LMFDB - Embedded newform 920.2.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [920,2,Mod(1,920)] 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("920.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	920.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$7.34623698596$
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:	$\Z[a_1, a_2, a_3]$
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	$\chi$	$=$	920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} -1.00000 q^{5} -2.56155 q^{7} -0.561553 q^{9} -2.00000 q^{11} -3.56155 q^{13} -1.56155 q^{15} +1.43845 q^{17} -2.00000 q^{19} -4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.56155 q^{27} -8.12311 q^{29} +0.123106 q^{31} -3.12311 q^{33} +2.56155 q^{35} +0.561553 q^{37} -5.56155 q^{39} +4.12311 q^{41} +6.24621 q^{43} +0.561553 q^{45} -8.68466 q^{47} -0.438447 q^{49} +2.24621 q^{51} +8.56155 q^{53} +2.00000 q^{55} -3.12311 q^{57} -1.43845 q^{59} -0.876894 q^{61} +1.43845 q^{63} +3.56155 q^{65} -7.43845 q^{67} -1.56155 q^{69} -5.00000 q^{71} +7.56155 q^{73} +1.56155 q^{75} +5.12311 q^{77} -0.876894 q^{79} -7.00000 q^{81} +7.68466 q^{83} -1.43845 q^{85} -12.6847 q^{87} +8.00000 q^{89} +9.12311 q^{91} +0.192236 q^{93} +2.00000 q^{95} +5.12311 q^{97} +1.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} - q^{7} + 3 q^{9} - 4 q^{11} - 3 q^{13} + q^{15} + 7 q^{17} - 4 q^{19} - 8 q^{21} - 2 q^{23} + 2 q^{25} - 7 q^{27} - 8 q^{29} - 8 q^{31} + 2 q^{33} + q^{35} - 3 q^{37} - 7 q^{39}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 - q^7 + 3 * q^9 - 4 * q^11 - 3 * q^13 + q^15 + 7 * q^17 - 4 * q^19 - 8 * q^21 - 2 * q^23 + 2 * q^25 - 7 * q^27 - 8 * q^29 - 8 * q^31 + 2 * q^33 + q^35 - 3 * q^37 - 7 * q^39 - 4 * q^43 - 3 * q^45 - 5 * q^47 - 5 * q^49 - 12 * q^51 + 13 * q^53 + 4 * q^55 + 2 * q^57 - 7 * q^59 - 10 * q^61 + 7 * q^63 + 3 * q^65 - 19 * q^67 + q^69 - 10 * q^71 + 11 * q^73 - q^75 + 2 * q^77 - 10 * q^79 - 14 * q^81 + 3 * q^83 - 7 * q^85 - 13 * q^87 + 16 * q^89 + 10 * q^91 + 21 * q^93 + 4 * q^95 + 2 * q^97 - 6 * 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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.56155	−0.968176	−0.484088	−	0.875019i	$-0.660849\pi$
$7$	−2.56155	−0.968176	−0.484088	+	0.875019i	$0.660849\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	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
$17$	1.43845	0.348875	0.174437	−	0.984668i	$-0.444189\pi$
$17$	1.43845	0.348875	0.174437	+	0.984668i	$0.444189\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	−4.00000	−0.872872
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	−8.12311	−1.50842	−0.754211	−	0.656632i	$-0.771981\pi$
$29$	−8.12311	−1.50842	−0.754211	+	0.656632i	$0.771981\pi$
$30$	0	0
$31$	0.123106	0.0221104	0.0110552	−	0.999939i	$-0.496481\pi$
$31$	0.123106	0.0221104	0.0110552	+	0.999939i	$0.496481\pi$
$32$	0	0
$33$	−3.12311	−0.543663
$34$	0	0
$35$	2.56155	0.432981
$36$	0	0
$37$	0.561553	0.0923187	0.0461594	−	0.998934i	$-0.485302\pi$
$37$	0.561553	0.0923187	0.0461594	+	0.998934i	$0.485302\pi$
$38$	0	0
$39$	−5.56155	−0.890561
$40$	0	0
$41$	4.12311	0.643921	0.321960	−	0.946753i	$-0.395658\pi$
$41$	4.12311	0.643921	0.321960	+	0.946753i	$0.395658\pi$
$42$	0	0
$43$	6.24621	0.952538	0.476269	−	0.879300i	$-0.341989\pi$
$43$	6.24621	0.952538	0.476269	+	0.879300i	$0.341989\pi$
$44$	0	0
$45$	0.561553	0.0837114
$46$	0	0
$47$	−8.68466	−1.26679	−0.633394	−	0.773830i	$-0.718339\pi$
$47$	−8.68466	−1.26679	−0.633394	+	0.773830i	$0.718339\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	2.24621	0.314532
$52$	0	0
$53$	8.56155	1.17602	0.588010	−	0.808854i	$-0.299912\pi$
$53$	8.56155	1.17602	0.588010	+	0.808854i	$0.299912\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	−3.12311	−0.413665
$58$	0	0
$59$	−1.43845	−0.187270	−0.0936349	−	0.995607i	$-0.529849\pi$
$59$	−1.43845	−0.187270	−0.0936349	+	0.995607i	$0.529849\pi$
$60$	0	0
$61$	−0.876894	−0.112275	−0.0561374	−	0.998423i	$-0.517878\pi$
$61$	−0.876894	−0.112275	−0.0561374	+	0.998423i	$0.517878\pi$
$62$	0	0
$63$	1.43845	0.181227
$64$	0	0
$65$	3.56155	0.441756
$66$	0	0
$67$	−7.43845	−0.908751	−0.454375	−	0.890810i	$-0.650138\pi$
$67$	−7.43845	−0.908751	−0.454375	+	0.890810i	$0.650138\pi$
$68$	0	0
$69$	−1.56155	−0.187989
$70$	0	0
$71$	−5.00000	−0.593391	−0.296695	−	0.954972i	$-0.595885\pi$
$71$	−5.00000	−0.593391	−0.296695	+	0.954972i	$0.595885\pi$
$72$	0	0
$73$	7.56155	0.885013	0.442506	−	0.896765i	$-0.354089\pi$
$73$	7.56155	0.885013	0.442506	+	0.896765i	$0.354089\pi$
$74$	0	0
$75$	1.56155	0.180313
$76$	0	0
$77$	5.12311	0.583832
