Properties

Label 2025.2.b.o.649.8
Level $2025$
Weight $2$
Character 2025.649
Analytic conductor $16.170$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2025,2,Mod(649,2025)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2025.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.1697064093\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.34810603776.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 12x^{6} + 42x^{4} + 49x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 225)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.8
Root \(2.63372i\) of defining polynomial
Character \(\chi\) \(=\) 2025.649
Dual form 2025.2.b.o.649.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.63372i q^{2} -4.93650 q^{4} +1.79743i q^{7} -7.73393i q^{8} +O(q^{10})\) \(q+2.63372i q^{2} -4.93650 q^{4} +1.79743i q^{7} -7.73393i q^{8} -1.80812 q^{11} +1.97183i q^{13} -4.73393 q^{14} +10.4960 q^{16} +4.80812i q^{17} -2.96467 q^{19} -4.76210i q^{22} +1.73393i q^{23} -5.19325 q^{26} -8.87300i q^{28} -7.36765 q^{29} -2.62303 q^{31} +12.1758i q^{32} -12.6633 q^{34} -11.6351i q^{37} -7.80812i q^{38} +2.46648 q^{41} -7.27814i q^{43} +8.92580 q^{44} -4.56668 q^{46} +6.29208i q^{47} +3.76926 q^{49} -9.73393i q^{52} -1.72540i q^{53} +13.9012 q^{56} -19.4044i q^{58} +11.0260 q^{59} -12.6704 q^{61} -6.90833i q^{62} -11.0756 q^{64} -9.10374i q^{67} -23.7353i q^{68} -1.27460 q^{71} -3.58770i q^{73} +30.6436 q^{74} +14.6351 q^{76} -3.24997i q^{77} -2.11090 q^{79} +6.49602i q^{82} -1.09883i q^{83} +19.1686 q^{86} +13.9839i q^{88} -13.2935 q^{89} -3.54422 q^{91} -8.55953i q^{92} -16.5716 q^{94} +3.83276i q^{97} +9.92718i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 2 q^{11} + 6 q^{14} + 8 q^{16} - 4 q^{19} + 20 q^{26} + 2 q^{29} - 8 q^{31} - 18 q^{34} + 10 q^{41} + 44 q^{44} + 6 q^{49} + 60 q^{56} + 34 q^{59} - 26 q^{61} - 38 q^{64} + 16 q^{71} + 80 q^{74} + 22 q^{76} + 14 q^{79} + 68 q^{86} - 18 q^{89} - 34 q^{91} - 6 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(1702\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.63372i 1.86232i 0.364606 + 0.931162i \(0.381204\pi\)
−0.364606 + 0.931162i \(0.618796\pi\)
\(3\) 0 0
\(4\) −4.93650 −2.46825
\(5\) 0 0
\(6\) 0 0
\(7\) 1.79743i 0.679364i 0.940540 + 0.339682i \(0.110319\pi\)
−0.940540 + 0.339682i \(0.889681\pi\)
\(8\) − 7.73393i − 2.73436i
\(9\) 0 0
\(10\) 0 0
\(11\) −1.80812 −0.545170 −0.272585 0.962132i \(-0.587879\pi\)
−0.272585 + 0.962132i \(0.587879\pi\)
\(12\) 0 0
\(13\) 1.97183i 0.546887i 0.961888 + 0.273443i \(0.0881626\pi\)
−0.961888 + 0.273443i \(0.911837\pi\)
\(14\) −4.73393 −1.26520
\(15\) 0 0
\(16\) 10.4960 2.62401
\(17\) 4.80812i 1.16614i 0.812421 + 0.583071i \(0.198149\pi\)
−0.812421 + 0.583071i \(0.801851\pi\)
\(18\) 0 0
\(19\) −2.96467 −0.680142 −0.340071 0.940400i \(-0.610451\pi\)
−0.340071 + 0.940400i \(0.610451\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 4.76210i − 1.01528i
\(23\) 1.73393i 0.361549i 0.983525 + 0.180774i \(0.0578604\pi\)
−0.983525 + 0.180774i \(0.942140\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.19325 −1.01848
\(27\) 0 0
\(28\) − 8.87300i − 1.67684i
\(29\) −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 \(0.575691\pi\)
\(32\) 12.1758i 2.15239i
\(33\) 0 0
\(34\) −12.6633 −2.17173
\(35\) 0 0
\(36\) 0 0
\(37\) − 11.6351i − 1.91280i −0.292063 0.956399i \(-0.594342\pi\)
0.292063 0.956399i \(-0.405658\pi\)
\(38\) − 7.80812i − 1.26664i
\(39\) 0 0
\(40\) 0 0
\(41\) 2.46648 0.385199 0.192600 0.981277i \(-0.438308\pi\)
0.192600 + 0.981277i \(0.438308\pi\)
\(42\) 0 0
\(43\) − 7.27814i − 1.10991i −0.831882 0.554953i \(-0.812736\pi\)
0.831882 0.554953i \(-0.187264\pi\)
\(44\) 8.92580 1.34562
\(45\) 0 0
\(46\) −4.56668 −0.673321
\(47\) 6.29208i 0.917794i 0.888489 + 0.458897i \(0.151755\pi\)
−0.888489 + 0.458897i \(0.848245\pi\)
\(48\) 0 0
\(49\) 3.76926 0.538465
\(50\) 0 0
\(51\) 0 0
\(52\) − 9.73393i − 1.34985i
\(53\) − 1.72540i − 0.237001i −0.992954 0.118501i \(-0.962191\pi\)
0.992954 0.118501i \(-0.0378088\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 13.9012 1.85762
\(57\) 0 0
\(58\) − 19.4044i − 2.54792i
\(59\) 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 \(0.801153\pi\)
\(62\) − 6.90833i − 0.877358i
\(63\) 0 0
\(64\) −11.0756 −1.38445
\(65\) 0 0
\(66\) 0 0
\(67\) − 9.10374i − 1.11220i −0.831116 0.556100i \(-0.812297\pi\)
