LMFDB - Embedded newform 189.3.b.c.134.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,3,Mod(134,189)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, 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("189.134");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	189.b (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:	$5.14987699641$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.1997017344.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 $ x^8 + 14x^6 + 53x^4 + 56*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			134.8
Root			$-1.92812i$ of defining polynomial
Character	$\chi$	$=$	189.134
Dual form			189.3.b.c.134.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.92812i q^{2} -11.4301 q^{4} +7.06404i q^{5} -2.64575 q^{7} -29.1863i q^{8} +O(q^{10})$	\(q+3.92812i q^{2} -11.4301 q^{4} +7.06404i q^{5} -2.64575 q^{7} -29.1863i q^{8} -27.7484 q^{10} +4.96808i q^{11} +5.02234 q^{13} -10.3928i q^{14} +68.9268 q^{16} -5.51380i q^{17} -18.2779 q^{19} -80.7427i q^{20} -19.5152 q^{22} -10.6882i q^{23} -24.9006 q^{25} +19.7284i q^{26} +30.2412 q^{28} +39.9307i q^{29} -2.71654 q^{31} +154.007i q^{32} +21.6589 q^{34} -18.6897i q^{35} -13.0417 q^{37} -71.7976i q^{38} +206.173 q^{40} +33.1988i q^{41} -61.4003 q^{43} -56.7856i q^{44} +41.9845 q^{46} +73.8862i q^{47} +7.00000 q^{49} -97.8125i q^{50} -57.4059 q^{52} -5.94700i q^{53} -35.0947 q^{55} +77.2197i q^{56} -156.852 q^{58} -44.6314i q^{59} -13.3171 q^{61} -10.6709i q^{62} -329.252 q^{64} +35.4780i q^{65} -105.774 q^{67} +63.0233i q^{68} +73.4153 q^{70} +10.1978i q^{71} +133.727 q^{73} -51.2294i q^{74} +208.918 q^{76} -13.1443i q^{77} +50.4748 q^{79} +486.902i q^{80} -130.409 q^{82} +41.5651i q^{83} +38.9497 q^{85} -241.188i q^{86} +145.000 q^{88} -53.3482i q^{89} -13.2879 q^{91} +122.167i q^{92} -290.234 q^{94} -129.115i q^{95} +97.8506 q^{97} +27.4968i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{4}+O(q^{10}) $ 8 * q - 12 * q^4	$ 8 q - 12 q^{4} - 52 q^{10} + 36 q^{13} + 132 q^{16} + 12 q^{19} - 136 q^{22} - 108 q^{25} + 56 q^{28} - 28 q^{31} - 12 q^{34} - 4 q^{37} + 336 q^{40} - 152 q^{43} + 108 q^{46} + 56 q^{49} - 272 q^{52} + 196 q^{55} - 220 q^{58} + 180 q^{61} - 700 q^{64} - 132 q^{67} + 196 q^{70} + 272 q^{73} + 544 q^{76} + 316 q^{79} + 28 q^{82} - 228 q^{85} - 56 q^{91} - 348 q^{94} - 364 q^{97}+O(q^{100}) $ 8 * q - 12 * q^4 - 52 * q^10 + 36 * q^13 + 132 * q^16 + 12 * q^19 - 136 * q^22 - 108 * q^25 + 56 * q^28 - 28 * q^31 - 12 * q^34 - 4 * q^37 + 336 * q^40 - 152 * q^43 + 108 * q^46 + 56 * q^49 - 272 * q^52 + 196 * q^55 - 220 * q^58 + 180 * q^61 - 700 * q^64 - 132 * q^67 + 196 * q^70 + 272 * q^73 + 544 * q^76 + 316 * q^79 + 28 * q^82 - 228 * q^85 - 56 * q^91 - 348 * q^94 - 364 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$-1$	$1$

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 $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.92812i	1.96406i	0.188729	+	0.982029i	$0.439563\pi$
$2$	3.92812i	1.96406i	−0.188729	+	0.982029i	$0.560437\pi$
$3$	0	0
$3$	0	0
$4$	−11.4301	−2.85753
$5$	7.06404i	1.41281i	0.707809	+	0.706404i	$0.249684\pi$
$5$	7.06404i	1.41281i	−0.707809	+	0.706404i	$0.750316\pi$
$6$	0	0
$7$	−2.64575	−0.377964
$7$	−2.64575	−0.377964
$8$	− 29.1863i	− 3.64829i
$9$	0	0
$10$	−27.7484	−2.77484
$11$	4.96808i	0.451643i	0.974169	+	0.225822i	$0.0725067\pi$
$11$	4.96808i	0.451643i	−0.974169	+	0.225822i	$0.927493\pi$
$12$	0	0
$13$	5.02234	0.386334	0.193167	−	0.981166i	$-0.438124\pi$
$13$	5.02234	0.386334	0.193167	+	0.981166i	$0.438124\pi$
$14$	− 10.3928i	− 0.742344i
$15$	0	0
$16$	68.9268	4.30793
$17$	− 5.51380i	− 0.324341i	−0.986763	−	0.162171i	$-0.948150\pi$
$17$	− 5.51380i	− 0.324341i	0.986763	−	0.162171i	$-0.0518495\pi$
$18$	0	0
$19$	−18.2779	−0.961992	−0.480996	−	0.876723i	$-0.659725\pi$
$19$	−18.2779	−0.961992	−0.480996	+	0.876723i	$0.659725\pi$
$20$	− 80.7427i	− 4.03713i
$21$	0	0
$22$	−19.5152	−0.887054
$23$	− 10.6882i	− 0.464704i	−0.972632	−	0.232352i	$-0.925358\pi$
$23$	− 10.6882i	− 0.464704i	0.972632	−	0.232352i	$-0.0746422\pi$
$24$	0	0
$25$	−24.9006	−0.996024
$26$	19.7284i	0.758783i
$27$	0	0
$28$	30.2412	1.08004
$29$	39.9307i	1.37692i	0.725274	+	0.688460i	$0.241713\pi$
$29$	39.9307i	1.37692i	−0.725274	+	0.688460i	$0.758287\pi$
$30$	0	0
$31$	−2.71654	−0.0876302	−0.0438151	−	0.999040i	$-0.513951\pi$
$31$	−2.71654	−0.0876302	−0.0438151	+	0.999040i	$0.513951\pi$
$32$	154.007i	4.81273i
$33$	0	0
$34$	21.6589	0.637025
$35$	− 18.6897i	− 0.533991i
$36$	0	0
$37$	−13.0417	−0.352479	−0.176239	−	0.984347i	$-0.556393\pi$
$37$	−13.0417	−0.352479	−0.176239	+	0.984347i	$0.556393\pi$
$38$	− 71.7976i	− 1.88941i
$39$	0	0
$40$	206.173	5.15433
$41$	33.1988i	0.809726i	0.914377	+	0.404863i	$0.132681\pi$
$41$	33.1988i	0.809726i	−0.914377	+	0.404863i	$0.867319\pi$
$42$	0	0
$43$	−61.4003	−1.42791	−0.713957	−	0.700190i	$-0.753099\pi$
$43$	−61.4003	−1.42791	−0.713957	+	0.700190i	$0.753099\pi$
$44$	− 56.7856i	− 1.29058i
$45$	0	0
$46$	41.9845	0.912706
$47$	73.8862i	1.57205i	0.618196	+	0.786024i	$0.287864\pi$
$47$	73.8862i	1.57205i	−0.618196	+	0.786024i	$0.712136\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	− 97.8125i	− 1.95625i
$51$	0	0
$52$	−57.4059	−1.10396
$53$	− 5.94700i	− 0.112208i	−0.998425	−	0.0561038i	$-0.982132\pi$
