LMFDB - Embedded newform 6012.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$48.0060616952$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.826865.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 6x - 1 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 6x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 668)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.873948$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{5} -1.42676 q^{7} +O(q^{10})$	$q-2.00000 q^{5} -1.42676 q^{7} -2.19055 q^{11} +1.74790 q^{13} -3.26510 q^{17} +5.89213 q^{19} -6.12900 q^{23} -1.00000 q^{25} +0.321133 q^{29} +1.75308 q^{31} +2.85353 q^{35} -2.47243 q^{37} +5.98963 q^{41} -3.49579 q^{43} -2.39634 q^{47} -4.96434 q^{49} +1.51720 q^{53} +4.38110 q^{55} -5.65526 q^{59} -4.51463 q^{61} -3.49579 q^{65} -9.61442 q^{67} +6.66956 q^{71} +2.86390 q^{73} +3.12540 q^{77} +10.4997 q^{79} +7.24369 q^{83} +6.53020 q^{85} +0.256634 q^{89} -2.49384 q^{91} -11.7843 q^{95} +5.83544 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{5} + 9 q^{7}+O(q^{10}) $ 5 * q - 10 * q^5 + 9 * q^7	$ 5 q - 10 q^{5} + 9 q^{7} - 5 q^{11} - 4 q^{13} + 2 q^{17} + 5 q^{19} - 6 q^{23} - 5 q^{25} + 5 q^{29} + 9 q^{31} - 18 q^{35} + 8 q^{37} + 4 q^{41} + 8 q^{43} - 13 q^{47} + 14 q^{49} + 2 q^{53} + 10 q^{55} - 4 q^{59} + 11 q^{61} + 8 q^{65} + 28 q^{67} - 2 q^{71} + 8 q^{73} + 12 q^{77} - 10 q^{79} - 2 q^{83} - 4 q^{85} + 17 q^{89} - 12 q^{91} - 10 q^{95} - 27 q^{97}+O(q^{100}) $ 5 * q - 10 * q^5 + 9 * q^7 - 5 * q^11 - 4 * q^13 + 2 * q^17 + 5 * q^19 - 6 * q^23 - 5 * q^25 + 5 * q^29 + 9 * q^31 - 18 * q^35 + 8 * q^37 + 4 * q^41 + 8 * q^43 - 13 * q^47 + 14 * q^49 + 2 * q^53 + 10 * q^55 - 4 * q^59 + 11 * q^61 + 8 * q^65 + 28 * q^67 - 2 * q^71 + 8 * q^73 + 12 * q^77 - 10 * q^79 - 2 * q^83 - 4 * q^85 + 17 * q^89 - 12 * q^91 - 10 * q^95 - 27 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−1.42676	−0.539266	−0.269633	−	0.962963i	$-0.586902\pi$
$7$	−1.42676	−0.539266	−0.269633	+	0.962963i	$0.586902\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.19055	−0.660476	−0.330238	−	0.943898i	$-0.607129\pi$
$11$	−2.19055	−0.660476	−0.330238	+	0.943898i	$0.607129\pi$
$12$	0	0
$13$	1.74790	0.484779	0.242390	−	0.970179i	$-0.422069\pi$
$13$	1.74790	0.484779	0.242390	+	0.970179i	$0.422069\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.26510	−0.791903	−0.395951	−	0.918271i	$-0.629585\pi$
$17$	−3.26510	−0.791903	−0.395951	+	0.918271i	$0.629585\pi$
$18$	0	0
$19$	5.89213	1.35175	0.675874	−	0.737018i	$-0.263767\pi$
$19$	5.89213	1.35175	0.675874	+	0.737018i	$0.263767\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.12900	−1.27798	−0.638992	−	0.769213i	$-0.720648\pi$
$23$	−6.12900	−1.27798	−0.638992	+	0.769213i	$0.720648\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.321133	0.0596329	0.0298164	−	0.999555i	$-0.490508\pi$
$29$	0.321133	0.0596329	0.0298164	+	0.999555i	$0.490508\pi$
$30$	0	0
$31$	1.75308	0.314863	0.157431	−	0.987530i	$-0.449679\pi$
$31$	1.75308	0.314863	0.157431	+	0.987530i	$0.449679\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.85353	0.482334
$36$	0	0
$37$	−2.47243	−0.406465	−0.203232	−	0.979131i	$-0.565145\pi$
$37$	−2.47243	−0.406465	−0.203232	+	0.979131i	$0.565145\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.98963	0.935423	0.467712	−	0.883881i	$-0.345079\pi$
$41$	5.98963	0.935423	0.467712	+	0.883881i	$0.345079\pi$
$42$	0	0
$43$	−3.49579	−0.533104	−0.266552	−	0.963821i	$-0.585884\pi$
$43$	−3.49579	−0.533104	−0.266552	+	0.963821i	$0.585884\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.39634	−0.349541	−0.174771	−	0.984609i	$-0.555918\pi$
$47$	−2.39634	−0.349541	−0.174771	+	0.984609i	$0.555918\pi$
$48$	0	0
$49$	−4.96434	−0.709192
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.51720	0.208404	0.104202	−	0.994556i	$-0.466771\pi$
$53$	1.51720	0.208404	0.104202	+	0.994556i	$0.466771\pi$
$54$	0	0
$55$	4.38110	0.590747
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.65526	−0.736252	−0.368126	−	0.929776i	$-0.620001\pi$
$59$	−5.65526	−0.736252	−0.368126	+	0.929776i	$0.620001\pi$
$60$	0	0
$61$	−4.51463	−0.578039	−0.289019	−	0.957323i	$-0.593329\pi$
$61$	−4.51463	−0.578039	−0.289019	+	0.957323i	$0.593329\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.49579	−0.433600
$66$	0	0
$67$	−9.61442	−1.17459	−0.587294	−	0.809374i	$-0.699807\pi$
$67$	−9.61442	−1.17459	−0.587294	+	0.809374i	$0.699807\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.66956	0.791532	0.395766	−	0.918351i	$-0.370479\pi$
