Properties

Label 241.2.a.b.1.9
Level $241$
Weight $2$
Character 241.1
Self dual yes
Analytic conductor $1.924$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [241,2,Mod(1,241)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(241, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("241.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 241.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: \(1.92439468871\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} - 328 x^{3} - 45 x^{2} + 18 x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(1.63125\) of defining polynomial
Character \(\chi\) \(=\) 241.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.63125 q^{2} -1.16790 q^{3} +0.660992 q^{4} +1.75438 q^{5} -1.90514 q^{6} +5.06139 q^{7} -2.18426 q^{8} -1.63601 q^{9} +O(q^{10})\) \(q+1.63125 q^{2} -1.16790 q^{3} +0.660992 q^{4} +1.75438 q^{5} -1.90514 q^{6} +5.06139 q^{7} -2.18426 q^{8} -1.63601 q^{9} +2.86183 q^{10} +1.08118 q^{11} -0.771973 q^{12} +3.01110 q^{13} +8.25641 q^{14} -2.04893 q^{15} -4.88507 q^{16} +2.47710 q^{17} -2.66875 q^{18} -7.12459 q^{19} +1.15963 q^{20} -5.91119 q^{21} +1.76369 q^{22} +5.33139 q^{23} +2.55100 q^{24} -1.92217 q^{25} +4.91187 q^{26} +5.41439 q^{27} +3.34554 q^{28} -6.80248 q^{29} -3.34233 q^{30} -8.37871 q^{31} -3.60028 q^{32} -1.26271 q^{33} +4.04078 q^{34} +8.87957 q^{35} -1.08139 q^{36} -2.09398 q^{37} -11.6220 q^{38} -3.51666 q^{39} -3.83202 q^{40} -10.6564 q^{41} -9.64266 q^{42} -5.49791 q^{43} +0.714654 q^{44} -2.87018 q^{45} +8.69686 q^{46} +4.90386 q^{47} +5.70528 q^{48} +18.6176 q^{49} -3.13555 q^{50} -2.89301 q^{51} +1.99032 q^{52} -4.04410 q^{53} +8.83226 q^{54} +1.89680 q^{55} -11.0554 q^{56} +8.32081 q^{57} -11.0966 q^{58} +5.64831 q^{59} -1.35433 q^{60} -2.03784 q^{61} -13.6678 q^{62} -8.28048 q^{63} +3.89718 q^{64} +5.28260 q^{65} -2.05981 q^{66} +7.65097 q^{67} +1.63735 q^{68} -6.22653 q^{69} +14.4848 q^{70} +12.2631 q^{71} +3.57348 q^{72} +0.733715 q^{73} -3.41582 q^{74} +2.24490 q^{75} -4.70930 q^{76} +5.47229 q^{77} -5.73657 q^{78} +6.86216 q^{79} -8.57025 q^{80} -1.41544 q^{81} -17.3833 q^{82} +5.58812 q^{83} -3.90725 q^{84} +4.34576 q^{85} -8.96849 q^{86} +7.94461 q^{87} -2.36159 q^{88} +9.62798 q^{89} -4.68199 q^{90} +15.2403 q^{91} +3.52401 q^{92} +9.78549 q^{93} +7.99945 q^{94} -12.4992 q^{95} +4.20476 q^{96} -10.9185 q^{97} +30.3701 q^{98} -1.76883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 3 q^{2} + q^{3} + 13 q^{4} + 6 q^{5} - q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 3 q^{2} + q^{3} + 13 q^{4} + 6 q^{5} - q^{6} + 3 q^{7} + 9 q^{8} + 15 q^{9} - 7 q^{10} + 22 q^{11} - 7 q^{12} - 5 q^{13} + 6 q^{14} + 13 q^{15} + 15 q^{16} - 4 q^{17} - q^{18} - 6 q^{19} + 10 q^{20} - 14 q^{21} - 12 q^{22} + 32 q^{23} - 15 q^{24} + 4 q^{25} + 8 q^{26} - 5 q^{27} - 11 q^{28} + 6 q^{29} - 19 q^{30} + 8 q^{31} + q^{32} - 24 q^{33} - 19 q^{34} + 15 q^{35} - 8 q^{36} - 8 q^{37} - 10 q^{38} + 31 q^{39} - 52 q^{40} - q^{41} - 49 q^{42} - 2 q^{43} + 42 q^{44} - 15 q^{45} - 25 q^{46} + 34 q^{47} - 49 q^{48} - 9 q^{49} - 27 q^{50} - 3 q^{51} - 41 q^{52} + 5 q^{53} - 40 q^{54} - 3 q^{55} + q^{56} - 22 q^{57} - 33 q^{58} + 26 q^{59} - 57 q^{60} - 26 q^{61} - 17 q^{62} - 4 q^{63} + 13 q^{64} - 25 q^{65} - 2 q^{66} + 6 q^{67} - 35 q^{68} - 2 q^{69} - 4 q^{70} + 94 q^{71} + 17 q^{72} - 22 q^{73} + 26 q^{74} - 20 q^{76} - 7 q^{77} + 54 q^{78} + 9 q^{79} + 19 q^{80} + 4 q^{81} + 15 q^{82} - 8 q^{83} + 2 q^{84} + 4 q^{85} + 9 q^{86} + 4 q^{87} + 6 q^{88} - 3 q^{89} + 11 q^{90} - 20 q^{91} + 36 q^{92} + 12 q^{93} + 48 q^{94} + 33 q^{95} - 23 q^{96} - 29 q^{97} + 28 q^{98} + 36 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.63125 1.15347 0.576736 0.816931i \(-0.304326\pi\)
0.576736 + 0.816931i \(0.304326\pi\)
\(3\) −1.16790 −0.674287 −0.337144 0.941453i \(-0.609461\pi\)
−0.337144 + 0.941453i \(0.609461\pi\)
\(4\) 0.660992 0.330496
\(5\) 1.75438 0.784580 0.392290 0.919842i \(-0.371683\pi\)
0.392290 + 0.919842i \(0.371683\pi\)
\(6\) −1.90514 −0.777771
\(7\) 5.06139 1.91302 0.956512 0.291693i \(-0.0942184\pi\)
0.956512 + 0.291693i \(0.0942184\pi\)
\(8\) −2.18426 −0.772253
\(9\) −1.63601 −0.545337
\(10\) 2.86183 0.904991
\(11\) 1.08118 0.325989 0.162995 0.986627i \(-0.447885\pi\)
0.162995 + 0.986627i \(0.447885\pi\)
\(12\) −0.771973 −0.222849
\(13\) 3.01110 0.835129 0.417565 0.908647i \(-0.362884\pi\)
0.417565 + 0.908647i \(0.362884\pi\)
\(14\) 8.25641 2.20662
\(15\) −2.04893 −0.529032
\(16\) −4.88507 −1.22127
\(17\) 2.47710 0.600785 0.300393 0.953816i \(-0.402882\pi\)
0.300393 + 0.953816i \(0.402882\pi\)
\(18\) −2.66875 −0.629030
\(19\) −7.12459 −1.63449 −0.817247 0.576287i \(-0.804501\pi\)
−0.817247 + 0.576287i \(0.804501\pi\)
\(20\) 1.15963 0.259301
\(21\) −5.91119 −1.28993
\(22\) 1.76369 0.376019
\(23\) 5.33139 1.11167 0.555836 0.831292i \(-0.312398\pi\)
0.555836 + 0.831292i \(0.312398\pi\)
\(24\) 2.55100 0.520721
\(25\) −1.92217 −0.384434
\(26\) 4.91187 0.963298
\(27\) 5.41439 1.04200
\(28\) 3.34554 0.632247
\(29\) −6.80248 −1.26319 −0.631595 0.775299i \(-0.717599\pi\)
−0.631595 + 0.775299i \(0.717599\pi\)
\(30\) −3.34233 −0.610224
\(31\) −8.37871 −1.50486 −0.752430 0.658672i \(-0.771119\pi\)
−0.752430 + 0.658672i \(0.771119\pi\)
\(32\) −3.60028 −0.636445
\(33\) −1.26271 −0.219810
