LMFDB - Embedded newform 8379.2.a.x.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8379,2,Mod(1,8379)] 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("8379.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8379, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-4,0,0,0,0,0,0,0,0,-4,0,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.9066518536$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 1197)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{4} -5.19615 q^{11} -2.00000 q^{13} +4.00000 q^{16} -5.19615 q^{17} -1.00000 q^{19} -5.19615 q^{23} -5.00000 q^{25} -10.3923 q^{29} -2.00000 q^{31} +8.00000 q^{37} -10.3923 q^{41} -4.00000 q^{43} +10.3923 q^{44} +5.19615 q^{47} +4.00000 q^{52} +10.3923 q^{59} -5.00000 q^{61} -8.00000 q^{64} +2.00000 q^{67} +10.3923 q^{68} +10.3923 q^{71} +7.00000 q^{73} +2.00000 q^{76} -4.00000 q^{79} +15.5885 q^{83} -10.3923 q^{89} +10.3923 q^{92} -14.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 4 q^{13} + 8 q^{16} - 2 q^{19} - 10 q^{25} - 4 q^{31} + 16 q^{37} - 8 q^{43} + 8 q^{52} - 10 q^{61} - 16 q^{64} + 4 q^{67} + 14 q^{73} + 4 q^{76} - 8 q^{79} - 28 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 - 4 * q^13 + 8 * q^16 - 2 * q^19 - 10 * q^25 - 4 * q^31 + 16 * q^37 - 8 * q^43 + 8 * q^52 - 10 * q^61 - 16 * q^64 + 4 * q^67 + 14 * q^73 + 4 * q^76 - 8 * q^79 - 28 * 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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.19615	−1.56670	−0.783349	−	0.621582i	$-0.786490\pi$
$11$	−5.19615	−1.56670	−0.783349	+	0.621582i	$0.786490\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	−5.19615	−1.26025	−0.630126	−	0.776493i	$-0.716997\pi$
$17$	−5.19615	−1.26025	−0.630126	+	0.776493i	$0.716997\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.19615	−1.08347	−0.541736	−	0.840548i	$-0.682233\pi$
$23$	−5.19615	−1.08347	−0.541736	+	0.840548i	$0.682233\pi$
$24$	0	0
$25$	−5.00000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−10.3923	−1.92980	−0.964901	−	0.262613i	$-0.915416\pi$
$29$	−10.3923	−1.92980	−0.964901	+	0.262613i	$0.915416\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.3923	−1.62301	−0.811503	−	0.584349i	$-0.801350\pi$
$41$	−10.3923	−1.62301	−0.811503	+	0.584349i	$0.801350\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	10.3923	1.56670
$45$	0	0
$46$	0	0
$47$	5.19615	0.757937	0.378968	−	0.925410i	$-0.376279\pi$
$47$	5.19615	0.757937	0.378968	+	0.925410i	$0.376279\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	4.00000	0.554700
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.3923	1.35296	0.676481	−	0.736460i	$-0.263504\pi$
$59$	10.3923	1.35296	0.676481	+	0.736460i	$0.263504\pi$
$60$	0	0
$61$	−5.00000	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$61$	−5.00000	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	10.3923	1.26025
$69$	0	0
$70$	0	0
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	0	0
$76$	2.00000	0.229416
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	15.5885	1.71106	0.855528	−	0.517757i	$-0.173233\pi$
$83$	15.5885	1.71106	0.855528	+	0.517757i	$0.173233\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.3923	−1.10158	−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923	−1.10158	−0.550791	+	0.834643i	$0.685674\pi$
$90$	0	0
$91$	0	0
$92$	10.3923	1.08347
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	0	0
$100$	10.0000	1.00000
