LMFDB - Embedded newform 440.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [440,2,Mod(1,440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(440, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 440 = 2^{3} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	440.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$3.51341768894$
Analytic rank:	$0$
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			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{3} -1.00000 q^{5} +3.56155 q^{7} -0.561553 q^{9} +O(q^{10})$	$q-1.56155 q^{3} -1.00000 q^{5} +3.56155 q^{7} -0.561553 q^{9} +1.00000 q^{11} -3.12311 q^{13} +1.56155 q^{15} +5.56155 q^{17} +2.43845 q^{19} -5.56155 q^{21} +7.12311 q^{23} +1.00000 q^{25} +5.56155 q^{27} -0.438447 q^{29} +8.68466 q^{31} -1.56155 q^{33} -3.56155 q^{35} +9.80776 q^{37} +4.87689 q^{39} -10.0000 q^{41} +5.12311 q^{43} +0.561553 q^{45} -7.12311 q^{47} +5.68466 q^{49} -8.68466 q^{51} -4.43845 q^{53} -1.00000 q^{55} -3.80776 q^{57} -13.3693 q^{59} -3.56155 q^{61} -2.00000 q^{63} +3.12311 q^{65} -11.1231 q^{69} -2.43845 q^{71} +4.87689 q^{73} -1.56155 q^{75} +3.56155 q^{77} +0.876894 q^{79} -7.00000 q^{81} +10.0000 q^{83} -5.56155 q^{85} +0.684658 q^{87} +9.80776 q^{89} -11.1231 q^{91} -13.5616 q^{93} -2.43845 q^{95} +17.1231 q^{97} -0.561553 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q + q^3 - 2 * q^5 + 3 * q^7 + 3 * q^9	$ 2 q + q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - q^{15} + 7 q^{17} + 9 q^{19} - 7 q^{21} + 6 q^{23} + 2 q^{25} + 7 q^{27} - 5 q^{29} + 5 q^{31} + q^{33} - 3 q^{35} - q^{37} + 18 q^{39} - 20 q^{41} + 2 q^{43} - 3 q^{45} - 6 q^{47} - q^{49} - 5 q^{51} - 13 q^{53} - 2 q^{55} + 13 q^{57} - 2 q^{59} - 3 q^{61} - 4 q^{63} - 2 q^{65} - 14 q^{69} - 9 q^{71} + 18 q^{73} + q^{75} + 3 q^{77} + 10 q^{79} - 14 q^{81} + 20 q^{83} - 7 q^{85} - 11 q^{87} - q^{89} - 14 q^{91} - 23 q^{93} - 9 q^{95} + 26 q^{97} + 3 q^{99}+O(q^{100}) $ 2 * q + q^3 - 2 * q^5 + 3 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - q^15 + 7 * q^17 + 9 * q^19 - 7 * q^21 + 6 * q^23 + 2 * q^25 + 7 * q^27 - 5 * q^29 + 5 * q^31 + q^33 - 3 * q^35 - q^37 + 18 * q^39 - 20 * q^41 + 2 * q^43 - 3 * q^45 - 6 * q^47 - q^49 - 5 * q^51 - 13 * q^53 - 2 * q^55 + 13 * q^57 - 2 * q^59 - 3 * q^61 - 4 * q^63 - 2 * q^65 - 14 * q^69 - 9 * q^71 + 18 * q^73 + q^75 + 3 * q^77 + 10 * q^79 - 14 * q^81 + 20 * q^83 - 7 * q^85 - 11 * q^87 - q^89 - 14 * q^91 - 23 * q^93 - 9 * q^95 + 26 * q^97 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.56155	−0.901563	−0.450781	−	0.892634i	$-0.648855\pi$
$3$	−1.56155	−0.901563	−0.450781	+	0.892634i	$0.648855\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.56155	1.34614	0.673070	−	0.739579i	$-0.264975\pi$
$7$	3.56155	1.34614	0.673070	+	0.739579i	$0.264975\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.12311	−0.866194	−0.433097	−	0.901347i	$-0.642579\pi$
$13$	−3.12311	−0.866194	−0.433097	+	0.901347i	$0.642579\pi$
$14$	0	0
$15$	1.56155	0.403191
$16$	0	0
$17$	5.56155	1.34887	0.674437	−	0.738332i	$-0.264386\pi$
$17$	5.56155	1.34887	0.674437	+	0.738332i	$0.264386\pi$
$18$	0	0
$19$	2.43845	0.559418	0.279709	−	0.960085i	$-0.409762\pi$
$19$	2.43845	0.559418	0.279709	+	0.960085i	$0.409762\pi$
$20$	0	0
$21$	−5.56155	−1.21363
$22$	0	0
$23$	7.12311	1.48527	0.742635	−	0.669696i	$-0.233576\pi$
$23$	7.12311	1.48527	0.742635	+	0.669696i	$0.233576\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.56155	1.07032
$28$	0	0
$29$	−0.438447	−0.0814176	−0.0407088	−	0.999171i	$-0.512962\pi$
$29$	−0.438447	−0.0814176	−0.0407088	+	0.999171i	$0.512962\pi$
$30$	0	0
$31$	8.68466	1.55981	0.779905	−	0.625897i	$-0.215267\pi$
$31$	8.68466	1.55981	0.779905	+	0.625897i	$0.215267\pi$
$32$	0	0
$33$	−1.56155	−0.271831
$34$	0	0
$35$	−3.56155	−0.602012
$36$	0	0
$37$	9.80776	1.61239	0.806193	−	0.591652i	$-0.201524\pi$
$37$	9.80776	1.61239	0.806193	+	0.591652i	$0.201524\pi$
$38$	0	0
$39$	4.87689	0.780928
$40$	0	0
$41$	−10.0000	−1.56174	−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000	−1.56174	−0.780869	+	0.624695i	$0.785223\pi$
$42$	0	0
$43$	5.12311	0.781266	0.390633	−	0.920546i	$-0.372256\pi$
$43$	5.12311	0.781266	0.390633	+	0.920546i	$0.372256\pi$
$44$	0	0
$45$	0.561553	0.0837114
$46$	0	0
$47$	−7.12311	−1.03901	−0.519506	−	0.854467i	$-0.673884\pi$
$47$	−7.12311	−1.03901	−0.519506	+	0.854467i	$0.673884\pi$
$48$	0	0
$49$	5.68466	0.812094
$50$	0	0
$51$	−8.68466	−1.21610
$52$	0	0
$53$	−4.43845	−0.609668	−0.304834	−	0.952406i	$-0.598601\pi$
$53$	−4.43845	−0.609668	−0.304834	+	0.952406i	$0.598601\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	−3.80776	−0.504351
$58$	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
