LMFDB - Embedded newform 8020.2.a.f.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, 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("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.332188 q^{3} +1.00000 q^{5} +0.507569 q^{7} -2.88965 q^{9} +O(q^{10})$	$q+0.332188 q^{3} +1.00000 q^{5} +0.507569 q^{7} -2.88965 q^{9} -0.290591 q^{11} -5.96145 q^{13} +0.332188 q^{15} -2.16316 q^{17} +8.41629 q^{19} +0.168608 q^{21} -3.58103 q^{23} +1.00000 q^{25} -1.95647 q^{27} +0.682488 q^{29} +4.02052 q^{31} -0.0965309 q^{33} +0.507569 q^{35} -5.47167 q^{37} -1.98032 q^{39} -9.91624 q^{41} +12.1241 q^{43} -2.88965 q^{45} +6.17194 q^{47} -6.74237 q^{49} -0.718578 q^{51} +10.7032 q^{53} -0.290591 q^{55} +2.79580 q^{57} +5.15732 q^{59} -4.13842 q^{61} -1.46670 q^{63} -5.96145 q^{65} +9.13610 q^{67} -1.18958 q^{69} +7.62587 q^{71} -1.80101 q^{73} +0.332188 q^{75} -0.147495 q^{77} -10.2955 q^{79} +8.01903 q^{81} +5.72854 q^{83} -2.16316 q^{85} +0.226715 q^{87} -9.17481 q^{89} -3.02584 q^{91} +1.33557 q^{93} +8.41629 q^{95} +5.10705 q^{97} +0.839706 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	$ 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) $ 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * 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
$3$	0.332188	0.191789	0.0958945	−	0.995391i	$-0.469429\pi$
$3$	0.332188	0.191789	0.0958945	+	0.995391i	$0.469429\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.507569	0.191843	0.0959215	−	0.995389i	$-0.469420\pi$
$7$	0.507569	0.191843	0.0959215	+	0.995389i	$0.469420\pi$
$8$	0	0
$9$	−2.88965	−0.963217
$10$	0	0
$11$	−0.290591	−0.0876164	−0.0438082	−	0.999040i	$-0.513949\pi$
$11$	−0.290591	−0.0876164	−0.0438082	+	0.999040i	$0.513949\pi$
$12$	0	0
$13$	−5.96145	−1.65341	−0.826704	−	0.562637i	$-0.809787\pi$
$13$	−5.96145	−1.65341	−0.826704	+	0.562637i	$0.809787\pi$
$14$	0	0
$15$	0.332188	0.0857707
$16$	0	0
$17$	−2.16316	−0.524644	−0.262322	−	0.964980i	$-0.584488\pi$
$17$	−2.16316	−0.524644	−0.262322	+	0.964980i	$0.584488\pi$
$18$	0	0
$19$	8.41629	1.93083	0.965415	−	0.260717i	$-0.0839590\pi$
$19$	8.41629	1.93083	0.965415	+	0.260717i	$0.0839590\pi$
$20$	0	0
$21$	0.168608	0.0367934
$22$	0	0
$23$	−3.58103	−0.746697	−0.373349	−	0.927691i	$-0.621790\pi$
$23$	−3.58103	−0.746697	−0.373349	+	0.927691i	$0.621790\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.95647	−0.376524
$28$	0	0
$29$	0.682488	0.126735	0.0633674	−	0.997990i	$-0.479816\pi$
$29$	0.682488	0.126735	0.0633674	+	0.997990i	$0.479816\pi$
$30$	0	0
$31$	4.02052	0.722106	0.361053	−	0.932545i	$-0.382417\pi$
$31$	4.02052	0.722106	0.361053	+	0.932545i	$0.382417\pi$
$32$	0	0
$33$	−0.0965309	−0.0168039
$34$	0	0
$35$	0.507569	0.0857948
$36$	0	0
$37$	−5.47167	−0.899537	−0.449768	−	0.893145i	$-0.648493\pi$
$37$	−5.47167	−0.899537	−0.449768	+	0.893145i	$0.648493\pi$
$38$	0	0
$39$	−1.98032	−0.317106
$40$	0	0
$41$	−9.91624	−1.54866	−0.774328	−	0.632784i	$-0.781912\pi$
$41$	−9.91624	−1.54866	−0.774328	+	0.632784i	$0.781912\pi$
$42$	0	0
$43$	12.1241	1.84891	0.924456	−	0.381289i	$-0.124520\pi$
$43$	12.1241	1.84891	0.924456	+	0.381289i	$0.124520\pi$
$44$	0	0
$45$	−2.88965	−0.430764
$46$	0	0
$47$	6.17194	0.900269	0.450135	−	0.892961i	$-0.351376\pi$
$47$	6.17194	0.900269	0.450135	+	0.892961i	$0.351376\pi$
$48$	0	0
$49$	−6.74237	−0.963196
$50$	0	0
$51$	−0.718578	−0.100621
$52$	0	0
$53$	10.7032	1.47020	0.735102	−	0.677956i	$-0.237134\pi$
$53$	10.7032	1.47020	0.735102	+	0.677956i	$0.237134\pi$
$54$	0	0
$55$	−0.290591	−0.0391833
$56$	0	0
$57$	2.79580	0.370312
$58$	0	0
$59$	5.15732	0.671425	0.335713	−	0.941964i	$-0.391023\pi$
$59$	5.15732	0.671425	0.335713	+	0.941964i	$0.391023\pi$
$60$	0	0
$61$	−4.13842	−0.529870	−0.264935	−	0.964266i	$-0.585351\pi$
$61$	−4.13842	−0.529870	−0.264935	+	0.964266i	$0.585351\pi$
$62$	0	0
$63$	−1.46670	−0.184786
$64$	0	0
$65$	−5.96145	−0.739427
$66$	0	0
$67$	9.13610	1.11615	0.558076	−	0.829790i	$-0.311540\pi$
$67$	9.13610	1.11615	0.558076	+	0.829790i	$0.311540\pi$
$68$	0	0
$69$	−1.18958	−0.143208
$70$	0	0
$71$	7.62587	0.905024	0.452512	−	0.891758i	$-0.350528\pi$
$71$	7.62587	0.905024	0.452512	+	0.891758i	$0.350528\pi$
$72$	0	0
$73$	−1.80101	−0.210793	−0.105396	−	0.994430i	$-0.533611\pi$
$73$	−1.80101	−0.210793	−0.105396	+	0.994430i	$0.533611\pi$
$74$	0	0
$75$	0.332188	0.0383578
$76$	0	0
$77$	−0.147495	−0.0168086
$78$	0	0
$79$	−10.2955	−1.15834	−0.579168	−	0.815208i	$-0.696623\pi$
$79$	−10.2955	−1.15834	−0.579168	+	0.815208i	$0.696623\pi$
$80$	0	0
$81$	8.01903	0.891004
$82$	0	0
$83$	5.72854	0.628789	0.314394	−	0.949292i	$-0.398199\pi$
