LMFDB - Embedded newform 9408.2.a.dh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9408,2,Mod(1,9408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9408, 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("9408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9408 = 2^{6} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9408.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:	$75.1232582216$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1176)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9408.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.585786 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.585786 q^{5} +1.00000 q^{9} -4.82843 q^{11} -4.24264 q^{13} +0.585786 q^{15} +4.58579 q^{17} +1.17157 q^{19} -0.828427 q^{23} -4.65685 q^{25} -1.00000 q^{27} +2.82843 q^{29} -2.82843 q^{31} +4.82843 q^{33} -9.65685 q^{37} +4.24264 q^{39} +1.75736 q^{41} +11.3137 q^{43} -0.585786 q^{45} -12.4853 q^{47} -4.58579 q^{51} +2.00000 q^{53} +2.82843 q^{55} -1.17157 q^{57} -8.48528 q^{59} +3.07107 q^{61} +2.48528 q^{65} -11.3137 q^{67} +0.828427 q^{69} -6.48528 q^{71} +16.2426 q^{73} +4.65685 q^{75} -2.34315 q^{79} +1.00000 q^{81} -4.00000 q^{83} -2.68629 q^{85} -2.82843 q^{87} +14.2426 q^{89} +2.82843 q^{93} -0.686292 q^{95} +8.24264 q^{97} -4.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} - 4 q^{11} + 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{33} - 8 q^{37} + 12 q^{41} - 4 q^{45} - 8 q^{47} - 12 q^{51} + 4 q^{53} - 8 q^{57} - 8 q^{61} - 12 q^{65} - 4 q^{69} + 4 q^{71} + 24 q^{73} - 2 q^{75} - 16 q^{79} + 2 q^{81} - 8 q^{83} - 28 q^{85} + 20 q^{89} - 24 q^{95} + 8 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 - 4 * q^11 + 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^33 - 8 * q^37 + 12 * q^41 - 4 * q^45 - 8 * q^47 - 12 * q^51 + 4 * q^53 - 8 * q^57 - 8 * q^61 - 12 * q^65 - 4 * q^69 + 4 * q^71 + 24 * q^73 - 2 * q^75 - 16 * q^79 + 2 * q^81 - 8 * q^83 - 28 * q^85 + 20 * q^89 - 24 * q^95 + 8 * 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
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.82843	−1.45583	−0.727913	−	0.685670i	$-0.759509\pi$
$11$	−4.82843	−1.45583	−0.727913	+	0.685670i	$0.759509\pi$
$12$	0	0
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	0	0
$15$	0.585786	0.151249
$16$	0	0
$17$	4.58579	1.11222	0.556108	−	0.831110i	$-0.312294\pi$
$17$	4.58579	1.11222	0.556108	+	0.831110i	$0.312294\pi$
$18$	0	0
$19$	1.17157	0.268777	0.134389	−	0.990929i	$-0.457093\pi$
$19$	1.17157	0.268777	0.134389	+	0.990929i	$0.457093\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.828427	−0.172739	−0.0863695	−	0.996263i	$-0.527527\pi$
$23$	−0.828427	−0.172739	−0.0863695	+	0.996263i	$0.527527\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.82843	0.525226	0.262613	−	0.964901i	$-0.415416\pi$
$29$	2.82843	0.525226	0.262613	+	0.964901i	$0.415416\pi$
$30$	0	0
$31$	−2.82843	−0.508001	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843	−0.508001	−0.254000	+	0.967204i	$0.581746\pi$
$32$	0	0
$33$	4.82843	0.840521
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.65685	−1.58758	−0.793789	−	0.608194i	$-0.791894\pi$
$37$	−9.65685	−1.58758	−0.793789	+	0.608194i	$0.791894\pi$
$38$	0	0
$39$	4.24264	0.679366
$40$	0	0
$41$	1.75736	0.274453	0.137227	−	0.990540i	$-0.456181\pi$
$41$	1.75736	0.274453	0.137227	+	0.990540i	$0.456181\pi$
$42$	0	0
$43$	11.3137	1.72532	0.862662	−	0.505781i	$-0.168795\pi$
$43$	11.3137	1.72532	0.862662	+	0.505781i	$0.168795\pi$
$44$	0	0
$45$	−0.585786	−0.0873239
$46$	0	0
$47$	−12.4853	−1.82117	−0.910583	−	0.413327i	$-0.864367\pi$
$47$	−12.4853	−1.82117	−0.910583	+	0.413327i	$0.864367\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.58579	−0.642139
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	−1.17157	−0.155179
$58$	0	0
$59$	−8.48528	−1.10469	−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528	−1.10469	−0.552345	+	0.833616i	$0.686267\pi$
$60$	0	0
$61$	3.07107	0.393210	0.196605	−	0.980483i	$-0.437008\pi$
$61$	3.07107	0.393210	0.196605	+	0.980483i	$0.437008\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.48528	0.308261
$66$	0	0
$67$	−11.3137	−1.38219	−0.691095	−	0.722764i	$-0.742871\pi$
$67$	−11.3137	−1.38219	−0.691095	+	0.722764i	$0.742871\pi$
$68$	0	0
$69$	0.828427	0.0997309
$70$	0	0
$71$	−6.48528	−0.769661	−0.384831	−	0.922987i	$-0.625740\pi$
$71$	−6.48528	−0.769661	−0.384831	+	0.922987i	$0.625740\pi$
$72$	0	0
$73$	16.2426	1.90106	0.950529	−	0.310637i	$-0.100542\pi$
$73$	16.2426	1.90106	0.950529	+	0.310637i	$0.100542\pi$
$74$	0	0
$75$	4.65685	0.537727
