Properties

Label 1368.2.e.b.379.2
Level $1368$
Weight $2$
Character 1368.379
Analytic conductor $10.924$
Analytic rank $0$
Dimension $4$
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] = [1368,2,Mod(379,1368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("1368.379");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.e (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: \(10.9235349965\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 379.2
Root \(0.809017 + 1.40126i\) of defining polynomial
Character \(\chi\) \(=\) 1368.379
Dual form 1368.2.e.b.379.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.11803 + 0.866025i) q^{2} +(0.500000 - 1.93649i) q^{4} +3.87298i q^{7} +(1.11803 + 2.59808i) q^{8} +O(q^{10})\) \(q+(-1.11803 + 0.866025i) q^{2} +(0.500000 - 1.93649i) q^{4} +3.87298i q^{7} +(1.11803 + 2.59808i) q^{8} +4.00000 q^{11} +2.23607 q^{13} +(-3.35410 - 4.33013i) q^{14} +(-3.50000 - 1.93649i) q^{16} +1.00000 q^{17} +(4.00000 + 1.73205i) q^{19} +(-4.47214 + 3.46410i) q^{22} -3.87298i q^{23} +5.00000 q^{25} +(-2.50000 + 1.93649i) q^{26} +(7.50000 + 1.93649i) q^{28} -2.23607 q^{29} -8.94427 q^{31} +(5.59017 - 0.866025i) q^{32} +(-1.11803 + 0.866025i) q^{34} -4.47214 q^{37} +(-5.97214 + 1.52761i) q^{38} -6.92820i q^{41} +(2.00000 - 7.74597i) q^{44} +(3.35410 + 4.33013i) q^{46} +7.74597i q^{47} -8.00000 q^{49} +(-5.59017 + 4.33013i) q^{50} +(1.11803 - 4.33013i) q^{52} +6.70820 q^{53} +(-10.0623 + 4.33013i) q^{56} +(2.50000 - 1.93649i) q^{58} +8.66025i q^{59} +(10.0000 - 7.74597i) q^{62} +(-5.50000 + 5.80948i) q^{64} +12.1244i q^{67} +(0.500000 - 1.93649i) q^{68} +8.94427 q^{71} +3.00000 q^{73} +(5.00000 - 3.87298i) q^{74} +(5.35410 - 6.87994i) q^{76} +15.4919i q^{77} +8.94427 q^{79} +(6.00000 + 7.74597i) q^{82} +8.00000 q^{83} +(4.47214 + 10.3923i) q^{88} +13.8564i q^{89} +8.66025i q^{91} +(-7.50000 - 1.93649i) q^{92} +(-6.70820 - 8.66025i) q^{94} -6.92820i q^{97} +(8.94427 - 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} + 16 q^{11} - 14 q^{16} + 4 q^{17} + 16 q^{19} + 20 q^{25} - 10 q^{26} + 30 q^{28} - 6 q^{38} + 8 q^{44} - 32 q^{49} + 10 q^{58} + 40 q^{62} - 22 q^{64} + 2 q^{68} + 12 q^{73} + 20 q^{74} + 8 q^{76} + 24 q^{82} + 32 q^{83} - 30 q^{92}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(685\) \(1009\) \(1217\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.11803 + 0.866025i −0.790569 + 0.612372i
\(3\) 0 0
\(4\) 0.500000 1.93649i 0.250000 0.968246i
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 3.87298i 1.46385i 0.681385 + 0.731925i \(0.261378\pi\)
−0.681385 + 0.731925i \(0.738622\pi\)
\(8\) 1.11803 + 2.59808i 0.395285 + 0.918559i
\(9\) 0 0
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 2.23607 0.620174 0.310087 0.950708i \(-0.399642\pi\)
0.310087 + 0.950708i \(0.399642\pi\)
\(14\) −3.35410 4.33013i −0.896421 1.15728i
\(15\) 0 0
\(16\) −3.50000 1.93649i −0.875000 0.484123i
\(17\) 1.00000 0.242536 0.121268 0.992620i \(-0.461304\pi\)
0.121268 + 0.992620i \(0.461304\pi\)
\(18\) 0 0
\(19\) 4.00000 + 1.73205i 0.917663 + 0.397360i
\(20\) 0 0
\(21\) 0 0
\(22\) −4.47214 + 3.46410i −0.953463 + 0.738549i
\(23\) 3.87298i 0.807573i −0.914853 0.403786i \(-0.867694\pi\)
0.914853 0.403786i \(-0.132306\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) −2.50000 + 1.93649i −0.490290 + 0.379777i
\(27\) 0 0
\(28\) 7.50000 + 1.93649i 1.41737 + 0.365963i
\(29\) −2.23607 −0.415227 −0.207614 0.978211i \(-0.566570\pi\)
−0.207614 + 0.978211i \(0.566570\pi\)
\(30\) 0 0
\(31\) −8.94427 −1.60644 −0.803219 0.595683i \(-0.796881\pi\)
−0.803219 + 0.595683i \(0.796881\pi\)
\(32\) 5.59017 0.866025i 0.988212 0.153093i
\(33\) 0 0
\(34\) −1.11803 + 0.866025i −0.191741 + 0.148522i
\(35\) 0 0
\(36\) 0 0
\(37\) −4.47214 −0.735215 −0.367607 0.929981i \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) −5.97214 + 1.52761i −0.968808 + 0.247811i
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820i 1.08200i −0.841021 0.541002i \(-0.818045\pi\)
0.841021 0.541002i \(-0.181955\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 2.00000 7.74597i 0.301511 1.16775i
\(45\) 0 0
\(46\) 3.35410 + 4.33013i 0.494535 + 0.638442i
\(47\) 7.74597i 1.12987i 0.825137 + 0.564933i \(0.191098\pi\)
−0.825137 + 0.564933i \(0.808902\pi\)
\(48\) 0 0
\(49\) −8.00000 −1.14286
\(50\) −5.59017 + 4.33013i −0.790569 + 0.612372i
\(51\) 0 0
\(52\) 1.11803 4.33013i 0.155043 0.600481i
\(53\) 6.70820 0.921443 0.460721 0.887545i \(-0.347591\pi\)
0.460721 + 0.887545i \(0.347591\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −10.0623 + 4.33013i −1.34463 + 0.578638i
\(57\) 0 0
\(58\) 2.50000 1.93649i 0.328266 0.254274i
\(59\) 8.66025i 1.12747i 0.825956 + 0.563735i \(0.190636\pi\)
−0.825956 + 0.563735i \(0.809364\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 10.0000 7.74597i 1.27000 0.983739i
\(63\) 0 0
\(64\) −5.50000 + 5.80948i −0.687500 + 0.726184i
\(65\) 0 0
\(66\) 0 0
\(67\) 12.1244i 1.48123i 0.671932 + 0.740613i \(0.265465\pi\)
−0.671932 + 0.740613i \(0.734535\pi\)