$71$	6.66956	0.791532	0.395766	+	0.918351i	$0.370479\pi$
$72$	0	0
$73$	2.86390	0.335194	0.167597	−	0.985856i	$-0.446399\pi$
$73$	2.86390	0.335194	0.167597	+	0.985856i	$0.446399\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.12540	0.356172
$78$	0	0
$79$	10.4997	1.18131	0.590656	−	0.806924i	$-0.298869\pi$
$79$	10.4997	1.18131	0.590656	+	0.806924i	$0.298869\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.24369	0.795098	0.397549	−	0.917581i	$-0.369861\pi$
$83$	7.24369	0.795098	0.397549	+	0.917581i	$0.369861\pi$
$84$	0	0
$85$	6.53020	0.708299
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.256634	0.0272032	0.0136016	−	0.999907i	$-0.495670\pi$
$89$	0.256634	0.0272032	0.0136016	+	0.999907i	$0.495670\pi$
$90$	0	0
$91$	−2.49384	−0.261425
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−11.7843	−1.20904
$96$	0	0
$97$	5.83544	0.592499	0.296250	−	0.955111i	$-0.404264\pi$
$97$	5.83544	0.592499	0.296250	+	0.955111i	$0.404264\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.58843	0.556069	0.278035	−	0.960571i	$-0.410317\pi$
$101$	5.58843	0.556069	0.278035	+	0.960571i	$0.410317\pi$
$102$	0	0
$103$	6.17359	0.608302	0.304151	−	0.952624i	$-0.401627\pi$
$103$	6.17359	0.608302	0.304151	+	0.952624i	$0.401627\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.9570	1.15592	0.577962	−	0.816064i	$-0.303848\pi$
$107$	11.9570	1.15592	0.577962	+	0.816064i	$0.303848\pi$
$108$	0	0
$109$	−15.1784	−1.45382	−0.726911	−	0.686731i	$-0.759045\pi$
$109$	−15.1784	−1.45382	−0.726911	+	0.686731i	$0.759045\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.710405	0.0668293	0.0334146	−	0.999442i	$-0.489362\pi$
$113$	0.710405	0.0668293	0.0334146	+	0.999442i	$0.489362\pi$
$114$	0	0
$115$	12.2580	1.14306
$116$	0	0
$117$	0	0
$118$	0	0
$119$	4.65853	0.427046
$120$	0	0
$121$	−6.20149	−0.563772
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	11.5165	1.02192	0.510962	−	0.859603i	$-0.329289\pi$
$127$	11.5165	1.02192	0.510962	+	0.859603i	$0.329289\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.02599	−0.701234	−0.350617	−	0.936519i	$-0.614028\pi$
$131$	−8.02599	−0.701234	−0.350617	+	0.936519i	$0.614028\pi$
$132$	0	0
$133$	−8.40668	−0.728951
$134$	0	0
$135$	0	0
$136$	0	0
$137$	15.6656	1.33840	0.669201	−	0.743081i	$-0.266636\pi$
$137$	15.6656	1.33840	0.669201	+	0.743081i	$0.266636\pi$
$138$	0	0
$139$	−2.22163	−0.188436	−0.0942182	−	0.995552i	$-0.530035\pi$
$139$	−2.22163	−0.188436	−0.0942182	+	0.995552i	$0.530035\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3.82886	−0.320185
$144$	0	0
$145$	−0.642266	−0.0533373
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.02599	−0.493668	−0.246834	−	0.969058i	$-0.579390\pi$
$149$	−6.02599	−0.493668	−0.246834	+	0.969058i	$0.579390\pi$
$150$	0	0
$151$	−10.9684	−0.892596	−0.446298	−	0.894884i	$-0.647258\pi$
$151$	−10.9684	−0.892596	−0.446298	+	0.894884i	$0.647258\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3.50616	−0.281622
$156$	0	0
$157$	20.0474	1.59996	0.799980	−	0.600027i	$-0.204843\pi$
$157$	20.0474	1.59996	0.799980	+	0.600027i	$0.204843\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.74463	0.689174
$162$	0	0
$163$	5.21770	0.408682	0.204341	−	0.978900i	$-0.434495\pi$
$163$	5.21770	0.408682	0.204341	+	0.978900i	$0.434495\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−9.94486	−0.764989
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.5165	1.63587	0.817935	−	0.575311i	$-0.195119\pi$
$173$	21.5165	1.63587	0.817935	+	0.575311i	$0.195119\pi$
$174$	0	0
$175$	1.42676	0.107853
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−10.1501	−0.758656	−0.379328	−	0.925262i	$-0.623845\pi$
$179$	−10.1501	−0.758656	−0.379328	+	0.925262i	$0.623845\pi$
$180$	0	0
$181$	−3.64003	−0.270561	−0.135281	−	0.990807i	$-0.543194\pi$
$181$	−3.64003	−0.270561	−0.135281	+	0.990807i	$0.543194\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.94486	0.363553
$186$	0	0
$187$	7.15236	0.523033
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.4861	1.26525	0.632626	−	0.774457i	$-0.281977\pi$
$191$	17.4861	1.26525	0.632626	+	0.774457i	$0.281977\pi$
$192$	0	0
$193$	4.75052	0.341950	0.170975	−	0.985275i	$-0.445308\pi$
$193$	4.75052	0.341950	0.170975	+	0.985275i	$0.445308\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.67780	0.190785	0.0953927	−	0.995440i	$-0.469589\pi$
$197$	2.67780	0.190785	0.0953927	+	0.995440i	$0.469589\pi$
$198$	0	0
$199$	25.6656	1.81939	0.909693	−	0.415281i	$-0.136317\pi$
