LMFDB - Embedded newform 9072.2.a.ce.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:	$1$
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			$1.45106$ 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)

$f(q)$	$=$	$q-3.74893 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-3.74893 q^{5} -1.00000 q^{7} +3.65105 q^{11} -5.54548 q^{13} -7.20767 q^{17} +3.30555 q^{19} +4.98886 q^{23} +9.05449 q^{25} +0.490993 q^{29} +3.89443 q^{31} +3.74893 q^{35} +7.89443 q^{37} +4.76429 q^{41} +1.60343 q^{43} -9.63990 q^{47} +1.00000 q^{49} -8.03647 q^{53} -13.6875 q^{55} +1.50901 q^{59} -2.08674 q^{61} +20.7896 q^{65} -3.40344 q^{67} +10.7301 q^{71} +9.83567 q^{73} -3.65105 q^{77} -3.72782 q^{79} +11.3812 q^{83} +27.0211 q^{85} +7.14891 q^{89} +5.54548 q^{91} -12.3923 q^{95} -10.9063 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−3.74893	−1.67657	−0.838287	−	0.545230i	$-0.816442\pi$
$5$	−3.74893	−1.67657	−0.838287	+	0.545230i	$0.816442\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.65105	1.10083	0.550416	−	0.834891i	$-0.314469\pi$
$11$	3.65105	1.10083	0.550416	+	0.834891i	$0.314469\pi$
$12$	0	0
$13$	−5.54548	−1.53804	−0.769020	−	0.639225i	$-0.779255\pi$
$13$	−5.54548	−1.53804	−0.769020	+	0.639225i	$0.779255\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.20767	−1.74812	−0.874058	−	0.485821i	$-0.838521\pi$
$17$	−7.20767	−1.74812	−0.874058	+	0.485821i	$0.838521\pi$
$18$	0	0
$19$	3.30555	0.758346	0.379173	−	0.925326i	$-0.376208\pi$
$19$	3.30555	0.758346	0.379173	+	0.925326i	$0.376208\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.98886	1.04025	0.520124	−	0.854091i	$-0.325886\pi$
$23$	4.98886	1.04025	0.520124	+	0.854091i	$0.325886\pi$
$24$	0	0
$25$	9.05449	1.81090
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.490993	0.0911752	0.0455876	−	0.998960i	$-0.485484\pi$
$29$	0.490993	0.0911752	0.0455876	+	0.998960i	$0.485484\pi$
$30$	0	0
$31$	3.89443	0.699461	0.349730	−	0.936850i	$-0.386273\pi$
$31$	3.89443	0.699461	0.349730	+	0.936850i	$0.386273\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.74893	0.633685
$36$	0	0
$37$	7.89443	1.29784	0.648918	−	0.760858i	$-0.275222\pi$
$37$	7.89443	1.29784	0.648918	+	0.760858i	$0.275222\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.76429	0.744057	0.372029	−	0.928221i	$-0.378662\pi$
$41$	4.76429	0.744057	0.372029	+	0.928221i	$0.378662\pi$
$42$	0	0
$43$	1.60343	0.244521	0.122260	−	0.992498i	$-0.460986\pi$
$43$	1.60343	0.244521	0.122260	+	0.992498i	$0.460986\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.63990	−1.40612	−0.703062	−	0.711129i	$-0.748184\pi$
$47$	−9.63990	−1.40612	−0.703062	+	0.711129i	$0.748184\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.03647	−1.10389	−0.551947	−	0.833879i	$-0.686115\pi$
$53$	−8.03647	−1.10389	−0.551947	+	0.833879i	$0.686115\pi$
$54$	0	0
$55$	−13.6875	−1.84562
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.50901	0.196456	0.0982280	−	0.995164i	$-0.468683\pi$
$59$	1.50901	0.196456	0.0982280	+	0.995164i	$0.468683\pi$
$60$	0	0
$61$	−2.08674	−0.267180	−0.133590	−	0.991037i	$-0.542651\pi$
$61$	−2.08674	−0.267180	−0.133590	+	0.991037i	$0.542651\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	20.7896	2.57863
$66$	0	0
$67$	−3.40344	−0.415796	−0.207898	−	0.978150i	$-0.566662\pi$
$67$	−3.40344	−0.415796	−0.207898	+	0.978150i	$0.566662\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.7301	1.27343	0.636715	−	0.771099i	$-0.280293\pi$
$71$	10.7301	1.27343	0.636715	+	0.771099i	$0.280293\pi$
$72$	0	0
$73$	9.83567	1.15118	0.575589	−	0.817739i	$-0.304773\pi$
$73$	9.83567	1.15118	0.575589	+	0.817739i	$0.304773\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.65105	−0.416075
$78$	0	0
$79$	−3.72782	−0.419412	−0.209706	−	0.977764i	$-0.567251\pi$
$79$	−3.72782	−0.419412	−0.209706	+	0.977764i	$0.567251\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.3812	1.24924	0.624622	−	0.780927i	$-0.285253\pi$
$83$	11.3812	1.24924	0.624622	+	0.780927i	$0.285253\pi$
$84$	0	0
$85$	27.0211	2.93084
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.14891	0.757783	0.378891	−	0.925441i	$-0.376305\pi$
$89$	7.14891	0.757783	0.378891	+	0.925441i	$0.376305\pi$
$90$	0	0
$91$	5.54548	0.581324
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−12.3923	−1.27142
$96$	0	0
$97$	−10.9063	−1.10737	−0.553685	−	0.832726i	$-0.686779\pi$
