LMFDB - Embedded newform 9450.2.a.ep.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9450,2,Mod(1,9450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9450, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9450.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:	$75.4586299101$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1890)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ of defining polynomial
Character	$\chi$	$=$	9450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +2.44949 q^{11} -3.44949 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +0.449490 q^{19} +2.44949 q^{22} -3.44949 q^{23} -3.44949 q^{26} -1.00000 q^{28} -7.44949 q^{29} +3.44949 q^{31} +1.00000 q^{32} +1.00000 q^{34} -11.3485 q^{37} +0.449490 q^{38} +4.89898 q^{41} +5.89898 q^{43} +2.44949 q^{44} -3.44949 q^{46} -7.79796 q^{47} +1.00000 q^{49} -3.44949 q^{52} -0.550510 q^{53} -1.00000 q^{56} -7.44949 q^{58} -9.89898 q^{59} +1.10102 q^{61} +3.44949 q^{62} +1.00000 q^{64} -5.00000 q^{67} +1.00000 q^{68} +0.550510 q^{71} +7.79796 q^{73} -11.3485 q^{74} +0.449490 q^{76} -2.44949 q^{77} -1.55051 q^{79} +4.89898 q^{82} -1.10102 q^{83} +5.89898 q^{86} +2.44949 q^{88} -11.0000 q^{89} +3.44949 q^{91} -3.44949 q^{92} -7.79796 q^{94} -14.4495 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{19} - 2 q^{23} - 2 q^{26} - 2 q^{28} - 10 q^{29} + 2 q^{31} + 2 q^{32} + 2 q^{34} - 8 q^{37} - 4 q^{38} + 2 q^{43} - 2 q^{46} + 4 q^{47} + 2 q^{49} - 2 q^{52} - 6 q^{53} - 2 q^{56} - 10 q^{58} - 10 q^{59} + 12 q^{61} + 2 q^{62} + 2 q^{64} - 10 q^{67} + 2 q^{68} + 6 q^{71} - 4 q^{73} - 8 q^{74} - 4 q^{76} - 8 q^{79} - 12 q^{83} + 2 q^{86} - 22 q^{89} + 2 q^{91} - 2 q^{92} + 4 q^{94} - 24 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 - 2 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 - 4 * q^19 - 2 * q^23 - 2 * q^26 - 2 * q^28 - 10 * q^29 + 2 * q^31 + 2 * q^32 + 2 * q^34 - 8 * q^37 - 4 * q^38 + 2 * q^43 - 2 * q^46 + 4 * q^47 + 2 * q^49 - 2 * q^52 - 6 * q^53 - 2 * q^56 - 10 * q^58 - 10 * q^59 + 12 * q^61 + 2 * q^62 + 2 * q^64 - 10 * q^67 + 2 * q^68 + 6 * q^71 - 4 * q^73 - 8 * q^74 - 4 * q^76 - 8 * q^79 - 12 * q^83 + 2 * q^86 - 22 * q^89 + 2 * q^91 - 2 * q^92 + 4 * q^94 - 24 * q^97 + 2 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	2.44949	0.738549	0.369274	−	0.929320i	$-0.379606\pi$
$11$	2.44949	0.738549	0.369274	+	0.929320i	$0.379606\pi$
$12$	0	0
$13$	−3.44949	−0.956716	−0.478358	−	0.878165i	$-0.658768\pi$
$13$	−3.44949	−0.956716	−0.478358	+	0.878165i	$0.658768\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	0.449490	0.103120	0.0515600	−	0.998670i	$-0.483581\pi$
$19$	0.449490	0.103120	0.0515600	+	0.998670i	$0.483581\pi$
$20$	0	0
$21$	0	0
$22$	2.44949	0.522233
$23$	−3.44949	−0.719268	−0.359634	−	0.933093i	$-0.617099\pi$
$23$	−3.44949	−0.719268	−0.359634	+	0.933093i	$0.617099\pi$
$24$	0	0
$25$	0	0
$26$	−3.44949	−0.676501
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−7.44949	−1.38334	−0.691668	−	0.722216i	$-0.743124\pi$
$29$	−7.44949	−1.38334	−0.691668	+	0.722216i	$0.743124\pi$
$30$	0	0
$31$	3.44949	0.619547	0.309773	−	0.950810i	$-0.399747\pi$
$31$	3.44949	0.619547	0.309773	+	0.950810i	$0.399747\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	0	0
$37$	−11.3485	−1.86568	−0.932838	−	0.360295i	$-0.882676\pi$
$37$	−11.3485	−1.86568	−0.932838	+	0.360295i	$0.882676\pi$
$38$	0.449490	0.0729169
$39$	0	0
$40$	0	0
$41$	4.89898	0.765092	0.382546	−	0.923936i	$-0.375047\pi$
$41$	4.89898	0.765092	0.382546	+	0.923936i	$0.375047\pi$
$42$	0	0
$43$	5.89898	0.899586	0.449793	−	0.893133i	$-0.351498\pi$
$43$	5.89898	0.899586	0.449793	+	0.893133i	$0.351498\pi$
$44$	2.44949	0.369274
$45$	0	0
$46$	−3.44949	−0.508600
$47$	−7.79796	−1.13745	−0.568725	−	0.822528i	$-0.692563\pi$
$47$	−7.79796	−1.13745	−0.568725	+	0.822528i	$0.692563\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	−3.44949	−0.478358
$53$	−0.550510	−0.0756184	−0.0378092	−	0.999285i	$-0.512038\pi$
$53$	−0.550510	−0.0756184	−0.0378092	+	0.999285i	$0.512038\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−7.44949	−0.978166
$59$	−9.89898	−1.28874	−0.644369	−	0.764715i	$-0.722880\pi$
$59$	−9.89898	−1.28874	−0.644369	+	0.764715i	$0.722880\pi$
$60$	0	0
$61$	1.10102	0.140971	0.0704856	−	0.997513i	$-0.477545\pi$
$61$	1.10102	0.140971	0.0704856	+	0.997513i	$0.477545\pi$
$62$	3.44949	0.438086
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−5.00000	−0.610847	−0.305424	−	0.952217i	$-0.598798\pi$
$67$	−5.00000	−0.610847	−0.305424	+	0.952217i	$0.598798\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	0.550510	0.0653335	0.0326668	−	0.999466i	$-0.489600\pi$
$71$	0.550510	0.0653335	0.0326668	+	0.999466i	$0.489600\pi$
$72$	0	0
