LMFDB - Embedded newform 864.3.h.c.593.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(593,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("864.593");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	864.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.5422948407$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.121670000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 4x^{4} - 6x^{3} + 16x^{2} - 16x + 64 $ x^6 - x^5 + 4x^4 - 6x^3 + 16x^2 - 16x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{9}\cdot 3 $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			593.1
Root			$-1.25395 - 1.55808i$ of defining polynomial
Character	$\chi$	$=$	864.593
Dual form			864.3.h.c.593.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.79748 q^{5} +10.8133 q^{7} +O(q^{10})$	$q-3.79748 q^{5} +10.8133 q^{7} -0.420847 q^{11} -21.1820i q^{13} +30.5797i q^{17} -13.1450i q^{19} -22.5427i q^{23} -10.5792 q^{25} +15.4208 q^{29} +22.7024 q^{31} -41.0632 q^{35} +15.0482i q^{37} -10.4236i q^{41} -6.13381i q^{43} -36.2302i q^{47} +67.9271 q^{49} +22.1232 q^{53} +1.59816 q^{55} +101.728 q^{59} -94.9438i q^{61} +80.4382i q^{65} -90.8618i q^{67} -63.5460i q^{71} +81.1356 q^{73} -4.55074 q^{77} +38.0408 q^{79} -111.054 q^{83} -116.126i q^{85} -70.0146i q^{89} -229.047i q^{91} +49.9177i q^{95} +144.646 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{5} + 10 q^{7}+O(q^{10}) $ 6 * q - 2 * q^5 + 10 * q^7	$ 6 q - 2 q^{5} + 10 q^{7} - 10 q^{11} - 56 q^{25} + 100 q^{29} - 6 q^{31} - 110 q^{35} - 24 q^{49} - 2 q^{53} - 170 q^{55} + 20 q^{59} + 130 q^{73} + 50 q^{77} + 76 q^{79} + 38 q^{83} - 70 q^{97}+O(q^{100}) $ 6 * q - 2 * q^5 + 10 * q^7 - 10 * q^11 - 56 * q^25 + 100 * q^29 - 6 * q^31 - 110 * q^35 - 24 * q^49 - 2 * q^53 - 170 * q^55 + 20 * q^59 + 130 * q^73 + 50 * q^77 + 76 * q^79 + 38 * q^83 - 70 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$-1$	$1$

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 $a_p$)

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.79748	−0.759496	−0.379748	−	0.925090i	$-0.623989\pi$
$5$	−3.79748	−0.759496	−0.379748	+	0.925090i	$0.623989\pi$
$6$	0	0
$7$	10.8133	1.54475	0.772377	−	0.635164i	$-0.219067\pi$
$7$	10.8133	1.54475	0.772377	+	0.635164i	$0.219067\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.420847	−0.0382589	−0.0191294	−	0.999817i	$-0.506089\pi$
$11$	−0.420847	−0.0382589	−0.0191294	+	0.999817i	$0.506089\pi$
$12$	0	0
$13$	− 21.1820i	− 1.62938i	−0.579893	−	0.814692i	$-0.696906\pi$
$13$	− 21.1820i	− 1.62938i	0.579893	−	0.814692i	$-0.303094\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	30.5797i	1.79881i	0.437119	+	0.899403i	$0.355999\pi$
$17$	30.5797i	1.79881i	−0.437119	+	0.899403i	$0.644001\pi$
$18$	0	0
$19$	− 13.1450i	− 0.691840i	−0.938264	−	0.345920i	$-0.887567\pi$
$19$	− 13.1450i	− 0.691840i	0.938264	−	0.345920i	$-0.112433\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 22.5427i	− 0.980116i	−0.871690	−	0.490058i	$-0.836976\pi$
$23$	− 22.5427i	− 0.980116i	0.871690	−	0.490058i	$-0.163024\pi$
$24$	0	0
$25$	−10.5792	−0.423166
$26$	0	0
$27$	0	0
$28$	0	0
$29$	15.4208	0.531753	0.265877	−	0.964007i	$-0.414339\pi$
$29$	15.4208	0.531753	0.265877	+	0.964007i	$0.414339\pi$
$30$	0	0
$31$	22.7024	0.732335	0.366168	−	0.930549i	$-0.380670\pi$
$31$	22.7024	0.732335	0.366168	+	0.930549i	$0.380670\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−41.0632	−1.17323
$36$	0	0
$37$	15.0482i	0.406708i	0.979105	+	0.203354i	$0.0651842\pi$
$37$	15.0482i	0.406708i	−0.979105	+	0.203354i	$0.934816\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 10.4236i	− 0.254234i	−0.991888	−	0.127117i	$-0.959428\pi$
$41$	− 10.4236i	− 0.254234i	0.991888	−	0.127117i	$-0.0405724\pi$
$42$	0	0
$43$	− 6.13381i	− 0.142647i	−0.997453	−	0.0713233i	$-0.977278\pi$
$43$	− 6.13381i	− 0.142647i	0.997453	−	0.0713233i	$-0.0227222\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 36.2302i	− 0.770855i	−0.922738	−	0.385428i	$-0.874054\pi$
$47$	− 36.2302i	− 0.770855i	0.922738	−	0.385428i	$-0.125946\pi$
$48$	0	0
$49$	67.9271	1.38627
$50$	0	0
$51$	0	0
$52$	0	0
$53$	22.1232	0.417420	0.208710	−	0.977978i	$-0.433074\pi$
$53$	22.1232	0.417420	0.208710	+	0.977978i	$0.433074\pi$
$54$	0	0
$55$	1.59816	0.0290574
$56$	0	0
$57$	0	0
$58$	0	0
$59$	101.728	1.72420	0.862100	−	0.506738i	$-0.169149\pi$
$59$	101.728	1.72420	0.862100	+	0.506738i	$0.169149\pi$
$60$	0	0
$61$	− 94.9438i	− 1.55646i	−0.627982	−	0.778228i	$-0.716119\pi$
$61$	− 94.9438i	− 1.55646i	0.627982	−	0.778228i	$-0.283881\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	80.4382i	1.23751i
$66$	0	0
$67$	− 90.8618i	− 1.35615i	−0.734994	−	0.678073i	$-0.762815\pi$