$97$	−10.9063	−1.10737	−0.553685	+	0.832726i	$0.686779\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.8034	−1.27399	−0.636994	−	0.770869i	$-0.719822\pi$
$101$	−12.8034	−1.27399	−0.636994	+	0.770869i	$0.719822\pi$
$102$	0	0
$103$	9.53433	0.939446	0.469723	−	0.882814i	$-0.344354\pi$
$103$	9.53433	0.939446	0.469723	+	0.882814i	$0.344354\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.43223	−0.815175	−0.407587	−	0.913166i	$-0.633630\pi$
$107$	−8.43223	−0.815175	−0.407587	+	0.913166i	$0.633630\pi$
$108$	0	0
$109$	16.5574	1.58591	0.792954	−	0.609281i	$-0.208542\pi$
$109$	16.5574	1.58591	0.792954	+	0.609281i	$0.208542\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.0656299	0.00617394	0.00308697	−	0.999995i	$-0.499017\pi$
$113$	0.0656299	0.00617394	0.00308697	+	0.999995i	$0.499017\pi$
$114$	0	0
$115$	−18.7029	−1.74405
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.20767	0.660726
$120$	0	0
$121$	2.33013	0.211830
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−15.2000	−1.35953
$126$	0	0
$127$	−3.20080	−0.284025	−0.142012	−	0.989865i	$-0.545357\pi$
$127$	−3.20080	−0.284025	−0.142012	+	0.989865i	$0.545357\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.3412	−1.34037	−0.670184	−	0.742195i	$-0.733785\pi$
$131$	−15.3412	−1.34037	−0.670184	+	0.742195i	$0.733785\pi$
$132$	0	0
$133$	−3.30555	−0.286628
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.685955	0.0586051	0.0293025	−	0.999571i	$-0.490671\pi$
$137$	0.685955	0.0586051	0.0293025	+	0.999571i	$0.490671\pi$
$138$	0	0
$139$	20.2007	1.71340	0.856702	−	0.515811i	$-0.172509\pi$
$139$	20.2007	1.71340	0.856702	+	0.515811i	$0.172509\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−20.2468	−1.69312
$144$	0	0
$145$	−1.84070	−0.152862
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.63303	−0.707246	−0.353623	−	0.935388i	$-0.615050\pi$
$149$	−8.63303	−0.707246	−0.353623	+	0.935388i	$0.615050\pi$
$150$	0	0
$151$	10.8767	0.885135	0.442568	−	0.896735i	$-0.354068\pi$
$151$	10.8767	0.885135	0.442568	+	0.896735i	$0.354068\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−14.6000	−1.17270
$156$	0	0
$157$	−10.9923	−0.877278	−0.438639	−	0.898663i	$-0.644539\pi$
$157$	−10.9923	−0.877278	−0.438639	+	0.898663i	$0.644539\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.98886	−0.393177
$162$	0	0
$163$	−4.79233	−0.375364	−0.187682	−	0.982230i	$-0.560098\pi$
$163$	−4.79233	−0.375364	−0.187682	+	0.982230i	$0.560098\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.93437	−0.227068	−0.113534	−	0.993534i	$-0.536217\pi$
$167$	−2.93437	−0.227068	−0.113534	+	0.993534i	$0.536217\pi$
$168$	0	0
$169$	17.7523	1.36556
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.9897	−0.911557	−0.455779	−	0.890093i	$-0.650639\pi$
$173$	−11.9897	−0.911557	−0.455779	+	0.890093i	$0.650639\pi$
$174$	0	0
$175$	−9.05449	−0.684455
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−24.9266	−1.86310	−0.931552	−	0.363608i	$-0.881545\pi$
$179$	−24.9266	−1.86310	−0.931552	+	0.363608i	$0.881545\pi$
$180$	0	0
$181$	−21.4203	−1.59216	−0.796078	−	0.605194i	$-0.793096\pi$
$181$	−21.4203	−1.59216	−0.796078	+	0.605194i	$0.793096\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−29.5957	−2.17592
$186$	0	0
$187$	−26.3155	−1.92438
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−17.9601	−1.29954	−0.649772	−	0.760129i	$-0.725136\pi$
$191$	−17.9601	−1.29954	−0.649772	+	0.760129i	$0.725136\pi$
$192$	0	0
$193$	9.37886	0.675105	0.337553	−	0.941307i	$-0.390401\pi$
$193$	9.37886	0.675105	0.337553	+	0.941307i	$0.390401\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.206917	−0.0147422	−0.00737110	−	0.999973i	$-0.502346\pi$
$197$	−0.206917	−0.0147422	−0.00737110	+	0.999973i	$0.502346\pi$
$198$	0	0
$199$	−12.6169	−0.894386	−0.447193	−	0.894438i	$-0.647576\pi$
$199$	−12.6169	−0.894386	−0.447193	+	0.894438i	$0.647576\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.490993	−0.0344610
$204$	0	0
$205$	−17.8610	−1.24747
$206$	0	0
$207$	0	0
$208$	0	0
$209$	12.0687	0.834811
$210$	0	0
$211$	5.93010	0.408245	0.204122	−	0.978945i	$-0.434566\pi$
$211$	5.93010	0.408245	0.204122	+	0.978945i	$0.434566\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.01114	−0.409957
$216$	0	0
$217$	−3.89443	−0.264371
$218$	0	0
$219$	0	0
$220$	0	0
$221$	39.9700	2.68867
$222$	0	0
$223$	3.60999	0.241743	0.120871	−	0.992668i	$-0.461431\pi$
