LMFDB - Embedded newform 6384.2.a.bv.1.3

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.81361$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +5.62721 q^{11} -2.57834 q^{13} +6.20555 q^{17} +1.00000 q^{19} -1.00000 q^{21} -7.83276 q^{23} -5.00000 q^{25} -1.00000 q^{27} -3.42166 q^{29} -5.04888 q^{31} -5.62721 q^{33} -7.04888 q^{37} +2.57834 q^{39} -9.25443 q^{41} -7.25443 q^{43} -3.62721 q^{47} +1.00000 q^{49} -6.20555 q^{51} -8.57834 q^{53} -1.00000 q^{57} +8.41110 q^{59} -14.4111 q^{61} +1.00000 q^{63} +12.8816 q^{67} +7.83276 q^{69} +13.8328 q^{71} +9.25443 q^{73} +5.00000 q^{75} +5.62721 q^{77} +6.67609 q^{79} +1.00000 q^{81} +12.6761 q^{83} +3.42166 q^{87} -7.15667 q^{89} -2.57834 q^{91} +5.04888 q^{93} +4.78389 q^{97} +5.62721 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 4 q^{11} - 6 q^{13} + 4 q^{17} + 3 q^{19} - 3 q^{21} + 4 q^{23} - 15 q^{25} - 3 q^{27} - 12 q^{29} - 4 q^{31} - 4 q^{33} - 10 q^{37} + 6 q^{39} - 2 q^{41} + 4 q^{43} + 2 q^{47} + 3 q^{49} - 4 q^{51} - 24 q^{53} - 3 q^{57} - 4 q^{59} - 14 q^{61} + 3 q^{63} - 4 q^{69} + 14 q^{71} + 2 q^{73} + 15 q^{75} + 4 q^{77} - 4 q^{79} + 3 q^{81} + 14 q^{83} + 12 q^{87} - 18 q^{89} - 6 q^{91} + 4 q^{93} - 2 q^{97} + 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 + 4 * q^11 - 6 * q^13 + 4 * q^17 + 3 * q^19 - 3 * q^21 + 4 * q^23 - 15 * q^25 - 3 * q^27 - 12 * q^29 - 4 * q^31 - 4 * q^33 - 10 * q^37 + 6 * q^39 - 2 * q^41 + 4 * q^43 + 2 * q^47 + 3 * q^49 - 4 * q^51 - 24 * q^53 - 3 * q^57 - 4 * q^59 - 14 * q^61 + 3 * q^63 - 4 * q^69 + 14 * q^71 + 2 * q^73 + 15 * q^75 + 4 * q^77 - 4 * q^79 + 3 * q^81 + 14 * q^83 + 12 * q^87 - 18 * q^89 - 6 * q^91 + 4 * q^93 - 2 * q^97 + 4 * 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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.62721	1.69667	0.848334	−	0.529461i	$-0.177606\pi$
$11$	5.62721	1.69667	0.848334	+	0.529461i	$0.177606\pi$
$12$	0	0
$13$	−2.57834	−0.715102	−0.357551	−	0.933894i	$-0.616388\pi$
$13$	−2.57834	−0.715102	−0.357551	+	0.933894i	$0.616388\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.20555	1.50507	0.752533	−	0.658554i	$-0.228832\pi$
$17$	6.20555	1.50507	0.752533	+	0.658554i	$0.228832\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−7.83276	−1.63324	−0.816622	−	0.577173i	$-0.804156\pi$
$23$	−7.83276	−1.63324	−0.816622	+	0.577173i	$0.804156\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.42166	−0.635387	−0.317693	−	0.948193i	$-0.602908\pi$
$29$	−3.42166	−0.635387	−0.317693	+	0.948193i	$0.602908\pi$
$30$	0	0
$31$	−5.04888	−0.906805	−0.453402	−	0.891306i	$-0.649790\pi$
$31$	−5.04888	−0.906805	−0.453402	+	0.891306i	$0.649790\pi$
$32$	0	0
$33$	−5.62721	−0.979572
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.04888	−1.15883	−0.579414	−	0.815033i	$-0.696719\pi$
$37$	−7.04888	−1.15883	−0.579414	+	0.815033i	$0.696719\pi$
$38$	0	0
$39$	2.57834	0.412864
$40$	0	0
$41$	−9.25443	−1.44530	−0.722649	−	0.691215i	$-0.757076\pi$
$41$	−9.25443	−1.44530	−0.722649	+	0.691215i	$0.757076\pi$
$42$	0	0
$43$	−7.25443	−1.10629	−0.553145	−	0.833085i	$-0.686572\pi$
$43$	−7.25443	−1.10629	−0.553145	+	0.833085i	$0.686572\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.62721	−0.529083	−0.264542	−	0.964374i	$-0.585221\pi$
$47$	−3.62721	−0.529083	−0.264542	+	0.964374i	$0.585221\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.20555	−0.868951
$52$	0	0
$53$	−8.57834	−1.17833	−0.589163	−	0.808014i	$-0.700542\pi$
$53$	−8.57834	−1.17833	−0.589163	+	0.808014i	$0.700542\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	8.41110	1.09503	0.547516	−	0.836795i	$-0.315574\pi$
$59$	8.41110	1.09503	0.547516	+	0.836795i	$0.315574\pi$
$60$	0	0
$61$	−14.4111	−1.84515	−0.922576	−	0.385815i	$-0.873920\pi$
$61$	−14.4111	−1.84515	−0.922576	+	0.385815i	$0.873920\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.8816	1.57374	0.786871	−	0.617117i	$-0.211700\pi$
$67$	12.8816	1.57374	0.786871	+	0.617117i	$0.211700\pi$
$68$	0	0
$69$	7.83276	0.942954
$70$	0	0
$71$	13.8328	1.64165	0.820823	−	0.571182i	$-0.193515\pi$
$71$	13.8328	1.64165	0.820823	+	0.571182i	$0.193515\pi$
$72$	0	0
$73$	9.25443	1.08315	0.541574	−	0.840653i	$-0.317828\pi$
$73$	9.25443	1.08315	0.541574	+	0.840653i	$0.317828\pi$
$74$	0	0
$75$	5.00000	0.577350
$76$	0	0
$77$	5.62721	0.641280
$78$	0	0
$79$	6.67609	0.751119	0.375559	−	0.926798i	$-0.377451\pi$
$79$	6.67609	0.751119	0.375559	+	0.926798i	$0.377451\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.6761	1.39138	0.695691	−	0.718341i	$-0.255098\pi$
