LMFDB - Embedded newform 9072.2.a.cl.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, 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("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9072 = 2^{4} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9072.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:	$72.4402847137$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.22545.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 5x + 4 $ x^4 - x^3 - 6x^2 + 5x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 504)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.33866$ of defining polynomial
Character	$\chi$	$=$	9072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43628 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-2.43628 q^{5} -1.00000 q^{7} +0.758955 q^{11} +2.22829 q^{13} +7.04904 q^{17} +1.37172 q^{19} +7.03629 q^{23} +0.935443 q^{25} -0.836267 q^{29} +0.530665 q^{31} +2.43628 q^{35} +4.53066 q^{37} -8.85704 q^{41} -7.40322 q^{43} -6.79524 q^{47} +1.00000 q^{49} +0.607978 q^{53} -1.84902 q^{55} -1.16373 q^{59} +11.7136 q^{61} -5.42873 q^{65} +0.305602 q^{67} +12.6192 q^{71} -10.1499 q^{73} -0.758955 q^{77} -15.2491 q^{79} +16.3782 q^{83} -17.1734 q^{85} +9.63151 q^{89} -2.22829 q^{91} -3.34189 q^{95} -10.9344 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 - 4 * q^7	$ 4 q + 4 q^{5} - 4 q^{7} + 6 q^{11} + 3 q^{13} + 8 q^{17} + 2 q^{19} + 5 q^{23} + 14 q^{25} + q^{29} + 11 q^{31} - 4 q^{35} + 27 q^{37} + 2 q^{41} - 11 q^{43} - 7 q^{47} + 4 q^{49} + 4 q^{53} - 6 q^{55} - 9 q^{59} + 7 q^{61} - 9 q^{65} - 12 q^{67} - 12 q^{71} + 13 q^{73} - 6 q^{77} - 22 q^{79} + 6 q^{83} + 11 q^{85} + 14 q^{89} - 3 q^{91} + 23 q^{95} + q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^7 + 6 * q^11 + 3 * q^13 + 8 * q^17 + 2 * q^19 + 5 * q^23 + 14 * q^25 + q^29 + 11 * q^31 - 4 * q^35 + 27 * q^37 + 2 * q^41 - 11 * q^43 - 7 * q^47 + 4 * q^49 + 4 * q^53 - 6 * q^55 - 9 * q^59 + 7 * q^61 - 9 * q^65 - 12 * q^67 - 12 * q^71 + 13 * q^73 - 6 * q^77 - 22 * q^79 + 6 * q^83 + 11 * q^85 + 14 * q^89 - 3 * q^91 + 23 * q^95 + q^97

Display 100 coefficients 10 coefficients

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
$5$	−2.43628	−1.08954	−0.544768	−	0.838587i	$-0.683382\pi$
$5$	−2.43628	−1.08954	−0.544768	+	0.838587i	$0.683382\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.758955	0.228833	0.114417	−	0.993433i	$-0.463500\pi$
$11$	0.758955	0.228833	0.114417	+	0.993433i	$0.463500\pi$
$12$	0	0
$13$	2.22829	0.618016	0.309008	−	0.951059i	$-0.400003\pi$
$13$	2.22829	0.618016	0.309008	+	0.951059i	$0.400003\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.04904	1.70964	0.854822	−	0.518922i	$-0.173666\pi$
$17$	7.04904	1.70964	0.854822	+	0.518922i	$0.173666\pi$
$18$	0	0
$19$	1.37172	0.314694	0.157347	−	0.987543i	$-0.449706\pi$
$19$	1.37172	0.314694	0.157347	+	0.987543i	$0.449706\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.03629	1.46717	0.733583	−	0.679599i	$-0.237846\pi$
$23$	7.03629	1.46717	0.733583	+	0.679599i	$0.237846\pi$
$24$	0	0
$25$	0.935443	0.187089
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.836267	−0.155291	−0.0776455	−	0.996981i	$-0.524740\pi$
$29$	−0.836267	−0.155291	−0.0776455	+	0.996981i	$0.524740\pi$
$30$	0	0
$31$	0.530665	0.0953102	0.0476551	−	0.998864i	$-0.484825\pi$
$31$	0.530665	0.0953102	0.0476551	+	0.998864i	$0.484825\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.43628	0.411806
$36$	0	0
$37$	4.53066	0.744837	0.372418	−	0.928065i	$-0.378529\pi$
$37$	4.53066	0.744837	0.372418	+	0.928065i	$0.378529\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.85704	−1.38324	−0.691618	−	0.722263i	$-0.743102\pi$
$41$	−8.85704	−1.38324	−0.691618	+	0.722263i	$0.743102\pi$
$42$	0	0
$43$	−7.40322	−1.12898	−0.564490	−	0.825440i	$-0.690927\pi$
$43$	−7.40322	−1.12898	−0.564490	+	0.825440i	$0.690927\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.79524	−0.991188	−0.495594	−	0.868554i	$-0.665050\pi$
$47$	−6.79524	−0.991188	−0.495594	+	0.868554i	$0.665050\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.607978	0.0835122	0.0417561	−	0.999128i	$-0.486705\pi$
$53$	0.607978	0.0835122	0.0417561	+	0.999128i	$0.486705\pi$
$54$	0	0
$55$	−1.84902	−0.249322
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.16373	−0.151505	−0.0757525	−	0.997127i	$-0.524136\pi$
$59$	−1.16373	−0.151505	−0.0757525	+	0.997127i	$0.524136\pi$
$60$	0	0
$61$	11.7136	1.49977	0.749887	−	0.661566i	$-0.230108\pi$
$61$	11.7136	1.49977	0.749887	+	0.661566i	$0.230108\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.42873	−0.673351
$66$	0	0
$67$	0.305602	0.0373353	0.0186676	−	0.999826i	$-0.494058\pi$