$223$	3.60999	0.241743	0.120871	+	0.992668i	$0.461431\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.6142	1.23547	0.617734	−	0.786387i	$-0.288051\pi$
$227$	18.6142	1.23547	0.617734	+	0.786387i	$0.288051\pi$
$228$	0	0
$229$	22.9931	1.51942	0.759712	−	0.650259i	$-0.225340\pi$
$229$	22.9931	1.51942	0.759712	+	0.650259i	$0.225340\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.07066	−0.266678	−0.133339	−	0.991071i	$-0.542570\pi$
$233$	−4.07066	−0.266678	−0.133339	+	0.991071i	$0.542570\pi$
$234$	0	0
$235$	36.1393	2.35747
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.00000	−0.323423	−0.161712	−	0.986838i	$-0.551701\pi$
$239$	−5.00000	−0.323423	−0.161712	+	0.986838i	$0.551701\pi$
$240$	0	0
$241$	14.6169	0.941555	0.470777	−	0.882252i	$-0.343973\pi$
$241$	14.6169	0.941555	0.470777	+	0.882252i	$0.343973\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.74893	−0.239510
$246$	0	0
$247$	−18.3309	−1.16637
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−8.74206	−0.551794	−0.275897	−	0.961187i	$-0.588975\pi$
$251$	−8.74206	−0.551794	−0.275897	+	0.961187i	$0.588975\pi$
$252$	0	0
$253$	18.2145	1.14514
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.2798	1.32740	0.663699	−	0.748000i	$-0.268986\pi$
$257$	21.2798	1.32740	0.663699	+	0.748000i	$0.268986\pi$
$258$	0	0
$259$	−7.89443	−0.490536
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.6103	−0.839247	−0.419623	−	0.907698i	$-0.637838\pi$
$263$	−13.6103	−0.839247	−0.419623	+	0.907698i	$0.637838\pi$
$264$	0	0
$265$	30.1282	1.85076
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−18.9343	−1.15445	−0.577223	−	0.816587i	$-0.695864\pi$
$269$	−18.9343	−1.15445	−0.577223	+	0.816587i	$0.695864\pi$
$270$	0	0
$271$	−14.1669	−0.860579	−0.430290	−	0.902691i	$-0.641589\pi$
$271$	−14.1669	−0.860579	−0.430290	+	0.902691i	$0.641589\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	33.0583	1.99349
$276$	0	0
$277$	1.27218	0.0764379	0.0382190	−	0.999269i	$-0.487832\pi$
$277$	1.27218	0.0764379	0.0382190	+	0.999269i	$0.487832\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−9.33781	−0.557047	−0.278524	−	0.960429i	$-0.589845\pi$
$281$	−9.33781	−0.557047	−0.278524	+	0.960429i	$0.589845\pi$
$282$	0	0
$283$	−9.77965	−0.581340	−0.290670	−	0.956823i	$-0.593878\pi$
$283$	−9.77965	−0.581340	−0.290670	+	0.956823i	$0.593878\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.76429	−0.281227
$288$	0	0
$289$	34.9505	2.05591
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.1344	−1.40994	−0.704972	−	0.709235i	$-0.749041\pi$
$293$	−24.1344	−1.40994	−0.704972	+	0.709235i	$0.749041\pi$
$294$	0	0
$295$	−5.65716	−0.329373
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−27.6656	−1.59994
$300$	0	0
$301$	−1.60343	−0.0924201
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.82305	0.447947
$306$	0	0
$307$	−20.0425	−1.14389	−0.571944	−	0.820293i	$-0.693810\pi$
$307$	−20.0425	−1.14389	−0.571944	+	0.820293i	$0.693810\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−13.1352	−0.744827	−0.372414	−	0.928067i	$-0.621470\pi$
$311$	−13.1352	−0.744827	−0.372414	+	0.928067i	$0.621470\pi$
$312$	0	0
$313$	12.5862	0.711416	0.355708	−	0.934597i	$-0.384240\pi$
$313$	12.5862	0.711416	0.355708	+	0.934597i	$0.384240\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.7336	−0.602857	−0.301429	−	0.953489i	$-0.597464\pi$
$317$	−10.7336	−0.602857	−0.301429	+	0.953489i	$0.597464\pi$
$318$	0	0
$319$	1.79264	0.100368
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−23.8253	−1.32568
$324$	0	0
$325$	−50.2115	−2.78523
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.63990	0.531465
$330$	0	0
$331$	11.2384	0.617718	0.308859	−	0.951108i	$-0.400053\pi$
$331$	11.2384	0.617718	0.308859	+	0.951108i	$0.400053\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.7593	0.697113
$336$	0	0
$337$	2.70905	0.147572	0.0737858	−	0.997274i	$-0.476492\pi$
$337$	2.70905	0.147572	0.0737858	+	0.997274i	$0.476492\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	14.2188	0.769989
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.54629	0.512472	0.256236	−	0.966614i	$-0.417518\pi$
$347$	9.54629	0.512472	0.256236	+	0.966614i	$0.417518\pi$
$348$	0	0
$349$	−2.02229	−0.108251	−0.0541253	−	0.998534i	$-0.517237\pi$
$349$	−2.02229	−0.108251	−0.0541253	+	0.998534i	$0.517237\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.8767	0.632134	0.316067	−	0.948737i	$-0.397637\pi$
