LMFDB - Embedded newform 8024.2.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.97688 q^{3} +0.00923108 q^{5} -2.57666 q^{7} +5.86179 q^{9} +O(q^{10})$	$q-2.97688 q^{3} +0.00923108 q^{5} -2.57666 q^{7} +5.86179 q^{9} +4.00063 q^{11} +4.58909 q^{13} -0.0274798 q^{15} +1.00000 q^{17} -7.62753 q^{19} +7.67041 q^{21} -2.77323 q^{23} -4.99991 q^{25} -8.51918 q^{27} +4.02572 q^{29} +1.76853 q^{31} -11.9094 q^{33} -0.0237854 q^{35} +2.00539 q^{37} -13.6611 q^{39} -7.96984 q^{41} +8.14488 q^{43} +0.0541106 q^{45} -6.63037 q^{47} -0.360803 q^{49} -2.97688 q^{51} +12.2437 q^{53} +0.0369302 q^{55} +22.7062 q^{57} +1.00000 q^{59} -3.65084 q^{61} -15.1039 q^{63} +0.0423623 q^{65} -12.0152 q^{67} +8.25556 q^{69} +4.08472 q^{71} -9.83430 q^{73} +14.8841 q^{75} -10.3083 q^{77} +4.61055 q^{79} +7.77518 q^{81} -8.66454 q^{83} +0.00923108 q^{85} -11.9841 q^{87} -2.63066 q^{89} -11.8245 q^{91} -5.26469 q^{93} -0.0704104 q^{95} +18.6887 q^{97} +23.4509 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * 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
$3$	−2.97688	−1.71870	−0.859350	−	0.511388i	$-0.829131\pi$
$3$	−2.97688	−1.71870	−0.859350	+	0.511388i	$0.829131\pi$
$4$	0	0
$5$	0.00923108	0.00412827	0.00206413	−	0.999998i	$-0.499343\pi$
$5$	0.00923108	0.00412827	0.00206413	+	0.999998i	$0.499343\pi$
$6$	0	0
$7$	−2.57666	−0.973887	−0.486944	−	0.873433i	$-0.661888\pi$
$7$	−2.57666	−0.973887	−0.486944	+	0.873433i	$0.661888\pi$
$8$	0	0
$9$	5.86179	1.95393
$10$	0	0
$11$	4.00063	1.20624	0.603118	−	0.797652i	$-0.293925\pi$
$11$	4.00063	1.20624	0.603118	+	0.797652i	$0.293925\pi$
$12$	0	0
$13$	4.58909	1.27278	0.636392	−	0.771366i	$-0.280426\pi$
$13$	4.58909	1.27278	0.636392	+	0.771366i	$0.280426\pi$
$14$	0	0
$15$	−0.0274798	−0.00709525
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−7.62753	−1.74988	−0.874938	−	0.484235i	$-0.839098\pi$
$19$	−7.62753	−1.74988	−0.874938	+	0.484235i	$0.839098\pi$
$20$	0	0
$21$	7.67041	1.67382
$22$	0	0
$23$	−2.77323	−0.578258	−0.289129	−	0.957290i	$-0.593366\pi$
$23$	−2.77323	−0.578258	−0.289129	+	0.957290i	$0.593366\pi$
$24$	0	0
$25$	−4.99991	−0.999983
$26$	0	0
$27$	−8.51918	−1.63952
$28$	0	0
$29$	4.02572	0.747558	0.373779	−	0.927518i	$-0.378062\pi$
$29$	4.02572	0.747558	0.373779	+	0.927518i	$0.378062\pi$
$30$	0	0
$31$	1.76853	0.317637	0.158819	−	0.987308i	$-0.449231\pi$
$31$	1.76853	0.317637	0.158819	+	0.987308i	$0.449231\pi$
$32$	0	0
$33$	−11.9094	−2.07316
$34$	0	0
$35$	−0.0237854	−0.00402047
$36$	0	0
$37$	2.00539	0.329685	0.164842	−	0.986320i	$-0.447288\pi$
$37$	2.00539	0.329685	0.164842	+	0.986320i	$0.447288\pi$
$38$	0	0
$39$	−13.6611	−2.18753
$40$	0	0
$41$	−7.96984	−1.24468	−0.622340	−	0.782747i	$-0.713818\pi$
$41$	−7.96984	−1.24468	−0.622340	+	0.782747i	$0.713818\pi$
$42$	0	0
$43$	8.14488	1.24208	0.621041	−	0.783778i	$-0.286710\pi$
$43$	8.14488	1.24208	0.621041	+	0.783778i	$0.286710\pi$
$44$	0	0
$45$	0.0541106	0.00806634
$46$	0	0
$47$	−6.63037	−0.967139	−0.483569	−	0.875306i	$-0.660660\pi$
$47$	−6.63037	−0.967139	−0.483569	+	0.875306i	$0.660660\pi$
$48$	0	0
$49$	−0.360803	−0.0515432
$50$	0	0
$51$	−2.97688	−0.416846
$52$	0	0
$53$	12.2437	1.68180	0.840898	−	0.541194i	$-0.182028\pi$
$53$	12.2437	1.68180	0.840898	+	0.541194i	$0.182028\pi$
$54$	0	0
$55$	0.0369302	0.00497966
$56$	0	0
$57$	22.7062	3.00751
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−3.65084	−0.467442	−0.233721	−	0.972304i	$-0.575090\pi$
$61$	−3.65084	−0.467442	−0.233721	+	0.972304i	$0.575090\pi$
$62$	0	0
$63$	−15.1039	−1.90291
$64$	0	0
$65$	0.0423623	0.00525439
$66$	0	0
$67$	−12.0152	−1.46789	−0.733947	−	0.679206i	$-0.762324\pi$
$67$	−12.0152	−1.46789	−0.733947	+	0.679206i	$0.762324\pi$
$68$	0	0
$69$	8.25556	0.993852
$70$	0	0
$71$	4.08472	0.484767	0.242383	−	0.970181i	$-0.422071\pi$
$71$	4.08472	0.484767	0.242383	+	0.970181i	$0.422071\pi$
$72$	0	0
$73$	−9.83430	−1.15102	−0.575509	−	0.817795i	$-0.695196\pi$
$73$	−9.83430	−1.15102	−0.575509	+	0.817795i	$0.695196\pi$
$74$	0	0
$75$	14.8841	1.71867
$76$	0	0
$77$	−10.3083	−1.17474
$78$	0	0
$79$	4.61055	0.518727	0.259364	−	0.965780i	$-0.416487\pi$
$79$	4.61055	0.518727	0.259364	+	0.965780i	$0.416487\pi$
$80$	0	0
$81$	7.77518	0.863909
$82$	0	0
$83$	−8.66454	−0.951057	−0.475529	−	0.879700i	$-0.657743\pi$
$83$	−8.66454	−0.951057	−0.475529	+	0.879700i	$0.657743\pi$
$84$	0	0
$85$	0.00923108	0.00100125
$86$	0	0
$87$	−11.9841	−1.28483
$88$	0	0
$89$	−2.63066	−0.278849	−0.139424	−	0.990233i	$-0.544525\pi$
