Properties

Label 2025.2.a.y.1.4
Level $2025$
Weight $2$
Character 2025.1
Self dual yes
Analytic conductor $16.170$
Analytic rank $0$
Dimension $4$
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] = [2025,2,Mod(1,2025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2025, 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("2025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.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: \(16.1697064093\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.11661.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 225)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.63372\) of defining polynomial
Character \(\chi\) \(=\) 2025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.63372 q^{2} +4.93650 q^{4} -1.79743 q^{7} +7.73393 q^{8} +O(q^{10})\) \(q+2.63372 q^{2} +4.93650 q^{4} -1.79743 q^{7} +7.73393 q^{8} +1.80812 q^{11} +1.97183 q^{13} -4.73393 q^{14} +10.4960 q^{16} +4.80812 q^{17} +2.96467 q^{19} +4.76210 q^{22} -1.73393 q^{23} +5.19325 q^{26} -8.87300 q^{28} -7.36765 q^{29} -2.62303 q^{31} +12.1758 q^{32} +12.6633 q^{34} +11.6351 q^{37} +7.80812 q^{38} -2.46648 q^{41} -7.27814 q^{43} +8.92580 q^{44} -4.56668 q^{46} +6.29208 q^{47} -3.76926 q^{49} +9.73393 q^{52} +1.72540 q^{53} -13.9012 q^{56} -19.4044 q^{58} +11.0260 q^{59} -12.6704 q^{61} -6.90833 q^{62} +11.0756 q^{64} +9.10374 q^{67} +23.7353 q^{68} +1.27460 q^{71} -3.58770 q^{73} +30.6436 q^{74} +14.6351 q^{76} -3.24997 q^{77} +2.11090 q^{79} -6.49602 q^{82} +1.09883 q^{83} -19.1686 q^{86} +13.9839 q^{88} -13.2935 q^{89} -3.54422 q^{91} -8.55953 q^{92} +16.5716 q^{94} -3.83276 q^{97} -9.92718 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 4 q^{4} - q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 4 q^{4} - q^{7} + 9 q^{8} - q^{11} + 2 q^{13} + 3 q^{14} + 4 q^{16} + 11 q^{17} + 2 q^{19} + 3 q^{22} + 15 q^{23} - 10 q^{26} - 4 q^{28} + q^{29} - 4 q^{31} + 10 q^{32} + 9 q^{34} - q^{37} + 23 q^{38} - 5 q^{41} - 10 q^{43} + 22 q^{44} + 20 q^{47} - 3 q^{49} + 17 q^{52} + 20 q^{53} - 30 q^{56} - 18 q^{58} + 17 q^{59} - 13 q^{61} - 6 q^{62} + 19 q^{64} + 17 q^{67} + 34 q^{68} - 8 q^{71} + 2 q^{73} + 40 q^{74} + 11 q^{76} + 12 q^{77} - 7 q^{79} + 12 q^{82} + 30 q^{83} - 34 q^{86} + 9 q^{88} - 9 q^{89} - 17 q^{91} - 12 q^{92} + 3 q^{94} - 19 q^{97} + 13 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.63372 1.86232 0.931162 0.364606i \(-0.118796\pi\)
0.931162 + 0.364606i \(0.118796\pi\)
\(3\) 0 0
\(4\) 4.93650 2.46825
\(5\) 0 0
\(6\) 0 0
\(7\) −1.79743 −0.679364 −0.339682 0.940540i \(-0.610319\pi\)
−0.339682 + 0.940540i \(0.610319\pi\)
\(8\) 7.73393 2.73436
\(9\) 0 0
\(10\) 0 0
\(11\) 1.80812 0.545170 0.272585 0.962132i \(-0.412121\pi\)
0.272585 + 0.962132i \(0.412121\pi\)
\(12\) 0 0
\(13\) 1.97183 0.546887 0.273443 0.961888i \(-0.411837\pi\)
0.273443 + 0.961888i \(0.411837\pi\)
\(14\) −4.73393 −1.26520
\(15\) 0 0
\(16\) 10.4960 2.62401
\(17\) 4.80812 1.16614 0.583071 0.812421i \(-0.301851\pi\)
0.583071 + 0.812421i \(0.301851\pi\)
\(18\) 0 0
\(19\) 2.96467 0.680142 0.340071 0.940400i \(-0.389549\pi\)
0.340071 + 0.940400i \(0.389549\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 4.76210 1.01528
\(23\) −1.73393 −0.361549 −0.180774 0.983525i \(-0.557860\pi\)
−0.180774 + 0.983525i \(0.557860\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 5.19325 1.01848
\(27\) 0 0
\(28\) −8.87300 −1.67684
\(29\) −7.36765 −1.36814 −0.684069 0.729417i \(-0.739791\pi\)
−0.684069 + 0.729417i \(0.739791\pi\)
\(30\) 0 0
\(31\) −2.62303 −0.471110 −0.235555 0.971861i \(-0.575691\pi\)
−0.235555 + 0.971861i \(0.575691\pi\)
\(32\) 12.1758 2.15239
\(33\) 0 0
\(34\) 12.6633 2.17173
\(35\) 0 0
\(36\) 0 0
\(37\) 11.6351 1.91280 0.956399 0.292063i \(-0.0943417\pi\)
0.956399 + 0.292063i \(0.0943417\pi\)
\(38\) 7.80812 1.26664
\(39\) 0 0
\(40\) 0 0
\(41\) −2.46648 −0.385199 −0.192600 0.981277i \(-0.561692\pi\)
−0.192600 + 0.981277i \(0.561692\pi\)
\(42\) 0 0
\(43\) −7.27814 −1.10991 −0.554953 0.831882i \(-0.687264\pi\)
−0.554953 + 0.831882i \(0.687264\pi\)
\(44\) 8.92580 1.34562
\(45\) 0 0
\(46\) −4.56668 −0.673321
\(47\) 6.29208 0.917794 0.458897 0.888489i \(-0.348245\pi\)
0.458897 + 0.888489i \(0.348245\pi\)
\(48\) 0 0
\(49\) −3.76926 −0.538465
\(50\) 0 0
\(51\) 0 0
\(52\) 9.73393 1.34985
\(53\) 1.72540 0.237001 0.118501 0.992954i \(-0.462191\pi\)
0.118501 + 0.992954i \(0.462191\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −13.9012 −1.85762
\(57\) 0 0
\(58\) −19.4044 −2.54792
