Properties

Label 3025.2.a.bj.1.2
Level $3025$
Weight $2$
Character 3025.1
Self dual yes
Analytic conductor $24.155$
Analytic rank $1$
Dimension $8$
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] = [3025,2,Mod(1,3025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.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: \(24.1547466114\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 7x^{6} + 18x^{5} + 16x^{4} - 30x^{3} - 12x^{2} + 12x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56247\) of defining polynomial
Character \(\chi\) \(=\) 3025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.56247 q^{2} +1.79260 q^{3} +4.56626 q^{4} -4.59347 q^{6} -3.80088 q^{7} -6.57595 q^{8} +0.213399 q^{9} +O(q^{10})\) \(q-2.56247 q^{2} +1.79260 q^{3} +4.56626 q^{4} -4.59347 q^{6} -3.80088 q^{7} -6.57595 q^{8} +0.213399 q^{9} +8.18545 q^{12} +4.49758 q^{13} +9.73964 q^{14} +7.71818 q^{16} -5.01207 q^{17} -0.546828 q^{18} +3.46053 q^{19} -6.81344 q^{21} +0.105727 q^{23} -11.7880 q^{24} -11.5249 q^{26} -4.99525 q^{27} -17.3558 q^{28} +2.35008 q^{29} +4.74526 q^{31} -6.62570 q^{32} +12.8433 q^{34} +0.974434 q^{36} -5.60088 q^{37} -8.86750 q^{38} +8.06233 q^{39} -0.916922 q^{41} +17.4592 q^{42} +4.46162 q^{43} -0.270922 q^{46} +1.25360 q^{47} +13.8356 q^{48} +7.44667 q^{49} -8.98461 q^{51} +20.5371 q^{52} -5.49861 q^{53} +12.8002 q^{54} +24.9944 q^{56} +6.20333 q^{57} -6.02200 q^{58} -1.18305 q^{59} +7.32605 q^{61} -12.1596 q^{62} -0.811103 q^{63} +1.54180 q^{64} +7.84414 q^{67} -22.8864 q^{68} +0.189526 q^{69} -2.18022 q^{71} -1.40330 q^{72} -8.95263 q^{73} +14.3521 q^{74} +15.8017 q^{76} -20.6595 q^{78} -14.3825 q^{79} -9.59466 q^{81} +2.34959 q^{82} +4.71142 q^{83} -31.1119 q^{84} -11.4328 q^{86} +4.21274 q^{87} -0.172993 q^{89} -17.0947 q^{91} +0.482777 q^{92} +8.50634 q^{93} -3.21232 q^{94} -11.8772 q^{96} -4.46043 q^{97} -19.0819 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 5 q^{2} + q^{3} + 9 q^{4} - q^{6} - 8 q^{7} - 12 q^{8} + 9 q^{9} + 3 q^{12} - 9 q^{13} + 9 q^{14} + 23 q^{16} - 19 q^{17} - 22 q^{18} - q^{19} - 5 q^{21} + 2 q^{23} - q^{24} - 2 q^{26} - 2 q^{27} - 9 q^{28} - 7 q^{29} - 5 q^{31} - 29 q^{32} + 10 q^{34} - 16 q^{36} - 8 q^{37} - 37 q^{38} + q^{39} + 41 q^{42} - 14 q^{43} + 20 q^{46} + 11 q^{47} - 27 q^{48} - 12 q^{49} + 25 q^{51} + 7 q^{52} + 11 q^{53} - 30 q^{54} - 10 q^{56} + 2 q^{57} + 27 q^{58} + 17 q^{59} + 2 q^{61} - 25 q^{62} - 41 q^{63} + 30 q^{64} + 7 q^{67} - 66 q^{68} + 17 q^{71} + 19 q^{72} - 34 q^{73} + 6 q^{74} + 31 q^{76} - 17 q^{78} - 23 q^{79} - 4 q^{81} - 17 q^{82} - 41 q^{83} - 83 q^{84} + q^{86} - 25 q^{87} - 11 q^{89} - 7 q^{91} + 33 q^{92} - 59 q^{93} - 50 q^{94} + 61 q^{96} + 2 q^{97} - 26 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.56247 −1.81194 −0.905970 0.423342i \(-0.860857\pi\)
−0.905970 + 0.423342i \(0.860857\pi\)
\(3\) 1.79260 1.03496 0.517478 0.855697i \(-0.326871\pi\)
0.517478 + 0.855697i \(0.326871\pi\)
\(4\) 4.56626 2.28313
\(5\) 0 0
\(6\) −4.59347 −1.87528
\(7\) −3.80088 −1.43660 −0.718298 0.695735i \(-0.755079\pi\)
−0.718298 + 0.695735i \(0.755079\pi\)
\(8\) −6.57595 −2.32495
\(9\) 0.213399 0.0711330
\(10\) 0 0
\(11\) 0 0
\(12\) 8.18545 2.36294
\(13\) 4.49758 1.24740 0.623702 0.781663i \(-0.285628\pi\)
0.623702 + 0.781663i \(0.285628\pi\)
\(14\) 9.73964 2.60303
\(15\) 0 0
\(16\) 7.71818 1.92955
\(17\) −5.01207 −1.21560 −0.607802 0.794088i \(-0.707949\pi\)
−0.607802 + 0.794088i \(0.707949\pi\)
\(18\) −0.546828 −0.128889
\(19\) 3.46053 0.793900 0.396950 0.917840i \(-0.370069\pi\)
0.396950 + 0.917840i \(0.370069\pi\)
\(20\) 0 0
\(21\) −6.81344 −1.48681
\(22\) 0 0
\(23\) 0.105727 0.0220456 0.0110228 0.999939i \(-0.496491\pi\)
0.0110228 + 0.999939i \(0.496491\pi\)
\(24\) −11.7880 −2.40622
\(25\) 0 0
\(26\) −11.5249 −2.26022
\(27\) −4.99525 −0.961336
\(28\) −17.3558 −3.27993
\(29\) 2.35008 0.436398 0.218199 0.975904i \(-0.429982\pi\)
0.218199 + 0.975904i \(0.429982\pi\)
\(30\) 0 0
\(31\) 4.74526 0.852274 0.426137 0.904659i \(-0.359874\pi\)
0.426137 + 0.904659i \(0.359874\pi\)
\(32\) −6.62570 −1.17127
\(33\) 0 0
\(34\) 12.8433 2.20260
\(35\) 0 0
\(36\) 0.974434 0.162406
\(37\) −5.60088 −0.920779 −0.460390 0.887717i \(-0.652290\pi\)
−0.460390 + 0.887717i \(0.652290\pi\)
\(38\) −8.86750 −1.43850
\(39\) 8.06233 1.29101
\(40\) 0 0
\(41\) −0.916922 −0.143199 −0.0715996 0.997433i \(-0.522810\pi\)
−0.0715996 + 0.997433i \(0.522810\pi\)
\(42\) 17.4592 2.69402
\(43\) 4.46162 0.680390 0.340195 0.940355i \(-0.389507\pi\)
0.340195 + 0.940355i \(0.389507\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.270922 −0.0399453
\(47\) 1.25360 0.182857 0.0914283 0.995812i \(-0.470857\pi\)
0.0914283 + 0.995812i \(0.470857\pi\)
\(48\) 13.8356 1.99699
\(49\) 7.44667 1.06381
\(50\) 0 0
