LMFDB - Embedded newform 9072.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9072 = 2^{4} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9072.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:	$72.4402847137$
Analytic rank:	$1$
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 126)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	9072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.37228 q^{5} -1.00000 q^{7} +O(q^{10})$	$q-4.37228 q^{5} -1.00000 q^{7} -1.37228 q^{11} +2.00000 q^{13} -1.37228 q^{17} -5.00000 q^{19} -1.62772 q^{23} +14.1168 q^{25} +8.74456 q^{29} -2.00000 q^{31} +4.37228 q^{35} +2.00000 q^{37} -4.62772 q^{41} +8.11684 q^{43} +1.00000 q^{49} +8.74456 q^{53} +6.00000 q^{55} -10.1168 q^{59} +3.11684 q^{61} -8.74456 q^{65} +2.11684 q^{67} -7.11684 q^{71} +12.1168 q^{73} +1.37228 q^{77} +5.11684 q^{79} +17.4891 q^{83} +6.00000 q^{85} -14.7446 q^{89} -2.00000 q^{91} +21.8614 q^{95} -8.11684 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 3 * q^5 - 2 * q^7	$ 2 q - 3 q^{5} - 2 q^{7} + 3 q^{11} + 4 q^{13} + 3 q^{17} - 10 q^{19} - 9 q^{23} + 11 q^{25} + 6 q^{29} - 4 q^{31} + 3 q^{35} + 4 q^{37} - 15 q^{41} - q^{43} + 2 q^{49} + 6 q^{53} + 12 q^{55} - 3 q^{59} - 11 q^{61} - 6 q^{65} - 13 q^{67} + 3 q^{71} + 7 q^{73} - 3 q^{77} - 7 q^{79} + 12 q^{83} + 12 q^{85} - 18 q^{89} - 4 q^{91} + 15 q^{95} + q^{97}+O(q^{100}) $ 2 * q - 3 * q^5 - 2 * q^7 + 3 * q^11 + 4 * q^13 + 3 * q^17 - 10 * q^19 - 9 * q^23 + 11 * q^25 + 6 * q^29 - 4 * q^31 + 3 * q^35 + 4 * q^37 - 15 * q^41 - q^43 + 2 * q^49 + 6 * q^53 + 12 * q^55 - 3 * q^59 - 11 * q^61 - 6 * q^65 - 13 * q^67 + 3 * q^71 + 7 * q^73 - 3 * q^77 - 7 * q^79 + 12 * q^83 + 12 * q^85 - 18 * q^89 - 4 * q^91 + 15 * q^95 + q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−4.37228	−1.95534	−0.977672	−	0.210138i	$-0.932609\pi$
$5$	−4.37228	−1.95534	−0.977672	+	0.210138i	$0.932609\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.37228	−0.413758	−0.206879	−	0.978366i	$-0.566331\pi$
$11$	−1.37228	−0.413758	−0.206879	+	0.978366i	$0.566331\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.37228	−0.332827	−0.166414	−	0.986056i	$-0.553219\pi$
$17$	−1.37228	−0.332827	−0.166414	+	0.986056i	$0.553219\pi$
$18$	0	0
$19$	−5.00000	−1.14708	−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000	−1.14708	−0.573539	+	0.819178i	$0.694430\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.62772	−0.339403	−0.169701	−	0.985496i	$-0.554280\pi$
$23$	−1.62772	−0.339403	−0.169701	+	0.985496i	$0.554280\pi$
$24$	0	0
$25$	14.1168	2.82337
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.74456	1.62382	0.811912	−	0.583779i	$-0.198427\pi$
$29$	8.74456	1.62382	0.811912	+	0.583779i	$0.198427\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.37228	0.739050
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.62772	−0.722728	−0.361364	−	0.932425i	$-0.617689\pi$
$41$	−4.62772	−0.722728	−0.361364	+	0.932425i	$0.617689\pi$
$42$	0	0
$43$	8.11684	1.23781	0.618904	−	0.785467i	$-0.287577\pi$
$43$	8.11684	1.23781	0.618904	+	0.785467i	$0.287577\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.74456	1.20116	0.600579	−	0.799565i	$-0.294937\pi$
$53$	8.74456	1.20116	0.600579	+	0.799565i	$0.294937\pi$
$54$	0	0
$55$	6.00000	0.809040
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.1168	−1.31710	−0.658550	−	0.752537i	$-0.728830\pi$
$59$	−10.1168	−1.31710	−0.658550	+	0.752537i	$0.728830\pi$
$60$	0	0
$61$	3.11684	0.399071	0.199535	−	0.979891i	$-0.436057\pi$
$61$	3.11684	0.399071	0.199535	+	0.979891i	$0.436057\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−8.74456	−1.08463
$66$	0	0
$67$	2.11684	0.258614	0.129307	−	0.991605i	$-0.458725\pi$
$67$	2.11684	0.258614	0.129307	+	0.991605i	$0.458725\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.11684	−0.844614	−0.422307	−	0.906453i	$-0.638780\pi$
$71$	−7.11684	−0.844614	−0.422307	+	0.906453i	$0.638780\pi$
$72$	0	0
$73$	12.1168	1.41817	0.709085	−	0.705123i	$-0.249108\pi$
$73$	12.1168	1.41817	0.709085	+	0.705123i	$0.249108\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.37228	0.156386
$78$	0	0
$79$	5.11684	0.575690	0.287845	−	0.957677i	$-0.407061\pi$
$79$	5.11684	0.575690	0.287845	+	0.957677i	$0.407061\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	17.4891	1.91968	0.959840	−	0.280546i	$-0.0905157\pi$