$53$	− 5.94700i	− 0.112208i	0.998425	−	0.0561038i	$-0.0178678\pi$
$54$	0	0
$55$	−35.0947	−0.638085
$56$	77.2197i	1.37892i
$57$	0	0
$58$	−156.852	−2.70435
$59$	− 44.6314i	− 0.756465i	−0.925711	−	0.378232i	$-0.876532\pi$
$59$	− 44.6314i	− 0.756465i	0.925711	−	0.378232i	$-0.123468\pi$
$60$	0	0
$61$	−13.3171	−0.218312	−0.109156	−	0.994025i	$-0.534815\pi$
$61$	−13.3171	−0.218312	−0.109156	+	0.994025i	$0.534815\pi$
$62$	− 10.6709i	− 0.172111i
$63$	0	0
$64$	−329.252	−5.14456
$65$	35.4780i	0.545816i
$66$	0	0
$67$	−105.774	−1.57872	−0.789362	−	0.613928i	$-0.789588\pi$
$67$	−105.774	−1.57872	−0.789362	+	0.613928i	$0.789588\pi$
$68$	63.0233i	0.926813i
$69$	0	0
$70$	73.4153	1.04879
$71$	10.1978i	0.143631i	0.997418	+	0.0718157i	$0.0228793\pi$
$71$	10.1978i	0.143631i	−0.997418	+	0.0718157i	$0.977121\pi$
$72$	0	0
$73$	133.727	1.83187	0.915937	−	0.401322i	$-0.131449\pi$
$73$	133.727	1.83187	0.915937	+	0.401322i	$0.131449\pi$
$74$	− 51.2294i	− 0.692289i
$75$	0	0
$76$	208.918	2.74892
$77$	− 13.1443i	− 0.170705i
$78$	0	0
$79$	50.4748	0.638921	0.319461	−	0.947600i	$-0.396498\pi$
$79$	50.4748	0.638921	0.319461	+	0.947600i	$0.396498\pi$
$80$	486.902i	6.08627i
$81$	0	0
$82$	−130.409	−1.59035
$83$	41.5651i	0.500784i	0.968144	+	0.250392i	$0.0805596\pi$
$83$	41.5651i	0.500784i	−0.968144	+	0.250392i	$0.919440\pi$
$84$	0	0
$85$	38.9497	0.458232
$86$	− 241.188i	− 2.80451i
$87$	0	0
$88$	145.000	1.64773
$89$	− 53.3482i	− 0.599418i	−0.954031	−	0.299709i	$-0.903110\pi$
$89$	− 53.3482i	− 0.599418i	0.954031	−	0.299709i	$-0.0968895\pi$
$90$	0	0
$91$	−13.2879	−0.146021
$92$	122.167i	1.32790i
$93$	0	0
$94$	−290.234	−3.08759
$95$	− 129.115i	− 1.35911i
$96$	0	0
$97$	97.8506	1.00877	0.504384	−	0.863479i	$-0.331719\pi$
$97$	97.8506	1.00877	0.504384	+	0.863479i	$0.331719\pi$
$98$	27.4968i	0.280580i
$99$	0	0
$100$	284.616	2.84616
$101$	122.551i	1.21338i	0.794940	+	0.606688i	$0.207502\pi$
$101$	122.551i	1.21338i	−0.794940	+	0.606688i	$0.792498\pi$
$102$	0	0
$103$	134.540	1.30621	0.653106	−	0.757267i	$-0.273466\pi$
$103$	134.540	1.30621	0.653106	+	0.757267i	$0.273466\pi$
$104$	− 146.584i	− 1.40946i
$105$	0	0
$106$	23.3605	0.220382
$107$	77.5589i	0.724850i	0.932013	+	0.362425i	$0.118051\pi$
$107$	77.5589i	0.724850i	−0.932013	+	0.362425i	$0.881949\pi$
$108$	0	0
$109$	−79.8733	−0.732783	−0.366391	−	0.930461i	$-0.619407\pi$
$109$	−79.8733	−0.732783	−0.366391	+	0.930461i	$0.619407\pi$
$110$	− 137.856i	− 1.25324i
$111$	0	0
$112$	−182.363	−1.62824
$113$	130.734i	1.15693i	0.815706	+	0.578467i	$0.196349\pi$
$113$	130.734i	1.15693i	−0.815706	+	0.578467i	$0.803651\pi$
$114$	0	0
$115$	75.5018	0.656538
$116$	− 456.412i	− 3.93459i
$117$	0	0
$118$	175.317	1.48574
$119$	14.5881i	0.122589i
$120$	0	0
$121$	96.3182	0.796018
$122$	− 52.3110i	− 0.428778i
$123$	0	0
$124$	31.0503	0.250406
$125$	0.702126i	0.00561701i
$126$	0	0
$127$	194.848	1.53424	0.767119	−	0.641505i	$-0.221689\pi$
$127$	194.848	1.53424	0.767119	+	0.641505i	$0.221689\pi$
$128$	− 677.309i	− 5.29148i
$129$	0	0
$130$	−139.362	−1.07201
$131$	− 135.214i	− 1.03217i	−0.856537	−	0.516085i	$-0.827389\pi$
$131$	− 135.214i	− 1.03217i	0.856537	−	0.516085i	$-0.172611\pi$
$132$	0	0
$133$	48.3587	0.363599
$134$	− 415.495i	− 3.10071i
$135$	0	0
$136$	−160.928	−1.18329
$137$	204.990i	1.49628i	0.663542	+	0.748139i	$0.269052\pi$
$137$	204.990i	1.49628i	−0.663542	+	0.748139i	$0.730948\pi$
$138$	0	0
$139$	190.556	1.37091	0.685455	−	0.728115i	$-0.259604\pi$
$139$	190.556	1.37091	0.685455	+	0.728115i	$0.259604\pi$
$140$	213.625i	1.52589i
$141$	0	0
$142$	−40.0583	−0.282101
$143$	24.9514i	0.174485i
$144$	0	0
$145$	−282.072	−1.94532
$146$	525.295i	3.59791i
$147$	0	0
$148$	149.068	1.00722
$149$	109.036i	0.731784i	0.930657	+	0.365892i	$0.119236\pi$
$149$	109.036i	0.731784i	−0.930657	+	0.365892i	$0.880764\pi$
$150$	0	0
$151$	−171.846	−1.13805	−0.569027	−	0.822319i	$-0.692680\pi$
$151$	−171.846	−1.13805	−0.569027	+	0.822319i	$0.692680\pi$
$152$	533.463i	3.50963i
$153$	0	0
$154$	51.6323	0.335275
$155$	− 19.1897i	− 0.123805i
$156$	0	0
$157$	250.791	1.59740	0.798699	−	0.601731i	$-0.205522\pi$
$157$	250.791	1.59740	0.798699	+	0.601731i	$0.205522\pi$
$158$	198.271i	1.25488i
$159$	0	0
$160$	−1087.91	−6.79946
$161$	28.2783i	0.175642i
$162$	0	0
$163$	−182.763	−1.12124	−0.560622	−	0.828072i	$-0.689438\pi$
$163$	−182.763	−1.12124	−0.560622	+	0.828072i	$0.689438\pi$
$164$	− 379.465i	− 2.31381i
$165$	0	0
$166$	−163.273	−0.983570
$167$	− 195.611i	− 1.17133i	−0.810555	−	0.585663i	$-0.800834\pi$
$167$	− 195.611i	− 1.17133i	0.810555	−	0.585663i	$-0.199166\pi$
$168$	0	0
$169$	−143.776	−0.850746
$170$	152.999i	0.899994i
$171$	0	0
$172$	701.812	4.08030
$173$	− 240.704i	− 1.39135i	−0.718355	−	0.695677i	$-0.755105\pi$
$173$	− 240.704i	− 1.39135i	0.718355	−	0.695677i	$-0.244895\pi$
$174$	0	0
$175$	65.8808	0.376462
$176$	342.434i	1.94565i
$177$	0	0
$178$	209.558	1.17729
$179$	105.930i	0.591787i	0.955221	+	0.295894i	$0.0956174\pi$
$179$	105.930i	0.591787i	−0.955221	+	0.295894i	$0.904383\pi$
$180$	0	0
$181$	−135.031	−0.746027	−0.373013	−	0.927826i	$-0.621675\pi$
$181$	−135.031	−0.746027	−0.373013	+	0.927826i	$0.621675\pi$
$182$	− 52.1963i	− 0.286793i
$183$	0	0
$184$	−311.949	−1.69538
$185$	− 92.1271i	− 0.497985i
$186$	0	0
$187$	27.3930	0.146487
$188$	− 844.527i	− 4.49217i
$189$	0	0
