LMFDB - Embedded newform 6664.2.a.o.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6664,2,Mod(1,6664)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6664.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6664 = 2^{3} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6664.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-3,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$53.2123079070$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5225.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + x + 11 $ x^4 - x^3 - 8*x^2 + x + 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.88301$ of defining polynomial
Character	$\chi$	$=$	6664.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.88301 q^{3} +1.78180 q^{5} +5.31175 q^{9} +3.23607 q^{11} -2.47214 q^{13} -5.13695 q^{15} -1.00000 q^{17} -7.76602 q^{19} -5.32962 q^{23} -1.82519 q^{25} -6.66481 q^{27} +10.5657 q^{29} +9.64137 q^{31} -9.32962 q^{33} -1.62142 q^{37} +7.12720 q^{39} -2.82519 q^{41} +0.587035 q^{43} +9.46448 q^{45} -12.1871 q^{47} +2.88301 q^{51} +3.40321 q^{53} +5.76602 q^{55} +22.3895 q^{57} +7.32962 q^{59} +3.82310 q^{61} -4.40485 q^{65} +0.394361 q^{67} +15.3654 q^{69} -2.15137 q^{71} +14.3572 q^{73} +5.26205 q^{75} +4.18710 q^{79} +3.27946 q^{81} -8.42108 q^{83} -1.78180 q^{85} -30.4610 q^{87} +17.0446 q^{89} -27.7962 q^{93} -13.8375 q^{95} +7.72398 q^{97} +17.1892 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + q^{5} + 7 q^{9} + 4 q^{11} + 8 q^{13} + 12 q^{15} - 4 q^{17} - 14 q^{19} + 8 q^{23} + 11 q^{25} - 12 q^{27} + 4 q^{29} - 5 q^{31} - 8 q^{33} - 4 q^{37} + 4 q^{39} + 7 q^{41} + 19 q^{43}+ \cdots + 2 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + q^5 + 7 * q^9 + 4 * q^11 + 8 * q^13 + 12 * q^15 - 4 * q^17 - 14 * q^19 + 8 * q^23 + 11 * q^25 - 12 * q^27 + 4 * q^29 - 5 * q^31 - 8 * q^33 - 4 * q^37 + 4 * q^39 + 7 * q^41 + 19 * q^43 + 2 * q^45 - 8 * q^47 + 3 * q^51 + 5 * q^53 + 6 * q^55 + 44 * q^57 + 23 * q^61 - 8 * q^65 + 15 * q^67 + 2 * q^69 + 2 * q^71 + 5 * q^73 - 10 * q^75 - 24 * q^79 - 8 * q^81 - 10 * q^83 - q^85 - 16 * q^87 + 16 * q^89 - 20 * q^93 + 22 * q^95 + 15 * q^97 + 2 * q^99

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
$3$	−2.88301	−1.66451	−0.832254	−	0.554395i	$-0.812950\pi$
$3$	−2.88301	−1.66451	−0.832254	+	0.554395i	$0.812950\pi$
$4$	0	0
$5$	1.78180	0.796845	0.398422	−	0.917202i	$-0.369558\pi$
$5$	1.78180	0.796845	0.398422	+	0.917202i	$0.369558\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.31175	1.77058
$10$	0	0
$11$	3.23607	0.975711	0.487856	−	0.872924i	$-0.337779\pi$
$11$	3.23607	0.975711	0.487856	+	0.872924i	$0.337779\pi$
$12$	0	0
$13$	−2.47214	−0.685647	−0.342824	−	0.939400i	$-0.611383\pi$
$13$	−2.47214	−0.685647	−0.342824	+	0.939400i	$0.611383\pi$
$14$	0	0
$15$	−5.13695	−1.32635
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−7.76602	−1.78165	−0.890824	−	0.454349i	$-0.849872\pi$
$19$	−7.76602	−1.78165	−0.890824	+	0.454349i	$0.849872\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.32962	−1.11130	−0.555651	−	0.831415i	$-0.687531\pi$
$23$	−5.32962	−1.11130	−0.555651	+	0.831415i	$0.687531\pi$
$24$	0	0
$25$	−1.82519	−0.365039
$26$	0	0
$27$	−6.66481	−1.28264
$28$	0	0
$29$	10.5657	1.96200	0.980999	−	0.194010i	$-0.0621495\pi$
$29$	10.5657	1.96200	0.980999	+	0.194010i	$0.0621495\pi$
$30$	0	0
$31$	9.64137	1.73164	0.865821	−	0.500354i	$-0.166797\pi$
$31$	9.64137	1.73164	0.865821	+	0.500354i	$0.166797\pi$
$32$	0	0
$33$	−9.32962	−1.62408
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.62142	−0.266559	−0.133280	−	0.991078i	$-0.542551\pi$
$37$	−1.62142	−0.266559	−0.133280	+	0.991078i	$0.542551\pi$
$38$	0	0
$39$	7.12720	1.14126
$40$	0	0
$41$	−2.82519	−0.441221	−0.220610	−	0.975362i	$-0.570805\pi$
$41$	−2.82519	−0.441221	−0.220610	+	0.975362i	$0.570805\pi$
$42$	0	0
$43$	0.587035	0.0895219	0.0447610	−	0.998998i	$-0.485747\pi$
$43$	0.587035	0.0895219	0.0447610	+	0.998998i	$0.485747\pi$
$44$	0	0
$45$	9.46448	1.41088
$46$	0	0
$47$	−12.1871	−1.77767	−0.888836	−	0.458226i	$-0.848485\pi$
$47$	−12.1871	−1.77767	−0.888836	+	0.458226i	$0.848485\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.88301	0.403702
$52$	0	0
$53$	3.40321	0.467468	0.233734	−	0.972301i	$-0.424906\pi$
$53$	3.40321	0.467468	0.233734	+	0.972301i	$0.424906\pi$
$54$	0	0
$55$	5.76602	0.777490
$56$	0	0
$57$	22.3895	2.96557
$58$	0	0
$59$	7.32962	0.954235	0.477118	−	0.878839i	$-0.341682\pi$
$59$	7.32962	0.954235	0.477118	+	0.878839i	$0.341682\pi$
$60$	0	0
$61$	3.82310	0.489498	0.244749	−	0.969586i	$-0.421294\pi$
$61$	3.82310	0.489498	0.244749	+	0.969586i	$0.421294\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.40485	−0.546354
