LMFDB - Embedded newform 4788.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4788,2,Mod(1,4788)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4788, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4788.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4788.a (trivial)

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:	yes
Analytic conductor:	$38.2323724878$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.32448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} + 12 $ x^4 - 10*x^2 + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.18087$ of defining polynomial
Character	$\chi$	$=$	4788.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.71927 q^{5} +1.00000 q^{7} -2.36174 q^{11} +6.60555 q^{13} +5.08101 q^{17} +1.00000 q^{19} -2.36174 q^{23} +2.39445 q^{25} +5.08101 q^{29} +0.605551 q^{31} +2.71927 q^{35} +2.00000 q^{37} -2.36174 q^{41} +0.605551 q^{43} -5.08101 q^{47} +1.00000 q^{49} +2.71927 q^{53} -6.42221 q^{55} -4.72347 q^{59} +2.00000 q^{61} +17.9623 q^{65} +5.21110 q^{67} +2.00420 q^{71} -7.21110 q^{73} -2.36174 q^{77} -4.00000 q^{79} -2.71927 q^{83} +13.8167 q^{85} -13.2388 q^{89} +6.60555 q^{91} +2.71927 q^{95} +11.2111 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} + 12 q^{13} + 4 q^{19} + 24 q^{25} - 12 q^{31} + 8 q^{37} - 12 q^{43} + 4 q^{49} + 32 q^{55} + 8 q^{61} - 8 q^{67} - 16 q^{79} + 12 q^{85} + 12 q^{91} + 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 + 12 * q^13 + 4 * q^19 + 24 * q^25 - 12 * q^31 + 8 * q^37 - 12 * q^43 + 4 * q^49 + 32 * q^55 + 8 * q^61 - 8 * q^67 - 16 * q^79 + 12 * q^85 + 12 * q^91 + 16 * q^97

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
$5$	2.71927	1.21610	0.608048	−	0.793900i	$-0.291953\pi$
$5$	2.71927	1.21610	0.608048	+	0.793900i	$0.291953\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.36174	−0.712090	−0.356045	−	0.934469i	$-0.615875\pi$
$11$	−2.36174	−0.712090	−0.356045	+	0.934469i	$0.615875\pi$
$12$	0	0
$13$	6.60555	1.83205	0.916025	−	0.401121i	$-0.131379\pi$
$13$	6.60555	1.83205	0.916025	+	0.401121i	$0.131379\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.08101	1.23233	0.616163	−	0.787619i	$-0.288686\pi$
$17$	5.08101	1.23233	0.616163	+	0.787619i	$0.288686\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.36174	−0.492456	−0.246228	−	0.969212i	$-0.579191\pi$
$23$	−2.36174	−0.492456	−0.246228	+	0.969212i	$0.579191\pi$
$24$	0	0
$25$	2.39445	0.478890
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.08101	0.943520	0.471760	−	0.881727i	$-0.343619\pi$
$29$	5.08101	0.943520	0.471760	+	0.881727i	$0.343619\pi$
$30$	0	0
$31$	0.605551	0.108760	0.0543801	−	0.998520i	$-0.482682\pi$
$31$	0.605551	0.108760	0.0543801	+	0.998520i	$0.482682\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.71927	0.459641
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.36174	−0.368841	−0.184421	−	0.982847i	$-0.559041\pi$
$41$	−2.36174	−0.368841	−0.184421	+	0.982847i	$0.559041\pi$
$42$	0	0
$43$	0.605551	0.0923457	0.0461729	−	0.998933i	$-0.485297\pi$
$43$	0.605551	0.0923457	0.0461729	+	0.998933i	$0.485297\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.08101	−0.741141	−0.370571	−	0.928804i	$-0.620838\pi$
$47$	−5.08101	−0.741141	−0.370571	+	0.928804i	$0.620838\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.71927	0.373521	0.186760	−	0.982405i	$-0.440201\pi$
$53$	2.71927	0.373521	0.186760	+	0.982405i	$0.440201\pi$
$54$	0	0
$55$	−6.42221	−0.865970
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−4.72347	−0.614944	−0.307472	−	0.951557i	$-0.599483\pi$
$59$	−4.72347	−0.614944	−0.307472	+	0.951557i	$0.599483\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	17.9623	2.22795
$66$	0	0
$67$	5.21110	0.636638	0.318319	−	0.947984i	$-0.396882\pi$
$67$	5.21110	0.636638	0.318319	+	0.947984i	$0.396882\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.00420	0.237854	0.118927	−	0.992903i	$-0.462054\pi$
$71$	2.00420	0.237854	0.118927	+	0.992903i	$0.462054\pi$
$72$	0	0
$73$	−7.21110	−0.843996	−0.421998	−	0.906597i	$-0.638671\pi$
$73$	−7.21110	−0.843996	−0.421998	+	0.906597i	$0.638671\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.36174	−0.269145
