LMFDB - Embedded newform 6012.2.a.e.1.2

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:	$1$
Dimension:	$5$
Coefficient field:	5.5.149169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - 3x^{2} + 4x + 1 $ x^5 - 6x^3 - 3x^2 + 4*x + 1
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			1.2
Root			$2.54970$ 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)

$f(q)$	$=$	$q-0.951283 q^{5} -1.28171 q^{7} +O(q^{10})$	$q-0.951283 q^{5} -1.28171 q^{7} +0.951283 q^{11} -2.82707 q^{13} -1.31500 q^{17} +5.94020 q^{19} +2.56927 q^{23} -4.09506 q^{25} +2.48910 q^{29} +0.984571 q^{31} +1.21927 q^{35} -7.81553 q^{37} +7.41944 q^{41} +7.65633 q^{43} -10.1096 q^{47} -5.35721 q^{49} +3.23734 q^{53} -0.904940 q^{55} +10.3093 q^{59} -11.8766 q^{61} +2.68935 q^{65} +8.14745 q^{67} -1.71178 q^{71} -0.502340 q^{73} -1.21927 q^{77} -14.6747 q^{79} +11.3212 q^{83} +1.25094 q^{85} -6.15777 q^{89} +3.62349 q^{91} -5.65081 q^{95} +4.75139 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{5} - 2 q^{7}+O(q^{10}) $ 5 * q + 3 * q^5 - 2 * q^7	$ 5 q + 3 q^{5} - 2 q^{7} - 3 q^{11} - 4 q^{13} + 7 q^{17} - 2 q^{19} - 13 q^{23} - 2 q^{25} + 3 q^{29} - 12 q^{31} - 10 q^{35} - 7 q^{37} + 16 q^{41} - q^{47} - 17 q^{49} - 3 q^{53} - 23 q^{55} - q^{59} - 22 q^{61} + 20 q^{65} + 2 q^{67} - 9 q^{71} - 28 q^{73} + 10 q^{77} - 28 q^{79} - 7 q^{83} - 11 q^{85} + 30 q^{89} - 13 q^{91} - 3 q^{95} - 33 q^{97}+O(q^{100}) $ 5 * q + 3 * q^5 - 2 * q^7 - 3 * q^11 - 4 * q^13 + 7 * q^17 - 2 * q^19 - 13 * q^23 - 2 * q^25 + 3 * q^29 - 12 * q^31 - 10 * q^35 - 7 * q^37 + 16 * q^41 - q^47 - 17 * q^49 - 3 * q^53 - 23 * q^55 - q^59 - 22 * q^61 + 20 * q^65 + 2 * q^67 - 9 * q^71 - 28 * q^73 + 10 * q^77 - 28 * q^79 - 7 * q^83 - 11 * q^85 + 30 * q^89 - 13 * q^91 - 3 * q^95 - 33 * 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
$5$	−0.951283	−0.425427	−0.212713	−	0.977115i	$-0.568230\pi$
$5$	−0.951283	−0.425427	−0.212713	+	0.977115i	$0.568230\pi$
$6$	0	0
$7$	−1.28171	−0.484442	−0.242221	−	0.970221i	$-0.577876\pi$
$7$	−1.28171	−0.484442	−0.242221	+	0.970221i	$0.577876\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.951283	0.286823	0.143411	−	0.989663i	$-0.454193\pi$
$11$	0.951283	0.286823	0.143411	+	0.989663i	$0.454193\pi$
$12$	0	0
$13$	−2.82707	−0.784088	−0.392044	−	0.919946i	$-0.628232\pi$
$13$	−2.82707	−0.784088	−0.392044	+	0.919946i	$0.628232\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.31500	−0.318935	−0.159467	−	0.987203i	$-0.550978\pi$
$17$	−1.31500	−0.318935	−0.159467	+	0.987203i	$0.550978\pi$
$18$	0	0
$19$	5.94020	1.36278	0.681388	−	0.731923i	$-0.261377\pi$
$19$	5.94020	1.36278	0.681388	+	0.731923i	$0.261377\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.56927	0.535729	0.267865	−	0.963457i	$-0.413682\pi$
$23$	2.56927	0.535729	0.267865	+	0.963457i	$0.413682\pi$
$24$	0	0
$25$	−4.09506	−0.819012
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.48910	0.462214	0.231107	−	0.972928i	$-0.425765\pi$
$29$	2.48910	0.462214	0.231107	+	0.972928i	$0.425765\pi$
$30$	0	0
$31$	0.984571	0.176834	0.0884171	−	0.996084i	$-0.471819\pi$
$31$	0.984571	0.176834	0.0884171	+	0.996084i	$0.471819\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.21927	0.206095
$36$	0	0
$37$	−7.81553	−1.28487	−0.642433	−	0.766342i	$-0.722075\pi$
$37$	−7.81553	−1.28487	−0.642433	+	0.766342i	$0.722075\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.41944	1.15872	0.579361	−	0.815071i	$-0.303302\pi$
$41$	7.41944	1.15872	0.579361	+	0.815071i	$0.303302\pi$
$42$	0	0
$43$	7.65633	1.16758	0.583789	−	0.811905i	$-0.301569\pi$
$43$	7.65633	1.16758	0.583789	+	0.811905i	$0.301569\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.1096	−1.47464	−0.737318	−	0.675546i	$-0.763908\pi$
$47$	−10.1096	−1.47464	−0.737318	+	0.675546i	$0.763908\pi$
$48$	0	0
$49$	−5.35721	−0.765316
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.23734	0.444683	0.222342	−	0.974969i	$-0.428630\pi$
$53$	3.23734	0.444683	0.222342	+	0.974969i	$0.428630\pi$
$54$	0	0
$55$	−0.904940	−0.122022
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.3093	1.34216	0.671078	−	0.741386i	$-0.265831\pi$
$59$	10.3093	1.34216	0.671078	+	0.741386i	$0.265831\pi$
$60$	0	0
$61$	−11.8766	−1.52064	−0.760321	−	0.649547i	$-0.774958\pi$
$61$	−11.8766	−1.52064	−0.760321	+	0.649547i	$0.774958\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.68935	0.333572
$66$	0	0
$67$	8.14745	0.995370	0.497685	−	0.867358i	$-0.334184\pi$
$67$	8.14745	0.995370	0.497685	+	0.867358i	$0.334184\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−1.71178	−0.203151	−0.101575	−	0.994828i	$-0.532388\pi$