$83$	17.4891	1.91968	0.959840	+	0.280546i	$0.0905157\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.7446	−1.56292	−0.781460	−	0.623955i	$-0.785525\pi$
$89$	−14.7446	−1.56292	−0.781460	+	0.623955i	$0.785525\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	21.8614	2.24293
$96$	0	0
$97$	−8.11684	−0.824141	−0.412070	−	0.911152i	$-0.635194\pi$
$97$	−8.11684	−0.824141	−0.412070	+	0.911152i	$0.635194\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.62772	−0.161964	−0.0809820	−	0.996716i	$-0.525806\pi$
$101$	−1.62772	−0.161964	−0.0809820	+	0.996716i	$0.525806\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.37228	−0.712705	−0.356353	−	0.934352i	$-0.615980\pi$
$107$	−7.37228	−0.712705	−0.356353	+	0.934352i	$0.615980\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.37228	0.411310	0.205655	−	0.978625i	$-0.434068\pi$
$113$	4.37228	0.411310	0.205655	+	0.978625i	$0.434068\pi$
$114$	0	0
$115$	7.11684	0.663649
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.37228	0.125797
$120$	0	0
$121$	−9.11684	−0.828804
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−39.8614	−3.56531
$126$	0	0
$127$	−3.11684	−0.276575	−0.138288	−	0.990392i	$-0.544160\pi$
$127$	−3.11684	−0.276575	−0.138288	+	0.990392i	$0.544160\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.62772	−0.142214	−0.0711072	−	0.997469i	$-0.522653\pi$
$131$	−1.62772	−0.142214	−0.0711072	+	0.997469i	$0.522653\pi$
$132$	0	0
$133$	5.00000	0.433555
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.6277	−0.907987	−0.453994	−	0.891005i	$-0.650001\pi$
$137$	−10.6277	−0.907987	−0.453994	+	0.891005i	$0.650001\pi$
$138$	0	0
$139$	−13.2337	−1.12247	−0.561233	−	0.827658i	$-0.689673\pi$
$139$	−13.2337	−1.12247	−0.561233	+	0.827658i	$0.689673\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.74456	−0.229512
$144$	0	0
$145$	−38.2337	−3.17513
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.25544	−0.266696	−0.133348	−	0.991069i	$-0.542573\pi$
$149$	−3.25544	−0.266696	−0.133348	+	0.991069i	$0.542573\pi$
$150$	0	0
$151$	−9.11684	−0.741918	−0.370959	−	0.928649i	$-0.620971\pi$
$151$	−9.11684	−0.741918	−0.370959	+	0.928649i	$0.620971\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.74456	0.702380
$156$	0	0
$157$	9.11684	0.727603	0.363802	−	0.931476i	$-0.381479\pi$
$157$	9.11684	0.727603	0.363802	+	0.931476i	$0.381479\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.62772	0.128282
$162$	0	0
$163$	18.2337	1.42817	0.714086	−	0.700058i	$-0.246842\pi$
$163$	18.2337	1.42817	0.714086	+	0.700058i	$0.246842\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.48913	−0.424761	−0.212381	−	0.977187i	$-0.568122\pi$
$167$	−5.48913	−0.424761	−0.212381	+	0.977187i	$0.568122\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−14.1168	−1.06713
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.25544	0.243323	0.121661	−	0.992572i	$-0.461178\pi$
$179$	3.25544	0.243323	0.121661	+	0.992572i	$0.461178\pi$
$180$	0	0
$181$	0.883156	0.0656445	0.0328222	−	0.999461i	$-0.489550\pi$
$181$	0.883156	0.0656445	0.0328222	+	0.999461i	$0.489550\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.74456	−0.642913
$186$	0	0
$187$	1.88316	0.137710
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−19.1168	−1.38325	−0.691623	−	0.722259i	$-0.743104\pi$
$191$	−19.1168	−1.38325	−0.691623	+	0.722259i	$0.743104\pi$
$192$	0	0
$193$	−7.00000	−0.503871	−0.251936	−	0.967744i	$-0.581067\pi$
$193$	−7.00000	−0.503871	−0.251936	+	0.967744i	$0.581067\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−8.74456	−0.613748
$204$	0	0
$205$	20.2337	1.41318
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.86141	0.474613
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−35.4891	−2.42034
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.74456	−0.184619
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.2554	−0.813422	−0.406711	−	0.913557i	$-0.633324\pi$
$227$	−12.2554	−0.813422	−0.406711	+	0.913557i	$0.633324\pi$
$228$	0	0
$229$	−2.88316	−0.190524	−0.0952622	−	0.995452i	$-0.530369\pi$
$229$	−2.88316	−0.190524	−0.0952622	+	0.995452i	$0.530369\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.255437	0.0167343	0.00836713	−	0.999965i	$-0.497337\pi$
$233$	0.255437	0.0167343	0.00836713	+	0.999965i	$0.497337\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.86141	0.637881	0.318941	−	0.947775i	$-0.396673\pi$
$239$	9.86141	0.637881	0.318941	+	0.947775i	$0.396673\pi$
