LMFDB - Embedded newform 3192.2.a.q.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3192,2,Mod(1,3192)] 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("3192.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3192, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3192.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,2,0,2,0,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.4882483252$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3192.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.73205 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.46410 q^{13} -2.73205 q^{15} -1.26795 q^{17} -1.00000 q^{19} -1.00000 q^{21} +2.46410 q^{25} -1.00000 q^{27} -9.66025 q^{29} +2.00000 q^{31} +2.73205 q^{35} -10.0000 q^{37} +5.46410 q^{39} -8.92820 q^{41} +4.92820 q^{43} +2.73205 q^{45} -11.6603 q^{47} +1.00000 q^{49} +1.26795 q^{51} -6.73205 q^{53} +1.00000 q^{57} +8.00000 q^{59} -2.00000 q^{61} +1.00000 q^{63} -14.9282 q^{65} -13.4641 q^{67} +4.73205 q^{71} +10.3923 q^{73} -2.46410 q^{75} +4.00000 q^{79} +1.00000 q^{81} +10.1962 q^{83} -3.46410 q^{85} +9.66025 q^{87} +4.92820 q^{89} -5.46410 q^{91} -2.00000 q^{93} -2.73205 q^{95} +2.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} + 2 q^{35} - 20 q^{37} + 4 q^{39} - 4 q^{41} - 4 q^{43} + 2 q^{45}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^13 - 2 * q^15 - 6 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^25 - 2 * q^27 - 2 * q^29 + 4 * q^31 + 2 * q^35 - 20 * q^37 + 4 * q^39 - 4 * q^41 - 4 * q^43 + 2 * q^45 - 6 * q^47 + 2 * q^49 + 6 * q^51 - 10 * q^53 + 2 * q^57 + 16 * q^59 - 4 * q^61 + 2 * q^63 - 16 * q^65 - 20 * q^67 + 6 * q^71 + 2 * q^75 + 8 * q^79 + 2 * q^81 + 10 * q^83 + 2 * q^87 - 4 * q^89 - 4 * q^91 - 4 * q^93 - 2 * q^95 + 4 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.73205	1.22181	0.610905	−	0.791704i	$-0.290806\pi$
$5$	2.73205	1.22181	0.610905	+	0.791704i	$0.290806\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−5.46410	−1.51547	−0.757735	−	0.652563i	$-0.773694\pi$
$13$	−5.46410	−1.51547	−0.757735	+	0.652563i	$0.773694\pi$
$14$	0	0
$15$	−2.73205	−0.705412
$16$	0	0
$17$	−1.26795	−0.307523	−0.153761	−	0.988108i	$-0.549139\pi$
$17$	−1.26795	−0.307523	−0.153761	+	0.988108i	$0.549139\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−9.66025	−1.79386	−0.896932	−	0.442168i	$-0.854209\pi$
$29$	−9.66025	−1.79386	−0.896932	+	0.442168i	$0.854209\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.73205	0.461801
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	0	0
$39$	5.46410	0.874957
$40$	0	0
$41$	−8.92820	−1.39435	−0.697176	−	0.716900i	$-0.745560\pi$
$41$	−8.92820	−1.39435	−0.697176	+	0.716900i	$0.745560\pi$
$42$	0	0
$43$	4.92820	0.751544	0.375772	−	0.926712i	$-0.377378\pi$
$43$	4.92820	0.751544	0.375772	+	0.926712i	$0.377378\pi$
$44$	0	0
$45$	2.73205	0.407270
$46$	0	0
$47$	−11.6603	−1.70082	−0.850411	−	0.526118i	$-0.823647\pi$
$47$	−11.6603	−1.70082	−0.850411	+	0.526118i	$0.823647\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.26795	0.177548
$52$	0	0
$53$	−6.73205	−0.924718	−0.462359	−	0.886693i	$-0.652997\pi$
$53$	−6.73205	−0.924718	−0.462359	+	0.886693i	$0.652997\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−14.9282	−1.85162
$66$	0	0
$67$	−13.4641	−1.64490	−0.822451	−	0.568836i	$-0.807394\pi$
$67$	−13.4641	−1.64490	−0.822451	+	0.568836i	$0.807394\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.73205	0.561591	0.280796	−	0.959768i	$-0.409402\pi$
$71$	4.73205	0.561591	0.280796	+	0.959768i	$0.409402\pi$
$72$	0	0
$73$	10.3923	1.21633	0.608164	−	0.793812i	$-0.291906\pi$
$73$	10.3923	1.21633	0.608164	+	0.793812i	$0.291906\pi$
$74$	0	0
$75$	−2.46410	−0.284530
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.1962	1.11917	0.559587	−	0.828772i	$-0.310960\pi$
$83$	10.1962	1.11917	0.559587	+	0.828772i	$0.310960\pi$
$84$	0	0
$85$	−3.46410	−0.375735
$86$	0	0
$87$	9.66025	1.03569
$88$	0	0
$89$	4.92820	0.522388	0.261194	−	0.965286i	$-0.415884\pi$
$89$	4.92820	0.522388	0.261194	+	0.965286i	$0.415884\pi$
$90$	0	0
$91$	−5.46410	−0.572793
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	−2.73205	−0.280302
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.26795	0.126166	0.0630828	−	0.998008i	$-0.479907\pi$