0.831116 0.556100i \(-0.187703\pi\)
\(68\) − 23.7353i − 2.87833i
\(69\) 0 0
\(70\) 0 0
\(71\) −1.27460 −0.151268 −0.0756338 0.997136i \(-0.524098\pi\)
−0.0756338 + 0.997136i \(0.524098\pi\)
\(72\) 0 0
\(73\) − 3.58770i − 0.419908i −0.977711 0.209954i \(-0.932669\pi\)
0.977711 0.209954i \(-0.0673315\pi\)
\(74\) 30.6436 3.56225
\(75\) 0 0
\(76\) 14.6351 1.67876
\(77\) − 3.24997i − 0.370369i
\(78\) 0 0
\(79\) −2.11090 −0.237495 −0.118747 0.992924i \(-0.537888\pi\)
−0.118747 + 0.992924i \(0.537888\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.49602i 0.717366i
\(83\) − 1.09883i − 0.120612i −0.998180 0.0603061i \(-0.980792\pi\)
0.998180 0.0603061i \(-0.0192077\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 19.1686 2.06701
\(87\) 0 0
\(88\) 13.9839i 1.49069i
\(89\) −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
\(92\) − 8.55953i − 0.892392i
\(93\) 0 0
\(94\) −16.5716 −1.70923
\(95\) 0 0
\(96\) 0 0
\(97\) 3.83276i 0.389157i 0.980887 + 0.194579i \(0.0623340\pi\)
−0.980887 + 0.194579i \(0.937666\pi\)
\(98\) 9.92718i 1.00280i
\(99\) 0 0
\(100\) 0 0
\(101\) −6.55237 −0.651985 −0.325993 0.945372i \(-0.605698\pi\)
−0.325993 + 0.945372i \(0.605698\pi\)
\(102\) 0 0
\(103\) − 8.07557i − 0.795710i −0.917448 0.397855i \(-0.869755\pi\)
0.917448 0.397855i \(-0.130245\pi\)
\(104\) 15.2500 1.49538
\(105\) 0 0
\(106\) 4.54422 0.441373
\(107\) − 8.97674i − 0.867814i −0.900958 0.433907i \(-0.857135\pi\)
0.900958 0.433907i \(-0.142865\pi\)
\(108\) 0 0
\(109\) 6.34164 0.607419 0.303710 0.952765i \(-0.401775\pi\)
0.303710 + 0.952765i \(0.401775\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 18.8658i 1.78265i
\(113\) 14.9025i 1.40191i 0.713204 + 0.700957i \(0.247243\pi\)
−0.713204 + 0.700957i \(0.752757\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 36.3704 3.37691
\(117\) 0 0
\(118\) 29.0394i 2.67330i
\(119\) −8.64225 −0.792234
\(120\) 0 0
\(121\) −7.73069 −0.702790
\(122\) − 33.3704i − 3.02121i
\(123\) 0 0
\(124\) 12.9486 1.16282
\(125\) 0 0
\(126\) 0 0
\(127\) − 3.62303i − 0.321492i −0.986996 0.160746i \(-0.948610\pi\)
0.986996 0.160746i \(-0.0513899\pi\)
\(128\) − 4.81844i − 0.425894i
\(129\) 0 0
\(130\) 0 0
\(131\) −7.29345 −0.637232 −0.318616 0.947884i \(-0.603218\pi\)
−0.318616 + 0.947884i \(0.603218\pi\)
\(132\) 0 0
\(133\) − 5.32878i − 0.462064i
\(134\) 23.9767 2.07127
\(135\) 0 0
\(136\) 37.1857 3.18865
\(137\) − 7.12621i − 0.608833i −0.952539 0.304417i \(-0.901539\pi\)
0.952539 0.304417i \(-0.0984615\pi\)
\(138\) 0 0
\(139\) 14.7107 1.24774 0.623871 0.781527i \(-0.285559\pi\)
0.623871 + 0.781527i \(0.285559\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 3.35695i − 0.281709i
\(143\) − 3.56531i − 0.298146i
\(144\) 0 0
\(145\) 0 0
\(146\) 9.44900 0.782005
\(147\) 0 0
\(148\) 57.4366i 4.72126i
\(149\) −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 \(0.498013\pi\)
\(152\) 22.9285i 1.85975i
\(153\) 0 0
\(154\) 8.55953 0.689746
\(155\) 0 0
\(156\) 0 0
\(157\) 11.4607i 0.914663i 0.889296 + 0.457332i \(0.151195\pi\)
−0.889296 + 0.457332i \(0.848805\pi\)
\(158\) − 5.55953i − 0.442292i
\(159\) 0 0
\(160\) 0 0
\(161\) −3.11661 −0.245623
\(162\) 0 0
\(163\) − 22.0595i − 1.72783i −0.503637 0.863915i \(-0.668005\pi\)
0.503637 0.863915i \(-0.331995\pi\)
\(164\) −12.1758 −0.950768
\(165\) 0 0
\(166\) 2.89401 0.224619
\(167\) 17.0684i 1.32079i 0.750917 + 0.660397i \(0.229612\pi\)
−0.750917 + 0.660397i \(0.770388\pi\)
\(168\) 0 0
\(169\) 9.11189 0.700915
\(170\) 0 0
\(171\) 0 0
\(172\) 35.9285i 2.73953i
\(173\) − 11.9447i − 0.908135i −0.890967 0.454067i \(-0.849972\pi\)
0.890967 0.454067i \(-0.150028\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −18.9781 −1.43053
\(177\) 0 0
\(178\) − 35.0113i − 2.62421i
\(179\) 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 \(0.628277\pi\)
\(182\) − 9.33449i − 0.691918i
\(183\) 0 0
\(184\) 13.4101 0.988603
\(185\) 0 0
\(186\) 0 0
\(187\) − 8.69368i − 0.635745i
\(188\) − 31.0608i − 2.26534i
\(189\) 0 0
\(190\) 0 0
\(191\) 17.3372 1.25448 0.627239 0.778827i \(-0.284185\pi\)
0.627239 + 0.778827i \(0.284185\pi\)
\(192\) 0 0
\(193\) 1.55953i 0.112257i 0.998424 + 0.0561286i \(0.0178757\pi\)
−0.998424 + 0.0561286i \(0.982124\pi\)
\(194\) −10.0944 −0.724737
\(195\) 0 0
\(196\) −18.6069 −1.32907
\(197\) 17.9767i 1.28079i 0.768046 + 0.640395i \(0.221229\pi\)
