LMFDB - Embedded newform 8550.2.a.cw.1.1

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:	$1$
Dimension:	$6$
Coefficient field:	6.6.3356224.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 6x^{4} + 8x^{2} - 1 $ x^6 - 6x^4 + 8x^2 - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1710)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.02852$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -4.05705 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -4.05705 q^{7} -1.00000 q^{8} -0.466185 q^{11} -0.985939 q^{13} +4.05705 q^{14} +1.00000 q^{16} -4.94567 q^{17} +1.00000 q^{19} +0.466185 q^{22} +2.22982 q^{23} +0.985939 q^{26} -4.05705 q^{28} +2.43806 q^{29} +0.715853 q^{31} -1.00000 q^{32} +4.94567 q^{34} +10.0324 q^{37} -1.00000 q^{38} +10.1993 q^{41} -0.466185 q^{43} -0.466185 q^{44} -2.22982 q^{46} -8.79811 q^{47} +9.45963 q^{49} -0.985939 q^{52} +3.89134 q^{53} +4.05705 q^{56} -2.43806 q^{58} +7.42748 q^{59} -6.45963 q^{61} -0.715853 q^{62} +1.00000 q^{64} +13.9226 q^{67} -4.94567 q^{68} +6.14222 q^{71} -7.07459 q^{73} -10.0324 q^{74} +1.00000 q^{76} +1.89134 q^{77} +1.85244 q^{79} -10.1993 q^{82} -4.37737 q^{83} +0.466185 q^{86} +0.466185 q^{88} -6.02892 q^{89} +4.00000 q^{91} +2.22982 q^{92} +8.79811 q^{94} -6.60840 q^{97} -9.45963 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8}+O(q^{10}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8	$ 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} + 6 q^{16} - 8 q^{17} + 6 q^{19} - 12 q^{23} + 8 q^{31} - 6 q^{32} + 8 q^{34} - 6 q^{38} + 12 q^{46} - 20 q^{47} + 6 q^{49} - 20 q^{53} + 12 q^{61} - 8 q^{62} + 6 q^{64} - 8 q^{68} + 6 q^{76} - 32 q^{77} - 12 q^{83} + 24 q^{91} - 12 q^{92} + 20 q^{94} - 6 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8 + 6 * q^16 - 8 * q^17 + 6 * q^19 - 12 * q^23 + 8 * q^31 - 6 * q^32 + 8 * q^34 - 6 * q^38 + 12 * q^46 - 20 * q^47 + 6 * q^49 - 20 * q^53 + 12 * q^61 - 8 * q^62 + 6 * q^64 - 8 * q^68 + 6 * q^76 - 32 * q^77 - 12 * q^83 + 24 * q^91 - 12 * q^92 + 20 * q^94 - 6 * 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
$7$	−4.05705	−1.53342	−0.766710	−	0.641994i	$-0.778107\pi$
$7$	−4.05705	−1.53342	−0.766710	+	0.641994i	$0.778107\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−0.466185	−0.140560	−0.0702801	−	0.997527i	$-0.522389\pi$
$11$	−0.466185	−0.140560	−0.0702801	+	0.997527i	$0.522389\pi$
$12$	0	0
$13$	−0.985939	−0.273450	−0.136725	−	0.990609i	$-0.543658\pi$
$13$	−0.985939	−0.273450	−0.136725	+	0.990609i	$0.543658\pi$
$14$	4.05705	1.08429
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.94567	−1.19950	−0.599750	−	0.800187i	$-0.704733\pi$
$17$	−4.94567	−1.19950	−0.599750	+	0.800187i	$0.704733\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0.466185	0.0993910
$23$	2.22982	0.464949	0.232474	−	0.972603i	$-0.425318\pi$
$23$	2.22982	0.464949	0.232474	+	0.972603i	$0.425318\pi$
$24$	0	0
$25$	0	0
$26$	0.985939	0.193359
$27$	0	0
$28$	−4.05705	−0.766710
$29$	2.43806	0.452737	0.226368	−	0.974042i	$-0.427315\pi$
$29$	2.43806	0.452737	0.226368	+	0.974042i	$0.427315\pi$
$30$	0	0
$31$	0.715853	0.128571	0.0642855	−	0.997932i	$-0.479523\pi$
$31$	0.715853	0.128571	0.0642855	+	0.997932i	$0.479523\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	4.94567	0.848175
$35$	0	0
$36$	0	0
$37$	10.0324	1.64932	0.824658	−	0.565631i	$-0.191367\pi$
$37$	10.0324	1.64932	0.824658	+	0.565631i	$0.191367\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	0	0
$41$	10.1993	1.59286	0.796429	−	0.604732i	$-0.206720\pi$
$41$	10.1993	1.59286	0.796429	+	0.604732i	$0.206720\pi$
$42$	0	0
$43$	−0.466185	−0.0710926	−0.0355463	−	0.999368i	$-0.511317\pi$
$43$	−0.466185	−0.0710926	−0.0355463	+	0.999368i	$0.511317\pi$
$44$	−0.466185	−0.0702801
$45$	0	0
$46$	−2.22982	−0.328768
$47$	−8.79811	−1.28334	−0.641668	−	0.766982i	$-0.721757\pi$
$47$	−8.79811	−1.28334	−0.641668	+	0.766982i	$0.721757\pi$
$48$	0	0
$49$	9.45963	1.35138
$50$	0	0
$51$	0	0
$52$	−0.985939	−0.136725
$53$	3.89134	0.534516	0.267258	−	0.963625i	$-0.413882\pi$
$53$	3.89134	0.534516	0.267258	+	0.963625i	$0.413882\pi$
$54$	0	0
$55$	0	0
$56$	4.05705	0.542146
$57$	0	0
$58$	−2.43806	−0.320133
$59$	7.42748	0.966976	0.483488	−	0.875351i	$-0.339370\pi$
$59$	7.42748	0.966976	0.483488	+	0.875351i	$0.339370\pi$
$60$	0	0
$61$	−6.45963	−0.827071	−0.413535	−	0.910488i	$-0.635706\pi$
$61$	−6.45963	−0.827071	−0.413535	+	0.910488i	$0.635706\pi$
$62$	−0.715853	−0.0909134
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	13.9226	1.70092	0.850458	−	0.526044i	$-0.176325\pi$
$67$	13.9226	1.70092	0.850458	+	0.526044i	$0.176325\pi$
$68$	−4.94567	−0.599750
$69$	0	0
$70$	0	0
$71$	6.14222	0.728947	0.364473	−	0.931214i	$-0.381249\pi$