\(34\) 4.04078 0.692989
\(35\) 8.87957 1.50092
\(36\) −1.08139 −0.180232
\(37\) −2.09398 −0.344249 −0.172124 0.985075i \(-0.555063\pi\)
−0.172124 + 0.985075i \(0.555063\pi\)
\(38\) −11.6220 −1.88534
\(39\) −3.51666 −0.563117
\(40\) −3.83202 −0.605895
\(41\) −10.6564 −1.66425 −0.832124 0.554589i \(-0.812875\pi\)
−0.832124 + 0.554589i \(0.812875\pi\)
\(42\) −9.64266 −1.48789
\(43\) −5.49791 −0.838423 −0.419211 0.907889i \(-0.637693\pi\)
−0.419211 + 0.907889i \(0.637693\pi\)
\(44\) 0.714654 0.107738
\(45\) −2.87018 −0.427861
\(46\) 8.69686 1.28228
\(47\) 4.90386 0.715302 0.357651 0.933855i \(-0.383578\pi\)
0.357651 + 0.933855i \(0.383578\pi\)
\(48\) 5.70528 0.823486
\(49\) 18.6176 2.65966
\(50\) −3.13555 −0.443433
\(51\) −2.89301 −0.405102
\(52\) 1.99032 0.276007
\(53\) −4.04410 −0.555500 −0.277750 0.960653i \(-0.589589\pi\)
−0.277750 + 0.960653i \(0.589589\pi\)
\(54\) 8.83226 1.20192
\(55\) 1.89680 0.255765
\(56\) −11.0554 −1.47734
\(57\) 8.32081 1.10212
\(58\) −11.0966 −1.45705
\(59\) 5.64831 0.735347 0.367673 0.929955i \(-0.380154\pi\)
0.367673 + 0.929955i \(0.380154\pi\)
\(60\) −1.35433 −0.174843
\(61\) −2.03784 −0.260919 −0.130459 0.991454i \(-0.541645\pi\)
−0.130459 + 0.991454i \(0.541645\pi\)
\(62\) −13.6678 −1.73581
\(63\) −8.28048 −1.04324
\(64\) 3.89718 0.487148
\(65\) 5.28260 0.655226
\(66\) −2.05981 −0.253545
\(67\) 7.65097 0.934715 0.467358 0.884068i \(-0.345206\pi\)
0.467358 + 0.884068i \(0.345206\pi\)
\(68\) 1.63735 0.198557
\(69\) −6.22653 −0.749586
\(70\) 14.4848 1.73127
\(71\) 12.2631 1.45536 0.727682 0.685915i \(-0.240598\pi\)
0.727682 + 0.685915i \(0.240598\pi\)
\(72\) 3.57348 0.421138
\(73\) 0.733715 0.0858748 0.0429374 0.999078i \(-0.486328\pi\)
0.0429374 + 0.999078i \(0.486328\pi\)
\(74\) −3.41582 −0.397081
\(75\) 2.24490 0.259219
\(76\) −4.70930 −0.540194
\(77\) 5.47229 0.623625
\(78\) −5.73657 −0.649539
\(79\) 6.86216 0.772053 0.386027 0.922488i \(-0.373847\pi\)
0.386027 + 0.922488i \(0.373847\pi\)
\(80\) −8.57025 −0.958183
\(81\) −1.41544 −0.157271
\(82\) −17.3833 −1.91966
\(83\) 5.58812 0.613376 0.306688 0.951810i \(-0.400779\pi\)
0.306688 + 0.951810i \(0.400779\pi\)
\(84\) −3.90725 −0.426316
\(85\) 4.34576 0.471364
\(86\) −8.96849 −0.967097
\(87\) 7.94461 0.851752
\(88\) −2.36159 −0.251746
\(89\) 9.62798 1.02056 0.510282 0.860007i \(-0.329541\pi\)
0.510282 + 0.860007i \(0.329541\pi\)
\(90\) −4.68199 −0.493525
\(91\) 15.2403 1.59762
\(92\) 3.52401 0.367403
\(93\) 9.78549 1.01471
\(94\) 7.99945 0.825080
\(95\) −12.4992 −1.28239
\(96\) 4.20476 0.429147
\(97\) −10.9185 −1.10861 −0.554303 0.832315i \(-0.687015\pi\)
−0.554303 + 0.832315i \(0.687015\pi\)
\(98\) 30.3701 3.06784
\(99\) −1.76883 −0.177774
\(100\) −1.27054 −0.127054
\(101\) −13.9401 −1.38709 −0.693545 0.720413i \(-0.743952\pi\)
−0.693545 + 0.720413i \(0.743952\pi\)
\(102\) −4.71923 −0.467273
\(103\) −7.14621 −0.704137 −0.352068 0.935974i \(-0.614522\pi\)
−0.352068 + 0.935974i \(0.614522\pi\)
\(104\) −6.57703 −0.644931
\(105\) −10.3704 −1.01205
\(106\) −6.59695 −0.640753
\(107\) 17.8929 1.72977 0.864886 0.501968i \(-0.167390\pi\)
0.864886 + 0.501968i \(0.167390\pi\)
\(108\) 3.57887 0.344377
\(109\) −14.9297 −1.43001 −0.715004 0.699120i \(-0.753575\pi\)
−0.715004 + 0.699120i \(0.753575\pi\)
\(110\) 3.09417 0.295017
\(111\) 2.44556 0.232123
\(112\) −24.7252 −2.33632
\(113\) 0.859306 0.0808367 0.0404184 0.999183i \(-0.487131\pi\)
0.0404184 + 0.999183i \(0.487131\pi\)
\(114\) 13.5734 1.27126
\(115\) 9.35326 0.872196
\(116\) −4.49639 −0.417479
\(117\) −4.92619 −0.455427
\(118\) 9.21383 0.848201
\(119\) 12.5376 1.14932
\(120\) 4.47541 0.408547
\(121\) −9.83104 −0.893731
\(122\) −3.32424 −0.300962
\(123\) 12.4456 1.12218
\(124\) −5.53826 −0.497351
\(125\) −12.1441 −1.08620
\(126\) −13.5076 −1.20335
\(127\) −3.01886 −0.267880 −0.133940 0.990989i \(-0.542763\pi\)
−0.133940 + 0.990989i \(0.542763\pi\)
\(128\) 13.5578 1.19836
\(129\) 6.42100 0.565338
\(130\) 8.61727 0.755784
\(131\) −2.17519 −0.190047 −0.0950234 0.995475i \(-0.530293\pi\)
−0.0950234 + 0.995475i \(0.530293\pi\)
\(132\) −0.834644 −0.0726464
\(133\) −36.0603 −3.12683
\(134\) 12.4807 1.07817
\(135\) 9.49888 0.817533
\(136\) −5.41064 −0.463958
\(137\) 15.9150 1.35971 0.679854 0.733347i \(-0.262043\pi\)
0.679854 + 0.733347i \(0.262043\pi\)
\(138\) −10.1571 −0.864626
\(139\) −15.0721 −1.27840 −0.639201 0.769040i \(-0.720735\pi\)
−0.639201 + 0.769040i \(0.720735\pi\)
\(140\) 5.86933 0.496049
\(141\) −5.72722 −0.482319
\(142\) 20.0043 1.67872
\(143\) 3.25555 0.272243
\(144\) 7.99203 0.666003
\(145\) −11.9341 −0.991073
\(146\) 1.19688 0.0990542
\(147\) −21.7435 −1.79338
\(148\) −1.38411 −0.113773
\(149\) −1.36774 −0.112050 −0.0560250 0.998429i \(-0.517843\pi\)
−0.0560250 + 0.998429i \(0.517843\pi\)
\(150\) 3.66200 0.299001
\(151\) 17.1854 1.39853 0.699264 0.714864i \(-0.253511\pi\)
0.699264 + 0.714864i \(0.253511\pi\)
\(152\) 15.5620 1.26224
\(153\) −4.05256 −0.327630
\(154\) 8.92669 0.719334
\(155\) −14.6994 −1.18068
\(156\) −2.32449 −0.186108
\(157\) 12.1279 0.967909 0.483955 0.875093i \(-0.339200\pi\)
0.483955 + 0.875093i \(0.339200\pi\)
\(158\) 11.1939 0.890541
\(159\) 4.72310 0.374566
\(160\) −6.31623 −0.499342
\(161\) 26.9842 2.12665
\(162\) −2.30894 −0.181407
\(163\) 10.7083 0.838737 0.419369 0.907816i \(-0.362251\pi\)
