LMFDB - Embedded newform 8112.2.a.cp.1.1

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:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.04892 q^{5} -0.554958 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.04892 q^{5} -0.554958 q^{7} +1.00000 q^{9} -2.91185 q^{11} -1.04892 q^{15} -4.85086 q^{17} +0.753020 q^{19} -0.554958 q^{21} -5.76271 q^{23} -3.89977 q^{25} +1.00000 q^{27} -1.91185 q^{29} +9.51573 q^{31} -2.91185 q^{33} +0.582105 q^{35} +5.75302 q^{37} +4.91185 q^{41} +11.0978 q^{43} -1.04892 q^{45} -0.753020 q^{47} -6.69202 q^{49} -4.85086 q^{51} -7.58211 q^{53} +3.05429 q^{55} +0.753020 q^{57} -4.09783 q^{59} -3.42327 q^{61} -0.554958 q^{63} +1.87263 q^{67} -5.76271 q^{69} +10.5036 q^{71} +10.4765 q^{73} -3.89977 q^{75} +1.61596 q^{77} -1.33513 q^{79} +1.00000 q^{81} -2.64310 q^{83} +5.08815 q^{85} -1.91185 q^{87} +9.92692 q^{89} +9.51573 q^{93} -0.789856 q^{95} +17.0737 q^{97} -2.91185 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 6 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 6 * q^5 - 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 6 q^{5} - 2 q^{7} + 3 q^{9} - 5 q^{11} + 6 q^{15} - q^{17} + 7 q^{19} - 2 q^{21} + 11 q^{25} + 3 q^{27} - 2 q^{29} + 16 q^{31} - 5 q^{33} - 4 q^{35} + 22 q^{37} + 11 q^{41} + 15 q^{43} + 6 q^{45} - 7 q^{47} - 15 q^{49} - q^{51} - 17 q^{53} - 3 q^{55} + 7 q^{57} + 6 q^{59} - 13 q^{61} - 2 q^{63} - 11 q^{67} + 6 q^{73} + 11 q^{75} + 15 q^{77} - 3 q^{79} + 3 q^{81} - 12 q^{83} + 19 q^{85} - 2 q^{87} + q^{89} + 16 q^{93} + 21 q^{95} - 5 q^{97} - 5 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 6 * q^5 - 2 * q^7 + 3 * q^9 - 5 * q^11 + 6 * q^15 - q^17 + 7 * q^19 - 2 * q^21 + 11 * q^25 + 3 * q^27 - 2 * q^29 + 16 * q^31 - 5 * q^33 - 4 * q^35 + 22 * q^37 + 11 * q^41 + 15 * q^43 + 6 * q^45 - 7 * q^47 - 15 * q^49 - q^51 - 17 * q^53 - 3 * q^55 + 7 * q^57 + 6 * q^59 - 13 * q^61 - 2 * q^63 - 11 * q^67 + 6 * q^73 + 11 * q^75 + 15 * q^77 - 3 * q^79 + 3 * q^81 - 12 * q^83 + 19 * q^85 - 2 * q^87 + q^89 + 16 * q^93 + 21 * q^95 - 5 * q^97 - 5 * 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
$5$	−1.04892	−0.469090	−0.234545	−	0.972105i	$-0.575360\pi$
$5$	−1.04892	−0.469090	−0.234545	+	0.972105i	$0.575360\pi$
$6$	0	0
$7$	−0.554958	−0.209754	−0.104877	−	0.994485i	$-0.533445\pi$
$7$	−0.554958	−0.209754	−0.104877	+	0.994485i	$0.533445\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.91185	−0.877957	−0.438979	−	0.898498i	$-0.644660\pi$
$11$	−2.91185	−0.877957	−0.438979	+	0.898498i	$0.644660\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.04892	−0.270829
$16$	0	0
$17$	−4.85086	−1.17651	−0.588253	−	0.808677i	$-0.700184\pi$
$17$	−4.85086	−1.17651	−0.588253	+	0.808677i	$0.700184\pi$
$18$	0	0
$19$	0.753020	0.172755	0.0863774	−	0.996262i	$-0.472471\pi$
$19$	0.753020	0.172755	0.0863774	+	0.996262i	$0.472471\pi$
$20$	0	0
$21$	−0.554958	−0.121102
$22$	0	0
$23$	−5.76271	−1.20161	−0.600804	−	0.799396i	$-0.705153\pi$
$23$	−5.76271	−1.20161	−0.600804	+	0.799396i	$0.705153\pi$
$24$	0	0
$25$	−3.89977	−0.779954
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.91185	−0.355022	−0.177511	−	0.984119i	$-0.556805\pi$
$29$	−1.91185	−0.355022	−0.177511	+	0.984119i	$0.556805\pi$
$30$	0	0
$31$	9.51573	1.70908	0.854538	−	0.519389i	$-0.173841\pi$
$31$	9.51573	1.70908	0.854538	+	0.519389i	$0.173841\pi$
$32$	0	0
$33$	−2.91185	−0.506889
$34$	0	0
$35$	0.582105	0.0983937
$36$	0	0
$37$	5.75302	0.945791	0.472895	−	0.881119i	$-0.343209\pi$
$37$	5.75302	0.945791	0.472895	+	0.881119i	$0.343209\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.91185	0.767103	0.383551	−	0.923520i	$-0.374701\pi$
$41$	4.91185	0.767103	0.383551	+	0.923520i	$0.374701\pi$
$42$	0	0
$43$	11.0978	1.69240	0.846202	−	0.532862i	$-0.178884\pi$
$43$	11.0978	1.69240	0.846202	+	0.532862i	$0.178884\pi$
$44$	0	0
$45$	−1.04892	−0.156363
$46$	0	0
$47$	−0.753020	−0.109839	−0.0549197	−	0.998491i	$-0.517490\pi$
$47$	−0.753020	−0.109839	−0.0549197	+	0.998491i	$0.517490\pi$
$48$	0	0
$49$	−6.69202	−0.956003
$50$	0	0
$51$	−4.85086	−0.679256
$52$	0	0
$53$	−7.58211	−1.04148	−0.520741	−	0.853715i	$-0.674344\pi$
$53$	−7.58211	−1.04148	−0.520741	+	0.853715i	$0.674344\pi$
$54$	0	0
$55$	3.05429	0.411841
$56$	0	0
$57$	0.753020	0.0997400
$58$	0	0
$59$	−4.09783	−0.533493	−0.266746	−	0.963767i	$-0.585949\pi$
$59$	−4.09783	−0.533493	−0.266746	+	0.963767i	$0.585949\pi$
$60$	0	0
$61$	−3.42327	−0.438305	−0.219153	−	0.975691i	$-0.570329\pi$
$61$	−3.42327	−0.438305	−0.219153	+	0.975691i	$0.570329\pi$
$62$	0	0
$63$	−0.554958	−0.0699182
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.87263	0.228778	0.114389	−	0.993436i	$-0.463509\pi$
$67$	1.87263	0.228778	0.114389	+	0.993436i	$0.463509\pi$
$68$	0	0
$69$	−5.76271	−0.693749
$70$	0	0
$71$	10.5036	1.24655	0.623277	−	0.782001i	$-0.285801\pi$
