LMFDB - Embedded newform 8032.2.a.i.1.18

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			1.18
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)

$f(q)$	$=$	$q+0.700584 q^{3} -3.32459 q^{5} -0.693647 q^{7} -2.50918 q^{9} +O(q^{10})$	$q+0.700584 q^{3} -3.32459 q^{5} -0.693647 q^{7} -2.50918 q^{9} -1.62924 q^{11} -1.46836 q^{13} -2.32915 q^{15} +3.60016 q^{17} -4.96695 q^{19} -0.485957 q^{21} +5.69938 q^{23} +6.05289 q^{25} -3.85964 q^{27} -7.60846 q^{29} -1.49158 q^{31} -1.14142 q^{33} +2.30609 q^{35} +2.17288 q^{37} -1.02871 q^{39} -6.26654 q^{41} -7.63700 q^{43} +8.34200 q^{45} +7.10161 q^{47} -6.51885 q^{49} +2.52221 q^{51} -5.44645 q^{53} +5.41654 q^{55} -3.47976 q^{57} -12.3331 q^{59} +7.01122 q^{61} +1.74049 q^{63} +4.88170 q^{65} -7.91134 q^{67} +3.99289 q^{69} -8.52164 q^{71} +3.34475 q^{73} +4.24055 q^{75} +1.13012 q^{77} -4.45172 q^{79} +4.82354 q^{81} -3.19698 q^{83} -11.9690 q^{85} -5.33036 q^{87} -1.90238 q^{89} +1.01852 q^{91} -1.04498 q^{93} +16.5131 q^{95} +3.67875 q^{97} +4.08806 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.700584	0.404482	0.202241	−	0.979336i	$-0.435178\pi$
$3$	0.700584	0.404482	0.202241	+	0.979336i	$0.435178\pi$
$4$	0	0
$5$	−3.32459	−1.48680	−0.743400	−	0.668847i	$-0.766788\pi$
$5$	−3.32459	−1.48680	−0.743400	+	0.668847i	$0.766788\pi$
$6$	0	0
$7$	−0.693647	−0.262174	−0.131087	−	0.991371i	$-0.541847\pi$
$7$	−0.693647	−0.262174	−0.131087	+	0.991371i	$0.541847\pi$
$8$	0	0
$9$	−2.50918	−0.836394
$10$	0	0
$11$	−1.62924	−0.491234	−0.245617	−	0.969367i	$-0.578991\pi$
$11$	−1.62924	−0.491234	−0.245617	+	0.969367i	$0.578991\pi$
$12$	0	0
$13$	−1.46836	−0.407250	−0.203625	−	0.979049i	$-0.565272\pi$
$13$	−1.46836	−0.407250	−0.203625	+	0.979049i	$0.565272\pi$
$14$	0	0
$15$	−2.32915	−0.601385
$16$	0	0
$17$	3.60016	0.873166	0.436583	−	0.899664i	$-0.356189\pi$
$17$	3.60016	0.873166	0.436583	+	0.899664i	$0.356189\pi$
$18$	0	0
$19$	−4.96695	−1.13950	−0.569748	−	0.821819i	$-0.692959\pi$
$19$	−4.96695	−1.13950	−0.569748	+	0.821819i	$0.692959\pi$
$20$	0	0
$21$	−0.485957	−0.106045
$22$	0	0
$23$	5.69938	1.18840	0.594201	−	0.804316i	$-0.297468\pi$
$23$	5.69938	1.18840	0.594201	+	0.804316i	$0.297468\pi$
$24$	0	0
$25$	6.05289	1.21058
$26$	0	0
$27$	−3.85964	−0.742789
$28$	0	0
$29$	−7.60846	−1.41285	−0.706427	−	0.707785i	$-0.749694\pi$
$29$	−7.60846	−1.41285	−0.706427	+	0.707785i	$0.749694\pi$
$30$	0	0
$31$	−1.49158	−0.267896	−0.133948	−	0.990988i	$-0.542766\pi$
$31$	−1.49158	−0.267896	−0.133948	+	0.990988i	$0.542766\pi$
$32$	0	0
$33$	−1.14142	−0.198695
$34$	0	0
$35$	2.30609	0.389800
$36$	0	0
$37$	2.17288	0.357220	0.178610	−	0.983920i	$-0.442840\pi$
$37$	2.17288	0.357220	0.178610	+	0.983920i	$0.442840\pi$
$38$	0	0
$39$	−1.02871	−0.164725
$40$	0	0
$41$	−6.26654	−0.978669	−0.489334	−	0.872096i	$-0.662760\pi$
$41$	−6.26654	−0.978669	−0.489334	+	0.872096i	$0.662760\pi$
$42$	0	0
$43$	−7.63700	−1.16463	−0.582315	−	0.812963i	$-0.697853\pi$
$43$	−7.63700	−1.16463	−0.582315	+	0.812963i	$0.697853\pi$
$44$	0	0
$45$	8.34200	1.24355
$46$	0	0
$47$	7.10161	1.03588	0.517938	−	0.855418i	$-0.326700\pi$
$47$	7.10161	1.03588	0.517938	+	0.855418i	$0.326700\pi$
$48$	0	0
$49$	−6.51885	−0.931265
$50$	0	0
$51$	2.52221	0.353180
$52$	0	0
$53$	−5.44645	−0.748128	−0.374064	−	0.927403i	$-0.622036\pi$
$53$	−5.44645	−0.748128	−0.374064	+	0.927403i	$0.622036\pi$
$54$	0	0
$55$	5.41654	0.730367
$56$	0	0
$57$	−3.47976	−0.460906
$58$	0	0
$59$	−12.3331	−1.60563	−0.802814	−	0.596229i	$-0.796665\pi$
$59$	−12.3331	−1.60563	−0.802814	+	0.596229i	$0.796665\pi$
$60$	0	0
$61$	7.01122	0.897694	0.448847	−	0.893609i	$-0.351835\pi$
$61$	7.01122	0.897694	0.448847	+	0.893609i	$0.351835\pi$
$62$	0	0
$63$	1.74049	0.219281
$64$	0	0
$65$	4.88170	0.605500
$66$	0	0
$67$	−7.91134	−0.966524	−0.483262	−	0.875476i	$-0.660548\pi$
$67$	−7.91134	−0.966524	−0.483262	+	0.875476i	$0.660548\pi$
$68$	0	0
$69$	3.99289	0.480688
$70$	0	0
$71$	−8.52164	−1.01133	−0.505667	−	0.862729i	$-0.668753\pi$
$71$	−8.52164	−1.01133	−0.505667	+	0.862729i	$0.668753\pi$
$72$	0	0
$73$	3.34475	0.391473	0.195737	−	0.980656i	$-0.437290\pi$
$73$	3.34475	0.391473	0.195737	+	0.980656i	$0.437290\pi$
$74$	0	0
$75$	4.24055	0.489657
$76$	0	0
$77$	1.13012	0.128789
$78$	0	0
$79$	−4.45172	−0.500858	−0.250429	−	0.968135i	$-0.580572\pi$
$79$	−4.45172	−0.500858	−0.250429	+	0.968135i	$0.580572\pi$
$80$	0	0
$81$	4.82354	0.535949
$82$	0	0
$83$	−3.19698	−0.350914	−0.175457	−	0.984487i	$-0.556140\pi$