$83$	12.6761	1.39138	0.695691	+	0.718341i	$0.255098\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.42166	0.366841
$88$	0	0
$89$	−7.15667	−0.758606	−0.379303	−	0.925273i	$-0.623836\pi$
$89$	−7.15667	−0.758606	−0.379303	+	0.925273i	$0.623836\pi$
$90$	0	0
$91$	−2.57834	−0.270283
$92$	0	0
$93$	5.04888	0.523544
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.78389	0.485730	0.242865	−	0.970060i	$-0.421913\pi$
$97$	4.78389	0.485730	0.242865	+	0.970060i	$0.421913\pi$
$98$	0	0
$99$	5.62721	0.565556
$100$	0	0
$101$	−8.00000	−0.796030	−0.398015	−	0.917379i	$-0.630301\pi$
$101$	−8.00000	−0.796030	−0.398015	+	0.917379i	$0.630301\pi$
$102$	0	0
$103$	−9.45998	−0.932119	−0.466060	−	0.884753i	$-0.654327\pi$
$103$	−9.45998	−0.932119	−0.466060	+	0.884753i	$0.654327\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.9894	1.06239	0.531195	−	0.847250i	$-0.321743\pi$
$107$	10.9894	1.06239	0.531195	+	0.847250i	$0.321743\pi$
$108$	0	0
$109$	−16.6167	−1.59159	−0.795793	−	0.605568i	$-0.792946\pi$
$109$	−16.6167	−1.59159	−0.795793	+	0.605568i	$0.792946\pi$
$110$	0	0
$111$	7.04888	0.669050
$112$	0	0
$113$	−10.6761	−1.00432	−0.502161	−	0.864774i	$-0.667462\pi$
$113$	−10.6761	−1.00432	−0.502161	+	0.864774i	$0.667462\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.57834	−0.238367
$118$	0	0
$119$	6.20555	0.568862
$120$	0	0
$121$	20.6655	1.87868
$122$	0	0
$123$	9.25443	0.834443
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.8328	−1.04999	−0.524994	−	0.851106i	$-0.675932\pi$
$127$	−11.8328	−1.04999	−0.524994	+	0.851106i	$0.675932\pi$
$128$	0	0
$129$	7.25443	0.638717
$130$	0	0
$131$	−18.2439	−1.59397	−0.796987	−	0.603997i	$-0.793574\pi$
$131$	−18.2439	−1.59397	−0.796987	+	0.603997i	$0.793574\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.66553	0.825782	0.412891	−	0.910781i	$-0.364519\pi$
$137$	9.66553	0.825782	0.412891	+	0.910781i	$0.364519\pi$
$138$	0	0
$139$	−3.25443	−0.276037	−0.138018	−	0.990430i	$-0.544073\pi$
$139$	−3.25443	−0.276037	−0.138018	+	0.990430i	$0.544073\pi$
$140$	0	0
$141$	3.62721	0.305466
$142$	0	0
$143$	−14.5089	−1.21329
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−15.4600	−1.26653	−0.633265	−	0.773935i	$-0.718286\pi$
$149$	−15.4600	−1.26653	−0.633265	+	0.773935i	$0.718286\pi$
$150$	0	0
$151$	−12.9894	−1.05707	−0.528533	−	0.848913i	$-0.677258\pi$
$151$	−12.9894	−1.05707	−0.528533	+	0.848913i	$0.677258\pi$
$152$	0	0
$153$	6.20555	0.501689
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.15667	0.571165	0.285582	−	0.958354i	$-0.407813\pi$
$157$	7.15667	0.571165	0.285582	+	0.958354i	$0.407813\pi$
$158$	0	0
$159$	8.57834	0.680306
$160$	0	0
$161$	−7.83276	−0.617308
$162$	0	0
$163$	−15.2544	−1.19482	−0.597409	−	0.801936i	$-0.703803\pi$
$163$	−15.2544	−1.19482	−0.597409	+	0.801936i	$0.703803\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.25443	0.561364	0.280682	−	0.959801i	$-0.409439\pi$
$167$	7.25443	0.561364	0.280682	+	0.959801i	$0.409439\pi$
$168$	0	0
$169$	−6.35218	−0.488629
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	5.66553	0.430742	0.215371	−	0.976532i	$-0.430904\pi$
$173$	5.66553	0.430742	0.215371	+	0.976532i	$0.430904\pi$
$174$	0	0
$175$	−5.00000	−0.377964
$176$	0	0
$177$	−8.41110	−0.632217
$178$	0	0
$179$	−16.6761	−1.24643	−0.623215	−	0.782051i	$-0.714174\pi$
$179$	−16.6761	−1.24643	−0.623215	+	0.782051i	$0.714174\pi$
$180$	0	0
$181$	−6.57834	−0.488964	−0.244482	−	0.969654i	$-0.578618\pi$
$181$	−6.57834	−0.488964	−0.244482	+	0.969654i	$0.578618\pi$
$182$	0	0
$183$	14.4111	1.06530
$184$	0	0
$185$	0	0
$186$	0	0
$187$	34.9200	2.55360
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	6.67609	0.483065	0.241532	−	0.970393i	$-0.422350\pi$
$191$	6.67609	0.483065	0.241532	+	0.970393i	$0.422350\pi$
$192$	0	0
$193$	12.0978	0.870815	0.435408	−	0.900233i	$-0.356604\pi$
$193$	12.0978	0.870815	0.435408	+	0.900233i	$0.356604\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.7144	0.763370	0.381685	−	0.924293i	$-0.375344\pi$
$197$	10.7144	0.763370	0.381685	+	0.924293i	$0.375344\pi$
$198$	0	0
$199$	19.6655	1.39405	0.697026	−	0.717046i	$-0.254506\pi$
$199$	19.6655	1.39405	0.697026	+	0.717046i	$0.254506\pi$
$200$	0	0
$201$	−12.8816	−0.908601
$202$	0	0
$203$	−3.42166	−0.240154
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−7.83276	−0.544415
$208$	0	0
$209$	5.62721	0.389242
$210$	0	0
$211$	−10.3728	−0.714092	−0.357046	−	0.934087i	$-0.616216\pi$