$61$	−3.56155	−0.456010	−0.228005	−	0.973660i	$-0.573220\pi$
$61$	−3.56155	−0.456010	−0.228005	+	0.973660i	$0.573220\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	3.12311	0.387374
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−11.1231	−1.33906
$70$	0	0
$71$	−2.43845	−0.289390	−0.144695	−	0.989476i	$-0.546220\pi$
$71$	−2.43845	−0.289390	−0.144695	+	0.989476i	$0.546220\pi$
$72$	0	0
$73$	4.87689	0.570797	0.285399	−	0.958409i	$-0.407874\pi$
$73$	4.87689	0.570797	0.285399	+	0.958409i	$0.407874\pi$
$74$	0	0
$75$	−1.56155	−0.180313
$76$	0	0
$77$	3.56155	0.405877
$78$	0	0
$79$	0.876894	0.0986583	0.0493292	−	0.998783i	$-0.484292\pi$
$79$	0.876894	0.0986583	0.0493292	+	0.998783i	$0.484292\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	0	0
$85$	−5.56155	−0.603235
$86$	0	0
$87$	0.684658	0.0734031
$88$	0	0
$89$	9.80776	1.03962	0.519810	−	0.854282i	$-0.326003\pi$
$89$	9.80776	1.03962	0.519810	+	0.854282i	$0.326003\pi$
$90$	0	0
$91$	−11.1231	−1.16602
$92$	0	0
$93$	−13.5616	−1.40627
$94$	0	0
$95$	−2.43845	−0.250179
$96$	0	0
$97$	17.1231	1.73859	0.869294	−	0.494295i	$-0.164574\pi$
$97$	17.1231	1.73859	0.869294	+	0.494295i	$0.164574\pi$
$98$	0	0
$99$	−0.561553	−0.0564382
$100$	0	0
$101$	−16.2462	−1.61656	−0.808279	−	0.588799i	$-0.799601\pi$
$101$	−16.2462	−1.61656	−0.808279	+	0.588799i	$0.799601\pi$
$102$	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
$105$	5.56155	0.542752
$106$	0	0
$107$	5.12311	0.495269	0.247635	−	0.968853i	$-0.420347\pi$
$107$	5.12311	0.495269	0.247635	+	0.968853i	$0.420347\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−15.3153	−1.45367
$112$	0	0
$113$	1.12311	0.105653	0.0528264	−	0.998604i	$-0.483177\pi$
$113$	1.12311	0.105653	0.0528264	+	0.998604i	$0.483177\pi$
$114$	0	0
$115$	−7.12311	−0.664233
$116$	0	0
$117$	1.75379	0.162138
$118$	0	0
$119$	19.8078	1.81577
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	15.6155	1.40800
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.36932	−0.653921	−0.326961	−	0.945038i	$-0.606024\pi$
$127$	−7.36932	−0.653921	−0.326961	+	0.945038i	$0.606024\pi$
$128$	0	0
$129$	−8.00000	−0.704361
$130$	0	0
$131$	0.684658	0.0598189	0.0299094	−	0.999553i	$-0.490478\pi$
$131$	0.684658	0.0598189	0.0299094	+	0.999553i	$0.490478\pi$
$132$	0	0
$133$	8.68466	0.753055
$134$	0	0
$135$	−5.56155	−0.478662
$136$	0	0
$137$	−19.3693	−1.65483	−0.827416	−	0.561589i	$-0.810190\pi$
$137$	−19.3693	−1.65483	−0.827416	+	0.561589i	$0.810190\pi$
$138$	0	0
$139$	16.4924	1.39887	0.699435	−	0.714697i	$-0.253435\pi$
$139$	16.4924	1.39887	0.699435	+	0.714697i	$0.253435\pi$
$140$	0	0
$141$	11.1231	0.936734
$142$	0	0
$143$	−3.12311	−0.261167
$144$	0	0
$145$	0.438447	0.0364111
$146$	0	0
$147$	−8.87689	−0.732154
$148$	0	0
$149$	−1.31534	−0.107757	−0.0538785	−	0.998547i	$-0.517158\pi$
$149$	−1.31534	−0.107757	−0.0538785	+	0.998547i	$0.517158\pi$
$150$	0	0
$151$	−0.876894	−0.0713607	−0.0356803	−	0.999363i	$-0.511360\pi$
$151$	−0.876894	−0.0713607	−0.0356803	+	0.999363i	$0.511360\pi$
$152$	0	0
$153$	−3.12311	−0.252488
$154$	0	0
$155$	−8.68466	−0.697569
$156$	0	0
$157$	−12.9309	−1.03200	−0.515998	−	0.856590i	$-0.672579\pi$
$157$	−12.9309	−1.03200	−0.515998	+	0.856590i	$0.672579\pi$
$158$	0	0
$159$	6.93087	0.549654
$160$	0	0
$161$	25.3693	1.99938
$162$	0	0
$163$	0.192236	0.0150571	0.00752854	−	0.999972i	$-0.497604\pi$
$163$	0.192236	0.0150571	0.00752854	+	0.999972i	$0.497604\pi$
$164$	0	0
$165$	1.56155	0.121567
$166$	0	0
$167$	1.31534	0.101784	0.0508921	−	0.998704i	$-0.483794\pi$
$167$	1.31534	0.101784	0.0508921	+	0.998704i	$0.483794\pi$
$168$	0	0
$169$	−3.24621	−0.249709
$170$	0	0
$171$	−1.36932	−0.104714
$172$	0	0
$173$	−21.3693	−1.62468	−0.812340	−	0.583185i	$-0.801806\pi$
$173$	−21.3693	−1.62468	−0.812340	+	0.583185i	$0.801806\pi$
$174$	0	0
$175$	3.56155	0.269228
$176$	0	0
$177$	20.8769	1.56920
$178$	0	0
$179$	8.49242	0.634753	0.317377	−	0.948300i	$-0.397198\pi$
$179$	8.49242	0.634753	0.317377	+	0.948300i	$0.397198\pi$
$180$	0	0
$181$	14.4924	1.07721	0.538607	−	0.842557i	$-0.318951\pi$
$181$	14.4924	1.07721	0.538607	+	0.842557i	$0.318951\pi$
$182$	0	0
$183$	5.56155	0.411122
$184$	0	0
$185$	−9.80776	−0.721081
$186$	0	0
$187$	5.56155	0.406701
$188$	0	0
$189$	19.8078	1.44080
$190$	0	0
$191$	1.75379	0.126900	0.0634499	−	0.997985i	$-0.479790\pi$
$191$	1.75379	0.126900	0.0634499	+	0.997985i	$0.479790\pi$
$192$	0	0
$193$	19.8078	1.42579	0.712897	−	0.701269i	$-0.247383\pi$
$193$	19.8078	1.42579	0.712897	+	0.701269i	$0.247383\pi$
$194$	0	0
$195$	−4.87689	−0.349242
$196$	0	0
$197$	18.2462	1.29999	0.649994	−	0.759939i	$-0.274771\pi$
$197$	18.2462	1.29999	0.649994	+	0.759939i	$0.274771\pi$
$198$	0	0
