LMFDB - Embedded newform 8112.2.a.cm.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:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1014)
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)

$f(q)$	$=$	$q+1.00000 q^{3} -4.29590 q^{5} -4.35690 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -4.29590 q^{5} -4.35690 q^{7} +1.00000 q^{9} +1.15883 q^{11} -4.29590 q^{15} +0.493959 q^{17} +1.78017 q^{19} -4.35690 q^{21} -3.38404 q^{23} +13.4547 q^{25} +1.00000 q^{27} +6.93900 q^{29} -2.22521 q^{31} +1.15883 q^{33} +18.7168 q^{35} +3.87800 q^{37} +6.31767 q^{41} -7.38404 q^{43} -4.29590 q^{45} -1.78017 q^{47} +11.9825 q^{49} +0.493959 q^{51} -2.51573 q^{53} -4.97823 q^{55} +1.78017 q^{57} +6.63102 q^{59} +10.4940 q^{61} -4.35690 q^{63} +4.09783 q^{67} -3.38404 q^{69} +9.38404 q^{71} -0.374354 q^{73} +13.4547 q^{75} -5.04892 q^{77} -2.65519 q^{79} +1.00000 q^{81} -14.2784 q^{83} -2.12200 q^{85} +6.93900 q^{87} -0.835790 q^{89} -2.22521 q^{93} -7.64742 q^{95} -18.3937 q^{97} +1.15883 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + q^{5} - 9 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + q^5 - 9 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} + q^{5} - 9 q^{7} + 3 q^{9} - 5 q^{11} + q^{15} - 8 q^{17} + 4 q^{19} - 9 q^{21} + 18 q^{25} + 3 q^{27} + 11 q^{29} - 5 q^{31} - 5 q^{33} + 4 q^{35} - 8 q^{37} + 2 q^{41} - 12 q^{43} + q^{45} - 4 q^{47} + 20 q^{49} - 8 q^{51} + 5 q^{53} - 18 q^{55} + 4 q^{57} + 5 q^{59} + 22 q^{61} - 9 q^{63} - 6 q^{67} + 18 q^{71} - 13 q^{73} + 18 q^{75} - 6 q^{77} - 31 q^{79} + 3 q^{81} - 13 q^{83} - 26 q^{85} + 11 q^{87} - 14 q^{89} - 5 q^{93} - 8 q^{95} - 23 q^{97} - 5 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + q^5 - 9 * q^7 + 3 * q^9 - 5 * q^11 + q^15 - 8 * q^17 + 4 * q^19 - 9 * q^21 + 18 * q^25 + 3 * q^27 + 11 * q^29 - 5 * q^31 - 5 * q^33 + 4 * q^35 - 8 * q^37 + 2 * q^41 - 12 * q^43 + q^45 - 4 * q^47 + 20 * q^49 - 8 * q^51 + 5 * q^53 - 18 * q^55 + 4 * q^57 + 5 * q^59 + 22 * q^61 - 9 * q^63 - 6 * q^67 + 18 * q^71 - 13 * q^73 + 18 * q^75 - 6 * q^77 - 31 * q^79 + 3 * q^81 - 13 * q^83 - 26 * q^85 + 11 * q^87 - 14 * q^89 - 5 * q^93 - 8 * q^95 - 23 * 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
$5$	−4.29590	−1.92118	−0.960592	−	0.277963i	$-0.910341\pi$
$5$	−4.29590	−1.92118	−0.960592	+	0.277963i	$0.910341\pi$
$6$	0	0
$7$	−4.35690	−1.64675	−0.823376	−	0.567496i	$-0.807912\pi$
$7$	−4.35690	−1.64675	−0.823376	+	0.567496i	$0.807912\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.15883	0.349401	0.174701	−	0.984622i	$-0.444104\pi$
$11$	1.15883	0.349401	0.174701	+	0.984622i	$0.444104\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−4.29590	−1.10920
$16$	0	0
$17$	0.493959	0.119803	0.0599014	−	0.998204i	$-0.480921\pi$
$17$	0.493959	0.119803	0.0599014	+	0.998204i	$0.480921\pi$
$18$	0	0
$19$	1.78017	0.408398	0.204199	−	0.978929i	$-0.434541\pi$
$19$	1.78017	0.408398	0.204199	+	0.978929i	$0.434541\pi$
$20$	0	0
$21$	−4.35690	−0.950753
$22$	0	0
$23$	−3.38404	−0.705622	−0.352811	−	0.935695i	$-0.614774\pi$
$23$	−3.38404	−0.705622	−0.352811	+	0.935695i	$0.614774\pi$
$24$	0	0
$25$	13.4547	2.69095
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.93900	1.28854	0.644270	−	0.764798i	$-0.277161\pi$
$29$	6.93900	1.28854	0.644270	+	0.764798i	$0.277161\pi$
$30$	0	0
$31$	−2.22521	−0.399659	−0.199830	−	0.979831i	$-0.564039\pi$
$31$	−2.22521	−0.399659	−0.199830	+	0.979831i	$0.564039\pi$
$32$	0	0
$33$	1.15883	0.201727
$34$	0	0
$35$	18.7168	3.16371
$36$	0	0
$37$	3.87800	0.637540	0.318770	−	0.947832i	$-0.396730\pi$
$37$	3.87800	0.637540	0.318770	+	0.947832i	$0.396730\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.31767	0.986654	0.493327	−	0.869844i	$-0.335781\pi$
$41$	6.31767	0.986654	0.493327	+	0.869844i	$0.335781\pi$
$42$	0	0
$43$	−7.38404	−1.12606	−0.563028	−	0.826438i	$-0.690364\pi$
$43$	−7.38404	−1.12606	−0.563028	+	0.826438i	$0.690364\pi$
$44$	0	0
$45$	−4.29590	−0.640395
$46$	0	0
$47$	−1.78017	−0.259664	−0.129832	−	0.991536i	$-0.541444\pi$
$47$	−1.78017	−0.259664	−0.129832	+	0.991536i	$0.541444\pi$
$48$	0	0
$49$	11.9825	1.71179
$50$	0	0
$51$	0.493959	0.0691681
$52$	0	0
$53$	−2.51573	−0.345562	−0.172781	−	0.984960i	$-0.555275\pi$
$53$	−2.51573	−0.345562	−0.172781	+	0.984960i	$0.555275\pi$
$54$	0	0
$55$	−4.97823	−0.671264
$56$	0	0
$57$	1.78017	0.235789
$58$	0	0
$59$	6.63102	0.863286	0.431643	−	0.902045i	$-0.357934\pi$
$59$	6.63102	0.863286	0.431643	+	0.902045i	$0.357934\pi$
$60$	0	0
$61$	10.4940	1.34361	0.671807	−	0.740726i	$-0.265518\pi$
$61$	10.4940	1.34361	0.671807	+	0.740726i	$0.265518\pi$
$62$	0	0
$63$	−4.35690	−0.548917
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.09783	0.500630	0.250315	−	0.968164i	$-0.419466\pi$
$67$	4.09783	0.500630	0.250315	+	0.968164i	$0.419466\pi$
$68$	0	0