$211$	−10.3728	−0.714092	−0.357046	+	0.934087i	$0.616216\pi$
$212$	0	0
$213$	−13.8328	−0.947805
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5.04888	−0.342740
$218$	0	0
$219$	−9.25443	−0.625356
$220$	0	0
$221$	−16.0000	−1.07628
$222$	0	0
$223$	8.51890	0.570468	0.285234	−	0.958458i	$-0.407929\pi$
$223$	8.51890	0.570468	0.285234	+	0.958458i	$0.407929\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	0	0
$227$	−19.2544	−1.27796	−0.638981	−	0.769223i	$-0.720644\pi$
$227$	−19.2544	−1.27796	−0.638981	+	0.769223i	$0.720644\pi$
$228$	0	0
$229$	3.15667	0.208599	0.104299	−	0.994546i	$-0.466740\pi$
$229$	3.15667	0.208599	0.104299	+	0.994546i	$0.466740\pi$
$230$	0	0
$231$	−5.62721	−0.370243
$232$	0	0
$233$	−14.4111	−0.944103	−0.472051	−	0.881571i	$-0.656486\pi$
$233$	−14.4111	−0.944103	−0.472051	+	0.881571i	$0.656486\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6.67609	−0.433659
$238$	0	0
$239$	−8.57834	−0.554886	−0.277443	−	0.960742i	$-0.589487\pi$
$239$	−8.57834	−0.554886	−0.277443	+	0.960742i	$0.589487\pi$
$240$	0	0
$241$	7.21611	0.464831	0.232415	−	0.972617i	$-0.425337\pi$
$241$	7.21611	0.464831	0.232415	+	0.972617i	$0.425337\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−2.57834	−0.164056
$248$	0	0
$249$	−12.6761	−0.803314
$250$	0	0
$251$	−18.2439	−1.15154	−0.575771	−	0.817611i	$-0.695298\pi$
$251$	−18.2439	−1.15154	−0.575771	+	0.817611i	$0.695298\pi$
$252$	0	0
$253$	−44.0766	−2.77107
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.25443	0.0782489	0.0391245	−	0.999234i	$-0.487543\pi$
$257$	1.25443	0.0782489	0.0391245	+	0.999234i	$0.487543\pi$
$258$	0	0
$259$	−7.04888	−0.437996
$260$	0	0
$261$	−3.42166	−0.211796
$262$	0	0
$263$	2.67609	0.165015	0.0825074	−	0.996590i	$-0.473707\pi$
$263$	2.67609	0.165015	0.0825074	+	0.996590i	$0.473707\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.15667	0.437981
$268$	0	0
$269$	−18.4111	−1.12254	−0.561272	−	0.827631i	$-0.689688\pi$
$269$	−18.4111	−1.12254	−0.561272	+	0.827631i	$0.689688\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	2.57834	0.156048
$274$	0	0
$275$	−28.1361	−1.69667
$276$	0	0
$277$	25.6655	1.54209	0.771046	−	0.636779i	$-0.219734\pi$
$277$	25.6655	1.54209	0.771046	+	0.636779i	$0.219734\pi$
$278$	0	0
$279$	−5.04888	−0.302268
$280$	0	0
$281$	−28.9894	−1.72936	−0.864682	−	0.502319i	$-0.832480\pi$
$281$	−28.9894	−1.72936	−0.864682	+	0.502319i	$0.832480\pi$
$282$	0	0
$283$	−21.5678	−1.28207	−0.641036	−	0.767511i	$-0.721495\pi$
$283$	−21.5678	−1.28207	−0.641036	+	0.767511i	$0.721495\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.25443	−0.546271
$288$	0	0
$289$	21.5089	1.26523
$290$	0	0
$291$	−4.78389	−0.280436
$292$	0	0
$293$	14.9411	0.872867	0.436434	−	0.899736i	$-0.356241\pi$
$293$	14.9411	0.872867	0.436434	+	0.899736i	$0.356241\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.62721	−0.326524
$298$	0	0
$299$	20.1955	1.16794
$300$	0	0
$301$	−7.25443	−0.418138
$302$	0	0
$303$	8.00000	0.459588
$304$	0	0
$305$	0	0
$306$	0	0
$307$	19.6655	1.12237	0.561185	−	0.827690i	$-0.310345\pi$
$307$	19.6655	1.12237	0.561185	+	0.827690i	$0.310345\pi$
$308$	0	0
$309$	9.45998	0.538159
$310$	0	0
$311$	22.1361	1.25522	0.627611	−	0.778527i	$-0.284033\pi$
$311$	22.1361	1.25522	0.627611	+	0.778527i	$0.284033\pi$
$312$	0	0
$313$	−0.0977518	−0.00552526	−0.00276263	−	0.999996i	$-0.500879\pi$
$313$	−0.0977518	−0.00552526	−0.00276263	+	0.999996i	$0.500879\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.7738	−0.942113	−0.471056	−	0.882103i	$-0.656127\pi$
$317$	−16.7738	−0.942113	−0.471056	+	0.882103i	$0.656127\pi$
$318$	0	0
$319$	−19.2544	−1.07804
$320$	0	0
$321$	−10.9894	−0.613371
$322$	0	0
$323$	6.20555	0.345286
$324$	0	0
$325$	12.8917	0.715102
$326$	0	0
$327$	16.6167	0.918903
$328$	0	0
$329$	−3.62721	−0.199975
$330$	0	0
$331$	−4.05944	−0.223127	−0.111563	−	0.993757i	$-0.535586\pi$
$331$	−4.05944	−0.223127	−0.111563	+	0.993757i	$0.535586\pi$
$332$	0	0
$333$	−7.04888	−0.386276
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.0978	0.876900	0.438450	−	0.898755i	$-0.355528\pi$
$337$	16.0978	0.876900	0.438450	+	0.898755i	$0.355528\pi$
$338$	0	0
$339$	10.6761	0.579846
$340$	0	0
$341$	−28.4111	−1.53855
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−30.9794	−1.66306	−0.831530	−	0.555479i	$-0.812535\pi$
$347$	−30.9794	−1.66306	−0.831530	+	0.555479i	$0.812535\pi$
$348$	0	0
$349$	−0.843326	−0.0451422	−0.0225711	−	0.999745i	$-0.507185\pi$
$349$	−0.843326	−0.0451422	−0.0225711	+	0.999745i	$0.507185\pi$
