LMFDB - Embedded newform 576.3.g.e.127.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,3,Mod(127,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("576.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	576.g (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:	$15.6948632272$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 12)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	576.127
Dual form			576.3.g.e.127.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{5} -6.92820i q^{7} +O(q^{10})$	$q-2.00000 q^{5} -6.92820i q^{7} +6.92820i q^{11} -2.00000 q^{13} -10.0000 q^{17} -20.7846i q^{19} +27.7128i q^{23} -21.0000 q^{25} -26.0000 q^{29} +6.92820i q^{31} +13.8564i q^{35} -26.0000 q^{37} -58.0000 q^{41} +48.4974i q^{43} -69.2820i q^{47} +1.00000 q^{49} -74.0000 q^{53} -13.8564i q^{55} -90.0666i q^{59} -26.0000 q^{61} +4.00000 q^{65} -6.92820i q^{67} -46.0000 q^{73} +48.0000 q^{77} +117.779i q^{79} +48.4974i q^{83} +20.0000 q^{85} -82.0000 q^{89} +13.8564i q^{91} +41.5692i q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} - 4 q^{13} - 20 q^{17} - 42 q^{25} - 52 q^{29} - 52 q^{37} - 116 q^{41} + 2 q^{49} - 148 q^{53} - 52 q^{61} + 8 q^{65} - 92 q^{73} + 96 q^{77} + 40 q^{85} - 164 q^{89} + 4 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 4 * q^13 - 20 * q^17 - 42 * q^25 - 52 * q^29 - 52 * q^37 - 116 * q^41 + 2 * q^49 - 148 * q^53 - 52 * q^61 + 8 * q^65 - 92 * q^73 + 96 * q^77 + 40 * q^85 - 164 * q^89 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	−2.00000	−0.400000	−0.200000	−	0.979796i	$-0.564094\pi$
$5$	−2.00000	−0.400000	−0.200000	+	0.979796i	$0.564094\pi$
$6$	0	0
$7$	− 6.92820i	− 0.989743i	−0.868966	−	0.494872i	$-0.835215\pi$
$7$	− 6.92820i	− 0.989743i	0.868966	−	0.494872i	$-0.164785\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.92820i	0.629837i	0.949119	+	0.314918i	$0.101977\pi$
$11$	6.92820i	0.629837i	−0.949119	+	0.314918i	$0.898023\pi$
$12$	0	0
$13$	−2.00000	−0.153846	−0.0769231	−	0.997037i	$-0.524510\pi$
$13$	−2.00000	−0.153846	−0.0769231	+	0.997037i	$0.524510\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−10.0000	−0.588235	−0.294118	−	0.955769i	$-0.595026\pi$
$17$	−10.0000	−0.588235	−0.294118	+	0.955769i	$0.595026\pi$
$18$	0	0
$19$	− 20.7846i	− 1.09393i	−0.837157	−	0.546963i	$-0.815784\pi$
$19$	− 20.7846i	− 1.09393i	0.837157	−	0.546963i	$-0.184216\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	27.7128i	1.20490i	0.798155	+	0.602452i	$0.205810\pi$
$23$	27.7128i	1.20490i	−0.798155	+	0.602452i	$0.794190\pi$
$24$	0	0
$25$	−21.0000	−0.840000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−26.0000	−0.896552	−0.448276	−	0.893895i	$-0.647962\pi$
$29$	−26.0000	−0.896552	−0.448276	+	0.893895i	$0.647962\pi$
$30$	0	0
$31$	6.92820i	0.223490i	0.993737	+	0.111745i	$0.0356441\pi$
$31$	6.92820i	0.223490i	−0.993737	+	0.111745i	$0.964356\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	13.8564i	0.395897i
$36$	0	0
$37$	−26.0000	−0.702703	−0.351351	−	0.936244i	$-0.614278\pi$
$37$	−26.0000	−0.702703	−0.351351	+	0.936244i	$0.614278\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−58.0000	−1.41463	−0.707317	−	0.706896i	$-0.750095\pi$
$41$	−58.0000	−1.41463	−0.707317	+	0.706896i	$0.750095\pi$
$42$	0	0
$43$	48.4974i	1.12785i	0.825827	+	0.563924i	$0.190709\pi$
$43$	48.4974i	1.12785i	−0.825827	+	0.563924i	$0.809291\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 69.2820i	− 1.47409i	−0.675846	−	0.737043i	$-0.736222\pi$
$47$	− 69.2820i	− 1.47409i	0.675846	−	0.737043i	$-0.263778\pi$
$48$	0	0
$49$	1.00000	0.0204082
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−74.0000	−1.39623	−0.698113	−	0.715987i	$-0.745977\pi$
$53$	−74.0000	−1.39623	−0.698113	+	0.715987i	$0.745977\pi$
$54$	0	0
$55$	− 13.8564i	− 0.251935i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 90.0666i	− 1.52655i	−0.646072	−	0.763277i	$-0.723589\pi$
$59$	− 90.0666i	− 1.52655i	0.646072	−	0.763277i	$-0.276411\pi$
$60$	0	0
$61$	−26.0000	−0.426230	−0.213115	−	0.977027i	$-0.568361\pi$
$61$	−26.0000	−0.426230	−0.213115	+	0.977027i	$0.568361\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.00000	0.0615385
$66$	0	0
$67$	− 6.92820i	− 0.103406i	−0.998663	−	0.0517030i	$-0.983535\pi$
$67$	− 6.92820i	− 0.103406i	0.998663	−	0.0517030i	$-0.0164649\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−46.0000	−0.630137	−0.315068	−	0.949069i	$-0.602027\pi$
$73$	−46.0000	−0.630137	−0.315068	+	0.949069i	$0.602027\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	48.0000	0.623377
$78$	0	0
$79$	117.779i	1.49088i	0.666573	+	0.745440i	$0.267760\pi$
$79$	117.779i	1.49088i	−0.666573	+	0.745440i	$0.732240\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	48.4974i	0.584306i	0.956372	+	0.292153i	$0.0943717\pi$
$83$	48.4974i	0.584306i	−0.956372	+	0.292153i	$0.905628\pi$
$84$	0	0
