LMFDB - Embedded newform 4560.2.a.bu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(1,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.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:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1524.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 1 $ x^3 - x^2 - 7*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1140)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.13264$ of defining polynomial
Character	$\chi$	$=$	4560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +4.81342 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +4.81342 q^{7} +1.00000 q^{9} +1.45186 q^{11} -2.81342 q^{13} +1.00000 q^{15} +6.26527 q^{17} -1.00000 q^{19} +4.81342 q^{21} -6.26527 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.548143 q^{29} -8.26527 q^{31} +1.45186 q^{33} +4.81342 q^{35} +6.81342 q^{37} -2.81342 q^{39} +4.54814 q^{41} +7.71713 q^{43} +1.00000 q^{45} +10.2653 q^{47} +16.1690 q^{49} +6.26527 q^{51} -0.265275 q^{53} +1.45186 q^{55} -1.00000 q^{57} -2.90371 q^{59} +11.3616 q^{61} +4.81342 q^{63} -2.81342 q^{65} -6.26527 q^{69} -6.90371 q^{71} -6.53055 q^{73} +1.00000 q^{75} +6.98840 q^{77} -10.7231 q^{79} +1.00000 q^{81} -9.16899 q^{83} +6.26527 q^{85} +0.548143 q^{87} -18.7055 q^{89} -13.5422 q^{91} -8.26527 q^{93} -1.00000 q^{95} -6.81342 q^{97} +1.45186 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + 2 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} - 3 q^{19} - 2 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 8 q^{31} + 2 q^{33} + 6 q^{37} + 6 q^{39} + 16 q^{41} + 4 q^{43} + 3 q^{45} + 14 q^{47} + 27 q^{49} + 2 q^{51} + 16 q^{53} + 2 q^{55} - 3 q^{57} - 4 q^{59} + 22 q^{61} + 6 q^{65} - 2 q^{69} - 16 q^{71} + 14 q^{73} + 3 q^{75} - 20 q^{77} - 8 q^{79} + 3 q^{81} - 6 q^{83} + 2 q^{85} + 4 q^{87} + 4 q^{89} - 48 q^{91} - 8 q^{93} - 3 q^{95} - 6 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9 + 2 * q^11 + 6 * q^13 + 3 * q^15 + 2 * q^17 - 3 * q^19 - 2 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 8 * q^31 + 2 * q^33 + 6 * q^37 + 6 * q^39 + 16 * q^41 + 4 * q^43 + 3 * q^45 + 14 * q^47 + 27 * q^49 + 2 * q^51 + 16 * q^53 + 2 * q^55 - 3 * q^57 - 4 * q^59 + 22 * q^61 + 6 * q^65 - 2 * q^69 - 16 * q^71 + 14 * q^73 + 3 * q^75 - 20 * q^77 - 8 * q^79 + 3 * q^81 - 6 * q^83 + 2 * q^85 + 4 * q^87 + 4 * q^89 - 48 * q^91 - 8 * q^93 - 3 * q^95 - 6 * q^97 + 2 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.81342	1.81930	0.909650	−	0.415375i	$-0.136350\pi$
$7$	4.81342	1.81930	0.909650	+	0.415375i	$0.136350\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.45186	0.437751	0.218876	−	0.975753i	$-0.429761\pi$
$11$	1.45186	0.437751	0.218876	+	0.975753i	$0.429761\pi$
$12$	0	0
$13$	−2.81342	−0.780302	−0.390151	−	0.920751i	$-0.627577\pi$
$13$	−2.81342	−0.780302	−0.390151	+	0.920751i	$0.627577\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	6.26527	1.51955	0.759776	−	0.650185i	$-0.225308\pi$
$17$	6.26527	1.51955	0.759776	+	0.650185i	$0.225308\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	4.81342	1.05037
$22$	0	0
$23$	−6.26527	−1.30640	−0.653200	−	0.757185i	$-0.726574\pi$
$23$	−6.26527	−1.30640	−0.653200	+	0.757185i	$0.726574\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.548143	0.101788	0.0508938	−	0.998704i	$-0.483793\pi$
$29$	0.548143	0.101788	0.0508938	+	0.998704i	$0.483793\pi$
$30$	0	0
$31$	−8.26527	−1.48449	−0.742244	−	0.670130i	$-0.766238\pi$
$31$	−8.26527	−1.48449	−0.742244	+	0.670130i	$0.766238\pi$
$32$	0	0
$33$	1.45186	0.252736
$34$	0	0
$35$	4.81342	0.813616
