LMFDB - Embedded newform 6384.2.a.cf.1.4

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:	$5$
Coefficient field:	5.5.368464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.14884$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.68348 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.68348 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.52202 q^{11} -3.43609 q^{13} +3.68348 q^{15} +2.09219 q^{17} +1.00000 q^{19} -1.00000 q^{21} +5.53273 q^{23} +8.56799 q^{25} +1.00000 q^{27} +5.84926 q^{29} +1.79896 q^{31} -4.52202 q^{33} -3.68348 q^{35} +3.59129 q^{37} -3.43609 q^{39} +10.0733 q^{41} -4.80304 q^{43} +3.68348 q^{45} +6.20550 q^{47} +1.00000 q^{49} +2.09219 q^{51} -7.62492 q^{53} -16.6568 q^{55} +1.00000 q^{57} -10.0548 q^{59} -11.1426 q^{61} -1.00000 q^{63} -12.6568 q^{65} -6.23913 q^{67} +5.53273 q^{69} +3.32724 q^{71} +10.2791 q^{73} +8.56799 q^{75} +4.52202 q^{77} +14.1867 q^{79} +1.00000 q^{81} +12.8075 q^{83} +7.70653 q^{85} +5.84926 q^{87} +12.9921 q^{89} +3.43609 q^{91} +1.79896 q^{93} +3.68348 q^{95} +0.872180 q^{97} -4.52202 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9} - 8 q^{11} - 6 q^{13} + 4 q^{15} + 12 q^{17} + 5 q^{19} - 5 q^{21} - 12 q^{23} + 15 q^{25} + 5 q^{27} + 4 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{35} + 2 q^{37} - 6 q^{39} + 10 q^{41} + 16 q^{43} + 4 q^{45} + 2 q^{47} + 5 q^{49} + 12 q^{51} + 12 q^{55} + 5 q^{57} + 4 q^{59} - 14 q^{61} - 5 q^{63} + 32 q^{65} + 20 q^{67} - 12 q^{69} + 6 q^{71} + 10 q^{73} + 15 q^{75} + 8 q^{77} + 5 q^{81} - 6 q^{83} + 8 q^{85} + 4 q^{87} + 26 q^{89} + 6 q^{91} + 8 q^{93} + 4 q^{95} - 18 q^{97} - 8 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9 - 8 * q^11 - 6 * q^13 + 4 * q^15 + 12 * q^17 + 5 * q^19 - 5 * q^21 - 12 * q^23 + 15 * q^25 + 5 * q^27 + 4 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^35 + 2 * q^37 - 6 * q^39 + 10 * q^41 + 16 * q^43 + 4 * q^45 + 2 * q^47 + 5 * q^49 + 12 * q^51 + 12 * q^55 + 5 * q^57 + 4 * q^59 - 14 * q^61 - 5 * q^63 + 32 * q^65 + 20 * q^67 - 12 * q^69 + 6 * q^71 + 10 * q^73 + 15 * q^75 + 8 * q^77 + 5 * q^81 - 6 * q^83 + 8 * q^85 + 4 * q^87 + 26 * q^89 + 6 * q^91 + 8 * q^93 + 4 * q^95 - 18 * q^97 - 8 * 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
$5$	3.68348	1.64730	0.823650	−	0.567098i	$-0.191934\pi$
$5$	3.68348	1.64730	0.823650	+	0.567098i	$0.191934\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.52202	−1.36344	−0.681720	−	0.731613i	$-0.738768\pi$
$11$	−4.52202	−1.36344	−0.681720	+	0.731613i	$0.738768\pi$
$12$	0	0
$13$	−3.43609	−0.953000	−0.476500	−	0.879175i	$-0.658095\pi$
$13$	−3.43609	−0.953000	−0.476500	+	0.879175i	$0.658095\pi$
$14$	0	0
$15$	3.68348	0.951069
$16$	0	0
$17$	2.09219	0.507430	0.253715	−	0.967279i	$-0.418347\pi$
$17$	2.09219	0.507430	0.253715	+	0.967279i	$0.418347\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	5.53273	1.15365	0.576827	−	0.816866i	$-0.304291\pi$
$23$	5.53273	1.15365	0.576827	+	0.816866i	$0.304291\pi$
$24$	0	0
$25$	8.56799	1.71360
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.84926	1.08618	0.543090	−	0.839675i	$-0.317254\pi$
$29$	5.84926	1.08618	0.543090	+	0.839675i	$0.317254\pi$
$30$	0	0
$31$	1.79896	0.323102	0.161551	−	0.986864i	$-0.448350\pi$
$31$	1.79896	0.323102	0.161551	+	0.986864i	$0.448350\pi$
$32$	0	0
$33$	−4.52202	−0.787183
$34$	0	0
$35$	−3.68348	−0.622621
$36$	0	0
$37$	3.59129	0.590404	0.295202	−	0.955435i	$-0.404613\pi$
$37$	3.59129	0.590404	0.295202	+	0.955435i	$0.404613\pi$
$38$	0	0
$39$	−3.43609	−0.550215
$40$	0	0
$41$	10.0733	1.57319	0.786596	−	0.617467i	$-0.211841\pi$
$41$	10.0733	1.57319	0.786596	+	0.617467i	$0.211841\pi$
$42$	0	0
$43$	−4.80304	−0.732457	−0.366228	−	0.930525i	$-0.619351\pi$
$43$	−4.80304	−0.732457	−0.366228	+	0.930525i	$0.619351\pi$
$44$	0	0
$45$	3.68348	0.549100
$46$	0	0
$47$	6.20550	0.905165	0.452582	−	0.891723i	$-0.350503\pi$
$47$	6.20550	0.905165	0.452582	+	0.891723i	$0.350503\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.09219	0.292965
$52$	0	0
$53$	−7.62492	−1.04736	−0.523682	−	0.851914i	$-0.675442\pi$
$53$	−7.62492	−1.04736	−0.523682	+	0.851914i	$0.675442\pi$
$54$	0	0
$55$	−16.6568	−2.24600
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−10.0548	−1.30902	−0.654509	−	0.756054i	$-0.727124\pi$
$59$	−10.0548	−1.30902	−0.654509	+	0.756054i	$0.727124\pi$
$60$	0	0
$61$	−11.1426	−1.42667	−0.713333	−	0.700825i	$-0.752815\pi$
$61$	−11.1426	−1.42667	−0.713333	+	0.700825i	$0.752815\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−12.6568	−1.56988
$66$	0	0
$67$	−6.23913	−0.762231	−0.381116	−	0.924527i	$-0.624460\pi$
$67$	−6.23913	−0.762231	−0.381116	+	0.924527i	$0.624460\pi$
$68$	0	0
$69$	5.53273	0.666063
$70$	0	0
