LMFDB - Embedded newform 6864.2.a.bi.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:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.732051 q^{15} -6.19615 q^{17} +5.46410 q^{19} -2.00000 q^{23} -4.46410 q^{25} +1.00000 q^{27} -2.19615 q^{29} -4.19615 q^{31} -1.00000 q^{33} +10.3923 q^{37} -1.00000 q^{39} -4.53590 q^{41} +2.73205 q^{43} +0.732051 q^{45} -6.92820 q^{47} -7.00000 q^{49} -6.19615 q^{51} -4.92820 q^{53} -0.732051 q^{55} +5.46410 q^{57} -2.53590 q^{59} -7.46410 q^{61} -0.732051 q^{65} +10.7321 q^{67} -2.00000 q^{69} -1.46410 q^{71} -9.46410 q^{73} -4.46410 q^{75} +9.26795 q^{79} +1.00000 q^{81} +6.92820 q^{83} -4.53590 q^{85} -2.19615 q^{87} -5.80385 q^{89} -4.19615 q^{93} +4.00000 q^{95} -14.3923 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 2 q^{11} - 2 q^{13} - 2 q^{15} - 2 q^{17} + 4 q^{19} - 4 q^{23} - 2 q^{25} + 2 q^{27} + 6 q^{29} + 2 q^{31} - 2 q^{33} - 2 q^{39} - 16 q^{41} + 2 q^{43} - 2 q^{45} - 14 q^{49} - 2 q^{51} + 4 q^{53} + 2 q^{55} + 4 q^{57} - 12 q^{59} - 8 q^{61} + 2 q^{65} + 18 q^{67} - 4 q^{69} + 4 q^{71} - 12 q^{73} - 2 q^{75} + 22 q^{79} + 2 q^{81} - 16 q^{85} + 6 q^{87} - 22 q^{89} + 2 q^{93} + 8 q^{95} - 8 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 - 2 * q^11 - 2 * q^13 - 2 * q^15 - 2 * q^17 + 4 * q^19 - 4 * q^23 - 2 * q^25 + 2 * q^27 + 6 * q^29 + 2 * q^31 - 2 * q^33 - 2 * q^39 - 16 * q^41 + 2 * q^43 - 2 * q^45 - 14 * q^49 - 2 * q^51 + 4 * q^53 + 2 * q^55 + 4 * q^57 - 12 * q^59 - 8 * q^61 + 2 * q^65 + 18 * q^67 - 4 * q^69 + 4 * q^71 - 12 * q^73 - 2 * q^75 + 22 * q^79 + 2 * q^81 - 16 * q^85 + 6 * q^87 - 22 * q^89 + 2 * q^93 + 8 * 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
$5$	0.732051	0.327383	0.163692	−	0.986512i	$-0.447660\pi$
$5$	0.732051	0.327383	0.163692	+	0.986512i	$0.447660\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	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
$15$	0.732051	0.189015
$16$	0	0
$17$	−6.19615	−1.50279	−0.751394	−	0.659854i	$-0.770618\pi$
$17$	−6.19615	−1.50279	−0.751394	+	0.659854i	$0.770618\pi$
$18$	0	0
$19$	5.46410	1.25355	0.626775	−	0.779200i	$-0.284374\pi$
$19$	5.46410	1.25355	0.626775	+	0.779200i	$0.284374\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	−4.46410	−0.892820
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.19615	−0.407815	−0.203908	−	0.978990i	$-0.565364\pi$
$29$	−2.19615	−0.407815	−0.203908	+	0.978990i	$0.565364\pi$
$30$	0	0
$31$	−4.19615	−0.753651	−0.376826	−	0.926284i	$-0.622984\pi$
$31$	−4.19615	−0.753651	−0.376826	+	0.926284i	$0.622984\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.3923	1.70848	0.854242	−	0.519875i	$-0.174022\pi$
$37$	10.3923	1.70848	0.854242	+	0.519875i	$0.174022\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−4.53590	−0.708388	−0.354194	−	0.935172i	$-0.615245\pi$
$41$	−4.53590	−0.708388	−0.354194	+	0.935172i	$0.615245\pi$
$42$	0	0
$43$	2.73205	0.416634	0.208317	−	0.978061i	$-0.433201\pi$
$43$	2.73205	0.416634	0.208317	+	0.978061i	$0.433201\pi$
$44$	0	0
$45$	0.732051	0.109128
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−6.19615	−0.867635
$52$	0	0
$53$	−4.92820	−0.676941	−0.338470	−	0.940977i	$-0.609909\pi$
$53$	−4.92820	−0.676941	−0.338470	+	0.940977i	$0.609909\pi$
$54$	0	0
$55$	−0.732051	−0.0987097
$56$	0	0
$57$	5.46410	0.723738
$58$	0	0
$59$	−2.53590	−0.330146	−0.165073	−	0.986281i	$-0.552786\pi$
$59$	−2.53590	−0.330146	−0.165073	+	0.986281i	$0.552786\pi$
$60$	0	0
$61$	−7.46410	−0.955680	−0.477840	−	0.878447i	$-0.658580\pi$
$61$	−7.46410	−0.955680	−0.477840	+	0.878447i	$0.658580\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.732051	−0.0907997
$66$	0	0
$67$	10.7321	1.31113	0.655564	−	0.755139i	$-0.272431\pi$
$67$	10.7321	1.31113	0.655564	+	0.755139i	$0.272431\pi$
$68$	0	0
$69$	−2.00000	−0.240772
$70$	0	0
$71$	−1.46410	−0.173757	−0.0868784	−	0.996219i	$-0.527689\pi$
$71$	−1.46410	−0.173757	−0.0868784	+	0.996219i	$0.527689\pi$
$72$	0	0
$73$	−9.46410	−1.10769	−0.553845	−	0.832620i	$-0.686840\pi$
$73$	−9.46410	−1.10769	−0.553845	+	0.832620i	$0.686840\pi$
$74$	0	0
$75$	−4.46410	−0.515470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	9.26795	1.04273	0.521363	−	0.853335i	$-0.325424\pi$
$79$	9.26795	1.04273	0.521363	+	0.853335i	$0.325424\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.92820	0.760469	0.380235	−	0.924890i	$-0.375843\pi$
$83$	6.92820	0.760469	0.380235	+	0.924890i	$0.375843\pi$
