LMFDB - Embedded newform 6384.2.a.bp.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1596)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.41421 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.41421 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -2.82843 q^{13} -1.41421 q^{15} +4.24264 q^{17} -1.00000 q^{19} -1.00000 q^{21} +3.65685 q^{23} -3.00000 q^{25} +1.00000 q^{27} +0.585786 q^{29} +4.82843 q^{31} -2.00000 q^{33} +1.41421 q^{35} +10.4853 q^{37} -2.82843 q^{39} -1.65685 q^{41} -2.00000 q^{43} -1.41421 q^{45} -7.07107 q^{47} +1.00000 q^{49} +4.24264 q^{51} +11.8995 q^{53} +2.82843 q^{55} -1.00000 q^{57} -13.6569 q^{59} +0.828427 q^{61} -1.00000 q^{63} +4.00000 q^{65} +4.48528 q^{67} +3.65685 q^{69} -15.4142 q^{71} -11.6569 q^{73} -3.00000 q^{75} +2.00000 q^{77} -9.17157 q^{79} +1.00000 q^{81} -1.41421 q^{83} -6.00000 q^{85} +0.585786 q^{87} -4.00000 q^{89} +2.82843 q^{91} +4.82843 q^{93} +1.41421 q^{95} -3.65685 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{19} - 2 q^{21} - 4 q^{23} - 6 q^{25} + 2 q^{27} + 4 q^{29} + 4 q^{31} - 4 q^{33} + 4 q^{37} + 8 q^{41} - 4 q^{43} + 2 q^{49} + 4 q^{53} - 2 q^{57} - 16 q^{59} - 4 q^{61} - 2 q^{63} + 8 q^{65} - 8 q^{67} - 4 q^{69} - 28 q^{71} - 12 q^{73} - 6 q^{75} + 4 q^{77} - 24 q^{79} + 2 q^{81} - 12 q^{85} + 4 q^{87} - 8 q^{89} + 4 q^{93} + 4 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^19 - 2 * q^21 - 4 * q^23 - 6 * q^25 + 2 * q^27 + 4 * q^29 + 4 * q^31 - 4 * q^33 + 4 * q^37 + 8 * q^41 - 4 * q^43 + 2 * q^49 + 4 * q^53 - 2 * q^57 - 16 * q^59 - 4 * q^61 - 2 * q^63 + 8 * q^65 - 8 * q^67 - 4 * q^69 - 28 * q^71 - 12 * q^73 - 6 * q^75 + 4 * q^77 - 24 * q^79 + 2 * q^81 - 12 * q^85 + 4 * q^87 - 8 * q^89 + 4 * q^93 + 4 * 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
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−2.82843	−0.784465	−0.392232	−	0.919866i	$-0.628297\pi$
$13$	−2.82843	−0.784465	−0.392232	+	0.919866i	$0.628297\pi$
$14$	0	0
$15$	−1.41421	−0.365148
$16$	0	0
$17$	4.24264	1.02899	0.514496	−	0.857493i	$-0.327979\pi$
$17$	4.24264	1.02899	0.514496	+	0.857493i	$0.327979\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	3.65685	0.762507	0.381253	−	0.924471i	$-0.375493\pi$
$23$	3.65685	0.762507	0.381253	+	0.924471i	$0.375493\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.585786	0.108778	0.0543889	−	0.998520i	$-0.482679\pi$
$29$	0.585786	0.108778	0.0543889	+	0.998520i	$0.482679\pi$
$30$	0	0
$31$	4.82843	0.867211	0.433606	−	0.901103i	$-0.357241\pi$
$31$	4.82843	0.867211	0.433606	+	0.901103i	$0.357241\pi$
$32$	0	0
$33$	−2.00000	−0.348155
$34$	0	0
$35$	1.41421	0.239046
$36$	0	0
$37$	10.4853	1.72377	0.861885	−	0.507104i	$-0.169284\pi$
$37$	10.4853	1.72377	0.861885	+	0.507104i	$0.169284\pi$
$38$	0	0
$39$	−2.82843	−0.452911
$40$	0	0
$41$	−1.65685	−0.258757	−0.129379	−	0.991595i	$-0.541298\pi$
$41$	−1.65685	−0.258757	−0.129379	+	0.991595i	$0.541298\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−1.41421	−0.210819
$46$	0	0
$47$	−7.07107	−1.03142	−0.515711	−	0.856763i	$-0.672472\pi$
$47$	−7.07107	−1.03142	−0.515711	+	0.856763i	$0.672472\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.24264	0.594089
$52$	0	0
$53$	11.8995	1.63452	0.817261	−	0.576268i	$-0.195492\pi$
$53$	11.8995	1.63452	0.817261	+	0.576268i	$0.195492\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−13.6569	−1.77797	−0.888985	−	0.457935i	$-0.848589\pi$
$59$	−13.6569	−1.77797	−0.888985	+	0.457935i	$0.848589\pi$
$60$	0	0
$61$	0.828427	0.106069	0.0530346	−	0.998593i	$-0.483111\pi$
$61$	0.828427	0.106069	0.0530346	+	0.998593i	$0.483111\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	4.48528	0.547964	0.273982	−	0.961735i	$-0.411659\pi$
$67$	4.48528	0.547964	0.273982	+	0.961735i	$0.411659\pi$
$68$	0	0
$69$	3.65685	0.440234
$70$	0	0
$71$	−15.4142	−1.82933	−0.914665	−	0.404212i	$-0.867546\pi$
$71$	−15.4142	−1.82933	−0.914665	+	0.404212i	$0.867546\pi$
$72$	0	0
$73$	−11.6569	−1.36433	−0.682166	−	0.731198i	$-0.738962\pi$
$73$	−11.6569	−1.36433	−0.682166	+	0.731198i	$0.738962\pi$
$74$	0	0
$75$	−3.00000	−0.346410
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	−9.17157	−1.03188	−0.515941	−	0.856624i	$-0.672558\pi$
$79$	−9.17157	−1.03188	−0.515941	+	0.856624i	$0.672558\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.41421	−0.155230	−0.0776151	−	0.996983i	$-0.524731\pi$
$83$	−1.41421	−0.155230	−0.0776151	+	0.996983i	$0.524731\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	0	0