$67$	− 90.8618i	− 1.35615i	0.734994	−	0.678073i	$-0.237185\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 63.5460i	− 0.895014i	−0.894280	−	0.447507i	$-0.852312\pi$
$71$	− 63.5460i	− 0.895014i	0.894280	−	0.447507i	$-0.147688\pi$
$72$	0	0
$73$	81.1356	1.11145	0.555723	−	0.831367i	$-0.312441\pi$
$73$	81.1356	1.11145	0.555723	+	0.831367i	$0.312441\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.55074	−0.0591006
$78$	0	0
$79$	38.0408	0.481529	0.240764	−	0.970584i	$-0.422602\pi$
$79$	38.0408	0.481529	0.240764	+	0.970584i	$0.422602\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−111.054	−1.33799	−0.668997	−	0.743265i	$-0.733276\pi$
$83$	−111.054	−1.33799	−0.668997	+	0.743265i	$0.733276\pi$
$84$	0	0
$85$	− 116.126i	− 1.36619i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 70.0146i	− 0.786681i	−0.919393	−	0.393340i	$-0.871319\pi$
$89$	− 70.0146i	− 0.786681i	0.919393	−	0.393340i	$-0.128681\pi$
$90$	0	0
$91$	− 229.047i	− 2.51700i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	49.9177i	0.525450i
$96$	0	0
$97$	144.646	1.49119	0.745597	−	0.666397i	$-0.232165\pi$
$97$	144.646	1.49119	0.745597	+	0.666397i	$0.232165\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−129.560	−1.28277	−0.641384	−	0.767220i	$-0.721640\pi$
$101$	−129.560	−1.28277	−0.641384	+	0.767220i	$0.721640\pi$
$102$	0	0
$103$	92.7688	0.900668	0.450334	−	0.892860i	$-0.351305\pi$
$103$	92.7688	0.900668	0.450334	+	0.892860i	$0.351305\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	100.031	0.934873	0.467436	−	0.884027i	$-0.345178\pi$
$107$	100.031	0.934873	0.467436	+	0.884027i	$0.345178\pi$
$108$	0	0
$109$	97.7244i	0.896554i	0.893895	+	0.448277i	$0.147962\pi$
$109$	97.7244i	0.896554i	−0.893895	+	0.448277i	$0.852038\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 150.453i	− 1.33144i	−0.746201	−	0.665720i	$-0.768124\pi$
$113$	− 150.453i	− 1.33144i	0.746201	−	0.665720i	$-0.231876\pi$
$114$	0	0
$115$	85.6054i	0.744394i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	330.667i	2.77872i
$120$	0	0
$121$	−120.823	−0.998536
$122$	0	0
$123$	0	0
$124$	0	0
$125$	135.111	1.08089
$126$	0	0
$127$	53.9528	0.424825	0.212413	−	0.977180i	$-0.431868\pi$
$127$	53.9528	0.424825	0.212413	+	0.977180i	$0.431868\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−89.5512	−0.683597	−0.341798	−	0.939773i	$-0.611036\pi$
$131$	−89.5512	−0.683597	−0.341798	+	0.939773i	$0.611036\pi$
$132$	0	0
$133$	− 142.140i	− 1.06872i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	129.419i	0.944667i	0.881420	+	0.472333i	$0.156588\pi$
$137$	129.419i	0.944667i	−0.881420	+	0.472333i	$0.843412\pi$
$138$	0	0
$139$	241.916i	1.74041i	0.492694	+	0.870203i	$0.336012\pi$
$139$	241.916i	1.74041i	−0.492694	+	0.870203i	$0.663988\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.91439i	0.0623384i
$144$	0	0
$145$	−58.5603	−0.403864
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.3576	0.0762253	0.0381127	−	0.999273i	$-0.487865\pi$
$149$	11.3576	0.0762253	0.0381127	+	0.999273i	$0.487865\pi$
$150$	0	0
$151$	10.0692	0.0666833	0.0333416	−	0.999444i	$-0.489385\pi$
$151$	10.0692	0.0666833	0.0333416	+	0.999444i	$0.489385\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−86.2119	−0.556206
$156$	0	0
$157$	42.3640i	0.269834i	0.990857	+	0.134917i	$0.0430768\pi$
$157$	42.3640i	0.269834i	−0.990857	+	0.134917i	$0.956923\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 243.760i	− 1.51404i
$162$	0	0
$163$	− 47.3234i	− 0.290328i	−0.989408	−	0.145164i	$-0.953629\pi$
$163$	− 47.3234i	− 0.290328i	0.989408	−	0.145164i	$-0.0463709\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 222.727i	− 1.33369i	−0.745195	−	0.666847i	$-0.767643\pi$
$167$	− 222.727i	− 1.33369i	0.745195	−	0.666847i	$-0.232357\pi$
$168$	0	0
$169$	−279.677	−1.65489
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−171.158	−0.989351	−0.494676	−	0.869078i	$-0.664713\pi$
$173$	−171.158	−0.989351	−0.494676	+	0.869078i	$0.664713\pi$
$174$	0	0
$175$	−114.395	−0.653688
$176$	0	0
$177$	0	0
$178$	0	0
$179$	224.917	1.25652	0.628261	−	0.778003i	$-0.283767\pi$
$179$	224.917	1.25652	0.628261	+	0.778003i	$0.283767\pi$
$180$	0	0
$181$	52.5798i	0.290496i	0.989395	+	0.145248i	$0.0463981\pi$
$181$	52.5798i	0.290496i	−0.989395	+	0.145248i	$0.953602\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 57.1452i	− 0.308893i
$186$	0	0
$187$	− 12.8694i	− 0.0688203i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	185.746i	0.972494i	0.873821	+	0.486247i	$0.161635\pi$
$191$	185.746i	0.972494i	−0.873821	+	0.486247i	$0.838365\pi$
$192$	0	0
$193$	50.3205	0.260728	0.130364	−	0.991466i	$-0.458385\pi$
$193$	50.3205	0.260728	0.130364	+	0.991466i	$0.458385\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−211.096	−1.07155	−0.535775	−	0.844361i	$-0.679981\pi$