$101$	−15.5885	−1.55111	−0.775555	−	0.631280i	$-0.782530\pi$
$101$	−15.5885	−1.55111	−0.775555	+	0.631280i	$0.782530\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.3923	−1.00466	−0.502331	−	0.864675i	$-0.667524\pi$
$107$	−10.3923	−1.00466	−0.502331	+	0.864675i	$0.667524\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	20.7846	1.92980
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	16.0000	1.45455
$122$	0	0
$123$	0	0
$124$	4.00000	0.359211
$125$	0	0
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.5885	−1.36197	−0.680985	−	0.732297i	$-0.738448\pi$
$131$	−15.5885	−1.36197	−0.680985	+	0.732297i	$0.738448\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.5885	−1.33181	−0.665906	−	0.746036i	$-0.731955\pi$
$137$	−15.5885	−1.33181	−0.665906	+	0.746036i	$0.731955\pi$
$138$	0	0
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.3923	0.869048
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	−16.0000	−1.31519
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	−4.00000	−0.325515	−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000	−0.325515	−0.162758	+	0.986666i	$0.552039\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13.0000	1.03751	0.518756	−	0.854922i	$-0.326395\pi$
$157$	13.0000	1.03751	0.518756	+	0.854922i	$0.326395\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1.00000	−0.0783260	−0.0391630	−	0.999233i	$-0.512469\pi$
$163$	−1.00000	−0.0783260	−0.0391630	+	0.999233i	$0.512469\pi$
$164$	20.7846	1.62301
$165$	0	0
$166$	0	0
$167$	10.3923	0.804181	0.402090	−	0.915600i	$-0.368284\pi$
$167$	10.3923	0.804181	0.402090	+	0.915600i	$0.368284\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	8.00000	0.609994
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	−20.7846	−1.56670
$177$	0	0
$178$	0	0
$179$	10.3923	0.776757	0.388379	−	0.921500i	$-0.373035\pi$
$179$	10.3923	0.776757	0.388379	+	0.921500i	$0.373035\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	27.0000	1.97444
$188$	−10.3923	−0.757937
$189$	0	0
$190$	0	0
$191$	−10.3923	−0.751961	−0.375980	−	0.926628i	$-0.622694\pi$
$191$	−10.3923	−0.751961	−0.375980	+	0.926628i	$0.622694\pi$
$192$	0	0
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.5885	1.11063	0.555316	−	0.831640i	$-0.312597\pi$
$197$	15.5885	1.11063	0.555316	+	0.831640i	$0.312597\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	−8.00000	−0.554700
$209$	5.19615	0.359425
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	10.3923	0.699062
$222$	0	0
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−10.3923	−0.689761	−0.344881	−	0.938647i	$-0.612081\pi$
$227$	−10.3923	−0.689761	−0.344881	+	0.938647i	$0.612081\pi$
$228$	0	0
$229$	25.0000	1.65205	0.826023	−	0.563636i	$-0.190598\pi$
$229$	25.0000	1.65205	0.826023	+	0.563636i	$0.190598\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.5885	1.02123	0.510617	−	0.859808i	$-0.329417\pi$
$233$	15.5885	1.02123	0.510617	+	0.859808i	$0.329417\pi$
$234$	0	0
$235$	0	0
$236$	−20.7846	−1.35296
$237$	0	0
$238$	0	0
$239$	−15.5885	−1.00833	−0.504167	−	0.863606i	$-0.668200\pi$
$239$	−15.5885	−1.00833	−0.504167	+	0.863606i	$0.668200\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	0	0
$244$	10.0000	0.640184
$245$	0	0
$246$	0	0
$247$	2.00000	0.127257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.3923	0.655956	0.327978	−	0.944685i	$-0.393633\pi$
$251$	10.3923	0.655956	0.327978	+	0.944685i	$0.393633\pi$
$252$	0	0
$253$	27.0000	1.69748