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.71927	−0.298479	−0.149240	−	0.988801i	$-0.547683\pi$
$83$	−2.71927	−0.298479	−0.149240	+	0.988801i	$0.547683\pi$
$84$	0	0
$85$	13.8167	1.49863
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.2388	−1.40331	−0.701657	−	0.712515i	$-0.747556\pi$
$89$	−13.2388	−1.40331	−0.701657	+	0.712515i	$0.747556\pi$
$90$	0	0
$91$	6.60555	0.692450
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.71927	0.278992
$96$	0	0
$97$	11.2111	1.13831	0.569157	−	0.822229i	$-0.307269\pi$
$97$	11.2111	1.13831	0.569157	+	0.822229i	$0.307269\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.2430	1.51674	0.758369	−	0.651826i	$-0.225997\pi$
$101$	15.2430	1.51674	0.758369	+	0.651826i	$0.225997\pi$
$102$	0	0
$103$	−10.4222	−1.02693	−0.513465	−	0.858110i	$-0.671638\pi$
$103$	−10.4222	−1.02693	−0.513465	+	0.858110i	$0.671638\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.08101	0.491200	0.245600	−	0.969371i	$-0.421015\pi$
$107$	5.08101	0.491200	0.245600	+	0.969371i	$0.421015\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	9.80448	0.922328	0.461164	−	0.887315i	$-0.347432\pi$
$113$	9.80448	0.922328	0.461164	+	0.887315i	$0.347432\pi$
$114$	0	0
$115$	−6.42221	−0.598874
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.08101	0.465775
$120$	0	0
$121$	−5.42221	−0.492928
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.08521	−0.633720
$126$	0	0
$127$	−1.21110	−0.107468	−0.0537340	−	0.998555i	$-0.517112\pi$
$127$	−1.21110	−0.107468	−0.0537340	+	0.998555i	$0.517112\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.15782	−0.712752	−0.356376	−	0.934343i	$-0.615988\pi$
$131$	−8.15782	−0.712752	−0.356376	+	0.934343i	$0.615988\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	17.2111	1.45983	0.729913	−	0.683540i	$-0.239560\pi$
$139$	17.2111	1.45983	0.729913	+	0.683540i	$0.239560\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−15.6006	−1.30458
$144$	0	0
$145$	13.8167	1.14741
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.43855	0.445543	0.222772	−	0.974871i	$-0.428490\pi$
$149$	5.43855	0.445543	0.222772	+	0.974871i	$0.428490\pi$
$150$	0	0
$151$	14.4222	1.17366	0.586831	−	0.809709i	$-0.300375\pi$
$151$	14.4222	1.17366	0.586831	+	0.809709i	$0.300375\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.64666	0.132263
$156$	0	0
$157$	−7.21110	−0.575509	−0.287754	−	0.957704i	$-0.592909\pi$
$157$	−7.21110	−0.575509	−0.287754	+	0.957704i	$0.592909\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.36174	−0.186131
$162$	0	0
$163$	9.81665	0.768900	0.384450	−	0.923146i	$-0.374391\pi$
$163$	9.81665	0.768900	0.384450	+	0.923146i	$0.374391\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.0391	−1.62806	−0.814028	−	0.580826i	$-0.802730\pi$
$167$	−21.0391	−1.62806	−0.814028	+	0.580826i	$0.802730\pi$
$168$	0	0
$169$	30.6333	2.35641
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−17.9623	−1.36565	−0.682824	−	0.730583i	$-0.739249\pi$
$173$	−17.9623	−1.36565	−0.682824	+	0.730583i	$0.739249\pi$
$174$	0	0
$175$	2.39445	0.181003
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−5.08101	−0.379772	−0.189886	−	0.981806i	$-0.560812\pi$
$179$	−5.08101	−0.379772	−0.189886	+	0.981806i	$0.560812\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.43855	0.399850
$186$	0	0
$187$	−12.0000	−0.877527
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.36174	0.170889	0.0854446	−	0.996343i	$-0.472769\pi$
$191$	2.36174	0.170889	0.0854446	+	0.996343i	$0.472769\pi$
$192$	0	0
$193$	−0.788897	−0.0567861	−0.0283930	−	0.999597i	$-0.509039\pi$
$193$	−0.788897	−0.0567861	−0.0283930	+	0.999597i	$0.509039\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.8771	−0.774961	−0.387480	−	0.921878i	$-0.626655\pi$
$197$	−10.8771	−0.774961	−0.387480	+	0.921878i	$0.626655\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5.08101	0.356617
$204$	0	0
$205$	−6.42221	−0.448546
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.36174	−0.163365
$210$	0	0
$211$	10.7889	0.742738	0.371369	−	0.928485i	$-0.378888\pi$
$211$	10.7889	0.742738	0.371369	+	0.928485i	$0.378888\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.64666	0.112301
$216$	0	0
$217$	0.605551	0.0411075
