Properties

Label 36.3.d.c.19.1
Level $36$
Weight $3$
Character 36.19
Analytic conductor $0.981$
Analytic rank $0$
Dimension $2$
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] = [36,3,Mod(19,36)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(36, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("36.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 36.d (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: \(0.980928951697\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 19.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 36.19
Dual form 36.3.d.c.19.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +2.00000 q^{5} +6.92820i q^{7} -8.00000 q^{8} +O(q^{10})\) \(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +2.00000 q^{5} +6.92820i q^{7} -8.00000 q^{8} +(2.00000 - 3.46410i) q^{10} +6.92820i q^{11} +2.00000 q^{13} +(12.0000 + 6.92820i) q^{14} +(-8.00000 + 13.8564i) q^{16} -10.0000 q^{17} -20.7846i q^{19} +(-4.00000 - 6.92820i) q^{20} +(12.0000 + 6.92820i) q^{22} -27.7128i q^{23} -21.0000 q^{25} +(2.00000 - 3.46410i) q^{26} +(24.0000 - 13.8564i) q^{28} +26.0000 q^{29} -6.92820i q^{31} +(16.0000 + 27.7128i) q^{32} +(-10.0000 + 17.3205i) q^{34} +13.8564i q^{35} +26.0000 q^{37} +(-36.0000 - 20.7846i) q^{38} -16.0000 q^{40} -58.0000 q^{41} +48.4974i q^{43} +(24.0000 - 13.8564i) q^{44} +(-48.0000 - 27.7128i) q^{46} +69.2820i q^{47} +1.00000 q^{49} +(-21.0000 + 36.3731i) q^{50} +(-4.00000 - 6.92820i) q^{52} +74.0000 q^{53} +13.8564i q^{55} -55.4256i q^{56} +(26.0000 - 45.0333i) q^{58} -90.0666i q^{59} +26.0000 q^{61} +(-12.0000 - 6.92820i) q^{62} +64.0000 q^{64} +4.00000 q^{65} -6.92820i q^{67} +(20.0000 + 34.6410i) q^{68} +(24.0000 + 13.8564i) q^{70} -46.0000 q^{73} +(26.0000 - 45.0333i) q^{74} +(-72.0000 + 41.5692i) q^{76} -48.0000 q^{77} -117.779i q^{79} +(-16.0000 + 27.7128i) q^{80} +(-58.0000 + 100.459i) q^{82} +48.4974i q^{83} -20.0000 q^{85} +(84.0000 + 48.4974i) q^{86} -55.4256i q^{88} -82.0000 q^{89} +13.8564i q^{91} +(-96.0000 + 55.4256i) q^{92} +(120.000 + 69.2820i) q^{94} -41.5692i q^{95} +2.00000 q^{97} +(1.00000 - 1.73205i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 4 q^{4} + 4 q^{5} - 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 4 q^{4} + 4 q^{5} - 16 q^{8} + 4 q^{10} + 4 q^{13} + 24 q^{14} - 16 q^{16} - 20 q^{17} - 8 q^{20} + 24 q^{22} - 42 q^{25} + 4 q^{26} + 48 q^{28} + 52 q^{29} + 32 q^{32} - 20 q^{34} + 52 q^{37} - 72 q^{38} - 32 q^{40} - 116 q^{41} + 48 q^{44} - 96 q^{46} + 2 q^{49} - 42 q^{50} - 8 q^{52} + 148 q^{53} + 52 q^{58} + 52 q^{61} - 24 q^{62} + 128 q^{64} + 8 q^{65} + 40 q^{68} + 48 q^{70} - 92 q^{73} + 52 q^{74} - 144 q^{76} - 96 q^{77} - 32 q^{80} - 116 q^{82} - 40 q^{85} + 168 q^{86} - 164 q^{89} - 192 q^{92} + 240 q^{94} + 4 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 1.73205i 0.500000 0.866025i
\(3\) 0 0
\(4\) −2.00000 3.46410i −0.500000 0.866025i
\(5\) 2.00000 0.400000 0.200000 0.979796i \(-0.435906\pi\)
0.200000 + 0.979796i \(0.435906\pi\)
\(6\) 0 0
\(7\) 6.92820i 0.989743i 0.868966 + 0.494872i \(0.164785\pi\)
−0.868966 + 0.494872i \(0.835215\pi\)
\(8\) −8.00000 −1.00000
\(9\) 0 0
\(10\) 2.00000 3.46410i 0.200000 0.346410i
\(11\) 6.92820i 0.629837i 0.949119 + 0.314918i \(0.101977\pi\)
−0.949119 + 0.314918i \(0.898023\pi\)
\(12\) 0 0
\(13\) 2.00000 0.153846 0.0769231 0.997037i \(-0.475490\pi\)
0.0769231 + 0.997037i \(0.475490\pi\)
\(14\) 12.0000 + 6.92820i 0.857143 + 0.494872i
\(15\) 0 0
\(16\) −8.00000 + 13.8564i −0.500000 + 0.866025i
\(17\) −10.0000 −0.588235 −0.294118 0.955769i \(-0.595026\pi\)
−0.294118 + 0.955769i \(0.595026\pi\)
\(18\) 0 0
\(19\) 20.7846i 1.09393i −0.837157 0.546963i \(-0.815784\pi\)
0.837157 0.546963i \(-0.184216\pi\)
\(20\) −4.00000 6.92820i −0.200000 0.346410i
\(21\) 0 0
\(22\) 12.0000 + 6.92820i 0.545455 + 0.314918i
\(23\) 27.7128i 1.20490i −0.798155 0.602452i \(-0.794190\pi\)
0.798155 0.602452i \(-0.205810\pi\)
\(24\) 0 0
\(25\) −21.0000 −0.840000
\(26\) 2.00000 3.46410i 0.0769231 0.133235i
\(27\) 0 0
\(28\) 24.0000 13.8564i 0.857143 0.494872i
\(29\) 26.0000 0.896552 0.448276 0.893895i \(-0.352038\pi\)
0.448276 + 0.893895i \(0.352038\pi\)
\(30\) 0 0
\(31\) 6.92820i 0.223490i −0.993737 0.111745i \(-0.964356\pi\)
0.993737 0.111745i \(-0.0356441\pi\)
\(32\) 16.0000 + 27.7128i 0.500000 + 0.866025i
\(33\) 0 0
\(34\) −10.0000 + 17.3205i −0.294118 + 0.509427i
\(35\) 13.8564i 0.395897i
\(36\) 0 0
\(37\) 26.0000 0.702703 0.351351 0.936244i \(-0.385722\pi\)
0.351351 + 0.936244i \(0.385722\pi\)
\(38\) −36.0000 20.7846i −0.947368 0.546963i
\(39\) 0 0
\(40\) −16.0000 −0.400000
\(41\) −58.0000 −1.41463 −0.707317 0.706896i \(-0.750095\pi\)
−0.707317 + 0.706896i \(0.750095\pi\)
\(42\) 0 0
\(43\) 48.4974i 1.12785i 0.825827 + 0.563924i \(0.190709\pi\)
−0.825827 + 0.563924i \(0.809291\pi\)
\(44\) 24.0000 13.8564i 0.545455 0.314918i
\(45\) 0 0
\(46\) −48.0000 27.7128i −1.04348 0.602452i
\(47\) 69.2820i 1.47409i 0.675846 + 0.737043i \(0.263778\pi\)
−0.675846 + 0.737043i \(0.736222\pi\)
\(48\) 0 0
\(49\) 1.00000 0.0204082
\(50\) −21.0000 + 36.3731i −0.420000 + 0.727461i
\(51\) 0 0
\(52\) −4.00000 6.92820i −0.0769231 0.133235i
\(53\) 74.0000 1.39623 0.698113 0.715987i \(-0.254023\pi\)
0.698113 + 0.715987i \(0.254023\pi\)
\(54\) 0 0
\(55\) 13.8564i 0.251935i
\(56\) 55.4256i 0.989743i
\(57\) 0 0
\(58\) 26.0000 45.0333i 0.448276 0.776437i
\(59\) 90.0666i 1.52655i −0.646072 0.763277i \(-0.723589\pi\)
0.646072 0.763277i \(-0.276411\pi\)
\(60\) 0 0
\(61\) 26.0000 0.426230 0.213115 0.977027i \(-0.431639\pi\)
0.213115 + 0.977027i \(0.431639\pi\)
\(62\) −12.0000 6.92820i −0.193548 0.111745i
