Properties

Label 504.2.c.e.253.6
Level $504$
Weight $2$
Character 504.253
Analytic conductor $4.024$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,2,Mod(253,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("504.253");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.c (of order \(2\), degree \(1\), 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.72339481600.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} - 2x^{4} - 4x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 253.6
Root \(0.569745 + 1.29437i\) of defining polynomial
Character \(\chi\) \(=\) 504.253
Dual form 504.2.c.e.253.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.569745 + 1.29437i) q^{2} +(-1.35078 + 1.47492i) q^{4} -0.772577i q^{5} -1.00000 q^{7} +(-2.67869 - 0.908080i) q^{8} +O(q^{10})\) \(q+(0.569745 + 1.29437i) q^{2} +(-1.35078 + 1.47492i) q^{4} -0.772577i q^{5} -1.00000 q^{7} +(-2.67869 - 0.908080i) q^{8} +(1.00000 - 0.440172i) q^{10} +4.40490i q^{11} +5.89968i q^{13} +(-0.569745 - 1.29437i) q^{14} +(-0.350781 - 3.98459i) q^{16} -4.21789 q^{17} +5.01934i q^{19} +(1.13949 + 1.04358i) q^{20} +(-5.70156 + 2.50967i) q^{22} -8.77585 q^{23} +4.40312 q^{25} +(-7.63636 + 3.36131i) q^{26} +(1.35078 - 1.47492i) q^{28} -3.63232i q^{29} +7.40312 q^{31} +(4.95767 - 2.72424i) q^{32} +(-2.40312 - 5.45951i) q^{34} +0.772577i q^{35} -5.01934i q^{37} +(-6.49687 + 2.85974i) q^{38} +(-0.701562 + 2.06950i) q^{40} +8.77585 q^{41} +0.880344i q^{43} +(-6.49687 - 5.95005i) q^{44} +(-5.00000 - 11.3592i) q^{46} +1.00000 q^{49} +(2.50866 + 5.69927i) q^{50} +(-8.70156 - 7.96918i) q^{52} -6.72263i q^{53} +3.40312 q^{55} +(2.67869 + 0.908080i) q^{56} +(4.70156 - 2.06950i) q^{58} +10.3550i q^{59} +5.89968i q^{61} +(4.21789 + 9.58237i) q^{62} +(6.35078 + 4.86493i) q^{64} +4.55796 q^{65} +0.880344i q^{67} +(5.69745 - 6.22106i) q^{68} +(-1.00000 + 0.440172i) q^{70} +4.21789 q^{71} -6.00000 q^{73} +(6.49687 - 2.85974i) q^{74} +(-7.40312 - 6.78003i) q^{76} -4.40490i q^{77} +(-3.07840 + 0.271006i) q^{80} +(5.00000 + 11.3592i) q^{82} +1.54515i q^{83} +3.25865i q^{85} +(-1.13949 + 0.501572i) q^{86} +(4.00000 - 11.7994i) q^{88} -8.77585 q^{89} -5.89968i q^{91} +(11.8543 - 12.9437i) q^{92} +3.87783 q^{95} +16.8062 q^{97} +(0.569745 + 1.29437i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} - 8 q^{7} + 8 q^{10} + 10 q^{16} - 20 q^{22} - 16 q^{25} - 2 q^{28} + 8 q^{31} + 32 q^{34} + 20 q^{40} - 40 q^{46} + 8 q^{49} - 44 q^{52} - 24 q^{55} + 12 q^{58} + 38 q^{64} - 8 q^{70} - 48 q^{73} - 8 q^{76} + 40 q^{82} + 32 q^{88} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.569745 + 1.29437i 0.402871 + 0.915257i
\(3\) 0 0
\(4\) −1.35078 + 1.47492i −0.675391 + 0.737460i
\(5\) 0.772577i 0.345507i −0.984965 0.172754i \(-0.944734\pi\)
0.984965 0.172754i \(-0.0552664\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.67869 0.908080i −0.947061 0.321055i
\(9\) 0 0
\(10\) 1.00000 0.440172i 0.316228 0.139195i
\(11\) 4.40490i 1.32813i 0.747676 + 0.664063i \(0.231169\pi\)
−0.747676 + 0.664063i \(0.768831\pi\)
\(12\) 0 0
\(13\) 5.89968i 1.63628i 0.575021 + 0.818139i \(0.304994\pi\)
−0.575021 + 0.818139i \(0.695006\pi\)
\(14\) −0.569745 1.29437i −0.152271 0.345935i
\(15\) 0 0
\(16\) −0.350781 3.98459i −0.0876953 0.996147i
\(17\) −4.21789 −1.02299 −0.511495 0.859286i \(-0.670908\pi\)
−0.511495 + 0.859286i \(0.670908\pi\)
\(18\) 0 0
\(19\) 5.01934i 1.15152i 0.817620 + 0.575758i \(0.195293\pi\)
−0.817620 + 0.575758i \(0.804707\pi\)
\(20\) 1.13949 + 1.04358i 0.254798 + 0.233352i
\(21\) 0 0
\(22\) −5.70156 + 2.50967i −1.21558 + 0.535063i
\(23\) −8.77585 −1.82989 −0.914946 0.403576i \(-0.867767\pi\)
−0.914946 + 0.403576i \(0.867767\pi\)
\(24\) 0 0
\(25\) 4.40312 0.880625
\(26\) −7.63636 + 3.36131i −1.49761 + 0.659208i
\(27\) 0 0
\(28\) 1.35078 1.47492i 0.255274 0.278734i
\(29\) 3.63232i 0.674505i −0.941414 0.337252i \(-0.890502\pi\)
0.941414 0.337252i \(-0.109498\pi\)
\(30\) 0 0
\(31\) 7.40312 1.32964 0.664820 0.747003i \(-0.268508\pi\)
0.664820 + 0.747003i \(0.268508\pi\)
\(32\) 4.95767 2.72424i 0.876401 0.481582i
\(33\) 0 0
\(34\) −2.40312 5.45951i −0.412132 0.936298i
\(35\) 0.772577i 0.130589i
\(36\) 0 0
\(37\) 5.01934i 0.825174i −0.910918 0.412587i \(-0.864625\pi\)
0.910918 0.412587i \(-0.135375\pi\)
\(38\) −6.49687 + 2.85974i −1.05393 + 0.463912i
\(39\) 0 0
\(40\) −0.701562 + 2.06950i −0.110927 + 0.327216i
\(41\) 8.77585 1.37056 0.685279 0.728281i \(-0.259680\pi\)
0.685279 + 0.728281i \(0.259680\pi\)
\(42\) 0 0
\(43\) 0.880344i 0.134251i 0.997745 + 0.0671256i \(0.0213828\pi\)
−0.997745 + 0.0671256i \(0.978617\pi\)
\(44\) −6.49687 5.95005i −0.979441 0.897004i
\(45\) 0 0
\(46\) −5.00000 11.3592i −0.737210 1.67482i
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 2.50866 + 5.69927i 0.354778 + 0.805998i
\(51\) 0 0
\(52\) −8.70156 7.96918i −1.20669 1.10513i
\(53\) 6.72263i 0.923424i −0.887030 0.461712i \(-0.847235\pi\)
0.887030 0.461712i \(-0.152765\pi\)
\(54\) 0 0
\(55\) 3.40312 0.458877
\(56\) 2.67869 + 0.908080i 0.357955 + 0.121347i
\(57\) 0 0
\(58\) 4.70156 2.06950i 0.617345 0.271738i
\(59\) 10.3550i 1.34810i 0.738686 + 0.674050i \(0.235447\pi\)
−0.738686 + 0.674050i \(0.764553\pi\)
\(60\) 0 0
\(61\) 5.89968i 0.755377i 0.925933 + 0.377688i \(0.123281\pi\)
−0.925933 + 0.377688i \(0.876719\pi\)
\(62\) 4.21789 + 9.58237i 0.535673 + 1.21696i
