LMFDB - Embedded newform 8020.2.a.d.1.12

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:	$29$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-0.697718 q^{3} +1.00000 q^{5} -4.99015 q^{7} -2.51319 q^{9} +O(q^{10})$	$q-0.697718 q^{3} +1.00000 q^{5} -4.99015 q^{7} -2.51319 q^{9} +4.56555 q^{11} -0.548024 q^{13} -0.697718 q^{15} -4.13086 q^{17} +4.54848 q^{19} +3.48172 q^{21} -3.56979 q^{23} +1.00000 q^{25} +3.84665 q^{27} +4.58491 q^{29} -3.18898 q^{31} -3.18547 q^{33} -4.99015 q^{35} +11.2323 q^{37} +0.382366 q^{39} +1.69549 q^{41} +7.02275 q^{43} -2.51319 q^{45} -7.52910 q^{47} +17.9016 q^{49} +2.88218 q^{51} -10.0493 q^{53} +4.56555 q^{55} -3.17356 q^{57} -2.20177 q^{59} -7.93669 q^{61} +12.5412 q^{63} -0.548024 q^{65} +12.4305 q^{67} +2.49071 q^{69} +14.9070 q^{71} -10.5529 q^{73} -0.697718 q^{75} -22.7828 q^{77} -13.3864 q^{79} +4.85569 q^{81} +6.00371 q^{83} -4.13086 q^{85} -3.19898 q^{87} +1.91289 q^{89} +2.73472 q^{91} +2.22501 q^{93} +4.54848 q^{95} +16.2928 q^{97} -11.4741 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	$ 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) $ 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * 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
$3$	−0.697718	−0.402828	−0.201414	−	0.979506i	$-0.564554\pi$
$3$	−0.697718	−0.402828	−0.201414	+	0.979506i	$0.564554\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−4.99015	−1.88610	−0.943050	−	0.332650i	$-0.892057\pi$
$7$	−4.99015	−1.88610	−0.943050	+	0.332650i	$0.892057\pi$
$8$	0	0
$9$	−2.51319	−0.837730
$10$	0	0
$11$	4.56555	1.37657	0.688283	−	0.725442i	$-0.258365\pi$
$11$	4.56555	1.37657	0.688283	+	0.725442i	$0.258365\pi$
$12$	0	0
$13$	−0.548024	−0.151995	−0.0759973	−	0.997108i	$-0.524214\pi$
$13$	−0.548024	−0.151995	−0.0759973	+	0.997108i	$0.524214\pi$
$14$	0	0
$15$	−0.697718	−0.180150
$16$	0	0
$17$	−4.13086	−1.00188	−0.500940	−	0.865482i	$-0.667012\pi$
$17$	−4.13086	−1.00188	−0.500940	+	0.865482i	$0.667012\pi$
$18$	0	0
$19$	4.54848	1.04349	0.521747	−	0.853101i	$-0.325281\pi$
$19$	4.54848	1.04349	0.521747	+	0.853101i	$0.325281\pi$
$20$	0	0
$21$	3.48172	0.759774
$22$	0	0
$23$	−3.56979	−0.744353	−0.372177	−	0.928162i	$-0.621388\pi$
$23$	−3.56979	−0.744353	−0.372177	+	0.928162i	$0.621388\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	3.84665	0.740289
$28$	0	0
$29$	4.58491	0.851397	0.425698	−	0.904865i	$-0.360028\pi$
$29$	4.58491	0.851397	0.425698	+	0.904865i	$0.360028\pi$
$30$	0	0
$31$	−3.18898	−0.572758	−0.286379	−	0.958116i	$-0.592452\pi$
$31$	−3.18898	−0.572758	−0.286379	+	0.958116i	$0.592452\pi$
$32$	0	0
$33$	−3.18547	−0.554519
$34$	0	0
$35$	−4.99015	−0.843490
$36$	0	0
$37$	11.2323	1.84658	0.923289	−	0.384107i	$-0.125491\pi$
$37$	11.2323	1.84658	0.923289	+	0.384107i	$0.125491\pi$
$38$	0	0
$39$	0.382366	0.0612276
$40$	0	0
$41$	1.69549	0.264791	0.132396	−	0.991197i	$-0.457733\pi$
$41$	1.69549	0.264791	0.132396	+	0.991197i	$0.457733\pi$
$42$	0	0
$43$	7.02275	1.07096	0.535480	−	0.844548i	$-0.320131\pi$
$43$	7.02275	1.07096	0.535480	+	0.844548i	$0.320131\pi$
$44$	0	0
$45$	−2.51319	−0.374644
$46$	0	0
$47$	−7.52910	−1.09823	−0.549116	−	0.835746i	$-0.685035\pi$
$47$	−7.52910	−1.09823	−0.549116	+	0.835746i	$0.685035\pi$
$48$	0	0
$49$	17.9016	2.55738
$50$	0	0
$51$	2.88218	0.403585
$52$	0	0
$53$	−10.0493	−1.38037	−0.690185	−	0.723633i	$-0.742471\pi$
$53$	−10.0493	−1.38037	−0.690185	+	0.723633i	$0.742471\pi$
$54$	0	0
$55$	4.56555	0.615619
$56$	0	0
$57$	−3.17356	−0.420348
$58$	0	0
$59$	−2.20177	−0.286647	−0.143323	−	0.989676i	$-0.545779\pi$
$59$	−2.20177	−0.286647	−0.143323	+	0.989676i	$0.545779\pi$
$60$	0	0
$61$	−7.93669	−1.01619	−0.508094	−	0.861301i	$-0.669650\pi$
$61$	−7.93669	−1.01619	−0.508094	+	0.861301i	$0.669650\pi$
$62$	0	0
$63$	12.5412	1.58004
$64$	0	0
$65$	−0.548024	−0.0679740
$66$	0	0
$67$	12.4305	1.51863	0.759315	−	0.650724i	$-0.225534\pi$
$67$	12.4305	1.51863	0.759315	+	0.650724i	$0.225534\pi$
$68$	0	0
$69$	2.49071	0.299846
$70$	0	0
$71$	14.9070	1.76914	0.884570	−	0.466407i	$-0.154452\pi$
$71$	14.9070	1.76914	0.884570	+	0.466407i	$0.154452\pi$
$72$	0	0
$73$	−10.5529	−1.23512	−0.617562	−	0.786522i	$-0.711880\pi$
$73$	−10.5529	−1.23512	−0.617562	+	0.786522i	$0.711880\pi$
$74$	0	0
$75$	−0.697718	−0.0805656
$76$	0	0
$77$	−22.7828	−2.59634
$78$	0	0
$79$	−13.3864	−1.50609	−0.753046	−	0.657968i	$-0.771416\pi$
$79$	−13.3864	−1.50609	−0.753046	+	0.657968i	$0.771416\pi$
$80$	0	0
$81$	4.85569	0.539521
$82$	0	0
$83$	6.00371	0.658992	0.329496	−	0.944157i	$-0.393121\pi$
$83$	6.00371	0.658992	0.329496	+	0.944157i	$0.393121\pi$
$84$	0	0