$69$	−3.38404	−0.407391
$70$	0	0
$71$	9.38404	1.11368	0.556841	−	0.830619i	$-0.312013\pi$
$71$	9.38404	1.11368	0.556841	+	0.830619i	$0.312013\pi$
$72$	0	0
$73$	−0.374354	−0.0438149	−0.0219074	−	0.999760i	$-0.506974\pi$
$73$	−0.374354	−0.0438149	−0.0219074	+	0.999760i	$0.506974\pi$
$74$	0	0
$75$	13.4547	1.55362
$76$	0	0
$77$	−5.04892	−0.575378
$78$	0	0
$79$	−2.65519	−0.298732	−0.149366	−	0.988782i	$-0.547723\pi$
$79$	−2.65519	−0.298732	−0.149366	+	0.988782i	$0.547723\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.2784	−1.56726	−0.783631	−	0.621226i	$-0.786635\pi$
$83$	−14.2784	−1.56726	−0.783631	+	0.621226i	$0.786635\pi$
$84$	0	0
$85$	−2.12200	−0.230163
$86$	0	0
$87$	6.93900	0.743939
$88$	0	0
$89$	−0.835790	−0.0885935	−0.0442968	−	0.999018i	$-0.514105\pi$
$89$	−0.835790	−0.0885935	−0.0442968	+	0.999018i	$0.514105\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.22521	−0.230743
$94$	0	0
$95$	−7.64742	−0.784608
$96$	0	0
$97$	−18.3937	−1.86760	−0.933800	−	0.357795i	$-0.883529\pi$
$97$	−18.3937	−1.86760	−0.933800	+	0.357795i	$0.883529\pi$
$98$	0	0
$99$	1.15883	0.116467
$100$	0	0
$101$	−5.46681	−0.543968	−0.271984	−	0.962302i	$-0.587680\pi$
$101$	−5.46681	−0.543968	−0.271984	+	0.962302i	$0.587680\pi$
$102$	0	0
$103$	−7.00969	−0.690685	−0.345343	−	0.938477i	$-0.612237\pi$
$103$	−7.00969	−0.690685	−0.345343	+	0.938477i	$0.612237\pi$
$104$	0	0
$105$	18.7168	1.82657
$106$	0	0
$107$	1.91723	0.185346	0.0926728	−	0.995697i	$-0.470459\pi$
$107$	1.91723	0.185346	0.0926728	+	0.995697i	$0.470459\pi$
$108$	0	0
$109$	−18.7681	−1.79766	−0.898828	−	0.438301i	$-0.855580\pi$
$109$	−18.7681	−1.79766	−0.898828	+	0.438301i	$0.855580\pi$
$110$	0	0
$111$	3.87800	0.368084
$112$	0	0
$113$	10.4155	0.979808	0.489904	−	0.871776i	$-0.337032\pi$
$113$	10.4155	0.979808	0.489904	+	0.871776i	$0.337032\pi$
$114$	0	0
$115$	14.5375	1.35563
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.15213	−0.197285
$120$	0	0
$121$	−9.65710	−0.877919
$122$	0	0
$123$	6.31767	0.569645
$124$	0	0
$125$	−36.3207	−3.24862
$126$	0	0
$127$	−8.39373	−0.744823	−0.372412	−	0.928068i	$-0.621469\pi$
$127$	−8.39373	−0.744823	−0.372412	+	0.928068i	$0.621469\pi$
$128$	0	0
$129$	−7.38404	−0.650129
$130$	0	0
$131$	13.5036	1.17982	0.589910	−	0.807469i	$-0.299163\pi$
$131$	13.5036	1.17982	0.589910	+	0.807469i	$0.299163\pi$
$132$	0	0
$133$	−7.75600	−0.672531
$134$	0	0
$135$	−4.29590	−0.369732
$136$	0	0
$137$	−4.00000	−0.341743	−0.170872	−	0.985293i	$-0.554658\pi$
$137$	−4.00000	−0.341743	−0.170872	+	0.985293i	$0.554658\pi$
$138$	0	0
$139$	15.0315	1.27495	0.637476	−	0.770470i	$-0.279979\pi$
$139$	15.0315	1.27495	0.637476	+	0.770470i	$0.279979\pi$
$140$	0	0
$141$	−1.78017	−0.149917
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−29.8092	−2.47552
$146$	0	0
$147$	11.9825	0.988303
$148$	0	0
$149$	3.92692	0.321706	0.160853	−	0.986978i	$-0.448576\pi$
$149$	3.92692	0.321706	0.160853	+	0.986978i	$0.448576\pi$
$150$	0	0
$151$	9.62863	0.783567	0.391783	−	0.920057i	$-0.371858\pi$
$151$	9.62863	0.783567	0.391783	+	0.920057i	$0.371858\pi$
$152$	0	0
$153$	0.493959	0.0399342
$154$	0	0
$155$	9.55927	0.767819
$156$	0	0
$157$	−17.3840	−1.38740	−0.693699	−	0.720265i	$-0.744020\pi$
$157$	−17.3840	−1.38740	−0.693699	+	0.720265i	$0.744020\pi$
$158$	0	0
$159$	−2.51573	−0.199510
$160$	0	0
$161$	14.7439	1.16198
$162$	0	0
$163$	−22.4698	−1.75997	−0.879985	−	0.475001i	$-0.842448\pi$
$163$	−22.4698	−1.75997	−0.879985	+	0.475001i	$0.842448\pi$
$164$	0	0
$165$	−4.97823	−0.387555
$166$	0	0
$167$	−0.835790	−0.0646753	−0.0323377	−	0.999477i	$-0.510295\pi$
$167$	−0.835790	−0.0646753	−0.0323377	+	0.999477i	$0.510295\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1.78017	0.136133
$172$	0	0
$173$	−4.17092	−0.317109	−0.158554	−	0.987350i	$-0.550683\pi$
$173$	−4.17092	−0.317109	−0.158554	+	0.987350i	$0.550683\pi$
$174$	0	0
$175$	−58.6209	−4.43132
$176$	0	0
$177$	6.63102	0.498418
$178$	0	0
$179$	−22.8877	−1.71071	−0.855353	−	0.518045i	$-0.826660\pi$
$179$	−22.8877	−1.71071	−0.855353	+	0.518045i	$0.826660\pi$
$180$	0	0
$181$	11.6039	0.862509	0.431255	−	0.902230i	$-0.358071\pi$
$181$	11.6039	0.862509	0.431255	+	0.902230i	$0.358071\pi$
$182$	0	0
$183$	10.4940	0.775736
$184$	0	0
$185$	−16.6595	−1.22483
$186$	0	0
$187$	0.572417	0.0418592
$188$	0	0
$189$	−4.35690	−0.316918
$190$	0	0
$191$	−16.6896	−1.20762	−0.603810	−	0.797129i	$-0.706351\pi$
$191$	−16.6896	−1.20762	−0.603810	+	0.797129i	$0.706351\pi$
$192$	0	0
$193$	9.64742	0.694436	0.347218	−	0.937784i	$-0.387126\pi$
$193$	9.64742	0.694436	0.347218	+	0.937784i	$0.387126\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.6383	1.39917	0.699586	−	0.714548i	$-0.253368\pi$
$197$	19.6383	1.39917	0.699586	+	0.714548i	$0.253368\pi$
$198$	0	0
$199$	18.4209	1.30582	0.652911	−	0.757435i	$-0.273548\pi$