$71$	−1.71178	−0.203151	−0.101575	+	0.994828i	$0.532388\pi$
$72$	0	0
$73$	−0.502340	−0.0587944	−0.0293972	−	0.999568i	$-0.509359\pi$
$73$	−0.502340	−0.0587944	−0.0293972	+	0.999568i	$0.509359\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.21927	−0.138949
$78$	0	0
$79$	−14.6747	−1.65104	−0.825518	−	0.564376i	$-0.809117\pi$
$79$	−14.6747	−1.65104	−0.825518	+	0.564376i	$0.809117\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11.3212	1.24266	0.621331	−	0.783548i	$-0.286592\pi$
$83$	11.3212	1.24266	0.621331	+	0.783548i	$0.286592\pi$
$84$	0	0
$85$	1.25094	0.135683
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.15777	−0.652722	−0.326361	−	0.945245i	$-0.605823\pi$
$89$	−6.15777	−0.652722	−0.326361	+	0.945245i	$0.605823\pi$
$90$	0	0
$91$	3.62349	0.379845
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.65081	−0.579761
$96$	0	0
$97$	4.75139	0.482430	0.241215	−	0.970472i	$-0.422454\pi$
$97$	4.75139	0.482430	0.241215	+	0.970472i	$0.422454\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.52463	0.251210	0.125605	−	0.992080i	$-0.459913\pi$
$101$	2.52463	0.251210	0.125605	+	0.992080i	$0.459913\pi$
$102$	0	0
$103$	−10.1843	−1.00349	−0.501743	−	0.865016i	$-0.667308\pi$
$103$	−10.1843	−1.00349	−0.501743	+	0.865016i	$0.667308\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.1161	−1.26798	−0.633991	−	0.773341i	$-0.718584\pi$
$107$	−13.1161	−1.26798	−0.633991	+	0.773341i	$0.718584\pi$
$108$	0	0
$109$	−0.209627	−0.0200786	−0.0100393	−	0.999950i	$-0.503196\pi$
$109$	−0.209627	−0.0200786	−0.0100393	+	0.999950i	$0.503196\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−0.554507	−0.0521636	−0.0260818	−	0.999660i	$-0.508303\pi$
$113$	−0.554507	−0.0521636	−0.0260818	+	0.999660i	$0.508303\pi$
$114$	0	0
$115$	−2.44410	−0.227914
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.68546	0.154505
$120$	0	0
$121$	−10.0951	−0.917733
$122$	0	0
$123$	0	0
$124$	0	0
$125$	8.65198	0.773857
$126$	0	0
$127$	−17.5210	−1.55474	−0.777370	−	0.629044i	$-0.783447\pi$
$127$	−17.5210	−1.55474	−0.777370	+	0.629044i	$0.783447\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.30379	−0.550765	−0.275382	−	0.961335i	$-0.588804\pi$
$131$	−6.30379	−0.550765	−0.275382	+	0.961335i	$0.588804\pi$
$132$	0	0
$133$	−7.61364	−0.660186
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.12238	0.523070	0.261535	−	0.965194i	$-0.415771\pi$
$137$	6.12238	0.523070	0.261535	+	0.965194i	$0.415771\pi$
$138$	0	0
$139$	−7.59742	−0.644405	−0.322203	−	0.946671i	$-0.604423\pi$
$139$	−7.59742	−0.644405	−0.322203	+	0.946671i	$0.604423\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.68935	−0.224894
$144$	0	0
$145$	−2.36784	−0.196638
$146$	0	0
$147$	0	0
$148$	0	0
$149$	17.8638	1.46346	0.731730	−	0.681594i	$-0.238713\pi$
$149$	17.8638	1.46346	0.731730	+	0.681594i	$0.238713\pi$
$150$	0	0
$151$	−4.74233	−0.385925	−0.192962	−	0.981206i	$-0.561810\pi$
$151$	−4.74233	−0.385925	−0.192962	+	0.981206i	$0.561810\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.936606	−0.0752300
$156$	0	0
$157$	−9.54024	−0.761394	−0.380697	−	0.924700i	$-0.624316\pi$
$157$	−9.54024	−0.761394	−0.380697	+	0.924700i	$0.624316\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.29306	−0.259530
$162$	0	0
$163$	−1.89939	−0.148771	−0.0743857	−	0.997230i	$-0.523700\pi$
$163$	−1.89939	−0.148771	−0.0743857	+	0.997230i	$0.523700\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−5.00767	−0.385206
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.8969	−1.05656	−0.528280	−	0.849070i	$-0.677163\pi$
$173$	−13.8969	−1.05656	−0.528280	+	0.849070i	$0.677163\pi$
$174$	0	0
$175$	5.24869	0.396764
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.87857	0.439385	0.219692	−	0.975569i	$-0.429495\pi$
$179$	5.87857	0.439385	0.219692	+	0.975569i	$0.429495\pi$
$180$	0	0
$181$	−20.6955	−1.53828	−0.769140	−	0.639080i	$-0.779315\pi$
$181$	−20.6955	−1.53828	−0.769140	+	0.639080i	$0.779315\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.43478	0.546616
$186$	0	0
$187$	−1.25094	−0.0914777
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.8727	1.00380	0.501898	−	0.864927i	$-0.332635\pi$
$191$	13.8727	1.00380	0.501898	+	0.864927i	$0.332635\pi$
$192$	0	0
$193$	−0.344963	−0.0248310	−0.0124155	−	0.999923i	$-0.503952\pi$
$193$	−0.344963	−0.0248310	−0.0124155	+	0.999923i	$0.503952\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−24.1397	−1.71988	−0.859942	−	0.510392i	$-0.829500\pi$
$197$	−24.1397	−1.71988	−0.859942	+	0.510392i	$0.829500\pi$
