LMFDB - Embedded newform 8032.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8032,2,Mod(1,8032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8032, 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("8032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8032 = 2^{5} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8032.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:	$64.1358429035$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	8032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.01325 q^{3} +0.418153 q^{5} -0.933114 q^{7} +6.07967 q^{9} +O(q^{10})$	$q-3.01325 q^{3} +0.418153 q^{5} -0.933114 q^{7} +6.07967 q^{9} +4.47616 q^{11} -4.34244 q^{13} -1.26000 q^{15} -7.93008 q^{17} -4.27244 q^{19} +2.81171 q^{21} +7.70210 q^{23} -4.82515 q^{25} -9.27981 q^{27} +4.27667 q^{29} -6.35554 q^{31} -13.4878 q^{33} -0.390184 q^{35} +3.56456 q^{37} +13.0849 q^{39} -8.23603 q^{41} +8.01732 q^{43} +2.54223 q^{45} +12.1557 q^{47} -6.12930 q^{49} +23.8953 q^{51} -2.39762 q^{53} +1.87172 q^{55} +12.8739 q^{57} -12.5524 q^{59} -0.530712 q^{61} -5.67302 q^{63} -1.81580 q^{65} +7.39267 q^{67} -23.2083 q^{69} -7.19753 q^{71} +4.19444 q^{73} +14.5394 q^{75} -4.17677 q^{77} -8.89554 q^{79} +9.72337 q^{81} +0.852904 q^{83} -3.31599 q^{85} -12.8867 q^{87} -10.8846 q^{89} +4.05199 q^{91} +19.1508 q^{93} -1.78653 q^{95} +17.1700 q^{97} +27.2136 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	$ 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) $ 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * 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
$3$	−3.01325	−1.73970	−0.869850	−	0.493316i	$-0.835785\pi$
$3$	−3.01325	−1.73970	−0.869850	+	0.493316i	$0.835785\pi$
$4$	0	0
$5$	0.418153	0.187004	0.0935018	−	0.995619i	$-0.470194\pi$
$5$	0.418153	0.187004	0.0935018	+	0.995619i	$0.470194\pi$
$6$	0	0
$7$	−0.933114	−0.352684	−0.176342	−	0.984329i	$-0.556426\pi$
$7$	−0.933114	−0.352684	−0.176342	+	0.984329i	$0.556426\pi$
$8$	0	0
$9$	6.07967	2.02656
$10$	0	0
$11$	4.47616	1.34961	0.674806	−	0.737995i	$-0.264227\pi$
$11$	4.47616	1.34961	0.674806	+	0.737995i	$0.264227\pi$
$12$	0	0
$13$	−4.34244	−1.20438	−0.602188	−	0.798354i	$-0.705704\pi$
$13$	−4.34244	−1.20438	−0.602188	+	0.798354i	$0.705704\pi$
$14$	0	0
$15$	−1.26000	−0.325330
$16$	0	0
$17$	−7.93008	−1.92333	−0.961664	−	0.274232i	$-0.911576\pi$
$17$	−7.93008	−1.92333	−0.961664	+	0.274232i	$0.911576\pi$
$18$	0	0
$19$	−4.27244	−0.980165	−0.490083	−	0.871676i	$-0.663033\pi$
$19$	−4.27244	−0.980165	−0.490083	+	0.871676i	$0.663033\pi$
$20$	0	0
$21$	2.81171	0.613564
$22$	0	0
$23$	7.70210	1.60600	0.802999	−	0.595980i	$-0.203236\pi$
$23$	7.70210	1.60600	0.802999	+	0.595980i	$0.203236\pi$
$24$	0	0
$25$	−4.82515	−0.965030
$26$	0	0
$27$	−9.27981	−1.78590
$28$	0	0
$29$	4.27667	0.794158	0.397079	−	0.917784i	$-0.370024\pi$
$29$	4.27667	0.794158	0.397079	+	0.917784i	$0.370024\pi$
$30$	0	0
$31$	−6.35554	−1.14149	−0.570744	−	0.821128i	$-0.693345\pi$
$31$	−6.35554	−1.14149	−0.570744	+	0.821128i	$0.693345\pi$
$32$	0	0
$33$	−13.4878	−2.34792
$34$	0	0
$35$	−0.390184	−0.0659532
$36$	0	0
$37$	3.56456	0.586010	0.293005	−	0.956111i	$-0.405345\pi$
$37$	3.56456	0.586010	0.293005	+	0.956111i	$0.405345\pi$
$38$	0	0
$39$	13.0849	2.09525
$40$	0	0
$41$	−8.23603	−1.28625	−0.643126	−	0.765760i	$-0.722363\pi$
$41$	−8.23603	−1.28625	−0.643126	+	0.765760i	$0.722363\pi$
$42$	0	0
$43$	8.01732	1.22263	0.611315	−	0.791388i	$-0.290641\pi$
$43$	8.01732	1.22263	0.611315	+	0.791388i	$0.290641\pi$
$44$	0	0
$45$	2.54223	0.378973
$46$	0	0
$47$	12.1557	1.77309	0.886545	−	0.462642i	$-0.153098\pi$
$47$	12.1557	1.77309	0.886545	+	0.462642i	$0.153098\pi$
$48$	0	0
$49$	−6.12930	−0.875614
$50$	0	0
$51$	23.8953	3.34601
$52$	0	0
$53$	−2.39762	−0.329338	−0.164669	−	0.986349i	$-0.552656\pi$
$53$	−2.39762	−0.329338	−0.164669	+	0.986349i	$0.552656\pi$
$54$	0	0
$55$	1.87172	0.252382
$56$	0	0
$57$	12.8739	1.70519
$58$	0	0
$59$	−12.5524	−1.63419	−0.817093	−	0.576506i	$-0.804416\pi$
$59$	−12.5524	−1.63419	−0.817093	+	0.576506i	$0.804416\pi$
$60$	0	0
$61$	−0.530712	−0.0679507	−0.0339754	−	0.999423i	$-0.510817\pi$
$61$	−0.530712	−0.0679507	−0.0339754	+	0.999423i	$0.510817\pi$
$62$	0	0
$63$	−5.67302	−0.714734
$64$	0	0
$65$	−1.81580	−0.225223
$66$	0	0
$67$	7.39267	0.903158	0.451579	−	0.892231i	$-0.350861\pi$
$67$	7.39267	0.903158	0.451579	+	0.892231i	$0.350861\pi$
$68$	0	0
$69$	−23.2083	−2.79396
$70$	0	0
$71$	−7.19753	−0.854190	−0.427095	−	0.904207i	$-0.640463\pi$
$71$	−7.19753	−0.854190	−0.427095	+	0.904207i	$0.640463\pi$
$72$	0	0
$73$	4.19444	0.490922	0.245461	−	0.969406i	$-0.421061\pi$
