LMFDB - Embedded newform 1323.2.a.x.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1323,2,Mod(1,1323)] 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("1323.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1323, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-2,0,4,-1,0,0,-9,0,-1,-7,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$10.5642081874$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.46050$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.46050 q^{2} +0.133074 q^{4} +0.593579 q^{5} -2.72665 q^{8} +0.866926 q^{10} -4.46050 q^{11} -4.51459 q^{13} -4.24844 q^{16} +0.273346 q^{17} +2.86693 q^{19} +0.0789903 q^{20} -6.51459 q^{22} -5.05408 q^{23} -4.64766 q^{25} -6.59358 q^{26} -0.352336 q^{29} +2.51459 q^{31} -0.751560 q^{32} +0.399223 q^{34} -6.64766 q^{37} +4.18716 q^{38} -1.61849 q^{40} +10.8961 q^{41} +3.38151 q^{43} -0.593579 q^{44} -7.38151 q^{46} -12.4356 q^{47} -6.78794 q^{50} -0.600777 q^{52} -11.3274 q^{53} -2.64766 q^{55} -0.514589 q^{58} +8.05408 q^{59} -2.73385 q^{61} +3.67257 q^{62} +7.39922 q^{64} -2.67977 q^{65} +5.86693 q^{67} +0.0363754 q^{68} -2.60078 q^{71} +11.1154 q^{73} -9.70895 q^{74} +0.381515 q^{76} +11.1623 q^{79} -2.52179 q^{80} +15.9138 q^{82} -16.5438 q^{83} +0.162253 q^{85} +4.93872 q^{86} +12.1623 q^{88} +5.37432 q^{89} -0.672570 q^{92} -18.1623 q^{94} +1.70175 q^{95} -2.26615 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 4 q^{4} - q^{5} - 9 q^{8} - q^{10} - 7 q^{11} + 2 q^{13} + 10 q^{16} + 5 q^{19} + 13 q^{20} - 4 q^{22} - 6 q^{23} - 2 q^{25} - 17 q^{26} - 13 q^{29} - 8 q^{31} - 25 q^{32} + 12 q^{34}+ \cdots - 14 q^{97}+O(q^{100}) $ 3 * q - 2 * q^2 + 4 * q^4 - q^5 - 9 * q^8 - q^10 - 7 * q^11 + 2 * q^13 + 10 * q^16 + 5 * q^19 + 13 * q^20 - 4 * q^22 - 6 * q^23 - 2 * q^25 - 17 * q^26 - 13 * q^29 - 8 * q^31 - 25 * q^32 + 12 * q^34 - 8 * q^37 + 7 * q^38 - 24 * q^40 - 2 * q^41 - 9 * q^43 + q^44 - 3 * q^46 - 9 * q^47 - 4 * q^50 + 9 * q^52 - 24 * q^53 + 4 * q^55 + 14 * q^58 + 15 * q^59 - q^61 + 21 * q^62 + 33 * q^64 - 10 * q^65 + 14 * q^67 - 39 * q^68 + 3 * q^71 + 7 * q^73 - 18 * q^76 + 6 * q^79 + 16 * q^80 + 43 * q^82 - 3 * q^83 - 27 * q^85 + 32 * q^86 + 9 * q^88 + 5 * q^89 - 12 * q^92 - 27 * q^94 - 16 * q^95 - 14 * q^97

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 $

$2$	1.46050	1.03273	0.516366	−	0.856368i	$-0.327284\pi$
$2$	1.46050	1.03273	0.516366	+	0.856368i	$0.327284\pi$
$3$	0	0
$3$	0	0
$4$	0.133074	0.0665372
$5$	0.593579	0.265457	0.132728	−	0.991152i	$-0.457626\pi$
$5$	0.593579	0.265457	0.132728	+	0.991152i	$0.457626\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.72665	−0.964018
$9$	0	0
$10$	0.866926	0.274146
$11$	−4.46050	−1.34489	−0.672446	−	0.740146i	$-0.734756\pi$
$11$	−4.46050	−1.34489	−0.672446	+	0.740146i	$0.734756\pi$
$12$	0	0
$13$	−4.51459	−1.25212	−0.626061	−	0.779774i	$-0.715334\pi$
$13$	−4.51459	−1.25212	−0.626061	+	0.779774i	$0.715334\pi$
$14$	0	0
$15$	0	0
$16$	−4.24844	−1.06211
$17$	0.273346	0.0662962	0.0331481	−	0.999450i	$-0.489447\pi$
$17$	0.273346	0.0662962	0.0331481	+	0.999450i	$0.489447\pi$
$18$	0	0
$19$	2.86693	0.657718	0.328859	−	0.944379i	$-0.393336\pi$
$19$	2.86693	0.657718	0.328859	+	0.944379i	$0.393336\pi$
$20$	0.0789903	0.0176628
$21$	0	0
$22$	−6.51459	−1.38892
$23$	−5.05408	−1.05385	−0.526925	−	0.849912i	$-0.676655\pi$
$23$	−5.05408	−1.05385	−0.526925	+	0.849912i	$0.676655\pi$
$24$	0	0
$25$	−4.64766	−0.929533
$26$	−6.59358	−1.29311
$27$	0	0
$28$	0	0
$29$	−0.352336	−0.0654272	−0.0327136	−	0.999465i	$-0.510415\pi$
$29$	−0.352336	−0.0654272	−0.0327136	+	0.999465i	$0.510415\pi$
$30$	0	0
$31$	2.51459	0.451634	0.225817	−	0.974170i	$-0.427495\pi$
$31$	2.51459	0.451634	0.225817	+	0.974170i	$0.427495\pi$
$32$	−0.751560	−0.132858
$33$	0	0
$34$	0.399223	0.0684663
$35$	0	0
$36$	0	0
$37$	−6.64766	−1.09287	−0.546435	−	0.837502i	$-0.684015\pi$
$37$	−6.64766	−1.09287	−0.546435	+	0.837502i	$0.684015\pi$
$38$	4.18716	0.679247
$39$	0	0
$40$	−1.61849	−0.255905
$41$	10.8961	1.70169	0.850843	−	0.525420i	$-0.176092\pi$
$41$	10.8961	1.70169	0.850843	+	0.525420i	$0.176092\pi$
$42$	0	0
$43$	3.38151	0.515676	0.257838	−	0.966188i	$-0.416990\pi$
$43$	3.38151	0.515676	0.257838	+	0.966188i	$0.416990\pi$
$44$	−0.593579	−0.0894855
$45$	0	0
$46$	−7.38151	−1.08834
$47$	−12.4356	−1.81392	−0.906959	−	0.421218i	$-0.861603\pi$
$47$	−12.4356	−1.81392	−0.906959	+	0.421218i	$0.861603\pi$
$48$	0	0
$49$	0	0
$50$	−6.78794	−0.959959
$51$	0	0
$52$	−0.600777	−0.0833127
$53$	−11.3274	−1.55594	−0.777971	−	0.628300i	$-0.783751\pi$
$53$	−11.3274	−1.55594	−0.777971	+	0.628300i	$0.783751\pi$
$54$	0	0
$55$	−2.64766	−0.357011
$56$	0	0
$57$	0	0
$58$	−0.514589	−0.0675689
$59$	8.05408	1.04855	0.524276	−	0.851548i	$-0.324336\pi$
$59$	8.05408	1.04855	0.524276	+	0.851548i	$0.324336\pi$
$60$	0	0
$61$	−2.73385	−0.350034	−0.175017	−	0.984565i	$-0.555998\pi$
$61$	−2.73385	−0.350034	−0.175017	+	0.984565i	$0.555998\pi$
$62$	3.67257	0.466417