$71$	3.32724	0.394870	0.197435	−	0.980316i	$-0.436739\pi$
$71$	3.32724	0.394870	0.197435	+	0.980316i	$0.436739\pi$
$72$	0	0
$73$	10.2791	1.20308	0.601538	−	0.798844i	$-0.294555\pi$
$73$	10.2791	1.20308	0.601538	+	0.798844i	$0.294555\pi$
$74$	0	0
$75$	8.56799	0.989347
$76$	0	0
$77$	4.52202	0.515332
$78$	0	0
$79$	14.1867	1.59612	0.798062	−	0.602576i	$-0.205859\pi$
$79$	14.1867	1.59612	0.798062	+	0.602576i	$0.205859\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.8075	1.40581	0.702903	−	0.711286i	$-0.251887\pi$
$83$	12.8075	1.40581	0.702903	+	0.711286i	$0.251887\pi$
$84$	0	0
$85$	7.70653	0.835890
$86$	0	0
$87$	5.84926	0.627106
$88$	0	0
$89$	12.9921	1.37716	0.688581	−	0.725160i	$-0.258234\pi$
$89$	12.9921	1.37716	0.688581	+	0.725160i	$0.258234\pi$
$90$	0	0
$91$	3.43609	0.360200
$92$	0	0
$93$	1.79896	0.186543
$94$	0	0
$95$	3.68348	0.377917
$96$	0	0
$97$	0.872180	0.0885564	0.0442782	−	0.999019i	$-0.485901\pi$
$97$	0.872180	0.0885564	0.0442782	+	0.999019i	$0.485901\pi$
$98$	0	0
$99$	−4.52202	−0.454480
$100$	0	0
$101$	4.86377	0.483963	0.241982	−	0.970281i	$-0.422203\pi$
$101$	4.86377	0.483963	0.241982	+	0.970281i	$0.422203\pi$
$102$	0	0
$103$	9.04404	0.891136	0.445568	−	0.895248i	$-0.353002\pi$
$103$	9.04404	0.891136	0.445568	+	0.895248i	$0.353002\pi$
$104$	0	0
$105$	−3.68348	−0.359470
$106$	0	0
$107$	−9.32061	−0.901057	−0.450529	−	0.892762i	$-0.648764\pi$
$107$	−9.32061	−0.901057	−0.450529	+	0.892762i	$0.648764\pi$
$108$	0	0
$109$	10.7797	1.03251	0.516256	−	0.856434i	$-0.327325\pi$
$109$	10.7797	1.03251	0.516256	+	0.856434i	$0.327325\pi$
$110$	0	0
$111$	3.59129	0.340870
$112$	0	0
$113$	9.84926	0.926540	0.463270	−	0.886217i	$-0.346676\pi$
$113$	9.84926	0.926540	0.463270	+	0.886217i	$0.346676\pi$
$114$	0	0
$115$	20.3797	1.90042
$116$	0	0
$117$	−3.43609	−0.317667
$118$	0	0
$119$	−2.09219	−0.191791
$120$	0	0
$121$	9.44867	0.858970
$122$	0	0
$123$	10.0733	0.908283
$124$	0	0
$125$	13.1426	1.17551
$126$	0	0
$127$	6.68780	0.593447	0.296723	−	0.954964i	$-0.404106\pi$
$127$	6.68780	0.593447	0.296723	+	0.954964i	$0.404106\pi$
$128$	0	0
$129$	−4.80304	−0.422884
$130$	0	0
$131$	10.8533	0.948261	0.474130	−	0.880455i	$-0.342763\pi$
$131$	10.8533	0.948261	0.474130	+	0.880455i	$0.342763\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	3.68348	0.317023
$136$	0	0
$137$	−23.1907	−1.98132	−0.990659	−	0.136360i	$-0.956460\pi$
$137$	−23.1907	−1.98132	−0.990659	+	0.136360i	$0.956460\pi$
$138$	0	0
$139$	−5.80899	−0.492712	−0.246356	−	0.969179i	$-0.579233\pi$
$139$	−5.80899	−0.492712	−0.246356	+	0.969179i	$0.579233\pi$
$140$	0	0
$141$	6.20550	0.522597
$142$	0	0
$143$	15.5381	1.29936
$144$	0	0
$145$	21.5456	1.78926
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	23.4217	1.91878	0.959390	−	0.282082i	$-0.0910249\pi$
$149$	23.4217	1.91878	0.959390	+	0.282082i	$0.0910249\pi$
$150$	0	0
$151$	−17.5536	−1.42849	−0.714246	−	0.699895i	$-0.753230\pi$
$151$	−17.5536	−1.42849	−0.714246	+	0.699895i	$0.753230\pi$
$152$	0	0
$153$	2.09219	0.169143
$154$	0	0
$155$	6.62642	0.532247
$156$	0	0
$157$	−13.6460	−1.08907	−0.544536	−	0.838737i	$-0.683294\pi$
$157$	−13.6460	−1.08907	−0.544536	+	0.838737i	$0.683294\pi$
$158$	0	0
$159$	−7.62492	−0.604696
$160$	0	0
$161$	−5.53273	−0.436040
$162$	0	0
$163$	−3.07985	−0.241233	−0.120616	−	0.992699i	$-0.538487\pi$
$163$	−3.07985	−0.241233	−0.120616	+	0.992699i	$0.538487\pi$
$164$	0	0
$165$	−16.6568	−1.29673
$166$	0	0
$167$	14.2391	1.10186	0.550929	−	0.834552i	$-0.314274\pi$
$167$	14.2391	1.10186	0.550929	+	0.834552i	$0.314274\pi$
$168$	0	0
$169$	−1.19329	−0.0917913
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−15.0841	−1.14682	−0.573410	−	0.819269i	$-0.694380\pi$
$173$	−15.0841	−1.14682	−0.573410	+	0.819269i	$0.694380\pi$
$174$	0	0
$175$	−8.56799	−0.647679
$176$	0	0
$177$	−10.0548	−0.755762
$178$	0	0
$179$	2.04634	0.152951	0.0764755	−	0.997071i	$-0.475633\pi$
$179$	2.04634	0.152951	0.0764755	+	0.997071i	$0.475633\pi$
$180$	0	0
$181$	−21.7779	−1.61874	−0.809371	−	0.587298i	$-0.800192\pi$
$181$	−21.7779	−1.61874	−0.809371	+	0.587298i	$0.800192\pi$
$182$	0	0
$183$	−11.1426	−0.823686
$184$	0	0
$185$	13.2284	0.972573
$186$	0	0
$187$	−9.46092	−0.691851
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	14.5302	1.05137	0.525684	−	0.850680i	$-0.323810\pi$
$191$	14.5302	1.05137	0.525684	+	0.850680i	$0.323810\pi$
$192$	0	0
$193$	−1.08786	−0.0783060	−0.0391530	−	0.999233i	$-0.512466\pi$
$193$	−1.08786	−0.0783060	−0.0391530	+	0.999233i	$0.512466\pi$
$194$	0	0
$195$	−12.6568	−0.906369
$196$	0	0
$197$	25.3293	1.80464	0.902318	−	0.431071i	$-0.141864\pi$
$197$	25.3293	1.80464	0.902318	+	0.431071i	$0.141864\pi$
$198$	0	0
$199$	−12.4110	−0.879792	−0.439896	−	0.898049i	$-0.644985\pi$
