LMFDB - Embedded newform 8112.2.a.bo.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8112,2,Mod(1,8112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8112.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.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:	$64.7746461197$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	8112.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.56155 q^{5} +0.561553 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +3.56155 q^{5} +0.561553 q^{7} +1.00000 q^{9} -2.00000 q^{11} -3.56155 q^{15} -1.56155 q^{17} +7.12311 q^{19} -0.561553 q^{21} -2.00000 q^{23} +7.68466 q^{25} -1.00000 q^{27} +6.68466 q^{29} +2.56155 q^{31} +2.00000 q^{33} +2.00000 q^{35} -7.56155 q^{37} +1.56155 q^{41} -4.56155 q^{43} +3.56155 q^{45} +8.24621 q^{47} -6.68466 q^{49} +1.56155 q^{51} -0.684658 q^{53} -7.12311 q^{55} -7.12311 q^{57} -2.87689 q^{59} +3.87689 q^{61} +0.561553 q^{63} +4.56155 q^{67} +2.00000 q^{69} +14.0000 q^{71} +10.1231 q^{73} -7.68466 q^{75} -1.12311 q^{77} -5.43845 q^{79} +1.00000 q^{81} -0.876894 q^{83} -5.56155 q^{85} -6.68466 q^{87} -4.87689 q^{89} -2.56155 q^{93} +25.3693 q^{95} +8.56155 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 4 q^{11} - 3 q^{15} + q^{17} + 6 q^{19} + 3 q^{21} - 4 q^{23} + 3 q^{25} - 2 q^{27} + q^{29} + q^{31} + 4 q^{33} + 4 q^{35} - 11 q^{37} - q^{41} - 5 q^{43} + 3 q^{45} - q^{49} - q^{51} + 11 q^{53} - 6 q^{55} - 6 q^{57} - 14 q^{59} + 16 q^{61} - 3 q^{63} + 5 q^{67} + 4 q^{69} + 28 q^{71} + 12 q^{73} - 3 q^{75} + 6 q^{77} - 15 q^{79} + 2 q^{81} - 10 q^{83} - 7 q^{85} - q^{87} - 18 q^{89} - q^{93} + 26 q^{95} + 13 q^{97} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9 - 4 * q^11 - 3 * q^15 + q^17 + 6 * q^19 + 3 * q^21 - 4 * q^23 + 3 * q^25 - 2 * q^27 + q^29 + q^31 + 4 * q^33 + 4 * q^35 - 11 * q^37 - q^41 - 5 * q^43 + 3 * q^45 - q^49 - q^51 + 11 * q^53 - 6 * q^55 - 6 * q^57 - 14 * q^59 + 16 * q^61 - 3 * q^63 + 5 * q^67 + 4 * q^69 + 28 * q^71 + 12 * q^73 - 3 * q^75 + 6 * q^77 - 15 * q^79 + 2 * q^81 - 10 * q^83 - 7 * q^85 - q^87 - 18 * q^89 - q^93 + 26 * q^95 + 13 * q^97 - 4 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.56155	1.59277	0.796387	−	0.604787i	$-0.206742\pi$
$5$	3.56155	1.59277	0.796387	+	0.604787i	$0.206742\pi$
$6$	0	0
$7$	0.561553	0.212247	0.106124	−	0.994353i	$-0.466156\pi$
$7$	0.561553	0.212247	0.106124	+	0.994353i	$0.466156\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−3.56155	−0.919589
$16$	0	0
$17$	−1.56155	−0.378732	−0.189366	−	0.981907i	$-0.560643\pi$
$17$	−1.56155	−0.378732	−0.189366	+	0.981907i	$0.560643\pi$
$18$	0	0
$19$	7.12311	1.63415	0.817076	−	0.576530i	$-0.195593\pi$
$19$	7.12311	1.63415	0.817076	+	0.576530i	$0.195593\pi$
$20$	0	0
$21$	−0.561553	−0.122541
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.68466	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$29$	6.68466	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$30$	0	0
$31$	2.56155	0.460068	0.230034	−	0.973183i	$-0.426116\pi$
$31$	2.56155	0.460068	0.230034	+	0.973183i	$0.426116\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−7.56155	−1.24311	−0.621556	−	0.783370i	$-0.713499\pi$
$37$	−7.56155	−1.24311	−0.621556	+	0.783370i	$0.713499\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.56155	0.243874	0.121937	−	0.992538i	$-0.461089\pi$
$41$	1.56155	0.243874	0.121937	+	0.992538i	$0.461089\pi$
$42$	0	0
$43$	−4.56155	−0.695630	−0.347815	−	0.937563i	$-0.613076\pi$
$43$	−4.56155	−0.695630	−0.347815	+	0.937563i	$0.613076\pi$
$44$	0	0
$45$	3.56155	0.530925
$46$	0	0
$47$	8.24621	1.20283	0.601417	−	0.798935i	$-0.294603\pi$
$47$	8.24621	1.20283	0.601417	+	0.798935i	$0.294603\pi$
$48$	0	0
$49$	−6.68466	−0.954951
$50$	0	0
$51$	1.56155	0.218661
$52$	0	0
$53$	−0.684658	−0.0940451	−0.0470225	−	0.998894i	$-0.514973\pi$
$53$	−0.684658	−0.0940451	−0.0470225	+	0.998894i	$0.514973\pi$
$54$	0	0
$55$	−7.12311	−0.960479
$56$	0	0
$57$	−7.12311	−0.943478
$58$	0	0
$59$	−2.87689	−0.374540	−0.187270	−	0.982309i	$-0.559964\pi$
$59$	−2.87689	−0.374540	−0.187270	+	0.982309i	$0.559964\pi$
$60$	0	0
$61$	3.87689	0.496385	0.248193	−	0.968711i	$-0.420163\pi$
$61$	3.87689	0.496385	0.248193	+	0.968711i	$0.420163\pi$
$62$	0	0
$63$	0.561553	0.0707490
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.56155	0.557282	0.278641	−	0.960395i	$-0.410116\pi$
$67$	4.56155	0.557282	0.278641	+	0.960395i	$0.410116\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	0	0
$73$	10.1231	1.18482	0.592410	−	0.805637i	$-0.298177\pi$
$73$	10.1231	1.18482	0.592410	+	0.805637i	$0.298177\pi$
$74$	0	0
$75$	−7.68466	−0.887348
$76$	0	0
$77$	−1.12311	−0.127990
$78$	0	0
$79$	−5.43845	−0.611873	−0.305937	−	0.952052i	$-0.598970\pi$