$89$	−2.63066	−0.278849	−0.139424	+	0.990233i	$0.544525\pi$
$90$	0	0
$91$	−11.8245	−1.23955
$92$	0	0
$93$	−5.26469	−0.545923
$94$	0	0
$95$	−0.0704104	−0.00722396
$96$	0	0
$97$	18.6887	1.89755	0.948776	−	0.315950i	$-0.102323\pi$
$97$	18.6887	1.89755	0.948776	+	0.315950i	$0.102323\pi$
$98$	0	0
$99$	23.4509	2.35690
$100$	0	0
$101$	−3.08753	−0.307221	−0.153611	−	0.988131i	$-0.549090\pi$
$101$	−3.08753	−0.307221	−0.153611	+	0.988131i	$0.549090\pi$
$102$	0	0
$103$	6.31290	0.622028	0.311014	−	0.950405i	$-0.399331\pi$
$103$	6.31290	0.622028	0.311014	+	0.950405i	$0.399331\pi$
$104$	0	0
$105$	0.0708062	0.00690997
$106$	0	0
$107$	−11.4611	−1.10799	−0.553995	−	0.832520i	$-0.686898\pi$
$107$	−11.4611	−1.10799	−0.553995	+	0.832520i	$0.686898\pi$
$108$	0	0
$109$	11.9506	1.14466	0.572330	−	0.820024i	$-0.306040\pi$
$109$	11.9506	1.14466	0.572330	+	0.820024i	$0.306040\pi$
$110$	0	0
$111$	−5.96981	−0.566629
$112$	0	0
$113$	8.73701	0.821909	0.410954	−	0.911656i	$-0.365196\pi$
$113$	8.73701	0.821909	0.410954	+	0.911656i	$0.365196\pi$
$114$	0	0
$115$	−0.0255999	−0.00238720
$116$	0	0
$117$	26.9003	2.48693
$118$	0	0
$119$	−2.57666	−0.236202
$120$	0	0
$121$	5.00506	0.455006
$122$	0	0
$123$	23.7252	2.13923
$124$	0	0
$125$	−0.0923100	−0.00825646
$126$	0	0
$127$	19.4931	1.72974	0.864868	−	0.501999i	$-0.167402\pi$
$127$	19.4931	1.72974	0.864868	+	0.501999i	$0.167402\pi$
$128$	0	0
$129$	−24.2463	−2.13477
$130$	0	0
$131$	−21.8899	−1.91253	−0.956265	−	0.292503i	$-0.905512\pi$
$131$	−21.8899	−1.91253	−0.956265	+	0.292503i	$0.905512\pi$
$132$	0	0
$133$	19.6536	1.70418
$134$	0	0
$135$	−0.0786413	−0.00676836
$136$	0	0
$137$	14.0225	1.19803	0.599013	−	0.800739i	$-0.295560\pi$
$137$	14.0225	1.19803	0.599013	+	0.800739i	$0.295560\pi$
$138$	0	0
$139$	8.77705	0.744460	0.372230	−	0.928141i	$-0.378593\pi$
$139$	8.77705	0.744460	0.372230	+	0.928141i	$0.378593\pi$
$140$	0	0
$141$	19.7378	1.66222
$142$	0	0
$143$	18.3593	1.53528
$144$	0	0
$145$	0.0371618	0.00308612
$146$	0	0
$147$	1.07406	0.0885873
$148$	0	0
$149$	2.50832	0.205490	0.102745	−	0.994708i	$-0.467237\pi$
$149$	2.50832	0.205490	0.102745	+	0.994708i	$0.467237\pi$
$150$	0	0
$151$	19.4930	1.58632	0.793158	−	0.609016i	$-0.208436\pi$
$151$	19.4930	1.58632	0.793158	+	0.609016i	$0.208436\pi$
$152$	0	0
$153$	5.86179	0.473897
$154$	0	0
$155$	0.0163254	0.00131129
$156$	0	0
$157$	17.3835	1.38735	0.693676	−	0.720287i	$-0.255990\pi$
$157$	17.3835	1.38735	0.693676	+	0.720287i	$0.255990\pi$
$158$	0	0
$159$	−36.4478	−2.89050
$160$	0	0
$161$	7.14568	0.563158
$162$	0	0
$163$	0.532321	0.0416946	0.0208473	−	0.999783i	$-0.493364\pi$
$163$	0.532321	0.0416946	0.0208473	+	0.999783i	$0.493364\pi$
$164$	0	0
$165$	−0.109937	−0.00855855
$166$	0	0
$167$	8.24887	0.638317	0.319159	−	0.947701i	$-0.396600\pi$
$167$	8.24887	0.638317	0.319159	+	0.947701i	$0.396600\pi$
$168$	0	0
$169$	8.05974	0.619980
$170$	0	0
$171$	−44.7110	−3.41913
$172$	0	0
$173$	−4.14918	−0.315456	−0.157728	−	0.987483i	$-0.550417\pi$
$173$	−4.14918	−0.315456	−0.157728	+	0.987483i	$0.550417\pi$
$174$	0	0
$175$	12.8831	0.973871
$176$	0	0
$177$	−2.97688	−0.223756
$178$	0	0
$179$	13.7070	1.02451	0.512253	−	0.858835i	$-0.328811\pi$
$179$	13.7070	1.02451	0.512253	+	0.858835i	$0.328811\pi$
$180$	0	0
$181$	−20.0349	−1.48918	−0.744591	−	0.667521i	$-0.767356\pi$
$181$	−20.0349	−1.48918	−0.744591	+	0.667521i	$0.767356\pi$
$182$	0	0
$183$	10.8681	0.803392
$184$	0	0
$185$	0.0185120	0.00136103
$186$	0	0
$187$	4.00063	0.292555
$188$	0	0
$189$	21.9511	1.59671
$190$	0	0
$191$	−4.10777	−0.297228	−0.148614	−	0.988895i	$-0.547481\pi$
$191$	−4.10777	−0.297228	−0.148614	+	0.988895i	$0.547481\pi$
$192$	0	0
$193$	−21.6245	−1.55657	−0.778284	−	0.627913i	$-0.783909\pi$
$193$	−21.6245	−1.55657	−0.778284	+	0.627913i	$0.783909\pi$
$194$	0	0
$195$	−0.126107	−0.00903072
$196$	0	0
$197$	2.19412	0.156325	0.0781624	−	0.996941i	$-0.475095\pi$
$197$	2.19412	0.156325	0.0781624	+	0.996941i	$0.475095\pi$
$198$	0	0
$199$	−7.35210	−0.521176	−0.260588	−	0.965450i	$-0.583916\pi$
$199$	−7.35210	−0.521176	−0.260588	+	0.965450i	$0.583916\pi$
$200$	0	0
$201$	35.7679	2.52287
$202$	0	0
$203$	−10.3729	−0.728038
$204$	0	0
$205$	−0.0735703	−0.00513837
$206$	0	0
$207$	−16.2561	−1.12988
$208$	0	0
$209$	−30.5150	−2.11076
$210$	0	0
$211$	−15.7852	−1.08670	−0.543349	−	0.839507i	$-0.682844\pi$
$211$	−15.7852	−1.08670	−0.543349	+	0.839507i	$0.682844\pi$
$212$	0	0
$213$	−12.1597	−0.833169