$66$	0	0
$67$	0.394361	0.0481788	0.0240894	−	0.999710i	$-0.492331\pi$
$67$	0.394361	0.0481788	0.0240894	+	0.999710i	$0.492331\pi$
$68$	0	0
$69$	15.3654	1.84977
$70$	0	0
$71$	−2.15137	−0.255321	−0.127660	−	0.991818i	$-0.540747\pi$
$71$	−2.15137	−0.255321	−0.127660	+	0.991818i	$0.540747\pi$
$72$	0	0
$73$	14.3572	1.68039	0.840194	−	0.542286i	$-0.182441\pi$
$73$	14.3572	1.68039	0.840194	+	0.542286i	$0.182441\pi$
$74$	0	0
$75$	5.26205	0.607609
$76$	0	0
$77$	0	0
$78$	0	0
$79$	4.18710	0.471086	0.235543	−	0.971864i	$-0.424313\pi$
$79$	4.18710	0.471086	0.235543	+	0.971864i	$0.424313\pi$
$80$	0	0
$81$	3.27946	0.364385
$82$	0	0
$83$	−8.42108	−0.924334	−0.462167	−	0.886793i	$-0.652928\pi$
$83$	−8.42108	−0.924334	−0.462167	+	0.886793i	$0.652928\pi$
$84$	0	0
$85$	−1.78180	−0.193263
$86$	0	0
$87$	−30.4610	−3.26576
$88$	0	0
$89$	17.0446	1.80672	0.903361	−	0.428880i	$-0.141092\pi$
$89$	17.0446	1.80672	0.903361	+	0.428880i	$0.141092\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−27.7962	−2.88233
$94$	0	0
$95$	−13.8375	−1.41970
$96$	0	0
$97$	7.72398	0.784251	0.392126	−	0.919912i	$-0.371740\pi$
$97$	7.72398	0.784251	0.392126	+	0.919912i	$0.371740\pi$
$98$	0	0
$99$	17.1892	1.72758
$100$	0	0
$101$	13.6834	1.36155	0.680775	−	0.732492i	$-0.261643\pi$
$101$	13.6834	1.36155	0.680775	+	0.732492i	$0.261643\pi$
$102$	0	0
$103$	−6.90854	−0.680718	−0.340359	−	0.940295i	$-0.610549\pi$
$103$	−6.90854	−0.680718	−0.340359	+	0.940295i	$0.610549\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−14.9868	−1.44883	−0.724413	−	0.689366i	$-0.757889\pi$
$107$	−14.9868	−1.44883	−0.724413	+	0.689366i	$0.757889\pi$
$108$	0	0
$109$	−19.3874	−1.85698	−0.928490	−	0.371358i	$-0.878892\pi$
$109$	−19.3874	−1.85698	−0.928490	+	0.371358i	$0.878892\pi$
$110$	0	0
$111$	4.67456	0.443690
$112$	0	0
$113$	6.26971	0.589805	0.294902	−	0.955527i	$-0.404713\pi$
$113$	6.26971	0.589805	0.294902	+	0.955527i	$0.404713\pi$
$114$	0	0
$115$	−9.49631	−0.885536
$116$	0	0
$117$	−13.1314	−1.21400
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	8.14506	0.734416
$124$	0	0
$125$	−12.1611	−1.08772
$126$	0	0
$127$	0.362807	0.0321939	0.0160970	−	0.999870i	$-0.494876\pi$
$127$	0.362807	0.0321939	0.0160970	+	0.999870i	$0.494876\pi$
$128$	0	0
$129$	−1.69243	−0.149010
$130$	0	0
$131$	1.94218	0.169689	0.0848446	−	0.996394i	$-0.472961\pi$
$131$	1.94218	0.169689	0.0848446	+	0.996394i	$0.472961\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−11.8754	−1.02207
$136$	0	0
$137$	5.17481	0.442114	0.221057	−	0.975261i	$-0.429049\pi$
$137$	5.17481	0.442114	0.221057	+	0.975261i	$0.429049\pi$
$138$	0	0
$139$	−4.13559	−0.350776	−0.175388	−	0.984499i	$-0.556118\pi$
$139$	−4.13559	−0.350776	−0.175388	+	0.984499i	$0.556118\pi$
$140$	0	0
$141$	35.1356	2.95895
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	18.8259	1.56341
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.17013	−0.177784	−0.0888921	−	0.996041i	$-0.528333\pi$
$149$	−2.17013	−0.177784	−0.0888921	+	0.996041i	$0.528333\pi$
$150$	0	0
$151$	3.05917	0.248952	0.124476	−	0.992223i	$-0.460275\pi$
$151$	3.05917	0.248952	0.124476	+	0.992223i	$0.460275\pi$
$152$	0	0
$153$	−5.31175	−0.429430
$154$	0	0
$155$	17.1790	1.37985
$156$	0	0
$157$	−9.95313	−0.794346	−0.397173	−	0.917744i	$-0.630009\pi$
$157$	−9.95313	−0.794346	−0.397173	+	0.917744i	$0.630009\pi$
$158$	0	0
$159$	−9.81151	−0.778103
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.6613	1.22669	0.613345	−	0.789815i	$-0.289824\pi$
$163$	15.6613	1.22669	0.613345	+	0.789815i	$0.289824\pi$
$164$	0	0
$165$	−16.6235	−1.29414
$166$	0	0
$167$	8.11908	0.628273	0.314137	−	0.949378i	$-0.398285\pi$
$167$	8.11908	0.628273	0.314137	+	0.949378i	$0.398285\pi$
$168$	0	0
$169$	−6.88854	−0.529888
$170$	0	0
$171$	−41.2512	−3.15456
$172$	0	0
$173$	−12.3640	−0.940018	−0.470009	−	0.882662i	$-0.655749\pi$
$173$	−12.3640	−0.940018	−0.470009	+	0.882662i	$0.655749\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−21.1314	−1.58833
$178$	0	0
$179$	−18.1233	−1.35460	−0.677298	−	0.735709i	$-0.736849\pi$
$179$	−18.1233	−1.35460	−0.677298	+	0.735709i	$0.736849\pi$
$180$	0	0
$181$	18.4143	1.36873	0.684363	−	0.729142i	$-0.260081\pi$
$181$	18.4143	1.36873	0.684363	+	0.729142i	$0.260081\pi$
$182$	0	0
$183$	−11.0220	−0.814773
$184$	0	0
$185$	−2.88904	−0.212406
$186$	0	0
$187$	−3.23607	−0.236645
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.7013	0.774318	0.387159	−	0.922013i	$-0.373457\pi$
$191$	10.7013	0.774318	0.387159	+	0.922013i	$0.373457\pi$
$192$	0	0
$193$	−1.51254	−0.108875	−0.0544376	−	0.998517i	$-0.517337\pi$
