LMFDB - Embedded newform 8550.2.a.bx.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8550,2,Mod(1,8550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8550, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8550.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:	$68.2720937282$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1710)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	8550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.12311 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +3.12311 q^{7} +1.00000 q^{8} -2.00000 q^{11} -4.00000 q^{13} +3.12311 q^{14} +1.00000 q^{16} +3.12311 q^{17} -1.00000 q^{19} -2.00000 q^{22} -4.00000 q^{26} +3.12311 q^{28} +2.00000 q^{29} +9.12311 q^{31} +1.00000 q^{32} +3.12311 q^{34} -1.00000 q^{38} +5.12311 q^{41} -10.2462 q^{43} -2.00000 q^{44} +10.2462 q^{47} +2.75379 q^{49} -4.00000 q^{52} -4.24621 q^{53} +3.12311 q^{56} +2.00000 q^{58} +3.12311 q^{59} +12.2462 q^{61} +9.12311 q^{62} +1.00000 q^{64} -6.24621 q^{67} +3.12311 q^{68} -6.24621 q^{71} -6.00000 q^{73} -1.00000 q^{76} -6.24621 q^{77} +9.12311 q^{79} +5.12311 q^{82} +6.87689 q^{83} -10.2462 q^{86} -2.00000 q^{88} +11.3693 q^{89} -12.4924 q^{91} +10.2462 q^{94} +6.00000 q^{97} +2.75379 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - 4 q^{11} - 8 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{19} - 4 q^{22} - 8 q^{26} - 2 q^{28} + 4 q^{29} + 10 q^{31} + 2 q^{32} - 2 q^{34} - 2 q^{38} + 2 q^{41} - 4 q^{43} - 4 q^{44} + 4 q^{47} + 22 q^{49} - 8 q^{52} + 8 q^{53} - 2 q^{56} + 4 q^{58} - 2 q^{59} + 8 q^{61} + 10 q^{62} + 2 q^{64} + 4 q^{67} - 2 q^{68} + 4 q^{71} - 12 q^{73} - 2 q^{76} + 4 q^{77} + 10 q^{79} + 2 q^{82} + 22 q^{83} - 4 q^{86} - 4 q^{88} - 2 q^{89} + 8 q^{91} + 4 q^{94} + 12 q^{97} + 22 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 - 4 * q^11 - 8 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 - 2 * q^19 - 4 * q^22 - 8 * q^26 - 2 * q^28 + 4 * q^29 + 10 * q^31 + 2 * q^32 - 2 * q^34 - 2 * q^38 + 2 * q^41 - 4 * q^43 - 4 * q^44 + 4 * q^47 + 22 * q^49 - 8 * q^52 + 8 * q^53 - 2 * q^56 + 4 * q^58 - 2 * q^59 + 8 * q^61 + 10 * q^62 + 2 * q^64 + 4 * q^67 - 2 * q^68 + 4 * q^71 - 12 * q^73 - 2 * q^76 + 4 * q^77 + 10 * q^79 + 2 * q^82 + 22 * q^83 - 4 * q^86 - 4 * q^88 - 2 * q^89 + 8 * q^91 + 4 * q^94 + 12 * q^97 + 22 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.12311	1.18042	0.590211	−	0.807249i	$-0.299044\pi$
$7$	3.12311	1.18042	0.590211	+	0.807249i	$0.299044\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	3.12311	0.834685
$15$	0	0
$16$	1.00000	0.250000
$17$	3.12311	0.757464	0.378732	−	0.925506i	$-0.376360\pi$
$17$	3.12311	0.757464	0.378732	+	0.925506i	$0.376360\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−2.00000	−0.426401
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	0	0
$26$	−4.00000	−0.784465
$27$	0	0
$28$	3.12311	0.590211
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	9.12311	1.63856	0.819279	−	0.573395i	$-0.194374\pi$
$31$	9.12311	1.63856	0.819279	+	0.573395i	$0.194374\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.12311	0.535608
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	5.12311	0.800095	0.400047	−	0.916494i	$-0.368994\pi$
$41$	5.12311	0.800095	0.400047	+	0.916494i	$0.368994\pi$
$42$	0	0
$43$	−10.2462	−1.56253	−0.781266	−	0.624198i	$-0.785426\pi$
$43$	−10.2462	−1.56253	−0.781266	+	0.624198i	$0.785426\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	0	0
$47$	10.2462	1.49456	0.747282	−	0.664507i	$-0.231359\pi$
$47$	10.2462	1.49456	0.747282	+	0.664507i	$0.231359\pi$
$48$	0	0
$49$	2.75379	0.393398
$50$	0	0
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−4.24621	−0.583262	−0.291631	−	0.956531i	$-0.594198\pi$
$53$	−4.24621	−0.583262	−0.291631	+	0.956531i	$0.594198\pi$
$54$	0	0
$55$	0	0
$56$	3.12311	0.417343
$57$	0	0
$58$	2.00000	0.262613
$59$	3.12311	0.406594	0.203297	−	0.979117i	$-0.434834\pi$
$59$	3.12311	0.406594	0.203297	+	0.979117i	$0.434834\pi$
$60$	0	0
$61$	12.2462	1.56797	0.783983	−	0.620782i	$-0.213185\pi$
$61$	12.2462	1.56797	0.783983	+	0.620782i	$0.213185\pi$
$62$	9.12311	1.15864
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−6.24621	−0.763096	−0.381548	−	0.924349i	$-0.624609\pi$
$67$	−6.24621	−0.763096	−0.381548	+	0.924349i	$0.624609\pi$
$68$	3.12311	0.378732
$69$	0	0
$70$	0	0
$71$	−6.24621	−0.741289	−0.370644	−	0.928775i	$-0.620863\pi$
