LMFDB - Embedded newform 6384.2.a.bw.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6384,2,Mod(1,6384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6384.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6384.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:	$50.9764966504$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.12489$ of defining polynomial
Character	$\chi$	$=$	6384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.64002 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.64002 q^{5} +1.00000 q^{7} +1.00000 q^{9} +6.24977 q^{11} +4.00000 q^{13} -2.64002 q^{15} -2.64002 q^{17} +1.00000 q^{19} +1.00000 q^{21} +6.24977 q^{23} +1.96972 q^{25} +1.00000 q^{27} +3.60975 q^{29} +8.24977 q^{31} +6.24977 q^{33} -2.64002 q^{35} +2.00000 q^{37} +4.00000 q^{39} -5.21949 q^{41} -4.24977 q^{43} -2.64002 q^{45} -1.60975 q^{47} +1.00000 q^{49} -2.64002 q^{51} -12.8898 q^{53} -16.4995 q^{55} +1.00000 q^{57} -1.93945 q^{59} +0.0605522 q^{61} +1.00000 q^{63} -10.5601 q^{65} -6.24977 q^{67} +6.24977 q^{69} -0.329700 q^{71} -1.21949 q^{73} +1.96972 q^{75} +6.24977 q^{77} -0.969724 q^{79} +1.00000 q^{81} -3.67030 q^{83} +6.96972 q^{85} +3.60975 q^{87} +9.21949 q^{89} +4.00000 q^{91} +8.24977 q^{93} -2.64002 q^{95} +9.03028 q^{97} +6.24977 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 12 q^{13} + 3 q^{19} + 3 q^{21} + 2 q^{23} + 5 q^{25} + 3 q^{27} + 2 q^{29} + 8 q^{31} + 2 q^{33} + 6 q^{37} + 12 q^{39} + 2 q^{41} + 4 q^{43} + 4 q^{47} + 3 q^{49} - 14 q^{53} - 16 q^{55} + 3 q^{57} - 4 q^{59} + 2 q^{61} + 3 q^{63} - 2 q^{67} + 2 q^{69} - 8 q^{71} + 14 q^{73} + 5 q^{75} + 2 q^{77} - 2 q^{79} + 3 q^{81} - 4 q^{83} + 20 q^{85} + 2 q^{87} + 10 q^{89} + 12 q^{91} + 8 q^{93} + 28 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 + 2 * q^11 + 12 * q^13 + 3 * q^19 + 3 * q^21 + 2 * q^23 + 5 * q^25 + 3 * q^27 + 2 * q^29 + 8 * q^31 + 2 * q^33 + 6 * q^37 + 12 * q^39 + 2 * q^41 + 4 * q^43 + 4 * q^47 + 3 * q^49 - 14 * q^53 - 16 * q^55 + 3 * q^57 - 4 * q^59 + 2 * q^61 + 3 * q^63 - 2 * q^67 + 2 * q^69 - 8 * q^71 + 14 * q^73 + 5 * q^75 + 2 * q^77 - 2 * q^79 + 3 * q^81 - 4 * q^83 + 20 * q^85 + 2 * q^87 + 10 * q^89 + 12 * q^91 + 8 * q^93 + 28 * q^97 + 2 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.64002	−1.18065	−0.590327	−	0.807164i	$-0.701001\pi$
$5$	−2.64002	−1.18065	−0.590327	+	0.807164i	$0.701001\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	6.24977	1.88438	0.942188	−	0.335084i	$-0.108765\pi$
$11$	6.24977	1.88438	0.942188	+	0.335084i	$0.108765\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	−2.64002	−0.681651
$16$	0	0
$17$	−2.64002	−0.640300	−0.320150	−	0.947367i	$-0.603733\pi$
$17$	−2.64002	−0.640300	−0.320150	+	0.947367i	$0.603733\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	6.24977	1.30317	0.651584	−	0.758577i	$-0.274105\pi$
$23$	6.24977	1.30317	0.651584	+	0.758577i	$0.274105\pi$
$24$	0	0
$25$	1.96972	0.393945
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.60975	0.670313	0.335157	−	0.942162i	$-0.391211\pi$
$29$	3.60975	0.670313	0.335157	+	0.942162i	$0.391211\pi$
$30$	0	0
$31$	8.24977	1.48170	0.740851	−	0.671669i	$-0.234422\pi$
$31$	8.24977	1.48170	0.740851	+	0.671669i	$0.234422\pi$
$32$	0	0
$33$	6.24977	1.08795
$34$	0	0
$35$	−2.64002	−0.446245
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−5.21949	−0.815148	−0.407574	−	0.913172i	$-0.633625\pi$
$41$	−5.21949	−0.815148	−0.407574	+	0.913172i	$0.633625\pi$
$42$	0	0
$43$	−4.24977	−0.648084	−0.324042	−	0.946043i	$-0.605042\pi$
$43$	−4.24977	−0.648084	−0.324042	+	0.946043i	$0.605042\pi$
$44$	0	0
$45$	−2.64002	−0.393551
$46$	0	0
$47$	−1.60975	−0.234806	−0.117403	−	0.993084i	$-0.537457\pi$
$47$	−1.60975	−0.234806	−0.117403	+	0.993084i	$0.537457\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.64002	−0.369677
$52$	0	0
$53$	−12.8898	−1.77055	−0.885275	−	0.465068i	$-0.846030\pi$
$53$	−12.8898	−1.77055	−0.885275	+	0.465068i	$0.846030\pi$
$54$	0	0
$55$	−16.4995	−2.22480
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−1.93945	−0.252495	−0.126247	−	0.991999i	$-0.540293\pi$
$59$	−1.93945	−0.252495	−0.126247	+	0.991999i	$0.540293\pi$
$60$	0	0
$61$	0.0605522	0.00775291	0.00387646	−	0.999992i	$-0.498766\pi$
$61$	0.0605522	0.00775291	0.00387646	+	0.999992i	$0.498766\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−10.5601	−1.30982
$66$	0	0
$67$	−6.24977	−0.763531	−0.381766	−	0.924259i	$-0.624684\pi$
$67$	−6.24977	−0.763531	−0.381766	+	0.924259i	$0.624684\pi$
$68$	0	0
$69$	6.24977	0.752384
$70$	0	0
$71$	−0.329700	−0.0391282	−0.0195641	−	0.999809i	$-0.506228\pi$