$87$	0.585786	0.0628029
$88$	0	0
$89$	−4.00000	−0.423999	−0.212000	−	0.977270i	$-0.567998\pi$
$89$	−4.00000	−0.423999	−0.212000	+	0.977270i	$0.567998\pi$
$90$	0	0
$91$	2.82843	0.296500
$92$	0	0
$93$	4.82843	0.500685
$94$	0	0
$95$	1.41421	0.145095
$96$	0	0
$97$	−3.65685	−0.371297	−0.185649	−	0.982616i	$-0.559439\pi$
$97$	−3.65685	−0.371297	−0.185649	+	0.982616i	$0.559439\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−0.242641	−0.0241437	−0.0120718	−	0.999927i	$-0.503843\pi$
$101$	−0.242641	−0.0241437	−0.0120718	+	0.999927i	$0.503843\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	1.41421	0.138013
$106$	0	0
$107$	−18.2426	−1.76358	−0.881791	−	0.471640i	$-0.843662\pi$
$107$	−18.2426	−1.76358	−0.881791	+	0.471640i	$0.843662\pi$
$108$	0	0
$109$	14.4853	1.38744	0.693719	−	0.720246i	$-0.255971\pi$
$109$	14.4853	1.38744	0.693719	+	0.720246i	$0.255971\pi$
$110$	0	0
$111$	10.4853	0.995219
$112$	0	0
$113$	−6.72792	−0.632910	−0.316455	−	0.948608i	$-0.602493\pi$
$113$	−6.72792	−0.632910	−0.316455	+	0.948608i	$0.602493\pi$
$114$	0	0
$115$	−5.17157	−0.482252
$116$	0	0
$117$	−2.82843	−0.261488
$118$	0	0
$119$	−4.24264	−0.388922
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−1.65685	−0.149394
$124$	0	0
$125$	11.3137	1.01193
$126$	0	0
$127$	−19.7990	−1.75688	−0.878438	−	0.477856i	$-0.841414\pi$
$127$	−19.7990	−1.75688	−0.878438	+	0.477856i	$0.841414\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	−0.928932	−0.0811612	−0.0405806	−	0.999176i	$-0.512921\pi$
$131$	−0.928932	−0.0811612	−0.0405806	+	0.999176i	$0.512921\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−1.41421	−0.121716
$136$	0	0
$137$	−11.6569	−0.995912	−0.497956	−	0.867202i	$-0.665916\pi$
$137$	−11.6569	−0.995912	−0.497956	+	0.867202i	$0.665916\pi$
$138$	0	0
$139$	−3.31371	−0.281065	−0.140533	−	0.990076i	$-0.544881\pi$
$139$	−3.31371	−0.281065	−0.140533	+	0.990076i	$0.544881\pi$
$140$	0	0
$141$	−7.07107	−0.595491
$142$	0	0
$143$	5.65685	0.473050
$144$	0	0
$145$	−0.828427	−0.0687971
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	22.4853	1.84207	0.921033	−	0.389485i	$-0.127347\pi$
$149$	22.4853	1.84207	0.921033	+	0.389485i	$0.127347\pi$
$150$	0	0
$151$	−3.31371	−0.269666	−0.134833	−	0.990868i	$-0.543050\pi$
$151$	−3.31371	−0.269666	−0.134833	+	0.990868i	$0.543050\pi$
$152$	0	0
$153$	4.24264	0.342997
$154$	0	0
$155$	−6.82843	−0.548472
$156$	0	0
$157$	−22.4853	−1.79452	−0.897260	−	0.441502i	$-0.854446\pi$
$157$	−22.4853	−1.79452	−0.897260	+	0.441502i	$0.854446\pi$
$158$	0	0
$159$	11.8995	0.943691
$160$	0	0
$161$	−3.65685	−0.288200
$162$	0	0
$163$	−4.34315	−0.340181	−0.170091	−	0.985428i	$-0.554406\pi$
$163$	−4.34315	−0.340181	−0.170091	+	0.985428i	$0.554406\pi$
$164$	0	0
$165$	2.82843	0.220193
$166$	0	0
$167$	8.48528	0.656611	0.328305	−	0.944572i	$-0.393522\pi$
$167$	8.48528	0.656611	0.328305	+	0.944572i	$0.393522\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−1.65685	−0.125968	−0.0629841	−	0.998015i	$-0.520062\pi$
$173$	−1.65685	−0.125968	−0.0629841	+	0.998015i	$0.520062\pi$
$174$	0	0
$175$	3.00000	0.226779
$176$	0	0
$177$	−13.6569	−1.02651
$178$	0	0
$179$	2.24264	0.167623	0.0838114	−	0.996482i	$-0.473291\pi$
$179$	2.24264	0.167623	0.0838114	+	0.996482i	$0.473291\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	0.828427	0.0612391
$184$	0	0
$185$	−14.8284	−1.09021
$186$	0	0
$187$	−8.48528	−0.620505
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−7.65685	−0.554031	−0.277015	−	0.960866i	$-0.589345\pi$
$191$	−7.65685	−0.554031	−0.277015	+	0.960866i	$0.589345\pi$
$192$	0	0
$193$	−22.4853	−1.61853	−0.809263	−	0.587447i	$-0.800133\pi$
$193$	−22.4853	−1.61853	−0.809263	+	0.587447i	$0.800133\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	4.34315	0.309436	0.154718	−	0.987959i	$-0.450553\pi$
$197$	4.34315	0.309436	0.154718	+	0.987959i	$0.450553\pi$
$198$	0	0
$199$	8.48528	0.601506	0.300753	−	0.953702i	$-0.402762\pi$
$199$	8.48528	0.601506	0.300753	+	0.953702i	$0.402762\pi$
$200$	0	0
$201$	4.48528	0.316367
$202$	0	0
$203$	−0.585786	−0.0411141
$204$	0	0
$205$	2.34315	0.163652
$206$	0	0
$207$	3.65685	0.254169
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	−10.3431	−0.712052	−0.356026	−	0.934476i	$-0.615868\pi$
$211$	−10.3431	−0.712052	−0.356026	+	0.934476i	$0.615868\pi$
$212$	0	0
$213$	−15.4142	−1.05616
$214$	0	0
$215$	2.82843	0.192897
$216$	0	0
$217$	−4.82843	−0.327775
$218$	0	0
$219$	−11.6569	−0.787697
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−2.48528	−0.166427	−0.0832134	−	0.996532i	$-0.526518\pi$