$240$	0	0
$241$	18.1168	1.16701	0.583504	−	0.812110i	$-0.301681\pi$
$241$	18.1168	1.16701	0.583504	+	0.812110i	$0.301681\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.37228	−0.279335
$246$	0	0
$247$	−10.0000	−0.636285
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	0	0
$253$	2.23369	0.140431
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.86141	0.428003	0.214001	−	0.976833i	$-0.431350\pi$
$257$	6.86141	0.428003	0.214001	+	0.976833i	$0.431350\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7.62772	0.470345	0.235173	−	0.971954i	$-0.424434\pi$
$263$	7.62772	0.470345	0.235173	+	0.971954i	$0.424434\pi$
$264$	0	0
$265$	−38.2337	−2.34868
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.62772	0.0992438	0.0496219	−	0.998768i	$-0.484198\pi$
$269$	1.62772	0.0992438	0.0496219	+	0.998768i	$0.484198\pi$
$270$	0	0
$271$	−16.2337	−0.986126	−0.493063	−	0.869994i	$-0.664123\pi$
$271$	−16.2337	−0.986126	−0.493063	+	0.869994i	$0.664123\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−19.3723	−1.16819
$276$	0	0
$277$	−12.2337	−0.735051	−0.367526	−	0.930013i	$-0.619795\pi$
$277$	−12.2337	−0.735051	−0.367526	+	0.930013i	$0.619795\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.3723	−0.976688	−0.488344	−	0.872651i	$-0.662399\pi$
$281$	−16.3723	−0.976688	−0.488344	+	0.872651i	$0.662399\pi$
$282$	0	0
$283$	−27.1168	−1.61193	−0.805965	−	0.591964i	$-0.798353\pi$
$283$	−27.1168	−1.61193	−0.805965	+	0.591964i	$0.798353\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.62772	0.273166
$288$	0	0
$289$	−15.1168	−0.889226
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.3723	0.605955	0.302978	−	0.952998i	$-0.402019\pi$
$293$	10.3723	0.605955	0.302978	+	0.952998i	$0.402019\pi$
$294$	0	0
$295$	44.2337	2.57538
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.25544	−0.188267
$300$	0	0
$301$	−8.11684	−0.467847
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−13.6277	−0.780321
$306$	0	0
$307$	13.0000	0.741949	0.370975	−	0.928643i	$-0.379024\pi$
$307$	13.0000	0.741949	0.370975	+	0.928643i	$0.379024\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.23369	0.466890	0.233445	−	0.972370i	$-0.425000\pi$
$311$	8.23369	0.466890	0.233445	+	0.972370i	$0.425000\pi$
$312$	0	0
$313$	−20.1168	−1.13707	−0.568536	−	0.822659i	$-0.692490\pi$
$313$	−20.1168	−1.13707	−0.568536	+	0.822659i	$0.692490\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	0	0
$319$	−12.0000	−0.671871
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.86141	0.381779
$324$	0	0
$325$	28.2337	1.56612
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−22.2337	−1.22207	−0.611037	−	0.791602i	$-0.709247\pi$
$331$	−22.2337	−1.22207	−0.611037	+	0.791602i	$0.709247\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.25544	−0.505679
$336$	0	0
$337$	−8.11684	−0.442153	−0.221076	−	0.975257i	$-0.570957\pi$
$337$	−8.11684	−0.442153	−0.221076	+	0.975257i	$0.570957\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.74456	0.148626
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.1168	−0.543101	−0.271550	−	0.962424i	$-0.587536\pi$
$347$	−10.1168	−0.543101	−0.271550	+	0.962424i	$0.587536\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	13.3723	0.711735	0.355867	−	0.934536i	$-0.384185\pi$
$353$	13.3723	0.711735	0.355867	+	0.934536i	$0.384185\pi$
$354$	0	0
$355$	31.1168	1.65151
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.8614	1.15380	0.576900	−	0.816814i	$-0.304262\pi$
$359$	21.8614	1.15380	0.576900	+	0.816814i	$0.304262\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−52.9783	−2.77301
$366$	0	0
$367$	12.2337	0.638593	0.319297	−	0.947655i	$-0.396553\pi$
$367$	12.2337	0.638593	0.319297	+	0.947655i	$0.396553\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.74456	−0.453995
$372$	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
$375$	0	0
$376$	0	0
$377$	17.4891	0.900736
$378$	0	0
$379$	8.11684	0.416934	0.208467	−	0.978029i	$-0.433153\pi$
$379$	8.11684	0.416934	0.208467	+	0.978029i	$0.433153\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.7446	−1.67317	−0.836584	−	0.547838i	$-0.815451\pi$
$383$	−32.7446	−1.67317	−0.836584	+	0.547838i	$0.815451\pi$
$384$	0	0
$385$	−6.00000	−0.305788
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10.9783	−0.556619	−0.278310	−	0.960491i	$-0.589774\pi$
$389$	−10.9783	−0.556619	−0.278310	+	0.960491i	$0.589774\pi$
$390$	0	0