$83$	5.72854	0.628789	0.314394	+	0.949292i	$0.398199\pi$
$84$	0	0
$85$	−2.16316	−0.234628
$86$	0	0
$87$	0.226715	0.0243064
$88$	0	0
$89$	−9.17481	−0.972528	−0.486264	−	0.873812i	$-0.661641\pi$
$89$	−9.17481	−0.972528	−0.486264	+	0.873812i	$0.661641\pi$
$90$	0	0
$91$	−3.02584	−0.317195
$92$	0	0
$93$	1.33557	0.138492
$94$	0	0
$95$	8.41629	0.863494
$96$	0	0
$97$	5.10705	0.518543	0.259271	−	0.965805i	$-0.416518\pi$
$97$	5.10705	0.518543	0.259271	+	0.965805i	$0.416518\pi$
$98$	0	0
$99$	0.839706	0.0843936
$100$	0	0
$101$	7.94523	0.790580	0.395290	−	0.918556i	$-0.370644\pi$
$101$	7.94523	0.790580	0.395290	+	0.918556i	$0.370644\pi$
$102$	0	0
$103$	−2.73176	−0.269169	−0.134584	−	0.990902i	$-0.542970\pi$
$103$	−2.73176	−0.269169	−0.134584	+	0.990902i	$0.542970\pi$
$104$	0	0
$105$	0.168608	0.0164545
$106$	0	0
$107$	11.1307	1.07605	0.538024	−	0.842929i	$-0.319171\pi$
$107$	11.1307	1.07605	0.538024	+	0.842929i	$0.319171\pi$
$108$	0	0
$109$	15.7161	1.50533	0.752664	−	0.658405i	$-0.228768\pi$
$109$	15.7161	1.50533	0.752664	+	0.658405i	$0.228768\pi$
$110$	0	0
$111$	−1.81763	−0.172521
$112$	0	0
$113$	−2.57309	−0.242056	−0.121028	−	0.992649i	$-0.538619\pi$
$113$	−2.57309	−0.242056	−0.121028	+	0.992649i	$0.538619\pi$
$114$	0	0
$115$	−3.58103	−0.333933
$116$	0	0
$117$	17.2265	1.59259
$118$	0	0
$119$	−1.09795	−0.100649
$120$	0	0
$121$	−10.9156	−0.992323
$122$	0	0
$123$	−3.29406	−0.297015
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−1.18536	−0.105184	−0.0525919	−	0.998616i	$-0.516748\pi$
$127$	−1.18536	−0.105184	−0.0525919	+	0.998616i	$0.516748\pi$
$128$	0	0
$129$	4.02750	0.354601
$130$	0	0
$131$	7.98192	0.697384	0.348692	−	0.937237i	$-0.386626\pi$
$131$	7.98192	0.697384	0.348692	+	0.937237i	$0.386626\pi$
$132$	0	0
$133$	4.27185	0.370416
$134$	0	0
$135$	−1.95647	−0.168386
$136$	0	0
$137$	3.17238	0.271034	0.135517	−	0.990775i	$-0.456730\pi$
$137$	3.17238	0.271034	0.135517	+	0.990775i	$0.456730\pi$
$138$	0	0
$139$	−15.2230	−1.29120	−0.645598	−	0.763677i	$-0.723392\pi$
$139$	−15.2230	−1.29120	−0.645598	+	0.763677i	$0.723392\pi$
$140$	0	0
$141$	2.05025	0.172662
$142$	0	0
$143$	1.73234	0.144866
$144$	0	0
$145$	0.682488	0.0566775
$146$	0	0
$147$	−2.23974	−0.184731
$148$	0	0
$149$	13.4084	1.09846	0.549229	−	0.835672i	$-0.314922\pi$
$149$	13.4084	1.09846	0.549229	+	0.835672i	$0.314922\pi$
$150$	0	0
$151$	−8.79975	−0.716114	−0.358057	−	0.933700i	$-0.616561\pi$
$151$	−8.79975	−0.716114	−0.358057	+	0.933700i	$0.616561\pi$
$152$	0	0
$153$	6.25079	0.505346
$154$	0	0
$155$	4.02052	0.322936
$156$	0	0
$157$	11.8997	0.949703	0.474851	−	0.880066i	$-0.342502\pi$
$157$	11.8997	0.949703	0.474851	+	0.880066i	$0.342502\pi$
$158$	0	0
$159$	3.55550	0.281969
$160$	0	0
$161$	−1.81762	−0.143249
$162$	0	0
$163$	17.7612	1.39117	0.695584	−	0.718445i	$-0.255146\pi$
$163$	17.7612	1.39117	0.695584	+	0.718445i	$0.255146\pi$
$164$	0	0
$165$	−0.0965309	−0.00751492
$166$	0	0
$167$	18.3378	1.41902	0.709509	−	0.704696i	$-0.248917\pi$
$167$	18.3378	1.41902	0.709509	+	0.704696i	$0.248917\pi$
$168$	0	0
$169$	22.5389	1.73376
$170$	0	0
$171$	−24.3202	−1.85981
$172$	0	0
$173$	8.42556	0.640584	0.320292	−	0.947319i	$-0.396219\pi$
$173$	8.42556	0.640584	0.320292	+	0.947319i	$0.396219\pi$
$174$	0	0
$175$	0.507569	0.0383686
$176$	0	0
$177$	1.71320	0.128772
$178$	0	0
$179$	2.29448	0.171497	0.0857486	−	0.996317i	$-0.472672\pi$
$179$	2.29448	0.171497	0.0857486	+	0.996317i	$0.472672\pi$
$180$	0	0
$181$	−16.6333	−1.23634	−0.618171	−	0.786044i	$-0.712126\pi$
$181$	−16.6333	−1.23634	−0.618171	+	0.786044i	$0.712126\pi$
$182$	0	0
$183$	−1.37474	−0.101623
$184$	0	0
$185$	−5.47167	−0.402285
$186$	0	0
$187$	0.628595	0.0459674
$188$	0	0
$189$	−0.993045	−0.0722334
$190$	0	0
$191$	−10.0711	−0.728722	−0.364361	−	0.931258i	$-0.618713\pi$
$191$	−10.0711	−0.728722	−0.364361	+	0.931258i	$0.618713\pi$
$192$	0	0
$193$	−0.158369	−0.0113997	−0.00569983	−	0.999984i	$-0.501814\pi$
$193$	−0.158369	−0.0113997	−0.00569983	+	0.999984i	$0.501814\pi$
$194$	0	0
$195$	−1.98032	−0.141814
$196$	0	0
$197$	9.88767	0.704467	0.352234	−	0.935912i	$-0.385422\pi$
$197$	9.88767	0.704467	0.352234	+	0.935912i	$0.385422\pi$
$198$	0	0
$199$	28.0256	1.98669	0.993343	−	0.115197i	$-0.0367498\pi$
$199$	28.0256	1.98669	0.993343	+	0.115197i	$0.0367498\pi$
$200$	0	0
$201$	3.03491	0.214066
$202$	0	0
$203$	0.346409	0.0243132
$204$	0	0
$205$	−9.91624	−0.692580
$206$	0	0
$207$	10.3479	0.719231
$208$	0	0
$209$	−2.44570	−0.169172
$210$	0	0
$211$	19.5433	1.34542	0.672708	−	0.739908i	$-0.265131\pi$
$211$	19.5433	1.34542	0.672708	+	0.739908i	$0.265131\pi$
$212$	0	0
$213$	2.53322	0.173574
$214$	0	0