$63$	0	0
$64$	7.39922	0.924903
$65$	−2.67977	−0.332384
$66$	0	0
$67$	5.86693	0.716759	0.358380	−	0.933576i	$-0.383329\pi$
$67$	5.86693	0.716759	0.358380	+	0.933576i	$0.383329\pi$
$68$	0.0363754	0.00441117
$69$	0	0
$70$	0	0
$71$	−2.60078	−0.308655	−0.154328	−	0.988020i	$-0.549321\pi$
$71$	−2.60078	−0.308655	−0.154328	+	0.988020i	$0.549321\pi$
$72$	0	0
$73$	11.1154	1.30096	0.650478	−	0.759525i	$-0.274569\pi$
$73$	11.1154	1.30096	0.650478	+	0.759525i	$0.274569\pi$
$74$	−9.70895	−1.12864
$75$	0	0
$76$	0.381515	0.0437627
$77$	0	0
$78$	0	0
$79$	11.1623	1.25585	0.627926	−	0.778273i	$-0.283904\pi$
$79$	11.1623	1.25585	0.627926	+	0.778273i	$0.283904\pi$
$80$	−2.52179	−0.281944
$81$	0	0
$82$	15.9138	1.75739
$83$	−16.5438	−1.81591	−0.907957	−	0.419063i	$-0.862359\pi$
$83$	−16.5438	−1.81591	−0.907957	+	0.419063i	$0.862359\pi$
$84$	0	0
$85$	0.162253	0.0175988
$86$	4.93872	0.532556
$87$	0	0
$88$	12.1623	1.29650
$89$	5.37432	0.569677	0.284838	−	0.958576i	$-0.408060\pi$
$89$	5.37432	0.569677	0.284838	+	0.958576i	$0.408060\pi$
$90$	0	0
$91$	0	0
$92$	−0.672570	−0.0701202
$93$	0	0
$94$	−18.1623	−1.87329
$95$	1.70175	0.174596
$96$	0	0
$97$	−2.26615	−0.230093	−0.115046	−	0.993360i	$-0.536702\pi$
$97$	−2.26615	−0.230093	−0.115046	+	0.993360i	$0.536702\pi$
$98$	0	0
$99$	0	0
$100$	−0.618485	−0.0618485
$101$	−9.35661	−0.931017	−0.465509	−	0.885043i	$-0.654129\pi$
$101$	−9.35661	−0.931017	−0.465509	+	0.885043i	$0.654129\pi$
$102$	0	0
$103$	−15.7630	−1.55318	−0.776589	−	0.630008i	$-0.783052\pi$
$103$	−15.7630	−1.55318	−0.776589	+	0.630008i	$0.783052\pi$
$104$	12.3097	1.20707
$105$	0	0
$106$	−16.5438	−1.60687
$107$	−1.02491	−0.0990814	−0.0495407	−	0.998772i	$-0.515776\pi$
$107$	−1.02491	−0.0990814	−0.0495407	+	0.998772i	$0.515776\pi$
$108$	0	0
$109$	1.29533	0.124070	0.0620349	−	0.998074i	$-0.480241\pi$
$109$	1.29533	0.124070	0.0620349	+	0.998074i	$0.480241\pi$
$110$	−3.86693	−0.368697
$111$	0	0
$112$	0	0
$113$	−14.2953	−1.34479	−0.672396	−	0.740192i	$-0.734735\pi$
$113$	−14.2953	−1.34479	−0.672396	+	0.740192i	$0.734735\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	−0.0468870	−0.00435335
$117$	0	0
$118$	11.7630	1.08287
$119$	0	0
$120$	0	0
$121$	8.89610	0.808737
$122$	−3.99280	−0.361491
$123$	0	0
$124$	0.334628	0.0300504
$125$	−5.72665	−0.512207
$126$	0	0
$127$	12.3346	1.09452	0.547261	−	0.836962i	$-0.315671\pi$
$127$	12.3346	1.09452	0.547261	+	0.836962i	$0.315671\pi$
$128$	12.3097	1.08804
$129$	0	0
$130$	−3.91381	−0.343264
$131$	3.19436	0.279092	0.139546	−	0.990216i	$-0.455436\pi$
$131$	3.19436	0.279092	0.139546	+	0.990216i	$0.455436\pi$
$132$	0	0
$133$	0	0
$134$	8.56867	0.740221
$135$	0	0
$136$	−0.745320	−0.0639107
$137$	−10.1082	−0.863599	−0.431800	−	0.901970i	$-0.642121\pi$
$137$	−10.1082	−0.863599	−0.431800	+	0.901970i	$0.642121\pi$
$138$	0	0
$139$	18.0761	1.53319	0.766596	−	0.642130i	$-0.221949\pi$
$139$	18.0761	1.53319	0.766596	+	0.642130i	$0.221949\pi$
$140$	0	0
$141$	0	0
$142$	−3.79845	−0.318759
$143$	20.1373	1.68397
$144$	0	0
$145$	−0.209140	−0.0173681
$146$	16.2340	1.34354
$147$	0	0
$148$	−0.884634	−0.0727165
$149$	−14.0541	−1.15136	−0.575678	−	0.817677i	$-0.695262\pi$
$149$	−14.0541	−1.15136	−0.575678	+	0.817677i	$0.695262\pi$
$150$	0	0
$151$	0.381515	0.0310472	0.0155236	−	0.999880i	$-0.495058\pi$
$151$	0.381515	0.0310472	0.0155236	+	0.999880i	$0.495058\pi$
$152$	−7.81711	−0.634052
$153$	0	0
$154$	0	0
$155$	1.49261	0.119889
$156$	0	0
$157$	−7.51459	−0.599729	−0.299865	−	0.953982i	$-0.596942\pi$
$157$	−7.51459	−0.599729	−0.299865	+	0.953982i	$0.596942\pi$
$158$	16.3025	1.29696
$159$	0	0
$160$	−0.446110	−0.0352681
$161$	0	0
$162$	0	0
$163$	15.1914	1.18988	0.594942	−	0.803768i	$-0.297175\pi$
$163$	15.1914	1.18988	0.594942	+	0.803768i	$0.297175\pi$
$164$	1.44999	0.113225
$165$	0	0
$166$	−24.1623	−1.87535
$167$	−8.95311	−0.692813	−0.346406	−	0.938085i	$-0.612598\pi$
$167$	−8.95311	−0.692813	−0.346406	+	0.938085i	$0.612598\pi$
$168$	0	0
$169$	7.38151	0.567809
$170$	0.236971	0.0181748
$171$	0	0
$172$	0.449993	0.0343117
$173$	10.4605	0.795297	0.397649	−	0.917538i	$-0.369826\pi$
$173$	10.4605	0.795297	0.397649	+	0.917538i	$0.369826\pi$
$174$	0	0
$175$	0	0
$176$	18.9502	1.42842
$177$	0	0
$178$	7.84922	0.588324
$179$	8.96790	0.670292	0.335146	−	0.942166i	$-0.391214\pi$
$179$	8.96790	0.670292	0.335146	+	0.942166i	$0.391214\pi$
$180$	0	0
$181$	5.04689	0.375132	0.187566	−	0.982252i	$-0.439940\pi$
$181$	5.04689	0.375132	0.187566	+	0.982252i	$0.439940\pi$
$182$	0	0
$183$	0	0
$184$	13.7807	1.01593
$185$	−3.94592	−0.290109
$186$	0	0
$187$	−1.21926	−0.0891613
$188$	−1.65486	−0.120693
$189$	0	0
$190$	2.48541	0.180311
$191$	−12.1301	−0.877707	−0.438853	−	0.898559i	$-0.644615\pi$
$191$	−12.1301	−0.877707	−0.438853	+	0.898559i	$0.644615\pi$
$192$	0	0
$193$	17.1623	1.23537	0.617683	−	0.786427i	$-0.288071\pi$
$193$	17.1623	1.23537	0.617683	+	0.786427i	$0.288071\pi$
$194$	−3.30972	−0.237624
$195$	0	0
$196$	0	0