$67$	0.305602	0.0373353	0.0186676	+	0.999826i	$0.494058\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.6192	1.49763	0.748813	−	0.662781i	$-0.230624\pi$
$71$	12.6192	1.49763	0.748813	+	0.662781i	$0.230624\pi$
$72$	0	0
$73$	−10.1499	−1.18795	−0.593977	−	0.804482i	$-0.702443\pi$
$73$	−10.1499	−1.18795	−0.593977	+	0.804482i	$0.702443\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.758955	−0.0864909
$78$	0	0
$79$	−15.2491	−1.71565	−0.857827	−	0.513939i	$-0.828186\pi$
$79$	−15.2491	−1.71565	−0.857827	+	0.513939i	$0.828186\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.3782	1.79774	0.898869	−	0.438217i	$-0.144390\pi$
$83$	16.3782	1.79774	0.898869	+	0.438217i	$0.144390\pi$
$84$	0	0
$85$	−17.1734	−1.86272
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.63151	1.02094	0.510469	−	0.859896i	$-0.329472\pi$
$89$	9.63151	1.02094	0.510469	+	0.859896i	$0.329472\pi$
$90$	0	0
$91$	−2.22829	−0.233588
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.34189	−0.342870
$96$	0	0
$97$	−10.9344	−1.11022	−0.555108	−	0.831779i	$-0.687323\pi$
$97$	−10.9344	−1.11022	−0.555108	+	0.831779i	$0.687323\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.50083	−0.149339	−0.0746693	−	0.997208i	$-0.523790\pi$
$101$	−1.50083	−0.149339	−0.0746693	+	0.997208i	$0.523790\pi$
$102$	0	0
$103$	−10.2646	−1.01140	−0.505699	−	0.862710i	$-0.668765\pi$
$103$	−10.2646	−1.01140	−0.505699	+	0.862710i	$0.668765\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.84428	−0.758335	−0.379168	−	0.925328i	$-0.623790\pi$
$107$	−7.84428	−0.758335	−0.379168	+	0.925328i	$0.623790\pi$
$108$	0	0
$109$	12.1754	1.16619	0.583096	−	0.812403i	$-0.301841\pi$
$109$	12.1754	1.16619	0.583096	+	0.812403i	$0.301841\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.97173	−0.373629	−0.186814	−	0.982395i	$-0.559816\pi$
$113$	−3.97173	−0.373629	−0.186814	+	0.982395i	$0.559816\pi$
$114$	0	0
$115$	−17.1423	−1.59853
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−7.04904	−0.646185
$120$	0	0
$121$	−10.4240	−0.947635
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.90238	0.885696
$126$	0	0
$127$	−15.7579	−1.39828	−0.699142	−	0.714983i	$-0.746435\pi$
$127$	−15.7579	−1.39828	−0.699142	+	0.714983i	$0.746435\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−11.8758	−1.03759	−0.518796	−	0.854898i	$-0.673620\pi$
$131$	−11.8758	−1.03759	−0.518796	+	0.854898i	$0.673620\pi$
$132$	0	0
$133$	−1.37172	−0.118943
$134$	0	0
$135$	0	0
$136$	0	0
$137$	20.3739	1.74066	0.870328	−	0.492473i	$-0.163907\pi$
$137$	20.3739	1.74066	0.870328	+	0.492473i	$0.163907\pi$
$138$	0	0
$139$	6.26978	0.531796	0.265898	−	0.964001i	$-0.414332\pi$
$139$	6.26978	0.531796	0.265898	+	0.964001i	$0.414332\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.69117	0.141423
$144$	0	0
$145$	2.03738	0.169195
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.91358	0.402536	0.201268	−	0.979536i	$-0.435494\pi$
$149$	4.91358	0.402536	0.201268	+	0.979536i	$0.435494\pi$
$150$	0	0
$151$	5.61755	0.457150	0.228575	−	0.973526i	$-0.426593\pi$
$151$	5.61755	0.457150	0.228575	+	0.973526i	$0.426593\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.29285	−0.103844
$156$	0	0
$157$	20.6347	1.64683	0.823416	−	0.567439i	$-0.192066\pi$
$157$	20.6347	1.64683	0.823416	+	0.567439i	$0.192066\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.03629	−0.554537
$162$	0	0
$163$	−4.95096	−0.387789	−0.193895	−	0.981022i	$-0.562112\pi$
$163$	−4.95096	−0.387789	−0.193895	+	0.981022i	$0.562112\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.971729	−0.0751946	−0.0375973	−	0.999293i	$-0.511970\pi$
$167$	−0.971729	−0.0751946	−0.0375973	+	0.999293i	$0.511970\pi$
$168$	0	0
$169$	−8.03473	−0.618056
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.8192	1.35477	0.677384	−	0.735630i	$-0.263114\pi$
$173$	17.8192	1.35477	0.677384	+	0.735630i	$0.263114\pi$
$174$	0	0
$175$	−0.935443	−0.0707128
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.6065	−0.792764	−0.396382	−	0.918086i	$-0.629734\pi$
$179$	−10.6065	−0.792764	−0.396382	+	0.918086i	$0.629734\pi$
$180$	0	0
$181$	24.7360	1.83862	0.919308	−	0.393539i	$-0.128749\pi$
$181$	24.7360	1.83862	0.919308	+	0.393539i	$0.128749\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−11.0380	−0.811526
$186$	0	0
$187$	5.34990	0.391224
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.5024	1.33879	0.669393	−	0.742908i	$-0.266554\pi$
$191$	18.5024	1.33879	0.669393	+	0.742908i	$0.266554\pi$
$192$	0	0
$193$	16.4901	1.18698	0.593492	−	0.804840i	$-0.297749\pi$
$193$	16.4901	1.18698	0.593492	+	0.804840i	$0.297749\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.68164	0.618541	0.309271	−	0.950974i	$-0.399915\pi$
