LMFDB - Embedded newform 6044.2.a.a.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$1$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.40035 q^{3} +2.30377 q^{5} -2.94550 q^{7} +2.76170 q^{9} +O(q^{10})$	$q-2.40035 q^{3} +2.30377 q^{5} -2.94550 q^{7} +2.76170 q^{9} -3.41291 q^{11} +1.04042 q^{13} -5.52986 q^{15} -2.59999 q^{17} +6.32932 q^{19} +7.07025 q^{21} -2.42786 q^{23} +0.307358 q^{25} +0.572008 q^{27} +4.20379 q^{29} -3.27508 q^{31} +8.19218 q^{33} -6.78577 q^{35} +0.0632276 q^{37} -2.49738 q^{39} +8.12799 q^{41} +7.14407 q^{43} +6.36232 q^{45} -7.63034 q^{47} +1.67599 q^{49} +6.24089 q^{51} +10.3176 q^{53} -7.86255 q^{55} -15.1926 q^{57} +1.46279 q^{59} -0.926970 q^{61} -8.13459 q^{63} +2.39689 q^{65} +6.38725 q^{67} +5.82773 q^{69} -11.5382 q^{71} +9.22529 q^{73} -0.737768 q^{75} +10.0527 q^{77} -12.8352 q^{79} -9.65812 q^{81} +3.91224 q^{83} -5.98977 q^{85} -10.0906 q^{87} -15.0491 q^{89} -3.06457 q^{91} +7.86135 q^{93} +14.5813 q^{95} -4.99634 q^{97} -9.42542 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * q^99

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
$3$	−2.40035	−1.38584	−0.692922	−	0.721012i	$-0.743677\pi$
$3$	−2.40035	−1.38584	−0.692922	+	0.721012i	$0.743677\pi$
$4$	0	0
$5$	2.30377	1.03028	0.515139	−	0.857107i	$-0.327740\pi$
$5$	2.30377	1.03028	0.515139	+	0.857107i	$0.327740\pi$
$6$	0	0
$7$	−2.94550	−1.11330	−0.556648	−	0.830748i	$-0.687913\pi$
$7$	−2.94550	−1.11330	−0.556648	+	0.830748i	$0.687913\pi$
$8$	0	0
$9$	2.76170	0.920566
$10$	0	0
$11$	−3.41291	−1.02903	−0.514515	−	0.857481i	$-0.672028\pi$
$11$	−3.41291	−1.02903	−0.514515	+	0.857481i	$0.672028\pi$
$12$	0	0
$13$	1.04042	0.288561	0.144281	−	0.989537i	$-0.453913\pi$
$13$	1.04042	0.288561	0.144281	+	0.989537i	$0.453913\pi$
$14$	0	0
$15$	−5.52986	−1.42780
$16$	0	0
$17$	−2.59999	−0.630589	−0.315295	−	0.948994i	$-0.602103\pi$
$17$	−2.59999	−0.630589	−0.315295	+	0.948994i	$0.602103\pi$
$18$	0	0
$19$	6.32932	1.45205	0.726023	−	0.687670i	$-0.241367\pi$
$19$	6.32932	1.45205	0.726023	+	0.687670i	$0.241367\pi$
$20$	0	0
$21$	7.07025	1.54286
$22$	0	0
$23$	−2.42786	−0.506244	−0.253122	−	0.967434i	$-0.581457\pi$
$23$	−2.42786	−0.506244	−0.253122	+	0.967434i	$0.581457\pi$
$24$	0	0
$25$	0.307358	0.0614716
$26$	0	0
$27$	0.572008	0.110083
$28$	0	0
$29$	4.20379	0.780624	0.390312	−	0.920683i	$-0.372367\pi$
$29$	4.20379	0.780624	0.390312	+	0.920683i	$0.372367\pi$
$30$	0	0
$31$	−3.27508	−0.588222	−0.294111	−	0.955771i	$-0.595024\pi$
$31$	−3.27508	−0.588222	−0.294111	+	0.955771i	$0.595024\pi$
$32$	0	0
$33$	8.19218	1.42608
$34$	0	0
$35$	−6.78577	−1.14700
$36$	0	0
$37$	0.0632276	0.0103946	0.00519728	−	0.999986i	$-0.498346\pi$
$37$	0.0632276	0.0103946	0.00519728	+	0.999986i	$0.498346\pi$
$38$	0	0
$39$	−2.49738	−0.399901
$40$	0	0
$41$	8.12799	1.26938	0.634689	−	0.772768i	$-0.281128\pi$
$41$	8.12799	1.26938	0.634689	+	0.772768i	$0.281128\pi$
$42$	0	0
$43$	7.14407	1.08946	0.544730	−	0.838611i	$-0.316632\pi$
$43$	7.14407	1.08946	0.544730	+	0.838611i	$0.316632\pi$
$44$	0	0
$45$	6.36232	0.948439
$46$	0	0
$47$	−7.63034	−1.11300	−0.556500	−	0.830848i	$-0.687856\pi$
$47$	−7.63034	−1.11300	−0.556500	+	0.830848i	$0.687856\pi$
$48$	0	0
$49$	1.67599	0.239428
$50$	0	0
$51$	6.24089	0.873899
$52$	0	0
$53$	10.3176	1.41723	0.708613	−	0.705597i	$-0.249321\pi$
$53$	10.3176	1.41723	0.708613	+	0.705597i	$0.249321\pi$
$54$	0	0
$55$	−7.86255	−1.06019
$56$	0	0
$57$	−15.1926	−2.01231
$58$	0	0
$59$	1.46279	0.190439	0.0952197	−	0.995456i	$-0.469645\pi$
$59$	1.46279	0.190439	0.0952197	+	0.995456i	$0.469645\pi$
$60$	0	0
$61$	−0.926970	−0.118686	−0.0593432	−	0.998238i	$-0.518901\pi$
$61$	−0.926970	−0.118686	−0.0593432	+	0.998238i	$0.518901\pi$
$62$	0	0
$63$	−8.13459	−1.02486
$64$	0	0
$65$	2.39689	0.297298
$66$	0	0
$67$	6.38725	0.780327	0.390164	−	0.920746i	$-0.372418\pi$
$67$	6.38725	0.780327	0.390164	+	0.920746i	$0.372418\pi$
$68$	0	0
$69$	5.82773	0.701576
$70$	0	0
$71$	−11.5382	−1.36933	−0.684663	−	0.728859i	$-0.740051\pi$
$71$	−11.5382	−1.36933	−0.684663	+	0.728859i	$0.740051\pi$
$72$	0	0
$73$	9.22529	1.07974	0.539869	−	0.841749i	$-0.318474\pi$
$73$	9.22529	1.07974	0.539869	+	0.841749i	$0.318474\pi$
$74$	0	0
$75$	−0.737768	−0.0851901
$76$	0	0
$77$	10.0527	1.14561
$78$	0	0
$79$	−12.8352	−1.44407	−0.722037	−	0.691854i	$-0.756794\pi$
$79$	−12.8352	−1.44407	−0.722037	+	0.691854i	$0.756794\pi$
$80$	0	0
$81$	−9.65812	−1.07312
$82$	0	0