0.419369 + 0.907816i \(0.362251\pi\)
\(164\) −7.04379 −0.550028
\(165\) −2.21527 −0.172459
\(166\) 9.11564 0.707511
\(167\) −10.5539 −0.816684 −0.408342 0.912829i \(-0.633893\pi\)
−0.408342 + 0.912829i \(0.633893\pi\)
\(168\) 12.9116 0.996151
\(169\) −3.93327 −0.302559
\(170\) 7.08905 0.543705
\(171\) 11.6559 0.891350
\(172\) −3.63407 −0.277096
\(173\) −2.07621 −0.157851 −0.0789255 0.996881i \(-0.525149\pi\)
−0.0789255 + 0.996881i \(0.525149\pi\)
\(174\) 12.9597 0.982472
\(175\) −9.72883 −0.735431
\(176\) −5.28166 −0.398120
\(177\) −6.59665 −0.495835
\(178\) 15.7057 1.17719
\(179\) −3.06922 −0.229405 −0.114702 0.993400i \(-0.536591\pi\)
−0.114702 + 0.993400i \(0.536591\pi\)
\(180\) −1.89717 −0.141406
\(181\) −0.391403 −0.0290927 −0.0145464 0.999894i \(-0.504630\pi\)
−0.0145464 + 0.999894i \(0.504630\pi\)
\(182\) 24.8609 1.84281
\(183\) 2.37999 0.175934
\(184\) −11.6452 −0.858492
\(185\) −3.67363 −0.270091
\(186\) 15.9626 1.17044
\(187\) 2.67820 0.195849
\(188\) 3.24142 0.236405
\(189\) 27.4043 1.99337
\(190\) −20.3894 −1.47920
\(191\) 4.99018 0.361077 0.180538 0.983568i \(-0.442216\pi\)
0.180538 + 0.983568i \(0.442216\pi\)
\(192\) −4.55152 −0.328477
\(193\) 13.5209 0.973254 0.486627 0.873610i \(-0.338227\pi\)
0.486627 + 0.873610i \(0.338227\pi\)
\(194\) −17.8109 −1.27874
\(195\) −6.16955 −0.441810
\(196\) 12.3061 0.879008
\(197\) 5.86751 0.418043 0.209021 0.977911i \(-0.432972\pi\)
0.209021 + 0.977911i \(0.432972\pi\)
\(198\) −2.88541 −0.205057
\(199\) 12.5774 0.891585 0.445792 0.895136i \(-0.352922\pi\)
0.445792 + 0.895136i \(0.352922\pi\)
\(200\) 4.19852 0.296880
\(201\) −8.93557 −0.630266
\(202\) −22.7398 −1.59997
\(203\) −34.4300 −2.41651
\(204\) −1.91225 −0.133885
\(205\) −18.6953 −1.30574
\(206\) −11.6573 −0.812201
\(207\) −8.72221 −0.606236
\(208\) −14.7095 −1.01992
\(209\) −7.70299 −0.532827
\(210\) −16.9168 −1.16737
\(211\) 6.00657 0.413509 0.206755 0.978393i \(-0.433710\pi\)
0.206755 + 0.978393i \(0.433710\pi\)
\(212\) −2.67312 −0.183591
\(213\) −14.3221 −0.981333
\(214\) 29.1879 1.99524
\(215\) −9.64539 −0.657810
\(216\) −11.8265 −0.804689
\(217\) −42.4079 −2.87884
\(218\) −24.3542 −1.64947
\(219\) −0.856905 −0.0579043
\(220\) 1.25377 0.0845292
\(221\) 7.45880 0.501733
\(222\) 3.98934 0.267747
\(223\) −25.7624 −1.72518 −0.862590 0.505904i \(-0.831159\pi\)
−0.862590 + 0.505904i \(0.831159\pi\)
\(224\) −18.2224 −1.21753
\(225\) 3.14469 0.209646
\(226\) 1.40175 0.0932429
\(227\) 5.03720 0.334331 0.167165 0.985929i \(-0.446539\pi\)
0.167165 + 0.985929i \(0.446539\pi\)
\(228\) 5.49999 0.364246
\(229\) 21.1245 1.39594 0.697972 0.716125i \(-0.254086\pi\)
0.697972 + 0.716125i \(0.254086\pi\)
\(230\) 15.2575 1.00605
\(231\) −6.39108 −0.420502
\(232\) 14.8584 0.975502
\(233\) 20.9468 1.37227 0.686135 0.727474i \(-0.259306\pi\)
0.686135 + 0.727474i \(0.259306\pi\)
\(234\) −8.03588 −0.525322
\(235\) 8.60322 0.561212
\(236\) 3.73349 0.243029
\(237\) −8.01431 −0.520586
\(238\) 20.4520 1.32570
\(239\) −7.83541 −0.506831 −0.253415 0.967358i \(-0.581554\pi\)
−0.253415 + 0.967358i \(0.581554\pi\)
\(240\) 10.0092 0.646091
\(241\) 1.00000 0.0644157
\(242\) −16.0369 −1.03089
\(243\) −14.5901 −0.935955
\(244\) −1.34700 −0.0862326
\(245\) 32.6623 2.08672
\(246\) 20.3019 1.29440
\(247\) −21.4529 −1.36501
\(248\) 18.3013 1.16213
\(249\) −6.52636 −0.413591
\(250\) −19.8101 −1.25290
\(251\) −7.01648 −0.442876 −0.221438 0.975174i \(-0.571075\pi\)
−0.221438 + 0.975174i \(0.571075\pi\)
\(252\) −5.47334 −0.344788
\(253\) 5.76421 0.362393
\(254\) −4.92453 −0.308992
\(255\) −5.07542 −0.317835
\(256\) 14.3219 0.895121
\(257\) 11.3057 0.705229 0.352615 0.935769i \(-0.385293\pi\)
0.352615 + 0.935769i \(0.385293\pi\)
\(258\) 10.4743 0.652101
\(259\) −10.5985 −0.658556
\(260\) 3.49176 0.216550
\(261\) 11.1289 0.688864
\(262\) −3.54828 −0.219214
\(263\) 19.8667 1.22503 0.612516 0.790458i \(-0.290158\pi\)
0.612516 + 0.790458i \(0.290158\pi\)
\(264\) 2.75810 0.169749
\(265\) −7.09487 −0.435834
\(266\) −58.8236 −3.60671
\(267\) −11.2445 −0.688153
\(268\) 5.05724 0.308920
\(269\) −20.2533 −1.23486 −0.617431 0.786625i \(-0.711827\pi\)
−0.617431 + 0.786625i \(0.711827\pi\)
\(270\) 15.4951 0.943001
\(271\) −4.23892 −0.257496 −0.128748 0.991677i \(-0.541096\pi\)
−0.128748 + 0.991677i \(0.541096\pi\)
\(272\) −12.1008 −0.733720
\(273\) −17.7992 −1.07726
\(274\) 25.9614 1.56839
\(275\) −2.07822 −0.125321
\(276\) −4.11569 −0.247735
\(277\) −0.369469 −0.0221992 −0.0110996 0.999938i \(-0.503533\pi\)
−0.0110996 + 0.999938i \(0.503533\pi\)
\(278\) −24.5865 −1.47460
\(279\) 13.7077 0.820656
\(280\) −19.3953 −1.15909
\(281\) 24.6918 1.47299 0.736494 0.676444i \(-0.236480\pi\)
0.736494 + 0.676444i \(0.236480\pi\)
\(282\) −9.34255 −0.556341
\(283\) 0.0696775 0.00414190 0.00207095 0.999998i \(-0.499341\pi\)
0.00207095 + 0.999998i \(0.499341\pi\)
\(284\) 8.10583 0.480992
\(285\) 14.5978 0.864700
\(286\) 5.31064 0.314024
\(287\) −53.9361 −3.18375
\(288\) 5.89009 0.347077
\(289\) −10.8640 −0.639057
\(290\) −19.4676 −1.14317
\(291\) 12.7517 0.747518
\(292\) 0.484980 0.0283813
\(293\) −7.59140 −0.443494 −0.221747 0.975104i \(-0.571176\pi\)
−0.221747 + 0.975104i \(0.571176\pi\)
\(294\) −35.4692 −2.06861
\(295\) 9.90925 0.576939
\(296\) 4.57381 0.265847
\(297\) 5.85395 0.339681
\(298\) −2.23114 −0.129246
\(299\) 16.0534 0.928389