$71$	10.5036	1.24655	0.623277	+	0.782001i	$0.285801\pi$
$72$	0	0
$73$	10.4765	1.22618	0.613091	−	0.790012i	$-0.289926\pi$
$73$	10.4765	1.22618	0.613091	+	0.790012i	$0.289926\pi$
$74$	0	0
$75$	−3.89977	−0.450307
$76$	0	0
$77$	1.61596	0.184155
$78$	0	0
$79$	−1.33513	−0.150213	−0.0751067	−	0.997176i	$-0.523930\pi$
$79$	−1.33513	−0.150213	−0.0751067	+	0.997176i	$0.523930\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.64310	−0.290118	−0.145059	−	0.989423i	$-0.546337\pi$
$83$	−2.64310	−0.290118	−0.145059	+	0.989423i	$0.546337\pi$
$84$	0	0
$85$	5.08815	0.551887
$86$	0	0
$87$	−1.91185	−0.204972
$88$	0	0
$89$	9.92692	1.05225	0.526126	−	0.850407i	$-0.323644\pi$
$89$	9.92692	1.05225	0.526126	+	0.850407i	$0.323644\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	9.51573	0.986735
$94$	0	0
$95$	−0.789856	−0.0810375
$96$	0	0
$97$	17.0737	1.73357	0.866784	−	0.498683i	$-0.166183\pi$
$97$	17.0737	1.73357	0.866784	+	0.498683i	$0.166183\pi$
$98$	0	0
$99$	−2.91185	−0.292652
$100$	0	0
$101$	7.32304	0.728670	0.364335	−	0.931268i	$-0.381296\pi$
$101$	7.32304	0.728670	0.364335	+	0.931268i	$0.381296\pi$
$102$	0	0
$103$	4.21983	0.415792	0.207896	−	0.978151i	$-0.433338\pi$
$103$	4.21983	0.415792	0.207896	+	0.978151i	$0.433338\pi$
$104$	0	0
$105$	0.582105	0.0568077
$106$	0	0
$107$	−6.39373	−0.618105	−0.309053	−	0.951045i	$-0.600012\pi$
$107$	−6.39373	−0.618105	−0.309053	+	0.951045i	$0.600012\pi$
$108$	0	0
$109$	−3.46011	−0.331418	−0.165709	−	0.986175i	$-0.552991\pi$
$109$	−3.46011	−0.331418	−0.165709	+	0.986175i	$0.552991\pi$
$110$	0	0
$111$	5.75302	0.546053
$112$	0	0
$113$	9.35690	0.880223	0.440111	−	0.897943i	$-0.354939\pi$
$113$	9.35690	0.880223	0.440111	+	0.897943i	$0.354939\pi$
$114$	0	0
$115$	6.04461	0.563662
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.69202	0.246777
$120$	0	0
$121$	−2.52111	−0.229191
$122$	0	0
$123$	4.91185	0.442887
$124$	0	0
$125$	9.33513	0.834959
$126$	0	0
$127$	4.48188	0.397702	0.198851	−	0.980030i	$-0.436279\pi$
$127$	4.48188	0.397702	0.198851	+	0.980030i	$0.436279\pi$
$128$	0	0
$129$	11.0978	0.977110
$130$	0	0
$131$	9.21744	0.805331	0.402666	−	0.915347i	$-0.368084\pi$
$131$	9.21744	0.805331	0.402666	+	0.915347i	$0.368084\pi$
$132$	0	0
$133$	−0.417895	−0.0362361
$134$	0	0
$135$	−1.04892	−0.0902764
$136$	0	0
$137$	−7.46980	−0.638188	−0.319094	−	0.947723i	$-0.603379\pi$
$137$	−7.46980	−0.638188	−0.319094	+	0.947723i	$0.603379\pi$
$138$	0	0
$139$	17.9976	1.52654	0.763269	−	0.646081i	$-0.223593\pi$
$139$	17.9976	1.52654	0.763269	+	0.646081i	$0.223593\pi$
$140$	0	0
$141$	−0.753020	−0.0634158
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.00538	0.166537
$146$	0	0
$147$	−6.69202	−0.551949
$148$	0	0
$149$	15.3351	1.25630	0.628151	−	0.778091i	$-0.283812\pi$
$149$	15.3351	1.25630	0.628151	+	0.778091i	$0.283812\pi$
$150$	0	0
$151$	−2.53079	−0.205953	−0.102977	−	0.994684i	$-0.532837\pi$
$151$	−2.53079	−0.205953	−0.102977	+	0.994684i	$0.532837\pi$
$152$	0	0
$153$	−4.85086	−0.392168
$154$	0	0
$155$	−9.98121	−0.801710
$156$	0	0
$157$	17.2392	1.37584	0.687919	−	0.725787i	$-0.258524\pi$
$157$	17.2392	1.37584	0.687919	+	0.725787i	$0.258524\pi$
$158$	0	0
$159$	−7.58211	−0.601300
$160$	0	0
$161$	3.19806	0.252043
$162$	0	0
$163$	−15.7071	−1.23027	−0.615137	−	0.788420i	$-0.710899\pi$
$163$	−15.7071	−1.23027	−0.615137	+	0.788420i	$0.710899\pi$
$164$	0	0
$165$	3.05429	0.237776
$166$	0	0
$167$	−5.39612	−0.417565	−0.208782	−	0.977962i	$-0.566950\pi$
$167$	−5.39612	−0.417565	−0.208782	+	0.977962i	$0.566950\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0.753020	0.0575849
$172$	0	0
$173$	23.9420	1.82028	0.910138	−	0.414306i	$-0.135976\pi$
$173$	23.9420	1.82028	0.910138	+	0.414306i	$0.135976\pi$
$174$	0	0
$175$	2.16421	0.163599
$176$	0	0
$177$	−4.09783	−0.308012
$178$	0	0
$179$	−18.4088	−1.37594	−0.687969	−	0.725740i	$-0.741498\pi$
$179$	−18.4088	−1.37594	−0.687969	+	0.725740i	$0.741498\pi$
$180$	0	0
$181$	−3.63342	−0.270070	−0.135035	−	0.990841i	$-0.543115\pi$
$181$	−3.63342	−0.270070	−0.135035	+	0.990841i	$0.543115\pi$
$182$	0	0
$183$	−3.42327	−0.253056
$184$	0	0
$185$	−6.03444	−0.443661
$186$	0	0
$187$	14.1250	1.03292
$188$	0	0
$189$	−0.554958	−0.0403673
$190$	0	0
$191$	21.1782	1.53240	0.766201	−	0.642601i	$-0.222145\pi$
$191$	21.1782	1.53240	0.766201	+	0.642601i	$0.222145\pi$
$192$	0	0
$193$	−17.6112	−1.26768	−0.633840	−	0.773464i	$-0.718522\pi$
$193$	−17.6112	−1.26768	−0.633840	+	0.773464i	$0.718522\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.66248	−0.332188	−0.166094	−	0.986110i	$-0.553116\pi$
$197$	−4.66248	−0.332188	−0.166094	+	0.986110i	$0.553116\pi$
$198$	0	0
$199$	−15.0368	−1.06593	−0.532967	−	0.846136i	$-0.678923\pi$
$199$	−15.0368	−1.06593	−0.532967	+	0.846136i	$0.678923\pi$
$200$	0	0
$201$	1.87263	0.132085