\(68\) 0.500000 1.93649i 0.0606339 0.234834i
\(69\) 0 0
\(70\) 0 0
\(71\) 8.94427 1.06149 0.530745 0.847532i \(-0.321912\pi\)
0.530745 + 0.847532i \(0.321912\pi\)
\(72\) 0 0
\(73\) 3.00000 0.351123 0.175562 0.984468i \(-0.443826\pi\)
0.175562 + 0.984468i \(0.443826\pi\)
\(74\) 5.00000 3.87298i 0.581238 0.450225i
\(75\) 0 0
\(76\) 5.35410 6.87994i 0.614158 0.789183i
\(77\) 15.4919i 1.76547i
\(78\) 0 0
\(79\) 8.94427 1.00631 0.503155 0.864196i \(-0.332173\pi\)
0.503155 + 0.864196i \(0.332173\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.00000 + 7.74597i 0.662589 + 0.855399i
\(83\) 8.00000 0.878114 0.439057 0.898459i \(-0.355313\pi\)
0.439057 + 0.898459i \(0.355313\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 4.47214 + 10.3923i 0.476731 + 1.10782i
\(89\) 13.8564i 1.46878i 0.678730 + 0.734388i \(0.262531\pi\)
−0.678730 + 0.734388i \(0.737469\pi\)
\(90\) 0 0
\(91\) 8.66025i 0.907841i
\(92\) −7.50000 1.93649i −0.781929 0.201893i
\(93\) 0 0
\(94\) −6.70820 8.66025i −0.691898 0.893237i
\(95\) 0 0
\(96\) 0 0
\(97\) 6.92820i 0.703452i −0.936103 0.351726i \(-0.885595\pi\)
0.936103 0.351726i \(-0.114405\pi\)
\(98\) 8.94427 6.92820i 0.903508 0.699854i
\(99\) 0 0
\(100\) 2.50000 9.68246i 0.250000 0.968246i
\(101\) 15.4919i 1.54150i 0.637135 + 0.770752i \(0.280120\pi\)
−0.637135 + 0.770752i \(0.719880\pi\)
\(102\) 0 0
\(103\) 17.8885 1.76261 0.881305 0.472547i \(-0.156665\pi\)
0.881305 + 0.472547i \(0.156665\pi\)
\(104\) 2.50000 + 5.80948i 0.245145 + 0.569666i
\(105\) 0 0
\(106\) −7.50000 + 5.80948i −0.728464 + 0.564266i
\(107\) 5.19615i 0.502331i −0.967944 0.251166i \(-0.919186\pi\)
0.967944 0.251166i \(-0.0808138\pi\)
\(108\) 0 0
\(109\) −15.6525 −1.49924 −0.749618 0.661871i \(-0.769763\pi\)
−0.749618 + 0.661871i \(0.769763\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 7.50000 13.5554i 0.708683 1.28087i
\(113\) 6.92820i 0.651751i −0.945413 0.325875i \(-0.894341\pi\)
0.945413 0.325875i \(-0.105659\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.11803 + 4.33013i −0.103807 + 0.402042i
\(117\) 0 0
\(118\) −7.50000 9.68246i −0.690431 0.891343i
\(119\) 3.87298i 0.355036i
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) −4.47214 + 17.3205i −0.401610 + 1.55543i
\(125\) 0 0
\(126\) 0 0
\(127\) 17.8885 1.58735 0.793676 0.608341i \(-0.208165\pi\)
0.793676 + 0.608341i \(0.208165\pi\)
\(128\) 1.11803 11.2583i 0.0988212 0.995105i
\(129\) 0 0
\(130\) 0 0
\(131\) −8.00000 −0.698963 −0.349482 0.936943i \(-0.613642\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) −6.70820 + 15.4919i −0.581675 + 1.34332i
\(134\) −10.5000 13.5554i −0.907062 1.17101i
\(135\) 0 0
\(136\) 1.11803 + 2.59808i 0.0958706 + 0.222783i
\(137\) −19.0000 −1.62328 −0.811640 0.584158i \(-0.801425\pi\)
−0.811640 + 0.584158i \(0.801425\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −10.0000 + 7.74597i −0.839181 + 0.650027i
\(143\) 8.94427 0.747958
\(144\) 0 0
\(145\) 0 0
\(146\) −3.35410 + 2.59808i −0.277587 + 0.215018i
\(147\) 0 0
\(148\) −2.23607 + 8.66025i −0.183804 + 0.711868i
\(149\) 15.4919i 1.26915i −0.772862 0.634574i \(-0.781175\pi\)
0.772862 0.634574i \(-0.218825\pi\)
\(150\) 0 0
\(151\) −8.94427 −0.727875 −0.363937 0.931423i \(-0.618568\pi\)
−0.363937 + 0.931423i \(0.618568\pi\)
\(152\) −0.0278640 + 12.3288i −0.00226007 + 0.999997i
\(153\) 0 0
\(154\) −13.4164 17.3205i −1.08112 1.39573i
\(155\) 0 0
\(156\) 0 0
\(157\) 15.4919i 1.23639i 0.786024 + 0.618195i \(0.212136\pi\)
−0.786024 + 0.618195i \(0.787864\pi\)
\(158\) −10.0000 + 7.74597i −0.795557 + 0.616236i
\(159\) 0 0
\(160\) 0 0
\(161\) 15.0000 1.18217
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) −13.4164 3.46410i −1.04765 0.270501i
\(165\) 0 0
\(166\) −8.94427 + 6.92820i −0.694210 + 0.537733i
\(167\) −8.94427 −0.692129 −0.346064 0.938211i \(-0.612482\pi\)
−0.346064 + 0.938211i \(0.612482\pi\)
\(168\) 0 0
\(169\) −8.00000 −0.615385
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −13.4164 −1.02003 −0.510015 0.860165i \(-0.670360\pi\)
−0.510015 + 0.860165i \(0.670360\pi\)
\(174\) 0 0
\(175\) 19.3649i 1.46385i
\(176\) −14.0000 7.74597i −1.05529 0.583874i
\(177\) 0 0
\(178\) −12.0000 15.4919i −0.899438 1.16117i
\(179\) 3.46410i 0.258919i −0.991585 0.129460i \(-0.958676\pi\)
0.991585 0.129460i \(-0.0413242\pi\)
\(180\) 0 0
\(181\) −4.47214 −0.332411 −0.166206 0.986091i \(-0.553152\pi\)
−0.166206 + 0.986091i \(0.553152\pi\)
\(182\) −7.50000 9.68246i −0.555937 0.717712i
\(183\) 0 0
\(184\) 10.0623 4.33013i 0.741803 0.319221i
\(185\) 0 0
\(186\) 0 0
\(187\) 4.00000 0.292509
\(188\) 15.0000 + 3.87298i 1.09399 + 0.282466i
\(189\) 0 0
\(190\) 0 0
\(191\) 11.6190i 0.840718i 0.907358 + 0.420359i \(0.138096\pi\)
−0.907358 + 0.420359i \(0.861904\pi\)
\(192\) 0 0
\(193\) 6.92820i 0.498703i 0.968413 + 0.249351i \(0.0802174\pi\)
−0.968413 + 0.249351i \(0.919783\pi\)
\(194\) 6.00000 + 7.74597i 0.430775 + 0.556128i
\(195\) 0 0
\(196\) −4.00000 + 15.4919i −0.285714 + 1.10657i
\(197\) 15.4919i 1.10375i 0.833925 + 0.551877i \(0.186088\pi\)