$101$	1.26795	0.126166	0.0630828	+	0.998008i	$0.479907\pi$
$102$	0	0
$103$	−2.92820	−0.288524	−0.144262	−	0.989539i	$-0.546081\pi$
$103$	−2.92820	−0.288524	−0.144262	+	0.989539i	$0.546081\pi$
$104$	0	0
$105$	−2.73205	−0.266621
$106$	0	0
$107$	−4.73205	−0.457465	−0.228732	−	0.973489i	$-0.573458\pi$
$107$	−4.73205	−0.457465	−0.228732	+	0.973489i	$0.573458\pi$
$108$	0	0
$109$	2.39230	0.229141	0.114571	−	0.993415i	$-0.463451\pi$
$109$	2.39230	0.229141	0.114571	+	0.993415i	$0.463451\pi$
$110$	0	0
$111$	10.0000	0.949158
$112$	0	0
$113$	0.196152	0.0184525	0.00922623	−	0.999957i	$-0.497063\pi$
$113$	0.196152	0.0184525	0.00922623	+	0.999957i	$0.497063\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.46410	−0.505156
$118$	0	0
$119$	−1.26795	−0.116233
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	8.92820	0.805029
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	20.3923	1.80952	0.904762	−	0.425917i	$-0.140048\pi$
$127$	20.3923	1.80952	0.904762	+	0.425917i	$0.140048\pi$
$128$	0	0
$129$	−4.92820	−0.433904
$130$	0	0
$131$	14.5885	1.27460	0.637300	−	0.770616i	$-0.280051\pi$
$131$	14.5885	1.27460	0.637300	+	0.770616i	$0.280051\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	−2.73205	−0.235137
$136$	0	0
$137$	12.9282	1.10453	0.552265	−	0.833668i	$-0.313763\pi$
$137$	12.9282	1.10453	0.552265	+	0.833668i	$0.313763\pi$
$138$	0	0
$139$	−15.3205	−1.29947	−0.649734	−	0.760161i	$-0.725120\pi$
$139$	−15.3205	−1.29947	−0.649734	+	0.760161i	$0.725120\pi$
$140$	0	0
$141$	11.6603	0.981971
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−26.3923	−2.19176
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	14.3923	1.17906	0.589532	−	0.807745i	$-0.299312\pi$
$149$	14.3923	1.17906	0.589532	+	0.807745i	$0.299312\pi$
$150$	0	0
$151$	−2.53590	−0.206368	−0.103184	−	0.994662i	$-0.532903\pi$
$151$	−2.53590	−0.206368	−0.103184	+	0.994662i	$0.532903\pi$
$152$	0	0
$153$	−1.26795	−0.102508
$154$	0	0
$155$	5.46410	0.438887
$156$	0	0
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	0	0
$159$	6.73205	0.533886
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−7.07180	−0.553906	−0.276953	−	0.960883i	$-0.589325\pi$
$163$	−7.07180	−0.553906	−0.276953	+	0.960883i	$0.589325\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.3205	0.876007	0.438004	−	0.898973i	$-0.355686\pi$
$167$	11.3205	0.876007	0.438004	+	0.898973i	$0.355686\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−16.9282	−1.28703	−0.643514	−	0.765435i	$-0.722524\pi$
$173$	−16.9282	−1.28703	−0.643514	+	0.765435i	$0.722524\pi$
$174$	0	0
$175$	2.46410	0.186269
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	−14.1962	−1.06107	−0.530535	−	0.847663i	$-0.678009\pi$
$179$	−14.1962	−1.06107	−0.530535	+	0.847663i	$0.678009\pi$
$180$	0	0
$181$	4.92820	0.366310	0.183155	−	0.983084i	$-0.441369\pi$
$181$	4.92820	0.366310	0.183155	+	0.983084i	$0.441369\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−27.3205	−2.00864
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−14.9282	−1.08017	−0.540083	−	0.841611i	$-0.681607\pi$
$191$	−14.9282	−1.08017	−0.540083	+	0.841611i	$0.681607\pi$
$192$	0	0
$193$	−23.8564	−1.71722	−0.858611	−	0.512628i	$-0.828672\pi$
$193$	−23.8564	−1.71722	−0.858611	+	0.512628i	$0.828672\pi$
$194$	0	0
$195$	14.9282	1.06903
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	−12.3923	−0.878467	−0.439234	−	0.898373i	$-0.644750\pi$
$199$	−12.3923	−0.878467	−0.439234	+	0.898373i	$0.644750\pi$
$200$	0	0
$201$	13.4641	0.949685
$202$	0	0
$203$	−9.66025	−0.678017
$204$	0	0
$205$	−24.3923	−1.70363
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	−4.73205	−0.324235
$214$	0	0
$215$	13.4641	0.918244
$216$	0	0
$217$	2.00000	0.135769
$218$	0	0
$219$	−10.3923	−0.702247
$220$	0	0
$221$	6.92820	0.466041
$222$	0	0
$223$	−20.9282	−1.40146	−0.700728	−	0.713428i	$-0.747141\pi$
$223$	−20.9282	−1.40146	−0.700728	+	0.713428i	$0.747141\pi$
$224$	0	0
$225$	2.46410	0.164273
$226$	0	0
$227$	16.3923	1.08800	0.543998	−	0.839087i	$-0.316910\pi$
$227$	16.3923	1.08800	0.543998	+	0.839087i	$0.316910\pi$
$228$	0	0
$229$	28.2487	1.86673	0.933364	−	0.358932i	$-0.116859\pi$