$79$	−5.43845	−0.611873	−0.305937	+	0.952052i	$0.598970\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.876894	−0.0962517	−0.0481258	−	0.998841i	$-0.515325\pi$
$83$	−0.876894	−0.0962517	−0.0481258	+	0.998841i	$0.515325\pi$
$84$	0	0
$85$	−5.56155	−0.603235
$86$	0	0
$87$	−6.68466	−0.716671
$88$	0	0
$89$	−4.87689	−0.516950	−0.258475	−	0.966018i	$-0.583220\pi$
$89$	−4.87689	−0.516950	−0.258475	+	0.966018i	$0.583220\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.56155	−0.265621
$94$	0	0
$95$	25.3693	2.60284
$96$	0	0
$97$	8.56155	0.869294	0.434647	−	0.900601i	$-0.356873\pi$
$97$	8.56155	0.869294	0.434647	+	0.900601i	$0.356873\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−7.56155	−0.752403	−0.376201	−	0.926538i	$-0.622770\pi$
$101$	−7.56155	−0.752403	−0.376201	+	0.926538i	$0.622770\pi$
$102$	0	0
$103$	3.43845	0.338800	0.169400	−	0.985547i	$-0.445817\pi$
$103$	3.43845	0.338800	0.169400	+	0.985547i	$0.445817\pi$
$104$	0	0
$105$	−2.00000	−0.195180
$106$	0	0
$107$	8.24621	0.797191	0.398596	−	0.917127i	$-0.369498\pi$
$107$	8.24621	0.797191	0.398596	+	0.917127i	$0.369498\pi$
$108$	0	0
$109$	2.80776	0.268935	0.134468	−	0.990918i	$-0.457068\pi$
$109$	2.80776	0.268935	0.134468	+	0.990918i	$0.457068\pi$
$110$	0	0
$111$	7.56155	0.717711
$112$	0	0
$113$	5.80776	0.546348	0.273174	−	0.961965i	$-0.411926\pi$
$113$	5.80776	0.546348	0.273174	+	0.961965i	$0.411926\pi$
$114$	0	0
$115$	−7.12311	−0.664233
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.876894	−0.0803848
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−1.56155	−0.140800
$124$	0	0
$125$	9.56155	0.855211
$126$	0	0
$127$	−5.43845	−0.482584	−0.241292	−	0.970453i	$-0.577571\pi$
$127$	−5.43845	−0.482584	−0.241292	+	0.970453i	$0.577571\pi$
$128$	0	0
$129$	4.56155	0.401622
$130$	0	0
$131$	−7.36932	−0.643860	−0.321930	−	0.946763i	$-0.604332\pi$
$131$	−7.36932	−0.643860	−0.321930	+	0.946763i	$0.604332\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	−3.56155	−0.306530
$136$	0	0
$137$	5.56155	0.475156	0.237578	−	0.971369i	$-0.423647\pi$
$137$	5.56155	0.475156	0.237578	+	0.971369i	$0.423647\pi$
$138$	0	0
$139$	17.9309	1.52088	0.760438	−	0.649410i	$-0.224984\pi$
$139$	17.9309	1.52088	0.760438	+	0.649410i	$0.224984\pi$
$140$	0	0
$141$	−8.24621	−0.694456
$142$	0	0
$143$	0	0
$144$	0	0
$145$	23.8078	1.97713
$146$	0	0
$147$	6.68466	0.551341
$148$	0	0
$149$	2.43845	0.199765	0.0998827	−	0.994999i	$-0.468153\pi$
$149$	2.43845	0.199765	0.0998827	+	0.994999i	$0.468153\pi$
$150$	0	0
$151$	−9.36932	−0.762464	−0.381232	−	0.924479i	$-0.624500\pi$
$151$	−9.36932	−0.762464	−0.381232	+	0.924479i	$0.624500\pi$
$152$	0	0
$153$	−1.56155	−0.126244
$154$	0	0
$155$	9.12311	0.732785
$156$	0	0
$157$	20.3693	1.62565	0.812824	−	0.582509i	$-0.197929\pi$
$157$	20.3693	1.62565	0.812824	+	0.582509i	$0.197929\pi$
$158$	0	0
$159$	0.684658	0.0542969
$160$	0	0
$161$	−1.12311	−0.0885131
$162$	0	0
$163$	−4.80776	−0.376573	−0.188287	−	0.982114i	$-0.560293\pi$
$163$	−4.80776	−0.376573	−0.188287	+	0.982114i	$0.560293\pi$
$164$	0	0
$165$	7.12311	0.554533
$166$	0	0
$167$	10.2462	0.792876	0.396438	−	0.918062i	$-0.370246\pi$
$167$	10.2462	0.792876	0.396438	+	0.918062i	$0.370246\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	7.12311	0.544718
$172$	0	0
$173$	20.2462	1.53929	0.769645	−	0.638471i	$-0.220433\pi$
$173$	20.2462	1.53929	0.769645	+	0.638471i	$0.220433\pi$
$174$	0	0
$175$	4.31534	0.326209
$176$	0	0
$177$	2.87689	0.216241
$178$	0	0
$179$	4.87689	0.364516	0.182258	−	0.983251i	$-0.441659\pi$
$179$	4.87689	0.364516	0.182258	+	0.983251i	$0.441659\pi$
$180$	0	0
$181$	−2.68466	−0.199549	−0.0997745	−	0.995010i	$-0.531812\pi$
$181$	−2.68466	−0.199549	−0.0997745	+	0.995010i	$0.531812\pi$
$182$	0	0
$183$	−3.87689	−0.286588
$184$	0	0
$185$	−26.9309	−1.98000
$186$	0	0
$187$	3.12311	0.228384
$188$	0	0
$189$	−0.561553	−0.0408470
$190$	0	0
$191$	9.12311	0.660125	0.330062	−	0.943959i	$-0.392930\pi$
$191$	9.12311	0.660125	0.330062	+	0.943959i	$0.392930\pi$
$192$	0	0
$193$	13.4924	0.971206	0.485603	−	0.874180i	$-0.338600\pi$
$193$	13.4924	0.971206	0.485603	+	0.874180i	$0.338600\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.3693	−0.952524	−0.476262	−	0.879303i	$-0.658009\pi$
$197$	−13.3693	−0.952524	−0.476262	+	0.879303i	$0.658009\pi$
$198$	0	0
$199$	−22.1771	−1.57209	−0.786046	−	0.618168i	$-0.787875\pi$
$199$	−22.1771	−1.57209	−0.786046	+	0.618168i	$0.787875\pi$
$200$	0	0
$201$	−4.56155	−0.321747
$202$	0	0
$203$	3.75379	0.263464
$204$	0	0
$205$	5.56155	0.388436
$206$	0	0
$207$	−2.00000	−0.139010
$208$	0	0
$209$	−14.2462	−0.985431
$210$	0	0
$211$	−19.6847	−1.35515	−0.677574	−	0.735455i	$-0.736969\pi$
$211$	−19.6847	−1.35515	−0.677574	+	0.735455i	$0.736969\pi$
$212$	0	0
$213$	−14.0000	−0.959264
$214$	0	0
$215$	−16.2462	−1.10798
$216$	0	0
