LMFDB - Embedded newform 4368.2.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, 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("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1092)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{13} +2.37228 q^{15} -6.37228 q^{19} +1.00000 q^{21} -4.37228 q^{23} +0.627719 q^{25} -1.00000 q^{27} -4.37228 q^{29} -2.37228 q^{31} +2.00000 q^{33} +2.37228 q^{35} +6.74456 q^{37} +1.00000 q^{39} +8.74456 q^{41} -11.1168 q^{43} -2.37228 q^{45} +6.37228 q^{47} +1.00000 q^{49} -0.372281 q^{53} +4.74456 q^{55} +6.37228 q^{57} +8.00000 q^{59} +6.00000 q^{61} -1.00000 q^{63} +2.37228 q^{65} -12.7446 q^{67} +4.37228 q^{69} -2.74456 q^{71} -13.8614 q^{73} -0.627719 q^{75} +2.00000 q^{77} -11.1168 q^{79} +1.00000 q^{81} -11.1168 q^{83} +4.37228 q^{87} -1.62772 q^{89} +1.00000 q^{91} +2.37228 q^{93} +15.1168 q^{95} +11.6277 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} - q^{15} - 7 q^{19} + 2 q^{21} - 3 q^{23} + 7 q^{25} - 2 q^{27} - 3 q^{29} + q^{31} + 4 q^{33} - q^{35} + 2 q^{37} + 2 q^{39} + 6 q^{41} - 5 q^{43} + q^{45} + 7 q^{47} + 2 q^{49} + 5 q^{53} - 2 q^{55} + 7 q^{57} + 16 q^{59} + 12 q^{61} - 2 q^{63} - q^{65} - 14 q^{67} + 3 q^{69} + 6 q^{71} + q^{73} - 7 q^{75} + 4 q^{77} - 5 q^{79} + 2 q^{81} - 5 q^{83} + 3 q^{87} - 9 q^{89} + 2 q^{91} - q^{93} + 13 q^{95} + 29 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^13 - q^15 - 7 * q^19 + 2 * q^21 - 3 * q^23 + 7 * q^25 - 2 * q^27 - 3 * q^29 + q^31 + 4 * q^33 - q^35 + 2 * q^37 + 2 * q^39 + 6 * q^41 - 5 * q^43 + q^45 + 7 * q^47 + 2 * q^49 + 5 * q^53 - 2 * q^55 + 7 * q^57 + 16 * q^59 + 12 * q^61 - 2 * q^63 - q^65 - 14 * q^67 + 3 * q^69 + 6 * q^71 + q^73 - 7 * q^75 + 4 * q^77 - 5 * q^79 + 2 * q^81 - 5 * q^83 + 3 * q^87 - 9 * q^89 + 2 * q^91 - q^93 + 13 * q^95 + 29 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.37228	−1.06092	−0.530458	−	0.847711i	$-0.677980\pi$
$5$	−2.37228	−1.06092	−0.530458	+	0.847711i	$0.677980\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	2.37228	0.612520
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−6.37228	−1.46190	−0.730951	−	0.682430i	$-0.760923\pi$
$19$	−6.37228	−1.46190	−0.730951	+	0.682430i	$0.760923\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−4.37228	−0.911684	−0.455842	−	0.890061i	$-0.650662\pi$
$23$	−4.37228	−0.911684	−0.455842	+	0.890061i	$0.650662\pi$
$24$	0	0
$25$	0.627719	0.125544
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.37228	−0.811912	−0.405956	−	0.913893i	$-0.633061\pi$
$29$	−4.37228	−0.811912	−0.405956	+	0.913893i	$0.633061\pi$
$30$	0	0
$31$	−2.37228	−0.426074	−0.213037	−	0.977044i	$-0.568336\pi$
$31$	−2.37228	−0.426074	−0.213037	+	0.977044i	$0.568336\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	2.37228	0.400989
$36$	0	0
$37$	6.74456	1.10880	0.554400	−	0.832251i	$-0.312948\pi$
$37$	6.74456	1.10880	0.554400	+	0.832251i	$0.312948\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	8.74456	1.36567	0.682836	−	0.730572i	$-0.260747\pi$
$41$	8.74456	1.36567	0.682836	+	0.730572i	$0.260747\pi$
$42$	0	0
$43$	−11.1168	−1.69530	−0.847651	−	0.530554i	$-0.821984\pi$
$43$	−11.1168	−1.69530	−0.847651	+	0.530554i	$0.821984\pi$
$44$	0	0
$45$	−2.37228	−0.353639
$46$	0	0
$47$	6.37228	0.929493	0.464746	−	0.885444i	$-0.346146\pi$
$47$	6.37228	0.929493	0.464746	+	0.885444i	$0.346146\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.372281	−0.0511368	−0.0255684	−	0.999673i	$-0.508140\pi$
$53$	−0.372281	−0.0511368	−0.0255684	+	0.999673i	$0.508140\pi$
$54$	0	0
$55$	4.74456	0.639757
$56$	0	0
$57$	6.37228	0.844029
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	2.37228	0.294245
$66$	0	0
$67$	−12.7446	−1.55700	−0.778498	−	0.627647i	$-0.784018\pi$
$67$	−12.7446	−1.55700	−0.778498	+	0.627647i	$0.784018\pi$
$68$	0	0
$69$	4.37228	0.526361
$70$	0	0
$71$	−2.74456	−0.325720	−0.162860	−	0.986649i	$-0.552072\pi$
$71$	−2.74456	−0.325720	−0.162860	+	0.986649i	$0.552072\pi$
$72$	0	0
$73$	−13.8614	−1.62235	−0.811177	−	0.584800i	$-0.801173\pi$
$73$	−13.8614	−1.62235	−0.811177	+	0.584800i	$0.801173\pi$
$74$	0	0
$75$	−0.627719	−0.0724827
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	−11.1168	−1.25074	−0.625371	−	0.780327i	$-0.715052\pi$
$79$	−11.1168	−1.25074	−0.625371	+	0.780327i	$0.715052\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.1168	−1.22023	−0.610116	−	0.792312i	$-0.708877\pi$