$223$	−2.48528	−0.166427	−0.0832134	+	0.996532i	$0.526518\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	6.82843	0.453219	0.226609	−	0.973986i	$-0.427236\pi$
$227$	6.82843	0.453219	0.226609	+	0.973986i	$0.427236\pi$
$228$	0	0
$229$	25.7990	1.70485	0.852423	−	0.522853i	$-0.175132\pi$
$229$	25.7990	1.70485	0.852423	+	0.522853i	$0.175132\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	12.1421	0.795458	0.397729	−	0.917503i	$-0.369798\pi$
$233$	12.1421	0.795458	0.397729	+	0.917503i	$0.369798\pi$
$234$	0	0
$235$	10.0000	0.652328
$236$	0	0
$237$	−9.17157	−0.595758
$238$	0	0
$239$	13.7990	0.892582	0.446291	−	0.894888i	$-0.352745\pi$
$239$	13.7990	0.892582	0.446291	+	0.894888i	$0.352745\pi$
$240$	0	0
$241$	−12.3431	−0.795092	−0.397546	−	0.917582i	$-0.630138\pi$
$241$	−12.3431	−0.795092	−0.397546	+	0.917582i	$0.630138\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.41421	−0.0903508
$246$	0	0
$247$	2.82843	0.179969
$248$	0	0
$249$	−1.41421	−0.0896221
$250$	0	0
$251$	16.2426	1.02523	0.512613	−	0.858620i	$-0.328677\pi$
$251$	16.2426	1.02523	0.512613	+	0.858620i	$0.328677\pi$
$252$	0	0
$253$	−7.31371	−0.459809
$254$	0	0
$255$	−6.00000	−0.375735
$256$	0	0
$257$	8.48528	0.529297	0.264649	−	0.964345i	$-0.414744\pi$
$257$	8.48528	0.529297	0.264649	+	0.964345i	$0.414744\pi$
$258$	0	0
$259$	−10.4853	−0.651524
$260$	0	0
$261$	0.585786	0.0362593
$262$	0	0
$263$	11.1716	0.688869	0.344434	−	0.938810i	$-0.388071\pi$
$263$	11.1716	0.688869	0.344434	+	0.938810i	$0.388071\pi$
$264$	0	0
$265$	−16.8284	−1.03376
$266$	0	0
$267$	−4.00000	−0.244796
$268$	0	0
$269$	24.4853	1.49289	0.746447	−	0.665445i	$-0.231758\pi$
$269$	24.4853	1.49289	0.746447	+	0.665445i	$0.231758\pi$
$270$	0	0
$271$	−15.7990	−0.959720	−0.479860	−	0.877345i	$-0.659313\pi$
$271$	−15.7990	−0.959720	−0.479860	+	0.877345i	$0.659313\pi$
$272$	0	0
$273$	2.82843	0.171184
$274$	0	0
$275$	6.00000	0.361814
$276$	0	0
$277$	−28.9706	−1.74067	−0.870336	−	0.492458i	$-0.836099\pi$
$277$	−28.9706	−1.74067	−0.870336	+	0.492458i	$0.836099\pi$
$278$	0	0
$279$	4.82843	0.289070
$280$	0	0
$281$	22.0416	1.31489	0.657447	−	0.753501i	$-0.271636\pi$
$281$	22.0416	1.31489	0.657447	+	0.753501i	$0.271636\pi$
$282$	0	0
$283$	−31.1127	−1.84946	−0.924729	−	0.380626i	$-0.875708\pi$
$283$	−31.1127	−1.84946	−0.924729	+	0.380626i	$0.875708\pi$
$284$	0	0
$285$	1.41421	0.0837708
$286$	0	0
$287$	1.65685	0.0978010
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−3.65685	−0.214369
$292$	0	0
$293$	−8.00000	−0.467365	−0.233682	−	0.972313i	$-0.575078\pi$
$293$	−8.00000	−0.467365	−0.233682	+	0.972313i	$0.575078\pi$
$294$	0	0
$295$	19.3137	1.12449
$296$	0	0
$297$	−2.00000	−0.116052
$298$	0	0
$299$	−10.3431	−0.598160
$300$	0	0
$301$	2.00000	0.115278
$302$	0	0
$303$	−0.242641	−0.0139393
$304$	0	0
$305$	−1.17157	−0.0670841
$306$	0	0
$307$	15.1716	0.865887	0.432944	−	0.901421i	$-0.357475\pi$
$307$	15.1716	0.865887	0.432944	+	0.901421i	$0.357475\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−21.2132	−1.20289	−0.601445	−	0.798914i	$-0.705408\pi$
$311$	−21.2132	−1.20289	−0.601445	+	0.798914i	$0.705408\pi$
$312$	0	0
$313$	−24.1421	−1.36459	−0.682297	−	0.731075i	$-0.739019\pi$
$313$	−24.1421	−1.36459	−0.682297	+	0.731075i	$0.739019\pi$
$314$	0	0
$315$	1.41421	0.0796819
$316$	0	0
$317$	−22.7279	−1.27653	−0.638264	−	0.769818i	$-0.720347\pi$
$317$	−22.7279	−1.27653	−0.638264	+	0.769818i	$0.720347\pi$
$318$	0	0
$319$	−1.17157	−0.0655955
$320$	0	0
$321$	−18.2426	−1.01820
$322$	0	0
$323$	−4.24264	−0.236067
$324$	0	0
$325$	8.48528	0.470679
$326$	0	0
$327$	14.4853	0.801038
$328$	0	0
$329$	7.07107	0.389841
$330$	0	0
$331$	3.31371	0.182138	0.0910689	−	0.995845i	$-0.470972\pi$
$331$	3.31371	0.182138	0.0910689	+	0.995845i	$0.470972\pi$
$332$	0	0
$333$	10.4853	0.574590
$334$	0	0
$335$	−6.34315	−0.346563
$336$	0	0
$337$	0.142136	0.00774262	0.00387131	−	0.999993i	$-0.498768\pi$
$337$	0.142136	0.00774262	0.00387131	+	0.999993i	$0.498768\pi$
$338$	0	0
$339$	−6.72792	−0.365411
$340$	0	0
$341$	−9.65685	−0.522948
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−5.17157	−0.278428
$346$	0	0
$347$	13.5147	0.725508	0.362754	−	0.931885i	$-0.381837\pi$
$347$	13.5147	0.725508	0.362754	+	0.931885i	$0.381837\pi$
$348$	0	0
$349$	−22.9706	−1.22959	−0.614793	−	0.788688i	$-0.710760\pi$
$349$	−22.9706	−1.22959	−0.614793	+	0.788688i	$0.710760\pi$
$350$	0	0
$351$	−2.82843	−0.150970
$352$	0	0
$353$	1.41421	0.0752710	0.0376355	−	0.999292i	$-0.488017\pi$