$199$	25.6656	1.81939	0.909693	+	0.415281i	$0.136317\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.458181	−0.0321580
$204$	0	0
$205$	−11.9793	−0.836668
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−12.9070	−0.892796
$210$	0	0
$211$	9.73234	0.670002	0.335001	−	0.942218i	$-0.391263\pi$
$211$	9.73234	0.670002	0.335001	+	0.942218i	$0.391263\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.99159	0.476822
$216$	0	0
$217$	−2.50123	−0.169795
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−5.70706	−0.383898
$222$	0	0
$223$	21.5995	1.44641	0.723205	−	0.690633i	$-0.242668\pi$
$223$	21.5995	1.44641	0.723205	+	0.690633i	$0.242668\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.0401	0.931875	0.465938	−	0.884818i	$-0.345717\pi$
$227$	14.0401	0.931875	0.465938	+	0.884818i	$0.345717\pi$
$228$	0	0
$229$	5.77020	0.381305	0.190653	−	0.981658i	$-0.438940\pi$
$229$	5.77020	0.381305	0.190653	+	0.981658i	$0.438940\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.4705	1.27556	0.637778	−	0.770221i	$-0.279854\pi$
$233$	19.4705	1.27556	0.637778	+	0.770221i	$0.279854\pi$
$234$	0	0
$235$	4.79267	0.312639
$236$	0	0
$237$	0	0
$238$	0	0
$239$	14.7014	0.950954	0.475477	−	0.879728i	$-0.342275\pi$
$239$	14.7014	0.950954	0.475477	+	0.879728i	$0.342275\pi$
$240$	0	0
$241$	−21.6813	−1.39661	−0.698306	−	0.715799i	$-0.746063\pi$
$241$	−21.6813	−1.39661	−0.698306	+	0.715799i	$0.746063\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9.92869	0.634321
$246$	0	0
$247$	10.2988	0.655299
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6.71130	0.423613	0.211807	−	0.977312i	$-0.432065\pi$
$251$	6.71130	0.423613	0.211807	+	0.977312i	$0.432065\pi$
$252$	0	0
$253$	13.4259	0.844077
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.1000	1.62807	0.814037	−	0.580813i	$-0.197265\pi$
$257$	26.1000	1.62807	0.814037	+	0.580813i	$0.197265\pi$
$258$	0	0
$259$	3.52757	0.219193
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.1595	−0.996439	−0.498219	−	0.867051i	$-0.666013\pi$
$263$	−16.1595	−0.996439	−0.498219	+	0.867051i	$0.666013\pi$
$264$	0	0
$265$	−3.03440	−0.186402
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.9418	−0.667131	−0.333566	−	0.942727i	$-0.608252\pi$
$269$	−10.9418	−0.667131	−0.333566	+	0.942727i	$0.608252\pi$
$270$	0	0
$271$	9.49579	0.576828	0.288414	−	0.957506i	$-0.406872\pi$
$271$	9.49579	0.576828	0.288414	+	0.957506i	$0.406872\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.19055	0.132095
$276$	0	0
$277$	1.86063	0.111795	0.0558973	−	0.998437i	$-0.482198\pi$
$277$	1.86063	0.111795	0.0558973	+	0.998437i	$0.482198\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.01042	−0.239242	−0.119621	−	0.992820i	$-0.538168\pi$
$281$	−4.01042	−0.239242	−0.119621	+	0.992820i	$0.538168\pi$
$282$	0	0
$283$	−7.71234	−0.458451	−0.229225	−	0.973373i	$-0.573619\pi$
$283$	−7.71234	−0.458451	−0.229225	+	0.973373i	$0.573619\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.54579	−0.504442
$288$	0	0
$289$	−6.33913	−0.372890
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.05585	−0.529048	−0.264524	−	0.964379i	$-0.585215\pi$
$293$	−9.05585	−0.529048	−0.264524	+	0.964379i	$0.585215\pi$
$294$	0	0
$295$	11.3105	0.658524
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−10.7129	−0.619540
$300$	0	0
$301$	4.98767	0.287485
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.02926	0.517014
$306$	0	0
$307$	12.1632	0.694192	0.347096	−	0.937830i	$-0.387168\pi$
$307$	12.1632	0.694192	0.347096	+	0.937830i	$0.387168\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.4952	−1.38900	−0.694499	−	0.719494i	$-0.744374\pi$
$311$	−24.4952	−1.38900	−0.694499	+	0.719494i	$0.744374\pi$
$312$	0	0
$313$	22.5593	1.27513	0.637563	−	0.770398i	$-0.279943\pi$
$313$	22.5593	1.27513	0.637563	+	0.770398i	$0.279943\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.9546	−0.783770	−0.391885	−	0.920014i	$-0.628177\pi$
$317$	−13.9546	−0.783770	−0.391885	+	0.920014i	$0.628177\pi$
$318$	0	0
$319$	−0.703457	−0.0393861
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−19.2384	−1.07045
$324$	0	0
$325$	−1.74790	−0.0969559
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.41901	0.188496
$330$	0	0
$331$	−21.5826	−1.18629	−0.593145	−	0.805096i	$-0.702114\pi$
$331$	−21.5826	−1.18629	−0.593145	+	0.805096i	$0.702114\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	19.2288	1.05058
$336$	0	0
$337$	16.7494	0.912399	0.456199	−	0.889878i	$-0.349210\pi$
$337$	16.7494	0.912399	0.456199	+	0.889878i	$0.349210\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.84021	−0.207959