$199$	−12.4110	−0.879792	−0.439896	+	0.898049i	$0.644985\pi$
$200$	0	0
$201$	−6.23913	−0.440074
$202$	0	0
$203$	−5.84926	−0.410537
$204$	0	0
$205$	37.1049	2.59152
$206$	0	0
$207$	5.53273	0.384552
$208$	0	0
$209$	−4.52202	−0.312795
$210$	0	0
$211$	25.6061	1.76280	0.881398	−	0.472375i	$-0.156603\pi$
$211$	25.6061	1.76280	0.881398	+	0.472375i	$0.156603\pi$
$212$	0	0
$213$	3.32724	0.227979
$214$	0	0
$215$	−17.6919	−1.20658
$216$	0	0
$217$	−1.79896	−0.122121
$218$	0	0
$219$	10.2791	0.694597
$220$	0	0
$221$	−7.18895	−0.483581
$222$	0	0
$223$	14.0839	0.943130	0.471565	−	0.881831i	$-0.343689\pi$
$223$	14.0839	0.943130	0.471565	+	0.881831i	$0.343689\pi$
$224$	0	0
$225$	8.56799	0.571200
$226$	0	0
$227$	−5.22842	−0.347022	−0.173511	−	0.984832i	$-0.555511\pi$
$227$	−5.22842	−0.347022	−0.173511	+	0.984832i	$0.555511\pi$
$228$	0	0
$229$	−9.76903	−0.645556	−0.322778	−	0.946475i	$-0.604617\pi$
$229$	−9.76903	−0.645556	−0.322778	+	0.946475i	$0.604617\pi$
$230$	0	0
$231$	4.52202	0.297527
$232$	0	0
$233$	16.5495	1.08420	0.542098	−	0.840315i	$-0.317630\pi$
$233$	16.5495	1.08420	0.542098	+	0.840315i	$0.317630\pi$
$234$	0	0
$235$	22.8578	1.49108
$236$	0	0
$237$	14.1867	0.921522
$238$	0	0
$239$	−18.3125	−1.18454	−0.592268	−	0.805741i	$-0.701767\pi$
$239$	−18.3125	−1.18454	−0.592268	+	0.805741i	$0.701767\pi$
$240$	0	0
$241$	−6.59537	−0.424845	−0.212423	−	0.977178i	$-0.568135\pi$
$241$	−6.59537	−0.424845	−0.212423	+	0.977178i	$0.568135\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.68348	0.235329
$246$	0	0
$247$	−3.43609	−0.218633
$248$	0	0
$249$	12.8075	0.811642
$250$	0	0
$251$	12.2269	0.771756	0.385878	−	0.922550i	$-0.373898\pi$
$251$	12.2269	0.771756	0.385878	+	0.922550i	$0.373898\pi$
$252$	0	0
$253$	−25.0191	−1.57294
$254$	0	0
$255$	7.70653	0.482601
$256$	0	0
$257$	−11.7504	−0.732972	−0.366486	−	0.930424i	$-0.619439\pi$
$257$	−11.7504	−0.732972	−0.366486	+	0.930424i	$0.619439\pi$
$258$	0	0
$259$	−3.59129	−0.223152
$260$	0	0
$261$	5.84926	0.362060
$262$	0	0
$263$	−7.44030	−0.458789	−0.229394	−	0.973334i	$-0.573675\pi$
$263$	−7.44030	−0.458789	−0.229394	+	0.973334i	$0.573675\pi$
$264$	0	0
$265$	−28.0862	−1.72532
$266$	0	0
$267$	12.9921	0.795105
$268$	0	0
$269$	−29.6336	−1.80679	−0.903396	−	0.428808i	$-0.858934\pi$
$269$	−29.6336	−1.80679	−0.903396	+	0.428808i	$0.858934\pi$
$270$	0	0
$271$	−2.33819	−0.142035	−0.0710176	−	0.997475i	$-0.522625\pi$
$271$	−2.33819	−0.142035	−0.0710176	+	0.997475i	$0.522625\pi$
$272$	0	0
$273$	3.43609	0.207962
$274$	0	0
$275$	−38.7446	−2.33639
$276$	0	0
$277$	−7.16591	−0.430558	−0.215279	−	0.976553i	$-0.569066\pi$
$277$	−7.16591	−0.430558	−0.215279	+	0.976553i	$0.569066\pi$
$278$	0	0
$279$	1.79896	0.107701
$280$	0	0
$281$	0.891019	0.0531537	0.0265769	−	0.999647i	$-0.491539\pi$
$281$	0.891019	0.0531537	0.0265769	+	0.999647i	$0.491539\pi$
$282$	0	0
$283$	17.1951	1.02214	0.511071	−	0.859539i	$-0.329249\pi$
$283$	17.1951	1.02214	0.511071	+	0.859539i	$0.329249\pi$
$284$	0	0
$285$	3.68348	0.218190
$286$	0	0
$287$	−10.0733	−0.594611
$288$	0	0
$289$	−12.6227	−0.742515
$290$	0	0
$291$	0.872180	0.0511281
$292$	0	0
$293$	4.24521	0.248008	0.124004	−	0.992282i	$-0.460426\pi$
$293$	4.24521	0.248008	0.124004	+	0.992282i	$0.460426\pi$
$294$	0	0
$295$	−37.0364	−2.15634
$296$	0	0
$297$	−4.52202	−0.262394
$298$	0	0
$299$	−19.0110	−1.09943
$300$	0	0
$301$	4.80304	0.276843
$302$	0	0
$303$	4.86377	0.279416
$304$	0	0
$305$	−41.0436	−2.35015
$306$	0	0
$307$	9.15700	0.522618	0.261309	−	0.965255i	$-0.415846\pi$
$307$	9.15700	0.522618	0.261309	+	0.965255i	$0.415846\pi$
$308$	0	0
$309$	9.04404	0.514498
$310$	0	0
$311$	27.0800	1.53556	0.767782	−	0.640712i	$-0.221361\pi$
$311$	27.0800	1.53556	0.767782	+	0.640712i	$0.221361\pi$
$312$	0	0
$313$	−19.6675	−1.11167	−0.555835	−	0.831292i	$-0.687602\pi$
$313$	−19.6675	−1.11167	−0.555835	+	0.831292i	$0.687602\pi$
$314$	0	0
$315$	−3.68348	−0.207540
$316$	0	0
$317$	−3.19478	−0.179437	−0.0897185	−	0.995967i	$-0.528597\pi$
$317$	−3.19478	−0.179437	−0.0897185	+	0.995967i	$0.528597\pi$
$318$	0	0
$319$	−26.4505	−1.48094
$320$	0	0
$321$	−9.32061	−0.520226
$322$	0	0
$323$	2.09219	0.116412
$324$	0	0
$325$	−29.4404	−1.63306
$326$	0	0
$327$	10.7797	0.596121
$328$	0	0
$329$	−6.20550	−0.342120
$330$	0	0
$331$	27.7133	1.52326	0.761631	−	0.648011i	$-0.224399\pi$
$331$	27.7133	1.52326	0.761631	+	0.648011i	$0.224399\pi$
$332$	0	0
$333$	3.59129	0.196801
$334$	0	0
$335$	−22.9817	−1.25562
$336$	0	0
$337$	−10.1472	−0.552752	−0.276376	−	0.961050i	$-0.589134\pi$
$337$	−10.1472	−0.552752	−0.276376	+	0.961050i	$0.589134\pi$
$338$	0	0
$339$	9.84926	0.534938
$340$	0	0
$341$	−8.13493	−0.440531
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	20.3797	1.09721
$346$	0	0