$78$	0	0
$79$	−0.876894	−0.0986583	−0.0493292	−	0.998783i	$-0.515708\pi$
$79$	−0.876894	−0.0986583	−0.0493292	+	0.998783i	$0.515708\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	7.68466	0.843501	0.421750	−	0.906712i	$-0.361416\pi$
$83$	7.68466	0.843501	0.421750	+	0.906712i	$0.361416\pi$
$84$	0	0
$85$	−1.43845	−0.156022
$86$	0	0
$87$	−12.6847	−1.35994
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	9.12311	0.956361
$92$	0	0
$93$	0.192236	0.0199339
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	5.12311	0.520173	0.260086	−	0.965585i	$-0.416249\pi$
$97$	5.12311	0.520173	0.260086	+	0.965585i	$0.416249\pi$
$98$	0	0
$99$	1.12311	0.112876
$100$	0	0
$101$	10.8078	1.07541	0.537706	−	0.843132i	$-0.319291\pi$
$101$	10.8078	1.07541	0.537706	+	0.843132i	$0.319291\pi$
$102$	0	0
$103$	−14.2462	−1.40372	−0.701860	−	0.712314i	$-0.747647\pi$
$103$	−14.2462	−1.40372	−0.701860	+	0.712314i	$0.747647\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	0	0
$107$	−1.68466	−0.162862	−0.0814310	−	0.996679i	$-0.525949\pi$
$107$	−1.68466	−0.162862	−0.0814310	+	0.996679i	$0.525949\pi$
$108$	0	0
$109$	−1.12311	−0.107574	−0.0537870	−	0.998552i	$-0.517129\pi$
$109$	−1.12311	−0.107574	−0.0537870	+	0.998552i	$0.517129\pi$
$110$	0	0
$111$	0.876894	0.0832311
$112$	0	0
$113$	5.43845	0.511606	0.255803	−	0.966729i	$-0.417660\pi$
$113$	5.43845	0.511606	0.255803	+	0.966729i	$0.417660\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−3.68466	−0.337772
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	6.43845	0.580535
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.8078	−1.75765	−0.878827	−	0.477140i	$-0.841674\pi$
$127$	−19.8078	−1.75765	−0.878827	+	0.477140i	$0.841674\pi$
$128$	0	0
$129$	9.75379	0.858773
$130$	0	0
$131$	−18.9309	−1.65400	−0.826999	−	0.562204i	$-0.809954\pi$
$131$	−18.9309	−1.65400	−0.826999	+	0.562204i	$0.809954\pi$
$132$	0	0
$133$	5.12311	0.444230
$134$	0	0
$135$	5.56155	0.478662
$136$	0	0
$137$	21.6155	1.84674	0.923370	−	0.383912i	$-0.125423\pi$
$137$	21.6155	1.84674	0.923370	+	0.383912i	$0.125423\pi$
$138$	0	0
$139$	−10.3693	−0.879514	−0.439757	−	0.898117i	$-0.644935\pi$
$139$	−10.3693	−0.879514	−0.439757	+	0.898117i	$0.644935\pi$
$140$	0	0
$141$	−13.5616	−1.14209
$142$	0	0
$143$	7.12311	0.595664
$144$	0	0
$145$	8.12311	0.674587
$146$	0	0
$147$	−0.684658	−0.0564697
$148$	0	0
$149$	0.876894	0.0718380	0.0359190	−	0.999355i	$-0.488564\pi$
$149$	0.876894	0.0718380	0.0359190	+	0.999355i	$0.488564\pi$
$150$	0	0
$151$	1.56155	0.127077	0.0635387	−	0.997979i	$-0.479761\pi$
$151$	1.56155	0.127077	0.0635387	+	0.997979i	$0.479761\pi$
$152$	0	0
$153$	−0.807764	−0.0653039
$154$	0	0
$155$	−0.123106	−0.00988808
$156$	0	0
$157$	−15.6847	−1.25177	−0.625886	−	0.779915i	$-0.715262\pi$
$157$	−15.6847	−1.25177	−0.625886	+	0.779915i	$0.715262\pi$
$158$	0	0
$159$	13.3693	1.06026
$160$	0	0
$161$	2.56155	0.201879
$162$	0	0
$163$	−6.68466	−0.523583	−0.261791	−	0.965124i	$-0.584313\pi$
$163$	−6.68466	−0.523583	−0.261791	+	0.965124i	$0.584313\pi$
$164$	0	0
$165$	3.12311	0.243133
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	0	0
$171$	1.12311	0.0858860
$172$	0	0
$173$	−7.12311	−0.541560	−0.270780	−	0.962641i	$-0.587282\pi$
$173$	−7.12311	−0.541560	−0.270780	+	0.962641i	$0.587282\pi$
$174$	0	0
$175$	−2.56155	−0.193635
$176$	0	0
$177$	−2.24621	−0.168836
$178$	0	0
$179$	−7.31534	−0.546774	−0.273387	−	0.961904i	$-0.588144\pi$
$179$	−7.31534	−0.546774	−0.273387	+	0.961904i	$0.588144\pi$
$180$	0	0
$181$	−8.87689	−0.659814	−0.329907	−	0.944013i	$-0.607017\pi$
$181$	−8.87689	−0.659814	−0.329907	+	0.944013i	$0.607017\pi$
$182$	0	0
$183$	−1.36932	−0.101223
$184$	0	0
$185$	−0.561553	−0.0412862
$186$	0	0
$187$	−2.87689	−0.210379
$188$	0	0
$189$	14.2462	1.03626
$190$	0	0
$191$	−3.12311	−0.225980	−0.112990	−	0.993596i	$-0.536043\pi$
$191$	−3.12311	−0.225980	−0.112990	+	0.993596i	$0.536043\pi$
$192$	0	0
$193$	18.9309	1.36267	0.681337	−	0.731970i	$-0.261399\pi$
$193$	18.9309	1.36267	0.681337	+	0.731970i	$0.261399\pi$
$194$	0	0
$195$	5.56155	0.398271
$196$	0	0
$197$	17.8078	1.26875	0.634375	−	0.773025i	$-0.281257\pi$
$197$	17.8078	1.26875	0.634375	+	0.773025i	$0.281257\pi$
$198$	0	0
$199$	18.0000	1.27599	0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000	1.27599	0.637993	+	0.770042i	$0.279765\pi$
$200$	0	0
$201$	−11.6155	−0.819296
$202$	0	0
$203$	20.8078	1.46042
$204$	0	0
$205$	−4.12311	−0.287970
$206$	0	0
$207$	0.561553	0.0390306
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−9.93087	−0.683669	−0.341835	−	0.939760i	$-0.611048\pi$