$217$	1.43845	0.0976482
$218$	0	0
$219$	−10.1231	−0.684056
$220$	0	0
$221$	0	0
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	7.68466	0.512311
$226$	0	0
$227$	7.12311	0.472777	0.236389	−	0.971659i	$-0.424036\pi$
$227$	7.12311	0.472777	0.236389	+	0.971659i	$0.424036\pi$
$228$	0	0
$229$	16.2462	1.07358	0.536790	−	0.843716i	$-0.319637\pi$
$229$	16.2462	1.07358	0.536790	+	0.843716i	$0.319637\pi$
$230$	0	0
$231$	1.12311	0.0738949
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	29.3693	1.91584
$236$	0	0
$237$	5.43845	0.353265
$238$	0	0
$239$	−25.3693	−1.64100	−0.820502	−	0.571643i	$-0.806306\pi$
$239$	−25.3693	−1.64100	−0.820502	+	0.571643i	$0.806306\pi$
$240$	0	0
$241$	17.8078	1.14710	0.573549	−	0.819171i	$-0.305566\pi$
$241$	17.8078	1.14710	0.573549	+	0.819171i	$0.305566\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−23.8078	−1.52102
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.876894	0.0555709
$250$	0	0
$251$	−18.7386	−1.18277	−0.591386	−	0.806389i	$-0.701419\pi$
$251$	−18.7386	−1.18277	−0.591386	+	0.806389i	$0.701419\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	0	0
$255$	5.56155	0.348278
$256$	0	0
$257$	29.1771	1.82002	0.910008	−	0.414590i	$-0.136075\pi$
$257$	29.1771	1.82002	0.910008	+	0.414590i	$0.136075\pi$
$258$	0	0
$259$	−4.24621	−0.263847
$260$	0	0
$261$	6.68466	0.413770
$262$	0	0
$263$	−9.36932	−0.577737	−0.288868	−	0.957369i	$-0.593279\pi$
$263$	−9.36932	−0.577737	−0.288868	+	0.957369i	$0.593279\pi$
$264$	0	0
$265$	−2.43845	−0.149793
$266$	0	0
$267$	4.87689	0.298461
$268$	0	0
$269$	−21.3693	−1.30291	−0.651455	−	0.758687i	$-0.725841\pi$
$269$	−21.3693	−1.30291	−0.651455	+	0.758687i	$0.725841\pi$
$270$	0	0
$271$	−29.9309	−1.81817	−0.909085	−	0.416610i	$-0.863218\pi$
$271$	−29.9309	−1.81817	−0.909085	+	0.416610i	$0.863218\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−15.3693	−0.926805
$276$	0	0
$277$	5.31534	0.319368	0.159684	−	0.987168i	$-0.448952\pi$
$277$	5.31534	0.319368	0.159684	+	0.987168i	$0.448952\pi$
$278$	0	0
$279$	2.56155	0.153356
$280$	0	0
$281$	−17.8078	−1.06232	−0.531161	−	0.847271i	$-0.678244\pi$
$281$	−17.8078	−1.06232	−0.531161	+	0.847271i	$0.678244\pi$
$282$	0	0
$283$	13.6847	0.813469	0.406734	−	0.913547i	$-0.366667\pi$
$283$	13.6847	0.813469	0.406734	+	0.913547i	$0.366667\pi$
$284$	0	0
$285$	−25.3693	−1.50275
$286$	0	0
$287$	0.876894	0.0517614
$288$	0	0
$289$	−14.5616	−0.856562
$290$	0	0
$291$	−8.56155	−0.501887
$292$	0	0
$293$	−20.4384	−1.19403	−0.597013	−	0.802231i	$-0.703646\pi$
$293$	−20.4384	−1.19403	−0.597013	+	0.802231i	$0.703646\pi$
$294$	0	0
$295$	−10.2462	−0.596557
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−2.56155	−0.147645
$302$	0	0
$303$	7.56155	0.434400
$304$	0	0
$305$	13.8078	0.790630
$306$	0	0
$307$	30.8078	1.75829	0.879146	−	0.476553i	$-0.158114\pi$
$307$	30.8078	1.75829	0.879146	+	0.476553i	$0.158114\pi$
$308$	0	0
$309$	−3.43845	−0.195606
$310$	0	0
$311$	19.1231	1.08437	0.542186	−	0.840259i	$-0.317597\pi$
$311$	19.1231	1.08437	0.542186	+	0.840259i	$0.317597\pi$
$312$	0	0
$313$	−13.6847	−0.773503	−0.386751	−	0.922184i	$-0.626403\pi$
$313$	−13.6847	−0.773503	−0.386751	+	0.922184i	$0.626403\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	−14.0540	−0.789350	−0.394675	−	0.918821i	$-0.629143\pi$
$317$	−14.0540	−0.789350	−0.394675	+	0.918821i	$0.629143\pi$
$318$	0	0
$319$	−13.3693	−0.748538
$320$	0	0
$321$	−8.24621	−0.460259
$322$	0	0
$323$	−11.1231	−0.618906
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.80776	−0.155270
$328$	0	0
$329$	4.63068	0.255298
$330$	0	0
$331$	−3.19224	−0.175461	−0.0877306	−	0.996144i	$-0.527961\pi$
$331$	−3.19224	−0.175461	−0.0877306	+	0.996144i	$0.527961\pi$
$332$	0	0
$333$	−7.56155	−0.414371
$334$	0	0
$335$	16.2462	0.887625
$336$	0	0
$337$	−6.12311	−0.333547	−0.166773	−	0.985995i	$-0.553335\pi$
$337$	−6.12311	−0.333547	−0.166773	+	0.985995i	$0.553335\pi$
$338$	0	0
$339$	−5.80776	−0.315434
$340$	0	0
$341$	−5.12311	−0.277432
$342$	0	0
$343$	−7.68466	−0.414933
$344$	0	0
$345$	7.12311	0.383495
$346$	0	0
$347$	−27.6155	−1.48248	−0.741240	−	0.671241i	$-0.765762\pi$
$347$	−27.6155	−1.48248	−0.741240	+	0.671241i	$0.765762\pi$
$348$	0	0
$349$	−6.80776	−0.364411	−0.182206	−	0.983260i	$-0.558324\pi$
$349$	−6.80776	−0.364411	−0.182206	+	0.983260i	$0.558324\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−5.31534	−0.282907	−0.141454	−	0.989945i	$-0.545178\pi$
$353$	−5.31534	−0.282907	−0.141454	+	0.989945i	$0.545178\pi$
$354$	0	0
$355$	49.8617	2.64639
$356$	0	0
$357$	0.876894	0.0464102
$358$	0	0
$359$	−9.36932	−0.494494	−0.247247	−	0.968953i	$-0.579526\pi$
$359$	−9.36932	−0.494494	−0.247247	+	0.968953i	$0.579526\pi$
$360$	0	0
$361$	31.7386	1.67045
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	36.0540	1.88715
