LMFDB - Embedded newform 8032.2.a.i.1.17

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			1.17
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)

$f(q)$	$=$	$q+0.679540 q^{3} -1.36350 q^{5} -2.99119 q^{7} -2.53823 q^{9} +O(q^{10})$	$q+0.679540 q^{3} -1.36350 q^{5} -2.99119 q^{7} -2.53823 q^{9} -4.20862 q^{11} +1.86960 q^{13} -0.926556 q^{15} +0.227597 q^{17} +3.54048 q^{19} -2.03263 q^{21} -2.07033 q^{23} -3.14086 q^{25} -3.76345 q^{27} -4.12050 q^{29} +2.59931 q^{31} -2.85993 q^{33} +4.07850 q^{35} -4.32332 q^{37} +1.27047 q^{39} -6.83717 q^{41} +3.66256 q^{43} +3.46088 q^{45} -7.50454 q^{47} +1.94722 q^{49} +0.154661 q^{51} +7.77747 q^{53} +5.73847 q^{55} +2.40590 q^{57} -3.60379 q^{59} -4.86472 q^{61} +7.59231 q^{63} -2.54921 q^{65} -7.83449 q^{67} -1.40687 q^{69} +15.5925 q^{71} -14.6380 q^{73} -2.13434 q^{75} +12.5888 q^{77} +2.78478 q^{79} +5.05726 q^{81} +3.68148 q^{83} -0.310329 q^{85} -2.80004 q^{87} -0.715053 q^{89} -5.59234 q^{91} +1.76634 q^{93} -4.82746 q^{95} -0.611218 q^{97} +10.6824 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	$ 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) $ 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * 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
$3$	0.679540	0.392333	0.196166	−	0.980571i	$-0.437151\pi$
$3$	0.679540	0.392333	0.196166	+	0.980571i	$0.437151\pi$
$4$	0	0
$5$	−1.36350	−0.609778	−0.304889	−	0.952388i	$-0.598619\pi$
$5$	−1.36350	−0.609778	−0.304889	+	0.952388i	$0.598619\pi$
$6$	0	0
$7$	−2.99119	−1.13056	−0.565282	−	0.824898i	$-0.691233\pi$
$7$	−2.99119	−1.13056	−0.565282	+	0.824898i	$0.691233\pi$
$8$	0	0
$9$	−2.53823	−0.846075
$10$	0	0
$11$	−4.20862	−1.26895	−0.634473	−	0.772945i	$-0.718783\pi$
$11$	−4.20862	−1.26895	−0.634473	+	0.772945i	$0.718783\pi$
$12$	0	0
$13$	1.86960	0.518535	0.259267	−	0.965806i	$-0.416519\pi$
$13$	1.86960	0.518535	0.259267	+	0.965806i	$0.416519\pi$
$14$	0	0
$15$	−0.926556	−0.239236
$16$	0	0
$17$	0.227597	0.0552003	0.0276002	−	0.999619i	$-0.491213\pi$
$17$	0.227597	0.0552003	0.0276002	+	0.999619i	$0.491213\pi$
$18$	0	0
$19$	3.54048	0.812242	0.406121	−	0.913819i	$-0.366881\pi$
$19$	3.54048	0.812242	0.406121	+	0.913819i	$0.366881\pi$
$20$	0	0
$21$	−2.03263	−0.443557
$22$	0	0
$23$	−2.07033	−0.431693	−0.215847	−	0.976427i	$-0.569251\pi$
$23$	−2.07033	−0.431693	−0.215847	+	0.976427i	$0.569251\pi$
$24$	0	0
$25$	−3.14086	−0.628171
$26$	0	0
$27$	−3.76345	−0.724276
$28$	0	0
$29$	−4.12050	−0.765157	−0.382578	−	0.923923i	$-0.624964\pi$
$29$	−4.12050	−0.765157	−0.382578	+	0.923923i	$0.624964\pi$
$30$	0	0
$31$	2.59931	0.466850	0.233425	−	0.972375i	$-0.425007\pi$
$31$	2.59931	0.466850	0.233425	+	0.972375i	$0.425007\pi$
$32$	0	0
$33$	−2.85993	−0.497849
$34$	0	0
$35$	4.07850	0.689392
$36$	0	0
$37$	−4.32332	−0.710749	−0.355375	−	0.934724i	$-0.615647\pi$
$37$	−4.32332	−0.710749	−0.355375	+	0.934724i	$0.615647\pi$
$38$	0	0
$39$	1.27047	0.203438
$40$	0	0
$41$	−6.83717	−1.06779	−0.533893	−	0.845552i	$-0.679271\pi$
$41$	−6.83717	−1.06779	−0.533893	+	0.845552i	$0.679271\pi$
$42$	0	0
$43$	3.66256	0.558536	0.279268	−	0.960213i	$-0.409908\pi$
$43$	3.66256	0.558536	0.279268	+	0.960213i	$0.409908\pi$
$44$	0	0
$45$	3.46088	0.515918
$46$	0	0
$47$	−7.50454	−1.09465	−0.547325	−	0.836920i	$-0.684354\pi$
$47$	−7.50454	−1.09465	−0.547325	+	0.836920i	$0.684354\pi$
$48$	0	0
$49$	1.94722	0.278174
$50$	0	0
$51$	0.154661	0.0216569
$52$	0	0
$53$	7.77747	1.06832	0.534159	−	0.845384i	$-0.320628\pi$
$53$	7.77747	1.06832	0.534159	+	0.845384i	$0.320628\pi$
$54$	0	0
$55$	5.73847	0.773775
$56$	0	0
$57$	2.40590	0.318669
$58$	0	0
$59$	−3.60379	−0.469174	−0.234587	−	0.972095i	$-0.575374\pi$
$59$	−3.60379	−0.469174	−0.234587	+	0.972095i	$0.575374\pi$
$60$	0	0
$61$	−4.86472	−0.622864	−0.311432	−	0.950269i	$-0.600809\pi$
$61$	−4.86472	−0.622864	−0.311432	+	0.950269i	$0.600809\pi$
$62$	0	0
$63$	7.59231	0.956542
$64$	0	0
$65$	−2.54921	−0.316191
$66$	0	0
$67$	−7.83449	−0.957136	−0.478568	−	0.878051i	$-0.658844\pi$
$67$	−7.83449	−0.957136	−0.478568	+	0.878051i	$0.658844\pi$
$68$	0	0
$69$	−1.40687	−0.169367
$70$	0	0
$71$	15.5925	1.85049	0.925244	−	0.379372i	$-0.123860\pi$
$71$	15.5925	1.85049	0.925244	+	0.379372i	$0.123860\pi$
$72$	0	0
$73$	−14.6380	−1.71325	−0.856623	−	0.515943i	$-0.827442\pi$
$73$	−14.6380	−1.71325	−0.856623	+	0.515943i	$0.827442\pi$
$74$	0	0
$75$	−2.13434	−0.246452
$76$	0	0
$77$	12.5888	1.43463
$78$	0	0
$79$	2.78478	0.313312	0.156656	−	0.987653i	$-0.449929\pi$
$79$	2.78478	0.313312	0.156656	+	0.987653i	$0.449929\pi$
$80$	0	0
$81$	5.05726	0.561918
$82$	0	0