$71$	−6.24621	−0.741289	−0.370644	+	0.928775i	$0.620863\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−6.24621	−0.711822
$78$	0	0
$79$	9.12311	1.02643	0.513215	−	0.858260i	$-0.328454\pi$
$79$	9.12311	1.02643	0.513215	+	0.858260i	$0.328454\pi$
$80$	0	0
$81$	0	0
$82$	5.12311	0.565752
$83$	6.87689	0.754837	0.377419	−	0.926043i	$-0.376812\pi$
$83$	6.87689	0.754837	0.377419	+	0.926043i	$0.376812\pi$
$84$	0	0
$85$	0	0
$86$	−10.2462	−1.10488
$87$	0	0
$88$	−2.00000	−0.213201
$89$	11.3693	1.20515	0.602573	−	0.798064i	$-0.294142\pi$
$89$	11.3693	1.20515	0.602573	+	0.798064i	$0.294142\pi$
$90$	0	0
$91$	−12.4924	−1.30956
$92$	0	0
$93$	0	0
$94$	10.2462	1.05682
$95$	0	0
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	2.75379	0.278175
$99$	0	0
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	8.24621	0.812523	0.406262	−	0.913757i	$-0.366832\pi$
$103$	8.24621	0.812523	0.406262	+	0.913757i	$0.366832\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−4.24621	−0.412428
$107$	14.2462	1.37723	0.688617	−	0.725126i	$-0.258218\pi$
$107$	14.2462	1.37723	0.688617	+	0.725126i	$0.258218\pi$
$108$	0	0
$109$	4.87689	0.467122	0.233561	−	0.972342i	$-0.424962\pi$
$109$	4.87689	0.467122	0.233561	+	0.972342i	$0.424962\pi$
$110$	0	0
$111$	0	0
$112$	3.12311	0.295106
$113$	4.24621	0.399450	0.199725	−	0.979852i	$-0.435995\pi$
$113$	4.24621	0.399450	0.199725	+	0.979852i	$0.435995\pi$
$114$	0	0
$115$	0	0
$116$	2.00000	0.185695
$117$	0	0
$118$	3.12311	0.287505
$119$	9.75379	0.894128
$120$	0	0
$121$	−7.00000	−0.636364
$122$	12.2462	1.10872
$123$	0	0
$124$	9.12311	0.819279
$125$	0	0
$126$	0	0
$127$	3.75379	0.333095	0.166547	−	0.986033i	$-0.446738\pi$
$127$	3.75379	0.333095	0.166547	+	0.986033i	$0.446738\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	22.4924	1.96517	0.982586	−	0.185808i	$-0.0594903\pi$
$131$	22.4924	1.96517	0.982586	+	0.185808i	$0.0594903\pi$
$132$	0	0
$133$	−3.12311	−0.270808
$134$	−6.24621	−0.539590
$135$	0	0
$136$	3.12311	0.267804
$137$	−19.1231	−1.63380	−0.816899	−	0.576781i	$-0.804308\pi$
$137$	−19.1231	−1.63380	−0.816899	+	0.576781i	$0.804308\pi$
$138$	0	0
$139$	−16.4924	−1.39887	−0.699435	−	0.714697i	$-0.746565\pi$
$139$	−16.4924	−1.39887	−0.699435	+	0.714697i	$0.746565\pi$
$140$	0	0
$141$	0	0
$142$	−6.24621	−0.524170
$143$	8.00000	0.668994
$144$	0	0
$145$	0	0
$146$	−6.00000	−0.496564
$147$	0	0
$148$	0	0
$149$	22.0000	1.80231	0.901155	−	0.433497i	$-0.142720\pi$
$149$	22.0000	1.80231	0.901155	+	0.433497i	$0.142720\pi$
$150$	0	0
$151$	2.87689	0.234118	0.117059	−	0.993125i	$-0.462653\pi$
$151$	2.87689	0.234118	0.117059	+	0.993125i	$0.462653\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−6.24621	−0.503334
$155$	0	0
$156$	0	0
$157$	1.12311	0.0896336	0.0448168	−	0.998995i	$-0.485730\pi$
$157$	1.12311	0.0896336	0.0448168	+	0.998995i	$0.485730\pi$
$158$	9.12311	0.725795
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	5.12311	0.400047
$165$	0	0
$166$	6.87689	0.533751
$167$	−8.00000	−0.619059	−0.309529	−	0.950890i	$-0.600171\pi$
$167$	−8.00000	−0.619059	−0.309529	+	0.950890i	$0.600171\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	−10.2462	−0.781266
$173$	8.24621	0.626948	0.313474	−	0.949597i	$-0.398507\pi$
$173$	8.24621	0.626948	0.313474	+	0.949597i	$0.398507\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	11.3693	0.852166
$179$	8.87689	0.663490	0.331745	−	0.943369i	$-0.392363\pi$
$179$	8.87689	0.663490	0.331745	+	0.943369i	$0.392363\pi$
$180$	0	0
$181$	9.36932	0.696416	0.348208	−	0.937417i	$-0.386790\pi$
$181$	9.36932	0.696416	0.348208	+	0.937417i	$0.386790\pi$
$182$	−12.4924	−0.926000
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.24621	−0.456768
$188$	10.2462	0.747282
$189$	0	0
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	−18.4924	−1.33111	−0.665557	−	0.746347i	$-0.731806\pi$
$193$	−18.4924	−1.33111	−0.665557	+	0.746347i	$0.731806\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	2.75379	0.196699
$197$	−8.24621	−0.587518	−0.293759	−	0.955879i	$-0.594906\pi$
$197$	−8.24621	−0.587518	−0.293759	+	0.955879i	$0.594906\pi$
$198$	0	0
$199$	10.2462	0.726335	0.363167	−	0.931724i	$-0.381695\pi$
$199$	10.2462	0.726335	0.363167	+	0.931724i	$0.381695\pi$
$200$	0	0
$201$	0	0
$202$	10.0000	0.703598
$203$	6.24621	0.438398
$204$	0	0
$205$	0	0
$206$	8.24621	0.574541
$207$	0	0
$208$	−4.00000	−0.277350
$209$	2.00000	0.138343
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−4.24621	−0.291631
$213$	0	0
$214$	14.2462	0.973851
$215$	0	0
$216$	0	0
$217$	28.4924	1.93419
