LMFDB - Embedded newform 6012.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, 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("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.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:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 31x^{7} + 24x^{6} + 293x^{5} - 101x^{4} - 864x^{3} - 278x^{2} + 24x + 2 $ x^9 - x^8 - 31x^7 + 24x^6 + 293x^5 - 101x^4 - 864x^3 - 278x^2 + 24*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.94172$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.94172 q^{5} -2.78729 q^{7} +O(q^{10})$	$q-2.94172 q^{5} -2.78729 q^{7} -3.89000 q^{11} -0.470621 q^{13} -4.98914 q^{17} -1.49771 q^{19} -0.667875 q^{23} +3.65370 q^{25} -0.0707310 q^{29} -3.13165 q^{31} +8.19942 q^{35} -1.64481 q^{37} -8.08147 q^{41} -2.74446 q^{43} -4.54043 q^{47} +0.768985 q^{49} -12.1695 q^{53} +11.4433 q^{55} -11.9713 q^{59} +2.07730 q^{61} +1.38443 q^{65} +0.754501 q^{67} -1.65397 q^{71} -3.00915 q^{73} +10.8426 q^{77} -5.37256 q^{79} +11.1661 q^{83} +14.6766 q^{85} +9.06894 q^{89} +1.31176 q^{91} +4.40582 q^{95} +2.36503 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{5} + 2 q^{7}+O(q^{10}) $ 9 * q - q^5 + 2 * q^7	$ 9 q - q^{5} + 2 q^{7} + 9 q^{11} + 10 q^{13} - 7 q^{17} - 2 q^{19} + 3 q^{23} + 18 q^{25} - 5 q^{29} + 12 q^{31} + 6 q^{35} + 15 q^{37} - 14 q^{41} + 6 q^{43} + 3 q^{47} + 27 q^{49} - 9 q^{53} + 19 q^{55} + 9 q^{59} + 30 q^{61} - 28 q^{65} + 16 q^{67} + 3 q^{71} + 32 q^{73} - 18 q^{77} + 24 q^{79} + 3 q^{83} + 37 q^{85} - 46 q^{89} + 33 q^{91} - 11 q^{95} + 43 q^{97}+O(q^{100}) $ 9 * q - q^5 + 2 * q^7 + 9 * q^11 + 10 * q^13 - 7 * q^17 - 2 * q^19 + 3 * q^23 + 18 * q^25 - 5 * q^29 + 12 * q^31 + 6 * q^35 + 15 * q^37 - 14 * q^41 + 6 * q^43 + 3 * q^47 + 27 * q^49 - 9 * q^53 + 19 * q^55 + 9 * q^59 + 30 * q^61 - 28 * q^65 + 16 * q^67 + 3 * q^71 + 32 * q^73 - 18 * q^77 + 24 * q^79 + 3 * q^83 + 37 * q^85 - 46 * q^89 + 33 * q^91 - 11 * q^95 + 43 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.94172	−1.31558	−0.657788	−	0.753203i	$-0.728508\pi$
$5$	−2.94172	−1.31558	−0.657788	+	0.753203i	$0.728508\pi$
$6$	0	0
$7$	−2.78729	−1.05350	−0.526748	−	0.850021i	$-0.676589\pi$
$7$	−2.78729	−1.05350	−0.526748	+	0.850021i	$0.676589\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.89000	−1.17288	−0.586440	−	0.809993i	$-0.699471\pi$
$11$	−3.89000	−1.17288	−0.586440	+	0.809993i	$0.699471\pi$
$12$	0	0
$13$	−0.470621	−0.130527	−0.0652634	−	0.997868i	$-0.520789\pi$
$13$	−0.470621	−0.130527	−0.0652634	+	0.997868i	$0.520789\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.98914	−1.21004	−0.605022	−	0.796209i	$-0.706836\pi$
$17$	−4.98914	−1.21004	−0.605022	+	0.796209i	$0.706836\pi$
$18$	0	0
$19$	−1.49771	−0.343597	−0.171799	−	0.985132i	$-0.554958\pi$
$19$	−1.49771	−0.343597	−0.171799	+	0.985132i	$0.554958\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.667875	−0.139262	−0.0696308	−	0.997573i	$-0.522182\pi$
$23$	−0.667875	−0.139262	−0.0696308	+	0.997573i	$0.522182\pi$
$24$	0	0
$25$	3.65370	0.730740
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.0707310	−0.0131344	−0.00656721	−	0.999978i	$-0.502090\pi$
$29$	−0.0707310	−0.0131344	−0.00656721	+	0.999978i	$0.502090\pi$
$30$	0	0
$31$	−3.13165	−0.562462	−0.281231	−	0.959640i	$-0.590743\pi$
$31$	−3.13165	−0.562462	−0.281231	+	0.959640i	$0.590743\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.19942	1.38595
$36$	0	0
$37$	−1.64481	−0.270406	−0.135203	−	0.990818i	$-0.543169\pi$
$37$	−1.64481	−0.270406	−0.135203	+	0.990818i	$0.543169\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.08147	−1.26211	−0.631057	−	0.775736i	$-0.717379\pi$
$41$	−8.08147	−1.26211	−0.631057	+	0.775736i	$0.717379\pi$
$42$	0	0
$43$	−2.74446	−0.418527	−0.209263	−	0.977859i	$-0.567107\pi$
$43$	−2.74446	−0.418527	−0.209263	+	0.977859i	$0.567107\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.54043	−0.662290	−0.331145	−	0.943580i	$-0.607435\pi$
$47$	−4.54043	−0.662290	−0.331145	+	0.943580i	$0.607435\pi$
$48$	0	0
$49$	0.768985	0.109855
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.1695	−1.67161	−0.835803	−	0.549029i	$-0.814998\pi$
$53$	−12.1695	−1.67161	−0.835803	+	0.549029i	$0.814998\pi$
$54$	0	0
$55$	11.4433	1.54301
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.9713	−1.55852	−0.779262	−	0.626698i	$-0.784406\pi$
$59$	−11.9713	−1.55852	−0.779262	+	0.626698i	$0.784406\pi$
$60$	0	0
$61$	2.07730	0.265971	0.132986	−	0.991118i	$-0.457544\pi$
$61$	2.07730	0.265971	0.132986	+	0.991118i	$0.457544\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.38443	0.171718