\(300\) 1.48386 0.0856708
\(301\) −27.8270 −1.60392
\(302\) 28.0338 1.61316
\(303\) 16.2806 0.935297
\(304\) 34.8042 1.99616
\(305\) −3.57514 −0.204712
\(306\) −6.61076 −0.377912
\(307\) −20.6437 −1.17820 −0.589100 0.808060i \(-0.700518\pi\)
−0.589100 + 0.808060i \(0.700518\pi\)
\(308\) 3.61714 0.206106
\(309\) 8.34605 0.474790
\(310\) −23.9785 −1.36189
\(311\) 24.0882 1.36592 0.682959 0.730456i \(-0.260693\pi\)
0.682959 + 0.730456i \(0.260693\pi\)
\(312\) 7.68132 0.434869
\(313\) 16.3804 0.925876 0.462938 0.886391i \(-0.346795\pi\)
0.462938 + 0.886391i \(0.346795\pi\)
\(314\) 19.7836 1.11646
\(315\) −14.5271 −0.818508
\(316\) 4.53584 0.255161
\(317\) −1.59833 −0.0897714 −0.0448857 0.998992i \(-0.514292\pi\)
−0.0448857 + 0.998992i \(0.514292\pi\)
\(318\) 7.70458 0.432051
\(319\) −7.35473 −0.411786
\(320\) 6.83712 0.382206
\(321\) −20.8971 −1.16636
\(322\) 44.0181 2.45304
\(323\) −17.6483 −0.981980
\(324\) −0.935593 −0.0519774
\(325\) −5.78784 −0.321052
\(326\) 17.4679 0.967460
\(327\) 17.4364 0.964236
\(328\) 23.2764 1.28522
\(329\) 24.8203 1.36839
\(330\) −3.61368 −0.198926
\(331\) −26.9070 −1.47894 −0.739472 0.673188i \(-0.764925\pi\)
−0.739472 + 0.673188i \(0.764925\pi\)
\(332\) 3.69370 0.202718
\(333\) 3.42578 0.187732
\(334\) −17.2161 −0.942022
\(335\) 13.4227 0.733359
\(336\) 28.8766 1.57535
\(337\) −10.8794 −0.592640 −0.296320 0.955089i \(-0.595760\pi\)
−0.296320 + 0.955089i \(0.595760\pi\)
\(338\) −6.41617 −0.348994
\(339\) −1.00358 −0.0545072
\(340\) 2.87252 0.155784
\(341\) −9.05892 −0.490568
\(342\) 19.0138 1.02815
\(343\) 58.8013 3.17497
\(344\) 12.0089 0.647475
\(345\) −10.9237 −0.588110
\(346\) −3.38682 −0.182077
\(347\) 12.9007 0.692546 0.346273 0.938134i \(-0.387447\pi\)
0.346273 + 0.938134i \(0.387447\pi\)
\(348\) 5.25133 0.281501
\(349\) −10.7803 −0.577058 −0.288529 0.957471i \(-0.593166\pi\)
−0.288529 + 0.957471i \(0.593166\pi\)
\(350\) −15.8702 −0.848298
\(351\) 16.3033 0.870205
\(352\) −3.89256 −0.207474
\(353\) 6.56859 0.349611 0.174805 0.984603i \(-0.444070\pi\)
0.174805 + 0.984603i \(0.444070\pi\)
\(354\) −10.7608 −0.571931
\(355\) 21.5141 1.14185
\(356\) 6.36402 0.337292
\(357\) −14.6426 −0.774970
\(358\) −5.00669 −0.264612
\(359\) 15.8305 0.835504 0.417752 0.908561i \(-0.362818\pi\)
0.417752 + 0.908561i \(0.362818\pi\)
\(360\) 6.26922 0.330417
\(361\) 31.7598 1.67157
\(362\) −0.638477 −0.0335576
\(363\) 11.4817 0.602631
\(364\) 10.0738 0.528008
\(365\) 1.28721 0.0673757
\(366\) 3.88237 0.202935
\(367\) −5.63657 −0.294226 −0.147113 0.989120i \(-0.546998\pi\)
−0.147113 + 0.989120i \(0.546998\pi\)
\(368\) −26.0442 −1.35765
\(369\) 17.4340 0.907576
\(370\) −5.99263 −0.311542
\(371\) −20.4687 −1.06268
\(372\) 6.46814 0.335357
\(373\) 1.29504 0.0670548 0.0335274 0.999438i \(-0.489326\pi\)
0.0335274 + 0.999438i \(0.489326\pi\)
\(374\) 4.36883 0.225907
\(375\) 14.1831 0.732410
\(376\) −10.7113 −0.552394
\(377\) −20.4830 −1.05493
\(378\) 44.7035 2.29930
\(379\) −13.8155 −0.709656 −0.354828 0.934932i \(-0.615461\pi\)
−0.354828 + 0.934932i \(0.615461\pi\)
\(380\) −8.26188 −0.423826
\(381\) 3.52572 0.180628
\(382\) 8.14026 0.416492
\(383\) 12.0143 0.613900 0.306950 0.951726i \(-0.400692\pi\)
0.306950 + 0.951726i \(0.400692\pi\)
\(384\) −15.8342 −0.808036
\(385\) 9.60044 0.489284
\(386\) 22.0560 1.12262
\(387\) 8.99463 0.457223
\(388\) −7.21704 −0.366390
\(389\) 4.86222 0.246524 0.123262 0.992374i \(-0.460664\pi\)
0.123262 + 0.992374i \(0.460664\pi\)
\(390\) −10.0641 −0.509616
\(391\) 13.2064 0.667876
\(392\) −40.6658 −2.05393
\(393\) 2.54040 0.128146
\(394\) 9.57141 0.482201
\(395\) 12.0388 0.605738
\(396\) −1.16918 −0.0587536
\(397\) 9.17854 0.460658 0.230329 0.973113i \(-0.426020\pi\)
0.230329 + 0.973113i \(0.426020\pi\)
\(398\) 20.5169 1.02842
\(399\) 42.1148 2.10838
\(400\) 9.38993 0.469497
\(401\) −28.0569 −1.40109 −0.700547 0.713607i \(-0.747060\pi\)
−0.700547 + 0.713607i \(0.747060\pi\)
\(402\) −14.5762 −0.726994
\(403\) −25.2291 −1.25675
\(404\) −9.21429 −0.458428
\(405\) −2.48321 −0.123392
\(406\) −56.1641 −2.78738
\(407\) −2.26398 −0.112221
\(408\) 6.31908 0.312841
\(409\) −11.6170 −0.574422 −0.287211 0.957867i \(-0.592728\pi\)
−0.287211 + 0.957867i \(0.592728\pi\)
\(410\) −30.4968 −1.50613
\(411\) −18.5871 −0.916834
\(412\) −4.72359 −0.232714
\(413\) 28.5883 1.40674
\(414\) −14.2281 −0.699275
\(415\) 9.80365 0.481242
\(416\) −10.8408 −0.531514
\(417\) 17.6027 0.862010
\(418\) −12.5655 −0.614601
\(419\) 15.5789 0.761080 0.380540 0.924765i \(-0.375738\pi\)
0.380540 + 0.924765i \(0.375738\pi\)
\(420\) −6.85479 −0.334479
\(421\) 7.53596 0.367280 0.183640 0.982994i \(-0.441212\pi\)
0.183640 + 0.982994i \(0.441212\pi\)
\(422\) 9.79824 0.476971
\(423\) −8.02277 −0.390081
\(424\) 8.83337 0.428986
\(425\) −4.76140 −0.230962
\(426\) −23.3630 −1.13194
\(427\) −10.3143 −0.499144
\(428\) 11.8271 0.571683
\(429\) −3.80216 −0.183570
\(430\) −15.7341 −0.758765
\(431\) 24.2144 1.16636 0.583182 0.812341i \(-0.301807\pi\)
0.583182 + 0.812341i \(0.301807\pi\)
\(432\) −26.4497 −1.27256
\(433\) 1.90821 0.0917025 0.0458512 0.998948i \(-0.485400\pi\)
0.0458512 + 0.998948i \(0.485400\pi\)
\(434\) −69.1781 −3.32065
\(435\) 13.9378 0.668268
\(436\) −9.86844 −0.472612
\(437\) −37.9840 −1.81702
\(438\) −1.39783 −0.0667910
\(439\) −14.9135 −0.711784 −0.355892 0.934527i \(-0.615823\pi\)