$71$	6.14222	0.728947	0.364473	+	0.931214i	$0.381249\pi$
$72$	0	0
$73$	−7.07459	−0.828018	−0.414009	−	0.910273i	$-0.635872\pi$
$73$	−7.07459	−0.828018	−0.414009	+	0.910273i	$0.635872\pi$
$74$	−10.0324	−1.16624
$75$	0	0
$76$	1.00000	0.114708
$77$	1.89134	0.215538
$78$	0	0
$79$	1.85244	0.208416	0.104208	−	0.994556i	$-0.466769\pi$
$79$	1.85244	0.208416	0.104208	+	0.994556i	$0.466769\pi$
$80$	0	0
$81$	0	0
$82$	−10.1993	−1.12632
$83$	−4.37737	−0.480479	−0.240240	−	0.970714i	$-0.577226\pi$
$83$	−4.37737	−0.480479	−0.240240	+	0.970714i	$0.577226\pi$
$84$	0	0
$85$	0	0
$86$	0.466185	0.0502701
$87$	0	0
$88$	0.466185	0.0496955
$89$	−6.02892	−0.639065	−0.319532	−	0.947575i	$-0.603526\pi$
$89$	−6.02892	−0.639065	−0.319532	+	0.947575i	$0.603526\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	2.22982	0.232474
$93$	0	0
$94$	8.79811	0.907456
$95$	0	0
$96$	0	0
$97$	−6.60840	−0.670982	−0.335491	−	0.942043i	$-0.608902\pi$
$97$	−6.60840	−0.670982	−0.335491	+	0.942043i	$0.608902\pi$
$98$	−9.45963	−0.955567
$99$	0	0
$100$	0	0
$101$	−5.62246	−0.559456	−0.279728	−	0.960079i	$-0.590244\pi$
$101$	−5.62246	−0.559456	−0.279728	+	0.960079i	$0.590244\pi$
$102$	0	0
$103$	18.9464	1.86684	0.933422	−	0.358780i	$-0.116807\pi$
$103$	18.9464	1.86684	0.933422	+	0.358780i	$0.116807\pi$
$104$	0.985939	0.0966793
$105$	0	0
$106$	−3.89134	−0.377960
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	4.60719	0.441289	0.220644	−	0.975354i	$-0.429184\pi$
$109$	4.60719	0.441289	0.220644	+	0.975354i	$0.429184\pi$
$110$	0	0
$111$	0	0
$112$	−4.05705	−0.383355
$113$	−14.4596	−1.36025	−0.680124	−	0.733097i	$-0.738074\pi$
$113$	−14.4596	−1.36025	−0.680124	+	0.733097i	$0.738074\pi$
$114$	0	0
$115$	0	0
$116$	2.43806	0.226368
$117$	0	0
$118$	−7.42748	−0.683755
$119$	20.0648	1.83934
$120$	0	0
$121$	−10.7827	−0.980243
$122$	6.45963	0.584827
$123$	0	0
$124$	0.715853	0.0642855
$125$	0	0
$126$	0	0
$127$	4.69009	0.416178	0.208089	−	0.978110i	$-0.433276\pi$
$127$	4.69009	0.416178	0.208089	+	0.978110i	$0.433276\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−9.61979	−0.840485	−0.420242	−	0.907412i	$-0.638055\pi$
$131$	−9.61979	−0.840485	−0.420242	+	0.907412i	$0.638055\pi$
$132$	0	0
$133$	−4.05705	−0.351791
$134$	−13.9226	−1.20273
$135$	0	0
$136$	4.94567	0.424088
$137$	−18.8370	−1.60935	−0.804677	−	0.593713i	$-0.797661\pi$
$137$	−18.8370	−1.60935	−0.804677	+	0.593713i	$0.797661\pi$
$138$	0	0
$139$	14.3510	1.21723	0.608617	−	0.793465i	$-0.291725\pi$
$139$	14.3510	1.21723	0.608617	+	0.793465i	$0.291725\pi$
$140$	0	0
$141$	0	0
$142$	−6.14222	−0.515443
$143$	0.459630	0.0384362
$144$	0	0
$145$	0	0
$146$	7.07459	0.585497
$147$	0	0
$148$	10.0324	0.824658
$149$	−15.3747	−1.25955	−0.629773	−	0.776779i	$-0.716852\pi$
$149$	−15.3747	−1.25955	−0.629773	+	0.776779i	$0.716852\pi$
$150$	0	0
$151$	−5.17548	−0.421175	−0.210587	−	0.977575i	$-0.567538\pi$
$151$	−5.17548	−0.421175	−0.210587	+	0.977575i	$0.567538\pi$
$152$	−1.00000	−0.0811107
$153$	0	0
$154$	−1.89134	−0.152408
$155$	0	0
$156$	0	0
$157$	11.7050	0.934157	0.467079	−	0.884216i	$-0.345307\pi$
$157$	11.7050	0.934157	0.467079	+	0.884216i	$0.345307\pi$
$158$	−1.85244	−0.147372
$159$	0	0
$160$	0	0
$161$	−9.04646	−0.712961
$162$	0	0
$163$	16.6944	1.30760	0.653802	−	0.756666i	$-0.273173\pi$
$163$	16.6944	1.30760	0.653802	+	0.756666i	$0.273173\pi$
$164$	10.1993	0.796429
$165$	0	0
$166$	4.37737	0.339750
$167$	−19.7827	−1.53083	−0.765415	−	0.643538i	$-0.777466\pi$
$167$	−19.7827	−1.53083	−0.765415	+	0.643538i	$0.777466\pi$
$168$	0	0
$169$	−12.0279	−0.925225
$170$	0	0
$171$	0	0
$172$	−0.466185	−0.0355463
$173$	12.3510	0.939027	0.469513	−	0.882925i	$-0.344429\pi$
$173$	12.3510	0.939027	0.469513	+	0.882925i	$0.344429\pi$
$174$	0	0
$175$	0	0
$176$	−0.466185	−0.0351400
$177$	0	0
$178$	6.02892	0.451887
$179$	4.52323	0.338082	0.169041	−	0.985609i	$-0.445933\pi$
$179$	4.52323	0.338082	0.169041	+	0.985609i	$0.445933\pi$
$180$	0	0
$181$	0.147558	0.0109679	0.00548396	−	0.999985i	$-0.498254\pi$
$181$	0.147558	0.0109679	0.00548396	+	0.999985i	$0.498254\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	−2.22982	−0.164384
$185$	0	0
$186$	0	0
$187$	2.30560	0.168602
$188$	−8.79811	−0.641668
$189$	0	0
$190$	0	0
$191$	−1.91831	−0.138804	−0.0694020	−	0.997589i	$-0.522109\pi$
$191$	−1.91831	−0.138804	−0.0694020	+	0.997589i	$0.522109\pi$
$192$	0	0
$193$	−26.6732	−1.91998	−0.959990	−	0.280035i	$-0.909654\pi$
$193$	−26.6732	−1.91998	−0.959990	+	0.280035i	$0.909654\pi$
$194$	6.60840	0.474456
$195$	0	0
$196$	9.45963	0.675688
$197$	6.58078	0.468861	0.234431	−	0.972133i	$-0.424677\pi$
$197$	6.58078	0.468861	0.234431	+	0.972133i	$0.424677\pi$
$198$	0	0
$199$	−12.4596	−0.883240	−0.441620	−	0.897202i	$-0.645596\pi$
$199$	−12.4596	−0.883240	−0.441620	+	0.897202i	$0.645596\pi$