$350$	0	0
$351$	2.57834	0.137621
$352$	0	0
$353$	12.1078	0.644433	0.322217	−	0.946666i	$-0.395572\pi$
$353$	12.1078	0.644433	0.322217	+	0.946666i	$0.395572\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−6.20555	−0.328433
$358$	0	0
$359$	25.9305	1.36856	0.684280	−	0.729219i	$-0.260117\pi$
$359$	25.9305	1.36856	0.684280	+	0.729219i	$0.260117\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−20.6655	−1.08466
$364$	0	0
$365$	0	0
$366$	0	0
$367$	35.6655	1.86173	0.930863	−	0.365369i	$-0.119057\pi$
$367$	35.6655	1.86173	0.930863	+	0.365369i	$0.119057\pi$
$368$	0	0
$369$	−9.25443	−0.481766
$370$	0	0
$371$	−8.57834	−0.445365
$372$	0	0
$373$	18.7144	0.968995	0.484498	−	0.874793i	$-0.339002\pi$
$373$	18.7144	0.968995	0.484498	+	0.874793i	$0.339002\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.82220	0.454366
$378$	0	0
$379$	−36.8816	−1.89448	−0.947241	−	0.320521i	$-0.896142\pi$
$379$	−36.8816	−1.89448	−0.947241	+	0.320521i	$0.896142\pi$
$380$	0	0
$381$	11.8328	0.606211
$382$	0	0
$383$	−27.6655	−1.41364	−0.706821	−	0.707392i	$-0.749871\pi$
$383$	−27.6655	−1.41364	−0.706821	+	0.707392i	$0.749871\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.25443	−0.368763
$388$	0	0
$389$	10.5189	0.533329	0.266665	−	0.963789i	$-0.414078\pi$
$389$	10.5189	0.533329	0.266665	+	0.963789i	$0.414078\pi$
$390$	0	0
$391$	−48.6066	−2.45814
$392$	0	0
$393$	18.2439	0.920281
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.0766	1.30875	0.654374	−	0.756171i	$-0.272932\pi$
$397$	26.0766	1.30875	0.654374	+	0.756171i	$0.272932\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	−31.4983	−1.57295	−0.786475	−	0.617622i	$-0.788096\pi$
$401$	−31.4983	−1.57295	−0.786475	+	0.617622i	$0.788096\pi$
$402$	0	0
$403$	13.0177	0.648458
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−39.6655	−1.96615
$408$	0	0
$409$	13.3139	0.658328	0.329164	−	0.944273i	$-0.393233\pi$
$409$	13.3139	0.658328	0.329164	+	0.944273i	$0.393233\pi$
$410$	0	0
$411$	−9.66553	−0.476765
$412$	0	0
$413$	8.41110	0.413883
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3.25443	0.159370
$418$	0	0
$419$	20.3416	0.993753	0.496876	−	0.867821i	$-0.334480\pi$
$419$	20.3416	0.993753	0.496876	+	0.867821i	$0.334480\pi$
$420$	0	0
$421$	7.87108	0.383613	0.191806	−	0.981433i	$-0.438565\pi$
$421$	7.87108	0.383613	0.191806	+	0.981433i	$0.438565\pi$
$422$	0	0
$423$	−3.62721	−0.176361
$424$	0	0
$425$	−31.0278	−1.50507
$426$	0	0
$427$	−14.4111	−0.697402
$428$	0	0
$429$	14.5089	0.700494
$430$	0	0
$431$	−4.48059	−0.215822	−0.107911	−	0.994161i	$-0.534416\pi$
$431$	−4.48059	−0.215822	−0.107911	+	0.994161i	$0.534416\pi$
$432$	0	0
$433$	0.157190	0.00755409	0.00377705	−	0.999993i	$-0.498798\pi$
$433$	0.157190	0.00755409	0.00377705	+	0.999993i	$0.498798\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.83276	−0.374692
$438$	0	0
$439$	25.4600	1.21514	0.607569	−	0.794267i	$-0.292145\pi$
$439$	25.4600	1.21514	0.607569	+	0.794267i	$0.292145\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	27.6061	1.31160	0.655802	−	0.754933i	$-0.272330\pi$
$443$	27.6061	1.31160	0.655802	+	0.754933i	$0.272330\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	15.4600	0.731232
$448$	0	0
$449$	−39.8328	−1.87982	−0.939912	−	0.341416i	$-0.889093\pi$
$449$	−39.8328	−1.87982	−0.939912	+	0.341416i	$0.889093\pi$
$450$	0	0
$451$	−52.0766	−2.45219
$452$	0	0
$453$	12.9894	0.610297
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−25.3311	−1.18494	−0.592468	−	0.805594i	$-0.701846\pi$
$457$	−25.3311	−1.18494	−0.592468	+	0.805594i	$0.701846\pi$
$458$	0	0
$459$	−6.20555	−0.289650
$460$	0	0
$461$	4.41110	0.205445	0.102723	−	0.994710i	$-0.467245\pi$
$461$	4.41110	0.205445	0.102723	+	0.994710i	$0.467245\pi$
$462$	0	0
$463$	−1.49115	−0.0692995	−0.0346498	−	0.999400i	$-0.511032\pi$
$463$	−1.49115	−0.0692995	−0.0346498	+	0.999400i	$0.511032\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.67609	−0.216384	−0.108192	−	0.994130i	$-0.534506\pi$
$467$	−4.67609	−0.216384	−0.108192	+	0.994130i	$0.534506\pi$
$468$	0	0
$469$	12.8816	0.594819
$470$	0	0
$471$	−7.15667	−0.329762
$472$	0	0
$473$	−40.8222	−1.87701
$474$	0	0
$475$	−5.00000	−0.229416
$476$	0	0
$477$	−8.57834	−0.392775
$478$	0	0
$479$	−13.3139	−0.608326	−0.304163	−	0.952620i	$-0.598377\pi$
$479$	−13.3139	−0.608326	−0.304163	+	0.952620i	$0.598377\pi$
$480$	0	0
$481$	18.1744	0.828680
$482$	0	0
$483$	7.83276	0.356403