$347$	−22.3125	−1.19780	−0.598898	−	0.800825i	$-0.704395\pi$
$347$	−22.3125	−1.19780	−0.598898	+	0.800825i	$0.704395\pi$
$348$	0	0
$349$	−13.1360	−0.703153	−0.351577	−	0.936159i	$-0.614354\pi$
$349$	−13.1360	−0.703153	−0.351577	+	0.936159i	$0.614354\pi$
$350$	0	0
$351$	−3.43609	−0.183405
$352$	0	0
$353$	−15.2348	−0.810867	−0.405433	−	0.914125i	$-0.632879\pi$
$353$	−15.2348	−0.810867	−0.405433	+	0.914125i	$0.632879\pi$
$354$	0	0
$355$	12.2558	0.650470
$356$	0	0
$357$	−2.09219	−0.110730
$358$	0	0
$359$	21.7260	1.14666	0.573328	−	0.819326i	$-0.305652\pi$
$359$	21.7260	1.14666	0.573328	+	0.819326i	$0.305652\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	9.44867	0.495927
$364$	0	0
$365$	37.8628	1.98183
$366$	0	0
$367$	−16.3777	−0.854907	−0.427454	−	0.904037i	$-0.640589\pi$
$367$	−16.3777	−0.854907	−0.427454	+	0.904037i	$0.640589\pi$
$368$	0	0
$369$	10.0733	0.524398
$370$	0	0
$371$	7.62492	0.395866
$372$	0	0
$373$	−12.1930	−0.631331	−0.315666	−	0.948871i	$-0.602228\pi$
$373$	−12.1930	−0.631331	−0.315666	+	0.948871i	$0.602228\pi$
$374$	0	0
$375$	13.1426	0.678682
$376$	0	0
$377$	−20.0986	−1.03513
$378$	0	0
$379$	26.6348	1.36814	0.684070	−	0.729416i	$-0.260208\pi$
$379$	26.6348	1.36814	0.684070	+	0.729416i	$0.260208\pi$
$380$	0	0
$381$	6.68780	0.342627
$382$	0	0
$383$	5.04839	0.257961	0.128980	−	0.991647i	$-0.458830\pi$
$383$	5.04839	0.257961	0.128980	+	0.991647i	$0.458830\pi$
$384$	0	0
$385$	16.6568	0.848907
$386$	0	0
$387$	−4.80304	−0.244152
$388$	0	0
$389$	−9.76903	−0.495310	−0.247655	−	0.968848i	$-0.579660\pi$
$389$	−9.76903	−0.495310	−0.247655	+	0.968848i	$0.579660\pi$
$390$	0	0
$391$	11.5755	0.585399
$392$	0	0
$393$	10.8533	0.547479
$394$	0	0
$395$	52.2562	2.62930
$396$	0	0
$397$	33.4760	1.68011	0.840055	−	0.542501i	$-0.182522\pi$
$397$	33.4760	1.68011	0.840055	+	0.542501i	$0.182522\pi$
$398$	0	0
$399$	−1.00000	−0.0500626
$400$	0	0
$401$	20.2997	1.01372	0.506860	−	0.862028i	$-0.330806\pi$
$401$	20.2997	1.01372	0.506860	+	0.862028i	$0.330806\pi$
$402$	0	0
$403$	−6.18138	−0.307917
$404$	0	0
$405$	3.68348	0.183033
$406$	0	0
$407$	−16.2399	−0.804981
$408$	0	0
$409$	−16.1914	−0.800614	−0.400307	−	0.916381i	$-0.631097\pi$
$409$	−16.1914	−0.800614	−0.400307	+	0.916381i	$0.631097\pi$
$410$	0	0
$411$	−23.1907	−1.14391
$412$	0	0
$413$	10.0548	0.494762
$414$	0	0
$415$	47.1761	2.31578
$416$	0	0
$417$	−5.80899	−0.284468
$418$	0	0
$419$	−34.9608	−1.70795	−0.853974	−	0.520316i	$-0.825814\pi$
$419$	−34.9608	−1.70795	−0.853974	+	0.520316i	$0.825814\pi$
$420$	0	0
$421$	18.3163	0.892681	0.446340	−	0.894863i	$-0.352727\pi$
$421$	18.3163	0.892681	0.446340	+	0.894863i	$0.352727\pi$
$422$	0	0
$423$	6.20550	0.301722
$424$	0	0
$425$	17.9259	0.869532
$426$	0	0
$427$	11.1426	0.539229
$428$	0	0
$429$	15.5381	0.750185
$430$	0	0
$431$	−37.8788	−1.82456	−0.912278	−	0.409570i	$-0.865679\pi$
$431$	−37.8788	−1.82456	−0.912278	+	0.409570i	$0.865679\pi$
$432$	0	0
$433$	26.9698	1.29609	0.648043	−	0.761604i	$-0.275588\pi$
$433$	26.9698	1.29609	0.648043	+	0.761604i	$0.275588\pi$
$434$	0	0
$435$	21.5456	1.03303
$436$	0	0
$437$	5.53273	0.264667
$438$	0	0
$439$	6.80967	0.325008	0.162504	−	0.986708i	$-0.448043\pi$
$439$	6.80967	0.325008	0.162504	+	0.986708i	$0.448043\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	4.29106	0.203874	0.101937	−	0.994791i	$-0.467496\pi$
$443$	4.29106	0.203874	0.101937	+	0.994791i	$0.467496\pi$
$444$	0	0
$445$	47.8561	2.26860
$446$	0	0
$447$	23.4217	1.10781
$448$	0	0
$449$	−0.295649	−0.0139525	−0.00697627	−	0.999976i	$-0.502221\pi$
$449$	−0.295649	−0.0139525	−0.00697627	+	0.999976i	$0.502221\pi$
$450$	0	0
$451$	−45.5519	−2.14495
$452$	0	0
$453$	−17.5536	−0.824740
$454$	0	0
$455$	12.6568	0.593358
$456$	0	0
$457$	28.5415	1.33512	0.667558	−	0.744558i	$-0.267340\pi$
$457$	28.5415	1.33512	0.667558	+	0.744558i	$0.267340\pi$
$458$	0	0
$459$	2.09219	0.0976550
$460$	0	0
$461$	−24.7090	−1.15081	−0.575406	−	0.817868i	$-0.695156\pi$
$461$	−24.7090	−1.15081	−0.575406	+	0.817868i	$0.695156\pi$
$462$	0	0
$463$	−1.52990	−0.0711007	−0.0355503	−	0.999368i	$-0.511318\pi$
$463$	−1.52990	−0.0711007	−0.0355503	+	0.999368i	$0.511318\pi$
$464$	0	0
$465$	6.62642	0.307293
$466$	0	0
$467$	−38.1153	−1.76377	−0.881884	−	0.471467i	$-0.843725\pi$
$467$	−38.1153	−1.76377	−0.881884	+	0.471467i	$0.843725\pi$
$468$	0	0
$469$	6.23913	0.288096
$470$	0	0
$471$	−13.6460	−0.628776
$472$	0	0
$473$	21.7195	0.998661
$474$	0	0
$475$	8.56799	0.393126
$476$	0	0
$477$	−7.62492	−0.349121
$478$	0	0
$479$	−11.2867	−0.515704	−0.257852	−	0.966184i	$-0.583015\pi$
$479$	−11.2867	−0.515704	−0.257852	+	0.966184i	$0.583015\pi$
$480$	0	0
$481$	−12.3400	−0.562655
$482$	0	0
$483$	−5.53273	−0.251748
$484$	0	0
$485$	3.21265	0.145879
$486$	0	0