−0.355892 + 0.934527i \(0.615823\pi\)
\(440\) −4.14311 −0.197515
\(441\) −30.4586 −1.45041
\(442\) 12.1672 0.578735
\(443\) −23.0698 −1.09608 −0.548039 0.836453i \(-0.684625\pi\)
−0.548039 + 0.836453i \(0.684625\pi\)
\(444\) 1.61650 0.0767156
\(445\) 16.8911 0.800714
\(446\) −42.0251 −1.98995
\(447\) 1.59739 0.0755539
\(448\) 19.7251 0.931925
\(449\) 14.5420 0.686277 0.343139 0.939285i \(-0.388510\pi\)
0.343139 + 0.939285i \(0.388510\pi\)
\(450\) 5.12979 0.241820
\(451\) −11.5215 −0.542527
\(452\) 0.567995 0.0267162
\(453\) −20.0708 −0.943009
\(454\) 8.21696 0.385641
\(455\) 26.7373 1.25346
\(456\) −18.1748 −0.851115
\(457\) 11.8002 0.551991 0.275995 0.961159i \(-0.410993\pi\)
0.275995 + 0.961159i \(0.410993\pi\)
\(458\) 34.4594 1.61018
\(459\) 13.4120 0.626019
\(460\) 6.18243 0.288257
\(461\) 15.0722 0.701985 0.350992 0.936378i \(-0.385844\pi\)
0.350992 + 0.936378i \(0.385844\pi\)
\(462\) −10.4255 −0.485037
\(463\) 24.3295 1.13069 0.565344 0.824855i \(-0.308743\pi\)
0.565344 + 0.824855i \(0.308743\pi\)
\(464\) 33.2306 1.54269
\(465\) 17.1674 0.796120
\(466\) 34.1695 1.58287
\(467\) 29.7743 1.37779 0.688895 0.724862i \(-0.258096\pi\)
0.688895 + 0.724862i \(0.258096\pi\)
\(468\) −3.25618 −0.150517
\(469\) 38.7245 1.78813
\(470\) 14.0340 0.647342
\(471\) −14.1641 −0.652649
\(472\) −12.3374 −0.567874
\(473\) −5.94424 −0.273317
\(474\) −13.0734 −0.600481
\(475\) 13.6947 0.628354
\(476\) 8.28724 0.379845
\(477\) 6.61619 0.302934
\(478\) −12.7816 −0.584615
\(479\) −10.1987 −0.465991 −0.232996 0.972478i \(-0.574853\pi\)
−0.232996 + 0.972478i \(0.574853\pi\)
\(480\) 7.37673 0.336700
\(481\) −6.30520 −0.287492
\(482\) 1.63125 0.0743016
\(483\) −31.5149 −1.43398
\(484\) −6.49825 −0.295375
\(485\) −19.1551 −0.869790
\(486\) −23.8002 −1.07960
\(487\) −24.9246 −1.12944 −0.564720 0.825282i \(-0.691016\pi\)
−0.564720 + 0.825282i \(0.691016\pi\)
\(488\) 4.45118 0.201495
\(489\) −12.5062 −0.565550
\(490\) 53.2806 2.40697
\(491\) −26.8710 −1.21267 −0.606335 0.795209i \(-0.707361\pi\)
−0.606335 + 0.795209i \(0.707361\pi\)
\(492\) 8.22644 0.370877
\(493\) −16.8504 −0.758905
\(494\) −34.9951 −1.57450
\(495\) −3.10319 −0.139478
\(496\) 40.9306 1.83784
\(497\) 62.0684 2.78415
\(498\) −10.6462 −0.477066
\(499\) −37.6147 −1.68387 −0.841933 0.539582i \(-0.818582\pi\)
−0.841933 + 0.539582i \(0.818582\pi\)
\(500\) −8.02714 −0.358985
\(501\) 12.3259 0.550680
\(502\) −11.4457 −0.510845
\(503\) −38.0897 −1.69834 −0.849168 0.528123i \(-0.822896\pi\)
−0.849168 + 0.528123i \(0.822896\pi\)
\(504\) 18.0867 0.805648
\(505\) −24.4561 −1.08828
\(506\) 9.40290 0.418010
\(507\) 4.59367 0.204012
\(508\) −1.99544 −0.0885334
\(509\) −37.4493 −1.65991 −0.829954 0.557831i \(-0.811634\pi\)
−0.829954 + 0.557831i \(0.811634\pi\)
\(510\) −8.27930 −0.366613
\(511\) 3.71362 0.164281
\(512\) −3.75296 −0.165859
\(513\) −38.5754 −1.70314
\(514\) 18.4424 0.813462
\(515\) −12.5371 −0.552452
\(516\) 4.24423 0.186842
\(517\) 5.30198 0.233181
\(518\) −17.2888 −0.759626
\(519\) 2.42480 0.106437
\(520\) −11.5386 −0.506001
\(521\) 14.8226 0.649391 0.324695 0.945819i \(-0.394738\pi\)
0.324695 + 0.945819i \(0.394738\pi\)
\(522\) 18.1541 0.794584
\(523\) 23.8624 1.04343 0.521715 0.853120i \(-0.325292\pi\)
0.521715 + 0.853120i \(0.325292\pi\)
\(524\) −1.43778 −0.0628098
\(525\) 11.3623 0.495891
\(526\) 32.4076 1.41304
\(527\) −20.7549 −0.904098
\(528\) 6.16845 0.268447
\(529\) 5.42372 0.235814
\(530\) −11.5735 −0.502722
\(531\) −9.24069 −0.401012
\(532\) −23.8356 −1.03340
\(533\) −32.0875 −1.38986
\(534\) −18.3427 −0.793765
\(535\) 31.3909 1.35715
\(536\) −16.7117 −0.721837
\(537\) 3.58454 0.154685
\(538\) −33.0382 −1.42438
\(539\) 20.1291 0.867021
\(540\) 6.27869 0.270192
\(541\) −12.9272 −0.555782 −0.277891 0.960613i \(-0.589635\pi\)
−0.277891 + 0.960613i \(0.589635\pi\)
\(542\) −6.91477 −0.297015
\(543\) 0.457119 0.0196168
\(544\) −8.91825 −0.382367
\(545\) −26.1923 −1.12196
\(546\) −29.0350 −1.24258
\(547\) −9.03128 −0.386150 −0.193075 0.981184i \(-0.561846\pi\)
−0.193075 + 0.981184i \(0.561846\pi\)
\(548\) 10.5197 0.449379
\(549\) 3.33393 0.142289
\(550\) −3.39010 −0.144554
\(551\) 48.4649 2.06467
\(552\) 13.6004 0.578870
\(553\) 34.7320 1.47696
\(554\) −0.602698 −0.0256062
\(555\) 4.29043 0.182119
\(556\) −9.96256 −0.422507
\(557\) −4.33616 −0.183729 −0.0918644 0.995772i \(-0.529283\pi\)
−0.0918644 + 0.995772i \(0.529283\pi\)
\(558\) 22.3607 0.946603
\(559\) −16.5547 −0.700191
\(560\) −43.3774 −1.83303
\(561\) −3.12787 −0.132059
\(562\) 40.2786 1.69905
\(563\) 6.56701 0.276767 0.138383 0.990379i \(-0.455809\pi\)
0.138383 + 0.990379i \(0.455809\pi\)
\(564\) −3.78565 −0.159405
\(565\) 1.50755 0.0634229
\(566\) 0.113662 0.00477756
\(567\) −7.16407 −0.300863
\(568\) −26.7859 −1.12391
\(569\) 18.1137 0.759366 0.379683 0.925117i \(-0.376033\pi\)
0.379683 + 0.925117i \(0.376033\pi\)
\(570\) 23.8128 0.997407
\(571\) 6.26298 0.262098 0.131049 0.991376i \(-0.458166\pi\)
0.131049 + 0.991376i \(0.458166\pi\)
\(572\) 2.15190 0.0899753
\(573\) −5.82803 −0.243470
\(574\) −87.9835 −3.67236
\(575\) −10.2478 −0.427364
\(576\) −6.37583 −0.265660
\(577\) −1.76988 −0.0736810 −0.0368405 0.999321i \(-0.511729\pi\)
−0.0368405 + 0.999321i \(0.511729\pi\)
\(578\) −17.7219 −0.737134
\(579\) −15.7910 −0.656253
