LMFDB - Embedded newform 408.2.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [408,2,Mod(1,408)] 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("408.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 408 = 2^{3} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	408.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,-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:	$3.25789640247$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	408.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.56155 q^{5} -5.12311 q^{7} +1.00000 q^{9} -2.43845 q^{11} -3.56155 q^{13} -1.56155 q^{15} -1.00000 q^{17} +4.68466 q^{19} +5.12311 q^{21} -7.56155 q^{23} -2.56155 q^{25} -1.00000 q^{27} -7.12311 q^{29} +8.24621 q^{31} +2.43845 q^{33} -8.00000 q^{35} -4.00000 q^{37} +3.56155 q^{39} -2.68466 q^{41} -4.68466 q^{43} +1.56155 q^{45} -0.876894 q^{47} +19.2462 q^{49} +1.00000 q^{51} +6.00000 q^{53} -3.80776 q^{55} -4.68466 q^{57} -13.3693 q^{59} -4.00000 q^{61} -5.12311 q^{63} -5.56155 q^{65} +12.0000 q^{67} +7.56155 q^{69} +11.3693 q^{71} +8.24621 q^{73} +2.56155 q^{75} +12.4924 q^{77} +2.00000 q^{79} +1.00000 q^{81} +7.12311 q^{83} -1.56155 q^{85} +7.12311 q^{87} +9.12311 q^{89} +18.2462 q^{91} -8.24621 q^{93} +7.31534 q^{95} +1.12311 q^{97} -2.43845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} - 9 q^{11} - 3 q^{13} + q^{15} - 2 q^{17} - 3 q^{19} + 2 q^{21} - 11 q^{23} - q^{25} - 2 q^{27} - 6 q^{29} + 9 q^{33} - 16 q^{35} - 8 q^{37} + 3 q^{39} + 7 q^{41}+ \cdots - 9 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 - 2 * q^7 + 2 * q^9 - 9 * q^11 - 3 * q^13 + q^15 - 2 * q^17 - 3 * q^19 + 2 * q^21 - 11 * q^23 - q^25 - 2 * q^27 - 6 * q^29 + 9 * q^33 - 16 * q^35 - 8 * q^37 + 3 * q^39 + 7 * q^41 + 3 * q^43 - q^45 - 10 * q^47 + 22 * q^49 + 2 * q^51 + 12 * q^53 + 13 * q^55 + 3 * q^57 - 2 * q^59 - 8 * q^61 - 2 * q^63 - 7 * q^65 + 24 * q^67 + 11 * q^69 - 2 * q^71 + q^75 - 8 * q^77 + 4 * q^79 + 2 * q^81 + 6 * q^83 + q^85 + 6 * q^87 + 10 * q^89 + 20 * q^91 + 27 * q^95 - 6 * q^97 - 9 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.56155	0.698348	0.349174	−	0.937058i	$-0.386462\pi$
$5$	1.56155	0.698348	0.349174	+	0.937058i	$0.386462\pi$
$6$	0	0
$7$	−5.12311	−1.93635	−0.968176	−	0.250270i	$-0.919480\pi$
$7$	−5.12311	−1.93635	−0.968176	+	0.250270i	$0.919480\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.43845	−0.735219	−0.367610	−	0.929980i	$-0.619824\pi$
$11$	−2.43845	−0.735219	−0.367610	+	0.929980i	$0.619824\pi$
$12$	0	0
$13$	−3.56155	−0.987797	−0.493899	−	0.869520i	$-0.664429\pi$
$13$	−3.56155	−0.987797	−0.493899	+	0.869520i	$0.664429\pi$
$14$	0	0
$15$	−1.56155	−0.403191
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	4.68466	1.07473	0.537367	−	0.843348i	$-0.319419\pi$
$19$	4.68466	1.07473	0.537367	+	0.843348i	$0.319419\pi$
$20$	0	0
$21$	5.12311	1.11795
$22$	0	0
$23$	−7.56155	−1.57669	−0.788346	−	0.615232i	$-0.789062\pi$
$23$	−7.56155	−1.57669	−0.788346	+	0.615232i	$0.789062\pi$
$24$	0	0
$25$	−2.56155	−0.512311
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.12311	−1.32273	−0.661364	−	0.750065i	$-0.730022\pi$
$29$	−7.12311	−1.32273	−0.661364	+	0.750065i	$0.730022\pi$
$30$	0	0
$31$	8.24621	1.48106	0.740532	−	0.672022i	$-0.234574\pi$
$31$	8.24621	1.48106	0.740532	+	0.672022i	$0.234574\pi$
$32$	0	0
$33$	2.43845	0.424479
$34$	0	0
$35$	−8.00000	−1.35225
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	3.56155	0.570305
$40$	0	0
$41$	−2.68466	−0.419273	−0.209637	−	0.977779i	$-0.567228\pi$
$41$	−2.68466	−0.419273	−0.209637	+	0.977779i	$0.567228\pi$
$42$	0	0
$43$	−4.68466	−0.714404	−0.357202	−	0.934027i	$-0.616269\pi$
$43$	−4.68466	−0.714404	−0.357202	+	0.934027i	$0.616269\pi$
$44$	0	0
$45$	1.56155	0.232783
$46$	0	0
$47$	−0.876894	−0.127908	−0.0639541	−	0.997953i	$-0.520371\pi$
$47$	−0.876894	−0.127908	−0.0639541	+	0.997953i	$0.520371\pi$
$48$	0	0
$49$	19.2462	2.74946
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	−3.80776	−0.513439
$56$	0	0
$57$	−4.68466	−0.620498
$58$	0	0
$59$	−13.3693	−1.74054	−0.870268	−	0.492578i	$-0.836055\pi$
$59$	−13.3693	−1.74054	−0.870268	+	0.492578i	$0.836055\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	−5.12311	−0.645451
$64$	0	0
$65$	−5.56155	−0.689826
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	0	0
$69$	7.56155	0.910304
$70$	0	0
$71$	11.3693	1.34929	0.674645	−	0.738142i	$-0.264297\pi$
$71$	11.3693	1.34929	0.674645	+	0.738142i	$0.264297\pi$
$72$	0	0
$73$	8.24621	0.965146	0.482573	−	0.875856i	$-0.339702\pi$