−0.768046 + 0.640395i \(0.778771\pi\)
\(198\) 0 0
\(199\) −11.0225 −0.781362 −0.390681 0.920526i \(-0.627760\pi\)
−0.390681 + 0.920526i \(0.627760\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 17.2571i − 1.21421i
\(203\) − 13.2428i − 0.929463i
\(204\) 0 0
\(205\) 0 0
\(206\) 21.2688 1.48187
\(207\) 0 0
\(208\) 20.6964i 1.43503i
\(209\) 5.36049 0.370793
\(210\) 0 0
\(211\) −23.9285 −1.64731 −0.823655 0.567092i \(-0.808068\pi\)
−0.823655 + 0.567092i \(0.808068\pi\)
\(212\) 8.51742i 0.584979i
\(213\) 0 0
\(214\) 23.6423 1.61615
\(215\) 0 0
\(216\) 0 0
\(217\) − 4.71470i − 0.320055i
\(218\) 16.7021i 1.13121i
\(219\) 0 0
\(220\) 0 0
\(221\) −9.48079 −0.637747
\(222\) 0 0
\(223\) 21.7107i 1.45385i 0.686715 + 0.726927i \(0.259052\pi\)
−0.686715 + 0.726927i \(0.740948\pi\)
\(224\) −21.8851 −1.46226
\(225\) 0 0
\(226\) −39.2492 −2.61082
\(227\) 14.1002i 0.935863i 0.883765 + 0.467932i \(0.155001\pi\)
−0.883765 + 0.467932i \(0.844999\pi\)
\(228\) 0 0
\(229\) −3.67758 −0.243021 −0.121511 0.992590i \(-0.538774\pi\)
−0.121511 + 0.992590i \(0.538774\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 56.9809i 3.74098i
\(233\) 5.34164i 0.349943i 0.984574 + 0.174971i \(0.0559833\pi\)
−0.984574 + 0.174971i \(0.944017\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −54.4299 −3.54308
\(237\) 0 0
\(238\) − 22.7613i − 1.47540i
\(239\) −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 \(0.705066\pi\)
\(242\) − 20.3605i − 1.30882i
\(243\) 0 0
\(244\) 62.5475 4.00420
\(245\) 0 0
\(246\) 0 0
\(247\) − 5.84582i − 0.371961i
\(248\) 20.2863i 1.28818i
\(249\) 0 0
\(250\) 0 0
\(251\) −14.6929 −0.927407 −0.463704 0.885990i \(-0.653480\pi\)
−0.463704 + 0.885990i \(0.653480\pi\)
\(252\) 0 0
\(253\) − 3.13515i − 0.197105i
\(254\) 9.54205 0.598721
\(255\) 0 0
\(256\) −9.46070 −0.591294
\(257\) 22.2089i 1.38536i 0.721247 + 0.692678i \(0.243569\pi\)
−0.721247 + 0.692678i \(0.756431\pi\)
\(258\) 0 0
\(259\) 20.9132 1.29949
\(260\) 0 0
\(261\) 0 0
\(262\) − 19.2089i − 1.18673i
\(263\) − 5.74001i − 0.353944i −0.984216 0.176972i \(-0.943370\pi\)
0.984216 0.176972i \(-0.0566302\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 14.0345 0.860513
\(267\) 0 0
\(268\) 44.9406i 2.74519i
\(269\) −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 \(0.565756\pi\)
\(272\) 50.4662i 3.05996i
\(273\) 0 0
\(274\) 18.7685 1.13384
\(275\) 0 0
\(276\) 0 0
\(277\) − 30.2966i − 1.82034i −0.414230 0.910172i \(-0.635949\pi\)
0.414230 0.910172i \(-0.364051\pi\)
\(278\) 38.7438i 2.32370i
\(279\) 0 0
\(280\) 0 0
\(281\) −18.6351 −1.11168 −0.555838 0.831290i \(-0.687603\pi\)
−0.555838 + 0.831290i \(0.687603\pi\)
\(282\) 0 0
\(283\) − 5.67366i − 0.337264i −0.985679 0.168632i \(-0.946065\pi\)
0.985679 0.168632i \(-0.0539350\pi\)
\(284\) 6.29208 0.373366
\(285\) 0 0
\(286\) 9.39004 0.555245
\(287\) 4.43332i 0.261690i
\(288\) 0 0
\(289\) −6.11806 −0.359886
\(290\) 0 0
\(291\) 0 0
\(292\) 17.7107i 1.03644i
\(293\) − 18.2773i − 1.06777i −0.845556 0.533887i \(-0.820731\pi\)
0.845556 0.533887i \(-0.179269\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −89.9850 −5.23027
\(297\) 0 0
\(298\) − 1.48887i − 0.0862478i
\(299\) −3.41900 −0.197726
\(300\) 0 0
\(301\) 13.0819 0.754030
\(302\) 0.403974i 0.0232461i
\(303\) 0 0
\(304\) −31.1173 −1.78470
\(305\) 0 0
\(306\) 0 0
\(307\) 15.5050i 0.884915i 0.896789 + 0.442458i \(0.145893\pi\)
−0.896789 + 0.442458i \(0.854107\pi\)
\(308\) 16.0435i 0.914162i
\(309\) 0 0
\(310\) 0 0
\(311\) −30.4464 −1.72646 −0.863228 0.504814i \(-0.831561\pi\)
−0.863228 + 0.504814i \(0.831561\pi\)
\(312\) 0 0
\(313\) − 6.94936i − 0.392801i −0.980524 0.196401i \(-0.937075\pi\)
0.980524 0.196401i \(-0.0629253\pi\)
\(314\) −30.1843 −1.70340
\(315\) 0 0
\(316\) 10.4205 0.586196
\(317\) − 16.1451i − 0.906797i −0.891308 0.453398i \(-0.850212\pi\)
0.891308 0.453398i \(-0.149788\pi\)
\(318\) 0 0
\(319\) 13.3216 0.745868
\(320\) 0 0
\(321\) 0 0
\(322\) − 8.20828i − 0.457430i
\(323\) − 14.2545i − 0.793142i
\(324\) 0 0
\(325\) 0 0
\(326\) 58.0985 3.21778
\(327\) 0 0
\(328\) − 19.0756i − 1.05327i
\(329\) −11.3096 −0.623516
\(330\) 0 0
\(331\) 12.6222 0.693781 0.346890 0.937906i \(-0.387238\pi\)
0.346890 + 0.937906i \(0.387238\pi\)
\(332\) 5.42437i 0.297701i
\(333\) 0 0
\(334\) −44.9535 −2.45975
\(335\) 0 0
\(336\) 0 0