$83$	−3.19698	−0.350914	−0.175457	+	0.984487i	$0.556140\pi$
$84$	0	0
$85$	−11.9690	−1.29822
$86$	0	0
$87$	−5.33036	−0.571475
$88$	0	0
$89$	−1.90238	−0.201652	−0.100826	−	0.994904i	$-0.532148\pi$
$89$	−1.90238	−0.201652	−0.100826	+	0.994904i	$0.532148\pi$
$90$	0	0
$91$	1.01852	0.106770
$92$	0	0
$93$	−1.04498	−0.108359
$94$	0	0
$95$	16.5131	1.69420
$96$	0	0
$97$	3.67875	0.373521	0.186760	−	0.982406i	$-0.440201\pi$
$97$	3.67875	0.373521	0.186760	+	0.982406i	$0.440201\pi$
$98$	0	0
$99$	4.08806	0.410865
$100$	0	0
$101$	15.9256	1.58465	0.792326	−	0.610098i	$-0.208870\pi$
$101$	15.9256	1.58465	0.792326	+	0.610098i	$0.208870\pi$
$102$	0	0
$103$	8.72064	0.859270	0.429635	−	0.903003i	$-0.358642\pi$
$103$	8.72064	0.859270	0.429635	+	0.903003i	$0.358642\pi$
$104$	0	0
$105$	1.61561	0.157667
$106$	0	0
$107$	−11.2577	−1.08833	−0.544163	−	0.838980i	$-0.683153\pi$
$107$	−11.2577	−1.08833	−0.544163	+	0.838980i	$0.683153\pi$
$108$	0	0
$109$	13.0955	1.25432	0.627162	−	0.778889i	$-0.284216\pi$
$109$	13.0955	1.25432	0.627162	+	0.778889i	$0.284216\pi$
$110$	0	0
$111$	1.52229	0.144489
$112$	0	0
$113$	−1.75471	−0.165069	−0.0825346	−	0.996588i	$-0.526301\pi$
$113$	−1.75471	−0.165069	−0.0825346	+	0.996588i	$0.526301\pi$
$114$	0	0
$115$	−18.9481	−1.76692
$116$	0	0
$117$	3.68439	0.340622
$118$	0	0
$119$	−2.49724	−0.228921
$120$	0	0
$121$	−8.34558	−0.758689
$122$	0	0
$123$	−4.39024	−0.395854
$124$	0	0
$125$	−3.50041	−0.313086
$126$	0	0
$127$	−8.39339	−0.744793	−0.372396	−	0.928074i	$-0.621464\pi$
$127$	−8.39339	−0.744793	−0.372396	+	0.928074i	$0.621464\pi$
$128$	0	0
$129$	−5.35035	−0.471072
$130$	0	0
$131$	15.2658	1.33378	0.666891	−	0.745155i	$-0.267625\pi$
$131$	15.2658	1.33378	0.666891	+	0.745155i	$0.267625\pi$
$132$	0	0
$133$	3.44531	0.298746
$134$	0	0
$135$	12.8317	1.10438
$136$	0	0
$137$	1.95972	0.167430	0.0837150	−	0.996490i	$-0.473321\pi$
$137$	1.95972	0.167430	0.0837150	+	0.996490i	$0.473321\pi$
$138$	0	0
$139$	20.6887	1.75480	0.877398	−	0.479764i	$-0.159278\pi$
$139$	20.6887	1.75480	0.877398	+	0.479764i	$0.159278\pi$
$140$	0	0
$141$	4.97527	0.418993
$142$	0	0
$143$	2.39231	0.200055
$144$	0	0
$145$	25.2950	2.10063
$146$	0	0
$147$	−4.56700	−0.376680
$148$	0	0
$149$	12.7977	1.04843	0.524213	−	0.851587i	$-0.324360\pi$
$149$	12.7977	1.04843	0.524213	+	0.851587i	$0.324360\pi$
$150$	0	0
$151$	−10.9584	−0.891785	−0.445892	−	0.895087i	$-0.647114\pi$
$151$	−10.9584	−0.891785	−0.445892	+	0.895087i	$0.647114\pi$
$152$	0	0
$153$	−9.03345	−0.730311
$154$	0	0
$155$	4.95890	0.398308
$156$	0	0
$157$	−1.75527	−0.140086	−0.0700430	−	0.997544i	$-0.522314\pi$
$157$	−1.75527	−0.140086	−0.0700430	+	0.997544i	$0.522314\pi$
$158$	0	0
$159$	−3.81570	−0.302604
$160$	0	0
$161$	−3.95335	−0.311568
$162$	0	0
$163$	11.5521	0.904833	0.452416	−	0.891807i	$-0.350562\pi$
$163$	11.5521	0.904833	0.452416	+	0.891807i	$0.350562\pi$
$164$	0	0
$165$	3.79474	0.295420
$166$	0	0
$167$	20.7368	1.60467	0.802333	−	0.596877i	$-0.203592\pi$
$167$	20.7368	1.60467	0.802333	+	0.596877i	$0.203592\pi$
$168$	0	0
$169$	−10.8439	−0.834147
$170$	0	0
$171$	12.4630	0.953068
$172$	0	0
$173$	−21.9040	−1.66533	−0.832664	−	0.553779i	$-0.813185\pi$
$173$	−21.9040	−1.66533	−0.832664	+	0.553779i	$0.813185\pi$
$174$	0	0
$175$	−4.19856	−0.317382
$176$	0	0
$177$	−8.64035	−0.649448
$178$	0	0
$179$	13.1066	0.979630	0.489815	−	0.871826i	$-0.337064\pi$
$179$	13.1066	0.979630	0.489815	+	0.871826i	$0.337064\pi$
$180$	0	0
$181$	10.7769	0.801043	0.400522	−	0.916287i	$-0.368829\pi$
$181$	10.7769	0.801043	0.400522	+	0.916287i	$0.368829\pi$
$182$	0	0
$183$	4.91194	0.363101
$184$	0	0
$185$	−7.22394	−0.531114
$186$	0	0
$187$	−5.86551	−0.428929
$188$	0	0
$189$	2.67723	0.194740
$190$	0	0
$191$	19.1757	1.38751	0.693753	−	0.720213i	$-0.255956\pi$
$191$	19.1757	1.38751	0.693753	+	0.720213i	$0.255956\pi$
$192$	0	0
$193$	15.7014	1.13021	0.565107	−	0.825018i	$-0.308835\pi$
$193$	15.7014	1.13021	0.565107	+	0.825018i	$0.308835\pi$
$194$	0	0
$195$	3.42004	0.244914
$196$	0	0
$197$	−13.5400	−0.964688	−0.482344	−	0.875982i	$-0.660214\pi$
$197$	−13.5400	−0.964688	−0.482344	+	0.875982i	$0.660214\pi$
$198$	0	0
$199$	7.16825	0.508144	0.254072	−	0.967185i	$-0.418230\pi$
$199$	7.16825	0.508144	0.254072	+	0.967185i	$0.418230\pi$
$200$	0	0
$201$	−5.54255	−0.390942
$202$	0	0
$203$	5.27758	0.370413
$204$	0	0
$205$	20.8337	1.45509
$206$	0	0
$207$	−14.3008	−0.993973
$208$	0	0
$209$	8.09234	0.559759