$83$	3.91224	0.429424	0.214712	−	0.976677i	$-0.431119\pi$
$83$	3.91224	0.429424	0.214712	+	0.976677i	$0.431119\pi$
$84$	0	0
$85$	−5.98977	−0.649682
$86$	0	0
$87$	−10.0906	−1.08182
$88$	0	0
$89$	−15.0491	−1.59520	−0.797601	−	0.603186i	$-0.793898\pi$
$89$	−15.0491	−1.59520	−0.797601	+	0.603186i	$0.793898\pi$
$90$	0	0
$91$	−3.06457	−0.321254
$92$	0	0
$93$	7.86135	0.815184
$94$	0	0
$95$	14.5813	1.49601
$96$	0	0
$97$	−4.99634	−0.507302	−0.253651	−	0.967296i	$-0.581631\pi$
$97$	−4.99634	−0.507302	−0.253651	+	0.967296i	$0.581631\pi$
$98$	0	0
$99$	−9.42542	−0.947290
$100$	0	0
$101$	3.94763	0.392804	0.196402	−	0.980523i	$-0.437074\pi$
$101$	3.94763	0.392804	0.196402	+	0.980523i	$0.437074\pi$
$102$	0	0
$103$	−10.4728	−1.03191	−0.515957	−	0.856615i	$-0.672563\pi$
$103$	−10.4728	−1.03191	−0.515957	+	0.856615i	$0.672563\pi$
$104$	0	0
$105$	16.2882	1.58957
$106$	0	0
$107$	2.69840	0.260864	0.130432	−	0.991457i	$-0.458364\pi$
$107$	2.69840	0.260864	0.130432	+	0.991457i	$0.458364\pi$
$108$	0	0
$109$	−7.53747	−0.721959	−0.360979	−	0.932574i	$-0.617558\pi$
$109$	−7.53747	−0.721959	−0.360979	+	0.932574i	$0.617558\pi$
$110$	0	0
$111$	−0.151769	−0.0144053
$112$	0	0
$113$	15.8568	1.49168	0.745839	−	0.666126i	$-0.232049\pi$
$113$	15.8568	1.49168	0.745839	+	0.666126i	$0.232049\pi$
$114$	0	0
$115$	−5.59324	−0.521572
$116$	0	0
$117$	2.87333	0.265640
$118$	0	0
$119$	7.65827	0.702033
$120$	0	0
$121$	0.647931	0.0589029
$122$	0	0
$123$	−19.5100	−1.75916
$124$	0	0
$125$	−10.8108	−0.966945
$126$	0	0
$127$	5.32011	0.472084	0.236042	−	0.971743i	$-0.424150\pi$
$127$	5.32011	0.472084	0.236042	+	0.971743i	$0.424150\pi$
$128$	0	0
$129$	−17.1483	−1.50982
$130$	0	0
$131$	13.1756	1.15116	0.575580	−	0.817746i	$-0.304776\pi$
$131$	13.1756	1.15116	0.575580	+	0.817746i	$0.304776\pi$
$132$	0	0
$133$	−18.6430	−1.61656
$134$	0	0
$135$	1.31777	0.113416
$136$	0	0
$137$	17.5124	1.49618	0.748091	−	0.663596i	$-0.230970\pi$
$137$	17.5124	1.49618	0.748091	+	0.663596i	$0.230970\pi$
$138$	0	0
$139$	9.70451	0.823126	0.411563	−	0.911381i	$-0.364983\pi$
$139$	9.70451	0.823126	0.411563	+	0.911381i	$0.364983\pi$
$140$	0	0
$141$	18.3155	1.54245
$142$	0	0
$143$	−3.55086	−0.296938
$144$	0	0
$145$	9.68456	0.804259
$146$	0	0
$147$	−4.02298	−0.331810
$148$	0	0
$149$	−19.0950	−1.56432	−0.782161	−	0.623077i	$-0.785882\pi$
$149$	−19.0950	−1.56432	−0.782161	+	0.623077i	$0.785882\pi$
$150$	0	0
$151$	−8.78046	−0.714544	−0.357272	−	0.934000i	$-0.616293\pi$
$151$	−8.78046	−0.714544	−0.357272	+	0.934000i	$0.616293\pi$
$152$	0	0
$153$	−7.18038	−0.580499
$154$	0	0
$155$	−7.54504	−0.606032
$156$	0	0
$157$	0.556589	0.0444207	0.0222103	−	0.999753i	$-0.492930\pi$
$157$	0.556589	0.0444207	0.0222103	+	0.999753i	$0.492930\pi$
$158$	0	0
$159$	−24.7658	−1.96406
$160$	0	0
$161$	7.15128	0.563600
$162$	0	0
$163$	−22.7664	−1.78320	−0.891601	−	0.452823i	$-0.850417\pi$
$163$	−22.7664	−1.78320	−0.891601	+	0.452823i	$0.850417\pi$
$164$	0	0
$165$	18.8729	1.46925
$166$	0	0
$167$	−11.1004	−0.858976	−0.429488	−	0.903073i	$-0.641306\pi$
$167$	−11.1004	−0.858976	−0.429488	+	0.903073i	$0.641306\pi$
$168$	0	0
$169$	−11.9175	−0.916732
$170$	0	0
$171$	17.4797	1.33670
$172$	0	0
$173$	10.3901	0.789945	0.394972	−	0.918693i	$-0.370754\pi$
$173$	10.3901	0.789945	0.394972	+	0.918693i	$0.370754\pi$
$174$	0	0
$175$	−0.905324	−0.0684361
$176$	0	0
$177$	−3.51122	−0.263919
$178$	0	0
$179$	2.29765	0.171735	0.0858673	−	0.996307i	$-0.472634\pi$
$179$	2.29765	0.171735	0.0858673	+	0.996307i	$0.472634\pi$
$180$	0	0
$181$	−10.8680	−0.807808	−0.403904	−	0.914801i	$-0.632347\pi$
$181$	−10.8680	−0.807808	−0.403904	+	0.914801i	$0.632347\pi$
$182$	0	0
$183$	2.22506	0.164481
$184$	0	0
$185$	0.145662	0.0107093
$186$	0	0
$187$	8.87351	0.648896
$188$	0	0
$189$	−1.68485	−0.122555
$190$	0	0
$191$	−11.8112	−0.854631	−0.427315	−	0.904103i	$-0.640541\pi$
$191$	−11.8112	−0.854631	−0.427315	+	0.904103i	$0.640541\pi$
$192$	0	0
$193$	24.5661	1.76831	0.884153	−	0.467197i	$-0.154736\pi$
$193$	24.5661	1.76831	0.884153	+	0.467197i	$0.154736\pi$
$194$	0	0
$195$	−5.75339	−0.412009
$196$	0	0
$197$	20.8971	1.48886	0.744429	−	0.667702i	$-0.232722\pi$
$197$	20.8971	1.48886	0.744429	+	0.667702i	$0.232722\pi$
$198$	0	0
$199$	8.72870	0.618761	0.309381	−	0.950938i	$-0.399878\pi$
$199$	8.72870	0.618761	0.309381	+	0.950938i	$0.399878\pi$
$200$	0	0
$201$	−15.3317	−1.08141
$202$	0	0
$203$	−12.3823	−0.869065
$204$	0	0
$205$	18.7250	1.30781
$206$	0	0
$207$	−6.70502	−0.466031