$353$	1.41421	0.0752710	0.0376355	+	0.999292i	$0.488017\pi$
$354$	0	0
$355$	21.7990	1.15697
$356$	0	0
$357$	−4.24264	−0.224544
$358$	0	0
$359$	−10.0000	−0.527780	−0.263890	−	0.964553i	$-0.585006\pi$
$359$	−10.0000	−0.527780	−0.263890	+	0.964553i	$0.585006\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	16.4853	0.862879
$366$	0	0
$367$	28.4853	1.48692	0.743460	−	0.668781i	$-0.233183\pi$
$367$	28.4853	1.48692	0.743460	+	0.668781i	$0.233183\pi$
$368$	0	0
$369$	−1.65685	−0.0862524
$370$	0	0
$371$	−11.8995	−0.617791
$372$	0	0
$373$	−18.9706	−0.982259	−0.491129	−	0.871087i	$-0.663416\pi$
$373$	−18.9706	−0.982259	−0.491129	+	0.871087i	$0.663416\pi$
$374$	0	0
$375$	11.3137	0.584237
$376$	0	0
$377$	−1.65685	−0.0853323
$378$	0	0
$379$	24.2843	1.24740	0.623700	−	0.781664i	$-0.285629\pi$
$379$	24.2843	1.24740	0.623700	+	0.781664i	$0.285629\pi$
$380$	0	0
$381$	−19.7990	−1.01433
$382$	0	0
$383$	5.17157	0.264255	0.132128	−	0.991233i	$-0.457819\pi$
$383$	5.17157	0.264255	0.132128	+	0.991233i	$0.457819\pi$
$384$	0	0
$385$	−2.82843	−0.144150
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	−24.1421	−1.22405	−0.612027	−	0.790837i	$-0.709646\pi$
$389$	−24.1421	−1.22405	−0.612027	+	0.790837i	$0.709646\pi$
$390$	0	0
$391$	15.5147	0.784613
$392$	0	0
$393$	−0.928932	−0.0468584
$394$	0	0
$395$	12.9706	0.652620
$396$	0	0
$397$	9.31371	0.467442	0.233721	−	0.972304i	$-0.424910\pi$
$397$	9.31371	0.467442	0.233721	+	0.972304i	$0.424910\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	28.5858	1.42751	0.713753	−	0.700397i	$-0.246994\pi$
$401$	28.5858	1.42751	0.713753	+	0.700397i	$0.246994\pi$
$402$	0	0
$403$	−13.6569	−0.680296
$404$	0	0
$405$	−1.41421	−0.0702728
$406$	0	0
$407$	−20.9706	−1.03947
$408$	0	0
$409$	−9.17157	−0.453505	−0.226753	−	0.973952i	$-0.572811\pi$
$409$	−9.17157	−0.453505	−0.226753	+	0.973952i	$0.572811\pi$
$410$	0	0
$411$	−11.6569	−0.574990
$412$	0	0
$413$	13.6569	0.672010
$414$	0	0
$415$	2.00000	0.0981761
$416$	0	0
$417$	−3.31371	−0.162273
$418$	0	0
$419$	−8.24264	−0.402679	−0.201340	−	0.979521i	$-0.564530\pi$
$419$	−8.24264	−0.402679	−0.201340	+	0.979521i	$0.564530\pi$
$420$	0	0
$421$	34.2843	1.67091	0.835457	−	0.549556i	$-0.185203\pi$
$421$	34.2843	1.67091	0.835457	+	0.549556i	$0.185203\pi$
$422$	0	0
$423$	−7.07107	−0.343807
$424$	0	0
$425$	−12.7279	−0.617395
$426$	0	0
$427$	−0.828427	−0.0400904
$428$	0	0
$429$	5.65685	0.273115
$430$	0	0
$431$	31.4142	1.51317	0.756585	−	0.653896i	$-0.226866\pi$
$431$	31.4142	1.51317	0.756585	+	0.653896i	$0.226866\pi$
$432$	0	0
$433$	−10.6863	−0.513550	−0.256775	−	0.966471i	$-0.582660\pi$
$433$	−10.6863	−0.513550	−0.256775	+	0.966471i	$0.582660\pi$
$434$	0	0
$435$	−0.828427	−0.0397200
$436$	0	0
$437$	−3.65685	−0.174931
$438$	0	0
$439$	−26.4853	−1.26407	−0.632037	−	0.774938i	$-0.717781\pi$
$439$	−26.4853	−1.26407	−0.632037	+	0.774938i	$0.717781\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	13.7990	0.655610	0.327805	−	0.944745i	$-0.393691\pi$
$443$	13.7990	0.655610	0.327805	+	0.944745i	$0.393691\pi$
$444$	0	0
$445$	5.65685	0.268161
$446$	0	0
$447$	22.4853	1.06352
$448$	0	0
$449$	14.2426	0.672152	0.336076	−	0.941835i	$-0.390900\pi$
$449$	14.2426	0.672152	0.336076	+	0.941835i	$0.390900\pi$
$450$	0	0
$451$	3.31371	0.156036
$452$	0	0
$453$	−3.31371	−0.155692
$454$	0	0
$455$	−4.00000	−0.187523
$456$	0	0
$457$	19.3137	0.903457	0.451729	−	0.892155i	$-0.350808\pi$
$457$	19.3137	0.903457	0.451729	+	0.892155i	$0.350808\pi$
$458$	0	0
$459$	4.24264	0.198030
$460$	0	0
$461$	−4.92893	−0.229563	−0.114782	−	0.993391i	$-0.536617\pi$
$461$	−4.92893	−0.229563	−0.114782	+	0.993391i	$0.536617\pi$
$462$	0	0
$463$	−7.31371	−0.339897	−0.169948	−	0.985453i	$-0.554360\pi$
$463$	−7.31371	−0.339897	−0.169948	+	0.985453i	$0.554360\pi$
$464$	0	0
$465$	−6.82843	−0.316661
$466$	0	0
$467$	−14.8701	−0.688104	−0.344052	−	0.938951i	$-0.611800\pi$
$467$	−14.8701	−0.688104	−0.344052	+	0.938951i	$0.611800\pi$
$468$	0	0
$469$	−4.48528	−0.207111
$470$	0	0
$471$	−22.4853	−1.03607
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	3.00000	0.137649
$476$	0	0
$477$	11.8995	0.544840
$478$	0	0
$479$	−0.928932	−0.0424440	−0.0212220	−	0.999775i	$-0.506756\pi$
$479$	−0.928932	−0.0424440	−0.0212220	+	0.999775i	$0.506756\pi$
$480$	0	0
$481$	−29.6569	−1.35224
$482$	0	0
$483$	−3.65685	−0.166393
$484$	0	0
$485$	5.17157	0.234829
$486$	0	0
$487$	−36.2843	−1.64420	−0.822099	−	0.569345i	$-0.807197\pi$
$487$	−36.2843	−1.64420	−0.822099	+	0.569345i	$0.807197\pi$
$488$	0	0