$73$	8.24621	0.965146	0.482573	+	0.875856i	$0.339702\pi$
$74$	0	0
$75$	2.56155	0.295783
$76$	0	0
$77$	12.4924	1.42364
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.12311	0.781862	0.390931	−	0.920420i	$-0.372153\pi$
$83$	7.12311	0.781862	0.390931	+	0.920420i	$0.372153\pi$
$84$	0	0
$85$	−1.56155	−0.169374
$86$	0	0
$87$	7.12311	0.763677
$88$	0	0
$89$	9.12311	0.967047	0.483524	−	0.875331i	$-0.339357\pi$
$89$	9.12311	0.967047	0.483524	+	0.875331i	$0.339357\pi$
$90$	0	0
$91$	18.2462	1.91272
$92$	0	0
$93$	−8.24621	−0.855092
$94$	0	0
$95$	7.31534	0.750538
$96$	0	0
$97$	1.12311	0.114034	0.0570170	−	0.998373i	$-0.481841\pi$
$97$	1.12311	0.114034	0.0570170	+	0.998373i	$0.481841\pi$
$98$	0	0
$99$	−2.43845	−0.245073
$100$	0	0
$101$	−5.12311	−0.509768	−0.254884	−	0.966972i	$-0.582037\pi$
$101$	−5.12311	−0.509768	−0.254884	+	0.966972i	$0.582037\pi$
$102$	0	0
$103$	−12.6847	−1.24986	−0.624928	−	0.780682i	$-0.714872\pi$
$103$	−12.6847	−1.24986	−0.624928	+	0.780682i	$0.714872\pi$
$104$	0	0
$105$	8.00000	0.780720
$106$	0	0
$107$	−8.68466	−0.839578	−0.419789	−	0.907622i	$-0.637896\pi$
$107$	−8.68466	−0.839578	−0.419789	+	0.907622i	$0.637896\pi$
$108$	0	0
$109$	−6.24621	−0.598279	−0.299139	−	0.954209i	$-0.596700\pi$
$109$	−6.24621	−0.598279	−0.299139	+	0.954209i	$0.596700\pi$
$110$	0	0
$111$	4.00000	0.379663
$112$	0	0
$113$	4.43845	0.417534	0.208767	−	0.977965i	$-0.433055\pi$
$113$	4.43845	0.417534	0.208767	+	0.977965i	$0.433055\pi$
$114$	0	0
$115$	−11.8078	−1.10108
$116$	0	0
$117$	−3.56155	−0.329266
$118$	0	0
$119$	5.12311	0.469634
$120$	0	0
$121$	−5.05398	−0.459452
$122$	0	0
$123$	2.68466	0.242067
$124$	0	0
$125$	−11.8078	−1.05612
$126$	0	0
$127$	14.9309	1.32490	0.662450	−	0.749106i	$-0.269517\pi$
$127$	14.9309	1.32490	0.662450	+	0.749106i	$0.269517\pi$
$128$	0	0
$129$	4.68466	0.412461
$130$	0	0
$131$	−13.5616	−1.18488	−0.592439	−	0.805615i	$-0.701835\pi$
$131$	−13.5616	−1.18488	−0.592439	+	0.805615i	$0.701835\pi$
$132$	0	0
$133$	−24.0000	−2.08106
$134$	0	0
$135$	−1.56155	−0.134397
$136$	0	0
$137$	0.246211	0.0210352	0.0105176	−	0.999945i	$-0.496652\pi$
$137$	0.246211	0.0210352	0.0105176	+	0.999945i	$0.496652\pi$
$138$	0	0
$139$	−3.12311	−0.264898	−0.132449	−	0.991190i	$-0.542284\pi$
$139$	−3.12311	−0.264898	−0.132449	+	0.991190i	$0.542284\pi$
$140$	0	0
$141$	0.876894	0.0738478
$142$	0	0
$143$	8.68466	0.726248
$144$	0	0
$145$	−11.1231	−0.923724
$146$	0	0
$147$	−19.2462	−1.58740
$148$	0	0
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	12.8769	1.03430
$156$	0	0
$157$	−7.56155	−0.603478	−0.301739	−	0.953391i	$-0.597567\pi$
$157$	−7.56155	−0.603478	−0.301739	+	0.953391i	$0.597567\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	38.7386	3.05303
$162$	0	0
$163$	13.3693	1.04717	0.523583	−	0.851975i	$-0.324595\pi$
$163$	13.3693	1.04717	0.523583	+	0.851975i	$0.324595\pi$
$164$	0	0
$165$	3.80776	0.296434
$166$	0	0
$167$	−21.8078	−1.68754	−0.843768	−	0.536709i	$-0.819667\pi$
$167$	−21.8078	−1.68754	−0.843768	+	0.536709i	$0.819667\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	0	0
$171$	4.68466	0.358245
$172$	0	0
$173$	−4.68466	−0.356168	−0.178084	−	0.984015i	$-0.556990\pi$
$173$	−4.68466	−0.356168	−0.178084	+	0.984015i	$0.556990\pi$
$174$	0	0
$175$	13.1231	0.992014
$176$	0	0
$177$	13.3693	1.00490
$178$	0	0
$179$	−15.1231	−1.13035	−0.565177	−	0.824970i	$-0.691192\pi$
$179$	−15.1231	−1.13035	−0.565177	+	0.824970i	$0.691192\pi$
$180$	0	0
$181$	7.12311	0.529456	0.264728	−	0.964323i	$-0.414718\pi$
$181$	7.12311	0.529456	0.264728	+	0.964323i	$0.414718\pi$
$182$	0	0
$183$	4.00000	0.295689
$184$	0	0
$185$	−6.24621	−0.459231
$186$	0	0
$187$	2.43845	0.178317
$188$	0	0
$189$	5.12311	0.372651
$190$	0	0
$191$	−5.36932	−0.388510	−0.194255	−	0.980951i	$-0.562229\pi$
$191$	−5.36932	−0.388510	−0.194255	+	0.980951i	$0.562229\pi$
$192$	0	0
$193$	12.2462	0.881502	0.440751	−	0.897630i	$-0.354712\pi$
$193$	12.2462	0.881502	0.440751	+	0.897630i	$0.354712\pi$
$194$	0	0
$195$	5.56155	0.398271
$196$	0	0
$197$	−24.6847	−1.75871	−0.879355	−	0.476168i	$-0.842026\pi$
$197$	−24.6847	−1.75871	−0.879355	+	0.476168i	$0.842026\pi$
$198$	0	0
$199$	−14.8769	−1.05460	−0.527298	−	0.849681i	$-0.676795\pi$
$199$	−14.8769	−1.05460	−0.527298	+	0.849681i	$0.676795\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	0	0
$203$	36.4924	2.56127
$204$	0	0
$205$	−4.19224	−0.292798
$206$	0	0
$207$	−7.56155	−0.525564
$208$	0	0
$209$	−11.4233	−0.790166
$210$	0	0
$211$	19.1231	1.31649	0.658244	−	0.752804i	$-0.271299\pi$
$211$	19.1231	1.31649	0.658244	+	0.752804i	$0.271299\pi$