$197$	−0.751560	−0.0535464	−0.0267732	−	0.999642i	$-0.508523\pi$
$197$	−0.751560	−0.0535464	−0.0267732	+	0.999642i	$0.508523\pi$
$198$	0	0
$199$	10.2953	0.729816	0.364908	−	0.931044i	$-0.381100\pi$
$199$	10.2953	0.729816	0.364908	+	0.931044i	$0.381100\pi$
$200$	12.6726	0.896086
$201$	0	0
$202$	−13.6654	−0.961492
$203$	0	0
$204$	0	0
$205$	6.46770	0.451724
$206$	−23.0220	−1.60402
$207$	0	0
$208$	19.1800	1.32989
$209$	−12.7879	−0.884560
$210$	0	0
$211$	−16.1154	−1.10943	−0.554714	−	0.832041i	$-0.687172\pi$
$211$	−16.1154	−1.10943	−0.554714	+	0.832041i	$0.687172\pi$
$212$	−1.50739	−0.103528
$213$	0	0
$214$	−1.49688	−0.102325
$215$	2.00720	0.136890
$216$	0	0
$217$	0	0
$218$	1.89183	0.128131
$219$	0	0
$220$	−0.352336	−0.0237545
$221$	−1.23405	−0.0830109
$222$	0	0
$223$	−6.95311	−0.465615	−0.232807	−	0.972523i	$-0.574791\pi$
$223$	−6.95311	−0.465615	−0.232807	+	0.972523i	$0.574791\pi$
$224$	0	0
$225$	0	0
$226$	−20.8784	−1.38881
$227$	5.29105	0.351180	0.175590	−	0.984463i	$-0.443817\pi$
$227$	5.29105	0.351180	0.175590	+	0.984463i	$0.443817\pi$
$228$	0	0
$229$	11.7237	0.774726	0.387363	−	0.921927i	$-0.373386\pi$
$229$	11.7237	0.774726	0.387363	+	0.921927i	$0.373386\pi$
$230$	−4.38151	−0.288909
$231$	0	0
$232$	0.960699	0.0630730
$233$	3.87120	0.253611	0.126805	−	0.991928i	$-0.459528\pi$
$233$	3.87120	0.253611	0.126805	+	0.991928i	$0.459528\pi$
$234$	0	0
$235$	−7.38151	−0.481517
$236$	1.07179	0.0697678
$237$	0	0
$238$	0	0
$239$	12.3992	0.802039	0.401020	−	0.916069i	$-0.368656\pi$
$239$	12.3992	0.802039	0.401020	+	0.916069i	$0.368656\pi$
$240$	0	0
$241$	−16.5615	−1.06682	−0.533409	−	0.845857i	$-0.679089\pi$
$241$	−16.5615	−1.06682	−0.533409	+	0.845857i	$0.679089\pi$
$242$	12.9928	0.835209
$243$	0	0
$244$	−0.363806	−0.0232903
$245$	0	0
$246$	0	0
$247$	−12.9430	−0.823543
$248$	−6.85641	−0.435383
$249$	0	0
$250$	−8.36381	−0.528974
$251$	1.84922	0.116722	0.0583608	−	0.998296i	$-0.481413\pi$
$251$	1.84922	0.116722	0.0583608	+	0.998296i	$0.481413\pi$
$252$	0	0
$253$	22.5438	1.41731
$254$	18.0148	1.13035
$255$	0	0
$256$	3.17996	0.198748
$257$	26.8420	1.67436	0.837180	−	0.546928i	$-0.184203\pi$
$257$	26.8420	1.67436	0.837180	+	0.546928i	$0.184203\pi$
$258$	0	0
$259$	0	0
$260$	−0.356609	−0.0221159
$261$	0	0
$262$	4.66537	0.288228
$263$	20.2848	1.25082	0.625408	−	0.780298i	$-0.284933\pi$
$263$	20.2848	1.25082	0.625408	+	0.780298i	$0.284933\pi$
$264$	0	0
$265$	−6.72373	−0.413035
$266$	0	0
$267$	0	0
$268$	0.780738	0.0476912
$269$	−8.72665	−0.532073	−0.266037	−	0.963963i	$-0.585714\pi$
$269$	−8.72665	−0.532073	−0.266037	+	0.963963i	$0.585714\pi$
$270$	0	0
$271$	24.1914	1.46952	0.734762	−	0.678325i	$-0.237294\pi$
$271$	24.1914	1.46952	0.734762	+	0.678325i	$0.237294\pi$
$272$	−1.16129	−0.0704138
$273$	0	0
$274$	−14.7630	−0.891867
$275$	20.7309	1.25012
$276$	0	0
$277$	−7.11537	−0.427521	−0.213760	−	0.976886i	$-0.568571\pi$
$277$	−7.11537	−0.427521	−0.213760	+	0.976886i	$0.568571\pi$
$278$	26.4002	1.58338
$279$	0	0
$280$	0	0
$281$	7.89610	0.471042	0.235521	−	0.971869i	$-0.424320\pi$
$281$	7.89610	0.471042	0.235521	+	0.971869i	$0.424320\pi$
$282$	0	0
$283$	2.20914	0.131320	0.0656599	−	0.997842i	$-0.479085\pi$
$283$	2.20914	0.131320	0.0656599	+	0.997842i	$0.479085\pi$
$284$	−0.346097	−0.0205371
$285$	0	0
$286$	29.4107	1.73909
$287$	0	0
$288$	0	0
$289$	−16.9253	−0.995605
$290$	−0.305449	−0.0179366
$291$	0	0
$292$	1.47917	0.0865620
$293$	−19.1914	−1.12118	−0.560588	−	0.828095i	$-0.689425\pi$
$293$	−19.1914	−1.12118	−0.560588	+	0.828095i	$0.689425\pi$
$294$	0	0
$295$	4.78074	0.278345
$296$	18.1259	1.05355
$297$	0	0
$298$	−20.5261	−1.18904
$299$	22.8171	1.31955
$300$	0	0
$301$	0	0
$302$	0.557204	0.0320635
$303$	0	0
$304$	−12.1800	−0.698569
$305$	−1.62276	−0.0929188
$306$	0	0
$307$	−13.9138	−0.794103	−0.397052	−	0.917796i	$-0.629967\pi$
$307$	−13.9138	−0.794103	−0.397052	+	0.917796i	$0.629967\pi$
$308$	0	0
$309$	0	0
$310$	2.17996	0.123813
$311$	10.6549	0.604182	0.302091	−	0.953279i	$-0.402315\pi$
$311$	10.6549	0.604182	0.302091	+	0.953279i	$0.402315\pi$
$312$	0	0
$313$	−16.5615	−0.936110	−0.468055	−	0.883699i	$-0.655045\pi$
$313$	−16.5615	−0.936110	−0.468055	+	0.883699i	$0.655045\pi$
$314$	−10.9751	−0.619360
$315$	0	0
$316$	1.48541	0.0835609
$317$	−26.6372	−1.49609	−0.748046	−	0.663647i	$-0.769008\pi$
$317$	−26.6372	−1.49609	−0.748046	+	0.663647i	$0.769008\pi$
$318$	0	0
$319$	1.57160	0.0879926
$320$	4.39203	0.245522
$321$	0	0
$322$	0	0
$323$	0.783663	0.0436042
$324$	0	0
$325$	20.9823	1.16389
$326$	22.1872	1.22883
$327$	0	0
$328$	−29.7099	−1.64045
$329$	0	0
$330$	0	0
$331$	−23.3068	−1.28106	−0.640529	−	0.767934i	$-0.721285\pi$
$331$	−23.3068	−1.28106	−0.640529	+	0.767934i	$0.721285\pi$
$332$	−2.20155	−0.120826
$333$	0	0
$334$	−13.0761	−0.715490
$335$	3.48249	0.190269
$336$	0	0
$337$	−23.2383	−1.26587	−0.632936	−	0.774204i	$-0.718150\pi$
$337$	−23.2383	−1.26587	−0.632936	+	0.774204i	$0.718150\pi$
$338$	10.7807	0.586395
$339$	0	0
$340$	0.0215917	0.00117097