$73$	4.19444	0.490922	0.245461	+	0.969406i	$0.421061\pi$
$74$	0	0
$75$	14.5394	1.67886
$76$	0	0
$77$	−4.17677	−0.475987
$78$	0	0
$79$	−8.89554	−1.00083	−0.500413	−	0.865787i	$-0.666818\pi$
$79$	−8.89554	−1.00083	−0.500413	+	0.865787i	$0.666818\pi$
$80$	0	0
$81$	9.72337	1.08037
$82$	0	0
$83$	0.852904	0.0936183	0.0468092	−	0.998904i	$-0.485095\pi$
$83$	0.852904	0.0936183	0.0468092	+	0.998904i	$0.485095\pi$
$84$	0	0
$85$	−3.31599	−0.359669
$86$	0	0
$87$	−12.8867	−1.38160
$88$	0	0
$89$	−10.8846	−1.15377	−0.576883	−	0.816827i	$-0.695731\pi$
$89$	−10.8846	−1.15377	−0.576883	+	0.816827i	$0.695731\pi$
$90$	0	0
$91$	4.05199	0.424764
$92$	0	0
$93$	19.1508	1.98585
$94$	0	0
$95$	−1.78653	−0.183294
$96$	0	0
$97$	17.1700	1.74334	0.871672	−	0.490089i	$-0.163036\pi$
$97$	17.1700	1.74334	0.871672	+	0.490089i	$0.163036\pi$
$98$	0	0
$99$	27.2136	2.73507
$100$	0	0
$101$	7.14631	0.711084	0.355542	−	0.934660i	$-0.384296\pi$
$101$	7.14631	0.711084	0.355542	+	0.934660i	$0.384296\pi$
$102$	0	0
$103$	15.5382	1.53102	0.765512	−	0.643421i	$-0.222486\pi$
$103$	15.5382	1.53102	0.765512	+	0.643421i	$0.222486\pi$
$104$	0	0
$105$	1.17572	0.114739
$106$	0	0
$107$	−6.15099	−0.594639	−0.297319	−	0.954778i	$-0.596093\pi$
$107$	−6.15099	−0.594639	−0.297319	+	0.954778i	$0.596093\pi$
$108$	0	0
$109$	11.3532	1.08744	0.543719	−	0.839267i	$-0.317016\pi$
$109$	11.3532	1.08744	0.543719	+	0.839267i	$0.317016\pi$
$110$	0	0
$111$	−10.7409	−1.01948
$112$	0	0
$113$	−5.92781	−0.557642	−0.278821	−	0.960343i	$-0.589944\pi$
$113$	−5.92781	−0.557642	−0.278821	+	0.960343i	$0.589944\pi$
$114$	0	0
$115$	3.22065	0.300327
$116$	0	0
$117$	−26.4006	−2.44074
$118$	0	0
$119$	7.39967	0.678327
$120$	0	0
$121$	9.03600	0.821455
$122$	0	0
$123$	24.8172	2.23769
$124$	0	0
$125$	−4.10841	−0.367468
$126$	0	0
$127$	−8.67992	−0.770218	−0.385109	−	0.922871i	$-0.625836\pi$
$127$	−8.67992	−0.770218	−0.385109	+	0.922871i	$0.625836\pi$
$128$	0	0
$129$	−24.1582	−2.12701
$130$	0	0
$131$	22.0811	1.92923	0.964617	−	0.263653i	$-0.0849275\pi$
$131$	22.0811	1.92923	0.964617	+	0.263653i	$0.0849275\pi$
$132$	0	0
$133$	3.98667	0.345689
$134$	0	0
$135$	−3.88038	−0.333970
$136$	0	0
$137$	−13.0859	−1.11800	−0.559001	−	0.829167i	$-0.688815\pi$
$137$	−13.0859	−1.11800	−0.559001	+	0.829167i	$0.688815\pi$
$138$	0	0
$139$	2.28307	0.193648	0.0968239	−	0.995302i	$-0.469132\pi$
$139$	2.28307	0.193648	0.0968239	+	0.995302i	$0.469132\pi$
$140$	0	0
$141$	−36.6281	−3.08465
$142$	0	0
$143$	−19.4375	−1.62544
$144$	0	0
$145$	1.78830	0.148510
$146$	0	0
$147$	18.4691	1.52331
$148$	0	0
$149$	−13.8643	−1.13580	−0.567902	−	0.823096i	$-0.692245\pi$
$149$	−13.8643	−1.13580	−0.567902	+	0.823096i	$0.692245\pi$
$150$	0	0
$151$	−22.7518	−1.85151	−0.925756	−	0.378122i	$-0.876570\pi$
$151$	−22.7518	−1.85151	−0.925756	+	0.378122i	$0.876570\pi$
$152$	0	0
$153$	−48.2123	−3.89773
$154$	0	0
$155$	−2.65759	−0.213462
$156$	0	0
$157$	1.48399	0.118436	0.0592178	−	0.998245i	$-0.481139\pi$
$157$	1.48399	0.118436	0.0592178	+	0.998245i	$0.481139\pi$
$158$	0	0
$159$	7.22461	0.572949
$160$	0	0
$161$	−7.18694	−0.566410
$162$	0	0
$163$	17.2563	1.35162	0.675808	−	0.737078i	$-0.263795\pi$
$163$	17.2563	1.35162	0.675808	+	0.737078i	$0.263795\pi$
$164$	0	0
$165$	−5.63995	−0.439070
$166$	0	0
$167$	−15.3789	−1.19006	−0.595029	−	0.803704i	$-0.702859\pi$
$167$	−15.3789	−1.19006	−0.595029	+	0.803704i	$0.702859\pi$
$168$	0	0
$169$	5.85680	0.450523
$170$	0	0
$171$	−25.9750	−1.98636
$172$	0	0
$173$	−1.41669	−0.107709	−0.0538546	−	0.998549i	$-0.517151\pi$
$173$	−1.41669	−0.107709	−0.0538546	+	0.998549i	$0.517151\pi$
$174$	0	0
$175$	4.50241	0.340350
$176$	0	0
$177$	37.8236	2.84299
$178$	0	0
$179$	−16.1649	−1.20822	−0.604109	−	0.796902i	$-0.706471\pi$
$179$	−16.1649	−1.20822	−0.604109	+	0.796902i	$0.706471\pi$
$180$	0	0
$181$	−14.2985	−1.06280	−0.531401	−	0.847120i	$-0.678334\pi$
$181$	−14.2985	−1.06280	−0.531401	+	0.847120i	$0.678334\pi$
$182$	0	0
$183$	1.59917	0.118214
$184$	0	0
$185$	1.49053	0.109586
$186$	0	0
$187$	−35.4963	−2.59575
$188$	0	0
$189$	8.65912	0.629858
$190$	0	0
$191$	0.122145	0.00883807	0.00441903	−	0.999990i	$-0.498593\pi$
$191$	0.122145	0.00883807	0.00441903	+	0.999990i	$0.498593\pi$
$192$	0	0
$193$	−15.9444	−1.14770	−0.573850	−	0.818960i	$-0.694551\pi$
$193$	−15.9444	−1.14770	−0.573850	+	0.818960i	$0.694551\pi$
$194$	0	0
$195$	5.47147	0.391820
$196$	0	0
$197$	−20.2584	−1.44335	−0.721677	−	0.692230i	$-0.756628\pi$
$197$	−20.2584	−1.44335	−0.721677	+	0.692230i	$0.756628\pi$
$198$	0	0
$199$	4.32896	0.306872	0.153436	−	0.988159i	$-0.450966\pi$
$199$	4.32896	0.306872	0.153436	+	0.988159i	$0.450966\pi$
$200$	0	0
$201$	−22.2760	−1.57122
$202$	0	0