$218$	0	0
$219$	0	0
$220$	0	0
$221$	33.5629	2.25768
$222$	0	0
$223$	28.2389	1.89101	0.945507	−	0.325602i	$-0.105567\pi$
$223$	28.2389	1.89101	0.945507	+	0.325602i	$0.105567\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.1620	0.674477	0.337238	−	0.941419i	$-0.390507\pi$
$227$	10.1620	0.674477	0.337238	+	0.941419i	$0.390507\pi$
$228$	0	0
$229$	11.2111	0.740851	0.370425	−	0.928862i	$-0.379212\pi$
$229$	11.2111	0.740851	0.370425	+	0.928862i	$0.379212\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.6090	−1.28463	−0.642313	−	0.766443i	$-0.722025\pi$
$233$	−19.6090	−1.28463	−0.642313	+	0.766443i	$0.722025\pi$
$234$	0	0
$235$	−13.8167	−0.901299
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.5238	−0.810094	−0.405047	−	0.914296i	$-0.632745\pi$
$239$	−12.5238	−0.810094	−0.405047	+	0.914296i	$0.632745\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.71927	0.173728
$246$	0	0
$247$	6.60555	0.420301
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−11.2346	−0.709124	−0.354562	−	0.935033i	$-0.615370\pi$
$251$	−11.2346	−0.709124	−0.354562	+	0.935033i	$0.615370\pi$
$252$	0	0
$253$	5.57779	0.350673
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.80028	−0.486568	−0.243284	−	0.969955i	$-0.578225\pi$
$257$	−7.80028	−0.486568	−0.243284	+	0.969955i	$0.578225\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.07681	−0.189724	−0.0948622	−	0.995490i	$-0.530241\pi$
$263$	−3.07681	−0.189724	−0.0948622	+	0.995490i	$0.530241\pi$
$264$	0	0
$265$	7.39445	0.454237
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−28.1243	−1.71477	−0.857385	−	0.514676i	$-0.827912\pi$
$269$	−28.1243	−1.71477	−0.857385	+	0.514676i	$0.827912\pi$
$270$	0	0
$271$	−13.2111	−0.802517	−0.401259	−	0.915965i	$-0.631427\pi$
$271$	−13.2111	−0.802517	−0.401259	+	0.915965i	$0.631427\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.65505	−0.341013
$276$	0	0
$277$	−9.02776	−0.542425	−0.271213	−	0.962519i	$-0.587425\pi$
$277$	−9.02776	−0.542425	−0.271213	+	0.962519i	$0.587425\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−29.1969	−1.74174	−0.870872	−	0.491511i	$-0.836445\pi$
$281$	−29.1969	−1.74174	−0.870872	+	0.491511i	$0.836445\pi$
$282$	0	0
$283$	23.6333	1.40485	0.702427	−	0.711756i	$-0.252100\pi$
$283$	23.6333	1.40485	0.702427	+	0.711756i	$0.252100\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.36174	−0.139409
$288$	0	0
$289$	8.81665	0.518627
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7.08521	−0.413922	−0.206961	−	0.978349i	$-0.566357\pi$
$293$	−7.08521	−0.413922	−0.206961	+	0.978349i	$0.566357\pi$
$294$	0	0
$295$	−12.8444	−0.747830
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−15.6006	−0.902204
$300$	0	0
$301$	0.605551	0.0349034
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.43855	0.311410
$306$	0	0
$307$	−15.0278	−0.857679	−0.428840	−	0.903381i	$-0.641077\pi$
$307$	−15.0278	−0.857679	−0.428840	+	0.903381i	$0.641077\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	32.4903	1.84235	0.921177	−	0.389145i	$-0.127229\pi$
$311$	32.4903	1.84235	0.921177	+	0.389145i	$0.127229\pi$
$312$	0	0
$313$	4.78890	0.270684	0.135342	−	0.990799i	$-0.456787\pi$
$313$	4.78890	0.270684	0.135342	+	0.990799i	$0.456787\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.5196	−0.590837	−0.295419	−	0.955368i	$-0.595459\pi$
$317$	−10.5196	−0.590837	−0.295419	+	0.955368i	$0.595459\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.08101	0.282715
$324$	0	0
$325$	15.8167	0.877350
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.08101	−0.280125
$330$	0	0
$331$	−6.78890	−0.373152	−0.186576	−	0.982441i	$-0.559739\pi$
$331$	−6.78890	−0.373152	−0.186576	+	0.982441i	$0.559739\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	14.1704	0.774212
$336$	0	0
$337$	−22.8444	−1.24441	−0.622207	−	0.782853i	$-0.713764\pi$
$337$	−22.8444	−1.24441	−0.622207	+	0.782853i	$0.713764\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.43015	−0.0774471
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.5545	1.58657	0.793284	−	0.608852i	$-0.208370\pi$
$347$	29.5545	1.58657	0.793284	+	0.608852i	$0.208370\pi$
$348$	0	0
$349$	4.78890	0.256344	0.128172	−	0.991752i	$-0.459089\pi$