$84$	0	0
$85$	−4.53590	−0.491987
$86$	0	0
$87$	−2.19615	−0.235452
$88$	0	0
$89$	−5.80385	−0.615207	−0.307603	−	0.951515i	$-0.599527\pi$
$89$	−5.80385	−0.615207	−0.307603	+	0.951515i	$0.599527\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.19615	−0.435121
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	−14.3923	−1.46132	−0.730659	−	0.682743i	$-0.760787\pi$
$97$	−14.3923	−1.46132	−0.730659	+	0.682743i	$0.760787\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−1.80385	−0.179490	−0.0897448	−	0.995965i	$-0.528605\pi$
$101$	−1.80385	−0.179490	−0.0897448	+	0.995965i	$0.528605\pi$
$102$	0	0
$103$	−8.39230	−0.826918	−0.413459	−	0.910523i	$-0.635680\pi$
$103$	−8.39230	−0.826918	−0.413459	+	0.910523i	$0.635680\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.53590	−0.631849	−0.315925	−	0.948784i	$-0.602315\pi$
$107$	−6.53590	−0.631849	−0.315925	+	0.948784i	$0.602315\pi$
$108$	0	0
$109$	−5.46410	−0.523366	−0.261683	−	0.965154i	$-0.584277\pi$
$109$	−5.46410	−0.523366	−0.261683	+	0.965154i	$0.584277\pi$
$110$	0	0
$111$	10.3923	0.986394
$112$	0	0
$113$	−3.07180	−0.288970	−0.144485	−	0.989507i	$-0.546153\pi$
$113$	−3.07180	−0.288970	−0.144485	+	0.989507i	$0.546153\pi$
$114$	0	0
$115$	−1.46410	−0.136528
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.53590	−0.408988
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	−2.73205	−0.242430	−0.121215	−	0.992626i	$-0.538679\pi$
$127$	−2.73205	−0.242430	−0.121215	+	0.992626i	$0.538679\pi$
$128$	0	0
$129$	2.73205	0.240544
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.732051	0.0630049
$136$	0	0
$137$	2.19615	0.187630	0.0938150	−	0.995590i	$-0.470094\pi$
$137$	2.19615	0.187630	0.0938150	+	0.995590i	$0.470094\pi$
$138$	0	0
$139$	1.26795	0.107546	0.0537730	−	0.998553i	$-0.482875\pi$
$139$	1.26795	0.107546	0.0537730	+	0.998553i	$0.482875\pi$
$140$	0	0
$141$	−6.92820	−0.583460
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−1.60770	−0.133512
$146$	0	0
$147$	−7.00000	−0.577350
$148$	0	0
$149$	4.92820	0.403734	0.201867	−	0.979413i	$-0.435299\pi$
$149$	4.92820	0.403734	0.201867	+	0.979413i	$0.435299\pi$
$150$	0	0
$151$	9.46410	0.770178	0.385089	−	0.922880i	$-0.374171\pi$
$151$	9.46410	0.770178	0.385089	+	0.922880i	$0.374171\pi$
$152$	0	0
$153$	−6.19615	−0.500929
$154$	0	0
$155$	−3.07180	−0.246733
$156$	0	0
$157$	−6.53590	−0.521621	−0.260811	−	0.965390i	$-0.583990\pi$
$157$	−6.53590	−0.521621	−0.260811	+	0.965390i	$0.583990\pi$
$158$	0	0
$159$	−4.92820	−0.390832
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.33975	−0.496567	−0.248284	−	0.968687i	$-0.579866\pi$
$163$	−6.33975	−0.496567	−0.248284	+	0.968687i	$0.579866\pi$
$164$	0	0
$165$	−0.732051	−0.0569901
$166$	0	0
$167$	−5.07180	−0.392467	−0.196234	−	0.980557i	$-0.562871\pi$
$167$	−5.07180	−0.392467	−0.196234	+	0.980557i	$0.562871\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	5.46410	0.417850
$172$	0	0
$173$	−1.80385	−0.137144	−0.0685720	−	0.997646i	$-0.521844\pi$
$173$	−1.80385	−0.137144	−0.0685720	+	0.997646i	$0.521844\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.53590	−0.190610
$178$	0	0
$179$	−19.8564	−1.48414	−0.742069	−	0.670324i	$-0.766155\pi$
$179$	−19.8564	−1.48414	−0.742069	+	0.670324i	$0.766155\pi$
$180$	0	0
$181$	−13.4641	−1.00078	−0.500389	−	0.865800i	$-0.666810\pi$
$181$	−13.4641	−1.00078	−0.500389	+	0.865800i	$0.666810\pi$
$182$	0	0
$183$	−7.46410	−0.551762
$184$	0	0
$185$	7.60770	0.559329
$186$	0	0
$187$	6.19615	0.453108
$188$	0	0
$189$	0	0
$190$	0	0
$191$	13.8564	1.00261	0.501307	−	0.865269i	$-0.332853\pi$
$191$	13.8564	1.00261	0.501307	+	0.865269i	$0.332853\pi$
$192$	0	0
$193$	9.46410	0.681241	0.340620	−	0.940201i	$-0.389363\pi$
$193$	9.46410	0.681241	0.340620	+	0.940201i	$0.389363\pi$
$194$	0	0
$195$	−0.732051	−0.0524232
$196$	0	0
$197$	26.7846	1.90832	0.954162	−	0.299290i	$-0.0967498\pi$
$197$	26.7846	1.90832	0.954162	+	0.299290i	$0.0967498\pi$
$198$	0	0
$199$	14.5359	1.03042	0.515211	−	0.857063i	$-0.327713\pi$
$199$	14.5359	1.03042	0.515211	+	0.857063i	$0.327713\pi$
$200$	0	0
$201$	10.7321	0.756980
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.32051	−0.231914
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	−5.46410	−0.377960
$210$	0	0
$211$	14.0526	0.967418	0.483709	−	0.875229i	$-0.339289\pi$
$211$	14.0526	0.967418	0.483709	+	0.875229i	$0.339289\pi$
$212$	0	0
$213$	−1.46410	−0.100319