$487$	−27.2921	−1.23672	−0.618361	−	0.785894i	$-0.712203\pi$
$487$	−27.2921	−1.23672	−0.618361	+	0.785894i	$0.712203\pi$
$488$	0	0
$489$	−3.07985	−0.139276
$490$	0	0
$491$	−2.86610	−0.129345	−0.0646726	−	0.997907i	$-0.520600\pi$
$491$	−2.86610	−0.129345	−0.0646726	+	0.997907i	$0.520600\pi$
$492$	0	0
$493$	12.2377	0.551161
$494$	0	0
$495$	−16.6568	−0.748665
$496$	0	0
$497$	−3.32724	−0.149247
$498$	0	0
$499$	−13.1490	−0.588630	−0.294315	−	0.955709i	$-0.595091\pi$
$499$	−13.1490	−0.588630	−0.294315	+	0.955709i	$0.595091\pi$
$500$	0	0
$501$	14.2391	0.636157
$502$	0	0
$503$	−32.3456	−1.44222	−0.721109	−	0.692822i	$-0.756367\pi$
$503$	−32.3456	−1.44222	−0.721109	+	0.692822i	$0.756367\pi$
$504$	0	0
$505$	17.9156	0.797233
$506$	0	0
$507$	−1.19329	−0.0529957
$508$	0	0
$509$	−2.45101	−0.108639	−0.0543196	−	0.998524i	$-0.517299\pi$
$509$	−2.45101	−0.108639	−0.0543196	+	0.998524i	$0.517299\pi$
$510$	0	0
$511$	−10.2791	−0.454720
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	33.3135	1.46797
$516$	0	0
$517$	−28.0614	−1.23414
$518$	0	0
$519$	−15.0841	−0.662117
$520$	0	0
$521$	−14.0357	−0.614914	−0.307457	−	0.951562i	$-0.599478\pi$
$521$	−14.0357	−0.614914	−0.307457	+	0.951562i	$0.599478\pi$
$522$	0	0
$523$	22.3105	0.975572	0.487786	−	0.872963i	$-0.337805\pi$
$523$	22.3105	0.975572	0.487786	+	0.872963i	$0.337805\pi$
$524$	0	0
$525$	−8.56799	−0.373938
$526$	0	0
$527$	3.76376	0.163952
$528$	0	0
$529$	7.61114	0.330919
$530$	0	0
$531$	−10.0548	−0.436339
$532$	0	0
$533$	−34.6129	−1.49925
$534$	0	0
$535$	−34.3322	−1.48431
$536$	0	0
$537$	2.04634	0.0883063
$538$	0	0
$539$	−4.52202	−0.194777
$540$	0	0
$541$	−13.8113	−0.593793	−0.296897	−	0.954910i	$-0.595952\pi$
$541$	−13.8113	−0.593793	−0.296897	+	0.954910i	$0.595952\pi$
$542$	0	0
$543$	−21.7779	−0.934581
$544$	0	0
$545$	39.7069	1.70086
$546$	0	0
$547$	6.19992	0.265089	0.132545	−	0.991177i	$-0.457685\pi$
$547$	6.19992	0.265089	0.132545	+	0.991177i	$0.457685\pi$
$548$	0	0
$549$	−11.1426	−0.475555
$550$	0	0
$551$	5.84926	0.249187
$552$	0	0
$553$	−14.1867	−0.603278
$554$	0	0
$555$	13.2284	0.561515
$556$	0	0
$557$	−3.33337	−0.141239	−0.0706196	−	0.997503i	$-0.522498\pi$
$557$	−3.33337	−0.141239	−0.0706196	+	0.997503i	$0.522498\pi$
$558$	0	0
$559$	16.5037	0.698031
$560$	0	0
$561$	−9.46092	−0.399440
$562$	0	0
$563$	21.8571	0.921168	0.460584	−	0.887616i	$-0.347640\pi$
$563$	21.8571	0.921168	0.460584	+	0.887616i	$0.347640\pi$
$564$	0	0
$565$	36.2795	1.52629
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	7.85914	0.329472	0.164736	−	0.986338i	$-0.447323\pi$
$569$	7.85914	0.329472	0.164736	+	0.986338i	$0.447323\pi$
$570$	0	0
$571$	27.0522	1.13210	0.566050	−	0.824371i	$-0.308471\pi$
$571$	27.0522	1.13210	0.566050	+	0.824371i	$0.308471\pi$
$572$	0	0
$573$	14.5302	0.607007
$574$	0	0
$575$	47.4044	1.97690
$576$	0	0
$577$	−6.66062	−0.277285	−0.138643	−	0.990342i	$-0.544274\pi$
$577$	−6.66062	−0.277285	−0.138643	+	0.990342i	$0.544274\pi$
$578$	0	0
$579$	−1.08786	−0.0452100
$580$	0	0
$581$	−12.8075	−0.531344
$582$	0	0
$583$	34.4801	1.42802
$584$	0	0
$585$	−12.6568	−0.523292
$586$	0	0
$587$	7.28673	0.300755	0.150378	−	0.988629i	$-0.451951\pi$
$587$	7.28673	0.300755	0.150378	+	0.988629i	$0.451951\pi$
$588$	0	0
$589$	1.79896	0.0741248
$590$	0	0
$591$	25.3293	1.04191
$592$	0	0
$593$	−32.2172	−1.32300	−0.661502	−	0.749944i	$-0.730081\pi$
$593$	−32.2172	−1.32300	−0.661502	+	0.749944i	$0.730081\pi$
$594$	0	0
$595$	−7.70653	−0.315937
$596$	0	0
$597$	−12.4110	−0.507948
$598$	0	0
$599$	39.2916	1.60541	0.802705	−	0.596376i	$-0.203393\pi$
$599$	39.2916	1.60541	0.802705	+	0.596376i	$0.203393\pi$
$600$	0	0
$601$	−10.7336	−0.437835	−0.218917	−	0.975743i	$-0.570252\pi$
$601$	−10.7336	−0.437835	−0.218917	+	0.975743i	$0.570252\pi$
$602$	0	0
$603$	−6.23913	−0.254077
$604$	0	0
$605$	34.8040	1.41498
$606$	0	0
$607$	−34.2348	−1.38955	−0.694773	−	0.719229i	$-0.744495\pi$
$607$	−34.2348	−1.38955	−0.694773	+	0.719229i	$0.744495\pi$
$608$	0	0
$609$	−5.84926	−0.237024
$610$	0	0
$611$	−21.3226	−0.862622
$612$	0	0
$613$	−0.800761	−0.0323424	−0.0161712	−	0.999869i	$-0.505148\pi$
$613$	−0.800761	−0.0323424	−0.0161712	+	0.999869i	$0.505148\pi$
$614$	0	0
$615$	37.1049	1.49622
$616$	0	0
$617$	12.5034	0.503369	0.251684	−	0.967809i	$-0.419015\pi$
$617$	12.5034	0.503369	0.251684	+	0.967809i	$0.419015\pi$
$618$	0	0
$619$	12.2124	0.490859	0.245430	−	0.969414i	$-0.421071\pi$
$619$	12.2124	0.490859	0.245430	+	0.969414i	$0.421071\pi$
$620$	0	0
$621$	5.53273	0.222021
$622$	0	0
$623$	−12.9921	−0.520518
$624$	0	0
$625$	5.57054	0.222822
$626$	0	0
$627$	−4.52202	−0.180592
$628$	0	0
$629$	7.51365	0.299589
$630$	0	0
$631$	−33.5705	−1.33642	−0.668211	−	0.743972i	$-0.732940\pi$
$631$	−33.5705	−1.33642	−0.668211	+	0.743972i	$0.732940\pi$
