 0 0
\(760\) 0 0
\(761\) 13.8252 0.501162 0.250581 0.968096i \(-0.419378\pi\)
0.250581 + 0.968096i \(0.419378\pi\)
\(762\) 0 0
\(763\) 48.5531 1.75774
\(764\) −1.10430 −0.0399520
\(765\) 0 0
\(766\) −22.6811 −0.819500
\(767\) 9.26517 0.334546
\(768\) 0 0
\(769\) −38.9688 −1.40525 −0.702626 0.711559i \(-0.747989\pi\)
−0.702626 + 0.711559i \(0.747989\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.10430 0.255689
\(773\) 0.445992 0.0160412 0.00802061 0.999968i \(-0.497447\pi\)
0.00802061 + 0.999968i \(0.497447\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 55.3274 1.98614
\(777\) 0 0
\(778\) 1.95851 0.0702161
\(779\) −6.75557 −0.242043
\(780\) 0 0
\(781\) −1.30327 −0.0466347
\(782\) −36.6035 −1.30894
\(783\) 0 0
\(784\) −3.81885 −0.136388
\(785\) 0 0
\(786\) 0 0
\(787\) −33.9037 −1.20854 −0.604268 0.796781i \(-0.706534\pi\)
−0.604268 + 0.796781i \(0.706534\pi\)
\(788\) −1.05086 −0.0374352
\(789\) 0 0
\(790\) 0 0
\(791\) −3.43801 −0.122241
\(792\) 0 0
\(793\) −0.280996 −0.00997847
\(794\) −7.98571 −0.283402
\(795\) 0 0
\(796\) −11.6346 −0.412379
\(797\) −10.2953 −0.364678 −0.182339 0.983236i \(-0.558367\pi\)
−0.182339 + 0.983236i \(0.558367\pi\)
\(798\) 0 0
\(799\) 7.05086 0.249441
\(800\) 0 0
\(801\) 0 0
\(802\) −26.6035 −0.939402
\(803\) −2.20342 −0.0777570
\(804\) 0 0
\(805\) 0 0
\(806\) 6.79060 0.239189
\(807\) 0 0
\(808\) −12.0820 −0.425044
\(809\) −7.94422 −0.279304 −0.139652 0.990201i \(-0.544598\pi\)
−0.139652 + 0.990201i \(0.544598\pi\)
\(810\) 0 0
\(811\) −8.12245 −0.285218 −0.142609 0.989779i \(-0.545549\pi\)
−0.142609 + 0.989779i \(0.545549\pi\)
\(812\) −13.2859 −0.466244
\(813\) 0 0
\(814\) −0.593635 −0.0208069
\(815\) 0 0
\(816\) 0 0
\(817\) −14.0874 −0.492856
\(818\) 12.3783 0.432796
\(819\) 0 0
\(820\) 0 0
\(821\) 22.2065 0.775012 0.387506 0.921867i \(-0.373337\pi\)
0.387506 + 0.921867i \(0.373337\pi\)
\(822\) 0 0
\(823\) −11.1175 −0.387533 −0.193766 0.981048i \(-0.562070\pi\)
−0.193766 + 0.981048i \(0.562070\pi\)
\(824\) −8.65491 −0.301508
\(825\) 0 0
\(826\) 32.6637 1.13652
\(827\) 23.1570 0.805248 0.402624 0.915365i \(-0.368098\pi\)
0.402624 + 0.915365i \(0.368098\pi\)
\(828\) 0 0
\(829\) −27.1195 −0.941901 −0.470950 0.882160i \(-0.656089\pi\)
−0.470950 + 0.882160i \(0.656089\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 8.85236 0.306900
\(833\) 9.18421 0.318214
\(834\) 0 0
\(835\) 0 0
\(836\) 0.249353 0.00862407
\(837\) 0 0
\(838\) 8.88586 0.306957