$71$	−0.329700	−0.0391282	−0.0195641	+	0.999809i	$0.506228\pi$
$72$	0	0
$73$	−1.21949	−0.142731	−0.0713655	−	0.997450i	$-0.522736\pi$
$73$	−1.21949	−0.142731	−0.0713655	+	0.997450i	$0.522736\pi$
$74$	0	0
$75$	1.96972	0.227444
$76$	0	0
$77$	6.24977	0.712227
$78$	0	0
$79$	−0.969724	−0.109102	−0.0545512	−	0.998511i	$-0.517373\pi$
$79$	−0.969724	−0.109102	−0.0545512	+	0.998511i	$0.517373\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.67030	−0.402868	−0.201434	−	0.979502i	$-0.564560\pi$
$83$	−3.67030	−0.402868	−0.201434	+	0.979502i	$0.564560\pi$
$84$	0	0
$85$	6.96972	0.755973
$86$	0	0
$87$	3.60975	0.387006
$88$	0	0
$89$	9.21949	0.977264	0.488632	−	0.872490i	$-0.337496\pi$
$89$	9.21949	0.977264	0.488632	+	0.872490i	$0.337496\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	8.24977	0.855461
$94$	0	0
$95$	−2.64002	−0.270861
$96$	0	0
$97$	9.03028	0.916886	0.458443	−	0.888724i	$-0.348407\pi$
$97$	9.03028	0.916886	0.458443	+	0.888724i	$0.348407\pi$
$98$	0	0
$99$	6.24977	0.628126
$100$	0	0
$101$	2.64002	0.262692	0.131346	−	0.991337i	$-0.458070\pi$
$101$	2.64002	0.262692	0.131346	+	0.991337i	$0.458070\pi$
$102$	0	0
$103$	1.93945	0.191099	0.0955497	−	0.995425i	$-0.469539\pi$
$103$	1.93945	0.191099	0.0955497	+	0.995425i	$0.469539\pi$
$104$	0	0
$105$	−2.64002	−0.257640
$106$	0	0
$107$	8.82924	0.853555	0.426778	−	0.904357i	$-0.359649\pi$
$107$	8.82924	0.853555	0.426778	+	0.904357i	$0.359649\pi$
$108$	0	0
$109$	7.28005	0.697302	0.348651	−	0.937253i	$-0.386640\pi$
$109$	7.28005	0.697302	0.348651	+	0.937253i	$0.386640\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−4.88979	−0.459993	−0.229997	−	0.973191i	$-0.573872\pi$
$113$	−4.88979	−0.459993	−0.229997	+	0.973191i	$0.573872\pi$
$114$	0	0
$115$	−16.4995	−1.53859
$116$	0	0
$117$	4.00000	0.369800
$118$	0	0
$119$	−2.64002	−0.242011
$120$	0	0
$121$	28.0596	2.55088
$122$	0	0
$123$	−5.21949	−0.470626
$124$	0	0
$125$	8.00000	0.715542
$126$	0	0
$127$	14.7493	1.30879	0.654395	−	0.756153i	$-0.272923\pi$
$127$	14.7493	1.30879	0.654395	+	0.756153i	$0.272923\pi$
$128$	0	0
$129$	−4.24977	−0.374171
$130$	0	0
$131$	−20.9503	−1.83044	−0.915220	−	0.402954i	$-0.867983\pi$
$131$	−20.9503	−1.83044	−0.915220	+	0.402954i	$0.867983\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−2.64002	−0.227217
$136$	0	0
$137$	6.49954	0.555293	0.277647	−	0.960683i	$-0.410446\pi$
$137$	6.49954	0.555293	0.277647	+	0.960683i	$0.410446\pi$
$138$	0	0
$139$	−7.21949	−0.612350	−0.306175	−	0.951975i	$-0.599049\pi$
$139$	−7.21949	−0.612350	−0.306175	+	0.951975i	$0.599049\pi$
$140$	0	0
$141$	−1.60975	−0.135565
$142$	0	0
$143$	24.9991	2.09053
$144$	0	0
$145$	−9.52982	−0.791408
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	21.8401	1.78921	0.894607	−	0.446854i	$-0.147456\pi$
$149$	21.8401	1.78921	0.894607	+	0.446854i	$0.147456\pi$
$150$	0	0
$151$	2.24977	0.183084	0.0915419	−	0.995801i	$-0.470820\pi$
$151$	2.24977	0.183084	0.0915419	+	0.995801i	$0.470820\pi$
$152$	0	0
$153$	−2.64002	−0.213433
$154$	0	0
$155$	−21.7796	−1.74938
$156$	0	0
$157$	−6.49954	−0.518720	−0.259360	−	0.965781i	$-0.583512\pi$
$157$	−6.49954	−0.518720	−0.259360	+	0.965781i	$0.583512\pi$
$158$	0	0
$159$	−12.8898	−1.02223
$160$	0	0
$161$	6.24977	0.492551
$162$	0	0
$163$	14.8099	1.16000	0.579999	−	0.814617i	$-0.303053\pi$
$163$	14.8099	1.16000	0.579999	+	0.814617i	$0.303053\pi$
$164$	0	0
$165$	−16.4995	−1.28449
$166$	0	0
$167$	−19.7190	−1.52590	−0.762952	−	0.646455i	$-0.776251\pi$
$167$	−19.7190	−1.52590	−0.762952	+	0.646455i	$0.776251\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−3.28005	−0.249377	−0.124689	−	0.992196i	$-0.539793\pi$
$173$	−3.28005	−0.249377	−0.124689	+	0.992196i	$0.539793\pi$
$174$	0	0
$175$	1.96972	0.148897
$176$	0	0
$177$	−1.93945	−0.145778
$178$	0	0
$179$	−19.3893	−1.44923	−0.724614	−	0.689155i	$-0.757982\pi$
$179$	−19.3893	−1.44923	−0.724614	+	0.689155i	$0.757982\pi$
$180$	0	0
$181$	6.47018	0.480925	0.240462	−	0.970658i	$-0.422701\pi$
$181$	6.47018	0.480925	0.240462	+	0.970658i	$0.422701\pi$
$182$	0	0
$183$	0.0605522	0.00447615
$184$	0	0
$185$	−5.28005	−0.388197
$186$	0	0
$187$	−16.4995	−1.20657
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	8.31032	0.601314	0.300657	−	0.953732i	$-0.402794\pi$
$191$	8.31032	0.601314	0.300657	+	0.953732i	$0.402794\pi$
$192$	0	0
$193$	−8.56009	−0.616169	−0.308085	−	0.951359i	$-0.599688\pi$
$193$	−8.56009	−0.616169	−0.308085	+	0.951359i	$0.599688\pi$
$194$	0	0
$195$	−10.5601	−0.756224
$196$	0	0
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	0	0
$199$	11.2195	0.795329	0.397664	−	0.917531i	$-0.369821\pi$
$199$	11.2195	0.795329	0.397664	+	0.917531i	$0.369821\pi$