$197$	−211.096	−1.07155	−0.535775	+	0.844361i	$0.679981\pi$
$198$	0	0
$199$	−25.3764	−0.127520	−0.0637598	−	0.997965i	$-0.520309\pi$
$199$	−25.3764	−0.127520	−0.0637598	+	0.997965i	$0.520309\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	166.750	0.821429
$204$	0	0
$205$	39.5834i	0.193090i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.53202i	0.0264690i
$210$	0	0
$211$	− 59.3154i	− 0.281116i	−0.990072	−	0.140558i	$-0.955110\pi$
$211$	− 59.3154i	− 0.281116i	0.990072	−	0.140558i	$-0.0448896\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	23.2930i	0.108340i
$216$	0	0
$217$	245.487	1.13128
$218$	0	0
$219$	0	0
$220$	0	0
$221$	647.740	2.93095
$222$	0	0
$223$	87.9725	0.394496	0.197248	−	0.980354i	$-0.436800\pi$
$223$	87.9725	0.394496	0.197248	+	0.980354i	$0.436800\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	13.5198	0.0595587	0.0297794	−	0.999556i	$-0.490520\pi$
$227$	13.5198	0.0595587	0.0297794	+	0.999556i	$0.490520\pi$
$228$	0	0
$229$	270.534i	1.18137i	0.806902	+	0.590685i	$0.201142\pi$
$229$	270.534i	1.18137i	−0.806902	+	0.590685i	$0.798858\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	121.628i	0.522008i	0.965338	+	0.261004i	$0.0840535\pi$
$233$	121.628i	0.522008i	−0.965338	+	0.261004i	$0.915946\pi$
$234$	0	0
$235$	137.583i	0.585461i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 281.500i	− 1.17782i	−0.808197	−	0.588912i	$-0.799557\pi$
$239$	− 281.500i	− 1.17782i	0.808197	−	0.588912i	$-0.200443\pi$
$240$	0	0
$241$	134.190	0.556804	0.278402	−	0.960465i	$-0.410195\pi$
$241$	134.190	0.556804	0.278402	+	0.960465i	$0.410195\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−257.952	−1.05286
$246$	0	0
$247$	−278.437	−1.12727
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−357.962	−1.42614	−0.713072	−	0.701090i	$-0.752697\pi$
$251$	−357.962	−1.42614	−0.713072	+	0.701090i	$0.752697\pi$
$252$	0	0
$253$	9.48703i	0.0374981i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	93.4939i	0.363789i	0.983318	+	0.181895i	$0.0582230\pi$
$257$	93.4939i	0.363789i	−0.983318	+	0.181895i	$0.941777\pi$
$258$	0	0
$259$	162.720i	0.628264i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 203.003i	− 0.771873i	−0.922525	−	0.385936i	$-0.873878\pi$
$263$	− 203.003i	− 0.771873i	0.922525	−	0.385936i	$-0.126122\pi$
$264$	0	0
$265$	−84.0126	−0.317029
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.0032	−0.0371864	−0.0185932	−	0.999827i	$-0.505919\pi$
$269$	−10.0032	−0.0371864	−0.0185932	+	0.999827i	$0.505919\pi$
$270$	0	0
$271$	−417.829	−1.54180	−0.770902	−	0.636954i	$-0.780194\pi$
$271$	−417.829	−1.54180	−0.770902	+	0.636954i	$0.780194\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.45221	0.0161898
$276$	0	0
$277$	466.004i	1.68232i	0.540782	+	0.841162i	$0.318128\pi$
$277$	466.004i	1.68232i	−0.540782	+	0.841162i	$0.681872\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	360.497i	1.28291i	0.767162	+	0.641453i	$0.221668\pi$
$281$	360.497i	1.28291i	−0.767162	+	0.641453i	$0.778332\pi$
$282$	0	0
$283$	90.2600i	0.318940i	0.987203	+	0.159470i	$0.0509785\pi$
$283$	90.2600i	0.318940i	−0.987203	+	0.159470i	$0.949021\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 112.713i	− 0.392730i
$288$	0	0
$289$	−646.119	−2.23571
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−123.339	−0.420953	−0.210477	−	0.977599i	$-0.567502\pi$
$293$	−123.339	−0.420953	−0.210477	+	0.977599i	$0.567502\pi$
$294$	0	0
$295$	−386.309	−1.30952
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−477.499	−1.59699
$300$	0	0
$301$	− 66.3266i	− 0.220354i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	360.547i	1.18212i
$306$	0	0
$307$	432.957i	1.41028i	0.709066	+	0.705142i	$0.249117\pi$
$307$	432.957i	1.41028i	−0.709066	+	0.705142i	$0.750883\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 53.1902i	− 0.171030i	−0.996337	−	0.0855148i	$-0.972747\pi$
$311$	− 53.1902i	− 0.171030i	0.996337	−	0.0855148i	$-0.0272535\pi$
$312$	0	0
$313$	−234.021	−0.747672	−0.373836	−	0.927495i	$-0.621958\pi$
$313$	−234.021	−0.747672	−0.373836	+	0.927495i	$0.621958\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	331.680	1.04631	0.523155	−	0.852238i	$-0.324755\pi$
$317$	331.680	1.04631	0.523155	+	0.852238i	$0.324755\pi$
$318$	0	0
$319$	−6.48982	−0.0203443
$320$	0	0
$321$	0	0
$322$	0	0
$323$	401.969	1.24449
$324$	0	0
$325$	224.088i	0.689500i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 391.767i	− 1.19078i
$330$	0	0
$331$	56.4078i	0.170416i	0.996363	+	0.0852082i	$0.0271555\pi$
$331$	56.4078i	0.170416i	−0.996363	+	0.0852082i	$0.972844\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	345.046i	1.02999i