$229$	28.2487	1.86673	0.933364	+	0.358932i	$0.116859\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.39230	−0.156725	−0.0783626	−	0.996925i	$-0.524969\pi$
$233$	−2.39230	−0.156725	−0.0783626	+	0.996925i	$0.524969\pi$
$234$	0	0
$235$	−31.8564	−2.07808
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	−4.39230	−0.284115	−0.142057	−	0.989858i	$-0.545372\pi$
$239$	−4.39230	−0.284115	−0.142057	+	0.989858i	$0.545372\pi$
$240$	0	0
$241$	−12.9282	−0.832779	−0.416389	−	0.909186i	$-0.636705\pi$
$241$	−12.9282	−0.832779	−0.416389	+	0.909186i	$0.636705\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.73205	0.174544
$246$	0	0
$247$	5.46410	0.347672
$248$	0	0
$249$	−10.1962	−0.646155
$250$	0	0
$251$	0.339746	0.0214446	0.0107223	−	0.999943i	$-0.496587\pi$
$251$	0.339746	0.0214446	0.0107223	+	0.999943i	$0.496587\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	3.46410	0.216930
$256$	0	0
$257$	11.4641	0.715111	0.357556	−	0.933892i	$-0.383610\pi$
$257$	11.4641	0.715111	0.357556	+	0.933892i	$0.383610\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	−9.66025	−0.597955
$262$	0	0
$263$	−19.3205	−1.19135	−0.595677	−	0.803224i	$-0.703116\pi$
$263$	−19.3205	−1.19135	−0.595677	+	0.803224i	$0.703116\pi$
$264$	0	0
$265$	−18.3923	−1.12983
$266$	0	0
$267$	−4.92820	−0.301601
$268$	0	0
$269$	31.1769	1.90089	0.950445	−	0.310893i	$-0.100628\pi$
$269$	31.1769	1.90089	0.950445	+	0.310893i	$0.100628\pi$
$270$	0	0
$271$	−28.7846	−1.74854	−0.874270	−	0.485440i	$-0.838660\pi$
$271$	−28.7846	−1.74854	−0.874270	+	0.485440i	$0.838660\pi$
$272$	0	0
$273$	5.46410	0.330702
$274$	0	0
$275$	0	0
$276$	0	0
$277$	12.3923	0.744581	0.372291	−	0.928116i	$-0.378572\pi$
$277$	12.3923	0.744581	0.372291	+	0.928116i	$0.378572\pi$
$278$	0	0
$279$	2.00000	0.119737
$280$	0	0
$281$	−4.19615	−0.250321	−0.125161	−	0.992136i	$-0.539945\pi$
$281$	−4.19615	−0.250321	−0.125161	+	0.992136i	$0.539945\pi$
$282$	0	0
$283$	−20.3923	−1.21220	−0.606098	−	0.795390i	$-0.707266\pi$
$283$	−20.3923	−1.21220	−0.606098	+	0.795390i	$0.707266\pi$
$284$	0	0
$285$	2.73205	0.161833
$286$	0	0
$287$	−8.92820	−0.527015
$288$	0	0
$289$	−15.3923	−0.905430
$290$	0	0
$291$	−2.00000	−0.117242
$292$	0	0
$293$	−6.00000	−0.350524	−0.175262	−	0.984522i	$-0.556077\pi$
$293$	−6.00000	−0.350524	−0.175262	+	0.984522i	$0.556077\pi$
$294$	0	0
$295$	21.8564	1.27253
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	4.92820	0.284057
$302$	0	0
$303$	−1.26795	−0.0728418
$304$	0	0
$305$	−5.46410	−0.312874
$306$	0	0
$307$	18.7846	1.07209	0.536047	−	0.844188i	$-0.319917\pi$
$307$	18.7846	1.07209	0.536047	+	0.844188i	$0.319917\pi$
$308$	0	0
$309$	2.92820	0.166580
$310$	0	0
$311$	18.5885	1.05405	0.527027	−	0.849848i	$-0.323307\pi$
$311$	18.5885	1.05405	0.527027	+	0.849848i	$0.323307\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	0	0
$315$	2.73205	0.153934
$316$	0	0
$317$	12.1962	0.685004	0.342502	−	0.939517i	$-0.388726\pi$
$317$	12.1962	0.685004	0.342502	+	0.939517i	$0.388726\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	4.73205	0.264117
$322$	0	0
$323$	1.26795	0.0705506
$324$	0	0
$325$	−13.4641	−0.746854
$326$	0	0
$327$	−2.39230	−0.132295
$328$	0	0
$329$	−11.6603	−0.642851
$330$	0	0
$331$	22.2487	1.22290	0.611450	−	0.791283i	$-0.290587\pi$
$331$	22.2487	1.22290	0.611450	+	0.791283i	$0.290587\pi$
$332$	0	0
$333$	−10.0000	−0.547997
$334$	0	0
$335$	−36.7846	−2.00976
$336$	0	0
$337$	30.3923	1.65557	0.827787	−	0.561042i	$-0.189599\pi$
$337$	30.3923	1.65557	0.827787	+	0.561042i	$0.189599\pi$
$338$	0	0
$339$	−0.196152	−0.0106535
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.3923	−0.879985	−0.439993	−	0.898001i	$-0.645019\pi$
$347$	−16.3923	−0.879985	−0.439993	+	0.898001i	$0.645019\pi$
$348$	0	0
$349$	−19.0718	−1.02089	−0.510445	−	0.859910i	$-0.670519\pi$
$349$	−19.0718	−1.02089	−0.510445	+	0.859910i	$0.670519\pi$
$350$	0	0
$351$	5.46410	0.291652
$352$	0	0
$353$	16.1962	0.862034	0.431017	−	0.902344i	$-0.358155\pi$
$353$	16.1962	0.862034	0.431017	+	0.902344i	$0.358155\pi$
$354$	0	0
$355$	12.9282	0.686158
$356$	0	0
$357$	1.26795	0.0671070
$358$	0	0
$359$	30.9282	1.63233	0.816164	−	0.577820i	$-0.196097\pi$
$359$	30.9282	1.63233	0.816164	+	0.577820i	$0.196097\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	28.3923	1.48612