$199$	−6.93087	−0.491316	−0.245658	−	0.969357i	$-0.579004\pi$
$199$	−6.93087	−0.491316	−0.245658	+	0.969357i	$0.579004\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1.56155	−0.109600
$204$	0	0
$205$	10.0000	0.698430
$206$	0	0
$207$	−4.00000	−0.278019
$208$	0	0
$209$	2.43845	0.168671
$210$	0	0
$211$	0.684658	0.0471338	0.0235669	−	0.999722i	$-0.492498\pi$
$211$	0.684658	0.0471338	0.0235669	+	0.999722i	$0.492498\pi$
$212$	0	0
$213$	3.80776	0.260904
$214$	0	0
$215$	−5.12311	−0.349393
$216$	0	0
$217$	30.9309	2.09972
$218$	0	0
$219$	−7.61553	−0.514610
$220$	0	0
$221$	−17.3693	−1.16839
$222$	0	0
$223$	−23.1231	−1.54844	−0.774219	−	0.632918i	$-0.781857\pi$
$223$	−23.1231	−1.54844	−0.774219	+	0.632918i	$0.781857\pi$
$224$	0	0
$225$	−0.561553	−0.0374369
$226$	0	0
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	0	0
$229$	−6.49242	−0.429031	−0.214516	−	0.976721i	$-0.568817\pi$
$229$	−6.49242	−0.429031	−0.214516	+	0.976721i	$0.568817\pi$
$230$	0	0
$231$	−5.56155	−0.365923
$232$	0	0
$233$	11.3153	0.741293	0.370646	−	0.928774i	$-0.379136\pi$
$233$	11.3153	0.741293	0.370646	+	0.928774i	$0.379136\pi$
$234$	0	0
$235$	7.12311	0.464660
$236$	0	0
$237$	−1.36932	−0.0889467
$238$	0	0
$239$	0.876894	0.0567216	0.0283608	−	0.999598i	$-0.490971\pi$
$239$	0.876894	0.0567216	0.0283608	+	0.999598i	$0.490971\pi$
$240$	0	0
$241$	−28.7386	−1.85122	−0.925609	−	0.378481i	$-0.876447\pi$
$241$	−28.7386	−1.85122	−0.925609	+	0.378481i	$0.876447\pi$
$242$	0	0
$243$	−5.75379	−0.369106
$244$	0	0
$245$	−5.68466	−0.363180
$246$	0	0
$247$	−7.61553	−0.484564
$248$	0	0
$249$	−15.6155	−0.989594
$250$	0	0
$251$	23.1231	1.45952	0.729759	−	0.683705i	$-0.239632\pi$
$251$	23.1231	1.45952	0.729759	+	0.683705i	$0.239632\pi$
$252$	0	0
$253$	7.12311	0.447826
$254$	0	0
$255$	8.68466	0.543854
$256$	0	0
$257$	−7.75379	−0.483668	−0.241834	−	0.970318i	$-0.577749\pi$
$257$	−7.75379	−0.483668	−0.241834	+	0.970318i	$0.577749\pi$
$258$	0	0
$259$	34.9309	2.17050
$260$	0	0
$261$	0.246211	0.0152401
$262$	0	0
$263$	24.4384	1.50694	0.753470	−	0.657483i	$-0.228379\pi$
$263$	24.4384	1.50694	0.753470	+	0.657483i	$0.228379\pi$
$264$	0	0
$265$	4.43845	0.272652
$266$	0	0
$267$	−15.3153	−0.937284
$268$	0	0
$269$	−22.8769	−1.39483	−0.697414	−	0.716668i	$-0.745666\pi$
$269$	−22.8769	−1.39483	−0.697414	+	0.716668i	$0.745666\pi$
$270$	0	0
$271$	12.0000	0.728948	0.364474	−	0.931214i	$-0.381249\pi$
$271$	12.0000	0.728948	0.364474	+	0.931214i	$0.381249\pi$
$272$	0	0
$273$	17.3693	1.05124
$274$	0	0
$275$	1.00000	0.0603023
$276$	0	0
$277$	18.7386	1.12590	0.562948	−	0.826493i	$-0.309667\pi$
$277$	18.7386	1.12590	0.562948	+	0.826493i	$0.309667\pi$
$278$	0	0
$279$	−4.87689	−0.291972
$280$	0	0
$281$	18.4924	1.10317	0.551583	−	0.834120i	$-0.314024\pi$
$281$	18.4924	1.10317	0.551583	+	0.834120i	$0.314024\pi$
$282$	0	0
$283$	−18.4924	−1.09926	−0.549630	−	0.835408i	$-0.685231\pi$
$283$	−18.4924	−1.09926	−0.549630	+	0.835408i	$0.685231\pi$
$284$	0	0
$285$	3.80776	0.225552
$286$	0	0
$287$	−35.6155	−2.10232
$288$	0	0
$289$	13.9309	0.819463
$290$	0	0
$291$	−26.7386	−1.56745
$292$	0	0
$293$	−11.1231	−0.649819	−0.324909	−	0.945745i	$-0.605334\pi$
$293$	−11.1231	−0.649819	−0.324909	+	0.945745i	$0.605334\pi$
$294$	0	0
$295$	13.3693	0.778392
$296$	0	0
$297$	5.56155	0.322714
$298$	0	0
$299$	−22.2462	−1.28653
$300$	0	0
$301$	18.2462	1.05169
$302$	0	0
$303$	25.3693	1.45743
$304$	0	0
$305$	3.56155	0.203934
$306$	0	0
$307$	20.7386	1.18362	0.591808	−	0.806079i	$-0.298414\pi$
$307$	20.7386	1.18362	0.591808	+	0.806079i	$0.298414\pi$
$308$	0	0
$309$	22.2462	1.26554
$310$	0	0
$311$	21.1771	1.20084	0.600421	−	0.799684i	$-0.295000\pi$
$311$	21.1771	1.20084	0.600421	+	0.799684i	$0.295000\pi$
$312$	0	0
$313$	16.2462	0.918290	0.459145	−	0.888361i	$-0.348156\pi$
$313$	16.2462	0.918290	0.459145	+	0.888361i	$0.348156\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	−3.56155	−0.200037	−0.100018	−	0.994986i	$-0.531890\pi$
$317$	−3.56155	−0.200037	−0.100018	+	0.994986i	$0.531890\pi$
$318$	0	0
$319$	−0.438447	−0.0245483
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	13.5616	0.754585
$324$	0	0
$325$	−3.12311	−0.173239
$326$	0	0
$327$	3.12311	0.172708
$328$	0	0
$329$	−25.3693	−1.39866
$330$	0	0
$331$	−2.24621	−0.123463	−0.0617315	−	0.998093i	$-0.519662\pi$
$331$	−2.24621	−0.123463	−0.0617315	+	0.998093i	$0.519662\pi$
$332$	0	0
$333$	−5.50758	−0.301813
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.93087	0.377549	0.188774	−	0.982021i	$-0.439549\pi$
$337$	6.93087	0.377549	0.188774	+	0.982021i	$0.439549\pi$
$338$	0	0
$339$	−1.75379	−0.0952527
$340$	0	0
$341$	8.68466	0.470301
$342$	0	0
$343$	−4.68466	−0.252948
$344$	0	0
$345$	11.1231	0.598848
$346$	0	0