$200$	0	0
$201$	−6.24977	−0.440825
$202$	0	0
$203$	3.60975	0.253355
$204$	0	0
$205$	13.7796	0.962408
$206$	0	0
$207$	6.24977	0.434389
$208$	0	0
$209$	6.24977	0.432306
$210$	0	0
$211$	18.0899	1.24536	0.622680	−	0.782476i	$-0.286044\pi$
$211$	18.0899	1.24536	0.622680	+	0.782476i	$0.286044\pi$
$212$	0	0
$213$	−0.329700	−0.0225907
$214$	0	0
$215$	11.2195	0.765163
$216$	0	0
$217$	8.24977	0.560031
$218$	0	0
$219$	−1.21949	−0.0824058
$220$	0	0
$221$	−10.5601	−0.710349
$222$	0	0
$223$	−6.31032	−0.422570	−0.211285	−	0.977424i	$-0.567765\pi$
$223$	−6.31032	−0.422570	−0.211285	+	0.977424i	$0.567765\pi$
$224$	0	0
$225$	1.96972	0.131315
$226$	0	0
$227$	−7.21949	−0.479175	−0.239587	−	0.970875i	$-0.577012\pi$
$227$	−7.21949	−0.479175	−0.239587	+	0.970875i	$0.577012\pi$
$228$	0	0
$229$	21.8401	1.44324	0.721619	−	0.692291i	$-0.243398\pi$
$229$	21.8401	1.44324	0.721619	+	0.692291i	$0.243398\pi$
$230$	0	0
$231$	6.24977	0.411205
$232$	0	0
$233$	−21.7190	−1.42286	−0.711431	−	0.702756i	$-0.751952\pi$
$233$	−21.7190	−1.42286	−0.711431	+	0.702756i	$0.751952\pi$
$234$	0	0
$235$	4.24977	0.277224
$236$	0	0
$237$	−0.969724	−0.0629903
$238$	0	0
$239$	28.0294	1.81307	0.906534	−	0.422132i	$-0.138718\pi$
$239$	28.0294	1.81307	0.906534	+	0.422132i	$0.138718\pi$
$240$	0	0
$241$	−5.52982	−0.356207	−0.178103	−	0.984012i	$-0.556996\pi$
$241$	−5.52982	−0.356207	−0.178103	+	0.984012i	$0.556996\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−2.64002	−0.168665
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	−3.67030	−0.232596
$250$	0	0
$251$	12.9503	0.817419	0.408709	−	0.912665i	$-0.365979\pi$
$251$	12.9503	0.817419	0.408709	+	0.912665i	$0.365979\pi$
$252$	0	0
$253$	39.0596	2.45566
$254$	0	0
$255$	6.96972	0.436461
$256$	0	0
$257$	−6.49954	−0.405430	−0.202715	−	0.979238i	$-0.564977\pi$
$257$	−6.49954	−0.405430	−0.202715	+	0.979238i	$0.564977\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	3.60975	0.223438
$262$	0	0
$263$	−28.5289	−1.75917	−0.879584	−	0.475744i	$-0.842179\pi$
$263$	−28.5289	−1.75917	−0.879584	+	0.475744i	$0.842179\pi$
$264$	0	0
$265$	34.0294	2.09041
$266$	0	0
$267$	9.21949	0.564224
$268$	0	0
$269$	24.9385	1.52053	0.760265	−	0.649614i	$-0.225069\pi$
$269$	24.9385	1.52053	0.760265	+	0.649614i	$0.225069\pi$
$270$	0	0
$271$	−14.4390	−0.877106	−0.438553	−	0.898705i	$-0.644509\pi$
$271$	−14.4390	−0.877106	−0.438553	+	0.898705i	$0.644509\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	12.3103	0.742340
$276$	0	0
$277$	25.5904	1.53758	0.768788	−	0.639504i	$-0.220860\pi$
$277$	25.5904	1.53758	0.768788	+	0.639504i	$0.220860\pi$
$278$	0	0
$279$	8.24977	0.493901
$280$	0	0
$281$	−26.1698	−1.56116	−0.780581	−	0.625055i	$-0.785077\pi$
$281$	−26.1698	−1.56116	−0.780581	+	0.625055i	$0.785077\pi$
$282$	0	0
$283$	−17.2800	−1.02719	−0.513596	−	0.858032i	$-0.671687\pi$
$283$	−17.2800	−1.02719	−0.513596	+	0.858032i	$0.671687\pi$
$284$	0	0
$285$	−2.64002	−0.156381
$286$	0	0
$287$	−5.21949	−0.308097
$288$	0	0
$289$	−10.0303	−0.590016
$290$	0	0
$291$	9.03028	0.529364
$292$	0	0
$293$	11.4012	0.666062	0.333031	−	0.942916i	$-0.391929\pi$
$293$	11.4012	0.666062	0.333031	+	0.942916i	$0.391929\pi$
$294$	0	0
$295$	5.12019	0.298109
$296$	0	0
$297$	6.24977	0.362648
$298$	0	0
$299$	24.9991	1.44573
$300$	0	0
$301$	−4.24977	−0.244953
$302$	0	0
$303$	2.64002	0.151665
$304$	0	0
$305$	−0.159859	−0.00915351
$306$	0	0
$307$	12.7493	0.727642	0.363821	−	0.931469i	$-0.381472\pi$
$307$	12.7493	0.727642	0.363821	+	0.931469i	$0.381472\pi$
$308$	0	0
$309$	1.93945	0.110331
$310$	0	0
$311$	−18.8898	−1.07114	−0.535571	−	0.844490i	$-0.679904\pi$
$311$	−18.8898	−1.07114	−0.535571	+	0.844490i	$0.679904\pi$
$312$	0	0
$313$	16.4390	0.929187	0.464593	−	0.885524i	$-0.346201\pi$
$313$	16.4390	0.929187	0.464593	+	0.885524i	$0.346201\pi$
$314$	0	0
$315$	−2.64002	−0.148748
$316$	0	0
$317$	4.23039	0.237603	0.118801	−	0.992918i	$-0.462095\pi$
$317$	4.23039	0.237603	0.118801	+	0.992918i	$0.462095\pi$
$318$	0	0
$319$	22.5601	1.26312
$320$	0	0
$321$	8.82924	0.492800
$322$	0	0
$323$	−2.64002	−0.146895
$324$	0	0
$325$	7.87890	0.437042
$326$	0	0
$327$	7.28005	0.402588
$328$	0	0
$329$	−1.60975	−0.0887482
$330$	0	0
$331$	20.1892	1.10970	0.554850	−	0.831950i	$-0.312776\pi$
$331$	20.1892	1.10970	0.554850	+	0.831950i	$0.312776\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	16.4995	0.901466
$336$	0	0
$337$	23.1589	1.26155	0.630774	−	0.775967i	$-0.282738\pi$
$337$	23.1589	1.26155	0.630774	+	0.775967i	$0.282738\pi$
$338$	0	0
$339$	−4.88979	−0.265577
$340$	0	0
$341$	51.5592	2.79209
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−16.4995	−0.888305