$36$	0	0
$37$	6.81342	1.12012	0.560059	−	0.828452i	$-0.310778\pi$
$37$	6.81342	1.12012	0.560059	+	0.828452i	$0.310778\pi$
$38$	0	0
$39$	−2.81342	−0.450507
$40$	0	0
$41$	4.54814	0.710301	0.355150	−	0.934809i	$-0.384430\pi$
$41$	4.54814	0.710301	0.355150	+	0.934809i	$0.384430\pi$
$42$	0	0
$43$	7.71713	1.17685	0.588426	−	0.808551i	$-0.299748\pi$
$43$	7.71713	1.17685	0.588426	+	0.808551i	$0.299748\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	10.2653	1.49734	0.748672	−	0.662940i	$-0.230692\pi$
$47$	10.2653	1.49734	0.748672	+	0.662940i	$0.230692\pi$
$48$	0	0
$49$	16.1690	2.30986
$50$	0	0
$51$	6.26527	0.877314
$52$	0	0
$53$	−0.265275	−0.0364383	−0.0182192	−	0.999834i	$-0.505800\pi$
$53$	−0.265275	−0.0364383	−0.0182192	+	0.999834i	$0.505800\pi$
$54$	0	0
$55$	1.45186	0.195768
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−2.90371	−0.378031	−0.189016	−	0.981974i	$-0.560530\pi$
$59$	−2.90371	−0.378031	−0.189016	+	0.981974i	$0.560530\pi$
$60$	0	0
$61$	11.3616	1.45470	0.727349	−	0.686267i	$-0.240752\pi$
$61$	11.3616	1.45470	0.727349	+	0.686267i	$0.240752\pi$
$62$	0	0
$63$	4.81342	0.606434
$64$	0	0
$65$	−2.81342	−0.348962
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−6.26527	−0.754250
$70$	0	0
$71$	−6.90371	−0.819320	−0.409660	−	0.912238i	$-0.634353\pi$
$71$	−6.90371	−0.819320	−0.409660	+	0.912238i	$0.634353\pi$
$72$	0	0
$73$	−6.53055	−0.764343	−0.382172	−	0.924091i	$-0.624824\pi$
$73$	−6.53055	−0.764343	−0.382172	+	0.924091i	$0.624824\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	6.98840	0.796402
$78$	0	0
$79$	−10.7231	−1.20645	−0.603223	−	0.797573i	$-0.706117\pi$
$79$	−10.7231	−1.20645	−0.603223	+	0.797573i	$0.706117\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.16899	−1.00643	−0.503214	−	0.864162i	$-0.667849\pi$
$83$	−9.16899	−1.00643	−0.503214	+	0.864162i	$0.667849\pi$
$84$	0	0
$85$	6.26527	0.679564
$86$	0	0
$87$	0.548143	0.0587671
$88$	0	0
$89$	−18.7055	−1.98278	−0.991391	−	0.130935i	$-0.958202\pi$
$89$	−18.7055	−1.98278	−0.991391	+	0.130935i	$0.958202\pi$
$90$	0	0
$91$	−13.5422	−1.41960
$92$	0	0
$93$	−8.26527	−0.857069
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−6.81342	−0.691798	−0.345899	−	0.938272i	$-0.612426\pi$
$97$	−6.81342	−0.691798	−0.345899	+	0.938272i	$0.612426\pi$
$98$	0	0
$99$	1.45186	0.145917
$100$	0	0
$101$	17.4343	1.73477	0.867387	−	0.497634i	$-0.165798\pi$
$101$	17.4343	1.73477	0.867387	+	0.497634i	$0.165798\pi$
$102$	0	0
$103$	−16.5305	−1.62880	−0.814402	−	0.580301i	$-0.802935\pi$
$103$	−16.5305	−1.62880	−0.814402	+	0.580301i	$0.802935\pi$
$104$	0	0
$105$	4.81342	0.469741
$106$	0	0
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	3.09629	0.296570	0.148285	−	0.988945i	$-0.452625\pi$
$109$	3.09629	0.296570	0.148285	+	0.988945i	$0.452625\pi$
$110$	0	0
$111$	6.81342	0.646701
$112$	0	0
$113$	13.3616	1.25695	0.628475	−	0.777830i	$-0.283679\pi$
$113$	13.3616	1.25695	0.628475	+	0.777830i	$0.283679\pi$
$114$	0	0
$115$	−6.26527	−0.584240
$116$	0	0
$117$	−2.81342	−0.260101
$118$	0	0
$119$	30.1574	2.76452
$120$	0	0
$121$	−8.89211	−0.808374
$122$	0	0
$123$	4.54814	0.410092
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	7.71713	0.679456
$130$	0	0
$131$	4.35557	0.380548	0.190274	−	0.981731i	$-0.439062\pi$
$131$	4.35557	0.380548	0.190274	+	0.981731i	$0.439062\pi$
$132$	0	0
$133$	−4.81342	−0.417376
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	2.90371	0.246290	0.123145	−	0.992389i	$-0.460702\pi$
$139$	2.90371	0.246290	0.123145	+	0.992389i	$0.460702\pi$
$140$	0	0
$141$	10.2653	0.864492
$142$	0	0