$190$	507.180	2.66937
$191$	− 182.449i	− 0.955231i	−0.878569	−	0.477616i	$-0.841501\pi$
$191$	− 182.449i	− 0.955231i	0.878569	−	0.477616i	$-0.158499\pi$
$192$	0	0
$193$	−63.3343	−0.328157	−0.164078	−	0.986447i	$-0.552465\pi$
$193$	−63.3343	−0.328157	−0.164078	+	0.986447i	$0.552465\pi$
$194$	384.369i	1.98128i
$195$	0	0
$196$	−80.0107	−0.408218
$197$	0.838016i	0.00425389i	0.999998	+	0.00212695i	$0.000677028\pi$
$197$	0.838016i	0.00425389i	−0.999998	+	0.00212695i	$0.999323\pi$
$198$	0	0
$199$	142.365	0.715403	0.357701	−	0.933836i	$-0.383561\pi$
$199$	142.365	0.715403	0.357701	+	0.933836i	$0.383561\pi$
$200$	726.757i	3.63378i
$201$	0	0
$202$	−481.395	−2.38314
$203$	− 105.647i	− 0.520427i
$204$	0	0
$205$	−234.517	−1.14399
$206$	528.488i	2.56548i
$207$	0	0
$208$	346.174	1.66430
$209$	− 90.8058i	− 0.434478i
$210$	0	0
$211$	50.9828	0.241625	0.120812	−	0.992675i	$-0.461450\pi$
$211$	50.9828	0.241625	0.120812	+	0.992675i	$0.461450\pi$
$212$	67.9748i	0.320636i
$213$	0	0
$214$	−304.661	−1.42365
$215$	− 433.734i	− 2.01737i
$216$	0	0
$217$	7.18728	0.0331211
$218$	− 313.752i	− 1.43923i
$219$	0	0
$220$	401.136	1.82334
$221$	− 27.6922i	− 0.125304i
$222$	0	0
$223$	348.849	1.56435	0.782174	−	0.623061i	$-0.214111\pi$
$223$	348.849	1.56435	0.782174	+	0.623061i	$0.214111\pi$
$224$	− 407.465i	− 1.81904i
$225$	0	0
$226$	−513.537	−2.27229
$227$	− 22.8796i	− 0.100791i	−0.998729	−	0.0503957i	$-0.983952\pi$
$227$	− 22.8796i	− 0.100791i	0.998729	−	0.0503957i	$-0.0160482\pi$
$228$	0	0
$229$	−212.433	−0.927653	−0.463827	−	0.885926i	$-0.653524\pi$
$229$	−212.433	−0.927653	−0.463827	+	0.885926i	$0.653524\pi$
$230$	296.580i	1.28948i
$231$	0	0
$232$	1165.43	5.02340
$233$	108.142i	0.464129i	0.972700	+	0.232064i	$0.0745480\pi$
$233$	108.142i	0.464129i	−0.972700	+	0.232064i	$0.925452\pi$
$234$	0	0
$235$	−521.935	−2.22100
$236$	510.142i	2.16162i
$237$	0	0
$238$	−57.3040	−0.240773
$239$	11.1464i	0.0466377i	0.999728	+	0.0233189i	$0.00742330\pi$
$239$	11.1464i	0.0466377i	−0.999728	+	0.0233189i	$0.992577\pi$
$240$	0	0
$241$	−156.224	−0.648231	−0.324116	−	0.946018i	$-0.605067\pi$
$241$	−156.224	−0.648231	−0.324116	+	0.946018i	$0.605067\pi$
$242$	378.349i	1.56343i
$243$	0	0
$244$	152.215	0.623833
$245$	49.4483i	0.201830i
$246$	0	0
$247$	−91.7977	−0.371651
$248$	79.2857i	0.319700i
$249$	0	0
$250$	−2.75803	−0.0110321
$251$	− 203.128i	− 0.809276i	−0.914477	−	0.404638i	$-0.867398\pi$
$251$	− 203.128i	− 0.809276i	0.914477	−	0.404638i	$-0.132602\pi$
$252$	0	0
$253$	53.0998	0.209881
$254$	765.387i	3.01333i
$255$	0	0
$256$	1343.54	5.24822
$257$	373.487i	1.45326i	0.687031	+	0.726628i	$0.258914\pi$
$257$	373.487i	1.45326i	−0.687031	+	0.726628i	$0.741086\pi$
$258$	0	0
$259$	34.5051	0.133224
$260$	− 405.517i	− 1.55968i
$261$	0	0
$262$	531.138	2.02724
$263$	4.29255i	0.0163215i	0.999967	+	0.00816075i	$0.00259768\pi$
$263$	4.29255i	0.0163215i	−0.999967	+	0.00816075i	$0.997402\pi$
$264$	0	0
$265$	42.0098	0.158528
$266$	189.958i	0.714130i
$267$	0	0
$268$	1209.01	4.51124
$269$	100.201i	0.372494i	0.982503	+	0.186247i	$0.0596324\pi$
$269$	100.201i	0.372494i	−0.982503	+	0.186247i	$0.940368\pi$
$270$	0	0
$271$	11.4367	0.0422020	0.0211010	−	0.999777i	$-0.493283\pi$
$271$	11.4367	0.0422020	0.0211010	+	0.999777i	$0.493283\pi$
$272$	− 380.049i	− 1.39724i
$273$	0	0
$274$	−805.225	−2.93878
$275$	− 123.708i	− 0.449848i
$276$	0	0
$277$	−48.4806	−0.175020	−0.0875102	−	0.996164i	$-0.527891\pi$
$277$	−48.4806	−0.175020	−0.0875102	+	0.996164i	$0.527891\pi$
$278$	748.528i	2.69255i
$279$	0	0
$280$	−545.483	−1.94815
$281$	78.0377i	0.277714i	0.990312	+	0.138857i	$0.0443429\pi$
$281$	78.0377i	0.277714i	−0.990312	+	0.138857i	$0.955657\pi$
$282$	0	0
$283$	−225.740	−0.797669	−0.398835	−	0.917023i	$-0.630585\pi$
$283$	−225.740	−0.797669	−0.398835	+	0.917023i	$0.630585\pi$
$284$	− 116.562i	− 0.410430i
$285$	0	0
$286$	−98.0120	−0.342699
$287$	− 87.8357i	− 0.306048i
$288$	0	0
$289$	258.598	0.894803
$290$	− 1108.01i	− 3.82073i
$291$	0	0
$292$	−1528.51	−5.23463
$293$	344.740i	1.17659i	0.808647	+	0.588294i	$0.200200\pi$
$293$	344.740i	1.17659i	−0.808647	+	0.588294i	$0.799800\pi$
$294$	0	0
$295$	315.278	1.06874
$296$	380.640i	1.28594i
$297$	0	0
$298$	−428.305	−1.43727
$299$	− 53.6798i	− 0.179531i
$300$	0	0
$301$	162.450	0.539701
$302$	− 675.032i	− 2.23521i
$303$	0	0
$304$	−1259.83	−4.14419
$305$	− 94.0722i	− 0.308433i
$306$	0	0
$307$	40.5120	0.131961	0.0659804	−	0.997821i	$-0.478983\pi$
$307$	40.5120	0.131961	0.0659804	+	0.997821i	$0.478983\pi$
$308$	150.241i	0.487794i
$309$	0	0
$310$	75.3794	0.243160
$311$	− 237.140i	− 0.762507i	−0.924471	−	0.381253i	$-0.875493\pi$
$311$	− 237.140i	− 0.762507i	0.924471	−	0.381253i	$-0.124507\pi$
$312$	0	0
$313$	−603.792	−1.92905	−0.964523	−	0.263998i	$-0.914959\pi$
$313$	−603.792	−1.92905	−0.964523	+	0.263998i	$0.914959\pi$
$314$	985.138i	3.13738i
$315$	0	0
$316$	−576.932	−1.82573
$317$	− 423.396i	− 1.33563i	−0.744326	−	0.667817i	$-0.767229\pi$
$317$	− 423.396i	− 1.33563i	0.744326	−	0.667817i	$-0.232771\pi$
$318$	0	0
$319$	−198.379	−0.621877
$320$	− 2325.85i	− 7.26827i
$321$	0	0
$322$	−111.081	−0.344971
$323$	100.780i	0.312014i
$324$	0	0
$325$	−125.059	−0.384798
$326$	− 717.913i	− 2.20219i
$327$	0	0
$328$	968.950	2.95412
$329$	− 195.485i	− 0.594178i
$330$	0	0
$331$	−395.632	−1.19526	−0.597631	−	0.801771i	$-0.703891\pi$
