LMFDB - Embedded newform 1176.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,2,Mod(1,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, 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("1176.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1176.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:	$9.39040727770$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.27492$ of defining polynomial
Character	$\chi$	$=$	1176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.27492 q^{5} +1.00000 q^{9} +3.27492 q^{11} +6.27492 q^{13} -3.27492 q^{15} -4.00000 q^{17} -6.27492 q^{19} +4.00000 q^{23} +5.72508 q^{25} -1.00000 q^{27} +5.27492 q^{29} -1.00000 q^{31} -3.27492 q^{33} -2.27492 q^{37} -6.27492 q^{39} -4.54983 q^{41} +0.274917 q^{43} +3.27492 q^{45} -6.00000 q^{47} +4.00000 q^{51} +9.27492 q^{53} +10.7251 q^{55} +6.27492 q^{57} -1.27492 q^{59} +10.0000 q^{61} +20.5498 q^{65} -0.274917 q^{67} -4.00000 q^{69} +2.00000 q^{71} -4.27492 q^{73} -5.72508 q^{75} +11.5498 q^{79} +1.00000 q^{81} +7.27492 q^{83} -13.0997 q^{85} -5.27492 q^{87} -10.5498 q^{89} +1.00000 q^{93} -20.5498 q^{95} +8.72508 q^{97} +3.27492 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} + 2 q^{9} - q^{11} + 5 q^{13} + q^{15} - 8 q^{17} - 5 q^{19} + 8 q^{23} + 19 q^{25} - 2 q^{27} + 3 q^{29} - 2 q^{31} + q^{33} + 3 q^{37} - 5 q^{39} + 6 q^{41} - 7 q^{43} - q^{45}+ \cdots - q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 + 2 * q^9 - q^11 + 5 * q^13 + q^15 - 8 * q^17 - 5 * q^19 + 8 * q^23 + 19 * q^25 - 2 * q^27 + 3 * q^29 - 2 * q^31 + q^33 + 3 * q^37 - 5 * q^39 + 6 * q^41 - 7 * q^43 - q^45 - 12 * q^47 + 8 * q^51 + 11 * q^53 + 29 * q^55 + 5 * q^57 + 5 * q^59 + 20 * q^61 + 26 * q^65 + 7 * q^67 - 8 * q^69 + 4 * q^71 - q^73 - 19 * q^75 + 8 * q^79 + 2 * q^81 + 7 * q^83 + 4 * q^85 - 3 * q^87 - 6 * q^89 + 2 * q^93 - 26 * q^95 + 25 * q^97 - q^99

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
$5$	3.27492	1.46459	0.732294	−	0.680989i	$-0.238450\pi$
$5$	3.27492	1.46459	0.732294	+	0.680989i	$0.238450\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.27492	0.987425	0.493712	−	0.869625i	$-0.335640\pi$
$11$	3.27492	0.987425	0.493712	+	0.869625i	$0.335640\pi$
$12$	0	0
$13$	6.27492	1.74035	0.870174	−	0.492744i	$-0.164006\pi$
$13$	6.27492	1.74035	0.870174	+	0.492744i	$0.164006\pi$
$14$	0	0
$15$	−3.27492	−0.845580
$16$	0	0
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−6.27492	−1.43956	−0.719782	−	0.694200i	$-0.755758\pi$
$19$	−6.27492	−1.43956	−0.719782	+	0.694200i	$0.755758\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	5.72508	1.14502
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.27492	0.979528	0.489764	−	0.871855i	$-0.337083\pi$
$29$	5.27492	0.979528	0.489764	+	0.871855i	$0.337083\pi$
$30$	0	0
$31$	−1.00000	−0.179605	−0.0898027	−	0.995960i	$-0.528624\pi$
$31$	−1.00000	−0.179605	−0.0898027	+	0.995960i	$0.528624\pi$
$32$	0	0
$33$	−3.27492	−0.570090
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.27492	−0.373994	−0.186997	−	0.982360i	$-0.559875\pi$
$37$	−2.27492	−0.373994	−0.186997	+	0.982360i	$0.559875\pi$
$38$	0	0
$39$	−6.27492	−1.00479
$40$	0	0
$41$	−4.54983	−0.710565	−0.355282	−	0.934759i	$-0.615615\pi$
$41$	−4.54983	−0.710565	−0.355282	+	0.934759i	$0.615615\pi$
$42$	0	0
$43$	0.274917	0.0419245	0.0209622	−	0.999780i	$-0.493327\pi$
$43$	0.274917	0.0419245	0.0209622	+	0.999780i	$0.493327\pi$
$44$	0	0
$45$	3.27492	0.488196
$46$	0	0
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	9.27492	1.27401	0.637004	−	0.770861i	$-0.280173\pi$
$53$	9.27492	1.27401	0.637004	+	0.770861i	$0.280173\pi$
$54$	0	0
$55$	10.7251	1.44617
$56$	0	0
$57$	6.27492	0.831133
$58$	0	0
$59$	−1.27492	−0.165980	−0.0829900	−	0.996550i	$-0.526447\pi$
$59$	−1.27492	−0.165980	−0.0829900	+	0.996550i	$0.526447\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	20.5498	2.54889
$66$	0	0
$67$	−0.274917	−0.0335865	−0.0167932	−	0.999859i	$-0.505346\pi$
$67$	−0.274917	−0.0335865	−0.0167932	+	0.999859i	$0.505346\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	0	0
$73$	−4.27492	−0.500341	−0.250171	−	0.968202i	$-0.580487\pi$
$73$	−4.27492	−0.500341	−0.250171	+	0.968202i	$0.580487\pi$
$74$	0	0
$75$	−5.72508	−0.661076
$76$	0	0
$77$	0	0
$78$	0	0
$79$	11.5498	1.29946	0.649729	−	0.760166i	$-0.274882\pi$
$79$	11.5498	1.29946	0.649729	+	0.760166i	$0.274882\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.27492	0.798526	0.399263	−	0.916836i	$-0.369266\pi$
$83$	7.27492	0.798526	0.399263	+	0.916836i	$0.369266\pi$
$84$	0	0
$85$	−13.0997	−1.42086
$86$	0	0
$87$	−5.27492	−0.565530
$88$	0	0