$336$	0	0
$337$	−84.7226	−0.251402	−0.125701	−	0.992068i	$-0.540118\pi$
$337$	−84.7226	−0.251402	−0.125701	+	0.992068i	$0.540118\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−9.55424	−0.0280183
$342$	0	0
$343$	204.664	0.596689
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.5865	0.0564451	0.0282226	−	0.999602i	$-0.491015\pi$
$347$	19.5865	0.0564451	0.0282226	+	0.999602i	$0.491015\pi$
$348$	0	0
$349$	348.356i	0.998155i	0.866557	+	0.499077i	$0.166328\pi$
$349$	348.356i	0.998155i	−0.866557	+	0.499077i	$0.833672\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	658.821i	1.86635i	0.359425	+	0.933174i	$0.382973\pi$
$353$	658.821i	1.86635i	−0.359425	+	0.933174i	$0.617027\pi$
$354$	0	0
$355$	241.315i	0.679759i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 132.556i	− 0.369237i	−0.982810	−	0.184619i	$-0.940895\pi$
$359$	− 132.556i	− 0.369237i	0.982810	−	0.184619i	$-0.0591050\pi$
$360$	0	0
$361$	188.210	0.521357
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−308.111	−0.844139
$366$	0	0
$367$	−106.494	−0.290175	−0.145088	−	0.989419i	$-0.546346\pi$
$367$	−106.494	−0.290175	−0.145088	+	0.989419i	$0.546346\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	239.225	0.644811
$372$	0	0
$373$	− 12.4668i	− 0.0334231i	−0.999860	−	0.0167115i	$-0.994680\pi$
$373$	− 12.4668i	− 0.0334231i	0.999860	−	0.0167115i	$-0.00531969\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 326.644i	− 0.866431i
$378$	0	0
$379$	489.691i	1.29206i	0.763312	+	0.646030i	$0.223572\pi$
$379$	489.691i	1.29206i	−0.763312	+	0.646030i	$0.776428\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 420.838i	− 1.09879i	−0.835562	−	0.549397i	$-0.814858\pi$
$383$	− 420.838i	− 1.09879i	0.835562	−	0.549397i	$-0.185142\pi$
$384$	0	0
$385$	17.2813	0.0448866
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−697.237	−1.79238	−0.896192	−	0.443667i	$-0.853677\pi$
$389$	−697.237	−1.79238	−0.896192	+	0.443667i	$0.853677\pi$
$390$	0	0
$391$	689.349	1.76304
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−144.459	−0.365719
$396$	0	0
$397$	− 416.734i	− 1.04971i	−0.851192	−	0.524854i	$-0.824120\pi$
$397$	− 416.734i	− 1.04971i	0.851192	−	0.524854i	$-0.175880\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	46.7216i	0.116513i	0.998302	+	0.0582564i	$0.0185541\pi$
$401$	46.7216i	0.116513i	−0.998302	+	0.0582564i	$0.981446\pi$
$402$	0	0
$403$	− 480.882i	− 1.19326i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.33299i	− 0.0155602i
$408$	0	0
$409$	−47.7336	−0.116708	−0.0583540	−	0.998296i	$-0.518585\pi$
$409$	−47.7336	−0.116708	−0.0583540	+	0.998296i	$0.518585\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1100.01	2.66347
$414$	0	0
$415$	421.724	1.01620
$416$	0	0
$417$	0	0
$418$	0	0
$419$	225.611	0.538452	0.269226	−	0.963077i	$-0.413232\pi$
$419$	225.611	0.538452	0.269226	+	0.963077i	$0.413232\pi$
$420$	0	0
$421$	− 573.215i	− 1.36156i	−0.732489	−	0.680778i	$-0.761642\pi$
$421$	− 573.215i	− 1.36156i	0.732489	−	0.680778i	$-0.238358\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 323.508i	− 0.761194i
$426$	0	0
$427$	− 1026.65i	− 2.40434i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 362.875i	− 0.841936i	−0.907075	−	0.420968i	$-0.861690\pi$
$431$	− 362.875i	− 0.841936i	0.907075	−	0.420968i	$-0.138310\pi$
$432$	0	0
$433$	−456.563	−1.05442	−0.527209	−	0.849736i	$-0.676761\pi$
$433$	−456.563	−1.05442	−0.527209	+	0.849736i	$0.676761\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−296.323	−0.678084
$438$	0	0
$439$	−251.319	−0.572481	−0.286241	−	0.958158i	$-0.592406\pi$
$439$	−251.319	−0.572481	−0.286241	+	0.958158i	$0.592406\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−385.248	−0.869633	−0.434817	−	0.900519i	$-0.643187\pi$
$443$	−385.248	−0.869633	−0.434817	+	0.900519i	$0.643187\pi$
$444$	0	0
$445$	265.879i	0.597481i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 631.937i	− 1.40743i	−0.710482	−	0.703716i	$-0.751523\pi$
$449$	− 631.937i	− 1.40743i	0.710482	−	0.703716i	$-0.248477\pi$
$450$	0	0
$451$	4.38675i	0.00972671i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	869.801i	1.91165i
$456$	0	0
$457$	428.473	0.937579	0.468789	−	0.883310i	$-0.344690\pi$
$457$	428.473	0.937579	0.468789	+	0.883310i	$0.344690\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	493.068	1.06956	0.534780	−	0.844991i	$-0.320394\pi$
$461$	493.068	1.06956	0.534780	+	0.844991i	$0.320394\pi$
$462$	0	0
$463$	−292.653	−0.632081	−0.316040	−	0.948746i	$-0.602354\pi$
$463$	−292.653	−0.632081	−0.316040	+	0.948746i	$0.602354\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.9329	0.0512481	0.0256241	−	0.999672i	$-0.491843\pi$
$467$	23.9329	0.0512481	0.0256241	+	0.999672i	$0.491843\pi$