$208$	0	0
$209$	−21.6014	−1.49420
$210$	0	0
$211$	19.8497	1.36651	0.683255	−	0.730180i	$-0.260564\pi$
$211$	19.8497	1.36651	0.683255	+	0.730180i	$0.260564\pi$
$212$	0	0
$213$	27.6956	1.89767
$214$	0	0
$215$	16.4583	1.12245
$216$	0	0
$217$	9.64677	0.654865
$218$	0	0
$219$	−22.1440	−1.49635
$220$	0	0
$221$	−2.70508	−0.181964
$222$	0	0
$223$	−20.4526	−1.36961	−0.684804	−	0.728727i	$-0.740112\pi$
$223$	−20.4526	−1.36961	−0.684804	+	0.728727i	$0.740112\pi$
$224$	0	0
$225$	0.848830	0.0565887
$226$	0	0
$227$	−19.8922	−1.32029	−0.660146	−	0.751138i	$-0.729505\pi$
$227$	−19.8922	−1.32029	−0.660146	+	0.751138i	$0.729505\pi$
$228$	0	0
$229$	−19.7812	−1.30718	−0.653589	−	0.756849i	$-0.726738\pi$
$229$	−19.7812	−1.30718	−0.653589	+	0.756849i	$0.726738\pi$
$230$	0	0
$231$	−24.1301	−1.58764
$232$	0	0
$233$	−8.56387	−0.561038	−0.280519	−	0.959848i	$-0.590507\pi$
$233$	−8.56387	−0.561038	−0.280519	+	0.959848i	$0.590507\pi$
$234$	0	0
$235$	−17.5786	−1.14670
$236$	0	0
$237$	30.8090	2.00126
$238$	0	0
$239$	−8.95456	−0.579222	−0.289611	−	0.957144i	$-0.593526\pi$
$239$	−8.95456	−0.579222	−0.289611	+	0.957144i	$0.593526\pi$
$240$	0	0
$241$	−8.68299	−0.559320	−0.279660	−	0.960099i	$-0.590222\pi$
$241$	−8.68299	−0.559320	−0.279660	+	0.960099i	$0.590222\pi$
$242$	0	0
$243$	21.4669	1.37710
$244$	0	0
$245$	3.86111	0.246677
$246$	0	0
$247$	6.58516	0.419004
$248$	0	0
$249$	−9.39077	−0.595116
$250$	0	0
$251$	−6.16064	−0.388856	−0.194428	−	0.980917i	$-0.562285\pi$
$251$	−6.16064	−0.388856	−0.194428	+	0.980917i	$0.562285\pi$
$252$	0	0
$253$	8.28607	0.520941
$254$	0	0
$255$	14.3776	0.900359
$256$	0	0
$257$	−10.2629	−0.640182	−0.320091	−	0.947387i	$-0.603713\pi$
$257$	−10.2629	−0.640182	−0.320091	+	0.947387i	$0.603713\pi$
$258$	0	0
$259$	−0.186237	−0.0115722
$260$	0	0
$261$	11.6096	0.718616
$262$	0	0
$263$	−23.6992	−1.46136	−0.730679	−	0.682721i	$-0.760796\pi$
$263$	−23.6992	−1.46136	−0.730679	+	0.682721i	$0.760796\pi$
$264$	0	0
$265$	23.7693	1.46014
$266$	0	0
$267$	36.1232	2.21070
$268$	0	0
$269$	−19.6278	−1.19673	−0.598365	−	0.801224i	$-0.704183\pi$
$269$	−19.6278	−1.19673	−0.598365	+	0.801224i	$0.704183\pi$
$270$	0	0
$271$	23.7227	1.44105	0.720526	−	0.693428i	$-0.243900\pi$
$271$	23.7227	1.44105	0.720526	+	0.693428i	$0.243900\pi$
$272$	0	0
$273$	7.35604	0.445208
$274$	0	0
$275$	−1.04898	−0.0632561
$276$	0	0
$277$	−4.41848	−0.265481	−0.132740	−	0.991151i	$-0.542378\pi$
$277$	−4.41848	−0.265481	−0.132740	+	0.991151i	$0.542378\pi$
$278$	0	0
$279$	−9.04479	−0.541497
$280$	0	0
$281$	20.6673	1.23291	0.616455	−	0.787390i	$-0.288568\pi$
$281$	20.6673	1.23291	0.616455	+	0.787390i	$0.288568\pi$
$282$	0	0
$283$	18.3006	1.08786	0.543929	−	0.839131i	$-0.316936\pi$
$283$	18.3006	1.08786	0.543929	+	0.839131i	$0.316936\pi$
$284$	0	0
$285$	−35.0003	−2.07324
$286$	0	0
$287$	−23.9410	−1.41319
$288$	0	0
$289$	−10.2401	−0.602357
$290$	0	0
$291$	11.9930	0.703042
$292$	0	0
$293$	9.53365	0.556962	0.278481	−	0.960442i	$-0.410169\pi$
$293$	9.53365	0.556962	0.278481	+	0.960442i	$0.410169\pi$
$294$	0	0
$295$	3.36994	0.196205
$296$	0	0
$297$	−1.95221	−0.113279
$298$	0	0
$299$	−2.52600	−0.146082
$300$	0	0
$301$	−21.0429	−1.21289
$302$	0	0
$303$	−9.47570	−0.544365
$304$	0	0
$305$	−2.13553	−0.122280
$306$	0	0
$307$	−7.98168	−0.455538	−0.227769	−	0.973715i	$-0.573143\pi$
$307$	−7.98168	−0.455538	−0.227769	+	0.973715i	$0.573143\pi$
$308$	0	0
$309$	25.1384	1.43007
$310$	0	0
$311$	5.84052	0.331185	0.165593	−	0.986194i	$-0.447046\pi$
$311$	5.84052	0.331185	0.165593	+	0.986194i	$0.447046\pi$
$312$	0	0
$313$	−3.37080	−0.190529	−0.0952644	−	0.995452i	$-0.530370\pi$
$313$	−3.37080	−0.190529	−0.0952644	+	0.995452i	$0.530370\pi$
$314$	0	0
$315$	−18.7402	−1.05589
$316$	0	0
$317$	15.4103	0.865527	0.432763	−	0.901508i	$-0.357539\pi$
$317$	15.4103	0.865527	0.432763	+	0.901508i	$0.357539\pi$
$318$	0	0
$319$	−14.3471	−0.803285
$320$	0	0
$321$	−6.47711	−0.361517
$322$	0	0
$323$	−16.4562	−0.915645
$324$	0	0
$325$	0.319782	0.0177383
$326$	0	0
$327$	18.0926	1.00052
$328$	0	0
$329$	22.4752	1.23910
$330$	0	0
$331$	−23.2871	−1.27997	−0.639987	−	0.768386i	$-0.721060\pi$
$331$	−23.2871	−1.27997	−0.639987	+	0.768386i	$0.721060\pi$
$332$	0	0
$333$	0.174616	0.00956888
$334$	0	0
$335$	14.7148	0.803953
$336$	0	0
$337$	−3.98849	−0.217267	−0.108633	−	0.994082i	$-0.534647\pi$
$337$	−3.98849	−0.217267	−0.108633	+	0.994082i	$0.534647\pi$
$338$	0	0
$339$	−38.0618	−2.06724
$340$	0	0
$341$	11.1775	0.605298