$366$	0	0
$367$	17.0540	0.890210	0.445105	−	0.895478i	$-0.353166\pi$
$367$	17.0540	0.890210	0.445105	+	0.895478i	$0.353166\pi$
$368$	0	0
$369$	1.56155	0.0812912
$370$	0	0
$371$	−0.384472	−0.0199608
$372$	0	0
$373$	28.3693	1.46891	0.734454	−	0.678659i	$-0.237438\pi$
$373$	28.3693	1.46891	0.734454	+	0.678659i	$0.237438\pi$
$374$	0	0
$375$	−9.56155	−0.493756
$376$	0	0
$377$	0	0
$378$	0	0
$379$	23.6847	1.21660	0.608300	−	0.793708i	$-0.291852\pi$
$379$	23.6847	1.21660	0.608300	+	0.793708i	$0.291852\pi$
$380$	0	0
$381$	5.43845	0.278620
$382$	0	0
$383$	−22.7386	−1.16189	−0.580945	−	0.813943i	$-0.697317\pi$
$383$	−22.7386	−1.16189	−0.580945	+	0.813943i	$0.697317\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	−4.56155	−0.231877
$388$	0	0
$389$	34.0540	1.72661	0.863303	−	0.504687i	$-0.168392\pi$
$389$	34.0540	1.72661	0.863303	+	0.504687i	$0.168392\pi$
$390$	0	0
$391$	3.12311	0.157942
$392$	0	0
$393$	7.36932	0.371733
$394$	0	0
$395$	−19.3693	−0.974576
$396$	0	0
$397$	−25.0540	−1.25742	−0.628711	−	0.777639i	$-0.716417\pi$
$397$	−25.0540	−1.25742	−0.628711	+	0.777639i	$0.716417\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	−14.4384	−0.721022	−0.360511	−	0.932755i	$-0.617398\pi$
$401$	−14.4384	−0.721022	−0.360511	+	0.932755i	$0.617398\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.56155	0.176975
$406$	0	0
$407$	15.1231	0.749625
$408$	0	0
$409$	−6.36932	−0.314942	−0.157471	−	0.987524i	$-0.550334\pi$
$409$	−6.36932	−0.314942	−0.157471	+	0.987524i	$0.550334\pi$
$410$	0	0
$411$	−5.56155	−0.274331
$412$	0	0
$413$	−1.61553	−0.0794949
$414$	0	0
$415$	−3.12311	−0.153307
$416$	0	0
$417$	−17.9309	−0.878078
$418$	0	0
$419$	34.2462	1.67304	0.836518	−	0.547939i	$-0.184587\pi$
$419$	34.2462	1.67304	0.836518	+	0.547939i	$0.184587\pi$
$420$	0	0
$421$	−31.2462	−1.52285	−0.761424	−	0.648255i	$-0.775499\pi$
$421$	−31.2462	−1.52285	−0.761424	+	0.648255i	$0.775499\pi$
$422$	0	0
$423$	8.24621	0.400945
$424$	0	0
$425$	−12.0000	−0.582086
$426$	0	0
$427$	2.17708	0.105356
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.1231	−0.535781	−0.267891	−	0.963449i	$-0.586327\pi$
$431$	−11.1231	−0.535781	−0.267891	+	0.963449i	$0.586327\pi$
$432$	0	0
$433$	8.75379	0.420680	0.210340	−	0.977628i	$-0.432543\pi$
$433$	8.75379	0.420680	0.210340	+	0.977628i	$0.432543\pi$
$434$	0	0
$435$	−23.8078	−1.14149
$436$	0	0
$437$	−14.2462	−0.681489
$438$	0	0
$439$	13.6847	0.653133	0.326567	−	0.945174i	$-0.394108\pi$
$439$	13.6847	0.653133	0.326567	+	0.945174i	$0.394108\pi$
$440$	0	0
$441$	−6.68466	−0.318317
$442$	0	0
$443$	34.7386	1.65048	0.825241	−	0.564781i	$-0.191039\pi$
$443$	34.7386	1.65048	0.825241	+	0.564781i	$0.191039\pi$
$444$	0	0
$445$	−17.3693	−0.823385
$446$	0	0
$447$	−2.43845	−0.115335
$448$	0	0
$449$	8.24621	0.389163	0.194581	−	0.980886i	$-0.437665\pi$
$449$	8.24621	0.389163	0.194581	+	0.980886i	$0.437665\pi$
$450$	0	0
$451$	−3.12311	−0.147061
$452$	0	0
$453$	9.36932	0.440209
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−12.6155	−0.590130	−0.295065	−	0.955477i	$-0.595341\pi$
$457$	−12.6155	−0.590130	−0.295065	+	0.955477i	$0.595341\pi$
$458$	0	0
$459$	1.56155	0.0728870
$460$	0	0
$461$	16.1922	0.754148	0.377074	−	0.926183i	$-0.376930\pi$
$461$	16.1922	0.754148	0.377074	+	0.926183i	$0.376930\pi$
$462$	0	0
$463$	−14.3153	−0.665290	−0.332645	−	0.943052i	$-0.607941\pi$
$463$	−14.3153	−0.665290	−0.332645	+	0.943052i	$0.607941\pi$
$464$	0	0
$465$	−9.12311	−0.423074
$466$	0	0
$467$	26.0000	1.20314	0.601568	−	0.798821i	$-0.294543\pi$
$467$	26.0000	1.20314	0.601568	+	0.798821i	$0.294543\pi$
$468$	0	0
$469$	2.56155	0.118282
$470$	0	0
$471$	−20.3693	−0.938569
$472$	0	0
$473$	9.12311	0.419481
$474$	0	0
$475$	54.7386	2.51158
$476$	0	0
$477$	−0.684658	−0.0313484
$478$	0	0
$479$	10.2462	0.468161	0.234081	−	0.972217i	$-0.424792\pi$
$479$	10.2462	0.468161	0.234081	+	0.972217i	$0.424792\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	1.12311	0.0511031
$484$	0	0
$485$	30.4924	1.38459
$486$	0	0
$487$	7.12311	0.322779	0.161389	−	0.986891i	$-0.448403\pi$
$487$	7.12311	0.322779	0.161389	+	0.986891i	$0.448403\pi$
$488$	0	0
$489$	4.80776	0.217415
$490$	0	0
$491$	36.2462	1.63577	0.817884	−	0.575383i	$-0.195147\pi$
$491$	36.2462	1.63577	0.817884	+	0.575383i	$0.195147\pi$
$492$	0	0
$493$	−10.4384	−0.470124
$494$	0	0
$495$	−7.12311	−0.320160
$496$	0	0
$497$	7.86174	0.352647
$498$	0	0
$499$	4.49242	0.201108	0.100554	−	0.994932i	$-0.467938\pi$
$499$	4.49242	0.201108	0.100554	+	0.994932i	$0.467938\pi$
$500$	0	0
$501$	−10.2462	−0.457767
$502$	0	0
$503$	28.2462	1.25944	0.629718	−	0.776824i	$-0.283170\pi$
$503$	28.2462	1.25944	0.629718	+	0.776824i	$0.283170\pi$
$504$	0	0
$505$	−26.9309	−1.19841
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13.8078	0.612018	0.306009	−	0.952029i	$-0.401006\pi$