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−9.46410	−0.639525
$220$	0	0
$221$	6.19615	0.416798
$222$	0	0
$223$	9.26795	0.620628	0.310314	−	0.950634i	$-0.399566\pi$
$223$	9.26795	0.620628	0.310314	+	0.950634i	$0.399566\pi$
$224$	0	0
$225$	−4.46410	−0.297607
$226$	0	0
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−27.6603	−1.81208	−0.906042	−	0.423188i	$-0.860911\pi$
$233$	−27.6603	−1.81208	−0.906042	+	0.423188i	$0.860911\pi$
$234$	0	0
$235$	−5.07180	−0.330848
$236$	0	0
$237$	9.26795	0.602018
$238$	0	0
$239$	7.07180	0.457437	0.228718	−	0.973493i	$-0.426547\pi$
$239$	7.07180	0.457437	0.228718	+	0.973493i	$0.426547\pi$
$240$	0	0
$241$	−11.8564	−0.763738	−0.381869	−	0.924216i	$-0.624719\pi$
$241$	−11.8564	−0.763738	−0.381869	+	0.924216i	$0.624719\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−5.12436	−0.327383
$246$	0	0
$247$	−5.46410	−0.347672
$248$	0	0
$249$	6.92820	0.439057
$250$	0	0
$251$	−25.8564	−1.63204	−0.816021	−	0.578022i	$-0.803825\pi$
$251$	−25.8564	−1.63204	−0.816021	+	0.578022i	$0.803825\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	−4.53590	−0.284049
$256$	0	0
$257$	18.3923	1.14728	0.573640	−	0.819107i	$-0.305531\pi$
$257$	18.3923	1.14728	0.573640	+	0.819107i	$0.305531\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.19615	−0.135938
$262$	0	0
$263$	−21.4641	−1.32353	−0.661767	−	0.749710i	$-0.730193\pi$
$263$	−21.4641	−1.32353	−0.661767	+	0.749710i	$0.730193\pi$
$264$	0	0
$265$	−3.60770	−0.221619
$266$	0	0
$267$	−5.80385	−0.355190
$268$	0	0
$269$	−19.4641	−1.18675	−0.593374	−	0.804927i	$-0.702204\pi$
$269$	−19.4641	−1.18675	−0.593374	+	0.804927i	$0.702204\pi$
$270$	0	0
$271$	−3.60770	−0.219152	−0.109576	−	0.993978i	$-0.534949\pi$
$271$	−3.60770	−0.219152	−0.109576	+	0.993978i	$0.534949\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.46410	0.269195
$276$	0	0
$277$	0.143594	0.00862770	0.00431385	−	0.999991i	$-0.498627\pi$
$277$	0.143594	0.00862770	0.00431385	+	0.999991i	$0.498627\pi$
$278$	0	0
$279$	−4.19615	−0.251217
$280$	0	0
$281$	14.3923	0.858573	0.429286	−	0.903168i	$-0.358765\pi$
$281$	14.3923	0.858573	0.429286	+	0.903168i	$0.358765\pi$
$282$	0	0
$283$	−28.9808	−1.72273	−0.861364	−	0.507989i	$-0.830389\pi$
$283$	−28.9808	−1.72273	−0.861364	+	0.507989i	$0.830389\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	0	0
$288$	0	0
$289$	21.3923	1.25837
$290$	0	0
$291$	−14.3923	−0.843692
$292$	0	0
$293$	14.7846	0.863726	0.431863	−	0.901939i	$-0.357856\pi$
$293$	14.7846	0.863726	0.431863	+	0.901939i	$0.357856\pi$
$294$	0	0
$295$	−1.85641	−0.108084
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−1.80385	−0.103628
$304$	0	0
$305$	−5.46410	−0.312874
$306$	0	0
$307$	6.53590	0.373023	0.186512	−	0.982453i	$-0.440282\pi$
$307$	6.53590	0.373023	0.186512	+	0.982453i	$0.440282\pi$
$308$	0	0
$309$	−8.39230	−0.477422
$310$	0	0
$311$	10.7846	0.611539	0.305770	−	0.952106i	$-0.401086\pi$
$311$	10.7846	0.611539	0.305770	+	0.952106i	$0.401086\pi$
$312$	0	0
$313$	8.39230	0.474361	0.237181	−	0.971466i	$-0.423777\pi$
$313$	8.39230	0.474361	0.237181	+	0.971466i	$0.423777\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.0526	1.57559	0.787794	−	0.615938i	$-0.211223\pi$
$317$	28.0526	1.57559	0.787794	+	0.615938i	$0.211223\pi$
$318$	0	0
$319$	2.19615	0.122961
$320$	0	0
$321$	−6.53590	−0.364798
$322$	0	0
$323$	−33.8564	−1.88382
$324$	0	0
$325$	4.46410	0.247624
$326$	0	0
$327$	−5.46410	−0.302166
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−20.9808	−1.15321	−0.576603	−	0.817024i	$-0.695622\pi$
$331$	−20.9808	−1.15321	−0.576603	+	0.817024i	$0.695622\pi$
$332$	0	0
$333$	10.3923	0.569495
$334$	0	0
$335$	7.85641	0.429241
$336$	0	0
$337$	9.60770	0.523365	0.261682	−	0.965154i	$-0.415723\pi$
$337$	9.60770	0.523365	0.261682	+	0.965154i	$0.415723\pi$
$338$	0	0
$339$	−3.07180	−0.166837
$340$	0	0
$341$	4.19615	0.227234
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.46410	−0.0788246
$346$	0	0
$347$	27.7128	1.48770	0.743851	−	0.668346i	$-0.232997\pi$
$347$	27.7128	1.48770	0.743851	+	0.668346i	$0.232997\pi$
$348$	0	0
$349$	−7.07180	−0.378545	−0.189272	−	0.981925i	$-0.560613\pi$
$349$	−7.07180	−0.378545	−0.189272	+	0.981925i	$0.560613\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−25.5167	−1.35811	−0.679057	−	0.734085i	$-0.737611\pi$
$353$	−25.5167	−1.35811	−0.679057	+	0.734085i	$0.737611\pi$
$354$	0	0
$355$	−1.07180	−0.0568851
