LMFDB - Embedded newform 8032.2.a.g.1.13

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:	$1$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
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)

$f(q)$	$=$	$q-0.933880 q^{3} +0.0426876 q^{5} +2.12724 q^{7} -2.12787 q^{9} +O(q^{10})$	$q-0.933880 q^{3} +0.0426876 q^{5} +2.12724 q^{7} -2.12787 q^{9} +1.32260 q^{11} -5.21113 q^{13} -0.0398651 q^{15} +3.02963 q^{17} -2.11385 q^{19} -1.98659 q^{21} +8.05538 q^{23} -4.99818 q^{25} +4.78881 q^{27} -4.11166 q^{29} -5.14940 q^{31} -1.23515 q^{33} +0.0908069 q^{35} +7.42245 q^{37} +4.86657 q^{39} +8.58655 q^{41} -3.62140 q^{43} -0.0908335 q^{45} +3.26900 q^{47} -2.47483 q^{49} -2.82931 q^{51} +2.40266 q^{53} +0.0564586 q^{55} +1.97408 q^{57} +0.729143 q^{59} -7.70480 q^{61} -4.52649 q^{63} -0.222450 q^{65} -7.09779 q^{67} -7.52276 q^{69} -14.3674 q^{71} +15.4741 q^{73} +4.66770 q^{75} +2.81350 q^{77} +7.89798 q^{79} +1.91142 q^{81} -2.92015 q^{83} +0.129328 q^{85} +3.83980 q^{87} -5.29992 q^{89} -11.0853 q^{91} +4.80892 q^{93} -0.0902350 q^{95} -3.16032 q^{97} -2.81432 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.933880	−0.539176	−0.269588	−	0.962976i	$-0.586888\pi$
$3$	−0.933880	−0.539176	−0.269588	+	0.962976i	$0.586888\pi$
$4$	0	0
$5$	0.0426876	0.0190905	0.00954523	−	0.999954i	$-0.496962\pi$
$5$	0.0426876	0.0190905	0.00954523	+	0.999954i	$0.496962\pi$
$6$	0	0
$7$	2.12724	0.804023	0.402011	−	0.915635i	$-0.368311\pi$
$7$	2.12724	0.804023	0.402011	+	0.915635i	$0.368311\pi$
$8$	0	0
$9$	−2.12787	−0.709289
$10$	0	0
$11$	1.32260	0.398779	0.199390	−	0.979920i	$-0.436104\pi$
$11$	1.32260	0.398779	0.199390	+	0.979920i	$0.436104\pi$
$12$	0	0
$13$	−5.21113	−1.44531	−0.722653	−	0.691211i	$-0.757078\pi$
$13$	−5.21113	−1.44531	−0.722653	+	0.691211i	$0.757078\pi$
$14$	0	0
$15$	−0.0398651	−0.0102931
$16$	0	0
$17$	3.02963	0.734793	0.367397	−	0.930064i	$-0.380249\pi$
$17$	3.02963	0.734793	0.367397	+	0.930064i	$0.380249\pi$
$18$	0	0
$19$	−2.11385	−0.484950	−0.242475	−	0.970158i	$-0.577959\pi$
$19$	−2.11385	−0.484950	−0.242475	+	0.970158i	$0.577959\pi$
$20$	0	0
$21$	−1.98659	−0.433510
$22$	0	0
$23$	8.05538	1.67966	0.839831	−	0.542848i	$-0.182654\pi$
$23$	8.05538	1.67966	0.839831	+	0.542848i	$0.182654\pi$
$24$	0	0
$25$	−4.99818	−0.999636
$26$	0	0
$27$	4.78881	0.921608
$28$	0	0
$29$	−4.11166	−0.763516	−0.381758	−	0.924262i	$-0.624681\pi$
$29$	−4.11166	−0.763516	−0.381758	+	0.924262i	$0.624681\pi$
$30$	0	0
$31$	−5.14940	−0.924860	−0.462430	−	0.886656i	$-0.653022\pi$
$31$	−5.14940	−0.924860	−0.462430	+	0.886656i	$0.653022\pi$
$32$	0	0
$33$	−1.23515	−0.215012
$34$	0	0
$35$	0.0908069	0.0153492
$36$	0	0
$37$	7.42245	1.22024	0.610122	−	0.792308i	$-0.291121\pi$
$37$	7.42245	1.22024	0.610122	+	0.792308i	$0.291121\pi$
$38$	0	0
$39$	4.86657	0.779275
$40$	0	0
$41$	8.58655	1.34099	0.670497	−	0.741913i	$-0.266081\pi$
$41$	8.58655	1.34099	0.670497	+	0.741913i	$0.266081\pi$
$42$	0	0
$43$	−3.62140	−0.552259	−0.276129	−	0.961121i	$-0.589052\pi$
$43$	−3.62140	−0.552259	−0.276129	+	0.961121i	$0.589052\pi$
$44$	0	0
$45$	−0.0908335	−0.0135407
$46$	0	0
$47$	3.26900	0.476833	0.238417	−	0.971163i	$-0.423372\pi$
$47$	3.26900	0.476833	0.238417	+	0.971163i	$0.423372\pi$
$48$	0	0
$49$	−2.47483	−0.353547
$50$	0	0
$51$	−2.82931	−0.396183
$52$	0	0
$53$	2.40266	0.330031	0.165016	−	0.986291i	$-0.447233\pi$
$53$	2.40266	0.330031	0.165016	+	0.986291i	$0.447233\pi$
$54$	0	0
$55$	0.0564586	0.00761288
$56$	0	0
$57$	1.97408	0.261473
$58$	0	0
$59$	0.729143	0.0949264	0.0474632	−	0.998873i	$-0.484886\pi$
$59$	0.729143	0.0949264	0.0474632	+	0.998873i	$0.484886\pi$
$60$	0	0
$61$	−7.70480	−0.986498	−0.493249	−	0.869888i	$-0.664191\pi$
$61$	−7.70480	−0.986498	−0.493249	+	0.869888i	$0.664191\pi$
$62$	0	0
$63$	−4.52649	−0.570285
$64$	0	0
$65$	−0.222450	−0.0275916
$66$	0	0
$67$	−7.09779	−0.867133	−0.433566	−	0.901122i	$-0.642745\pi$
$67$	−7.09779	−0.867133	−0.433566	+	0.901122i	$0.642745\pi$
$68$	0	0
$69$	−7.52276	−0.905634
$70$	0	0
$71$	−14.3674	−1.70510	−0.852550	−	0.522646i	$-0.824945\pi$
$71$	−14.3674	−1.70510	−0.852550	+	0.522646i	$0.824945\pi$
$72$	0	0
$73$	15.4741	1.81111	0.905554	−	0.424231i	$-0.139456\pi$
$73$	15.4741	1.81111	0.905554	+	0.424231i	$0.139456\pi$
$74$	0	0
$75$	4.66770	0.538980
$76$	0	0
$77$	2.81350	0.320628
$78$	0	0
$79$	7.89798	0.888592	0.444296	−	0.895880i	$-0.353454\pi$
$79$	7.89798	0.888592	0.444296	+	0.895880i	$0.353454\pi$
$80$	0	0
$81$	1.91142	0.212380
$82$	0	0