$89$	−10.5498	−1.11828	−0.559140	−	0.829073i	$-0.688869\pi$
$89$	−10.5498	−1.11828	−0.559140	+	0.829073i	$0.688869\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−20.5498	−2.10837
$96$	0	0
$97$	8.72508	0.885898	0.442949	−	0.896547i	$-0.353932\pi$
$97$	8.72508	0.885898	0.442949	+	0.896547i	$0.353932\pi$
$98$	0	0
$99$	3.27492	0.329142
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	10.8248	1.06659	0.533297	−	0.845928i	$-0.320953\pi$
$103$	10.8248	1.06659	0.533297	+	0.845928i	$0.320953\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.8248	−1.52984	−0.764918	−	0.644127i	$-0.777221\pi$
$107$	−15.8248	−1.52984	−0.764918	+	0.644127i	$0.777221\pi$
$108$	0	0
$109$	16.8248	1.61152	0.805759	−	0.592243i	$-0.201757\pi$
$109$	16.8248	1.61152	0.805759	+	0.592243i	$0.201757\pi$
$110$	0	0
$111$	2.27492	0.215926
$112$	0	0
$113$	−4.54983	−0.428012	−0.214006	−	0.976832i	$-0.568651\pi$
$113$	−4.54983	−0.428012	−0.214006	+	0.976832i	$0.568651\pi$
$114$	0	0
$115$	13.0997	1.22155
$116$	0	0
$117$	6.27492	0.580116
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−0.274917	−0.0249925
$122$	0	0
$123$	4.54983	0.410245
$124$	0	0
$125$	2.37459	0.212389
$126$	0	0
$127$	−6.45017	−0.572360	−0.286180	−	0.958176i	$-0.592385\pi$
$127$	−6.45017	−0.572360	−0.286180	+	0.958176i	$0.592385\pi$
$128$	0	0
$129$	−0.274917	−0.0242051
$130$	0	0
$131$	7.27492	0.635612	0.317806	−	0.948156i	$-0.397054\pi$
$131$	7.27492	0.635612	0.317806	+	0.948156i	$0.397054\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.27492	−0.281860
$136$	0	0
$137$	1.45017	0.123896	0.0619480	−	0.998079i	$-0.480269\pi$
$137$	1.45017	0.123896	0.0619480	+	0.998079i	$0.480269\pi$
$138$	0	0
$139$	8.27492	0.701869	0.350935	−	0.936400i	$-0.385864\pi$
$139$	8.27492	0.701869	0.350935	+	0.936400i	$0.385864\pi$
$140$	0	0
$141$	6.00000	0.505291
$142$	0	0
$143$	20.5498	1.71846
$144$	0	0
$145$	17.2749	1.43460
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.5498	−1.19197	−0.595984	−	0.802996i	$-0.703238\pi$
$149$	−14.5498	−1.19197	−0.595984	+	0.802996i	$0.703238\pi$
$150$	0	0
$151$	22.3746	1.82082	0.910409	−	0.413709i	$-0.135767\pi$
$151$	22.3746	1.82082	0.910409	+	0.413709i	$0.135767\pi$
$152$	0	0
$153$	−4.00000	−0.323381
$154$	0	0
$155$	−3.27492	−0.263048
$156$	0	0
$157$	−0.549834	−0.0438816	−0.0219408	−	0.999759i	$-0.506985\pi$
$157$	−0.549834	−0.0438816	−0.0219408	+	0.999759i	$0.506985\pi$
$158$	0	0
$159$	−9.27492	−0.735549
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	0	0
$165$	−10.7251	−0.834947
$166$	0	0
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	0	0
$169$	26.3746	2.02881
$170$	0	0
$171$	−6.27492	−0.479855
$172$	0	0
$173$	−22.5498	−1.71443	−0.857216	−	0.514957i	$-0.827808\pi$
$173$	−22.5498	−1.71443	−0.857216	+	0.514957i	$0.827808\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.27492	0.0958286
$178$	0	0
$179$	−16.5498	−1.23699	−0.618496	−	0.785788i	$-0.712258\pi$
$179$	−16.5498	−1.23699	−0.618496	+	0.785788i	$0.712258\pi$
$180$	0	0
$181$	−18.8248	−1.39923	−0.699616	−	0.714519i	$-0.746646\pi$
$181$	−18.8248	−1.39923	−0.699616	+	0.714519i	$0.746646\pi$
$182$	0	0
$183$	−10.0000	−0.739221
$184$	0	0
$185$	−7.45017	−0.547747
$186$	0	0
$187$	−13.0997	−0.957943
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.54983	0.329214	0.164607	−	0.986359i	$-0.447364\pi$
$191$	4.54983	0.329214	0.164607	+	0.986359i	$0.447364\pi$
$192$	0	0
$193$	0.450166	0.0324036	0.0162018	−	0.999869i	$-0.494843\pi$
$193$	0.450166	0.0324036	0.0162018	+	0.999869i	$0.494843\pi$
$194$	0	0
$195$	−20.5498	−1.47160
$196$	0	0
$197$	−1.45017	−0.103320	−0.0516600	−	0.998665i	$-0.516451\pi$
$197$	−1.45017	−0.103320	−0.0516600	+	0.998665i	$0.516451\pi$
$198$	0	0
$199$	−5.09967	−0.361506	−0.180753	−	0.983529i	$-0.557853\pi$
$199$	−5.09967	−0.361506	−0.180753	+	0.983529i	$0.557853\pi$
$200$	0	0
$201$	0.274917	0.0193912
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−14.9003	−1.04068
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	−20.5498	−1.42146
$210$	0	0
$211$	−27.6495	−1.90347	−0.951735	−	0.306921i	$-0.900701\pi$
$211$	−27.6495	−1.90347	−0.951735	+	0.306921i	$0.900701\pi$
$212$	0	0
$213$	−2.00000	−0.137038
$214$	0	0
$215$	0.900331	0.0614021
$216$	0	0
$217$	0	0
$218$	0	0
$219$	4.27492	0.288872
$220$	0	0
$221$	−25.0997	−1.68839
$222$	0	0
$223$	−1.27492	−0.0853748	−0.0426874	−	0.999088i	$-0.513592\pi$
$223$	−1.27492	−0.0853748	−0.0426874	+	0.999088i	$0.513592\pi$
$224$	0	0
$225$	5.72508	0.381672
$226$	0	0
$227$	−11.2749	−0.748343	−0.374171	−	0.927360i	$-0.622073\pi$
$227$	−11.2749	−0.748343	−0.374171	+	0.927360i	$0.622073\pi$
$228$	0	0