$198$	0	0
$199$	−3.97139	−0.281524	−0.140762	−	0.990043i	$-0.544955\pi$
$199$	−3.97139	−0.281524	−0.140762	+	0.990043i	$0.544955\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.19031	−0.223916
$204$	0	0
$205$	−7.05799	−0.492951
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.65081	0.390875
$210$	0	0
$211$	−21.3487	−1.46970	−0.734852	−	0.678227i	$-0.762749\pi$
$211$	−21.3487	−1.46970	−0.734852	+	0.678227i	$0.762749\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−7.28334	−0.496719
$216$	0	0
$217$	−1.26194	−0.0856660
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.71760	0.250073
$222$	0	0
$223$	11.6936	0.783063	0.391531	−	0.920165i	$-0.371945\pi$
$223$	11.6936	0.783063	0.391531	+	0.920165i	$0.371945\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.3838	−0.954685	−0.477342	−	0.878717i	$-0.658400\pi$
$227$	−14.3838	−0.954685	−0.477342	+	0.878717i	$0.658400\pi$
$228$	0	0
$229$	9.30406	0.614830	0.307415	−	0.951576i	$-0.400536\pi$
$229$	9.30406	0.614830	0.307415	+	0.951576i	$0.400536\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.32737	0.0869591	0.0434795	−	0.999054i	$-0.486156\pi$
$233$	1.32737	0.0869591	0.0434795	+	0.999054i	$0.486156\pi$
$234$	0	0
$235$	9.61709	0.627350
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6.67440	0.431731	0.215865	−	0.976423i	$-0.430743\pi$
$239$	6.67440	0.431731	0.215865	+	0.976423i	$0.430743\pi$
$240$	0	0
$241$	21.3838	1.37745	0.688726	−	0.725021i	$-0.258170\pi$
$241$	21.3838	1.37745	0.688726	+	0.725021i	$0.258170\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5.09622	0.325586
$246$	0	0
$247$	−16.7934	−1.06854
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−27.1243	−1.71207	−0.856034	−	0.516919i	$-0.827079\pi$
$251$	−27.1243	−1.71207	−0.856034	+	0.516919i	$0.827079\pi$
$252$	0	0
$253$	2.44410	0.153659
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.2287	−0.700429	−0.350214	−	0.936670i	$-0.613891\pi$
$257$	−11.2287	−0.700429	−0.350214	+	0.936670i	$0.613891\pi$
$258$	0	0
$259$	10.0173	0.622443
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.5610	−1.57616	−0.788079	−	0.615574i	$-0.788924\pi$
$263$	−25.5610	−1.57616	−0.788079	+	0.615574i	$0.788924\pi$
$264$	0	0
$265$	−3.07963	−0.189180
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.1581	−1.22906	−0.614532	−	0.788892i	$-0.710655\pi$
$269$	−20.1581	−1.22906	−0.614532	+	0.788892i	$0.710655\pi$
$270$	0	0
$271$	−1.90126	−0.115493	−0.0577467	−	0.998331i	$-0.518392\pi$
$271$	−1.90126	−0.115493	−0.0577467	+	0.998331i	$0.518392\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.89556	−0.234911
$276$	0	0
$277$	31.1953	1.87435	0.937173	−	0.348865i	$-0.113433\pi$
$277$	31.1953	1.87435	0.937173	+	0.348865i	$0.113433\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	31.9013	1.90307	0.951535	−	0.307540i	$-0.0995058\pi$
$281$	31.9013	1.90307	0.951535	+	0.307540i	$0.0995058\pi$
$282$	0	0
$283$	−12.2935	−0.730775	−0.365387	−	0.930856i	$-0.619063\pi$
$283$	−12.2935	−0.730775	−0.365387	+	0.930856i	$0.619063\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.50960	−0.561334
$288$	0	0
$289$	−15.2708	−0.898281
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.78588	0.279594	0.139797	−	0.990180i	$-0.455355\pi$
$293$	4.78588	0.279594	0.139797	+	0.990180i	$0.455355\pi$
$294$	0	0
$295$	−9.80707	−0.570990
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7.26350	−0.420059
$300$	0	0
$301$	−9.81322	−0.565624
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.2980	0.646922
$306$	0	0
$307$	−19.7957	−1.12980	−0.564900	−	0.825159i	$-0.691085\pi$
$307$	−19.7957	−1.12980	−0.564900	+	0.825159i	$0.691085\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	25.3327	1.43648	0.718242	−	0.695794i	$-0.244947\pi$
$311$	25.3327	1.43648	0.718242	+	0.695794i	$0.244947\pi$
$312$	0	0
$313$	2.86377	0.161870	0.0809349	−	0.996719i	$-0.474209\pi$
$313$	2.86377	0.161870	0.0809349	+	0.996719i	$0.474209\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4.73056	−0.265695	−0.132847	−	0.991137i	$-0.542412\pi$
$317$	−4.73056	−0.265695	−0.132847	+	0.991137i	$0.542412\pi$
$318$	0	0
$319$	2.36784	0.132573
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−7.81137	−0.434636
$324$	0	0
$325$	11.5770	0.642178
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.9576	0.714376
$330$	0	0
$331$	−15.1256	−0.831379	−0.415690	−	0.909506i	$-0.636460\pi$
$331$	−15.1256	−0.831379	−0.415690	+	0.909506i	$0.636460\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.75054	−0.423457
$336$	0	0
$337$	−4.31419	−0.235009	−0.117504	−	0.993072i	$-0.537489\pi$