$489$	−4.34315	−0.196404
$490$	0	0
$491$	0.828427	0.0373864	0.0186932	−	0.999825i	$-0.494049\pi$
$491$	0.828427	0.0373864	0.0186932	+	0.999825i	$0.494049\pi$
$492$	0	0
$493$	2.48528	0.111931
$494$	0	0
$495$	2.82843	0.127128
$496$	0	0
$497$	15.4142	0.691422
$498$	0	0
$499$	7.31371	0.327407	0.163703	−	0.986510i	$-0.447656\pi$
$499$	7.31371	0.327407	0.163703	+	0.986510i	$0.447656\pi$
$500$	0	0
$501$	8.48528	0.379094
$502$	0	0
$503$	23.7574	1.05929	0.529644	−	0.848220i	$-0.322325\pi$
$503$	23.7574	1.05929	0.529644	+	0.848220i	$0.322325\pi$
$504$	0	0
$505$	0.343146	0.0152698
$506$	0	0
$507$	−5.00000	−0.222058
$508$	0	0
$509$	−31.1127	−1.37905	−0.689523	−	0.724264i	$-0.742180\pi$
$509$	−31.1127	−1.37905	−0.689523	+	0.724264i	$0.742180\pi$
$510$	0	0
$511$	11.6569	0.515669
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	11.3137	0.498542
$516$	0	0
$517$	14.1421	0.621970
$518$	0	0
$519$	−1.65685	−0.0727278
$520$	0	0
$521$	−41.4558	−1.81621	−0.908107	−	0.418739i	$-0.862472\pi$
$521$	−41.4558	−1.81621	−0.908107	+	0.418739i	$0.862472\pi$
$522$	0	0
$523$	5.79899	0.253572	0.126786	−	0.991930i	$-0.459534\pi$
$523$	5.79899	0.253572	0.126786	+	0.991930i	$0.459534\pi$
$524$	0	0
$525$	3.00000	0.130931
$526$	0	0
$527$	20.4853	0.892353
$528$	0	0
$529$	−9.62742	−0.418583
$530$	0	0
$531$	−13.6569	−0.592657
$532$	0	0
$533$	4.68629	0.202986
$534$	0	0
$535$	25.7990	1.11539
$536$	0	0
$537$	2.24264	0.0967771
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−5.31371	−0.228454	−0.114227	−	0.993455i	$-0.536439\pi$
$541$	−5.31371	−0.228454	−0.114227	+	0.993455i	$0.536439\pi$
$542$	0	0
$543$	−6.00000	−0.257485
$544$	0	0
$545$	−20.4853	−0.877493
$546$	0	0
$547$	−24.4853	−1.04692	−0.523458	−	0.852052i	$-0.675358\pi$
$547$	−24.4853	−1.04692	−0.523458	+	0.852052i	$0.675358\pi$
$548$	0	0
$549$	0.828427	0.0353564
$550$	0	0
$551$	−0.585786	−0.0249553
$552$	0	0
$553$	9.17157	0.390015
$554$	0	0
$555$	−14.8284	−0.629432
$556$	0	0
$557$	9.79899	0.415197	0.207598	−	0.978214i	$-0.433435\pi$
$557$	9.79899	0.415197	0.207598	+	0.978214i	$0.433435\pi$
$558$	0	0
$559$	5.65685	0.239259
$560$	0	0
$561$	−8.48528	−0.358249
$562$	0	0
$563$	−34.1421	−1.43892	−0.719460	−	0.694534i	$-0.755611\pi$
$563$	−34.1421	−1.43892	−0.719460	+	0.694534i	$0.755611\pi$
$564$	0	0
$565$	9.51472	0.400287
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−16.5858	−0.695312	−0.347656	−	0.937622i	$-0.613022\pi$
$569$	−16.5858	−0.695312	−0.347656	+	0.937622i	$0.613022\pi$
$570$	0	0
$571$	20.2843	0.848870	0.424435	−	0.905458i	$-0.360473\pi$
$571$	20.2843	0.848870	0.424435	+	0.905458i	$0.360473\pi$
$572$	0	0
$573$	−7.65685	−0.319870
$574$	0	0
$575$	−10.9706	−0.457504
$576$	0	0
$577$	6.48528	0.269986	0.134993	−	0.990847i	$-0.456899\pi$
$577$	6.48528	0.269986	0.134993	+	0.990847i	$0.456899\pi$
$578$	0	0
$579$	−22.4853	−0.934456
$580$	0	0
$581$	1.41421	0.0586715
$582$	0	0
$583$	−23.7990	−0.985653
$584$	0	0
$585$	4.00000	0.165380
$586$	0	0
$587$	12.9289	0.533634	0.266817	−	0.963747i	$-0.414028\pi$
$587$	12.9289	0.533634	0.266817	+	0.963747i	$0.414028\pi$
$588$	0	0
$589$	−4.82843	−0.198952
$590$	0	0
$591$	4.34315	0.178653
$592$	0	0
$593$	42.8701	1.76046	0.880231	−	0.474545i	$-0.157387\pi$
$593$	42.8701	1.76046	0.880231	+	0.474545i	$0.157387\pi$
$594$	0	0
$595$	6.00000	0.245976
$596$	0	0
$597$	8.48528	0.347279
$598$	0	0
$599$	−22.7279	−0.928638	−0.464319	−	0.885668i	$-0.653701\pi$
$599$	−22.7279	−0.928638	−0.464319	+	0.885668i	$0.653701\pi$
$600$	0	0
$601$	1.31371	0.0535873	0.0267936	−	0.999641i	$-0.491470\pi$
$601$	1.31371	0.0535873	0.0267936	+	0.999641i	$0.491470\pi$
$602$	0	0
$603$	4.48528	0.182655
$604$	0	0
$605$	9.89949	0.402472
$606$	0	0
$607$	22.6274	0.918419	0.459209	−	0.888328i	$-0.348133\pi$
$607$	22.6274	0.918419	0.459209	+	0.888328i	$0.348133\pi$
$608$	0	0
$609$	−0.585786	−0.0237373
$610$	0	0
$611$	20.0000	0.809113
$612$	0	0
$613$	−36.2843	−1.46551	−0.732754	−	0.680494i	$-0.761765\pi$
$613$	−36.2843	−1.46551	−0.732754	+	0.680494i	$0.761765\pi$
$614$	0	0
$615$	2.34315	0.0944848
$616$	0	0
$617$	−13.3137	−0.535990	−0.267995	−	0.963420i	$-0.586361\pi$
$617$	−13.3137	−0.535990	−0.267995	+	0.963420i	$0.586361\pi$
$618$	0	0
$619$	−43.3137	−1.74092	−0.870462	−	0.492235i	$-0.836180\pi$
$619$	−43.3137	−1.74092	−0.870462	+	0.492235i	$0.836180\pi$
$620$	0	0
$621$	3.65685	0.146745
$622$	0	0
$623$	4.00000	0.160257
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	2.00000	0.0798723
$628$	0	0
$629$	44.4853	1.77374
$630$	0	0