$349$	4.78890	0.256344	0.128172	+	0.991752i	$0.459089\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	34.6355	1.84346	0.921730	−	0.387831i	$-0.126776\pi$
$353$	34.6355	1.84346	0.921730	+	0.387831i	$0.126776\pi$
$354$	0	0
$355$	5.44996	0.289254
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.9623	−0.948014	−0.474007	−	0.880521i	$-0.657193\pi$
$359$	−17.9623	−0.948014	−0.474007	+	0.880521i	$0.657193\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−19.6090	−1.02638
$366$	0	0
$367$	17.2111	0.898412	0.449206	−	0.893428i	$-0.351707\pi$
$367$	17.2111	0.898412	0.449206	+	0.893428i	$0.351707\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.71927	0.141178
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	33.5629	1.72858
$378$	0	0
$379$	20.8444	1.07071	0.535353	−	0.844629i	$-0.320179\pi$
$379$	20.8444	1.07071	0.535353	+	0.844629i	$0.320179\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−16.3156	−0.833690	−0.416845	−	0.908978i	$-0.636864\pi$
$383$	−16.3156	−0.833690	−0.416845	+	0.908978i	$0.636864\pi$
$384$	0	0
$385$	−6.42221	−0.327306
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.3156	0.827236	0.413618	−	0.910451i	$-0.364265\pi$
$389$	16.3156	0.827236	0.413618	+	0.910451i	$0.364265\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.8771	−0.547286
$396$	0	0
$397$	−4.42221	−0.221944	−0.110972	−	0.993824i	$-0.535396\pi$
$397$	−4.42221	−0.221944	−0.110972	+	0.993824i	$0.535396\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.7668	1.38661	0.693303	−	0.720646i	$-0.256155\pi$
$401$	27.7668	1.38661	0.693303	+	0.720646i	$0.256155\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.72347	−0.234134
$408$	0	0
$409$	0.183346	0.00906588	0.00453294	−	0.999990i	$-0.498557\pi$
$409$	0.183346	0.00906588	0.00453294	+	0.999990i	$0.498557\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.72347	−0.232427
$414$	0	0
$415$	−7.39445	−0.362979
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.6816	1.01036	0.505181	−	0.863014i	$-0.331426\pi$
$419$	20.6816	1.01036	0.505181	+	0.863014i	$0.331426\pi$
$420$	0	0
$421$	4.78890	0.233397	0.116698	−	0.993167i	$-0.462769\pi$
$421$	4.78890	0.233397	0.116698	+	0.993167i	$0.462769\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12.1662	0.590148
$426$	0	0
$427$	2.00000	0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.07261	−0.0516660	−0.0258330	−	0.999666i	$-0.508224\pi$
$431$	−1.07261	−0.0516660	−0.0258330	+	0.999666i	$0.508224\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.36174	−0.112977
$438$	0	0
$439$	−3.02776	−0.144507	−0.0722535	−	0.997386i	$-0.523019\pi$
$439$	−3.02776	−0.144507	−0.0722535	+	0.997386i	$0.523019\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	39.0014	1.85301	0.926507	−	0.376279i	$-0.122796\pi$
$443$	39.0014	1.85301	0.926507	+	0.376279i	$0.122796\pi$
$444$	0	0
$445$	−36.0000	−1.70656
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.71927	0.128330	0.0641652	−	0.997939i	$-0.479562\pi$
$449$	2.71927	0.128330	0.0641652	+	0.997939i	$0.479562\pi$
$450$	0	0
$451$	5.57779	0.262648
$452$	0	0
$453$	0	0
$454$	0	0
$455$	17.9623	0.842086
$456$	0	0
$457$	1.02776	0.0480764	0.0240382	−	0.999711i	$-0.492348\pi$
$457$	1.02776	0.0480764	0.0240382	+	0.999711i	$0.492348\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	16.6732	0.776547	0.388274	−	0.921544i	$-0.373072\pi$
$461$	16.6732	0.776547	0.388274	+	0.921544i	$0.373072\pi$
$462$	0	0
$463$	−9.57779	−0.445118	−0.222559	−	0.974919i	$-0.571441\pi$
$463$	−9.57779	−0.445118	−0.222559	+	0.974919i	$0.571441\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−2.00420	−0.0927432	−0.0463716	−	0.998924i	$-0.514766\pi$
$467$	−2.00420	−0.0927432	−0.0463716	+	0.998924i	$0.514766\pi$
$468$	0	0
$469$	5.21110	0.240626
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.43015	−0.0657585
$474$	0	0
$475$	2.39445	0.109865
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.2430	−0.696472	−0.348236	−	0.937407i	$-0.613219\pi$
$479$	−15.2430	−0.696472	−0.348236	+	0.937407i	$0.613219\pi$
$480$	0	0
$481$	13.2111	0.602374
$482$	0	0
$483$	0	0
$484$	0	0
$485$	30.4861	1.38430
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	39.0014	1.76011	0.880055	−	0.474873i	$-0.157506\pi$