$215$	12.1241	0.826859
$216$	0	0
$217$	2.04069	0.138531
$218$	0	0
$219$	−0.598276	−0.0404278
$220$	0	0
$221$	12.8956	0.867451
$222$	0	0
$223$	28.6841	1.92083	0.960414	−	0.278578i	$-0.0898630\pi$
$223$	28.6841	1.92083	0.960414	+	0.278578i	$0.0898630\pi$
$224$	0	0
$225$	−2.88965	−0.192643
$226$	0	0
$227$	−3.48680	−0.231427	−0.115713	−	0.993283i	$-0.536915\pi$
$227$	−3.48680	−0.231427	−0.115713	+	0.993283i	$0.536915\pi$
$228$	0	0
$229$	−4.41192	−0.291548	−0.145774	−	0.989318i	$-0.546567\pi$
$229$	−4.41192	−0.291548	−0.145774	+	0.989318i	$0.546567\pi$
$230$	0	0
$231$	−0.0489961	−0.00322370
$232$	0	0
$233$	8.63548	0.565729	0.282865	−	0.959160i	$-0.408715\pi$
$233$	8.63548	0.565729	0.282865	+	0.959160i	$0.408715\pi$
$234$	0	0
$235$	6.17194	0.402613
$236$	0	0
$237$	−3.42005	−0.222156
$238$	0	0
$239$	10.9720	0.709719	0.354859	−	0.934920i	$-0.384529\pi$
$239$	10.9720	0.709719	0.354859	+	0.934920i	$0.384529\pi$
$240$	0	0
$241$	−10.4351	−0.672186	−0.336093	−	0.941829i	$-0.609106\pi$
$241$	−10.4351	−0.672186	−0.336093	+	0.941829i	$0.609106\pi$
$242$	0	0
$243$	8.53325	0.547408
$244$	0	0
$245$	−6.74237	−0.430754
$246$	0	0
$247$	−50.1733	−3.19245
$248$	0	0
$249$	1.90295	0.120595
$250$	0	0
$251$	−2.28939	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$251$	−2.28939	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$252$	0	0
$253$	1.04062	0.0654229
$254$	0	0
$255$	−0.718578	−0.0449991
$256$	0	0
$257$	−6.14610	−0.383383	−0.191692	−	0.981455i	$-0.561397\pi$
$257$	−6.14610	−0.383383	−0.191692	+	0.981455i	$0.561397\pi$
$258$	0	0
$259$	−2.77725	−0.172570
$260$	0	0
$261$	−1.97215	−0.122073
$262$	0	0
$263$	31.8319	1.96284	0.981418	−	0.191880i	$-0.0614585\pi$
$263$	31.8319	1.96284	0.981418	+	0.191880i	$0.0614585\pi$
$264$	0	0
$265$	10.7032	0.657495
$266$	0	0
$267$	−3.04777	−0.186520
$268$	0	0
$269$	20.5640	1.25381	0.626905	−	0.779096i	$-0.284322\pi$
$269$	20.5640	1.25381	0.626905	+	0.779096i	$0.284322\pi$
$270$	0	0
$271$	−20.4542	−1.24251	−0.621253	−	0.783610i	$-0.713376\pi$
$271$	−20.4542	−1.24251	−0.621253	+	0.783610i	$0.713376\pi$
$272$	0	0
$273$	−1.00515	−0.0608345
$274$	0	0
$275$	−0.290591	−0.0175233
$276$	0	0
$277$	15.2108	0.913927	0.456964	−	0.889485i	$-0.348937\pi$
$277$	15.2108	0.913927	0.456964	+	0.889485i	$0.348937\pi$
$278$	0	0
$279$	−11.6179	−0.695545
$280$	0	0
$281$	14.4264	0.860605	0.430302	−	0.902685i	$-0.358407\pi$
$281$	14.4264	0.860605	0.430302	+	0.902685i	$0.358407\pi$
$282$	0	0
$283$	−13.9703	−0.830446	−0.415223	−	0.909720i	$-0.636297\pi$
$283$	−13.9703	−0.830446	−0.415223	+	0.909720i	$0.636297\pi$
$284$	0	0
$285$	2.79580	0.165609
$286$	0	0
$287$	−5.03317	−0.297099
$288$	0	0
$289$	−12.3207	−0.724749
$290$	0	0
$291$	1.69650	0.0994509
$292$	0	0
$293$	−20.6021	−1.20359	−0.601794	−	0.798651i	$-0.705547\pi$
$293$	−20.6021	−1.20359	−0.601794	+	0.798651i	$0.705547\pi$
$294$	0	0
$295$	5.15732	0.300271
$296$	0	0
$297$	0.568533	0.0329897
$298$	0	0
$299$	21.3481	1.23460
$300$	0	0
$301$	6.15383	0.354701
$302$	0	0
$303$	2.63931	0.151625
$304$	0	0
$305$	−4.13842	−0.236965
$306$	0	0
$307$	−5.18541	−0.295947	−0.147974	−	0.988991i	$-0.547275\pi$
$307$	−5.18541	−0.295947	−0.147974	+	0.988991i	$0.547275\pi$
$308$	0	0
$309$	−0.907460	−0.0516236
$310$	0	0
$311$	−27.0816	−1.53566	−0.767829	−	0.640655i	$-0.778663\pi$
$311$	−27.0816	−1.53566	−0.767829	+	0.640655i	$0.778663\pi$
$312$	0	0
$313$	−20.7090	−1.17054	−0.585271	−	0.810838i	$-0.699012\pi$
$313$	−20.7090	−1.17054	−0.585271	+	0.810838i	$0.699012\pi$
$314$	0	0
$315$	−1.46670	−0.0826390
$316$	0	0
$317$	−24.6487	−1.38441	−0.692204	−	0.721701i	$-0.743360\pi$
$317$	−24.6487	−1.38441	−0.692204	+	0.721701i	$0.743360\pi$
$318$	0	0
$319$	−0.198325	−0.0111041
$320$	0	0
$321$	3.69750	0.206374
$322$	0	0
$323$	−18.2058	−1.01300
$324$	0	0
$325$	−5.96145	−0.330682
$326$	0	0
$327$	5.22070	0.288706
$328$	0	0
$329$	3.13268	0.172710
$330$	0	0
$331$	−13.8099	−0.759060	−0.379530	−	0.925179i	$-0.623914\pi$
$331$	−13.8099	−0.759060	−0.379530	+	0.925179i	$0.623914\pi$
$332$	0	0
$333$	15.8112	0.866449
$334$	0	0
$335$	9.13610	0.499158
$336$	0	0
$337$	−19.3401	−1.05352	−0.526762	−	0.850013i	$-0.676594\pi$
$337$	−19.3401	−1.05352	−0.526762	+	0.850013i	$0.676594\pi$
$338$	0	0
$339$	−0.854750	−0.0464237
$340$	0	0
$341$	−1.16832	−0.0632683
$342$	0	0
$343$	−6.97520	−0.376625
$344$	0	0
$345$	−1.18958	−0.0640447
$346$	0	0
$347$	−30.5705	−1.64111	−0.820556	−	0.571566i	$-0.806336\pi$
$347$	−30.5705	−1.64111	−0.820556	+	0.571566i	$0.806336\pi$
$348$	0	0
$349$	−9.91262	−0.530610	−0.265305	−	0.964165i	$-0.585473\pi$
$349$	−9.91262	−0.530610	−0.265305	+	0.964165i	$0.585473\pi$