$197$	8.68164	0.618541	0.309271	+	0.950974i	$0.399915\pi$
$198$	0	0
$199$	19.2352	1.36355	0.681774	−	0.731563i	$-0.261209\pi$
$199$	19.2352	1.36355	0.681774	+	0.731563i	$0.261209\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.836267	0.0586945
$204$	0	0
$205$	21.5782	1.50709
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.04107	0.0720125
$210$	0	0
$211$	−22.7168	−1.56389	−0.781946	−	0.623347i	$-0.785773\pi$
$211$	−22.7168	−1.56389	−0.781946	+	0.623347i	$0.785773\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	18.0363	1.23006
$216$	0	0
$217$	−0.530665	−0.0360239
$218$	0	0
$219$	0	0
$220$	0	0
$221$	15.7073	1.05659
$222$	0	0
$223$	−15.5264	−1.03972	−0.519862	−	0.854250i	$-0.674017\pi$
$223$	−15.5264	−1.03972	−0.519862	+	0.854250i	$0.674017\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.26936	0.150623	0.0753114	−	0.997160i	$-0.476005\pi$
$227$	2.26936	0.150623	0.0753114	+	0.997160i	$0.476005\pi$
$228$	0	0
$229$	9.22074	0.609324	0.304662	−	0.952461i	$-0.401457\pi$
$229$	9.22074	0.609324	0.304662	+	0.952461i	$0.401457\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.0842	1.57781	0.788905	−	0.614515i	$-0.210648\pi$
$233$	24.0842	1.57781	0.788905	+	0.614515i	$0.210648\pi$
$234$	0	0
$235$	16.5551	1.07993
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.00000	0.323423	0.161712	−	0.986838i	$-0.448299\pi$
$239$	5.00000	0.323423	0.161712	+	0.986838i	$0.448299\pi$
$240$	0	0
$241$	−17.2352	−1.11022	−0.555109	−	0.831778i	$-0.687323\pi$
$241$	−17.2352	−1.11022	−0.555109	+	0.831778i	$0.687323\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.43628	−0.155648
$246$	0	0
$247$	3.05659	0.194486
$248$	0	0
$249$	0	0
$250$	0	0
$251$	15.2725	0.963994	0.481997	−	0.876173i	$-0.339912\pi$
$251$	15.2725	0.963994	0.481997	+	0.876173i	$0.339912\pi$
$252$	0	0
$253$	5.34022	0.335737
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.5905	0.722994	0.361497	−	0.932373i	$-0.382266\pi$
$257$	11.5905	0.722994	0.361497	+	0.932373i	$0.382266\pi$
$258$	0	0
$259$	−4.53066	−0.281522
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.11204	−0.500210	−0.250105	−	0.968219i	$-0.580465\pi$
$263$	−8.11204	−0.500210	−0.250105	+	0.968219i	$0.580465\pi$
$264$	0	0
$265$	−1.48120	−0.0909895
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.4598	−0.698717	−0.349358	−	0.936989i	$-0.613600\pi$
$269$	−11.4598	−0.698717	−0.349358	+	0.936989i	$0.613600\pi$
$270$	0	0
$271$	3.30404	0.200706	0.100353	−	0.994952i	$-0.468003\pi$
$271$	3.30404	0.200706	0.100353	+	0.994952i	$0.468003\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.709959	0.0428121
$276$	0	0
$277$	−10.2491	−0.615806	−0.307903	−	0.951418i	$-0.599627\pi$
$277$	−10.2491	−0.615806	−0.307903	+	0.951418i	$0.599627\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.72267	0.102766	0.0513829	−	0.998679i	$-0.483637\pi$
$281$	1.72267	0.102766	0.0513829	+	0.998679i	$0.483637\pi$
$282$	0	0
$283$	−24.1504	−1.43559	−0.717795	−	0.696255i	$-0.754848\pi$
$283$	−24.1504	−1.43559	−0.717795	+	0.696255i	$0.754848\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.85704	0.522814
$288$	0	0
$289$	32.6890	1.92288
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.9307	0.872258	0.436129	−	0.899884i	$-0.356349\pi$
$293$	14.9307	0.872258	0.436129	+	0.899884i	$0.356349\pi$
$294$	0	0
$295$	2.83517	0.165070
$296$	0	0
$297$	0	0
$298$	0	0
$299$	15.6789	0.906733
$300$	0	0
$301$	7.40322	0.426714
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−28.5376	−1.63406
$306$	0	0
$307$	17.9563	1.02482	0.512411	−	0.858741i	$-0.328753\pi$
$307$	17.9563	1.02482	0.512411	+	0.858741i	$0.328753\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.7861	1.23538	0.617689	−	0.786422i	$-0.288069\pi$
$311$	21.7861	1.23538	0.617689	+	0.786422i	$0.288069\pi$
$312$	0	0
$313$	22.1248	1.25057	0.625284	−	0.780398i	$-0.284983\pi$
$313$	22.1248	1.25057	0.625284	+	0.780398i	$0.284983\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.72959	−0.321806	−0.160903	−	0.986970i	$-0.551441\pi$
$317$	−5.72959	−0.321806	−0.160903	+	0.986970i	$0.551441\pi$
$318$	0	0
$319$	−0.634689	−0.0355358
$320$	0	0
$321$	0	0
$322$	0	0
$323$	9.66931	0.538015
$324$	0	0
$325$	2.08444	0.115624
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.79524	0.374634
$330$	0	0
$331$	31.6357	1.73885	0.869427	−	0.494062i	$-0.164488\pi$
$331$	31.6357	1.73885	0.869427	+	0.494062i	$0.164488\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−0.744532	−0.0406781
$336$	0	0
$337$	23.5542	1.28308	0.641539	−	0.767090i	$-0.278296\pi$