$200$	0	0
$201$	0	0
$202$	5.62246	0.395595
$203$	−9.89134	−0.694236
$204$	0	0
$205$	0	0
$206$	−18.9464	−1.32006
$207$	0	0
$208$	−0.985939	−0.0683626
$209$	−0.466185	−0.0322467
$210$	0	0
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	3.89134	0.267258
$213$	0	0
$214$	8.00000	0.546869
$215$	0	0
$216$	0	0
$217$	−2.90425	−0.197153
$218$	−4.60719	−0.312038
$219$	0	0
$220$	0	0
$221$	4.87613	0.328004
$222$	0	0
$223$	−3.75772	−0.251636	−0.125818	−	0.992053i	$-0.540155\pi$
$223$	−3.75772	−0.251636	−0.125818	+	0.992053i	$0.540155\pi$
$224$	4.05705	0.271073
$225$	0	0
$226$	14.4596	0.961840
$227$	−17.8913	−1.18749	−0.593745	−	0.804653i	$-0.702351\pi$
$227$	−17.8913	−1.18749	−0.593745	+	0.804653i	$0.702351\pi$
$228$	0	0
$229$	−5.32304	−0.351756	−0.175878	−	0.984412i	$-0.556276\pi$
$229$	−5.32304	−0.351756	−0.175878	+	0.984412i	$0.556276\pi$
$230$	0	0
$231$	0	0
$232$	−2.43806	−0.160067
$233$	3.51396	0.230207	0.115104	−	0.993353i	$-0.463280\pi$
$233$	3.51396	0.230207	0.115104	+	0.993353i	$0.463280\pi$
$234$	0	0
$235$	0	0
$236$	7.42748	0.483488
$237$	0	0
$238$	−20.0648	−1.30061
$239$	−29.4986	−1.90810	−0.954052	−	0.299642i	$-0.903133\pi$
$239$	−29.4986	−1.90810	−0.954052	+	0.299642i	$0.903133\pi$
$240$	0	0
$241$	29.2702	1.88546	0.942731	−	0.333555i	$-0.108248\pi$
$241$	29.2702	1.88546	0.942731	+	0.333555i	$0.108248\pi$
$242$	10.7827	0.693136
$243$	0	0
$244$	−6.45963	−0.413535
$245$	0	0
$246$	0	0
$247$	−0.985939	−0.0627338
$248$	−0.715853	−0.0454567
$249$	0	0
$250$	0	0
$251$	−13.7901	−0.870425	−0.435212	−	0.900328i	$-0.643327\pi$
$251$	−13.7901	−0.870425	−0.435212	+	0.900328i	$0.643327\pi$
$252$	0	0
$253$	−1.03951	−0.0653532
$254$	−4.69009	−0.294283
$255$	0	0
$256$	1.00000	0.0625000
$257$	−4.10866	−0.256291	−0.128146	−	0.991755i	$-0.540903\pi$
$257$	−4.10866	−0.256291	−0.128146	+	0.991755i	$0.540903\pi$
$258$	0	0
$259$	−40.7019	−2.52909
$260$	0	0
$261$	0	0
$262$	9.61979	0.594312
$263$	−2.90677	−0.179239	−0.0896197	−	0.995976i	$-0.528565\pi$
$263$	−2.90677	−0.179239	−0.0896197	+	0.995976i	$0.528565\pi$
$264$	0	0
$265$	0	0
$266$	4.05705	0.248754
$267$	0	0
$268$	13.9226	0.850458
$269$	−6.60840	−0.402921	−0.201461	−	0.979497i	$-0.564569\pi$
$269$	−6.60840	−0.402921	−0.201461	+	0.979497i	$0.564569\pi$
$270$	0	0
$271$	29.3789	1.78464	0.892320	−	0.451403i	$-0.149076\pi$
$271$	29.3789	1.78464	0.892320	+	0.451403i	$0.149076\pi$
$272$	−4.94567	−0.299875
$273$	0	0
$274$	18.8370	1.13798
$275$	0	0
$276$	0	0
$277$	−17.8472	−1.07233	−0.536166	−	0.844112i	$-0.680128\pi$
$277$	−17.8472	−1.07233	−0.536166	+	0.844112i	$0.680128\pi$
$278$	−14.3510	−0.860714
$279$	0	0
$280$	0	0
$281$	−10.9051	−0.650541	−0.325270	−	0.945621i	$-0.605455\pi$
$281$	−10.9051	−0.650541	−0.325270	+	0.945621i	$0.605455\pi$
$282$	0	0
$283$	6.27468	0.372991	0.186496	−	0.982456i	$-0.440287\pi$
$283$	6.27468	0.372991	0.186496	+	0.982456i	$0.440287\pi$
$284$	6.14222	0.364473
$285$	0	0
$286$	−0.459630	−0.0271785
$287$	−41.3789	−2.44252
$288$	0	0
$289$	7.45963	0.438802
$290$	0	0
$291$	0	0
$292$	−7.07459	−0.414009
$293$	−22.9193	−1.33896	−0.669479	−	0.742831i	$-0.733482\pi$
$293$	−22.9193	−1.33896	−0.669479	+	0.742831i	$0.733482\pi$
$294$	0	0
$295$	0	0
$296$	−10.0324	−0.583122
$297$	0	0
$298$	15.3747	0.890633
$299$	−2.19846	−0.127140
$300$	0	0
$301$	1.89134	0.109015
$302$	5.17548	0.297816
$303$	0	0
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	−4.17034	−0.238014	−0.119007	−	0.992893i	$-0.537971\pi$
$307$	−4.17034	−0.238014	−0.119007	+	0.992893i	$0.537971\pi$
$308$	1.89134	0.107769
$309$	0	0
$310$	0	0
$311$	12.2309	0.693549	0.346774	−	0.937949i	$-0.387277\pi$
$311$	12.2309	0.693549	0.346774	+	0.937949i	$0.387277\pi$
$312$	0	0
$313$	8.71274	0.492473	0.246237	−	0.969210i	$-0.420806\pi$
$313$	8.71274	0.492473	0.246237	+	0.969210i	$0.420806\pi$
$314$	−11.7050	−0.660549
$315$	0	0
$316$	1.85244	0.104208
$317$	9.78267	0.549450	0.274725	−	0.961523i	$-0.411413\pi$
$317$	9.78267	0.549450	0.274725	+	0.961523i	$0.411413\pi$
$318$	0	0
$319$	−1.13659	−0.0636368
$320$	0	0
$321$	0	0
$322$	9.04646	0.504140
$323$	−4.94567	−0.275184
$324$	0	0
$325$	0	0
$326$	−16.6944	−0.924616
$327$	0	0
$328$	−10.1993	−0.563160
$329$	35.6943	1.96789
$330$	0	0
$331$	12.4596	0.684843	0.342422	−	0.939546i	$-0.388753\pi$
$331$	12.4596	0.684843	0.342422	+	0.939546i	$0.388753\pi$
$332$	−4.37737	−0.240240
$333$	0	0
$334$	19.7827	1.08246
$335$	0	0
$336$	0	0
$337$	−10.5522	−0.574813	−0.287406	−	0.957809i	$-0.592793\pi$
$337$	−10.5522	−0.574813	−0.287406	+	0.957809i	$0.592793\pi$
$338$	12.0279	0.654233
$339$	0	0
$340$	0	0
$341$	−0.333720	−0.0180720
$342$	0	0
$343$	−9.97884	−0.538806
$344$	0.466185	0.0251350
$345$	0	0
$346$	−12.3510	−0.663992