$353$	11.8767	0.632134	0.316067	+	0.948737i	$0.397637\pi$
$354$	0	0
$355$	−40.2264	−2.13500
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13.9796	0.737817	0.368909	−	0.929466i	$-0.379732\pi$
$359$	13.9796	0.737817	0.368909	+	0.929466i	$0.379732\pi$
$360$	0	0
$361$	−8.07331	−0.424911
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−36.8733	−1.93003
$366$	0	0
$367$	2.28598	0.119327	0.0596635	−	0.998219i	$-0.480997\pi$
$367$	2.28598	0.119327	0.0596635	+	0.998219i	$0.480997\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.03647	0.417233
$372$	0	0
$373$	−34.1054	−1.76591	−0.882957	−	0.469455i	$-0.844451\pi$
$373$	−34.1054	−1.76591	−0.882957	+	0.469455i	$0.844451\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.72279	−0.140231
$378$	0	0
$379$	6.50473	0.334126	0.167063	−	0.985946i	$-0.446572\pi$
$379$	6.50473	0.334126	0.167063	+	0.985946i	$0.446572\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−22.4695	−1.14814	−0.574069	−	0.818807i	$-0.694636\pi$
$383$	−22.4695	−1.14814	−0.574069	+	0.818807i	$0.694636\pi$
$384$	0	0
$385$	13.6875	0.697580
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.14631	−0.260928	−0.130464	−	0.991453i	$-0.541647\pi$
$389$	−5.14631	−0.260928	−0.130464	+	0.991453i	$0.541647\pi$
$390$	0	0
$391$	−35.9580	−1.81848
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13.9753	0.703176
$396$	0	0
$397$	−21.0214	−1.05503	−0.527517	−	0.849544i	$-0.676877\pi$
$397$	−21.0214	−1.05503	−0.527517	+	0.849544i	$0.676877\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.60770	−0.0802849	−0.0401424	−	0.999194i	$-0.512781\pi$
$401$	−1.60770	−0.0802849	−0.0401424	+	0.999194i	$0.512781\pi$
$402$	0	0
$403$	−21.5965	−1.07580
$404$	0	0
$405$	0	0
$406$	0	0
$407$	28.8229	1.42870
$408$	0	0
$409$	21.4111	1.05871	0.529354	−	0.848401i	$-0.322434\pi$
$409$	21.4111	1.05871	0.529354	+	0.848401i	$0.322434\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.50901	−0.0742534
$414$	0	0
$415$	−42.6672	−2.09445
$416$	0	0
$417$	0	0
$418$	0	0
$419$	34.6994	1.69517	0.847587	−	0.530656i	$-0.178054\pi$
$419$	34.6994	1.69517	0.847587	+	0.530656i	$0.178054\pi$
$420$	0	0
$421$	−23.2467	−1.13298	−0.566488	−	0.824070i	$-0.691698\pi$
$421$	−23.2467	−1.13298	−0.566488	+	0.824070i	$0.691698\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−65.2617	−3.16566
$426$	0	0
$427$	2.08674	0.100985
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.5201	−0.506735	−0.253368	−	0.967370i	$-0.581538\pi$
$431$	−10.5201	−0.506735	−0.253368	+	0.967370i	$0.581538\pi$
$432$	0	0
$433$	−29.8296	−1.43352	−0.716758	−	0.697322i	$-0.754375\pi$
$433$	−29.8296	−1.43352	−0.716758	+	0.697322i	$0.754375\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.4909	0.788868
$438$	0	0
$439$	−4.56924	−0.218078	−0.109039	−	0.994037i	$-0.534777\pi$
$439$	−4.56924	−0.218078	−0.109039	+	0.994037i	$0.534777\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10.0291	0.476497	0.238248	−	0.971204i	$-0.423427\pi$
$443$	10.0291	0.476497	0.238248	+	0.971204i	$0.423427\pi$
$444$	0	0
$445$	−26.8008	−1.27048
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.1908	−0.858476	−0.429238	−	0.903192i	$-0.641218\pi$
$449$	−18.1908	−0.858476	−0.429238	+	0.903192i	$0.641218\pi$
$450$	0	0
$451$	17.3946	0.819082
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−20.7896	−0.974632
$456$	0	0
$457$	−5.25417	−0.245780	−0.122890	−	0.992420i	$-0.539216\pi$
$457$	−5.25417	−0.245780	−0.122890	+	0.992420i	$0.539216\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.65526	0.403116	0.201558	−	0.979477i	$-0.435400\pi$
$461$	8.65526	0.403116	0.201558	+	0.979477i	$0.435400\pi$
$462$	0	0
$463$	1.01958	0.0473837	0.0236919	−	0.999719i	$-0.492458\pi$
$463$	1.01958	0.0473837	0.0236919	+	0.999719i	$0.492458\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.7167	−0.866104	−0.433052	−	0.901369i	$-0.642563\pi$
$467$	−18.7167	−0.866104	−0.433052	+	0.901369i	$0.642563\pi$
$468$	0	0
$469$	3.40344	0.157156
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.85419	0.269176
$474$	0	0
$475$	29.9301	1.37329
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.6510	1.67463	0.837313	−	0.546724i	$-0.184125\pi$
$479$	36.6510	1.67463	0.837313	+	0.546724i	$0.184125\pi$
$480$	0	0
$481$	−43.7784	−1.99612
$482$	0	0
$483$	0	0
$484$	0	0
$485$	40.8871	1.85659
$486$	0	0
$487$	35.1877	1.59451	0.797253	−	0.603646i	$-0.206286\pi$