$193$	−1.51254	−0.108875	−0.0544376	+	0.998517i	$0.517337\pi$
$194$	0	0
$195$	12.6992	0.909411
$196$	0	0
$197$	−7.27180	−0.518094	−0.259047	−	0.965865i	$-0.583409\pi$
$197$	−7.27180	−0.518094	−0.259047	+	0.965865i	$0.583409\pi$
$198$	0	0
$199$	23.1832	1.64341	0.821706	−	0.569912i	$-0.193023\pi$
$199$	23.1832	1.64341	0.821706	+	0.569912i	$0.193023\pi$
$200$	0	0
$201$	−1.13695	−0.0801940
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−5.03393	−0.351585
$206$	0	0
$207$	−28.3096	−1.96765
$208$	0	0
$209$	−25.1314	−1.73837
$210$	0	0
$211$	11.1535	0.767836	0.383918	−	0.923367i	$-0.374575\pi$
$211$	11.1535	0.767836	0.383918	+	0.923367i	$0.374575\pi$
$212$	0	0
$213$	6.20242	0.424983
$214$	0	0
$215$	1.04598	0.0713351
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−41.3921	−2.79702
$220$	0	0
$221$	2.47214	0.166294
$222$	0	0
$223$	15.5636	1.04222	0.521108	−	0.853491i	$-0.325519\pi$
$223$	15.5636	1.04222	0.521108	+	0.853491i	$0.325519\pi$
$224$	0	0
$225$	−9.69497	−0.646332
$226$	0	0
$227$	−15.3649	−1.01980	−0.509902	−	0.860232i	$-0.670318\pi$
$227$	−15.3649	−1.01980	−0.509902	+	0.860232i	$0.670318\pi$
$228$	0	0
$229$	14.9085	0.985184	0.492592	−	0.870260i	$-0.336050\pi$
$229$	14.9085	0.985184	0.492592	+	0.870260i	$0.336050\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.9085	0.845666	0.422833	−	0.906208i	$-0.361036\pi$
$233$	12.9085	0.845666	0.422833	+	0.906208i	$0.361036\pi$
$234$	0	0
$235$	−21.7150	−1.41653
$236$	0	0
$237$	−12.0715	−0.784126
$238$	0	0
$239$	7.05917	0.456620	0.228310	−	0.973589i	$-0.426680\pi$
$239$	7.05917	0.456620	0.228310	+	0.973589i	$0.426680\pi$
$240$	0	0
$241$	22.0309	1.41914	0.709568	−	0.704637i	$-0.248890\pi$
$241$	22.0309	1.41914	0.709568	+	0.704637i	$0.248890\pi$
$242$	0	0
$243$	10.5397	0.676122
$244$	0	0
$245$	0	0
$246$	0	0
$247$	19.1987	1.22158
$248$	0	0
$249$	24.2781	1.53856
$250$	0	0
$251$	−28.1267	−1.77534	−0.887671	−	0.460479i	$-0.847678\pi$
$251$	−28.1267	−1.77534	−0.887671	+	0.460479i	$0.847678\pi$
$252$	0	0
$253$	−17.2470	−1.08431
$254$	0	0
$255$	5.13695	0.321688
$256$	0	0
$257$	−9.04459	−0.564186	−0.282093	−	0.959387i	$-0.591029\pi$
$257$	−9.04459	−0.564186	−0.282093	+	0.959387i	$0.591029\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	56.1223	3.47388
$262$	0	0
$263$	14.6592	0.903927	0.451964	−	0.892036i	$-0.350724\pi$
$263$	14.6592	0.903927	0.451964	+	0.892036i	$0.350724\pi$
$264$	0	0
$265$	6.06384	0.372499
$266$	0	0
$267$	−49.1397	−3.00730
$268$	0	0
$269$	9.75926	0.595032	0.297516	−	0.954717i	$-0.403842\pi$
$269$	9.75926	0.595032	0.297516	+	0.954717i	$0.403842\pi$
$270$	0	0
$271$	−15.1602	−0.920918	−0.460459	−	0.887681i	$-0.652315\pi$
$271$	−15.1602	−0.920918	−0.460459	+	0.887681i	$0.652315\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.90645	−0.356172
$276$	0	0
$277$	10.7324	0.644846	0.322423	−	0.946596i	$-0.395503\pi$
$277$	10.7324	0.644846	0.322423	+	0.946596i	$0.395503\pi$
$278$	0	0
$279$	51.2126	3.06602
$280$	0	0
$281$	−2.06245	−0.123036	−0.0615179	−	0.998106i	$-0.519594\pi$
$281$	−2.06245	−0.123036	−0.0615179	+	0.998106i	$0.519594\pi$
$282$	0	0
$283$	15.4364	0.917597	0.458798	−	0.888540i	$-0.348280\pi$
$283$	15.4364	0.917597	0.458798	+	0.888540i	$0.348280\pi$
$284$	0	0
$285$	39.8936	2.36310
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−22.2683	−1.30539
$292$	0	0
$293$	−2.28503	−0.133493	−0.0667465	−	0.997770i	$-0.521262\pi$
$293$	−2.28503	−0.133493	−0.0667465	+	0.997770i	$0.521262\pi$
$294$	0	0
$295$	13.0599	0.760377
$296$	0	0
$297$	−21.5678	−1.25149
$298$	0	0
$299$	13.1755	0.761961
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−39.4494	−2.26631
$304$	0	0
$305$	6.81200	0.390054
$306$	0	0
$307$	20.6277	1.17728	0.588642	−	0.808394i	$-0.299663\pi$
$307$	20.6277	1.17728	0.588642	+	0.808394i	$0.299663\pi$
$308$	0	0
$309$	19.9174	1.13306
$310$	0	0
$311$	−18.2250	−1.03344	−0.516721	−	0.856154i	$-0.672848\pi$
$311$	−18.2250	−1.03344	−0.516721	+	0.856154i	$0.672848\pi$
$312$	0	0
$313$	15.6609	0.885205	0.442602	−	0.896718i	$-0.354055\pi$
$313$	15.6609	0.885205	0.442602	+	0.896718i	$0.354055\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.75508	0.0985750	0.0492875	−	0.998785i	$-0.484305\pi$
$317$	1.75508	0.0985750	0.0492875	+	0.998785i	$0.484305\pi$
$318$	0	0
$319$	34.1913	1.91434
$320$	0	0
$321$	43.2070	2.41158
$322$	0	0
$323$	7.76602	0.432113
$324$	0	0
$325$	4.51212	0.250288
$326$	0	0
$327$	55.8942	3.09096
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−14.2829	−0.785059	−0.392530	−	0.919739i	$-0.628400\pi$
$331$	−14.2829	−0.785059	−0.392530	+	0.919739i	$0.628400\pi$
$332$	0	0
$333$	−8.61256	−0.471965