$83$	−2.92015	−0.320528	−0.160264	−	0.987074i	$-0.551235\pi$
$83$	−2.92015	−0.320528	−0.160264	+	0.987074i	$0.551235\pi$
$84$	0	0
$85$	0.129328	0.0140275
$86$	0	0
$87$	3.83980	0.411670
$88$	0	0
$89$	−5.29992	−0.561790	−0.280895	−	0.959738i	$-0.590631\pi$
$89$	−5.29992	−0.561790	−0.280895	+	0.959738i	$0.590631\pi$
$90$	0	0
$91$	−11.0853	−1.16206
$92$	0	0
$93$	4.80892	0.498662
$94$	0	0
$95$	−0.0902350	−0.00925791
$96$	0	0
$97$	−3.16032	−0.320882	−0.160441	−	0.987045i	$-0.551292\pi$
$97$	−3.16032	−0.320882	−0.160441	+	0.987045i	$0.551292\pi$
$98$	0	0
$99$	−2.81432	−0.282850
$100$	0	0
$101$	16.0214	1.59419	0.797093	−	0.603857i	$-0.206370\pi$
$101$	16.0214	1.59419	0.797093	+	0.603857i	$0.206370\pi$
$102$	0	0
$103$	−9.17467	−0.904007	−0.452004	−	0.892016i	$-0.649291\pi$
$103$	−9.17467	−0.904007	−0.452004	+	0.892016i	$0.649291\pi$
$104$	0	0
$105$	−0.0848028	−0.00827590
$106$	0	0
$107$	0.705430	0.0681965	0.0340983	−	0.999418i	$-0.489144\pi$
$107$	0.705430	0.0681965	0.0340983	+	0.999418i	$0.489144\pi$
$108$	0	0
$109$	−12.9961	−1.24480	−0.622401	−	0.782699i	$-0.713843\pi$
$109$	−12.9961	−1.24480	−0.622401	+	0.782699i	$0.713843\pi$
$110$	0	0
$111$	−6.93168	−0.657926
$112$	0	0
$113$	8.89185	0.836475	0.418237	−	0.908338i	$-0.362648\pi$
$113$	8.89185	0.836475	0.418237	+	0.908338i	$0.362648\pi$
$114$	0	0
$115$	0.343864	0.0320655
$116$	0	0
$117$	11.0886	1.02514
$118$	0	0
$119$	6.44476	0.590790
$120$	0	0
$121$	−9.25073	−0.840975
$122$	0	0
$123$	−8.01881	−0.723032
$124$	0	0
$125$	−0.426798	−0.0381740
$126$	0	0
$127$	−14.0860	−1.24993	−0.624966	−	0.780652i	$-0.714887\pi$
$127$	−14.0860	−1.24993	−0.624966	+	0.780652i	$0.714887\pi$
$128$	0	0
$129$	3.38196	0.297765
$130$	0	0
$131$	−21.9377	−1.91670	−0.958352	−	0.285591i	$-0.907810\pi$
$131$	−21.9377	−1.91670	−0.958352	+	0.285591i	$0.907810\pi$
$132$	0	0
$133$	−4.49667	−0.389911
$134$	0	0
$135$	0.204423	0.0175939
$136$	0	0
$137$	−4.92052	−0.420389	−0.210194	−	0.977660i	$-0.567410\pi$
$137$	−4.92052	−0.420389	−0.210194	+	0.977660i	$0.567410\pi$
$138$	0	0
$139$	−19.0769	−1.61808	−0.809041	−	0.587752i	$-0.800013\pi$
$139$	−19.0769	−1.61808	−0.809041	+	0.587752i	$0.800013\pi$
$140$	0	0
$141$	−3.05286	−0.257097
$142$	0	0
$143$	−6.89224	−0.576358
$144$	0	0
$145$	−0.175517	−0.0145759
$146$	0	0
$147$	2.31120	0.190624
$148$	0	0
$149$	−6.82554	−0.559170	−0.279585	−	0.960121i	$-0.590197\pi$
$149$	−6.82554	−0.559170	−0.279585	+	0.960121i	$0.590197\pi$
$150$	0	0
$151$	15.4860	1.26024	0.630118	−	0.776499i	$-0.283006\pi$
$151$	15.4860	1.26024	0.630118	+	0.776499i	$0.283006\pi$
$152$	0	0
$153$	−6.44665	−0.521181
$154$	0	0
$155$	−0.219815	−0.0176560
$156$	0	0
$157$	−20.7646	−1.65719	−0.828597	−	0.559846i	$-0.810860\pi$
$157$	−20.7646	−1.65719	−0.828597	+	0.559846i	$0.810860\pi$
$158$	0	0
$159$	−2.24380	−0.177945
$160$	0	0
$161$	17.1358	1.35049
$162$	0	0
$163$	15.7853	1.23640	0.618198	−	0.786022i	$-0.287863\pi$
$163$	15.7853	1.23640	0.618198	+	0.786022i	$0.287863\pi$
$164$	0	0
$165$	−0.0527256	−0.00410468
$166$	0	0
$167$	22.2023	1.71807	0.859033	−	0.511920i	$-0.171066\pi$
$167$	22.2023	1.71807	0.859033	+	0.511920i	$0.171066\pi$
$168$	0	0
$169$	14.1558	1.08891
$170$	0	0
$171$	4.49799	0.343970
$172$	0	0
$173$	−4.82792	−0.367060	−0.183530	−	0.983014i	$-0.558752\pi$
$173$	−4.82792	−0.367060	−0.183530	+	0.983014i	$0.558752\pi$
$174$	0	0
$175$	−10.6323	−0.803730
$176$	0	0
$177$	−0.680933	−0.0511820
$178$	0	0
$179$	−1.32300	−0.0988860	−0.0494430	−	0.998777i	$-0.515745\pi$
$179$	−1.32300	−0.0988860	−0.0494430	+	0.998777i	$0.515745\pi$
$180$	0	0
$181$	25.6059	1.90327	0.951636	−	0.307226i	$-0.0994009\pi$
$181$	25.6059	1.90327	0.951636	+	0.307226i	$0.0994009\pi$
$182$	0	0
$183$	7.19536	0.531896
$184$	0	0
$185$	0.316846	0.0232950
$186$	0	0
$187$	4.00699	0.293020
$188$	0	0
$189$	10.1870	0.740994
$190$	0	0
$191$	−10.0312	−0.725829	−0.362915	−	0.931822i	$-0.618218\pi$
$191$	−10.0312	−0.725829	−0.362915	+	0.931822i	$0.618218\pi$
$192$	0	0
$193$	14.5059	1.04416	0.522080	−	0.852896i	$-0.325156\pi$
$193$	14.5059	1.04416	0.522080	+	0.852896i	$0.325156\pi$
$194$	0	0
$195$	0.207742	0.0148767
$196$	0	0
$197$	8.51286	0.606516	0.303258	−	0.952908i	$-0.401926\pi$
$197$	8.51286	0.606516	0.303258	+	0.952908i	$0.401926\pi$
$198$	0	0
$199$	−20.7516	−1.47104	−0.735521	−	0.677502i	$-0.763062\pi$
$199$	−20.7516	−1.47104	−0.735521	+	0.677502i	$0.763062\pi$
$200$	0	0
$201$	6.62848	0.467537
$202$	0	0
$203$	−8.74651	−0.613884
$204$	0	0
$205$	0.366539	0.0256002
$206$	0	0
$207$	−17.1408	−1.19137