$211$	−9.93087	−0.683669	−0.341835	+	0.939760i	$0.611048\pi$
$212$	0	0
$213$	−7.80776	−0.534979
$214$	0	0
$215$	−6.24621	−0.425988
$216$	0	0
$217$	−0.315342	−0.0214068
$218$	0	0
$219$	11.8078	0.797895
$220$	0	0
$221$	−5.12311	−0.344617
$222$	0	0
$223$	18.8769	1.26409	0.632045	−	0.774932i	$-0.282216\pi$
$223$	18.8769	1.26409	0.632045	+	0.774932i	$0.282216\pi$
$224$	0	0
$225$	−0.561553	−0.0374369
$226$	0	0
$227$	18.2462	1.21104	0.605522	−	0.795829i	$-0.292964\pi$
$227$	18.2462	1.21104	0.605522	+	0.795829i	$0.292964\pi$
$228$	0	0
$229$	−5.75379	−0.380221	−0.190111	−	0.981763i	$-0.560885\pi$
$229$	−5.75379	−0.380221	−0.190111	+	0.981763i	$0.560885\pi$
$230$	0	0
$231$	8.00000	0.526361
$232$	0	0
$233$	28.3002	1.85401	0.927003	−	0.375053i	$-0.122375\pi$
$233$	28.3002	1.85401	0.927003	+	0.375053i	$0.122375\pi$
$234$	0	0
$235$	8.68466	0.566525
$236$	0	0
$237$	−1.36932	−0.0889467
$238$	0	0
$239$	−7.87689	−0.509514	−0.254757	−	0.967005i	$-0.581995\pi$
$239$	−7.87689	−0.509514	−0.254757	+	0.967005i	$0.581995\pi$
$240$	0	0
$241$	−18.4924	−1.19120	−0.595601	−	0.803281i	$-0.703086\pi$
$241$	−18.4924	−1.19120	−0.595601	+	0.803281i	$0.703086\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	0.438447	0.0280114
$246$	0	0
$247$	7.12311	0.453232
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	9.12311	0.575845	0.287923	−	0.957654i	$-0.407035\pi$
$251$	9.12311	0.575845	0.287923	+	0.957654i	$0.407035\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	0	0
$255$	−2.24621	−0.140663
$256$	0	0
$257$	−0.684658	−0.0427078	−0.0213539	−	0.999772i	$-0.506798\pi$
$257$	−0.684658	−0.0427078	−0.0213539	+	0.999772i	$0.506798\pi$
$258$	0	0
$259$	−1.43845	−0.0893808
$260$	0	0
$261$	4.56155	0.282353
$262$	0	0
$263$	−14.3153	−0.882722	−0.441361	−	0.897330i	$-0.645504\pi$
$263$	−14.3153	−0.882722	−0.441361	+	0.897330i	$0.645504\pi$
$264$	0	0
$265$	−8.56155	−0.525932
$266$	0	0
$267$	12.4924	0.764524
$268$	0	0
$269$	−27.4924	−1.67624	−0.838121	−	0.545484i	$-0.816346\pi$
$269$	−27.4924	−1.67624	−0.838121	+	0.545484i	$0.816346\pi$
$270$	0	0
$271$	24.1771	1.46865	0.734327	−	0.678796i	$-0.237498\pi$
$271$	24.1771	1.46865	0.734327	+	0.678796i	$0.237498\pi$
$272$	0	0
$273$	14.2462	0.862220
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−20.4384	−1.22803	−0.614014	−	0.789295i	$-0.710446\pi$
$277$	−20.4384	−1.22803	−0.614014	+	0.789295i	$0.710446\pi$
$278$	0	0
$279$	−0.0691303	−0.00413872
$280$	0	0
$281$	−9.12311	−0.544239	−0.272119	−	0.962263i	$-0.587725\pi$
$281$	−9.12311	−0.544239	−0.272119	+	0.962263i	$0.587725\pi$
$282$	0	0
$283$	3.93087	0.233666	0.116833	−	0.993152i	$-0.462726\pi$
$283$	3.93087	0.233666	0.116833	+	0.993152i	$0.462726\pi$
$284$	0	0
$285$	3.12311	0.184997
$286$	0	0
$287$	−10.5616	−0.623429
$288$	0	0
$289$	−14.9309	−0.878286
$290$	0	0
$291$	8.00000	0.468968
$292$	0	0
$293$	14.5616	0.850695	0.425347	−	0.905030i	$-0.360152\pi$
$293$	14.5616	0.850695	0.425347	+	0.905030i	$0.360152\pi$
$294$	0	0
$295$	1.43845	0.0837496
$296$	0	0
$297$	11.1231	0.645428
$298$	0	0
$299$	3.56155	0.205970
$300$	0	0
$301$	−16.0000	−0.922225
$302$	0	0
$303$	16.8769	0.969552
$304$	0	0
$305$	0.876894	0.0502108
$306$	0	0
$307$	−15.3693	−0.877173	−0.438587	−	0.898689i	$-0.644521\pi$
$307$	−15.3693	−0.877173	−0.438587	+	0.898689i	$0.644521\pi$
$308$	0	0
$309$	−22.2462	−1.26554
$310$	0	0
$311$	−4.19224	−0.237720	−0.118860	−	0.992911i	$-0.537924\pi$
$311$	−4.19224	−0.237720	−0.118860	+	0.992911i	$0.537924\pi$
$312$	0	0
$313$	−13.9309	−0.787419	−0.393710	−	0.919235i	$-0.628808\pi$
$313$	−13.9309	−0.787419	−0.393710	+	0.919235i	$0.628808\pi$
$314$	0	0
$315$	−1.43845	−0.0810473
$316$	0	0
$317$	25.3693	1.42488	0.712441	−	0.701732i	$-0.247589\pi$
$317$	25.3693	1.42488	0.712441	+	0.701732i	$0.247589\pi$
$318$	0	0
$319$	16.2462	0.909613
$320$	0	0
$321$	−2.63068	−0.146830
$322$	0	0
$323$	−2.87689	−0.160075
$324$	0	0
$325$	−3.56155	−0.197559
$326$	0	0
$327$	−1.75379	−0.0969847
$328$	0	0
$329$	22.2462	1.22647
$330$	0	0
$331$	−3.49242	−0.191961	−0.0959805	−	0.995383i	$-0.530599\pi$
$331$	−3.49242	−0.191961	−0.0959805	+	0.995383i	$0.530599\pi$
$332$	0	0
$333$	−0.315342	−0.0172806
$334$	0	0
$335$	7.43845	0.406406
$336$	0	0
$337$	13.1231	0.714861	0.357431	−	0.933940i	$-0.383653\pi$
$337$	13.1231	0.714861	0.357431	+	0.933940i	$0.383653\pi$
$338$	0	0
$339$	8.49242	0.461245
$340$	0	0
$341$	−0.246211	−0.0133331
$342$	0	0
$343$	19.0540	1.02882
$344$	0	0
$345$	1.56155	0.0840712
$346$	0	0
$347$	−14.2462	−0.764777	−0.382388	−	0.924002i	$-0.624898\pi$
$347$	−14.2462	−0.764777	−0.382388	+	0.924002i	$0.624898\pi$
$348$	0	0