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 4.00000 0.0615385
\(66\) 0 0
\(67\) 6.92820i 0.103406i −0.998663 0.0517030i \(-0.983535\pi\)
0.998663 0.0517030i \(-0.0164649\pi\)
\(68\) 20.0000 + 34.6410i 0.294118 + 0.509427i
\(69\) 0 0
\(70\) 24.0000 + 13.8564i 0.342857 + 0.197949i
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −46.0000 −0.630137 −0.315068 0.949069i \(-0.602027\pi\)
−0.315068 + 0.949069i \(0.602027\pi\)
\(74\) 26.0000 45.0333i 0.351351 0.608558i
\(75\) 0 0
\(76\) −72.0000 + 41.5692i −0.947368 + 0.546963i
\(77\) −48.0000 −0.623377
\(78\) 0 0
\(79\) 117.779i 1.49088i −0.666573 0.745440i \(-0.732240\pi\)
0.666573 0.745440i \(-0.267760\pi\)
\(80\) −16.0000 + 27.7128i −0.200000 + 0.346410i
\(81\) 0 0
\(82\) −58.0000 + 100.459i −0.707317 + 1.22511i
\(83\) 48.4974i 0.584306i 0.956372 + 0.292153i \(0.0943717\pi\)
−0.956372 + 0.292153i \(0.905628\pi\)
\(84\) 0 0
\(85\) −20.0000 −0.235294
\(86\) 84.0000 + 48.4974i 0.976744 + 0.563924i
\(87\) 0 0
\(88\) 55.4256i 0.629837i
\(89\) −82.0000 −0.921348 −0.460674 0.887569i \(-0.652392\pi\)
−0.460674 + 0.887569i \(0.652392\pi\)
\(90\) 0 0
\(91\) 13.8564i 0.152268i
\(92\) −96.0000 + 55.4256i −1.04348 + 0.602452i
\(93\) 0 0
\(94\) 120.000 + 69.2820i 1.27660 + 0.737043i
\(95\) 41.5692i 0.437571i
\(96\) 0 0
\(97\) 2.00000 0.0206186 0.0103093 0.999947i \(-0.496718\pi\)
0.0103093 + 0.999947i \(0.496718\pi\)
\(98\) 1.00000 1.73205i 0.0102041 0.0176740i
\(99\) 0 0
\(100\) 42.0000 + 72.7461i 0.420000 + 0.727461i
\(101\) 74.0000 0.732673 0.366337 0.930482i \(-0.380612\pi\)
0.366337 + 0.930482i \(0.380612\pi\)
\(102\) 0 0
\(103\) 76.2102i 0.739905i 0.929051 + 0.369953i \(0.120626\pi\)
−0.929051 + 0.369953i \(0.879374\pi\)
\(104\) −16.0000 −0.153846
\(105\) 0 0
\(106\) 74.0000 128.172i 0.698113 1.20917i
\(107\) 20.7846i 0.194249i 0.995272 + 0.0971243i \(0.0309645\pi\)
−0.995272 + 0.0971243i \(0.969036\pi\)
\(108\) 0 0
\(109\) −46.0000 −0.422018 −0.211009 0.977484i \(-0.567675\pi\)
−0.211009 + 0.977484i \(0.567675\pi\)
\(110\) 24.0000 + 13.8564i 0.218182 + 0.125967i
\(111\) 0 0
\(112\) −96.0000 55.4256i −0.857143 0.494872i
\(113\) 110.000 0.973451 0.486726 0.873555i \(-0.338191\pi\)
0.486726 + 0.873555i \(0.338191\pi\)
\(114\) 0 0
\(115\) 55.4256i 0.481962i
\(116\) −52.0000 90.0666i −0.448276 0.776437i
\(117\) 0 0
\(118\) −156.000 90.0666i −1.32203 0.763277i
\(119\) 69.2820i 0.582202i
\(120\) 0 0
\(121\) 73.0000 0.603306
\(122\) 26.0000 45.0333i 0.213115 0.369126i
\(123\) 0 0
\(124\) −24.0000 + 13.8564i −0.193548 + 0.111745i
\(125\) −92.0000 −0.736000
\(126\) 0 0
\(127\) 145.492i 1.14561i 0.819692 + 0.572804i \(0.194144\pi\)
−0.819692 + 0.572804i \(0.805856\pi\)
\(128\) 64.0000 110.851i 0.500000 0.866025i
\(129\) 0 0
\(130\) 4.00000 6.92820i 0.0307692 0.0532939i
\(131\) 117.779i 0.899080i 0.893260 + 0.449540i \(0.148412\pi\)
−0.893260 + 0.449540i \(0.851588\pi\)
\(132\) 0 0
\(133\) 144.000 1.08271
\(134\) −12.0000 6.92820i −0.0895522 0.0517030i
\(135\) 0 0
\(136\) 80.0000 0.588235
\(137\) −10.0000 −0.0729927 −0.0364964 0.999334i \(-0.511620\pi\)
−0.0364964 + 0.999334i \(0.511620\pi\)
\(138\) 0 0
\(139\) 48.4974i 0.348902i −0.984666 0.174451i \(-0.944185\pi\)
0.984666 0.174451i \(-0.0558151\pi\)
\(140\) 48.0000 27.7128i 0.342857 0.197949i
\(141\) 0 0
\(142\) 0 0
\(143\) 13.8564i 0.0968979i
\(144\) 0 0
\(145\) 52.0000 0.358621
\(146\) −46.0000 + 79.6743i −0.315068 + 0.545715i
\(147\) 0 0
\(148\) −52.0000 90.0666i −0.351351 0.608558i
\(149\) 2.00000 0.0134228 0.00671141 0.999977i \(-0.497864\pi\)
0.00671141 + 0.999977i \(0.497864\pi\)
\(150\) 0 0
\(151\) 90.0666i 0.596468i 0.954493 + 0.298234i \(0.0963975\pi\)
−0.954493 + 0.298234i \(0.903602\pi\)
\(152\) 166.277i 1.09393i
\(153\) 0 0
\(154\) −48.0000 + 83.1384i −0.311688 + 0.539860i
\(155\) 13.8564i 0.0893962i
\(156\) 0 0
\(157\) −214.000 −1.36306 −0.681529 0.731791i \(-0.738685\pi\)
−0.681529 + 0.731791i \(0.738685\pi\)
\(158\) −204.000 117.779i −1.29114 0.745440i
\(159\) 0 0
\(160\) 32.0000 + 55.4256i 0.200000 + 0.346410i
\(161\) 192.000 1.19255
\(162\) 0 0
\(163\) 20.7846i 0.127513i −0.997965 0.0637565i \(-0.979692\pi\)
0.997965 0.0637565i \(-0.0203081\pi\)
\(164\) 116.000 + 200.918i 0.707317 + 1.22511i
\(165\) 0 0
\(166\) 84.0000 + 48.4974i 0.506024 + 0.292153i
\(167\) 96.9948i 0.580807i 0.956904 + 0.290404i \(0.0937896\pi\)
−0.956904 + 0.290404i \(0.906210\pi\)
\(168\) 0 0
\(169\) −165.000 −0.976331
\(170\) −20.0000 + 34.6410i −0.117647 + 0.203771i
\(171\) 0 0
\(172\) 168.000 96.9948i 0.976744 0.563924i
\(173\) −334.000 −1.93064 −0.965318 0.261077i \(-0.915922\pi\)
−0.965318 + 0.261077i \(0.915922\pi\)
\(174\) 0 0
\(175\) 145.492i 0.831384i
\(176\) −96.0000 55.4256i −0.545455 0.314918i
\(177\) 0 0
\(178\) −82.0000 + 142.028i −0.460674 + 0.797911i
\(179\) 187.061i 1.04504i −0.852628 0.522518i \(-0.824993\pi\)
0.852628 0.522518i \(-0.175007\pi\)
\(180\) 0 0
\(181\) 2.00000 0.0110497 0.00552486 0.999985i \(-0.498241\pi\)
0.00552486 + 0.999985i \(0.498241\pi\)
\(182\) 24.0000 + 13.8564i 0.131868 + 0.0761341i
\(183\) 0 0
\(184\) 221.703i 1.20490i
\(185\) 52.0000 0.281081
\(186\) 0 0
\(187\) 69.2820i 0.370492i
\(188\) 240.000 138.564i 1.27660 0.737043i
\(189\) 0 0
\(190\) −72.0000 41.5692i −0.378947 0.218785i
\(191\) 221.703i 1.16075i −0.814351 0.580373i \(-0.802907\pi\)
0.814351 0.580373i \(-0.197093\pi\)
\(192\) 0 0
\(193\) 290.000 1.50259 0.751295 0.659966i \(-0.229429\pi\)
0.751295 + 0.659966i \(0.229429\pi\)
\(194\) 2.00000 3.46410i 0.0103093 0.0178562i
\(195\) 0 0
\(196\) −2.00000 3.46410i −0.0102041 0.0176740i
\(197\) 26.0000 0.131980 0.0659898 0.997820i \(-0.478980\pi\)
