LMFDB - Embedded newform 9216.2.a.bc.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9216,2,Mod(1,9216)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9216.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9216 = 2^{10} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9216.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,16,0,0,0,0,0,12,0,0,0,0, 0,0,0,0,0,24,0,0,0,0,0,-32,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,-24,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(67)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$73.5901305028$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 768)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.517638$ of defining polynomial
Character	$\chi$	$=$	9216.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.03528 q^{5} +2.44949 q^{7} -5.46410 q^{11} +4.24264 q^{13} -3.46410 q^{17} +0.535898 q^{19} +2.82843 q^{23} -3.92820 q^{25} +5.93426 q^{29} +7.34847 q^{31} +2.53590 q^{35} -9.14162 q^{37} -11.4641 q^{41} +3.46410 q^{43} -2.82843 q^{47} -1.00000 q^{49} +9.52056 q^{53} -5.65685 q^{55} +13.8564 q^{59} +9.14162 q^{61} +4.39230 q^{65} +1.07180 q^{67} +16.2127 q^{71} +4.00000 q^{73} -13.3843 q^{77} -2.44949 q^{79} +1.46410 q^{83} -3.58630 q^{85} -8.92820 q^{89} +10.3923 q^{91} +0.554803 q^{95} +14.9282 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{11} + 16 q^{19} + 12 q^{25} + 24 q^{35} - 32 q^{41} - 4 q^{49} - 24 q^{65} + 32 q^{67} + 16 q^{73} - 8 q^{83} - 8 q^{89} + 32 q^{97}+O(q^{100}) $ 4 * q - 8 * q^11 + 16 * q^19 + 12 * q^25 + 24 * q^35 - 32 * q^41 - 4 * q^49 - 24 * q^65 + 32 * q^67 + 16 * q^73 - 8 * q^83 - 8 * q^89 + 32 * q^97

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
$5$	1.03528	0.462990	0.231495	−	0.972836i	$-0.425638\pi$
$5$	1.03528	0.462990	0.231495	+	0.972836i	$0.425638\pi$
$6$	0	0
$7$	2.44949	0.925820	0.462910	−	0.886405i	$-0.346805\pi$
$7$	2.44949	0.925820	0.462910	+	0.886405i	$0.346805\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.46410	−1.64749	−0.823744	−	0.566961i	$-0.808119\pi$
$11$	−5.46410	−1.64749	−0.823744	+	0.566961i	$0.808119\pi$
$12$	0	0
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	0.535898	0.122944	0.0614718	−	0.998109i	$-0.480421\pi$
$19$	0.535898	0.122944	0.0614718	+	0.998109i	$0.480421\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	−3.92820	−0.785641
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.93426	1.10196	0.550982	−	0.834517i	$-0.314253\pi$
$29$	5.93426	1.10196	0.550982	+	0.834517i	$0.314253\pi$
$30$	0	0
$31$	7.34847	1.31982	0.659912	−	0.751343i	$-0.270594\pi$
$31$	7.34847	1.31982	0.659912	+	0.751343i	$0.270594\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.53590	0.428645
$36$	0	0
$37$	−9.14162	−1.50287	−0.751437	−	0.659805i	$-0.770639\pi$
$37$	−9.14162	−1.50287	−0.751437	+	0.659805i	$0.770639\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.4641	−1.79039	−0.895196	−	0.445673i	$-0.852964\pi$
$41$	−11.4641	−1.79039	−0.895196	+	0.445673i	$0.852964\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.52056	1.30775	0.653875	−	0.756603i	$-0.273142\pi$
$53$	9.52056	1.30775	0.653875	+	0.756603i	$0.273142\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.8564	1.80395	0.901975	−	0.431788i	$-0.142117\pi$
$59$	13.8564	1.80395	0.901975	+	0.431788i	$0.142117\pi$
$60$	0	0
$61$	9.14162	1.17046	0.585232	−	0.810866i	$-0.301003\pi$
$61$	9.14162	1.17046	0.585232	+	0.810866i	$0.301003\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.39230	0.544798
$66$	0	0
$67$	1.07180	0.130941	0.0654704	−	0.997855i	$-0.479145\pi$
$67$	1.07180	0.130941	0.0654704	+	0.997855i	$0.479145\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	16.2127	1.92409	0.962046	−	0.272887i	$-0.0879786\pi$
$71$	16.2127	1.92409	0.962046	+	0.272887i	$0.0879786\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13.3843	−1.52528
$78$	0	0
$79$	−2.44949	−0.275589	−0.137795	−	0.990461i	$-0.544001\pi$
$79$	−2.44949	−0.275589	−0.137795	+	0.990461i	$0.544001\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.46410	0.160706	0.0803530	−	0.996766i	$-0.474395\pi$
$83$	1.46410	0.160706	0.0803530	+	0.996766i	$0.474395\pi$
$84$	0	0
$85$	−3.58630	−0.388989
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.92820	−0.946388	−0.473194	−	0.880958i	$-0.656899\pi$
$89$	−8.92820	−0.946388	−0.473194	+	0.880958i	$0.656899\pi$
$90$	0	0