$468$	0	0
$469$	− 982.515i	− 2.09491i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.58140i	0.00545750i
$474$	0	0
$475$	139.063i	0.292763i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	551.499i	1.15135i	0.817677	+	0.575677i	$0.195261\pi$
$479$	551.499i	1.15135i	−0.817677	+	0.575677i	$0.804739\pi$
$480$	0	0
$481$	318.751	0.662684
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−549.289	−1.13256
$486$	0	0
$487$	−185.067	−0.380014	−0.190007	−	0.981783i	$-0.560851\pi$
$487$	−185.067	−0.380014	−0.190007	+	0.981783i	$0.560851\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	724.436	1.47543	0.737715	−	0.675112i	$-0.235905\pi$
$491$	724.436	1.47543	0.737715	+	0.675112i	$0.235905\pi$
$492$	0	0
$493$	471.565i	0.956522i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 687.141i	− 1.38258i
$498$	0	0
$499$	110.191i	0.220824i	0.993886	+	0.110412i	$0.0352170\pi$
$499$	110.191i	0.220824i	−0.993886	+	0.110412i	$0.964783\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	90.1115i	0.179148i	0.995980	+	0.0895740i	$0.0285506\pi$
$503$	90.1115i	0.179148i	−0.995980	+	0.0895740i	$0.971449\pi$
$504$	0	0
$505$	492.000	0.974258
$506$	0	0
$507$	0	0
$508$	0	0
$509$	513.395	1.00863	0.504317	−	0.863519i	$-0.331744\pi$
$509$	513.395	1.00863	0.504317	+	0.863519i	$0.331744\pi$
$510$	0	0
$511$	877.342	1.71691
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−352.288	−0.684054
$516$	0	0
$517$	15.2474i	0.0294920i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	101.150i	0.194145i	0.995277	+	0.0970727i	$0.0309480\pi$
$521$	101.150i	0.194145i	−0.995277	+	0.0970727i	$0.969052\pi$
$522$	0	0
$523$	370.415i	0.708251i	0.935198	+	0.354126i	$0.115221\pi$
$523$	370.415i	0.708251i	−0.935198	+	0.354126i	$0.884779\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	694.233i	1.31733i
$528$	0	0
$529$	20.8276	0.0393717
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−220.793	−0.414245
$534$	0	0
$535$	−379.867	−0.710032
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−28.5870	−0.0530370
$540$	0	0
$541$	23.3899i	0.0432347i	0.999766	+	0.0216173i	$0.00688154\pi$
$541$	23.3899i	0.0432347i	−0.999766	+	0.0216173i	$0.993118\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 371.106i	− 0.680929i
$546$	0	0
$547$	− 617.029i	− 1.12802i	−0.825766	−	0.564012i	$-0.809257\pi$
$547$	− 617.029i	− 1.12802i	0.825766	−	0.564012i	$-0.190743\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 202.706i	− 0.367888i
$552$	0	0
$553$	411.346	0.743844
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−277.478	−0.498165	−0.249083	−	0.968482i	$-0.580129\pi$
$557$	−277.478	−0.498165	−0.249083	+	0.968482i	$0.580129\pi$
$558$	0	0
$559$	−129.926	−0.232426
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−593.792	−1.05469	−0.527347	−	0.849650i	$-0.676813\pi$
$563$	−593.792	−1.05469	−0.527347	+	0.849650i	$0.676813\pi$
$564$	0	0
$565$	571.341i	1.01122i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.8200i	0.0278032i	0.999903	+	0.0139016i	$0.00442516\pi$
$569$	15.8200i	0.0278032i	−0.999903	+	0.0139016i	$0.995575\pi$
$570$	0	0
$571$	5.86588i	0.0102730i	0.999987	+	0.00513650i	$0.00163500\pi$
$571$	5.86588i	0.0102730i	−0.999987	+	0.00513650i	$0.998365\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	238.482i	0.414752i
$576$	0	0
$577$	611.669	1.06009	0.530043	−	0.847971i	$-0.322176\pi$
$577$	611.669	1.06009	0.530043	+	0.847971i	$0.322176\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1200.85	−2.06687
$582$	0	0
$583$	−9.31051	−0.0159700
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1129.91	1.92489	0.962444	−	0.271479i	$-0.0875127\pi$
$587$	1129.91	1.92489	0.962444	+	0.271479i	$0.0875127\pi$
$588$	0	0
$589$	− 298.422i	− 0.506659i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	664.034i	1.11979i	0.828564	+	0.559894i	$0.189158\pi$
$593$	664.034i	1.11979i	−0.828564	+	0.559894i	$0.810842\pi$
$594$	0	0
$595$	− 1255.70i	− 2.11042i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 459.201i	− 0.766612i	−0.923621	−	0.383306i	$-0.874785\pi$
$599$	− 459.201i	− 0.766612i	0.923621	−	0.383306i	$-0.125215\pi$
$600$	0	0
$601$	316.211	0.526141	0.263071	−	0.964777i	$-0.415265\pi$
$601$	316.211	0.526141	0.263071	+	0.964777i	$0.415265\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	458.822	0.758384
$606$	0	0
$607$	−530.344	−0.873713	−0.436857	−	0.899531i	$-0.643908\pi$
$607$	−530.344	−0.873713	−0.436857	+	0.899531i	$0.643908\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−767.428	−1.25602
$612$	0	0
$613$	− 78.5942i	− 0.128212i	−0.997943	−	0.0641062i	$-0.979580\pi$
$613$	− 78.5942i	− 0.128212i	0.997943	−	0.0641062i	$-0.0204196\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	745.482i	1.20824i	0.796895	+	0.604118i	$0.206475\pi$
