LMFDB - Embedded newform 8020.2.a.f.1.9

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			1.9
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)

$f(q)$	$=$	$q-2.16998 q^{3} +1.00000 q^{5} +4.79835 q^{7} +1.70880 q^{9} +O(q^{10})$	$q-2.16998 q^{3} +1.00000 q^{5} +4.79835 q^{7} +1.70880 q^{9} -0.223557 q^{11} -0.368302 q^{13} -2.16998 q^{15} +0.909198 q^{17} +1.25372 q^{19} -10.4123 q^{21} -1.59805 q^{23} +1.00000 q^{25} +2.80188 q^{27} -9.95847 q^{29} +6.88740 q^{31} +0.485113 q^{33} +4.79835 q^{35} -7.91334 q^{37} +0.799207 q^{39} +8.50498 q^{41} -1.29410 q^{43} +1.70880 q^{45} +6.99851 q^{47} +16.0241 q^{49} -1.97294 q^{51} +3.68538 q^{53} -0.223557 q^{55} -2.72054 q^{57} +2.79984 q^{59} -3.82129 q^{61} +8.19940 q^{63} -0.368302 q^{65} -6.04882 q^{67} +3.46774 q^{69} +0.670127 q^{71} -4.80893 q^{73} -2.16998 q^{75} -1.07270 q^{77} +2.02262 q^{79} -11.2064 q^{81} -6.22957 q^{83} +0.909198 q^{85} +21.6096 q^{87} +9.78651 q^{89} -1.76724 q^{91} -14.9455 q^{93} +1.25372 q^{95} +17.4544 q^{97} -0.382013 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.16998	−1.25284	−0.626418	−	0.779487i	$-0.715480\pi$
$3$	−2.16998	−1.25284	−0.626418	+	0.779487i	$0.715480\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.79835	1.81361	0.906803	−	0.421555i	$-0.138516\pi$
$7$	4.79835	1.81361	0.906803	+	0.421555i	$0.138516\pi$
$8$	0	0
$9$	1.70880	0.569599
$10$	0	0
$11$	−0.223557	−0.0674050	−0.0337025	−	0.999432i	$-0.510730\pi$
$11$	−0.223557	−0.0674050	−0.0337025	+	0.999432i	$0.510730\pi$
$12$	0	0
$13$	−0.368302	−0.102149	−0.0510744	−	0.998695i	$-0.516265\pi$
$13$	−0.368302	−0.102149	−0.0510744	+	0.998695i	$0.516265\pi$
$14$	0	0
$15$	−2.16998	−0.560285
$16$	0	0
$17$	0.909198	0.220513	0.110256	−	0.993903i	$-0.464833\pi$
$17$	0.909198	0.220513	0.110256	+	0.993903i	$0.464833\pi$
$18$	0	0
$19$	1.25372	0.287623	0.143812	−	0.989605i	$-0.454064\pi$
$19$	1.25372	0.287623	0.143812	+	0.989605i	$0.454064\pi$
$20$	0	0
$21$	−10.4123	−2.27215
$22$	0	0
$23$	−1.59805	−0.333217	−0.166609	−	0.986023i	$-0.553282\pi$
$23$	−1.59805	−0.333217	−0.166609	+	0.986023i	$0.553282\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.80188	0.539222
$28$	0	0
$29$	−9.95847	−1.84924	−0.924621	−	0.380889i	$-0.875618\pi$
$29$	−9.95847	−1.84924	−0.924621	+	0.380889i	$0.875618\pi$
$30$	0	0
$31$	6.88740	1.23701	0.618507	−	0.785779i	$-0.287738\pi$
$31$	6.88740	1.23701	0.618507	+	0.785779i	$0.287738\pi$
$32$	0	0
$33$	0.485113	0.0844474
$34$	0	0
$35$	4.79835	0.811069
$36$	0	0
$37$	−7.91334	−1.30095	−0.650473	−	0.759529i	$-0.725429\pi$
$37$	−7.91334	−1.30095	−0.650473	+	0.759529i	$0.725429\pi$
$38$	0	0
$39$	0.799207	0.127976
$40$	0	0
$41$	8.50498	1.32825	0.664127	−	0.747620i	$-0.268803\pi$
$41$	8.50498	1.32825	0.664127	+	0.747620i	$0.268803\pi$
$42$	0	0
$43$	−1.29410	−0.197348	−0.0986740	−	0.995120i	$-0.531460\pi$
$43$	−1.29410	−0.197348	−0.0986740	+	0.995120i	$0.531460\pi$
$44$	0	0
$45$	1.70880	0.254732
$46$	0	0
$47$	6.99851	1.02084	0.510419	−	0.859926i	$-0.329491\pi$
$47$	6.99851	1.02084	0.510419	+	0.859926i	$0.329491\pi$
$48$	0	0
$49$	16.0241	2.28916
$50$	0	0
$51$	−1.97294	−0.276266
$52$	0	0
$53$	3.68538	0.506226	0.253113	−	0.967437i	$-0.418546\pi$
$53$	3.68538	0.506226	0.253113	+	0.967437i	$0.418546\pi$
$54$	0	0
$55$	−0.223557	−0.0301444
$56$	0	0
$57$	−2.72054	−0.360345
$58$	0	0
$59$	2.79984	0.364508	0.182254	−	0.983251i	$-0.441661\pi$
$59$	2.79984	0.364508	0.182254	+	0.983251i	$0.441661\pi$
$60$	0	0
$61$	−3.82129	−0.489266	−0.244633	−	0.969616i	$-0.578667\pi$
$61$	−3.82129	−0.489266	−0.244633	+	0.969616i	$0.578667\pi$
$62$	0	0
$63$	8.19940	1.03303
$64$	0	0
$65$	−0.368302	−0.0456823
$66$	0	0
$67$	−6.04882	−0.738981	−0.369491	−	0.929234i	$-0.620468\pi$
$67$	−6.04882	−0.738981	−0.369491	+	0.929234i	$0.620468\pi$
$68$	0	0
$69$	3.46774	0.417467
$70$	0	0
$71$	0.670127	0.0795294	0.0397647	−	0.999209i	$-0.487339\pi$
$71$	0.670127	0.0795294	0.0397647	+	0.999209i	$0.487339\pi$
$72$	0	0
$73$	−4.80893	−0.562842	−0.281421	−	0.959584i	$-0.590806\pi$
$73$	−4.80893	−0.562842	−0.281421	+	0.959584i	$0.590806\pi$
$74$	0	0
$75$	−2.16998	−0.250567
$76$	0	0
$77$	−1.07270	−0.122246
$78$	0	0
$79$	2.02262	0.227562	0.113781	−	0.993506i	$-0.463704\pi$
$79$	2.02262	0.227562	0.113781	+	0.993506i	$0.463704\pi$
$80$	0	0
$81$	−11.2064	−1.24516
$82$	0	0
$83$	−6.22957	−0.683784	−0.341892	−	0.939739i	$-0.611068\pi$
$83$	−6.22957	−0.683784	−0.341892	+	0.939739i	$0.611068\pi$
$84$	0	0