$143$	−4.08468	−0.341578
$144$	0	0
$145$	0.548143	0.0455208
$146$	0	0
$147$	16.1690	1.33360
$148$	0	0
$149$	6.53055	0.535003	0.267502	−	0.963557i	$-0.413802\pi$
$149$	6.53055	0.535003	0.267502	+	0.963557i	$0.413802\pi$
$150$	0	0
$151$	−18.9884	−1.54525	−0.772627	−	0.634860i	$-0.781058\pi$
$151$	−18.9884	−1.54525	−0.772627	+	0.634860i	$0.781058\pi$
$152$	0	0
$153$	6.26527	0.506517
$154$	0	0
$155$	−8.26527	−0.663883
$156$	0	0
$157$	16.1574	1.28950	0.644750	−	0.764394i	$-0.276962\pi$
$157$	16.1574	1.28950	0.644750	+	0.764394i	$0.276962\pi$
$158$	0	0
$159$	−0.265275	−0.0210377
$160$	0	0
$161$	−30.1574	−2.37673
$162$	0	0
$163$	−10.4403	−0.817744	−0.408872	−	0.912592i	$-0.634078\pi$
$163$	−10.4403	−0.817744	−0.408872	+	0.912592i	$0.634078\pi$
$164$	0	0
$165$	1.45186	0.113027
$166$	0	0
$167$	13.8921	1.07500	0.537502	−	0.843263i	$-0.319368\pi$
$167$	13.8921	1.07500	0.537502	+	0.843263i	$0.319368\pi$
$168$	0	0
$169$	−5.08468	−0.391129
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−2.63844	−0.200597	−0.100298	−	0.994957i	$-0.531980\pi$
$173$	−2.63844	−0.200597	−0.100298	+	0.994957i	$0.531980\pi$
$174$	0	0
$175$	4.81342	0.363860
$176$	0	0
$177$	−2.90371	−0.218257
$178$	0	0
$179$	2.37316	0.177379	0.0886893	−	0.996059i	$-0.471732\pi$
$179$	2.37316	0.177379	0.0886893	+	0.996059i	$0.471732\pi$
$180$	0	0
$181$	0.373165	0.0277371	0.0138686	−	0.999904i	$-0.495585\pi$
$181$	0.373165	0.0277371	0.0138686	+	0.999904i	$0.495585\pi$
$182$	0	0
$183$	11.3616	0.839871
$184$	0	0
$185$	6.81342	0.500932
$186$	0	0
$187$	9.09629	0.665186
$188$	0	0
$189$	4.81342	0.350125
$190$	0	0
$191$	−4.17498	−0.302091	−0.151045	−	0.988527i	$-0.548264\pi$
$191$	−4.17498	−0.302091	−0.151045	+	0.988527i	$0.548264\pi$
$192$	0	0
$193$	15.3440	1.10448	0.552241	−	0.833684i	$-0.313773\pi$
$193$	15.3440	1.10448	0.552241	+	0.833684i	$0.313773\pi$
$194$	0	0
$195$	−2.81342	−0.201473
$196$	0	0
$197$	−22.2653	−1.58634	−0.793168	−	0.609003i	$-0.791570\pi$
$197$	−22.2653	−1.58634	−0.793168	+	0.609003i	$0.791570\pi$
$198$	0	0
$199$	−6.15739	−0.436485	−0.218243	−	0.975895i	$-0.570032\pi$
$199$	−6.15739	−0.436485	−0.218243	+	0.975895i	$0.570032\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.63844	0.185182
$204$	0	0
$205$	4.54814	0.317656
$206$	0	0
$207$	−6.26527	−0.435467
$208$	0	0
$209$	−1.45186	−0.100427
$210$	0	0
$211$	23.7842	1.63737	0.818687	−	0.574241i	$-0.194703\pi$
$211$	23.7842	1.63737	0.818687	+	0.574241i	$0.194703\pi$
$212$	0	0
$213$	−6.90371	−0.473035
$214$	0	0
$215$	7.71713	0.526304
$216$	0	0
$217$	−39.7842	−2.70073
$218$	0	0
$219$	−6.53055	−0.441294
$220$	0	0
$221$	−17.6268	−1.18571
$222$	0	0
$223$	−16.5305	−1.10697	−0.553484	−	0.832860i	$-0.686702\pi$
$223$	−16.5305	−1.10697	−0.553484	+	0.832860i	$0.686702\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	20.0847	1.33307	0.666534	−	0.745475i	$-0.267777\pi$
$227$	20.0847	1.33307	0.666534	+	0.745475i	$0.267777\pi$
$228$	0	0
$229$	−27.8921	−1.84316	−0.921581	−	0.388185i	$-0.873102\pi$
$229$	−27.8921	−1.84316	−0.921581	+	0.388185i	$0.873102\pi$
$230$	0	0
$231$	6.98840	0.459803
$232$	0	0
$233$	20.3380	1.33239	0.666193	−	0.745780i	$-0.267923\pi$
$233$	20.3380	1.33239	0.666193	+	0.745780i	$0.267923\pi$
$234$	0	0
$235$	10.2653	0.669633
$236$	0	0
$237$	−10.7231	−0.696542
$238$	0	0
$239$	0.921307	0.0595944	0.0297972	−	0.999556i	$-0.490514\pi$
$239$	0.921307	0.0595944	0.0297972	+	0.999556i	$0.490514\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	16.1690	1.03300
$246$	0	0
$247$	2.81342	0.179013
$248$	0	0
$249$	−9.16899	−0.581061
$250$	0	0