$85$	−4.13086	−0.448055
$86$	0	0
$87$	−3.19898	−0.342966
$88$	0	0
$89$	1.91289	0.202766	0.101383	−	0.994847i	$-0.467673\pi$
$89$	1.91289	0.202766	0.101383	+	0.994847i	$0.467673\pi$
$90$	0	0
$91$	2.73472	0.286677
$92$	0	0
$93$	2.22501	0.230723
$94$	0	0
$95$	4.54848	0.466664
$96$	0	0
$97$	16.2928	1.65428	0.827139	−	0.561997i	$-0.189967\pi$
$97$	16.2928	1.65428	0.827139	+	0.561997i	$0.189967\pi$
$98$	0	0
$99$	−11.4741	−1.15319
$100$	0	0
$101$	−14.8742	−1.48004	−0.740020	−	0.672585i	$-0.765184\pi$
$101$	−14.8742	−1.48004	−0.740020	+	0.672585i	$0.765184\pi$
$102$	0	0
$103$	−16.5300	−1.62875	−0.814375	−	0.580340i	$-0.802920\pi$
$103$	−16.5300	−1.62875	−0.814375	+	0.580340i	$0.802920\pi$
$104$	0	0
$105$	3.48172	0.339781
$106$	0	0
$107$	−17.2945	−1.67192	−0.835961	−	0.548789i	$-0.815089\pi$
$107$	−17.2945	−1.67192	−0.835961	+	0.548789i	$0.815089\pi$
$108$	0	0
$109$	8.84224	0.846933	0.423467	−	0.905912i	$-0.360813\pi$
$109$	8.84224	0.846933	0.423467	+	0.905912i	$0.360813\pi$
$110$	0	0
$111$	−7.83697	−0.743853
$112$	0	0
$113$	−14.3522	−1.35014	−0.675070	−	0.737754i	$-0.735887\pi$
$113$	−14.3522	−1.35014	−0.675070	+	0.737754i	$0.735887\pi$
$114$	0	0
$115$	−3.56979	−0.332885
$116$	0	0
$117$	1.37729	0.127330
$118$	0	0
$119$	20.6136	1.88965
$120$	0	0
$121$	9.84429	0.894935
$122$	0	0
$123$	−1.18298	−0.106665
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	20.2646	1.79820	0.899098	−	0.437747i	$-0.144224\pi$
$127$	20.2646	1.79820	0.899098	+	0.437747i	$0.144224\pi$
$128$	0	0
$129$	−4.89990	−0.431412
$130$	0	0
$131$	21.9761	1.92006	0.960030	−	0.279896i	$-0.0903000\pi$
$131$	21.9761	1.92006	0.960030	+	0.279896i	$0.0903000\pi$
$132$	0	0
$133$	−22.6976	−1.96813
$134$	0	0
$135$	3.84665	0.331067
$136$	0	0
$137$	6.42332	0.548781	0.274391	−	0.961618i	$-0.411524\pi$
$137$	6.42332	0.548781	0.274391	+	0.961618i	$0.411524\pi$
$138$	0	0
$139$	13.4502	1.14083	0.570413	−	0.821358i	$-0.306783\pi$
$139$	13.4502	1.14083	0.570413	+	0.821358i	$0.306783\pi$
$140$	0	0
$141$	5.25319	0.442398
$142$	0	0
$143$	−2.50203	−0.209231
$144$	0	0
$145$	4.58491	0.380756
$146$	0	0
$147$	−12.4903	−1.03018
$148$	0	0
$149$	−13.4930	−1.10539	−0.552694	−	0.833384i	$-0.686400\pi$
$149$	−13.4930	−1.10539	−0.552694	+	0.833384i	$0.686400\pi$
$150$	0	0
$151$	−17.3274	−1.41008	−0.705040	−	0.709167i	$-0.749071\pi$
$151$	−17.3274	−1.41008	−0.705040	+	0.709167i	$0.749071\pi$
$152$	0	0
$153$	10.3816	0.839305
$154$	0	0
$155$	−3.18898	−0.256145
$156$	0	0
$157$	−8.19254	−0.653836	−0.326918	−	0.945053i	$-0.606010\pi$
$157$	−8.19254	−0.653836	−0.326918	+	0.945053i	$0.606010\pi$
$158$	0	0
$159$	7.01155	0.556052
$160$	0	0
$161$	17.8138	1.40393
$162$	0	0
$163$	−19.8963	−1.55840	−0.779199	−	0.626777i	$-0.784374\pi$
$163$	−19.8963	−1.55840	−0.779199	+	0.626777i	$0.784374\pi$
$164$	0	0
$165$	−3.18547	−0.247989
$166$	0	0
$167$	−4.16900	−0.322607	−0.161303	−	0.986905i	$-0.551570\pi$
$167$	−4.16900	−0.322607	−0.161303	+	0.986905i	$0.551570\pi$
$168$	0	0
$169$	−12.6997	−0.976898
$170$	0	0
$171$	−11.4312	−0.874165
$172$	0	0
$173$	−12.3243	−0.936997	−0.468499	−	0.883464i	$-0.655205\pi$
$173$	−12.3243	−0.936997	−0.468499	+	0.883464i	$0.655205\pi$
$174$	0	0
$175$	−4.99015	−0.377220
$176$	0	0
$177$	1.53622	0.115469
$178$	0	0
$179$	0.179185	0.0133929	0.00669646	−	0.999978i	$-0.497868\pi$
$179$	0.179185	0.0133929	0.00669646	+	0.999978i	$0.497868\pi$
$180$	0	0
$181$	3.37487	0.250852	0.125426	−	0.992103i	$-0.459970\pi$
$181$	3.37487	0.250852	0.125426	+	0.992103i	$0.459970\pi$
$182$	0	0
$183$	5.53757	0.409349
$184$	0	0
$185$	11.2323	0.825814
$186$	0	0
$187$	−18.8597	−1.37916
$188$	0	0
$189$	−19.1954	−1.39626
$190$	0	0
$191$	−5.23347	−0.378681	−0.189340	−	0.981912i	$-0.560635\pi$
$191$	−5.23347	−0.378681	−0.189340	+	0.981912i	$0.560635\pi$
$192$	0	0
$193$	−18.2352	−1.31260	−0.656299	−	0.754501i	$-0.727879\pi$
$193$	−18.2352	−1.31260	−0.656299	+	0.754501i	$0.727879\pi$
$194$	0	0
$195$	0.382366	0.0273818
$196$	0	0
$197$	−15.6472	−1.11481	−0.557407	−	0.830240i	$-0.688204\pi$
$197$	−15.6472	−1.11481	−0.557407	+	0.830240i	$0.688204\pi$
$198$	0	0
$199$	4.12965	0.292743	0.146372	−	0.989230i	$-0.453240\pi$
$199$	4.12965	0.292743	0.146372	+	0.989230i	$0.453240\pi$
$200$	0	0
$201$	−8.67300	−0.611746
$202$	0	0
$203$	−22.8794	−1.60582
$204$	0	0
$205$	1.69549	0.118418
$206$	0	0
$207$	8.97157	0.623567
$208$	0	0
$209$	20.7663	1.43644
$210$	0	0
$211$	−11.2562	−0.774909	−0.387454	−	0.921889i	$-0.626646\pi$
$211$	−11.2562	−0.774909	−0.387454	+	0.921889i	$0.626646\pi$
$212$	0	0
$213$	−10.4009	−0.712659
$214$	0	0
$215$	7.02275	0.478948
$216$	0	0
$217$	15.9135	1.08028