$334$	0	0
$335$	0.702671	0.0383910
$336$	0	0
$337$	24.9290	1.35797	0.678983	−	0.734154i	$-0.262421\pi$
$337$	24.9290	1.35797	0.678983	+	0.734154i	$0.262421\pi$
$338$	0	0
$339$	−18.0756	−0.981734
$340$	0	0
$341$	31.2001	1.68958
$342$	0	0
$343$	0	0
$344$	0	0
$345$	27.3780	1.47398
$346$	0	0
$347$	35.1934	1.88928	0.944640	−	0.328110i	$-0.106412\pi$
$347$	35.1934	1.88928	0.944640	+	0.328110i	$0.106412\pi$
$348$	0	0
$349$	−2.71029	−0.145079	−0.0725394	−	0.997366i	$-0.523110\pi$
$349$	−2.71029	−0.145079	−0.0725394	+	0.997366i	$0.523110\pi$
$350$	0	0
$351$	16.4763	0.879441
$352$	0	0
$353$	12.3580	0.657749	0.328874	−	0.944374i	$-0.393331\pi$
$353$	12.3580	0.657749	0.328874	+	0.944374i	$0.393331\pi$
$354$	0	0
$355$	−3.83331	−0.203451
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.04942	−0.266498	−0.133249	−	0.991083i	$-0.542541\pi$
$359$	−5.04942	−0.266498	−0.133249	+	0.991083i	$0.542541\pi$
$360$	0	0
$361$	41.3111	2.17427
$362$	0	0
$363$	1.52184	0.0798758
$364$	0	0
$365$	25.5817	1.33901
$366$	0	0
$367$	−25.9566	−1.35492	−0.677461	−	0.735559i	$-0.736920\pi$
$367$	−25.9566	−1.35492	−0.677461	+	0.735559i	$0.736920\pi$
$368$	0	0
$369$	−15.0067	−0.781219
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−8.23890	−0.426594	−0.213297	−	0.976987i	$-0.568420\pi$
$373$	−8.23890	−0.426594	−0.213297	+	0.976987i	$0.568420\pi$
$374$	0	0
$375$	35.0606	1.81052
$376$	0	0
$377$	−26.1198	−1.34524
$378$	0	0
$379$	11.7755	0.604866	0.302433	−	0.953171i	$-0.402201\pi$
$379$	11.7755	0.604866	0.302433	+	0.953171i	$0.402201\pi$
$380$	0	0
$381$	−1.04598	−0.0535871
$382$	0	0
$383$	23.2827	1.18969	0.594846	−	0.803839i	$-0.297213\pi$
$383$	23.2827	1.18969	0.594846	+	0.803839i	$0.297213\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.11818	0.158506
$388$	0	0
$389$	−11.3717	−0.576566	−0.288283	−	0.957545i	$-0.593084\pi$
$389$	−11.3717	−0.576566	−0.288283	+	0.957545i	$0.593084\pi$
$390$	0	0
$391$	5.32962	0.269530
$392$	0	0
$393$	−5.59933	−0.282449
$394$	0	0
$395$	7.46058	0.375382
$396$	0	0
$397$	−25.0645	−1.25795	−0.628977	−	0.777424i	$-0.716526\pi$
$397$	−25.0645	−1.25795	−0.628977	+	0.777424i	$0.716526\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.49631	0.374348	0.187174	−	0.982327i	$-0.440067\pi$
$401$	7.49631	0.374348	0.187174	+	0.982327i	$0.440067\pi$
$402$	0	0
$403$	−23.8348	−1.18730
$404$	0	0
$405$	5.84334	0.290358
$406$	0	0
$407$	−5.24701	−0.260085
$408$	0	0
$409$	11.4657	0.566941	0.283470	−	0.958981i	$-0.408514\pi$
$409$	11.4657	0.566941	0.283470	+	0.958981i	$0.408514\pi$
$410$	0	0
$411$	−14.9190	−0.735901
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−15.0047	−0.736550
$416$	0	0
$417$	11.9230	0.583870
$418$	0	0
$419$	−3.40998	−0.166588	−0.0832942	−	0.996525i	$-0.526544\pi$
$419$	−3.40998	−0.166588	−0.0832942	+	0.996525i	$0.526544\pi$
$420$	0	0
$421$	13.9524	0.679998	0.339999	−	0.940426i	$-0.389573\pi$
$421$	13.9524	0.679998	0.339999	+	0.940426i	$0.389573\pi$
$422$	0	0
$423$	−64.7349	−3.14752
$424$	0	0
$425$	1.82519	0.0885349
$426$	0	0
$427$	0	0
$428$	0	0
$429$	23.0641	1.11354
$430$	0	0
$431$	27.7702	1.33764	0.668822	−	0.743423i	$-0.266799\pi$
$431$	27.7702	1.33764	0.668822	+	0.743423i	$0.266799\pi$
$432$	0	0
$433$	16.1913	0.778103	0.389052	−	0.921216i	$-0.372803\pi$
$433$	16.1913	0.778103	0.389052	+	0.921216i	$0.372803\pi$
$434$	0	0
$435$	−54.2754	−2.60230
$436$	0	0
$437$	41.3899	1.97995
$438$	0	0
$439$	14.5266	0.693318	0.346659	−	0.937991i	$-0.387316\pi$
$439$	14.5266	0.693318	0.346659	+	0.937991i	$0.387316\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	33.5993	1.59635	0.798176	−	0.602424i	$-0.205798\pi$
$443$	33.5993	1.59635	0.798176	+	0.602424i	$0.205798\pi$
$444$	0	0
$445$	30.3700	1.43968
$446$	0	0
$447$	6.25652	0.295923
$448$	0	0
$449$	−13.7191	−0.647447	−0.323723	−	0.946152i	$-0.604935\pi$
$449$	−13.7191	−0.647447	−0.323723	+	0.946152i	$0.604935\pi$
$450$	0	0
$451$	−9.14252	−0.430504
$452$	0	0
$453$	−8.81962	−0.414382
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9.18874	0.429831	0.214916	−	0.976633i	$-0.431052\pi$
$457$	9.18874	0.429831	0.214916	+	0.976633i	$0.431052\pi$
$458$	0	0
$459$	6.66481	0.311087
$460$	0	0
$461$	27.3338	1.27306	0.636531	−	0.771251i	$-0.280369\pi$
$461$	27.3338	1.27306	0.636531	+	0.771251i	$0.280369\pi$
$462$	0	0
$463$	7.36191	0.342137	0.171069	−	0.985259i	$-0.445278\pi$
$463$	7.36191	0.342137	0.171069	+	0.985259i	$0.445278\pi$
$464$	0	0
$465$	−49.5272	−2.29677
$466$	0	0
$467$	28.4805	1.31792	0.658960	−	0.752178i	$-0.270997\pi$
$467$	28.4805	1.31792	0.658960	+	0.752178i	$0.270997\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	28.6950	1.32219