\(580\) −7.88835 −0.327546
\(581\) 28.2836 1.17340
\(582\) 20.8013 0.862241
\(583\) −4.37241 −0.181087
\(584\) −1.60263 −0.0663171
\(585\) −8.64239 −0.357319
\(586\) −12.3835 −0.511558
\(587\) 11.9082 0.491502 0.245751 0.969333i \(-0.420965\pi\)
0.245751 + 0.969333i \(0.420965\pi\)
\(588\) −14.3723 −0.592704
\(589\) 59.6949 2.45969
\(590\) 16.1645 0.665482
\(591\) −6.85266 −0.281881
\(592\) 10.2293 0.420420
\(593\) −34.2932 −1.40825 −0.704126 0.710075i \(-0.748661\pi\)
−0.704126 + 0.710075i \(0.748661\pi\)
\(594\) 9.54929 0.391812
\(595\) 21.9956 0.901731
\(596\) −0.904069 −0.0370321
\(597\) −14.6891 −0.601184
\(598\) 26.1871 1.07087
\(599\) −18.2245 −0.744632 −0.372316 0.928106i \(-0.621436\pi\)
−0.372316 + 0.928106i \(0.621436\pi\)
\(600\) −4.90345 −0.200182
\(601\) −19.2591 −0.785595 −0.392797 0.919625i \(-0.628493\pi\)
−0.392797 + 0.919625i \(0.628493\pi\)
\(602\) −45.3930 −1.85008
\(603\) −12.5171 −0.509735
\(604\) 11.3594 0.462208
\(605\) −17.2473 −0.701204
\(606\) 26.5578 1.07884
\(607\) 39.9799 1.62273 0.811367 0.584536i \(-0.198724\pi\)
0.811367 + 0.584536i \(0.198724\pi\)
\(608\) 25.6505 1.04027
\(609\) 40.2108 1.62942
\(610\) −5.83196 −0.236129
\(611\) 14.7660 0.597369
\(612\) −2.67871 −0.108281
\(613\) −18.3233 −0.740070 −0.370035 0.929018i \(-0.620654\pi\)
−0.370035 + 0.929018i \(0.620654\pi\)
\(614\) −33.6752 −1.35902
\(615\) 21.8342 0.880442
\(616\) −11.9529 −0.481597
\(617\) 12.3769 0.498273 0.249137 0.968468i \(-0.419853\pi\)
0.249137 + 0.968468i \(0.419853\pi\)
\(618\) 13.6145 0.547657
\(619\) −8.87878 −0.356868 −0.178434 0.983952i \(-0.557103\pi\)
−0.178434 + 0.983952i \(0.557103\pi\)
\(620\) −9.71619 −0.390212
\(621\) 28.8662 1.15836
\(622\) 39.2941 1.57555
\(623\) 48.7309 1.95236
\(624\) 17.1792 0.687717
\(625\) −11.6944 −0.467777
\(626\) 26.7206 1.06797
\(627\) 8.99632 0.359278
\(628\) 8.01643 0.319890
\(629\) −5.18701 −0.206820
\(630\) −23.6974 −0.944125
\(631\) 34.5564 1.37567 0.687835 0.725867i \(-0.258561\pi\)
0.687835 + 0.725867i \(0.258561\pi\)
\(632\) −14.9888 −0.596221
\(633\) −7.01507 −0.278824
\(634\) −2.60729 −0.103549
\(635\) −5.29621 −0.210174
\(636\) 3.12193 0.123793
\(637\) 56.0596 2.22116
\(638\) −11.9974 −0.474983
\(639\) −20.0626 −0.793664
\(640\) 23.7855 0.940206
\(641\) −1.70265 −0.0672505 −0.0336253 0.999435i \(-0.510705\pi\)
−0.0336253 + 0.999435i \(0.510705\pi\)
\(642\) −34.0885 −1.34537
\(643\) 40.7018 1.60512 0.802561 0.596571i \(-0.203470\pi\)
0.802561 + 0.596571i \(0.203470\pi\)
\(644\) 17.8364 0.702851
\(645\) 11.2648 0.443553
\(646\) −28.7889 −1.13269
\(647\) −30.3915 −1.19481 −0.597406 0.801939i \(-0.703802\pi\)
−0.597406 + 0.801939i \(0.703802\pi\)
\(648\) 3.09169 0.121453
\(649\) 6.10685 0.239715
\(650\) −9.44144 −0.370324
\(651\) 49.5282 1.94116
\(652\) 7.07809 0.277200
\(653\) 6.95225 0.272062 0.136031 0.990705i \(-0.456565\pi\)
0.136031 + 0.990705i \(0.456565\pi\)
\(654\) 28.4432 1.11222
\(655\) −3.81609 −0.149107
\(656\) 52.0573 2.03249
\(657\) −1.20037 −0.0468307
\(658\) 40.4883 1.57840
\(659\) −38.2304 −1.48924 −0.744622 0.667486i \(-0.767370\pi\)
−0.744622 + 0.667486i \(0.767370\pi\)
\(660\) −1.46428 −0.0569970
\(661\) 35.0225 1.36222 0.681109 0.732182i \(-0.261498\pi\)
0.681109 + 0.732182i \(0.261498\pi\)
\(662\) −43.8922 −1.70592
\(663\) −8.71113 −0.338312
\(664\) −12.2059 −0.473681
\(665\) −63.2633 −2.45325
\(666\) 5.58832 0.216543
\(667\) −36.2667 −1.40425
\(668\) −6.97604 −0.269911
\(669\) 30.0879 1.16327
\(670\) 21.8958 0.845909
\(671\) −2.20328 −0.0850566
\(672\) 21.2819 0.820968
\(673\) −19.3765 −0.746909 −0.373454 0.927649i \(-0.621827\pi\)
−0.373454 + 0.927649i \(0.621827\pi\)
\(674\) −17.7471 −0.683594
\(675\) −10.4074 −0.400580
\(676\) −2.59986 −0.0999947
\(677\) −3.01738 −0.115967 −0.0579837 0.998318i \(-0.518467\pi\)
−0.0579837 + 0.998318i \(0.518467\pi\)
\(678\) −1.63710 −0.0628725
\(679\) −55.2627 −2.12079
\(680\) −9.49229 −0.364013
\(681\) −5.88294 −0.225435
\(682\) −14.7774 −0.565856
\(683\) −5.60316 −0.214399 −0.107199 0.994238i \(-0.534188\pi\)
−0.107199 + 0.994238i \(0.534188\pi\)
\(684\) 7.70447 0.294588
\(685\) 27.9209 1.06680
\(686\) 95.9200 3.66224
\(687\) −24.6713 −0.941267
\(688\) 26.8577 1.02394
\(689\) −12.1772 −0.463914
\(690\) −17.8193 −0.678368
\(691\) 49.0976 1.86776 0.933881 0.357583i \(-0.116399\pi\)
0.933881 + 0.357583i \(0.116399\pi\)
\(692\) −1.37236 −0.0521691
\(693\) −8.95272 −0.340086
\(694\) 21.0443 0.798832
\(695\) −26.4422 −1.00301
\(696\) −17.3531 −0.657768
\(697\) −26.3970 −0.999856
\(698\) −17.5855 −0.665619
\(699\) −24.4637 −0.925304
\(700\) −6.43069 −0.243057
\(701\) 14.0248 0.529709 0.264855 0.964288i \(-0.414676\pi\)
0.264855 + 0.964288i \(0.414676\pi\)
\(702\) 26.5948 1.00376
\(703\) 14.9188 0.562673
\(704\) 4.21357 0.158805
\(705\) −10.0477 −0.378418
\(706\) 10.7150 0.403266
\(707\) −70.5561 −2.65354
\(708\) −4.36034 −0.163872
\(709\) −10.0074 −0.375835 −0.187918 0.982185i \(-0.560174\pi\)
−0.187918 + 0.982185i \(0.560174\pi\)
\(710\) 35.0950 1.31709
\(711\) −11.2266 −0.421029
\(712\) −21.0300 −0.788134
\(713\) −44.6702 −1.67291
\(714\) −23.8858 −0.893905
\(715\) 5.71146 0.213596
\(716\) −2.02873 −0.0758173
\(717\) 9.15098 0.341749
\(718\) 25.8236 0.963730
\(719\) 32.3086 1.20491 0.602454 0.798154i \(-0.294190\pi\)
0.602454 + 0.798154i \(0.294190\pi\)