$66$	0	0
$67$	0.754501	0.0921770	0.0460885	−	0.998937i	$-0.485324\pi$
$67$	0.754501	0.0921770	0.0460885	+	0.998937i	$0.485324\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.65397	−0.196290	−0.0981448	−	0.995172i	$-0.531291\pi$
$71$	−1.65397	−0.196290	−0.0981448	+	0.995172i	$0.531291\pi$
$72$	0	0
$73$	−3.00915	−0.352195	−0.176097	−	0.984373i	$-0.556347\pi$
$73$	−3.00915	−0.352195	−0.176097	+	0.984373i	$0.556347\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.8426	1.23563
$78$	0	0
$79$	−5.37256	−0.604461	−0.302230	−	0.953235i	$-0.597731\pi$
$79$	−5.37256	−0.604461	−0.302230	+	0.953235i	$0.597731\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.1661	1.22564	0.612820	−	0.790222i	$-0.290035\pi$
$83$	11.1661	1.22564	0.612820	+	0.790222i	$0.290035\pi$
$84$	0	0
$85$	14.6766	1.59191
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.06894	0.961306	0.480653	−	0.876911i	$-0.340400\pi$
$89$	9.06894	0.961306	0.480653	+	0.876911i	$0.340400\pi$
$90$	0	0
$91$	1.31176	0.137510
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.40582	0.452028
$96$	0	0
$97$	2.36503	0.240133	0.120066	−	0.992766i	$-0.461689\pi$
$97$	2.36503	0.240133	0.120066	+	0.992766i	$0.461689\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.3197	−1.72337	−0.861687	−	0.507440i	$-0.830592\pi$
$101$	−17.3197	−1.72337	−0.861687	+	0.507440i	$0.830592\pi$
$102$	0	0
$103$	1.21898	0.120110	0.0600548	−	0.998195i	$-0.480872\pi$
$103$	1.21898	0.120110	0.0600548	+	0.998195i	$0.480872\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.335485	−0.0324325	−0.0162163	−	0.999869i	$-0.505162\pi$
$107$	−0.335485	−0.0324325	−0.0162163	+	0.999869i	$0.505162\pi$
$108$	0	0
$109$	17.4356	1.67002	0.835012	−	0.550232i	$-0.185461\pi$
$109$	17.4356	1.67002	0.835012	+	0.550232i	$0.185461\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.41636	0.321384	0.160692	−	0.987005i	$-0.448627\pi$
$113$	3.41636	0.321384	0.160692	+	0.987005i	$0.448627\pi$
$114$	0	0
$115$	1.96470	0.183209
$116$	0	0
$117$	0	0
$118$	0	0
$119$	13.9062	1.27478
$120$	0	0
$121$	4.13213	0.375649
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.96043	0.354232
$126$	0	0
$127$	14.9752	1.32883	0.664417	−	0.747362i	$-0.268680\pi$
$127$	14.9752	1.32883	0.664417	+	0.747362i	$0.268680\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.5029	0.917641	0.458820	−	0.888529i	$-0.348272\pi$
$131$	10.5029	0.917641	0.458820	+	0.888529i	$0.348272\pi$
$132$	0	0
$133$	4.17454	0.361978
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.5392	−1.15673	−0.578366	−	0.815778i	$-0.696309\pi$
$137$	−13.5392	−1.15673	−0.578366	+	0.815778i	$0.696309\pi$
$138$	0	0
$139$	11.0800	0.939797	0.469899	−	0.882720i	$-0.344290\pi$
$139$	11.0800	0.939797	0.469899	+	0.882720i	$0.344290\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.83072	0.153092
$144$	0	0
$145$	0.208071	0.0172793
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−15.4865	−1.26870	−0.634351	−	0.773045i	$-0.718733\pi$
$149$	−15.4865	−1.26870	−0.634351	+	0.773045i	$0.718733\pi$
$150$	0	0
$151$	19.5686	1.59247	0.796235	−	0.604987i	$-0.206822\pi$
$151$	19.5686	1.59247	0.796235	+	0.604987i	$0.206822\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	9.21244	0.739961
$156$	0	0
$157$	−0.359659	−0.0287039	−0.0143520	−	0.999897i	$-0.504569\pi$
$157$	−0.359659	−0.0287039	−0.0143520	+	0.999897i	$0.504569\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.86156	0.146712
$162$	0	0
$163$	−1.12275	−0.0879403	−0.0439701	−	0.999033i	$-0.514001\pi$
$163$	−1.12275	−0.0879403	−0.0439701	+	0.999033i	$0.514001\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.7785	−0.982963
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.0970	−0.919719	−0.459859	−	0.887992i	$-0.652100\pi$
$173$	−12.0970	−0.919719	−0.459859	+	0.887992i	$0.652100\pi$
$174$	0	0
$175$	−10.1839	−0.769832
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−0.710737	−0.0531230	−0.0265615	−	0.999647i	$-0.508456\pi$
$179$	−0.710737	−0.0531230	−0.0265615	+	0.999647i	$0.508456\pi$
$180$	0	0
$181$	16.0045	1.18960	0.594801	−	0.803873i	$-0.297231\pi$
$181$	16.0045	1.18960	0.594801	+	0.803873i	$0.297231\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.83858	0.355739
$186$	0	0
$187$	19.4078	1.41924
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−5.85478	−0.423637	−0.211819	−	0.977309i	$-0.567939\pi$
$191$	−5.85478	−0.423637	−0.211819	+	0.977309i	$0.567939\pi$
$192$	0	0
$193$	−8.19532	−0.589912	−0.294956	−	0.955511i	$-0.595305\pi$