$203$	−3.99062	−0.280087
$204$	0	0
$205$	−3.44392	−0.240534
$206$	0	0
$207$	46.8262	3.25465
$208$	0	0
$209$	−19.1241	−1.32284
$210$	0	0
$211$	13.1506	0.905327	0.452664	−	0.891681i	$-0.350474\pi$
$211$	13.1506	0.905327	0.452664	+	0.891681i	$0.350474\pi$
$212$	0	0
$213$	21.6880	1.48603
$214$	0	0
$215$	3.35246	0.228636
$216$	0	0
$217$	5.93044	0.402585
$218$	0	0
$219$	−12.6389	−0.854058
$220$	0	0
$221$	34.4359	2.31641
$222$	0	0
$223$	−4.22613	−0.283003	−0.141501	−	0.989938i	$-0.545193\pi$
$223$	−4.22613	−0.283003	−0.141501	+	0.989938i	$0.545193\pi$
$224$	0	0
$225$	−29.3353	−1.95569
$226$	0	0
$227$	−2.32837	−0.154539	−0.0772697	−	0.997010i	$-0.524620\pi$
$227$	−2.32837	−0.154539	−0.0772697	+	0.997010i	$0.524620\pi$
$228$	0	0
$229$	12.6254	0.834310	0.417155	−	0.908835i	$-0.363027\pi$
$229$	12.6254	0.834310	0.417155	+	0.908835i	$0.363027\pi$
$230$	0	0
$231$	12.5856	0.828074
$232$	0	0
$233$	−5.24323	−0.343496	−0.171748	−	0.985141i	$-0.554941\pi$
$233$	−5.24323	−0.343496	−0.171748	+	0.985141i	$0.554941\pi$
$234$	0	0
$235$	5.08294	0.331574
$236$	0	0
$237$	26.8045	1.74114
$238$	0	0
$239$	3.83017	0.247753	0.123876	−	0.992298i	$-0.460467\pi$
$239$	3.83017	0.247753	0.123876	+	0.992298i	$0.460467\pi$
$240$	0	0
$241$	28.5801	1.84101	0.920503	−	0.390735i	$-0.127779\pi$
$241$	28.5801	1.84101	0.920503	+	0.390735i	$0.127779\pi$
$242$	0	0
$243$	−1.45950	−0.0936271
$244$	0	0
$245$	−2.56298	−0.163743
$246$	0	0
$247$	18.5528	1.18049
$248$	0	0
$249$	−2.57001	−0.162868
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	34.4758	2.16748
$254$	0	0
$255$	9.99189	0.625716
$256$	0	0
$257$	−9.67986	−0.603813	−0.301907	−	0.953337i	$-0.597623\pi$
$257$	−9.67986	−0.603813	−0.301907	+	0.953337i	$0.597623\pi$
$258$	0	0
$259$	−3.32614	−0.206676
$260$	0	0
$261$	26.0007	1.60941
$262$	0	0
$263$	16.0621	0.990432	0.495216	−	0.868770i	$-0.335089\pi$
$263$	16.0621	0.990432	0.495216	+	0.868770i	$0.335089\pi$
$264$	0	0
$265$	−1.00257	−0.0615874
$266$	0	0
$267$	32.7980	2.00721
$268$	0	0
$269$	−3.97388	−0.242292	−0.121146	−	0.992635i	$-0.538657\pi$
$269$	−3.97388	−0.242292	−0.121146	+	0.992635i	$0.538657\pi$
$270$	0	0
$271$	−25.3611	−1.54058	−0.770289	−	0.637695i	$-0.779888\pi$
$271$	−25.3611	−1.54058	−0.770289	+	0.637695i	$0.779888\pi$
$272$	0	0
$273$	−12.2097	−0.738963
$274$	0	0
$275$	−21.5981	−1.30242
$276$	0	0
$277$	11.5967	0.696780	0.348390	−	0.937350i	$-0.386728\pi$
$277$	11.5967	0.696780	0.348390	+	0.937350i	$0.386728\pi$
$278$	0	0
$279$	−38.6396	−2.31329
$280$	0	0
$281$	7.38170	0.440355	0.220178	−	0.975460i	$-0.429336\pi$
$281$	7.38170	0.440355	0.220178	+	0.975460i	$0.429336\pi$
$282$	0	0
$283$	2.64212	0.157058	0.0785288	−	0.996912i	$-0.474978\pi$
$283$	2.64212	0.157058	0.0785288	+	0.996912i	$0.474978\pi$
$284$	0	0
$285$	5.38327	0.318877
$286$	0	0
$287$	7.68516	0.453641
$288$	0	0
$289$	45.8862	2.69919
$290$	0	0
$291$	−51.7374	−3.03290
$292$	0	0
$293$	21.4296	1.25193	0.625966	−	0.779851i	$-0.284705\pi$
$293$	21.4296	1.25193	0.625966	+	0.779851i	$0.284705\pi$
$294$	0	0
$295$	−5.24883	−0.305599
$296$	0	0
$297$	−41.5379	−2.41027
$298$	0	0
$299$	−33.4459	−1.93423
$300$	0	0
$301$	−7.48107	−0.431202
$302$	0	0
$303$	−21.5336	−1.23707
$304$	0	0
$305$	−0.221919	−0.0127070
$306$	0	0
$307$	−0.399105	−0.0227781	−0.0113890	−	0.999935i	$-0.503625\pi$
$307$	−0.399105	−0.0227781	−0.0113890	+	0.999935i	$0.503625\pi$
$308$	0	0
$309$	−46.8205	−2.66352
$310$	0	0
$311$	25.4690	1.44422	0.722108	−	0.691781i	$-0.243174\pi$
$311$	25.4690	1.44422	0.722108	+	0.691781i	$0.243174\pi$
$312$	0	0
$313$	−0.544720	−0.0307894	−0.0153947	−	0.999881i	$-0.504900\pi$
$313$	−0.544720	−0.0307894	−0.0153947	+	0.999881i	$0.504900\pi$
$314$	0	0
$315$	−2.37219	−0.133658
$316$	0	0
$317$	30.5387	1.71523	0.857613	−	0.514296i	$-0.171947\pi$
$317$	30.5387	1.71523	0.857613	+	0.514296i	$0.171947\pi$
$318$	0	0
$319$	19.1431	1.07181
$320$	0	0
$321$	18.5345	1.03449
$322$	0	0
$323$	33.8808	1.88518
$324$	0	0
$325$	20.9529	1.16226
$326$	0	0
$327$	−34.2100	−1.89182
$328$	0	0
$329$	−11.3427	−0.625341
$330$	0	0
$331$	0.997975	0.0548537	0.0274268	−	0.999624i	$-0.491269\pi$
$331$	0.997975	0.0548537	0.0274268	+	0.999624i	$0.491269\pi$
$332$	0	0
$333$	21.6714	1.18758
$334$	0	0
$335$	3.09126	0.168894
$336$	0	0
$337$	4.87921	0.265788	0.132894	−	0.991130i	$-0.457573\pi$
$337$	4.87921	0.265788	0.132894	+	0.991130i	$0.457573\pi$
$338$	0	0
$339$	17.8620	0.970129
$340$	0	0
$341$	−28.4484	−1.54057
$342$	0	0
$343$	12.2511	0.661499
$344$	0	0
$345$	−9.70463	−0.522480
$346$	0	0
$347$	17.4301	0.935695	0.467848	−	0.883809i	$-0.345030\pi$
$347$	17.4301	0.935695	0.467848	+	0.883809i	$0.345030\pi$
$348$	0	0