$337$	−4.31419	−0.235009	−0.117504	+	0.993072i	$0.537489\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.936606	0.0507201
$342$	0	0
$343$	15.8384	0.855193
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−23.0258	−1.23609	−0.618045	−	0.786143i	$-0.712075\pi$
$347$	−23.0258	−1.23609	−0.618045	+	0.786143i	$0.712075\pi$
$348$	0	0
$349$	16.1595	0.864998	0.432499	−	0.901634i	$-0.357632\pi$
$349$	16.1595	0.864998	0.432499	+	0.901634i	$0.357632\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−29.4163	−1.56567	−0.782835	−	0.622229i	$-0.786227\pi$
$353$	−29.4163	−1.56567	−0.782835	+	0.622229i	$0.786227\pi$
$354$	0	0
$355$	1.62839	0.0864258
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−27.2507	−1.43824	−0.719118	−	0.694888i	$-0.755454\pi$
$359$	−27.2507	−1.43824	−0.719118	+	0.694888i	$0.755454\pi$
$360$	0	0
$361$	16.2860	0.857157
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.477867	0.0250127
$366$	0	0
$367$	2.46015	0.128419	0.0642095	−	0.997936i	$-0.479547\pi$
$367$	2.46015	0.128419	0.0642095	+	0.997936i	$0.479547\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4.14935	−0.215423
$372$	0	0
$373$	−2.75950	−0.142881	−0.0714407	−	0.997445i	$-0.522760\pi$
$373$	−2.75950	−0.142881	−0.0714407	+	0.997445i	$0.522760\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.03685	−0.362416
$378$	0	0
$379$	29.6373	1.52237	0.761183	−	0.648537i	$-0.224619\pi$
$379$	29.6373	1.52237	0.761183	+	0.648537i	$0.224619\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.3376	0.988104	0.494052	−	0.869432i	$-0.335515\pi$
$383$	19.3376	0.988104	0.494052	+	0.869432i	$0.335515\pi$
$384$	0	0
$385$	1.15987	0.0591127
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0.877537	0.0444929	0.0222465	−	0.999753i	$-0.492918\pi$
$389$	0.877537	0.0444929	0.0222465	+	0.999753i	$0.492918\pi$
$390$	0	0
$391$	−3.37859	−0.170863
$392$	0	0
$393$	0	0
$394$	0	0
$395$	13.9598	0.702395
$396$	0	0
$397$	−24.0236	−1.20571	−0.602855	−	0.797851i	$-0.705970\pi$
$397$	−24.0236	−1.20571	−0.602855	+	0.797851i	$0.705970\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.0892	−1.10308	−0.551541	−	0.834148i	$-0.685960\pi$
$401$	−22.0892	−1.10308	−0.551541	+	0.834148i	$0.685960\pi$
$402$	0	0
$403$	−2.78345	−0.138654
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−7.43478	−0.368529
$408$	0	0
$409$	9.50296	0.469891	0.234946	−	0.972009i	$-0.424509\pi$
$409$	9.50296	0.469891	0.234946	+	0.972009i	$0.424509\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13.2136	−0.650197
$414$	0	0
$415$	−10.7697	−0.528662
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−24.0099	−1.17296	−0.586479	−	0.809964i	$-0.699486\pi$
$419$	−24.0099	−1.17296	−0.586479	+	0.809964i	$0.699486\pi$
$420$	0	0
$421$	39.8756	1.94342	0.971708	−	0.236187i	$-0.0758976\pi$
$421$	39.8756	1.94342	0.971708	+	0.236187i	$0.0758976\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.38501	0.261211
$426$	0	0
$427$	15.2224	0.736663
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.6800	−0.899785	−0.449893	−	0.893083i	$-0.648538\pi$
$431$	−18.6800	−0.899785	−0.449893	+	0.893083i	$0.648538\pi$
$432$	0	0
$433$	−19.5622	−0.940099	−0.470049	−	0.882640i	$-0.655764\pi$
$433$	−19.5622	−0.940099	−0.470049	+	0.882640i	$0.655764\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	15.2620	0.730079
$438$	0	0
$439$	11.5272	0.550163	0.275082	−	0.961421i	$-0.411295\pi$
$439$	11.5272	0.550163	0.275082	+	0.961421i	$0.411295\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−37.3814	−1.77604	−0.888022	−	0.459800i	$-0.847921\pi$
$443$	−37.3814	−1.77604	−0.888022	+	0.459800i	$0.847921\pi$
$444$	0	0
$445$	5.85778	0.277686
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.0879786	0.00415197	0.00207598	−	0.999998i	$-0.499339\pi$
$449$	0.0879786	0.00415197	0.00207598	+	0.999998i	$0.499339\pi$
$450$	0	0
$451$	7.05799	0.332348
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.44697	−0.161596
$456$	0	0
$457$	−35.8125	−1.67524	−0.837619	−	0.546254i	$-0.816053\pi$
$457$	−35.8125	−1.67524	−0.837619	+	0.546254i	$0.816053\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−19.4523	−0.905985	−0.452993	−	0.891514i	$-0.649644\pi$
$461$	−19.4523	−0.905985	−0.452993	+	0.891514i	$0.649644\pi$
$462$	0	0
$463$	−7.12623	−0.331184	−0.165592	−	0.986194i	$-0.552953\pi$
$463$	−7.12623	−0.331184	−0.165592	+	0.986194i	$0.552953\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.30038	−0.0601746	−0.0300873	−	0.999547i	$-0.509579\pi$
$467$	−1.30038	−0.0601746	−0.0300873	+	0.999547i	$0.509579\pi$