$218$	0	0
$219$	7.36296	0.497543
$220$	0	0
$221$	2.26381	0.152280
$222$	0	0
$223$	6.58075	0.440680	0.220340	−	0.975423i	$-0.429283\pi$
$223$	6.58075	0.440680	0.220340	+	0.975423i	$0.429283\pi$
$224$	0	0
$225$	−2.51319	−0.167546
$226$	0	0
$227$	12.1047	0.803416	0.401708	−	0.915768i	$-0.368417\pi$
$227$	12.1047	0.803416	0.401708	+	0.915768i	$0.368417\pi$
$228$	0	0
$229$	14.6007	0.964844	0.482422	−	0.875939i	$-0.339757\pi$
$229$	14.6007	0.964844	0.482422	+	0.875939i	$0.339757\pi$
$230$	0	0
$231$	15.8960	1.04588
$232$	0	0
$233$	−21.1756	−1.38726	−0.693631	−	0.720330i	$-0.743990\pi$
$233$	−21.1756	−1.38726	−0.693631	+	0.720330i	$0.743990\pi$
$234$	0	0
$235$	−7.52910	−0.491144
$236$	0	0
$237$	9.33996	0.606696
$238$	0	0
$239$	−8.84401	−0.572071	−0.286036	−	0.958219i	$-0.592338\pi$
$239$	−8.84401	−0.572071	−0.286036	+	0.958219i	$0.592338\pi$
$240$	0	0
$241$	1.33374	0.0859137	0.0429568	−	0.999077i	$-0.486322\pi$
$241$	1.33374	0.0859137	0.0429568	+	0.999077i	$0.486322\pi$
$242$	0	0
$243$	−14.9279	−0.957623
$244$	0	0
$245$	17.9016	1.14369
$246$	0	0
$247$	−2.49268	−0.158605
$248$	0	0
$249$	−4.18890	−0.265460
$250$	0	0
$251$	2.77021	0.174854	0.0874270	−	0.996171i	$-0.472136\pi$
$251$	2.77021	0.174854	0.0874270	+	0.996171i	$0.472136\pi$
$252$	0	0
$253$	−16.2981	−1.02465
$254$	0	0
$255$	2.88218	0.180489
$256$	0	0
$257$	23.6988	1.47829	0.739146	−	0.673545i	$-0.235229\pi$
$257$	23.6988	1.47829	0.739146	+	0.673545i	$0.235229\pi$
$258$	0	0
$259$	−56.0508	−3.48283
$260$	0	0
$261$	−11.5228	−0.713241
$262$	0	0
$263$	−2.75281	−0.169745	−0.0848727	−	0.996392i	$-0.527048\pi$
$263$	−2.75281	−0.169745	−0.0848727	+	0.996392i	$0.527048\pi$
$264$	0	0
$265$	−10.0493	−0.617321
$266$	0	0
$267$	−1.33466	−0.0816798
$268$	0	0
$269$	−12.9419	−0.789083	−0.394542	−	0.918878i	$-0.629097\pi$
$269$	−12.9419	−0.789083	−0.394542	+	0.918878i	$0.629097\pi$
$270$	0	0
$271$	−17.4439	−1.05964	−0.529821	−	0.848109i	$-0.677741\pi$
$271$	−17.4439	−1.05964	−0.529821	+	0.848109i	$0.677741\pi$
$272$	0	0
$273$	−1.90807	−0.115481
$274$	0	0
$275$	4.56555	0.275313
$276$	0	0
$277$	0.467160	0.0280689	0.0140345	−	0.999902i	$-0.495533\pi$
$277$	0.467160	0.0280689	0.0140345	+	0.999902i	$0.495533\pi$
$278$	0	0
$279$	8.01451	0.479816
$280$	0	0
$281$	−10.5882	−0.631638	−0.315819	−	0.948819i	$-0.602279\pi$
$281$	−10.5882	−0.631638	−0.315819	+	0.948819i	$0.602279\pi$
$282$	0	0
$283$	1.04021	0.0618341	0.0309170	−	0.999522i	$-0.490157\pi$
$283$	1.04021	0.0618341	0.0309170	+	0.999522i	$0.490157\pi$
$284$	0	0
$285$	−3.17356	−0.187985
$286$	0	0
$287$	−8.46076	−0.499423
$288$	0	0
$289$	0.0639995	0.00376468
$290$	0	0
$291$	−11.3678	−0.666390
$292$	0	0
$293$	−8.14758	−0.475986	−0.237993	−	0.971267i	$-0.576490\pi$
$293$	−8.14758	−0.475986	−0.237993	+	0.971267i	$0.576490\pi$
$294$	0	0
$295$	−2.20177	−0.128192
$296$	0	0
$297$	17.5621	1.01906
$298$	0	0
$299$	1.95633	0.113138
$300$	0	0
$301$	−35.0446	−2.01994
$302$	0	0
$303$	10.3780	0.596201
$304$	0	0
$305$	−7.93669	−0.454453
$306$	0	0
$307$	20.6144	1.17653	0.588263	−	0.808670i	$-0.299812\pi$
$307$	20.6144	1.17653	0.588263	+	0.808670i	$0.299812\pi$
$308$	0	0
$309$	11.5333	0.656106
$310$	0	0
$311$	21.6208	1.22600	0.613000	−	0.790083i	$-0.289962\pi$
$311$	21.6208	1.22600	0.613000	+	0.790083i	$0.289962\pi$
$312$	0	0
$313$	−31.0213	−1.75343	−0.876714	−	0.481013i	$-0.840269\pi$
$313$	−31.0213	−1.75343	−0.876714	+	0.481013i	$0.840269\pi$
$314$	0	0
$315$	12.5412	0.706616
$316$	0	0
$317$	−13.0114	−0.730794	−0.365397	−	0.930852i	$-0.619067\pi$
$317$	−13.0114	−0.730794	−0.365397	+	0.930852i	$0.619067\pi$
$318$	0	0
$319$	20.9327	1.17200
$320$	0	0
$321$	12.0667	0.673497
$322$	0	0
$323$	−18.7891	−1.04546
$324$	0	0
$325$	−0.548024	−0.0303989
$326$	0	0
$327$	−6.16939	−0.341168
$328$	0	0
$329$	37.5714	2.07138
$330$	0	0
$331$	−17.8404	−0.980599	−0.490299	−	0.871554i	$-0.663113\pi$
$331$	−17.8404	−0.980599	−0.490299	+	0.871554i	$0.663113\pi$
$332$	0	0
$333$	−28.2289	−1.54693
$334$	0	0
$335$	12.4305	0.679152
$336$	0	0
$337$	−9.47692	−0.516241	−0.258120	−	0.966113i	$-0.583103\pi$
$337$	−9.47692	−0.516241	−0.258120	+	0.966113i	$0.583103\pi$
$338$	0	0
$339$	10.0138	0.543874
$340$	0	0
$341$	−14.5595	−0.788439
$342$	0	0
$343$	−54.4008	−2.93737
$344$	0	0
$345$	2.49071	0.134095
$346$	0	0
$347$	−14.5254	−0.779766	−0.389883	−	0.920864i	$-0.627485\pi$
$347$	−14.5254	−0.779766	−0.389883	+	0.920864i	$0.627485\pi$
$348$	0	0
$349$	14.1938	0.759776	0.379888	−	0.925032i	$-0.375962\pi$
$349$	14.1938	0.759776	0.379888	+	0.925032i	$0.375962\pi$
$350$	0	0
$351$	−2.10806	−0.112520
$352$	0	0