$472$	0	0
$473$	1.89968	0.0873476
$474$	0	0
$475$	14.1745	0.650370
$476$	0	0
$477$	18.0770	0.827691
$478$	0	0
$479$	−7.72259	−0.352854	−0.176427	−	0.984314i	$-0.556454\pi$
$479$	−7.72259	−0.352854	−0.176427	+	0.984314i	$0.556454\pi$
$480$	0	0
$481$	4.00836	0.182766
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13.7626	0.624927
$486$	0	0
$487$	29.0172	1.31490	0.657448	−	0.753500i	$-0.271636\pi$
$487$	29.0172	1.31490	0.657448	+	0.753500i	$0.271636\pi$
$488$	0	0
$489$	−45.1518	−2.04183
$490$	0	0
$491$	14.2250	0.641964	0.320982	−	0.947085i	$-0.395987\pi$
$491$	14.2250	0.641964	0.320982	+	0.947085i	$0.395987\pi$
$492$	0	0
$493$	−10.5657	−0.475855
$494$	0	0
$495$	30.6277	1.37661
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−21.6971	−0.971294	−0.485647	−	0.874155i	$-0.661416\pi$
$499$	−21.6971	−0.971294	−0.485647	+	0.874155i	$0.661416\pi$
$500$	0	0
$501$	−23.4074	−1.04577
$502$	0	0
$503$	26.4700	1.18024	0.590120	−	0.807316i	$-0.299081\pi$
$503$	26.4700	1.18024	0.590120	+	0.807316i	$0.299081\pi$
$504$	0	0
$505$	24.3811	1.08494
$506$	0	0
$507$	19.8597	0.882002
$508$	0	0
$509$	−37.3380	−1.65498	−0.827488	−	0.561483i	$-0.810231\pi$
$509$	−37.3380	−1.65498	−0.827488	+	0.561483i	$0.810231\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	51.7591	2.28522
$514$	0	0
$515$	−12.3096	−0.542427
$516$	0	0
$517$	−39.4383	−1.73449
$518$	0	0
$519$	35.6456	1.56467
$520$	0	0
$521$	−40.7287	−1.78436	−0.892179	−	0.451681i	$-0.850824\pi$
$521$	−40.7287	−1.78436	−0.892179	+	0.451681i	$0.850824\pi$
$522$	0	0
$523$	−29.7056	−1.29894	−0.649468	−	0.760389i	$-0.725008\pi$
$523$	−29.7056	−1.29894	−0.649468	+	0.760389i	$0.725008\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.64137	−0.419985
$528$	0	0
$529$	5.40485	0.234993
$530$	0	0
$531$	38.9331	1.68955
$532$	0	0
$533$	6.98426	0.302522
$534$	0	0
$535$	−26.7034	−1.15449
$536$	0	0
$537$	52.2496	2.25473
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−45.4631	−1.95461	−0.977305	−	0.211835i	$-0.932056\pi$
$541$	−45.4631	−1.95461	−0.977305	+	0.211835i	$0.932056\pi$
$542$	0	0
$543$	−53.0887	−2.27825
$544$	0	0
$545$	−34.5445	−1.47972
$546$	0	0
$547$	−10.3113	−0.440879	−0.220440	−	0.975401i	$-0.570749\pi$
$547$	−10.3113	−0.440879	−0.220440	+	0.975401i	$0.570749\pi$
$548$	0	0
$549$	20.3074	0.866698
$550$	0	0
$551$	−82.0534	−3.49559
$552$	0	0
$553$	0	0
$554$	0	0
$555$	8.32913	0.353552
$556$	0	0
$557$	16.8770	0.715101	0.357550	−	0.933894i	$-0.383612\pi$
$557$	16.8770	0.715101	0.357550	+	0.933894i	$0.383612\pi$
$558$	0	0
$559$	−1.45123	−0.0613805
$560$	0	0
$561$	9.32962	0.393897
$562$	0	0
$563$	35.6197	1.50119	0.750597	−	0.660761i	$-0.229766\pi$
$563$	35.6197	1.50119	0.750597	+	0.660761i	$0.229766\pi$
$564$	0	0
$565$	11.1714	0.469983
$566$	0	0
$567$	0	0
$568$	0	0
$569$	32.2578	1.35232	0.676159	−	0.736755i	$-0.263643\pi$
$569$	32.2578	1.35232	0.676159	+	0.736755i	$0.263643\pi$
$570$	0	0
$571$	35.4946	1.48540	0.742702	−	0.669622i	$-0.233544\pi$
$571$	35.4946	1.48540	0.742702	+	0.669622i	$0.233544\pi$
$572$	0	0
$573$	−30.8519	−1.28886
$574$	0	0
$575$	9.72758	0.405668
$576$	0	0
$577$	−33.1356	−1.37945	−0.689726	−	0.724071i	$-0.742269\pi$
$577$	−33.1356	−1.37945	−0.689726	+	0.724071i	$0.742269\pi$
$578$	0	0
$579$	4.36068	0.181224
$580$	0	0
$581$	0	0
$582$	0	0
$583$	11.0130	0.456113
$584$	0	0
$585$	−23.3975	−0.967366
$586$	0	0
$587$	35.1397	1.45037	0.725186	−	0.688553i	$-0.241754\pi$
$587$	35.1397	1.45037	0.725186	+	0.688553i	$0.241754\pi$
$588$	0	0
$589$	−74.8751	−3.08518
$590$	0	0
$591$	20.9647	0.862372
$592$	0	0
$593$	30.0729	1.23495	0.617474	−	0.786591i	$-0.288156\pi$
$593$	30.0729	1.23495	0.617474	+	0.786591i	$0.288156\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−66.8373	−2.73547
$598$	0	0
$599$	20.6464	0.843591	0.421796	−	0.906691i	$-0.361400\pi$
$599$	20.6464	0.843591	0.421796	+	0.906691i	$0.361400\pi$
$600$	0	0
$601$	20.5961	0.840134	0.420067	−	0.907493i	$-0.362007\pi$
$601$	20.5961	0.840134	0.420067	+	0.907493i	$0.362007\pi$
$602$	0	0
$603$	2.09475	0.0853047
$604$	0	0
$605$	−0.940548	−0.0382387
$606$	0	0
$607$	14.9199	0.605582	0.302791	−	0.953057i	$-0.402082\pi$
$607$	14.9199	0.605582	0.302791	+	0.953057i	$0.402082\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.1282	1.21886
$612$	0	0
$613$	−24.9575	−1.00802	−0.504011	−	0.863697i	$-0.668143\pi$
$613$	−24.9575	−1.00802	−0.504011	+	0.863697i	$0.668143\pi$
$614$	0	0
$615$	14.5129	0.585215
$616$	0	0
$617$	−7.19596	−0.289698	−0.144849	−	0.989454i	$-0.546270\pi$
$617$	−7.19596	−0.289698	−0.144849	+	0.989454i	$0.546270\pi$
$618$	0	0
$619$	17.7770	0.714517	0.357258	−	0.934006i	$-0.383712\pi$
