LMFDB - Embedded newform 6864.2.a.bq.1.3

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:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.21432 q^{5} +2.90321 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.21432 q^{5} +2.90321 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.21432 q^{15} -5.73975 q^{17} +4.14764 q^{19} -2.90321 q^{21} -5.33185 q^{23} -0.0967881 q^{25} -1.00000 q^{27} -6.49532 q^{29} +1.16346 q^{31} -1.00000 q^{33} +6.42864 q^{35} +10.2351 q^{37} +1.00000 q^{39} -2.14764 q^{41} +9.11753 q^{43} +2.21432 q^{45} +10.8573 q^{47} +1.42864 q^{49} +5.73975 q^{51} -6.42864 q^{53} +2.21432 q^{55} -4.14764 q^{57} -7.18421 q^{59} +8.85728 q^{61} +2.90321 q^{63} -2.21432 q^{65} +6.96989 q^{67} +5.33185 q^{69} +10.9906 q^{71} +2.28100 q^{73} +0.0967881 q^{75} +2.90321 q^{77} +1.93332 q^{79} +1.00000 q^{81} +10.2351 q^{83} -12.7096 q^{85} +6.49532 q^{87} +2.70318 q^{89} -2.90321 q^{91} -1.16346 q^{93} +9.18421 q^{95} +5.37778 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{13} - 4 q^{17} + 6 q^{19} - 2 q^{21} + 4 q^{23} - 7 q^{25} - 3 q^{27} - 6 q^{29} + 10 q^{31} - 3 q^{33} + 6 q^{35} + 4 q^{37} + 3 q^{39} + 14 q^{43} + 6 q^{47} - 9 q^{49} + 4 q^{51} - 6 q^{53} - 6 q^{57} - 8 q^{59} + 2 q^{63} + 14 q^{67} - 4 q^{69} + 6 q^{71} + 7 q^{75} + 2 q^{77} + 6 q^{79} + 3 q^{81} + 4 q^{83} - 18 q^{85} + 6 q^{87} + 2 q^{89} - 2 q^{91} - 10 q^{93} + 14 q^{95} + 16 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^13 - 4 * q^17 + 6 * q^19 - 2 * q^21 + 4 * q^23 - 7 * q^25 - 3 * q^27 - 6 * q^29 + 10 * q^31 - 3 * q^33 + 6 * q^35 + 4 * q^37 + 3 * q^39 + 14 * q^43 + 6 * q^47 - 9 * q^49 + 4 * q^51 - 6 * q^53 - 6 * q^57 - 8 * q^59 + 2 * q^63 + 14 * q^67 - 4 * q^69 + 6 * q^71 + 7 * q^75 + 2 * q^77 + 6 * q^79 + 3 * q^81 + 4 * q^83 - 18 * q^85 + 6 * q^87 + 2 * q^89 - 2 * q^91 - 10 * q^93 + 14 * q^95 + 16 * q^97 + 3 * 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
$5$	2.21432	0.990274	0.495137	−	0.868815i	$-0.335118\pi$
$5$	2.21432	0.990274	0.495137	+	0.868815i	$0.335118\pi$
$6$	0	0
$7$	2.90321	1.09731	0.548655	−	0.836049i	$-0.315140\pi$
$7$	2.90321	1.09731	0.548655	+	0.836049i	$0.315140\pi$
$8$	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
$15$	−2.21432	−0.571735
$16$	0	0
$17$	−5.73975	−1.39209	−0.696047	−	0.717997i	$-0.745059\pi$
$17$	−5.73975	−1.39209	−0.696047	+	0.717997i	$0.745059\pi$
$18$	0	0
$19$	4.14764	0.951535	0.475767	−	0.879571i	$-0.342170\pi$
$19$	4.14764	0.951535	0.475767	+	0.879571i	$0.342170\pi$
$20$	0	0
$21$	−2.90321	−0.633533
$22$	0	0
$23$	−5.33185	−1.11177	−0.555884	−	0.831260i	$-0.687620\pi$
$23$	−5.33185	−1.11177	−0.555884	+	0.831260i	$0.687620\pi$
$24$	0	0
$25$	−0.0967881	−0.0193576
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.49532	−1.20615	−0.603075	−	0.797685i	$-0.706058\pi$
$29$	−6.49532	−1.20615	−0.603075	+	0.797685i	$0.706058\pi$
$30$	0	0
$31$	1.16346	0.208964	0.104482	−	0.994527i	$-0.466681\pi$
$31$	1.16346	0.208964	0.104482	+	0.994527i	$0.466681\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	6.42864	1.08664
$36$	0	0
$37$	10.2351	1.68263	0.841317	−	0.540542i	$-0.181781\pi$
$37$	10.2351	1.68263	0.841317	+	0.540542i	$0.181781\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−2.14764	−0.335406	−0.167703	−	0.985838i	$-0.553635\pi$
$41$	−2.14764	−0.335406	−0.167703	+	0.985838i	$0.553635\pi$
$42$	0	0
$43$	9.11753	1.39041	0.695205	−	0.718811i	$-0.255313\pi$
$43$	9.11753	1.39041	0.695205	+	0.718811i	$0.255313\pi$
$44$	0	0
$45$	2.21432	0.330091
$46$	0	0
$47$	10.8573	1.58370	0.791848	−	0.610718i	$-0.209119\pi$
$47$	10.8573	1.58370	0.791848	+	0.610718i	$0.209119\pi$
$48$	0	0
$49$	1.42864	0.204091
$50$	0	0
$51$	5.73975	0.803725
$52$	0	0
$53$	−6.42864	−0.883042	−0.441521	−	0.897251i	$-0.645561\pi$
$53$	−6.42864	−0.883042	−0.441521	+	0.897251i	$0.645561\pi$
$54$	0	0
$55$	2.21432	0.298579
$56$	0	0
$57$	−4.14764	−0.549369
$58$	0	0
$59$	−7.18421	−0.935304	−0.467652	−	0.883913i	$-0.654900\pi$
$59$	−7.18421	−0.935304	−0.467652	+	0.883913i	$0.654900\pi$
$60$	0	0
$61$	8.85728	1.13406	0.567029	−	0.823698i	$-0.308093\pi$
$61$	8.85728	1.13406	0.567029	+	0.823698i	$0.308093\pi$
$62$	0	0
$63$	2.90321	0.365770
$64$	0	0
$65$	−2.21432	−0.274653
$66$	0	0
$67$	6.96989	0.851507	0.425754	−	0.904839i	$-0.360009\pi$
$67$	6.96989	0.851507	0.425754	+	0.904839i	$0.360009\pi$
$68$	0	0
$69$	5.33185	0.641879
$70$	0	0
$71$	10.9906	1.30435	0.652174	−	0.758069i	$-0.273857\pi$
$71$	10.9906	1.30435	0.652174	+	0.758069i	$0.273857\pi$
$72$	0	0
$73$	2.28100	0.266970	0.133485	−	0.991051i	$-0.457383\pi$
$73$	2.28100	0.266970	0.133485	+	0.991051i	$0.457383\pi$