$468$	0	0
$469$	−10.4427	−0.482199
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.28334	0.334888
$474$	0	0
$475$	−24.3255	−1.11613
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−13.3269	−0.608921	−0.304461	−	0.952525i	$-0.598476\pi$
$479$	−13.3269	−0.608921	−0.304461	+	0.952525i	$0.598476\pi$
$480$	0	0
$481$	22.0951	1.00745
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.51991	−0.205239
$486$	0	0
$487$	−9.08890	−0.411858	−0.205929	−	0.978567i	$-0.566022\pi$
$487$	−9.08890	−0.411858	−0.205929	+	0.978567i	$0.566022\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	28.7483	1.29739	0.648697	−	0.761047i	$-0.275314\pi$
$491$	28.7483	1.29739	0.648697	+	0.761047i	$0.275314\pi$
$492$	0	0
$493$	−3.27316	−0.147416
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.19401	0.0984148
$498$	0	0
$499$	19.4053	0.868702	0.434351	−	0.900744i	$-0.356978\pi$
$499$	19.4053	0.868702	0.434351	+	0.900744i	$0.356978\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	11.9825	0.534271	0.267136	−	0.963659i	$-0.413923\pi$
$503$	11.9825	0.534271	0.267136	+	0.963659i	$0.413923\pi$
$504$	0	0
$505$	−2.40164	−0.106871
$506$	0	0
$507$	0	0
$508$	0	0
$509$	7.58018	0.335986	0.167993	−	0.985788i	$-0.446271\pi$
$509$	7.58018	0.335986	0.167993	+	0.985788i	$0.446271\pi$
$510$	0	0
$511$	0.643856	0.0284825
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9.68814	0.426910
$516$	0	0
$517$	−9.61709	−0.422959
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.31483	0.408090	0.204045	−	0.978962i	$-0.434591\pi$
$521$	9.31483	0.408090	0.204045	+	0.978962i	$0.434591\pi$
$522$	0	0
$523$	8.16544	0.357050	0.178525	−	0.983935i	$-0.442867\pi$
$523$	8.16544	0.357050	0.178525	+	0.983935i	$0.442867\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.29471	−0.0563986
$528$	0	0
$529$	−16.3989	−0.712994
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−20.9753	−0.908540
$534$	0	0
$535$	12.4771	0.539433
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.09622	−0.219510
$540$	0	0
$541$	31.2426	1.34322	0.671611	−	0.740904i	$-0.265603\pi$
$541$	31.2426	1.34322	0.671611	+	0.740904i	$0.265603\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.199414	0.00854198
$546$	0	0
$547$	−23.9718	−1.02496	−0.512481	−	0.858699i	$-0.671273\pi$
$547$	−23.9718	−1.02496	−0.512481	+	0.858699i	$0.671273\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	14.7857	0.629893
$552$	0	0
$553$	18.8088	0.799831
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.3707	0.608907	0.304454	−	0.952527i	$-0.401526\pi$
$557$	14.3707	0.608907	0.304454	+	0.952527i	$0.401526\pi$
$558$	0	0
$559$	−21.6450	−0.915485
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−37.5575	−1.58286	−0.791431	−	0.611259i	$-0.790663\pi$
$563$	−37.5575	−1.58286	−0.791431	+	0.611259i	$0.790663\pi$
$564$	0	0
$565$	0.527493	0.0221918
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.9670	0.878983	0.439491	−	0.898247i	$-0.355159\pi$
$569$	20.9670	0.878983	0.439491	+	0.898247i	$0.355159\pi$
$570$	0	0
$571$	17.0570	0.713815	0.356907	−	0.934140i	$-0.383831\pi$
$571$	17.0570	0.713815	0.356907	+	0.934140i	$0.383831\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−10.5213	−0.438769
$576$	0	0
$577$	20.0594	0.835082	0.417541	−	0.908658i	$-0.362892\pi$
$577$	20.0594	0.835082	0.417541	+	0.908658i	$0.362892\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−14.5105	−0.601998
$582$	0	0
$583$	3.07963	0.127545
$584$	0	0
$585$	0	0
$586$	0	0
$587$	42.1301	1.73890	0.869449	−	0.494023i	$-0.164474\pi$
$587$	42.1301	1.73890	0.869449	+	0.494023i	$0.164474\pi$
$588$	0	0
$589$	5.84855	0.240985
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.4857	1.25190	0.625950	−	0.779863i	$-0.284712\pi$
$593$	30.4857	1.25190	0.625950	+	0.779863i	$0.284712\pi$
$594$	0	0
$595$	−1.60335	−0.0657308
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9.37998	0.383256	0.191628	−	0.981468i	$-0.438623\pi$
$599$	9.37998	0.383256	0.191628	+	0.981468i	$0.438623\pi$
$600$	0	0
$601$	−11.7008	−0.477285	−0.238642	−	0.971108i	$-0.576702\pi$
$601$	−11.7008	−0.477285	−0.238642	+	0.971108i	$0.576702\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	9.60326	0.390428
$606$	0	0
$607$	−12.2337	−0.496549	−0.248275	−	0.968690i	$-0.579864\pi$
$607$	−12.2337	−0.496549	−0.248275	+	0.968690i	$0.579864\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	28.5805	1.15624
$612$	0	0
$613$	−44.6472	−1.80328	−0.901642	−	0.432483i	$-0.857637\pi$
$613$	−44.6472	−1.80328	−0.901642	+	0.432483i	$0.857637\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−36.7226	−1.47840	−0.739198	−	0.673488i	$-0.764795\pi$