$212$	0	0
$213$	−11.3693	−0.779013
$214$	0	0
$215$	−7.31534	−0.498902
$216$	0	0
$217$	−42.2462	−2.86786
$218$	0	0
$219$	−8.24621	−0.557227
$220$	0	0
$221$	3.56155	0.239576
$222$	0	0
$223$	−25.5616	−1.71173	−0.855864	−	0.517201i	$-0.826974\pi$
$223$	−25.5616	−1.71173	−0.855864	+	0.517201i	$0.826974\pi$
$224$	0	0
$225$	−2.56155	−0.170770
$226$	0	0
$227$	−3.31534	−0.220047	−0.110023	−	0.993929i	$-0.535093\pi$
$227$	−3.31534	−0.220047	−0.110023	+	0.993929i	$0.535093\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	−12.4924	−0.821941
$232$	0	0
$233$	7.56155	0.495374	0.247687	−	0.968840i	$-0.420330\pi$
$233$	7.56155	0.495374	0.247687	+	0.968840i	$0.420330\pi$
$234$	0	0
$235$	−1.36932	−0.0893244
$236$	0	0
$237$	−2.00000	−0.129914
$238$	0	0
$239$	5.75379	0.372182	0.186091	−	0.982533i	$-0.440418\pi$
$239$	5.75379	0.372182	0.186091	+	0.982533i	$0.440418\pi$
$240$	0	0
$241$	−19.3693	−1.24769	−0.623844	−	0.781549i	$-0.714430\pi$
$241$	−19.3693	−1.24769	−0.623844	+	0.781549i	$0.714430\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	30.0540	1.92008
$246$	0	0
$247$	−16.6847	−1.06162
$248$	0	0
$249$	−7.12311	−0.451408
$250$	0	0
$251$	10.2462	0.646735	0.323368	−	0.946273i	$-0.395185\pi$
$251$	10.2462	0.646735	0.323368	+	0.946273i	$0.395185\pi$
$252$	0	0
$253$	18.4384	1.15922
$254$	0	0
$255$	1.56155	0.0977882
$256$	0	0
$257$	13.1231	0.818597	0.409298	−	0.912401i	$-0.365773\pi$
$257$	13.1231	0.818597	0.409298	+	0.912401i	$0.365773\pi$
$258$	0	0
$259$	20.4924	1.27334
$260$	0	0
$261$	−7.12311	−0.440909
$262$	0	0
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	9.36932	0.575553
$266$	0	0
$267$	−9.12311	−0.558325
$268$	0	0
$269$	−11.3153	−0.689909	−0.344954	−	0.938619i	$-0.612106\pi$
$269$	−11.3153	−0.689909	−0.344954	+	0.938619i	$0.612106\pi$
$270$	0	0
$271$	−18.0540	−1.09670	−0.548350	−	0.836249i	$-0.684744\pi$
$271$	−18.0540	−1.09670	−0.548350	+	0.836249i	$0.684744\pi$
$272$	0	0
$273$	−18.2462	−1.10431
$274$	0	0
$275$	6.24621	0.376661
$276$	0	0
$277$	−12.8769	−0.773698	−0.386849	−	0.922143i	$-0.626436\pi$
$277$	−12.8769	−0.773698	−0.386849	+	0.922143i	$0.626436\pi$
$278$	0	0
$279$	8.24621	0.493688
$280$	0	0
$281$	2.87689	0.171621	0.0858106	−	0.996311i	$-0.472652\pi$
$281$	2.87689	0.171621	0.0858106	+	0.996311i	$0.472652\pi$
$282$	0	0
$283$	4.87689	0.289901	0.144951	−	0.989439i	$-0.453698\pi$
$283$	4.87689	0.289901	0.144951	+	0.989439i	$0.453698\pi$
$284$	0	0
$285$	−7.31534	−0.433323
$286$	0	0
$287$	13.7538	0.811860
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−1.12311	−0.0658376
$292$	0	0
$293$	5.12311	0.299295	0.149648	−	0.988739i	$-0.452186\pi$
$293$	5.12311	0.299295	0.149648	+	0.988739i	$0.452186\pi$
$294$	0	0
$295$	−20.8769	−1.21550
$296$	0	0
$297$	2.43845	0.141493
$298$	0	0
$299$	26.9309	1.55745
$300$	0	0
$301$	24.0000	1.38334
$302$	0	0
$303$	5.12311	0.294315
$304$	0	0
$305$	−6.24621	−0.357657
$306$	0	0
$307$	−20.0000	−1.14146	−0.570730	−	0.821138i	$-0.693340\pi$
$307$	−20.0000	−1.14146	−0.570730	+	0.821138i	$0.693340\pi$
$308$	0	0
$309$	12.6847	0.721605
$310$	0	0
$311$	31.3693	1.77879	0.889395	−	0.457139i	$-0.151126\pi$
$311$	31.3693	1.77879	0.889395	+	0.457139i	$0.151126\pi$
$312$	0	0
$313$	29.6155	1.67397	0.836984	−	0.547227i	$-0.184317\pi$
$313$	29.6155	1.67397	0.836984	+	0.547227i	$0.184317\pi$
$314$	0	0
$315$	−8.00000	−0.450749
$316$	0	0
$317$	−19.1231	−1.07406	−0.537030	−	0.843563i	$-0.680454\pi$
$317$	−19.1231	−1.07406	−0.537030	+	0.843563i	$0.680454\pi$
$318$	0	0
$319$	17.3693	0.972495
$320$	0	0
$321$	8.68466	0.484730
$322$	0	0
$323$	−4.68466	−0.260661
$324$	0	0
$325$	9.12311	0.506059
$326$	0	0
$327$	6.24621	0.345416
$328$	0	0
$329$	4.49242	0.247675
$330$	0	0
$331$	−1.56155	−0.0858307	−0.0429154	−	0.999079i	$-0.513665\pi$
$331$	−1.56155	−0.0858307	−0.0429154	+	0.999079i	$0.513665\pi$
$332$	0	0
$333$	−4.00000	−0.219199
$334$	0	0
$335$	18.7386	1.02380
$336$	0	0
$337$	−2.49242	−0.135771	−0.0678855	−	0.997693i	$-0.521625\pi$
$337$	−2.49242	−0.135771	−0.0678855	+	0.997693i	$0.521625\pi$
$338$	0	0
$339$	−4.43845	−0.241063
$340$	0	0
$341$	−20.1080	−1.08891
$342$	0	0
$343$	−62.7386	−3.38757
$344$	0	0
$345$	11.8078	0.635709
$346$	0	0
$347$	−16.4924	−0.885360	−0.442680	−	0.896680i	$-0.645972\pi$
$347$	−16.4924	−0.885360	−0.442680	+	0.896680i	$0.645972\pi$
$348$	0	0
$349$	15.5616	0.832991	0.416495	−	0.909138i	$-0.363258\pi$
$349$	15.5616	0.832991	0.416495	+	0.909138i	$0.363258\pi$
$350$	0	0
$351$	3.56155	0.190102
$352$	0	0
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	0	0