$353$	23.2665	1.23835	0.619175	−	0.785253i	$-0.287467\pi$
$353$	23.2665	1.23835	0.619175	+	0.785253i	$0.287467\pi$
$354$	0	0
$355$	14.9070	0.791184
$356$	0	0
$357$	−14.3825	−0.761203
$358$	0	0
$359$	4.42627	0.233610	0.116805	−	0.993155i	$-0.462735\pi$
$359$	4.42627	0.233610	0.116805	+	0.993155i	$0.462735\pi$
$360$	0	0
$361$	1.68868	0.0888779
$362$	0	0
$363$	−6.86854	−0.360505
$364$	0	0
$365$	−10.5529	−0.552364
$366$	0	0
$367$	−4.38989	−0.229150	−0.114575	−	0.993415i	$-0.536551\pi$
$367$	−4.38989	−0.229150	−0.114575	+	0.993415i	$0.536551\pi$
$368$	0	0
$369$	−4.26109	−0.221823
$370$	0	0
$371$	50.1473	2.60352
$372$	0	0
$373$	7.44712	0.385597	0.192799	−	0.981238i	$-0.438244\pi$
$373$	7.44712	0.385597	0.192799	+	0.981238i	$0.438244\pi$
$374$	0	0
$375$	−0.697718	−0.0360300
$376$	0	0
$377$	−2.51264	−0.129408
$378$	0	0
$379$	−0.814760	−0.0418514	−0.0209257	−	0.999781i	$-0.506661\pi$
$379$	−0.814760	−0.0418514	−0.0209257	+	0.999781i	$0.506661\pi$
$380$	0	0
$381$	−14.1390	−0.724363
$382$	0	0
$383$	13.9709	0.713880	0.356940	−	0.934127i	$-0.383820\pi$
$383$	13.9709	0.713880	0.356940	+	0.934127i	$0.383820\pi$
$384$	0	0
$385$	−22.7828	−1.16112
$386$	0	0
$387$	−17.6495	−0.897175
$388$	0	0
$389$	−15.3468	−0.778115	−0.389057	−	0.921214i	$-0.627199\pi$
$389$	−15.3468	−0.778115	−0.389057	+	0.921214i	$0.627199\pi$
$390$	0	0
$391$	14.7463	0.745753
$392$	0	0
$393$	−15.3331	−0.773454
$394$	0	0
$395$	−13.3864	−0.673544
$396$	0	0
$397$	−9.30377	−0.466943	−0.233471	−	0.972364i	$-0.575009\pi$
$397$	−9.30377	−0.466943	−0.233471	+	0.972364i	$0.575009\pi$
$398$	0	0
$399$	15.8365	0.792819
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	1.74764	0.0870561
$404$	0	0
$405$	4.85569	0.241281
$406$	0	0
$407$	51.2816	2.54194
$408$	0	0
$409$	−28.5209	−1.41027	−0.705134	−	0.709074i	$-0.749113\pi$
$409$	−28.5209	−1.41027	−0.705134	+	0.709074i	$0.749113\pi$
$410$	0	0
$411$	−4.48167	−0.221064
$412$	0	0
$413$	10.9872	0.540644
$414$	0	0
$415$	6.00371	0.294710
$416$	0	0
$417$	−9.38442	−0.459557
$418$	0	0
$419$	8.27236	0.404131	0.202066	−	0.979372i	$-0.435235\pi$
$419$	8.27236	0.404131	0.202066	+	0.979372i	$0.435235\pi$
$420$	0	0
$421$	−18.8214	−0.917299	−0.458649	−	0.888617i	$-0.651667\pi$
$421$	−18.8214	−0.917299	−0.458649	+	0.888617i	$0.651667\pi$
$422$	0	0
$423$	18.9220	0.920022
$424$	0	0
$425$	−4.13086	−0.200376
$426$	0	0
$427$	39.6053	1.91663
$428$	0	0
$429$	1.74571	0.0842839
$430$	0	0
$431$	6.12220	0.294896	0.147448	−	0.989070i	$-0.452894\pi$
$431$	6.12220	0.294896	0.147448	+	0.989070i	$0.452894\pi$
$432$	0	0
$433$	24.1671	1.16140	0.580699	−	0.814119i	$-0.302779\pi$
$433$	24.1671	1.16140	0.580699	+	0.814119i	$0.302779\pi$
$434$	0	0
$435$	−3.19898	−0.153379
$436$	0	0
$437$	−16.2371	−0.776728
$438$	0	0
$439$	16.4362	0.784456	0.392228	−	0.919868i	$-0.371704\pi$
$439$	16.4362	0.784456	0.392228	+	0.919868i	$0.371704\pi$
$440$	0	0
$441$	−44.9902	−2.14239
$442$	0	0
$443$	−26.7296	−1.26996	−0.634980	−	0.772529i	$-0.718992\pi$
$443$	−26.7296	−1.26996	−0.634980	+	0.772529i	$0.718992\pi$
$444$	0	0
$445$	1.91289	0.0906797
$446$	0	0
$447$	9.41431	0.445281
$448$	0	0
$449$	−26.8731	−1.26822	−0.634109	−	0.773244i	$-0.718633\pi$
$449$	−26.8731	−1.26822	−0.634109	+	0.773244i	$0.718633\pi$
$450$	0	0
$451$	7.74086	0.364503
$452$	0	0
$453$	12.0896	0.568020
$454$	0	0
$455$	2.73472	0.128206
$456$	0	0
$457$	−22.4744	−1.05131	−0.525655	−	0.850698i	$-0.676180\pi$
$457$	−22.4744	−1.05131	−0.525655	+	0.850698i	$0.676180\pi$
$458$	0	0
$459$	−15.8900	−0.741681
$460$	0	0
$461$	−9.79293	−0.456102	−0.228051	−	0.973649i	$-0.573235\pi$
$461$	−9.79293	−0.456102	−0.228051	+	0.973649i	$0.573235\pi$
$462$	0	0
$463$	−5.00697	−0.232694	−0.116347	−	0.993209i	$-0.537118\pi$
$463$	−5.00697	−0.232694	−0.116347	+	0.993209i	$0.537118\pi$
$464$	0	0
$465$	2.22501	0.103182
$466$	0	0
$467$	1.95978	0.0906877	0.0453438	−	0.998971i	$-0.485562\pi$
$467$	1.95978	0.0906877	0.0453438	+	0.998971i	$0.485562\pi$
$468$	0	0
$469$	−62.0302	−2.86429
$470$	0	0
$471$	5.71609	0.263383
$472$	0	0
$473$	32.0628	1.47425
$474$	0	0
$475$	4.54848	0.208699
$476$	0	0
$477$	25.2557	1.15638
$478$	0	0
$479$	5.44853	0.248950	0.124475	−	0.992223i	$-0.460275\pi$
$479$	5.44853	0.248950	0.124475	+	0.992223i	$0.460275\pi$
$480$	0	0
$481$	−6.15557	−0.280670
$482$	0	0
$483$	−12.4290	−0.565540
$484$	0	0
$485$	16.2928	0.739816
$486$	0	0
$487$	13.3274	0.603920	0.301960	−	0.953321i	$-0.402359\pi$
$487$	13.3274	0.603920	0.301960	+	0.953321i	$0.402359\pi$
$488$	0	0
$489$	13.8820	0.627766
$490$	0	0
$491$	22.2863	1.00577	0.502883	−	0.864355i	$-0.332273\pi$
$491$	22.2863	1.00577	0.502883	+	0.864355i	$0.332273\pi$