$251$	−15.6092	−0.985247	−0.492623	−	0.870243i	$-0.663962\pi$
$251$	−15.6092	−0.985247	−0.492623	+	0.870243i	$0.663962\pi$
$252$	0	0
$253$	−9.09629	−0.571879
$254$	0	0
$255$	6.26527	0.392347
$256$	0	0
$257$	7.16899	0.447189	0.223595	−	0.974682i	$-0.428221\pi$
$257$	7.16899	0.447189	0.223595	+	0.974682i	$0.428221\pi$
$258$	0	0
$259$	32.7958	2.03783
$260$	0	0
$261$	0.548143	0.0339292
$262$	0	0
$263$	24.4227	1.50597	0.752983	−	0.658040i	$-0.228614\pi$
$263$	24.4227	1.50597	0.752983	+	0.658040i	$0.228614\pi$
$264$	0	0
$265$	−0.265275	−0.0162957
$266$	0	0
$267$	−18.7055	−1.14476
$268$	0	0
$269$	−10.1750	−0.620379	−0.310190	−	0.950675i	$-0.600393\pi$
$269$	−10.1750	−0.620379	−0.310190	+	0.950675i	$0.600393\pi$
$270$	0	0
$271$	−2.37316	−0.144159	−0.0720797	−	0.997399i	$-0.522964\pi$
$271$	−2.37316	−0.144159	−0.0720797	+	0.997399i	$0.522964\pi$
$272$	0	0
$273$	−13.5422	−0.819608
$274$	0	0
$275$	1.45186	0.0875503
$276$	0	0
$277$	−16.3380	−0.981654	−0.490827	−	0.871257i	$-0.663305\pi$
$277$	−16.3380	−0.981654	−0.490827	+	0.871257i	$0.663305\pi$
$278$	0	0
$279$	−8.26527	−0.494829
$280$	0	0
$281$	5.64443	0.336718	0.168359	−	0.985726i	$-0.446153\pi$
$281$	5.64443	0.336718	0.168359	+	0.985726i	$0.446153\pi$
$282$	0	0
$283$	21.3440	1.26877	0.634384	−	0.773018i	$-0.281254\pi$
$283$	21.3440	1.26877	0.634384	+	0.773018i	$0.281254\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	21.8921	1.29225
$288$	0	0
$289$	22.2537	1.30904
$290$	0	0
$291$	−6.81342	−0.399410
$292$	0	0
$293$	10.6384	0.621504	0.310752	−	0.950491i	$-0.399419\pi$
$293$	10.6384	0.621504	0.310752	+	0.950491i	$0.399419\pi$
$294$	0	0
$295$	−2.90371	−0.169061
$296$	0	0
$297$	1.45186	0.0842453
$298$	0	0
$299$	17.6268	1.01939
$300$	0	0
$301$	37.1458	2.14105
$302$	0	0
$303$	17.4343	1.00157
$304$	0	0
$305$	11.3616	0.650561
$306$	0	0
$307$	−22.1574	−1.26459	−0.632294	−	0.774728i	$-0.717887\pi$
$307$	−22.1574	−1.26459	−0.632294	+	0.774728i	$0.717887\pi$
$308$	0	0
$309$	−16.5305	−0.940390
$310$	0	0
$311$	−4.17498	−0.236741	−0.118371	−	0.992969i	$-0.537767\pi$
$311$	−4.17498	−0.236741	−0.118371	+	0.992969i	$0.537767\pi$
$312$	0	0
$313$	−12.3732	−0.699373	−0.349686	−	0.936867i	$-0.613712\pi$
$313$	−12.3732	−0.699373	−0.349686	+	0.936867i	$0.613712\pi$
$314$	0	0
$315$	4.81342	0.271205
$316$	0	0
$317$	−3.73473	−0.209763	−0.104882	−	0.994485i	$-0.533446\pi$
$317$	−3.73473	−0.209763	−0.104882	+	0.994485i	$0.533446\pi$
$318$	0	0
$319$	0.795825	0.0445576
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	−6.26527	−0.348609
$324$	0	0
$325$	−2.81342	−0.156060
$326$	0	0
$327$	3.09629	0.171225
$328$	0	0
$329$	49.4111	2.72412
$330$	0	0
$331$	−14.9884	−0.823837	−0.411918	−	0.911221i	$-0.635141\pi$
$331$	−14.9884	−0.823837	−0.411918	+	0.911221i	$0.635141\pi$
$332$	0	0
$333$	6.81342	0.373373
$334$	0	0
$335$	0	0
$336$	0	0
$337$	32.9708	1.79603	0.898017	−	0.439961i	$-0.145008\pi$
$337$	32.9708	1.79603	0.898017	+	0.439961i	$0.145008\pi$
$338$	0	0
$339$	13.3616	0.725700
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	44.1342	2.38302
$344$	0	0
$345$	−6.26527	−0.337311
$346$	0	0
$347$	−35.1458	−1.88672	−0.943362	−	0.331765i	$-0.892356\pi$
$347$	−35.1458	−1.88672	−0.943362	+	0.331765i	$0.892356\pi$
$348$	0	0
$349$	−4.19257	−0.224423	−0.112212	−	0.993684i	$-0.535793\pi$
$349$	−4.19257	−0.224423	−0.112212	+	0.993684i	$0.535793\pi$
$350$	0	0
$351$	−2.81342	−0.150169
$352$	0	0
$353$	−3.80743	−0.202649	−0.101325	−	0.994853i	$-0.532308\pi$
$353$	−3.80743	−0.202649	−0.101325	+	0.994853i	$0.532308\pi$
$354$	0	0
$355$	−6.90371	−0.366411
$356$	0	0
$357$	30.1574	1.59610
$358$	0	0