$347$	24.2462	1.30160	0.650802	−	0.759247i	$-0.274433\pi$
$347$	24.2462	1.30160	0.650802	+	0.759247i	$0.274433\pi$
$348$	0	0
$349$	−24.7386	−1.32423	−0.662114	−	0.749403i	$-0.730341\pi$
$349$	−24.7386	−1.32423	−0.662114	+	0.749403i	$0.730341\pi$
$350$	0	0
$351$	−17.3693	−0.927106
$352$	0	0
$353$	3.75379	0.199794	0.0998970	−	0.994998i	$-0.468149\pi$
$353$	3.75379	0.199794	0.0998970	+	0.994998i	$0.468149\pi$
$354$	0	0
$355$	2.43845	0.129419
$356$	0	0
$357$	−30.9309	−1.63704
$358$	0	0
$359$	−8.49242	−0.448213	−0.224106	−	0.974565i	$-0.571946\pi$
$359$	−8.49242	−0.448213	−0.224106	+	0.974565i	$0.571946\pi$
$360$	0	0
$361$	−13.0540	−0.687051
$362$	0	0
$363$	−1.56155	−0.0819603
$364$	0	0
$365$	−4.87689	−0.255268
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	0	0
$369$	5.61553	0.292333
$370$	0	0
$371$	−15.8078	−0.820698
$372$	0	0
$373$	−29.3693	−1.52069	−0.760343	−	0.649522i	$-0.774969\pi$
$373$	−29.3693	−1.52069	−0.760343	+	0.649522i	$0.774969\pi$
$374$	0	0
$375$	1.56155	0.0806382
$376$	0	0
$377$	1.36932	0.0705234
$378$	0	0
$379$	27.6155	1.41851	0.709257	−	0.704950i	$-0.249030\pi$
$379$	27.6155	1.41851	0.709257	+	0.704950i	$0.249030\pi$
$380$	0	0
$381$	11.5076	0.589551
$382$	0	0
$383$	−29.8617	−1.52586	−0.762932	−	0.646479i	$-0.776241\pi$
$383$	−29.8617	−1.52586	−0.762932	+	0.646479i	$0.776241\pi$
$384$	0	0
$385$	−3.56155	−0.181514
$386$	0	0
$387$	−2.87689	−0.146241
$388$	0	0
$389$	−32.2462	−1.63495	−0.817474	−	0.575966i	$-0.804626\pi$
$389$	−32.2462	−1.63495	−0.817474	+	0.575966i	$0.804626\pi$
$390$	0	0
$391$	39.6155	2.00344
$392$	0	0
$393$	−1.06913	−0.0539305
$394$	0	0
$395$	−0.876894	−0.0441213
$396$	0	0
$397$	5.50758	0.276417	0.138209	−	0.990403i	$-0.455866\pi$
$397$	5.50758	0.276417	0.138209	+	0.990403i	$0.455866\pi$
$398$	0	0
$399$	−13.5616	−0.678927
$400$	0	0
$401$	−21.8078	−1.08903	−0.544514	−	0.838752i	$-0.683286\pi$
$401$	−21.8078	−1.08903	−0.544514	+	0.838752i	$0.683286\pi$
$402$	0	0
$403$	−27.1231	−1.35110
$404$	0	0
$405$	7.00000	0.347833
$406$	0	0
$407$	9.80776	0.486153
$408$	0	0
$409$	−17.1231	−0.846683	−0.423342	−	0.905970i	$-0.639143\pi$
$409$	−17.1231	−0.846683	−0.423342	+	0.905970i	$0.639143\pi$
$410$	0	0
$411$	30.2462	1.49194
$412$	0	0
$413$	−47.6155	−2.34301
$414$	0	0
$415$	−10.0000	−0.490881
$416$	0	0
$417$	−25.7538	−1.26117
$418$	0	0
$419$	−5.75379	−0.281091	−0.140545	−	0.990074i	$-0.544886\pi$
$419$	−5.75379	−0.281091	−0.140545	+	0.990074i	$0.544886\pi$
$420$	0	0
$421$	21.1231	1.02948	0.514739	−	0.857347i	$-0.327889\pi$
$421$	21.1231	1.02948	0.514739	+	0.857347i	$0.327889\pi$
$422$	0	0
$423$	4.00000	0.194487
$424$	0	0
$425$	5.56155	0.269775
$426$	0	0
$427$	−12.6847	−0.613854
$428$	0	0
$429$	4.87689	0.235459
$430$	0	0
$431$	20.4924	0.987085	0.493543	−	0.869722i	$-0.335702\pi$
$431$	20.4924	0.987085	0.493543	+	0.869722i	$0.335702\pi$
$432$	0	0
$433$	−23.8617	−1.14672	−0.573361	−	0.819303i	$-0.694361\pi$
$433$	−23.8617	−1.14672	−0.573361	+	0.819303i	$0.694361\pi$
$434$	0	0
$435$	−0.684658	−0.0328269
$436$	0	0
$437$	17.3693	0.830887
$438$	0	0
$439$	−24.9848	−1.19246	−0.596231	−	0.802813i	$-0.703336\pi$
$439$	−24.9848	−1.19246	−0.596231	+	0.802813i	$0.703336\pi$
$440$	0	0
$441$	−3.19224	−0.152011
$442$	0	0
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	0	0
$445$	−9.80776	−0.464933
$446$	0	0
$447$	2.05398	0.0971497
$448$	0	0
$449$	−28.7386	−1.35626	−0.678130	−	0.734942i	$-0.737209\pi$
$449$	−28.7386	−1.35626	−0.678130	+	0.734942i	$0.737209\pi$
$450$	0	0
$451$	−10.0000	−0.470882
$452$	0	0
$453$	1.36932	0.0643361
$454$	0	0
$455$	11.1231	0.521459
$456$	0	0
$457$	−38.0540	−1.78009	−0.890045	−	0.455873i	$-0.849327\pi$
$457$	−38.0540	−1.78009	−0.890045	+	0.455873i	$0.849327\pi$
$458$	0	0
$459$	30.9309	1.44373
$460$	0	0
$461$	28.9309	1.34744	0.673722	−	0.738984i	$-0.264694\pi$
$461$	28.9309	1.34744	0.673722	+	0.738984i	$0.264694\pi$
$462$	0	0
$463$	−0.876894	−0.0407527	−0.0203764	−	0.999792i	$-0.506486\pi$
$463$	−0.876894	−0.0407527	−0.0203764	+	0.999792i	$0.506486\pi$
$464$	0	0
$465$	13.5616	0.628902
$466$	0	0
$467$	8.68466	0.401878	0.200939	−	0.979604i	$-0.435601\pi$
$467$	8.68466	0.401878	0.200939	+	0.979604i	$0.435601\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	20.1922	0.930409
$472$	0	0
$473$	5.12311	0.235561
$474$	0	0
$475$	2.43845	0.111884
$476$	0	0
$477$	2.49242	0.114120
$478$	0	0
$479$	−12.0000	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$479$	−12.0000	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$480$	0	0
$481$	−30.6307	−1.39664
$482$	0	0
$483$	−39.6155	−1.80257
$484$	0	0
$485$	−17.1231	−0.777520
$486$	0	0
$487$	−34.2462	−1.55184	−0.775922	−	0.630829i	$-0.782715\pi$
$487$	−34.2462	−1.55184	−0.775922	+	0.630829i	$0.782715\pi$
$488$	0	0
