LMFDB - Embedded newform 8020.2.a.c.1.4

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:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.32143 q^{3} -1.00000 q^{5} +0.515797 q^{7} +2.38904 q^{9} +O(q^{10})$	$q-2.32143 q^{3} -1.00000 q^{5} +0.515797 q^{7} +2.38904 q^{9} +4.54079 q^{11} +1.42147 q^{13} +2.32143 q^{15} +3.21357 q^{17} -2.56977 q^{19} -1.19739 q^{21} -0.925577 q^{23} +1.00000 q^{25} +1.41829 q^{27} -1.44774 q^{29} -6.16275 q^{31} -10.5411 q^{33} -0.515797 q^{35} -5.88040 q^{37} -3.29984 q^{39} -9.02106 q^{41} -7.54662 q^{43} -2.38904 q^{45} -1.06959 q^{47} -6.73395 q^{49} -7.46009 q^{51} +3.39722 q^{53} -4.54079 q^{55} +5.96555 q^{57} +8.53271 q^{59} +4.29821 q^{61} +1.23226 q^{63} -1.42147 q^{65} -7.75453 q^{67} +2.14866 q^{69} +7.31767 q^{71} +6.48265 q^{73} -2.32143 q^{75} +2.34213 q^{77} +6.18911 q^{79} -10.4596 q^{81} +16.0002 q^{83} -3.21357 q^{85} +3.36084 q^{87} +13.5007 q^{89} +0.733188 q^{91} +14.3064 q^{93} +2.56977 q^{95} +7.67511 q^{97} +10.8482 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * 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
$3$	−2.32143	−1.34028	−0.670140	−	0.742235i	$-0.733766\pi$
$3$	−2.32143	−1.34028	−0.670140	+	0.742235i	$0.733766\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.515797	0.194953	0.0974765	−	0.995238i	$-0.468923\pi$
$7$	0.515797	0.194953	0.0974765	+	0.995238i	$0.468923\pi$
$8$	0	0
$9$	2.38904	0.796348
$10$	0	0
$11$	4.54079	1.36910	0.684550	−	0.728966i	$-0.259999\pi$
$11$	4.54079	1.36910	0.684550	+	0.728966i	$0.259999\pi$
$12$	0	0
$13$	1.42147	0.394244	0.197122	−	0.980379i	$-0.436841\pi$
$13$	1.42147	0.394244	0.197122	+	0.980379i	$0.436841\pi$
$14$	0	0
$15$	2.32143	0.599391
$16$	0	0
$17$	3.21357	0.779406	0.389703	−	0.920941i	$-0.372578\pi$
$17$	3.21357	0.779406	0.389703	+	0.920941i	$0.372578\pi$
$18$	0	0
$19$	−2.56977	−0.589546	−0.294773	−	0.955567i	$-0.595244\pi$
$19$	−2.56977	−0.589546	−0.294773	+	0.955567i	$0.595244\pi$
$20$	0	0
$21$	−1.19739	−0.261291
$22$	0	0
$23$	−0.925577	−0.192996	−0.0964981	−	0.995333i	$-0.530764\pi$
$23$	−0.925577	−0.192996	−0.0964981	+	0.995333i	$0.530764\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.41829	0.272950
$28$	0	0
$29$	−1.44774	−0.268839	−0.134420	−	0.990925i	$-0.542917\pi$
$29$	−1.44774	−0.268839	−0.134420	+	0.990925i	$0.542917\pi$
$30$	0	0
$31$	−6.16275	−1.10686	−0.553431	−	0.832895i	$-0.686682\pi$
$31$	−6.16275	−1.10686	−0.553431	+	0.832895i	$0.686682\pi$
$32$	0	0
$33$	−10.5411	−1.83498
$34$	0	0
$35$	−0.515797	−0.0871856
$36$	0	0
$37$	−5.88040	−0.966732	−0.483366	−	0.875418i	$-0.660586\pi$
$37$	−5.88040	−0.966732	−0.483366	+	0.875418i	$0.660586\pi$
$38$	0	0
$39$	−3.29984	−0.528397
$40$	0	0
$41$	−9.02106	−1.40885	−0.704426	−	0.709777i	$-0.748796\pi$
$41$	−9.02106	−1.40885	−0.704426	+	0.709777i	$0.748796\pi$
$42$	0	0
$43$	−7.54662	−1.15085	−0.575424	−	0.817855i	$-0.695163\pi$
$43$	−7.54662	−1.15085	−0.575424	+	0.817855i	$0.695163\pi$
$44$	0	0
$45$	−2.38904	−0.356138
$46$	0	0
$47$	−1.06959	−0.156016	−0.0780082	−	0.996953i	$-0.524856\pi$
$47$	−1.06959	−0.156016	−0.0780082	+	0.996953i	$0.524856\pi$
$48$	0	0
$49$	−6.73395	−0.961993
$50$	0	0
$51$	−7.46009	−1.04462
$52$	0	0
$53$	3.39722	0.466644	0.233322	−	0.972400i	$-0.425040\pi$
$53$	3.39722	0.466644	0.233322	+	0.972400i	$0.425040\pi$
$54$	0	0
$55$	−4.54079	−0.612280
$56$	0	0
$57$	5.96555	0.790156
$58$	0	0
$59$	8.53271	1.11086	0.555432	−	0.831562i	$-0.312553\pi$
$59$	8.53271	1.11086	0.555432	+	0.831562i	$0.312553\pi$
$60$	0	0
$61$	4.29821	0.550330	0.275165	−	0.961397i	$-0.411268\pi$
$61$	4.29821	0.550330	0.275165	+	0.961397i	$0.411268\pi$
$62$	0	0
$63$	1.23226	0.155250
$64$	0	0
$65$	−1.42147	−0.176311
$66$	0	0
$67$	−7.75453	−0.947367	−0.473683	−	0.880695i	$-0.657076\pi$
$67$	−7.75453	−0.947367	−0.473683	+	0.880695i	$0.657076\pi$
$68$	0	0
$69$	2.14866	0.258669
$70$	0	0
$71$	7.31767	0.868448	0.434224	−	0.900805i	$-0.357023\pi$
$71$	7.31767	0.868448	0.434224	+	0.900805i	$0.357023\pi$
$72$	0	0
$73$	6.48265	0.758737	0.379368	−	0.925246i	$-0.376141\pi$
$73$	6.48265	0.758737	0.379368	+	0.925246i	$0.376141\pi$
$74$	0	0
$75$	−2.32143	−0.268056
$76$	0	0
$77$	2.34213	0.266910
$78$	0	0
$79$	6.18911	0.696330	0.348165	−	0.937433i	$-0.386805\pi$
$79$	6.18911	0.696330	0.348165	+	0.937433i	$0.386805\pi$
$80$	0	0
$81$	−10.4596	−1.16218
$82$	0	0
$83$	16.0002	1.75625	0.878125	−	0.478432i	$-0.158795\pi$
$83$	16.0002	1.75625	0.878125	+	0.478432i	$0.158795\pi$