$492$	0	0
$493$	−18.9396	−0.852998
$494$	0	0
$495$	−11.4741	−0.515723
$496$	0	0
$497$	−74.3884	−3.33678
$498$	0	0
$499$	−20.3897	−0.912767	−0.456383	−	0.889783i	$-0.650855\pi$
$499$	−20.3897	−0.912767	−0.456383	+	0.889783i	$0.650855\pi$
$500$	0	0
$501$	2.90879	0.129955
$502$	0	0
$503$	27.3810	1.22086	0.610430	−	0.792070i	$-0.290997\pi$
$503$	27.3810	1.22086	0.610430	+	0.792070i	$0.290997\pi$
$504$	0	0
$505$	−14.8742	−0.661894
$506$	0	0
$507$	8.86079	0.393522
$508$	0	0
$509$	−30.7635	−1.36357	−0.681784	−	0.731553i	$-0.738796\pi$
$509$	−30.7635	−1.36357	−0.681784	+	0.731553i	$0.738796\pi$
$510$	0	0
$511$	52.6606	2.32957
$512$	0	0
$513$	17.4964	0.772486
$514$	0	0
$515$	−16.5300	−0.728399
$516$	0	0
$517$	−34.3745	−1.51179
$518$	0	0
$519$	8.59887	0.377449
$520$	0	0
$521$	10.7657	0.471653	0.235827	−	0.971795i	$-0.424220\pi$
$521$	10.7657	0.471653	0.235827	+	0.971795i	$0.424220\pi$
$522$	0	0
$523$	−2.78470	−0.121767	−0.0608833	−	0.998145i	$-0.519392\pi$
$523$	−2.78470	−0.121767	−0.0608833	+	0.998145i	$0.519392\pi$
$524$	0	0
$525$	3.48172	0.151955
$526$	0	0
$527$	13.1732	0.573835
$528$	0	0
$529$	−10.2566	−0.445938
$530$	0	0
$531$	5.53348	0.240132
$532$	0	0
$533$	−0.929170	−0.0402468
$534$	0	0
$535$	−17.2945	−0.747706
$536$	0	0
$537$	−0.125021	−0.00539504
$538$	0	0
$539$	81.7309	3.52040
$540$	0	0
$541$	5.66695	0.243641	0.121821	−	0.992552i	$-0.461127\pi$
$541$	5.66695	0.243641	0.121821	+	0.992552i	$0.461127\pi$
$542$	0	0
$543$	−2.35471	−0.101050
$544$	0	0
$545$	8.84224	0.378760
$546$	0	0
$547$	8.98936	0.384357	0.192179	−	0.981360i	$-0.438445\pi$
$547$	8.98936	0.384357	0.192179	+	0.981360i	$0.438445\pi$
$548$	0	0
$549$	19.9464	0.851292
$550$	0	0
$551$	20.8544	0.888427
$552$	0	0
$553$	66.8003	2.84064
$554$	0	0
$555$	−7.83697	−0.332661
$556$	0	0
$557$	−29.8109	−1.26313	−0.631564	−	0.775324i	$-0.717587\pi$
$557$	−29.8109	−1.26313	−0.631564	+	0.775324i	$0.717587\pi$
$558$	0	0
$559$	−3.84864	−0.162780
$560$	0	0
$561$	13.1587	0.555562
$562$	0	0
$563$	0.0962347	0.00405581	0.00202790	−	0.999998i	$-0.499354\pi$
$563$	0.0962347	0.00405581	0.00202790	+	0.999998i	$0.499354\pi$
$564$	0	0
$565$	−14.3522	−0.603801
$566$	0	0
$567$	−24.2306	−1.01759
$568$	0	0
$569$	1.08780	0.0456031	0.0228015	−	0.999740i	$-0.492741\pi$
$569$	1.08780	0.0456031	0.0228015	+	0.999740i	$0.492741\pi$
$570$	0	0
$571$	29.5377	1.23611	0.618057	−	0.786133i	$-0.287920\pi$
$571$	29.5377	1.23611	0.618057	+	0.786133i	$0.287920\pi$
$572$	0	0
$573$	3.65149	0.152543
$574$	0	0
$575$	−3.56979	−0.148871
$576$	0	0
$577$	−20.2343	−0.842367	−0.421183	−	0.906976i	$-0.638385\pi$
$577$	−20.2343	−0.842367	−0.421183	+	0.906976i	$0.638385\pi$
$578$	0	0
$579$	12.7230	0.528751
$580$	0	0
$581$	−29.9594	−1.24293
$582$	0	0
$583$	−45.8804	−1.90017
$584$	0	0
$585$	1.37729	0.0569439
$586$	0	0
$587$	−25.6160	−1.05729	−0.528643	−	0.848844i	$-0.677299\pi$
$587$	−25.6160	−1.05729	−0.528643	+	0.848844i	$0.677299\pi$
$588$	0	0
$589$	−14.5050	−0.597669
$590$	0	0
$591$	10.9173	0.449078
$592$	0	0
$593$	42.2589	1.73536	0.867682	−	0.497120i	$-0.165609\pi$
$593$	42.2589	1.73536	0.867682	+	0.497120i	$0.165609\pi$
$594$	0	0
$595$	20.6136	0.845076
$596$	0	0
$597$	−2.88133	−0.117925
$598$	0	0
$599$	−6.23343	−0.254691	−0.127346	−	0.991858i	$-0.540646\pi$
$599$	−6.23343	−0.254691	−0.127346	+	0.991858i	$0.540646\pi$
$600$	0	0
$601$	−37.8441	−1.54369	−0.771846	−	0.635809i	$-0.780667\pi$
$601$	−37.8441	−1.54369	−0.771846	+	0.635809i	$0.780667\pi$
$602$	0	0
$603$	−31.2402	−1.27220
$604$	0	0
$605$	9.84429	0.400227
$606$	0	0
$607$	34.0252	1.38104	0.690519	−	0.723314i	$-0.257382\pi$
$607$	34.0252	1.38104	0.690519	+	0.723314i	$0.257382\pi$
$608$	0	0
$609$	15.9634	0.646869
$610$	0	0
$611$	4.12613	0.166925
$612$	0	0
$613$	−35.4634	−1.43235	−0.716177	−	0.697919i	$-0.754110\pi$
$613$	−35.4634	−1.43235	−0.716177	+	0.697919i	$0.754110\pi$
$614$	0	0
$615$	−1.18298	−0.0477022
$616$	0	0
$617$	22.5760	0.908875	0.454438	−	0.890779i	$-0.349840\pi$
$617$	22.5760	0.908875	0.454438	+	0.890779i	$0.349840\pi$
$618$	0	0
$619$	2.47196	0.0993565	0.0496782	−	0.998765i	$-0.484180\pi$
$619$	2.47196	0.0993565	0.0496782	+	0.998765i	$0.484180\pi$
$620$	0	0
$621$	−13.7318	−0.551036
$622$	0	0
$623$	−9.54561	−0.382437
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−14.4891	−0.578637
$628$	0	0
$629$	−46.3990	−1.85005
$630$	0	0
$631$	1.62866	0.0648361	0.0324181	−	0.999474i	$-0.489679\pi$
$631$	1.62866	0.0648361	0.0324181	+	0.999474i	$0.489679\pi$
$632$	0	0
$633$	7.85366	0.312155
$634$	0	0
$635$	20.2646	0.804178
$636$	0	0
$637$	−9.81052	−0.388707
$638$	0	0