$350$	0	0
$351$	11.6634	0.622547
$352$	0	0
$353$	−23.7375	−1.26342	−0.631711	−	0.775204i	$-0.717647\pi$
$353$	−23.7375	−1.26342	−0.631711	+	0.775204i	$0.717647\pi$
$354$	0	0
$355$	7.62587	0.404739
$356$	0	0
$357$	−0.364728	−0.0193034
$358$	0	0
$359$	1.32642	0.0700058	0.0350029	−	0.999387i	$-0.488856\pi$
$359$	1.32642	0.0700058	0.0350029	+	0.999387i	$0.488856\pi$
$360$	0	0
$361$	51.8340	2.72811
$362$	0	0
$363$	−3.62602	−0.190317
$364$	0	0
$365$	−1.80101	−0.0942694
$366$	0	0
$367$	12.5918	0.657284	0.328642	−	0.944455i	$-0.393409\pi$
$367$	12.5918	0.657284	0.328642	+	0.944455i	$0.393409\pi$
$368$	0	0
$369$	28.6545	1.49169
$370$	0	0
$371$	5.43263	0.282048
$372$	0	0
$373$	29.1720	1.51047	0.755235	−	0.655454i	$-0.227523\pi$
$373$	29.1720	1.51047	0.755235	+	0.655454i	$0.227523\pi$
$374$	0	0
$375$	0.332188	0.0171541
$376$	0	0
$377$	−4.06862	−0.209544
$378$	0	0
$379$	13.4938	0.693129	0.346564	−	0.938026i	$-0.387348\pi$
$379$	13.4938	0.693129	0.346564	+	0.938026i	$0.387348\pi$
$380$	0	0
$381$	−0.393763	−0.0201731
$382$	0	0
$383$	−0.333688	−0.0170506	−0.00852532	−	0.999964i	$-0.502714\pi$
$383$	−0.333688	−0.0170506	−0.00852532	+	0.999964i	$0.502714\pi$
$384$	0	0
$385$	−0.147495	−0.00751703
$386$	0	0
$387$	−35.0345	−1.78090
$388$	0	0
$389$	−11.1556	−0.565610	−0.282805	−	0.959177i	$-0.591265\pi$
$389$	−11.1556	−0.565610	−0.282805	+	0.959177i	$0.591265\pi$
$390$	0	0
$391$	7.74636	0.391750
$392$	0	0
$393$	2.65150	0.133751
$394$	0	0
$395$	−10.2955	−0.518024
$396$	0	0
$397$	32.3137	1.62178	0.810890	−	0.585199i	$-0.198984\pi$
$397$	32.3137	1.62178	0.810890	+	0.585199i	$0.198984\pi$
$398$	0	0
$399$	1.41906	0.0710418
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−23.9681	−1.19394
$404$	0	0
$405$	8.01903	0.398469
$406$	0	0
$407$	1.59002	0.0788142
$408$	0	0
$409$	1.53092	0.0756990	0.0378495	−	0.999283i	$-0.487949\pi$
$409$	1.53092	0.0756990	0.0378495	+	0.999283i	$0.487949\pi$
$410$	0	0
$411$	1.05383	0.0519814
$412$	0	0
$413$	2.61769	0.128808
$414$	0	0
$415$	5.72854	0.281203
$416$	0	0
$417$	−5.05690	−0.247637
$418$	0	0
$419$	19.1279	0.934461	0.467230	−	0.884136i	$-0.345252\pi$
$419$	19.1279	0.934461	0.467230	+	0.884136i	$0.345252\pi$
$420$	0	0
$421$	9.46136	0.461118	0.230559	−	0.973058i	$-0.425944\pi$
$421$	9.46136	0.461118	0.230559	+	0.973058i	$0.425944\pi$
$422$	0	0
$423$	−17.8347	−0.867155
$424$	0	0
$425$	−2.16316	−0.104929
$426$	0	0
$427$	−2.10053	−0.101652
$428$	0	0
$429$	0.575464	0.0277837
$430$	0	0
$431$	1.89354	0.0912086	0.0456043	−	0.998960i	$-0.485479\pi$
$431$	1.89354	0.0912086	0.0456043	+	0.998960i	$0.485479\pi$
$432$	0	0
$433$	15.1403	0.727596	0.363798	−	0.931478i	$-0.381480\pi$
$433$	15.1403	0.727596	0.363798	+	0.931478i	$0.381480\pi$
$434$	0	0
$435$	0.226715	0.0108701
$436$	0	0
$437$	−30.1390	−1.44175
$438$	0	0
$439$	−33.9088	−1.61838	−0.809190	−	0.587548i	$-0.800094\pi$
$439$	−33.9088	−1.61838	−0.809190	+	0.587548i	$0.800094\pi$
$440$	0	0
$441$	19.4831	0.927767
$442$	0	0
$443$	6.27126	0.297957	0.148978	−	0.988840i	$-0.452402\pi$
$443$	6.27126	0.297957	0.148978	+	0.988840i	$0.452402\pi$
$444$	0	0
$445$	−9.17481	−0.434928
$446$	0	0
$447$	4.45411	0.210672
$448$	0	0
$449$	−32.0989	−1.51484	−0.757421	−	0.652927i	$-0.773541\pi$
$449$	−32.0989	−1.51484	−0.757421	+	0.652927i	$0.773541\pi$
$450$	0	0
$451$	2.88157	0.135688
$452$	0	0
$453$	−2.92318	−0.137343
$454$	0	0
$455$	−3.02584	−0.141854
$456$	0	0
$457$	2.55444	0.119492	0.0597458	−	0.998214i	$-0.480971\pi$
$457$	2.55444	0.119492	0.0597458	+	0.998214i	$0.480971\pi$
$458$	0	0
$459$	4.23217	0.197541
$460$	0	0
$461$	−28.5913	−1.33163	−0.665814	−	0.746118i	$-0.731916\pi$
$461$	−28.5913	−1.33163	−0.665814	+	0.746118i	$0.731916\pi$
$462$	0	0
$463$	−10.3325	−0.480193	−0.240097	−	0.970749i	$-0.577179\pi$
$463$	−10.3325	−0.480193	−0.240097	+	0.970749i	$0.577179\pi$
$464$	0	0
$465$	1.33557	0.0619355
$466$	0	0
$467$	−36.6486	−1.69590	−0.847948	−	0.530080i	$-0.822162\pi$
$467$	−36.6486	−1.69590	−0.847948	+	0.530080i	$0.822162\pi$
$468$	0	0
$469$	4.63720	0.214126
$470$	0	0
$471$	3.95296	0.182143
$472$	0	0
$473$	−3.52316	−0.161995
$474$	0	0
$475$	8.41629	0.386166
$476$	0	0
$477$	−30.9287	−1.41613
$478$	0	0
$479$	18.3553	0.838675	0.419338	−	0.907830i	$-0.362262\pi$
$479$	18.3553	0.838675	0.419338	+	0.907830i	$0.362262\pi$
$480$	0	0
$481$	32.6191	1.48730
$482$	0	0
$483$	−0.603793	−0.0274735
$484$	0	0
$485$	5.10705	0.231899
$486$	0	0
$487$	−0.237708	−0.0107716	−0.00538579	−	0.999985i	$-0.501714\pi$
$487$	−0.237708	−0.0107716	−0.00538579	+	0.999985i	$0.501714\pi$
$488$	0	0
$489$	5.90008	0.266811
$490$	0	0
$491$	19.4851	0.879350	0.439675	−	0.898157i	$-0.355094\pi$