$85$	0.909198	0.0986163
$86$	0	0
$87$	21.6096	2.31680
$88$	0	0
$89$	9.78651	1.03737	0.518684	−	0.854966i	$-0.326422\pi$
$89$	9.78651	1.03737	0.518684	+	0.854966i	$0.326422\pi$
$90$	0	0
$91$	−1.76724	−0.185257
$92$	0	0
$93$	−14.9455	−1.54978
$94$	0	0
$95$	1.25372	0.128629
$96$	0	0
$97$	17.4544	1.77222	0.886111	−	0.463473i	$-0.153397\pi$
$97$	17.4544	1.77222	0.886111	+	0.463473i	$0.153397\pi$
$98$	0	0
$99$	−0.382013	−0.0383938
$100$	0	0
$101$	15.1445	1.50693	0.753467	−	0.657486i	$-0.228380\pi$
$101$	15.1445	1.50693	0.753467	+	0.657486i	$0.228380\pi$
$102$	0	0
$103$	13.3017	1.31066	0.655329	−	0.755344i	$-0.272530\pi$
$103$	13.3017	1.31066	0.655329	+	0.755344i	$0.272530\pi$
$104$	0	0
$105$	−10.4123	−1.01614
$106$	0	0
$107$	5.98249	0.578349	0.289174	−	0.957276i	$-0.406619\pi$
$107$	5.98249	0.578349	0.289174	+	0.957276i	$0.406619\pi$
$108$	0	0
$109$	17.2162	1.64901	0.824505	−	0.565855i	$-0.191454\pi$
$109$	17.2162	1.64901	0.824505	+	0.565855i	$0.191454\pi$
$110$	0	0
$111$	17.1718	1.62987
$112$	0	0
$113$	19.3012	1.81571	0.907854	−	0.419287i	$-0.137720\pi$
$113$	19.3012	1.81571	0.907854	+	0.419287i	$0.137720\pi$
$114$	0	0
$115$	−1.59805	−0.149019
$116$	0	0
$117$	−0.629354	−0.0581838
$118$	0	0
$119$	4.36265	0.399923
$120$	0	0
$121$	−10.9500	−0.995457
$122$	0	0
$123$	−18.4556	−1.66408
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−19.4240	−1.72360	−0.861800	−	0.507248i	$-0.830663\pi$
$127$	−19.4240	−1.72360	−0.861800	+	0.507248i	$0.830663\pi$
$128$	0	0
$129$	2.80816	0.247245
$130$	0	0
$131$	1.25705	0.109829	0.0549146	−	0.998491i	$-0.482511\pi$
$131$	1.25705	0.109829	0.0549146	+	0.998491i	$0.482511\pi$
$132$	0	0
$133$	6.01579	0.521635
$134$	0	0
$135$	2.80188	0.241148
$136$	0	0
$137$	12.3574	1.05576	0.527881	−	0.849318i	$-0.322987\pi$
$137$	12.3574	1.05576	0.527881	+	0.849318i	$0.322987\pi$
$138$	0	0
$139$	−7.04811	−0.597813	−0.298906	−	0.954282i	$-0.596622\pi$
$139$	−7.04811	−0.597813	−0.298906	+	0.954282i	$0.596622\pi$
$140$	0	0
$141$	−15.1866	−1.27894
$142$	0	0
$143$	0.0823366	0.00688533
$144$	0	0
$145$	−9.95847	−0.827006
$146$	0	0
$147$	−34.7720	−2.86795
$148$	0	0
$149$	−0.781230	−0.0640009	−0.0320004	−	0.999488i	$-0.510188\pi$
$149$	−0.781230	−0.0640009	−0.0320004	+	0.999488i	$0.510188\pi$
$150$	0	0
$151$	−17.9128	−1.45772	−0.728862	−	0.684661i	$-0.759950\pi$
$151$	−17.9128	−1.45772	−0.728862	+	0.684661i	$0.759950\pi$
$152$	0	0
$153$	1.55363	0.125604
$154$	0	0
$155$	6.88740	0.553210
$156$	0	0
$157$	−8.24709	−0.658189	−0.329095	−	0.944297i	$-0.606743\pi$
$157$	−8.24709	−0.658189	−0.329095	+	0.944297i	$0.606743\pi$
$158$	0	0
$159$	−7.99718	−0.634218
$160$	0	0
$161$	−7.66802	−0.604324
$162$	0	0
$163$	−11.2712	−0.882828	−0.441414	−	0.897304i	$-0.645523\pi$
$163$	−11.2712	−0.882828	−0.441414	+	0.897304i	$0.645523\pi$
$164$	0	0
$165$	0.485113	0.0377660
$166$	0	0
$167$	4.40254	0.340679	0.170339	−	0.985385i	$-0.445514\pi$
$167$	4.40254	0.340679	0.170339	+	0.985385i	$0.445514\pi$
$168$	0	0
$169$	−12.8644	−0.989566
$170$	0	0
$171$	2.14235	0.163830
$172$	0	0
$173$	−11.9438	−0.908074	−0.454037	−	0.890983i	$-0.650017\pi$
$173$	−11.9438	−0.908074	−0.454037	+	0.890983i	$0.650017\pi$
$174$	0	0
$175$	4.79835	0.362721
$176$	0	0
$177$	−6.07559	−0.456669
$178$	0	0
$179$	−1.62526	−0.121478	−0.0607390	−	0.998154i	$-0.519346\pi$
$179$	−1.62526	−0.121478	−0.0607390	+	0.998154i	$0.519346\pi$
$180$	0	0
$181$	−10.0231	−0.745014	−0.372507	−	0.928029i	$-0.621502\pi$
$181$	−10.0231	−0.745014	−0.372507	+	0.928029i	$0.621502\pi$
$182$	0	0
$183$	8.29210	0.612970
$184$	0	0
$185$	−7.91334	−0.581801
$186$	0	0
$187$	−0.203258	−0.0148637
$188$	0	0
$189$	13.4444	0.977936
$190$	0	0
$191$	−3.61151	−0.261319	−0.130660	−	0.991427i	$-0.541710\pi$
$191$	−3.61151	−0.261319	−0.130660	+	0.991427i	$0.541710\pi$
$192$	0	0
$193$	−13.3317	−0.959638	−0.479819	−	0.877367i	$-0.659298\pi$
$193$	−13.3317	−0.959638	−0.479819	+	0.877367i	$0.659298\pi$
$194$	0	0
$195$	0.799207	0.0572324
$196$	0	0
$197$	2.15168	0.153301	0.0766504	−	0.997058i	$-0.475577\pi$
$197$	2.15168	0.153301	0.0766504	+	0.997058i	$0.475577\pi$
$198$	0	0
$199$	7.71456	0.546870	0.273435	−	0.961890i	$-0.411840\pi$
$199$	7.71456	0.546870	0.273435	+	0.961890i	$0.411840\pi$
$200$	0	0
$201$	13.1258	0.925823
$202$	0	0
$203$	−47.7842	−3.35379
$204$	0	0
$205$	8.50498	0.594013
$206$	0	0
$207$	−2.73075	−0.189800
$208$	0	0
$209$	−0.280278	−0.0193872
$210$	0	0
$211$	−16.3532	−1.12580	−0.562899	−	0.826526i	$-0.690314\pi$