$73$	7.79796	0.912682	0.456341	−	0.889805i	$-0.349160\pi$
$73$	7.79796	0.912682	0.456341	+	0.889805i	$0.349160\pi$
$74$	−11.3485	−1.31923
$75$	0	0
$76$	0.449490	0.0515600
$77$	−2.44949	−0.279145
$78$	0	0
$79$	−1.55051	−0.174446	−0.0872230	−	0.996189i	$-0.527799\pi$
$79$	−1.55051	−0.174446	−0.0872230	+	0.996189i	$0.527799\pi$
$80$	0	0
$81$	0	0
$82$	4.89898	0.541002
$83$	−1.10102	−0.120853	−0.0604264	−	0.998173i	$-0.519246\pi$
$83$	−1.10102	−0.120853	−0.0604264	+	0.998173i	$0.519246\pi$
$84$	0	0
$85$	0	0
$86$	5.89898	0.636103
$87$	0	0
$88$	2.44949	0.261116
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	0	0
$91$	3.44949	0.361605
$92$	−3.44949	−0.359634
$93$	0	0
$94$	−7.79796	−0.804298
$95$	0	0
$96$	0	0
$97$	−14.4495	−1.46712	−0.733562	−	0.679623i	$-0.762143\pi$
$97$	−14.4495	−1.46712	−0.733562	+	0.679623i	$0.762143\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	0	0
$101$	12.2474	1.21867	0.609333	−	0.792914i	$-0.291437\pi$
$101$	12.2474	1.21867	0.609333	+	0.792914i	$0.291437\pi$
$102$	0	0
$103$	−10.5505	−1.03957	−0.519786	−	0.854296i	$-0.673988\pi$
$103$	−10.5505	−1.03957	−0.519786	+	0.854296i	$0.673988\pi$
$104$	−3.44949	−0.338250
$105$	0	0
$106$	−0.550510	−0.0534703
$107$	−11.3485	−1.09710	−0.548549	−	0.836118i	$-0.684820\pi$
$107$	−11.3485	−1.09710	−0.548549	+	0.836118i	$0.684820\pi$
$108$	0	0
$109$	19.1464	1.83390	0.916948	−	0.399008i	$-0.130645\pi$
$109$	19.1464	1.83390	0.916948	+	0.399008i	$0.130645\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−1.55051	−0.145860	−0.0729299	−	0.997337i	$-0.523235\pi$
$113$	−1.55051	−0.145860	−0.0729299	+	0.997337i	$0.523235\pi$
$114$	0	0
$115$	0	0
$116$	−7.44949	−0.691668
$117$	0	0
$118$	−9.89898	−0.911275
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−5.00000	−0.454545
$122$	1.10102	0.0996817
$123$	0	0
$124$	3.44949	0.309773
$125$	0	0
$126$	0	0
$127$	0.898979	0.0797715	0.0398858	−	0.999204i	$-0.487301\pi$
$127$	0.898979	0.0797715	0.0398858	+	0.999204i	$0.487301\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−2.79796	−0.244459	−0.122229	−	0.992502i	$-0.539004\pi$
$131$	−2.79796	−0.244459	−0.122229	+	0.992502i	$0.539004\pi$
$132$	0	0
$133$	−0.449490	−0.0389757
$134$	−5.00000	−0.431934
$135$	0	0
$136$	1.00000	0.0857493
$137$	14.8990	1.27291	0.636453	−	0.771316i	$-0.280401\pi$
$137$	14.8990	1.27291	0.636453	+	0.771316i	$0.280401\pi$
$138$	0	0
$139$	−2.24745	−0.190626	−0.0953131	−	0.995447i	$-0.530385\pi$
$139$	−2.24745	−0.190626	−0.0953131	+	0.995447i	$0.530385\pi$
$140$	0	0
$141$	0	0
$142$	0.550510	0.0461978
$143$	−8.44949	−0.706582
$144$	0	0
$145$	0	0
$146$	7.79796	0.645364
$147$	0	0
$148$	−11.3485	−0.932838
$149$	−5.44949	−0.446440	−0.223220	−	0.974768i	$-0.571657\pi$
$149$	−5.44949	−0.446440	−0.223220	+	0.974768i	$0.571657\pi$
$150$	0	0
$151$	−12.2474	−0.996683	−0.498342	−	0.866981i	$-0.666057\pi$
$151$	−12.2474	−0.996683	−0.498342	+	0.866981i	$0.666057\pi$
$152$	0.449490	0.0364584
$153$	0	0
$154$	−2.44949	−0.197386
$155$	0	0
$156$	0	0
$157$	−0.550510	−0.0439355	−0.0219678	−	0.999759i	$-0.506993\pi$
$157$	−0.550510	−0.0439355	−0.0219678	+	0.999759i	$0.506993\pi$
$158$	−1.55051	−0.123352
$159$	0	0
$160$	0	0
$161$	3.44949	0.271858
$162$	0	0
$163$	−13.8990	−1.08865	−0.544326	−	0.838874i	$-0.683215\pi$
$163$	−13.8990	−1.08865	−0.544326	+	0.838874i	$0.683215\pi$
$164$	4.89898	0.382546
$165$	0	0
$166$	−1.10102	−0.0854558
$167$	8.44949	0.653841	0.326921	−	0.945052i	$-0.393989\pi$
$167$	8.44949	0.653841	0.326921	+	0.945052i	$0.393989\pi$
$168$	0	0
$169$	−1.10102	−0.0846939
$170$	0	0
$171$	0	0
$172$	5.89898	0.449793
$173$	23.1464	1.75979	0.879895	−	0.475168i	$-0.157613\pi$
$173$	23.1464	1.75979	0.879895	+	0.475168i	$0.157613\pi$
$174$	0	0
$175$	0	0
$176$	2.44949	0.184637
$177$	0	0
$178$	−11.0000	−0.824485
$179$	−8.89898	−0.665141	−0.332570	−	0.943078i	$-0.607916\pi$
$179$	−8.89898	−0.665141	−0.332570	+	0.943078i	$0.607916\pi$
$180$	0	0
$181$	2.55051	0.189578	0.0947890	−	0.995497i	$-0.469782\pi$
$181$	2.55051	0.189578	0.0947890	+	0.995497i	$0.469782\pi$
$182$	3.44949	0.255693
$183$	0	0
$184$	−3.44949	−0.254300
$185$	0	0
$186$	0	0
$187$	2.44949	0.179124
$188$	−7.79796	−0.568725
$189$	0	0
$190$	0	0
$191$	−6.69694	−0.484573	−0.242287	−	0.970205i	$-0.577897\pi$
$191$	−6.69694	−0.484573	−0.242287	+	0.970205i	$0.577897\pi$
$192$	0	0
$193$	−26.7980	−1.92896	−0.964480	−	0.264157i	$-0.914906\pi$
$193$	−26.7980	−1.92896	−0.964480	+	0.264157i	$0.914906\pi$
$194$	−14.4495	−1.03741
$195$	0	0
$196$	1.00000	0.0714286
$197$	−3.10102	−0.220939	−0.110469	−	0.993880i	$-0.535235\pi$
$197$	−3.10102	−0.220939	−0.110469	+	0.993880i	$0.535235\pi$
$198$	0	0
$199$	−15.2474	−1.08086	−0.540431	−	0.841388i	$-0.681739\pi$
$199$	−15.2474	−1.08086	−0.540431	+	0.841388i	$0.681739\pi$