$489$	−0.300187	−0.0135749
$490$	0	0
$491$	−15.8078	−0.713394	−0.356697	−	0.934220i	$-0.616097\pi$
$491$	−15.8078	−0.713394	−0.356697	+	0.934220i	$0.616097\pi$
$492$	0	0
$493$	−2.43845	−0.109822
$494$	0	0
$495$	0.561553	0.0252399
$496$	0	0
$497$	−8.68466	−0.389560
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	−2.05398	−0.0917648
$502$	0	0
$503$	29.1231	1.29854	0.649268	−	0.760560i	$-0.275076\pi$
$503$	29.1231	1.29854	0.649268	+	0.760560i	$0.275076\pi$
$504$	0	0
$505$	16.2462	0.722947
$506$	0	0
$507$	5.06913	0.225128
$508$	0	0
$509$	−35.3693	−1.56772	−0.783859	−	0.620939i	$-0.786751\pi$
$509$	−35.3693	−1.56772	−0.783859	+	0.620939i	$0.786751\pi$
$510$	0	0
$511$	17.3693	0.768373
$512$	0	0
$513$	13.5616	0.598757
$514$	0	0
$515$	14.2462	0.627763
$516$	0	0
$517$	−7.12311	−0.313274
$518$	0	0
$519$	33.3693	1.46475
$520$	0	0
$521$	9.50758	0.416535	0.208267	−	0.978072i	$-0.433218\pi$
$521$	9.50758	0.416535	0.208267	+	0.978072i	$0.433218\pi$
$522$	0	0
$523$	23.3693	1.02187	0.510934	−	0.859620i	$-0.329299\pi$
$523$	23.3693	1.02187	0.510934	+	0.859620i	$0.329299\pi$
$524$	0	0
$525$	−5.56155	−0.242726
$526$	0	0
$527$	48.3002	2.10399
$528$	0	0
$529$	27.7386	1.20603
$530$	0	0
$531$	7.50758	0.325801
$532$	0	0
$533$	31.2311	1.35277
$534$	0	0
$535$	−5.12311	−0.221491
$536$	0	0
$537$	−13.2614	−0.572270
$538$	0	0
$539$	5.68466	0.244856
$540$	0	0
$541$	11.5616	0.497070	0.248535	−	0.968623i	$-0.420051\pi$
$541$	11.5616	0.497070	0.248535	+	0.968623i	$0.420051\pi$
$542$	0	0
$543$	−22.6307	−0.971176
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	−13.1231	−0.561103	−0.280552	−	0.959839i	$-0.590517\pi$
$547$	−13.1231	−0.561103	−0.280552	+	0.959839i	$0.590517\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	−1.06913	−0.0455465
$552$	0	0
$553$	3.12311	0.132808
$554$	0	0
$555$	15.3153	0.650100
$556$	0	0
$557$	7.12311	0.301816	0.150908	−	0.988548i	$-0.451780\pi$
$557$	7.12311	0.301816	0.150908	+	0.988548i	$0.451780\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	0	0
$561$	−8.68466	−0.366667
$562$	0	0
$563$	−8.24621	−0.347536	−0.173768	−	0.984787i	$-0.555594\pi$
$563$	−8.24621	−0.347536	−0.173768	+	0.984787i	$0.555594\pi$
$564$	0	0
$565$	−1.12311	−0.0472494
$566$	0	0
$567$	−24.9309	−1.04700
$568$	0	0
$569$	−2.87689	−0.120606	−0.0603028	−	0.998180i	$-0.519207\pi$
$569$	−2.87689	−0.120606	−0.0603028	+	0.998180i	$0.519207\pi$
$570$	0	0
$571$	31.8078	1.33111	0.665557	−	0.746347i	$-0.268194\pi$
$571$	31.8078	1.33111	0.665557	+	0.746347i	$0.268194\pi$
$572$	0	0
$573$	−2.73863	−0.114408
$574$	0	0
$575$	7.12311	0.297054
$576$	0	0
$577$	−30.4924	−1.26942	−0.634708	−	0.772752i	$-0.718880\pi$
$577$	−30.4924	−1.26942	−0.634708	+	0.772752i	$0.718880\pi$
$578$	0	0
$579$	−30.9309	−1.28544
$580$	0	0
$581$	35.6155	1.47758
$582$	0	0
$583$	−4.43845	−0.183822
$584$	0	0
$585$	−1.75379	−0.0725102
$586$	0	0
$587$	21.5616	0.889941	0.444970	−	0.895545i	$-0.353214\pi$
$587$	21.5616	0.889941	0.444970	+	0.895545i	$0.353214\pi$
$588$	0	0
$589$	21.1771	0.872586
$590$	0	0
$591$	−28.4924	−1.17202
$592$	0	0
$593$	−7.61553	−0.312732	−0.156366	−	0.987699i	$-0.549978\pi$
$593$	−7.61553	−0.312732	−0.156366	+	0.987699i	$0.549978\pi$
$594$	0	0
$595$	−19.8078	−0.812039
$596$	0	0
$597$	10.8229	0.442953
$598$	0	0
$599$	−15.3153	−0.625768	−0.312884	−	0.949791i	$-0.601295\pi$
$599$	−15.3153	−0.625768	−0.312884	+	0.949791i	$0.601295\pi$
$600$	0	0
$601$	37.2311	1.51869	0.759343	−	0.650690i	$-0.225520\pi$
$601$	37.2311	1.51869	0.759343	+	0.650690i	$0.225520\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	34.6847	1.40781	0.703903	−	0.710296i	$-0.251439\pi$
$607$	34.6847	1.40781	0.703903	+	0.710296i	$0.251439\pi$
$608$	0	0
$609$	2.43845	0.0988109
$610$	0	0
$611$	22.2462	0.899985
$612$	0	0
$613$	27.6155	1.11538	0.557690	−	0.830049i	$-0.311688\pi$
$613$	27.6155	1.11538	0.557690	+	0.830049i	$0.311688\pi$
$614$	0	0
$615$	−15.6155	−0.629679
$616$	0	0
$617$	2.87689	0.115819	0.0579097	−	0.998322i	$-0.481556\pi$
$617$	2.87689	0.115819	0.0579097	+	0.998322i	$0.481556\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	39.6155	1.58972
$622$	0	0
$623$	34.9309	1.39948
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.80776	−0.152067
$628$	0	0
$629$	54.5464	2.17491
$630$	0	0
$631$	−2.43845	−0.0970730	−0.0485365	−	0.998821i	$-0.515456\pi$
$631$	−2.43845	−0.0970730	−0.0485365	+	0.998821i	$0.515456\pi$
$632$	0	0
$633$	−1.06913	−0.0424941
$634$	0	0
$635$	7.36932	0.292442
$636$	0	0
$637$	−17.7538	−0.703431
$638$	0	0
$639$	1.36932	0.0541693
$640$	0	0
$641$	3.94602	0.155859	0.0779293	−	0.996959i	$-0.475169\pi$
$641$	3.94602	0.155859	0.0779293	+	0.996959i	$0.475169\pi$
$642$	0	0
$643$	−19.3153	−0.761723	−0.380861	−	0.924632i	$-0.624372\pi$