$218$	4.87689	0.330305
$219$	0	0
$220$	0	0
$221$	−12.4924	−0.840331
$222$	0	0
$223$	−10.0000	−0.669650	−0.334825	−	0.942280i	$-0.608677\pi$
$223$	−10.0000	−0.669650	−0.334825	+	0.942280i	$0.608677\pi$
$224$	3.12311	0.208671
$225$	0	0
$226$	4.24621	0.282454
$227$	−20.4924	−1.36013	−0.680065	−	0.733152i	$-0.738048\pi$
$227$	−20.4924	−1.36013	−0.680065	+	0.733152i	$0.738048\pi$
$228$	0	0
$229$	18.4924	1.22201	0.611007	−	0.791625i	$-0.290765\pi$
$229$	18.4924	1.22201	0.611007	+	0.791625i	$0.290765\pi$
$230$	0	0
$231$	0	0
$232$	2.00000	0.131306
$233$	−1.36932	−0.0897069	−0.0448535	−	0.998994i	$-0.514282\pi$
$233$	−1.36932	−0.0897069	−0.0448535	+	0.998994i	$0.514282\pi$
$234$	0	0
$235$	0	0
$236$	3.12311	0.203297
$237$	0	0
$238$	9.75379	0.632244
$239$	1.75379	0.113443	0.0567216	−	0.998390i	$-0.481935\pi$
$239$	1.75379	0.113443	0.0567216	+	0.998390i	$0.481935\pi$
$240$	0	0
$241$	0.246211	0.0158599	0.00792993	−	0.999969i	$-0.497476\pi$
$241$	0.246211	0.0158599	0.00792993	+	0.999969i	$0.497476\pi$
$242$	−7.00000	−0.449977
$243$	0	0
$244$	12.2462	0.783983
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	9.12311	0.579318
$249$	0	0
$250$	0	0
$251$	−16.2462	−1.02545	−0.512726	−	0.858552i	$-0.671364\pi$
$251$	−16.2462	−1.02545	−0.512726	+	0.858552i	$0.671364\pi$
$252$	0	0
$253$	0	0
$254$	3.75379	0.235534
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.0000	0.623783	0.311891	−	0.950118i	$-0.399037\pi$
$257$	10.0000	0.623783	0.311891	+	0.950118i	$0.399037\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	22.4924	1.38959
$263$	12.4924	0.770316	0.385158	−	0.922851i	$-0.374147\pi$
$263$	12.4924	0.770316	0.385158	+	0.922851i	$0.374147\pi$
$264$	0	0
$265$	0	0
$266$	−3.12311	−0.191490
$267$	0	0
$268$	−6.24621	−0.381548
$269$	−20.7386	−1.26446	−0.632228	−	0.774782i	$-0.717860\pi$
$269$	−20.7386	−1.26446	−0.632228	+	0.774782i	$0.717860\pi$
$270$	0	0
$271$	2.24621	0.136448	0.0682238	−	0.997670i	$-0.478267\pi$
$271$	2.24621	0.136448	0.0682238	+	0.997670i	$0.478267\pi$
$272$	3.12311	0.189366
$273$	0	0
$274$	−19.1231	−1.15527
$275$	0	0
$276$	0	0
$277$	9.61553	0.577741	0.288871	−	0.957368i	$-0.406720\pi$
$277$	9.61553	0.577741	0.288871	+	0.957368i	$0.406720\pi$
$278$	−16.4924	−0.989150
$279$	0	0
$280$	0	0
$281$	−19.3693	−1.15548	−0.577738	−	0.816222i	$-0.696065\pi$
$281$	−19.3693	−1.15548	−0.577738	+	0.816222i	$0.696065\pi$
$282$	0	0
$283$	−8.49242	−0.504822	−0.252411	−	0.967620i	$-0.581224\pi$
$283$	−8.49242	−0.504822	−0.252411	+	0.967620i	$0.581224\pi$
$284$	−6.24621	−0.370644
$285$	0	0
$286$	8.00000	0.473050
$287$	16.0000	0.944450
$288$	0	0
$289$	−7.24621	−0.426248
$290$	0	0
$291$	0	0
$292$	−6.00000	−0.351123
$293$	−18.4924	−1.08034	−0.540169	−	0.841556i	$-0.681640\pi$
$293$	−18.4924	−1.08034	−0.540169	+	0.841556i	$0.681640\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	22.0000	1.27443
$299$	0	0
$300$	0	0
$301$	−32.0000	−1.84445
$302$	2.87689	0.165547
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	5.75379	0.328386	0.164193	−	0.986428i	$-0.447498\pi$
$307$	5.75379	0.328386	0.164193	+	0.986428i	$0.447498\pi$
$308$	−6.24621	−0.355911
$309$	0	0
$310$	0	0
$311$	−20.0000	−1.13410	−0.567048	−	0.823685i	$-0.691915\pi$
$311$	−20.0000	−1.13410	−0.567048	+	0.823685i	$0.691915\pi$
$312$	0	0
$313$	24.7386	1.39831	0.699155	−	0.714970i	$-0.253560\pi$
$313$	24.7386	1.39831	0.699155	+	0.714970i	$0.253560\pi$
$314$	1.12311	0.0633805
$315$	0	0
$316$	9.12311	0.513215
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.12311	−0.173774
$324$	0	0
$325$	0	0
$326$	−12.0000	−0.664619
$327$	0	0
$328$	5.12311	0.282876
$329$	32.0000	1.76422
$330$	0	0
$331$	−24.4924	−1.34623	−0.673113	−	0.739540i	$-0.735043\pi$
$331$	−24.4924	−1.34623	−0.673113	+	0.739540i	$0.735043\pi$
$332$	6.87689	0.377419
$333$	0	0
$334$	−8.00000	−0.437741
$335$	0	0
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	3.00000	0.163178
$339$	0	0
$340$	0	0
$341$	−18.2462	−0.988088
$342$	0	0
$343$	−13.2614	−0.716046
$344$	−10.2462	−0.552439
$345$	0	0
$346$	8.24621	0.443319
$347$	8.63068	0.463319	0.231660	−	0.972797i	$-0.425584\pi$
$347$	8.63068	0.463319	0.231660	+	0.972797i	$0.425584\pi$
$348$	0	0
$349$	−10.4924	−0.561646	−0.280823	−	0.959760i	$-0.590607\pi$
$349$	−10.4924	−0.561646	−0.280823	+	0.959760i	$0.590607\pi$
$350$	0	0
$351$	0	0
$352$	−2.00000	−0.106600
$353$	25.8617	1.37648	0.688241	−	0.725482i	$-0.258383\pi$
$353$	25.8617	1.37648	0.688241	+	0.725482i	$0.258383\pi$
$354$	0	0