$74$	0	0
$75$	0.0967881	0.0111761
$76$	0	0
$77$	2.90321	0.330852
$78$	0	0
$79$	1.93332	0.217516	0.108758	−	0.994068i	$-0.465313\pi$
$79$	1.93332	0.217516	0.108758	+	0.994068i	$0.465313\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.2351	1.12344	0.561722	−	0.827326i	$-0.310139\pi$
$83$	10.2351	1.12344	0.561722	+	0.827326i	$0.310139\pi$
$84$	0	0
$85$	−12.7096	−1.37855
$86$	0	0
$87$	6.49532	0.696371
$88$	0	0
$89$	2.70318	0.286537	0.143268	−	0.989684i	$-0.454239\pi$
$89$	2.70318	0.286537	0.143268	+	0.989684i	$0.454239\pi$
$90$	0	0
$91$	−2.90321	−0.304339
$92$	0	0
$93$	−1.16346	−0.120646
$94$	0	0
$95$	9.18421	0.942280
$96$	0	0
$97$	5.37778	0.546031	0.273016	−	0.962010i	$-0.411979\pi$
$97$	5.37778	0.546031	0.273016	+	0.962010i	$0.411979\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	19.1590	1.90639	0.953197	−	0.302351i	$-0.0977713\pi$
$101$	19.1590	1.90639	0.953197	+	0.302351i	$0.0977713\pi$
$102$	0	0
$103$	6.19358	0.610271	0.305136	−	0.952309i	$-0.401298\pi$
$103$	6.19358	0.610271	0.305136	+	0.952309i	$0.401298\pi$
$104$	0	0
$105$	−6.42864	−0.627371
$106$	0	0
$107$	5.93978	0.574220	0.287110	−	0.957898i	$-0.407305\pi$
$107$	5.93978	0.574220	0.287110	+	0.957898i	$0.407305\pi$
$108$	0	0
$109$	−5.03657	−0.482415	−0.241208	−	0.970474i	$-0.577544\pi$
$109$	−5.03657	−0.482415	−0.241208	+	0.970474i	$0.577544\pi$
$110$	0	0
$111$	−10.2351	−0.971469
$112$	0	0
$113$	9.47949	0.891756	0.445878	−	0.895094i	$-0.352892\pi$
$113$	9.47949	0.891756	0.445878	+	0.895094i	$0.352892\pi$
$114$	0	0
$115$	−11.8064	−1.10095
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	−16.6637	−1.52756
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.14764	0.193646
$124$	0	0
$125$	−11.2859	−1.00944
$126$	0	0
$127$	10.6889	0.948486	0.474243	−	0.880394i	$-0.342722\pi$
$127$	10.6889	0.948486	0.474243	+	0.880394i	$0.342722\pi$
$128$	0	0
$129$	−9.11753	−0.802754
$130$	0	0
$131$	7.05086	0.616036	0.308018	−	0.951381i	$-0.400334\pi$
$131$	7.05086	0.616036	0.308018	+	0.951381i	$0.400334\pi$
$132$	0	0
$133$	12.0415	1.04413
$134$	0	0
$135$	−2.21432	−0.190578
$136$	0	0
$137$	13.0716	1.11678	0.558391	−	0.829578i	$-0.311419\pi$
$137$	13.0716	1.11678	0.558391	+	0.829578i	$0.311419\pi$
$138$	0	0
$139$	−1.60639	−0.136253	−0.0681263	−	0.997677i	$-0.521702\pi$
$139$	−1.60639	−0.136253	−0.0681263	+	0.997677i	$0.521702\pi$
$140$	0	0
$141$	−10.8573	−0.914348
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−14.3827	−1.19442
$146$	0	0
$147$	−1.42864	−0.117832
$148$	0	0
$149$	−9.46520	−0.775420	−0.387710	−	0.921781i	$-0.626734\pi$
$149$	−9.46520	−0.775420	−0.387710	+	0.921781i	$0.626734\pi$
$150$	0	0
$151$	0.534795	0.0435210	0.0217605	−	0.999763i	$-0.493073\pi$
$151$	0.534795	0.0435210	0.0217605	+	0.999763i	$0.493073\pi$
$152$	0	0
$153$	−5.73975	−0.464031
$154$	0	0
$155$	2.57628	0.206932
$156$	0	0
$157$	−15.7605	−1.25782	−0.628912	−	0.777476i	$-0.716499\pi$
$157$	−15.7605	−1.25782	−0.628912	+	0.777476i	$0.716499\pi$
$158$	0	0
$159$	6.42864	0.509824
$160$	0	0
$161$	−15.4795	−1.21996
$162$	0	0
$163$	−6.21432	−0.486743	−0.243372	−	0.969933i	$-0.578253\pi$
$163$	−6.21432	−0.486743	−0.243372	+	0.969933i	$0.578253\pi$
$164$	0	0
$165$	−2.21432	−0.172385
$166$	0	0
$167$	−14.6637	−1.13471	−0.567356	−	0.823473i	$-0.692034\pi$
$167$	−14.6637	−1.13471	−0.567356	+	0.823473i	$0.692034\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	4.14764	0.317178
$172$	0	0
$173$	4.49532	0.341773	0.170886	−	0.985291i	$-0.445337\pi$
$173$	4.49532	0.341773	0.170886	+	0.985291i	$0.445337\pi$
$174$	0	0
$175$	−0.280996	−0.0212413
$176$	0	0
$177$	7.18421	0.539998
$178$	0	0
$179$	−9.46520	−0.707463	−0.353731	−	0.935347i	$-0.615087\pi$
$179$	−9.46520	−0.707463	−0.353731	+	0.935347i	$0.615087\pi$
$180$	0	0
$181$	−17.5669	−1.30574	−0.652869	−	0.757471i	$-0.726435\pi$
$181$	−17.5669	−1.30574	−0.652869	+	0.757471i	$0.726435\pi$
$182$	0	0
$183$	−8.85728	−0.654749
$184$	0	0
$185$	22.6637	1.66627
$186$	0	0
$187$	−5.73975	−0.419732
$188$	0	0
$189$	−2.90321	−0.211178
$190$	0	0
$191$	12.7239	0.920671	0.460335	−	0.887745i	$-0.347729\pi$
$191$	12.7239	0.920671	0.460335	+	0.887745i	$0.347729\pi$
$192$	0	0
$193$	15.4336	1.11093	0.555466	−	0.831539i	$-0.312540\pi$
$193$	15.4336	1.11093	0.555466	+	0.831539i	$0.312540\pi$
$194$	0	0
$195$	2.21432	0.158571
$196$	0	0
$197$	−22.4844	−1.60195	−0.800974	−	0.598699i	$-0.795685\pi$
$197$	−22.4844	−1.60195	−0.800974	+	0.598699i	$0.795685\pi$
$198$	0	0
$199$	1.33630	0.0947276	0.0473638	−	0.998878i	$-0.484918\pi$
$199$	1.33630	0.0947276	0.0473638	+	0.998878i	$0.484918\pi$
$200$	0	0
$201$	−6.96989	−0.491618
$202$	0	0
$203$	−18.8573	−1.32352