$617$	745.482i	1.20824i	−0.796895	+	0.604118i	$0.793525\pi$
$618$	0	0
$619$	− 1060.51i	− 1.71326i	−0.515933	−	0.856629i	$-0.672554\pi$
$619$	− 1060.51i	− 1.71326i	0.515933	−	0.856629i	$-0.327446\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 757.088i	− 1.21523i
$624$	0	0
$625$	−248.603	−0.397764
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−460.170	−0.731589
$630$	0	0
$631$	875.000	1.38669	0.693344	−	0.720607i	$-0.256137\pi$
$631$	875.000	1.38669	0.693344	+	0.720607i	$0.256137\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−204.885	−0.322653
$636$	0	0
$637$	− 1438.83i	− 2.25876i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 379.908i	− 0.592680i	−0.955083	−	0.296340i	$-0.904234\pi$
$641$	− 379.908i	− 0.592680i	0.955083	−	0.296340i	$-0.0957661\pi$
$642$	0	0
$643$	401.454i	0.624345i	0.950025	+	0.312172i	$0.101057\pi$
$643$	401.454i	0.624345i	−0.950025	+	0.312172i	$0.898943\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1008.57i	1.55884i	0.626499	+	0.779422i	$0.284487\pi$
$647$	1008.57i	1.55884i	−0.626499	+	0.779422i	$0.715513\pi$
$648$	0	0
$649$	−42.8119	−0.0659659
$650$	0	0
$651$	0	0
$652$	0	0
$653$	733.986	1.12402	0.562011	−	0.827130i	$-0.310028\pi$
$653$	733.986	1.12402	0.562011	+	0.827130i	$0.310028\pi$
$654$	0	0
$655$	340.069	0.519189
$656$	0	0
$657$	0	0
$658$	0	0
$659$	663.270	1.00648	0.503240	−	0.864147i	$-0.332141\pi$
$659$	663.270	1.00648	0.503240	+	0.864147i	$0.332141\pi$
$660$	0	0
$661$	− 623.566i	− 0.943367i	−0.881768	−	0.471684i	$-0.843646\pi$
$661$	− 623.566i	− 0.943367i	0.881768	−	0.471684i	$-0.156354\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	539.774i	0.811691i
$666$	0	0
$667$	− 347.627i	− 0.521180i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	39.9569i	0.0595482i
$672$	0	0
$673$	−273.223	−0.405977	−0.202989	−	0.979181i	$-0.565065\pi$
$673$	−273.223	−0.405977	−0.202989	+	0.979181i	$0.565065\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	619.407	0.914929	0.457464	−	0.889228i	$-0.348758\pi$
$677$	619.407	0.914929	0.457464	+	0.889228i	$0.348758\pi$
$678$	0	0
$679$	1564.10	2.30353
$680$	0	0
$681$	0	0
$682$	0	0
$683$	529.761	0.775638	0.387819	−	0.921736i	$-0.373229\pi$
$683$	529.761	0.775638	0.387819	+	0.921736i	$0.373229\pi$
$684$	0	0
$685$	− 491.467i	− 0.717470i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 468.615i	− 0.680137i
$690$	0	0
$691$	437.068i	0.632515i	0.948673	+	0.316258i	$0.102426\pi$
$691$	437.068i	0.632515i	−0.948673	+	0.316258i	$0.897574\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 918.672i	− 1.32183i
$696$	0	0
$697$	318.751	0.457318
$698$	0	0
$699$	0	0
$700$	0	0
$701$	98.4102	0.140385	0.0701927	−	0.997533i	$-0.477639\pi$
$701$	98.4102	0.140385	0.0701927	+	0.997533i	$0.477639\pi$
$702$	0	0
$703$	197.808	0.281377
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1400.97	−1.98156
$708$	0	0
$709$	− 639.012i	− 0.901287i	−0.892704	−	0.450643i	$-0.851195\pi$
$709$	− 639.012i	− 0.901287i	0.892704	−	0.450643i	$-0.148805\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 511.773i	− 0.717774i
$714$	0	0
$715$	− 33.8522i	− 0.0473457i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1066.48i	1.48328i	0.670800	+	0.741638i	$0.265951\pi$
$719$	1066.48i	1.48328i	−0.670800	+	0.741638i	$0.734049\pi$
$720$	0	0
$721$	1003.14	1.39131
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−163.139	−0.225020
$726$	0	0
$727$	−349.694	−0.481010	−0.240505	−	0.970648i	$-0.577313\pi$
$727$	−349.694	−0.481010	−0.240505	+	0.970648i	$0.577313\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	187.570	0.256594
$732$	0	0
$733$	− 183.359i	− 0.250149i	−0.992147	−	0.125074i	$-0.960083\pi$
$733$	− 183.359i	− 0.250149i	0.992147	−	0.125074i	$-0.0399169\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	38.2390i	0.0518846i
$738$	0	0
$739$	57.2852i	0.0775171i	0.999249	+	0.0387586i	$0.0123403\pi$
$739$	57.2852i	0.0775171i	−0.999249	+	0.0387586i	$0.987660\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 673.995i	− 0.907127i	−0.891224	−	0.453563i	$-0.850153\pi$
$743$	− 673.995i	− 0.907127i	0.891224	−	0.453563i	$-0.149847\pi$
$744$	0	0
$745$	−43.1302	−0.0578928
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1081.67	1.44415
$750$	0	0
$751$	364.398	0.485218	0.242609	−	0.970124i	$-0.421997\pi$
$751$	364.398	0.485218	0.242609	+	0.970124i	$0.421997\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−38.2375	−0.0506457
$756$	0	0
$757$	1053.83i	1.39211i	0.717987	+	0.696057i	$0.245064\pi$
$757$	1053.83i	1.39211i	−0.717987	+	0.696057i	$0.754936\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1078.27i	1.41691i	0.705756	+	0.708455i	$0.250607\pi$