$202$	0	0
$203$	1.06100	0.0744675
$204$	0	0
$205$	−5.15213	−0.359840
$206$	0	0
$207$	−5.76271	−0.400536
$208$	0	0
$209$	−2.19269	−0.151671
$210$	0	0
$211$	0.460107	0.0316751	0.0158375	−	0.999875i	$-0.494959\pi$
$211$	0.460107	0.0316751	0.0158375	+	0.999875i	$0.494959\pi$
$212$	0	0
$213$	10.5036	0.719698
$214$	0	0
$215$	−11.6407	−0.793890
$216$	0	0
$217$	−5.28083	−0.358486
$218$	0	0
$219$	10.4765	0.707936
$220$	0	0
$221$	0	0
$222$	0	0
$223$	16.3502	1.09489	0.547445	−	0.836842i	$-0.315601\pi$
$223$	16.3502	1.09489	0.547445	+	0.836842i	$0.315601\pi$
$224$	0	0
$225$	−3.89977	−0.259985
$226$	0	0
$227$	−6.56033	−0.435425	−0.217712	−	0.976013i	$-0.569859\pi$
$227$	−6.56033	−0.435425	−0.217712	+	0.976013i	$0.569859\pi$
$228$	0	0
$229$	−3.95539	−0.261380	−0.130690	−	0.991423i	$-0.541719\pi$
$229$	−3.95539	−0.261380	−0.130690	+	0.991423i	$0.541719\pi$
$230$	0	0
$231$	1.61596	0.106322
$232$	0	0
$233$	8.35690	0.547478	0.273739	−	0.961804i	$-0.411739\pi$
$233$	8.35690	0.547478	0.273739	+	0.961804i	$0.411739\pi$
$234$	0	0
$235$	0.789856	0.0515245
$236$	0	0
$237$	−1.33513	−0.0867257
$238$	0	0
$239$	−20.1008	−1.30021	−0.650107	−	0.759843i	$-0.725276\pi$
$239$	−20.1008	−1.30021	−0.650107	+	0.759843i	$0.725276\pi$
$240$	0	0
$241$	−19.0127	−1.22471	−0.612357	−	0.790581i	$-0.709778\pi$
$241$	−19.0127	−1.22471	−0.612357	+	0.790581i	$0.709778\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	7.01938	0.448452
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−2.64310	−0.167500
$250$	0	0
$251$	0.763774	0.0482090	0.0241045	−	0.999709i	$-0.492327\pi$
$251$	0.763774	0.0482090	0.0241045	+	0.999709i	$0.492327\pi$
$252$	0	0
$253$	16.7802	1.05496
$254$	0	0
$255$	5.08815	0.318632
$256$	0	0
$257$	−13.0911	−0.816602	−0.408301	−	0.912847i	$-0.633879\pi$
$257$	−13.0911	−0.816602	−0.408301	+	0.912847i	$0.633879\pi$
$258$	0	0
$259$	−3.19269	−0.198384
$260$	0	0
$261$	−1.91185	−0.118341
$262$	0	0
$263$	−18.3773	−1.13320	−0.566598	−	0.823995i	$-0.691741\pi$
$263$	−18.3773	−1.13320	−0.566598	+	0.823995i	$0.691741\pi$
$264$	0	0
$265$	7.95300	0.488549
$266$	0	0
$267$	9.92692	0.607518
$268$	0	0
$269$	23.6625	1.44273	0.721363	−	0.692557i	$-0.243516\pi$
$269$	23.6625	1.44273	0.721363	+	0.692557i	$0.243516\pi$
$270$	0	0
$271$	19.7530	1.19991	0.599955	−	0.800034i	$-0.295185\pi$
$271$	19.7530	1.19991	0.599955	+	0.800034i	$0.295185\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	11.3556	0.684767
$276$	0	0
$277$	1.77777	0.106816	0.0534081	−	0.998573i	$-0.482992\pi$
$277$	1.77777	0.106816	0.0534081	+	0.998573i	$0.482992\pi$
$278$	0	0
$279$	9.51573	0.569692
$280$	0	0
$281$	1.62133	0.0967207	0.0483603	−	0.998830i	$-0.484600\pi$
$281$	1.62133	0.0967207	0.0483603	+	0.998830i	$0.484600\pi$
$282$	0	0
$283$	−4.98361	−0.296245	−0.148122	−	0.988969i	$-0.547323\pi$
$283$	−4.98361	−0.296245	−0.148122	+	0.988969i	$0.547323\pi$
$284$	0	0
$285$	−0.789856	−0.0467870
$286$	0	0
$287$	−2.72587	−0.160903
$288$	0	0
$289$	6.53079	0.384164
$290$	0	0
$291$	17.0737	1.00088
$292$	0	0
$293$	−0.0717525	−0.00419183	−0.00209591	−	0.999998i	$-0.500667\pi$
$293$	−0.0717525	−0.00419183	−0.00209591	+	0.999998i	$0.500667\pi$
$294$	0	0
$295$	4.29829	0.250256
$296$	0	0
$297$	−2.91185	−0.168963
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−6.15883	−0.354989
$302$	0	0
$303$	7.32304	0.420698
$304$	0	0
$305$	3.59073	0.205605
$306$	0	0
$307$	5.19806	0.296669	0.148335	−	0.988937i	$-0.452609\pi$
$307$	5.19806	0.296669	0.148335	+	0.988937i	$0.452609\pi$
$308$	0	0
$309$	4.21983	0.240058
$310$	0	0
$311$	22.5429	1.27829	0.639145	−	0.769087i	$-0.279289\pi$
$311$	22.5429	1.27829	0.639145	+	0.769087i	$0.279289\pi$
$312$	0	0
$313$	22.6612	1.28088	0.640442	−	0.768006i	$-0.278751\pi$
$313$	22.6612	1.28088	0.640442	+	0.768006i	$0.278751\pi$
$314$	0	0
$315$	0.582105	0.0327979
$316$	0	0
$317$	26.3424	1.47954	0.739769	−	0.672861i	$-0.234935\pi$
$317$	26.3424	1.47954	0.739769	+	0.672861i	$0.234935\pi$
$318$	0	0
$319$	5.56704	0.311694
$320$	0	0
$321$	−6.39373	−0.356863
$322$	0	0
$323$	−3.65279	−0.203247
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.46011	−0.191344
$328$	0	0
$329$	0.417895	0.0230393
$330$	0	0
$331$	11.2295	0.617230	0.308615	−	0.951187i	$-0.400134\pi$
$331$	11.2295	0.617230	0.308615	+	0.951187i	$0.400134\pi$
$332$	0	0
$333$	5.75302	0.315264
$334$	0	0
$335$	−1.96423	−0.107317
$336$	0	0
$337$	2.30798	0.125724	0.0628618	−	0.998022i	$-0.479977\pi$
$337$	2.30798	0.125724	0.0628618	+	0.998022i	$0.479977\pi$
$338$	0	0
$339$	9.35690	0.508197
$340$	0	0
$341$	−27.7084	−1.50049
$342$	0	0
$343$	7.59850	0.410280
$344$	0	0
$345$	6.04461	0.325431
$346$	0	0
$347$	9.13706	0.490503	0.245252	−	0.969459i	$-0.421129\pi$
$347$	9.13706	0.490503	0.245252	+	0.969459i	$0.421129\pi$
$348$	0	0
$349$	23.9758	1.28340	0.641699	−	0.766957i	$-0.278230\pi$