\(839\) −25.3955 −0.876749 −0.438374 0.898792i \(-0.644446\pi\)
−0.438374 + 0.898792i \(0.644446\pi\)
\(840\) 0 0
\(841\) 46.8578 1.61578
\(842\) 9.55262 0.329205
\(843\) 0 0
\(844\) −10.3269 −0.355468
\(845\) 0 0
\(846\) 0 0
\(847\) 31.8020 1.09273
\(848\) 16.6668 0.572339
\(849\) 0 0
\(850\) 0 0
\(851\) −10.6953 −0.366632
\(852\) 0 0
\(853\) −25.0651 −0.858214 −0.429107 0.903254i \(-0.641172\pi\)
−0.429107 + 0.903254i \(0.641172\pi\)
\(854\) −0.990632 −0.0338987
\(855\) 0 0
\(856\) 52.4939 1.79421
\(857\) 7.61285 0.260050 0.130025 0.991511i \(-0.458494\pi\)
0.130025 + 0.991511i \(0.458494\pi\)
\(858\) 0 0
\(859\) 42.1432 1.43791 0.718954 0.695058i \(-0.244621\pi\)
0.718954 + 0.695058i \(0.244621\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −47.2607 −1.60971
\(863\) 51.5768 1.75569 0.877847 0.478942i \(-0.158980\pi\)
0.877847 + 0.478942i \(0.158980\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 24.5334 0.833679
\(867\) 0 0
\(868\) −8.53035 −0.289539
\(869\) 3.05086 0.103493
\(870\) 0 0
\(871\) 7.76049 0.262954
\(872\) −51.2869 −1.73679
\(873\) 0 0
\(874\) −12.6079 −0.426469
\(875\) 0 0
\(876\) 0 0
\(877\) 34.0701 1.15046 0.575232 0.817990i \(-0.304912\pi\)
0.575232 + 0.817990i \(0.304912\pi\)
\(878\) −13.2226 −0.446242
\(879\) 0 0
\(880\) 0 0
\(881\) −3.71900 −0.125296 −0.0626482 0.998036i \(-0.519955\pi\)
−0.0626482 + 0.998036i \(0.519955\pi\)
\(882\) 0 0
\(883\) 42.0163 1.41396 0.706981 0.707233i \(-0.250057\pi\)
0.706981 + 0.707233i \(0.250057\pi\)
\(884\) −3.37778 −0.113607
\(885\) 0 0
\(886\) −34.7644 −1.16793
\(887\) −40.3116 −1.35353 −0.676765 0.736199i \(-0.736619\pi\)
−0.676765 + 0.736199i \(0.736619\pi\)
\(888\) 0 0
\(889\) 6.68244 0.224122
\(890\) 0 0
\(891\) 0 0
\(892\) −10.3344 −0.346023
\(893\) 2.42864 0.0812713
\(894\) 0 0
\(895\) 0 0
\(896\) 14.4429 0.482504
\(897\) 0 0
\(898\) −13.2730 −0.442926
\(899\) 48.7052 1.62441
\(900\) 0 0
\(901\) −40.0830 −1.33536
\(902\) −0.793993 −0.0264371
\(903\) 0 0
\(904\) 3.63158 0.120785
\(905\) 0 0
\(906\) 0 0
\(907\) −34.8419 −1.15691 −0.578454 0.815715i \(-0.696344\pi\)
−0.578454 + 0.815715i \(0.696344\pi\)
\(908\) −6.97328 −0.231416
\(909\) 0 0
\(910\) 0 0
\(911\) −23.2672 −0.770876 −0.385438 0.922734i \(-0.625950\pi\)
−0.385438 + 0.922734i \(0.625950\pi\)
\(912\) 0 0
\(913\) −2.04149 −0.0675634
\(914\) 13.8510 0.458149
\(915\) 0 0
\(916\) 1.27607 0.0421627
\(917\) −38.9590 −1.28654
\(918\) 0 0