$639$	−37.4642	−1.48206
$640$	0	0
$641$	−2.32381	−0.0917851	−0.0458925	−	0.998946i	$-0.514613\pi$
$641$	−2.32381	−0.0917851	−0.0458925	+	0.998946i	$0.514613\pi$
$642$	0	0
$643$	12.1880	0.480648	0.240324	−	0.970693i	$-0.422746\pi$
$643$	12.1880	0.480648	0.240324	+	0.970693i	$0.422746\pi$
$644$	0	0
$645$	−4.89990	−0.192934
$646$	0	0
$647$	−23.9669	−0.942235	−0.471117	−	0.882071i	$-0.656149\pi$
$647$	−23.9669	−0.942235	−0.471117	+	0.882071i	$0.656149\pi$
$648$	0	0
$649$	−10.0523	−0.394588
$650$	0	0
$651$	−11.1031	−0.435167
$652$	0	0
$653$	22.6144	0.884969	0.442484	−	0.896776i	$-0.354097\pi$
$653$	22.6144	0.884969	0.442484	+	0.896776i	$0.354097\pi$
$654$	0	0
$655$	21.9761	0.858677
$656$	0	0
$657$	26.5215	1.03470
$658$	0	0
$659$	24.9672	0.972583	0.486292	−	0.873797i	$-0.338349\pi$
$659$	24.9672	0.972583	0.486292	+	0.873797i	$0.338349\pi$
$660$	0	0
$661$	1.22121	0.0474994	0.0237497	−	0.999718i	$-0.492440\pi$
$661$	1.22121	0.0474994	0.0237497	+	0.999718i	$0.492440\pi$
$662$	0	0
$663$	−1.57950	−0.0613428
$664$	0	0
$665$	−22.6976	−0.880176
$666$	0	0
$667$	−16.3672	−0.633740
$668$	0	0
$669$	−4.59151	−0.177518
$670$	0	0
$671$	−36.2354	−1.39885
$672$	0	0
$673$	24.9339	0.961132	0.480566	−	0.876959i	$-0.340431\pi$
$673$	24.9339	0.961132	0.480566	+	0.876959i	$0.340431\pi$
$674$	0	0
$675$	3.84665	0.148058
$676$	0	0
$677$	23.4653	0.901846	0.450923	−	0.892563i	$-0.351095\pi$
$677$	23.4653	0.901846	0.450923	+	0.892563i	$0.351095\pi$
$678$	0	0
$679$	−81.3034	−3.12014
$680$	0	0
$681$	−8.44566	−0.323639
$682$	0	0
$683$	−15.1953	−0.581430	−0.290715	−	0.956810i	$-0.593893\pi$
$683$	−15.1953	−0.581430	−0.290715	+	0.956810i	$0.593893\pi$
$684$	0	0
$685$	6.42332	0.245422
$686$	0	0
$687$	−10.1872	−0.388666
$688$	0	0
$689$	5.50723	0.209809
$690$	0	0
$691$	2.94672	0.112098	0.0560492	−	0.998428i	$-0.482150\pi$
$691$	2.94672	0.112098	0.0560492	+	0.998428i	$0.482150\pi$
$692$	0	0
$693$	57.2575	2.17503
$694$	0	0
$695$	13.4502	0.510193
$696$	0	0
$697$	−7.00383	−0.265289
$698$	0	0
$699$	14.7746	0.558828
$700$	0	0
$701$	−32.8194	−1.23957	−0.619785	−	0.784771i	$-0.712780\pi$
$701$	−32.8194	−1.23957	−0.619785	+	0.784771i	$0.712780\pi$
$702$	0	0
$703$	51.0899	1.92689
$704$	0	0
$705$	5.25319	0.197847
$706$	0	0
$707$	74.2246	2.79150
$708$	0	0
$709$	−5.22699	−0.196304	−0.0981518	−	0.995171i	$-0.531293\pi$
$709$	−5.22699	−0.196304	−0.0981518	+	0.995171i	$0.531293\pi$
$710$	0	0
$711$	33.6426	1.26170
$712$	0	0
$713$	11.3840	0.426334
$714$	0	0
$715$	−2.50203	−0.0935708
$716$	0	0
$717$	6.17063	0.230446
$718$	0	0
$719$	−18.5480	−0.691722	−0.345861	−	0.938286i	$-0.612413\pi$
$719$	−18.5480	−0.691722	−0.345861	+	0.938286i	$0.612413\pi$
$720$	0	0
$721$	82.4872	3.07198
$722$	0	0
$723$	−0.930574	−0.0346084
$724$	0	0
$725$	4.58491	0.170279
$726$	0	0
$727$	13.2247	0.490475	0.245238	−	0.969463i	$-0.421134\pi$
$727$	13.2247	0.490475	0.245238	+	0.969463i	$0.421134\pi$
$728$	0	0
$729$	−4.15162	−0.153764
$730$	0	0
$731$	−29.0100	−1.07297
$732$	0	0
$733$	−23.2496	−0.858744	−0.429372	−	0.903128i	$-0.641265\pi$
$733$	−23.2496	−0.858744	−0.429372	+	0.903128i	$0.641265\pi$
$734$	0	0
$735$	−12.4903	−0.460711
$736$	0	0
$737$	56.7522	2.09049
$738$	0	0
$739$	10.3587	0.381050	0.190525	−	0.981682i	$-0.438981\pi$
$739$	10.3587	0.381050	0.190525	+	0.981682i	$0.438981\pi$
$740$	0	0
$741$	1.73919	0.0638906
$742$	0	0
$743$	−30.8112	−1.13035	−0.565177	−	0.824970i	$-0.691192\pi$
$743$	−30.8112	−1.13035	−0.565177	+	0.824970i	$0.691192\pi$
$744$	0	0
$745$	−13.4930	−0.494345
$746$	0	0
$747$	−15.0884	−0.552057
$748$	0	0
$749$	86.3021	3.15341
$750$	0	0
$751$	4.76202	0.173769	0.0868843	−	0.996218i	$-0.472309\pi$
$751$	4.76202	0.173769	0.0868843	+	0.996218i	$0.472309\pi$
$752$	0	0
$753$	−1.93283	−0.0704361
$754$	0	0
$755$	−17.3274	−0.630607
$756$	0	0
$757$	47.6885	1.73327	0.866635	−	0.498943i	$-0.166278\pi$
$757$	47.6885	1.73327	0.866635	+	0.498943i	$0.166278\pi$
$758$	0	0
$759$	11.3715	0.412758
$760$	0	0
$761$	−19.0883	−0.691949	−0.345975	−	0.938244i	$-0.612452\pi$
$761$	−19.0883	−0.691949	−0.345975	+	0.938244i	$0.612452\pi$
$762$	0	0
$763$	−44.1241	−1.59740
$764$	0	0
$765$	10.3816	0.375349
$766$	0	0
$767$	1.20663	0.0435687
$768$	0	0
$769$	−51.9879	−1.87473	−0.937365	−	0.348349i	$-0.886743\pi$
$769$	−51.9879	−1.87473	−0.937365	+	0.348349i	$0.886743\pi$
$770$	0	0
$771$	−16.5351	−0.595497
$772$	0	0
$773$	−3.49852	−0.125833	−0.0629165	−	0.998019i	$-0.520040\pi$
$773$	−3.49852	−0.125833	−0.0629165	+	0.998019i	$0.520040\pi$
$774$	0	0
$775$	−3.18898	−0.114552
$776$	0	0
$777$	39.1077	1.40298
$778$	0	0
$779$	7.71191	0.276308
$780$	0	0