\(51\) −8.98461 −1.25810
\(52\) 20.5371 2.84798
\(53\) −5.49861 −0.755292 −0.377646 0.925950i \(-0.623266\pi\)
−0.377646 + 0.925950i \(0.623266\pi\)
\(54\) 12.8002 1.74188
\(55\) 0 0
\(56\) 24.9944 3.34002
\(57\) 6.20333 0.821651
\(58\) −6.02200 −0.790728
\(59\) −1.18305 −0.154020 −0.0770099 0.997030i \(-0.524537\pi\)
−0.0770099 + 0.997030i \(0.524537\pi\)
\(60\) 0 0
\(61\) 7.32605 0.938004 0.469002 0.883197i \(-0.344614\pi\)
0.469002 + 0.883197i \(0.344614\pi\)
\(62\) −12.1596 −1.54427
\(63\) −0.811103 −0.102189
\(64\) 1.54180 0.192725
\(65\) 0 0
\(66\) 0 0
\(67\) 7.84414 0.958315 0.479157 0.877729i \(-0.340942\pi\)
0.479157 + 0.877729i \(0.340942\pi\)
\(68\) −22.8864 −2.77538
\(69\) 0.189526 0.0228162
\(70\) 0 0
\(71\) −2.18022 −0.258745 −0.129372 0.991596i \(-0.541296\pi\)
−0.129372 + 0.991596i \(0.541296\pi\)
\(72\) −1.40330 −0.165381
\(73\) −8.95263 −1.04783 −0.523913 0.851772i \(-0.675528\pi\)
−0.523913 + 0.851772i \(0.675528\pi\)
\(74\) 14.3521 1.66840
\(75\) 0 0
\(76\) 15.8017 1.81257
\(77\) 0 0
\(78\) −20.6595 −2.33923
\(79\) −14.3825 −1.61815 −0.809077 0.587702i \(-0.800033\pi\)
−0.809077 + 0.587702i \(0.800033\pi\)
\(80\) 0 0
\(81\) −9.59466 −1.06607
\(82\) 2.34959 0.259468
\(83\) 4.71142 0.517146 0.258573 0.965992i \(-0.416748\pi\)
0.258573 + 0.965992i \(0.416748\pi\)
\(84\) −31.1119 −3.39459
\(85\) 0 0
\(86\) −11.4328 −1.23283
\(87\) 4.21274 0.451653
\(88\) 0 0
\(89\) −0.172993 −0.0183372 −0.00916862 0.999958i \(-0.502919\pi\)
−0.00916862 + 0.999958i \(0.502919\pi\)
\(90\) 0 0
\(91\) −17.0947 −1.79202
\(92\) 0.482777 0.0503330
\(93\) 8.50634 0.882066
\(94\) −3.21232 −0.331325
\(95\) 0 0
\(96\) −11.8772 −1.21221
\(97\) −4.46043 −0.452888 −0.226444 0.974024i \(-0.572710\pi\)
−0.226444 + 0.974024i \(0.572710\pi\)
\(98\) −19.0819 −1.92756
\(99\) 0 0
\(100\) 0 0
\(101\) −8.80519 −0.876149 −0.438074 0.898939i \(-0.644339\pi\)
−0.438074 + 0.898939i \(0.644339\pi\)
\(102\) 23.0228 2.27960
\(103\) −16.4740 −1.62323 −0.811615 0.584193i \(-0.801411\pi\)
−0.811615 + 0.584193i \(0.801411\pi\)
\(104\) −29.5759 −2.90015
\(105\) 0 0
\(106\) 14.0900 1.36854
\(107\) −17.1724 −1.66012 −0.830060 0.557674i \(-0.811694\pi\)
−0.830060 + 0.557674i \(0.811694\pi\)
\(108\) −22.8096 −2.19485
\(109\) −17.9060 −1.71508 −0.857542 0.514414i \(-0.828009\pi\)
−0.857542 + 0.514414i \(0.828009\pi\)
\(110\) 0 0
\(111\) −10.0401 −0.952965
\(112\) −29.3359 −2.77198
\(113\) −14.3161 −1.34675 −0.673374 0.739302i \(-0.735156\pi\)
−0.673374 + 0.739302i \(0.735156\pi\)
\(114\) −15.8958 −1.48878
\(115\) 0 0
\(116\) 10.7311 0.996353
\(117\) 0.959778 0.0887315
\(118\) 3.03153 0.279075
\(119\) 19.0503 1.74633
\(120\) 0 0
\(121\) 0 0
\(122\) −18.7728 −1.69961
\(123\) −1.64367 −0.148205
\(124\) 21.6681 1.94585
\(125\) 0 0
\(126\) 2.07843 0.185161
\(127\) 5.57000 0.494257 0.247129 0.968983i \(-0.420513\pi\)
0.247129 + 0.968983i \(0.420513\pi\)
\(128\) 9.30058 0.822063
\(129\) 7.99788 0.704174
\(130\) 0 0
\(131\) 13.8367 1.20892 0.604458 0.796637i \(-0.293390\pi\)
0.604458 + 0.796637i \(0.293390\pi\)
\(132\) 0 0
\(133\) −13.1530 −1.14051
\(134\) −20.1004 −1.73641
\(135\) 0 0
\(136\) 32.9591 2.82622
\(137\) −9.94447 −0.849614 −0.424807 0.905284i \(-0.639658\pi\)
−0.424807 + 0.905284i \(0.639658\pi\)
\(138\) −0.485654 −0.0413417
\(139\) −2.03061 −0.172234 −0.0861170 0.996285i \(-0.527446\pi\)
−0.0861170 + 0.996285i \(0.527446\pi\)
\(140\) 0 0
\(141\) 2.24720 0.189248
\(142\) 5.58676 0.468830
\(143\) 0 0
\(144\) 1.64705 0.137254
\(145\) 0 0
\(146\) 22.9408 1.89860
\(147\) 13.3489 1.10100
\(148\) −25.5751 −2.10226
\(149\) −10.6201 −0.870033 −0.435017 0.900422i \(-0.643258\pi\)
−0.435017 + 0.900422i \(0.643258\pi\)
\(150\) 0 0
\(151\) 3.43448 0.279494 0.139747 0.990187i \(-0.455371\pi\)
0.139747 + 0.990187i \(0.455371\pi\)
\(152\) −22.7563 −1.84578
\(153\) −1.06957 −0.0864696
\(154\) 0 0
\(155\) 0 0
\(156\) 36.8147 2.94753
\(157\) −0.298585 −0.0238296 −0.0119148 0.999929i \(-0.503793\pi\)
−0.0119148 + 0.999929i \(0.503793\pi\)
\(158\) 36.8547 2.93200
\(159\) −9.85678 −0.781693
\(160\) 0 0
\(161\) −0.401856 −0.0316707
\(162\) 24.5860 1.93166
\(163\) 14.2597 1.11691 0.558454 0.829536i \(-0.311395\pi\)
0.558454 + 0.829536i \(0.311395\pi\)
\(164\) −4.18690 −0.326942
\(165\) 0 0
\(166\) −12.0729 −0.937037
\(167\) −17.9977 −1.39270 −0.696352 0.717700i \(-0.745195\pi\)
−0.696352 + 0.717700i \(0.745195\pi\)
\(168\) 44.8049 3.45677
\(169\) 7.22819 0.556014
\(170\) 0 0
\(171\) 0.738473 0.0564724
\(172\) 20.3729 1.55342
\(173\) 8.75353 0.665519 0.332759 0.943012i \(-0.392020\pi\)
0.332759 + 0.943012i \(0.392020\pi\)
\(174\) −10.7950 −0.818368
\(175\) 0 0
\(176\) 0 0
\(177\) −2.12073 −0.159404
\(178\) 0.443290 0.0332260
\(179\) −22.8999 −1.71162 −0.855811 0.517289i \(-0.826941\pi\)
−0.855811 + 0.517289i \(0.826941\pi\)