$342$	0	0
$343$	15.6819	0.846742
$344$	0	0
$345$	13.4257	0.722818
$346$	0	0
$347$	−29.0556	−1.55979	−0.779894	−	0.625912i	$-0.784727\pi$
$347$	−29.0556	−1.55979	−0.779894	+	0.625912i	$0.784727\pi$
$348$	0	0
$349$	−27.9033	−1.49363	−0.746816	−	0.665031i	$-0.768418\pi$
$349$	−27.9033	−1.49363	−0.746816	+	0.665031i	$0.768418\pi$
$350$	0	0
$351$	0.595129	0.0317657
$352$	0	0
$353$	23.4912	1.25031	0.625156	−	0.780500i	$-0.285035\pi$
$353$	23.4912	1.25031	0.625156	+	0.780500i	$0.285035\pi$
$354$	0	0
$355$	−26.5813	−1.41079
$356$	0	0
$357$	−18.3826	−0.972908
$358$	0	0
$359$	−4.34380	−0.229257	−0.114628	−	0.993408i	$-0.536568\pi$
$359$	−4.34380	−0.229257	−0.114628	+	0.993408i	$0.536568\pi$
$360$	0	0
$361$	21.0603	1.10844
$362$	0	0
$363$	−1.55526	−0.0816302
$364$	0	0
$365$	21.2529	1.11243
$366$	0	0
$367$	−28.0665	−1.46506	−0.732531	−	0.680734i	$-0.761661\pi$
$367$	−28.0665	−1.46506	−0.732531	+	0.680734i	$0.761661\pi$
$368$	0	0
$369$	22.4470	1.16855
$370$	0	0
$371$	−30.3904	−1.57779
$372$	0	0
$373$	10.8099	0.559716	0.279858	−	0.960041i	$-0.409713\pi$
$373$	10.8099	0.559716	0.279858	+	0.960041i	$0.409713\pi$
$374$	0	0
$375$	25.9497	1.34004
$376$	0	0
$377$	4.37371	0.225258
$378$	0	0
$379$	7.68440	0.394721	0.197361	−	0.980331i	$-0.436763\pi$
$379$	7.68440	0.394721	0.197361	+	0.980331i	$0.436763\pi$
$380$	0	0
$381$	−12.7702	−0.654235
$382$	0	0
$383$	−26.5437	−1.35632	−0.678161	−	0.734914i	$-0.737223\pi$
$383$	−26.5437	−1.35632	−0.678161	+	0.734914i	$0.737223\pi$
$384$	0	0
$385$	23.1592	1.18030
$386$	0	0
$387$	19.7298	1.00292
$388$	0	0
$389$	−33.3114	−1.68895	−0.844477	−	0.535592i	$-0.820088\pi$
$389$	−33.3114	−1.68895	−0.844477	+	0.535592i	$0.820088\pi$
$390$	0	0
$391$	6.31241	0.319232
$392$	0	0
$393$	−31.6262	−1.59533
$394$	0	0
$395$	−29.5694	−1.48780
$396$	0	0
$397$	−25.4875	−1.27918	−0.639590	−	0.768716i	$-0.720896\pi$
$397$	−25.4875	−1.27918	−0.639590	+	0.768716i	$0.720896\pi$
$398$	0	0
$399$	44.7499	2.24030
$400$	0	0
$401$	−20.9349	−1.04544	−0.522719	−	0.852505i	$-0.675082\pi$
$401$	−20.9349	−1.04544	−0.522719	+	0.852505i	$0.675082\pi$
$402$	0	0
$403$	−3.40747	−0.169738
$404$	0	0
$405$	−22.2501	−1.10562
$406$	0	0
$407$	−0.215790	−0.0106963
$408$	0	0
$409$	1.15427	0.0570750	0.0285375	−	0.999593i	$-0.490915\pi$
$409$	1.15427	0.0570750	0.0285375	+	0.999593i	$0.490915\pi$
$410$	0	0
$411$	−42.0359	−2.07348
$412$	0	0
$413$	−4.30866	−0.212015
$414$	0	0
$415$	9.01291	0.442426
$416$	0	0
$417$	−23.2943	−1.14072
$418$	0	0
$419$	−18.4279	−0.900263	−0.450131	−	0.892962i	$-0.648623\pi$
$419$	−18.4279	−0.900263	−0.450131	+	0.892962i	$0.648623\pi$
$420$	0	0
$421$	−16.7220	−0.814980	−0.407490	−	0.913210i	$-0.633596\pi$
$421$	−16.7220	−0.814980	−0.407490	+	0.913210i	$0.633596\pi$
$422$	0	0
$423$	−21.0727	−1.02459
$424$	0	0
$425$	−0.799127	−0.0387634
$426$	0	0
$427$	2.73039	0.132133
$428$	0	0
$429$	8.52333	0.411510
$430$	0	0
$431$	−29.3463	−1.41356	−0.706780	−	0.707433i	$-0.749853\pi$
$431$	−29.3463	−1.41356	−0.706780	+	0.707433i	$0.749853\pi$
$432$	0	0
$433$	−29.8973	−1.43677	−0.718387	−	0.695644i	$-0.755119\pi$
$433$	−29.8973	−1.43677	−0.718387	+	0.695644i	$0.755119\pi$
$434$	0	0
$435$	−23.2464	−1.11458
$436$	0	0
$437$	−15.3667	−0.735090
$438$	0	0
$439$	−30.4429	−1.45296	−0.726479	−	0.687189i	$-0.758845\pi$
$439$	−30.4429	−1.45296	−0.726479	+	0.687189i	$0.758845\pi$
$440$	0	0
$441$	4.62859	0.220409
$442$	0	0
$443$	−8.97648	−0.426485	−0.213243	−	0.976999i	$-0.568403\pi$
$443$	−8.97648	−0.426485	−0.213243	+	0.976999i	$0.568403\pi$
$444$	0	0
$445$	−34.6697	−1.64350
$446$	0	0
$447$	45.8347	2.16791
$448$	0	0
$449$	27.8704	1.31529	0.657643	−	0.753330i	$-0.271554\pi$
$449$	27.8704	1.31529	0.657643	+	0.753330i	$0.271554\pi$
$450$	0	0
$451$	−27.7401	−1.30623
$452$	0	0
$453$	21.0762	0.990247
$454$	0	0
$455$	−7.06006	−0.330981
$456$	0	0
$457$	−12.7650	−0.597122	−0.298561	−	0.954391i	$-0.596507\pi$
$457$	−12.7650	−0.597122	−0.298561	+	0.954391i	$0.596507\pi$
$458$	0	0
$459$	−1.48721	−0.0694172
$460$	0	0
$461$	−17.1213	−0.797416	−0.398708	−	0.917078i	$-0.630541\pi$
$461$	−17.1213	−0.797416	−0.398708	+	0.917078i	$0.630541\pi$
$462$	0	0
$463$	32.1504	1.49416	0.747078	−	0.664737i	$-0.231456\pi$
$463$	32.1504	1.49416	0.747078	+	0.664737i	$0.231456\pi$
$464$	0	0
$465$	18.1108	0.839866
$466$	0	0
$467$	39.2607	1.81677	0.908385	−	0.418134i	$-0.137316\pi$
$467$	39.2607	1.81677	0.908385	+	0.418134i	$0.137316\pi$
$468$	0	0
$469$	−18.8137	−0.868735
