LMFDB - Embedded newform 8550.2.a.br.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:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.56155 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} +2.56155 q^{7} -1.00000 q^{8} -4.00000 q^{11} -5.68466 q^{13} -2.56155 q^{14} +1.00000 q^{16} +3.43845 q^{17} -1.00000 q^{19} +4.00000 q^{22} +7.68466 q^{23} +5.68466 q^{26} +2.56155 q^{28} +5.68466 q^{29} -5.12311 q^{31} -1.00000 q^{32} -3.43845 q^{34} +6.00000 q^{37} +1.00000 q^{38} -12.2462 q^{41} +2.87689 q^{43} -4.00000 q^{44} -7.68466 q^{46} +6.24621 q^{47} -0.438447 q^{49} -5.68466 q^{52} -4.56155 q^{53} -2.56155 q^{56} -5.68466 q^{58} -2.56155 q^{59} +11.1231 q^{61} +5.12311 q^{62} +1.00000 q^{64} +2.56155 q^{67} +3.43845 q^{68} -10.2462 q^{71} +1.68466 q^{73} -6.00000 q^{74} -1.00000 q^{76} -10.2462 q^{77} -5.12311 q^{79} +12.2462 q^{82} +2.87689 q^{83} -2.87689 q^{86} +4.00000 q^{88} -2.00000 q^{89} -14.5616 q^{91} +7.68466 q^{92} -6.24621 q^{94} -6.00000 q^{97} +0.438447 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + q^{7} - 2 q^{8} - 8 q^{11} + q^{13} - q^{14} + 2 q^{16} + 11 q^{17} - 2 q^{19} + 8 q^{22} + 3 q^{23} - q^{26} + q^{28} - q^{29} - 2 q^{31} - 2 q^{32} - 11 q^{34} + 12 q^{37} + 2 q^{38} - 8 q^{41} + 14 q^{43} - 8 q^{44} - 3 q^{46} - 4 q^{47} - 5 q^{49} + q^{52} - 5 q^{53} - q^{56} + q^{58} - q^{59} + 14 q^{61} + 2 q^{62} + 2 q^{64} + q^{67} + 11 q^{68} - 4 q^{71} - 9 q^{73} - 12 q^{74} - 2 q^{76} - 4 q^{77} - 2 q^{79} + 8 q^{82} + 14 q^{83} - 14 q^{86} + 8 q^{88} - 4 q^{89} - 25 q^{91} + 3 q^{92} + 4 q^{94} - 12 q^{97} + 5 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + q^7 - 2 * q^8 - 8 * q^11 + q^13 - q^14 + 2 * q^16 + 11 * q^17 - 2 * q^19 + 8 * q^22 + 3 * q^23 - q^26 + q^28 - q^29 - 2 * q^31 - 2 * q^32 - 11 * q^34 + 12 * q^37 + 2 * q^38 - 8 * q^41 + 14 * q^43 - 8 * q^44 - 3 * q^46 - 4 * q^47 - 5 * q^49 + q^52 - 5 * q^53 - q^56 + q^58 - q^59 + 14 * q^61 + 2 * q^62 + 2 * q^64 + q^67 + 11 * q^68 - 4 * q^71 - 9 * q^73 - 12 * q^74 - 2 * q^76 - 4 * q^77 - 2 * q^79 + 8 * q^82 + 14 * q^83 - 14 * q^86 + 8 * q^88 - 4 * q^89 - 25 * q^91 + 3 * q^92 + 4 * q^94 - 12 * q^97 + 5 * 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
$7$	2.56155	0.968176	0.484088	−	0.875019i	$-0.339151\pi$
$7$	2.56155	0.968176	0.484088	+	0.875019i	$0.339151\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−5.68466	−1.57664	−0.788320	−	0.615265i	$-0.789049\pi$
$13$	−5.68466	−1.57664	−0.788320	+	0.615265i	$0.789049\pi$
$14$	−2.56155	−0.684604
$15$	0	0
$16$	1.00000	0.250000
$17$	3.43845	0.833946	0.416973	−	0.908919i	$-0.363091\pi$
$17$	3.43845	0.833946	0.416973	+	0.908919i	$0.363091\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	4.00000	0.852803
$23$	7.68466	1.60236	0.801181	−	0.598422i	$-0.204205\pi$
$23$	7.68466	1.60236	0.801181	+	0.598422i	$0.204205\pi$
$24$	0	0
$25$	0	0
$26$	5.68466	1.11485
$27$	0	0
$28$	2.56155	0.484088
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	−5.12311	−0.920137	−0.460068	−	0.887883i	$-0.652175\pi$
$31$	−5.12311	−0.920137	−0.460068	+	0.887883i	$0.652175\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.43845	−0.589689
$35$	0	0
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	0	0
$41$	−12.2462	−1.91254	−0.956268	−	0.292490i	$-0.905516\pi$
$41$	−12.2462	−1.91254	−0.956268	+	0.292490i	$0.905516\pi$
$42$	0	0
$43$	2.87689	0.438722	0.219361	−	0.975644i	$-0.429603\pi$
$43$	2.87689	0.438722	0.219361	+	0.975644i	$0.429603\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	−7.68466	−1.13304
$47$	6.24621	0.911104	0.455552	−	0.890209i	$-0.349442\pi$
$47$	6.24621	0.911104	0.455552	+	0.890209i	$0.349442\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	0	0
$52$	−5.68466	−0.788320
$53$	−4.56155	−0.626577	−0.313289	−	0.949658i	$-0.601431\pi$
$53$	−4.56155	−0.626577	−0.313289	+	0.949658i	$0.601431\pi$
$54$	0	0
$55$	0	0
$56$	−2.56155	−0.342302
$57$	0	0
$58$	−5.68466	−0.746432
$59$	−2.56155	−0.333486	−0.166743	−	0.986000i	$-0.553325\pi$
$59$	−2.56155	−0.333486	−0.166743	+	0.986000i	$0.553325\pi$
$60$	0	0
$61$	11.1231	1.42417	0.712084	−	0.702094i	$-0.247752\pi$
$61$	11.1231	1.42417	0.712084	+	0.702094i	$0.247752\pi$
$62$	5.12311	0.650635
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	2.56155	0.312943	0.156472	−	0.987682i	$-0.449988\pi$
$67$	2.56155	0.312943	0.156472	+	0.987682i	$0.449988\pi$
$68$	3.43845	0.416973
$69$	0	0
$70$	0	0
$71$	−10.2462	−1.21600	−0.608001	−	0.793936i	$-0.708028\pi$
$71$	−10.2462	−1.21600	−0.608001	+	0.793936i	$0.708028\pi$
$72$	0	0
$73$	1.68466	0.197174	0.0985872	−	0.995128i	$-0.468568\pi$