$366$	0	0
$367$	18.5359	0.967566	0.483783	−	0.875188i	$-0.339262\pi$
$367$	18.5359	0.967566	0.483783	+	0.875188i	$0.339262\pi$
$368$	0	0
$369$	−8.92820	−0.464784
$370$	0	0
$371$	−6.73205	−0.349511
$372$	0	0
$373$	0.928203	0.0480605	0.0240303	−	0.999711i	$-0.492350\pi$
$373$	0.928203	0.0480605	0.0240303	+	0.999711i	$0.492350\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	52.7846	2.71855
$378$	0	0
$379$	−15.3205	−0.786962	−0.393481	−	0.919333i	$-0.628729\pi$
$379$	−15.3205	−0.786962	−0.393481	+	0.919333i	$0.628729\pi$
$380$	0	0
$381$	−20.3923	−1.04473
$382$	0	0
$383$	−11.3205	−0.578451	−0.289225	−	0.957261i	$-0.593398\pi$
$383$	−11.3205	−0.578451	−0.289225	+	0.957261i	$0.593398\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.92820	0.250515
$388$	0	0
$389$	−20.5359	−1.04121	−0.520606	−	0.853797i	$-0.674294\pi$
$389$	−20.5359	−1.04121	−0.520606	+	0.853797i	$0.674294\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−14.5885	−0.735890
$394$	0	0
$395$	10.9282	0.549858
$396$	0	0
$397$	−21.7128	−1.08973	−0.544867	−	0.838522i	$-0.683420\pi$
$397$	−21.7128	−1.08973	−0.544867	+	0.838522i	$0.683420\pi$
$398$	0	0
$399$	1.00000	0.0500626
$400$	0	0
$401$	0.196152	0.00979538	0.00489769	−	0.999988i	$-0.498441\pi$
$401$	0.196152	0.00979538	0.00489769	+	0.999988i	$0.498441\pi$
$402$	0	0
$403$	−10.9282	−0.544373
$404$	0	0
$405$	2.73205	0.135757
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−28.3923	−1.40391	−0.701955	−	0.712222i	$-0.747689\pi$
$409$	−28.3923	−1.40391	−0.701955	+	0.712222i	$0.747689\pi$
$410$	0	0
$411$	−12.9282	−0.637701
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	27.8564	1.36742
$416$	0	0
$417$	15.3205	0.750249
$418$	0	0
$419$	20.7321	1.01283	0.506413	−	0.862291i	$-0.330971\pi$
$419$	20.7321	1.01283	0.506413	+	0.862291i	$0.330971\pi$
$420$	0	0
$421$	−32.2487	−1.57171	−0.785853	−	0.618413i	$-0.787776\pi$
$421$	−32.2487	−1.57171	−0.785853	+	0.618413i	$0.787776\pi$
$422$	0	0
$423$	−11.6603	−0.566941
$424$	0	0
$425$	−3.12436	−0.151554
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0.0525589	0.00253167	0.00126584	−	0.999999i	$-0.499597\pi$
$431$	0.0525589	0.00253167	0.00126584	+	0.999999i	$0.499597\pi$
$432$	0	0
$433$	−15.8564	−0.762010	−0.381005	−	0.924573i	$-0.624422\pi$
$433$	−15.8564	−0.762010	−0.381005	+	0.924573i	$0.624422\pi$
$434$	0	0
$435$	26.3923	1.26541
$436$	0	0
$437$	0	0
$438$	0	0
$439$	35.8564	1.71133	0.855666	−	0.517528i	$-0.173148\pi$
$439$	35.8564	1.71133	0.855666	+	0.517528i	$0.173148\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−34.5359	−1.64085	−0.820425	−	0.571754i	$-0.806263\pi$
$443$	−34.5359	−1.64085	−0.820425	+	0.571754i	$0.806263\pi$
$444$	0	0
$445$	13.4641	0.638260
$446$	0	0
$447$	−14.3923	−0.680733
$448$	0	0
$449$	5.26795	0.248610	0.124305	−	0.992244i	$-0.460330\pi$
$449$	5.26795	0.248610	0.124305	+	0.992244i	$0.460330\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	2.53590	0.119147
$454$	0	0
$455$	−14.9282	−0.699845
$456$	0	0
$457$	−20.3923	−0.953912	−0.476956	−	0.878927i	$-0.658260\pi$
$457$	−20.3923	−0.953912	−0.476956	+	0.878927i	$0.658260\pi$
$458$	0	0
$459$	1.26795	0.0591828
$460$	0	0
$461$	14.3397	0.667869	0.333934	−	0.942596i	$-0.391624\pi$
$461$	14.3397	0.667869	0.333934	+	0.942596i	$0.391624\pi$
$462$	0	0
$463$	21.8564	1.01575	0.507877	−	0.861430i	$-0.330431\pi$
$463$	21.8564	1.01575	0.507877	+	0.861430i	$0.330431\pi$
$464$	0	0
$465$	−5.46410	−0.253392
$466$	0	0
$467$	−0.0525589	−0.00243214	−0.00121607	−	0.999999i	$-0.500387\pi$
$467$	−0.0525589	−0.00243214	−0.00121607	+	0.999999i	$0.500387\pi$
$468$	0	0
$469$	−13.4641	−0.621714
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.46410	−0.113061
$476$	0	0
$477$	−6.73205	−0.308239
$478$	0	0
$479$	27.6603	1.26383	0.631915	−	0.775038i	$-0.282269\pi$
$479$	27.6603	1.26383	0.631915	+	0.775038i	$0.282269\pi$
$480$	0	0
$481$	54.6410	2.49142
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.46410	0.248112
$486$	0	0
$487$	25.8564	1.17167	0.585833	−	0.810432i	$-0.300768\pi$
$487$	25.8564	1.17167	0.585833	+	0.810432i	$0.300768\pi$
$488$	0	0
$489$	7.07180	0.319798
$490$	0	0
$491$	19.3205	0.871922	0.435961	−	0.899965i	$-0.356409\pi$