$83$	−11.1168	−1.22023	−0.610116	+	0.792312i	$0.708877\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.37228	0.468758
$88$	0	0
$89$	−1.62772	−0.172538	−0.0862689	−	0.996272i	$-0.527494\pi$
$89$	−1.62772	−0.172538	−0.0862689	+	0.996272i	$0.527494\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	2.37228	0.245994
$94$	0	0
$95$	15.1168	1.55096
$96$	0	0
$97$	11.6277	1.18062	0.590308	−	0.807178i	$-0.299006\pi$
$97$	11.6277	1.18062	0.590308	+	0.807178i	$0.299006\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−16.7446	−1.66615	−0.833073	−	0.553163i	$-0.813421\pi$
$101$	−16.7446	−1.66615	−0.833073	+	0.553163i	$0.813421\pi$
$102$	0	0
$103$	17.4891	1.72325	0.861627	−	0.507541i	$-0.169446\pi$
$103$	17.4891	1.72325	0.861627	+	0.507541i	$0.169446\pi$
$104$	0	0
$105$	−2.37228	−0.231511
$106$	0	0
$107$	10.7446	1.03872	0.519358	−	0.854557i	$-0.326171\pi$
$107$	10.7446	1.03872	0.519358	+	0.854557i	$0.326171\pi$
$108$	0	0
$109$	−11.4891	−1.10046	−0.550229	−	0.835014i	$-0.685460\pi$
$109$	−11.4891	−1.10046	−0.550229	+	0.835014i	$0.685460\pi$
$110$	0	0
$111$	−6.74456	−0.640166
$112$	0	0
$113$	−5.11684	−0.481352	−0.240676	−	0.970605i	$-0.577369\pi$
$113$	−5.11684	−0.481352	−0.240676	+	0.970605i	$0.577369\pi$
$114$	0	0
$115$	10.3723	0.967220
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−8.74456	−0.788471
$124$	0	0
$125$	10.3723	0.927725
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	11.1168	0.978784
$130$	0	0
$131$	13.4891	1.17855	0.589275	−	0.807932i	$-0.299413\pi$
$131$	13.4891	1.17855	0.589275	+	0.807932i	$0.299413\pi$
$132$	0	0
$133$	6.37228	0.552547
$134$	0	0
$135$	2.37228	0.204173
$136$	0	0
$137$	12.2337	1.04519	0.522597	−	0.852580i	$-0.324963\pi$
$137$	12.2337	1.04519	0.522597	+	0.852580i	$0.324963\pi$
$138$	0	0
$139$	8.74456	0.741704	0.370852	−	0.928692i	$-0.379066\pi$
$139$	8.74456	0.741704	0.370852	+	0.928692i	$0.379066\pi$
$140$	0	0
$141$	−6.37228	−0.536643
$142$	0	0
$143$	2.00000	0.167248
$144$	0	0
$145$	10.3723	0.861371
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	1.48913	0.121183	0.0605916	−	0.998163i	$-0.480701\pi$
$151$	1.48913	0.121183	0.0605916	+	0.998163i	$0.480701\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.62772	0.452029
$156$	0	0
$157$	2.74456	0.219040	0.109520	−	0.993985i	$-0.465069\pi$
$157$	2.74456	0.219040	0.109520	+	0.993985i	$0.465069\pi$
$158$	0	0
$159$	0.372281	0.0295238
$160$	0	0
$161$	4.37228	0.344584
$162$	0	0
$163$	−9.48913	−0.743246	−0.371623	−	0.928384i	$-0.621199\pi$
$163$	−9.48913	−0.743246	−0.371623	+	0.928384i	$0.621199\pi$
$164$	0	0
$165$	−4.74456	−0.369364
$166$	0	0
$167$	19.1168	1.47931	0.739653	−	0.672989i	$-0.234990\pi$
$167$	19.1168	1.47931	0.739653	+	0.672989i	$0.234990\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.37228	−0.487301
$172$	0	0
$173$	22.9783	1.74700	0.873502	−	0.486821i	$-0.161843\pi$
$173$	22.9783	1.74700	0.873502	+	0.486821i	$0.161843\pi$
$174$	0	0
$175$	−0.627719	−0.0474511
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	−9.86141	−0.737076	−0.368538	−	0.929613i	$-0.620142\pi$
$179$	−9.86141	−0.737076	−0.368538	+	0.929613i	$0.620142\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	0	0
$185$	−16.0000	−1.17634
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	14.7446	1.06688	0.533440	−	0.845838i	$-0.320899\pi$
$191$	14.7446	1.06688	0.533440	+	0.845838i	$0.320899\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	−2.37228	−0.169883
$196$	0	0
$197$	18.0000	1.28245	0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000	1.28245	0.641223	+	0.767354i	$0.278427\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	12.7446	0.898932
$202$	0	0
$203$	4.37228	0.306874
$204$	0	0
$205$	−20.7446	−1.44886
$206$	0	0
$207$	−4.37228	−0.303895
$208$	0	0
$209$	12.7446	0.881560
$210$	0	0
$211$	−1.62772	−0.112057	−0.0560284	−	0.998429i	$-0.517844\pi$
$211$	−1.62772	−0.112057	−0.0560284	+	0.998429i	$0.517844\pi$
$212$	0	0
$213$	2.74456	0.188054
$214$	0	0
$215$	26.3723	1.79857
$216$	0	0
$217$	2.37228	0.161041
$218$	0	0
$219$	13.8614	0.936667
$220$	0	0
$221$	0	0
$222$	0	0
$223$	11.8614	0.794299	0.397149	−	0.917754i	$-0.370000\pi$
$223$	11.8614	0.794299	0.397149	+	0.917754i	$0.370000\pi$
$224$	0	0
$225$	0.627719	0.0418479
$226$	0	0
$227$	6.51087	0.432142	0.216071	−	0.976378i	$-0.430676\pi$
$227$	6.51087	0.432142	0.216071	+	0.976378i	$0.430676\pi$