$85$	20.0000	0.235294
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−82.0000	−0.921348	−0.460674	−	0.887569i	$-0.652392\pi$
$89$	−82.0000	−0.921348	−0.460674	+	0.887569i	$0.652392\pi$
$90$	0	0
$91$	13.8564i	0.152268i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	41.5692i	0.437571i
$96$	0	0
$97$	2.00000	0.0206186	0.0103093	−	0.999947i	$-0.496718\pi$
$97$	2.00000	0.0206186	0.0103093	+	0.999947i	$0.496718\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−74.0000	−0.732673	−0.366337	−	0.930482i	$-0.619388\pi$
$101$	−74.0000	−0.732673	−0.366337	+	0.930482i	$0.619388\pi$
$102$	0	0
$103$	− 76.2102i	− 0.739905i	−0.929051	−	0.369953i	$-0.879374\pi$
$103$	− 76.2102i	− 0.739905i	0.929051	−	0.369953i	$-0.120626\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.7846i	0.194249i	0.995272	+	0.0971243i	$0.0309645\pi$
$107$	20.7846i	0.194249i	−0.995272	+	0.0971243i	$0.969036\pi$
$108$	0	0
$109$	46.0000	0.422018	0.211009	−	0.977484i	$-0.432325\pi$
$109$	46.0000	0.422018	0.211009	+	0.977484i	$0.432325\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	110.000	0.973451	0.486726	−	0.873555i	$-0.338191\pi$
$113$	110.000	0.973451	0.486726	+	0.873555i	$0.338191\pi$
$114$	0	0
$115$	− 55.4256i	− 0.481962i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	69.2820i	0.582202i
$120$	0	0
$121$	73.0000	0.603306
$122$	0	0
$123$	0	0
$124$	0	0
$125$	92.0000	0.736000
$126$	0	0
$127$	− 145.492i	− 1.14561i	−0.819692	−	0.572804i	$-0.805856\pi$
$127$	− 145.492i	− 1.14561i	0.819692	−	0.572804i	$-0.194144\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	117.779i	0.899080i	0.893260	+	0.449540i	$0.148412\pi$
$131$	117.779i	0.899080i	−0.893260	+	0.449540i	$0.851588\pi$
$132$	0	0
$133$	−144.000	−1.08271
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.0000	−0.0729927	−0.0364964	−	0.999334i	$-0.511620\pi$
$137$	−10.0000	−0.0729927	−0.0364964	+	0.999334i	$0.511620\pi$
$138$	0	0
$139$	− 48.4974i	− 0.348902i	−0.984666	−	0.174451i	$-0.944185\pi$
$139$	− 48.4974i	− 0.348902i	0.984666	−	0.174451i	$-0.0558151\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 13.8564i	− 0.0968979i
$144$	0	0
$145$	52.0000	0.358621
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.00000	−0.0134228	−0.00671141	−	0.999977i	$-0.502136\pi$
$149$	−2.00000	−0.0134228	−0.00671141	+	0.999977i	$0.502136\pi$
$150$	0	0
$151$	− 90.0666i	− 0.596468i	−0.954493	−	0.298234i	$-0.903602\pi$
$151$	− 90.0666i	− 0.596468i	0.954493	−	0.298234i	$-0.0963975\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 13.8564i	− 0.0893962i
$156$	0	0
$157$	214.000	1.36306	0.681529	−	0.731791i	$-0.261315\pi$
$157$	214.000	1.36306	0.681529	+	0.731791i	$0.261315\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	192.000	1.19255
$162$	0	0
$163$	− 20.7846i	− 0.127513i	−0.997965	−	0.0637565i	$-0.979692\pi$
$163$	− 20.7846i	− 0.127513i	0.997965	−	0.0637565i	$-0.0203081\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 96.9948i	− 0.580807i	−0.956904	−	0.290404i	$-0.906210\pi$
$167$	− 96.9948i	− 0.580807i	0.956904	−	0.290404i	$-0.0937896\pi$
$168$	0	0
$169$	−165.000	−0.976331
$170$	0	0
$171$	0	0
$172$	0	0
$173$	334.000	1.93064	0.965318	−	0.261077i	$-0.0840777\pi$
$173$	334.000	1.93064	0.965318	+	0.261077i	$0.0840777\pi$
$174$	0	0
$175$	145.492i	0.831384i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 187.061i	− 1.04504i	−0.852628	−	0.522518i	$-0.824993\pi$
$179$	− 187.061i	− 1.04504i	0.852628	−	0.522518i	$-0.175007\pi$
$180$	0	0
$181$	−2.00000	−0.0110497	−0.00552486	−	0.999985i	$-0.501759\pi$
$181$	−2.00000	−0.0110497	−0.00552486	+	0.999985i	$0.501759\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	52.0000	0.281081
$186$	0	0
$187$	− 69.2820i	− 0.370492i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	221.703i	1.16075i	0.814351	+	0.580373i	$0.197093\pi$
$191$	221.703i	1.16075i	−0.814351	+	0.580373i	$0.802907\pi$
$192$	0	0
$193$	290.000	1.50259	0.751295	−	0.659966i	$-0.229429\pi$
$193$	290.000	1.50259	0.751295	+	0.659966i	$0.229429\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.0000	−0.131980	−0.0659898	−	0.997820i	$-0.521020\pi$
$197$	−26.0000	−0.131980	−0.0659898	+	0.997820i	$0.521020\pi$
$198$	0	0
$199$	394.908i	1.98446i	0.124416	+	0.992230i	$0.460294\pi$
$199$	394.908i	1.98446i	−0.124416	+	0.992230i	$0.539706\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	180.133i	0.887356i
$204$	0	0
$205$	116.000	0.565854
$206$	0	0
$207$	0	0
$208$	0	0
$209$	144.000	0.688995
$210$	0	0
$211$	242.487i	1.14923i	0.818425	+	0.574614i	$0.194848\pi$
$211$	242.487i	1.14923i	−0.818425	+	0.574614i	$0.805152\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 96.9948i	− 0.451139i
$216$	0	0
$217$	48.0000	0.221198
$218$	0	0
$219$	0	0
$220$	0	0
$221$	20.0000	0.0904977
$222$	0	0
$223$	− 339.482i	− 1.52234i	−0.648552	−	0.761170i	$-0.724625\pi$