$208$	0	0
$209$	−2.79578	−0.193388
$210$	0	0
$211$	−0.0423444	−0.00291511	−0.00145755	−	0.999999i	$-0.500464\pi$
$211$	−0.0423444	−0.00291511	−0.00145755	+	0.999999i	$0.500464\pi$
$212$	0	0
$213$	13.4175	0.919349
$214$	0	0
$215$	−0.154589	−0.0105429
$216$	0	0
$217$	−10.9540	−0.743608
$218$	0	0
$219$	−14.4510	−0.976506
$220$	0	0
$221$	−15.7878	−1.06200
$222$	0	0
$223$	−9.44799	−0.632684	−0.316342	−	0.948645i	$-0.602455\pi$
$223$	−9.44799	−0.632684	−0.316342	+	0.948645i	$0.602455\pi$
$224$	0	0
$225$	10.6355	0.709031
$226$	0	0
$227$	20.3404	1.35004	0.675019	−	0.737801i	$-0.264136\pi$
$227$	20.3404	1.35004	0.675019	+	0.737801i	$0.264136\pi$
$228$	0	0
$229$	−24.3796	−1.61105	−0.805524	−	0.592563i	$-0.798116\pi$
$229$	−24.3796	−1.61105	−0.805524	+	0.592563i	$0.798116\pi$
$230$	0	0
$231$	−2.62747	−0.172875
$232$	0	0
$233$	−13.6100	−0.891618	−0.445809	−	0.895128i	$-0.647084\pi$
$233$	−13.6100	−0.891618	−0.445809	+	0.895128i	$0.647084\pi$
$234$	0	0
$235$	0.139546	0.00910296
$236$	0	0
$237$	−7.37577	−0.479108
$238$	0	0
$239$	−6.69540	−0.433089	−0.216545	−	0.976273i	$-0.569479\pi$
$239$	−6.69540	−0.433089	−0.216545	+	0.976273i	$0.569479\pi$
$240$	0	0
$241$	−15.9997	−1.03063	−0.515315	−	0.857001i	$-0.672325\pi$
$241$	−15.9997	−1.03063	−0.515315	+	0.857001i	$0.672325\pi$
$242$	0	0
$243$	−16.1515	−1.03612
$244$	0	0
$245$	−0.105645	−0.00674938
$246$	0	0
$247$	11.0155	0.700901
$248$	0	0
$249$	2.72707	0.172821
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	10.6541	0.669815
$254$	0	0
$255$	−0.120776	−0.00756331
$256$	0	0
$257$	17.7417	1.10670	0.553348	−	0.832950i	$-0.313350\pi$
$257$	17.7417	1.10670	0.553348	+	0.832950i	$0.313350\pi$
$258$	0	0
$259$	15.7894	0.981103
$260$	0	0
$261$	8.74907	0.541554
$262$	0	0
$263$	−26.7857	−1.65168	−0.825838	−	0.563907i	$-0.809298\pi$
$263$	−26.7857	−1.65168	−0.825838	+	0.563907i	$0.809298\pi$
$264$	0	0
$265$	0.102564	0.00630045
$266$	0	0
$267$	4.94949	0.302904
$268$	0	0
$269$	2.13473	0.130157	0.0650785	−	0.997880i	$-0.479270\pi$
$269$	2.13473	0.130157	0.0650785	+	0.997880i	$0.479270\pi$
$270$	0	0
$271$	−26.5682	−1.61390	−0.806951	−	0.590618i	$-0.798884\pi$
$271$	−26.5682	−1.61390	−0.806951	+	0.590618i	$0.798884\pi$
$272$	0	0
$273$	10.3524	0.626555
$274$	0	0
$275$	−6.61060	−0.398634
$276$	0	0
$277$	3.36094	0.201939	0.100970	−	0.994890i	$-0.467806\pi$
$277$	3.36094	0.201939	0.100970	+	0.994890i	$0.467806\pi$
$278$	0	0
$279$	10.9572	0.655993
$280$	0	0
$281$	0.637685	0.0380411	0.0190206	−	0.999819i	$-0.493945\pi$
$281$	0.637685	0.0380411	0.0190206	+	0.999819i	$0.493945\pi$
$282$	0	0
$283$	−11.2224	−0.667104	−0.333552	−	0.942732i	$-0.608247\pi$
$283$	−11.2224	−0.667104	−0.333552	+	0.942732i	$0.608247\pi$
$284$	0	0
$285$	0.0842687	0.00499164
$286$	0	0
$287$	18.2657	1.07819
$288$	0	0
$289$	−7.82134	−0.460079
$290$	0	0
$291$	2.95137	0.173012
$292$	0	0
$293$	4.69621	0.274356	0.137178	−	0.990546i	$-0.456197\pi$
$293$	4.69621	0.274356	0.137178	+	0.990546i	$0.456197\pi$
$294$	0	0
$295$	0.0311254	0.00181219
$296$	0	0
$297$	6.33369	0.367518
$298$	0	0
$299$	−41.9776	−2.42763
$300$	0	0
$301$	−7.70361	−0.444028
$302$	0	0
$303$	−14.9620	−0.859547
$304$	0	0
$305$	−0.328899	−0.0188327
$306$	0	0
$307$	−26.9586	−1.53861	−0.769304	−	0.638883i	$-0.779397\pi$
$307$	−26.9586	−1.53861	−0.769304	+	0.638883i	$0.779397\pi$
$308$	0	0
$309$	8.56805	0.487419
$310$	0	0
$311$	20.7147	1.17462	0.587311	−	0.809361i	$-0.300187\pi$
$311$	20.7147	1.17462	0.587311	+	0.809361i	$0.300187\pi$
$312$	0	0
$313$	19.5850	1.10701	0.553506	−	0.832845i	$-0.313290\pi$
$313$	19.5850	1.10701	0.553506	+	0.832845i	$0.313290\pi$
$314$	0	0
$315$	−0.193225	−0.0108870
$316$	0	0
$317$	8.42245	0.473052	0.236526	−	0.971625i	$-0.423991\pi$
$317$	8.42245	0.473052	0.236526	+	0.971625i	$0.423991\pi$
$318$	0	0
$319$	−5.43809	−0.304474
$320$	0	0
$321$	−0.658787	−0.0367699
$322$	0	0
$323$	−6.40417	−0.356338
$324$	0	0
$325$	26.0461	1.44478
$326$	0	0
$327$	12.1368	0.671167
$328$	0	0
$329$	6.95397	0.383385
$330$	0	0
$331$	−26.6901	−1.46702	−0.733511	−	0.679678i	$-0.762120\pi$
$331$	−26.6901	−1.46702	−0.733511	+	0.679678i	$0.762120\pi$
$332$	0	0
$333$	−15.7940	−0.865505
$334$	0	0
$335$	−0.302987	−0.0165540
$336$	0	0
$337$	3.26187	0.177685	0.0888427	−	0.996046i	$-0.471683\pi$
$337$	3.26187	0.177685	0.0888427	+	0.996046i	$0.471683\pi$
$338$	0	0
$339$	−8.30393	−0.451007
$340$	0	0
$341$	−6.81060	−0.368815
$342$	0	0
$343$	−20.1553	−1.08828
$344$	0	0
$345$	−0.321128	−0.0172890
$346$	0	0