$355$	17.7538	0.942273
$356$	0	0
$357$	−5.12311	−0.271144
$358$	0	0
$359$	−28.4924	−1.50377	−0.751886	−	0.659293i	$-0.770856\pi$
$359$	−28.4924	−1.50377	−0.751886	+	0.659293i	$0.770856\pi$
$360$	0	0
$361$	2.94602	0.155054
$362$	0	0
$363$	5.05398	0.265265
$364$	0	0
$365$	12.8769	0.674007
$366$	0	0
$367$	−2.87689	−0.150173	−0.0750863	−	0.997177i	$-0.523923\pi$
$367$	−2.87689	−0.150173	−0.0750863	+	0.997177i	$0.523923\pi$
$368$	0	0
$369$	−2.68466	−0.139758
$370$	0	0
$371$	−30.7386	−1.59587
$372$	0	0
$373$	30.4924	1.57884	0.789419	−	0.613855i	$-0.210382\pi$
$373$	30.4924	1.57884	0.789419	+	0.613855i	$0.210382\pi$
$374$	0	0
$375$	11.8078	0.609750
$376$	0	0
$377$	25.3693	1.30659
$378$	0	0
$379$	−0.492423	−0.0252940	−0.0126470	−	0.999920i	$-0.504026\pi$
$379$	−0.492423	−0.0252940	−0.0126470	+	0.999920i	$0.504026\pi$
$380$	0	0
$381$	−14.9309	−0.764932
$382$	0	0
$383$	18.2462	0.932338	0.466169	−	0.884696i	$-0.345634\pi$
$383$	18.2462	0.932338	0.466169	+	0.884696i	$0.345634\pi$
$384$	0	0
$385$	19.5076	0.994198
$386$	0	0
$387$	−4.68466	−0.238135
$388$	0	0
$389$	27.3693	1.38768	0.693840	−	0.720129i	$-0.255918\pi$
$389$	27.3693	1.38768	0.693840	+	0.720129i	$0.255918\pi$
$390$	0	0
$391$	7.56155	0.382404
$392$	0	0
$393$	13.5616	0.684090
$394$	0	0
$395$	3.12311	0.157140
$396$	0	0
$397$	10.2462	0.514243	0.257121	−	0.966379i	$-0.417226\pi$
$397$	10.2462	0.514243	0.257121	+	0.966379i	$0.417226\pi$
$398$	0	0
$399$	24.0000	1.20150
$400$	0	0
$401$	−32.9309	−1.64449	−0.822245	−	0.569134i	$-0.807278\pi$
$401$	−32.9309	−1.64449	−0.822245	+	0.569134i	$0.807278\pi$
$402$	0	0
$403$	−29.3693	−1.46299
$404$	0	0
$405$	1.56155	0.0775942
$406$	0	0
$407$	9.75379	0.483477
$408$	0	0
$409$	−12.4384	−0.615042	−0.307521	−	0.951541i	$-0.599499\pi$
$409$	−12.4384	−0.615042	−0.307521	+	0.951541i	$0.599499\pi$
$410$	0	0
$411$	−0.246211	−0.0121447
$412$	0	0
$413$	68.4924	3.37029
$414$	0	0
$415$	11.1231	0.546012
$416$	0	0
$417$	3.12311	0.152939
$418$	0	0
$419$	−28.0000	−1.36789	−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000	−1.36789	−0.683945	+	0.729534i	$0.739737\pi$
$420$	0	0
$421$	29.4233	1.43400	0.717002	−	0.697071i	$-0.245514\pi$
$421$	29.4233	1.43400	0.717002	+	0.697071i	$0.245514\pi$
$422$	0	0
$423$	−0.876894	−0.0426361
$424$	0	0
$425$	2.56155	0.124254
$426$	0	0
$427$	20.4924	0.991698
$428$	0	0
$429$	−8.68466	−0.419299
$430$	0	0
$431$	−25.6155	−1.23386	−0.616928	−	0.787019i	$-0.711623\pi$
$431$	−25.6155	−1.23386	−0.616928	+	0.787019i	$0.711623\pi$
$432$	0	0
$433$	−12.0540	−0.579277	−0.289639	−	0.957136i	$-0.593535\pi$
$433$	−12.0540	−0.579277	−0.289639	+	0.957136i	$0.593535\pi$
$434$	0	0
$435$	11.1231	0.533312
$436$	0	0
$437$	−35.4233	−1.69453
$438$	0	0
$439$	5.61553	0.268015	0.134007	−	0.990980i	$-0.457215\pi$
$439$	5.61553	0.268015	0.134007	+	0.990980i	$0.457215\pi$
$440$	0	0
$441$	19.2462	0.916486
$442$	0	0
$443$	5.36932	0.255104	0.127552	−	0.991832i	$-0.459288\pi$
$443$	5.36932	0.255104	0.127552	+	0.991832i	$0.459288\pi$
$444$	0	0
$445$	14.2462	0.675335
$446$	0	0
$447$	14.0000	0.662177
$448$	0	0
$449$	−0.246211	−0.0116194	−0.00580971	−	0.999983i	$-0.501849\pi$
$449$	−0.246211	−0.0116194	−0.00580971	+	0.999983i	$0.501849\pi$
$450$	0	0
$451$	6.54640	0.308258
$452$	0	0
$453$	−16.0000	−0.751746
$454$	0	0
$455$	28.4924	1.33575
$456$	0	0
$457$	−8.93087	−0.417768	−0.208884	−	0.977940i	$-0.566983\pi$
$457$	−8.93087	−0.417768	−0.208884	+	0.977940i	$0.566983\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	38.9848	1.81571	0.907853	−	0.419289i	$-0.137721\pi$
$461$	38.9848	1.81571	0.907853	+	0.419289i	$0.137721\pi$
$462$	0	0
$463$	26.2462	1.21976	0.609882	−	0.792492i	$-0.291217\pi$
$463$	26.2462	1.21976	0.609882	+	0.792492i	$0.291217\pi$
$464$	0	0
$465$	−12.8769	−0.597152
$466$	0	0
$467$	−0.876894	−0.0405778	−0.0202889	−	0.999794i	$-0.506459\pi$
$467$	−0.876894	−0.0405778	−0.0202889	+	0.999794i	$0.506459\pi$
$468$	0	0
$469$	−61.4773	−2.83876
$470$	0	0
$471$	7.56155	0.348418
$472$	0	0
$473$	11.4233	0.525244
$474$	0	0
$475$	−12.0000	−0.550598
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	−16.0540	−0.733525	−0.366762	−	0.930315i	$-0.619534\pi$
$479$	−16.0540	−0.733525	−0.366762	+	0.930315i	$0.619534\pi$
$480$	0	0
$481$	14.2462	0.649571
$482$	0	0
$483$	−38.7386	−1.76267
$484$	0	0
$485$	1.75379	0.0796354
$486$	0	0
$487$	20.7386	0.939757	0.469879	−	0.882731i	$-0.344298\pi$
$487$	20.7386	0.939757	0.469879	+	0.882731i	$0.344298\pi$
$488$	0	0
$489$	−13.3693	−0.604581
$490$	0	0
$491$	27.6155	1.24627	0.623136	−	0.782114i	$-0.285858\pi$