$631$	4.62742	0.184215	0.0921073	−	0.995749i	$-0.470640\pi$
$631$	4.62742	0.184215	0.0921073	+	0.995749i	$0.470640\pi$
$632$	0	0
$633$	−10.3431	−0.411103
$634$	0	0
$635$	28.0000	1.11115
$636$	0	0
$637$	−2.82843	−0.112066
$638$	0	0
$639$	−15.4142	−0.609777
$640$	0	0
$641$	17.0711	0.674267	0.337133	−	0.941457i	$-0.390543\pi$
$641$	17.0711	0.674267	0.337133	+	0.941457i	$0.390543\pi$
$642$	0	0
$643$	24.4853	0.965605	0.482803	−	0.875729i	$-0.339619\pi$
$643$	24.4853	0.965605	0.482803	+	0.875729i	$0.339619\pi$
$644$	0	0
$645$	2.82843	0.111369
$646$	0	0
$647$	−39.0711	−1.53604	−0.768021	−	0.640425i	$-0.778758\pi$
$647$	−39.0711	−1.53604	−0.768021	+	0.640425i	$0.778758\pi$
$648$	0	0
$649$	27.3137	1.07216
$650$	0	0
$651$	−4.82843	−0.189241
$652$	0	0
$653$	31.9411	1.24995	0.624976	−	0.780644i	$-0.285109\pi$
$653$	31.9411	1.24995	0.624976	+	0.780644i	$0.285109\pi$
$654$	0	0
$655$	1.31371	0.0513308
$656$	0	0
$657$	−11.6569	−0.454777
$658$	0	0
$659$	26.7279	1.04117	0.520586	−	0.853809i	$-0.325714\pi$
$659$	26.7279	1.04117	0.520586	+	0.853809i	$0.325714\pi$
$660$	0	0
$661$	27.7990	1.08126	0.540628	−	0.841262i	$-0.318187\pi$
$661$	27.7990	1.08126	0.540628	+	0.841262i	$0.318187\pi$
$662$	0	0
$663$	−12.0000	−0.466041
$664$	0	0
$665$	−1.41421	−0.0548408
$666$	0	0
$667$	2.14214	0.0829438
$668$	0	0
$669$	−2.48528	−0.0960865
$670$	0	0
$671$	−1.65685	−0.0639621
$672$	0	0
$673$	26.4853	1.02093	0.510466	−	0.859898i	$-0.329473\pi$
$673$	26.4853	1.02093	0.510466	+	0.859898i	$0.329473\pi$
$674$	0	0
$675$	−3.00000	−0.115470
$676$	0	0
$677$	−27.1127	−1.04203	−0.521013	−	0.853549i	$-0.674446\pi$
$677$	−27.1127	−1.04203	−0.521013	+	0.853549i	$0.674446\pi$
$678$	0	0
$679$	3.65685	0.140337
$680$	0	0
$681$	6.82843	0.261666
$682$	0	0
$683$	51.2132	1.95962	0.979809	−	0.199934i	$-0.0640727\pi$
$683$	51.2132	1.95962	0.979809	+	0.199934i	$0.0640727\pi$
$684$	0	0
$685$	16.4853	0.629870
$686$	0	0
$687$	25.7990	0.984293
$688$	0	0
$689$	−33.6569	−1.28222
$690$	0	0
$691$	−19.7990	−0.753189	−0.376595	−	0.926378i	$-0.622905\pi$
$691$	−19.7990	−0.753189	−0.376595	+	0.926378i	$0.622905\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	4.68629	0.177761
$696$	0	0
$697$	−7.02944	−0.266259
$698$	0	0
$699$	12.1421	0.459258
$700$	0	0
$701$	2.68629	0.101460	0.0507299	−	0.998712i	$-0.483845\pi$
$701$	2.68629	0.101460	0.0507299	+	0.998712i	$0.483845\pi$
$702$	0	0
$703$	−10.4853	−0.395460
$704$	0	0
$705$	10.0000	0.376622
$706$	0	0
$707$	0.242641	0.00912544
$708$	0	0
$709$	43.5980	1.63736	0.818678	−	0.574252i	$-0.194707\pi$
$709$	43.5980	1.63736	0.818678	+	0.574252i	$0.194707\pi$
$710$	0	0
$711$	−9.17157	−0.343961
$712$	0	0
$713$	17.6569	0.661254
$714$	0	0
$715$	−8.00000	−0.299183
$716$	0	0
$717$	13.7990	0.515333
$718$	0	0
$719$	−13.4142	−0.500266	−0.250133	−	0.968212i	$-0.580474\pi$
$719$	−13.4142	−0.500266	−0.250133	+	0.968212i	$0.580474\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	−12.3431	−0.459047
$724$	0	0
$725$	−1.75736	−0.0652667
$726$	0	0
$727$	33.4558	1.24081	0.620404	−	0.784282i	$-0.286969\pi$
$727$	33.4558	1.24081	0.620404	+	0.784282i	$0.286969\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.48528	−0.313839
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	−1.41421	−0.0521641
$736$	0	0
$737$	−8.97056	−0.330435
$738$	0	0
$739$	−23.6569	−0.870231	−0.435116	−	0.900375i	$-0.643293\pi$
$739$	−23.6569	−0.870231	−0.435116	+	0.900375i	$0.643293\pi$
$740$	0	0
$741$	2.82843	0.103905
$742$	0	0
$743$	−30.2426	−1.10949	−0.554747	−	0.832019i	$-0.687185\pi$
$743$	−30.2426	−1.10949	−0.554747	+	0.832019i	$0.687185\pi$
$744$	0	0
$745$	−31.7990	−1.16502
$746$	0	0
$747$	−1.41421	−0.0517434
$748$	0	0
$749$	18.2426	0.666572
$750$	0	0
$751$	−22.6274	−0.825686	−0.412843	−	0.910802i	$-0.635464\pi$
$751$	−22.6274	−0.825686	−0.412843	+	0.910802i	$0.635464\pi$
$752$	0	0
$753$	16.2426	0.591915
$754$	0	0
$755$	4.68629	0.170552
$756$	0	0
$757$	26.9706	0.980262	0.490131	−	0.871649i	$-0.336949\pi$
$757$	26.9706	0.980262	0.490131	+	0.871649i	$0.336949\pi$
$758$	0	0
$759$	−7.31371	−0.265471
$760$	0	0
$761$	2.58579	0.0937347	0.0468673	−	0.998901i	$-0.485076\pi$
$761$	2.58579	0.0937347	0.0468673	+	0.998901i	$0.485076\pi$
$762$	0	0
$763$	−14.4853	−0.524402
$764$	0	0
$765$	−6.00000	−0.216930
$766$	0	0
$767$	38.6274	1.39476
$768$	0	0
$769$	13.1127	0.472856	0.236428	−	0.971649i	$-0.424023\pi$
$769$	13.1127	0.472856	0.236428	+	0.971649i	$0.424023\pi$
$770$	0	0
$771$	8.48528	0.305590
$772$	0	0
$773$	10.6274	0.382242	0.191121	−	0.981567i	$-0.438788\pi$