\(63\) 0 0
\(64\) 6.35078 + 4.86493i 0.793848 + 0.608117i
\(65\) 4.55796 0.565345
\(66\) 0 0
\(67\) 0.880344i 0.107551i 0.998553 + 0.0537756i \(0.0171256\pi\)
−0.998553 + 0.0537756i \(0.982874\pi\)
\(68\) 5.69745 6.22106i 0.690917 0.754414i
\(69\) 0 0
\(70\) −1.00000 + 0.440172i −0.119523 + 0.0526106i
\(71\) 4.21789 0.500572 0.250286 0.968172i \(-0.419475\pi\)
0.250286 + 0.968172i \(0.419475\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 6.49687 2.85974i 0.755246 0.332438i
\(75\) 0 0
\(76\) −7.40312 6.78003i −0.849197 0.777722i
\(77\) 4.40490i 0.501985i
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) −3.07840 + 0.271006i −0.344176 + 0.0302993i
\(81\) 0 0
\(82\) 5.00000 + 11.3592i 0.552158 + 1.25441i
\(83\) 1.54515i 0.169603i 0.996398 + 0.0848014i \(0.0270256\pi\)
−0.996398 + 0.0848014i \(0.972974\pi\)
\(84\) 0 0
\(85\) 3.25865i 0.353450i
\(86\) −1.13949 + 0.501572i −0.122874 + 0.0540859i
\(87\) 0 0
\(88\) 4.00000 11.7994i 0.426401 1.25782i
\(89\) −8.77585 −0.930239 −0.465119 0.885248i \(-0.653989\pi\)
−0.465119 + 0.885248i \(0.653989\pi\)
\(90\) 0 0
\(91\) 5.89968i 0.618455i
\(92\) 11.8543 12.9437i 1.23589 1.34947i
\(93\) 0 0
\(94\) 0 0
\(95\) 3.87783 0.397857
\(96\) 0 0
\(97\) 16.8062 1.70642 0.853208 0.521571i \(-0.174654\pi\)
0.853208 + 0.521571i \(0.174654\pi\)
\(98\) 0.569745 + 1.29437i 0.0575529 + 0.130751i
\(99\) 0 0
\(100\) −5.94766 + 6.49426i −0.594766 + 0.649426i
\(101\) 11.1275i 1.10723i −0.832773 0.553615i \(-0.813248\pi\)
0.832773 0.553615i \(-0.186752\pi\)
\(102\) 0 0
\(103\) −3.40312 −0.335320 −0.167660 0.985845i \(-0.553621\pi\)
−0.167660 + 0.985845i \(0.553621\pi\)
\(104\) 5.35738 15.8034i 0.525335 1.54965i
\(105\) 0 0
\(106\) 8.70156 3.83019i 0.845170 0.372020i
\(107\) 13.2147i 1.27751i 0.769409 + 0.638756i \(0.220551\pi\)
−0.769409 + 0.638756i \(0.779449\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 1.93891 + 4.40490i 0.184868 + 0.419991i
\(111\) 0 0
\(112\) 0.350781 + 3.98459i 0.0331457 + 0.376508i
\(113\) 12.9937 1.22235 0.611175 0.791496i \(-0.290697\pi\)
0.611175 + 0.791496i \(0.290697\pi\)
\(114\) 0 0
\(115\) 6.78003i 0.632241i
\(116\) 5.35738 + 4.90647i 0.497421 + 0.455554i
\(117\) 0 0
\(118\) −13.4031 + 5.89968i −1.23386 + 0.543110i
\(119\) 4.21789 0.386654
\(120\) 0 0
\(121\) −8.40312 −0.763920
\(122\) −7.63636 + 3.36131i −0.691364 + 0.304319i
\(123\) 0 0
\(124\) −10.0000 + 10.9190i −0.898027 + 0.980557i
\(125\) 7.26464i 0.649769i
\(126\) 0 0
\(127\) −6.80625 −0.603957 −0.301978 0.953315i \(-0.597647\pi\)
−0.301978 + 0.953315i \(0.597647\pi\)
\(128\) −2.67869 + 10.9920i −0.236765 + 0.971567i
\(129\) 0 0
\(130\) 2.59688 + 5.89968i 0.227761 + 0.517436i
\(131\) 1.54515i 0.135001i −0.997719 0.0675004i \(-0.978498\pi\)
0.997719 0.0675004i \(-0.0215024\pi\)
\(132\) 0 0
\(133\) 5.01934i 0.435232i
\(134\) −1.13949 + 0.501572i −0.0984370 + 0.0433292i
\(135\) 0 0
\(136\) 11.2984 + 3.83019i 0.968833 + 0.328436i
\(137\) 17.5517 1.49954 0.749772 0.661696i \(-0.230163\pi\)
0.749772 + 0.661696i \(0.230163\pi\)
\(138\) 0 0
\(139\) 21.8380i 1.85228i 0.377182 + 0.926139i \(0.376893\pi\)
−0.377182 + 0.926139i \(0.623107\pi\)
\(140\) −1.13949 1.04358i −0.0963045 0.0881988i
\(141\) 0 0
\(142\) 2.40312 + 5.45951i 0.201666 + 0.458152i
\(143\) −25.9875 −2.17318
\(144\) 0 0
\(145\) −2.80625 −0.233046
\(146\) −3.41847 7.76621i −0.282915 0.642736i
\(147\) 0 0
\(148\) 7.40312 + 6.78003i 0.608533 + 0.557315i
\(149\) 17.0776i 1.39905i −0.714608 0.699525i \(-0.753395\pi\)
0.714608 0.699525i \(-0.246605\pi\)
\(150\) 0 0
\(151\) 6.80625 0.553885 0.276942 0.960887i \(-0.410679\pi\)
0.276942 + 0.960887i \(0.410679\pi\)
\(152\) 4.55796 13.4453i 0.369699 1.09055i
\(153\) 0 0
\(154\) 5.70156 2.50967i 0.459445 0.202235i
\(155\) 5.71949i 0.459400i
\(156\) 0 0
\(157\) 5.89968i 0.470846i −0.971893 0.235423i \(-0.924352\pi\)
0.971893 0.235423i \(-0.0756475\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −2.10469 3.83019i −0.166390 0.302803i
\(161\) 8.77585 0.691634
\(162\) 0 0
\(163\) 24.4791i 1.91735i −0.284505 0.958674i \(-0.591829\pi\)
0.284505 0.958674i \(-0.408171\pi\)
\(164\) −11.8543 + 12.9437i −0.925662 + 1.01073i
\(165\) 0 0
\(166\) −2.00000 + 0.880344i −0.155230 + 0.0683280i
\(167\) 8.43579 0.652781 0.326390 0.945235i \(-0.394168\pi\)
0.326390 + 0.945235i \(0.394168\pi\)
\(168\) 0 0
\(169\) −21.8062 −1.67740
\(170\) −4.21789 + 1.85660i −0.323498 + 0.142395i
\(171\) 0 0
\(172\) −1.29844 1.18915i −0.0990050 0.0906720i
\(173\) 21.4825i 1.63328i 0.577146 + 0.816641i \(0.304167\pi\)
−0.577146 + 0.816641i \(0.695833\pi\)
\(174\) 0 0
\(175\) −4.40312 −0.332845
\(176\) 17.5517 1.54515i 1.32301 0.116470i
\(177\) 0 0
\(178\) −5.00000 11.3592i −0.374766 0.851407i
\(179\) 7.49521i 0.560218i 0.959968 + 0.280109i \(0.0903706\pi\)
−0.959968 + 0.280109i \(0.909629\pi\)
\(180\) 0 0
\(181\) 7.66037i 0.569391i 0.958618 + 0.284695i \(0.0918925\pi\)
−0.958618 + 0.284695i \(0.908108\pi\)
\(182\) 7.63636 3.36131i 0.566045 0.249157i
\(183\) 0 0
\(184\) 23.5078 + 7.96918i 1.73302 + 0.587496i
\(185\) −3.87783 −0.285103
\(186\) 0 0
\(187\) 18.5794i 1.35866i
\(188\) 0 0
\(189\) 0 0
\(190\) 2.20937 + 5.01934i 0.160285 + 0.364141i
\(191\) 4.21789 0.305196 0.152598 0.988288i \(-0.451236\pi\)
0.152598 + 0.988288i \(0.451236\pi\)
\(192\) 0 0
\(193\) −1.40312 −0.100999 −0.0504995 0.998724i \(-0.516081\pi\)