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3.08719	−0.139894	−0.0699469	−	0.997551i	$-0.522283\pi$
$487$	−3.08719	−0.139894	−0.0699469	+	0.997551i	$0.522283\pi$
$488$	0	0
$489$	15.2544	0.689829
$490$	0	0
$491$	−4.88164	−0.220305	−0.110153	−	0.993915i	$-0.535134\pi$
$491$	−4.88164	−0.220305	−0.110153	+	0.993915i	$0.535134\pi$
$492$	0	0
$493$	−21.2333	−0.956300
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.8328	0.620484
$498$	0	0
$499$	4.94108	0.221193	0.110597	−	0.993865i	$-0.464724\pi$
$499$	4.94108	0.221193	0.110597	+	0.993865i	$0.464724\pi$
$500$	0	0
$501$	−7.25443	−0.324104
$502$	0	0
$503$	22.1361	0.986998	0.493499	−	0.869746i	$-0.335718\pi$
$503$	22.1361	0.986998	0.493499	+	0.869746i	$0.335718\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.35218	0.282110
$508$	0	0
$509$	−38.6066	−1.71121	−0.855604	−	0.517631i	$-0.826814\pi$
$509$	−38.6066	−1.71121	−0.855604	+	0.517631i	$0.826814\pi$
$510$	0	0
$511$	9.25443	0.409392
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−20.4111	−0.897679
$518$	0	0
$519$	−5.66553	−0.248689
$520$	0	0
$521$	27.0177	1.18367	0.591834	−	0.806060i	$-0.298404\pi$
$521$	27.0177	1.18367	0.591834	+	0.806060i	$0.298404\pi$
$522$	0	0
$523$	37.9789	1.66070	0.830350	−	0.557242i	$-0.188140\pi$
$523$	37.9789	1.66070	0.830350	+	0.557242i	$0.188140\pi$
$524$	0	0
$525$	5.00000	0.218218
$526$	0	0
$527$	−31.3311	−1.36480
$528$	0	0
$529$	38.3522	1.66749
$530$	0	0
$531$	8.41110	0.365011
$532$	0	0
$533$	23.8610	1.03354
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.6761	0.719626
$538$	0	0
$539$	5.62721	0.242381
$540$	0	0
$541$	−1.66553	−0.0716066	−0.0358033	−	0.999359i	$-0.511399\pi$
$541$	−1.66553	−0.0716066	−0.0358033	+	0.999359i	$0.511399\pi$
$542$	0	0
$543$	6.57834	0.282303
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−46.1149	−1.97173	−0.985866	−	0.167534i	$-0.946419\pi$
$547$	−46.1149	−1.97173	−0.985866	+	0.167534i	$0.946419\pi$
$548$	0	0
$549$	−14.4111	−0.615051
$550$	0	0
$551$	−3.42166	−0.145768
$552$	0	0
$553$	6.67609	0.283896
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−43.6555	−1.84974	−0.924871	−	0.380281i	$-0.875827\pi$
$557$	−43.6555	−1.84974	−0.924871	+	0.380281i	$0.875827\pi$
$558$	0	0
$559$	18.7044	0.791110
$560$	0	0
$561$	−34.9200	−1.47432
$562$	0	0
$563$	−21.5678	−0.908973	−0.454487	−	0.890754i	$-0.650177\pi$
$563$	−21.5678	−0.908973	−0.454487	+	0.890754i	$0.650177\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−3.01056	−0.126209	−0.0631047	−	0.998007i	$-0.520100\pi$
$569$	−3.01056	−0.126209	−0.0631047	+	0.998007i	$0.520100\pi$
$570$	0	0
$571$	−9.68665	−0.405374	−0.202687	−	0.979244i	$-0.564967\pi$
$571$	−9.68665	−0.405374	−0.202687	+	0.979244i	$0.564967\pi$
$572$	0	0
$573$	−6.67609	−0.278898
$574$	0	0
$575$	39.1638	1.63324
$576$	0	0
$577$	−4.50885	−0.187706	−0.0938530	−	0.995586i	$-0.529918\pi$
$577$	−4.50885	−0.187706	−0.0938530	+	0.995586i	$0.529918\pi$
$578$	0	0
$579$	−12.0978	−0.502765
$580$	0	0
$581$	12.6761	0.525893
$582$	0	0
$583$	−48.2721	−1.99923
$584$	0	0
$585$	0	0
$586$	0	0
$587$	40.5572	1.67398	0.836988	−	0.547222i	$-0.184315\pi$
$587$	40.5572	1.67398	0.836988	+	0.547222i	$0.184315\pi$
$588$	0	0
$589$	−5.04888	−0.208035
$590$	0	0
$591$	−10.7144	−0.440732
$592$	0	0
$593$	16.7144	0.686378	0.343189	−	0.939266i	$-0.388493\pi$
$593$	16.7144	0.686378	0.343189	+	0.939266i	$0.388493\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−19.6655	−0.804856
$598$	0	0
$599$	−27.5194	−1.12441	−0.562206	−	0.826997i	$-0.690047\pi$
$599$	−27.5194	−1.12441	−0.562206	+	0.826997i	$0.690047\pi$
$600$	0	0
$601$	−19.8227	−0.808585	−0.404293	−	0.914630i	$-0.632482\pi$
$601$	−19.8227	−0.808585	−0.404293	+	0.914630i	$0.632482\pi$
$602$	0	0
$603$	12.8816	0.524581
$604$	0	0
$605$	0	0
$606$	0	0
$607$	25.7944	1.04696	0.523482	−	0.852037i	$-0.324633\pi$
$607$	25.7944	1.04696	0.523482	+	0.852037i	$0.324633\pi$
$608$	0	0
$609$	3.42166	0.138653
$610$	0	0
$611$	9.35218	0.378349
$612$	0	0
$613$	21.2544	0.858458	0.429229	−	0.903196i	$-0.358785\pi$
$613$	21.2544	0.858458	0.429229	+	0.903196i	$0.358785\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.94108	0.118403	0.0592017	−	0.998246i	$-0.481144\pi$
$617$	2.94108	0.118403	0.0592017	+	0.998246i	$0.481144\pi$
$618$	0	0
$619$	9.76328	0.392419	0.196210	−	0.980562i	$-0.437137\pi$
$619$	9.76328	0.392419	0.196210	+	0.980562i	$0.437137\pi$
$620$	0	0
$621$	7.83276	0.314318
$622$	0	0
$623$	−7.15667	−0.286726
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	−5.62721	−0.224729
$628$	0	0