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.07180	−0.162123	−0.0810616	−	0.996709i	$-0.525831\pi$
$359$	−3.07180	−0.162123	−0.0810616	+	0.996709i	$0.525831\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−6.92820	−0.362639
$366$	0	0
$367$	−22.9282	−1.19684	−0.598421	−	0.801182i	$-0.704205\pi$
$367$	−22.9282	−1.19684	−0.598421	+	0.801182i	$0.704205\pi$
$368$	0	0
$369$	−4.53590	−0.236129
$370$	0	0
$371$	0	0
$372$	0	0
$373$	21.3205	1.10393	0.551967	−	0.833866i	$-0.313877\pi$
$373$	21.3205	1.10393	0.551967	+	0.833866i	$0.313877\pi$
$374$	0	0
$375$	−6.92820	−0.357771
$376$	0	0
$377$	2.19615	0.113108
$378$	0	0
$379$	28.9808	1.48864	0.744321	−	0.667822i	$-0.232773\pi$
$379$	28.9808	1.48864	0.744321	+	0.667822i	$0.232773\pi$
$380$	0	0
$381$	−2.73205	−0.139967
$382$	0	0
$383$	−10.5359	−0.538359	−0.269180	−	0.963090i	$-0.586753\pi$
$383$	−10.5359	−0.538359	−0.269180	+	0.963090i	$0.586753\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.73205	0.138878
$388$	0	0
$389$	34.3923	1.74376	0.871880	−	0.489720i	$-0.162901\pi$
$389$	34.3923	1.74376	0.871880	+	0.489720i	$0.162901\pi$
$390$	0	0
$391$	12.3923	0.626706
$392$	0	0
$393$	0	0
$394$	0	0
$395$	6.78461	0.341371
$396$	0	0
$397$	−33.3205	−1.67231	−0.836154	−	0.548494i	$-0.815201\pi$
$397$	−33.3205	−1.67231	−0.836154	+	0.548494i	$0.815201\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.26795	0.362944	0.181472	−	0.983396i	$-0.441914\pi$
$401$	7.26795	0.362944	0.181472	+	0.983396i	$0.441914\pi$
$402$	0	0
$403$	4.19615	0.209025
$404$	0	0
$405$	0.732051	0.0363759
$406$	0	0
$407$	−10.3923	−0.515127
$408$	0	0
$409$	−3.60770	−0.178389	−0.0891945	−	0.996014i	$-0.528429\pi$
$409$	−3.60770	−0.178389	−0.0891945	+	0.996014i	$0.528429\pi$
$410$	0	0
$411$	2.19615	0.108328
$412$	0	0
$413$	0	0
$414$	0	0
$415$	5.07180	0.248965
$416$	0	0
$417$	1.26795	0.0620917
$418$	0	0
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	8.92820	0.435134	0.217567	−	0.976045i	$-0.430188\pi$
$421$	8.92820	0.435134	0.217567	+	0.976045i	$0.430188\pi$
$422$	0	0
$423$	−6.92820	−0.336861
$424$	0	0
$425$	27.6603	1.34172
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−19.7128	−0.949533	−0.474766	−	0.880112i	$-0.657467\pi$
$431$	−19.7128	−0.949533	−0.474766	+	0.880112i	$0.657467\pi$
$432$	0	0
$433$	27.8564	1.33869	0.669347	−	0.742950i	$-0.266574\pi$
$433$	27.8564	1.33869	0.669347	+	0.742950i	$0.266574\pi$
$434$	0	0
$435$	−1.60770	−0.0770831
$436$	0	0
$437$	−10.9282	−0.522767
$438$	0	0
$439$	15.5167	0.740570	0.370285	−	0.928918i	$-0.379260\pi$
$439$	15.5167	0.740570	0.370285	+	0.928918i	$0.379260\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	−4.24871	−0.201408
$446$	0	0
$447$	4.92820	0.233096
$448$	0	0
$449$	−28.0526	−1.32388	−0.661941	−	0.749556i	$-0.730267\pi$
$449$	−28.0526	−1.32388	−0.661941	+	0.749556i	$0.730267\pi$
$450$	0	0
$451$	4.53590	0.213587
$452$	0	0
$453$	9.46410	0.444662
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−18.7846	−0.878707	−0.439353	−	0.898314i	$-0.644792\pi$
$457$	−18.7846	−0.878707	−0.439353	+	0.898314i	$0.644792\pi$
$458$	0	0
$459$	−6.19615	−0.289212
$460$	0	0
$461$	29.3205	1.36559	0.682796	−	0.730609i	$-0.260764\pi$
$461$	29.3205	1.36559	0.682796	+	0.730609i	$0.260764\pi$
$462$	0	0
$463$	26.7321	1.24234	0.621172	−	0.783674i	$-0.286657\pi$
$463$	26.7321	1.24234	0.621172	+	0.783674i	$0.286657\pi$
$464$	0	0
$465$	−3.07180	−0.142451
$466$	0	0
$467$	−19.8564	−0.918845	−0.459422	−	0.888218i	$-0.651944\pi$
$467$	−19.8564	−0.918845	−0.459422	+	0.888218i	$0.651944\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6.53590	−0.301158
$472$	0	0
$473$	−2.73205	−0.125620
$474$	0	0
$475$	−24.3923	−1.11920
$476$	0	0
$477$	−4.92820	−0.225647
$478$	0	0
$479$	−29.7128	−1.35761	−0.678807	−	0.734317i	$-0.737503\pi$
$479$	−29.7128	−1.35761	−0.678807	+	0.734317i	$0.737503\pi$
$480$	0	0
$481$	−10.3923	−0.473848
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.5359	−0.478411
$486$	0	0
$487$	−21.6603	−0.981520	−0.490760	−	0.871295i	$-0.663281\pi$
$487$	−21.6603	−0.981520	−0.490760	+	0.871295i	$0.663281\pi$
$488$	0	0
$489$	−6.33975	−0.286693
$490$	0	0
$491$	−21.8564	−0.986366	−0.493183	−	0.869926i	$-0.664167\pi$
$491$	−21.8564	−0.986366	−0.493183	+	0.869926i	$0.664167\pi$
$492$	0	0
$493$	13.6077	0.612860
$494$	0	0
$495$	−0.732051	−0.0329032
$496$	0	0
$497$	0	0
$498$	0	0
$499$	21.6603	0.969646	0.484823	−	0.874612i	$-0.338884\pi$