$619$	17.7770	0.714517	0.357258	+	0.934006i	$0.383712\pi$
$620$	0	0
$621$	35.5209	1.42540
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−12.5427	−0.501708
$626$	0	0
$627$	72.4540	2.89354
$628$	0	0
$629$	1.62142	0.0646501
$630$	0	0
$631$	35.3428	1.40698	0.703488	−	0.710708i	$-0.251625\pi$
$631$	35.3428	1.40698	0.703488	+	0.710708i	$0.251625\pi$
$632$	0	0
$633$	−32.1556	−1.27807
$634$	0	0
$635$	0.646450	0.0256536
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−11.4275	−0.452067
$640$	0	0
$641$	−3.66153	−0.144622	−0.0723108	−	0.997382i	$-0.523037\pi$
$641$	−3.66153	−0.144622	−0.0723108	+	0.997382i	$0.523037\pi$
$642$	0	0
$643$	−29.6783	−1.17040	−0.585199	−	0.810890i	$-0.698984\pi$
$643$	−29.6783	−1.17040	−0.585199	+	0.810890i	$0.698984\pi$
$644$	0	0
$645$	−3.01557	−0.118738
$646$	0	0
$647$	−28.9758	−1.13916	−0.569579	−	0.821937i	$-0.692894\pi$
$647$	−28.9758	−1.13916	−0.569579	+	0.821937i	$0.692894\pi$
$648$	0	0
$649$	23.7191	0.931058
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.2003	0.751367	0.375684	−	0.926748i	$-0.377408\pi$
$653$	19.2003	0.751367	0.375684	+	0.926748i	$0.377408\pi$
$654$	0	0
$655$	3.46058	0.135216
$656$	0	0
$657$	76.2621	2.97527
$658$	0	0
$659$	−5.13350	−0.199973	−0.0999865	−	0.994989i	$-0.531880\pi$
$659$	−5.13350	−0.199973	−0.0999865	+	0.994989i	$0.531880\pi$
$660$	0	0
$661$	34.8839	1.35683	0.678413	−	0.734681i	$-0.262668\pi$
$661$	34.8839	1.35683	0.678413	+	0.734681i	$0.262668\pi$
$662$	0	0
$663$	−7.12720	−0.276797
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−56.3111	−2.18037
$668$	0	0
$669$	−44.8700	−1.73478
$670$	0	0
$671$	12.3718	0.477609
$672$	0	0
$673$	−46.2976	−1.78464	−0.892320	−	0.451403i	$-0.850924\pi$
$673$	−46.2976	−1.78464	−0.892320	+	0.451403i	$0.850924\pi$
$674$	0	0
$675$	12.1646	0.468214
$676$	0	0
$677$	8.29450	0.318784	0.159392	−	0.987215i	$-0.449047\pi$
$677$	8.29450	0.318784	0.159392	+	0.987215i	$0.449047\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	44.2972	1.69747
$682$	0	0
$683$	−43.0198	−1.64611	−0.823053	−	0.567964i	$-0.807731\pi$
$683$	−43.0198	−1.64611	−0.823053	+	0.567964i	$0.807731\pi$
$684$	0	0
$685$	9.22047	0.352296
$686$	0	0
$687$	−42.9815	−1.63985
$688$	0	0
$689$	−8.41321	−0.320518
$690$	0	0
$691$	36.3817	1.38403	0.692013	−	0.721885i	$-0.256724\pi$
$691$	36.3817	1.38403	0.692013	+	0.721885i	$0.256724\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.36880	−0.279514
$696$	0	0
$697$	2.82519	0.107012
$698$	0	0
$699$	−37.2155	−1.40762
$700$	0	0
$701$	18.5920	0.702208	0.351104	−	0.936336i	$-0.385806\pi$
$701$	18.5920	0.702208	0.351104	+	0.936336i	$0.385806\pi$
$702$	0	0
$703$	12.5920	0.474915
$704$	0	0
$705$	62.6045	2.35782
$706$	0	0
$707$	0	0
$708$	0	0
$709$	11.0871	0.416384	0.208192	−	0.978088i	$-0.433242\pi$
$709$	11.0871	0.416384	0.208192	+	0.978088i	$0.433242\pi$
$710$	0	0
$711$	22.2409	0.834098
$712$	0	0
$713$	−51.3849	−1.92438
$714$	0	0
$715$	−14.2544	−0.533084
$716$	0	0
$717$	−20.3517	−0.760047
$718$	0	0
$719$	−0.357237	−0.0133227	−0.00666135	−	0.999978i	$-0.502120\pi$
$719$	−0.357237	−0.0133227	−0.00666135	+	0.999978i	$0.502120\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−63.5153	−2.36216
$724$	0	0
$725$	−19.2844	−0.716205
$726$	0	0
$727$	3.07614	0.114088	0.0570439	−	0.998372i	$-0.481833\pi$
$727$	3.07614	0.114088	0.0570439	+	0.998372i	$0.481833\pi$
$728$	0	0
$729$	−40.2245	−1.48980
$730$	0	0
$731$	−0.587035	−0.0217123
$732$	0	0
$733$	−17.9930	−0.664588	−0.332294	−	0.943176i	$-0.607823\pi$
$733$	−17.9930	−0.664588	−0.332294	+	0.943176i	$0.607823\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.27618	0.0470086
$738$	0	0
$739$	−8.47140	−0.311625	−0.155813	−	0.987787i	$-0.549800\pi$
$739$	−8.47140	−0.311625	−0.155813	+	0.987787i	$0.549800\pi$
$740$	0	0
$741$	−55.3500	−2.03333
$742$	0	0
$743$	23.1587	0.849612	0.424806	−	0.905284i	$-0.360342\pi$
$743$	23.1587	0.849612	0.424806	+	0.905284i	$0.360342\pi$
$744$	0	0
$745$	−3.86674	−0.141666
$746$	0	0
$747$	−44.7307	−1.63661
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−19.3849	−0.707363	−0.353682	−	0.935366i	$-0.615070\pi$
$751$	−19.3849	−0.707363	−0.353682	+	0.935366i	$0.615070\pi$
$752$	0	0
$753$	81.0896	2.95507
$754$	0	0
$755$	5.45083	0.198376
$756$	0	0
$757$	48.9669	1.77973	0.889866	−	0.456222i	$-0.150798\pi$
$757$	48.9669	1.77973	0.889866	+	0.456222i	$0.150798\pi$
$758$	0	0
$759$	49.7233	1.80484
$760$	0	0
$761$	25.9354	0.940158	0.470079	−	0.882624i	$-0.344225\pi$
$761$	25.9354	0.940158	0.470079	+	0.882624i	$0.344225\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−9.46448	−0.342189
$766$	0	0