$347$	−19.8649	−1.06641	−0.533203	−	0.845988i	$-0.679012\pi$
$347$	−19.8649	−1.06641	−0.533203	+	0.845988i	$0.679012\pi$
$348$	0	0
$349$	−32.3510	−1.73171	−0.865854	−	0.500297i	$-0.833224\pi$
$349$	−32.3510	−1.73171	−0.865854	+	0.500297i	$0.833224\pi$
$350$	0	0
$351$	0	0
$352$	0.466185	0.0248478
$353$	17.4053	0.926391	0.463195	−	0.886256i	$-0.346703\pi$
$353$	17.4053	0.926391	0.463195	+	0.886256i	$0.346703\pi$
$354$	0	0
$355$	0	0
$356$	−6.02892	−0.319532
$357$	0	0
$358$	−4.52323	−0.239060
$359$	−32.5099	−1.71581	−0.857905	−	0.513809i	$-0.828234\pi$
$359$	−32.5099	−1.71581	−0.857905	+	0.513809i	$0.828234\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−0.147558	−0.00775549
$363$	0	0
$364$	4.00000	0.209657
$365$	0	0
$366$	0	0
$367$	−35.2473	−1.83990	−0.919948	−	0.392041i	$-0.871769\pi$
$367$	−35.2473	−1.83990	−0.919948	+	0.392041i	$0.871769\pi$
$368$	2.22982	0.116237
$369$	0	0
$370$	0	0
$371$	−15.7873	−0.819637
$372$	0	0
$373$	16.0675	0.831943	0.415971	−	0.909378i	$-0.363442\pi$
$373$	16.0675	0.831943	0.415971	+	0.909378i	$0.363442\pi$
$374$	−2.30560	−0.119220
$375$	0	0
$376$	8.79811	0.453728
$377$	−2.40378	−0.123801
$378$	0	0
$379$	8.24230	0.423379	0.211689	−	0.977337i	$-0.432104\pi$
$379$	8.24230	0.423379	0.211689	+	0.977337i	$0.432104\pi$
$380$	0	0
$381$	0	0
$382$	1.91831	0.0981492
$383$	12.9193	0.660143	0.330072	−	0.943956i	$-0.392927\pi$
$383$	12.9193	0.660143	0.330072	+	0.943956i	$0.392927\pi$
$384$	0	0
$385$	0	0
$386$	26.6732	1.35763
$387$	0	0
$388$	−6.60840	−0.335491
$389$	−22.7830	−1.15515	−0.577573	−	0.816339i	$-0.696000\pi$
$389$	−22.7830	−1.15515	−0.577573	+	0.816339i	$0.696000\pi$
$390$	0	0
$391$	−11.0279	−0.557706
$392$	−9.45963	−0.477783
$393$	0	0
$394$	−6.58078	−0.331535
$395$	0	0
$396$	0	0
$397$	−20.4177	−1.02473	−0.512367	−	0.858766i	$-0.671231\pi$
$397$	−20.4177	−1.02473	−0.512367	+	0.858766i	$0.671231\pi$
$398$	12.4596	0.624545
$399$	0	0
$400$	0	0
$401$	−21.8163	−1.08945	−0.544726	−	0.838614i	$-0.683366\pi$
$401$	−21.8163	−1.08945	−0.544726	+	0.838614i	$0.683366\pi$
$402$	0	0
$403$	−0.705787	−0.0351578
$404$	−5.62246	−0.279728
$405$	0	0
$406$	9.89134	0.490899
$407$	−4.67696	−0.231828
$408$	0	0
$409$	19.3789	0.958224	0.479112	−	0.877754i	$-0.340959\pi$
$409$	19.3789	0.958224	0.479112	+	0.877754i	$0.340959\pi$
$410$	0	0
$411$	0	0
$412$	18.9464	0.933422
$413$	−30.1336	−1.48278
$414$	0	0
$415$	0	0
$416$	0.985939	0.0483396
$417$	0	0
$418$	0.466185	0.0228019
$419$	28.7522	1.40464	0.702319	−	0.711862i	$-0.252148\pi$
$419$	28.7522	1.40464	0.702319	+	0.711862i	$0.252148\pi$
$420$	0	0
$421$	−14.7159	−0.717207	−0.358603	−	0.933490i	$-0.616747\pi$
$421$	−14.7159	−0.717207	−0.358603	+	0.933490i	$0.616747\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	−3.89134	−0.188980
$425$	0	0
$426$	0	0
$427$	26.2070	1.26825
$428$	−8.00000	−0.386695
$429$	0	0
$430$	0	0
$431$	−18.2001	−0.876666	−0.438333	−	0.898813i	$-0.644431\pi$
$431$	−18.2001	−0.876666	−0.438333	+	0.898813i	$0.644431\pi$
$432$	0	0
$433$	−2.77178	−0.133203	−0.0666017	−	0.997780i	$-0.521216\pi$
$433$	−2.77178	−0.133203	−0.0666017	+	0.997780i	$0.521216\pi$
$434$	2.90425	0.139408
$435$	0	0
$436$	4.60719	0.220644
$437$	2.22982	0.106667
$438$	0	0
$439$	−12.2034	−0.582437	−0.291218	−	0.956657i	$-0.594061\pi$
$439$	−12.2034	−0.582437	−0.291218	+	0.956657i	$0.594061\pi$
$440$	0	0
$441$	0	0
$442$	−4.87613	−0.231934
$443$	21.9736	1.04400	0.521998	−	0.852946i	$-0.325187\pi$
$443$	21.9736	1.04400	0.521998	+	0.852946i	$0.325187\pi$
$444$	0	0
$445$	0	0
$446$	3.75772	0.177933
$447$	0	0
$448$	−4.05705	−0.191677
$449$	23.1895	1.09438	0.547190	−	0.837009i	$-0.315698\pi$
$449$	23.1895	1.09438	0.547190	+	0.837009i	$0.315698\pi$
$450$	0	0
$451$	−4.75475	−0.223892
$452$	−14.4596	−0.680124
$453$	0	0
$454$	17.8913	0.839682
$455$	0	0
$456$	0	0
$457$	34.3211	1.60547	0.802737	−	0.596333i	$-0.203376\pi$
$457$	34.3211	1.60547	0.802737	+	0.596333i	$0.203376\pi$
$458$	5.32304	0.248729
$459$	0	0
$460$	0	0
$461$	23.5959	1.09897	0.549486	−	0.835503i	$-0.314823\pi$
$461$	23.5959	1.09897	0.549486	+	0.835503i	$0.314823\pi$
$462$	0	0
$463$	−3.94991	−0.183568	−0.0917840	−	0.995779i	$-0.529257\pi$
$463$	−3.94991	−0.183568	−0.0917840	+	0.995779i	$0.529257\pi$
$464$	2.43806	0.113184
$465$	0	0
$466$	−3.51396	−0.162781
$467$	−9.51396	−0.440254	−0.220127	−	0.975471i	$-0.570647\pi$
$467$	−9.51396	−0.440254	−0.220127	+	0.975471i	$0.570647\pi$
$468$	0	0
$469$	−56.4846	−2.60822
$470$	0	0
$471$	0	0
$472$	−7.42748	−0.341877
$473$	0.217329	0.00999279
$474$	0	0
$475$	0	0
$476$	20.0648	0.919669
$477$	0	0
$478$	29.4986	1.34923
$479$	28.4591	1.30033	0.650164	−	0.759794i	$-0.274700\pi$
$479$	28.4591	1.30033	0.650164	+	0.759794i	$0.274700\pi$
$480$	0	0
$481$	−9.89134	−0.451006
$482$	−29.2702	−1.33322