$73$	1.68466	0.197174	0.0985872	+	0.995128i	$0.468568\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−1.00000	−0.114708
$77$	−10.2462	−1.16766
$78$	0	0
$79$	−5.12311	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$79$	−5.12311	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$80$	0	0
$81$	0	0
$82$	12.2462	1.35237
$83$	2.87689	0.315780	0.157890	−	0.987457i	$-0.449531\pi$
$83$	2.87689	0.315780	0.157890	+	0.987457i	$0.449531\pi$
$84$	0	0
$85$	0	0
$86$	−2.87689	−0.310223
$87$	0	0
$88$	4.00000	0.426401
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−14.5616	−1.52647
$92$	7.68466	0.801181
$93$	0	0
$94$	−6.24621	−0.644247
$95$	0	0
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0.438447	0.0442899
$99$	0	0
$100$	0	0
$101$	17.3693	1.72831	0.864156	−	0.503224i	$-0.167853\pi$
$101$	17.3693	1.72831	0.864156	+	0.503224i	$0.167853\pi$
$102$	0	0
$103$	−2.24621	−0.221326	−0.110663	−	0.993858i	$-0.535297\pi$
$103$	−2.24621	−0.221326	−0.110663	+	0.993858i	$0.535297\pi$
$104$	5.68466	0.557427
$105$	0	0
$106$	4.56155	0.443057
$107$	−5.43845	−0.525755	−0.262877	−	0.964829i	$-0.584671\pi$
$107$	−5.43845	−0.525755	−0.262877	+	0.964829i	$0.584671\pi$
$108$	0	0
$109$	−0.561553	−0.0537870	−0.0268935	−	0.999638i	$-0.508561\pi$
$109$	−0.561553	−0.0537870	−0.0268935	+	0.999638i	$0.508561\pi$
$110$	0	0
$111$	0	0
$112$	2.56155	0.242044
$113$	8.87689	0.835068	0.417534	−	0.908661i	$-0.362894\pi$
$113$	8.87689	0.835068	0.417534	+	0.908661i	$0.362894\pi$
$114$	0	0
$115$	0	0
$116$	5.68466	0.527807
$117$	0	0
$118$	2.56155	0.235810
$119$	8.80776	0.807406
$120$	0	0
$121$	5.00000	0.454545
$122$	−11.1231	−1.00704
$123$	0	0
$124$	−5.12311	−0.460068
$125$	0	0
$126$	0	0
$127$	−13.1231	−1.16449	−0.582244	−	0.813014i	$-0.697825\pi$
$127$	−13.1231	−1.16449	−0.582244	+	0.813014i	$0.697825\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	16.4924	1.44095	0.720475	−	0.693481i	$-0.243924\pi$
$131$	16.4924	1.44095	0.720475	+	0.693481i	$0.243924\pi$
$132$	0	0
$133$	−2.56155	−0.222115
$134$	−2.56155	−0.221284
$135$	0	0
$136$	−3.43845	−0.294844
$137$	−14.8078	−1.26511	−0.632556	−	0.774514i	$-0.717994\pi$
$137$	−14.8078	−1.26511	−0.632556	+	0.774514i	$0.717994\pi$
$138$	0	0
$139$	16.4924	1.39887	0.699435	−	0.714697i	$-0.253435\pi$
$139$	16.4924	1.39887	0.699435	+	0.714697i	$0.253435\pi$
$140$	0	0
$141$	0	0
$142$	10.2462	0.859843
$143$	22.7386	1.90150
$144$	0	0
$145$	0	0
$146$	−1.68466	−0.139423
$147$	0	0
$148$	6.00000	0.493197
$149$	−13.3693	−1.09526	−0.547629	−	0.836722i	$-0.684469\pi$
$149$	−13.3693	−1.09526	−0.547629	+	0.836722i	$0.684469\pi$
$150$	0	0
$151$	5.12311	0.416912	0.208456	−	0.978032i	$-0.433156\pi$
$151$	5.12311	0.416912	0.208456	+	0.978032i	$0.433156\pi$
$152$	1.00000	0.0811107
$153$	0	0
$154$	10.2462	0.825663
$155$	0	0
$156$	0	0
$157$	20.2462	1.61582	0.807912	−	0.589303i	$-0.200598\pi$
$157$	20.2462	1.61582	0.807912	+	0.589303i	$0.200598\pi$
$158$	5.12311	0.407572
$159$	0	0
$160$	0	0
$161$	19.6847	1.55137
$162$	0	0
$163$	15.3693	1.20382	0.601909	−	0.798565i	$-0.294407\pi$
$163$	15.3693	1.20382	0.601909	+	0.798565i	$0.294407\pi$
$164$	−12.2462	−0.956268
$165$	0	0
$166$	−2.87689	−0.223290
$167$	−7.36932	−0.570255	−0.285127	−	0.958490i	$-0.592036\pi$
$167$	−7.36932	−0.570255	−0.285127	+	0.958490i	$0.592036\pi$
$168$	0	0
$169$	19.3153	1.48580
$170$	0	0
$171$	0	0
$172$	2.87689	0.219361
$173$	20.2462	1.53929	0.769645	−	0.638471i	$-0.220433\pi$
$173$	20.2462	1.53929	0.769645	+	0.638471i	$0.220433\pi$
$174$	0	0
$175$	0	0
$176$	−4.00000	−0.301511
$177$	0	0
$178$	2.00000	0.149906
$179$	22.2462	1.66276	0.831380	−	0.555704i	$-0.187551\pi$
$179$	22.2462	1.66276	0.831380	+	0.555704i	$0.187551\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	14.5616	1.07937
$183$	0	0
$184$	−7.68466	−0.566521
$185$	0	0
$186$	0	0
$187$	−13.7538	−1.00578
$188$	6.24621	0.455552
$189$	0	0
$190$	0	0
$191$	3.68466	0.266613	0.133306	−	0.991075i	$-0.457441\pi$
$191$	3.68466	0.266613	0.133306	+	0.991075i	$0.457441\pi$
$192$	0	0
$193$	14.4924	1.04319	0.521594	−	0.853194i	$-0.325338\pi$
$193$	14.4924	1.04319	0.521594	+	0.853194i	$0.325338\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	−0.438447	−0.0313177
$197$	−20.2462	−1.44248	−0.721241	−	0.692684i	$-0.756428\pi$
$197$	−20.2462	−1.44248	−0.721241	+	0.692684i	$0.756428\pi$
$198$	0	0
$199$	16.8078	1.19147	0.595735	−	0.803181i	$-0.296861\pi$
$199$	16.8078	1.19147	0.595735	+	0.803181i	$0.296861\pi$
$200$	0	0
$201$	0	0
$202$	−17.3693	−1.22210
$203$	14.5616	1.02202
$204$	0	0
$205$	0	0
$206$	2.24621	0.156501
$207$	0	0
$208$	−5.68466	−0.394160
$209$	4.00000	0.276686
$210$	0	0
$211$	8.31534	0.572452	0.286226	−	0.958162i	$-0.407599\pi$
$211$	8.31534	0.572452	0.286226	+	0.958162i	$0.407599\pi$