$470$	0	0
$471$	−1.33601	−0.0615601
$472$	0	0
$473$	−24.3820	−1.12109
$474$	0	0
$475$	1.94537	0.0892596
$476$	0	0
$477$	28.4940	1.30465
$478$	0	0
$479$	27.6006	1.26110	0.630551	−	0.776148i	$-0.282829\pi$
$479$	27.6006	1.26110	0.630551	+	0.776148i	$0.282829\pi$
$480$	0	0
$481$	0.0657834	0.00299947
$482$	0	0
$483$	−17.1656	−0.781062
$484$	0	0
$485$	−11.5104	−0.522662
$486$	0	0
$487$	−26.1048	−1.18292	−0.591461	−	0.806334i	$-0.701449\pi$
$487$	−26.1048	−1.18292	−0.591461	+	0.806334i	$0.701449\pi$
$488$	0	0
$489$	54.6474	2.47124
$490$	0	0
$491$	−12.8115	−0.578174	−0.289087	−	0.957303i	$-0.593352\pi$
$491$	−12.8115	−0.578174	−0.289087	+	0.957303i	$0.593352\pi$
$492$	0	0
$493$	−10.9298	−0.492253
$494$	0	0
$495$	−21.7140	−0.975972
$496$	0	0
$497$	33.9857	1.52447
$498$	0	0
$499$	16.0569	0.718804	0.359402	−	0.933183i	$-0.382981\pi$
$499$	16.0569	0.718804	0.359402	+	0.933183i	$0.382981\pi$
$500$	0	0
$501$	26.6449	1.19041
$502$	0	0
$503$	−26.9719	−1.20262	−0.601309	−	0.799017i	$-0.705354\pi$
$503$	−26.9719	−1.20262	−0.601309	+	0.799017i	$0.705354\pi$
$504$	0	0
$505$	9.09443	0.404697
$506$	0	0
$507$	28.6063	1.27045
$508$	0	0
$509$	−18.3676	−0.814131	−0.407066	−	0.913399i	$-0.633448\pi$
$509$	−18.3676	−0.814131	−0.407066	+	0.913399i	$0.633448\pi$
$510$	0	0
$511$	−27.1731	−1.20207
$512$	0	0
$513$	3.62042	0.159846
$514$	0	0
$515$	−24.1269	−1.06316
$516$	0	0
$517$	26.0417	1.14531
$518$	0	0
$519$	−24.9399	−1.09474
$520$	0	0
$521$	−38.3435	−1.67986	−0.839930	−	0.542694i	$-0.817404\pi$
$521$	−38.3435	−1.67986	−0.839930	+	0.542694i	$0.817404\pi$
$522$	0	0
$523$	36.5526	1.59833	0.799166	−	0.601110i	$-0.205275\pi$
$523$	36.5526	1.59833	0.799166	+	0.601110i	$0.205275\pi$
$524$	0	0
$525$	2.17310	0.0948418
$526$	0	0
$527$	8.51517	0.370927
$528$	0	0
$529$	−17.1055	−0.743717
$530$	0	0
$531$	4.03979	0.175312
$532$	0	0
$533$	8.45653	0.366293
$534$	0	0
$535$	6.21649	0.268762
$536$	0	0
$537$	−5.51518	−0.237998
$538$	0	0
$539$	−5.72001	−0.246378
$540$	0	0
$541$	39.4045	1.69413	0.847066	−	0.531488i	$-0.178367\pi$
$541$	39.4045	1.69413	0.847066	+	0.531488i	$0.178367\pi$
$542$	0	0
$543$	26.0869	1.11950
$544$	0	0
$545$	−17.3646	−0.743818
$546$	0	0
$547$	23.9085	1.02225	0.511127	−	0.859505i	$-0.329228\pi$
$547$	23.9085	1.02225	0.511127	+	0.859505i	$0.329228\pi$
$548$	0	0
$549$	−2.56001	−0.109259
$550$	0	0
$551$	26.6071	1.13350
$552$	0	0
$553$	37.8062	1.60768
$554$	0	0
$555$	−0.349640	−0.0148414
$556$	0	0
$557$	13.1113	0.555544	0.277772	−	0.960647i	$-0.410404\pi$
$557$	13.1113	0.555544	0.277772	+	0.960647i	$0.410404\pi$
$558$	0	0
$559$	7.43285	0.314376
$560$	0	0
$561$	−21.2996	−0.899269
$562$	0	0
$563$	3.35586	0.141433	0.0707163	−	0.997496i	$-0.477472\pi$
$563$	3.35586	0.141433	0.0707163	+	0.997496i	$0.477472\pi$
$564$	0	0
$565$	36.5303	1.53684
$566$	0	0
$567$	28.4480	1.19470
$568$	0	0
$569$	−40.3629	−1.69210	−0.846051	−	0.533102i	$-0.821026\pi$
$569$	−40.3629	−1.69210	−0.846051	+	0.533102i	$0.821026\pi$
$570$	0	0
$571$	7.60285	0.318169	0.159085	−	0.987265i	$-0.449146\pi$
$571$	7.60285	0.318169	0.159085	+	0.987265i	$0.449146\pi$
$572$	0	0
$573$	28.3511	1.18439
$574$	0	0
$575$	−0.746223	−0.0311196
$576$	0	0
$577$	−44.7881	−1.86455	−0.932277	−	0.361746i	$-0.882181\pi$
$577$	−44.7881	−1.86455	−0.932277	+	0.361746i	$0.882181\pi$
$578$	0	0
$579$	−58.9673	−2.45060
$580$	0	0
$581$	−11.5235	−0.478076
$582$	0	0
$583$	−35.2129	−1.45837
$584$	0	0
$585$	6.61950	0.273682
$586$	0	0
$587$	21.7505	0.897737	0.448869	−	0.893598i	$-0.351827\pi$
$587$	21.7505	0.897737	0.448869	+	0.893598i	$0.351827\pi$
$588$	0	0
$589$	−20.7290	−0.854125
$590$	0	0
$591$	−50.1605	−2.06333
$592$	0	0
$593$	−47.2789	−1.94151	−0.970756	−	0.240067i	$-0.922831\pi$
$593$	−47.2789	−1.94151	−0.970756	+	0.240067i	$0.922831\pi$
$594$	0	0
$595$	17.6429	0.723288
$596$	0	0
$597$	−20.9520	−0.857507
$598$	0	0
$599$	22.2276	0.908197	0.454098	−	0.890952i	$-0.349961\pi$
$599$	22.2276	0.908197	0.454098	+	0.890952i	$0.349961\pi$
$600$	0	0
$601$	−7.18310	−0.293005	−0.146502	−	0.989210i	$-0.546802\pi$
$601$	−7.18310	−0.293005	−0.146502	+	0.989210i	$0.546802\pi$
$602$	0	0
$603$	17.6397	0.718343
$604$	0	0
$605$	1.49269	0.0606863
$606$	0	0
$607$	10.0166	0.406562	0.203281	−	0.979120i	$-0.434839\pi$
$607$	10.0166	0.406562	0.203281	+	0.979120i	$0.434839\pi$
$608$	0	0
$609$	29.7218	1.20439
$610$	0	0
$611$	−7.93878	−0.321169
$612$	0	0
$613$	−9.69174	−0.391446	−0.195723	−	0.980659i	$-0.562705\pi$