$204$	0	0
$205$	−4.75557	−0.332143
$206$	0	0
$207$	−5.33185	−0.370589
$208$	0	0
$209$	4.14764	0.286898
$210$	0	0
$211$	8.00645	0.551187	0.275593	−	0.961274i	$-0.411126\pi$
$211$	8.00645	0.551187	0.275593	+	0.961274i	$0.411126\pi$
$212$	0	0
$213$	−10.9906	−0.753066
$214$	0	0
$215$	20.1891	1.37689
$216$	0	0
$217$	3.37778	0.229299
$218$	0	0
$219$	−2.28100	−0.154135
$220$	0	0
$221$	5.73975	0.386097
$222$	0	0
$223$	−6.02074	−0.403179	−0.201589	−	0.979470i	$-0.564611\pi$
$223$	−6.02074	−0.403179	−0.201589	+	0.979470i	$0.564611\pi$
$224$	0	0
$225$	−0.0967881	−0.00645254
$226$	0	0
$227$	−13.2716	−0.880869	−0.440434	−	0.897785i	$-0.645176\pi$
$227$	−13.2716	−0.880869	−0.440434	+	0.897785i	$0.645176\pi$
$228$	0	0
$229$	14.7556	0.975075	0.487538	−	0.873102i	$-0.337895\pi$
$229$	14.7556	0.975075	0.487538	+	0.873102i	$0.337895\pi$
$230$	0	0
$231$	−2.90321	−0.191017
$232$	0	0
$233$	19.6479	1.28718	0.643588	−	0.765372i	$-0.277445\pi$
$233$	19.6479	1.28718	0.643588	+	0.765372i	$0.277445\pi$
$234$	0	0
$235$	24.0415	1.56829
$236$	0	0
$237$	−1.93332	−0.125583
$238$	0	0
$239$	−14.8113	−0.958066	−0.479033	−	0.877797i	$-0.659013\pi$
$239$	−14.8113	−0.958066	−0.479033	+	0.877797i	$0.659013\pi$
$240$	0	0
$241$	23.5526	1.51716	0.758579	−	0.651581i	$-0.225894\pi$
$241$	23.5526	1.51716	0.758579	+	0.651581i	$0.225894\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.16346	0.202106
$246$	0	0
$247$	−4.14764	−0.263908
$248$	0	0
$249$	−10.2351	−0.648621
$250$	0	0
$251$	6.23506	0.393554	0.196777	−	0.980448i	$-0.436953\pi$
$251$	6.23506	0.393554	0.196777	+	0.980448i	$0.436953\pi$
$252$	0	0
$253$	−5.33185	−0.335211
$254$	0	0
$255$	12.7096	0.795908
$256$	0	0
$257$	1.73329	0.108120	0.0540599	−	0.998538i	$-0.482784\pi$
$257$	1.73329	0.108120	0.0540599	+	0.998538i	$0.482784\pi$
$258$	0	0
$259$	29.7146	1.84637
$260$	0	0
$261$	−6.49532	−0.402050
$262$	0	0
$263$	7.18421	0.442997	0.221499	−	0.975161i	$-0.428905\pi$
$263$	7.18421	0.442997	0.221499	+	0.975161i	$0.428905\pi$
$264$	0	0
$265$	−14.2351	−0.874453
$266$	0	0
$267$	−2.70318	−0.165432
$268$	0	0
$269$	−14.3368	−0.874129	−0.437064	−	0.899430i	$-0.643982\pi$
$269$	−14.3368	−0.874129	−0.437064	+	0.899430i	$0.643982\pi$
$270$	0	0
$271$	6.47457	0.393302	0.196651	−	0.980474i	$-0.436993\pi$
$271$	6.47457	0.393302	0.196651	+	0.980474i	$0.436993\pi$
$272$	0	0
$273$	2.90321	0.175710
$274$	0	0
$275$	−0.0967881	−0.00583654
$276$	0	0
$277$	4.19358	0.251968	0.125984	−	0.992032i	$-0.459791\pi$
$277$	4.19358	0.251968	0.125984	+	0.992032i	$0.459791\pi$
$278$	0	0
$279$	1.16346	0.0696548
$280$	0	0
$281$	18.8430	1.12408	0.562039	−	0.827111i	$-0.310017\pi$
$281$	18.8430	1.12408	0.562039	+	0.827111i	$0.310017\pi$
$282$	0	0
$283$	−7.15902	−0.425559	−0.212780	−	0.977100i	$-0.568252\pi$
$283$	−7.15902	−0.425559	−0.212780	+	0.977100i	$0.568252\pi$
$284$	0	0
$285$	−9.18421	−0.544026
$286$	0	0
$287$	−6.23506	−0.368044
$288$	0	0
$289$	15.9447	0.937923
$290$	0	0
$291$	−5.37778	−0.315251
$292$	0	0
$293$	27.0464	1.58007	0.790034	−	0.613063i	$-0.210063\pi$
$293$	27.0464	1.58007	0.790034	+	0.613063i	$0.210063\pi$
$294$	0	0
$295$	−15.9081	−0.926207
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	5.33185	0.308349
$300$	0	0
$301$	26.4701	1.52571
$302$	0	0
$303$	−19.1590	−1.10066
$304$	0	0
$305$	19.6128	1.12303
$306$	0	0
$307$	6.60793	0.377134	0.188567	−	0.982060i	$-0.439616\pi$
$307$	6.60793	0.377134	0.188567	+	0.982060i	$0.439616\pi$
$308$	0	0
$309$	−6.19358	−0.352340
$310$	0	0
$311$	32.5575	1.84617	0.923085	−	0.384597i	$-0.125660\pi$
$311$	32.5575	1.84617	0.923085	+	0.384597i	$0.125660\pi$
$312$	0	0
$313$	−29.9037	−1.69026	−0.845128	−	0.534564i	$-0.820476\pi$
$313$	−29.9037	−1.69026	−0.845128	+	0.534564i	$0.820476\pi$
$314$	0	0
$315$	6.42864	0.362213
$316$	0	0
$317$	−28.8464	−1.62017	−0.810087	−	0.586310i	$-0.800580\pi$
$317$	−28.8464	−1.62017	−0.810087	+	0.586310i	$0.800580\pi$
$318$	0	0
$319$	−6.49532	−0.363668
$320$	0	0
$321$	−5.93978	−0.331526
$322$	0	0
$323$	−23.8064	−1.32462
$324$	0	0
$325$	0.0967881	0.00536884
$326$	0	0
$327$	5.03657	0.278523
$328$	0	0
$329$	31.5210	1.73781
$330$	0	0
$331$	18.3477	1.00848	0.504240	−	0.863564i	$-0.331773\pi$
$331$	18.3477	1.00848	0.504240	+	0.863564i	$0.331773\pi$
$332$	0	0
$333$	10.2351	0.560878
$334$	0	0
$335$	15.4336	0.843226
$336$	0	0
$337$	−26.0415	−1.41857	−0.709285	−	0.704922i	$-0.750982\pi$
$337$	−26.0415	−1.41857	−0.709285	+	0.704922i	$0.750982\pi$
$338$	0	0
$339$	−9.47949	−0.514855
$340$	0	0
$341$	1.16346	0.0630051
$342$	0	0
$343$	−16.1748	−0.873359
$344$	0	0
$345$	11.8064	0.635636
$346$	0	0
$347$	−13.7462	−0.737935	−0.368967	−	0.929442i	$-0.620289\pi$