$491$	19.4851	0.879350	0.439675	+	0.898157i	$0.355094\pi$
$492$	0	0
$493$	−1.47633	−0.0664907
$494$	0	0
$495$	0.839706	0.0377420
$496$	0	0
$497$	3.87065	0.173622
$498$	0	0
$499$	20.9852	0.939426	0.469713	−	0.882819i	$-0.344357\pi$
$499$	20.9852	0.939426	0.469713	+	0.882819i	$0.344357\pi$
$500$	0	0
$501$	6.09159	0.272152
$502$	0	0
$503$	31.0942	1.38642	0.693211	−	0.720735i	$-0.256195\pi$
$503$	31.0942	1.38642	0.693211	+	0.720735i	$0.256195\pi$
$504$	0	0
$505$	7.94523	0.353558
$506$	0	0
$507$	7.48715	0.332516
$508$	0	0
$509$	20.8302	0.923282	0.461641	−	0.887067i	$-0.347261\pi$
$509$	20.8302	0.923282	0.461641	+	0.887067i	$0.347261\pi$
$510$	0	0
$511$	−0.914138	−0.0404391
$512$	0	0
$513$	−16.4663	−0.727003
$514$	0	0
$515$	−2.73176	−0.120376
$516$	0	0
$517$	−1.79351	−0.0788784
$518$	0	0
$519$	2.79887	0.122857
$520$	0	0
$521$	33.9340	1.48668	0.743338	−	0.668916i	$-0.233241\pi$
$521$	33.9340	1.48668	0.743338	+	0.668916i	$0.233241\pi$
$522$	0	0
$523$	39.1519	1.71199	0.855997	−	0.516980i	$-0.172944\pi$
$523$	39.1519	1.71199	0.855997	+	0.516980i	$0.172944\pi$
$524$	0	0
$525$	0.168608	0.00735868
$526$	0	0
$527$	−8.69703	−0.378849
$528$	0	0
$529$	−10.1762	−0.442443
$530$	0	0
$531$	−14.9028	−0.646728
$532$	0	0
$533$	59.1151	2.56056
$534$	0	0
$535$	11.1307	0.481224
$536$	0	0
$537$	0.762199	0.0328913
$538$	0	0
$539$	1.95927	0.0843918
$540$	0	0
$541$	−14.5538	−0.625718	−0.312859	−	0.949800i	$-0.601287\pi$
$541$	−14.5538	−0.625718	−0.312859	+	0.949800i	$0.601287\pi$
$542$	0	0
$543$	−5.52539	−0.237117
$544$	0	0
$545$	15.7161	0.673203
$546$	0	0
$547$	−25.7447	−1.10076	−0.550382	−	0.834913i	$-0.685518\pi$
$547$	−25.7447	−1.10076	−0.550382	+	0.834913i	$0.685518\pi$
$548$	0	0
$549$	11.9586	0.510380
$550$	0	0
$551$	5.74402	0.244703
$552$	0	0
$553$	−5.22568	−0.222219
$554$	0	0
$555$	−1.81763	−0.0771539
$556$	0	0
$557$	13.7701	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$557$	13.7701	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$558$	0	0
$559$	−72.2773	−3.05701
$560$	0	0
$561$	0.208812	0.00881605
$562$	0	0
$563$	7.92364	0.333942	0.166971	−	0.985962i	$-0.446601\pi$
$563$	7.92364	0.333942	0.166971	+	0.985962i	$0.446601\pi$
$564$	0	0
$565$	−2.57309	−0.108251
$566$	0	0
$567$	4.07021	0.170933
$568$	0	0
$569$	−6.03695	−0.253082	−0.126541	−	0.991961i	$-0.540388\pi$
$569$	−6.03695	−0.253082	−0.126541	+	0.991961i	$0.540388\pi$
$570$	0	0
$571$	−8.46175	−0.354113	−0.177057	−	0.984201i	$-0.556658\pi$
$571$	−8.46175	−0.354113	−0.177057	+	0.984201i	$0.556658\pi$
$572$	0	0
$573$	−3.34552	−0.139761
$574$	0	0
$575$	−3.58103	−0.149339
$576$	0	0
$577$	12.9460	0.538948	0.269474	−	0.963008i	$-0.413150\pi$
$577$	12.9460	0.538948	0.269474	+	0.963008i	$0.413150\pi$
$578$	0	0
$579$	−0.0526084	−0.00218633
$580$	0	0
$581$	2.90763	0.120629
$582$	0	0
$583$	−3.11027	−0.128814
$584$	0	0
$585$	17.2265	0.712228
$586$	0	0
$587$	−12.2620	−0.506105	−0.253053	−	0.967453i	$-0.581435\pi$
$587$	−12.2620	−0.506105	−0.253053	+	0.967453i	$0.581435\pi$
$588$	0	0
$589$	33.8378	1.39426
$590$	0	0
$591$	3.28457	0.135109
$592$	0	0
$593$	3.16129	0.129819	0.0649094	−	0.997891i	$-0.479324\pi$
$593$	3.16129	0.129819	0.0649094	+	0.997891i	$0.479324\pi$
$594$	0	0
$595$	−1.09795	−0.0450117
$596$	0	0
$597$	9.30980	0.381025
$598$	0	0
$599$	4.88676	0.199667	0.0998337	−	0.995004i	$-0.468169\pi$
$599$	4.88676	0.199667	0.0998337	+	0.995004i	$0.468169\pi$
$600$	0	0
$601$	24.2916	0.990876	0.495438	−	0.868643i	$-0.335008\pi$
$601$	24.2916	0.990876	0.495438	+	0.868643i	$0.335008\pi$
$602$	0	0
$603$	−26.4001	−1.07510
$604$	0	0
$605$	−10.9156	−0.443780
$606$	0	0
$607$	−37.8765	−1.53736	−0.768680	−	0.639634i	$-0.779086\pi$
$607$	−37.8765	−1.53736	−0.768680	+	0.639634i	$0.779086\pi$
$608$	0	0
$609$	0.115073	0.00466300
$610$	0	0
$611$	−36.7937	−1.48851
$612$	0	0
$613$	40.5352	1.63720	0.818600	−	0.574364i	$-0.194751\pi$
$613$	40.5352	1.63720	0.818600	+	0.574364i	$0.194751\pi$
$614$	0	0
$615$	−3.29406	−0.132829
$616$	0	0
$617$	23.4616	0.944526	0.472263	−	0.881458i	$-0.343437\pi$
$617$	23.4616	0.944526	0.472263	+	0.881458i	$0.343437\pi$
$618$	0	0
$619$	−14.6415	−0.588490	−0.294245	−	0.955730i	$-0.595068\pi$
$619$	−14.6415	−0.588490	−0.294245	+	0.955730i	$0.595068\pi$
$620$	0	0
$621$	7.00620	0.281149
$622$	0	0
$623$	−4.65685	−0.186573
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.812433	−0.0324454
$628$	0	0
$629$	11.8361	0.471937
$630$	0	0
$631$	−0.800973	−0.0318862	−0.0159431	−	0.999873i	$-0.505075\pi$
$631$	−0.800973	−0.0318862	−0.0159431	+	0.999873i	$0.505075\pi$
$632$	0	0
$633$	6.49206	0.258036
$634$	0	0