$632$	0	0
$633$	25.6061	1.01775
$634$	0	0
$635$	24.6344	0.977585
$636$	0	0
$637$	−3.43609	−0.136143
$638$	0	0
$639$	3.32724	0.131623
$640$	0	0
$641$	−30.3529	−1.19887	−0.599434	−	0.800424i	$-0.704608\pi$
$641$	−30.3529	−1.19887	−0.599434	+	0.800424i	$0.704608\pi$
$642$	0	0
$643$	31.4553	1.24048	0.620238	−	0.784414i	$-0.287036\pi$
$643$	31.4553	1.24048	0.620238	+	0.784414i	$0.287036\pi$
$644$	0	0
$645$	−17.6919	−0.696617
$646$	0	0
$647$	−41.6565	−1.63769	−0.818844	−	0.574017i	$-0.805385\pi$
$647$	−41.6565	−1.63769	−0.818844	+	0.574017i	$0.805385\pi$
$648$	0	0
$649$	45.4678	1.78477
$650$	0	0
$651$	−1.79896	−0.0705067
$652$	0	0
$653$	−35.6823	−1.39635	−0.698177	−	0.715925i	$-0.746005\pi$
$653$	−35.6823	−1.39635	−0.698177	+	0.715925i	$0.746005\pi$
$654$	0	0
$655$	39.9780	1.56207
$656$	0	0
$657$	10.2791	0.401025
$658$	0	0
$659$	−9.63738	−0.375419	−0.187709	−	0.982225i	$-0.560106\pi$
$659$	−9.63738	−0.375419	−0.187709	+	0.982225i	$0.560106\pi$
$660$	0	0
$661$	18.7654	0.729888	0.364944	−	0.931029i	$-0.381088\pi$
$661$	18.7654	0.729888	0.364944	+	0.931029i	$0.381088\pi$
$662$	0	0
$663$	−7.18895	−0.279196
$664$	0	0
$665$	−3.68348	−0.142839
$666$	0	0
$667$	32.3624	1.25308
$668$	0	0
$669$	14.0839	0.544516
$670$	0	0
$671$	50.3871	1.94517
$672$	0	0
$673$	25.6593	0.989093	0.494547	−	0.869151i	$-0.335334\pi$
$673$	25.6593	0.989093	0.494547	+	0.869151i	$0.335334\pi$
$674$	0	0
$675$	8.56799	0.329782
$676$	0	0
$677$	0.450757	0.0173240	0.00866200	−	0.999962i	$-0.497243\pi$
$677$	0.450757	0.0173240	0.00866200	+	0.999962i	$0.497243\pi$
$678$	0	0
$679$	−0.872180	−0.0334712
$680$	0	0
$681$	−5.22842	−0.200353
$682$	0	0
$683$	−27.9796	−1.07061	−0.535306	−	0.844658i	$-0.679804\pi$
$683$	−27.9796	−1.07061	−0.535306	+	0.844658i	$0.679804\pi$
$684$	0	0
$685$	−85.4225	−3.26383
$686$	0	0
$687$	−9.76903	−0.372712
$688$	0	0
$689$	26.1999	0.998137
$690$	0	0
$691$	−15.6965	−0.597123	−0.298561	−	0.954390i	$-0.596507\pi$
$691$	−15.6965	−0.597123	−0.298561	+	0.954390i	$0.596507\pi$
$692$	0	0
$693$	4.52202	0.171777
$694$	0	0
$695$	−21.3973	−0.811645
$696$	0	0
$697$	21.0753	0.798286
$698$	0	0
$699$	16.5495	0.625961
$700$	0	0
$701$	−32.4653	−1.22620	−0.613098	−	0.790007i	$-0.710077\pi$
$701$	−32.4653	−1.22620	−0.613098	+	0.790007i	$0.710077\pi$
$702$	0	0
$703$	3.59129	0.135448
$704$	0	0
$705$	22.8578	0.860874
$706$	0	0
$707$	−4.86377	−0.182921
$708$	0	0
$709$	−18.1343	−0.681049	−0.340525	−	0.940236i	$-0.610605\pi$
$709$	−18.1343	−0.681049	−0.340525	+	0.940236i	$0.610605\pi$
$710$	0	0
$711$	14.1867	0.532041
$712$	0	0
$713$	9.95316	0.372749
$714$	0	0
$715$	57.2341	2.14043
$716$	0	0
$717$	−18.3125	−0.683892
$718$	0	0
$719$	12.2361	0.456328	0.228164	−	0.973623i	$-0.426728\pi$
$719$	12.2361	0.456328	0.228164	+	0.973623i	$0.426728\pi$
$720$	0	0
$721$	−9.04404	−0.336818
$722$	0	0
$723$	−6.59537	−0.245284
$724$	0	0
$725$	50.1164	1.86128
$726$	0	0
$727$	2.06389	0.0765455	0.0382727	−	0.999267i	$-0.487814\pi$
$727$	2.06389	0.0765455	0.0382727	+	0.999267i	$0.487814\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.0489	−0.371671
$732$	0	0
$733$	−35.8386	−1.32373	−0.661864	−	0.749624i	$-0.730234\pi$
$733$	−35.8386	−1.32373	−0.661864	+	0.749624i	$0.730234\pi$
$734$	0	0
$735$	3.68348	0.135867
$736$	0	0
$737$	28.2135	1.03926
$738$	0	0
$739$	5.47833	0.201524	0.100762	−	0.994911i	$-0.467872\pi$
$739$	5.47833	0.201524	0.100762	+	0.994911i	$0.467872\pi$
$740$	0	0
$741$	−3.43609	−0.126228
$742$	0	0
$743$	2.94320	0.107976	0.0539878	−	0.998542i	$-0.482807\pi$
$743$	2.94320	0.107976	0.0539878	+	0.998542i	$0.482807\pi$
$744$	0	0
$745$	86.2733	3.16081
$746$	0	0
$747$	12.8075	0.468602
$748$	0	0
$749$	9.32061	0.340568
$750$	0	0
$751$	−23.6667	−0.863611	−0.431805	−	0.901967i	$-0.642123\pi$
$751$	−23.6667	−0.863611	−0.431805	+	0.901967i	$0.642123\pi$
$752$	0	0
$753$	12.2269	0.445574
$754$	0	0
$755$	−64.6583	−2.35316
$756$	0	0
$757$	15.9126	0.578354	0.289177	−	0.957276i	$-0.406618\pi$
$757$	15.9126	0.578354	0.289177	+	0.957276i	$0.406618\pi$
$758$	0	0
$759$	−25.0191	−0.908137
$760$	0	0
$761$	5.05908	0.183392	0.0916958	−	0.995787i	$-0.470771\pi$
$761$	5.05908	0.183392	0.0916958	+	0.995787i	$0.470771\pi$
$762$	0	0
$763$	−10.7797	−0.390253
$764$	0	0
$765$	7.70653	0.278630
$766$	0	0
$767$	34.5490	1.24749
$768$	0	0
$769$	46.8368	1.68898	0.844489	−	0.535573i	$-0.179904\pi$
$769$	46.8368	1.68898	0.844489	+	0.535573i	$0.179904\pi$
$770$	0	0
$771$	−11.7504	−0.423182
$772$	0	0
$773$	−50.8248	−1.82804	−0.914021	−	0.405667i	$-0.867039\pi$
$773$	−50.8248	−1.82804	−0.914021	+	0.405667i	$0.867039\pi$
$774$	0	0
$775$	15.4135	0.553668
$776$	0	0
$777$	−3.59129	−0.128837
$778$	0	0
$779$	10.0733	0.360915
$780$	0	0
$781$	−15.0458	−0.538382