$509$	13.8078	0.612018	0.306009	+	0.952029i	$0.401006\pi$
$510$	0	0
$511$	5.68466	0.251474
$512$	0	0
$513$	−7.12311	−0.314493
$514$	0	0
$515$	12.2462	0.539633
$516$	0	0
$517$	−16.4924	−0.725336
$518$	0	0
$519$	−20.2462	−0.888710
$520$	0	0
$521$	−9.06913	−0.397326	−0.198663	−	0.980068i	$-0.563660\pi$
$521$	−9.06913	−0.397326	−0.198663	+	0.980068i	$0.563660\pi$
$522$	0	0
$523$	−33.8617	−1.48067	−0.740335	−	0.672238i	$-0.765333\pi$
$523$	−33.8617	−1.48067	−0.740335	+	0.672238i	$0.765333\pi$
$524$	0	0
$525$	−4.31534	−0.188337
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−2.87689	−0.124847
$532$	0	0
$533$	0	0
$534$	0	0
$535$	29.3693	1.26975
$536$	0	0
$537$	−4.87689	−0.210454
$538$	0	0
$539$	13.3693	0.575857
$540$	0	0
$541$	−19.7386	−0.848630	−0.424315	−	0.905515i	$-0.639485\pi$
$541$	−19.7386	−0.848630	−0.424315	+	0.905515i	$0.639485\pi$
$542$	0	0
$543$	2.68466	0.115210
$544$	0	0
$545$	10.0000	0.428353
$546$	0	0
$547$	−3.93087	−0.168072	−0.0840359	−	0.996463i	$-0.526781\pi$
$547$	−3.93087	−0.168072	−0.0840359	+	0.996463i	$0.526781\pi$
$548$	0	0
$549$	3.87689	0.165462
$550$	0	0
$551$	47.6155	2.02849
$552$	0	0
$553$	−3.05398	−0.129868
$554$	0	0
$555$	26.9309	1.14315
$556$	0	0
$557$	42.9309	1.81904	0.909520	−	0.415661i	$-0.136450\pi$
$557$	42.9309	1.81904	0.909520	+	0.415661i	$0.136450\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−3.12311	−0.131858
$562$	0	0
$563$	23.3693	0.984899	0.492450	−	0.870341i	$-0.336102\pi$
$563$	23.3693	0.984899	0.492450	+	0.870341i	$0.336102\pi$
$564$	0	0
$565$	20.6847	0.870210
$566$	0	0
$567$	0.561553	0.0235830
$568$	0	0
$569$	−8.73863	−0.366343	−0.183171	−	0.983081i	$-0.558636\pi$
$569$	−8.73863	−0.366343	−0.183171	+	0.983081i	$0.558636\pi$
$570$	0	0
$571$	5.36932	0.224699	0.112349	−	0.993669i	$-0.464162\pi$
$571$	5.36932	0.224699	0.112349	+	0.993669i	$0.464162\pi$
$572$	0	0
$573$	−9.12311	−0.381123
$574$	0	0
$575$	−15.3693	−0.640945
$576$	0	0
$577$	17.3153	0.720847	0.360424	−	0.932789i	$-0.382632\pi$
$577$	17.3153	0.720847	0.360424	+	0.932789i	$0.382632\pi$
$578$	0	0
$579$	−13.4924	−0.560726
$580$	0	0
$581$	−0.492423	−0.0204291
$582$	0	0
$583$	1.36932	0.0567113
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.3693	1.62495	0.812473	−	0.582999i	$-0.198121\pi$
$587$	39.3693	1.62495	0.812473	+	0.582999i	$0.198121\pi$
$588$	0	0
$589$	18.2462	0.751822
$590$	0	0
$591$	13.3693	0.549940
$592$	0	0
$593$	17.4233	0.715489	0.357744	−	0.933820i	$-0.383546\pi$
$593$	17.4233	0.715489	0.357744	+	0.933820i	$0.383546\pi$
$594$	0	0
$595$	−3.12311	−0.128035
$596$	0	0
$597$	22.1771	0.907648
$598$	0	0
$599$	41.6155	1.70036	0.850182	−	0.526489i	$-0.176492\pi$
$599$	41.6155	1.70036	0.850182	+	0.526489i	$0.176492\pi$
$600$	0	0
$601$	−7.06913	−0.288356	−0.144178	−	0.989552i	$-0.546054\pi$
$601$	−7.06913	−0.288356	−0.144178	+	0.989552i	$0.546054\pi$
$602$	0	0
$603$	4.56155	0.185761
$604$	0	0
$605$	−24.9309	−1.01358
$606$	0	0
$607$	16.0000	0.649420	0.324710	−	0.945814i	$-0.394733\pi$
$607$	16.0000	0.649420	0.324710	+	0.945814i	$0.394733\pi$
$608$	0	0
$609$	−3.75379	−0.152111
$610$	0	0
$611$	0	0
$612$	0	0
$613$	34.8617	1.40805	0.704026	−	0.710174i	$-0.251384\pi$
$613$	34.8617	1.40805	0.704026	+	0.710174i	$0.251384\pi$
$614$	0	0
$615$	−5.56155	−0.224263
$616$	0	0
$617$	−9.80776	−0.394846	−0.197423	−	0.980318i	$-0.563257\pi$
$617$	−9.80776	−0.394846	−0.197423	+	0.980318i	$0.563257\pi$
$618$	0	0
$619$	29.3002	1.17767	0.588837	−	0.808252i	$-0.299586\pi$
$619$	29.3002	1.17767	0.588837	+	0.808252i	$0.299586\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	0	0
$623$	−2.73863	−0.109721
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	14.2462	0.568939
$628$	0	0
$629$	11.8078	0.470806
$630$	0	0
$631$	−18.5616	−0.738924	−0.369462	−	0.929246i	$-0.620458\pi$
$631$	−18.5616	−0.738924	−0.369462	+	0.929246i	$0.620458\pi$
$632$	0	0
$633$	19.6847	0.782395
$634$	0	0
$635$	−19.3693	−0.768648
$636$	0	0
$637$	0	0
$638$	0	0
$639$	14.0000	0.553831
$640$	0	0
$641$	−19.1771	−0.757449	−0.378725	−	0.925509i	$-0.623637\pi$
$641$	−19.1771	−0.757449	−0.378725	+	0.925509i	$0.623637\pi$
$642$	0	0
$643$	−31.5464	−1.24407	−0.622034	−	0.782990i	$-0.713694\pi$
$643$	−31.5464	−1.24407	−0.622034	+	0.782990i	$0.713694\pi$
$644$	0	0
$645$	16.2462	0.639694
$646$	0	0
$647$	6.38447	0.250999	0.125500	−	0.992094i	$-0.459947\pi$
$647$	6.38447	0.250999	0.125500	+	0.992094i	$0.459947\pi$
$648$	0	0
$649$	5.75379	0.225856
$650$	0	0
$651$	−1.43845	−0.0563772
$652$	0	0
$653$	23.1231	0.904877	0.452439	−	0.891796i	$-0.350554\pi$
$653$	23.1231	0.904877	0.452439	+	0.891796i	$0.350554\pi$
$654$	0	0
$655$	−26.2462	−1.02552
$656$	0	0
$657$	10.1231	0.394940
$658$	0	0
$659$	2.24621	0.0875000	0.0437500	−	0.999043i	$-0.486070\pi$