\(337\) 6.92040i 0.376978i 0.982075 + 0.188489i \(0.0603590\pi\)
−0.982075 + 0.188489i \(0.939641\pi\)
\(338\) 23.9982i 1.30533i
\(339\) 0 0
\(340\) 0 0
\(341\) 4.74276 0.256835
\(342\) 0 0
\(343\) 19.3570i 1.04518i
\(344\) −56.2886 −3.03488
\(345\) 0 0
\(346\) 31.4589 1.69124
\(347\) 13.8063i 0.741163i 0.928800 + 0.370581i \(0.120842\pi\)
−0.928800 + 0.370581i \(0.879158\pi\)
\(348\) 0 0
\(349\) −6.56768 −0.351560 −0.175780 0.984429i \(-0.556245\pi\)
−0.175780 + 0.984429i \(0.556245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 22.0153i − 1.17342i
\(353\) 3.52499i 0.187616i 0.995590 + 0.0938082i \(0.0299040\pi\)
−0.995590 + 0.0938082i \(0.970096\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 65.6231 3.47802
\(357\) 0 0
\(358\) 22.5162i 1.19002i
\(359\) 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
\(362\) − 27.7920i − 1.46072i
\(363\) 0 0
\(364\) 17.4960 0.917041
\(365\) 0 0
\(366\) 0 0
\(367\) − 4.17931i − 0.218158i −0.994033 0.109079i \(-0.965210\pi\)
0.994033 0.109079i \(-0.0347902\pi\)
\(368\) 18.1993i 0.948706i
\(369\) 0 0
\(370\) 0 0
\(371\) 3.10127 0.161010
\(372\) 0 0
\(373\) 6.84091i 0.354209i 0.984192 + 0.177104i \(0.0566730\pi\)
−0.984192 + 0.177104i \(0.943327\pi\)
\(374\) 22.8968 1.18396
\(375\) 0 0
\(376\) 48.6625 2.50958
\(377\) − 14.5277i − 0.748217i
\(378\) 0 0
\(379\) 12.7764 0.656280 0.328140 0.944629i \(-0.393578\pi\)
0.328140 + 0.944629i \(0.393578\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 45.6615i 2.33624i
\(383\) − 7.53459i − 0.385000i −0.981297 0.192500i \(-0.938341\pi\)
0.981297 0.192500i \(-0.0616595\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.10736 −0.209059
\(387\) 0 0
\(388\) − 18.9204i − 0.960538i
\(389\) 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
\(392\) − 29.1511i − 1.47235i
\(393\) 0 0
\(394\) −47.3458 −2.38525
\(395\) 0 0
\(396\) 0 0
\(397\) 5.64549i 0.283339i 0.989914 + 0.141670i \(0.0452470\pi\)
−0.989914 + 0.141670i \(0.954753\pi\)
\(398\) − 29.0301i − 1.45515i
\(399\) 0 0
\(400\) 0 0
\(401\) 5.50418 0.274865 0.137433 0.990511i \(-0.456115\pi\)
0.137433 + 0.990511i \(0.456115\pi\)
\(402\) 0 0
\(403\) − 5.17216i − 0.257643i
\(404\) 32.3458 1.60926
\(405\) 0 0
\(406\) 34.8779 1.73096
\(407\) 21.0377i 1.04280i
\(408\) 0 0
\(409\) −32.8530 −1.62448 −0.812238 0.583327i \(-0.801751\pi\)
−0.812238 + 0.583327i \(0.801751\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 39.8650i 1.96401i
\(413\) 19.8184i 0.975202i
\(414\) 0 0
\(415\) 0 0
\(416\) −24.0085 −1.17712
\(417\) 0 0
\(418\) 14.1181i 0.690537i
\(419\) 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 \(0.355968\pi\)
\(422\) − 63.0212i − 3.06782i
\(423\) 0 0
\(424\) −13.3441 −0.648046
\(425\) 0 0
\(426\) 0 0
\(427\) − 22.7742i − 1.10212i
\(428\) 44.3137i 2.14198i
\(429\) 0 0
\(430\) 0 0
\(431\) 6.18871 0.298100 0.149050 0.988830i \(-0.452378\pi\)
0.149050 + 0.988830i \(0.452378\pi\)
\(432\) 0 0
\(433\) − 3.11806i − 0.149844i −0.997189 0.0749221i \(-0.976129\pi\)
0.997189 0.0749221i \(-0.0238708\pi\)
\(434\) 12.4172 0.596045
\(435\) 0 0
\(436\) −31.3055 −1.49926
\(437\) − 5.14052i − 0.245904i
\(438\) 0 0
\(439\) −13.5099 −0.644792 −0.322396 0.946605i \(-0.604488\pi\)
−0.322396 + 0.946605i \(0.604488\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 24.9698i − 1.18769i
\(443\) 24.2773i 1.15345i 0.816938 + 0.576726i \(0.195670\pi\)
−0.816938 + 0.576726i \(0.804330\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −57.1799 −2.70755
\(447\) 0 0
\(448\) − 19.9075i − 0.940543i
\(449\) −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
\(452\) − 73.5664i − 3.46027i
\(453\) 0 0
\(454\) −37.1360 −1.74288
\(455\) 0 0
\(456\) 0 0
\(457\) 2.82157i 0.131987i 0.997820 + 0.0659937i \(0.0210217\pi\)
−0.997820 + 0.0659937i \(0.978978\pi\)
\(458\) − 9.68573i − 0.452585i
\(459\) 0 0
\(460\) 0 0
\(461\) −21.4572 −0.999363 −0.499681 0.866209i \(-0.666550\pi\)
−0.499681 + 0.866209i \(0.666550\pi\)
\(462\) 0 0
\(463\) − 19.8033i − 0.920339i −0.887831 0.460170i \(-0.847789\pi\)
0.887831 0.460170i \(-0.152211\pi\)
\(464\) −77.3310 −3.59000
\(465\) 0 0
\(466\) −14.0684 −0.651707
\(467\) 22.7210i 1.05140i 0.850669 + 0.525701i \(0.176197\pi\)
−0.850669 + 0.525701i \(0.823803\pi\)
\(468\) 0 0
\(469\) 16.3633 0.755588
\(470\) 0 0
\(471\) 0 0
\(472\) − 85.2743i − 3.92507i
\(473\) 13.1598i 0.605088i