$200$	0	0
$201$	0	0
$202$	12.2474	0.861727
$203$	7.44949	0.522852
$204$	0	0
$205$	0	0
$206$	−10.5505	−0.735089
$207$	0	0
$208$	−3.44949	−0.239179
$209$	1.10102	0.0761592
$210$	0	0
$211$	−15.0000	−1.03264	−0.516321	−	0.856395i	$-0.672699\pi$
$211$	−15.0000	−1.03264	−0.516321	+	0.856395i	$0.672699\pi$
$212$	−0.550510	−0.0378092
$213$	0	0
$214$	−11.3485	−0.775765
$215$	0	0
$216$	0	0
$217$	−3.44949	−0.234167
$218$	19.1464	1.29676
$219$	0	0
$220$	0	0
$221$	−3.44949	−0.232038
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−1.55051	−0.103138
$227$	17.6969	1.17459	0.587294	−	0.809374i	$-0.300193\pi$
$227$	17.6969	1.17459	0.587294	+	0.809374i	$0.300193\pi$
$228$	0	0
$229$	−12.8990	−0.852389	−0.426194	−	0.904632i	$-0.640146\pi$
$229$	−12.8990	−0.852389	−0.426194	+	0.904632i	$0.640146\pi$
$230$	0	0
$231$	0	0
$232$	−7.44949	−0.489083
$233$	21.1464	1.38535	0.692674	−	0.721251i	$-0.256432\pi$
$233$	21.1464	1.38535	0.692674	+	0.721251i	$0.256432\pi$
$234$	0	0
$235$	0	0
$236$	−9.89898	−0.644369
$237$	0	0
$238$	−1.00000	−0.0648204
$239$	28.4949	1.84318	0.921591	−	0.388163i	$-0.126890\pi$
$239$	28.4949	1.84318	0.921591	+	0.388163i	$0.126890\pi$
$240$	0	0
$241$	25.3485	1.63284	0.816419	−	0.577460i	$-0.195956\pi$
$241$	25.3485	1.63284	0.816419	+	0.577460i	$0.195956\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	1.10102	0.0704856
$245$	0	0
$246$	0	0
$247$	−1.55051	−0.0986566
$248$	3.44949	0.219043
$249$	0	0
$250$	0	0
$251$	11.5959	0.731928	0.365964	−	0.930629i	$-0.380739\pi$
$251$	11.5959	0.731928	0.365964	+	0.930629i	$0.380739\pi$
$252$	0	0
$253$	−8.44949	−0.531215
$254$	0.898979	0.0564070
$255$	0	0
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	0	0
$259$	11.3485	0.705160
$260$	0	0
$261$	0	0
$262$	−2.79796	−0.172858
$263$	−29.9444	−1.84645	−0.923225	−	0.384260i	$-0.874457\pi$
$263$	−29.9444	−1.84645	−0.923225	+	0.384260i	$0.874457\pi$
$264$	0	0
$265$	0	0
$266$	−0.449490	−0.0275600
$267$	0	0
$268$	−5.00000	−0.305424
$269$	−12.8990	−0.786465	−0.393232	−	0.919439i	$-0.628643\pi$
$269$	−12.8990	−0.786465	−0.393232	+	0.919439i	$0.628643\pi$
$270$	0	0
$271$	10.3485	0.628625	0.314312	−	0.949320i	$-0.398226\pi$
$271$	10.3485	0.628625	0.314312	+	0.949320i	$0.398226\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	14.8990	0.900080
$275$	0	0
$276$	0	0
$277$	−25.7980	−1.55005	−0.775025	−	0.631931i	$-0.782263\pi$
$277$	−25.7980	−1.55005	−0.775025	+	0.631931i	$0.782263\pi$
$278$	−2.24745	−0.134793
$279$	0	0
$280$	0	0
$281$	−1.34847	−0.0804429	−0.0402215	−	0.999191i	$-0.512806\pi$
$281$	−1.34847	−0.0804429	−0.0402215	+	0.999191i	$0.512806\pi$
$282$	0	0
$283$	−22.4495	−1.33448	−0.667242	−	0.744841i	$-0.732525\pi$
$283$	−22.4495	−1.33448	−0.667242	+	0.744841i	$0.732525\pi$
$284$	0.550510	0.0326668
$285$	0	0
$286$	−8.44949	−0.499629
$287$	−4.89898	−0.289178
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	0	0
$292$	7.79796	0.456341
$293$	−18.4949	−1.08048	−0.540242	−	0.841510i	$-0.681667\pi$
$293$	−18.4949	−1.08048	−0.540242	+	0.841510i	$0.681667\pi$
$294$	0	0
$295$	0	0
$296$	−11.3485	−0.659616
$297$	0	0
$298$	−5.44949	−0.315680
$299$	11.8990	0.688136
$300$	0	0
$301$	−5.89898	−0.340012
$302$	−12.2474	−0.704761
$303$	0	0
$304$	0.449490	0.0257800
$305$	0	0
$306$	0	0
$307$	19.5505	1.11581	0.557903	−	0.829906i	$-0.311606\pi$
$307$	19.5505	1.11581	0.557903	+	0.829906i	$0.311606\pi$
$308$	−2.44949	−0.139573
$309$	0	0
$310$	0	0
$311$	−1.34847	−0.0764647	−0.0382323	−	0.999269i	$-0.512173\pi$
$311$	−1.34847	−0.0764647	−0.0382323	+	0.999269i	$0.512173\pi$
$312$	0	0
$313$	24.9444	1.40994	0.704970	−	0.709237i	$-0.250960\pi$
$313$	24.9444	1.40994	0.704970	+	0.709237i	$0.250960\pi$
$314$	−0.550510	−0.0310671
$315$	0	0
$316$	−1.55051	−0.0872230
$317$	−23.7980	−1.33663	−0.668313	−	0.743880i	$-0.732983\pi$
$317$	−23.7980	−1.33663	−0.668313	+	0.743880i	$0.732983\pi$
$318$	0	0
$319$	−18.2474	−1.02166
$320$	0	0
$321$	0	0
$322$	3.44949	0.192233
$323$	0.449490	0.0250103
$324$	0	0
$325$	0	0
$326$	−13.8990	−0.769793
$327$	0	0
$328$	4.89898	0.270501
$329$	7.79796	0.429915
$330$	0	0
$331$	−13.0000	−0.714545	−0.357272	−	0.934000i	$-0.616293\pi$
$331$	−13.0000	−0.714545	−0.357272	+	0.934000i	$0.616293\pi$
$332$	−1.10102	−0.0604264
$333$	0	0
$334$	8.44949	0.462336
$335$	0	0
$336$	0	0
$337$	−3.20204	−0.174426	−0.0872131	−	0.996190i	$-0.527796\pi$
$337$	−3.20204	−0.174426	−0.0872131	+	0.996190i	$0.527796\pi$
$338$	−1.10102	−0.0598876
$339$	0	0
$340$	0	0
$341$	8.44949	0.457566
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	5.89898	0.318052
$345$	0	0
$346$	23.1464	1.24436
$347$	25.5959	1.37406	0.687030	−	0.726629i	$-0.258914\pi$
$347$	25.5959	1.37406	0.687030	+	0.726629i	$0.258914\pi$
$348$	0	0