$210$	0	0
$211$	16.1740	1.11346	0.556732	−	0.830692i	$-0.312055\pi$
$211$	16.1740	1.11346	0.556732	+	0.830692i	$0.312055\pi$
$212$	0	0
$213$	−5.97013	−0.409066
$214$	0	0
$215$	25.3899	1.73157
$216$	0	0
$217$	1.03463	0.0702353
$218$	0	0
$219$	2.34328	0.158344
$220$	0	0
$221$	−5.28633	−0.355597
$222$	0	0
$223$	3.81415	0.255415	0.127707	−	0.991812i	$-0.459238\pi$
$223$	3.81415	0.255415	0.127707	+	0.991812i	$0.459238\pi$
$224$	0	0
$225$	−15.1878	−1.01252
$226$	0	0
$227$	11.0689	0.734671	0.367335	−	0.930089i	$-0.380270\pi$
$227$	11.0689	0.734671	0.367335	+	0.930089i	$0.380270\pi$
$228$	0	0
$229$	−4.88432	−0.322765	−0.161382	−	0.986892i	$-0.551595\pi$
$229$	−4.88432	−0.322765	−0.161382	+	0.986892i	$0.551595\pi$
$230$	0	0
$231$	0.791740	0.0520927
$232$	0	0
$233$	17.5555	1.15010	0.575049	−	0.818119i	$-0.304983\pi$
$233$	17.5555	1.15010	0.575049	+	0.818119i	$0.304983\pi$
$234$	0	0
$235$	−23.6099	−1.54014
$236$	0	0
$237$	−3.11880	−0.202588
$238$	0	0
$239$	13.5766	0.878196	0.439098	−	0.898439i	$-0.355298\pi$
$239$	13.5766	0.878196	0.439098	+	0.898439i	$0.355298\pi$
$240$	0	0
$241$	10.8939	0.701738	0.350869	−	0.936425i	$-0.385886\pi$
$241$	10.8939	0.701738	0.350869	+	0.936425i	$0.385886\pi$
$242$	0	0
$243$	14.9582	0.959571
$244$	0	0
$245$	21.6725	1.38461
$246$	0	0
$247$	7.29327	0.464060
$248$	0	0
$249$	−2.23975	−0.141939
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−9.28564	−0.583783
$254$	0	0
$255$	−8.38531	−0.525108
$256$	0	0
$257$	−19.5360	−1.21862	−0.609312	−	0.792931i	$-0.708554\pi$
$257$	−19.5360	−1.21862	−0.609312	+	0.792931i	$0.708554\pi$
$258$	0	0
$259$	−1.50721	−0.0936536
$260$	0	0
$261$	19.0910	1.18170
$262$	0	0
$263$	13.9219	0.858461	0.429230	−	0.903195i	$-0.358785\pi$
$263$	13.9219	0.858461	0.429230	+	0.903195i	$0.358785\pi$
$264$	0	0
$265$	18.1072	1.11232
$266$	0	0
$267$	−1.33278	−0.0815645
$268$	0	0
$269$	−16.0001	−0.975545	−0.487773	−	0.872971i	$-0.662190\pi$
$269$	−16.0001	−0.975545	−0.487773	+	0.872971i	$0.662190\pi$
$270$	0	0
$271$	30.6996	1.86487	0.932435	−	0.361337i	$-0.117680\pi$
$271$	30.6996	1.86487	0.932435	+	0.361337i	$0.117680\pi$
$272$	0	0
$273$	0.713561	0.0431867
$274$	0	0
$275$	−9.86159	−0.594676
$276$	0	0
$277$	−10.7480	−0.645782	−0.322891	−	0.946436i	$-0.604655\pi$
$277$	−10.7480	−0.645782	−0.322891	+	0.946436i	$0.604655\pi$
$278$	0	0
$279$	3.74265	0.224067
$280$	0	0
$281$	−1.21486	−0.0724723	−0.0362361	−	0.999343i	$-0.511537\pi$
$281$	−1.21486	−0.0724723	−0.0362361	+	0.999343i	$0.511537\pi$
$282$	0	0
$283$	9.84210	0.585052	0.292526	−	0.956258i	$-0.405504\pi$
$283$	9.84210	0.585052	0.292526	+	0.956258i	$0.405504\pi$
$284$	0	0
$285$	11.5688	0.685275
$286$	0	0
$287$	4.34676	0.256581
$288$	0	0
$289$	−4.03888	−0.237581
$290$	0	0
$291$	2.57727	0.151082
$292$	0	0
$293$	14.9487	0.873312	0.436656	−	0.899628i	$-0.356163\pi$
$293$	14.9487	0.873312	0.436656	+	0.899628i	$0.356163\pi$
$294$	0	0
$295$	41.0024	2.38725
$296$	0	0
$297$	6.28828	0.364883
$298$	0	0
$299$	−8.36874	−0.483977
$300$	0	0
$301$	5.29738	0.305336
$302$	0	0
$303$	11.1572	0.640964
$304$	0	0
$305$	−23.3094	−1.33469
$306$	0	0
$307$	−21.6363	−1.23485	−0.617425	−	0.786630i	$-0.711824\pi$
$307$	−21.6363	−1.23485	−0.617425	+	0.786630i	$0.711824\pi$
$308$	0	0
$309$	6.10954	0.347560
$310$	0	0
$311$	−7.14392	−0.405094	−0.202547	−	0.979272i	$-0.564922\pi$
$311$	−7.14392	−0.405094	−0.202547	+	0.979272i	$0.564922\pi$
$312$	0	0
$313$	−26.5702	−1.50184	−0.750919	−	0.660394i	$-0.770389\pi$
$313$	−26.5702	−1.50184	−0.750919	+	0.660394i	$0.770389\pi$
$314$	0	0
$315$	−5.78640	−0.326027
$316$	0	0
$317$	−1.31690	−0.0739647	−0.0369824	−	0.999316i	$-0.511775\pi$
$317$	−1.31690	−0.0739647	−0.0369824	+	0.999316i	$0.511775\pi$
$318$	0	0
$319$	12.3960	0.694042
$320$	0	0
$321$	−7.88698	−0.440208
$322$	0	0
$323$	−17.8818	−0.994969
$324$	0	0
$325$	−8.88782	−0.493008
$326$	0	0
$327$	9.17452	0.507352
$328$	0	0
$329$	−4.92600	−0.271579
$330$	0	0
$331$	−5.69516	−0.313034	−0.156517	−	0.987675i	$-0.550027\pi$
$331$	−5.69516	−0.313034	−0.156517	+	0.987675i	$0.550027\pi$
$332$	0	0
$333$	−5.45216	−0.298776
$334$	0	0
$335$	26.3019	1.43703
$336$	0	0
$337$	−21.0292	−1.14553	−0.572766	−	0.819719i	$-0.694130\pi$
$337$	−21.0292	−1.14553	−0.572766	+	0.819719i	$0.694130\pi$
$338$	0	0
$339$	−1.22932	−0.0667675
$340$	0	0
$341$	2.43014	0.131600
$342$	0	0
$343$	9.37731	0.506327
$344$	0	0
$345$	−13.2747	−0.714687
$346$	0	0
$347$	−26.6894	−1.43276	−0.716381	−	0.697709i	$-0.754203\pi$