$613$	−9.69174	−0.391446	−0.195723	+	0.980659i	$0.562705\pi$
$614$	0	0
$615$	−44.9467	−1.81242
$616$	0	0
$617$	18.4275	0.741863	0.370931	−	0.928660i	$-0.379039\pi$
$617$	18.4275	0.741863	0.370931	+	0.928660i	$0.379039\pi$
$618$	0	0
$619$	11.5626	0.464740	0.232370	−	0.972627i	$-0.425352\pi$
$619$	11.5626	0.464740	0.232370	+	0.972627i	$0.425352\pi$
$620$	0	0
$621$	−1.38876	−0.0557289
$622$	0	0
$623$	44.3272	1.77593
$624$	0	0
$625$	−26.4423	−1.05769
$626$	0	0
$627$	51.8510	2.07073
$628$	0	0
$629$	−0.164391	−0.00655470
$630$	0	0
$631$	1.97823	0.0787520	0.0393760	−	0.999224i	$-0.487463\pi$
$631$	1.97823	0.0787520	0.0393760	+	0.999224i	$0.487463\pi$
$632$	0	0
$633$	−47.6463	−1.89377
$634$	0	0
$635$	12.2563	0.486377
$636$	0	0
$637$	1.74374	0.0690896
$638$	0	0
$639$	−31.8649	−1.26056
$640$	0	0
$641$	−19.8431	−0.783754	−0.391877	−	0.920018i	$-0.628174\pi$
$641$	−19.8431	−0.783754	−0.391877	+	0.920018i	$0.628174\pi$
$642$	0	0
$643$	6.90686	0.272380	0.136190	−	0.990683i	$-0.456514\pi$
$643$	6.90686	0.272380	0.136190	+	0.990683i	$0.456514\pi$
$644$	0	0
$645$	−39.5057	−1.55554
$646$	0	0
$647$	9.01410	0.354381	0.177190	−	0.984177i	$-0.443299\pi$
$647$	9.01410	0.354381	0.177190	+	0.984177i	$0.443299\pi$
$648$	0	0
$649$	−4.99237	−0.195968
$650$	0	0
$651$	−23.1557	−0.907542
$652$	0	0
$653$	−1.64536	−0.0643879	−0.0321939	−	0.999482i	$-0.510249\pi$
$653$	−1.64536	−0.0643879	−0.0321939	+	0.999482i	$0.510249\pi$
$654$	0	0
$655$	30.3536	1.18601
$656$	0	0
$657$	25.4775	0.993971
$658$	0	0
$659$	−42.8092	−1.66761	−0.833806	−	0.552058i	$-0.813843\pi$
$659$	−42.8092	−1.66761	−0.833806	+	0.552058i	$0.813843\pi$
$660$	0	0
$661$	−40.7938	−1.58669	−0.793347	−	0.608769i	$-0.791663\pi$
$661$	−40.7938	−1.58669	−0.793347	+	0.608769i	$0.791663\pi$
$662$	0	0
$663$	6.49316	0.252173
$664$	0	0
$665$	−42.9493	−1.66550
$666$	0	0
$667$	−10.2062	−0.395186
$668$	0	0
$669$	49.0935	1.89806
$670$	0	0
$671$	3.16366	0.122132
$672$	0	0
$673$	22.8734	0.881706	0.440853	−	0.897579i	$-0.354676\pi$
$673$	22.8734	0.881706	0.440853	+	0.897579i	$0.354676\pi$
$674$	0	0
$675$	0.175811	0.00676698
$676$	0	0
$677$	−44.4857	−1.70972	−0.854862	−	0.518856i	$-0.826358\pi$
$677$	−44.4857	−1.70972	−0.854862	+	0.518856i	$0.826358\pi$
$678$	0	0
$679$	14.7168	0.564777
$680$	0	0
$681$	47.7483	1.82972
$682$	0	0
$683$	9.28692	0.355354	0.177677	−	0.984089i	$-0.443142\pi$
$683$	9.28692	0.355354	0.177677	+	0.984089i	$0.443142\pi$
$684$	0	0
$685$	40.3445	1.54148
$686$	0	0
$687$	47.4819	1.81155
$688$	0	0
$689$	10.7346	0.408956
$690$	0	0
$691$	−10.8568	−0.413011	−0.206506	−	0.978445i	$-0.566209\pi$
$691$	−10.8568	−0.413011	−0.206506	+	0.978445i	$0.566209\pi$
$692$	0	0
$693$	27.7626	1.05461
$694$	0	0
$695$	22.3570	0.848048
$696$	0	0
$697$	−21.1327	−0.800457
$698$	0	0
$699$	20.5563	0.777512
$700$	0	0
$701$	44.0210	1.66265	0.831325	−	0.555787i	$-0.187583\pi$
$701$	44.0210	1.66265	0.831325	+	0.555787i	$0.187583\pi$
$702$	0	0
$703$	0.400188	0.0150934
$704$	0	0
$705$	42.1948	1.58915
$706$	0	0
$707$	−11.6278	−0.437307
$708$	0	0
$709$	−18.4620	−0.693355	−0.346677	−	0.937984i	$-0.612690\pi$
$709$	−18.4620	−0.693355	−0.346677	+	0.937984i	$0.612690\pi$
$710$	0	0
$711$	−35.4470	−1.32937
$712$	0	0
$713$	7.95145	0.297784
$714$	0	0
$715$	−8.18037	−0.305929
$716$	0	0
$717$	21.4941	0.802712
$718$	0	0
$719$	52.8796	1.97208	0.986038	−	0.166522i	$-0.0532538\pi$
$719$	52.8796	1.97208	0.986038	+	0.166522i	$0.0532538\pi$
$720$	0	0
$721$	30.8476	1.14883
$722$	0	0
$723$	20.8422	0.775131
$724$	0	0
$725$	1.29207	0.0479862
$726$	0	0
$727$	−29.7014	−1.10156	−0.550781	−	0.834650i	$-0.685670\pi$
$727$	−29.7014	−1.10156	−0.550781	+	0.834650i	$0.685670\pi$
$728$	0	0
$729$	−22.5537	−0.835324
$730$	0	0
$731$	−18.5745	−0.687002
$732$	0	0
$733$	4.14968	0.153272	0.0766359	−	0.997059i	$-0.475582\pi$
$733$	4.14968	0.153272	0.0766359	+	0.997059i	$0.475582\pi$
$734$	0	0
$735$	−9.26802	−0.341856
$736$	0	0
$737$	−21.7991	−0.802980
$738$	0	0
$739$	4.09076	0.150481	0.0752404	−	0.997165i	$-0.476028\pi$
$739$	4.09076	0.150481	0.0752404	+	0.997165i	$0.476028\pi$
$740$	0	0
$741$	−15.8067	−0.580675
$742$	0	0
$743$	−25.6904	−0.942488	−0.471244	−	0.882003i	$-0.656195\pi$
$743$	−25.6904	−0.942488	−0.471244	+	0.882003i	$0.656195\pi$
$744$	0	0
$745$	−43.9904	−1.61168
$746$	0	0
$747$	10.8044	0.395314
$748$	0	0
$749$	−7.94814	−0.290419
$750$	0	0
$751$	−1.56779	−0.0572093	−0.0286047	−	0.999591i	$-0.509106\pi$