$84$	0	0
$85$	−3.21357	−0.348561
$86$	0	0
$87$	3.36084	0.360319
$88$	0	0
$89$	13.5007	1.43108	0.715538	−	0.698574i	$-0.246182\pi$
$89$	13.5007	1.43108	0.715538	+	0.698574i	$0.246182\pi$
$90$	0	0
$91$	0.733188	0.0768590
$92$	0	0
$93$	14.3064	1.48350
$94$	0	0
$95$	2.56977	0.263653
$96$	0	0
$97$	7.67511	0.779289	0.389645	−	0.920965i	$-0.372598\pi$
$97$	7.67511	0.779289	0.389645	+	0.920965i	$0.372598\pi$
$98$	0	0
$99$	10.8482	1.09028
$100$	0	0
$101$	−15.5710	−1.54938	−0.774688	−	0.632343i	$-0.782093\pi$
$101$	−15.5710	−1.54938	−0.774688	+	0.632343i	$0.782093\pi$
$102$	0	0
$103$	−3.28758	−0.323934	−0.161967	−	0.986796i	$-0.551784\pi$
$103$	−3.28758	−0.323934	−0.161967	+	0.986796i	$0.551784\pi$
$104$	0	0
$105$	1.19739	0.116853
$106$	0	0
$107$	3.65515	0.353357	0.176678	−	0.984269i	$-0.443465\pi$
$107$	3.65515	0.353357	0.176678	+	0.984269i	$0.443465\pi$
$108$	0	0
$109$	−0.780421	−0.0747507	−0.0373754	−	0.999301i	$-0.511900\pi$
$109$	−0.780421	−0.0747507	−0.0373754	+	0.999301i	$0.511900\pi$
$110$	0	0
$111$	13.6509	1.29569
$112$	0	0
$113$	−3.57304	−0.336123	−0.168062	−	0.985777i	$-0.553751\pi$
$113$	−3.57304	−0.336123	−0.168062	+	0.985777i	$0.553751\pi$
$114$	0	0
$115$	0.925577	0.0863105
$116$	0	0
$117$	3.39595	0.313955
$118$	0	0
$119$	1.65755	0.151948
$120$	0	0
$121$	9.61879	0.874435
$122$	0	0
$123$	20.9418	1.88826
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.3561	1.45136	0.725682	−	0.688030i	$-0.241524\pi$
$127$	16.3561	1.45136	0.725682	+	0.688030i	$0.241524\pi$
$128$	0	0
$129$	17.5190	1.54246
$130$	0	0
$131$	−17.7380	−1.54978	−0.774889	−	0.632098i	$-0.782194\pi$
$131$	−17.7380	−1.54978	−0.774889	+	0.632098i	$0.782194\pi$
$132$	0	0
$133$	−1.32548	−0.114934
$134$	0	0
$135$	−1.41829	−0.122067
$136$	0	0
$137$	−13.2460	−1.13168	−0.565839	−	0.824516i	$-0.691448\pi$
$137$	−13.2460	−1.13168	−0.565839	+	0.824516i	$0.691448\pi$
$138$	0	0
$139$	−7.27964	−0.617451	−0.308726	−	0.951151i	$-0.599902\pi$
$139$	−7.27964	−0.617451	−0.308726	+	0.951151i	$0.599902\pi$
$140$	0	0
$141$	2.48299	0.209106
$142$	0	0
$143$	6.45458	0.539759
$144$	0	0
$145$	1.44774	0.120229
$146$	0	0
$147$	15.6324	1.28934
$148$	0	0
$149$	14.1910	1.16257	0.581287	−	0.813699i	$-0.302549\pi$
$149$	14.1910	1.16257	0.581287	+	0.813699i	$0.302549\pi$
$150$	0	0
$151$	−11.8909	−0.967664	−0.483832	−	0.875161i	$-0.660755\pi$
$151$	−11.8909	−0.967664	−0.483832	+	0.875161i	$0.660755\pi$
$152$	0	0
$153$	7.67737	0.620678
$154$	0	0
$155$	6.16275	0.495004
$156$	0	0
$157$	−0.709493	−0.0566237	−0.0283118	−	0.999599i	$-0.509013\pi$
$157$	−0.709493	−0.0566237	−0.0283118	+	0.999599i	$0.509013\pi$
$158$	0	0
$159$	−7.88642	−0.625433
$160$	0	0
$161$	−0.477410	−0.0376252
$162$	0	0
$163$	−17.2640	−1.35222	−0.676112	−	0.736799i	$-0.736336\pi$
$163$	−17.2640	−1.35222	−0.676112	+	0.736799i	$0.736336\pi$
$164$	0	0
$165$	10.5411	0.820626
$166$	0	0
$167$	8.82603	0.682978	0.341489	−	0.939886i	$-0.389069\pi$
$167$	8.82603	0.682978	0.341489	+	0.939886i	$0.389069\pi$
$168$	0	0
$169$	−10.9794	−0.844572
$170$	0	0
$171$	−6.13930	−0.469484
$172$	0	0
$173$	0.823790	0.0626316	0.0313158	−	0.999510i	$-0.490030\pi$
$173$	0.823790	0.0626316	0.0313158	+	0.999510i	$0.490030\pi$
$174$	0	0
$175$	0.515797	0.0389906
$176$	0	0
$177$	−19.8081	−1.48887
$178$	0	0
$179$	−0.190010	−0.0142020	−0.00710100	−	0.999975i	$-0.502260\pi$
$179$	−0.190010	−0.0142020	−0.00710100	+	0.999975i	$0.502260\pi$
$180$	0	0
$181$	−14.5591	−1.08217	−0.541083	−	0.840969i	$-0.681986\pi$
$181$	−14.5591	−1.08217	−0.541083	+	0.840969i	$0.681986\pi$
$182$	0	0
$183$	−9.97801	−0.737596
$184$	0	0
$185$	5.88040	0.432336
$186$	0	0
$187$	14.5922	1.06708
$188$	0	0
$189$	0.731551	0.0532125
$190$	0	0
$191$	−21.9084	−1.58523	−0.792617	−	0.609720i	$-0.791282\pi$
$191$	−21.9084	−1.58523	−0.792617	+	0.609720i	$0.791282\pi$
$192$	0	0
$193$	−6.18409	−0.445140	−0.222570	−	0.974917i	$-0.571445\pi$
$193$	−6.18409	−0.445140	−0.222570	+	0.974917i	$0.571445\pi$
$194$	0	0
$195$	3.29984	0.236306
$196$	0	0
$197$	−17.4254	−1.24151	−0.620756	−	0.784004i	$-0.713174\pi$
$197$	−17.4254	−1.24151	−0.620756	+	0.784004i	$0.713174\pi$
$198$	0	0
$199$	−14.2398	−1.00943	−0.504717	−	0.863285i	$-0.668403\pi$
$199$	−14.2398	−1.00943	−0.504717	+	0.863285i	$0.668403\pi$
$200$	0	0
$201$	18.0016	1.26974
$202$	0	0
$203$	−0.746742	−0.0524110
$204$	0	0
$205$	9.02106	0.630058
$206$	0	0
$207$	−2.21125	−0.153692
$208$	0	0
$209$	−11.6688	−0.807148