$223$	− 339.482i	− 1.52234i	0.648552	−	0.761170i	$-0.275375\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 284.056i	− 1.25135i	−0.780084	−	0.625675i	$-0.784824\pi$
$227$	− 284.056i	− 1.25135i	0.780084	−	0.625675i	$-0.215176\pi$
$228$	0	0
$229$	142.000	0.620087	0.310044	−	0.950722i	$-0.399656\pi$
$229$	142.000	0.620087	0.310044	+	0.950722i	$0.399656\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−82.0000	−0.351931	−0.175966	−	0.984396i	$-0.556305\pi$
$233$	−82.0000	−0.351931	−0.175966	+	0.984396i	$0.556305\pi$
$234$	0	0
$235$	138.564i	0.589634i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 387.979i	− 1.62334i	−0.584113	−	0.811672i	$-0.698558\pi$
$239$	− 387.979i	− 1.62334i	0.584113	−	0.811672i	$-0.301442\pi$
$240$	0	0
$241$	−46.0000	−0.190871	−0.0954357	−	0.995436i	$-0.530424\pi$
$241$	−46.0000	−0.190871	−0.0954357	+	0.995436i	$0.530424\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.00000	−0.00816327
$246$	0	0
$247$	41.5692i	0.168296i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	145.492i	0.579650i	0.957080	+	0.289825i	$0.0935972\pi$
$251$	145.492i	0.579650i	−0.957080	+	0.289825i	$0.906403\pi$
$252$	0	0
$253$	−192.000	−0.758893
$254$	0	0
$255$	0	0
$256$	0	0
$257$	254.000	0.988327	0.494163	−	0.869369i	$-0.335474\pi$
$257$	254.000	0.988327	0.494163	+	0.869369i	$0.335474\pi$
$258$	0	0
$259$	180.133i	0.695495i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 152.420i	− 0.579546i	−0.957095	−	0.289773i	$-0.906420\pi$
$263$	− 152.420i	− 0.579546i	0.957095	−	0.289773i	$-0.0935797\pi$
$264$	0	0
$265$	148.000	0.558491
$266$	0	0
$267$	0	0
$268$	0	0
$269$	262.000	0.973978	0.486989	−	0.873408i	$-0.338095\pi$
$269$	262.000	0.973978	0.486989	+	0.873408i	$0.338095\pi$
$270$	0	0
$271$	20.7846i	0.0766960i	0.999264	+	0.0383480i	$0.0122095\pi$
$271$	20.7846i	0.0766960i	−0.999264	+	0.0383480i	$0.987790\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 145.492i	− 0.529063i
$276$	0	0
$277$	−290.000	−1.04693	−0.523466	−	0.852047i	$-0.675361\pi$
$277$	−290.000	−1.04693	−0.523466	+	0.852047i	$0.675361\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−226.000	−0.804270	−0.402135	−	0.915580i	$-0.631732\pi$
$281$	−226.000	−0.804270	−0.402135	+	0.915580i	$0.631732\pi$
$282$	0	0
$283$	− 297.913i	− 1.05270i	−0.850269	−	0.526348i	$-0.823561\pi$
$283$	− 297.913i	− 1.05270i	0.850269	−	0.526348i	$-0.176439\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	401.836i	1.40012i
$288$	0	0
$289$	−189.000	−0.653979
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−362.000	−1.23549	−0.617747	−	0.786377i	$-0.711955\pi$
$293$	−362.000	−1.23549	−0.617747	+	0.786377i	$0.711955\pi$
$294$	0	0
$295$	180.133i	0.610621i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 55.4256i	− 0.185370i
$300$	0	0
$301$	336.000	1.11628
$302$	0	0
$303$	0	0
$304$	0	0
$305$	52.0000	0.170492
$306$	0	0
$307$	− 145.492i	− 0.473916i	−0.971520	−	0.236958i	$-0.923850\pi$
$307$	− 145.492i	− 0.473916i	0.971520	−	0.236958i	$-0.0761504\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	235.559i	0.757424i	0.925515	+	0.378712i	$0.123633\pi$
$311$	235.559i	0.757424i	−0.925515	+	0.378712i	$0.876367\pi$
$312$	0	0
$313$	−478.000	−1.52716	−0.763578	−	0.645715i	$-0.776559\pi$
$313$	−478.000	−1.52716	−0.763578	+	0.645715i	$0.776559\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−170.000	−0.536278	−0.268139	−	0.963380i	$-0.586409\pi$
$317$	−170.000	−0.536278	−0.268139	+	0.963380i	$0.586409\pi$
$318$	0	0
$319$	− 180.133i	− 0.564681i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	207.846i	0.643486i
$324$	0	0
$325$	42.0000	0.129231
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−480.000	−1.45897
$330$	0	0
$331$	− 408.764i	− 1.23494i	−0.786596	−	0.617468i	$-0.788158\pi$
$331$	− 408.764i	− 1.23494i	0.786596	−	0.617468i	$-0.211842\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	13.8564i	0.0413624i
$336$	0	0
$337$	338.000	1.00297	0.501484	−	0.865167i	$-0.332788\pi$
$337$	338.000	1.00297	0.501484	+	0.865167i	$0.332788\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−48.0000	−0.140762
$342$	0	0
$343$	− 346.410i	− 1.00994i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	200.918i	0.579014i	0.957176	+	0.289507i	$0.0934914\pi$
$347$	200.918i	0.579014i	−0.957176	+	0.289507i	$0.906509\pi$
$348$	0	0
$349$	−506.000	−1.44986	−0.724928	−	0.688824i	$-0.758127\pi$
$349$	−506.000	−1.44986	−0.724928	+	0.688824i	$0.758127\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−178.000	−0.504249	−0.252125	−	0.967695i	$-0.581129\pi$
$353$	−178.000	−0.504249	−0.252125	+	0.967695i	$0.581129\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 166.277i	− 0.463167i	−0.972815	−	0.231583i	$-0.925609\pi$
$359$	− 166.277i	− 0.463167i	0.972815	−	0.231583i	$-0.0743906\pi$
$360$	0	0
$361$	−71.0000	−0.196676
$362$	0	0
$363$	0	0
$364$	0	0
$365$	92.0000	0.252055