$91$	10.3923	1.08941
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.554803	0.0569216
$96$	0	0
$97$	14.9282	1.51573	0.757865	−	0.652412i	$-0.226243\pi$
$97$	14.9282	1.51573	0.757865	+	0.652412i	$0.226243\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.277401	−0.0276025	−0.0138012	−	0.999905i	$-0.504393\pi$
$101$	−0.277401	−0.0276025	−0.0138012	+	0.999905i	$0.504393\pi$
$102$	0	0
$103$	8.10634	0.798742	0.399371	−	0.916789i	$-0.369229\pi$
$103$	8.10634	0.798742	0.399371	+	0.916789i	$0.369229\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	1.41421	0.135457	0.0677285	−	0.997704i	$-0.478425\pi$
$109$	1.41421	0.135457	0.0677285	+	0.997704i	$0.478425\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	2.92820	0.273056
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−8.48528	−0.777844
$120$	0	0
$121$	18.8564	1.71422
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.24316	−0.826733
$126$	0	0
$127$	−19.4201	−1.72325	−0.861625	−	0.507545i	$-0.830553\pi$
$127$	−19.4201	−1.72325	−0.861625	+	0.507545i	$0.830553\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.07180	0.0936433	0.0468217	−	0.998903i	$-0.485091\pi$
$131$	1.07180	0.0936433	0.0468217	+	0.998903i	$0.485091\pi$
$132$	0	0
$133$	1.31268	0.113824
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.39230	0.546131	0.273066	−	0.961995i	$-0.411962\pi$
$137$	6.39230	0.546131	0.273066	+	0.961995i	$0.411962\pi$
$138$	0	0
$139$	−6.92820	−0.587643	−0.293821	−	0.955860i	$-0.594927\pi$
$139$	−6.92820	−0.587643	−0.293821	+	0.955860i	$0.594927\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−23.1822	−1.93859
$144$	0	0
$145$	6.14359	0.510198
$146$	0	0
$147$	0	0
$148$	0	0
$149$	20.0764	1.64472	0.822361	−	0.568966i	$-0.192656\pi$
$149$	20.0764	1.64472	0.822361	+	0.568966i	$0.192656\pi$
$150$	0	0
$151$	17.9043	1.45703	0.728516	−	0.685029i	$-0.240211\pi$
$151$	17.9043	1.45703	0.728516	+	0.685029i	$0.240211\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	7.60770	0.611065
$156$	0	0
$157$	11.9700	0.955314	0.477657	−	0.878546i	$-0.341486\pi$
$157$	11.9700	0.955314	0.477657	+	0.878546i	$0.341486\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.92820	0.546019
$162$	0	0
$163$	11.4641	0.897938	0.448969	−	0.893547i	$-0.351791\pi$
$163$	11.4641	0.897938	0.448969	+	0.893547i	$0.351791\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.07055	0.160224	0.0801121	−	0.996786i	$-0.474472\pi$
$167$	2.07055	0.160224	0.0801121	+	0.996786i	$0.474472\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2.55103	0.193951	0.0969754	−	0.995287i	$-0.469083\pi$
$173$	2.55103	0.193951	0.0969754	+	0.995287i	$0.469083\pi$
$174$	0	0
$175$	−9.62209	−0.727362
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.07180	0.379084	0.189542	−	0.981873i	$-0.439300\pi$
$179$	5.07180	0.379084	0.189542	+	0.981873i	$0.439300\pi$
$180$	0	0
$181$	7.07107	0.525588	0.262794	−	0.964852i	$-0.415356\pi$
$181$	7.07107	0.525588	0.262794	+	0.964852i	$0.415356\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−9.46410	−0.695815
$186$	0	0
$187$	18.9282	1.38417
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.07055	0.149820	0.0749100	−	0.997190i	$-0.476133\pi$
$191$	2.07055	0.149820	0.0749100	+	0.997190i	$0.476133\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.93426	−0.422798	−0.211399	−	0.977400i	$-0.567802\pi$
$197$	−5.93426	−0.422798	−0.211399	+	0.977400i	$0.567802\pi$
$198$	0	0
$199$	−3.96524	−0.281088	−0.140544	−	0.990074i	$-0.544885\pi$
$199$	−3.96524	−0.281088	−0.140544	+	0.990074i	$0.544885\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	14.5359	1.02022
$204$	0	0
$205$	−11.8685	−0.828933
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.92820	−0.202548
$210$	0	0
$211$	−6.92820	−0.476957	−0.238479	−	0.971148i	$-0.576649\pi$
$211$	−6.92820	−0.476957	−0.238479	+	0.971148i	$0.576649\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.58630	0.244584
$216$	0	0
$217$	18.0000	1.22192
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−14.6969	−0.988623
$222$	0	0
$223$	19.4201	1.30046	0.650231	−	0.759736i	$-0.274672\pi$
$223$	19.4201	1.30046	0.650231	+	0.759736i	$0.274672\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.3205	1.54784	0.773918	−	0.633286i	$-0.218294\pi$
$227$	23.3205	1.54784	0.773918	+	0.633286i	$0.218294\pi$
$228$	0	0
$229$	2.72689	0.180198	0.0900990	−	0.995933i	$-0.471282\pi$