$228$	0	0
$229$	11.4891	0.759223	0.379611	−	0.925146i	$-0.376058\pi$
$229$	11.4891	0.759223	0.379611	+	0.925146i	$0.376058\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	−4.37228	−0.286438	−0.143219	−	0.989691i	$-0.545745\pi$
$233$	−4.37228	−0.286438	−0.143219	+	0.989691i	$0.545745\pi$
$234$	0	0
$235$	−15.1168	−0.986114
$236$	0	0
$237$	11.1168	0.722117
$238$	0	0
$239$	26.7446	1.72996	0.864981	−	0.501805i	$-0.167330\pi$
$239$	26.7446	1.72996	0.864981	+	0.501805i	$0.167330\pi$
$240$	0	0
$241$	−4.37228	−0.281643	−0.140822	−	0.990035i	$-0.544974\pi$
$241$	−4.37228	−0.281643	−0.140822	+	0.990035i	$0.544974\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.37228	−0.151559
$246$	0	0
$247$	6.37228	0.405459
$248$	0	0
$249$	11.1168	0.704501
$250$	0	0
$251$	−4.74456	−0.299474	−0.149737	−	0.988726i	$-0.547843\pi$
$251$	−4.74456	−0.299474	−0.149737	+	0.988726i	$0.547843\pi$
$252$	0	0
$253$	8.74456	0.549766
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−26.2337	−1.63641	−0.818206	−	0.574925i	$-0.805031\pi$
$257$	−26.2337	−1.63641	−0.818206	+	0.574925i	$0.805031\pi$
$258$	0	0
$259$	−6.74456	−0.419087
$260$	0	0
$261$	−4.37228	−0.270637
$262$	0	0
$263$	16.3723	1.00956	0.504779	−	0.863249i	$-0.331574\pi$
$263$	16.3723	1.00956	0.504779	+	0.863249i	$0.331574\pi$
$264$	0	0
$265$	0.883156	0.0542518
$266$	0	0
$267$	1.62772	0.0996148
$268$	0	0
$269$	−24.0000	−1.46331	−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000	−1.46331	−0.731653	+	0.681677i	$0.761251\pi$
$270$	0	0
$271$	−18.9783	−1.15285	−0.576423	−	0.817151i	$-0.695552\pi$
$271$	−18.9783	−1.15285	−0.576423	+	0.817151i	$0.695552\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	−1.25544	−0.0757057
$276$	0	0
$277$	−5.11684	−0.307441	−0.153721	−	0.988114i	$-0.549126\pi$
$277$	−5.11684	−0.307441	−0.153721	+	0.988114i	$0.549126\pi$
$278$	0	0
$279$	−2.37228	−0.142025
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	2.51087	0.149256	0.0746280	−	0.997211i	$-0.476223\pi$
$283$	2.51087	0.149256	0.0746280	+	0.997211i	$0.476223\pi$
$284$	0	0
$285$	−15.1168	−0.895445
$286$	0	0
$287$	−8.74456	−0.516175
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−11.6277	−0.681629
$292$	0	0
$293$	−7.86141	−0.459268	−0.229634	−	0.973277i	$-0.573753\pi$
$293$	−7.86141	−0.459268	−0.229634	+	0.973277i	$0.573753\pi$
$294$	0	0
$295$	−18.9783	−1.10496
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	4.37228	0.252856
$300$	0	0
$301$	11.1168	0.640764
$302$	0	0
$303$	16.7446	0.961950
$304$	0	0
$305$	−14.2337	−0.815019
$306$	0	0
$307$	15.8614	0.905258	0.452629	−	0.891699i	$-0.350486\pi$
$307$	15.8614	0.905258	0.452629	+	0.891699i	$0.350486\pi$
$308$	0	0
$309$	−17.4891	−0.994922
$310$	0	0
$311$	7.25544	0.411418	0.205709	−	0.978613i	$-0.434050\pi$
$311$	7.25544	0.411418	0.205709	+	0.978613i	$0.434050\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	2.37228	0.133663
$316$	0	0
$317$	14.7446	0.828137	0.414069	−	0.910246i	$-0.364107\pi$
$317$	14.7446	0.828137	0.414069	+	0.910246i	$0.364107\pi$
$318$	0	0
$319$	8.74456	0.489602
$320$	0	0
$321$	−10.7446	−0.599703
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−0.627719	−0.0348196
$326$	0	0
$327$	11.4891	0.635350
$328$	0	0
$329$	−6.37228	−0.351315
$330$	0	0
$331$	10.2337	0.562494	0.281247	−	0.959635i	$-0.409252\pi$
$331$	10.2337	0.562494	0.281247	+	0.959635i	$0.409252\pi$
$332$	0	0
$333$	6.74456	0.369600
$334$	0	0
$335$	30.2337	1.65184
$336$	0	0
$337$	−14.6060	−0.795638	−0.397819	−	0.917464i	$-0.630233\pi$
$337$	−14.6060	−0.795638	−0.397819	+	0.917464i	$0.630233\pi$
$338$	0	0
$339$	5.11684	0.277909
$340$	0	0
$341$	4.74456	0.256932
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−10.3723	−0.558425
$346$	0	0
$347$	6.74456	0.362067	0.181034	−	0.983477i	$-0.442056\pi$
$347$	6.74456	0.362067	0.181034	+	0.983477i	$0.442056\pi$
$348$	0	0
$349$	14.6060	0.781840	0.390920	−	0.920425i	$-0.372157\pi$
$349$	14.6060	0.781840	0.390920	+	0.920425i	$0.372157\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−3.25544	−0.173269	−0.0866347	−	0.996240i	$-0.527611\pi$
$353$	−3.25544	−0.173269	−0.0866347	+	0.996240i	$0.527611\pi$
$354$	0	0
$355$	6.51087	0.345561
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.74456	0.355964	0.177982	−	0.984034i	$-0.443043\pi$
$359$	6.74456	0.355964	0.177982	+	0.984034i	$0.443043\pi$
$360$	0	0
$361$	21.6060	1.13716