$331$	−395.632	−1.19526	−0.597631	+	0.801771i	$0.703891\pi$
$332$	− 475.093i	− 1.43100i
$333$	0	0
$334$	768.384	2.30055
$335$	− 747.195i	− 2.23043i
$336$	0	0
$337$	441.604	1.31040	0.655199	−	0.755456i	$-0.272585\pi$
$337$	441.604	1.31040	0.655199	+	0.755456i	$0.272585\pi$
$338$	− 564.769i	− 1.67091i
$339$	0	0
$340$	−445.199	−1.30941
$341$	− 13.4960i	− 0.0395776i
$342$	0	0
$343$	−18.5203	−0.0539949
$344$	1792.05i	5.20944i
$345$	0	0
$346$	945.514	2.73270
$347$	− 98.7133i	− 0.284476i	−0.989832	−	0.142238i	$-0.954570\pi$
$347$	− 98.7133i	− 0.284476i	0.989832	−	0.142238i	$-0.0454299\pi$
$348$	0	0
$349$	118.678	0.340051	0.170025	−	0.985440i	$-0.445615\pi$
$349$	118.678	0.340051	0.170025	+	0.985440i	$0.445615\pi$
$350$	258.788i	0.739393i
$351$	0	0
$352$	−765.121	−2.17364
$353$	567.205i	1.60681i	0.595431	+	0.803406i	$0.296981\pi$
$353$	567.205i	1.60681i	−0.595431	+	0.803406i	$0.703019\pi$
$354$	0	0
$355$	−72.0379	−0.202924
$356$	609.775i	1.71285i
$357$	0	0
$358$	−416.105	−1.16230
$359$	− 200.744i	− 0.559175i	−0.960120	−	0.279587i	$-0.909802\pi$
$359$	− 200.744i	− 0.559175i	0.960120	−	0.279587i	$-0.0901977\pi$
$360$	0	0
$361$	−26.9200	−0.0745707
$362$	− 530.417i	− 1.46524i
$363$	0	0
$364$	151.882	0.417258
$365$	944.651i	2.58809i
$366$	0	0
$367$	37.5360	0.102278	0.0511389	−	0.998692i	$-0.483715\pi$
$367$	37.5360	0.102278	0.0511389	+	0.998692i	$0.483715\pi$
$368$	− 736.704i	− 2.00191i
$369$	0	0
$370$	361.886	0.978071
$371$	15.7343i	0.0424105i
$372$	0	0
$373$	164.448	0.440880	0.220440	−	0.975401i	$-0.429251\pi$
$373$	164.448	0.440880	0.220440	+	0.975401i	$0.429251\pi$
$374$	107.603i	0.287708i
$375$	0	0
$376$	2156.47	5.73528
$377$	200.546i	0.531951i
$378$	0	0
$379$	−60.7949	−0.160409	−0.0802044	−	0.996778i	$-0.525557\pi$
$379$	−60.7949	−0.160409	−0.0802044	+	0.996778i	$0.525557\pi$
$380$	1475.80i	3.88369i
$381$	0	0
$382$	716.681	1.87613
$383$	492.378i	1.28558i	0.766042	+	0.642791i	$0.222224\pi$
$383$	492.378i	1.28558i	−0.766042	+	0.642791i	$0.777776\pi$
$384$	0	0
$385$	92.8518	0.241174
$386$	− 248.784i	− 0.644519i
$387$	0	0
$388$	−1118.44	−2.88258
$389$	− 336.545i	− 0.865155i	−0.901597	−	0.432578i	$-0.857604\pi$
$389$	− 336.545i	− 0.865155i	0.901597	−	0.432578i	$-0.142396\pi$
$390$	0	0
$391$	−58.9326	−0.150723
$392$	− 204.304i	− 0.521184i
$393$	0	0
$394$	−3.29183	−0.00835489
$395$	356.556i	0.902673i
$396$	0	0
$397$	216.470	0.545265	0.272632	−	0.962118i	$-0.412106\pi$
$397$	216.470	0.545265	0.272632	+	0.962118i	$0.412106\pi$
$398$	559.227i	1.40509i
$399$	0	0
$400$	−1716.32	−4.29080
$401$	− 585.530i	− 1.46017i	−0.683354	−	0.730087i	$-0.739479\pi$
$401$	− 585.530i	− 1.46017i	0.683354	−	0.730087i	$-0.260521\pi$
$402$	0	0
$403$	−13.6434	−0.0338546
$404$	− 1400.77i	− 3.46725i
$405$	0	0
$406$	414.993	1.02215
$407$	− 64.7923i	− 0.159195i
$408$	0	0
$409$	298.365	0.729500	0.364750	−	0.931106i	$-0.381155\pi$
$409$	298.365	0.729500	0.364750	+	0.931106i	$0.381155\pi$
$410$	− 921.212i	− 2.24686i
$411$	0	0
$412$	−1537.80	−3.73253
$413$	118.084i	0.285917i
$414$	0	0
$415$	−293.617	−0.707512
$416$	773.478i	1.85932i
$417$	0	0
$418$	356.696	0.853339
$419$	452.293i	1.07946i	0.841839	+	0.539729i	$0.181473\pi$
$419$	452.293i	1.07946i	−0.841839	+	0.539729i	$0.818527\pi$
$420$	0	0
$421$	41.2249	0.0979215	0.0489607	−	0.998801i	$-0.484409\pi$
$421$	41.2249	0.0979215	0.0489607	+	0.998801i	$0.484409\pi$
$422$	200.266i	0.474565i
$423$	0	0
$424$	−173.571	−0.409366
$425$	137.297i	0.323052i
$426$	0	0
$427$	35.2336	0.0825144
$428$	− 886.507i	− 2.07128i
$429$	0	0
$430$	1703.76	3.96223
$431$	415.390i	0.963781i	0.876231	+	0.481891i	$0.160050\pi$
$431$	415.390i	0.963781i	−0.876231	+	0.481891i	$0.839950\pi$
$432$	0	0
$433$	758.050	1.75069	0.875346	−	0.483497i	$-0.160633\pi$
$433$	758.050	1.75069	0.875346	+	0.483497i	$0.160633\pi$
$434$	28.2325i	0.0650518i
$435$	0	0
$436$	912.960	2.09395
$437$	195.357i	0.447042i
$438$	0	0
$439$	449.621	1.02419	0.512097	−	0.858928i	$-0.328869\pi$
$439$	449.621	1.02419	0.512097	+	0.858928i	$0.328869\pi$
$440$	1024.28i	2.32792i
$441$	0	0
$442$	108.778	0.246105
$443$	− 432.433i	− 0.976146i	−0.872803	−	0.488073i	$-0.837700\pi$
$443$	− 432.433i	− 0.976146i	0.872803	−	0.488073i	$-0.162300\pi$
$444$	0	0
$445$	376.853	0.846862
$446$	1370.32i	3.07247i
$447$	0	0
$448$	871.118	1.94446
$449$	218.358i	0.486320i	0.969986	+	0.243160i	$0.0781840\pi$
$449$	218.358i	0.486320i	−0.969986	+	0.243160i	$0.921816\pi$
$450$	0	0
$451$	−164.934	−0.365708
$452$	− 1494.30i	− 3.30597i
$453$	0	0
$454$	89.8739	0.197960
$455$	− 93.8660i	− 0.206299i
$456$	0	0
$457$	−89.4367	−0.195704	−0.0978519	−	0.995201i	$-0.531197\pi$
$457$	−89.4367	−0.195704	−0.0978519	+	0.995201i	$0.531197\pi$
$458$	− 834.460i	− 1.82196i
$459$	0	0
$460$	−862.994	−1.87607
$461$	− 277.872i	− 0.602760i	−0.953504	−	0.301380i	$-0.902553\pi$
$461$	− 277.872i	− 0.602760i	0.953504	−	0.301380i	$-0.0974473\pi$
$462$	0	0
$463$	85.2627	0.184153	0.0920763	−	0.995752i	$-0.470650\pi$
$463$	85.2627	0.184153	0.0920763	+	0.995752i	$0.470650\pi$
$464$	2752.30i	5.93167i
$465$	0	0
$466$	−424.795	−0.911576
$467$	− 384.319i	− 0.822952i	−0.911421	−	0.411476i	$-0.865013\pi$
$467$	− 384.319i	− 0.822952i	0.911421	−	0.411476i	$-0.134987\pi$
$468$	0	0
$469$	279.853	0.596702
$470$	− 2050.22i	− 4.36217i
$471$	0	0