$391$	2.23369	0.112962
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−22.3723	−1.12567
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.7446	0.586495	0.293248	−	0.956036i	$-0.405264\pi$
$401$	11.7446	0.586495	0.293248	+	0.956036i	$0.405264\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.74456	−0.136043
$408$	0	0
$409$	−22.3505	−1.10516	−0.552581	−	0.833459i	$-0.686357\pi$
$409$	−22.3505	−1.10516	−0.552581	+	0.833459i	$0.686357\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	10.1168	0.497817
$414$	0	0
$415$	−76.4674	−3.75364
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.6060	−0.615842	−0.307921	−	0.951412i	$-0.599633\pi$
$419$	−12.6060	−0.615842	−0.307921	+	0.951412i	$0.599633\pi$
$420$	0	0
$421$	34.2337	1.66845	0.834224	−	0.551426i	$-0.185916\pi$
$421$	34.2337	1.66845	0.834224	+	0.551426i	$0.185916\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−19.3723	−0.939694
$426$	0	0
$427$	−3.11684	−0.150835
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.51087	−0.313618	−0.156809	−	0.987629i	$-0.550121\pi$
$431$	−6.51087	−0.313618	−0.156809	+	0.987629i	$0.550121\pi$
$432$	0	0
$433$	−20.1168	−0.966754	−0.483377	−	0.875412i	$-0.660590\pi$
$433$	−20.1168	−0.966754	−0.483377	+	0.875412i	$0.660590\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.13859	0.389322
$438$	0	0
$439$	−8.00000	−0.381819	−0.190910	−	0.981608i	$-0.561144\pi$
$439$	−8.00000	−0.381819	−0.190910	+	0.981608i	$0.561144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−40.1168	−1.90601	−0.953004	−	0.302956i	$-0.902026\pi$
$443$	−40.1168	−1.90601	−0.953004	+	0.302956i	$0.902026\pi$
$444$	0	0
$445$	64.4674	3.05605
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−33.0000	−1.55737	−0.778683	−	0.627417i	$-0.784112\pi$
$449$	−33.0000	−1.55737	−0.778683	+	0.627417i	$0.784112\pi$
$450$	0	0
$451$	6.35053	0.299035
$452$	0	0
$453$	0	0
$454$	0	0
$455$	8.74456	0.409951
$456$	0	0
$457$	−35.4674	−1.65909	−0.829547	−	0.558437i	$-0.811401\pi$
$457$	−35.4674	−1.65909	−0.829547	+	0.558437i	$0.811401\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.13859	−0.0996042	−0.0498021	−	0.998759i	$-0.515859\pi$
$461$	−2.13859	−0.0996042	−0.0498021	+	0.998759i	$0.515859\pi$
$462$	0	0
$463$	23.1168	1.07433	0.537165	−	0.843477i	$-0.319495\pi$
$463$	23.1168	1.07433	0.537165	+	0.843477i	$0.319495\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.0951	1.53146	0.765729	−	0.643163i	$-0.222378\pi$
$467$	33.0951	1.53146	0.765729	+	0.643163i	$0.222378\pi$
$468$	0	0
$469$	−2.11684	−0.0977468
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11.1386	−0.512153
$474$	0	0
$475$	−70.5842	−3.23863
$476$	0	0
$477$	0	0
$478$	0	0
$479$	32.7446	1.49614	0.748069	−	0.663621i	$-0.230981\pi$
$479$	32.7446	1.49614	0.748069	+	0.663621i	$0.230981\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	35.4891	1.61148
$486$	0	0
$487$	−35.3505	−1.60189	−0.800943	−	0.598741i	$-0.795668\pi$
$487$	−35.3505	−1.60189	−0.800943	+	0.598741i	$0.795668\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−25.3723	−1.14504	−0.572518	−	0.819892i	$-0.694033\pi$
$491$	−25.3723	−1.14504	−0.572518	+	0.819892i	$0.694033\pi$
$492$	0	0
$493$	−12.0000	−0.540453
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.11684	0.319234
$498$	0	0
$499$	−18.1168	−0.811021	−0.405511	−	0.914090i	$-0.632906\pi$
$499$	−18.1168	−0.811021	−0.405511	+	0.914090i	$0.632906\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−32.2337	−1.43723	−0.718615	−	0.695409i	$-0.755223\pi$
$503$	−32.2337	−1.43723	−0.718615	+	0.695409i	$0.755223\pi$
$504$	0	0
$505$	7.11684	0.316695
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.9783	1.28444	0.642219	−	0.766521i	$-0.278014\pi$
$509$	28.9783	1.28444	0.642219	+	0.766521i	$0.278014\pi$
$510$	0	0
$511$	−12.1168	−0.536018
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−43.7228	−1.92666
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	24.8614	1.08920	0.544599	−	0.838697i	$-0.316682\pi$
$521$	24.8614	1.08920	0.544599	+	0.838697i	$0.316682\pi$
$522$	0	0
$523$	35.1168	1.53555	0.767776	−	0.640718i	$-0.221363\pi$
$523$	35.1168	1.53555	0.767776	+	0.640718i	$0.221363\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.74456	0.119555
$528$	0	0
$529$	−20.3505	−0.884806
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9.25544	−0.400897
$534$	0	0
$535$	32.2337	1.39358