$635$	−1.18536	−0.0470396
$636$	0	0
$637$	40.1943	1.59256
$638$	0	0
$639$	−22.0361	−0.871734
$640$	0	0
$641$	−19.3623	−0.764765	−0.382383	−	0.924004i	$-0.624896\pi$
$641$	−19.3623	−0.764765	−0.382383	+	0.924004i	$0.624896\pi$
$642$	0	0
$643$	29.4236	1.16035	0.580177	−	0.814490i	$-0.302983\pi$
$643$	29.4236	1.16035	0.580177	+	0.814490i	$0.302983\pi$
$644$	0	0
$645$	4.02750	0.158582
$646$	0	0
$647$	19.3828	0.762017	0.381008	−	0.924572i	$-0.375577\pi$
$647$	19.3828	0.762017	0.381008	+	0.924572i	$0.375577\pi$
$648$	0	0
$649$	−1.49867	−0.0588279
$650$	0	0
$651$	0.677893	0.0265687
$652$	0	0
$653$	−39.6291	−1.55081	−0.775403	−	0.631467i	$-0.782453\pi$
$653$	−39.6291	−1.55081	−0.775403	+	0.631467i	$0.782453\pi$
$654$	0	0
$655$	7.98192	0.311879
$656$	0	0
$657$	5.20430	0.203039
$658$	0	0
$659$	49.4376	1.92582	0.962908	−	0.269828i	$-0.0869670\pi$
$659$	49.4376	1.92582	0.962908	+	0.269828i	$0.0869670\pi$
$660$	0	0
$661$	−12.5252	−0.487175	−0.243587	−	0.969879i	$-0.578324\pi$
$661$	−12.5252	−0.487175	−0.243587	+	0.969879i	$0.578324\pi$
$662$	0	0
$663$	4.28376	0.166368
$664$	0	0
$665$	4.27185	0.165655
$666$	0	0
$667$	−2.44401	−0.0946326
$668$	0	0
$669$	9.52852	0.368394
$670$	0	0
$671$	1.20259	0.0464254
$672$	0	0
$673$	33.6555	1.29732	0.648662	−	0.761077i	$-0.275329\pi$
$673$	33.6555	1.29732	0.648662	+	0.761077i	$0.275329\pi$
$674$	0	0
$675$	−1.95647	−0.0753047
$676$	0	0
$677$	−18.5811	−0.714131	−0.357065	−	0.934079i	$-0.616223\pi$
$677$	−18.5811	−0.714131	−0.357065	+	0.934079i	$0.616223\pi$
$678$	0	0
$679$	2.59218	0.0994788
$680$	0	0
$681$	−1.15827	−0.0443851
$682$	0	0
$683$	18.7015	0.715593	0.357797	−	0.933800i	$-0.383528\pi$
$683$	18.7015	0.715593	0.357797	+	0.933800i	$0.383528\pi$
$684$	0	0
$685$	3.17238	0.121210
$686$	0	0
$687$	−1.46559	−0.0559157
$688$	0	0
$689$	−63.8069	−2.43085
$690$	0	0
$691$	29.7443	1.13153	0.565763	−	0.824568i	$-0.308582\pi$
$691$	29.7443	1.13153	0.565763	+	0.824568i	$0.308582\pi$
$692$	0	0
$693$	0.426208	0.0161903
$694$	0	0
$695$	−15.2230	−0.577441
$696$	0	0
$697$	21.4504	0.812494
$698$	0	0
$699$	2.86861	0.108501
$700$	0	0
$701$	9.43071	0.356193	0.178096	−	0.984013i	$-0.443006\pi$
$701$	9.43071	0.356193	0.178096	+	0.984013i	$0.443006\pi$
$702$	0	0
$703$	−46.0512	−1.73685
$704$	0	0
$705$	2.05025	0.0772167
$706$	0	0
$707$	4.03275	0.151667
$708$	0	0
$709$	46.3698	1.74145	0.870727	−	0.491766i	$-0.163649\pi$
$709$	46.3698	1.74145	0.870727	+	0.491766i	$0.163649\pi$
$710$	0	0
$711$	29.7505	1.11573
$712$	0	0
$713$	−14.3976	−0.539195
$714$	0	0
$715$	1.73234	0.0647859
$716$	0	0
$717$	3.64477	0.136116
$718$	0	0
$719$	44.1940	1.64816	0.824078	−	0.566476i	$-0.191694\pi$
$719$	44.1940	1.64816	0.824078	+	0.566476i	$0.191694\pi$
$720$	0	0
$721$	−1.38656	−0.0516381
$722$	0	0
$723$	−3.46643	−0.128918
$724$	0	0
$725$	0.682488	0.0253470
$726$	0	0
$727$	10.0168	0.371503	0.185751	−	0.982597i	$-0.440528\pi$
$727$	10.0168	0.371503	0.185751	+	0.982597i	$0.440528\pi$
$728$	0	0
$729$	−21.2225	−0.786017
$730$	0	0
$731$	−26.2265	−0.970021
$732$	0	0
$733$	44.6414	1.64887	0.824434	−	0.565958i	$-0.191493\pi$
$733$	44.6414	1.64887	0.824434	+	0.565958i	$0.191493\pi$
$734$	0	0
$735$	−2.23974	−0.0826140
$736$	0	0
$737$	−2.65487	−0.0977932
$738$	0	0
$739$	11.0054	0.404840	0.202420	−	0.979299i	$-0.435119\pi$
$739$	11.0054	0.404840	0.202420	+	0.979299i	$0.435119\pi$
$740$	0	0
$741$	−16.6670	−0.612277
$742$	0	0
$743$	44.0115	1.61463	0.807314	−	0.590123i	$-0.200921\pi$
$743$	44.0115	1.61463	0.807314	+	0.590123i	$0.200921\pi$
$744$	0	0
$745$	13.4084	0.491245
$746$	0	0
$747$	−16.5535	−0.605660
$748$	0	0
$749$	5.64961	0.206432
$750$	0	0
$751$	−37.0492	−1.35195	−0.675973	−	0.736927i	$-0.736276\pi$
$751$	−37.0492	−1.35195	−0.675973	+	0.736927i	$0.736276\pi$
$752$	0	0
$753$	−0.760509	−0.0277145
$754$	0	0
$755$	−8.79975	−0.320256
$756$	0	0
$757$	−16.4666	−0.598490	−0.299245	−	0.954176i	$-0.596735\pi$
$757$	−16.4666	−0.598490	−0.299245	+	0.954176i	$0.596735\pi$
$758$	0	0
$759$	0.345681	0.0125474
$760$	0	0
$761$	49.6134	1.79849	0.899243	−	0.437450i	$-0.144118\pi$
$761$	49.6134	1.79849	0.899243	+	0.437450i	$0.144118\pi$
$762$	0	0
$763$	7.97699	0.288787
$764$	0	0
$765$	6.25079	0.225998
$766$	0	0
$767$	−30.7451	−1.11014
$768$	0	0
$769$	−37.4152	−1.34923	−0.674613	−	0.738171i	$-0.735690\pi$
$769$	−37.4152	−1.34923	−0.674613	+	0.738171i	$0.735690\pi$
$770$	0	0
$771$	−2.04166	−0.0735287
$772$	0	0
$773$	43.4633	1.56327	0.781634	−	0.623738i	$-0.214387\pi$
$773$	43.4633	1.56327	0.781634	+	0.623738i	$0.214387\pi$
$774$	0	0
$775$	4.02052	0.144421
$776$	0	0
$777$	−0.922570	−0.0330970
$778$	0	0
$779$	−83.4580	−2.99019
$780$	0	0