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	20.7846	1.29651	0.648254	−	0.761424i	$-0.275499\pi$
$257$	20.7846	1.29651	0.648254	+	0.761424i	$0.275499\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.19615	0.320408	0.160204	−	0.987084i	$-0.448785\pi$
$263$	5.19615	0.320408	0.160204	+	0.987084i	$0.448785\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−4.00000	−0.244339
$269$	−20.7846	−1.26726	−0.633630	−	0.773636i	$-0.718436\pi$
$269$	−20.7846	−1.26726	−0.633630	+	0.773636i	$0.718436\pi$
$270$	0	0
$271$	1.00000	0.0607457	0.0303728	−	0.999539i	$-0.490331\pi$
$271$	1.00000	0.0607457	0.0303728	+	0.999539i	$0.490331\pi$
$272$	−20.7846	−1.26025
$273$	0	0
$274$	0	0
$275$	25.9808	1.56670
$276$	0	0
$277$	23.0000	1.38194	0.690968	−	0.722885i	$-0.257185\pi$
$277$	23.0000	1.38194	0.690968	+	0.722885i	$0.257185\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	20.7846	1.23991	0.619953	−	0.784639i	$-0.287152\pi$
$281$	20.7846	1.23991	0.619953	+	0.784639i	$0.287152\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	−20.7846	−1.23334
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	10.0000	0.588235
$290$	0	0
$291$	0	0
$292$	−14.0000	−0.819288
$293$	−31.1769	−1.82137	−0.910687	−	0.413096i	$-0.864447\pi$
$293$	−31.1769	−1.82137	−0.910687	+	0.413096i	$0.864447\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10.3923	0.601003
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	−4.00000	−0.229416
$305$	0	0
$306$	0	0
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.19615	0.294647	0.147323	−	0.989088i	$-0.452934\pi$
$311$	5.19615	0.294647	0.147323	+	0.989088i	$0.452934\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	0	0
$315$	0	0
$316$	8.00000	0.450035
$317$	10.3923	0.583690	0.291845	−	0.956466i	$-0.405731\pi$
$317$	10.3923	0.583690	0.291845	+	0.956466i	$0.405731\pi$
$318$	0	0
$319$	54.0000	3.02342
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.19615	0.289122
$324$	0	0
$325$	10.0000	0.554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	−31.1769	−1.71106
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	32.0000	1.74315	0.871576	−	0.490261i	$-0.163099\pi$
$337$	32.0000	1.74315	0.871576	+	0.490261i	$0.163099\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.3923	0.562775
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.3923	−0.557888	−0.278944	−	0.960307i	$-0.589984\pi$
$347$	−10.3923	−0.557888	−0.278944	+	0.960307i	$0.589984\pi$
$348$	0	0
$349$	−5.00000	−0.267644	−0.133822	−	0.991005i	$-0.542725\pi$
$349$	−5.00000	−0.267644	−0.133822	+	0.991005i	$0.542725\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.7846	1.10625	0.553127	−	0.833097i	$-0.313435\pi$
$353$	20.7846	1.10625	0.553127	+	0.833097i	$0.313435\pi$
$354$	0	0
$355$	0	0
$356$	20.7846	1.10158
$357$	0	0
$358$	0	0
$359$	−15.5885	−0.822727	−0.411364	−	0.911471i	$-0.634947\pi$
$359$	−15.5885	−0.822727	−0.411364	+	0.911471i	$0.634947\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	−20.7846	−1.08347
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	20.7846	1.07046
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−31.1769	−1.59307	−0.796533	−	0.604595i	$-0.793335\pi$
$383$	−31.1769	−1.59307	−0.796533	+	0.604595i	$0.793335\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	28.0000	1.42148
$389$	15.5885	0.790366	0.395183	−	0.918602i	$-0.370681\pi$
$389$	15.5885	0.790366	0.395183	+	0.918602i	$0.370681\pi$