$199$	18.4209	1.30582	0.652911	+	0.757435i	$0.273548\pi$
$200$	0	0
$201$	4.09783	0.289039
$202$	0	0
$203$	−30.2325	−2.12191
$204$	0	0
$205$	−27.1400	−1.89554
$206$	0	0
$207$	−3.38404	−0.235207
$208$	0	0
$209$	2.06292	0.142695
$210$	0	0
$211$	−15.9215	−1.09608	−0.548042	−	0.836451i	$-0.684627\pi$
$211$	−15.9215	−1.09608	−0.548042	+	0.836451i	$0.684627\pi$
$212$	0	0
$213$	9.38404	0.642984
$214$	0	0
$215$	31.7211	2.16336
$216$	0	0
$217$	9.69501	0.658140
$218$	0	0
$219$	−0.374354	−0.0252965
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5.18359	0.347119	0.173559	−	0.984823i	$-0.444473\pi$
$223$	5.18359	0.347119	0.173559	+	0.984823i	$0.444473\pi$
$224$	0	0
$225$	13.4547	0.896982
$226$	0	0
$227$	−14.8877	−0.988131	−0.494065	−	0.869425i	$-0.664490\pi$
$227$	−14.8877	−0.988131	−0.494065	+	0.869425i	$0.664490\pi$
$228$	0	0
$229$	23.4577	1.55013	0.775065	−	0.631882i	$-0.217717\pi$
$229$	23.4577	1.55013	0.775065	+	0.631882i	$0.217717\pi$
$230$	0	0
$231$	−5.04892	−0.332194
$232$	0	0
$233$	−11.8780	−0.778154	−0.389077	−	0.921205i	$-0.627206\pi$
$233$	−11.8780	−0.778154	−0.389077	+	0.921205i	$0.627206\pi$
$234$	0	0
$235$	7.64742	0.498862
$236$	0	0
$237$	−2.65519	−0.172473
$238$	0	0
$239$	25.3599	1.64039	0.820197	−	0.572081i	$-0.193864\pi$
$239$	25.3599	1.64039	0.820197	+	0.572081i	$0.193864\pi$
$240$	0	0
$241$	12.0218	0.774390	0.387195	−	0.921998i	$-0.373444\pi$
$241$	12.0218	0.774390	0.387195	+	0.921998i	$0.373444\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−51.4758	−3.28867
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−14.2784	−0.904859
$250$	0	0
$251$	15.3448	0.968556	0.484278	−	0.874914i	$-0.339082\pi$
$251$	15.3448	0.968556	0.484278	+	0.874914i	$0.339082\pi$
$252$	0	0
$253$	−3.92154	−0.246545
$254$	0	0
$255$	−2.12200	−0.132885
$256$	0	0
$257$	−7.50604	−0.468214	−0.234107	−	0.972211i	$-0.575217\pi$
$257$	−7.50604	−0.468214	−0.234107	+	0.972211i	$0.575217\pi$
$258$	0	0
$259$	−16.8961	−1.04987
$260$	0	0
$261$	6.93900	0.429513
$262$	0	0
$263$	12.2306	0.754170	0.377085	−	0.926179i	$-0.376926\pi$
$263$	12.2306	0.754170	0.377085	+	0.926179i	$0.376926\pi$
$264$	0	0
$265$	10.8073	0.663888
$266$	0	0
$267$	−0.835790	−0.0511495
$268$	0	0
$269$	2.71618	0.165609	0.0828044	−	0.996566i	$-0.473612\pi$
$269$	2.71618	0.165609	0.0828044	+	0.996566i	$0.473612\pi$
$270$	0	0
$271$	−12.3937	−0.752866	−0.376433	−	0.926444i	$-0.622849\pi$
$271$	−12.3937	−0.752866	−0.376433	+	0.926444i	$0.622849\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	15.5918	0.940221
$276$	0	0
$277$	−24.1280	−1.44971	−0.724854	−	0.688902i	$-0.758093\pi$
$277$	−24.1280	−1.44971	−0.724854	+	0.688902i	$0.758093\pi$
$278$	0	0
$279$	−2.22521	−0.133220
$280$	0	0
$281$	12.8358	0.765719	0.382860	−	0.923807i	$-0.374939\pi$
$281$	12.8358	0.765719	0.382860	+	0.923807i	$0.374939\pi$
$282$	0	0
$283$	−5.64742	−0.335704	−0.167852	−	0.985812i	$-0.553683\pi$
$283$	−5.64742	−0.335704	−0.167852	+	0.985812i	$0.553683\pi$
$284$	0	0
$285$	−7.64742	−0.452994
$286$	0	0
$287$	−27.5254	−1.62477
$288$	0	0
$289$	−16.7560	−0.985647
$290$	0	0
$291$	−18.3937	−1.07826
$292$	0	0
$293$	21.3653	1.24817	0.624086	−	0.781356i	$-0.285472\pi$
$293$	21.3653	1.24817	0.624086	+	0.781356i	$0.285472\pi$
$294$	0	0
$295$	−28.4862	−1.65853
$296$	0	0
$297$	1.15883	0.0672423
$298$	0	0
$299$	0	0
$300$	0	0
$301$	32.1715	1.85433
$302$	0	0
$303$	−5.46681	−0.314060
$304$	0	0
$305$	−45.0810	−2.58133
$306$	0	0
$307$	28.8853	1.64857	0.824286	−	0.566174i	$-0.191577\pi$
$307$	28.8853	1.64857	0.824286	+	0.566174i	$0.191577\pi$
$308$	0	0
$309$	−7.00969	−0.398767
$310$	0	0
$311$	−3.38404	−0.191891	−0.0959457	−	0.995387i	$-0.530588\pi$
$311$	−3.38404	−0.191891	−0.0959457	+	0.995387i	$0.530588\pi$
$312$	0	0
$313$	14.3502	0.811121	0.405560	−	0.914068i	$-0.367076\pi$
$313$	14.3502	0.811121	0.405560	+	0.914068i	$0.367076\pi$
$314$	0	0
$315$	18.7168	1.05457
$316$	0	0
$317$	−18.5332	−1.04093	−0.520464	−	0.853884i	$-0.674241\pi$
$317$	−18.5332	−1.04093	−0.520464	+	0.853884i	$0.674241\pi$
$318$	0	0
$319$	8.04115	0.450218
$320$	0	0
$321$	1.91723	0.107009
$322$	0	0
$323$	0.879330	0.0489272
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−18.7681	−1.03788
$328$	0	0
$329$	7.75600	0.427602
$330$	0	0
$331$	−7.87800	−0.433014	−0.216507	−	0.976281i	$-0.569466\pi$
$331$	−7.87800	−0.433014	−0.216507	+	0.976281i	$0.569466\pi$
$332$	0	0
$333$	3.87800	0.212513
$334$	0	0
$335$	−17.6039	−0.961802
$336$	0	0
$337$	−13.7265	−0.747728	−0.373864	−	0.927484i	$-0.621967\pi$
$337$	−13.7265	−0.747728	−0.373864	+	0.927484i	$0.621967\pi$
$338$	0	0
$339$	10.4155	0.565692
$340$	0	0
$341$	−2.57865	−0.139642
$342$	0	0
$343$	−21.7084	−1.17214
$344$	0	0
$345$	14.5375	0.782673
$346$	0	0
$347$	4.85086	0.260408	0.130204	−	0.991487i	$-0.458437\pi$
$347$	4.85086	0.260408	0.130204	+	0.991487i	$0.458437\pi$
$348$	0	0