−0.0504995 + 0.998724i \(0.516081\pi\)
\(194\) 9.57528 + 21.7535i 0.687465 + 1.56181i
\(195\) 0 0
\(196\) −1.35078 + 1.47492i −0.0964844 + 0.105351i
\(197\) 12.4421i 0.886464i 0.896407 + 0.443232i \(0.146168\pi\)
−0.896407 + 0.443232i \(0.853832\pi\)
\(198\) 0 0
\(199\) −10.8062 −0.766035 −0.383017 0.923741i \(-0.625115\pi\)
−0.383017 + 0.923741i \(0.625115\pi\)
\(200\) −11.7946 3.99839i −0.834005 0.282729i
\(201\) 0 0
\(202\) 14.4031 6.33985i 1.01340 0.446071i
\(203\) 3.63232i 0.254939i
\(204\) 0 0
\(205\) 6.78003i 0.473538i
\(206\) −1.93891 4.40490i −0.135090 0.306904i
\(207\) 0 0
\(208\) 23.5078 2.06950i 1.62997 0.143494i
\(209\) −22.1097 −1.52936
\(210\) 0 0
\(211\) 10.9190i 0.751696i −0.926681 0.375848i \(-0.877351\pi\)
0.926681 0.375848i \(-0.122649\pi\)
\(212\) 9.91534 + 9.08080i 0.680989 + 0.623672i
\(213\) 0 0
\(214\) −17.1047 + 7.52901i −1.16925 + 0.514672i
\(215\) 0.680134 0.0463848
\(216\) 0 0
\(217\) −7.40312 −0.502557
\(218\) 0 0
\(219\) 0 0
\(220\) −4.59688 + 5.01934i −0.309921 + 0.338404i
\(221\) 24.8842i 1.67389i
\(222\) 0 0
\(223\) 10.8062 0.723640 0.361820 0.932248i \(-0.382156\pi\)
0.361820 + 0.932248i \(0.382156\pi\)
\(224\) −4.95767 + 2.72424i −0.331248 + 0.182021i
\(225\) 0 0
\(226\) 7.40312 + 16.8187i 0.492448 + 1.11876i
\(227\) 5.71949i 0.379616i 0.981821 + 0.189808i \(0.0607865\pi\)
−0.981821 + 0.189808i \(0.939214\pi\)
\(228\) 0 0
\(229\) 15.9384i 1.05324i −0.850102 0.526618i \(-0.823460\pi\)
0.850102 0.526618i \(-0.176540\pi\)
\(230\) −8.77585 + 3.86289i −0.578663 + 0.254711i
\(231\) 0 0
\(232\) −3.29844 + 9.72987i −0.216553 + 0.638797i
\(233\) −21.4295 −1.40390 −0.701948 0.712228i \(-0.747686\pi\)
−0.701948 + 0.712228i \(0.747686\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −15.2727 13.9873i −0.994170 0.910494i
\(237\) 0 0
\(238\) 2.40312 + 5.45951i 0.155771 + 0.353887i
\(239\) 13.3338 0.862493 0.431246 0.902234i \(-0.358074\pi\)
0.431246 + 0.902234i \(0.358074\pi\)
\(240\) 0 0
\(241\) 24.8062 1.59791 0.798955 0.601390i \(-0.205386\pi\)
0.798955 + 0.601390i \(0.205386\pi\)
\(242\) −4.78764 10.8767i −0.307761 0.699183i
\(243\) 0 0
\(244\) −8.70156 7.96918i −0.557060 0.510174i
\(245\) 0.772577i 0.0493582i
\(246\) 0 0
\(247\) −29.6125 −1.88420
\(248\) −19.8307 6.72263i −1.25925 0.426887i
\(249\) 0 0
\(250\) 9.40312 4.13899i 0.594706 0.261773i
\(251\) 11.9001i 0.751128i −0.926796 0.375564i \(-0.877449\pi\)
0.926796 0.375564i \(-0.122551\pi\)
\(252\) 0 0
\(253\) 38.6567i 2.43033i
\(254\) −3.87783 8.80980i −0.243316 0.552776i
\(255\) 0 0
\(256\) −15.7539 + 2.79544i −0.984619 + 0.174715i
\(257\) −0.340067 −0.0212128 −0.0106064 0.999944i \(-0.503376\pi\)
−0.0106064 + 0.999944i \(0.503376\pi\)
\(258\) 0 0
\(259\) 5.01934i 0.311886i
\(260\) −6.15681 + 6.72263i −0.381829 + 0.416920i
\(261\) 0 0
\(262\) 2.00000 0.880344i 0.123560 0.0543879i
\(263\) −17.2116 −1.06132 −0.530658 0.847586i \(-0.678055\pi\)
−0.530658 + 0.847586i \(0.678055\pi\)
\(264\) 0 0
\(265\) −5.19375 −0.319050
\(266\) 6.49687 2.85974i 0.398349 0.175342i
\(267\) 0 0
\(268\) −1.29844 1.18915i −0.0793147 0.0726390i
\(269\) 25.6568i 1.56432i 0.623076 + 0.782162i \(0.285883\pi\)
−0.623076 + 0.782162i \(0.714117\pi\)
\(270\) 0 0
\(271\) −19.4031 −1.17866 −0.589328 0.807894i \(-0.700607\pi\)
−0.589328 + 0.807894i \(0.700607\pi\)
\(272\) 1.47956 + 16.8066i 0.0897113 + 1.01905i
\(273\) 0 0
\(274\) 10.0000 + 22.7184i 0.604122 + 1.37247i
\(275\) 19.3953i 1.16958i
\(276\) 0 0
\(277\) 16.8187i 1.01054i 0.862962 + 0.505269i \(0.168607\pi\)
−0.862962 + 0.505269i \(0.831393\pi\)
\(278\) −28.2665 + 12.4421i −1.69531 + 0.746229i
\(279\) 0 0
\(280\) 0.701562 2.06950i 0.0419264 0.123676i
\(281\) −13.6739 −0.815715 −0.407858 0.913046i \(-0.633724\pi\)
−0.407858 + 0.913046i \(0.633724\pi\)
\(282\) 0 0
\(283\) 16.8187i 0.999768i −0.866092 0.499884i \(-0.833376\pi\)
0.866092 0.499884i \(-0.166624\pi\)
\(284\) −5.69745 + 6.22106i −0.338082 + 0.369152i
\(285\) 0 0
\(286\) −14.8062 33.6374i −0.875512 1.98902i
\(287\) −8.77585 −0.518022
\(288\) 0 0
\(289\) 0.790627 0.0465075
\(290\) −1.59885 3.63232i −0.0938875 0.213297i
\(291\) 0 0
\(292\) 8.10469 8.84952i 0.474291 0.517879i
\(293\) 15.3019i 0.893944i 0.894548 + 0.446972i \(0.147498\pi\)
−0.894548 + 0.446972i \(0.852502\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) −4.55796 + 13.4453i −0.264926 + 0.781490i
\(297\) 0 0
\(298\) 22.1047 9.72987i 1.28049 0.563636i
\(299\) 51.7748i 2.99421i
\(300\) 0 0
\(301\) 0.880344i 0.0507422i
\(302\) 3.87783 + 8.80980i 0.223144 + 0.506947i
\(303\) 0 0
\(304\) 20.0000 1.76069i 1.14708 0.100982i
\(305\) 4.55796 0.260988
\(306\) 0 0
\(307\) 5.01934i 0.286469i −0.989689 0.143234i \(-0.954250\pi\)
0.989689 0.143234i \(-0.0457503\pi\)
\(308\) 6.49687 + 5.95005i 0.370194 + 0.339036i
\(309\) 0 0
\(310\) 7.40312 3.25865i 0.420469 0.185079i
\(311\) −9.11592 −0.516916 −0.258458 0.966022i \(-0.583214\pi\)
−0.258458 + 0.966022i \(0.583214\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 7.63636 3.36131i 0.430945 0.189690i
\(315\) 0 0
\(316\) 0 0
\(317\) 3.63232i 0.204011i 0.994784 + 0.102006i \(0.0325260\pi\)
−0.994784 + 0.102006i \(0.967474\pi\)
\(318\) 0 0
\(319\) 16.0000 0.895828
\(320\) 3.75854 4.90647i 0.210109 0.274280i
\(321\) 0 0
\(322\) 5.00000 + 11.3592i 0.278639 + 0.633023i
\(323\) 21.1710i 1.17799i
\(324\) 0 0