\(59\) 11.0260 1.43546 0.717732 0.696320i \(-0.245180\pi\)
0.717732 + 0.696320i \(0.245180\pi\)
\(60\) 0 0
\(61\) −12.6704 −1.62228 −0.811141 0.584851i \(-0.801153\pi\)
−0.811141 + 0.584851i \(0.801153\pi\)
\(62\) −6.90833 −0.877358
\(63\) 0 0
\(64\) 11.0756 1.38445
\(65\) 0 0
\(66\) 0 0
\(67\) 9.10374 1.11220 0.556100 0.831116i \(-0.312297\pi\)
0.556100 + 0.831116i \(0.312297\pi\)
\(68\) 23.7353 2.87833
\(69\) 0 0
\(70\) 0 0
\(71\) 1.27460 0.151268 0.0756338 0.997136i \(-0.475902\pi\)
0.0756338 + 0.997136i \(0.475902\pi\)
\(72\) 0 0
\(73\) −3.58770 −0.419908 −0.209954 0.977711i \(-0.567331\pi\)
−0.209954 + 0.977711i \(0.567331\pi\)
\(74\) 30.6436 3.56225
\(75\) 0 0
\(76\) 14.6351 1.67876
\(77\) −3.24997 −0.370369
\(78\) 0 0
\(79\) 2.11090 0.237495 0.118747 0.992924i \(-0.462112\pi\)
0.118747 + 0.992924i \(0.462112\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −6.49602 −0.717366
\(83\) 1.09883 0.120612 0.0603061 0.998180i \(-0.480792\pi\)
0.0603061 + 0.998180i \(0.480792\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −19.1686 −2.06701
\(87\) 0 0
\(88\) 13.9839 1.49069
\(89\) −13.2935 −1.40910 −0.704552 0.709653i \(-0.748852\pi\)
−0.704552 + 0.709653i \(0.748852\pi\)
\(90\) 0 0
\(91\) −3.54422 −0.371535
\(92\) −8.55953 −0.892392
\(93\) 0 0
\(94\) 16.5716 1.70923
\(95\) 0 0
\(96\) 0 0
\(97\) −3.83276 −0.389157 −0.194579 0.980887i \(-0.562334\pi\)
−0.194579 + 0.980887i \(0.562334\pi\)
\(98\) −9.92718 −1.00280
\(99\) 0 0
\(100\) 0 0
\(101\) 6.55237 0.651985 0.325993 0.945372i \(-0.394302\pi\)
0.325993 + 0.945372i \(0.394302\pi\)
\(102\) 0 0
\(103\) −8.07557 −0.795710 −0.397855 0.917448i \(-0.630245\pi\)
−0.397855 + 0.917448i \(0.630245\pi\)
\(104\) 15.2500 1.49538
\(105\) 0 0
\(106\) 4.54422 0.441373
\(107\) −8.97674 −0.867814 −0.433907 0.900958i \(-0.642865\pi\)
−0.433907 + 0.900958i \(0.642865\pi\)
\(108\) 0 0
\(109\) −6.34164 −0.607419 −0.303710 0.952765i \(-0.598225\pi\)
−0.303710 + 0.952765i \(0.598225\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −18.8658 −1.78265
\(113\) −14.9025 −1.40191 −0.700957 0.713204i \(-0.747243\pi\)
−0.700957 + 0.713204i \(0.747243\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −36.3704 −3.37691
\(117\) 0 0
\(118\) 29.0394 2.67330
\(119\) −8.64225 −0.792234
\(120\) 0 0
\(121\) −7.73069 −0.702790
\(122\) −33.3704 −3.02121
\(123\) 0 0
\(124\) −12.9486 −1.16282
\(125\) 0 0
\(126\) 0 0
\(127\) 3.62303 0.321492 0.160746 0.986996i \(-0.448610\pi\)
0.160746 + 0.986996i \(0.448610\pi\)
\(128\) 4.81844 0.425894
\(129\) 0 0
\(130\) 0 0
\(131\) 7.29345 0.637232 0.318616 0.947884i \(-0.396782\pi\)
0.318616 + 0.947884i \(0.396782\pi\)
\(132\) 0 0
\(133\) −5.32878 −0.462064
\(134\) 23.9767 2.07127
\(135\) 0 0
\(136\) 37.1857 3.18865
\(137\) −7.12621 −0.608833 −0.304417 0.952539i \(-0.598461\pi\)
−0.304417 + 0.952539i \(0.598461\pi\)
\(138\) 0 0
\(139\) −14.7107 −1.24774 −0.623871 0.781527i \(-0.714441\pi\)
−0.623871 + 0.781527i \(0.714441\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 3.35695 0.281709
\(143\) 3.56531 0.298146
\(144\) 0 0
\(145\) 0 0
\(146\) −9.44900 −0.782005
\(147\) 0 0
\(148\) 57.4366 4.72126
\(149\) −0.565309 −0.0463119 −0.0231560 0.999732i \(-0.507371\pi\)
−0.0231560 + 0.999732i \(0.507371\pi\)
\(150\) 0 0
\(151\) 0.153385 0.0124823 0.00624115 0.999981i \(-0.498013\pi\)
0.00624115 + 0.999981i \(0.498013\pi\)
\(152\) 22.9285 1.85975
\(153\) 0 0
\(154\) −8.55953 −0.689746
\(155\) 0 0
\(156\) 0 0
\(157\) −11.4607 −0.914663 −0.457332 0.889296i \(-0.651195\pi\)
−0.457332 + 0.889296i \(0.651195\pi\)
\(158\) 5.55953 0.442292
\(159\) 0 0
\(160\) 0 0
\(161\) 3.11661 0.245623
\(162\) 0 0
\(163\) −22.0595 −1.72783 −0.863915 0.503637i \(-0.831995\pi\)
−0.863915 + 0.503637i \(0.831995\pi\)
\(164\) −12.1758 −0.950768
\(165\) 0 0
\(166\) 2.89401 0.224619
\(167\) 17.0684 1.32079 0.660397 0.750917i \(-0.270388\pi\)
0.660397 + 0.750917i \(0.270388\pi\)
\(168\) 0 0
\(169\) −9.11189 −0.700915
\(170\) 0 0
\(171\) 0 0
\(172\) −35.9285 −2.73953
\(173\) 11.9447 0.908135 0.454067 0.890967i \(-0.349972\pi\)
0.454067 + 0.890967i \(0.349972\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 18.9781 1.43053
\(177\) 0 0
\(178\) −35.0113 −2.62421
\(179\) 8.54921 0.638998 0.319499 0.947587i \(-0.396485\pi\)
0.319499 + 0.947587i \(0.396485\pi\)
\(180\) 0 0
\(181\) −10.5524 −0.784351 −0.392176 0.919890i \(-0.628277\pi\)
−0.392176 + 0.919890i \(0.628277\pi\)
\(182\) −9.33449 −0.691918
\(183\) 0 0
\(184\) −13.4101 −0.988603
\(185\) 0 0
\(186\) 0 0
\(187\) 8.69368 0.635745
\(188\) 31.0608 2.26534