$349$	−28.3478	−1.51742	−0.758711	−	0.651427i	$-0.774171\pi$
$349$	−28.3478	−1.51742	−0.758711	+	0.651427i	$0.774171\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−28.4456	−1.51401	−0.757004	−	0.653410i	$-0.773338\pi$
$353$	−28.4456	−1.51401	−0.757004	+	0.653410i	$0.773338\pi$
$354$	0	0
$355$	−40.3129	−2.13959
$356$	0	0
$357$	−2.15213	−0.113903
$358$	0	0
$359$	−11.2271	−0.592545	−0.296273	−	0.955103i	$-0.595744\pi$
$359$	−11.2271	−0.592545	−0.296273	+	0.955103i	$0.595744\pi$
$360$	0	0
$361$	−15.8310	−0.833211
$362$	0	0
$363$	−9.65710	−0.506867
$364$	0	0
$365$	1.60819	0.0841764
$366$	0	0
$367$	−16.8267	−0.878346	−0.439173	−	0.898402i	$-0.644729\pi$
$367$	−16.8267	−0.878346	−0.439173	+	0.898402i	$0.644729\pi$
$368$	0	0
$369$	6.31767	0.328885
$370$	0	0
$371$	10.9608	0.569055
$372$	0	0
$373$	−26.5327	−1.37381	−0.686906	−	0.726746i	$-0.741032\pi$
$373$	−26.5327	−1.37381	−0.686906	+	0.726746i	$0.741032\pi$
$374$	0	0
$375$	−36.3207	−1.87559
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−21.9758	−1.12882	−0.564411	−	0.825494i	$-0.690897\pi$
$379$	−21.9758	−1.12882	−0.564411	+	0.825494i	$0.690897\pi$
$380$	0	0
$381$	−8.39373	−0.430024
$382$	0	0
$383$	0.615957	0.0314739	0.0157370	−	0.999876i	$-0.494991\pi$
$383$	0.615957	0.0314739	0.0157370	+	0.999876i	$0.494991\pi$
$384$	0	0
$385$	21.6896	1.10541
$386$	0	0
$387$	−7.38404	−0.375352
$388$	0	0
$389$	−35.5260	−1.80124	−0.900620	−	0.434607i	$-0.856887\pi$
$389$	−35.5260	−1.80124	−0.900620	+	0.434607i	$0.856887\pi$
$390$	0	0
$391$	−1.67158	−0.0845354
$392$	0	0
$393$	13.5036	0.681169
$394$	0	0
$395$	11.4064	0.573918
$396$	0	0
$397$	−16.0785	−0.806955	−0.403477	−	0.914990i	$-0.632199\pi$
$397$	−16.0785	−0.806955	−0.403477	+	0.914990i	$0.632199\pi$
$398$	0	0
$399$	−7.75600	−0.388286
$400$	0	0
$401$	22.9095	1.14404	0.572022	−	0.820238i	$-0.306159\pi$
$401$	22.9095	1.14404	0.572022	+	0.820238i	$0.306159\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−4.29590	−0.213465
$406$	0	0
$407$	4.49396	0.222757
$408$	0	0
$409$	−1.24698	−0.0616592	−0.0308296	−	0.999525i	$-0.509815\pi$
$409$	−1.24698	−0.0616592	−0.0308296	+	0.999525i	$0.509815\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	0	0
$413$	−28.8907	−1.42162
$414$	0	0
$415$	61.3387	3.01100
$416$	0	0
$417$	15.0315	0.736094
$418$	0	0
$419$	−29.4631	−1.43937	−0.719683	−	0.694303i	$-0.755713\pi$
$419$	−29.4631	−1.43937	−0.719683	+	0.694303i	$0.755713\pi$
$420$	0	0
$421$	−17.3491	−0.845545	−0.422772	−	0.906236i	$-0.638943\pi$
$421$	−17.3491	−0.845545	−0.422772	+	0.906236i	$0.638943\pi$
$422$	0	0
$423$	−1.78017	−0.0865547
$424$	0	0
$425$	6.64609	0.322383
$426$	0	0
$427$	−45.7211	−2.21260
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.3840	−1.22271	−0.611353	−	0.791358i	$-0.709374\pi$
$431$	−25.3840	−1.22271	−0.611353	+	0.791358i	$0.709374\pi$
$432$	0	0
$433$	20.0054	0.961397	0.480699	−	0.876886i	$-0.340383\pi$
$433$	20.0054	0.961397	0.480699	+	0.876886i	$0.340383\pi$
$434$	0	0
$435$	−29.8092	−1.42924
$436$	0	0
$437$	−6.02416	−0.288175
$438$	0	0
$439$	−0.543941	−0.0259609	−0.0129805	−	0.999916i	$-0.504132\pi$
$439$	−0.543941	−0.0259609	−0.0129805	+	0.999916i	$0.504132\pi$
$440$	0	0
$441$	11.9825	0.570597
$442$	0	0
$443$	18.1360	0.861667	0.430834	−	0.902431i	$-0.358220\pi$
$443$	18.1360	0.861667	0.430834	+	0.902431i	$0.358220\pi$
$444$	0	0
$445$	3.59047	0.170204
$446$	0	0
$447$	3.92692	0.185737
$448$	0	0
$449$	29.3793	1.38649	0.693246	−	0.720701i	$-0.256180\pi$
$449$	29.3793	1.38649	0.693246	+	0.720701i	$0.256180\pi$
$450$	0	0
$451$	7.32113	0.344738
$452$	0	0
$453$	9.62863	0.452392
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.34721	−0.390466	−0.195233	−	0.980757i	$-0.562546\pi$
$457$	−8.34721	−0.390466	−0.195233	+	0.980757i	$0.562546\pi$
$458$	0	0
$459$	0.493959	0.0230560
$460$	0	0
$461$	21.4969	1.00121	0.500606	−	0.865675i	$-0.333110\pi$
$461$	21.4969	1.00121	0.500606	+	0.865675i	$0.333110\pi$
$462$	0	0
$463$	−26.1715	−1.21629	−0.608147	−	0.793825i	$-0.708087\pi$
$463$	−26.1715	−1.21629	−0.608147	+	0.793825i	$0.708087\pi$
$464$	0	0
$465$	9.55927	0.443301
$466$	0	0
$467$	9.81269	0.454077	0.227039	−	0.973886i	$-0.427096\pi$
$467$	9.81269	0.454077	0.227039	+	0.973886i	$0.427096\pi$
$468$	0	0
$469$	−17.8538	−0.824414
$470$	0	0
$471$	−17.3840	−0.801014
$472$	0	0
$473$	−8.55688	−0.393446
$474$	0	0
$475$	23.9517	1.09898
$476$	0	0
$477$	−2.51573	−0.115187
$478$	0	0
$479$	−17.2814	−0.789608	−0.394804	−	0.918765i	$-0.629188\pi$
$479$	−17.2814	−0.789608	−0.394804	+	0.918765i	$0.629188\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	14.7439	0.670872
$484$	0	0
$485$	79.0176	3.58800
$486$	0	0
$487$	31.6886	1.43595	0.717973	−	0.696071i	$-0.245070\pi$
$487$	31.6886	1.43595	0.717973	+	0.696071i	$0.245070\pi$
$488$	0	0
$489$	−22.4698	−1.01612
$490$	0	0