$346$	0	0
$347$	21.4693	1.15253	0.576265	−	0.817263i	$-0.304510\pi$
$347$	21.4693	1.15253	0.576265	+	0.817263i	$0.304510\pi$
$348$	0	0
$349$	−14.9991	−0.802883	−0.401441	−	0.915885i	$-0.631491\pi$
$349$	−14.9991	−0.802883	−0.401441	+	0.915885i	$0.631491\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	−34.4802	−1.83519	−0.917597	−	0.397512i	$-0.869874\pi$
$353$	−34.4802	−1.83519	−0.917597	+	0.397512i	$0.869874\pi$
$354$	0	0
$355$	0.870417	0.0461969
$356$	0	0
$357$	−2.64002	−0.139725
$358$	0	0
$359$	8.68876	0.458575	0.229288	−	0.973359i	$-0.426360\pi$
$359$	8.68876	0.458575	0.229288	+	0.973359i	$0.426360\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	28.0596	1.47275
$364$	0	0
$365$	3.21949	0.168516
$366$	0	0
$367$	−27.7190	−1.44692	−0.723461	−	0.690365i	$-0.757450\pi$
$367$	−27.7190	−1.44692	−0.723461	+	0.690365i	$0.757450\pi$
$368$	0	0
$369$	−5.21949	−0.271716
$370$	0	0
$371$	−12.8898	−0.669205
$372$	0	0
$373$	16.5601	0.857449	0.428725	−	0.903435i	$-0.358963\pi$
$373$	16.5601	0.857449	0.428725	+	0.903435i	$0.358963\pi$
$374$	0	0
$375$	8.00000	0.413118
$376$	0	0
$377$	14.4390	0.743646
$378$	0	0
$379$	8.80986	0.452532	0.226266	−	0.974066i	$-0.427348\pi$
$379$	8.80986	0.452532	0.226266	+	0.974066i	$0.427348\pi$
$380$	0	0
$381$	14.7493	0.755630
$382$	0	0
$383$	−30.2791	−1.54719	−0.773596	−	0.633680i	$-0.781544\pi$
$383$	−30.2791	−1.54719	−0.773596	+	0.633680i	$0.781544\pi$
$384$	0	0
$385$	−16.4995	−0.840895
$386$	0	0
$387$	−4.24977	−0.216028
$388$	0	0
$389$	−34.2186	−1.73495	−0.867475	−	0.497480i	$-0.834259\pi$
$389$	−34.2186	−1.73495	−0.867475	+	0.497480i	$0.834259\pi$
$390$	0	0
$391$	−16.4995	−0.834418
$392$	0	0
$393$	−20.9503	−1.05681
$394$	0	0
$395$	2.56009	0.128812
$396$	0	0
$397$	12.0606	0.605302	0.302651	−	0.953101i	$-0.402128\pi$
$397$	12.0606	0.605302	0.302651	+	0.953101i	$0.402128\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	−24.8898	−1.24294	−0.621469	−	0.783439i	$-0.713464\pi$
$401$	−24.8898	−1.24294	−0.621469	+	0.783439i	$0.713464\pi$
$402$	0	0
$403$	32.9991	1.64380
$404$	0	0
$405$	−2.64002	−0.131184
$406$	0	0
$407$	12.4995	0.619579
$408$	0	0
$409$	18.5601	0.917738	0.458869	−	0.888504i	$-0.348255\pi$
$409$	18.5601	0.917738	0.458869	+	0.888504i	$0.348255\pi$
$410$	0	0
$411$	6.49954	0.320599
$412$	0	0
$413$	−1.93945	−0.0954340
$414$	0	0
$415$	9.68968	0.475648
$416$	0	0
$417$	−7.21949	−0.353540
$418$	0	0
$419$	13.6097	0.664880	0.332440	−	0.943124i	$-0.392128\pi$
$419$	13.6097	0.664880	0.332440	+	0.943124i	$0.392128\pi$
$420$	0	0
$421$	−11.6585	−0.568200	−0.284100	−	0.958795i	$-0.591695\pi$
$421$	−11.6585	−0.568200	−0.284100	+	0.958795i	$0.591695\pi$
$422$	0	0
$423$	−1.60975	−0.0782686
$424$	0	0
$425$	−5.20012	−0.252243
$426$	0	0
$427$	0.0605522	0.00293033
$428$	0	0
$429$	24.9991	1.20697
$430$	0	0
$431$	22.7687	1.09673	0.548365	−	0.836239i	$-0.315251\pi$
$431$	22.7687	1.09673	0.548365	+	0.836239i	$0.315251\pi$
$432$	0	0
$433$	−36.5895	−1.75838	−0.879188	−	0.476474i	$-0.841915\pi$
$433$	−36.5895	−1.75838	−0.879188	+	0.476474i	$0.841915\pi$
$434$	0	0
$435$	−9.52982	−0.456920
$436$	0	0
$437$	6.24977	0.298967
$438$	0	0
$439$	10.0681	0.480525	0.240262	−	0.970708i	$-0.422766\pi$
$439$	10.0681	0.480525	0.240262	+	0.970708i	$0.422766\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	23.6509	1.12369	0.561845	−	0.827243i	$-0.310092\pi$
$443$	23.6509	1.12369	0.561845	+	0.827243i	$0.310092\pi$
$444$	0	0
$445$	−24.3397	−1.15381
$446$	0	0
$447$	21.8401	1.03300
$448$	0	0
$449$	10.2909	0.485660	0.242830	−	0.970069i	$-0.421924\pi$
$449$	10.2909	0.485660	0.242830	+	0.970069i	$0.421924\pi$
$450$	0	0
$451$	−32.6206	−1.53605
$452$	0	0
$453$	2.24977	0.105703
$454$	0	0
$455$	−10.5601	−0.495065
$456$	0	0
$457$	26.5895	1.24380	0.621901	−	0.783096i	$-0.286361\pi$
$457$	26.5895	1.24380	0.621901	+	0.783096i	$0.286361\pi$
$458$	0	0
$459$	−2.64002	−0.123226
$460$	0	0
$461$	13.2001	0.614791	0.307395	−	0.951582i	$-0.400543\pi$
$461$	13.2001	0.614791	0.307395	+	0.951582i	$0.400543\pi$
$462$	0	0
$463$	−36.8780	−1.71387	−0.856933	−	0.515429i	$-0.827633\pi$
$463$	−36.8780	−1.71387	−0.856933	+	0.515429i	$0.827633\pi$
$464$	0	0
$465$	−21.7796	−1.01000
$466$	0	0
$467$	−18.7687	−0.868511	−0.434256	−	0.900790i	$-0.642989\pi$
$467$	−18.7687	−0.868511	−0.434256	+	0.900790i	$0.642989\pi$
$468$	0	0
$469$	−6.24977	−0.288588
$470$	0	0
$471$	−6.49954	−0.299483
$472$	0	0
$473$	−26.5601	−1.22123
$474$	0	0
$475$	1.96972	0.0903771
$476$	0	0
$477$	−12.8898	−0.590183
$478$	0	0
$479$	−9.32878	−0.426243	−0.213122	−	0.977026i	$-0.568363\pi$
$479$	−9.32878	−0.426243	−0.213122	+	0.977026i	$0.568363\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	6.24977	0.284374