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.34315	−0.263624	−0.131812	−	0.991275i	$-0.542080\pi$
$79$	−2.34315	−0.263624	−0.131812	+	0.991275i	$0.542080\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−2.68629	−0.291369
$86$	0	0
$87$	−2.82843	−0.303239
$88$	0	0
$89$	14.2426	1.50972	0.754858	−	0.655888i	$-0.227706\pi$
$89$	14.2426	1.50972	0.754858	+	0.655888i	$0.227706\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.82843	0.293294
$94$	0	0
$95$	−0.686292	−0.0704120
$96$	0	0
$97$	8.24264	0.836913	0.418457	−	0.908237i	$-0.362571\pi$
$97$	8.24264	0.836913	0.418457	+	0.908237i	$0.362571\pi$
$98$	0	0
$99$	−4.82843	−0.485275
$100$	0	0
$101$	−14.7279	−1.46548	−0.732742	−	0.680507i	$-0.761760\pi$
$101$	−14.7279	−1.46548	−0.732742	+	0.680507i	$0.761760\pi$
$102$	0	0
$103$	14.1421	1.39347	0.696733	−	0.717331i	$-0.254636\pi$
$103$	14.1421	1.39347	0.696733	+	0.717331i	$0.254636\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.17157	0.693302	0.346651	−	0.937994i	$-0.387319\pi$
$107$	7.17157	0.693302	0.346651	+	0.937994i	$0.387319\pi$
$108$	0	0
$109$	−19.3137	−1.84992	−0.924959	−	0.380067i	$-0.875901\pi$
$109$	−19.3137	−1.84992	−0.924959	+	0.380067i	$0.875901\pi$
$110$	0	0
$111$	9.65685	0.916588
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	0.485281	0.0452527
$116$	0	0
$117$	−4.24264	−0.392232
$118$	0	0
$119$	0	0
$120$	0	0
$121$	12.3137	1.11943
$122$	0	0
$123$	−1.75736	−0.158456
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	0	0
$129$	−11.3137	−0.996116
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.585786	0.0504165
$136$	0	0
$137$	−14.8284	−1.26688	−0.633439	−	0.773793i	$-0.718357\pi$
$137$	−14.8284	−1.26688	−0.633439	+	0.773793i	$0.718357\pi$
$138$	0	0
$139$	12.9706	1.10015	0.550074	−	0.835116i	$-0.314599\pi$
$139$	12.9706	1.10015	0.550074	+	0.835116i	$0.314599\pi$
$140$	0	0
$141$	12.4853	1.05145
$142$	0	0
$143$	20.4853	1.71307
$144$	0	0
$145$	−1.65685	−0.137594
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.3137	1.74609	0.873044	−	0.487642i	$-0.162143\pi$
$149$	21.3137	1.74609	0.873044	+	0.487642i	$0.162143\pi$
$150$	0	0
$151$	−1.65685	−0.134833	−0.0674164	−	0.997725i	$-0.521476\pi$
$151$	−1.65685	−0.134833	−0.0674164	+	0.997725i	$0.521476\pi$
$152$	0	0
$153$	4.58579	0.370739
$154$	0	0
$155$	1.65685	0.133082
$156$	0	0
$157$	8.24264	0.657834	0.328917	−	0.944359i	$-0.393316\pi$
$157$	8.24264	0.657834	0.328917	+	0.944359i	$0.393316\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−5.65685	−0.443079	−0.221540	−	0.975151i	$-0.571108\pi$
$163$	−5.65685	−0.443079	−0.221540	+	0.975151i	$0.571108\pi$
$164$	0	0
$165$	−2.82843	−0.220193
$166$	0	0
$167$	9.17157	0.709718	0.354859	−	0.934920i	$-0.384529\pi$
$167$	9.17157	0.709718	0.354859	+	0.934920i	$0.384529\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	1.17157	0.0895924
$172$	0	0
$173$	19.4142	1.47604	0.738018	−	0.674781i	$-0.235762\pi$
$173$	19.4142	1.47604	0.738018	+	0.674781i	$0.235762\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8.48528	0.637793
$178$	0	0
$179$	10.4853	0.783707	0.391853	−	0.920028i	$-0.371834\pi$
$179$	10.4853	0.783707	0.391853	+	0.920028i	$0.371834\pi$
$180$	0	0
$181$	−7.07107	−0.525588	−0.262794	−	0.964852i	$-0.584644\pi$
$181$	−7.07107	−0.525588	−0.262794	+	0.964852i	$0.584644\pi$
$182$	0	0
$183$	−3.07107	−0.227020
$184$	0	0
$185$	5.65685	0.415900
$186$	0	0
$187$	−22.1421	−1.61919
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.1421	1.16800	0.584002	−	0.811752i	$-0.301486\pi$
$191$	16.1421	1.16800	0.584002	+	0.811752i	$0.301486\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	0	0
$195$	−2.48528	−0.177975
$196$	0	0
$197$	25.3137	1.80353	0.901764	−	0.432230i	$-0.142273\pi$
$197$	25.3137	1.80353	0.901764	+	0.432230i	$0.142273\pi$
$198$	0	0
$199$	−5.65685	−0.401004	−0.200502	−	0.979693i	$-0.564257\pi$
$199$	−5.65685	−0.401004	−0.200502	+	0.979693i	$0.564257\pi$
$200$	0	0
$201$	11.3137	0.798007
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1.02944	−0.0718990
$206$	0	0
$207$	−0.828427	−0.0575797
$208$	0	0
$209$	−5.65685	−0.391293
$210$	0	0
$211$	−9.65685	−0.664805	−0.332403	−	0.943138i	$-0.607859\pi$
$211$	−9.65685	−0.664805	−0.332403	+	0.943138i	$0.607859\pi$
$212$	0	0
$213$	6.48528	0.444364
$214$	0	0
$215$	−6.62742	−0.451986
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−16.2426	−1.09758
$220$	0	0
$221$	−19.4558	−1.30874
$222$	0	0
$223$	−2.34315	−0.156909	−0.0784543	−	0.996918i	$-0.524998\pi$
$223$	−2.34315	−0.156909	−0.0784543	+	0.996918i	$0.524998\pi$
$224$	0	0
$225$	−4.65685	−0.310457
$226$	0	0
$227$	−10.8284	−0.718708	−0.359354	−	0.933201i	$-0.617003\pi$
$227$	−10.8284	−0.718708	−0.359354	+	0.933201i	$0.617003\pi$