$499$	21.6603	0.969646	0.484823	+	0.874612i	$0.338884\pi$
$500$	0	0
$501$	−5.07180	−0.226591
$502$	0	0
$503$	35.3205	1.57486	0.787432	−	0.616402i	$-0.211410\pi$
$503$	35.3205	1.57486	0.787432	+	0.616402i	$0.211410\pi$
$504$	0	0
$505$	−1.32051	−0.0587618
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−29.1244	−1.29091	−0.645457	−	0.763796i	$-0.723333\pi$
$509$	−29.1244	−1.29091	−0.645457	+	0.763796i	$0.723333\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	5.46410	0.241246
$514$	0	0
$515$	−6.14359	−0.270719
$516$	0	0
$517$	6.92820	0.304702
$518$	0	0
$519$	−1.80385	−0.0791801
$520$	0	0
$521$	−18.7846	−0.822969	−0.411484	−	0.911417i	$-0.634990\pi$
$521$	−18.7846	−0.822969	−0.411484	+	0.911417i	$0.634990\pi$
$522$	0	0
$523$	−16.5885	−0.725363	−0.362681	−	0.931913i	$-0.618139\pi$
$523$	−16.5885	−0.725363	−0.362681	+	0.931913i	$0.618139\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	26.0000	1.13258
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−2.53590	−0.110049
$532$	0	0
$533$	4.53590	0.196472
$534$	0	0
$535$	−4.78461	−0.206857
$536$	0	0
$537$	−19.8564	−0.856867
$538$	0	0
$539$	7.00000	0.301511
$540$	0	0
$541$	21.7128	0.933507	0.466753	−	0.884388i	$-0.345424\pi$
$541$	21.7128	0.933507	0.466753	+	0.884388i	$0.345424\pi$
$542$	0	0
$543$	−13.4641	−0.577800
$544$	0	0
$545$	−4.00000	−0.171341
$546$	0	0
$547$	39.5167	1.68961	0.844805	−	0.535074i	$-0.179716\pi$
$547$	39.5167	1.68961	0.844805	+	0.535074i	$0.179716\pi$
$548$	0	0
$549$	−7.46410	−0.318560
$550$	0	0
$551$	−12.0000	−0.511217
$552$	0	0
$553$	0	0
$554$	0	0
$555$	7.60770	0.322929
$556$	0	0
$557$	15.0718	0.638613	0.319306	−	0.947652i	$-0.396550\pi$
$557$	15.0718	0.638613	0.319306	+	0.947652i	$0.396550\pi$
$558$	0	0
$559$	−2.73205	−0.115553
$560$	0	0
$561$	6.19615	0.261602
$562$	0	0
$563$	17.8564	0.752558	0.376279	−	0.926506i	$-0.377203\pi$
$563$	17.8564	0.752558	0.376279	+	0.926506i	$0.377203\pi$
$564$	0	0
$565$	−2.24871	−0.0946040
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−44.4449	−1.86323	−0.931613	−	0.363452i	$-0.881598\pi$
$569$	−44.4449	−1.86323	−0.931613	+	0.363452i	$0.881598\pi$
$570$	0	0
$571$	−44.3013	−1.85395	−0.926975	−	0.375123i	$-0.877601\pi$
$571$	−44.3013	−1.85395	−0.926975	+	0.375123i	$0.877601\pi$
$572$	0	0
$573$	13.8564	0.578860
$574$	0	0
$575$	8.92820	0.372332
$576$	0	0
$577$	4.92820	0.205164	0.102582	−	0.994725i	$-0.467290\pi$
$577$	4.92820	0.205164	0.102582	+	0.994725i	$0.467290\pi$
$578$	0	0
$579$	9.46410	0.393315
$580$	0	0
$581$	0	0
$582$	0	0
$583$	4.92820	0.204105
$584$	0	0
$585$	−0.732051	−0.0302666
$586$	0	0
$587$	−31.3205	−1.29274	−0.646368	−	0.763026i	$-0.723713\pi$
$587$	−31.3205	−1.29274	−0.646368	+	0.763026i	$0.723713\pi$
$588$	0	0
$589$	−22.9282	−0.944740
$590$	0	0
$591$	26.7846	1.10177
$592$	0	0
$593$	−25.7128	−1.05590	−0.527949	−	0.849276i	$-0.677039\pi$
$593$	−25.7128	−1.05590	−0.527949	+	0.849276i	$0.677039\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14.5359	0.594915
$598$	0	0
$599$	−35.7128	−1.45919	−0.729593	−	0.683882i	$-0.760290\pi$
$599$	−35.7128	−1.45919	−0.729593	+	0.683882i	$0.760290\pi$
$600$	0	0
$601$	38.7846	1.58206	0.791029	−	0.611779i	$-0.209546\pi$
$601$	38.7846	1.58206	0.791029	+	0.611779i	$0.209546\pi$
$602$	0	0
$603$	10.7321	0.437043
$604$	0	0
$605$	0.732051	0.0297621
$606$	0	0
$607$	26.3397	1.06910	0.534549	−	0.845138i	$-0.320482\pi$
$607$	26.3397	1.06910	0.534549	+	0.845138i	$0.320482\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	6.92820	0.280285
$612$	0	0
$613$	−40.3923	−1.63143	−0.815715	−	0.578454i	$-0.803656\pi$
$613$	−40.3923	−1.63143	−0.815715	+	0.578454i	$0.803656\pi$
$614$	0	0
$615$	−3.32051	−0.133896
$616$	0	0
$617$	−28.0526	−1.12935	−0.564677	−	0.825312i	$-0.690999\pi$
$617$	−28.0526	−1.12935	−0.564677	+	0.825312i	$0.690999\pi$
$618$	0	0
$619$	−35.8038	−1.43908	−0.719539	−	0.694452i	$-0.755647\pi$
$619$	−35.8038	−1.43908	−0.719539	+	0.694452i	$0.755647\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	0	0
$623$	0	0
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	−5.46410	−0.218215
$628$	0	0
$629$	−64.3923	−2.56749
$630$	0	0
$631$	19.5167	0.776946	0.388473	−	0.921460i	$-0.373003\pi$
$631$	19.5167	0.776946	0.388473	+	0.921460i	$0.373003\pi$
$632$	0	0
$633$	14.0526	0.558539
$634$	0	0
$635$	−2.00000	−0.0793676
$636$	0	0
$637$	7.00000	0.277350
$638$	0	0
$639$	−1.46410	−0.0579190
$640$	0	0