$193$	−8.19532	−0.589912	−0.294956	+	0.955511i	$0.595305\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.74323	0.409188	0.204594	−	0.978847i	$-0.434413\pi$
$197$	5.74323	0.409188	0.204594	+	0.978847i	$0.434413\pi$
$198$	0	0
$199$	−12.8466	−0.910671	−0.455335	−	0.890320i	$-0.650481\pi$
$199$	−12.8466	−0.910671	−0.455335	+	0.890320i	$0.650481\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.197148	0.0138371
$204$	0	0
$205$	23.7734	1.66041
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.82608	0.402998
$210$	0	0
$211$	−3.76354	−0.259093	−0.129547	−	0.991573i	$-0.541352\pi$
$211$	−3.76354	−0.259093	−0.129547	+	0.991573i	$0.541352\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.07343	0.550604
$216$	0	0
$217$	8.72883	0.592551
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.34800	0.157943
$222$	0	0
$223$	0.189252	0.0126733	0.00633663	−	0.999980i	$-0.497983\pi$
$223$	0.189252	0.0126733	0.00633663	+	0.999980i	$0.497983\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.8597	1.18539	0.592696	−	0.805427i	$-0.298064\pi$
$227$	17.8597	1.18539	0.592696	+	0.805427i	$0.298064\pi$
$228$	0	0
$229$	16.7945	1.10981	0.554906	−	0.831913i	$-0.312754\pi$
$229$	16.7945	1.10981	0.554906	+	0.831913i	$0.312754\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	13.0060	0.852052	0.426026	−	0.904711i	$-0.359913\pi$
$233$	13.0060	0.852052	0.426026	+	0.904711i	$0.359913\pi$
$234$	0	0
$235$	13.3567	0.871292
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4.64337	−0.300355	−0.150177	−	0.988659i	$-0.547985\pi$
$239$	−4.64337	−0.300355	−0.150177	+	0.988659i	$0.547985\pi$
$240$	0	0
$241$	−4.63503	−0.298568	−0.149284	−	0.988794i	$-0.547697\pi$
$241$	−4.63503	−0.298568	−0.149284	+	0.988794i	$0.547697\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.26214	−0.144523
$246$	0	0
$247$	0.704852	0.0448486
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−27.1447	−1.71336	−0.856680	−	0.515848i	$-0.827477\pi$
$251$	−27.1447	−1.71336	−0.856680	+	0.515848i	$0.827477\pi$
$252$	0	0
$253$	2.59804	0.163337
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.4905	−0.779137	−0.389569	−	0.920997i	$-0.627376\pi$
$257$	−12.4905	−0.779137	−0.389569	+	0.920997i	$0.627376\pi$
$258$	0	0
$259$	4.58457	0.284872
$260$	0	0
$261$	0	0
$262$	0	0
$263$	20.8246	1.28410	0.642051	−	0.766662i	$-0.278084\pi$
$263$	20.8246	1.28410	0.642051	+	0.766662i	$0.278084\pi$
$264$	0	0
$265$	35.7992	2.19913
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.2227	0.684259	0.342130	−	0.939653i	$-0.388852\pi$
$269$	11.2227	0.684259	0.342130	+	0.939653i	$0.388852\pi$
$270$	0	0
$271$	−2.75633	−0.167435	−0.0837177	−	0.996490i	$-0.526679\pi$
$271$	−2.75633	−0.167435	−0.0837177	+	0.996490i	$0.526679\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−14.2129	−0.857071
$276$	0	0
$277$	−20.6207	−1.23898	−0.619489	−	0.785005i	$-0.712660\pi$
$277$	−20.6207	−1.23898	−0.619489	+	0.785005i	$0.712660\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.8580	1.00566	0.502832	−	0.864384i	$-0.332291\pi$
$281$	16.8580	1.00566	0.502832	+	0.864384i	$0.332291\pi$
$282$	0	0
$283$	−28.1576	−1.67379	−0.836897	−	0.547361i	$-0.815633\pi$
$283$	−28.1576	−1.67379	−0.836897	+	0.547361i	$0.815633\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	22.5254	1.32963
$288$	0	0
$289$	7.89154	0.464208
$290$	0	0
$291$	0	0
$292$	0	0
$293$	8.79839	0.514008	0.257004	−	0.966410i	$-0.417265\pi$
$293$	8.79839	0.514008	0.257004	+	0.966410i	$0.417265\pi$
$294$	0	0
$295$	35.2160	2.05036
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.314316	0.0181774
$300$	0	0
$301$	7.64961	0.440916
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.11083	−0.349905
$306$	0	0
$307$	−14.9804	−0.854977	−0.427489	−	0.904021i	$-0.640602\pi$
$307$	−14.9804	−0.854977	−0.427489	+	0.904021i	$0.640602\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.4300	−1.15848	−0.579240	−	0.815157i	$-0.696650\pi$
$311$	−20.4300	−1.15848	−0.579240	+	0.815157i	$0.696650\pi$
$312$	0	0
$313$	−23.9100	−1.35147	−0.675737	−	0.737143i	$-0.736174\pi$
$313$	−23.9100	−1.35147	−0.675737	+	0.737143i	$0.736174\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.2140	1.86548	0.932742	−	0.360545i	$-0.117409\pi$
$317$	33.2140	1.86548	0.932742	+	0.360545i	$0.117409\pi$
$318$	0	0
$319$	0.275144	0.0154051
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.47226	0.415768
$324$	0	0
$325$	−1.71951	−0.0953812
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.6555	0.697720
$330$	0	0
$331$	7.42341	0.408028	0.204014	−	0.978968i	$-0.434601\pi$
$331$	7.42341	0.408028	0.204014	+	0.978968i	$0.434601\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.21953	−0.121266