$483$	0	0
$484$	−10.7827	−0.490121
$485$	0	0
$486$	0	0
$487$	−21.5169	−0.975025	−0.487513	−	0.873116i	$-0.662096\pi$
$487$	−21.5169	−0.975025	−0.487513	+	0.873116i	$0.662096\pi$
$488$	6.45963	0.292414
$489$	0	0
$490$	0	0
$491$	20.3044	0.916325	0.458163	−	0.888868i	$-0.348508\pi$
$491$	20.3044	0.916325	0.458163	+	0.888868i	$0.348508\pi$
$492$	0	0
$493$	−12.0578	−0.543058
$494$	0.985939	0.0443595
$495$	0	0
$496$	0.715853	0.0321427
$497$	−24.9193	−1.11778
$498$	0	0
$499$	−12.9193	−0.578346	−0.289173	−	0.957277i	$-0.593380\pi$
$499$	−12.9193	−0.578346	−0.289173	+	0.957277i	$0.593380\pi$
$500$	0	0
$501$	0	0
$502$	13.7901	0.615483
$503$	18.4721	0.823631	0.411815	−	0.911267i	$-0.364895\pi$
$503$	18.4721	0.823631	0.411815	+	0.911267i	$0.364895\pi$
$504$	0	0
$505$	0	0
$506$	1.03951	0.0462117
$507$	0	0
$508$	4.69009	0.208089
$509$	41.2632	1.82896	0.914480	−	0.404630i	$-0.132600\pi$
$509$	41.2632	1.82896	0.914480	+	0.404630i	$0.132600\pi$
$510$	0	0
$511$	28.7019	1.26970
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	4.10866	0.181225
$515$	0	0
$516$	0	0
$517$	4.10155	0.180386
$518$	40.7019	1.78834
$519$	0	0
$520$	0	0
$521$	27.3598	1.19866	0.599328	−	0.800504i	$-0.295435\pi$
$521$	27.3598	1.19866	0.599328	+	0.800504i	$0.295435\pi$
$522$	0	0
$523$	5.31698	0.232495	0.116248	−	0.993220i	$-0.462913\pi$
$523$	5.31698	0.232495	0.116248	+	0.993220i	$0.462913\pi$
$524$	−9.61979	−0.420242
$525$	0	0
$526$	2.90677	0.126741
$527$	−3.54037	−0.154221
$528$	0	0
$529$	−18.0279	−0.783823
$530$	0	0
$531$	0	0
$532$	−4.05705	−0.175895
$533$	−10.0558	−0.435567
$534$	0	0
$535$	0	0
$536$	−13.9226	−0.601364
$537$	0	0
$538$	6.60840	0.284908
$539$	−4.40994	−0.189950
$540$	0	0
$541$	−10.2951	−0.442622	−0.221311	−	0.975203i	$-0.571034\pi$
$541$	−10.2951	−0.442622	−0.221311	+	0.975203i	$0.571034\pi$
$542$	−29.3789	−1.26193
$543$	0	0
$544$	4.94567	0.212044
$545$	0	0
$546$	0	0
$547$	−3.50290	−0.149773	−0.0748866	−	0.997192i	$-0.523859\pi$
$547$	−3.50290	−0.149773	−0.0748866	+	0.997192i	$0.523859\pi$
$548$	−18.8370	−0.804677
$549$	0	0
$550$	0	0
$551$	2.43806	0.103865
$552$	0	0
$553$	−7.51544	−0.319589
$554$	17.8472	0.758254
$555$	0	0
$556$	14.3510	0.608617
$557$	15.2577	0.646491	0.323246	−	0.946315i	$-0.395226\pi$
$557$	15.2577	0.646491	0.323246	+	0.946315i	$0.395226\pi$
$558$	0	0
$559$	0.459630	0.0194403
$560$	0	0
$561$	0	0
$562$	10.9051	0.460002
$563$	27.5653	1.16174	0.580870	−	0.813996i	$-0.302712\pi$
$563$	27.5653	1.16174	0.580870	+	0.813996i	$0.302712\pi$
$564$	0	0
$565$	0	0
$566$	−6.27468	−0.263745
$567$	0	0
$568$	−6.14222	−0.257722
$569$	−27.3598	−1.14698	−0.573492	−	0.819211i	$-0.694412\pi$
$569$	−27.3598	−1.14698	−0.573492	+	0.819211i	$0.694412\pi$
$570$	0	0
$571$	−40.1895	−1.68188	−0.840939	−	0.541130i	$-0.817997\pi$
$571$	−40.1895	−1.68188	−0.840939	+	0.541130i	$0.817997\pi$
$572$	0.459630	0.0192181
$573$	0	0
$574$	41.3789	1.72712
$575$	0	0
$576$	0	0
$577$	−18.7987	−0.782601	−0.391300	−	0.920263i	$-0.627975\pi$
$577$	−18.7987	−0.782601	−0.391300	+	0.920263i	$0.627975\pi$
$578$	−7.45963	−0.310280
$579$	0	0
$580$	0	0
$581$	17.7592	0.736776
$582$	0	0
$583$	−1.81408	−0.0751317
$584$	7.07459	0.292748
$585$	0	0
$586$	22.9193	0.946786
$587$	25.7563	1.06307	0.531537	−	0.847035i	$-0.321615\pi$
$587$	25.7563	1.06307	0.531537	+	0.847035i	$0.321615\pi$
$588$	0	0
$589$	0.715853	0.0294962
$590$	0	0
$591$	0	0
$592$	10.0324	0.412329
$593$	−36.4332	−1.49613	−0.748067	−	0.663624i	$-0.769018\pi$
$593$	−36.4332	−1.49613	−0.748067	+	0.663624i	$0.769018\pi$
$594$	0	0
$595$	0	0
$596$	−15.3747	−0.629773
$597$	0	0
$598$	2.19846	0.0899018
$599$	−11.3521	−0.463833	−0.231916	−	0.972736i	$-0.574500\pi$
$599$	−11.3521	−0.463833	−0.231916	+	0.972736i	$0.574500\pi$
$600$	0	0
$601$	−14.9721	−0.610724	−0.305362	−	0.952236i	$-0.598777\pi$
$601$	−14.9721	−0.610724	−0.305362	+	0.952236i	$0.598777\pi$
$602$	−1.89134	−0.0770851
$603$	0	0
$604$	−5.17548	−0.210587
$605$	0	0
$606$	0	0
$607$	28.3649	1.15130	0.575649	−	0.817697i	$-0.304750\pi$
$607$	28.3649	1.15130	0.575649	+	0.817697i	$0.304750\pi$
$608$	−1.00000	−0.0405554
$609$	0	0
$610$	0	0
$611$	8.67440	0.350929
$612$	0	0
$613$	22.8304	0.922113	0.461056	−	0.887371i	$-0.347471\pi$
$613$	22.8304	0.922113	0.461056	+	0.887371i	$0.347471\pi$
$614$	4.17034	0.168301
$615$	0	0
$616$	−1.89134	−0.0762041
$617$	9.62263	0.387392	0.193696	−	0.981062i	$-0.437952\pi$
$617$	9.62263	0.387392	0.193696	+	0.981062i	$0.437952\pi$
$618$	0	0
$619$	2.56829	0.103228	0.0516142	−	0.998667i	$-0.483563\pi$
$619$	2.56829	0.103228	0.0516142	+	0.998667i	$0.483563\pi$
$620$	0	0
$621$	0	0
$622$	−12.2309	−0.490413
$623$	24.4596	0.979954
$624$	0	0
$625$	0	0
$626$	−8.71274	−0.348231
$627$	0	0
$628$	11.7050	0.467079
$629$	−49.6169	−1.97836