$643$	−19.3153	−0.761723	−0.380861	+	0.924632i	$0.624372\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	−9.36932	−0.368346	−0.184173	−	0.982894i	$-0.558961\pi$
$647$	−9.36932	−0.368346	−0.184173	+	0.982894i	$0.558961\pi$
$648$	0	0
$649$	−13.3693	−0.524792
$650$	0	0
$651$	−48.3002	−1.89303
$652$	0	0
$653$	10.1922	0.398853	0.199427	−	0.979913i	$-0.436092\pi$
$653$	10.1922	0.398853	0.199427	+	0.979913i	$0.436092\pi$
$654$	0	0
$655$	−0.684658	−0.0267518
$656$	0	0
$657$	−2.73863	−0.106844
$658$	0	0
$659$	5.06913	0.197465	0.0987326	−	0.995114i	$-0.468521\pi$
$659$	5.06913	0.197465	0.0987326	+	0.995114i	$0.468521\pi$
$660$	0	0
$661$	−14.4924	−0.563690	−0.281845	−	0.959460i	$-0.590946\pi$
$661$	−14.4924	−0.563690	−0.281845	+	0.959460i	$0.590946\pi$
$662$	0	0
$663$	27.1231	1.05337
$664$	0	0
$665$	−8.68466	−0.336777
$666$	0	0
$667$	−3.12311	−0.120927
$668$	0	0
$669$	36.1080	1.39601
$670$	0	0
$671$	−3.56155	−0.137492
$672$	0	0
$673$	−21.1771	−0.816316	−0.408158	−	0.912911i	$-0.633829\pi$
$673$	−21.1771	−0.816316	−0.408158	+	0.912911i	$0.633829\pi$
$674$	0	0
$675$	5.56155	0.214064
$676$	0	0
$677$	−6.24621	−0.240061	−0.120031	−	0.992770i	$-0.538299\pi$
$677$	−6.24621	−0.240061	−0.120031	+	0.992770i	$0.538299\pi$
$678$	0	0
$679$	60.9848	2.34038
$680$	0	0
$681$	28.1080	1.07710
$682$	0	0
$683$	3.80776	0.145700	0.0728500	−	0.997343i	$-0.476791\pi$
$683$	3.80776	0.145700	0.0728500	+	0.997343i	$0.476791\pi$
$684$	0	0
$685$	19.3693	0.740064
$686$	0	0
$687$	10.1383	0.386799
$688$	0	0
$689$	13.8617	0.528090
$690$	0	0
$691$	−35.6155	−1.35488	−0.677439	−	0.735579i	$-0.736910\pi$
$691$	−35.6155	−1.35488	−0.677439	+	0.735579i	$0.736910\pi$
$692$	0	0
$693$	−2.00000	−0.0759737
$694$	0	0
$695$	−16.4924	−0.625593
$696$	0	0
$697$	−55.6155	−2.10659
$698$	0	0
$699$	−17.6695	−0.668322
$700$	0	0
$701$	−36.5464	−1.38034	−0.690169	−	0.723648i	$-0.742464\pi$
$701$	−36.5464	−1.38034	−0.690169	+	0.723648i	$0.742464\pi$
$702$	0	0
$703$	23.9157	0.901998
$704$	0	0
$705$	−11.1231	−0.418920
$706$	0	0
$707$	−57.8617	−2.17611
$708$	0	0
$709$	48.2462	1.81192	0.905962	−	0.423358i	$-0.139149\pi$
$709$	48.2462	1.81192	0.905962	+	0.423358i	$0.139149\pi$
$710$	0	0
$711$	−0.492423	−0.0184673
$712$	0	0
$713$	61.8617	2.31674
$714$	0	0
$715$	3.12311	0.116798
$716$	0	0
$717$	−1.36932	−0.0511381
$718$	0	0
$719$	−42.0540	−1.56835	−0.784174	−	0.620541i	$-0.786913\pi$
$719$	−42.0540	−1.56835	−0.784174	+	0.620541i	$0.786913\pi$
$720$	0	0
$721$	−50.7386	−1.88961
$722$	0	0
$723$	44.8769	1.66899
$724$	0	0
$725$	−0.438447	−0.0162835
$726$	0	0
$727$	23.6155	0.875851	0.437926	−	0.899011i	$-0.355713\pi$
$727$	23.6155	0.875851	0.437926	+	0.899011i	$0.355713\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	28.4924	1.05383
$732$	0	0
$733$	−4.00000	−0.147743	−0.0738717	−	0.997268i	$-0.523536\pi$
$733$	−4.00000	−0.147743	−0.0738717	+	0.997268i	$0.523536\pi$
$734$	0	0
$735$	8.87689	0.327429
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−46.7386	−1.71931	−0.859654	−	0.510876i	$-0.829321\pi$
$739$	−46.7386	−1.71931	−0.859654	+	0.510876i	$0.829321\pi$
$740$	0	0
$741$	11.8920	0.436865
$742$	0	0
$743$	−28.4384	−1.04331	−0.521653	−	0.853158i	$-0.674684\pi$
$743$	−28.4384	−1.04331	−0.521653	+	0.853158i	$0.674684\pi$
$744$	0	0
$745$	1.31534	0.0481904
$746$	0	0
$747$	−5.61553	−0.205461
$748$	0	0
$749$	18.2462	0.666702
$750$	0	0
$751$	18.4384	0.672828	0.336414	−	0.941714i	$-0.390786\pi$
$751$	18.4384	0.672828	0.336414	+	0.941714i	$0.390786\pi$
$752$	0	0
$753$	−36.1080	−1.31585
$754$	0	0
$755$	0.876894	0.0319135
$756$	0	0
$757$	−26.4924	−0.962883	−0.481442	−	0.876478i	$-0.659887\pi$
$757$	−26.4924	−0.962883	−0.481442	+	0.876478i	$0.659887\pi$
$758$	0	0
$759$	−11.1231	−0.403743
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	0	0
$763$	−7.12311	−0.257874
$764$	0	0
$765$	3.12311	0.112916
$766$	0	0
$767$	41.7538	1.50764
$768$	0	0
$769$	30.9848	1.11734	0.558671	−	0.829389i	$-0.311311\pi$
$769$	30.9848	1.11734	0.558671	+	0.829389i	$0.311311\pi$
$770$	0	0
$771$	12.1080	0.436057
$772$	0	0
$773$	0.822919	0.0295983	0.0147992	−	0.999890i	$-0.495289\pi$
$773$	0.822919	0.0295983	0.0147992	+	0.999890i	$0.495289\pi$
$774$	0	0
$775$	8.68466	0.311962
$776$	0	0
$777$	−54.5464	−1.95684
$778$	0	0
$779$	−24.3845	−0.873664
$780$	0	0
$781$	−2.43845	−0.0872545
$782$	0	0
$783$	−2.43845	−0.0871430
$784$	0	0
$785$	12.9309	0.461523
$786$	0	0
$787$	−26.8769	−0.958058	−0.479029	−	0.877799i	$-0.659011\pi$
$787$	−26.8769	−0.958058	−0.479029	+	0.877799i	$0.659011\pi$
$788$	0	0
$789$	−38.1619	−1.35860
$790$	0	0
$791$	4.00000	0.142224
$792$	0	0
$793$	11.1231	0.394993
$794$	0	0
$795$	−6.93087	−0.245813
$796$	0	0