$781$	68.0589	2.43534
$782$	0	0
$783$	17.6366	0.630280
$784$	0	0
$785$	−8.19254	−0.292404
$786$	0	0
$787$	−46.8785	−1.67104	−0.835519	−	0.549462i	$-0.814833\pi$
$787$	−46.8785	−1.67104	−0.835519	+	0.549462i	$0.814833\pi$
$788$	0	0
$789$	1.92068	0.0683782
$790$	0	0
$791$	71.6196	2.54650
$792$	0	0
$793$	4.34950	0.154455
$794$	0	0
$795$	7.01155	0.248674
$796$	0	0
$797$	29.2090	1.03464	0.517318	−	0.855793i	$-0.326930\pi$
$797$	29.2090	1.03464	0.517318	+	0.855793i	$0.326930\pi$
$798$	0	0
$799$	31.1016	1.10030
$800$	0	0
$801$	−4.80745	−0.169863
$802$	0	0
$803$	−48.1799	−1.70023
$804$	0	0
$805$	17.8138	0.627854
$806$	0	0
$807$	9.02982	0.317865
$808$	0	0
$809$	−8.85720	−0.311402	−0.155701	−	0.987804i	$-0.549764\pi$
$809$	−8.85720	−0.311402	−0.155701	+	0.987804i	$0.549764\pi$
$810$	0	0
$811$	0.504850	0.0177277	0.00886383	−	0.999961i	$-0.497179\pi$
$811$	0.504850	0.0177277	0.00886383	+	0.999961i	$0.497179\pi$
$812$	0	0
$813$	12.1709	0.426853
$814$	0	0
$815$	−19.8963	−0.696937
$816$	0	0
$817$	31.9429	1.11754
$818$	0	0
$819$	−6.87288	−0.240158
$820$	0	0
$821$	−12.5751	−0.438874	−0.219437	−	0.975627i	$-0.570422\pi$
$821$	−12.5751	−0.438874	−0.219437	+	0.975627i	$0.570422\pi$
$822$	0	0
$823$	2.42672	0.0845901	0.0422950	−	0.999105i	$-0.486533\pi$
$823$	2.42672	0.0845901	0.0422950	+	0.999105i	$0.486533\pi$
$824$	0	0
$825$	−3.18547	−0.110904
$826$	0	0
$827$	9.06124	0.315090	0.157545	−	0.987512i	$-0.449642\pi$
$827$	9.06124	0.315090	0.157545	+	0.987512i	$0.449642\pi$
$828$	0	0
$829$	2.50814	0.0871113	0.0435557	−	0.999051i	$-0.486131\pi$
$829$	2.50814	0.0871113	0.0435557	+	0.999051i	$0.486131\pi$
$830$	0	0
$831$	−0.325946	−0.0113070
$832$	0	0
$833$	−73.9491	−2.56218
$834$	0	0
$835$	−4.16900	−0.144274
$836$	0	0
$837$	−12.2669	−0.424006
$838$	0	0
$839$	−46.2478	−1.59665	−0.798325	−	0.602227i	$-0.794280\pi$
$839$	−46.2478	−1.59665	−0.798325	+	0.602227i	$0.794280\pi$
$840$	0	0
$841$	−7.97857	−0.275123
$842$	0	0
$843$	7.38757	0.254441
$844$	0	0
$845$	−12.6997	−0.436882
$846$	0	0
$847$	−49.1245	−1.68794
$848$	0	0
$849$	−0.725774	−0.0249085
$850$	0	0
$851$	−40.0970	−1.37451
$852$	0	0
$853$	23.6714	0.810494	0.405247	−	0.914207i	$-0.367186\pi$
$853$	23.6714	0.810494	0.405247	+	0.914207i	$0.367186\pi$
$854$	0	0
$855$	−11.4312	−0.390939
$856$	0	0
$857$	−5.52297	−0.188661	−0.0943305	−	0.995541i	$-0.530071\pi$
$857$	−5.52297	−0.188661	−0.0943305	+	0.995541i	$0.530071\pi$
$858$	0	0
$859$	7.61440	0.259800	0.129900	−	0.991527i	$-0.458534\pi$
$859$	7.61440	0.259800	0.129900	+	0.991527i	$0.458534\pi$
$860$	0	0
$861$	5.90323	0.201181
$862$	0	0
$863$	−3.94149	−0.134170	−0.0670850	−	0.997747i	$-0.521370\pi$
$863$	−3.94149	−0.134170	−0.0670850	+	0.997747i	$0.521370\pi$
$864$	0	0
$865$	−12.3243	−0.419038
$866$	0	0
$867$	−0.0446536	−0.00151652
$868$	0	0
$869$	−61.1165	−2.07323
$870$	0	0
$871$	−6.81222	−0.230823
$872$	0	0
$873$	−40.9468	−1.38584
$874$	0	0
$875$	−4.99015	−0.168698
$876$	0	0
$877$	−26.3452	−0.889613	−0.444806	−	0.895627i	$-0.646728\pi$
$877$	−26.3452	−0.889613	−0.444806	+	0.895627i	$0.646728\pi$
$878$	0	0
$879$	5.68471	0.191741
$880$	0	0
$881$	42.0772	1.41762	0.708809	−	0.705401i	$-0.249233\pi$
$881$	42.0772	1.41762	0.708809	+	0.705401i	$0.249233\pi$
$882$	0	0
$883$	16.2009	0.545203	0.272602	−	0.962127i	$-0.412116\pi$
$883$	16.2009	0.545203	0.272602	+	0.962127i	$0.412116\pi$
$884$	0	0
$885$	1.53622	0.0516394
$886$	0	0
$887$	−53.6183	−1.80033	−0.900163	−	0.435553i	$-0.856553\pi$
$887$	−53.6183	−1.80033	−0.900163	+	0.435553i	$0.856553\pi$
$888$	0	0
$889$	−101.124	−3.39158
$890$	0	0
$891$	22.1689	0.742686
$892$	0	0
$893$	−34.2460	−1.14600
$894$	0	0
$895$	0.179185	0.00598949
$896$	0	0
$897$	−1.36497	−0.0455750
$898$	0	0
$899$	−14.6212	−0.487644
$900$	0	0
$901$	41.5120	1.38297
$902$	0	0
$903$	24.4513	0.813687
$904$	0	0
$905$	3.37487	0.112184
$906$	0	0
$907$	−44.4505	−1.47595	−0.737977	−	0.674826i	$-0.764219\pi$
$907$	−44.4505	−1.47595	−0.737977	+	0.674826i	$0.764219\pi$
$908$	0	0
$909$	37.3817	1.23987
$910$	0	0
$911$	−7.64826	−0.253398	−0.126699	−	0.991941i	$-0.540438\pi$
$911$	−7.64826	−0.253398	−0.126699	+	0.991941i	$0.540438\pi$
$912$	0	0
$913$	27.4102	0.907147
$914$	0	0
$915$	5.53757	0.183067
$916$	0	0
$917$	−109.664	−3.62143
$918$	0	0
$919$	24.3778	0.804150	0.402075	−	0.915607i	$-0.368289\pi$
$919$	24.3778	0.804150	0.402075	+	0.915607i	$0.368289\pi$
$920$	0	0
$921$	−14.3830	−0.473937
$922$	0	0
$923$	−8.16942	−0.268900
$924$	0	0
$925$	11.2323	0.369315
$926$	0	0
$927$	41.5430	1.36445
$928$	0	0
$929$	43.8872	1.43989	0.719947	−	0.694029i	$-0.244166\pi$
$929$	43.8872	1.43989	0.719947	+	0.694029i	$0.244166\pi$