$341$	−11.2163	−0.607399
$342$	0	0
$343$	0	0
$344$	−9.22022	−0.497121
$345$	0	0
$346$	15.2776	0.821330
$347$	−17.1373	−0.919981	−0.459990	−	0.887924i	$-0.652147\pi$
$347$	−17.1373	−0.919981	−0.459990	+	0.887924i	$0.652147\pi$
$348$	0	0
$349$	−19.5146	−1.04459	−0.522296	−	0.852764i	$-0.674924\pi$
$349$	−19.5146	−1.04459	−0.522296	+	0.852764i	$0.674924\pi$
$350$	0	0
$351$	0	0
$352$	3.35234	0.178680
$353$	−16.0613	−0.854856	−0.427428	−	0.904049i	$-0.640580\pi$
$353$	−16.0613	−0.854856	−0.427428	+	0.904049i	$0.640580\pi$
$354$	0	0
$355$	−1.54377	−0.0819347
$356$	0.715184	0.0379047
$357$	0	0
$358$	13.0977	0.692233
$359$	−13.8640	−0.731714	−0.365857	−	0.930671i	$-0.619224\pi$
$359$	−13.8640	−0.731714	−0.365857	+	0.930671i	$0.619224\pi$
$360$	0	0
$361$	−10.7807	−0.567407
$362$	7.37100	0.387411
$363$	0	0
$364$	0	0
$365$	6.59785	0.345347
$366$	0	0
$367$	25.2953	1.32041	0.660203	−	0.751088i	$-0.270470\pi$
$367$	25.2953	1.32041	0.660203	+	0.751088i	$0.270470\pi$
$368$	21.4720	1.11930
$369$	0	0
$370$	−5.76303	−0.299606
$371$	0	0
$372$	0	0
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	−1.78074	−0.0920798
$375$	0	0
$376$	33.9076	1.74865
$377$	1.59065	0.0819229
$378$	0	0
$379$	−18.8099	−0.966200	−0.483100	−	0.875565i	$-0.660489\pi$
$379$	−18.8099	−0.966200	−0.483100	+	0.875565i	$0.660489\pi$
$380$	0.226459	0.0116171
$381$	0	0
$382$	−17.7161	−0.906437
$383$	−35.1416	−1.79565	−0.897826	−	0.440350i	$-0.854855\pi$
$383$	−35.1416	−1.79565	−0.897826	+	0.440350i	$0.854855\pi$
$384$	0	0
$385$	0	0
$386$	25.0656	1.27580
$387$	0	0
$388$	−0.301567	−0.0153097
$389$	8.37859	0.424811	0.212406	−	0.977182i	$-0.431870\pi$
$389$	8.37859	0.424811	0.212406	+	0.977182i	$0.431870\pi$
$390$	0	0
$391$	−1.38151	−0.0698662
$392$	0	0
$393$	0	0
$394$	−1.09766	−0.0552992
$395$	6.62568	0.333374
$396$	0	0
$397$	−9.24844	−0.464166	−0.232083	−	0.972696i	$-0.574554\pi$
$397$	−9.24844	−0.464166	−0.232083	+	0.972696i	$0.574554\pi$
$398$	15.0364	0.753705
$399$	0	0
$400$	19.7453	0.987266
$401$	0.147469	0.00736425	0.00368212	−	0.999993i	$-0.498828\pi$
$401$	0.147469	0.00736425	0.00368212	+	0.999993i	$0.498828\pi$
$402$	0	0
$403$	−11.3523	−0.565500
$404$	−1.24513	−0.0619473
$405$	0	0
$406$	0	0
$407$	29.6519	1.46979
$408$	0	0
$409$	−3.92528	−0.194093	−0.0970463	−	0.995280i	$-0.530940\pi$
$409$	−3.92528	−0.194093	−0.0970463	+	0.995280i	$0.530940\pi$
$410$	9.44611	0.466510
$411$	0	0
$412$	−2.09766	−0.103344
$413$	0	0
$414$	0	0
$415$	−9.82004	−0.482047
$416$	3.39298	0.166355
$417$	0	0
$418$	−18.6768	−0.913514
$419$	−21.0000	−1.02592	−0.512959	−	0.858413i	$-0.671451\pi$
$419$	−21.0000	−1.02592	−0.512959	+	0.858413i	$0.671451\pi$
$420$	0	0
$421$	−24.6883	−1.20323	−0.601617	−	0.798784i	$-0.705477\pi$
$421$	−24.6883	−1.20323	−0.601617	+	0.798784i	$0.705477\pi$
$422$	−23.5366	−1.14574
$423$	0	0
$424$	30.8860	1.49996
$425$	−1.27042	−0.0616245
$426$	0	0
$427$	0	0
$428$	−0.136389	−0.00659260
$429$	0	0
$430$	2.93152	0.141371
$431$	5.47490	0.263717	0.131858	−	0.991269i	$-0.457906\pi$
$431$	5.47490	0.263717	0.131858	+	0.991269i	$0.457906\pi$
$432$	0	0
$433$	23.6300	1.13558	0.567792	−	0.823172i	$-0.307798\pi$
$433$	23.6300	1.13558	0.567792	+	0.823172i	$0.307798\pi$
$434$	0	0
$435$	0	0
$436$	0.172375	0.00825526
$437$	−14.4897	−0.693136
$438$	0	0
$439$	−5.26615	−0.251340	−0.125670	−	0.992072i	$-0.540108\pi$
$439$	−5.26615	−0.251340	−0.125670	+	0.992072i	$0.540108\pi$
$440$	7.21926	0.344165
$441$	0	0
$442$	−1.80233	−0.0857281
$443$	−6.03638	−0.286797	−0.143398	−	0.989665i	$-0.545803\pi$
$443$	−6.03638	−0.286797	−0.143398	+	0.989665i	$0.545803\pi$
$444$	0	0
$445$	3.19008	0.151224
$446$	−10.1551	−0.480856
$447$	0	0
$448$	0	0
$449$	−22.1445	−1.04507	−0.522533	−	0.852619i	$-0.675013\pi$
$449$	−22.1445	−1.04507	−0.522533	+	0.852619i	$0.675013\pi$
$450$	0	0
$451$	−48.6021	−2.28858
$452$	−1.90234	−0.0894787
$453$	0	0
$454$	7.72761	0.362675
$455$	0	0
$456$	0	0
$457$	−5.33463	−0.249543	−0.124772	−	0.992185i	$-0.539820\pi$
$457$	−5.33463	−0.249543	−0.124772	+	0.992185i	$0.539820\pi$
$458$	17.1226	0.800085
$459$	0	0
$460$	−0.399223	−0.0186139
$461$	29.6946	1.38301	0.691507	−	0.722370i	$-0.256947\pi$
$461$	29.6946	1.38301	0.691507	+	0.722370i	$0.256947\pi$
$462$	0	0
$463$	18.5907	0.863981	0.431990	−	0.901878i	$-0.357811\pi$
$463$	18.5907	0.863981	0.431990	+	0.901878i	$0.357811\pi$
$464$	1.49688	0.0694909
$465$	0	0
$466$	5.65390	0.261912
$467$	24.6126	1.13894	0.569468	−	0.822013i	$-0.307149\pi$
$467$	24.6126	1.13894	0.569468	+	0.822013i	$0.307149\pi$
$468$	0	0
$469$	0	0
$470$	−10.7807	−0.497278
$471$	0	0
$472$	−21.9607	−1.01082
$473$	−15.0833	−0.693529
$474$	0	0
$475$	−13.3245	−0.611370
$476$	0	0
$477$	0	0
$478$	18.1091	0.828292
$479$	−0.356609	−0.0162939	−0.00814693	−	0.999967i	$-0.502593\pi$
$479$	−0.356609	−0.0162939	−0.00814693	+	0.999967i	$0.502593\pi$
$480$	0	0
$481$	30.0115	1.36841
$482$	−24.1881	−1.10174
$483$	0	0
$484$	1.18384	0.0538111