$214$	0	0
$215$	0.0751861	0.00512765
$216$	0	0
$217$	−4.55691	−0.309343
$218$	0	0
$219$	29.2755	1.97825
$220$	0	0
$221$	4.58909	0.308696
$222$	0	0
$223$	14.4791	0.969594	0.484797	−	0.874627i	$-0.338893\pi$
$223$	14.4791	0.969594	0.484797	+	0.874627i	$0.338893\pi$
$224$	0	0
$225$	−29.3084	−1.95390
$226$	0	0
$227$	−11.5070	−0.763749	−0.381874	−	0.924214i	$-0.624721\pi$
$227$	−11.5070	−0.763749	−0.381874	+	0.924214i	$0.624721\pi$
$228$	0	0
$229$	−11.5071	−0.760412	−0.380206	−	0.924902i	$-0.624147\pi$
$229$	−11.5071	−0.760412	−0.380206	+	0.924902i	$0.624147\pi$
$230$	0	0
$231$	30.6865	2.01902
$232$	0	0
$233$	−21.3002	−1.39542	−0.697712	−	0.716379i	$-0.745798\pi$
$233$	−21.3002	−1.39542	−0.697712	+	0.716379i	$0.745798\pi$
$234$	0	0
$235$	−0.0612055	−0.00399261
$236$	0	0
$237$	−13.7250	−0.891537
$238$	0	0
$239$	−19.6245	−1.26940	−0.634701	−	0.772757i	$-0.718877\pi$
$239$	−19.6245	−1.26940	−0.634701	+	0.772757i	$0.718877\pi$
$240$	0	0
$241$	−6.80735	−0.438500	−0.219250	−	0.975669i	$-0.570361\pi$
$241$	−6.80735	−0.438500	−0.219250	+	0.975669i	$0.570361\pi$
$242$	0	0
$243$	2.41180	0.154717
$244$	0	0
$245$	−0.00333060	−0.000212784 0
$246$	0	0
$247$	−35.0034	−2.22722
$248$	0	0
$249$	25.7933	1.63458
$250$	0	0
$251$	−0.819576	−0.0517312	−0.0258656	−	0.999665i	$-0.508234\pi$
$251$	−0.819576	−0.0517312	−0.0258656	+	0.999665i	$0.508234\pi$
$252$	0	0
$253$	−11.0947	−0.697516
$254$	0	0
$255$	−0.0274798	−0.00172085
$256$	0	0
$257$	10.4320	0.650730	0.325365	−	0.945589i	$-0.394513\pi$
$257$	10.4320	0.650730	0.325365	+	0.945589i	$0.394513\pi$
$258$	0	0
$259$	−5.16723	−0.321076
$260$	0	0
$261$	23.5979	1.46068
$262$	0	0
$263$	14.2506	0.878728	0.439364	−	0.898309i	$-0.355204\pi$
$263$	14.2506	0.878728	0.439364	+	0.898309i	$0.355204\pi$
$264$	0	0
$265$	0.113022	0.00694290
$266$	0	0
$267$	7.83113	0.479258
$268$	0	0
$269$	−4.12622	−0.251580	−0.125790	−	0.992057i	$-0.540147\pi$
$269$	−4.12622	−0.251580	−0.125790	+	0.992057i	$0.540147\pi$
$270$	0	0
$271$	−23.5057	−1.42787	−0.713936	−	0.700211i	$-0.753089\pi$
$271$	−23.5057	−1.42787	−0.713936	+	0.700211i	$0.753089\pi$
$272$	0	0
$273$	35.2002	2.13041
$274$	0	0
$275$	−20.0028	−1.20622
$276$	0	0
$277$	−20.5440	−1.23437	−0.617183	−	0.786819i	$-0.711726\pi$
$277$	−20.5440	−1.23437	−0.617183	+	0.786819i	$0.711726\pi$
$278$	0	0
$279$	10.3667	0.620640
$280$	0	0
$281$	−7.68784	−0.458618	−0.229309	−	0.973354i	$-0.573647\pi$
$281$	−7.68784	−0.458618	−0.229309	+	0.973354i	$0.573647\pi$
$282$	0	0
$283$	−1.97383	−0.117332	−0.0586660	−	0.998278i	$-0.518685\pi$
$283$	−1.97383	−0.117332	−0.0586660	+	0.998278i	$0.518685\pi$
$284$	0	0
$285$	0.209603	0.0124158
$286$	0	0
$287$	20.5356	1.21218
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−55.6340	−3.26132
$292$	0	0
$293$	−5.30002	−0.309631	−0.154815	−	0.987943i	$-0.549478\pi$
$293$	−5.30002	−0.309631	−0.154815	+	0.987943i	$0.549478\pi$
$294$	0	0
$295$	0.00923108	0.000537454 0
$296$	0	0
$297$	−34.0821	−1.97764
$298$	0	0
$299$	−12.7266	−0.735998
$300$	0	0
$301$	−20.9866	−1.20965
$302$	0	0
$303$	9.19120	0.528021
$304$	0	0
$305$	−0.0337012	−0.00192972
$306$	0	0
$307$	27.7168	1.58188	0.790940	−	0.611893i	$-0.209592\pi$
$307$	27.7168	1.58188	0.790940	+	0.611893i	$0.209592\pi$
$308$	0	0
$309$	−18.7927	−1.06908
$310$	0	0
$311$	11.4997	0.652089	0.326044	−	0.945354i	$-0.394284\pi$
$311$	11.4997	0.652089	0.326044	+	0.945354i	$0.394284\pi$
$312$	0	0
$313$	11.4194	0.645463	0.322732	−	0.946491i	$-0.395399\pi$
$313$	11.4194	0.645463	0.322732	+	0.946491i	$0.395399\pi$
$314$	0	0
$315$	−0.139425	−0.00785570
$316$	0	0
$317$	−9.89542	−0.555782	−0.277891	−	0.960613i	$-0.589635\pi$
$317$	−9.89542	−0.555782	−0.277891	+	0.960613i	$0.589635\pi$
$318$	0	0
$319$	16.1054	0.901732
$320$	0	0
$321$	34.1184	1.90430
$322$	0	0
$323$	−7.62753	−0.424407
$324$	0	0
$325$	−22.9451	−1.27276
$326$	0	0
$327$	−35.5754	−1.96733
$328$	0	0
$329$	17.0842	0.941884
$330$	0	0
$331$	3.17132	0.174311	0.0871557	−	0.996195i	$-0.472222\pi$
$331$	3.17132	0.174311	0.0871557	+	0.996195i	$0.472222\pi$
$332$	0	0
$333$	11.7552	0.644180
$334$	0	0
$335$	−0.110914	−0.00605986
$336$	0	0
$337$	−14.8893	−0.811069	−0.405535	−	0.914080i	$-0.632915\pi$
$337$	−14.8893	−0.811069	−0.405535	+	0.914080i	$0.632915\pi$
$338$	0	0
$339$	−26.0090	−1.41261
$340$	0	0
$341$	7.07523	0.383145
$342$	0	0
$343$	18.9663	1.02408
$344$	0	0
$345$	0.0762077	0.00410289
$346$	0	0
$347$	−20.2276	−1.08587	−0.542936	−	0.839774i	$-0.682687\pi$