−0.833925 + 0.551877i \(0.813912\pi\)
\(198\) 0 0
\(199\) 27.1109i 1.92184i −0.276827 0.960920i \(-0.589283\pi\)
0.276827 0.960920i \(-0.410717\pi\)
\(200\) 5.59017 + 12.9904i 0.395285 + 0.918559i
\(201\) 0 0
\(202\) −13.4164 17.3205i −0.943975 1.21867i
\(203\) 8.66025i 0.607831i
\(204\) 0 0
\(205\) 0 0
\(206\) −20.0000 + 15.4919i −1.39347 + 1.07937i
\(207\) 0 0
\(208\) −7.82624 4.33013i −0.542652 0.300240i
\(209\) 16.0000 + 6.92820i 1.10674 + 0.479234i
\(210\) 0 0
\(211\) 15.5885i 1.07315i −0.843851 0.536577i \(-0.819717\pi\)
0.843851 0.536577i \(-0.180283\pi\)
\(212\) 3.35410 12.9904i 0.230361 0.892183i
\(213\) 0 0
\(214\) 4.50000 + 5.80948i 0.307614 + 0.397128i
\(215\) 0 0
\(216\) 0 0
\(217\) 34.6410i 2.35159i
\(218\) 17.5000 13.5554i 1.18525 0.918090i
\(219\) 0 0
\(220\) 0 0
\(221\) 2.23607 0.150414
\(222\) 0 0
\(223\) 17.8885 1.19791 0.598953 0.800784i \(-0.295584\pi\)
0.598953 + 0.800784i \(0.295584\pi\)
\(224\) 3.35410 + 21.6506i 0.224105 + 1.44659i
\(225\) 0 0
\(226\) 6.00000 + 7.74597i 0.399114 + 0.515254i
\(227\) 15.5885i 1.03464i 0.855791 + 0.517321i \(0.173071\pi\)
−0.855791 + 0.517321i \(0.826929\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.50000 5.80948i −0.164133 0.381411i
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 16.7705 + 4.33013i 1.09167 + 0.281867i
\(237\) 0 0
\(238\) −3.35410 4.33013i −0.217414 0.280680i
\(239\) 11.6190i 0.751567i 0.926707 + 0.375784i \(0.122626\pi\)
−0.926707 + 0.375784i \(0.877374\pi\)
\(240\) 0 0
\(241\) 6.92820i 0.446285i 0.974786 + 0.223142i \(0.0716315\pi\)
−0.974786 + 0.223142i \(0.928369\pi\)
\(242\) −5.59017 + 4.33013i −0.359350 + 0.278351i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 8.94427 + 3.87298i 0.569110 + 0.246432i
\(248\) −10.0000 23.2379i −0.635001 1.47561i
\(249\) 0 0
\(250\) 0 0
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 15.4919i 0.973970i
\(254\) −20.0000 + 15.4919i −1.25491 + 0.972050i
\(255\) 0 0
\(256\) 8.50000 + 13.5554i 0.531250 + 0.847215i
\(257\) 27.7128i 1.72868i −0.502910 0.864339i \(-0.667737\pi\)
0.502910 0.864339i \(-0.332263\pi\)
\(258\) 0 0
\(259\) 17.3205i 1.07624i
\(260\) 0 0
\(261\) 0 0
\(262\) 8.94427 6.92820i 0.552579 0.428026i
\(263\) 23.2379i 1.43291i −0.697633 0.716455i \(-0.745763\pi\)
0.697633 0.716455i \(-0.254237\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −5.91641 23.1300i −0.362758 1.41819i
\(267\) 0 0
\(268\) 23.4787 + 6.06218i 1.43419 + 0.370306i
\(269\) −13.4164 −0.818013 −0.409006 0.912532i \(-0.634125\pi\)
−0.409006 + 0.912532i \(0.634125\pi\)
\(270\) 0 0
\(271\) 11.6190i 0.705801i −0.935661 0.352900i \(-0.885195\pi\)
0.935661 0.352900i \(-0.114805\pi\)
\(272\) −3.50000 1.93649i −0.212219 0.117417i
\(273\) 0 0
\(274\) 21.2426 16.4545i 1.28332 0.994052i
\(275\) 20.0000 1.20605
\(276\) 0 0
\(277\) 15.4919i 0.930820i −0.885095 0.465410i \(-0.845907\pi\)
0.885095 0.465410i \(-0.154093\pi\)
\(278\) 13.4164 10.3923i 0.804663 0.623289i
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) 4.47214 17.3205i 0.265372 1.02778i
\(285\) 0 0
\(286\) −10.0000 + 7.74597i −0.591312 + 0.458029i
\(287\) 26.8328 1.58389
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 0 0
\(292\) 1.50000 5.80948i 0.0877809 0.339974i
\(293\) 6.70820 0.391897 0.195949 0.980614i \(-0.437221\pi\)
0.195949 + 0.980614i \(0.437221\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −5.00000 11.6190i −0.290619 0.675338i
\(297\) 0 0
\(298\) 13.4164 + 17.3205i 0.777192 + 1.00335i
\(299\) 8.66025i 0.500835i
\(300\) 0 0
\(301\) 0 0
\(302\) 10.0000 7.74597i 0.575435 0.445730i
\(303\) 0 0
\(304\) −10.6459 13.8081i −0.610584 0.791951i
\(305\) 0 0
\(306\) 0 0
\(307\) 3.46410i 0.197707i 0.995102 + 0.0988534i \(0.0315175\pi\)
−0.995102 + 0.0988534i \(0.968483\pi\)
\(308\) 30.0000 + 7.74597i 1.70941 + 0.441367i
\(309\) 0 0
\(310\) 0 0
\(311\) 3.87298i 0.219617i −0.993953 0.109808i \(-0.964976\pi\)
0.993953 0.109808i \(-0.0350237\pi\)
\(312\) 0 0
\(313\) −5.00000 −0.282617 −0.141308 0.989966i \(-0.545131\pi\)
−0.141308 + 0.989966i \(0.545131\pi\)
\(314\) −13.4164 17.3205i −0.757132 0.977453i
\(315\) 0 0
\(316\) 4.47214 17.3205i 0.251577 0.974355i
\(317\) 33.5410 1.88385 0.941926 0.335821i \(-0.109014\pi\)
0.941926 + 0.335821i \(0.109014\pi\)
\(318\) 0 0
\(319\) −8.94427 −0.500783
\(320\) 0 0
\(321\) 0 0
\(322\) −16.7705 + 12.9904i −0.934584 + 0.723926i
\(323\) 4.00000 + 1.73205i 0.222566 + 0.0963739i
\(324\) 0 0
\(325\) 11.1803 0.620174
\(326\) −13.4164 + 10.3923i −0.743066 + 0.575577i
\(327\) 0 0
\(328\) 18.0000 7.74597i 0.993884 0.427699i
\(329\) −30.0000 −1.65395
\(330\) 0 0
\(331\) 32.9090i 1.80884i 0.426643 + 0.904420i \(0.359696\pi\)
−0.426643 + 0.904420i \(0.640304\pi\)
\(332\) 4.00000 15.4919i 0.219529 0.850230i
\(333\) 0 0
\(334\) 10.0000 7.74597i 0.547176 0.423840i
\(335\) 0 0
\(336\) 0 0
\(337\) 13.8564i 0.754807i −0.926049 0.377403i \(-0.876817\pi\)
0.926049 0.377403i \(-0.123183\pi\)
\(338\) 8.94427 6.92820i 0.486504 0.376845i
\(339\) 0 0
\(340\) 0 0
\(341\) −35.7771 −1.93744
\(342\) 0 0