$349$	14.1464	0.757241	0.378620	−	0.925552i	$-0.376399\pi$
$349$	14.1464	0.757241	0.378620	+	0.925552i	$0.376399\pi$
$350$	0	0
$351$	0	0
$352$	2.44949	0.130558
$353$	−33.4949	−1.78275	−0.891377	−	0.453263i	$-0.850260\pi$
$353$	−33.4949	−1.78275	−0.891377	+	0.453263i	$0.850260\pi$
$354$	0	0
$355$	0	0
$356$	−11.0000	−0.582999
$357$	0	0
$358$	−8.89898	−0.470326
$359$	−16.5505	−0.873503	−0.436751	−	0.899582i	$-0.643871\pi$
$359$	−16.5505	−0.873503	−0.436751	+	0.899582i	$0.643871\pi$
$360$	0	0
$361$	−18.7980	−0.989366
$362$	2.55051	0.134052
$363$	0	0
$364$	3.44949	0.180802
$365$	0	0
$366$	0	0
$367$	−29.0454	−1.51616	−0.758079	−	0.652163i	$-0.773862\pi$
$367$	−29.0454	−1.51616	−0.758079	+	0.652163i	$0.773862\pi$
$368$	−3.44949	−0.179817
$369$	0	0
$370$	0	0
$371$	0.550510	0.0285811
$372$	0	0
$373$	−12.6515	−0.655071	−0.327536	−	0.944839i	$-0.606218\pi$
$373$	−12.6515	−0.655071	−0.327536	+	0.944839i	$0.606218\pi$
$374$	2.44949	0.126660
$375$	0	0
$376$	−7.79796	−0.402149
$377$	25.6969	1.32346
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	0	0
$381$	0	0
$382$	−6.69694	−0.342645
$383$	16.8990	0.863498	0.431749	−	0.901994i	$-0.357897\pi$
$383$	16.8990	0.863498	0.431749	+	0.901994i	$0.357897\pi$
$384$	0	0
$385$	0	0
$386$	−26.7980	−1.36398
$387$	0	0
$388$	−14.4495	−0.733562
$389$	19.1010	0.968460	0.484230	−	0.874941i	$-0.339100\pi$
$389$	19.1010	0.968460	0.484230	+	0.874941i	$0.339100\pi$
$390$	0	0
$391$	−3.44949	−0.174448
$392$	1.00000	0.0505076
$393$	0	0
$394$	−3.10102	−0.156227
$395$	0	0
$396$	0	0
$397$	−6.69694	−0.336110	−0.168055	−	0.985778i	$-0.553749\pi$
$397$	−6.69694	−0.336110	−0.168055	+	0.985778i	$0.553749\pi$
$398$	−15.2474	−0.764286
$399$	0	0
$400$	0	0
$401$	−20.6969	−1.03356	−0.516778	−	0.856120i	$-0.672869\pi$
$401$	−20.6969	−1.03356	−0.516778	+	0.856120i	$0.672869\pi$
$402$	0	0
$403$	−11.8990	−0.592730
$404$	12.2474	0.609333
$405$	0	0
$406$	7.44949	0.369712
$407$	−27.7980	−1.37789
$408$	0	0
$409$	29.5959	1.46342	0.731712	−	0.681614i	$-0.238722\pi$
$409$	29.5959	1.46342	0.731712	+	0.681614i	$0.238722\pi$
$410$	0	0
$411$	0	0
$412$	−10.5505	−0.519786
$413$	9.89898	0.487097
$414$	0	0
$415$	0	0
$416$	−3.44949	−0.169125
$417$	0	0
$418$	1.10102	0.0538527
$419$	−0.595918	−0.0291125	−0.0145562	−	0.999894i	$-0.504634\pi$
$419$	−0.595918	−0.0291125	−0.0145562	+	0.999894i	$0.504634\pi$
$420$	0	0
$421$	0.202041	0.00984688	0.00492344	−	0.999988i	$-0.498433\pi$
$421$	0.202041	0.00984688	0.00492344	+	0.999988i	$0.498433\pi$
$422$	−15.0000	−0.730189
$423$	0	0
$424$	−0.550510	−0.0267351
$425$	0	0
$426$	0	0
$427$	−1.10102	−0.0532821
$428$	−11.3485	−0.548549
$429$	0	0
$430$	0	0
$431$	−11.7980	−0.568288	−0.284144	−	0.958782i	$-0.591709\pi$
$431$	−11.7980	−0.568288	−0.284144	+	0.958782i	$0.591709\pi$
$432$	0	0
$433$	15.5959	0.749492	0.374746	−	0.927128i	$-0.377730\pi$
$433$	15.5959	0.749492	0.374746	+	0.927128i	$0.377730\pi$
$434$	−3.44949	−0.165581
$435$	0	0
$436$	19.1464	0.916948
$437$	−1.55051	−0.0741710
$438$	0	0
$439$	−8.34847	−0.398451	−0.199225	−	0.979954i	$-0.563843\pi$
$439$	−8.34847	−0.398451	−0.199225	+	0.979954i	$0.563843\pi$
$440$	0	0
$441$	0	0
$442$	−3.44949	−0.164075
$443$	32.4495	1.54172	0.770861	−	0.637004i	$-0.219826\pi$
$443$	32.4495	1.54172	0.770861	+	0.637004i	$0.219826\pi$
$444$	0	0
$445$	0	0
$446$	−4.00000	−0.189405
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−32.0000	−1.51017	−0.755087	−	0.655625i	$-0.772405\pi$
$449$	−32.0000	−1.51017	−0.755087	+	0.655625i	$0.772405\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	−1.55051	−0.0729299
$453$	0	0
$454$	17.6969	0.830558
$455$	0	0
$456$	0	0
$457$	−9.89898	−0.463055	−0.231527	−	0.972828i	$-0.574372\pi$
$457$	−9.89898	−0.463055	−0.231527	+	0.972828i	$0.574372\pi$
$458$	−12.8990	−0.602730
$459$	0	0
$460$	0	0
$461$	−10.0000	−0.465746	−0.232873	−	0.972507i	$-0.574813\pi$
$461$	−10.0000	−0.465746	−0.232873	+	0.972507i	$0.574813\pi$
$462$	0	0
$463$	−28.2474	−1.31277	−0.656385	−	0.754426i	$-0.727915\pi$
$463$	−28.2474	−1.31277	−0.656385	+	0.754426i	$0.727915\pi$
$464$	−7.44949	−0.345834
$465$	0	0
$466$	21.1464	0.979589
$467$	31.7980	1.47143	0.735717	−	0.677289i	$-0.236845\pi$
$467$	31.7980	1.47143	0.735717	+	0.677289i	$0.236845\pi$
$468$	0	0
$469$	5.00000	0.230879
$470$	0	0
$471$	0	0
$472$	−9.89898	−0.455637
$473$	14.4495	0.664388
$474$	0	0
$475$	0	0
$476$	−1.00000	−0.0458349
$477$	0	0
$478$	28.4949	1.30333
$479$	36.0000	1.64488	0.822441	−	0.568850i	$-0.192612\pi$
$479$	36.0000	1.64488	0.822441	+	0.568850i	$0.192612\pi$
$480$	0	0
$481$	39.1464	1.78492
$482$	25.3485	1.15459
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	19.3939	0.878820	0.439410	−	0.898287i	$-0.355187\pi$
$487$	19.3939	0.878820	0.439410	+	0.898287i	$0.355187\pi$
$488$	1.10102	0.0498409