$229$	18.2749	1.20764	0.603820	−	0.797121i	$-0.293644\pi$
$229$	18.2749	1.20764	0.603820	+	0.797121i	$0.293644\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.549834	−0.0360209	−0.0180104	−	0.999838i	$-0.505733\pi$
$233$	−0.549834	−0.0360209	−0.0180104	+	0.999838i	$0.505733\pi$
$234$	0	0
$235$	−19.6495	−1.28179
$236$	0	0
$237$	−11.5498	−0.750242
$238$	0	0
$239$	15.4502	0.999388	0.499694	−	0.866202i	$-0.333446\pi$
$239$	15.4502	0.999388	0.499694	+	0.866202i	$0.333446\pi$
$240$	0	0
$241$	9.82475	0.632868	0.316434	−	0.948615i	$-0.397514\pi$
$241$	9.82475	0.632868	0.316434	+	0.948615i	$0.397514\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−39.3746	−2.50534
$248$	0	0
$249$	−7.27492	−0.461029
$250$	0	0
$251$	−18.3746	−1.15979	−0.579897	−	0.814690i	$-0.696907\pi$
$251$	−18.3746	−1.15979	−0.579897	+	0.814690i	$0.696907\pi$
$252$	0	0
$253$	13.0997	0.823569
$254$	0	0
$255$	13.0997	0.820333
$256$	0	0
$257$	11.0997	0.692378	0.346189	−	0.938165i	$-0.387476\pi$
$257$	11.0997	0.692378	0.346189	+	0.938165i	$0.387476\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5.27492	0.326509
$262$	0	0
$263$	−9.45017	−0.582722	−0.291361	−	0.956613i	$-0.594108\pi$
$263$	−9.45017	−0.582722	−0.291361	+	0.956613i	$0.594108\pi$
$264$	0	0
$265$	30.3746	1.86590
$266$	0	0
$267$	10.5498	0.645639
$268$	0	0
$269$	20.7251	1.26363	0.631815	−	0.775119i	$-0.282310\pi$
$269$	20.7251	1.26363	0.631815	+	0.775119i	$0.282310\pi$
$270$	0	0
$271$	1.27492	0.0774457	0.0387229	−	0.999250i	$-0.487671\pi$
$271$	1.27492	0.0774457	0.0387229	+	0.999250i	$0.487671\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.7492	1.13062
$276$	0	0
$277$	−26.8248	−1.61174	−0.805872	−	0.592090i	$-0.798303\pi$
$277$	−26.8248	−1.61174	−0.805872	+	0.592090i	$0.798303\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−26.5498	−1.58383	−0.791915	−	0.610631i	$-0.790916\pi$
$281$	−26.5498	−1.58383	−0.791915	+	0.610631i	$0.790916\pi$
$282$	0	0
$283$	−25.9244	−1.54105	−0.770523	−	0.637412i	$-0.780005\pi$
$283$	−25.9244	−1.54105	−0.770523	+	0.637412i	$0.780005\pi$
$284$	0	0
$285$	20.5498	1.21727
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−8.72508	−0.511473
$292$	0	0
$293$	−27.8248	−1.62554	−0.812770	−	0.582585i	$-0.802041\pi$
$293$	−27.8248	−1.62554	−0.812770	+	0.582585i	$0.802041\pi$
$294$	0	0
$295$	−4.17525	−0.243092
$296$	0	0
$297$	−3.27492	−0.190030
$298$	0	0
$299$	25.0997	1.45155
$300$	0	0
$301$	0	0
$302$	0	0
$303$	6.00000	0.344691
$304$	0	0
$305$	32.7492	1.87521
$306$	0	0
$307$	11.3746	0.649182	0.324591	−	0.945854i	$-0.394773\pi$
$307$	11.3746	0.649182	0.324591	+	0.945854i	$0.394773\pi$
$308$	0	0
$309$	−10.8248	−0.615799
$310$	0	0
$311$	−4.54983	−0.257997	−0.128999	−	0.991645i	$-0.541176\pi$
$311$	−4.54983	−0.257997	−0.128999	+	0.991645i	$0.541176\pi$
$312$	0	0
$313$	−19.5498	−1.10502	−0.552511	−	0.833506i	$-0.686330\pi$
$313$	−19.5498	−1.10502	−0.552511	+	0.833506i	$0.686330\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.8248	−1.56279	−0.781397	−	0.624034i	$-0.785493\pi$
$317$	−27.8248	−1.56279	−0.781397	+	0.624034i	$0.785493\pi$
$318$	0	0
$319$	17.2749	0.967210
$320$	0	0
$321$	15.8248	0.883252
$322$	0	0
$323$	25.0997	1.39658
$324$	0	0
$325$	35.9244	1.99273
$326$	0	0
$327$	−16.8248	−0.930411
$328$	0	0
$329$	0	0
$330$	0	0
$331$	1.17525	0.0645975	0.0322987	−	0.999478i	$-0.489717\pi$
$331$	1.17525	0.0645975	0.0322987	+	0.999478i	$0.489717\pi$
$332$	0	0
$333$	−2.27492	−0.124665
$334$	0	0
$335$	−0.900331	−0.0491903
$336$	0	0
$337$	−24.0997	−1.31279	−0.656396	−	0.754416i	$-0.727920\pi$
$337$	−24.0997	−1.31279	−0.656396	+	0.754416i	$0.727920\pi$
$338$	0	0
$339$	4.54983	0.247113
$340$	0	0
$341$	−3.27492	−0.177347
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−13.0997	−0.705262
$346$	0	0
$347$	30.1993	1.62119	0.810593	−	0.585610i	$-0.199145\pi$
$347$	30.1993	1.62119	0.810593	+	0.585610i	$0.199145\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	−6.27492	−0.334930
$352$	0	0
$353$	−20.5498	−1.09376	−0.546879	−	0.837212i	$-0.684184\pi$
$353$	−20.5498	−1.09376	−0.546879	+	0.837212i	$0.684184\pi$
$354$	0	0
$355$	6.54983	0.347629
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.6495	−1.03706	−0.518531	−	0.855059i	$-0.673521\pi$
$359$	−19.6495	−1.03706	−0.518531	+	0.855059i	$0.673521\pi$
$360$	0	0
$361$	20.3746	1.07235
$362$	0	0
$363$	0.274917	0.0144294
$364$	0	0
$365$	−14.0000	−0.732793
$366$	0	0
$367$	22.0997	1.15359	0.576797	−	0.816888i	$-0.304302\pi$
$367$	22.0997	1.15359	0.576797	+	0.816888i	$0.304302\pi$
$368$	0	0
$369$	−4.54983	−0.236855
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6.27492	−0.324903	−0.162451	−	0.986717i	$-0.551940\pi$