\(343\) 3.87298i 0.209121i
\(344\) 0 0
\(345\) 0 0
\(346\) 15.0000 11.6190i 0.806405 0.624639i
\(347\) −16.0000 −0.858925 −0.429463 0.903085i \(-0.641297\pi\)
−0.429463 + 0.903085i \(0.641297\pi\)
\(348\) 0 0
\(349\) 15.4919i 0.829264i 0.909989 + 0.414632i \(0.136090\pi\)
−0.909989 + 0.414632i \(0.863910\pi\)
\(350\) −16.7705 21.6506i −0.896421 1.15728i
\(351\) 0 0
\(352\) 22.3607 3.46410i 1.19183 0.184637i
\(353\) 25.0000 1.33062 0.665308 0.746569i \(-0.268300\pi\)
0.665308 + 0.746569i \(0.268300\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 26.8328 + 6.92820i 1.42214 + 0.367194i
\(357\) 0 0
\(358\) 3.00000 + 3.87298i 0.158555 + 0.204694i
\(359\) 3.87298i 0.204408i −0.994763 0.102204i \(-0.967411\pi\)
0.994763 0.102204i \(-0.0325895\pi\)
\(360\) 0 0
\(361\) 13.0000 + 13.8564i 0.684211 + 0.729285i
\(362\) 5.00000 3.87298i 0.262794 0.203559i
\(363\) 0 0
\(364\) 16.7705 + 4.33013i 0.879014 + 0.226960i
\(365\) 0 0
\(366\) 0 0
\(367\) 7.74597i 0.404336i −0.979351 0.202168i \(-0.935201\pi\)
0.979351 0.202168i \(-0.0647987\pi\)
\(368\) −7.50000 + 13.5554i −0.390965 + 0.706626i
\(369\) 0 0
\(370\) 0 0
\(371\) 25.9808i 1.34885i
\(372\) 0 0
\(373\) 11.1803 0.578896 0.289448 0.957194i \(-0.406528\pi\)
0.289448 + 0.957194i \(0.406528\pi\)
\(374\) −4.47214 + 3.46410i −0.231249 + 0.179124i
\(375\) 0 0
\(376\) −20.1246 + 8.66025i −1.03785 + 0.446619i
\(377\) −5.00000 −0.257513
\(378\) 0 0
\(379\) 5.19615i 0.266908i 0.991055 + 0.133454i \(0.0426069\pi\)
−0.991055 + 0.133454i \(0.957393\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −10.0623 12.9904i −0.514832 0.664646i
\(383\) 17.8885 0.914062 0.457031 0.889451i \(-0.348913\pi\)
0.457031 + 0.889451i \(0.348913\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.00000 7.74597i −0.305392 0.394259i
\(387\) 0 0
\(388\) −13.4164 3.46410i −0.681115 0.175863i
\(389\) 15.4919i 0.785472i −0.919651 0.392736i \(-0.871529\pi\)
0.919651 0.392736i \(-0.128471\pi\)
\(390\) 0 0
\(391\) 3.87298i 0.195865i
\(392\) −8.94427 20.7846i −0.451754 1.04978i
\(393\) 0 0
\(394\) −13.4164 17.3205i −0.675909 0.872595i
\(395\) 0 0
\(396\) 0 0
\(397\) 15.4919i 0.777518i 0.921340 + 0.388759i \(0.127096\pi\)
−0.921340 + 0.388759i \(0.872904\pi\)
\(398\) 23.4787 + 30.3109i 1.17688 + 1.51935i
\(399\) 0 0
\(400\) −17.5000 9.68246i −0.875000 0.484123i
\(401\) 27.7128i 1.38391i −0.721940 0.691956i \(-0.756749\pi\)
0.721940 0.691956i \(-0.243251\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) 30.0000 + 7.74597i 1.49256 + 0.385376i
\(405\) 0 0
\(406\) 7.50000 + 9.68246i 0.372219 + 0.480532i
\(407\) −17.8885 −0.886702
\(408\) 0 0
\(409\) 20.7846i 1.02773i −0.857870 0.513866i \(-0.828213\pi\)
0.857870 0.513866i \(-0.171787\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 8.94427 34.6410i 0.440653 1.70664i
\(413\) −33.5410 −1.65045
\(414\) 0 0
\(415\) 0 0
\(416\) 12.5000 1.93649i 0.612863 0.0949443i
\(417\) 0 0
\(418\) −23.8885 + 6.11044i −1.16843 + 0.298871i
\(419\) −8.00000 −0.390826 −0.195413 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 0 0
\(421\) 11.1803 0.544896 0.272448 0.962170i \(-0.412167\pi\)
0.272448 + 0.962170i \(0.412167\pi\)
\(422\) 13.5000 + 17.4284i 0.657170 + 0.848402i
\(423\) 0 0
\(424\) 7.50000 + 17.4284i 0.364232 + 0.846399i
\(425\) 5.00000 0.242536
\(426\) 0 0
\(427\) 0 0
\(428\) −10.0623 2.59808i −0.486380 0.125583i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 27.7128i 1.33179i −0.746044 0.665896i \(-0.768049\pi\)
0.746044 0.665896i \(-0.231951\pi\)
\(434\) 30.0000 + 38.7298i 1.44005 + 1.85909i
\(435\) 0 0
\(436\) −7.82624 + 30.3109i −0.374809 + 1.45163i
\(437\) 6.70820 15.4919i 0.320897 0.741080i
\(438\) 0 0
\(439\) −17.8885 −0.853774 −0.426887 0.904305i \(-0.640390\pi\)
−0.426887 + 0.904305i \(0.640390\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2.50000 + 1.93649i −0.118913 + 0.0921095i
\(443\) −28.0000 −1.33032 −0.665160 0.746701i \(-0.731637\pi\)
−0.665160 + 0.746701i \(0.731637\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −20.0000 + 15.4919i −0.947027 + 0.733564i
\(447\) 0 0
\(448\) −22.5000 21.3014i −1.06303 1.00640i
\(449\) 20.7846i 0.980886i −0.871473 0.490443i \(-0.836835\pi\)
0.871473 0.490443i \(-0.163165\pi\)
\(450\) 0 0
\(451\) 27.7128i 1.30495i
\(452\) −13.4164 3.46410i −0.631055 0.162938i
\(453\) 0 0
\(454\) −13.5000 17.4284i −0.633586 0.817957i
\(455\) 0 0
\(456\) 0 0
\(457\) −29.0000 −1.35656 −0.678281 0.734802i \(-0.737275\pi\)
−0.678281 + 0.734802i \(0.737275\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 30.9839i 1.44306i −0.692382 0.721531i \(-0.743439\pi\)
0.692382 0.721531i \(-0.256561\pi\)
\(462\) 0 0
\(463\) 7.74597i 0.359986i −0.983668 0.179993i \(-0.942393\pi\)
0.983668 0.179993i \(-0.0576074\pi\)
\(464\) 7.82624 + 4.33013i 0.363324 + 0.201021i
\(465\) 0 0
\(466\) 11.1803 8.66025i 0.517919 0.401179i
\(467\) 20.0000 0.925490 0.462745 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) 0 0
\(469\) −46.9574 −2.16829
\(470\) 0 0
\(471\) 0 0
\(472\) −22.5000 + 9.68246i −1.03565 + 0.445671i