\(325\) 25.9770i 1.44095i
\(326\) 31.6849 13.9468i 1.75487 0.772444i
\(327\) 0 0
\(328\) −23.5078 7.96918i −1.29800 0.440024i
\(329\) 0 0
\(330\) 0 0
\(331\) 24.4791i 1.34549i 0.739874 + 0.672746i \(0.234885\pi\)
−0.739874 + 0.672746i \(0.765115\pi\)
\(332\) −2.27898 2.08717i −0.125075 0.114548i
\(333\) 0 0
\(334\) 4.80625 + 10.9190i 0.262986 + 0.597462i
\(335\) 0.680134 0.0371597
\(336\) 0 0
\(337\) 3.19375 0.173975 0.0869874 0.996209i \(-0.472276\pi\)
0.0869874 + 0.996209i \(0.472276\pi\)
\(338\) −12.4240 28.2253i −0.675777 1.53526i
\(339\) 0 0
\(340\) −4.80625 4.40172i −0.260655 0.238717i
\(341\) 32.6100i 1.76593i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0.799423 2.35817i 0.0431020 0.127144i
\(345\) 0 0
\(346\) −27.8062 + 12.2395i −1.49487 + 0.658002i
\(347\) 10.5855i 0.568260i 0.958786 + 0.284130i \(0.0917048\pi\)
−0.958786 + 0.284130i \(0.908295\pi\)
\(348\) 0 0
\(349\) 27.7377i 1.48477i 0.669976 + 0.742383i \(0.266305\pi\)
−0.669976 + 0.742383i \(0.733695\pi\)
\(350\) −2.50866 5.69927i −0.134093 0.304639i
\(351\) 0 0
\(352\) 12.0000 + 21.8380i 0.639602 + 1.16397i
\(353\) −8.77585 −0.467092 −0.233546 0.972346i \(-0.575033\pi\)
−0.233546 + 0.972346i \(0.575033\pi\)
\(354\) 0 0
\(355\) 3.25865i 0.172951i
\(356\) 11.8543 12.9437i 0.628274 0.686014i
\(357\) 0 0
\(358\) −9.70156 + 4.27036i −0.512743 + 0.225695i
\(359\) −4.21789 −0.222612 −0.111306 0.993786i \(-0.535503\pi\)
−0.111306 + 0.993786i \(0.535503\pi\)
\(360\) 0 0
\(361\) −6.19375 −0.325987
\(362\) −9.91534 + 4.36446i −0.521139 + 0.229391i
\(363\) 0 0
\(364\) 8.70156 + 7.96918i 0.456086 + 0.417698i
\(365\) 4.63546i 0.242631i
\(366\) 0 0
\(367\) 2.80625 0.146485 0.0732425 0.997314i \(-0.476665\pi\)
0.0732425 + 0.997314i \(0.476665\pi\)
\(368\) 3.07840 + 34.9682i 0.160473 + 1.82284i
\(369\) 0 0
\(370\) −2.20937 5.01934i −0.114860 0.260943i
\(371\) 6.72263i 0.349022i
\(372\) 0 0
\(373\) 11.7994i 0.610948i −0.952200 0.305474i \(-0.901185\pi\)
0.952200 0.305474i \(-0.0988149\pi\)
\(374\) 24.0486 10.5855i 1.24352 0.547364i
\(375\) 0 0
\(376\) 0 0
\(377\) 21.4295 1.10368
\(378\) 0 0
\(379\) 12.6797i 0.651313i −0.945488 0.325656i \(-0.894415\pi\)
0.945488 0.325656i \(-0.105585\pi\)
\(380\) −5.23809 + 5.71949i −0.268709 + 0.293403i
\(381\) 0 0
\(382\) 2.40312 + 5.45951i 0.122955 + 0.279333i
\(383\) 26.6676 1.36265 0.681326 0.731980i \(-0.261404\pi\)
0.681326 + 0.731980i \(0.261404\pi\)
\(384\) 0 0
\(385\) −3.40312 −0.173439
\(386\) −0.799423 1.81616i −0.0406896 0.0924401i
\(387\) 0 0
\(388\) −22.7016 + 24.7879i −1.15250 + 1.25841i
\(389\) 17.0776i 0.865868i 0.901426 + 0.432934i \(0.142522\pi\)
−0.901426 + 0.432934i \(0.857478\pi\)
\(390\) 0 0
\(391\) 37.0156 1.87196
\(392\) −2.67869 0.908080i −0.135294 0.0458650i
\(393\) 0 0
\(394\) −16.1047 + 7.08883i −0.811342 + 0.357130i
\(395\) 0 0
\(396\) 0 0
\(397\) 19.4597i 0.976656i −0.872660 0.488328i \(-0.837607\pi\)
0.872660 0.488328i \(-0.162393\pi\)
\(398\) −6.15681 13.9873i −0.308613 0.701119i
\(399\) 0 0
\(400\) −1.54453 17.5446i −0.0772266 0.877232i
\(401\) 17.5517 0.876491 0.438245 0.898855i \(-0.355600\pi\)
0.438245 + 0.898855i \(0.355600\pi\)
\(402\) 0 0
\(403\) 43.6761i 2.17566i
\(404\) 16.4122 + 15.0309i 0.816538 + 0.747813i
\(405\) 0 0
\(406\) −4.70156 + 2.06950i −0.233335 + 0.102707i
\(407\) 22.1097 1.09594
\(408\) 0 0
\(409\) −10.0000 −0.494468 −0.247234 0.968956i \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 8.77585 3.86289i 0.433409 0.190774i
\(411\) 0 0
\(412\) 4.59688 5.01934i 0.226472 0.247285i
\(413\) 10.3550i 0.509534i
\(414\) 0 0
\(415\) 1.19375 0.0585990
\(416\) 16.0722 + 29.2487i 0.788002 + 1.43404i
\(417\) 0 0
\(418\) −12.5969 28.6181i −0.616133 1.39976i
\(419\) 10.3550i 0.505872i −0.967483 0.252936i \(-0.918604\pi\)
0.967483 0.252936i \(-0.0813963\pi\)
\(420\) 0 0
\(421\) 6.78003i 0.330438i 0.986257 + 0.165219i \(0.0528331\pi\)
−0.986257 + 0.165219i \(0.947167\pi\)
\(422\) 14.1332 6.22106i 0.687995 0.302836i
\(423\) 0 0
\(424\) −6.10469 + 18.0079i −0.296470 + 0.874539i
\(425\) −18.5719 −0.900870
\(426\) 0 0
\(427\) 5.89968i 0.285506i
\(428\) −19.4906 17.8502i −0.942115 0.862820i
\(429\) 0 0
\(430\) 0.387503 + 0.880344i 0.0186871 + 0.0424540i
\(431\) 25.6474 1.23539 0.617697 0.786417i \(-0.288066\pi\)
0.617697 + 0.786417i \(0.288066\pi\)
\(432\) 0 0
\(433\) 28.8062 1.38434 0.692170 0.721735i \(-0.256655\pi\)
0.692170 + 0.721735i \(0.256655\pi\)
\(434\) −4.21789 9.58237i −0.202465 0.459969i
\(435\) 0 0
\(436\) 0 0
\(437\) 44.0490i 2.10715i
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) −9.11592 3.09031i −0.434585 0.147325i
\(441\) 0 0
\(442\) 32.2094 14.1777i 1.53204 0.674363i
\(443\) 16.3050i 0.774674i −0.921938 0.387337i \(-0.873395\pi\)
0.921938 0.387337i \(-0.126605\pi\)
\(444\) 0 0
\(445\) 6.78003i 0.321404i
\(446\) 6.15681 + 13.9873i 0.291533 + 0.662316i
\(447\) 0 0
\(448\) −6.35078 4.86493i −0.300046 0.229847i
\(449\) −22.1097 −1.04342 −0.521710 0.853123i \(-0.674706\pi\)
−0.521710 + 0.853123i \(0.674706\pi\)
\(450\) 0 0
\(451\) 38.6567i 1.82027i
\(452\) −17.5517 + 19.1647i −0.825563 + 0.901434i
\(453\) 0 0
\(454\) −7.40312 + 3.25865i −0.347446 + 0.152936i
\(455\) −4.55796 −0.213680
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 20.6301 9.08080i 0.963982 0.424318i
\(459\) 0 0
\(460\) −10.0000 9.15833i −0.466252 0.427009i