$336$	0	0
$337$	9.69696	0.528227	0.264113	−	0.964492i	$-0.414921\pi$
$337$	9.69696	0.528227	0.264113	+	0.964492i	$0.414921\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	12.1821	0.659700
$342$	0	0
$343$	17.3676	0.937765
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0.199217	0.0106945	0.00534727	−	0.999986i	$-0.498298\pi$
$347$	0.199217	0.0106945	0.00534727	+	0.999986i	$0.498298\pi$
$348$	0	0
$349$	−9.73705	−0.521213	−0.260606	−	0.965445i	$-0.583922\pi$
$349$	−9.73705	−0.521213	−0.260606	+	0.965445i	$0.583922\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.40523	0.181242	0.0906211	−	0.995885i	$-0.471115\pi$
$353$	3.40523	0.181242	0.0906211	+	0.995885i	$0.471115\pi$
$354$	0	0
$355$	4.86550	0.258234
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.89035	0.416437	0.208218	−	0.978082i	$-0.433234\pi$
$359$	7.89035	0.416437	0.208218	+	0.978082i	$0.433234\pi$
$360$	0	0
$361$	−16.7569	−0.881941
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.85208	0.463339
$366$	0	0
$367$	19.6686	1.02669	0.513347	−	0.858181i	$-0.328405\pi$
$367$	19.6686	1.02669	0.513347	+	0.858181i	$0.328405\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	33.9199	1.76103
$372$	0	0
$373$	−25.4806	−1.31934	−0.659668	−	0.751557i	$-0.729303\pi$
$373$	−25.4806	−1.31934	−0.659668	+	0.751557i	$0.729303\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.0332875	0.00171439
$378$	0	0
$379$	−6.19297	−0.318111	−0.159056	−	0.987270i	$-0.550845\pi$
$379$	−6.19297	−0.318111	−0.159056	+	0.987270i	$0.550845\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.93978	0.303509	0.151754	−	0.988418i	$-0.451508\pi$
$383$	5.93978	0.303509	0.151754	+	0.988418i	$0.451508\pi$
$384$	0	0
$385$	−31.8958	−1.62556
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.9031	0.552810	0.276405	−	0.961041i	$-0.410857\pi$
$389$	10.9031	0.552810	0.276405	+	0.961041i	$0.410857\pi$
$390$	0	0
$391$	3.33212	0.168513
$392$	0	0
$393$	0	0
$394$	0	0
$395$	15.8046	0.795214
$396$	0	0
$397$	0.455424	0.0228571	0.0114285	−	0.999935i	$-0.496362\pi$
$397$	0.455424	0.0228571	0.0114285	+	0.999935i	$0.496362\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.94773	0.0972652	0.0486326	−	0.998817i	$-0.484514\pi$
$401$	1.94773	0.0972652	0.0486326	+	0.998817i	$0.484514\pi$
$402$	0	0
$403$	1.47382	0.0734163
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.39833	0.317154
$408$	0	0
$409$	−4.36458	−0.215815	−0.107907	−	0.994161i	$-0.534415\pi$
$409$	−4.36458	−0.215815	−0.107907	+	0.994161i	$0.534415\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	33.3674	1.64190
$414$	0	0
$415$	−32.8475	−1.61242
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−22.0315	−1.07631	−0.538154	−	0.842847i	$-0.680878\pi$
$419$	−22.0315	−1.07631	−0.538154	+	0.842847i	$0.680878\pi$
$420$	0	0
$421$	−32.2357	−1.57107	−0.785535	−	0.618817i	$-0.787612\pi$
$421$	−32.2357	−1.57107	−0.785535	+	0.618817i	$0.787612\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−18.2288	−0.884228
$426$	0	0
$427$	−5.79004	−0.280200
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.42440	0.261284	0.130642	−	0.991430i	$-0.458296\pi$
$431$	5.42440	0.261284	0.130642	+	0.991430i	$0.458296\pi$
$432$	0	0
$433$	−9.20893	−0.442553	−0.221276	−	0.975211i	$-0.571022\pi$
$433$	−9.20893	−0.442553	−0.221276	+	0.975211i	$0.571022\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.00028	0.0478499
$438$	0	0
$439$	−32.4748	−1.54994	−0.774969	−	0.632000i	$-0.782234\pi$
$439$	−32.4748	−1.54994	−0.774969	+	0.632000i	$0.782234\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.613271	−0.0291374	−0.0145687	−	0.999894i	$-0.504638\pi$
$443$	−0.613271	−0.0291374	−0.0145687	+	0.999894i	$0.504638\pi$
$444$	0	0
$445$	−26.6783	−1.26467
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.30511	0.203171	0.101585	−	0.994827i	$-0.467608\pi$
$449$	4.30511	0.203171	0.101585	+	0.994827i	$0.467608\pi$
$450$	0	0
$451$	31.4370	1.48031
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.85882	−0.180904
$456$	0	0
$457$	−5.40530	−0.252849	−0.126425	−	0.991976i	$-0.540350\pi$
$457$	−5.40530	−0.252849	−0.126425	+	0.991976i	$0.540350\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17.5751	0.818552	0.409276	−	0.912411i	$-0.365781\pi$
$461$	17.5751	0.818552	0.409276	+	0.912411i	$0.365781\pi$
$462$	0	0
$463$	−22.3864	−1.04038	−0.520192	−	0.854049i	$-0.674140\pi$
$463$	−22.3864	−1.04038	−0.520192	+	0.854049i	$0.674140\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.8757	0.919736	0.459868	−	0.887987i	$-0.347897\pi$
$467$	19.8757	0.919736	0.459868	+	0.887987i	$0.347897\pi$