\(919\) −3.22570 −0.106406 −0.0532029 0.998584i \(-0.516943\pi\)
−0.0532029 + 0.998584i \(0.516943\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −31.7342 −1.04511
\(923\) 6.08097 0.200157
\(924\) 0 0
\(925\) 0 0
\(926\) 9.62036 0.316145
\(927\) 0 0
\(928\) 25.1481 0.825527
\(929\) 39.3461 1.29091 0.645453 0.763800i \(-0.276669\pi\)
0.645453 + 0.763800i \(0.276669\pi\)
\(930\) 0 0
\(931\) 3.16346 0.103678
\(932\) 8.49871 0.278384
\(933\) 0 0
\(934\) 13.2268 0.432792
\(935\) 0 0
\(936\) 0 0
\(937\) 51.6040 1.68583 0.842914 0.538048i \(-0.180838\pi\)
0.842914 + 0.538048i \(0.180838\pi\)
\(938\) 27.3590 0.893305
\(939\) 0 0
\(940\) 0 0
\(941\) −37.5081 −1.22273 −0.611364 0.791349i \(-0.709379\pi\)
−0.611364 + 0.791349i \(0.709379\pi\)
\(942\) 0 0
\(943\) −14.3051 −0.465839
\(944\) −24.7665 −0.806080
\(945\) 0 0
\(946\) −1.65572 −0.0538320
\(947\) 38.1160 1.23860 0.619302 0.785153i \(-0.287416\pi\)
0.619302 + 0.785153i \(0.287416\pi\)
\(948\) 0 0
\(949\) 10.2810 0.333735
\(950\) 0 0
\(951\) 0 0
\(952\) −57.2355 −1.85501
\(953\) 28.7368 0.930877 0.465439 0.885080i \(-0.345897\pi\)
0.465439 + 0.885080i \(0.345897\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −6.71303 −0.217115
\(957\) 0 0
\(958\) −11.0890 −0.358268
\(959\) −55.6040 −1.79555
\(960\) 0 0
\(961\) 0.271628 0.00876221
\(962\) 2.76986 0.0893038
\(963\) 0 0
\(964\) 3.09912 0.0998161
\(965\) 0 0
\(966\) 0 0
\(967\) 29.0593 0.934485 0.467242 0.884129i \(-0.345248\pi\)
0.467242 + 0.884129i \(0.345248\pi\)
\(968\) −33.5926 −1.07971
\(969\) 0 0
\(970\) 0 0
\(971\) 39.8578 1.27910 0.639548 0.768751i \(-0.279121\pi\)
0.639548 + 0.768751i \(0.279121\pi\)
\(972\) 0 0
\(973\) −55.4291 −1.77698
\(974\) 19.6588 0.629908
\(975\) 0 0
\(976\) 0.751123 0.0240429
\(977\) 12.8617 0.411483 0.205742 0.978606i \(-0.434039\pi\)
0.205742 + 0.978606i \(0.434039\pi\)
\(978\) 0 0
\(979\) −1.20294 −0.0384463
\(980\) 0 0
\(981\) 0 0
\(982\) 31.8925 1.01773
\(983\) −45.4880 −1.45084 −0.725420 0.688306i \(-0.758355\pi\)
−0.725420 + 0.688306i \(0.758355\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 67.9911 2.16528
\(987\) 0 0
\(988\) −1.16346 −0.0370147
\(989\) −29.8306 −0.948557
\(990\) 0 0
\(991\) 8.07007 0.256354 0.128177 0.991751i \(-0.459087\pi\)
0.128177 + 0.991751i \(0.459087\pi\)
\(992\) 16.1466 0.512655
\(993\) 0 0
\(994\) 21.4380 0.679972
\(995\) 0 0
\(996\) 0 0
\(997\) −32.8158 −1.03929 −0.519643 0.854383i \(-0.673935\pi\)
−0.519643 + 0.854383i \(0.673935\pi\)