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	32.8832	1.72118
$366$	0	0
$367$	−4.74456	−0.247664	−0.123832	−	0.992303i	$-0.539518\pi$
$367$	−4.74456	−0.247664	−0.123832	+	0.992303i	$0.539518\pi$
$368$	0	0
$369$	8.74456	0.455224
$370$	0	0
$371$	0.372281	0.0193279
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	−10.3723	−0.535622
$376$	0	0
$377$	4.37228	0.225184
$378$	0	0
$379$	10.2337	0.525669	0.262835	−	0.964841i	$-0.415343\pi$
$379$	10.2337	0.525669	0.262835	+	0.964841i	$0.415343\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	−9.48913	−0.484872	−0.242436	−	0.970167i	$-0.577946\pi$
$383$	−9.48913	−0.484872	−0.242436	+	0.970167i	$0.577946\pi$
$384$	0	0
$385$	−4.74456	−0.241805
$386$	0	0
$387$	−11.1168	−0.565101
$388$	0	0
$389$	−2.00000	−0.101404	−0.0507020	−	0.998714i	$-0.516146\pi$
$389$	−2.00000	−0.101404	−0.0507020	+	0.998714i	$0.516146\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−13.4891	−0.680436
$394$	0	0
$395$	26.3723	1.32693
$396$	0	0
$397$	−28.3723	−1.42396	−0.711982	−	0.702198i	$-0.752202\pi$
$397$	−28.3723	−1.42396	−0.711982	+	0.702198i	$0.752202\pi$
$398$	0	0
$399$	−6.37228	−0.319013
$400$	0	0
$401$	9.25544	0.462194	0.231097	−	0.972931i	$-0.425768\pi$
$401$	9.25544	0.462194	0.231097	+	0.972931i	$0.425768\pi$
$402$	0	0
$403$	2.37228	0.118172
$404$	0	0
$405$	−2.37228	−0.117880
$406$	0	0
$407$	−13.4891	−0.668631
$408$	0	0
$409$	−5.86141	−0.289828	−0.144914	−	0.989444i	$-0.546291\pi$
$409$	−5.86141	−0.289828	−0.144914	+	0.989444i	$0.546291\pi$
$410$	0	0
$411$	−12.2337	−0.603443
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	26.3723	1.29456
$416$	0	0
$417$	−8.74456	−0.428223
$418$	0	0
$419$	−12.7446	−0.622613	−0.311306	−	0.950310i	$-0.600767\pi$
$419$	−12.7446	−0.622613	−0.311306	+	0.950310i	$0.600767\pi$
$420$	0	0
$421$	14.7446	0.718606	0.359303	−	0.933221i	$-0.383014\pi$
$421$	14.7446	0.718606	0.359303	+	0.933221i	$0.383014\pi$
$422$	0	0
$423$	6.37228	0.309831
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.00000	−0.290360
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	−39.4891	−1.90212	−0.951062	−	0.309000i	$-0.900006\pi$
$431$	−39.4891	−1.90212	−0.951062	+	0.309000i	$0.900006\pi$
$432$	0	0
$433$	−10.7446	−0.516351	−0.258175	−	0.966098i	$-0.583121\pi$
$433$	−10.7446	−0.516351	−0.258175	+	0.966098i	$0.583121\pi$
$434$	0	0
$435$	−10.3723	−0.497313
$436$	0	0
$437$	27.8614	1.33279
$438$	0	0
$439$	−12.7446	−0.608265	−0.304132	−	0.952630i	$-0.598367\pi$
$439$	−12.7446	−0.608265	−0.304132	+	0.952630i	$0.598367\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−40.3723	−1.91814	−0.959072	−	0.283161i	$-0.908617\pi$
$443$	−40.3723	−1.91814	−0.959072	+	0.283161i	$0.908617\pi$
$444$	0	0
$445$	3.86141	0.183048
$446$	0	0
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	5.25544	0.248019	0.124010	−	0.992281i	$-0.460425\pi$
$449$	5.25544	0.248019	0.124010	+	0.992281i	$0.460425\pi$
$450$	0	0
$451$	−17.4891	−0.823531
$452$	0	0
$453$	−1.48913	−0.0699652
$454$	0	0
$455$	−2.37228	−0.111214
$456$	0	0
$457$	24.2337	1.13360	0.566802	−	0.823854i	$-0.308180\pi$
$457$	24.2337	1.13360	0.566802	+	0.823854i	$0.308180\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	24.7446	1.15247	0.576235	−	0.817284i	$-0.304521\pi$
$461$	24.7446	1.15247	0.576235	+	0.817284i	$0.304521\pi$
$462$	0	0
$463$	−8.74456	−0.406394	−0.203197	−	0.979138i	$-0.565133\pi$
$463$	−8.74456	−0.406394	−0.203197	+	0.979138i	$0.565133\pi$
$464$	0	0
$465$	−5.62772	−0.260979
$466$	0	0
$467$	−14.2337	−0.658657	−0.329328	−	0.944215i	$-0.606822\pi$
$467$	−14.2337	−0.658657	−0.329328	+	0.944215i	$0.606822\pi$
$468$	0	0
$469$	12.7446	0.588489
$470$	0	0
$471$	−2.74456	−0.126463
$472$	0	0
$473$	22.2337	1.02231
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	−0.372281	−0.0170456
$478$	0	0
$479$	35.1168	1.60453	0.802265	−	0.596968i	$-0.203628\pi$
$479$	35.1168	1.60453	0.802265	+	0.596968i	$0.203628\pi$
$480$	0	0
$481$	−6.74456	−0.307526
$482$	0	0
$483$	−4.37228	−0.198946
$484$	0	0
$485$	−27.5842	−1.25253
$486$	0	0
$487$	−11.2554	−0.510033	−0.255016	−	0.966937i	$-0.582081\pi$
$487$	−11.2554	−0.510033	−0.255016	+	0.966937i	$0.582081\pi$
$488$	0	0
$489$	9.48913	0.429113
$490$	0	0
$491$	−18.7446	−0.845930	−0.422965	−	0.906146i	$-0.639011\pi$
$491$	−18.7446	−0.845930	−0.422965	+	0.906146i	$0.639011\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4.74456	0.213252
$496$	0	0