$349$	23.9758	1.28340	0.641699	+	0.766957i	$0.278230\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−27.1239	−1.44366	−0.721830	−	0.692070i	$-0.756699\pi$
$353$	−27.1239	−1.44366	−0.721830	+	0.692070i	$0.756699\pi$
$354$	0	0
$355$	−11.0175	−0.584746
$356$	0	0
$357$	2.69202	0.142477
$358$	0	0
$359$	26.0790	1.37640	0.688200	−	0.725521i	$-0.258401\pi$
$359$	26.0790	1.37640	0.688200	+	0.725521i	$0.258401\pi$
$360$	0	0
$361$	−18.4330	−0.970156
$362$	0	0
$363$	−2.52111	−0.132324
$364$	0	0
$365$	−10.9890	−0.575190
$366$	0	0
$367$	−9.57434	−0.499776	−0.249888	−	0.968275i	$-0.580394\pi$
$367$	−9.57434	−0.499776	−0.249888	+	0.968275i	$0.580394\pi$
$368$	0	0
$369$	4.91185	0.255701
$370$	0	0
$371$	4.20775	0.218456
$372$	0	0
$373$	28.1497	1.45754	0.728769	−	0.684760i	$-0.240093\pi$
$373$	28.1497	1.45754	0.728769	+	0.684760i	$0.240093\pi$
$374$	0	0
$375$	9.33513	0.482064
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−16.0465	−0.824255	−0.412127	−	0.911126i	$-0.635214\pi$
$379$	−16.0465	−0.824255	−0.412127	+	0.911126i	$0.635214\pi$
$380$	0	0
$381$	4.48188	0.229614
$382$	0	0
$383$	24.6165	1.25785	0.628923	−	0.777467i	$-0.283496\pi$
$383$	24.6165	1.25785	0.628923	+	0.777467i	$0.283496\pi$
$384$	0	0
$385$	−1.69501	−0.0863855
$386$	0	0
$387$	11.0978	0.564135
$388$	0	0
$389$	−17.2198	−0.873080	−0.436540	−	0.899685i	$-0.643796\pi$
$389$	−17.2198	−0.873080	−0.436540	+	0.899685i	$0.643796\pi$
$390$	0	0
$391$	27.9541	1.41370
$392$	0	0
$393$	9.21744	0.464958
$394$	0	0
$395$	1.40044	0.0704636
$396$	0	0
$397$	−2.03923	−0.102346	−0.0511730	−	0.998690i	$-0.516296\pi$
$397$	−2.03923	−0.102346	−0.0511730	+	0.998690i	$0.516296\pi$
$398$	0	0
$399$	−0.417895	−0.0209209
$400$	0	0
$401$	−1.46144	−0.0729806	−0.0364903	−	0.999334i	$-0.511618\pi$
$401$	−1.46144	−0.0729806	−0.0364903	+	0.999334i	$0.511618\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.04892	−0.0521211
$406$	0	0
$407$	−16.7520	−0.830364
$408$	0	0
$409$	−29.9390	−1.48039	−0.740194	−	0.672393i	$-0.765266\pi$
$409$	−29.9390	−1.48039	−0.740194	+	0.672393i	$0.765266\pi$
$410$	0	0
$411$	−7.46980	−0.368458
$412$	0	0
$413$	2.27413	0.111902
$414$	0	0
$415$	2.77240	0.136092
$416$	0	0
$417$	17.9976	0.881347
$418$	0	0
$419$	−6.64742	−0.324748	−0.162374	−	0.986729i	$-0.551915\pi$
$419$	−6.64742	−0.324748	−0.162374	+	0.986729i	$0.551915\pi$
$420$	0	0
$421$	13.5646	0.661100	0.330550	−	0.943788i	$-0.392766\pi$
$421$	13.5646	0.661100	0.330550	+	0.943788i	$0.392766\pi$
$422$	0	0
$423$	−0.753020	−0.0366131
$424$	0	0
$425$	18.9172	0.917620
$426$	0	0
$427$	1.89977	0.0919364
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−35.9463	−1.73147	−0.865736	−	0.500501i	$-0.833149\pi$
$431$	−35.9463	−1.73147	−0.865736	+	0.500501i	$0.833149\pi$
$432$	0	0
$433$	−32.4741	−1.56061	−0.780303	−	0.625402i	$-0.784935\pi$
$433$	−32.4741	−1.56061	−0.780303	+	0.625402i	$0.784935\pi$
$434$	0	0
$435$	2.00538	0.0961505
$436$	0	0
$437$	−4.33944	−0.207583
$438$	0	0
$439$	−12.8321	−0.612441	−0.306221	−	0.951961i	$-0.599065\pi$
$439$	−12.8321	−0.612441	−0.306221	+	0.951961i	$0.599065\pi$
$440$	0	0
$441$	−6.69202	−0.318668
$442$	0	0
$443$	−11.9608	−0.568273	−0.284137	−	0.958784i	$-0.591707\pi$
$443$	−11.9608	−0.568273	−0.284137	+	0.958784i	$0.591707\pi$
$444$	0	0
$445$	−10.4125	−0.493601
$446$	0	0
$447$	15.3351	0.725327
$448$	0	0
$449$	−12.4789	−0.588915	−0.294458	−	0.955665i	$-0.595139\pi$
$449$	−12.4789	−0.588915	−0.294458	+	0.955665i	$0.595139\pi$
$450$	0	0
$451$	−14.3026	−0.673483
$452$	0	0
$453$	−2.53079	−0.118907
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.4523	1.51806	0.759028	−	0.651058i	$-0.225674\pi$
$457$	32.4523	1.51806	0.759028	+	0.651058i	$0.225674\pi$
$458$	0	0
$459$	−4.85086	−0.226419
$460$	0	0
$461$	24.4034	1.13658	0.568290	−	0.822828i	$-0.307605\pi$
$461$	24.4034	1.13658	0.568290	+	0.822828i	$0.307605\pi$
$462$	0	0
$463$	33.1836	1.54217	0.771086	−	0.636731i	$-0.219714\pi$
$463$	33.1836	1.54217	0.771086	+	0.636731i	$0.219714\pi$
$464$	0	0
$465$	−9.98121	−0.462868
$466$	0	0
$467$	38.5206	1.78252	0.891261	−	0.453490i	$-0.149821\pi$
$467$	38.5206	1.78252	0.891261	+	0.453490i	$0.149821\pi$
$468$	0	0
$469$	−1.03923	−0.0479871
$470$	0	0
$471$	17.2392	0.794341
$472$	0	0
$473$	−32.3153	−1.48586
$474$	0	0
$475$	−2.93661	−0.134741
$476$	0	0
$477$	−7.58211	−0.347161
$478$	0	0
$479$	8.34481	0.381284	0.190642	−	0.981660i	$-0.438943\pi$
$479$	8.34481	0.381284	0.190642	+	0.981660i	$0.438943\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.19806	0.145517
$484$	0	0
$485$	−17.9089	−0.813200
$486$	0	0
$487$	14.8586	0.673309	0.336654	−	0.941628i	$-0.390705\pi$
$487$	14.8586	0.673309	0.336654	+	0.941628i	$0.390705\pi$
$488$	0	0
$489$	−15.7071	−0.710299
$490$	0	0
$491$	−4.99894	−0.225599	−0.112799	−	0.993618i	$-0.535982\pi$