\(180\) 0 0
\(181\) 17.2705 1.28370 0.641852 0.766828i \(-0.278166\pi\)
0.641852 + 0.766828i \(0.278166\pi\)
\(182\) 43.8048 3.24702
\(183\) 13.1326 0.970793
\(184\) −0.695256 −0.0512550
\(185\) 0 0
\(186\) −21.7972 −1.59825
\(187\) 0 0
\(188\) 5.72427 0.417485
\(189\) 18.9863 1.38105
\(190\) 0 0
\(191\) −18.2017 −1.31703 −0.658513 0.752569i \(-0.728814\pi\)
−0.658513 + 0.752569i \(0.728814\pi\)
\(192\) 2.76383 0.199462
\(193\) 11.0361 0.794397 0.397199 0.917733i \(-0.369982\pi\)
0.397199 + 0.917733i \(0.369982\pi\)
\(194\) 11.4297 0.820607
\(195\) 0 0
\(196\) 34.0034 2.42882
\(197\) 15.7213 1.12009 0.560047 0.828461i \(-0.310783\pi\)
0.560047 + 0.828461i \(0.310783\pi\)
\(198\) 0 0
\(199\) 4.44523 0.315114 0.157557 0.987510i \(-0.449638\pi\)
0.157557 + 0.987510i \(0.449638\pi\)
\(200\) 0 0
\(201\) 14.0614 0.991813
\(202\) 22.5630 1.58753
\(203\) −8.93236 −0.626929
\(204\) −41.0260 −2.87240
\(205\) 0 0
\(206\) 42.2141 2.94119
\(207\) 0.0225620 0.00156817
\(208\) 34.7131 2.40692
\(209\) 0 0
\(210\) 0 0
\(211\) −2.31213 −0.159173 −0.0795866 0.996828i \(-0.525360\pi\)
−0.0795866 + 0.996828i \(0.525360\pi\)
\(212\) −25.1080 −1.72443
\(213\) −3.90826 −0.267790
\(214\) 44.0038 3.00804
\(215\) 0 0
\(216\) 32.8485 2.23506
\(217\) −18.0362 −1.22437
\(218\) 45.8836 3.10763
\(219\) −16.0484 −1.08445
\(220\) 0 0
\(221\) −22.5422 −1.51635
\(222\) 25.7275 1.72672
\(223\) 10.2973 0.689555 0.344777 0.938684i \(-0.387954\pi\)
0.344777 + 0.938684i \(0.387954\pi\)
\(224\) 25.1835 1.68264
\(225\) 0 0
\(226\) 36.6847 2.44023
\(227\) 7.17833 0.476442 0.238221 0.971211i \(-0.423436\pi\)
0.238221 + 0.971211i \(0.423436\pi\)
\(228\) 28.3260 1.87593
\(229\) 2.05937 0.136087 0.0680435 0.997682i \(-0.478324\pi\)
0.0680435 + 0.997682i \(0.478324\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −15.4540 −1.01461
\(233\) −17.5865 −1.15213 −0.576065 0.817404i \(-0.695413\pi\)
−0.576065 + 0.817404i \(0.695413\pi\)
\(234\) −2.45940 −0.160776
\(235\) 0 0
\(236\) −5.40210 −0.351647
\(237\) −25.7820 −1.67472
\(238\) −48.8157 −3.16425
\(239\) −18.6889 −1.20888 −0.604442 0.796649i \(-0.706604\pi\)
−0.604442 + 0.796649i \(0.706604\pi\)
\(240\) 0 0
\(241\) −30.1916 −1.94481 −0.972406 0.233296i \(-0.925049\pi\)
−0.972406 + 0.233296i \(0.925049\pi\)
\(242\) 0 0
\(243\) −2.21359 −0.142002
\(244\) 33.4526 2.14158
\(245\) 0 0
\(246\) 4.21186 0.268538
\(247\) 15.5640 0.990313
\(248\) −31.2046 −1.98150
\(249\) 8.44568 0.535223
\(250\) 0 0
\(251\) −25.5114 −1.61026 −0.805132 0.593096i \(-0.797906\pi\)
−0.805132 + 0.593096i \(0.797906\pi\)
\(252\) −3.70370 −0.233311
\(253\) 0 0
\(254\) −14.2730 −0.895565
\(255\) 0 0
\(256\) −26.9161 −1.68225
\(257\) 23.9417 1.49344 0.746721 0.665137i \(-0.231627\pi\)
0.746721 + 0.665137i \(0.231627\pi\)
\(258\) −20.4943 −1.27592
\(259\) 21.2883 1.32279
\(260\) 0 0
\(261\) 0.501504 0.0310423
\(262\) −35.4561 −2.19048
\(263\) −26.7696 −1.65069 −0.825343 0.564631i \(-0.809018\pi\)
−0.825343 + 0.564631i \(0.809018\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 33.7043 2.06654
\(267\) −0.310107 −0.0189782
\(268\) 35.8184 2.18796
\(269\) −5.96585 −0.363744 −0.181872 0.983322i \(-0.558216\pi\)
−0.181872 + 0.983322i \(0.558216\pi\)
\(270\) 0 0
\(271\) 7.49365 0.455207 0.227603 0.973754i \(-0.426911\pi\)
0.227603 + 0.973754i \(0.426911\pi\)
\(272\) −38.6840 −2.34556
\(273\) −30.6439 −1.85466
\(274\) 25.4824 1.53945
\(275\) 0 0
\(276\) 0.865424 0.0520924
\(277\) −1.44255 −0.0866742 −0.0433371 0.999061i \(-0.513799\pi\)
−0.0433371 + 0.999061i \(0.513799\pi\)
\(278\) 5.20338 0.312078
\(279\) 1.01263 0.0606248
\(280\) 0 0
\(281\) 0.239273 0.0142738 0.00713691 0.999975i \(-0.497728\pi\)
0.00713691 + 0.999975i \(0.497728\pi\)
\(282\) −5.75839 −0.342907
\(283\) −5.27885 −0.313795 −0.156898 0.987615i \(-0.550149\pi\)
−0.156898 + 0.987615i \(0.550149\pi\)
\(284\) −9.95546 −0.590748
\(285\) 0 0
\(286\) 0 0
\(287\) 3.48511 0.205720
\(288\) −1.41392 −0.0833159
\(289\) 8.12082 0.477695
\(290\) 0 0
\(291\) −7.99575 −0.468719
\(292\) −40.8800 −2.39232
\(293\) −14.9959 −0.876069 −0.438035 0.898958i \(-0.644325\pi\)
−0.438035 + 0.898958i \(0.644325\pi\)
\(294\) −34.2061 −1.99494
\(295\) 0 0
\(296\) 36.8311 2.14077
\(297\) 0 0
\(298\) 27.2137 1.57645
\(299\) 0.475515 0.0274998
\(300\) 0 0
\(301\) −16.9581 −0.977447
\(302\) −8.80076 −0.506427
\(303\) −15.7841 −0.906775
\(304\) 26.7090 1.53187
\(305\) 0 0
\(306\) 2.74074 0.156678
\(307\) 2.11034 0.120443 0.0602217 0.998185i \(-0.480819\pi\)
0.0602217 + 0.998185i \(0.480819\pi\)
\(308\) 0 0
\(309\) −29.5312 −1.67997
\(310\) 0 0
\(311\) −16.6612 −0.944768 −0.472384 0.881393i \(-0.656606\pi\)
−0.472384 + 0.881393i \(0.656606\pi\)
\(312\) −53.0175 −3.00153
\(313\) 29.0799 1.64369 0.821845 0.569710i \(-0.192945\pi\)