\(461\) 27.2020i 1.26692i 0.773775 + 0.633461i \(0.218366\pi\)
−0.773775 + 0.633461i \(0.781634\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −14.4733 + 1.27415i −0.671906 + 0.0591509i
\(465\) 0 0
\(466\) −12.2094 27.7377i −0.565588 1.28493i
\(467\) 40.9587i 1.89534i −0.319250 0.947671i \(-0.603431\pi\)
0.319250 0.947671i \(-0.396569\pi\)
\(468\) 0 0
\(469\) 0.880344i 0.0406505i
\(470\) 0 0
\(471\) 0 0
\(472\) 9.40312 27.7377i 0.432814 1.27673i
\(473\) −3.87783 −0.178303
\(474\) 0 0
\(475\) 22.1008i 1.01405i
\(476\) −5.69745 + 6.22106i −0.261142 + 0.285142i
\(477\) 0 0
\(478\) 7.59688 + 17.2589i 0.347473 + 0.789403i
\(479\) 35.1034 1.60392 0.801958 0.597380i \(-0.203792\pi\)
0.801958 + 0.597380i \(0.203792\pi\)
\(480\) 0 0
\(481\) 29.6125 1.35021
\(482\) 14.1332 + 32.1084i 0.643751 + 1.46250i
\(483\) 0 0
\(484\) 11.3508 12.3939i 0.515945 0.563361i
\(485\) 12.9841i 0.589579i
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) 5.35738 15.8034i 0.242517 0.715388i
\(489\) 0 0
\(490\) 1.00000 0.440172i 0.0451754 0.0198850i
\(491\) 15.8439i 0.715024i −0.933909 0.357512i \(-0.883625\pi\)
0.933909 0.357512i \(-0.116375\pi\)
\(492\) 0 0
\(493\) 15.3207i 0.690011i
\(494\) −16.8716 38.3295i −0.759088 1.72453i
\(495\) 0 0
\(496\) −2.59688 29.4984i −0.116603 1.32452i
\(497\) −4.21789 −0.189198
\(498\) 0 0
\(499\) 34.5177i 1.54523i −0.634877 0.772613i \(-0.718949\pi\)
0.634877 0.772613i \(-0.281051\pi\)
\(500\) 10.7148 + 9.81294i 0.479179 + 0.438848i
\(501\) 0 0
\(502\) 15.4031 6.78003i 0.687475 0.302607i
\(503\) 17.5517 0.782592 0.391296 0.920265i \(-0.372027\pi\)
0.391296 + 0.920265i \(0.372027\pi\)
\(504\) 0 0
\(505\) −8.59688 −0.382556
\(506\) 50.0361 22.0245i 2.22438 0.979108i
\(507\) 0 0
\(508\) 9.19375 10.0387i 0.407907 0.445394i
\(509\) 37.5569i 1.66468i 0.554265 + 0.832340i \(0.313000\pi\)
−0.554265 + 0.832340i \(0.687000\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) −12.5940 18.7987i −0.556583 0.830792i
\(513\) 0 0
\(514\) −0.193752 0.440172i −0.00854601 0.0194152i
\(515\) 2.62918i 0.115855i
\(516\) 0 0
\(517\) 0 0
\(518\) −6.49687 + 2.85974i −0.285456 + 0.125650i
\(519\) 0 0
\(520\) −12.2094 4.13899i −0.535416 0.181507i
\(521\) 21.0895 0.923946 0.461973 0.886894i \(-0.347142\pi\)
0.461973 + 0.886894i \(0.347142\pi\)
\(522\) 0 0
\(523\) 20.0774i 0.877921i −0.898506 0.438961i \(-0.855347\pi\)
0.898506 0.438961i \(-0.144653\pi\)
\(524\) 2.27898 + 2.08717i 0.0995577 + 0.0911783i
\(525\) 0 0
\(526\) −9.80625 22.2782i −0.427573 0.971376i
\(527\) −31.2256 −1.36021
\(528\) 0 0
\(529\) 54.0156 2.34851
\(530\) −2.95911 6.72263i −0.128536 0.292012i
\(531\) 0 0
\(532\) 7.40312 + 6.78003i 0.320966 + 0.293951i
\(533\) 51.7748i 2.24261i
\(534\) 0 0
\(535\) 10.2094 0.441390
\(536\) 0.799423 2.35817i 0.0345298 0.101857i
\(537\) 0 0
\(538\) −33.2094 + 14.6178i −1.43176 + 0.630220i
\(539\) 4.40490i 0.189732i
\(540\) 0 0
\(541\) 16.8187i 0.723092i 0.932354 + 0.361546i \(0.117751\pi\)
−0.932354 + 0.361546i \(0.882249\pi\)
\(542\) −11.0548 25.1148i −0.474846 1.07877i
\(543\) 0 0
\(544\) −20.9109 + 11.4906i −0.896549 + 0.492654i
\(545\) 0 0
\(546\) 0 0
\(547\) 14.4404i 0.617427i −0.951155 0.308713i \(-0.900102\pi\)
0.951155 0.308713i \(-0.0998984\pi\)
\(548\) −23.7085 + 25.8874i −1.01278 + 1.10585i
\(549\) 0 0
\(550\) −25.1047 + 11.0504i −1.07047 + 0.471190i
\(551\) 18.2318 0.776703
\(552\) 0 0
\(553\) 0 0
\(554\) −21.7696 + 9.58237i −0.924902 + 0.407116i
\(555\) 0 0
\(556\) −32.2094 29.4984i −1.36598 1.25101i
\(557\) 33.6131i 1.42423i −0.702060 0.712117i \(-0.747736\pi\)
0.702060 0.712117i \(-0.252264\pi\)
\(558\) 0 0
\(559\) −5.19375 −0.219672
\(560\) 3.07840 0.271006i 0.130086 0.0114521i
\(561\) 0 0
\(562\) −7.79063 17.6990i −0.328628 0.746589i
\(563\) 13.4453i 0.566650i 0.959024 + 0.283325i \(0.0914375\pi\)
−0.959024 + 0.283325i \(0.908562\pi\)
\(564\) 0 0
\(565\) 10.0387i 0.422330i
\(566\) 21.7696 9.58237i 0.915045 0.402777i
\(567\) 0 0
\(568\) −11.2984 3.83019i −0.474072 0.160711i
\(569\) 47.4170 1.98783 0.993913 0.110171i \(-0.0351397\pi\)
0.993913 + 0.110171i \(0.0351397\pi\)
\(570\) 0 0
\(571\) 0.880344i 0.0368413i −0.999830 0.0184206i \(-0.994136\pi\)
0.999830 0.0184206i \(-0.00586380\pi\)
\(572\) 35.1034 38.3295i 1.46775 1.60264i
\(573\) 0 0
\(574\) −5.00000 11.3592i −0.208696 0.474124i
\(575\) −38.6412 −1.61145
\(576\) 0 0
\(577\) 3.19375 0.132958 0.0664788 0.997788i \(-0.478824\pi\)
0.0664788 + 0.997788i \(0.478824\pi\)
\(578\) 0.450456 + 1.02336i 0.0187365 + 0.0425663i
\(579\) 0 0
\(580\) 3.79063 4.13899i 0.157397 0.171862i
\(581\) 1.54515i 0.0641038i
\(582\) 0 0
\(583\) 29.6125 1.22642
\(584\) 16.0722 + 5.44848i 0.665070 + 0.225460i
\(585\) 0 0
\(586\) −19.8062 + 8.71816i −0.818189 + 0.360144i
\(587\) 18.0807i 0.746271i −0.927777 0.373136i \(-0.878283\pi\)
0.927777 0.373136i \(-0.121717\pi\)
\(588\) 0 0
\(589\) 37.1588i 1.53110i
\(590\) 4.55796 + 10.3550i 0.187648 + 0.426307i
\(591\) 0 0
\(592\) −20.0000 + 1.76069i −0.821995 + 0.0723639i
\(593\) −40.0014 −1.64266 −0.821331 0.570452i \(-0.806768\pi\)
−0.821331 + 0.570452i \(0.806768\pi\)
\(594\) 0 0
\(595\) 3.25865i 0.133592i
\(596\) 25.1881 + 23.0681i 1.03174 + 0.944905i
\(597\) 0 0
\(598\) 67.0156 29.4984i 2.74047 1.20628i
\(599\) −21.7696 −0.889482 −0.444741 0.895659i \(-0.646704\pi\)
−0.444741 + 0.895659i \(0.646704\pi\)
\(600\) 0 0