\(473\) 0 0
\(474\) 0 0
\(475\) 20.0000 + 8.66025i 0.917663 + 0.397360i
\(476\) 7.50000 + 1.93649i 0.343762 + 0.0887590i
\(477\) 0 0
\(478\) −10.0623 12.9904i −0.460239 0.594166i
\(479\) 7.74597i 0.353922i 0.984218 + 0.176961i \(0.0566267\pi\)
−0.984218 + 0.176961i \(0.943373\pi\)
\(480\) 0 0
\(481\) −10.0000 −0.455961
\(482\) −6.00000 7.74597i −0.273293 0.352819i
\(483\) 0 0
\(484\) 2.50000 9.68246i 0.113636 0.440112i
\(485\) 0 0
\(486\) 0 0
\(487\) −17.8885 −0.810607 −0.405304 0.914182i \(-0.632834\pi\)
−0.405304 + 0.914182i \(0.632834\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 32.0000 1.44414 0.722070 0.691820i \(-0.243191\pi\)
0.722070 + 0.691820i \(0.243191\pi\)
\(492\) 0 0
\(493\) −2.23607 −0.100707
\(494\) −13.3541 + 3.41584i −0.600829 + 0.153686i
\(495\) 0 0
\(496\) 31.3050 + 17.3205i 1.40563 + 0.777714i
\(497\) 34.6410i 1.55386i
\(498\) 0 0
\(499\) 16.0000 0.716258 0.358129 0.933672i \(-0.383415\pi\)
0.358129 + 0.933672i \(0.383415\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 4.47214 3.46410i 0.199601 0.154610i
\(503\) 34.8569i 1.55419i −0.629383 0.777095i \(-0.716692\pi\)
0.629383 0.777095i \(-0.283308\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 13.4164 + 17.3205i 0.596432 + 0.769991i
\(507\) 0 0
\(508\) 8.94427 34.6410i 0.396838 1.53695i
\(509\) 22.3607 0.991120 0.495560 0.868574i \(-0.334963\pi\)
0.495560 + 0.868574i \(0.334963\pi\)
\(510\) 0 0
\(511\) 11.6190i 0.513992i
\(512\) −21.2426 7.79423i −0.938801 0.344459i
\(513\) 0 0
\(514\) 24.0000 + 30.9839i 1.05859 + 1.36664i
\(515\) 0 0
\(516\) 0 0
\(517\) 30.9839i 1.36267i
\(518\) 15.0000 + 19.3649i 0.659062 + 0.850846i
\(519\) 0 0
\(520\) 0 0
\(521\) 6.92820i 0.303530i −0.988417 0.151765i \(-0.951504\pi\)
0.988417 0.151765i \(-0.0484957\pi\)
\(522\) 0 0
\(523\) 19.0526i 0.833110i 0.909110 + 0.416555i \(0.136763\pi\)
−0.909110 + 0.416555i \(0.863237\pi\)
\(524\) −4.00000 + 15.4919i −0.174741 + 0.676768i
\(525\) 0 0
\(526\) 20.1246 + 25.9808i 0.877475 + 1.13282i
\(527\) −8.94427 −0.389619
\(528\) 0 0
\(529\) 8.00000 0.347826
\(530\) 0 0
\(531\) 0 0
\(532\) 26.6459 + 20.7363i 1.15525 + 0.899035i
\(533\) 15.4919i 0.671030i
\(534\) 0 0
\(535\) 0 0
\(536\) −31.5000 + 13.5554i −1.36059 + 0.585506i
\(537\) 0 0
\(538\) 15.0000 11.6190i 0.646696 0.500929i
\(539\) −32.0000 −1.37834
\(540\) 0 0
\(541\) 30.9839i 1.33210i −0.745907 0.666050i \(-0.767984\pi\)
0.745907 0.666050i \(-0.232016\pi\)
\(542\) 10.0623 + 12.9904i 0.432213 + 0.557985i
\(543\) 0 0
\(544\) 5.59017 0.866025i 0.239677 0.0371305i
\(545\) 0 0
\(546\) 0 0
\(547\) 31.1769i 1.33303i 0.745492 + 0.666514i \(0.232214\pi\)
−0.745492 + 0.666514i \(0.767786\pi\)
\(548\) −9.50000 + 36.7933i −0.405820 + 1.57173i
\(549\) 0 0
\(550\) −22.3607 + 17.3205i −0.953463 + 0.738549i
\(551\) −8.94427 3.87298i −0.381039 0.164995i
\(552\) 0 0
\(553\) 34.6410i 1.47309i
\(554\) 13.4164 + 17.3205i 0.570009 + 0.735878i
\(555\) 0 0
\(556\) −6.00000 + 23.2379i −0.254457 + 0.985506i
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3.46410i 0.145994i −0.997332 0.0729972i \(-0.976744\pi\)
0.997332 0.0729972i \(-0.0232564\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −8.94427 + 6.92820i −0.375956 + 0.291214i
\(567\) 0 0
\(568\) 10.0000 + 23.2379i 0.419591 + 0.975041i
\(569\) 20.7846i 0.871336i 0.900107 + 0.435668i \(0.143488\pi\)
−0.900107 + 0.435668i \(0.856512\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) 4.47214 17.3205i 0.186989 0.724207i
\(573\) 0 0
\(574\) −30.0000 + 23.2379i −1.25218 + 0.969931i
\(575\) 19.3649i 0.807573i
\(576\) 0 0
\(577\) −9.00000 −0.374675 −0.187337 0.982296i \(-0.559986\pi\)
−0.187337 + 0.982296i \(0.559986\pi\)
\(578\) 17.8885 13.8564i 0.744065 0.576351i
\(579\) 0 0
\(580\) 0 0
\(581\) 30.9839i 1.28543i
\(582\) 0 0
\(583\) 26.8328 1.11130
\(584\) 3.35410 + 7.79423i 0.138794 + 0.322527i
\(585\) 0 0
\(586\) −7.50000 + 5.80948i −0.309822 + 0.239987i
\(587\) 16.0000 0.660391 0.330195 0.943913i \(-0.392885\pi\)
0.330195 + 0.943913i \(0.392885\pi\)
\(588\) 0 0
\(589\) −35.7771 15.4919i −1.47417 0.638334i
\(590\) 0 0
\(591\) 0 0
\(592\) 15.6525 + 8.66025i 0.643313 + 0.355934i
\(593\) −2.00000 −0.0821302 −0.0410651 0.999156i \(-0.513075\pi\)
−0.0410651 + 0.999156i \(0.513075\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −30.0000 7.74597i −1.22885 0.317287i
\(597\) 0 0
\(598\) 7.50000 + 9.68246i 0.306698 + 0.395945i
\(599\) 17.8885 0.730906 0.365453 0.930830i \(-0.380914\pi\)
0.365453 + 0.930830i \(0.380914\pi\)
\(600\) 0 0
\(601\) 6.92820i 0.282607i −0.989966 0.141304i \(-0.954871\pi\)
0.989966 0.141304i \(-0.0451294\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −4.47214 + 17.3205i −0.181969 + 0.704761i
\(605\) 0 0
\(606\) 0 0
\(607\) 8.94427 0.363037 0.181518 0.983388i \(-0.441899\pi\)
0.181518 + 0.983388i \(0.441899\pi\)
\(608\) 23.8607 + 6.21836i 0.967678 + 0.252188i
\(609\) 0 0
\(610\) 0 0
\(611\) 17.3205i 0.700713i
\(612\) 0 0
\(613\) 30.9839i 1.25143i −0.780053 0.625713i \(-0.784808\pi\)