$229$	2.72689	0.180198	0.0900990	+	0.995933i	$0.471282\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.928203	0.0608086	0.0304043	−	0.999538i	$-0.490321\pi$
$233$	0.928203	0.0608086	0.0304043	+	0.999538i	$0.490321\pi$
$234$	0	0
$235$	−2.92820	−0.191015
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.07055	−0.133933	−0.0669664	−	0.997755i	$-0.521332\pi$
$239$	−2.07055	−0.133933	−0.0669664	+	0.997755i	$0.521332\pi$
$240$	0	0
$241$	14.7846	0.952360	0.476180	−	0.879348i	$-0.342021\pi$
$241$	14.7846	0.952360	0.476180	+	0.879348i	$0.342021\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.03528	−0.0661414
$246$	0	0
$247$	2.27362	0.144667
$248$	0	0
$249$	0	0
$250$	0	0
$251$	13.4641	0.849847	0.424923	−	0.905229i	$-0.360301\pi$
$251$	13.4641	0.849847	0.424923	+	0.905229i	$0.360301\pi$
$252$	0	0
$253$	−15.4548	−0.971636
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.7846	−1.17175	−0.585876	−	0.810401i	$-0.699249\pi$
$257$	−18.7846	−1.17175	−0.585876	+	0.810401i	$0.699249\pi$
$258$	0	0
$259$	−22.3923	−1.39139
$260$	0	0
$261$	0	0
$262$	0	0
$263$	22.6274	1.39527	0.697633	−	0.716455i	$-0.254237\pi$
$263$	22.6274	1.39527	0.697633	+	0.716455i	$0.254237\pi$
$264$	0	0
$265$	9.85641	0.605474
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.1774	−0.925383	−0.462692	−	0.886519i	$-0.653116\pi$
$269$	−15.1774	−0.925383	−0.462692	+	0.886519i	$0.653116\pi$
$270$	0	0
$271$	8.10634	0.492425	0.246213	−	0.969216i	$-0.420814\pi$
$271$	8.10634	0.492425	0.246213	+	0.969216i	$0.420814\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	21.4641	1.29433
$276$	0	0
$277$	−26.6670	−1.60226	−0.801132	−	0.598488i	$-0.795769\pi$
$277$	−26.6670	−1.60226	−0.801132	+	0.598488i	$0.795769\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.9282	0.771232	0.385616	−	0.922659i	$-0.373989\pi$
$281$	12.9282	0.771232	0.385616	+	0.922659i	$0.373989\pi$
$282$	0	0
$283$	−17.8564	−1.06145	−0.530727	−	0.847543i	$-0.678081\pi$
$283$	−17.8564	−1.06145	−0.530727	+	0.847543i	$0.678081\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−28.0812	−1.65758
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	13.6617	0.798123	0.399061	−	0.916924i	$-0.369336\pi$
$293$	13.6617	0.798123	0.399061	+	0.916924i	$0.369336\pi$
$294$	0	0
$295$	14.3452	0.835210
$296$	0	0
$297$	0	0
$298$	0	0
$299$	12.0000	0.693978
$300$	0	0
$301$	8.48528	0.489083
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.46410	0.541913
$306$	0	0
$307$	9.07180	0.517755	0.258877	−	0.965910i	$-0.416647\pi$
$307$	9.07180	0.517755	0.258877	+	0.965910i	$0.416647\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	12.8295	0.727492	0.363746	−	0.931498i	$-0.381498\pi$
$311$	12.8295	0.727492	0.363746	+	0.931498i	$0.381498\pi$
$312$	0	0
$313$	−23.8564	−1.34844	−0.674222	−	0.738529i	$-0.735521\pi$
$313$	−23.8564	−1.34844	−0.674222	+	0.738529i	$0.735521\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.79315	−0.100713	−0.0503567	−	0.998731i	$-0.516036\pi$
$317$	−1.79315	−0.100713	−0.0503567	+	0.998731i	$0.516036\pi$
$318$	0	0
$319$	−32.4254	−1.81547
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.85641	−0.103293
$324$	0	0
$325$	−16.6660	−0.924461
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−6.92820	−0.381964
$330$	0	0
$331$	14.9282	0.820528	0.410264	−	0.911967i	$-0.365437\pi$
$331$	14.9282	0.820528	0.410264	+	0.911967i	$0.365437\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.10961	0.0606242
$336$	0	0
$337$	−17.8564	−0.972700	−0.486350	−	0.873764i	$-0.661672\pi$
$337$	−17.8564	−0.972700	−0.486350	+	0.873764i	$0.661672\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−40.1528	−2.17440
$342$	0	0
$343$	−19.5959	−1.05808
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.46410	−0.508060	−0.254030	−	0.967196i	$-0.581756\pi$
$347$	−9.46410	−0.508060	−0.254030	+	0.967196i	$0.581756\pi$
$348$	0	0
$349$	−1.96902	−0.105399	−0.0526995	−	0.998610i	$-0.516783\pi$
$349$	−1.96902	−0.105399	−0.0526995	+	0.998610i	$0.516783\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−12.9282	−0.688099	−0.344049	−	0.938952i	$-0.611799\pi$
$353$	−12.9282	−0.688099	−0.344049	+	0.938952i	$0.611799\pi$
$354$	0	0
$355$	16.7846	0.890835
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.1518	−1.59135	−0.795674	−	0.605725i	$-0.792883\pi$
$359$	−30.1518	−1.59135	−0.795674	+	0.605725i	$0.792883\pi$
$360$	0	0
$361$	−18.7128	−0.984885