$337$	23.5542	1.28308	0.641539	+	0.767090i	$0.278296\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.402751	0.0218102
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−19.6272	−1.05364	−0.526821	−	0.849976i	$-0.676616\pi$
$347$	−19.6272	−1.05364	−0.526821	+	0.849976i	$0.676616\pi$
$348$	0	0
$349$	−26.0726	−1.39563	−0.697816	−	0.716277i	$-0.745845\pi$
$349$	−26.0726	−1.39563	−0.697816	+	0.716277i	$0.745845\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6.61755	−0.352217	−0.176108	−	0.984371i	$-0.556351\pi$
$353$	−6.61755	−0.352217	−0.176108	+	0.984371i	$0.556351\pi$
$354$	0	0
$355$	−30.7439	−1.63172
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−23.0527	−1.21668	−0.608338	−	0.793678i	$-0.708164\pi$
$359$	−23.0527	−1.21668	−0.608338	+	0.793678i	$0.708164\pi$
$360$	0	0
$361$	−17.1184	−0.900968
$362$	0	0
$363$	0	0
$364$	0	0
$365$	24.7279	1.29432
$366$	0	0
$367$	−8.17862	−0.426921	−0.213460	−	0.976952i	$-0.568473\pi$
$367$	−8.17862	−0.426921	−0.213460	+	0.976952i	$0.568473\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−0.607978	−0.0315646
$372$	0	0
$373$	15.5068	0.802913	0.401456	−	0.915878i	$-0.368504\pi$
$373$	15.5068	0.802913	0.401456	+	0.915878i	$0.368504\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.86345	−0.0959723
$378$	0	0
$379$	−18.5814	−0.954461	−0.477231	−	0.878778i	$-0.658359\pi$
$379$	−18.5814	−0.954461	−0.477231	+	0.878778i	$0.658359\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.6192	0.849200	0.424600	−	0.905381i	$-0.360415\pi$
$383$	16.6192	0.849200	0.424600	+	0.905381i	$0.360415\pi$
$384$	0	0
$385$	1.84902	0.0942349
$386$	0	0
$387$	0	0
$388$	0	0
$389$	25.8224	1.30925	0.654624	−	0.755954i	$-0.272827\pi$
$389$	25.8224	1.30925	0.654624	+	0.755954i	$0.272827\pi$
$390$	0	0
$391$	49.5991	2.50833
$392$	0	0
$393$	0	0
$394$	0	0
$395$	37.1509	1.86927
$396$	0	0
$397$	−0.729016	−0.0365883	−0.0182941	−	0.999833i	$-0.505824\pi$
$397$	−0.729016	−0.0365883	−0.0182941	+	0.999833i	$0.505824\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.3419	0.866013	0.433006	−	0.901391i	$-0.357453\pi$
$401$	17.3419	0.866013	0.433006	+	0.901391i	$0.357453\pi$
$402$	0	0
$403$	1.18248	0.0589033
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.43857	0.170444
$408$	0	0
$409$	−3.64702	−0.180334	−0.0901668	−	0.995927i	$-0.528740\pi$
$409$	−3.64702	−0.180334	−0.0901668	+	0.995927i	$0.528740\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1.16373	0.0572635
$414$	0	0
$415$	−39.9018	−1.95870
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.235369	0.0114985	0.00574927	−	0.999983i	$-0.498170\pi$
$419$	0.235369	0.0114985	0.00574927	+	0.999983i	$0.498170\pi$
$420$	0	0
$421$	21.7969	1.06232	0.531158	−	0.847273i	$-0.321757\pi$
$421$	21.7969	1.06232	0.531158	+	0.847273i	$0.321757\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.59398	0.319855
$426$	0	0
$427$	−11.7136	−0.566861
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.28806	−0.206549	−0.103274	−	0.994653i	$-0.532932\pi$
$431$	−4.28806	−0.206549	−0.103274	+	0.994653i	$0.532932\pi$
$432$	0	0
$433$	−13.9263	−0.669257	−0.334628	−	0.942350i	$-0.608611\pi$
$433$	−13.9263	−0.669257	−0.334628	+	0.942350i	$0.608611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.65181	0.461709
$438$	0	0
$439$	31.8795	1.52152	0.760762	−	0.649031i	$-0.224825\pi$
$439$	31.8795	1.52152	0.760762	+	0.649031i	$0.224825\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.12433	0.243464	0.121732	−	0.992563i	$-0.461155\pi$
$443$	5.12433	0.243464	0.121732	+	0.992563i	$0.461155\pi$
$444$	0	0
$445$	−23.4650	−1.11235
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.9914	1.60415	0.802077	−	0.597221i	$-0.203728\pi$
$449$	33.9914	1.60415	0.802077	+	0.597221i	$0.203728\pi$
$450$	0	0
$451$	−6.72209	−0.316531
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.42873	0.254503
$456$	0	0
$457$	5.57652	0.260859	0.130429	−	0.991458i	$-0.458364\pi$
$457$	5.57652	0.260859	0.130429	+	0.991458i	$0.458364\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.49807	−0.116347	−0.0581734	−	0.998306i	$-0.518528\pi$
$461$	−2.49807	−0.116347	−0.0581734	+	0.998306i	$0.518528\pi$
$462$	0	0
$463$	9.55034	0.443842	0.221921	−	0.975065i	$-0.428767\pi$
$463$	9.55034	0.443842	0.221921	+	0.975065i	$0.428767\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.2128	0.842787	0.421393	−	0.906878i	$-0.361541\pi$
$467$	18.2128	0.842787	0.421393	+	0.906878i	$0.361541\pi$
$468$	0	0
$469$	−0.305602	−0.0141114
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−5.61871	−0.258348
$474$	0	0
$475$	1.28317	0.0588757
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.75297	−0.262860	−0.131430	−	0.991325i	$-0.541957\pi$