$497$	2.74456	0.123110
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	−19.1168	−0.854078
$502$	0	0
$503$	−31.7228	−1.41445	−0.707225	−	0.706988i	$-0.750053\pi$
$503$	−31.7228	−1.41445	−0.707225	+	0.706988i	$0.750053\pi$
$504$	0	0
$505$	39.7228	1.76764
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−29.3505	−1.30094	−0.650470	−	0.759532i	$-0.725428\pi$
$509$	−29.3505	−1.30094	−0.650470	+	0.759532i	$0.725428\pi$
$510$	0	0
$511$	13.8614	0.613193
$512$	0	0
$513$	6.37228	0.281343
$514$	0	0
$515$	−41.4891	−1.82823
$516$	0	0
$517$	−12.7446	−0.560505
$518$	0	0
$519$	−22.9783	−1.00863
$520$	0	0
$521$	16.0000	0.700973	0.350486	−	0.936568i	$-0.386016\pi$
$521$	16.0000	0.700973	0.350486	+	0.936568i	$0.386016\pi$
$522$	0	0
$523$	27.7228	1.21223	0.606117	−	0.795376i	$-0.292726\pi$
$523$	27.7228	1.21223	0.606117	+	0.795376i	$0.292726\pi$
$524$	0	0
$525$	0.627719	0.0273959
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−3.88316	−0.168833
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	−8.74456	−0.378769
$534$	0	0
$535$	−25.4891	−1.10199
$536$	0	0
$537$	9.86141	0.425551
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−18.7446	−0.805892	−0.402946	−	0.915224i	$-0.632014\pi$
$541$	−18.7446	−0.805892	−0.402946	+	0.915224i	$0.632014\pi$
$542$	0	0
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	27.2554	1.16749
$546$	0	0
$547$	−34.3723	−1.46965	−0.734826	−	0.678255i	$-0.762736\pi$
$547$	−34.3723	−1.46965	−0.734826	+	0.678255i	$0.762736\pi$
$548$	0	0
$549$	6.00000	0.256074
$550$	0	0
$551$	27.8614	1.18694
$552$	0	0
$553$	11.1168	0.472736
$554$	0	0
$555$	16.0000	0.679162
$556$	0	0
$557$	−21.7228	−0.920425	−0.460213	−	0.887809i	$-0.652227\pi$
$557$	−21.7228	−0.920425	−0.460213	+	0.887809i	$0.652227\pi$
$558$	0	0
$559$	11.1168	0.470192
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.4674	1.19976	0.599878	−	0.800091i	$-0.295216\pi$
$563$	28.4674	1.19976	0.599878	+	0.800091i	$0.295216\pi$
$564$	0	0
$565$	12.1386	0.510674
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	14.6060	0.612314	0.306157	−	0.951981i	$-0.400957\pi$
$569$	14.6060	0.612314	0.306157	+	0.951981i	$0.400957\pi$
$570$	0	0
$571$	−29.6277	−1.23988	−0.619941	−	0.784649i	$-0.712843\pi$
$571$	−29.6277	−1.23988	−0.619941	+	0.784649i	$0.712843\pi$
$572$	0	0
$573$	−14.7446	−0.615963
$574$	0	0
$575$	−2.74456	−0.114456
$576$	0	0
$577$	−24.9783	−1.03986	−0.519929	−	0.854209i	$-0.674042\pi$
$577$	−24.9783	−1.03986	−0.519929	+	0.854209i	$0.674042\pi$
$578$	0	0
$579$	−6.00000	−0.249351
$580$	0	0
$581$	11.1168	0.461204
$582$	0	0
$583$	0.744563	0.0308366
$584$	0	0
$585$	2.37228	0.0980818
$586$	0	0
$587$	−21.6277	−0.892671	−0.446336	−	0.894866i	$-0.647271\pi$
$587$	−21.6277	−0.892671	−0.446336	+	0.894866i	$0.647271\pi$
$588$	0	0
$589$	15.1168	0.622879
$590$	0	0
$591$	−18.0000	−0.740421
$592$	0	0
$593$	42.0951	1.72864	0.864319	−	0.502944i	$-0.167750\pi$
$593$	42.0951	1.72864	0.864319	+	0.502944i	$0.167750\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.0000	−0.654836
$598$	0	0
$599$	−17.8614	−0.729797	−0.364899	−	0.931047i	$-0.618896\pi$
$599$	−17.8614	−0.729797	−0.364899	+	0.931047i	$0.618896\pi$
$600$	0	0
$601$	8.51087	0.347166	0.173583	−	0.984819i	$-0.444466\pi$
$601$	8.51087	0.347166	0.173583	+	0.984819i	$0.444466\pi$
$602$	0	0
$603$	−12.7446	−0.518999
$604$	0	0
$605$	16.6060	0.675129
$606$	0	0
$607$	36.7446	1.49142	0.745708	−	0.666273i	$-0.232111\pi$
$607$	36.7446	1.49142	0.745708	+	0.666273i	$0.232111\pi$
$608$	0	0
$609$	−4.37228	−0.177174
$610$	0	0
$611$	−6.37228	−0.257795
$612$	0	0
$613$	−14.7446	−0.595527	−0.297764	−	0.954640i	$-0.596241\pi$
$613$	−14.7446	−0.595527	−0.297764	+	0.954640i	$0.596241\pi$
$614$	0	0
$615$	20.7446	0.836502
$616$	0	0
$617$	12.5109	0.503669	0.251834	−	0.967770i	$-0.418966\pi$
$617$	12.5109	0.503669	0.251834	+	0.967770i	$0.418966\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	4.37228	0.175454
$622$	0	0
$623$	1.62772	0.0652132
$624$	0	0
$625$	−27.7446	−1.10978
$626$	0	0
$627$	−12.7446	−0.508969
$628$	0	0
$629$	0	0
$630$	0	0
$631$	45.2119	1.79986	0.899929	−	0.436036i	$-0.143618\pi$
$631$	45.2119	1.79986	0.899929	+	0.436036i	$0.143618\pi$
$632$	0	0
$633$	1.62772	0.0646960
$634$	0	0
$635$	−9.48913	−0.376564
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−2.74456	−0.108573
$640$	0	0
$641$	−1.11684	−0.0441127	−0.0220563	−	0.999757i	$-0.507021\pi$