$629$	−43.7422	−1.74411
$630$	0	0
$631$	−25.1567	−1.00147	−0.500736	−	0.865600i	$-0.666937\pi$
$631$	−25.1567	−1.00147	−0.500736	+	0.865600i	$0.666937\pi$
$632$	0	0
$633$	10.3728	0.412281
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.57834	−0.102157
$638$	0	0
$639$	13.8328	0.547216
$640$	0	0
$641$	1.65838	0.0655023	0.0327511	−	0.999464i	$-0.489573\pi$
$641$	1.65838	0.0655023	0.0327511	+	0.999464i	$0.489573\pi$
$642$	0	0
$643$	−32.4877	−1.28119	−0.640595	−	0.767879i	$-0.721312\pi$
$643$	−32.4877	−1.28119	−0.640595	+	0.767879i	$0.721312\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.4116	0.605893	0.302947	−	0.953008i	$-0.402030\pi$
$647$	15.4116	0.605893	0.302947	+	0.953008i	$0.402030\pi$
$648$	0	0
$649$	47.3311	1.85791
$650$	0	0
$651$	5.04888	0.197881
$652$	0	0
$653$	43.6555	1.70837	0.854185	−	0.519968i	$-0.174056\pi$
$653$	43.6555	1.70837	0.854185	+	0.519968i	$0.174056\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.25443	0.361050
$658$	0	0
$659$	5.83276	0.227212	0.113606	−	0.993526i	$-0.463760\pi$
$659$	5.83276	0.227212	0.113606	+	0.993526i	$0.463760\pi$
$660$	0	0
$661$	−28.3416	−1.10236	−0.551181	−	0.834386i	$-0.685822\pi$
$661$	−28.3416	−1.10236	−0.551181	+	0.834386i	$0.685822\pi$
$662$	0	0
$663$	16.0000	0.621389
$664$	0	0
$665$	0	0
$666$	0	0
$667$	26.8011	1.03774
$668$	0	0
$669$	−8.51890	−0.329360
$670$	0	0
$671$	−81.0943	−3.13061
$672$	0	0
$673$	13.4700	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$673$	13.4700	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$674$	0	0
$675$	5.00000	0.192450
$676$	0	0
$677$	−15.1567	−0.582518	−0.291259	−	0.956644i	$-0.594074\pi$
$677$	−15.1567	−0.582518	−0.291259	+	0.956644i	$0.594074\pi$
$678$	0	0
$679$	4.78389	0.183589
$680$	0	0
$681$	19.2544	0.737831
$682$	0	0
$683$	−19.9305	−0.762620	−0.381310	−	0.924447i	$-0.624527\pi$
$683$	−19.9305	−0.762620	−0.381310	+	0.924447i	$0.624527\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3.15667	−0.120435
$688$	0	0
$689$	22.1178	0.842623
$690$	0	0
$691$	34.5089	1.31278	0.656389	−	0.754422i	$-0.272083\pi$
$691$	34.5089	1.31278	0.656389	+	0.754422i	$0.272083\pi$
$692$	0	0
$693$	5.62721	0.213760
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−57.4288	−2.17527
$698$	0	0
$699$	14.4111	0.545078
$700$	0	0
$701$	−43.4600	−1.64146	−0.820730	−	0.571316i	$-0.806433\pi$
$701$	−43.4600	−1.64146	−0.820730	+	0.571316i	$0.806433\pi$
$702$	0	0
$703$	−7.04888	−0.265853
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8.00000	−0.300871
$708$	0	0
$709$	31.2333	1.17299	0.586496	−	0.809952i	$-0.300507\pi$
$709$	31.2333	1.17299	0.586496	+	0.809952i	$0.300507\pi$
$710$	0	0
$711$	6.67609	0.250373
$712$	0	0
$713$	39.5466	1.48103
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.57834	0.320364
$718$	0	0
$719$	31.9194	1.19039	0.595197	−	0.803580i	$-0.297074\pi$
$719$	31.9194	1.19039	0.595197	+	0.803580i	$0.297074\pi$
$720$	0	0
$721$	−9.45998	−0.352308
$722$	0	0
$723$	−7.21611	−0.268370
$724$	0	0
$725$	17.1083	0.635387
$726$	0	0
$727$	−6.72445	−0.249396	−0.124698	−	0.992195i	$-0.539796\pi$
$727$	−6.72445	−0.249396	−0.124698	+	0.992195i	$0.539796\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−45.0177	−1.66504
$732$	0	0
$733$	−45.3311	−1.67434	−0.837170	−	0.546942i	$-0.815792\pi$
$733$	−45.3311	−1.67434	−0.837170	+	0.546942i	$0.815792\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	72.4877	2.67012
$738$	0	0
$739$	−26.5089	−0.975144	−0.487572	−	0.873083i	$-0.662117\pi$
$739$	−26.5089	−0.975144	−0.487572	+	0.873083i	$0.662117\pi$
$740$	0	0
$741$	2.57834	0.0947176
$742$	0	0
$743$	2.36274	0.0866805	0.0433403	−	0.999060i	$-0.486200\pi$
$743$	2.36274	0.0866805	0.0433403	+	0.999060i	$0.486200\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.6761	0.463794
$748$	0	0
$749$	10.9894	0.401545
$750$	0	0
$751$	18.4605	0.673633	0.336816	−	0.941570i	$-0.390650\pi$
$751$	18.4605	0.673633	0.336816	+	0.941570i	$0.390650\pi$
$752$	0	0
$753$	18.2439	0.664843
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−34.6066	−1.25780	−0.628899	−	0.777487i	$-0.716494\pi$
$757$	−34.6066	−1.25780	−0.628899	+	0.777487i	$0.716494\pi$
$758$	0	0
$759$	44.0766	1.59988
$760$	0	0
$761$	−14.4011	−0.522038	−0.261019	−	0.965334i	$-0.584058\pi$
$761$	−14.4011	−0.522038	−0.261019	+	0.965334i	$0.584058\pi$
$762$	0	0
$763$	−16.6167	−0.601563
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−21.6867	−0.783060
$768$	0	0
$769$	40.7244	1.46856	0.734281	−	0.678846i	$-0.237520\pi$