0.0659898 + 0.997820i \(0.478980\pi\)
\(198\) 0 0
\(199\) 394.908i 1.98446i −0.124416 0.992230i \(-0.539706\pi\)
0.124416 0.992230i \(-0.460294\pi\)
\(200\) 168.000 0.840000
\(201\) 0 0
\(202\) 74.0000 128.172i 0.366337 0.634514i
\(203\) 180.133i 0.887356i
\(204\) 0 0
\(205\) −116.000 −0.565854
\(206\) 132.000 + 76.2102i 0.640777 + 0.369953i
\(207\) 0 0
\(208\) −16.0000 + 27.7128i −0.0769231 + 0.133235i
\(209\) 144.000 0.688995
\(210\) 0 0
\(211\) 242.487i 1.14923i 0.818425 + 0.574614i \(0.194848\pi\)
−0.818425 + 0.574614i \(0.805152\pi\)
\(212\) −148.000 256.344i −0.698113 1.20917i
\(213\) 0 0
\(214\) 36.0000 + 20.7846i 0.168224 + 0.0971243i
\(215\) 96.9948i 0.451139i
\(216\) 0 0
\(217\) 48.0000 0.221198
\(218\) −46.0000 + 79.6743i −0.211009 + 0.365479i
\(219\) 0 0
\(220\) 48.0000 27.7128i 0.218182 0.125967i
\(221\) −20.0000 −0.0904977
\(222\) 0 0
\(223\) 339.482i 1.52234i 0.648552 + 0.761170i \(0.275375\pi\)
−0.648552 + 0.761170i \(0.724625\pi\)
\(224\) −192.000 + 110.851i −0.857143 + 0.494872i
\(225\) 0 0
\(226\) 110.000 190.526i 0.486726 0.843034i
\(227\) 284.056i 1.25135i −0.780084 0.625675i \(-0.784824\pi\)
0.780084 0.625675i \(-0.215176\pi\)
\(228\) 0 0
\(229\) −142.000 −0.620087 −0.310044 0.950722i \(-0.600344\pi\)
−0.310044 + 0.950722i \(0.600344\pi\)
\(230\) −96.0000 55.4256i −0.417391 0.240981i
\(231\) 0 0
\(232\) −208.000 −0.896552
\(233\) −82.0000 −0.351931 −0.175966 0.984396i \(-0.556305\pi\)
−0.175966 + 0.984396i \(0.556305\pi\)
\(234\) 0 0
\(235\) 138.564i 0.589634i
\(236\) −312.000 + 180.133i −1.32203 + 0.763277i
\(237\) 0 0
\(238\) −120.000 69.2820i −0.504202 0.291101i
\(239\) 387.979i 1.62334i 0.584113 + 0.811672i \(0.301442\pi\)
−0.584113 + 0.811672i \(0.698558\pi\)
\(240\) 0 0
\(241\) −46.0000 −0.190871 −0.0954357 0.995436i \(-0.530424\pi\)
−0.0954357 + 0.995436i \(0.530424\pi\)
\(242\) 73.0000 126.440i 0.301653 0.522478i
\(243\) 0 0
\(244\) −52.0000 90.0666i −0.213115 0.369126i
\(245\) 2.00000 0.00816327
\(246\) 0 0
\(247\) 41.5692i 0.168296i
\(248\) 55.4256i 0.223490i
\(249\) 0 0
\(250\) −92.0000 + 159.349i −0.368000 + 0.637395i
\(251\) 145.492i 0.579650i 0.957080 + 0.289825i \(0.0935972\pi\)
−0.957080 + 0.289825i \(0.906403\pi\)
\(252\) 0 0
\(253\) 192.000 0.758893
\(254\) 252.000 + 145.492i 0.992126 + 0.572804i
\(255\) 0 0
\(256\) −128.000 221.703i −0.500000 0.866025i
\(257\) 254.000 0.988327 0.494163 0.869369i \(-0.335474\pi\)
0.494163 + 0.869369i \(0.335474\pi\)
\(258\) 0 0
\(259\) 180.133i 0.695495i
\(260\) −8.00000 13.8564i −0.0307692 0.0532939i
\(261\) 0 0
\(262\) 204.000 + 117.779i 0.778626 + 0.449540i
\(263\) 152.420i 0.579546i 0.957095 + 0.289773i \(0.0935797\pi\)
−0.957095 + 0.289773i \(0.906420\pi\)
\(264\) 0 0
\(265\) 148.000 0.558491
\(266\) 144.000 249.415i 0.541353 0.937652i
\(267\) 0 0
\(268\) −24.0000 + 13.8564i −0.0895522 + 0.0517030i
\(269\) −262.000 −0.973978 −0.486989 0.873408i \(-0.661905\pi\)
−0.486989 + 0.873408i \(0.661905\pi\)
\(270\) 0 0
\(271\) 20.7846i 0.0766960i −0.999264 0.0383480i \(-0.987790\pi\)
0.999264 0.0383480i \(-0.0122095\pi\)
\(272\) 80.0000 138.564i 0.294118 0.509427i
\(273\) 0 0
\(274\) −10.0000 + 17.3205i −0.0364964 + 0.0632135i
\(275\) 145.492i 0.529063i
\(276\) 0 0
\(277\) 290.000 1.04693 0.523466 0.852047i \(-0.324639\pi\)
0.523466 + 0.852047i \(0.324639\pi\)
\(278\) −84.0000 48.4974i −0.302158 0.174451i
\(279\) 0 0
\(280\) 110.851i 0.395897i
\(281\) −226.000 −0.804270 −0.402135 0.915580i \(-0.631732\pi\)
−0.402135 + 0.915580i \(0.631732\pi\)
\(282\) 0 0
\(283\) 297.913i 1.05270i −0.850269 0.526348i \(-0.823561\pi\)
0.850269 0.526348i \(-0.176439\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 24.0000 + 13.8564i 0.0839161 + 0.0484490i
\(287\) 401.836i 1.40012i
\(288\) 0 0
\(289\) −189.000 −0.653979
\(290\) 52.0000 90.0666i 0.179310 0.310575i
\(291\) 0 0
\(292\) 92.0000 + 159.349i 0.315068 + 0.545715i
\(293\) 362.000 1.23549 0.617747 0.786377i \(-0.288045\pi\)
0.617747 + 0.786377i \(0.288045\pi\)
\(294\) 0 0
\(295\) 180.133i 0.610621i
\(296\) −208.000 −0.702703
\(297\) 0 0
\(298\) 2.00000 3.46410i 0.00671141 0.0116245i
\(299\) 55.4256i 0.185370i
\(300\) 0 0
\(301\) −336.000 −1.11628
\(302\) 156.000 + 90.0666i 0.516556 + 0.298234i
\(303\) 0 0
\(304\) 288.000 + 166.277i 0.947368 + 0.546963i
\(305\) 52.0000 0.170492
\(306\) 0 0
\(307\) 145.492i 0.473916i −0.971520 0.236958i \(-0.923850\pi\)
0.971520 0.236958i \(-0.0761504\pi\)
\(308\) 96.0000 + 166.277i 0.311688 + 0.539860i
\(309\) 0 0
\(310\) −24.0000 13.8564i −0.0774194 0.0446981i
\(311\) 235.559i 0.757424i −0.925515 0.378712i \(-0.876367\pi\)
0.925515 0.378712i \(-0.123633\pi\)
\(312\) 0 0
\(313\) −478.000 −1.52716 −0.763578 0.645715i \(-0.776559\pi\)
−0.763578 + 0.645715i \(0.776559\pi\)
\(314\) −214.000 + 370.659i −0.681529 + 1.18044i
\(315\) 0 0
\(316\) −408.000 + 235.559i −1.29114 + 0.745440i
\(317\) 170.000 0.536278 0.268139 0.963380i \(-0.413591\pi\)
0.268139 + 0.963380i \(0.413591\pi\)
\(318\) 0 0
\(319\) 180.133i 0.564681i
\(320\) 128.000 0.400000
\(321\) 0 0
\(322\) 192.000 332.554i 0.596273 1.03278i
\(323\) 207.846i 0.643486i
\(324\) 0 0
\(325\) −42.0000 −0.129231
\(326\) −36.0000 20.7846i −0.110429 0.0637565i
\(327\) 0 0
\(328\) 464.000 1.41463
\(329\) −480.000 −1.45897
\(330\) 0 0
\(331\) 408.764i 1.23494i −0.786596 0.617468i \(-0.788158\pi\)
0.786596 0.617468i \(-0.211842\pi\)
\(332\) 168.000 96.9948i 0.506024 0.292153i
\(333\) 0 0
\(334\) 168.000 + 96.9948i 0.502994 + 0.290404i
\(335\) 13.8564i 0.0413624i
\(336\) 0 0
\(337\) 338.000 1.00297 0.501484 0.865167i \(-0.332788\pi\)
0.501484 + 0.865167i \(0.332788\pi\)
\(338\) −165.000 + 285.788i −0.488166 + 0.845528i