$781$	−2.21601	−0.0792949
$782$	0	0
$783$	−1.33527	−0.0477187
$784$	0	0
$785$	11.8997	0.424720
$786$	0	0
$787$	−9.33181	−0.332643	−0.166322	−	0.986072i	$-0.553189\pi$
$787$	−9.33181	−0.332643	−0.166322	+	0.986072i	$0.553189\pi$
$788$	0	0
$789$	10.5742	0.376451
$790$	0	0
$791$	−1.30602	−0.0464367
$792$	0	0
$793$	24.6710	0.876092
$794$	0	0
$795$	3.55550	0.126100
$796$	0	0
$797$	−37.0822	−1.31352	−0.656759	−	0.754100i	$-0.728073\pi$
$797$	−37.0822	−1.31352	−0.656759	+	0.754100i	$0.728073\pi$
$798$	0	0
$799$	−13.3509	−0.472321
$800$	0	0
$801$	26.5120	0.936755
$802$	0	0
$803$	0.523358	0.0184689
$804$	0	0
$805$	−1.81762	−0.0640627
$806$	0	0
$807$	6.83113	0.240467
$808$	0	0
$809$	−44.9789	−1.58138	−0.790688	−	0.612220i	$-0.790277\pi$
$809$	−44.9789	−1.58138	−0.790688	+	0.612220i	$0.790277\pi$
$810$	0	0
$811$	−14.6032	−0.512787	−0.256393	−	0.966572i	$-0.582534\pi$
$811$	−14.6032	−0.512787	−0.256393	+	0.966572i	$0.582534\pi$
$812$	0	0
$813$	−6.79466	−0.238299
$814$	0	0
$815$	17.7612	0.622149
$816$	0	0
$817$	102.040	3.56994
$818$	0	0
$819$	8.74363	0.305527
$820$	0	0
$821$	−4.84639	−0.169140	−0.0845702	−	0.996418i	$-0.526952\pi$
$821$	−4.84639	−0.169140	−0.0845702	+	0.996418i	$0.526952\pi$
$822$	0	0
$823$	39.2104	1.36679	0.683394	−	0.730050i	$-0.260503\pi$
$823$	39.2104	1.36679	0.683394	+	0.730050i	$0.260503\pi$
$824$	0	0
$825$	−0.0965309	−0.00336077
$826$	0	0
$827$	49.0245	1.70475	0.852374	−	0.522933i	$-0.175162\pi$
$827$	49.0245	1.70475	0.852374	+	0.522933i	$0.175162\pi$
$828$	0	0
$829$	−34.2207	−1.18853	−0.594267	−	0.804268i	$-0.702558\pi$
$829$	−34.2207	−1.18853	−0.594267	+	0.804268i	$0.702558\pi$
$830$	0	0
$831$	5.05285	0.175281
$832$	0	0
$833$	14.5849	0.505335
$834$	0	0
$835$	18.3378	0.634604
$836$	0	0
$837$	−7.86604	−0.271890
$838$	0	0
$839$	50.1243	1.73048	0.865241	−	0.501356i	$-0.167165\pi$
$839$	50.1243	1.73048	0.865241	+	0.501356i	$0.167165\pi$
$840$	0	0
$841$	−28.5342	−0.983938
$842$	0	0
$843$	4.79227	0.165055
$844$	0	0
$845$	22.5389	0.775360
$846$	0	0
$847$	−5.54039	−0.190370
$848$	0	0
$849$	−4.64076	−0.159271
$850$	0	0
$851$	19.5942	0.671682
$852$	0	0
$853$	−1.57768	−0.0540189	−0.0270095	−	0.999635i	$-0.508598\pi$
$853$	−1.57768	−0.0540189	−0.0270095	+	0.999635i	$0.508598\pi$
$854$	0	0
$855$	−24.3202	−0.831732
$856$	0	0
$857$	−52.2493	−1.78480	−0.892401	−	0.451244i	$-0.850980\pi$
$857$	−52.2493	−1.78480	−0.892401	+	0.451244i	$0.850980\pi$
$858$	0	0
$859$	−14.9239	−0.509196	−0.254598	−	0.967047i	$-0.581943\pi$
$859$	−14.9239	−0.509196	−0.254598	+	0.967047i	$0.581943\pi$
$860$	0	0
$861$	−1.67196	−0.0569803
$862$	0	0
$863$	6.51601	0.221808	0.110904	−	0.993831i	$-0.464625\pi$
$863$	6.51601	0.221808	0.110904	+	0.993831i	$0.464625\pi$
$864$	0	0
$865$	8.42556	0.286478
$866$	0	0
$867$	−4.09280	−0.138999
$868$	0	0
$869$	2.99178	0.101489
$870$	0	0
$871$	−54.4643	−1.84545
$872$	0	0
$873$	−14.7576	−0.499469
$874$	0	0
$875$	0.507569	0.0171590
$876$	0	0
$877$	21.6754	0.731927	0.365964	−	0.930629i	$-0.380739\pi$
$877$	21.6754	0.731927	0.365964	+	0.930629i	$0.380739\pi$
$878$	0	0
$879$	−6.84378	−0.230835
$880$	0	0
$881$	−31.2525	−1.05292	−0.526462	−	0.850199i	$-0.676482\pi$
$881$	−31.2525	−1.05292	−0.526462	+	0.850199i	$0.676482\pi$
$882$	0	0
$883$	21.2940	0.716599	0.358299	−	0.933607i	$-0.383357\pi$
$883$	21.2940	0.716599	0.358299	+	0.933607i	$0.383357\pi$
$884$	0	0
$885$	1.71320	0.0575886
$886$	0	0
$887$	−8.75899	−0.294098	−0.147049	−	0.989129i	$-0.546978\pi$
$887$	−8.75899	−0.294098	−0.147049	+	0.989129i	$0.546978\pi$
$888$	0	0
$889$	−0.601652	−0.0201788
$890$	0	0
$891$	−2.33026	−0.0780666
$892$	0	0
$893$	51.9448	1.73827
$894$	0	0
$895$	2.29448	0.0766959
$896$	0	0
$897$	7.09161	0.236782
$898$	0	0
$899$	2.74395	0.0915160
$900$	0	0
$901$	−23.1529	−0.771334
$902$	0	0
$903$	2.04423	0.0680277
$904$	0	0
$905$	−16.6333	−0.552909
$906$	0	0
$907$	−44.9456	−1.49239	−0.746197	−	0.665725i	$-0.768123\pi$
$907$	−44.9456	−1.49239	−0.746197	+	0.665725i	$0.768123\pi$
$908$	0	0
$909$	−22.9589	−0.761500
$910$	0	0
$911$	−43.6359	−1.44572	−0.722861	−	0.690993i	$-0.757173\pi$
$911$	−43.6359	−1.44572	−0.722861	+	0.690993i	$0.757173\pi$
$912$	0	0
$913$	−1.66466	−0.0550922
$914$	0	0
$915$	−1.37474	−0.0454474
$916$	0	0
$917$	4.05137	0.133788
$918$	0	0
$919$	−0.193736	−0.00639077	−0.00319539	−	0.999995i	$-0.501017\pi$
$919$	−0.193736	−0.00639077	−0.00319539	+	0.999995i	$0.501017\pi$
$920$	0	0
$921$	−1.72253	−0.0567594
$922$	0	0
$923$	−45.4612	−1.49637
$924$	0	0
$925$	−5.47167	−0.179907
$926$	0	0
$927$	7.89384	0.259268
$928$	0	0
$929$	−32.7280	−1.07377	−0.536886	−	0.843655i	$-0.680400\pi$