$83$	3.68148	0.404095	0.202047	−	0.979376i	$-0.435241\pi$
$83$	3.68148	0.404095	0.202047	+	0.979376i	$0.435241\pi$
$84$	0	0
$85$	−0.310329	−0.0336599
$86$	0	0
$87$	−2.80004	−0.300196
$88$	0	0
$89$	−0.715053	−0.0757954	−0.0378977	−	0.999282i	$-0.512066\pi$
$89$	−0.715053	−0.0757954	−0.0378977	+	0.999282i	$0.512066\pi$
$90$	0	0
$91$	−5.59234	−0.586237
$92$	0	0
$93$	1.76634	0.183161
$94$	0	0
$95$	−4.82746	−0.495287
$96$	0	0
$97$	−0.611218	−0.0620598	−0.0310299	−	0.999518i	$-0.509879\pi$
$97$	−0.611218	−0.0620598	−0.0310299	+	0.999518i	$0.509879\pi$
$98$	0	0
$99$	10.6824	1.07362
$100$	0	0
$101$	−4.78674	−0.476299	−0.238149	−	0.971229i	$-0.576541\pi$
$101$	−4.78674	−0.476299	−0.238149	+	0.971229i	$0.576541\pi$
$102$	0	0
$103$	−6.52719	−0.643143	−0.321572	−	0.946885i	$-0.604211\pi$
$103$	−6.52719	−0.643143	−0.321572	+	0.946885i	$0.604211\pi$
$104$	0	0
$105$	2.77151	0.270471
$106$	0	0
$107$	8.75917	0.846781	0.423391	−	0.905947i	$-0.360840\pi$
$107$	8.75917	0.846781	0.423391	+	0.905947i	$0.360840\pi$
$108$	0	0
$109$	2.55359	0.244590	0.122295	−	0.992494i	$-0.460975\pi$
$109$	2.55359	0.244590	0.122295	+	0.992494i	$0.460975\pi$
$110$	0	0
$111$	−2.93787	−0.278850
$112$	0	0
$113$	3.75811	0.353533	0.176767	−	0.984253i	$-0.443436\pi$
$113$	3.75811	0.353533	0.176767	+	0.984253i	$0.443436\pi$
$114$	0	0
$115$	2.82290	0.263237
$116$	0	0
$117$	−4.74548	−0.438719
$118$	0	0
$119$	−0.680785	−0.0624075
$120$	0	0
$121$	6.71249	0.610226
$122$	0	0
$123$	−4.64613	−0.418927
$124$	0	0
$125$	11.1001	0.992822
$126$	0	0
$127$	−8.60626	−0.763682	−0.381841	−	0.924228i	$-0.624710\pi$
$127$	−8.60626	−0.763682	−0.381841	+	0.924228i	$0.624710\pi$
$128$	0	0
$129$	2.48886	0.219132
$130$	0	0
$131$	1.96398	0.171594	0.0857968	−	0.996313i	$-0.472656\pi$
$131$	1.96398	0.171594	0.0857968	+	0.996313i	$0.472656\pi$
$132$	0	0
$133$	−10.5903	−0.918291
$134$	0	0
$135$	5.13148	0.441647
$136$	0	0
$137$	22.5122	1.92335	0.961674	−	0.274195i	$-0.0884113\pi$
$137$	22.5122	1.92335	0.961674	+	0.274195i	$0.0884113\pi$
$138$	0	0
$139$	−0.338030	−0.0286713	−0.0143357	−	0.999897i	$-0.504563\pi$
$139$	−0.338030	−0.0286713	−0.0143357	+	0.999897i	$0.504563\pi$
$140$	0	0
$141$	−5.09964	−0.429467
$142$	0	0
$143$	−7.86845	−0.657993
$144$	0	0
$145$	5.61831	0.466576
$146$	0	0
$147$	1.32321	0.109137
$148$	0	0
$149$	−16.3561	−1.33995	−0.669973	−	0.742385i	$-0.733695\pi$
$149$	−16.3561	−1.33995	−0.669973	+	0.742385i	$0.733695\pi$
$150$	0	0
$151$	6.69657	0.544959	0.272479	−	0.962162i	$-0.412156\pi$
$151$	6.69657	0.544959	0.272479	+	0.962162i	$0.412156\pi$
$152$	0	0
$153$	−0.577692	−0.0467036
$154$	0	0
$155$	−3.54417	−0.284675
$156$	0	0
$157$	21.6728	1.72968	0.864840	−	0.502047i	$-0.167420\pi$
$157$	21.6728	1.72968	0.864840	+	0.502047i	$0.167420\pi$
$158$	0	0
$159$	5.28510	0.419136
$160$	0	0
$161$	6.19275	0.488057
$162$	0	0
$163$	−10.1372	−0.794003	−0.397002	−	0.917818i	$-0.629949\pi$
$163$	−10.1372	−0.794003	−0.397002	+	0.917818i	$0.629949\pi$
$164$	0	0
$165$	3.89952	0.303577
$166$	0	0
$167$	21.1348	1.63546	0.817731	−	0.575600i	$-0.195232\pi$
$167$	21.1348	1.63546	0.817731	+	0.575600i	$0.195232\pi$
$168$	0	0
$169$	−9.50458	−0.731122
$170$	0	0
$171$	−8.98654	−0.687218
$172$	0	0
$173$	15.3989	1.17076	0.585380	−	0.810759i	$-0.300945\pi$
$173$	15.3989	1.17076	0.585380	+	0.810759i	$0.300945\pi$
$174$	0	0
$175$	9.39490	0.710188
$176$	0	0
$177$	−2.44892	−0.184072
$178$	0	0
$179$	3.01859	0.225620	0.112810	−	0.993617i	$-0.464015\pi$
$179$	3.01859	0.225620	0.112810	+	0.993617i	$0.464015\pi$
$180$	0	0
$181$	3.68158	0.273650	0.136825	−	0.990595i	$-0.456310\pi$
$181$	3.68158	0.273650	0.136825	+	0.990595i	$0.456310\pi$
$182$	0	0
$183$	−3.30577	−0.244370
$184$	0	0
$185$	5.89487	0.433399
$186$	0	0
$187$	−0.957868	−0.0700463
$188$	0	0
$189$	11.2572	0.818840
$190$	0	0
$191$	24.1589	1.74808	0.874040	−	0.485854i	$-0.161491\pi$
$191$	24.1589	1.74808	0.874040	+	0.485854i	$0.161491\pi$
$192$	0	0
$193$	−14.0374	−1.01043	−0.505216	−	0.862993i	$-0.668587\pi$
$193$	−14.0374	−1.01043	−0.505216	+	0.862993i	$0.668587\pi$
$194$	0	0
$195$	−1.73229	−0.124052
$196$	0	0
$197$	14.7083	1.04793	0.523963	−	0.851741i	$-0.324453\pi$
$197$	14.7083	1.04793	0.523963	+	0.851741i	$0.324453\pi$
$198$	0	0
$199$	3.25012	0.230395	0.115197	−	0.993343i	$-0.463250\pi$
$199$	3.25012	0.230395	0.115197	+	0.993343i	$0.463250\pi$
$200$	0	0
$201$	−5.32385	−0.375516
$202$	0	0
$203$	12.3252	0.865059
$204$	0	0
$205$	9.32251	0.651112
$206$	0	0
$207$	5.25496	0.365245
$208$	0	0
$209$	−14.9005	−1.03069
$210$	0	0
$211$	−7.19360	−0.495228	−0.247614	−	0.968859i	$-0.579646\pi$
$211$	−7.19360	−0.495228	−0.247614	+	0.968859i	$0.579646\pi$