\(339\) 0 0
\(340\) 40.0000 + 69.2820i 0.117647 + 0.203771i
\(341\) 48.0000 0.140762
\(342\) 0 0
\(343\) 346.410i 1.00994i
\(344\) 387.979i 1.12785i
\(345\) 0 0
\(346\) −334.000 + 578.505i −0.965318 + 1.67198i
\(347\) 200.918i 0.579014i 0.957176 + 0.289507i \(0.0934914\pi\)
−0.957176 + 0.289507i \(0.906509\pi\)
\(348\) 0 0
\(349\) 506.000 1.44986 0.724928 0.688824i \(-0.241873\pi\)
0.724928 + 0.688824i \(0.241873\pi\)
\(350\) −252.000 145.492i −0.720000 0.415692i
\(351\) 0 0
\(352\) −192.000 + 110.851i −0.545455 + 0.314918i
\(353\) −178.000 −0.504249 −0.252125 0.967695i \(-0.581129\pi\)
−0.252125 + 0.967695i \(0.581129\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 164.000 + 284.056i 0.460674 + 0.797911i
\(357\) 0 0
\(358\) −324.000 187.061i −0.905028 0.522518i
\(359\) 166.277i 0.463167i 0.972815 + 0.231583i \(0.0743906\pi\)
−0.972815 + 0.231583i \(0.925609\pi\)
\(360\) 0 0
\(361\) −71.0000 −0.196676
\(362\) 2.00000 3.46410i 0.00552486 0.00956934i
\(363\) 0 0
\(364\) 48.0000 27.7128i 0.131868 0.0761341i
\(365\) −92.0000 −0.252055
\(366\) 0 0
\(367\) 200.918i 0.547460i −0.961807 0.273730i \(-0.911742\pi\)
0.961807 0.273730i \(-0.0882575\pi\)
\(368\) 384.000 + 221.703i 1.04348 + 0.602452i
\(369\) 0 0
\(370\) 52.0000 90.0666i 0.140541 0.243423i
\(371\) 512.687i 1.38191i
\(372\) 0 0
\(373\) −310.000 −0.831099 −0.415550 0.909571i \(-0.636411\pi\)
−0.415550 + 0.909571i \(0.636411\pi\)
\(374\) −120.000 69.2820i −0.320856 0.185246i
\(375\) 0 0
\(376\) 554.256i 1.47409i
\(377\) 52.0000 0.137931
\(378\) 0 0
\(379\) 436.477i 1.15165i 0.817572 + 0.575827i \(0.195320\pi\)
−0.817572 + 0.575827i \(0.804680\pi\)
\(380\) −144.000 + 83.1384i −0.378947 + 0.218785i
\(381\) 0 0
\(382\) −384.000 221.703i −1.00524 0.580373i
\(383\) 609.682i 1.59186i −0.605390 0.795929i \(-0.706983\pi\)
0.605390 0.795929i \(-0.293017\pi\)
\(384\) 0 0
\(385\) −96.0000 −0.249351
\(386\) 290.000 502.295i 0.751295 1.30128i
\(387\) 0 0
\(388\) −4.00000 6.92820i −0.0103093 0.0178562i
\(389\) 578.000 1.48586 0.742931 0.669368i \(-0.233435\pi\)
0.742931 + 0.669368i \(0.233435\pi\)
\(390\) 0 0
\(391\) 277.128i 0.708768i
\(392\) −8.00000 −0.0204082
\(393\) 0 0
\(394\) 26.0000 45.0333i 0.0659898 0.114298i
\(395\) 235.559i 0.596352i
\(396\) 0 0
\(397\) 26.0000 0.0654912 0.0327456 0.999464i \(-0.489575\pi\)
0.0327456 + 0.999464i \(0.489575\pi\)
\(398\) −684.000 394.908i −1.71859 0.992230i
\(399\) 0 0
\(400\) 168.000 290.985i 0.420000 0.727461i
\(401\) −250.000 −0.623441 −0.311721 0.950174i \(-0.600905\pi\)
−0.311721 + 0.950174i \(0.600905\pi\)
\(402\) 0 0
\(403\) 13.8564i 0.0343831i
\(404\) −148.000 256.344i −0.366337 0.634514i
\(405\) 0 0
\(406\) 312.000 + 180.133i 0.768473 + 0.443678i
\(407\) 180.133i 0.442588i
\(408\) 0 0
\(409\) 290.000 0.709046 0.354523 0.935047i \(-0.384643\pi\)
0.354523 + 0.935047i \(0.384643\pi\)
\(410\) −116.000 + 200.918i −0.282927 + 0.490044i
\(411\) 0 0
\(412\) 264.000 152.420i 0.640777 0.369953i
\(413\) 624.000 1.51090
\(414\) 0 0
\(415\) 96.9948i 0.233723i
\(416\) 32.0000 + 55.4256i 0.0769231 + 0.133235i
\(417\) 0 0
\(418\) 144.000 249.415i 0.344498 0.596687i
\(419\) 339.482i 0.810219i −0.914268 0.405110i \(-0.867233\pi\)
0.914268 0.405110i \(-0.132767\pi\)
\(420\) 0 0
\(421\) 674.000 1.60095 0.800475 0.599366i \(-0.204581\pi\)
0.800475 + 0.599366i \(0.204581\pi\)
\(422\) 420.000 + 242.487i 0.995261 + 0.574614i
\(423\) 0 0
\(424\) −592.000 −1.39623
\(425\) 210.000 0.494118
\(426\) 0 0
\(427\) 180.133i 0.421858i
\(428\) 72.0000 41.5692i 0.168224 0.0971243i
\(429\) 0 0
\(430\) 168.000 + 96.9948i 0.390698 + 0.225569i
\(431\) 540.400i 1.25383i 0.779088 + 0.626914i \(0.215682\pi\)
−0.779088 + 0.626914i \(0.784318\pi\)
\(432\) 0 0
\(433\) −334.000 −0.771363 −0.385681 0.922632i \(-0.626034\pi\)
−0.385681 + 0.922632i \(0.626034\pi\)
\(434\) 48.0000 83.1384i 0.110599 0.191563i
\(435\) 0 0
\(436\) 92.0000 + 159.349i 0.211009 + 0.365479i
\(437\) −576.000 −1.31808
\(438\) 0 0
\(439\) 117.779i 0.268290i −0.990962 0.134145i \(-0.957171\pi\)
0.990962 0.134145i \(-0.0428288\pi\)
\(440\) 110.851i 0.251935i
\(441\) 0 0
\(442\) −20.0000 + 34.6410i −0.0452489 + 0.0783733i
\(443\) 76.2102i 0.172032i −0.996294 0.0860161i \(-0.972586\pi\)
0.996294 0.0860161i \(-0.0274136\pi\)
\(444\) 0 0
\(445\) −164.000 −0.368539
\(446\) 588.000 + 339.482i 1.31839 + 0.761170i
\(447\) 0 0
\(448\) 443.405i 0.989743i
\(449\) −394.000 −0.877506 −0.438753 0.898608i \(-0.644580\pi\)
−0.438753 + 0.898608i \(0.644580\pi\)
\(450\) 0 0
\(451\) 401.836i 0.890988i
\(452\) −220.000 381.051i −0.486726 0.843034i
\(453\) 0 0
\(454\) −492.000 284.056i −1.08370 0.625675i
\(455\) 27.7128i 0.0609073i
\(456\) 0 0
\(457\) −478.000 −1.04595 −0.522976 0.852347i \(-0.675178\pi\)
−0.522976 + 0.852347i \(0.675178\pi\)
\(458\) −142.000 + 245.951i −0.310044 + 0.537011i
\(459\) 0 0
\(460\) −192.000 + 110.851i −0.417391 + 0.240981i
\(461\) −142.000 −0.308026 −0.154013 0.988069i \(-0.549220\pi\)
−0.154013 + 0.988069i \(0.549220\pi\)
\(462\) 0 0
\(463\) 630.466i 1.36170i −0.732423 0.680849i \(-0.761611\pi\)
0.732423 0.680849i \(-0.238389\pi\)
\(464\) −208.000 + 360.267i −0.448276 + 0.776437i
\(465\) 0 0
\(466\) −82.0000 + 142.028i −0.175966 + 0.304781i
\(467\) 20.7846i 0.0445067i 0.999752 + 0.0222533i \(0.00708404\pi\)
−0.999752 + 0.0222533i \(0.992916\pi\)
\(468\) 0 0
\(469\) 48.0000 0.102345
\(470\) 240.000 + 138.564i 0.510638 + 0.294817i
\(471\) 0 0
\(472\) 720.533i 1.52655i
\(473\) −336.000 −0.710359
\(474\) 0 0
\(475\) 436.477i 0.918899i
\(476\) −240.000 + 138.564i −0.504202 + 0.291101i
\(477\) 0 0
\(478\) 672.000 + 387.979i 1.40586 + 0.811672i