$659$	2.24621	0.0875000	0.0437500	+	0.999043i	$0.486070\pi$
$660$	0	0
$661$	−5.63068	−0.219008	−0.109504	−	0.993986i	$-0.534926\pi$
$661$	−5.63068	−0.219008	−0.109504	+	0.993986i	$0.534926\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	14.2462	0.552444
$666$	0	0
$667$	−13.3693	−0.517662
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	−7.75379	−0.299332
$672$	0	0
$673$	−23.2462	−0.896076	−0.448038	−	0.894015i	$-0.647877\pi$
$673$	−23.2462	−0.896076	−0.448038	+	0.894015i	$0.647877\pi$
$674$	0	0
$675$	−7.68466	−0.295783
$676$	0	0
$677$	−15.6155	−0.600153	−0.300077	−	0.953915i	$-0.597012\pi$
$677$	−15.6155	−0.600153	−0.300077	+	0.953915i	$0.597012\pi$
$678$	0	0
$679$	4.80776	0.184505
$680$	0	0
$681$	−7.12311	−0.272958
$682$	0	0
$683$	−38.1080	−1.45816	−0.729080	−	0.684428i	$-0.760052\pi$
$683$	−38.1080	−1.45816	−0.729080	+	0.684428i	$0.760052\pi$
$684$	0	0
$685$	19.8078	0.756816
$686$	0	0
$687$	−16.2462	−0.619832
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−51.3002	−1.95155	−0.975776	−	0.218774i	$-0.929794\pi$
$691$	−51.3002	−1.95155	−0.975776	+	0.218774i	$0.929794\pi$
$692$	0	0
$693$	−1.12311	−0.0426633
$694$	0	0
$695$	63.8617	2.42241
$696$	0	0
$697$	−2.43845	−0.0923628
$698$	0	0
$699$	−26.0000	−0.983410
$700$	0	0
$701$	−5.36932	−0.202796	−0.101398	−	0.994846i	$-0.532332\pi$
$701$	−5.36932	−0.202796	−0.101398	+	0.994846i	$0.532332\pi$
$702$	0	0
$703$	−53.8617	−2.03143
$704$	0	0
$705$	−29.3693	−1.10611
$706$	0	0
$707$	−4.24621	−0.159695
$708$	0	0
$709$	7.49242	0.281384	0.140692	−	0.990053i	$-0.455067\pi$
$709$	7.49242	0.281384	0.140692	+	0.990053i	$0.455067\pi$
$710$	0	0
$711$	−5.43845	−0.203958
$712$	0	0
$713$	−5.12311	−0.191862
$714$	0	0
$715$	0	0
$716$	0	0
$717$	25.3693	0.947435
$718$	0	0
$719$	23.3693	0.871528	0.435764	−	0.900061i	$-0.356478\pi$
$719$	23.3693	0.871528	0.435764	+	0.900061i	$0.356478\pi$
$720$	0	0
$721$	1.93087	0.0719093
$722$	0	0
$723$	−17.8078	−0.662278
$724$	0	0
$725$	51.3693	1.90781
$726$	0	0
$727$	38.6695	1.43417	0.717086	−	0.696984i	$-0.245475\pi$
$727$	38.6695	1.43417	0.717086	+	0.696984i	$0.245475\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	7.12311	0.263458
$732$	0	0
$733$	−20.5076	−0.757465	−0.378732	−	0.925506i	$-0.623640\pi$
$733$	−20.5076	−0.757465	−0.378732	+	0.925506i	$0.623640\pi$
$734$	0	0
$735$	23.8078	0.878163
$736$	0	0
$737$	−9.12311	−0.336054
$738$	0	0
$739$	10.2462	0.376913	0.188456	−	0.982082i	$-0.439652\pi$
$739$	10.2462	0.376913	0.188456	+	0.982082i	$0.439652\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−12.6307	−0.463375	−0.231687	−	0.972790i	$-0.574425\pi$
$743$	−12.6307	−0.463375	−0.231687	+	0.972790i	$0.574425\pi$
$744$	0	0
$745$	8.68466	0.318181
$746$	0	0
$747$	−0.876894	−0.0320839
$748$	0	0
$749$	4.63068	0.169201
$750$	0	0
$751$	−44.1080	−1.60952	−0.804761	−	0.593599i	$-0.797707\pi$
$751$	−44.1080	−1.60952	−0.804761	+	0.593599i	$0.797707\pi$
$752$	0	0
$753$	18.7386	0.682874
$754$	0	0
$755$	−33.3693	−1.21443
$756$	0	0
$757$	30.0000	1.09037	0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000	1.09037	0.545184	+	0.838316i	$0.316460\pi$
$758$	0	0
$759$	−4.00000	−0.145191
$760$	0	0
$761$	−9.36932	−0.339637	−0.169819	−	0.985475i	$-0.554318\pi$
$761$	−9.36932	−0.339637	−0.169819	+	0.985475i	$0.554318\pi$
$762$	0	0
$763$	1.57671	0.0570807
$764$	0	0
$765$	−5.56155	−0.201078
$766$	0	0
$767$	0	0
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	−29.1771	−1.05079
$772$	0	0
$773$	24.2462	0.872076	0.436038	−	0.899928i	$-0.356381\pi$
$773$	24.2462	0.872076	0.436038	+	0.899928i	$0.356381\pi$
$774$	0	0
$775$	19.6847	0.707094
$776$	0	0
$777$	4.24621	0.152332
$778$	0	0
$779$	11.1231	0.398527
$780$	0	0
$781$	−28.0000	−1.00192
$782$	0	0
$783$	−6.68466	−0.238890
$784$	0	0
$785$	72.5464	2.58929
$786$	0	0
$787$	44.1771	1.57474	0.787371	−	0.616479i	$-0.211441\pi$
$787$	44.1771	1.57474	0.787371	+	0.616479i	$0.211441\pi$
$788$	0	0
$789$	9.36932	0.333557
$790$	0	0
$791$	3.26137	0.115961
$792$	0	0
$793$	0	0
$794$	0	0
$795$	2.43845	0.0864828
$796$	0	0
$797$	−0.384472	−0.0136187	−0.00680935	−	0.999977i	$-0.502167\pi$
$797$	−0.384472	−0.0136187	−0.00680935	+	0.999977i	$0.502167\pi$
$798$	0	0
$799$	−12.8769	−0.455552
$800$	0	0
$801$	−4.87689	−0.172317
$802$	0	0
$803$	−20.2462	−0.714473
$804$	0	0
$805$	−4.00000	−0.140981
$806$	0	0
$807$	21.3693	0.752236
$808$	0	0
$809$	−16.3002	−0.573084	−0.286542	−	0.958068i	$-0.592506\pi$
$809$	−16.3002	−0.573084	−0.286542	+	0.958068i	$0.592506\pi$
$810$	0	0
$811$	2.56155	0.0899483	0.0449741	−	0.998988i	$-0.485679\pi$
$811$	2.56155	0.0899483	0.0449741	+	0.998988i	$0.485679\pi$
$812$	0	0
$813$	29.9309	1.04972
$814$	0	0
$815$	−17.1231	−0.599796
$816$	0	0
$817$	−32.4924	−1.13677