$767$	−18.1198	−0.654269
$768$	0	0
$769$	5.37649	0.193881	0.0969407	−	0.995290i	$-0.469094\pi$
$769$	5.37649	0.193881	0.0969407	+	0.995290i	$0.469094\pi$
$770$	0	0
$771$	26.0756	0.939092
$772$	0	0
$773$	3.82642	0.137627	0.0688135	−	0.997630i	$-0.478079\pi$
$773$	3.82642	0.137627	0.0688135	+	0.997630i	$0.478079\pi$
$774$	0	0
$775$	−17.5974	−0.632116
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21.9405	0.786100
$780$	0	0
$781$	−6.96198	−0.249119
$782$	0	0
$783$	−70.4183	−2.51654
$784$	0	0
$785$	−17.7345	−0.632970
$786$	0	0
$787$	1.50487	0.0536427	0.0268214	−	0.999640i	$-0.491461\pi$
$787$	1.50487	0.0536427	0.0268214	+	0.999640i	$0.491461\pi$
$788$	0	0
$789$	−42.2628	−1.50459
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−9.45123	−0.335623
$794$	0	0
$795$	−17.4821	−0.620027
$796$	0	0
$797$	−28.4448	−1.00757	−0.503783	−	0.863830i	$-0.668059\pi$
$797$	−28.4448	−1.00757	−0.503783	+	0.863830i	$0.668059\pi$
$798$	0	0
$799$	12.1871	0.431149
$800$	0	0
$801$	90.5366	3.19896
$802$	0	0
$803$	46.4610	1.63957
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−28.1360	−0.990436
$808$	0	0
$809$	−29.5952	−1.04051	−0.520255	−	0.854011i	$-0.674163\pi$
$809$	−29.5952	−1.04051	−0.520255	+	0.854011i	$0.674163\pi$
$810$	0	0
$811$	−49.2818	−1.73052	−0.865259	−	0.501325i	$-0.832846\pi$
$811$	−49.2818	−1.73052	−0.865259	+	0.501325i	$0.832846\pi$
$812$	0	0
$813$	43.7071	1.53287
$814$	0	0
$815$	27.9053	0.977481
$816$	0	0
$817$	−4.55892	−0.159497
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−39.0114	−1.36151	−0.680753	−	0.732513i	$-0.738347\pi$
$821$	−39.0114	−1.36151	−0.680753	+	0.732513i	$0.738347\pi$
$822$	0	0
$823$	−36.4772	−1.27152	−0.635758	−	0.771888i	$-0.719312\pi$
$823$	−36.4772	−1.27152	−0.635758	+	0.771888i	$0.719312\pi$
$824$	0	0
$825$	17.0284	0.592851
$826$	0	0
$827$	−7.06846	−0.245795	−0.122897	−	0.992419i	$-0.539219\pi$
$827$	−7.06846	−0.245795	−0.122897	+	0.992419i	$0.539219\pi$
$828$	0	0
$829$	19.5952	0.680568	0.340284	−	0.940323i	$-0.389477\pi$
$829$	19.5952	0.680568	0.340284	+	0.940323i	$0.389477\pi$
$830$	0	0
$831$	−30.9416	−1.07335
$832$	0	0
$833$	0	0
$834$	0	0
$835$	14.4666	0.500636
$836$	0	0
$837$	−64.2579	−2.22108
$838$	0	0
$839$	−5.42576	−0.187318	−0.0936589	−	0.995604i	$-0.529856\pi$
$839$	−5.42576	−0.187318	−0.0936589	+	0.995604i	$0.529856\pi$
$840$	0	0
$841$	82.6338	2.84944
$842$	0	0
$843$	5.94608	0.204794
$844$	0	0
$845$	−12.2740	−0.422238
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−44.5032	−1.52735
$850$	0	0
$851$	8.64153	0.296228
$852$	0	0
$853$	−8.52577	−0.291917	−0.145958	−	0.989291i	$-0.546627\pi$
$853$	−8.52577	−0.291917	−0.145958	+	0.989291i	$0.546627\pi$
$854$	0	0
$855$	−73.5013	−2.51369
$856$	0	0
$857$	−28.9710	−0.989630	−0.494815	−	0.868998i	$-0.664764\pi$
$857$	−28.9710	−0.989630	−0.494815	+	0.868998i	$0.664764\pi$
$858$	0	0
$859$	−9.67776	−0.330201	−0.165100	−	0.986277i	$-0.552795\pi$
$859$	−9.67776	−0.330201	−0.165100	+	0.986277i	$0.552795\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.67293	0.329270	0.164635	−	0.986355i	$-0.447355\pi$
$863$	9.67293	0.329270	0.164635	+	0.986355i	$0.447355\pi$
$864$	0	0
$865$	−22.0302	−0.749048
$866$	0	0
$867$	−2.88301	−0.0979122
$868$	0	0
$869$	13.5498	0.459644
$870$	0	0
$871$	−0.974913	−0.0330337
$872$	0	0
$873$	41.0279	1.38858
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−40.8080	−1.37799	−0.688995	−	0.724766i	$-0.741948\pi$
$877$	−40.8080	−1.37799	−0.688995	+	0.724766i	$0.741948\pi$
$878$	0	0
$879$	6.58777	0.222200
$880$	0	0
$881$	−0.544255	−0.0183364	−0.00916821	−	0.999958i	$-0.502918\pi$
$881$	−0.544255	−0.0183364	−0.00916821	+	0.999958i	$0.502918\pi$
$882$	0	0
$883$	7.31126	0.246043	0.123022	−	0.992404i	$-0.460742\pi$
$883$	7.31126	0.246043	0.123022	+	0.992404i	$0.460742\pi$
$884$	0	0
$885$	−37.6519	−1.26565
$886$	0	0
$887$	−13.8600	−0.465374	−0.232687	−	0.972552i	$-0.574752\pi$
$887$	−13.8600	−0.465374	−0.232687	+	0.972552i	$0.574752\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	10.6126	0.355534
$892$	0	0
$893$	94.6453	3.16718
$894$	0	0
$895$	−32.2920	−1.07940
$896$	0	0
$897$	−37.9852	−1.26829
$898$	0	0
$899$	101.868	3.39748
$900$	0	0
$901$	−3.40321	−0.113378
$902$	0	0
$903$	0	0
$904$	0	0
$905$	32.8106	1.09066
$906$	0	0
$907$	50.8411	1.68815	0.844075	−	0.536225i	$-0.180150\pi$
$907$	50.8411	1.68815	0.844075	+	0.536225i	$0.180150\pi$
$908$	0	0
$909$	72.6829	2.41074
$910$	0	0
$911$	25.8459	0.856311	0.428156	−	0.903705i	$-0.359164\pi$
$911$	25.8459	0.856311	0.428156	+	0.903705i	$0.359164\pi$
$912$	0	0
$913$	−27.2512	−0.901883
$914$	0	0
$915$	−19.6391	−0.649248
$916$	0	0
$917$	0	0