$212$	0	0
$213$	10.5957	0.726007
$214$	0	0
$215$	−4.99392	−0.340583
$216$	0	0
$217$	−7.77503	−0.527804
$218$	0	0
$219$	−9.94710	−0.672162
$220$	0	0
$221$	0.425516	0.0286233
$222$	0	0
$223$	18.6736	1.25048	0.625239	−	0.780433i	$-0.285002\pi$
$223$	18.6736	1.25048	0.625239	+	0.780433i	$0.285002\pi$
$224$	0	0
$225$	7.97220	0.531480
$226$	0	0
$227$	22.5440	1.49630	0.748148	−	0.663531i	$-0.230943\pi$
$227$	22.5440	1.49630	0.748148	+	0.663531i	$0.230943\pi$
$228$	0	0
$229$	−3.97996	−0.263003	−0.131502	−	0.991316i	$-0.541980\pi$
$229$	−3.97996	−0.263003	−0.131502	+	0.991316i	$0.541980\pi$
$230$	0	0
$231$	8.55459	0.562850
$232$	0	0
$233$	−13.3681	−0.875773	−0.437886	−	0.899030i	$-0.644273\pi$
$233$	−13.3681	−0.875773	−0.437886	+	0.899030i	$0.644273\pi$
$234$	0	0
$235$	10.2325	0.667493
$236$	0	0
$237$	1.89237	0.122923
$238$	0	0
$239$	26.7495	1.73028	0.865140	−	0.501530i	$-0.167229\pi$
$239$	26.7495	1.73028	0.865140	+	0.501530i	$0.167229\pi$
$240$	0	0
$241$	14.5753	0.938877	0.469438	−	0.882965i	$-0.344456\pi$
$241$	14.5753	0.938877	0.469438	+	0.882965i	$0.344456\pi$
$242$	0	0
$243$	14.7270	0.944734
$244$	0	0
$245$	−2.65504	−0.169624
$246$	0	0
$247$	6.61930	0.421176
$248$	0	0
$249$	2.50171	0.158540
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	8.71323	0.547796
$254$	0	0
$255$	−0.210881	−0.0132059
$256$	0	0
$257$	11.3840	0.710112	0.355056	−	0.934845i	$-0.384462\pi$
$257$	11.3840	0.710112	0.355056	+	0.934845i	$0.384462\pi$
$258$	0	0
$259$	12.9319	0.803548
$260$	0	0
$261$	10.4587	0.647380
$262$	0	0
$263$	−0.792727	−0.0488817	−0.0244408	−	0.999701i	$-0.507781\pi$
$263$	−0.792727	−0.0488817	−0.0244408	+	0.999701i	$0.507781\pi$
$264$	0	0
$265$	−10.6046	−0.651436
$266$	0	0
$267$	−0.485907	−0.0297370
$268$	0	0
$269$	3.65585	0.222901	0.111451	−	0.993770i	$-0.464450\pi$
$269$	3.65585	0.222901	0.111451	+	0.993770i	$0.464450\pi$
$270$	0	0
$271$	5.58681	0.339375	0.169687	−	0.985498i	$-0.445724\pi$
$271$	5.58681	0.339375	0.169687	+	0.985498i	$0.445724\pi$
$272$	0	0
$273$	−3.80022	−0.230000
$274$	0	0
$275$	13.2187	0.797116
$276$	0	0
$277$	8.90053	0.534781	0.267390	−	0.963588i	$-0.413839\pi$
$277$	8.90053	0.534781	0.267390	+	0.963588i	$0.413839\pi$
$278$	0	0
$279$	−6.59764	−0.394990
$280$	0	0
$281$	2.00303	0.119490	0.0597452	−	0.998214i	$-0.480971\pi$
$281$	2.00303	0.119490	0.0597452	+	0.998214i	$0.480971\pi$
$282$	0	0
$283$	−27.9941	−1.66408	−0.832039	−	0.554717i	$-0.812827\pi$
$283$	−27.9941	−1.66408	−0.832039	+	0.554717i	$0.812827\pi$
$284$	0	0
$285$	−3.28045	−0.194317
$286$	0	0
$287$	20.4513	1.20720
$288$	0	0
$289$	−16.9482	−0.996953
$290$	0	0
$291$	−0.415347	−0.0243481
$292$	0	0
$293$	−24.5569	−1.43463	−0.717316	−	0.696748i	$-0.754629\pi$
$293$	−24.5569	−1.43463	−0.717316	+	0.696748i	$0.754629\pi$
$294$	0	0
$295$	4.91378	0.286092
$296$	0	0
$297$	15.8389	0.919067
$298$	0	0
$299$	−3.87069	−0.223848
$300$	0	0
$301$	−10.9554	−0.631460
$302$	0	0
$303$	−3.25279	−0.186868
$304$	0	0
$305$	6.63307	0.379808
$306$	0	0
$307$	15.2155	0.868393	0.434197	−	0.900818i	$-0.357032\pi$
$307$	15.2155	0.868393	0.434197	+	0.900818i	$0.357032\pi$
$308$	0	0
$309$	−4.43549	−0.252326
$310$	0	0
$311$	19.1508	1.08594	0.542971	−	0.839751i	$-0.317299\pi$
$311$	19.1508	1.08594	0.542971	+	0.839751i	$0.317299\pi$
$312$	0	0
$313$	−6.62198	−0.374296	−0.187148	−	0.982332i	$-0.559924\pi$
$313$	−6.62198	−0.374296	−0.187148	+	0.982332i	$0.559924\pi$
$314$	0	0
$315$	−10.3522	−0.583278
$316$	0	0
$317$	−9.74674	−0.547432	−0.273716	−	0.961811i	$-0.588253\pi$
$317$	−9.74674	−0.547432	−0.273716	+	0.961811i	$0.588253\pi$
$318$	0	0
$319$	17.3416	0.970943
$320$	0	0
$321$	5.95221	0.332220
$322$	0	0
$323$	0.805802	0.0448360
$324$	0	0
$325$	−5.87216	−0.325729
$326$	0	0
$327$	1.73527	0.0959605
$328$	0	0
$329$	22.4475	1.23757
$330$	0	0
$331$	15.9147	0.874754	0.437377	−	0.899278i	$-0.355908\pi$
$331$	15.9147	0.874754	0.437377	+	0.899278i	$0.355908\pi$
$332$	0	0
$333$	10.9736	0.601347
$334$	0	0
$335$	10.6824	0.583640
$336$	0	0
$337$	36.5040	1.98850	0.994249	−	0.107097i	$-0.0341554\pi$
$337$	36.5040	1.98850	0.994249	+	0.107097i	$0.0341554\pi$
$338$	0	0
$339$	2.55379	0.138703
$340$	0	0
$341$	−10.9395	−0.592408
$342$	0	0
$343$	15.1138	0.816070
$344$	0	0
$345$	1.91828	0.103276
$346$	0	0
$347$	−21.5077	−1.15459	−0.577296	−	0.816535i	$-0.695892\pi$
$347$	−21.5077	−1.15459	−0.577296	+	0.816535i	$0.695892\pi$
$348$	0	0
$349$	24.1242	1.29134	0.645669	−	0.763618i	$-0.276579\pi$
$349$	24.1242	1.29134	0.645669	+	0.763618i	$0.276579\pi$
$350$	0	0
$351$	−7.03615	−0.375562
$352$	0	0
$353$	−26.8474	−1.42894	−0.714470	−	0.699666i	$-0.753332\pi$