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.49242	−0.226587	−0.113294	−	0.993562i	$-0.536140\pi$
$821$	−6.49242	−0.226587	−0.113294	+	0.993562i	$0.536140\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	0	0
$825$	15.3693	0.535091
$826$	0	0
$827$	−14.7386	−0.512513	−0.256256	−	0.966609i	$-0.582489\pi$
$827$	−14.7386	−0.512513	−0.256256	+	0.966609i	$0.582489\pi$
$828$	0	0
$829$	13.4924	0.468611	0.234306	−	0.972163i	$-0.424718\pi$
$829$	13.4924	0.468611	0.234306	+	0.972163i	$0.424718\pi$
$830$	0	0
$831$	−5.31534	−0.184387
$832$	0	0
$833$	10.4384	0.361671
$834$	0	0
$835$	36.4924	1.26287
$836$	0	0
$837$	−2.56155	−0.0885402
$838$	0	0
$839$	21.6155	0.746251	0.373125	−	0.927781i	$-0.378286\pi$
$839$	21.6155	0.746251	0.373125	+	0.927781i	$0.378286\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	0	0
$843$	17.8078	0.613332
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.93087	−0.135066
$848$	0	0
$849$	−13.6847	−0.469656
$850$	0	0
$851$	15.1231	0.518413
$852$	0	0
$853$	−2.12311	−0.0726938	−0.0363469	−	0.999339i	$-0.511572\pi$
$853$	−2.12311	−0.0726938	−0.0363469	+	0.999339i	$0.511572\pi$
$854$	0	0
$855$	25.3693	0.867612
$856$	0	0
$857$	35.5616	1.21476	0.607380	−	0.794412i	$-0.292221\pi$
$857$	35.5616	1.21476	0.607380	+	0.794412i	$0.292221\pi$
$858$	0	0
$859$	−24.5616	−0.838029	−0.419015	−	0.907979i	$-0.637624\pi$
$859$	−24.5616	−0.838029	−0.419015	+	0.907979i	$0.637624\pi$
$860$	0	0
$861$	−0.876894	−0.0298845
$862$	0	0
$863$	30.4924	1.03797	0.518987	−	0.854782i	$-0.326309\pi$
$863$	30.4924	1.03797	0.518987	+	0.854782i	$0.326309\pi$
$864$	0	0
$865$	72.1080	2.45174
$866$	0	0
$867$	14.5616	0.494536
$868$	0	0
$869$	10.8769	0.368973
$870$	0	0
$871$	0	0
$872$	0	0
$873$	8.56155	0.289765
$874$	0	0
$875$	5.36932	0.181516
$876$	0	0
$877$	−23.5616	−0.795617	−0.397809	−	0.917468i	$-0.630229\pi$
$877$	−23.5616	−0.795617	−0.397809	+	0.917468i	$0.630229\pi$
$878$	0	0
$879$	20.4384	0.689372
$880$	0	0
$881$	−9.06913	−0.305547	−0.152773	−	0.988261i	$-0.548820\pi$
$881$	−9.06913	−0.305547	−0.152773	+	0.988261i	$0.548820\pi$
$882$	0	0
$883$	−8.80776	−0.296405	−0.148202	−	0.988957i	$-0.547349\pi$
$883$	−8.80776	−0.296405	−0.148202	+	0.988957i	$0.547349\pi$
$884$	0	0
$885$	10.2462	0.344423
$886$	0	0
$887$	−24.6307	−0.827017	−0.413509	−	0.910500i	$-0.635697\pi$
$887$	−24.6307	−0.827017	−0.413509	+	0.910500i	$0.635697\pi$
$888$	0	0
$889$	−3.05398	−0.102427
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	58.7386	1.96561
$894$	0	0
$895$	17.3693	0.580592
$896$	0	0
$897$	0	0
$898$	0	0
$899$	17.1231	0.571088
$900$	0	0
$901$	1.06913	0.0356179
$902$	0	0
$903$	2.56155	0.0852431
$904$	0	0
$905$	−9.56155	−0.317837
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	−7.56155	−0.250801
$910$	0	0
$911$	−38.7386	−1.28347	−0.641734	−	0.766927i	$-0.721785\pi$
$911$	−38.7386	−1.28347	−0.641734	+	0.766927i	$0.721785\pi$
$912$	0	0
$913$	1.75379	0.0580419
$914$	0	0
$915$	−13.8078	−0.456471
$916$	0	0
$917$	−4.13826	−0.136657
$918$	0	0
$919$	−11.5076	−0.379600	−0.189800	−	0.981823i	$-0.560784\pi$
$919$	−11.5076	−0.379600	−0.189800	+	0.981823i	$0.560784\pi$
$920$	0	0
$921$	−30.8078	−1.01515
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−58.1080	−1.91058
$926$	0	0
$927$	3.43845	0.112933
$928$	0	0
$929$	7.80776	0.256164	0.128082	−	0.991764i	$-0.459118\pi$
$929$	7.80776	0.256164	0.128082	+	0.991764i	$0.459118\pi$
$930$	0	0
$931$	−47.6155	−1.56054
$932$	0	0
$933$	−19.1231	−0.626062
$934$	0	0
$935$	11.1231	0.363764
$936$	0	0
$937$	−7.56155	−0.247025	−0.123513	−	0.992343i	$-0.539416\pi$
$937$	−7.56155	−0.247025	−0.123513	+	0.992343i	$0.539416\pi$
$938$	0	0
$939$	13.6847	0.446582
$940$	0	0
$941$	−30.4924	−0.994025	−0.497012	−	0.867744i	$-0.665570\pi$
$941$	−30.4924	−0.994025	−0.497012	+	0.867744i	$0.665570\pi$
$942$	0	0
$943$	−3.12311	−0.101702
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	−38.7386	−1.25884	−0.629418	−	0.777067i	$-0.716707\pi$
$947$	−38.7386	−1.25884	−0.629418	+	0.777067i	$0.716707\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	14.0540	0.455731
$952$	0	0
$953$	30.9848	1.00370	0.501849	−	0.864955i	$-0.332653\pi$
$953$	30.9848	1.00370	0.501849	+	0.864955i	$0.332653\pi$
$954$	0	0
$955$	32.4924	1.05143
$956$	0	0
$957$	13.3693	0.432169
$958$	0	0
$959$	3.12311	0.100850
$960$	0	0
$961$	−24.4384	−0.788337
$962$	0	0
$963$	8.24621	0.265730
$964$	0	0
$965$	48.0540	1.54691
$966$	0	0
$967$	−0.876894	−0.0281990	−0.0140995	−	0.999901i	$-0.504488\pi$
$967$	−0.876894	−0.0281990	−0.0140995	+	0.999901i	$0.504488\pi$
$968$	0	0
$969$	11.1231	0.357326
$970$	0	0
$971$	12.9848	0.416704	0.208352	−	0.978054i	$-0.433190\pi$
$971$	12.9848	0.416704	0.208352	+	0.978054i	$0.433190\pi$
$972$	0	0
$973$	10.0691	0.322801
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−61.1771	−1.95723	−0.978614	−	0.205705i	$-0.934051\pi$