$487$	35.1877	1.59451	0.797253	+	0.603646i	$0.206286\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−18.3098	−0.826308	−0.413154	−	0.910661i	$-0.635573\pi$
$491$	−18.3098	−0.826308	−0.413154	+	0.910661i	$0.635573\pi$
$492$	0	0
$493$	−3.53892	−0.159385
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10.7301	−0.481311
$498$	0	0
$499$	−24.5581	−1.09937	−0.549686	−	0.835372i	$-0.685253\pi$
$499$	−24.5581	−1.09937	−0.549686	+	0.835372i	$0.685253\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17.0176	0.758779	0.379390	−	0.925237i	$-0.376134\pi$
$503$	17.0176	0.758779	0.379390	+	0.925237i	$0.376134\pi$
$504$	0	0
$505$	47.9991	2.13593
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.7635	−0.920325	−0.460163	−	0.887835i	$-0.652209\pi$
$509$	−20.7635	−0.920325	−0.460163	+	0.887835i	$0.652209\pi$
$510$	0	0
$511$	−9.83567	−0.435105
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−35.7436	−1.57505
$516$	0	0
$517$	−35.1957	−1.54791
$518$	0	0
$519$	0	0
$520$	0	0
$521$	29.9915	1.31395	0.656977	−	0.753911i	$-0.271835\pi$
$521$	29.9915	1.31395	0.656977	+	0.753911i	$0.271835\pi$
$522$	0	0
$523$	−2.57036	−0.112394	−0.0561970	−	0.998420i	$-0.517898\pi$
$523$	−2.57036	−0.112394	−0.0561970	+	0.998420i	$0.517898\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−28.0698	−1.22274
$528$	0	0
$529$	1.88868	0.0821166
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−26.4203	−1.14439
$534$	0	0
$535$	31.6119	1.36670
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.65105	0.157262
$540$	0	0
$541$	24.7962	1.06607	0.533036	−	0.846093i	$-0.321051\pi$
$541$	24.7962	1.06607	0.533036	+	0.846093i	$0.321051\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−62.0725	−2.65889
$546$	0	0
$547$	11.4391	0.489101	0.244550	−	0.969637i	$-0.421360\pi$
$547$	11.4391	0.489101	0.244550	+	0.969637i	$0.421360\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.62300	0.0691423
$552$	0	0
$553$	3.72782	0.158523
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.8445	0.586609	0.293305	−	0.956019i	$-0.405245\pi$
$557$	13.8445	0.586609	0.293305	+	0.956019i	$0.405245\pi$
$558$	0	0
$559$	−8.89178	−0.376082
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−35.0387	−1.47671	−0.738353	−	0.674415i	$-0.764396\pi$
$563$	−35.0387	−1.47671	−0.738353	+	0.674415i	$0.764396\pi$
$564$	0	0
$565$	−0.246042	−0.0103511
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.45880	0.312689	0.156344	−	0.987703i	$-0.450029\pi$
$569$	7.45880	0.312689	0.156344	+	0.987703i	$0.450029\pi$
$570$	0	0
$571$	−42.8349	−1.79258	−0.896292	−	0.443464i	$-0.853749\pi$
$571$	−42.8349	−1.79258	−0.896292	+	0.443464i	$0.853749\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	45.1715	1.88378
$576$	0	0
$577$	−16.1386	−0.671860	−0.335930	−	0.941887i	$-0.609051\pi$
$577$	−16.1386	−0.671860	−0.335930	+	0.941887i	$0.609051\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−11.3812	−0.472170
$582$	0	0
$583$	−29.3415	−1.21520
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−45.8187	−1.89114	−0.945570	−	0.325417i	$-0.894495\pi$
$587$	−45.8187	−1.89114	−0.945570	+	0.325417i	$0.894495\pi$
$588$	0	0
$589$	12.8733	0.530434
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−22.4176	−0.920579	−0.460290	−	0.887769i	$-0.652254\pi$
$593$	−22.4176	−0.920579	−0.460290	+	0.887769i	$0.652254\pi$
$594$	0	0
$595$	−27.0211	−1.10776
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.24451	−0.0508491	−0.0254246	−	0.999677i	$-0.508094\pi$
$599$	−1.24451	−0.0508491	−0.0254246	+	0.999677i	$0.508094\pi$
$600$	0	0
$601$	−37.8022	−1.54199	−0.770993	−	0.636844i	$-0.780240\pi$
$601$	−37.8022	−1.54199	−0.770993	+	0.636844i	$0.780240\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.73550	−0.355149
$606$	0	0
$607$	−12.0253	−0.488093	−0.244047	−	0.969764i	$-0.578475\pi$
$607$	−12.0253	−0.488093	−0.244047	+	0.969764i	$0.578475\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	53.4579	2.16267
$612$	0	0
$613$	−29.5048	−1.19169	−0.595843	−	0.803101i	$-0.703182\pi$
$613$	−29.5048	−1.19169	−0.595843	+	0.803101i	$0.703182\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12.5378	−0.504753	−0.252376	−	0.967629i	$-0.581212\pi$
$617$	−12.5378	−0.504753	−0.252376	+	0.967629i	$0.581212\pi$
$618$	0	0
$619$	30.4306	1.22311	0.611555	−	0.791202i	$-0.290544\pi$
$619$	30.4306	1.22311	0.611555	+	0.791202i	$0.290544\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.14891	−0.286415