$366$	0	0
$367$	200.918i	0.547460i	0.961807	+	0.273730i	$0.0882575\pi$
$367$	200.918i	0.547460i	−0.961807	+	0.273730i	$0.911742\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	512.687i	1.38191i
$372$	0	0
$373$	310.000	0.831099	0.415550	−	0.909571i	$-0.363589\pi$
$373$	310.000	0.831099	0.415550	+	0.909571i	$0.363589\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	52.0000	0.137931
$378$	0	0
$379$	436.477i	1.15165i	0.817572	+	0.575827i	$0.195320\pi$
$379$	436.477i	1.15165i	−0.817572	+	0.575827i	$0.804680\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	609.682i	1.59186i	0.605390	+	0.795929i	$0.293017\pi$
$383$	609.682i	1.59186i	−0.605390	+	0.795929i	$0.706983\pi$
$384$	0	0
$385$	−96.0000	−0.249351
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−578.000	−1.48586	−0.742931	−	0.669368i	$-0.766565\pi$
$389$	−578.000	−1.48586	−0.742931	+	0.669368i	$0.766565\pi$
$390$	0	0
$391$	− 277.128i	− 0.708768i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 235.559i	− 0.596352i
$396$	0	0
$397$	−26.0000	−0.0654912	−0.0327456	−	0.999464i	$-0.510425\pi$
$397$	−26.0000	−0.0654912	−0.0327456	+	0.999464i	$0.510425\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−250.000	−0.623441	−0.311721	−	0.950174i	$-0.600905\pi$
$401$	−250.000	−0.623441	−0.311721	+	0.950174i	$0.600905\pi$
$402$	0	0
$403$	− 13.8564i	− 0.0343831i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 180.133i	− 0.442588i
$408$	0	0
$409$	290.000	0.709046	0.354523	−	0.935047i	$-0.384643\pi$
$409$	290.000	0.709046	0.354523	+	0.935047i	$0.384643\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−624.000	−1.51090
$414$	0	0
$415$	− 96.9948i	− 0.233723i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 339.482i	− 0.810219i	−0.914268	−	0.405110i	$-0.867233\pi$
$419$	− 339.482i	− 0.810219i	0.914268	−	0.405110i	$-0.132767\pi$
$420$	0	0
$421$	−674.000	−1.60095	−0.800475	−	0.599366i	$-0.795419\pi$
$421$	−674.000	−1.60095	−0.800475	+	0.599366i	$0.795419\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	210.000	0.494118
$426$	0	0
$427$	180.133i	0.421858i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 540.400i	− 1.25383i	−0.779088	−	0.626914i	$-0.784318\pi$
$431$	− 540.400i	− 1.25383i	0.779088	−	0.626914i	$-0.215682\pi$
$432$	0	0
$433$	−334.000	−0.771363	−0.385681	−	0.922632i	$-0.626034\pi$
$433$	−334.000	−0.771363	−0.385681	+	0.922632i	$0.626034\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	576.000	1.31808
$438$	0	0
$439$	117.779i	0.268290i	0.990962	+	0.134145i	$0.0428288\pi$
$439$	117.779i	0.268290i	−0.990962	+	0.134145i	$0.957171\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 76.2102i	− 0.172032i	−0.996294	−	0.0860161i	$-0.972586\pi$
$443$	− 76.2102i	− 0.172032i	0.996294	−	0.0860161i	$-0.0274136\pi$
$444$	0	0
$445$	164.000	0.368539
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−394.000	−0.877506	−0.438753	−	0.898608i	$-0.644580\pi$
$449$	−394.000	−0.877506	−0.438753	+	0.898608i	$0.644580\pi$
$450$	0	0
$451$	− 401.836i	− 0.890988i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 27.7128i	− 0.0609073i
$456$	0	0
$457$	−478.000	−1.04595	−0.522976	−	0.852347i	$-0.675178\pi$
$457$	−478.000	−1.04595	−0.522976	+	0.852347i	$0.675178\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	142.000	0.308026	0.154013	−	0.988069i	$-0.450780\pi$
$461$	142.000	0.308026	0.154013	+	0.988069i	$0.450780\pi$
$462$	0	0
$463$	630.466i	1.36170i	0.732423	+	0.680849i	$0.238389\pi$
$463$	630.466i	1.36170i	−0.732423	+	0.680849i	$0.761611\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.7846i	0.0445067i	0.999752	+	0.0222533i	$0.00708404\pi$
$467$	20.7846i	0.0445067i	−0.999752	+	0.0222533i	$0.992916\pi$
$468$	0	0
$469$	−48.0000	−0.102345
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−336.000	−0.710359
$474$	0	0
$475$	436.477i	0.918899i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 734.390i	− 1.53317i	−0.642141	−	0.766586i	$-0.721954\pi$
$479$	− 734.390i	− 1.53317i	0.642141	−	0.766586i	$-0.278046\pi$
$480$	0	0
$481$	52.0000	0.108108
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.00000	−0.00824742
$486$	0	0
$487$	103.923i	0.213394i	0.994292	+	0.106697i	$0.0340275\pi$
$487$	103.923i	0.213394i	−0.994292	+	0.106697i	$0.965972\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 921.451i	− 1.87668i	−0.345711	−	0.938341i	$-0.612362\pi$
$491$	− 921.451i	− 1.87668i	0.345711	−	0.938341i	$-0.387638\pi$
$492$	0	0
$493$	260.000	0.527383
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	76.2102i	0.152726i	0.997080	+	0.0763630i	$0.0243308\pi$
$499$	76.2102i	0.152726i	−0.997080	+	0.0763630i	$0.975669\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	581.969i	1.15700i	0.815684	+	0.578498i	$0.196361\pi$
$503$	581.969i	1.15700i	−0.815684	+	0.578498i	$0.803639\pi$
$504$	0	0
$505$	148.000	0.293069