$347$	−20.2276	−1.08587	−0.542936	+	0.839774i	$0.682687\pi$
$348$	0	0
$349$	−16.6433	−0.890895	−0.445447	−	0.895308i	$-0.646955\pi$
$349$	−16.6433	−0.890895	−0.445447	+	0.895308i	$0.646955\pi$
$350$	0	0
$351$	−39.0953	−2.08675
$352$	0	0
$353$	15.0426	0.800638	0.400319	−	0.916376i	$-0.368899\pi$
$353$	15.0426	0.800638	0.400319	+	0.916376i	$0.368899\pi$
$354$	0	0
$355$	0.0377064	0.00200125
$356$	0	0
$357$	7.67041	0.405961
$358$	0	0
$359$	−5.45105	−0.287695	−0.143848	−	0.989600i	$-0.545948\pi$
$359$	−5.45105	−0.287695	−0.143848	+	0.989600i	$0.545948\pi$
$360$	0	0
$361$	39.1793	2.06207
$362$	0	0
$363$	−14.8994	−0.782018
$364$	0	0
$365$	−0.0907812	−0.00475171
$366$	0	0
$367$	−31.0689	−1.62178	−0.810890	−	0.585198i	$-0.801017\pi$
$367$	−31.0689	−1.62178	−0.810890	+	0.585198i	$0.801017\pi$
$368$	0	0
$369$	−46.7175	−2.43202
$370$	0	0
$371$	−31.5478	−1.63788
$372$	0	0
$373$	−7.08670	−0.366935	−0.183468	−	0.983026i	$-0.558732\pi$
$373$	−7.08670	−0.366935	−0.183468	+	0.983026i	$0.558732\pi$
$374$	0	0
$375$	0.274795	0.0141904
$376$	0	0
$377$	18.4744	0.951480
$378$	0	0
$379$	−3.60455	−0.185153	−0.0925767	−	0.995706i	$-0.529510\pi$
$379$	−3.60455	−0.185153	−0.0925767	+	0.995706i	$0.529510\pi$
$380$	0	0
$381$	−58.0286	−2.97290
$382$	0	0
$383$	22.5000	1.14970	0.574849	−	0.818260i	$-0.305061\pi$
$383$	22.5000	1.14970	0.574849	+	0.818260i	$0.305061\pi$
$384$	0	0
$385$	−0.0951566	−0.00484963
$386$	0	0
$387$	47.7436	2.42694
$388$	0	0
$389$	−2.18323	−0.110694	−0.0553470	−	0.998467i	$-0.517626\pi$
$389$	−2.18323	−0.110694	−0.0553470	+	0.998467i	$0.517626\pi$
$390$	0	0
$391$	−2.77323	−0.140248
$392$	0	0
$393$	65.1635	3.28706
$394$	0	0
$395$	0.0425604	0.00214144
$396$	0	0
$397$	−20.2851	−1.01808	−0.509039	−	0.860743i	$-0.669999\pi$
$397$	−20.2851	−1.01808	−0.509039	+	0.860743i	$0.669999\pi$
$398$	0	0
$399$	−58.5063	−2.92898
$400$	0	0
$401$	−26.2533	−1.31103	−0.655513	−	0.755184i	$-0.727548\pi$
$401$	−26.2533	−1.31103	−0.655513	+	0.755184i	$0.727548\pi$
$402$	0	0
$403$	8.11594	0.404284
$404$	0	0
$405$	0.0717733	0.00356645
$406$	0	0
$407$	8.02284	0.397678
$408$	0	0
$409$	1.52530	0.0754214	0.0377107	−	0.999289i	$-0.487993\pi$
$409$	1.52530	0.0754214	0.0377107	+	0.999289i	$0.487993\pi$
$410$	0	0
$411$	−41.7434	−2.05905
$412$	0	0
$413$	−2.57666	−0.126789
$414$	0	0
$415$	−0.0799831	−0.00392622
$416$	0	0
$417$	−26.1282	−1.27950
$418$	0	0
$419$	−22.4923	−1.09882	−0.549410	−	0.835553i	$-0.685148\pi$
$419$	−22.4923	−1.09882	−0.549410	+	0.835553i	$0.685148\pi$
$420$	0	0
$421$	21.0137	1.02414	0.512071	−	0.858943i	$-0.328878\pi$
$421$	21.0137	1.02414	0.512071	+	0.858943i	$0.328878\pi$
$422$	0	0
$423$	−38.8658	−1.88972
$424$	0	0
$425$	−4.99991	−0.242531
$426$	0	0
$427$	9.40698	0.455236
$428$	0	0
$429$	−54.6532	−2.63868
$430$	0	0
$431$	−14.8664	−0.716091	−0.358045	−	0.933704i	$-0.616557\pi$
$431$	−14.8664	−0.716091	−0.358045	+	0.933704i	$0.616557\pi$
$432$	0	0
$433$	21.9231	1.05356	0.526779	−	0.850003i	$-0.323400\pi$
$433$	21.9231	1.05356	0.526779	+	0.850003i	$0.323400\pi$
$434$	0	0
$435$	−0.110626	−0.00530411
$436$	0	0
$437$	21.1529	1.01188
$438$	0	0
$439$	−39.1945	−1.87065	−0.935325	−	0.353789i	$-0.884893\pi$
$439$	−39.1945	−1.87065	−0.935325	+	0.353789i	$0.884893\pi$
$440$	0	0
$441$	−2.11495	−0.100712
$442$	0	0
$443$	−9.10578	−0.432629	−0.216314	−	0.976324i	$-0.569404\pi$
$443$	−9.10578	−0.432629	−0.216314	+	0.976324i	$0.569404\pi$
$444$	0	0
$445$	−0.0242838	−0.00115116
$446$	0	0
$447$	−7.46696	−0.353175
$448$	0	0
$449$	19.0355	0.898342	0.449171	−	0.893446i	$-0.351719\pi$
$449$	19.0355	0.898342	0.449171	+	0.893446i	$0.351719\pi$
$450$	0	0
$451$	−31.8844	−1.50138
$452$	0	0
$453$	−58.0281	−2.72640
$454$	0	0
$455$	−0.109153	−0.00511719
$456$	0	0
$457$	23.5264	1.10052	0.550259	−	0.834994i	$-0.314529\pi$
$457$	23.5264	1.10052	0.550259	+	0.834994i	$0.314529\pi$
$458$	0	0
$459$	−8.51918	−0.397641
$460$	0	0
$461$	−3.05821	−0.142435	−0.0712175	−	0.997461i	$-0.522688\pi$
$461$	−3.05821	−0.142435	−0.0712175	+	0.997461i	$0.522688\pi$
$462$	0	0
$463$	−20.8433	−0.968668	−0.484334	−	0.874883i	$-0.660938\pi$
$463$	−20.8433	−0.968668	−0.484334	+	0.874883i	$0.660938\pi$
$464$	0	0
$465$	−0.0485988	−0.00225372
$466$	0	0
$467$	−32.3327	−1.49618	−0.748089	−	0.663599i	$-0.769028\pi$
$467$	−32.3327	−1.49618	−0.748089	+	0.663599i	$0.769028\pi$
$468$	0	0
$469$	30.9592	1.42956
$470$	0	0
$471$	−51.7484	−2.38444
$472$	0	0
$473$	32.5847	1.49825
$474$	0	0
$475$	38.1370	1.74985