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.37228	−0.0591083
$540$	0	0
$541$	−6.23369	−0.268007	−0.134004	−	0.990981i	$-0.542783\pi$
$541$	−6.23369	−0.268007	−0.134004	+	0.990981i	$0.542783\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−61.2119	−2.62203
$546$	0	0
$547$	−18.1168	−0.774620	−0.387310	−	0.921949i	$-0.626596\pi$
$547$	−18.1168	−0.774620	−0.387310	+	0.921949i	$0.626596\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−43.7228	−1.86265
$552$	0	0
$553$	−5.11684	−0.217590
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−29.4891	−1.24949	−0.624747	−	0.780827i	$-0.714798\pi$
$557$	−29.4891	−1.24949	−0.624747	+	0.780827i	$0.714798\pi$
$558$	0	0
$559$	16.2337	0.686612
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	0	0
$565$	−19.1168	−0.804252
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−16.1168	−0.675653	−0.337827	−	0.941208i	$-0.609692\pi$
$569$	−16.1168	−0.675653	−0.337827	+	0.941208i	$0.609692\pi$
$570$	0	0
$571$	22.3505	0.935341	0.467670	−	0.883903i	$-0.345093\pi$
$571$	22.3505	0.935341	0.467670	+	0.883903i	$0.345093\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−22.9783	−0.958259
$576$	0	0
$577$	9.88316	0.411441	0.205721	−	0.978611i	$-0.434046\pi$
$577$	9.88316	0.411441	0.205721	+	0.978611i	$0.434046\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−17.4891	−0.725571
$582$	0	0
$583$	−12.0000	−0.496989
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.4891	0.598030	0.299015	−	0.954248i	$-0.403342\pi$
$587$	14.4891	0.598030	0.299015	+	0.954248i	$0.403342\pi$
$588$	0	0
$589$	10.0000	0.412043
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14.7446	−0.605487	−0.302743	−	0.953072i	$-0.597902\pi$
$593$	−14.7446	−0.605487	−0.302743	+	0.953072i	$0.597902\pi$
$594$	0	0
$595$	−6.00000	−0.245976
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	24.1168	0.983747	0.491873	−	0.870667i	$-0.336312\pi$
$601$	24.1168	0.983747	0.491873	+	0.870667i	$0.336312\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	39.8614	1.62060
$606$	0	0
$607$	−22.2337	−0.902438	−0.451219	−	0.892413i	$-0.649011\pi$
$607$	−22.2337	−0.902438	−0.451219	+	0.892413i	$0.649011\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−36.2337	−1.46346	−0.731732	−	0.681592i	$-0.761288\pi$
$613$	−36.2337	−1.46346	−0.731732	+	0.681592i	$0.761288\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−18.8614	−0.759332	−0.379666	−	0.925124i	$-0.623961\pi$
$617$	−18.8614	−0.759332	−0.379666	+	0.925124i	$0.623961\pi$
$618$	0	0
$619$	−45.4674	−1.82749	−0.913744	−	0.406290i	$-0.866822\pi$
$619$	−45.4674	−1.82749	−0.913744	+	0.406290i	$0.866822\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	14.7446	0.590728
$624$	0	0
$625$	103.701	4.14804
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2.74456	−0.109433
$630$	0	0
$631$	37.3505	1.48690	0.743451	−	0.668791i	$-0.233188\pi$
$631$	37.3505	1.48690	0.743451	+	0.668791i	$0.233188\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	13.6277	0.540800
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−34.2119	−1.35129	−0.675645	−	0.737227i	$-0.736135\pi$
$641$	−34.2119	−1.35129	−0.675645	+	0.737227i	$0.736135\pi$
$642$	0	0
$643$	−26.3505	−1.03916	−0.519582	−	0.854421i	$-0.673912\pi$
$643$	−26.3505	−1.03916	−0.519582	+	0.854421i	$0.673912\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.48913	0.215800	0.107900	−	0.994162i	$-0.465587\pi$
$647$	5.48913	0.215800	0.107900	+	0.994162i	$0.465587\pi$
$648$	0	0
$649$	13.8832	0.544962
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.7446	1.04660	0.523298	−	0.852150i	$-0.324702\pi$
$653$	26.7446	1.04660	0.523298	+	0.852150i	$0.324702\pi$
$654$	0	0
$655$	7.11684	0.278078
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−20.7446	−0.808093	−0.404047	−	0.914738i	$-0.632397\pi$
$659$	−20.7446	−0.808093	−0.404047	+	0.914738i	$0.632397\pi$
$660$	0	0
$661$	27.1168	1.05472	0.527361	−	0.849641i	$-0.323181\pi$
$661$	27.1168	1.05472	0.527361	+	0.849641i	$0.323181\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−21.8614	−0.847749
$666$	0	0
$667$	−14.2337	−0.551131
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.27719	−0.165119
$672$	0	0
$673$	−2.88316	−0.111137	−0.0555687	−	0.998455i	$-0.517697\pi$
$673$	−2.88316	−0.111137	−0.0555687	+	0.998455i	$0.517697\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−34.4674	−1.32469	−0.662344	−	0.749199i	$-0.730438\pi$