\(474\) 0 0
\(475\) 0 0
\(476\) 42.6625 1.95543
\(477\) 0 0
\(478\) − 58.0300i − 2.65423i
\(479\) 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
\(482\) − 49.1115i − 2.23697i
\(483\) 0 0
\(484\) 38.1625 1.73466
\(485\) 0 0
\(486\) 0 0
\(487\) 9.58690i 0.434424i 0.976124 + 0.217212i \(0.0696963\pi\)
−0.976124 + 0.217212i \(0.930304\pi\)
\(488\) 97.9921i 4.43590i
\(489\) 0 0
\(490\) 0 0
\(491\) 37.8443 1.70789 0.853945 0.520363i \(-0.174203\pi\)
0.853945 + 0.520363i \(0.174203\pi\)
\(492\) 0 0
\(493\) − 35.4246i − 1.59544i
\(494\) 15.3963 0.692711
\(495\) 0 0
\(496\) −27.5314 −1.23619
\(497\) − 2.29101i − 0.102766i
\(498\) 0 0
\(499\) −16.9253 −0.757681 −0.378840 0.925462i \(-0.623677\pi\)
−0.378840 + 0.925462i \(0.623677\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 38.6970i − 1.72713i
\(503\) − 40.4168i − 1.80210i −0.433719 0.901048i \(-0.642799\pi\)
0.433719 0.901048i \(-0.357201\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 8.25713 0.367074
\(507\) 0 0
\(508\) 17.8851i 0.793522i
\(509\) −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
\(512\) − 34.5537i − 1.52707i
\(513\) 0 0
\(514\) −58.4922 −2.57998
\(515\) 0 0
\(516\) 0 0
\(517\) − 11.3769i − 0.500354i
\(518\) 55.0797i 2.42006i
\(519\) 0 0
\(520\) 0 0
\(521\) 17.0301 0.746103 0.373052 0.927811i \(-0.378311\pi\)
0.373052 + 0.927811i \(0.378311\pi\)
\(522\) 0 0
\(523\) 9.57651i 0.418751i 0.977835 + 0.209376i \(0.0671432\pi\)
−0.977835 + 0.209376i \(0.932857\pi\)
\(524\) 36.0041 1.57285
\(525\) 0 0
\(526\) 15.1176 0.659159
\(527\) − 12.6118i − 0.549380i
\(528\) 0 0
\(529\) 19.9935 0.869283
\(530\) 0 0
\(531\) 0 0
\(532\) 26.3055i 1.14049i
\(533\) 4.86347i 0.210660i
\(534\) 0 0
\(535\) 0 0
\(536\) −70.4077 −3.04115
\(537\) 0 0
\(538\) − 41.1289i − 1.77319i
\(539\) −6.81528 −0.293555
\(540\) 0 0
\(541\) −0.833751 −0.0358458 −0.0179229 0.999839i \(-0.505705\pi\)
−0.0179229 + 0.999839i \(0.505705\pi\)
\(542\) − 17.7859i − 0.763971i
\(543\) 0 0
\(544\) −58.5426 −2.50999
\(545\) 0 0
\(546\) 0 0
\(547\) 28.3270i 1.21117i 0.795779 + 0.605587i \(0.207062\pi\)
−0.795779 + 0.605587i \(0.792938\pi\)
\(548\) 35.1785i 1.50275i
\(549\) 0 0
\(550\) 0 0
\(551\) 21.8427 0.930529
\(552\) 0 0
\(553\) − 3.79419i − 0.161345i
\(554\) 79.7928 3.39007
\(555\) 0 0
\(556\) −72.6192 −3.07974
\(557\) 11.5042i 0.487448i 0.969845 + 0.243724i \(0.0783690\pi\)
−0.969845 + 0.243724i \(0.921631\pi\)
\(558\) 0 0
\(559\) 14.3512 0.606993
\(560\) 0 0
\(561\) 0 0
\(562\) − 49.0797i − 2.07030i
\(563\) − 33.0059i − 1.39103i −0.718510 0.695517i \(-0.755176\pi\)
0.718510 0.695517i \(-0.244824\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 14.9429 0.628095
\(567\) 0 0
\(568\) 9.85769i 0.413619i
\(569\) −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 \(0.670749\pi\)
\(572\) 17.6001i 0.735899i
\(573\) 0 0
\(574\) −11.6761 −0.487352
\(575\) 0 0
\(576\) 0 0
\(577\) − 14.7976i − 0.616033i −0.951381 0.308017i \(-0.900335\pi\)
0.951381 0.308017i \(-0.0996653\pi\)
\(578\) − 16.1133i − 0.670224i
\(579\) 0 0
\(580\) 0 0
\(581\) 1.97507 0.0819396
\(582\) 0 0
\(583\) 3.11973i 0.129206i
\(584\) −27.7470 −1.14818
\(585\) 0 0
\(586\) 48.1375 1.98854
\(587\) 30.5780i 1.26209i 0.775747 + 0.631044i \(0.217373\pi\)
−0.775747 + 0.631044i \(0.782627\pi\)
\(588\) 0 0
\(589\) 7.77641 0.320421
\(590\) 0 0
\(591\) 0 0
\(592\) − 122.122i − 5.01919i
\(593\) 5.09990i 0.209428i 0.994502 + 0.104714i \(0.0333927\pi\)
−0.994502 + 0.104714i \(0.966607\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 2.79065 0.114309
\(597\) 0 0
\(598\) − 9.00471i − 0.368230i
\(599\) −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 \(0.572090\pi\)
\(602\) 34.4542i 1.40425i
\(603\) 0 0
\(604\) −0.757185 −0.0308094
\(605\) 0 0
\(606\) 0 0
\(607\) 19.0983i 0.775174i 0.921833 + 0.387587i \(0.126691\pi\)
−0.921833 + 0.387587i \(0.873309\pi\)
\(608\) − 36.0972i − 1.46393i
\(609\) 0 0
\(610\) 0 0
\(611\) −12.4069 −0.501929
\(612\) 0 0
\(613\) 9.33918i 0.377206i 0.982053 + 0.188603i \(0.0603959\pi\)
−0.982053 + 0.188603i \(0.939604\pi\)
\(614\) −40.8358 −1.64800
\(615\) 0 0
\(616\) −25.1350 −1.01272
\(617\) − 24.4154i − 0.982928i −0.870898 0.491464i \(-0.836462\pi\)
0.870898 0.491464i \(-0.163538\pi\)
\(618\) 0 0
\(619\) −39.4863 −1.58709 −0.793544 0.608513i \(-0.791766\pi\)