$491$	27.6155	1.24627	0.623136	+	0.782114i	$0.285858\pi$
$492$	0	0
$493$	7.12311	0.320809
$494$	0	0
$495$	−3.80776	−0.171146
$496$	0	0
$497$	−58.2462	−2.61270
$498$	0	0
$499$	−36.1080	−1.61641	−0.808207	−	0.588899i	$-0.799562\pi$
$499$	−36.1080	−1.61641	−0.808207	+	0.588899i	$0.799562\pi$
$500$	0	0
$501$	21.8078	0.974299
$502$	0	0
$503$	−2.68466	−0.119703	−0.0598515	−	0.998207i	$-0.519063\pi$
$503$	−2.68466	−0.119703	−0.0598515	+	0.998207i	$0.519063\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	0	0
$507$	0.315342	0.0140048
$508$	0	0
$509$	−12.6307	−0.559845	−0.279923	−	0.960023i	$-0.590309\pi$
$509$	−12.6307	−0.559845	−0.279923	+	0.960023i	$0.590309\pi$
$510$	0	0
$511$	−42.2462	−1.86886
$512$	0	0
$513$	−4.68466	−0.206833
$514$	0	0
$515$	−19.8078	−0.872834
$516$	0	0
$517$	2.13826	0.0940406
$518$	0	0
$519$	4.68466	0.205634
$520$	0	0
$521$	38.3002	1.67796	0.838981	−	0.544161i	$-0.183152\pi$
$521$	38.3002	1.67796	0.838981	+	0.544161i	$0.183152\pi$
$522$	0	0
$523$	−8.49242	−0.371348	−0.185674	−	0.982611i	$-0.559447\pi$
$523$	−8.49242	−0.371348	−0.185674	+	0.982611i	$0.559447\pi$
$524$	0	0
$525$	−13.1231	−0.572739
$526$	0	0
$527$	−8.24621	−0.359211
$528$	0	0
$529$	34.1771	1.48596
$530$	0	0
$531$	−13.3693	−0.580179
$532$	0	0
$533$	9.56155	0.414157
$534$	0	0
$535$	−13.5616	−0.586317
$536$	0	0
$537$	15.1231	0.652610
$538$	0	0
$539$	−46.9309	−2.02146
$540$	0	0
$541$	22.7386	0.977610	0.488805	−	0.872393i	$-0.337433\pi$
$541$	22.7386	0.977610	0.488805	+	0.872393i	$0.337433\pi$
$542$	0	0
$543$	−7.12311	−0.305682
$544$	0	0
$545$	−9.75379	−0.417806
$546$	0	0
$547$	26.7386	1.14326	0.571631	−	0.820511i	$-0.306311\pi$
$547$	26.7386	1.14326	0.571631	+	0.820511i	$0.306311\pi$
$548$	0	0
$549$	−4.00000	−0.170716
$550$	0	0
$551$	−33.3693	−1.42158
$552$	0	0
$553$	−10.2462	−0.435713
$554$	0	0
$555$	6.24621	0.265137
$556$	0	0
$557$	−28.7386	−1.21769	−0.608847	−	0.793287i	$-0.708368\pi$
$557$	−28.7386	−1.21769	−0.608847	+	0.793287i	$0.708368\pi$
$558$	0	0
$559$	16.6847	0.705686
$560$	0	0
$561$	−2.43845	−0.102951
$562$	0	0
$563$	5.36932	0.226290	0.113145	−	0.993579i	$-0.463908\pi$
$563$	5.36932	0.226290	0.113145	+	0.993579i	$0.463908\pi$
$564$	0	0
$565$	6.93087	0.291584
$566$	0	0
$567$	−5.12311	−0.215150
$568$	0	0
$569$	−17.1231	−0.717838	−0.358919	−	0.933369i	$-0.616855\pi$
$569$	−17.1231	−0.717838	−0.358919	+	0.933369i	$0.616855\pi$
$570$	0	0
$571$	−0.492423	−0.0206072	−0.0103036	−	0.999947i	$-0.503280\pi$
$571$	−0.492423	−0.0206072	−0.0103036	+	0.999947i	$0.503280\pi$
$572$	0	0
$573$	5.36932	0.224306
$574$	0	0
$575$	19.3693	0.807756
$576$	0	0
$577$	−3.94602	−0.164275	−0.0821376	−	0.996621i	$-0.526175\pi$
$577$	−3.94602	−0.164275	−0.0821376	+	0.996621i	$0.526175\pi$
$578$	0	0
$579$	−12.2462	−0.508935
$580$	0	0
$581$	−36.4924	−1.51396
$582$	0	0
$583$	−14.6307	−0.605941
$584$	0	0
$585$	−5.56155	−0.229942
$586$	0	0
$587$	−7.50758	−0.309871	−0.154935	−	0.987925i	$-0.549517\pi$
$587$	−7.50758	−0.309871	−0.154935	+	0.987925i	$0.549517\pi$
$588$	0	0
$589$	38.6307	1.59175
$590$	0	0
$591$	24.6847	1.01539
$592$	0	0
$593$	−42.4924	−1.74495	−0.872477	−	0.488655i	$-0.837488\pi$
$593$	−42.4924	−1.74495	−0.872477	+	0.488655i	$0.837488\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	0	0
$597$	14.8769	0.608871
$598$	0	0
$599$	33.3693	1.36343	0.681717	−	0.731616i	$-0.261234\pi$
$599$	33.3693	1.36343	0.681717	+	0.731616i	$0.261234\pi$
$600$	0	0
$601$	−24.7386	−1.00911	−0.504555	−	0.863380i	$-0.668343\pi$
$601$	−24.7386	−1.00911	−0.504555	+	0.863380i	$0.668343\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	0	0
$605$	−7.89205	−0.320857
$606$	0	0
$607$	11.7538	0.477072	0.238536	−	0.971134i	$-0.423333\pi$
$607$	11.7538	0.477072	0.238536	+	0.971134i	$0.423333\pi$
$608$	0	0
$609$	−36.4924	−1.47875
$610$	0	0
$611$	3.12311	0.126347
$612$	0	0
$613$	16.4384	0.663943	0.331971	−	0.943289i	$-0.392286\pi$
$613$	16.4384	0.663943	0.331971	+	0.943289i	$0.392286\pi$
$614$	0	0
$615$	4.19224	0.169047
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−24.8769	−0.999887	−0.499943	−	0.866058i	$-0.666646\pi$
$619$	−24.8769	−0.999887	−0.499943	+	0.866058i	$0.666646\pi$
$620$	0	0
$621$	7.56155	0.303435
$622$	0	0
$623$	−46.7386	−1.87254
$624$	0	0
$625$	−5.63068	−0.225227
$626$	0	0
$627$	11.4233	0.456202
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	−23.8078	−0.947772	−0.473886	−	0.880586i	$-0.657149\pi$
$631$	−23.8078	−0.947772	−0.473886	+	0.880586i	$0.657149\pi$
$632$	0	0
$633$	−19.1231	−0.760075
$634$	0	0
$635$	23.3153	0.925241
$636$	0	0