$390$	0	0
$391$	27.0000	1.36545
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−17.0000	−0.853206	−0.426603	−	0.904439i	$-0.640290\pi$
$397$	−17.0000	−0.853206	−0.426603	+	0.904439i	$0.640290\pi$
$398$	0	0
$399$	0	0
$400$	−20.0000	−1.00000
$401$	−31.1769	−1.55690	−0.778450	−	0.627706i	$-0.783994\pi$
$401$	−31.1769	−1.55690	−0.778450	+	0.627706i	$0.783994\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	31.1769	1.55111
$405$	0	0
$406$	0	0
$407$	−41.5692	−2.06051
$408$	0	0
$409$	−20.0000	−0.988936	−0.494468	−	0.869196i	$-0.664637\pi$
$409$	−20.0000	−0.988936	−0.494468	+	0.869196i	$0.664637\pi$
$410$	0	0
$411$	0	0
$412$	−32.0000	−1.57653
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−15.5885	−0.761546	−0.380773	−	0.924669i	$-0.624342\pi$
$419$	−15.5885	−0.761546	−0.380773	+	0.924669i	$0.624342\pi$
$420$	0	0
$421$	8.00000	0.389896	0.194948	−	0.980814i	$-0.437546\pi$
$421$	8.00000	0.389896	0.194948	+	0.980814i	$0.437546\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	25.9808	1.26025
$426$	0	0
$427$	0	0
$428$	20.7846	1.00466
$429$	0	0
$430$	0	0
$431$	−31.1769	−1.50174	−0.750870	−	0.660451i	$-0.770365\pi$
$431$	−31.1769	−1.50174	−0.750870	+	0.660451i	$0.770365\pi$
$432$	0	0
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	0	0
$436$	20.0000	0.957826
$437$	5.19615	0.248566
$438$	0	0
$439$	−26.0000	−1.24091	−0.620456	−	0.784241i	$-0.713053\pi$
$439$	−26.0000	−1.24091	−0.620456	+	0.784241i	$0.713053\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.19615	−0.246877	−0.123438	−	0.992352i	$-0.539392\pi$
$443$	−5.19615	−0.246877	−0.123438	+	0.992352i	$0.539392\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−31.1769	−1.47133	−0.735665	−	0.677346i	$-0.763130\pi$
$449$	−31.1769	−1.47133	−0.735665	+	0.677346i	$0.763130\pi$
$450$	0	0
$451$	54.0000	2.54276
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.0000	1.07589	0.537947	−	0.842978i	$-0.319200\pi$
$457$	23.0000	1.07589	0.537947	+	0.842978i	$0.319200\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.19615	0.242009	0.121004	−	0.992652i	$-0.461388\pi$
$461$	5.19615	0.242009	0.121004	+	0.992652i	$0.461388\pi$
$462$	0	0
$463$	23.0000	1.06890	0.534450	−	0.845200i	$-0.320519\pi$
$463$	23.0000	1.06890	0.534450	+	0.845200i	$0.320519\pi$
$464$	−41.5692	−1.92980
$465$	0	0
$466$	0	0
$467$	−5.19615	−0.240449	−0.120225	−	0.992747i	$-0.538361\pi$
$467$	−5.19615	−0.240449	−0.120225	+	0.992747i	$0.538361\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	20.7846	0.955677
$474$	0	0
$475$	5.00000	0.229416
$476$	0	0
$477$	0	0
$478$	0	0
$479$	15.5885	0.712255	0.356127	−	0.934437i	$-0.384097\pi$
$479$	15.5885	0.712255	0.356127	+	0.934437i	$0.384097\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	0	0
$484$	−32.0000	−1.45455
$485$	0	0
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.19615	−0.234499	−0.117250	−	0.993102i	$-0.537408\pi$
$491$	−5.19615	−0.234499	−0.117250	+	0.993102i	$0.537408\pi$
$492$	0	0
$493$	54.0000	2.43204
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.19615	−0.231685	−0.115842	−	0.993268i	$-0.536957\pi$
$503$	−5.19615	−0.231685	−0.115842	+	0.993268i	$0.536957\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	−16.0000	−0.709885
$509$	−10.3923	−0.460631	−0.230315	−	0.973116i	$-0.573976\pi$
$509$	−10.3923	−0.460631	−0.230315	+	0.973116i	$0.573976\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−27.0000	−1.18746