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−842.000	−1.65422	−0.827112	−	0.562037i	$-0.810018\pi$
$509$	−842.000	−1.65422	−0.827112	+	0.562037i	$0.810018\pi$
$510$	0	0
$511$	318.697i	0.623674i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	152.420i	0.295962i
$516$	0	0
$517$	480.000	0.928433
$518$	0	0
$519$	0	0
$520$	0	0
$521$	326.000	0.625720	0.312860	−	0.949799i	$-0.398713\pi$
$521$	326.000	0.625720	0.312860	+	0.949799i	$0.398713\pi$
$522$	0	0
$523$	− 311.769i	− 0.596117i	−0.954548	−	0.298058i	$-0.903661\pi$
$523$	− 311.769i	− 0.596117i	0.954548	−	0.298058i	$-0.0963390\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 69.2820i	− 0.131465i
$528$	0	0
$529$	−239.000	−0.451796
$530$	0	0
$531$	0	0
$532$	0	0
$533$	116.000	0.217636
$534$	0	0
$535$	− 41.5692i	− 0.0776995i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.92820i	0.0128538i
$540$	0	0
$541$	−530.000	−0.979667	−0.489834	−	0.871816i	$-0.662942\pi$
$541$	−530.000	−0.979667	−0.489834	+	0.871816i	$0.662942\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−92.0000	−0.168807
$546$	0	0
$547$	339.482i	0.620625i	0.950635	+	0.310313i	$0.100434\pi$
$547$	339.482i	0.620625i	−0.950635	+	0.310313i	$0.899566\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	540.400i	0.980762i
$552$	0	0
$553$	816.000	1.47559
$554$	0	0
$555$	0	0
$556$	0	0
$557$	766.000	1.37522	0.687612	−	0.726078i	$-0.258659\pi$
$557$	766.000	1.37522	0.687612	+	0.726078i	$0.258659\pi$
$558$	0	0
$559$	− 96.9948i	− 0.173515i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	491.902i	0.873717i	0.899530	+	0.436858i	$0.143909\pi$
$563$	491.902i	0.873717i	−0.899530	+	0.436858i	$0.856091\pi$
$564$	0	0
$565$	−220.000	−0.389381
$566$	0	0
$567$	0	0
$568$	0	0
$569$	422.000	0.741652	0.370826	−	0.928702i	$-0.379075\pi$
$569$	422.000	0.741652	0.370826	+	0.928702i	$0.379075\pi$
$570$	0	0
$571$	284.056i	0.497472i	0.968571	+	0.248736i	$0.0800151\pi$
$571$	284.056i	0.497472i	−0.968571	+	0.248736i	$0.919985\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 581.969i	− 1.01212i
$576$	0	0
$577$	−46.0000	−0.0797227	−0.0398614	−	0.999205i	$-0.512692\pi$
$577$	−46.0000	−0.0797227	−0.0398614	+	0.999205i	$0.512692\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	336.000	0.578313
$582$	0	0
$583$	− 512.687i	− 0.879395i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	630.466i	1.07405i	0.843567	+	0.537024i	$0.180452\pi$
$587$	630.466i	1.07405i	−0.843567	+	0.537024i	$0.819548\pi$
$588$	0	0
$589$	144.000	0.244482
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−82.0000	−0.138280	−0.0691400	−	0.997607i	$-0.522026\pi$
$593$	−82.0000	−0.138280	−0.0691400	+	0.997607i	$0.522026\pi$
$594$	0	0
$595$	− 138.564i	− 0.232881i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 55.4256i	− 0.0925303i	−0.998929	−	0.0462651i	$-0.985268\pi$
$599$	− 55.4256i	− 0.0925303i	0.998929	−	0.0462651i	$-0.0147319\pi$
$600$	0	0
$601$	−334.000	−0.555740	−0.277870	−	0.960619i	$-0.589629\pi$
$601$	−334.000	−0.555740	−0.277870	+	0.960619i	$0.589629\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−146.000	−0.241322
$606$	0	0
$607$	− 367.195i	− 0.604934i	−0.953160	−	0.302467i	$-0.902190\pi$
$607$	− 367.195i	− 0.604934i	0.953160	−	0.302467i	$-0.0978102\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	138.564i	0.226782i
$612$	0	0
$613$	214.000	0.349103	0.174551	−	0.984648i	$-0.444152\pi$
$613$	214.000	0.349103	0.174551	+	0.984648i	$0.444152\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1118.00	1.81199	0.905997	−	0.423285i	$-0.139123\pi$
$617$	1118.00	1.81199	0.905997	+	0.423285i	$0.139123\pi$
$618$	0	0
$619$	672.036i	1.08568i	0.839836	+	0.542840i	$0.182651\pi$
$619$	672.036i	1.08568i	−0.839836	+	0.542840i	$0.817349\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	568.113i	0.911898i
$624$	0	0
$625$	341.000	0.545600
$626$	0	0
$627$	0	0
$628$	0	0
$629$	260.000	0.413355
$630$	0	0
$631$	− 145.492i	− 0.230574i	−0.993332	−	0.115287i	$-0.963221\pi$
$631$	− 145.492i	− 0.230574i	0.993332	−	0.115287i	$-0.0367788\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	290.985i	0.458243i
$636$	0	0
$637$	−2.00000	−0.00313972
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10.0000	−0.0156006	−0.00780031	−	0.999970i	$-0.502483\pi$
$641$	−10.0000	−0.0156006	−0.00780031	+	0.999970i	$0.502483\pi$
$642$	0	0
$643$	− 1212.44i	− 1.88559i	−0.333370	−	0.942796i	$-0.608186\pi$
$643$	− 1212.44i	− 1.88559i	0.333370	−	0.942796i	$-0.391814\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	332.554i	0.513993i	0.966412	+	0.256997i	$0.0827330\pi$
$647$	332.554i	0.513993i	−0.966412	+	0.256997i	$0.917267\pi$
$648$	0	0
$649$	624.000	0.961479
$650$	0	0
$651$	0	0
$652$	0	0
$653$	670.000	1.02603	0.513017	−	0.858379i	$-0.328528\pi$
$653$	670.000	1.02603	0.513017	+	0.858379i	$0.328528\pi$
$654$	0	0