$617$	−36.7226	−1.47840	−0.739198	+	0.673488i	$0.764795\pi$
$618$	0	0
$619$	5.35966	0.215423	0.107711	−	0.994182i	$-0.465648\pi$
$619$	5.35966	0.215423	0.107711	+	0.994182i	$0.465648\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.89250	0.316206
$624$	0	0
$625$	12.2448	0.489793
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.2774	0.409788
$630$	0	0
$631$	−7.52358	−0.299509	−0.149754	−	0.988723i	$-0.547848\pi$
$631$	−7.52358	−0.299509	−0.149754	+	0.988723i	$0.547848\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	16.6675	0.661428
$636$	0	0
$637$	15.1452	0.600075
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.3947	0.608056	0.304028	−	0.952663i	$-0.401668\pi$
$641$	15.3947	0.608056	0.304028	+	0.952663i	$0.401668\pi$
$642$	0	0
$643$	40.2115	1.58579	0.792893	−	0.609361i	$-0.208574\pi$
$643$	40.2115	1.58579	0.792893	+	0.609361i	$0.208574\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.1639	1.50038	0.750189	−	0.661224i	$-0.229963\pi$
$647$	38.1639	1.50038	0.750189	+	0.661224i	$0.229963\pi$
$648$	0	0
$649$	9.80707	0.384961
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.62256	0.0634956	0.0317478	−	0.999496i	$-0.489893\pi$
$653$	1.62256	0.0634956	0.0317478	+	0.999496i	$0.489893\pi$
$654$	0	0
$655$	5.99669	0.234310
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12.5684	0.489595	0.244798	−	0.969574i	$-0.421278\pi$
$659$	12.5684	0.489595	0.244798	+	0.969574i	$0.421278\pi$
$660$	0	0
$661$	−35.0118	−1.36180	−0.680901	−	0.732375i	$-0.738412\pi$
$661$	−35.0118	−1.36180	−0.680901	+	0.732375i	$0.738412\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.24273	0.280861
$666$	0	0
$667$	6.39515	0.247621
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.2980	−0.436155
$672$	0	0
$673$	15.1295	0.583199	0.291599	−	0.956541i	$-0.405813\pi$
$673$	15.1295	0.583199	0.291599	+	0.956541i	$0.405813\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	33.9274	1.30393	0.651967	−	0.758247i	$-0.273944\pi$
$677$	33.9274	1.30393	0.651967	+	0.758247i	$0.273944\pi$
$678$	0	0
$679$	−6.08992	−0.233710
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−46.4231	−1.77633	−0.888164	−	0.459526i	$-0.848019\pi$
$683$	−46.4231	−1.77633	−0.888164	+	0.459526i	$0.848019\pi$
$684$	0	0
$685$	−5.82412	−0.222528
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−9.15220	−0.348671
$690$	0	0
$691$	17.4612	0.664256	0.332128	−	0.943234i	$-0.392233\pi$
$691$	17.4612	0.664256	0.332128	+	0.943234i	$0.392233\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.22730	0.274147
$696$	0	0
$697$	−9.75657	−0.369557
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−33.9740	−1.28318	−0.641591	−	0.767047i	$-0.721725\pi$
$701$	−33.9740	−1.28318	−0.641591	+	0.767047i	$0.721725\pi$
$702$	0	0
$703$	−46.4258	−1.75098
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.23585	−0.121697
$708$	0	0
$709$	−23.0747	−0.866587	−0.433293	−	0.901253i	$-0.642649\pi$
$709$	−23.0747	−0.866587	−0.433293	+	0.901253i	$0.642649\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.52963	0.0947352
$714$	0	0
$715$	2.55833	0.0956761
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.8956	−0.779273	−0.389637	−	0.920969i	$-0.627399\pi$
$719$	−20.8956	−0.779273	−0.389637	+	0.920969i	$0.627399\pi$
$720$	0	0
$721$	13.0533	0.486131
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−10.1930	−0.378558
$726$	0	0
$727$	8.15415	0.302421	0.151210	−	0.988502i	$-0.451683\pi$
$727$	8.15415	0.302421	0.151210	+	0.988502i	$0.451683\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−10.0681	−0.372381
$732$	0	0
$733$	−19.2185	−0.709853	−0.354927	−	0.934894i	$-0.615494\pi$
$733$	−19.2185	−0.709853	−0.354927	+	0.934894i	$0.615494\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.75054	0.285495
$738$	0	0
$739$	11.4850	0.422482	0.211241	−	0.977434i	$-0.432250\pi$
$739$	11.4850	0.422482	0.211241	+	0.977434i	$0.432250\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−44.3551	−1.62723	−0.813615	−	0.581404i	$-0.802504\pi$
$743$	−44.3551	−1.62723	−0.813615	+	0.581404i	$0.802504\pi$
$744$	0	0
$745$	−16.9935	−0.622595
$746$	0	0
$747$	0	0
$748$	0	0
$749$	16.8111	0.614264
$750$	0	0
$751$	31.2517	1.14039	0.570195	−	0.821509i	$-0.306868\pi$
$751$	31.2517	1.14039	0.570195	+	0.821509i	$0.306868\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4.51130	0.164183
$756$	0	0
$757$	33.1738	1.20572	0.602862	−	0.797846i	$-0.294027\pi$
$757$	33.1738	1.20572	0.602862	+	0.797846i	$0.294027\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	50.1683	1.81860	0.909299	−	0.416143i	$-0.136618\pi$