$349$	−4.12311	−0.220705	−0.110352	−	0.993893i	$-0.535198\pi$
$349$	−4.12311	−0.220705	−0.110352	+	0.993893i	$0.535198\pi$
$350$	0	0
$351$	19.8078	1.05726
$352$	0	0
$353$	4.19224	0.223130	0.111565	−	0.993757i	$-0.464414\pi$
$353$	4.19224	0.223130	0.111565	+	0.993757i	$0.464414\pi$
$354$	0	0
$355$	5.00000	0.265372
$356$	0	0
$357$	−5.75379	−0.304523
$358$	0	0
$359$	11.1231	0.587055	0.293528	−	0.955951i	$-0.405171\pi$
$359$	11.1231	0.587055	0.293528	+	0.955951i	$0.405171\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	−10.9309	−0.573722
$364$	0	0
$365$	−7.56155	−0.395790
$366$	0	0
$367$	−11.1922	−0.584230	−0.292115	−	0.956383i	$-0.594359\pi$
$367$	−11.1922	−0.584230	−0.292115	+	0.956383i	$0.594359\pi$
$368$	0	0
$369$	−2.31534	−0.120532
$370$	0	0
$371$	−21.9309	−1.13859
$372$	0	0
$373$	24.7386	1.28092	0.640459	−	0.767992i	$-0.278744\pi$
$373$	24.7386	1.28092	0.640459	+	0.767992i	$0.278744\pi$
$374$	0	0
$375$	−1.56155	−0.0806382
$376$	0	0
$377$	28.9309	1.49002
$378$	0	0
$379$	−32.9848	−1.69432	−0.847159	−	0.531340i	$-0.821689\pi$
$379$	−32.9848	−1.69432	−0.847159	+	0.531340i	$0.821689\pi$
$380$	0	0
$381$	−30.9309	−1.58464
$382$	0	0
$383$	−13.9309	−0.711834	−0.355917	−	0.934518i	$-0.615831\pi$
$383$	−13.9309	−0.711834	−0.355917	+	0.934518i	$0.615831\pi$
$384$	0	0
$385$	−5.12311	−0.261098
$386$	0	0
$387$	−3.50758	−0.178300
$388$	0	0
$389$	−18.8769	−0.957097	−0.478548	−	0.878061i	$-0.658837\pi$
$389$	−18.8769	−0.957097	−0.478548	+	0.878061i	$0.658837\pi$
$390$	0	0
$391$	−1.43845	−0.0727454
$392$	0	0
$393$	−29.5616	−1.49118
$394$	0	0
$395$	0.876894	0.0441213
$396$	0	0
$397$	−21.1771	−1.06285	−0.531424	−	0.847106i	$-0.678343\pi$
$397$	−21.1771	−1.06285	−0.531424	+	0.847106i	$0.678343\pi$
$398$	0	0
$399$	8.00000	0.400501
$400$	0	0
$401$	−32.7386	−1.63489	−0.817445	−	0.576007i	$-0.804610\pi$
$401$	−32.7386	−1.63489	−0.817445	+	0.576007i	$0.804610\pi$
$402$	0	0
$403$	−0.438447	−0.0218406
$404$	0	0
$405$	7.00000	0.347833
$406$	0	0
$407$	−1.12311	−0.0556703
$408$	0	0
$409$	5.00000	0.247234	0.123617	−	0.992330i	$-0.460551\pi$
$409$	5.00000	0.247234	0.123617	+	0.992330i	$0.460551\pi$
$410$	0	0
$411$	33.7538	1.66495
$412$	0	0
$413$	3.68466	0.181310
$414$	0	0
$415$	−7.68466	−0.377225
$416$	0	0
$417$	−16.1922	−0.792937
$418$	0	0
$419$	−21.6155	−1.05599	−0.527994	−	0.849248i	$-0.677056\pi$
$419$	−21.6155	−1.05599	−0.527994	+	0.849248i	$0.677056\pi$
$420$	0	0
$421$	−30.8769	−1.50485	−0.752424	−	0.658679i	$-0.771115\pi$
$421$	−30.8769	−1.50485	−0.752424	+	0.658679i	$0.771115\pi$
$422$	0	0
$423$	4.87689	0.237123
$424$	0	0
$425$	1.43845	0.0697749
$426$	0	0
$427$	2.24621	0.108702
$428$	0	0
$429$	11.1231	0.537029
$430$	0	0
$431$	30.2462	1.45691	0.728454	−	0.685094i	$-0.240239\pi$
$431$	30.2462	1.45691	0.728454	+	0.685094i	$0.240239\pi$
$432$	0	0
$433$	−13.6847	−0.657643	−0.328821	−	0.944392i	$-0.606651\pi$
$433$	−13.6847	−0.657643	−0.328821	+	0.944392i	$0.606651\pi$
$434$	0	0
$435$	12.6847	0.608183
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	−3.80776	−0.181735	−0.0908673	−	0.995863i	$-0.528964\pi$
$439$	−3.80776	−0.181735	−0.0908673	+	0.995863i	$0.528964\pi$
$440$	0	0
$441$	0.246211	0.0117243
$442$	0	0
$443$	22.0540	1.04782	0.523908	−	0.851775i	$-0.324474\pi$
$443$	22.0540	1.04782	0.523908	+	0.851775i	$0.324474\pi$
$444$	0	0
$445$	−8.00000	−0.379236
$446$	0	0
$447$	1.36932	0.0647665
$448$	0	0
$449$	18.3153	0.864354	0.432177	−	0.901789i	$-0.357745\pi$
$449$	18.3153	0.864354	0.432177	+	0.901789i	$0.357745\pi$
$450$	0	0
$451$	−8.24621	−0.388299
$452$	0	0
$453$	2.43845	0.114568
$454$	0	0
$455$	−9.12311	−0.427698
$456$	0	0
$457$	−17.9309	−0.838771	−0.419385	−	0.907808i	$-0.637754\pi$
$457$	−17.9309	−0.838771	−0.419385	+	0.907808i	$0.637754\pi$
$458$	0	0
$459$	−8.00000	−0.373408
$460$	0	0
$461$	−39.1771	−1.82466	−0.912329	−	0.409457i	$-0.865718\pi$
$461$	−39.1771	−1.82466	−0.912329	+	0.409457i	$0.865718\pi$
$462$	0	0
$463$	−29.1231	−1.35347	−0.676733	−	0.736229i	$-0.736605\pi$
$463$	−29.1231	−1.35347	−0.676733	+	0.736229i	$0.736605\pi$
$464$	0	0
$465$	−0.192236	−0.00891473
$466$	0	0
$467$	−9.68466	−0.448153	−0.224076	−	0.974572i	$-0.571936\pi$
$467$	−9.68466	−0.448153	−0.224076	+	0.974572i	$0.571936\pi$
$468$	0	0
$469$	19.0540	0.879831
$470$	0	0
$471$	−24.4924	−1.12855
$472$	0	0
$473$	−12.4924	−0.574402
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	−4.80776	−0.220132
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	−5.12311	−0.232628
$486$	0	0
$487$	0.684658	0.0310248	0.0155124	−	0.999880i	$-0.495062\pi$
$487$	0.684658	0.0310248	0.0155124	+	0.999880i	$0.495062\pi$