0.821845 + 0.569710i \(0.192945\pi\)
\(314\) 0.765114 0.0431779
\(315\) 0 0
\(316\) −65.6740 −3.69445
\(317\) 18.3165 1.02876 0.514379 0.857563i \(-0.328023\pi\)
0.514379 + 0.857563i \(0.328023\pi\)
\(318\) 25.2577 1.41638
\(319\) 0 0
\(320\) 0 0
\(321\) −30.7832 −1.71815
\(322\) 1.02974 0.0573854
\(323\) −17.3444 −0.965068
\(324\) −43.8117 −2.43398
\(325\) 0 0
\(326\) −36.5401 −2.02377
\(327\) −32.0982 −1.77504
\(328\) 6.02964 0.332931
\(329\) −4.76479 −0.262691
\(330\) 0 0
\(331\) 21.5599 1.18504 0.592520 0.805556i \(-0.298133\pi\)
0.592520 + 0.805556i \(0.298133\pi\)
\(332\) 21.5136 1.18071
\(333\) −1.19522 −0.0654977
\(334\) 46.1186 2.52350
\(335\) 0 0
\(336\) −52.5873 −2.86887
\(337\) 14.0614 0.765973 0.382986 0.923754i \(-0.374896\pi\)
0.382986 + 0.923754i \(0.374896\pi\)
\(338\) −18.5220 −1.00746
\(339\) −25.6630 −1.39382
\(340\) 0 0
\(341\) 0 0
\(342\) −1.89232 −0.102325
\(343\) −1.69775 −0.0916699
\(344\) −29.3394 −1.58187
\(345\) 0 0
\(346\) −22.4307 −1.20588
\(347\) −28.6497 −1.53800 −0.768999 0.639250i \(-0.779245\pi\)
−0.768999 + 0.639250i \(0.779245\pi\)
\(348\) 19.2364 1.03118
\(349\) 17.9222 0.959353 0.479676 0.877446i \(-0.340754\pi\)
0.479676 + 0.877446i \(0.340754\pi\)
\(350\) 0 0
\(351\) −22.4665 −1.19917
\(352\) 0 0
\(353\) 24.7757 1.31868 0.659339 0.751845i \(-0.270836\pi\)
0.659339 + 0.751845i \(0.270836\pi\)
\(354\) 5.43430 0.288830
\(355\) 0 0
\(356\) −0.789931 −0.0418663
\(357\) 34.1494 1.80738
\(358\) 58.6804 3.10136
\(359\) 10.4523 0.551651 0.275826 0.961208i \(-0.411049\pi\)
0.275826 + 0.961208i \(0.411049\pi\)
\(360\) 0 0
\(361\) −7.02474 −0.369723
\(362\) −44.2551 −2.32600
\(363\) 0 0
\(364\) −78.0589 −4.09140
\(365\) 0 0
\(366\) −33.6520 −1.75902
\(367\) −15.1951 −0.793176 −0.396588 0.917997i \(-0.629806\pi\)
−0.396588 + 0.917997i \(0.629806\pi\)
\(368\) 0.816020 0.0425380
\(369\) −0.195670 −0.0101862
\(370\) 0 0
\(371\) 20.8995 1.08505
\(372\) 38.8421 2.01387
\(373\) 21.1774 1.09653 0.548263 0.836306i \(-0.315289\pi\)
0.548263 + 0.836306i \(0.315289\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −8.24363 −0.425133
\(377\) 10.5697 0.544365
\(378\) −48.6519 −2.50238
\(379\) −35.1644 −1.80627 −0.903137 0.429351i \(-0.858742\pi\)
−0.903137 + 0.429351i \(0.858742\pi\)
\(380\) 0 0
\(381\) 9.98475 0.511534
\(382\) 46.6412 2.38637
\(383\) −1.98986 −0.101677 −0.0508385 0.998707i \(-0.516189\pi\)
−0.0508385 + 0.998707i \(0.516189\pi\)
\(384\) 16.6722 0.850799
\(385\) 0 0
\(386\) −28.2797 −1.43940
\(387\) 0.952104 0.0483982
\(388\) −20.3675 −1.03400
\(389\) −6.05849 −0.307178 −0.153589 0.988135i \(-0.549083\pi\)
−0.153589 + 0.988135i \(0.549083\pi\)
\(390\) 0 0
\(391\) −0.529911 −0.0267988
\(392\) −48.9690 −2.47331
\(393\) 24.8036 1.25118
\(394\) −40.2853 −2.02954
\(395\) 0 0
\(396\) 0 0
\(397\) 35.0663 1.75993 0.879964 0.475040i \(-0.157566\pi\)
0.879964 + 0.475040i \(0.157566\pi\)
\(398\) −11.3908 −0.570968
\(399\) −23.5781 −1.18038
\(400\) 0 0
\(401\) 16.4747 0.822708 0.411354 0.911476i \(-0.365056\pi\)
0.411354 + 0.911476i \(0.365056\pi\)
\(402\) −36.0319 −1.79711
\(403\) 21.3422 1.06313
\(404\) −40.2067 −2.00036
\(405\) 0 0
\(406\) 22.8889 1.13596
\(407\) 0 0
\(408\) 59.0824 2.92501
\(409\) −4.31034 −0.213133 −0.106566 0.994306i \(-0.533986\pi\)
−0.106566 + 0.994306i \(0.533986\pi\)
\(410\) 0 0
\(411\) −17.8264 −0.879312
\(412\) −75.2244 −3.70604
\(413\) 4.49662 0.221264
\(414\) −0.0578146 −0.00284143
\(415\) 0 0
\(416\) −29.7996 −1.46104
\(417\) −3.64006 −0.178255
\(418\) 0 0
\(419\) −18.7114 −0.914110 −0.457055 0.889438i \(-0.651096\pi\)
−0.457055 + 0.889438i \(0.651096\pi\)
\(420\) 0 0
\(421\) −30.5078 −1.48686 −0.743429 0.668815i \(-0.766802\pi\)
−0.743429 + 0.668815i \(0.766802\pi\)
\(422\) 5.92475 0.288412
\(423\) 0.267517 0.0130071
\(424\) 36.1586 1.75602
\(425\) 0 0
\(426\) 10.0148 0.485219
\(427\) −27.8454 −1.34753
\(428\) −78.4136 −3.79027
\(429\) 0 0
\(430\) 0 0
\(431\) −8.31340 −0.400442 −0.200221 0.979751i \(-0.564166\pi\)
−0.200221 + 0.979751i \(0.564166\pi\)
\(432\) −38.5542 −1.85494
\(433\) −27.4497 −1.31915 −0.659574 0.751640i \(-0.729263\pi\)
−0.659574 + 0.751640i \(0.729263\pi\)
\(434\) 46.2171 2.21849
\(435\) 0 0
\(436\) −81.7634 −3.91576
\(437\) 0.365872 0.0175020
\(438\) 41.1237 1.96496
\(439\) 23.1527 1.10502 0.552510 0.833506i \(-0.313670\pi\)
0.552510 + 0.833506i \(0.313670\pi\)
\(440\) 0 0
\(441\) 1.58911 0.0756720
\(442\) 57.7636 2.74753
\(443\) −20.1835 −0.958947 −0.479474 0.877556i \(-0.659172\pi\)
−0.479474 + 0.877556i \(0.659172\pi\)
\(444\) −45.8457 −2.17574
\(445\) 0 0
\(446\) −26.3864 −1.24943
\(447\) −19.0376 −0.900446
\(448\) −5.86020 −0.276868
\(449\) 14.0341 0.662312 0.331156 0.943576i \(-0.392561\pi\)
0.331156 + 0.943576i \(0.392561\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −65.3711 −3.07480