$228$	0	0
$229$	22.5858	1.49251	0.746255	−	0.665660i	$-0.231850\pi$
$229$	22.5858	1.49251	0.746255	+	0.665660i	$0.231850\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.17157	−0.600850	−0.300425	−	0.953805i	$-0.597128\pi$
$233$	−9.17157	−0.600850	−0.300425	+	0.953805i	$0.597128\pi$
$234$	0	0
$235$	7.31371	0.477094
$236$	0	0
$237$	2.34315	0.152204
$238$	0	0
$239$	7.17157	0.463890	0.231945	−	0.972729i	$-0.425491\pi$
$239$	7.17157	0.463890	0.231945	+	0.972729i	$0.425491\pi$
$240$	0	0
$241$	−5.89949	−0.380020	−0.190010	−	0.981782i	$-0.560852\pi$
$241$	−5.89949	−0.380020	−0.190010	+	0.981782i	$0.560852\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−4.97056	−0.316269
$248$	0	0
$249$	4.00000	0.253490
$250$	0	0
$251$	22.1421	1.39760	0.698800	−	0.715317i	$-0.253718\pi$
$251$	22.1421	1.39760	0.698800	+	0.715317i	$0.253718\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	2.68629	0.168222
$256$	0	0
$257$	−0.585786	−0.0365404	−0.0182702	−	0.999833i	$-0.505816\pi$
$257$	−0.585786	−0.0365404	−0.0182702	+	0.999833i	$0.505816\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	2.82843	0.175075
$262$	0	0
$263$	−8.14214	−0.502066	−0.251033	−	0.967979i	$-0.580770\pi$
$263$	−8.14214	−0.502066	−0.251033	+	0.967979i	$0.580770\pi$
$264$	0	0
$265$	−1.17157	−0.0719691
$266$	0	0
$267$	−14.2426	−0.871635
$268$	0	0
$269$	19.8995	1.21329	0.606647	−	0.794971i	$-0.292514\pi$
$269$	19.8995	1.21329	0.606647	+	0.794971i	$0.292514\pi$
$270$	0	0
$271$	22.1421	1.34504	0.672519	−	0.740079i	$-0.265212\pi$
$271$	22.1421	1.34504	0.672519	+	0.740079i	$0.265212\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	22.4853	1.35591
$276$	0	0
$277$	−16.6274	−0.999045	−0.499522	−	0.866301i	$-0.666491\pi$
$277$	−16.6274	−0.999045	−0.499522	+	0.866301i	$0.666491\pi$
$278$	0	0
$279$	−2.82843	−0.169334
$280$	0	0
$281$	6.82843	0.407350	0.203675	−	0.979039i	$-0.434711\pi$
$281$	6.82843	0.407350	0.203675	+	0.979039i	$0.434711\pi$
$282$	0	0
$283$	2.14214	0.127337	0.0636684	−	0.997971i	$-0.479720\pi$
$283$	2.14214	0.127337	0.0636684	+	0.997971i	$0.479720\pi$
$284$	0	0
$285$	0.686292	0.0406524
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4.02944	0.237026
$290$	0	0
$291$	−8.24264	−0.483192
$292$	0	0
$293$	10.2426	0.598381	0.299191	−	0.954193i	$-0.403283\pi$
$293$	10.2426	0.598381	0.299191	+	0.954193i	$0.403283\pi$
$294$	0	0
$295$	4.97056	0.289397
$296$	0	0
$297$	4.82843	0.280174
$298$	0	0
$299$	3.51472	0.203261
$300$	0	0
$301$	0	0
$302$	0	0
$303$	14.7279	0.846097
$304$	0	0
$305$	−1.79899	−0.103010
$306$	0	0
$307$	−28.4853	−1.62574	−0.812870	−	0.582445i	$-0.802096\pi$
$307$	−28.4853	−1.62574	−0.812870	+	0.582445i	$0.802096\pi$
$308$	0	0
$309$	−14.1421	−0.804518
$310$	0	0
$311$	15.7990	0.895879	0.447939	−	0.894064i	$-0.352158\pi$
$311$	15.7990	0.895879	0.447939	+	0.894064i	$0.352158\pi$
$312$	0	0
$313$	3.27208	0.184949	0.0924744	−	0.995715i	$-0.470522\pi$
$313$	3.27208	0.184949	0.0924744	+	0.995715i	$0.470522\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	22.0000	1.23564	0.617822	−	0.786318i	$-0.288015\pi$
$317$	22.0000	1.23564	0.617822	+	0.786318i	$0.288015\pi$
$318$	0	0
$319$	−13.6569	−0.764637
$320$	0	0
$321$	−7.17157	−0.400278
$322$	0	0
$323$	5.37258	0.298939
$324$	0	0
$325$	19.7574	1.09594
$326$	0	0
$327$	19.3137	1.06805
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−15.3137	−0.841718	−0.420859	−	0.907126i	$-0.638271\pi$
$331$	−15.3137	−0.841718	−0.420859	+	0.907126i	$0.638271\pi$
$332$	0	0
$333$	−9.65685	−0.529192
$334$	0	0
$335$	6.62742	0.362094
$336$	0	0
$337$	21.6569	1.17972	0.589862	−	0.807504i	$-0.299182\pi$
$337$	21.6569	1.17972	0.589862	+	0.807504i	$0.299182\pi$
$338$	0	0
$339$	10.0000	0.543125
$340$	0	0
$341$	13.6569	0.739560
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.485281	−0.0261267
$346$	0	0
$347$	14.4853	0.777611	0.388805	−	0.921320i	$-0.372888\pi$
$347$	14.4853	0.777611	0.388805	+	0.921320i	$0.372888\pi$
$348$	0	0
$349$	−13.4142	−0.718046	−0.359023	−	0.933329i	$-0.616890\pi$
$349$	−13.4142	−0.718046	−0.359023	+	0.933329i	$0.616890\pi$
$350$	0	0
$351$	4.24264	0.226455
$352$	0	0
$353$	31.6985	1.68714	0.843570	−	0.537019i	$-0.180450\pi$
$353$	31.6985	1.68714	0.843570	+	0.537019i	$0.180450\pi$
$354$	0	0
$355$	3.79899	0.201629
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−23.1716	−1.22295	−0.611474	−	0.791264i	$-0.709423\pi$
$359$	−23.1716	−1.22295	−0.611474	+	0.791264i	$0.709423\pi$
$360$	0	0
$361$	−17.6274	−0.927759
$362$	0	0
$363$	−12.3137	−0.646302
$364$	0	0
$365$	−9.51472	−0.498023
$366$	0	0
$367$	−27.3137	−1.42576	−0.712882	−	0.701284i	$-0.752610\pi$