$491$	−2.30367	−0.103963	−0.0519815	−	0.998648i	$-0.516554\pi$
$491$	−2.30367	−0.103963	−0.0519815	+	0.998648i	$0.516554\pi$
$492$	0	0
$493$	3.42758	0.154371
$494$	0	0
$495$	−4.97823	−0.223755
$496$	0	0
$497$	−40.8853	−1.83396
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	−0.835790	−0.0373403
$502$	0	0
$503$	35.3685	1.57700	0.788502	−	0.615032i	$-0.210857\pi$
$503$	35.3685	1.57700	0.788502	+	0.615032i	$0.210857\pi$
$504$	0	0
$505$	23.4849	1.04506
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.6112	1.26817	0.634084	−	0.773264i	$-0.281377\pi$
$509$	28.6112	1.26817	0.634084	+	0.773264i	$0.281377\pi$
$510$	0	0
$511$	1.63102	0.0721522
$512$	0	0
$513$	1.78017	0.0785963
$514$	0	0
$515$	30.1129	1.32693
$516$	0	0
$517$	−2.06292	−0.0907270
$518$	0	0
$519$	−4.17092	−0.183083
$520$	0	0
$521$	−17.1594	−0.751768	−0.375884	−	0.926667i	$-0.622661\pi$
$521$	−17.1594	−0.751768	−0.375884	+	0.926667i	$0.622661\pi$
$522$	0	0
$523$	−16.5133	−0.722078	−0.361039	−	0.932551i	$-0.617578\pi$
$523$	−16.5133	−0.722078	−0.361039	+	0.932551i	$0.617578\pi$
$524$	0	0
$525$	−58.6209	−2.55842
$526$	0	0
$527$	−1.09916	−0.0478803
$528$	0	0
$529$	−11.5483	−0.502098
$530$	0	0
$531$	6.63102	0.287762
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−8.23623	−0.356083
$536$	0	0
$537$	−22.8877	−0.987677
$538$	0	0
$539$	13.8858	0.598103
$540$	0	0
$541$	−25.0858	−1.07852	−0.539260	−	0.842139i	$-0.681296\pi$
$541$	−25.0858	−1.07852	−0.539260	+	0.842139i	$0.681296\pi$
$542$	0	0
$543$	11.6039	0.497970
$544$	0	0
$545$	80.6258	3.45363
$546$	0	0
$547$	−31.1594	−1.33228	−0.666140	−	0.745826i	$-0.732055\pi$
$547$	−31.1594	−1.33228	−0.666140	+	0.745826i	$0.732055\pi$
$548$	0	0
$549$	10.4940	0.447871
$550$	0	0
$551$	12.3526	0.526238
$552$	0	0
$553$	11.5684	0.491937
$554$	0	0
$555$	−16.6595	−0.707156
$556$	0	0
$557$	3.97584	0.168462	0.0842308	−	0.996446i	$-0.473157\pi$
$557$	3.97584	0.168462	0.0842308	+	0.996446i	$0.473157\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.572417	0.0241674
$562$	0	0
$563$	−0.602811	−0.0254054	−0.0127027	−	0.999919i	$-0.504044\pi$
$563$	−0.602811	−0.0254054	−0.0127027	+	0.999919i	$0.504044\pi$
$564$	0	0
$565$	−44.7439	−1.88239
$566$	0	0
$567$	−4.35690	−0.182972
$568$	0	0
$569$	2.21983	0.0930602	0.0465301	−	0.998917i	$-0.485184\pi$
$569$	2.21983	0.0930602	0.0465301	+	0.998917i	$0.485184\pi$
$570$	0	0
$571$	−24.4155	−1.02176	−0.510878	−	0.859653i	$-0.670680\pi$
$571$	−24.4155	−1.02176	−0.510878	+	0.859653i	$0.670680\pi$
$572$	0	0
$573$	−16.6896	−0.697219
$574$	0	0
$575$	−45.5314	−1.89879
$576$	0	0
$577$	−20.7375	−0.863313	−0.431656	−	0.902038i	$-0.642071\pi$
$577$	−20.7375	−0.863313	−0.431656	+	0.902038i	$0.642071\pi$
$578$	0	0
$579$	9.64742	0.400933
$580$	0	0
$581$	62.2097	2.58089
$582$	0	0
$583$	−2.91531	−0.120740
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.3679	0.469204	0.234602	−	0.972092i	$-0.424621\pi$
$587$	11.3679	0.469204	0.234602	+	0.972092i	$0.424621\pi$
$588$	0	0
$589$	−3.96125	−0.163220
$590$	0	0
$591$	19.6383	0.807812
$592$	0	0
$593$	36.7198	1.50790	0.753950	−	0.656932i	$-0.228146\pi$
$593$	36.7198	1.50790	0.753950	+	0.656932i	$0.228146\pi$
$594$	0	0
$595$	9.24532	0.379021
$596$	0	0
$597$	18.4209	0.753916
$598$	0	0
$599$	47.4965	1.94065	0.970327	−	0.241798i	$-0.0777372\pi$
$599$	47.4965	1.94065	0.970327	+	0.241798i	$0.0777372\pi$
$600$	0	0
$601$	7.75063	0.316155	0.158077	−	0.987427i	$-0.449470\pi$
$601$	7.75063	0.316155	0.158077	+	0.987427i	$0.449470\pi$
$602$	0	0
$603$	4.09783	0.166877
$604$	0	0
$605$	41.4859	1.68664
$606$	0	0
$607$	−3.64310	−0.147869	−0.0739345	−	0.997263i	$-0.523556\pi$
$607$	−3.64310	−0.147869	−0.0739345	+	0.997263i	$0.523556\pi$
$608$	0	0
$609$	−30.2325	−1.22508
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−30.4263	−1.22890	−0.614452	−	0.788954i	$-0.710623\pi$
$613$	−30.4263	−1.22890	−0.614452	+	0.788954i	$0.710623\pi$
$614$	0	0
$615$	−27.1400	−1.09439
$616$	0	0
$617$	8.16554	0.328732	0.164366	−	0.986399i	$-0.447442\pi$
$617$	8.16554	0.328732	0.164366	+	0.986399i	$0.447442\pi$
$618$	0	0
$619$	−7.95646	−0.319797	−0.159899	−	0.987133i	$-0.551117\pi$
$619$	−7.95646	−0.319797	−0.159899	+	0.987133i	$0.551117\pi$
$620$	0	0
$621$	−3.38404	−0.135797
$622$	0	0
$623$	3.64145	0.145892
$624$	0	0
$625$	88.7561	3.55024
$626$	0	0
$627$	2.06292	0.0823850
$628$	0	0
$629$	1.91557	0.0763790
$630$	0	0
$631$	15.2336	0.606439	0.303219	−	0.952921i	$-0.401938\pi$
$631$	15.2336	0.606439	0.303219	+	0.952921i	$0.401938\pi$
$632$	0	0
$633$	−15.9215	−0.632825
$634$	0	0
$635$	36.0586	1.43094
$636$	0	0
$637$	0	0
$638$	0	0
$639$	9.38404	0.371227
$640$	0	0
$641$	−14.1909	−0.560506	−0.280253	−	0.959926i	$-0.590418\pi$
$641$	−14.1909	−0.560506	−0.280253	+	0.959926i	$0.590418\pi$
$642$	0	0
$643$	22.5569	0.889556	0.444778	−	0.895641i	$-0.353283\pi$