$342$	0	0
$343$	17.0703	0.921709
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.7273	0.683239	0.341620	−	0.939838i	$-0.389025\pi$
$347$	12.7273	0.683239	0.341620	+	0.939838i	$0.389025\pi$
$348$	0	0
$349$	21.8665	1.17049	0.585244	−	0.810857i	$-0.300999\pi$
$349$	21.8665	1.17049	0.585244	+	0.810857i	$0.300999\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−27.7352	−1.47620	−0.738099	−	0.674692i	$-0.764276\pi$
$353$	−27.7352	−1.47620	−0.738099	+	0.674692i	$0.764276\pi$
$354$	0	0
$355$	−13.3391	−0.707967
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.30402	0.438269	0.219135	−	0.975695i	$-0.429677\pi$
$359$	8.30402	0.438269	0.219135	+	0.975695i	$0.429677\pi$
$360$	0	0
$361$	15.7172	0.827220
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.72780	−0.299807
$366$	0	0
$367$	12.4528	0.650031	0.325015	−	0.945709i	$-0.394631\pi$
$367$	12.4528	0.650031	0.325015	+	0.945709i	$0.394631\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.16469	−0.112385
$372$	0	0
$373$	17.6189	0.912272	0.456136	−	0.889910i	$-0.349233\pi$
$373$	17.6189	0.912272	0.456136	+	0.889910i	$0.349233\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.561307	0.0289088
$378$	0	0
$379$	12.1749	0.625383	0.312691	−	0.949855i	$-0.398769\pi$
$379$	12.1749	0.625383	0.312691	+	0.949855i	$0.398769\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−16.9672	−0.866981	−0.433491	−	0.901158i	$-0.642718\pi$
$383$	−16.9672	−0.866981	−0.433491	+	0.901158i	$0.642718\pi$
$384$	0	0
$385$	−6.25080	−0.318570
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−14.9358	−0.757275	−0.378637	−	0.925545i	$-0.623607\pi$
$389$	−14.9358	−0.757275	−0.378637	+	0.925545i	$0.623607\pi$
$390$	0	0
$391$	20.0118	1.01204
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−20.9995	−1.05660
$396$	0	0
$397$	27.3638	1.37335	0.686674	−	0.726966i	$-0.259070\pi$
$397$	27.3638	1.37335	0.686674	+	0.726966i	$0.259070\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−16.3411	−0.816035	−0.408017	−	0.912974i	$-0.633780\pi$
$401$	−16.3411	−0.816035	−0.408017	+	0.912974i	$0.633780\pi$
$402$	0	0
$403$	3.06421	0.152639
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.41598	0.268460
$408$	0	0
$409$	−19.2236	−0.950544	−0.475272	−	0.879839i	$-0.657650\pi$
$409$	−19.2236	−0.950544	−0.475272	+	0.879839i	$0.657650\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.06872	0.397036
$414$	0	0
$415$	−14.4874	−0.711158
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.15541	−0.0564456	−0.0282228	−	0.999602i	$-0.508985\pi$
$419$	−1.15541	−0.0564456	−0.0282228	+	0.999602i	$0.508985\pi$
$420$	0	0
$421$	4.80899	0.234376	0.117188	−	0.993110i	$-0.462612\pi$
$421$	4.80899	0.234376	0.117188	+	0.993110i	$0.462612\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.26510	0.158381
$426$	0	0
$427$	6.44131	0.311717
$428$	0	0
$429$	0	0
$430$	0	0
$431$	34.3708	1.65558	0.827791	−	0.561036i	$-0.189597\pi$
$431$	34.3708	1.65558	0.827791	+	0.561036i	$0.189597\pi$
$432$	0	0
$433$	−20.0626	−0.964145	−0.482072	−	0.876131i	$-0.660116\pi$
$433$	−20.0626	−0.964145	−0.482072	+	0.876131i	$0.660116\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−36.1128	−1.72751
$438$	0	0
$439$	1.90270	0.0908108	0.0454054	−	0.998969i	$-0.485542\pi$
$439$	1.90270	0.0908108	0.0454054	+	0.998969i	$0.485542\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	13.8782	0.659373	0.329687	−	0.944090i	$-0.393057\pi$
$443$	13.8782	0.659373	0.329687	+	0.944090i	$0.393057\pi$
$444$	0	0
$445$	−0.513269	−0.0243313
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−14.6316	−0.690509	−0.345254	−	0.938509i	$-0.612207\pi$
$449$	−14.6316	−0.690509	−0.345254	+	0.938509i	$0.612207\pi$
$450$	0	0
$451$	−13.1206	−0.617824
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.98767	0.233826
$456$	0	0
$457$	−15.1270	−0.707613	−0.353807	−	0.935319i	$-0.615113\pi$
$457$	−15.1270	−0.707613	−0.353807	+	0.935319i	$0.615113\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.617968	0.0287816	0.0143908	−	0.999896i	$-0.495419\pi$
$461$	0.617968	0.0287816	0.0143908	+	0.999896i	$0.495419\pi$
$462$	0	0
$463$	−6.12321	−0.284570	−0.142285	−	0.989826i	$-0.545445\pi$
$463$	−6.12321	−0.284570	−0.142285	+	0.989826i	$0.545445\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.0319	1.01952	0.509758	−	0.860318i	$-0.329735\pi$
$467$	22.0319	1.01952	0.509758	+	0.860318i	$0.329735\pi$
$468$	0	0
$469$	13.7175	0.633416
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.65771	0.352102