\(189\) 0 0
\(190\) 0 0
\(191\) −17.3372 −1.25448 −0.627239 0.778827i \(-0.715815\pi\)
−0.627239 + 0.778827i \(0.715815\pi\)
\(192\) 0 0
\(193\) 1.55953 0.112257 0.0561286 0.998424i \(-0.482124\pi\)
0.0561286 + 0.998424i \(0.482124\pi\)
\(194\) −10.0944 −0.724737
\(195\) 0 0
\(196\) −18.6069 −1.32907
\(197\) 17.9767 1.28079 0.640395 0.768046i \(-0.278771\pi\)
0.640395 + 0.768046i \(0.278771\pi\)
\(198\) 0 0
\(199\) 11.0225 0.781362 0.390681 0.920526i \(-0.372240\pi\)
0.390681 + 0.920526i \(0.372240\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 17.2571 1.21421
\(203\) 13.2428 0.929463
\(204\) 0 0
\(205\) 0 0
\(206\) −21.2688 −1.48187
\(207\) 0 0
\(208\) 20.6964 1.43503
\(209\) 5.36049 0.370793
\(210\) 0 0
\(211\) −23.9285 −1.64731 −0.823655 0.567092i \(-0.808068\pi\)
−0.823655 + 0.567092i \(0.808068\pi\)
\(212\) 8.51742 0.584979
\(213\) 0 0
\(214\) −23.6423 −1.61615
\(215\) 0 0
\(216\) 0 0
\(217\) 4.71470 0.320055
\(218\) −16.7021 −1.13121
\(219\) 0 0
\(220\) 0 0
\(221\) 9.48079 0.637747
\(222\) 0 0
\(223\) 21.7107 1.45385 0.726927 0.686715i \(-0.240948\pi\)
0.726927 + 0.686715i \(0.240948\pi\)
\(224\) −21.8851 −1.46226
\(225\) 0 0
\(226\) −39.2492 −2.61082
\(227\) 14.1002 0.935863 0.467932 0.883765i \(-0.344999\pi\)
0.467932 + 0.883765i \(0.344999\pi\)
\(228\) 0 0
\(229\) 3.67758 0.243021 0.121511 0.992590i \(-0.461226\pi\)
0.121511 + 0.992590i \(0.461226\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −56.9809 −3.74098
\(233\) −5.34164 −0.349943 −0.174971 0.984574i \(-0.555983\pi\)
−0.174971 + 0.984574i \(0.555983\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 54.4299 3.54308
\(237\) 0 0
\(238\) −22.7613 −1.47540
\(239\) −22.0335 −1.42523 −0.712613 0.701557i \(-0.752488\pi\)
−0.712613 + 0.701557i \(0.752488\pi\)
\(240\) 0 0
\(241\) −18.6472 −1.20117 −0.600585 0.799561i \(-0.705066\pi\)
−0.600585 + 0.799561i \(0.705066\pi\)
\(242\) −20.3605 −1.30882
\(243\) 0 0
\(244\) −62.5475 −4.00420
\(245\) 0 0
\(246\) 0 0
\(247\) 5.84582 0.371961
\(248\) −20.2863 −1.28818
\(249\) 0 0
\(250\) 0 0
\(251\) 14.6929 0.927407 0.463704 0.885990i \(-0.346520\pi\)
0.463704 + 0.885990i \(0.346520\pi\)
\(252\) 0 0
\(253\) −3.13515 −0.197105
\(254\) 9.54205 0.598721
\(255\) 0 0
\(256\) −9.46070 −0.591294
\(257\) 22.2089 1.38536 0.692678 0.721247i \(-0.256431\pi\)
0.692678 + 0.721247i \(0.256431\pi\)
\(258\) 0 0
\(259\) −20.9132 −1.29949
\(260\) 0 0
\(261\) 0 0
\(262\) 19.2089 1.18673
\(263\) 5.74001 0.353944 0.176972 0.984216i \(-0.443370\pi\)
0.176972 + 0.984216i \(0.443370\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −14.0345 −0.860513
\(267\) 0 0
\(268\) 44.9406 2.74519
\(269\) −15.6162 −0.952139 −0.476070 0.879408i \(-0.657939\pi\)
−0.476070 + 0.879408i \(0.657939\pi\)
\(270\) 0 0
\(271\) −6.75315 −0.410225 −0.205112 0.978738i \(-0.565756\pi\)
−0.205112 + 0.978738i \(0.565756\pi\)
\(272\) 50.4662 3.05996
\(273\) 0 0
\(274\) −18.7685 −1.13384
\(275\) 0 0
\(276\) 0 0
\(277\) 30.2966 1.82034 0.910172 0.414230i \(-0.135949\pi\)
0.910172 + 0.414230i \(0.135949\pi\)
\(278\) −38.7438 −2.32370
\(279\) 0 0
\(280\) 0 0
\(281\) 18.6351 1.11168 0.555838 0.831290i \(-0.312397\pi\)
0.555838 + 0.831290i \(0.312397\pi\)
\(282\) 0 0
\(283\) −5.67366 −0.337264 −0.168632 0.985679i \(-0.553935\pi\)
−0.168632 + 0.985679i \(0.553935\pi\)
\(284\) 6.29208 0.373366
\(285\) 0 0
\(286\) 9.39004 0.555245
\(287\) 4.43332 0.261690
\(288\) 0 0
\(289\) 6.11806 0.359886
\(290\) 0 0
\(291\) 0 0
\(292\) −17.7107 −1.03644
\(293\) 18.2773 1.06777 0.533887 0.845556i \(-0.320731\pi\)
0.533887 + 0.845556i \(0.320731\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 89.9850 5.23027
\(297\) 0 0
\(298\) −1.48887 −0.0862478
\(299\) −3.41900 −0.197726
\(300\) 0 0
\(301\) 13.0819 0.754030
\(302\) 0.403974 0.0232461
\(303\) 0 0
\(304\) 31.1173 1.78470
\(305\) 0 0
\(306\) 0 0
\(307\) −15.5050 −0.884915 −0.442458 0.896789i \(-0.645893\pi\)
−0.442458 + 0.896789i \(0.645893\pi\)
\(308\) −16.0435 −0.914162
\(309\) 0 0
\(310\) 0 0
\(311\) 30.4464 1.72646 0.863228 0.504814i \(-0.168439\pi\)
0.863228 + 0.504814i \(0.168439\pi\)
\(312\) 0 0
\(313\) −6.94936 −0.392801 −0.196401 0.980524i \(-0.562925\pi\)
−0.196401 + 0.980524i \(0.562925\pi\)
\(314\) −30.1843 −1.70340
\(315\) 0 0
\(316\) 10.4205 0.586196
\(317\) −16.1451 −0.906797 −0.453398 0.891308i \(-0.649788\pi\)
−0.453398 + 0.891308i \(0.649788\pi\)
\(318\) 0 0
\(319\) −13.3216 −0.745868
\(320\) 0 0
\(321\) 0 0
\(322\) 8.20828 0.457430
\(323\) 14.2545 0.793142
\(324\) 0 0
\(325\) 0 0
\(326\) −58.0985 −3.21778