$347$	6.21023	0.333383	0.166691	−	0.986009i	$-0.446692\pi$
$347$	6.21023	0.333383	0.166691	+	0.986009i	$0.446692\pi$
$348$	0	0
$349$	3.21986	0.172355	0.0861777	−	0.996280i	$-0.472535\pi$
$349$	3.21986	0.172355	0.0861777	+	0.996280i	$0.472535\pi$
$350$	0	0
$351$	−24.9551	−1.33201
$352$	0	0
$353$	−14.9307	−0.794682	−0.397341	−	0.917671i	$-0.630067\pi$
$353$	−14.9307	−0.794682	−0.397341	+	0.917671i	$0.630067\pi$
$354$	0	0
$355$	−0.613310	−0.0325511
$356$	0	0
$357$	−6.01864	−0.318540
$358$	0	0
$359$	29.9317	1.57974	0.789868	−	0.613276i	$-0.210149\pi$
$359$	29.9317	1.57974	0.789868	+	0.613276i	$0.210149\pi$
$360$	0	0
$361$	−14.5317	−0.764824
$362$	0	0
$363$	8.63907	0.453434
$364$	0	0
$365$	0.660552	0.0345749
$366$	0	0
$367$	34.5213	1.80200	0.900998	−	0.433822i	$-0.142835\pi$
$367$	34.5213	1.80200	0.900998	+	0.433822i	$0.142835\pi$
$368$	0	0
$369$	−18.2710	−0.951152
$370$	0	0
$371$	5.11105	0.265353
$372$	0	0
$373$	−34.6219	−1.79266	−0.896328	−	0.443392i	$-0.853775\pi$
$373$	−34.6219	−1.79266	−0.896328	+	0.443392i	$0.853775\pi$
$374$	0	0
$375$	0.398578	0.0205825
$376$	0	0
$377$	21.4264	1.10351
$378$	0	0
$379$	−13.0725	−0.671490	−0.335745	−	0.941953i	$-0.608988\pi$
$379$	−13.0725	−0.671490	−0.335745	+	0.941953i	$0.608988\pi$
$380$	0	0
$381$	13.1547	0.673933
$382$	0	0
$383$	18.8109	0.961191	0.480595	−	0.876943i	$-0.340421\pi$
$383$	18.8109	0.961191	0.480595	+	0.876943i	$0.340421\pi$
$384$	0	0
$385$	0.120101	0.00612093
$386$	0	0
$387$	7.70586	0.391711
$388$	0	0
$389$	−25.1136	−1.27331	−0.636655	−	0.771149i	$-0.719682\pi$
$389$	−25.1136	−1.27331	−0.636655	+	0.771149i	$0.719682\pi$
$390$	0	0
$391$	24.4048	1.23420
$392$	0	0
$393$	20.4872	1.03344
$394$	0	0
$395$	0.337146	0.0169636
$396$	0	0
$397$	32.3823	1.62522	0.812609	−	0.582809i	$-0.198046\pi$
$397$	32.3823	1.62522	0.812609	+	0.582809i	$0.198046\pi$
$398$	0	0
$399$	4.19935	0.210230
$400$	0	0
$401$	−25.4636	−1.27159	−0.635797	−	0.771857i	$-0.719328\pi$
$401$	−25.4636	−1.27159	−0.635797	+	0.771857i	$0.719328\pi$
$402$	0	0
$403$	26.8342	1.33671
$404$	0	0
$405$	0.0815939	0.00405443
$406$	0	0
$407$	9.81694	0.486608
$408$	0	0
$409$	4.45654	0.220362	0.110181	−	0.993912i	$-0.464857\pi$
$409$	4.45654	0.220362	0.110181	+	0.993912i	$0.464857\pi$
$410$	0	0
$411$	4.59518	0.226664
$412$	0	0
$413$	1.55107	0.0763230
$414$	0	0
$415$	−0.124654	−0.00611902
$416$	0	0
$417$	17.8155	0.872431
$418$	0	0
$419$	−14.4137	−0.704154	−0.352077	−	0.935971i	$-0.614524\pi$
$419$	−14.4137	−0.704154	−0.352077	+	0.935971i	$0.614524\pi$
$420$	0	0
$421$	23.8887	1.16427	0.582133	−	0.813094i	$-0.302218\pi$
$421$	23.8887	1.16427	0.582133	+	0.813094i	$0.302218\pi$
$422$	0	0
$423$	−6.95600	−0.338212
$424$	0	0
$425$	−15.1426	−0.734525
$426$	0	0
$427$	−16.3900	−0.793167
$428$	0	0
$429$	6.43653	0.310759
$430$	0	0
$431$	5.25262	0.253010	0.126505	−	0.991966i	$-0.459624\pi$
$431$	5.25262	0.253010	0.126505	+	0.991966i	$0.459624\pi$
$432$	0	0
$433$	−12.6843	−0.609569	−0.304784	−	0.952421i	$-0.598584\pi$
$433$	−12.6843	−0.609569	−0.304784	+	0.952421i	$0.598584\pi$
$434$	0	0
$435$	0.163912	0.00785896
$436$	0	0
$437$	−17.0278	−0.814552
$438$	0	0
$439$	−28.5669	−1.36342	−0.681711	−	0.731622i	$-0.738764\pi$
$439$	−28.5669	−1.36342	−0.681711	+	0.731622i	$0.738764\pi$
$440$	0	0
$441$	5.26611	0.250767
$442$	0	0
$443$	−15.7713	−0.749316	−0.374658	−	0.927163i	$-0.622240\pi$
$443$	−15.7713	−0.749316	−0.374658	+	0.927163i	$0.622240\pi$
$444$	0	0
$445$	−0.226241	−0.0107248
$446$	0	0
$447$	6.37424	0.301491
$448$	0	0
$449$	−8.04821	−0.379818	−0.189909	−	0.981802i	$-0.560819\pi$
$449$	−8.04821	−0.379818	−0.189909	+	0.981802i	$0.560819\pi$
$450$	0	0
$451$	11.3566	0.534760
$452$	0	0
$453$	−14.4621	−0.679489
$454$	0	0
$455$	−0.473206	−0.0221842
$456$	0	0
$457$	11.0151	0.515263	0.257632	−	0.966243i	$-0.417058\pi$
$457$	11.0151	0.515263	0.257632	+	0.966243i	$0.417058\pi$
$458$	0	0
$459$	14.5083	0.677191
$460$	0	0
$461$	−23.8300	−1.10987	−0.554936	−	0.831893i	$-0.687257\pi$
$461$	−23.8300	−1.10987	−0.554936	+	0.831893i	$0.687257\pi$
$462$	0	0
$463$	27.4926	1.27769	0.638846	−	0.769335i	$-0.279412\pi$
$463$	27.4926	1.27769	0.638846	+	0.769335i	$0.279412\pi$
$464$	0	0
$465$	0.205281	0.00951969
$466$	0	0
$467$	−20.3669	−0.942466	−0.471233	−	0.882009i	$-0.656191\pi$
$467$	−20.3669	−0.942466	−0.471233	+	0.882009i	$0.656191\pi$
$468$	0	0
$469$	−15.0987	−0.697194
$470$	0	0
$471$	19.3916	0.893519
$472$	0	0
$473$	−4.78967	−0.220229
$474$	0	0
$475$	10.5654	0.484773
$476$	0	0
$477$	−5.11255	−0.234088
$478$	0	0