$359$	1.27126	0.0670947	0.0335474	−	0.999437i	$-0.489320\pi$
$359$	1.27126	0.0670947	0.0335474	+	0.999437i	$0.489320\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−8.89211	−0.466715
$364$	0	0
$365$	−6.53055	−0.341825
$366$	0	0
$367$	−10.0903	−0.526709	−0.263355	−	0.964699i	$-0.584829\pi$
$367$	−10.0903	−0.526709	−0.263355	+	0.964699i	$0.584829\pi$
$368$	0	0
$369$	4.54814	0.236767
$370$	0	0
$371$	−1.27688	−0.0662923
$372$	0	0
$373$	−14.2829	−0.739539	−0.369769	−	0.929124i	$-0.620563\pi$
$373$	−14.2829	−0.739539	−0.369769	+	0.929124i	$0.620563\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−1.54215	−0.0794250
$378$	0	0
$379$	−20.7958	−1.06821	−0.534105	−	0.845418i	$-0.679351\pi$
$379$	−20.7958	−1.06821	−0.534105	+	0.845418i	$0.679351\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	−3.46945	−0.177281	−0.0886403	−	0.996064i	$-0.528252\pi$
$383$	−3.46945	−0.177281	−0.0886403	+	0.996064i	$0.528252\pi$
$384$	0	0
$385$	6.98840	0.356162
$386$	0	0
$387$	7.71713	0.392284
$388$	0	0
$389$	8.72312	0.442280	0.221140	−	0.975242i	$-0.429022\pi$
$389$	8.72312	0.442280	0.221140	+	0.975242i	$0.429022\pi$
$390$	0	0
$391$	−39.2537	−1.98514
$392$	0	0
$393$	4.35557	0.219710
$394$	0	0
$395$	−10.7231	−0.539539
$396$	0	0
$397$	19.4462	0.975979	0.487989	−	0.872850i	$-0.337730\pi$
$397$	19.4462	0.975979	0.487989	+	0.872850i	$0.337730\pi$
$398$	0	0
$399$	−4.81342	−0.240972
$400$	0	0
$401$	27.9824	1.39737	0.698687	−	0.715427i	$-0.253768\pi$
$401$	27.9824	1.39737	0.698687	+	0.715427i	$0.253768\pi$
$402$	0	0
$403$	23.2537	1.15835
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	9.89211	0.490334
$408$	0	0
$409$	−25.9648	−1.28388	−0.641939	−	0.766756i	$-0.721870\pi$
$409$	−25.9648	−1.28388	−0.641939	+	0.766756i	$0.721870\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	−13.9768	−0.687753
$414$	0	0
$415$	−9.16899	−0.450088
$416$	0	0
$417$	2.90371	0.142196
$418$	0	0
$419$	28.7055	1.40236	0.701178	−	0.712986i	$-0.252658\pi$
$419$	28.7055	1.40236	0.701178	+	0.712986i	$0.252658\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	10.2653	0.499115
$424$	0	0
$425$	6.26527	0.303910
$426$	0	0
$427$	54.6879	2.64653
$428$	0	0
$429$	−4.08468	−0.197210
$430$	0	0
$431$	−22.6879	−1.09284	−0.546420	−	0.837512i	$-0.684010\pi$
$431$	−22.6879	−1.09284	−0.546420	+	0.837512i	$0.684010\pi$
$432$	0	0
$433$	31.3440	1.50629	0.753147	−	0.657852i	$-0.228535\pi$
$433$	31.3440	1.50629	0.753147	+	0.657852i	$0.228535\pi$
$434$	0	0
$435$	0.548143	0.0262814
$436$	0	0
$437$	6.26527	0.299709
$438$	0	0
$439$	10.1926	0.486465	0.243232	−	0.969968i	$-0.421792\pi$
$439$	10.1926	0.486465	0.243232	+	0.969968i	$0.421792\pi$
$440$	0	0
$441$	16.1690	0.769952
$442$	0	0
$443$	13.5189	0.642304	0.321152	−	0.947028i	$-0.395930\pi$
$443$	13.5189	0.642304	0.321152	+	0.947028i	$0.395930\pi$
$444$	0	0
$445$	−18.7055	−0.886727
$446$	0	0
$447$	6.53055	0.308884
$448$	0	0
$449$	−15.9824	−0.754256	−0.377128	−	0.926161i	$-0.623088\pi$
$449$	−15.9824	−0.754256	−0.377128	+	0.926161i	$0.623088\pi$
$450$	0	0
$451$	6.60325	0.310935
$452$	0	0
$453$	−18.9884	−0.892153
$454$	0	0
$455$	−13.5422	−0.634866
$456$	0	0
$457$	−23.0963	−1.08040	−0.540199	−	0.841537i	$-0.681651\pi$
$457$	−23.0963	−1.08040	−0.540199	+	0.841537i	$0.681651\pi$
$458$	0	0
$459$	6.26527	0.292438
$460$	0	0
$461$	−29.2537	−1.36248	−0.681240	−	0.732060i	$-0.738559\pi$
$461$	−29.2537	−1.36248	−0.681240	+	0.732060i	$0.738559\pi$
$462$	0	0
$463$	−15.5365	−0.722044	−0.361022	−	0.932557i	$-0.617572\pi$
$463$	−15.5365	−0.722044	−0.361022	+	0.932557i	$0.617572\pi$
$464$	0	0
