LMFDB - Embedded newform 108.6.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,6,Mod(1,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("108.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	108.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:	$17.3214525398$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	108.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+57.6281 q^{5} +156.884 q^{7} +O(q^{10})$	$q+57.6281 q^{5} +156.884 q^{7} -218.025 q^{11} -587.537 q^{13} +2124.56 q^{17} +29.7687 q^{19} +3171.49 q^{23} +196.000 q^{25} +3642.13 q^{29} +6331.88 q^{31} +9040.95 q^{35} -8435.69 q^{37} +10146.4 q^{41} -6855.76 q^{43} +24447.8 q^{47} +7805.70 q^{49} -25483.2 q^{53} -12564.4 q^{55} -20696.2 q^{59} +45580.2 q^{61} -33858.7 q^{65} +19099.3 q^{67} +23055.0 q^{71} -89651.6 q^{73} -34204.7 q^{77} -34023.6 q^{79} -108497. q^{83} +122435. q^{85} -110055. q^{89} -92175.4 q^{91} +1715.51 q^{95} -28832.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{7}+O(q^{10}) $ 2 * q - 32 * q^7	$ 2 q - 32 q^{7} + 486 q^{11} + 208 q^{13} + 1944 q^{17} - 632 q^{19} + 6804 q^{23} + 392 q^{25} + 11664 q^{29} + 3328 q^{31} + 19926 q^{35} - 9956 q^{37} + 13608 q^{41} + 4960 q^{43} + 18468 q^{47} + 26676 q^{49} - 11664 q^{53} - 53136 q^{55} + 1944 q^{59} + 8176 q^{61} - 79704 q^{65} + 90064 q^{67} - 44712 q^{71} - 121214 q^{73} - 167184 q^{77} + 28768 q^{79} - 15066 q^{83} + 132840 q^{85} - 178848 q^{89} - 242440 q^{91} + 39852 q^{95} + 88942 q^{97}+O(q^{100}) $ 2 * q - 32 * q^7 + 486 * q^11 + 208 * q^13 + 1944 * q^17 - 632 * q^19 + 6804 * q^23 + 392 * q^25 + 11664 * q^29 + 3328 * q^31 + 19926 * q^35 - 9956 * q^37 + 13608 * q^41 + 4960 * q^43 + 18468 * q^47 + 26676 * q^49 - 11664 * q^53 - 53136 * q^55 + 1944 * q^59 + 8176 * q^61 - 79704 * q^65 + 90064 * q^67 - 44712 * q^71 - 121214 * q^73 - 167184 * q^77 + 28768 * q^79 - 15066 * q^83 + 132840 * q^85 - 178848 * q^89 - 242440 * q^91 + 39852 * q^95 + 88942 * 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
$5$	57.6281	1.03088	0.515442	−	0.856925i	$-0.327628\pi$
$5$	57.6281	1.03088	0.515442	+	0.856925i	$0.327628\pi$
$6$	0	0
$7$	156.884	1.21014	0.605068	−	0.796173i	$-0.293146\pi$
$7$	156.884	1.21014	0.605068	+	0.796173i	$0.293146\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−218.025	−0.543281	−0.271640	−	0.962399i	$-0.587566\pi$
$11$	−218.025	−0.543281	−0.271640	+	0.962399i	$0.587566\pi$
$12$	0	0
$13$	−587.537	−0.964222	−0.482111	−	0.876110i	$-0.660130\pi$
$13$	−587.537	−0.964222	−0.482111	+	0.876110i	$0.660130\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2124.56	1.78298	0.891491	−	0.453038i	$-0.149660\pi$
$17$	2124.56	1.78298	0.891491	+	0.453038i	$0.149660\pi$
$18$	0	0
$19$	29.7687	0.0189180	0.00945902	−	0.999955i	$-0.496989\pi$
$19$	29.7687	0.0189180	0.00945902	+	0.999955i	$0.496989\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3171.49	1.25010	0.625048	−	0.780586i	$-0.285079\pi$
$23$	3171.49	1.25010	0.625048	+	0.780586i	$0.285079\pi$
$24$	0	0
$25$	196.000	0.0627200
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3642.13	0.804194	0.402097	−	0.915597i	$-0.368282\pi$
$29$	3642.13	0.804194	0.402097	+	0.915597i	$0.368282\pi$
$30$	0	0
$31$	6331.88	1.18339	0.591696	−	0.806162i	$-0.298459\pi$
$31$	6331.88	1.18339	0.591696	+	0.806162i	$0.298459\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9040.95	1.24751
$36$	0	0
$37$	−8435.69	−1.01302	−0.506508	−	0.862235i	$-0.669064\pi$
$37$	−8435.69	−1.01302	−0.506508	+	0.862235i	$0.669064\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	10146.4	0.942657	0.471328	−	0.881958i	$-0.343775\pi$
$41$	10146.4	0.942657	0.471328	+	0.881958i	$0.343775\pi$
$42$	0	0
$43$	−6855.76	−0.565437	−0.282718	−	0.959203i	$-0.591236\pi$
$43$	−6855.76	−0.565437	−0.282718	+	0.959203i	$0.591236\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	24447.8	1.61434	0.807171	−	0.590318i	$-0.200998\pi$
$47$	24447.8	1.61434	0.807171	+	0.590318i	$0.200998\pi$
$48$	0	0
$49$	7805.70	0.464432
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−25483.2	−1.24613	−0.623066	−	0.782169i	$-0.714113\pi$
$53$	−25483.2	−1.24613	−0.623066	+	0.782169i	$0.714113\pi$
$54$	0	0
$55$	−12564.4	−0.560059
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−20696.2	−0.774034	−0.387017	−	0.922073i	$-0.626495\pi$
$59$	−20696.2	−0.774034	−0.387017	+	0.922073i	$0.626495\pi$
$60$	0	0
$61$	45580.2	1.56838	0.784191	−	0.620519i	$-0.213078\pi$
$61$	45580.2	1.56838	0.784191	+	0.620519i	$0.213078\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−33858.7	−0.994000
$66$	0	0
$67$	19099.3	0.519794	0.259897	−	0.965636i	$-0.416311\pi$
$67$	19099.3	0.519794	0.259897	+	0.965636i	$0.416311\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	23055.0	0.542773	0.271387	−	0.962470i	$-0.412518\pi$