$761$	1078.27i	1.41691i	−0.705756	+	0.708455i	$0.749393\pi$
$762$	0	0
$763$	1056.72i	1.38496i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2154.80i	− 2.80939i
$768$	0	0
$769$	−499.223	−0.649185	−0.324593	−	0.945854i	$-0.605227\pi$
$769$	−499.223	−0.649185	−0.324593	+	0.945854i	$0.605227\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1132.91	−1.46560	−0.732799	−	0.680445i	$-0.761787\pi$
$773$	−1132.91	−1.46560	−0.732799	+	0.680445i	$0.761787\pi$
$774$	0	0
$775$	−240.172	−0.309899
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−137.018	−0.175889
$780$	0	0
$781$	26.7432i	0.0342422i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 160.876i	− 0.204938i
$786$	0	0
$787$	− 5.90125i	− 0.00749841i	−0.999993	−	0.00374921i	$-0.998807\pi$
$787$	− 5.90125i	− 0.00749841i	0.999993	−	0.00374921i	$-0.00119341\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1626.89i	− 2.05675i
$792$	0	0
$793$	−2011.10	−2.53607
$794$	0	0
$795$	0	0
$796$	0	0
$797$	692.893	0.869377	0.434688	−	0.900581i	$-0.356858\pi$
$797$	692.893	0.869377	0.434688	+	0.900581i	$0.356858\pi$
$798$	0	0
$799$	1107.91	1.38662
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−34.1457	−0.0425227
$804$	0	0
$805$	925.675i	1.14991i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	158.566i	0.196003i	0.995186	+	0.0980014i	$0.0312450\pi$
$809$	158.566i	0.196003i	−0.995186	+	0.0980014i	$0.968755\pi$
$810$	0	0
$811$	− 832.294i	− 1.02626i	−0.858312	−	0.513128i	$-0.828486\pi$
$811$	− 832.294i	− 1.02626i	0.858312	−	0.513128i	$-0.171514\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	179.710i	0.220503i
$816$	0	0
$817$	−80.6286	−0.0986887
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−498.242	−0.606872	−0.303436	−	0.952852i	$-0.598134\pi$
$821$	−498.242	−0.606872	−0.303436	+	0.952852i	$0.598134\pi$
$822$	0	0
$823$	−1050.59	−1.27654	−0.638271	−	0.769811i	$-0.720350\pi$
$823$	−1050.59	−1.27654	−0.638271	+	0.769811i	$0.720350\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−246.790	−0.298416	−0.149208	−	0.988806i	$-0.547672\pi$
$827$	−246.790	−0.298416	−0.149208	+	0.988806i	$0.547672\pi$
$828$	0	0
$829$	647.155i	0.780645i	0.920678	+	0.390323i	$0.127637\pi$
$829$	647.155i	0.780645i	−0.920678	+	0.390323i	$0.872363\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2077.19i	2.49363i
$834$	0	0
$835$	845.801i	1.01294i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1525.10i	1.81776i	0.417060	+	0.908879i	$0.363060\pi$
$839$	1525.10i	1.81776i	−0.417060	+	0.908879i	$0.636940\pi$
$840$	0	0
$841$	−603.197	−0.717238
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1062.07	1.25689
$846$	0	0
$847$	−1306.49	−1.54249
$848$	0	0
$849$	0	0
$850$	0	0
$851$	339.227	0.398621
$852$	0	0
$853$	564.718i	0.662037i	0.943624	+	0.331018i	$0.107392\pi$
$853$	564.718i	0.662037i	−0.943624	+	0.331018i	$0.892608\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 773.565i	− 0.902643i	−0.892361	−	0.451322i	$-0.850953\pi$
$857$	− 773.565i	− 0.902643i	0.892361	−	0.451322i	$-0.149047\pi$
$858$	0	0
$859$	− 263.001i	− 0.306171i	−0.988213	−	0.153085i	$-0.951079\pi$
$859$	− 263.001i	− 0.306171i	0.988213	−	0.153085i	$-0.0489209\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 481.738i	− 0.558213i	−0.960260	−	0.279107i	$-0.909962\pi$
$863$	− 481.738i	− 0.558213i	0.960260	−	0.279107i	$-0.0900383\pi$
$864$	0	0
$865$	649.968	0.751408
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−16.0094	−0.0184227
$870$	0	0
$871$	−1924.63	−2.20968
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1460.99	1.66971
$876$	0	0
$877$	196.573i	0.224142i	0.993700	+	0.112071i	$0.0357484\pi$
$877$	196.573i	0.224142i	−0.993700	+	0.112071i	$0.964252\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	506.427i	0.574832i	0.957806	+	0.287416i	$0.0927962\pi$
$881$	506.427i	0.574832i	−0.957806	+	0.287416i	$0.907204\pi$
$882$	0	0
$883$	962.075i	1.08955i	0.838581	+	0.544776i	$0.183386\pi$
$883$	962.075i	1.08955i	−0.838581	+	0.544776i	$0.816614\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	747.127i	0.842308i	0.906989	+	0.421154i	$0.138375\pi$
$887$	747.127i	0.842308i	−0.906989	+	0.421154i	$0.861625\pi$
$888$	0	0
$889$	583.407	0.656251
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−476.244	−0.533308
$894$	0	0
$895$	−854.119	−0.954323
$896$	0	0
$897$	0	0
$898$	0	0
$899$	350.090	0.389422
$900$	0	0
$901$	676.523i	0.750857i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 199.671i	− 0.220631i
$906$	0	0
$907$	1350.58i	1.48906i	0.667587	+	0.744532i	$0.267327\pi$
$907$	1350.58i	1.48906i	−0.667587	+	0.744532i	$0.732673\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 58.3583i	− 0.0640596i	−0.999487	−	0.0320298i	$-0.989803\pi$