$212$	−4.56155	−0.313289
$213$	0	0
$214$	5.43845	0.371765
$215$	0	0
$216$	0	0
$217$	−13.1231	−0.890854
$218$	0.561553	0.0380332
$219$	0	0
$220$	0	0
$221$	−19.5464	−1.31483
$222$	0	0
$223$	23.3693	1.56493	0.782463	−	0.622698i	$-0.213963\pi$
$223$	23.3693	1.56493	0.782463	+	0.622698i	$0.213963\pi$
$224$	−2.56155	−0.171151
$225$	0	0
$226$	−8.87689	−0.590482
$227$	25.9309	1.72109	0.860546	−	0.509373i	$-0.170122\pi$
$227$	25.9309	1.72109	0.860546	+	0.509373i	$0.170122\pi$
$228$	0	0
$229$	−14.4924	−0.957686	−0.478843	−	0.877900i	$-0.658944\pi$
$229$	−14.4924	−0.957686	−0.478843	+	0.877900i	$0.658944\pi$
$230$	0	0
$231$	0	0
$232$	−5.68466	−0.373216
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	0	0
$236$	−2.56155	−0.166743
$237$	0	0
$238$	−8.80776	−0.570923
$239$	1.43845	0.0930454	0.0465227	−	0.998917i	$-0.485186\pi$
$239$	1.43845	0.0930454	0.0465227	+	0.998917i	$0.485186\pi$
$240$	0	0
$241$	23.1231	1.48949	0.744745	−	0.667349i	$-0.232571\pi$
$241$	23.1231	1.48949	0.744745	+	0.667349i	$0.232571\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	11.1231	0.712084
$245$	0	0
$246$	0	0
$247$	5.68466	0.361706
$248$	5.12311	0.325318
$249$	0	0
$250$	0	0
$251$	−6.24621	−0.394257	−0.197129	−	0.980378i	$-0.563162\pi$
$251$	−6.24621	−0.394257	−0.197129	+	0.980378i	$0.563162\pi$
$252$	0	0
$253$	−30.7386	−1.93252
$254$	13.1231	0.823417
$255$	0	0
$256$	1.00000	0.0625000
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	15.3693	0.955003
$260$	0	0
$261$	0	0
$262$	−16.4924	−1.01891
$263$	−22.2462	−1.37176	−0.685880	−	0.727715i	$-0.740583\pi$
$263$	−22.2462	−1.37176	−0.685880	+	0.727715i	$0.740583\pi$
$264$	0	0
$265$	0	0
$266$	2.56155	0.157059
$267$	0	0
$268$	2.56155	0.156472
$269$	26.0000	1.58525	0.792624	−	0.609711i	$-0.208714\pi$
$269$	26.0000	1.58525	0.792624	+	0.609711i	$0.208714\pi$
$270$	0	0
$271$	−21.9309	−1.33221	−0.666103	−	0.745860i	$-0.732039\pi$
$271$	−21.9309	−1.33221	−0.666103	+	0.745860i	$0.732039\pi$
$272$	3.43845	0.208486
$273$	0	0
$274$	14.8078	0.894570
$275$	0	0
$276$	0	0
$277$	−0.876894	−0.0526875	−0.0263437	−	0.999653i	$-0.508386\pi$
$277$	−0.876894	−0.0526875	−0.0263437	+	0.999653i	$0.508386\pi$
$278$	−16.4924	−0.989150
$279$	0	0
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	21.1231	1.25564	0.627819	−	0.778359i	$-0.283948\pi$
$283$	21.1231	1.25564	0.627819	+	0.778359i	$0.283948\pi$
$284$	−10.2462	−0.608001
$285$	0	0
$286$	−22.7386	−1.34456
$287$	−31.3693	−1.85167
$288$	0	0
$289$	−5.17708	−0.304534
$290$	0	0
$291$	0	0
$292$	1.68466	0.0985872
$293$	−22.1771	−1.29560	−0.647799	−	0.761811i	$-0.724311\pi$
$293$	−22.1771	−1.29560	−0.647799	+	0.761811i	$0.724311\pi$
$294$	0	0
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	13.3693	0.774464
$299$	−43.6847	−2.52635
$300$	0	0
$301$	7.36932	0.424760
$302$	−5.12311	−0.294802
$303$	0	0
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	0	0
$307$	−32.4924	−1.85444	−0.927220	−	0.374516i	$-0.877809\pi$
$307$	−32.4924	−1.85444	−0.927220	+	0.374516i	$0.877809\pi$
$308$	−10.2462	−0.583832
$309$	0	0
$310$	0	0
$311$	3.68466	0.208938	0.104469	−	0.994528i	$-0.466686\pi$
$311$	3.68466	0.208938	0.104469	+	0.994528i	$0.466686\pi$
$312$	0	0
$313$	−5.05398	−0.285668	−0.142834	−	0.989747i	$-0.545621\pi$
$313$	−5.05398	−0.285668	−0.142834	+	0.989747i	$0.545621\pi$
$314$	−20.2462	−1.14256
$315$	0	0
$316$	−5.12311	−0.288197
$317$	13.0540	0.733184	0.366592	−	0.930382i	$-0.380524\pi$
$317$	13.0540	0.733184	0.366592	+	0.930382i	$0.380524\pi$
$318$	0	0
$319$	−22.7386	−1.27312
$320$	0	0
$321$	0	0
$322$	−19.6847	−1.09698
$323$	−3.43845	−0.191320
$324$	0	0
$325$	0	0
$326$	−15.3693	−0.851228
$327$	0	0
$328$	12.2462	0.676184
$329$	16.0000	0.882109
$330$	0	0
$331$	−2.56155	−0.140796	−0.0703978	−	0.997519i	$-0.522427\pi$
$331$	−2.56155	−0.140796	−0.0703978	+	0.997519i	$0.522427\pi$
$332$	2.87689	0.157890
$333$	0	0
$334$	7.36932	0.403231
$335$	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	$0.249528\pi$
$338$	−19.3153	−1.05062
$339$	0	0
$340$	0	0
$341$	20.4924	1.10973
$342$	0	0
$343$	−19.0540	−1.02882
$344$	−2.87689	−0.155112
$345$	0	0
$346$	−20.2462	−1.08844
$347$	−8.63068	−0.463319	−0.231660	−	0.972797i	$-0.574416\pi$
$347$	−8.63068	−0.463319	−0.231660	+	0.972797i	$0.574416\pi$
$348$	0	0
$349$	3.75379	0.200936	0.100468	−	0.994940i	$-0.467966\pi$
$349$	3.75379	0.200936	0.100468	+	0.994940i	$0.467966\pi$
$350$	0	0
$351$	0	0
$352$	4.00000	0.213201
$353$	−3.93087	−0.209219	−0.104610	−	0.994513i	$-0.533359\pi$
$353$	−3.93087	−0.209219	−0.104610	+	0.994513i	$0.533359\pi$
$354$	0	0
$355$	0	0
$356$	−2.00000	−0.106000
$357$	0	0
$358$	−22.2462	−1.17575