$468$	0	0
$469$	−2.10301	−0.0971081
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10.6760	0.490882
$474$	0	0
$475$	−5.47217	−0.251080
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−19.2738	−0.880643	−0.440321	−	0.897840i	$-0.645135\pi$
$479$	−19.2738	−0.880643	−0.440321	+	0.897840i	$0.645135\pi$
$480$	0	0
$481$	0.774084	0.0352952
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.95725	−0.315913
$486$	0	0
$487$	−20.9329	−0.948558	−0.474279	−	0.880375i	$-0.657291\pi$
$487$	−20.9329	−0.948558	−0.474279	+	0.880375i	$0.657291\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	20.4989	0.925104	0.462552	−	0.886592i	$-0.346934\pi$
$491$	20.4989	0.925104	0.462552	+	0.886592i	$0.346934\pi$
$492$	0	0
$493$	0.352887	0.0158932
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.61008	0.206791
$498$	0	0
$499$	34.8985	1.56227	0.781136	−	0.624361i	$-0.214640\pi$
$499$	34.8985	1.56227	0.781136	+	0.624361i	$0.214640\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−13.7974	−0.615196	−0.307598	−	0.951516i	$-0.599525\pi$
$503$	−13.7974	−0.615196	−0.307598	+	0.951516i	$0.599525\pi$
$504$	0	0
$505$	50.9497	2.26723
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−36.0627	−1.59845	−0.799225	−	0.601031i	$-0.794757\pi$
$509$	−36.0627	−1.59845	−0.799225	+	0.601031i	$0.794757\pi$
$510$	0	0
$511$	8.38738	0.371036
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.58589	−0.158013
$516$	0	0
$517$	17.6623	0.776787
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−17.3865	−0.761716	−0.380858	−	0.924634i	$-0.624371\pi$
$521$	−17.3865	−0.761716	−0.380858	+	0.924634i	$0.624371\pi$
$522$	0	0
$523$	−4.03478	−0.176429	−0.0882143	−	0.996102i	$-0.528116\pi$
$523$	−4.03478	−0.176429	−0.0882143	+	0.996102i	$0.528116\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15.6243	0.680604
$528$	0	0
$529$	−22.5539	−0.980606
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.80331	0.164740
$534$	0	0
$535$	0.986901	0.0426674
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.99135	−0.128847
$540$	0	0
$541$	23.0954	0.992951	0.496475	−	0.868051i	$-0.334627\pi$
$541$	23.0954	0.992951	0.496475	+	0.868051i	$0.334627\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−51.2905	−2.19704
$546$	0	0
$547$	−14.7888	−0.632322	−0.316161	−	0.948706i	$-0.602394\pi$
$547$	−14.7888	−0.632322	−0.316161	+	0.948706i	$0.602394\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.105934	0.00451295
$552$	0	0
$553$	14.9749	0.636797
$554$	0	0
$555$	0	0
$556$	0	0
$557$	40.2860	1.70697	0.853486	−	0.521116i	$-0.174484\pi$
$557$	40.2860	1.70697	0.853486	+	0.521116i	$0.174484\pi$
$558$	0	0
$559$	1.29160	0.0546290
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0.767368	0.0323407	0.0161704	−	0.999869i	$-0.494853\pi$
$563$	0.767368	0.0323407	0.0161704	+	0.999869i	$0.494853\pi$
$564$	0	0
$565$	−10.0500	−0.422805
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.5129	−0.985713	−0.492857	−	0.870110i	$-0.664047\pi$
$569$	−23.5129	−0.985713	−0.492857	+	0.870110i	$0.664047\pi$
$570$	0	0
$571$	0.332900	0.0139314	0.00696571	−	0.999976i	$-0.497783\pi$
$571$	0.332900	0.0139314	0.00696571	+	0.999976i	$0.497783\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.44022	−0.101764
$576$	0	0
$577$	9.29804	0.387082	0.193541	−	0.981092i	$-0.438003\pi$
$577$	9.29804	0.387082	0.193541	+	0.981092i	$0.438003\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−31.1232	−1.29121
$582$	0	0
$583$	47.3393	1.96059
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.6733	0.770729	0.385364	−	0.922764i	$-0.374076\pi$
$587$	18.6733	0.770729	0.385364	+	0.922764i	$0.374076\pi$
$588$	0	0
$589$	4.69029	0.193260
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6.71101	−0.275588	−0.137794	−	0.990461i	$-0.544001\pi$
$593$	−6.71101	−0.275588	−0.137794	+	0.990461i	$0.544001\pi$
$594$	0	0
$595$	−40.9081	−1.67707
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−7.80833	−0.319040	−0.159520	−	0.987195i	$-0.550995\pi$
$599$	−7.80833	−0.319040	−0.159520	+	0.987195i	$0.550995\pi$
$600$	0	0
$601$	−23.6693	−0.965493	−0.482746	−	0.875760i	$-0.660361\pi$
$601$	−23.6693	−0.965493	−0.482746	+	0.875760i	$0.660361\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−12.1556	−0.494194
$606$	0	0
$607$	19.9861	0.811210	0.405605	−	0.914048i	$-0.367061\pi$
$607$	19.9861	0.811210	0.405605	+	0.914048i	$0.367061\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.13682	0.0864466
$612$	0	0
$613$	25.0681	1.01249	0.506246	−	0.862389i	$-0.331033\pi$