$474$	0	0
$475$	−5.89213	−0.270349
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.88989	0.314807	0.157404	−	0.987534i	$-0.449688\pi$
$479$	6.88989	0.314807	0.157404	+	0.987534i	$0.449688\pi$
$480$	0	0
$481$	−4.32155	−0.197046
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−11.6709	−0.529948
$486$	0	0
$487$	−20.9486	−0.949272	−0.474636	−	0.880182i	$-0.657420\pi$
$487$	−20.9486	−0.949272	−0.474636	+	0.880182i	$0.657420\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−13.3184	−0.601051	−0.300525	−	0.953774i	$-0.597162\pi$
$491$	−13.3184	−0.601051	−0.300525	+	0.953774i	$0.597162\pi$
$492$	0	0
$493$	−1.04853	−0.0472234
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.51589	−0.426846
$498$	0	0
$499$	15.2496	0.682665	0.341333	−	0.939943i	$-0.389122\pi$
$499$	15.2496	0.682665	0.341333	+	0.939943i	$0.389122\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	26.7387	1.19222	0.596110	−	0.802903i	$-0.296712\pi$
$503$	26.7387	1.19222	0.596110	+	0.802903i	$0.296712\pi$
$504$	0	0
$505$	−11.1769	−0.497364
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−0.156638	−0.00694286	−0.00347143	−	0.999994i	$-0.501105\pi$
$509$	−0.156638	−0.00694286	−0.00347143	+	0.999994i	$0.501105\pi$
$510$	0	0
$511$	−4.08611	−0.180759
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−12.3472	−0.544082
$516$	0	0
$517$	5.24929	0.230864
$518$	0	0
$519$	0	0
$520$	0	0
$521$	32.1200	1.40720	0.703602	−	0.710594i	$-0.251574\pi$
$521$	32.1200	1.40720	0.703602	+	0.710594i	$0.251574\pi$
$522$	0	0
$523$	4.73691	0.207131	0.103565	−	0.994623i	$-0.466975\pi$
$523$	4.73691	0.207131	0.103565	+	0.994623i	$0.466975\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.72399	−0.249341
$528$	0	0
$529$	14.5646	0.633244
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.4693	0.453474
$534$	0	0
$535$	−23.9139	−1.03389
$536$	0	0
$537$	0	0
$538$	0	0
$539$	10.8746	0.468404
$540$	0	0
$541$	15.3824	0.661342	0.330671	−	0.943746i	$-0.392725\pi$
$541$	15.3824	0.661342	0.330671	+	0.943746i	$0.392725\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	30.3567	1.30034
$546$	0	0
$547$	19.2682	0.823848	0.411924	−	0.911218i	$-0.364857\pi$
$547$	19.2682	0.823848	0.411924	+	0.911218i	$0.364857\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.89216	0.0806086
$552$	0	0
$553$	−14.9806	−0.637041
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.0313	0.425039	0.212519	−	0.977157i	$-0.431833\pi$
$557$	10.0313	0.425039	0.212519	+	0.977157i	$0.431833\pi$
$558$	0	0
$559$	−6.11029	−0.258438
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.61733	0.363177	0.181589	−	0.983375i	$-0.441876\pi$
$563$	8.61733	0.363177	0.181589	+	0.983375i	$0.441876\pi$
$564$	0	0
$565$	−1.42081	−0.0597739
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.8782	0.959105	0.479552	−	0.877513i	$-0.340799\pi$
$569$	22.8782	0.959105	0.479552	+	0.877513i	$0.340799\pi$
$570$	0	0
$571$	−36.9981	−1.54832	−0.774162	−	0.632987i	$-0.781829\pi$
$571$	−36.9981	−1.54832	−0.774162	+	0.632987i	$0.781829\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.12900	0.255597
$576$	0	0
$577$	9.42378	0.392317	0.196158	−	0.980572i	$-0.437153\pi$
$577$	9.42378	0.392317	0.196158	+	0.980572i	$0.437153\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−10.3350	−0.428770
$582$	0	0
$583$	−3.32351	−0.137646
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.06497	0.250328	0.125164	−	0.992136i	$-0.460054\pi$
$587$	6.06497	0.250328	0.125164	+	0.992136i	$0.460054\pi$
$588$	0	0
$589$	10.3294	0.425615
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−30.0669	−1.23470	−0.617351	−	0.786688i	$-0.711794\pi$
$593$	−30.0669	−1.23470	−0.617351	+	0.786688i	$0.711794\pi$
$594$	0	0
$595$	−9.31705	−0.381962
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.74168	0.357175	0.178588	−	0.983924i	$-0.442847\pi$
$599$	8.74168	0.357175	0.178588	+	0.983924i	$0.442847\pi$
$600$	0	0
$601$	−5.47200	−0.223208	−0.111604	−	0.993753i	$-0.535599\pi$
$601$	−5.47200	−0.223208	−0.111604	+	0.993753i	$0.535599\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	12.4030	0.504253
$606$	0	0
$607$	8.28118	0.336123	0.168061	−	0.985777i	$-0.446249\pi$
$607$	8.28118	0.336123	0.168061	+	0.985777i	$0.446249\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.18855	−0.169450
$612$	0	0
$613$	46.5063	1.87837	0.939186	−	0.343408i	$-0.111581\pi$
$613$	46.5063	1.87837	0.939186	+	0.343408i	$0.111581\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−34.1883	−1.37637	−0.688185	−	0.725535i	$-0.741592\pi$