$71$	23055.0	0.542773	0.271387	+	0.962470i	$0.412518\pi$
$72$	0	0
$73$	−89651.6	−1.96902	−0.984511	−	0.175320i	$-0.943904\pi$
$73$	−89651.6	−1.96902	−0.984511	+	0.175320i	$0.943904\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−34204.7	−0.657444
$78$	0	0
$79$	−34023.6	−0.613356	−0.306678	−	0.951813i	$-0.599218\pi$
$79$	−34023.6	−0.613356	−0.306678	+	0.951813i	$0.599218\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−108497.	−1.72872	−0.864359	−	0.502875i	$-0.832276\pi$
$83$	−108497.	−1.72872	−0.864359	+	0.502875i	$0.832276\pi$
$84$	0	0
$85$	122435.	1.83805
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−110055.	−1.47277	−0.736384	−	0.676564i	$-0.763468\pi$
$89$	−110055.	−1.47277	−0.736384	+	0.676564i	$0.763468\pi$
$90$	0	0
$91$	−92175.4	−1.16684
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1715.51	0.0195023
$96$	0	0
$97$	−28832.0	−0.311132	−0.155566	−	0.987825i	$-0.549720\pi$
$97$	−28832.0	−0.311132	−0.155566	+	0.987825i	$0.549720\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−40290.3	−0.393004	−0.196502	−	0.980503i	$-0.562958\pi$
$101$	−40290.3	−0.393004	−0.196502	+	0.980503i	$0.562958\pi$
$102$	0	0
$103$	204547.	1.89977	0.949885	−	0.312600i	$-0.101200\pi$
$103$	204547.	1.89977	0.949885	+	0.312600i	$0.101200\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	164042.	1.38515	0.692573	−	0.721348i	$-0.256477\pi$
$107$	164042.	1.38515	0.692573	+	0.721348i	$0.256477\pi$
$108$	0	0
$109$	−131554.	−1.06057	−0.530285	−	0.847819i	$-0.677915\pi$
$109$	−131554.	−1.06057	−0.530285	+	0.847819i	$0.677915\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−94322.0	−0.694891	−0.347446	−	0.937700i	$-0.612951\pi$
$113$	−94322.0	−0.694891	−0.347446	+	0.937700i	$0.612951\pi$
$114$	0	0
$115$	182767.	1.28870
$116$	0	0
$117$	0	0
$118$	0	0
$119$	333311.	2.15765
$120$	0	0
$121$	−113516.	−0.704846
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−168793.	−0.966226
$126$	0	0
$127$	−303144.	−1.66778	−0.833891	−	0.551929i	$-0.813892\pi$
$127$	−303144.	−1.66778	−0.833891	+	0.551929i	$0.813892\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	316573.	1.61174	0.805871	−	0.592091i	$-0.201697\pi$
$131$	316573.	1.61174	0.805871	+	0.592091i	$0.201697\pi$
$132$	0	0
$133$	4670.24	0.0228934
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−982.957	−0.00447438	−0.00223719	−	0.999997i	$-0.500712\pi$
$137$	−982.957	−0.00447438	−0.00223719	+	0.999997i	$0.500712\pi$
$138$	0	0
$139$	77533.8	0.340372	0.170186	−	0.985412i	$-0.445563\pi$
$139$	77533.8	0.340372	0.170186	+	0.985412i	$0.445563\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	128098.	0.523844
$144$	0	0
$145$	209889.	0.829030
$146$	0	0
$147$	0	0
$148$	0	0
$149$	119209.	0.439890	0.219945	−	0.975512i	$-0.429412\pi$
$149$	119209.	0.439890	0.219945	+	0.975512i	$0.429412\pi$
$150$	0	0
$151$	−59568.9	−0.212607	−0.106303	−	0.994334i	$-0.533901\pi$
$151$	−59568.9	−0.212607	−0.106303	+	0.994334i	$0.533901\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	364894.	1.21994
$156$	0	0
$157$	297561.	0.963446	0.481723	−	0.876324i	$-0.340011\pi$
$157$	297561.	0.963446	0.481723	+	0.876324i	$0.340011\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	497557.	1.51279
$162$	0	0
$163$	−98681.9	−0.290917	−0.145458	−	0.989364i	$-0.546466\pi$
$163$	−98681.9	−0.290917	−0.145458	+	0.989364i	$0.546466\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18553.0	−0.0514782	−0.0257391	−	0.999669i	$-0.508194\pi$
$167$	−18553.0	−0.0514782	−0.0257391	+	0.999669i	$0.508194\pi$
$168$	0	0
$169$	−26092.8	−0.0702755
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−496652.	−1.26164	−0.630822	−	0.775927i	$-0.717282\pi$
$173$	−496652.	−1.26164	−0.630822	+	0.775927i	$0.717282\pi$
$174$	0	0
$175$	30749.3	0.0758998
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−150072.	−0.350080	−0.175040	−	0.984561i	$-0.556005\pi$
$179$	−150072.	−0.350080	−0.175040	+	0.984561i	$0.556005\pi$
$180$	0	0
$181$	−73515.6	−0.166795	−0.0833976	−	0.996516i	$-0.526577\pi$
$181$	−73515.6	−0.166795	−0.0833976	+	0.996516i	$0.526577\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−486133.	−1.04430
$186$	0	0
$187$	−463208.	−0.968661
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−599319.	−1.18871	−0.594353	−	0.804204i	$-0.702592\pi$
$191$	−599319.	−1.18871	−0.594353	+	0.804204i	$0.702592\pi$
$192$	0	0
$193$	−723746.	−1.39860	−0.699299	−	0.714829i	$-0.746504\pi$
$193$	−723746.	−1.39860	−0.699299	+	0.714829i	$0.746504\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−416589.	−0.764790	−0.382395	−	0.923999i	$-0.624901\pi$
$197$	−416589.	−0.764790	−0.382395	+	0.923999i	$0.624901\pi$
$198$	0	0