$613$	25.0681	1.01249	0.506246	+	0.862389i	$0.331033\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−48.1170	−1.93712	−0.968559	−	0.248784i	$-0.919969\pi$
$617$	−48.1170	−1.93712	−0.968559	+	0.248784i	$0.919969\pi$
$618$	0	0
$619$	−24.1162	−0.969311	−0.484655	−	0.874705i	$-0.661055\pi$
$619$	−24.1162	−0.969311	−0.484655	+	0.874705i	$0.661055\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−25.2778	−1.01273
$624$	0	0
$625$	−29.9190	−1.19676
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.20621	0.327203
$630$	0	0
$631$	42.8844	1.70720	0.853601	−	0.520927i	$-0.174414\pi$
$631$	42.8844	1.70720	0.853601	+	0.520927i	$0.174414\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−44.0528	−1.74818
$636$	0	0
$637$	−0.361901	−0.0143390
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−36.9879	−1.46094	−0.730468	−	0.682947i	$-0.760698\pi$
$641$	−36.9879	−1.46094	−0.730468	+	0.682947i	$0.760698\pi$
$642$	0	0
$643$	−7.90122	−0.311594	−0.155797	−	0.987789i	$-0.549795\pi$
$643$	−7.90122	−0.311594	−0.155797	+	0.987789i	$0.549795\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.01646	0.315160	0.157580	−	0.987506i	$-0.449631\pi$
$647$	8.01646	0.315160	0.157580	+	0.987506i	$0.449631\pi$
$648$	0	0
$649$	46.5682	1.82796
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−17.2813	−0.676271	−0.338136	−	0.941097i	$-0.609796\pi$
$653$	−17.2813	−0.676271	−0.338136	+	0.941097i	$0.609796\pi$
$654$	0	0
$655$	−30.8965	−1.20723
$656$	0	0
$657$	0	0
$658$	0	0
$659$	25.9189	1.00966	0.504829	−	0.863220i	$-0.331556\pi$
$659$	25.9189	1.00966	0.504829	+	0.863220i	$0.331556\pi$
$660$	0	0
$661$	−21.8718	−0.850715	−0.425357	−	0.905026i	$-0.639852\pi$
$661$	−21.8718	−0.850715	−0.425357	+	0.905026i	$0.639852\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−12.2803	−0.476210
$666$	0	0
$667$	0.0472395	0.00182912
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.08071	−0.311952
$672$	0	0
$673$	−18.1707	−0.700429	−0.350214	−	0.936670i	$-0.613891\pi$
$673$	−18.1707	−0.700429	−0.350214	+	0.936670i	$0.613891\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27.6866	1.06408	0.532041	−	0.846719i	$-0.321425\pi$
$677$	27.6866	1.06408	0.532041	+	0.846719i	$0.321425\pi$
$678$	0	0
$679$	−6.59203	−0.252979
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7.55246	0.288987	0.144493	−	0.989506i	$-0.453845\pi$
$683$	7.55246	0.288987	0.144493	+	0.989506i	$0.453845\pi$
$684$	0	0
$685$	39.8285	1.52177
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.72721	0.218189
$690$	0	0
$691$	−18.7496	−0.713270	−0.356635	−	0.934244i	$-0.616076\pi$
$691$	−18.7496	−0.713270	−0.356635	+	0.934244i	$0.616076\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−32.5944	−1.23637
$696$	0	0
$697$	40.3196	1.52721
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7.15605	0.270280	0.135140	−	0.990826i	$-0.456852\pi$
$701$	7.15605	0.270280	0.135140	+	0.990826i	$0.456852\pi$
$702$	0	0
$703$	2.46345	0.0929107
$704$	0	0
$705$	0	0
$706$	0	0
$707$	48.2750	1.81557
$708$	0	0
$709$	7.03676	0.264271	0.132136	−	0.991232i	$-0.457817\pi$
$709$	7.03676	0.264271	0.132136	+	0.991232i	$0.457817\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.09155	0.0783293
$714$	0	0
$715$	−5.38546	−0.201405
$716$	0	0
$717$	0	0
$718$	0	0
$719$	35.0939	1.30878	0.654392	−	0.756156i	$-0.272925\pi$
$719$	35.0939	1.30878	0.654392	+	0.756156i	$0.272925\pi$
$720$	0	0
$721$	−3.39765	−0.126535
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.258430	−0.00959784
$726$	0	0
$727$	12.9195	0.479160	0.239580	−	0.970877i	$-0.422990\pi$
$727$	12.9195	0.479160	0.239580	+	0.970877i	$0.422990\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	13.6925	0.506436
$732$	0	0
$733$	−52.6248	−1.94374	−0.971871	−	0.235516i	$-0.924322\pi$
$733$	−52.6248	−1.94374	−0.971871	+	0.235516i	$0.924322\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.93501	−0.108113
$738$	0	0
$739$	−13.3792	−0.492161	−0.246081	−	0.969249i	$-0.579143\pi$
$739$	−13.3792	−0.492161	−0.246081	+	0.969249i	$0.579143\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.76762	0.248280	0.124140	−	0.992265i	$-0.460383\pi$
$743$	6.76762	0.248280	0.124140	+	0.992265i	$0.460383\pi$
$744$	0	0
$745$	45.5569	1.66908
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.935093	0.0341676
$750$	0	0
$751$	22.8195	0.832694	0.416347	−	0.909206i	$-0.363310\pi$
$751$	22.8195	0.832694	0.416347	+	0.909206i	$0.363310\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−57.5653	−2.09502
$756$	0	0
$757$	−5.27356	−0.191671	−0.0958353	−	0.995397i	$-0.530552\pi$
$757$	−5.27356	−0.191671	−0.0958353	+	0.995397i	$0.530552\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−15.9457	−0.578030	−0.289015	−	0.957325i	$-0.593328\pi$