\(327\) 0 0
\(328\) −19.0756 −1.05327
\(329\) −11.3096 −0.623516
\(330\) 0 0
\(331\) 12.6222 0.693781 0.346890 0.937906i \(-0.387238\pi\)
0.346890 + 0.937906i \(0.387238\pi\)
\(332\) 5.42437 0.297701
\(333\) 0 0
\(334\) 44.9535 2.45975
\(335\) 0 0
\(336\) 0 0
\(337\) −6.92040 −0.376978 −0.188489 0.982075i \(-0.560359\pi\)
−0.188489 + 0.982075i \(0.560359\pi\)
\(338\) −23.9982 −1.30533
\(339\) 0 0
\(340\) 0 0
\(341\) −4.74276 −0.256835
\(342\) 0 0
\(343\) 19.3570 1.04518
\(344\) −56.2886 −3.03488
\(345\) 0 0
\(346\) 31.4589 1.69124
\(347\) 13.8063 0.741163 0.370581 0.928800i \(-0.379158\pi\)
0.370581 + 0.928800i \(0.379158\pi\)
\(348\) 0 0
\(349\) 6.56768 0.351560 0.175780 0.984429i \(-0.443755\pi\)
0.175780 + 0.984429i \(0.443755\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 22.0153 1.17342
\(353\) −3.52499 −0.187616 −0.0938082 0.995590i \(-0.529904\pi\)
−0.0938082 + 0.995590i \(0.529904\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −65.6231 −3.47802
\(357\) 0 0
\(358\) 22.5162 1.19002
\(359\) 22.9285 1.21012 0.605061 0.796179i \(-0.293149\pi\)
0.605061 + 0.796179i \(0.293149\pi\)
\(360\) 0 0
\(361\) −10.2107 −0.537407
\(362\) −27.7920 −1.46072
\(363\) 0 0
\(364\) −17.4960 −0.917041
\(365\) 0 0
\(366\) 0 0
\(367\) 4.17931 0.218158 0.109079 0.994033i \(-0.465210\pi\)
0.109079 + 0.994033i \(0.465210\pi\)
\(368\) −18.1993 −0.948706
\(369\) 0 0
\(370\) 0 0
\(371\) −3.10127 −0.161010
\(372\) 0 0
\(373\) 6.84091 0.354209 0.177104 0.984192i \(-0.443327\pi\)
0.177104 + 0.984192i \(0.443327\pi\)
\(374\) 22.8968 1.18396
\(375\) 0 0
\(376\) 48.6625 2.50958
\(377\) −14.5277 −0.748217
\(378\) 0 0
\(379\) −12.7764 −0.656280 −0.328140 0.944629i \(-0.606422\pi\)
−0.328140 + 0.944629i \(0.606422\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −45.6615 −2.33624
\(383\) 7.53459 0.385000 0.192500 0.981297i \(-0.438341\pi\)
0.192500 + 0.981297i \(0.438341\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.10736 0.209059
\(387\) 0 0
\(388\) −18.9204 −0.960538
\(389\) 5.45175 0.276415 0.138207 0.990403i \(-0.455866\pi\)
0.138207 + 0.990403i \(0.455866\pi\)
\(390\) 0 0
\(391\) −8.33693 −0.421617
\(392\) −29.1511 −1.47235
\(393\) 0 0
\(394\) 47.3458 2.38525
\(395\) 0 0
\(396\) 0 0
\(397\) −5.64549 −0.283339 −0.141670 0.989914i \(-0.545247\pi\)
−0.141670 + 0.989914i \(0.545247\pi\)
\(398\) 29.0301 1.45515
\(399\) 0 0
\(400\) 0 0
\(401\) −5.50418 −0.274865 −0.137433 0.990511i \(-0.543885\pi\)
−0.137433 + 0.990511i \(0.543885\pi\)
\(402\) 0 0
\(403\) −5.17216 −0.257643
\(404\) 32.3458 1.60926
\(405\) 0 0
\(406\) 34.8779 1.73096
\(407\) 21.0377 1.04280
\(408\) 0 0
\(409\) 32.8530 1.62448 0.812238 0.583327i \(-0.198249\pi\)
0.812238 + 0.583327i \(0.198249\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −39.8650 −1.96401
\(413\) −19.8184 −0.975202
\(414\) 0 0
\(415\) 0 0
\(416\) 24.0085 1.17712
\(417\) 0 0
\(418\) 14.1181 0.690537
\(419\) 22.8591 1.11674 0.558369 0.829593i \(-0.311427\pi\)
0.558369 + 0.829593i \(0.311427\pi\)
\(420\) 0 0
\(421\) 17.9414 0.874411 0.437205 0.899362i \(-0.355968\pi\)
0.437205 + 0.899362i \(0.355968\pi\)
\(422\) −63.0212 −3.06782
\(423\) 0 0
\(424\) 13.3441 0.648046
\(425\) 0 0
\(426\) 0 0
\(427\) 22.7742 1.10212
\(428\) −44.3137 −2.14198
\(429\) 0 0
\(430\) 0 0
\(431\) −6.18871 −0.298100 −0.149050 0.988830i \(-0.547622\pi\)
−0.149050 + 0.988830i \(0.547622\pi\)
\(432\) 0 0
\(433\) −3.11806 −0.149844 −0.0749221 0.997189i \(-0.523871\pi\)
−0.0749221 + 0.997189i \(0.523871\pi\)
\(434\) 12.4172 0.596045
\(435\) 0 0
\(436\) −31.3055 −1.49926
\(437\) −5.14052 −0.245904
\(438\) 0 0
\(439\) 13.5099 0.644792 0.322396 0.946605i \(-0.395512\pi\)
0.322396 + 0.946605i \(0.395512\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 24.9698 1.18769
\(443\) −24.2773 −1.15345 −0.576726 0.816938i \(-0.695670\pi\)
−0.576726 + 0.816938i \(0.695670\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 57.1799 2.70755
\(447\) 0 0
\(448\) −19.9075 −0.940543
\(449\) −24.1437 −1.13941 −0.569705 0.821849i \(-0.692943\pi\)
−0.569705 + 0.821849i \(0.692943\pi\)
\(450\) 0 0
\(451\) −4.45970 −0.209999
\(452\) −73.5664 −3.46027
\(453\) 0 0
\(454\) 37.1360 1.74288
\(455\) 0 0
\(456\) 0 0
\(457\) −2.82157 −0.131987 −0.0659937 0.997820i \(-0.521022\pi\)
−0.0659937 + 0.997820i \(0.521022\pi\)
\(458\) 9.68573 0.452585
\(459\) 0 0
\(460\) 0 0
\(461\) 21.4572 0.999363 0.499681 0.866209i \(-0.333450\pi\)
0.499681 + 0.866209i \(0.333450\pi\)
\(462\) 0 0
\(463\) −19.8033 −0.920339 −0.460170 0.887831i \(-0.652211\pi\)