$655$	− 235.559i	− 0.359632i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 824.456i	− 1.25107i	−0.780195	−	0.625536i	$-0.784880\pi$
$659$	− 824.456i	− 1.25107i	0.780195	−	0.625536i	$-0.215120\pi$
$660$	0	0
$661$	1222.00	1.84871	0.924357	−	0.381529i	$-0.124602\pi$
$661$	1222.00	1.84871	0.924357	+	0.381529i	$0.124602\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	288.000	0.433083
$666$	0	0
$667$	− 720.533i	− 1.08026i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 180.133i	− 0.268455i
$672$	0	0
$673$	−334.000	−0.496285	−0.248143	−	0.968724i	$-0.579820\pi$
$673$	−334.000	−0.496285	−0.248143	+	0.968724i	$0.579820\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1006.00	1.48597	0.742984	−	0.669309i	$-0.233410\pi$
$677$	1006.00	1.48597	0.742984	+	0.669309i	$0.233410\pi$
$678$	0	0
$679$	− 13.8564i	− 0.0204071i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 187.061i	− 0.273882i	−0.990579	−	0.136941i	$-0.956273\pi$
$683$	− 187.061i	− 0.273882i	0.990579	−	0.136941i	$-0.0437271\pi$
$684$	0	0
$685$	20.0000	0.0291971
$686$	0	0
$687$	0	0
$688$	0	0
$689$	148.000	0.214804
$690$	0	0
$691$	− 990.733i	− 1.43377i	−0.697193	−	0.716884i	$-0.745568\pi$
$691$	− 990.733i	− 1.43377i	0.697193	−	0.716884i	$-0.254432\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	96.9948i	0.139561i
$696$	0	0
$697$	580.000	0.832138
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1034.00	−1.47504	−0.737518	−	0.675328i	$-0.764002\pi$
$701$	−1034.00	−1.47504	−0.737518	+	0.675328i	$0.764002\pi$
$702$	0	0
$703$	540.400i	0.768705i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	512.687i	0.725158i
$708$	0	0
$709$	−530.000	−0.747532	−0.373766	−	0.927523i	$-0.621934\pi$
$709$	−530.000	−0.747532	−0.373766	+	0.927523i	$0.621934\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−192.000	−0.269285
$714$	0	0
$715$	27.7128i	0.0387592i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	706.677i	0.982861i	0.870917	+	0.491430i	$0.163526\pi$
$719$	706.677i	0.982861i	−0.870917	+	0.491430i	$0.836474\pi$
$720$	0	0
$721$	−528.000	−0.732316
$722$	0	0
$723$	0	0
$724$	0	0
$725$	546.000	0.753103
$726$	0	0
$727$	242.487i	0.333545i	0.985995	+	0.166772i	$0.0533345\pi$
$727$	242.487i	0.333545i	−0.985995	+	0.166772i	$0.946665\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 484.974i	− 0.663439i
$732$	0	0
$733$	−194.000	−0.264666	−0.132333	−	0.991205i	$-0.542247\pi$
$733$	−194.000	−0.264666	−0.132333	+	0.991205i	$0.542247\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	48.0000	0.0651289
$738$	0	0
$739$	1351.00i	1.82815i	0.405550	+	0.914073i	$0.367080\pi$
$739$	1351.00i	1.82815i	−0.405550	+	0.914073i	$0.632920\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 678.964i	− 0.913814i	−0.889514	−	0.456907i	$-0.848957\pi$
$743$	− 678.964i	− 0.913814i	0.889514	−	0.456907i	$-0.151043\pi$
$744$	0	0
$745$	4.00000	0.00536913
$746$	0	0
$747$	0	0
$748$	0	0
$749$	144.000	0.192256
$750$	0	0
$751$	− 658.179i	− 0.876404i	−0.898877	−	0.438202i	$-0.855615\pi$
$751$	− 658.179i	− 0.876404i	0.898877	−	0.438202i	$-0.144385\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	180.133i	0.238587i
$756$	0	0
$757$	1006.00	1.32893	0.664465	−	0.747319i	$-0.268659\pi$
$757$	1006.00	1.32893	0.664465	+	0.747319i	$0.268659\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	758.000	0.996058	0.498029	−	0.867160i	$-0.334057\pi$
$761$	758.000	0.996058	0.498029	+	0.867160i	$0.334057\pi$
$762$	0	0
$763$	− 318.697i	− 0.417690i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	180.133i	0.234854i
$768$	0	0
$769$	2.00000	0.00260078	0.00130039	−	0.999999i	$-0.499586\pi$
$769$	2.00000	0.00260078	0.00130039	+	0.999999i	$0.499586\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	262.000	0.338939	0.169470	−	0.985535i	$-0.445795\pi$
$773$	262.000	0.338939	0.169470	+	0.985535i	$0.445795\pi$
$774$	0	0
$775$	− 145.492i	− 0.187732i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1205.51i	1.54751i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−428.000	−0.545223
$786$	0	0
$787$	1447.99i	1.83989i	0.392046	+	0.919946i	$0.371767\pi$
$787$	1447.99i	1.83989i	−0.392046	+	0.919946i	$0.628233\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 762.102i	− 0.963467i
$792$	0	0
$793$	52.0000	0.0655738
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−866.000	−1.08657	−0.543287	−	0.839547i	$-0.682821\pi$
$797$	−866.000	−1.08657	−0.543287	+	0.839547i	$0.682821\pi$
$798$	0	0
$799$	692.820i	0.867109i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 318.697i	− 0.396883i
$804$	0	0
$805$	−384.000	−0.477019
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−10.0000	−0.0123609	−0.00618047	−	0.999981i	$-0.501967\pi$
$809$	−10.0000	−0.0123609	−0.00618047	+	0.999981i	$0.501967\pi$
$810$	0	0
$811$	436.477i	0.538196i	0.963113	+	0.269098i	$0.0867255\pi$