$761$	−15.9457	−0.578030	−0.289015	+	0.957325i	$0.593328\pi$
$762$	0	0
$763$	−48.5980	−1.75936
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.63392	0.203429
$768$	0	0
$769$	−1.46687	−0.0528968	−0.0264484	−	0.999650i	$-0.508420\pi$
$769$	−1.46687	−0.0528968	−0.0264484	+	0.999650i	$0.508420\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	21.0752	0.758022	0.379011	−	0.925392i	$-0.376264\pi$
$773$	21.0752	0.758022	0.379011	+	0.925392i	$0.376264\pi$
$774$	0	0
$775$	−11.4421	−0.411013
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.1037	0.433659
$780$	0	0
$781$	6.43394	0.230224
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.05802	0.0377622
$786$	0	0
$787$	27.1679	0.968431	0.484216	−	0.874949i	$-0.339105\pi$
$787$	27.1679	0.968431	0.484216	+	0.874949i	$0.339105\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.52239	−0.338577
$792$	0	0
$793$	−0.977622	−0.0347164
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−28.1742	−0.997982	−0.498991	−	0.866607i	$-0.666296\pi$
$797$	−28.1742	−0.997982	−0.498991	+	0.866607i	$0.666296\pi$
$798$	0	0
$799$	22.6528	0.801400
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.7056	0.413082
$804$	0	0
$805$	−5.47619	−0.193010
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.25972	0.255238	0.127619	−	0.991823i	$-0.459267\pi$
$809$	7.25972	0.255238	0.127619	+	0.991823i	$0.459267\pi$
$810$	0	0
$811$	6.87841	0.241534	0.120767	−	0.992681i	$-0.461465\pi$
$811$	6.87841	0.241534	0.120767	+	0.992681i	$0.461465\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.30280	0.115692
$816$	0	0
$817$	4.11040	0.143805
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−39.5748	−1.38117	−0.690585	−	0.723251i	$-0.742647\pi$
$821$	−39.5748	−1.38117	−0.690585	+	0.723251i	$0.742647\pi$
$822$	0	0
$823$	−37.1407	−1.29464	−0.647322	−	0.762217i	$-0.724111\pi$
$823$	−37.1407	−1.29464	−0.647322	+	0.762217i	$0.724111\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.4433	0.884750	0.442375	−	0.896830i	$-0.354136\pi$
$827$	25.4433	0.884750	0.442375	+	0.896830i	$0.354136\pi$
$828$	0	0
$829$	−25.2955	−0.878549	−0.439275	−	0.898353i	$-0.644764\pi$
$829$	−25.2955	−0.878549	−0.439275	+	0.898353i	$0.644764\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.83657	−0.132929
$834$	0	0
$835$	−2.94172	−0.101802
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−29.1828	−1.00750	−0.503751	−	0.863849i	$-0.668047\pi$
$839$	−29.1828	−1.00750	−0.503751	+	0.863849i	$0.668047\pi$
$840$	0	0
$841$	−28.9950	−0.999827
$842$	0	0
$843$	0	0
$844$	0	0
$845$	37.5908	1.29316
$846$	0	0
$847$	−11.5175	−0.395744
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.09853	0.0376572
$852$	0	0
$853$	−13.2876	−0.454957	−0.227479	−	0.973783i	$-0.573048\pi$
$853$	−13.2876	−0.454957	−0.227479	+	0.973783i	$0.573048\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13.4648	−0.459949	−0.229975	−	0.973197i	$-0.573864\pi$
$857$	−13.4648	−0.459949	−0.229975	+	0.973197i	$0.573864\pi$
$858$	0	0
$859$	−21.3283	−0.727711	−0.363856	−	0.931455i	$-0.618540\pi$
$859$	−21.3283	−0.727711	−0.363856	+	0.931455i	$0.618540\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32.5371	1.10758	0.553788	−	0.832658i	$-0.313182\pi$
$863$	32.5371	1.10758	0.553788	+	0.832658i	$0.313182\pi$
$864$	0	0
$865$	35.5860	1.20996
$866$	0	0
$867$	0	0
$868$	0	0
$869$	20.8993	0.708960
$870$	0	0
$871$	−0.355084	−0.0120316
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−11.0389	−0.373182
$876$	0	0
$877$	26.0045	0.878109	0.439054	−	0.898460i	$-0.355314\pi$
$877$	26.0045	0.878109	0.439054	+	0.898460i	$0.355314\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	42.9012	1.44538	0.722689	−	0.691173i	$-0.242906\pi$
$881$	42.9012	1.44538	0.722689	+	0.691173i	$0.242906\pi$
$882$	0	0
$883$	40.0829	1.34890	0.674449	−	0.738322i	$-0.264381\pi$
$883$	40.0829	1.34890	0.674449	+	0.738322i	$0.264381\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5.80878	−0.195040	−0.0975198	−	0.995234i	$-0.531091\pi$
$887$	−5.80878	−0.195040	−0.0975198	+	0.995234i	$0.531091\pi$
$888$	0	0
$889$	−41.7402	−1.39992
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.80022	0.227561
$894$	0	0
$895$	2.09079	0.0698873
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.221505	0.00738760
$900$	0	0
$901$	60.7153	2.02272
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−47.0806	−1.56501
$906$	0	0
$907$	27.8347	0.924235	0.462117	−	0.886819i	$-0.347090\pi$