$617$	−34.1883	−1.37637	−0.688185	+	0.725535i	$0.741592\pi$
$618$	0	0
$619$	−23.3216	−0.937373	−0.468686	−	0.883365i	$-0.655273\pi$
$619$	−23.3216	−0.937373	−0.468686	+	0.883365i	$0.655273\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.366157	−0.0146698
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.07272	0.321881
$630$	0	0
$631$	−29.0863	−1.15791	−0.578953	−	0.815361i	$-0.696538\pi$
$631$	−29.0863	−1.15791	−0.578953	+	0.815361i	$0.696538\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−23.0330	−0.914037
$636$	0	0
$637$	−8.67716	−0.343802
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−31.0603	−1.22681	−0.613404	−	0.789769i	$-0.710200\pi$
$641$	−31.0603	−1.22681	−0.613404	+	0.789769i	$0.710200\pi$
$642$	0	0
$643$	14.9145	0.588169	0.294085	−	0.955779i	$-0.404985\pi$
$643$	14.9145	0.588169	0.294085	+	0.955779i	$0.404985\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28.9951	−1.13991	−0.569957	−	0.821675i	$-0.693040\pi$
$647$	−28.9951	−1.13991	−0.569957	+	0.821675i	$0.693040\pi$
$648$	0	0
$649$	12.3881	0.486277
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.42152	−0.290426	−0.145213	−	0.989400i	$-0.546387\pi$
$653$	−7.42152	−0.290426	−0.145213	+	0.989400i	$0.546387\pi$
$654$	0	0
$655$	16.0520	0.627203
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.40447	0.0547102	0.0273551	−	0.999626i	$-0.491292\pi$
$659$	1.40447	0.0547102	0.0273551	+	0.999626i	$0.491292\pi$
$660$	0	0
$661$	13.1593	0.511837	0.255919	−	0.966698i	$-0.417622\pi$
$661$	13.1593	0.511837	0.255919	+	0.966698i	$0.417622\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.8134	0.651994
$666$	0	0
$667$	−1.96822	−0.0762099
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.88952	0.381781
$672$	0	0
$673$	−24.1775	−0.931974	−0.465987	−	0.884791i	$-0.654301\pi$
$673$	−24.1775	−0.931974	−0.465987	+	0.884791i	$0.654301\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−13.6811	−0.525806	−0.262903	−	0.964822i	$-0.584680\pi$
$677$	−13.6811	−0.525806	−0.262903	+	0.964822i	$0.584680\pi$
$678$	0	0
$679$	−8.32580	−0.319515
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.3047	0.547355	0.273678	−	0.961822i	$-0.411760\pi$
$683$	14.3047	0.547355	0.273678	+	0.961822i	$0.411760\pi$
$684$	0	0
$685$	−31.3312	−1.19710
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.65191	0.101030
$690$	0	0
$691$	27.8109	1.05798	0.528989	−	0.848629i	$-0.322571\pi$
$691$	27.8109	1.05798	0.528989	+	0.848629i	$0.322571\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.44326	0.168543
$696$	0	0
$697$	−19.5567	−0.740764
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−10.7440	−0.405796	−0.202898	−	0.979200i	$-0.565036\pi$
$701$	−10.7440	−0.405796	−0.202898	+	0.979200i	$0.565036\pi$
$702$	0	0
$703$	−14.5679	−0.549437
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7.97337	−0.299869
$708$	0	0
$709$	−39.8043	−1.49488	−0.747440	−	0.664329i	$-0.768717\pi$
$709$	−39.8043	−1.49488	−0.747440	+	0.664329i	$0.768717\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10.7446	−0.402390
$714$	0	0
$715$	7.65771	0.286382
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−23.3327	−0.870162	−0.435081	−	0.900391i	$-0.643280\pi$
$719$	−23.3327	−0.870162	−0.435081	+	0.900391i	$0.643280\pi$
$720$	0	0
$721$	−8.80826	−0.328037
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.321133	−0.0119266
$726$	0	0
$727$	−21.2593	−0.788464	−0.394232	−	0.919011i	$-0.628989\pi$
$727$	−21.2593	−0.788464	−0.394232	+	0.919011i	$0.628989\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11.4141	0.422166
$732$	0	0
$733$	−40.8733	−1.50969	−0.754846	−	0.655902i	$-0.772288\pi$
$733$	−40.8733	−1.50969	−0.754846	+	0.655902i	$0.772288\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.0609	0.775787
$738$	0	0
$739$	38.5912	1.41960	0.709799	−	0.704404i	$-0.248786\pi$
$739$	38.5912	1.41960	0.709799	+	0.704404i	$0.248786\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−18.3330	−0.672572	−0.336286	−	0.941760i	$-0.609171\pi$
$743$	−18.3330	−0.672572	−0.336286	+	0.941760i	$0.609171\pi$
$744$	0	0
$745$	12.0520	0.441551
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−17.0598	−0.623350
$750$	0	0
$751$	−41.3572	−1.50915	−0.754573	−	0.656216i	$-0.772156\pi$
$751$	−41.3572	−1.50915	−0.754573	+	0.656216i	$0.772156\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	21.9368	0.798362
$756$	0	0
$757$	14.2655	0.518490	0.259245	−	0.965812i	$-0.416526\pi$
$757$	14.2655	0.518490	0.259245	+	0.965812i	$0.416526\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	41.8123	1.51569	0.757847	−	0.652432i	$-0.226251\pi$