$479$	21.5881	0.986387	0.493193	−	0.869920i	$-0.335829\pi$
$479$	21.5881	0.986387	0.493193	+	0.869920i	$0.335829\pi$
$480$	0	0
$481$	−38.6793	−1.76363
$482$	0	0
$483$	−16.0027	−0.728150
$484$	0	0
$485$	−0.134907	−0.00612579
$486$	0	0
$487$	2.79284	0.126556	0.0632779	−	0.997996i	$-0.479845\pi$
$487$	2.79284	0.126556	0.0632779	+	0.997996i	$0.479845\pi$
$488$	0	0
$489$	−14.7415	−0.666636
$490$	0	0
$491$	0.901316	0.0406758	0.0203379	−	0.999793i	$-0.493526\pi$
$491$	0.901316	0.0406758	0.0203379	+	0.999793i	$0.493526\pi$
$492$	0	0
$493$	−12.4568	−0.561027
$494$	0	0
$495$	−0.120136	−0.00539973
$496$	0	0
$497$	−30.5630	−1.37094
$498$	0	0
$499$	7.02014	0.314264	0.157132	−	0.987578i	$-0.449775\pi$
$499$	7.02014	0.314264	0.157132	+	0.987578i	$0.449775\pi$
$500$	0	0
$501$	−20.7343	−0.926340
$502$	0	0
$503$	−34.1866	−1.52431	−0.762153	−	0.647398i	$-0.775857\pi$
$503$	−34.1866	−1.52431	−0.762153	+	0.647398i	$0.775857\pi$
$504$	0	0
$505$	0.683913	0.0304337
$506$	0	0
$507$	−13.2199	−0.587114
$508$	0	0
$509$	−16.1114	−0.714126	−0.357063	−	0.934080i	$-0.616222\pi$
$509$	−16.1114	−0.714126	−0.357063	+	0.934080i	$0.616222\pi$
$510$	0	0
$511$	32.9172	1.45617
$512$	0	0
$513$	−10.1228	−0.446933
$514$	0	0
$515$	−0.391644	−0.0172579
$516$	0	0
$517$	4.32359	0.190151
$518$	0	0
$519$	4.50870	0.197910
$520$	0	0
$521$	10.8224	0.474139	0.237070	−	0.971493i	$-0.423813\pi$
$521$	10.8224	0.474139	0.237070	+	0.971493i	$0.423813\pi$
$522$	0	0
$523$	−40.9126	−1.78898	−0.894492	−	0.447083i	$-0.852463\pi$
$523$	−40.9126	−1.78898	−0.894492	+	0.447083i	$0.852463\pi$
$524$	0	0
$525$	9.92934	0.433352
$526$	0	0
$527$	−15.6008	−0.679580
$528$	0	0
$529$	41.8891	1.82127
$530$	0	0
$531$	−1.55152	−0.0673303
$532$	0	0
$533$	−44.7456	−1.93815
$534$	0	0
$535$	0.0301131	0.00130190
$536$	0	0
$537$	1.23553	0.0533170
$538$	0	0
$539$	−3.27322	−0.140987
$540$	0	0
$541$	−31.8939	−1.37123	−0.685614	−	0.727965i	$-0.740466\pi$
$541$	−31.8939	−1.37123	−0.685614	+	0.727965i	$0.740466\pi$
$542$	0	0
$543$	−23.9129	−1.02620
$544$	0	0
$545$	−0.554772	−0.0237638
$546$	0	0
$547$	−32.1087	−1.37287	−0.686435	−	0.727191i	$-0.740825\pi$
$547$	−32.1087	−1.37287	−0.686435	+	0.727191i	$0.740825\pi$
$548$	0	0
$549$	16.3948	0.699712
$550$	0	0
$551$	8.69142	0.370267
$552$	0	0
$553$	16.8009	0.714448
$554$	0	0
$555$	−0.295897	−0.0125601
$556$	0	0
$557$	−34.9765	−1.48200	−0.741000	−	0.671505i	$-0.765648\pi$
$557$	−34.9765	−1.48200	−0.741000	+	0.671505i	$0.765648\pi$
$558$	0	0
$559$	18.8716	0.798183
$560$	0	0
$561$	−3.74205	−0.157990
$562$	0	0
$563$	−19.4630	−0.820268	−0.410134	−	0.912025i	$-0.634518\pi$
$563$	−19.4630	−0.820268	−0.410134	+	0.912025i	$0.634518\pi$
$564$	0	0
$565$	0.379571	0.0159687
$566$	0	0
$567$	4.06606	0.170758
$568$	0	0
$569$	−13.9843	−0.586252	−0.293126	−	0.956074i	$-0.594696\pi$
$569$	−13.9843	−0.586252	−0.293126	+	0.956074i	$0.594696\pi$
$570$	0	0
$571$	−11.3182	−0.473651	−0.236826	−	0.971552i	$-0.576107\pi$
$571$	−11.3182	−0.473651	−0.236826	+	0.971552i	$0.576107\pi$
$572$	0	0
$573$	9.36790	0.391350
$574$	0	0
$575$	−40.2622	−1.67905
$576$	0	0
$577$	−5.51727	−0.229687	−0.114844	−	0.993384i	$-0.536637\pi$
$577$	−5.51727	−0.229687	−0.114844	+	0.993384i	$0.536637\pi$
$578$	0	0
$579$	−13.5468	−0.562987
$580$	0	0
$581$	−6.21186	−0.257712
$582$	0	0
$583$	3.17777	0.131610
$584$	0	0
$585$	0.473345	0.0195704
$586$	0	0
$587$	−31.3504	−1.29397	−0.646985	−	0.762503i	$-0.723970\pi$
$587$	−31.3504	−1.29397	−0.646985	+	0.762503i	$0.723970\pi$
$588$	0	0
$589$	10.8850	0.448510
$590$	0	0
$591$	−7.95000	−0.327019
$592$	0	0
$593$	39.3479	1.61583	0.807913	−	0.589302i	$-0.200597\pi$
$593$	39.3479	1.61583	0.807913	+	0.589302i	$0.200597\pi$
$594$	0	0
$595$	0.275111	0.0112785
$596$	0	0
$597$	19.3795	0.793150
$598$	0	0
$599$	−13.8238	−0.564825	−0.282413	−	0.959293i	$-0.591135\pi$
$599$	−13.8238	−0.564825	−0.282413	+	0.959293i	$0.591135\pi$
$600$	0	0
$601$	−43.6787	−1.78169	−0.890846	−	0.454306i	$-0.849887\pi$
$601$	−43.6787	−1.78169	−0.890846	+	0.454306i	$0.849887\pi$
$602$	0	0
$603$	15.1031	0.615048
$604$	0	0
$605$	−0.394891	−0.0160546
$606$	0	0
$607$	17.5233	0.711250	0.355625	−	0.934629i	$-0.384268\pi$
$607$	17.5233	0.711250	0.355625	+	0.934629i	$0.384268\pi$
$608$	0	0
$609$	8.16819	0.330992
$610$	0	0
$611$	−17.0352	−0.689170
$612$	0	0
$613$	23.6646	0.955804	0.477902	−	0.878413i	$-0.341397\pi$
$613$	23.6646	0.955804	0.477902	+	0.878413i	$0.341397\pi$
$614$	0	0
$615$	−0.342303	−0.0138030
$616$	0	0
$617$	2.35702	0.0948903	0.0474451	−	0.998874i	$-0.484892\pi$