$930$	0	0
$931$	81.4252	2.66860
$932$	0	0
$933$	−15.0852	−0.493867
$934$	0	0
$935$	−18.8597	−0.616777
$936$	0	0
$937$	−46.9562	−1.53399	−0.766996	−	0.641652i	$-0.778249\pi$
$937$	−46.9562	−1.53399	−0.766996	+	0.641652i	$0.778249\pi$
$938$	0	0
$939$	21.6441	0.706329
$940$	0	0
$941$	−42.2965	−1.37882	−0.689412	−	0.724369i	$-0.742131\pi$
$941$	−42.2965	−1.37882	−0.689412	+	0.724369i	$0.742131\pi$
$942$	0	0
$943$	−6.05255	−0.197098
$944$	0	0
$945$	−19.1954	−0.624426
$946$	0	0
$947$	−5.63082	−0.182977	−0.0914886	−	0.995806i	$-0.529162\pi$
$947$	−5.63082	−0.182977	−0.0914886	+	0.995806i	$0.529162\pi$
$948$	0	0
$949$	5.78325	0.187732
$950$	0	0
$951$	9.07830	0.294384
$952$	0	0
$953$	−11.3994	−0.369262	−0.184631	−	0.982808i	$-0.559109\pi$
$953$	−11.3994	−0.369262	−0.184631	+	0.982808i	$0.559109\pi$
$954$	0	0
$955$	−5.23347	−0.169351
$956$	0	0
$957$	−14.6051	−0.472116
$958$	0	0
$959$	−32.0534	−1.03506
$960$	0	0
$961$	−20.8304	−0.671948
$962$	0	0
$963$	43.4643	1.40062
$964$	0	0
$965$	−18.2352	−0.587012
$966$	0	0
$967$	−34.4145	−1.10670	−0.553348	−	0.832950i	$-0.686650\pi$
$967$	−34.4145	−1.10670	−0.553348	+	0.832950i	$0.686650\pi$
$968$	0	0
$969$	13.1095	0.421139
$970$	0	0
$971$	0.270339	0.00867559	0.00433779	−	0.999991i	$-0.498619\pi$
$971$	0.270339	0.00867559	0.00433779	+	0.999991i	$0.498619\pi$
$972$	0	0
$973$	−67.1183	−2.15171
$974$	0	0
$975$	0.382366	0.0122455
$976$	0	0
$977$	−35.4319	−1.13357	−0.566783	−	0.823867i	$-0.691812\pi$
$977$	−35.4319	−1.13357	−0.566783	+	0.823867i	$0.691812\pi$
$978$	0	0
$979$	8.73340	0.279121
$980$	0	0
$981$	−22.2222	−0.709501
$982$	0	0
$983$	58.8991	1.87859	0.939294	−	0.343112i	$-0.111481\pi$
$983$	58.8991	1.87859	0.939294	+	0.343112i	$0.111481\pi$
$984$	0	0
$985$	−15.6472	−0.498560
$986$	0	0
$987$	−26.2142	−0.834408
$988$	0	0
$989$	−25.0698	−0.797172
$990$	0	0
$991$	47.6875	1.51484	0.757422	−	0.652926i	$-0.226459\pi$
$991$	47.6875	1.51484	0.757422	+	0.652926i	$0.226459\pi$
$992$	0	0
$993$	12.4476	0.395013
$994$	0	0
$995$	4.12965	0.130919
$996$	0	0
$997$	−35.0644	−1.11050	−0.555251	−	0.831683i	$-0.687378\pi$
$997$	−35.0644	−1.11050	−0.555251	+	0.831683i	$0.687378\pi$
$998$	0	0
$999$	43.2067	1.36700

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.d.1.12	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.d.1.12	✓	29	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)

\(f(q)\)	\(=\)	\(q-0.697718 q^{3} +1.00000 q^{5} -4.99015 q^{7} -2.51319 q^{9} +O(q^{10})\)	\(q-0.697718 q^{3} +1.00000 q^{5} -4.99015 q^{7} -2.51319 q^{9} +4.56555 q^{11} -0.548024 q^{13} -0.697718 q^{15} -4.13086 q^{17} +4.54848 q^{19} +3.48172 q^{21} -3.56979 q^{23} +1.00000 q^{25} +3.84665 q^{27} +4.58491 q^{29} -3.18898 q^{31} -3.18547 q^{33} -4.99015 q^{35} +11.2323 q^{37} +0.382366 q^{39} +1.69549 q^{41} +7.02275 q^{43} -2.51319 q^{45} -7.52910 q^{47} +17.9016 q^{49} +2.88218 q^{51} -10.0493 q^{53} +4.56555 q^{55} -3.17356 q^{57} -2.20177 q^{59} -7.93669 q^{61} +12.5412 q^{63} -0.548024 q^{65} +12.4305 q^{67} +2.49071 q^{69} +14.9070 q^{71} -10.5529 q^{73} -0.697718 q^{75} -22.7828 q^{77} -13.3864 q^{79} +4.85569 q^{81} +6.00371 q^{83} -4.13086 q^{85} -3.19898 q^{87} +1.91289 q^{89} +2.73472 q^{91} +2.22501 q^{93} +4.54848 q^{95} +16.2928 q^{97} -11.4741 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9	\( 29 q - 3 q^{3} + 29 q^{5} - 8 q^{7} + 10 q^{9} + 2 q^{11} - 23 q^{13} - 3 q^{15} - 30 q^{17} - 6 q^{19} - 16 q^{21} - 21 q^{23} + 29 q^{25} - 15 q^{27} - 35 q^{29} - 7 q^{31} - 36 q^{33} - 8 q^{35} - 31 q^{37} - 11 q^{39} - 24 q^{41} - 17 q^{43} + 10 q^{45} - 17 q^{47} + q^{49} + 8 q^{51} - 57 q^{53} + 2 q^{55} - 46 q^{57} - 9 q^{59} - 27 q^{61} - 34 q^{63} - 23 q^{65} - 21 q^{67} - 28 q^{69} - 19 q^{71} - 81 q^{73} - 3 q^{75} - 66 q^{77} - 17 q^{79} - 39 q^{81} - 30 q^{83} - 30 q^{85} - 20 q^{87} - 38 q^{89} + q^{91} - 75 q^{93} - 6 q^{95} - 48 q^{97} - 7 q^{99}+O(q^{100}) \) 29 * q - 3 * q^3 + 29 * q^5 - 8 * q^7 + 10 * q^9 + 2 * q^11 - 23 * q^13 - 3 * q^15 - 30 * q^17 - 6 * q^19 - 16 * q^21 - 21 * q^23 + 29 * q^25 - 15 * q^27 - 35 * q^29 - 7 * q^31 - 36 * q^33 - 8 * q^35 - 31 * q^37 - 11 * q^39 - 24 * q^41 - 17 * q^43 + 10 * q^45 - 17 * q^47 + q^49 + 8 * q^51 - 57 * q^53 + 2 * q^55 - 46 * q^57 - 9 * q^59 - 27 * q^61 - 34 * q^63 - 23 * q^65 - 21 * q^67 - 28 * q^69 - 19 * q^71 - 81 * q^73 - 3 * q^75 - 66 * q^77 - 17 * q^79 - 39 * q^81 - 30 * q^83 - 30 * q^85 - 20 * q^87 - 38 * q^89 + q^91 - 75 * q^93 - 6 * q^95 - 48 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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