−0.460170 + 0.887831i \(0.652211\pi\)
\(464\) −77.3310 −3.59000
\(465\) 0 0
\(466\) −14.0684 −0.651707
\(467\) 22.7210 1.05140 0.525701 0.850669i \(-0.323803\pi\)
0.525701 + 0.850669i \(0.323803\pi\)
\(468\) 0 0
\(469\) −16.3633 −0.755588
\(470\) 0 0
\(471\) 0 0
\(472\) 85.2743 3.92507
\(473\) −13.1598 −0.605088
\(474\) 0 0
\(475\) 0 0
\(476\) −42.6625 −1.95543
\(477\) 0 0
\(478\) −58.0300 −2.65423
\(479\) 21.2880 0.972672 0.486336 0.873772i \(-0.338333\pi\)
0.486336 + 0.873772i \(0.338333\pi\)
\(480\) 0 0
\(481\) 22.9424 1.04608
\(482\) −49.1115 −2.23697
\(483\) 0 0
\(484\) −38.1625 −1.73466
\(485\) 0 0
\(486\) 0 0
\(487\) −9.58690 −0.434424 −0.217212 0.976124i \(-0.569696\pi\)
−0.217212 + 0.976124i \(0.569696\pi\)
\(488\) −97.9921 −4.43590
\(489\) 0 0
\(490\) 0 0
\(491\) −37.8443 −1.70789 −0.853945 0.520363i \(-0.825797\pi\)
−0.853945 + 0.520363i \(0.825797\pi\)
\(492\) 0 0
\(493\) −35.4246 −1.59544
\(494\) 15.3963 0.692711
\(495\) 0 0
\(496\) −27.5314 −1.23619
\(497\) −2.29101 −0.102766
\(498\) 0 0
\(499\) 16.9253 0.757681 0.378840 0.925462i \(-0.376323\pi\)
0.378840 + 0.925462i \(0.376323\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 38.6970 1.72713
\(503\) 40.4168 1.80210 0.901048 0.433719i \(-0.142799\pi\)
0.901048 + 0.433719i \(0.142799\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −8.25713 −0.367074
\(507\) 0 0
\(508\) 17.8851 0.793522
\(509\) −41.4067 −1.83532 −0.917660 0.397366i \(-0.869924\pi\)
−0.917660 + 0.397366i \(0.869924\pi\)
\(510\) 0 0
\(511\) 6.44863 0.285270
\(512\) −34.5537 −1.52707
\(513\) 0 0
\(514\) 58.4922 2.57998
\(515\) 0 0
\(516\) 0 0
\(517\) 11.3769 0.500354
\(518\) −55.0797 −2.42006
\(519\) 0 0
\(520\) 0 0
\(521\) −17.0301 −0.746103 −0.373052 0.927811i \(-0.621689\pi\)
−0.373052 + 0.927811i \(0.621689\pi\)
\(522\) 0 0
\(523\) 9.57651 0.418751 0.209376 0.977835i \(-0.432857\pi\)
0.209376 + 0.977835i \(0.432857\pi\)
\(524\) 36.0041 1.57285
\(525\) 0 0
\(526\) 15.1176 0.659159
\(527\) −12.6118 −0.549380
\(528\) 0 0
\(529\) −19.9935 −0.869283
\(530\) 0 0
\(531\) 0 0
\(532\) −26.3055 −1.14049
\(533\) −4.86347 −0.210660
\(534\) 0 0
\(535\) 0 0
\(536\) 70.4077 3.04115
\(537\) 0 0
\(538\) −41.1289 −1.77319
\(539\) −6.81528 −0.293555
\(540\) 0 0
\(541\) −0.833751 −0.0358458 −0.0179229 0.999839i \(-0.505705\pi\)
−0.0179229 + 0.999839i \(0.505705\pi\)
\(542\) −17.7859 −0.763971
\(543\) 0 0
\(544\) 58.5426 2.50999
\(545\) 0 0
\(546\) 0 0
\(547\) −28.3270 −1.21117 −0.605587 0.795779i \(-0.707062\pi\)
−0.605587 + 0.795779i \(0.707062\pi\)
\(548\) −35.1785 −1.50275
\(549\) 0 0
\(550\) 0 0
\(551\) −21.8427 −0.930529
\(552\) 0 0
\(553\) −3.79419 −0.161345
\(554\) 79.7928 3.39007
\(555\) 0 0
\(556\) −72.6192 −3.07974
\(557\) 11.5042 0.487448 0.243724 0.969845i \(-0.421631\pi\)
0.243724 + 0.969845i \(0.421631\pi\)
\(558\) 0 0
\(559\) −14.3512 −0.606993
\(560\) 0 0
\(561\) 0 0
\(562\) 49.0797 2.07030
\(563\) 33.0059 1.39103 0.695517 0.718510i \(-0.255176\pi\)
0.695517 + 0.718510i \(0.255176\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −14.9429 −0.628095
\(567\) 0 0
\(568\) 9.85769 0.413619
\(569\) −27.0088 −1.13227 −0.566135 0.824313i \(-0.691562\pi\)
−0.566135 + 0.824313i \(0.691562\pi\)
\(570\) 0 0
\(571\) −24.4244 −1.02213 −0.511064 0.859543i \(-0.670749\pi\)
−0.511064 + 0.859543i \(0.670749\pi\)
\(572\) 17.6001 0.735899
\(573\) 0 0
\(574\) 11.6761 0.487352
\(575\) 0 0
\(576\) 0 0
\(577\) 14.7976 0.616033 0.308017 0.951381i \(-0.400335\pi\)
0.308017 + 0.951381i \(0.400335\pi\)
\(578\) 16.1133 0.670224
\(579\) 0 0
\(580\) 0 0
\(581\) −1.97507 −0.0819396
\(582\) 0 0
\(583\) 3.11973 0.129206
\(584\) −27.7470 −1.14818
\(585\) 0 0
\(586\) 48.1375 1.98854
\(587\) 30.5780 1.26209 0.631044 0.775747i \(-0.282627\pi\)
0.631044 + 0.775747i \(0.282627\pi\)
\(588\) 0 0
\(589\) −7.77641 −0.320421
\(590\) 0 0
\(591\) 0 0
\(592\) 122.122 5.01919
\(593\) −5.09990 −0.209428 −0.104714 0.994502i \(-0.533393\pi\)
−0.104714 + 0.994502i \(0.533393\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2.79065 −0.114309
\(597\) 0 0
\(598\) −9.00471 −0.368230
\(599\) −0.565309 −0.0230979 −0.0115490 0.999933i \(-0.503676\pi\)
−0.0115490 + 0.999933i \(0.503676\pi\)
\(600\) 0 0
\(601\) −11.0096 −0.449091 −0.224546 0.974464i \(-0.572090\pi\)
−0.224546 + 0.974464i \(0.572090\pi\)
\(602\) 34.4542 1.40425
\(603\) 0 0
\(604\) 0.757185 0.0308094
\(605\) 0 0
\(606\) 0 0
\(607\) −19.0983 −0.775174 −0.387587 0.921833i \(-0.626691\pi\)
−0.387587 + 0.921833i \(0.626691\pi\)
\(608\) 36.0972 1.46393
\(609\) 0 0
\(610\) 0 0