$491$	19.3205	0.871922	0.435961	+	0.899965i	$0.356409\pi$
$492$	0	0
$493$	12.2487	0.551654
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.73205	0.212261
$498$	0	0
$499$	−9.85641	−0.441233	−0.220617	−	0.975361i	$-0.570807\pi$
$499$	−9.85641	−0.441233	−0.220617	+	0.975361i	$0.570807\pi$
$500$	0	0
$501$	−11.3205	−0.505763
$502$	0	0
$503$	4.73205	0.210992	0.105496	−	0.994420i	$-0.466357\pi$
$503$	4.73205	0.210992	0.105496	+	0.994420i	$0.466357\pi$
$504$	0	0
$505$	3.46410	0.154150
$506$	0	0
$507$	−16.8564	−0.748619
$508$	0	0
$509$	−8.24871	−0.365618	−0.182809	−	0.983148i	$-0.558519\pi$
$509$	−8.24871	−0.365618	−0.182809	+	0.983148i	$0.558519\pi$
$510$	0	0
$511$	10.3923	0.459728
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	0	0
$518$	0	0
$519$	16.9282	0.743066
$520$	0	0
$521$	36.2487	1.58808	0.794042	−	0.607862i	$-0.207973\pi$
$521$	36.2487	1.58808	0.794042	+	0.607862i	$0.207973\pi$
$522$	0	0
$523$	−22.0000	−0.961993	−0.480996	−	0.876723i	$-0.659725\pi$
$523$	−22.0000	−0.961993	−0.480996	+	0.876723i	$0.659725\pi$
$524$	0	0
$525$	−2.46410	−0.107542
$526$	0	0
$527$	−2.53590	−0.110465
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	48.7846	2.11310
$534$	0	0
$535$	−12.9282	−0.558935
$536$	0	0
$537$	14.1962	0.612609
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	0	0
$543$	−4.92820	−0.211489
$544$	0	0
$545$	6.53590	0.279967
$546$	0	0
$547$	−11.7128	−0.500804	−0.250402	−	0.968142i	$-0.580563\pi$
$547$	−11.7128	−0.500804	−0.250402	+	0.968142i	$0.580563\pi$
$548$	0	0
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	9.66025	0.411541
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	27.3205	1.15969
$556$	0	0
$557$	20.5359	0.870134	0.435067	−	0.900398i	$-0.356725\pi$
$557$	20.5359	0.870134	0.435067	+	0.900398i	$0.356725\pi$
$558$	0	0
$559$	−26.9282	−1.13894
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−25.4641	−1.07318	−0.536592	−	0.843842i	$-0.680289\pi$
$563$	−25.4641	−1.07318	−0.536592	+	0.843842i	$0.680289\pi$
$564$	0	0
$565$	0.535898	0.0225454
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	28.5885	1.19849	0.599245	−	0.800566i	$-0.295467\pi$
$569$	28.5885	1.19849	0.599245	+	0.800566i	$0.295467\pi$
$570$	0	0
$571$	−25.8564	−1.08206	−0.541028	−	0.841004i	$-0.681965\pi$
$571$	−25.8564	−1.08206	−0.541028	+	0.841004i	$0.681965\pi$
$572$	0	0
$573$	14.9282	0.623635
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	0	0
$579$	23.8564	0.991438
$580$	0	0
$581$	10.1962	0.423008
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−14.9282	−0.617205
$586$	0	0
$587$	25.9090	1.06938	0.534689	−	0.845049i	$-0.320429\pi$
$587$	25.9090	1.06938	0.534689	+	0.845049i	$0.320429\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	18.0000	0.740421
$592$	0	0
$593$	27.8038	1.14177	0.570884	−	0.821031i	$-0.306601\pi$
$593$	27.8038	1.14177	0.570884	+	0.821031i	$0.306601\pi$
$594$	0	0
$595$	−3.46410	−0.142014
$596$	0	0
$597$	12.3923	0.507183
$598$	0	0
$599$	−7.26795	−0.296960	−0.148480	−	0.988915i	$-0.547438\pi$
$599$	−7.26795	−0.296960	−0.148480	+	0.988915i	$0.547438\pi$
$600$	0	0
$601$	34.0000	1.38689	0.693444	−	0.720510i	$-0.256092\pi$
$601$	34.0000	1.38689	0.693444	+	0.720510i	$0.256092\pi$
$602$	0	0
$603$	−13.4641	−0.548301
$604$	0	0
$605$	−30.0526	−1.22181
$606$	0	0
$607$	−13.0718	−0.530568	−0.265284	−	0.964170i	$-0.585466\pi$
$607$	−13.0718	−0.530568	−0.265284	+	0.964170i	$0.585466\pi$
$608$	0	0
$609$	9.66025	0.391453
$610$	0	0
$611$	63.7128	2.57754
$612$	0	0
$613$	−18.2487	−0.737059	−0.368529	−	0.929616i	$-0.620139\pi$
$613$	−18.2487	−0.737059	−0.368529	+	0.929616i	$0.620139\pi$
$614$	0	0
$615$	24.3923	0.983593
$616$	0	0
$617$	−24.6410	−0.992010	−0.496005	−	0.868320i	$-0.665200\pi$
$617$	−24.6410	−0.992010	−0.496005	+	0.868320i	$0.665200\pi$
$618$	0	0
$619$	−13.1769	−0.529625	−0.264812	−	0.964300i	$-0.585310\pi$
$619$	−13.1769	−0.529625	−0.264812	+	0.964300i	$0.585310\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.92820	0.197444
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.6795	0.505564
$630$	0	0
$631$	40.9282	1.62933	0.814663	−	0.579935i	$-0.196922\pi$
$631$	40.9282	1.62933	0.814663	+	0.579935i	$0.196922\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	55.7128	2.21090