$491$	−4.99894	−0.225599	−0.112799	+	0.993618i	$0.535982\pi$
$492$	0	0
$493$	9.27413	0.417686
$494$	0	0
$495$	3.05429	0.137280
$496$	0	0
$497$	−5.82908	−0.261470
$498$	0	0
$499$	0.385371	0.0172516	0.00862579	−	0.999963i	$-0.497254\pi$
$499$	0.385371	0.0172516	0.00862579	+	0.999963i	$0.497254\pi$
$500$	0	0
$501$	−5.39612	−0.241081
$502$	0	0
$503$	−22.6179	−1.00848	−0.504241	−	0.863563i	$-0.668228\pi$
$503$	−22.6179	−1.00848	−0.504241	+	0.863563i	$0.668228\pi$
$504$	0	0
$505$	−7.68127	−0.341812
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11.6039	−0.514333	−0.257166	−	0.966367i	$-0.582789\pi$
$509$	−11.6039	−0.514333	−0.257166	+	0.966367i	$0.582789\pi$
$510$	0	0
$511$	−5.81402	−0.257197
$512$	0	0
$513$	0.753020	0.0332467
$514$	0	0
$515$	−4.42626	−0.195044
$516$	0	0
$517$	2.19269	0.0964342
$518$	0	0
$519$	23.9420	1.05094
$520$	0	0
$521$	−1.62671	−0.0712675	−0.0356337	−	0.999365i	$-0.511345\pi$
$521$	−1.62671	−0.0712675	−0.0356337	+	0.999365i	$0.511345\pi$
$522$	0	0
$523$	−10.0718	−0.440407	−0.220203	−	0.975454i	$-0.570672\pi$
$523$	−10.0718	−0.440407	−0.220203	+	0.975454i	$0.570672\pi$
$524$	0	0
$525$	2.16421	0.0944539
$526$	0	0
$527$	−46.1594	−2.01074
$528$	0	0
$529$	10.2088	0.443862
$530$	0	0
$531$	−4.09783	−0.177831
$532$	0	0
$533$	0	0
$534$	0	0
$535$	6.70650	0.289947
$536$	0	0
$537$	−18.4088	−0.794398
$538$	0	0
$539$	19.4862	0.839330
$540$	0	0
$541$	−20.4674	−0.879962	−0.439981	−	0.898007i	$-0.645015\pi$
$541$	−20.4674	−0.879962	−0.439981	+	0.898007i	$0.645015\pi$
$542$	0	0
$543$	−3.63342	−0.155925
$544$	0	0
$545$	3.62937	0.155465
$546$	0	0
$547$	27.5478	1.17786	0.588929	−	0.808185i	$-0.299550\pi$
$547$	27.5478	1.17786	0.588929	+	0.808185i	$0.299550\pi$
$548$	0	0
$549$	−3.42327	−0.146102
$550$	0	0
$551$	−1.43967	−0.0613318
$552$	0	0
$553$	0.740939	0.0315079
$554$	0	0
$555$	−6.03444	−0.256148
$556$	0	0
$557$	37.9855	1.60950	0.804749	−	0.593615i	$-0.202300\pi$
$557$	37.9855	1.60950	0.804749	+	0.593615i	$0.202300\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	14.1250	0.596357
$562$	0	0
$563$	29.7724	1.25476	0.627378	−	0.778714i	$-0.284128\pi$
$563$	29.7724	1.25476	0.627378	+	0.778714i	$0.284128\pi$
$564$	0	0
$565$	−9.81461	−0.412904
$566$	0	0
$567$	−0.554958	−0.0233061
$568$	0	0
$569$	−21.9541	−0.920362	−0.460181	−	0.887825i	$-0.652216\pi$
$569$	−21.9541	−0.920362	−0.460181	+	0.887825i	$0.652216\pi$
$570$	0	0
$571$	−2.46575	−0.103188	−0.0515942	−	0.998668i	$-0.516430\pi$
$571$	−2.46575	−0.103188	−0.0515942	+	0.998668i	$0.516430\pi$
$572$	0	0
$573$	21.1782	0.884732
$574$	0	0
$575$	22.4733	0.937199
$576$	0	0
$577$	17.4547	0.726650	0.363325	−	0.931662i	$-0.381641\pi$
$577$	17.4547	0.726650	0.363325	+	0.931662i	$0.381641\pi$
$578$	0	0
$579$	−17.6112	−0.731895
$580$	0	0
$581$	1.46681	0.0608536
$582$	0	0
$583$	22.0780	0.914377
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.26337	−0.258517	−0.129259	−	0.991611i	$-0.541260\pi$
$587$	−6.26337	−0.258517	−0.129259	+	0.991611i	$0.541260\pi$
$588$	0	0
$589$	7.16554	0.295251
$590$	0	0
$591$	−4.66248	−0.191789
$592$	0	0
$593$	22.8745	0.939345	0.469672	−	0.882841i	$-0.344372\pi$
$593$	22.8745	0.939345	0.469672	+	0.882841i	$0.344372\pi$
$594$	0	0
$595$	−2.82371	−0.115761
$596$	0	0
$597$	−15.0368	−0.615417
$598$	0	0
$599$	1.05621	0.0431557	0.0215778	−	0.999767i	$-0.493131\pi$
$599$	1.05621	0.0431557	0.0215778	+	0.999767i	$0.493131\pi$
$600$	0	0
$601$	−33.3236	−1.35930	−0.679650	−	0.733537i	$-0.737868\pi$
$601$	−33.3236	−1.35930	−0.679650	+	0.733537i	$0.737868\pi$
$602$	0	0
$603$	1.87263	0.0762592
$604$	0	0
$605$	2.64443	0.107511
$606$	0	0
$607$	−16.2403	−0.659172	−0.329586	−	0.944125i	$-0.606909\pi$
$607$	−16.2403	−0.659172	−0.329586	+	0.944125i	$0.606909\pi$
$608$	0	0
$609$	1.06100	0.0429938
$610$	0	0
$611$	0	0
$612$	0	0
$613$	11.0479	0.446219	0.223109	−	0.974793i	$-0.428379\pi$
$613$	11.0479	0.446219	0.223109	+	0.974793i	$0.428379\pi$
$614$	0	0
$615$	−5.15213	−0.207754
$616$	0	0
$617$	−4.65950	−0.187584	−0.0937922	−	0.995592i	$-0.529899\pi$
$617$	−4.65950	−0.187584	−0.0937922	+	0.995592i	$0.529899\pi$
$618$	0	0
$619$	−31.9259	−1.28321	−0.641604	−	0.767036i	$-0.721731\pi$
$619$	−31.9259	−1.28321	−0.641604	+	0.767036i	$0.721731\pi$
$620$	0	0
$621$	−5.76271	−0.231250
$622$	0	0
$623$	−5.50902	−0.220714
$624$	0	0
$625$	9.70709	0.388283
$626$	0	0
$627$	−2.19269	−0.0875674
$628$	0	0
$629$	−27.9071	−1.11273
$630$	0	0
$631$	39.4413	1.57013	0.785067	−	0.619411i	$-0.212628\pi$
$631$	39.4413	1.57013	0.785067	+	0.619411i	$0.212628\pi$
$632$	0	0
$633$	0.460107	0.0182876
$634$	0	0
$635$	−4.70112	−0.186558
$636$	0	0
$637$	0	0
$638$	0	0
$639$	10.5036	0.415518
$640$	0	0
$641$	45.4510	1.79521	0.897603	−	0.440804i	$-0.145307\pi$
$641$	45.4510	1.79521	0.897603	+	0.440804i	$0.145307\pi$
$642$	0	0
$643$	29.7469	1.17310	0.586552	−	0.809912i	$-0.300485\pi$