$210$	0	0
$211$	21.9263	1.50947	0.754734	−	0.656031i	$-0.227766\pi$
$211$	21.9263	1.50947	0.754734	+	0.656031i	$0.227766\pi$
$212$	0	0
$213$	−16.9875	−1.16396
$214$	0	0
$215$	7.54662	0.514675
$216$	0	0
$217$	−3.17873	−0.215786
$218$	0	0
$219$	−15.0490	−1.01692
$220$	0	0
$221$	4.56798	0.307276
$222$	0	0
$223$	7.60831	0.509490	0.254745	−	0.967008i	$-0.418008\pi$
$223$	7.60831	0.509490	0.254745	+	0.967008i	$0.418008\pi$
$224$	0	0
$225$	2.38904	0.159270
$226$	0	0
$227$	−0.536195	−0.0355885	−0.0177943	−	0.999842i	$-0.505664\pi$
$227$	−0.536195	−0.0355885	−0.0177943	+	0.999842i	$0.505664\pi$
$228$	0	0
$229$	16.1704	1.06857	0.534286	−	0.845304i	$-0.320581\pi$
$229$	16.1704	1.06857	0.534286	+	0.845304i	$0.320581\pi$
$230$	0	0
$231$	−5.43709	−0.357734
$232$	0	0
$233$	9.59466	0.628567	0.314284	−	0.949329i	$-0.398236\pi$
$233$	9.59466	0.628567	0.314284	+	0.949329i	$0.398236\pi$
$234$	0	0
$235$	1.06959	0.0697727
$236$	0	0
$237$	−14.3676	−0.933276
$238$	0	0
$239$	25.1698	1.62810	0.814048	−	0.580798i	$-0.197259\pi$
$239$	25.1698	1.62810	0.814048	+	0.580798i	$0.197259\pi$
$240$	0	0
$241$	−11.4350	−0.736591	−0.368295	−	0.929709i	$-0.620058\pi$
$241$	−11.4350	−0.736591	−0.368295	+	0.929709i	$0.620058\pi$
$242$	0	0
$243$	20.0264	1.28469
$244$	0	0
$245$	6.73395	0.430216
$246$	0	0
$247$	−3.65284	−0.232425
$248$	0	0
$249$	−37.1434	−2.35386
$250$	0	0
$251$	−11.0793	−0.699322	−0.349661	−	0.936876i	$-0.613703\pi$
$251$	−11.0793	−0.699322	−0.349661	+	0.936876i	$0.613703\pi$
$252$	0	0
$253$	−4.20285	−0.264231
$254$	0	0
$255$	7.46009	0.467169
$256$	0	0
$257$	−9.82315	−0.612751	−0.306376	−	0.951911i	$-0.599116\pi$
$257$	−9.82315	−0.612751	−0.306376	+	0.951911i	$0.599116\pi$
$258$	0	0
$259$	−3.03309	−0.188467
$260$	0	0
$261$	−3.45872	−0.214090
$262$	0	0
$263$	9.01282	0.555754	0.277877	−	0.960617i	$-0.410369\pi$
$263$	9.01282	0.555754	0.277877	+	0.960617i	$0.410369\pi$
$264$	0	0
$265$	−3.39722	−0.208690
$266$	0	0
$267$	−31.3410	−1.91804
$268$	0	0
$269$	−31.4219	−1.91583	−0.957914	−	0.287055i	$-0.907324\pi$
$269$	−31.4219	−1.91583	−0.957914	+	0.287055i	$0.907324\pi$
$270$	0	0
$271$	−10.7742	−0.654484	−0.327242	−	0.944941i	$-0.606119\pi$
$271$	−10.7742	−0.654484	−0.327242	+	0.944941i	$0.606119\pi$
$272$	0	0
$273$	−1.70205	−0.103013
$274$	0	0
$275$	4.54079	0.273820
$276$	0	0
$277$	30.6572	1.84202	0.921008	−	0.389544i	$-0.127367\pi$
$277$	30.6572	1.84202	0.921008	+	0.389544i	$0.127367\pi$
$278$	0	0
$279$	−14.7231	−0.881448
$280$	0	0
$281$	−9.39866	−0.560677	−0.280339	−	0.959901i	$-0.590447\pi$
$281$	−9.39866	−0.560677	−0.280339	+	0.959901i	$0.590447\pi$
$282$	0	0
$283$	18.4967	1.09951	0.549757	−	0.835325i	$-0.314720\pi$
$283$	18.4967	1.09951	0.549757	+	0.835325i	$0.314720\pi$
$284$	0	0
$285$	−5.96555	−0.353369
$286$	0	0
$287$	−4.65304	−0.274660
$288$	0	0
$289$	−6.67295	−0.392527
$290$	0	0
$291$	−17.8172	−1.04446
$292$	0	0
$293$	−10.3951	−0.607288	−0.303644	−	0.952786i	$-0.598203\pi$
$293$	−10.3951	−0.607288	−0.303644	+	0.952786i	$0.598203\pi$
$294$	0	0
$295$	−8.53271	−0.496794
$296$	0	0
$297$	6.44017	0.373696
$298$	0	0
$299$	−1.31568	−0.0760876
$300$	0	0
$301$	−3.89253	−0.224361
$302$	0	0
$303$	36.1471	2.07660
$304$	0	0
$305$	−4.29821	−0.246115
$306$	0	0
$307$	21.6878	1.23779	0.618894	−	0.785474i	$-0.287581\pi$
$307$	21.6878	1.23779	0.618894	+	0.785474i	$0.287581\pi$
$308$	0	0
$309$	7.63188	0.434163
$310$	0	0
$311$	15.7859	0.895134	0.447567	−	0.894250i	$-0.352291\pi$
$311$	15.7859	0.895134	0.447567	+	0.894250i	$0.352291\pi$
$312$	0	0
$313$	13.9086	0.786160	0.393080	−	0.919504i	$-0.371410\pi$
$313$	13.9086	0.786160	0.393080	+	0.919504i	$0.371410\pi$
$314$	0	0
$315$	−1.23226	−0.0694301
$316$	0	0
$317$	−4.52800	−0.254318	−0.127159	−	0.991882i	$-0.540586\pi$
$317$	−4.52800	−0.254318	−0.127159	+	0.991882i	$0.540586\pi$
$318$	0	0
$319$	−6.57390	−0.368068
$320$	0	0
$321$	−8.48519	−0.473597
$322$	0	0
$323$	−8.25815	−0.459496
$324$	0	0
$325$	1.42147	0.0788488
$326$	0	0
$327$	1.81169	0.100187
$328$	0	0
$329$	−0.551694	−0.0304159
$330$	0	0
$331$	−28.9864	−1.59324	−0.796618	−	0.604483i	$-0.793380\pi$
$331$	−28.9864	−1.59324	−0.796618	+	0.604483i	$0.793380\pi$
$332$	0	0
$333$	−14.0485	−0.769855
$334$	0	0
$335$	7.75453	0.423675
$336$	0	0
$337$	−4.05627	−0.220959	−0.110480	−	0.993878i	$-0.535239\pi$
$337$	−4.05627	−0.220959	−0.110480	+	0.993878i	$0.535239\pi$
$338$	0	0
$339$	8.29456	0.450499
$340$	0	0
$341$	−27.9838	−1.51541
$342$	0	0