$199$	688177.	1.23188	0.615938	−	0.787794i	$-0.288777\pi$
$199$	688177.	1.23188	0.615938	+	0.787794i	$0.288777\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	571393.	0.973184
$204$	0	0
$205$	584720.	0.971769
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−6490.32	−0.0102778
$210$	0	0
$211$	−321874.	−0.497714	−0.248857	−	0.968540i	$-0.580055\pi$
$211$	−321874.	−0.497714	−0.248857	+	0.968540i	$0.580055\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−395084.	−0.582899
$216$	0	0
$217$	993373.	1.43207
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.24826e6	−1.71919
$222$	0	0
$223$	−354474.	−0.477334	−0.238667	−	0.971101i	$-0.576710\pi$
$223$	−354474.	−0.477334	−0.238667	+	0.971101i	$0.576710\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−226523.	−0.291774	−0.145887	−	0.989301i	$-0.546604\pi$
$227$	−226523.	−0.291774	−0.145887	+	0.989301i	$0.546604\pi$
$228$	0	0
$229$	−1.45431e6	−1.83260	−0.916299	−	0.400494i	$-0.868839\pi$
$229$	−1.45431e6	−1.83260	−0.916299	+	0.400494i	$0.868839\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−423642.	−0.511221	−0.255611	−	0.966780i	$-0.582277\pi$
$233$	−423642.	−0.511221	−0.255611	+	0.966780i	$0.582277\pi$
$234$	0	0
$235$	1.40888e6	1.66420
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−747052.	−0.845972	−0.422986	−	0.906136i	$-0.639018\pi$
$239$	−747052.	−0.845972	−0.422986	+	0.906136i	$0.639018\pi$
$240$	0	0
$241$	611874.	0.678609	0.339305	−	0.940677i	$-0.389808\pi$
$241$	611874.	0.678609	0.339305	+	0.940677i	$0.389808\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	449828.	0.478775
$246$	0	0
$247$	−17490.2	−0.0182412
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−985152.	−0.987004	−0.493502	−	0.869745i	$-0.664283\pi$
$251$	−985152.	−0.987004	−0.493502	+	0.869745i	$0.664283\pi$
$252$	0	0
$253$	−691463.	−0.679153
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.38778e6	1.31065	0.655327	−	0.755345i	$-0.272531\pi$
$257$	1.38778e6	1.31065	0.655327	+	0.755345i	$0.272531\pi$
$258$	0	0
$259$	−1.32343e6	−1.22589
$260$	0	0
$261$	0	0
$262$	0	0
$263$	877536.	0.782304	0.391152	−	0.920326i	$-0.372077\pi$
$263$	877536.	0.782304	0.391152	+	0.920326i	$0.372077\pi$
$264$	0	0
$265$	−1.46855e6	−1.28462
$266$	0	0
$267$	0	0
$268$	0	0
$269$	911089.	0.767680	0.383840	−	0.923400i	$-0.374601\pi$
$269$	911089.	0.767680	0.383840	+	0.923400i	$0.374601\pi$
$270$	0	0
$271$	620129.	0.512931	0.256465	−	0.966553i	$-0.417442\pi$
$271$	620129.	0.512931	0.256465	+	0.966553i	$0.417442\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−42732.9	−0.0340746
$276$	0	0
$277$	−236728.	−0.185374	−0.0926871	−	0.995695i	$-0.529546\pi$
$277$	−236728.	−0.185374	−0.0926871	+	0.995695i	$0.529546\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.07691e6	0.813609	0.406804	−	0.913515i	$-0.366643\pi$
$281$	1.07691e6	0.813609	0.406804	+	0.913515i	$0.366643\pi$
$282$	0	0
$283$	2.05317e6	1.52391	0.761953	−	0.647633i	$-0.224241\pi$
$283$	2.05317e6	1.52391	0.761953	+	0.647633i	$0.224241\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.59182e6	1.14074
$288$	0	0
$289$	3.09391e6	2.17903
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.12866e6	1.44856	0.724280	−	0.689506i	$-0.242172\pi$
$293$	2.12866e6	1.44856	0.724280	+	0.689506i	$0.242172\pi$
$294$	0	0
$295$	−1.19268e6	−0.797939
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.86337e6	−1.20537
$300$	0	0
$301$	−1.07556e6	−0.684256
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.62670e6	1.61682
$306$	0	0
$307$	−1.54154e6	−0.933489	−0.466744	−	0.884392i	$-0.654573\pi$
$307$	−1.54154e6	−0.933489	−0.466744	+	0.884392i	$0.654573\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.73639e6	−1.01800	−0.508999	−	0.860767i	$-0.669984\pi$
$311$	−1.73639e6	−1.01800	−0.508999	+	0.860767i	$0.669984\pi$
$312$	0	0
$313$	2.27414e6	1.31207	0.656034	−	0.754732i	$-0.272233\pi$
$313$	2.27414e6	1.31207	0.656034	+	0.754732i	$0.272233\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−316836.	−0.177087	−0.0885435	−	0.996072i	$-0.528221\pi$
$317$	−316836.	−0.177087	−0.0885435	+	0.996072i	$0.528221\pi$
$318$	0	0
$319$	−794076.	−0.436903
$320$	0	0
$321$	0	0
$322$	0	0
$323$	63245.5	0.0337305
$324$	0	0
$325$	−115157.	−0.0604760
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.83548e6	1.95357
$330$	0	0
$331$	−448791.	−0.225151	−0.112576	−	0.993643i	$-0.535910\pi$
$331$	−448791.	−0.225151	−0.112576	+	0.993643i	$0.535910\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.10066e6	0.535847
$336$	0	0
$337$	−1.85869e6	−0.891524	−0.445762	−	0.895151i	$-0.647067\pi$
$337$	−1.85869e6	−0.891524	−0.445762	+	0.895151i	$0.647067\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.38051e6	−0.642914
$342$	0	0
$343$	−1.41216e6	−0.648111