$485$	−1.34514	−0.0610796
$486$	0	0
$487$	−12.8784	−0.583576	−0.291788	−	0.956483i	$-0.594250\pi$
$487$	−12.8784	−0.583576	−0.291788	+	0.956483i	$0.594250\pi$
$488$	7.45427	0.337439
$489$	0	0
$490$	0	0
$491$	5.55389	0.250644	0.125322	−	0.992116i	$-0.460004\pi$
$491$	5.55389	0.250644	0.125322	+	0.992116i	$0.460004\pi$
$492$	0	0
$493$	−0.0963098	−0.00433758
$494$	−18.9033	−0.850500
$495$	0	0
$496$	−10.6831	−0.479685
$497$	0	0
$498$	0	0
$499$	−28.1154	−1.25862	−0.629308	−	0.777156i	$-0.716662\pi$
$499$	−28.1154	−1.25862	−0.629308	+	0.777156i	$0.716662\pi$
$500$	−0.762071	−0.0340809
$501$	0	0
$502$	2.70079	0.120542
$503$	16.9430	0.755451	0.377725	−	0.925918i	$-0.376706\pi$
$503$	16.9430	0.755451	0.377725	+	0.925918i	$0.376706\pi$
$504$	0	0
$505$	−5.55389	−0.247145
$506$	32.9253	1.46371
$507$	0	0
$508$	1.64142	0.0728264
$509$	−22.3025	−0.988542	−0.494271	−	0.869308i	$-0.664565\pi$
$509$	−22.3025	−0.988542	−0.494271	+	0.869308i	$0.664565\pi$
$510$	0	0
$511$	0	0
$512$	−19.9751	−0.882783
$513$	0	0
$514$	39.2029	1.72917
$515$	−9.35661	−0.412301
$516$	0	0
$517$	55.4690	2.43953
$518$	0	0
$519$	0	0
$520$	7.30680	0.320424
$521$	12.3609	0.541540	0.270770	−	0.962644i	$-0.412722\pi$
$521$	12.3609	0.541540	0.270770	+	0.962644i	$0.412722\pi$
$522$	0	0
$523$	6.19143	0.270732	0.135366	−	0.990796i	$-0.456779\pi$
$523$	6.19143	0.270732	0.135366	+	0.990796i	$0.456779\pi$
$524$	0.425087	0.0185700
$525$	0	0
$526$	29.6261	1.29176
$527$	0.687353	0.0299416
$528$	0	0
$529$	2.54377	0.110599
$530$	−9.82004	−0.426555
$531$	0	0
$532$	0	0
$533$	−49.1914	−2.13072
$534$	0	0
$535$	−0.608363	−0.0263018
$536$	−15.9971	−0.690968
$537$	0	0
$538$	−12.7453	−0.549490
$539$	0	0
$540$	0	0
$541$	26.8200	1.15308	0.576542	−	0.817068i	$-0.304402\pi$
$541$	26.8200	1.15308	0.576542	+	0.817068i	$0.304402\pi$
$542$	35.3317	1.51763
$543$	0	0
$544$	−0.205436	−0.00880800
$545$	0.768879	0.0329352
$546$	0	0
$547$	−14.6591	−0.626779	−0.313390	−	0.949625i	$-0.601465\pi$
$547$	−14.6591	−0.626779	−0.313390	+	0.949625i	$0.601465\pi$
$548$	−1.34514	−0.0574615
$549$	0	0
$550$	30.2776	1.29104
$551$	−1.01012	−0.0430327
$552$	0	0
$553$	0	0
$554$	−10.3920	−0.441515
$555$	0	0
$556$	2.40546	0.102014
$557$	−23.6798	−1.00334	−0.501672	−	0.865058i	$-0.667282\pi$
$557$	−23.6798	−1.00334	−0.501672	+	0.865058i	$0.667282\pi$
$558$	0	0
$559$	−15.2661	−0.645689
$560$	0	0
$561$	0	0
$562$	11.5323	0.486461
$563$	−16.3858	−0.690579	−0.345289	−	0.938496i	$-0.612219\pi$
$563$	−16.3858	−0.690579	−0.345289	+	0.938496i	$0.612219\pi$
$564$	0	0
$565$	−8.48541	−0.356984
$566$	3.22646	0.135618
$567$	0	0
$568$	7.09142	0.297549
$569$	−15.7879	−0.661865	−0.330932	−	0.943654i	$-0.607363\pi$
$569$	−15.7879	−0.661865	−0.330932	+	0.943654i	$0.607363\pi$
$570$	0	0
$571$	6.38151	0.267058	0.133529	−	0.991045i	$-0.457369\pi$
$571$	6.38151	0.267058	0.133529	+	0.991045i	$0.457369\pi$
$572$	2.67977	0.112047
$573$	0	0
$574$	0	0
$575$	23.4897	0.979587
$576$	0	0
$577$	−37.0406	−1.54202	−0.771011	−	0.636822i	$-0.780249\pi$
$577$	−37.0406	−1.54202	−0.771011	+	0.636822i	$0.780249\pi$
$578$	−24.7195	−1.02819
$579$	0	0
$580$	−0.0278311	−0.00115563
$581$	0	0
$582$	0	0
$583$	50.5261	2.09258
$584$	−30.3078	−1.25414
$585$	0	0
$586$	−28.0292	−1.15787
$587$	−12.0938	−0.499163	−0.249582	−	0.968354i	$-0.580293\pi$
$587$	−12.0938	−0.499163	−0.249582	+	0.968354i	$0.580293\pi$
$588$	0	0
$589$	7.20914	0.297047
$590$	6.98229	0.287456
$591$	0	0
$592$	28.2422	1.16075
$593$	16.5290	0.678764	0.339382	−	0.940649i	$-0.389782\pi$
$593$	16.5290	0.678764	0.339382	+	0.940649i	$0.389782\pi$
$594$	0	0
$595$	0	0
$596$	−1.87024	−0.0766080
$597$	0	0
$598$	33.3245	1.36274
$599$	−8.74825	−0.357444	−0.178722	−	0.983900i	$-0.557196\pi$
$599$	−8.74825	−0.357444	−0.178722	+	0.983900i	$0.557196\pi$
$600$	0	0
$601$	−5.92393	−0.241642	−0.120821	−	0.992674i	$-0.538553\pi$
$601$	−5.92393	−0.241642	−0.120821	+	0.992674i	$0.538553\pi$
$602$	0	0
$603$	0	0
$604$	0.0507699	0.00206580
$605$	5.28054	0.214685
$606$	0	0
$607$	−0.741438	−0.0300940	−0.0150470	−	0.999887i	$-0.504790\pi$
$607$	−0.741438	−0.0300940	−0.0150470	+	0.999887i	$0.504790\pi$
$608$	−2.15467	−0.0873833
$609$	0	0
$610$	−2.37005	−0.0959603
$611$	56.1416	2.27125
$612$	0	0
$613$	4.50700	0.182036	0.0910181	−	0.995849i	$-0.470988\pi$
$613$	4.50700	0.182036	0.0910181	+	0.995849i	$0.470988\pi$
$614$	−20.3212	−0.820097
$615$	0	0
$616$	0	0
$617$	−17.2016	−0.692508	−0.346254	−	0.938141i	$-0.612547\pi$
$617$	−17.2016	−0.692508	−0.346254	+	0.938141i	$0.612547\pi$
$618$	0	0
$619$	4.48541	0.180284	0.0901419	−	0.995929i	$-0.471268\pi$
$619$	4.48541	0.180284	0.0901419	+	0.995929i	$0.471268\pi$
$620$	0.198628	0.00797709
$621$	0	0
$622$	15.5615	0.623958
$623$	0	0
$624$	0	0
$625$	19.8391	0.793564
$626$	−24.1881	−0.966752
$627$	0	0
$628$	−1.00000	−0.0399043
$629$	−1.81711	−0.0724531
$630$	0	0
$631$	−17.3068	−0.688973	−0.344486	−	0.938791i	$-0.611947\pi$
$631$	−17.3068	−0.688973	−0.344486	+	0.938791i	$0.611947\pi$