$918$	0	0
$919$	44.8471	1.47937	0.739684	−	0.672954i	$-0.234975\pi$
$919$	44.8471	1.47937	0.739684	+	0.672954i	$0.234975\pi$
$920$	0	0
$921$	−59.4699	−1.95960
$922$	0	0
$923$	5.31848	0.175060
$924$	0	0
$925$	2.95940	0.0973044
$926$	0	0
$927$	−36.6964	−1.20527
$928$	0	0
$929$	44.2402	1.45147	0.725737	−	0.687972i	$-0.241499\pi$
$929$	44.2402	1.45147	0.725737	+	0.687972i	$0.241499\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	52.5428	1.72017
$934$	0	0
$935$	−5.76602	−0.188569
$936$	0	0
$937$	41.2813	1.34860	0.674300	−	0.738457i	$-0.264445\pi$
$937$	41.2813	1.34860	0.674300	+	0.738457i	$0.264445\pi$
$938$	0	0
$939$	−45.1505	−1.47343
$940$	0	0
$941$	11.1816	0.364509	0.182254	−	0.983251i	$-0.441661\pi$
$941$	11.1816	0.364509	0.182254	+	0.983251i	$0.441661\pi$
$942$	0	0
$943$	15.0572	0.490330
$944$	0	0
$945$	0	0
$946$	0	0
$947$	18.5810	0.603802	0.301901	−	0.953339i	$-0.402379\pi$
$947$	18.5810	0.603802	0.301901	+	0.953339i	$0.402379\pi$
$948$	0	0
$949$	−35.4930	−1.15215
$950$	0	0
$951$	−5.05991	−0.164079
$952$	0	0
$953$	13.3359	0.431993	0.215997	−	0.976394i	$-0.430700\pi$
$953$	13.3359	0.431993	0.215997	+	0.976394i	$0.430700\pi$
$954$	0	0
$955$	19.0675	0.617011
$956$	0	0
$957$	−98.5739	−3.18644
$958$	0	0
$959$	0	0
$960$	0	0
$961$	61.9561	1.99858
$962$	0	0
$963$	−79.6060	−2.56527
$964$	0	0
$965$	−2.69505	−0.0867567
$966$	0	0
$967$	10.9877	0.353341	0.176670	−	0.984270i	$-0.443467\pi$
$967$	10.9877	0.353341	0.176670	+	0.984270i	$0.443467\pi$
$968$	0	0
$969$	−22.3895	−0.719255
$970$	0	0
$971$	−47.3831	−1.52059	−0.760297	−	0.649575i	$-0.774947\pi$
$971$	−47.3831	−1.52059	−0.760297	+	0.649575i	$0.774947\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−13.0085	−0.416606
$976$	0	0
$977$	1.49148	0.0477166	0.0238583	−	0.999715i	$-0.492405\pi$
$977$	1.49148	0.0477166	0.0238583	+	0.999715i	$0.492405\pi$
$978$	0	0
$979$	55.1574	1.76284
$980$	0	0
$981$	−102.981	−3.28794
$982$	0	0
$983$	53.9441	1.72055	0.860275	−	0.509830i	$-0.170292\pi$
$983$	53.9441	1.72055	0.860275	+	0.509830i	$0.170292\pi$
$984$	0	0
$985$	−12.9569	−0.412841
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.12867	−0.0994860
$990$	0	0
$991$	−51.2401	−1.62769	−0.813847	−	0.581079i	$-0.802631\pi$
$991$	−51.2401	−1.62769	−0.813847	+	0.581079i	$0.802631\pi$
$992$	0	0
$993$	41.1778	1.30674
$994$	0	0
$995$	41.3077	1.30954
$996$	0	0
$997$	12.0544	0.381766	0.190883	−	0.981613i	$-0.438865\pi$
$997$	12.0544	0.381766	0.190883	+	0.981613i	$0.438865\pi$
$998$	0	0
$999$	10.8064	0.341900

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6664.2.a.o.1.1		4
7.6	odd	2		952.2.a.g.1.4	✓	4
21.20	even	2		8568.2.a.bj.1.3		4
28.27	even	2		1904.2.a.q.1.1		4
56.13	odd	2		7616.2.a.bj.1.1		4
56.27	even	2		7616.2.a.bp.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.g.1.4	✓	4	7.6	odd	2
1904.2.a.q.1.1		4	28.27	even	2
6664.2.a.o.1.1		4	1.1	even	1	trivial
7616.2.a.bj.1.1		4	56.13	odd	2
7616.2.a.bp.1.4		4	56.27	even	2
8568.2.a.bj.1.3		4	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6664 = 2^{3} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6664.a (trivial)

\(f(q)\)	\(=\)	\(q-2.88301 q^{3} +1.78180 q^{5} +5.31175 q^{9} +3.23607 q^{11} -2.47214 q^{13} -5.13695 q^{15} -1.00000 q^{17} -7.76602 q^{19} -5.32962 q^{23} -1.82519 q^{25} -6.66481 q^{27} +10.5657 q^{29} +9.64137 q^{31} -9.32962 q^{33} -1.62142 q^{37} +7.12720 q^{39} -2.82519 q^{41} +0.587035 q^{43} +9.46448 q^{45} -12.1871 q^{47} +2.88301 q^{51} +3.40321 q^{53} +5.76602 q^{55} +22.3895 q^{57} +7.32962 q^{59} +3.82310 q^{61} -4.40485 q^{65} +0.394361 q^{67} +15.3654 q^{69} -2.15137 q^{71} +14.3572 q^{73} +5.26205 q^{75} +4.18710 q^{79} +3.27946 q^{81} -8.42108 q^{83} -1.78180 q^{85} -30.4610 q^{87} +17.0446 q^{89} -27.7962 q^{93} -13.8375 q^{95} +7.72398 q^{97} +17.1892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + q^{5} + 7 q^{9} + 4 q^{11} + 8 q^{13} + 12 q^{15} - 4 q^{17} - 14 q^{19} + 8 q^{23} + 11 q^{25} - 12 q^{27} + 4 q^{29} - 5 q^{31} - 8 q^{33} - 4 q^{37} + 4 q^{39} + 7 q^{41} + 19 q^{43}+ \cdots + 2 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + q^5 + 7 * q^9 + 4 * q^11 + 8 * q^13 + 12 * q^15 - 4 * q^17 - 14 * q^19 + 8 * q^23 + 11 * q^25 - 12 * q^27 + 4 * q^29 - 5 * q^31 - 8 * q^33 - 4 * q^37 + 4 * q^39 + 7 * q^41 + 19 * q^43 + 2 * q^45 - 8 * q^47 + 3 * q^51 + 5 * q^53 + 6 * q^55 + 44 * q^57 + 23 * q^61 - 8 * q^65 + 15 * q^67 + 2 * q^69 + 2 * q^71 + 5 * q^73 - 10 * q^75 - 24 * q^79 - 8 * q^81 - 10 * q^83 - q^85 - 16 * q^87 + 16 * q^89 - 20 * q^93 + 22 * q^95 + 15 * q^97 + 2 * q^99

Introduction

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