$349$	−0.0386115	−0.00206683	−0.00103341	−	0.999999i	$-0.500329\pi$
$349$	−0.0386115	−0.00206683	−0.00103341	+	0.999999i	$0.500329\pi$
$350$	0	0
$351$	40.2970	2.15090
$352$	0	0
$353$	33.4199	1.77876	0.889381	−	0.457167i	$-0.151136\pi$
$353$	33.4199	1.77876	0.889381	+	0.457167i	$0.151136\pi$
$354$	0	0
$355$	−3.00967	−0.159737
$356$	0	0
$357$	−22.2970	−1.18008
$358$	0	0
$359$	17.2935	0.912717	0.456359	−	0.889796i	$-0.349153\pi$
$359$	17.2935	0.912717	0.456359	+	0.889796i	$0.349153\pi$
$360$	0	0
$361$	−0.746248	−0.0392762
$362$	0	0
$363$	−27.2277	−1.42909
$364$	0	0
$365$	1.75392	0.0918042
$366$	0	0
$367$	−19.5743	−1.02177	−0.510886	−	0.859649i	$-0.670683\pi$
$367$	−19.5743	−1.02177	−0.510886	+	0.859649i	$0.670683\pi$
$368$	0	0
$369$	−50.0724	−2.60666
$370$	0	0
$371$	2.23725	0.116152
$372$	0	0
$373$	2.73303	0.141511	0.0707554	−	0.997494i	$-0.477459\pi$
$373$	2.73303	0.141511	0.0707554	+	0.997494i	$0.477459\pi$
$374$	0	0
$375$	12.3797	0.639283
$376$	0	0
$377$	−18.5712	−0.956465
$378$	0	0
$379$	4.54929	0.233681	0.116841	−	0.993151i	$-0.462723\pi$
$379$	4.54929	0.233681	0.116841	+	0.993151i	$0.462723\pi$
$380$	0	0
$381$	26.1548	1.33995
$382$	0	0
$383$	9.41490	0.481079	0.240540	−	0.970639i	$-0.422676\pi$
$383$	9.41490	0.481079	0.240540	+	0.970639i	$0.422676\pi$
$384$	0	0
$385$	−1.74653	−0.0890112
$386$	0	0
$387$	48.7426	2.47773
$388$	0	0
$389$	0.599468	0.0303943	0.0151971	−	0.999885i	$-0.495162\pi$
$389$	0.599468	0.0303943	0.0151971	+	0.999885i	$0.495162\pi$
$390$	0	0
$391$	−61.0783	−3.08886
$392$	0	0
$393$	−66.5359	−3.35629
$394$	0	0
$395$	−3.71969	−0.187158
$396$	0	0
$397$	16.7026	0.838278	0.419139	−	0.907922i	$-0.362332\pi$
$397$	16.7026	0.838278	0.419139	+	0.907922i	$0.362332\pi$
$398$	0	0
$399$	−12.0128	−0.601394
$400$	0	0
$401$	39.3625	1.96567	0.982834	−	0.184491i	$-0.0590635\pi$
$401$	39.3625	1.96567	0.982834	+	0.184491i	$0.0590635\pi$
$402$	0	0
$403$	27.5986	1.37478
$404$	0	0
$405$	4.06585	0.202034
$406$	0	0
$407$	15.9555	0.790887
$408$	0	0
$409$	−18.2381	−0.901816	−0.450908	−	0.892570i	$-0.648900\pi$
$409$	−18.2381	−0.901816	−0.450908	+	0.892570i	$0.648900\pi$
$410$	0	0
$411$	39.4310	1.94499
$412$	0	0
$413$	11.7128	0.576351
$414$	0	0
$415$	0.356644	0.0175070
$416$	0	0
$417$	−6.87947	−0.336889
$418$	0	0
$419$	6.43629	0.314434	0.157217	−	0.987564i	$-0.449748\pi$
$419$	6.43629	0.314434	0.157217	+	0.987564i	$0.449748\pi$
$420$	0	0
$421$	20.9916	1.02307	0.511535	−	0.859263i	$-0.329077\pi$
$421$	20.9916	1.02307	0.511535	+	0.859263i	$0.329077\pi$
$422$	0	0
$423$	73.9026	3.59327
$424$	0	0
$425$	38.2638	1.85607
$426$	0	0
$427$	0.495215	0.0239651
$428$	0	0
$429$	58.5699	2.82778
$430$	0	0
$431$	−32.2580	−1.55381	−0.776907	−	0.629615i	$-0.783212\pi$
$431$	−32.2580	−1.55381	−0.776907	+	0.629615i	$0.783212\pi$
$432$	0	0
$433$	−10.8084	−0.519417	−0.259709	−	0.965687i	$-0.583627\pi$
$433$	−10.8084	−0.519417	−0.259709	+	0.965687i	$0.583627\pi$
$434$	0	0
$435$	−5.38860	−0.258364
$436$	0	0
$437$	−32.9068	−1.57414
$438$	0	0
$439$	29.7528	1.42002	0.710011	−	0.704190i	$-0.248690\pi$
$439$	29.7528	1.42002	0.710011	+	0.704190i	$0.248690\pi$
$440$	0	0
$441$	−37.2641	−1.77448
$442$	0	0
$443$	9.11522	0.433077	0.216539	−	0.976274i	$-0.430523\pi$
$443$	9.11522	0.433077	0.216539	+	0.976274i	$0.430523\pi$
$444$	0	0
$445$	−4.55142	−0.215758
$446$	0	0
$447$	41.7765	1.97596
$448$	0	0
$449$	27.1042	1.27913	0.639564	−	0.768738i	$-0.279115\pi$
$449$	27.1042	1.27913	0.639564	+	0.768738i	$0.279115\pi$
$450$	0	0
$451$	−36.8658	−1.73594
$452$	0	0
$453$	68.5567	3.22108
$454$	0	0
$455$	1.69435	0.0794325
$456$	0	0
$457$	14.5848	0.682250	0.341125	−	0.940018i	$-0.389192\pi$
$457$	14.5848	0.682250	0.341125	+	0.940018i	$0.389192\pi$
$458$	0	0
$459$	73.5896	3.43487
$460$	0	0
$461$	−33.9705	−1.58216	−0.791081	−	0.611711i	$-0.790481\pi$
$461$	−33.9705	−1.58216	−0.791081	+	0.611711i	$0.790481\pi$
$462$	0	0
$463$	−1.35142	−0.0628056	−0.0314028	−	0.999507i	$-0.509997\pi$
$463$	−1.35142	−0.0628056	−0.0314028	+	0.999507i	$0.509997\pi$
$464$	0	0
$465$	8.00797	0.371361
$466$	0	0
$467$	0.0709469	0.00328303	0.00164152	−	0.999999i	$-0.499477\pi$
$467$	0.0709469	0.00328303	0.00164152	+	0.999999i	$0.499477\pi$
$468$	0	0
$469$	−6.89820	−0.318529
$470$	0	0
$471$	−4.47164	−0.206042
$472$	0	0
$473$	35.8868	1.65008
$474$	0	0
$475$	20.6152	0.945888
$476$	0	0
$477$	−14.5767	−0.667422
$478$	0	0
$479$	−19.9006	−0.909283	−0.454641	−	0.890675i	$-0.650233\pi$
$479$	−19.9006	−0.909283	−0.454641	+	0.890675i	$0.650233\pi$
$480$	0	0
$481$	−15.4789	−0.705777
$482$	0	0
$483$	21.6560	0.985384
$484$	0	0
$485$	7.17966	0.326012
$486$	0	0
$487$	34.1818	1.54893	0.774463	−	0.632619i	$-0.218020\pi$