$632$	−30.4356	−1.21066
$633$	0	0
$634$	−38.9037	−1.54506
$635$	7.32158	0.290548
$636$	0	0
$637$	0	0
$638$	2.29533	0.0908729
$639$	0	0
$640$	7.30680	0.288826
$641$	43.3216	1.71110	0.855550	−	0.517721i	$-0.173219\pi$
$641$	43.3216	1.71110	0.855550	+	0.517721i	$0.173219\pi$
$642$	0	0
$643$	29.9823	1.18239	0.591193	−	0.806530i	$-0.298657\pi$
$643$	29.9823	1.18239	0.591193	+	0.806530i	$0.298657\pi$
$644$	0	0
$645$	0	0
$646$	1.14454	0.0450315
$647$	14.1694	0.557059	0.278529	−	0.960428i	$-0.410153\pi$
$647$	14.1694	0.557059	0.278529	+	0.960428i	$0.410153\pi$
$648$	0	0
$649$	−35.9253	−1.41019
$650$	30.6447	1.20199
$651$	0	0
$652$	2.02159	0.0791716
$653$	−28.3963	−1.11123	−0.555617	−	0.831439i	$-0.687518\pi$
$653$	−28.3963	−1.11123	−0.555617	+	0.831439i	$0.687518\pi$
$654$	0	0
$655$	1.89610	0.0740869
$656$	−46.2914	−1.80738
$657$	0	0
$658$	0	0
$659$	9.39922	0.366142	0.183071	−	0.983100i	$-0.441396\pi$
$659$	9.39922	0.366142	0.183071	+	0.983100i	$0.441396\pi$
$660$	0	0
$661$	−12.7161	−0.494601	−0.247300	−	0.968939i	$-0.579543\pi$
$661$	−12.7161	−0.494601	−0.247300	+	0.968939i	$0.579543\pi$
$662$	−34.0397	−1.32299
$663$	0	0
$664$	45.1091	1.75057
$665$	0	0
$666$	0	0
$667$	1.78074	0.0689505
$668$	−1.19143	−0.0460978
$669$	0	0
$670$	5.08619	0.196497
$671$	12.1944	0.470758
$672$	0	0
$673$	35.7922	1.37969	0.689844	−	0.723958i	$-0.257679\pi$
$673$	35.7922	1.37969	0.689844	+	0.723958i	$0.257679\pi$
$674$	−33.9397	−1.30731
$675$	0	0
$676$	0.982291	0.0377804
$677$	−10.8918	−0.418607	−0.209304	−	0.977851i	$-0.567120\pi$
$677$	−10.8918	−0.418607	−0.209304	+	0.977851i	$0.567120\pi$
$678$	0	0
$679$	0	0
$680$	−0.442407	−0.0169655
$681$	0	0
$682$	−16.3815	−0.627281
$683$	−35.0157	−1.33984	−0.669920	−	0.742433i	$-0.733672\pi$
$683$	−35.0157	−1.33984	−0.669920	+	0.742433i	$0.733672\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	0	0
$688$	−14.3662	−0.547705
$689$	51.1387	1.94823
$690$	0	0
$691$	8.42082	0.320343	0.160171	−	0.987089i	$-0.448795\pi$
$691$	8.42082	0.320343	0.160171	+	0.987089i	$0.448795\pi$
$692$	1.39203	0.0529169
$693$	0	0
$694$	−25.0292	−0.950095
$695$	10.7296	0.406996
$696$	0	0
$697$	2.97841	0.112815
$698$	−28.5012	−1.07878
$699$	0	0
$700$	0	0
$701$	−42.7453	−1.61447	−0.807234	−	0.590231i	$-0.799037\pi$
$701$	−42.7453	−1.61447	−0.807234	+	0.590231i	$0.799037\pi$
$702$	0	0
$703$	−19.0584	−0.718800
$704$	−33.0043	−1.24390
$705$	0	0
$706$	−23.4576	−0.882838
$707$	0	0
$708$	0	0
$709$	−24.0862	−0.904576	−0.452288	−	0.891872i	$-0.649392\pi$
$709$	−24.0862	−0.904576	−0.452288	+	0.891872i	$0.649392\pi$
$710$	−2.25468	−0.0846166
$711$	0	0
$712$	−14.6539	−0.549178
$713$	−12.7089	−0.475954
$714$	0	0
$715$	11.9531	0.447021
$716$	1.19340	0.0445994
$717$	0	0
$718$	−20.2484	−0.755665
$719$	−42.1023	−1.57015	−0.785076	−	0.619400i	$-0.787376\pi$
$719$	−42.1023	−1.57015	−0.785076	+	0.619400i	$0.787376\pi$
$720$	0	0
$721$	0	0
$722$	−15.7453	−0.585980
$723$	0	0
$724$	0.671612	0.0249603
$725$	1.63754	0.0608168
$726$	0	0
$727$	36.0698	1.33776	0.668878	−	0.743372i	$-0.266775\pi$
$727$	36.0698	1.33776	0.668878	+	0.743372i	$0.266775\pi$
$728$	0	0
$729$	0	0
$730$	9.63619	0.356652
$731$	0.924324	0.0341874
$732$	0	0
$733$	34.1331	1.26073	0.630367	−	0.776297i	$-0.282904\pi$
$733$	34.1331	1.26073	0.630367	+	0.776297i	$0.282904\pi$
$734$	36.9439	1.36363
$735$	0	0
$736$	3.79845	0.140013
$737$	−26.1694	−0.963964
$738$	0	0
$739$	21.8961	0.805461	0.402731	−	0.915319i	$-0.368061\pi$
$739$	21.8961	0.805461	0.402731	+	0.915319i	$0.368061\pi$
$740$	−0.525101	−0.0193031
$741$	0	0
$742$	0	0
$743$	28.2852	1.03768	0.518842	−	0.854870i	$-0.326363\pi$
$743$	28.2852	1.03768	0.518842	+	0.854870i	$0.326363\pi$
$744$	0	0
$745$	−8.34221	−0.305635
$746$	2.92101	0.106946
$747$	0	0
$748$	−0.162253	−0.00593254
$749$	0	0
$750$	0	0
$751$	−14.4969	−0.528999	−0.264499	−	0.964386i	$-0.585207\pi$
$751$	−14.4969	−0.528999	−0.264499	+	0.964386i	$0.585207\pi$
$752$	52.8319	1.92658
$753$	0	0
$754$	2.32316	0.0846044
$755$	0.226459	0.00824169
$756$	0	0
$757$	−38.9646	−1.41619	−0.708096	−	0.706116i	$-0.750446\pi$
$757$	−38.9646	−1.41619	−0.708096	+	0.706116i	$0.750446\pi$
$758$	−27.4720	−0.997827
$759$	0	0
$760$	−4.64008	−0.168313
$761$	−1.25564	−0.0455168	−0.0227584	−	0.999741i	$-0.507245\pi$
$761$	−1.25564	−0.0455168	−0.0227584	+	0.999741i	$0.507245\pi$
$762$	0	0
$763$	0	0
$764$	−1.61421	−0.0584002
$765$	0	0
$766$	−51.3245	−1.85443
$767$	−36.3609	−1.31292
$768$	0	0
$769$	−27.6883	−0.998466	−0.499233	−	0.866468i	$-0.666385\pi$
$769$	−27.6883	−0.998466	−0.499233	+	0.866468i	$0.666385\pi$
$770$	0	0
$771$	0	0
$772$	2.28386	0.0821978
$773$	7.91089	0.284535	0.142267	−	0.989828i	$-0.454561\pi$
$773$	7.91089	0.284535	0.142267	+	0.989828i	$0.454561\pi$
$774$	0	0
$775$	−11.6870	−0.419808
$776$	6.17900	0.221813
$777$	0	0
$778$	12.2370	0.438717
$779$	31.2383	1.11923
$780$	0	0
$781$	11.6008	0.415108
$782$	−2.01771	−0.0721531
$783$	0	0
$784$	0	0
$785$	−4.46050	−0.159202
$786$	0	0