$630$	0	0
$631$	−20.6241	−0.821034	−0.410517	−	0.911853i	$-0.634652\pi$
$631$	−20.6241	−0.821034	−0.410517	+	0.911853i	$0.634652\pi$
$632$	−1.85244	−0.0736862
$633$	0	0
$634$	−9.78267	−0.388520
$635$	0	0
$636$	0	0
$637$	−9.32662	−0.369534
$638$	1.13659	0.0449980
$639$	0	0
$640$	0	0
$641$	27.1332	1.07170	0.535849	−	0.844314i	$-0.319992\pi$
$641$	27.1332	1.07170	0.535849	+	0.844314i	$0.319992\pi$
$642$	0	0
$643$	35.0016	1.38033	0.690164	−	0.723653i	$-0.257539\pi$
$643$	35.0016	1.38033	0.690164	+	0.723653i	$0.257539\pi$
$644$	−9.04646	−0.356481
$645$	0	0
$646$	4.94567	0.194585
$647$	−5.77018	−0.226849	−0.113425	−	0.993547i	$-0.536182\pi$
$647$	−5.77018	−0.226849	−0.113425	+	0.993547i	$0.536182\pi$
$648$	0	0
$649$	−3.46258	−0.135918
$650$	0	0
$651$	0	0
$652$	16.6944	0.653802
$653$	−41.6087	−1.62827	−0.814137	−	0.580673i	$-0.802790\pi$
$653$	−41.6087	−1.62827	−0.814137	+	0.580673i	$0.802790\pi$
$654$	0	0
$655$	0	0
$656$	10.1993	0.398214
$657$	0	0
$658$	−35.6943	−1.39151
$659$	−6.16139	−0.240014	−0.120007	−	0.992773i	$-0.538292\pi$
$659$	−6.16139	−0.240014	−0.120007	+	0.992773i	$0.538292\pi$
$660$	0	0
$661$	33.3091	1.29557	0.647787	−	0.761821i	$-0.275695\pi$
$661$	33.3091	1.29557	0.647787	+	0.761821i	$0.275695\pi$
$662$	−12.4596	−0.484257
$663$	0	0
$664$	4.37737	0.169875
$665$	0	0
$666$	0	0
$667$	5.43643	0.210499
$668$	−19.7827	−0.765415
$669$	0	0
$670$	0	0
$671$	3.01138	0.116253
$672$	0	0
$673$	−20.7576	−0.800146	−0.400073	−	0.916483i	$-0.631015\pi$
$673$	−20.7576	−0.800146	−0.400073	+	0.916483i	$0.631015\pi$
$674$	10.5522	0.406454
$675$	0	0
$676$	−12.0279	−0.462612
$677$	−0.403781	−0.0155186	−0.00775928	−	0.999970i	$-0.502470\pi$
$677$	−0.403781	−0.0155186	−0.00775928	+	0.999970i	$0.502470\pi$
$678$	0	0
$679$	26.8106	1.02890
$680$	0	0
$681$	0	0
$682$	0.333720	0.0127788
$683$	−29.8913	−1.14376	−0.571880	−	0.820337i	$-0.693786\pi$
$683$	−29.8913	−1.14376	−0.571880	+	0.820337i	$0.693786\pi$
$684$	0	0
$685$	0	0
$686$	9.97884	0.380994
$687$	0	0
$688$	−0.466185	−0.0177731
$689$	−3.83662	−0.146164
$690$	0	0
$691$	18.1336	0.689836	0.344918	−	0.938633i	$-0.387907\pi$
$691$	18.1336	0.689836	0.344918	+	0.938633i	$0.387907\pi$
$692$	12.3510	0.469513
$693$	0	0
$694$	19.8649	0.754062
$695$	0	0
$696$	0	0
$697$	−50.4422	−1.91063
$698$	32.3510	1.22450
$699$	0	0
$700$	0	0
$701$	−27.1676	−1.02611	−0.513054	−	0.858357i	$-0.671486\pi$
$701$	−27.1676	−1.02611	−0.513054	+	0.858357i	$0.671486\pi$
$702$	0	0
$703$	10.0324	0.378379
$704$	−0.466185	−0.0175700
$705$	0	0
$706$	−17.4053	−0.655057
$707$	22.8106	0.857881
$708$	0	0
$709$	−16.1087	−0.604974	−0.302487	−	0.953154i	$-0.597817\pi$
$709$	−16.1087	−0.604974	−0.302487	+	0.953154i	$0.597817\pi$
$710$	0	0
$711$	0	0
$712$	6.02892	0.225944
$713$	1.59622	0.0597789
$714$	0	0
$715$	0	0
$716$	4.52323	0.169041
$717$	0	0
$718$	32.5099	1.21326
$719$	−38.2113	−1.42504	−0.712521	−	0.701651i	$-0.752446\pi$
$719$	−38.2113	−1.42504	−0.712521	+	0.701651i	$0.752446\pi$
$720$	0	0
$721$	−76.8664	−2.86266
$722$	−1.00000	−0.0372161
$723$	0	0
$724$	0.147558	0.00548396
$725$	0	0
$726$	0	0
$727$	−26.3203	−0.976166	−0.488083	−	0.872797i	$-0.662304\pi$
$727$	−26.3203	−0.976166	−0.488083	+	0.872797i	$0.662304\pi$
$728$	−4.00000	−0.148250
$729$	0	0
$730$	0	0
$731$	2.30560	0.0852756
$732$	0	0
$733$	−31.1023	−1.14879	−0.574395	−	0.818578i	$-0.694763\pi$
$733$	−31.1023	−1.14879	−0.574395	+	0.818578i	$0.694763\pi$
$734$	35.2473	1.30100
$735$	0	0
$736$	−2.22982	−0.0821921
$737$	−6.49051	−0.239081
$738$	0	0
$739$	−6.86341	−0.252475	−0.126237	−	0.992000i	$-0.540290\pi$
$739$	−6.86341	−0.252475	−0.126237	+	0.992000i	$0.540290\pi$
$740$	0	0
$741$	0	0
$742$	15.7873	0.579571
$743$	−14.1865	−0.520450	−0.260225	−	0.965548i	$-0.583797\pi$
$743$	−14.1865	−0.520450	−0.260225	+	0.965548i	$0.583797\pi$
$744$	0	0
$745$	0	0
$746$	−16.0675	−0.588272
$747$	0	0
$748$	2.30560	0.0843010
$749$	32.4564	1.18593
$750$	0	0
$751$	−27.3091	−0.996524	−0.498262	−	0.867027i	$-0.666028\pi$
$751$	−27.3091	−0.996524	−0.498262	+	0.867027i	$0.666028\pi$
$752$	−8.79811	−0.320834
$753$	0	0
$754$	2.40378	0.0875405
$755$	0	0
$756$	0	0
$757$	1.72612	0.0627369	0.0313685	−	0.999508i	$-0.490013\pi$
$757$	1.72612	0.0627369	0.0313685	+	0.999508i	$0.490013\pi$
$758$	−8.24230	−0.299374
$759$	0	0
$760$	0	0
$761$	−2.63932	−0.0956752	−0.0478376	−	0.998855i	$-0.515233\pi$
$761$	−2.63932	−0.0956752	−0.0478376	+	0.998855i	$0.515233\pi$
$762$	0	0
$763$	−18.6916	−0.676681
$764$	−1.91831	−0.0694020
$765$	0	0
$766$	−12.9193	−0.466792
$767$	−7.32304	−0.264420
$768$	0	0
$769$	26.6241	0.960091	0.480046	−	0.877244i	$-0.340620\pi$
$769$	26.6241	0.960091	0.480046	+	0.877244i	$0.340620\pi$
$770$	0	0
$771$	0	0
$772$	−26.6732	−0.959990