\(601\) 11.6125 0.473684 0.236842 0.971548i \(-0.423888\pi\)
0.236842 + 0.971548i \(0.423888\pi\)
\(602\) 1.13949 0.501572i 0.0464422 0.0204425i
\(603\) 0 0
\(604\) −9.19375 + 10.0387i −0.374088 + 0.408468i
\(605\) 6.49206i 0.263940i
\(606\) 0 0
\(607\) 2.80625 0.113902 0.0569511 0.998377i \(-0.481862\pi\)
0.0569511 + 0.998377i \(0.481862\pi\)
\(608\) 13.6739 + 24.8842i 0.554549 + 1.00919i
\(609\) 0 0
\(610\) 2.59688 + 5.89968i 0.105144 + 0.238871i
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 6.49687 2.85974i 0.262192 0.115410i
\(615\) 0 0
\(616\) −4.00000 + 11.7994i −0.161165 + 0.475410i
\(617\) −8.43579 −0.339612 −0.169806 0.985478i \(-0.554314\pi\)
−0.169806 + 0.985478i \(0.554314\pi\)
\(618\) 0 0
\(619\) 3.52138i 0.141536i −0.997493 0.0707681i \(-0.977455\pi\)
0.997493 0.0707681i \(-0.0225450\pi\)
\(620\) 8.43579 + 7.72577i 0.338789 + 0.310275i
\(621\) 0 0
\(622\) −5.19375 11.7994i −0.208250 0.473111i
\(623\) 8.77585 0.351597
\(624\) 0 0
\(625\) 16.4031 0.656125
\(626\) −7.97643 18.1212i −0.318802 0.724267i
\(627\) 0 0
\(628\) 8.70156 + 7.96918i 0.347230 + 0.318005i
\(629\) 21.1710i 0.844144i
\(630\) 0 0
\(631\) −14.8062 −0.589427 −0.294714 0.955586i \(-0.595224\pi\)
−0.294714 + 0.955586i \(0.595224\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −4.70156 + 2.06950i −0.186723 + 0.0821902i
\(635\) 5.25835i 0.208671i
\(636\) 0 0
\(637\) 5.89968i 0.233754i
\(638\) 9.11592 + 20.7099i 0.360903 + 0.819913i
\(639\) 0 0
\(640\) 8.49219 + 2.06950i 0.335683 + 0.0818040i
\(641\) −8.43579 −0.333194 −0.166597 0.986025i \(-0.553278\pi\)
−0.166597 + 0.986025i \(0.553278\pi\)
\(642\) 0 0
\(643\) 26.8574i 1.05915i 0.848263 + 0.529576i \(0.177649\pi\)
−0.848263 + 0.529576i \(0.822351\pi\)
\(644\) −11.8543 + 12.9437i −0.467123 + 0.510053i
\(645\) 0 0
\(646\) 27.4031 12.0621i 1.07816 0.474577i
\(647\) 34.4233 1.35332 0.676659 0.736296i \(-0.263427\pi\)
0.676659 + 0.736296i \(0.263427\pi\)
\(648\) 0 0
\(649\) −45.6125 −1.79045
\(650\) −33.6239 + 14.8003i −1.31884 + 0.580515i
\(651\) 0 0
\(652\) 36.1047 + 33.0659i 1.41397 + 1.29496i
\(653\) 18.6227i 0.728764i −0.931250 0.364382i \(-0.881280\pi\)
0.931250 0.364382i \(-0.118720\pi\)
\(654\) 0 0
\(655\) −1.19375 −0.0466437
\(656\) −3.07840 34.9682i −0.120191 1.36528i
\(657\) 0 0
\(658\) 0 0
\(659\) 40.1052i 1.56228i −0.624358 0.781139i \(-0.714639\pi\)
0.624358 0.781139i \(-0.285361\pi\)
\(660\) 0 0
\(661\) 36.0157i 1.40085i −0.713727 0.700424i \(-0.752994\pi\)
0.713727 0.700424i \(-0.247006\pi\)
\(662\) −31.6849 + 13.9468i −1.23147 + 0.542059i
\(663\) 0 0
\(664\) 1.40312 4.13899i 0.0544518 0.160624i
\(665\) −3.87783 −0.150376
\(666\) 0 0
\(667\) 31.8767i 1.23427i
\(668\) −11.3949 + 12.4421i −0.440882 + 0.481400i
\(669\) 0 0
\(670\) 0.387503 + 0.880344i 0.0149705 + 0.0340107i
\(671\) −25.9875 −1.00324
\(672\) 0 0
\(673\) 33.4031 1.28760 0.643798 0.765196i \(-0.277358\pi\)
0.643798 + 0.765196i \(0.277358\pi\)
\(674\) 1.81962 + 4.13389i 0.0700893 + 0.159232i
\(675\) 0 0
\(676\) 29.4555 32.1625i 1.13290 1.23702i
\(677\) 26.1179i 1.00379i 0.864927 + 0.501897i \(0.167364\pi\)
−0.864927 + 0.501897i \(0.832636\pi\)
\(678\) 0 0
\(679\) −16.8062 −0.644965
\(680\) 2.95911 8.72892i 0.113477 0.334739i
\(681\) 0 0
\(682\) −42.2094 + 18.5794i −1.61628 + 0.711442i
\(683\) 42.7344i 1.63519i 0.575797 + 0.817593i \(0.304692\pi\)
−0.575797 + 0.817593i \(0.695308\pi\)
\(684\) 0 0
\(685\) 13.5601i 0.518103i
\(686\) −0.569745 1.29437i −0.0217530 0.0494192i
\(687\) 0 0
\(688\) 3.50781 0.308808i 0.133734 0.0117732i
\(689\) 39.6614 1.51098
\(690\) 0 0
\(691\) 21.8380i 0.830758i 0.909648 + 0.415379i \(0.136351\pi\)
−0.909648 + 0.415379i \(0.863649\pi\)
\(692\) −31.6849 29.0181i −1.20448 1.10310i
\(693\) 0 0
\(694\) −13.7016 + 6.03105i −0.520104 + 0.228935i
\(695\) 16.8716 0.639975
\(696\) 0 0
\(697\) −37.0156 −1.40207
\(698\) −35.9028 + 15.8034i −1.35894 + 0.598169i
\(699\) 0 0
\(700\) 5.94766 6.49426i 0.224800 0.245460i
\(701\) 6.26150i 0.236493i 0.992984 + 0.118247i \(0.0377274\pi\)
−0.992984 + 0.118247i \(0.962273\pi\)
\(702\) 0 0
\(703\) 25.1938 0.950200
\(704\) −21.4295 + 27.9745i −0.807656 + 1.05433i
\(705\) 0 0
\(706\) −5.00000 11.3592i −0.188177 0.427509i
\(707\) 11.1275i 0.418494i
\(708\) 0 0
\(709\) 3.52138i 0.132248i −0.997811 0.0661240i \(-0.978937\pi\)
0.997811 0.0661240i \(-0.0210633\pi\)
\(710\) 4.21789 1.85660i 0.158295 0.0696769i
\(711\) 0 0
\(712\) 23.5078 + 7.96918i 0.880992 + 0.298658i
\(713\) −64.9687 −2.43310
\(714\) 0 0
\(715\) 20.0774i 0.750850i
\(716\) −11.0548 10.1244i −0.413139 0.378366i
\(717\) 0 0
\(718\) −2.40312 5.45951i −0.0896838 0.203747i
\(719\) 26.6676 0.994535 0.497267 0.867597i \(-0.334337\pi\)
0.497267 + 0.867597i \(0.334337\pi\)
\(720\) 0 0
\(721\) 3.40312 0.126739
\(722\) −3.52886 8.01700i −0.131331 0.298362i
\(723\) 0 0
\(724\) −11.2984 10.3475i −0.419903 0.384561i
\(725\) 15.9936i 0.593986i
\(726\) 0 0
\(727\) −25.0156 −0.927778 −0.463889 0.885893i \(-0.653546\pi\)
−0.463889 + 0.885893i \(0.653546\pi\)
\(728\) −5.35738 + 15.8034i −0.198558 + 0.585714i
\(729\) 0 0
\(730\) −6.00000 + 2.64103i −0.222070 + 0.0977490i
\(731\) 3.71320i 0.137338i
\(732\) 0 0
\(733\) 37.7764i 1.39530i 0.716437 + 0.697652i \(0.245772\pi\)
−0.716437 + 0.697652i \(0.754228\pi\)
\(734\) 1.59885 + 3.63232i 0.0590145 + 0.134071i
\(735\) 0 0