$484$	0	0
$485$	−23.8401	−1.08253
$486$	0	0
$487$	25.4693	1.15412	0.577061	−	0.816701i	$-0.304199\pi$
$487$	25.4693	1.15412	0.577061	+	0.816701i	$0.304199\pi$
$488$	0	0
$489$	14.8099	0.669725
$490$	0	0
$491$	−7.65092	−0.345281	−0.172641	−	0.984985i	$-0.555230\pi$
$491$	−7.65092	−0.345281	−0.172641	+	0.984985i	$0.555230\pi$
$492$	0	0
$493$	−9.52982	−0.429201
$494$	0	0
$495$	−16.4995	−0.741599
$496$	0	0
$497$	−0.329700	−0.0147891
$498$	0	0
$499$	36.6206	1.63937	0.819683	−	0.572818i	$-0.194150\pi$
$499$	36.6206	1.63937	0.819683	+	0.572818i	$0.194150\pi$
$500$	0	0
$501$	−19.7190	−0.880982
$502$	0	0
$503$	15.6703	0.698704	0.349352	−	0.936992i	$-0.386402\pi$
$503$	15.6703	0.698704	0.349352	+	0.936992i	$0.386402\pi$
$504$	0	0
$505$	−6.96972	−0.310149
$506$	0	0
$507$	3.00000	0.133235
$508$	0	0
$509$	−10.6206	−0.470752	−0.235376	−	0.971904i	$-0.575632\pi$
$509$	−10.6206	−0.470752	−0.235376	+	0.971904i	$0.575632\pi$
$510$	0	0
$511$	−1.21949	−0.0539473
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−5.12019	−0.225622
$516$	0	0
$517$	−10.0606	−0.442463
$518$	0	0
$519$	−3.28005	−0.143978
$520$	0	0
$521$	−34.0000	−1.48957	−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	0	0
$523$	0.870417	0.0380607	0.0190303	−	0.999819i	$-0.493942\pi$
$523$	0.870417	0.0380607	0.0190303	+	0.999819i	$0.493942\pi$
$524$	0	0
$525$	1.96972	0.0859658
$526$	0	0
$527$	−21.7796	−0.948734
$528$	0	0
$529$	16.0596	0.698245
$530$	0	0
$531$	−1.93945	−0.0841649
$532$	0	0
$533$	−20.8780	−0.904326
$534$	0	0
$535$	−23.3094	−1.00775
$536$	0	0
$537$	−19.3893	−0.836712
$538$	0	0
$539$	6.24977	0.269197
$540$	0	0
$541$	−2.12110	−0.0911934	−0.0455967	−	0.998960i	$-0.514519\pi$
$541$	−2.12110	−0.0911934	−0.0455967	+	0.998960i	$0.514519\pi$
$542$	0	0
$543$	6.47018	0.277662
$544$	0	0
$545$	−19.2195	−0.823273
$546$	0	0
$547$	25.8477	1.10517	0.552584	−	0.833457i	$-0.313642\pi$
$547$	25.8477	1.10517	0.552584	+	0.833457i	$0.313642\pi$
$548$	0	0
$549$	0.0605522	0.00258430
$550$	0	0
$551$	3.60975	0.153780
$552$	0	0
$553$	−0.969724	−0.0412369
$554$	0	0
$555$	−5.28005	−0.224126
$556$	0	0
$557$	26.2186	1.11092	0.555458	−	0.831544i	$-0.312543\pi$
$557$	26.2186	1.11092	0.555458	+	0.831544i	$0.312543\pi$
$558$	0	0
$559$	−16.9991	−0.718985
$560$	0	0
$561$	−16.4995	−0.696611
$562$	0	0
$563$	−35.8401	−1.51048	−0.755241	−	0.655447i	$-0.772480\pi$
$563$	−35.8401	−1.51048	−0.755241	+	0.655447i	$0.772480\pi$
$564$	0	0
$565$	12.9092	0.543093
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	24.4490	1.02495	0.512477	−	0.858701i	$-0.328728\pi$
$569$	24.4490	1.02495	0.512477	+	0.858701i	$0.328728\pi$
$570$	0	0
$571$	22.5601	0.944111	0.472055	−	0.881569i	$-0.343512\pi$
$571$	22.5601	0.944111	0.472055	+	0.881569i	$0.343512\pi$
$572$	0	0
$573$	8.31032	0.347169
$574$	0	0
$575$	12.3103	0.513376
$576$	0	0
$577$	−24.5601	−1.02245	−0.511225	−	0.859447i	$-0.670808\pi$
$577$	−24.5601	−1.02245	−0.511225	+	0.859447i	$0.670808\pi$
$578$	0	0
$579$	−8.56009	−0.355745
$580$	0	0
$581$	−3.67030	−0.152270
$582$	0	0
$583$	−80.5583	−3.33638
$584$	0	0
$585$	−10.5601	−0.436606
$586$	0	0
$587$	35.7678	1.47629	0.738147	−	0.674640i	$-0.235701\pi$
$587$	35.7678	1.47629	0.738147	+	0.674640i	$0.235701\pi$
$588$	0	0
$589$	8.24977	0.339926
$590$	0	0
$591$	14.0000	0.575883
$592$	0	0
$593$	0.0411748	0.00169085	0.000845423	−	1.00000i	$-0.499731\pi$
$593$	0.0411748	0.00169085	0.000845423		1.00000i	$0.499731\pi$
$594$	0	0
$595$	6.96972	0.285731
$596$	0	0
$597$	11.2195	0.459183
$598$	0	0
$599$	−35.8889	−1.46638	−0.733190	−	0.680024i	$-0.761969\pi$
$599$	−35.8889	−1.46638	−0.733190	+	0.680024i	$0.761969\pi$
$600$	0	0
$601$	−9.52982	−0.388729	−0.194365	−	0.980929i	$-0.562265\pi$
$601$	−9.52982	−0.388729	−0.194365	+	0.980929i	$0.562265\pi$
$602$	0	0
$603$	−6.24977	−0.254510
$604$	0	0
$605$	−74.0781	−3.01170
$606$	0	0
$607$	−5.12019	−0.207822	−0.103911	−	0.994587i	$-0.533136\pi$
$607$	−5.12019	−0.207822	−0.103911	+	0.994587i	$0.533136\pi$
$608$	0	0
$609$	3.60975	0.146274
$610$	0	0
$611$	−6.43899	−0.260494
$612$	0	0
$613$	−1.21193	−0.0489495	−0.0244747	−	0.999700i	$-0.507791\pi$
$613$	−1.21193	−0.0489495	−0.0244747	+	0.999700i	$0.507791\pi$
$614$	0	0
$615$	13.7796	0.555647
$616$	0	0
$617$	30.9991	1.24798	0.623988	−	0.781434i	$-0.285511\pi$
$617$	30.9991	1.24798	0.623988	+	0.781434i	$0.285511\pi$
$618$	0	0
$619$	−3.34060	−0.134270	−0.0671350	−	0.997744i	$-0.521386\pi$
$619$	−3.34060	−0.134270	−0.0671350	+	0.997744i	$0.521386\pi$
$620$	0	0
$621$	6.24977	0.250795
$622$	0	0
$623$	9.21949	0.369371
$624$	0	0
$625$	−30.9688	−1.23875
$626$	0	0