$489$	0	0
$490$	0	0
$491$	43.5959	1.96746	0.983728	−	0.179664i	$-0.0575009\pi$
$491$	43.5959	1.96746	0.983728	+	0.179664i	$0.0575009\pi$
$492$	0	0
$493$	−7.44949	−0.335508
$494$	−1.55051	−0.0697608
$495$	0	0
$496$	3.44949	0.154887
$497$	−0.550510	−0.0246938
$498$	0	0
$499$	−6.89898	−0.308841	−0.154420	−	0.988005i	$-0.549351\pi$
$499$	−6.89898	−0.308841	−0.154420	+	0.988005i	$0.549351\pi$
$500$	0	0
$501$	0	0
$502$	11.5959	0.517551
$503$	−21.3485	−0.951881	−0.475941	−	0.879477i	$-0.657892\pi$
$503$	−21.3485	−0.951881	−0.475941	+	0.879477i	$0.657892\pi$
$504$	0	0
$505$	0	0
$506$	−8.44949	−0.375626
$507$	0	0
$508$	0.898979	0.0398858
$509$	21.1464	0.937299	0.468649	−	0.883384i	$-0.344741\pi$
$509$	21.1464	0.937299	0.468649	+	0.883384i	$0.344741\pi$
$510$	0	0
$511$	−7.79796	−0.344961
$512$	1.00000	0.0441942
$513$	0	0
$514$	−12.0000	−0.529297
$515$	0	0
$516$	0	0
$517$	−19.1010	−0.840062
$518$	11.3485	0.498623
$519$	0	0
$520$	0	0
$521$	−3.00000	−0.131432	−0.0657162	−	0.997838i	$-0.520933\pi$
$521$	−3.00000	−0.131432	−0.0657162	+	0.997838i	$0.520933\pi$
$522$	0	0
$523$	−35.5959	−1.55650	−0.778250	−	0.627954i	$-0.783893\pi$
$523$	−35.5959	−1.55650	−0.778250	+	0.627954i	$0.783893\pi$
$524$	−2.79796	−0.122229
$525$	0	0
$526$	−29.9444	−1.30564
$527$	3.44949	0.150262
$528$	0	0
$529$	−11.1010	−0.482653
$530$	0	0
$531$	0	0
$532$	−0.449490	−0.0194879
$533$	−16.8990	−0.731976
$534$	0	0
$535$	0	0
$536$	−5.00000	−0.215967
$537$	0	0
$538$	−12.8990	−0.556114
$539$	2.44949	0.105507
$540$	0	0
$541$	−34.2474	−1.47241	−0.736206	−	0.676757i	$-0.763385\pi$
$541$	−34.2474	−1.47241	−0.736206	+	0.676757i	$0.763385\pi$
$542$	10.3485	0.444505
$543$	0	0
$544$	1.00000	0.0428746
$545$	0	0
$546$	0	0
$547$	28.6969	1.22699	0.613496	−	0.789698i	$-0.289763\pi$
$547$	28.6969	1.22699	0.613496	+	0.789698i	$0.289763\pi$
$548$	14.8990	0.636453
$549$	0	0
$550$	0	0
$551$	−3.34847	−0.142650
$552$	0	0
$553$	1.55051	0.0659344
$554$	−25.7980	−1.09605
$555$	0	0
$556$	−2.24745	−0.0953131
$557$	−11.0454	−0.468009	−0.234004	−	0.972236i	$-0.575183\pi$
$557$	−11.0454	−0.468009	−0.234004	+	0.972236i	$0.575183\pi$
$558$	0	0
$559$	−20.3485	−0.860649
$560$	0	0
$561$	0	0
$562$	−1.34847	−0.0568817
$563$	19.6969	0.830127	0.415063	−	0.909792i	$-0.363759\pi$
$563$	19.6969	0.830127	0.415063	+	0.909792i	$0.363759\pi$
$564$	0	0
$565$	0	0
$566$	−22.4495	−0.943622
$567$	0	0
$568$	0.550510	0.0230989
$569$	31.8434	1.33494	0.667472	−	0.744635i	$-0.267377\pi$
$569$	31.8434	1.33494	0.667472	+	0.744635i	$0.267377\pi$
$570$	0	0
$571$	29.4949	1.23432	0.617162	−	0.786836i	$-0.288283\pi$
$571$	29.4949	1.23432	0.617162	+	0.786836i	$0.288283\pi$
$572$	−8.44949	−0.353291
$573$	0	0
$574$	−4.89898	−0.204479
$575$	0	0
$576$	0	0
$577$	24.4495	1.01785	0.508923	−	0.860812i	$-0.330044\pi$
$577$	24.4495	1.01785	0.508923	+	0.860812i	$0.330044\pi$
$578$	−16.0000	−0.665512
$579$	0	0
$580$	0	0
$581$	1.10102	0.0456780
$582$	0	0
$583$	−1.34847	−0.0558479
$584$	7.79796	0.322682
$585$	0	0
$586$	−18.4949	−0.764017
$587$	13.4949	0.556994	0.278497	−	0.960437i	$-0.410164\pi$
$587$	13.4949	0.556994	0.278497	+	0.960437i	$0.410164\pi$
$588$	0	0
$589$	1.55051	0.0638877
$590$	0	0
$591$	0	0
$592$	−11.3485	−0.466419
$593$	7.10102	0.291604	0.145802	−	0.989314i	$-0.453424\pi$
$593$	7.10102	0.291604	0.145802	+	0.989314i	$0.453424\pi$
$594$	0	0
$595$	0	0
$596$	−5.44949	−0.223220
$597$	0	0
$598$	11.8990	0.486585
$599$	0.550510	0.0224932	0.0112466	−	0.999937i	$-0.496420\pi$
$599$	0.550510	0.0224932	0.0112466	+	0.999937i	$0.496420\pi$
$600$	0	0
$601$	−42.6969	−1.74164	−0.870822	−	0.491598i	$-0.836413\pi$
$601$	−42.6969	−1.74164	−0.870822	+	0.491598i	$0.836413\pi$
$602$	−5.89898	−0.240424
$603$	0	0
$604$	−12.2474	−0.498342
$605$	0	0
$606$	0	0
$607$	−21.0454	−0.854207	−0.427103	−	0.904203i	$-0.640466\pi$
$607$	−21.0454	−0.854207	−0.427103	+	0.904203i	$0.640466\pi$
$608$	0.449490	0.0182292
$609$	0	0
$610$	0	0
$611$	26.8990	1.08822
$612$	0	0
$613$	23.7526	0.959357	0.479678	−	0.877444i	$-0.340753\pi$
$613$	23.7526	0.959357	0.479678	+	0.877444i	$0.340753\pi$
$614$	19.5505	0.788994
$615$	0	0
$616$	−2.44949	−0.0986928
$617$	−29.1010	−1.17156	−0.585781	−	0.810469i	$-0.699212\pi$
$617$	−29.1010	−1.17156	−0.585781	+	0.810469i	$0.699212\pi$
$618$	0	0
$619$	24.2020	0.972762	0.486381	−	0.873747i	$-0.338317\pi$
$619$	24.2020	0.972762	0.486381	+	0.873747i	$0.338317\pi$
$620$	0	0
$621$	0	0
$622$	−1.34847	−0.0540687
$623$	11.0000	0.440706
$624$	0	0
$625$	0	0
$626$	24.9444	0.996978
$627$	0	0
$628$	−0.550510	−0.0219678
$629$	−11.3485	−0.452493
$630$	0	0
$631$	−4.89898	−0.195025	−0.0975126	−	0.995234i	$-0.531089\pi$
$631$	−4.89898	−0.195025	−0.0975126	+	0.995234i	$0.531089\pi$
$632$	−1.55051	−0.0616760
$633$	0	0
$634$	−23.7980	−0.945138