\(736\) −43.5078 + 23.9075i −1.60372 + 0.881243i
\(737\) −3.87783 −0.142842
\(738\) 0 0
\(739\) 41.0350i 1.50950i −0.656013 0.754749i \(-0.727759\pi\)
0.656013 0.754749i \(-0.272241\pi\)
\(740\) 5.23809 5.71949i 0.192556 0.210252i
\(741\) 0 0
\(742\) −8.70156 + 3.83019i −0.319444 + 0.140611i
\(743\) −21.0895 −0.773698 −0.386849 0.922143i \(-0.626436\pi\)
−0.386849 + 0.922143i \(0.626436\pi\)
\(744\) 0 0
\(745\) −13.1938 −0.483382
\(746\) 15.2727 6.72263i 0.559174 0.246133i
\(747\) 0 0
\(748\) 27.4031 + 25.0967i 1.00196 + 0.917626i
\(749\) 13.2147i 0.482854i
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 12.2094 + 27.7377i 0.444639 + 1.01015i
\(755\) 5.25835i 0.191371i
\(756\) 0 0
\(757\) 20.0774i 0.729724i −0.931062 0.364862i \(-0.881116\pi\)
0.931062 0.364862i \(-0.118884\pi\)
\(758\) 16.4122 7.22420i 0.596119 0.262395i
\(759\) 0 0
\(760\) −10.3875 3.52138i −0.376794 0.127734i
\(761\) −4.21789 −0.152899 −0.0764493 0.997073i \(-0.524358\pi\)
−0.0764493 + 0.997073i \(0.524358\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −5.69745 + 6.22106i −0.206127 + 0.225070i
\(765\) 0 0
\(766\) 15.1938 + 34.5177i 0.548973 + 1.24718i
\(767\) −61.0909 −2.20587
\(768\) 0 0
\(769\) −8.80625 −0.317561 −0.158781 0.987314i \(-0.550756\pi\)
−0.158781 + 0.987314i \(0.550756\pi\)
\(770\) −1.93891 4.40490i −0.0698736 0.158741i
\(771\) 0 0
\(772\) 1.89531 2.06950i 0.0682138 0.0744828i
\(773\) 22.5665i 0.811661i −0.913948 0.405830i \(-0.866982\pi\)
0.913948 0.405830i \(-0.133018\pi\)
\(774\) 0 0
\(775\) 32.5969 1.17091
\(776\) −45.0188 15.2614i −1.61608 0.547853i
\(777\) 0 0
\(778\) −22.1047 + 9.72987i −0.792491 + 0.348833i
\(779\) 44.0490i 1.57822i
\(780\) 0 0
\(781\) 18.5794i 0.664823i
\(782\) 21.0895 + 47.9119i 0.754158 + 1.71332i
\(783\) 0 0
\(784\) −0.350781 3.98459i −0.0125279 0.142307i
\(785\) −4.55796 −0.162681
\(786\) 0 0
\(787\) 21.8380i 0.778442i −0.921144 0.389221i \(-0.872744\pi\)
0.921144 0.389221i \(-0.127256\pi\)
\(788\) −18.3511 16.8066i −0.653732 0.598709i
\(789\) 0 0
\(790\) 0 0
\(791\) −12.9937 −0.462004
\(792\) 0 0
\(793\) −34.8062 −1.23601
\(794\) 25.1881 11.0871i 0.893891 0.393466i
\(795\) 0 0
\(796\) 14.5969 15.9384i 0.517373 0.564920i
\(797\) 37.5569i 1.33033i −0.746695 0.665167i \(-0.768360\pi\)
0.746695 0.665167i \(-0.231640\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 21.8292 11.9952i 0.771780 0.424093i
\(801\) 0 0
\(802\) 10.0000 + 22.7184i 0.353112 + 0.802214i
\(803\) 26.4294i 0.932673i
\(804\) 0 0
\(805\) 6.78003i 0.238965i
\(806\) −56.5330 + 24.8842i −1.99129 + 0.876510i
\(807\) 0 0
\(808\) −10.1047 + 29.8072i −0.355482 + 1.04861i
\(809\) −12.9937 −0.456836 −0.228418 0.973563i \(-0.573355\pi\)
−0.228418 + 0.973563i \(0.573355\pi\)
\(810\) 0 0
\(811\) 23.5987i 0.828663i −0.910126 0.414332i \(-0.864015\pi\)
0.910126 0.414332i \(-0.135985\pi\)
\(812\) −5.35738 4.90647i −0.188007 0.172183i
\(813\) 0 0
\(814\) 12.5969 + 28.6181i 0.441520 + 1.00306i
\(815\) −18.9120 −0.662458
\(816\) 0 0
\(817\) −4.41875 −0.154592
\(818\) −5.69745 12.9437i −0.199207 0.452565i
\(819\) 0 0
\(820\) 10.0000 + 9.15833i 0.349215 + 0.319823i
\(821\) 24.3422i 0.849549i −0.905299 0.424775i \(-0.860353\pi\)
0.905299 0.424775i \(-0.139647\pi\)
\(822\) 0 0
\(823\) −30.8062 −1.07384 −0.536919 0.843634i \(-0.680412\pi\)
−0.536919 + 0.843634i \(0.680412\pi\)
\(824\) 9.11592 + 3.09031i 0.317568 + 0.107656i
\(825\) 0 0
\(826\) 13.4031 5.89968i 0.466354 0.205276i
\(827\) 48.9150i 1.70094i 0.526022 + 0.850471i \(0.323683\pi\)
−0.526022 + 0.850471i \(0.676317\pi\)
\(828\) 0 0
\(829\) 19.4597i 0.675865i −0.941170 0.337932i \(-0.890273\pi\)
0.941170 0.337932i \(-0.109727\pi\)
\(830\) 0.680134 + 1.54515i 0.0236078 + 0.0536331i
\(831\) 0 0
\(832\) −28.7016 + 37.4676i −0.995048 + 1.29895i
\(833\) −4.21789 −0.146141
\(834\) 0 0
\(835\) 6.51730i 0.225540i
\(836\) 29.8653 32.6100i 1.03291 1.12784i
\(837\) 0 0
\(838\) 13.4031 5.89968i 0.463003 0.203801i
\(839\) −25.3074 −0.873707 −0.436854 0.899533i \(-0.643907\pi\)
−0.436854 + 0.899533i \(0.643907\pi\)
\(840\) 0 0
\(841\) 15.8062 0.545043
\(842\) −8.77585 + 3.86289i −0.302436 + 0.133124i
\(843\) 0 0
\(844\) 16.1047 + 14.7492i 0.554346 + 0.507689i
\(845\) 16.8470i 0.579555i
\(846\) 0 0
\(847\) 8.40312 0.288735
\(848\) −26.7869 + 2.35817i −0.919867 + 0.0809799i
\(849\) 0 0
\(850\) −10.5813 24.0389i −0.362934 0.824527i
\(851\) 44.0490i 1.50998i
\(852\) 0 0
\(853\) 12.4170i 0.425149i 0.977145 + 0.212575i \(0.0681849\pi\)
−0.977145 + 0.212575i \(0.931815\pi\)
\(854\) 7.63636 3.36131i 0.261311 0.115022i
\(855\) 0 0
\(856\) 12.0000 35.3981i 0.410152 1.20988i
\(857\) 4.89803 0.167313 0.0836567 0.996495i \(-0.473340\pi\)
0.0836567 + 0.996495i \(0.473340\pi\)
\(858\) 0 0
\(859\) 38.6567i 1.31895i 0.751726 + 0.659476i \(0.229222\pi\)
−0.751726 + 0.659476i \(0.770778\pi\)
\(860\) −0.918712 + 1.00314i −0.0313278 + 0.0342069i
\(861\) 0 0
\(862\) 14.6125 + 33.1972i 0.497704 + 1.13070i
\(863\) 21.7696 0.741046 0.370523 0.928823i \(-0.379178\pi\)
0.370523 + 0.928823i \(0.379178\pi\)
\(864\) 0 0
\(865\) 16.5969 0.564311
\(866\) 16.4122 + 37.2859i 0.557710 + 1.26703i
\(867\) 0 0
\(868\) 10.0000 10.9190i 0.339422 0.370616i
\(869\) 0 0
\(870\) 0 0
\(871\) −5.19375 −0.175984
\(872\) 0 0
\(873\) 0 0
\(874\) 57.0156 25.0967i 1.92858 0.848908i
\(875\) 7.26464i 0.245590i
\(876\) 0 0