$211$	−16.3532	−1.12580	−0.562899	+	0.826526i	$0.690314\pi$
$212$	0	0
$213$	−1.45416	−0.0996373
$214$	0	0
$215$	−1.29410	−0.0882567
$216$	0	0
$217$	33.0482	2.24346
$218$	0	0
$219$	10.4353	0.705149
$220$	0	0
$221$	−0.334860	−0.0225251
$222$	0	0
$223$	−7.34999	−0.492192	−0.246096	−	0.969245i	$-0.579148\pi$
$223$	−7.34999	−0.492192	−0.246096	+	0.969245i	$0.579148\pi$
$224$	0	0
$225$	1.70880	0.113920
$226$	0	0
$227$	20.5167	1.36174	0.680869	−	0.732405i	$-0.261602\pi$
$227$	20.5167	1.36174	0.680869	+	0.732405i	$0.261602\pi$
$228$	0	0
$229$	21.0329	1.38989	0.694945	−	0.719063i	$-0.255429\pi$
$229$	21.0329	1.38989	0.694945	+	0.719063i	$0.255429\pi$
$230$	0	0
$231$	2.32774	0.153154
$232$	0	0
$233$	7.22543	0.473354	0.236677	−	0.971588i	$-0.423942\pi$
$233$	7.22543	0.473354	0.236677	+	0.971588i	$0.423942\pi$
$234$	0	0
$235$	6.99851	0.456532
$236$	0	0
$237$	−4.38903	−0.285098
$238$	0	0
$239$	18.7122	1.21039	0.605196	−	0.796076i	$-0.293095\pi$
$239$	18.7122	1.21039	0.605196	+	0.796076i	$0.293095\pi$
$240$	0	0
$241$	−27.9993	−1.80359	−0.901796	−	0.432162i	$-0.857751\pi$
$241$	−27.9993	−1.80359	−0.901796	+	0.432162i	$0.857751\pi$
$242$	0	0
$243$	15.9120	1.02075
$244$	0	0
$245$	16.0241	1.02375
$246$	0	0
$247$	−0.461748	−0.0293804
$248$	0	0
$249$	13.5180	0.856669
$250$	0	0
$251$	−11.1855	−0.706020	−0.353010	−	0.935620i	$-0.614842\pi$
$251$	−11.1855	−0.706020	−0.353010	+	0.935620i	$0.614842\pi$
$252$	0	0
$253$	0.357256	0.0224605
$254$	0	0
$255$	−1.97294	−0.123550
$256$	0	0
$257$	−4.15214	−0.259003	−0.129502	−	0.991579i	$-0.541338\pi$
$257$	−4.15214	−0.259003	−0.129502	+	0.991579i	$0.541338\pi$
$258$	0	0
$259$	−37.9710	−2.35940
$260$	0	0
$261$	−17.0170	−1.05333
$262$	0	0
$263$	−23.5710	−1.45345	−0.726726	−	0.686928i	$-0.758959\pi$
$263$	−23.5710	−1.45345	−0.726726	+	0.686928i	$0.758959\pi$
$264$	0	0
$265$	3.68538	0.226391
$266$	0	0
$267$	−21.2365	−1.29965
$268$	0	0
$269$	25.5302	1.55661	0.778303	−	0.627889i	$-0.216081\pi$
$269$	25.5302	1.55661	0.778303	+	0.627889i	$0.216081\pi$
$270$	0	0
$271$	4.76310	0.289338	0.144669	−	0.989480i	$-0.453788\pi$
$271$	4.76310	0.289338	0.144669	+	0.989480i	$0.453788\pi$
$272$	0	0
$273$	3.83488	0.232097
$274$	0	0
$275$	−0.223557	−0.0134810
$276$	0	0
$277$	26.8973	1.61610	0.808051	−	0.589113i	$-0.200523\pi$
$277$	26.8973	1.61610	0.808051	+	0.589113i	$0.200523\pi$
$278$	0	0
$279$	11.7692	0.704602
$280$	0	0
$281$	13.1339	0.783503	0.391751	−	0.920071i	$-0.371869\pi$
$281$	13.1339	0.783503	0.391751	+	0.920071i	$0.371869\pi$
$282$	0	0
$283$	12.4028	0.737272	0.368636	−	0.929574i	$-0.379825\pi$
$283$	12.4028	0.737272	0.368636	+	0.929574i	$0.379825\pi$
$284$	0	0
$285$	−2.72054	−0.161151
$286$	0	0
$287$	40.8098	2.40893
$288$	0	0
$289$	−16.1734	−0.951374
$290$	0	0
$291$	−37.8756	−2.22030
$292$	0	0
$293$	−17.1888	−1.00418	−0.502089	−	0.864816i	$-0.667435\pi$
$293$	−17.1888	−1.00418	−0.502089	+	0.864816i	$0.667435\pi$
$294$	0	0
$295$	2.79984	0.163013
$296$	0	0
$297$	−0.626380	−0.0363463
$298$	0	0
$299$	0.588567	0.0340377
$300$	0	0
$301$	−6.20953	−0.357912
$302$	0	0
$303$	−32.8632	−1.88794
$304$	0	0
$305$	−3.82129	−0.218806
$306$	0	0
$307$	−0.535064	−0.0305377	−0.0152689	−	0.999883i	$-0.504860\pi$
$307$	−0.535064	−0.0305377	−0.0152689	+	0.999883i	$0.504860\pi$
$308$	0	0
$309$	−28.8644	−1.64204
$310$	0	0
$311$	19.5478	1.10845	0.554227	−	0.832366i	$-0.313014\pi$
$311$	19.5478	1.10845	0.554227	+	0.832366i	$0.313014\pi$
$312$	0	0
$313$	8.72689	0.493273	0.246636	−	0.969108i	$-0.420675\pi$
$313$	8.72689	0.493273	0.246636	+	0.969108i	$0.420675\pi$
$314$	0	0
$315$	8.19940	0.461984
$316$	0	0
$317$	27.8720	1.56545	0.782725	−	0.622368i	$-0.213829\pi$
$317$	27.8720	1.56545	0.782725	+	0.622368i	$0.213829\pi$
$318$	0	0
$319$	2.22629	0.124648
$320$	0	0
$321$	−12.9819	−0.724576
$322$	0	0
$323$	1.13988	0.0634246
$324$	0	0
$325$	−0.368302	−0.0204297
$326$	0	0
$327$	−37.3587	−2.06594
$328$	0	0
$329$	33.5813	1.85140
$330$	0	0
$331$	−0.189268	−0.0104031	−0.00520154	−	0.999986i	$-0.501656\pi$
$331$	−0.189268	−0.0104031	−0.00520154	+	0.999986i	$0.501656\pi$
$332$	0	0
$333$	−13.5223	−0.741017
$334$	0	0
$335$	−6.04882	−0.330482
$336$	0	0
$337$	31.6237	1.72265	0.861327	−	0.508051i	$-0.169634\pi$
$337$	31.6237	1.72265	0.861327	+	0.508051i	$0.169634\pi$
$338$	0	0
$339$	−41.8832	−2.27478
$340$	0	0
$341$	−1.53973	−0.0833809
$342$	0	0
$343$	43.3010	2.33804
$344$	0	0
$345$	3.46774	0.186697
$346$	0	0