$751$	−1.56779	−0.0572093	−0.0286047	+	0.999591i	$0.509106\pi$
$752$	0	0
$753$	14.7877	0.538895
$754$	0	0
$755$	−20.2282	−0.736179
$756$	0	0
$757$	−16.4328	−0.597260	−0.298630	−	0.954369i	$-0.596530\pi$
$757$	−16.4328	−0.597260	−0.298630	+	0.954369i	$0.596530\pi$
$758$	0	0
$759$	−19.8895	−0.721943
$760$	0	0
$761$	−8.76559	−0.317752	−0.158876	−	0.987299i	$-0.550787\pi$
$761$	−8.76559	−0.317752	−0.158876	+	0.987299i	$0.550787\pi$
$762$	0	0
$763$	22.2016	0.803754
$764$	0	0
$765$	−16.5419	−0.598075
$766$	0	0
$767$	1.52192	0.0549534
$768$	0	0
$769$	9.93842	0.358389	0.179194	−	0.983814i	$-0.442651\pi$
$769$	9.93842	0.358389	0.179194	+	0.983814i	$0.442651\pi$
$770$	0	0
$771$	24.6346	0.887192
$772$	0	0
$773$	48.9989	1.76237	0.881184	−	0.472774i	$-0.156747\pi$
$773$	48.9989	1.76237	0.881184	+	0.472774i	$0.156747\pi$
$774$	0	0
$775$	−1.00662	−0.0361590
$776$	0	0
$777$	0.447035	0.0160373
$778$	0	0
$779$	51.4446	1.84320
$780$	0	0
$781$	39.3786	1.40908
$782$	0	0
$783$	2.40460	0.0859334
$784$	0	0
$785$	1.28225	0.0457656
$786$	0	0
$787$	4.57115	0.162944	0.0814719	−	0.996676i	$-0.474038\pi$
$787$	4.57115	0.162944	0.0814719	+	0.996676i	$0.474038\pi$
$788$	0	0
$789$	56.8866	2.02522
$790$	0	0
$791$	−46.7062	−1.66068
$792$	0	0
$793$	−0.964440	−0.0342483
$794$	0	0
$795$	−57.0547	−2.02352
$796$	0	0
$797$	−47.9962	−1.70011	−0.850057	−	0.526691i	$-0.823432\pi$
$797$	−47.9962	−1.70011	−0.850057	+	0.526691i	$0.823432\pi$
$798$	0	0
$799$	19.8388	0.701846
$800$	0	0
$801$	−41.5611	−1.46849
$802$	0	0
$803$	−31.4851	−1.11108
$804$	0	0
$805$	16.4749	0.580664
$806$	0	0
$807$	47.1138	1.65848
$808$	0	0
$809$	3.76583	0.132399	0.0661997	−	0.997806i	$-0.478913\pi$
$809$	3.76583	0.132399	0.0661997	+	0.997806i	$0.478913\pi$
$810$	0	0
$811$	32.5761	1.14390	0.571950	−	0.820288i	$-0.306187\pi$
$811$	32.5761	1.14390	0.571950	+	0.820288i	$0.306187\pi$
$812$	0	0
$813$	−56.9429	−1.99708
$814$	0	0
$815$	−52.4485	−1.83719
$816$	0	0
$817$	45.2171	1.58195
$818$	0	0
$819$	−8.46341	−0.295735
$820$	0	0
$821$	22.8064	0.795949	0.397975	−	0.917396i	$-0.369713\pi$
$821$	22.8064	0.795949	0.397975	+	0.917396i	$0.369713\pi$
$822$	0	0
$823$	31.6948	1.10481	0.552406	−	0.833575i	$-0.313710\pi$
$823$	31.6948	1.10481	0.552406	+	0.833575i	$0.313710\pi$
$824$	0	0
$825$	2.51793	0.0876632
$826$	0	0
$827$	−9.64566	−0.335412	−0.167706	−	0.985837i	$-0.553636\pi$
$827$	−9.64566	−0.335412	−0.167706	+	0.985837i	$0.553636\pi$
$828$	0	0
$829$	1.75029	0.0607902	0.0303951	−	0.999538i	$-0.490323\pi$
$829$	1.75029	0.0607902	0.0303951	+	0.999538i	$0.490323\pi$
$830$	0	0
$831$	10.6059	0.367915
$832$	0	0
$833$	−4.35756	−0.150981
$834$	0	0
$835$	−25.5728	−0.884983
$836$	0	0
$837$	−1.87337	−0.0647532
$838$	0	0
$839$	26.7116	0.922187	0.461093	−	0.887352i	$-0.347457\pi$
$839$	26.7116	0.922187	0.461093	+	0.887352i	$0.347457\pi$
$840$	0	0
$841$	−11.3282	−0.390626
$842$	0	0
$843$	−49.6090	−1.70862
$844$	0	0
$845$	−27.4552	−0.944489
$846$	0	0
$847$	−1.90848	−0.0655763
$848$	0	0
$849$	−43.9279	−1.50760
$850$	0	0
$851$	−0.153508	−0.00526219
$852$	0	0
$853$	−0.566770	−0.0194058	−0.00970292	−	0.999953i	$-0.503089\pi$
$853$	−0.566770	−0.0194058	−0.00970292	+	0.999953i	$0.503089\pi$
$854$	0	0
$855$	40.2692	1.37718
$856$	0	0
$857$	40.0586	1.36838	0.684188	−	0.729305i	$-0.260157\pi$
$857$	40.0586	1.36838	0.684188	+	0.729305i	$0.260157\pi$
$858$	0	0
$859$	26.4049	0.900923	0.450461	−	0.892796i	$-0.351260\pi$
$859$	26.4049	0.900923	0.450461	+	0.892796i	$0.351260\pi$
$860$	0	0
$861$	57.4669	1.95847
$862$	0	0
$863$	26.9331	0.916815	0.458407	−	0.888742i	$-0.348420\pi$
$863$	26.9331	0.916815	0.458407	+	0.888742i	$0.348420\pi$
$864$	0	0
$865$	23.9364	0.813862
$866$	0	0
$867$	24.5798	0.834773
$868$	0	0
$869$	43.8054	1.48600
$870$	0	0
$871$	6.64544	0.225172
$872$	0	0
$873$	−13.7984	−0.467005
$874$	0	0
$875$	31.8432	1.07650
$876$	0	0
$877$	1.61198	0.0544326	0.0272163	−	0.999630i	$-0.491336\pi$
$877$	1.61198	0.0544326	0.0272163	+	0.999630i	$0.491336\pi$
$878$	0	0
$879$	−22.8841	−0.771863
$880$	0	0
$881$	−25.8817	−0.871977	−0.435989	−	0.899952i	$-0.643601\pi$
$881$	−25.8817	−0.871977	−0.435989	+	0.899952i	$0.643601\pi$
$882$	0	0
$883$	−0.497944	−0.0167572	−0.00837858	−	0.999965i	$-0.502667\pi$
$883$	−0.497944	−0.0167572	−0.00837858	+	0.999965i	$0.502667\pi$
$884$	0	0
$885$	−8.08904	−0.271910
$886$	0	0
$887$	−40.8391	−1.37124	−0.685622	−	0.727958i	$-0.740470\pi$
$887$	−40.8391	−1.37124	−0.685622	+	0.727958i	$0.740470\pi$
$888$	0	0