$353$	−26.8474	−1.42894	−0.714470	+	0.699666i	$0.753332\pi$
$354$	0	0
$355$	−21.2604	−1.12839
$356$	0	0
$357$	−0.462621	−0.0244845
$358$	0	0
$359$	18.8590	0.995338	0.497669	−	0.867367i	$-0.334189\pi$
$359$	18.8590	0.995338	0.497669	+	0.867367i	$0.334189\pi$
$360$	0	0
$361$	−6.46499	−0.340263
$362$	0	0
$363$	4.56140	0.239412
$364$	0	0
$365$	19.9589	1.04470
$366$	0	0
$367$	−21.2398	−1.10871	−0.554353	−	0.832282i	$-0.687034\pi$
$367$	−21.2398	−1.10871	−0.554353	+	0.832282i	$0.687034\pi$
$368$	0	0
$369$	17.3543	0.903427
$370$	0	0
$371$	−23.2639	−1.20780
$372$	0	0
$373$	−24.3452	−1.26055	−0.630273	−	0.776374i	$-0.717057\pi$
$373$	−24.3452	−1.26055	−0.630273	+	0.776374i	$0.717057\pi$
$374$	0	0
$375$	7.54296	0.389517
$376$	0	0
$377$	−7.70369	−0.396760
$378$	0	0
$379$	−11.6765	−0.599782	−0.299891	−	0.953973i	$-0.596950\pi$
$379$	−11.6765	−0.599782	−0.299891	+	0.953973i	$0.596950\pi$
$380$	0	0
$381$	−5.84830	−0.299618
$382$	0	0
$383$	11.3577	0.580350	0.290175	−	0.956974i	$-0.406287\pi$
$383$	11.3577	0.580350	0.290175	+	0.956974i	$0.406287\pi$
$384$	0	0
$385$	−17.1649	−0.874802
$386$	0	0
$387$	−9.29641	−0.472563
$388$	0	0
$389$	−33.6374	−1.70548	−0.852742	−	0.522332i	$-0.825062\pi$
$389$	−33.6374	−1.70548	−0.852742	+	0.522332i	$0.825062\pi$
$390$	0	0
$391$	−0.471200	−0.0238296
$392$	0	0
$393$	1.33460	0.0673218
$394$	0	0
$395$	−3.79706	−0.191051
$396$	0	0
$397$	−1.44145	−0.0723442	−0.0361721	−	0.999346i	$-0.511516\pi$
$397$	−1.44145	−0.0723442	−0.0361721	+	0.999346i	$0.511516\pi$
$398$	0	0
$399$	−7.19650	−0.360276
$400$	0	0
$401$	−17.4372	−0.870772	−0.435386	−	0.900244i	$-0.643388\pi$
$401$	−17.4372	−0.870772	−0.435386	+	0.900244i	$0.643388\pi$
$402$	0	0
$403$	4.85968	0.242078
$404$	0	0
$405$	−6.89560	−0.342645
$406$	0	0
$407$	18.1952	0.901903
$408$	0	0
$409$	−12.0268	−0.594687	−0.297343	−	0.954771i	$-0.596101\pi$
$409$	−12.0268	−0.594687	−0.297343	+	0.954771i	$0.596101\pi$
$410$	0	0
$411$	15.2980	0.754592
$412$	0	0
$413$	10.7796	0.530431
$414$	0	0
$415$	−5.01971	−0.246408
$416$	0	0
$417$	−0.229705	−0.0112487
$418$	0	0
$419$	3.31694	0.162043	0.0810215	−	0.996712i	$-0.474182\pi$
$419$	3.31694	0.162043	0.0810215	+	0.996712i	$0.474182\pi$
$420$	0	0
$421$	−5.05446	−0.246339	−0.123170	−	0.992386i	$-0.539306\pi$
$421$	−5.05446	−0.246339	−0.123170	+	0.992386i	$0.539306\pi$
$422$	0	0
$423$	19.0482	0.926156
$424$	0	0
$425$	−0.714849	−0.0346753
$426$	0	0
$427$	14.5513	0.704187
$428$	0	0
$429$	−5.34693	−0.258152
$430$	0	0
$431$	25.2555	1.21651	0.608257	−	0.793740i	$-0.291869\pi$
$431$	25.2555	1.21651	0.608257	+	0.793740i	$0.291869\pi$
$432$	0	0
$433$	−6.14271	−0.295200	−0.147600	−	0.989047i	$-0.547155\pi$
$433$	−6.14271	−0.295200	−0.147600	+	0.989047i	$0.547155\pi$
$434$	0	0
$435$	3.81787	0.183053
$436$	0	0
$437$	−7.32996	−0.350640
$438$	0	0
$439$	−11.3925	−0.543737	−0.271868	−	0.962334i	$-0.587641\pi$
$439$	−11.3925	−0.543737	−0.271868	+	0.962334i	$0.587641\pi$
$440$	0	0
$441$	−4.94248	−0.235356
$442$	0	0
$443$	−11.6474	−0.553385	−0.276693	−	0.960959i	$-0.589238\pi$
$443$	−11.6474	−0.553385	−0.276693	+	0.960959i	$0.589238\pi$
$444$	0	0
$445$	0.974978	0.0462184
$446$	0	0
$447$	−11.1147	−0.525705
$448$	0	0
$449$	−6.07118	−0.286517	−0.143258	−	0.989685i	$-0.545758\pi$
$449$	−6.07118	−0.286517	−0.143258	+	0.989685i	$0.545758\pi$
$450$	0	0
$451$	28.7750	1.35496
$452$	0	0
$453$	4.55059	0.213805
$454$	0	0
$455$	7.62518	0.357474
$456$	0	0
$457$	−6.35362	−0.297210	−0.148605	−	0.988897i	$-0.547478\pi$
$457$	−6.35362	−0.297210	−0.148605	+	0.988897i	$0.547478\pi$
$458$	0	0
$459$	−0.856548	−0.0399802
$460$	0	0
$461$	−27.9227	−1.30049	−0.650246	−	0.759724i	$-0.725334\pi$
$461$	−27.9227	−1.30049	−0.650246	+	0.759724i	$0.725334\pi$
$462$	0	0
$463$	−5.52184	−0.256622	−0.128311	−	0.991734i	$-0.540956\pi$
$463$	−5.52184	−0.256622	−0.128311	+	0.991734i	$0.540956\pi$
$464$	0	0
$465$	−2.40841	−0.111687
$466$	0	0
$467$	19.0946	0.883592	0.441796	−	0.897115i	$-0.354341\pi$
$467$	19.0946	0.883592	0.441796	+	0.897115i	$0.354341\pi$
$468$	0	0
$469$	23.4345	1.08210
$470$	0	0
$471$	14.7276	0.678610
$472$	0	0
$473$	−15.4143	−0.708752
$474$	0	0
$475$	−11.1201	−0.510227
$476$	0	0
$477$	−19.7410	−0.903877
$478$	0	0
$479$	−13.7312	−0.627393	−0.313696	−	0.949523i	$-0.601567\pi$
$479$	−13.7312	−0.627393	−0.313696	+	0.949523i	$0.601567\pi$
$480$	0	0
$481$	−8.08290	−0.368548
$482$	0	0
$483$	4.20822	0.191481
$484$	0	0
$485$	0.833399	0.0378427
$486$	0	0
$487$	20.1790	0.914397	0.457199	−	0.889365i	$-0.348853\pi$
$487$	20.1790	0.914397	0.457199	+	0.889365i	$0.348853\pi$
$488$	0	0
$489$	−6.88860	−0.311513