$491$	39.0014	1.76011	0.880055	+	0.474873i	$0.157506\pi$
$492$	0	0
$493$	25.8167	1.16272
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.00420	0.0899005
$498$	0	0
$499$	−10.4222	−0.466562	−0.233281	−	0.972409i	$-0.574946\pi$
$499$	−10.4222	−0.466562	−0.233281	+	0.972409i	$0.574946\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−23.7584	−1.05933	−0.529667	−	0.848206i	$-0.677683\pi$
$503$	−23.7584	−1.05933	−0.529667	+	0.848206i	$0.677683\pi$
$504$	0	0
$505$	41.4500	1.84450
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.37013	0.282351	0.141176	−	0.989985i	$-0.454912\pi$
$509$	6.37013	0.282351	0.141176	+	0.989985i	$0.454912\pi$
$510$	0	0
$511$	−7.21110	−0.319000
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−28.3408	−1.24885
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	0	0
$520$	0	0
$521$	28.1243	1.23215	0.616074	−	0.787688i	$-0.288722\pi$
$521$	28.1243	1.23215	0.616074	+	0.787688i	$0.288722\pi$
$522$	0	0
$523$	−2.18335	−0.0954711	−0.0477355	−	0.998860i	$-0.515200\pi$
$523$	−2.18335	−0.0954711	−0.0477355	+	0.998860i	$0.515200\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.07681	0.134028
$528$	0	0
$529$	−17.4222	−0.757487
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−15.6006	−0.675735
$534$	0	0
$535$	13.8167	0.597346
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.36174	−0.101727
$540$	0	0
$541$	23.2111	0.997923	0.498962	−	0.866624i	$-0.333715\pi$
$541$	23.2111	0.997923	0.498962	+	0.866624i	$0.333715\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.43855	0.232962
$546$	0	0
$547$	−16.0000	−0.684111	−0.342055	−	0.939680i	$-0.611123\pi$
$547$	−16.0000	−0.684111	−0.342055	+	0.939680i	$0.611123\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.08101	0.216458
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.6090	−0.830858	−0.415429	−	0.909626i	$-0.636369\pi$
$557$	−19.6090	−0.830858	−0.415429	+	0.909626i	$0.636369\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4.00840	−0.168934	−0.0844669	−	0.996426i	$-0.526919\pi$
$563$	−4.00840	−0.168934	−0.0844669	+	0.996426i	$0.526919\pi$
$564$	0	0
$565$	26.6611	1.12164
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.44274	−0.312016	−0.156008	−	0.987756i	$-0.549863\pi$
$569$	−7.44274	−0.312016	−0.156008	+	0.987756i	$0.549863\pi$
$570$	0	0
$571$	−46.4222	−1.94271	−0.971354	−	0.237635i	$-0.923628\pi$
$571$	−46.4222	−1.94271	−0.971354	+	0.237635i	$0.923628\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.65505	−0.235832
$576$	0	0
$577$	41.6333	1.73322	0.866609	−	0.498988i	$-0.166295\pi$
$577$	41.6333	1.73322	0.866609	+	0.498988i	$0.166295\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.71927	−0.112814
$582$	0	0
$583$	−6.42221	−0.265981
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.08940	0.375160	0.187580	−	0.982249i	$-0.439936\pi$
$587$	9.08940	0.375160	0.187580	+	0.982249i	$0.439936\pi$
$588$	0	0
$589$	0.605551	0.0249513
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−28.4819	−1.16961	−0.584805	−	0.811174i	$-0.698829\pi$
$593$	−28.4819	−1.16961	−0.584805	+	0.811174i	$0.698829\pi$
$594$	0	0
$595$	13.8167	0.566428
$596$	0	0
$597$	0	0
$598$	0	0
$599$	36.9972	1.51167	0.755833	−	0.654765i	$-0.227232\pi$
$599$	36.9972	1.51167	0.755833	+	0.654765i	$0.227232\pi$
$600$	0	0
$601$	47.2111	1.92578	0.962891	−	0.269892i	$-0.0869881\pi$
$601$	47.2111	1.92578	0.962891	+	0.269892i	$0.0869881\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−14.7445	−0.599448
$606$	0	0
$607$	−40.8444	−1.65782	−0.828912	−	0.559379i	$-0.811039\pi$
$607$	−40.8444	−1.65782	−0.828912	+	0.559379i	$0.811039\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−33.5629	−1.35781
$612$	0	0
$613$	−26.6056	−1.07459	−0.537294	−	0.843395i	$-0.680553\pi$
$613$	−26.6056	−1.07459	−0.537294	+	0.843395i	$0.680553\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15.6006	−0.628055	−0.314028	−	0.949414i	$-0.601678\pi$
$617$	−15.6006	−0.628055	−0.314028	+	0.949414i	$0.601678\pi$
$618$	0	0
$619$	22.7889	0.915963	0.457982	−	0.888962i	$-0.348573\pi$
$619$	22.7889	0.915963	0.457982	+	0.888962i	$0.348573\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−13.2388	−0.530403