$373$	−6.27492	−0.324903	−0.162451	+	0.986717i	$0.551940\pi$
$374$	0	0
$375$	−2.37459	−0.122623
$376$	0	0
$377$	33.0997	1.70472
$378$	0	0
$379$	13.1752	0.676767	0.338384	−	0.941008i	$-0.390120\pi$
$379$	13.1752	0.676767	0.338384	+	0.941008i	$0.390120\pi$
$380$	0	0
$381$	6.45017	0.330452
$382$	0	0
$383$	10.5498	0.539071	0.269536	−	0.962990i	$-0.413130\pi$
$383$	10.5498	0.539071	0.269536	+	0.962990i	$0.413130\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.274917	0.0139748
$388$	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
$391$	−16.0000	−0.809155
$392$	0	0
$393$	−7.27492	−0.366971
$394$	0	0
$395$	37.8248	1.90317
$396$	0	0
$397$	−3.37459	−0.169366	−0.0846828	−	0.996408i	$-0.526988\pi$
$397$	−3.37459	−0.169366	−0.0846828	+	0.996408i	$0.526988\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.0000	1.19850	0.599251	−	0.800561i	$-0.295465\pi$
$401$	24.0000	1.19850	0.599251	+	0.800561i	$0.295465\pi$
$402$	0	0
$403$	−6.27492	−0.312576
$404$	0	0
$405$	3.27492	0.162732
$406$	0	0
$407$	−7.45017	−0.369291
$408$	0	0
$409$	10.4502	0.516727	0.258364	−	0.966048i	$-0.416817\pi$
$409$	10.4502	0.516727	0.258364	+	0.966048i	$0.416817\pi$
$410$	0	0
$411$	−1.45017	−0.0715314
$412$	0	0
$413$	0	0
$414$	0	0
$415$	23.8248	1.16951
$416$	0	0
$417$	−8.27492	−0.405224
$418$	0	0
$419$	28.5498	1.39475	0.697375	−	0.716706i	$-0.254351\pi$
$419$	28.5498	1.39475	0.697375	+	0.716706i	$0.254351\pi$
$420$	0	0
$421$	8.82475	0.430092	0.215046	−	0.976604i	$-0.431010\pi$
$421$	8.82475	0.430092	0.215046	+	0.976604i	$0.431010\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	−22.9003	−1.11083
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−20.5498	−0.992155
$430$	0	0
$431$	−17.6495	−0.850147	−0.425073	−	0.905159i	$-0.639752\pi$
$431$	−17.6495	−0.850147	−0.425073	+	0.905159i	$0.639752\pi$
$432$	0	0
$433$	3.17525	0.152593	0.0762963	−	0.997085i	$-0.475690\pi$
$433$	3.17525	0.152593	0.0762963	+	0.997085i	$0.475690\pi$
$434$	0	0
$435$	−17.2749	−0.828269
$436$	0	0
$437$	−25.0997	−1.20068
$438$	0	0
$439$	17.2749	0.824487	0.412243	−	0.911074i	$-0.364745\pi$
$439$	17.2749	0.824487	0.412243	+	0.911074i	$0.364745\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.37459	−0.302866	−0.151433	−	0.988468i	$-0.548389\pi$
$443$	−6.37459	−0.302866	−0.151433	+	0.988468i	$0.548389\pi$
$444$	0	0
$445$	−34.5498	−1.63782
$446$	0	0
$447$	14.5498	0.688184
$448$	0	0
$449$	20.5498	0.969807	0.484903	−	0.874568i	$-0.338855\pi$
$449$	20.5498	0.969807	0.484903	+	0.874568i	$0.338855\pi$
$450$	0	0
$451$	−14.9003	−0.701629
$452$	0	0
$453$	−22.3746	−1.05125
$454$	0	0
$455$	0	0
$456$	0	0
$457$	36.6495	1.71439	0.857196	−	0.514991i	$-0.172205\pi$
$457$	36.6495	1.71439	0.857196	+	0.514991i	$0.172205\pi$
$458$	0	0
$459$	4.00000	0.186704
$460$	0	0
$461$	−3.64950	−0.169974	−0.0849872	−	0.996382i	$-0.527085\pi$
$461$	−3.64950	−0.169974	−0.0849872	+	0.996382i	$0.527085\pi$
$462$	0	0
$463$	13.1752	0.612306	0.306153	−	0.951982i	$-0.400958\pi$
$463$	13.1752	0.612306	0.306153	+	0.951982i	$0.400958\pi$
$464$	0	0
$465$	3.27492	0.151871
$466$	0	0
$467$	40.5498	1.87642	0.938211	−	0.346063i	$-0.112482\pi$
$467$	40.5498	1.87642	0.938211	+	0.346063i	$0.112482\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0.549834	0.0253350
$472$	0	0
$473$	0.900331	0.0413973
$474$	0	0
$475$	−35.9244	−1.64833
$476$	0	0
$477$	9.27492	0.424669
$478$	0	0
$479$	−5.45017	−0.249024	−0.124512	−	0.992218i	$-0.539737\pi$
$479$	−5.45017	−0.249024	−0.124512	+	0.992218i	$0.539737\pi$
$480$	0	0
$481$	−14.2749	−0.650880
$482$	0	0
$483$	0	0
$484$	0	0
$485$	28.5739	1.29748
$486$	0	0
$487$	1.00000	0.0453143	0.0226572	−	0.999743i	$-0.492787\pi$
$487$	1.00000	0.0453143	0.0226572	+	0.999743i	$0.492787\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	36.9244	1.66638	0.833188	−	0.552990i	$-0.186513\pi$
$491$	36.9244	1.66638	0.833188	+	0.552990i	$0.186513\pi$
$492$	0	0
$493$	−21.0997	−0.950281
$494$	0	0
$495$	10.7251	0.482057
$496$	0	0
$497$	0	0
$498$	0	0
$499$	32.2749	1.44482	0.722412	−	0.691463i	$-0.243034\pi$
$499$	32.2749	1.44482	0.722412	+	0.691463i	$0.243034\pi$
$500$	0	0
$501$	6.00000	0.268060
$502$	0	0
$503$	−37.6495	−1.67871	−0.839354	−	0.543585i	$-0.817067\pi$
$503$	−37.6495	−1.67871	−0.839354	+	0.543585i	$0.817067\pi$
$504$	0	0
$505$	−19.6495	−0.874391
$506$	0	0
$507$	−26.3746	−1.17134
$508$	0	0
$509$	−11.2749	−0.499752	−0.249876	−	0.968278i	$-0.580390\pi$
$509$	−11.2749	−0.499752	−0.249876	+	0.968278i	$0.580390\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	6.27492	0.277044
$514$	0	0
$515$	35.4502	1.56212
$516$	0	0