$490$	0	0
$491$	−10.6305	−0.479747	−0.239873	−	0.970804i	$-0.577106\pi$
$491$	−10.6305	−0.479747	−0.239873	+	0.970804i	$0.577106\pi$
$492$	0	0
$493$	−0.937812	−0.0422369
$494$	0	0
$495$	−14.5655	−0.654672
$496$	0	0
$497$	−46.6401	−2.09210
$498$	0	0
$499$	−6.16748	−0.276094	−0.138047	−	0.990426i	$-0.544083\pi$
$499$	−6.16748	−0.276094	−0.138047	+	0.990426i	$0.544083\pi$
$500$	0	0
$501$	14.3620	0.641645
$502$	0	0
$503$	15.3841	0.685943	0.342972	−	0.939346i	$-0.388567\pi$
$503$	15.3841	0.685943	0.342972	+	0.939346i	$0.388567\pi$
$504$	0	0
$505$	6.52675	0.290436
$506$	0	0
$507$	−6.45875	−0.286843
$508$	0	0
$509$	−20.7461	−0.919554	−0.459777	−	0.888034i	$-0.652071\pi$
$509$	−20.7461	−0.919554	−0.459777	+	0.888034i	$0.652071\pi$
$510$	0	0
$511$	43.7850	1.93693
$512$	0	0
$513$	−13.3244	−0.588287
$514$	0	0
$515$	8.89985	0.392174
$516$	0	0
$517$	31.5838	1.38905
$518$	0	0
$519$	10.4642	0.459327
$520$	0	0
$521$	13.0882	0.573406	0.286703	−	0.958020i	$-0.407441\pi$
$521$	13.0882	0.573406	0.286703	+	0.958020i	$0.407441\pi$
$522$	0	0
$523$	−35.5058	−1.55256	−0.776279	−	0.630389i	$-0.782895\pi$
$523$	−35.5058	−1.55256	−0.776279	+	0.630389i	$0.782895\pi$
$524$	0	0
$525$	6.38421	0.278630
$526$	0	0
$527$	0.591595	0.0257703
$528$	0	0
$529$	−18.7137	−0.813641
$530$	0	0
$531$	9.14723	0.396956
$532$	0	0
$533$	−12.7828	−0.553684
$534$	0	0
$535$	−11.9432	−0.516348
$536$	0	0
$537$	2.05125	0.0885180
$538$	0	0
$539$	−8.19511	−0.352988
$540$	0	0
$541$	−0.598123	−0.0257153	−0.0128577	−	0.999917i	$-0.504093\pi$
$541$	−0.598123	−0.0257153	−0.0128577	+	0.999917i	$0.504093\pi$
$542$	0	0
$543$	2.50178	0.107362
$544$	0	0
$545$	−3.48183	−0.149145
$546$	0	0
$547$	6.85299	0.293013	0.146506	−	0.989210i	$-0.453197\pi$
$547$	6.85299	0.293013	0.146506	+	0.989210i	$0.453197\pi$
$548$	0	0
$549$	12.3478	0.526989
$550$	0	0
$551$	−14.5885	−0.621493
$552$	0	0
$553$	−8.32981	−0.354220
$554$	0	0
$555$	4.00580	0.170037
$556$	0	0
$557$	−23.9715	−1.01571	−0.507853	−	0.861444i	$-0.669561\pi$
$557$	−23.9715	−1.01571	−0.507853	+	0.861444i	$0.669561\pi$
$558$	0	0
$559$	6.84754	0.289620
$560$	0	0
$561$	−0.650910	−0.0274814
$562$	0	0
$563$	−9.17913	−0.386854	−0.193427	−	0.981115i	$-0.561960\pi$
$563$	−9.17913	−0.386854	−0.193427	+	0.981115i	$0.561960\pi$
$564$	0	0
$565$	−5.12420	−0.215577
$566$	0	0
$567$	−15.1272	−0.635284
$568$	0	0
$569$	45.9680	1.92708	0.963540	−	0.267566i	$-0.0862192\pi$
$569$	45.9680	1.92708	0.963540	+	0.267566i	$0.0862192\pi$
$570$	0	0
$571$	−8.73574	−0.365579	−0.182790	−	0.983152i	$-0.558513\pi$
$571$	−8.73574	−0.365579	−0.182790	+	0.983152i	$0.558513\pi$
$572$	0	0
$573$	16.4170	0.685829
$574$	0	0
$575$	6.50260	0.271177
$576$	0	0
$577$	8.25261	0.343561	0.171780	−	0.985135i	$-0.445048\pi$
$577$	8.25261	0.343561	0.171780	+	0.985135i	$0.445048\pi$
$578$	0	0
$579$	−9.53896	−0.396426
$580$	0	0
$581$	−11.0120	−0.456855
$582$	0	0
$583$	−32.7324	−1.35564
$584$	0	0
$585$	6.47048	0.267521
$586$	0	0
$587$	−11.7878	−0.486533	−0.243266	−	0.969960i	$-0.578219\pi$
$587$	−11.7878	−0.486533	−0.243266	+	0.969960i	$0.578219\pi$
$588$	0	0
$589$	9.20281	0.379195
$590$	0	0
$591$	9.99490	0.411135
$592$	0	0
$593$	44.0021	1.80695	0.903476	−	0.428639i	$-0.141007\pi$
$593$	44.0021	1.80695	0.903476	+	0.428639i	$0.141007\pi$
$594$	0	0
$595$	0.928254	0.0380547
$596$	0	0
$597$	2.20858	0.0903914
$598$	0	0
$599$	−31.7252	−1.29626	−0.648129	−	0.761531i	$-0.724448\pi$
$599$	−31.7252	−1.29626	−0.648129	+	0.761531i	$0.724448\pi$
$600$	0	0
$601$	−42.9575	−1.75227	−0.876137	−	0.482062i	$-0.839888\pi$
$601$	−42.9575	−1.75227	−0.876137	+	0.482062i	$0.839888\pi$
$602$	0	0
$603$	19.8857	0.809809
$604$	0	0
$605$	−9.15251	−0.372102
$606$	0	0
$607$	−41.0420	−1.66584	−0.832922	−	0.553391i	$-0.813334\pi$
$607$	−41.0420	−1.66584	−0.832922	+	0.553391i	$0.813334\pi$
$608$	0	0
$609$	8.37546	0.339391
$610$	0	0
$611$	−14.0305	−0.567614
$612$	0	0
$613$	17.4336	0.704137	0.352069	−	0.935974i	$-0.385478\pi$
$613$	17.4336	0.704137	0.352069	+	0.935974i	$0.385478\pi$
$614$	0	0
$615$	6.33502	0.255453
$616$	0	0
$617$	−14.1336	−0.568998	−0.284499	−	0.958676i	$-0.591827\pi$
$617$	−14.1336	−0.568998	−0.284499	+	0.958676i	$0.591827\pi$
$618$	0	0
$619$	33.8743	1.36153	0.680763	−	0.732504i	$-0.261649\pi$
$619$	33.8743	1.36153	0.680763	+	0.732504i	$0.261649\pi$
$620$	0	0
$621$	7.79157	0.312665
$622$	0	0
$623$	2.13886	0.0856916
$624$	0	0
$625$	0.569256	0.0227702
$626$	0	0
$627$	−10.1255	−0.404374
$628$	0	0
$629$	−0.983974	−0.0392336
$630$	0	0
$631$	22.0157	0.876432	0.438216	−	0.898870i	$-0.355611\pi$
$631$	22.0157	0.876432	0.438216	+	0.898870i	$0.355611\pi$
$632$	0	0