$347$	−0.800586	−0.0429777	−0.0214888	−	0.999769i	$-0.506841\pi$
$347$	−0.800586	−0.0429777	−0.0214888	+	0.999769i	$0.506841\pi$
$348$	0	0
$349$	33.2549	1.78010	0.890048	−	0.455866i	$-0.150670\pi$
$349$	33.2549	1.78010	0.890048	+	0.455866i	$0.150670\pi$
$350$	0	0
$351$	−1.03194	−0.0550809
$352$	0	0
$353$	21.7139	1.15571	0.577856	−	0.816138i	$-0.303889\pi$
$353$	21.7139	1.15571	0.577856	+	0.816138i	$0.303889\pi$
$354$	0	0
$355$	0.670127	0.0355666
$356$	0	0
$357$	−9.46684	−0.501038
$358$	0	0
$359$	−35.1314	−1.85416	−0.927081	−	0.374860i	$-0.877691\pi$
$359$	−35.1314	−1.85416	−0.927081	+	0.374860i	$0.877691\pi$
$360$	0	0
$361$	−17.4282	−0.917273
$362$	0	0
$363$	23.7613	1.24714
$364$	0	0
$365$	−4.80893	−0.251711
$366$	0	0
$367$	7.43897	0.388311	0.194156	−	0.980971i	$-0.437803\pi$
$367$	7.43897	0.388311	0.194156	+	0.980971i	$0.437803\pi$
$368$	0	0
$369$	14.5333	0.756572
$370$	0	0
$371$	17.6837	0.918093
$372$	0	0
$373$	24.2150	1.25381	0.626903	−	0.779098i	$-0.284322\pi$
$373$	24.2150	1.25381	0.626903	+	0.779098i	$0.284322\pi$
$374$	0	0
$375$	−2.16998	−0.112057
$376$	0	0
$377$	3.66773	0.188898
$378$	0	0
$379$	−5.86907	−0.301474	−0.150737	−	0.988574i	$-0.548165\pi$
$379$	−5.86907	−0.301474	−0.150737	+	0.988574i	$0.548165\pi$
$380$	0	0
$381$	42.1496	2.15939
$382$	0	0
$383$	9.97780	0.509842	0.254921	−	0.966962i	$-0.417951\pi$
$383$	9.97780	0.509842	0.254921	+	0.966962i	$0.417951\pi$
$384$	0	0
$385$	−1.07270	−0.0546701
$386$	0	0
$387$	−2.21135	−0.112409
$388$	0	0
$389$	22.8908	1.16061	0.580305	−	0.814399i	$-0.302933\pi$
$389$	22.8908	1.16061	0.580305	+	0.814399i	$0.302933\pi$
$390$	0	0
$391$	−1.45295	−0.0734786
$392$	0	0
$393$	−2.72778	−0.137598
$394$	0	0
$395$	2.02262	0.101769
$396$	0	0
$397$	−8.06508	−0.404775	−0.202387	−	0.979306i	$-0.564870\pi$
$397$	−8.06508	−0.404775	−0.202387	+	0.979306i	$0.564870\pi$
$398$	0	0
$399$	−13.0541	−0.653524
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	−2.53665	−0.126359
$404$	0	0
$405$	−11.2064	−0.556851
$406$	0	0
$407$	1.76908	0.0876902
$408$	0	0
$409$	30.2993	1.49821	0.749103	−	0.662454i	$-0.230485\pi$
$409$	30.2993	1.49821	0.749103	+	0.662454i	$0.230485\pi$
$410$	0	0
$411$	−26.8152	−1.32270
$412$	0	0
$413$	13.4346	0.661074
$414$	0	0
$415$	−6.22957	−0.305797
$416$	0	0
$417$	15.2942	0.748962
$418$	0	0
$419$	35.8701	1.75237	0.876185	−	0.481974i	$-0.160080\pi$
$419$	35.8701	1.75237	0.876185	+	0.481974i	$0.160080\pi$
$420$	0	0
$421$	−3.05658	−0.148969	−0.0744844	−	0.997222i	$-0.523731\pi$
$421$	−3.05658	−0.148969	−0.0744844	+	0.997222i	$0.523731\pi$
$422$	0	0
$423$	11.9590	0.581468
$424$	0	0
$425$	0.909198	0.0441026
$426$	0	0
$427$	−18.3359	−0.887335
$428$	0	0
$429$	−0.178668	−0.00862619
$430$	0	0
$431$	−9.84504	−0.474219	−0.237109	−	0.971483i	$-0.576200\pi$
$431$	−9.84504	−0.474219	−0.237109	+	0.971483i	$0.576200\pi$
$432$	0	0
$433$	−13.6932	−0.658055	−0.329028	−	0.944320i	$-0.606721\pi$
$433$	−13.6932	−0.658055	−0.329028	+	0.944320i	$0.606721\pi$
$434$	0	0
$435$	21.6096	1.03610
$436$	0	0
$437$	−2.00351	−0.0958410
$438$	0	0
$439$	−30.8160	−1.47077	−0.735383	−	0.677651i	$-0.762998\pi$
$439$	−30.8160	−1.47077	−0.735383	+	0.677651i	$0.762998\pi$
$440$	0	0
$441$	27.3820	1.30391
$442$	0	0
$443$	4.31181	0.204860	0.102430	−	0.994740i	$-0.467338\pi$
$443$	4.31181	0.204860	0.102430	+	0.994740i	$0.467338\pi$
$444$	0	0
$445$	9.78651	0.463925
$446$	0	0
$447$	1.69525	0.0801826
$448$	0	0
$449$	−38.0148	−1.79403	−0.897015	−	0.442001i	$-0.854269\pi$
$449$	−38.0148	−1.79403	−0.897015	+	0.442001i	$0.854269\pi$
$450$	0	0
$451$	−1.90135	−0.0895309
$452$	0	0
$453$	38.8704	1.82629
$454$	0	0
$455$	−1.76724	−0.0828496
$456$	0	0
$457$	18.3317	0.857520	0.428760	−	0.903419i	$-0.358951\pi$
$457$	18.3317	0.857520	0.428760	+	0.903419i	$0.358951\pi$
$458$	0	0
$459$	2.54746	0.118905
$460$	0	0
$461$	6.12118	0.285092	0.142546	−	0.989788i	$-0.454471\pi$
$461$	6.12118	0.285092	0.142546	+	0.989788i	$0.454471\pi$
$462$	0	0
$463$	25.9890	1.20781	0.603906	−	0.797056i	$-0.293610\pi$
$463$	25.9890	1.20781	0.603906	+	0.797056i	$0.293610\pi$
$464$	0	0
$465$	−14.9455	−0.693081
$466$	0	0
$467$	11.9413	0.552575	0.276288	−	0.961075i	$-0.410896\pi$
$467$	11.9413	0.552575	0.276288	+	0.961075i	$0.410896\pi$
$468$	0	0
$469$	−29.0244	−1.34022
$470$	0	0
$471$	17.8960	0.824603
$472$	0	0
$473$	0.289305	0.0133022
$474$	0	0
$475$	1.25372	0.0575247
$476$	0	0
$477$	6.29756	0.288345
$478$	0	0
$479$	26.5046	1.21103	0.605513	−	0.795835i	$-0.292968\pi$