$641$	−1.11684	−0.0441127	−0.0220563	+	0.999757i	$0.507021\pi$
$642$	0	0
$643$	−37.4891	−1.47843	−0.739213	−	0.673471i	$-0.764803\pi$
$643$	−37.4891	−1.47843	−0.739213	+	0.673471i	$0.764803\pi$
$644$	0	0
$645$	−26.3723	−1.03841
$646$	0	0
$647$	−16.7446	−0.658297	−0.329148	−	0.944278i	$-0.606762\pi$
$647$	−16.7446	−0.658297	−0.329148	+	0.944278i	$0.606762\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	−2.37228	−0.0929770
$652$	0	0
$653$	26.4674	1.03575	0.517874	−	0.855457i	$-0.326724\pi$
$653$	26.4674	1.03575	0.517874	+	0.855457i	$0.326724\pi$
$654$	0	0
$655$	−32.0000	−1.25034
$656$	0	0
$657$	−13.8614	−0.540785
$658$	0	0
$659$	9.86141	0.384146	0.192073	−	0.981381i	$-0.438479\pi$
$659$	9.86141	0.384146	0.192073	+	0.981381i	$0.438479\pi$
$660$	0	0
$661$	−29.8614	−1.16147	−0.580737	−	0.814091i	$-0.697236\pi$
$661$	−29.8614	−1.16147	−0.580737	+	0.814091i	$0.697236\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−15.1168	−0.586206
$666$	0	0
$667$	19.1168	0.740207
$668$	0	0
$669$	−11.8614	−0.458588
$670$	0	0
$671$	−12.0000	−0.463255
$672$	0	0
$673$	−36.0951	−1.39136	−0.695682	−	0.718350i	$-0.744898\pi$
$673$	−36.0951	−1.39136	−0.695682	+	0.718350i	$0.744898\pi$
$674$	0	0
$675$	−0.627719	−0.0241609
$676$	0	0
$677$	13.7663	0.529082	0.264541	−	0.964374i	$-0.414779\pi$
$677$	13.7663	0.529082	0.264541	+	0.964374i	$0.414779\pi$
$678$	0	0
$679$	−11.6277	−0.446231
$680$	0	0
$681$	−6.51087	−0.249497
$682$	0	0
$683$	32.2337	1.23339	0.616694	−	0.787203i	$-0.288472\pi$
$683$	32.2337	1.23339	0.616694	+	0.787203i	$0.288472\pi$
$684$	0	0
$685$	−29.0217	−1.10886
$686$	0	0
$687$	−11.4891	−0.438337
$688$	0	0
$689$	0.372281	0.0141828
$690$	0	0
$691$	1.35053	0.0513767	0.0256883	−	0.999670i	$-0.491822\pi$
$691$	1.35053	0.0513767	0.0256883	+	0.999670i	$0.491822\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	−20.7446	−0.786886
$696$	0	0
$697$	0	0
$698$	0	0
$699$	4.37228	0.165375
$700$	0	0
$701$	41.8614	1.58108	0.790542	−	0.612408i	$-0.209799\pi$
$701$	41.8614	1.58108	0.790542	+	0.612408i	$0.209799\pi$
$702$	0	0
$703$	−42.9783	−1.62096
$704$	0	0
$705$	15.1168	0.569333
$706$	0	0
$707$	16.7446	0.629744
$708$	0	0
$709$	−35.4891	−1.33282	−0.666411	−	0.745585i	$-0.732170\pi$
$709$	−35.4891	−1.33282	−0.666411	+	0.745585i	$0.732170\pi$
$710$	0	0
$711$	−11.1168	−0.416914
$712$	0	0
$713$	10.3723	0.388445
$714$	0	0
$715$	−4.74456	−0.177437
$716$	0	0
$717$	−26.7446	−0.998794
$718$	0	0
$719$	17.7663	0.662572	0.331286	−	0.943530i	$-0.392518\pi$
$719$	17.7663	0.662572	0.331286	+	0.943530i	$0.392518\pi$
$720$	0	0
$721$	−17.4891	−0.651329
$722$	0	0
$723$	4.37228	0.162607
$724$	0	0
$725$	−2.74456	−0.101930
$726$	0	0
$727$	−6.23369	−0.231195	−0.115597	−	0.993296i	$-0.536878\pi$
$727$	−6.23369	−0.231195	−0.115597	+	0.993296i	$0.536878\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	33.1168	1.22320	0.611599	−	0.791168i	$-0.290526\pi$
$733$	33.1168	1.22320	0.611599	+	0.791168i	$0.290526\pi$
$734$	0	0
$735$	2.37228	0.0875029
$736$	0	0
$737$	25.4891	0.938904
$738$	0	0
$739$	33.2119	1.22172	0.610860	−	0.791738i	$-0.290824\pi$
$739$	33.2119	1.22172	0.610860	+	0.791738i	$0.290824\pi$
$740$	0	0
$741$	−6.37228	−0.234092
$742$	0	0
$743$	16.2337	0.595556	0.297778	−	0.954635i	$-0.403754\pi$
$743$	16.2337	0.595556	0.297778	+	0.954635i	$0.403754\pi$
$744$	0	0
$745$	−4.74456	−0.173827
$746$	0	0
$747$	−11.1168	−0.406744
$748$	0	0
$749$	−10.7446	−0.392598
$750$	0	0
$751$	−34.0951	−1.24415	−0.622074	−	0.782959i	$-0.713710\pi$
$751$	−34.0951	−1.24415	−0.622074	+	0.782959i	$0.713710\pi$
$752$	0	0
$753$	4.74456	0.172901
$754$	0	0
$755$	−3.53262	−0.128565
$756$	0	0
$757$	19.3505	0.703307	0.351654	−	0.936130i	$-0.385620\pi$
$757$	19.3505	0.703307	0.351654	+	0.936130i	$0.385620\pi$
$758$	0	0
$759$	−8.74456	−0.317408
$760$	0	0
$761$	−31.8614	−1.15498	−0.577488	−	0.816399i	$-0.695967\pi$
$761$	−31.8614	−1.15498	−0.577488	+	0.816399i	$0.695967\pi$
$762$	0	0
$763$	11.4891	0.415934
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−21.1168	−0.761493	−0.380746	−	0.924679i	$-0.624333\pi$
$769$	−21.1168	−0.761493	−0.380746	+	0.924679i	$0.624333\pi$
$770$	0	0
$771$	26.2337	0.944783
$772$	0	0
$773$	−19.7228	−0.709380	−0.354690	−	0.934984i	$-0.615414\pi$
$773$	−19.7228	−0.709380	−0.354690	+	0.934984i	$0.615414\pi$
$774$	0	0
$775$	−1.48913	−0.0534910
$776$	0	0
$777$	6.74456	0.241960
$778$	0	0
$779$	−55.7228	−1.99648