$633$	−4.88834	−0.194294
$634$	0	0
$635$	11.7347	0.465676
$636$	0	0
$637$	3.64053	0.144243
$638$	0	0
$639$	−39.5773	−1.56565
$640$	0	0
$641$	−7.67040	−0.302962	−0.151481	−	0.988460i	$-0.548404\pi$
$641$	−7.67040	−0.302962	−0.151481	+	0.988460i	$0.548404\pi$
$642$	0	0
$643$	−23.7858	−0.938019	−0.469009	−	0.883193i	$-0.655389\pi$
$643$	−23.7858	−0.938019	−0.469009	+	0.883193i	$0.655389\pi$
$644$	0	0
$645$	−3.39357	−0.133622
$646$	0	0
$647$	39.8081	1.56502	0.782509	−	0.622639i	$-0.213940\pi$
$647$	39.8081	1.56502	0.782509	+	0.622639i	$0.213940\pi$
$648$	0	0
$649$	15.1670	0.595356
$650$	0	0
$651$	−5.28345	−0.207075
$652$	0	0
$653$	−27.5966	−1.07994	−0.539968	−	0.841685i	$-0.681564\pi$
$653$	−27.5966	−1.07994	−0.539968	+	0.841685i	$0.681564\pi$
$654$	0	0
$655$	−2.67789	−0.104634
$656$	0	0
$657$	37.1545	1.44953
$658$	0	0
$659$	−7.37969	−0.287472	−0.143736	−	0.989616i	$-0.545912\pi$
$659$	−7.37969	−0.287472	−0.143736	+	0.989616i	$0.545912\pi$
$660$	0	0
$661$	0.752600	0.0292727	0.0146364	−	0.999893i	$-0.495341\pi$
$661$	0.752600	0.0292727	0.0146364	+	0.999893i	$0.495341\pi$
$662$	0	0
$663$	0.289155	0.0112299
$664$	0	0
$665$	14.4399	0.559954
$666$	0	0
$667$	8.53078	0.330313
$668$	0	0
$669$	12.6895	0.490603
$670$	0	0
$671$	20.4738	0.790381
$672$	0	0
$673$	14.9415	0.575953	0.287977	−	0.957637i	$-0.407017\pi$
$673$	14.9415	0.575953	0.287977	+	0.957637i	$0.407017\pi$
$674$	0	0
$675$	11.8204	0.454969
$676$	0	0
$677$	−35.3094	−1.35705	−0.678525	−	0.734577i	$-0.737381\pi$
$677$	−35.3094	−1.35705	−0.678525	+	0.734577i	$0.737381\pi$
$678$	0	0
$679$	1.82827	0.0701626
$680$	0	0
$681$	15.3195	0.587046
$682$	0	0
$683$	−7.29495	−0.279133	−0.139567	−	0.990213i	$-0.544571\pi$
$683$	−7.29495	−0.279133	−0.139567	+	0.990213i	$0.544571\pi$
$684$	0	0
$685$	−30.6955	−1.17281
$686$	0	0
$687$	−2.70454	−0.103185
$688$	0	0
$689$	14.5408	0.553960
$690$	0	0
$691$	29.9939	1.14102	0.570510	−	0.821290i	$-0.306745\pi$
$691$	29.9939	1.14102	0.570510	+	0.821290i	$0.306745\pi$
$692$	0	0
$693$	−31.9532	−1.21380
$694$	0	0
$695$	0.460905	0.0174831
$696$	0	0
$697$	−1.55612	−0.0589421
$698$	0	0
$699$	−9.08415	−0.343594
$700$	0	0
$701$	−7.52887	−0.284362	−0.142181	−	0.989841i	$-0.545411\pi$
$701$	−7.52887	−0.284362	−0.142181	+	0.989841i	$0.545411\pi$
$702$	0	0
$703$	−15.3066	−0.577301
$704$	0	0
$705$	6.95338	0.261879
$706$	0	0
$707$	14.3181	0.538486
$708$	0	0
$709$	4.23645	0.159103	0.0795517	−	0.996831i	$-0.474651\pi$
$709$	4.23645	0.159103	0.0795517	+	0.996831i	$0.474651\pi$
$710$	0	0
$711$	−7.06840	−0.265086
$712$	0	0
$713$	−5.38143	−0.201536
$714$	0	0
$715$	10.7287	0.401229
$716$	0	0
$717$	18.1773	0.678845
$718$	0	0
$719$	21.0902	0.786530	0.393265	−	0.919425i	$-0.371345\pi$
$719$	21.0902	0.786530	0.393265	+	0.919425i	$0.371345\pi$
$720$	0	0
$721$	19.5241	0.727114
$722$	0	0
$723$	9.90449	0.368352
$724$	0	0
$725$	12.9419	0.480650
$726$	0	0
$727$	−2.04832	−0.0759678	−0.0379839	−	0.999278i	$-0.512094\pi$
$727$	−2.04832	−0.0759678	−0.0379839	+	0.999278i	$0.512094\pi$
$728$	0	0
$729$	−5.16423	−0.191268
$730$	0	0
$731$	0.833588	0.0308314
$732$	0	0
$733$	−11.5788	−0.427673	−0.213837	−	0.976869i	$-0.568596\pi$
$733$	−11.5788	−0.427673	−0.213837	+	0.976869i	$0.568596\pi$
$734$	0	0
$735$	−1.80421	−0.0665492
$736$	0	0
$737$	32.9724	1.21455
$738$	0	0
$739$	8.40400	0.309146	0.154573	−	0.987981i	$-0.450600\pi$
$739$	8.40400	0.309146	0.154573	+	0.987981i	$0.450600\pi$
$740$	0	0
$741$	4.49808	0.165241
$742$	0	0
$743$	46.0904	1.69089	0.845447	−	0.534060i	$-0.179334\pi$
$743$	46.0904	1.69089	0.845447	+	0.534060i	$0.179334\pi$
$744$	0	0
$745$	22.3017	0.817070
$746$	0	0
$747$	−9.34442	−0.341894
$748$	0	0
$749$	−26.2004	−0.957340
$750$	0	0
$751$	30.5479	1.11471	0.557354	−	0.830275i	$-0.311817\pi$
$751$	30.5479	1.11471	0.557354	+	0.830275i	$0.311817\pi$
$752$	0	0
$753$	0.679540	0.0247638
$754$	0	0
$755$	−9.13080	−0.332304
$756$	0	0
$757$	14.5569	0.529079	0.264540	−	0.964375i	$-0.414780\pi$
$757$	14.5569	0.529079	0.264540	+	0.964375i	$0.414780\pi$
$758$	0	0
$759$	5.92099	0.214918
$760$	0	0
$761$	44.6971	1.62027	0.810135	−	0.586244i	$-0.199394\pi$
$761$	44.6971	1.62027	0.810135	+	0.586244i	$0.199394\pi$
$762$	0	0
$763$	−7.63828	−0.276524
$764$	0	0
$765$	0.787685	0.0284788
$766$	0	0
$767$	−6.73766	−0.243283
$768$	0	0
$769$	−14.8380	−0.535073	−0.267537	−	0.963548i	$-0.586210\pi$
$769$	−14.8380	−0.535073	−0.267537	+	0.963548i	$0.586210\pi$
$770$	0	0
$771$	7.73586	0.278600
$772$	0	0
$773$	41.6561	1.49826	0.749132	−	0.662421i	$-0.230471\pi$
$773$	41.6561	1.49826	0.749132	+	0.662421i	$0.230471\pi$
$774$	0	0
$775$	−8.16406	−0.293262
