LMFDB - Embedded newform 6864.2.a.bd.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, 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("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.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:	$54.8093159474$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 858)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.70156 q^{5} -4.70156 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.70156 q^{5} -4.70156 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.70156 q^{15} +4.00000 q^{17} -6.00000 q^{19} +4.70156 q^{21} +6.70156 q^{23} +2.29844 q^{25} -1.00000 q^{27} -6.70156 q^{29} +7.40312 q^{31} -1.00000 q^{33} -12.7016 q^{35} -5.40312 q^{37} +1.00000 q^{39} -6.70156 q^{41} +10.1047 q^{43} +2.70156 q^{45} -4.00000 q^{47} +15.1047 q^{49} -4.00000 q^{51} -5.40312 q^{53} +2.70156 q^{55} +6.00000 q^{57} +7.29844 q^{59} +10.7016 q^{61} -4.70156 q^{63} -2.70156 q^{65} -14.1047 q^{67} -6.70156 q^{69} -1.40312 q^{71} +4.70156 q^{73} -2.29844 q^{75} -4.70156 q^{77} -9.40312 q^{79} +1.00000 q^{81} -2.59688 q^{83} +10.8062 q^{85} +6.70156 q^{87} -11.4031 q^{89} +4.70156 q^{91} -7.40312 q^{93} -16.2094 q^{95} -16.8062 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} + q^{15} + 8 q^{17} - 12 q^{19} + 3 q^{21} + 7 q^{23} + 11 q^{25} - 2 q^{27} - 7 q^{29} + 2 q^{31} - 2 q^{33} - 19 q^{35} + 2 q^{37} + 2 q^{39} - 7 q^{41} + q^{43} - q^{45} - 8 q^{47} + 11 q^{49} - 8 q^{51} + 2 q^{53} - q^{55} + 12 q^{57} + 21 q^{59} + 15 q^{61} - 3 q^{63} + q^{65} - 9 q^{67} - 7 q^{69} + 10 q^{71} + 3 q^{73} - 11 q^{75} - 3 q^{77} - 6 q^{79} + 2 q^{81} - 18 q^{83} - 4 q^{85} + 7 q^{87} - 10 q^{89} + 3 q^{91} - 2 q^{93} + 6 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + q^15 + 8 * q^17 - 12 * q^19 + 3 * q^21 + 7 * q^23 + 11 * q^25 - 2 * q^27 - 7 * q^29 + 2 * q^31 - 2 * q^33 - 19 * q^35 + 2 * q^37 + 2 * q^39 - 7 * q^41 + q^43 - q^45 - 8 * q^47 + 11 * q^49 - 8 * q^51 + 2 * q^53 - q^55 + 12 * q^57 + 21 * q^59 + 15 * q^61 - 3 * q^63 + q^65 - 9 * q^67 - 7 * q^69 + 10 * q^71 + 3 * q^73 - 11 * q^75 - 3 * q^77 - 6 * q^79 + 2 * q^81 - 18 * q^83 - 4 * q^85 + 7 * q^87 - 10 * q^89 + 3 * q^91 - 2 * q^93 + 6 * q^95 - 8 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.70156	1.20818	0.604088	−	0.796918i	$-0.293538\pi$
$5$	2.70156	1.20818	0.604088	+	0.796918i	$0.293538\pi$
$6$	0	0
$7$	−4.70156	−1.77702	−0.888512	−	0.458854i	$-0.848260\pi$
$7$	−4.70156	−1.77702	−0.888512	+	0.458854i	$0.848260\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−2.70156	−0.697540
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	4.70156	1.02596
$22$	0	0
$23$	6.70156	1.39737	0.698686	−	0.715428i	$-0.253768\pi$
$23$	6.70156	1.39737	0.698686	+	0.715428i	$0.253768\pi$
$24$	0	0
$25$	2.29844	0.459688
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.70156	−1.24445	−0.622224	−	0.782839i	$-0.713771\pi$
$29$	−6.70156	−1.24445	−0.622224	+	0.782839i	$0.713771\pi$
$30$	0	0
$31$	7.40312	1.32964	0.664820	−	0.747003i	$-0.268508\pi$
$31$	7.40312	1.32964	0.664820	+	0.747003i	$0.268508\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−12.7016	−2.14696
$36$	0	0
$37$	−5.40312	−0.888268	−0.444134	−	0.895960i	$-0.646489\pi$
$37$	−5.40312	−0.888268	−0.444134	+	0.895960i	$0.646489\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−6.70156	−1.04661	−0.523304	−	0.852146i	$-0.675301\pi$
$41$	−6.70156	−1.04661	−0.523304	+	0.852146i	$0.675301\pi$
$42$	0	0
$43$	10.1047	1.54095	0.770475	−	0.637470i	$-0.220019\pi$
$43$	10.1047	1.54095	0.770475	+	0.637470i	$0.220019\pi$
$44$	0	0
$45$	2.70156	0.402725
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	15.1047	2.15781
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−5.40312	−0.742176	−0.371088	−	0.928598i	$-0.621015\pi$
$53$	−5.40312	−0.742176	−0.371088	+	0.928598i	$0.621015\pi$
$54$	0	0
$55$	2.70156	0.364279
$56$	0	0
$57$	6.00000	0.794719
$58$	0	0
$59$	7.29844	0.950176	0.475088	−	0.879938i	$-0.342416\pi$
$59$	7.29844	0.950176	0.475088	+	0.879938i	$0.342416\pi$
$60$	0	0
$61$	10.7016	1.37019	0.685097	−	0.728452i	$-0.259760\pi$
$61$	10.7016	1.37019	0.685097	+	0.728452i	$0.259760\pi$
$62$	0	0
$63$	−4.70156	−0.592341
$64$	0	0
$65$	−2.70156	−0.335088
$66$	0	0
$67$	−14.1047	−1.72316	−0.861581	−	0.507620i	$-0.830525\pi$
$67$	−14.1047	−1.72316	−0.861581	+	0.507620i	$0.830525\pi$
$68$	0	0
$69$	−6.70156	−0.806773
$70$	0	0
$71$	−1.40312	−0.166520	−0.0832601	−	0.996528i	$-0.526533\pi$