$359$	−1.43845	−0.0759183	−0.0379592	−	0.999279i	$-0.512086\pi$
$359$	−1.43845	−0.0759183	−0.0379592	+	0.999279i	$0.512086\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	18.0000	0.946059
$363$	0	0
$364$	−14.5616	−0.763233
$365$	0	0
$366$	0	0
$367$	−6.24621	−0.326050	−0.163025	−	0.986622i	$-0.552125\pi$
$367$	−6.24621	−0.326050	−0.163025	+	0.986622i	$0.552125\pi$
$368$	7.68466	0.400591
$369$	0	0
$370$	0	0
$371$	−11.6847	−0.606637
$372$	0	0
$373$	23.4384	1.21360	0.606798	−	0.794856i	$-0.292454\pi$
$373$	23.4384	1.21360	0.606798	+	0.794856i	$0.292454\pi$
$374$	13.7538	0.711191
$375$	0	0
$376$	−6.24621	−0.322124
$377$	−32.3153	−1.66432
$378$	0	0
$379$	10.5616	0.542511	0.271255	−	0.962507i	$-0.412561\pi$
$379$	10.5616	0.542511	0.271255	+	0.962507i	$0.412561\pi$
$380$	0	0
$381$	0	0
$382$	−3.68466	−0.188524
$383$	13.7538	0.702786	0.351393	−	0.936228i	$-0.385708\pi$
$383$	13.7538	0.702786	0.351393	+	0.936228i	$0.385708\pi$
$384$	0	0
$385$	0	0
$386$	−14.4924	−0.737645
$387$	0	0
$388$	−6.00000	−0.304604
$389$	7.12311	0.361156	0.180578	−	0.983561i	$-0.442203\pi$
$389$	7.12311	0.361156	0.180578	+	0.983561i	$0.442203\pi$
$390$	0	0
$391$	26.4233	1.33628
$392$	0.438447	0.0221449
$393$	0	0
$394$	20.2462	1.01999
$395$	0	0
$396$	0	0
$397$	7.12311	0.357498	0.178749	−	0.983895i	$-0.442795\pi$
$397$	7.12311	0.357498	0.178749	+	0.983895i	$0.442795\pi$
$398$	−16.8078	−0.842497
$399$	0	0
$400$	0	0
$401$	3.75379	0.187455	0.0937276	−	0.995598i	$-0.470122\pi$
$401$	3.75379	0.187455	0.0937276	+	0.995598i	$0.470122\pi$
$402$	0	0
$403$	29.1231	1.45073
$404$	17.3693	0.864156
$405$	0	0
$406$	−14.5616	−0.722678
$407$	−24.0000	−1.18964
$408$	0	0
$409$	24.7386	1.22325	0.611623	−	0.791149i	$-0.290517\pi$
$409$	24.7386	1.22325	0.611623	+	0.791149i	$0.290517\pi$
$410$	0	0
$411$	0	0
$412$	−2.24621	−0.110663
$413$	−6.56155	−0.322873
$414$	0	0
$415$	0	0
$416$	5.68466	0.278713
$417$	0	0
$418$	−4.00000	−0.195646
$419$	−23.8617	−1.16572	−0.582861	−	0.812572i	$-0.698067\pi$
$419$	−23.8617	−1.16572	−0.582861	+	0.812572i	$0.698067\pi$
$420$	0	0
$421$	−23.9309	−1.16632	−0.583160	−	0.812358i	$-0.698184\pi$
$421$	−23.9309	−1.16632	−0.583160	+	0.812358i	$0.698184\pi$
$422$	−8.31534	−0.404784
$423$	0	0
$424$	4.56155	0.221529
$425$	0	0
$426$	0	0
$427$	28.4924	1.37884
$428$	−5.43845	−0.262877
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	−14.6307	−0.703106	−0.351553	−	0.936168i	$-0.614346\pi$
$433$	−14.6307	−0.703106	−0.351553	+	0.936168i	$0.614346\pi$
$434$	13.1231	0.629929
$435$	0	0
$436$	−0.561553	−0.0268935
$437$	−7.68466	−0.367607
$438$	0	0
$439$	−13.1231	−0.626332	−0.313166	−	0.949698i	$-0.601390\pi$
$439$	−13.1231	−0.626332	−0.313166	+	0.949698i	$0.601390\pi$
$440$	0	0
$441$	0	0
$442$	19.5464	0.929727
$443$	2.24621	0.106721	0.0533604	−	0.998575i	$-0.483007\pi$
$443$	2.24621	0.106721	0.0533604	+	0.998575i	$0.483007\pi$
$444$	0	0
$445$	0	0
$446$	−23.3693	−1.10657
$447$	0	0
$448$	2.56155	0.121022
$449$	28.7386	1.35626	0.678130	−	0.734942i	$-0.262791\pi$
$449$	28.7386	1.35626	0.678130	+	0.734942i	$0.262791\pi$
$450$	0	0
$451$	48.9848	2.30661
$452$	8.87689	0.417534
$453$	0	0
$454$	−25.9309	−1.21700
$455$	0	0
$456$	0	0
$457$	−6.31534	−0.295419	−0.147710	−	0.989031i	$-0.547190\pi$
$457$	−6.31534	−0.295419	−0.147710	+	0.989031i	$0.547190\pi$
$458$	14.4924	0.677186
$459$	0	0
$460$	0	0
$461$	−3.75379	−0.174831	−0.0874157	−	0.996172i	$-0.527861\pi$
$461$	−3.75379	−0.174831	−0.0874157	+	0.996172i	$0.527861\pi$
$462$	0	0
$463$	30.2462	1.40566	0.702830	−	0.711358i	$-0.251919\pi$
$463$	30.2462	1.40566	0.702830	+	0.711358i	$0.251919\pi$
$464$	5.68466	0.263904
$465$	0	0
$466$	−10.0000	−0.463241
$467$	18.2462	0.844334	0.422167	−	0.906518i	$-0.361270\pi$
$467$	18.2462	0.844334	0.422167	+	0.906518i	$0.361270\pi$
$468$	0	0
$469$	6.56155	0.302984
$470$	0	0
$471$	0	0
$472$	2.56155	0.117905
$473$	−11.5076	−0.529119
$474$	0	0
$475$	0	0
$476$	8.80776	0.403703
$477$	0	0
$478$	−1.43845	−0.0657930
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	−34.1080	−1.55519
$482$	−23.1231	−1.05323
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	0	0
$487$	17.6155	0.798236	0.399118	−	0.916900i	$-0.369316\pi$
$487$	17.6155	0.798236	0.399118	+	0.916900i	$0.369316\pi$
$488$	−11.1231	−0.503519
$489$	0	0
$490$	0	0
$491$	1.12311	0.0506850	0.0253425	−	0.999679i	$-0.491932\pi$
$491$	1.12311	0.0506850	0.0253425	+	0.999679i	$0.491932\pi$
$492$	0	0
$493$	19.5464	0.880325
$494$	−5.68466	−0.255765
$495$	0	0
$496$	−5.12311	−0.230034
$497$	−26.2462	−1.17730
$498$	0	0
$499$	42.1080	1.88501	0.942505	−	0.334191i	$-0.108463\pi$
$499$	42.1080	1.88501	0.942505	+	0.334191i	$0.108463\pi$
$500$	0	0
$501$	0	0
$502$	6.24621	0.278782