$769$	40.7244	1.46856	0.734281	+	0.678846i	$0.237520\pi$
$770$	0	0
$771$	−1.25443	−0.0451770
$772$	0	0
$773$	6.94108	0.249653	0.124827	−	0.992179i	$-0.460163\pi$
$773$	6.94108	0.249653	0.124827	+	0.992179i	$0.460163\pi$
$774$	0	0
$775$	25.2444	0.906805
$776$	0	0
$777$	7.04888	0.252877
$778$	0	0
$779$	−9.25443	−0.331574
$780$	0	0
$781$	77.8399	2.78533
$782$	0	0
$783$	3.42166	0.122280
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−20.9411	−0.746469	−0.373234	−	0.927737i	$-0.621751\pi$
$787$	−20.9411	−0.746469	−0.373234	+	0.927737i	$0.621751\pi$
$788$	0	0
$789$	−2.67609	−0.0952713
$790$	0	0
$791$	−10.6761	−0.379598
$792$	0	0
$793$	37.1567	1.31947
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.33447	−0.0826913	−0.0413457	−	0.999145i	$-0.513164\pi$
$797$	−2.33447	−0.0826913	−0.0413457	+	0.999145i	$0.513164\pi$
$798$	0	0
$799$	−22.5089	−0.796306
$800$	0	0
$801$	−7.15667	−0.252869
$802$	0	0
$803$	52.0766	1.83774
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18.4111	0.648101
$808$	0	0
$809$	22.4111	0.787932	0.393966	−	0.919125i	$-0.371103\pi$
$809$	22.4111	0.787932	0.393966	+	0.919125i	$0.371103\pi$
$810$	0	0
$811$	−10.8433	−0.380761	−0.190380	−	0.981710i	$-0.560972\pi$
$811$	−10.8433	−0.380761	−0.190380	+	0.981710i	$0.560972\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−7.25443	−0.253800
$818$	0	0
$819$	−2.57834	−0.0900944
$820$	0	0
$821$	52.2822	1.82466	0.912330	−	0.409455i	$-0.134281\pi$
$821$	52.2822	1.82466	0.912330	+	0.409455i	$0.134281\pi$
$822$	0	0
$823$	8.82220	0.307523	0.153761	−	0.988108i	$-0.450861\pi$
$823$	8.82220	0.307523	0.153761	+	0.988108i	$0.450861\pi$
$824$	0	0
$825$	28.1361	0.979572
$826$	0	0
$827$	−10.4595	−0.363711	−0.181856	−	0.983325i	$-0.558210\pi$
$827$	−10.4595	−0.363711	−0.181856	+	0.983325i	$0.558210\pi$
$828$	0	0
$829$	17.3028	0.600951	0.300475	−	0.953790i	$-0.402855\pi$
$829$	17.3028	0.600951	0.300475	+	0.953790i	$0.402855\pi$
$830$	0	0
$831$	−25.6655	−0.890327
$832$	0	0
$833$	6.20555	0.215010
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.04888	0.174515
$838$	0	0
$839$	−40.6066	−1.40190	−0.700948	−	0.713213i	$-0.747239\pi$
$839$	−40.6066	−1.40190	−0.700948	+	0.713213i	$0.747239\pi$
$840$	0	0
$841$	−17.2922	−0.596284
$842$	0	0
$843$	28.9894	0.998449
$844$	0	0
$845$	0	0
$846$	0	0
$847$	20.6655	0.710076
$848$	0	0
$849$	21.5678	0.740204
$850$	0	0
$851$	55.2122	1.89265
$852$	0	0
$853$	−0.724449	−0.0248046	−0.0124023	−	0.999923i	$-0.503948\pi$
$853$	−0.724449	−0.0248046	−0.0124023	+	0.999923i	$0.503948\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.0388	1.26522	0.632611	−	0.774470i	$-0.281983\pi$
$857$	37.0388	1.26522	0.632611	+	0.774470i	$0.281983\pi$
$858$	0	0
$859$	−39.7422	−1.35598	−0.677992	−	0.735069i	$-0.737150\pi$
$859$	−39.7422	−1.35598	−0.677992	+	0.735069i	$0.737150\pi$
$860$	0	0
$861$	9.25443	0.315390
$862$	0	0
$863$	25.1638	0.856586	0.428293	−	0.903640i	$-0.359115\pi$
$863$	25.1638	0.856586	0.428293	+	0.903640i	$0.359115\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−21.5089	−0.730479
$868$	0	0
$869$	37.5678	1.27440
$870$	0	0
$871$	−33.2132	−1.12539
$872$	0	0
$873$	4.78389	0.161910
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.1844	−1.15433	−0.577163	−	0.816629i	$-0.695840\pi$
$877$	−34.1844	−1.15433	−0.577163	+	0.816629i	$0.695840\pi$
$878$	0	0
$879$	−14.9411	−0.503950
$880$	0	0
$881$	−7.69670	−0.259308	−0.129654	−	0.991559i	$-0.541387\pi$
$881$	−7.69670	−0.259308	−0.129654	+	0.991559i	$0.541387\pi$
$882$	0	0
$883$	1.56777	0.0527598	0.0263799	−	0.999652i	$-0.491602\pi$
$883$	1.56777	0.0527598	0.0263799	+	0.999652i	$0.491602\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−15.2544	−0.512193	−0.256097	−	0.966651i	$-0.582437\pi$
$887$	−15.2544	−0.512193	−0.256097	+	0.966651i	$0.582437\pi$
$888$	0	0
$889$	−11.8328	−0.396858
$890$	0	0
$891$	5.62721	0.188519
$892$	0	0
$893$	−3.62721	−0.121380
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−20.1955	−0.674308
$898$	0	0
$899$	17.2756	0.576172
$900$	0	0
$901$	−53.2333	−1.77346
$902$	0	0
$903$	7.25443	0.241412
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−42.8605	−1.42316	−0.711580	−	0.702605i	$-0.752020\pi$
$907$	−42.8605	−1.42316	−0.711580	+	0.702605i	$0.752020\pi$
$908$	0	0
$909$	−8.00000	−0.265343
$910$	0	0
$911$	−26.0283	−0.862355	−0.431177	−	0.902267i	$-0.641902\pi$
$911$	−26.0283	−0.862355	−0.431177	+	0.902267i	$0.641902\pi$
$912$	0	0
$913$	71.3311	2.36071