\(720\) 14.0210 0.522533
\(721\) −36.1697 −1.34703
\(722\) 51.8084 1.92811
\(723\) −1.16790 −0.0434347
\(724\) −0.258714 −0.00961504
\(725\) 13.0755 0.485612
\(726\) 18.7295 0.695118
\(727\) 14.7153 0.545762 0.272881 0.962048i \(-0.412023\pi\)
0.272881 + 0.962048i \(0.412023\pi\)
\(728\) −33.2889 −1.23377
\(729\) 21.2861 0.788373
\(730\) 2.09977 0.0777160
\(731\) −13.6189 −0.503712
\(732\) 1.57316 0.0581455
\(733\) 28.2634 1.04393 0.521966 0.852966i \(-0.325199\pi\)
0.521966 + 0.852966i \(0.325199\pi\)
\(734\) −9.19468 −0.339382
\(735\) −38.1463 −1.40705
\(736\) −19.1945 −0.707518
\(737\) 8.27210 0.304707
\(738\) 28.4392 1.04686
\(739\) −34.0536 −1.25268 −0.626340 0.779550i \(-0.715448\pi\)
−0.626340 + 0.779550i \(0.715448\pi\)
\(740\) −2.42824 −0.0892640
\(741\) 25.0548 0.920411
\(742\) −33.3897 −1.22578
\(743\) −47.2361 −1.73293 −0.866463 0.499241i \(-0.833612\pi\)
−0.866463 + 0.499241i \(0.833612\pi\)
\(744\) −21.3741 −0.783612
\(745\) −2.39954 −0.0879122
\(746\) 2.11255 0.0773458
\(747\) −9.14222 −0.334496
\(748\) 1.77027 0.0647275
\(749\) 90.5629 3.30910
\(750\) 23.1362 0.844814
\(751\) 36.7588 1.34135 0.670674 0.741752i \(-0.266005\pi\)
0.670674 + 0.741752i \(0.266005\pi\)
\(752\) −23.9557 −0.873576
\(753\) 8.19454 0.298626
\(754\) −33.4129 −1.21683
\(755\) 30.1496 1.09726
\(756\) 18.1141 0.658802
\(757\) 7.43315 0.270162 0.135081 0.990835i \(-0.456870\pi\)
0.135081 + 0.990835i \(0.456870\pi\)
\(758\) −22.5367 −0.818568
\(759\) −6.73202 −0.244357
\(760\) 27.3016 0.990332
\(761\) 1.06982 0.0387809 0.0193905 0.999812i \(-0.493827\pi\)
0.0193905 + 0.999812i \(0.493827\pi\)
\(762\) 5.75135 0.208350
\(763\) −75.5651 −2.73564
\(764\) 3.29847 0.119335
\(765\) −7.10972 −0.257052
\(766\) 19.5983 0.708116
\(767\) 17.0076 0.614109
\(768\) −16.7266 −0.603569
\(769\) −26.2212 −0.945562 −0.472781 0.881180i \(-0.656750\pi\)
−0.472781 + 0.881180i \(0.656750\pi\)
\(770\) 15.6608 0.564375
\(771\) −13.2039 −0.475527
\(772\) 8.93720 0.321657
\(773\) −15.9051 −0.572065 −0.286033 0.958220i \(-0.592337\pi\)
−0.286033 + 0.958220i \(0.592337\pi\)
\(774\) 14.6725 0.527394
\(775\) 16.1053 0.578519
\(776\) 23.8489 0.856124
\(777\) 12.3779 0.444056
\(778\) 7.93152 0.284359
\(779\) 75.9225 2.72020
\(780\) −4.07802 −0.146017
\(781\) 13.2587 0.474433
\(782\) 21.5430 0.770376
\(783\) −36.8313 −1.31624
\(784\) −90.9485 −3.24816
\(785\) 21.2768 0.759403
\(786\) 4.14404 0.147813
\(787\) −44.7474 −1.59507 −0.797536 0.603272i \(-0.793863\pi\)
−0.797536 + 0.603272i \(0.793863\pi\)
\(788\) 3.87838 0.138162
\(789\) −23.2023 −0.826023
\(790\) 19.6384 0.698701
\(791\) 4.34928 0.154643
\(792\) 3.86358 0.137286
\(793\) −6.13614 −0.217901
\(794\) 14.9725 0.531356
\(795\) 8.28609 0.293877
\(796\) 8.31354 0.294665
\(797\) −20.2003 −0.715530 −0.357765 0.933812i \(-0.616461\pi\)
−0.357765 + 0.933812i \(0.616461\pi\)
\(798\) 68.7000 2.43195
\(799\) 12.1474 0.429743
\(800\) 6.92033 0.244671
\(801\) −15.7515 −0.556551
\(802\) −45.7679 −1.61612
\(803\) 0.793280 0.0279943
\(804\) −5.90634 −0.208301
\(805\) 47.3405 1.66853
\(806\) −41.1552 −1.44963
\(807\) 23.6538 0.832652
\(808\) 30.4488 1.07118
\(809\) 31.3338 1.10164 0.550818 0.834625i \(-0.314316\pi\)
0.550818 + 0.834625i \(0.314316\pi\)
\(810\) −4.05074 −0.142329
\(811\) −28.2192 −0.990909 −0.495454 0.868634i \(-0.664998\pi\)
−0.495454 + 0.868634i \(0.664998\pi\)
\(812\) −22.7580 −0.798648
\(813\) 4.95064 0.173626
\(814\) −3.69313 −0.129444
\(815\) 18.7863 0.658057
\(816\) 14.1325 0.494738
\(817\) 39.1704 1.37040
\(818\) −18.9502 −0.662579
\(819\) −24.9334 −0.871242
\(820\) −12.3575 −0.431541
\(821\) −36.8115 −1.28473 −0.642365 0.766399i \(-0.722047\pi\)
−0.642365 + 0.766399i \(0.722047\pi\)
\(822\) −30.3203 −1.05754
\(823\) −34.8680 −1.21542 −0.607711 0.794158i \(-0.707912\pi\)
−0.607711 + 0.794158i \(0.707912\pi\)
\(824\) 15.6092 0.543772
\(825\) 2.42715 0.0845024
\(826\) 46.6347 1.62263
\(827\) −11.6305 −0.404431 −0.202216 0.979341i \(-0.564814\pi\)
−0.202216 + 0.979341i \(0.564814\pi\)
\(828\) −5.76532 −0.200359
\(829\) −45.8232 −1.59150 −0.795752 0.605622i \(-0.792924\pi\)
−0.795752 + 0.605622i \(0.792924\pi\)
\(830\) 15.9923 0.555099
\(831\) 0.431502 0.0149686
\(832\) 11.7348 0.406831
\(833\) 46.1178 1.59789
\(834\) 28.7145 0.994303
\(835\) −18.5155 −0.640755
\(836\) −5.09162 −0.176097
\(837\) −45.3656 −1.56807
\(838\) 25.4132 0.877884
\(839\) −3.48632 −0.120361 −0.0601806 0.998188i \(-0.519168\pi\)
−0.0601806 + 0.998188i \(0.519168\pi\)
\(840\) 22.6518 0.781561
\(841\) 17.2737 0.595646
\(842\) 12.2931 0.423647
\(843\) −28.8375 −0.993216
\(844\) 3.97030 0.136663
\(845\) −6.90043 −0.237382
\(846\) −13.0872 −0.449947
\(847\) −49.7587 −1.70973
\(848\) 19.7557 0.678414
\(849\) −0.0813764 −0.00279283
\(850\) −7.76706 −0.266408
\(851\) −11.1638 −0.382692
\(852\) −9.46679 −0.324327
\(853\) −44.6130 −1.52752 −0.763760 0.645500i \(-0.776649\pi\)
−0.763760 + 0.645500i \(0.776649\pi\)
\(854\) −16.8252 −0.575748
\(855\) 20.4488 0.699336
\(856\) −39.0828 −1.33582
\(857\) 21.1188 0.721404 0.360702 0.932681i \(-0.382537\pi\)
0.360702 + 0.932681i \(0.382537\pi\)
\(858\) −6.20229 −0.211743
\(859\) −36.4020 −1.24202 −0.621011 0.783802i \(-0.713278\pi\)
−0.621011 + 0.783802i \(0.713278\pi\)
\(860\) −6.37553 −0.217404
\(861\) 62.9920 2.14676
\(862\) 39.4998 1.34537