$761$	50.1683	1.81860	0.909299	+	0.416143i	$0.136618\pi$
$762$	0	0
$763$	0.268682	0.00972692
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−29.1451	−1.05237
$768$	0	0
$769$	39.9272	1.43981	0.719905	−	0.694073i	$-0.244185\pi$
$769$	39.9272	1.43981	0.719905	+	0.694073i	$0.244185\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.0108	0.791675	0.395838	−	0.918321i	$-0.370454\pi$
$773$	22.0108	0.791675	0.395838	+	0.918321i	$0.370454\pi$
$774$	0	0
$775$	−4.03188	−0.144829
$776$	0	0
$777$	0	0
$778$	0	0
$779$	44.0730	1.57908
$780$	0	0
$781$	−1.62839	−0.0582682
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.07547	0.323918
$786$	0	0
$787$	−27.7640	−0.989681	−0.494840	−	0.868984i	$-0.664773\pi$
$787$	−27.7640	−0.989681	−0.494840	+	0.868984i	$0.664773\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.710719	0.0252702
$792$	0	0
$793$	33.5760	1.19232
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.03196	0.319928	0.159964	−	0.987123i	$-0.448862\pi$
$797$	9.03196	0.319928	0.159964	+	0.987123i	$0.448862\pi$
$798$	0	0
$799$	13.2941	0.470312
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−0.477867	−0.0168636
$804$	0	0
$805$	3.13264	0.110411
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−27.3478	−0.961496	−0.480748	−	0.876859i	$-0.659635\pi$
$809$	−27.3478	−0.961496	−0.480748	+	0.876859i	$0.659635\pi$
$810$	0	0
$811$	−42.6726	−1.49844	−0.749219	−	0.662323i	$-0.769571\pi$
$811$	−42.6726	−1.49844	−0.749219	+	0.662323i	$0.769571\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.80685	0.0632913
$816$	0	0
$817$	45.4801	1.59115
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.5232	−0.995468	−0.497734	−	0.867330i	$-0.665834\pi$
$821$	−28.5232	−0.995468	−0.497734	+	0.867330i	$0.665834\pi$
$822$	0	0
$823$	−4.05092	−0.141206	−0.0706031	−	0.997504i	$-0.522492\pi$
$823$	−4.05092	−0.141206	−0.0706031	+	0.997504i	$0.522492\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.63277	0.230644	0.115322	−	0.993328i	$-0.463210\pi$
$827$	6.63277	0.230644	0.115322	+	0.993328i	$0.463210\pi$
$828$	0	0
$829$	−7.97201	−0.276879	−0.138440	−	0.990371i	$-0.544209\pi$
$829$	−7.97201	−0.276879	−0.138440	+	0.990371i	$0.544209\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.04474	0.244086
$834$	0	0
$835$	0.951283	0.0329205
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.98285	−0.0684556	−0.0342278	−	0.999414i	$-0.510897\pi$
$839$	−1.98285	−0.0684556	−0.0342278	+	0.999414i	$0.510897\pi$
$840$	0	0
$841$	−22.8044	−0.786359
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.76372	0.163877
$846$	0	0
$847$	12.9390	0.444588
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−20.0802	−0.688340
$852$	0	0
$853$	−40.0040	−1.36971	−0.684856	−	0.728678i	$-0.740135\pi$
$853$	−40.0040	−1.36971	−0.684856	+	0.728678i	$0.740135\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.9386	1.36428	0.682138	−	0.731224i	$-0.261050\pi$
$857$	39.9386	1.36428	0.682138	+	0.731224i	$0.261050\pi$
$858$	0	0
$859$	−12.2669	−0.418541	−0.209271	−	0.977858i	$-0.567109\pi$
$859$	−12.2669	−0.418541	−0.209271	+	0.977858i	$0.567109\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	40.6235	1.38284	0.691419	−	0.722454i	$-0.256986\pi$
$863$	40.6235	1.38284	0.691419	+	0.722454i	$0.256986\pi$
$864$	0	0
$865$	13.2199	0.449489
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−13.9598	−0.473555
$870$	0	0
$871$	−23.0334	−0.780458
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−11.0894	−0.374889
$876$	0	0
$877$	11.9961	0.405080	0.202540	−	0.979274i	$-0.435080\pi$
$877$	11.9961	0.405080	0.202540	+	0.979274i	$0.435080\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−13.4368	−0.452697	−0.226348	−	0.974046i	$-0.572679\pi$
$881$	−13.4368	−0.452697	−0.226348	+	0.974046i	$0.572679\pi$
$882$	0	0
$883$	32.1980	1.08355	0.541775	−	0.840524i	$-0.317752\pi$
$883$	32.1980	1.08355	0.541775	+	0.840524i	$0.317752\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−13.6383	−0.457928	−0.228964	−	0.973435i	$-0.573534\pi$
$887$	−13.6383	−0.457928	−0.228964	+	0.973435i	$0.573534\pi$
$888$	0	0
$889$	22.4569	0.753182
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−60.0530	−2.00960
$894$	0	0
$895$	−5.59218	−0.186926
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.45069	0.0817352
$900$	0	0
$901$	−4.25711	−0.141825
$902$	0	0
$903$	0	0
$904$	0	0
$905$	19.6872	0.654426
$906$	0	0
$907$	15.0800	0.500723	0.250362	−	0.968152i	$-0.419450\pi$