$479$	26.5046	1.21103	0.605513	+	0.795835i	$0.292968\pi$
$480$	0	0
$481$	2.91450	0.132890
$482$	0	0
$483$	16.6394	0.757119
$484$	0	0
$485$	17.4544	0.792562
$486$	0	0
$487$	12.4822	0.565621	0.282811	−	0.959176i	$-0.408733\pi$
$487$	12.4822	0.565621	0.282811	+	0.959176i	$0.408733\pi$
$488$	0	0
$489$	24.4582	1.10604
$490$	0	0
$491$	−16.7741	−0.757004	−0.378502	−	0.925601i	$-0.623561\pi$
$491$	−16.7741	−0.757004	−0.378502	+	0.925601i	$0.623561\pi$
$492$	0	0
$493$	−9.05422	−0.407781
$494$	0	0
$495$	−0.382013	−0.0171702
$496$	0	0
$497$	3.21550	0.144235
$498$	0	0
$499$	−32.9038	−1.47297	−0.736487	−	0.676452i	$-0.763517\pi$
$499$	−32.9038	−1.47297	−0.736487	+	0.676452i	$0.763517\pi$
$500$	0	0
$501$	−9.55341	−0.426815
$502$	0	0
$503$	41.0953	1.83235	0.916174	−	0.400782i	$-0.131261\pi$
$503$	41.0953	1.83235	0.916174	+	0.400782i	$0.131261\pi$
$504$	0	0
$505$	15.1445	0.673921
$506$	0	0
$507$	27.9153	1.23976
$508$	0	0
$509$	27.1992	1.20558	0.602792	−	0.797898i	$-0.294055\pi$
$509$	27.1992	1.20558	0.602792	+	0.797898i	$0.294055\pi$
$510$	0	0
$511$	−23.0749	−1.02077
$512$	0	0
$513$	3.51278	0.155093
$514$	0	0
$515$	13.3017	0.586144
$516$	0	0
$517$	−1.56457	−0.0688095
$518$	0	0
$519$	25.9179	1.13767
$520$	0	0
$521$	−25.7466	−1.12798	−0.563990	−	0.825781i	$-0.690735\pi$
$521$	−25.7466	−1.12798	−0.563990	+	0.825781i	$0.690735\pi$
$522$	0	0
$523$	21.9940	0.961732	0.480866	−	0.876794i	$-0.340322\pi$
$523$	21.9940	0.961732	0.480866	+	0.876794i	$0.340322\pi$
$524$	0	0
$525$	−10.4123	−0.454430
$526$	0	0
$527$	6.26201	0.272777
$528$	0	0
$529$	−20.4462	−0.888966
$530$	0	0
$531$	4.78436	0.207624
$532$	0	0
$533$	−3.13240	−0.135679
$534$	0	0
$535$	5.98249	0.258645
$536$	0	0
$537$	3.52678	0.152192
$538$	0	0
$539$	−3.58231	−0.154301
$540$	0	0
$541$	−10.7567	−0.462468	−0.231234	−	0.972898i	$-0.574276\pi$
$541$	−10.7567	−0.462468	−0.231234	+	0.972898i	$0.574276\pi$
$542$	0	0
$543$	21.7500	0.933381
$544$	0	0
$545$	17.2162	0.737460
$546$	0	0
$547$	1.52552	0.0652267	0.0326133	−	0.999468i	$-0.489617\pi$
$547$	1.52552	0.0652267	0.0326133	+	0.999468i	$0.489617\pi$
$548$	0	0
$549$	−6.52980	−0.278685
$550$	0	0
$551$	−12.4851	−0.531885
$552$	0	0
$553$	9.70523	0.412708
$554$	0	0
$555$	17.1718	0.728901
$556$	0	0
$557$	−21.9441	−0.929800	−0.464900	−	0.885363i	$-0.653910\pi$
$557$	−21.9441	−0.929800	−0.464900	+	0.885363i	$0.653910\pi$
$558$	0	0
$559$	0.476619	0.0201588
$560$	0	0
$561$	0.441064	0.0186217
$562$	0	0
$563$	31.6023	1.33188	0.665940	−	0.746005i	$-0.268031\pi$
$563$	31.6023	1.33188	0.665940	+	0.746005i	$0.268031\pi$
$564$	0	0
$565$	19.3012	0.812009
$566$	0	0
$567$	−53.7722	−2.25822
$568$	0	0
$569$	−3.90125	−0.163549	−0.0817744	−	0.996651i	$-0.526059\pi$
$569$	−3.90125	−0.163549	−0.0817744	+	0.996651i	$0.526059\pi$
$570$	0	0
$571$	14.3471	0.600407	0.300204	−	0.953875i	$-0.402945\pi$
$571$	14.3471	0.600407	0.300204	+	0.953875i	$0.402945\pi$
$572$	0	0
$573$	7.83688	0.327391
$574$	0	0
$575$	−1.59805	−0.0666434
$576$	0	0
$577$	−2.31968	−0.0965697	−0.0482849	−	0.998834i	$-0.515376\pi$
$577$	−2.31968	−0.0965697	−0.0482849	+	0.998834i	$0.515376\pi$
$578$	0	0
$579$	28.9295	1.20227
$580$	0	0
$581$	−29.8916	−1.24011
$582$	0	0
$583$	−0.823892	−0.0341221
$584$	0	0
$585$	−0.629354	−0.0260206
$586$	0	0
$587$	34.3263	1.41680	0.708399	−	0.705812i	$-0.249418\pi$
$587$	34.3263	1.41680	0.708399	+	0.705812i	$0.249418\pi$
$588$	0	0
$589$	8.63488	0.355794
$590$	0	0
$591$	−4.66909	−0.192061
$592$	0	0
$593$	18.9849	0.779618	0.389809	−	0.920896i	$-0.372541\pi$
$593$	18.9849	0.779618	0.389809	+	0.920896i	$0.372541\pi$
$594$	0	0
$595$	4.36265	0.178851
$596$	0	0
$597$	−16.7404	−0.685139
$598$	0	0
$599$	5.30330	0.216687	0.108343	−	0.994114i	$-0.465445\pi$
$599$	5.30330	0.216687	0.108343	+	0.994114i	$0.465445\pi$
$600$	0	0
$601$	−46.7705	−1.90781	−0.953904	−	0.300113i	$-0.902976\pi$
$601$	−46.7705	−1.90781	−0.953904	+	0.300113i	$0.902976\pi$
$602$	0	0
$603$	−10.3362	−0.420923
$604$	0	0
$605$	−10.9500	−0.445182
$606$	0	0
$607$	25.8825	1.05054	0.525270	−	0.850936i	$-0.323965\pi$
$607$	25.8825	1.05054	0.525270	+	0.850936i	$0.323965\pi$
$608$	0	0
$609$	103.691	4.20175
$610$	0	0
$611$	−2.57757	−0.104277
$612$	0	0
$613$	−23.3504	−0.943112	−0.471556	−	0.881836i	$-0.656307\pi$
$613$	−23.3504	−0.943112	−0.471556	+	0.881836i	$0.656307\pi$
$614$	0	0
$615$	−18.4556	−0.744201
$616$	0	0
$617$	20.0537	0.807330	0.403665	−	0.914907i	$-0.367736\pi$
$617$	20.0537	0.807330	0.403665	+	0.914907i	$0.367736\pi$