$624$	0	0
$625$	−31.2389	−1.24955
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.1620	0.405186
$630$	0	0
$631$	−11.3944	−0.453606	−0.226803	−	0.973941i	$-0.572827\pi$
$631$	−11.3944	−0.453606	−0.226803	+	0.973941i	$0.572827\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.29332	−0.130691
$636$	0	0
$637$	6.60555	0.261721
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−25.4050	−1.00344	−0.501719	−	0.865030i	$-0.667299\pi$
$641$	−25.4050	−1.00344	−0.501719	+	0.865030i	$0.667299\pi$
$642$	0	0
$643$	−26.0555	−1.02753	−0.513765	−	0.857931i	$-0.671750\pi$
$643$	−26.0555	−1.02753	−0.513765	+	0.857931i	$0.671750\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−5.08101	−0.199755	−0.0998775	−	0.995000i	$-0.531845\pi$
$647$	−5.08101	−0.199755	−0.0998775	+	0.995000i	$0.531845\pi$
$648$	0	0
$649$	11.1556	0.437895
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−26.4777	−1.03615	−0.518075	−	0.855335i	$-0.673351\pi$
$653$	−26.4777	−1.03615	−0.518075	+	0.855335i	$0.673351\pi$
$654$	0	0
$655$	−22.1833	−0.866775
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12.8813	−0.501784	−0.250892	−	0.968015i	$-0.580724\pi$
$659$	−12.8813	−0.501784	−0.250892	+	0.968015i	$0.580724\pi$
$660$	0	0
$661$	−20.1833	−0.785041	−0.392521	−	0.919743i	$-0.628397\pi$
$661$	−20.1833	−0.785041	−0.392521	+	0.919743i	$0.628397\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.71927	0.105449
$666$	0	0
$667$	−12.0000	−0.464642
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.72347	−0.182348
$672$	0	0
$673$	−37.6333	−1.45066	−0.725329	−	0.688403i	$-0.758312\pi$
$673$	−37.6333	−1.45066	−0.725329	+	0.688403i	$0.758312\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.37013	−0.244824	−0.122412	−	0.992479i	$-0.539063\pi$
$677$	−6.37013	−0.244824	−0.122412	+	0.992479i	$0.539063\pi$
$678$	0	0
$679$	11.2111	0.430243
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.08101	−0.194419	−0.0972097	−	0.995264i	$-0.530992\pi$
$683$	−5.08101	−0.194419	−0.0972097	+	0.995264i	$0.530992\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	17.9623	0.684309
$690$	0	0
$691$	−49.2111	−1.87208	−0.936039	−	0.351896i	$-0.885537\pi$
$691$	−49.2111	−1.87208	−0.936039	+	0.351896i	$0.885537\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	46.8017	1.77529
$696$	0	0
$697$	−12.0000	−0.454532
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−20.3240	−0.767628	−0.383814	−	0.923410i	$-0.625390\pi$
$701$	−20.3240	−0.767628	−0.383814	+	0.923410i	$0.625390\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.2430	0.573273
$708$	0	0
$709$	31.4500	1.18113	0.590564	−	0.806991i	$-0.298905\pi$
$709$	31.4500	1.18113	0.590564	+	0.806991i	$0.298905\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.43015	−0.0535596
$714$	0	0
$715$	−42.4222	−1.58650
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11.4511	0.427055	0.213528	−	0.976937i	$-0.431505\pi$
$719$	11.4511	0.427055	0.213528	+	0.976937i	$0.431505\pi$
$720$	0	0
$721$	−10.4222	−0.388143
$722$	0	0
$723$	0	0
$724$	0	0
$725$	12.1662	0.451842
$726$	0	0
$727$	14.4222	0.534890	0.267445	−	0.963573i	$-0.413821\pi$
$727$	14.4222	0.534890	0.267445	+	0.963573i	$0.413821\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.07681	0.113800
$732$	0	0
$733$	−43.2111	−1.59604	−0.798019	−	0.602632i	$-0.794119\pi$
$733$	−43.2111	−1.59604	−0.798019	+	0.602632i	$0.794119\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.3072	−0.453343
$738$	0	0
$739$	4.23886	0.155929	0.0779645	−	0.996956i	$-0.475158\pi$
$739$	4.23886	0.155929	0.0779645	+	0.996956i	$0.475158\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0.357538	0.0131168	0.00655840	−	0.999978i	$-0.497912\pi$
$743$	0.357538	0.0131168	0.00655840	+	0.999978i	$0.497912\pi$
$744$	0	0
$745$	14.7889	0.541823
$746$	0	0
$747$	0	0
$748$	0	0
$749$	5.08101	0.185656
$750$	0	0
$751$	8.84441	0.322737	0.161369	−	0.986894i	$-0.448409\pi$
$751$	8.84441	0.322737	0.161369	+	0.986894i	$0.448409\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	39.2179	1.42729
$756$	0	0
$757$	28.7889	1.04635	0.523175	−	0.852225i	$-0.324747\pi$
$757$	28.7889	1.04635	0.523175	+	0.852225i	$0.324747\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.6355	−1.25554	−0.627768	−	0.778401i	$-0.716031\pi$