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.7846	0.910590	0.455295	−	0.890341i	$-0.349534\pi$
$521$	20.7846	0.910590	0.455295	+	0.890341i	$0.349534\pi$
$522$	0	0
$523$	10.0000	0.437269	0.218635	−	0.975807i	$-0.429840\pi$
$523$	10.0000	0.437269	0.218635	+	0.975807i	$0.429840\pi$
$524$	31.1769	1.36197
$525$	0	0
$526$	0	0
$527$	10.3923	0.452696
$528$	0	0
$529$	4.00000	0.173913
$530$	0	0
$531$	0	0
$532$	0	0
$533$	20.7846	0.900281
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	31.1769	1.33181
$549$	0	0
$550$	0	0
$551$	10.3923	0.442727
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	22.0000	0.933008
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.3923	−0.437983	−0.218992	−	0.975727i	$-0.570277\pi$
$563$	−10.3923	−0.437983	−0.218992	+	0.975727i	$0.570277\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.7846	0.871336	0.435668	−	0.900107i	$-0.356512\pi$
$569$	20.7846	0.871336	0.435668	+	0.900107i	$0.356512\pi$
$570$	0	0
$571$	23.0000	0.962520	0.481260	−	0.876578i	$-0.340179\pi$
$571$	23.0000	0.962520	0.481260	+	0.876578i	$0.340179\pi$
$572$	−20.7846	−0.869048
$573$	0	0
$574$	0	0
$575$	25.9808	1.08347
$576$	0	0
$577$	19.0000	0.790980	0.395490	−	0.918470i	$-0.370575\pi$
$577$	19.0000	0.790980	0.395490	+	0.918470i	$0.370575\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25.9808	−1.07234	−0.536170	−	0.844110i	$-0.680130\pi$
$587$	−25.9808	−1.07234	−0.536170	+	0.844110i	$0.680130\pi$
$588$	0	0
$589$	2.00000	0.0824086
$590$	0	0
$591$	0	0
$592$	32.0000	1.31519
$593$	5.19615	0.213380	0.106690	−	0.994292i	$-0.465975\pi$
$593$	5.19615	0.213380	0.106690	+	0.994292i	$0.465975\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	40.0000	1.63163	0.815817	−	0.578310i	$-0.196288\pi$
$601$	40.0000	1.63163	0.815817	+	0.578310i	$0.196288\pi$
$602$	0	0
$603$	0	0
$604$	8.00000	0.325515
$605$	0	0
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.3923	−0.420428
$612$	0	0
$613$	−37.0000	−1.49442	−0.747208	−	0.664590i	$-0.768606\pi$
$613$	−37.0000	−1.49442	−0.747208	+	0.664590i	$0.768606\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20.7846	0.836757	0.418378	−	0.908273i	$-0.362599\pi$
$617$	20.7846	0.836757	0.418378	+	0.908273i	$0.362599\pi$
$618$	0	0
$619$	−41.0000	−1.64793	−0.823965	−	0.566641i	$-0.808243\pi$
$619$	−41.0000	−1.64793	−0.823965	+	0.566641i	$0.808243\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	−26.0000	−1.03751
$629$	−41.5692	−1.65747
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10.3923	−0.410471	−0.205236	−	0.978713i	$-0.565796\pi$
$641$	−10.3923	−0.410471	−0.205236	+	0.978713i	$0.565796\pi$
$642$	0	0
$643$	25.0000	0.985904	0.492952	−	0.870057i	$-0.335918\pi$
$643$	25.0000	0.985904	0.492952	+	0.870057i	$0.335918\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.3923	−0.408564	−0.204282	−	0.978912i	$-0.565486\pi$
$647$	−10.3923	−0.408564	−0.204282	+	0.978912i	$0.565486\pi$
$648$	0	0
$649$	−54.0000	−2.11969
$650$	0	0
$651$	0	0
$652$	2.00000	0.0783260
$653$	36.3731	1.42339	0.711694	−	0.702490i	$-0.247928\pi$
$653$	36.3731	1.42339	0.711694	+	0.702490i	$0.247928\pi$
$654$	0	0
$655$	0	0
$656$	−41.5692	−1.62301
$657$	0	0
$658$	0	0
$659$	−10.3923	−0.404827	−0.202413	−	0.979300i	$-0.564878\pi$
$659$	−10.3923	−0.404827	−0.202413	+	0.979300i	$0.564878\pi$
$660$	0	0
$661$	−32.0000	−1.24466	−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000	−1.24466	−0.622328	+	0.782757i	$0.713813\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	54.0000	2.09089