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−223643.	−0.0997084	−0.0498542	−	0.998757i	$-0.515876\pi$
$347$	−223643.	−0.0997084	−0.0498542	+	0.998757i	$0.515876\pi$
$348$	0	0
$349$	1.33981e6	0.588815	0.294407	−	0.955680i	$-0.404878\pi$
$349$	1.33981e6	0.588815	0.294407	+	0.955680i	$0.404878\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.39683e6	1.45090	0.725449	−	0.688276i	$-0.241632\pi$
$353$	3.39683e6	1.45090	0.725449	+	0.688276i	$0.241632\pi$
$354$	0	0
$355$	1.32861e6	0.559536
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−761965.	−0.312032	−0.156016	−	0.987755i	$-0.549865\pi$
$359$	−761965.	−0.312032	−0.156016	+	0.987755i	$0.549865\pi$
$360$	0	0
$361$	−2.47521e6	−0.999642
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.16645e6	−2.02983
$366$	0	0
$367$	1.39712e6	0.541461	0.270730	−	0.962655i	$-0.412735\pi$
$367$	1.39712e6	0.541461	0.270730	+	0.962655i	$0.412735\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3.99791e6	−1.50799
$372$	0	0
$373$	3.25243e6	1.21042	0.605209	−	0.796067i	$-0.293090\pi$
$373$	3.25243e6	1.21042	0.605209	+	0.796067i	$0.293090\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.13989e6	−0.775421
$378$	0	0
$379$	−700188.	−0.250390	−0.125195	−	0.992132i	$-0.539956\pi$
$379$	−700188.	−0.250390	−0.125195	+	0.992132i	$0.539956\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.30721e6	−0.803693	−0.401847	−	0.915707i	$-0.631631\pi$
$383$	−2.30721e6	−0.803693	−0.401847	+	0.915707i	$0.631631\pi$
$384$	0	0
$385$	−1.97115e6	−0.677748
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.01126e6	−0.673899	−0.336950	−	0.941523i	$-0.609395\pi$
$389$	−2.01126e6	−0.673899	−0.336950	+	0.941523i	$0.609395\pi$
$390$	0	0
$391$	6.73802e6	2.22890
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.96072e6	−0.632299
$396$	0	0
$397$	−165619.	−0.0527392	−0.0263696	−	0.999652i	$-0.508395\pi$
$397$	−165619.	−0.0527392	−0.0263696	+	0.999652i	$0.508395\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.09454e6	0.339915	0.169958	−	0.985451i	$-0.445637\pi$
$401$	1.09454e6	0.339915	0.169958	+	0.985451i	$0.445637\pi$
$402$	0	0
$403$	−3.72021e6	−1.14105
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.83919e6	0.550352
$408$	0	0
$409$	−1.07293e6	−0.317149	−0.158574	−	0.987347i	$-0.550690\pi$
$409$	−1.07293e6	−0.317149	−0.158574	+	0.987347i	$0.550690\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−3.24691e6	−0.936687
$414$	0	0
$415$	−6.25250e6	−1.78211
$416$	0	0
$417$	0	0
$418$	0	0
$419$	852181.	0.237135	0.118568	−	0.992946i	$-0.462170\pi$
$419$	852181.	0.237135	0.118568	+	0.992946i	$0.462170\pi$
$420$	0	0
$421$	3.18784e6	0.876579	0.438290	−	0.898834i	$-0.355584\pi$
$421$	3.18784e6	0.876579	0.438290	+	0.898834i	$0.355584\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	416414.	0.111829
$426$	0	0
$427$	7.15083e6	1.89796
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.54409e6	−1.17829	−0.589147	−	0.808026i	$-0.700536\pi$
$431$	−4.54409e6	−1.17829	−0.589147	+	0.808026i	$0.700536\pi$
$432$	0	0
$433$	−6.16131e6	−1.57926	−0.789629	−	0.613585i	$-0.789727\pi$
$433$	−6.16131e6	−1.57926	−0.789629	+	0.613585i	$0.789727\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	94411.1	0.0236494
$438$	0	0
$439$	902834.	0.223587	0.111793	−	0.993731i	$-0.464340\pi$
$439$	902834.	0.223587	0.111793	+	0.993731i	$0.464340\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.68712e6	−0.892642	−0.446321	−	0.894873i	$-0.647266\pi$
$443$	−3.68712e6	−0.892642	−0.446321	+	0.894873i	$0.647266\pi$
$444$	0	0
$445$	−6.34225e6	−1.51825
$446$	0	0
$447$	0	0
$448$	0	0
$449$	7.71368e6	1.80570	0.902851	−	0.429954i	$-0.141470\pi$
$449$	7.71368e6	1.80570	0.902851	+	0.429954i	$0.141470\pi$
$450$	0	0
$451$	−2.21218e6	−0.512128
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.31190e6	−1.20288
$456$	0	0
$457$	3.79267e6	0.849484	0.424742	−	0.905315i	$-0.360365\pi$
$457$	3.79267e6	0.849484	0.424742	+	0.905315i	$0.360365\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.77898e6	−1.04733	−0.523664	−	0.851925i	$-0.675435\pi$
$461$	−4.77898e6	−1.04733	−0.523664	+	0.851925i	$0.675435\pi$
$462$	0	0
$463$	2.55903e6	0.554782	0.277391	−	0.960757i	$-0.410530\pi$
$463$	2.55903e6	0.554782	0.277391	+	0.960757i	$0.410530\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.22563e6	1.74533	0.872664	−	0.488321i	$-0.162390\pi$
$467$	8.22563e6	1.74533	0.872664	+	0.488321i	$0.162390\pi$
$468$	0	0
$469$	2.99639e6	0.629022
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.49473e6	0.307191
$474$	0	0
$475$	5834.67	0.00118654
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.54369e6	0.506554	0.253277	−	0.967394i	$-0.418492\pi$
$479$	2.54369e6	0.506554	0.253277	+	0.967394i	$0.418492\pi$