$907$	27.8347	0.924235	0.462117	+	0.886819i	$0.347090\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−7.44183	−0.246559	−0.123279	−	0.992372i	$-0.539341\pi$
$911$	−7.44183	−0.246559	−0.123279	+	0.992372i	$0.539341\pi$
$912$	0	0
$913$	−43.4362	−1.43753
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−29.2746	−0.966731
$918$	0	0
$919$	28.8849	0.952823	0.476412	−	0.879222i	$-0.341937\pi$
$919$	28.8849	0.952823	0.476412	+	0.879222i	$0.341937\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0.778391	0.0256211
$924$	0	0
$925$	−6.00966	−0.197596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6.25814	−0.205323	−0.102661	−	0.994716i	$-0.532736\pi$
$929$	−6.25814	−0.205323	−0.102661	+	0.994716i	$0.532736\pi$
$930$	0	0
$931$	−1.15171	−0.0377459
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−57.0922	−1.86712
$936$	0	0
$937$	−39.4567	−1.28899	−0.644497	−	0.764607i	$-0.722933\pi$
$937$	−39.4567	−1.28899	−0.644497	+	0.764607i	$0.722933\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	51.6685	1.68434	0.842172	−	0.539209i	$-0.181277\pi$
$941$	51.6685	1.68434	0.842172	+	0.539209i	$0.181277\pi$
$942$	0	0
$943$	5.39742	0.175764
$944$	0	0
$945$	0	0
$946$	0	0
$947$	31.2141	1.01432	0.507161	−	0.861851i	$-0.330695\pi$
$947$	31.2141	1.01432	0.507161	+	0.861851i	$0.330695\pi$
$948$	0	0
$949$	1.41617	0.0459709
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−48.5537	−1.57281	−0.786404	−	0.617712i	$-0.788060\pi$
$953$	−48.5537	−1.57281	−0.786404	+	0.617712i	$0.788060\pi$
$954$	0	0
$955$	17.2231	0.557327
$956$	0	0
$957$	0	0
$958$	0	0
$959$	37.7377	1.21861
$960$	0	0
$961$	−21.1927	−0.683637
$962$	0	0
$963$	0	0
$964$	0	0
$965$	24.1083	0.776074
$966$	0	0
$967$	−19.2557	−0.619221	−0.309610	−	0.950864i	$-0.600199\pi$
$967$	−19.2557	−0.619221	−0.309610	+	0.950864i	$0.600199\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−23.8442	−0.765195	−0.382598	−	0.923915i	$-0.624970\pi$
$971$	−23.8442	−0.765195	−0.382598	+	0.923915i	$0.624970\pi$
$972$	0	0
$973$	−30.8833	−0.990073
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25.7167	−0.822749	−0.411374	−	0.911466i	$-0.634951\pi$
$977$	−25.7167	−0.822749	−0.411374	+	0.911466i	$0.634951\pi$
$978$	0	0
$979$	−35.2782	−1.12750
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−29.0660	−0.927061	−0.463531	−	0.886081i	$-0.653418\pi$
$983$	−29.0660	−0.927061	−0.463531	+	0.886081i	$0.653418\pi$
$984$	0	0
$985$	−16.8950	−0.538318
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.83296	0.0582847
$990$	0	0
$991$	−18.0081	−0.572047	−0.286024	−	0.958223i	$-0.592334\pi$
$991$	−18.0081	−0.572047	−0.286024	+	0.958223i	$0.592334\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	37.7910	1.19806
$996$	0	0
$997$	−13.7841	−0.436547	−0.218274	−	0.975888i	$-0.570043\pi$
$997$	−13.7841	−0.436547	−0.218274	+	0.975888i	$0.570043\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.i.1.3		9
3.2	odd	2		2004.2.a.c.1.7	✓	9
12.11	even	2		8016.2.a.bc.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.c.1.7	✓	9	3.2	odd	2
6012.2.a.i.1.3		9	1.1	even	1	trivial
8016.2.a.bc.1.7		9	12.11	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.94172 q^{5} -2.78729 q^{7} +O(q^{10})\)	\(q-2.94172 q^{5} -2.78729 q^{7} -3.89000 q^{11} -0.470621 q^{13} -4.98914 q^{17} -1.49771 q^{19} -0.667875 q^{23} +3.65370 q^{25} -0.0707310 q^{29} -3.13165 q^{31} +8.19942 q^{35} -1.64481 q^{37} -8.08147 q^{41} -2.74446 q^{43} -4.54043 q^{47} +0.768985 q^{49} -12.1695 q^{53} +11.4433 q^{55} -11.9713 q^{59} +2.07730 q^{61} +1.38443 q^{65} +0.754501 q^{67} -1.65397 q^{71} -3.00915 q^{73} +10.8426 q^{77} -5.37256 q^{79} +11.1661 q^{83} +14.6766 q^{85} +9.06894 q^{89} +1.31176 q^{91} +4.40582 q^{95} +2.36503 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{5} + 2 q^{7}+O(q^{10}) \) 9 * q - q^5 + 2 * q^7	\( 9 q - q^{5} + 2 q^{7} + 9 q^{11} + 10 q^{13} - 7 q^{17} - 2 q^{19} + 3 q^{23} + 18 q^{25} - 5 q^{29} + 12 q^{31} + 6 q^{35} + 15 q^{37} - 14 q^{41} + 6 q^{43} + 3 q^{47} + 27 q^{49} - 9 q^{53} + 19 q^{55} + 9 q^{59} + 30 q^{61} - 28 q^{65} + 16 q^{67} + 3 q^{71} + 32 q^{73} - 18 q^{77} + 24 q^{79} + 3 q^{83} + 37 q^{85} - 46 q^{89} + 33 q^{91} - 11 q^{95} + 43 q^{97}+O(q^{100}) \) 9 * q - q^5 + 2 * q^7 + 9 * q^11 + 10 * q^13 - 7 * q^17 - 2 * q^19 + 3 * q^23 + 18 * q^25 - 5 * q^29 + 12 * q^31 + 6 * q^35 + 15 * q^37 - 14 * q^41 + 6 * q^43 + 3 * q^47 + 27 * q^49 - 9 * q^53 + 19 * q^55 + 9 * q^59 + 30 * q^61 - 28 * q^65 + 16 * q^67 + 3 * q^71 + 32 * q^73 - 18 * q^77 + 24 * q^79 + 3 * q^83 + 37 * q^85 - 46 * q^89 + 33 * q^91 - 11 * q^95 + 43 * q^97

Introduction

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