$367$	−27.3137	−1.42576	−0.712882	+	0.701284i	$0.752610\pi$
$368$	0	0
$369$	1.75736	0.0914845
$370$	0	0
$371$	0	0
$372$	0	0
$373$	17.3137	0.896470	0.448235	−	0.893916i	$-0.352053\pi$
$373$	17.3137	0.896470	0.448235	+	0.893916i	$0.352053\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−0.686292	−0.0352524	−0.0176262	−	0.999845i	$-0.505611\pi$
$379$	−0.686292	−0.0352524	−0.0176262	+	0.999845i	$0.505611\pi$
$380$	0	0
$381$	20.0000	1.02463
$382$	0	0
$383$	−24.9706	−1.27594	−0.637968	−	0.770063i	$-0.720225\pi$
$383$	−24.9706	−1.27594	−0.637968	+	0.770063i	$0.720225\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.3137	0.575108
$388$	0	0
$389$	−3.79899	−0.192616	−0.0963082	−	0.995352i	$-0.530703\pi$
$389$	−3.79899	−0.192616	−0.0963082	+	0.995352i	$0.530703\pi$
$390$	0	0
$391$	−3.79899	−0.192123
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	1.37258	0.0690621
$396$	0	0
$397$	7.75736	0.389331	0.194665	−	0.980870i	$-0.437638\pi$
$397$	7.75736	0.389331	0.194665	+	0.980870i	$0.437638\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.82843	0.340995	0.170498	−	0.985358i	$-0.445462\pi$
$401$	6.82843	0.340995	0.170498	+	0.985358i	$0.445462\pi$
$402$	0	0
$403$	12.0000	0.597763
$404$	0	0
$405$	−0.585786	−0.0291080
$406$	0	0
$407$	46.6274	2.31124
$408$	0	0
$409$	0.242641	0.0119978	0.00599890	−	0.999982i	$-0.498090\pi$
$409$	0.242641	0.0119978	0.00599890	+	0.999982i	$0.498090\pi$
$410$	0	0
$411$	14.8284	0.731432
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2.34315	0.115021
$416$	0	0
$417$	−12.9706	−0.635171
$418$	0	0
$419$	24.4853	1.19618	0.598092	−	0.801427i	$-0.295926\pi$
$419$	24.4853	1.19618	0.598092	+	0.801427i	$0.295926\pi$
$420$	0	0
$421$	2.68629	0.130922	0.0654609	−	0.997855i	$-0.479148\pi$
$421$	2.68629	0.130922	0.0654609	+	0.997855i	$0.479148\pi$
$422$	0	0
$423$	−12.4853	−0.607055
$424$	0	0
$425$	−21.3553	−1.03589
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−20.4853	−0.989039
$430$	0	0
$431$	−20.1421	−0.970213	−0.485106	−	0.874455i	$-0.661219\pi$
$431$	−20.1421	−0.970213	−0.485106	+	0.874455i	$0.661219\pi$
$432$	0	0
$433$	15.0711	0.724269	0.362135	−	0.932126i	$-0.382048\pi$
$433$	15.0711	0.724269	0.362135	+	0.932126i	$0.382048\pi$
$434$	0	0
$435$	1.65685	0.0794401
$436$	0	0
$437$	−0.970563	−0.0464283
$438$	0	0
$439$	−35.3137	−1.68543	−0.842716	−	0.538359i	$-0.819044\pi$
$439$	−35.3137	−1.68543	−0.842716	+	0.538359i	$0.819044\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−16.8284	−0.799543	−0.399771	−	0.916615i	$-0.630910\pi$
$443$	−16.8284	−0.799543	−0.399771	+	0.916615i	$0.630910\pi$
$444$	0	0
$445$	−8.34315	−0.395503
$446$	0	0
$447$	−21.3137	−1.00810
$448$	0	0
$449$	28.6274	1.35101	0.675506	−	0.737355i	$-0.263925\pi$
$449$	28.6274	1.35101	0.675506	+	0.737355i	$0.263925\pi$
$450$	0	0
$451$	−8.48528	−0.399556
$452$	0	0
$453$	1.65685	0.0778458
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.6274	0.964910	0.482455	−	0.875921i	$-0.339745\pi$
$457$	20.6274	0.964910	0.482455	+	0.875921i	$0.339745\pi$
$458$	0	0
$459$	−4.58579	−0.214046
$460$	0	0
$461$	20.3848	0.949414	0.474707	−	0.880144i	$-0.342554\pi$
$461$	20.3848	0.949414	0.474707	+	0.880144i	$0.342554\pi$
$462$	0	0
$463$	9.65685	0.448792	0.224396	−	0.974498i	$-0.427959\pi$
$463$	9.65685	0.448792	0.224396	+	0.974498i	$0.427959\pi$
$464$	0	0
$465$	−1.65685	−0.0768348
$466$	0	0
$467$	−13.1716	−0.609508	−0.304754	−	0.952431i	$-0.598574\pi$
$467$	−13.1716	−0.609508	−0.304754	+	0.952431i	$0.598574\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−8.24264	−0.379801
$472$	0	0
$473$	−54.6274	−2.51177
$474$	0	0
$475$	−5.45584	−0.250331
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−17.1716	−0.784589	−0.392295	−	0.919840i	$-0.628319\pi$
$479$	−17.1716	−0.784589	−0.392295	+	0.919840i	$0.628319\pi$
$480$	0	0
$481$	40.9706	1.86810
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.82843	−0.219248
$486$	0	0
$487$	28.9706	1.31278	0.656391	−	0.754421i	$-0.272082\pi$
$487$	28.9706	1.31278	0.656391	+	0.754421i	$0.272082\pi$
$488$	0	0
$489$	5.65685	0.255812
$490$	0	0
$491$	24.8284	1.12049	0.560246	−	0.828327i	$-0.310707\pi$
$491$	24.8284	1.12049	0.560246	+	0.828327i	$0.310707\pi$
$492$	0	0
$493$	12.9706	0.584165
$494$	0	0
$495$	2.82843	0.127128
$496$	0	0
$497$	0	0
$498$	0	0
$499$	8.97056	0.401578	0.200789	−	0.979635i	$-0.435650\pi$
$499$	8.97056	0.401578	0.200789	+	0.979635i	$0.435650\pi$
$500$	0	0
$501$	−9.17157	−0.409756
$502$	0	0
$503$	−3.31371	−0.147751	−0.0738755	−	0.997267i	$-0.523537\pi$
$503$	−3.31371	−0.147751	−0.0738755	+	0.997267i	$0.523537\pi$
$504$	0	0
$505$	8.62742	0.383915
$506$	0	0
$507$	−5.00000	−0.222058
$508$	0	0