$641$	−10.0000	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$641$	−10.0000	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$642$	0	0
$643$	35.9090	1.41611	0.708056	−	0.706157i	$-0.249573\pi$
$643$	35.9090	1.41611	0.708056	+	0.706157i	$0.249573\pi$
$644$	0	0
$645$	2.00000	0.0787499
$646$	0	0
$647$	27.7128	1.08950	0.544752	−	0.838597i	$-0.316624\pi$
$647$	27.7128	1.08950	0.544752	+	0.838597i	$0.316624\pi$
$648$	0	0
$649$	2.53590	0.0995427
$650$	0	0
$651$	0	0
$652$	0	0
$653$	32.5359	1.27323	0.636614	−	0.771183i	$-0.280334\pi$
$653$	32.5359	1.27323	0.636614	+	0.771183i	$0.280334\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−9.46410	−0.369230
$658$	0	0
$659$	25.4641	0.991941	0.495970	−	0.868339i	$-0.334812\pi$
$659$	25.4641	0.991941	0.495970	+	0.868339i	$0.334812\pi$
$660$	0	0
$661$	5.60770	0.218114	0.109057	−	0.994035i	$-0.465217\pi$
$661$	5.60770	0.218114	0.109057	+	0.994035i	$0.465217\pi$
$662$	0	0
$663$	6.19615	0.240639
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.39230	0.170071
$668$	0	0
$669$	9.26795	0.358320
$670$	0	0
$671$	7.46410	0.288148
$672$	0	0
$673$	41.7128	1.60791	0.803955	−	0.594690i	$-0.202725\pi$
$673$	41.7128	1.60791	0.803955	+	0.594690i	$0.202725\pi$
$674$	0	0
$675$	−4.46410	−0.171823
$676$	0	0
$677$	25.9090	0.995762	0.497881	−	0.867245i	$-0.334112\pi$
$677$	25.9090	0.995762	0.497881	+	0.867245i	$0.334112\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−14.0000	−0.536481
$682$	0	0
$683$	−19.3205	−0.739279	−0.369639	−	0.929175i	$-0.620519\pi$
$683$	−19.3205	−0.739279	−0.369639	+	0.929175i	$0.620519\pi$
$684$	0	0
$685$	1.60770	0.0614269
$686$	0	0
$687$	−14.0000	−0.534133
$688$	0	0
$689$	4.92820	0.187750
$690$	0	0
$691$	−4.98076	−0.189477	−0.0947386	−	0.995502i	$-0.530202\pi$
$691$	−4.98076	−0.189477	−0.0947386	+	0.995502i	$0.530202\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0.928203	0.0352088
$696$	0	0
$697$	28.1051	1.06456
$698$	0	0
$699$	−27.6603	−1.04621
$700$	0	0
$701$	12.0526	0.455219	0.227609	−	0.973753i	$-0.426909\pi$
$701$	12.0526	0.455219	0.227609	+	0.973753i	$0.426909\pi$
$702$	0	0
$703$	56.7846	2.14167
$704$	0	0
$705$	−5.07180	−0.191015
$706$	0	0
$707$	0	0
$708$	0	0
$709$	35.5692	1.33583	0.667915	−	0.744238i	$-0.267187\pi$
$709$	35.5692	1.33583	0.667915	+	0.744238i	$0.267187\pi$
$710$	0	0
$711$	9.26795	0.347575
$712$	0	0
$713$	8.39230	0.314294
$714$	0	0
$715$	0.732051	0.0273771
$716$	0	0
$717$	7.07180	0.264101
$718$	0	0
$719$	10.1436	0.378292	0.189146	−	0.981949i	$-0.439428\pi$
$719$	10.1436	0.378292	0.189146	+	0.981949i	$0.439428\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−11.8564	−0.440945
$724$	0	0
$725$	9.80385	0.364106
$726$	0	0
$727$	1.46410	0.0543005	0.0271503	−	0.999631i	$-0.491357\pi$
$727$	1.46410	0.0543005	0.0271503	+	0.999631i	$0.491357\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.9282	−0.626112
$732$	0	0
$733$	11.6077	0.428740	0.214370	−	0.976753i	$-0.431230\pi$
$733$	11.6077	0.428740	0.214370	+	0.976753i	$0.431230\pi$
$734$	0	0
$735$	−5.12436	−0.189015
$736$	0	0
$737$	−10.7321	−0.395320
$738$	0	0
$739$	1.85641	0.0682890	0.0341445	−	0.999417i	$-0.489129\pi$
$739$	1.85641	0.0682890	0.0341445	+	0.999417i	$0.489129\pi$
$740$	0	0
$741$	−5.46410	−0.200729
$742$	0	0
$743$	34.9282	1.28139	0.640696	−	0.767795i	$-0.278646\pi$
$743$	34.9282	1.28139	0.640696	+	0.767795i	$0.278646\pi$
$744$	0	0
$745$	3.60770	0.132176
$746$	0	0
$747$	6.92820	0.253490
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−7.21539	−0.263293	−0.131647	−	0.991297i	$-0.542026\pi$
$751$	−7.21539	−0.263293	−0.131647	+	0.991297i	$0.542026\pi$
$752$	0	0
$753$	−25.8564	−0.942260
$754$	0	0
$755$	6.92820	0.252143
$756$	0	0
$757$	39.3205	1.42913	0.714564	−	0.699570i	$-0.246625\pi$
$757$	39.3205	1.42913	0.714564	+	0.699570i	$0.246625\pi$
$758$	0	0
$759$	2.00000	0.0725954
$760$	0	0
$761$	−13.2154	−0.479058	−0.239529	−	0.970889i	$-0.576993\pi$
$761$	−13.2154	−0.479058	−0.239529	+	0.970889i	$0.576993\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−4.53590	−0.163996
$766$	0	0
$767$	2.53590	0.0915660
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	18.3923	0.662383
$772$	0	0
$773$	−17.8038	−0.640360	−0.320180	−	0.947357i	$-0.603743\pi$
$773$	−17.8038	−0.640360	−0.320180	+	0.947357i	$0.603743\pi$
$774$	0	0
$775$	18.7321	0.672875
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.7846	−0.888001
$780$	0	0
$781$	1.46410	0.0523897