$347$	−13.7462	−0.737935	−0.368967	+	0.929442i	$0.620289\pi$
$348$	0	0
$349$	−21.0005	−1.12413	−0.562065	−	0.827093i	$-0.689993\pi$
$349$	−21.0005	−1.12413	−0.562065	+	0.827093i	$0.689993\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	9.78568	0.520839	0.260420	−	0.965496i	$-0.416139\pi$
$353$	9.78568	0.520839	0.260420	+	0.965496i	$0.416139\pi$
$354$	0	0
$355$	24.3368	1.29166
$356$	0	0
$357$	16.6637	0.881937
$358$	0	0
$359$	−32.2306	−1.70107	−0.850533	−	0.525921i	$-0.823721\pi$
$359$	−32.2306	−1.70107	−0.850533	+	0.525921i	$0.823721\pi$
$360$	0	0
$361$	−1.79706	−0.0945819
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	5.05086	0.264374
$366$	0	0
$367$	−0.695346	−0.0362967	−0.0181484	−	0.999835i	$-0.505777\pi$
$367$	−0.695346	−0.0362967	−0.0181484	+	0.999835i	$0.505777\pi$
$368$	0	0
$369$	−2.14764	−0.111802
$370$	0	0
$371$	−18.6637	−0.968971
$372$	0	0
$373$	−4.91750	−0.254619	−0.127309	−	0.991863i	$-0.540634\pi$
$373$	−4.91750	−0.254619	−0.127309	+	0.991863i	$0.540634\pi$
$374$	0	0
$375$	11.2859	0.582802
$376$	0	0
$377$	6.49532	0.334526
$378$	0	0
$379$	19.5417	1.00379	0.501896	−	0.864928i	$-0.332636\pi$
$379$	19.5417	1.00379	0.501896	+	0.864928i	$0.332636\pi$
$380$	0	0
$381$	−10.6889	−0.547609
$382$	0	0
$383$	23.0923	1.17996	0.589982	−	0.807417i	$-0.299135\pi$
$383$	23.0923	1.17996	0.589982	+	0.807417i	$0.299135\pi$
$384$	0	0
$385$	6.42864	0.327634
$386$	0	0
$387$	9.11753	0.463470
$388$	0	0
$389$	26.6321	1.35030	0.675150	−	0.737681i	$-0.264079\pi$
$389$	26.6321	1.35030	0.675150	+	0.737681i	$0.264079\pi$
$390$	0	0
$391$	30.6035	1.54768
$392$	0	0
$393$	−7.05086	−0.355669
$394$	0	0
$395$	4.28100	0.215400
$396$	0	0
$397$	−21.5812	−1.08313	−0.541565	−	0.840659i	$-0.682168\pi$
$397$	−21.5812	−1.08313	−0.541565	+	0.840659i	$0.682168\pi$
$398$	0	0
$399$	−12.0415	−0.602828
$400$	0	0
$401$	−22.5926	−1.12822	−0.564110	−	0.825700i	$-0.690781\pi$
$401$	−22.5926	−1.12822	−0.564110	+	0.825700i	$0.690781\pi$
$402$	0	0
$403$	−1.16346	−0.0579563
$404$	0	0
$405$	2.21432	0.110030
$406$	0	0
$407$	10.2351	0.507333
$408$	0	0
$409$	9.42372	0.465973	0.232986	−	0.972480i	$-0.425150\pi$
$409$	9.42372	0.465973	0.232986	+	0.972480i	$0.425150\pi$
$410$	0	0
$411$	−13.0716	−0.644774
$412$	0	0
$413$	−20.8573	−1.02632
$414$	0	0
$415$	22.6637	1.11252
$416$	0	0
$417$	1.60639	0.0786655
$418$	0	0
$419$	−28.6909	−1.40164	−0.700821	−	0.713337i	$-0.747183\pi$
$419$	−28.6909	−1.40164	−0.700821	+	0.713337i	$0.747183\pi$
$420$	0	0
$421$	9.90813	0.482893	0.241446	−	0.970414i	$-0.422378\pi$
$421$	9.90813	0.482893	0.241446	+	0.970414i	$0.422378\pi$
$422$	0	0
$423$	10.8573	0.527899
$424$	0	0
$425$	0.555539	0.0269476
$426$	0	0
$427$	25.7146	1.24441
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−2.46028	−0.118508	−0.0592538	−	0.998243i	$-0.518872\pi$
$431$	−2.46028	−0.118508	−0.0592538	+	0.998243i	$0.518872\pi$
$432$	0	0
$433$	−13.2257	−0.635586	−0.317793	−	0.948160i	$-0.602942\pi$
$433$	−13.2257	−0.635586	−0.317793	+	0.948160i	$0.602942\pi$
$434$	0	0
$435$	14.3827	0.689598
$436$	0	0
$437$	−22.1146	−1.05789
$438$	0	0
$439$	39.2192	1.87183	0.935916	−	0.352223i	$-0.114574\pi$
$439$	39.2192	1.87183	0.935916	+	0.352223i	$0.114574\pi$
$440$	0	0
$441$	1.42864	0.0680305
$442$	0	0
$443$	7.87955	0.374369	0.187184	−	0.982325i	$-0.440064\pi$
$443$	7.87955	0.374369	0.187184	+	0.982325i	$0.440064\pi$
$444$	0	0
$445$	5.98571	0.283750
$446$	0	0
$447$	9.46520	0.447689
$448$	0	0
$449$	39.6336	1.87042	0.935212	−	0.354087i	$-0.115208\pi$
$449$	39.6336	1.87042	0.935212	+	0.354087i	$0.115208\pi$
$450$	0	0
$451$	−2.14764	−0.101129
$452$	0	0
$453$	−0.534795	−0.0251269
$454$	0	0
$455$	−6.42864	−0.301379
$456$	0	0
$457$	−20.9304	−0.979083	−0.489542	−	0.871980i	$-0.662836\pi$
$457$	−20.9304	−0.979083	−0.489542	+	0.871980i	$0.662836\pi$
$458$	0	0
$459$	5.73975	0.267908
$460$	0	0
$461$	−33.8435	−1.57625	−0.788124	−	0.615517i	$-0.788947\pi$
$461$	−33.8435	−1.57625	−0.788124	+	0.615517i	$0.788947\pi$
$462$	0	0
$463$	2.73483	0.127098	0.0635491	−	0.997979i	$-0.479758\pi$
$463$	2.73483	0.127098	0.0635491	+	0.997979i	$0.479758\pi$
$464$	0	0
$465$	−2.57628	−0.119472
$466$	0	0
$467$	−26.9634	−1.24772	−0.623859	−	0.781537i	$-0.714436\pi$
$467$	−26.9634	−1.24772	−0.623859	+	0.781537i	$0.714436\pi$
$468$	0	0
$469$	20.2351	0.934368
$470$	0	0
$471$	15.7605	0.726205
$472$	0	0
$473$	9.11753	0.419225
$474$	0	0
$475$	−0.401442	−0.0184194
$476$	0	0
$477$	−6.42864	−0.294347
$478$	0	0
$479$	−2.63651	−0.120465	−0.0602325	−	0.998184i	$-0.519184\pi$
$479$	−2.63651	−0.120465	−0.0602325	+	0.998184i	$0.519184\pi$
$480$	0	0
$481$	−10.2351	−0.466679
$482$	0	0