$517$	−19.6495	−0.864184
$518$	0	0
$519$	22.5498	0.989828
$520$	0	0
$521$	−14.5498	−0.637440	−0.318720	−	0.947849i	$-0.603253\pi$
$521$	−14.5498	−0.637440	−0.318720	+	0.947849i	$0.603253\pi$
$522$	0	0
$523$	−17.7251	−0.775064	−0.387532	−	0.921856i	$-0.626672\pi$
$523$	−17.7251	−0.775064	−0.387532	+	0.921856i	$0.626672\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.00000	0.174243
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−1.27492	−0.0553267
$532$	0	0
$533$	−28.5498	−1.23663
$534$	0	0
$535$	−51.8248	−2.24058
$536$	0	0
$537$	16.5498	0.714178
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8.27492	−0.355766	−0.177883	−	0.984052i	$-0.556925\pi$
$541$	−8.27492	−0.355766	−0.177883	+	0.984052i	$0.556925\pi$
$542$	0	0
$543$	18.8248	0.807847
$544$	0	0
$545$	55.0997	2.36021
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	−33.0997	−1.41009
$552$	0	0
$553$	0	0
$554$	0	0
$555$	7.45017	0.316242
$556$	0	0
$557$	−29.8248	−1.26372	−0.631858	−	0.775084i	$-0.717707\pi$
$557$	−29.8248	−1.26372	−0.631858	+	0.775084i	$0.717707\pi$
$558$	0	0
$559$	1.72508	0.0729632
$560$	0	0
$561$	13.0997	0.553068
$562$	0	0
$563$	15.2749	0.643761	0.321881	−	0.946780i	$-0.395685\pi$
$563$	15.2749	0.643761	0.321881	+	0.946780i	$0.395685\pi$
$564$	0	0
$565$	−14.9003	−0.626862
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.4502	0.480016	0.240008	−	0.970771i	$-0.422850\pi$
$569$	11.4502	0.480016	0.240008	+	0.970771i	$0.422850\pi$
$570$	0	0
$571$	8.27492	0.346295	0.173147	−	0.984896i	$-0.444606\pi$
$571$	8.27492	0.346295	0.173147	+	0.984896i	$0.444606\pi$
$572$	0	0
$573$	−4.54983	−0.190072
$574$	0	0
$575$	22.9003	0.955010
$576$	0	0
$577$	−25.0000	−1.04076	−0.520382	−	0.853934i	$-0.674210\pi$
$577$	−25.0000	−1.04076	−0.520382	+	0.853934i	$0.674210\pi$
$578$	0	0
$579$	−0.450166	−0.0187082
$580$	0	0
$581$	0	0
$582$	0	0
$583$	30.3746	1.25799
$584$	0	0
$585$	20.5498	0.849631
$586$	0	0
$587$	9.27492	0.382817	0.191408	−	0.981510i	$-0.438695\pi$
$587$	9.27492	0.382817	0.191408	+	0.981510i	$0.438695\pi$
$588$	0	0
$589$	6.27492	0.258553
$590$	0	0
$591$	1.45017	0.0596518
$592$	0	0
$593$	−0.549834	−0.0225790	−0.0112895	−	0.999936i	$-0.503594\pi$
$593$	−0.549834	−0.0225790	−0.0112895	+	0.999936i	$0.503594\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.09967	0.208716
$598$	0	0
$599$	22.5498	0.921361	0.460681	−	0.887566i	$-0.347605\pi$
$599$	22.5498	0.921361	0.460681	+	0.887566i	$0.347605\pi$
$600$	0	0
$601$	4.09967	0.167229	0.0836145	−	0.996498i	$-0.473354\pi$
$601$	4.09967	0.167229	0.0836145	+	0.996498i	$0.473354\pi$
$602$	0	0
$603$	−0.274917	−0.0111955
$604$	0	0
$605$	−0.900331	−0.0366037
$606$	0	0
$607$	−7.00000	−0.284121	−0.142061	−	0.989858i	$-0.545373\pi$
$607$	−7.00000	−0.284121	−0.142061	+	0.989858i	$0.545373\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−37.6495	−1.52314
$612$	0	0
$613$	−8.54983	−0.345325	−0.172662	−	0.984981i	$-0.555237\pi$
$613$	−8.54983	−0.345325	−0.172662	+	0.984981i	$0.555237\pi$
$614$	0	0
$615$	14.9003	0.600839
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	28.8248	1.15856	0.579282	−	0.815127i	$-0.303333\pi$
$619$	28.8248	1.15856	0.579282	+	0.815127i	$0.303333\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−20.8488	−0.833954
$626$	0	0
$627$	20.5498	0.820681
$628$	0	0
$629$	9.09967	0.362828
$630$	0	0
$631$	19.8248	0.789211	0.394605	−	0.918851i	$-0.370881\pi$
$631$	19.8248	0.789211	0.394605	+	0.918851i	$0.370881\pi$
$632$	0	0
$633$	27.6495	1.09897
$634$	0	0
$635$	−21.1238	−0.838271
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.00000	0.0791188
$640$	0	0
$641$	−3.64950	−0.144147	−0.0720734	−	0.997399i	$-0.522962\pi$
$641$	−3.64950	−0.144147	−0.0720734	+	0.997399i	$0.522962\pi$
$642$	0	0
$643$	5.37459	0.211953	0.105976	−	0.994369i	$-0.466203\pi$
$643$	5.37459	0.211953	0.105976	+	0.994369i	$0.466203\pi$
$644$	0	0
$645$	−0.900331	−0.0354505
$646$	0	0
$647$	−34.0000	−1.33668	−0.668339	−	0.743857i	$-0.732994\pi$
$647$	−34.0000	−1.33668	−0.668339	+	0.743857i	$0.732994\pi$
$648$	0	0
$649$	−4.17525	−0.163893
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6.92442	−0.270974	−0.135487	−	0.990779i	$-0.543260\pi$
$653$	−6.92442	−0.270974	−0.135487	+	0.990779i	$0.543260\pi$
$654$	0	0
$655$	23.8248	0.930910
$656$	0	0
$657$	−4.27492	−0.166780
$658$	0	0
$659$	−42.1993	−1.64385	−0.821926	−	0.569594i	$-0.807101\pi$
$659$	−42.1993	−1.64385	−0.821926	+	0.569594i	$0.807101\pi$
$660$	0	0
$661$	7.17525	0.279085	0.139542	−	0.990216i	$-0.455437\pi$
$661$	7.17525	0.279085	0.139542	+	0.990216i	$0.455437\pi$
$662$	0	0
$663$	25.0997	0.974790
$664$	0	0
$665$	0	0
$666$	0	0