$782$	0	0
$783$	−2.19615	−0.0784841
$784$	0	0
$785$	−4.78461	−0.170770
$786$	0	0
$787$	−42.6410	−1.51999	−0.759994	−	0.649930i	$-0.774798\pi$
$787$	−42.6410	−1.51999	−0.759994	+	0.649930i	$0.774798\pi$
$788$	0	0
$789$	−21.4641	−0.764142
$790$	0	0
$791$	0	0
$792$	0	0
$793$	7.46410	0.265058
$794$	0	0
$795$	−3.60770	−0.127952
$796$	0	0
$797$	−13.3205	−0.471837	−0.235918	−	0.971773i	$-0.575810\pi$
$797$	−13.3205	−0.471837	−0.235918	+	0.971773i	$0.575810\pi$
$798$	0	0
$799$	42.9282	1.51869
$800$	0	0
$801$	−5.80385	−0.205069
$802$	0	0
$803$	9.46410	0.333981
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19.4641	−0.685169
$808$	0	0
$809$	−42.3013	−1.48723	−0.743617	−	0.668606i	$-0.766891\pi$
$809$	−42.3013	−1.48723	−0.743617	+	0.668606i	$0.766891\pi$
$810$	0	0
$811$	39.3205	1.38073	0.690365	−	0.723461i	$-0.257450\pi$
$811$	39.3205	1.38073	0.690365	+	0.723461i	$0.257450\pi$
$812$	0	0
$813$	−3.60770	−0.126527
$814$	0	0
$815$	−4.64102	−0.162568
$816$	0	0
$817$	14.9282	0.522272
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−46.1051	−1.60908	−0.804540	−	0.593899i	$-0.797588\pi$
$821$	−46.1051	−1.60908	−0.804540	+	0.593899i	$0.797588\pi$
$822$	0	0
$823$	−8.39230	−0.292537	−0.146269	−	0.989245i	$-0.546726\pi$
$823$	−8.39230	−0.292537	−0.146269	+	0.989245i	$0.546726\pi$
$824$	0	0
$825$	4.46410	0.155420
$826$	0	0
$827$	39.8564	1.38594	0.692972	−	0.720965i	$-0.256301\pi$
$827$	39.8564	1.38594	0.692972	+	0.720965i	$0.256301\pi$
$828$	0	0
$829$	−9.71281	−0.337340	−0.168670	−	0.985673i	$-0.553947\pi$
$829$	−9.71281	−0.337340	−0.168670	+	0.985673i	$0.553947\pi$
$830$	0	0
$831$	0.143594	0.00498120
$832$	0	0
$833$	43.3731	1.50279
$834$	0	0
$835$	−3.71281	−0.128487
$836$	0	0
$837$	−4.19615	−0.145040
$838$	0	0
$839$	−4.78461	−0.165183	−0.0825915	−	0.996583i	$-0.526320\pi$
$839$	−4.78461	−0.165183	−0.0825915	+	0.996583i	$0.526320\pi$
$840$	0	0
$841$	−24.1769	−0.833687
$842$	0	0
$843$	14.3923	0.495697
$844$	0	0
$845$	0.732051	0.0251833
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−28.9808	−0.994617
$850$	0	0
$851$	−20.7846	−0.712487
$852$	0	0
$853$	−8.92820	−0.305696	−0.152848	−	0.988250i	$-0.548844\pi$
$853$	−8.92820	−0.305696	−0.152848	+	0.988250i	$0.548844\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	28.7321	0.981468	0.490734	−	0.871309i	$-0.336729\pi$
$857$	28.7321	0.981468	0.490734	+	0.871309i	$0.336729\pi$
$858$	0	0
$859$	16.6795	0.569097	0.284548	−	0.958662i	$-0.408156\pi$
$859$	16.6795	0.569097	0.284548	+	0.958662i	$0.408156\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14.2487	−0.485032	−0.242516	−	0.970147i	$-0.577973\pi$
$863$	−14.2487	−0.485032	−0.242516	+	0.970147i	$0.577973\pi$
$864$	0	0
$865$	−1.32051	−0.0448986
$866$	0	0
$867$	21.3923	0.726521
$868$	0	0
$869$	−9.26795	−0.314394
$870$	0	0
$871$	−10.7321	−0.363642
$872$	0	0
$873$	−14.3923	−0.487106
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.8564	0.670503	0.335252	−	0.942129i	$-0.391179\pi$
$877$	19.8564	0.670503	0.335252	+	0.942129i	$0.391179\pi$
$878$	0	0
$879$	14.7846	0.498673
$880$	0	0
$881$	−23.0718	−0.777309	−0.388654	−	0.921384i	$-0.627060\pi$
$881$	−23.0718	−0.777309	−0.388654	+	0.921384i	$0.627060\pi$
$882$	0	0
$883$	17.1769	0.578049	0.289025	−	0.957322i	$-0.406669\pi$
$883$	17.1769	0.578049	0.289025	+	0.957322i	$0.406669\pi$
$884$	0	0
$885$	−1.85641	−0.0624024
$886$	0	0
$887$	−15.3205	−0.514412	−0.257206	−	0.966357i	$-0.582802\pi$
$887$	−15.3205	−0.514412	−0.257206	+	0.966357i	$0.582802\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−37.8564	−1.26682
$894$	0	0
$895$	−14.5359	−0.485881
$896$	0	0
$897$	2.00000	0.0667781
$898$	0	0
$899$	9.21539	0.307350
$900$	0	0
$901$	30.5359	1.01730
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.85641	−0.327638
$906$	0	0
$907$	−21.0718	−0.699678	−0.349839	−	0.936810i	$-0.613764\pi$
$907$	−21.0718	−0.699678	−0.349839	+	0.936810i	$0.613764\pi$
$908$	0	0
$909$	−1.80385	−0.0598298
$910$	0	0
$911$	−45.8564	−1.51929	−0.759645	−	0.650338i	$-0.774627\pi$
$911$	−45.8564	−1.51929	−0.759645	+	0.650338i	$0.774627\pi$
$912$	0	0
$913$	−6.92820	−0.229290
$914$	0	0
$915$	−5.46410	−0.180638
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−18.7321	−0.617913	−0.308957	−	0.951076i	$-0.599980\pi$
$919$	−18.7321	−0.617913	−0.308957	+	0.951076i	$0.599980\pi$
$920$	0	0
$921$	6.53590	0.215365
$922$	0	0
$923$	1.46410	0.0481915
$924$	0	0
$925$	−46.3923	−1.52537