$889$	−15.6704	−0.525569
$890$	0	0
$891$	32.9623	1.10428
$892$	0	0
$893$	−48.2949	−1.61613
$894$	0	0
$895$	5.29326	0.176934
$896$	0	0
$897$	6.06330	0.202448
$898$	0	0
$899$	−13.7677	−0.459180
$900$	0	0
$901$	−26.8255	−0.893688
$902$	0	0
$903$	50.5104	1.68088
$904$	0	0
$905$	−25.0373	−0.832267
$906$	0	0
$907$	10.2676	0.340931	0.170466	−	0.985364i	$-0.445473\pi$
$907$	10.2676	0.340931	0.170466	+	0.985364i	$0.445473\pi$
$908$	0	0
$909$	10.9022	0.361602
$910$	0	0
$911$	18.6524	0.617980	0.308990	−	0.951065i	$-0.400009\pi$
$911$	18.6524	0.617980	0.308990	+	0.951065i	$0.400009\pi$
$912$	0	0
$913$	−13.3521	−0.441891
$914$	0	0
$915$	5.12602	0.169461
$916$	0	0
$917$	−38.8089	−1.28158
$918$	0	0
$919$	−44.7651	−1.47666	−0.738332	−	0.674438i	$-0.764386\pi$
$919$	−44.7651	−1.47666	−0.738332	+	0.674438i	$0.764386\pi$
$920$	0	0
$921$	19.1589	0.631306
$922$	0	0
$923$	−12.0045	−0.395134
$924$	0	0
$925$	0.0194335	0.000638970 0
$926$	0	0
$927$	−28.9227	−0.949945
$928$	0	0
$929$	2.76238	0.0906307	0.0453153	−	0.998973i	$-0.485571\pi$
$929$	2.76238	0.0906307	0.0453153	+	0.998973i	$0.485571\pi$
$930$	0	0
$931$	10.6079	0.347660
$932$	0	0
$933$	−14.0193	−0.458972
$934$	0	0
$935$	20.4425	0.668542
$936$	0	0
$937$	1.87528	0.0612626	0.0306313	−	0.999531i	$-0.490248\pi$
$937$	1.87528	0.0612626	0.0306313	+	0.999531i	$0.490248\pi$
$938$	0	0
$939$	8.09111	0.264043
$940$	0	0
$941$	3.85501	0.125670	0.0628349	−	0.998024i	$-0.479986\pi$
$941$	3.85501	0.125670	0.0628349	+	0.998024i	$0.479986\pi$
$942$	0	0
$943$	−19.7336	−0.642615
$944$	0	0
$945$	−3.88151	−0.126266
$946$	0	0
$947$	−8.33621	−0.270890	−0.135445	−	0.990785i	$-0.543246\pi$
$947$	−8.33621	−0.270890	−0.135445	+	0.990785i	$0.543246\pi$
$948$	0	0
$949$	9.59819	0.311571
$950$	0	0
$951$	−36.9901	−1.19949
$952$	0	0
$953$	−30.2383	−0.979516	−0.489758	−	0.871859i	$-0.662915\pi$
$953$	−30.2383	−0.979516	−0.489758	+	0.871859i	$0.662915\pi$
$954$	0	0
$955$	−27.2104	−0.880507
$956$	0	0
$957$	34.4382	1.11323
$958$	0	0
$959$	−51.5828	−1.66569
$960$	0	0
$961$	−20.2738	−0.653995
$962$	0	0
$963$	7.45216	0.240142
$964$	0	0
$965$	56.5947	1.82185
$966$	0	0
$967$	−7.14371	−0.229726	−0.114863	−	0.993381i	$-0.536643\pi$
$967$	−7.14371	−0.229726	−0.114863	+	0.993381i	$0.536643\pi$
$968$	0	0
$969$	39.5006	1.26894
$970$	0	0
$971$	−27.4510	−0.880944	−0.440472	−	0.897766i	$-0.645189\pi$
$971$	−27.4510	−0.880944	−0.440472	+	0.897766i	$0.645189\pi$
$972$	0	0
$973$	−28.5847	−0.916382
$974$	0	0
$975$	−0.767590	−0.0245826
$976$	0	0
$977$	30.4480	0.974118	0.487059	−	0.873369i	$-0.338070\pi$
$977$	30.4480	0.974118	0.487059	+	0.873369i	$0.338070\pi$
$978$	0	0
$979$	51.3612	1.64151
$980$	0	0
$981$	−20.8162	−0.664611
$982$	0	0
$983$	10.1430	0.323512	0.161756	−	0.986831i	$-0.448284\pi$
$983$	10.1430	0.323512	0.161756	+	0.986831i	$0.448284\pi$
$984$	0	0
$985$	48.1421	1.53394
$986$	0	0
$987$	−53.9485	−1.71720
$988$	0	0
$989$	−17.3448	−0.551533
$990$	0	0
$991$	−55.8752	−1.77493	−0.887467	−	0.460872i	$-0.847537\pi$
$991$	−55.8752	−1.77493	−0.887467	+	0.460872i	$0.847537\pi$
$992$	0	0
$993$	55.8972	1.77385
$994$	0	0
$995$	20.1089	0.637496
$996$	0	0
$997$	−16.3180	−0.516796	−0.258398	−	0.966039i	$-0.583195\pi$
$997$	−16.3180	−0.516796	−0.258398	+	0.966039i	$0.583195\pi$
$998$	0	0
$999$	0.0361667	0.00114426

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.a.1.12	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.a.1.12	✓	63	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.40035 q^{3} +2.30377 q^{5} -2.94550 q^{7} +2.76170 q^{9} +O(q^{10})\)	\(q-2.40035 q^{3} +2.30377 q^{5} -2.94550 q^{7} +2.76170 q^{9} -3.41291 q^{11} +1.04042 q^{13} -5.52986 q^{15} -2.59999 q^{17} +6.32932 q^{19} +7.07025 q^{21} -2.42786 q^{23} +0.307358 q^{25} +0.572008 q^{27} +4.20379 q^{29} -3.27508 q^{31} +8.19218 q^{33} -6.78577 q^{35} +0.0632276 q^{37} -2.49738 q^{39} +8.12799 q^{41} +7.14407 q^{43} +6.36232 q^{45} -7.63034 q^{47} +1.67599 q^{49} +6.24089 q^{51} +10.3176 q^{53} -7.86255 q^{55} -15.1926 q^{57} +1.46279 q^{59} -0.926970 q^{61} -8.13459 q^{63} +2.39689 q^{65} +6.38725 q^{67} +5.82773 q^{69} -11.5382 q^{71} +9.22529 q^{73} -0.737768 q^{75} +10.0527 q^{77} -12.8352 q^{79} -9.65812 q^{81} +3.91224 q^{83} -5.98977 q^{85} -10.0906 q^{87} -15.0491 q^{89} -3.06457 q^{91} +7.86135 q^{93} +14.5813 q^{95} -4.99634 q^{97} -9.42542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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