$907$	15.0800	0.500723	0.250362	+	0.968152i	$0.419450\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	28.4366	0.942147	0.471074	−	0.882094i	$-0.343867\pi$
$911$	28.4366	0.942147	0.471074	+	0.882094i	$0.343867\pi$
$912$	0	0
$913$	10.7697	0.356424
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.07965	0.266814
$918$	0	0
$919$	−26.8458	−0.885562	−0.442781	−	0.896630i	$-0.646008\pi$
$919$	−26.8458	−0.885562	−0.442781	+	0.896630i	$0.646008\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.83932	0.159288
$924$	0	0
$925$	32.0051	1.05232
$926$	0	0
$927$	0	0
$928$	0	0
$929$	16.0592	0.526885	0.263442	−	0.964675i	$-0.415142\pi$
$929$	16.0592	0.526885	0.263442	+	0.964675i	$0.415142\pi$
$930$	0	0
$931$	−31.8229	−1.04295
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.19000	0.0389171
$936$	0	0
$937$	−21.3748	−0.698283	−0.349141	−	0.937070i	$-0.613527\pi$
$937$	−21.3748	−0.698283	−0.349141	+	0.937070i	$0.613527\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−2.32835	−0.0759020	−0.0379510	−	0.999280i	$-0.512083\pi$
$941$	−2.32835	−0.0759020	−0.0379510	+	0.999280i	$0.512083\pi$
$942$	0	0
$943$	19.0625	0.620761
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42.7080	−1.38782	−0.693912	−	0.720059i	$-0.744115\pi$
$947$	−42.7080	−1.38782	−0.693912	+	0.720059i	$0.744115\pi$
$948$	0	0
$949$	1.42015	0.0461000
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−40.9000	−1.32488	−0.662440	−	0.749115i	$-0.730479\pi$
$953$	−40.9000	−1.32488	−0.662440	+	0.749115i	$0.730479\pi$
$954$	0	0
$955$	−13.1969	−0.427042
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−7.84714	−0.253397
$960$	0	0
$961$	−30.0306	−0.968730
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0.328157	0.0105638
$966$	0	0
$967$	−54.1145	−1.74021	−0.870103	−	0.492870i	$-0.835948\pi$
$967$	−54.1145	−1.74021	−0.870103	+	0.492870i	$0.835948\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	53.6916	1.72305	0.861523	−	0.507719i	$-0.169511\pi$
$971$	53.6916	1.72305	0.861523	+	0.507719i	$0.169511\pi$
$972$	0	0
$973$	9.73772	0.312177
$974$	0	0
$975$	0	0
$976$	0	0
$977$	51.4253	1.64524	0.822621	−	0.568590i	$-0.192511\pi$
$977$	51.4253	1.64524	0.822621	+	0.568590i	$0.192511\pi$
$978$	0	0
$979$	−5.85778	−0.187216
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−27.2398	−0.868816	−0.434408	−	0.900716i	$-0.643042\pi$
$983$	−27.2398	−0.868816	−0.434408	+	0.900716i	$0.643042\pi$
$984$	0	0
$985$	22.9637	0.731685
$986$	0	0
$987$	0	0
$988$	0	0
$989$	19.6711	0.625506
$990$	0	0
$991$	−36.0301	−1.14453	−0.572267	−	0.820067i	$-0.693936\pi$
$991$	−36.0301	−1.14453	−0.572267	+	0.820067i	$0.693936\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.77791	0.119768
$996$	0	0
$997$	−1.57982	−0.0500334	−0.0250167	−	0.999687i	$-0.507964\pi$
$997$	−1.57982	−0.0500334	−0.0250167	+	0.999687i	$0.507964\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.e.1.2		5
3.2	odd	2		2004.2.a.a.1.4	✓	5
12.11	even	2		8016.2.a.t.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.a.1.4	✓	5	3.2	odd	2
6012.2.a.e.1.2		5	1.1	even	1	trivial
8016.2.a.t.1.4		5	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)

\(f(q)\)	\(=\)	\(q-0.951283 q^{5} -1.28171 q^{7} +O(q^{10})\)	\(q-0.951283 q^{5} -1.28171 q^{7} +0.951283 q^{11} -2.82707 q^{13} -1.31500 q^{17} +5.94020 q^{19} +2.56927 q^{23} -4.09506 q^{25} +2.48910 q^{29} +0.984571 q^{31} +1.21927 q^{35} -7.81553 q^{37} +7.41944 q^{41} +7.65633 q^{43} -10.1096 q^{47} -5.35721 q^{49} +3.23734 q^{53} -0.904940 q^{55} +10.3093 q^{59} -11.8766 q^{61} +2.68935 q^{65} +8.14745 q^{67} -1.71178 q^{71} -0.502340 q^{73} -1.21927 q^{77} -14.6747 q^{79} +11.3212 q^{83} +1.25094 q^{85} -6.15777 q^{89} +3.62349 q^{91} -5.65081 q^{95} +4.75139 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{5} - 2 q^{7}+O(q^{10}) \) 5 * q + 3 * q^5 - 2 * q^7	\( 5 q + 3 q^{5} - 2 q^{7} - 3 q^{11} - 4 q^{13} + 7 q^{17} - 2 q^{19} - 13 q^{23} - 2 q^{25} + 3 q^{29} - 12 q^{31} - 10 q^{35} - 7 q^{37} + 16 q^{41} - q^{47} - 17 q^{49} - 3 q^{53} - 23 q^{55} - q^{59} - 22 q^{61} + 20 q^{65} + 2 q^{67} - 9 q^{71} - 28 q^{73} + 10 q^{77} - 28 q^{79} - 7 q^{83} - 11 q^{85} + 30 q^{89} - 13 q^{91} - 3 q^{95} - 33 q^{97}+O(q^{100}) \) 5 * q + 3 * q^5 - 2 * q^7 - 3 * q^11 - 4 * q^13 + 7 * q^17 - 2 * q^19 - 13 * q^23 - 2 * q^25 + 3 * q^29 - 12 * q^31 - 10 * q^35 - 7 * q^37 + 16 * q^41 - q^47 - 17 * q^49 - 3 * q^53 - 23 * q^55 - q^59 - 22 * q^61 + 20 * q^65 + 2 * q^67 - 9 * q^71 - 28 * q^73 + 10 * q^77 - 28 * q^79 - 7 * q^83 - 11 * q^85 + 30 * q^89 - 13 * q^91 - 3 * q^95 - 33 * q^97

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