$465$	−8.26527	−0.383293
$466$	0	0
$467$	11.5422	0.534107	0.267054	−	0.963682i	$-0.413950\pi$
$467$	11.5422	0.534107	0.267054	+	0.963682i	$0.413950\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	16.1574	0.744493
$472$	0	0
$473$	11.2042	0.515169
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−0.265275	−0.0121461
$478$	0	0
$479$	16.3556	0.747305	0.373653	−	0.927569i	$-0.378105\pi$
$479$	16.3556	0.747305	0.373653	+	0.927569i	$0.378105\pi$
$480$	0	0
$481$	−19.1690	−0.874031
$482$	0	0
$483$	−30.1574	−1.37221
$484$	0	0
$485$	−6.81342	−0.309381
$486$	0	0
$487$	−41.4111	−1.87651	−0.938257	−	0.345939i	$-0.887560\pi$
$487$	−41.4111	−1.87651	−0.938257	+	0.345939i	$0.887560\pi$
$488$	0	0
$489$	−10.4403	−0.472125
$490$	0	0
$491$	11.0787	0.499974	0.249987	−	0.968249i	$-0.419574\pi$
$491$	11.0787	0.499974	0.249987	+	0.968249i	$0.419574\pi$
$492$	0	0
$493$	3.43426	0.154671
$494$	0	0
$495$	1.45186	0.0652561
$496$	0	0
$497$	−33.2305	−1.49059
$498$	0	0
$499$	16.3500	0.731925	0.365962	−	0.930630i	$-0.380740\pi$
$499$	16.3500	0.731925	0.365962	+	0.930630i	$0.380740\pi$
$500$	0	0
$501$	13.8921	0.620654
$502$	0	0
$503$	−11.8921	−0.530243	−0.265121	−	0.964215i	$-0.585412\pi$
$503$	−11.8921	−0.530243	−0.265121	+	0.964215i	$0.585412\pi$
$504$	0	0
$505$	17.4343	0.775815
$506$	0	0
$507$	−5.08468	−0.225819
$508$	0	0
$509$	32.8981	1.45818	0.729091	−	0.684416i	$-0.239943\pi$
$509$	32.8981	1.45818	0.729091	+	0.684416i	$0.239943\pi$
$510$	0	0
$511$	−31.4343	−1.39057
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−16.5305	−0.728423
$516$	0	0
$517$	14.9037	0.655465
$518$	0	0
$519$	−2.63844	−0.115815
$520$	0	0
$521$	−25.0787	−1.09872	−0.549359	−	0.835587i	$-0.685128\pi$
$521$	−25.0787	−1.09872	−0.549359	+	0.835587i	$0.685128\pi$
$522$	0	0
$523$	−23.9648	−1.04791	−0.523954	−	0.851747i	$-0.675544\pi$
$523$	−23.9648	−1.04791	−0.523954	+	0.851747i	$0.675544\pi$
$524$	0	0
$525$	4.81342	0.210075
$526$	0	0
$527$	−51.7842	−2.25576
$528$	0	0
$529$	16.2537	0.706681
$530$	0	0
$531$	−2.90371	−0.126010
$532$	0	0
$533$	−12.7958	−0.554249
$534$	0	0
$535$	16.0000	0.691740
$536$	0	0
$537$	2.37316	0.102410
$538$	0	0
$539$	23.4751	1.01114
$540$	0	0
$541$	11.0611	0.475554	0.237777	−	0.971320i	$-0.423581\pi$
$541$	11.0611	0.475554	0.237777	+	0.971320i	$0.423581\pi$
$542$	0	0
$543$	0.373165	0.0160140
$544$	0	0
$545$	3.09629	0.132630
$546$	0	0
$547$	−21.0963	−0.902012	−0.451006	−	0.892521i	$-0.648935\pi$
$547$	−21.0963	−0.902012	−0.451006	+	0.892521i	$0.648935\pi$
$548$	0	0
$549$	11.3616	0.484900
$550$	0	0
$551$	−0.548143	−0.0233517
$552$	0	0
$553$	−51.6149	−2.19489
$554$	0	0
$555$	6.81342	0.289213
$556$	0	0
$557$	−39.5916	−1.67755	−0.838776	−	0.544477i	$-0.816728\pi$
$557$	−39.5916	−1.67755	−0.838776	+	0.544477i	$0.816728\pi$
$558$	0	0
$559$	−21.7115	−0.918299
$560$	0	0
$561$	9.09629	0.384045
$562$	0	0
$563$	28.0847	1.18363	0.591814	−	0.806074i	$-0.298412\pi$
$563$	28.0847	1.18363	0.591814	+	0.806074i	$0.298412\pi$
$564$	0	0
$565$	13.3616	0.562125
$566$	0	0
$567$	4.81342	0.202145
$568$	0	0
$569$	41.0787	1.72211	0.861054	−	0.508513i	$-0.169805\pi$
$569$	41.0787	1.72211	0.861054	+	0.508513i	$0.169805\pi$
$570$	0	0
$571$	11.8194	0.494627	0.247313	−	0.968936i	$-0.420452\pi$
$571$	11.8194	0.494627	0.247313	+	0.968936i	$0.420452\pi$
$572$	0	0
$573$	−4.17498	−0.174412
$574$	0	0
$575$	−6.26527	−0.261280
$576$	0	0
$577$	−29.4343	−1.22536	−0.612682	−	0.790329i	$-0.709909\pi$
$577$	−29.4343	−1.22536	−0.612682	+	0.790329i	$0.709909\pi$
$578$	0	0
$579$	15.3440	0.637674
$580$	0	0
$581$	−44.1342	−1.83099
$582$	0	0
$583$	−0.385141	−0.0159509