$977$	−61.1771	−1.95723	−0.978614	+	0.205705i	$0.934051\pi$
$978$	0	0
$979$	9.75379	0.311732
$980$	0	0
$981$	2.80776	0.0896450
$982$	0	0
$983$	13.6155	0.434268	0.217134	−	0.976142i	$-0.430329\pi$
$983$	13.6155	0.434268	0.217134	+	0.976142i	$0.430329\pi$
$984$	0	0
$985$	−47.6155	−1.51716
$986$	0	0
$987$	−4.63068	−0.147396
$988$	0	0
$989$	9.12311	0.290098
$990$	0	0
$991$	50.3542	1.59955	0.799776	−	0.600298i	$-0.204951\pi$
$991$	50.3542	1.59955	0.799776	+	0.600298i	$0.204951\pi$
$992$	0	0
$993$	3.19224	0.101303
$994$	0	0
$995$	−78.9848	−2.50399
$996$	0	0
$997$	−20.6155	−0.652900	−0.326450	−	0.945214i	$-0.605853\pi$
$997$	−20.6155	−0.652900	−0.326450	+	0.945214i	$0.605853\pi$
$998$	0	0
$999$	7.56155	0.239237

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.bo.1.2		2
4.3	odd	2		507.2.a.d.1.2		2
12.11	even	2		1521.2.a.m.1.1		2
13.4	even	6		624.2.q.h.289.1		4
13.10	even	6		624.2.q.h.529.1		4
13.12	even	2		8112.2.a.bk.1.1		2
39.17	odd	6		1872.2.t.r.289.2		4
39.23	odd	6		1872.2.t.r.1153.2		4
52.3	odd	6		507.2.e.g.22.1		4
52.7	even	12		507.2.j.g.361.3		8
52.11	even	12		507.2.j.g.316.2		8
52.15	even	12		507.2.j.g.316.3		8
52.19	even	12		507.2.j.g.361.2		8
52.23	odd	6		39.2.e.b.22.2	yes	4
52.31	even	4		507.2.b.d.337.2		4
52.35	odd	6		507.2.e.g.484.1		4
52.43	odd	6		39.2.e.b.16.2	✓	4
52.47	even	4		507.2.b.d.337.3		4
52.51	odd	2		507.2.a.g.1.1		2
156.23	even	6		117.2.g.c.100.1		4
156.47	odd	4		1521.2.b.h.1351.2		4
156.83	odd	4		1521.2.b.h.1351.3		4
156.95	even	6		117.2.g.c.55.1		4
156.155	even	2		1521.2.a.g.1.2		2
260.23	even	12		975.2.bb.i.724.2		8
260.43	even	12		975.2.bb.i.874.3		8
260.127	even	12		975.2.bb.i.724.3		8
260.147	even	12		975.2.bb.i.874.2		8
260.179	odd	6		975.2.i.k.451.1		4
260.199	odd	6		975.2.i.k.601.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.2.e.b.16.2	✓	4	52.43	odd	6
39.2.e.b.22.2	yes	4	52.23	odd	6
117.2.g.c.55.1		4	156.95	even	6
117.2.g.c.100.1		4	156.23	even	6
507.2.a.d.1.2		2	4.3	odd	2
507.2.a.g.1.1		2	52.51	odd	2
507.2.b.d.337.2		4	52.31	even	4
507.2.b.d.337.3		4	52.47	even	4
507.2.e.g.22.1		4	52.3	odd	6
507.2.e.g.484.1		4	52.35	odd	6
507.2.j.g.316.2		8	52.11	even	12
507.2.j.g.316.3		8	52.15	even	12
507.2.j.g.361.2		8	52.19	even	12
507.2.j.g.361.3		8	52.7	even	12
624.2.q.h.289.1		4	13.4	even	6
624.2.q.h.529.1		4	13.10	even	6
975.2.i.k.451.1		4	260.179	odd	6
975.2.i.k.601.1		4	260.199	odd	6
975.2.bb.i.724.2		8	260.23	even	12
975.2.bb.i.724.3		8	260.127	even	12
975.2.bb.i.874.2		8	260.147	even	12
975.2.bb.i.874.3		8	260.43	even	12
1521.2.a.g.1.2		2	156.155	even	2
1521.2.a.m.1.1		2	12.11	even	2
1521.2.b.h.1351.2		4	156.47	odd	4
1521.2.b.h.1351.3		4	156.83	odd	4
1872.2.t.r.289.2		4	39.17	odd	6
1872.2.t.r.1153.2		4	39.23	odd	6
8112.2.a.bk.1.1		2	13.12	even	2
8112.2.a.bo.1.2		2	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:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.56155 q^{5} +0.561553 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +3.56155 q^{5} +0.561553 q^{7} +1.00000 q^{9} -2.00000 q^{11} -3.56155 q^{15} -1.56155 q^{17} +7.12311 q^{19} -0.561553 q^{21} -2.00000 q^{23} +7.68466 q^{25} -1.00000 q^{27} +6.68466 q^{29} +2.56155 q^{31} +2.00000 q^{33} +2.00000 q^{35} -7.56155 q^{37} +1.56155 q^{41} -4.56155 q^{43} +3.56155 q^{45} +8.24621 q^{47} -6.68466 q^{49} +1.56155 q^{51} -0.684658 q^{53} -7.12311 q^{55} -7.12311 q^{57} -2.87689 q^{59} +3.87689 q^{61} +0.561553 q^{63} +4.56155 q^{67} +2.00000 q^{69} +14.0000 q^{71} +10.1231 q^{73} -7.68466 q^{75} -1.12311 q^{77} -5.43845 q^{79} +1.00000 q^{81} -0.876894 q^{83} -5.56155 q^{85} -6.68466 q^{87} -4.87689 q^{89} -2.56155 q^{93} +25.3693 q^{95} +8.56155 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 3 q^{5} - 3 q^{7} + 2 q^{9} - 4 q^{11} - 3 q^{15} + q^{17} + 6 q^{19} + 3 q^{21} - 4 q^{23} + 3 q^{25} - 2 q^{27} + q^{29} + q^{31} + 4 q^{33} + 4 q^{35} - 11 q^{37} - q^{41} - 5 q^{43} + 3 q^{45} - q^{49} - q^{51} + 11 q^{53} - 6 q^{55} - 6 q^{57} - 14 q^{59} + 16 q^{61} - 3 q^{63} + 5 q^{67} + 4 q^{69} + 28 q^{71} + 12 q^{73} - 3 q^{75} + 6 q^{77} - 15 q^{79} + 2 q^{81} - 10 q^{83} - 7 q^{85} - q^{87} - 18 q^{89} - q^{93} + 26 q^{95} + 13 q^{97} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 3 * q^5 - 3 * q^7 + 2 * q^9 - 4 * q^11 - 3 * q^15 + q^17 + 6 * q^19 + 3 * q^21 - 4 * q^23 + 3 * q^25 - 2 * q^27 + q^29 + q^31 + 4 * q^33 + 4 * q^35 - 11 * q^37 - q^41 - 5 * q^43 + 3 * q^45 - q^49 - q^51 + 11 * q^53 - 6 * q^55 - 6 * q^57 - 14 * q^59 + 16 * q^61 - 3 * q^63 + 5 * q^67 + 4 * q^69 + 28 * q^71 + 12 * q^73 - 3 * q^75 + 6 * q^77 - 15 * q^79 + 2 * q^81 - 10 * q^83 - 7 * q^85 - q^87 - 18 * q^89 - q^93 + 26 * q^95 + 13 * q^97 - 4 * q^99

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