$617$	2.35702	0.0948903	0.0474451	+	0.998874i	$0.484892\pi$
$618$	0	0
$619$	20.7292	0.833177	0.416589	−	0.909095i	$-0.363226\pi$
$619$	20.7292	0.833177	0.416589	+	0.909095i	$0.363226\pi$
$620$	0	0
$621$	38.5757	1.54799
$622$	0	0
$623$	−11.2742	−0.451692
$624$	0	0
$625$	24.9727	0.998907
$626$	0	0
$627$	2.61092	0.104270
$628$	0	0
$629$	22.4873	0.896626
$630$	0	0
$631$	6.74378	0.268466	0.134233	−	0.990950i	$-0.457143\pi$
$631$	6.74378	0.268466	0.134233	+	0.990950i	$0.457143\pi$
$632$	0	0
$633$	0.0395446	0.00157176
$634$	0	0
$635$	−0.601298	−0.0238618
$636$	0	0
$637$	12.8967	0.510984
$638$	0	0
$639$	30.5720	1.20941
$640$	0	0
$641$	−36.7633	−1.45206	−0.726032	−	0.687661i	$-0.758638\pi$
$641$	−36.7633	−1.45206	−0.726032	+	0.687661i	$0.758638\pi$
$642$	0	0
$643$	−32.4941	−1.28144	−0.640721	−	0.767774i	$-0.721364\pi$
$643$	−32.4941	−1.28144	−0.640721	+	0.767774i	$0.721364\pi$
$644$	0	0
$645$	0.144367	0.00568446
$646$	0	0
$647$	−10.7061	−0.420899	−0.210449	−	0.977605i	$-0.567493\pi$
$647$	−10.7061	−0.420899	−0.210449	+	0.977605i	$0.567493\pi$
$648$	0	0
$649$	0.964366	0.0378547
$650$	0	0
$651$	10.2298	0.400936
$652$	0	0
$653$	−14.8390	−0.580696	−0.290348	−	0.956921i	$-0.593771\pi$
$653$	−14.8390	−0.580696	−0.290348	+	0.956921i	$0.593771\pi$
$654$	0	0
$655$	−0.936466	−0.0365907
$656$	0	0
$657$	−32.9269	−1.28460
$658$	0	0
$659$	−17.7696	−0.692207	−0.346103	−	0.938196i	$-0.612495\pi$
$659$	−17.7696	−0.692207	−0.346103	+	0.938196i	$0.612495\pi$
$660$	0	0
$661$	−17.1922	−0.668698	−0.334349	−	0.942449i	$-0.608516\pi$
$661$	−17.1922	−0.668698	−0.334349	+	0.942449i	$0.608516\pi$
$662$	0	0
$663$	14.7439	0.572606
$664$	0	0
$665$	−0.191952	−0.00744357
$666$	0	0
$667$	−33.1210	−1.28245
$668$	0	0
$669$	8.82329	0.341128
$670$	0	0
$671$	−10.1904	−0.393395
$672$	0	0
$673$	−10.1821	−0.392492	−0.196246	−	0.980555i	$-0.562875\pi$
$673$	−10.1821	−0.392492	−0.196246	+	0.980555i	$0.562875\pi$
$674$	0	0
$675$	−23.9353	−0.921272
$676$	0	0
$677$	27.7753	1.06749	0.533746	−	0.845645i	$-0.320784\pi$
$677$	27.7753	1.06749	0.533746	+	0.845645i	$0.320784\pi$
$678$	0	0
$679$	−6.72278	−0.257997
$680$	0	0
$681$	−18.9955	−0.727908
$682$	0	0
$683$	−50.4494	−1.93039	−0.965197	−	0.261525i	$-0.915774\pi$
$683$	−50.4494	−1.93039	−0.965197	+	0.261525i	$0.915774\pi$
$684$	0	0
$685$	−0.210045	−0.00802541
$686$	0	0
$687$	22.7676	0.868639
$688$	0	0
$689$	−12.5206	−0.476996
$690$	0	0
$691$	−38.9659	−1.48233	−0.741167	−	0.671321i	$-0.765727\pi$
$691$	−38.9659	−1.48233	−0.741167	+	0.671321i	$0.765727\pi$
$692$	0	0
$693$	−5.98675	−0.227418
$694$	0	0
$695$	−0.814347	−0.0308899
$696$	0	0
$697$	26.0141	0.985353
$698$	0	0
$699$	12.7101	0.480739
$700$	0	0
$701$	22.9145	0.865470	0.432735	−	0.901521i	$-0.357549\pi$
$701$	22.9145	0.865470	0.432735	+	0.901521i	$0.357549\pi$
$702$	0	0
$703$	−15.6899	−0.591757
$704$	0	0
$705$	−0.130319	−0.00490810
$706$	0	0
$707$	34.0814	1.28176
$708$	0	0
$709$	17.0445	0.640121	0.320060	−	0.947397i	$-0.396297\pi$
$709$	17.0445	0.640121	0.320060	+	0.947397i	$0.396297\pi$
$710$	0	0
$711$	−16.8059	−0.630269
$712$	0	0
$713$	−41.4804	−1.55345
$714$	0	0
$715$	−0.294213	−0.0110029
$716$	0	0
$717$	6.25270	0.233511
$718$	0	0
$719$	−20.4652	−0.763224	−0.381612	−	0.924323i	$-0.624631\pi$
$719$	−20.4652	−0.763224	−0.381612	+	0.924323i	$0.624631\pi$
$720$	0	0
$721$	−19.5168	−0.726842
$722$	0	0
$723$	14.9418	0.555691
$724$	0	0
$725$	20.5508	0.763238
$726$	0	0
$727$	47.1147	1.74739	0.873693	−	0.486477i	$-0.161718\pi$
$727$	47.1147	1.74739	0.873693	+	0.486477i	$0.161718\pi$
$728$	0	0
$729$	9.34929	0.346270
$730$	0	0
$731$	−10.9715	−0.405796
$732$	0	0
$733$	5.80003	0.214229	0.107114	−	0.994247i	$-0.465839\pi$
$733$	5.80003	0.214229	0.107114	+	0.994247i	$0.465839\pi$
$734$	0	0
$735$	0.0986594	0.00363911
$736$	0	0
$737$	−9.38754	−0.345795
$738$	0	0
$739$	15.6216	0.574652	0.287326	−	0.957833i	$-0.407234\pi$
$739$	15.6216	0.574652	0.287326	+	0.957833i	$0.407234\pi$
$740$	0	0
$741$	−10.2872	−0.377909
$742$	0	0
$743$	−4.91148	−0.180185	−0.0900924	−	0.995933i	$-0.528716\pi$
$743$	−4.91148	−0.180185	−0.0900924	+	0.995933i	$0.528716\pi$
$744$	0	0
$745$	−0.291366	−0.0106748
$746$	0	0
$747$	6.21368	0.227347
$748$	0	0
$749$	1.50062	0.0548315
$750$	0	0
$751$	50.0234	1.82538	0.912689	−	0.408654i	$-0.134002\pi$
$751$	50.0234	1.82538	0.912689	+	0.408654i	$0.134002\pi$
$752$	0	0
$753$	−0.933880	−0.0340325
$754$	0	0
$755$	0.661061	0.0240585
$756$	0	0
$757$	5.99117	0.217753	0.108876	−	0.994055i	$-0.465275\pi$