$636$	0	0
$637$	−5.46410	−0.216496
$638$	0	0
$639$	4.73205	0.187197
$640$	0	0
$641$	−6.33975	−0.250405	−0.125202	−	0.992131i	$-0.539958\pi$
$641$	−6.33975	−0.250405	−0.125202	+	0.992131i	$0.539958\pi$
$642$	0	0
$643$	20.3923	0.804194	0.402097	−	0.915597i	$-0.368281\pi$
$643$	20.3923	0.804194	0.402097	+	0.915597i	$0.368281\pi$
$644$	0	0
$645$	−13.4641	−0.530148
$646$	0	0
$647$	−3.66025	−0.143899	−0.0719497	−	0.997408i	$-0.522922\pi$
$647$	−3.66025	−0.143899	−0.0719497	+	0.997408i	$0.522922\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.00000	−0.0783862
$652$	0	0
$653$	−42.7846	−1.67429	−0.837146	−	0.546980i	$-0.815777\pi$
$653$	−42.7846	−1.67429	−0.837146	+	0.546980i	$0.815777\pi$
$654$	0	0
$655$	39.8564	1.55732
$656$	0	0
$657$	10.3923	0.405442
$658$	0	0
$659$	32.0526	1.24859	0.624295	−	0.781189i	$-0.285386\pi$
$659$	32.0526	1.24859	0.624295	+	0.781189i	$0.285386\pi$
$660$	0	0
$661$	−6.53590	−0.254217	−0.127108	−	0.991889i	$-0.540570\pi$
$661$	−6.53590	−0.254217	−0.127108	+	0.991889i	$0.540570\pi$
$662$	0	0
$663$	−6.92820	−0.269069
$664$	0	0
$665$	−2.73205	−0.105944
$666$	0	0
$667$	0	0
$668$	0	0
$669$	20.9282	0.809131
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−7.07180	−0.272598	−0.136299	−	0.990668i	$-0.543521\pi$
$673$	−7.07180	−0.272598	−0.136299	+	0.990668i	$0.543521\pi$
$674$	0	0
$675$	−2.46410	−0.0948433
$676$	0	0
$677$	29.3205	1.12688	0.563439	−	0.826157i	$-0.309478\pi$
$677$	29.3205	1.12688	0.563439	+	0.826157i	$0.309478\pi$
$678$	0	0
$679$	2.00000	0.0767530
$680$	0	0
$681$	−16.3923	−0.628154
$682$	0	0
$683$	4.73205	0.181067	0.0905334	−	0.995893i	$-0.471143\pi$
$683$	4.73205	0.181067	0.0905334	+	0.995893i	$0.471143\pi$
$684$	0	0
$685$	35.3205	1.34953
$686$	0	0
$687$	−28.2487	−1.07776
$688$	0	0
$689$	36.7846	1.40138
$690$	0	0
$691$	−8.78461	−0.334182	−0.167091	−	0.985941i	$-0.553437\pi$
$691$	−8.78461	−0.334182	−0.167091	+	0.985941i	$0.553437\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−41.8564	−1.58770
$696$	0	0
$697$	11.3205	0.428795
$698$	0	0
$699$	2.39230	0.0904853
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	10.0000	0.377157
$704$	0	0
$705$	31.8564	1.19978
$706$	0	0
$707$	1.26795	0.0476861
$708$	0	0
$709$	0.392305	0.0147333	0.00736666	−	0.999973i	$-0.497655\pi$
$709$	0.392305	0.0147333	0.00736666	+	0.999973i	$0.497655\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.39230	0.164034
$718$	0	0
$719$	46.9808	1.75209	0.876043	−	0.482232i	$-0.160174\pi$
$719$	46.9808	1.75209	0.876043	+	0.482232i	$0.160174\pi$
$720$	0	0
$721$	−2.92820	−0.109052
$722$	0	0
$723$	12.9282	0.480805
$724$	0	0
$725$	−23.8038	−0.884053
$726$	0	0
$727$	18.9282	0.702008	0.351004	−	0.936374i	$-0.385840\pi$
$727$	18.9282	0.702008	0.351004	+	0.936374i	$0.385840\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.24871	−0.231117
$732$	0	0
$733$	4.24871	0.156930	0.0784649	−	0.996917i	$-0.474998\pi$
$733$	4.24871	0.156930	0.0784649	+	0.996917i	$0.474998\pi$
$734$	0	0
$735$	−2.73205	−0.100773
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−20.1436	−0.740994	−0.370497	−	0.928834i	$-0.620813\pi$
$739$	−20.1436	−0.740994	−0.370497	+	0.928834i	$0.620813\pi$
$740$	0	0
$741$	−5.46410	−0.200729
$742$	0	0
$743$	−8.73205	−0.320348	−0.160174	−	0.987089i	$-0.551206\pi$
$743$	−8.73205	−0.320348	−0.160174	+	0.987089i	$0.551206\pi$
$744$	0	0
$745$	39.3205	1.44059
$746$	0	0
$747$	10.1962	0.373058
$748$	0	0
$749$	−4.73205	−0.172905
$750$	0	0
$751$	−49.1769	−1.79449	−0.897246	−	0.441532i	$-0.854435\pi$
$751$	−49.1769	−1.79449	−0.897246	+	0.441532i	$0.854435\pi$
$752$	0	0
$753$	−0.339746	−0.0123810
$754$	0	0
$755$	−6.92820	−0.252143
$756$	0	0
$757$	−22.7846	−0.828121	−0.414060	−	0.910249i	$-0.635890\pi$
$757$	−22.7846	−0.828121	−0.414060	+	0.910249i	$0.635890\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.6603	1.22018	0.610092	−	0.792331i	$-0.291133\pi$
$761$	33.6603	1.22018	0.610092	+	0.792331i	$0.291133\pi$
$762$	0	0
$763$	2.39230	0.0866073
$764$	0	0
$765$	−3.46410	−0.125245
$766$	0	0
$767$	−43.7128	−1.57838
$768$	0	0
$769$	0.143594	0.00517812	0.00258906	−	0.999997i	$-0.499176\pi$
$769$	0.143594	0.00517812	0.00258906	+	0.999997i	$0.499176\pi$
$770$	0	0