$71$	−1.40312	−0.166520	−0.0832601	+	0.996528i	$0.526533\pi$
$72$	0	0
$73$	4.70156	0.550276	0.275138	−	0.961405i	$-0.411276\pi$
$73$	4.70156	0.550276	0.275138	+	0.961405i	$0.411276\pi$
$74$	0	0
$75$	−2.29844	−0.265401
$76$	0	0
$77$	−4.70156	−0.535793
$78$	0	0
$79$	−9.40312	−1.05793	−0.528967	−	0.848642i	$-0.677421\pi$
$79$	−9.40312	−1.05793	−0.528967	+	0.848642i	$0.677421\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.59688	−0.285044	−0.142522	−	0.989792i	$-0.545521\pi$
$83$	−2.59688	−0.285044	−0.142522	+	0.989792i	$0.545521\pi$
$84$	0	0
$85$	10.8062	1.17210
$86$	0	0
$87$	6.70156	0.718483
$88$	0	0
$89$	−11.4031	−1.20873	−0.604364	−	0.796708i	$-0.706573\pi$
$89$	−11.4031	−1.20873	−0.604364	+	0.796708i	$0.706573\pi$
$90$	0	0
$91$	4.70156	0.492858
$92$	0	0
$93$	−7.40312	−0.767668
$94$	0	0
$95$	−16.2094	−1.66305
$96$	0	0
$97$	−16.8062	−1.70642	−0.853208	−	0.521571i	$-0.825346\pi$
$97$	−16.8062	−1.70642	−0.853208	+	0.521571i	$0.825346\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	16.7016	1.64565	0.822827	−	0.568292i	$-0.192396\pi$
$103$	16.7016	1.64565	0.822827	+	0.568292i	$0.192396\pi$
$104$	0	0
$105$	12.7016	1.23955
$106$	0	0
$107$	−5.29844	−0.512219	−0.256110	−	0.966648i	$-0.582441\pi$
$107$	−5.29844	−0.512219	−0.256110	+	0.966648i	$0.582441\pi$
$108$	0	0
$109$	11.4031	1.09222	0.546111	−	0.837713i	$-0.316108\pi$
$109$	11.4031	1.09222	0.546111	+	0.837713i	$0.316108\pi$
$110$	0	0
$111$	5.40312	0.512842
$112$	0	0
$113$	−0.104686	−0.00984806	−0.00492403	−	0.999988i	$-0.501567\pi$
$113$	−0.104686	−0.00984806	−0.00492403	+	0.999988i	$0.501567\pi$
$114$	0	0
$115$	18.1047	1.68827
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−18.8062	−1.72397
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.70156	0.604260
$124$	0	0
$125$	−7.29844	−0.652792
$126$	0	0
$127$	−9.40312	−0.834392	−0.417196	−	0.908816i	$-0.636987\pi$
$127$	−9.40312	−0.834392	−0.417196	+	0.908816i	$0.636987\pi$
$128$	0	0
$129$	−10.1047	−0.889668
$130$	0	0
$131$	0.104686	0.00914649	0.00457324	−	0.999990i	$-0.498544\pi$
$131$	0.104686	0.00914649	0.00457324	+	0.999990i	$0.498544\pi$
$132$	0	0
$133$	28.2094	2.44606
$134$	0	0
$135$	−2.70156	−0.232513
$136$	0	0
$137$	12.8062	1.09411	0.547056	−	0.837096i	$-0.315749\pi$
$137$	12.8062	1.09411	0.547056	+	0.837096i	$0.315749\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−18.1047	−1.50351
$146$	0	0
$147$	−15.1047	−1.24581
$148$	0	0
$149$	−22.2094	−1.81946	−0.909731	−	0.415197i	$-0.863713\pi$
$149$	−22.2094	−1.81946	−0.909731	+	0.415197i	$0.863713\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	20.0000	1.60644
$156$	0	0
$157$	12.8062	1.02205	0.511025	−	0.859566i	$-0.329266\pi$
$157$	12.8062	1.02205	0.511025	+	0.859566i	$0.329266\pi$
$158$	0	0
$159$	5.40312	0.428496
$160$	0	0
$161$	−31.5078	−2.48316
$162$	0	0
$163$	2.10469	0.164852	0.0824259	−	0.996597i	$-0.473733\pi$
$163$	2.10469	0.164852	0.0824259	+	0.996597i	$0.473733\pi$
$164$	0	0
$165$	−2.70156	−0.210316
$166$	0	0
$167$	−4.70156	−0.363818	−0.181909	−	0.983315i	$-0.558228\pi$
$167$	−4.70156	−0.363818	−0.181909	+	0.983315i	$0.558228\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.00000	−0.458831
$172$	0	0
$173$	9.50781	0.722865	0.361433	−	0.932398i	$-0.382288\pi$
$173$	9.50781	0.722865	0.361433	+	0.932398i	$0.382288\pi$
$174$	0	0
$175$	−10.8062	−0.816876
$176$	0	0
$177$	−7.29844	−0.548584
$178$	0	0
$179$	−14.8062	−1.10667	−0.553335	−	0.832958i	$-0.686645\pi$
$179$	−14.8062	−1.10667	−0.553335	+	0.832958i	$0.686645\pi$
$180$	0	0
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	0	0
$183$	−10.7016	−0.791082
$184$	0	0
$185$	−14.5969	−1.07318
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	4.70156	0.341988
$190$	0	0
$191$	−21.5078	−1.55625	−0.778125	−	0.628109i	$-0.783829\pi$
$191$	−21.5078	−1.55625	−0.778125	+	0.628109i	$0.783829\pi$
$192$	0	0
$193$	−21.4031	−1.54063	−0.770315	−	0.637663i	$-0.779901\pi$
$193$	−21.4031	−1.54063	−0.770315	+	0.637663i	$0.779901\pi$
$194$	0	0
$195$	2.70156	0.193463
$196$	0	0
$197$	−19.4031	−1.38242	−0.691208	−	0.722656i	$-0.742921\pi$
$197$	−19.4031	−1.38242	−0.691208	+	0.722656i	$0.742921\pi$
$198$	0	0
$199$	22.1047	1.56696	0.783480	−	0.621417i	$-0.213443\pi$
$199$	22.1047	1.56696	0.783480	+	0.621417i	$0.213443\pi$
$200$	0	0
$201$	14.1047	0.994868
$202$	0	0
$203$	31.5078	2.21141
$204$	0	0
$205$	−18.1047	−1.26449
$206$	0	0
$207$	6.70156	0.465791
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	−18.8062	−1.29468	−0.647338	−	0.762203i	$-0.724118\pi$