$757$	5.99117	0.217753	0.108876	+	0.994055i	$0.465275\pi$
$758$	0	0
$759$	−9.94961	−0.361148
$760$	0	0
$761$	−36.1259	−1.30956	−0.654781	−	0.755819i	$-0.727239\pi$
$761$	−36.1259	−1.30956	−0.654781	+	0.755819i	$0.727239\pi$
$762$	0	0
$763$	−27.6459	−1.00085
$764$	0	0
$765$	−0.275192	−0.00994958
$766$	0	0
$767$	−3.79966	−0.137198
$768$	0	0
$769$	22.7803	0.821480	0.410740	−	0.911752i	$-0.365270\pi$
$769$	22.7803	0.821480	0.410740	+	0.911752i	$0.365270\pi$
$770$	0	0
$771$	−16.5686	−0.596704
$772$	0	0
$773$	−26.1664	−0.941139	−0.470569	−	0.882363i	$-0.655951\pi$
$773$	−26.1664	−0.941139	−0.470569	+	0.882363i	$0.655951\pi$
$774$	0	0
$775$	25.7376	0.924522
$776$	0	0
$777$	−14.7454	−0.528988
$778$	0	0
$779$	−18.1506	−0.650314
$780$	0	0
$781$	−19.0024	−0.679958
$782$	0	0
$783$	−19.6900	−0.703663
$784$	0	0
$785$	−0.886389	−0.0316366
$786$	0	0
$787$	−28.1832	−1.00462	−0.502312	−	0.864687i	$-0.667517\pi$
$787$	−28.1832	−1.00462	−0.502312	+	0.864687i	$0.667517\pi$
$788$	0	0
$789$	25.0146	0.890544
$790$	0	0
$791$	18.9151	0.672545
$792$	0	0
$793$	40.1507	1.42579
$794$	0	0
$795$	−0.0957824	−0.00339705
$796$	0	0
$797$	46.9756	1.66396	0.831980	−	0.554805i	$-0.187207\pi$
$797$	46.9756	1.66396	0.831980	+	0.554805i	$0.187207\pi$
$798$	0	0
$799$	9.90387	0.350374
$800$	0	0
$801$	11.2775	0.398472
$802$	0	0
$803$	20.4661	0.722232
$804$	0	0
$805$	0.731484	0.0257814
$806$	0	0
$807$	−1.99359	−0.0701775
$808$	0	0
$809$	−47.9346	−1.68529	−0.842646	−	0.538468i	$-0.819003\pi$
$809$	−47.9346	−1.68529	−0.842646	+	0.538468i	$0.819003\pi$
$810$	0	0
$811$	6.23321	0.218878	0.109439	−	0.993994i	$-0.465095\pi$
$811$	6.23321	0.218878	0.109439	+	0.993994i	$0.465095\pi$
$812$	0	0
$813$	24.8115	0.870177
$814$	0	0
$815$	0.673834	0.0236034
$816$	0	0
$817$	7.65509	0.267818
$818$	0	0
$819$	23.5881	0.824236
$820$	0	0
$821$	50.8087	1.77323	0.886617	−	0.462504i	$-0.153049\pi$
$821$	50.8087	1.77323	0.886617	+	0.462504i	$0.153049\pi$
$822$	0	0
$823$	−55.0394	−1.91855	−0.959276	−	0.282471i	$-0.908846\pi$
$823$	−55.0394	−1.91855	−0.959276	+	0.282471i	$0.908846\pi$
$824$	0	0
$825$	6.17351	0.214934
$826$	0	0
$827$	17.9643	0.624681	0.312341	−	0.949970i	$-0.398887\pi$
$827$	17.9643	0.624681	0.312341	+	0.949970i	$0.398887\pi$
$828$	0	0
$829$	−1.38808	−0.0482100	−0.0241050	−	0.999709i	$-0.507674\pi$
$829$	−1.38808	−0.0482100	−0.0241050	+	0.999709i	$0.507674\pi$
$830$	0	0
$831$	−3.13872	−0.108881
$832$	0	0
$833$	−7.49782	−0.259784
$834$	0	0
$835$	0.947762	0.0327987
$836$	0	0
$837$	−24.6595	−0.852358
$838$	0	0
$839$	16.2891	0.562362	0.281181	−	0.959655i	$-0.409274\pi$
$839$	16.2891	0.562362	0.281181	+	0.959655i	$0.409274\pi$
$840$	0	0
$841$	−12.0942	−0.417043
$842$	0	0
$843$	−0.595522	−0.0205109
$844$	0	0
$845$	0.604278	0.0207878
$846$	0	0
$847$	−19.6786	−0.676163
$848$	0	0
$849$	10.4804	0.359686
$850$	0	0
$851$	59.7906	2.04960
$852$	0	0
$853$	43.9144	1.50360	0.751800	−	0.659391i	$-0.229186\pi$
$853$	43.9144	1.50360	0.751800	+	0.659391i	$0.229186\pi$
$854$	0	0
$855$	0.192008	0.00656654
$856$	0	0
$857$	19.7541	0.674786	0.337393	−	0.941364i	$-0.390455\pi$
$857$	19.7541	0.674786	0.337393	+	0.941364i	$0.390455\pi$
$858$	0	0
$859$	7.69544	0.262565	0.131283	−	0.991345i	$-0.458090\pi$
$859$	7.69544	0.262565	0.131283	+	0.991345i	$0.458090\pi$
$860$	0	0
$861$	−17.0580	−0.581334
$862$	0	0
$863$	−49.1936	−1.67457	−0.837284	−	0.546768i	$-0.815858\pi$
$863$	−49.1936	−1.67457	−0.837284	+	0.546768i	$0.815858\pi$
$864$	0	0
$865$	−0.206092	−0.00700734
$866$	0	0
$867$	7.30420	0.248064
$868$	0	0
$869$	10.4459	0.354352
$870$	0	0
$871$	36.9875	1.25327
$872$	0	0
$873$	6.72475	0.227598
$874$	0	0
$875$	−0.907903	−0.0306927
$876$	0	0
$877$	19.7677	0.667509	0.333755	−	0.942660i	$-0.391684\pi$
$877$	19.7677	0.667509	0.333755	+	0.942660i	$0.391684\pi$
$878$	0	0
$879$	−4.38570	−0.147926
$880$	0	0
$881$	−3.28078	−0.110532	−0.0552661	−	0.998472i	$-0.517601\pi$
$881$	−3.28078	−0.110532	−0.0552661	+	0.998472i	$0.517601\pi$
$882$	0	0
$883$	4.51248	0.151857	0.0759286	−	0.997113i	$-0.475808\pi$
$883$	4.51248	0.151857	0.0759286	+	0.997113i	$0.475808\pi$
$884$	0	0
$885$	−0.0290674	−0.000977089 0
$886$	0	0
$887$	−8.04559	−0.270145	−0.135072	−	0.990836i	$-0.543127\pi$
$887$	−8.04559	−0.270145	−0.135072	+	0.990836i	$0.543127\pi$
$888$	0	0
$889$	−29.9644	−1.00497
$890$	0	0
$891$	2.52805	0.0846928
$892$	0	0
$893$	−6.91017	−0.231240
$894$	0	0
$895$	−0.0564758	−0.00188778
$896$	0	0
$897$	39.2020	1.30892
$898$	0	0
$899$	21.1726	0.706145