$618$	0	0
$619$	41.7345	1.67745	0.838725	−	0.544556i	$-0.183302\pi$
$619$	41.7345	1.67745	0.838725	+	0.544556i	$0.183302\pi$
$620$	0	0
$621$	−4.47756	−0.179678
$622$	0	0
$623$	46.9591	1.88138
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0.608197	0.0242890
$628$	0	0
$629$	−7.19479	−0.286875
$630$	0	0
$631$	20.6772	0.823148	0.411574	−	0.911376i	$-0.364979\pi$
$631$	20.6772	0.823148	0.411574	+	0.911376i	$0.364979\pi$
$632$	0	0
$633$	35.4860	1.41044
$634$	0	0
$635$	−19.4240	−0.770817
$636$	0	0
$637$	−5.90173	−0.233835
$638$	0	0
$639$	1.14511	0.0452999
$640$	0	0
$641$	19.7241	0.779055	0.389527	−	0.921015i	$-0.372638\pi$
$641$	19.7241	0.779055	0.389527	+	0.921015i	$0.372638\pi$
$642$	0	0
$643$	−33.6910	−1.32865	−0.664323	−	0.747446i	$-0.731280\pi$
$643$	−33.6910	−1.32865	−0.664323	+	0.747446i	$0.731280\pi$
$644$	0	0
$645$	2.80816	0.110571
$646$	0	0
$647$	11.2254	0.441314	0.220657	−	0.975351i	$-0.429180\pi$
$647$	11.2254	0.441314	0.220657	+	0.975351i	$0.429180\pi$
$648$	0	0
$649$	−0.625924	−0.0245697
$650$	0	0
$651$	−71.7137	−2.81068
$652$	0	0
$653$	28.1161	1.10027	0.550134	−	0.835077i	$-0.314577\pi$
$653$	28.1161	1.10027	0.550134	+	0.835077i	$0.314577\pi$
$654$	0	0
$655$	1.25705	0.0491171
$656$	0	0
$657$	−8.21748	−0.320594
$658$	0	0
$659$	0.897838	0.0349748	0.0174874	−	0.999847i	$-0.494433\pi$
$659$	0.897838	0.0349748	0.0174874	+	0.999847i	$0.494433\pi$
$660$	0	0
$661$	−32.3998	−1.26020	−0.630102	−	0.776512i	$-0.716987\pi$
$661$	−32.3998	−1.26020	−0.630102	+	0.776512i	$0.716987\pi$
$662$	0	0
$663$	0.726637	0.0282203
$664$	0	0
$665$	6.01579	0.233282
$666$	0	0
$667$	15.9142	0.616199
$668$	0	0
$669$	15.9493	0.616636
$670$	0	0
$671$	0.854276	0.0329789
$672$	0	0
$673$	40.3989	1.55726	0.778632	−	0.627481i	$-0.215914\pi$
$673$	40.3989	1.55726	0.778632	+	0.627481i	$0.215914\pi$
$674$	0	0
$675$	2.80188	0.107844
$676$	0	0
$677$	−25.0583	−0.963068	−0.481534	−	0.876427i	$-0.659920\pi$
$677$	−25.0583	−0.963068	−0.481534	+	0.876427i	$0.659920\pi$
$678$	0	0
$679$	83.7521	3.21411
$680$	0	0
$681$	−44.5207	−1.70604
$682$	0	0
$683$	−33.5847	−1.28508	−0.642541	−	0.766251i	$-0.722120\pi$
$683$	−33.5847	−1.28508	−0.642541	+	0.766251i	$0.722120\pi$
$684$	0	0
$685$	12.3574	0.472151
$686$	0	0
$687$	−45.6408	−1.74131
$688$	0	0
$689$	−1.35733	−0.0517103
$690$	0	0
$691$	−35.7217	−1.35892	−0.679458	−	0.733714i	$-0.737785\pi$
$691$	−35.7217	−1.35892	−0.679458	+	0.733714i	$0.737785\pi$
$692$	0	0
$693$	−1.83303	−0.0696312
$694$	0	0
$695$	−7.04811	−0.267350
$696$	0	0
$697$	7.73270	0.292897
$698$	0	0
$699$	−15.6790	−0.593035
$700$	0	0
$701$	9.60519	0.362783	0.181392	−	0.983411i	$-0.441940\pi$
$701$	9.60519	0.362783	0.181392	+	0.983411i	$0.441940\pi$
$702$	0	0
$703$	−9.92113	−0.374182
$704$	0	0
$705$	−15.1866	−0.571960
$706$	0	0
$707$	72.6686	2.73298
$708$	0	0
$709$	−0.113314	−0.00425561	−0.00212780	−	0.999998i	$-0.500677\pi$
$709$	−0.113314	−0.00425561	−0.00212780	+	0.999998i	$0.500677\pi$
$710$	0	0
$711$	3.45624	0.129619
$712$	0	0
$713$	−11.0064	−0.412194
$714$	0	0
$715$	0.0823366	0.00307921
$716$	0	0
$717$	−40.6051	−1.51642
$718$	0	0
$719$	−52.8548	−1.97115	−0.985577	−	0.169230i	$-0.945872\pi$
$719$	−52.8548	−1.97115	−0.985577	+	0.169230i	$0.945872\pi$
$720$	0	0
$721$	63.8263	2.37701
$722$	0	0
$723$	60.7578	2.25961
$724$	0	0
$725$	−9.95847	−0.369848
$726$	0	0
$727$	−25.5445	−0.947393	−0.473697	−	0.880688i	$-0.657081\pi$
$727$	−25.5445	−0.947393	−0.473697	+	0.880688i	$0.657081\pi$
$728$	0	0
$729$	−0.909418	−0.0336821
$730$	0	0
$731$	−1.17659	−0.0435178
$732$	0	0
$733$	26.5807	0.981781	0.490890	−	0.871221i	$-0.336672\pi$
$733$	26.5807	0.981781	0.490890	+	0.871221i	$0.336672\pi$
$734$	0	0
$735$	−34.7720	−1.28259
$736$	0	0
$737$	1.35226	0.0498110
$738$	0	0
$739$	−43.6418	−1.60539	−0.802695	−	0.596390i	$-0.796601\pi$
$739$	−43.6418	−1.60539	−0.802695	+	0.596390i	$0.796601\pi$
$740$	0	0
$741$	1.00198	0.0368088
$742$	0	0
$743$	−38.6934	−1.41952	−0.709762	−	0.704441i	$-0.751198\pi$
$743$	−38.6934	−1.41952	−0.709762	+	0.704441i	$0.751198\pi$
$744$	0	0
$745$	−0.781230	−0.0286221
$746$	0	0
$747$	−10.6451	−0.389482
$748$	0	0
$749$	28.7061	1.04890
$750$	0	0
$751$	−1.30463	−0.0476065	−0.0238032	−	0.999717i	$-0.507578\pi$
$751$	−1.30463	−0.0476065	−0.0238032	+	0.999717i	$0.507578\pi$
$752$	0	0
$753$	24.2722	0.884528
$754$	0	0
$755$	−17.9128	−0.651914
$756$	0	0
$757$	16.8719	0.613218	0.306609	−	0.951836i	$-0.400806\pi$
$757$	16.8719	0.613218	0.306609	+	0.951836i	$0.400806\pi$