$667$	21.0997	0.816982
$668$	0	0
$669$	1.27492	0.0492911
$670$	0	0
$671$	32.7492	1.26427
$672$	0	0
$673$	41.5498	1.60163	0.800814	−	0.598913i	$-0.204400\pi$
$673$	41.5498	1.60163	0.800814	+	0.598913i	$0.204400\pi$
$674$	0	0
$675$	−5.72508	−0.220359
$676$	0	0
$677$	−13.2749	−0.510197	−0.255098	−	0.966915i	$-0.582108\pi$
$677$	−13.2749	−0.510197	−0.255098	+	0.966915i	$0.582108\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	11.2749	0.432056
$682$	0	0
$683$	12.1752	0.465873	0.232936	−	0.972492i	$-0.425167\pi$
$683$	12.1752	0.465873	0.232936	+	0.972492i	$0.425167\pi$
$684$	0	0
$685$	4.74917	0.181457
$686$	0	0
$687$	−18.2749	−0.697232
$688$	0	0
$689$	58.1993	2.21722
$690$	0	0
$691$	−10.8248	−0.411793	−0.205896	−	0.978574i	$-0.566011\pi$
$691$	−10.8248	−0.411793	−0.205896	+	0.978574i	$0.566011\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	27.0997	1.02795
$696$	0	0
$697$	18.1993	0.689349
$698$	0	0
$699$	0.549834	0.0207966
$700$	0	0
$701$	−12.9244	−0.488149	−0.244074	−	0.969757i	$-0.578484\pi$
$701$	−12.9244	−0.488149	−0.244074	+	0.969757i	$0.578484\pi$
$702$	0	0
$703$	14.2749	0.538389
$704$	0	0
$705$	19.6495	0.740043
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−28.1993	−1.05905	−0.529524	−	0.848295i	$-0.677630\pi$
$709$	−28.1993	−1.05905	−0.529524	+	0.848295i	$0.677630\pi$
$710$	0	0
$711$	11.5498	0.433153
$712$	0	0
$713$	−4.00000	−0.149801
$714$	0	0
$715$	67.2990	2.51684
$716$	0	0
$717$	−15.4502	−0.576997
$718$	0	0
$719$	28.1993	1.05166	0.525829	−	0.850590i	$-0.323755\pi$
$719$	28.1993	1.05166	0.525829	+	0.850590i	$0.323755\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−9.82475	−0.365386
$724$	0	0
$725$	30.1993	1.12158
$726$	0	0
$727$	31.5498	1.17012	0.585059	−	0.810991i	$-0.301071\pi$
$727$	31.5498	1.17012	0.585059	+	0.810991i	$0.301071\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.09967	−0.0406727
$732$	0	0
$733$	5.92442	0.218823	0.109412	−	0.993997i	$-0.465103\pi$
$733$	5.92442	0.218823	0.109412	+	0.993997i	$0.465103\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.900331	−0.0331641
$738$	0	0
$739$	1.37459	0.0505650	0.0252825	−	0.999680i	$-0.491951\pi$
$739$	1.37459	0.0505650	0.0252825	+	0.999680i	$0.491951\pi$
$740$	0	0
$741$	39.3746	1.44646
$742$	0	0
$743$	−44.1993	−1.62152	−0.810758	−	0.585381i	$-0.800945\pi$
$743$	−44.1993	−1.62152	−0.810758	+	0.585381i	$0.800945\pi$
$744$	0	0
$745$	−47.6495	−1.74574
$746$	0	0
$747$	7.27492	0.266175
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−10.4502	−0.381332	−0.190666	−	0.981655i	$-0.561065\pi$
$751$	−10.4502	−0.381332	−0.190666	+	0.981655i	$0.561065\pi$
$752$	0	0
$753$	18.3746	0.669607
$754$	0	0
$755$	73.2749	2.66675
$756$	0	0
$757$	−22.0000	−0.799604	−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000	−0.799604	−0.399802	+	0.916602i	$0.630921\pi$
$758$	0	0
$759$	−13.0997	−0.475488
$760$	0	0
$761$	−25.0997	−0.909862	−0.454931	−	0.890527i	$-0.650336\pi$
$761$	−25.0997	−0.909862	−0.454931	+	0.890527i	$0.650336\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−13.0997	−0.473620
$766$	0	0
$767$	−8.00000	−0.288863
$768$	0	0
$769$	−32.6495	−1.17737	−0.588686	−	0.808362i	$-0.700354\pi$
$769$	−32.6495	−1.17737	−0.588686	+	0.808362i	$0.700354\pi$
$770$	0	0
$771$	−11.0997	−0.399745
$772$	0	0
$773$	3.09967	0.111487	0.0557437	−	0.998445i	$-0.482247\pi$
$773$	3.09967	0.111487	0.0557437	+	0.998445i	$0.482247\pi$
$774$	0	0
$775$	−5.72508	−0.205651
$776$	0	0
$777$	0	0
$778$	0	0
$779$	28.5498	1.02290
$780$	0	0
$781$	6.54983	0.234372
$782$	0	0
$783$	−5.27492	−0.188510
$784$	0	0
$785$	−1.80066	−0.0642684
$786$	0	0
$787$	2.54983	0.0908918	0.0454459	−	0.998967i	$-0.485529\pi$
$787$	2.54983	0.0908918	0.0454459	+	0.998967i	$0.485529\pi$
$788$	0	0
$789$	9.45017	0.336435
$790$	0	0
$791$	0	0
$792$	0	0
$793$	62.7492	2.22829
$794$	0	0
$795$	−30.3746	−1.07728
$796$	0	0
$797$	−31.4743	−1.11488	−0.557438	−	0.830219i	$-0.688215\pi$
$797$	−31.4743	−1.11488	−0.557438	+	0.830219i	$0.688215\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	−10.5498	−0.372760
$802$	0	0
$803$	−14.0000	−0.494049
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−20.7251	−0.729557
$808$	0	0
$809$	−29.6495	−1.04242	−0.521211	−	0.853428i	$-0.674519\pi$
$809$	−29.6495	−1.04242	−0.521211	+	0.853428i	$0.674519\pi$
$810$	0	0
$811$	−42.5498	−1.49413	−0.747063	−	0.664753i	$-0.768537\pi$
$811$	−42.5498	−1.49413	−0.747063	+	0.664753i	$0.768537\pi$
$812$	0	0
$813$	−1.27492	−0.0447133
$814$	0	0
$815$	−39.2990	−1.37658
$816$	0	0
$817$	−1.72508	−0.0603530
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12.7251	0.444108	0.222054	−	0.975034i	$-0.428724\pi$