$627$	6.24977	0.249592
$628$	0	0
$629$	−5.28005	−0.210529
$630$	0	0
$631$	−35.8089	−1.42553	−0.712766	−	0.701402i	$-0.752558\pi$
$631$	−35.8089	−1.42553	−0.712766	+	0.701402i	$0.752558\pi$
$632$	0	0
$633$	18.0899	0.719009
$634$	0	0
$635$	−38.9385	−1.54523
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	−0.329700	−0.0130427
$640$	0	0
$641$	−4.55011	−0.179719	−0.0898593	−	0.995954i	$-0.528642\pi$
$641$	−4.55011	−0.179719	−0.0898593	+	0.995954i	$0.528642\pi$
$642$	0	0
$643$	−44.8392	−1.76829	−0.884143	−	0.467216i	$-0.845257\pi$
$643$	−44.8392	−1.76829	−0.884143	+	0.467216i	$0.845257\pi$
$644$	0	0
$645$	11.2195	0.441767
$646$	0	0
$647$	9.32878	0.366752	0.183376	−	0.983043i	$-0.441297\pi$
$647$	9.32878	0.366752	0.183376	+	0.983043i	$0.441297\pi$
$648$	0	0
$649$	−12.1211	−0.475795
$650$	0	0
$651$	8.24977	0.323334
$652$	0	0
$653$	−13.1807	−0.515802	−0.257901	−	0.966171i	$-0.583031\pi$
$653$	−13.1807	−0.515802	−0.257901	+	0.966171i	$0.583031\pi$
$654$	0	0
$655$	55.3094	2.16112
$656$	0	0
$657$	−1.21949	−0.0475770
$658$	0	0
$659$	−29.2900	−1.14098	−0.570489	−	0.821305i	$-0.693246\pi$
$659$	−29.2900	−1.14098	−0.570489	+	0.821305i	$0.693246\pi$
$660$	0	0
$661$	31.4381	1.22280	0.611400	−	0.791322i	$-0.290607\pi$
$661$	31.4381	1.22280	0.611400	+	0.791322i	$0.290607\pi$
$662$	0	0
$663$	−10.5601	−0.410120
$664$	0	0
$665$	−2.64002	−0.102376
$666$	0	0
$667$	22.5601	0.873530
$668$	0	0
$669$	−6.31032	−0.243971
$670$	0	0
$671$	0.378437	0.0146094
$672$	0	0
$673$	16.0606	0.619089	0.309544	−	0.950885i	$-0.399823\pi$
$673$	16.0606	0.619089	0.309544	+	0.950885i	$0.399823\pi$
$674$	0	0
$675$	1.96972	0.0758147
$676$	0	0
$677$	30.4995	1.17219	0.586096	−	0.810241i	$-0.300664\pi$
$677$	30.4995	1.17219	0.586096	+	0.810241i	$0.300664\pi$
$678$	0	0
$679$	9.03028	0.346550
$680$	0	0
$681$	−7.21949	−0.276652
$682$	0	0
$683$	18.1093	0.692933	0.346466	−	0.938062i	$-0.387381\pi$
$683$	18.1093	0.692933	0.346466	+	0.938062i	$0.387381\pi$
$684$	0	0
$685$	−17.1589	−0.655609
$686$	0	0
$687$	21.8401	0.833253
$688$	0	0
$689$	−51.5592	−1.96425
$690$	0	0
$691$	6.31789	0.240344	0.120172	−	0.992753i	$-0.461655\pi$
$691$	6.31789	0.240344	0.120172	+	0.992753i	$0.461655\pi$
$692$	0	0
$693$	6.24977	0.237409
$694$	0	0
$695$	19.0596	0.722973
$696$	0	0
$697$	13.7796	0.521939
$698$	0	0
$699$	−21.7190	−0.821489
$700$	0	0
$701$	−19.4399	−0.734235	−0.367118	−	0.930175i	$-0.619655\pi$
$701$	−19.4399	−0.734235	−0.367118	+	0.930175i	$0.619655\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	4.24977	0.160056
$706$	0	0
$707$	2.64002	0.0992883
$708$	0	0
$709$	−25.8477	−0.970731	−0.485365	−	0.874311i	$-0.661313\pi$
$709$	−25.8477	−0.970731	−0.485365	+	0.874311i	$0.661313\pi$
$710$	0	0
$711$	−0.969724	−0.0363675
$712$	0	0
$713$	51.5592	1.93091
$714$	0	0
$715$	−65.9982	−2.46819
$716$	0	0
$717$	28.0294	1.04678
$718$	0	0
$719$	8.32970	0.310645	0.155323	−	0.987864i	$-0.450358\pi$
$719$	8.32970	0.310645	0.155323	+	0.987864i	$0.450358\pi$
$720$	0	0
$721$	1.93945	0.0722288
$722$	0	0
$723$	−5.52982	−0.205656
$724$	0	0
$725$	7.11021	0.264066
$726$	0	0
$727$	29.9394	1.11039	0.555196	−	0.831719i	$-0.312643\pi$
$727$	29.9394	1.11039	0.555196	+	0.831719i	$0.312643\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	11.2195	0.414968
$732$	0	0
$733$	12.7200	0.469822	0.234911	−	0.972017i	$-0.424520\pi$
$733$	12.7200	0.469822	0.234911	+	0.972017i	$0.424520\pi$
$734$	0	0
$735$	−2.64002	−0.0973787
$736$	0	0
$737$	−39.0596	−1.43878
$738$	0	0
$739$	−27.8089	−1.02297	−0.511484	−	0.859293i	$-0.670904\pi$
$739$	−27.8089	−1.02297	−0.511484	+	0.859293i	$0.670904\pi$
$740$	0	0
$741$	4.00000	0.146944
$742$	0	0
$743$	−1.60975	−0.0590559	−0.0295280	−	0.999564i	$-0.509400\pi$
$743$	−1.60975	−0.0590559	−0.0295280	+	0.999564i	$0.509400\pi$
$744$	0	0
$745$	−57.6585	−2.11244
$746$	0	0
$747$	−3.67030	−0.134289
$748$	0	0
$749$	8.82924	0.322613
$750$	0	0
$751$	39.7484	1.45044	0.725220	−	0.688517i	$-0.241738\pi$
$751$	39.7484	1.45044	0.725220	+	0.688517i	$0.241738\pi$
$752$	0	0
$753$	12.9503	0.471937
$754$	0	0
$755$	−5.93945	−0.216159
$756$	0	0
$757$	−11.4986	−0.417925	−0.208962	−	0.977924i	$-0.567009\pi$
$757$	−11.4986	−0.417925	−0.208962	+	0.977924i	$0.567009\pi$
$758$	0	0
$759$	39.0596	1.41777
$760$	0	0
$761$	−19.7602	−0.716307	−0.358154	−	0.933663i	$-0.616594\pi$
$761$	−19.7602	−0.716307	−0.358154	+	0.933663i	$0.616594\pi$
$762$	0	0
$763$	7.28005	0.263555
$764$	0	0
$765$	6.96972	0.251991
$766$	0	0
$767$	−7.75779	−0.280118
$768$	0	0
$769$	33.4381	1.20581	0.602904	−	0.797814i	$-0.294010\pi$
$769$	33.4381	1.20581	0.602904	+	0.797814i	$0.294010\pi$
$770$	0	0
$771$	−6.49954	−0.234075