$761$	41.8123	1.51569	0.757847	+	0.652432i	$0.226251\pi$
$762$	0	0
$763$	21.6559	0.783997
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.88481	−0.356920
$768$	0	0
$769$	3.27230	0.118002	0.0590010	−	0.998258i	$-0.481208\pi$
$769$	3.27230	0.118002	0.0590010	+	0.998258i	$0.481208\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−23.3460	−0.839696	−0.419848	−	0.907594i	$-0.637917\pi$
$773$	−23.3460	−0.839696	−0.419848	+	0.907594i	$0.637917\pi$
$774$	0	0
$775$	−1.75308	−0.0629726
$776$	0	0
$777$	0	0
$778$	0	0
$779$	35.2917	1.26446
$780$	0	0
$781$	−14.6100	−0.522787
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−40.0949	−1.43105
$786$	0	0
$787$	44.5309	1.58736	0.793678	−	0.608338i	$-0.208163\pi$
$787$	44.5309	1.58736	0.793678	+	0.608338i	$0.208163\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.01358	−0.0360388
$792$	0	0
$793$	−7.89110	−0.280221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−7.60786	−0.269484	−0.134742	−	0.990881i	$-0.543021\pi$
$797$	−7.60786	−0.269484	−0.134742	+	0.990881i	$0.543021\pi$
$798$	0	0
$799$	7.82427	0.276803
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.27351	−0.221387
$804$	0	0
$805$	−17.4893	−0.616416
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−7.40132	−0.260217	−0.130108	−	0.991500i	$-0.541532\pi$
$809$	−7.40132	−0.260217	−0.130108	+	0.991500i	$0.541532\pi$
$810$	0	0
$811$	−5.69610	−0.200017	−0.100009	−	0.994987i	$-0.531887\pi$
$811$	−5.69610	−0.200017	−0.100009	+	0.994987i	$0.531887\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10.4354	−0.365536
$816$	0	0
$817$	−20.5977	−0.720621
$818$	0	0
$819$	0	0
$820$	0	0
$821$	45.1962	1.57736	0.788680	−	0.614804i	$-0.210765\pi$
$821$	45.1962	1.57736	0.788680	+	0.614804i	$0.210765\pi$
$822$	0	0
$823$	13.9450	0.486093	0.243047	−	0.970015i	$-0.421853\pi$
$823$	13.9450	0.486093	0.243047	+	0.970015i	$0.421853\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.90541	−0.101031	−0.0505155	−	0.998723i	$-0.516086\pi$
$827$	−2.90541	−0.101031	−0.0505155	+	0.998723i	$0.516086\pi$
$828$	0	0
$829$	31.2634	1.08582	0.542912	−	0.839790i	$-0.317322\pi$
$829$	31.2634	1.08582	0.542912	+	0.839790i	$0.317322\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	16.2091	0.561611
$834$	0	0
$835$	−2.00000	−0.0692129
$836$	0	0
$837$	0	0
$838$	0	0
$839$	44.4886	1.53592	0.767958	−	0.640500i	$-0.221273\pi$
$839$	44.4886	1.53592	0.767958	+	0.640500i	$0.221273\pi$
$840$	0	0
$841$	−28.8969	−0.996444
$842$	0	0
$843$	0	0
$844$	0	0
$845$	19.8897	0.684227
$846$	0	0
$847$	8.84806	0.304023
$848$	0	0
$849$	0	0
$850$	0	0
$851$	15.1535	0.519455
$852$	0	0
$853$	−17.2717	−0.591371	−0.295686	−	0.955285i	$-0.595548\pi$
$853$	−17.2717	−0.591371	−0.295686	+	0.955285i	$0.595548\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35.0334	1.19672	0.598359	−	0.801228i	$-0.295820\pi$
$857$	35.0334	1.19672	0.598359	+	0.801228i	$0.295820\pi$
$858$	0	0
$859$	38.9649	1.32947	0.664733	−	0.747081i	$-0.268546\pi$
$859$	38.9649	1.32947	0.664733	+	0.747081i	$0.268546\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−32.0638	−1.09146	−0.545732	−	0.837960i	$-0.683748\pi$
$863$	−32.0638	−1.09146	−0.545732	+	0.837960i	$0.683748\pi$
$864$	0	0
$865$	−43.0330	−1.46317
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−23.0002	−0.780228
$870$	0	0
$871$	−16.8050	−0.569416
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−17.1212	−0.578801
$876$	0	0
$877$	53.9033	1.82018	0.910092	−	0.414406i	$-0.136011\pi$
$877$	53.9033	1.82018	0.910092	+	0.414406i	$0.136011\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−49.8614	−1.67987	−0.839937	−	0.542684i	$-0.817408\pi$
$881$	−49.8614	−1.67987	−0.839937	+	0.542684i	$0.817408\pi$
$882$	0	0
$883$	12.9835	0.436930	0.218465	−	0.975845i	$-0.429895\pi$
$883$	12.9835	0.436930	0.218465	+	0.975845i	$0.429895\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11.8245	−0.397027	−0.198513	−	0.980098i	$-0.563611\pi$
$887$	−11.8245	−0.397027	−0.198513	+	0.980098i	$0.563611\pi$
$888$	0	0
$889$	−16.4313	−0.551089
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−14.1195	−0.472492
$894$	0	0
$895$	20.3002	0.678562
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0.562972	0.0187762
$900$	0	0
$901$	−4.95381	−0.165036
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7.28005	0.241997
$906$	0	0
$907$	1.08780	0.0361198	0.0180599	−	0.999837i	$-0.494251\pi$