$584$	0	0
$585$	−2.81342	−0.116321
$586$	0	0
$587$	−31.7115	−1.30887	−0.654437	−	0.756116i	$-0.727094\pi$
$587$	−31.7115	−1.30887	−0.654437	+	0.756116i	$0.727094\pi$
$588$	0	0
$589$	8.26527	0.340565
$590$	0	0
$591$	−22.2653	−0.915871
$592$	0	0
$593$	14.5305	0.596698	0.298349	−	0.954457i	$-0.403564\pi$
$593$	14.5305	0.596698	0.298349	+	0.954457i	$0.403564\pi$
$594$	0	0
$595$	30.1574	1.23633
$596$	0	0
$597$	−6.15739	−0.252005
$598$	0	0
$599$	−17.8074	−0.727592	−0.363796	−	0.931479i	$-0.618519\pi$
$599$	−17.8074	−0.727592	−0.363796	+	0.931479i	$0.618519\pi$
$600$	0	0
$601$	20.6879	0.843878	0.421939	−	0.906624i	$-0.361350\pi$
$601$	20.6879	0.843878	0.421939	+	0.906624i	$0.361350\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8.89211	−0.361516
$606$	0	0
$607$	17.2417	0.699819	0.349909	−	0.936784i	$-0.386212\pi$
$607$	17.2417	0.699819	0.349909	+	0.936784i	$0.386212\pi$
$608$	0	0
$609$	2.63844	0.106915
$610$	0	0
$611$	−28.8805	−1.16838
$612$	0	0
$613$	−3.62684	−0.146486	−0.0732432	−	0.997314i	$-0.523335\pi$
$613$	−3.62684	−0.146486	−0.0732432	+	0.997314i	$0.523335\pi$
$614$	0	0
$615$	4.54814	0.183399
$616$	0	0
$617$	23.3264	0.939084	0.469542	−	0.882910i	$-0.344419\pi$
$617$	23.3264	0.939084	0.469542	+	0.882910i	$0.344419\pi$
$618$	0	0
$619$	24.8805	1.00003	0.500016	−	0.866016i	$-0.333327\pi$
$619$	24.8805	1.00003	0.500016	+	0.866016i	$0.333327\pi$
$620$	0	0
$621$	−6.26527	−0.251417
$622$	0	0
$623$	−90.0375	−3.60728
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.45186	−0.0579816
$628$	0	0
$629$	42.6879	1.70208
$630$	0	0
$631$	36.3148	1.44567	0.722834	−	0.691022i	$-0.242839\pi$
$631$	36.3148	1.44567	0.722834	+	0.691022i	$0.242839\pi$
$632$	0	0
$633$	23.7842	0.945338
$634$	0	0
$635$	−4.00000	−0.158735
$636$	0	0
$637$	−45.4901	−1.80238
$638$	0	0
$639$	−6.90371	−0.273107
$640$	0	0
$641$	14.3556	0.567011	0.283506	−	0.958971i	$-0.408503\pi$
$641$	14.3556	0.567011	0.283506	+	0.958971i	$0.408503\pi$
$642$	0	0
$643$	−37.1634	−1.46558	−0.732790	−	0.680455i	$-0.761782\pi$
$643$	−37.1634	−1.46558	−0.732790	+	0.680455i	$0.761782\pi$
$644$	0	0
$645$	7.71713	0.303862
$646$	0	0
$647$	−40.9532	−1.61004	−0.805018	−	0.593250i	$-0.797845\pi$
$647$	−40.9532	−1.61004	−0.805018	+	0.593250i	$0.797845\pi$
$648$	0	0
$649$	−4.21578	−0.165484
$650$	0	0
$651$	−39.7842	−1.55927
$652$	0	0
$653$	−18.7958	−0.735537	−0.367769	−	0.929917i	$-0.619878\pi$
$653$	−18.7958	−0.735537	−0.367769	+	0.929917i	$0.619878\pi$
$654$	0	0
$655$	4.35557	0.170186
$656$	0	0
$657$	−6.53055	−0.254781
$658$	0	0
$659$	−8.53055	−0.332303	−0.166152	−	0.986100i	$-0.553134\pi$
$659$	−8.53055	−0.332303	−0.166152	+	0.986100i	$0.553134\pi$
$660$	0	0
$661$	13.4343	0.522532	0.261266	−	0.965267i	$-0.415860\pi$
$661$	13.4343	0.522532	0.261266	+	0.965267i	$0.415860\pi$
$662$	0	0
$663$	−17.6268	−0.684570
$664$	0	0
$665$	−4.81342	−0.186656
$666$	0	0
$667$	−3.43426	−0.132975
$668$	0	0
$669$	−16.5305	−0.639108
$670$	0	0
$671$	16.4954	0.636796
$672$	0	0
$673$	−6.81342	−0.262638	−0.131319	−	0.991340i	$-0.541921\pi$
$673$	−6.81342	−0.262638	−0.131319	+	0.991340i	$0.541921\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	47.3032	1.81801	0.909004	−	0.416787i	$-0.136844\pi$
$677$	47.3032	1.81801	0.909004	+	0.416787i	$0.136844\pi$
$678$	0	0
$679$	−32.7958	−1.25859
$680$	0	0
$681$	20.0847	0.769647
$682$	0	0
$683$	11.2537	0.430610	0.215305	−	0.976547i	$-0.430925\pi$
$683$	11.2537	0.430610	0.215305	+	0.976547i	$0.430925\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	−27.8921	−1.06415
$688$	0	0
$689$	0.746329	0.0284329
$690$	0	0