$776$	0	0
$777$	8.78773	0.315258
$778$	0	0
$779$	−24.2069	−0.867301
$780$	0	0
$781$	−65.6229	−2.34817
$782$	0	0
$783$	15.5073	0.554184
$784$	0	0
$785$	−29.5510	−1.05472
$786$	0	0
$787$	−37.4389	−1.33455	−0.667276	−	0.744811i	$-0.732540\pi$
$787$	−37.4389	−1.33455	−0.667276	+	0.744811i	$0.732540\pi$
$788$	0	0
$789$	−0.538690	−0.0191779
$790$	0	0
$791$	−11.2412	−0.399692
$792$	0	0
$793$	−9.09510	−0.322976
$794$	0	0
$795$	−7.20626	−0.255580
$796$	0	0
$797$	6.94145	0.245879	0.122939	−	0.992414i	$-0.460768\pi$
$797$	6.94145	0.245879	0.122939	+	0.992414i	$0.460768\pi$
$798$	0	0
$799$	−1.70801	−0.0604250
$800$	0	0
$801$	1.81496	0.0641286
$802$	0	0
$803$	61.6057	2.17402
$804$	0	0
$805$	−8.44384	−0.297606
$806$	0	0
$807$	2.48430	0.0874515
$808$	0	0
$809$	34.2665	1.20474	0.602372	−	0.798215i	$-0.294222\pi$
$809$	34.2665	1.20474	0.602372	+	0.798215i	$0.294222\pi$
$810$	0	0
$811$	−12.8888	−0.452589	−0.226294	−	0.974059i	$-0.572661\pi$
$811$	−12.8888	−0.452589	−0.226294	+	0.974059i	$0.572661\pi$
$812$	0	0
$813$	3.79646	0.133148
$814$	0	0
$815$	13.8221	0.484165
$816$	0	0
$817$	12.9672	0.453666
$818$	0	0
$819$	14.1946	0.496000
$820$	0	0
$821$	−45.5118	−1.58837	−0.794186	−	0.607675i	$-0.792102\pi$
$821$	−45.5118	−1.58837	−0.794186	+	0.607675i	$0.792102\pi$
$822$	0	0
$823$	−2.59330	−0.0903969	−0.0451984	−	0.998978i	$-0.514392\pi$
$823$	−2.59330	−0.0903969	−0.0451984	+	0.998978i	$0.514392\pi$
$824$	0	0
$825$	8.98262	0.312735
$826$	0	0
$827$	15.9105	0.553262	0.276631	−	0.960976i	$-0.410782\pi$
$827$	15.9105	0.553262	0.276631	+	0.960976i	$0.410782\pi$
$828$	0	0
$829$	28.3558	0.984837	0.492418	−	0.870359i	$-0.336113\pi$
$829$	28.3558	0.984837	0.492418	+	0.870359i	$0.336113\pi$
$830$	0	0
$831$	6.04827	0.209812
$832$	0	0
$833$	0.443181	0.0153553
$834$	0	0
$835$	−28.8174	−0.997268
$836$	0	0
$837$	−9.78237	−0.338128
$838$	0	0
$839$	7.35538	0.253936	0.126968	−	0.991907i	$-0.459475\pi$
$839$	7.35538	0.253936	0.126968	+	0.991907i	$0.459475\pi$
$840$	0	0
$841$	−12.0215	−0.414535
$842$	0	0
$843$	1.36114	0.0468800
$844$	0	0
$845$	12.9595	0.445822
$846$	0	0
$847$	−20.0783	−0.689899
$848$	0	0
$849$	−19.0231	−0.652872
$850$	0	0
$851$	8.95069	0.306826
$852$	0	0
$853$	−26.0336	−0.891373	−0.445686	−	0.895189i	$-0.647040\pi$
$853$	−26.0336	−0.891373	−0.445686	+	0.895189i	$0.647040\pi$
$854$	0	0
$855$	12.2532	0.419050
$856$	0	0
$857$	−48.2591	−1.64850	−0.824250	−	0.566226i	$-0.808403\pi$
$857$	−48.2591	−1.64850	−0.824250	+	0.566226i	$0.808403\pi$
$858$	0	0
$859$	29.1397	0.994234	0.497117	−	0.867684i	$-0.334392\pi$
$859$	29.1397	0.994234	0.497117	+	0.867684i	$0.334392\pi$
$860$	0	0
$861$	13.8975	0.473624
$862$	0	0
$863$	4.04525	0.137702	0.0688510	−	0.997627i	$-0.478067\pi$
$863$	4.04525	0.137702	0.0688510	+	0.997627i	$0.478067\pi$
$864$	0	0
$865$	−20.9965	−0.713903
$866$	0	0
$867$	−11.5170	−0.391137
$868$	0	0
$869$	−11.7201	−0.397577
$870$	0	0
$871$	−14.6474	−0.496308
$872$	0	0
$873$	1.55141	0.0525073
$874$	0	0
$875$	−33.2025	−1.12245
$876$	0	0
$877$	10.1319	0.342128	0.171064	−	0.985260i	$-0.445279\pi$
$877$	10.1319	0.342128	0.171064	+	0.985260i	$0.445279\pi$
$878$	0	0
$879$	−16.6874	−0.562853
$880$	0	0
$881$	−23.0262	−0.775773	−0.387886	−	0.921707i	$-0.626795\pi$
$881$	−23.0262	−0.775773	−0.387886	+	0.921707i	$0.626795\pi$
$882$	0	0
$883$	−31.3915	−1.05641	−0.528204	−	0.849117i	$-0.677134\pi$
$883$	−31.3915	−1.05641	−0.528204	+	0.849117i	$0.677134\pi$
$884$	0	0
$885$	3.33911	0.112243
$886$	0	0
$887$	8.76122	0.294173	0.147087	−	0.989124i	$-0.453010\pi$
$887$	8.76122	0.294173	0.147087	+	0.989124i	$0.453010\pi$
$888$	0	0
$889$	25.7430	0.863392
$890$	0	0
$891$	−21.2841	−0.713044
$892$	0	0
$893$	−26.5697	−0.889121
$894$	0	0
$895$	−4.11585	−0.137578
$896$	0	0
$897$	−2.63029	−0.0878229
$898$	0	0
$899$	−10.7104	−0.357213
$900$	0	0
$901$	1.77013	0.0589715
$902$	0	0
$903$	−7.44465	−0.247743
$904$	0	0
$905$	−5.01986	−0.166866
$906$	0	0
$907$	−22.8893	−0.760028	−0.380014	−	0.924981i	$-0.624081\pi$
$907$	−22.8893	−0.760028	−0.380014	+	0.924981i	$0.624081\pi$
$908$	0	0
$909$	12.1498	0.402985
$910$	0	0
$911$	22.8379	0.756654	0.378327	−	0.925672i	$-0.376499\pi$
$911$	22.8379	0.756654	0.378327	+	0.925672i	$0.376499\pi$
$912$	0	0
$913$	−15.4939	−0.512775
$914$	0	0
$915$	4.50744	0.149011
$916$	0	0
$917$	−5.87464	−0.193998
$918$	0	0
$919$	43.5651	1.43708	0.718540	−	0.695486i	$-0.244811\pi$
$919$	43.5651	1.43708	0.718540	+	0.695486i	$0.244811\pi$
$920$	0	0
$921$	10.3395	0.340699
$922$	0	0
$923$	29.1518	0.959543
$924$	0	0
$925$	13.5789	0.446472
$926$	0	0
$927$	16.5675	0.544147
$928$	0	0