$472$	−1302.63	−2.75980
$473$	− 305.041i	− 0.644908i
$474$	0	0
$475$	455.130	0.958168
$476$	− 166.744i	− 0.350303i
$477$	0	0
$478$	−43.7844	−0.0915992
$479$	− 335.146i	− 0.699678i	−0.936810	−	0.349839i	$-0.886236\pi$
$479$	− 335.146i	− 0.699678i	0.936810	−	0.349839i	$-0.113764\pi$
$480$	0	0
$481$	−65.5000	−0.136175
$482$	− 613.665i	− 1.27316i
$483$	0	0
$484$	−1100.93	−2.27464
$485$	691.220i	1.42520i
$486$	0	0
$487$	99.6350	0.204589	0.102295	−	0.994754i	$-0.467382\pi$
$487$	99.6350	0.204589	0.102295	+	0.994754i	$0.467382\pi$
$488$	388.676i	0.796467i
$489$	0	0
$490$	−194.239	−0.396405
$491$	522.971i	1.06511i	0.846394	+	0.532557i	$0.178769\pi$
$491$	522.971i	1.06511i	−0.846394	+	0.532557i	$0.821231\pi$
$492$	0	0
$493$	220.170	0.446592
$494$	− 360.592i	− 0.729943i
$495$	0	0
$496$	−187.242	−0.377505
$497$	− 26.9809i	− 0.0542876i
$498$	0	0
$499$	84.1197	0.168576	0.0842882	−	0.996441i	$-0.473138\pi$
$499$	84.1197	0.168576	0.0842882	+	0.996441i	$0.473138\pi$
$500$	− 8.02537i	− 0.0160507i
$501$	0	0
$502$	797.911	1.58946
$503$	− 675.278i	− 1.34250i	−0.741231	−	0.671250i	$-0.765758\pi$
$503$	− 675.278i	− 1.34250i	0.741231	−	0.671250i	$-0.234242\pi$
$504$	0	0
$505$	−865.705	−1.71427
$506$	208.582i	0.412218i
$507$	0	0
$508$	−2227.14	−4.38413
$509$	− 57.3769i	− 0.112725i	−0.998410	−	0.0563623i	$-0.982050\pi$
$509$	− 57.3769i	− 0.112725i	0.998410	−	0.0563623i	$-0.0179502\pi$
$510$	0	0
$511$	−353.808	−0.692383
$512$	2568.36i	5.01633i
$513$	0	0
$514$	−1467.10	−2.85428
$515$	950.394i	1.84543i
$516$	0	0
$517$	−367.073	−0.710005
$518$	135.540i	0.261661i
$519$	0	0
$520$	1035.47	1.99129
$521$	193.458i	0.371321i	0.982614	+	0.185660i	$0.0594424\pi$
$521$	193.458i	0.371321i	−0.982614	+	0.185660i	$0.940558\pi$
$522$	0	0
$523$	818.858	1.56569	0.782847	−	0.622214i	$-0.213766\pi$
$523$	818.858	1.56569	0.782847	+	0.622214i	$0.213766\pi$
$524$	1545.51i	2.94945i
$525$	0	0
$526$	−16.8617	−0.0320564
$527$	14.9784i	0.0284221i
$528$	0	0
$529$	414.762	0.784050
$530$	165.020i	0.311358i
$531$	0	0
$532$	−552.744	−1.03899
$533$	166.736i	0.312825i
$534$	0	0
$535$	−547.879	−1.02407
$536$	3087.17i	5.75964i
$537$	0	0
$538$	−393.600	−0.731599
$539$	34.7765i	0.0645205i
$540$	0	0
$541$	930.468	1.71990	0.859952	−	0.510375i	$-0.170494\pi$
$541$	930.468	1.71990	0.859952	+	0.510375i	$0.170494\pi$
$542$	44.9249i	0.0828872i
$543$	0	0
$544$	849.166	1.56097
$545$	− 564.228i	− 1.03528i
$546$	0	0
$547$	854.384	1.56195	0.780973	−	0.624565i	$-0.214724\pi$
$547$	854.384	1.56195	0.780973	+	0.624565i	$0.214724\pi$
$548$	− 2343.06i	− 4.27565i
$549$	0	0
$550$	485.940	0.883527
$551$	− 729.847i	− 1.32459i
$552$	0	0
$553$	−133.544	−0.241490
$554$	− 190.438i	− 0.343750i
$555$	0	0
$556$	−2178.08	−3.91741
$557$	734.064i	1.31789i	0.752192	+	0.658945i	$0.228997\pi$
$557$	734.064i	1.31789i	−0.752192	+	0.658945i	$0.771003\pi$
$558$	0	0
$559$	−308.373	−0.551652
$560$	− 1288.22i	− 2.30039i
$561$	0	0
$562$	−306.541	−0.545447
$563$	254.910i	0.452771i	0.974038	+	0.226385i	$0.0726909\pi$
$563$	254.910i	0.452771i	−0.974038	+	0.226385i	$0.927309\pi$
$564$	0	0
$565$	−923.507	−1.63452
$566$	− 886.734i	− 1.56667i
$567$	0	0
$568$	297.637	0.524009
$569$	218.544i	0.384084i	0.981387	+	0.192042i	$0.0615110\pi$
$569$	218.544i	0.384084i	−0.981387	+	0.192042i	$0.938489\pi$
$570$	0	0
$571$	−215.722	−0.377797	−0.188899	−	0.981997i	$-0.560492\pi$
$571$	−215.722	−0.377797	−0.188899	+	0.981997i	$0.560492\pi$
$572$	− 285.197i	− 0.498596i
$573$	0	0
$574$	345.029	0.601096
$575$	266.143i	0.462857i
$576$	0	0
$577$	825.876	1.43133	0.715664	−	0.698445i	$-0.246124\pi$
$577$	825.876	1.43133	0.715664	+	0.698445i	$0.246124\pi$
$578$	1015.80i	1.75744i
$579$	0	0
$580$	3224.11	5.55881
$581$	− 109.971i	− 0.189279i
$582$	0	0
$583$	29.5452	0.0506778
$584$	− 3902.99i	− 6.68321i
$585$	0	0
$586$	−1354.18	−2.31089
$587$	503.828i	0.858311i	0.903231	+	0.429155i	$0.141189\pi$
$587$	503.828i	0.858311i	−0.903231	+	0.429155i	$0.858811\pi$
$588$	0	0
$589$	49.6525	0.0842996
$590$	1238.45i	2.09907i
$591$	0	0
$592$	−898.924	−1.51845
$593$	− 771.173i	− 1.30046i	−0.759737	−	0.650230i	$-0.774673\pi$
$593$	− 771.173i	− 1.30046i	0.759737	−	0.650230i	$-0.225327\pi$
$594$	0	0
$595$	−103.051	−0.173195
$596$	− 1246.29i	− 2.09109i
$597$	0	0
$598$	210.861	0.352610
$599$	− 377.218i	− 0.629746i	−0.949134	−	0.314873i	$-0.898038\pi$
$599$	− 377.218i	− 0.629746i	0.949134	−	0.314873i	$-0.101962\pi$
$600$	0	0
$601$	−731.960	−1.21790	−0.608952	−	0.793207i	$-0.708410\pi$
$601$	−731.960	−1.21790	−0.608952	+	0.793207i	$0.708410\pi$
$602$	638.122i	1.06000i
$603$	0	0
$604$	1964.22	3.25202
$605$	680.395i	1.12462i
$606$	0	0
$607$	−209.500	−0.345140	−0.172570	−	0.984997i	$-0.555207\pi$
$607$	−209.500	−0.345140	−0.172570	+	0.984997i	$0.555207\pi$
$608$	− 2814.92i	− 4.62981i
$609$	0	0
$610$	369.527	0.605781
$611$	371.082i	0.607336i
$612$	0	0
$613$	−334.191	−0.545173	−0.272587	−	0.962131i	$-0.587879\pi$
$613$	−334.191	−0.545173	−0.272587	+	0.962131i	$0.587879\pi$
$614$	159.136i	0.259179i
$615$	0	0
$616$	−383.634	−0.622782
$617$	103.652i	0.167993i	0.996466	+	0.0839966i	$0.0267685\pi$
$617$	103.652i	0.167993i	−0.996466	+	0.0839966i	$0.973232\pi$
$618$	0	0
$619$	−30.9836	−0.0500543	−0.0250272	−	0.999687i	$-0.507967\pi$
$619$	−30.9836	−0.0500543	−0.0250272	+	0.999687i	$0.507967\pi$