$480$	0	0
$481$	4.95628e6	0.976772
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1.66153e6	−0.320741
$486$	0	0
$487$	1.25686e6	0.240139	0.120070	−	0.992765i	$-0.461688\pi$
$487$	1.25686e6	0.240139	0.120070	+	0.992765i	$0.461688\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.89966e6	1.85318	0.926588	−	0.376078i	$-0.122727\pi$
$491$	9.89966e6	1.85318	0.926588	+	0.376078i	$0.122727\pi$
$492$	0	0
$493$	7.73794e6	1.43386
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.61696e6	0.656830
$498$	0	0
$499$	−3.04151e6	−0.546812	−0.273406	−	0.961899i	$-0.588150\pi$
$499$	−3.04151e6	−0.546812	−0.273406	+	0.961899i	$0.588150\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2.75985e6	0.486368	0.243184	−	0.969980i	$-0.421808\pi$
$503$	2.75985e6	0.486368	0.243184	+	0.969980i	$0.421808\pi$
$504$	0	0
$505$	−2.32186e6	−0.405142
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.44508e6	−0.247227	−0.123614	−	0.992330i	$-0.539448\pi$
$509$	−1.44508e6	−0.247227	−0.123614	+	0.992330i	$0.539448\pi$
$510$	0	0
$511$	−1.40649e7	−2.38279
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.17877e7	1.95844
$516$	0	0
$517$	−5.33024e6	−0.877041
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.28492e6	0.207388	0.103694	−	0.994609i	$-0.466934\pi$
$521$	1.28492e6	0.207388	0.103694	+	0.994609i	$0.466934\pi$
$522$	0	0
$523$	−2.38275e6	−0.380912	−0.190456	−	0.981696i	$-0.560997\pi$
$523$	−2.38275e6	−0.380912	−0.190456	+	0.981696i	$0.560997\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.34525e7	2.10997
$528$	0	0
$529$	3.62199e6	0.562740
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.96141e6	−0.908931
$534$	0	0
$535$	9.45342e6	1.42792
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.70184e6	−0.252317
$540$	0	0
$541$	2.22158e6	0.326339	0.163170	−	0.986598i	$-0.447828\pi$
$541$	2.22158e6	0.326339	0.163170	+	0.986598i	$0.447828\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.58124e6	−1.09332
$546$	0	0
$547$	−1.81689e6	−0.259634	−0.129817	−	0.991538i	$-0.541439\pi$
$547$	−1.81689e6	−0.259634	−0.129817	+	0.991538i	$0.541439\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	108422.	0.0152138
$552$	0	0
$553$	−5.33777e6	−0.742245
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3.53109e6	−0.482248	−0.241124	−	0.970494i	$-0.577516\pi$
$557$	−3.53109e6	−0.482248	−0.241124	+	0.970494i	$0.577516\pi$
$558$	0	0
$559$	4.02801e6	0.545207
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−327456.	−0.0435393	−0.0217697	−	0.999763i	$-0.506930\pi$
$563$	−327456.	−0.0435393	−0.0217697	+	0.999763i	$0.506930\pi$
$564$	0	0
$565$	−5.43560e6	−0.716351
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.06773e6	1.04465	0.522325	−	0.852747i	$-0.325065\pi$
$569$	8.06773e6	1.04465	0.522325	+	0.852747i	$0.325065\pi$
$570$	0	0
$571$	−211349.	−0.0271275	−0.0135637	−	0.999908i	$-0.504318\pi$
$571$	−211349.	−0.0271275	−0.0135637	+	0.999908i	$0.504318\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	621612.	0.0784060
$576$	0	0
$577$	−1.60363e6	−0.200523	−0.100261	−	0.994961i	$-0.531968\pi$
$577$	−1.60363e6	−0.200523	−0.100261	+	0.994961i	$0.531968\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.70216e7	−2.09199
$582$	0	0
$583$	5.55597e6	0.677000
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.92480e6	1.18885	0.594424	−	0.804151i	$-0.297380\pi$
$587$	9.92480e6	1.18885	0.594424	+	0.804151i	$0.297380\pi$
$588$	0	0
$589$	188492.	0.0223874
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.74316e6	0.670678	0.335339	−	0.942098i	$-0.391149\pi$
$593$	5.74316e6	0.670678	0.335339	+	0.942098i	$0.391149\pi$
$594$	0	0
$595$	1.92081e7	2.22429
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.27574e6	0.600781	0.300391	−	0.953816i	$-0.402883\pi$
$599$	5.27574e6	0.600781	0.300391	+	0.953816i	$0.402883\pi$
$600$	0	0
$601$	4.14952e6	0.468610	0.234305	−	0.972163i	$-0.424719\pi$
$601$	4.14952e6	0.468610	0.234305	+	0.972163i	$0.424719\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.54172e6	−0.726614
$606$	0	0
$607$	−1.58501e7	−1.74606	−0.873031	−	0.487665i	$-0.837849\pi$
$607$	−1.58501e7	−1.74606	−0.873031	+	0.487665i	$0.837849\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.43640e7	−1.55658
$612$	0	0
$613$	1.48963e7	1.60113	0.800565	−	0.599246i	$-0.204533\pi$
$613$	1.48963e7	1.60113	0.800565	+	0.599246i	$0.204533\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.44790e7	1.53118	0.765591	−	0.643328i	$-0.222447\pi$
$617$	1.44790e7	1.53118	0.765591	+	0.643328i	$0.222447\pi$
$618$	0	0
$619$	−1.05892e7	−1.11080	−0.555402	−	0.831582i	$-0.687435\pi$
$619$	−1.05892e7	−1.11080	−0.555402	+	0.831582i	$0.687435\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.72659e7	−1.78225
$624$	0	0