$643$	22.5569	0.889556	0.444778	+	0.895641i	$0.353283\pi$
$644$	0	0
$645$	31.7211	1.24902
$646$	0	0
$647$	36.9047	1.45087	0.725436	−	0.688289i	$-0.241638\pi$
$647$	36.9047	1.45087	0.725436	+	0.688289i	$0.241638\pi$
$648$	0	0
$649$	7.68425	0.301633
$650$	0	0
$651$	9.69501	0.379977
$652$	0	0
$653$	−1.84953	−0.0723776	−0.0361888	−	0.999345i	$-0.511522\pi$
$653$	−1.84953	−0.0723776	−0.0361888	+	0.999345i	$0.511522\pi$
$654$	0	0
$655$	−58.0103	−2.26665
$656$	0	0
$657$	−0.374354	−0.0146050
$658$	0	0
$659$	−18.2784	−0.712027	−0.356013	−	0.934481i	$-0.615864\pi$
$659$	−18.2784	−0.712027	−0.356013	+	0.934481i	$0.615864\pi$
$660$	0	0
$661$	−3.63401	−0.141346	−0.0706732	−	0.997500i	$-0.522515\pi$
$661$	−3.63401	−0.141346	−0.0706732	+	0.997500i	$0.522515\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	33.3190	1.29206
$666$	0	0
$667$	−23.4819	−0.909222
$668$	0	0
$669$	5.18359	0.200409
$670$	0	0
$671$	12.1608	0.469461
$672$	0	0
$673$	−6.14782	−0.236981	−0.118490	−	0.992955i	$-0.537806\pi$
$673$	−6.14782	−0.236981	−0.118490	+	0.992955i	$0.537806\pi$
$674$	0	0
$675$	13.4547	0.517873
$676$	0	0
$677$	−18.2892	−0.702911	−0.351455	−	0.936205i	$-0.614313\pi$
$677$	−18.2892	−0.702911	−0.351455	+	0.936205i	$0.614313\pi$
$678$	0	0
$679$	80.1396	3.07547
$680$	0	0
$681$	−14.8877	−0.570498
$682$	0	0
$683$	3.59286	0.137477	0.0687385	−	0.997635i	$-0.478103\pi$
$683$	3.59286	0.137477	0.0687385	+	0.997635i	$0.478103\pi$
$684$	0	0
$685$	17.1836	0.656551
$686$	0	0
$687$	23.4577	0.894968
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−19.8189	−0.753947	−0.376974	−	0.926224i	$-0.623035\pi$
$691$	−19.8189	−0.753947	−0.376974	+	0.926224i	$0.623035\pi$
$692$	0	0
$693$	−5.04892	−0.191793
$694$	0	0
$695$	−64.5736	−2.44942
$696$	0	0
$697$	3.12067	0.118204
$698$	0	0
$699$	−11.8780	−0.449267
$700$	0	0
$701$	5.25608	0.198519	0.0992596	−	0.995062i	$-0.468353\pi$
$701$	5.25608	0.198519	0.0992596	+	0.995062i	$0.468353\pi$
$702$	0	0
$703$	6.90349	0.260370
$704$	0	0
$705$	7.64742	0.288018
$706$	0	0
$707$	23.8183	0.895781
$708$	0	0
$709$	−26.0978	−0.980125	−0.490062	−	0.871687i	$-0.663026\pi$
$709$	−26.0978	−0.980125	−0.490062	+	0.871687i	$0.663026\pi$
$710$	0	0
$711$	−2.65519	−0.0995772
$712$	0	0
$713$	7.53020	0.282008
$714$	0	0
$715$	0	0
$716$	0	0
$717$	25.3599	0.947082
$718$	0	0
$719$	−50.8310	−1.89568	−0.947838	−	0.318752i	$-0.896736\pi$
$719$	−50.8310	−1.89568	−0.947838	+	0.318752i	$0.896736\pi$
$720$	0	0
$721$	30.5405	1.13739
$722$	0	0
$723$	12.0218	0.447094
$724$	0	0
$725$	93.3624	3.46739
$726$	0	0
$727$	20.5042	0.760460	0.380230	−	0.924892i	$-0.375845\pi$
$727$	20.5042	0.760460	0.380230	+	0.924892i	$0.375845\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.64742	−0.134905
$732$	0	0
$733$	−3.26205	−0.120486	−0.0602432	−	0.998184i	$-0.519188\pi$
$733$	−3.26205	−0.120486	−0.0602432	+	0.998184i	$0.519188\pi$
$734$	0	0
$735$	−51.4758	−1.89871
$736$	0	0
$737$	4.74871	0.174921
$738$	0	0
$739$	−32.8203	−1.20731	−0.603656	−	0.797245i	$-0.706290\pi$
$739$	−32.8203	−1.20731	−0.603656	+	0.797245i	$0.706290\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	34.9530	1.28230	0.641151	−	0.767415i	$-0.278457\pi$
$743$	34.9530	1.28230	0.641151	+	0.767415i	$0.278457\pi$
$744$	0	0
$745$	−16.8696	−0.618056
$746$	0	0
$747$	−14.2784	−0.522421
$748$	0	0
$749$	−8.35317	−0.305218
$750$	0	0
$751$	10.1661	0.370967	0.185484	−	0.982647i	$-0.440615\pi$
$751$	10.1661	0.370967	0.185484	+	0.982647i	$0.440615\pi$
$752$	0	0
$753$	15.3448	0.559196
$754$	0	0
$755$	−41.3636	−1.50538
$756$	0	0
$757$	−2.46383	−0.0895494	−0.0447747	−	0.998997i	$-0.514257\pi$
$757$	−2.46383	−0.0895494	−0.0447747	+	0.998997i	$0.514257\pi$
$758$	0	0
$759$	−3.92154	−0.142343
$760$	0	0
$761$	−42.8068	−1.55175	−0.775873	−	0.630889i	$-0.782690\pi$
$761$	−42.8068	−1.55175	−0.775873	+	0.630889i	$0.782690\pi$
$762$	0	0
$763$	81.7706	2.96029
$764$	0	0
$765$	−2.12200	−0.0767210
$766$	0	0
$767$	0	0
$768$	0	0
$769$	25.4034	0.916071	0.458035	−	0.888934i	$-0.348553\pi$
$769$	25.4034	0.916071	0.458035	+	0.888934i	$0.348553\pi$
$770$	0	0
$771$	−7.50604	−0.270323
$772$	0	0
$773$	4.47889	0.161095	0.0805473	−	0.996751i	$-0.474333\pi$
$773$	4.47889	0.161095	0.0805473	+	0.996751i	$0.474333\pi$
$774$	0	0
$775$	−29.9396	−1.07546
$776$	0	0
$777$	−16.8961	−0.606142
$778$	0	0
$779$	11.2465	0.402948
$780$	0	0
$781$	10.8745	0.389122
$782$	0	0
$783$	6.93900	0.247980
$784$	0	0
$785$	74.6801	2.66545
$786$	0	0
$787$	−9.43834	−0.336440	−0.168220	−	0.985749i	$-0.553802\pi$
$787$	−9.43834	−0.336440	−0.168220	+	0.985749i	$0.553802\pi$
$788$	0	0
$789$	12.2306	0.435420
$790$	0	0
$791$	−45.3793	−1.61350
$792$	0	0
$793$	0	0
$794$	0	0
$795$	10.8073	0.383296
$796$	0	0
$797$	−25.2459	−0.894256	−0.447128	−	0.894470i	$-0.647553\pi$
$797$	−25.2459	−0.894256	−0.447128	+	0.894470i	$0.647553\pi$
$798$	0	0
$799$	−0.879330	−0.0311085