$761$	−34.6355	−1.25554	−0.627768	+	0.778401i	$0.716031\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−31.2011	−1.12661
$768$	0	0
$769$	−0.788897	−0.0284484	−0.0142242	−	0.999899i	$-0.504528\pi$
$769$	−0.788897	−0.0284484	−0.0142242	+	0.999899i	$0.504528\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−50.5936	−1.81972	−0.909862	−	0.414910i	$-0.863813\pi$
$773$	−50.5936	−1.81972	−0.909862	+	0.414910i	$0.863813\pi$
$774$	0	0
$775$	1.44996	0.0520842
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.36174	−0.0846180
$780$	0	0
$781$	−4.73338	−0.169374
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−19.6090	−0.699874
$786$	0	0
$787$	−35.3944	−1.26168	−0.630838	−	0.775915i	$-0.717289\pi$
$787$	−35.3944	−1.26168	−0.630838	+	0.775915i	$0.717289\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.80448	0.348607
$792$	0	0
$793$	13.2111	0.469140
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.79189	0.134316	0.0671578	−	0.997742i	$-0.478607\pi$
$797$	3.79189	0.134316	0.0671578	+	0.997742i	$0.478607\pi$
$798$	0	0
$799$	−25.8167	−0.913328
$800$	0	0
$801$	0	0
$802$	0	0
$803$	17.0307	0.601001
$804$	0	0
$805$	−6.42221	−0.226353
$806$	0	0
$807$	0	0
$808$	0	0
$809$	40.6481	1.42911	0.714555	−	0.699579i	$-0.246629\pi$
$809$	40.6481	1.42911	0.714555	+	0.699579i	$0.246629\pi$
$810$	0	0
$811$	−46.4222	−1.63010	−0.815052	−	0.579388i	$-0.803292\pi$
$811$	−46.4222	−1.63010	−0.815052	+	0.579388i	$0.803292\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	26.6942	0.935056
$816$	0	0
$817$	0.605551	0.0211856
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22.4693	−0.784183	−0.392091	−	0.919926i	$-0.628248\pi$
$821$	−22.4693	−0.784183	−0.392091	+	0.919926i	$0.628248\pi$
$822$	0	0
$823$	−29.8167	−1.03934	−0.519672	−	0.854366i	$-0.673946\pi$
$823$	−29.8167	−1.03934	−0.519672	+	0.854366i	$0.673946\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.4819	0.990411	0.495206	−	0.868776i	$-0.335093\pi$
$827$	28.4819	0.990411	0.495206	+	0.868776i	$0.335093\pi$
$828$	0	0
$829$	−13.6333	−0.473504	−0.236752	−	0.971570i	$-0.576083\pi$
$829$	−13.6333	−0.473504	−0.236752	+	0.971570i	$0.576083\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.08101	0.176047
$834$	0	0
$835$	−57.2111	−1.97987
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−52.9553	−1.82822	−0.914110	−	0.405466i	$-0.867109\pi$
$839$	−52.9553	−1.82822	−0.914110	+	0.405466i	$0.867109\pi$
$840$	0	0
$841$	−3.18335	−0.109771
$842$	0	0
$843$	0	0
$844$	0	0
$845$	83.3003	2.86562
$846$	0	0
$847$	−5.42221	−0.186309
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.72347	−0.161919
$852$	0	0
$853$	10.3667	0.354949	0.177474	−	0.984125i	$-0.443207\pi$
$853$	10.3667	0.354949	0.177474	+	0.984125i	$0.443207\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.36174	0.0806754	0.0403377	−	0.999186i	$-0.487157\pi$
$857$	2.36174	0.0806754	0.0403377	+	0.999186i	$0.487157\pi$
$858$	0	0
$859$	−13.2111	−0.450757	−0.225379	−	0.974271i	$-0.572362\pi$
$859$	−13.2111	−0.450757	−0.225379	+	0.974271i	$0.572362\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.357538	−0.0121707	−0.00608537	−	0.999981i	$-0.501937\pi$
$863$	−0.357538	−0.0121707	−0.00608537	+	0.999981i	$0.501937\pi$
$864$	0	0
$865$	−48.8444	−1.66076
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.44694	0.320466
$870$	0	0
$871$	34.4222	1.16635
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−7.08521	−0.239524
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−38.6439	−1.30195	−0.650973	−	0.759101i	$-0.725639\pi$
$881$	−38.6439	−1.30195	−0.650973	+	0.759101i	$0.725639\pi$
$882$	0	0
$883$	−28.8444	−0.970692	−0.485346	−	0.874322i	$-0.661306\pi$
$883$	−28.8444	−0.970692	−0.485346	+	0.874322i	$0.661306\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−50.8101	−1.70604	−0.853018	−	0.521882i	$-0.825230\pi$
$887$	−50.8101	−1.70604	−0.853018	+	0.521882i	$0.825230\pi$
$888$	0	0
$889$	−1.21110	−0.0406191
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.08101	−0.170029
$894$	0	0
$895$	−13.8167	−0.461840
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.07681	0.102617
$900$	0	0
$901$	13.8167	0.460299