$476$	0	0
$477$	71.7697	3.28611
$478$	0	0
$479$	−13.3170	−0.608471	−0.304236	−	0.952597i	$-0.598401\pi$
$479$	−13.3170	−0.608471	−0.304236	+	0.952597i	$0.598401\pi$
$480$	0	0
$481$	9.20293	0.419617
$482$	0	0
$483$	−21.2718	−0.967900
$484$	0	0
$485$	0.172517	0.00783360
$486$	0	0
$487$	1.10294	0.0499789	0.0249894	−	0.999688i	$-0.492045\pi$
$487$	1.10294	0.0499789	0.0249894	+	0.999688i	$0.492045\pi$
$488$	0	0
$489$	−1.58465	−0.0716605
$490$	0	0
$491$	29.2326	1.31925	0.659624	−	0.751596i	$-0.270716\pi$
$491$	29.2326	1.31925	0.659624	+	0.751596i	$0.270716\pi$
$492$	0	0
$493$	4.02572	0.181309
$494$	0	0
$495$	0.216477	0.00972991
$496$	0	0
$497$	−10.5249	−0.472108
$498$	0	0
$499$	16.9267	0.757743	0.378871	−	0.925449i	$-0.376312\pi$
$499$	16.9267	0.757743	0.378871	+	0.925449i	$0.376312\pi$
$500$	0	0
$501$	−24.5559	−1.09708
$502$	0	0
$503$	−16.2420	−0.724196	−0.362098	−	0.932140i	$-0.617939\pi$
$503$	−16.2420	−0.724196	−0.362098	+	0.932140i	$0.617939\pi$
$504$	0	0
$505$	−0.0285013	−0.00126829
$506$	0	0
$507$	−23.9928	−1.06556
$508$	0	0
$509$	−37.8856	−1.67925	−0.839624	−	0.543168i	$-0.817225\pi$
$509$	−37.8856	−1.67925	−0.839624	+	0.543168i	$0.817225\pi$
$510$	0	0
$511$	25.3397	1.12096
$512$	0	0
$513$	64.9803	2.86895
$514$	0	0
$515$	0.0582749	0.00256790
$516$	0	0
$517$	−26.5257	−1.16660
$518$	0	0
$519$	12.3516	0.542175
$520$	0	0
$521$	38.0179	1.66559	0.832797	−	0.553578i	$-0.186738\pi$
$521$	38.0179	1.66559	0.832797	+	0.553578i	$0.186738\pi$
$522$	0	0
$523$	37.1402	1.62403	0.812014	−	0.583637i	$-0.198371\pi$
$523$	37.1402	1.62403	0.812014	+	0.583637i	$0.198371\pi$
$524$	0	0
$525$	−38.3514	−1.67379
$526$	0	0
$527$	1.76853	0.0770383
$528$	0	0
$529$	−15.3092	−0.665618
$530$	0	0
$531$	5.86179	0.254380
$532$	0	0
$533$	−36.5743	−1.58421
$534$	0	0
$535$	−0.105799	−0.00457408
$536$	0	0
$537$	−40.8039	−1.76082
$538$	0	0
$539$	−1.44344	−0.0621733
$540$	0	0
$541$	−44.4691	−1.91188	−0.955939	−	0.293566i	$-0.905158\pi$
$541$	−44.4691	−1.91188	−0.955939	+	0.293566i	$0.905158\pi$
$542$	0	0
$543$	59.6414	2.55946
$544$	0	0
$545$	0.110317	0.00472546
$546$	0	0
$547$	−21.0090	−0.898279	−0.449140	−	0.893462i	$-0.648269\pi$
$547$	−21.0090	−0.898279	−0.449140	+	0.893462i	$0.648269\pi$
$548$	0	0
$549$	−21.4004	−0.913348
$550$	0	0
$551$	−30.7063	−1.30813
$552$	0	0
$553$	−11.8798	−0.505182
$554$	0	0
$555$	−0.0551078	−0.00233919
$556$	0	0
$557$	−2.06484	−0.0874902	−0.0437451	−	0.999043i	$-0.513929\pi$
$557$	−2.06484	−0.0874902	−0.0437451	+	0.999043i	$0.513929\pi$
$558$	0	0
$559$	37.3776	1.58090
$560$	0	0
$561$	−11.9094	−0.502815
$562$	0	0
$563$	−21.2060	−0.893725	−0.446862	−	0.894603i	$-0.647459\pi$
$563$	−21.2060	−0.893725	−0.446862	+	0.894603i	$0.647459\pi$
$564$	0	0
$565$	0.0806521	0.00339306
$566$	0	0
$567$	−20.0340	−0.841350
$568$	0	0
$569$	4.74102	0.198754	0.0993770	−	0.995050i	$-0.468315\pi$
$569$	4.74102	0.198754	0.0993770	+	0.995050i	$0.468315\pi$
$570$	0	0
$571$	−38.1404	−1.59613	−0.798063	−	0.602575i	$-0.794142\pi$
$571$	−38.1404	−1.59613	−0.798063	+	0.602575i	$0.794142\pi$
$572$	0	0
$573$	12.2283	0.510845
$574$	0	0
$575$	13.8659	0.578248
$576$	0	0
$577$	−43.2148	−1.79905	−0.899527	−	0.436865i	$-0.856089\pi$
$577$	−43.2148	−1.79905	−0.899527	+	0.436865i	$0.856089\pi$
$578$	0	0
$579$	64.3735	2.67527
$580$	0	0
$581$	22.3256	0.926223
$582$	0	0
$583$	48.9824	2.02864
$584$	0	0
$585$	0.248319	0.0102667
$586$	0	0
$587$	12.9494	0.534477	0.267239	−	0.963630i	$-0.413889\pi$
$587$	12.9494	0.534477	0.267239	+	0.963630i	$0.413889\pi$
$588$	0	0
$589$	−13.4895	−0.555826
$590$	0	0
$591$	−6.53163	−0.268676
$592$	0	0
$593$	20.5803	0.845132	0.422566	−	0.906332i	$-0.361129\pi$
$593$	20.5803	0.845132	0.422566	+	0.906332i	$0.361129\pi$
$594$	0	0
$595$	−0.0237854	−0.000975106 0
$596$	0	0
$597$	21.8863	0.895746
$598$	0	0
$599$	2.72173	0.111207	0.0556035	−	0.998453i	$-0.482292\pi$
$599$	2.72173	0.111207	0.0556035	+	0.998453i	$0.482292\pi$
$600$	0	0
$601$	6.81846	0.278131	0.139065	−	0.990283i	$-0.455590\pi$
$601$	6.81846	0.278131	0.139065	+	0.990283i	$0.455590\pi$
$602$	0	0
$603$	−70.4308	−2.86816
$604$	0	0
$605$	0.0462021	0.00187838
$606$	0	0
$607$	31.8845	1.29415	0.647076	−	0.762426i	$-0.275992\pi$
$607$	31.8845	1.29415	0.647076	+	0.762426i	$0.275992\pi$
$608$	0	0
$609$	30.8789	1.25128
$610$	0	0
$611$	−30.4274	−1.23096
$612$	0	0
$613$	−22.0788	−0.891753	−0.445876	−	0.895095i	$-0.647108\pi$
$613$	−22.0788	−0.891753	−0.445876	+	0.895095i	$0.647108\pi$