$677$	−34.4674	−1.32469	−0.662344	+	0.749199i	$0.730438\pi$
$678$	0	0
$679$	8.11684	0.311496
$680$	0	0
$681$	0	0
$682$	0	0
$683$	29.8397	1.14178	0.570891	−	0.821026i	$-0.306598\pi$
$683$	29.8397	1.14178	0.570891	+	0.821026i	$0.306598\pi$
$684$	0	0
$685$	46.4674	1.77543
$686$	0	0
$687$	0	0
$688$	0	0
$689$	17.4891	0.666283
$690$	0	0
$691$	23.1168	0.879406	0.439703	−	0.898143i	$-0.355084\pi$
$691$	23.1168	0.879406	0.439703	+	0.898143i	$0.355084\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	57.8614	2.19481
$696$	0	0
$697$	6.35053	0.240544
$698$	0	0
$699$	0	0
$700$	0	0
$701$	38.2337	1.44407	0.722033	−	0.691858i	$-0.243208\pi$
$701$	38.2337	1.44407	0.722033	+	0.691858i	$0.243208\pi$
$702$	0	0
$703$	−10.0000	−0.377157
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.62772	0.0612167
$708$	0	0
$709$	44.0000	1.65245	0.826227	−	0.563337i	$-0.190483\pi$
$709$	44.0000	1.65245	0.826227	+	0.563337i	$0.190483\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.25544	0.121917
$714$	0	0
$715$	12.0000	0.448775
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.74456	0.102355	0.0511775	−	0.998690i	$-0.483703\pi$
$719$	2.74456	0.102355	0.0511775	+	0.998690i	$0.483703\pi$
$720$	0	0
$721$	−10.0000	−0.372419
$722$	0	0
$723$	0	0
$724$	0	0
$725$	123.446	4.58466
$726$	0	0
$727$	36.2337	1.34383	0.671917	−	0.740627i	$-0.265471\pi$
$727$	36.2337	1.34383	0.671917	+	0.740627i	$0.265471\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−11.1386	−0.411976
$732$	0	0
$733$	−41.1168	−1.51869	−0.759343	−	0.650691i	$-0.774479\pi$
$733$	−41.1168	−1.51869	−0.759343	+	0.650691i	$0.774479\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.90491	−0.107004
$738$	0	0
$739$	8.11684	0.298583	0.149291	−	0.988793i	$-0.452301\pi$
$739$	8.11684	0.298583	0.149291	+	0.988793i	$0.452301\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.7228	−0.503441	−0.251721	−	0.967800i	$-0.580996\pi$
$743$	−13.7228	−0.503441	−0.251721	+	0.967800i	$0.580996\pi$
$744$	0	0
$745$	14.2337	0.521482
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.37228	0.269377
$750$	0	0
$751$	17.1168	0.624603	0.312301	−	0.949983i	$-0.398900\pi$
$751$	17.1168	0.624603	0.312301	+	0.949983i	$0.398900\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	39.8614	1.45071
$756$	0	0
$757$	46.2337	1.68039	0.840196	−	0.542283i	$-0.182440\pi$
$757$	46.2337	1.68039	0.840196	+	0.542283i	$0.182440\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35.4891	−1.28648	−0.643240	−	0.765665i	$-0.722410\pi$
$761$	−35.4891	−1.28648	−0.643240	+	0.765665i	$0.722410\pi$
$762$	0	0
$763$	−14.0000	−0.506834
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−20.2337	−0.730596
$768$	0	0
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	39.8614	1.43372	0.716858	−	0.697220i	$-0.245580\pi$
$773$	39.8614	1.43372	0.716858	+	0.697220i	$0.245580\pi$
$774$	0	0
$775$	−28.2337	−1.01418
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23.1386	0.829026
$780$	0	0
$781$	9.76631	0.349466
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−39.8614	−1.42271
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.37228	−0.155460
$792$	0	0
$793$	6.23369	0.221365
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.13859	0.288284	0.144142	−	0.989557i	$-0.453958\pi$
$797$	8.13859	0.288284	0.144142	+	0.989557i	$0.453958\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−16.6277	−0.586779
$804$	0	0
$805$	−7.11684	−0.250836
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.86141	0.241234	0.120617	−	0.992699i	$-0.461513\pi$
$809$	6.86141	0.241234	0.120617	+	0.992699i	$0.461513\pi$
$810$	0	0
$811$	−42.1168	−1.47892	−0.739461	−	0.673199i	$-0.764920\pi$
$811$	−42.1168	−1.47892	−0.739461	+	0.673199i	$0.764920\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−79.7228	−2.79257
$816$	0	0
$817$	−40.5842	−1.41986
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.76631	0.131445	0.0657226	−	0.997838i	$-0.479065\pi$
$821$	3.76631	0.131445	0.0657226	+	0.997838i	$0.479065\pi$
$822$	0	0
$823$	12.2337	0.426440	0.213220	−	0.977004i	$-0.431605\pi$
$823$	12.2337	0.426440	0.213220	+	0.977004i	$0.431605\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	−13.7663	−0.478124	−0.239062	−	0.971004i	$-0.576840\pi$
$829$	−13.7663	−0.478124	−0.239062	+	0.971004i	$0.576840\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.37228	−0.0475467
$834$	0	0
$835$	24.0000	0.830554