$347$	−26.6894	−1.43276	−0.716381	+	0.697709i	$0.754203\pi$
$348$	0	0
$349$	−13.4997	−0.722623	−0.361311	−	0.932445i	$-0.617671\pi$
$349$	−13.4997	−0.722623	−0.361311	+	0.932445i	$0.617671\pi$
$350$	0	0
$351$	5.66735	0.302501
$352$	0	0
$353$	3.46064	0.184191	0.0920956	−	0.995750i	$-0.470643\pi$
$353$	3.46064	0.184191	0.0920956	+	0.995750i	$0.470643\pi$
$354$	0	0
$355$	28.3310	1.50365
$356$	0	0
$357$	−1.74952	−0.0925945
$358$	0	0
$359$	20.8235	1.09902	0.549512	−	0.835486i	$-0.314814\pi$
$359$	20.8235	1.09902	0.549512	+	0.835486i	$0.314814\pi$
$360$	0	0
$361$	5.67057	0.298451
$362$	0	0
$363$	−5.84678	−0.306876
$364$	0	0
$365$	−11.1199	−0.582043
$366$	0	0
$367$	−18.7632	−0.979433	−0.489717	−	0.871882i	$-0.662900\pi$
$367$	−18.7632	−0.979433	−0.489717	+	0.871882i	$0.662900\pi$
$368$	0	0
$369$	15.7239	0.818553
$370$	0	0
$371$	3.77791	0.196139
$372$	0	0
$373$	−37.3370	−1.93324	−0.966618	−	0.256220i	$-0.917523\pi$
$373$	−37.3370	−1.93324	−0.966618	+	0.256220i	$0.917523\pi$
$374$	0	0
$375$	−2.45233	−0.126638
$376$	0	0
$377$	11.1720	0.575385
$378$	0	0
$379$	−7.50914	−0.385718	−0.192859	−	0.981226i	$-0.561776\pi$
$379$	−7.50914	−0.385718	−0.192859	+	0.981226i	$0.561776\pi$
$380$	0	0
$381$	−5.88027	−0.301255
$382$	0	0
$383$	19.9796	1.02091	0.510455	−	0.859904i	$-0.329477\pi$
$383$	19.9796	1.02091	0.510455	+	0.859904i	$0.329477\pi$
$384$	0	0
$385$	−3.75717	−0.191483
$386$	0	0
$387$	19.1626	0.974090
$388$	0	0
$389$	8.09123	0.410242	0.205121	−	0.978737i	$-0.434241\pi$
$389$	8.09123	0.410242	0.205121	+	0.978737i	$0.434241\pi$
$390$	0	0
$391$	20.5186	1.03767
$392$	0	0
$393$	10.6950	0.539491
$394$	0	0
$395$	14.8001	0.744676
$396$	0	0
$397$	27.6858	1.38951	0.694755	−	0.719246i	$-0.255513\pi$
$397$	27.6858	1.38951	0.694755	+	0.719246i	$0.255513\pi$
$398$	0	0
$399$	2.41373	0.120837
$400$	0	0
$401$	18.5461	0.926148	0.463074	−	0.886320i	$-0.346746\pi$
$401$	18.5461	0.926148	0.463074	+	0.886320i	$0.346746\pi$
$402$	0	0
$403$	2.19018	0.109101
$404$	0	0
$405$	−16.0363	−0.796850
$406$	0	0
$407$	−3.54014	−0.175478
$408$	0	0
$409$	11.5741	0.572300	0.286150	−	0.958185i	$-0.407624\pi$
$409$	11.5741	0.572300	0.286150	+	0.958185i	$0.407624\pi$
$410$	0	0
$411$	1.37295	0.0677224
$412$	0	0
$413$	8.55479	0.420954
$414$	0	0
$415$	10.6286	0.521739
$416$	0	0
$417$	14.4942	0.709784
$418$	0	0
$419$	2.02256	0.0988083	0.0494042	−	0.998779i	$-0.484268\pi$
$419$	2.02256	0.0988083	0.0494042	+	0.998779i	$0.484268\pi$
$420$	0	0
$421$	−6.02710	−0.293743	−0.146871	−	0.989156i	$-0.546920\pi$
$421$	−6.02710	−0.293743	−0.146871	+	0.989156i	$0.546920\pi$
$422$	0	0
$423$	−17.8192	−0.866400
$424$	0	0
$425$	21.7913	1.05703
$426$	0	0
$427$	−4.86331	−0.235352
$428$	0	0
$429$	1.67601	0.0809187
$430$	0	0
$431$	15.1741	0.730910	0.365455	−	0.930829i	$-0.380913\pi$
$431$	15.1741	0.730910	0.365455	+	0.930829i	$0.380913\pi$
$432$	0	0
$433$	−29.9376	−1.43871	−0.719355	−	0.694642i	$-0.755563\pi$
$433$	−29.9376	−1.43871	−0.719355	+	0.694642i	$0.755563\pi$
$434$	0	0
$435$	17.7213	0.849669
$436$	0	0
$437$	−28.3085	−1.35418
$438$	0	0
$439$	−37.1581	−1.77346	−0.886729	−	0.462289i	$-0.847028\pi$
$439$	−37.1581	−1.77346	−0.886729	+	0.462289i	$0.847028\pi$
$440$	0	0
$441$	16.3570	0.778905
$442$	0	0
$443$	−30.7462	−1.46080	−0.730398	−	0.683022i	$-0.760665\pi$
$443$	−30.7462	−1.46080	−0.730398	+	0.683022i	$0.760665\pi$
$444$	0	0
$445$	6.32462	0.299816
$446$	0	0
$447$	8.96583	0.424069
$448$	0	0
$449$	30.4323	1.43619	0.718095	−	0.695945i	$-0.245014\pi$
$449$	30.4323	1.43619	0.718095	+	0.695945i	$0.245014\pi$
$450$	0	0
$451$	10.2097	0.480755
$452$	0	0
$453$	−7.67730	−0.360711
$454$	0	0
$455$	−3.38617	−0.158746
$456$	0	0
$457$	13.0144	0.608788	0.304394	−	0.952546i	$-0.401546\pi$
$457$	13.0144	0.608788	0.304394	+	0.952546i	$0.401546\pi$
$458$	0	0
$459$	−13.8953	−0.648578
$460$	0	0
$461$	5.92937	0.276158	0.138079	−	0.990421i	$-0.455907\pi$
$461$	5.92937	0.276158	0.138079	+	0.990421i	$0.455907\pi$
$462$	0	0
$463$	2.81059	0.130619	0.0653096	−	0.997865i	$-0.479196\pi$
$463$	2.81059	0.130619	0.0653096	+	0.997865i	$0.479196\pi$
$464$	0	0
$465$	3.47412	0.161109
$466$	0	0
$467$	0.929065	0.0429920	0.0214960	−	0.999769i	$-0.493157\pi$
$467$	0.929065	0.0429920	0.0214960	+	0.999769i	$0.493157\pi$
$468$	0	0
$469$	5.48767	0.253397
$470$	0	0
$471$	−1.22972	−0.0566623
$472$	0	0
$473$	12.4425	0.572106
$474$	0	0
$475$	−30.0644	−1.37945
$476$	0	0
$477$	13.6661	0.625730