$211$	−18.8062	−1.29468	−0.647338	+	0.762203i	$0.724118\pi$
$212$	0	0
$213$	1.40312	0.0961405
$214$	0	0
$215$	27.2984	1.86174
$216$	0	0
$217$	−34.8062	−2.36280
$218$	0	0
$219$	−4.70156	−0.317702
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−8.59688	−0.575689	−0.287845	−	0.957677i	$-0.592939\pi$
$223$	−8.59688	−0.575689	−0.287845	+	0.957677i	$0.592939\pi$
$224$	0	0
$225$	2.29844	0.153229
$226$	0	0
$227$	2.80625	0.186257	0.0931286	−	0.995654i	$-0.470313\pi$
$227$	2.80625	0.186257	0.0931286	+	0.995654i	$0.470313\pi$
$228$	0	0
$229$	0.701562	0.0463605	0.0231803	−	0.999731i	$-0.492621\pi$
$229$	0.701562	0.0463605	0.0231803	+	0.999731i	$0.492621\pi$
$230$	0	0
$231$	4.70156	0.309340
$232$	0	0
$233$	6.59688	0.432176	0.216088	−	0.976374i	$-0.430670\pi$
$233$	6.59688	0.432176	0.216088	+	0.976374i	$0.430670\pi$
$234$	0	0
$235$	−10.8062	−0.704922
$236$	0	0
$237$	9.40312	0.610799
$238$	0	0
$239$	4.70156	0.304119	0.152059	−	0.988371i	$-0.451410\pi$
$239$	4.70156	0.304119	0.152059	+	0.988371i	$0.451410\pi$
$240$	0	0
$241$	−14.5969	−0.940267	−0.470134	−	0.882595i	$-0.655794\pi$
$241$	−14.5969	−0.940267	−0.470134	+	0.882595i	$0.655794\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	40.8062	2.60702
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	2.59688	0.164570
$250$	0	0
$251$	−5.40312	−0.341042	−0.170521	−	0.985354i	$-0.554545\pi$
$251$	−5.40312	−0.341042	−0.170521	+	0.985354i	$0.554545\pi$
$252$	0	0
$253$	6.70156	0.421324
$254$	0	0
$255$	−10.8062	−0.676714
$256$	0	0
$257$	−2.70156	−0.168519	−0.0842594	−	0.996444i	$-0.526852\pi$
$257$	−2.70156	−0.168519	−0.0842594	+	0.996444i	$0.526852\pi$
$258$	0	0
$259$	25.4031	1.57847
$260$	0	0
$261$	−6.70156	−0.414816
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−14.5969	−0.896679
$266$	0	0
$267$	11.4031	0.697860
$268$	0	0
$269$	5.19375	0.316669	0.158334	−	0.987386i	$-0.449388\pi$
$269$	5.19375	0.316669	0.158334	+	0.987386i	$0.449388\pi$
$270$	0	0
$271$	−1.19375	−0.0725152	−0.0362576	−	0.999342i	$-0.511544\pi$
$271$	−1.19375	−0.0725152	−0.0362576	+	0.999342i	$0.511544\pi$
$272$	0	0
$273$	−4.70156	−0.284551
$274$	0	0
$275$	2.29844	0.138601
$276$	0	0
$277$	−4.10469	−0.246627	−0.123313	−	0.992368i	$-0.539352\pi$
$277$	−4.10469	−0.246627	−0.123313	+	0.992368i	$0.539352\pi$
$278$	0	0
$279$	7.40312	0.443213
$280$	0	0
$281$	−13.2984	−0.793318	−0.396659	−	0.917966i	$-0.629831\pi$
$281$	−13.2984	−0.793318	−0.396659	+	0.917966i	$0.629831\pi$
$282$	0	0
$283$	−0.701562	−0.0417035	−0.0208518	−	0.999783i	$-0.506638\pi$
$283$	−0.701562	−0.0417035	−0.0208518	+	0.999783i	$0.506638\pi$
$284$	0	0
$285$	16.2094	0.960160
$286$	0	0
$287$	31.5078	1.85985
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	16.8062	0.985200
$292$	0	0
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	0	0
$295$	19.7172	1.14798
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−6.70156	−0.387561
$300$	0	0
$301$	−47.5078	−2.73830
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	28.9109	1.65544
$306$	0	0
$307$	−10.0000	−0.570730	−0.285365	−	0.958419i	$-0.592115\pi$
$307$	−10.0000	−0.570730	−0.285365	+	0.958419i	$0.592115\pi$
$308$	0	0
$309$	−16.7016	−0.950119
$310$	0	0
$311$	12.5969	0.714303	0.357152	−	0.934046i	$-0.383748\pi$
$311$	12.5969	0.714303	0.357152	+	0.934046i	$0.383748\pi$
$312$	0	0
$313$	14.9109	0.842816	0.421408	−	0.906871i	$-0.361536\pi$
$313$	14.9109	0.842816	0.421408	+	0.906871i	$0.361536\pi$
$314$	0	0
$315$	−12.7016	−0.715652
$316$	0	0
$317$	30.9109	1.73613	0.868066	−	0.496450i	$-0.165363\pi$
$317$	30.9109	1.73613	0.868066	+	0.496450i	$0.165363\pi$
$318$	0	0
$319$	−6.70156	−0.375215
$320$	0	0
$321$	5.29844	0.295730
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	−2.29844	−0.127494
$326$	0	0
$327$	−11.4031	−0.630594
$328$	0	0
$329$	18.8062	1.03682
$330$	0	0
$331$	7.50781	0.412667	0.206333	−	0.978482i	$-0.433847\pi$
$331$	7.50781	0.412667	0.206333	+	0.978482i	$0.433847\pi$
$332$	0	0
$333$	−5.40312	−0.296089
$334$	0	0
$335$	−38.1047	−2.08188
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	0.104686	0.00568578
$340$	0	0
$341$	7.40312	0.400902
$342$	0	0
$343$	−38.1047	−2.05746
$344$	0	0
$345$	−18.1047	−0.974724
$346$	0	0
$347$	−0.596876	−0.0320420	−0.0160210	−	0.999872i	$-0.505100\pi$
$347$	−0.596876	−0.0320420	−0.0160210	+	0.999872i	$0.505100\pi$
$348$	0	0
$349$	−30.2094	−1.61707	−0.808535	−	0.588448i	$-0.799739\pi$
$349$	−30.2094	−1.61707	−0.808535	+	0.588448i	$0.799739\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	−3.79063	−0.201186