$911$	− 58.3583i	− 0.0640596i	0.999487	−	0.0320298i	$-0.0101971\pi$
$912$	0	0
$913$	46.7366	0.0511901
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−968.343	−1.05599
$918$	0	0
$919$	580.796	0.631987	0.315994	−	0.948761i	$-0.397662\pi$
$919$	580.796	0.631987	0.315994	+	0.948761i	$0.397662\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1346.03	−1.45832
$924$	0	0
$925$	− 159.197i	− 0.172105i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 643.641i	− 0.692832i	−0.938081	−	0.346416i	$-0.887399\pi$
$929$	− 643.641i	− 0.692832i	0.938081	−	0.346416i	$-0.112601\pi$
$930$	0	0
$931$	− 892.899i	− 0.959076i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	48.8713i	0.0522687i
$936$	0	0
$937$	−799.455	−0.853207	−0.426603	−	0.904439i	$-0.640290\pi$
$937$	−799.455	−0.853207	−0.426603	+	0.904439i	$0.640290\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1003.19	−1.06609	−0.533045	−	0.846087i	$-0.678952\pi$
$941$	−1003.19	−1.06609	−0.533045	+	0.846087i	$0.678952\pi$
$942$	0	0
$943$	−234.976	−0.249179
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−421.475	−0.445063	−0.222532	−	0.974925i	$-0.571432\pi$
$947$	−421.475	−0.445063	−0.222532	+	0.974925i	$0.571432\pi$
$948$	0	0
$949$	− 1718.61i	− 1.81097i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1411.08i	− 1.48068i	−0.672234	−	0.740338i	$-0.734665\pi$
$953$	− 1411.08i	− 1.48068i	0.672234	−	0.740338i	$-0.265335\pi$
$954$	0	0
$955$	− 705.368i	− 0.738605i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1399.45i	1.45928i
$960$	0	0
$961$	−445.601	−0.463685
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−191.091	−0.198022
$966$	0	0
$967$	1708.76	1.76707	0.883537	−	0.468361i	$-0.155155\pi$
$967$	1708.76	1.76707	0.883537	+	0.468361i	$0.155155\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−77.9821	−0.0803111	−0.0401556	−	0.999193i	$-0.512785\pi$
$971$	−77.9821	−0.0803111	−0.0401556	+	0.999193i	$0.512785\pi$
$972$	0	0
$973$	2615.91i	2.68850i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	805.683i	0.824649i	0.911037	+	0.412325i	$0.135283\pi$
$977$	805.683i	0.824649i	−0.911037	+	0.412325i	$0.864717\pi$
$978$	0	0
$979$	29.4655i	0.0300975i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 190.223i	− 0.193513i	−0.995308	−	0.0967566i	$-0.969153\pi$
$983$	− 190.223i	− 0.193513i	0.995308	−	0.0967566i	$-0.0308468\pi$
$984$	0	0
$985$	801.631	0.813838
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−138.272	−0.139810
$990$	0	0
$991$	−893.333	−0.901446	−0.450723	−	0.892664i	$-0.648834\pi$
$991$	−893.333	−0.901446	−0.450723	+	0.892664i	$0.648834\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	96.3663	0.0968505
$996$	0	0
$997$	− 174.818i	− 0.175344i	−0.996149	−	0.0876720i	$-0.972057\pi$
$997$	− 174.818i	− 0.175344i	0.996149	−	0.0876720i	$-0.0279427\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.3.h.c.593.1		6
3.2	odd	2		864.3.h.d.593.5		6
4.3	odd	2		216.3.h.c.53.6	yes	6
8.3	odd	2		216.3.h.d.53.2	yes	6
8.5	even	2		864.3.h.d.593.6		6
12.11	even	2		216.3.h.d.53.1	yes	6
24.5	odd	2	inner	864.3.h.c.593.2		6
24.11	even	2		216.3.h.c.53.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.h.c.53.5	✓	6	24.11	even	2
216.3.h.c.53.6	yes	6	4.3	odd	2
216.3.h.d.53.1	yes	6	12.11	even	2
216.3.h.d.53.2	yes	6	8.3	odd	2
864.3.h.c.593.1		6	1.1	even	1	trivial
864.3.h.c.593.2		6	24.5	odd	2	inner
864.3.h.d.593.5		6	3.2	odd	2
864.3.h.d.593.6		6	8.5	even	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	864.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.79748 q^{5} +10.8133 q^{7} +O(q^{10})\)	\(q-3.79748 q^{5} +10.8133 q^{7} -0.420847 q^{11} -21.1820i q^{13} +30.5797i q^{17} -13.1450i q^{19} -22.5427i q^{23} -10.5792 q^{25} +15.4208 q^{29} +22.7024 q^{31} -41.0632 q^{35} +15.0482i q^{37} -10.4236i q^{41} -6.13381i q^{43} -36.2302i q^{47} +67.9271 q^{49} +22.1232 q^{53} +1.59816 q^{55} +101.728 q^{59} -94.9438i q^{61} +80.4382i q^{65} -90.8618i q^{67} -63.5460i q^{71} +81.1356 q^{73} -4.55074 q^{77} +38.0408 q^{79} -111.054 q^{83} -116.126i q^{85} -70.0146i q^{89} -229.047i q^{91} +49.9177i q^{95} +144.646 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{5} + 10 q^{7}+O(q^{10}) \) 6 * q - 2 * q^5 + 10 * q^7	\( 6 q - 2 q^{5} + 10 q^{7} - 10 q^{11} - 56 q^{25} + 100 q^{29} - 6 q^{31} - 110 q^{35} - 24 q^{49} - 2 q^{53} - 170 q^{55} + 20 q^{59} + 130 q^{73} + 50 q^{77} + 76 q^{79} + 38 q^{83} - 70 q^{97}+O(q^{100}) \) 6 * q - 2 * q^5 + 10 * q^7 - 10 * q^11 - 56 * q^25 + 100 * q^29 - 6 * q^31 - 110 * q^35 - 24 * q^49 - 2 * q^53 - 170 * q^55 + 20 * q^59 + 130 * q^73 + 50 * q^77 + 76 * q^79 + 38 * q^83 - 70 * q^97

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