$821$	12.7251	0.444108	0.222054	+	0.975034i	$0.428724\pi$
$822$	0	0
$823$	−34.1993	−1.19211	−0.596057	−	0.802942i	$-0.703267\pi$
$823$	−34.1993	−1.19211	−0.596057	+	0.802942i	$0.703267\pi$
$824$	0	0
$825$	−18.7492	−0.652762
$826$	0	0
$827$	−44.0241	−1.53087	−0.765434	−	0.643515i	$-0.777476\pi$
$827$	−44.0241	−1.53087	−0.765434	+	0.643515i	$0.777476\pi$
$828$	0	0
$829$	30.2749	1.05149	0.525746	−	0.850642i	$-0.323786\pi$
$829$	30.2749	1.05149	0.525746	+	0.850642i	$0.323786\pi$
$830$	0	0
$831$	26.8248	0.930540
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−19.6495	−0.679999
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	−30.1993	−1.04260	−0.521298	−	0.853374i	$-0.674552\pi$
$839$	−30.1993	−1.04260	−0.521298	+	0.853374i	$0.674552\pi$
$840$	0	0
$841$	−1.17525	−0.0405258
$842$	0	0
$843$	26.5498	0.914425
$844$	0	0
$845$	86.3746	2.97138
$846$	0	0
$847$	0	0
$848$	0	0
$849$	25.9244	0.889724
$850$	0	0
$851$	−9.09967	−0.311933
$852$	0	0
$853$	13.3746	0.457937	0.228969	−	0.973434i	$-0.426465\pi$
$853$	13.3746	0.457937	0.228969	+	0.973434i	$0.426465\pi$
$854$	0	0
$855$	−20.5498	−0.702790
$856$	0	0
$857$	−13.4502	−0.459449	−0.229724	−	0.973256i	$-0.573782\pi$
$857$	−13.4502	−0.459449	−0.229724	+	0.973256i	$0.573782\pi$
$858$	0	0
$859$	−55.6495	−1.89874	−0.949368	−	0.314165i	$-0.898275\pi$
$859$	−55.6495	−1.89874	−0.949368	+	0.314165i	$0.898275\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19.0997	0.650160	0.325080	−	0.945686i	$-0.394609\pi$
$863$	19.0997	0.650160	0.325080	+	0.945686i	$0.394609\pi$
$864$	0	0
$865$	−73.8488	−2.51094
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	37.8248	1.28312
$870$	0	0
$871$	−1.72508	−0.0584522
$872$	0	0
$873$	8.72508	0.295299
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.7492	1.57861	0.789304	−	0.614003i	$-0.210442\pi$
$877$	46.7492	1.57861	0.789304	+	0.614003i	$0.210442\pi$
$878$	0	0
$879$	27.8248	0.938506
$880$	0	0
$881$	9.45017	0.318384	0.159192	−	0.987248i	$-0.449111\pi$
$881$	9.45017	0.318384	0.159192	+	0.987248i	$0.449111\pi$
$882$	0	0
$883$	7.37459	0.248175	0.124087	−	0.992271i	$-0.460400\pi$
$883$	7.37459	0.248175	0.124087	+	0.992271i	$0.460400\pi$
$884$	0	0
$885$	4.17525	0.140349
$886$	0	0
$887$	33.6495	1.12984	0.564920	−	0.825146i	$-0.308907\pi$
$887$	33.6495	1.12984	0.564920	+	0.825146i	$0.308907\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.27492	0.109714
$892$	0	0
$893$	37.6495	1.25989
$894$	0	0
$895$	−54.1993	−1.81168
$896$	0	0
$897$	−25.0997	−0.838054
$898$	0	0
$899$	−5.27492	−0.175928
$900$	0	0
$901$	−37.0997	−1.23597
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−61.6495	−2.04930
$906$	0	0
$907$	−29.7251	−0.987005	−0.493503	−	0.869744i	$-0.664284\pi$
$907$	−29.7251	−0.987005	−0.493503	+	0.869744i	$0.664284\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	53.8488	1.78409	0.892046	−	0.451945i	$-0.149270\pi$
$911$	53.8488	1.78409	0.892046	+	0.451945i	$0.149270\pi$
$912$	0	0
$913$	23.8248	0.788484
$914$	0	0
$915$	−32.7492	−1.08265
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−2.82475	−0.0931800	−0.0465900	−	0.998914i	$-0.514835\pi$
$919$	−2.82475	−0.0931800	−0.0465900	+	0.998914i	$0.514835\pi$
$920$	0	0
$921$	−11.3746	−0.374805
$922$	0	0
$923$	12.5498	0.413083
$924$	0	0
$925$	−13.0241	−0.428229
$926$	0	0
$927$	10.8248	0.355531
$928$	0	0
$929$	−40.1993	−1.31890	−0.659449	−	0.751750i	$-0.729210\pi$
$929$	−40.1993	−1.31890	−0.659449	+	0.751750i	$0.729210\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	4.54983	0.148955
$934$	0	0
$935$	−42.9003	−1.40299
$936$	0	0
$937$	−6.09967	−0.199267	−0.0996337	−	0.995024i	$-0.531767\pi$
$937$	−6.09967	−0.199267	−0.0996337	+	0.995024i	$0.531767\pi$
$938$	0	0
$939$	19.5498	0.637985
$940$	0	0
$941$	51.8248	1.68944	0.844719	−	0.535210i	$-0.179767\pi$
$941$	51.8248	1.68944	0.844719	+	0.535210i	$0.179767\pi$
$942$	0	0
$943$	−18.1993	−0.592652
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.4502	−0.632045	−0.316023	−	0.948752i	$-0.602348\pi$
$947$	−19.4502	−0.632045	−0.316023	+	0.948752i	$0.602348\pi$
$948$	0	0
$949$	−26.8248	−0.870768
$950$	0	0
$951$	27.8248	0.902279
$952$	0	0
$953$	55.6495	1.80266	0.901332	−	0.433129i	$-0.142590\pi$
$953$	55.6495	1.80266	0.901332	+	0.433129i	$0.142590\pi$
$954$	0	0
$955$	14.9003	0.482163
$956$	0	0
$957$	−17.2749	−0.558419
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−15.8248	−0.509945
$964$	0	0
$965$	1.47425	0.0474579
$966$	0	0
$967$	−53.5498	−1.72205	−0.861023	−	0.508566i	$-0.830176\pi$
$967$	−53.5498	−1.72205	−0.861023	+	0.508566i	$0.830176\pi$
$968$	0	0
$969$	−25.0997	−0.806318
$970$	0	0