$488$	0	0
$489$	−10.4384	−0.472043
$490$	0	0
$491$	−12.6155	−0.569331	−0.284665	−	0.958627i	$-0.591882\pi$
$491$	−12.6155	−0.569331	−0.284665	+	0.958627i	$0.591882\pi$
$492$	0	0
$493$	−11.6847	−0.526251
$494$	0	0
$495$	−1.12311	−0.0504798
$496$	0	0
$497$	12.8078	0.574507
$498$	0	0
$499$	−39.7386	−1.77895	−0.889473	−	0.456988i	$-0.848928\pi$
$499$	−39.7386	−1.77895	−0.889473	+	0.456988i	$0.848928\pi$
$500$	0	0
$501$	12.4924	0.558120
$502$	0	0
$503$	15.4384	0.688366	0.344183	−	0.938903i	$-0.388156\pi$
$503$	15.4384	0.688366	0.344183	+	0.938903i	$0.388156\pi$
$504$	0	0
$505$	−10.8078	−0.480939
$506$	0	0
$507$	−0.492423	−0.0218693
$508$	0	0
$509$	−24.4384	−1.08322	−0.541608	−	0.840631i	$-0.682184\pi$
$509$	−24.4384	−1.08322	−0.541608	+	0.840631i	$0.682184\pi$
$510$	0	0
$511$	−19.3693	−0.856848
$512$	0	0
$513$	11.1231	0.491097
$514$	0	0
$515$	14.2462	0.627763
$516$	0	0
$517$	17.3693	0.763902
$518$	0	0
$519$	−11.1231	−0.488250
$520$	0	0
$521$	23.8617	1.04540	0.522701	−	0.852516i	$-0.324924\pi$
$521$	23.8617	1.04540	0.522701	+	0.852516i	$0.324924\pi$
$522$	0	0
$523$	10.2462	0.448036	0.224018	−	0.974585i	$-0.428083\pi$
$523$	10.2462	0.448036	0.224018	+	0.974585i	$0.428083\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	0	0
$527$	0.177081	0.00771377
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.807764	0.0350540
$532$	0	0
$533$	−14.6847	−0.636063
$534$	0	0
$535$	1.68466	0.0728341
$536$	0	0
$537$	−11.4233	−0.492951
$538$	0	0
$539$	0.876894	0.0377705
$540$	0	0
$541$	16.0540	0.690214	0.345107	−	0.938563i	$-0.387843\pi$
$541$	16.0540	0.690214	0.345107	+	0.938563i	$0.387843\pi$
$542$	0	0
$543$	−13.8617	−0.594864
$544$	0	0
$545$	1.12311	0.0481086
$546$	0	0
$547$	−7.31534	−0.312781	−0.156391	−	0.987695i	$-0.549986\pi$
$547$	−7.31534	−0.312781	−0.156391	+	0.987695i	$0.549986\pi$
$548$	0	0
$549$	0.492423	0.0210161
$550$	0	0
$551$	16.2462	0.692112
$552$	0	0
$553$	2.24621	0.0955186
$554$	0	0
$555$	−0.876894	−0.0372221
$556$	0	0
$557$	−21.3002	−0.902518	−0.451259	−	0.892393i	$-0.649025\pi$
$557$	−21.3002	−0.902518	−0.451259	+	0.892393i	$0.649025\pi$
$558$	0	0
$559$	−22.2462	−0.940914
$560$	0	0
$561$	−4.49242	−0.189670
$562$	0	0
$563$	−11.1922	−0.471697	−0.235848	−	0.971790i	$-0.575787\pi$
$563$	−11.1922	−0.471697	−0.235848	+	0.971790i	$0.575787\pi$
$564$	0	0
$565$	−5.43845	−0.228797
$566$	0	0
$567$	17.9309	0.753026
$568$	0	0
$569$	44.7386	1.87554	0.937771	−	0.347256i	$-0.112886\pi$
$569$	44.7386	1.87554	0.937771	+	0.347256i	$0.112886\pi$
$570$	0	0
$571$	−30.7386	−1.28637	−0.643186	−	0.765710i	$-0.722388\pi$
$571$	−30.7386	−1.28637	−0.643186	+	0.765710i	$0.722388\pi$
$572$	0	0
$573$	−4.87689	−0.203735
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−7.31534	−0.304542	−0.152271	−	0.988339i	$-0.548659\pi$
$577$	−7.31534	−0.304542	−0.152271	+	0.988339i	$0.548659\pi$
$578$	0	0
$579$	29.5616	1.22854
$580$	0	0
$581$	−19.6847	−0.816657
$582$	0	0
$583$	−17.1231	−0.709167
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−10.3002	−0.425134	−0.212567	−	0.977146i	$-0.568182\pi$
$587$	−10.3002	−0.425134	−0.212567	+	0.977146i	$0.568182\pi$
$588$	0	0
$589$	−0.246211	−0.0101450
$590$	0	0
$591$	27.8078	1.14386
$592$	0	0
$593$	−42.3542	−1.73928	−0.869638	−	0.493689i	$-0.835648\pi$
$593$	−42.3542	−1.73928	−0.869638	+	0.493689i	$0.835648\pi$
$594$	0	0
$595$	3.68466	0.151056
$596$	0	0
$597$	28.1080	1.15038
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	16.8617	0.687805	0.343902	−	0.939005i	$-0.388251\pi$
$601$	16.8617	0.687805	0.343902	+	0.939005i	$0.388251\pi$
$602$	0	0
$603$	4.17708	0.170104
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	36.0000	1.46119	0.730597	−	0.682808i	$-0.239242\pi$
$607$	36.0000	1.46119	0.730597	+	0.682808i	$0.239242\pi$
$608$	0	0
$609$	32.4924	1.31666
$610$	0	0
$611$	30.9309	1.25133
$612$	0	0
$613$	−9.61553	−0.388368	−0.194184	−	0.980965i	$-0.562206\pi$
$613$	−9.61553	−0.388368	−0.194184	+	0.980965i	$0.562206\pi$
$614$	0	0
$615$	−6.43845	−0.259623
$616$	0	0
$617$	−7.30019	−0.293894	−0.146947	−	0.989144i	$-0.546945\pi$
$617$	−7.30019	−0.293894	−0.146947	+	0.989144i	$0.546945\pi$
$618$	0	0
$619$	44.4924	1.78830	0.894151	−	0.447766i	$-0.147780\pi$
$619$	44.4924	1.78830	0.894151	+	0.447766i	$0.147780\pi$
$620$	0	0
$621$	5.56155	0.223177
$622$	0	0
$623$	−20.4924	−0.821012
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.24621	0.249450
$628$	0	0
$629$	0.807764	0.0322077
$630$	0	0
$631$	14.7386	0.586736	0.293368	−	0.956000i	$-0.405224\pi$
$631$	14.7386	0.586736	0.293368	+	0.956000i	$0.405224\pi$
$632$	0	0
$633$	−15.5076	−0.616371