$487$	34.1818	1.54893	0.774463	+	0.632619i	$0.218020\pi$
$488$	0	0
$489$	−51.9975	−2.35141
$490$	0	0
$491$	12.0615	0.544327	0.272164	−	0.962251i	$-0.412261\pi$
$491$	12.0615	0.544327	0.272164	+	0.962251i	$0.412261\pi$
$492$	0	0
$493$	−33.9144	−1.52743
$494$	0	0
$495$	11.3794	0.511467
$496$	0	0
$497$	6.71612	0.301259
$498$	0	0
$499$	22.0276	0.986089	0.493044	−	0.870004i	$-0.335884\pi$
$499$	22.0276	0.986089	0.493044	+	0.870004i	$0.335884\pi$
$500$	0	0
$501$	46.3405	2.07034
$502$	0	0
$503$	10.2824	0.458471	0.229236	−	0.973371i	$-0.426377\pi$
$503$	10.2824	0.458471	0.229236	+	0.973371i	$0.426377\pi$
$504$	0	0
$505$	2.98825	0.132975
$506$	0	0
$507$	−17.6480	−0.783776
$508$	0	0
$509$	36.3470	1.61105	0.805525	−	0.592561i	$-0.201883\pi$
$509$	36.3470	1.61105	0.805525	+	0.592561i	$0.201883\pi$
$510$	0	0
$511$	−3.91389	−0.173140
$512$	0	0
$513$	39.6474	1.75048
$514$	0	0
$515$	6.49734	0.286307
$516$	0	0
$517$	54.4108	2.39299
$518$	0	0
$519$	4.26885	0.187382
$520$	0	0
$521$	15.9878	0.700436	0.350218	−	0.936668i	$-0.386107\pi$
$521$	15.9878	0.700436	0.350218	+	0.936668i	$0.386107\pi$
$522$	0	0
$523$	19.1152	0.835850	0.417925	−	0.908481i	$-0.362757\pi$
$523$	19.1152	0.835850	0.417925	+	0.908481i	$0.362757\pi$
$524$	0	0
$525$	−13.5669	−0.592108
$526$	0	0
$527$	50.3999	2.19546
$528$	0	0
$529$	36.3223	1.57923
$530$	0	0
$531$	−76.3146	−3.31177
$532$	0	0
$533$	35.7645	1.54913
$534$	0	0
$535$	−2.57205	−0.111200
$536$	0	0
$537$	48.7087	2.10194
$538$	0	0
$539$	−27.4357	−1.18174
$540$	0	0
$541$	−45.0836	−1.93830	−0.969148	−	0.246478i	$-0.920727\pi$
$541$	−45.0836	−1.93830	−0.969148	+	0.246478i	$0.920727\pi$
$542$	0	0
$543$	43.0851	1.84896
$544$	0	0
$545$	4.74737	0.203355
$546$	0	0
$547$	3.28952	0.140650	0.0703248	−	0.997524i	$-0.477596\pi$
$547$	3.28952	0.140650	0.0703248	+	0.997524i	$0.477596\pi$
$548$	0	0
$549$	−3.22655	−0.137706
$550$	0	0
$551$	−18.2718	−0.778406
$552$	0	0
$553$	8.30055	0.352975
$554$	0	0
$555$	−4.49134	−0.190647
$556$	0	0
$557$	−35.2107	−1.49193	−0.745963	−	0.665987i	$-0.768011\pi$
$557$	−35.2107	−1.49193	−0.745963	+	0.665987i	$0.768011\pi$
$558$	0	0
$559$	−34.8147	−1.47251
$560$	0	0
$561$	106.959	4.51582
$562$	0	0
$563$	1.65541	0.0697672	0.0348836	−	0.999391i	$-0.488894\pi$
$563$	1.65541	0.0697672	0.0348836	+	0.999391i	$0.488894\pi$
$564$	0	0
$565$	−2.47873	−0.104281
$566$	0	0
$567$	−9.07301	−0.381031
$568$	0	0
$569$	22.2467	0.932631	0.466316	−	0.884618i	$-0.345581\pi$
$569$	22.2467	0.932631	0.466316	+	0.884618i	$0.345581\pi$
$570$	0	0
$571$	−39.1676	−1.63911	−0.819557	−	0.572997i	$-0.805780\pi$
$571$	−39.1676	−1.63911	−0.819557	+	0.572997i	$0.805780\pi$
$572$	0	0
$573$	−0.368052	−0.0153756
$574$	0	0
$575$	−37.1638	−1.54984
$576$	0	0
$577$	−45.3008	−1.88590	−0.942949	−	0.332938i	$-0.891960\pi$
$577$	−45.3008	−1.88590	−0.942949	+	0.332938i	$0.891960\pi$
$578$	0	0
$579$	48.0443	1.99665
$580$	0	0
$581$	−0.795856	−0.0330177
$582$	0	0
$583$	−10.7321	−0.444479
$584$	0	0
$585$	−11.0395	−0.456427
$586$	0	0
$587$	29.3804	1.21266	0.606329	−	0.795214i	$-0.292642\pi$
$587$	29.3804	1.21266	0.606329	+	0.795214i	$0.292642\pi$
$588$	0	0
$589$	27.1537	1.11885
$590$	0	0
$591$	61.0437	2.51100
$592$	0	0
$593$	−29.5645	−1.21407	−0.607034	−	0.794676i	$-0.707641\pi$
$593$	−29.5645	−1.21407	−0.607034	+	0.794676i	$0.707641\pi$
$594$	0	0
$595$	3.09419	0.126850
$596$	0	0
$597$	−13.0442	−0.533865
$598$	0	0
$599$	−23.0938	−0.943586	−0.471793	−	0.881709i	$-0.656393\pi$
$599$	−23.0938	−0.943586	−0.471793	+	0.881709i	$0.656393\pi$
$600$	0	0
$601$	−3.91150	−0.159553	−0.0797766	−	0.996813i	$-0.525421\pi$
$601$	−3.91150	−0.159553	−0.0797766	+	0.996813i	$0.525421\pi$
$602$	0	0
$603$	44.9450	1.83030
$604$	0	0
$605$	3.77843	0.153615
$606$	0	0
$607$	−4.81004	−0.195233	−0.0976167	−	0.995224i	$-0.531122\pi$
$607$	−4.81004	−0.195233	−0.0976167	+	0.995224i	$0.531122\pi$
$608$	0	0
$609$	12.0247	0.487267
$610$	0	0
$611$	−52.7854	−2.13547
$612$	0	0
$613$	−5.30388	−0.214221	−0.107111	−	0.994247i	$-0.534160\pi$
$613$	−5.30388	−0.214221	−0.107111	+	0.994247i	$0.534160\pi$
$614$	0	0
$615$	10.3774	0.418457
$616$	0	0
$617$	33.4492	1.34661	0.673306	−	0.739364i	$-0.264874\pi$
$617$	33.4492	1.34661	0.673306	+	0.739364i	$0.264874\pi$
$618$	0	0
$619$	28.6864	1.15300	0.576501	−	0.817096i	$-0.304418\pi$
$619$	28.6864	1.15300	0.576501	+	0.817096i	$0.304418\pi$
$620$	0	0
$621$	−71.4740	−2.86815
$622$	0	0
$623$	10.1566	0.406914
$624$	0	0
$625$	22.4078	0.896312
$626$	0	0
$627$	57.6258	2.30135
$628$	0	0
$629$	−28.2673	−1.12709
$630$	0	0
$631$	−1.79526	−0.0714681	−0.0357341	−	0.999361i	$-0.511377\pi$
$631$	−1.79526	−0.0714681	−0.0357341	+	0.999361i	$0.511377\pi$