$971$	56.5739	1.81554	0.907772	−	0.419464i	$-0.137782\pi$
$971$	56.5739	1.81554	0.907772	+	0.419464i	$0.137782\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−35.9244	−1.15050
$976$	0	0
$977$	8.90033	0.284747	0.142373	−	0.989813i	$-0.454527\pi$
$977$	8.90033	0.284747	0.142373	+	0.989813i	$0.454527\pi$
$978$	0	0
$979$	−34.5498	−1.10422
$980$	0	0
$981$	16.8248	0.537173
$982$	0	0
$983$	39.2990	1.25344	0.626722	−	0.779243i	$-0.284396\pi$
$983$	39.2990	1.25344	0.626722	+	0.779243i	$0.284396\pi$
$984$	0	0
$985$	−4.74917	−0.151321
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.09967	0.0349674
$990$	0	0
$991$	−14.0997	−0.447891	−0.223945	−	0.974602i	$-0.571894\pi$
$991$	−14.0997	−0.447891	−0.223945	+	0.974602i	$0.571894\pi$
$992$	0	0
$993$	−1.17525	−0.0372954
$994$	0	0
$995$	−16.7010	−0.529457
$996$	0	0
$997$	−44.8248	−1.41961	−0.709807	−	0.704396i	$-0.751218\pi$
$997$	−44.8248	−1.41961	−0.709807	+	0.704396i	$0.751218\pi$
$998$	0	0
$999$	2.27492	0.0719752

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.2.a.k.1.2		2
3.2	odd	2		3528.2.a.bk.1.1		2
4.3	odd	2		2352.2.a.bf.1.2		2
7.2	even	3		168.2.q.c.25.1	✓	4
7.3	odd	6		1176.2.q.l.961.2		4
7.4	even	3		168.2.q.c.121.1	yes	4
7.5	odd	6		1176.2.q.l.361.2		4
7.6	odd	2		1176.2.a.n.1.1		2
8.3	odd	2		9408.2.a.dp.1.1		2
8.5	even	2		9408.2.a.ec.1.1		2
12.11	even	2		7056.2.a.cu.1.1		2
21.2	odd	6		504.2.s.i.361.2		4
21.5	even	6		3528.2.s.bk.361.1		4
21.11	odd	6		504.2.s.i.289.2		4
21.17	even	6		3528.2.s.bk.3313.1		4
21.20	even	2		3528.2.a.bd.1.2		2
28.3	even	6		2352.2.q.bf.961.2		4
28.11	odd	6		336.2.q.g.289.1		4
28.19	even	6		2352.2.q.bf.1537.2		4
28.23	odd	6		336.2.q.g.193.1		4
28.27	even	2		2352.2.a.ba.1.1		2
56.11	odd	6		1344.2.q.x.961.2		4
56.13	odd	2		9408.2.a.dj.1.2		2
56.27	even	2		9408.2.a.dw.1.2		2
56.37	even	6		1344.2.q.w.193.2		4
56.51	odd	6		1344.2.q.x.193.2		4
56.53	even	6		1344.2.q.w.961.2		4
84.11	even	6		1008.2.s.r.289.2		4
84.23	even	6		1008.2.s.r.865.2		4
84.83	odd	2		7056.2.a.ch.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.q.c.25.1	✓	4	7.2	even	3
168.2.q.c.121.1	yes	4	7.4	even	3
336.2.q.g.193.1		4	28.23	odd	6
336.2.q.g.289.1		4	28.11	odd	6
504.2.s.i.289.2		4	21.11	odd	6
504.2.s.i.361.2		4	21.2	odd	6
1008.2.s.r.289.2		4	84.11	even	6
1008.2.s.r.865.2		4	84.23	even	6
1176.2.a.k.1.2		2	1.1	even	1	trivial
1176.2.a.n.1.1		2	7.6	odd	2
1176.2.q.l.361.2		4	7.5	odd	6
1176.2.q.l.961.2		4	7.3	odd	6
1344.2.q.w.193.2		4	56.37	even	6
1344.2.q.w.961.2		4	56.53	even	6
1344.2.q.x.193.2		4	56.51	odd	6
1344.2.q.x.961.2		4	56.11	odd	6
2352.2.a.ba.1.1		2	28.27	even	2
2352.2.a.bf.1.2		2	4.3	odd	2
2352.2.q.bf.961.2		4	28.3	even	6
2352.2.q.bf.1537.2		4	28.19	even	6
3528.2.a.bd.1.2		2	21.20	even	2
3528.2.a.bk.1.1		2	3.2	odd	2
3528.2.s.bk.361.1		4	21.5	even	6
3528.2.s.bk.3313.1		4	21.17	even	6
7056.2.a.ch.1.2		2	84.83	odd	2
7056.2.a.cu.1.1		2	12.11	even	2
9408.2.a.dj.1.2		2	56.13	odd	2
9408.2.a.dp.1.1		2	8.3	odd	2
9408.2.a.dw.1.2		2	56.27	even	2
9408.2.a.ec.1.1		2	8.5	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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1176.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.27492 q^{5} +1.00000 q^{9} +3.27492 q^{11} +6.27492 q^{13} -3.27492 q^{15} -4.00000 q^{17} -6.27492 q^{19} +4.00000 q^{23} +5.72508 q^{25} -1.00000 q^{27} +5.27492 q^{29} -1.00000 q^{31} -3.27492 q^{33} -2.27492 q^{37} -6.27492 q^{39} -4.54983 q^{41} +0.274917 q^{43} +3.27492 q^{45} -6.00000 q^{47} +4.00000 q^{51} +9.27492 q^{53} +10.7251 q^{55} +6.27492 q^{57} -1.27492 q^{59} +10.0000 q^{61} +20.5498 q^{65} -0.274917 q^{67} -4.00000 q^{69} +2.00000 q^{71} -4.27492 q^{73} -5.72508 q^{75} +11.5498 q^{79} +1.00000 q^{81} +7.27492 q^{83} -13.0997 q^{85} -5.27492 q^{87} -10.5498 q^{89} +1.00000 q^{93} -20.5498 q^{95} +8.72508 q^{97} +3.27492 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} + 2 q^{9} - q^{11} + 5 q^{13} + q^{15} - 8 q^{17} - 5 q^{19} + 8 q^{23} + 19 q^{25} - 2 q^{27} + 3 q^{29} - 2 q^{31} + q^{33} + 3 q^{37} - 5 q^{39} + 6 q^{41} - 7 q^{43} - q^{45}+ \cdots - q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 + 2 * q^9 - q^11 + 5 * q^13 + q^15 - 8 * q^17 - 5 * q^19 + 8 * q^23 + 19 * q^25 - 2 * q^27 + 3 * q^29 - 2 * q^31 + q^33 + 3 * q^37 - 5 * q^39 + 6 * q^41 - 7 * q^43 - q^45 - 12 * q^47 + 8 * q^51 + 11 * q^53 + 29 * q^55 + 5 * q^57 + 5 * q^59 + 20 * q^61 + 26 * q^65 + 7 * q^67 - 8 * q^69 + 4 * q^71 - q^73 - 19 * q^75 + 8 * q^79 + 2 * q^81 + 7 * q^83 + 4 * q^85 - 3 * q^87 - 6 * q^89 + 2 * q^93 - 26 * q^95 + 25 * q^97 - q^99

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