$634$	0	0
$635$	19.8078	0.786047
$636$	0	0
$637$	1.56155	0.0618710
$638$	0	0
$639$	2.80776	0.111073
$640$	0	0
$641$	2.87689	0.113630	0.0568152	−	0.998385i	$-0.481905\pi$
$641$	2.87689	0.113630	0.0568152	+	0.998385i	$0.481905\pi$
$642$	0	0
$643$	−22.1771	−0.874579	−0.437289	−	0.899321i	$-0.644061\pi$
$643$	−22.1771	−0.874579	−0.437289	+	0.899321i	$0.644061\pi$
$644$	0	0
$645$	−9.75379	−0.384055
$646$	0	0
$647$	11.1771	0.439416	0.219708	−	0.975566i	$-0.429489\pi$
$647$	11.1771	0.439416	0.219708	+	0.975566i	$0.429489\pi$
$648$	0	0
$649$	2.87689	0.112928
$650$	0	0
$651$	−0.492423	−0.0192996
$652$	0	0
$653$	29.4233	1.15142	0.575711	−	0.817653i	$-0.304725\pi$
$653$	29.4233	1.15142	0.575711	+	0.817653i	$0.304725\pi$
$654$	0	0
$655$	18.9309	0.739690
$656$	0	0
$657$	−4.24621	−0.165660
$658$	0	0
$659$	−29.6155	−1.15366	−0.576829	−	0.816865i	$-0.695710\pi$
$659$	−29.6155	−1.15366	−0.576829	+	0.816865i	$0.695710\pi$
$660$	0	0
$661$	27.7538	1.07950	0.539749	−	0.841826i	$-0.318519\pi$
$661$	27.7538	1.07950	0.539749	+	0.841826i	$0.318519\pi$
$662$	0	0
$663$	−8.00000	−0.310694
$664$	0	0
$665$	−5.12311	−0.198666
$666$	0	0
$667$	8.12311	0.314528
$668$	0	0
$669$	29.4773	1.13966
$670$	0	0
$671$	1.75379	0.0677043
$672$	0	0
$673$	−16.1922	−0.624165	−0.312082	−	0.950055i	$-0.601027\pi$
$673$	−16.1922	−0.624165	−0.312082	+	0.950055i	$0.601027\pi$
$674$	0	0
$675$	−5.56155	−0.214064
$676$	0	0
$677$	5.68466	0.218479	0.109240	−	0.994015i	$-0.465158\pi$
$677$	5.68466	0.218479	0.109240	+	0.994015i	$0.465158\pi$
$678$	0	0
$679$	−13.1231	−0.503619
$680$	0	0
$681$	28.4924	1.09183
$682$	0	0
$683$	6.93087	0.265202	0.132601	−	0.991169i	$-0.457667\pi$
$683$	6.93087	0.265202	0.132601	+	0.991169i	$0.457667\pi$
$684$	0	0
$685$	−21.6155	−0.825887
$686$	0	0
$687$	−8.98485	−0.342793
$688$	0	0
$689$	−30.4924	−1.16167
$690$	0	0
$691$	−0.492423	−0.0187326	−0.00936632	−	0.999956i	$-0.502981\pi$
$691$	−0.492423	−0.0187326	−0.00936632	+	0.999956i	$0.502981\pi$
$692$	0	0
$693$	−2.87689	−0.109284
$694$	0	0
$695$	10.3693	0.393331
$696$	0	0
$697$	5.93087	0.224648
$698$	0	0
$699$	44.1922	1.67150
$700$	0	0
$701$	9.75379	0.368396	0.184198	−	0.982889i	$-0.441031\pi$
$701$	9.75379	0.368396	0.184198	+	0.982889i	$0.441031\pi$
$702$	0	0
$703$	−1.12311	−0.0423587
$704$	0	0
$705$	13.5616	0.510758
$706$	0	0
$707$	−27.6847	−1.04119
$708$	0	0
$709$	−16.0000	−0.600893	−0.300446	−	0.953799i	$-0.597136\pi$
$709$	−16.0000	−0.600893	−0.300446	+	0.953799i	$0.597136\pi$
$710$	0	0
$711$	0.492423	0.0184673
$712$	0	0
$713$	−0.123106	−0.00461034
$714$	0	0
$715$	−7.12311	−0.266389
$716$	0	0
$717$	−12.3002	−0.459359
$718$	0	0
$719$	−3.68466	−0.137415	−0.0687073	−	0.997637i	$-0.521887\pi$
$719$	−3.68466	−0.137415	−0.0687073	+	0.997637i	$0.521887\pi$
$720$	0	0
$721$	36.4924	1.35905
$722$	0	0
$723$	−28.8769	−1.07394
$724$	0	0
$725$	−8.12311	−0.301685
$726$	0	0
$727$	11.4384	0.424229	0.212114	−	0.977245i	$-0.431965\pi$
$727$	11.4384	0.424229	0.212114	+	0.977245i	$0.431965\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	8.98485	0.332316
$732$	0	0
$733$	40.1771	1.48397	0.741987	−	0.670414i	$-0.233884\pi$
$733$	40.1771	1.48397	0.741987	+	0.670414i	$0.233884\pi$
$734$	0	0
$735$	0.684658	0.0252540
$736$	0	0
$737$	14.8769	0.547997
$738$	0	0
$739$	1.49242	0.0548996	0.0274498	−	0.999623i	$-0.491261\pi$
$739$	1.49242	0.0548996	0.0274498	+	0.999623i	$0.491261\pi$
$740$	0	0
$741$	11.1231	0.408617
$742$	0	0
$743$	−46.2462	−1.69661	−0.848304	−	0.529509i	$-0.822376\pi$
$743$	−46.2462	−1.69661	−0.848304	+	0.529509i	$0.822376\pi$
$744$	0	0
$745$	−0.876894	−0.0321269
$746$	0	0
$747$	−4.31534	−0.157890
$748$	0	0
$749$	4.31534	0.157679
$750$	0	0
$751$	44.3542	1.61851	0.809253	−	0.587460i	$-0.199872\pi$
$751$	44.3542	1.61851	0.809253	+	0.587460i	$0.199872\pi$
$752$	0	0
$753$	14.2462	0.519161
$754$	0	0
$755$	−1.56155	−0.0568307
$756$	0	0
$757$	−25.5464	−0.928500	−0.464250	−	0.885704i	$-0.653676\pi$
$757$	−25.5464	−0.928500	−0.464250	+	0.885704i	$0.653676\pi$
$758$	0	0
$759$	3.12311	0.113362
$760$	0	0
$761$	21.9848	0.796950	0.398475	−	0.917179i	$-0.369540\pi$
$761$	21.9848	0.796950	0.398475	+	0.917179i	$0.369540\pi$
$762$	0	0
$763$	2.87689	0.104151
$764$	0	0
$765$	0.807764	0.0292048
$766$	0	0
$767$	5.12311	0.184985
$768$	0	0
$769$	−7.61553	−0.274623	−0.137311	−	0.990528i	$-0.543846\pi$
$769$	−7.61553	−0.274623	−0.137311	+	0.990528i	$0.543846\pi$
$770$	0	0
$771$	−1.06913	−0.0385038
$772$	0	0
$773$	−29.1231	−1.04749	−0.523743	−	0.851877i	$-0.675465\pi$
$773$	−29.1231	−1.04749	−0.523743	+	0.851877i	$0.675465\pi$
$774$	0	0
$775$	0.123106	0.00442208