$483$	15.4795	0.704341
$484$	0	0
$485$	11.9081	0.540721
$486$	0	0
$487$	23.3383	1.05756	0.528780	−	0.848759i	$-0.322650\pi$
$487$	23.3383	1.05756	0.528780	+	0.848759i	$0.322650\pi$
$488$	0	0
$489$	6.21432	0.281021
$490$	0	0
$491$	−6.89877	−0.311337	−0.155668	−	0.987809i	$-0.549753\pi$
$491$	−6.89877	−0.311337	−0.155668	+	0.987809i	$0.549753\pi$
$492$	0	0
$493$	37.2815	1.67907
$494$	0	0
$495$	2.21432	0.0995263
$496$	0	0
$497$	31.9081	1.43128
$498$	0	0
$499$	5.16346	0.231148	0.115574	−	0.993299i	$-0.463129\pi$
$499$	5.16346	0.231148	0.115574	+	0.993299i	$0.463129\pi$
$500$	0	0
$501$	14.6637	0.655126
$502$	0	0
$503$	−2.91750	−0.130085	−0.0650425	−	0.997882i	$-0.520718\pi$
$503$	−2.91750	−0.130085	−0.0650425	+	0.997882i	$0.520718\pi$
$504$	0	0
$505$	42.4242	1.88785
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	27.0114	1.19726	0.598629	−	0.801026i	$-0.295712\pi$
$509$	27.0114	1.19726	0.598629	+	0.801026i	$0.295712\pi$
$510$	0	0
$511$	6.62222	0.292950
$512$	0	0
$513$	−4.14764	−0.183123
$514$	0	0
$515$	13.7146	0.604336
$516$	0	0
$517$	10.8573	0.477503
$518$	0	0
$519$	−4.49532	−0.197322
$520$	0	0
$521$	11.3907	0.499035	0.249518	−	0.968370i	$-0.419728\pi$
$521$	11.3907	0.499035	0.249518	+	0.968370i	$0.419728\pi$
$522$	0	0
$523$	41.1876	1.80101	0.900504	−	0.434848i	$-0.143198\pi$
$523$	41.1876	1.80101	0.900504	+	0.434848i	$0.143198\pi$
$524$	0	0
$525$	0.280996	0.0122637
$526$	0	0
$527$	−6.67799	−0.290898
$528$	0	0
$529$	5.42864	0.236028
$530$	0	0
$531$	−7.18421	−0.311768
$532$	0	0
$533$	2.14764	0.0930248
$534$	0	0
$535$	13.1526	0.568635
$536$	0	0
$537$	9.46520	0.408454
$538$	0	0
$539$	1.42864	0.0615359
$540$	0	0
$541$	37.3461	1.60564	0.802818	−	0.596224i	$-0.203333\pi$
$541$	37.3461	1.60564	0.802818	+	0.596224i	$0.203333\pi$
$542$	0	0
$543$	17.5669	0.753868
$544$	0	0
$545$	−11.1526	−0.477723
$546$	0	0
$547$	−4.76202	−0.203609	−0.101805	−	0.994804i	$-0.532462\pi$
$547$	−4.76202	−0.203609	−0.101805	+	0.994804i	$0.532462\pi$
$548$	0	0
$549$	8.85728	0.378019
$550$	0	0
$551$	−26.9403	−1.14769
$552$	0	0
$553$	5.61285	0.238683
$554$	0	0
$555$	−22.6637	−0.962021
$556$	0	0
$557$	−23.5669	−0.998562	−0.499281	−	0.866440i	$-0.666403\pi$
$557$	−23.5669	−0.998562	−0.499281	+	0.866440i	$0.666403\pi$
$558$	0	0
$559$	−9.11753	−0.385631
$560$	0	0
$561$	5.73975	0.242332
$562$	0	0
$563$	−13.1526	−0.554315	−0.277157	−	0.960825i	$-0.589392\pi$
$563$	−13.1526	−0.554315	−0.277157	+	0.960825i	$0.589392\pi$
$564$	0	0
$565$	20.9906	0.883083
$566$	0	0
$567$	2.90321	0.121923
$568$	0	0
$569$	22.5970	0.947317	0.473658	−	0.880709i	$-0.342933\pi$
$569$	22.5970	0.947317	0.473658	+	0.880709i	$0.342933\pi$
$570$	0	0
$571$	−23.3941	−0.979012	−0.489506	−	0.872000i	$-0.662823\pi$
$571$	−23.3941	−0.979012	−0.489506	+	0.872000i	$0.662823\pi$
$572$	0	0
$573$	−12.7239	−0.531550
$574$	0	0
$575$	0.516060	0.0215212
$576$	0	0
$577$	−45.7877	−1.90617	−0.953083	−	0.302708i	$-0.902109\pi$
$577$	−45.7877	−1.90617	−0.953083	+	0.302708i	$0.902109\pi$
$578$	0	0
$579$	−15.4336	−0.641397
$580$	0	0
$581$	29.7146	1.23277
$582$	0	0
$583$	−6.42864	−0.266247
$584$	0	0
$585$	−2.21432	−0.0915509
$586$	0	0
$587$	−7.30465	−0.301495	−0.150748	−	0.988572i	$-0.548168\pi$
$587$	−7.30465	−0.301495	−0.150748	+	0.988572i	$0.548168\pi$
$588$	0	0
$589$	4.82564	0.198837
$590$	0	0
$591$	22.4844	0.924885
$592$	0	0
$593$	−2.28100	−0.0936693	−0.0468346	−	0.998903i	$-0.514913\pi$
$593$	−2.28100	−0.0936693	−0.0468346	+	0.998903i	$0.514913\pi$
$594$	0	0
$595$	−36.8988	−1.51270
$596$	0	0
$597$	−1.33630	−0.0546910
$598$	0	0
$599$	−40.6035	−1.65901	−0.829507	−	0.558497i	$-0.811378\pi$
$599$	−40.6035	−1.65901	−0.829507	+	0.558497i	$0.811378\pi$
$600$	0	0
$601$	5.73329	0.233866	0.116933	−	0.993140i	$-0.462694\pi$
$601$	5.73329	0.233866	0.116933	+	0.993140i	$0.462694\pi$
$602$	0	0
$603$	6.96989	0.283836
$604$	0	0
$605$	2.21432	0.0900249
$606$	0	0
$607$	−5.31111	−0.215571	−0.107786	−	0.994174i	$-0.534376\pi$
$607$	−5.31111	−0.215571	−0.107786	+	0.994174i	$0.534376\pi$
$608$	0	0
$609$	18.8573	0.764136
$610$	0	0
$611$	−10.8573	−0.439238
$612$	0	0
$613$	41.4434	1.67388	0.836942	−	0.547292i	$-0.184341\pi$
$613$	41.4434	1.67388	0.836942	+	0.547292i	$0.184341\pi$
$614$	0	0
$615$	4.75557	0.191763
$616$	0	0
$617$	−13.6938	−0.551292	−0.275646	−	0.961259i	$-0.588892\pi$
$617$	−13.6938	−0.551292	−0.275646	+	0.961259i	$0.588892\pi$
$618$	0	0
$619$	8.40790	0.337942	0.168971	−	0.985621i	$-0.445956\pi$
$619$	8.40790	0.337942	0.168971	+	0.985621i	$0.445956\pi$
$620$	0	0
$621$	5.33185	0.213960
$622$	0	0
$623$	7.84791	0.314420
$624$	0	0
$625$	−24.5067	−0.980268
$626$	0	0
$627$	−4.14764	−0.165641