$929$	11.0815	0.363573	0.181787	−	0.983338i	$-0.441812\pi$
$929$	11.0815	0.363573	0.181787	+	0.983338i	$0.441812\pi$
$930$	0	0
$931$	6.89409	0.225945
$932$	0	0
$933$	13.0137	0.426051
$934$	0	0
$935$	1.30606	0.0427127
$936$	0	0
$937$	28.4013	0.927829	0.463915	−	0.885880i	$-0.346444\pi$
$937$	28.4013	0.927829	0.463915	+	0.885880i	$0.346444\pi$
$938$	0	0
$939$	−4.49990	−0.146849
$940$	0	0
$941$	−33.8076	−1.10210	−0.551048	−	0.834473i	$-0.685772\pi$
$941$	−33.8076	−1.10210	−0.551048	+	0.834473i	$0.685772\pi$
$942$	0	0
$943$	14.1552	0.460956
$944$	0	0
$945$	−15.3492	−0.499310
$946$	0	0
$947$	−23.7460	−0.771641	−0.385820	−	0.922574i	$-0.626082\pi$
$947$	−23.7460	−0.771641	−0.385820	+	0.922574i	$0.626082\pi$
$948$	0	0
$949$	−27.3672	−0.888378
$950$	0	0
$951$	−6.62330	−0.214775
$952$	0	0
$953$	−16.0896	−0.521193	−0.260596	−	0.965448i	$-0.583919\pi$
$953$	−16.0896	−0.521193	−0.260596	+	0.965448i	$0.583919\pi$
$954$	0	0
$955$	−32.9408	−1.06594
$956$	0	0
$957$	11.7843	0.380933
$958$	0	0
$959$	−67.3383	−2.17447
$960$	0	0
$961$	−24.2436	−0.782051
$962$	0	0
$963$	−22.2328	−0.716440
$964$	0	0
$965$	19.1400	0.616139
$966$	0	0
$967$	−23.7350	−0.763267	−0.381633	−	0.924314i	$-0.624638\pi$
$967$	−23.7350	−0.763267	−0.381633	+	0.924314i	$0.624638\pi$
$968$	0	0
$969$	0.547575	0.0175906
$970$	0	0
$971$	22.0347	0.707128	0.353564	−	0.935410i	$-0.384970\pi$
$971$	22.0347	0.707128	0.353564	+	0.935410i	$0.384970\pi$
$972$	0	0
$973$	1.01111	0.0324148
$974$	0	0
$975$	−3.99037	−0.127794
$976$	0	0
$977$	7.54883	0.241508	0.120754	−	0.992682i	$-0.461469\pi$
$977$	7.54883	0.241508	0.120754	+	0.992682i	$0.461469\pi$
$978$	0	0
$979$	3.00939	0.0961804
$980$	0	0
$981$	−6.48159	−0.206941
$982$	0	0
$983$	−13.6908	−0.436669	−0.218334	−	0.975874i	$-0.570062\pi$
$983$	−13.6908	−0.436669	−0.218334	+	0.975874i	$0.570062\pi$
$984$	0	0
$985$	−20.0549	−0.639001
$986$	0	0
$987$	15.2540	0.485540
$988$	0	0
$989$	−7.58271	−0.241116
$990$	0	0
$991$	−28.4502	−0.903751	−0.451876	−	0.892081i	$-0.649245\pi$
$991$	−28.4502	−0.903751	−0.451876	+	0.892081i	$0.649245\pi$
$992$	0	0
$993$	10.8147	0.343194
$994$	0	0
$995$	−4.43155	−0.140490
$996$	0	0
$997$	−31.5032	−0.997716	−0.498858	−	0.866684i	$-0.666247\pi$
$997$	−31.5032	−0.997716	−0.498858	+	0.866684i	$0.666247\pi$
$998$	0	0
$999$	16.2706	0.514778

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.i.1.17	yes	30
4.3	odd	2		8032.2.a.h.1.14	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.14	✓	30	4.3	odd	2
8032.2.a.i.1.17	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)

\(f(q)\)	\(=\)	\(q+0.679540 q^{3} -1.36350 q^{5} -2.99119 q^{7} -2.53823 q^{9} +O(q^{10})\)	\(q+0.679540 q^{3} -1.36350 q^{5} -2.99119 q^{7} -2.53823 q^{9} -4.20862 q^{11} +1.86960 q^{13} -0.926556 q^{15} +0.227597 q^{17} +3.54048 q^{19} -2.03263 q^{21} -2.07033 q^{23} -3.14086 q^{25} -3.76345 q^{27} -4.12050 q^{29} +2.59931 q^{31} -2.85993 q^{33} +4.07850 q^{35} -4.32332 q^{37} +1.27047 q^{39} -6.83717 q^{41} +3.66256 q^{43} +3.46088 q^{45} -7.50454 q^{47} +1.94722 q^{49} +0.154661 q^{51} +7.77747 q^{53} +5.73847 q^{55} +2.40590 q^{57} -3.60379 q^{59} -4.86472 q^{61} +7.59231 q^{63} -2.54921 q^{65} -7.83449 q^{67} -1.40687 q^{69} +15.5925 q^{71} -14.6380 q^{73} -2.13434 q^{75} +12.5888 q^{77} +2.78478 q^{79} +5.05726 q^{81} +3.68148 q^{83} -0.310329 q^{85} -2.80004 q^{87} -0.715053 q^{89} -5.59234 q^{91} +1.76634 q^{93} -4.82746 q^{95} -0.611218 q^{97} +10.6824 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9} + 13 q^{11} - 7 q^{13} + 28 q^{15} - 9 q^{17} - 17 q^{19} - 6 q^{21} + 43 q^{23} + 34 q^{25} + 12 q^{27} - q^{29} + 39 q^{31} + 17 q^{35} - q^{37} + 48 q^{39} - 3 q^{41} - 19 q^{43} + 66 q^{47} + 25 q^{49} - 14 q^{51} + 3 q^{53} + 50 q^{55} - 14 q^{57} + 27 q^{59} + 15 q^{61} + 75 q^{63} - 6 q^{65} + 8 q^{67} + 18 q^{69} + 64 q^{71} - 15 q^{73} + 9 q^{75} + 71 q^{79} + 6 q^{81} + 60 q^{83} + 15 q^{85} + 64 q^{87} - 32 q^{89} - 26 q^{91} - 4 q^{93} + 72 q^{95} - 4 q^{97} + 13 q^{99}+O(q^{100}) \) 30 * q + 3 * q^3 + 13 * q^7 + 35 * q^9 + 13 * q^11 - 7 * q^13 + 28 * q^15 - 9 * q^17 - 17 * q^19 - 6 * q^21 + 43 * q^23 + 34 * q^25 + 12 * q^27 - q^29 + 39 * q^31 + 17 * q^35 - q^37 + 48 * q^39 - 3 * q^41 - 19 * q^43 + 66 * q^47 + 25 * q^49 - 14 * q^51 + 3 * q^53 + 50 * q^55 - 14 * q^57 + 27 * q^59 + 15 * q^61 + 75 * q^63 - 6 * q^65 + 8 * q^67 + 18 * q^69 + 64 * q^71 - 15 * q^73 + 9 * q^75 + 71 * q^79 + 6 * q^81 + 60 * q^83 + 15 * q^85 + 64 * q^87 - 32 * q^89 - 26 * q^91 - 4 * q^93 + 72 * q^95 - 4 * q^97 + 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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