$797$	12.7386	0.451226	0.225613	−	0.974217i	$-0.427562\pi$
$797$	12.7386	0.451226	0.225613	+	0.974217i	$0.427562\pi$
$798$	0	0
$799$	−39.6155	−1.40150
$800$	0	0
$801$	−5.50758	−0.194601
$802$	0	0
$803$	4.87689	0.172102
$804$	0	0
$805$	−25.3693	−0.894151
$806$	0	0
$807$	35.7235	1.25753
$808$	0	0
$809$	−52.7386	−1.85419	−0.927096	−	0.374824i	$-0.877703\pi$
$809$	−52.7386	−1.85419	−0.927096	+	0.374824i	$0.877703\pi$
$810$	0	0
$811$	−46.4384	−1.63067	−0.815337	−	0.578986i	$-0.803448\pi$
$811$	−46.4384	−1.63067	−0.815337	+	0.578986i	$0.803448\pi$
$812$	0	0
$813$	−18.7386	−0.657193
$814$	0	0
$815$	−0.192236	−0.00673373
$816$	0	0
$817$	12.4924	0.437055
$818$	0	0
$819$	6.24621	0.218260
$820$	0	0
$821$	19.7538	0.689412	0.344706	−	0.938711i	$-0.387979\pi$
$821$	19.7538	0.689412	0.344706	+	0.938711i	$0.387979\pi$
$822$	0	0
$823$	18.7386	0.653188	0.326594	−	0.945165i	$-0.394099\pi$
$823$	18.7386	0.653188	0.326594	+	0.945165i	$0.394099\pi$
$824$	0	0
$825$	−1.56155	−0.0543663
$826$	0	0
$827$	10.3845	0.361103	0.180552	−	0.983565i	$-0.442212\pi$
$827$	10.3845	0.361103	0.180552	+	0.983565i	$0.442212\pi$
$828$	0	0
$829$	47.8617	1.66231	0.831153	−	0.556043i	$-0.187681\pi$
$829$	47.8617	1.66231	0.831153	+	0.556043i	$0.187681\pi$
$830$	0	0
$831$	−29.2614	−1.01507
$832$	0	0
$833$	31.6155	1.09541
$834$	0	0
$835$	−1.31534	−0.0455193
$836$	0	0
$837$	48.3002	1.66950
$838$	0	0
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	−28.8078	−0.993371
$842$	0	0
$843$	−28.8769	−0.994573
$844$	0	0
$845$	3.24621	0.111673
$846$	0	0
$847$	3.56155	0.122376
$848$	0	0
$849$	28.8769	0.991052
$850$	0	0
$851$	69.8617	2.39483
$852$	0	0
$853$	−40.1080	−1.37327	−0.686635	−	0.727002i	$-0.740913\pi$
$853$	−40.1080	−1.37327	−0.686635	+	0.727002i	$0.740913\pi$
$854$	0	0
$855$	1.36932	0.0468296
$856$	0	0
$857$	50.0540	1.70981	0.854906	−	0.518784i	$-0.173615\pi$
$857$	50.0540	1.70981	0.854906	+	0.518784i	$0.173615\pi$
$858$	0	0
$859$	21.3693	0.729112	0.364556	−	0.931182i	$-0.381221\pi$
$859$	21.3693	0.729112	0.364556	+	0.931182i	$0.381221\pi$
$860$	0	0
$861$	55.6155	1.89537
$862$	0	0
$863$	9.36932	0.318935	0.159468	−	0.987203i	$-0.449022\pi$
$863$	9.36932	0.318935	0.159468	+	0.987203i	$0.449022\pi$
$864$	0	0
$865$	21.3693	0.726579
$866$	0	0
$867$	−21.7538	−0.738797
$868$	0	0
$869$	0.876894	0.0297466
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−9.61553	−0.325436
$874$	0	0
$875$	−3.56155	−0.120402
$876$	0	0
$877$	39.1231	1.32109	0.660547	−	0.750785i	$-0.270325\pi$
$877$	39.1231	1.32109	0.660547	+	0.750785i	$0.270325\pi$
$878$	0	0
$879$	17.3693	0.585853
$880$	0	0
$881$	−34.4924	−1.16208	−0.581040	−	0.813875i	$-0.697354\pi$
$881$	−34.4924	−1.16208	−0.581040	+	0.813875i	$0.697354\pi$
$882$	0	0
$883$	−17.9460	−0.603932	−0.301966	−	0.953319i	$-0.597643\pi$
$883$	−17.9460	−0.603932	−0.301966	+	0.953319i	$0.597643\pi$
$884$	0	0
$885$	−20.8769	−0.701769
$886$	0	0
$887$	−55.3693	−1.85912	−0.929560	−	0.368671i	$-0.879813\pi$
$887$	−55.3693	−1.85912	−0.929560	+	0.368671i	$0.879813\pi$
$888$	0	0
$889$	−26.2462	−0.880270
$890$	0	0
$891$	−7.00000	−0.234509
$892$	0	0
$893$	−17.3693	−0.581242
$894$	0	0
$895$	−8.49242	−0.283870
$896$	0	0
$897$	34.7386	1.15989
$898$	0	0
$899$	−3.80776	−0.126996
$900$	0	0
$901$	−24.6847	−0.822365
$902$	0	0
$903$	−28.4924	−0.948168
$904$	0	0
$905$	−14.4924	−0.481744
$906$	0	0
$907$	45.1771	1.50008	0.750040	−	0.661392i	$-0.230034\pi$
$907$	45.1771	1.50008	0.750040	+	0.661392i	$0.230034\pi$
$908$	0	0
$909$	9.12311	0.302594
$910$	0	0
$911$	−41.6695	−1.38057	−0.690286	−	0.723537i	$-0.742515\pi$
$911$	−41.6695	−1.38057	−0.690286	+	0.723537i	$0.742515\pi$
$912$	0	0
$913$	10.0000	0.330952
$914$	0	0
$915$	−5.56155	−0.183859
$916$	0	0
$917$	2.43845	0.0805246
$918$	0	0
$919$	−9.75379	−0.321748	−0.160874	−	0.986975i	$-0.551431\pi$
$919$	−9.75379	−0.321748	−0.160874	+	0.986975i	$0.551431\pi$
$920$	0	0
$921$	−32.3845	−1.06710
$922$	0	0
$923$	7.61553	0.250668
$924$	0	0
$925$	9.80776	0.322477
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	0.930870	0.0305408	0.0152704	−	0.999883i	$-0.495139\pi$
$929$	0.930870	0.0305408	0.0152704	+	0.999883i	$0.495139\pi$
$930$	0	0
$931$	13.8617	0.454300
$932$	0	0
$933$	−33.0691	−1.08263
$934$	0	0
$935$	−5.56155	−0.181882
$936$	0	0
$937$	−29.3693	−0.959454	−0.479727	−	0.877418i	$-0.659264\pi$
$937$	−29.3693	−0.959454	−0.479727	+	0.877418i	$0.659264\pi$
$938$	0	0
$939$	−25.3693	−0.827896
$940$	0	0
$941$	−28.4384	−0.927067	−0.463533	−	0.886079i	$-0.653419\pi$
$941$	−28.4384	−0.927067	−0.463533	+	0.886079i	$0.653419\pi$
$942$	0	0
$943$	−71.2311	−2.31960
$944$	0	0
$945$	−19.8078	−0.644347
$946$	0	0
$947$	50.0540	1.62654	0.813268	−	0.581890i	$-0.197686\pi$