$772$	0	0
$773$	−12.8411	−0.461861	−0.230930	−	0.972970i	$-0.574177\pi$
$773$	−12.8411	−0.461861	−0.230930	+	0.972970i	$0.574177\pi$
$774$	0	0
$775$	16.2498	0.583709
$776$	0	0
$777$	2.00000	0.0717496
$778$	0	0
$779$	−5.21949	−0.187008
$780$	0	0
$781$	−2.06055	−0.0737324
$782$	0	0
$783$	3.60975	0.129002
$784$	0	0
$785$	17.1589	0.612429
$786$	0	0
$787$	−0.870417	−0.0310270	−0.0155135	−	0.999880i	$-0.504938\pi$
$787$	−0.870417	−0.0310270	−0.0155135	+	0.999880i	$0.504938\pi$
$788$	0	0
$789$	−28.5289	−1.01566
$790$	0	0
$791$	−4.88979	−0.173861
$792$	0	0
$793$	0.242209	0.00860109
$794$	0	0
$795$	34.0294	1.20690
$796$	0	0
$797$	31.7796	1.12569	0.562845	−	0.826562i	$-0.309707\pi$
$797$	31.7796	1.12569	0.562845	+	0.826562i	$0.309707\pi$
$798$	0	0
$799$	4.24977	0.150346
$800$	0	0
$801$	9.21949	0.325755
$802$	0	0
$803$	−7.62156	−0.268959
$804$	0	0
$805$	−16.4995	−0.581532
$806$	0	0
$807$	24.9385	0.877878
$808$	0	0
$809$	−43.9394	−1.54483	−0.772414	−	0.635119i	$-0.780951\pi$
$809$	−43.9394	−1.54483	−0.772414	+	0.635119i	$0.780951\pi$
$810$	0	0
$811$	−3.37935	−0.118665	−0.0593326	−	0.998238i	$-0.518897\pi$
$811$	−3.37935	−0.118665	−0.0593326	+	0.998238i	$0.518897\pi$
$812$	0	0
$813$	−14.4390	−0.506397
$814$	0	0
$815$	−39.0984	−1.36956
$816$	0	0
$817$	−4.24977	−0.148681
$818$	0	0
$819$	4.00000	0.139771
$820$	0	0
$821$	−8.59885	−0.300102	−0.150051	−	0.988678i	$-0.547944\pi$
$821$	−8.59885	−0.300102	−0.150051	+	0.988678i	$0.547944\pi$
$822$	0	0
$823$	12.0076	0.418557	0.209279	−	0.977856i	$-0.432888\pi$
$823$	12.0076	0.418557	0.209279	+	0.977856i	$0.432888\pi$
$824$	0	0
$825$	12.3103	0.428590
$826$	0	0
$827$	−27.3893	−0.952421	−0.476210	−	0.879331i	$-0.657990\pi$
$827$	−27.3893	−0.952421	−0.476210	+	0.879331i	$0.657990\pi$
$828$	0	0
$829$	0.909172	0.0315768	0.0157884	−	0.999875i	$-0.494974\pi$
$829$	0.909172	0.0315768	0.0157884	+	0.999875i	$0.494974\pi$
$830$	0	0
$831$	25.5904	0.887720
$832$	0	0
$833$	−2.64002	−0.0914714
$834$	0	0
$835$	52.0587	1.80157
$836$	0	0
$837$	8.24977	0.285154
$838$	0	0
$839$	40.1798	1.38716	0.693581	−	0.720379i	$-0.256032\pi$
$839$	40.1798	1.38716	0.693581	+	0.720379i	$0.256032\pi$
$840$	0	0
$841$	−15.9697	−0.550680
$842$	0	0
$843$	−26.1698	−0.901337
$844$	0	0
$845$	−7.92007	−0.272459
$846$	0	0
$847$	28.0596	0.964140
$848$	0	0
$849$	−17.2800	−0.593050
$850$	0	0
$851$	12.4995	0.428479
$852$	0	0
$853$	2.37844	0.0814361	0.0407181	−	0.999171i	$-0.487035\pi$
$853$	2.37844	0.0814361	0.0407181	+	0.999171i	$0.487035\pi$
$854$	0	0
$855$	−2.64002	−0.0902869
$856$	0	0
$857$	−39.2800	−1.34178	−0.670890	−	0.741556i	$-0.734088\pi$
$857$	−39.2800	−1.34178	−0.670890	+	0.741556i	$0.734088\pi$
$858$	0	0
$859$	45.7408	1.56066	0.780329	−	0.625370i	$-0.215052\pi$
$859$	45.7408	1.56066	0.780329	+	0.625370i	$0.215052\pi$
$860$	0	0
$861$	−5.21949	−0.177880
$862$	0	0
$863$	1.60975	0.0547964	0.0273982	−	0.999625i	$-0.491278\pi$
$863$	1.60975	0.0547964	0.0273982	+	0.999625i	$0.491278\pi$
$864$	0	0
$865$	8.65940	0.294428
$866$	0	0
$867$	−10.0303	−0.340646
$868$	0	0
$869$	−6.06055	−0.205590
$870$	0	0
$871$	−24.9991	−0.847062
$872$	0	0
$873$	9.03028	0.305629
$874$	0	0
$875$	8.00000	0.270449
$876$	0	0
$877$	−50.0587	−1.69036	−0.845181	−	0.534480i	$-0.820508\pi$
$877$	−50.0587	−1.69036	−0.845181	+	0.534480i	$0.820508\pi$
$878$	0	0
$879$	11.4012	0.384551
$880$	0	0
$881$	17.0790	0.575407	0.287703	−	0.957720i	$-0.407108\pi$
$881$	17.0790	0.575407	0.287703	+	0.957720i	$0.407108\pi$
$882$	0	0
$883$	−1.06147	−0.0357213	−0.0178606	−	0.999840i	$-0.505686\pi$
$883$	−1.06147	−0.0357213	−0.0178606	+	0.999840i	$0.505686\pi$
$884$	0	0
$885$	5.12019	0.172113
$886$	0	0
$887$	−21.7796	−0.731287	−0.365644	−	0.930755i	$-0.619151\pi$
$887$	−21.7796	−0.731287	−0.365644	+	0.930755i	$0.619151\pi$
$888$	0	0
$889$	14.7493	0.494676
$890$	0	0
$891$	6.24977	0.209375
$892$	0	0
$893$	−1.60975	−0.0538681
$894$	0	0
$895$	51.1883	1.71104
$896$	0	0
$897$	24.9991	0.834695
$898$	0	0
$899$	29.7796	0.993205
$900$	0	0
$901$	34.0294	1.13368
$902$	0	0
$903$	−4.24977	−0.141424
$904$	0	0
$905$	−17.0814	−0.567806
$906$	0	0
$907$	−44.9073	−1.49112	−0.745562	−	0.666436i	$-0.767819\pi$
$907$	−44.9073	−1.49112	−0.745562	+	0.666436i	$0.767819\pi$
$908$	0	0
$909$	2.64002	0.0875641
$910$	0	0
$911$	19.8889	0.658948	0.329474	−	0.944165i	$-0.393129\pi$
$911$	19.8889	0.658948	0.329474	+	0.944165i	$0.393129\pi$
$912$	0	0
$913$	−22.9385	−0.759155
$914$	0	0
$915$	−0.159859	−0.00528478
$916$	0	0
$917$	−20.9503	−0.691841
$918$	0	0
$919$	−32.9991	−1.08854	−0.544270	−	0.838910i	$-0.683193\pi$
$919$	−32.9991	−1.08854	−0.544270	+	0.838910i	$0.683193\pi$
$920$	0	0