\(453\) 6.15664 0.289264
\(454\) −18.3943 −0.863285
\(455\) 0 0
\(456\) −40.7928 −1.91030
\(457\) 15.6236 0.730843 0.365421 0.930842i \(-0.380925\pi\)
0.365421 + 0.930842i \(0.380925\pi\)
\(458\) −5.27707 −0.246581
\(459\) 25.0365 1.16860
\(460\) 0 0
\(461\) −29.6082 −1.37899 −0.689496 0.724289i \(-0.742168\pi\)
−0.689496 + 0.724289i \(0.742168\pi\)
\(462\) 0 0
\(463\) 1.31629 0.0611730 0.0305865 0.999532i \(-0.490262\pi\)
0.0305865 + 0.999532i \(0.490262\pi\)
\(464\) 18.1383 0.842050
\(465\) 0 0
\(466\) 45.0649 2.08759
\(467\) 12.0995 0.559898 0.279949 0.960015i \(-0.409682\pi\)
0.279949 + 0.960015i \(0.409682\pi\)
\(468\) 4.38259 0.202585
\(469\) −29.8146 −1.37671
\(470\) 0 0
\(471\) −0.535242 −0.0246626
\(472\) 7.77967 0.358088
\(473\) 0 0
\(474\) 66.0655 3.03449
\(475\) 0 0
\(476\) 86.9883 3.98710
\(477\) −1.17340 −0.0537261
\(478\) 47.8897 2.19043
\(479\) −3.05400 −0.139541 −0.0697703 0.997563i \(-0.522227\pi\)
−0.0697703 + 0.997563i \(0.522227\pi\)
\(480\) 0 0
\(481\) −25.1904 −1.14858
\(482\) 77.3651 3.52388
\(483\) −0.720365 −0.0327777
\(484\) 0 0
\(485\) 0 0
\(486\) 5.67227 0.257299
\(487\) −24.2398 −1.09841 −0.549205 0.835688i \(-0.685069\pi\)
−0.549205 + 0.835688i \(0.685069\pi\)
\(488\) −48.1758 −2.18081
\(489\) 25.5619 1.15595
\(490\) 0 0
\(491\) 32.6796 1.47481 0.737404 0.675452i \(-0.236051\pi\)
0.737404 + 0.675452i \(0.236051\pi\)
\(492\) −7.50542 −0.338370
\(493\) −11.7787 −0.530488
\(494\) −39.8823 −1.79439
\(495\) 0 0
\(496\) 36.6248 1.64450
\(497\) 8.28676 0.371712
\(498\) −21.6418 −0.969792
\(499\) −30.0059 −1.34325 −0.671625 0.740891i \(-0.734403\pi\)
−0.671625 + 0.740891i \(0.734403\pi\)
\(500\) 0 0
\(501\) −32.2626 −1.44139
\(502\) 65.3721 2.91770
\(503\) 13.5516 0.604236 0.302118 0.953271i \(-0.402306\pi\)
0.302118 + 0.953271i \(0.402306\pi\)
\(504\) 5.33378 0.237585
\(505\) 0 0
\(506\) 0 0
\(507\) 12.9572 0.575450
\(508\) 25.4340 1.12845
\(509\) 38.0843 1.68806 0.844028 0.536299i \(-0.180178\pi\)
0.844028 + 0.536299i \(0.180178\pi\)
\(510\) 0 0
\(511\) 34.0278 1.50530
\(512\) 50.3705 2.22608
\(513\) −17.2862 −0.763204
\(514\) −61.3499 −2.70603
\(515\) 0 0
\(516\) 36.5203 1.60772
\(517\) 0 0
\(518\) −54.5505 −2.39681
\(519\) 15.6915 0.688782
\(520\) 0 0
\(521\) 2.20872 0.0967658 0.0483829 0.998829i \(-0.484593\pi\)
0.0483829 + 0.998829i \(0.484593\pi\)
\(522\) −1.28509 −0.0562468
\(523\) −2.40261 −0.105059 −0.0525294 0.998619i \(-0.516728\pi\)
−0.0525294 + 0.998619i \(0.516728\pi\)
\(524\) 63.1818 2.76011
\(525\) 0 0
\(526\) 68.5964 2.99095
\(527\) −23.7836 −1.03603
\(528\) 0 0
\(529\) −22.9888 −0.999514
\(530\) 0 0
\(531\) −0.252461 −0.0109559
\(532\) −60.0602 −2.60394
\(533\) −4.12393 −0.178627
\(534\) 0.794639 0.0343874
\(535\) 0 0
\(536\) −51.5827 −2.22804
\(537\) −41.0503 −1.77145
\(538\) 15.2873 0.659083
\(539\) 0 0
\(540\) 0 0
\(541\) 23.5101 1.01078 0.505388 0.862892i \(-0.331349\pi\)
0.505388 + 0.862892i \(0.331349\pi\)
\(542\) −19.2023 −0.824807
\(543\) 30.9590 1.32858
\(544\) 33.2085 1.42380
\(545\) 0 0
\(546\) 78.5242 3.36053
\(547\) −1.41425 −0.0604688 −0.0302344 0.999543i \(-0.509625\pi\)
−0.0302344 + 0.999543i \(0.509625\pi\)
\(548\) −45.4090 −1.93978
\(549\) 1.56337 0.0667230
\(550\) 0 0
\(551\) 8.13251 0.346457
\(552\) −1.24631 −0.0530466
\(553\) 54.6660 2.32464
\(554\) 3.69648 0.157049
\(555\) 0 0
\(556\) −9.27228 −0.393232
\(557\) −4.96462 −0.210358 −0.105179 0.994453i \(-0.533542\pi\)
−0.105179 + 0.994453i \(0.533542\pi\)
\(558\) −2.59484 −0.109849
\(559\) 20.0665 0.848721
\(560\) 0 0
\(561\) 0 0
\(562\) −0.613129 −0.0258633
\(563\) −16.6024 −0.699707 −0.349853 0.936804i \(-0.613769\pi\)
−0.349853 + 0.936804i \(0.613769\pi\)
\(564\) 10.2613 0.432078
\(565\) 0 0
\(566\) 13.5269 0.568578
\(567\) 36.4681 1.53152
\(568\) 14.3371 0.601569
\(569\) 27.0868 1.13554 0.567768 0.823189i \(-0.307807\pi\)
0.567768 + 0.823189i \(0.307807\pi\)
\(570\) 0 0
\(571\) 25.9648 1.08659 0.543296 0.839541i \(-0.317176\pi\)
0.543296 + 0.839541i \(0.317176\pi\)
\(572\) 0 0
\(573\) −32.6282 −1.36306
\(574\) −8.93049 −0.372751
\(575\) 0 0
\(576\) 0.329019 0.0137091
\(577\) −0.746507 −0.0310775 −0.0155388 0.999879i \(-0.504946\pi\)
−0.0155388 + 0.999879i \(0.504946\pi\)
\(578\) −20.8094 −0.865555
\(579\) 19.7833 0.822166
\(580\) 0 0
\(581\) −17.9075 −0.742930
\(582\) 20.4889 0.849291
\(583\) 0 0
\(584\) 58.8721 2.43614
\(585\) 0 0
\(586\) 38.4265 1.58738
\(587\) 17.9002 0.738819 0.369409 0.929267i \(-0.379560\pi\)
0.369409 + 0.929267i \(0.379560\pi\)
\(588\) 60.9544 2.51372
\(589\) 16.4211 0.676620
\(590\) 0 0
\(591\) 28.1819 1.15925
\(592\) −43.2286 −1.77668
\(593\) −11.1992 −0.459895 −0.229948 0.973203i \(-0.573855\pi\)
−0.229948 + 0.973203i \(0.573855\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −48.4941 −1.98640