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.48913	0.189506	0.0947528	−	0.995501i	$-0.469794\pi$
$839$	5.48913	0.189506	0.0947528	+	0.995501i	$0.469794\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	0	0
$843$	0	0
$844$	0	0
$845$	39.3505	1.35370
$846$	0	0
$847$	9.11684	0.313258
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3.25544	−0.111595
$852$	0	0
$853$	−35.1168	−1.20238	−0.601189	−	0.799107i	$-0.705306\pi$
$853$	−35.1168	−1.20238	−0.601189	+	0.799107i	$0.705306\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.9565	1.36489	0.682444	−	0.730938i	$-0.260917\pi$
$857$	39.9565	1.36489	0.682444	+	0.730938i	$0.260917\pi$
$858$	0	0
$859$	−33.8832	−1.15608	−0.578039	−	0.816009i	$-0.696182\pi$
$859$	−33.8832	−1.15608	−0.578039	+	0.816009i	$0.696182\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.86141	0.335686	0.167843	−	0.985814i	$-0.446320\pi$
$863$	9.86141	0.335686	0.167843	+	0.985814i	$0.446320\pi$
$864$	0	0
$865$	−26.2337	−0.891972
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7.02175	−0.238197
$870$	0	0
$871$	4.23369	0.143453
$872$	0	0
$873$	0	0
$874$	0	0
$875$	39.8614	1.34756
$876$	0	0
$877$	−58.7011	−1.98219	−0.991097	−	0.133141i	$-0.957494\pi$
$877$	−58.7011	−1.98219	−0.991097	+	0.133141i	$0.957494\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.2337	0.681690	0.340845	−	0.940119i	$-0.389287\pi$
$881$	20.2337	0.681690	0.340845	+	0.940119i	$0.389287\pi$
$882$	0	0
$883$	40.3505	1.35790	0.678952	−	0.734183i	$-0.262435\pi$
$883$	40.3505	1.35790	0.678952	+	0.734183i	$0.262435\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.7228	0.863688	0.431844	−	0.901948i	$-0.357863\pi$
$887$	25.7228	0.863688	0.431844	+	0.901948i	$0.357863\pi$
$888$	0	0
$889$	3.11684	0.104536
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−14.2337	−0.475780
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−17.4891	−0.583295
$900$	0	0
$901$	−12.0000	−0.399778
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−3.86141	−0.128357
$906$	0	0
$907$	26.1168	0.867196	0.433598	−	0.901107i	$-0.357244\pi$
$907$	26.1168	0.867196	0.433598	+	0.901107i	$0.357244\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−37.6277	−1.24666	−0.623331	−	0.781958i	$-0.714221\pi$
$911$	−37.6277	−1.24666	−0.623331	+	0.781958i	$0.714221\pi$
$912$	0	0
$913$	−24.0000	−0.794284
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.62772	0.0537520
$918$	0	0
$919$	47.1168	1.55424	0.777121	−	0.629352i	$-0.216679\pi$
$919$	47.1168	1.55424	0.777121	+	0.629352i	$0.216679\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−14.2337	−0.468508
$924$	0	0
$925$	28.2337	0.928318
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−44.2337	−1.45126	−0.725630	−	0.688085i	$-0.758452\pi$
$929$	−44.2337	−1.45126	−0.725630	+	0.688085i	$0.758452\pi$
$930$	0	0
$931$	−5.00000	−0.163868
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.23369	−0.269270
$936$	0	0
$937$	30.4674	0.995326	0.497663	−	0.867371i	$-0.334192\pi$
$937$	30.4674	0.995326	0.497663	+	0.867371i	$0.334192\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−19.1168	−0.623191	−0.311596	−	0.950215i	$-0.600863\pi$
$941$	−19.1168	−0.623191	−0.311596	+	0.950215i	$0.600863\pi$
$942$	0	0
$943$	7.53262	0.245296
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.1168	1.10865	0.554324	−	0.832301i	$-0.312977\pi$
$947$	34.1168	1.10865	0.554324	+	0.832301i	$0.312977\pi$
$948$	0	0
$949$	24.2337	0.786659
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−28.1168	−0.910794	−0.455397	−	0.890288i	$-0.650503\pi$
$953$	−28.1168	−0.910794	−0.455397	+	0.890288i	$0.650503\pi$
$954$	0	0
$955$	83.5842	2.70472
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.6277	0.343187
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	30.6060	0.985241
$966$	0	0
$967$	−30.8832	−0.993135	−0.496568	−	0.867998i	$-0.665407\pi$
$967$	−30.8832	−0.993135	−0.496568	+	0.867998i	$0.665407\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.62772	0.0522360	0.0261180	−	0.999659i	$-0.491685\pi$
$971$	1.62772	0.0522360	0.0261180	+	0.999659i	$0.491685\pi$
$972$	0	0
$973$	13.2337	0.424253
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−40.1168	−1.28345	−0.641726	−	0.766934i	$-0.721781\pi$
$977$	−40.1168	−1.28345	−0.641726	+	0.766934i	$0.721781\pi$
$978$	0	0
$979$	20.2337	0.646671
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−39.2554	−1.25205	−0.626027	−	0.779801i	$-0.715320\pi$