$800$	0	0
$801$	−0.835790	−0.0295312
$802$	0	0
$803$	−0.433814	−0.0153090
$804$	0	0
$805$	−63.3384	−2.23238
$806$	0	0
$807$	2.71618	0.0956142
$808$	0	0
$809$	11.7065	0.411578	0.205789	−	0.978596i	$-0.434024\pi$
$809$	11.7065	0.411578	0.205789	+	0.978596i	$0.434024\pi$
$810$	0	0
$811$	−30.5628	−1.07321	−0.536603	−	0.843835i	$-0.680293\pi$
$811$	−30.5628	−1.07321	−0.536603	+	0.843835i	$0.680293\pi$
$812$	0	0
$813$	−12.3937	−0.434667
$814$	0	0
$815$	96.5279	3.38123
$816$	0	0
$817$	−13.1448	−0.459879
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22.5593	−0.787324	−0.393662	−	0.919255i	$-0.628792\pi$
$821$	−22.5593	−0.787324	−0.393662	+	0.919255i	$0.628792\pi$
$822$	0	0
$823$	14.5767	0.508113	0.254056	−	0.967189i	$-0.418235\pi$
$823$	14.5767	0.508113	0.254056	+	0.967189i	$0.418235\pi$
$824$	0	0
$825$	15.5918	0.542837
$826$	0	0
$827$	−20.3666	−0.708216	−0.354108	−	0.935205i	$-0.615215\pi$
$827$	−20.3666	−0.708216	−0.354108	+	0.935205i	$0.615215\pi$
$828$	0	0
$829$	32.4107	1.12567	0.562835	−	0.826569i	$-0.309711\pi$
$829$	32.4107	1.12567	0.562835	+	0.826569i	$0.309711\pi$
$830$	0	0
$831$	−24.1280	−0.836990
$832$	0	0
$833$	5.91889	0.205077
$834$	0	0
$835$	3.59047	0.124253
$836$	0	0
$837$	−2.22521	−0.0769145
$838$	0	0
$839$	29.9409	1.03368	0.516838	−	0.856083i	$-0.327109\pi$
$839$	29.9409	1.03368	0.516838	+	0.856083i	$0.327109\pi$
$840$	0	0
$841$	19.1497	0.660336
$842$	0	0
$843$	12.8358	0.442088
$844$	0	0
$845$	0	0
$846$	0	0
$847$	42.0750	1.44571
$848$	0	0
$849$	−5.64742	−0.193819
$850$	0	0
$851$	−13.1233	−0.449862
$852$	0	0
$853$	−41.8780	−1.43388	−0.716938	−	0.697137i	$-0.754457\pi$
$853$	−41.8780	−1.43388	−0.716938	+	0.697137i	$0.754457\pi$
$854$	0	0
$855$	−7.64742	−0.261536
$856$	0	0
$857$	25.2137	0.861284	0.430642	−	0.902523i	$-0.358287\pi$
$857$	25.2137	0.861284	0.430642	+	0.902523i	$0.358287\pi$
$858$	0	0
$859$	46.9939	1.60341	0.801705	−	0.597719i	$-0.203926\pi$
$859$	46.9939	1.60341	0.801705	+	0.597719i	$0.203926\pi$
$860$	0	0
$861$	−27.5254	−0.938064
$862$	0	0
$863$	43.6969	1.48746	0.743730	−	0.668480i	$-0.233055\pi$
$863$	43.6969	1.48746	0.743730	+	0.668480i	$0.233055\pi$
$864$	0	0
$865$	17.9178	0.609224
$866$	0	0
$867$	−16.7560	−0.569064
$868$	0	0
$869$	−3.07692	−0.104377
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−18.3937	−0.622533
$874$	0	0
$875$	158.245	5.34967
$876$	0	0
$877$	−12.8901	−0.435267	−0.217634	−	0.976031i	$-0.569834\pi$
$877$	−12.8901	−0.435267	−0.217634	+	0.976031i	$0.569834\pi$
$878$	0	0
$879$	21.3653	0.720632
$880$	0	0
$881$	−14.9578	−0.503941	−0.251970	−	0.967735i	$-0.581079\pi$
$881$	−14.9578	−0.503941	−0.251970	+	0.967735i	$0.581079\pi$
$882$	0	0
$883$	20.6896	0.696261	0.348131	−	0.937446i	$-0.386817\pi$
$883$	20.6896	0.696261	0.348131	+	0.937446i	$0.386817\pi$
$884$	0	0
$885$	−28.4862	−0.957553
$886$	0	0
$887$	−24.1849	−0.812050	−0.406025	−	0.913862i	$-0.633085\pi$
$887$	−24.1849	−0.812050	−0.406025	+	0.913862i	$0.633085\pi$
$888$	0	0
$889$	36.5706	1.22654
$890$	0	0
$891$	1.15883	0.0388224
$892$	0	0
$893$	−3.16900	−0.106046
$894$	0	0
$895$	98.3232	3.28658
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−15.4407	−0.514977
$900$	0	0
$901$	−1.24267	−0.0413993
$902$	0	0
$903$	32.1715	1.07060
$904$	0	0
$905$	−49.8491	−1.65704
$906$	0	0
$907$	−27.2707	−0.905508	−0.452754	−	0.891636i	$-0.649558\pi$
$907$	−27.2707	−0.905508	−0.452754	+	0.891636i	$0.649558\pi$
$908$	0	0
$909$	−5.46681	−0.181323
$910$	0	0
$911$	−45.2766	−1.50008	−0.750041	−	0.661391i	$-0.769966\pi$
$911$	−45.2766	−1.50008	−0.750041	+	0.661391i	$0.769966\pi$
$912$	0	0
$913$	−16.5463	−0.547604
$914$	0	0
$915$	−45.0810	−1.49033
$916$	0	0
$917$	−58.8340	−1.94287
$918$	0	0
$919$	−4.68127	−0.154421	−0.0772104	−	0.997015i	$-0.524601\pi$
$919$	−4.68127	−0.154421	−0.0772104	+	0.997015i	$0.524601\pi$
$920$	0	0
$921$	28.8853	0.951803
$922$	0	0
$923$	0	0
$924$	0	0
$925$	52.1775	1.71558
$926$	0	0
$927$	−7.00969	−0.230228
$928$	0	0
$929$	3.29099	0.107974	0.0539870	−	0.998542i	$-0.482807\pi$
$929$	3.29099	0.107974	0.0539870	+	0.998542i	$0.482807\pi$
$930$	0	0
$931$	21.3309	0.699093
$932$	0	0
$933$	−3.38404	−0.110789
$934$	0	0
$935$	−2.45904	−0.0804193
$936$	0	0
$937$	6.67563	0.218083	0.109042	−	0.994037i	$-0.465222\pi$
$937$	6.67563	0.218083	0.109042	+	0.994037i	$0.465222\pi$
$938$	0	0
$939$	14.3502	0.468301
$940$	0	0
$941$	38.9778	1.27064	0.635319	−	0.772250i	$-0.280869\pi$
$941$	38.9778	1.27064	0.635319	+	0.772250i	$0.280869\pi$
$942$	0	0
$943$	−21.3793	−0.696204
$944$	0	0
$945$	18.7168	0.608857
$946$	0	0
$947$	40.6679	1.32153	0.660764	−	0.750594i	$-0.270232\pi$
$947$	40.6679	1.32153	0.660764	+	0.750594i	$0.270232\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−18.5332	−0.600980
$952$	0	0
$953$	−44.8504	−1.45285	−0.726423	−	0.687248i	$-0.758819\pi$