$343$	−7.08393	−0.382497
$344$	0	0
$345$	−2.14866	−0.115680
$346$	0	0
$347$	16.8017	0.901960	0.450980	−	0.892534i	$-0.351075\pi$
$347$	16.8017	0.901960	0.450980	+	0.892534i	$0.351075\pi$
$348$	0	0
$349$	13.4514	0.720037	0.360018	−	0.932945i	$-0.382770\pi$
$349$	13.4514	0.720037	0.360018	+	0.932945i	$0.382770\pi$
$350$	0	0
$351$	2.01605	0.107609
$352$	0	0
$353$	−21.1803	−1.12731	−0.563656	−	0.826010i	$-0.690606\pi$
$353$	−21.1803	−1.12731	−0.563656	+	0.826010i	$0.690606\pi$
$354$	0	0
$355$	−7.31767	−0.388382
$356$	0	0
$357$	−3.84789	−0.203652
$358$	0	0
$359$	−14.2466	−0.751905	−0.375953	−	0.926639i	$-0.622684\pi$
$359$	−14.2466	−0.751905	−0.375953	+	0.926639i	$0.622684\pi$
$360$	0	0
$361$	−12.3963	−0.652435
$362$	0	0
$363$	−22.3294	−1.17199
$364$	0	0
$365$	−6.48265	−0.339317
$366$	0	0
$367$	2.98613	0.155874	0.0779372	−	0.996958i	$-0.475167\pi$
$367$	2.98613	0.155874	0.0779372	+	0.996958i	$0.475167\pi$
$368$	0	0
$369$	−21.5517	−1.12194
$370$	0	0
$371$	1.75228	0.0909737
$372$	0	0
$373$	−14.7039	−0.761341	−0.380671	−	0.924711i	$-0.624307\pi$
$373$	−14.7039	−0.761341	−0.380671	+	0.924711i	$0.624307\pi$
$374$	0	0
$375$	2.32143	0.119878
$376$	0	0
$377$	−2.05792	−0.105988
$378$	0	0
$379$	30.3896	1.56101	0.780503	−	0.625151i	$-0.214963\pi$
$379$	30.3896	1.56101	0.780503	+	0.625151i	$0.214963\pi$
$380$	0	0
$381$	−37.9695	−1.94523
$382$	0	0
$383$	−30.0848	−1.53726	−0.768630	−	0.639693i	$-0.779061\pi$
$383$	−30.0848	−1.53726	−0.768630	+	0.639693i	$0.779061\pi$
$384$	0	0
$385$	−2.34213	−0.119366
$386$	0	0
$387$	−18.0292	−0.916476
$388$	0	0
$389$	−22.4816	−1.13986	−0.569930	−	0.821693i	$-0.693030\pi$
$389$	−22.4816	−1.13986	−0.569930	+	0.821693i	$0.693030\pi$
$390$	0	0
$391$	−2.97441	−0.150422
$392$	0	0
$393$	41.1776	2.07713
$394$	0	0
$395$	−6.18911	−0.311408
$396$	0	0
$397$	−15.0925	−0.757469	−0.378734	−	0.925505i	$-0.623641\pi$
$397$	−15.0925	−0.757469	−0.378734	+	0.925505i	$0.623641\pi$
$398$	0	0
$399$	3.07701	0.154043
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−8.76014	−0.436374
$404$	0	0
$405$	10.4596	0.519742
$406$	0	0
$407$	−26.7017	−1.32355
$408$	0	0
$409$	14.9349	0.738483	0.369241	−	0.929334i	$-0.379618\pi$
$409$	14.9349	0.738483	0.369241	+	0.929334i	$0.379618\pi$
$410$	0	0
$411$	30.7496	1.51676
$412$	0	0
$413$	4.40115	0.216566
$414$	0	0
$415$	−16.0002	−0.785419
$416$	0	0
$417$	16.8992	0.827557
$418$	0	0
$419$	−4.91475	−0.240101	−0.120051	−	0.992768i	$-0.538306\pi$
$419$	−4.91475	−0.240101	−0.120051	+	0.992768i	$0.538306\pi$
$420$	0	0
$421$	−14.1682	−0.690516	−0.345258	−	0.938508i	$-0.612209\pi$
$421$	−14.1682	−0.690516	−0.345258	+	0.938508i	$0.612209\pi$
$422$	0	0
$423$	−2.55531	−0.124243
$424$	0	0
$425$	3.21357	0.155881
$426$	0	0
$427$	2.21701	0.107289
$428$	0	0
$429$	−14.9839	−0.723428
$430$	0	0
$431$	−5.27141	−0.253915	−0.126957	−	0.991908i	$-0.540521\pi$
$431$	−5.27141	−0.253915	−0.126957	+	0.991908i	$0.540521\pi$
$432$	0	0
$433$	15.7421	0.756517	0.378259	−	0.925700i	$-0.376523\pi$
$433$	15.7421	0.756517	0.378259	+	0.925700i	$0.376523\pi$
$434$	0	0
$435$	−3.36084	−0.161140
$436$	0	0
$437$	2.37852	0.113780
$438$	0	0
$439$	−21.2511	−1.01426	−0.507129	−	0.861870i	$-0.669293\pi$
$439$	−21.2511	−1.01426	−0.507129	+	0.861870i	$0.669293\pi$
$440$	0	0
$441$	−16.0877	−0.766082
$442$	0	0
$443$	−20.6058	−0.979012	−0.489506	−	0.872000i	$-0.662823\pi$
$443$	−20.6058	−0.979012	−0.489506	+	0.872000i	$0.662823\pi$
$444$	0	0
$445$	−13.5007	−0.639996
$446$	0	0
$447$	−32.9435	−1.55817
$448$	0	0
$449$	−28.0196	−1.32233	−0.661164	−	0.750241i	$-0.729937\pi$
$449$	−28.0196	−1.32233	−0.661164	+	0.750241i	$0.729937\pi$
$450$	0	0
$451$	−40.9627	−1.92886
$452$	0	0
$453$	27.6038	1.29694
$454$	0	0
$455$	−0.733188	−0.0343724
$456$	0	0
$457$	−11.5111	−0.538466	−0.269233	−	0.963075i	$-0.586770\pi$
$457$	−11.5111	−0.538466	−0.269233	+	0.963075i	$0.586770\pi$
$458$	0	0
$459$	4.55778	0.212739
$460$	0	0
$461$	−3.58908	−0.167160	−0.0835800	−	0.996501i	$-0.526635\pi$
$461$	−3.58908	−0.167160	−0.0835800	+	0.996501i	$0.526635\pi$
$462$	0	0
$463$	−7.30656	−0.339565	−0.169782	−	0.985482i	$-0.554306\pi$
$463$	−7.30656	−0.339565	−0.169782	+	0.985482i	$0.554306\pi$
$464$	0	0
$465$	−14.3064	−0.663444
$466$	0	0
$467$	42.6213	1.97228	0.986140	−	0.165918i	$-0.0530586\pi$
$467$	42.6213	1.97228	0.986140	+	0.165918i	$0.0530586\pi$
$468$	0	0
$469$	−3.99976	−0.184692
$470$	0	0
$471$	1.64704	0.0758915
$472$	0	0
$473$	−34.2676	−1.57563