$356$	0	0
$357$	18.8062	0.995332
$358$	0	0
$359$	−20.7016	−1.09259	−0.546293	−	0.837594i	$-0.683962\pi$
$359$	−20.7016	−1.09259	−0.546293	+	0.837594i	$0.683962\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	12.7016	0.664830
$366$	0	0
$367$	22.8062	1.19048	0.595238	−	0.803549i	$-0.297058\pi$
$367$	22.8062	1.19048	0.595238	+	0.803549i	$0.297058\pi$
$368$	0	0
$369$	−6.70156	−0.348869
$370$	0	0
$371$	25.4031	1.31886
$372$	0	0
$373$	24.3141	1.25893	0.629467	−	0.777027i	$-0.283273\pi$
$373$	24.3141	1.25893	0.629467	+	0.777027i	$0.283273\pi$
$374$	0	0
$375$	7.29844	0.376890
$376$	0	0
$377$	6.70156	0.345148
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	9.40312	0.481737
$382$	0	0
$383$	−21.4031	−1.09365	−0.546824	−	0.837247i	$-0.684163\pi$
$383$	−21.4031	−1.09365	−0.546824	+	0.837247i	$0.684163\pi$
$384$	0	0
$385$	−12.7016	−0.647332
$386$	0	0
$387$	10.1047	0.513650
$388$	0	0
$389$	5.40312	0.273949	0.136975	−	0.990575i	$-0.456262\pi$
$389$	5.40312	0.273949	0.136975	+	0.990575i	$0.456262\pi$
$390$	0	0
$391$	26.8062	1.35565
$392$	0	0
$393$	−0.104686	−0.00528073
$394$	0	0
$395$	−25.4031	−1.27817
$396$	0	0
$397$	−32.9109	−1.65175	−0.825876	−	0.563851i	$-0.809319\pi$
$397$	−32.9109	−1.65175	−0.825876	+	0.563851i	$0.809319\pi$
$398$	0	0
$399$	−28.2094	−1.41224
$400$	0	0
$401$	−26.2094	−1.30883	−0.654417	−	0.756134i	$-0.727086\pi$
$401$	−26.2094	−1.30883	−0.654417	+	0.756134i	$0.727086\pi$
$402$	0	0
$403$	−7.40312	−0.368776
$404$	0	0
$405$	2.70156	0.134242
$406$	0	0
$407$	−5.40312	−0.267823
$408$	0	0
$409$	−27.5078	−1.36017	−0.680087	−	0.733132i	$-0.738058\pi$
$409$	−27.5078	−1.36017	−0.680087	+	0.733132i	$0.738058\pi$
$410$	0	0
$411$	−12.8062	−0.631686
$412$	0	0
$413$	−34.3141	−1.68848
$414$	0	0
$415$	−7.01562	−0.344383
$416$	0	0
$417$	16.0000	0.783523
$418$	0	0
$419$	−14.8062	−0.723333	−0.361666	−	0.932308i	$-0.617792\pi$
$419$	−14.8062	−0.723333	−0.361666	+	0.932308i	$0.617792\pi$
$420$	0	0
$421$	3.50781	0.170960	0.0854801	−	0.996340i	$-0.472758\pi$
$421$	3.50781	0.170960	0.0854801	+	0.996340i	$0.472758\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	9.19375	0.445962
$426$	0	0
$427$	−50.3141	−2.43487
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	21.6125	1.04104	0.520519	−	0.853850i	$-0.325739\pi$
$431$	21.6125	1.04104	0.520519	+	0.853850i	$0.325739\pi$
$432$	0	0
$433$	7.89531	0.379425	0.189712	−	0.981840i	$-0.439244\pi$
$433$	7.89531	0.379425	0.189712	+	0.981840i	$0.439244\pi$
$434$	0	0
$435$	18.1047	0.868053
$436$	0	0
$437$	−40.2094	−1.92348
$438$	0	0
$439$	10.8062	0.515754	0.257877	−	0.966178i	$-0.416977\pi$
$439$	10.8062	0.515754	0.257877	+	0.966178i	$0.416977\pi$
$440$	0	0
$441$	15.1047	0.719271
$442$	0	0
$443$	−5.19375	−0.246763	−0.123381	−	0.992359i	$-0.539374\pi$
$443$	−5.19375	−0.246763	−0.123381	+	0.992359i	$0.539374\pi$
$444$	0	0
$445$	−30.8062	−1.46036
$446$	0	0
$447$	22.2094	1.05047
$448$	0	0
$449$	33.0156	1.55810	0.779052	−	0.626960i	$-0.215701\pi$
$449$	33.0156	1.55810	0.779052	+	0.626960i	$0.215701\pi$
$450$	0	0
$451$	−6.70156	−0.315564
$452$	0	0
$453$	0	0
$454$	0	0
$455$	12.7016	0.595458
$456$	0	0
$457$	28.9109	1.35240	0.676198	−	0.736720i	$-0.263626\pi$
$457$	28.9109	1.35240	0.676198	+	0.736720i	$0.263626\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−34.2094	−1.59329	−0.796645	−	0.604448i	$-0.793394\pi$
$461$	−34.2094	−1.59329	−0.796645	+	0.604448i	$0.793394\pi$
$462$	0	0
$463$	4.80625	0.223365	0.111683	−	0.993744i	$-0.464376\pi$
$463$	4.80625	0.223365	0.111683	+	0.993744i	$0.464376\pi$
$464$	0	0
$465$	−20.0000	−0.927478
$466$	0	0
$467$	−40.4187	−1.87036	−0.935178	−	0.354177i	$-0.884761\pi$
$467$	−40.4187	−1.87036	−0.935178	+	0.354177i	$0.884761\pi$
$468$	0	0
$469$	66.3141	3.06210
$470$	0	0
$471$	−12.8062	−0.590081
$472$	0	0
$473$	10.1047	0.464614
$474$	0	0
$475$	−13.7906	−0.632757
$476$	0	0
$477$	−5.40312	−0.247392
$478$	0	0
$479$	−39.5078	−1.80516	−0.902579	−	0.430524i	$-0.858329\pi$
$479$	−39.5078	−1.80516	−0.902579	+	0.430524i	$0.858329\pi$
$480$	0	0
$481$	5.40312	0.246361
$482$	0	0
$483$	31.5078	1.43366
$484$	0	0
$485$	−45.4031	−2.06165
$486$	0	0
$487$	−11.6125	−0.526212	−0.263106	−	0.964767i	$-0.584747\pi$
$487$	−11.6125	−0.526212	−0.263106	+	0.964767i	$0.584747\pi$
$488$	0	0
$489$	−2.10469	−0.0951772
$490$	0	0
$491$	16.1047	0.726794	0.363397	−	0.931634i	$-0.381617\pi$
$491$	16.1047	0.726794	0.363397	+	0.931634i	$0.381617\pi$
$492$	0	0
$493$	−26.8062	−1.20729
$494$	0	0