$355$	0	0
$356$	11.3693	0.602573
$357$	0	0
$358$	8.87689	0.469158
$359$	10.2462	0.540774	0.270387	−	0.962752i	$-0.412848\pi$
$359$	10.2462	0.540774	0.270387	+	0.962752i	$0.412848\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	9.36932	0.492440
$363$	0	0
$364$	−12.4924	−0.654781
$365$	0	0
$366$	0	0
$367$	0.876894	0.0457735	0.0228868	−	0.999738i	$-0.492714\pi$
$367$	0.876894	0.0457735	0.0228868	+	0.999738i	$0.492714\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−13.2614	−0.688496
$372$	0	0
$373$	14.2462	0.737641	0.368820	−	0.929501i	$-0.379762\pi$
$373$	14.2462	0.737641	0.368820	+	0.929501i	$0.379762\pi$
$374$	−6.24621	−0.322984
$375$	0	0
$376$	10.2462	0.528408
$377$	−8.00000	−0.412021
$378$	0	0
$379$	8.49242	0.436226	0.218113	−	0.975923i	$-0.430010\pi$
$379$	8.49242	0.436226	0.218113	+	0.975923i	$0.430010\pi$
$380$	0	0
$381$	0	0
$382$	4.00000	0.204658
$383$	36.4924	1.86468	0.932338	−	0.361588i	$-0.117765\pi$
$383$	36.4924	1.86468	0.932338	+	0.361588i	$0.117765\pi$
$384$	0	0
$385$	0	0
$386$	−18.4924	−0.941240
$387$	0	0
$388$	6.00000	0.304604
$389$	34.4924	1.74884	0.874418	−	0.485174i	$-0.161244\pi$
$389$	34.4924	1.74884	0.874418	+	0.485174i	$0.161244\pi$
$390$	0	0
$391$	0	0
$392$	2.75379	0.139087
$393$	0	0
$394$	−8.24621	−0.415438
$395$	0	0
$396$	0	0
$397$	−13.1231	−0.658630	−0.329315	−	0.944220i	$-0.606818\pi$
$397$	−13.1231	−0.658630	−0.329315	+	0.944220i	$0.606818\pi$
$398$	10.2462	0.513596
$399$	0	0
$400$	0	0
$401$	31.8617	1.59110	0.795550	−	0.605888i	$-0.207182\pi$
$401$	31.8617	1.59110	0.795550	+	0.605888i	$0.207182\pi$
$402$	0	0
$403$	−36.4924	−1.81782
$404$	10.0000	0.497519
$405$	0	0
$406$	6.24621	0.309994
$407$	0	0
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	0	0
$412$	8.24621	0.406262
$413$	9.75379	0.479953
$414$	0	0
$415$	0	0
$416$	−4.00000	−0.196116
$417$	0	0
$418$	2.00000	0.0978232
$419$	12.2462	0.598267	0.299133	−	0.954211i	$-0.403302\pi$
$419$	12.2462	0.598267	0.299133	+	0.954211i	$0.403302\pi$
$420$	0	0
$421$	−15.1231	−0.737055	−0.368528	−	0.929617i	$-0.620138\pi$
$421$	−15.1231	−0.737055	−0.368528	+	0.929617i	$0.620138\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	−4.24621	−0.206214
$425$	0	0
$426$	0	0
$427$	38.2462	1.85086
$428$	14.2462	0.688617
$429$	0	0
$430$	0	0
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	0	0
$433$	18.0000	0.865025	0.432512	−	0.901628i	$-0.357627\pi$
$433$	18.0000	0.865025	0.432512	+	0.901628i	$0.357627\pi$
$434$	28.4924	1.36768
$435$	0	0
$436$	4.87689	0.233561
$437$	0	0
$438$	0	0
$439$	−10.8769	−0.519126	−0.259563	−	0.965726i	$-0.583578\pi$
$439$	−10.8769	−0.519126	−0.259563	+	0.965726i	$0.583578\pi$
$440$	0	0
$441$	0	0
$442$	−12.4924	−0.594204
$443$	−19.8617	−0.943660	−0.471830	−	0.881690i	$-0.656406\pi$
$443$	−19.8617	−0.943660	−0.471830	+	0.881690i	$0.656406\pi$
$444$	0	0
$445$	0	0
$446$	−10.0000	−0.473514
$447$	0	0
$448$	3.12311	0.147553
$449$	4.63068	0.218535	0.109268	−	0.994012i	$-0.465149\pi$
$449$	4.63068	0.218535	0.109268	+	0.994012i	$0.465149\pi$
$450$	0	0
$451$	−10.2462	−0.482475
$452$	4.24621	0.199725
$453$	0	0
$454$	−20.4924	−0.961757
$455$	0	0
$456$	0	0
$457$	−26.0000	−1.21623	−0.608114	−	0.793849i	$-0.708074\pi$
$457$	−26.0000	−1.21623	−0.608114	+	0.793849i	$0.708074\pi$
$458$	18.4924	0.864094
$459$	0	0
$460$	0	0
$461$	−22.4924	−1.04758	−0.523788	−	0.851848i	$-0.675482\pi$
$461$	−22.4924	−1.04758	−0.523788	+	0.851848i	$0.675482\pi$
$462$	0	0
$463$	23.1231	1.07462	0.537311	−	0.843384i	$-0.319440\pi$
$463$	23.1231	1.07462	0.537311	+	0.843384i	$0.319440\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−1.36932	−0.0634324
$467$	30.8769	1.42881	0.714406	−	0.699731i	$-0.246697\pi$
$467$	30.8769	1.42881	0.714406	+	0.699731i	$0.246697\pi$
$468$	0	0
$469$	−19.5076	−0.900776
$470$	0	0
$471$	0	0
$472$	3.12311	0.143753
$473$	20.4924	0.942243
$474$	0	0
$475$	0	0
$476$	9.75379	0.447064
$477$	0	0
$478$	1.75379	0.0802164
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	0	0
$482$	0.246211	0.0112146
$483$	0	0
$484$	−7.00000	−0.318182
$485$	0	0
$486$	0	0
$487$	15.7538	0.713872	0.356936	−	0.934129i	$-0.383821\pi$
$487$	15.7538	0.713872	0.356936	+	0.934129i	$0.383821\pi$
$488$	12.2462	0.554360
$489$	0	0
$490$	0	0
$491$	22.4924	1.01507	0.507534	−	0.861631i	$-0.330557\pi$
$491$	22.4924	1.01507	0.507534	+	0.861631i	$0.330557\pi$
$492$	0	0
$493$	6.24621	0.281315
$494$	4.00000	0.179969
$495$	0	0
$496$	9.12311	0.409640
$497$	−19.5076	−0.875034