$632$	0	0
$633$	−39.6261	−1.57500
$634$	0	0
$635$	−3.62953	−0.144034
$636$	0	0
$637$	26.6161	1.05457
$638$	0	0
$639$	−43.7586	−1.73106
$640$	0	0
$641$	34.9404	1.38006	0.690032	−	0.723779i	$-0.257596\pi$
$641$	34.9404	1.38006	0.690032	+	0.723779i	$0.257596\pi$
$642$	0	0
$643$	41.5176	1.63729	0.818647	−	0.574298i	$-0.194725\pi$
$643$	41.5176	1.63729	0.818647	+	0.574298i	$0.194725\pi$
$644$	0	0
$645$	−10.1018	−0.397758
$646$	0	0
$647$	31.0899	1.22227	0.611136	−	0.791526i	$-0.290713\pi$
$647$	31.0899	1.22227	0.611136	+	0.791526i	$0.290713\pi$
$648$	0	0
$649$	−56.1866	−2.20552
$650$	0	0
$651$	−17.8699	−0.700377
$652$	0	0
$653$	−11.6902	−0.457472	−0.228736	−	0.973489i	$-0.573459\pi$
$653$	−11.6902	−0.457472	−0.228736	+	0.973489i	$0.573459\pi$
$654$	0	0
$655$	9.23327	0.360774
$656$	0	0
$657$	25.5008	0.994882
$658$	0	0
$659$	35.3713	1.37787	0.688934	−	0.724824i	$-0.258079\pi$
$659$	35.3713	1.37787	0.688934	+	0.724824i	$0.258079\pi$
$660$	0	0
$661$	−6.97252	−0.271199	−0.135600	−	0.990764i	$-0.543296\pi$
$661$	−6.97252	−0.271199	−0.135600	+	0.990764i	$0.543296\pi$
$662$	0	0
$663$	−103.764	−4.02986
$664$	0	0
$665$	1.66704	0.0646450
$666$	0	0
$667$	32.9393	1.27542
$668$	0	0
$669$	12.7344	0.492340
$670$	0	0
$671$	−2.37555	−0.0917071
$672$	0	0
$673$	−45.3299	−1.74734	−0.873669	−	0.486520i	$-0.838266\pi$
$673$	−45.3299	−1.74734	−0.873669	+	0.486520i	$0.838266\pi$
$674$	0	0
$675$	44.7765	1.72345
$676$	0	0
$677$	23.1277	0.888870	0.444435	−	0.895811i	$-0.353404\pi$
$677$	23.1277	0.888870	0.444435	+	0.895811i	$0.353404\pi$
$678$	0	0
$679$	−16.0215	−0.614850
$680$	0	0
$681$	7.01596	0.268852
$682$	0	0
$683$	11.0050	0.421096	0.210548	−	0.977584i	$-0.432475\pi$
$683$	11.0050	0.421096	0.210548	+	0.977584i	$0.432475\pi$
$684$	0	0
$685$	−5.47190	−0.209071
$686$	0	0
$687$	−38.0435	−1.45145
$688$	0	0
$689$	10.4115	0.396647
$690$	0	0
$691$	−14.6589	−0.557650	−0.278825	−	0.960342i	$-0.589945\pi$
$691$	−14.6589	−0.557650	−0.278825	+	0.960342i	$0.589945\pi$
$692$	0	0
$693$	−25.3934	−0.964614
$694$	0	0
$695$	0.954674	0.0362128
$696$	0	0
$697$	65.3124	2.47388
$698$	0	0
$699$	15.7992	0.597579
$700$	0	0
$701$	11.8897	0.449069	0.224534	−	0.974466i	$-0.427914\pi$
$701$	11.8897	0.449069	0.224534	+	0.974466i	$0.427914\pi$
$702$	0	0
$703$	−15.2294	−0.574387
$704$	0	0
$705$	−15.3162	−0.576840
$706$	0	0
$707$	−6.66832	−0.250788
$708$	0	0
$709$	−33.1765	−1.24597	−0.622986	−	0.782233i	$-0.714080\pi$
$709$	−33.1765	−1.24597	−0.622986	+	0.782233i	$0.714080\pi$
$710$	0	0
$711$	−54.0819	−2.02823
$712$	0	0
$713$	−48.9510	−1.83323
$714$	0	0
$715$	−8.12783	−0.303964
$716$	0	0
$717$	−11.5412	−0.431016
$718$	0	0
$719$	13.4511	0.501643	0.250821	−	0.968033i	$-0.419299\pi$
$719$	13.4511	0.501643	0.250821	+	0.968033i	$0.419299\pi$
$720$	0	0
$721$	−14.4989	−0.539968
$722$	0	0
$723$	−86.1190	−3.20280
$724$	0	0
$725$	−20.6356	−0.766386
$726$	0	0
$727$	6.67753	0.247656	0.123828	−	0.992304i	$-0.460483\pi$
$727$	6.67753	0.247656	0.123828	+	0.992304i	$0.460483\pi$
$728$	0	0
$729$	−24.7723	−0.917491
$730$	0	0
$731$	−63.5780	−2.35152
$732$	0	0
$733$	45.3474	1.67495	0.837473	−	0.546479i	$-0.184032\pi$
$733$	45.3474	1.67495	0.837473	+	0.546479i	$0.184032\pi$
$734$	0	0
$735$	7.72290	0.284864
$736$	0	0
$737$	33.0908	1.21891
$738$	0	0
$739$	12.8841	0.473950	0.236975	−	0.971516i	$-0.423844\pi$
$739$	12.8841	0.473950	0.236975	+	0.971516i	$0.423844\pi$
$740$	0	0
$741$	−55.9043	−2.05370
$742$	0	0
$743$	17.3010	0.634711	0.317356	−	0.948307i	$-0.397205\pi$
$743$	17.3010	0.634711	0.317356	+	0.948307i	$0.397205\pi$
$744$	0	0
$745$	−5.79738	−0.212400
$746$	0	0
$747$	5.18537	0.189723
$748$	0	0
$749$	5.73958	0.209720
$750$	0	0
$751$	−21.8108	−0.795888	−0.397944	−	0.917410i	$-0.630276\pi$
$751$	−21.8108	−0.795888	−0.397944	+	0.917410i	$0.630276\pi$
$752$	0	0
$753$	3.01325	0.109809
$754$	0	0
$755$	−9.51371	−0.346239
$756$	0	0
$757$	−32.9853	−1.19887	−0.599435	−	0.800423i	$-0.704608\pi$
$757$	−32.9853	−1.19887	−0.599435	+	0.800423i	$0.704608\pi$
$758$	0	0
$759$	−103.884	−3.77076
$760$	0	0
$761$	−11.0105	−0.399129	−0.199564	−	0.979885i	$-0.563953\pi$
$761$	−11.0105	−0.399129	−0.199564	+	0.979885i	$0.563953\pi$
$762$	0	0
$763$	−10.5938	−0.383522
$764$	0	0
$765$	−20.1601	−0.728890
$766$	0	0
$767$	54.5082	1.96818
$768$	0	0
$769$	24.0723	0.868071	0.434036	−	0.900896i	$-0.357089\pi$
$769$	24.0723	0.868071	0.434036	+	0.900896i	$0.357089\pi$
$770$	0	0
$771$	29.1678	1.05045
$772$	0	0
$773$	−40.9623	−1.47331	−0.736656	−	0.676267i	$-0.763597\pi$
$773$	−40.9623	−1.47331	−0.736656	+	0.676267i	$0.763597\pi$
$774$	0	0
$775$	30.6664	1.10157
$776$	0	0
$777$	10.0225	0.359555
$778$	0	0