$620$	219.340i	0.353775i
$621$	0	0
$622$	931.512	1.49761
$623$	141.146i	0.226559i
$624$	0	0
$625$	−627.475	−1.00396
$626$	− 2371.76i	− 3.78876i
$627$	0	0
$628$	−2866.57	−4.56461
$629$	71.9094i	0.114323i
$630$	0	0
$631$	−1063.05	−1.68471	−0.842353	−	0.538926i	$-0.818830\pi$
$631$	−1063.05	−1.68471	−0.842353	+	0.538926i	$0.818830\pi$
$632$	− 1473.17i	− 2.33097i
$633$	0	0
$634$	1663.15	2.62326
$635$	1376.42i	2.16758i
$636$	0	0
$637$	35.1564	0.0551906
$638$	− 779.255i	− 1.22140i
$639$	0	0
$640$	4784.54	7.47584
$641$	633.821i	0.988801i	0.869234	+	0.494400i	$0.164612\pi$
$641$	633.821i	0.988801i	−0.869234	+	0.494400i	$0.835388\pi$
$642$	0	0
$643$	−796.623	−1.23892	−0.619458	−	0.785030i	$-0.712648\pi$
$643$	−796.623	−1.23892	−0.619458	+	0.785030i	$0.712648\pi$
$644$	− 323.224i	− 0.501901i
$645$	0	0
$646$	−395.877	−0.612813
$647$	88.2332i	0.136373i	0.997673	+	0.0681864i	$0.0217213\pi$
$647$	88.2332i	0.136373i	−0.997673	+	0.0681864i	$0.978279\pi$
$648$	0	0
$649$	221.732	0.341652
$650$	− 491.248i	− 0.755766i
$651$	0	0
$652$	2089.00	3.20398
$653$	− 703.447i	− 1.07725i	−0.842544	−	0.538627i	$-0.818943\pi$
$653$	− 703.447i	− 1.07725i	0.842544	−	0.538627i	$-0.181057\pi$
$654$	0	0
$655$	955.159	1.45826
$656$	2288.29i	3.48824i
$657$	0	0
$658$	767.886	1.16700
$659$	− 547.027i	− 0.830086i	−0.909802	−	0.415043i	$-0.863767\pi$
$659$	− 547.027i	− 0.830086i	0.909802	−	0.415043i	$-0.136233\pi$
$660$	0	0
$661$	−981.778	−1.48529	−0.742646	−	0.669684i	$-0.766429\pi$
$661$	−981.778	−1.48529	−0.742646	+	0.669684i	$0.766429\pi$
$662$	− 1554.09i	− 2.34756i
$663$	0	0
$664$	1213.13	1.82701
$665$	341.607i	0.513695i
$666$	0	0
$667$	426.787	0.639861
$668$	2235.86i	3.34709i
$669$	0	0
$670$	2935.07	4.38070
$671$	− 66.1602i	− 0.0985994i
$672$	0	0
$673$	−906.243	−1.34657	−0.673286	−	0.739382i	$-0.735118\pi$
$673$	−906.243	−1.34657	−0.673286	+	0.739382i	$0.735118\pi$
$674$	1734.67i	2.57370i
$675$	0	0
$676$	1643.38	2.43103
$677$	− 944.531i	− 1.39517i	−0.716501	−	0.697586i	$-0.754258\pi$
$677$	− 944.531i	− 1.39517i	0.716501	−	0.697586i	$-0.245742\pi$
$678$	0	0
$679$	−258.888	−0.381279
$680$	− 1136.80i	− 1.67176i
$681$	0	0
$682$	53.0137	0.0777328
$683$	1111.31i	1.62710i	0.581492	+	0.813552i	$0.302469\pi$
$683$	1111.31i	1.62710i	−0.581492	+	0.813552i	$0.697531\pi$
$684$	0	0
$685$	−1448.06	−2.11395
$686$	− 72.7497i	− 0.106049i
$687$	0	0
$688$	−4232.13	−6.15135
$689$	− 29.8679i	− 0.0433496i
$690$	0	0
$691$	199.562	0.288802	0.144401	−	0.989519i	$-0.453875\pi$
$691$	199.562	0.288802	0.144401	+	0.989519i	$0.453875\pi$
$692$	2751.27i	3.97583i
$693$	0	0
$694$	387.757	0.558728
$695$	1346.10i	1.93683i
$696$	0	0
$697$	183.052	0.262628
$698$	466.180i	0.667880i
$699$	0	0
$700$	−753.024	−1.07575
$701$	− 989.769i	− 1.41194i	−0.708243	−	0.705969i	$-0.750512\pi$
$701$	− 989.769i	− 1.41194i	0.708243	−	0.705969i	$-0.249488\pi$
$702$	0	0
$703$	238.375	0.339082
$704$	− 1635.75i	− 2.32351i
$705$	0	0
$706$	−2228.05	−3.15587
$707$	− 324.240i	− 0.458613i
$708$	0	0
$709$	−56.0482	−0.0790524	−0.0395262	−	0.999219i	$-0.512585\pi$
$709$	−56.0482	−0.0790524	−0.0395262	+	0.999219i	$0.512585\pi$
$710$	− 282.973i	− 0.398554i
$711$	0	0
$712$	−1557.04	−2.18685
$713$	29.0349i	0.0407221i
$714$	0	0
$715$	−176.258	−0.246514
$716$	− 1210.79i	− 1.69105i
$717$	0	0
$718$	788.545	1.09825
$719$	− 1020.35i	− 1.41912i	−0.704644	−	0.709561i	$-0.748893\pi$
$719$	− 1020.35i	− 1.41912i	0.704644	−	0.709561i	$-0.251107\pi$
$720$	0	0
$721$	−355.959	−0.493702
$722$	− 105.745i	− 0.146461i
$723$	0	0
$724$	1543.42	2.13179
$725$	− 994.299i	− 1.37145i
$726$	0	0
$727$	−72.2895	−0.0994353	−0.0497176	−	0.998763i	$-0.515832\pi$
$727$	−72.2895	−0.0994353	−0.0497176	+	0.998763i	$0.515832\pi$
$728$	387.824i	0.532725i
$729$	0	0
$730$	−3710.70	−5.08315
$731$	338.549i	0.463131i
$732$	0	0
$733$	1004.66	1.37061	0.685304	−	0.728257i	$-0.259669\pi$
$733$	1004.66	1.37061	0.685304	+	0.728257i	$0.259669\pi$
$734$	147.446i	0.200880i
$735$	0	0
$736$	1646.06	2.23650
$737$	− 525.496i	− 0.713020i
$738$	0	0
$739$	−22.3057	−0.0301836	−0.0150918	−	0.999886i	$-0.504804\pi$
$739$	−22.3057	−0.0301836	−0.0150918	+	0.999886i	$0.504804\pi$
$740$	1053.02i	1.42300i
$741$	0	0
$742$	−61.8061	−0.0832966
$743$	− 112.997i	− 0.152083i	−0.997105	−	0.0760414i	$-0.975772\pi$
$743$	− 112.997i	− 0.152083i	0.997105	−	0.0760414i	$-0.0242281\pi$
$744$	0	0
$745$	−770.233	−1.03387
$746$	645.972i	0.865914i
$747$	0	0
$748$	−313.105	−0.418589
$749$	− 205.202i	− 0.273968i
$750$	0	0
$751$	−665.342	−0.885941	−0.442971	−	0.896536i	$-0.646075\pi$
$751$	−665.342	−0.885941	−0.442971	+	0.896536i	$0.646075\pi$
$752$	5092.74i	6.77226i
$753$	0	0
$754$	−787.767	−1.04478
$755$	− 1213.93i	− 1.60785i
$756$	0	0
$757$	−206.225	−0.272425	−0.136212	−	0.990680i	$-0.543493\pi$
$757$	−206.225	−0.272425	−0.136212	+	0.990680i	$0.543493\pi$
$758$	− 238.810i	− 0.315052i
$759$	0	0
$760$	−3768.40	−4.95842
$761$	− 490.229i	− 0.644191i	−0.946707	−	0.322095i	$-0.895613\pi$
$761$	− 490.229i	− 0.644191i	0.946707	−	0.322095i	$-0.104387\pi$
$762$	0	0
$763$	211.325	0.276966
$764$	2085.41i	2.72960i
$765$	0	0
$766$	−1934.12	−2.52496
$767$	− 224.154i	− 0.292248i
$768$	0	0
$769$	769.195	1.00025	0.500127	−	0.865952i	$-0.333287\pi$
$769$	769.195	1.00025	0.500127	+	0.865952i	$0.333287\pi$
$770$	364.733i	0.473679i
$771$	0	0