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	0	0
$502$	−16.2462	−0.725104
$503$	32.0000	1.42681	0.713405	−	0.700752i	$-0.247152\pi$
$503$	32.0000	1.42681	0.713405	+	0.700752i	$0.247152\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	3.75379	0.166547
$509$	−42.4924	−1.88344	−0.941722	−	0.336393i	$-0.890793\pi$
$509$	−42.4924	−1.88344	−0.941722	+	0.336393i	$0.890793\pi$
$510$	0	0
$511$	−18.7386	−0.828948
$512$	1.00000	0.0441942
$513$	0	0
$514$	10.0000	0.441081
$515$	0	0
$516$	0	0
$517$	−20.4924	−0.901256
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−13.6155	−0.596507	−0.298254	−	0.954487i	$-0.596404\pi$
$521$	−13.6155	−0.596507	−0.298254	+	0.954487i	$0.596404\pi$
$522$	0	0
$523$	−42.7386	−1.86883	−0.934415	−	0.356186i	$-0.884077\pi$
$523$	−42.7386	−1.86883	−0.934415	+	0.356186i	$0.884077\pi$
$524$	22.4924	0.982586
$525$	0	0
$526$	12.4924	0.544696
$527$	28.4924	1.24115
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	−3.12311	−0.135404
$533$	−20.4924	−0.887625
$534$	0	0
$535$	0	0
$536$	−6.24621	−0.269795
$537$	0	0
$538$	−20.7386	−0.894106
$539$	−5.50758	−0.237228
$540$	0	0
$541$	−1.50758	−0.0648158	−0.0324079	−	0.999475i	$-0.510318\pi$
$541$	−1.50758	−0.0648158	−0.0324079	+	0.999475i	$0.510318\pi$
$542$	2.24621	0.0964830
$543$	0	0
$544$	3.12311	0.133902
$545$	0	0
$546$	0	0
$547$	−2.24621	−0.0960411	−0.0480205	−	0.998846i	$-0.515291\pi$
$547$	−2.24621	−0.0960411	−0.0480205	+	0.998846i	$0.515291\pi$
$548$	−19.1231	−0.816899
$549$	0	0
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	0	0
$553$	28.4924	1.21162
$554$	9.61553	0.408525
$555$	0	0
$556$	−16.4924	−0.699435
$557$	16.2462	0.688374	0.344187	−	0.938901i	$-0.388155\pi$
$557$	16.2462	0.688374	0.344187	+	0.938901i	$0.388155\pi$
$558$	0	0
$559$	40.9848	1.73347
$560$	0	0
$561$	0	0
$562$	−19.3693	−0.817045
$563$	−43.2311	−1.82197	−0.910986	−	0.412438i	$-0.864678\pi$
$563$	−43.2311	−1.82197	−0.910986	+	0.412438i	$0.864678\pi$
$564$	0	0
$565$	0	0
$566$	−8.49242	−0.356963
$567$	0	0
$568$	−6.24621	−0.262085
$569$	41.1231	1.72397	0.861985	−	0.506934i	$-0.169221\pi$
$569$	41.1231	1.72397	0.861985	+	0.506934i	$0.169221\pi$
$570$	0	0
$571$	32.4924	1.35977	0.679883	−	0.733321i	$-0.262031\pi$
$571$	32.4924	1.35977	0.679883	+	0.733321i	$0.262031\pi$
$572$	8.00000	0.334497
$573$	0	0
$574$	16.0000	0.667827
$575$	0	0
$576$	0	0
$577$	−24.7386	−1.02988	−0.514941	−	0.857225i	$-0.672186\pi$
$577$	−24.7386	−1.02988	−0.514941	+	0.857225i	$0.672186\pi$
$578$	−7.24621	−0.301403
$579$	0	0
$580$	0	0
$581$	21.4773	0.891027
$582$	0	0
$583$	8.49242	0.351720
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−18.4924	−0.763915
$587$	−35.3693	−1.45985	−0.729924	−	0.683528i	$-0.760445\pi$
$587$	−35.3693	−1.45985	−0.729924	+	0.683528i	$0.760445\pi$
$588$	0	0
$589$	−9.12311	−0.375911
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−15.6155	−0.641253	−0.320626	−	0.947206i	$-0.603893\pi$
$593$	−15.6155	−0.641253	−0.320626	+	0.947206i	$0.603893\pi$
$594$	0	0
$595$	0	0
$596$	22.0000	0.901155
$597$	0	0
$598$	0	0
$599$	12.4924	0.510427	0.255213	−	0.966885i	$-0.417854\pi$
$599$	12.4924	0.510427	0.255213	+	0.966885i	$0.417854\pi$
$600$	0	0
$601$	36.7386	1.49860	0.749300	−	0.662231i	$-0.230390\pi$
$601$	36.7386	1.49860	0.749300	+	0.662231i	$0.230390\pi$
$602$	−32.0000	−1.30422
$603$	0	0
$604$	2.87689	0.117059
$605$	0	0
$606$	0	0
$607$	4.24621	0.172348	0.0861742	−	0.996280i	$-0.472536\pi$
$607$	4.24621	0.172348	0.0861742	+	0.996280i	$0.472536\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	−40.9848	−1.65807
$612$	0	0
$613$	−16.6307	−0.671707	−0.335853	−	0.941914i	$-0.609025\pi$
$613$	−16.6307	−0.671707	−0.335853	+	0.941914i	$0.609025\pi$
$614$	5.75379	0.232204
$615$	0	0
$616$	−6.24621	−0.251667
$617$	0.876894	0.0353024	0.0176512	−	0.999844i	$-0.494381\pi$
$617$	0.876894	0.0353024	0.0176512	+	0.999844i	$0.494381\pi$
$618$	0	0
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	0	0
$621$	0	0
$622$	−20.0000	−0.801927
$623$	35.5076	1.42258
$624$	0	0
$625$	0	0
$626$	24.7386	0.988755
$627$	0	0
$628$	1.12311	0.0448168
$629$	0	0
$630$	0	0
$631$	−36.9848	−1.47234	−0.736172	−	0.676795i	$-0.763368\pi$
$631$	−36.9848	−1.47234	−0.736172	+	0.676795i	$0.763368\pi$
$632$	9.12311	0.362898
$633$	0	0
$634$	−14.0000	−0.556011
$635$	0	0
$636$	0	0
$637$	−11.0152	−0.436436
$638$	−4.00000	−0.158362
$639$	0	0
$640$	0	0
$641$	−11.8617	−0.468511	−0.234255	−	0.972175i	$-0.575265\pi$
$641$	−11.8617	−0.468511	−0.234255	+	0.972175i	$0.575265\pi$