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.43855	0.180783
$906$	0	0
$907$	−35.2666	−1.17101	−0.585504	−	0.810669i	$-0.699103\pi$
$907$	−35.2666	−1.17101	−0.585504	+	0.810669i	$0.699103\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−8.15782	−0.270281	−0.135140	−	0.990826i	$-0.543149\pi$
$911$	−8.15782	−0.270281	−0.135140	+	0.990826i	$0.543149\pi$
$912$	0	0
$913$	6.42221	0.212544
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−8.15782	−0.269395
$918$	0	0
$919$	−10.4222	−0.343797	−0.171898	−	0.985115i	$-0.554990\pi$
$919$	−10.4222	−0.343797	−0.171898	+	0.985115i	$0.554990\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13.2388	0.435761
$924$	0	0
$925$	4.78890	0.157458
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.07261	−0.0351913	−0.0175957	−	0.999845i	$-0.505601\pi$
$929$	−1.07261	−0.0351913	−0.0175957	+	0.999845i	$0.505601\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−32.6313	−1.06716
$936$	0	0
$937$	44.4222	1.45121	0.725605	−	0.688111i	$-0.241560\pi$
$937$	44.4222	1.45121	0.725605	+	0.688111i	$0.241560\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−20.1075	−0.655487	−0.327743	−	0.944767i	$-0.606288\pi$
$941$	−20.1075	−0.655487	−0.327743	+	0.944767i	$0.606288\pi$
$942$	0	0
$943$	5.57779	0.181638
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.08521	0.230238	0.115119	−	0.993352i	$-0.463275\pi$
$947$	7.08521	0.230238	0.115119	+	0.993352i	$0.463275\pi$
$948$	0	0
$949$	−47.6333	−1.54624
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21.6132	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$953$	21.6132	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$954$	0	0
$955$	6.42221	0.207818
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.6333	−0.988171
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.14523	−0.0690573
$966$	0	0
$967$	58.6611	1.88641	0.943206	−	0.332208i	$-0.107793\pi$
$967$	58.6611	1.88641	0.943206	+	0.332208i	$0.107793\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	57.6788	1.85100	0.925500	−	0.378747i	$-0.123645\pi$
$971$	57.6788	1.85100	0.925500	+	0.378747i	$0.123645\pi$
$972$	0	0
$973$	17.2111	0.551763
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25.1885	−0.805853	−0.402926	−	0.915232i	$-0.632007\pi$
$977$	−25.1885	−0.805853	−0.402926	+	0.915232i	$0.632007\pi$
$978$	0	0
$979$	31.2666	0.999285
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−54.8185	−1.74844	−0.874219	−	0.485532i	$-0.838626\pi$
$983$	−54.8185	−1.74844	−0.874219	+	0.485532i	$0.838626\pi$
$984$	0	0
$985$	−29.5778	−0.942427
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.43015	−0.0454762
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	21.7542	0.689654
$996$	0	0
$997$	5.63331	0.178409	0.0892043	−	0.996013i	$-0.471568\pi$
$997$	5.63331	0.178409	0.0892043	+	0.996013i	$0.471568\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4788.2.a.s.1.3	yes	4
3.2	odd	2	inner	4788.2.a.s.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4788.2.a.s.1.2	✓	4	3.2	odd	2	inner
4788.2.a.s.1.3	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4788 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4788.a (trivial)

\(f(q)\)	\(=\)	\(q+2.71927 q^{5} +1.00000 q^{7} -2.36174 q^{11} +6.60555 q^{13} +5.08101 q^{17} +1.00000 q^{19} -2.36174 q^{23} +2.39445 q^{25} +5.08101 q^{29} +0.605551 q^{31} +2.71927 q^{35} +2.00000 q^{37} -2.36174 q^{41} +0.605551 q^{43} -5.08101 q^{47} +1.00000 q^{49} +2.71927 q^{53} -6.42221 q^{55} -4.72347 q^{59} +2.00000 q^{61} +17.9623 q^{65} +5.21110 q^{67} +2.00420 q^{71} -7.21110 q^{73} -2.36174 q^{77} -4.00000 q^{79} -2.71927 q^{83} +13.8167 q^{85} -13.2388 q^{89} +6.60555 q^{91} +2.71927 q^{95} +11.2111 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} + 12 q^{13} + 4 q^{19} + 24 q^{25} - 12 q^{31} + 8 q^{37} - 12 q^{43} + 4 q^{49} + 32 q^{55} + 8 q^{61} - 8 q^{67} - 16 q^{79} + 12 q^{85} + 12 q^{91} + 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 + 12 * q^13 + 4 * q^19 + 24 * q^25 - 12 * q^31 + 8 * q^37 - 12 * q^43 + 4 * q^49 + 32 * q^55 + 8 * q^61 - 8 * q^67 - 16 * q^79 + 12 * q^85 + 12 * q^91 + 16 * q^97

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