\(998\) −36.4667 −1.15433
\(999\) 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 2925.2.a.bj.1.1 3
3.2 odd 2 325.2.a.j.1.3 3
5.2 odd 4 585.2.c.b.469.3 6
5.3 odd 4 585.2.c.b.469.4 6
5.4 even 2 2925.2.a.bf.1.3 3
12.11 even 2 5200.2.a.cj.1.2 3
15.2 even 4 65.2.b.a.14.4 yes 6
15.8 even 4 65.2.b.a.14.3 6
15.14 odd 2 325.2.a.k.1.1 3
39.38 odd 2 4225.2.a.bh.1.1 3
60.23 odd 4 1040.2.d.c.209.5 6
60.47 odd 4 1040.2.d.c.209.2 6
60.59 even 2 5200.2.a.cb.1.2 3
195.2 odd 12 845.2.l.e.654.1 12
195.8 odd 4 845.2.d.a.844.5 6
195.17 even 12 845.2.n.g.484.3 12
195.23 even 12 845.2.n.g.529.3 12
195.32 odd 12 845.2.l.e.699.2 12
195.38 even 4 845.2.b.c.339.4 6
195.47 odd 4 845.2.d.b.844.2 6
195.62 even 12 845.2.n.g.529.4 12
195.68 even 12 845.2.n.f.529.4 12
195.77 even 4 845.2.b.c.339.3 6
195.83 odd 4 845.2.d.b.844.1 6
195.98 odd 12 845.2.l.e.699.1 12
195.107 even 12 845.2.n.f.529.3 12
195.113 even 12 845.2.n.f.484.3 12
195.122 odd 4 845.2.d.a.844.6 6
195.128 odd 12 845.2.l.e.654.2 12
195.137 odd 12 845.2.l.d.699.6 12
195.152 even 12 845.2.n.f.484.4 12
195.158 odd 12 845.2.l.d.654.6 12
195.167 odd 12 845.2.l.d.654.5 12
195.173 even 12 845.2.n.g.484.4 12
195.188 odd 12 845.2.l.d.699.5 12
195.194 odd 2 4225.2.a.ba.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.b.a.14.3 6 15.8 even 4
65.2.b.a.14.4 yes 6 15.2 even 4
325.2.a.j.1.3 3 3.2 odd 2
325.2.a.k.1.1 3 15.14 odd 2
585.2.c.b.469.3 6 5.2 odd 4
585.2.c.b.469.4 6 5.3 odd 4
845.2.b.c.339.3 6 195.77 even 4
845.2.b.c.339.4 6 195.38 even 4
845.2.d.a.844.5 6 195.8 odd 4
845.2.d.a.844.6 6 195.122 odd 4
845.2.d.b.844.1 6 195.83 odd 4
845.2.d.b.844.2 6 195.47 odd 4
845.2.l.d.654.5 12 195.167 odd 12
845.2.l.d.654.6 12 195.158 odd 12
845.2.l.d.699.5 12 195.188 odd 12
845.2.l.d.699.6 12 195.137 odd 12
845.2.l.e.654.1 12 195.2 odd 12
845.2.l.e.654.2 12 195.128 odd 12
845.2.l.e.699.1 12 195.98 odd 12
845.2.l.e.699.2 12 195.32 odd 12
845.2.n.f.484.3 12 195.113 even 12
845.2.n.f.484.4 12 195.152 even 12
845.2.n.f.529.3 12 195.107 even 12
845.2.n.f.529.4 12 195.68 even 12
845.2.n.g.484.3 12 195.17 even 12
845.2.n.g.484.4 12 195.173 even 12
845.2.n.g.529.3 12 195.23 even 12
845.2.n.g.529.4 12 195.62 even 12
1040.2.d.c.209.2 6 60.47 odd 4
1040.2.d.c.209.5 6 60.23 odd 4
2925.2.a.bf.1.3 3 5.4 even 2
2925.2.a.bj.1.1 3 1.1 even 1 trivial
4225.2.a.ba.1.3 3 195.194 odd 2
4225.2.a.bh.1.1 3 39.38 odd 2
5200.2.a.cb.1.2 3 60.59 even 2
5200.2.a.cj.1.2 3 12.11 even 2