$495$	2.70156	0.121426
$496$	0	0
$497$	6.59688	0.295910
$498$	0	0
$499$	32.9109	1.47330	0.736648	−	0.676276i	$-0.236408\pi$
$499$	32.9109	1.47330	0.736648	+	0.676276i	$0.236408\pi$
$500$	0	0
$501$	4.70156	0.210050
$502$	0	0
$503$	−32.0000	−1.42681	−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000	−1.42681	−0.713405	+	0.700752i	$0.752848\pi$
$504$	0	0
$505$	16.2094	0.721308
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−7.61250	−0.337418	−0.168709	−	0.985666i	$-0.553960\pi$
$509$	−7.61250	−0.337418	−0.168709	+	0.985666i	$0.553960\pi$
$510$	0	0
$511$	−22.1047	−0.977854
$512$	0	0
$513$	6.00000	0.264906
$514$	0	0
$515$	45.1203	1.98824
$516$	0	0
$517$	−4.00000	−0.175920
$518$	0	0
$519$	−9.50781	−0.417347
$520$	0	0
$521$	0.104686	0.00458639	0.00229320	−	0.999997i	$-0.499270\pi$
$521$	0.104686	0.00458639	0.00229320	+	0.999997i	$0.499270\pi$
$522$	0	0
$523$	−25.6125	−1.11996	−0.559978	−	0.828507i	$-0.689190\pi$
$523$	−25.6125	−1.11996	−0.559978	+	0.828507i	$0.689190\pi$
$524$	0	0
$525$	10.8062	0.471623
$526$	0	0
$527$	29.6125	1.28994
$528$	0	0
$529$	21.9109	0.952649
$530$	0	0
$531$	7.29844	0.316725
$532$	0	0
$533$	6.70156	0.290277
$534$	0	0
$535$	−14.3141	−0.618851
$536$	0	0
$537$	14.8062	0.638937
$538$	0	0
$539$	15.1047	0.650605
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	30.8062	1.31959
$546$	0	0
$547$	−4.70156	−0.201024	−0.100512	−	0.994936i	$-0.532048\pi$
$547$	−4.70156	−0.201024	−0.100512	+	0.994936i	$0.532048\pi$
$548$	0	0
$549$	10.7016	0.456732
$550$	0	0
$551$	40.2094	1.71298
$552$	0	0
$553$	44.2094	1.87997
$554$	0	0
$555$	14.5969	0.619603
$556$	0	0
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	0	0
$559$	−10.1047	−0.427383
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	−12.5969	−0.530895	−0.265448	−	0.964125i	$-0.585520\pi$
$563$	−12.5969	−0.530895	−0.265448	+	0.964125i	$0.585520\pi$
$564$	0	0
$565$	−0.282817	−0.0118982
$566$	0	0
$567$	−4.70156	−0.197447
$568$	0	0
$569$	−17.4031	−0.729577	−0.364788	−	0.931090i	$-0.618859\pi$
$569$	−17.4031	−0.729577	−0.364788	+	0.931090i	$0.618859\pi$
$570$	0	0
$571$	4.49219	0.187992	0.0939961	−	0.995573i	$-0.470036\pi$
$571$	4.49219	0.187992	0.0939961	+	0.995573i	$0.470036\pi$
$572$	0	0
$573$	21.5078	0.898502
$574$	0	0
$575$	15.4031	0.642355
$576$	0	0
$577$	−30.2094	−1.25763	−0.628816	−	0.777554i	$-0.716460\pi$
$577$	−30.2094	−1.25763	−0.628816	+	0.777554i	$0.716460\pi$
$578$	0	0
$579$	21.4031	0.889483
$580$	0	0
$581$	12.2094	0.506530
$582$	0	0
$583$	−5.40312	−0.223775
$584$	0	0
$585$	−2.70156	−0.111696
$586$	0	0
$587$	−31.2984	−1.29182	−0.645912	−	0.763412i	$-0.723523\pi$
$587$	−31.2984	−1.29182	−0.645912	+	0.763412i	$0.723523\pi$
$588$	0	0
$589$	−44.4187	−1.83024
$590$	0	0
$591$	19.4031	0.798138
$592$	0	0
$593$	20.8062	0.854410	0.427205	−	0.904155i	$-0.359498\pi$
$593$	20.8062	0.854410	0.427205	+	0.904155i	$0.359498\pi$
$594$	0	0
$595$	−50.8062	−2.08285
$596$	0	0
$597$	−22.1047	−0.904685
$598$	0	0
$599$	−18.9109	−0.772680	−0.386340	−	0.922356i	$-0.626261\pi$
$599$	−18.9109	−0.772680	−0.386340	+	0.922356i	$0.626261\pi$
$600$	0	0
$601$	12.5969	0.513837	0.256919	−	0.966433i	$-0.417293\pi$
$601$	12.5969	0.513837	0.256919	+	0.966433i	$0.417293\pi$
$602$	0	0
$603$	−14.1047	−0.574387
$604$	0	0
$605$	2.70156	0.109834
$606$	0	0
$607$	−20.2094	−0.820273	−0.410137	−	0.912024i	$-0.634519\pi$
$607$	−20.2094	−0.820273	−0.410137	+	0.912024i	$0.634519\pi$
$608$	0	0
$609$	−31.5078	−1.27676
$610$	0	0
$611$	4.00000	0.161823
$612$	0	0
$613$	−2.20937	−0.0892357	−0.0446179	−	0.999004i	$-0.514207\pi$
$613$	−2.20937	−0.0892357	−0.0446179	+	0.999004i	$0.514207\pi$
$614$	0	0
$615$	18.1047	0.730051
$616$	0	0
$617$	11.6125	0.467502	0.233751	−	0.972297i	$-0.424900\pi$
$617$	11.6125	0.467502	0.233751	+	0.972297i	$0.424900\pi$
$618$	0	0
$619$	8.70156	0.349745	0.174873	−	0.984591i	$-0.444049\pi$
$619$	8.70156	0.349745	0.174873	+	0.984591i	$0.444049\pi$
$620$	0	0
$621$	−6.70156	−0.268924
$622$	0	0
$623$	53.6125	2.14794
$624$	0	0
$625$	−31.2094	−1.24837
$626$	0	0
$627$	6.00000	0.239617
$628$	0	0
$629$	−21.6125	−0.861747
$630$	0	0
$631$	−23.1938	−0.923329	−0.461664	−	0.887055i	$-0.652748\pi$
$631$	−23.1938	−0.923329	−0.461664	+	0.887055i	$0.652748\pi$
$632$	0	0
$633$	18.8062	0.747481
$634$	0	0
$635$	−25.4031	−1.00809
$636$	0	0
$637$	−15.1047	−0.598469
$638$	0	0
$639$	−1.40312	−0.0555067
$640$	0	0
$641$	−6.70156	−0.264696	−0.132348	−	0.991203i	$-0.542252\pi$
$641$	−6.70156	−0.264696	−0.132348	+	0.991203i	$0.542252\pi$