$628$	0	0
$629$	−58.7467	−2.34238
$630$	0	0
$631$	−17.3481	−0.690619	−0.345309	−	0.938489i	$-0.612226\pi$
$631$	−17.3481	−0.690619	−0.345309	+	0.938489i	$0.612226\pi$
$632$	0	0
$633$	−8.00645	−0.318228
$634$	0	0
$635$	23.6686	0.939261
$636$	0	0
$637$	−1.42864	−0.0566048
$638$	0	0
$639$	10.9906	0.434783
$640$	0	0
$641$	−2.09187	−0.0826237	−0.0413119	−	0.999146i	$-0.513154\pi$
$641$	−2.09187	−0.0826237	−0.0413119	+	0.999146i	$0.513154\pi$
$642$	0	0
$643$	−37.5417	−1.48050	−0.740251	−	0.672331i	$-0.765293\pi$
$643$	−37.5417	−1.48050	−0.740251	+	0.672331i	$0.765293\pi$
$644$	0	0
$645$	−20.1891	−0.794946
$646$	0	0
$647$	−5.84791	−0.229905	−0.114953	−	0.993371i	$-0.536672\pi$
$647$	−5.84791	−0.229905	−0.114953	+	0.993371i	$0.536672\pi$
$648$	0	0
$649$	−7.18421	−0.282005
$650$	0	0
$651$	−3.37778	−0.132386
$652$	0	0
$653$	4.28544	0.167702	0.0838512	−	0.996478i	$-0.473278\pi$
$653$	4.28544	0.167702	0.0838512	+	0.996478i	$0.473278\pi$
$654$	0	0
$655$	15.6128	0.610044
$656$	0	0
$657$	2.28100	0.0889901
$658$	0	0
$659$	−3.63158	−0.141466	−0.0707332	−	0.997495i	$-0.522534\pi$
$659$	−3.63158	−0.141466	−0.0707332	+	0.997495i	$0.522534\pi$
$660$	0	0
$661$	−26.3368	−1.02438	−0.512191	−	0.858872i	$-0.671166\pi$
$661$	−26.3368	−1.02438	−0.512191	+	0.858872i	$0.671166\pi$
$662$	0	0
$663$	−5.73975	−0.222913
$664$	0	0
$665$	26.6637	1.03397
$666$	0	0
$667$	34.6321	1.34096
$668$	0	0
$669$	6.02074	0.232775
$670$	0	0
$671$	8.85728	0.341931
$672$	0	0
$673$	28.8702	1.11286	0.556432	−	0.830893i	$-0.312170\pi$
$673$	28.8702	1.11286	0.556432	+	0.830893i	$0.312170\pi$
$674$	0	0
$675$	0.0967881	0.00372537
$676$	0	0
$677$	−15.9935	−0.614682	−0.307341	−	0.951599i	$-0.599439\pi$
$677$	−15.9935	−0.614682	−0.307341	+	0.951599i	$0.599439\pi$
$678$	0	0
$679$	15.6128	0.599166
$680$	0	0
$681$	13.2716	0.508570
$682$	0	0
$683$	24.3180	0.930504	0.465252	−	0.885178i	$-0.345964\pi$
$683$	24.3180	0.930504	0.465252	+	0.885178i	$0.345964\pi$
$684$	0	0
$685$	28.9447	1.10592
$686$	0	0
$687$	−14.7556	−0.562960
$688$	0	0
$689$	6.42864	0.244912
$690$	0	0
$691$	−47.2464	−1.79734	−0.898670	−	0.438626i	$-0.855465\pi$
$691$	−47.2464	−1.79734	−0.898670	+	0.438626i	$0.855465\pi$
$692$	0	0
$693$	2.90321	0.110284
$694$	0	0
$695$	−3.55707	−0.134927
$696$	0	0
$697$	12.3269	0.466916
$698$	0	0
$699$	−19.6479	−0.743151
$700$	0	0
$701$	33.1403	1.25169	0.625846	−	0.779947i	$-0.284754\pi$
$701$	33.1403	1.25169	0.625846	+	0.779947i	$0.284754\pi$
$702$	0	0
$703$	42.4514	1.60108
$704$	0	0
$705$	−24.0415	−0.905455
$706$	0	0
$707$	55.6227	2.09191
$708$	0	0
$709$	−17.8894	−0.671850	−0.335925	−	0.941889i	$-0.609049\pi$
$709$	−17.8894	−0.671850	−0.335925	+	0.941889i	$0.609049\pi$
$710$	0	0
$711$	1.93332	0.0725053
$712$	0	0
$713$	−6.20342	−0.232320
$714$	0	0
$715$	−2.21432	−0.0828109
$716$	0	0
$717$	14.8113	0.553140
$718$	0	0
$719$	34.8385	1.29926	0.649629	−	0.760251i	$-0.274924\pi$
$719$	34.8385	1.29926	0.649629	+	0.760251i	$0.274924\pi$
$720$	0	0
$721$	17.9813	0.669657
$722$	0	0
$723$	−23.5526	−0.875932
$724$	0	0
$725$	0.628669	0.0233482
$726$	0	0
$727$	−25.5812	−0.948754	−0.474377	−	0.880322i	$-0.657327\pi$
$727$	−25.5812	−0.948754	−0.474377	+	0.880322i	$0.657327\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−52.3323	−1.93558
$732$	0	0
$733$	40.4657	1.49463	0.747317	−	0.664468i	$-0.231342\pi$
$733$	40.4657	1.49463	0.747317	+	0.664468i	$0.231342\pi$
$734$	0	0
$735$	−3.16346	−0.116686
$736$	0	0
$737$	6.96989	0.256739
$738$	0	0
$739$	−27.1699	−0.999462	−0.499731	−	0.866181i	$-0.666568\pi$
$739$	−27.1699	−0.999462	−0.499731	+	0.866181i	$0.666568\pi$
$740$	0	0
$741$	4.14764	0.152367
$742$	0	0
$743$	−51.6543	−1.89501	−0.947507	−	0.319735i	$-0.896406\pi$
$743$	−51.6543	−1.89501	−0.947507	+	0.319735i	$0.896406\pi$
$744$	0	0
$745$	−20.9590	−0.767878
$746$	0	0
$747$	10.2351	0.374481
$748$	0	0
$749$	17.2444	0.630098
$750$	0	0
$751$	33.9398	1.23848	0.619240	−	0.785202i	$-0.287441\pi$
$751$	33.9398	1.23848	0.619240	+	0.785202i	$0.287441\pi$
$752$	0	0
$753$	−6.23506	−0.227218
$754$	0	0
$755$	1.18421	0.0430977
$756$	0	0
$757$	17.3002	0.628787	0.314393	−	0.949293i	$-0.398199\pi$
$757$	17.3002	0.628787	0.314393	+	0.949293i	$0.398199\pi$
$758$	0	0
$759$	5.33185	0.193534
$760$	0	0
$761$	10.4014	0.377052	0.188526	−	0.982068i	$-0.439629\pi$
$761$	10.4014	0.377052	0.188526	+	0.982068i	$0.439629\pi$
$762$	0	0
$763$	−14.6222	−0.529360
$764$	0	0
$765$	−12.7096	−0.459518
$766$	0	0
$767$	7.18421	0.259407
$768$	0	0
$769$	0.755569	0.0272465	0.0136233	−	0.999907i	$-0.495663\pi$
$769$	0.755569	0.0272465	0.0136233	+	0.999907i	$0.495663\pi$
$770$	0	0
$771$	−1.73329	−0.0624231