$503$	−7.05398	−0.314521	−0.157261	−	0.987557i	$-0.550266\pi$
$503$	−7.05398	−0.314521	−0.157261	+	0.987557i	$0.550266\pi$
$504$	0	0
$505$	0	0
$506$	30.7386	1.36650
$507$	0	0
$508$	−13.1231	−0.582244
$509$	−2.49242	−0.110475	−0.0552373	−	0.998473i	$-0.517592\pi$
$509$	−2.49242	−0.110475	−0.0552373	+	0.998473i	$0.517592\pi$
$510$	0	0
$511$	4.31534	0.190899
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−14.0000	−0.617514
$515$	0	0
$516$	0	0
$517$	−24.9848	−1.09883
$518$	−15.3693	−0.675289
$519$	0	0
$520$	0	0
$521$	3.12311	0.136826	0.0684129	−	0.997657i	$-0.478206\pi$
$521$	3.12311	0.136826	0.0684129	+	0.997657i	$0.478206\pi$
$522$	0	0
$523$	31.6847	1.38547	0.692737	−	0.721191i	$-0.256405\pi$
$523$	31.6847	1.38547	0.692737	+	0.721191i	$0.256405\pi$
$524$	16.4924	0.720475
$525$	0	0
$526$	22.2462	0.969981
$527$	−17.6155	−0.767344
$528$	0	0
$529$	36.0540	1.56756
$530$	0	0
$531$	0	0
$532$	−2.56155	−0.111057
$533$	69.6155	3.01538
$534$	0	0
$535$	0	0
$536$	−2.56155	−0.110642
$537$	0	0
$538$	−26.0000	−1.12094
$539$	1.75379	0.0755410
$540$	0	0
$541$	−0.384472	−0.0165297	−0.00826487	−	0.999966i	$-0.502631\pi$
$541$	−0.384472	−0.0165297	−0.00826487	+	0.999966i	$0.502631\pi$
$542$	21.9309	0.942012
$543$	0	0
$544$	−3.43845	−0.147422
$545$	0	0
$546$	0	0
$547$	−16.4924	−0.705165	−0.352583	−	0.935781i	$-0.614696\pi$
$547$	−16.4924	−0.705165	−0.352583	+	0.935781i	$0.614696\pi$
$548$	−14.8078	−0.632556
$549$	0	0
$550$	0	0
$551$	−5.68466	−0.242175
$552$	0	0
$553$	−13.1231	−0.558051
$554$	0.876894	0.0372557
$555$	0	0
$556$	16.4924	0.699435
$557$	39.6155	1.67856	0.839282	−	0.543697i	$-0.182976\pi$
$557$	39.6155	1.67856	0.839282	+	0.543697i	$0.182976\pi$
$558$	0	0
$559$	−16.3542	−0.691707
$560$	0	0
$561$	0	0
$562$	2.00000	0.0843649
$563$	−8.49242	−0.357913	−0.178956	−	0.983857i	$-0.557272\pi$
$563$	−8.49242	−0.357913	−0.178956	+	0.983857i	$0.557272\pi$
$564$	0	0
$565$	0	0
$566$	−21.1231	−0.887870
$567$	0	0
$568$	10.2462	0.429921
$569$	3.12311	0.130927	0.0654637	−	0.997855i	$-0.479147\pi$
$569$	3.12311	0.130927	0.0654637	+	0.997855i	$0.479147\pi$
$570$	0	0
$571$	21.6155	0.904582	0.452291	−	0.891870i	$-0.350607\pi$
$571$	21.6155	0.904582	0.452291	+	0.891870i	$0.350607\pi$
$572$	22.7386	0.950750
$573$	0	0
$574$	31.3693	1.30933
$575$	0	0
$576$	0	0
$577$	10.3153	0.429433	0.214717	−	0.976676i	$-0.431117\pi$
$577$	10.3153	0.429433	0.214717	+	0.976676i	$0.431117\pi$
$578$	5.17708	0.215338
$579$	0	0
$580$	0	0
$581$	7.36932	0.305731
$582$	0	0
$583$	18.2462	0.755681
$584$	−1.68466	−0.0697117
$585$	0	0
$586$	22.1771	0.916127
$587$	−7.36932	−0.304164	−0.152082	−	0.988368i	$-0.548598\pi$
$587$	−7.36932	−0.304164	−0.152082	+	0.988368i	$0.548598\pi$
$588$	0	0
$589$	5.12311	0.211094
$590$	0	0
$591$	0	0
$592$	6.00000	0.246598
$593$	7.75379	0.318410	0.159205	−	0.987246i	$-0.449107\pi$
$593$	7.75379	0.318410	0.159205	+	0.987246i	$0.449107\pi$
$594$	0	0
$595$	0	0
$596$	−13.3693	−0.547629
$597$	0	0
$598$	43.6847	1.78640
$599$	11.8617	0.484658	0.242329	−	0.970194i	$-0.422089\pi$
$599$	11.8617	0.484658	0.242329	+	0.970194i	$0.422089\pi$
$600$	0	0
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	−7.36932	−0.300351
$603$	0	0
$604$	5.12311	0.208456
$605$	0	0
$606$	0	0
$607$	21.1231	0.857360	0.428680	−	0.903456i	$-0.358979\pi$
$607$	21.1231	0.857360	0.428680	+	0.903456i	$0.358979\pi$
$608$	1.00000	0.0405554
$609$	0	0
$610$	0	0
$611$	−35.5076	−1.43648
$612$	0	0
$613$	−5.36932	−0.216865	−0.108432	−	0.994104i	$-0.534583\pi$
$613$	−5.36932	−0.216865	−0.108432	+	0.994104i	$0.534583\pi$
$614$	32.4924	1.31129
$615$	0	0
$616$	10.2462	0.412832
$617$	12.2462	0.493014	0.246507	−	0.969141i	$-0.420717\pi$
$617$	12.2462	0.493014	0.246507	+	0.969141i	$0.420717\pi$
$618$	0	0
$619$	−36.0000	−1.44696	−0.723481	−	0.690344i	$-0.757459\pi$
$619$	−36.0000	−1.44696	−0.723481	+	0.690344i	$0.757459\pi$
$620$	0	0
$621$	0	0
$622$	−3.68466	−0.147741
$623$	−5.12311	−0.205253
$624$	0	0
$625$	0	0
$626$	5.05398	0.201997
$627$	0	0
$628$	20.2462	0.807912
$629$	20.6307	0.822599
$630$	0	0
$631$	−20.4924	−0.815790	−0.407895	−	0.913029i	$-0.633737\pi$
$631$	−20.4924	−0.815790	−0.407895	+	0.913029i	$0.633737\pi$
$632$	5.12311	0.203786
$633$	0	0
$634$	−13.0540	−0.518440
$635$	0	0
$636$	0	0
$637$	2.49242	0.0987534
$638$	22.7386	0.900231
$639$	0	0
$640$	0	0
$641$	−21.8617	−0.863487	−0.431743	−	0.901996i	$-0.642101\pi$
$641$	−21.8617	−0.863487	−0.431743	+	0.901996i	$0.642101\pi$
$642$	0	0
$643$	33.6155	1.32567	0.662834	−	0.748767i	$-0.269354\pi$
$643$	33.6155	1.32567	0.662834	+	0.748767i	$0.269354\pi$
$644$	19.6847	0.775684
$645$	0	0
$646$	3.43845	0.135284
$647$	5.43845	0.213807	0.106904	−	0.994269i	$-0.465906\pi$