$773$	−10.1645	−0.365592	−0.182796	−	0.983151i	$-0.558515\pi$
$773$	−10.1645	−0.365592	−0.182796	+	0.983151i	$0.558515\pi$
$774$	0	0
$775$	0	0
$776$	6.60840	0.237228
$777$	0	0
$778$	22.7830	0.816811
$779$	10.1993	0.365427
$780$	0	0
$781$	−2.86341	−0.102461
$782$	11.0279	0.394358
$783$	0	0
$784$	9.45963	0.337844
$785$	0	0
$786$	0	0
$787$	20.1719	0.719052	0.359526	−	0.933135i	$-0.382938\pi$
$787$	20.1719	0.719052	0.359526	+	0.933135i	$0.382938\pi$
$788$	6.58078	0.234431
$789$	0	0
$790$	0	0
$791$	58.6634	2.08583
$792$	0	0
$793$	6.36880	0.226163
$794$	20.4177	0.724597
$795$	0	0
$796$	−12.4596	−0.441620
$797$	52.0808	1.84480	0.922399	−	0.386239i	$-0.126226\pi$
$797$	52.0808	1.84480	0.922399	+	0.386239i	$0.126226\pi$
$798$	0	0
$799$	43.5125	1.53936
$800$	0	0
$801$	0	0
$802$	21.8163	0.770359
$803$	3.29807	0.116386
$804$	0	0
$805$	0	0
$806$	0.705787	0.0248603
$807$	0	0
$808$	5.62246	0.197798
$809$	−11.4592	−0.402884	−0.201442	−	0.979500i	$-0.564563\pi$
$809$	−11.4592	−0.402884	−0.201442	+	0.979500i	$0.564563\pi$
$810$	0	0
$811$	13.5962	0.477428	0.238714	−	0.971090i	$-0.423274\pi$
$811$	13.5962	0.477428	0.238714	+	0.971090i	$0.423274\pi$
$812$	−9.89134	−0.347118
$813$	0	0
$814$	4.67696	0.163927
$815$	0	0
$816$	0	0
$817$	−0.466185	−0.0163098
$818$	−19.3789	−0.677567
$819$	0	0
$820$	0	0
$821$	43.6731	1.52420	0.762100	−	0.647459i	$-0.224168\pi$
$821$	43.6731	1.52420	0.762100	+	0.647459i	$0.224168\pi$
$822$	0	0
$823$	−42.2148	−1.47151	−0.735757	−	0.677245i	$-0.763174\pi$
$823$	−42.2148	−1.47151	−0.735757	+	0.677245i	$0.763174\pi$
$824$	−18.9464	−0.660029
$825$	0	0
$826$	30.1336	1.04848
$827$	−53.8385	−1.87215	−0.936074	−	0.351802i	$-0.885569\pi$
$827$	−53.8385	−1.87215	−0.936074	+	0.351802i	$0.885569\pi$
$828$	0	0
$829$	−55.1755	−1.91632	−0.958162	−	0.286227i	$-0.907599\pi$
$829$	−55.1755	−1.91632	−0.958162	+	0.286227i	$0.907599\pi$
$830$	0	0
$831$	0	0
$832$	−0.985939	−0.0341813
$833$	−46.7842	−1.62098
$834$	0	0
$835$	0	0
$836$	−0.466185	−0.0161234
$837$	0	0
$838$	−28.7522	−0.993229
$839$	17.0534	0.588750	0.294375	−	0.955690i	$-0.404889\pi$
$839$	17.0534	0.588750	0.294375	+	0.955690i	$0.404889\pi$
$840$	0	0
$841$	−23.0558	−0.795029
$842$	14.7159	0.507142
$843$	0	0
$844$	12.0000	0.413057
$845$	0	0
$846$	0	0
$847$	43.7458	1.50312
$848$	3.89134	0.133629
$849$	0	0
$850$	0	0
$851$	22.3704	0.766848
$852$	0	0
$853$	−14.0106	−0.479712	−0.239856	−	0.970808i	$-0.577100\pi$
$853$	−14.0106	−0.479712	−0.239856	+	0.970808i	$0.577100\pi$
$854$	−26.2070	−0.896786
$855$	0	0
$856$	8.00000	0.273434
$857$	−48.5933	−1.65991	−0.829957	−	0.557827i	$-0.811635\pi$
$857$	−48.5933	−1.65991	−0.829957	+	0.557827i	$0.811635\pi$
$858$	0	0
$859$	57.5434	1.96336	0.981678	−	0.190548i	$-0.0610266\pi$
$859$	57.5434	1.96336	0.981678	+	0.190548i	$0.0610266\pi$
$860$	0	0
$861$	0	0
$862$	18.2001	0.619897
$863$	−30.8634	−1.05060	−0.525301	−	0.850916i	$-0.676047\pi$
$863$	−30.8634	−1.05060	−0.525301	+	0.850916i	$0.676047\pi$
$864$	0	0
$865$	0	0
$866$	2.77178	0.0941890
$867$	0	0
$868$	−2.90425	−0.0985766
$869$	−0.863581	−0.0292950
$870$	0	0
$871$	−13.7268	−0.465116
$872$	−4.60719	−0.156019
$873$	0	0
$874$	−2.22982	−0.0754246
$875$	0	0
$876$	0	0
$877$	−42.4888	−1.43474	−0.717372	−	0.696690i	$-0.754655\pi$
$877$	−42.4888	−1.43474	−0.717372	+	0.696690i	$0.754655\pi$
$878$	12.2034	0.411845
$879$	0	0
$880$	0	0
$881$	−30.8182	−1.03829	−0.519146	−	0.854686i	$-0.673750\pi$
$881$	−30.8182	−1.03829	−0.519146	+	0.854686i	$0.673750\pi$
$882$	0	0
$883$	39.8900	1.34241	0.671203	−	0.741274i	$-0.265778\pi$
$883$	39.8900	1.34241	0.671203	+	0.741274i	$0.265778\pi$
$884$	4.87613	0.164002
$885$	0	0
$886$	−21.9736	−0.738217
$887$	39.1924	1.31595	0.657977	−	0.753038i	$-0.271413\pi$
$887$	39.1924	1.31595	0.657977	+	0.753038i	$0.271413\pi$
$888$	0	0
$889$	−19.0279	−0.638176
$890$	0	0
$891$	0	0
$892$	−3.75772	−0.125818
$893$	−8.79811	−0.294418
$894$	0	0
$895$	0	0
$896$	4.05705	0.135536
$897$	0	0
$898$	−23.1895	−0.773843
$899$	1.74529	0.0582088
$900$	0	0
$901$	−19.2453	−0.641152
$902$	4.75475	0.158316
$903$	0	0
$904$	14.4596	0.480920
$905$	0	0
$906$	0	0
$907$	49.5098	1.64395	0.821973	−	0.569527i	$-0.192873\pi$
$907$	49.5098	1.64395	0.821973	+	0.569527i	$0.192873\pi$
$908$	−17.8913	−0.593745
$909$	0	0
$910$	0	0
$911$	−14.3634	−0.475882	−0.237941	−	0.971280i	$-0.576473\pi$
$911$	−14.3634	−0.475882	−0.237941	+	0.971280i	$0.576473\pi$
$912$	0	0
$913$	2.04067	0.0675362
$914$	−34.3211	−1.13524
$915$	0	0
$916$	−5.32304	−0.175878
$917$	39.0279	1.28882
$918$	0	0
$919$	−54.1336	−1.78570	−0.892852	−	0.450350i	$-0.851299\pi$
$919$	−54.1336	−1.78570	−0.892852	+	0.450350i	$0.851299\pi$
$920$	0	0
$921$	0	0
$922$	−23.5959	−0.777091
$923$	−6.05585	−0.199331
$924$	0	0
$925$	0	0
$926$	3.94991	0.129802
$927$	0	0