$787$	31.6693	1.12889	0.564444	−	0.825472i	$-0.309091\pi$
$787$	31.6693	1.12889	0.564444	+	0.825472i	$0.309091\pi$
$788$	−0.100013	−0.00356283
$789$	0	0
$790$	9.67684	0.344287
$791$	0	0
$792$	0	0
$793$	12.3422	0.438285
$794$	−13.5074	−0.479359
$795$	0	0
$796$	1.37005	0.0485600
$797$	33.7922	1.19698	0.598491	−	0.801130i	$-0.295767\pi$
$797$	33.7922	1.19698	0.598491	+	0.801130i	$0.295767\pi$
$798$	0	0
$799$	−3.39922	−0.120256
$800$	3.49300	0.123496
$801$	0	0
$802$	0.215379	0.00760530
$803$	−49.5801	−1.74965
$804$	0	0
$805$	0	0
$806$	−16.5801	−0.584011
$807$	0	0
$808$	25.5122	0.897517
$809$	36.7601	1.29242	0.646208	−	0.763161i	$-0.276354\pi$
$809$	36.7601	1.29242	0.646208	+	0.763161i	$0.276354\pi$
$810$	0	0
$811$	3.54377	0.124438	0.0622192	−	0.998063i	$-0.480182\pi$
$811$	3.54377	0.124438	0.0622192	+	0.998063i	$0.480182\pi$
$812$	0	0
$813$	0	0
$814$	43.3068	1.51790
$815$	9.01732	0.315863
$816$	0	0
$817$	9.69455	0.339169
$818$	−5.73289	−0.200446
$819$	0	0
$820$	0.860686	0.0300565
$821$	20.8636	0.728145	0.364073	−	0.931371i	$-0.381386\pi$
$821$	20.8636	0.728145	0.364073	+	0.931371i	$0.381386\pi$
$822$	0	0
$823$	45.4006	1.58256	0.791282	−	0.611451i	$-0.209414\pi$
$823$	45.4006	1.58256	0.791282	+	0.611451i	$0.209414\pi$
$824$	42.9803	1.49729
$825$	0	0
$826$	0	0
$827$	−5.34221	−0.185767	−0.0928835	−	0.995677i	$-0.529608\pi$
$827$	−5.34221	−0.185767	−0.0928835	+	0.995677i	$0.529608\pi$
$828$	0	0
$829$	16.9037	0.587089	0.293545	−	0.955945i	$-0.405165\pi$
$829$	16.9037	0.587089	0.293545	+	0.955945i	$0.405165\pi$
$830$	−14.3422	−0.497826
$831$	0	0
$832$	−33.4045	−1.15809
$833$	0	0
$834$	0	0
$835$	−5.31438	−0.183912
$836$	−1.70175	−0.0588562
$837$	0	0
$838$	−30.6706	−1.05950
$839$	−9.83909	−0.339683	−0.169842	−	0.985471i	$-0.554326\pi$
$839$	−9.83909	−0.339683	−0.169842	+	0.985471i	$0.554326\pi$
$840$	0	0
$841$	−28.8759	−0.995719
$842$	−36.0574	−1.24262
$843$	0	0
$844$	−2.14454	−0.0738182
$845$	4.38151	0.150729
$846$	0	0
$847$	0	0
$848$	48.1239	1.65258
$849$	0	0
$850$	−1.85546	−0.0636416
$851$	33.5979	1.15172
$852$	0	0
$853$	−55.4868	−1.89983	−0.949915	−	0.312508i	$-0.898831\pi$
$853$	−55.4868	−1.89983	−0.949915	+	0.312508i	$0.898831\pi$
$854$	0	0
$855$	0	0
$856$	2.79456	0.0955162
$857$	−56.3465	−1.92476	−0.962380	−	0.271708i	$-0.912411\pi$
$857$	−56.3465	−1.92476	−0.962380	+	0.271708i	$0.912411\pi$
$858$	0	0
$859$	43.9253	1.49871	0.749356	−	0.662168i	$-0.230363\pi$
$859$	43.9253	1.49871	0.749356	+	0.662168i	$0.230363\pi$
$860$	0.267107	0.00910826
$861$	0	0
$862$	7.99612	0.272349
$863$	11.3274	0.385590	0.192795	−	0.981239i	$-0.438245\pi$
$863$	11.3274	0.385590	0.192795	+	0.981239i	$0.438245\pi$
$864$	0	0
$865$	6.20914	0.211117
$866$	34.5117	1.17275
$867$	0	0
$868$	0	0
$869$	−49.7893	−1.68899
$870$	0	0
$871$	−26.4868	−0.897470
$872$	−3.53191	−0.119606
$873$	0	0
$874$	−21.1623	−0.715824
$875$	0	0
$876$	0	0
$877$	−46.4615	−1.56889	−0.784446	−	0.620197i	$-0.787053\pi$
$877$	−46.4615	−1.56889	−0.784446	+	0.620197i	$0.787053\pi$
$878$	−7.69124	−0.259567
$879$	0	0
$880$	11.2484	0.379185
$881$	21.6578	0.729669	0.364835	−	0.931072i	$-0.381126\pi$
$881$	21.6578	0.729669	0.364835	+	0.931072i	$0.381126\pi$
$882$	0	0
$883$	11.4868	0.386560	0.193280	−	0.981144i	$-0.438087\pi$
$883$	11.4868	0.386560	0.193280	+	0.981144i	$0.438087\pi$
$884$	−0.164220	−0.00552332
$885$	0	0
$886$	−8.81616	−0.296185
$887$	17.8420	0.599076	0.299538	−	0.954084i	$-0.403167\pi$
$887$	17.8420	0.599076	0.299538	+	0.954084i	$0.403167\pi$
$888$	0	0
$889$	0	0
$890$	4.65913	0.156174
$891$	0	0
$892$	−0.925282	−0.0309807
$893$	−35.6519	−1.19305
$894$	0	0
$895$	5.32316	0.177934
$896$	0	0
$897$	0	0
$898$	−32.3422	−1.07927
$899$	−0.885981	−0.0295491
$900$	0	0
$901$	−3.09631	−0.103153
$902$	−70.9836	−2.36350
$903$	0	0
$904$	38.9784	1.29640
$905$	2.99573	0.0995814
$906$	0	0
$907$	−51.8506	−1.72167	−0.860835	−	0.508884i	$-0.830058\pi$
$907$	−51.8506	−1.72167	−0.860835	+	0.508884i	$0.830058\pi$
$908$	0.704104	0.0233665
$909$	0	0
$910$	0	0
$911$	−6.48676	−0.214916	−0.107458	−	0.994210i	$-0.534271\pi$
$911$	−6.48676	−0.214916	−0.107458	+	0.994210i	$0.534271\pi$
$912$	0	0
$913$	73.7936	2.44221
$914$	−7.79125	−0.257712
$915$	0	0
$916$	1.56013	0.0515481
$917$	0	0
$918$	0	0
$919$	−11.6844	−0.385434	−0.192717	−	0.981254i	$-0.561730\pi$
$919$	−11.6844	−0.385434	−0.192717	+	0.981254i	$0.561730\pi$
$920$	8.17996	0.269685
$921$	0	0
$922$	43.3690	1.42828
$923$	11.7414	0.386474
$924$	0	0
$925$	30.8961	1.01586
$926$	27.1517	0.892262
$927$	0	0
$928$	0.264802	0.00869255
$929$	1.46050	0.0479176	0.0239588	−	0.999713i	$-0.492373\pi$
$929$	1.46050	0.0479176	0.0239588	+	0.999713i	$0.492373\pi$
$930$	0	0
$931$	0	0
$932$	0.515158	0.0168745
$933$	0	0
$934$	35.9469	1.17622
$935$	−0.723729	−0.0236685
$936$	0	0
$937$	9.87451	0.322586	0.161293	−	0.986907i	$-0.448434\pi$
$937$	9.87451	0.322586	0.161293	+	0.986907i	$0.448434\pi$
$938$	0	0
$939$	0	0
$940$	−0.982291	−0.0320388