$647$	5.43845	0.213807	0.106904	+	0.994269i	$0.465906\pi$
$648$	0	0
$649$	10.2462	0.402199
$650$	0	0
$651$	0	0
$652$	15.3693	0.601909
$653$	−7.12311	−0.278749	−0.139374	−	0.990240i	$-0.544509\pi$
$653$	−7.12311	−0.278749	−0.139374	+	0.990240i	$0.544509\pi$
$654$	0	0
$655$	0	0
$656$	−12.2462	−0.478134
$657$	0	0
$658$	−16.0000	−0.623745
$659$	14.0691	0.548056	0.274028	−	0.961722i	$-0.411644\pi$
$659$	14.0691	0.548056	0.274028	+	0.961722i	$0.411644\pi$
$660$	0	0
$661$	−10.8078	−0.420373	−0.210187	−	0.977661i	$-0.567407\pi$
$661$	−10.8078	−0.420373	−0.210187	+	0.977661i	$0.567407\pi$
$662$	2.56155	0.0995576
$663$	0	0
$664$	−2.87689	−0.111645
$665$	0	0
$666$	0	0
$667$	43.6847	1.69148
$668$	−7.36932	−0.285127
$669$	0	0
$670$	0	0
$671$	−44.4924	−1.71761
$672$	0	0
$673$	−21.3693	−0.823727	−0.411863	−	0.911246i	$-0.635122\pi$
$673$	−21.3693	−0.823727	−0.411863	+	0.911246i	$0.635122\pi$
$674$	−26.0000	−1.00148
$675$	0	0
$676$	19.3153	0.742898
$677$	40.5616	1.55891	0.779454	−	0.626460i	$-0.215497\pi$
$677$	40.5616	1.55891	0.779454	+	0.626460i	$0.215497\pi$
$678$	0	0
$679$	−15.3693	−0.589820
$680$	0	0
$681$	0	0
$682$	−20.4924	−0.784695
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	0	0
$686$	19.0540	0.727484
$687$	0	0
$688$	2.87689	0.109681
$689$	25.9309	0.987887
$690$	0	0
$691$	−17.1231	−0.651394	−0.325697	−	0.945474i	$-0.605599\pi$
$691$	−17.1231	−0.651394	−0.325697	+	0.945474i	$0.605599\pi$
$692$	20.2462	0.769645
$693$	0	0
$694$	8.63068	0.327616
$695$	0	0
$696$	0	0
$697$	−42.1080	−1.59495
$698$	−3.75379	−0.142083
$699$	0	0
$700$	0	0
$701$	20.8769	0.788509	0.394255	−	0.919001i	$-0.371003\pi$
$701$	20.8769	0.788509	0.394255	+	0.919001i	$0.371003\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	−4.00000	−0.150756
$705$	0	0
$706$	3.93087	0.147940
$707$	44.4924	1.67331
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	0	0
$712$	2.00000	0.0749532
$713$	−39.3693	−1.47439
$714$	0	0
$715$	0	0
$716$	22.2462	0.831380
$717$	0	0
$718$	1.43845	0.0536824
$719$	25.4384	0.948694	0.474347	−	0.880338i	$-0.342684\pi$
$719$	25.4384	0.948694	0.474347	+	0.880338i	$0.342684\pi$
$720$	0	0
$721$	−5.75379	−0.214282
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	−18.0000	−0.668965
$725$	0	0
$726$	0	0
$727$	24.3153	0.901806	0.450903	−	0.892573i	$-0.351102\pi$
$727$	24.3153	0.901806	0.450903	+	0.892573i	$0.351102\pi$
$728$	14.5616	0.539687
$729$	0	0
$730$	0	0
$731$	9.89205	0.365871
$732$	0	0
$733$	36.8769	1.36208	0.681040	−	0.732247i	$-0.261528\pi$
$733$	36.8769	1.36208	0.681040	+	0.732247i	$0.261528\pi$
$734$	6.24621	0.230552
$735$	0	0
$736$	−7.68466	−0.283260
$737$	−10.2462	−0.377424
$738$	0	0
$739$	25.1231	0.924168	0.462084	−	0.886836i	$-0.347102\pi$
$739$	25.1231	0.924168	0.462084	+	0.886836i	$0.347102\pi$
$740$	0	0
$741$	0	0
$742$	11.6847	0.428957
$743$	18.8769	0.692526	0.346263	−	0.938137i	$-0.387450\pi$
$743$	18.8769	0.692526	0.346263	+	0.938137i	$0.387450\pi$
$744$	0	0
$745$	0	0
$746$	−23.4384	−0.858143
$747$	0	0
$748$	−13.7538	−0.502888
$749$	−13.9309	−0.509023
$750$	0	0
$751$	−34.8769	−1.27268	−0.636338	−	0.771410i	$-0.719552\pi$
$751$	−34.8769	−1.27268	−0.636338	+	0.771410i	$0.719552\pi$
$752$	6.24621	0.227776
$753$	0	0
$754$	32.3153	1.17686
$755$	0	0
$756$	0	0
$757$	−42.4924	−1.54441	−0.772207	−	0.635371i	$-0.780847\pi$
$757$	−42.4924	−1.54441	−0.772207	+	0.635371i	$0.780847\pi$
$758$	−10.5616	−0.383613
$759$	0	0
$760$	0	0
$761$	−41.5464	−1.50606	−0.753028	−	0.657989i	$-0.771407\pi$
$761$	−41.5464	−1.50606	−0.753028	+	0.657989i	$0.771407\pi$
$762$	0	0
$763$	−1.43845	−0.0520753
$764$	3.68466	0.133306
$765$	0	0
$766$	−13.7538	−0.496945
$767$	14.5616	0.525787
$768$	0	0
$769$	27.4384	0.989456	0.494728	−	0.869048i	$-0.335268\pi$
$769$	27.4384	0.989456	0.494728	+	0.869048i	$0.335268\pi$
$770$	0	0
$771$	0	0
$772$	14.4924	0.521594
$773$	10.8078	0.388728	0.194364	−	0.980929i	$-0.437736\pi$
$773$	10.8078	0.388728	0.194364	+	0.980929i	$0.437736\pi$
$774$	0	0
$775$	0	0
$776$	6.00000	0.215387
$777$	0	0
$778$	−7.12311	−0.255376
$779$	12.2462	0.438766
$780$	0	0
$781$	40.9848	1.46655
$782$	−26.4233	−0.944895
$783$	0	0
$784$	−0.438447	−0.0156588
$785$	0	0
$786$	0	0
$787$	−11.1922	−0.398960	−0.199480	−	0.979902i	$-0.563925\pi$
$787$	−11.1922	−0.398960	−0.199480	+	0.979902i	$0.563925\pi$
$788$	−20.2462	−0.721241
$789$	0	0
$790$	0	0
$791$	22.7386	0.808493
$792$	0	0
$793$	−63.2311	−2.24540
$794$	−7.12311	−0.252790
$795$	0	0
$796$	16.8078	0.595735
$797$	−11.3002	−0.400273	−0.200137	−	0.979768i	$-0.564139\pi$
$797$	−11.3002	−0.400273	−0.200137	+	0.979768i	$0.564139\pi$
$798$	0	0
$799$	21.4773	0.759811
$800$	0	0
$801$	0	0
$802$	−3.75379	−0.132551