$509$	−0.384776	−0.0170549	−0.00852746	−	0.999964i	$-0.502714\pi$
$509$	−0.384776	−0.0170549	−0.00852746	+	0.999964i	$0.502714\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.17157	−0.0517262
$514$	0	0
$515$	−8.28427	−0.365049
$516$	0	0
$517$	60.2843	2.65130
$518$	0	0
$519$	−19.4142	−0.852189
$520$	0	0
$521$	5.75736	0.252234	0.126117	−	0.992015i	$-0.459748\pi$
$521$	5.75736	0.252234	0.126117	+	0.992015i	$0.459748\pi$
$522$	0	0
$523$	28.9706	1.26679	0.633397	−	0.773827i	$-0.281660\pi$
$523$	28.9706	1.26679	0.633397	+	0.773827i	$0.281660\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.9706	−0.565007
$528$	0	0
$529$	−22.3137	−0.970161
$530$	0	0
$531$	−8.48528	−0.368230
$532$	0	0
$533$	−7.45584	−0.322948
$534$	0	0
$535$	−4.20101	−0.181626
$536$	0	0
$537$	−10.4853	−0.452473
$538$	0	0
$539$	0	0
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	0	0
$543$	7.07107	0.303449
$544$	0	0
$545$	11.3137	0.484626
$546$	0	0
$547$	−36.9706	−1.58075	−0.790374	−	0.612625i	$-0.790114\pi$
$547$	−36.9706	−1.58075	−0.790374	+	0.612625i	$0.790114\pi$
$548$	0	0
$549$	3.07107	0.131070
$550$	0	0
$551$	3.31371	0.141169
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−5.65685	−0.240120
$556$	0	0
$557$	12.6274	0.535041	0.267520	−	0.963552i	$-0.413796\pi$
$557$	12.6274	0.535041	0.267520	+	0.963552i	$0.413796\pi$
$558$	0	0
$559$	−48.0000	−2.03018
$560$	0	0
$561$	22.1421	0.934842
$562$	0	0
$563$	30.1421	1.27034	0.635170	−	0.772373i	$-0.280930\pi$
$563$	30.1421	1.27034	0.635170	+	0.772373i	$0.280930\pi$
$564$	0	0
$565$	5.85786	0.246442
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−34.1421	−1.43131	−0.715656	−	0.698453i	$-0.753872\pi$
$569$	−34.1421	−1.43131	−0.715656	+	0.698453i	$0.753872\pi$
$570$	0	0
$571$	−30.3431	−1.26982	−0.634911	−	0.772586i	$-0.718963\pi$
$571$	−30.3431	−1.26982	−0.634911	+	0.772586i	$0.718963\pi$
$572$	0	0
$573$	−16.1421	−0.674347
$574$	0	0
$575$	3.85786	0.160884
$576$	0	0
$577$	−29.2132	−1.21616	−0.608081	−	0.793875i	$-0.708060\pi$
$577$	−29.2132	−1.21616	−0.608081	+	0.793875i	$0.708060\pi$
$578$	0	0
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−9.65685	−0.399946
$584$	0	0
$585$	2.48528	0.102754
$586$	0	0
$587$	3.79899	0.156801	0.0784005	−	0.996922i	$-0.475019\pi$
$587$	3.79899	0.156801	0.0784005	+	0.996922i	$0.475019\pi$
$588$	0	0
$589$	−3.31371	−0.136539
$590$	0	0
$591$	−25.3137	−1.04127
$592$	0	0
$593$	44.5858	1.83092	0.915459	−	0.402410i	$-0.131827\pi$
$593$	44.5858	1.83092	0.915459	+	0.402410i	$0.131827\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.65685	0.231520
$598$	0	0
$599$	27.4558	1.12182	0.560908	−	0.827878i	$-0.310452\pi$
$599$	27.4558	1.12182	0.560908	+	0.827878i	$0.310452\pi$
$600$	0	0
$601$	−3.75736	−0.153266	−0.0766329	−	0.997059i	$-0.524417\pi$
$601$	−3.75736	−0.153266	−0.0766329	+	0.997059i	$0.524417\pi$
$602$	0	0
$603$	−11.3137	−0.460730
$604$	0	0
$605$	−7.21320	−0.293258
$606$	0	0
$607$	8.97056	0.364104	0.182052	−	0.983289i	$-0.441726\pi$
$607$	8.97056	0.364104	0.182052	+	0.983289i	$0.441726\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	52.9706	2.14296
$612$	0	0
$613$	21.6569	0.874712	0.437356	−	0.899288i	$-0.355915\pi$
$613$	21.6569	0.874712	0.437356	+	0.899288i	$0.355915\pi$
$614$	0	0
$615$	1.02944	0.0415109
$616$	0	0
$617$	12.4853	0.502639	0.251319	−	0.967904i	$-0.419136\pi$
$617$	12.4853	0.502639	0.251319	+	0.967904i	$0.419136\pi$
$618$	0	0
$619$	22.3431	0.898047	0.449023	−	0.893520i	$-0.351772\pi$
$619$	22.3431	0.898047	0.449023	+	0.893520i	$0.351772\pi$
$620$	0	0
$621$	0.828427	0.0332436
$622$	0	0
$623$	0	0
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	5.65685	0.225913
$628$	0	0
$629$	−44.2843	−1.76573
$630$	0	0
$631$	36.9706	1.47177	0.735887	−	0.677104i	$-0.236765\pi$
$631$	36.9706	1.47177	0.735887	+	0.677104i	$0.236765\pi$
$632$	0	0
$633$	9.65685	0.383825
$634$	0	0
$635$	11.7157	0.464925
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.48528	−0.256554
$640$	0	0
$641$	−27.1127	−1.07089	−0.535444	−	0.844571i	$-0.679856\pi$
$641$	−27.1127	−1.07089	−0.535444	+	0.844571i	$0.679856\pi$
$642$	0	0
$643$	−7.79899	−0.307562	−0.153781	−	0.988105i	$-0.549145\pi$
$643$	−7.79899	−0.307562	−0.153781	+	0.988105i	$0.549145\pi$
$644$	0	0
$645$	6.62742	0.260954
$646$	0	0
$647$	39.7990	1.56466	0.782330	−	0.622864i	$-0.214031\pi$
$647$	39.7990	1.56466	0.782330	+	0.622864i	$0.214031\pi$
$648$	0	0
$649$	40.9706	1.60824
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−17.8579	−0.698832	−0.349416	−	0.936968i	$-0.613620\pi$
$653$	−17.8579	−0.698832	−0.349416	+	0.936968i	$0.613620\pi$
$654$	0	0
$655$	2.34315	0.0915543
$656$	0	0