$926$	0	0
$927$	−8.39230	−0.275639
$928$	0	0
$929$	26.1962	0.859468	0.429734	−	0.902956i	$-0.358607\pi$
$929$	26.1962	0.859468	0.429734	+	0.902956i	$0.358607\pi$
$930$	0	0
$931$	−38.2487	−1.25355
$932$	0	0
$933$	10.7846	0.353072
$934$	0	0
$935$	4.53590	0.148340
$936$	0	0
$937$	14.0000	0.457360	0.228680	−	0.973502i	$-0.426559\pi$
$937$	14.0000	0.457360	0.228680	+	0.973502i	$0.426559\pi$
$938$	0	0
$939$	8.39230	0.273873
$940$	0	0
$941$	12.2487	0.399297	0.199648	−	0.979868i	$-0.436020\pi$
$941$	12.2487	0.399297	0.199648	+	0.979868i	$0.436020\pi$
$942$	0	0
$943$	9.07180	0.295418
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.1436	−0.849553	−0.424776	−	0.905298i	$-0.639647\pi$
$947$	−26.1436	−0.849553	−0.424776	+	0.905298i	$0.639647\pi$
$948$	0	0
$949$	9.46410	0.307218
$950$	0	0
$951$	28.0526	0.909667
$952$	0	0
$953$	49.2295	1.59470	0.797350	−	0.603518i	$-0.206235\pi$
$953$	49.2295	1.59470	0.797350	+	0.603518i	$0.206235\pi$
$954$	0	0
$955$	10.1436	0.328239
$956$	0	0
$957$	2.19615	0.0709915
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−13.3923	−0.432010
$962$	0	0
$963$	−6.53590	−0.210616
$964$	0	0
$965$	6.92820	0.223027
$966$	0	0
$967$	9.07180	0.291729	0.145865	−	0.989305i	$-0.453404\pi$
$967$	9.07180	0.291729	0.145865	+	0.989305i	$0.453404\pi$
$968$	0	0
$969$	−33.8564	−1.08762
$970$	0	0
$971$	27.8564	0.893955	0.446977	−	0.894545i	$-0.352500\pi$
$971$	27.8564	0.893955	0.446977	+	0.894545i	$0.352500\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4.46410	0.142966
$976$	0	0
$977$	−28.0526	−0.897481	−0.448740	−	0.893662i	$-0.648127\pi$
$977$	−28.0526	−0.897481	−0.448740	+	0.893662i	$0.648127\pi$
$978$	0	0
$979$	5.80385	0.185492
$980$	0	0
$981$	−5.46410	−0.174455
$982$	0	0
$983$	26.6410	0.849716	0.424858	−	0.905260i	$-0.360324\pi$
$983$	26.6410	0.849716	0.424858	+	0.905260i	$0.360324\pi$
$984$	0	0
$985$	19.6077	0.624753
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5.46410	−0.173748
$990$	0	0
$991$	33.1769	1.05390	0.526950	−	0.849896i	$-0.323336\pi$
$991$	33.1769	1.05390	0.526950	+	0.849896i	$0.323336\pi$
$992$	0	0
$993$	−20.9808	−0.665804
$994$	0	0
$995$	10.6410	0.337343
$996$	0	0
$997$	−21.3205	−0.675227	−0.337614	−	0.941285i	$-0.609620\pi$
$997$	−21.3205	−0.675227	−0.337614	+	0.941285i	$0.609620\pi$
$998$	0	0
$999$	10.3923	0.328798

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bi.1.2		2
4.3	odd	2		1716.2.a.d.1.2	✓	2
12.11	even	2		5148.2.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.d.1.2	✓	2	4.3	odd	2
5148.2.a.i.1.1		2	12.11	even	2
6864.2.a.bi.1.2		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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.732051 q^{5} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.732051 q^{15} -6.19615 q^{17} +5.46410 q^{19} -2.00000 q^{23} -4.46410 q^{25} +1.00000 q^{27} -2.19615 q^{29} -4.19615 q^{31} -1.00000 q^{33} +10.3923 q^{37} -1.00000 q^{39} -4.53590 q^{41} +2.73205 q^{43} +0.732051 q^{45} -6.92820 q^{47} -7.00000 q^{49} -6.19615 q^{51} -4.92820 q^{53} -0.732051 q^{55} +5.46410 q^{57} -2.53590 q^{59} -7.46410 q^{61} -0.732051 q^{65} +10.7321 q^{67} -2.00000 q^{69} -1.46410 q^{71} -9.46410 q^{73} -4.46410 q^{75} +9.26795 q^{79} +1.00000 q^{81} +6.92820 q^{83} -4.53590 q^{85} -2.19615 q^{87} -5.80385 q^{89} -4.19615 q^{93} +4.00000 q^{95} -14.3923 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 2 q^{11} - 2 q^{13} - 2 q^{15} - 2 q^{17} + 4 q^{19} - 4 q^{23} - 2 q^{25} + 2 q^{27} + 6 q^{29} + 2 q^{31} - 2 q^{33} - 2 q^{39} - 16 q^{41} + 2 q^{43} - 2 q^{45} - 14 q^{49} - 2 q^{51} + 4 q^{53} + 2 q^{55} + 4 q^{57} - 12 q^{59} - 8 q^{61} + 2 q^{65} + 18 q^{67} - 4 q^{69} + 4 q^{71} - 12 q^{73} - 2 q^{75} + 22 q^{79} + 2 q^{81} - 16 q^{85} + 6 q^{87} - 22 q^{89} + 2 q^{93} + 8 q^{95} - 8 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 - 2 * q^11 - 2 * q^13 - 2 * q^15 - 2 * q^17 + 4 * q^19 - 4 * q^23 - 2 * q^25 + 2 * q^27 + 6 * q^29 + 2 * q^31 - 2 * q^33 - 2 * q^39 - 16 * q^41 + 2 * q^43 - 2 * q^45 - 14 * q^49 - 2 * q^51 + 4 * q^53 + 2 * q^55 + 4 * q^57 - 12 * q^59 - 8 * q^61 + 2 * q^65 + 18 * q^67 - 4 * q^69 + 4 * q^71 - 12 * q^73 - 2 * q^75 + 22 * q^79 + 2 * q^81 - 16 * q^85 + 6 * q^87 - 22 * q^89 + 2 * q^93 + 8 * q^95 - 8 * q^97 - 2 * q^99

Introduction

L-functions

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Database

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