\(611\) 12.4069 0.501929
\(612\) 0 0
\(613\) 9.33918 0.377206 0.188603 0.982053i \(-0.439604\pi\)
0.188603 + 0.982053i \(0.439604\pi\)
\(614\) −40.8358 −1.64800
\(615\) 0 0
\(616\) −25.1350 −1.01272
\(617\) −24.4154 −0.982928 −0.491464 0.870898i \(-0.663538\pi\)
−0.491464 + 0.870898i \(0.663538\pi\)
\(618\) 0 0
\(619\) 39.4863 1.58709 0.793544 0.608513i \(-0.208234\pi\)
0.793544 + 0.608513i \(0.208234\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 80.1874 3.21522
\(623\) 23.8940 0.957293
\(624\) 0 0
\(625\) 0 0
\(626\) −18.3027 −0.731523
\(627\) 0 0
\(628\) −56.5757 −2.25762
\(629\) 55.9430 2.23059
\(630\) 0 0
\(631\) 42.1634 1.67850 0.839249 0.543747i \(-0.182995\pi\)
0.839249 + 0.543747i \(0.182995\pi\)
\(632\) 16.3255 0.649395
\(633\) 0 0
\(634\) −42.5216 −1.68875
\(635\) 0 0
\(636\) 0 0
\(637\) −7.43232 −0.294479
\(638\) −35.0855 −1.38905
\(639\) 0 0
\(640\) 0 0
\(641\) 35.3155 1.39488 0.697438 0.716645i \(-0.254323\pi\)
0.697438 + 0.716645i \(0.254323\pi\)
\(642\) 0 0
\(643\) 14.1954 0.559813 0.279906 0.960027i \(-0.409697\pi\)
0.279906 + 0.960027i \(0.409697\pi\)
\(644\) 15.3851 0.606259
\(645\) 0 0
\(646\) 37.5424 1.47709
\(647\) 17.4897 0.687593 0.343796 0.939044i \(-0.388287\pi\)
0.343796 + 0.939044i \(0.388287\pi\)
\(648\) 0 0
\(649\) 19.9364 0.782572
\(650\) 0 0
\(651\) 0 0
\(652\) −108.897 −4.26472
\(653\) −10.9772 −0.429569 −0.214785 0.976661i \(-0.568905\pi\)
−0.214785 + 0.976661i \(0.568905\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −25.8882 −1.01077
\(657\) 0 0
\(658\) −29.7862 −1.16119
\(659\) −15.7876 −0.614998 −0.307499 0.951548i \(-0.599492\pi\)
−0.307499 + 0.951548i \(0.599492\pi\)
\(660\) 0 0
\(661\) 49.8932 1.94062 0.970311 0.241862i \(-0.0777581\pi\)
0.970311 + 0.241862i \(0.0777581\pi\)
\(662\) 33.2435 1.29204
\(663\) 0 0
\(664\) 8.49827 0.329797
\(665\) 0 0
\(666\) 0 0
\(667\) 12.7750 0.494649
\(668\) 84.2582 3.26005
\(669\) 0 0
\(670\) 0 0
\(671\) −22.9097 −0.884419
\(672\) 0 0
\(673\) 28.8395 1.11168 0.555840 0.831289i \(-0.312397\pi\)
0.555840 + 0.831289i \(0.312397\pi\)
\(674\) −18.2264 −0.702055
\(675\) 0 0
\(676\) −44.9809 −1.73003
\(677\) 10.4636 0.402150 0.201075 0.979576i \(-0.435557\pi\)
0.201075 + 0.979576i \(0.435557\pi\)
\(678\) 0 0
\(679\) 6.88910 0.264379
\(680\) 0 0
\(681\) 0 0
\(682\) −12.4911 −0.478309
\(683\) −16.1875 −0.619396 −0.309698 0.950835i \(-0.600228\pi\)
−0.309698 + 0.950835i \(0.600228\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 50.9809 1.94646
\(687\) 0 0
\(688\) −76.3916 −2.91240
\(689\) 3.40218 0.129613
\(690\) 0 0
\(691\) 9.88362 0.375991 0.187995 0.982170i \(-0.439801\pi\)
0.187995 + 0.982170i \(0.439801\pi\)
\(692\) 58.9648 2.24150
\(693\) 0 0
\(694\) 36.3621 1.38029
\(695\) 0 0
\(696\) 0 0
\(697\) −11.8591 −0.449197
\(698\) 17.2974 0.654718
\(699\) 0 0
\(700\) 0 0
\(701\) 43.9692 1.66069 0.830346 0.557248i \(-0.188143\pi\)
0.830346 + 0.557248i \(0.188143\pi\)
\(702\) 0 0
\(703\) 34.4942 1.30097
\(704\) 20.0260 0.754758
\(705\) 0 0
\(706\) −9.28385 −0.349402
\(707\) −11.7774 −0.442935
\(708\) 0 0
\(709\) 25.2260 0.947384 0.473692 0.880691i \(-0.342921\pi\)
0.473692 + 0.880691i \(0.342921\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −102.811 −3.85299
\(713\) 4.54814 0.170329
\(714\) 0 0
\(715\) 0 0
\(716\) 42.2032 1.57721
\(717\) 0 0
\(718\) 60.3875 2.25364
\(719\) −36.8600 −1.37465 −0.687323 0.726352i \(-0.741214\pi\)
−0.687323 + 0.726352i \(0.741214\pi\)
\(720\) 0 0
\(721\) 14.5153 0.540576
\(722\) −26.8922 −1.00082
\(723\) 0 0
\(724\) −52.0918 −1.93598
\(725\) 0 0
\(726\) 0 0
\(727\) −38.2451 −1.41843 −0.709217 0.704990i \(-0.750951\pi\)
−0.709217 + 0.704990i \(0.750951\pi\)
\(728\) −27.4107 −1.01591
\(729\) 0 0
\(730\) 0 0
\(731\) −34.9942 −1.29431
\(732\) 0 0
\(733\) 39.5832 1.46204 0.731020 0.682356i \(-0.239045\pi\)
0.731020 + 0.682356i \(0.239045\pi\)
\(734\) 11.0072 0.406282
\(735\) 0 0
\(736\) −21.1119 −0.778195
\(737\) 16.4607 0.606338
\(738\) 0 0
\(739\) −8.24773 −0.303398 −0.151699 0.988427i \(-0.548474\pi\)
−0.151699 + 0.988427i \(0.548474\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −8.16790 −0.299853
\(743\) 1.30818 0.0479925 0.0239963 0.999712i \(-0.492361\pi\)
0.0239963 + 0.999712i \(0.492361\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 18.0171 0.659651
\(747\) 0 0
\(748\) 42.9164 1.56918
\(749\) 16.1350 0.589562
\(750\) 0 0
\(751\) 28.4468 1.03804 0.519020 0.854762i \(-0.326297\pi\)
0.519020 + 0.854762i \(0.326297\pi\)
\(752\) 66.0418 2.40830
\(753\) 0 0
\(754\) −38.2620 −1.39342