$637$	−68.5464	−2.71591
$638$	0	0
$639$	11.3693	0.449763
$640$	0	0
$641$	47.6695	1.88283	0.941416	−	0.337247i	$-0.109496\pi$
$641$	47.6695	1.88283	0.941416	+	0.337247i	$0.109496\pi$
$642$	0	0
$643$	16.4924	0.650398	0.325199	−	0.945646i	$-0.394569\pi$
$643$	16.4924	0.650398	0.325199	+	0.945646i	$0.394569\pi$
$644$	0	0
$645$	7.31534	0.288041
$646$	0	0
$647$	−24.8769	−0.978012	−0.489006	−	0.872281i	$-0.662640\pi$
$647$	−24.8769	−0.978012	−0.489006	+	0.872281i	$0.662640\pi$
$648$	0	0
$649$	32.6004	1.27968
$650$	0	0
$651$	42.2462	1.65576
$652$	0	0
$653$	−31.4233	−1.22969	−0.614844	−	0.788649i	$-0.710781\pi$
$653$	−31.4233	−1.22969	−0.614844	+	0.788649i	$0.710781\pi$
$654$	0	0
$655$	−21.1771	−0.827457
$656$	0	0
$657$	8.24621	0.321715
$658$	0	0
$659$	−3.61553	−0.140841	−0.0704205	−	0.997517i	$-0.522434\pi$
$659$	−3.61553	−0.140841	−0.0704205	+	0.997517i	$0.522434\pi$
$660$	0	0
$661$	9.31534	0.362325	0.181162	−	0.983453i	$-0.442014\pi$
$661$	9.31534	0.362325	0.181162	+	0.983453i	$0.442014\pi$
$662$	0	0
$663$	−3.56155	−0.138319
$664$	0	0
$665$	−37.4773	−1.45331
$666$	0	0
$667$	53.8617	2.08553
$668$	0	0
$669$	25.5616	0.988267
$670$	0	0
$671$	9.75379	0.376541
$672$	0	0
$673$	−26.0000	−1.00223	−0.501113	−	0.865382i	$-0.667076\pi$
$673$	−26.0000	−1.00223	−0.501113	+	0.865382i	$0.667076\pi$
$674$	0	0
$675$	2.56155	0.0985942
$676$	0	0
$677$	−19.8078	−0.761274	−0.380637	−	0.924724i	$-0.624295\pi$
$677$	−19.8078	−0.761274	−0.380637	+	0.924724i	$0.624295\pi$
$678$	0	0
$679$	−5.75379	−0.220810
$680$	0	0
$681$	3.31534	0.127044
$682$	0	0
$683$	−7.80776	−0.298756	−0.149378	−	0.988780i	$-0.547727\pi$
$683$	−7.80776	−0.298756	−0.149378	+	0.988780i	$0.547727\pi$
$684$	0	0
$685$	0.384472	0.0146899
$686$	0	0
$687$	18.0000	0.686743
$688$	0	0
$689$	−21.3693	−0.814106
$690$	0	0
$691$	9.75379	0.371052	0.185526	−	0.982639i	$-0.440601\pi$
$691$	9.75379	0.371052	0.185526	+	0.982639i	$0.440601\pi$
$692$	0	0
$693$	12.4924	0.474548
$694$	0	0
$695$	−4.87689	−0.184991
$696$	0	0
$697$	2.68466	0.101689
$698$	0	0
$699$	−7.56155	−0.286004
$700$	0	0
$701$	−37.1231	−1.40212	−0.701060	−	0.713102i	$-0.747290\pi$
$701$	−37.1231	−1.40212	−0.701060	+	0.713102i	$0.747290\pi$
$702$	0	0
$703$	−18.7386	−0.706741
$704$	0	0
$705$	1.36932	0.0515715
$706$	0	0
$707$	26.2462	0.987090
$708$	0	0
$709$	−11.1231	−0.417737	−0.208869	−	0.977944i	$-0.566978\pi$
$709$	−11.1231	−0.417737	−0.208869	+	0.977944i	$0.566978\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	0	0
$713$	−62.3542	−2.33518
$714$	0	0
$715$	13.5616	0.507173
$716$	0	0
$717$	−5.75379	−0.214879
$718$	0	0
$719$	38.7926	1.44672	0.723360	−	0.690471i	$-0.242597\pi$
$719$	38.7926	1.44672	0.723360	+	0.690471i	$0.242597\pi$
$720$	0	0
$721$	64.9848	2.42016
$722$	0	0
$723$	19.3693	0.720353
$724$	0	0
$725$	18.2462	0.677647
$726$	0	0
$727$	3.50758	0.130089	0.0650444	−	0.997882i	$-0.479281\pi$
$727$	3.50758	0.130089	0.0650444	+	0.997882i	$0.479281\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.68466	0.173268
$732$	0	0
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	0	0
$735$	−30.0540	−1.10856
$736$	0	0
$737$	−29.2614	−1.07786
$738$	0	0
$739$	−39.8078	−1.46435	−0.732176	−	0.681115i	$-0.761495\pi$
$739$	−39.8078	−1.46435	−0.732176	+	0.681115i	$0.761495\pi$
$740$	0	0
$741$	16.6847	0.612926
$742$	0	0
$743$	−5.12311	−0.187949	−0.0939743	−	0.995575i	$-0.529957\pi$
$743$	−5.12311	−0.187949	−0.0939743	+	0.995575i	$0.529957\pi$
$744$	0	0
$745$	−21.8617	−0.800952
$746$	0	0
$747$	7.12311	0.260621
$748$	0	0
$749$	44.4924	1.62572
$750$	0	0
$751$	14.9848	0.546805	0.273402	−	0.961900i	$-0.411851\pi$
$751$	14.9848	0.546805	0.273402	+	0.961900i	$0.411851\pi$
$752$	0	0
$753$	−10.2462	−0.373393
$754$	0	0
$755$	24.9848	0.909292
$756$	0	0
$757$	14.3002	0.519749	0.259875	−	0.965642i	$-0.416319\pi$
$757$	14.3002	0.519749	0.259875	+	0.965642i	$0.416319\pi$
$758$	0	0
$759$	−18.4384	−0.669273
$760$	0	0
$761$	−28.2462	−1.02392	−0.511962	−	0.859008i	$-0.671081\pi$
$761$	−28.2462	−1.02392	−0.511962	+	0.859008i	$0.671081\pi$
$762$	0	0
$763$	32.0000	1.15848
$764$	0	0
$765$	−1.56155	−0.0564581
$766$	0	0
$767$	47.6155	1.71930
$768$	0	0
$769$	52.0540	1.87711	0.938557	−	0.345124i	$-0.112163\pi$
$769$	52.0540	1.87711	0.938557	+	0.345124i	$0.112163\pi$
$770$	0	0
$771$	−13.1231	−0.472617
$772$	0	0
$773$	22.8769	0.822825	0.411412	−	0.911449i	$-0.365036\pi$
$773$	22.8769	0.822825	0.411412	+	0.911449i	$0.365036\pi$
$774$	0	0
$775$	−21.1231	−0.758764
$776$	0	0
$777$	−20.4924	−0.735162
$778$	0	0
$779$	−12.5767	−0.450607
$780$	0	0