0.780053 0.625713i \(-0.215192\pi\)
\(614\) −3.00000 3.87298i −0.121070 0.156301i
\(615\) 0 0
\(616\) −40.2492 + 17.3205i −1.62169 + 0.697863i
\(617\) −10.0000 −0.402585 −0.201292 0.979531i \(-0.564514\pi\)
−0.201292 + 0.979531i \(0.564514\pi\)
\(618\) 0 0
\(619\) −16.0000 −0.643094 −0.321547 0.946894i \(-0.604203\pi\)
−0.321547 + 0.946894i \(0.604203\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 3.35410 + 4.33013i 0.134487 + 0.173622i
\(623\) −53.6656 −2.15007
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 5.59017 4.33013i 0.223428 0.173067i
\(627\) 0 0
\(628\) 30.0000 + 7.74597i 1.19713 + 0.309098i
\(629\) −4.47214 −0.178316
\(630\) 0 0
\(631\) 38.7298i 1.54181i −0.636950 0.770905i \(-0.719804\pi\)
0.636950 0.770905i \(-0.280196\pi\)
\(632\) 10.0000 + 23.2379i 0.397779 + 0.924354i
\(633\) 0 0
\(634\) −37.5000 + 29.0474i −1.48932 + 1.15362i
\(635\) 0 0
\(636\) 0 0
\(637\) −17.8885 −0.708770
\(638\) 10.0000 7.74597i 0.395904 0.306666i
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 7.50000 29.0474i 0.295541 1.14463i
\(645\) 0 0
\(646\) −5.97214 + 1.52761i −0.234971 + 0.0601030i
\(647\) 27.1109i 1.06584i 0.846166 + 0.532919i \(0.178905\pi\)
−0.846166 + 0.532919i \(0.821095\pi\)
\(648\) 0 0
\(649\) 34.6410i 1.35978i
\(650\) −12.5000 + 9.68246i −0.490290 + 0.379777i
\(651\) 0 0
\(652\) 6.00000 23.2379i 0.234978 0.910066i
\(653\) 46.4758i 1.81874i −0.415990 0.909369i \(-0.636565\pi\)
0.415990 0.909369i \(-0.363435\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −13.4164 + 24.2487i −0.523823 + 0.946753i
\(657\) 0 0
\(658\) 33.5410 25.9808i 1.30757 1.01284i
\(659\) 39.8372i 1.55184i −0.630834 0.775918i \(-0.717287\pi\)
0.630834 0.775918i \(-0.282713\pi\)
\(660\) 0 0
\(661\) 11.1803 0.434865 0.217432 0.976075i \(-0.430232\pi\)
0.217432 + 0.976075i \(0.430232\pi\)
\(662\) −28.5000 36.7933i −1.10768 1.43001i
\(663\) 0 0
\(664\) 8.94427 + 20.7846i 0.347105 + 0.806599i
\(665\) 0 0
\(666\) 0 0
\(667\) 8.66025i 0.335326i
\(668\) −4.47214 + 17.3205i −0.173032 + 0.670151i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 20.7846i 0.801188i −0.916256 0.400594i \(-0.868804\pi\)
0.916256 0.400594i \(-0.131196\pi\)
\(674\) 12.0000 + 15.4919i 0.462223 + 0.596727i
\(675\) 0 0
\(676\) −4.00000 + 15.4919i −0.153846 + 0.595844i
\(677\) 6.70820 0.257817 0.128909 0.991656i \(-0.458853\pi\)
0.128909 + 0.991656i \(0.458853\pi\)
\(678\) 0 0
\(679\) 26.8328 1.02975
\(680\) 0 0
\(681\) 0 0
\(682\) 40.0000 30.9839i 1.53168 1.18643i
\(683\) 10.3923i 0.397650i 0.980035 + 0.198825i \(0.0637126\pi\)
−0.980035 + 0.198825i \(0.936287\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 3.35410 + 4.33013i 0.128060 + 0.165325i
\(687\) 0 0
\(688\) 0 0
\(689\) 15.0000 0.571454
\(690\) 0 0
\(691\) −24.0000 −0.913003 −0.456502 0.889723i \(-0.650898\pi\)
−0.456502 + 0.889723i \(0.650898\pi\)
\(692\) −6.70820 + 25.9808i −0.255008 + 0.987640i
\(693\) 0 0
\(694\) 17.8885 13.8564i 0.679040 0.525982i
\(695\) 0 0
\(696\) 0 0
\(697\) 6.92820i 0.262424i
\(698\) −13.4164 17.3205i −0.507819 0.655591i
\(699\) 0 0
\(700\) 37.5000 + 9.68246i 1.41737 + 0.365963i
\(701\) 30.9839i 1.17024i 0.810945 + 0.585122i \(0.198953\pi\)
−0.810945 + 0.585122i \(0.801047\pi\)
\(702\) 0 0
\(703\) −17.8885 7.74597i −0.674679 0.292145i
\(704\) −22.0000 + 23.2379i −0.829156 + 0.875811i
\(705\) 0 0
\(706\) −27.9508 + 21.6506i −1.05194 + 0.814832i
\(707\) −60.0000 −2.25653
\(708\) 0 0
\(709\) 30.9839i 1.16362i 0.813323 + 0.581812i \(0.197656\pi\)
−0.813323 + 0.581812i \(0.802344\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −36.0000 + 15.4919i −1.34916 + 0.580585i
\(713\) 34.6410i 1.29732i
\(714\) 0 0
\(715\) 0 0
\(716\) −6.70820 1.73205i −0.250697 0.0647298i
\(717\) 0 0
\(718\) 3.35410 + 4.33013i 0.125174 + 0.161599i
\(719\) 19.3649i 0.722190i −0.932529 0.361095i \(-0.882403\pi\)
0.932529 0.361095i \(-0.117597\pi\)
\(720\) 0 0
\(721\) 69.2820i 2.58020i
\(722\) −26.5344 4.23360i −0.987510 0.157558i
\(723\) 0 0
\(724\) −2.23607 + 8.66025i −0.0831028 + 0.321856i
\(725\) −11.1803 −0.415227
\(726\) 0 0
\(727\) 3.87298i 0.143641i 0.997418 + 0.0718205i \(0.0228809\pi\)
−0.997418 + 0.0718205i \(0.977119\pi\)
\(728\) −22.5000 + 9.68246i −0.833905 + 0.358856i
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 30.9839i 1.14442i 0.820109 + 0.572208i \(0.193913\pi\)
−0.820109 + 0.572208i \(0.806087\pi\)
\(734\) 6.70820 + 8.66025i 0.247604 + 0.319656i
\(735\) 0 0
\(736\) −3.35410 21.6506i −0.123634 0.798053i
\(737\) 48.4974i 1.78643i
\(738\) 0 0
\(739\) −24.0000 −0.882854 −0.441427 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −22.5000 29.0474i −0.826001 1.06636i
\(743\) 26.8328 0.984401 0.492200 0.870482i \(-0.336193\pi\)
0.492200 + 0.870482i \(0.336193\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −12.5000 + 9.68246i −0.457658 + 0.354500i
\(747\) 0 0
\(748\) 2.00000 7.74597i 0.0731272 0.283221i
\(749\) 20.1246 0.735337
\(750\) 0 0
\(751\) −8.94427 −0.326381 −0.163191 0.986595i \(-0.552179\pi\)
−0.163191 + 0.986595i \(0.552179\pi\)