\(597\) 7.96851 0.326129
\(598\) −1.21849 −0.0498279
\(599\) −1.81590 −0.0741958 −0.0370979 0.999312i \(-0.511811\pi\)
−0.0370979 + 0.999312i \(0.511811\pi\)
\(600\) 0 0
\(601\) −46.8125 −1.90952 −0.954761 0.297373i \(-0.903889\pi\)
−0.954761 + 0.297373i \(0.903889\pi\)
\(602\) 43.4545 1.77107
\(603\) 1.67393 0.0681678
\(604\) 15.6827 0.638121
\(605\) 0 0
\(606\) 40.4464 1.64302
\(607\) −32.8999 −1.33537 −0.667683 0.744446i \(-0.732714\pi\)
−0.667683 + 0.744446i \(0.732714\pi\)
\(608\) −22.9284 −0.929870
\(609\) −16.0121 −0.648843
\(610\) 0 0
\(611\) 5.63817 0.228096
\(612\) −4.88393 −0.197421
\(613\) −14.8860 −0.601240 −0.300620 0.953744i \(-0.597194\pi\)
−0.300620 + 0.953744i \(0.597194\pi\)
\(614\) −5.40768 −0.218236
\(615\) 0 0
\(616\) 0 0
\(617\) 37.4741 1.50865 0.754324 0.656502i \(-0.227965\pi\)
0.754324 + 0.656502i \(0.227965\pi\)
\(618\) 75.6728 3.04401
\(619\) −12.3920 −0.498077 −0.249039 0.968494i \(-0.580115\pi\)
−0.249039 + 0.968494i \(0.580115\pi\)
\(620\) 0 0
\(621\) −0.528133 −0.0211932
\(622\) 42.6937 1.71186
\(623\) 0.657526 0.0263432
\(624\) 62.2265 2.49106
\(625\) 0 0
\(626\) −74.5163 −2.97827
\(627\) 0 0
\(628\) −1.36341 −0.0544061
\(629\) 28.0720 1.11930
\(630\) 0 0
\(631\) 1.54279 0.0614174 0.0307087 0.999528i \(-0.490224\pi\)
0.0307087 + 0.999528i \(0.490224\pi\)
\(632\) 94.5785 3.76213
\(633\) −4.14471 −0.164737
\(634\) −46.9355 −1.86405
\(635\) 0 0
\(636\) −45.0086 −1.78471
\(637\) 33.4920 1.32700
\(638\) 0 0
\(639\) −0.465257 −0.0184053
\(640\) 0 0
\(641\) 31.2172 1.23300 0.616502 0.787353i \(-0.288549\pi\)
0.616502 + 0.787353i \(0.288549\pi\)
\(642\) 78.8810 3.11319
\(643\) 20.3183 0.801275 0.400638 0.916237i \(-0.368789\pi\)
0.400638 + 0.916237i \(0.368789\pi\)
\(644\) −1.83498 −0.0723082
\(645\) 0 0
\(646\) 44.4445 1.74865
\(647\) −22.1687 −0.871541 −0.435771 0.900058i \(-0.643524\pi\)
−0.435771 + 0.900058i \(0.643524\pi\)
\(648\) 63.0940 2.47857
\(649\) 0 0
\(650\) 0 0
\(651\) −32.3316 −1.26717
\(652\) 65.1135 2.55004
\(653\) −11.8719 −0.464583 −0.232292 0.972646i \(-0.574622\pi\)
−0.232292 + 0.972646i \(0.574622\pi\)
\(654\) 82.2507 3.21626
\(655\) 0 0
\(656\) −7.07697 −0.276309
\(657\) −1.91048 −0.0745349
\(658\) 12.2096 0.475981
\(659\) −13.3368 −0.519527 −0.259764 0.965672i \(-0.583645\pi\)
−0.259764 + 0.965672i \(0.583645\pi\)
\(660\) 0 0
\(661\) −44.8817 −1.74570 −0.872848 0.487992i \(-0.837730\pi\)
−0.872848 + 0.487992i \(0.837730\pi\)
\(662\) −55.2467 −2.14722
\(663\) −40.4090 −1.56935
\(664\) −30.9821 −1.20234
\(665\) 0 0
\(666\) 3.06272 0.118678
\(667\) 0.248467 0.00962067
\(668\) −82.1821 −3.17972
\(669\) 18.4588 0.713659
\(670\) 0 0
\(671\) 0 0
\(672\) 45.1438 1.74146
\(673\) −5.88475 −0.226840 −0.113420 0.993547i \(-0.536181\pi\)
−0.113420 + 0.993547i \(0.536181\pi\)
\(674\) −36.0319 −1.38790
\(675\) 0 0
\(676\) 33.0057 1.26945
\(677\) −29.6008 −1.13765 −0.568825 0.822458i \(-0.692602\pi\)
−0.568825 + 0.822458i \(0.692602\pi\)
\(678\) 65.7608 2.52553
\(679\) 16.9536 0.650618
\(680\) 0 0
\(681\) 12.8678 0.493097
\(682\) 0 0
\(683\) 13.1257 0.502239 0.251120 0.967956i \(-0.419201\pi\)
0.251120 + 0.967956i \(0.419201\pi\)
\(684\) 3.37206 0.128934
\(685\) 0 0
\(686\) 4.35043 0.166100
\(687\) 3.69162 0.140844
\(688\) 34.4356 1.31284
\(689\) −24.7304 −0.942153
\(690\) 0 0
\(691\) −7.04276 −0.267919 −0.133960 0.990987i \(-0.542769\pi\)
−0.133960 + 0.990987i \(0.542769\pi\)
\(692\) 39.9709 1.51946
\(693\) 0 0
\(694\) 73.4141 2.78676
\(695\) 0 0
\(696\) −27.7028 −1.05007
\(697\) 4.59568 0.174074
\(698\) −45.9251 −1.73829
\(699\) −31.5255 −1.19240
\(700\) 0 0
\(701\) −44.3634 −1.67558 −0.837791 0.545992i \(-0.816153\pi\)
−0.837791 + 0.545992i \(0.816153\pi\)
\(702\) 57.5698 2.17283
\(703\) −19.3820 −0.731006
\(704\) 0 0
\(705\) 0 0
\(706\) −63.4871 −2.38937
\(707\) 33.4674 1.25867
\(708\) −9.68378 −0.363939
\(709\) −19.5161 −0.732943 −0.366471 0.930429i \(-0.619434\pi\)
−0.366471 + 0.930429i \(0.619434\pi\)
\(710\) 0 0
\(711\) −3.06920 −0.115104
\(712\) 1.13760 0.0426332
\(713\) 0.501703 0.0187889
\(714\) −87.5068 −3.27486
\(715\) 0 0
\(716\) −104.567 −3.90785
\(717\) −33.5016 −1.25114
\(718\) −26.7837 −0.999560
\(719\) 47.4024 1.76781 0.883906 0.467665i \(-0.154905\pi\)
0.883906 + 0.467665i \(0.154905\pi\)
\(720\) 0 0
\(721\) 62.6156 2.33193
\(722\) 18.0007 0.669916
\(723\) −54.1213 −2.01279
\(724\) 78.8614 2.93086
\(725\) 0 0
\(726\) 0 0
\(727\) −37.2483 −1.38146 −0.690731 0.723112i \(-0.742711\pi\)
−0.690731 + 0.723112i \(0.742711\pi\)
\(728\) 112.414 4.16635
\(729\) 24.8159 0.919107
\(730\) 0 0
\(731\) −22.3619 −0.827086
\(732\) 59.9670 2.21644
\(733\) 11.6209 0.429226 0.214613 0.976699i \(-0.431151\pi\)
0.214613 + 0.976699i \(0.431151\pi\)
\(734\) 38.9369 1.43719
\(735\) 0 0