$642$	0	0
$643$	34.8062	1.37262	0.686312	−	0.727307i	$-0.259228\pi$
$643$	34.8062	1.37262	0.686312	+	0.727307i	$0.259228\pi$
$644$	0	0
$645$	−27.2984	−1.07487
$646$	0	0
$647$	41.0156	1.61249	0.806245	−	0.591581i	$-0.201496\pi$
$647$	41.0156	1.61249	0.806245	+	0.591581i	$0.201496\pi$
$648$	0	0
$649$	7.29844	0.286489
$650$	0	0
$651$	34.8062	1.36416
$652$	0	0
$653$	−43.0156	−1.68333	−0.841666	−	0.539999i	$-0.818425\pi$
$653$	−43.0156	−1.68333	−0.841666	+	0.539999i	$0.818425\pi$
$654$	0	0
$655$	0.282817	0.0110506
$656$	0	0
$657$	4.70156	0.183425
$658$	0	0
$659$	−20.5969	−0.802340	−0.401170	−	0.916004i	$-0.631396\pi$
$659$	−20.5969	−0.802340	−0.401170	+	0.916004i	$0.631396\pi$
$660$	0	0
$661$	2.59688	0.101007	0.0505034	−	0.998724i	$-0.483917\pi$
$661$	2.59688	0.101007	0.0505034	+	0.998724i	$0.483917\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	76.2094	2.95527
$666$	0	0
$667$	−44.9109	−1.73896
$668$	0	0
$669$	8.59688	0.332374
$670$	0	0
$671$	10.7016	0.413129
$672$	0	0
$673$	8.59688	0.331385	0.165693	−	0.986177i	$-0.447014\pi$
$673$	8.59688	0.331385	0.165693	+	0.986177i	$0.447014\pi$
$674$	0	0
$675$	−2.29844	−0.0884669
$676$	0	0
$677$	−30.0000	−1.15299	−0.576497	−	0.817099i	$-0.695581\pi$
$677$	−30.0000	−1.15299	−0.576497	+	0.817099i	$0.695581\pi$
$678$	0	0
$679$	79.0156	3.03234
$680$	0	0
$681$	−2.80625	−0.107536
$682$	0	0
$683$	−8.70156	−0.332956	−0.166478	−	0.986045i	$-0.553239\pi$
$683$	−8.70156	−0.332956	−0.166478	+	0.986045i	$0.553239\pi$
$684$	0	0
$685$	34.5969	1.32188
$686$	0	0
$687$	−0.701562	−0.0267663
$688$	0	0
$689$	5.40312	0.205843
$690$	0	0
$691$	−30.8062	−1.17192	−0.585962	−	0.810338i	$-0.699283\pi$
$691$	−30.8062	−1.17192	−0.585962	+	0.810338i	$0.699283\pi$
$692$	0	0
$693$	−4.70156	−0.178598
$694$	0	0
$695$	−43.2250	−1.63962
$696$	0	0
$697$	−26.8062	−1.01536
$698$	0	0
$699$	−6.59688	−0.249517
$700$	0	0
$701$	−40.3141	−1.52264	−0.761320	−	0.648376i	$-0.775449\pi$
$701$	−40.3141	−1.52264	−0.761320	+	0.648376i	$0.775449\pi$
$702$	0	0
$703$	32.4187	1.22270
$704$	0	0
$705$	10.8062	0.406987
$706$	0	0
$707$	−28.2094	−1.06092
$708$	0	0
$709$	8.70156	0.326794	0.163397	−	0.986560i	$-0.447755\pi$
$709$	8.70156	0.326794	0.163397	+	0.986560i	$0.447755\pi$
$710$	0	0
$711$	−9.40312	−0.352645
$712$	0	0
$713$	49.6125	1.85800
$714$	0	0
$715$	−2.70156	−0.101033
$716$	0	0
$717$	−4.70156	−0.175583
$718$	0	0
$719$	38.9109	1.45113	0.725567	−	0.688152i	$-0.241578\pi$
$719$	38.9109	1.45113	0.725567	+	0.688152i	$0.241578\pi$
$720$	0	0
$721$	−78.5234	−2.92437
$722$	0	0
$723$	14.5969	0.542864
$724$	0	0
$725$	−15.4031	−0.572058
$726$	0	0
$727$	9.61250	0.356508	0.178254	−	0.983985i	$-0.442955\pi$
$727$	9.61250	0.356508	0.178254	+	0.983985i	$0.442955\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	40.4187	1.49494
$732$	0	0
$733$	15.1938	0.561194	0.280597	−	0.959826i	$-0.409468\pi$
$733$	15.1938	0.561194	0.280597	+	0.959826i	$0.409468\pi$
$734$	0	0
$735$	−40.8062	−1.50516
$736$	0	0
$737$	−14.1047	−0.519553
$738$	0	0
$739$	−3.40312	−0.125186	−0.0625930	−	0.998039i	$-0.519937\pi$
$739$	−3.40312	−0.125186	−0.0625930	+	0.998039i	$0.519937\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	−10.3141	−0.378386	−0.189193	−	0.981940i	$-0.560587\pi$
$743$	−10.3141	−0.378386	−0.189193	+	0.981940i	$0.560587\pi$
$744$	0	0
$745$	−60.0000	−2.19823
$746$	0	0
$747$	−2.59688	−0.0950147
$748$	0	0
$749$	24.9109	0.910226
$750$	0	0
$751$	8.49219	0.309884	0.154942	−	0.987924i	$-0.450481\pi$
$751$	8.49219	0.309884	0.154942	+	0.987924i	$0.450481\pi$
$752$	0	0
$753$	5.40312	0.196901
$754$	0	0
$755$	0	0
$756$	0	0
$757$	47.6125	1.73051	0.865253	−	0.501336i	$-0.167158\pi$
$757$	47.6125	1.73051	0.865253	+	0.501336i	$0.167158\pi$
$758$	0	0
$759$	−6.70156	−0.243251
$760$	0	0
$761$	−29.2984	−1.06207	−0.531034	−	0.847351i	$-0.678196\pi$
$761$	−29.2984	−1.06207	−0.531034	+	0.847351i	$0.678196\pi$
$762$	0	0
$763$	−53.6125	−1.94090
$764$	0	0
$765$	10.8062	0.390701
$766$	0	0
$767$	−7.29844	−0.263531
$768$	0	0
$769$	−2.10469	−0.0758969	−0.0379485	−	0.999280i	$-0.512082\pi$
$769$	−2.10469	−0.0758969	−0.0379485	+	0.999280i	$0.512082\pi$
$770$	0	0
$771$	2.70156	0.0972944
$772$	0	0
$773$	31.6125	1.13702	0.568511	−	0.822675i	$-0.307520\pi$
$773$	31.6125	1.13702	0.568511	+	0.822675i	$0.307520\pi$
$774$	0	0
$775$	17.0156	0.611219
$776$	0	0
$777$	−25.4031	−0.911332
$778$	0	0
$779$	40.2094	1.44065
$780$	0	0
$781$	−1.40312	−0.0502077
$782$	0	0
$783$	6.70156	0.239494
$784$	0	0
$785$	34.5969	1.23482