$625$	−1.03397e7	−1.05879
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1.79221e7	−1.80619
$630$	0	0
$631$	−4.93118e6	−0.493035	−0.246517	−	0.969138i	$-0.579286\pi$
$631$	−4.93118e6	−0.493035	−0.246517	+	0.969138i	$0.579286\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1.74696e7	−1.71929
$636$	0	0
$637$	−4.58614e6	−0.447815
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.90060e7	−1.82703	−0.913514	−	0.406807i	$-0.866642\pi$
$641$	−1.90060e7	−1.82703	−0.913514	+	0.406807i	$0.866642\pi$
$642$	0	0
$643$	−6.69738e6	−0.638818	−0.319409	−	0.947617i	$-0.603484\pi$
$643$	−6.69738e6	−0.638818	−0.319409	+	0.947617i	$0.603484\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.62821e7	1.52915	0.764574	−	0.644536i	$-0.222949\pi$
$647$	1.62821e7	1.52915	0.764574	+	0.644536i	$0.222949\pi$
$648$	0	0
$649$	4.51228e6	0.420518
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−5.52086e6	−0.506668	−0.253334	−	0.967379i	$-0.581527\pi$
$653$	−5.52086e6	−0.506668	−0.253334	+	0.967379i	$0.581527\pi$
$654$	0	0
$655$	1.82435e7	1.66152
$656$	0	0
$657$	0	0
$658$	0	0
$659$	139176.	0.0124839	0.00624197	−	0.999981i	$-0.498013\pi$
$659$	139176.	0.0124839	0.00624197	+	0.999981i	$0.498013\pi$
$660$	0	0
$661$	−1.94550e7	−1.73192	−0.865960	−	0.500113i	$-0.833292\pi$
$661$	−1.94550e7	−1.73192	−0.865960	+	0.500113i	$0.833292\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	269137.	0.0236004
$666$	0	0
$667$	1.15510e7	1.00532
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.93763e6	−0.852072
$672$	0	0
$673$	−9.15012e6	−0.778734	−0.389367	−	0.921083i	$-0.627306\pi$
$673$	−9.15012e6	−0.778734	−0.389367	+	0.921083i	$0.627306\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.92716e6	−0.832441	−0.416220	−	0.909264i	$-0.636646\pi$
$677$	−9.92716e6	−0.832441	−0.416220	+	0.909264i	$0.636646\pi$
$678$	0	0
$679$	−4.52328e6	−0.376513
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.27969e7	1.04967	0.524835	−	0.851204i	$-0.324127\pi$
$683$	1.27969e7	1.04967	0.524835	+	0.851204i	$0.324127\pi$
$684$	0	0
$685$	−56646.0	−0.00461257
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.49723e7	1.20155
$690$	0	0
$691$	8.05692e6	0.641909	0.320955	−	0.947095i	$-0.395996\pi$
$691$	8.05692e6	0.641909	0.320955	+	0.947095i	$0.395996\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.46813e6	0.350884
$696$	0	0
$697$	2.15567e7	1.68074
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.05658e7	−0.812100	−0.406050	−	0.913851i	$-0.633094\pi$
$701$	−1.05658e7	−0.812100	−0.406050	+	0.913851i	$0.633094\pi$
$702$	0	0
$703$	−251120.	−0.0191643
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.32092e6	−0.475589
$708$	0	0
$709$	−1.51822e7	−1.13428	−0.567138	−	0.823623i	$-0.691949\pi$
$709$	−1.51822e7	−1.13428	−0.567138	+	0.823623i	$0.691949\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.00815e7	1.47935
$714$	0	0
$715$	7.38204e6	0.540021
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.93525e6	0.211750	0.105875	−	0.994379i	$-0.466236\pi$
$719$	2.93525e6	0.211750	0.105875	+	0.994379i	$0.466236\pi$
$720$	0	0
$721$	3.20903e7	2.29898
$722$	0	0
$723$	0	0
$724$	0	0
$725$	713858.	0.0504390
$726$	0	0
$727$	−9.33363e6	−0.654960	−0.327480	−	0.944858i	$-0.606199\pi$
$727$	−9.33363e6	−0.654960	−0.327480	+	0.944858i	$0.606199\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.45655e7	−1.00816
$732$	0	0
$733$	−4.73250e6	−0.325335	−0.162667	−	0.986681i	$-0.552010\pi$
$733$	−4.73250e6	−0.325335	−0.162667	+	0.986681i	$0.552010\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.16413e6	−0.282394
$738$	0	0
$739$	−2.42162e7	−1.63115	−0.815575	−	0.578651i	$-0.803579\pi$
$739$	−2.42162e7	−1.63115	−0.815575	+	0.578651i	$0.803579\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.47448e7	1.64441	0.822207	−	0.569189i	$-0.192743\pi$
$743$	2.47448e7	1.64441	0.822207	+	0.569189i	$0.192743\pi$
$744$	0	0
$745$	6.86981e6	0.453476
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.57356e7	1.67622
$750$	0	0
$751$	−5.57065e6	−0.360418	−0.180209	−	0.983628i	$-0.557677\pi$
$751$	−5.57065e6	−0.360418	−0.180209	+	0.983628i	$0.557677\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.43284e6	−0.219173
$756$	0	0
$757$	−2.72203e6	−0.172645	−0.0863224	−	0.996267i	$-0.527511\pi$
$757$	−2.72203e6	−0.172645	−0.0863224	+	0.996267i	$0.527511\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.50617e7	−0.942787	−0.471393	−	0.881923i	$-0.656249\pi$
$761$	−1.50617e7	−0.942787	−0.471393	+	0.881923i	$0.656249\pi$
$762$	0	0
$763$	−2.06388e7	−1.28344
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.21598e7	0.746341
$768$	0	0
$769$	9.13893e6	0.557288	0.278644	−	0.960395i	$-0.410115\pi$