$614$	0	0
$615$	0.219010	0.00883132
$616$	0	0
$617$	−39.2632	−1.58068	−0.790338	−	0.612670i	$-0.790095\pi$
$617$	−39.2632	−1.58068	−0.790338	+	0.612670i	$0.790095\pi$
$618$	0	0
$619$	−9.72362	−0.390825	−0.195413	−	0.980721i	$-0.562605\pi$
$619$	−9.72362	−0.390825	−0.195413	+	0.980721i	$0.562605\pi$
$620$	0	0
$621$	23.6256	0.948064
$622$	0	0
$623$	6.77832	0.271567
$624$	0	0
$625$	24.9987	0.999949
$626$	0	0
$627$	90.8392	3.62777
$628$	0	0
$629$	2.00539	0.0799603
$630$	0	0
$631$	4.23835	0.168726	0.0843631	−	0.996435i	$-0.473114\pi$
$631$	4.23835	0.168726	0.0843631	+	0.996435i	$0.473114\pi$
$632$	0	0
$633$	46.9906	1.86771
$634$	0	0
$635$	0.179943	0.00714081
$636$	0	0
$637$	−1.65576	−0.0656034
$638$	0	0
$639$	23.9437	0.947200
$640$	0	0
$641$	−8.39203	−0.331465	−0.165733	−	0.986171i	$-0.552999\pi$
$641$	−8.39203	−0.331465	−0.165733	+	0.986171i	$0.552999\pi$
$642$	0	0
$643$	20.7479	0.818216	0.409108	−	0.912486i	$-0.365840\pi$
$643$	20.7479	0.818216	0.409108	+	0.912486i	$0.365840\pi$
$644$	0	0
$645$	−0.223820	−0.00881289
$646$	0	0
$647$	−11.3454	−0.446035	−0.223017	−	0.974814i	$-0.571591\pi$
$647$	−11.3454	−0.446035	−0.223017	+	0.974814i	$0.571591\pi$
$648$	0	0
$649$	4.00063	0.157039
$650$	0	0
$651$	13.5653	0.531668
$652$	0	0
$653$	36.6446	1.43401	0.717006	−	0.697067i	$-0.245512\pi$
$653$	36.6446	1.43401	0.717006	+	0.697067i	$0.245512\pi$
$654$	0	0
$655$	−0.202067	−0.00789543
$656$	0	0
$657$	−57.6466	−2.24901
$658$	0	0
$659$	−23.1937	−0.903500	−0.451750	−	0.892145i	$-0.649200\pi$
$659$	−23.1937	−0.903500	−0.451750	+	0.892145i	$0.649200\pi$
$660$	0	0
$661$	14.3758	0.559154	0.279577	−	0.960123i	$-0.409806\pi$
$661$	14.3758	0.559154	0.279577	+	0.960123i	$0.409806\pi$
$662$	0	0
$663$	−13.6611	−0.530555
$664$	0	0
$665$	0.181424	0.00703532
$666$	0	0
$667$	−11.1643	−0.432282
$668$	0	0
$669$	−43.1026	−1.66644
$670$	0	0
$671$	−14.6057	−0.563845
$672$	0	0
$673$	0.0925812	0.00356874	0.00178437	−	0.999998i	$-0.499432\pi$
$673$	0.0925812	0.00356874	0.00178437	+	0.999998i	$0.499432\pi$
$674$	0	0
$675$	42.5952	1.63949
$676$	0	0
$677$	−26.6759	−1.02524	−0.512620	−	0.858616i	$-0.671325\pi$
$677$	−26.6759	−1.02524	−0.512620	+	0.858616i	$0.671325\pi$
$678$	0	0
$679$	−48.1545	−1.84800
$680$	0	0
$681$	34.2550	1.31265
$682$	0	0
$683$	−23.6343	−0.904342	−0.452171	−	0.891931i	$-0.649350\pi$
$683$	−23.6343	−0.904342	−0.452171	+	0.891931i	$0.649350\pi$
$684$	0	0
$685$	0.129443	0.00494577
$686$	0	0
$687$	34.2553	1.30692
$688$	0	0
$689$	56.1872	2.14056
$690$	0	0
$691$	19.7922	0.752930	0.376465	−	0.926431i	$-0.377140\pi$
$691$	19.7922	0.752930	0.376465	+	0.926431i	$0.377140\pi$
$692$	0	0
$693$	−60.4250	−2.29535
$694$	0	0
$695$	0.0810217	0.00307333
$696$	0	0
$697$	−7.96984	−0.301879
$698$	0	0
$699$	63.4081	2.39831
$700$	0	0
$701$	−45.9332	−1.73487	−0.867436	−	0.497548i	$-0.834234\pi$
$701$	−45.9332	−1.73487	−0.867436	+	0.497548i	$0.834234\pi$
$702$	0	0
$703$	−15.2962	−0.576907
$704$	0	0
$705$	0.182201	0.00686209
$706$	0	0
$707$	7.95554	0.299199
$708$	0	0
$709$	−3.92405	−0.147371	−0.0736855	−	0.997282i	$-0.523476\pi$
$709$	−3.92405	−0.147371	−0.0736855	+	0.997282i	$0.523476\pi$
$710$	0	0
$711$	27.0261	1.01356
$712$	0	0
$713$	−4.90454	−0.183676
$714$	0	0
$715$	0.169476	0.00633804
$716$	0	0
$717$	58.4197	2.18172
$718$	0	0
$719$	30.4266	1.13472	0.567361	−	0.823470i	$-0.307965\pi$
$719$	30.4266	1.13472	0.567361	+	0.823470i	$0.307965\pi$
$720$	0	0
$721$	−16.2662	−0.605786
$722$	0	0
$723$	20.2646	0.753650
$724$	0	0
$725$	−20.1283	−0.747545
$726$	0	0
$727$	−41.3266	−1.53272	−0.766359	−	0.642413i	$-0.777933\pi$
$727$	−41.3266	−1.53272	−0.766359	+	0.642413i	$0.777933\pi$
$728$	0	0
$729$	−30.5052	−1.12982
$730$	0	0
$731$	8.14488	0.301249
$732$	0	0
$733$	−48.0265	−1.77390	−0.886950	−	0.461865i	$-0.847181\pi$
$733$	−48.0265	−1.77390	−0.886950	+	0.461865i	$0.847181\pi$
$734$	0	0
$735$	0.00991478	0.000365712 0
$736$	0	0
$737$	−48.0685	−1.77063
$738$	0	0
$739$	9.24932	0.340242	0.170121	−	0.985423i	$-0.445584\pi$
$739$	9.24932	0.340242	0.170121	+	0.985423i	$0.445584\pi$
$740$	0	0
$741$	104.201	3.82791
$742$	0	0
$743$	−32.0169	−1.17459	−0.587294	−	0.809374i	$-0.699807\pi$
$743$	−32.0169	−1.17459	−0.587294	+	0.809374i	$0.699807\pi$
$744$	0	0
$745$	0.0231545	0.000848316 0
$746$	0	0
$747$	−50.7897	−1.85830
$748$	0	0
$749$	29.5315	1.07906
$750$	0	0
$751$	−19.4578	−0.710025	−0.355013	−	0.934861i	$-0.615523\pi$
$751$	−19.4578	−0.710025	−0.355013	+	0.934861i	$0.615523\pi$