$478$	0	0
$479$	−0.0250906	−0.00114642	−0.000573210	−	1.00000i	$-0.500182\pi$
$479$	−0.0250906	−0.00114642	−0.000573210		1.00000i	$0.500182\pi$
$480$	0	0
$481$	−3.19057	−0.145478
$482$	0	0
$483$	−2.76966	−0.126024
$484$	0	0
$485$	−12.2303	−0.555351
$486$	0	0
$487$	−2.56746	−0.116343	−0.0581713	−	0.998307i	$-0.518527\pi$
$487$	−2.56746	−0.116343	−0.0581713	+	0.998307i	$0.518527\pi$
$488$	0	0
$489$	8.09323	0.365989
$490$	0	0
$491$	11.4814	0.518148	0.259074	−	0.965858i	$-0.416583\pi$
$491$	11.4814	0.518148	0.259074	+	0.965858i	$0.416583\pi$
$492$	0	0
$493$	−27.3916	−1.23366
$494$	0	0
$495$	−13.5911	−0.610874
$496$	0	0
$497$	5.91101	0.265145
$498$	0	0
$499$	38.9826	1.74510	0.872551	−	0.488523i	$-0.162464\pi$
$499$	38.9826	1.74510	0.872551	+	0.488523i	$0.162464\pi$
$500$	0	0
$501$	14.5279	0.649059
$502$	0	0
$503$	−21.7427	−0.969458	−0.484729	−	0.874664i	$-0.661082\pi$
$503$	−21.7427	−0.969458	−0.484729	+	0.874664i	$0.661082\pi$
$504$	0	0
$505$	−52.9459	−2.35606
$506$	0	0
$507$	−7.59707	−0.337398
$508$	0	0
$509$	−18.1146	−0.802915	−0.401457	−	0.915878i	$-0.631496\pi$
$509$	−18.1146	−0.802915	−0.401457	+	0.915878i	$0.631496\pi$
$510$	0	0
$511$	−2.32007	−0.102634
$512$	0	0
$513$	19.1706	0.846405
$514$	0	0
$515$	−28.9925	−1.27756
$516$	0	0
$517$	−11.5702	−0.508857
$518$	0	0
$519$	−15.3456	−0.673595
$520$	0	0
$521$	−14.3056	−0.626741	−0.313371	−	0.949631i	$-0.601458\pi$
$521$	−14.3056	−0.626741	−0.313371	+	0.949631i	$0.601458\pi$
$522$	0	0
$523$	−11.3652	−0.496964	−0.248482	−	0.968637i	$-0.579932\pi$
$523$	−11.3652	−0.496964	−0.248482	+	0.968637i	$0.579932\pi$
$524$	0	0
$525$	−2.94144	−0.128375
$526$	0	0
$527$	−5.36993	−0.233918
$528$	0	0
$529$	9.48290	0.412300
$530$	0	0
$531$	30.9459	1.34294
$532$	0	0
$533$	9.20154	0.398563
$534$	0	0
$535$	37.4273	1.61812
$536$	0	0
$537$	9.18224	0.396243
$538$	0	0
$539$	10.6208	0.457469
$540$	0	0
$541$	42.6927	1.83550	0.917751	−	0.397156i	$-0.130003\pi$
$541$	42.6927	1.83550	0.917751	+	0.397156i	$0.130003\pi$
$542$	0	0
$543$	7.55014	0.324008
$544$	0	0
$545$	−43.5373	−1.86493
$546$	0	0
$547$	42.7839	1.82931	0.914654	−	0.404238i	$-0.132463\pi$
$547$	42.7839	1.82931	0.914654	+	0.404238i	$0.132463\pi$
$548$	0	0
$549$	−17.5924	−0.750826
$550$	0	0
$551$	37.7908	1.60994
$552$	0	0
$553$	3.08792	0.131312
$554$	0	0
$555$	−5.06097	−0.214826
$556$	0	0
$557$	−5.34165	−0.226333	−0.113166	−	0.993576i	$-0.536099\pi$
$557$	−5.34165	−0.226333	−0.113166	+	0.993576i	$0.536099\pi$
$558$	0	0
$559$	11.2139	0.474296
$560$	0	0
$561$	−4.10928	−0.173494
$562$	0	0
$563$	46.6179	1.96471	0.982354	−	0.187029i	$-0.0598857\pi$
$563$	46.6179	1.96471	0.982354	+	0.187029i	$0.0598857\pi$
$564$	0	0
$565$	5.83369	0.245425
$566$	0	0
$567$	−3.34583	−0.140512
$568$	0	0
$569$	−40.1399	−1.68275	−0.841377	−	0.540449i	$-0.818254\pi$
$569$	−40.1399	−1.68275	−0.841377	+	0.540449i	$0.818254\pi$
$570$	0	0
$571$	−24.0793	−1.00769	−0.503843	−	0.863795i	$-0.668081\pi$
$571$	−24.0793	−1.00769	−0.503843	+	0.863795i	$0.668081\pi$
$572$	0	0
$573$	13.4342	0.561222
$574$	0	0
$575$	34.4977	1.43865
$576$	0	0
$577$	−18.4939	−0.769909	−0.384955	−	0.922936i	$-0.625783\pi$
$577$	−18.4939	−0.769909	−0.384955	+	0.922936i	$0.625783\pi$
$578$	0	0
$579$	11.0002	0.457152
$580$	0	0
$581$	2.21757	0.0920005
$582$	0	0
$583$	8.87357	0.367506
$584$	0	0
$585$	−12.2491	−0.506436
$586$	0	0
$587$	8.12630	0.335408	0.167704	−	0.985837i	$-0.446365\pi$
$587$	8.12630	0.335408	0.167704	+	0.985837i	$0.446365\pi$
$588$	0	0
$589$	7.40861	0.305267
$590$	0	0
$591$	−9.48593	−0.390199
$592$	0	0
$593$	17.2602	0.708792	0.354396	−	0.935095i	$-0.384686\pi$
$593$	17.2602	0.708792	0.354396	+	0.935095i	$0.384686\pi$
$594$	0	0
$595$	8.30228	0.340360
$596$	0	0
$597$	5.02196	0.205535
$598$	0	0
$599$	36.2376	1.48063	0.740315	−	0.672260i	$-0.234676\pi$
$599$	36.2376	1.48063	0.740315	+	0.672260i	$0.234676\pi$
$600$	0	0
$601$	−5.60700	−0.228714	−0.114357	−	0.993440i	$-0.536481\pi$
$601$	−5.60700	−0.228714	−0.114357	+	0.993440i	$0.536481\pi$
$602$	0	0
$603$	19.8510	0.808395
$604$	0	0
$605$	27.7456	1.12802
$606$	0	0
$607$	−11.7557	−0.477151	−0.238576	−	0.971124i	$-0.576680\pi$
$607$	−11.7557	−0.477151	−0.238576	+	0.971124i	$0.576680\pi$
$608$	0	0
$609$	3.69739	0.149826
$610$	0	0
$611$	−10.4277	−0.421860
$612$	0	0
$613$	6.39257	0.258193	0.129097	−	0.991632i	$-0.458792\pi$
$613$	6.39257	0.258193	0.129097	+	0.991632i	$0.458792\pi$
$614$	0	0
$615$	14.5957	0.588556
$616$	0	0