$769$	9.13893e6	0.557288	0.278644	+	0.960395i	$0.410115\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.35738e7	0.817060	0.408530	−	0.912745i	$-0.366041\pi$
$773$	1.35738e7	0.817060	0.408530	+	0.912745i	$0.366041\pi$
$774$	0	0
$775$	1.24105e6	0.0742223
$776$	0	0
$777$	0	0
$778$	0	0
$779$	302046.	0.0178332
$780$	0	0
$781$	−5.02656e6	−0.294878
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.71479e7	0.993200
$786$	0	0
$787$	−2.04292e7	−1.17575	−0.587875	−	0.808952i	$-0.700035\pi$
$787$	−2.04292e7	−1.17575	−0.587875	+	0.808952i	$0.700035\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.47976e7	−0.840913
$792$	0	0
$793$	−2.67801e7	−1.51227
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.03966e7	1.69504	0.847518	−	0.530766i	$-0.178096\pi$
$797$	3.03966e7	1.69504	0.847518	+	0.530766i	$0.178096\pi$
$798$	0	0
$799$	5.19409e7	2.87834
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.95463e7	1.06973
$804$	0	0
$805$	2.86733e7	1.55951
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.50428e7	1.88247	0.941234	−	0.337755i	$-0.109667\pi$
$809$	3.50428e7	1.88247	0.941234	+	0.337755i	$0.109667\pi$
$810$	0	0
$811$	1.55069e7	0.827893	0.413946	−	0.910301i	$-0.364150\pi$
$811$	1.55069e7	0.827893	0.413946	+	0.910301i	$0.364150\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.68685e6	−0.299901
$816$	0	0
$817$	−204087.	−0.0106970
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.51617e7	−0.785038	−0.392519	−	0.919744i	$-0.628396\pi$
$821$	−1.51617e7	−0.785038	−0.392519	+	0.919744i	$0.628396\pi$
$822$	0	0
$823$	−1.26532e7	−0.651180	−0.325590	−	0.945511i	$-0.605563\pi$
$823$	−1.26532e7	−0.651180	−0.325590	+	0.945511i	$0.605563\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.21240e7	0.616427	0.308213	−	0.951317i	$-0.400269\pi$
$827$	1.21240e7	0.616427	0.308213	+	0.951317i	$0.400269\pi$
$828$	0	0
$829$	−1.79003e7	−0.904634	−0.452317	−	0.891857i	$-0.649402\pi$
$829$	−1.79003e7	−0.904634	−0.452317	+	0.891857i	$0.649402\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.65837e7	0.828073
$834$	0	0
$835$	−1.06918e6	−0.0530680
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.34365e7	−1.14945	−0.574723	−	0.818348i	$-0.694890\pi$
$839$	−2.34365e7	−1.14945	−0.574723	+	0.818348i	$0.694890\pi$
$840$	0	0
$841$	−7.24603e6	−0.353273
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.50368e6	−0.0724458
$846$	0	0
$847$	−1.78089e7	−0.852960
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.67537e7	−1.26637
$852$	0	0
$853$	2.89092e7	1.36039	0.680196	−	0.733030i	$-0.261895\pi$
$853$	2.89092e7	1.36039	0.680196	+	0.733030i	$0.261895\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.06961e7	0.497476	0.248738	−	0.968571i	$-0.419984\pi$
$857$	1.06961e7	0.497476	0.248738	+	0.968571i	$0.419984\pi$
$858$	0	0
$859$	3.69703e7	1.70950	0.854752	−	0.519037i	$-0.173709\pi$
$859$	3.69703e7	1.70950	0.854752	+	0.519037i	$0.173709\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.83720e7	−1.75383	−0.876914	−	0.480646i	$-0.840402\pi$
$863$	−3.83720e7	−1.75383	−0.876914	+	0.480646i	$0.840402\pi$
$864$	0	0
$865$	−2.86211e7	−1.30061
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.41800e6	0.333225
$870$	0	0
$871$	−1.12216e7	−0.501197
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.64809e7	−1.16927
$876$	0	0
$877$	3.66208e7	1.60779	0.803895	−	0.594771i	$-0.202757\pi$
$877$	3.66208e7	1.60779	0.803895	+	0.594771i	$0.202757\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.70223e7	−0.738889	−0.369445	−	0.929253i	$-0.620452\pi$
$881$	−1.70223e7	−0.738889	−0.369445	+	0.929253i	$0.620452\pi$
$882$	0	0
$883$	3.54409e7	1.52969	0.764844	−	0.644215i	$-0.222816\pi$
$883$	3.54409e7	1.52969	0.764844	+	0.644215i	$0.222816\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.20072e7	−0.512428	−0.256214	−	0.966620i	$-0.582475\pi$
$887$	−1.20072e7	−0.512428	−0.256214	+	0.966620i	$0.582475\pi$
$888$	0	0
$889$	−4.75585e7	−2.01824
$890$	0	0
$891$	0	0
$892$	0	0
$893$	727780.	0.0305402
$894$	0	0
$895$	−8.64837e6	−0.360892
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.30615e7	0.951676
$900$	0	0
$901$	−5.41406e7	−2.22183
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.23657e6	−0.171946
$906$	0	0
$907$	−8.42904e6	−0.340220	−0.170110	−	0.985425i	$-0.554412\pi$
$907$	−8.42904e6	−0.340220	−0.170110	+	0.985425i	$0.554412\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.48847e7	−0.993430	−0.496715	−	0.867914i	$-0.665461\pi$
$911$	−2.48847e7	−0.993430	−0.496715	+	0.867914i	$0.665461\pi$
$912$	0	0
$913$	2.36552e7	0.939180
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.96653e7	1.95043
$918$	0	0
$919$	−3.68789e7	−1.44042	−0.720210	−	0.693757i	$-0.755954\pi$
$919$	−3.68789e7	−1.44042	−0.720210	+	0.693757i	$0.755954\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.35456e7	−0.523354