\(479\) 734.390i 1.53317i 0.642141 + 0.766586i \(0.278046\pi\)
−0.642141 + 0.766586i \(0.721954\pi\)
\(480\) 0 0
\(481\) 52.0000 0.108108
\(482\) −46.0000 + 79.6743i −0.0954357 + 0.165299i
\(483\) 0 0
\(484\) −146.000 252.879i −0.301653 0.522478i
\(485\) 4.00000 0.00824742
\(486\) 0 0
\(487\) 103.923i 0.213394i −0.994292 0.106697i \(-0.965972\pi\)
0.994292 0.106697i \(-0.0340275\pi\)
\(488\) −208.000 −0.426230
\(489\) 0 0
\(490\) 2.00000 3.46410i 0.00408163 0.00706960i
\(491\) 921.451i 1.87668i −0.345711 0.938341i \(-0.612362\pi\)
0.345711 0.938341i \(-0.387638\pi\)
\(492\) 0 0
\(493\) −260.000 −0.527383
\(494\) −72.0000 41.5692i −0.145749 0.0841482i
\(495\) 0 0
\(496\) 96.0000 + 55.4256i 0.193548 + 0.111745i
\(497\) 0 0
\(498\) 0 0
\(499\) 76.2102i 0.152726i 0.997080 + 0.0763630i \(0.0243308\pi\)
−0.997080 + 0.0763630i \(0.975669\pi\)
\(500\) 184.000 + 318.697i 0.368000 + 0.637395i
\(501\) 0 0
\(502\) 252.000 + 145.492i 0.501992 + 0.289825i
\(503\) 581.969i 1.15700i −0.815684 0.578498i \(-0.803639\pi\)
0.815684 0.578498i \(-0.196361\pi\)
\(504\) 0 0
\(505\) 148.000 0.293069
\(506\) 192.000 332.554i 0.379447 0.657221i
\(507\) 0 0
\(508\) 504.000 290.985i 0.992126 0.572804i
\(509\) 842.000 1.65422 0.827112 0.562037i \(-0.189982\pi\)
0.827112 + 0.562037i \(0.189982\pi\)
\(510\) 0 0
\(511\) 318.697i 0.623674i
\(512\) −512.000 −1.00000
\(513\) 0 0
\(514\) 254.000 439.941i 0.494163 0.855916i
\(515\) 152.420i 0.295962i
\(516\) 0 0
\(517\) −480.000 −0.928433
\(518\) 312.000 + 180.133i 0.602317 + 0.347748i
\(519\) 0 0
\(520\) −32.0000 −0.0615385
\(521\) 326.000 0.625720 0.312860 0.949799i \(-0.398713\pi\)
0.312860 + 0.949799i \(0.398713\pi\)
\(522\) 0 0
\(523\) 311.769i 0.596117i −0.954548 0.298058i \(-0.903661\pi\)
0.954548 0.298058i \(-0.0963390\pi\)
\(524\) 408.000 235.559i 0.778626 0.449540i
\(525\) 0 0
\(526\) 264.000 + 152.420i 0.501901 + 0.289773i
\(527\) 69.2820i 0.131465i
\(528\) 0 0
\(529\) −239.000 −0.451796
\(530\) 148.000 256.344i 0.279245 0.483667i
\(531\) 0 0
\(532\) −288.000 498.831i −0.541353 0.937652i
\(533\) −116.000 −0.217636
\(534\) 0 0
\(535\) 41.5692i 0.0776995i
\(536\) 55.4256i 0.103406i
\(537\) 0 0
\(538\) −262.000 + 453.797i −0.486989 + 0.843489i
\(539\) 6.92820i 0.0128538i
\(540\) 0 0
\(541\) 530.000 0.979667 0.489834 0.871816i \(-0.337058\pi\)
0.489834 + 0.871816i \(0.337058\pi\)
\(542\) −36.0000 20.7846i −0.0664207 0.0383480i
\(543\) 0 0
\(544\) −160.000 277.128i −0.294118 0.509427i
\(545\) −92.0000 −0.168807
\(546\) 0 0
\(547\) 339.482i 0.620625i 0.950635 + 0.310313i \(0.100434\pi\)
−0.950635 + 0.310313i \(0.899566\pi\)
\(548\) 20.0000 + 34.6410i 0.0364964 + 0.0632135i
\(549\) 0 0
\(550\) −252.000 145.492i −0.458182 0.264531i
\(551\) 540.400i 0.980762i
\(552\) 0 0
\(553\) 816.000 1.47559
\(554\) 290.000 502.295i 0.523466 0.906669i
\(555\) 0 0
\(556\) −168.000 + 96.9948i −0.302158 + 0.174451i
\(557\) −766.000 −1.37522 −0.687612 0.726078i \(-0.741341\pi\)
−0.687612 + 0.726078i \(0.741341\pi\)
\(558\) 0 0
\(559\) 96.9948i 0.173515i
\(560\) −192.000 110.851i −0.342857 0.197949i
\(561\) 0 0
\(562\) −226.000 + 391.443i −0.402135 + 0.696519i
\(563\) 491.902i 0.873717i 0.899530 + 0.436858i \(0.143909\pi\)
−0.899530 + 0.436858i \(0.856091\pi\)
\(564\) 0 0
\(565\) 220.000 0.389381
\(566\) −516.000 297.913i −0.911661 0.526348i
\(567\) 0 0
\(568\) 0 0
\(569\) 422.000 0.741652 0.370826 0.928702i \(-0.379075\pi\)
0.370826 + 0.928702i \(0.379075\pi\)
\(570\) 0 0
\(571\) 284.056i 0.497472i 0.968571 + 0.248736i \(0.0800151\pi\)
−0.968571 + 0.248736i \(0.919985\pi\)
\(572\) 48.0000 27.7128i 0.0839161 0.0484490i
\(573\) 0 0
\(574\) −696.000 401.836i −1.21254 0.700062i
\(575\) 581.969i 1.01212i
\(576\) 0 0
\(577\) −46.0000 −0.0797227 −0.0398614 0.999205i \(-0.512692\pi\)
−0.0398614 + 0.999205i \(0.512692\pi\)
\(578\) −189.000 + 327.358i −0.326990 + 0.566363i
\(579\) 0 0
\(580\) −104.000 180.133i −0.179310 0.310575i
\(581\) −336.000 −0.578313
\(582\) 0 0
\(583\) 512.687i 0.879395i
\(584\) 368.000 0.630137
\(585\) 0 0
\(586\) 362.000 627.002i 0.617747 1.06997i
\(587\) 630.466i 1.07405i 0.843567 + 0.537024i \(0.180452\pi\)
−0.843567 + 0.537024i \(0.819548\pi\)
\(588\) 0 0
\(589\) −144.000 −0.244482
\(590\) −312.000 180.133i −0.528814 0.305311i
\(591\) 0 0
\(592\) −208.000 + 360.267i −0.351351 + 0.608558i
\(593\) −82.0000 −0.138280 −0.0691400 0.997607i \(-0.522026\pi\)
−0.0691400 + 0.997607i \(0.522026\pi\)
\(594\) 0 0
\(595\) 138.564i 0.232881i
\(596\) −4.00000 6.92820i −0.00671141 0.0116245i
\(597\) 0 0
\(598\) −96.0000 55.4256i −0.160535 0.0926850i
\(599\) 55.4256i 0.0925303i 0.998929 + 0.0462651i \(0.0147319\pi\)
−0.998929 + 0.0462651i \(0.985268\pi\)
\(600\) 0 0
\(601\) −334.000 −0.555740 −0.277870 0.960619i \(-0.589629\pi\)
−0.277870 + 0.960619i \(0.589629\pi\)
\(602\) −336.000 + 581.969i −0.558140 + 0.966726i
\(603\) 0 0
\(604\) 312.000 180.133i 0.516556 0.298234i
\(605\) 146.000 0.241322
\(606\) 0 0
\(607\) 367.195i 0.604934i 0.953160 + 0.302467i \(0.0978102\pi\)
−0.953160 + 0.302467i \(0.902190\pi\)
\(608\) 576.000 332.554i 0.947368 0.546963i
\(609\) 0 0
\(610\) 52.0000 90.0666i 0.0852459 0.147650i
\(611\) 138.564i 0.226782i
\(612\) 0 0
\(613\) −214.000 −0.349103 −0.174551 0.984648i \(-0.555848\pi\)
−0.174551 + 0.984648i \(0.555848\pi\)
\(614\) −252.000 145.492i −0.410423 0.236958i
\(615\) 0 0
\(616\) 384.000 0.623377
\(617\) 1118.00 1.81199 0.905997 0.423285i \(-0.139123\pi\)
0.905997 + 0.423285i \(0.139123\pi\)
\(618\) 0 0
\(619\) 672.036i 1.08568i 0.839836 + 0.542840i \(0.182651\pi\)
−0.839836 + 0.542840i \(0.817349\pi\)
\(620\) −48.0000 + 27.7128i −0.0774194 + 0.0446981i