\(755\) 0 0
\(756\) 0 0
\(757\) −38.2012 −1.38845 −0.694223 0.719760i \(-0.744252\pi\)
−0.694223 + 0.719760i \(0.744252\pi\)
\(758\) −33.6495 −1.22221
\(759\) 0 0
\(760\) 0 0
\(761\) 22.1904 0.804401 0.402200 0.915552i \(-0.368245\pi\)
0.402200 + 0.915552i \(0.368245\pi\)
\(762\) 0 0
\(763\) 11.3986 0.412659
\(764\) −85.5852 −3.09637
\(765\) 0 0
\(766\) 19.8440 0.716994
\(767\) 21.7414 0.785036
\(768\) 0 0
\(769\) −16.9130 −0.609900 −0.304950 0.952368i \(-0.598640\pi\)
−0.304950 + 0.952368i \(0.598640\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.69860 0.277079
\(773\) −38.6464 −1.39001 −0.695007 0.719003i \(-0.744599\pi\)
−0.695007 + 0.719003i \(0.744599\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −29.6423 −1.06409
\(777\) 0 0
\(778\) 14.3584 0.514774
\(779\) −7.31230 −0.261990
\(780\) 0 0
\(781\) 2.30464 0.0824665
\(782\) −21.9572 −0.785187
\(783\) 0 0
\(784\) −39.5622 −1.41294
\(785\) 0 0
\(786\) 0 0
\(787\) 10.9147 0.389067 0.194533 0.980896i \(-0.437681\pi\)
0.194533 + 0.980896i \(0.437681\pi\)
\(788\) 88.7422 3.16131
\(789\) 0 0
\(790\) 0 0
\(791\) 26.7862 0.952409
\(792\) 0 0
\(793\) −24.9839 −0.887204
\(794\) −14.8687 −0.527669
\(795\) 0 0
\(796\) 54.4124 1.92860
\(797\) −2.92550 −0.103627 −0.0518133 0.998657i \(-0.516500\pi\)
−0.0518133 + 0.998657i \(0.516500\pi\)
\(798\) 0 0
\(799\) 30.2531 1.07028
\(800\) 0 0
\(801\) 0 0
\(802\) −14.4965 −0.511888
\(803\) −6.48700 −0.228921
\(804\) 0 0
\(805\) 0 0
\(806\) −13.6220 −0.479816
\(807\) 0 0
\(808\) 50.6755 1.78276
\(809\) −37.9241 −1.33334 −0.666671 0.745352i \(-0.732281\pi\)
−0.666671 + 0.745352i \(0.732281\pi\)
\(810\) 0 0
\(811\) 19.5050 0.684912 0.342456 0.939534i \(-0.388741\pi\)
0.342456 + 0.939534i \(0.388741\pi\)
\(812\) 65.3731 2.29415
\(813\) 0 0
\(814\) 55.4075 1.94203
\(815\) 0 0
\(816\) 0 0
\(817\) −21.5773 −0.754894
\(818\) 86.5257 3.02530
\(819\) 0 0
\(820\) 0 0
\(821\) 44.7524 1.56187 0.780934 0.624613i \(-0.214743\pi\)
0.780934 + 0.624613i \(0.214743\pi\)
\(822\) 0 0
\(823\) −35.8994 −1.25137 −0.625687 0.780074i \(-0.715181\pi\)
−0.625687 + 0.780074i \(0.715181\pi\)
\(824\) −62.4559 −2.17575
\(825\) 0 0
\(826\) −52.1963 −1.81614
\(827\) −16.2717 −0.565822 −0.282911 0.959146i \(-0.591300\pi\)
−0.282911 + 0.959146i \(0.591300\pi\)
\(828\) 0 0
\(829\) −20.6592 −0.717525 −0.358762 0.933429i \(-0.616801\pi\)
−0.358762 + 0.933429i \(0.616801\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 21.8391 0.757135
\(833\) −18.1230 −0.627926
\(834\) 0 0
\(835\) 0 0
\(836\) 26.4621 0.915210
\(837\) 0 0
\(838\) 60.2044 2.07973
\(839\) −12.2367 −0.422459 −0.211229 0.977437i \(-0.567747\pi\)
−0.211229 + 0.977437i \(0.567747\pi\)
\(840\) 0 0
\(841\) 25.2823 0.871802
\(842\) 47.2527 1.62844
\(843\) 0 0
\(844\) −118.123 −4.06597
\(845\) 0 0
\(846\) 0 0
\(847\) 13.8953 0.477450
\(848\) 18.1098 0.621893
\(849\) 0 0
\(850\) 0 0
\(851\) −20.1744 −0.691570
\(852\) 0 0
\(853\) 16.4293 0.562528 0.281264 0.959630i \(-0.409246\pi\)
0.281264 + 0.959630i \(0.409246\pi\)
\(854\) 59.9809 2.05250
\(855\) 0 0
\(856\) −69.4255 −2.37291
\(857\) 49.0897 1.67687 0.838436 0.545000i \(-0.183470\pi\)
0.838436 + 0.545000i \(0.183470\pi\)
\(858\) 0 0
\(859\) 41.0908 1.40200 0.700999 0.713162i \(-0.252738\pi\)
0.700999 + 0.713162i \(0.252738\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −16.2994 −0.555158
\(863\) 47.8366 1.62838 0.814188 0.580601i \(-0.197183\pi\)
0.814188 + 0.580601i \(0.197183\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −8.21210 −0.279058
\(867\) 0 0
\(868\) 23.2741 0.789975
\(869\) 3.81677 0.129475
\(870\) 0 0
\(871\) 17.9510 0.608247
\(872\) −49.0458 −1.66090
\(873\) 0 0
\(874\) −13.5387 −0.457954
\(875\) 0 0
\(876\) 0 0
\(877\) −11.3772 −0.384180 −0.192090 0.981377i \(-0.561527\pi\)
−0.192090 + 0.981377i \(0.561527\pi\)
\(878\) 35.5813 1.20081
\(879\) 0 0
\(880\) 0 0
\(881\) −43.9924 −1.48214 −0.741071 0.671426i \(-0.765682\pi\)
−0.741071 + 0.671426i \(0.765682\pi\)
\(882\) 0 0
\(883\) 37.6820 1.26810 0.634051 0.773291i \(-0.281391\pi\)
0.634051 + 0.773291i \(0.281391\pi\)
\(884\) 46.8019 1.57412
\(885\) 0 0
\(886\) −63.9398 −2.14810
\(887\) −33.2833 −1.11754 −0.558771 0.829322i \(-0.688727\pi\)
−0.558771 + 0.829322i \(0.688727\pi\)
\(888\) 0 0
\(889\) −6.51213 −0.218410
\(890\) 0 0
\(891\) 0 0
\(892\) 107.175 3.58847
\(893\) 18.6539 0.624230
\(894\) 0 0
\(895\) 0 0
\(896\) −8.66080 −0.289337
\(897\) 0 0
\(898\) −63.5878 −2.12195
\(899\) 19.3255 0.644543
\(900\) 0 0
\(901\) 8.29592 0.276377
\(902\) −11.7456 −0.391086
\(903\) 0 0