$657$	16.2426	0.633686
$658$	0	0
$659$	38.4853	1.49917	0.749587	−	0.661906i	$-0.230252\pi$
$659$	38.4853	1.49917	0.749587	+	0.661906i	$0.230252\pi$
$660$	0	0
$661$	−19.0711	−0.741779	−0.370889	−	0.928677i	$-0.620947\pi$
$661$	−19.0711	−0.741779	−0.370889	+	0.928677i	$0.620947\pi$
$662$	0	0
$663$	19.4558	0.755602
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.34315	−0.0907270
$668$	0	0
$669$	2.34315	0.0905912
$670$	0	0
$671$	−14.8284	−0.572445
$672$	0	0
$673$	−0.686292	−0.0264546	−0.0132273	−	0.999913i	$-0.504211\pi$
$673$	−0.686292	−0.0264546	−0.0132273	+	0.999913i	$0.504211\pi$
$674$	0	0
$675$	4.65685	0.179242
$676$	0	0
$677$	31.6985	1.21827	0.609136	−	0.793066i	$-0.291516\pi$
$677$	31.6985	1.21827	0.609136	+	0.793066i	$0.291516\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	10.8284	0.414946
$682$	0	0
$683$	37.7990	1.44634	0.723169	−	0.690671i	$-0.242685\pi$
$683$	37.7990	1.44634	0.723169	+	0.690671i	$0.242685\pi$
$684$	0	0
$685$	8.68629	0.331886
$686$	0	0
$687$	−22.5858	−0.861701
$688$	0	0
$689$	−8.48528	−0.323263
$690$	0	0
$691$	31.3137	1.19123	0.595615	−	0.803270i	$-0.296908\pi$
$691$	31.3137	1.19123	0.595615	+	0.803270i	$0.296908\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.59798	−0.288208
$696$	0	0
$697$	8.05887	0.305252
$698$	0	0
$699$	9.17157	0.346901
$700$	0	0
$701$	−22.1421	−0.836297	−0.418148	−	0.908379i	$-0.637321\pi$
$701$	−22.1421	−0.836297	−0.418148	+	0.908379i	$0.637321\pi$
$702$	0	0
$703$	−11.3137	−0.426705
$704$	0	0
$705$	−7.31371	−0.275450
$706$	0	0
$707$	0	0
$708$	0	0
$709$	38.6274	1.45068	0.725342	−	0.688389i	$-0.241682\pi$
$709$	38.6274	1.45068	0.725342	+	0.688389i	$0.241682\pi$
$710$	0	0
$711$	−2.34315	−0.0878748
$712$	0	0
$713$	2.34315	0.0877515
$714$	0	0
$715$	−12.0000	−0.448775
$716$	0	0
$717$	−7.17157	−0.267827
$718$	0	0
$719$	−38.6274	−1.44056	−0.720280	−	0.693684i	$-0.755987\pi$
$719$	−38.6274	−1.44056	−0.720280	+	0.693684i	$0.755987\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	5.89949	0.219405
$724$	0	0
$725$	−13.1716	−0.489180
$726$	0	0
$727$	38.1421	1.41461	0.707307	−	0.706907i	$-0.249910\pi$
$727$	38.1421	1.41461	0.707307	+	0.706907i	$0.249910\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	51.8823	1.91893
$732$	0	0
$733$	−40.0416	−1.47897	−0.739486	−	0.673172i	$-0.764931\pi$
$733$	−40.0416	−1.47897	−0.739486	+	0.673172i	$0.764931\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	54.6274	2.01223
$738$	0	0
$739$	−0.970563	−0.0357027	−0.0178514	−	0.999841i	$-0.505683\pi$
$739$	−0.970563	−0.0357027	−0.0178514	+	0.999841i	$0.505683\pi$
$740$	0	0
$741$	4.97056	0.182598
$742$	0	0
$743$	−0.828427	−0.0303920	−0.0151960	−	0.999885i	$-0.504837\pi$
$743$	−0.828427	−0.0303920	−0.0151960	+	0.999885i	$0.504837\pi$
$744$	0	0
$745$	−12.4853	−0.457425
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	0	0
$750$	0	0
$751$	36.2843	1.32403	0.662016	−	0.749490i	$-0.269701\pi$
$751$	36.2843	1.32403	0.662016	+	0.749490i	$0.269701\pi$
$752$	0	0
$753$	−22.1421	−0.806904
$754$	0	0
$755$	0.970563	0.0353224
$756$	0	0
$757$	25.9411	0.942846	0.471423	−	0.881907i	$-0.343740\pi$
$757$	25.9411	0.942846	0.471423	+	0.881907i	$0.343740\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	−12.1005	−0.438643	−0.219321	−	0.975653i	$-0.570384\pi$
$761$	−12.1005	−0.438643	−0.219321	+	0.975653i	$0.570384\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2.68629	−0.0971231
$766$	0	0
$767$	36.0000	1.29988
$768$	0	0
$769$	−18.8701	−0.680472	−0.340236	−	0.940340i	$-0.610507\pi$
$769$	−18.8701	−0.680472	−0.340236	+	0.940340i	$0.610507\pi$
$770$	0	0
$771$	0.585786	0.0210966
$772$	0	0
$773$	37.3553	1.34358	0.671789	−	0.740742i	$-0.265526\pi$
$773$	37.3553	1.34358	0.671789	+	0.740742i	$0.265526\pi$
$774$	0	0
$775$	13.1716	0.473137
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.05887	0.0737668
$780$	0	0
$781$	31.3137	1.12049
$782$	0	0
$783$	−2.82843	−0.101080
$784$	0	0
$785$	−4.82843	−0.172334
$786$	0	0
$787$	−7.31371	−0.260706	−0.130353	−	0.991468i	$-0.541611\pi$
$787$	−7.31371	−0.260706	−0.130353	+	0.991468i	$0.541611\pi$
$788$	0	0
$789$	8.14214	0.289868
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−13.0294	−0.462689
$794$	0	0
$795$	1.17157	0.0415514
$796$	0	0
$797$	36.5858	1.29594	0.647968	−	0.761668i	$-0.275619\pi$
$797$	36.5858	1.29594	0.647968	+	0.761668i	$0.275619\pi$
$798$	0	0
$799$	−57.2548	−2.02553
$800$	0	0
$801$	14.2426	0.503239
$802$	0	0
$803$	−78.4264	−2.76761
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−19.8995	−0.700495
$808$	0	0
$809$	40.6274	1.42838	0.714192	−	0.699950i	$-0.246794\pi$
$809$	40.6274	1.42838	0.714192	+	0.699950i	$0.246794\pi$