$668$	−20.7846	−0.804181
$669$	0	0
$670$	0	0
$671$	25.9808	1.00298
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	0	0
$675$	0	0
$676$	18.0000	0.692308
$677$	−20.7846	−0.798817	−0.399409	−	0.916773i	$-0.630785\pi$
$677$	−20.7846	−0.798817	−0.399409	+	0.916773i	$0.630785\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	41.5692	1.59060	0.795301	−	0.606215i	$-0.207313\pi$
$683$	41.5692	1.59060	0.795301	+	0.606215i	$0.207313\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	−16.0000	−0.609994
$689$	0	0
$690$	0	0
$691$	−5.00000	−0.190209	−0.0951045	−	0.995467i	$-0.530319\pi$
$691$	−5.00000	−0.190209	−0.0951045	+	0.995467i	$0.530319\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	54.0000	2.04540
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.19615	−0.196256	−0.0981280	−	0.995174i	$-0.531285\pi$
$701$	−5.19615	−0.196256	−0.0981280	+	0.995174i	$0.531285\pi$
$702$	0	0
$703$	−8.00000	−0.301726
$704$	41.5692	1.56670
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	41.0000	1.53979	0.769894	−	0.638172i	$-0.220309\pi$
$709$	41.0000	1.53979	0.769894	+	0.638172i	$0.220309\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.3923	0.389195
$714$	0	0
$715$	0	0
$716$	−20.7846	−0.776757
$717$	0	0
$718$	0	0
$719$	46.7654	1.74405	0.872027	−	0.489458i	$-0.162805\pi$
$719$	46.7654	1.74405	0.872027	+	0.489458i	$0.162805\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	40.0000	1.48659
$725$	51.9615	1.92980
$726$	0	0
$727$	4.00000	0.148352	0.0741759	−	0.997245i	$-0.476367\pi$
$727$	4.00000	0.148352	0.0741759	+	0.997245i	$0.476367\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	20.7846	0.768747
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.3923	−0.382805
$738$	0	0
$739$	17.0000	0.625355	0.312678	−	0.949859i	$-0.398774\pi$
$739$	17.0000	0.625355	0.312678	+	0.949859i	$0.398774\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20.7846	−0.762513	−0.381257	−	0.924469i	$-0.624509\pi$
$743$	−20.7846	−0.762513	−0.381257	+	0.924469i	$0.624509\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	−54.0000	−1.97444
$749$	0	0
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	20.7846	0.757937
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	17.0000	0.617876	0.308938	−	0.951082i	$-0.400027\pi$
$757$	17.0000	0.617876	0.308938	+	0.951082i	$0.400027\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.7846	−0.753442	−0.376721	−	0.926327i	$-0.622948\pi$
$761$	−20.7846	−0.753442	−0.376721	+	0.926327i	$0.622948\pi$
$762$	0	0
$763$	0	0
$764$	20.7846	0.751961
$765$	0	0
$766$	0	0
$767$	−20.7846	−0.750489
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	0	0
$771$	0	0
$772$	−40.0000	−1.43963
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	10.0000	0.359211
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10.3923	0.372343
$780$	0	0
$781$	−54.0000	−1.93227
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−20.0000	−0.712923	−0.356462	−	0.934310i	$-0.616017\pi$
$787$	−20.0000	−0.712923	−0.356462	+	0.934310i	$0.616017\pi$
$788$	−31.1769	−1.11063
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	10.0000	0.355110
$794$	0	0
$795$	0	0
$796$	16.0000	0.567105
$797$	31.1769	1.10434	0.552171	−	0.833731i	$-0.313799\pi$
$797$	31.1769	1.10434	0.552171	+	0.833731i	$0.313799\pi$
$798$	0	0
$799$	−27.0000	−0.955191
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−36.3731	−1.28358