$635$	0	0
$636$	0	0
$637$	−3.44949	−0.136674
$638$	−18.2474	−0.722423
$639$	0	0
$640$	0	0
$641$	10.4495	0.412730	0.206365	−	0.978475i	$-0.433837\pi$
$641$	10.4495	0.412730	0.206365	+	0.978475i	$0.433837\pi$
$642$	0	0
$643$	−45.1464	−1.78040	−0.890201	−	0.455569i	$-0.849436\pi$
$643$	−45.1464	−1.78040	−0.890201	+	0.455569i	$0.849436\pi$
$644$	3.44949	0.135929
$645$	0	0
$646$	0.449490	0.0176849
$647$	−25.5959	−1.00628	−0.503140	−	0.864205i	$-0.667822\pi$
$647$	−25.5959	−1.00628	−0.503140	+	0.864205i	$0.667822\pi$
$648$	0	0
$649$	−24.2474	−0.951796
$650$	0	0
$651$	0	0
$652$	−13.8990	−0.544326
$653$	−9.24745	−0.361881	−0.180940	−	0.983494i	$-0.557914\pi$
$653$	−9.24745	−0.361881	−0.180940	+	0.983494i	$0.557914\pi$
$654$	0	0
$655$	0	0
$656$	4.89898	0.191273
$657$	0	0
$658$	7.79796	0.303996
$659$	48.2474	1.87945	0.939727	−	0.341926i	$-0.111079\pi$
$659$	48.2474	1.87945	0.939727	+	0.341926i	$0.111079\pi$
$660$	0	0
$661$	−14.4949	−0.563786	−0.281893	−	0.959446i	$-0.590962\pi$
$661$	−14.4949	−0.563786	−0.281893	+	0.959446i	$0.590962\pi$
$662$	−13.0000	−0.505259
$663$	0	0
$664$	−1.10102	−0.0427279
$665$	0	0
$666$	0	0
$667$	25.6969	0.994989
$668$	8.44949	0.326921
$669$	0	0
$670$	0	0
$671$	2.69694	0.104114
$672$	0	0
$673$	1.49490	0.0576241	0.0288120	−	0.999585i	$-0.490828\pi$
$673$	1.49490	0.0576241	0.0288120	+	0.999585i	$0.490828\pi$
$674$	−3.20204	−0.123338
$675$	0	0
$676$	−1.10102	−0.0423469
$677$	−5.79796	−0.222834	−0.111417	−	0.993774i	$-0.535539\pi$
$677$	−5.79796	−0.222834	−0.111417	+	0.993774i	$0.535539\pi$
$678$	0	0
$679$	14.4495	0.554521
$680$	0	0
$681$	0	0
$682$	8.44949	0.323548
$683$	−10.4495	−0.399839	−0.199919	−	0.979812i	$-0.564068\pi$
$683$	−10.4495	−0.399839	−0.199919	+	0.979812i	$0.564068\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	5.89898	0.224896
$689$	1.89898	0.0723454
$690$	0	0
$691$	50.2929	1.91323	0.956615	−	0.291354i	$-0.0941059\pi$
$691$	50.2929	1.91323	0.956615	+	0.291354i	$0.0941059\pi$
$692$	23.1464	0.879895
$693$	0	0
$694$	25.5959	0.971608
$695$	0	0
$696$	0	0
$697$	4.89898	0.185562
$698$	14.1464	0.535450
$699$	0	0
$700$	0	0
$701$	−19.3939	−0.732497	−0.366248	−	0.930517i	$-0.619358\pi$
$701$	−19.3939	−0.732497	−0.366248	+	0.930517i	$0.619358\pi$
$702$	0	0
$703$	−5.10102	−0.192389
$704$	2.44949	0.0923186
$705$	0	0
$706$	−33.4949	−1.26060
$707$	−12.2474	−0.460613
$708$	0	0
$709$	−11.1464	−0.418613	−0.209306	−	0.977850i	$-0.567121\pi$
$709$	−11.1464	−0.418613	−0.209306	+	0.977850i	$0.567121\pi$
$710$	0	0
$711$	0	0
$712$	−11.0000	−0.412242
$713$	−11.8990	−0.445620
$714$	0	0
$715$	0	0
$716$	−8.89898	−0.332570
$717$	0	0
$718$	−16.5505	−0.617660
$719$	−19.1464	−0.714041	−0.357021	−	0.934096i	$-0.616207\pi$
$719$	−19.1464	−0.714041	−0.357021	+	0.934096i	$0.616207\pi$
$720$	0	0
$721$	10.5505	0.392922
$722$	−18.7980	−0.699588
$723$	0	0
$724$	2.55051	0.0947890
$725$	0	0
$726$	0	0
$727$	18.3485	0.680507	0.340254	−	0.940334i	$-0.389487\pi$
$727$	18.3485	0.680507	0.340254	+	0.940334i	$0.389487\pi$
$728$	3.44949	0.127847
$729$	0	0
$730$	0	0
$731$	5.89898	0.218182
$732$	0	0
$733$	−32.5505	−1.20228	−0.601140	−	0.799144i	$-0.705287\pi$
$733$	−32.5505	−1.20228	−0.601140	+	0.799144i	$0.705287\pi$
$734$	−29.0454	−1.07209
$735$	0	0
$736$	−3.44949	−0.127150
$737$	−12.2474	−0.451141
$738$	0	0
$739$	24.2020	0.890286	0.445143	−	0.895459i	$-0.353153\pi$
$739$	24.2020	0.890286	0.445143	+	0.895459i	$0.353153\pi$
$740$	0	0
$741$	0	0
$742$	0.550510	0.0202099
$743$	−25.6515	−0.941063	−0.470532	−	0.882383i	$-0.655938\pi$
$743$	−25.6515	−0.941063	−0.470532	+	0.882383i	$0.655938\pi$
$744$	0	0
$745$	0	0
$746$	−12.6515	−0.463205
$747$	0	0
$748$	2.44949	0.0895622
$749$	11.3485	0.414664
$750$	0	0
$751$	−41.5505	−1.51620	−0.758100	−	0.652139i	$-0.773872\pi$
$751$	−41.5505	−1.51620	−0.758100	+	0.652139i	$0.773872\pi$
$752$	−7.79796	−0.284362
$753$	0	0
$754$	25.6969	0.935827
$755$	0	0
$756$	0	0
$757$	4.40408	0.160069	0.0800345	−	0.996792i	$-0.474497\pi$
$757$	4.40408	0.160069	0.0800345	+	0.996792i	$0.474497\pi$
$758$	10.0000	0.363216
$759$	0	0
$760$	0	0
$761$	−19.4949	−0.706689	−0.353345	−	0.935493i	$-0.614956\pi$
$761$	−19.4949	−0.706689	−0.353345	+	0.935493i	$0.614956\pi$
$762$	0	0
$763$	−19.1464	−0.693147
$764$	−6.69694	−0.242287
$765$	0	0
$766$	16.8990	0.610585
$767$	34.1464	1.23296
$768$	0	0
$769$	52.7423	1.90194	0.950969	−	0.309287i	$-0.100090\pi$
$769$	52.7423	1.90194	0.950969	+	0.309287i	$0.100090\pi$
$770$	0	0
$771$	0	0
$772$	−26.7980	−0.964480
$773$	−9.30306	−0.334608	−0.167304	−	0.985905i	$-0.553506\pi$
$773$	−9.30306	−0.334608	−0.167304	+	0.985905i	$0.553506\pi$
$774$	0	0
$775$	0	0
$776$	−14.4495	−0.518706
$777$	0	0
$778$	19.1010	0.684805
$779$	2.20204	0.0788963
$780$	0	0
$781$	1.34847	0.0482520