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.14110	0.216755
$366$	0	0
$367$	−27.7023	−1.44605	−0.723023	−	0.690824i	$-0.757248\pi$
$367$	−27.7023	−1.44605	−0.723023	+	0.690824i	$0.757248\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	23.3205	1.21074
$372$	0	0
$373$	16.1112	0.834204	0.417102	−	0.908860i	$-0.363046\pi$
$373$	16.1112	0.834204	0.417102	+	0.908860i	$0.363046\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	25.1769	1.29668
$378$	0	0
$379$	−21.3205	−1.09516	−0.547580	−	0.836753i	$-0.684451\pi$
$379$	−21.3205	−1.09516	−0.547580	+	0.836753i	$0.684451\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−0.554803	−0.0283491	−0.0141746	−	0.999900i	$-0.504512\pi$
$383$	−0.554803	−0.0283491	−0.0141746	+	0.999900i	$0.504512\pi$
$384$	0	0
$385$	−13.8564	−0.706188
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4.62158	−0.234323	−0.117162	−	0.993113i	$-0.537380\pi$
$389$	−4.62158	−0.234323	−0.117162	+	0.993113i	$0.537380\pi$
$390$	0	0
$391$	−9.79796	−0.495504
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.53590	−0.127595
$396$	0	0
$397$	25.9091	1.30034	0.650171	−	0.759788i	$-0.274697\pi$
$397$	25.9091	1.30034	0.650171	+	0.759788i	$0.274697\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.39230	−0.119466	−0.0597330	−	0.998214i	$-0.519025\pi$
$401$	−2.39230	−0.119466	−0.0597330	+	0.998214i	$0.519025\pi$
$402$	0	0
$403$	31.1769	1.55303
$404$	0	0
$405$	0	0
$406$	0	0
$407$	49.9507	2.47597
$408$	0	0
$409$	−22.9282	−1.13373	−0.566863	−	0.823812i	$-0.691843\pi$
$409$	−22.9282	−1.13373	−0.566863	+	0.823812i	$0.691843\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	33.9411	1.67013
$414$	0	0
$415$	1.51575	0.0744052
$416$	0	0
$417$	0	0
$418$	0	0
$419$	20.3923	0.996229	0.498115	−	0.867111i	$-0.334026\pi$
$419$	20.3923	0.996229	0.498115	+	0.867111i	$0.334026\pi$
$420$	0	0
$421$	0.101536	0.00494856	0.00247428	−	0.999997i	$-0.499212\pi$
$421$	0.101536	0.00494856	0.00247428	+	0.999997i	$0.499212\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13.6077	0.660070
$426$	0	0
$427$	22.3923	1.08364
$428$	0	0
$429$	0	0
$430$	0	0
$431$	23.9401	1.15315	0.576577	−	0.817043i	$-0.304388\pi$
$431$	23.9401	1.15315	0.576577	+	0.817043i	$0.304388\pi$
$432$	0	0
$433$	−2.14359	−0.103015	−0.0515073	−	0.998673i	$-0.516403\pi$
$433$	−2.14359	−0.103015	−0.0515073	+	0.998673i	$0.516403\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.51575	0.0725081
$438$	0	0
$439$	28.4601	1.35833	0.679164	−	0.733986i	$-0.262342\pi$
$439$	28.4601	1.35833	0.679164	+	0.733986i	$0.262342\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.679492	0.0322836	0.0161418	−	0.999870i	$-0.494862\pi$
$443$	0.679492	0.0322836	0.0161418	+	0.999870i	$0.494862\pi$
$444$	0	0
$445$	−9.24316	−0.438168
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14.3923	−0.679215	−0.339607	−	0.940567i	$-0.610294\pi$
$449$	−14.3923	−0.679215	−0.339607	+	0.940567i	$0.610294\pi$
$450$	0	0
$451$	62.6410	2.94965
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.7589	0.504385
$456$	0	0
$457$	20.7846	0.972263	0.486132	−	0.873886i	$-0.338408\pi$
$457$	20.7846	0.972263	0.486132	+	0.873886i	$0.338408\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.7017	1.05733	0.528663	−	0.848832i	$-0.322694\pi$
$461$	22.7017	1.05733	0.528663	+	0.848832i	$0.322694\pi$
$462$	0	0
$463$	4.72311	0.219502	0.109751	−	0.993959i	$-0.464995\pi$
$463$	4.72311	0.219502	0.109751	+	0.993959i	$0.464995\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.2487	−0.844450	−0.422225	−	0.906491i	$-0.638751\pi$
$467$	−18.2487	−0.844450	−0.422225	+	0.906491i	$0.638751\pi$
$468$	0	0
$469$	2.62536	0.121228
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−18.9282	−0.870320
$474$	0	0
$475$	−2.10512	−0.0965894
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−38.8401	−1.77465	−0.887325	−	0.461145i	$-0.847439\pi$
$479$	−38.8401	−1.77465	−0.887325	+	0.461145i	$0.847439\pi$
$480$	0	0
$481$	−38.7846	−1.76843
$482$	0	0
$483$	0	0
$484$	0	0
$485$	15.4548	0.701767
$486$	0	0
$487$	22.8033	1.03332	0.516658	−	0.856192i	$-0.327176\pi$
$487$	22.8033	1.03332	0.516658	+	0.856192i	$0.327176\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.8564	0.986366	0.493183	−	0.869926i	$-0.335833\pi$
$491$	21.8564	0.986366	0.493183	+	0.869926i	$0.335833\pi$
$492$	0	0
$493$	−20.5569	−0.925835
$494$	0	0
$495$	0	0
$496$	0	0
$497$	39.7128	1.78136