$624$	0	0
$625$	11.7113	0.468451
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−56.9005	−2.26877
$630$	0	0
$631$	20.6901	0.823660	0.411830	−	0.911261i	$-0.364890\pi$
$631$	20.6901	0.823660	0.411830	+	0.911261i	$0.364890\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.9996	0.476189
$636$	0	0
$637$	−5.54548	−0.219720
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10.4883	−0.414265	−0.207132	−	0.978313i	$-0.566413\pi$
$641$	−10.4883	−0.414265	−0.207132	+	0.978313i	$0.566413\pi$
$642$	0	0
$643$	40.4111	1.59366	0.796830	−	0.604204i	$-0.206509\pi$
$643$	40.4111	1.59366	0.796830	+	0.604204i	$0.206509\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.8906	0.664038	0.332019	−	0.943273i	$-0.392270\pi$
$647$	16.8906	0.664038	0.332019	+	0.943273i	$0.392270\pi$
$648$	0	0
$649$	5.50945	0.216265
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−38.1965	−1.49474	−0.747372	−	0.664406i	$-0.768684\pi$
$653$	−38.1965	−1.49474	−0.747372	+	0.664406i	$0.768684\pi$
$654$	0	0
$655$	57.5132	2.24723
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−30.5727	−1.19094	−0.595472	−	0.803376i	$-0.703035\pi$
$659$	−30.5727	−1.19094	−0.595472	+	0.803376i	$0.703035\pi$
$660$	0	0
$661$	24.1788	0.940447	0.470223	−	0.882547i	$-0.344173\pi$
$661$	24.1788	0.940447	0.470223	+	0.882547i	$0.344173\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12.3923	0.480553
$666$	0	0
$667$	2.44949	0.0948448
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7.61879	−0.294120
$672$	0	0
$673$	27.0970	1.04451	0.522257	−	0.852788i	$-0.325090\pi$
$673$	27.0970	1.04451	0.522257	+	0.852788i	$0.325090\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.4491	0.516890	0.258445	−	0.966026i	$-0.416790\pi$
$677$	13.4491	0.516890	0.258445	+	0.966026i	$0.416790\pi$
$678$	0	0
$679$	10.9063	0.418547
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−11.4449	−0.437925	−0.218963	−	0.975733i	$-0.570267\pi$
$683$	−11.4449	−0.437925	−0.218963	+	0.975733i	$0.570267\pi$
$684$	0	0
$685$	−2.57160	−0.0982557
$686$	0	0
$687$	0	0
$688$	0	0
$689$	44.5661	1.69783
$690$	0	0
$691$	−36.2832	−1.38028	−0.690139	−	0.723677i	$-0.742451\pi$
$691$	−36.2832	−1.38028	−0.690139	+	0.723677i	$0.742451\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−75.7312	−2.87265
$696$	0	0
$697$	−34.3394	−1.30070
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.80688	0.181553	0.0907767	−	0.995871i	$-0.471065\pi$
$701$	4.80688	0.181553	0.0907767	+	0.995871i	$0.471065\pi$
$702$	0	0
$703$	26.0955	0.984210
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.8034	0.481522
$708$	0	0
$709$	−12.3554	−0.464017	−0.232008	−	0.972714i	$-0.574530\pi$
$709$	−12.3554	−0.464017	−0.232008	+	0.972714i	$0.574530\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19.4288	0.727613
$714$	0	0
$715$	75.9038	2.83864
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.4488	0.389673	0.194837	−	0.980836i	$-0.437582\pi$
$719$	10.4488	0.389673	0.194837	+	0.980836i	$0.437582\pi$
$720$	0	0
$721$	−9.53433	−0.355077
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.44569	0.165109
$726$	0	0
$727$	18.2653	0.677423	0.338711	−	0.940890i	$-0.390009\pi$
$727$	18.2653	0.677423	0.338711	+	0.940890i	$0.390009\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−11.5570	−0.427450
$732$	0	0
$733$	−48.6491	−1.79689	−0.898447	−	0.439082i	$-0.855304\pi$
$733$	−48.6491	−1.79689	−0.898447	+	0.439082i	$0.855304\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.4261	−0.457722
$738$	0	0
$739$	−52.2935	−1.92365	−0.961823	−	0.273671i	$-0.911762\pi$
$739$	−52.2935	−1.92365	−0.961823	+	0.273671i	$0.911762\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.4231	−0.382386	−0.191193	−	0.981553i	$-0.561236\pi$
$743$	−10.4231	−0.382386	−0.191193	+	0.981553i	$0.561236\pi$
$744$	0	0
$745$	32.3646	1.18575
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.43223	0.308107
$750$	0	0
$751$	−10.2077	−0.372483	−0.186242	−	0.982504i	$-0.559631\pi$
$751$	−10.2077	−0.372483	−0.186242	+	0.982504i	$0.559631\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−40.7761	−1.48399
$756$	0	0
$757$	−6.13207	−0.222874	−0.111437	−	0.993772i	$-0.535545\pi$
$757$	−6.13207	−0.222874	−0.111437	+	0.993772i	$0.535545\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32.1934	1.16701	0.583505	−	0.812109i	$-0.301681\pi$