$617$	10.6638	0.429308	0.214654	−	0.976690i	$-0.431138\pi$
$617$	10.6638	0.429308	0.214654	+	0.976690i	$0.431138\pi$
$618$	0	0
$619$	−46.5546	−1.87119	−0.935594	−	0.353077i	$-0.885135\pi$
$619$	−46.5546	−1.87119	−0.935594	+	0.353077i	$0.885135\pi$
$620$	0	0
$621$	−21.9976	−0.882732
$622$	0	0
$623$	1.31958	0.0528678
$624$	0	0
$625$	−18.6270	−0.745080
$626$	0	0
$627$	5.66936	0.226412
$628$	0	0
$629$	7.82271	0.311912
$630$	0	0
$631$	−14.1029	−0.561428	−0.280714	−	0.959791i	$-0.590571\pi$
$631$	−14.1029	−0.561428	−0.280714	+	0.959791i	$0.590571\pi$
$632$	0	0
$633$	11.3312	0.450377
$634$	0	0
$635$	27.9046	1.10736
$636$	0	0
$637$	9.57203	0.379258
$638$	0	0
$639$	21.3824	0.845873
$640$	0	0
$641$	−0.986881	−0.0389795	−0.0194897	−	0.999810i	$-0.506204\pi$
$641$	−0.986881	−0.0389795	−0.0194897	+	0.999810i	$0.506204\pi$
$642$	0	0
$643$	−14.5506	−0.573822	−0.286911	−	0.957957i	$-0.592628\pi$
$643$	−14.5506	−0.573822	−0.286911	+	0.957957i	$0.592628\pi$
$644$	0	0
$645$	17.7877	0.700391
$646$	0	0
$647$	−34.5997	−1.36025	−0.680126	−	0.733095i	$-0.738075\pi$
$647$	−34.5997	−1.36025	−0.680126	+	0.733095i	$0.738075\pi$
$648$	0	0
$649$	20.0935	0.788739
$650$	0	0
$651$	0.724846	0.0284089
$652$	0	0
$653$	26.4546	1.03525	0.517623	−	0.855609i	$-0.326817\pi$
$653$	26.4546	1.03525	0.517623	+	0.855609i	$0.326817\pi$
$654$	0	0
$655$	−50.7526	−1.98307
$656$	0	0
$657$	−8.39259	−0.327426
$658$	0	0
$659$	0.921354	0.0358908	0.0179454	−	0.999839i	$-0.494287\pi$
$659$	0.921354	0.0358908	0.0179454	+	0.999839i	$0.494287\pi$
$660$	0	0
$661$	−32.2607	−1.25480	−0.627398	−	0.778699i	$-0.715880\pi$
$661$	−32.2607	−1.25480	−0.627398	+	0.778699i	$0.715880\pi$
$662$	0	0
$663$	−3.70351	−0.143833
$664$	0	0
$665$	−11.4542	−0.444176
$666$	0	0
$667$	−43.3635	−1.67904
$668$	0	0
$669$	2.67213	0.103311
$670$	0	0
$671$	−11.4229	−0.440978
$672$	0	0
$673$	−35.6824	−1.37546	−0.687729	−	0.725968i	$-0.741392\pi$
$673$	−35.6824	−1.37546	−0.687729	+	0.725968i	$0.741392\pi$
$674$	0	0
$675$	−23.3620	−0.899203
$676$	0	0
$677$	22.1725	0.852159	0.426079	−	0.904686i	$-0.359894\pi$
$677$	22.1725	0.852159	0.426079	+	0.904686i	$0.359894\pi$
$678$	0	0
$679$	−2.55175	−0.0979273
$680$	0	0
$681$	7.75471	0.297161
$682$	0	0
$683$	33.1329	1.26780	0.633898	−	0.773417i	$-0.281454\pi$
$683$	33.1329	1.26780	0.633898	+	0.773417i	$0.281454\pi$
$684$	0	0
$685$	−6.51525	−0.248935
$686$	0	0
$687$	−3.42187	−0.130553
$688$	0	0
$689$	7.99736	0.304675
$690$	0	0
$691$	22.6859	0.863012	0.431506	−	0.902110i	$-0.357982\pi$
$691$	22.6859	0.863012	0.431506	+	0.902110i	$0.357982\pi$
$692$	0	0
$693$	−2.83567	−0.107718
$694$	0	0
$695$	−68.7815	−2.60903
$696$	0	0
$697$	−22.5605	−0.854540
$698$	0	0
$699$	12.2991	0.465194
$700$	0	0
$701$	−14.9398	−0.564269	−0.282135	−	0.959375i	$-0.591042\pi$
$701$	−14.9398	−0.564269	−0.282135	+	0.959375i	$0.591042\pi$
$702$	0	0
$703$	−10.7926	−0.407050
$704$	0	0
$705$	−16.5407	−0.622960
$706$	0	0
$707$	−11.0467	−0.415454
$708$	0	0
$709$	−39.3345	−1.47724	−0.738618	−	0.674124i	$-0.764521\pi$
$709$	−39.3345	−1.47724	−0.738618	+	0.674124i	$0.764521\pi$
$710$	0	0
$711$	11.1702	0.418914
$712$	0	0
$713$	−8.50109	−0.318368
$714$	0	0
$715$	−7.95344	−0.297442
$716$	0	0
$717$	9.51153	0.355215
$718$	0	0
$719$	6.21327	0.231716	0.115858	−	0.993266i	$-0.463038\pi$
$719$	6.21327	0.231716	0.115858	+	0.993266i	$0.463038\pi$
$720$	0	0
$721$	−6.04904	−0.225278
$722$	0	0
$723$	7.63209	0.283840
$724$	0	0
$725$	−46.0531	−1.71037
$726$	0	0
$727$	5.11643	0.189758	0.0948789	−	0.995489i	$-0.469754\pi$
$727$	5.11643	0.189758	0.0948789	+	0.995489i	$0.469754\pi$
$728$	0	0
$729$	−3.99114	−0.147820
$730$	0	0
$731$	−27.4944	−1.01692
$732$	0	0
$733$	−42.9990	−1.58821	−0.794103	−	0.607783i	$-0.792059\pi$
$733$	−42.9990	−1.58821	−0.794103	+	0.607783i	$0.792059\pi$
$734$	0	0
$735$	15.1834	0.560048
$736$	0	0
$737$	12.8894	0.474789
$738$	0	0
$739$	−38.5655	−1.41866	−0.709328	−	0.704879i	$-0.751001\pi$
$739$	−38.5655	−1.41866	−0.709328	+	0.704879i	$0.751001\pi$
$740$	0	0
$741$	5.10955	0.187704
$742$	0	0
$743$	0.253761	0.00930960	0.00465480	−	0.999989i	$-0.498518\pi$
$743$	0.253761	0.00930960	0.00465480	+	0.999989i	$0.498518\pi$
$744$	0	0
$745$	−42.5470	−1.55880
$746$	0	0
$747$	8.02181	0.293503
$748$	0	0
$749$	7.80888	0.285330
$750$	0	0
$751$	−17.8609	−0.651752	−0.325876	−	0.945413i	$-0.605659\pi$
$751$	−17.8609	−0.651752	−0.325876	+	0.945413i	$0.605659\pi$
$752$	0	0
$753$	0.700584	0.0255307