$810$	0	0
$811$	14.3431	0.503656	0.251828	−	0.967772i	$-0.418968\pi$
$811$	14.3431	0.503656	0.251828	+	0.967772i	$0.418968\pi$
$812$	0	0
$813$	−22.1421	−0.776559
$814$	0	0
$815$	3.31371	0.116074
$816$	0	0
$817$	13.2548	0.463728
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33.3137	1.16266	0.581328	−	0.813669i	$-0.302533\pi$
$821$	33.3137	1.16266	0.581328	+	0.813669i	$0.302533\pi$
$822$	0	0
$823$	−8.97056	−0.312694	−0.156347	−	0.987702i	$-0.549972\pi$
$823$	−8.97056	−0.312694	−0.156347	+	0.987702i	$0.549972\pi$
$824$	0	0
$825$	−22.4853	−0.782837
$826$	0	0
$827$	11.8579	0.412338	0.206169	−	0.978516i	$-0.433900\pi$
$827$	11.8579	0.412338	0.206169	+	0.978516i	$0.433900\pi$
$828$	0	0
$829$	2.38478	0.0828267	0.0414134	−	0.999142i	$-0.486814\pi$
$829$	2.38478	0.0828267	0.0414134	+	0.999142i	$0.486814\pi$
$830$	0	0
$831$	16.6274	0.576799
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5.37258	−0.185926
$836$	0	0
$837$	2.82843	0.0977647
$838$	0	0
$839$	15.7990	0.545442	0.272721	−	0.962093i	$-0.412076\pi$
$839$	15.7990	0.545442	0.272721	+	0.962093i	$0.412076\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	−6.82843	−0.235184
$844$	0	0
$845$	−2.92893	−0.100758
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.14214	−0.0735179
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	−19.0711	−0.652981	−0.326490	−	0.945200i	$-0.605866\pi$
$853$	−19.0711	−0.652981	−0.326490	+	0.945200i	$0.605866\pi$
$854$	0	0
$855$	−0.686292	−0.0234707
$856$	0	0
$857$	31.2132	1.06622	0.533111	−	0.846045i	$-0.321023\pi$
$857$	31.2132	1.06622	0.533111	+	0.846045i	$0.321023\pi$
$858$	0	0
$859$	53.4558	1.82389	0.911945	−	0.410313i	$-0.134580\pi$
$859$	53.4558	1.82389	0.911945	+	0.410313i	$0.134580\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	42.0833	1.43253	0.716265	−	0.697828i	$-0.245850\pi$
$863$	42.0833	1.43253	0.716265	+	0.697828i	$0.245850\pi$
$864$	0	0
$865$	−11.3726	−0.386679
$866$	0	0
$867$	−4.02944	−0.136847
$868$	0	0
$869$	11.3137	0.383791
$870$	0	0
$871$	48.0000	1.62642
$872$	0	0
$873$	8.24264	0.278971
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−28.2843	−0.955092	−0.477546	−	0.878607i	$-0.658474\pi$
$877$	−28.2843	−0.955092	−0.477546	+	0.878607i	$0.658474\pi$
$878$	0	0
$879$	−10.2426	−0.345476
$880$	0	0
$881$	−18.0416	−0.607838	−0.303919	−	0.952698i	$-0.598295\pi$
$881$	−18.0416	−0.607838	−0.303919	+	0.952698i	$0.598295\pi$
$882$	0	0
$883$	−21.6569	−0.728811	−0.364406	−	0.931240i	$-0.618728\pi$
$883$	−21.6569	−0.728811	−0.364406	+	0.931240i	$0.618728\pi$
$884$	0	0
$885$	−4.97056	−0.167084
$886$	0	0
$887$	0.201010	0.00674926	0.00337463	−	0.999994i	$-0.498926\pi$
$887$	0.201010	0.00674926	0.00337463	+	0.999994i	$0.498926\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.82843	−0.161758
$892$	0	0
$893$	−14.6274	−0.489488
$894$	0	0
$895$	−6.14214	−0.205309
$896$	0	0
$897$	−3.51472	−0.117353
$898$	0	0
$899$	−8.00000	−0.266815
$900$	0	0
$901$	9.17157	0.305549
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.14214	0.137689
$906$	0	0
$907$	42.3431	1.40598	0.702991	−	0.711199i	$-0.251848\pi$
$907$	42.3431	1.40598	0.702991	+	0.711199i	$0.251848\pi$
$908$	0	0
$909$	−14.7279	−0.488494
$910$	0	0
$911$	10.4853	0.347393	0.173696	−	0.984799i	$-0.444429\pi$
$911$	10.4853	0.347393	0.173696	+	0.984799i	$0.444429\pi$
$912$	0	0
$913$	19.3137	0.639190
$914$	0	0
$915$	1.79899	0.0594728
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−8.68629	−0.286534	−0.143267	−	0.989684i	$-0.545761\pi$
$919$	−8.68629	−0.286534	−0.143267	+	0.989684i	$0.545761\pi$
$920$	0	0
$921$	28.4853	0.938622
$922$	0	0
$923$	27.5147	0.905658
$924$	0	0
$925$	44.9706	1.47862
$926$	0	0
$927$	14.1421	0.464489
$928$	0	0
$929$	3.89949	0.127938	0.0639691	−	0.997952i	$-0.479624\pi$
$929$	3.89949	0.127938	0.0639691	+	0.997952i	$0.479624\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−15.7990	−0.517236
$934$	0	0
$935$	12.9706	0.424183
$936$	0	0
$937$	−19.3553	−0.632311	−0.316156	−	0.948707i	$-0.602392\pi$
$937$	−19.3553	−0.632311	−0.316156	+	0.948707i	$0.602392\pi$
$938$	0	0
$939$	−3.27208	−0.106780
$940$	0	0
$941$	−10.7279	−0.349720	−0.174860	−	0.984593i	$-0.555947\pi$
$941$	−10.7279	−0.349720	−0.174860	+	0.984593i	$0.555947\pi$
$942$	0	0
$943$	−1.45584	−0.0474088
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.7696	0.479946	0.239973	−	0.970780i	$-0.422861\pi$
$947$	14.7696	0.479946	0.239973	+	0.970780i	$0.422861\pi$
$948$	0	0
$949$	−68.9117	−2.23697
$950$	0	0
$951$	−22.0000	−0.713399
$952$	0	0
$953$	2.00000	0.0647864	0.0323932	−	0.999475i	$-0.489687\pi$
$953$	2.00000	0.0647864	0.0323932	+	0.999475i	$0.489687\pi$
$954$	0	0