$900$	0	0
$901$	7.27918	0.242505
$902$	0	0
$903$	7.19425	0.239410
$904$	0	0
$905$	1.09305	0.0363344
$906$	0	0
$907$	45.3701	1.50649	0.753245	−	0.657740i	$-0.228487\pi$
$907$	45.3701	1.50649	0.753245	+	0.657740i	$0.228487\pi$
$908$	0	0
$909$	−34.0913	−1.13074
$910$	0	0
$911$	−14.3507	−0.475461	−0.237730	−	0.971331i	$-0.576404\pi$
$911$	−14.3507	−0.475461	−0.237730	+	0.971331i	$0.576404\pi$
$912$	0	0
$913$	−3.86219	−0.127820
$914$	0	0
$915$	0.307152	0.0101541
$916$	0	0
$917$	−46.6668	−1.54107
$918$	0	0
$919$	−16.2392	−0.535682	−0.267841	−	0.963463i	$-0.586310\pi$
$919$	−16.2392	−0.535682	−0.267841	+	0.963463i	$0.586310\pi$
$920$	0	0
$921$	25.1761	0.829580
$922$	0	0
$923$	74.8704	2.46439
$924$	0	0
$925$	−37.0987	−1.21980
$926$	0	0
$927$	19.5225	0.641203
$928$	0	0
$929$	5.90381	0.193698	0.0968488	−	0.995299i	$-0.469124\pi$
$929$	5.90381	0.193698	0.0968488	+	0.995299i	$0.469124\pi$
$930$	0	0
$931$	5.23141	0.171453
$932$	0	0
$933$	−19.3450	−0.633328
$934$	0	0
$935$	0.171049	0.00559389
$936$	0	0
$937$	30.1527	0.985045	0.492522	−	0.870300i	$-0.336075\pi$
$937$	30.1527	0.985045	0.492522	+	0.870300i	$0.336075\pi$
$938$	0	0
$939$	−18.2901	−0.596874
$940$	0	0
$941$	28.5490	0.930671	0.465335	−	0.885134i	$-0.345934\pi$
$941$	28.5490	0.930671	0.465335	+	0.885134i	$0.345934\pi$
$942$	0	0
$943$	69.1679	2.25242
$944$	0	0
$945$	0.434857	0.0141459
$946$	0	0
$947$	−9.95204	−0.323398	−0.161699	−	0.986840i	$-0.551697\pi$
$947$	−9.95204	−0.323398	−0.161699	+	0.986840i	$0.551697\pi$
$948$	0	0
$949$	−80.6375	−2.61761
$950$	0	0
$951$	−7.86556	−0.255058
$952$	0	0
$953$	−2.76818	−0.0896701	−0.0448351	−	0.998994i	$-0.514276\pi$
$953$	−2.76818	−0.0896701	−0.0448351	+	0.998994i	$0.514276\pi$
$954$	0	0
$955$	−0.428206	−0.0138564
$956$	0	0
$957$	5.07852	0.164165
$958$	0	0
$959$	−10.4672	−0.338002
$960$	0	0
$961$	−4.48368	−0.144635
$962$	0	0
$963$	−1.50106	−0.0483710
$964$	0	0
$965$	0.619223	0.0199335
$966$	0	0
$967$	50.0595	1.60980	0.804902	−	0.593408i	$-0.202218\pi$
$967$	50.0595	1.60980	0.804902	+	0.593408i	$0.202218\pi$
$968$	0	0
$969$	5.98073	0.192129
$970$	0	0
$971$	−14.7579	−0.473603	−0.236801	−	0.971558i	$-0.576099\pi$
$971$	−14.7579	−0.473603	−0.236801	+	0.971558i	$0.576099\pi$
$972$	0	0
$973$	−40.5812	−1.30097
$974$	0	0
$975$	−24.3240	−0.778991
$976$	0	0
$977$	−14.0606	−0.449838	−0.224919	−	0.974378i	$-0.572212\pi$
$977$	−14.0606	−0.449838	−0.224919	+	0.974378i	$0.572212\pi$
$978$	0	0
$979$	−7.00968	−0.224030
$980$	0	0
$981$	27.6540	0.882924
$982$	0	0
$983$	−3.20900	−0.102351	−0.0511756	−	0.998690i	$-0.516297\pi$
$983$	−3.20900	−0.102351	−0.0511756	+	0.998690i	$0.516297\pi$
$984$	0	0
$985$	0.363393	0.0115787
$986$	0	0
$987$	−6.49417	−0.206712
$988$	0	0
$989$	−29.1718	−0.927608
$990$	0	0
$991$	32.5165	1.03292	0.516460	−	0.856311i	$-0.327250\pi$
$991$	32.5165	1.03292	0.516460	+	0.856311i	$0.327250\pi$
$992$	0	0
$993$	24.9254	0.790983
$994$	0	0
$995$	−0.885835	−0.0280828
$996$	0	0
$997$	−57.4250	−1.81867	−0.909333	−	0.416069i	$-0.863408\pi$
$997$	−57.4250	−1.81867	−0.909333	+	0.416069i	$0.863408\pi$
$998$	0	0
$999$	35.5447	1.12459

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8032.2.a.g.1.13	✓	30
4.3	odd	2		8032.2.a.j.1.18	yes	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.g.1.13	✓	30	1.1	even	1	trivial
8032.2.a.j.1.18	yes	30	4.3	odd	2

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)

\(f(q)\)	\(=\)	\(q-0.933880 q^{3} +0.0426876 q^{5} +2.12724 q^{7} -2.12787 q^{9} +O(q^{10})\)	\(q-0.933880 q^{3} +0.0426876 q^{5} +2.12724 q^{7} -2.12787 q^{9} +1.32260 q^{11} -5.21113 q^{13} -0.0398651 q^{15} +3.02963 q^{17} -2.11385 q^{19} -1.98659 q^{21} +8.05538 q^{23} -4.99818 q^{25} +4.78881 q^{27} -4.11166 q^{29} -5.14940 q^{31} -1.23515 q^{33} +0.0908069 q^{35} +7.42245 q^{37} +4.86657 q^{39} +8.58655 q^{41} -3.62140 q^{43} -0.0908335 q^{45} +3.26900 q^{47} -2.47483 q^{49} -2.82931 q^{51} +2.40266 q^{53} +0.0564586 q^{55} +1.97408 q^{57} +0.729143 q^{59} -7.70480 q^{61} -4.52649 q^{63} -0.222450 q^{65} -7.09779 q^{67} -7.52276 q^{69} -14.3674 q^{71} +15.4741 q^{73} +4.66770 q^{75} +2.81350 q^{77} +7.89798 q^{79} +1.91142 q^{81} -2.92015 q^{83} +0.129328 q^{85} +3.83980 q^{87} -5.29992 q^{89} -11.0853 q^{91} +4.80892 q^{93} -0.0902350 q^{95} -3.16032 q^{97} -2.81432 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 9 q^{3} - 5 q^{7} + 35 q^{9}+O(q^{10}) \) 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

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