$776$	0	0
$777$	−2.24621	−0.0805824
$778$	0	0
$779$	−8.24621	−0.295451
$780$	0	0
$781$	10.0000	0.357828
$782$	0	0
$783$	45.1771	1.61450
$784$	0	0
$785$	15.6847	0.559809
$786$	0	0
$787$	−33.3002	−1.18702	−0.593512	−	0.804825i	$-0.702259\pi$
$787$	−33.3002	−1.18702	−0.593512	+	0.804825i	$0.702259\pi$
$788$	0	0
$789$	−22.3542	−0.795829
$790$	0	0
$791$	−13.9309	−0.495325
$792$	0	0
$793$	3.12311	0.110905
$794$	0	0
$795$	−13.3693	−0.474161
$796$	0	0
$797$	25.1922	0.892355	0.446177	−	0.894945i	$-0.352785\pi$
$797$	25.1922	0.892355	0.446177	+	0.894945i	$0.352785\pi$
$798$	0	0
$799$	−12.4924	−0.441950
$800$	0	0
$801$	−4.49242	−0.158732
$802$	0	0
$803$	−15.1231	−0.533683
$804$	0	0
$805$	−2.56155	−0.0902829
$806$	0	0
$807$	−42.9309	−1.51124
$808$	0	0
$809$	50.8078	1.78631	0.893153	−	0.449753i	$-0.148488\pi$
$809$	50.8078	1.78631	0.893153	+	0.449753i	$0.148488\pi$
$810$	0	0
$811$	14.3693	0.504575	0.252287	−	0.967652i	$-0.418817\pi$
$811$	14.3693	0.504575	0.252287	+	0.967652i	$0.418817\pi$
$812$	0	0
$813$	37.7538	1.32408
$814$	0	0
$815$	6.68466	0.234153
$816$	0	0
$817$	−12.4924	−0.437055
$818$	0	0
$819$	−5.12311	−0.179016
$820$	0	0
$821$	38.4924	1.34339	0.671697	−	0.740826i	$-0.265566\pi$
$821$	38.4924	1.34339	0.671697	+	0.740826i	$0.265566\pi$
$822$	0	0
$823$	−33.1771	−1.15648	−0.578240	−	0.815867i	$-0.696260\pi$
$823$	−33.1771	−1.15648	−0.578240	+	0.815867i	$0.696260\pi$
$824$	0	0
$825$	−3.12311	−0.108733
$826$	0	0
$827$	28.3153	0.984621	0.492310	−	0.870420i	$-0.336152\pi$
$827$	28.3153	0.984621	0.492310	+	0.870420i	$0.336152\pi$
$828$	0	0
$829$	−20.5616	−0.714132	−0.357066	−	0.934079i	$-0.616223\pi$
$829$	−20.5616	−0.714132	−0.357066	+	0.934079i	$0.616223\pi$
$830$	0	0
$831$	−31.9157	−1.10714
$832$	0	0
$833$	−0.630683	−0.0218519
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	−0.684658	−0.0236653
$838$	0	0
$839$	−33.6155	−1.16054	−0.580268	−	0.814425i	$-0.697052\pi$
$839$	−33.6155	−1.16054	−0.580268	+	0.814425i	$0.697052\pi$
$840$	0	0
$841$	36.9848	1.27534
$842$	0	0
$843$	−14.2462	−0.490666
$844$	0	0
$845$	0.315342	0.0108481
$846$	0	0
$847$	17.9309	0.616112
$848$	0	0
$849$	6.13826	0.210665
$850$	0	0
$851$	−0.561553	−0.0192498
$852$	0	0
$853$	−14.4924	−0.496211	−0.248106	−	0.968733i	$-0.579808\pi$
$853$	−14.4924	−0.496211	−0.248106	+	0.968733i	$0.579808\pi$
$854$	0	0
$855$	−1.12311	−0.0384094
$856$	0	0
$857$	−40.5464	−1.38504	−0.692519	−	0.721399i	$-0.743499\pi$
$857$	−40.5464	−1.38504	−0.692519	+	0.721399i	$0.743499\pi$
$858$	0	0
$859$	−0.369317	−0.0126009	−0.00630046	−	0.999980i	$-0.502006\pi$
$859$	−0.369317	−0.0126009	−0.00630046	+	0.999980i	$0.502006\pi$
$860$	0	0
$861$	−16.4924	−0.562060
$862$	0	0
$863$	9.31534	0.317098	0.158549	−	0.987351i	$-0.449318\pi$
$863$	9.31534	0.317098	0.158549	+	0.987351i	$0.449318\pi$
$864$	0	0
$865$	7.12311	0.242193
$866$	0	0
$867$	−23.3153	−0.791831
$868$	0	0
$869$	1.75379	0.0594932
$870$	0	0
$871$	26.4924	0.897661
$872$	0	0
$873$	−2.87689	−0.0973681
$874$	0	0
$875$	2.56155	0.0865963
$876$	0	0
$877$	−47.4773	−1.60319	−0.801597	−	0.597865i	$-0.796016\pi$
$877$	−47.4773	−1.60319	−0.801597	+	0.597865i	$0.796016\pi$
$878$	0	0
$879$	22.7386	0.766955
$880$	0	0
$881$	22.7386	0.766084	0.383042	−	0.923731i	$-0.374876\pi$
$881$	22.7386	0.766084	0.383042	+	0.923731i	$0.374876\pi$
$882$	0	0
$883$	44.9848	1.51386	0.756930	−	0.653496i	$-0.226698\pi$
$883$	44.9848	1.51386	0.756930	+	0.653496i	$0.226698\pi$
$884$	0	0
$885$	2.24621	0.0755056
$886$	0	0
$887$	43.1771	1.44974	0.724872	−	0.688883i	$-0.241899\pi$
$887$	43.1771	1.44974	0.724872	+	0.688883i	$0.241899\pi$
$888$	0	0
$889$	50.7386	1.70172
$890$	0	0
$891$	14.0000	0.469018
$892$	0	0
$893$	17.3693	0.581242
$894$	0	0
$895$	7.31534	0.244525
$896$	0	0
$897$	5.56155	0.185695
$898$	0	0
$899$	−1.00000	−0.0333519
$900$	0	0
$901$	12.3153	0.410284
$902$	0	0
$903$	−24.9848	−0.831444
$904$	0	0
$905$	8.87689	0.295078
$906$	0	0
$907$	−24.4233	−0.810962	−0.405481	−	0.914103i	$-0.632896\pi$
$907$	−24.4233	−0.810962	−0.405481	+	0.914103i	$0.632896\pi$
$908$	0	0
$909$	−6.06913	−0.201300
$910$	0	0
$911$	23.1231	0.766103	0.383051	−	0.923727i	$-0.374873\pi$
$911$	23.1231	0.766103	0.383051	+	0.923727i	$0.374873\pi$
$912$	0	0
$913$	−15.3693	−0.508650
$914$	0	0
$915$	1.36932	0.0452682
$916$	0	0
$917$	48.4924	1.60136
$918$	0	0
$919$	−20.4924	−0.675983	−0.337991	−	0.941149i	$-0.609747\pi$
$919$	−20.4924	−0.675983	−0.337991	+	0.941149i	$0.609747\pi$
$920$	0	0
$921$	−24.0000	−0.790827
$922$	0	0
$923$	17.8078	0.586150
$924$	0	0
$925$	0.561553	0.0184637