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−18.2439	−0.602465
$918$	0	0
$919$	−37.0177	−1.22110	−0.610551	−	0.791977i	$-0.709052\pi$
$919$	−37.0177	−1.22110	−0.610551	+	0.791977i	$0.709052\pi$
$920$	0	0
$921$	−19.6655	−0.648001
$922$	0	0
$923$	−35.6655	−1.17395
$924$	0	0
$925$	35.2444	1.15883
$926$	0	0
$927$	−9.45998	−0.310706
$928$	0	0
$929$	1.57885	0.0518005	0.0259002	−	0.999665i	$-0.491755\pi$
$929$	1.57885	0.0518005	0.0259002	+	0.999665i	$0.491755\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−22.1361	−0.724702
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−21.6655	−0.707782	−0.353891	−	0.935287i	$-0.615142\pi$
$937$	−21.6655	−0.707782	−0.353891	+	0.935287i	$0.615142\pi$
$938$	0	0
$939$	0.0977518	0.00319001
$940$	0	0
$941$	−27.9022	−0.909587	−0.454794	−	0.890597i	$-0.650287\pi$
$941$	−27.9022	−0.909587	−0.454794	+	0.890597i	$0.650287\pi$
$942$	0	0
$943$	72.4877	2.36053
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−8.54717	−0.277746	−0.138873	−	0.990310i	$-0.544348\pi$
$947$	−8.54717	−0.277746	−0.138873	+	0.990310i	$0.544348\pi$
$948$	0	0
$949$	−23.8610	−0.774562
$950$	0	0
$951$	16.7738	0.543929
$952$	0	0
$953$	−44.0283	−1.42622	−0.713108	−	0.701054i	$-0.752713\pi$
$953$	−44.0283	−1.42622	−0.713108	+	0.701054i	$0.752713\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	19.2544	0.622407
$958$	0	0
$959$	9.66553	0.312116
$960$	0	0
$961$	−5.50885	−0.177705
$962$	0	0
$963$	10.9894	0.354130
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40.4877	1.30200	0.650999	−	0.759079i	$-0.274350\pi$
$967$	40.4877	1.30200	0.650999	+	0.759079i	$0.274350\pi$
$968$	0	0
$969$	−6.20555	−0.199351
$970$	0	0
$971$	5.27555	0.169301	0.0846503	−	0.996411i	$-0.473023\pi$
$971$	5.27555	0.169301	0.0846503	+	0.996411i	$0.473023\pi$
$972$	0	0
$973$	−3.25443	−0.104332
$974$	0	0
$975$	−12.8917	−0.412864
$976$	0	0
$977$	12.4595	0.398613	0.199307	−	0.979937i	$-0.436131\pi$
$977$	12.4595	0.398613	0.199307	+	0.979937i	$0.436131\pi$
$978$	0	0
$979$	−40.2721	−1.28710
$980$	0	0
$981$	−16.6167	−0.530529
$982$	0	0
$983$	16.8222	0.536545	0.268272	−	0.963343i	$-0.413547\pi$
$983$	16.8222	0.536545	0.268272	+	0.963343i	$0.413547\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	3.62721	0.115455
$988$	0	0
$989$	56.8222	1.80684
$990$	0	0
$991$	3.08719	0.0980678	0.0490339	−	0.998797i	$-0.484386\pi$
$991$	3.08719	0.0980678	0.0490339	+	0.998797i	$0.484386\pi$
$992$	0	0
$993$	4.05944	0.128822
$994$	0	0
$995$	0	0
$996$	0	0
$997$	49.8610	1.57911	0.789557	−	0.613677i	$-0.210310\pi$
$997$	49.8610	1.57911	0.789557	+	0.613677i	$0.210310\pi$
$998$	0	0
$999$	7.04888	0.223017

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.v.1.1	✓	3	4.3	odd	2
6384.2.a.bv.1.3		3	1.1	even	1	trivial
9576.2.a.cc.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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{7} +1.00000 q^{9} +5.62721 q^{11} -2.57834 q^{13} +6.20555 q^{17} +1.00000 q^{19} -1.00000 q^{21} -7.83276 q^{23} -5.00000 q^{25} -1.00000 q^{27} -3.42166 q^{29} -5.04888 q^{31} -5.62721 q^{33} -7.04888 q^{37} +2.57834 q^{39} -9.25443 q^{41} -7.25443 q^{43} -3.62721 q^{47} +1.00000 q^{49} -6.20555 q^{51} -8.57834 q^{53} -1.00000 q^{57} +8.41110 q^{59} -14.4111 q^{61} +1.00000 q^{63} +12.8816 q^{67} +7.83276 q^{69} +13.8328 q^{71} +9.25443 q^{73} +5.00000 q^{75} +5.62721 q^{77} +6.67609 q^{79} +1.00000 q^{81} +12.6761 q^{83} +3.42166 q^{87} -7.15667 q^{89} -2.57834 q^{91} +5.04888 q^{93} +4.78389 q^{97} +5.62721 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{7} + 3 q^{9} + 4 q^{11} - 6 q^{13} + 4 q^{17} + 3 q^{19} - 3 q^{21} + 4 q^{23} - 15 q^{25} - 3 q^{27} - 12 q^{29} - 4 q^{31} - 4 q^{33} - 10 q^{37} + 6 q^{39} - 2 q^{41} + 4 q^{43} + 2 q^{47} + 3 q^{49} - 4 q^{51} - 24 q^{53} - 3 q^{57} - 4 q^{59} - 14 q^{61} + 3 q^{63} - 4 q^{69} + 14 q^{71} + 2 q^{73} + 15 q^{75} + 4 q^{77} - 4 q^{79} + 3 q^{81} + 14 q^{83} + 12 q^{87} - 18 q^{89} - 6 q^{91} + 4 q^{93} - 2 q^{97} + 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^7 + 3 * q^9 + 4 * q^11 - 6 * q^13 + 4 * q^17 + 3 * q^19 - 3 * q^21 + 4 * q^23 - 15 * q^25 - 3 * q^27 - 12 * q^29 - 4 * q^31 - 4 * q^33 - 10 * q^37 + 6 * q^39 - 2 * q^41 + 4 * q^43 + 2 * q^47 + 3 * q^49 - 4 * q^51 - 24 * q^53 - 3 * q^57 - 4 * q^59 - 14 * q^61 + 3 * q^63 - 4 * q^69 + 14 * q^71 + 2 * q^73 + 15 * q^75 + 4 * q^77 - 4 * q^79 + 3 * q^81 + 14 * q^83 + 12 * q^87 - 18 * q^89 - 6 * q^91 + 4 * q^93 - 2 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more