$479$	−5.75297	−0.262860	−0.131430	+	0.991325i	$0.541957\pi$
$480$	0	0
$481$	10.0956	0.460321
$482$	0	0
$483$	0	0
$484$	0	0
$485$	26.6391	1.20962
$486$	0	0
$487$	−23.8918	−1.08264	−0.541321	−	0.840816i	$-0.682075\pi$
$487$	−23.8918	−1.08264	−0.541321	+	0.840816i	$0.682075\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14.6287	0.660186	0.330093	−	0.943948i	$-0.392920\pi$
$491$	14.6287	0.660186	0.330093	+	0.943948i	$0.392920\pi$
$492$	0	0
$493$	−5.89488	−0.265492
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.6192	−0.566049
$498$	0	0
$499$	23.9158	1.07062	0.535308	−	0.844657i	$-0.320195\pi$
$499$	23.9158	1.07062	0.535308	+	0.844657i	$0.320195\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.57496	0.337751	0.168875	−	0.985637i	$-0.445986\pi$
$503$	7.57496	0.337751	0.168875	+	0.985637i	$0.445986\pi$
$504$	0	0
$505$	3.65645	0.162710
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.00156	0.310339	0.155169	−	0.987888i	$-0.450408\pi$
$509$	7.00156	0.310339	0.155169	+	0.987888i	$0.450408\pi$
$510$	0	0
$511$	10.1499	0.449004
$512$	0	0
$513$	0	0
$514$	0	0
$515$	25.0073	1.10196
$516$	0	0
$517$	−5.15728	−0.226817
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−42.4564	−1.86005	−0.930025	−	0.367497i	$-0.880215\pi$
$521$	−42.4564	−1.86005	−0.930025	+	0.367497i	$0.880215\pi$
$522$	0	0
$523$	18.6096	0.813743	0.406871	−	0.913485i	$-0.366620\pi$
$523$	18.6096	0.813743	0.406871	+	0.913485i	$0.366620\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.74068	0.162947
$528$	0	0
$529$	26.5093	1.15258
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−19.7360	−0.854863
$534$	0	0
$535$	19.1108	0.826234
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.758955	0.0326905
$540$	0	0
$541$	−0.694443	−0.0298564	−0.0149282	−	0.999889i	$-0.504752\pi$
$541$	−0.694443	−0.0298564	−0.0149282	+	0.999889i	$0.504752\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−29.6626	−1.27061
$546$	0	0
$547$	−17.5531	−0.750516	−0.375258	−	0.926920i	$-0.622446\pi$
$547$	−17.5531	−0.750516	−0.375258	+	0.926920i	$0.622446\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.14712	−0.0488691
$552$	0	0
$553$	15.2491	0.648456
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.2666	−0.604495	−0.302248	−	0.953229i	$-0.597737\pi$
$557$	−14.2666	−0.604495	−0.302248	+	0.953229i	$0.597737\pi$
$558$	0	0
$559$	−16.4965	−0.697728
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−33.7484	−1.42232	−0.711162	−	0.703028i	$-0.751831\pi$
$563$	−33.7484	−1.42232	−0.711162	+	0.703028i	$0.751831\pi$
$564$	0	0
$565$	9.67623	0.407082
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−39.9734	−1.67577	−0.837886	−	0.545845i	$-0.816209\pi$
$569$	−39.9734	−1.67577	−0.837886	+	0.545845i	$0.816209\pi$
$570$	0	0
$571$	−40.4532	−1.69291	−0.846457	−	0.532458i	$-0.821269\pi$
$571$	−40.4532	−1.69291	−0.846457	+	0.532458i	$0.821269\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.58204	0.274490
$576$	0	0
$577$	−31.6758	−1.31868	−0.659340	−	0.751845i	$-0.729164\pi$
$577$	−31.6758	−1.31868	−0.659340	+	0.751845i	$0.729164\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−16.3782	−0.679481
$582$	0	0
$583$	0.461427	0.0191104
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.3044	0.631680	0.315840	−	0.948812i	$-0.397714\pi$
$587$	15.3044	0.631680	0.315840	+	0.948812i	$0.397714\pi$
$588$	0	0
$589$	0.727923	0.0299936
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−39.2583	−1.61214	−0.806072	−	0.591817i	$-0.798411\pi$
$593$	−39.2583	−1.61214	−0.806072	+	0.591817i	$0.798411\pi$
$594$	0	0
$595$	17.1734	0.704041
$596$	0	0
$597$	0	0
$598$	0	0
$599$	17.5594	0.717460	0.358730	−	0.933441i	$-0.383210\pi$
$599$	17.5594	0.717460	0.358730	+	0.933441i	$0.383210\pi$
$600$	0	0
$601$	18.2587	0.744790	0.372395	−	0.928074i	$-0.378537\pi$
$601$	18.2587	0.744790	0.372395	+	0.928074i	$0.378537\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	25.3957	1.03248
$606$	0	0
$607$	7.42831	0.301506	0.150753	−	0.988571i	$-0.451830\pi$
$607$	7.42831	0.301506	0.150753	+	0.988571i	$0.451830\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.1418	−0.612570
$612$	0	0
$613$	−39.8772	−1.61062	−0.805312	−	0.592851i	$-0.798002\pi$
$613$	−39.8772	−1.61062	−0.805312	+	0.592851i	$0.798002\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.374947	−0.0150948	−0.00754739	−	0.999972i	$-0.502402\pi$
$617$	−0.374947	−0.0150948	−0.00754739	+	0.999972i	$0.502402\pi$
$618$	0	0
$619$	13.9033	0.558822	0.279411	−	0.960172i	$-0.409861\pi$
$619$	13.9033	0.558822	0.279411	+	0.960172i	$0.409861\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−9.63151	−0.385878
$624$	0	0
$625$	−28.8022	−1.15209