$761$	32.1934	1.16701	0.583505	+	0.812109i	$0.301681\pi$
$762$	0	0
$763$	−16.5574	−0.599417
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.36817	−0.302157
$768$	0	0
$769$	−10.7704	−0.388391	−0.194195	−	0.980963i	$-0.562210\pi$
$769$	−10.7704	−0.388391	−0.194195	+	0.980963i	$0.562210\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−26.6154	−0.957289	−0.478644	−	0.878009i	$-0.658872\pi$
$773$	−26.6154	−0.957289	−0.478644	+	0.878009i	$0.658872\pi$
$774$	0	0
$775$	35.2621	1.26665
$776$	0	0
$777$	0	0
$778$	0	0
$779$	15.7486	0.564253
$780$	0	0
$781$	39.1761	1.40183
$782$	0	0
$783$	0	0
$784$	0	0
$785$	41.2092	1.47082
$786$	0	0
$787$	8.00503	0.285348	0.142674	−	0.989770i	$-0.454430\pi$
$787$	8.00503	0.285348	0.142674	+	0.989770i	$0.454430\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−0.0656299	−0.00233353
$792$	0	0
$793$	11.5720	0.410933
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.9351	0.847823	0.423912	−	0.905704i	$-0.360657\pi$
$797$	23.9351	0.847823	0.423912	+	0.905704i	$0.360657\pi$
$798$	0	0
$799$	69.4812	2.45807
$800$	0	0
$801$	0	0
$802$	0	0
$803$	35.9105	1.26725
$804$	0	0
$805$	18.7029	0.659190
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26.9965	0.949148	0.474574	−	0.880216i	$-0.342602\pi$
$809$	26.9965	0.949148	0.474574	+	0.880216i	$0.342602\pi$
$810$	0	0
$811$	41.6011	1.46081	0.730406	−	0.683013i	$-0.239331\pi$
$811$	41.6011	1.46081	0.730406	+	0.683013i	$0.239331\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	17.9661	0.629326
$816$	0	0
$817$	5.30022	0.185431
$818$	0	0
$819$	0	0
$820$	0	0
$821$	18.6717	0.651645	0.325823	−	0.945431i	$-0.394359\pi$
$821$	18.6717	0.651645	0.325823	+	0.945431i	$0.394359\pi$
$822$	0	0
$823$	42.9511	1.49718	0.748591	−	0.663032i	$-0.230730\pi$
$823$	42.9511	1.49718	0.748591	+	0.663032i	$0.230730\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28.4602	−0.989657	−0.494828	−	0.868991i	$-0.664769\pi$
$827$	−28.4602	−0.989657	−0.494828	+	0.868991i	$0.664769\pi$
$828$	0	0
$829$	32.0061	1.11162	0.555809	−	0.831310i	$-0.312409\pi$
$829$	32.0061	1.11162	0.555809	+	0.831310i	$0.312409\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.20767	−0.249731
$834$	0	0
$835$	11.0008	0.380697
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−4.02382	−0.138918	−0.0694589	−	0.997585i	$-0.522127\pi$
$839$	−4.02382	−0.138918	−0.0694589	+	0.997585i	$0.522127\pi$
$840$	0	0
$841$	−28.7589	−0.991687
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−66.5523	−2.28947
$846$	0	0
$847$	−2.33013	−0.0800642
$848$	0	0
$849$	0	0
$850$	0	0
$851$	39.3842	1.35007
$852$	0	0
$853$	50.2428	1.72028	0.860141	−	0.510056i	$-0.170375\pi$
$853$	50.2428	1.72028	0.860141	+	0.510056i	$0.170375\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.7582	−0.367492	−0.183746	−	0.982974i	$-0.558822\pi$
$857$	−10.7582	−0.367492	−0.183746	+	0.982974i	$0.558822\pi$
$858$	0	0
$859$	−43.2253	−1.47483	−0.737413	−	0.675442i	$-0.763953\pi$
$859$	−43.2253	−1.47483	−0.737413	+	0.675442i	$0.763953\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−44.1446	−1.50270	−0.751349	−	0.659905i	$-0.770597\pi$
$863$	−44.1446	−1.50270	−0.751349	+	0.659905i	$0.770597\pi$
$864$	0	0
$865$	44.9484	1.52829
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−13.6104	−0.461702
$870$	0	0
$871$	18.8737	0.639511
$872$	0	0
$873$	0	0
$874$	0	0
$875$	15.2000	0.513853
$876$	0	0
$877$	21.9261	0.740393	0.370196	−	0.928954i	$-0.379290\pi$
$877$	21.9261	0.740393	0.370196	+	0.928954i	$0.379290\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	49.6969	1.67433	0.837166	−	0.546949i	$-0.184211\pi$
$881$	49.6969	1.67433	0.837166	+	0.546949i	$0.184211\pi$
$882$	0	0
$883$	−16.8441	−0.566848	−0.283424	−	0.958995i	$-0.591470\pi$
$883$	−16.8441	−0.566848	−0.283424	+	0.958995i	$0.591470\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−20.9900	−0.704774	−0.352387	−	0.935854i	$-0.614630\pi$
$887$	−20.9900	−0.704774	−0.352387	+	0.935854i	$0.614630\pi$
$888$	0	0
$889$	3.20080	0.107351
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−31.8652	−1.06633
$894$	0	0
$895$	93.4482	3.12363
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.91214	0.0637735
$900$	0	0
$901$	57.9242	1.92974
$902$	0	0
$903$	0	0
$904$	0	0
$905$	80.3031	2.66937
$906$	0	0
$907$	−9.11629	−0.302701	−0.151351	−	0.988480i	$-0.548362\pi$
$907$	−9.11629	−0.302701	−0.151351	+	0.988480i	$0.548362\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−18.3216	−0.607021	−0.303511	−	0.952828i	$-0.598159\pi$