$803$	−6.73863	−0.237801
$804$	0	0
$805$	0	0
$806$	−29.1231	−1.02582
$807$	0	0
$808$	−17.3693	−0.611050
$809$	12.5616	0.441641	0.220820	−	0.975315i	$-0.429127\pi$
$809$	12.5616	0.441641	0.220820	+	0.975315i	$0.429127\pi$
$810$	0	0
$811$	−20.8078	−0.730659	−0.365330	−	0.930878i	$-0.619044\pi$
$811$	−20.8078	−0.730659	−0.365330	+	0.930878i	$0.619044\pi$
$812$	14.5616	0.511010
$813$	0	0
$814$	24.0000	0.841200
$815$	0	0
$816$	0	0
$817$	−2.87689	−0.100650
$818$	−24.7386	−0.864966
$819$	0	0
$820$	0	0
$821$	17.3693	0.606193	0.303097	−	0.952960i	$-0.401979\pi$
$821$	17.3693	0.606193	0.303097	+	0.952960i	$0.401979\pi$
$822$	0	0
$823$	29.4384	1.02616	0.513080	−	0.858341i	$-0.328504\pi$
$823$	29.4384	1.02616	0.513080	+	0.858341i	$0.328504\pi$
$824$	2.24621	0.0782505
$825$	0	0
$826$	6.56155	0.228306
$827$	−10.5616	−0.367261	−0.183631	−	0.982995i	$-0.558785\pi$
$827$	−10.5616	−0.367261	−0.183631	+	0.982995i	$0.558785\pi$
$828$	0	0
$829$	−21.0540	−0.731235	−0.365617	−	0.930765i	$-0.619142\pi$
$829$	−21.0540	−0.731235	−0.365617	+	0.930765i	$0.619142\pi$
$830$	0	0
$831$	0	0
$832$	−5.68466	−0.197080
$833$	−1.50758	−0.0522345
$834$	0	0
$835$	0	0
$836$	4.00000	0.138343
$837$	0	0
$838$	23.8617	0.824290
$839$	−20.4924	−0.707477	−0.353738	−	0.935344i	$-0.615090\pi$
$839$	−20.4924	−0.707477	−0.353738	+	0.935344i	$0.615090\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	23.9309	0.824712
$843$	0	0
$844$	8.31534	0.286226
$845$	0	0
$846$	0	0
$847$	12.8078	0.440080
$848$	−4.56155	−0.156644
$849$	0	0
$850$	0	0
$851$	46.1080	1.58056
$852$	0	0
$853$	24.7386	0.847035	0.423517	−	0.905888i	$-0.360795\pi$
$853$	24.7386	0.847035	0.423517	+	0.905888i	$0.360795\pi$
$854$	−28.4924	−0.974991
$855$	0	0
$856$	5.43845	0.185882
$857$	14.6307	0.499775	0.249887	−	0.968275i	$-0.419606\pi$
$857$	14.6307	0.499775	0.249887	+	0.968275i	$0.419606\pi$
$858$	0	0
$859$	−52.9848	−1.80782	−0.903910	−	0.427723i	$-0.859316\pi$
$859$	−52.9848	−1.80782	−0.903910	+	0.427723i	$0.859316\pi$
$860$	0	0
$861$	0	0
$862$	−16.0000	−0.544962
$863$	2.24621	0.0764619	0.0382310	−	0.999269i	$-0.487828\pi$
$863$	2.24621	0.0764619	0.0382310	+	0.999269i	$0.487828\pi$
$864$	0	0
$865$	0	0
$866$	14.6307	0.497171
$867$	0	0
$868$	−13.1231	−0.445427
$869$	20.4924	0.695158
$870$	0	0
$871$	−14.5616	−0.493399
$872$	0.561553	0.0190166
$873$	0	0
$874$	7.68466	0.259937
$875$	0	0
$876$	0	0
$877$	3.93087	0.132736	0.0663680	−	0.997795i	$-0.478859\pi$
$877$	3.93087	0.132736	0.0663680	+	0.997795i	$0.478859\pi$
$878$	13.1231	0.442883
$879$	0	0
$880$	0	0
$881$	−42.9848	−1.44820	−0.724098	−	0.689697i	$-0.757744\pi$
$881$	−42.9848	−1.44820	−0.724098	+	0.689697i	$0.757744\pi$
$882$	0	0
$883$	6.38447	0.214855	0.107427	−	0.994213i	$-0.465739\pi$
$883$	6.38447	0.214855	0.107427	+	0.994213i	$0.465739\pi$
$884$	−19.5464	−0.657417
$885$	0	0
$886$	−2.24621	−0.0754629
$887$	28.4924	0.956682	0.478341	−	0.878174i	$-0.341238\pi$
$887$	28.4924	0.956682	0.478341	+	0.878174i	$0.341238\pi$
$888$	0	0
$889$	−33.6155	−1.12743
$890$	0	0
$891$	0	0
$892$	23.3693	0.782463
$893$	−6.24621	−0.209021
$894$	0	0
$895$	0	0
$896$	−2.56155	−0.0855755
$897$	0	0
$898$	−28.7386	−0.959021
$899$	−29.1231	−0.971310
$900$	0	0
$901$	−15.6847	−0.522532
$902$	−48.9848	−1.63102
$903$	0	0
$904$	−8.87689	−0.295241
$905$	0	0
$906$	0	0
$907$	20.1771	0.669969	0.334984	−	0.942224i	$-0.391269\pi$
$907$	20.1771	0.669969	0.334984	+	0.942224i	$0.391269\pi$
$908$	25.9309	0.860546
$909$	0	0
$910$	0	0
$911$	−4.49242	−0.148841	−0.0744203	−	0.997227i	$-0.523711\pi$
$911$	−4.49242	−0.148841	−0.0744203	+	0.997227i	$0.523711\pi$
$912$	0	0
$913$	−11.5076	−0.380845
$914$	6.31534	0.208893
$915$	0	0
$916$	−14.4924	−0.478843
$917$	42.2462	1.39509
$918$	0	0
$919$	−2.06913	−0.0682543	−0.0341272	−	0.999417i	$-0.510865\pi$
$919$	−2.06913	−0.0682543	−0.0341272	+	0.999417i	$0.510865\pi$
$920$	0	0
$921$	0	0
$922$	3.75379	0.123624
$923$	58.2462	1.91720
$924$	0	0
$925$	0	0
$926$	−30.2462	−0.993952
$927$	0	0
$928$	−5.68466	−0.186608
$929$	19.3002	0.633219	0.316609	−	0.948556i	$-0.397456\pi$
$929$	19.3002	0.633219	0.316609	+	0.948556i	$0.397456\pi$
$930$	0	0
$931$	0.438447	0.0143695
$932$	10.0000	0.327561
$933$	0	0
$934$	−18.2462	−0.597034
$935$	0	0
$936$	0	0
$937$	−40.5616	−1.32509	−0.662544	−	0.749023i	$-0.730523\pi$
$937$	−40.5616	−1.32509	−0.662544	+	0.749023i	$0.730523\pi$
$938$	−6.56155	−0.214242
$939$	0	0
$940$	0	0
$941$	−54.8078	−1.78668	−0.893341	−	0.449379i	$-0.851645\pi$
$941$	−54.8078	−1.78668	−0.893341	+	0.449379i	$0.851645\pi$
$942$	0	0
$943$	−94.1080	−3.06458
$944$	−2.56155	−0.0833714
$945$	0	0
$946$	11.5076	0.374144
$947$	−34.2462	−1.11285	−0.556426	−	0.830897i	$-0.687828\pi$