$772$	0	0
$773$	−12.6430	−0.454736	−0.227368	−	0.973809i	$-0.573012\pi$
$773$	−12.6430	−0.454736	−0.227368	+	0.973809i	$0.573012\pi$
$774$	0	0
$775$	−0.112610	−0.00404505
$776$	0	0
$777$	−29.7146	−1.06600
$778$	0	0
$779$	−8.90766	−0.319150
$780$	0	0
$781$	10.9906	0.393276
$782$	0	0
$783$	6.49532	0.232124
$784$	0	0
$785$	−34.8988	−1.24559
$786$	0	0
$787$	3.19850	0.114014	0.0570071	−	0.998374i	$-0.481844\pi$
$787$	3.19850	0.114014	0.0570071	+	0.998374i	$0.481844\pi$
$788$	0	0
$789$	−7.18421	−0.255765
$790$	0	0
$791$	27.5210	0.978533
$792$	0	0
$793$	−8.85728	−0.314531
$794$	0	0
$795$	14.2351	0.504866
$796$	0	0
$797$	13.3176	0.471732	0.235866	−	0.971786i	$-0.424207\pi$
$797$	13.3176	0.471732	0.235866	+	0.971786i	$0.424207\pi$
$798$	0	0
$799$	−62.3180	−2.20465
$800$	0	0
$801$	2.70318	0.0955122
$802$	0	0
$803$	2.28100	0.0804946
$804$	0	0
$805$	−34.2766	−1.20809
$806$	0	0
$807$	14.3368	0.504678
$808$	0	0
$809$	−38.9338	−1.36884	−0.684420	−	0.729088i	$-0.739944\pi$
$809$	−38.9338	−1.36884	−0.684420	+	0.729088i	$0.739944\pi$
$810$	0	0
$811$	9.09679	0.319431	0.159716	−	0.987163i	$-0.448942\pi$
$811$	9.09679	0.319431	0.159716	+	0.987163i	$0.448942\pi$
$812$	0	0
$813$	−6.47457	−0.227073
$814$	0	0
$815$	−13.7605	−0.482009
$816$	0	0
$817$	37.8163	1.32302
$818$	0	0
$819$	−2.90321	−0.101446
$820$	0	0
$821$	34.1891	1.19321	0.596604	−	0.802535i	$-0.296516\pi$
$821$	34.1891	1.19321	0.596604	+	0.802535i	$0.296516\pi$
$822$	0	0
$823$	−12.2163	−0.425834	−0.212917	−	0.977070i	$-0.568296\pi$
$823$	−12.2163	−0.425834	−0.212917	+	0.977070i	$0.568296\pi$
$824$	0	0
$825$	0.0967881	0.00336973
$826$	0	0
$827$	−8.14764	−0.283321	−0.141661	−	0.989915i	$-0.545244\pi$
$827$	−8.14764	−0.283321	−0.141661	+	0.989915i	$0.545244\pi$
$828$	0	0
$829$	14.8573	0.516015	0.258007	−	0.966143i	$-0.416934\pi$
$829$	14.8573	0.516015	0.258007	+	0.966143i	$0.416934\pi$
$830$	0	0
$831$	−4.19358	−0.145474
$832$	0	0
$833$	−8.20003	−0.284114
$834$	0	0
$835$	−32.4701	−1.12368
$836$	0	0
$837$	−1.16346	−0.0402152
$838$	0	0
$839$	−24.4415	−0.843816	−0.421908	−	0.906639i	$-0.638639\pi$
$839$	−24.4415	−0.843816	−0.421908	+	0.906639i	$0.638639\pi$
$840$	0	0
$841$	13.1891	0.454798
$842$	0	0
$843$	−18.8430	−0.648987
$844$	0	0
$845$	2.21432	0.0761749
$846$	0	0
$847$	2.90321	0.0997555
$848$	0	0
$849$	7.15902	0.245697
$850$	0	0
$851$	−54.5718	−1.87070
$852$	0	0
$853$	−30.9688	−1.06035	−0.530176	−	0.847887i	$-0.677874\pi$
$853$	−30.9688	−1.06035	−0.530176	+	0.847887i	$0.677874\pi$
$854$	0	0
$855$	9.18421	0.314093
$856$	0	0
$857$	0.463673	0.0158388	0.00791939	−	0.999969i	$-0.497479\pi$
$857$	0.463673	0.0158388	0.00791939	+	0.999969i	$0.497479\pi$
$858$	0	0
$859$	39.0291	1.33165	0.665827	−	0.746106i	$-0.268079\pi$
$859$	39.0291	1.33165	0.665827	+	0.746106i	$0.268079\pi$
$860$	0	0
$861$	6.23506	0.212490
$862$	0	0
$863$	−40.3368	−1.37308	−0.686540	−	0.727092i	$-0.740871\pi$
$863$	−40.3368	−1.37308	−0.686540	+	0.727092i	$0.740871\pi$
$864$	0	0
$865$	9.95407	0.338448
$866$	0	0
$867$	−15.9447	−0.541510
$868$	0	0
$869$	1.93332	0.0655835
$870$	0	0
$871$	−6.96989	−0.236166
$872$	0	0
$873$	5.37778	0.182010
$874$	0	0
$875$	−32.7654	−1.10767
$876$	0	0
$877$	−13.6543	−0.461074	−0.230537	−	0.973064i	$-0.574048\pi$
$877$	−13.6543	−0.461074	−0.230537	+	0.973064i	$0.574048\pi$
$878$	0	0
$879$	−27.0464	−0.912253
$880$	0	0
$881$	−16.5718	−0.558319	−0.279160	−	0.960245i	$-0.590056\pi$
$881$	−16.5718	−0.558319	−0.279160	+	0.960245i	$0.590056\pi$
$882$	0	0
$883$	12.6450	0.425537	0.212769	−	0.977103i	$-0.431752\pi$
$883$	12.6450	0.425537	0.212769	+	0.977103i	$0.431752\pi$
$884$	0	0
$885$	15.9081	0.534746
$886$	0	0
$887$	4.10171	0.137722	0.0688610	−	0.997626i	$-0.478064\pi$
$887$	4.10171	0.137722	0.0688610	+	0.997626i	$0.478064\pi$
$888$	0	0
$889$	31.0321	1.04078
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	45.0321	1.50694
$894$	0	0
$895$	−20.9590	−0.700582
$896$	0	0
$897$	−5.33185	−0.178025
$898$	0	0
$899$	−7.55707	−0.252042
$900$	0	0
$901$	36.8988	1.22928
$902$	0	0
$903$	−26.4701	−0.880871
$904$	0	0
$905$	−38.8988	−1.29304
$906$	0	0
$907$	−47.5910	−1.58023	−0.790117	−	0.612956i	$-0.789980\pi$
$907$	−47.5910	−1.58023	−0.790117	+	0.612956i	$0.789980\pi$
$908$	0	0
$909$	19.1590	0.635465
$910$	0	0
$911$	16.4001	0.543358	0.271679	−	0.962388i	$-0.412421\pi$
$911$	16.4001	0.543358	0.271679	+	0.962388i	$0.412421\pi$
$912$	0	0
$913$	10.2351	0.338731
$914$	0	0
$915$	−19.6128	−0.648381
$916$	0	0
$917$	20.4701	0.675983
$918$	0	0
$919$	36.8736	1.21635	0.608174	−	0.793804i	$-0.291902\pi$
$919$	36.8736	1.21635	0.608174	+	0.793804i	$0.291902\pi$
$920$	0	0
$921$	−6.60793	−0.217739
$922$	0	0