$782$	−3.44949	−0.123354
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	16.2929	0.580778	0.290389	−	0.956909i	$-0.406215\pi$
$787$	16.2929	0.580778	0.290389	+	0.956909i	$0.406215\pi$
$788$	−3.10102	−0.110469
$789$	0	0
$790$	0	0
$791$	1.55051	0.0551298
$792$	0	0
$793$	−3.79796	−0.134869
$794$	−6.69694	−0.237665
$795$	0	0
$796$	−15.2474	−0.540431
$797$	46.4949	1.64693	0.823467	−	0.567364i	$-0.192037\pi$
$797$	46.4949	1.64693	0.823467	+	0.567364i	$0.192037\pi$
$798$	0	0
$799$	−7.79796	−0.275872
$800$	0	0
$801$	0	0
$802$	−20.6969	−0.730834
$803$	19.1010	0.674060
$804$	0	0
$805$	0	0
$806$	−11.8990	−0.419124
$807$	0	0
$808$	12.2474	0.430864
$809$	−25.7980	−0.907008	−0.453504	−	0.891254i	$-0.649826\pi$
$809$	−25.7980	−0.907008	−0.453504	+	0.891254i	$0.649826\pi$
$810$	0	0
$811$	33.1918	1.16552	0.582761	−	0.812643i	$-0.301972\pi$
$811$	33.1918	1.16552	0.582761	+	0.812643i	$0.301972\pi$
$812$	7.44949	0.261426
$813$	0	0
$814$	−27.7980	−0.974318
$815$	0	0
$816$	0	0
$817$	2.65153	0.0927653
$818$	29.5959	1.03480
$819$	0	0
$820$	0	0
$821$	48.5505	1.69442	0.847212	−	0.531255i	$-0.178279\pi$
$821$	48.5505	1.69442	0.847212	+	0.531255i	$0.178279\pi$
$822$	0	0
$823$	7.84337	0.273403	0.136701	−	0.990612i	$-0.456350\pi$
$823$	7.84337	0.273403	0.136701	+	0.990612i	$0.456350\pi$
$824$	−10.5505	−0.367544
$825$	0	0
$826$	9.89898	0.344430
$827$	−32.2929	−1.12293	−0.561466	−	0.827500i	$-0.689763\pi$
$827$	−32.2929	−1.12293	−0.561466	+	0.827500i	$0.689763\pi$
$828$	0	0
$829$	2.89898	0.100686	0.0503429	−	0.998732i	$-0.483969\pi$
$829$	2.89898	0.100686	0.0503429	+	0.998732i	$0.483969\pi$
$830$	0	0
$831$	0	0
$832$	−3.44949	−0.119590
$833$	1.00000	0.0346479
$834$	0	0
$835$	0	0
$836$	1.10102	0.0380796
$837$	0	0
$838$	−0.595918	−0.0205856
$839$	37.5505	1.29639	0.648194	−	0.761475i	$-0.275525\pi$
$839$	37.5505	1.29639	0.648194	+	0.761475i	$0.275525\pi$
$840$	0	0
$841$	26.4949	0.913617
$842$	0.202041	0.00696279
$843$	0	0
$844$	−15.0000	−0.516321
$845$	0	0
$846$	0	0
$847$	5.00000	0.171802
$848$	−0.550510	−0.0189046
$849$	0	0
$850$	0	0
$851$	39.1464	1.34192
$852$	0	0
$853$	−11.2474	−0.385105	−0.192553	−	0.981287i	$-0.561677\pi$
$853$	−11.2474	−0.385105	−0.192553	+	0.981287i	$0.561677\pi$
$854$	−1.10102	−0.0376761
$855$	0	0
$856$	−11.3485	−0.387883
$857$	9.89898	0.338143	0.169071	−	0.985604i	$-0.445923\pi$
$857$	9.89898	0.338143	0.169071	+	0.985604i	$0.445923\pi$
$858$	0	0
$859$	−32.9444	−1.12405	−0.562024	−	0.827121i	$-0.689977\pi$
$859$	−32.9444	−1.12405	−0.562024	+	0.827121i	$0.689977\pi$
$860$	0	0
$861$	0	0
$862$	−11.7980	−0.401840
$863$	14.3485	0.488428	0.244214	−	0.969721i	$-0.421470\pi$
$863$	14.3485	0.488428	0.244214	+	0.969721i	$0.421470\pi$
$864$	0	0
$865$	0	0
$866$	15.5959	0.529971
$867$	0	0
$868$	−3.44949	−0.117083
$869$	−3.79796	−0.128837
$870$	0	0
$871$	17.2474	0.584408
$872$	19.1464	0.648380
$873$	0	0
$874$	−1.55051	−0.0524468
$875$	0	0
$876$	0	0
$877$	−42.6969	−1.44177	−0.720887	−	0.693053i	$-0.756265\pi$
$877$	−42.6969	−1.44177	−0.720887	+	0.693053i	$0.756265\pi$
$878$	−8.34847	−0.281747
$879$	0	0
$880$	0	0
$881$	55.2929	1.86286	0.931432	−	0.363916i	$-0.118560\pi$
$881$	55.2929	1.86286	0.931432	+	0.363916i	$0.118560\pi$
$882$	0	0
$883$	−11.8990	−0.400433	−0.200216	−	0.979752i	$-0.564164\pi$
$883$	−11.8990	−0.400433	−0.200216	+	0.979752i	$0.564164\pi$
$884$	−3.44949	−0.116019
$885$	0	0
$886$	32.4495	1.09016
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−0.898979	−0.0301508
$890$	0	0
$891$	0	0
$892$	−4.00000	−0.133930
$893$	−3.50510	−0.117294
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−32.0000	−1.06785
$899$	−25.6969	−0.857041
$900$	0	0
$901$	−0.550510	−0.0183402
$902$	12.0000	0.399556
$903$	0	0
$904$	−1.55051	−0.0515692
$905$	0	0
$906$	0	0
$907$	37.7980	1.25506	0.627530	−	0.778592i	$-0.284066\pi$
$907$	37.7980	1.25506	0.627530	+	0.778592i	$0.284066\pi$
$908$	17.6969	0.587294
$909$	0	0
$910$	0	0
$911$	25.7980	0.854725	0.427362	−	0.904080i	$-0.359443\pi$
$911$	25.7980	0.854725	0.427362	+	0.904080i	$0.359443\pi$
$912$	0	0
$913$	−2.69694	−0.0892556
$914$	−9.89898	−0.327429
$915$	0	0
$916$	−12.8990	−0.426194
$917$	2.79796	0.0923967
$918$	0	0
$919$	21.5959	0.712384	0.356192	−	0.934413i	$-0.384075\pi$
$919$	21.5959	0.712384	0.356192	+	0.934413i	$0.384075\pi$
$920$	0	0
$921$	0	0
$922$	−10.0000	−0.329332
$923$	−1.89898	−0.0625057
$924$	0	0
$925$	0	0
$926$	−28.2474	−0.928269
$927$	0	0
$928$	−7.44949	−0.244541
$929$	−7.59592	−0.249214	−0.124607	−	0.992206i	$-0.539767\pi$
$929$	−7.59592	−0.249214	−0.124607	+	0.992206i	$0.539767\pi$
$930$	0	0
$931$	0.449490	0.0147314
$932$	21.1464	0.692674
$933$	0	0
$934$	31.7980	1.04046
$935$	0	0
$936$	0	0
$937$	−18.9444	−0.618886	−0.309443	−	0.950918i	$-0.600143\pi$