\(877\) 43.6761i 1.47484i −0.675436 0.737418i \(-0.736045\pi\)
0.675436 0.737418i \(-0.263955\pi\)
\(878\) 9.11592 + 20.7099i 0.307647 + 0.698925i
\(879\) 0 0
\(880\) −1.19375 13.5601i −0.0402414 0.457109i
\(881\) 39.3213 1.32477 0.662384 0.749164i \(-0.269545\pi\)
0.662384 + 0.749164i \(0.269545\pi\)
\(882\) 0 0
\(883\) 10.9190i 0.367454i −0.982977 0.183727i \(-0.941184\pi\)
0.982977 0.183727i \(-0.0588163\pi\)
\(884\) 36.7023 + 33.6131i 1.23443 + 1.13053i
\(885\) 0 0
\(886\) 21.1047 9.28970i 0.709026 0.312093i
\(887\) −0.680134 −0.0228367 −0.0114183 0.999935i \(-0.503635\pi\)
−0.0114183 + 0.999935i \(0.503635\pi\)
\(888\) 0 0
\(889\) 6.80625 0.228274
\(890\) −8.77585 + 3.86289i −0.294167 + 0.129484i
\(891\) 0 0
\(892\) −14.5969 + 15.9384i −0.488740 + 0.533656i
\(893\) 0 0
\(894\) 0 0
\(895\) 5.79063 0.193559
\(896\) 2.67869 10.9920i 0.0894888 0.367218i
\(897\) 0 0
\(898\) −12.5969 28.6181i −0.420363 0.954997i
\(899\) 26.8905i 0.896849i
\(900\) 0 0
\(901\) 28.3553i 0.944653i
\(902\) −50.0361 + 22.0245i −1.66602 + 0.733335i
\(903\) 0 0
\(904\) −34.8062 11.7994i −1.15764 0.392441i
\(905\) 5.91823 0.196729
\(906\) 0 0
\(907\) 44.5564i 1.47947i −0.672897 0.739736i \(-0.734950\pi\)
0.672897 0.739736i \(-0.265050\pi\)
\(908\) −8.43579 7.72577i −0.279951 0.256389i
\(909\) 0 0
\(910\) −2.59688 5.89968i −0.0860856 0.195573i
\(911\) 43.8793 1.45379 0.726893 0.686751i \(-0.240964\pi\)
0.726893 + 0.686751i \(0.240964\pi\)
\(912\) 0 0
\(913\) −6.80625 −0.225254
\(914\) −12.5344 28.4761i −0.414601 0.941906i
\(915\) 0 0
\(916\) 23.5078 + 21.5292i 0.776720 + 0.711346i
\(917\) 1.54515i 0.0510255i
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) 6.15681 18.1616i 0.202984 0.598770i
\(921\) 0 0
\(922\) −35.2094 + 15.4982i −1.15956 + 0.510406i
\(923\) 24.8842i 0.819074i
\(924\) 0 0
\(925\) 22.1008i 0.726669i
\(926\) −13.6739 31.0649i −0.449352 1.02085i
\(927\) 0 0
\(928\) −9.89531 18.0079i −0.324830 0.591137i
\(929\) 34.0832 1.11823 0.559117 0.829089i \(-0.311140\pi\)
0.559117 + 0.829089i \(0.311140\pi\)
\(930\) 0 0
\(931\) 5.01934i 0.164502i
\(932\) 28.9466 31.6069i 0.948178 1.03532i
\(933\) 0 0
\(934\) 53.0156 23.3360i 1.73472 0.763577i
\(935\) −14.3540 −0.469427
\(936\) 0 0
\(937\) −47.6125 −1.55543 −0.777716 0.628616i \(-0.783622\pi\)
−0.777716 + 0.628616i \(0.783622\pi\)
\(938\) 1.13949 0.501572i 0.0372057 0.0163769i
\(939\) 0 0
\(940\) 0 0
\(941\) 46.3667i 1.51151i −0.654854 0.755756i \(-0.727270\pi\)
0.654854 0.755756i \(-0.272730\pi\)
\(942\) 0 0
\(943\) −77.0156 −2.50797
\(944\) 41.2602 3.63232i 1.34291 0.118222i
\(945\) 0 0
\(946\) −2.20937 5.01934i −0.0718329 0.163193i
\(947\) 43.1955i 1.40367i 0.712342 + 0.701833i \(0.247635\pi\)
−0.712342 + 0.701833i \(0.752365\pi\)
\(948\) 0 0
\(949\) 35.3981i 1.14907i
\(950\) −28.6065 + 12.5918i −0.928119 + 0.408532i
\(951\) 0 0
\(952\) −11.2984 3.83019i −0.366184 0.124137i
\(953\) 9.11592 0.295294 0.147647 0.989040i \(-0.452830\pi\)
0.147647 + 0.989040i \(0.452830\pi\)
\(954\) 0 0
\(955\) 3.25865i 0.105447i
\(956\) −18.0111 + 19.6663i −0.582519 + 0.636054i
\(957\) 0 0
\(958\) 20.0000 + 45.4368i 0.646171 + 1.46800i
\(959\) −17.5517 −0.566774
\(960\) 0 0
\(961\) 23.8062 0.767943
\(962\) 16.8716 + 38.3295i 0.543961 + 1.23579i
\(963\) 0 0
\(964\) −33.5078 + 36.5872i −1.07921 + 1.17840i
\(965\) 1.08402i 0.0348959i
\(966\) 0 0
\(967\) 17.1938 0.552914 0.276457 0.961026i \(-0.410840\pi\)
0.276457 + 0.961026i \(0.410840\pi\)
\(968\) 22.5094 + 7.63071i 0.723479 + 0.245260i
\(969\) 0 0
\(970\) 16.8062 7.39764i 0.539616 0.237524i
\(971\) 47.6004i 1.52757i −0.645471 0.763785i \(-0.723339\pi\)
0.645471 0.763785i \(-0.276661\pi\)
\(972\) 0 0
\(973\) 21.8380i 0.700095i
\(974\) −4.55796 10.3550i −0.146046 0.331794i
\(975\) 0 0
\(976\) 23.5078 2.06950i 0.752467 0.0662430i
\(977\) 22.7898 0.729110 0.364555 0.931182i \(-0.381221\pi\)
0.364555 + 0.931182i \(0.381221\pi\)
\(978\) 0 0
\(979\) 38.6567i 1.23547i
\(980\) 1.13949 + 1.04358i 0.0363997 + 0.0333360i
\(981\) 0 0
\(982\) 20.5078 9.02697i 0.654431 0.288062i
\(983\) −35.1034 −1.11963 −0.559813 0.828619i \(-0.689127\pi\)
−0.559813 + 0.828619i \(0.689127\pi\)
\(984\) 0 0
\(985\) 9.61250 0.306280
\(986\) −19.8307 + 8.72892i −0.631538 + 0.277985i
\(987\) 0 0
\(988\) 40.0000 43.6761i 1.27257 1.38952i
\(989\) 7.72577i 0.245665i
\(990\) 0 0
\(991\) 46.8062 1.48685 0.743425 0.668820i \(-0.233200\pi\)
0.743425 + 0.668820i \(0.233200\pi\)
\(992\) 36.7023 20.1679i 1.16530 0.640331i
\(993\) 0 0
\(994\) −2.40312 5.45951i −0.0762225 0.173165i
\(995\) 8.34866i 0.264670i
\(996\) 0 0
\(997\) 7.66037i 0.242606i 0.992616 + 0.121303i \(0.0387073\pi\)
−0.992616 + 0.121303i \(0.961293\pi\)
\(998\) 44.6787 19.6663i 1.41428 0.622526i
\(999\) 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 504.2.c.e.253.6 yes 8
3.2 odd 2 inner 504.2.c.e.253.3 8
4.3 odd 2 2016.2.c.f.1009.4 8
8.3 odd 2 2016.2.c.f.1009.5 8
8.5 even 2 inner 504.2.c.e.253.5 yes 8
12.11 even 2 2016.2.c.f.1009.6 8
24.5 odd 2 inner 504.2.c.e.253.4 yes 8
24.11 even 2 2016.2.c.f.1009.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.c.e.253.3 8 3.2 odd 2 inner
504.2.c.e.253.4 yes 8 24.5 odd 2 inner
504.2.c.e.253.5 yes 8 8.5 even 2 inner
504.2.c.e.253.6 yes 8 1.1 even 1 trivial
2016.2.c.f.1009.3 8 24.11 even 2
2016.2.c.f.1009.4 8 4.3 odd 2
2016.2.c.f.1009.5 8 8.3 odd 2
2016.2.c.f.1009.6 8 12.11 even 2