−0.793544 + 0.608513i \(0.791766\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 80.1874i − 3.21522i
\(623\) − 23.8940i − 0.957293i
\(624\) 0 0
\(625\) 0 0
\(626\) 18.3027 0.731523
\(627\) 0 0
\(628\) − 56.5757i − 2.25762i
\(629\) 55.9430 2.23059
\(630\) 0 0
\(631\) 42.1634 1.67850 0.839249 0.543747i \(-0.182995\pi\)
0.839249 + 0.543747i \(0.182995\pi\)
\(632\) 16.3255i 0.649395i
\(633\) 0 0
\(634\) 42.5216 1.68875
\(635\) 0 0
\(636\) 0 0
\(637\) 7.43232i 0.294479i
\(638\) 35.0855i 1.38905i
\(639\) 0 0
\(640\) 0 0
\(641\) −35.3155 −1.39488 −0.697438 0.716645i \(-0.745677\pi\)
−0.697438 + 0.716645i \(0.745677\pi\)
\(642\) 0 0
\(643\) 14.1954i 0.559813i 0.960027 + 0.279906i \(0.0903034\pi\)
−0.960027 + 0.279906i \(0.909697\pi\)
\(644\) 15.3851 0.606259
\(645\) 0 0
\(646\) 37.5424 1.47709
\(647\) 17.4897i 0.687593i 0.939044 + 0.343796i \(0.111713\pi\)
−0.939044 + 0.343796i \(0.888287\pi\)
\(648\) 0 0
\(649\) −19.9364 −0.782572
\(650\) 0 0
\(651\) 0 0
\(652\) 108.897i 4.26472i
\(653\) 10.9772i 0.429569i 0.976661 + 0.214785i \(0.0689050\pi\)
−0.976661 + 0.214785i \(0.931095\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 25.8882 1.01077
\(657\) 0 0
\(658\) − 29.7862i − 1.16119i
\(659\) −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 \(0.0777581\pi\)
\(662\) 33.2435i 1.29204i
\(663\) 0 0
\(664\) −8.49827 −0.329797
\(665\) 0 0
\(666\) 0 0
\(667\) − 12.7750i − 0.494649i
\(668\) − 84.2582i − 3.26005i
\(669\) 0 0
\(670\) 0 0
\(671\) 22.9097 0.884419
\(672\) 0 0
\(673\) 28.8395i 1.11168i 0.831289 + 0.555840i \(0.187603\pi\)
−0.831289 + 0.555840i \(0.812397\pi\)
\(674\) −18.2264 −0.702055
\(675\) 0 0
\(676\) −44.9809 −1.73003
\(677\) 10.4636i 0.402150i 0.979576 + 0.201075i \(0.0644434\pi\)
−0.979576 + 0.201075i \(0.935557\pi\)
\(678\) 0 0
\(679\) −6.88910 −0.264379
\(680\) 0 0
\(681\) 0 0
\(682\) 12.4911i 0.478309i
\(683\) 16.1875i 0.619396i 0.950835 + 0.309698i \(0.100228\pi\)
−0.950835 + 0.309698i \(0.899772\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −50.9809 −1.94646
\(687\) 0 0
\(688\) − 76.3916i − 2.91240i
\(689\) 3.40218 0.129613
\(690\) 0 0
\(691\) 9.88362 0.375991 0.187995 0.982170i \(-0.439801\pi\)
0.187995 + 0.982170i \(0.439801\pi\)
\(692\) 58.9648i 2.24150i
\(693\) 0 0
\(694\) −36.3621 −1.38029
\(695\) 0 0
\(696\) 0 0
\(697\) 11.8591i 0.449197i
\(698\) − 17.2974i − 0.654718i
\(699\) 0 0
\(700\) 0 0
\(701\) −43.9692 −1.66069 −0.830346 0.557248i \(-0.811857\pi\)
−0.830346 + 0.557248i \(0.811857\pi\)
\(702\) 0 0
\(703\) 34.4942i 1.30097i
\(704\) 20.0260 0.754758
\(705\) 0 0
\(706\) −9.28385 −0.349402
\(707\) − 11.7774i − 0.442935i
\(708\) 0 0
\(709\) −25.2260 −0.947384 −0.473692 0.880691i \(-0.657079\pi\)
−0.473692 + 0.880691i \(0.657079\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 102.811i 3.85299i
\(713\) − 4.54814i − 0.170329i
\(714\) 0 0
\(715\) 0 0
\(716\) −42.2032 −1.57721
\(717\) 0 0
\(718\) 60.3875i 2.25364i
\(719\) −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
\(722\) − 26.8922i − 1.00082i
\(723\) 0 0
\(724\) 52.0918 1.93598
\(725\) 0 0
\(726\) 0 0
\(727\) 38.2451i 1.41843i 0.704990 + 0.709217i \(0.250951\pi\)
−0.704990 + 0.709217i \(0.749049\pi\)
\(728\) 27.4107i 1.01591i
\(729\) 0 0
\(730\) 0 0
\(731\) 34.9942 1.29431
\(732\) 0 0
\(733\) 39.5832i 1.46204i 0.682356 + 0.731020i \(0.260955\pi\)
−0.682356 + 0.731020i \(0.739045\pi\)
\(734\) 11.0072 0.406282
\(735\) 0 0
\(736\) −21.1119 −0.778195
\(737\) 16.4607i 0.606338i
\(738\) 0 0
\(739\) 8.24773 0.303398 0.151699 0.988427i \(-0.451526\pi\)
0.151699 + 0.988427i \(0.451526\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 8.16790i 0.299853i
\(743\) − 1.30818i − 0.0479925i −0.999712 0.0239963i \(-0.992361\pi\)
0.999712 0.0239963i \(-0.00763898\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −18.0171 −0.659651
\(747\) 0 0
\(748\) 42.9164i 1.56918i
\(749\) 16.1350 0.589562
\(750\) 0 0
\(751\) 28.4468 1.03804 0.519020 0.854762i \(-0.326297\pi\)
0.519020 + 0.854762i \(0.326297\pi\)
\(752\) 66.0418i 2.40830i
\(753\) 0 0
\(754\) 38.2620 1.39342
\(755\) 0 0
\(756\) 0 0
\(757\) 38.2012i 1.38845i 0.719760 + 0.694223i \(0.244252\pi\)
−0.719760 + 0.694223i \(0.755748\pi\)
\(758\) 33.6495i 1.22221i
\(759\) 0 0
\(760\) 0 0
\(761\) −22.1904 −0.804401 −0.402200 0.915552i \(-0.631755\pi\)
−0.402200 + 0.915552i \(0.631755\pi\)
\(762\) 0 0
\(763\) 11.3986i 0.412659i