$643$	29.7469	1.17310	0.586552	+	0.809912i	$0.300485\pi$
$644$	0	0
$645$	−11.6407	−0.458353
$646$	0	0
$647$	16.9312	0.665635	0.332818	−	0.942991i	$-0.392001\pi$
$647$	16.9312	0.665635	0.332818	+	0.942991i	$0.392001\pi$
$648$	0	0
$649$	11.9323	0.468384
$650$	0	0
$651$	−5.28083	−0.206972
$652$	0	0
$653$	−33.1976	−1.29912	−0.649561	−	0.760309i	$-0.725047\pi$
$653$	−33.1976	−1.29912	−0.649561	+	0.760309i	$0.725047\pi$
$654$	0	0
$655$	−9.66833	−0.377773
$656$	0	0
$657$	10.4765	0.408727
$658$	0	0
$659$	42.6571	1.66168	0.830842	−	0.556508i	$-0.187859\pi$
$659$	42.6571	1.66168	0.830842	+	0.556508i	$0.187859\pi$
$660$	0	0
$661$	38.6902	1.50488	0.752438	−	0.658664i	$-0.228878\pi$
$661$	38.6902	1.50488	0.752438	+	0.658664i	$0.228878\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.438337	0.0169980
$666$	0	0
$667$	11.0175	0.426598
$668$	0	0
$669$	16.3502	0.632135
$670$	0	0
$671$	9.96807	0.384813
$672$	0	0
$673$	−2.59419	−0.0999986	−0.0499993	−	0.998749i	$-0.515922\pi$
$673$	−2.59419	−0.0999986	−0.0499993	+	0.998749i	$0.515922\pi$
$674$	0	0
$675$	−3.89977	−0.150102
$676$	0	0
$677$	1.75302	0.0673740	0.0336870	−	0.999432i	$-0.489275\pi$
$677$	1.75302	0.0673740	0.0336870	+	0.999432i	$0.489275\pi$
$678$	0	0
$679$	−9.47517	−0.363624
$680$	0	0
$681$	−6.56033	−0.251393
$682$	0	0
$683$	−16.3351	−0.625046	−0.312523	−	0.949910i	$-0.601174\pi$
$683$	−16.3351	−0.625046	−0.312523	+	0.949910i	$0.601174\pi$
$684$	0	0
$685$	7.83520	0.299368
$686$	0	0
$687$	−3.95539	−0.150908
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−15.9105	−0.605265	−0.302632	−	0.953107i	$-0.597865\pi$
$691$	−15.9105	−0.605265	−0.302632	+	0.953107i	$0.597865\pi$
$692$	0	0
$693$	1.61596	0.0613851
$694$	0	0
$695$	−18.8780	−0.716083
$696$	0	0
$697$	−23.8267	−0.902500
$698$	0	0
$699$	8.35690	0.316087
$700$	0	0
$701$	−20.8635	−0.788005	−0.394002	−	0.919109i	$-0.628910\pi$
$701$	−20.8635	−0.788005	−0.394002	+	0.919109i	$0.628910\pi$
$702$	0	0
$703$	4.33214	0.163390
$704$	0	0
$705$	0.789856	0.0297477
$706$	0	0
$707$	−4.06398	−0.152842
$708$	0	0
$709$	15.6485	0.587691	0.293846	−	0.955853i	$-0.405065\pi$
$709$	15.6485	0.587691	0.293846	+	0.955853i	$0.405065\pi$
$710$	0	0
$711$	−1.33513	−0.0500711
$712$	0	0
$713$	−54.8364	−2.05364
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−20.1008	−0.750679
$718$	0	0
$719$	27.1594	1.01288	0.506438	−	0.862276i	$-0.330962\pi$
$719$	27.1594	1.01288	0.506438	+	0.862276i	$0.330962\pi$
$720$	0	0
$721$	−2.34183	−0.0872143
$722$	0	0
$723$	−19.0127	−0.707089
$724$	0	0
$725$	7.45580	0.276901
$726$	0	0
$727$	−31.7784	−1.17859	−0.589297	−	0.807916i	$-0.700595\pi$
$727$	−31.7784	−1.17859	−0.589297	+	0.807916i	$0.700595\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−53.8340	−1.99112
$732$	0	0
$733$	46.8907	1.73195	0.865973	−	0.500090i	$-0.166700\pi$
$733$	46.8907	1.73195	0.865973	+	0.500090i	$0.166700\pi$
$734$	0	0
$735$	7.01938	0.258914
$736$	0	0
$737$	−5.45281	−0.200857
$738$	0	0
$739$	−17.0479	−0.627115	−0.313558	−	0.949569i	$-0.601521\pi$
$739$	−17.0479	−0.627115	−0.313558	+	0.949569i	$0.601521\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11.6324	0.426750	0.213375	−	0.976970i	$-0.431554\pi$
$743$	11.6324	0.426750	0.213375	+	0.976970i	$0.431554\pi$
$744$	0	0
$745$	−16.0853	−0.589319
$746$	0	0
$747$	−2.64310	−0.0967061
$748$	0	0
$749$	3.54825	0.129650
$750$	0	0
$751$	13.0295	0.475455	0.237727	−	0.971332i	$-0.423598\pi$
$751$	13.0295	0.475455	0.237727	+	0.971332i	$0.423598\pi$
$752$	0	0
$753$	0.763774	0.0278335
$754$	0	0
$755$	2.65459	0.0966106
$756$	0	0
$757$	22.7899	0.828311	0.414156	−	0.910206i	$-0.364077\pi$
$757$	22.7899	0.828311	0.414156	+	0.910206i	$0.364077\pi$
$758$	0	0
$759$	16.7802	0.609081
$760$	0	0
$761$	−38.3424	−1.38991	−0.694956	−	0.719052i	$-0.744576\pi$
$761$	−38.3424	−1.38991	−0.694956	+	0.719052i	$0.744576\pi$
$762$	0	0
$763$	1.92021	0.0695164
$764$	0	0
$765$	5.08815	0.183962
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−3.63879	−0.131218	−0.0656091	−	0.997845i	$-0.520899\pi$
$769$	−3.63879	−0.131218	−0.0656091	+	0.997845i	$0.520899\pi$
$770$	0	0
$771$	−13.0911	−0.471466
$772$	0	0
$773$	−39.3424	−1.41505	−0.707524	−	0.706689i	$-0.750188\pi$
$773$	−39.3424	−1.41505	−0.707524	+	0.706689i	$0.750188\pi$
$774$	0	0
$775$	−37.1092	−1.33300
$776$	0	0
$777$	−3.19269	−0.114537
$778$	0	0
$779$	3.69873	0.132521
$780$	0	0
$781$	−30.5851	−1.09442
$782$	0	0
$783$	−1.91185	−0.0683241
$784$	0	0
$785$	−18.0825	−0.645392
$786$	0	0
$787$	25.4252	0.906310	0.453155	−	0.891432i	$-0.350298\pi$
$787$	25.4252	0.906310	0.453155	+	0.891432i	$0.350298\pi$
$788$	0	0
$789$	−18.3773	−0.654251
$790$	0	0
$791$	−5.19269	−0.184631
$792$	0	0
$793$	0	0
$794$	0	0
$795$	7.95300	0.282064
$796$	0	0
$797$	20.7138	0.733720	0.366860	−	0.930276i	$-0.380433\pi$