\(736\) −0.700516 −0.0258214
\(737\) 0 0
\(738\) 0.501399 0.0184568
\(739\) −37.9801 −1.39712 −0.698560 0.715551i \(-0.746176\pi\)
−0.698560 + 0.715551i \(0.746176\pi\)
\(740\) 0 0
\(741\) 27.8999 1.02493
\(742\) −53.5544 −1.96605
\(743\) −7.71703 −0.283111 −0.141555 0.989930i \(-0.545210\pi\)
−0.141555 + 0.989930i \(0.545210\pi\)
\(744\) −55.9373 −2.05076
\(745\) 0 0
\(746\) −54.2665 −1.98684
\(747\) 1.00541 0.0367861
\(748\) 0 0
\(749\) 65.2703 2.38492
\(750\) 0 0
\(751\) 16.0280 0.584870 0.292435 0.956285i \(-0.405534\pi\)
0.292435 + 0.956285i \(0.405534\pi\)
\(752\) 9.67552 0.352830
\(753\) −45.7316 −1.66655
\(754\) −27.0844 −0.986356
\(755\) 0 0
\(756\) 86.6964 3.15312
\(757\) 6.37605 0.231742 0.115871 0.993264i \(-0.463034\pi\)
0.115871 + 0.993264i \(0.463034\pi\)
\(758\) 90.1078 3.27286
\(759\) 0 0
\(760\) 0 0
\(761\) 15.8646 0.575092 0.287546 0.957767i \(-0.407161\pi\)
0.287546 + 0.957767i \(0.407161\pi\)
\(762\) −25.5856 −0.926870
\(763\) 68.0585 2.46388
\(764\) −83.1135 −3.00694
\(765\) 0 0
\(766\) 5.09895 0.184233
\(767\) −5.32085 −0.192125
\(768\) −48.2496 −1.74106
\(769\) 51.7503 1.86616 0.933082 0.359663i \(-0.117108\pi\)
0.933082 + 0.359663i \(0.117108\pi\)
\(770\) 0 0
\(771\) 42.9178 1.54565
\(772\) 50.3937 1.81371
\(773\) −9.68605 −0.348383 −0.174192 0.984712i \(-0.555731\pi\)
−0.174192 + 0.984712i \(0.555731\pi\)
\(774\) −2.43974 −0.0876946
\(775\) 0 0
\(776\) 29.3316 1.05294
\(777\) 38.1612 1.36903
\(778\) 15.5247 0.556588
\(779\) −3.17304 −0.113686
\(780\) 0 0
\(781\) 0 0
\(782\) 1.35788 0.0485578
\(783\) −11.7392 −0.419526
\(784\) 57.4748 2.05267
\(785\) 0 0
\(786\) −63.5584 −2.26705
\(787\) −18.2226 −0.649566 −0.324783 0.945789i \(-0.605291\pi\)
−0.324783 + 0.945789i \(0.605291\pi\)
\(788\) 71.7874 2.55732
\(789\) −47.9871 −1.70839
\(790\) 0 0
\(791\) 54.4139 1.93473
\(792\) 0 0
\(793\) 32.9495 1.17007
\(794\) −89.8564 −3.18889
\(795\) 0 0
\(796\) 20.2981 0.719446
\(797\) 20.0132 0.708902 0.354451 0.935075i \(-0.384668\pi\)
0.354451 + 0.935075i \(0.384668\pi\)
\(798\) 60.4182 2.13878
\(799\) −6.28314 −0.222281
\(800\) 0 0
\(801\) −0.0369165 −0.00130438
\(802\) −42.2160 −1.49070
\(803\) 0 0
\(804\) 64.2079 2.26444
\(805\) 0 0
\(806\) −54.6887 −1.92633
\(807\) −10.6944 −0.376459
\(808\) 57.9025 2.03700
\(809\) 31.8062 1.11825 0.559123 0.829085i \(-0.311138\pi\)
0.559123 + 0.829085i \(0.311138\pi\)
\(810\) 0 0
\(811\) −3.65484 −0.128339 −0.0641694 0.997939i \(-0.520440\pi\)
−0.0641694 + 0.997939i \(0.520440\pi\)
\(812\) −40.7874 −1.43136
\(813\) 13.4331 0.471119
\(814\) 0 0
\(815\) 0 0
\(816\) −69.3448 −2.42755
\(817\) 15.4396 0.540162
\(818\) 11.0451 0.386184
\(819\) −3.64800 −0.127471
\(820\) 0 0
\(821\) −13.7678 −0.480501 −0.240250 0.970711i \(-0.577230\pi\)
−0.240250 + 0.970711i \(0.577230\pi\)
\(822\) 45.6797 1.59326
\(823\) 40.5760 1.41439 0.707195 0.707018i \(-0.249960\pi\)
0.707195 + 0.707018i \(0.249960\pi\)
\(824\) 108.332 3.77393
\(825\) 0 0
\(826\) −11.5225 −0.400918
\(827\) 15.5321 0.540105 0.270052 0.962846i \(-0.412959\pi\)
0.270052 + 0.962846i \(0.412959\pi\)
\(828\) 0.103024 0.00358033
\(829\) −29.1311 −1.01177 −0.505883 0.862602i \(-0.668833\pi\)
−0.505883 + 0.862602i \(0.668833\pi\)
\(830\) 0 0
\(831\) −2.58590 −0.0897040
\(832\) 6.93437 0.240406
\(833\) −37.3232 −1.29317
\(834\) 9.32755 0.322987
\(835\) 0 0
\(836\) 0 0
\(837\) −23.7038 −0.819322
\(838\) 47.9473 1.65631
\(839\) 21.7832 0.752041 0.376020 0.926611i \(-0.377292\pi\)
0.376020 + 0.926611i \(0.377292\pi\)
\(840\) 0 0
\(841\) −23.4771 −0.809556
\(842\) 78.1753 2.69410
\(843\) 0.428919 0.0147728
\(844\) −10.5578 −0.363413
\(845\) 0 0
\(846\) −0.685505 −0.0235681
\(847\) 0 0
\(848\) −42.4392 −1.45737
\(849\) −9.46285 −0.324764
\(850\) 0 0
\(851\) −0.592165 −0.0202991
\(852\) −17.8461 −0.611398
\(853\) 8.75293 0.299695 0.149847 0.988709i \(-0.452122\pi\)
0.149847 + 0.988709i \(0.452122\pi\)
\(854\) 71.3531 2.44165
\(855\) 0 0
\(856\) 112.925 3.85970
\(857\) 14.0273 0.479163 0.239582 0.970876i \(-0.422990\pi\)
0.239582 + 0.970876i \(0.422990\pi\)
\(858\) 0 0
\(859\) 31.8774 1.08764 0.543822 0.839201i \(-0.316977\pi\)
0.543822 + 0.839201i \(0.316977\pi\)
\(860\) 0 0
\(861\) 6.24739 0.212911
\(862\) 21.3028 0.725577
\(863\) −3.46523 −0.117958 −0.0589789 0.998259i \(-0.518784\pi\)
−0.0589789 + 0.998259i \(0.518784\pi\)
\(864\) 33.0970 1.12598
\(865\) 0 0
\(866\) 70.3390 2.39022
\(867\) 14.5573 0.494393
\(868\) −82.3577 −2.79540
\(869\) 0 0
\(870\) 0 0
\(871\) 35.2796 1.19540
\(872\) 117.749 3.98749
\(873\) −0.951851 −0.0322153
\(874\) −0.937535 −0.0317126
\(875\) 0 0
\(876\) −73.2813 −2.47594
\(877\) 19.7763 0.667799 0.333899 0.942609i \(-0.391635\pi\)
0.333899 + 0.942609i \(0.391635\pi\)
\(878\) −59.3282 −2.00223
\(879\) −26.8816 −0.906692
\(880\) 0 0