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−20.7846	−0.730748	−0.365374	−	0.930861i	$-0.619059\pi$
$809$	−20.7846	−0.730748	−0.365374	+	0.930861i	$0.619059\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.00000	0.139942
$818$	0	0
$819$	0	0
$820$	0	0
$821$	25.9808	0.906735	0.453367	−	0.891324i	$-0.350223\pi$
$821$	25.9808	0.906735	0.453367	+	0.891324i	$0.350223\pi$
$822$	0	0
$823$	−25.0000	−0.871445	−0.435723	−	0.900081i	$-0.643507\pi$
$823$	−25.0000	−0.871445	−0.435723	+	0.900081i	$0.643507\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.3923	−0.361376	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923	−0.361376	−0.180688	+	0.983540i	$0.557832\pi$
$828$	0	0
$829$	34.0000	1.18087	0.590434	−	0.807086i	$-0.298956\pi$
$829$	34.0000	1.18087	0.590434	+	0.807086i	$0.298956\pi$
$830$	0	0
$831$	0	0
$832$	16.0000	0.554700
$833$	0	0
$834$	0	0
$835$	0	0
$836$	−10.3923	−0.359425
$837$	0	0
$838$	0	0
$839$	−41.5692	−1.43513	−0.717564	−	0.696492i	$-0.754743\pi$
$839$	−41.5692	−1.43513	−0.717564	+	0.696492i	$0.754743\pi$
$840$	0	0
$841$	79.0000	2.72414
$842$	0	0
$843$	0	0
$844$	−40.0000	−1.37686
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−41.5692	−1.42497
$852$	0	0
$853$	−35.0000	−1.19838	−0.599189	−	0.800608i	$-0.704510\pi$
$853$	−35.0000	−1.19838	−0.599189	+	0.800608i	$0.704510\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−51.9615	−1.77497	−0.887486	−	0.460835i	$-0.847550\pi$
$857$	−51.9615	−1.77497	−0.887486	+	0.460835i	$0.847550\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.7846	0.707516	0.353758	−	0.935337i	$-0.384904\pi$
$863$	20.7846	0.707516	0.353758	+	0.935337i	$0.384904\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	20.7846	0.705070
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−16.0000	−0.540282	−0.270141	−	0.962821i	$-0.587070\pi$
$877$	−16.0000	−0.540282	−0.270141	+	0.962821i	$0.587070\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	46.7654	1.57557	0.787783	−	0.615953i	$-0.211229\pi$
$881$	46.7654	1.57557	0.787783	+	0.615953i	$0.211229\pi$
$882$	0	0
$883$	−7.00000	−0.235569	−0.117784	−	0.993039i	$-0.537579\pi$
$883$	−7.00000	−0.235569	−0.117784	+	0.993039i	$0.537579\pi$
$884$	−20.7846	−0.699062
$885$	0	0
$886$	0	0
$887$	51.9615	1.74470	0.872349	−	0.488884i	$-0.162596\pi$
$887$	51.9615	1.74470	0.872349	+	0.488884i	$0.162596\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	52.0000	1.74109
$893$	−5.19615	−0.173883
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	20.7846	0.693206
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−22.0000	−0.730498	−0.365249	−	0.930910i	$-0.619016\pi$
$907$	−22.0000	−0.730498	−0.365249	+	0.930910i	$0.619016\pi$
$908$	20.7846	0.689761
$909$	0	0
$910$	0	0
$911$	31.1769	1.03294	0.516469	−	0.856306i	$-0.327246\pi$
$911$	31.1769	1.03294	0.516469	+	0.856306i	$0.327246\pi$
$912$	0	0
$913$	−81.0000	−2.68071
$914$	0	0
$915$	0	0
$916$	−50.0000	−1.65205
$917$	0	0
$918$	0	0
$919$	−19.0000	−0.626752	−0.313376	−	0.949629i	$-0.601460\pi$
$919$	−19.0000	−0.626752	−0.313376	+	0.949629i	$0.601460\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−20.7846	−0.684134
$924$	0	0
$925$	−40.0000	−1.31519
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−15.5885	−0.511441	−0.255720	−	0.966751i	$-0.582313\pi$
$929$	−15.5885	−0.511441	−0.255720	+	0.966751i	$0.582313\pi$
$930$	0	0
$931$	0	0
$932$	−31.1769	−1.02123