$924$	0	0
$925$	−1.65339e6	−0.0635363
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.60910e7	−0.991863	−0.495932	−	0.868362i	$-0.665173\pi$
$929$	−2.60910e7	−0.991863	−0.495932	+	0.868362i	$0.665173\pi$
$930$	0	0
$931$	232366.	0.00878613
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.66938e7	−0.998576
$936$	0	0
$937$	7.84259e6	0.291817	0.145908	−	0.989298i	$-0.453390\pi$
$937$	7.84259e6	0.291817	0.145908	+	0.989298i	$0.453390\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.36164e7	−1.23759	−0.618796	−	0.785552i	$-0.712379\pi$
$941$	−3.36164e7	−1.23759	−0.618796	+	0.785552i	$0.712379\pi$
$942$	0	0
$943$	3.21793e7	1.17841
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.57753e7	0.571615	0.285808	−	0.958287i	$-0.407738\pi$
$947$	1.57753e7	0.571615	0.285808	+	0.958287i	$0.407738\pi$
$948$	0	0
$949$	5.26737e7	1.89858
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−2.32737e7	−0.830104	−0.415052	−	0.909798i	$-0.636237\pi$
$953$	−2.32737e7	−0.830104	−0.415052	+	0.909798i	$0.636237\pi$
$954$	0	0
$955$	−3.45376e7	−1.22542
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−154211.	−0.00541462
$960$	0	0
$961$	1.14635e7	0.400414
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.17081e7	−1.44179
$966$	0	0
$967$	1.95430e7	0.672085	0.336043	−	0.941847i	$-0.390911\pi$
$967$	1.95430e7	0.672085	0.336043	+	0.941847i	$0.390911\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5.59842e7	−1.90554	−0.952768	−	0.303700i	$-0.901778\pi$
$971$	−5.59842e7	−1.90554	−0.952768	+	0.303700i	$0.901778\pi$
$972$	0	0
$973$	1.21638e7	0.411897
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.20225e6	0.140846	0.0704231	−	0.997517i	$-0.477565\pi$
$977$	4.20225e6	0.140846	0.0704231	+	0.997517i	$0.477565\pi$
$978$	0	0
$979$	2.39947e7	0.800127
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.81575e7	0.599340	0.299670	−	0.954043i	$-0.403123\pi$
$983$	1.81575e7	0.599340	0.299670	+	0.954043i	$0.403123\pi$
$984$	0	0
$985$	−2.40072e7	−0.788409
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.17429e7	−0.706851
$990$	0	0
$991$	6.51450e6	0.210716	0.105358	−	0.994434i	$-0.466401\pi$
$991$	6.51450e6	0.210716	0.105358	+	0.994434i	$0.466401\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.96583e7	1.26992
$996$	0	0
$997$	−2.89170e7	−0.921329	−0.460664	−	0.887574i	$-0.652389\pi$
$997$	−2.89170e7	−0.921329	−0.460664	+	0.887574i	$0.652389\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.6.a.c.1.2	yes	2
3.2	odd	2		108.6.a.b.1.1	✓	2
4.3	odd	2		432.6.a.q.1.2		2
9.2	odd	6		324.6.e.g.109.2		4
9.4	even	3		324.6.e.f.217.1		4
9.5	odd	6		324.6.e.g.217.2		4
9.7	even	3		324.6.e.f.109.1		4
12.11	even	2		432.6.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.a.b.1.1	✓	2	3.2	odd	2
108.6.a.c.1.2	yes	2	1.1	even	1	trivial
324.6.e.f.109.1		4	9.7	even	3
324.6.e.f.217.1		4	9.4	even	3
324.6.e.g.109.2		4	9.2	odd	6
324.6.e.g.217.2		4	9.5	odd	6
432.6.a.q.1.2		2	4.3	odd	2
432.6.a.r.1.1		2	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	108.a (trivial)

\(f(q)\)	\(=\)	\(q+57.6281 q^{5} +156.884 q^{7} +O(q^{10})\)	\(q+57.6281 q^{5} +156.884 q^{7} -218.025 q^{11} -587.537 q^{13} +2124.56 q^{17} +29.7687 q^{19} +3171.49 q^{23} +196.000 q^{25} +3642.13 q^{29} +6331.88 q^{31} +9040.95 q^{35} -8435.69 q^{37} +10146.4 q^{41} -6855.76 q^{43} +24447.8 q^{47} +7805.70 q^{49} -25483.2 q^{53} -12564.4 q^{55} -20696.2 q^{59} +45580.2 q^{61} -33858.7 q^{65} +19099.3 q^{67} +23055.0 q^{71} -89651.6 q^{73} -34204.7 q^{77} -34023.6 q^{79} -108497. q^{83} +122435. q^{85} -110055. q^{89} -92175.4 q^{91} +1715.51 q^{95} -28832.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{7}+O(q^{10}) \) 2 * q - 32 * q^7	\( 2 q - 32 q^{7} + 486 q^{11} + 208 q^{13} + 1944 q^{17} - 632 q^{19} + 6804 q^{23} + 392 q^{25} + 11664 q^{29} + 3328 q^{31} + 19926 q^{35} - 9956 q^{37} + 13608 q^{41} + 4960 q^{43} + 18468 q^{47} + 26676 q^{49} - 11664 q^{53} - 53136 q^{55} + 1944 q^{59} + 8176 q^{61} - 79704 q^{65} + 90064 q^{67} - 44712 q^{71} - 121214 q^{73} - 167184 q^{77} + 28768 q^{79} - 15066 q^{83} + 132840 q^{85} - 178848 q^{89} - 242440 q^{91} + 39852 q^{95} + 88942 q^{97}+O(q^{100}) \) 2 * q - 32 * q^7 + 486 * q^11 + 208 * q^13 + 1944 * q^17 - 632 * q^19 + 6804 * q^23 + 392 * q^25 + 11664 * q^29 + 3328 * q^31 + 19926 * q^35 - 9956 * q^37 + 13608 * q^41 + 4960 * q^43 + 18468 * q^47 + 26676 * q^49 - 11664 * q^53 - 53136 * q^55 + 1944 * q^59 + 8176 * q^61 - 79704 * q^65 + 90064 * q^67 - 44712 * q^71 - 121214 * q^73 - 167184 * q^77 + 28768 * q^79 - 15066 * q^83 + 132840 * q^85 - 178848 * q^89 - 242440 * q^91 + 39852 * q^95 + 88942 * q^97

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