\(752\) 15.0000 27.1109i 0.546994 0.988632i
\(753\) 0 0
\(754\) 5.59017 4.33013i 0.203582 0.157694i
\(755\) 0 0
\(756\) 0 0
\(757\) 30.9839i 1.12613i −0.826413 0.563064i \(-0.809622\pi\)
0.826413 0.563064i \(-0.190378\pi\)
\(758\) −4.50000 5.80948i −0.163447 0.211010i
\(759\) 0 0
\(760\) 0 0
\(761\) −11.0000 −0.398750 −0.199375 0.979923i \(-0.563891\pi\)
−0.199375 + 0.979923i \(0.563891\pi\)
\(762\) 0 0
\(763\) 60.6218i 2.19466i
\(764\) 22.5000 + 5.80948i 0.814021 + 0.210179i
\(765\) 0 0
\(766\) −20.0000 + 15.4919i −0.722629 + 0.559746i
\(767\) 19.3649i 0.699227i
\(768\) 0 0
\(769\) 47.0000 1.69486 0.847432 0.530904i \(-0.178148\pi\)
0.847432 + 0.530904i \(0.178148\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 13.4164 + 3.46410i 0.482867 + 0.124676i
\(773\) −29.0689 −1.04554 −0.522768 0.852475i \(-0.675100\pi\)
−0.522768 + 0.852475i \(0.675100\pi\)
\(774\) 0 0
\(775\) −44.7214 −1.60644
\(776\) 18.0000 7.74597i 0.646162 0.278064i
\(777\) 0 0
\(778\) 13.4164 + 17.3205i 0.481002 + 0.620970i
\(779\) 12.0000 27.7128i 0.429945 0.992915i
\(780\) 0 0
\(781\) 35.7771 1.28020
\(782\) 3.35410 + 4.33013i 0.119942 + 0.154845i
\(783\) 0 0
\(784\) 28.0000 + 15.4919i 1.00000 + 0.553283i
\(785\) 0 0
\(786\) 0 0
\(787\) 1.73205i 0.0617409i −0.999523 0.0308705i \(-0.990172\pi\)
0.999523 0.0308705i \(-0.00982794\pi\)
\(788\) 30.0000 + 7.74597i 1.06871 + 0.275939i
\(789\) 0 0
\(790\) 0 0
\(791\) 26.8328 0.954065
\(792\) 0 0
\(793\) 0 0
\(794\) −13.4164 17.3205i −0.476130 0.614682i
\(795\) 0 0
\(796\) −52.5000 13.5554i −1.86081 0.480460i
\(797\) −2.23607 −0.0792056 −0.0396028 0.999216i \(-0.512609\pi\)
−0.0396028 + 0.999216i \(0.512609\pi\)
\(798\) 0 0
\(799\) 7.74597i 0.274033i
\(800\) 27.9508 4.33013i 0.988212 0.153093i
\(801\) 0 0
\(802\) 24.0000 + 30.9839i 0.847469 + 1.09408i
\(803\) 12.0000 0.423471
\(804\) 0 0
\(805\) 0 0
\(806\) 22.3607 17.3205i 0.787621 0.610089i
\(807\) 0 0
\(808\) −40.2492 + 17.3205i −1.41596 + 0.609333i
\(809\) −11.0000 −0.386739 −0.193370 0.981126i \(-0.561942\pi\)
−0.193370 + 0.981126i \(0.561942\pi\)
\(810\) 0 0
\(811\) 8.66025i 0.304103i −0.988373 0.152051i \(-0.951412\pi\)
0.988373 0.152051i \(-0.0485879\pi\)
\(812\) −16.7705 4.33013i −0.588530 0.151958i
\(813\) 0 0
\(814\) 20.0000 15.4919i 0.701000 0.542992i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 18.0000 + 23.2379i 0.629355 + 0.812494i
\(819\) 0 0
\(820\) 0 0
\(821\) 15.4919i 0.540672i −0.962766 0.270336i \(-0.912865\pi\)
0.962766 0.270336i \(-0.0871348\pi\)
\(822\) 0 0
\(823\) 34.8569i 1.21503i 0.794307 + 0.607517i \(0.207834\pi\)
−0.794307 + 0.607517i \(0.792166\pi\)
\(824\) 20.0000 + 46.4758i 0.696733 + 1.61906i
\(825\) 0 0
\(826\) 37.5000 29.0474i 1.30479 1.01069i
\(827\) 22.5167i 0.782981i 0.920182 + 0.391491i \(0.128040\pi\)
−0.920182 + 0.391491i \(0.871960\pi\)
\(828\) 0 0
\(829\) 38.0132 1.32025 0.660126 0.751155i \(-0.270503\pi\)
0.660126 + 0.751155i \(0.270503\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −12.2984 + 12.9904i −0.426369 + 0.450360i
\(833\) −8.00000 −0.277184
\(834\) 0 0
\(835\) 0 0
\(836\) 21.4164 27.5198i 0.740702 0.951791i
\(837\) 0 0
\(838\) 8.94427 6.92820i 0.308975 0.239331i
\(839\) −53.6656 −1.85274 −0.926372 0.376611i \(-0.877089\pi\)
−0.926372 + 0.376611i \(0.877089\pi\)
\(840\) 0 0
\(841\) −24.0000 −0.827586
\(842\) −12.5000 + 9.68246i −0.430778 + 0.333680i
\(843\) 0 0
\(844\) −30.1869 7.79423i −1.03908 0.268288i
\(845\) 0 0
\(846\) 0 0
\(847\) 19.3649i 0.665386i
\(848\) −23.4787 12.9904i −0.806262 0.446092i
\(849\) 0 0
\(850\) −5.59017 + 4.33013i −0.191741 + 0.148522i
\(851\) 17.3205i 0.593739i
\(852\) 0 0
\(853\) 30.9839i 1.06087i −0.847726 0.530434i \(-0.822029\pi\)
0.847726 0.530434i \(-0.177971\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 13.5000 5.80948i 0.461421 0.198564i
\(857\) 13.8564i 0.473326i −0.971592 0.236663i \(-0.923946\pi\)
0.971592 0.236663i \(-0.0760537\pi\)
\(858\) 0 0
\(859\) 8.00000 0.272956 0.136478 0.990643i \(-0.456422\pi\)
0.136478 + 0.990643i \(0.456422\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −53.6656 −1.82680 −0.913400 0.407064i \(-0.866553\pi\)
−0.913400 + 0.407064i \(0.866553\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 24.0000 + 30.9839i 0.815553 + 1.05287i
\(867\) 0 0
\(868\) −67.0820 17.3205i −2.27691 0.587896i
\(869\) 35.7771 1.21365
\(870\) 0 0
\(871\) 27.1109i 0.918617i
\(872\) −17.5000 40.6663i −0.592625 1.37714i
\(873\) 0 0
\(874\) 5.91641 + 23.1300i 0.200125 + 0.782383i
\(875\) 0 0
\(876\) 0 0
\(877\) 38.0132 1.28361 0.641807 0.766867i \(-0.278185\pi\)
0.641807 + 0.766867i \(0.278185\pi\)
\(878\) 20.0000 15.4919i 0.674967 0.522827i
\(879\) 0 0
\(880\) 0 0
\(881\) −2.00000 −0.0673817 −0.0336909 0.999432i \(-0.510726\pi\)
−0.0336909 + 0.999432i \(0.510726\pi\)
\(882\) 0 0
\(883\) −32.0000 −1.07689 −0.538443 0.842662i \(-0.680987\pi\)
−0.538443 + 0.842662i \(0.680987\pi\)
\(884\) 1.11803 4.33013i 0.0376036 0.145638i
\(885\) 0 0
\(886\) 31.3050 24.2487i 1.05171 0.814651i