$626$	0	0
$627$	0	0
$628$	0	0
$629$	31.9368	1.27341
$630$	0	0
$631$	−28.6049	−1.13874	−0.569372	−	0.822080i	$-0.692813\pi$
$631$	−28.6049	−1.13874	−0.569372	+	0.822080i	$0.692813\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	38.3905	1.52348
$636$	0	0
$637$	2.22829	0.0882880
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21.8021	0.861132	0.430566	−	0.902559i	$-0.358314\pi$
$641$	21.8021	0.861132	0.430566	+	0.902559i	$0.358314\pi$
$642$	0	0
$643$	41.8411	1.65005	0.825025	−	0.565096i	$-0.191161\pi$
$643$	41.8411	1.65005	0.825025	+	0.565096i	$0.191161\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.2614	0.678615	0.339308	−	0.940675i	$-0.389807\pi$
$647$	17.2614	0.678615	0.339308	+	0.940675i	$0.389807\pi$
$648$	0	0
$649$	−0.883220	−0.0346694
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.475322	−0.0186008	−0.00930039	−	0.999957i	$-0.502960\pi$
$653$	−0.475322	−0.0186008	−0.00930039	+	0.999957i	$0.502960\pi$
$654$	0	0
$655$	28.9327	1.13049
$656$	0	0
$657$	0	0
$658$	0	0
$659$	36.4687	1.42062	0.710310	−	0.703889i	$-0.248555\pi$
$659$	36.4687	1.42062	0.710310	+	0.703889i	$0.248555\pi$
$660$	0	0
$661$	10.0996	0.392831	0.196415	−	0.980521i	$-0.437070\pi$
$661$	10.0996	0.392831	0.196415	+	0.980521i	$0.437070\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.34189	0.129593
$666$	0	0
$667$	−5.88422	−0.227838
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.89010	0.343198
$672$	0	0
$673$	−19.0209	−0.733201	−0.366600	−	0.930378i	$-0.619478\pi$
$673$	−19.0209	−0.733201	−0.366600	+	0.930378i	$0.619478\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	45.2747	1.74005	0.870024	−	0.493009i	$-0.164103\pi$
$677$	45.2747	1.74005	0.870024	+	0.493009i	$0.164103\pi$
$678$	0	0
$679$	10.9344	0.419622
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−45.5318	−1.74222	−0.871112	−	0.491084i	$-0.836601\pi$
$683$	−45.5318	−1.74222	−0.871112	+	0.491084i	$0.836601\pi$
$684$	0	0
$685$	−49.6363	−1.89651
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.35475	0.0516119
$690$	0	0
$691$	16.1889	0.615856	0.307928	−	0.951410i	$-0.400364\pi$
$691$	16.1889	0.615856	0.307928	+	0.951410i	$0.400364\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.2749	−0.579411
$696$	0	0
$697$	−62.4336	−2.36484
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.61120	0.0986238	0.0493119	−	0.998783i	$-0.484297\pi$
$701$	2.61120	0.0986238	0.0493119	+	0.998783i	$0.484297\pi$
$702$	0	0
$703$	6.21480	0.234396
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.50083	0.0564447
$708$	0	0
$709$	46.3404	1.74035	0.870175	−	0.492743i	$-0.164006\pi$
$709$	46.3404	1.74035	0.870175	+	0.492743i	$0.164006\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.73391	0.139836
$714$	0	0
$715$	−4.12016	−0.154085
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−46.2069	−1.72323	−0.861614	−	0.507564i	$-0.830546\pi$
$719$	−46.2069	−1.72323	−0.861614	+	0.507564i	$0.830546\pi$
$720$	0	0
$721$	10.2646	0.382273
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.782280	−0.0290532
$726$	0	0
$727$	19.4598	0.721723	0.360861	−	0.932620i	$-0.382483\pi$
$727$	19.4598	0.721723	0.360861	+	0.932620i	$0.382483\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−52.1856	−1.93015
$732$	0	0
$733$	15.3723	0.567790	0.283895	−	0.958855i	$-0.408373\pi$
$733$	15.3723	0.567790	0.283895	+	0.958855i	$0.408373\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.231938	0.00854356
$738$	0	0
$739$	32.4962	1.19539	0.597696	−	0.801723i	$-0.296083\pi$
$739$	32.4962	1.19539	0.597696	+	0.801723i	$0.296083\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	41.7328	1.53103	0.765514	−	0.643419i	$-0.222485\pi$
$743$	41.7328	1.53103	0.765514	+	0.643419i	$0.222485\pi$
$744$	0	0
$745$	−11.9708	−0.438578
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.84428	0.286624
$750$	0	0
$751$	−10.0490	−0.366695	−0.183347	−	0.983048i	$-0.558693\pi$
$751$	−10.0490	−0.366695	−0.183347	+	0.983048i	$0.558693\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−13.6859	−0.498081
$756$	0	0
$757$	−31.7989	−1.15575	−0.577876	−	0.816125i	$-0.696118\pi$
$757$	−31.7989	−1.15575	−0.577876	+	0.816125i	$0.696118\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	54.8705	1.98905	0.994526	−	0.104487i	$-0.0333200\pi$
$761$	54.8705	1.98905	0.994526	+	0.104487i	$0.0333200\pi$
$762$	0	0
$763$	−12.1754	−0.440779
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.59313	−0.0936326
$768$	0	0
$769$	−10.7808	−0.388766	−0.194383	−	0.980926i	$-0.562270\pi$
$769$	−10.7808	−0.388766	−0.194383	+	0.980926i	$0.562270\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	47.4885	1.70804	0.854022	−	0.520237i	$-0.174156\pi$