$983$	−39.2554	−1.25205	−0.626027	+	0.779801i	$0.715320\pi$
$984$	0	0
$985$	−26.2337	−0.835875
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−13.2119	−0.420115
$990$	0	0
$991$	−48.4674	−1.53962	−0.769808	−	0.638275i	$-0.779648\pi$
$991$	−48.4674	−1.53962	−0.769808	+	0.638275i	$0.779648\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−43.7228	−1.38611
$996$	0	0
$997$	−5.11684	−0.162052	−0.0810260	−	0.996712i	$-0.525820\pi$
$997$	−5.11684	−0.162052	−0.0810260	+	0.996712i	$0.525820\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.bb.1.1		2
3.2	odd	2		9072.2.a.bm.1.2		2
4.3	odd	2		1134.2.a.n.1.1		2
9.2	odd	6		1008.2.r.f.337.2		4
9.4	even	3		3024.2.r.f.2017.2		4
9.5	odd	6		1008.2.r.f.673.2		4
9.7	even	3		3024.2.r.f.1009.2		4
12.11	even	2		1134.2.a.k.1.2		2
28.27	even	2		7938.2.a.bs.1.2		2
36.7	odd	6		378.2.f.c.253.2		4
36.11	even	6		126.2.f.d.85.1	yes	4
36.23	even	6		126.2.f.d.43.1	✓	4
36.31	odd	6		378.2.f.c.127.2		4
84.83	odd	2		7938.2.a.bh.1.1		2
252.11	even	6		882.2.e.l.373.2		4
252.23	even	6		882.2.e.l.655.1		4
252.31	even	6		2646.2.h.l.667.2		4
252.47	odd	6		882.2.h.n.67.1		4
252.59	odd	6		882.2.h.n.79.1		4
252.67	odd	6		2646.2.h.k.667.1		4
252.79	odd	6		2646.2.h.k.361.1		4
252.83	odd	6		882.2.f.k.589.2		4
252.95	even	6		882.2.h.m.79.2		4
252.103	even	6		2646.2.e.m.2125.1		4
252.115	even	6		2646.2.e.m.1549.1		4
252.131	odd	6		882.2.e.k.655.2		4
252.139	even	6		2646.2.f.j.883.1		4
252.151	odd	6		2646.2.e.n.1549.2		4
252.167	odd	6		882.2.f.k.295.2		4
252.187	even	6		2646.2.h.l.361.2		4
252.191	even	6		882.2.h.m.67.2		4
252.223	even	6		2646.2.f.j.1765.1		4
252.227	odd	6		882.2.e.k.373.1		4
252.247	odd	6		2646.2.e.n.2125.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.f.d.43.1	✓	4	36.23	even	6
126.2.f.d.85.1	yes	4	36.11	even	6
378.2.f.c.127.2		4	36.31	odd	6
378.2.f.c.253.2		4	36.7	odd	6
882.2.e.k.373.1		4	252.227	odd	6
882.2.e.k.655.2		4	252.131	odd	6
882.2.e.l.373.2		4	252.11	even	6
882.2.e.l.655.1		4	252.23	even	6
882.2.f.k.295.2		4	252.167	odd	6
882.2.f.k.589.2		4	252.83	odd	6
882.2.h.m.67.2		4	252.191	even	6
882.2.h.m.79.2		4	252.95	even	6
882.2.h.n.67.1		4	252.47	odd	6
882.2.h.n.79.1		4	252.59	odd	6
1008.2.r.f.337.2		4	9.2	odd	6
1008.2.r.f.673.2		4	9.5	odd	6
1134.2.a.k.1.2		2	12.11	even	2
1134.2.a.n.1.1		2	4.3	odd	2
2646.2.e.m.1549.1		4	252.115	even	6
2646.2.e.m.2125.1		4	252.103	even	6
2646.2.e.n.1549.2		4	252.151	odd	6
2646.2.e.n.2125.2		4	252.247	odd	6
2646.2.f.j.883.1		4	252.139	even	6
2646.2.f.j.1765.1		4	252.223	even	6
2646.2.h.k.361.1		4	252.79	odd	6
2646.2.h.k.667.1		4	252.67	odd	6
2646.2.h.l.361.2		4	252.187	even	6
2646.2.h.l.667.2		4	252.31	even	6
3024.2.r.f.1009.2		4	9.7	even	3
3024.2.r.f.2017.2		4	9.4	even	3
7938.2.a.bh.1.1		2	84.83	odd	2
7938.2.a.bs.1.2		2	28.27	even	2
9072.2.a.bb.1.1		2	1.1	even	1	trivial
9072.2.a.bm.1.2		2	3.2	odd	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:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q-4.37228 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-4.37228 q^{5} -1.00000 q^{7} -1.37228 q^{11} +2.00000 q^{13} -1.37228 q^{17} -5.00000 q^{19} -1.62772 q^{23} +14.1168 q^{25} +8.74456 q^{29} -2.00000 q^{31} +4.37228 q^{35} +2.00000 q^{37} -4.62772 q^{41} +8.11684 q^{43} +1.00000 q^{49} +8.74456 q^{53} +6.00000 q^{55} -10.1168 q^{59} +3.11684 q^{61} -8.74456 q^{65} +2.11684 q^{67} -7.11684 q^{71} +12.1168 q^{73} +1.37228 q^{77} +5.11684 q^{79} +17.4891 q^{83} +6.00000 q^{85} -14.7446 q^{89} -2.00000 q^{91} +21.8614 q^{95} -8.11684 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 3 * q^5 - 2 * q^7	\( 2 q - 3 q^{5} - 2 q^{7} + 3 q^{11} + 4 q^{13} + 3 q^{17} - 10 q^{19} - 9 q^{23} + 11 q^{25} + 6 q^{29} - 4 q^{31} + 3 q^{35} + 4 q^{37} - 15 q^{41} - q^{43} + 2 q^{49} + 6 q^{53} + 12 q^{55} - 3 q^{59} - 11 q^{61} - 6 q^{65} - 13 q^{67} + 3 q^{71} + 7 q^{73} - 3 q^{77} - 7 q^{79} + 12 q^{83} + 12 q^{85} - 18 q^{89} - 4 q^{91} + 15 q^{95} + q^{97}+O(q^{100}) \) 2 * q - 3 * q^5 - 2 * q^7 + 3 * q^11 + 4 * q^13 + 3 * q^17 - 10 * q^19 - 9 * q^23 + 11 * q^25 + 6 * q^29 - 4 * q^31 + 3 * q^35 + 4 * q^37 - 15 * q^41 - q^43 + 2 * q^49 + 6 * q^53 + 12 * q^55 - 3 * q^59 - 11 * q^61 - 6 * q^65 - 13 * q^67 + 3 * q^71 + 7 * q^73 - 3 * q^77 - 7 * q^79 + 12 * q^83 + 12 * q^85 - 18 * q^89 - 4 * q^91 + 15 * q^95 + q^97

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