$779$	35.1880	1.26074
$780$	0	0
$781$	−32.2173	−1.15283
$782$	0	0
$783$	−39.6867	−1.41829
$784$	0	0
$785$	0.620536	0.0221479
$786$	0	0
$787$	10.8766	0.387709	0.193854	−	0.981030i	$-0.437901\pi$
$787$	10.8766	0.387709	0.193854	+	0.981030i	$0.437901\pi$
$788$	0	0
$789$	−48.3991	−1.72305
$790$	0	0
$791$	5.53133	0.196671
$792$	0	0
$793$	2.30459	0.0818382
$794$	0	0
$795$	3.02099	0.107144
$796$	0	0
$797$	24.9631	0.884239	0.442119	−	0.896956i	$-0.354227\pi$
$797$	24.9631	0.884239	0.442119	+	0.896956i	$0.354227\pi$
$798$	0	0
$799$	−96.3957	−3.41023
$800$	0	0
$801$	−66.1748	−2.33817
$802$	0	0
$803$	18.7750	0.662555
$804$	0	0
$805$	−3.00524	−0.105921
$806$	0	0
$807$	11.9743	0.421515
$808$	0	0
$809$	56.2947	1.97922	0.989608	−	0.143788i	$-0.0459284\pi$
$809$	56.2947	1.97922	0.989608	+	0.143788i	$0.0459284\pi$
$810$	0	0
$811$	2.74348	0.0963364	0.0481682	−	0.998839i	$-0.484662\pi$
$811$	2.74348	0.0963364	0.0481682	+	0.998839i	$0.484662\pi$
$812$	0	0
$813$	76.4193	2.68014
$814$	0	0
$815$	7.21576	0.252757
$816$	0	0
$817$	−34.2535	−1.19838
$818$	0	0
$819$	24.6348	0.860809
$820$	0	0
$821$	28.4602	0.993267	0.496633	−	0.867960i	$-0.334569\pi$
$821$	28.4602	0.993267	0.496633	+	0.867960i	$0.334569\pi$
$822$	0	0
$823$	34.6651	1.20835	0.604174	−	0.796852i	$-0.293503\pi$
$823$	34.6651	1.20835	0.604174	+	0.796852i	$0.293503\pi$
$824$	0	0
$825$	65.0806	2.26581
$826$	0	0
$827$	−17.7336	−0.616659	−0.308329	−	0.951280i	$-0.599770\pi$
$827$	−17.7336	−0.616659	−0.308329	+	0.951280i	$0.599770\pi$
$828$	0	0
$829$	−44.6199	−1.54971	−0.774857	−	0.632136i	$-0.782178\pi$
$829$	−44.6199	−1.54971	−0.774857	+	0.632136i	$0.782178\pi$
$830$	0	0
$831$	−34.9438	−1.21219
$832$	0	0
$833$	48.6058	1.68409
$834$	0	0
$835$	−6.43074	−0.222545
$836$	0	0
$837$	58.9782	2.03858
$838$	0	0
$839$	33.7745	1.16602	0.583012	−	0.812463i	$-0.301874\pi$
$839$	33.7745	1.16602	0.583012	+	0.812463i	$0.301874\pi$
$840$	0	0
$841$	−10.7101	−0.369313
$842$	0	0
$843$	−22.2429	−0.766086
$844$	0	0
$845$	2.44904	0.0842495
$846$	0	0
$847$	−8.43162	−0.289714
$848$	0	0
$849$	−7.96136	−0.273233
$850$	0	0
$851$	27.4546	0.941132
$852$	0	0
$853$	54.5577	1.86802	0.934010	−	0.357248i	$-0.116285\pi$
$853$	54.5577	1.86802	0.934010	+	0.357248i	$0.116285\pi$
$854$	0	0
$855$	−10.8615	−0.371456
$856$	0	0
$857$	−16.4710	−0.562638	−0.281319	−	0.959614i	$-0.590772\pi$
$857$	−16.4710	−0.562638	−0.281319	+	0.959614i	$0.590772\pi$
$858$	0	0
$859$	34.6805	1.18328	0.591642	−	0.806201i	$-0.298480\pi$
$859$	34.6805	1.18328	0.591642	+	0.806201i	$0.298480\pi$
$860$	0	0
$861$	−23.1573	−0.789199
$862$	0	0
$863$	−1.67325	−0.0569582	−0.0284791	−	0.999594i	$-0.509066\pi$
$863$	−1.67325	−0.0569582	−0.0284791	+	0.999594i	$0.509066\pi$
$864$	0	0
$865$	−0.592394	−0.0201420
$866$	0	0
$867$	−138.267	−4.69578
$868$	0	0
$869$	−39.8178	−1.35073
$870$	0	0
$871$	−32.1022	−1.08774
$872$	0	0
$873$	104.388	3.53299
$874$	0	0
$875$	3.83362	0.129600
$876$	0	0
$877$	2.47346	0.0835227	0.0417613	−	0.999128i	$-0.486703\pi$
$877$	2.47346	0.0835227	0.0417613	+	0.999128i	$0.486703\pi$
$878$	0	0
$879$	−64.5728	−2.17798
$880$	0	0
$881$	−16.2844	−0.548637	−0.274318	−	0.961639i	$-0.588452\pi$
$881$	−16.2844	−0.548637	−0.274318	+	0.961639i	$0.588452\pi$
$882$	0	0
$883$	4.68697	0.157729	0.0788646	−	0.996885i	$-0.474871\pi$
$883$	4.68697	0.157729	0.0788646	+	0.996885i	$0.474871\pi$
$884$	0	0
$885$	15.8160	0.531650
$886$	0	0
$887$	−38.1078	−1.27953	−0.639767	−	0.768569i	$-0.720969\pi$
$887$	−38.1078	−1.27953	−0.639767	+	0.768569i	$0.720969\pi$
$888$	0	0
$889$	8.09935	0.271644
$890$	0	0
$891$	43.5233	1.45809
$892$	0	0
$893$	−51.9345	−1.73792
$894$	0	0
$895$	−6.75938	−0.225941
$896$	0	0
$897$	100.781	3.36498
$898$	0	0
$899$	−27.1806	−0.906522
$900$	0	0
$901$	19.0133	0.633425
$902$	0	0
$903$	22.5423	0.750162
$904$	0	0
$905$	−5.97898	−0.198748
$906$	0	0
$907$	6.44138	0.213882	0.106941	−	0.994265i	$-0.465894\pi$
$907$	6.44138	0.213882	0.106941	+	0.994265i	$0.465894\pi$
$908$	0	0
$909$	43.4472	1.44105
$910$	0	0
$911$	−15.5652	−0.515698	−0.257849	−	0.966185i	$-0.583014\pi$
$911$	−15.5652	−0.515698	−0.257849	+	0.966185i	$0.583014\pi$
$912$	0	0
$913$	3.81773	0.126349
$914$	0	0
$915$	0.668696	0.0221064
$916$	0	0
$917$	−20.6042	−0.680410
$918$	0	0
$919$	−46.6490	−1.53881	−0.769405	−	0.638761i	$-0.779447\pi$
$919$	−46.6490	−1.53881	−0.769405	+	0.638761i	$0.779447\pi$
$920$	0	0
$921$	1.20260	0.0396271
$922$	0	0
$923$	31.2549	1.02877
$924$	0	0
$925$	−17.1995	−0.565517
$926$	0	0
$927$	94.4671	3.10271
$928$	0	0
$929$	1.42255	0.0466723	0.0233361	−	0.999728i	$-0.492571\pi$
$929$	1.42255	0.0466723	0.0233361	+	0.999728i	$0.492571\pi$
$930$	0	0
$931$	26.1871	0.858246