$474$	0	0
$475$	−2.56977	−0.117909
$476$	0	0
$477$	8.11611	0.371611
$478$	0	0
$479$	−34.1882	−1.56210	−0.781049	−	0.624470i	$-0.785315\pi$
$479$	−34.1882	−1.56210	−0.781049	+	0.624470i	$0.785315\pi$
$480$	0	0
$481$	−8.35879	−0.381128
$482$	0	0
$483$	1.10828	0.0504283
$484$	0	0
$485$	−7.67511	−0.348509
$486$	0	0
$487$	−25.6718	−1.16330	−0.581650	−	0.813439i	$-0.697593\pi$
$487$	−25.6718	−1.16330	−0.581650	+	0.813439i	$0.697593\pi$
$488$	0	0
$489$	40.0773	1.81236
$490$	0	0
$491$	11.7593	0.530690	0.265345	−	0.964153i	$-0.414514\pi$
$491$	11.7593	0.530690	0.265345	+	0.964153i	$0.414514\pi$
$492$	0	0
$493$	−4.65243	−0.209535
$494$	0	0
$495$	−10.8482	−0.487588
$496$	0	0
$497$	3.77444	0.169307
$498$	0	0
$499$	24.1953	1.08313	0.541566	−	0.840658i	$-0.317832\pi$
$499$	24.1953	1.08313	0.541566	+	0.840658i	$0.317832\pi$
$500$	0	0
$501$	−20.4890	−0.915382
$502$	0	0
$503$	9.90898	0.441819	0.220910	−	0.975294i	$-0.429097\pi$
$503$	9.90898	0.441819	0.220910	+	0.975294i	$0.429097\pi$
$504$	0	0
$505$	15.5710	0.692902
$506$	0	0
$507$	25.4880	1.13196
$508$	0	0
$509$	31.1842	1.38221	0.691107	−	0.722753i	$-0.257123\pi$
$509$	31.1842	1.38221	0.691107	+	0.722753i	$0.257123\pi$
$510$	0	0
$511$	3.34373	0.147918
$512$	0	0
$513$	−3.64469	−0.160917
$514$	0	0
$515$	3.28758	0.144868
$516$	0	0
$517$	−4.85681	−0.213602
$518$	0	0
$519$	−1.91237	−0.0839439
$520$	0	0
$521$	−43.5489	−1.90791	−0.953955	−	0.299949i	$-0.903030\pi$
$521$	−43.5489	−1.90791	−0.953955	+	0.299949i	$0.903030\pi$
$522$	0	0
$523$	−22.4814	−0.983044	−0.491522	−	0.870865i	$-0.663559\pi$
$523$	−22.4814	−0.983044	−0.491522	+	0.870865i	$0.663559\pi$
$524$	0	0
$525$	−1.19739	−0.0522583
$526$	0	0
$527$	−19.8044	−0.862695
$528$	0	0
$529$	−22.1433	−0.962752
$530$	0	0
$531$	20.3850	0.884635
$532$	0	0
$533$	−12.8231	−0.555431
$534$	0	0
$535$	−3.65515	−0.158026
$536$	0	0
$537$	0.441095	0.0190346
$538$	0	0
$539$	−30.5775	−1.31707
$540$	0	0
$541$	−29.3186	−1.26051	−0.630253	−	0.776389i	$-0.717049\pi$
$541$	−29.3186	−1.26051	−0.630253	+	0.776389i	$0.717049\pi$
$542$	0	0
$543$	33.7979	1.45041
$544$	0	0
$545$	0.780421	0.0334295
$546$	0	0
$547$	38.1177	1.62980	0.814898	−	0.579604i	$-0.196793\pi$
$547$	38.1177	1.62980	0.814898	+	0.579604i	$0.196793\pi$
$548$	0	0
$549$	10.2686	0.438254
$550$	0	0
$551$	3.72037	0.158493
$552$	0	0
$553$	3.19233	0.135752
$554$	0	0
$555$	−13.6509	−0.579450
$556$	0	0
$557$	−0.325749	−0.0138024	−0.00690120	−	0.999976i	$-0.502197\pi$
$557$	−0.325749	−0.0138024	−0.00690120	+	0.999976i	$0.502197\pi$
$558$	0	0
$559$	−10.7273	−0.453715
$560$	0	0
$561$	−33.8747	−1.43019
$562$	0	0
$563$	35.3560	1.49008	0.745040	−	0.667020i	$-0.232431\pi$
$563$	35.3560	1.49008	0.745040	+	0.667020i	$0.232431\pi$
$564$	0	0
$565$	3.57304	0.150319
$566$	0	0
$567$	−5.39503	−0.226570
$568$	0	0
$569$	14.1728	0.594157	0.297078	−	0.954853i	$-0.403988\pi$
$569$	14.1728	0.594157	0.297078	+	0.954853i	$0.403988\pi$
$570$	0	0
$571$	−8.26903	−0.346048	−0.173024	−	0.984918i	$-0.555354\pi$
$571$	−8.26903	−0.346048	−0.173024	+	0.984918i	$0.555354\pi$
$572$	0	0
$573$	50.8588	2.12465
$574$	0	0
$575$	−0.925577	−0.0385992
$576$	0	0
$577$	10.4121	0.433462	0.216731	−	0.976231i	$-0.430461\pi$
$577$	10.4121	0.433462	0.216731	+	0.976231i	$0.430461\pi$
$578$	0	0
$579$	14.3559	0.596612
$580$	0	0
$581$	8.25286	0.342386
$582$	0	0
$583$	15.4261	0.638883
$584$	0	0
$585$	−3.39595	−0.140405
$586$	0	0
$587$	35.3519	1.45913	0.729564	−	0.683912i	$-0.239723\pi$
$587$	35.3519	1.45913	0.729564	+	0.683912i	$0.239723\pi$
$588$	0	0
$589$	15.8369	0.652547
$590$	0	0
$591$	40.4520	1.66397
$592$	0	0
$593$	−26.6716	−1.09527	−0.547636	−	0.836717i	$-0.684472\pi$
$593$	−26.6716	−1.09527	−0.547636	+	0.836717i	$0.684472\pi$
$594$	0	0
$595$	−1.65755	−0.0679530
$596$	0	0
$597$	33.0568	1.35292
$598$	0	0
$599$	−38.1497	−1.55876	−0.779378	−	0.626554i	$-0.784465\pi$
$599$	−38.1497	−1.55876	−0.779378	+	0.626554i	$0.784465\pi$
$600$	0	0
$601$	−37.2826	−1.52079	−0.760395	−	0.649461i	$-0.774995\pi$
$601$	−37.2826	−1.52079	−0.760395	+	0.649461i	$0.774995\pi$
$602$	0	0
$603$	−18.5259	−0.754434
$604$	0	0
$605$	−9.61879	−0.391059
$606$	0	0
$607$	−13.3600	−0.542266	−0.271133	−	0.962542i	$-0.587398\pi$
$607$	−13.3600	−0.542266	−0.271133	+	0.962542i	$0.587398\pi$
$608$	0	0
$609$	1.73351	0.0702454
$610$	0	0
$611$	−1.52039	−0.0615085
$612$	0	0
$613$	−26.5021	−1.07041	−0.535205	−	0.844722i	$-0.679766\pi$
$613$	−26.5021	−1.07041	−0.535205	+	0.844722i	$0.679766\pi$