$780$	0	0
$781$	5.48913	0.196416
$782$	0	0
$783$	4.37228	0.156253
$784$	0	0
$785$	−6.51087	−0.232383
$786$	0	0
$787$	12.6060	0.449354	0.224677	−	0.974433i	$-0.427867\pi$
$787$	12.6060	0.449354	0.224677	+	0.974433i	$0.427867\pi$
$788$	0	0
$789$	−16.3723	−0.582869
$790$	0	0
$791$	5.11684	0.181934
$792$	0	0
$793$	−6.00000	−0.213066
$794$	0	0
$795$	−0.883156	−0.0313223
$796$	0	0
$797$	43.7228	1.54874	0.774371	−	0.632732i	$-0.218067\pi$
$797$	43.7228	1.54874	0.774371	+	0.632732i	$0.218067\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−1.62772	−0.0575126
$802$	0	0
$803$	27.7228	0.978317
$804$	0	0
$805$	−10.3723	−0.365575
$806$	0	0
$807$	24.0000	0.844840
$808$	0	0
$809$	−43.6277	−1.53387	−0.766934	−	0.641725i	$-0.778219\pi$
$809$	−43.6277	−1.53387	−0.766934	+	0.641725i	$0.778219\pi$
$810$	0	0
$811$	−6.97825	−0.245040	−0.122520	−	0.992466i	$-0.539097\pi$
$811$	−6.97825	−0.245040	−0.122520	+	0.992466i	$0.539097\pi$
$812$	0	0
$813$	18.9783	0.665596
$814$	0	0
$815$	22.5109	0.788522
$816$	0	0
$817$	70.8397	2.47837
$818$	0	0
$819$	1.00000	0.0349428
$820$	0	0
$821$	44.9783	1.56975	0.784876	−	0.619653i	$-0.212727\pi$
$821$	44.9783	1.56975	0.784876	+	0.619653i	$0.212727\pi$
$822$	0	0
$823$	45.4891	1.58565	0.792826	−	0.609449i	$-0.208609\pi$
$823$	45.4891	1.58565	0.792826	+	0.609449i	$0.208609\pi$
$824$	0	0
$825$	1.25544	0.0437087
$826$	0	0
$827$	−51.2119	−1.78081	−0.890407	−	0.455166i	$-0.849580\pi$
$827$	−51.2119	−1.78081	−0.890407	+	0.455166i	$0.849580\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	5.11684	0.177501
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−45.3505	−1.56942
$836$	0	0
$837$	2.37228	0.0819980
$838$	0	0
$839$	−13.4891	−0.465696	−0.232848	−	0.972513i	$-0.574805\pi$
$839$	−13.4891	−0.465696	−0.232848	+	0.972513i	$0.574805\pi$
$840$	0	0
$841$	−9.88316	−0.340798
$842$	0	0
$843$	−22.0000	−0.757720
$844$	0	0
$845$	−2.37228	−0.0816090
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	−2.51087	−0.0861730
$850$	0	0
$851$	−29.4891	−1.01087
$852$	0	0
$853$	52.8397	1.80920	0.904598	−	0.426266i	$-0.140171\pi$
$853$	52.8397	1.80920	0.904598	+	0.426266i	$0.140171\pi$
$854$	0	0
$855$	15.1168	0.516985
$856$	0	0
$857$	−2.23369	−0.0763013	−0.0381507	−	0.999272i	$-0.512147\pi$
$857$	−2.23369	−0.0763013	−0.0381507	+	0.999272i	$0.512147\pi$
$858$	0	0
$859$	34.2337	1.16804	0.584019	−	0.811740i	$-0.301479\pi$
$859$	34.2337	1.16804	0.584019	+	0.811740i	$0.301479\pi$
$860$	0	0
$861$	8.74456	0.298014
$862$	0	0
$863$	12.2337	0.416440	0.208220	−	0.978082i	$-0.433233\pi$
$863$	12.2337	0.416440	0.208220	+	0.978082i	$0.433233\pi$
$864$	0	0
$865$	−54.5109	−1.85343
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	22.2337	0.754226
$870$	0	0
$871$	12.7446	0.431833
$872$	0	0
$873$	11.6277	0.393539
$874$	0	0
$875$	−10.3723	−0.350647
$876$	0	0
$877$	2.74456	0.0926773	0.0463386	−	0.998926i	$-0.485245\pi$
$877$	2.74456	0.0926773	0.0463386	+	0.998926i	$0.485245\pi$
$878$	0	0
$879$	7.86141	0.265159
$880$	0	0
$881$	−44.7446	−1.50748	−0.753741	−	0.657171i	$-0.771753\pi$
$881$	−44.7446	−1.50748	−0.753741	+	0.657171i	$0.771753\pi$
$882$	0	0
$883$	22.9783	0.773280	0.386640	−	0.922231i	$-0.373636\pi$
$883$	22.9783	0.773280	0.386640	+	0.922231i	$0.373636\pi$
$884$	0	0
$885$	18.9783	0.637947
$886$	0	0
$887$	46.9783	1.57738	0.788688	−	0.614794i	$-0.210761\pi$
$887$	46.9783	1.57738	0.788688	+	0.614794i	$0.210761\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	−40.6060	−1.35883
$894$	0	0
$895$	23.3940	0.781976
$896$	0	0
$897$	−4.37228	−0.145986
$898$	0	0
$899$	10.3723	0.345935
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−11.1168	−0.369945
$904$	0	0
$905$	−4.74456	−0.157715
$906$	0	0
$907$	−43.1168	−1.43167	−0.715836	−	0.698269i	$-0.753954\pi$
$907$	−43.1168	−1.43167	−0.715836	+	0.698269i	$0.753954\pi$
$908$	0	0
$909$	−16.7446	−0.555382
$910$	0	0
$911$	13.8614	0.459249	0.229624	−	0.973279i	$-0.426250\pi$
$911$	13.8614	0.459249	0.229624	+	0.973279i	$0.426250\pi$
$912$	0	0
$913$	22.2337	0.735828
$914$	0	0
$915$	14.2337	0.470551
$916$	0	0
$917$	−13.4891	−0.445450
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−15.8614	−0.522651
$922$	0	0
$923$	2.74456	0.0903384
$924$	0	0
$925$	4.23369	0.139203
$926$	0	0
$927$	17.4891	0.574418
$928$	0	0
$929$	−19.3940	−0.636298	−0.318149	−	0.948041i	$-0.603061\pi$
$929$	−19.3940	−0.636298	−0.318149	+	0.948041i	$0.603061\pi$