\(863\) −4.01817 −0.136780 −0.0683900 0.997659i \(-0.521786\pi\)
−0.0683900 + 0.997659i \(0.521786\pi\)
\(864\) −19.4933 −0.663176
\(865\) −3.64244 −0.123847
\(866\) 3.11277 0.105776
\(867\) 12.6880 0.430908
\(868\) −28.0313 −0.951444
\(869\) 7.41925 0.251681
\(870\) 22.7362 0.770828
\(871\) 23.0379 0.780608
\(872\) 32.6104 1.10433
\(873\) 17.8628 0.604563
\(874\) −61.9616 −2.09588
\(875\) −61.4659 −2.07793
\(876\) −0.566408 −0.0191372
\(877\) 48.0234 1.62163 0.810817 0.585300i \(-0.199023\pi\)
0.810817 + 0.585300i \(0.199023\pi\)
\(878\) −24.3278 −0.821022
\(879\) 8.86599 0.299042
\(880\) −9.26601 −0.312357
\(881\) −56.8752 −1.91617 −0.958087 0.286479i \(-0.907515\pi\)
−0.958087 + 0.286479i \(0.907515\pi\)
\(882\) −49.6858 −1.67301
\(883\) 44.7065 1.50449 0.752246 0.658882i \(-0.228970\pi\)
0.752246 + 0.658882i \(0.228970\pi\)
\(884\) 4.93021 0.165821
\(885\) −11.5730 −0.389022
\(886\) −37.6327 −1.26429
\(887\) 5.64866 0.189663 0.0948317 0.995493i \(-0.469769\pi\)
0.0948317 + 0.995493i \(0.469769\pi\)
\(888\) −5.34175 −0.179257
\(889\) −15.2796 −0.512462
\(890\) 27.5537 0.923601
\(891\) −1.53035 −0.0512686
\(892\) −17.0288 −0.570165
\(893\) −34.9380 −1.16916
\(894\) 2.60575 0.0871492
\(895\) −5.38457 −0.179986
\(896\) 68.6215 2.29248
\(897\) −18.7487 −0.626001
\(898\) 23.7216 0.791601
\(899\) 56.9960 1.90092
\(900\) 2.07861 0.0692871
\(901\) −10.0176 −0.333736
\(902\) −18.7945 −0.625789
\(903\) 32.4992 1.08150
\(904\) −1.87695 −0.0624265
\(905\) −0.686667 −0.0228256
\(906\) −32.7406 −1.08773
\(907\) 25.2839 0.839538 0.419769 0.907631i \(-0.362111\pi\)
0.419769 + 0.907631i \(0.362111\pi\)
\(908\) 3.32955 0.110495
\(909\) 22.8061 0.756431
\(910\) 43.6153 1.44583
\(911\) 27.3571 0.906380 0.453190 0.891414i \(-0.350286\pi\)
0.453190 + 0.891414i \(0.350286\pi\)
\(912\) −40.6478 −1.34598
\(913\) 6.04178 0.199954
\(914\) 19.2492 0.636705
\(915\) 4.17540 0.138034
\(916\) 13.9631 0.461354
\(917\) −11.0095 −0.363564
\(918\) 21.8784 0.722095
\(919\) −44.3463 −1.46285 −0.731424 0.681923i \(-0.761144\pi\)
−0.731424 + 0.681923i \(0.761144\pi\)
\(920\) −20.4300 −0.673556
\(921\) 24.1098 0.794445
\(922\) 24.5867 0.809719
\(923\) 36.9255 1.21542
\(924\) −4.22446 −0.138974
\(925\) 4.02499 0.132341
\(926\) 39.6876 1.30422
\(927\) 11.6913 0.383992
\(928\) 24.4908 0.803950
\(929\) −33.2552 −1.09107 −0.545535 0.838088i \(-0.683673\pi\)
−0.545535 + 0.838088i \(0.683673\pi\)
\(930\) 28.0044 0.918302
\(931\) −132.643 −4.34720
\(932\) 13.8457 0.453530
\(933\) −28.1326 −0.921021
\(934\) 48.5694 1.58924
\(935\) 4.69857 0.153660
\(936\) 10.7601 0.351705
\(937\) −51.9272 −1.69639 −0.848194 0.529686i \(-0.822310\pi\)
−0.848194 + 0.529686i \(0.822310\pi\)
\(938\) 63.1696 2.06256
\(939\) −19.1307 −0.624306
\(940\) 5.68666 0.185478
\(941\) 21.0604 0.686549 0.343274 0.939235i \(-0.388464\pi\)
0.343274 + 0.939235i \(0.388464\pi\)
\(942\) −23.1053 −0.752812
\(943\) −56.8134 −1.85010
\(944\) −27.5924 −0.898056
\(945\) 48.0775 1.56396
\(946\) −9.69658 −0.315263
\(947\) −22.4529 −0.729621 −0.364810 0.931082i \(-0.618866\pi\)
−0.364810 + 0.931082i \(0.618866\pi\)
\(948\) −5.29740 −0.172052
\(949\) 2.20929 0.0717166
\(950\) 22.3395 0.724789
\(951\) 1.86669 0.0605317
\(952\) −27.3853 −0.887564
\(953\) 29.3558 0.950927 0.475463 0.879736i \(-0.342280\pi\)
0.475463 + 0.879736i \(0.342280\pi\)
\(954\) 10.7927 0.349426
\(955\) 8.75465 0.283294
\(956\) −5.17915 −0.167506
\(957\) 8.58958 0.277662
\(958\) −16.6367 −0.537508
\(959\) 80.5519 2.60116
\(960\) −7.98507 −0.257717
\(961\) 39.2028 1.26461
\(962\) −10.2854 −0.331614
\(963\) −29.2730 −0.943309
\(964\) 0.660992 0.0212891
\(965\) 23.7207 0.763596
\(966\) −51.4088 −1.65405
\(967\) 26.9397 0.866324 0.433162 0.901316i \(-0.357398\pi\)
0.433162 + 0.901316i \(0.357398\pi\)
\(968\) 21.4736 0.690187
\(969\) 20.6115 0.662136
\(970\) −31.2469 −1.00328
\(971\) −38.3427 −1.23047 −0.615237 0.788342i \(-0.710940\pi\)
−0.615237 + 0.788342i \(0.710940\pi\)
\(972\) −9.64394 −0.309330
\(973\) −76.2859 −2.44561
\(974\) −40.6583 −1.30278
\(975\) 6.75962 0.216481
\(976\) 9.95500 0.318652
\(977\) −13.7841 −0.440991 −0.220496 0.975388i \(-0.570767\pi\)
−0.220496 + 0.975388i \(0.570767\pi\)
\(978\) −20.4008 −0.652346
\(979\) 10.4096 0.332693
\(980\) 21.5895 0.689653
\(981\) 24.4252 0.779836
\(982\) −43.8334 −1.39878
\(983\) 14.8201 0.472688 0.236344 0.971669i \(-0.424051\pi\)
0.236344 + 0.971669i \(0.424051\pi\)
\(984\) −27.1844 −0.866608
\(985\) 10.2938 0.327988
\(986\) −27.4874 −0.875376
\(987\) −28.9877 −0.922688
\(988\) −14.1802 −0.451132
\(989\) −29.3115 −0.932051
\(990\) −5.06209 −0.160884
\(991\) −52.3768 −1.66380 −0.831902 0.554923i \(-0.812748\pi\)
−0.831902 + 0.554923i \(0.812748\pi\)
\(992\) 30.1657 0.957761
\(993\) 31.4247 0.997233
\(994\) 101.249 3.21143
\(995\) 22.0654 0.699520
\(996\) −4.31387 −0.136690
\(997\) 23.1646 0.733630 0.366815 0.930294i \(-0.380448\pi\)
0.366815 + 0.930294i \(0.380448\pi\)
\(998\) −61.3592 −1.94229
\(999\) −11.3377 −0.358708
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 241.2.a.b.1.9 12
3.2 odd 2 2169.2.a.h.1.4 12
4.3 odd 2 3856.2.a.n.1.8 12
5.4 even 2 6025.2.a.h.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.b.1.9 12 1.1 even 1 trivial
2169.2.a.h.1.4 12 3.2 odd 2
3856.2.a.n.1.8 12 4.3 odd 2
6025.2.a.h.1.4 12 5.4 even 2