$928$	−2.43806	−0.0800333
$929$	−30.2579	−0.992730	−0.496365	−	0.868114i	$-0.665332\pi$
$929$	−30.2579	−0.992730	−0.496365	+	0.868114i	$0.665332\pi$
$930$	0	0
$931$	9.45963	0.310027
$932$	3.51396	0.115104
$933$	0	0
$934$	9.51396	0.311306
$935$	0	0
$936$	0	0
$937$	49.7747	1.62607	0.813035	−	0.582215i	$-0.197814\pi$
$937$	49.7747	1.62607	0.813035	+	0.582215i	$0.197814\pi$
$938$	56.4846	1.84429
$939$	0	0
$940$	0	0
$941$	−9.51265	−0.310104	−0.155052	−	0.987906i	$-0.549554\pi$
$941$	−9.51265	−0.310104	−0.155052	+	0.987906i	$0.549554\pi$
$942$	0	0
$943$	22.7425	0.740597
$944$	7.42748	0.241744
$945$	0	0
$946$	−0.217329	−0.00706597
$947$	2.26871	0.0737231	0.0368616	−	0.999320i	$-0.488264\pi$
$947$	2.26871	0.0737231	0.0368616	+	0.999320i	$0.488264\pi$
$948$	0	0
$949$	6.97511	0.226422
$950$	0	0
$951$	0	0
$952$	−20.0648	−0.650304
$953$	−1.75770	−0.0569374	−0.0284687	−	0.999595i	$-0.509063\pi$
$953$	−1.75770	−0.0569374	−0.0284687	+	0.999595i	$0.509063\pi$
$954$	0	0
$955$	0	0
$956$	−29.4986	−0.954052
$957$	0	0
$958$	−28.4591	−0.919470
$959$	76.4226	2.46781
$960$	0	0
$961$	−30.4876	−0.983470
$962$	9.89134	0.318909
$963$	0	0
$964$	29.2702	0.942731
$965$	0	0
$966$	0	0
$967$	−9.60061	−0.308735	−0.154367	−	0.988014i	$-0.549334\pi$
$967$	−9.60061	−0.308735	−0.154367	+	0.988014i	$0.549334\pi$
$968$	10.7827	0.346568
$969$	0	0
$970$	0	0
$971$	−48.8232	−1.56681	−0.783405	−	0.621511i	$-0.786519\pi$
$971$	−48.8232	−1.56681	−0.783405	+	0.621511i	$0.786519\pi$
$972$	0	0
$973$	−58.2225	−1.86653
$974$	21.5169	0.689447
$975$	0	0
$976$	−6.45963	−0.206768
$977$	−25.1057	−0.803203	−0.401601	−	0.915815i	$-0.631546\pi$
$977$	−25.1057	−0.803203	−0.401601	+	0.915815i	$0.631546\pi$
$978$	0	0
$979$	2.81060	0.0898270
$980$	0	0
$981$	0	0
$982$	−20.3044	−0.647940
$983$	27.7827	0.886130	0.443065	−	0.896490i	$-0.353891\pi$
$983$	27.7827	0.886130	0.443065	+	0.896490i	$0.353891\pi$
$984$	0	0
$985$	0	0
$986$	12.0578	0.384000
$987$	0	0
$988$	−0.985939	−0.0313669
$989$	−1.03951	−0.0330544
$990$	0	0
$991$	−51.0140	−1.62051	−0.810257	−	0.586075i	$-0.800672\pi$
$991$	−51.0140	−1.62051	−0.810257	+	0.586075i	$0.800672\pi$
$992$	−0.715853	−0.0227283
$993$	0	0
$994$	24.9193	0.790391
$995$	0	0
$996$	0	0
$997$	14.2371	0.450895	0.225447	−	0.974255i	$-0.427616\pi$
$997$	14.2371	0.450895	0.225447	+	0.974255i	$0.427616\pi$
$998$	12.9193	0.408952
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8550.2.a.cw.1.1		6
3.2	odd	2		8550.2.a.cx.1.1		6
5.2	odd	4		1710.2.d.h.1369.4	yes	12
5.3	odd	4		1710.2.d.h.1369.10	yes	12
5.4	even	2		8550.2.a.cx.1.6		6
15.2	even	4		1710.2.d.h.1369.9	yes	12
15.8	even	4		1710.2.d.h.1369.3	✓	12
15.14	odd	2	inner	8550.2.a.cw.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1710.2.d.h.1369.3	✓	12	15.8	even	4
1710.2.d.h.1369.4	yes	12	5.2	odd	4
1710.2.d.h.1369.9	yes	12	15.2	even	4
1710.2.d.h.1369.10	yes	12	5.3	odd	4
8550.2.a.cw.1.1		6	1.1	even	1	trivial
8550.2.a.cw.1.6		6	15.14	odd	2	inner
8550.2.a.cx.1.1		6	3.2	odd	2
8550.2.a.cx.1.6		6	5.4	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.05705 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.05705 q^{7} -1.00000 q^{8} -0.466185 q^{11} -0.985939 q^{13} +4.05705 q^{14} +1.00000 q^{16} -4.94567 q^{17} +1.00000 q^{19} +0.466185 q^{22} +2.22982 q^{23} +0.985939 q^{26} -4.05705 q^{28} +2.43806 q^{29} +0.715853 q^{31} -1.00000 q^{32} +4.94567 q^{34} +10.0324 q^{37} -1.00000 q^{38} +10.1993 q^{41} -0.466185 q^{43} -0.466185 q^{44} -2.22982 q^{46} -8.79811 q^{47} +9.45963 q^{49} -0.985939 q^{52} +3.89134 q^{53} +4.05705 q^{56} -2.43806 q^{58} +7.42748 q^{59} -6.45963 q^{61} -0.715853 q^{62} +1.00000 q^{64} +13.9226 q^{67} -4.94567 q^{68} +6.14222 q^{71} -7.07459 q^{73} -10.0324 q^{74} +1.00000 q^{76} +1.89134 q^{77} +1.85244 q^{79} -10.1993 q^{82} -4.37737 q^{83} +0.466185 q^{86} +0.466185 q^{88} -6.02892 q^{89} +4.00000 q^{91} +2.22982 q^{92} +8.79811 q^{94} -6.60840 q^{97} -9.45963 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8}+O(q^{10}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8	\( 6 q - 6 q^{2} + 6 q^{4} - 6 q^{8} + 6 q^{16} - 8 q^{17} + 6 q^{19} - 12 q^{23} + 8 q^{31} - 6 q^{32} + 8 q^{34} - 6 q^{38} + 12 q^{46} - 20 q^{47} + 6 q^{49} - 20 q^{53} + 12 q^{61} - 8 q^{62} + 6 q^{64} - 8 q^{68} + 6 q^{76} - 32 q^{77} - 12 q^{83} + 24 q^{91} - 12 q^{92} + 20 q^{94} - 6 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 6 * q^4 - 6 * q^8 + 6 * q^16 - 8 * q^17 + 6 * q^19 - 12 * q^23 + 8 * q^31 - 6 * q^32 + 8 * q^34 - 6 * q^38 + 12 * q^46 - 20 * q^47 + 6 * q^49 - 20 * q^53 + 12 * q^61 - 8 * q^62 + 6 * q^64 - 8 * q^68 + 6 * q^76 - 32 * q^77 - 12 * q^83 + 24 * q^91 - 12 * q^92 + 20 * q^94 - 6 * q^98

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