$781$	−27.7235	−0.992024
$782$	0	0
$783$	7.12311	0.254559
$784$	0	0
$785$	−11.8078	−0.421437
$786$	0	0
$787$	−47.2311	−1.68361	−0.841803	−	0.539785i	$-0.818505\pi$
$787$	−47.2311	−1.68361	−0.841803	+	0.539785i	$0.818505\pi$
$788$	0	0
$789$	−4.00000	−0.142404
$790$	0	0
$791$	−22.7386	−0.808493
$792$	0	0
$793$	14.2462	0.505898
$794$	0	0
$795$	−9.36932	−0.332295
$796$	0	0
$797$	3.86174	0.136790	0.0683949	−	0.997658i	$-0.478212\pi$
$797$	3.86174	0.136790	0.0683949	+	0.997658i	$0.478212\pi$
$798$	0	0
$799$	0.876894	0.0310223
$800$	0	0
$801$	9.12311	0.322349
$802$	0	0
$803$	−20.1080	−0.709594
$804$	0	0
$805$	60.4924	2.13208
$806$	0	0
$807$	11.3153	0.398319
$808$	0	0
$809$	41.8078	1.46988	0.734941	−	0.678131i	$-0.237210\pi$
$809$	41.8078	1.46988	0.734941	+	0.678131i	$0.237210\pi$
$810$	0	0
$811$	53.3693	1.87405	0.937025	−	0.349262i	$-0.113568\pi$
$811$	53.3693	1.87405	0.937025	+	0.349262i	$0.113568\pi$
$812$	0	0
$813$	18.0540	0.633181
$814$	0	0
$815$	20.8769	0.731286
$816$	0	0
$817$	−21.9460	−0.767794
$818$	0	0
$819$	18.2462	0.637574
$820$	0	0
$821$	52.7926	1.84247	0.921237	−	0.389001i	$-0.127180\pi$
$821$	52.7926	1.84247	0.921237	+	0.389001i	$0.127180\pi$
$822$	0	0
$823$	27.3693	0.954034	0.477017	−	0.878894i	$-0.341718\pi$
$823$	27.3693	0.954034	0.477017	+	0.878894i	$0.341718\pi$
$824$	0	0
$825$	−6.24621	−0.217465
$826$	0	0
$827$	−13.5616	−0.471581	−0.235791	−	0.971804i	$-0.575768\pi$
$827$	−13.5616	−0.471581	−0.235791	+	0.971804i	$0.575768\pi$
$828$	0	0
$829$	−22.4924	−0.781194	−0.390597	−	0.920562i	$-0.627731\pi$
$829$	−22.4924	−0.781194	−0.390597	+	0.920562i	$0.627731\pi$
$830$	0	0
$831$	12.8769	0.446695
$832$	0	0
$833$	−19.2462	−0.666842
$834$	0	0
$835$	−34.0540	−1.17849
$836$	0	0
$837$	−8.24621	−0.285031
$838$	0	0
$839$	−3.94602	−0.136232	−0.0681160	−	0.997677i	$-0.521699\pi$
$839$	−3.94602	−0.136232	−0.0681160	+	0.997677i	$0.521699\pi$
$840$	0	0
$841$	21.7386	0.749608
$842$	0	0
$843$	−2.87689	−0.0990855
$844$	0	0
$845$	−0.492423	−0.0169398
$846$	0	0
$847$	25.8920	0.889661
$848$	0	0
$849$	−4.87689	−0.167375
$850$	0	0
$851$	30.2462	1.03683
$852$	0	0
$853$	−7.61553	−0.260751	−0.130375	−	0.991465i	$-0.541618\pi$
$853$	−7.61553	−0.260751	−0.130375	+	0.991465i	$0.541618\pi$
$854$	0	0
$855$	7.31534	0.250179
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−16.4924	−0.562714	−0.281357	−	0.959603i	$-0.590785\pi$
$859$	−16.4924	−0.562714	−0.281357	+	0.959603i	$0.590785\pi$
$860$	0	0
$861$	−13.7538	−0.468728
$862$	0	0
$863$	−50.7386	−1.72716	−0.863582	−	0.504209i	$-0.831784\pi$
$863$	−50.7386	−1.72716	−0.863582	+	0.504209i	$0.831784\pi$
$864$	0	0
$865$	−7.31534	−0.248729
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−4.87689	−0.165437
$870$	0	0
$871$	−42.7386	−1.44814
$872$	0	0
$873$	1.12311	0.0380114
$874$	0	0
$875$	60.4924	2.04502
$876$	0	0
$877$	34.3542	1.16006	0.580029	−	0.814596i	$-0.303041\pi$
$877$	34.3542	1.16006	0.580029	+	0.814596i	$0.303041\pi$
$878$	0	0
$879$	−5.12311	−0.172798
$880$	0	0
$881$	34.0000	1.14549	0.572745	−	0.819734i	$-0.305879\pi$
$881$	34.0000	1.14549	0.572745	+	0.819734i	$0.305879\pi$
$882$	0	0
$883$	−25.5616	−0.860215	−0.430107	−	0.902778i	$-0.641524\pi$
$883$	−25.5616	−0.860215	−0.430107	+	0.902778i	$0.641524\pi$
$884$	0	0
$885$	20.8769	0.701769
$886$	0	0
$887$	47.5616	1.59696	0.798480	−	0.602021i	$-0.205638\pi$
$887$	47.5616	1.59696	0.798480	+	0.602021i	$0.205638\pi$
$888$	0	0
$889$	−76.4924	−2.56547
$890$	0	0
$891$	−2.43845	−0.0816911
$892$	0	0
$893$	−4.10795	−0.137467
$894$	0	0
$895$	−23.6155	−0.789380
$896$	0	0
$897$	−26.9309	−0.899196
$898$	0	0
$899$	−58.7386	−1.95904
$900$	0	0
$901$	−6.00000	−0.199889
$902$	0	0
$903$	−24.0000	−0.798670
$904$	0	0
$905$	11.1231	0.369745
$906$	0	0
$907$	41.3693	1.37365	0.686823	−	0.726825i	$-0.259005\pi$
$907$	41.3693	1.37365	0.686823	+	0.726825i	$0.259005\pi$
$908$	0	0
$909$	−5.12311	−0.169923
$910$	0	0
$911$	28.4384	0.942208	0.471104	−	0.882078i	$-0.343856\pi$
$911$	28.4384	0.942208	0.471104	+	0.882078i	$0.343856\pi$
$912$	0	0
$913$	−17.3693	−0.574840
$914$	0	0
$915$	6.24621	0.206493
$916$	0	0
$917$	69.4773	2.29434
$918$	0	0
$919$	38.9309	1.28421	0.642105	−	0.766616i	$-0.278061\pi$
$919$	38.9309	1.28421	0.642105	+	0.766616i	$0.278061\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	0	0
$923$	−40.4924	−1.33282
$924$	0	0
$925$	10.2462	0.336893
$926$	0	0
$927$	−12.6847	−0.416619
$928$	0	0
$929$	−30.7926	−1.01027	−0.505136	−	0.863040i	$-0.668558\pi$
$929$	−30.7926	−1.01027	−0.505136	+	0.863040i	$0.668558\pi$
$930$	0	0
$931$	90.1619	2.95494