$614$	0	0
$615$	−20.9418	−0.844453
$616$	0	0
$617$	18.6828	0.752142	0.376071	−	0.926591i	$-0.377275\pi$
$617$	18.6828	0.752142	0.376071	+	0.926591i	$0.377275\pi$
$618$	0	0
$619$	23.4388	0.942083	0.471042	−	0.882111i	$-0.343878\pi$
$619$	23.4388	0.942083	0.471042	+	0.882111i	$0.343878\pi$
$620$	0	0
$621$	−1.31274	−0.0526784
$622$	0	0
$623$	6.96364	0.278992
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	27.0883	1.08180
$628$	0	0
$629$	−18.8971	−0.753476
$630$	0	0
$631$	−23.8401	−0.949061	−0.474530	−	0.880239i	$-0.657382\pi$
$631$	−23.8401	−0.949061	−0.474530	+	0.880239i	$0.657382\pi$
$632$	0	0
$633$	−50.9004	−2.02311
$634$	0	0
$635$	−16.3561	−0.649070
$636$	0	0
$637$	−9.57209	−0.379260
$638$	0	0
$639$	17.4822	0.691587
$640$	0	0
$641$	−27.4559	−1.08444	−0.542221	−	0.840236i	$-0.682416\pi$
$641$	−27.4559	−1.08444	−0.542221	+	0.840236i	$0.682416\pi$
$642$	0	0
$643$	−27.7027	−1.09249	−0.546245	−	0.837626i	$-0.683943\pi$
$643$	−27.7027	−1.09249	−0.546245	+	0.837626i	$0.683943\pi$
$644$	0	0
$645$	−17.5190	−0.689808
$646$	0	0
$647$	−3.42407	−0.134614	−0.0673070	−	0.997732i	$-0.521441\pi$
$647$	−3.42407	−0.134614	−0.0673070	+	0.997732i	$0.521441\pi$
$648$	0	0
$649$	38.7453	1.52088
$650$	0	0
$651$	7.37920	0.289214
$652$	0	0
$653$	32.5176	1.27251	0.636255	−	0.771478i	$-0.280482\pi$
$653$	32.5176	1.27251	0.636255	+	0.771478i	$0.280482\pi$
$654$	0	0
$655$	17.7380	0.693082
$656$	0	0
$657$	15.4873	0.604219
$658$	0	0
$659$	34.5960	1.34767	0.673834	−	0.738882i	$-0.264646\pi$
$659$	34.5960	1.34767	0.673834	+	0.738882i	$0.264646\pi$
$660$	0	0
$661$	−2.35365	−0.0915465	−0.0457732	−	0.998952i	$-0.514575\pi$
$661$	−2.35365	−0.0915465	−0.0457732	+	0.998952i	$0.514575\pi$
$662$	0	0
$663$	−10.6043	−0.411835
$664$	0	0
$665$	1.32548	0.0514000
$666$	0	0
$667$	1.34000	0.0518849
$668$	0	0
$669$	−17.6622	−0.682859
$670$	0	0
$671$	19.5173	0.753457
$672$	0	0
$673$	−23.6612	−0.912071	−0.456035	−	0.889962i	$-0.650731\pi$
$673$	−23.6612	−0.912071	−0.456035	+	0.889962i	$0.650731\pi$
$674$	0	0
$675$	1.41829	0.0545901
$676$	0	0
$677$	16.3029	0.626572	0.313286	−	0.949659i	$-0.398570\pi$
$677$	16.3029	0.626572	0.313286	+	0.949659i	$0.398570\pi$
$678$	0	0
$679$	3.95880	0.151925
$680$	0	0
$681$	1.24474	0.0476986
$682$	0	0
$683$	3.14519	0.120347	0.0601737	−	0.998188i	$-0.480835\pi$
$683$	3.14519	0.120347	0.0601737	+	0.998188i	$0.480835\pi$
$684$	0	0
$685$	13.2460	0.506102
$686$	0	0
$687$	−37.5385	−1.43218
$688$	0	0
$689$	4.82903	0.183972
$690$	0	0
$691$	−3.58626	−0.136428	−0.0682139	−	0.997671i	$-0.521730\pi$
$691$	−3.58626	−0.136428	−0.0682139	+	0.997671i	$0.521730\pi$
$692$	0	0
$693$	5.59545	0.212553
$694$	0	0
$695$	7.27964	0.276133
$696$	0	0
$697$	−28.9898	−1.09807
$698$	0	0
$699$	−22.2733	−0.842455
$700$	0	0
$701$	−14.8783	−0.561945	−0.280972	−	0.959716i	$-0.590657\pi$
$701$	−14.8783	−0.561945	−0.280972	+	0.959716i	$0.590657\pi$
$702$	0	0
$703$	15.1113	0.569933
$704$	0	0
$705$	−2.48299	−0.0935149
$706$	0	0
$707$	−8.03150	−0.302056
$708$	0	0
$709$	4.46752	0.167781	0.0838906	−	0.996475i	$-0.473265\pi$
$709$	4.46752	0.167781	0.0838906	+	0.996475i	$0.473265\pi$
$710$	0	0
$711$	14.7861	0.554521
$712$	0	0
$713$	5.70410	0.213620
$714$	0	0
$715$	−6.45458	−0.241388
$716$	0	0
$717$	−58.4299	−2.18210
$718$	0	0
$719$	8.85410	0.330202	0.165101	−	0.986277i	$-0.447205\pi$
$719$	8.85410	0.330202	0.165101	+	0.986277i	$0.447205\pi$
$720$	0	0
$721$	−1.69572	−0.0631520
$722$	0	0
$723$	26.5455	0.987237
$724$	0	0
$725$	−1.44774	−0.0537678
$726$	0	0
$727$	25.5022	0.945823	0.472912	−	0.881110i	$-0.343203\pi$
$727$	25.5022	0.945823	0.472912	+	0.881110i	$0.343203\pi$
$728$	0	0
$729$	−15.1110	−0.559668
$730$	0	0
$731$	−24.2516	−0.896978
$732$	0	0
$733$	36.2118	1.33751	0.668756	−	0.743482i	$-0.266827\pi$
$733$	36.2118	1.33751	0.668756	+	0.743482i	$0.266827\pi$
$734$	0	0
$735$	−15.6324	−0.576610
$736$	0	0
$737$	−35.2117	−1.29704
$738$	0	0
$739$	−15.4633	−0.568825	−0.284413	−	0.958702i	$-0.591799\pi$
$739$	−15.4633	−0.568825	−0.284413	+	0.958702i	$0.591799\pi$
$740$	0	0
$741$	8.47983	0.311514
$742$	0	0
$743$	−49.2279	−1.80600	−0.902998	−	0.429645i	$-0.858639\pi$
$743$	−49.2279	−1.80600	−0.902998	+	0.429645i	$0.858639\pi$
$744$	0	0
$745$	−14.1910	−0.519919
$746$	0	0
$747$	38.2252	1.39859
$748$	0	0
$749$	1.88532	0.0688880
$750$	0	0
$751$	34.7967	1.26975	0.634875	−	0.772615i	$-0.281052\pi$
$751$	34.7967	1.26975	0.634875	+	0.772615i	$0.281052\pi$