$782$	0	0
$783$	5.84926	0.209035
$784$	0	0
$785$	−50.2649	−1.79403
$786$	0	0
$787$	11.7775	0.419824	0.209912	−	0.977720i	$-0.432682\pi$
$787$	11.7775	0.419824	0.209912	+	0.977720i	$0.432682\pi$
$788$	0	0
$789$	−7.44030	−0.264882
$790$	0	0
$791$	−9.84926	−0.350199
$792$	0	0
$793$	38.2870	1.35961
$794$	0	0
$795$	−28.0862	−0.996115
$796$	0	0
$797$	−6.14111	−0.217529	−0.108765	−	0.994068i	$-0.534689\pi$
$797$	−6.14111	−0.217529	−0.108765	+	0.994068i	$0.534689\pi$
$798$	0	0
$799$	12.9831	0.459308
$800$	0	0
$801$	12.9921	0.459054
$802$	0	0
$803$	−46.4823	−1.64032
$804$	0	0
$805$	−20.3797	−0.718290
$806$	0	0
$807$	−29.6336	−1.04315
$808$	0	0
$809$	20.4159	0.717785	0.358893	−	0.933379i	$-0.383154\pi$
$809$	20.4159	0.717785	0.358893	+	0.933379i	$0.383154\pi$
$810$	0	0
$811$	−23.7912	−0.835422	−0.417711	−	0.908580i	$-0.637168\pi$
$811$	−23.7912	−0.835422	−0.417711	+	0.908580i	$0.637168\pi$
$812$	0	0
$813$	−2.33819	−0.0820040
$814$	0	0
$815$	−11.3446	−0.397382
$816$	0	0
$817$	−4.80304	−0.168037
$818$	0	0
$819$	3.43609	0.120067
$820$	0	0
$821$	−49.1782	−1.71633	−0.858166	−	0.513372i	$-0.828396\pi$
$821$	−49.1782	−1.71633	−0.858166	+	0.513372i	$0.828396\pi$
$822$	0	0
$823$	−8.85349	−0.308613	−0.154307	−	0.988023i	$-0.549314\pi$
$823$	−8.85349	−0.308613	−0.154307	+	0.988023i	$0.549314\pi$
$824$	0	0
$825$	−38.7446	−1.34892
$826$	0	0
$827$	−16.2374	−0.564628	−0.282314	−	0.959322i	$-0.591102\pi$
$827$	−16.2374	−0.564628	−0.282314	+	0.959322i	$0.591102\pi$
$828$	0	0
$829$	−12.8854	−0.447530	−0.223765	−	0.974643i	$-0.571835\pi$
$829$	−12.8854	−0.447530	−0.223765	+	0.974643i	$0.571835\pi$
$830$	0	0
$831$	−7.16591	−0.248583
$832$	0	0
$833$	2.09219	0.0724900
$834$	0	0
$835$	52.4495	1.81509
$836$	0	0
$837$	1.79896	0.0621811
$838$	0	0
$839$	17.8403	0.615915	0.307958	−	0.951400i	$-0.400354\pi$
$839$	17.8403	0.615915	0.307958	+	0.951400i	$0.400354\pi$
$840$	0	0
$841$	5.21381	0.179787
$842$	0	0
$843$	0.891019	0.0306883
$844$	0	0
$845$	−4.39544	−0.151208
$846$	0	0
$847$	−9.44867	−0.324660
$848$	0	0
$849$	17.1951	0.590134
$850$	0	0
$851$	19.8696	0.681122
$852$	0	0
$853$	8.76723	0.300184	0.150092	−	0.988672i	$-0.452043\pi$
$853$	8.76723	0.300184	0.150092	+	0.988672i	$0.452043\pi$
$854$	0	0
$855$	3.68348	0.125972
$856$	0	0
$857$	28.3130	0.967153	0.483576	−	0.875302i	$-0.339338\pi$
$857$	28.3130	0.967153	0.483576	+	0.875302i	$0.339338\pi$
$858$	0	0
$859$	5.84030	0.199268	0.0996342	−	0.995024i	$-0.468233\pi$
$859$	5.84030	0.199268	0.0996342	+	0.995024i	$0.468233\pi$
$860$	0	0
$861$	−10.0733	−0.343299
$862$	0	0
$863$	−45.9582	−1.56444	−0.782218	−	0.623005i	$-0.785912\pi$
$863$	−45.9582	−1.56444	−0.782218	+	0.623005i	$0.785912\pi$
$864$	0	0
$865$	−55.5618	−1.88916
$866$	0	0
$867$	−12.6227	−0.428691
$868$	0	0
$869$	−64.1524	−2.17622
$870$	0	0
$871$	21.4382	0.726406
$872$	0	0
$873$	0.872180	0.0295188
$874$	0	0
$875$	−13.1426	−0.444302
$876$	0	0
$877$	56.3148	1.90162	0.950808	−	0.309780i	$-0.100255\pi$
$877$	56.3148	1.90162	0.950808	+	0.309780i	$0.100255\pi$
$878$	0	0
$879$	4.24521	0.143187
$880$	0	0
$881$	31.5287	1.06223	0.531114	−	0.847300i	$-0.321773\pi$
$881$	31.5287	1.06223	0.531114	+	0.847300i	$0.321773\pi$
$882$	0	0
$883$	0.159958	0.00538300	0.00269150	−	0.999996i	$-0.499143\pi$
$883$	0.159958	0.00538300	0.00269150	+	0.999996i	$0.499143\pi$
$884$	0	0
$885$	−37.0364	−1.24497
$886$	0	0
$887$	3.71177	0.124629	0.0623146	−	0.998057i	$-0.480152\pi$
$887$	3.71177	0.124629	0.0623146	+	0.998057i	$0.480152\pi$
$888$	0	0
$889$	−6.68780	−0.224302
$890$	0	0
$891$	−4.52202	−0.151493
$892$	0	0
$893$	6.20550	0.207659
$894$	0	0
$895$	7.53766	0.251956
$896$	0	0
$897$	−19.0110	−0.634758
$898$	0	0
$899$	10.5226	0.350947
$900$	0	0
$901$	−15.9528	−0.531464
$902$	0	0
$903$	4.80304	0.159835
$904$	0	0
$905$	−80.2185	−2.66655
$906$	0	0
$907$	8.46193	0.280974	0.140487	−	0.990083i	$-0.455133\pi$
$907$	8.46193	0.280974	0.140487	+	0.990083i	$0.455133\pi$
$908$	0	0
$909$	4.86377	0.161321
$910$	0	0
$911$	−48.0815	−1.59301	−0.796505	−	0.604632i	$-0.793320\pi$
$911$	−48.0815	−1.59301	−0.796505	+	0.604632i	$0.793320\pi$
$912$	0	0
$913$	−57.9158	−1.91673
$914$	0	0
$915$	−41.0436	−1.35686
$916$	0	0
$917$	−10.8533	−0.358409
$918$	0	0
$919$	51.0446	1.68381	0.841903	−	0.539629i	$-0.181435\pi$
$919$	51.0446	1.68381	0.841903	+	0.539629i	$0.181435\pi$
$920$	0	0
$921$	9.15700	0.301733
$922$	0	0
$923$	−11.4327	−0.376311
$924$	0	0
$925$	30.7701	1.01172
$926$	0	0
$927$	9.04404	0.297045
$928$	0	0
$929$	−1.11432	−0.0365596	−0.0182798	−	0.999833i	$-0.505819\pi$
$929$	−1.11432	−0.0365596	−0.0182798	+	0.999833i	$0.505819\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	27.0800	0.886558
$934$	0	0
$935$	−34.8491	−1.13969
$936$	0	0