\(881\) 57.2097 1.92744 0.963722 0.266910i \(-0.0860025\pi\)
0.963722 + 0.266910i \(0.0860025\pi\)
\(882\) −4.07205 −0.137113
\(883\) −57.2412 −1.92632 −0.963159 0.268931i \(-0.913329\pi\)
−0.963159 + 0.268931i \(0.913329\pi\)
\(884\) −102.933 −3.46202
\(885\) 0 0
\(886\) 51.7196 1.73756
\(887\) −39.7251 −1.33384 −0.666920 0.745129i \(-0.732388\pi\)
−0.666920 + 0.745129i \(0.732388\pi\)
\(888\) 66.0233 2.21560
\(889\) −21.1709 −0.710049
\(890\) 0 0
\(891\) 0 0
\(892\) 47.0199 1.57434
\(893\) 4.33812 0.145170
\(894\) 48.7832 1.63155
\(895\) 0 0
\(896\) −35.3504 −1.18097
\(897\) 0.852407 0.0284610
\(898\) −35.9621 −1.20007
\(899\) 11.1517 0.371931
\(900\) 0 0
\(901\) 27.5594 0.918136
\(902\) 0 0
\(903\) −30.3989 −1.01161
\(904\) 94.1422 3.13112
\(905\) 0 0
\(906\) −15.7762 −0.524129
\(907\) −26.6119 −0.883635 −0.441817 0.897105i \(-0.645666\pi\)
−0.441817 + 0.897105i \(0.645666\pi\)
\(908\) 32.7781 1.08778
\(909\) −1.87902 −0.0623230
\(910\) 0 0
\(911\) 7.78778 0.258021 0.129010 0.991643i \(-0.458820\pi\)
0.129010 + 0.991643i \(0.458820\pi\)
\(912\) 47.8784 1.58541
\(913\) 0 0
\(914\) −40.0351 −1.32424
\(915\) 0 0
\(916\) 9.40361 0.310704
\(917\) −52.5915 −1.73673
\(918\) −64.1554 −2.11744
\(919\) 15.9781 0.527069 0.263535 0.964650i \(-0.415112\pi\)
0.263535 + 0.964650i \(0.415112\pi\)
\(920\) 0 0
\(921\) 3.78298 0.124654
\(922\) 75.8702 2.49865
\(923\) −9.80572 −0.322759
\(924\) 0 0
\(925\) 0 0
\(926\) −3.37295 −0.110842
\(927\) −3.51553 −0.115465
\(928\) −15.5709 −0.511140
\(929\) 9.92533 0.325639 0.162820 0.986656i \(-0.447941\pi\)
0.162820 + 0.986656i \(0.447941\pi\)
\(930\) 0 0
\(931\) 25.7694 0.844559
\(932\) −80.3045 −2.63046
\(933\) −29.8667 −0.977793
\(934\) −31.0046 −1.01450
\(935\) 0 0
\(936\) −6.31145 −0.206296
\(937\) 8.91808 0.291341 0.145671 0.989333i \(-0.453466\pi\)
0.145671 + 0.989333i \(0.453466\pi\)
\(938\) 76.3991 2.49452
\(939\) 52.1284 1.70115
\(940\) 0 0
\(941\) −37.3215 −1.21665 −0.608323 0.793690i \(-0.708157\pi\)
−0.608323 + 0.793690i \(0.708157\pi\)
\(942\) 1.37154 0.0446872
\(943\) −0.0969435 −0.00315691
\(944\) −9.13098 −0.297188
\(945\) 0 0
\(946\) 0 0
\(947\) −1.62118 −0.0526814 −0.0263407 0.999653i \(-0.508385\pi\)
−0.0263407 + 0.999653i \(0.508385\pi\)
\(948\) −117.727 −3.82359
\(949\) −40.2651 −1.30706
\(950\) 0 0
\(951\) 32.8341 1.06472
\(952\) −125.274 −4.06014
\(953\) −25.3420 −0.820907 −0.410454 0.911881i \(-0.634630\pi\)
−0.410454 + 0.911881i \(0.634630\pi\)
\(954\) 3.00679 0.0973486
\(955\) 0 0
\(956\) −85.3383 −2.76004
\(957\) 0 0
\(958\) 7.82578 0.252839
\(959\) 37.7977 1.22055
\(960\) 0 0
\(961\) −8.48248 −0.273628
\(962\) 64.5496 2.08116
\(963\) −3.66457 −0.118089
\(964\) −137.863 −4.44025
\(965\) 0 0
\(966\) 1.84591 0.0593913
\(967\) 6.70594 0.215648 0.107824 0.994170i \(-0.465612\pi\)
0.107824 + 0.994170i \(0.465612\pi\)
\(968\) 0 0
\(969\) −31.0915 −0.998803
\(970\) 0 0
\(971\) 5.81959 0.186760 0.0933798 0.995631i \(-0.470233\pi\)
0.0933798 + 0.995631i \(0.470233\pi\)
\(972\) −10.1078 −0.324209
\(973\) 7.71810 0.247431
\(974\) 62.1137 1.99025
\(975\) 0 0
\(976\) 56.5438 1.80992
\(977\) 5.23201 0.167387 0.0836934 0.996492i \(-0.473328\pi\)
0.0836934 + 0.996492i \(0.473328\pi\)
\(978\) −65.5017 −2.09451
\(979\) 0 0
\(980\) 0 0
\(981\) −3.82112 −0.121999
\(982\) −83.7404 −2.67227
\(983\) −14.7913 −0.471768 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(984\) 10.8087 0.344569
\(985\) 0 0
\(986\) 30.1827 0.961213
\(987\) −8.54134 −0.271874
\(988\) 71.0692 2.26101
\(989\) 0.471714 0.0149996
\(990\) 0 0
\(991\) 24.9189 0.791575 0.395788 0.918342i \(-0.370472\pi\)
0.395788 + 0.918342i \(0.370472\pi\)
\(992\) −31.4407 −0.998243
\(993\) 38.6482 1.22646
\(994\) −21.2346 −0.673520
\(995\) 0 0
\(996\) 38.5651 1.22198
\(997\) −14.8384 −0.469937 −0.234969 0.972003i \(-0.575499\pi\)
−0.234969 + 0.972003i \(0.575499\pi\)
\(998\) 76.8893 2.43389
\(999\) 27.9778 0.885178
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 3025.2.a.bj.1.2 8
5.4 even 2 3025.2.a.bm.1.7 8
11.2 odd 10 275.2.h.e.26.1 yes 16
11.6 odd 10 275.2.h.e.201.1 yes 16
11.10 odd 2 3025.2.a.bn.1.7 8
55.2 even 20 275.2.z.c.224.8 32
55.13 even 20 275.2.z.c.224.1 32
55.17 even 20 275.2.z.c.124.1 32
55.24 odd 10 275.2.h.c.26.4 16
55.28 even 20 275.2.z.c.124.8 32
55.39 odd 10 275.2.h.c.201.4 yes 16
55.54 odd 2 3025.2.a.bi.1.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.h.c.26.4 16 55.24 odd 10
275.2.h.c.201.4 yes 16 55.39 odd 10
275.2.h.e.26.1 yes 16 11.2 odd 10
275.2.h.e.201.1 yes 16 11.6 odd 10
275.2.z.c.124.1 32 55.17 even 20
275.2.z.c.124.8 32 55.28 even 20
275.2.z.c.224.1 32 55.13 even 20
275.2.z.c.224.8 32 55.2 even 20
3025.2.a.bi.1.2 8 55.54 odd 2
3025.2.a.bj.1.2 8 1.1 even 1 trivial
3025.2.a.bm.1.7 8 5.4 even 2
3025.2.a.bn.1.7 8 11.10 odd 2