$786$	0	0
$787$	−7.40312	−0.263893	−0.131946	−	0.991257i	$-0.542123\pi$
$787$	−7.40312	−0.263893	−0.131946	+	0.991257i	$0.542123\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0.492189	0.0175002
$792$	0	0
$793$	−10.7016	−0.380024
$794$	0	0
$795$	14.5969	0.517698
$796$	0	0
$797$	20.0000	0.708436	0.354218	−	0.935163i	$-0.384747\pi$
$797$	20.0000	0.708436	0.354218	+	0.935163i	$0.384747\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−11.4031	−0.402910
$802$	0	0
$803$	4.70156	0.165915
$804$	0	0
$805$	−85.1203	−3.00010
$806$	0	0
$807$	−5.19375	−0.182829
$808$	0	0
$809$	15.0156	0.527921	0.263961	−	0.964533i	$-0.414971\pi$
$809$	15.0156	0.527921	0.263961	+	0.964533i	$0.414971\pi$
$810$	0	0
$811$	27.4031	0.962254	0.481127	−	0.876651i	$-0.340228\pi$
$811$	27.4031	0.962254	0.481127	+	0.876651i	$0.340228\pi$
$812$	0	0
$813$	1.19375	0.0418667
$814$	0	0
$815$	5.68594	0.199170
$816$	0	0
$817$	−60.6281	−2.12111
$818$	0	0
$819$	4.70156	0.164286
$820$	0	0
$821$	12.8062	0.446941	0.223471	−	0.974711i	$-0.428261\pi$
$821$	12.8062	0.446941	0.223471	+	0.974711i	$0.428261\pi$
$822$	0	0
$823$	−42.3141	−1.47498	−0.737488	−	0.675361i	$-0.763988\pi$
$823$	−42.3141	−1.47498	−0.737488	+	0.675361i	$0.763988\pi$
$824$	0	0
$825$	−2.29844	−0.0800213
$826$	0	0
$827$	−51.2250	−1.78127	−0.890634	−	0.454721i	$-0.849739\pi$
$827$	−51.2250	−1.78127	−0.890634	+	0.454721i	$0.849739\pi$
$828$	0	0
$829$	−4.38750	−0.152384	−0.0761921	−	0.997093i	$-0.524276\pi$
$829$	−4.38750	−0.152384	−0.0761921	+	0.997093i	$0.524276\pi$
$830$	0	0
$831$	4.10469	0.142390
$832$	0	0
$833$	60.4187	2.09339
$834$	0	0
$835$	−12.7016	−0.439556
$836$	0	0
$837$	−7.40312	−0.255889
$838$	0	0
$839$	−2.59688	−0.0896541	−0.0448271	−	0.998995i	$-0.514274\pi$
$839$	−2.59688	−0.0896541	−0.0448271	+	0.998995i	$0.514274\pi$
$840$	0	0
$841$	15.9109	0.548653
$842$	0	0
$843$	13.2984	0.458023
$844$	0	0
$845$	2.70156	0.0929366
$846$	0	0
$847$	−4.70156	−0.161548
$848$	0	0
$849$	0.701562	0.0240775
$850$	0	0
$851$	−36.2094	−1.24124
$852$	0	0
$853$	35.8219	1.22652	0.613259	−	0.789882i	$-0.289858\pi$
$853$	35.8219	1.22652	0.613259	+	0.789882i	$0.289858\pi$
$854$	0	0
$855$	−16.2094	−0.554349
$856$	0	0
$857$	−37.6125	−1.28482	−0.642409	−	0.766362i	$-0.722065\pi$
$857$	−37.6125	−1.28482	−0.642409	+	0.766362i	$0.722065\pi$
$858$	0	0
$859$	−9.19375	−0.313687	−0.156843	−	0.987623i	$-0.550132\pi$
$859$	−9.19375	−0.313687	−0.156843	+	0.987623i	$0.550132\pi$
$860$	0	0
$861$	−31.5078	−1.07378
$862$	0	0
$863$	−2.80625	−0.0955258	−0.0477629	−	0.998859i	$-0.515209\pi$
$863$	−2.80625	−0.0955258	−0.0477629	+	0.998859i	$0.515209\pi$
$864$	0	0
$865$	25.6859	0.873348
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−9.40312	−0.318979
$870$	0	0
$871$	14.1047	0.477919
$872$	0	0
$873$	−16.8062	−0.568805
$874$	0	0
$875$	34.3141	1.16003
$876$	0	0
$877$	−16.5969	−0.560437	−0.280218	−	0.959936i	$-0.590407\pi$
$877$	−16.5969	−0.560437	−0.280218	+	0.959936i	$0.590407\pi$
$878$	0	0
$879$	−14.0000	−0.472208
$880$	0	0
$881$	24.3141	0.819161	0.409581	−	0.912274i	$-0.365675\pi$
$881$	24.3141	0.819161	0.409581	+	0.912274i	$0.365675\pi$
$882$	0	0
$883$	−37.4031	−1.25872	−0.629358	−	0.777116i	$-0.716682\pi$
$883$	−37.4031	−1.25872	−0.629358	+	0.777116i	$0.716682\pi$
$884$	0	0
$885$	−19.7172	−0.662786
$886$	0	0
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	0	0
$889$	44.2094	1.48273
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	24.0000	0.803129
$894$	0	0
$895$	−40.0000	−1.33705
$896$	0	0
$897$	6.70156	0.223759
$898$	0	0
$899$	−49.6125	−1.65467
$900$	0	0
$901$	−21.6125	−0.720017
$902$	0	0
$903$	47.5078	1.58096
$904$	0	0
$905$	−59.4344	−1.97567
$906$	0	0
$907$	−30.8062	−1.02290	−0.511452	−	0.859312i	$-0.670892\pi$
$907$	−30.8062	−1.02290	−0.511452	+	0.859312i	$0.670892\pi$
$908$	0	0
$909$	6.00000	0.199007
$910$	0	0
$911$	−12.5969	−0.417353	−0.208677	−	0.977985i	$-0.566916\pi$
$911$	−12.5969	−0.417353	−0.208677	+	0.977985i	$0.566916\pi$
$912$	0	0
$913$	−2.59688	−0.0859440
$914$	0	0
$915$	−28.9109	−0.955766
$916$	0	0
$917$	−0.492189	−0.0162535
$918$	0	0
$919$	−2.80625	−0.0925696	−0.0462848	−	0.998928i	$-0.514738\pi$
$919$	−2.80625	−0.0925696	−0.0462848	+	0.998928i	$0.514738\pi$
$920$	0	0
$921$	10.0000	0.329511
$922$	0	0
$923$	1.40312	0.0461844
$924$	0	0
$925$	−12.4187	−0.408326
$926$	0	0
$927$	16.7016	0.548551
$928$	0	0
$929$	24.5969	0.806997	0.403499	−	0.914980i	$-0.367794\pi$
$929$	24.5969	0.806997	0.403499	+	0.914980i	$0.367794\pi$
$930$	0	0