$937$	−18.9444	−0.618886	−0.309443	+	0.950918i	$0.600143\pi$
$938$	5.00000	0.163256
$939$	0	0
$940$	0	0
$941$	−9.59592	−0.312818	−0.156409	−	0.987692i	$-0.549992\pi$
$941$	−9.59592	−0.312818	−0.156409	+	0.987692i	$0.549992\pi$
$942$	0	0
$943$	−16.8990	−0.550306
$944$	−9.89898	−0.322184
$945$	0	0
$946$	14.4495	0.469793
$947$	−16.0454	−0.521406	−0.260703	−	0.965419i	$-0.583954\pi$
$947$	−16.0454	−0.521406	−0.260703	+	0.965419i	$0.583954\pi$
$948$	0	0
$949$	−26.8990	−0.873178
$950$	0	0
$951$	0	0
$952$	−1.00000	−0.0324102
$953$	22.2929	0.722136	0.361068	−	0.932539i	$-0.382412\pi$
$953$	22.2929	0.722136	0.361068	+	0.932539i	$0.382412\pi$
$954$	0	0
$955$	0	0
$956$	28.4949	0.921591
$957$	0	0
$958$	36.0000	1.16311
$959$	−14.8990	−0.481113
$960$	0	0
$961$	−19.1010	−0.616162
$962$	39.1464	1.26213
$963$	0	0
$964$	25.3485	0.816419
$965$	0	0
$966$	0	0
$967$	−15.1010	−0.485616	−0.242808	−	0.970074i	$-0.578069\pi$
$967$	−15.1010	−0.485616	−0.242808	+	0.970074i	$0.578069\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	0	0
$971$	39.4949	1.26745	0.633726	−	0.773558i	$-0.281525\pi$
$971$	39.4949	1.26745	0.633726	+	0.773558i	$0.281525\pi$
$972$	0	0
$973$	2.24745	0.0720499
$974$	19.3939	0.621420
$975$	0	0
$976$	1.10102	0.0352428
$977$	−23.3939	−0.748436	−0.374218	−	0.927341i	$-0.622089\pi$
$977$	−23.3939	−0.748436	−0.374218	+	0.927341i	$0.622089\pi$
$978$	0	0
$979$	−26.9444	−0.861146
$980$	0	0
$981$	0	0
$982$	43.5959	1.39120
$983$	−5.30306	−0.169141	−0.0845707	−	0.996417i	$-0.526952\pi$
$983$	−5.30306	−0.169141	−0.0845707	+	0.996417i	$0.526952\pi$
$984$	0	0
$985$	0	0
$986$	−7.44949	−0.237240
$987$	0	0
$988$	−1.55051	−0.0493283
$989$	−20.3485	−0.647044
$990$	0	0
$991$	−31.3939	−0.997259	−0.498630	−	0.866815i	$-0.666163\pi$
$991$	−31.3939	−0.997259	−0.498630	+	0.866815i	$0.666163\pi$
$992$	3.44949	0.109521
$993$	0	0
$994$	−0.550510	−0.0174611
$995$	0	0
$996$	0	0
$997$	23.7423	0.751928	0.375964	−	0.926634i	$-0.377312\pi$
$997$	23.7423	0.751928	0.375964	+	0.926634i	$0.377312\pi$
$998$	−6.89898	−0.218383
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9450.2.a.ep.1.2		2
3.2	odd	2		9450.2.a.ea.1.1		2
5.2	odd	4		1890.2.g.m.379.4	yes	4
5.3	odd	4		1890.2.g.m.379.2	✓	4
5.4	even	2		9450.2.a.ek.1.2		2
15.2	even	4		1890.2.g.r.379.1	yes	4
15.8	even	4		1890.2.g.r.379.3	yes	4
15.14	odd	2		9450.2.a.ev.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1890.2.g.m.379.2	✓	4	5.3	odd	4
1890.2.g.m.379.4	yes	4	5.2	odd	4
1890.2.g.r.379.1	yes	4	15.2	even	4
1890.2.g.r.379.3	yes	4	15.8	even	4
9450.2.a.ea.1.1		2	3.2	odd	2
9450.2.a.ek.1.2		2	5.4	even	2
9450.2.a.ep.1.2		2	1.1	even	1	trivial
9450.2.a.ev.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9450 = 2 \cdot 3^{3} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9450.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{7} +1.00000 q^{8} +2.44949 q^{11} -3.44949 q^{13} -1.00000 q^{14} +1.00000 q^{16} +1.00000 q^{17} +0.449490 q^{19} +2.44949 q^{22} -3.44949 q^{23} -3.44949 q^{26} -1.00000 q^{28} -7.44949 q^{29} +3.44949 q^{31} +1.00000 q^{32} +1.00000 q^{34} -11.3485 q^{37} +0.449490 q^{38} +4.89898 q^{41} +5.89898 q^{43} +2.44949 q^{44} -3.44949 q^{46} -7.79796 q^{47} +1.00000 q^{49} -3.44949 q^{52} -0.550510 q^{53} -1.00000 q^{56} -7.44949 q^{58} -9.89898 q^{59} +1.10102 q^{61} +3.44949 q^{62} +1.00000 q^{64} -5.00000 q^{67} +1.00000 q^{68} +0.550510 q^{71} +7.79796 q^{73} -11.3485 q^{74} +0.449490 q^{76} -2.44949 q^{77} -1.55051 q^{79} +4.89898 q^{82} -1.10102 q^{83} +5.89898 q^{86} +2.44949 q^{88} -11.0000 q^{89} +3.44949 q^{91} -3.44949 q^{92} -7.79796 q^{94} -14.4495 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - 2 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{17} - 4 q^{19} - 2 q^{23} - 2 q^{26} - 2 q^{28} - 10 q^{29} + 2 q^{31} + 2 q^{32} + 2 q^{34} - 8 q^{37} - 4 q^{38} + 2 q^{43} - 2 q^{46} + 4 q^{47} + 2 q^{49} - 2 q^{52} - 6 q^{53} - 2 q^{56} - 10 q^{58} - 10 q^{59} + 12 q^{61} + 2 q^{62} + 2 q^{64} - 10 q^{67} + 2 q^{68} + 6 q^{71} - 4 q^{73} - 8 q^{74} - 4 q^{76} - 8 q^{79} - 12 q^{83} + 2 q^{86} - 22 q^{89} + 2 q^{91} - 2 q^{92} + 4 q^{94} - 24 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^7 + 2 * q^8 - 2 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^17 - 4 * q^19 - 2 * q^23 - 2 * q^26 - 2 * q^28 - 10 * q^29 + 2 * q^31 + 2 * q^32 + 2 * q^34 - 8 * q^37 - 4 * q^38 + 2 * q^43 - 2 * q^46 + 4 * q^47 + 2 * q^49 - 2 * q^52 - 6 * q^53 - 2 * q^56 - 10 * q^58 - 10 * q^59 + 12 * q^61 + 2 * q^62 + 2 * q^64 - 10 * q^67 + 2 * q^68 + 6 * q^71 - 4 * q^73 - 8 * q^74 - 4 * q^76 - 8 * q^79 - 12 * q^83 + 2 * q^86 - 22 * q^89 + 2 * q^91 - 2 * q^92 + 4 * q^94 - 24 * q^97 + 2 * q^98

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