$811$	436.477i	0.538196i	−0.963113	+	0.269098i	$0.913274\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	41.5692i	0.0510052i
$816$	0	0
$817$	1008.00	1.23378
$818$	0	0
$819$	0	0
$820$	0	0
$821$	838.000	1.02071	0.510353	−	0.859965i	$-0.329515\pi$
$821$	838.000	1.02071	0.510353	+	0.859965i	$0.329515\pi$
$822$	0	0
$823$	879.882i	1.06912i	0.845132	+	0.534558i	$0.179522\pi$
$823$	879.882i	1.06912i	−0.845132	+	0.534558i	$0.820478\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 727.461i	− 0.879639i	−0.898086	−	0.439819i	$-0.855042\pi$
$827$	− 727.461i	− 0.879639i	0.898086	−	0.439819i	$-0.144958\pi$
$828$	0	0
$829$	−1298.00	−1.56574	−0.782871	−	0.622184i	$-0.786246\pi$
$829$	−1298.00	−1.56574	−0.782871	+	0.622184i	$0.786246\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−10.0000	−0.0120048
$834$	0	0
$835$	193.990i	0.232323i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 193.990i	− 0.231215i	−0.993295	−	0.115608i	$-0.963118\pi$
$839$	− 193.990i	− 0.231215i	0.993295	−	0.115608i	$-0.0368815\pi$
$840$	0	0
$841$	−165.000	−0.196195
$842$	0	0
$843$	0	0
$844$	0	0
$845$	330.000	0.390533
$846$	0	0
$847$	− 505.759i	− 0.597118i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 720.533i	− 0.846690i
$852$	0	0
$853$	−506.000	−0.593200	−0.296600	−	0.955002i	$-0.595853\pi$
$853$	−506.000	−0.593200	−0.296600	+	0.955002i	$0.595853\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	998.000	1.16453	0.582264	−	0.813000i	$-0.302167\pi$
$857$	998.000	1.16453	0.582264	+	0.813000i	$0.302167\pi$
$858$	0	0
$859$	− 505.759i	− 0.588776i	−0.955686	−	0.294388i	$-0.904884\pi$
$859$	− 505.759i	− 0.588776i	0.955686	−	0.294388i	$-0.0951158\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 166.277i	− 0.192673i	−0.995349	−	0.0963365i	$-0.969287\pi$
$863$	− 166.277i	− 0.192673i	0.995349	−	0.0963365i	$-0.0307125\pi$
$864$	0	0
$865$	−668.000	−0.772254
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−816.000	−0.939010
$870$	0	0
$871$	13.8564i	0.0159086i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 637.395i	− 0.728451i
$876$	0	0
$877$	646.000	0.736602	0.368301	−	0.929707i	$-0.379940\pi$
$877$	646.000	0.736602	0.368301	+	0.929707i	$0.379940\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−898.000	−1.01930	−0.509648	−	0.860383i	$-0.670224\pi$
$881$	−898.000	−1.01930	−0.509648	+	0.860383i	$0.670224\pi$
$882$	0	0
$883$	727.461i	0.823852i	0.911217	+	0.411926i	$0.135144\pi$
$883$	727.461i	0.823852i	−0.911217	+	0.411926i	$0.864856\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 845.241i	− 0.952921i	−0.879196	−	0.476460i	$-0.841920\pi$
$887$	− 845.241i	− 0.952921i	0.879196	−	0.476460i	$-0.158080\pi$
$888$	0	0
$889$	−1008.00	−1.13386
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1440.00	−1.61254
$894$	0	0
$895$	374.123i	0.418014i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 180.133i	− 0.200371i
$900$	0	0
$901$	740.000	0.821310
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.00000	0.00441989
$906$	0	0
$907$	− 1364.86i	− 1.50480i	−0.658705	−	0.752401i	$-0.728895\pi$
$907$	− 1364.86i	− 1.50480i	0.658705	−	0.752401i	$-0.271105\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	387.979i	0.425883i	0.977065	+	0.212941i	$0.0683044\pi$
$911$	387.979i	0.425883i	−0.977065	+	0.212941i	$0.931696\pi$
$912$	0	0
$913$	−336.000	−0.368018
$914$	0	0
$915$	0	0
$916$	0	0
$917$	816.000	0.889858
$918$	0	0
$919$	602.754i	0.655880i	0.944699	+	0.327940i	$0.106354\pi$
$919$	602.754i	0.655880i	−0.944699	+	0.327940i	$0.893646\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	546.000	0.590270
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1594.00	−1.71582	−0.857912	−	0.513797i	$-0.828238\pi$
$929$	−1594.00	−1.71582	−0.857912	+	0.513797i	$0.828238\pi$
$930$	0	0
$931$	− 20.7846i	− 0.0223250i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	138.564i	0.148197i
$936$	0	0
$937$	674.000	0.719317	0.359658	−	0.933084i	$-0.382893\pi$
$937$	674.000	0.719317	0.359658	+	0.933084i	$0.382893\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	430.000	0.456961	0.228480	−	0.973549i	$-0.426624\pi$
$941$	430.000	0.456961	0.228480	+	0.973549i	$0.426624\pi$
$942$	0	0
$943$	− 1607.34i	− 1.70450i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 76.2102i	− 0.0804754i	−0.999190	−	0.0402377i	$-0.987188\pi$
$947$	− 76.2102i	− 0.0804754i	0.999190	−	0.0402377i	$-0.0128115\pi$
$948$	0	0
$949$	92.0000	0.0969442
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−730.000	−0.766002	−0.383001	−	0.923748i	$-0.625109\pi$
$953$	−730.000	−0.766002	−0.383001	+	0.923748i	$0.625109\pi$
$954$	0	0
$955$	− 443.405i	− 0.464298i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	69.2820i	0.0722440i
$960$	0	0
$961$	913.000	0.950052
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−580.000	−0.601036
$966$	0	0
$967$	921.451i	0.952897i	0.879202	+	0.476448i	$0.158076\pi$