$931$	−90.6281	−2.97022
$932$	0	0
$933$	−12.5969	−0.412403
$934$	0	0
$935$	10.8062	0.353402
$936$	0	0
$937$	32.5969	1.06489	0.532447	−	0.846463i	$-0.321273\pi$
$937$	32.5969	1.06489	0.532447	+	0.846463i	$0.321273\pi$
$938$	0	0
$939$	−14.9109	−0.486600
$940$	0	0
$941$	−1.79063	−0.0583728	−0.0291864	−	0.999574i	$-0.509292\pi$
$941$	−1.79063	−0.0583728	−0.0291864	+	0.999574i	$0.509292\pi$
$942$	0	0
$943$	−44.9109	−1.46250
$944$	0	0
$945$	12.7016	0.413182
$946$	0	0
$947$	49.6125	1.61219	0.806095	−	0.591786i	$-0.201577\pi$
$947$	49.6125	1.61219	0.806095	+	0.591786i	$0.201577\pi$
$948$	0	0
$949$	−4.70156	−0.152619
$950$	0	0
$951$	−30.9109	−1.00236
$952$	0	0
$953$	29.6125	0.959243	0.479621	−	0.877476i	$-0.340774\pi$
$953$	29.6125	0.959243	0.479621	+	0.877476i	$0.340774\pi$
$954$	0	0
$955$	−58.1047	−1.88022
$956$	0	0
$957$	6.70156	0.216631
$958$	0	0
$959$	−60.2094	−1.94426
$960$	0	0
$961$	23.8062	0.767943
$962$	0	0
$963$	−5.29844	−0.170740
$964$	0	0
$965$	−57.8219	−1.86135
$966$	0	0
$967$	50.3141	1.61799	0.808996	−	0.587815i	$-0.200012\pi$
$967$	50.3141	1.61799	0.808996	+	0.587815i	$0.200012\pi$
$968$	0	0
$969$	24.0000	0.770991
$970$	0	0
$971$	25.4031	0.815225	0.407613	−	0.913155i	$-0.366361\pi$
$971$	25.4031	0.815225	0.407613	+	0.913155i	$0.366361\pi$
$972$	0	0
$973$	75.2250	2.41160
$974$	0	0
$975$	2.29844	0.0736089
$976$	0	0
$977$	21.0156	0.672349	0.336175	−	0.941800i	$-0.390867\pi$
$977$	21.0156	0.672349	0.336175	+	0.941800i	$0.390867\pi$
$978$	0	0
$979$	−11.4031	−0.364445
$980$	0	0
$981$	11.4031	0.364074
$982$	0	0
$983$	47.0156	1.49957	0.749783	−	0.661684i	$-0.230158\pi$
$983$	47.0156	1.49957	0.749783	+	0.661684i	$0.230158\pi$
$984$	0	0
$985$	−52.4187	−1.67020
$986$	0	0
$987$	−18.8062	−0.598609
$988$	0	0
$989$	67.7172	2.15328
$990$	0	0
$991$	18.3141	0.581765	0.290883	−	0.956759i	$-0.406051\pi$
$991$	18.3141	0.581765	0.290883	+	0.956759i	$0.406051\pi$
$992$	0	0
$993$	−7.50781	−0.238253
$994$	0	0
$995$	59.7172	1.89316
$996$	0	0
$997$	33.7172	1.06783	0.533917	−	0.845537i	$-0.320719\pi$
$997$	33.7172	1.06783	0.533917	+	0.845537i	$0.320719\pi$
$998$	0	0
$999$	5.40312	0.170947

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bd.1.2		2
4.3	odd	2		858.2.a.p.1.2	✓	2
12.11	even	2		2574.2.a.be.1.1		2
44.43	even	2		9438.2.a.ch.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
858.2.a.p.1.2	✓	2	4.3	odd	2
2574.2.a.be.1.1		2	12.11	even	2
6864.2.a.bd.1.2		2	1.1	even	1	trivial
9438.2.a.ch.1.2		2	44.43	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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.70156 q^{5} -4.70156 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.70156 q^{5} -4.70156 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.70156 q^{15} +4.00000 q^{17} -6.00000 q^{19} +4.70156 q^{21} +6.70156 q^{23} +2.29844 q^{25} -1.00000 q^{27} -6.70156 q^{29} +7.40312 q^{31} -1.00000 q^{33} -12.7016 q^{35} -5.40312 q^{37} +1.00000 q^{39} -6.70156 q^{41} +10.1047 q^{43} +2.70156 q^{45} -4.00000 q^{47} +15.1047 q^{49} -4.00000 q^{51} -5.40312 q^{53} +2.70156 q^{55} +6.00000 q^{57} +7.29844 q^{59} +10.7016 q^{61} -4.70156 q^{63} -2.70156 q^{65} -14.1047 q^{67} -6.70156 q^{69} -1.40312 q^{71} +4.70156 q^{73} -2.29844 q^{75} -4.70156 q^{77} -9.40312 q^{79} +1.00000 q^{81} -2.59688 q^{83} +10.8062 q^{85} +6.70156 q^{87} -11.4031 q^{89} +4.70156 q^{91} -7.40312 q^{93} -16.2094 q^{95} -16.8062 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} + q^{15} + 8 q^{17} - 12 q^{19} + 3 q^{21} + 7 q^{23} + 11 q^{25} - 2 q^{27} - 7 q^{29} + 2 q^{31} - 2 q^{33} - 19 q^{35} + 2 q^{37} + 2 q^{39} - 7 q^{41} + q^{43} - q^{45} - 8 q^{47} + 11 q^{49} - 8 q^{51} + 2 q^{53} - q^{55} + 12 q^{57} + 21 q^{59} + 15 q^{61} - 3 q^{63} + q^{65} - 9 q^{67} - 7 q^{69} + 10 q^{71} + 3 q^{73} - 11 q^{75} - 3 q^{77} - 6 q^{79} + 2 q^{81} - 18 q^{83} - 4 q^{85} + 7 q^{87} - 10 q^{89} + 3 q^{91} - 2 q^{93} + 6 q^{95} - 8 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + q^15 + 8 * q^17 - 12 * q^19 + 3 * q^21 + 7 * q^23 + 11 * q^25 - 2 * q^27 - 7 * q^29 + 2 * q^31 - 2 * q^33 - 19 * q^35 + 2 * q^37 + 2 * q^39 - 7 * q^41 + q^43 - q^45 - 8 * q^47 + 11 * q^49 - 8 * q^51 + 2 * q^53 - q^55 + 12 * q^57 + 21 * q^59 + 15 * q^61 - 3 * q^63 + q^65 - 9 * q^67 - 7 * q^69 + 10 * q^71 + 3 * q^73 - 11 * q^75 - 3 * q^77 - 6 * q^79 + 2 * q^81 - 18 * q^83 - 4 * q^85 + 7 * q^87 - 10 * q^89 + 3 * q^91 - 2 * q^93 + 6 * q^95 - 8 * q^97 + 2 * q^99

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