\(621\) 0 0
\(622\) −408.000 235.559i −0.655949 0.378712i
\(623\) 568.113i 0.911898i
\(624\) 0 0
\(625\) 341.000 0.545600
\(626\) −478.000 + 827.920i −0.763578 + 1.32256i
\(627\) 0 0
\(628\) 428.000 + 741.318i 0.681529 + 1.18044i
\(629\) −260.000 −0.413355
\(630\) 0 0
\(631\) 145.492i 0.230574i 0.993332 + 0.115287i \(0.0367788\pi\)
−0.993332 + 0.115287i \(0.963221\pi\)
\(632\) 942.236i 1.49088i
\(633\) 0 0
\(634\) 170.000 294.449i 0.268139 0.464430i
\(635\) 290.985i 0.458243i
\(636\) 0 0
\(637\) 2.00000 0.00313972
\(638\) 312.000 + 180.133i 0.489028 + 0.282341i
\(639\) 0 0
\(640\) 128.000 221.703i 0.200000 0.346410i
\(641\) −10.0000 −0.0156006 −0.00780031 0.999970i \(-0.502483\pi\)
−0.00780031 + 0.999970i \(0.502483\pi\)
\(642\) 0 0
\(643\) 1212.44i 1.88559i −0.333370 0.942796i \(-0.608186\pi\)
0.333370 0.942796i \(-0.391814\pi\)
\(644\) −384.000 665.108i −0.596273 1.03278i
\(645\) 0 0
\(646\) 360.000 + 207.846i 0.557276 + 0.321743i
\(647\) 332.554i 0.513993i −0.966412 0.256997i \(-0.917267\pi\)
0.966412 0.256997i \(-0.0827330\pi\)
\(648\) 0 0
\(649\) 624.000 0.961479
\(650\) −42.0000 + 72.7461i −0.0646154 + 0.111917i
\(651\) 0 0
\(652\) −72.0000 + 41.5692i −0.110429 + 0.0637565i
\(653\) −670.000 −1.02603 −0.513017 0.858379i \(-0.671472\pi\)
−0.513017 + 0.858379i \(0.671472\pi\)
\(654\) 0 0
\(655\) 235.559i 0.359632i
\(656\) 464.000 803.672i 0.707317 1.22511i
\(657\) 0 0
\(658\) −480.000 + 831.384i −0.729483 + 1.26350i
\(659\) 824.456i 1.25107i −0.780195 0.625536i \(-0.784880\pi\)
0.780195 0.625536i \(-0.215120\pi\)
\(660\) 0 0
\(661\) −1222.00 −1.84871 −0.924357 0.381529i \(-0.875398\pi\)
−0.924357 + 0.381529i \(0.875398\pi\)
\(662\) −708.000 408.764i −1.06949 0.617468i
\(663\) 0 0
\(664\) 387.979i 0.584306i
\(665\) 288.000 0.433083
\(666\) 0 0
\(667\) 720.533i 1.08026i
\(668\) 336.000 193.990i 0.502994 0.290404i
\(669\) 0 0
\(670\) −24.0000 13.8564i −0.0358209 0.0206812i
\(671\) 180.133i 0.268455i
\(672\) 0 0
\(673\) −334.000 −0.496285 −0.248143 0.968724i \(-0.579820\pi\)
−0.248143 + 0.968724i \(0.579820\pi\)
\(674\) 338.000 585.433i 0.501484 0.868595i
\(675\) 0 0
\(676\) 330.000 + 571.577i 0.488166 + 0.845528i
\(677\) −1006.00 −1.48597 −0.742984 0.669309i \(-0.766590\pi\)
−0.742984 + 0.669309i \(0.766590\pi\)
\(678\) 0 0
\(679\) 13.8564i 0.0204071i
\(680\) 160.000 0.235294
\(681\) 0 0
\(682\) 48.0000 83.1384i 0.0703812 0.121904i
\(683\) 187.061i 0.273882i −0.990579 0.136941i \(-0.956273\pi\)
0.990579 0.136941i \(-0.0437271\pi\)
\(684\) 0 0
\(685\) −20.0000 −0.0291971
\(686\) 600.000 + 346.410i 0.874636 + 0.504971i
\(687\) 0 0
\(688\) −672.000 387.979i −0.976744 0.563924i
\(689\) 148.000 0.214804
\(690\) 0 0
\(691\) 990.733i 1.43377i −0.697193 0.716884i \(-0.745568\pi\)
0.697193 0.716884i \(-0.254432\pi\)
\(692\) 668.000 + 1157.01i 0.965318 + 1.67198i
\(693\) 0 0
\(694\) 348.000 + 200.918i 0.501441 + 0.289507i
\(695\) 96.9948i 0.139561i
\(696\) 0 0
\(697\) 580.000 0.832138
\(698\) 506.000 876.418i 0.724928 1.25561i
\(699\) 0 0
\(700\) −504.000 + 290.985i −0.720000 + 0.415692i
\(701\) 1034.00 1.47504 0.737518 0.675328i \(-0.235998\pi\)
0.737518 + 0.675328i \(0.235998\pi\)
\(702\) 0 0
\(703\) 540.400i 0.768705i
\(704\) 443.405i 0.629837i
\(705\) 0 0
\(706\) −178.000 + 308.305i −0.252125 + 0.436693i
\(707\) 512.687i 0.725158i
\(708\) 0 0
\(709\) 530.000 0.747532 0.373766 0.927523i \(-0.378066\pi\)
0.373766 + 0.927523i \(0.378066\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 656.000 0.921348
\(713\) −192.000 −0.269285
\(714\) 0 0
\(715\) 27.7128i 0.0387592i
\(716\) −648.000 + 374.123i −0.905028 + 0.522518i
\(717\) 0 0
\(718\) 288.000 + 166.277i 0.401114 + 0.231583i
\(719\) 706.677i 0.982861i −0.870917 0.491430i \(-0.836474\pi\)
0.870917 0.491430i \(-0.163526\pi\)
\(720\) 0 0
\(721\) −528.000 −0.732316
\(722\) −71.0000 + 122.976i −0.0983380 + 0.170326i
\(723\) 0 0
\(724\) −4.00000 6.92820i −0.00552486 0.00956934i
\(725\) −546.000 −0.753103
\(726\) 0 0
\(727\) 242.487i 0.333545i −0.985995 0.166772i \(-0.946665\pi\)
0.985995 0.166772i \(-0.0533345\pi\)
\(728\) 110.851i 0.152268i
\(729\) 0 0
\(730\) −92.0000 + 159.349i −0.126027 + 0.218286i
\(731\) 484.974i 0.663439i
\(732\) 0 0
\(733\) 194.000 0.264666 0.132333 0.991205i \(-0.457753\pi\)
0.132333 + 0.991205i \(0.457753\pi\)
\(734\) −348.000 200.918i −0.474114 0.273730i
\(735\) 0 0
\(736\) 768.000 443.405i 1.04348 0.602452i
\(737\) 48.0000 0.0651289
\(738\) 0 0
\(739\) 1351.00i 1.82815i 0.405550 + 0.914073i \(0.367080\pi\)
−0.405550 + 0.914073i \(0.632920\pi\)
\(740\) −104.000 180.133i −0.140541 0.243423i
\(741\) 0 0
\(742\) 888.000 + 512.687i 1.19677 + 0.690953i
\(743\) 678.964i 0.913814i 0.889514 + 0.456907i \(0.151043\pi\)
−0.889514 + 0.456907i \(0.848957\pi\)
\(744\) 0 0
\(745\) 4.00000 0.00536913
\(746\) −310.000 + 536.936i −0.415550 + 0.719753i
\(747\) 0 0
\(748\) −240.000 + 138.564i −0.320856 + 0.185246i
\(749\) −144.000 −0.192256
\(750\) 0 0
\(751\) 658.179i 0.876404i 0.898877 + 0.438202i \(0.144385\pi\)
−0.898877 + 0.438202i \(0.855615\pi\)
\(752\) −960.000 554.256i −1.27660 0.737043i
\(753\) 0 0
\(754\) 52.0000 90.0666i 0.0689655 0.119452i
\(755\) 180.133i 0.238587i
\(756\) 0 0
\(757\) −1006.00 −1.32893 −0.664465 0.747319i \(-0.731341\pi\)
−0.664465 + 0.747319i \(0.731341\pi\)
\(758\) 756.000 + 436.477i 0.997361 + 0.575827i
\(759\) 0 0
\(760\) 332.554i 0.437571i
\(761\) 758.000 0.996058 0.498029 0.867160i \(-0.334057\pi\)
0.498029 + 0.867160i \(0.334057\pi\)
\(762\) 0 0
\(763\) 318.697i 0.417690i
\(764\) −768.000 + 443.405i −1.00524 + 0.580373i
\(765\) 0 0
\(766\) −1056.00 609.682i −1.37859 0.795929i