$772$	723.917	0.937716
$773$	346.226i	0.447899i	0.974601	+	0.223950i	$0.0718951\pi$
$773$	346.226i	0.447899i	−0.974601	+	0.223950i	$0.928105\pi$
$774$	0	0
$775$	67.6434	0.0872818
$776$	− 2855.90i	− 3.68028i
$777$	0	0
$778$	1321.99	1.69922
$779$	− 606.803i	− 0.778951i
$780$	0	0
$781$	−50.6636	−0.0648702
$782$	− 231.494i	− 0.296028i
$783$	0	0
$784$	482.488	0.615418
$785$	1771.60i	2.25682i
$786$	0	0
$787$	676.003	0.858962	0.429481	−	0.903076i	$-0.358697\pi$
$787$	676.003	0.858962	0.429481	+	0.903076i	$0.358697\pi$
$788$	− 9.57861i	− 0.0121556i
$789$	0	0
$790$	−1400.59	−1.77290
$791$	− 345.888i	− 0.437280i
$792$	0	0
$793$	−66.8829	−0.0843416
$794$	850.320i	1.07093i
$795$	0	0
$796$	−1627.25	−2.04428
$797$	277.024i	0.347583i	0.984782	+	0.173792i	$0.0556020\pi$
$797$	277.024i	0.347583i	−0.984782	+	0.173792i	$0.944398\pi$
$798$	0	0
$799$	407.394	0.509880
$800$	− 3834.88i	− 4.79360i
$801$	0	0
$802$	2300.03	2.86787
$803$	664.365i	0.827354i
$804$	0	0
$805$	−199.759	−0.248148
$806$	− 53.5928i	− 0.0664923i
$807$	0	0
$808$	3576.81	4.42675
$809$	− 1413.32i	− 1.74699i	−0.486830	−	0.873496i	$-0.661847\pi$
$809$	− 1413.32i	− 1.74699i	0.486830	−	0.873496i	$-0.338153\pi$
$810$	0	0
$811$	−129.077	−0.159158	−0.0795792	−	0.996829i	$-0.525358\pi$
$811$	−129.077	−0.159158	−0.0795792	+	0.996829i	$0.525358\pi$
$812$	1207.55i	1.48713i
$813$	0	0
$814$	254.512	0.312668
$815$	− 1291.04i	− 1.58410i
$816$	0	0
$817$	1122.27	1.37364
$818$	1172.01i	1.43278i
$819$	0	0
$820$	2680.56	3.26897
$821$	368.471i	0.448808i	0.974496	+	0.224404i	$0.0720435\pi$
$821$	368.471i	0.448808i	−0.974496	+	0.224404i	$0.927957\pi$
$822$	0	0
$823$	21.6229	0.0262732	0.0131366	−	0.999914i	$-0.495818\pi$
$823$	21.6229	0.0262732	0.0131366	+	0.999914i	$0.495818\pi$
$824$	− 3926.72i	− 4.76544i
$825$	0	0
$826$	−463.846	−0.561557
$827$	1159.12i	1.40160i	0.713359	+	0.700799i	$0.247173\pi$
$827$	1159.12i	1.40160i	−0.713359	+	0.700799i	$0.752827\pi$
$828$	0	0
$829$	314.303	0.379135	0.189567	−	0.981868i	$-0.439291\pi$
$829$	314.303	0.379135	0.189567	+	0.981868i	$0.439291\pi$
$830$	− 1153.36i	− 1.38959i
$831$	0	0
$832$	−1653.62	−1.98752
$833$	− 38.5966i	− 0.0463345i
$834$	0	0
$835$	1381.81	1.65486
$836$	1037.92i	1.24153i
$837$	0	0
$838$	−1776.66	−2.12012
$839$	457.069i	0.544779i	0.962187	+	0.272389i	$0.0878138\pi$
$839$	457.069i	0.544779i	−0.962187	+	0.272389i	$0.912186\pi$
$840$	0	0
$841$	−753.461	−0.895910
$842$	161.936i	0.192324i
$843$	0	0
$844$	−582.739	−0.690449
$845$	− 1015.64i	− 1.20194i
$846$	0	0
$847$	−254.834	−0.300867
$848$	− 409.908i	− 0.483382i
$849$	0	0
$850$	−539.319	−0.634493
$851$	139.392i	0.163798i
$852$	0	0
$853$	−521.286	−0.611120	−0.305560	−	0.952173i	$-0.598844\pi$
$853$	−521.286	−0.611120	−0.305560	+	0.952173i	$0.598844\pi$
$854$	138.402i	0.162063i
$855$	0	0
$856$	2263.66	2.64446
$857$	91.5255i	0.106798i	0.998573	+	0.0533988i	$0.0170054\pi$
$857$	91.5255i	0.106798i	−0.998573	+	0.0533988i	$0.982995\pi$
$858$	0	0
$859$	−513.844	−0.598189	−0.299094	−	0.954224i	$-0.596685\pi$
$859$	−513.844	−0.598189	−0.299094	+	0.954224i	$0.596685\pi$
$860$	4957.62i	5.76468i
$861$	0	0
$862$	−1631.70	−1.89292
$863$	− 1131.40i	− 1.31100i	−0.755194	−	0.655501i	$-0.772457\pi$
$863$	− 1131.40i	− 1.31100i	0.755194	−	0.655501i	$-0.227543\pi$
$864$	0	0
$865$	1700.34	1.96571
$866$	2977.71i	3.43846i
$867$	0	0
$868$	−82.1514	−0.0946444
$869$	250.763i	0.288565i
$870$	0	0
$871$	−531.236	−0.609915
$872$	2331.21i	2.67340i
$873$	0	0
$874$	−767.387	−0.878017
$875$	− 1.85765i	− 0.00212303i
$876$	0	0
$877$	1253.65	1.42947	0.714735	−	0.699395i	$-0.246547\pi$
$877$	1253.65	1.42947	0.714735	+	0.699395i	$0.246547\pi$
$878$	1766.16i	2.01158i
$879$	0	0
$880$	−2418.96	−2.74882
$881$	852.645i	0.967816i	0.875119	+	0.483908i	$0.160783\pi$
$881$	852.645i	0.967816i	−0.875119	+	0.483908i	$0.839217\pi$
$882$	0	0
$883$	329.935	0.373653	0.186826	−	0.982393i	$-0.440180\pi$
$883$	329.935	0.373653	0.186826	+	0.982393i	$0.440180\pi$
$884$	316.525i	0.358060i
$885$	0	0
$886$	1698.65	1.91721
$887$	1524.20i	1.71838i	0.511657	+	0.859190i	$0.329032\pi$
$887$	1524.20i	1.71838i	−0.511657	+	0.859190i	$0.670968\pi$
$888$	0	0
$889$	−515.520	−0.579888
$890$	1480.32i	1.66329i
$891$	0	0
$892$	−3987.38	−4.47016
$893$	− 1350.48i	− 1.51230i
$894$	0	0
$895$	−748.293	−0.836081
$896$	1791.99i	1.99999i
$897$	0	0
$898$	−857.734	−0.955160
$899$	− 108.473i	− 0.120660i
$900$	0	0
$901$	−32.7906	−0.0363935
$902$	− 647.881i	− 0.718271i
$903$	0	0
$904$	3815.63	4.22083
$905$	− 953.863i	− 1.05399i
$906$	0	0
$907$	1301.37	1.43481	0.717404	−	0.696657i	$-0.245330\pi$
$907$	1301.37	1.43481	0.717404	+	0.696657i	$0.245330\pi$
$908$	261.517i	0.288014i
$909$	0	0
$910$	368.717	0.405183
$911$	− 403.136i	− 0.442521i	−0.975215	−	0.221260i	$-0.928983\pi$
$911$	− 403.136i	− 0.442521i	0.975215	−	0.221260i	$-0.0710170\pi$
$912$	0	0
$913$	−206.499	−0.226176
$914$	− 351.318i	− 0.384374i
$915$	0	0
$916$	2428.13	2.65079
$917$	357.744i	0.390124i
$918$	0	0
$919$	1249.88	1.36004	0.680021	−	0.733193i	$-0.261971\pi$
$919$	1249.88	1.36004	0.680021	+	0.733193i	$0.261971\pi$
$920$	− 2203.62i	− 2.39524i
$921$	0	0
$922$	1091.52	1.18386
$923$	51.2170i	0.0554897i
$924$	0	0
$925$	324.747	0.351077
$926$	334.922i	0.361687i
$927$	0	0
$928$	−6149.62	−6.62675
$929$	− 930.851i	− 1.00199i	−0.865449	−	0.500996i	$-0.832967\pi$