$911$	−18.3216	−0.607021	−0.303511	+	0.952828i	$0.598159\pi$
$912$	0	0
$913$	41.5531	1.37521
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.3412	0.506612
$918$	0	0
$919$	−55.7097	−1.83769	−0.918847	−	0.394613i	$-0.870879\pi$
$919$	−55.7097	−1.83769	−0.918847	+	0.394613i	$0.870879\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−59.5036	−1.95858
$924$	0	0
$925$	71.4800	2.35025
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20.8840	0.685183	0.342592	−	0.939484i	$-0.388695\pi$
$929$	20.8840	0.685183	0.342592	+	0.939484i	$0.388695\pi$
$930$	0	0
$931$	3.30555	0.108335
$932$	0	0
$933$	0	0
$934$	0	0
$935$	98.6551	3.22637
$936$	0	0
$937$	−6.56584	−0.214497	−0.107248	−	0.994232i	$-0.534204\pi$
$937$	−6.56584	−0.214497	−0.107248	+	0.994232i	$0.534204\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	10.4330	0.340106	0.170053	−	0.985435i	$-0.445606\pi$
$941$	10.4330	0.340106	0.170053	+	0.985435i	$0.445606\pi$
$942$	0	0
$943$	23.7684	0.774004
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31.0783	−1.00991	−0.504954	−	0.863146i	$-0.668491\pi$
$947$	−31.0783	−1.00991	−0.504954	+	0.863146i	$0.668491\pi$
$948$	0	0
$949$	−54.5435	−1.77056
$950$	0	0
$951$	0	0
$952$	0	0
$953$	27.2222	0.881814	0.440907	−	0.897553i	$-0.354657\pi$
$953$	27.2222	0.881814	0.440907	+	0.897553i	$0.354657\pi$
$954$	0	0
$955$	67.3310	2.17878
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.685955	−0.0221506
$960$	0	0
$961$	−15.8334	−0.510754
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−35.1607	−1.13186
$966$	0	0
$967$	−43.1419	−1.38735	−0.693675	−	0.720288i	$-0.744010\pi$
$967$	−43.1419	−1.38735	−0.693675	+	0.720288i	$0.744010\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−18.0357	−0.578794	−0.289397	−	0.957209i	$-0.593455\pi$
$971$	−18.0357	−0.578794	−0.289397	+	0.957209i	$0.593455\pi$
$972$	0	0
$973$	−20.2007	−0.647606
$974$	0	0
$975$	0	0
$976$	0	0
$977$	27.2031	0.870304	0.435152	−	0.900357i	$-0.356695\pi$
$977$	27.2031	0.870304	0.435152	+	0.900357i	$0.356695\pi$
$978$	0	0
$979$	26.1010	0.834191
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−18.0793	−0.576640	−0.288320	−	0.957534i	$-0.593097\pi$
$983$	−18.0793	−0.576640	−0.288320	+	0.957534i	$0.593097\pi$
$984$	0	0
$985$	0.775716	0.0247164
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.99927	0.254362
$990$	0	0
$991$	50.2203	1.59530	0.797650	−	0.603121i	$-0.206076\pi$
$991$	50.2203	1.59530	0.797650	+	0.603121i	$0.206076\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	47.2997	1.49950
$996$	0	0
$997$	−26.7513	−0.847222	−0.423611	−	0.905844i	$-0.639238\pi$
$997$	−26.7513	−0.847222	−0.423611	+	0.905844i	$0.639238\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.ce.1.1		4
3.2	odd	2		9072.2.a.cl.1.4		4
4.3	odd	2		4536.2.a.x.1.1		4
9.2	odd	6		3024.2.r.l.1009.1		8
9.4	even	3		1008.2.r.m.673.4		8
9.5	odd	6		3024.2.r.l.2017.1		8
9.7	even	3		1008.2.r.m.337.4		8
12.11	even	2		4536.2.a.ba.1.4		4
36.7	odd	6		504.2.r.d.337.1	yes	8
36.11	even	6		1512.2.r.d.1009.1		8
36.23	even	6		1512.2.r.d.505.1		8
36.31	odd	6		504.2.r.d.169.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.r.d.169.1	✓	8	36.31	odd	6
504.2.r.d.337.1	yes	8	36.7	odd	6
1008.2.r.m.337.4		8	9.7	even	3
1008.2.r.m.673.4		8	9.4	even	3
1512.2.r.d.505.1		8	36.23	even	6
1512.2.r.d.1009.1		8	36.11	even	6
3024.2.r.l.1009.1		8	9.2	odd	6
3024.2.r.l.2017.1		8	9.5	odd	6
4536.2.a.x.1.1		4	4.3	odd	2
4536.2.a.ba.1.4		4	12.11	even	2
9072.2.a.ce.1.1		4	1.1	even	1	trivial
9072.2.a.cl.1.4		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q-3.74893 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-3.74893 q^{5} -1.00000 q^{7} +3.65105 q^{11} -5.54548 q^{13} -7.20767 q^{17} +3.30555 q^{19} +4.98886 q^{23} +9.05449 q^{25} +0.490993 q^{29} +3.89443 q^{31} +3.74893 q^{35} +7.89443 q^{37} +4.76429 q^{41} +1.60343 q^{43} -9.63990 q^{47} +1.00000 q^{49} -8.03647 q^{53} -13.6875 q^{55} +1.50901 q^{59} -2.08674 q^{61} +20.7896 q^{65} -3.40344 q^{67} +10.7301 q^{71} +9.83567 q^{73} -3.65105 q^{77} -3.72782 q^{79} +11.3812 q^{83} +27.0211 q^{85} +7.14891 q^{89} +5.54548 q^{91} -12.3923 q^{95} -10.9063 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{7}+O(q^{10}) \) 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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