\(764\) −85.5852 −3.09637
\(765\) 0 0
\(766\) 19.8440 0.716994
\(767\) 21.7414i 0.785036i
\(768\) 0 0
\(769\) 16.9130 0.609900 0.304950 0.952368i \(-0.401360\pi\)
0.304950 + 0.952368i \(0.401360\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 7.69860i − 0.277079i
\(773\) 38.6464i 1.39001i 0.719003 + 0.695007i \(0.244599\pi\)
−0.719003 + 0.695007i \(0.755401\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 29.6423 1.06409
\(777\) 0 0
\(778\) 14.3584i 0.514774i
\(779\) −7.31230 −0.261990
\(780\) 0 0
\(781\) 2.30464 0.0824665
\(782\) − 21.9572i − 0.785187i
\(783\) 0 0
\(784\) 39.5622 1.41294
\(785\) 0 0
\(786\) 0 0
\(787\) − 10.9147i − 0.389067i −0.980896 0.194533i \(-0.937681\pi\)
0.980896 0.194533i \(-0.0623192\pi\)
\(788\) − 88.7422i − 3.16131i
\(789\) 0 0
\(790\) 0 0
\(791\) −26.7862 −0.952409
\(792\) 0 0
\(793\) − 24.9839i − 0.887204i
\(794\) −14.8687 −0.527669
\(795\) 0 0
\(796\) 54.4124 1.92860
\(797\) − 2.92550i − 0.103627i −0.998657 0.0518133i \(-0.983500\pi\)
0.998657 0.0518133i \(-0.0165001\pi\)
\(798\) 0 0
\(799\) −30.2531 −1.07028
\(800\) 0 0
\(801\) 0 0
\(802\) 14.4965i 0.511888i
\(803\) 6.48700i 0.228921i
\(804\) 0 0
\(805\) 0 0
\(806\) 13.6220 0.479816
\(807\) 0 0
\(808\) 50.6755i 1.78276i
\(809\) −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 \(0.388741\pi\)
\(812\) 65.3731i 2.29415i
\(813\) 0 0
\(814\) −55.4075 −1.94203
\(815\) 0 0
\(816\) 0 0
\(817\) 21.5773i 0.754894i
\(818\) − 86.5257i − 3.02530i
\(819\) 0 0
\(820\) 0 0
\(821\) −44.7524 −1.56187 −0.780934 0.624613i \(-0.785257\pi\)
−0.780934 + 0.624613i \(0.785257\pi\)
\(822\) 0 0
\(823\) − 35.8994i − 1.25137i −0.780074 0.625687i \(-0.784819\pi\)
0.780074 0.625687i \(-0.215181\pi\)
\(824\) −62.4559 −2.17575
\(825\) 0 0
\(826\) −52.1963 −1.81614
\(827\) − 16.2717i − 0.565822i −0.959146 0.282911i \(-0.908700\pi\)
0.959146 0.282911i \(-0.0913001\pi\)
\(828\) 0 0
\(829\) 20.6592 0.717525 0.358762 0.933429i \(-0.383199\pi\)
0.358762 + 0.933429i \(0.383199\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 21.8391i − 0.757135i
\(833\) 18.1230i 0.627926i
\(834\) 0 0
\(835\) 0 0
\(836\) −26.4621 −0.915210
\(837\) 0 0
\(838\) 60.2044i 2.07973i
\(839\) −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
\(842\) 47.2527i 1.62844i
\(843\) 0 0
\(844\) 118.123 4.06597
\(845\) 0 0
\(846\) 0 0
\(847\) − 13.8953i − 0.477450i
\(848\) − 18.1098i − 0.621893i
\(849\) 0 0
\(850\) 0 0
\(851\) 20.1744 0.691570
\(852\) 0 0
\(853\) 16.4293i 0.562528i 0.959630 + 0.281264i \(0.0907537\pi\)
−0.959630 + 0.281264i \(0.909246\pi\)
\(854\) 59.9809 2.05250
\(855\) 0 0
\(856\) −69.4255 −2.37291
\(857\) 49.0897i 1.67687i 0.545000 + 0.838436i \(0.316530\pi\)
−0.545000 + 0.838436i \(0.683470\pi\)
\(858\) 0 0
\(859\) −41.0908 −1.40200 −0.700999 0.713162i \(-0.747262\pi\)
−0.700999 + 0.713162i \(0.747262\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 16.2994i 0.555158i
\(863\) − 47.8366i − 1.62838i −0.580601 0.814188i \(-0.697183\pi\)
0.580601 0.814188i \(-0.302817\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 8.21210 0.279058
\(867\) 0 0
\(868\) 23.2741i 0.789975i
\(869\) 3.81677 0.129475
\(870\) 0 0
\(871\) 17.9510 0.608247
\(872\) − 49.0458i − 1.66090i
\(873\) 0 0
\(874\) 13.5387 0.457954
\(875\) 0 0
\(876\) 0 0
\(877\) 11.3772i 0.384180i 0.981377 + 0.192090i \(0.0615265\pi\)
−0.981377 + 0.192090i \(0.938473\pi\)
\(878\) − 35.5813i − 1.20081i
\(879\) 0 0
\(880\) 0 0
\(881\) 43.9924 1.48214 0.741071 0.671426i \(-0.234318\pi\)
0.741071 + 0.671426i \(0.234318\pi\)
\(882\) 0 0
\(883\) 37.6820i 1.26810i 0.773291 + 0.634051i \(0.218609\pi\)
−0.773291 + 0.634051i \(0.781391\pi\)
\(884\) 46.8019 1.57412
\(885\) 0 0
\(886\) −63.9398 −2.14810
\(887\) − 33.2833i − 1.11754i −0.829322 0.558771i \(-0.811273\pi\)
0.829322 0.558771i \(-0.188727\pi\)
\(888\) 0 0
\(889\) 6.51213 0.218410
\(890\) 0 0
\(891\) 0 0
\(892\) − 107.175i − 3.58847i
\(893\) − 18.6539i − 0.624230i
\(894\) 0 0
\(895\) 0 0
\(896\) 8.66080 0.289337
\(897\) 0 0
\(898\) − 63.5878i − 2.12195i
\(899\) 19.3255 0.644543
\(900\) 0 0
\(901\) 8.29592 0.276377
\(902\) − 11.7456i − 0.391086i
\(903\) 0 0
\(904\) 115.255 3.83333
\(905\) 0 0
\(906\) 0 0
\(907\) 14.0784i 0.467464i 0.972301 + 0.233732i \(0.0750939\pi\)
−0.972301 + 0.233732i \(0.924906\pi\)
\(908\) − 69.6056i − 2.30994i
\(909\) 0 0
\(910\) 0 0