$752$	0	0
$753$	25.7199	0.937286
$754$	0	0
$755$	11.8909	0.432753
$756$	0	0
$757$	−38.0393	−1.38256	−0.691282	−	0.722585i	$-0.742954\pi$
$757$	−38.0393	−1.38256	−0.691282	+	0.722585i	$0.742954\pi$
$758$	0	0
$759$	9.75664	0.354144
$760$	0	0
$761$	−13.8507	−0.502088	−0.251044	−	0.967976i	$-0.580774\pi$
$761$	−13.8507	−0.502088	−0.251044	+	0.967976i	$0.580774\pi$
$762$	0	0
$763$	−0.402539	−0.0145729
$764$	0	0
$765$	−7.67737	−0.277576
$766$	0	0
$767$	12.1290	0.437951
$768$	0	0
$769$	−7.18816	−0.259212	−0.129606	−	0.991566i	$-0.541371\pi$
$769$	−7.18816	−0.259212	−0.129606	+	0.991566i	$0.541371\pi$
$770$	0	0
$771$	22.8038	0.821258
$772$	0	0
$773$	−31.7005	−1.14019	−0.570093	−	0.821580i	$-0.693093\pi$
$773$	−31.7005	−1.14019	−0.570093	+	0.821580i	$0.693093\pi$
$774$	0	0
$775$	−6.16275	−0.221373
$776$	0	0
$777$	7.04112	0.252599
$778$	0	0
$779$	23.1821	0.830583
$780$	0	0
$781$	33.2280	1.18899
$782$	0	0
$783$	−2.05332	−0.0733798
$784$	0	0
$785$	0.709493	0.0253229
$786$	0	0
$787$	52.0064	1.85383	0.926915	−	0.375273i	$-0.122451\pi$
$787$	52.0064	1.85383	0.926915	+	0.375273i	$0.122451\pi$
$788$	0	0
$789$	−20.9226	−0.744866
$790$	0	0
$791$	−1.84296	−0.0655282
$792$	0	0
$793$	6.10977	0.216964
$794$	0	0
$795$	7.88642	0.279702
$796$	0	0
$797$	0.650091	0.0230274	0.0115137	−	0.999934i	$-0.496335\pi$
$797$	0.650091	0.0230274	0.0115137	+	0.999934i	$0.496335\pi$
$798$	0	0
$799$	−3.43722	−0.121600
$800$	0	0
$801$	32.2539	1.13963
$802$	0	0
$803$	29.4364	1.03879
$804$	0	0
$805$	0.477410	0.0168265
$806$	0	0
$807$	72.9438	2.56774
$808$	0	0
$809$	−0.529620	−0.0186205	−0.00931023	−	0.999957i	$-0.502964\pi$
$809$	−0.529620	−0.0186205	−0.00931023	+	0.999957i	$0.502964\pi$
$810$	0	0
$811$	4.69632	0.164910	0.0824551	−	0.996595i	$-0.473724\pi$
$811$	4.69632	0.164910	0.0824551	+	0.996595i	$0.473724\pi$
$812$	0	0
$813$	25.0115	0.877191
$814$	0	0
$815$	17.2640	0.604733
$816$	0	0
$817$	19.3931	0.678478
$818$	0	0
$819$	1.75162	0.0612065
$820$	0	0
$821$	17.7904	0.620889	0.310445	−	0.950591i	$-0.399522\pi$
$821$	17.7904	0.620889	0.310445	+	0.950591i	$0.399522\pi$
$822$	0	0
$823$	25.4813	0.888221	0.444111	−	0.895972i	$-0.353520\pi$
$823$	25.4813	0.888221	0.444111	+	0.895972i	$0.353520\pi$
$824$	0	0
$825$	−10.5411	−0.366995
$826$	0	0
$827$	−11.3759	−0.395580	−0.197790	−	0.980244i	$-0.563376\pi$
$827$	−11.3759	−0.395580	−0.197790	+	0.980244i	$0.563376\pi$
$828$	0	0
$829$	9.15670	0.318025	0.159013	−	0.987277i	$-0.449169\pi$
$829$	9.15670	0.318025	0.159013	+	0.987277i	$0.449169\pi$
$830$	0	0
$831$	−71.1687	−2.46881
$832$	0	0
$833$	−21.6400	−0.749783
$834$	0	0
$835$	−8.82603	−0.305437
$836$	0	0
$837$	−8.74058	−0.302119
$838$	0	0
$839$	−41.3930	−1.42904	−0.714522	−	0.699613i	$-0.753356\pi$
$839$	−41.3930	−1.42904	−0.714522	+	0.699613i	$0.753356\pi$
$840$	0	0
$841$	−26.9040	−0.927726
$842$	0	0
$843$	21.8184	0.751464
$844$	0	0
$845$	10.9794	0.377704
$846$	0	0
$847$	4.96134	0.170474
$848$	0	0
$849$	−42.9388	−1.47366
$850$	0	0
$851$	5.44277	0.186576
$852$	0	0
$853$	−26.0489	−0.891896	−0.445948	−	0.895059i	$-0.647133\pi$
$853$	−26.0489	−0.891896	−0.445948	+	0.895059i	$0.647133\pi$
$854$	0	0
$855$	6.13930	0.209960
$856$	0	0
$857$	−17.4390	−0.595706	−0.297853	−	0.954612i	$-0.596270\pi$
$857$	−17.4390	−0.595706	−0.297853	+	0.954612i	$0.596270\pi$
$858$	0	0
$859$	53.1349	1.81294	0.906470	−	0.422271i	$-0.138767\pi$
$859$	53.1349	1.81294	0.906470	+	0.422271i	$0.138767\pi$
$860$	0	0
$861$	10.8017	0.368121
$862$	0	0
$863$	−1.21959	−0.0415152	−0.0207576	−	0.999785i	$-0.506608\pi$
$863$	−1.21959	−0.0415152	−0.0207576	+	0.999785i	$0.506608\pi$
$864$	0	0
$865$	−0.823790	−0.0280097
$866$	0	0
$867$	15.4908	0.526095
$868$	0	0
$869$	28.1035	0.953345
$870$	0	0
$871$	−11.0228	−0.373493
$872$	0	0
$873$	18.3362	0.620585
$874$	0	0
$875$	−0.515797	−0.0174371
$876$	0	0
$877$	−58.3629	−1.97077	−0.985387	−	0.170331i	$-0.945516\pi$
$877$	−58.3629	−1.97077	−0.985387	+	0.170331i	$0.945516\pi$
$878$	0	0
$879$	24.1315	0.813935
$880$	0	0
$881$	−24.5870	−0.828357	−0.414178	−	0.910196i	$-0.635931\pi$
$881$	−24.5870	−0.828357	−0.414178	+	0.910196i	$0.635931\pi$
$882$	0	0
$883$	−31.9454	−1.07505	−0.537525	−	0.843248i	$-0.680641\pi$
$883$	−31.9454	−1.07505	−0.537525	+	0.843248i	$0.680641\pi$
$884$	0	0
$885$	19.8081	0.665842
$886$	0	0
$887$	27.0532	0.908359	0.454179	−	0.890910i	$-0.349933\pi$
$887$	27.0532	0.908359	0.454179	+	0.890910i	$0.349933\pi$
$888$	0	0
$889$	8.43641	0.282948