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	33.1833	1.47957	0.739784	−	0.672844i	$-0.234928\pi$
$503$	33.1833	1.47957	0.739784	+	0.672844i	$0.234928\pi$
$504$	0	0
$505$	−0.287187	−0.0127797
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−27.0459	−1.19879	−0.599395	−	0.800454i	$-0.704592\pi$
$509$	−27.0459	−1.19879	−0.599395	+	0.800454i	$0.704592\pi$
$510$	0	0
$511$	9.79796	0.433436
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.39230	0.369809
$516$	0	0
$517$	15.4548	0.679702
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−13.3205	−0.583582	−0.291791	−	0.956482i	$-0.594251\pi$
$521$	−13.3205	−0.583582	−0.291791	+	0.956482i	$0.594251\pi$
$522$	0	0
$523$	24.5359	1.07288	0.536440	−	0.843938i	$-0.319769\pi$
$523$	24.5359	1.07288	0.536440	+	0.843938i	$0.319769\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−25.4558	−1.10887
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−48.6381	−2.10675
$534$	0	0
$535$	4.14110	0.179036
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.46410	0.235356
$540$	0	0
$541$	−8.58682	−0.369176	−0.184588	−	0.982816i	$-0.559095\pi$
$541$	−8.58682	−0.369176	−0.184588	+	0.982816i	$0.559095\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.46410	0.0627152
$546$	0	0
$547$	−39.1769	−1.67508	−0.837542	−	0.546373i	$-0.816008\pi$
$547$	−39.1769	−1.67508	−0.837542	+	0.546373i	$0.816008\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.18016	0.135479
$552$	0	0
$553$	−6.00000	−0.255146
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.7038	1.80942	0.904709	−	0.426030i	$-0.140088\pi$
$557$	42.7038	1.80942	0.904709	+	0.426030i	$0.140088\pi$
$558$	0	0
$559$	14.6969	0.621614
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.46410	0.230284	0.115142	−	0.993349i	$-0.463268\pi$
$563$	5.46410	0.230284	0.115142	+	0.993349i	$0.463268\pi$
$564$	0	0
$565$	−18.6350	−0.783979
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.46410	−0.145223	−0.0726113	−	0.997360i	$-0.523133\pi$
$569$	−3.46410	−0.145223	−0.0726113	+	0.997360i	$0.523133\pi$
$570$	0	0
$571$	23.7128	0.992350	0.496175	−	0.868222i	$-0.334737\pi$
$571$	23.7128	0.992350	0.496175	+	0.868222i	$0.334737\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−11.1106	−0.463346
$576$	0	0
$577$	5.85641	0.243805	0.121903	−	0.992542i	$-0.461100\pi$
$577$	5.85641	0.243805	0.121903	+	0.992542i	$0.461100\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	3.58630	0.148785
$582$	0	0
$583$	−52.0213	−2.15450
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.0718	1.19992	0.599961	−	0.800029i	$-0.295183\pi$
$587$	29.0718	1.19992	0.599961	+	0.800029i	$0.295183\pi$
$588$	0	0
$589$	3.93803	0.162264
$590$	0	0
$591$	0	0
$592$	0	0
$593$	17.7128	0.727378	0.363689	−	0.931520i	$-0.381517\pi$
$593$	17.7128	0.727378	0.363689	+	0.931520i	$0.381517\pi$
$594$	0	0
$595$	−8.78461	−0.360134
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12.6264	−0.515900	−0.257950	−	0.966158i	$-0.583047\pi$
$599$	−12.6264	−0.515900	−0.257950	+	0.966158i	$0.583047\pi$
$600$	0	0
$601$	−35.7128	−1.45676	−0.728378	−	0.685176i	$-0.759725\pi$
$601$	−35.7128	−1.45676	−0.728378	+	0.685176i	$0.759725\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	19.5216	0.793665
$606$	0	0
$607$	−31.8434	−1.29248	−0.646241	−	0.763133i	$-0.723660\pi$
$607$	−31.8434	−1.29248	−0.646241	+	0.763133i	$0.723660\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	42.8797	1.73189	0.865947	−	0.500136i	$-0.166717\pi$
$613$	42.8797	1.73189	0.865947	+	0.500136i	$0.166717\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.1436	0.488883	0.244441	−	0.969664i	$-0.421395\pi$
$617$	12.1436	0.488883	0.244441	+	0.969664i	$0.421395\pi$
$618$	0	0
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−21.8695	−0.876185
$624$	0	0
$625$	10.0718	0.402872
$626$	0	0
$627$	0	0
$628$	0	0
$629$	31.6675	1.26267
$630$	0	0
$631$	−0.933740	−0.0371716	−0.0185858	−	0.999827i	$-0.505916\pi$
$631$	−0.933740	−0.0371716	−0.0185858	+	0.999827i	$0.505916\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−20.1051	−0.797847
$636$	0	0
$637$	−4.24264	−0.168100
$638$	0	0
$639$	0	0
$640$	0	0
$641$	46.3923	1.83239	0.916193	−	0.400737i	$-0.131246\pi$
$641$	46.3923	1.83239	0.916193	+	0.400737i	$0.131246\pi$
$642$	0	0
$643$	−35.4641	−1.39857	−0.699284	−	0.714844i	$-0.746498\pi$
$643$	−35.4641	−1.39857	−0.699284	+	0.714844i	$0.746498\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.3147	0.837969	0.418984	−	0.907993i	$-0.362386\pi$