$797$	20.7138	0.733720	0.366860	+	0.930276i	$0.380433\pi$
$798$	0	0
$799$	3.65279	0.129227
$800$	0	0
$801$	9.92692	0.350750
$802$	0	0
$803$	−30.5060	−1.07653
$804$	0	0
$805$	−3.35450	−0.118231
$806$	0	0
$807$	23.6625	0.832959
$808$	0	0
$809$	11.0978	0.390179	0.195090	−	0.980785i	$-0.437500\pi$
$809$	11.0978	0.390179	0.195090	+	0.980785i	$0.437500\pi$
$810$	0	0
$811$	4.84223	0.170034	0.0850169	−	0.996380i	$-0.472906\pi$
$811$	4.84223	0.170034	0.0850169	+	0.996380i	$0.472906\pi$
$812$	0	0
$813$	19.7530	0.692769
$814$	0	0
$815$	16.4754	0.577109
$816$	0	0
$817$	8.35690	0.292371
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−43.7381	−1.52647	−0.763235	−	0.646121i	$-0.776390\pi$
$821$	−43.7381	−1.52647	−0.763235	+	0.646121i	$0.776390\pi$
$822$	0	0
$823$	−12.0998	−0.421771	−0.210885	−	0.977511i	$-0.567635\pi$
$823$	−12.0998	−0.421771	−0.210885	+	0.977511i	$0.567635\pi$
$824$	0	0
$825$	11.3556	0.395350
$826$	0	0
$827$	35.3212	1.22824	0.614120	−	0.789213i	$-0.289511\pi$
$827$	35.3212	1.22824	0.614120	+	0.789213i	$0.289511\pi$
$828$	0	0
$829$	53.4946	1.85794	0.928971	−	0.370152i	$-0.120694\pi$
$829$	53.4946	1.85794	0.928971	+	0.370152i	$0.120694\pi$
$830$	0	0
$831$	1.77777	0.0616703
$832$	0	0
$833$	32.4620	1.12474
$834$	0	0
$835$	5.66009	0.195875
$836$	0	0
$837$	9.51573	0.328912
$838$	0	0
$839$	−23.1521	−0.799300	−0.399650	−	0.916668i	$-0.630868\pi$
$839$	−23.1521	−0.799300	−0.399650	+	0.916668i	$0.630868\pi$
$840$	0	0
$841$	−25.3448	−0.873959
$842$	0	0
$843$	1.62133	0.0558417
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.39911	0.0480739
$848$	0	0
$849$	−4.98361	−0.171037
$850$	0	0
$851$	−33.1530	−1.13647
$852$	0	0
$853$	26.7265	0.915097	0.457548	−	0.889185i	$-0.348728\pi$
$853$	26.7265	0.915097	0.457548	+	0.889185i	$0.348728\pi$
$854$	0	0
$855$	−0.789856	−0.0270125
$856$	0	0
$857$	42.6064	1.45541	0.727703	−	0.685892i	$-0.240588\pi$
$857$	42.6064	1.45541	0.727703	+	0.685892i	$0.240588\pi$
$858$	0	0
$859$	−33.6079	−1.14669	−0.573344	−	0.819315i	$-0.694354\pi$
$859$	−33.6079	−1.14669	−0.573344	+	0.819315i	$0.694354\pi$
$860$	0	0
$861$	−2.72587	−0.0928975
$862$	0	0
$863$	−18.7047	−0.636715	−0.318358	−	0.947971i	$-0.603131\pi$
$863$	−18.7047	−0.636715	−0.318358	+	0.947971i	$0.603131\pi$
$864$	0	0
$865$	−25.1132	−0.853873
$866$	0	0
$867$	6.53079	0.221797
$868$	0	0
$869$	3.88769	0.131881
$870$	0	0
$871$	0	0
$872$	0	0
$873$	17.0737	0.577856
$874$	0	0
$875$	−5.18060	−0.175136
$876$	0	0
$877$	36.8237	1.24345	0.621724	−	0.783236i	$-0.286433\pi$
$877$	36.8237	1.24345	0.621724	+	0.783236i	$0.286433\pi$
$878$	0	0
$879$	−0.0717525	−0.00242015
$880$	0	0
$881$	−41.1250	−1.38554	−0.692768	−	0.721161i	$-0.743609\pi$
$881$	−41.1250	−1.38554	−0.692768	+	0.721161i	$0.743609\pi$
$882$	0	0
$883$	−30.7482	−1.03476	−0.517380	−	0.855756i	$-0.673093\pi$
$883$	−30.7482	−1.03476	−0.517380	+	0.855756i	$0.673093\pi$
$884$	0	0
$885$	4.29829	0.144485
$886$	0	0
$887$	−7.58940	−0.254827	−0.127414	−	0.991850i	$-0.540668\pi$
$887$	−7.58940	−0.254827	−0.127414	+	0.991850i	$0.540668\pi$
$888$	0	0
$889$	−2.48725	−0.0834198
$890$	0	0
$891$	−2.91185	−0.0975508
$892$	0	0
$893$	−0.567040	−0.0189753
$894$	0	0
$895$	19.3093	0.645439
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−18.1927	−0.606760
$900$	0	0
$901$	36.7797	1.22531
$902$	0	0
$903$	−6.15883	−0.204953
$904$	0	0
$905$	3.81115	0.126687
$906$	0	0
$907$	−19.3333	−0.641952	−0.320976	−	0.947087i	$-0.604011\pi$
$907$	−19.3333	−0.641952	−0.320976	+	0.947087i	$0.604011\pi$
$908$	0	0
$909$	7.32304	0.242890
$910$	0	0
$911$	−22.7149	−0.752577	−0.376288	−	0.926503i	$-0.622800\pi$
$911$	−22.7149	−0.752577	−0.376288	+	0.926503i	$0.622800\pi$
$912$	0	0
$913$	7.69633	0.254711
$914$	0	0
$915$	3.59073	0.118706
$916$	0	0
$917$	−5.11529	−0.168922
$918$	0	0
$919$	−14.2911	−0.471420	−0.235710	−	0.971823i	$-0.575742\pi$
$919$	−14.2911	−0.471420	−0.235710	+	0.971823i	$0.575742\pi$
$920$	0	0
$921$	5.19806	0.171282
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−22.4355	−0.737674
$926$	0	0
$927$	4.21983	0.138597
$928$	0	0
$929$	32.7211	1.07354	0.536772	−	0.843727i	$-0.319644\pi$
$929$	32.7211	1.07354	0.536772	+	0.843727i	$0.319644\pi$
$930$	0	0
$931$	−5.03923	−0.165154
$932$	0	0
$933$	22.5429	0.738021
$934$	0	0
$935$	−14.8159	−0.484533
$936$	0	0
$937$	−4.01400	−0.131132	−0.0655658	−	0.997848i	$-0.520885\pi$
$937$	−4.01400	−0.131132	−0.0655658	+	0.997848i	$0.520885\pi$
$938$	0	0
$939$	22.6612	0.739519
$940$	0	0
$941$	−28.2669	−0.921476	−0.460738	−	0.887536i	$-0.652415\pi$
$941$	−28.2669	−0.921476	−0.460738	+	0.887536i	$0.652415\pi$
$942$	0	0
$943$	−28.3056	−0.921757
$944$	0	0
$945$	0.582105	0.0189359
$946$	0	0
$947$	37.8455	1.22981	0.614906	−	0.788600i	$-0.289194\pi$
$947$	37.8455	1.22981	0.614906	+	0.788600i	$0.289194\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	26.3424	0.854212