$771$	−11.4641	−0.412870
$772$	0	0
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	4.92820	0.177026
$776$	0	0
$777$	10.0000	0.358748
$778$	0	0
$779$	8.92820	0.319886
$780$	0	0
$781$	0	0
$782$	0	0
$783$	9.66025	0.345229
$784$	0	0
$785$	−38.2487	−1.36516
$786$	0	0
$787$	27.8564	0.992974	0.496487	−	0.868044i	$-0.334623\pi$
$787$	27.8564	0.992974	0.496487	+	0.868044i	$0.334623\pi$
$788$	0	0
$789$	19.3205	0.687828
$790$	0	0
$791$	0.196152	0.00697438
$792$	0	0
$793$	10.9282	0.388072
$794$	0	0
$795$	18.3923	0.652308
$796$	0	0
$797$	−10.7846	−0.382010	−0.191005	−	0.981589i	$-0.561175\pi$
$797$	−10.7846	−0.382010	−0.191005	+	0.981589i	$0.561175\pi$
$798$	0	0
$799$	14.7846	0.523042
$800$	0	0
$801$	4.92820	0.174129
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−31.1769	−1.09748
$808$	0	0
$809$	−10.7846	−0.379167	−0.189583	−	0.981865i	$-0.560714\pi$
$809$	−10.7846	−0.379167	−0.189583	+	0.981865i	$0.560714\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	0	0
$813$	28.7846	1.00952
$814$	0	0
$815$	−19.3205	−0.676768
$816$	0	0
$817$	−4.92820	−0.172416
$818$	0	0
$819$	−5.46410	−0.190931
$820$	0	0
$821$	35.4641	1.23771	0.618853	−	0.785507i	$-0.287598\pi$
$821$	35.4641	1.23771	0.618853	+	0.785507i	$0.287598\pi$
$822$	0	0
$823$	20.6410	0.719501	0.359750	−	0.933049i	$-0.382862\pi$
$823$	20.6410	0.719501	0.359750	+	0.933049i	$0.382862\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−53.9090	−1.87460	−0.937299	−	0.348526i	$-0.886682\pi$
$827$	−53.9090	−1.87460	−0.937299	+	0.348526i	$0.886682\pi$
$828$	0	0
$829$	−37.7128	−1.30982	−0.654910	−	0.755707i	$-0.727294\pi$
$829$	−37.7128	−1.30982	−0.654910	+	0.755707i	$0.727294\pi$
$830$	0	0
$831$	−12.3923	−0.429884
$832$	0	0
$833$	−1.26795	−0.0439318
$834$	0	0
$835$	30.9282	1.07031
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	−28.7846	−0.993755	−0.496878	−	0.867821i	$-0.665520\pi$
$839$	−28.7846	−0.993755	−0.496878	+	0.867821i	$0.665520\pi$
$840$	0	0
$841$	64.3205	2.21795
$842$	0	0
$843$	4.19615	0.144523
$844$	0	0
$845$	46.0526	1.58426
$846$	0	0
$847$	−11.0000	−0.377964
$848$	0	0
$849$	20.3923	0.699862
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−16.1436	−0.552746	−0.276373	−	0.961050i	$-0.589133\pi$
$853$	−16.1436	−0.552746	−0.276373	+	0.961050i	$0.589133\pi$
$854$	0	0
$855$	−2.73205	−0.0934342
$856$	0	0
$857$	16.9282	0.578256	0.289128	−	0.957290i	$-0.406635\pi$
$857$	16.9282	0.578256	0.289128	+	0.957290i	$0.406635\pi$
$858$	0	0
$859$	45.8564	1.56460	0.782300	−	0.622902i	$-0.214046\pi$
$859$	45.8564	1.56460	0.782300	+	0.622902i	$0.214046\pi$
$860$	0	0
$861$	8.92820	0.304272
$862$	0	0
$863$	−42.9808	−1.46308	−0.731541	−	0.681797i	$-0.761199\pi$
$863$	−42.9808	−1.46308	−0.731541	+	0.681797i	$0.761199\pi$
$864$	0	0
$865$	−46.2487	−1.57250
$866$	0	0
$867$	15.3923	0.522750
$868$	0	0
$869$	0	0
$870$	0	0
$871$	73.5692	2.49280
$872$	0	0
$873$	2.00000	0.0676897
$874$	0	0
$875$	−6.92820	−0.234216
$876$	0	0
$877$	−53.7128	−1.81375	−0.906876	−	0.421397i	$-0.861540\pi$
$877$	−53.7128	−1.81375	−0.906876	+	0.421397i	$0.861540\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	−2.73205	−0.0920451	−0.0460226	−	0.998940i	$-0.514655\pi$
$881$	−2.73205	−0.0920451	−0.0460226	+	0.998940i	$0.514655\pi$
$882$	0	0
$883$	9.07180	0.305290	0.152645	−	0.988281i	$-0.451221\pi$
$883$	9.07180	0.305290	0.152645	+	0.988281i	$0.451221\pi$
$884$	0	0
$885$	−21.8564	−0.734695
$886$	0	0
$887$	42.2487	1.41857	0.709286	−	0.704920i	$-0.249017\pi$
$887$	42.2487	1.41857	0.709286	+	0.704920i	$0.249017\pi$
$888$	0	0
$889$	20.3923	0.683936
$890$	0	0
$891$	0	0
$892$	0	0
$893$	11.6603	0.390196
$894$	0	0
$895$	−38.7846	−1.29643
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−19.3205	−0.644375
$900$	0	0
$901$	8.53590	0.284372
$902$	0	0
$903$	−4.92820	−0.164000
$904$	0	0
$905$	13.4641	0.447562
$906$	0	0
$907$	14.6410	0.486147	0.243073	−	0.970008i	$-0.421844\pi$
$907$	14.6410	0.486147	0.243073	+	0.970008i	$0.421844\pi$
$908$	0	0
$909$	1.26795	0.0420552
$910$	0	0
$911$	−12.7321	−0.421832	−0.210916	−	0.977504i	$-0.567645\pi$
$911$	−12.7321	−0.421832	−0.210916	+	0.977504i	$0.567645\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	5.46410	0.180638
$916$	0	0
$917$	14.5885	0.481753