\(887\) −17.8885 −0.600639 −0.300319 0.953839i \(-0.597093\pi\)
−0.300319 + 0.953839i \(0.597093\pi\)
\(888\) 0 0
\(889\) 69.2820i 2.32364i
\(890\) 0 0
\(891\) 0 0
\(892\) 8.94427 34.6410i 0.299476 1.15987i
\(893\) −13.4164 + 30.9839i −0.448963 + 1.03684i
\(894\) 0 0
\(895\) 0 0
\(896\) 43.6033 + 4.33013i 1.45668 + 0.144659i
\(897\) 0 0
\(898\) 18.0000 + 23.2379i 0.600668 + 0.775459i
\(899\) 20.0000 0.667037
\(900\) 0 0
\(901\) 6.70820 0.223483
\(902\) 24.0000 + 30.9839i 0.799113 + 1.03165i
\(903\) 0 0
\(904\) 18.0000 7.74597i 0.598671 0.257627i
\(905\) 0 0
\(906\) 0 0
\(907\) 36.3731i 1.20775i −0.797080 0.603874i \(-0.793623\pi\)
0.797080 0.603874i \(-0.206377\pi\)
\(908\) 30.1869 + 7.79423i 1.00179 + 0.258661i
\(909\) 0 0
\(910\) 0 0
\(911\) 35.7771 1.18535 0.592674 0.805443i \(-0.298072\pi\)
0.592674 + 0.805443i \(0.298072\pi\)
\(912\) 0 0
\(913\) 32.0000 1.05905
\(914\) 32.4230 25.1147i 1.07246 0.830722i
\(915\) 0 0
\(916\) 0 0
\(917\) 30.9839i 1.02318i
\(918\) 0 0
\(919\) 27.1109i 0.894306i −0.894458 0.447153i \(-0.852438\pi\)
0.894458 0.447153i \(-0.147562\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 26.8328 + 34.6410i 0.883692 + 1.14084i
\(923\) 20.0000 0.658308
\(924\) 0 0
\(925\) −22.3607 −0.735215
\(926\) 6.70820 + 8.66025i 0.220445 + 0.284594i
\(927\) 0 0
\(928\) −12.5000 + 1.93649i −0.410333 + 0.0635685i
\(929\) 1.00000 0.0328089 0.0164045 0.999865i \(-0.494778\pi\)
0.0164045 + 0.999865i \(0.494778\pi\)
\(930\) 0 0
\(931\) −32.0000 13.8564i −1.04876 0.454125i
\(932\) −5.00000 + 19.3649i −0.163780 + 0.634319i
\(933\) 0 0
\(934\) −22.3607 + 17.3205i −0.731664 + 0.566744i
\(935\) 0 0
\(936\) 0 0
\(937\) −21.0000 −0.686040 −0.343020 0.939328i \(-0.611450\pi\)
−0.343020 + 0.939328i \(0.611450\pi\)
\(938\) 52.5000 40.6663i 1.71419 1.32780i
\(939\) 0 0
\(940\) 0 0
\(941\) −38.0132 −1.23919 −0.619597 0.784920i \(-0.712704\pi\)
−0.619597 + 0.784920i \(0.712704\pi\)
\(942\) 0 0
\(943\) −26.8328 −0.873797
\(944\) 16.7705 30.3109i 0.545834 0.986535i
\(945\) 0 0
\(946\) 0 0
\(947\) −40.0000 −1.29983 −0.649913 0.760009i \(-0.725195\pi\)
−0.649913 + 0.760009i \(0.725195\pi\)
\(948\) 0 0
\(949\) 6.70820 0.217758
\(950\) −29.8607 + 7.63805i −0.968808 + 0.247811i
\(951\) 0 0
\(952\) −10.0623 + 4.33013i −0.326121 + 0.140340i
\(953\) 27.7128i 0.897706i −0.893606 0.448853i \(-0.851833\pi\)
0.893606 0.448853i \(-0.148167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 22.5000 + 5.80948i 0.727702 + 0.187892i
\(957\) 0 0
\(958\) −6.70820 8.66025i −0.216732 0.279800i
\(959\) 73.5867i 2.37624i
\(960\) 0 0
\(961\) 49.0000 1.58065
\(962\) 11.1803 8.66025i 0.360469 0.279218i
\(963\) 0 0
\(964\) 13.4164 + 3.46410i 0.432113 + 0.111571i
\(965\) 0 0
\(966\) 0 0
\(967\) 23.2379i 0.747280i 0.927574 + 0.373640i \(0.121891\pi\)
−0.927574 + 0.373640i \(0.878109\pi\)
\(968\) 5.59017 + 12.9904i 0.179675 + 0.417527i
\(969\) 0 0
\(970\) 0 0
\(971\) 10.3923i 0.333505i 0.985999 + 0.166752i \(0.0533281\pi\)
−0.985999 + 0.166752i \(0.946672\pi\)
\(972\) 0 0
\(973\) 46.4758i 1.48995i
\(974\) 20.0000 15.4919i 0.640841 0.496394i
\(975\) 0 0
\(976\) 0 0
\(977\) 20.7846i 0.664959i −0.943111 0.332479i \(-0.892115\pi\)
0.943111 0.332479i \(-0.107885\pi\)
\(978\) 0 0
\(979\) 55.4256i 1.77141i
\(980\) 0 0
\(981\) 0 0
\(982\) −35.7771 + 27.7128i −1.14169 + 0.884351i
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 2.50000 1.93649i 0.0796162 0.0616705i
\(987\) 0 0
\(988\) 11.9721 15.3840i 0.380884 0.489431i
\(989\) 0 0
\(990\) 0 0
\(991\) −17.8885 −0.568248 −0.284124 0.958787i \(-0.591703\pi\)
−0.284124 + 0.958787i \(0.591703\pi\)
\(992\) −50.0000 + 7.74597i −1.58750 + 0.245935i
\(993\) 0 0
\(994\) −30.0000 38.7298i −0.951542 1.22844i
\(995\) 0 0
\(996\) 0 0
\(997\) 15.4919i 0.490634i −0.969443 0.245317i \(-0.921108\pi\)
0.969443 0.245317i \(-0.0788921\pi\)
\(998\) −17.8885 + 13.8564i −0.566252 + 0.438617i
\(999\) 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 1368.2.e.b.379.2 4
3.2 odd 2 152.2.b.b.75.3 yes 4
4.3 odd 2 5472.2.e.b.5167.2 4
8.3 odd 2 inner 1368.2.e.b.379.4 4
8.5 even 2 5472.2.e.b.5167.3 4
12.11 even 2 608.2.b.b.303.3 4
19.18 odd 2 inner 1368.2.e.b.379.3 4
24.5 odd 2 608.2.b.b.303.4 4
24.11 even 2 152.2.b.b.75.1 4
57.56 even 2 152.2.b.b.75.2 yes 4
76.75 even 2 5472.2.e.b.5167.1 4
152.37 odd 2 5472.2.e.b.5167.4 4
152.75 even 2 inner 1368.2.e.b.379.1 4
228.227 odd 2 608.2.b.b.303.1 4
456.227 odd 2 152.2.b.b.75.4 yes 4
456.341 even 2 608.2.b.b.303.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.b.b.75.1 4 24.11 even 2
152.2.b.b.75.2 yes 4 57.56 even 2
152.2.b.b.75.3 yes 4 3.2 odd 2
152.2.b.b.75.4 yes 4 456.227 odd 2
608.2.b.b.303.1 4 228.227 odd 2
608.2.b.b.303.2 4 456.341 even 2
608.2.b.b.303.3 4 12.11 even 2
608.2.b.b.303.4 4 24.5 odd 2
1368.2.e.b.379.1 4 152.75 even 2 inner
1368.2.e.b.379.2 4 1.1 even 1 trivial
1368.2.e.b.379.3 4 19.18 odd 2 inner
1368.2.e.b.379.4 4 8.3 odd 2 inner
5472.2.e.b.5167.1 4 76.75 even 2
5472.2.e.b.5167.2 4 4.3 odd 2
5472.2.e.b.5167.3 4 8.5 even 2
5472.2.e.b.5167.4 4 152.37 odd 2