$929$	−32.7280	−1.07377	−0.536886	+	0.843655i	$0.680400\pi$
$930$	0	0
$931$	−56.7458	−1.85977
$932$	0	0
$933$	−8.99620	−0.294522
$934$	0	0
$935$	0.628595	0.0205573
$936$	0	0
$937$	55.0707	1.79908	0.899540	−	0.436839i	$-0.143902\pi$
$937$	55.0707	1.79908	0.899540	+	0.436839i	$0.143902\pi$
$938$	0	0
$939$	−6.87929	−0.224497
$940$	0	0
$941$	34.5317	1.12570	0.562851	−	0.826558i	$-0.309704\pi$
$941$	34.5317	1.12570	0.562851	+	0.826558i	$0.309704\pi$
$942$	0	0
$943$	35.5104	1.15638
$944$	0	0
$945$	−0.993045	−0.0323038
$946$	0	0
$947$	−34.4403	−1.11916	−0.559580	−	0.828777i	$-0.689037\pi$
$947$	−34.4403	−1.11916	−0.559580	+	0.828777i	$0.689037\pi$
$948$	0	0
$949$	10.7367	0.348526
$950$	0	0
$951$	−8.18801	−0.265515
$952$	0	0
$953$	21.5658	0.698584	0.349292	−	0.937014i	$-0.386422\pi$
$953$	21.5658	0.698584	0.349292	+	0.937014i	$0.386422\pi$
$954$	0	0
$955$	−10.0711	−0.325895
$956$	0	0
$957$	−0.0658812	−0.00212964
$958$	0	0
$959$	1.61020	0.0519960
$960$	0	0
$961$	−14.8355	−0.478563
$962$	0	0
$963$	−32.1639	−1.03647
$964$	0	0
$965$	−0.158369	−0.00509808
$966$	0	0
$967$	−50.8529	−1.63532	−0.817660	−	0.575702i	$-0.804729\pi$
$967$	−50.8529	−1.63532	−0.817660	+	0.575702i	$0.804729\pi$
$968$	0	0
$969$	−6.04776	−0.194282
$970$	0	0
$971$	−34.7494	−1.11516	−0.557580	−	0.830123i	$-0.688270\pi$
$971$	−34.7494	−1.11516	−0.557580	+	0.830123i	$0.688270\pi$
$972$	0	0
$973$	−7.72671	−0.247707
$974$	0	0
$975$	−1.98032	−0.0634211
$976$	0	0
$977$	39.9523	1.27819	0.639094	−	0.769129i	$-0.279309\pi$
$977$	39.9523	1.27819	0.639094	+	0.769129i	$0.279309\pi$
$978$	0	0
$979$	2.66611	0.0852094
$980$	0	0
$981$	−45.4140	−1.44996
$982$	0	0
$983$	−22.4958	−0.717503	−0.358752	−	0.933433i	$-0.616797\pi$
$983$	−22.4958	−0.717503	−0.358752	+	0.933433i	$0.616797\pi$
$984$	0	0
$985$	9.88767	0.315047
$986$	0	0
$987$	1.04064	0.0331240
$988$	0	0
$989$	−43.4169	−1.38058
$990$	0	0
$991$	27.1161	0.861372	0.430686	−	0.902502i	$-0.358272\pi$
$991$	27.1161	0.861372	0.430686	+	0.902502i	$0.358272\pi$
$992$	0	0
$993$	−4.58748	−0.145579
$994$	0	0
$995$	28.0256	0.888473
$996$	0	0
$997$	−29.0710	−0.920688	−0.460344	−	0.887741i	$-0.652274\pi$
$997$	−29.0710	−0.920688	−0.460344	+	0.887741i	$0.652274\pi$
$998$	0	0
$999$	10.7052	0.338697

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.f.1.19	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.f.1.19	✓	37	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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q+0.332188 q^{3} +1.00000 q^{5} +0.507569 q^{7} -2.88965 q^{9} +O(q^{10})\)	\(q+0.332188 q^{3} +1.00000 q^{5} +0.507569 q^{7} -2.88965 q^{9} -0.290591 q^{11} -5.96145 q^{13} +0.332188 q^{15} -2.16316 q^{17} +8.41629 q^{19} +0.168608 q^{21} -3.58103 q^{23} +1.00000 q^{25} -1.95647 q^{27} +0.682488 q^{29} +4.02052 q^{31} -0.0965309 q^{33} +0.507569 q^{35} -5.47167 q^{37} -1.98032 q^{39} -9.91624 q^{41} +12.1241 q^{43} -2.88965 q^{45} +6.17194 q^{47} -6.74237 q^{49} -0.718578 q^{51} +10.7032 q^{53} -0.290591 q^{55} +2.79580 q^{57} +5.15732 q^{59} -4.13842 q^{61} -1.46670 q^{63} -5.96145 q^{65} +9.13610 q^{67} -1.18958 q^{69} +7.62587 q^{71} -1.80101 q^{73} +0.332188 q^{75} -0.147495 q^{77} -10.2955 q^{79} +8.01903 q^{81} +5.72854 q^{83} -2.16316 q^{85} +0.226715 q^{87} -9.17481 q^{89} -3.02584 q^{91} +1.33557 q^{93} +8.41629 q^{95} +5.10705 q^{97} +0.839706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9} + 2 q^{11} + 27 q^{13} + 3 q^{15} + 36 q^{17} - 6 q^{19} + 20 q^{21} + 17 q^{23} + 37 q^{25} + 9 q^{27} + 29 q^{29} + 5 q^{31} + 36 q^{33} + 4 q^{35} + 35 q^{37} + 21 q^{39} + 24 q^{41} + 11 q^{43} + 50 q^{45} + 19 q^{47} + 57 q^{49} + 8 q^{51} + 65 q^{53} + 2 q^{55} + 62 q^{57} - 9 q^{59} + 13 q^{61} + 26 q^{63} + 27 q^{65} + 13 q^{67} + 20 q^{69} + 33 q^{71} + 67 q^{73} + 3 q^{75} + 62 q^{77} + 23 q^{79} + 97 q^{81} + 2 q^{83} + 36 q^{85} + 32 q^{87} + 34 q^{89} + q^{91} + 41 q^{93} - 6 q^{95} + 66 q^{97} - 7 q^{99}+O(q^{100}) \) 37 * q + 3 * q^3 + 37 * q^5 + 4 * q^7 + 50 * q^9 + 2 * q^11 + 27 * q^13 + 3 * q^15 + 36 * q^17 - 6 * q^19 + 20 * q^21 + 17 * q^23 + 37 * q^25 + 9 * q^27 + 29 * q^29 + 5 * q^31 + 36 * q^33 + 4 * q^35 + 35 * q^37 + 21 * q^39 + 24 * q^41 + 11 * q^43 + 50 * q^45 + 19 * q^47 + 57 * q^49 + 8 * q^51 + 65 * q^53 + 2 * q^55 + 62 * q^57 - 9 * q^59 + 13 * q^61 + 26 * q^63 + 27 * q^65 + 13 * q^67 + 20 * q^69 + 33 * q^71 + 67 * q^73 + 3 * q^75 + 62 * q^77 + 23 * q^79 + 97 * q^81 + 2 * q^83 + 36 * q^85 + 32 * q^87 + 34 * q^89 + q^91 + 41 * q^93 - 6 * q^95 + 66 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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