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10.0000	0.326686	0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000	0.326686	0.163343	+	0.986569i	$0.447772\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−31.1769	−1.01634	−0.508169	−	0.861257i	$-0.669678\pi$
$941$	−31.1769	−1.01634	−0.508169	+	0.861257i	$0.669678\pi$
$942$	0	0
$943$	54.0000	1.75848
$944$	41.5692	1.35296
$945$	0	0
$946$	0	0
$947$	−31.1769	−1.01311	−0.506557	−	0.862207i	$-0.669082\pi$
$947$	−31.1769	−1.01311	−0.506557	+	0.862207i	$0.669082\pi$
$948$	0	0
$949$	−14.0000	−0.454459
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10.3923	0.336640	0.168320	−	0.985732i	$-0.446166\pi$
$953$	10.3923	0.336640	0.168320	+	0.985732i	$0.446166\pi$
$954$	0	0
$955$	0	0
$956$	31.1769	1.00833
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	−8.00000	−0.257663
$965$	0	0
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−31.1769	−1.00051	−0.500257	−	0.865877i	$-0.666761\pi$
$971$	−31.1769	−1.00051	−0.500257	+	0.865877i	$0.666761\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	−20.0000	−0.640184
$977$	41.5692	1.32992	0.664959	−	0.746880i	$-0.268449\pi$
$977$	41.5692	1.32992	0.664959	+	0.746880i	$0.268449\pi$
$978$	0	0
$979$	54.0000	1.72585
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−20.7846	−0.662926	−0.331463	−	0.943468i	$-0.607542\pi$
$983$	−20.7846	−0.662926	−0.331463	+	0.943468i	$0.607542\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	−4.00000	−0.127257
$989$	20.7846	0.660912
$990$	0	0
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−23.0000	−0.728417	−0.364209	−	0.931317i	$-0.618661\pi$
$997$	−23.0000	−0.728417	−0.364209	+	0.931317i	$0.618661\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.x.1.1		2
3.2	odd	2	inner	8379.2.a.x.1.2		2
7.3	odd	6		1197.2.j.h.856.2	yes	4
7.5	odd	6		1197.2.j.h.172.2	yes	4
7.6	odd	2		8379.2.a.y.1.1		2
21.5	even	6		1197.2.j.h.172.1	✓	4
21.17	even	6		1197.2.j.h.856.1	yes	4
21.20	even	2		8379.2.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1197.2.j.h.172.1	✓	4	21.5	even	6
1197.2.j.h.172.2	yes	4	7.5	odd	6
1197.2.j.h.856.1	yes	4	21.17	even	6
1197.2.j.h.856.2	yes	4	7.3	odd	6
8379.2.a.x.1.1		2	1.1	even	1	trivial
8379.2.a.x.1.2		2	3.2	odd	2	inner
8379.2.a.y.1.1		2	7.6	odd	2
8379.2.a.y.1.2		2	21.20	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} -5.19615 q^{11} -2.00000 q^{13} +4.00000 q^{16} -5.19615 q^{17} -1.00000 q^{19} -5.19615 q^{23} -5.00000 q^{25} -10.3923 q^{29} -2.00000 q^{31} +8.00000 q^{37} -10.3923 q^{41} -4.00000 q^{43} +10.3923 q^{44} +5.19615 q^{47} +4.00000 q^{52} +10.3923 q^{59} -5.00000 q^{61} -8.00000 q^{64} +2.00000 q^{67} +10.3923 q^{68} +10.3923 q^{71} +7.00000 q^{73} +2.00000 q^{76} -4.00000 q^{79} +15.5885 q^{83} -10.3923 q^{89} +10.3923 q^{92} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 4 q^{13} + 8 q^{16} - 2 q^{19} - 10 q^{25} - 4 q^{31} + 16 q^{37} - 8 q^{43} + 8 q^{52} - 10 q^{61} - 16 q^{64} + 4 q^{67} + 14 q^{73} + 4 q^{76} - 8 q^{79} - 28 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 - 4 * q^13 + 8 * q^16 - 2 * q^19 - 10 * q^25 - 4 * q^31 + 16 * q^37 - 8 * q^43 + 8 * q^52 - 10 * q^61 - 16 * q^64 + 4 * q^67 + 14 * q^73 + 4 * q^76 - 8 * q^79 - 28 * q^97

Introduction

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Varieties

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Database

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