$642$	0	0
$643$	32.4924	1.28138	0.640688	−	0.767801i	$-0.278649\pi$
$643$	32.4924	1.28138	0.640688	+	0.767801i	$0.278649\pi$
$644$	0	0
$645$	0	0
$646$	−3.12311	−0.122877
$647$	4.49242	0.176615	0.0883077	−	0.996093i	$-0.471854\pi$
$647$	4.49242	0.176615	0.0883077	+	0.996093i	$0.471854\pi$
$648$	0	0
$649$	−6.24621	−0.245185
$650$	0	0
$651$	0	0
$652$	−12.0000	−0.469956
$653$	26.9848	1.05600	0.527999	−	0.849245i	$-0.322942\pi$
$653$	26.9848	1.05600	0.527999	+	0.849245i	$0.322942\pi$
$654$	0	0
$655$	0	0
$656$	5.12311	0.200024
$657$	0	0
$658$	32.0000	1.24749
$659$	0.384472	0.0149769	0.00748845	−	0.999972i	$-0.497616\pi$
$659$	0.384472	0.0149769	0.00748845	+	0.999972i	$0.497616\pi$
$660$	0	0
$661$	−31.6155	−1.22970	−0.614851	−	0.788643i	$-0.710784\pi$
$661$	−31.6155	−1.22970	−0.614851	+	0.788643i	$0.710784\pi$
$662$	−24.4924	−0.951925
$663$	0	0
$664$	6.87689	0.266875
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−8.00000	−0.309529
$669$	0	0
$670$	0	0
$671$	−24.4924	−0.945519
$672$	0	0
$673$	−6.00000	−0.231283	−0.115642	−	0.993291i	$-0.536892\pi$
$673$	−6.00000	−0.231283	−0.115642	+	0.993291i	$0.536892\pi$
$674$	26.0000	1.00148
$675$	0	0
$676$	3.00000	0.115385
$677$	34.9848	1.34458	0.672288	−	0.740289i	$-0.265311\pi$
$677$	34.9848	1.34458	0.672288	+	0.740289i	$0.265311\pi$
$678$	0	0
$679$	18.7386	0.719123
$680$	0	0
$681$	0	0
$682$	−18.2462	−0.698684
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	0	0
$685$	0	0
$686$	−13.2614	−0.506321
$687$	0	0
$688$	−10.2462	−0.390633
$689$	16.9848	0.647071
$690$	0	0
$691$	0.492423	0.0187326	0.00936632	−	0.999956i	$-0.497019\pi$
$691$	0.492423	0.0187326	0.00936632	+	0.999956i	$0.497019\pi$
$692$	8.24621	0.313474
$693$	0	0
$694$	8.63068	0.327616
$695$	0	0
$696$	0	0
$697$	16.0000	0.606043
$698$	−10.4924	−0.397144
$699$	0	0
$700$	0	0
$701$	34.9848	1.32136	0.660680	−	0.750668i	$-0.270268\pi$
$701$	34.9848	1.32136	0.660680	+	0.750668i	$0.270268\pi$
$702$	0	0
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	25.8617	0.973319
$707$	31.2311	1.17456
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	0	0
$711$	0	0
$712$	11.3693	0.426083
$713$	0	0
$714$	0	0
$715$	0	0
$716$	8.87689	0.331745
$717$	0	0
$718$	10.2462	0.382385
$719$	−10.2462	−0.382119	−0.191060	−	0.981578i	$-0.561192\pi$
$719$	−10.2462	−0.382119	−0.191060	+	0.981578i	$0.561192\pi$
$720$	0	0
$721$	25.7538	0.959121
$722$	1.00000	0.0372161
$723$	0	0
$724$	9.36932	0.348208
$725$	0	0
$726$	0	0
$727$	−6.63068	−0.245918	−0.122959	−	0.992412i	$-0.539238\pi$
$727$	−6.63068	−0.245918	−0.122959	+	0.992412i	$0.539238\pi$
$728$	−12.4924	−0.463000
$729$	0	0
$730$	0	0
$731$	−32.0000	−1.18356
$732$	0	0
$733$	−33.6155	−1.24162	−0.620809	−	0.783962i	$-0.713196\pi$
$733$	−33.6155	−1.24162	−0.620809	+	0.783962i	$0.713196\pi$
$734$	0.876894	0.0323668
$735$	0	0
$736$	0	0
$737$	12.4924	0.460164
$738$	0	0
$739$	−16.4924	−0.606684	−0.303342	−	0.952882i	$-0.598102\pi$
$739$	−16.4924	−0.606684	−0.303342	+	0.952882i	$0.598102\pi$
$740$	0	0
$741$	0	0
$742$	−13.2614	−0.486840
$743$	24.9848	0.916605	0.458303	−	0.888796i	$-0.348458\pi$
$743$	24.9848	0.916605	0.458303	+	0.888796i	$0.348458\pi$
$744$	0	0
$745$	0	0
$746$	14.2462	0.521591
$747$	0	0
$748$	−6.24621	−0.228384
$749$	44.4924	1.62572
$750$	0	0
$751$	−7.86174	−0.286879	−0.143439	−	0.989659i	$-0.545816\pi$
$751$	−7.86174	−0.286879	−0.143439	+	0.989659i	$0.545816\pi$
$752$	10.2462	0.373641
$753$	0	0
$754$	−8.00000	−0.291343
$755$	0	0
$756$	0	0
$757$	35.8617	1.30342	0.651709	−	0.758469i	$-0.274053\pi$
$757$	35.8617	1.30342	0.651709	+	0.758469i	$0.274053\pi$
$758$	8.49242	0.308459
$759$	0	0
$760$	0	0
$761$	46.2462	1.67642	0.838212	−	0.545345i	$-0.183601\pi$
$761$	46.2462	1.67642	0.838212	+	0.545345i	$0.183601\pi$
$762$	0	0
$763$	15.2311	0.551401
$764$	4.00000	0.144715
$765$	0	0
$766$	36.4924	1.31852
$767$	−12.4924	−0.451075
$768$	0	0
$769$	−23.7538	−0.856584	−0.428292	−	0.903641i	$-0.640884\pi$
$769$	−23.7538	−0.856584	−0.428292	+	0.903641i	$0.640884\pi$
$770$	0	0
$771$	0	0
$772$	−18.4924	−0.665557
$773$	31.7538	1.14210	0.571052	−	0.820914i	$-0.306535\pi$
$773$	31.7538	1.14210	0.571052	+	0.820914i	$0.306535\pi$
$774$	0	0
$775$	0	0
$776$	6.00000	0.215387
$777$	0	0
$778$	34.4924	1.23661
$779$	−5.12311	−0.183554
$780$	0	0
$781$	12.4924	0.447014
$782$	0	0
$783$	0	0
$784$	2.75379	0.0983496
$785$	0	0
$786$	0	0
$787$	−24.4924	−0.873061	−0.436530	−	0.899690i	$-0.643793\pi$
$787$	−24.4924	−0.873061	−0.436530	+	0.899690i	$0.643793\pi$