$647$	21.3147	0.837969	0.418984	+	0.907993i	$0.362386\pi$
$648$	0	0
$649$	−75.7128	−2.97199
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0.480473	0.0188024	0.00940119	−	0.999956i	$-0.497007\pi$
$653$	0.480473	0.0188024	0.00940119	+	0.999956i	$0.497007\pi$
$654$	0	0
$655$	1.10961	0.0433559
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	35.9101	1.39674	0.698371	−	0.715736i	$-0.253908\pi$
$661$	35.9101	1.39674	0.698371	+	0.715736i	$0.253908\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.35898	0.0526991
$666$	0	0
$667$	16.7846	0.649903
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−49.9507	−1.92833
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.8038	−1.06859	−0.534293	−	0.845299i	$-0.679422\pi$
$677$	−27.8038	−1.06859	−0.534293	+	0.845299i	$0.679422\pi$
$678$	0	0
$679$	36.5665	1.40329
$680$	0	0
$681$	0	0
$682$	0	0
$683$	37.1769	1.42254	0.711268	−	0.702921i	$-0.248121\pi$
$683$	37.1769	1.42254	0.711268	+	0.702921i	$0.248121\pi$
$684$	0	0
$685$	6.61780	0.252853
$686$	0	0
$687$	0	0
$688$	0	0
$689$	40.3923	1.53882
$690$	0	0
$691$	5.60770	0.213327	0.106663	−	0.994295i	$-0.465983\pi$
$691$	5.60770	0.213327	0.106663	+	0.994295i	$0.465983\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.17260	−0.272072
$696$	0	0
$697$	39.7128	1.50423
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−29.1165	−1.09971	−0.549857	−	0.835259i	$-0.685318\pi$
$701$	−29.1165	−1.09971	−0.549857	+	0.835259i	$0.685318\pi$
$702$	0	0
$703$	−4.89898	−0.184769
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.679492	−0.0255549
$708$	0	0
$709$	29.2923	1.10010	0.550048	−	0.835133i	$-0.314609\pi$
$709$	29.2923	1.10010	0.550048	+	0.835133i	$0.314609\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	20.7846	0.778390
$714$	0	0
$715$	−24.0000	−0.897549
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−31.6675	−1.18100	−0.590499	−	0.807038i	$-0.701069\pi$
$719$	−31.6675	−1.18100	−0.590499	+	0.807038i	$0.701069\pi$
$720$	0	0
$721$	19.8564	0.739491
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−23.3110	−0.865747
$726$	0	0
$727$	3.20736	0.118955	0.0594773	−	0.998230i	$-0.481057\pi$
$727$	3.20736	0.118955	0.0594773	+	0.998230i	$0.481057\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−12.0000	−0.443836
$732$	0	0
$733$	7.07107	0.261176	0.130588	−	0.991437i	$-0.458314\pi$
$733$	7.07107	0.261176	0.130588	+	0.991437i	$0.458314\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.85641	−0.215724
$738$	0	0
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.6665	−0.794866	−0.397433	−	0.917631i	$-0.630099\pi$
$743$	−21.6665	−0.794866	−0.397433	+	0.917631i	$0.630099\pi$
$744$	0	0
$745$	20.7846	0.761489
$746$	0	0
$747$	0	0
$748$	0	0
$749$	9.79796	0.358010
$750$	0	0
$751$	−21.6937	−0.791614	−0.395807	−	0.918334i	$-0.629535\pi$
$751$	−21.6937	−0.791614	−0.395807	+	0.918334i	$0.629535\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	18.5359	0.674590
$756$	0	0
$757$	25.1512	0.914137	0.457069	−	0.889431i	$-0.348899\pi$
$757$	25.1512	0.914137	0.457069	+	0.889431i	$0.348899\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.75129	0.135984	0.0679921	−	0.997686i	$-0.478341\pi$
$761$	3.75129	0.135984	0.0679921	+	0.997686i	$0.478341\pi$
$762$	0	0
$763$	3.46410	0.125409
$764$	0	0
$765$	0	0
$766$	0	0
$767$	58.7878	2.12270
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−4.82465	−0.173531	−0.0867653	−	0.996229i	$-0.527653\pi$
$773$	−4.82465	−0.173531	−0.0867653	+	0.996229i	$0.527653\pi$
$774$	0	0
$775$	−28.8663	−1.03691
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.14359	−0.220117
$780$	0	0
$781$	−88.5878	−3.16992
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12.3923	0.442300
$786$	0	0
$787$	−16.5359	−0.589441	−0.294721	−	0.955583i	$-0.595227\pi$
$787$	−16.5359	−0.589441	−0.294721	+	0.955583i	$0.595227\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−44.0908	−1.56769
$792$	0	0
$793$	38.7846	1.37728
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41.7429	1.47861	0.739304	−	0.673372i	$-0.235155\pi$
$797$	41.7429	1.47861	0.739304	+	0.673372i	$0.235155\pi$
$798$	0	0
$799$	9.79796	0.346627
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−21.8564	−0.771296
$804$	0	0
$805$	7.17260	0.252801
$806$	0	0
$807$	0	0
$808$	0	0
$809$	41.3205	1.45275	0.726376	−	0.687298i	$-0.241203\pi$