$907$	1.08780	0.0361198	0.0180599	+	0.999837i	$0.494251\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.16245	0.303566	0.151783	−	0.988414i	$-0.451499\pi$
$911$	9.16245	0.303566	0.151783	+	0.988414i	$0.451499\pi$
$912$	0	0
$913$	−15.8677	−0.525143
$914$	0	0
$915$	0	0
$916$	0	0
$917$	11.4512	0.378152
$918$	0	0
$919$	−23.2997	−0.768587	−0.384293	−	0.923211i	$-0.625555\pi$
$919$	−23.2997	−0.768587	−0.384293	+	0.923211i	$0.625555\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	11.6577	0.383718
$924$	0	0
$925$	2.47243	0.0812929
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−48.9106	−1.60471	−0.802353	−	0.596850i	$-0.796419\pi$
$929$	−48.9106	−1.60471	−0.802353	+	0.596850i	$0.796419\pi$
$930$	0	0
$931$	−29.2506	−0.958648
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−14.3047	−0.467815
$936$	0	0
$937$	29.3682	0.959418	0.479709	−	0.877428i	$-0.340742\pi$
$937$	29.3682	0.959418	0.479709	+	0.877428i	$0.340742\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	33.9716	1.10744	0.553721	−	0.832702i	$-0.313207\pi$
$941$	33.9716	1.10744	0.553721	+	0.832702i	$0.313207\pi$
$942$	0	0
$943$	−36.7104	−1.19546
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−36.7945	−1.19566	−0.597830	−	0.801623i	$-0.703970\pi$
$947$	−36.7945	−1.19566	−0.597830	+	0.801623i	$0.703970\pi$
$948$	0	0
$949$	5.00580	0.162495
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.19613	0.168319	0.0841596	−	0.996452i	$-0.473179\pi$
$953$	5.19613	0.168319	0.0841596	+	0.996452i	$0.473179\pi$
$954$	0	0
$955$	−34.9723	−1.13168
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−22.3511	−0.721755
$960$	0	0
$961$	−27.9267	−0.900861
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−9.50105	−0.305849
$966$	0	0
$967$	49.3421	1.58673	0.793367	−	0.608743i	$-0.208326\pi$
$967$	49.3421	1.58673	0.793367	+	0.608743i	$0.208326\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	43.3773	1.39204	0.696021	−	0.718021i	$-0.254952\pi$
$971$	43.3773	1.39204	0.696021	+	0.718021i	$0.254952\pi$
$972$	0	0
$973$	3.16975	0.101617
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−55.2452	−1.76745	−0.883725	−	0.468006i	$-0.844972\pi$
$977$	−55.2452	−1.76745	−0.883725	+	0.468006i	$0.844972\pi$
$978$	0	0
$979$	−0.562170	−0.0179670
$980$	0	0
$981$	0	0
$982$	0	0
$983$	12.0299	0.383694	0.191847	−	0.981425i	$-0.438552\pi$
$983$	12.0299	0.383694	0.191847	+	0.981425i	$0.438552\pi$
$984$	0	0
$985$	−5.35560	−0.170644
$986$	0	0
$987$	0	0
$988$	0	0
$989$	21.4257	0.681298
$990$	0	0
$991$	44.9737	1.42864	0.714318	−	0.699822i	$-0.246737\pi$
$991$	44.9737	1.42864	0.714318	+	0.699822i	$0.246737\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−51.3312	−1.62731
$996$	0	0
$997$	44.0823	1.39610	0.698050	−	0.716049i	$-0.254051\pi$
$997$	44.0823	1.39610	0.698050	+	0.716049i	$0.254051\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.d.1.1		5
3.2	odd	2		668.2.a.b.1.3	✓	5
12.11	even	2		2672.2.a.j.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.b.1.3	✓	5	3.2	odd	2
2672.2.a.j.1.3		5	12.11	even	2
6012.2.a.d.1.1		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{5} -1.42676 q^{7} +O(q^{10})\)	\(q-2.00000 q^{5} -1.42676 q^{7} -2.19055 q^{11} +1.74790 q^{13} -3.26510 q^{17} +5.89213 q^{19} -6.12900 q^{23} -1.00000 q^{25} +0.321133 q^{29} +1.75308 q^{31} +2.85353 q^{35} -2.47243 q^{37} +5.98963 q^{41} -3.49579 q^{43} -2.39634 q^{47} -4.96434 q^{49} +1.51720 q^{53} +4.38110 q^{55} -5.65526 q^{59} -4.51463 q^{61} -3.49579 q^{65} -9.61442 q^{67} +6.66956 q^{71} +2.86390 q^{73} +3.12540 q^{77} +10.4997 q^{79} +7.24369 q^{83} +6.53020 q^{85} +0.256634 q^{89} -2.49384 q^{91} -11.7843 q^{95} +5.83544 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{5} + 9 q^{7}+O(q^{10}) \) 5 * q - 10 * q^5 + 9 * q^7	\( 5 q - 10 q^{5} + 9 q^{7} - 5 q^{11} - 4 q^{13} + 2 q^{17} + 5 q^{19} - 6 q^{23} - 5 q^{25} + 5 q^{29} + 9 q^{31} - 18 q^{35} + 8 q^{37} + 4 q^{41} + 8 q^{43} - 13 q^{47} + 14 q^{49} + 2 q^{53} + 10 q^{55} - 4 q^{59} + 11 q^{61} + 8 q^{65} + 28 q^{67} - 2 q^{71} + 8 q^{73} + 12 q^{77} - 10 q^{79} - 2 q^{83} - 4 q^{85} + 17 q^{89} - 12 q^{91} - 10 q^{95} - 27 q^{97}+O(q^{100}) \) 5 * q - 10 * q^5 + 9 * q^7 - 5 * q^11 - 4 * q^13 + 2 * q^17 + 5 * q^19 - 6 * q^23 - 5 * q^25 + 5 * q^29 + 9 * q^31 - 18 * q^35 + 8 * q^37 + 4 * q^41 + 8 * q^43 - 13 * q^47 + 14 * q^49 + 2 * q^53 + 10 * q^55 - 4 * q^59 + 11 * q^61 + 8 * q^65 + 28 * q^67 - 2 * q^71 + 8 * q^73 + 12 * q^77 - 10 * q^79 - 2 * q^83 - 4 * q^85 + 17 * q^89 - 12 * q^91 - 10 * q^95 - 27 * q^97

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