$773$	10.6274	0.382242	0.191121	+	0.981567i	$0.438788\pi$
$774$	0	0
$775$	−14.4853	−0.520327
$776$	0	0
$777$	−10.4853	−0.376157
$778$	0	0
$779$	1.65685	0.0593630
$780$	0	0
$781$	30.8284	1.10313
$782$	0	0
$783$	0.585786	0.0209343
$784$	0	0
$785$	31.7990	1.13495
$786$	0	0
$787$	20.1421	0.717990	0.358995	−	0.933340i	$-0.383120\pi$
$787$	20.1421	0.717990	0.358995	+	0.933340i	$0.383120\pi$
$788$	0	0
$789$	11.1716	0.397719
$790$	0	0
$791$	6.72792	0.239217
$792$	0	0
$793$	−2.34315	−0.0832075
$794$	0	0
$795$	−16.8284	−0.596843
$796$	0	0
$797$	33.9411	1.20226	0.601128	−	0.799153i	$-0.294718\pi$
$797$	33.9411	1.20226	0.601128	+	0.799153i	$0.294718\pi$
$798$	0	0
$799$	−30.0000	−1.06132
$800$	0	0
$801$	−4.00000	−0.141333
$802$	0	0
$803$	23.3137	0.822723
$804$	0	0
$805$	5.17157	0.182274
$806$	0	0
$807$	24.4853	0.861923
$808$	0	0
$809$	−24.3431	−0.855859	−0.427930	−	0.903812i	$-0.640757\pi$
$809$	−24.3431	−0.855859	−0.427930	+	0.903812i	$0.640757\pi$
$810$	0	0
$811$	−10.6274	−0.373179	−0.186590	−	0.982438i	$-0.559743\pi$
$811$	−10.6274	−0.373179	−0.186590	+	0.982438i	$0.559743\pi$
$812$	0	0
$813$	−15.7990	−0.554095
$814$	0	0
$815$	6.14214	0.215150
$816$	0	0
$817$	2.00000	0.0699711
$818$	0	0
$819$	2.82843	0.0988332
$820$	0	0
$821$	0.544156	0.0189912	0.00949559	−	0.999955i	$-0.496977\pi$
$821$	0.544156	0.0189912	0.00949559	+	0.999955i	$0.496977\pi$
$822$	0	0
$823$	16.3431	0.569686	0.284843	−	0.958574i	$-0.408058\pi$
$823$	16.3431	0.569686	0.284843	+	0.958574i	$0.408058\pi$
$824$	0	0
$825$	6.00000	0.208893
$826$	0	0
$827$	10.2426	0.356172	0.178086	−	0.984015i	$-0.443010\pi$
$827$	10.2426	0.356172	0.178086	+	0.984015i	$0.443010\pi$
$828$	0	0
$829$	−11.9411	−0.414732	−0.207366	−	0.978263i	$-0.566489\pi$
$829$	−11.9411	−0.414732	−0.207366	+	0.978263i	$0.566489\pi$
$830$	0	0
$831$	−28.9706	−1.00498
$832$	0	0
$833$	4.24264	0.146999
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	4.82843	0.166895
$838$	0	0
$839$	20.9706	0.723984	0.361992	−	0.932181i	$-0.382097\pi$
$839$	20.9706	0.723984	0.361992	+	0.932181i	$0.382097\pi$
$840$	0	0
$841$	−28.6569	−0.988167
$842$	0	0
$843$	22.0416	0.759154
$844$	0	0
$845$	7.07107	0.243252
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	−31.1127	−1.06779
$850$	0	0
$851$	38.3431	1.31439
$852$	0	0
$853$	−26.2843	−0.899956	−0.449978	−	0.893040i	$-0.648568\pi$
$853$	−26.2843	−0.899956	−0.449978	+	0.893040i	$0.648568\pi$
$854$	0	0
$855$	1.41421	0.0483651
$856$	0	0
$857$	−11.3137	−0.386469	−0.193234	−	0.981153i	$-0.561898\pi$
$857$	−11.3137	−0.386469	−0.193234	+	0.981153i	$0.561898\pi$
$858$	0	0
$859$	−50.9117	−1.73708	−0.868542	−	0.495615i	$-0.834943\pi$
$859$	−50.9117	−1.73708	−0.868542	+	0.495615i	$0.834943\pi$
$860$	0	0
$861$	1.65685	0.0564654
$862$	0	0
$863$	30.2426	1.02947	0.514736	−	0.857349i	$-0.327890\pi$
$863$	30.2426	1.02947	0.514736	+	0.857349i	$0.327890\pi$
$864$	0	0
$865$	2.34315	0.0796693
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	18.3431	0.622249
$870$	0	0
$871$	−12.6863	−0.429859
$872$	0	0
$873$	−3.65685	−0.123766
$874$	0	0
$875$	−11.3137	−0.382473
$876$	0	0
$877$	−42.9706	−1.45101	−0.725506	−	0.688215i	$-0.758394\pi$
$877$	−42.9706	−1.45101	−0.725506	+	0.688215i	$0.758394\pi$
$878$	0	0
$879$	−8.00000	−0.269833
$880$	0	0
$881$	−22.1838	−0.747390	−0.373695	−	0.927552i	$-0.621909\pi$
$881$	−22.1838	−0.747390	−0.373695	+	0.927552i	$0.621909\pi$
$882$	0	0
$883$	−0.284271	−0.00956649	−0.00478324	−	0.999989i	$-0.501523\pi$
$883$	−0.284271	−0.00956649	−0.00478324	+	0.999989i	$0.501523\pi$
$884$	0	0
$885$	19.3137	0.649223
$886$	0	0
$887$	5.17157	0.173644	0.0868222	−	0.996224i	$-0.472329\pi$
$887$	5.17157	0.173644	0.0868222	+	0.996224i	$0.472329\pi$
$888$	0	0
$889$	19.7990	0.664037
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	7.07107	0.236624
$894$	0	0
$895$	−3.17157	−0.106014
$896$	0	0
$897$	−10.3431	−0.345348
$898$	0	0
$899$	2.82843	0.0943333
$900$	0	0
$901$	50.4853	1.68191
$902$	0	0
$903$	2.00000	0.0665558
$904$	0	0
$905$	8.48528	0.282060
$906$	0	0
$907$	−49.6569	−1.64883	−0.824414	−	0.565987i	$-0.808495\pi$
$907$	−49.6569	−1.64883	−0.824414	+	0.565987i	$0.808495\pi$
$908$	0	0
$909$	−0.242641	−0.00804788
$910$	0	0
$911$	10.4437	0.346014	0.173007	−	0.984921i	$-0.444652\pi$
$911$	10.4437	0.346014	0.173007	+	0.984921i	$0.444652\pi$
$912$	0	0
$913$	2.82843	0.0936073
$914$	0	0
$915$	−1.17157	−0.0387310
$916$	0	0
$917$	0.928932	0.0306760
$918$	0	0
$919$	14.6274	0.482514	0.241257	−	0.970461i	$-0.422440\pi$