\(767\) 180.133i 0.234854i
\(768\) 0 0
\(769\) 2.00000 0.00260078 0.00130039 0.999999i \(-0.499586\pi\)
0.00130039 + 0.999999i \(0.499586\pi\)
\(770\) −96.0000 + 166.277i −0.124675 + 0.215944i
\(771\) 0 0
\(772\) −580.000 1004.59i −0.751295 1.30128i
\(773\) −262.000 −0.338939 −0.169470 0.985535i \(-0.554205\pi\)
−0.169470 + 0.985535i \(0.554205\pi\)
\(774\) 0 0
\(775\) 145.492i 0.187732i
\(776\) −16.0000 −0.0206186
\(777\) 0 0
\(778\) 578.000 1001.13i 0.742931 1.28679i
\(779\) 1205.51i 1.54751i
\(780\) 0 0
\(781\) 0 0
\(782\) 480.000 + 277.128i 0.613811 + 0.354384i
\(783\) 0 0
\(784\) −8.00000 + 13.8564i −0.0102041 + 0.0176740i
\(785\) −428.000 −0.545223
\(786\) 0 0
\(787\) 1447.99i 1.83989i 0.392046 + 0.919946i \(0.371767\pi\)
−0.392046 + 0.919946i \(0.628233\pi\)
\(788\) −52.0000 90.0666i −0.0659898 0.114298i
\(789\) 0 0
\(790\) −408.000 235.559i −0.516456 0.298176i
\(791\) 762.102i 0.963467i
\(792\) 0 0
\(793\) 52.0000 0.0655738
\(794\) 26.0000 45.0333i 0.0327456 0.0567170i
\(795\) 0 0
\(796\) −1368.00 + 789.815i −1.71859 + 0.992230i
\(797\) 866.000 1.08657 0.543287 0.839547i \(-0.317179\pi\)
0.543287 + 0.839547i \(0.317179\pi\)
\(798\) 0 0
\(799\) 692.820i 0.867109i
\(800\) −336.000 581.969i −0.420000 0.727461i
\(801\) 0 0
\(802\) −250.000 + 433.013i −0.311721 + 0.539916i
\(803\) 318.697i 0.396883i
\(804\) 0 0
\(805\) 384.000 0.477019
\(806\) −24.0000 13.8564i −0.0297767 0.0171916i
\(807\) 0 0
\(808\) −592.000 −0.732673
\(809\) −10.0000 −0.0123609 −0.00618047 0.999981i \(-0.501967\pi\)
−0.00618047 + 0.999981i \(0.501967\pi\)
\(810\) 0 0
\(811\) 436.477i 0.538196i 0.963113 + 0.269098i \(0.0867255\pi\)
−0.963113 + 0.269098i \(0.913274\pi\)
\(812\) 624.000 360.267i 0.768473 0.443678i
\(813\) 0 0
\(814\) 312.000 + 180.133i 0.383292 + 0.221294i
\(815\) 41.5692i 0.0510052i
\(816\) 0 0
\(817\) 1008.00 1.23378
\(818\) 290.000 502.295i 0.354523 0.614052i
\(819\) 0 0
\(820\) 232.000 + 401.836i 0.282927 + 0.490044i
\(821\) −838.000 −1.02071 −0.510353 0.859965i \(-0.670485\pi\)
−0.510353 + 0.859965i \(0.670485\pi\)
\(822\) 0 0
\(823\) 879.882i 1.06912i −0.845132 0.534558i \(-0.820478\pi\)
0.845132 0.534558i \(-0.179522\pi\)
\(824\) 609.682i 0.739905i
\(825\) 0 0
\(826\) 624.000 1080.80i 0.755448 1.30847i
\(827\) 727.461i 0.879639i −0.898086 0.439819i \(-0.855042\pi\)
0.898086 0.439819i \(-0.144958\pi\)
\(828\) 0 0
\(829\) 1298.00 1.56574 0.782871 0.622184i \(-0.213754\pi\)
0.782871 + 0.622184i \(0.213754\pi\)
\(830\) 168.000 + 96.9948i 0.202410 + 0.116861i
\(831\) 0 0
\(832\) 128.000 0.153846
\(833\) −10.0000 −0.0120048
\(834\) 0 0
\(835\) 193.990i 0.232323i
\(836\) −288.000 498.831i −0.344498 0.596687i
\(837\) 0 0
\(838\) −588.000 339.482i −0.701671 0.405110i
\(839\) 193.990i 0.231215i 0.993295 + 0.115608i \(0.0368815\pi\)
−0.993295 + 0.115608i \(0.963118\pi\)
\(840\) 0 0
\(841\) −165.000 −0.196195
\(842\) 674.000 1167.40i 0.800475 1.38646i
\(843\) 0 0
\(844\) 840.000 484.974i 0.995261 0.574614i
\(845\) −330.000 −0.390533
\(846\) 0 0
\(847\) 505.759i 0.597118i
\(848\) −592.000 + 1025.37i −0.698113 + 1.20917i
\(849\) 0 0
\(850\) 210.000 363.731i 0.247059 0.427918i
\(851\) 720.533i 0.846690i
\(852\) 0 0
\(853\) 506.000 0.593200 0.296600 0.955002i \(-0.404147\pi\)
0.296600 + 0.955002i \(0.404147\pi\)
\(854\) 312.000 + 180.133i 0.365340 + 0.210929i
\(855\) 0 0
\(856\) 166.277i 0.194249i
\(857\) 998.000 1.16453 0.582264 0.813000i \(-0.302167\pi\)
0.582264 + 0.813000i \(0.302167\pi\)
\(858\) 0 0
\(859\) 505.759i 0.588776i −0.955686 0.294388i \(-0.904884\pi\)
0.955686 0.294388i \(-0.0951158\pi\)
\(860\) 336.000 193.990i 0.390698 0.225569i
\(861\) 0 0
\(862\) 936.000 + 540.400i 1.08585 + 0.626914i
\(863\) 166.277i 0.192673i 0.995349 + 0.0963365i \(0.0307125\pi\)
−0.995349 + 0.0963365i \(0.969287\pi\)
\(864\) 0 0
\(865\) −668.000 −0.772254
\(866\) −334.000 + 578.505i −0.385681 + 0.668020i
\(867\) 0 0
\(868\) −96.0000 166.277i −0.110599 0.191563i
\(869\) 816.000 0.939010
\(870\) 0 0
\(871\) 13.8564i 0.0159086i
\(872\) 368.000 0.422018
\(873\) 0 0
\(874\) −576.000 + 997.661i −0.659039 + 1.14149i
\(875\) 637.395i 0.728451i
\(876\) 0 0
\(877\) −646.000 −0.736602 −0.368301 0.929707i \(-0.620060\pi\)
−0.368301 + 0.929707i \(0.620060\pi\)
\(878\) −204.000 117.779i −0.232346 0.134145i
\(879\) 0 0
\(880\) −192.000 110.851i −0.218182 0.125967i
\(881\) −898.000 −1.01930 −0.509648 0.860383i \(-0.670224\pi\)
−0.509648 + 0.860383i \(0.670224\pi\)
\(882\) 0 0
\(883\) 727.461i 0.823852i 0.911217 + 0.411926i \(0.135144\pi\)
−0.911217 + 0.411926i \(0.864856\pi\)
\(884\) 40.0000 + 69.2820i 0.0452489 + 0.0783733i
\(885\) 0 0
\(886\) −132.000 76.2102i −0.148984 0.0860161i
\(887\) 845.241i 0.952921i 0.879196 + 0.476460i \(0.158080\pi\)
−0.879196 + 0.476460i \(0.841920\pi\)
\(888\) 0 0
\(889\) −1008.00 −1.13386
\(890\) −164.000 + 284.056i −0.184270 + 0.319164i
\(891\) 0 0
\(892\) 1176.00 678.964i 1.31839 0.761170i
\(893\) 1440.00 1.61254
\(894\) 0 0
\(895\) 374.123i 0.418014i
\(896\) 768.000 + 443.405i 0.857143 + 0.494872i
\(897\) 0 0
\(898\) −394.000 + 682.428i −0.438753 + 0.759942i
\(899\) 180.133i 0.200371i
\(900\) 0 0
\(901\) −740.000 −0.821310
\(902\) −696.000 401.836i −0.771619 0.445494i
\(903\) 0 0
\(904\) −880.000 −0.973451
\(905\) 4.00000 0.00441989
\(906\) 0 0
\(907\) 1364.86i 1.50480i −0.658705 0.752401i \(-0.728895\pi\)
0.658705 0.752401i \(-0.271105\pi\)
\(908\) −984.000 + 568.113i −1.08370 + 0.625675i
\(909\) 0 0
\(910\) 48.0000 + 27.7128i 0.0527473 + 0.0304536i
\(911\) 387.979i 0.425883i −0.977065 0.212941i \(-0.931696\pi\)
0.977065 0.212941i \(-0.0683044\pi\)
\(912\) 0 0
\(913\) −336.000 −0.368018