$754$	0	0
$755$	36.4323	1.32591
$756$	0	0
$757$	−5.45436	−0.198242	−0.0991211	−	0.995075i	$-0.531603\pi$
$757$	−5.45436	−0.198242	−0.0991211	+	0.995075i	$0.531603\pi$
$758$	0	0
$759$	−6.50537	−0.236130
$760$	0	0
$761$	44.2963	1.60574	0.802869	−	0.596155i	$-0.203306\pi$
$761$	44.2963	1.60574	0.802869	+	0.596155i	$0.203306\pi$
$762$	0	0
$763$	−9.08367	−0.328851
$764$	0	0
$765$	30.0325	1.08583
$766$	0	0
$767$	18.1094	0.653892
$768$	0	0
$769$	32.4474	1.17008	0.585041	−	0.811004i	$-0.301078\pi$
$769$	32.4474	1.17008	0.585041	+	0.811004i	$0.301078\pi$
$770$	0	0
$771$	−13.6866	−0.492911
$772$	0	0
$773$	−6.23192	−0.224146	−0.112073	−	0.993700i	$-0.535749\pi$
$773$	−6.23192	−0.224146	−0.112073	+	0.993700i	$0.535749\pi$
$774$	0	0
$775$	−9.02838	−0.324309
$776$	0	0
$777$	−1.05593	−0.0378812
$778$	0	0
$779$	31.1256	1.11519
$780$	0	0
$781$	13.8838	0.496801
$782$	0	0
$783$	29.3659	1.04945
$784$	0	0
$785$	5.83556	0.208280
$786$	0	0
$787$	32.9972	1.17622	0.588111	−	0.808780i	$-0.299872\pi$
$787$	32.9972	1.17622	0.588111	+	0.808780i	$0.299872\pi$
$788$	0	0
$789$	9.75345	0.347232
$790$	0	0
$791$	1.21715	0.0432768
$792$	0	0
$793$	−10.2950	−0.365586
$794$	0	0
$795$	12.6856	0.449912
$796$	0	0
$797$	18.9344	0.670689	0.335345	−	0.942096i	$-0.391147\pi$
$797$	18.9344	0.670689	0.335345	+	0.942096i	$0.391147\pi$
$798$	0	0
$799$	25.5669	0.904491
$800$	0	0
$801$	4.77341	0.168660
$802$	0	0
$803$	−5.44939	−0.192305
$804$	0	0
$805$	13.1433	0.463239
$806$	0	0
$807$	−11.2094	−0.394591
$808$	0	0
$809$	−11.0422	−0.388224	−0.194112	−	0.980979i	$-0.562183\pi$
$809$	−11.0422	−0.388224	−0.194112	+	0.980979i	$0.562183\pi$
$810$	0	0
$811$	−49.2377	−1.72897	−0.864485	−	0.502658i	$-0.832355\pi$
$811$	−49.2377	−1.72897	−0.864485	+	0.502658i	$0.832355\pi$
$812$	0	0
$813$	21.5077	0.754307
$814$	0	0
$815$	−38.4061	−1.34531
$816$	0	0
$817$	37.9326	1.32709
$818$	0	0
$819$	−2.55566	−0.0893020
$820$	0	0
$821$	−48.8077	−1.70340	−0.851700	−	0.524030i	$-0.824428\pi$
$821$	−48.8077	−1.70340	−0.851700	+	0.524030i	$0.824428\pi$
$822$	0	0
$823$	10.9742	0.382536	0.191268	−	0.981538i	$-0.438740\pi$
$823$	10.9742	0.382536	0.191268	+	0.981538i	$0.438740\pi$
$824$	0	0
$825$	−6.90887	−0.240536
$826$	0	0
$827$	−44.9509	−1.56310	−0.781549	−	0.623844i	$-0.785570\pi$
$827$	−44.9509	−1.56310	−0.781549	+	0.623844i	$0.785570\pi$
$828$	0	0
$829$	−20.8156	−0.722956	−0.361478	−	0.932381i	$-0.617728\pi$
$829$	−20.8156	−0.722956	−0.361478	+	0.932381i	$0.617728\pi$
$830$	0	0
$831$	−7.52984	−0.261207
$832$	0	0
$833$	−23.4689	−0.813149
$834$	0	0
$835$	−68.9415	−2.38582
$836$	0	0
$837$	5.75698	0.198990
$838$	0	0
$839$	1.10491	0.0381456	0.0190728	−	0.999818i	$-0.493929\pi$
$839$	1.10491	0.0381456	0.0190728	+	0.999818i	$0.493929\pi$
$840$	0	0
$841$	28.8886	0.996159
$842$	0	0
$843$	−0.851109	−0.0293137
$844$	0	0
$845$	36.0516	1.24021
$846$	0	0
$847$	5.78889	0.198908
$848$	0	0
$849$	6.89522	0.236643
$850$	0	0
$851$	12.3841	0.424521
$852$	0	0
$853$	−25.4961	−0.872969	−0.436485	−	0.899712i	$-0.643777\pi$
$853$	−25.4961	−0.872969	−0.436485	+	0.899712i	$0.643777\pi$
$854$	0	0
$855$	−41.4343	−1.41702
$856$	0	0
$857$	4.99902	0.170763	0.0853817	−	0.996348i	$-0.472789\pi$
$857$	4.99902	0.170763	0.0853817	+	0.996348i	$0.472789\pi$
$858$	0	0
$859$	−15.5639	−0.531033	−0.265517	−	0.964106i	$-0.585542\pi$
$859$	−15.5639	−0.531033	−0.265517	+	0.964106i	$0.585542\pi$
$860$	0	0
$861$	3.04527	0.103783
$862$	0	0
$863$	52.7977	1.79725	0.898627	−	0.438713i	$-0.144565\pi$
$863$	52.7977	1.79725	0.898627	+	0.438713i	$0.144565\pi$
$864$	0	0
$865$	72.8217	2.47601
$866$	0	0
$867$	−2.82958	−0.0960975
$868$	0	0
$869$	7.25291	0.246038
$870$	0	0
$871$	11.6167	0.393617
$872$	0	0
$873$	−9.23066	−0.312411
$874$	0	0
$875$	2.42805	0.0820830
$876$	0	0
$877$	25.0473	0.845787	0.422893	−	0.906179i	$-0.361014\pi$
$877$	25.0473	0.845787	0.422893	+	0.906179i	$0.361014\pi$
$878$	0	0
$879$	10.4728	0.353239
$880$	0	0
$881$	−3.27673	−0.110396	−0.0551978	−	0.998475i	$-0.517579\pi$
$881$	−3.27673	−0.110396	−0.0551978	+	0.998475i	$0.517579\pi$
$882$	0	0
$883$	21.2365	0.714665	0.357332	−	0.933977i	$-0.383686\pi$
$883$	21.2365	0.714665	0.357332	+	0.933977i	$0.383686\pi$
$884$	0	0
$885$	28.7256	0.965600
$886$	0	0
$887$	16.1903	0.543616	0.271808	−	0.962351i	$-0.412378\pi$
$887$	16.1903	0.543616	0.271808	+	0.962351i	$0.412378\pi$
$888$	0	0
$889$	5.82205	0.195265
$890$	0	0
$891$	−7.85870	−0.263276
$892$	0	0
$893$	−35.2733	−1.18038