$947$	50.0540	1.62654	0.813268	+	0.581890i	$0.197686\pi$
$948$	0	0
$949$	−15.2311	−0.494421
$950$	0	0
$951$	5.56155	0.180346
$952$	0	0
$953$	37.6695	1.22023	0.610117	−	0.792311i	$-0.291122\pi$
$953$	37.6695	1.22023	0.610117	+	0.792311i	$0.291122\pi$
$954$	0	0
$955$	−1.75379	−0.0567513
$956$	0	0
$957$	0.684658	0.0221319
$958$	0	0
$959$	−68.9848	−2.22764
$960$	0	0
$961$	44.4233	1.43301
$962$	0	0
$963$	−2.87689	−0.0927066
$964$	0	0
$965$	−19.8078	−0.637634
$966$	0	0
$967$	−8.05398	−0.258998	−0.129499	−	0.991580i	$-0.541337\pi$
$967$	−8.05398	−0.258998	−0.129499	+	0.991580i	$0.541337\pi$
$968$	0	0
$969$	−21.1771	−0.680306
$970$	0	0
$971$	2.24621	0.0720843	0.0360422	−	0.999350i	$-0.488525\pi$
$971$	2.24621	0.0720843	0.0360422	+	0.999350i	$0.488525\pi$
$972$	0	0
$973$	58.7386	1.88307
$974$	0	0
$975$	4.87689	0.156186
$976$	0	0
$977$	−51.3693	−1.64345	−0.821725	−	0.569884i	$-0.806988\pi$
$977$	−51.3693	−1.64345	−0.821725	+	0.569884i	$0.806988\pi$
$978$	0	0
$979$	9.80776	0.313457
$980$	0	0
$981$	1.12311	0.0358580
$982$	0	0
$983$	20.8769	0.665870	0.332935	−	0.942950i	$-0.391961\pi$
$983$	20.8769	0.665870	0.332935	+	0.942950i	$0.391961\pi$
$984$	0	0
$985$	−18.2462	−0.581373
$986$	0	0
$987$	39.6155	1.26098
$988$	0	0
$989$	36.4924	1.16039
$990$	0	0
$991$	−52.4924	−1.66748	−0.833738	−	0.552160i	$-0.813804\pi$
$991$	−52.4924	−1.66748	−0.833738	+	0.552160i	$0.813804\pi$
$992$	0	0
$993$	3.50758	0.111310
$994$	0	0
$995$	6.93087	0.219723
$996$	0	0
$997$	12.9848	0.411234	0.205617	−	0.978633i	$-0.434080\pi$
$997$	12.9848	0.411234	0.205617	+	0.978633i	$0.434080\pi$
$998$	0	0
$999$	54.5464	1.72577

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	440.2.a.f.1.1	✓	2
3.2	odd	2		3960.2.a.be.1.2		2
4.3	odd	2		880.2.a.l.1.2		2
5.2	odd	4		2200.2.b.h.1849.3		4
5.3	odd	4		2200.2.b.h.1849.2		4
5.4	even	2		2200.2.a.m.1.2		2
8.3	odd	2		3520.2.a.bs.1.1		2
8.5	even	2		3520.2.a.bl.1.2		2
11.10	odd	2		4840.2.a.n.1.1		2
12.11	even	2		7920.2.a.ca.1.1		2
20.3	even	4		4400.2.b.u.4049.3		4
20.7	even	4		4400.2.b.u.4049.2		4
20.19	odd	2		4400.2.a.br.1.1		2
44.43	even	2		9680.2.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.a.f.1.1	✓	2	1.1	even	1	trivial
880.2.a.l.1.2		2	4.3	odd	2
2200.2.a.m.1.2		2	5.4	even	2
2200.2.b.h.1849.2		4	5.3	odd	4
2200.2.b.h.1849.3		4	5.2	odd	4
3520.2.a.bl.1.2		2	8.5	even	2
3520.2.a.bs.1.1		2	8.3	odd	2
3960.2.a.be.1.2		2	3.2	odd	2
4400.2.a.br.1.1		2	20.19	odd	2
4400.2.b.u.4049.2		4	20.7	even	4
4400.2.b.u.4049.3		4	20.3	even	4
4840.2.a.n.1.1		2	11.10	odd	2
7920.2.a.ca.1.1		2	12.11	even	2
9680.2.a.bl.1.2		2	44.43	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{3} -1.00000 q^{5} +3.56155 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q-1.56155 q^{3} -1.00000 q^{5} +3.56155 q^{7} -0.561553 q^{9} +1.00000 q^{11} -3.12311 q^{13} +1.56155 q^{15} +5.56155 q^{17} +2.43845 q^{19} -5.56155 q^{21} +7.12311 q^{23} +1.00000 q^{25} +5.56155 q^{27} -0.438447 q^{29} +8.68466 q^{31} -1.56155 q^{33} -3.56155 q^{35} +9.80776 q^{37} +4.87689 q^{39} -10.0000 q^{41} +5.12311 q^{43} +0.561553 q^{45} -7.12311 q^{47} +5.68466 q^{49} -8.68466 q^{51} -4.43845 q^{53} -1.00000 q^{55} -3.80776 q^{57} -13.3693 q^{59} -3.56155 q^{61} -2.00000 q^{63} +3.12311 q^{65} -11.1231 q^{69} -2.43845 q^{71} +4.87689 q^{73} -1.56155 q^{75} +3.56155 q^{77} +0.876894 q^{79} -7.00000 q^{81} +10.0000 q^{83} -5.56155 q^{85} +0.684658 q^{87} +9.80776 q^{89} -11.1231 q^{91} -13.5616 q^{93} -2.43845 q^{95} +17.1231 q^{97} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q + q^3 - 2 * q^5 + 3 * q^7 + 3 * q^9	\( 2 q + q^{3} - 2 q^{5} + 3 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - q^{15} + 7 q^{17} + 9 q^{19} - 7 q^{21} + 6 q^{23} + 2 q^{25} + 7 q^{27} - 5 q^{29} + 5 q^{31} + q^{33} - 3 q^{35} - q^{37} + 18 q^{39} - 20 q^{41} + 2 q^{43} - 3 q^{45} - 6 q^{47} - q^{49} - 5 q^{51} - 13 q^{53} - 2 q^{55} + 13 q^{57} - 2 q^{59} - 3 q^{61} - 4 q^{63} - 2 q^{65} - 14 q^{69} - 9 q^{71} + 18 q^{73} + q^{75} + 3 q^{77} + 10 q^{79} - 14 q^{81} + 20 q^{83} - 7 q^{85} - 11 q^{87} - q^{89} - 14 q^{91} - 23 q^{93} - 9 q^{95} + 26 q^{97} + 3 q^{99}+O(q^{100}) \) 2 * q + q^3 - 2 * q^5 + 3 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - q^15 + 7 * q^17 + 9 * q^19 - 7 * q^21 + 6 * q^23 + 2 * q^25 + 7 * q^27 - 5 * q^29 + 5 * q^31 + q^33 - 3 * q^35 - q^37 + 18 * q^39 - 20 * q^41 + 2 * q^43 - 3 * q^45 - 6 * q^47 - q^49 - 5 * q^51 - 13 * q^53 - 2 * q^55 + 13 * q^57 - 2 * q^59 - 3 * q^61 - 4 * q^63 - 2 * q^65 - 14 * q^69 - 9 * q^71 + 18 * q^73 + q^75 + 3 * q^77 + 10 * q^79 - 14 * q^81 + 20 * q^83 - 7 * q^85 - 11 * q^87 - q^89 - 14 * q^91 - 23 * q^93 - 9 * q^95 + 26 * q^97 + 3 * q^99

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