\(911\) −7.31522 −0.242364 −0.121182 0.992630i \(-0.538668\pi\)
−0.121182 + 0.992630i \(0.538668\pi\)
\(912\) 0 0
\(913\) 1.98682i 0.0657542i
\(914\) −7.43123 −0.245803
\(915\) 0 0
\(916\) 18.1544 0.599838
\(917\) − 13.1094i − 0.432912i
\(918\) 0 0
\(919\) −4.44684 −0.146688 −0.0733438 0.997307i \(-0.523367\pi\)
−0.0733438 + 0.997307i \(0.523367\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 56.5124i − 1.86114i
\(923\) − 2.51330i − 0.0827262i
\(924\) 0 0
\(925\) 0 0
\(926\) 52.1565 1.71397
\(927\) 0 0
\(928\) − 89.7068i − 2.94477i
\(929\) 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
\(932\) − 26.3690i − 0.863746i
\(933\) 0 0
\(934\) −59.8408 −1.95805
\(935\) 0 0
\(936\) 0 0
\(937\) 33.5187i 1.09501i 0.836803 + 0.547504i \(0.184422\pi\)
−0.836803 + 0.547504i \(0.815578\pi\)
\(938\) 43.0964i 1.40715i
\(939\) 0 0
\(940\) 0 0
\(941\) −1.79076 −0.0583772 −0.0291886 0.999574i \(-0.509292\pi\)
−0.0291886 + 0.999574i \(0.509292\pi\)
\(942\) 0 0
\(943\) 4.27669i 0.139268i
\(944\) 115.729 3.76667
\(945\) 0 0
\(946\) −34.6592 −1.12687
\(947\) − 52.6350i − 1.71041i −0.518292 0.855204i \(-0.673432\pi\)
0.518292 0.855204i \(-0.326568\pi\)
\(948\) 0 0
\(949\) 7.07432 0.229642
\(950\) 0 0
\(951\) 0 0
\(952\) 66.8386i 2.16625i
\(953\) 18.4072i 0.596268i 0.954524 + 0.298134i \(0.0963642\pi\)
−0.954524 + 0.298134i \(0.903636\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 108.768 3.51781
\(957\) 0 0
\(958\) 56.0666i 1.81143i
\(959\) 12.8088 0.413619
\(960\) 0 0
\(961\) −24.1197 −0.778056
\(962\) 60.4240i 1.94815i
\(963\) 0 0
\(964\) 92.0517 2.96479
\(965\) 0 0
\(966\) 0 0
\(967\) 6.09579i 0.196027i 0.995185 + 0.0980137i \(0.0312489\pi\)
−0.995185 + 0.0980137i \(0.968751\pi\)
\(968\) 59.7886i 1.92168i
\(969\) 0 0
\(970\) 0 0
\(971\) 34.1077 1.09457 0.547283 0.836947i \(-0.315662\pi\)
0.547283 + 0.836947i \(0.315662\pi\)
\(972\) 0 0
\(973\) 26.4414i 0.847671i
\(974\) −25.2493 −0.809038
\(975\) 0 0
\(976\) −132.989 −4.25688
\(977\) 48.7914i 1.56098i 0.625171 + 0.780488i \(0.285029\pi\)
−0.625171 + 0.780488i \(0.714971\pi\)
\(978\) 0 0
\(979\) 24.0362 0.768201
\(980\) 0 0
\(981\) 0 0
\(982\) 99.6715i 3.18065i
\(983\) − 33.4172i − 1.06584i −0.846164 0.532922i \(-0.821094\pi\)
0.846164 0.532922i \(-0.178906\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 93.2985 2.97123
\(987\) 0 0
\(988\) 28.8579i 0.918092i
\(989\) 12.6198 0.401285
\(990\) 0 0
\(991\) −19.3512 −0.614713 −0.307356 0.951595i \(-0.599444\pi\)
−0.307356 + 0.951595i \(0.599444\pi\)
\(992\) − 31.9374i − 1.01401i
\(993\) 0 0
\(994\) 6.03388 0.191383
\(995\) 0 0
\(996\) 0 0
\(997\) − 39.5442i − 1.25238i −0.779671 0.626189i \(-0.784614\pi\)
0.779671 0.626189i \(-0.215386\pi\)
\(998\) − 44.5766i − 1.41105i
\(999\) 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.b.o.649.8 8
3.2 odd 2 2025.2.b.n.649.1 8
5.2 odd 4 2025.2.a.p.1.1 4
5.3 odd 4 2025.2.a.z.1.4 4
5.4 even 2 inner 2025.2.b.o.649.1 8
9.2 odd 6 225.2.k.c.49.1 16
9.4 even 3 675.2.k.c.424.1 16
9.5 odd 6 225.2.k.c.124.8 16
9.7 even 3 675.2.k.c.199.8 16
15.2 even 4 2025.2.a.y.1.4 4
15.8 even 4 2025.2.a.q.1.1 4
15.14 odd 2 2025.2.b.n.649.8 8
45.2 even 12 225.2.e.c.76.1 8
45.4 even 6 675.2.k.c.424.8 16
45.7 odd 12 675.2.e.e.226.4 8
45.13 odd 12 675.2.e.c.451.1 8
45.14 odd 6 225.2.k.c.124.1 16
45.22 odd 12 675.2.e.e.451.4 8
45.23 even 12 225.2.e.e.151.4 yes 8
45.29 odd 6 225.2.k.c.49.8 16
45.32 even 12 225.2.e.c.151.1 yes 8
45.34 even 6 675.2.k.c.199.1 16
45.38 even 12 225.2.e.e.76.4 yes 8
45.43 odd 12 675.2.e.c.226.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.2.e.c.76.1 8 45.2 even 12
225.2.e.c.151.1 yes 8 45.32 even 12
225.2.e.e.76.4 yes 8 45.38 even 12
225.2.e.e.151.4 yes 8 45.23 even 12
225.2.k.c.49.1 16 9.2 odd 6
225.2.k.c.49.8 16 45.29 odd 6
225.2.k.c.124.1 16 45.14 odd 6
225.2.k.c.124.8 16 9.5 odd 6
675.2.e.c.226.1 8 45.43 odd 12
675.2.e.c.451.1 8 45.13 odd 12
675.2.e.e.226.4 8 45.7 odd 12
675.2.e.e.451.4 8 45.22 odd 12
675.2.k.c.199.1 16 45.34 even 6
675.2.k.c.199.8 16 9.7 even 3
675.2.k.c.424.1 16 9.4 even 3
675.2.k.c.424.8 16 45.4 even 6
2025.2.a.p.1.1 4 5.2 odd 4
2025.2.a.q.1.1 4 15.8 even 4
2025.2.a.y.1.4 4 15.2 even 4
2025.2.a.z.1.4 4 5.3 odd 4
2025.2.b.n.649.1 8 3.2 odd 2
2025.2.b.n.649.8 8 15.14 odd 2
2025.2.b.o.649.1 8 5.4 even 2 inner
2025.2.b.o.649.8 8 1.1 even 1 trivial