$773$	47.4885	1.70804	0.854022	+	0.520237i	$0.174156\pi$
$774$	0	0
$775$	0.496407	0.0178315
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.1494	−0.435296
$780$	0	0
$781$	9.57741	0.342707
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−50.2719	−1.79428
$786$	0	0
$787$	24.1125	0.859518	0.429759	−	0.902944i	$-0.358599\pi$
$787$	24.1125	0.859518	0.429759	+	0.902944i	$0.358599\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.97173	0.141218
$792$	0	0
$793$	26.1013	0.926885
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.0924	0.534600	0.267300	−	0.963613i	$-0.413868\pi$
$797$	15.0924	0.534600	0.267300	+	0.963613i	$0.413868\pi$
$798$	0	0
$799$	−47.8999	−1.69458
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−7.70330	−0.271844
$804$	0	0
$805$	17.1423	0.604188
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−20.1104	−0.707043	−0.353522	−	0.935426i	$-0.615016\pi$
$809$	−20.1104	−0.707043	−0.353522	+	0.935426i	$0.615016\pi$
$810$	0	0
$811$	40.9770	1.43890	0.719448	−	0.694546i	$-0.244395\pi$
$811$	40.9770	1.43890	0.719448	+	0.694546i	$0.244395\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12.0619	0.422510
$816$	0	0
$817$	−10.1551	−0.355283
$818$	0	0
$819$	0	0
$820$	0	0
$821$	23.8854	0.833607	0.416803	−	0.908997i	$-0.363150\pi$
$821$	23.8854	0.833607	0.416803	+	0.908997i	$0.363150\pi$
$822$	0	0
$823$	−29.8902	−1.04191	−0.520954	−	0.853585i	$-0.674424\pi$
$823$	−29.8902	−1.04191	−0.520954	+	0.853585i	$0.674424\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−44.7265	−1.55529	−0.777647	−	0.628702i	$-0.783587\pi$
$827$	−44.7265	−1.55529	−0.777647	+	0.628702i	$0.783587\pi$
$828$	0	0
$829$	36.8943	1.28139	0.640695	−	0.767795i	$-0.278646\pi$
$829$	36.8943	1.28139	0.640695	+	0.767795i	$0.278646\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.04904	0.244235
$834$	0	0
$835$	2.36740	0.0819272
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.83689	0.0634166	0.0317083	−	0.999497i	$-0.489905\pi$
$839$	1.83689	0.0634166	0.0317083	+	0.999497i	$0.489905\pi$
$840$	0	0
$841$	−28.3007	−0.975885
$842$	0	0
$843$	0	0
$844$	0	0
$845$	19.5748	0.673394
$846$	0	0
$847$	10.4240	0.358172
$848$	0	0
$849$	0	0
$850$	0	0
$851$	31.8791	1.09280
$852$	0	0
$853$	−52.0770	−1.78308	−0.891542	−	0.452938i	$-0.850376\pi$
$853$	−52.0770	−1.78308	−0.891542	+	0.452938i	$0.850376\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.9333	0.646748	0.323374	−	0.946271i	$-0.395183\pi$
$857$	18.9333	0.646748	0.323374	+	0.946271i	$0.395183\pi$
$858$	0	0
$859$	8.01400	0.273434	0.136717	−	0.990610i	$-0.456345\pi$
$859$	8.01400	0.273434	0.136717	+	0.990610i	$0.456345\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−23.8647	−0.812364	−0.406182	−	0.913792i	$-0.633140\pi$
$863$	−23.8647	−0.812364	−0.406182	+	0.913792i	$0.633140\pi$
$864$	0	0
$865$	−43.4125	−1.47607
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−11.5733	−0.392599
$870$	0	0
$871$	0.680971	0.0230738
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.90238	−0.334762
$876$	0	0
$877$	−34.0476	−1.14971	−0.574853	−	0.818257i	$-0.694941\pi$
$877$	−34.0476	−1.14971	−0.574853	+	0.818257i	$0.694941\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	38.8018	1.30727	0.653633	−	0.756812i	$-0.273244\pi$
$881$	38.8018	1.30727	0.653633	+	0.756812i	$0.273244\pi$
$882$	0	0
$883$	42.0944	1.41659	0.708294	−	0.705917i	$-0.249465\pi$
$883$	42.0944	1.41659	0.708294	+	0.705917i	$0.249465\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.2335	0.813683	0.406841	−	0.913499i	$-0.366630\pi$
$887$	24.2335	0.813683	0.406841	+	0.913499i	$0.366630\pi$
$888$	0	0
$889$	15.7579	0.528502
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.32116	−0.311921
$894$	0	0
$895$	25.8403	0.863745
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.443778	−0.0148008
$900$	0	0
$901$	4.28566	0.142776
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−60.2638	−2.00324
$906$	0	0
$907$	25.8849	0.859494	0.429747	−	0.902949i	$-0.358603\pi$
$907$	25.8849	0.859494	0.429747	+	0.902949i	$0.358603\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−43.9142	−1.45494	−0.727471	−	0.686138i	$-0.759305\pi$
$911$	−43.9142	−1.45494	−0.727471	+	0.686138i	$0.759305\pi$
$912$	0	0
$913$	12.4303	0.411383
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.8758	0.392173
$918$	0	0
$919$	−41.4335	−1.36677	−0.683383	−	0.730060i	$-0.739492\pi$
$919$	−41.4335	−1.36677	−0.683383	+	0.730060i	$0.739492\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	28.1193	0.925557
$924$	0	0
$925$	4.23818	0.139350
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.13822	0.102962	0.0514808	−	0.998674i	$-0.483606\pi$