$752$	0	0
$753$	2.43978	0.0889104
$754$	0	0
$755$	0.179941	0.00654873
$756$	0	0
$757$	−17.5305	−0.637158	−0.318579	−	0.947896i	$-0.603206\pi$
$757$	−17.5305	−0.637158	−0.318579	+	0.947896i	$0.603206\pi$
$758$	0	0
$759$	33.0274	1.19882
$760$	0	0
$761$	−17.8681	−0.647717	−0.323858	−	0.946106i	$-0.604980\pi$
$761$	−17.8681	−0.647717	−0.323858	+	0.946106i	$0.604980\pi$
$762$	0	0
$763$	−30.7927	−1.11477
$764$	0	0
$765$	0.0541106	0.00195637
$766$	0	0
$767$	4.58909	0.165702
$768$	0	0
$769$	−16.1192	−0.581272	−0.290636	−	0.956834i	$-0.593867\pi$
$769$	−16.1192	−0.581272	−0.290636	+	0.956834i	$0.593867\pi$
$770$	0	0
$771$	−31.0547	−1.11841
$772$	0	0
$773$	12.9929	0.467323	0.233662	−	0.972318i	$-0.424929\pi$
$773$	12.9929	0.467323	0.233662	+	0.972318i	$0.424929\pi$
$774$	0	0
$775$	−8.84249	−0.317632
$776$	0	0
$777$	15.3822	0.551833
$778$	0	0
$779$	60.7903	2.17804
$780$	0	0
$781$	16.3415	0.584743
$782$	0	0
$783$	−34.2959	−1.22563
$784$	0	0
$785$	0.160468	0.00572736
$786$	0	0
$787$	−11.4857	−0.409420	−0.204710	−	0.978823i	$-0.565625\pi$
$787$	−11.4857	−0.409420	−0.204710	+	0.978823i	$0.565625\pi$
$788$	0	0
$789$	−42.4222	−1.51027
$790$	0	0
$791$	−22.5123	−0.800447
$792$	0	0
$793$	−16.7540	−0.594953
$794$	0	0
$795$	−0.336453	−0.0119328
$796$	0	0
$797$	26.8215	0.950067	0.475034	−	0.879968i	$-0.342436\pi$
$797$	26.8215	0.950067	0.475034	+	0.879968i	$0.342436\pi$
$798$	0	0
$799$	−6.63037	−0.234566
$800$	0	0
$801$	−15.4203	−0.544851
$802$	0	0
$803$	−39.3434	−1.38840
$804$	0	0
$805$	0.0659623	0.00232487
$806$	0	0
$807$	12.2832	0.432391
$808$	0	0
$809$	−0.646924	−0.0227446	−0.0113723	−	0.999935i	$-0.503620\pi$
$809$	−0.646924	−0.0227446	−0.0113723	+	0.999935i	$0.503620\pi$
$810$	0	0
$811$	0.799808	0.0280851	0.0140425	−	0.999901i	$-0.495530\pi$
$811$	0.799808	0.0280851	0.0140425	+	0.999901i	$0.495530\pi$
$812$	0	0
$813$	69.9736	2.45408
$814$	0	0
$815$	0.00491390	0.000172126 0
$816$	0	0
$817$	−62.1254	−2.17349
$818$	0	0
$819$	−69.3129	−2.42199
$820$	0	0
$821$	22.8031	0.795833	0.397917	−	0.917422i	$-0.369733\pi$
$821$	22.8031	0.795833	0.397917	+	0.917422i	$0.369733\pi$
$822$	0	0
$823$	−9.48845	−0.330747	−0.165373	−	0.986231i	$-0.552883\pi$
$823$	−9.48845	−0.330747	−0.165373	+	0.986231i	$0.552883\pi$
$824$	0	0
$825$	59.5459	2.07312
$826$	0	0
$827$	−26.5499	−0.923231	−0.461616	−	0.887080i	$-0.652730\pi$
$827$	−26.5499	−0.923231	−0.461616	+	0.887080i	$0.652730\pi$
$828$	0	0
$829$	1.26352	0.0438838	0.0219419	−	0.999759i	$-0.493015\pi$
$829$	1.26352	0.0438838	0.0219419	+	0.999759i	$0.493015\pi$
$830$	0	0
$831$	61.1568	2.12151
$832$	0	0
$833$	−0.360803	−0.0125011
$834$	0	0
$835$	0.0761460	0.00263514
$836$	0	0
$837$	−15.0664	−0.520772
$838$	0	0
$839$	−30.7726	−1.06239	−0.531193	−	0.847251i	$-0.678256\pi$
$839$	−30.7726	−1.06239	−0.531193	+	0.847251i	$0.678256\pi$
$840$	0	0
$841$	−12.7935	−0.441157
$842$	0	0
$843$	22.8857	0.788226
$844$	0	0
$845$	0.0744001	0.00255944
$846$	0	0
$847$	−12.8964	−0.443124
$848$	0	0
$849$	5.87585	0.201659
$850$	0	0
$851$	−5.56141	−0.190643
$852$	0	0
$853$	35.6806	1.22168	0.610841	−	0.791753i	$-0.290832\pi$
$853$	35.6806	1.22168	0.610841	+	0.791753i	$0.290832\pi$
$854$	0	0
$855$	−0.412731	−0.0141151
$856$	0	0
$857$	53.9755	1.84377	0.921884	−	0.387467i	$-0.126650\pi$
$857$	53.9755	1.84377	0.921884	+	0.387467i	$0.126650\pi$
$858$	0	0
$859$	53.4106	1.82235	0.911174	−	0.412022i	$-0.135177\pi$
$859$	53.4106	1.82235	0.911174	+	0.412022i	$0.135177\pi$
$860$	0	0
$861$	−61.1320	−2.08337
$862$	0	0
$863$	6.57524	0.223824	0.111912	−	0.993718i	$-0.464303\pi$
$863$	6.57524	0.223824	0.111912	+	0.993718i	$0.464303\pi$
$864$	0	0
$865$	−0.0383014	−0.00130229
$866$	0	0
$867$	−2.97688	−0.101100
$868$	0	0
$869$	18.4451	0.625708
$870$	0	0
$871$	−55.1390	−1.86831
$872$	0	0
$873$	109.549	3.70768
$874$	0	0
$875$	0.237852	0.00804086
$876$	0	0
$877$	23.1555	0.781906	0.390953	−	0.920411i	$-0.372146\pi$
$877$	23.1555	0.781906	0.390953	+	0.920411i	$0.372146\pi$
$878$	0	0
$879$	15.7775	0.532162
$880$	0	0
$881$	6.91834	0.233085	0.116542	−	0.993186i	$-0.462819\pi$
$881$	6.91834	0.233085	0.116542	+	0.993186i	$0.462819\pi$
$882$	0	0
$883$	0.0112153	0.000377425 0	0.000188713	−	1.00000i	$-0.499940\pi$
$883$	0.0112153	0.000377425 0	0.000188713		1.00000i	$0.499940\pi$
$884$	0	0
$885$	−0.0274798	−0.000923723 0
$886$	0	0
$887$	−36.2065	−1.21569	−0.607847	−	0.794054i	$-0.707967\pi$
$887$	−36.2065	−1.21569	−0.607847	+	0.794054i	$0.707967\pi$
$888$	0	0