$941$	−54.1944	−1.76669	−0.883343	−	0.468728i	$-0.844713\pi$
$941$	−54.1944	−1.76669	−0.883343	+	0.468728i	$0.844713\pi$
$942$	0	0
$943$	−55.0698	−1.79332
$944$	−34.2173	−1.11368
$945$	0	0
$946$	−22.0292	−0.716230
$947$	54.8903	1.78369	0.891847	−	0.452338i	$-0.149410\pi$
$947$	54.8903	1.78369	0.891847	+	0.452338i	$0.149410\pi$
$948$	0	0
$949$	−50.1813	−1.62895
$950$	−19.4605	−0.631382
$951$	0	0
$952$	0	0
$953$	27.0406	0.875932	0.437966	−	0.898991i	$-0.355699\pi$
$953$	27.0406	0.875932	0.437966	+	0.898991i	$0.355699\pi$
$954$	0	0
$955$	−7.20021	−0.232993
$956$	1.65002	0.0533655
$957$	0	0
$958$	−0.520829	−0.0168272
$959$	0	0
$960$	0	0
$961$	−24.6768	−0.796027
$962$	43.8319	1.41320
$963$	0	0
$964$	−2.20391	−0.0709832
$965$	10.1872	0.327936
$966$	0	0
$967$	−13.5146	−0.434600	−0.217300	−	0.976105i	$-0.569725\pi$
$967$	−13.5146	−0.434600	−0.217300	+	0.976105i	$0.569725\pi$
$968$	−24.2566	−0.779636
$969$	0	0
$970$	−1.96458	−0.0630789
$971$	−12.9311	−0.414980	−0.207490	−	0.978237i	$-0.566529\pi$
$971$	−12.9311	−0.414980	−0.207490	+	0.978237i	$0.566529\pi$
$972$	0	0
$973$	0	0
$974$	−18.8090	−0.602678
$975$	0	0
$976$	11.6146	0.371774
$977$	36.5979	1.17087	0.585435	−	0.810720i	$-0.300924\pi$
$977$	36.5979	1.17087	0.585435	+	0.810720i	$0.300924\pi$
$978$	0	0
$979$	−23.9722	−0.766154
$980$	0	0
$981$	0	0
$982$	8.11148	0.258848
$983$	17.2379	0.549805	0.274902	−	0.961472i	$-0.411354\pi$
$983$	17.2379	0.549805	0.274902	+	0.961472i	$0.411354\pi$
$984$	0	0
$985$	−0.446110	−0.0142143
$986$	−0.140661	−0.00447956
$987$	0	0
$988$	−1.72238	−0.0547963
$989$	−17.0905	−0.543445
$990$	0	0
$991$	−15.3422	−0.487361	−0.243681	−	0.969856i	$-0.578355\pi$
$991$	−15.3422	−0.487361	−0.243681	+	0.969856i	$0.578355\pi$
$992$	−1.88986	−0.0600032
$993$	0	0
$994$	0	0
$995$	6.11109	0.193735
$996$	0	0
$997$	−34.1154	−1.08044	−0.540222	−	0.841522i	$-0.681660\pi$
$997$	−34.1154	−1.08044	−0.540222	+	0.841522i	$0.681660\pi$
$998$	−41.0626	−1.29981
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.a.x.1.3		3
3.2	odd	2		1323.2.a.ba.1.1		3
7.2	even	3		189.2.e.f.109.1	yes	6
7.4	even	3		189.2.e.f.163.1	yes	6
7.6	odd	2		1323.2.a.y.1.3		3
21.2	odd	6		189.2.e.e.109.3	✓	6
21.11	odd	6		189.2.e.e.163.3	yes	6
21.20	even	2		1323.2.a.z.1.1		3
63.2	odd	6		567.2.g.h.109.3		6
63.4	even	3		567.2.g.i.541.1		6
63.11	odd	6		567.2.h.i.352.1		6
63.16	even	3		567.2.g.i.109.1		6
63.23	odd	6		567.2.h.i.298.1		6
63.25	even	3		567.2.h.h.352.3		6
63.32	odd	6		567.2.g.h.541.3		6
63.58	even	3		567.2.h.h.298.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.e.e.109.3	✓	6	21.2	odd	6
189.2.e.e.163.3	yes	6	21.11	odd	6
189.2.e.f.109.1	yes	6	7.2	even	3
189.2.e.f.163.1	yes	6	7.4	even	3
567.2.g.h.109.3		6	63.2	odd	6
567.2.g.h.541.3		6	63.32	odd	6
567.2.g.i.109.1		6	63.16	even	3
567.2.g.i.541.1		6	63.4	even	3
567.2.h.h.298.3		6	63.58	even	3
567.2.h.h.352.3		6	63.25	even	3
567.2.h.i.298.1		6	63.23	odd	6
567.2.h.i.352.1		6	63.11	odd	6
1323.2.a.x.1.3		3	1.1	even	1	trivial
1323.2.a.y.1.3		3	7.6	odd	2
1323.2.a.z.1.1		3	21.20	even	2
1323.2.a.ba.1.1		3	3.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q+1.46050 q^{2} +0.133074 q^{4} +0.593579 q^{5} -2.72665 q^{8} +0.866926 q^{10} -4.46050 q^{11} -4.51459 q^{13} -4.24844 q^{16} +0.273346 q^{17} +2.86693 q^{19} +0.0789903 q^{20} -6.51459 q^{22} -5.05408 q^{23} -4.64766 q^{25} -6.59358 q^{26} -0.352336 q^{29} +2.51459 q^{31} -0.751560 q^{32} +0.399223 q^{34} -6.64766 q^{37} +4.18716 q^{38} -1.61849 q^{40} +10.8961 q^{41} +3.38151 q^{43} -0.593579 q^{44} -7.38151 q^{46} -12.4356 q^{47} -6.78794 q^{50} -0.600777 q^{52} -11.3274 q^{53} -2.64766 q^{55} -0.514589 q^{58} +8.05408 q^{59} -2.73385 q^{61} +3.67257 q^{62} +7.39922 q^{64} -2.67977 q^{65} +5.86693 q^{67} +0.0363754 q^{68} -2.60078 q^{71} +11.1154 q^{73} -9.70895 q^{74} +0.381515 q^{76} +11.1623 q^{79} -2.52179 q^{80} +15.9138 q^{82} -16.5438 q^{83} +0.162253 q^{85} +4.93872 q^{86} +12.1623 q^{88} +5.37432 q^{89} -0.672570 q^{92} -18.1623 q^{94} +1.70175 q^{95} -2.26615 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 4 q^{4} - q^{5} - 9 q^{8} - q^{10} - 7 q^{11} + 2 q^{13} + 10 q^{16} + 5 q^{19} + 13 q^{20} - 4 q^{22} - 6 q^{23} - 2 q^{25} - 17 q^{26} - 13 q^{29} - 8 q^{31} - 25 q^{32} + 12 q^{34}+ \cdots - 14 q^{97}+O(q^{100}) \) 3 * q - 2 * q^2 + 4 * q^4 - q^5 - 9 * q^8 - q^10 - 7 * q^11 + 2 * q^13 + 10 * q^16 + 5 * q^19 + 13 * q^20 - 4 * q^22 - 6 * q^23 - 2 * q^25 - 17 * q^26 - 13 * q^29 - 8 * q^31 - 25 * q^32 + 12 * q^34 - 8 * q^37 + 7 * q^38 - 24 * q^40 - 2 * q^41 - 9 * q^43 + q^44 - 3 * q^46 - 9 * q^47 - 4 * q^50 + 9 * q^52 - 24 * q^53 + 4 * q^55 + 14 * q^58 + 15 * q^59 - q^61 + 21 * q^62 + 33 * q^64 - 10 * q^65 + 14 * q^67 - 39 * q^68 + 3 * q^71 + 7 * q^73 - 18 * q^76 + 6 * q^79 + 16 * q^80 + 43 * q^82 - 3 * q^83 - 27 * q^85 + 32 * q^86 + 9 * q^88 + 5 * q^89 - 12 * q^92 - 27 * q^94 - 16 * q^95 - 14 * q^97

Introduction

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