$929$	− 930.851i	− 1.00199i	0.865449	−	0.500996i	$-0.167033\pi$
$930$	0	0
$931$	−127.945	−0.137427
$932$	− 1236.07i	− 1.32626i
$933$	0	0
$934$	1509.65	1.61633
$935$	193.505i	0.206957i
$936$	0	0
$937$	717.981	0.766255	0.383127	−	0.923695i	$-0.374847\pi$
$937$	717.981	0.766255	0.383127	+	0.923695i	$0.374847\pi$
$938$	1099.30i	1.17196i
$939$	0	0
$940$	5965.77	6.34656
$941$	− 1089.25i	− 1.15754i	−0.815491	−	0.578770i	$-0.803533\pi$
$941$	− 1089.25i	− 1.15754i	0.815491	−	0.578770i	$-0.196467\pi$
$942$	0	0
$943$	354.835	0.376283
$944$	− 3076.30i	− 3.25879i
$945$	0	0
$946$	1198.24	1.26664
$947$	1660.50i	1.75343i	0.481007	+	0.876717i	$0.340271\pi$
$947$	1660.50i	1.75343i	−0.481007	+	0.876717i	$0.659729\pi$
$948$	0	0
$949$	671.622	0.707716
$950$	1787.80i	1.88190i
$951$	0	0
$952$	425.774	0.447242
$953$	− 1831.96i	− 1.92231i	−0.276003	−	0.961157i	$-0.589010\pi$
$953$	− 1831.96i	− 1.92231i	0.276003	−	0.961157i	$-0.410990\pi$
$954$	0	0
$955$	1288.83	1.34956
$956$	− 127.405i	− 0.133268i
$957$	0	0
$958$	1316.49	1.37421
$959$	− 542.353i	− 0.565540i
$960$	0	0
$961$	−953.620	−0.992321
$962$	− 257.292i	− 0.267455i
$963$	0	0
$964$	1785.65	1.85234
$965$	− 447.395i	− 0.463622i
$966$	0	0
$967$	−1361.09	−1.40753	−0.703767	−	0.710431i	$-0.748500\pi$
$967$	−1361.09	−1.40753	−0.703767	+	0.710431i	$0.748500\pi$
$968$	− 2811.17i	− 2.90410i
$969$	0	0
$970$	−2715.19	−2.79917
$971$	343.016i	0.353261i	0.984277	+	0.176630i	$0.0565197\pi$
$971$	343.016i	0.353261i	−0.984277	+	0.176630i	$0.943480\pi$
$972$	0	0
$973$	−504.165	−0.518155
$974$	391.378i	0.401825i
$975$	0	0
$976$	−917.903	−0.940474
$977$	− 1413.35i	− 1.44663i	−0.690521	−	0.723313i	$-0.742619\pi$
$977$	− 1413.35i	− 1.44663i	0.690521	−	0.723313i	$-0.257381\pi$
$978$	0	0
$979$	265.038	0.270723
$980$	− 565.199i	− 0.576733i
$981$	0	0
$982$	−2054.29	−2.09195
$983$	− 1153.38i	− 1.17333i	−0.809830	−	0.586665i	$-0.800441\pi$
$983$	− 1153.38i	− 1.17333i	0.809830	−	0.586665i	$-0.199559\pi$
$984$	0	0
$985$	−5.91978	−0.00600993
$986$	864.853i	0.877133i
$987$	0	0
$988$	1049.26	1.06200
$989$	656.259i	0.663558i
$990$	0	0
$991$	−238.933	−0.241103	−0.120551	−	0.992707i	$-0.538466\pi$
$991$	−238.933	−0.241103	−0.120551	+	0.992707i	$0.538466\pi$
$992$	− 418.367i	− 0.421741i
$993$	0	0
$994$	105.984	0.106624
$995$	1005.67i	1.01073i
$996$	0	0
$997$	−1328.25	−1.33224	−0.666121	−	0.745843i	$-0.732047\pi$
$997$	−1328.25	−1.33224	−0.666121	+	0.745843i	$0.732047\pi$
$998$	330.432i	0.331094i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.3.b.c.134.8	yes	8
3.2	odd	2	inner	189.3.b.c.134.1	✓	8
4.3	odd	2		3024.3.d.j.1457.7		8
9.2	odd	6		567.3.r.e.134.1		16
9.4	even	3		567.3.r.e.512.1		16
9.5	odd	6		567.3.r.e.512.8		16
9.7	even	3		567.3.r.e.134.8		16
12.11	even	2		3024.3.d.j.1457.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.3.b.c.134.1	✓	8	3.2	odd	2	inner
189.3.b.c.134.8	yes	8	1.1	even	1	trivial
567.3.r.e.134.1		16	9.2	odd	6
567.3.r.e.134.8		16	9.7	even	3
567.3.r.e.512.1		16	9.4	even	3
567.3.r.e.512.8		16	9.5	odd	6
3024.3.d.j.1457.2		8	12.11	even	2
3024.3.d.j.1457.7		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	189.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.92812i q^{2} -11.4301 q^{4} +7.06404i q^{5} -2.64575 q^{7} -29.1863i q^{8} +O(q^{10})\)	\(q+3.92812i q^{2} -11.4301 q^{4} +7.06404i q^{5} -2.64575 q^{7} -29.1863i q^{8} -27.7484 q^{10} +4.96808i q^{11} +5.02234 q^{13} -10.3928i q^{14} +68.9268 q^{16} -5.51380i q^{17} -18.2779 q^{19} -80.7427i q^{20} -19.5152 q^{22} -10.6882i q^{23} -24.9006 q^{25} +19.7284i q^{26} +30.2412 q^{28} +39.9307i q^{29} -2.71654 q^{31} +154.007i q^{32} +21.6589 q^{34} -18.6897i q^{35} -13.0417 q^{37} -71.7976i q^{38} +206.173 q^{40} +33.1988i q^{41} -61.4003 q^{43} -56.7856i q^{44} +41.9845 q^{46} +73.8862i q^{47} +7.00000 q^{49} -97.8125i q^{50} -57.4059 q^{52} -5.94700i q^{53} -35.0947 q^{55} +77.2197i q^{56} -156.852 q^{58} -44.6314i q^{59} -13.3171 q^{61} -10.6709i q^{62} -329.252 q^{64} +35.4780i q^{65} -105.774 q^{67} +63.0233i q^{68} +73.4153 q^{70} +10.1978i q^{71} +133.727 q^{73} -51.2294i q^{74} +208.918 q^{76} -13.1443i q^{77} +50.4748 q^{79} +486.902i q^{80} -130.409 q^{82} +41.5651i q^{83} +38.9497 q^{85} -241.188i q^{86} +145.000 q^{88} -53.3482i q^{89} -13.2879 q^{91} +122.167i q^{92} -290.234 q^{94} -129.115i q^{95} +97.8506 q^{97} +27.4968i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{4}+O(q^{10}) \) 8 * q - 12 * q^4	\( 8 q - 12 q^{4} - 52 q^{10} + 36 q^{13} + 132 q^{16} + 12 q^{19} - 136 q^{22} - 108 q^{25} + 56 q^{28} - 28 q^{31} - 12 q^{34} - 4 q^{37} + 336 q^{40} - 152 q^{43} + 108 q^{46} + 56 q^{49} - 272 q^{52} + 196 q^{55} - 220 q^{58} + 180 q^{61} - 700 q^{64} - 132 q^{67} + 196 q^{70} + 272 q^{73} + 544 q^{76} + 316 q^{79} + 28 q^{82} - 228 q^{85} - 56 q^{91} - 348 q^{94} - 364 q^{97}+O(q^{100}) \) 8 * q - 12 * q^4 - 52 * q^10 + 36 * q^13 + 132 * q^16 + 12 * q^19 - 136 * q^22 - 108 * q^25 + 56 * q^28 - 28 * q^31 - 12 * q^34 - 4 * q^37 + 336 * q^40 - 152 * q^43 + 108 * q^46 + 56 * q^49 - 272 * q^52 + 196 * q^55 - 220 * q^58 + 180 * q^61 - 700 * q^64 - 132 * q^67 + 196 * q^70 + 272 * q^73 + 544 * q^76 + 316 * q^79 + 28 * q^82 - 228 * q^85 - 56 * q^91 - 348 * q^94 - 364 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more