$758$	0	0
$759$	−0.775237	−0.0281393
$760$	0	0
$761$	−2.08540	−0.0755959	−0.0377979	−	0.999285i	$-0.512034\pi$
$761$	−2.08540	−0.0755959	−0.0377979	+	0.999285i	$0.512034\pi$
$762$	0	0
$763$	82.6092	2.99065
$764$	0	0
$765$	1.55363	0.0561717
$766$	0	0
$767$	−1.03119	−0.0372341
$768$	0	0
$769$	49.7922	1.79555	0.897777	−	0.440451i	$-0.145181\pi$
$769$	49.7922	1.79555	0.897777	+	0.440451i	$0.145181\pi$
$770$	0	0
$771$	9.01004	0.324489
$772$	0	0
$773$	−5.27167	−0.189609	−0.0948045	−	0.995496i	$-0.530223\pi$
$773$	−5.27167	−0.189609	−0.0948045	+	0.995496i	$0.530223\pi$
$774$	0	0
$775$	6.88740	0.247403
$776$	0	0
$777$	82.3961	2.95594
$778$	0	0
$779$	10.6629	0.382037
$780$	0	0
$781$	−0.149812	−0.00536068
$782$	0	0
$783$	−27.9024	−0.997152
$784$	0	0
$785$	−8.24709	−0.294351
$786$	0	0
$787$	24.7716	0.883011	0.441505	−	0.897259i	$-0.354445\pi$
$787$	24.7716	0.883011	0.441505	+	0.897259i	$0.354445\pi$
$788$	0	0
$789$	51.1485	1.82094
$790$	0	0
$791$	92.6140	3.29298
$792$	0	0
$793$	1.40739	0.0499779
$794$	0	0
$795$	−7.99718	−0.283631
$796$	0	0
$797$	−53.7524	−1.90401	−0.952003	−	0.306087i	$-0.900980\pi$
$797$	−53.7524	−1.90401	−0.952003	+	0.306087i	$0.900980\pi$
$798$	0	0
$799$	6.36302	0.225108
$800$	0	0
$801$	16.7231	0.590883
$802$	0	0
$803$	1.07507	0.0379384
$804$	0	0
$805$	−7.66802	−0.270262
$806$	0	0
$807$	−55.4000	−1.95017
$808$	0	0
$809$	−33.6101	−1.18167	−0.590834	−	0.806793i	$-0.701201\pi$
$809$	−33.6101	−1.18167	−0.590834	+	0.806793i	$0.701201\pi$
$810$	0	0
$811$	47.0336	1.65157	0.825786	−	0.563983i	$-0.190732\pi$
$811$	47.0336	1.65157	0.825786	+	0.563983i	$0.190732\pi$
$812$	0	0
$813$	−10.3358	−0.362493
$814$	0	0
$815$	−11.2712	−0.394813
$816$	0	0
$817$	−1.62244	−0.0567619
$818$	0	0
$819$	−3.01986	−0.105522
$820$	0	0
$821$	27.7270	0.967678	0.483839	−	0.875157i	$-0.339242\pi$
$821$	27.7270	0.967678	0.483839	+	0.875157i	$0.339242\pi$
$822$	0	0
$823$	−29.0334	−1.01204	−0.506021	−	0.862521i	$-0.668884\pi$
$823$	−29.0334	−1.01204	−0.506021	+	0.862521i	$0.668884\pi$
$824$	0	0
$825$	0.485113	0.0168895
$826$	0	0
$827$	20.4824	0.712244	0.356122	−	0.934439i	$-0.384099\pi$
$827$	20.4824	0.712244	0.356122	+	0.934439i	$0.384099\pi$
$828$	0	0
$829$	6.94310	0.241144	0.120572	−	0.992705i	$-0.461527\pi$
$829$	6.94310	0.241144	0.120572	+	0.992705i	$0.461527\pi$
$830$	0	0
$831$	−58.3665	−2.02471
$832$	0	0
$833$	14.5691	0.504790
$834$	0	0
$835$	4.40254	0.152356
$836$	0	0
$837$	19.2977	0.667026
$838$	0	0
$839$	−5.55615	−0.191820	−0.0959098	−	0.995390i	$-0.530576\pi$
$839$	−5.55615	−0.191820	−0.0959098	+	0.995390i	$0.530576\pi$
$840$	0	0
$841$	70.1711	2.41969
$842$	0	0
$843$	−28.5002	−0.981601
$844$	0	0
$845$	−12.8644	−0.442547
$846$	0	0
$847$	−52.5420	−1.80537
$848$	0	0
$849$	−26.9139	−0.923681
$850$	0	0
$851$	12.6459	0.433497
$852$	0	0
$853$	32.8942	1.12627	0.563137	−	0.826363i	$-0.309594\pi$
$853$	32.8942	1.12627	0.563137	+	0.826363i	$0.309594\pi$
$854$	0	0
$855$	2.14235	0.0732670
$856$	0	0
$857$	−28.0701	−0.958855	−0.479427	−	0.877582i	$-0.659156\pi$
$857$	−28.0701	−0.958855	−0.479427	+	0.877582i	$0.659156\pi$
$858$	0	0
$859$	−11.6818	−0.398579	−0.199290	−	0.979941i	$-0.563863\pi$
$859$	−11.6818	−0.398579	−0.199290	+	0.979941i	$0.563863\pi$
$860$	0	0
$861$	−88.5564	−3.01799
$862$	0	0
$863$	−21.5886	−0.734886	−0.367443	−	0.930046i	$-0.619767\pi$
$863$	−21.5886	−0.734886	−0.367443	+	0.930046i	$0.619767\pi$
$864$	0	0
$865$	−11.9438	−0.406103
$866$	0	0
$867$	35.0958	1.19192
$868$	0	0
$869$	−0.452171	−0.0153388
$870$	0	0
$871$	2.22780	0.0754860
$872$	0	0
$873$	29.8260	1.00946
$874$	0	0
$875$	4.79835	0.162214
$876$	0	0
$877$	27.5536	0.930420	0.465210	−	0.885200i	$-0.345979\pi$
$877$	27.5536	0.930420	0.465210	+	0.885200i	$0.345979\pi$
$878$	0	0
$879$	37.2992	1.25807
$880$	0	0
$881$	31.7622	1.07010	0.535049	−	0.844821i	$-0.320293\pi$
$881$	31.7622	1.07010	0.535049	+	0.844821i	$0.320293\pi$
$882$	0	0
$883$	−13.0718	−0.439900	−0.219950	−	0.975511i	$-0.570589\pi$
$883$	−13.0718	−0.439900	−0.219950	+	0.975511i	$0.570589\pi$
$884$	0	0
$885$	−6.07559	−0.204229
$886$	0	0
$887$	−27.4832	−0.922797	−0.461399	−	0.887193i	$-0.652652\pi$
$887$	−27.4832	−0.922797	−0.461399	+	0.887193i	$0.652652\pi$
$888$	0	0
$889$	−93.2031	−3.12593
$890$	0	0
$891$	2.50527	0.0839297
$892$	0	0
$893$	8.77417	0.293617
$894$	0	0
$895$	−1.62526	−0.0543266
$896$	0	0
$897$	−1.27718	−0.0426437
$898$	0	0
$899$	−68.5880	−2.28754
$900$	0	0
$901$	3.35074	0.111629