$919$	14.6274	0.482514	0.241257	+	0.970461i	$0.422440\pi$
$920$	0	0
$921$	15.1716	0.499920
$922$	0	0
$923$	43.5980	1.43504
$924$	0	0
$925$	−31.4558	−1.03426
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	5.89949	0.193556	0.0967781	−	0.995306i	$-0.469146\pi$
$929$	5.89949	0.193556	0.0967781	+	0.995306i	$0.469146\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−21.2132	−0.694489
$934$	0	0
$935$	12.0000	0.392442
$936$	0	0
$937$	−58.2843	−1.90406	−0.952032	−	0.305998i	$-0.901010\pi$
$937$	−58.2843	−1.90406	−0.952032	+	0.305998i	$0.901010\pi$
$938$	0	0
$939$	−24.1421	−0.787849
$940$	0	0
$941$	−2.14214	−0.0698316	−0.0349158	−	0.999390i	$-0.511116\pi$
$941$	−2.14214	−0.0698316	−0.0349158	+	0.999390i	$0.511116\pi$
$942$	0	0
$943$	−6.05887	−0.197304
$944$	0	0
$945$	1.41421	0.0460044
$946$	0	0
$947$	−20.6274	−0.670301	−0.335150	−	0.942165i	$-0.608787\pi$
$947$	−20.6274	−0.670301	−0.335150	+	0.942165i	$0.608787\pi$
$948$	0	0
$949$	32.9706	1.07027
$950$	0	0
$951$	−22.7279	−0.737003
$952$	0	0
$953$	−41.3553	−1.33963	−0.669815	−	0.742528i	$-0.733627\pi$
$953$	−41.3553	−1.33963	−0.669815	+	0.742528i	$0.733627\pi$
$954$	0	0
$955$	10.8284	0.350400
$956$	0	0
$957$	−1.17157	−0.0378716
$958$	0	0
$959$	11.6569	0.376419
$960$	0	0
$961$	−7.68629	−0.247945
$962$	0	0
$963$	−18.2426	−0.587861
$964$	0	0
$965$	31.7990	1.02365
$966$	0	0
$967$	40.6274	1.30649	0.653245	−	0.757147i	$-0.273407\pi$
$967$	40.6274	1.30649	0.653245	+	0.757147i	$0.273407\pi$
$968$	0	0
$969$	−4.24264	−0.136293
$970$	0	0
$971$	−19.5980	−0.628929	−0.314465	−	0.949269i	$-0.601825\pi$
$971$	−19.5980	−0.628929	−0.314465	+	0.949269i	$0.601825\pi$
$972$	0	0
$973$	3.31371	0.106233
$974$	0	0
$975$	8.48528	0.271746
$976$	0	0
$977$	2.24264	0.0717484	0.0358742	−	0.999356i	$-0.488578\pi$
$977$	2.24264	0.0717484	0.0358742	+	0.999356i	$0.488578\pi$
$978$	0	0
$979$	8.00000	0.255681
$980$	0	0
$981$	14.4853	0.462479
$982$	0	0
$983$	−20.0000	−0.637901	−0.318950	−	0.947771i	$-0.603330\pi$
$983$	−20.0000	−0.637901	−0.318950	+	0.947771i	$0.603330\pi$
$984$	0	0
$985$	−6.14214	−0.195705
$986$	0	0
$987$	7.07107	0.225075
$988$	0	0
$989$	−7.31371	−0.232562
$990$	0	0
$991$	−23.5980	−0.749615	−0.374807	−	0.927103i	$-0.622291\pi$
$991$	−23.5980	−0.749615	−0.374807	+	0.927103i	$0.622291\pi$
$992$	0	0
$993$	3.31371	0.105157
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	13.7990	0.437018	0.218509	−	0.975835i	$-0.429881\pi$
$997$	13.7990	0.437018	0.218509	+	0.975835i	$0.429881\pi$
$998$	0	0
$999$	10.4853	0.331740

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6384.2.a.bp.1.1		2
4.3	odd	2		1596.2.a.h.1.1	✓	2
12.11	even	2		4788.2.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1596.2.a.h.1.1	✓	2	4.3	odd	2
4788.2.a.h.1.2		2	12.11	even	2
6384.2.a.bp.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.41421 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.41421 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -2.82843 q^{13} -1.41421 q^{15} +4.24264 q^{17} -1.00000 q^{19} -1.00000 q^{21} +3.65685 q^{23} -3.00000 q^{25} +1.00000 q^{27} +0.585786 q^{29} +4.82843 q^{31} -2.00000 q^{33} +1.41421 q^{35} +10.4853 q^{37} -2.82843 q^{39} -1.65685 q^{41} -2.00000 q^{43} -1.41421 q^{45} -7.07107 q^{47} +1.00000 q^{49} +4.24264 q^{51} +11.8995 q^{53} +2.82843 q^{55} -1.00000 q^{57} -13.6569 q^{59} +0.828427 q^{61} -1.00000 q^{63} +4.00000 q^{65} +4.48528 q^{67} +3.65685 q^{69} -15.4142 q^{71} -11.6569 q^{73} -3.00000 q^{75} +2.00000 q^{77} -9.17157 q^{79} +1.00000 q^{81} -1.41421 q^{83} -6.00000 q^{85} +0.585786 q^{87} -4.00000 q^{89} +2.82843 q^{91} +4.82843 q^{93} +1.41421 q^{95} -3.65685 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{19} - 2 q^{21} - 4 q^{23} - 6 q^{25} + 2 q^{27} + 4 q^{29} + 4 q^{31} - 4 q^{33} + 4 q^{37} + 8 q^{41} - 4 q^{43} + 2 q^{49} + 4 q^{53} - 2 q^{57} - 16 q^{59} - 4 q^{61} - 2 q^{63} + 8 q^{65} - 8 q^{67} - 4 q^{69} - 28 q^{71} - 12 q^{73} - 6 q^{75} + 4 q^{77} - 24 q^{79} + 2 q^{81} - 12 q^{85} + 4 q^{87} - 8 q^{89} + 4 q^{93} + 4 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^19 - 2 * q^21 - 4 * q^23 - 6 * q^25 + 2 * q^27 + 4 * q^29 + 4 * q^31 - 4 * q^33 + 4 * q^37 + 8 * q^41 - 4 * q^43 + 2 * q^49 + 4 * q^53 - 2 * q^57 - 16 * q^59 - 4 * q^61 - 2 * q^63 + 8 * q^65 - 8 * q^67 - 4 * q^69 - 28 * q^71 - 12 * q^73 - 6 * q^75 + 4 * q^77 - 24 * q^79 + 2 * q^81 - 12 * q^85 + 4 * q^87 - 8 * q^89 + 4 * q^93 + 4 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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