$932$	0	0
$933$	−31.3693	−1.02699
$934$	0	0
$935$	3.80776	0.124527
$936$	0	0
$937$	12.2462	0.400066	0.200033	−	0.979789i	$-0.435895\pi$
$937$	12.2462	0.400066	0.200033	+	0.979789i	$0.435895\pi$
$938$	0	0
$939$	−29.6155	−0.966466
$940$	0	0
$941$	−47.6155	−1.55222	−0.776111	−	0.630596i	$-0.782810\pi$
$941$	−47.6155	−1.55222	−0.776111	+	0.630596i	$0.782810\pi$
$942$	0	0
$943$	20.3002	0.661065
$944$	0	0
$945$	8.00000	0.260240
$946$	0	0
$947$	14.2462	0.462940	0.231470	−	0.972842i	$-0.425647\pi$
$947$	14.2462	0.462940	0.231470	+	0.972842i	$0.425647\pi$
$948$	0	0
$949$	−29.3693	−0.953368
$950$	0	0
$951$	19.1231	0.620109
$952$	0	0
$953$	42.1080	1.36401	0.682005	−	0.731347i	$-0.261108\pi$
$953$	42.1080	1.36401	0.682005	+	0.731347i	$0.261108\pi$
$954$	0	0
$955$	−8.38447	−0.271315
$956$	0	0
$957$	−17.3693	−0.561470
$958$	0	0
$959$	−1.26137	−0.0407316
$960$	0	0
$961$	37.0000	1.19355
$962$	0	0
$963$	−8.68466	−0.279859
$964$	0	0
$965$	19.1231	0.615595
$966$	0	0
$967$	9.56155	0.307479	0.153739	−	0.988111i	$-0.450868\pi$
$967$	9.56155	0.307479	0.153739	+	0.988111i	$0.450868\pi$
$968$	0	0
$969$	4.68466	0.150493
$970$	0	0
$971$	−46.3542	−1.48758	−0.743788	−	0.668416i	$-0.766973\pi$
$971$	−46.3542	−1.48758	−0.743788	+	0.668416i	$0.766973\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	−9.12311	−0.292173
$976$	0	0
$977$	5.50758	0.176203	0.0881015	−	0.996112i	$-0.471920\pi$
$977$	5.50758	0.176203	0.0881015	+	0.996112i	$0.471920\pi$
$978$	0	0
$979$	−22.2462	−0.710992
$980$	0	0
$981$	−6.24621	−0.199426
$982$	0	0
$983$	55.5616	1.77214	0.886069	−	0.463553i	$-0.153426\pi$
$983$	55.5616	1.77214	0.886069	+	0.463553i	$0.153426\pi$
$984$	0	0
$985$	−38.5464	−1.22819
$986$	0	0
$987$	−4.49242	−0.142995
$988$	0	0
$989$	35.4233	1.12640
$990$	0	0
$991$	−24.6307	−0.782419	−0.391210	−	0.920302i	$-0.627943\pi$
$991$	−24.6307	−0.782419	−0.391210	+	0.920302i	$0.627943\pi$
$992$	0	0
$993$	1.56155	0.0495544
$994$	0	0
$995$	−23.2311	−0.736474
$996$	0	0
$997$	13.3693	0.423411	0.211705	−	0.977334i	$-0.432098\pi$
$997$	13.3693	0.423411	0.211705	+	0.977334i	$0.432098\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	408.2.a.e.1.2	✓	2
3.2	odd	2		1224.2.a.j.1.1		2
4.3	odd	2		816.2.a.l.1.2		2
8.3	odd	2		3264.2.a.bi.1.1		2
8.5	even	2		3264.2.a.bo.1.1		2
12.11	even	2		2448.2.a.ba.1.1		2
17.16	even	2		6936.2.a.y.1.1		2
24.5	odd	2		9792.2.a.cm.1.2		2
24.11	even	2		9792.2.a.cn.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.a.e.1.2	✓	2	1.1	even	1	trivial
816.2.a.l.1.2		2	4.3	odd	2
1224.2.a.j.1.1		2	3.2	odd	2
2448.2.a.ba.1.1		2	12.11	even	2
3264.2.a.bi.1.1		2	8.3	odd	2
3264.2.a.bo.1.1		2	8.5	even	2
6936.2.a.y.1.1		2	17.16	even	2
9792.2.a.cm.1.2		2	24.5	odd	2
9792.2.a.cn.1.2		2	24.11	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:	\( N \)	\(=\)	\( 408 = 2^{3} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	408.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.56155 q^{5} -5.12311 q^{7} +1.00000 q^{9} -2.43845 q^{11} -3.56155 q^{13} -1.56155 q^{15} -1.00000 q^{17} +4.68466 q^{19} +5.12311 q^{21} -7.56155 q^{23} -2.56155 q^{25} -1.00000 q^{27} -7.12311 q^{29} +8.24621 q^{31} +2.43845 q^{33} -8.00000 q^{35} -4.00000 q^{37} +3.56155 q^{39} -2.68466 q^{41} -4.68466 q^{43} +1.56155 q^{45} -0.876894 q^{47} +19.2462 q^{49} +1.00000 q^{51} +6.00000 q^{53} -3.80776 q^{55} -4.68466 q^{57} -13.3693 q^{59} -4.00000 q^{61} -5.12311 q^{63} -5.56155 q^{65} +12.0000 q^{67} +7.56155 q^{69} +11.3693 q^{71} +8.24621 q^{73} +2.56155 q^{75} +12.4924 q^{77} +2.00000 q^{79} +1.00000 q^{81} +7.12311 q^{83} -1.56155 q^{85} +7.12311 q^{87} +9.12311 q^{89} +18.2462 q^{91} -8.24621 q^{93} +7.31534 q^{95} +1.12311 q^{97} -2.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} - 2 q^{7} + 2 q^{9} - 9 q^{11} - 3 q^{13} + q^{15} - 2 q^{17} - 3 q^{19} + 2 q^{21} - 11 q^{23} - q^{25} - 2 q^{27} - 6 q^{29} + 9 q^{33} - 16 q^{35} - 8 q^{37} + 3 q^{39} + 7 q^{41}+ \cdots - 9 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 - 2 * q^7 + 2 * q^9 - 9 * q^11 - 3 * q^13 + q^15 - 2 * q^17 - 3 * q^19 + 2 * q^21 - 11 * q^23 - q^25 - 2 * q^27 - 6 * q^29 + 9 * q^33 - 16 * q^35 - 8 * q^37 + 3 * q^39 + 7 * q^41 + 3 * q^43 - q^45 - 10 * q^47 + 22 * q^49 + 2 * q^51 + 12 * q^53 + 13 * q^55 + 3 * q^57 - 2 * q^59 - 8 * q^61 - 2 * q^63 - 7 * q^65 + 24 * q^67 + 11 * q^69 - 2 * q^71 + q^75 - 8 * q^77 + 4 * q^79 + 2 * q^81 + 6 * q^83 + q^85 + 6 * q^87 + 10 * q^89 + 20 * q^91 + 27 * q^95 - 6 * q^97 - 9 * q^99

Introduction

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