$967$	921.451i	0.952897i	−0.879202	+	0.476448i	$0.841924\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1475.71i	1.51978i	0.650051	+	0.759890i	$0.274747\pi$
$971$	1475.71i	1.51978i	−0.650051	+	0.759890i	$0.725253\pi$
$972$	0	0
$973$	−336.000	−0.345324
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−346.000	−0.354145	−0.177073	−	0.984198i	$-0.556663\pi$
$977$	−346.000	−0.354145	−0.177073	+	0.984198i	$0.556663\pi$
$978$	0	0
$979$	− 568.113i	− 0.580299i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 734.390i	− 0.747090i	−0.927612	−	0.373545i	$-0.878142\pi$
$983$	− 734.390i	− 0.747090i	0.927612	−	0.373545i	$-0.121858\pi$
$984$	0	0
$985$	52.0000	0.0527919
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1344.00	−1.35895
$990$	0	0
$991$	976.877i	0.985748i	0.870101	+	0.492874i	$0.164054\pi$
$991$	976.877i	0.985748i	−0.870101	+	0.492874i	$0.835946\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 789.815i	− 0.793784i
$996$	0	0
$997$	−458.000	−0.459378	−0.229689	−	0.973264i	$-0.573771\pi$
$997$	−458.000	−0.459378	−0.229689	+	0.973264i	$0.573771\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.3.g.e.127.1		2
3.2	odd	2		192.3.g.b.127.1		2
4.3	odd	2	inner	576.3.g.e.127.2		2
8.3	odd	2		36.3.d.c.19.1		2
8.5	even	2		36.3.d.c.19.2		2
12.11	even	2		192.3.g.b.127.2		2
16.3	odd	4		2304.3.b.l.127.3		4
16.5	even	4		2304.3.b.l.127.2		4
16.11	odd	4		2304.3.b.l.127.1		4
16.13	even	4		2304.3.b.l.127.4		4
24.5	odd	2		12.3.d.a.7.1	✓	2
24.11	even	2		12.3.d.a.7.2	yes	2
40.3	even	4		900.3.f.c.199.1		4
40.13	odd	4		900.3.f.c.199.3		4
40.19	odd	2		900.3.c.e.451.2		2
40.27	even	4		900.3.f.c.199.4		4
40.29	even	2		900.3.c.e.451.1		2
40.37	odd	4		900.3.f.c.199.2		4
48.5	odd	4		768.3.b.c.127.2		4
48.11	even	4		768.3.b.c.127.4		4
48.29	odd	4		768.3.b.c.127.3		4
48.35	even	4		768.3.b.c.127.1		4
72.5	odd	6		324.3.f.j.55.1		2
72.11	even	6		324.3.f.j.271.1		2
72.13	even	6		324.3.f.a.55.1		2
72.29	odd	6		324.3.f.d.271.1		2
72.43	odd	6		324.3.f.a.271.1		2
72.59	even	6		324.3.f.d.55.1		2
72.61	even	6		324.3.f.g.271.1		2
72.67	odd	6		324.3.f.g.55.1		2
120.29	odd	2		300.3.c.b.151.2		2
120.53	even	4		300.3.f.a.199.2		4
120.59	even	2		300.3.c.b.151.1		2
120.77	even	4		300.3.f.a.199.3		4
120.83	odd	4		300.3.f.a.199.4		4
120.107	odd	4		300.3.f.a.199.1		4
168.83	odd	2		588.3.g.b.295.2		2
168.125	even	2		588.3.g.b.295.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
12.3.d.a.7.1	✓	2	24.5	odd	2
12.3.d.a.7.2	yes	2	24.11	even	2
36.3.d.c.19.1		2	8.3	odd	2
36.3.d.c.19.2		2	8.5	even	2
192.3.g.b.127.1		2	3.2	odd	2
192.3.g.b.127.2		2	12.11	even	2
300.3.c.b.151.1		2	120.59	even	2
300.3.c.b.151.2		2	120.29	odd	2
300.3.f.a.199.1		4	120.107	odd	4
300.3.f.a.199.2		4	120.53	even	4
300.3.f.a.199.3		4	120.77	even	4
300.3.f.a.199.4		4	120.83	odd	4
324.3.f.a.55.1		2	72.13	even	6
324.3.f.a.271.1		2	72.43	odd	6
324.3.f.d.55.1		2	72.59	even	6
324.3.f.d.271.1		2	72.29	odd	6
324.3.f.g.55.1		2	72.67	odd	6
324.3.f.g.271.1		2	72.61	even	6
324.3.f.j.55.1		2	72.5	odd	6
324.3.f.j.271.1		2	72.11	even	6
576.3.g.e.127.1		2	1.1	even	1	trivial
576.3.g.e.127.2		2	4.3	odd	2	inner
588.3.g.b.295.1		2	168.125	even	2
588.3.g.b.295.2		2	168.83	odd	2
768.3.b.c.127.1		4	48.35	even	4
768.3.b.c.127.2		4	48.5	odd	4
768.3.b.c.127.3		4	48.29	odd	4
768.3.b.c.127.4		4	48.11	even	4
900.3.c.e.451.1		2	40.29	even	2
900.3.c.e.451.2		2	40.19	odd	2
900.3.f.c.199.1		4	40.3	even	4
900.3.f.c.199.2		4	40.37	odd	4
900.3.f.c.199.3		4	40.13	odd	4
900.3.f.c.199.4		4	40.27	even	4
2304.3.b.l.127.1		4	16.11	odd	4
2304.3.b.l.127.2		4	16.5	even	4
2304.3.b.l.127.3		4	16.3	odd	4
2304.3.b.l.127.4		4	16.13	even	4

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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	576.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.00000 q^{5} -6.92820i q^{7} +O(q^{10})\)	\(q-2.00000 q^{5} -6.92820i q^{7} +6.92820i q^{11} -2.00000 q^{13} -10.0000 q^{17} -20.7846i q^{19} +27.7128i q^{23} -21.0000 q^{25} -26.0000 q^{29} +6.92820i q^{31} +13.8564i q^{35} -26.0000 q^{37} -58.0000 q^{41} +48.4974i q^{43} -69.2820i q^{47} +1.00000 q^{49} -74.0000 q^{53} -13.8564i q^{55} -90.0666i q^{59} -26.0000 q^{61} +4.00000 q^{65} -6.92820i q^{67} -46.0000 q^{73} +48.0000 q^{77} +117.779i q^{79} +48.4974i q^{83} +20.0000 q^{85} -82.0000 q^{89} +13.8564i q^{91} +41.5692i q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} - 4 q^{13} - 20 q^{17} - 42 q^{25} - 52 q^{29} - 52 q^{37} - 116 q^{41} + 2 q^{49} - 148 q^{53} - 52 q^{61} + 8 q^{65} - 92 q^{73} + 96 q^{77} + 40 q^{85} - 164 q^{89} + 4 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 4 * q^13 - 20 * q^17 - 42 * q^25 - 52 * q^29 - 52 * q^37 - 116 * q^41 + 2 * q^49 - 148 * q^53 - 52 * q^61 + 8 * q^65 - 92 * q^73 + 96 * q^77 + 40 * q^85 - 164 * q^89 + 4 * q^97

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