\(904\) −115.255 −3.83333
\(905\) 0 0
\(906\) 0 0
\(907\) −14.0784 −0.467464 −0.233732 0.972301i \(-0.575094\pi\)
−0.233732 + 0.972301i \(0.575094\pi\)
\(908\) 69.6056 2.30994
\(909\) 0 0
\(910\) 0 0
\(911\) 7.31522 0.242364 0.121182 0.992630i \(-0.461332\pi\)
0.121182 + 0.992630i \(0.461332\pi\)
\(912\) 0 0
\(913\) 1.98682 0.0657542
\(914\) −7.43123 −0.245803
\(915\) 0 0
\(916\) 18.1544 0.599838
\(917\) −13.1094 −0.432912
\(918\) 0 0
\(919\) 4.44684 0.146688 0.0733438 0.997307i \(-0.476633\pi\)
0.0733438 + 0.997307i \(0.476633\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 56.5124 1.86114
\(923\) 2.51330 0.0827262
\(924\) 0 0
\(925\) 0 0
\(926\) −52.1565 −1.71397
\(927\) 0 0
\(928\) −89.7068 −2.94477
\(929\) 13.1133 0.430232 0.215116 0.976588i \(-0.430987\pi\)
0.215116 + 0.976588i \(0.430987\pi\)
\(930\) 0 0
\(931\) −11.1746 −0.366233
\(932\) −26.3690 −0.863746
\(933\) 0 0
\(934\) 59.8408 1.95805
\(935\) 0 0
\(936\) 0 0
\(937\) −33.5187 −1.09501 −0.547504 0.836803i \(-0.684422\pi\)
−0.547504 + 0.836803i \(0.684422\pi\)
\(938\) −43.0964 −1.40715
\(939\) 0 0
\(940\) 0 0
\(941\) 1.79076 0.0583772 0.0291886 0.999574i \(-0.490708\pi\)
0.0291886 + 0.999574i \(0.490708\pi\)
\(942\) 0 0
\(943\) 4.27669 0.139268
\(944\) 115.729 3.76667
\(945\) 0 0
\(946\) −34.6592 −1.12687
\(947\) −52.6350 −1.71041 −0.855204 0.518292i \(-0.826568\pi\)
−0.855204 + 0.518292i \(0.826568\pi\)
\(948\) 0 0
\(949\) −7.07432 −0.229642
\(950\) 0 0
\(951\) 0 0
\(952\) −66.8386 −2.16625
\(953\) −18.4072 −0.596268 −0.298134 0.954524i \(-0.596364\pi\)
−0.298134 + 0.954524i \(0.596364\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −108.768 −3.51781
\(957\) 0 0
\(958\) 56.0666 1.81143
\(959\) 12.8088 0.413619
\(960\) 0 0
\(961\) −24.1197 −0.778056
\(962\) 60.4240 1.94815
\(963\) 0 0
\(964\) −92.0517 −2.96479
\(965\) 0 0
\(966\) 0 0
\(967\) −6.09579 −0.196027 −0.0980137 0.995185i \(-0.531249\pi\)
−0.0980137 + 0.995185i \(0.531249\pi\)
\(968\) −59.7886 −1.92168
\(969\) 0 0
\(970\) 0 0
\(971\) −34.1077 −1.09457 −0.547283 0.836947i \(-0.684338\pi\)
−0.547283 + 0.836947i \(0.684338\pi\)
\(972\) 0 0
\(973\) 26.4414 0.847671
\(974\) −25.2493 −0.809038
\(975\) 0 0
\(976\) −132.989 −4.25688
\(977\) 48.7914 1.56098 0.780488 0.625171i \(-0.214971\pi\)
0.780488 + 0.625171i \(0.214971\pi\)
\(978\) 0 0
\(979\) −24.0362 −0.768201
\(980\) 0 0
\(981\) 0 0
\(982\) −99.6715 −3.18065
\(983\) 33.4172 1.06584 0.532922 0.846164i \(-0.321094\pi\)
0.532922 + 0.846164i \(0.321094\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −93.2985 −2.97123
\(987\) 0 0
\(988\) 28.8579 0.918092
\(989\) 12.6198 0.401285
\(990\) 0 0
\(991\) −19.3512 −0.614713 −0.307356 0.951595i \(-0.599444\pi\)
−0.307356 + 0.951595i \(0.599444\pi\)
\(992\) −31.9374 −1.01401
\(993\) 0 0
\(994\) −6.03388 −0.191383
\(995\) 0 0
\(996\) 0 0
\(997\) 39.5442 1.25238 0.626189 0.779671i \(-0.284614\pi\)
0.626189 + 0.779671i \(0.284614\pi\)
\(998\) 44.5766 1.41105
\(999\) 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 2025.2.a.y.1.4 4
3.2 odd 2 2025.2.a.p.1.1 4
5.2 odd 4 2025.2.b.n.649.8 8
5.3 odd 4 2025.2.b.n.649.1 8
5.4 even 2 2025.2.a.q.1.1 4
9.2 odd 6 675.2.e.e.226.4 8
9.4 even 3 225.2.e.c.151.1 yes 8
9.5 odd 6 675.2.e.e.451.4 8
9.7 even 3 225.2.e.c.76.1 8
15.2 even 4 2025.2.b.o.649.1 8
15.8 even 4 2025.2.b.o.649.8 8
15.14 odd 2 2025.2.a.z.1.4 4
45.2 even 12 675.2.k.c.199.1 16
45.4 even 6 225.2.e.e.151.4 yes 8
45.7 odd 12 225.2.k.c.49.8 16
45.13 odd 12 225.2.k.c.124.8 16
45.14 odd 6 675.2.e.c.451.1 8
45.22 odd 12 225.2.k.c.124.1 16
45.23 even 12 675.2.k.c.424.1 16
45.29 odd 6 675.2.e.c.226.1 8
45.32 even 12 675.2.k.c.424.8 16
45.34 even 6 225.2.e.e.76.4 yes 8
45.38 even 12 675.2.k.c.199.8 16
45.43 odd 12 225.2.k.c.49.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.e.c.76.1 8 9.7 even 3
225.2.e.c.151.1 yes 8 9.4 even 3
225.2.e.e.76.4 yes 8 45.34 even 6
225.2.e.e.151.4 yes 8 45.4 even 6
225.2.k.c.49.1 16 45.43 odd 12
225.2.k.c.49.8 16 45.7 odd 12
225.2.k.c.124.1 16 45.22 odd 12
225.2.k.c.124.8 16 45.13 odd 12
675.2.e.c.226.1 8 45.29 odd 6
675.2.e.c.451.1 8 45.14 odd 6
675.2.e.e.226.4 8 9.2 odd 6
675.2.e.e.451.4 8 9.5 odd 6
675.2.k.c.199.1 16 45.2 even 12
675.2.k.c.199.8 16 45.38 even 12
675.2.k.c.424.1 16 45.23 even 12
675.2.k.c.424.8 16 45.32 even 12
2025.2.a.p.1.1 4 3.2 odd 2
2025.2.a.q.1.1 4 5.4 even 2
2025.2.a.y.1.4 4 1.1 even 1 trivial
2025.2.a.z.1.4 4 15.14 odd 2
2025.2.b.n.649.1 8 5.3 odd 4
2025.2.b.n.649.8 8 5.2 odd 4
2025.2.b.o.649.1 8 15.2 even 4
2025.2.b.o.649.8 8 15.8 even 4