$894$	0	0
$895$	−43.5739	−1.45651
$896$	0	0
$897$	−5.86301	−0.195760
$898$	0	0
$899$	11.3486	0.378498
$900$	0	0
$901$	−19.6081	−0.653240
$902$	0	0
$903$	3.71126	0.123503
$904$	0	0
$905$	−35.8289	−1.19099
$906$	0	0
$907$	25.7911	0.856378	0.428189	−	0.903689i	$-0.359152\pi$
$907$	25.7911	0.856378	0.428189	+	0.903689i	$0.359152\pi$
$908$	0	0
$909$	−39.9601	−1.32539
$910$	0	0
$911$	30.6634	1.01592	0.507961	−	0.861380i	$-0.330399\pi$
$911$	30.6634	1.01592	0.507961	+	0.861380i	$0.330399\pi$
$912$	0	0
$913$	5.20864	0.172381
$914$	0	0
$915$	−16.3302	−0.539860
$916$	0	0
$917$	−10.5891	−0.349683
$918$	0	0
$919$	−21.1469	−0.697571	−0.348786	−	0.937202i	$-0.613406\pi$
$919$	−21.1469	−0.697571	−0.348786	+	0.937202i	$0.613406\pi$
$920$	0	0
$921$	−15.1581	−0.499475
$922$	0	0
$923$	12.5129	0.411865
$924$	0	0
$925$	13.1522	0.432442
$926$	0	0
$927$	−21.8817	−0.718689
$928$	0	0
$929$	2.16376	0.0709907	0.0354953	−	0.999370i	$-0.488699\pi$
$929$	2.16376	0.0709907	0.0354953	+	0.999370i	$0.488699\pi$
$930$	0	0
$931$	32.3788	1.06117
$932$	0	0
$933$	−5.00491	−0.163853
$934$	0	0
$935$	19.5004	0.637731
$936$	0	0
$937$	−44.7658	−1.46244	−0.731218	−	0.682144i	$-0.761048\pi$
$937$	−44.7658	−1.46244	−0.731218	+	0.682144i	$0.761048\pi$
$938$	0	0
$939$	−18.6147	−0.607467
$940$	0	0
$941$	14.4963	0.472567	0.236283	−	0.971684i	$-0.424071\pi$
$941$	14.4963	0.472567	0.236283	+	0.971684i	$0.424071\pi$
$942$	0	0
$943$	−35.7154	−1.16305
$944$	0	0
$945$	−8.90068	−0.289539
$946$	0	0
$947$	−35.0699	−1.13962	−0.569809	−	0.821777i	$-0.692983\pi$
$947$	−35.0699	−1.13962	−0.569809	+	0.821777i	$0.692983\pi$
$948$	0	0
$949$	−4.91130	−0.159428
$950$	0	0
$951$	−0.922602	−0.0299174
$952$	0	0
$953$	5.83503	0.189015	0.0945075	−	0.995524i	$-0.469872\pi$
$953$	5.83503	0.189015	0.0945075	+	0.995524i	$0.469872\pi$
$954$	0	0
$955$	−63.7514	−2.06295
$956$	0	0
$957$	8.68443	0.280728
$958$	0	0
$959$	−1.35935	−0.0438957
$960$	0	0
$961$	−28.7752	−0.928232
$962$	0	0
$963$	28.2477	0.910269
$964$	0	0
$965$	−52.2008	−1.68040
$966$	0	0
$967$	18.1466	0.583555	0.291778	−	0.956486i	$-0.405753\pi$
$967$	18.1466	0.583555	0.291778	+	0.956486i	$0.405753\pi$
$968$	0	0
$969$	−12.5277	−0.402447
$970$	0	0
$971$	−5.08451	−0.163170	−0.0815848	−	0.996666i	$-0.525998\pi$
$971$	−5.08451	−0.163170	−0.0815848	+	0.996666i	$0.525998\pi$
$972$	0	0
$973$	−14.3507	−0.460061
$974$	0	0
$975$	−6.22666	−0.199413
$976$	0	0
$977$	45.6244	1.45965	0.729827	−	0.683632i	$-0.239601\pi$
$977$	45.6244	1.45965	0.729827	+	0.683632i	$0.239601\pi$
$978$	0	0
$979$	3.09943	0.0990581
$980$	0	0
$981$	−32.8591	−1.04911
$982$	0	0
$983$	14.8481	0.473581	0.236791	−	0.971561i	$-0.423904\pi$
$983$	14.8481	0.473581	0.236791	+	0.971561i	$0.423904\pi$
$984$	0	0
$985$	45.0151	1.43430
$986$	0	0
$987$	−3.45108	−0.109849
$988$	0	0
$989$	−43.5261	−1.38405
$990$	0	0
$991$	0.417201	0.0132528	0.00662641	−	0.999978i	$-0.497891\pi$
$991$	0.417201	0.0132528	0.00662641	+	0.999978i	$0.497891\pi$
$992$	0	0
$993$	−3.98993	−0.126617
$994$	0	0
$995$	−23.8315	−0.755509
$996$	0	0
$997$	40.8773	1.29460	0.647298	−	0.762237i	$-0.275899\pi$
$997$	40.8773	1.29460	0.647298	+	0.762237i	$0.275899\pi$
$998$	0	0
$999$	−8.38655	−0.265339

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8032.2.a.h.1.13	✓	30	4.3	odd	2
8032.2.a.i.1.18	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)

\(f(q)\)	\(=\)	\(q+0.700584 q^{3} -3.32459 q^{5} -0.693647 q^{7} -2.50918 q^{9} +O(q^{10})\)	\(q+0.700584 q^{3} -3.32459 q^{5} -0.693647 q^{7} -2.50918 q^{9} -1.62924 q^{11} -1.46836 q^{13} -2.32915 q^{15} +3.60016 q^{17} -4.96695 q^{19} -0.485957 q^{21} +5.69938 q^{23} +6.05289 q^{25} -3.85964 q^{27} -7.60846 q^{29} -1.49158 q^{31} -1.14142 q^{33} +2.30609 q^{35} +2.17288 q^{37} -1.02871 q^{39} -6.26654 q^{41} -7.63700 q^{43} +8.34200 q^{45} +7.10161 q^{47} -6.51885 q^{49} +2.52221 q^{51} -5.44645 q^{53} +5.41654 q^{55} -3.47976 q^{57} -12.3331 q^{59} +7.01122 q^{61} +1.74049 q^{63} +4.88170 q^{65} -7.91134 q^{67} +3.99289 q^{69} -8.52164 q^{71} +3.34475 q^{73} +4.24055 q^{75} +1.13012 q^{77} -4.45172 q^{79} +4.82354 q^{81} -3.19698 q^{83} -11.9690 q^{85} -5.33036 q^{87} -1.90238 q^{89} +1.01852 q^{91} -1.04498 q^{93} +16.5131 q^{95} +3.67875 q^{97} +4.08806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 3 q^{3} + 13 q^{7} + 35 q^{9}+O(q^{10}) \) 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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