$890$	0	0
$891$	−47.4949	−1.59114
$892$	0	0
$893$	2.74861	0.0919789
$894$	0	0
$895$	0.190010	0.00635132
$896$	0	0
$897$	3.05425	0.101979
$898$	0	0
$899$	8.92208	0.297568
$900$	0	0
$901$	10.9172	0.363705
$902$	0	0
$903$	9.03623	0.300707
$904$	0	0
$905$	14.5591	0.483960
$906$	0	0
$907$	1.34778	0.0447524	0.0223762	−	0.999750i	$-0.492877\pi$
$907$	1.34778	0.0447524	0.0223762	+	0.999750i	$0.492877\pi$
$908$	0	0
$909$	−37.1999	−1.23384
$910$	0	0
$911$	−54.8993	−1.81890	−0.909448	−	0.415818i	$-0.863495\pi$
$911$	−54.8993	−1.81890	−0.909448	+	0.415818i	$0.863495\pi$
$912$	0	0
$913$	72.6535	2.40448
$914$	0	0
$915$	9.97801	0.329863
$916$	0	0
$917$	−9.14922	−0.302134
$918$	0	0
$919$	9.58092	0.316045	0.158023	−	0.987435i	$-0.449488\pi$
$919$	9.58092	0.316045	0.158023	+	0.987435i	$0.449488\pi$
$920$	0	0
$921$	−50.3468	−1.65898
$922$	0	0
$923$	10.4018	0.342380
$924$	0	0
$925$	−5.88040	−0.193346
$926$	0	0
$927$	−7.85416	−0.257965
$928$	0	0
$929$	−10.5310	−0.345512	−0.172756	−	0.984965i	$-0.555267\pi$
$929$	−10.5310	−0.345512	−0.172756	+	0.984965i	$0.555267\pi$
$930$	0	0
$931$	17.3047	0.567139
$932$	0	0
$933$	−36.6458	−1.19973
$934$	0	0
$935$	−14.5922	−0.477215
$936$	0	0
$937$	−4.05730	−0.132546	−0.0662731	−	0.997802i	$-0.521111\pi$
$937$	−4.05730	−0.132546	−0.0662731	+	0.997802i	$0.521111\pi$
$938$	0	0
$939$	−32.2878	−1.05367
$940$	0	0
$941$	−14.4347	−0.470558	−0.235279	−	0.971928i	$-0.575600\pi$
$941$	−14.4347	−0.470558	−0.235279	+	0.971928i	$0.575600\pi$
$942$	0	0
$943$	8.34969	0.271903
$944$	0	0
$945$	−0.731551	−0.0237974
$946$	0	0
$947$	−60.1369	−1.95419	−0.977094	−	0.212810i	$-0.931739\pi$
$947$	−60.1369	−1.95419	−0.977094	+	0.212810i	$0.931739\pi$
$948$	0	0
$949$	9.21487	0.299127
$950$	0	0
$951$	10.5114	0.340857
$952$	0	0
$953$	−39.8898	−1.29216	−0.646078	−	0.763271i	$-0.723592\pi$
$953$	−39.8898	−1.29216	−0.646078	+	0.763271i	$0.723592\pi$
$954$	0	0
$955$	21.9084	0.708938
$956$	0	0
$957$	15.2609	0.493313
$958$	0	0
$959$	−6.83223	−0.220624
$960$	0	0
$961$	6.97950	0.225145
$962$	0	0
$963$	8.73232	0.281395
$964$	0	0
$965$	6.18409	0.199073
$966$	0	0
$967$	−22.2323	−0.714942	−0.357471	−	0.933924i	$-0.616361\pi$
$967$	−22.2323	−0.714942	−0.357471	+	0.933924i	$0.616361\pi$
$968$	0	0
$969$	19.1707	0.615852
$970$	0	0
$971$	−7.27517	−0.233471	−0.116736	−	0.993163i	$-0.537243\pi$
$971$	−7.27517	−0.233471	−0.116736	+	0.993163i	$0.537243\pi$
$972$	0	0
$973$	−3.75482	−0.120374
$974$	0	0
$975$	−3.29984	−0.105679
$976$	0	0
$977$	−41.3326	−1.32235	−0.661173	−	0.750233i	$-0.729941\pi$
$977$	−41.3326	−1.32235	−0.661173	+	0.750233i	$0.729941\pi$
$978$	0	0
$979$	61.3040	1.95929
$980$	0	0
$981$	−1.86446	−0.0595276
$982$	0	0
$983$	3.54531	0.113078	0.0565389	−	0.998400i	$-0.481993\pi$
$983$	3.54531	0.113078	0.0565389	+	0.998400i	$0.481993\pi$
$984$	0	0
$985$	17.4254	0.555221
$986$	0	0
$987$	1.28072	0.0407658
$988$	0	0
$989$	6.98498	0.222109
$990$	0	0
$991$	57.4573	1.82519	0.912596	−	0.408863i	$-0.134075\pi$
$991$	57.4573	1.82519	0.912596	+	0.408863i	$0.134075\pi$
$992$	0	0
$993$	67.2899	2.13538
$994$	0	0
$995$	14.2398	0.451433
$996$	0	0
$997$	−43.2684	−1.37032	−0.685162	−	0.728391i	$-0.740268\pi$
$997$	−43.2684	−1.37032	−0.685162	+	0.728391i	$0.740268\pi$
$998$	0	0
$999$	−8.34013	−0.263870

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.4	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.4	✓	28	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)

\(f(q)\)	\(=\)	\(q-2.32143 q^{3} -1.00000 q^{5} +0.515797 q^{7} +2.38904 q^{9} +O(q^{10})\)	\(q-2.32143 q^{3} -1.00000 q^{5} +0.515797 q^{7} +2.38904 q^{9} +4.54079 q^{11} +1.42147 q^{13} +2.32143 q^{15} +3.21357 q^{17} -2.56977 q^{19} -1.19739 q^{21} -0.925577 q^{23} +1.00000 q^{25} +1.41829 q^{27} -1.44774 q^{29} -6.16275 q^{31} -10.5411 q^{33} -0.515797 q^{35} -5.88040 q^{37} -3.29984 q^{39} -9.02106 q^{41} -7.54662 q^{43} -2.38904 q^{45} -1.06959 q^{47} -6.73395 q^{49} -7.46009 q^{51} +3.39722 q^{53} -4.54079 q^{55} +5.96555 q^{57} +8.53271 q^{59} +4.29821 q^{61} +1.23226 q^{63} -1.42147 q^{65} -7.75453 q^{67} +2.14866 q^{69} +7.31767 q^{71} +6.48265 q^{73} -2.32143 q^{75} +2.34213 q^{77} +6.18911 q^{79} -10.4596 q^{81} +16.0002 q^{83} -3.21357 q^{85} +3.36084 q^{87} +13.5007 q^{89} +0.733188 q^{91} +14.3064 q^{93} +2.56977 q^{95} +7.67511 q^{97} +10.8482 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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