$952$	0	0
$953$	−40.8256	−1.32247	−0.661236	−	0.750178i	$-0.729968\pi$
$953$	−40.8256	−1.32247	−0.661236	+	0.750178i	$0.729968\pi$
$954$	0	0
$955$	−22.2142	−0.718834
$956$	0	0
$957$	5.56704	0.179957
$958$	0	0
$959$	4.14542	0.133863
$960$	0	0
$961$	59.5491	1.92094
$962$	0	0
$963$	−6.39373	−0.206035
$964$	0	0
$965$	18.4727	0.594656
$966$	0	0
$967$	−10.2798	−0.330575	−0.165288	−	0.986245i	$-0.552855\pi$
$967$	−10.2798	−0.330575	−0.165288	+	0.986245i	$0.552855\pi$
$968$	0	0
$969$	−3.65279	−0.117345
$970$	0	0
$971$	19.4805	0.625161	0.312580	−	0.949891i	$-0.398807\pi$
$971$	19.4805	0.625161	0.312580	+	0.949891i	$0.398807\pi$
$972$	0	0
$973$	−9.98792	−0.320198
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.0538	1.50539	0.752693	−	0.658372i	$-0.228755\pi$
$977$	47.0538	1.50539	0.752693	+	0.658372i	$0.228755\pi$
$978$	0	0
$979$	−28.9057	−0.923831
$980$	0	0
$981$	−3.46011	−0.110473
$982$	0	0
$983$	−34.4295	−1.09813	−0.549065	−	0.835779i	$-0.685016\pi$
$983$	−34.4295	−1.09813	−0.549065	+	0.835779i	$0.685016\pi$
$984$	0	0
$985$	4.89056	0.155826
$986$	0	0
$987$	0.417895	0.0133017
$988$	0	0
$989$	−63.9536	−2.03361
$990$	0	0
$991$	−7.30798	−0.232146	−0.116073	−	0.993241i	$-0.537031\pi$
$991$	−7.30798	−0.232146	−0.116073	+	0.993241i	$0.537031\pi$
$992$	0	0
$993$	11.2295	0.356358
$994$	0	0
$995$	15.7724	0.500019
$996$	0	0
$997$	−35.6915	−1.13036	−0.565181	−	0.824967i	$-0.691194\pi$
$997$	−35.6915	−1.13036	−0.565181	+	0.824967i	$0.691194\pi$
$998$	0	0
$999$	5.75302	0.182018

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.cp.1.1		3
4.3	odd	2		507.2.a.l.1.3	yes	3
12.11	even	2		1521.2.a.n.1.1		3
13.12	even	2		8112.2.a.cg.1.3		3
52.3	odd	6		507.2.e.i.22.1		6
52.7	even	12		507.2.j.i.361.6		12
52.11	even	12		507.2.j.i.316.1		12
52.15	even	12		507.2.j.i.316.6		12
52.19	even	12		507.2.j.i.361.1		12
52.23	odd	6		507.2.e.l.22.3		6
52.31	even	4		507.2.b.f.337.1		6
52.35	odd	6		507.2.e.i.484.1		6
52.43	odd	6		507.2.e.l.484.3		6
52.47	even	4		507.2.b.f.337.6		6
52.51	odd	2		507.2.a.i.1.1	✓	3
156.47	odd	4		1521.2.b.k.1351.1		6
156.83	odd	4		1521.2.b.k.1351.6		6
156.155	even	2		1521.2.a.s.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.i.1.1	✓	3	52.51	odd	2
507.2.a.l.1.3	yes	3	4.3	odd	2
507.2.b.f.337.1		6	52.31	even	4
507.2.b.f.337.6		6	52.47	even	4
507.2.e.i.22.1		6	52.3	odd	6
507.2.e.i.484.1		6	52.35	odd	6
507.2.e.l.22.3		6	52.23	odd	6
507.2.e.l.484.3		6	52.43	odd	6
507.2.j.i.316.1		12	52.11	even	12
507.2.j.i.316.6		12	52.15	even	12
507.2.j.i.361.1		12	52.19	even	12
507.2.j.i.361.6		12	52.7	even	12
1521.2.a.n.1.1		3	12.11	even	2
1521.2.a.s.1.3		3	156.155	even	2
1521.2.b.k.1351.1		6	156.47	odd	4
1521.2.b.k.1351.6		6	156.83	odd	4
8112.2.a.cg.1.3		3	13.12	even	2
8112.2.a.cp.1.1		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.04892 q^{5} -0.554958 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.04892 q^{5} -0.554958 q^{7} +1.00000 q^{9} -2.91185 q^{11} -1.04892 q^{15} -4.85086 q^{17} +0.753020 q^{19} -0.554958 q^{21} -5.76271 q^{23} -3.89977 q^{25} +1.00000 q^{27} -1.91185 q^{29} +9.51573 q^{31} -2.91185 q^{33} +0.582105 q^{35} +5.75302 q^{37} +4.91185 q^{41} +11.0978 q^{43} -1.04892 q^{45} -0.753020 q^{47} -6.69202 q^{49} -4.85086 q^{51} -7.58211 q^{53} +3.05429 q^{55} +0.753020 q^{57} -4.09783 q^{59} -3.42327 q^{61} -0.554958 q^{63} +1.87263 q^{67} -5.76271 q^{69} +10.5036 q^{71} +10.4765 q^{73} -3.89977 q^{75} +1.61596 q^{77} -1.33513 q^{79} +1.00000 q^{81} -2.64310 q^{83} +5.08815 q^{85} -1.91185 q^{87} +9.92692 q^{89} +9.51573 q^{93} -0.789856 q^{95} +17.0737 q^{97} -2.91185 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 6 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 6 * q^5 - 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 6 q^{5} - 2 q^{7} + 3 q^{9} - 5 q^{11} + 6 q^{15} - q^{17} + 7 q^{19} - 2 q^{21} + 11 q^{25} + 3 q^{27} - 2 q^{29} + 16 q^{31} - 5 q^{33} - 4 q^{35} + 22 q^{37} + 11 q^{41} + 15 q^{43} + 6 q^{45} - 7 q^{47} - 15 q^{49} - q^{51} - 17 q^{53} - 3 q^{55} + 7 q^{57} + 6 q^{59} - 13 q^{61} - 2 q^{63} - 11 q^{67} + 6 q^{73} + 11 q^{75} + 15 q^{77} - 3 q^{79} + 3 q^{81} - 12 q^{83} + 19 q^{85} - 2 q^{87} + q^{89} + 16 q^{93} + 21 q^{95} - 5 q^{97} - 5 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 6 * q^5 - 2 * q^7 + 3 * q^9 - 5 * q^11 + 6 * q^15 - q^17 + 7 * q^19 - 2 * q^21 + 11 * q^25 + 3 * q^27 - 2 * q^29 + 16 * q^31 - 5 * q^33 - 4 * q^35 + 22 * q^37 + 11 * q^41 + 15 * q^43 + 6 * q^45 - 7 * q^47 - 15 * q^49 - q^51 - 17 * q^53 - 3 * q^55 + 7 * q^57 + 6 * q^59 - 13 * q^61 - 2 * q^63 - 11 * q^67 + 6 * q^73 + 11 * q^75 + 15 * q^77 - 3 * q^79 + 3 * q^81 - 12 * q^83 + 19 * q^85 - 2 * q^87 + q^89 + 16 * q^93 + 21 * q^95 - 5 * q^97 - 5 * q^99

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