$937$	23.5728	0.770088	0.385044	−	0.922898i	$-0.374186\pi$
$937$	23.5728	0.770088	0.385044	+	0.922898i	$0.374186\pi$
$938$	0	0
$939$	−19.6675	−0.641823
$940$	0	0
$941$	−5.05939	−0.164931	−0.0824656	−	0.996594i	$-0.526279\pi$
$941$	−5.05939	−0.164931	−0.0824656	+	0.996594i	$0.526279\pi$
$942$	0	0
$943$	55.7332	1.81492
$944$	0	0
$945$	−3.68348	−0.119823
$946$	0	0
$947$	4.99186	0.162214	0.0811068	−	0.996705i	$-0.474154\pi$
$947$	4.99186	0.162214	0.0811068	+	0.996705i	$0.474154\pi$
$948$	0	0
$949$	−35.3199	−1.14653
$950$	0	0
$951$	−3.19478	−0.103598
$952$	0	0
$953$	16.8294	0.545158	0.272579	−	0.962133i	$-0.412123\pi$
$953$	16.8294	0.545158	0.272579	+	0.962133i	$0.412123\pi$
$954$	0	0
$955$	53.5216	1.73192
$956$	0	0
$957$	−26.4505	−0.855022
$958$	0	0
$959$	23.1907	0.748868
$960$	0	0
$961$	−27.7637	−0.895605
$962$	0	0
$963$	−9.32061	−0.300352
$964$	0	0
$965$	−4.00711	−0.128994
$966$	0	0
$967$	−12.4249	−0.399558	−0.199779	−	0.979841i	$-0.564022\pi$
$967$	−12.4249	−0.399558	−0.199779	+	0.979841i	$0.564022\pi$
$968$	0	0
$969$	2.09219	0.0672108
$970$	0	0
$971$	−4.19329	−0.134569	−0.0672845	−	0.997734i	$-0.521434\pi$
$971$	−4.19329	−0.134569	−0.0672845	+	0.997734i	$0.521434\pi$
$972$	0	0
$973$	5.80899	0.186228
$974$	0	0
$975$	−29.4404	−0.942847
$976$	0	0
$977$	−36.6185	−1.17153	−0.585765	−	0.810481i	$-0.699206\pi$
$977$	−36.6185	−1.17153	−0.585765	+	0.810481i	$0.699206\pi$
$978$	0	0
$979$	−58.7506	−1.87768
$980$	0	0
$981$	10.7797	0.344171
$982$	0	0
$983$	10.4329	0.332758	0.166379	−	0.986062i	$-0.446792\pi$
$983$	10.4329	0.332758	0.166379	+	0.986062i	$0.446792\pi$
$984$	0	0
$985$	93.2998	2.97278
$986$	0	0
$987$	−6.20550	−0.197523
$988$	0	0
$989$	−26.5739	−0.845002
$990$	0	0
$991$	50.0556	1.59007	0.795034	−	0.606565i	$-0.207453\pi$
$991$	50.0556	1.59007	0.795034	+	0.606565i	$0.207453\pi$
$992$	0	0
$993$	27.7133	0.879455
$994$	0	0
$995$	−45.7156	−1.44928
$996$	0	0
$997$	51.6971	1.63726	0.818631	−	0.574320i	$-0.194733\pi$
$997$	51.6971	1.63726	0.818631	+	0.574320i	$0.194733\pi$
$998$	0	0
$999$	3.59129	0.113623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.cf.1.4		5
4.3	odd	2		399.2.a.g.1.1	✓	5
12.11	even	2		1197.2.a.o.1.5		5
20.19	odd	2		9975.2.a.bp.1.5		5
28.27	even	2		2793.2.a.bg.1.1		5
76.75	even	2		7581.2.a.w.1.5		5
84.83	odd	2		8379.2.a.cb.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.a.g.1.1	✓	5	4.3	odd	2
1197.2.a.o.1.5		5	12.11	even	2
2793.2.a.bg.1.1		5	28.27	even	2
6384.2.a.cf.1.4		5	1.1	even	1	trivial
7581.2.a.w.1.5		5	76.75	even	2
8379.2.a.cb.1.5		5	84.83	odd	2
9975.2.a.bp.1.5		5	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6384.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.68348 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.68348 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.52202 q^{11} -3.43609 q^{13} +3.68348 q^{15} +2.09219 q^{17} +1.00000 q^{19} -1.00000 q^{21} +5.53273 q^{23} +8.56799 q^{25} +1.00000 q^{27} +5.84926 q^{29} +1.79896 q^{31} -4.52202 q^{33} -3.68348 q^{35} +3.59129 q^{37} -3.43609 q^{39} +10.0733 q^{41} -4.80304 q^{43} +3.68348 q^{45} +6.20550 q^{47} +1.00000 q^{49} +2.09219 q^{51} -7.62492 q^{53} -16.6568 q^{55} +1.00000 q^{57} -10.0548 q^{59} -11.1426 q^{61} -1.00000 q^{63} -12.6568 q^{65} -6.23913 q^{67} +5.53273 q^{69} +3.32724 q^{71} +10.2791 q^{73} +8.56799 q^{75} +4.52202 q^{77} +14.1867 q^{79} +1.00000 q^{81} +12.8075 q^{83} +7.70653 q^{85} +5.84926 q^{87} +12.9921 q^{89} +3.43609 q^{91} +1.79896 q^{93} +3.68348 q^{95} +0.872180 q^{97} -4.52202 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} + 4 q^{5} - 5 q^{7} + 5 q^{9} - 8 q^{11} - 6 q^{13} + 4 q^{15} + 12 q^{17} + 5 q^{19} - 5 q^{21} - 12 q^{23} + 15 q^{25} + 5 q^{27} + 4 q^{29} + 8 q^{31} - 8 q^{33} - 4 q^{35} + 2 q^{37} - 6 q^{39} + 10 q^{41} + 16 q^{43} + 4 q^{45} + 2 q^{47} + 5 q^{49} + 12 q^{51} + 12 q^{55} + 5 q^{57} + 4 q^{59} - 14 q^{61} - 5 q^{63} + 32 q^{65} + 20 q^{67} - 12 q^{69} + 6 q^{71} + 10 q^{73} + 15 q^{75} + 8 q^{77} + 5 q^{81} - 6 q^{83} + 8 q^{85} + 4 q^{87} + 26 q^{89} + 6 q^{91} + 8 q^{93} + 4 q^{95} - 18 q^{97} - 8 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 4 * q^5 - 5 * q^7 + 5 * q^9 - 8 * q^11 - 6 * q^13 + 4 * q^15 + 12 * q^17 + 5 * q^19 - 5 * q^21 - 12 * q^23 + 15 * q^25 + 5 * q^27 + 4 * q^29 + 8 * q^31 - 8 * q^33 - 4 * q^35 + 2 * q^37 - 6 * q^39 + 10 * q^41 + 16 * q^43 + 4 * q^45 + 2 * q^47 + 5 * q^49 + 12 * q^51 + 12 * q^55 + 5 * q^57 + 4 * q^59 - 14 * q^61 - 5 * q^63 + 32 * q^65 + 20 * q^67 - 12 * q^69 + 6 * q^71 + 10 * q^73 + 15 * q^75 + 8 * q^77 + 5 * q^81 - 6 * q^83 + 8 * q^85 + 4 * q^87 + 26 * q^89 + 6 * q^91 + 8 * q^93 + 4 * q^95 - 18 * q^97 - 8 * q^99

Introduction

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