$809$	41.3205	1.45275	0.726376	+	0.687298i	$0.241203\pi$
$810$	0	0
$811$	33.6077	1.18013	0.590063	−	0.807357i	$-0.299103\pi$
$811$	33.6077	1.18013	0.590063	+	0.807357i	$0.299103\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	11.8685	0.415736
$816$	0	0
$817$	1.85641	0.0649474
$818$	0	0
$819$	0	0
$820$	0	0
$821$	39.8754	1.39166	0.695830	−	0.718206i	$-0.255037\pi$
$821$	39.8754	1.39166	0.695830	+	0.718206i	$0.255037\pi$
$822$	0	0
$823$	43.9149	1.53078	0.765389	−	0.643567i	$-0.222546\pi$
$823$	43.9149	1.53078	0.765389	+	0.643567i	$0.222546\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.21539	−0.111810	−0.0559050	−	0.998436i	$-0.517804\pi$
$827$	−3.21539	−0.111810	−0.0559050	+	0.998436i	$0.517804\pi$
$828$	0	0
$829$	−40.8091	−1.41736	−0.708680	−	0.705530i	$-0.750709\pi$
$829$	−40.8091	−1.41736	−0.708680	+	0.705530i	$0.750709\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.46410	0.120024
$834$	0	0
$835$	2.14359	0.0741821
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.38323	0.116802	0.0584010	−	0.998293i	$-0.481400\pi$
$839$	3.38323	0.116802	0.0584010	+	0.998293i	$0.481400\pi$
$840$	0	0
$841$	6.21539	0.214324
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.17638	0.178073
$846$	0	0
$847$	46.1886	1.58706
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−25.8564	−0.886346
$852$	0	0
$853$	−21.5649	−0.738369	−0.369185	−	0.929356i	$-0.620363\pi$
$853$	−21.5649	−0.738369	−0.369185	+	0.929356i	$0.620363\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.1769	0.655071	0.327535	−	0.944839i	$-0.393782\pi$
$857$	19.1769	0.655071	0.327535	+	0.944839i	$0.393782\pi$
$858$	0	0
$859$	16.2487	0.554399	0.277199	−	0.960812i	$-0.410594\pi$
$859$	16.2487	0.554399	0.277199	+	0.960812i	$0.410594\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.4644	1.07106	0.535531	−	0.844516i	$-0.320112\pi$
$863$	31.4644	1.07106	0.535531	+	0.844516i	$0.320112\pi$
$864$	0	0
$865$	2.64102	0.0897972
$866$	0	0
$867$	0	0
$868$	0	0
$869$	13.3843	0.454030
$870$	0	0
$871$	4.54725	0.154078
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−22.6410	−0.765406
$876$	0	0
$877$	32.8786	1.11023	0.555116	−	0.831773i	$-0.312674\pi$
$877$	32.8786	1.11023	0.555116	+	0.831773i	$0.312674\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−32.6410	−1.09970	−0.549852	−	0.835262i	$-0.685316\pi$
$881$	−32.6410	−1.09970	−0.549852	+	0.835262i	$0.685316\pi$
$882$	0	0
$883$	−26.3923	−0.888172	−0.444086	−	0.895984i	$-0.646471\pi$
$883$	−26.3923	−0.888172	−0.444086	+	0.895984i	$0.646471\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.62536	0.0881508	0.0440754	−	0.999028i	$-0.485966\pi$
$887$	2.62536	0.0881508	0.0440754	+	0.999028i	$0.485966\pi$
$888$	0	0
$889$	−47.5692	−1.59542
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.51575	−0.0507226
$894$	0	0
$895$	5.25071	0.175512
$896$	0	0
$897$	0	0
$898$	0	0
$899$	43.6077	1.45440
$900$	0	0
$901$	−32.9802	−1.09873
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.32051	0.243342
$906$	0	0
$907$	−21.6077	−0.717472	−0.358736	−	0.933439i	$-0.616792\pi$
$907$	−21.6077	−0.717472	−0.358736	+	0.933439i	$0.616792\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	51.4665	1.70516	0.852580	−	0.522596i	$-0.175036\pi$
$911$	51.4665	1.70516	0.852580	+	0.522596i	$0.175036\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.62536	0.0866969
$918$	0	0
$919$	−44.6728	−1.47362	−0.736810	−	0.676100i	$-0.763669\pi$
$919$	−44.6728	−1.47362	−0.736810	+	0.676100i	$0.763669\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	68.7846	2.26407
$924$	0	0
$925$	35.9101	1.18072
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10.6795	0.350383	0.175191	−	0.984534i	$-0.443946\pi$
$929$	10.6795	0.350383	0.175191	+	0.984534i	$0.443946\pi$
$930$	0	0
$931$	−0.535898	−0.0175634
$932$	0	0
$933$	0	0
$934$	0	0
$935$	19.5959	0.640855
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11.7942	0.384479	0.192240	−	0.981348i	$-0.438425\pi$
$941$	11.7942	0.384479	0.192240	+	0.981348i	$0.438425\pi$
$942$	0	0
$943$	−32.4254	−1.05592
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42.9282	−1.39498	−0.697490	−	0.716595i	$-0.745700\pi$
$947$	−42.9282	−1.39498	−0.697490	+	0.716595i	$0.745700\pi$
$948$	0	0
$949$	16.9706	0.550888
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−25.6077	−0.829515	−0.414757	−	0.909932i	$-0.636134\pi$