$955$	−9.45584	−0.305984
$956$	0	0
$957$	13.6569	0.441463
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−23.0000	−0.741935
$962$	0	0
$963$	7.17157	0.231101
$964$	0	0
$965$	−1.17157	−0.0377143
$966$	0	0
$967$	8.68629	0.279332	0.139666	−	0.990199i	$-0.455397\pi$
$967$	8.68629	0.279332	0.139666	+	0.990199i	$0.455397\pi$
$968$	0	0
$969$	−5.37258	−0.172592
$970$	0	0
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−19.7574	−0.632742
$976$	0	0
$977$	31.7990	1.01734	0.508670	−	0.860962i	$-0.330137\pi$
$977$	31.7990	1.01734	0.508670	+	0.860962i	$0.330137\pi$
$978$	0	0
$979$	−68.7696	−2.19788
$980$	0	0
$981$	−19.3137	−0.616639
$982$	0	0
$983$	−46.6274	−1.48718	−0.743592	−	0.668634i	$-0.766879\pi$
$983$	−46.6274	−1.48718	−0.743592	+	0.668634i	$0.766879\pi$
$984$	0	0
$985$	−14.8284	−0.472473
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.37258	−0.298031
$990$	0	0
$991$	−26.6274	−0.845848	−0.422924	−	0.906165i	$-0.638996\pi$
$991$	−26.6274	−0.845848	−0.422924	+	0.906165i	$0.638996\pi$
$992$	0	0
$993$	15.3137	0.485966
$994$	0	0
$995$	3.31371	0.105052
$996$	0	0
$997$	17.6152	0.557880	0.278940	−	0.960309i	$-0.410017\pi$
$997$	17.6152	0.557880	0.278940	+	0.960309i	$0.410017\pi$
$998$	0	0
$999$	9.65685	0.305529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9408.2.a.dh.1.2		2
4.3	odd	2		9408.2.a.dv.1.2		2
7.6	odd	2		9408.2.a.ed.1.1		2
8.3	odd	2		1176.2.a.l.1.1	✓	2
8.5	even	2		2352.2.a.bg.1.1		2
24.5	odd	2		7056.2.a.ce.1.2		2
24.11	even	2		3528.2.a.bc.1.2		2
28.27	even	2		9408.2.a.dr.1.1		2
56.3	even	6		1176.2.q.m.961.1		4
56.5	odd	6		2352.2.q.bg.1537.1		4
56.11	odd	6		1176.2.q.n.961.2		4
56.13	odd	2		2352.2.a.z.1.2		2
56.19	even	6		1176.2.q.m.361.1		4
56.27	even	2		1176.2.a.m.1.2	yes	2
56.37	even	6		2352.2.q.ba.1537.2		4
56.45	odd	6		2352.2.q.bg.961.1		4
56.51	odd	6		1176.2.q.n.361.2		4
56.53	even	6		2352.2.q.ba.961.2		4
168.11	even	6		3528.2.s.bl.3313.1		4
168.59	odd	6		3528.2.s.bc.3313.2		4
168.83	odd	2		3528.2.a.bm.1.1		2
168.107	even	6		3528.2.s.bl.361.1		4
168.125	even	2		7056.2.a.cw.1.1		2
168.131	odd	6		3528.2.s.bc.361.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.2.a.l.1.1	✓	2	8.3	odd	2
1176.2.a.m.1.2	yes	2	56.27	even	2
1176.2.q.m.361.1		4	56.19	even	6
1176.2.q.m.961.1		4	56.3	even	6
1176.2.q.n.361.2		4	56.51	odd	6
1176.2.q.n.961.2		4	56.11	odd	6
2352.2.a.z.1.2		2	56.13	odd	2
2352.2.a.bg.1.1		2	8.5	even	2
2352.2.q.ba.961.2		4	56.53	even	6
2352.2.q.ba.1537.2		4	56.37	even	6
2352.2.q.bg.961.1		4	56.45	odd	6
2352.2.q.bg.1537.1		4	56.5	odd	6
3528.2.a.bc.1.2		2	24.11	even	2
3528.2.a.bm.1.1		2	168.83	odd	2
3528.2.s.bc.361.2		4	168.131	odd	6
3528.2.s.bc.3313.2		4	168.59	odd	6
3528.2.s.bl.361.1		4	168.107	even	6
3528.2.s.bl.3313.1		4	168.11	even	6
7056.2.a.ce.1.2		2	24.5	odd	2
7056.2.a.cw.1.1		2	168.125	even	2
9408.2.a.dh.1.2		2	1.1	even	1	trivial
9408.2.a.dr.1.1		2	28.27	even	2
9408.2.a.dv.1.2		2	4.3	odd	2
9408.2.a.ed.1.1		2	7.6	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:	\( N \)	\(=\)	\( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9408.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.585786 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.585786 q^{5} +1.00000 q^{9} -4.82843 q^{11} -4.24264 q^{13} +0.585786 q^{15} +4.58579 q^{17} +1.17157 q^{19} -0.828427 q^{23} -4.65685 q^{25} -1.00000 q^{27} +2.82843 q^{29} -2.82843 q^{31} +4.82843 q^{33} -9.65685 q^{37} +4.24264 q^{39} +1.75736 q^{41} +11.3137 q^{43} -0.585786 q^{45} -12.4853 q^{47} -4.58579 q^{51} +2.00000 q^{53} +2.82843 q^{55} -1.17157 q^{57} -8.48528 q^{59} +3.07107 q^{61} +2.48528 q^{65} -11.3137 q^{67} +0.828427 q^{69} -6.48528 q^{71} +16.2426 q^{73} +4.65685 q^{75} -2.34315 q^{79} +1.00000 q^{81} -4.00000 q^{83} -2.68629 q^{85} -2.82843 q^{87} +14.2426 q^{89} +2.82843 q^{93} -0.686292 q^{95} +8.24264 q^{97} -4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} - 4 q^{11} + 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{27} + 4 q^{33} - 8 q^{37} + 12 q^{41} - 4 q^{45} - 8 q^{47} - 12 q^{51} + 4 q^{53} - 8 q^{57} - 8 q^{61} - 12 q^{65} - 4 q^{69} + 4 q^{71} + 24 q^{73} - 2 q^{75} - 16 q^{79} + 2 q^{81} - 8 q^{83} - 28 q^{85} + 20 q^{89} - 24 q^{95} + 8 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 - 4 * q^11 + 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 2 * q^25 - 2 * q^27 + 4 * q^33 - 8 * q^37 + 12 * q^41 - 4 * q^45 - 8 * q^47 - 12 * q^51 + 4 * q^53 - 8 * q^57 - 8 * q^61 - 12 * q^65 - 4 * q^69 + 4 * q^71 + 24 * q^73 - 2 * q^75 - 16 * q^79 + 2 * q^81 - 8 * q^83 - 28 * q^85 + 20 * q^89 - 24 * q^95 + 8 * q^97 - 4 * q^99

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