$788$	−8.24621	−0.293759
$789$	0	0
$790$	0	0
$791$	13.2614	0.471520
$792$	0	0
$793$	−48.9848	−1.73950
$794$	−13.1231	−0.465722
$795$	0	0
$796$	10.2462	0.363167
$797$	2.49242	0.0882861	0.0441431	−	0.999025i	$-0.485944\pi$
$797$	2.49242	0.0882861	0.0441431	+	0.999025i	$0.485944\pi$
$798$	0	0
$799$	32.0000	1.13208
$800$	0	0
$801$	0	0
$802$	31.8617	1.12508
$803$	12.0000	0.423471
$804$	0	0
$805$	0	0
$806$	−36.4924	−1.28539
$807$	0	0
$808$	10.0000	0.351799
$809$	2.24621	0.0789726	0.0394863	−	0.999220i	$-0.487428\pi$
$809$	2.24621	0.0789726	0.0394863	+	0.999220i	$0.487428\pi$
$810$	0	0
$811$	−44.9848	−1.57963	−0.789816	−	0.613344i	$-0.789824\pi$
$811$	−44.9848	−1.57963	−0.789816	+	0.613344i	$0.789824\pi$
$812$	6.24621	0.219199
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	10.2462	0.358470
$818$	−6.00000	−0.209785
$819$	0	0
$820$	0	0
$821$	−14.0000	−0.488603	−0.244302	−	0.969699i	$-0.578559\pi$
$821$	−14.0000	−0.488603	−0.244302	+	0.969699i	$0.578559\pi$
$822$	0	0
$823$	−1.86174	−0.0648962	−0.0324481	−	0.999473i	$-0.510330\pi$
$823$	−1.86174	−0.0648962	−0.0324481	+	0.999473i	$0.510330\pi$
$824$	8.24621	0.287270
$825$	0	0
$826$	9.75379	0.339378
$827$	13.7538	0.478266	0.239133	−	0.970987i	$-0.423137\pi$
$827$	13.7538	0.478266	0.239133	+	0.970987i	$0.423137\pi$
$828$	0	0
$829$	−44.1080	−1.53193	−0.765966	−	0.642881i	$-0.777739\pi$
$829$	−44.1080	−1.53193	−0.765966	+	0.642881i	$0.777739\pi$
$830$	0	0
$831$	0	0
$832$	−4.00000	−0.138675
$833$	8.60037	0.297985
$834$	0	0
$835$	0	0
$836$	2.00000	0.0691714
$837$	0	0
$838$	12.2462	0.423038
$839$	24.9848	0.862573	0.431286	−	0.902215i	$-0.358060\pi$
$839$	24.9848	0.862573	0.431286	+	0.902215i	$0.358060\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−15.1231	−0.521177
$843$	0	0
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	−21.8617	−0.751178
$848$	−4.24621	−0.145815
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−15.3693	−0.526235	−0.263118	−	0.964764i	$-0.584751\pi$
$853$	−15.3693	−0.526235	−0.263118	+	0.964764i	$0.584751\pi$
$854$	38.2462	1.30876
$855$	0	0
$856$	14.2462	0.486925
$857$	−55.4773	−1.89507	−0.947534	−	0.319656i	$-0.896433\pi$
$857$	−55.4773	−1.89507	−0.947534	+	0.319656i	$0.896433\pi$
$858$	0	0
$859$	−0.492423	−0.0168012	−0.00840062	−	0.999965i	$-0.502674\pi$
$859$	−0.492423	−0.0168012	−0.00840062	+	0.999965i	$0.502674\pi$
$860$	0	0
$861$	0	0
$862$	−16.0000	−0.544962
$863$	−28.4924	−0.969893	−0.484947	−	0.874544i	$-0.661161\pi$
$863$	−28.4924	−0.969893	−0.484947	+	0.874544i	$0.661161\pi$
$864$	0	0
$865$	0	0
$866$	18.0000	0.611665
$867$	0	0
$868$	28.4924	0.967096
$869$	−18.2462	−0.618960
$870$	0	0
$871$	24.9848	0.846579
$872$	4.87689	0.165152
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.2462	−1.29148	−0.645741	−	0.763556i	$-0.723452\pi$
$877$	−38.2462	−1.29148	−0.645741	+	0.763556i	$0.723452\pi$
$878$	−10.8769	−0.367077
$879$	0	0
$880$	0	0
$881$	32.4924	1.09470	0.547349	−	0.836905i	$-0.315637\pi$
$881$	32.4924	1.09470	0.547349	+	0.836905i	$0.315637\pi$
$882$	0	0
$883$	−40.4924	−1.36268	−0.681339	−	0.731968i	$-0.738602\pi$
$883$	−40.4924	−1.36268	−0.681339	+	0.731968i	$0.738602\pi$
$884$	−12.4924	−0.420166
$885$	0	0
$886$	−19.8617	−0.667268
$887$	−8.98485	−0.301682	−0.150841	−	0.988558i	$-0.548198\pi$
$887$	−8.98485	−0.301682	−0.150841	+	0.988558i	$0.548198\pi$
$888$	0	0
$889$	11.7235	0.393193
$890$	0	0
$891$	0	0
$892$	−10.0000	−0.334825
$893$	−10.2462	−0.342876
$894$	0	0
$895$	0	0
$896$	3.12311	0.104336
$897$	0	0
$898$	4.63068	0.154528
$899$	18.2462	0.608545
$900$	0	0
$901$	−13.2614	−0.441800
$902$	−10.2462	−0.341162
$903$	0	0
$904$	4.24621	0.141227
$905$	0	0
$906$	0	0
$907$	16.9848	0.563973	0.281986	−	0.959418i	$-0.409007\pi$
$907$	16.9848	0.563973	0.281986	+	0.959418i	$0.409007\pi$
$908$	−20.4924	−0.680065
$909$	0	0
$910$	0	0
$911$	8.98485	0.297681	0.148841	−	0.988861i	$-0.452446\pi$
$911$	8.98485	0.297681	0.148841	+	0.988861i	$0.452446\pi$
$912$	0	0
$913$	−13.7538	−0.455184
$914$	−26.0000	−0.860004
$915$	0	0
$916$	18.4924	0.611007
$917$	70.2462	2.31973
$918$	0	0
$919$	16.4924	0.544035	0.272017	−	0.962292i	$-0.412309\pi$
$919$	16.4924	0.544035	0.272017	+	0.962292i	$0.412309\pi$
$920$	0	0
$921$	0	0
$922$	−22.4924	−0.740748
$923$	24.9848	0.822386
$924$	0	0
$925$	0	0
$926$	23.1231	0.759872
$927$	0	0
$928$	2.00000	0.0656532
$929$	−16.4924	−0.541099	−0.270549	−	0.962706i	$-0.587205\pi$
$929$	−16.4924	−0.541099	−0.270549	+	0.962706i	$0.587205\pi$