$932$	0	0
$933$	−76.7445	−2.51250
$934$	0	0
$935$	−14.8429	−0.485414
$936$	0	0
$937$	−9.44143	−0.308438	−0.154219	−	0.988037i	$-0.549286\pi$
$937$	−9.44143	−0.308438	−0.154219	+	0.988037i	$0.549286\pi$
$938$	0	0
$939$	1.64138	0.0535643
$940$	0	0
$941$	−21.3994	−0.697599	−0.348800	−	0.937197i	$-0.613411\pi$
$941$	−21.3994	−0.697599	−0.348800	+	0.937197i	$0.613411\pi$
$942$	0	0
$943$	−63.4347	−2.06572
$944$	0	0
$945$	3.62083	0.117786
$946$	0	0
$947$	51.7696	1.68228	0.841142	−	0.540814i	$-0.181884\pi$
$947$	51.7696	1.68228	0.841142	+	0.540814i	$0.181884\pi$
$948$	0	0
$949$	−18.2141	−0.591255
$950$	0	0
$951$	−92.0208	−2.98398
$952$	0	0
$953$	−13.1552	−0.426138	−0.213069	−	0.977037i	$-0.568346\pi$
$953$	−13.1552	−0.426138	−0.213069	+	0.977037i	$0.568346\pi$
$954$	0	0
$955$	0.0510751	0.00165275
$956$	0	0
$957$	−57.6828	−1.86462
$958$	0	0
$959$	12.2106	0.394302
$960$	0	0
$961$	9.39288	0.302996
$962$	0	0
$963$	−37.3960	−1.20507
$964$	0	0
$965$	−6.66718	−0.214624
$966$	0	0
$967$	16.3834	0.526855	0.263427	−	0.964679i	$-0.415147\pi$
$967$	16.3834	0.526855	0.263427	+	0.964679i	$0.415147\pi$
$968$	0	0
$969$	−102.091	−3.27964
$970$	0	0
$971$	10.6149	0.340647	0.170324	−	0.985388i	$-0.445519\pi$
$971$	10.6149	0.340647	0.170324	+	0.985388i	$0.445519\pi$
$972$	0	0
$973$	−2.13037	−0.0682965
$974$	0	0
$975$	−63.1364	−2.02198
$976$	0	0
$977$	−59.5917	−1.90651	−0.953254	−	0.302171i	$-0.902289\pi$
$977$	−59.5917	−1.90651	−0.953254	+	0.302171i	$0.902289\pi$
$978$	0	0
$979$	−48.7212	−1.55714
$980$	0	0
$981$	69.0237	2.20376
$982$	0	0
$983$	−31.1275	−0.992814	−0.496407	−	0.868090i	$-0.665348\pi$
$983$	−31.1275	−0.992814	−0.496407	+	0.868090i	$0.665348\pi$
$984$	0	0
$985$	−8.47112	−0.269912
$986$	0	0
$987$	34.1782	1.08791
$988$	0	0
$989$	61.7502	1.96354
$990$	0	0
$991$	53.5834	1.70213	0.851066	−	0.525059i	$-0.175957\pi$
$991$	53.5834	1.70213	0.851066	+	0.525059i	$0.175957\pi$
$992$	0	0
$993$	−3.00715	−0.0954290
$994$	0	0
$995$	1.81017	0.0573862
$996$	0	0
$997$	24.8000	0.785424	0.392712	−	0.919662i	$-0.371537\pi$
$997$	24.8000	0.785424	0.392712	+	0.919662i	$0.371537\pi$
$998$	0	0
$999$	−33.0784	−1.04656

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.j.1.1	yes	30
4.3	odd	2		8032.2.a.g.1.30	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.30	✓	30	4.3	odd	2
8032.2.a.j.1.1	yes	30	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:	\( N \)	\(=\)	\( 8032 = 2^{5} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8032.a (trivial)

\(f(q)\)	\(=\)	\(q-3.01325 q^{3} +0.418153 q^{5} -0.933114 q^{7} +6.07967 q^{9} +O(q^{10})\)	\(q-3.01325 q^{3} +0.418153 q^{5} -0.933114 q^{7} +6.07967 q^{9} +4.47616 q^{11} -4.34244 q^{13} -1.26000 q^{15} -7.93008 q^{17} -4.27244 q^{19} +2.81171 q^{21} +7.70210 q^{23} -4.82515 q^{25} -9.27981 q^{27} +4.27667 q^{29} -6.35554 q^{31} -13.4878 q^{33} -0.390184 q^{35} +3.56456 q^{37} +13.0849 q^{39} -8.23603 q^{41} +8.01732 q^{43} +2.54223 q^{45} +12.1557 q^{47} -6.12930 q^{49} +23.8953 q^{51} -2.39762 q^{53} +1.87172 q^{55} +12.8739 q^{57} -12.5524 q^{59} -0.530712 q^{61} -5.67302 q^{63} -1.81580 q^{65} +7.39267 q^{67} -23.2083 q^{69} -7.19753 q^{71} +4.19444 q^{73} +14.5394 q^{75} -4.17677 q^{77} -8.89554 q^{79} +9.72337 q^{81} +0.852904 q^{83} -3.31599 q^{85} -12.8867 q^{87} -10.8846 q^{89} +4.05199 q^{91} +19.1508 q^{93} -1.78653 q^{95} +17.1700 q^{97} +27.2136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9	\( 30 q + 9 q^{3} + 5 q^{7} + 35 q^{9} + 31 q^{11} + 5 q^{13} + 8 q^{15} - q^{17} + 29 q^{19} + 6 q^{21} + 27 q^{23} + 34 q^{25} + 36 q^{27} + q^{29} - 9 q^{31} - 8 q^{33} + 51 q^{35} + 5 q^{37} - 8 q^{39} - 3 q^{41} + 43 q^{43} + 42 q^{47} + 41 q^{49} + 54 q^{51} - 11 q^{53} - 10 q^{55} + 2 q^{57} + 49 q^{59} + 21 q^{61} + 15 q^{63} - 14 q^{65} + 68 q^{67} - 10 q^{69} + 44 q^{71} + 25 q^{73} + 51 q^{75} - 20 q^{77} - 49 q^{79} + 6 q^{81} + 88 q^{83} - 5 q^{85} + 16 q^{87} - 16 q^{89} + 46 q^{91} - 4 q^{93} + 28 q^{95} - 4 q^{97} + 71 q^{99}+O(q^{100}) \) 30 * q + 9 * q^3 + 5 * q^7 + 35 * q^9 + 31 * q^11 + 5 * q^13 + 8 * q^15 - q^17 + 29 * q^19 + 6 * q^21 + 27 * q^23 + 34 * q^25 + 36 * q^27 + q^29 - 9 * q^31 - 8 * q^33 + 51 * q^35 + 5 * q^37 - 8 * q^39 - 3 * q^41 + 43 * q^43 + 42 * q^47 + 41 * q^49 + 54 * q^51 - 11 * q^53 - 10 * q^55 + 2 * q^57 + 49 * q^59 + 21 * q^61 + 15 * q^63 - 14 * q^65 + 68 * q^67 - 10 * q^69 + 44 * q^71 + 25 * q^73 + 51 * q^75 - 20 * q^77 - 49 * q^79 + 6 * q^81 + 88 * q^83 - 5 * q^85 + 16 * q^87 - 16 * q^89 + 46 * q^91 - 4 * q^93 + 28 * q^95 - 4 * q^97 + 71 * q^99

Introduction

L-functions

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