$947$	−34.2462	−1.11285	−0.556426	+	0.830897i	$0.687828\pi$
$948$	0	0
$949$	−9.57671	−0.310873
$950$	0	0
$951$	0	0
$952$	−8.80776	−0.285461
$953$	44.1080	1.42880	0.714398	−	0.699739i	$-0.246700\pi$
$953$	44.1080	1.42880	0.714398	+	0.699739i	$0.246700\pi$
$954$	0	0
$955$	0	0
$956$	1.43845	0.0465227
$957$	0	0
$958$	32.0000	1.03387
$959$	−37.9309	−1.22485
$960$	0	0
$961$	−4.75379	−0.153348
$962$	34.1080	1.09968
$963$	0	0
$964$	23.1231	0.744745
$965$	0	0
$966$	0	0
$967$	15.5076	0.498690	0.249345	−	0.968415i	$-0.419785\pi$
$967$	15.5076	0.498690	0.249345	+	0.968415i	$0.419785\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	0	0
$971$	56.4924	1.81293	0.906464	−	0.422283i	$-0.138771\pi$
$971$	56.4924	1.81293	0.906464	+	0.422283i	$0.138771\pi$
$972$	0	0
$973$	42.2462	1.35435
$974$	−17.6155	−0.564438
$975$	0	0
$976$	11.1231	0.356042
$977$	28.7386	0.919430	0.459715	−	0.888066i	$-0.347952\pi$
$977$	28.7386	0.919430	0.459715	+	0.888066i	$0.347952\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	0	0
$982$	−1.12311	−0.0358397
$983$	18.8769	0.602079	0.301040	−	0.953612i	$-0.402666\pi$
$983$	18.8769	0.602079	0.301040	+	0.953612i	$0.402666\pi$
$984$	0	0
$985$	0	0
$986$	−19.5464	−0.622484
$987$	0	0
$988$	5.68466	0.180853
$989$	22.1080	0.702992
$990$	0	0
$991$	2.87689	0.0913876	0.0456938	−	0.998955i	$-0.485450\pi$
$991$	2.87689	0.0913876	0.0456938	+	0.998955i	$0.485450\pi$
$992$	5.12311	0.162659
$993$	0	0
$994$	26.2462	0.832479
$995$	0	0
$996$	0	0
$997$	16.7386	0.530118	0.265059	−	0.964232i	$-0.414609\pi$
$997$	16.7386	0.530118	0.265059	+	0.964232i	$0.414609\pi$
$998$	−42.1080	−1.33290
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.br.1.2		2
3.2	odd	2		950.2.a.h.1.2		2
5.4	even	2		1710.2.a.w.1.1		2
12.11	even	2		7600.2.a.y.1.1		2
15.2	even	4		950.2.b.f.799.3		4
15.8	even	4		950.2.b.f.799.2		4
15.14	odd	2		190.2.a.d.1.1	✓	2
60.59	even	2		1520.2.a.n.1.2		2
105.104	even	2		9310.2.a.bc.1.2		2
120.29	odd	2		6080.2.a.bh.1.2		2
120.59	even	2		6080.2.a.bb.1.1		2
285.284	even	2		3610.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.a.d.1.1	✓	2	15.14	odd	2
950.2.a.h.1.2		2	3.2	odd	2
950.2.b.f.799.2		4	15.8	even	4
950.2.b.f.799.3		4	15.2	even	4
1520.2.a.n.1.2		2	60.59	even	2
1710.2.a.w.1.1		2	5.4	even	2
3610.2.a.t.1.2		2	285.284	even	2
6080.2.a.bb.1.1		2	120.59	even	2
6080.2.a.bh.1.2		2	120.29	odd	2
7600.2.a.y.1.1		2	12.11	even	2
8550.2.a.br.1.2		2	1.1	even	1	trivial
9310.2.a.bc.1.2		2	105.104	even	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:	\( N \)	\(=\)	\( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.56155 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.56155 q^{7} -1.00000 q^{8} -4.00000 q^{11} -5.68466 q^{13} -2.56155 q^{14} +1.00000 q^{16} +3.43845 q^{17} -1.00000 q^{19} +4.00000 q^{22} +7.68466 q^{23} +5.68466 q^{26} +2.56155 q^{28} +5.68466 q^{29} -5.12311 q^{31} -1.00000 q^{32} -3.43845 q^{34} +6.00000 q^{37} +1.00000 q^{38} -12.2462 q^{41} +2.87689 q^{43} -4.00000 q^{44} -7.68466 q^{46} +6.24621 q^{47} -0.438447 q^{49} -5.68466 q^{52} -4.56155 q^{53} -2.56155 q^{56} -5.68466 q^{58} -2.56155 q^{59} +11.1231 q^{61} +5.12311 q^{62} +1.00000 q^{64} +2.56155 q^{67} +3.43845 q^{68} -10.2462 q^{71} +1.68466 q^{73} -6.00000 q^{74} -1.00000 q^{76} -10.2462 q^{77} -5.12311 q^{79} +12.2462 q^{82} +2.87689 q^{83} -2.87689 q^{86} +4.00000 q^{88} -2.00000 q^{89} -14.5616 q^{91} +7.68466 q^{92} -6.24621 q^{94} -6.00000 q^{97} +0.438447 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + q^{7} - 2 q^{8} - 8 q^{11} + q^{13} - q^{14} + 2 q^{16} + 11 q^{17} - 2 q^{19} + 8 q^{22} + 3 q^{23} - q^{26} + q^{28} - q^{29} - 2 q^{31} - 2 q^{32} - 11 q^{34} + 12 q^{37} + 2 q^{38} - 8 q^{41} + 14 q^{43} - 8 q^{44} - 3 q^{46} - 4 q^{47} - 5 q^{49} + q^{52} - 5 q^{53} - q^{56} + q^{58} - q^{59} + 14 q^{61} + 2 q^{62} + 2 q^{64} + q^{67} + 11 q^{68} - 4 q^{71} - 9 q^{73} - 12 q^{74} - 2 q^{76} - 4 q^{77} - 2 q^{79} + 8 q^{82} + 14 q^{83} - 14 q^{86} + 8 q^{88} - 4 q^{89} - 25 q^{91} + 3 q^{92} + 4 q^{94} - 12 q^{97} + 5 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + q^7 - 2 * q^8 - 8 * q^11 + q^13 - q^14 + 2 * q^16 + 11 * q^17 - 2 * q^19 + 8 * q^22 + 3 * q^23 - q^26 + q^28 - q^29 - 2 * q^31 - 2 * q^32 - 11 * q^34 + 12 * q^37 + 2 * q^38 - 8 * q^41 + 14 * q^43 - 8 * q^44 - 3 * q^46 - 4 * q^47 - 5 * q^49 + q^52 - 5 * q^53 - q^56 + q^58 - q^59 + 14 * q^61 + 2 * q^62 + 2 * q^64 + q^67 + 11 * q^68 - 4 * q^71 - 9 * q^73 - 12 * q^74 - 2 * q^76 - 4 * q^77 - 2 * q^79 + 8 * q^82 + 14 * q^83 - 14 * q^86 + 8 * q^88 - 4 * q^89 - 25 * q^91 + 3 * q^92 + 4 * q^94 - 12 * q^97 + 5 * q^98

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