$930$	0	0
$931$	−2.75379	−0.0902518
$932$	−1.36932	−0.0448535
$933$	0	0
$934$	30.8769	1.01032
$935$	0	0
$936$	0	0
$937$	−31.7538	−1.03735	−0.518676	−	0.854971i	$-0.673575\pi$
$937$	−31.7538	−1.03735	−0.518676	+	0.854971i	$0.673575\pi$
$938$	−19.5076	−0.636945
$939$	0	0
$940$	0	0
$941$	−37.2311	−1.21370	−0.606849	−	0.794817i	$-0.707567\pi$
$941$	−37.2311	−1.21370	−0.606849	+	0.794817i	$0.707567\pi$
$942$	0	0
$943$	0	0
$944$	3.12311	0.101648
$945$	0	0
$946$	20.4924	0.666266
$947$	−9.61553	−0.312463	−0.156231	−	0.987720i	$-0.549935\pi$
$947$	−9.61553	−0.312463	−0.156231	+	0.987720i	$0.549935\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	0	0
$952$	9.75379	0.316122
$953$	−59.4773	−1.92666	−0.963329	−	0.268324i	$-0.913530\pi$
$953$	−59.4773	−1.92666	−0.963329	+	0.268324i	$0.913530\pi$
$954$	0	0
$955$	0	0
$956$	1.75379	0.0567216
$957$	0	0
$958$	−4.00000	−0.129234
$959$	−59.7235	−1.92857
$960$	0	0
$961$	52.2311	1.68487
$962$	0	0
$963$	0	0
$964$	0.246211	0.00792993
$965$	0	0
$966$	0	0
$967$	51.1231	1.64401	0.822004	−	0.569482i	$-0.192856\pi$
$967$	51.1231	1.64401	0.822004	+	0.569482i	$0.192856\pi$
$968$	−7.00000	−0.224989
$969$	0	0
$970$	0	0
$971$	−39.6155	−1.27132	−0.635661	−	0.771968i	$-0.719273\pi$
$971$	−39.6155	−1.27132	−0.635661	+	0.771968i	$0.719273\pi$
$972$	0	0
$973$	−51.5076	−1.65126
$974$	15.7538	0.504784
$975$	0	0
$976$	12.2462	0.391992
$977$	−58.0000	−1.85558	−0.927792	−	0.373097i	$-0.878296\pi$
$977$	−58.0000	−1.85558	−0.927792	+	0.373097i	$0.878296\pi$
$978$	0	0
$979$	−22.7386	−0.726730
$980$	0	0
$981$	0	0
$982$	22.4924	0.717762
$983$	−36.4924	−1.16393	−0.581964	−	0.813215i	$-0.697715\pi$
$983$	−36.4924	−1.16393	−0.581964	+	0.813215i	$0.697715\pi$
$984$	0	0
$985$	0	0
$986$	6.24621	0.198920
$987$	0	0
$988$	4.00000	0.127257
$989$	0	0
$990$	0	0
$991$	−1.61553	−0.0513189	−0.0256595	−	0.999671i	$-0.508169\pi$
$991$	−1.61553	−0.0513189	−0.0256595	+	0.999671i	$0.508169\pi$
$992$	9.12311	0.289659
$993$	0	0
$994$	−19.5076	−0.618743
$995$	0	0
$996$	0	0
$997$	−35.3693	−1.12016	−0.560079	−	0.828439i	$-0.689229\pi$
$997$	−35.3693	−1.12016	−0.560079	+	0.828439i	$0.689229\pi$
$998$	−4.00000	−0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.bx.1.2		2
3.2	odd	2		8550.2.a.bp.1.2		2
5.4	even	2		1710.2.a.v.1.1	✓	2
15.14	odd	2		1710.2.a.x.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1710.2.a.v.1.1	✓	2	5.4	even	2
1710.2.a.x.1.1	yes	2	15.14	odd	2
8550.2.a.bp.1.2		2	3.2	odd	2
8550.2.a.bx.1.2		2	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.12311 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.12311 q^{7} +1.00000 q^{8} -2.00000 q^{11} -4.00000 q^{13} +3.12311 q^{14} +1.00000 q^{16} +3.12311 q^{17} -1.00000 q^{19} -2.00000 q^{22} -4.00000 q^{26} +3.12311 q^{28} +2.00000 q^{29} +9.12311 q^{31} +1.00000 q^{32} +3.12311 q^{34} -1.00000 q^{38} +5.12311 q^{41} -10.2462 q^{43} -2.00000 q^{44} +10.2462 q^{47} +2.75379 q^{49} -4.00000 q^{52} -4.24621 q^{53} +3.12311 q^{56} +2.00000 q^{58} +3.12311 q^{59} +12.2462 q^{61} +9.12311 q^{62} +1.00000 q^{64} -6.24621 q^{67} +3.12311 q^{68} -6.24621 q^{71} -6.00000 q^{73} -1.00000 q^{76} -6.24621 q^{77} +9.12311 q^{79} +5.12311 q^{82} +6.87689 q^{83} -10.2462 q^{86} -2.00000 q^{88} +11.3693 q^{89} -12.4924 q^{91} +10.2462 q^{94} +6.00000 q^{97} +2.75379 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - 4 q^{11} - 8 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{19} - 4 q^{22} - 8 q^{26} - 2 q^{28} + 4 q^{29} + 10 q^{31} + 2 q^{32} - 2 q^{34} - 2 q^{38} + 2 q^{41} - 4 q^{43} - 4 q^{44} + 4 q^{47} + 22 q^{49} - 8 q^{52} + 8 q^{53} - 2 q^{56} + 4 q^{58} - 2 q^{59} + 8 q^{61} + 10 q^{62} + 2 q^{64} + 4 q^{67} - 2 q^{68} + 4 q^{71} - 12 q^{73} - 2 q^{76} + 4 q^{77} + 10 q^{79} + 2 q^{82} + 22 q^{83} - 4 q^{86} - 4 q^{88} - 2 q^{89} + 8 q^{91} + 4 q^{94} + 12 q^{97} + 22 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 - 4 * q^11 - 8 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 - 2 * q^19 - 4 * q^22 - 8 * q^26 - 2 * q^28 + 4 * q^29 + 10 * q^31 + 2 * q^32 - 2 * q^34 - 2 * q^38 + 2 * q^41 - 4 * q^43 - 4 * q^44 + 4 * q^47 + 22 * q^49 - 8 * q^52 + 8 * q^53 - 2 * q^56 + 4 * q^58 - 2 * q^59 + 8 * q^61 + 10 * q^62 + 2 * q^64 + 4 * q^67 - 2 * q^68 + 4 * q^71 - 12 * q^73 - 2 * q^76 + 4 * q^77 + 10 * q^79 + 2 * q^82 + 22 * q^83 - 4 * q^86 - 4 * q^88 - 2 * q^89 + 8 * q^91 + 4 * q^94 + 12 * q^97 + 22 * q^98

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