$953$	−25.6077	−0.829515	−0.414757	+	0.909932i	$0.636134\pi$
$954$	0	0
$955$	2.14359	0.0693651
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.6579	0.505619
$960$	0	0
$961$	23.0000	0.741935
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6.21166	0.199960
$966$	0	0
$967$	34.8749	1.12150	0.560750	−	0.827985i	$-0.310513\pi$
$967$	34.8749	1.12150	0.560750	+	0.827985i	$0.310513\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	26.2487	0.842361	0.421181	−	0.906977i	$-0.361616\pi$
$971$	26.2487	0.842361	0.421181	+	0.906977i	$0.361616\pi$
$972$	0	0
$973$	−16.9706	−0.544051
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27.1769	−0.869467	−0.434733	−	0.900559i	$-0.643157\pi$
$977$	−27.1769	−0.869467	−0.434733	+	0.900559i	$0.643157\pi$
$978$	0	0
$979$	48.7846	1.55916
$980$	0	0
$981$	0	0
$982$	0	0
$983$	35.0507	1.11794	0.558972	−	0.829186i	$-0.311196\pi$
$983$	35.0507	1.11794	0.558972	+	0.829186i	$0.311196\pi$
$984$	0	0
$985$	−6.14359	−0.195751
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9.79796	0.311557
$990$	0	0
$991$	12.2474	0.389053	0.194527	−	0.980897i	$-0.437683\pi$
$991$	12.2474	0.389053	0.194527	+	0.980897i	$0.437683\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−4.10512	−0.130141
$996$	0	0
$997$	7.82894	0.247945	0.123973	−	0.992286i	$-0.460437\pi$
$997$	7.82894	0.247945	0.123973	+	0.992286i	$0.460437\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9216.2.a.bc.1.3		4
3.2	odd	2		3072.2.a.r.1.2		4
4.3	odd	2		9216.2.a.bi.1.3		4
8.3	odd	2	inner	9216.2.a.bc.1.2		4
8.5	even	2		9216.2.a.bi.1.2		4
12.11	even	2		3072.2.a.l.1.2		4
24.5	odd	2		3072.2.a.l.1.3		4
24.11	even	2		3072.2.a.r.1.3		4
32.3	odd	8		2304.2.k.l.577.2		8
32.5	even	8		2304.2.k.e.1729.4		8
32.11	odd	8		2304.2.k.l.1729.1		8
32.13	even	8		2304.2.k.e.577.3		8
32.19	odd	8		2304.2.k.e.577.4		8
32.21	even	8		2304.2.k.l.1729.2		8
32.27	odd	8		2304.2.k.e.1729.3		8
32.29	even	8		2304.2.k.l.577.1		8
48.5	odd	4		3072.2.d.h.1537.2		8
48.11	even	4		3072.2.d.h.1537.6		8
48.29	odd	4		3072.2.d.h.1537.7		8
48.35	even	4		3072.2.d.h.1537.3		8
96.5	odd	8		768.2.j.f.193.1	yes	8
96.11	even	8		768.2.j.e.193.2	✓	8
96.29	odd	8		768.2.j.e.577.4	yes	8
96.35	even	8		768.2.j.e.577.2	yes	8
96.53	odd	8		768.2.j.e.193.4	yes	8
96.59	even	8		768.2.j.f.193.3	yes	8
96.77	odd	8		768.2.j.f.577.1	yes	8
96.83	even	8		768.2.j.f.577.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.2.j.e.193.2	✓	8	96.11	even	8
768.2.j.e.193.4	yes	8	96.53	odd	8
768.2.j.e.577.2	yes	8	96.35	even	8
768.2.j.e.577.4	yes	8	96.29	odd	8
768.2.j.f.193.1	yes	8	96.5	odd	8
768.2.j.f.193.3	yes	8	96.59	even	8
768.2.j.f.577.1	yes	8	96.77	odd	8
768.2.j.f.577.3	yes	8	96.83	even	8
2304.2.k.e.577.3		8	32.13	even	8
2304.2.k.e.577.4		8	32.19	odd	8
2304.2.k.e.1729.3		8	32.27	odd	8
2304.2.k.e.1729.4		8	32.5	even	8
2304.2.k.l.577.1		8	32.29	even	8
2304.2.k.l.577.2		8	32.3	odd	8
2304.2.k.l.1729.1		8	32.11	odd	8
2304.2.k.l.1729.2		8	32.21	even	8
3072.2.a.l.1.2		4	12.11	even	2
3072.2.a.l.1.3		4	24.5	odd	2
3072.2.a.r.1.2		4	3.2	odd	2
3072.2.a.r.1.3		4	24.11	even	2
3072.2.d.h.1537.2		8	48.5	odd	4
3072.2.d.h.1537.3		8	48.35	even	4
3072.2.d.h.1537.6		8	48.11	even	4
3072.2.d.h.1537.7		8	48.29	odd	4
9216.2.a.bc.1.2		4	8.3	odd	2	inner
9216.2.a.bc.1.3		4	1.1	even	1	trivial
9216.2.a.bi.1.2		4	8.5	even	2
9216.2.a.bi.1.3		4	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9216 = 2^{10} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9216.a (trivial)

\(f(q)\)	\(=\)	\(q+1.03528 q^{5} +2.44949 q^{7} -5.46410 q^{11} +4.24264 q^{13} -3.46410 q^{17} +0.535898 q^{19} +2.82843 q^{23} -3.92820 q^{25} +5.93426 q^{29} +7.34847 q^{31} +2.53590 q^{35} -9.14162 q^{37} -11.4641 q^{41} +3.46410 q^{43} -2.82843 q^{47} -1.00000 q^{49} +9.52056 q^{53} -5.65685 q^{55} +13.8564 q^{59} +9.14162 q^{61} +4.39230 q^{65} +1.07180 q^{67} +16.2127 q^{71} +4.00000 q^{73} -13.3843 q^{77} -2.44949 q^{79} +1.46410 q^{83} -3.58630 q^{85} -8.92820 q^{89} +10.3923 q^{91} +0.554803 q^{95} +14.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{11} + 16 q^{19} + 12 q^{25} + 24 q^{35} - 32 q^{41} - 4 q^{49} - 24 q^{65} + 32 q^{67} + 16 q^{73} - 8 q^{83} - 8 q^{89} + 32 q^{97}+O(q^{100}) \) 4 * q - 8 * q^11 + 16 * q^19 + 12 * q^25 + 24 * q^35 - 32 * q^41 - 4 * q^49 - 24 * q^65 + 32 * q^67 + 16 * q^73 - 8 * q^83 - 8 * q^89 + 32 * q^97

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