$889$	−50.2273	−1.68457
$890$	0	0
$891$	31.1056	1.04208
$892$	0	0
$893$	50.5734	1.69237
$894$	0	0
$895$	0.126530	0.00422943
$896$	0	0
$897$	37.8855	1.26496
$898$	0	0
$899$	7.11961	0.237452
$900$	0	0
$901$	12.2437	0.407895
$902$	0	0
$903$	62.4746	2.07902
$904$	0	0
$905$	−0.184944	−0.00614774
$906$	0	0
$907$	40.2459	1.33634	0.668171	−	0.744008i	$-0.267077\pi$
$907$	40.2459	1.33634	0.668171	+	0.744008i	$0.267077\pi$
$908$	0	0
$909$	−18.0985	−0.600288
$910$	0	0
$911$	18.4981	0.612871	0.306435	−	0.951891i	$-0.400864\pi$
$911$	18.4981	0.612871	0.306435	+	0.951891i	$0.400864\pi$
$912$	0	0
$913$	−34.6637	−1.14720
$914$	0	0
$915$	0.100324	0.00331662
$916$	0	0
$917$	56.4029	1.86259
$918$	0	0
$919$	−11.0088	−0.363146	−0.181573	−	0.983377i	$-0.558119\pi$
$919$	−11.0088	−0.363146	−0.181573	+	0.983377i	$0.558119\pi$
$920$	0	0
$921$	−82.5094	−2.71878
$922$	0	0
$923$	18.7451	0.617004
$924$	0	0
$925$	−10.0268	−0.329679
$926$	0	0
$927$	37.0049	1.21540
$928$	0	0
$929$	42.7178	1.40152	0.700762	−	0.713395i	$-0.252843\pi$
$929$	42.7178	1.40152	0.700762	+	0.713395i	$0.252843\pi$
$930$	0	0
$931$	2.75203	0.0901943
$932$	0	0
$933$	−34.2332	−1.12074
$934$	0	0
$935$	0.0369302	0.00120775
$936$	0	0
$937$	1.76695	0.0577236	0.0288618	−	0.999583i	$-0.490812\pi$
$937$	1.76695	0.0577236	0.0288618	+	0.999583i	$0.490812\pi$
$938$	0	0
$939$	−33.9942	−1.10936
$940$	0	0
$941$	−6.16330	−0.200918	−0.100459	−	0.994941i	$-0.532031\pi$
$941$	−6.16330	−0.200918	−0.100459	+	0.994941i	$0.532031\pi$
$942$	0	0
$943$	22.1022	0.719747
$944$	0	0
$945$	0.202632	0.00659162
$946$	0	0
$947$	41.3956	1.34518	0.672588	−	0.740017i	$-0.265183\pi$
$947$	41.3956	1.34518	0.672588	+	0.740017i	$0.265183\pi$
$948$	0	0
$949$	−45.1305	−1.46500
$950$	0	0
$951$	29.4574	0.955223
$952$	0	0
$953$	7.93417	0.257013	0.128507	−	0.991709i	$-0.458982\pi$
$953$	7.93417	0.257013	0.128507	+	0.991709i	$0.458982\pi$
$954$	0	0
$955$	−0.0379191	−0.00122703
$956$	0	0
$957$	−47.9439	−1.54981
$958$	0	0
$959$	−36.1314	−1.16674
$960$	0	0
$961$	−27.8723	−0.899107
$962$	0	0
$963$	−67.1828	−2.16494
$964$	0	0
$965$	−0.199618	−0.00642592
$966$	0	0
$967$	−31.2073	−1.00356	−0.501780	−	0.864995i	$-0.667321\pi$
$967$	−31.2073	−1.00356	−0.501780	+	0.864995i	$0.667321\pi$
$968$	0	0
$969$	22.7062	0.729429
$970$	0	0
$971$	−22.9865	−0.737672	−0.368836	−	0.929494i	$-0.620244\pi$
$971$	−22.9865	−0.737672	−0.368836	+	0.929494i	$0.620244\pi$
$972$	0	0
$973$	−22.6155	−0.725020
$974$	0	0
$975$	68.3046	2.18750
$976$	0	0
$977$	35.5944	1.13876	0.569382	−	0.822073i	$-0.307182\pi$
$977$	35.5944	1.13876	0.569382	+	0.822073i	$0.307182\pi$
$978$	0	0
$979$	−10.5243	−0.336358
$980$	0	0
$981$	70.0518	2.23658
$982$	0	0
$983$	−3.59056	−0.114521	−0.0572606	−	0.998359i	$-0.518237\pi$
$983$	−3.59056	−0.114521	−0.0572606	+	0.998359i	$0.518237\pi$
$984$	0	0
$985$	0.0202541	0.000645351 0
$986$	0	0
$987$	−50.8576	−1.61882
$988$	0	0
$989$	−22.5876	−0.718245
$990$	0	0
$991$	51.2375	1.62761	0.813807	−	0.581136i	$-0.197392\pi$
$991$	51.2375	1.62761	0.813807	+	0.581136i	$0.197392\pi$
$992$	0	0
$993$	−9.44061	−0.299589
$994$	0	0
$995$	−0.0678678	−0.00215155
$996$	0	0
$997$	35.1960	1.11467	0.557333	−	0.830289i	$-0.311825\pi$
$997$	35.1960	1.11467	0.557333	+	0.830289i	$0.311825\pi$
$998$	0	0
$999$	−17.0843	−0.540524

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.z.1.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.z.1.3	✓	24	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.97688 q^{3} +0.00923108 q^{5} -2.57666 q^{7} +5.86179 q^{9} +O(q^{10})\)	\(q-2.97688 q^{3} +0.00923108 q^{5} -2.57666 q^{7} +5.86179 q^{9} +4.00063 q^{11} +4.58909 q^{13} -0.0274798 q^{15} +1.00000 q^{17} -7.62753 q^{19} +7.67041 q^{21} -2.77323 q^{23} -4.99991 q^{25} -8.51918 q^{27} +4.02572 q^{29} +1.76853 q^{31} -11.9094 q^{33} -0.0237854 q^{35} +2.00539 q^{37} -13.6611 q^{39} -7.96984 q^{41} +8.14488 q^{43} +0.0541106 q^{45} -6.63037 q^{47} -0.360803 q^{49} -2.97688 q^{51} +12.2437 q^{53} +0.0369302 q^{55} +22.7062 q^{57} +1.00000 q^{59} -3.65084 q^{61} -15.1039 q^{63} +0.0423623 q^{65} -12.0152 q^{67} +8.25556 q^{69} +4.08472 q^{71} -9.83430 q^{73} +14.8841 q^{75} -10.3083 q^{77} +4.61055 q^{79} +7.77518 q^{81} -8.66454 q^{83} +0.00923108 q^{85} -11.9841 q^{87} -2.63066 q^{89} -11.8245 q^{91} -5.26469 q^{93} -0.0704104 q^{95} +18.6887 q^{97} +23.4509 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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