$918$	0	0
$919$	−29.8564	−0.984872	−0.492436	−	0.870349i	$-0.663893\pi$
$919$	−29.8564	−0.984872	−0.492436	+	0.870349i	$0.663893\pi$
$920$	0	0
$921$	−18.7846	−0.618974
$922$	0	0
$923$	−25.8564	−0.851074
$924$	0	0
$925$	−24.6410	−0.810192
$926$	0	0
$927$	−2.92820	−0.0961748
$928$	0	0
$929$	−4.98076	−0.163414	−0.0817068	−	0.996656i	$-0.526037\pi$
$929$	−4.98076	−0.163414	−0.0817068	+	0.996656i	$0.526037\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−18.5885	−0.608559
$934$	0	0
$935$	0	0
$936$	0	0
$937$	48.6410	1.58903	0.794516	−	0.607243i	$-0.207724\pi$
$937$	48.6410	1.58903	0.794516	+	0.607243i	$0.207724\pi$
$938$	0	0
$939$	2.00000	0.0652675
$940$	0	0
$941$	−51.4641	−1.67768	−0.838841	−	0.544377i	$-0.816766\pi$
$941$	−51.4641	−1.67768	−0.838841	+	0.544377i	$0.816766\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−2.73205	−0.0888736
$946$	0	0
$947$	9.07180	0.294794	0.147397	−	0.989077i	$-0.452911\pi$
$947$	9.07180	0.294794	0.147397	+	0.989077i	$0.452911\pi$
$948$	0	0
$949$	−56.7846	−1.84331
$950$	0	0
$951$	−12.1962	−0.395487
$952$	0	0
$953$	40.1962	1.30208	0.651041	−	0.759043i	$-0.274333\pi$
$953$	40.1962	1.30208	0.651041	+	0.759043i	$0.274333\pi$
$954$	0	0
$955$	−40.7846	−1.31976
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.9282	0.417473
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−4.73205	−0.152488
$964$	0	0
$965$	−65.1769	−2.09812
$966$	0	0
$967$	−36.6410	−1.17830	−0.589148	−	0.808025i	$-0.700536\pi$
$967$	−36.6410	−1.17830	−0.589148	+	0.808025i	$0.700536\pi$
$968$	0	0
$969$	−1.26795	−0.0407324
$970$	0	0
$971$	4.28719	0.137582	0.0687912	−	0.997631i	$-0.478086\pi$
$971$	4.28719	0.137582	0.0687912	+	0.997631i	$0.478086\pi$
$972$	0	0
$973$	−15.3205	−0.491153
$974$	0	0
$975$	13.4641	0.431196
$976$	0	0
$977$	−28.3013	−0.905438	−0.452719	−	0.891653i	$-0.649546\pi$
$977$	−28.3013	−0.905438	−0.452719	+	0.891653i	$0.649546\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	2.39230	0.0763804
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	−49.1769	−1.56691
$986$	0	0
$987$	11.6603	0.371150
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−12.7846	−0.406117	−0.203058	−	0.979167i	$-0.565088\pi$
$991$	−12.7846	−0.406117	−0.203058	+	0.979167i	$0.565088\pi$
$992$	0	0
$993$	−22.2487	−0.706042
$994$	0	0
$995$	−33.8564	−1.07332
$996$	0	0
$997$	−19.8564	−0.628859	−0.314429	−	0.949281i	$-0.601813\pi$
$997$	−19.8564	−0.628859	−0.314429	+	0.949281i	$0.601813\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.2.a.q.1.2	✓	2
3.2	odd	2		9576.2.a.bj.1.1		2
4.3	odd	2		6384.2.a.bs.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.q.1.2	✓	2	1.1	even	1	trivial
6384.2.a.bs.1.2		2	4.3	odd	2
9576.2.a.bj.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3192.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.73205 q^{5} +1.00000 q^{7} +1.00000 q^{9} -5.46410 q^{13} -2.73205 q^{15} -1.26795 q^{17} -1.00000 q^{19} -1.00000 q^{21} +2.46410 q^{25} -1.00000 q^{27} -9.66025 q^{29} +2.00000 q^{31} +2.73205 q^{35} -10.0000 q^{37} +5.46410 q^{39} -8.92820 q^{41} +4.92820 q^{43} +2.73205 q^{45} -11.6603 q^{47} +1.00000 q^{49} +1.26795 q^{51} -6.73205 q^{53} +1.00000 q^{57} +8.00000 q^{59} -2.00000 q^{61} +1.00000 q^{63} -14.9282 q^{65} -13.4641 q^{67} +4.73205 q^{71} +10.3923 q^{73} -2.46410 q^{75} +4.00000 q^{79} +1.00000 q^{81} +10.1962 q^{83} -3.46410 q^{85} +9.66025 q^{87} +4.92820 q^{89} -5.46410 q^{91} -2.00000 q^{93} -2.73205 q^{95} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{13} - 2 q^{15} - 6 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{25} - 2 q^{27} - 2 q^{29} + 4 q^{31} + 2 q^{35} - 20 q^{37} + 4 q^{39} - 4 q^{41} - 4 q^{43} + 2 q^{45}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^13 - 2 * q^15 - 6 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^25 - 2 * q^27 - 2 * q^29 + 4 * q^31 + 2 * q^35 - 20 * q^37 + 4 * q^39 - 4 * q^41 - 4 * q^43 + 2 * q^45 - 6 * q^47 + 2 * q^49 + 6 * q^51 - 10 * q^53 + 2 * q^57 + 16 * q^59 - 4 * q^61 + 2 * q^63 - 16 * q^65 - 20 * q^67 + 6 * q^71 + 2 * q^75 + 8 * q^79 + 2 * q^81 + 10 * q^83 + 2 * q^87 - 4 * q^89 - 4 * q^91 - 4 * q^93 - 2 * q^95 + 4 * q^97

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