$923$	−10.9906	−0.361761
$924$	0	0
$925$	−0.990632	−0.0325718
$926$	0	0
$927$	6.19358	0.203424
$928$	0	0
$929$	56.4721	1.85279	0.926395	−	0.376552i	$-0.122891\pi$
$929$	56.4721	1.85279	0.926395	+	0.376552i	$0.122891\pi$
$930$	0	0
$931$	5.92549	0.194200
$932$	0	0
$933$	−32.5575	−1.06589
$934$	0	0
$935$	−12.7096	−0.415650
$936$	0	0
$937$	15.7748	0.515340	0.257670	−	0.966233i	$-0.417045\pi$
$937$	15.7748	0.515340	0.257670	+	0.966233i	$0.417045\pi$
$938$	0	0
$939$	29.9037	0.975870
$940$	0	0
$941$	−16.3126	−0.531777	−0.265888	−	0.964004i	$-0.585665\pi$
$941$	−16.3126	−0.531777	−0.265888	+	0.964004i	$0.585665\pi$
$942$	0	0
$943$	11.4509	0.372893
$944$	0	0
$945$	−6.42864	−0.209124
$946$	0	0
$947$	12.5433	0.407601	0.203801	−	0.979012i	$-0.434671\pi$
$947$	12.5433	0.407601	0.203801	+	0.979012i	$0.434671\pi$
$948$	0	0
$949$	−2.28100	−0.0740443
$950$	0	0
$951$	28.8464	0.935408
$952$	0	0
$953$	−38.9369	−1.26129	−0.630644	−	0.776072i	$-0.717209\pi$
$953$	−38.9369	−1.26129	−0.630644	+	0.776072i	$0.717209\pi$
$954$	0	0
$955$	28.1748	0.911716
$956$	0	0
$957$	6.49532	0.209964
$958$	0	0
$959$	37.9496	1.22546
$960$	0	0
$961$	−29.6464	−0.956334
$962$	0	0
$963$	5.93978	0.191407
$964$	0	0
$965$	34.1748	1.10013
$966$	0	0
$967$	−24.6307	−0.792069	−0.396035	−	0.918236i	$-0.629614\pi$
$967$	−24.6307	−0.792069	−0.396035	+	0.918236i	$0.629614\pi$
$968$	0	0
$969$	23.8064	0.764773
$970$	0	0
$971$	−13.6904	−0.439347	−0.219673	−	0.975573i	$-0.570499\pi$
$971$	−13.6904	−0.439347	−0.219673	+	0.975573i	$0.570499\pi$
$972$	0	0
$973$	−4.66370	−0.149511
$974$	0	0
$975$	−0.0967881	−0.00309970
$976$	0	0
$977$	−0.0938736	−0.00300328	−0.00150164	−	0.999999i	$-0.500478\pi$
$977$	−0.0938736	−0.00300328	−0.00150164	+	0.999999i	$0.500478\pi$
$978$	0	0
$979$	2.70318	0.0863941
$980$	0	0
$981$	−5.03657	−0.160805
$982$	0	0
$983$	31.3176	0.998875	0.499438	−	0.866350i	$-0.333540\pi$
$983$	31.3176	0.998875	0.499438	+	0.866350i	$0.333540\pi$
$984$	0	0
$985$	−49.7877	−1.58637
$986$	0	0
$987$	−31.5210	−1.00332
$988$	0	0
$989$	−48.6133	−1.54581
$990$	0	0
$991$	43.5339	1.38290	0.691450	−	0.722425i	$-0.256972\pi$
$991$	43.5339	1.38290	0.691450	+	0.722425i	$0.256972\pi$
$992$	0	0
$993$	−18.3477	−0.582246
$994$	0	0
$995$	2.95899	0.0938063
$996$	0	0
$997$	−33.8578	−1.07229	−0.536143	−	0.844127i	$-0.680119\pi$
$997$	−33.8578	−1.07229	−0.536143	+	0.844127i	$0.680119\pi$
$998$	0	0
$999$	−10.2351	−0.323823

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bq.1.3		3
4.3	odd	2		3432.2.a.p.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.p.1.3	✓	3	4.3	odd	2
6864.2.a.bq.1.3		3	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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.21432 q^{5} +2.90321 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.21432 q^{5} +2.90321 q^{7} +1.00000 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.21432 q^{15} -5.73975 q^{17} +4.14764 q^{19} -2.90321 q^{21} -5.33185 q^{23} -0.0967881 q^{25} -1.00000 q^{27} -6.49532 q^{29} +1.16346 q^{31} -1.00000 q^{33} +6.42864 q^{35} +10.2351 q^{37} +1.00000 q^{39} -2.14764 q^{41} +9.11753 q^{43} +2.21432 q^{45} +10.8573 q^{47} +1.42864 q^{49} +5.73975 q^{51} -6.42864 q^{53} +2.21432 q^{55} -4.14764 q^{57} -7.18421 q^{59} +8.85728 q^{61} +2.90321 q^{63} -2.21432 q^{65} +6.96989 q^{67} +5.33185 q^{69} +10.9906 q^{71} +2.28100 q^{73} +0.0967881 q^{75} +2.90321 q^{77} +1.93332 q^{79} +1.00000 q^{81} +10.2351 q^{83} -12.7096 q^{85} +6.49532 q^{87} +2.70318 q^{89} -2.90321 q^{91} -1.16346 q^{93} +9.18421 q^{95} +5.37778 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 2 q^{7} + 3 q^{9} + 3 q^{11} - 3 q^{13} - 4 q^{17} + 6 q^{19} - 2 q^{21} + 4 q^{23} - 7 q^{25} - 3 q^{27} - 6 q^{29} + 10 q^{31} - 3 q^{33} + 6 q^{35} + 4 q^{37} + 3 q^{39} + 14 q^{43} + 6 q^{47} - 9 q^{49} + 4 q^{51} - 6 q^{53} - 6 q^{57} - 8 q^{59} + 2 q^{63} + 14 q^{67} - 4 q^{69} + 6 q^{71} + 7 q^{75} + 2 q^{77} + 6 q^{79} + 3 q^{81} + 4 q^{83} - 18 q^{85} + 6 q^{87} + 2 q^{89} - 2 q^{91} - 10 q^{93} + 14 q^{95} + 16 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 2 * q^7 + 3 * q^9 + 3 * q^11 - 3 * q^13 - 4 * q^17 + 6 * q^19 - 2 * q^21 + 4 * q^23 - 7 * q^25 - 3 * q^27 - 6 * q^29 + 10 * q^31 - 3 * q^33 + 6 * q^35 + 4 * q^37 + 3 * q^39 + 14 * q^43 + 6 * q^47 - 9 * q^49 + 4 * q^51 - 6 * q^53 - 6 * q^57 - 8 * q^59 + 2 * q^63 + 14 * q^67 - 4 * q^69 + 6 * q^71 + 7 * q^75 + 2 * q^77 + 6 * q^79 + 3 * q^81 + 4 * q^83 - 18 * q^85 + 6 * q^87 + 2 * q^89 - 2 * q^91 - 10 * q^93 + 14 * q^95 + 16 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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