\(914\) −478.000 + 827.920i −0.522976 + 0.905821i
\(915\) 0 0
\(916\) 284.000 + 491.902i 0.310044 + 0.537011i
\(917\) −816.000 −0.889858
\(918\) 0 0
\(919\) 602.754i 0.655880i −0.944699 0.327940i \(-0.893646\pi\)
0.944699 0.327940i \(-0.106354\pi\)
\(920\) 443.405i 0.481962i
\(921\) 0 0
\(922\) −142.000 + 245.951i −0.154013 + 0.266758i
\(923\) 0 0
\(924\) 0 0
\(925\) −546.000 −0.590270
\(926\) −1092.00 630.466i −1.17927 0.680849i
\(927\) 0 0
\(928\) 416.000 + 720.533i 0.448276 + 0.776437i
\(929\) −1594.00 −1.71582 −0.857912 0.513797i \(-0.828238\pi\)
−0.857912 + 0.513797i \(0.828238\pi\)
\(930\) 0 0
\(931\) 20.7846i 0.0223250i
\(932\) 164.000 + 284.056i 0.175966 + 0.304781i
\(933\) 0 0
\(934\) 36.0000 + 20.7846i 0.0385439 + 0.0222533i
\(935\) 138.564i 0.148197i
\(936\) 0 0
\(937\) 674.000 0.719317 0.359658 0.933084i \(-0.382893\pi\)
0.359658 + 0.933084i \(0.382893\pi\)
\(938\) 48.0000 83.1384i 0.0511727 0.0886337i
\(939\) 0 0
\(940\) 480.000 277.128i 0.510638 0.294817i
\(941\) −430.000 −0.456961 −0.228480 0.973549i \(-0.573376\pi\)
−0.228480 + 0.973549i \(0.573376\pi\)
\(942\) 0 0
\(943\) 1607.34i 1.70450i
\(944\) 1248.00 + 720.533i 1.32203 + 0.763277i
\(945\) 0 0
\(946\) −336.000 + 581.969i −0.355180 + 0.615189i
\(947\) 76.2102i 0.0804754i −0.999190 0.0402377i \(-0.987188\pi\)
0.999190 0.0402377i \(-0.0128115\pi\)
\(948\) 0 0
\(949\) −92.0000 −0.0969442
\(950\) 756.000 + 436.477i 0.795789 + 0.459449i
\(951\) 0 0
\(952\) 554.256i 0.582202i
\(953\) −730.000 −0.766002 −0.383001 0.923748i \(-0.625109\pi\)
−0.383001 + 0.923748i \(0.625109\pi\)
\(954\) 0 0
\(955\) 443.405i 0.464298i
\(956\) 1344.00 775.959i 1.40586 0.811672i
\(957\) 0 0
\(958\) 1272.00 + 734.390i 1.32777 + 0.766586i
\(959\) 69.2820i 0.0722440i
\(960\) 0 0
\(961\) 913.000 0.950052
\(962\) 52.0000 90.0666i 0.0540541 0.0936244i
\(963\) 0 0
\(964\) 92.0000 + 159.349i 0.0954357 + 0.165299i
\(965\) 580.000 0.601036
\(966\) 0 0
\(967\) 921.451i 0.952897i −0.879202 0.476448i \(-0.841924\pi\)
0.879202 0.476448i \(-0.158076\pi\)
\(968\) −584.000 −0.603306
\(969\) 0 0
\(970\) 4.00000 6.92820i 0.00412371 0.00714248i
\(971\) 1475.71i 1.51978i 0.650051 + 0.759890i \(0.274747\pi\)
−0.650051 + 0.759890i \(0.725253\pi\)
\(972\) 0 0
\(973\) 336.000 0.345324
\(974\) −180.000 103.923i −0.184805 0.106697i
\(975\) 0 0
\(976\) −208.000 + 360.267i −0.213115 + 0.369126i
\(977\) −346.000 −0.354145 −0.177073 0.984198i \(-0.556663\pi\)
−0.177073 + 0.984198i \(0.556663\pi\)
\(978\) 0 0
\(979\) 568.113i 0.580299i
\(980\) −4.00000 6.92820i −0.00408163 0.00706960i
\(981\) 0 0
\(982\) −1596.00 921.451i −1.62525 0.938341i
\(983\) 734.390i 0.747090i 0.927612 + 0.373545i \(0.121858\pi\)
−0.927612 + 0.373545i \(0.878142\pi\)
\(984\) 0 0
\(985\) 52.0000 0.0527919
\(986\) −260.000 + 450.333i −0.263692 + 0.456727i
\(987\) 0 0
\(988\) −144.000 + 83.1384i −0.145749 + 0.0841482i
\(989\) 1344.00 1.35895
\(990\) 0 0
\(991\) 976.877i 0.985748i −0.870101 0.492874i \(-0.835946\pi\)
0.870101 0.492874i \(-0.164054\pi\)
\(992\) 192.000 110.851i 0.193548 0.111745i
\(993\) 0 0
\(994\) 0 0
\(995\) 789.815i 0.793784i
\(996\) 0 0
\(997\) 458.000 0.459378 0.229689 0.973264i \(-0.426229\pi\)
0.229689 + 0.973264i \(0.426229\pi\)
\(998\) 132.000 + 76.2102i 0.132265 + 0.0763630i
\(999\) 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 36.3.d.c.19.1 2
3.2 odd 2 12.3.d.a.7.2 yes 2
4.3 odd 2 inner 36.3.d.c.19.2 2
5.2 odd 4 900.3.f.c.199.4 4
5.3 odd 4 900.3.f.c.199.1 4
5.4 even 2 900.3.c.e.451.2 2
8.3 odd 2 576.3.g.e.127.1 2
8.5 even 2 576.3.g.e.127.2 2
9.2 odd 6 324.3.f.j.271.1 2
9.4 even 3 324.3.f.g.55.1 2
9.5 odd 6 324.3.f.d.55.1 2
9.7 even 3 324.3.f.a.271.1 2
12.11 even 2 12.3.d.a.7.1 2
15.2 even 4 300.3.f.a.199.1 4
15.8 even 4 300.3.f.a.199.4 4
15.14 odd 2 300.3.c.b.151.1 2
16.3 odd 4 2304.3.b.l.127.2 4
16.5 even 4 2304.3.b.l.127.3 4
16.11 odd 4 2304.3.b.l.127.4 4
16.13 even 4 2304.3.b.l.127.1 4
20.3 even 4 900.3.f.c.199.3 4
20.7 even 4 900.3.f.c.199.2 4
20.19 odd 2 900.3.c.e.451.1 2
21.20 even 2 588.3.g.b.295.2 2
24.5 odd 2 192.3.g.b.127.2 2
24.11 even 2 192.3.g.b.127.1 2
36.7 odd 6 324.3.f.g.271.1 2
36.11 even 6 324.3.f.d.271.1 2
36.23 even 6 324.3.f.j.55.1 2
36.31 odd 6 324.3.f.a.55.1 2
48.5 odd 4 768.3.b.c.127.1 4
48.11 even 4 768.3.b.c.127.3 4
48.29 odd 4 768.3.b.c.127.4 4
48.35 even 4 768.3.b.c.127.2 4
60.23 odd 4 300.3.f.a.199.2 4
60.47 odd 4 300.3.f.a.199.3 4
60.59 even 2 300.3.c.b.151.2 2
84.83 odd 2 588.3.g.b.295.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.3.d.a.7.1 2 12.11 even 2
12.3.d.a.7.2 yes 2 3.2 odd 2
36.3.d.c.19.1 2 1.1 even 1 trivial
36.3.d.c.19.2 2 4.3 odd 2 inner
192.3.g.b.127.1 2 24.11 even 2
192.3.g.b.127.2 2 24.5 odd 2
300.3.c.b.151.1 2 15.14 odd 2
300.3.c.b.151.2 2 60.59 even 2
300.3.f.a.199.1 4 15.2 even 4
300.3.f.a.199.2 4 60.23 odd 4
300.3.f.a.199.3 4 60.47 odd 4
300.3.f.a.199.4 4 15.8 even 4
324.3.f.a.55.1 2 36.31 odd 6
324.3.f.a.271.1 2 9.7 even 3
324.3.f.d.55.1 2 9.5 odd 6
324.3.f.d.271.1 2 36.11 even 6
324.3.f.g.55.1 2 9.4 even 3
324.3.f.g.271.1 2 36.7 odd 6
324.3.f.j.55.1 2 36.23 even 6
324.3.f.j.271.1 2 9.2 odd 6
576.3.g.e.127.1 2 8.3 odd 2
576.3.g.e.127.2 2 8.5 even 2
588.3.g.b.295.1 2 84.83 odd 2
588.3.g.b.295.2 2 21.20 even 2
768.3.b.c.127.1 4 48.5 odd 4
768.3.b.c.127.2 4 48.35 even 4
768.3.b.c.127.3 4 48.11 even 4
768.3.b.c.127.4 4 48.29 odd 4
900.3.c.e.451.1 2 20.19 odd 2
900.3.c.e.451.2 2 5.4 even 2
900.3.f.c.199.1 4 5.3 odd 4
900.3.f.c.199.2 4 20.7 even 4
900.3.f.c.199.3 4 20.3 even 4
900.3.f.c.199.4 4 5.2 odd 4
2304.3.b.l.127.1 4 16.13 even 4
2304.3.b.l.127.2 4 16.3 odd 4
2304.3.b.l.127.3 4 16.5 even 4
2304.3.b.l.127.4 4 16.11 odd 4