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	0.261366	0.00857515	0.00428757	−	0.999991i	$-0.498635\pi$
$929$	0.261366	0.00857515	0.00428757	+	0.999991i	$0.498635\pi$
$930$	0	0
$931$	0.876894	0.0287391
$932$	0	0
$933$	−6.54640	−0.214319
$934$	0	0
$935$	2.87689	0.0940845
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	−21.7538	−0.709908
$940$	0	0
$941$	32.4924	1.05922	0.529611	−	0.848240i	$-0.322338\pi$
$941$	32.4924	1.05922	0.529611	+	0.848240i	$0.322338\pi$
$942$	0	0
$943$	−4.12311	−0.134267
$944$	0	0
$945$	−14.2462	−0.463429
$946$	0	0
$947$	17.5616	0.570674	0.285337	−	0.958427i	$-0.407895\pi$
$947$	17.5616	0.570674	0.285337	+	0.958427i	$0.407895\pi$
$948$	0	0
$949$	−26.9309	−0.874213
$950$	0	0
$951$	39.6155	1.28462
$952$	0	0
$953$	−44.2462	−1.43328	−0.716638	−	0.697446i	$-0.754320\pi$
$953$	−44.2462	−1.43328	−0.716638	+	0.697446i	$0.754320\pi$
$954$	0	0
$955$	3.12311	0.101061
$956$	0	0
$957$	25.3693	0.820074
$958$	0	0
$959$	−55.3693	−1.78797
$960$	0	0
$961$	−30.9848	−0.999511
$962$	0	0
$963$	0.946025	0.0304852
$964$	0	0
$965$	−18.9309	−0.609406
$966$	0	0
$967$	9.17708	0.295115	0.147558	−	0.989053i	$-0.452859\pi$
$967$	9.17708	0.295115	0.147558	+	0.989053i	$0.452859\pi$
$968$	0	0
$969$	−4.49242	−0.144317
$970$	0	0
$971$	−40.1080	−1.28713	−0.643563	−	0.765393i	$-0.722544\pi$
$971$	−40.1080	−1.28713	−0.643563	+	0.765393i	$0.722544\pi$
$972$	0	0
$973$	26.5616	0.851524
$974$	0	0
$975$	−5.56155	−0.178112
$976$	0	0
$977$	53.6847	1.71752	0.858762	−	0.512374i	$-0.171234\pi$
$977$	53.6847	1.71752	0.858762	+	0.512374i	$0.171234\pi$
$978$	0	0
$979$	−16.0000	−0.511362
$980$	0	0
$981$	0.630683	0.0201362
$982$	0	0
$983$	62.0388	1.97873	0.989366	−	0.145450i	$-0.0464631\pi$
$983$	62.0388	1.97873	0.989366	+	0.145450i	$0.0464631\pi$
$984$	0	0
$985$	−17.8078	−0.567403
$986$	0	0
$987$	34.7386	1.10574
$988$	0	0
$989$	−6.24621	−0.198618
$990$	0	0
$991$	22.5616	0.716691	0.358346	−	0.933589i	$-0.383341\pi$
$991$	22.5616	0.716691	0.358346	+	0.933589i	$0.383341\pi$
$992$	0	0
$993$	−5.45360	−0.173065
$994$	0	0
$995$	−18.0000	−0.570638
$996$	0	0
$997$	28.1080	0.890188	0.445094	−	0.895484i	$-0.353170\pi$
$997$	28.1080	0.890188	0.445094	+	0.895484i	$0.353170\pi$
$998$	0	0
$999$	−3.12311	−0.0988107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.2.a.e.1.2	✓	2
3.2	odd	2		8280.2.a.bf.1.1		2
4.3	odd	2		1840.2.a.o.1.1		2
5.2	odd	4		4600.2.e.n.4049.2		4
5.3	odd	4		4600.2.e.n.4049.3		4
5.4	even	2		4600.2.a.t.1.1		2
8.3	odd	2		7360.2.a.bl.1.2		2
8.5	even	2		7360.2.a.bp.1.1		2
20.19	odd	2		9200.2.a.bq.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.e.1.2	✓	2	1.1	even	1	trivial
1840.2.a.o.1.1		2	4.3	odd	2
4600.2.a.t.1.1		2	5.4	even	2
4600.2.e.n.4049.2		4	5.2	odd	4
4600.2.e.n.4049.3		4	5.3	odd	4
7360.2.a.bl.1.2		2	8.3	odd	2
7360.2.a.bp.1.1		2	8.5	even	2
8280.2.a.bf.1.1		2	3.2	odd	2
9200.2.a.bq.1.2		2	20.19	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:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} -1.00000 q^{5} -2.56155 q^{7} -0.561553 q^{9} -2.00000 q^{11} -3.56155 q^{13} -1.56155 q^{15} +1.43845 q^{17} -2.00000 q^{19} -4.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} -5.56155 q^{27} -8.12311 q^{29} +0.123106 q^{31} -3.12311 q^{33} +2.56155 q^{35} +0.561553 q^{37} -5.56155 q^{39} +4.12311 q^{41} +6.24621 q^{43} +0.561553 q^{45} -8.68466 q^{47} -0.438447 q^{49} +2.24621 q^{51} +8.56155 q^{53} +2.00000 q^{55} -3.12311 q^{57} -1.43845 q^{59} -0.876894 q^{61} +1.43845 q^{63} +3.56155 q^{65} -7.43845 q^{67} -1.56155 q^{69} -5.00000 q^{71} +7.56155 q^{73} +1.56155 q^{75} +5.12311 q^{77} -0.876894 q^{79} -7.00000 q^{81} +7.68466 q^{83} -1.43845 q^{85} -12.6847 q^{87} +8.00000 q^{89} +9.12311 q^{91} +0.192236 q^{93} +2.00000 q^{95} +5.12311 q^{97} +1.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} - q^{7} + 3 q^{9} - 4 q^{11} - 3 q^{13} + q^{15} + 7 q^{17} - 4 q^{19} - 8 q^{21} - 2 q^{23} + 2 q^{25} - 7 q^{27} - 8 q^{29} - 8 q^{31} + 2 q^{33} + q^{35} - 3 q^{37} - 7 q^{39}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 - q^7 + 3 * q^9 - 4 * q^11 - 3 * q^13 + q^15 + 7 * q^17 - 4 * q^19 - 8 * q^21 - 2 * q^23 + 2 * q^25 - 7 * q^27 - 8 * q^29 - 8 * q^31 + 2 * q^33 + q^35 - 3 * q^37 - 7 * q^39 - 4 * q^43 - 3 * q^45 - 5 * q^47 - 5 * q^49 - 12 * q^51 + 13 * q^53 + 4 * q^55 + 2 * q^57 - 7 * q^59 - 10 * q^61 + 7 * q^63 + 3 * q^65 - 19 * q^67 + q^69 - 10 * q^71 + 11 * q^73 - q^75 + 2 * q^77 - 10 * q^79 - 14 * q^81 + 3 * q^83 - 7 * q^85 - 13 * q^87 + 16 * q^89 + 10 * q^91 + 21 * q^93 + 4 * q^95 + 2 * q^97 - 6 * q^99

Introduction

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