$953$	−44.8504	−1.45285	−0.726423	+	0.687248i	$0.758819\pi$
$954$	0	0
$955$	71.6969	2.32006
$956$	0	0
$957$	8.04115	0.259933
$958$	0	0
$959$	17.4276	0.562766
$960$	0	0
$961$	−26.0484	−0.840272
$962$	0	0
$963$	1.91723	0.0617819
$964$	0	0
$965$	−41.4443	−1.33414
$966$	0	0
$967$	22.4704	0.722599	0.361299	−	0.932450i	$-0.382333\pi$
$967$	22.4704	0.722599	0.361299	+	0.932450i	$0.382333\pi$
$968$	0	0
$969$	0.879330	0.0282482
$970$	0	0
$971$	−33.8437	−1.08610	−0.543048	−	0.839702i	$-0.682730\pi$
$971$	−33.8437	−1.08610	−0.543048	+	0.839702i	$0.682730\pi$
$972$	0	0
$973$	−65.4905	−2.09953
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27.2137	−0.870644	−0.435322	−	0.900275i	$-0.643365\pi$
$977$	−27.2137	−0.870644	−0.435322	+	0.900275i	$0.643365\pi$
$978$	0	0
$979$	−0.968541	−0.0309547
$980$	0	0
$981$	−18.7681	−0.599219
$982$	0	0
$983$	−50.9965	−1.62654	−0.813269	−	0.581889i	$-0.802314\pi$
$983$	−50.9965	−1.62654	−0.813269	+	0.581889i	$0.802314\pi$
$984$	0	0
$985$	−84.3642	−2.68807
$986$	0	0
$987$	7.75600	0.246876
$988$	0	0
$989$	24.9879	0.794570
$990$	0	0
$991$	24.2804	0.771291	0.385645	−	0.922647i	$-0.373979\pi$
$991$	24.2804	0.771291	0.385645	+	0.922647i	$0.373979\pi$
$992$	0	0
$993$	−7.87800	−0.250001
$994$	0	0
$995$	−79.1342	−2.50872
$996$	0	0
$997$	−4.06505	−0.128741	−0.0643707	−	0.997926i	$-0.520504\pi$
$997$	−4.06505	−0.128741	−0.0643707	+	0.997926i	$0.520504\pi$
$998$	0	0
$999$	3.87800	0.122695

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.cm.1.1		3
4.3	odd	2		1014.2.a.n.1.1	yes	3
12.11	even	2		3042.2.a.ba.1.3		3
13.12	even	2		8112.2.a.cj.1.3		3
52.3	odd	6		1014.2.e.l.529.1		6
52.7	even	12		1014.2.i.h.361.4		12
52.11	even	12		1014.2.i.h.823.1		12
52.15	even	12		1014.2.i.h.823.6		12
52.19	even	12		1014.2.i.h.361.3		12
52.23	odd	6		1014.2.e.n.529.3		6
52.31	even	4		1014.2.b.f.337.3		6
52.35	odd	6		1014.2.e.l.991.1		6
52.43	odd	6		1014.2.e.n.991.3		6
52.47	even	4		1014.2.b.f.337.4		6
52.51	odd	2		1014.2.a.l.1.3	✓	3
156.47	odd	4		3042.2.b.o.1351.3		6
156.83	odd	4		3042.2.b.o.1351.4		6
156.155	even	2		3042.2.a.bh.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1014.2.a.l.1.3	✓	3	52.51	odd	2
1014.2.a.n.1.1	yes	3	4.3	odd	2
1014.2.b.f.337.3		6	52.31	even	4
1014.2.b.f.337.4		6	52.47	even	4
1014.2.e.l.529.1		6	52.3	odd	6
1014.2.e.l.991.1		6	52.35	odd	6
1014.2.e.n.529.3		6	52.23	odd	6
1014.2.e.n.991.3		6	52.43	odd	6
1014.2.i.h.361.3		12	52.19	even	12
1014.2.i.h.361.4		12	52.7	even	12
1014.2.i.h.823.1		12	52.11	even	12
1014.2.i.h.823.6		12	52.15	even	12
3042.2.a.ba.1.3		3	12.11	even	2
3042.2.a.bh.1.1		3	156.155	even	2
3042.2.b.o.1351.3		6	156.47	odd	4
3042.2.b.o.1351.4		6	156.83	odd	4
8112.2.a.cj.1.3		3	13.12	even	2
8112.2.a.cm.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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.29590 q^{5} -4.35690 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -4.29590 q^{5} -4.35690 q^{7} +1.00000 q^{9} +1.15883 q^{11} -4.29590 q^{15} +0.493959 q^{17} +1.78017 q^{19} -4.35690 q^{21} -3.38404 q^{23} +13.4547 q^{25} +1.00000 q^{27} +6.93900 q^{29} -2.22521 q^{31} +1.15883 q^{33} +18.7168 q^{35} +3.87800 q^{37} +6.31767 q^{41} -7.38404 q^{43} -4.29590 q^{45} -1.78017 q^{47} +11.9825 q^{49} +0.493959 q^{51} -2.51573 q^{53} -4.97823 q^{55} +1.78017 q^{57} +6.63102 q^{59} +10.4940 q^{61} -4.35690 q^{63} +4.09783 q^{67} -3.38404 q^{69} +9.38404 q^{71} -0.374354 q^{73} +13.4547 q^{75} -5.04892 q^{77} -2.65519 q^{79} +1.00000 q^{81} -14.2784 q^{83} -2.12200 q^{85} +6.93900 q^{87} -0.835790 q^{89} -2.22521 q^{93} -7.64742 q^{95} -18.3937 q^{97} +1.15883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + q^{5} - 9 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + q^5 - 9 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + q^{5} - 9 q^{7} + 3 q^{9} - 5 q^{11} + q^{15} - 8 q^{17} + 4 q^{19} - 9 q^{21} + 18 q^{25} + 3 q^{27} + 11 q^{29} - 5 q^{31} - 5 q^{33} + 4 q^{35} - 8 q^{37} + 2 q^{41} - 12 q^{43} + q^{45} - 4 q^{47} + 20 q^{49} - 8 q^{51} + 5 q^{53} - 18 q^{55} + 4 q^{57} + 5 q^{59} + 22 q^{61} - 9 q^{63} - 6 q^{67} + 18 q^{71} - 13 q^{73} + 18 q^{75} - 6 q^{77} - 31 q^{79} + 3 q^{81} - 13 q^{83} - 26 q^{85} + 11 q^{87} - 14 q^{89} - 5 q^{93} - 8 q^{95} - 23 q^{97} - 5 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + q^5 - 9 * q^7 + 3 * q^9 - 5 * q^11 + q^15 - 8 * q^17 + 4 * q^19 - 9 * q^21 + 18 * q^25 + 3 * q^27 + 11 * q^29 - 5 * q^31 - 5 * q^33 + 4 * q^35 - 8 * q^37 + 2 * q^41 - 12 * q^43 + q^45 - 4 * q^47 + 20 * q^49 - 8 * q^51 + 5 * q^53 - 18 * q^55 + 4 * q^57 + 5 * q^59 + 22 * q^61 - 9 * q^63 - 6 * q^67 + 18 * q^71 - 13 * q^73 + 18 * q^75 - 6 * q^77 - 31 * q^79 + 3 * q^81 - 13 * q^83 - 26 * q^85 + 11 * q^87 - 14 * q^89 - 5 * q^93 - 8 * q^95 - 23 * q^97 - 5 * q^99

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