$902$	0	0
$903$	13.4745	0.448405
$904$	0	0
$905$	−10.0231	−0.333181
$906$	0	0
$907$	−19.9841	−0.663560	−0.331780	−	0.943357i	$-0.607649\pi$
$907$	−19.9841	−0.663560	−0.331780	+	0.943357i	$0.607649\pi$
$908$	0	0
$909$	25.8789	0.858348
$910$	0	0
$911$	−8.03789	−0.266307	−0.133154	−	0.991095i	$-0.542510\pi$
$911$	−8.03789	−0.266307	−0.133154	+	0.991095i	$0.542510\pi$
$912$	0	0
$913$	1.39266	0.0460904
$914$	0	0
$915$	8.29210	0.274128
$916$	0	0
$917$	6.03178	0.199187
$918$	0	0
$919$	−49.0912	−1.61937	−0.809685	−	0.586864i	$-0.800362\pi$
$919$	−49.0912	−1.61937	−0.809685	+	0.586864i	$0.800362\pi$
$920$	0	0
$921$	1.16108	0.0382587
$922$	0	0
$923$	−0.246809	−0.00812383
$924$	0	0
$925$	−7.91334	−0.260189
$926$	0	0
$927$	22.7299	0.746549
$928$	0	0
$929$	54.4802	1.78744	0.893719	−	0.448626i	$-0.148087\pi$
$929$	54.4802	1.78744	0.893719	+	0.448626i	$0.148087\pi$
$930$	0	0
$931$	20.0898	0.658417
$932$	0	0
$933$	−42.4183	−1.38871
$934$	0	0
$935$	−0.203258	−0.00664723
$936$	0	0
$937$	37.8299	1.23585	0.617924	−	0.786238i	$-0.287974\pi$
$937$	37.8299	1.23585	0.617924	+	0.786238i	$0.287974\pi$
$938$	0	0
$939$	−18.9371	−0.617990
$940$	0	0
$941$	−50.4179	−1.64358	−0.821789	−	0.569792i	$-0.807024\pi$
$941$	−50.4179	−1.64358	−0.821789	+	0.569792i	$0.807024\pi$
$942$	0	0
$943$	−13.5914	−0.442597
$944$	0	0
$945$	13.4444	0.437346
$946$	0	0
$947$	−13.1681	−0.427907	−0.213954	−	0.976844i	$-0.568634\pi$
$947$	−13.1681	−0.427907	−0.213954	+	0.976844i	$0.568634\pi$
$948$	0	0
$949$	1.77114	0.0574936
$950$	0	0
$951$	−60.4816	−1.96125
$952$	0	0
$953$	30.4455	0.986226	0.493113	−	0.869965i	$-0.335859\pi$
$953$	30.4455	0.986226	0.493113	+	0.869965i	$0.335859\pi$
$954$	0	0
$955$	−3.61151	−0.116866
$956$	0	0
$957$	−4.83099	−0.156164
$958$	0	0
$959$	59.2950	1.91474
$960$	0	0
$961$	16.4363	0.530204
$962$	0	0
$963$	10.2229	0.329427
$964$	0	0
$965$	−13.3317	−0.429163
$966$	0	0
$967$	15.0760	0.484810	0.242405	−	0.970175i	$-0.422064\pi$
$967$	15.0760	0.484810	0.242405	+	0.970175i	$0.422064\pi$
$968$	0	0
$969$	−2.47351	−0.0794607
$970$	0	0
$971$	−24.9005	−0.799095	−0.399547	−	0.916713i	$-0.630833\pi$
$971$	−24.9005	−0.799095	−0.399547	+	0.916713i	$0.630833\pi$
$972$	0	0
$973$	−33.8193	−1.08420
$974$	0	0
$975$	0.799207	0.0255951
$976$	0	0
$977$	40.6001	1.29891	0.649456	−	0.760399i	$-0.274997\pi$
$977$	40.6001	1.29891	0.649456	+	0.760399i	$0.274997\pi$
$978$	0	0
$979$	−2.18784	−0.0699238
$980$	0	0
$981$	29.4189	0.939274
$982$	0	0
$983$	−5.88030	−0.187553	−0.0937763	−	0.995593i	$-0.529894\pi$
$983$	−5.88030	−0.187553	−0.0937763	+	0.995593i	$0.529894\pi$
$984$	0	0
$985$	2.15168	0.0685582
$986$	0	0
$987$	−72.8706	−2.31950
$988$	0	0
$989$	2.06804	0.0657598
$990$	0	0
$991$	−10.1696	−0.323050	−0.161525	−	0.986869i	$-0.551641\pi$
$991$	−10.1696	−0.323050	−0.161525	+	0.986869i	$0.551641\pi$
$992$	0	0
$993$	0.410706	0.0130334
$994$	0	0
$995$	7.71456	0.244568
$996$	0	0
$997$	−10.6880	−0.338493	−0.169246	−	0.985574i	$-0.554133\pi$
$997$	−10.6880	−0.338493	−0.169246	+	0.985574i	$0.554133\pi$
$998$	0	0
$999$	−22.1722	−0.701499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.f.1.9	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.f.1.9	✓	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)

\(f(q)\)	\(=\)	\(q-2.16998 q^{3} +1.00000 q^{5} +4.79835 q^{7} +1.70880 q^{9} +O(q^{10})\)	\(q-2.16998 q^{3} +1.00000 q^{5} +4.79835 q^{7} +1.70880 q^{9} -0.223557 q^{11} -0.368302 q^{13} -2.16998 q^{15} +0.909198 q^{17} +1.25372 q^{19} -10.4123 q^{21} -1.59805 q^{23} +1.00000 q^{25} +2.80188 q^{27} -9.95847 q^{29} +6.88740 q^{31} +0.485113 q^{33} +4.79835 q^{35} -7.91334 q^{37} +0.799207 q^{39} +8.50498 q^{41} -1.29410 q^{43} +1.70880 q^{45} +6.99851 q^{47} +16.0241 q^{49} -1.97294 q^{51} +3.68538 q^{53} -0.223557 q^{55} -2.72054 q^{57} +2.79984 q^{59} -3.82129 q^{61} +8.19940 q^{63} -0.368302 q^{65} -6.04882 q^{67} +3.46774 q^{69} +0.670127 q^{71} -4.80893 q^{73} -2.16998 q^{75} -1.07270 q^{77} +2.02262 q^{79} -11.2064 q^{81} -6.22957 q^{83} +0.909198 q^{85} +21.6096 q^{87} +9.78651 q^{89} -1.76724 q^{91} -14.9455 q^{93} +1.25372 q^{95} +17.4544 q^{97} -0.382013 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 3 q^{3} + 37 q^{5} + 4 q^{7} + 50 q^{9}+O(q^{10}) \) 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

Fields

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Groups

Database

Properties

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