$921$	12.7493	0.420104
$922$	0	0
$923$	−1.31880	−0.0434089
$924$	0	0
$925$	3.93945	0.129528
$926$	0	0
$927$	1.93945	0.0636998
$928$	0	0
$929$	−3.95883	−0.129885	−0.0649424	−	0.997889i	$-0.520686\pi$
$929$	−3.95883	−0.129885	−0.0649424	+	0.997889i	$0.520686\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−18.8898	−0.618424
$934$	0	0
$935$	43.5592	1.42454
$936$	0	0
$937$	−35.4986	−1.15969	−0.579845	−	0.814727i	$-0.696887\pi$
$937$	−35.4986	−1.15969	−0.579845	+	0.814727i	$0.696887\pi$
$938$	0	0
$939$	16.4390	0.536466
$940$	0	0
$941$	5.55918	0.181224	0.0906120	−	0.995886i	$-0.471118\pi$
$941$	5.55918	0.181224	0.0906120	+	0.995886i	$0.471118\pi$
$942$	0	0
$943$	−32.6206	−1.06227
$944$	0	0
$945$	−2.64002	−0.0858800
$946$	0	0
$947$	19.8108	0.643764	0.321882	−	0.946780i	$-0.395685\pi$
$947$	19.8108	0.643764	0.321882	+	0.946780i	$0.395685\pi$
$948$	0	0
$949$	−4.87798	−0.158346
$950$	0	0
$951$	4.23039	0.137180
$952$	0	0
$953$	57.7290	1.87003	0.935013	−	0.354613i	$-0.115387\pi$
$953$	57.7290	1.87003	0.935013	+	0.354613i	$0.115387\pi$
$954$	0	0
$955$	−21.9394	−0.709944
$956$	0	0
$957$	22.5601	0.729264
$958$	0	0
$959$	6.49954	0.209881
$960$	0	0
$961$	37.0587	1.19544
$962$	0	0
$963$	8.82924	0.284518
$964$	0	0
$965$	22.5988	0.727483
$966$	0	0
$967$	−51.3094	−1.65000	−0.825000	−	0.565133i	$-0.808825\pi$
$967$	−51.3094	−1.65000	−0.825000	+	0.565133i	$0.808825\pi$
$968$	0	0
$969$	−2.64002	−0.0848098
$970$	0	0
$971$	16.6206	0.533382	0.266691	−	0.963782i	$-0.414070\pi$
$971$	16.6206	0.533382	0.266691	+	0.963782i	$0.414070\pi$
$972$	0	0
$973$	−7.21949	−0.231446
$974$	0	0
$975$	7.87890	0.252327
$976$	0	0
$977$	34.6694	1.10917	0.554586	−	0.832126i	$-0.312877\pi$
$977$	34.6694	1.10917	0.554586	+	0.832126i	$0.312877\pi$
$978$	0	0
$979$	57.6197	1.84153
$980$	0	0
$981$	7.28005	0.232434
$982$	0	0
$983$	22.1193	0.705495	0.352748	−	0.935719i	$-0.385247\pi$
$983$	22.1193	0.705495	0.352748	+	0.935719i	$0.385247\pi$
$984$	0	0
$985$	−36.9603	−1.17765
$986$	0	0
$987$	−1.60975	−0.0512388
$988$	0	0
$989$	−26.5601	−0.844562
$990$	0	0
$991$	28.9697	0.920254	0.460127	−	0.887853i	$-0.347804\pi$
$991$	28.9697	0.920254	0.460127	+	0.887853i	$0.347804\pi$
$992$	0	0
$993$	20.1892	0.640685
$994$	0	0
$995$	−29.6197	−0.939009
$996$	0	0
$997$	−43.3775	−1.37378	−0.686890	−	0.726761i	$-0.741025\pi$
$997$	−43.3775	−1.37378	−0.686890	+	0.726761i	$0.741025\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.bw.1.1		3
4.3	odd	2		3192.2.a.u.1.1	✓	3
12.11	even	2		9576.2.a.cb.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.u.1.1	✓	3	4.3	odd	2
6384.2.a.bw.1.1		3	1.1	even	1	trivial
9576.2.a.cb.1.3		3	12.11	even	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:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.64002 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.64002 q^{5} +1.00000 q^{7} +1.00000 q^{9} +6.24977 q^{11} +4.00000 q^{13} -2.64002 q^{15} -2.64002 q^{17} +1.00000 q^{19} +1.00000 q^{21} +6.24977 q^{23} +1.96972 q^{25} +1.00000 q^{27} +3.60975 q^{29} +8.24977 q^{31} +6.24977 q^{33} -2.64002 q^{35} +2.00000 q^{37} +4.00000 q^{39} -5.21949 q^{41} -4.24977 q^{43} -2.64002 q^{45} -1.60975 q^{47} +1.00000 q^{49} -2.64002 q^{51} -12.8898 q^{53} -16.4995 q^{55} +1.00000 q^{57} -1.93945 q^{59} +0.0605522 q^{61} +1.00000 q^{63} -10.5601 q^{65} -6.24977 q^{67} +6.24977 q^{69} -0.329700 q^{71} -1.21949 q^{73} +1.96972 q^{75} +6.24977 q^{77} -0.969724 q^{79} +1.00000 q^{81} -3.67030 q^{83} +6.96972 q^{85} +3.60975 q^{87} +9.21949 q^{89} +4.00000 q^{91} +8.24977 q^{93} -2.64002 q^{95} +9.03028 q^{97} +6.24977 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 12 q^{13} + 3 q^{19} + 3 q^{21} + 2 q^{23} + 5 q^{25} + 3 q^{27} + 2 q^{29} + 8 q^{31} + 2 q^{33} + 6 q^{37} + 12 q^{39} + 2 q^{41} + 4 q^{43} + 4 q^{47} + 3 q^{49} - 14 q^{53} - 16 q^{55} + 3 q^{57} - 4 q^{59} + 2 q^{61} + 3 q^{63} - 2 q^{67} + 2 q^{69} - 8 q^{71} + 14 q^{73} + 5 q^{75} + 2 q^{77} - 2 q^{79} + 3 q^{81} - 4 q^{83} + 20 q^{85} + 2 q^{87} + 10 q^{89} + 12 q^{91} + 8 q^{93} + 28 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^7 + 3 * q^9 + 2 * q^11 + 12 * q^13 + 3 * q^19 + 3 * q^21 + 2 * q^23 + 5 * q^25 + 3 * q^27 + 2 * q^29 + 8 * q^31 + 2 * q^33 + 6 * q^37 + 12 * q^39 + 2 * q^41 + 4 * q^43 + 4 * q^47 + 3 * q^49 - 14 * q^53 - 16 * q^55 + 3 * q^57 - 4 * q^59 + 2 * q^61 + 3 * q^63 - 2 * q^67 + 2 * q^69 - 8 * q^71 + 14 * q^73 + 5 * q^75 + 2 * q^77 - 2 * q^79 + 3 * q^81 - 4 * q^83 + 20 * q^85 + 2 * q^87 + 10 * q^89 + 12 * q^91 + 8 * q^93 + 28 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

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