$930$	0	0
$931$	−6.37228	−0.208843
$932$	0	0
$933$	−7.25544	−0.237532
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.2337	0.530331	0.265166	−	0.964203i	$-0.414573\pi$
$937$	16.2337	0.530331	0.265166	+	0.964203i	$0.414573\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−31.1168	−1.01438	−0.507190	−	0.861834i	$-0.669316\pi$
$941$	−31.1168	−1.01438	−0.507190	+	0.861834i	$0.669316\pi$
$942$	0	0
$943$	−38.2337	−1.24506
$944$	0	0
$945$	−2.37228	−0.0771703
$946$	0	0
$947$	−7.02175	−0.228176	−0.114088	−	0.993471i	$-0.536395\pi$
$947$	−7.02175	−0.228176	−0.114088	+	0.993471i	$0.536395\pi$
$948$	0	0
$949$	13.8614	0.449960
$950$	0	0
$951$	−14.7446	−0.478125
$952$	0	0
$953$	20.3723	0.659923	0.329961	−	0.943994i	$-0.392964\pi$
$953$	20.3723	0.659923	0.329961	+	0.943994i	$0.392964\pi$
$954$	0	0
$955$	−34.9783	−1.13187
$956$	0	0
$957$	−8.74456	−0.282672
$958$	0	0
$959$	−12.2337	−0.395046
$960$	0	0
$961$	−25.3723	−0.818461
$962$	0	0
$963$	10.7446	0.346239
$964$	0	0
$965$	−14.2337	−0.458199
$966$	0	0
$967$	51.7228	1.66329	0.831647	−	0.555305i	$-0.187398\pi$
$967$	51.7228	1.66329	0.831647	+	0.555305i	$0.187398\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−0.744563	−0.0238942	−0.0119471	−	0.999929i	$-0.503803\pi$
$971$	−0.744563	−0.0238942	−0.0119471	+	0.999929i	$0.503803\pi$
$972$	0	0
$973$	−8.74456	−0.280338
$974$	0	0
$975$	0.627719	0.0201031
$976$	0	0
$977$	16.9783	0.543182	0.271591	−	0.962413i	$-0.412450\pi$
$977$	16.9783	0.543182	0.271591	+	0.962413i	$0.412450\pi$
$978$	0	0
$979$	3.25544	0.104044
$980$	0	0
$981$	−11.4891	−0.366820
$982$	0	0
$983$	26.0951	0.832304	0.416152	−	0.909295i	$-0.363378\pi$
$983$	26.0951	0.832304	0.416152	+	0.909295i	$0.363378\pi$
$984$	0	0
$985$	−42.7011	−1.36057
$986$	0	0
$987$	6.37228	0.202832
$988$	0	0
$989$	48.6060	1.54558
$990$	0	0
$991$	−33.4891	−1.06382	−0.531909	−	0.846802i	$-0.678525\pi$
$991$	−33.4891	−1.06382	−0.531909	+	0.846802i	$0.678525\pi$
$992$	0	0
$993$	−10.2337	−0.324756
$994$	0	0
$995$	−37.9565	−1.20330
$996$	0	0
$997$	10.7446	0.340284	0.170142	−	0.985420i	$-0.445577\pi$
$997$	10.7446	0.340284	0.170142	+	0.985420i	$0.445577\pi$
$998$	0	0
$999$	−6.74456	−0.213389

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bc.1.1		2
4.3	odd	2		1092.2.a.g.1.1	✓	2
12.11	even	2		3276.2.a.m.1.2		2
28.27	even	2		7644.2.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1092.2.a.g.1.1	✓	2	4.3	odd	2
3276.2.a.m.1.2		2	12.11	even	2
4368.2.a.bc.1.1		2	1.1	even	1	trivial
7644.2.a.n.1.2		2	28.27	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:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.37228 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} -1.00000 q^{13} +2.37228 q^{15} -6.37228 q^{19} +1.00000 q^{21} -4.37228 q^{23} +0.627719 q^{25} -1.00000 q^{27} -4.37228 q^{29} -2.37228 q^{31} +2.00000 q^{33} +2.37228 q^{35} +6.74456 q^{37} +1.00000 q^{39} +8.74456 q^{41} -11.1168 q^{43} -2.37228 q^{45} +6.37228 q^{47} +1.00000 q^{49} -0.372281 q^{53} +4.74456 q^{55} +6.37228 q^{57} +8.00000 q^{59} +6.00000 q^{61} -1.00000 q^{63} +2.37228 q^{65} -12.7446 q^{67} +4.37228 q^{69} -2.74456 q^{71} -13.8614 q^{73} -0.627719 q^{75} +2.00000 q^{77} -11.1168 q^{79} +1.00000 q^{81} -11.1168 q^{83} +4.37228 q^{87} -1.62772 q^{89} +1.00000 q^{91} +2.37228 q^{93} +15.1168 q^{95} +11.6277 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} - q^{15} - 7 q^{19} + 2 q^{21} - 3 q^{23} + 7 q^{25} - 2 q^{27} - 3 q^{29} + q^{31} + 4 q^{33} - q^{35} + 2 q^{37} + 2 q^{39} + 6 q^{41} - 5 q^{43} + q^{45} + 7 q^{47} + 2 q^{49} + 5 q^{53} - 2 q^{55} + 7 q^{57} + 16 q^{59} + 12 q^{61} - 2 q^{63} - q^{65} - 14 q^{67} + 3 q^{69} + 6 q^{71} + q^{73} - 7 q^{75} + 4 q^{77} - 5 q^{79} + 2 q^{81} - 5 q^{83} + 3 q^{87} - 9 q^{89} + 2 q^{91} - q^{93} + 13 q^{95} + 29 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + q^5 - 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^13 - q^15 - 7 * q^19 + 2 * q^21 - 3 * q^23 + 7 * q^25 - 2 * q^27 - 3 * q^29 + q^31 + 4 * q^33 - q^35 + 2 * q^37 + 2 * q^39 + 6 * q^41 - 5 * q^43 + q^45 + 7 * q^47 + 2 * q^49 + 5 * q^53 - 2 * q^55 + 7 * q^57 + 16 * q^59 + 12 * q^61 - 2 * q^63 - q^65 - 14 * q^67 + 3 * q^69 + 6 * q^71 + q^73 - 7 * q^75 + 4 * q^77 - 5 * q^79 + 2 * q^81 - 5 * q^83 + 3 * q^87 - 9 * q^89 + 2 * q^91 - q^93 + 13 * q^95 + 29 * q^97 - 4 * q^99

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