$929$	3.13822	0.102962	0.0514808	+	0.998674i	$0.483606\pi$
$930$	0	0
$931$	1.37172	0.0449563
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−13.0338	−0.426252
$936$	0	0
$937$	10.2810	0.335866	0.167933	−	0.985798i	$-0.446291\pi$
$937$	10.2810	0.335866	0.167933	+	0.985798i	$0.446291\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	14.4769	0.471933	0.235966	−	0.971761i	$-0.424174\pi$
$941$	14.4769	0.471933	0.235966	+	0.971761i	$0.424174\pi$
$942$	0	0
$943$	−62.3206	−2.02944
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.7428	0.966512	0.483256	−	0.875479i	$-0.339454\pi$
$947$	29.7428	0.966512	0.483256	+	0.875479i	$0.339454\pi$
$948$	0	0
$949$	−22.6169	−0.734175
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19.4869	−0.631242	−0.315621	−	0.948885i	$-0.602213\pi$
$953$	−19.4869	−0.631242	−0.315621	+	0.948885i	$0.602213\pi$
$954$	0	0
$955$	−45.0769	−1.45866
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20.3739	−0.657906
$960$	0	0
$961$	−30.7184	−0.990916
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−40.1745	−1.29326
$966$	0	0
$967$	13.8988	0.446957	0.223478	−	0.974709i	$-0.428259\pi$
$967$	13.8988	0.446957	0.223478	+	0.974709i	$0.428259\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.2406	0.617460	0.308730	−	0.951150i	$-0.400096\pi$
$971$	19.2406	0.617460	0.308730	+	0.951150i	$0.400096\pi$
$972$	0	0
$973$	−6.26978	−0.201000
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.88958	−0.156432	−0.0782158	−	0.996936i	$-0.524922\pi$
$977$	−4.88958	−0.156432	−0.0782158	+	0.996936i	$0.524922\pi$
$978$	0	0
$979$	7.30988	0.233625
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−29.9340	−0.954746	−0.477373	−	0.878701i	$-0.658411\pi$
$983$	−29.9340	−0.954746	−0.477373	+	0.878701i	$0.658411\pi$
$984$	0	0
$985$	−21.1509	−0.673923
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−52.0912	−1.65640
$990$	0	0
$991$	9.36157	0.297380	0.148690	−	0.988884i	$-0.452494\pi$
$991$	9.36157	0.297380	0.148690	+	0.988884i	$0.452494\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−46.8623	−1.48563
$996$	0	0
$997$	−47.6421	−1.50884	−0.754420	−	0.656392i	$-0.772082\pi$
$997$	−47.6421	−1.50884	−0.754420	+	0.656392i	$0.772082\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.cl.1.1		4
3.2	odd	2		9072.2.a.ce.1.4		4
4.3	odd	2		4536.2.a.ba.1.1		4
9.2	odd	6		1008.2.r.m.337.2		8
9.4	even	3		3024.2.r.l.2017.4		8
9.5	odd	6		1008.2.r.m.673.2		8
9.7	even	3		3024.2.r.l.1009.4		8
12.11	even	2		4536.2.a.x.1.4		4
36.7	odd	6		1512.2.r.d.1009.4		8
36.11	even	6		504.2.r.d.337.3	yes	8
36.23	even	6		504.2.r.d.169.3	✓	8
36.31	odd	6		1512.2.r.d.505.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.r.d.169.3	✓	8	36.23	even	6
504.2.r.d.337.3	yes	8	36.11	even	6
1008.2.r.m.337.2		8	9.2	odd	6
1008.2.r.m.673.2		8	9.5	odd	6
1512.2.r.d.505.4		8	36.31	odd	6
1512.2.r.d.1009.4		8	36.7	odd	6
3024.2.r.l.1009.4		8	9.7	even	3
3024.2.r.l.2017.4		8	9.4	even	3
4536.2.a.x.1.4		4	12.11	even	2
4536.2.a.ba.1.1		4	4.3	odd	2
9072.2.a.ce.1.4		4	3.2	odd	2
9072.2.a.cl.1.1		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:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q-2.43628 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-2.43628 q^{5} -1.00000 q^{7} +0.758955 q^{11} +2.22829 q^{13} +7.04904 q^{17} +1.37172 q^{19} +7.03629 q^{23} +0.935443 q^{25} -0.836267 q^{29} +0.530665 q^{31} +2.43628 q^{35} +4.53066 q^{37} -8.85704 q^{41} -7.40322 q^{43} -6.79524 q^{47} +1.00000 q^{49} +0.607978 q^{53} -1.84902 q^{55} -1.16373 q^{59} +11.7136 q^{61} -5.42873 q^{65} +0.305602 q^{67} +12.6192 q^{71} -10.1499 q^{73} -0.758955 q^{77} -15.2491 q^{79} +16.3782 q^{83} -17.1734 q^{85} +9.63151 q^{89} -2.22829 q^{91} -3.34189 q^{95} -10.9344 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 - 4 * q^7	\( 4 q + 4 q^{5} - 4 q^{7} + 6 q^{11} + 3 q^{13} + 8 q^{17} + 2 q^{19} + 5 q^{23} + 14 q^{25} + q^{29} + 11 q^{31} - 4 q^{35} + 27 q^{37} + 2 q^{41} - 11 q^{43} - 7 q^{47} + 4 q^{49} + 4 q^{53} - 6 q^{55} - 9 q^{59} + 7 q^{61} - 9 q^{65} - 12 q^{67} - 12 q^{71} + 13 q^{73} - 6 q^{77} - 22 q^{79} + 6 q^{83} + 11 q^{85} + 14 q^{89} - 3 q^{91} + 23 q^{95} + q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^7 + 6 * q^11 + 3 * q^13 + 8 * q^17 + 2 * q^19 + 5 * q^23 + 14 * q^25 + q^29 + 11 * q^31 - 4 * q^35 + 27 * q^37 + 2 * q^41 - 11 * q^43 - 7 * q^47 + 4 * q^49 + 4 * q^53 - 6 * q^55 - 9 * q^59 + 7 * q^61 - 9 * q^65 - 12 * q^67 - 12 * q^71 + 13 * q^73 - 6 * q^77 - 22 * q^79 + 6 * q^83 + 11 * q^85 + 14 * q^89 - 3 * q^91 + 23 * q^95 + q^97

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