$691$	−9.62684	−0.366222	−0.183111	−	0.983092i	$-0.558617\pi$
$691$	−9.62684	−0.366222	−0.183111	+	0.983092i	$0.558617\pi$
$692$	0	0
$693$	6.98840	0.265467
$694$	0	0
$695$	2.90371	0.110144
$696$	0	0
$697$	28.4954	1.07934
$698$	0	0
$699$	20.3380	0.769253
$700$	0	0
$701$	16.7231	0.631624	0.315812	−	0.948822i	$-0.397723\pi$
$701$	16.7231	0.631624	0.315812	+	0.948822i	$0.397723\pi$
$702$	0	0
$703$	−6.81342	−0.256973
$704$	0	0
$705$	10.2653	0.386613
$706$	0	0
$707$	83.9184	3.15608
$708$	0	0
$709$	−32.9532	−1.23758	−0.618792	−	0.785555i	$-0.712378\pi$
$709$	−32.9532	−1.23758	−0.618792	+	0.785555i	$0.712378\pi$
$710$	0	0
$711$	−10.7231	−0.402148
$712$	0	0
$713$	51.7842	1.93933
$714$	0	0
$715$	−4.08468	−0.152758
$716$	0	0
$717$	0.921307	0.0344069
$718$	0	0
$719$	15.6444	0.583439	0.291719	−	0.956504i	$-0.405773\pi$
$719$	15.6444	0.583439	0.291719	+	0.956504i	$0.405773\pi$
$720$	0	0
$721$	−79.5684	−2.96328
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	0.548143	0.0203575
$726$	0	0
$727$	−24.2829	−0.900602	−0.450301	−	0.892877i	$-0.648683\pi$
$727$	−24.2829	−0.900602	−0.450301	+	0.892877i	$0.648683\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	48.3500	1.78829
$732$	0	0
$733$	−18.5305	−0.684441	−0.342221	−	0.939620i	$-0.611179\pi$
$733$	−18.5305	−0.684441	−0.342221	+	0.939620i	$0.611179\pi$
$734$	0	0
$735$	16.1690	0.596402
$736$	0	0
$737$	0	0
$738$	0	0
$739$	9.44624	0.347486	0.173743	−	0.984791i	$-0.444414\pi$
$739$	9.44624	0.347486	0.173743	+	0.984791i	$0.444414\pi$
$740$	0	0
$741$	2.81342	0.103353
$742$	0	0
$743$	−41.9768	−1.53998	−0.769990	−	0.638056i	$-0.779739\pi$
$743$	−41.9768	−1.53998	−0.769990	+	0.638056i	$0.779739\pi$
$744$	0	0
$745$	6.53055	0.239261
$746$	0	0
$747$	−9.16899	−0.335476
$748$	0	0
$749$	77.0147	2.81406
$750$	0	0
$751$	−26.4578	−0.965461	−0.482730	−	0.875769i	$-0.660355\pi$
$751$	−26.4578	−0.965461	−0.482730	+	0.875769i	$0.660355\pi$
$752$	0	0
$753$	−15.6092	−0.568832
$754$	0	0
$755$	−18.9884	−0.691058
$756$	0	0
$757$	12.9037	0.468993	0.234497	−	0.972117i	$-0.424656\pi$
$757$	12.9037	0.468993	0.234497	+	0.972117i	$0.424656\pi$
$758$	0	0
$759$	−9.09629	−0.330174
$760$	0	0
$761$	−40.7231	−1.47621	−0.738106	−	0.674685i	$-0.764280\pi$
$761$	−40.7231	−1.47621	−0.738106	+	0.674685i	$0.764280\pi$
$762$	0	0
$763$	14.9037	0.539551
$764$	0	0
$765$	6.26527	0.226521
$766$	0	0
$767$	8.16936	0.294979
$768$	0	0
$769$	−5.25367	−0.189452	−0.0947261	−	0.995503i	$-0.530198\pi$
$769$	−5.25367	−0.189452	−0.0947261	+	0.995503i	$0.530198\pi$
$770$	0	0
$771$	7.16899	0.258185
$772$	0	0
$773$	−38.9884	−1.40232	−0.701158	−	0.713006i	$-0.747333\pi$
$773$	−38.9884	−1.40232	−0.701158	+	0.713006i	$0.747333\pi$
$774$	0	0
$775$	−8.26527	−0.296897
$776$	0	0
$777$	32.7958	1.17654
$778$	0	0
$779$	−4.54814	−0.162954
$780$	0	0
$781$	−10.0232	−0.358659
$782$	0	0
$783$	0.548143	0.0195890
$784$	0	0
$785$	16.1574	0.576682
$786$	0	0
$787$	11.2889	0.402404	0.201202	−	0.979550i	$-0.435515\pi$
$787$	11.2889	0.402404	0.201202	+	0.979550i	$0.435515\pi$
$788$	0	0
$789$	24.4227	0.869470
$790$	0	0
$791$	64.3148	2.28677
$792$	0	0
$793$	−31.9648	−1.13510
$794$	0	0
$795$	−0.265275	−0.00940833
$796$	0	0
$797$	−27.6995	−0.981168	−0.490584	−	0.871394i	$-0.663217\pi$
$797$	−27.6995	−0.981168	−0.490584	+	0.871394i	$0.663217\pi$
$798$	0	0
$799$	64.3148	2.27529
$800$	0	0
$801$	−18.7055	−0.660927
$802$	0	0
$803$	−9.48143	−0.334592
$804$	0	0
$805$	−30.1574	−1.06291
$806$	0	0
$807$	−10.1750	−0.358176
$808$	0	0
$809$	−0.723121	−0.0254236	−0.0127118	−	0.999919i	$-0.504046\pi$
$809$	−0.723121	−0.0254236	−0.0127118	+	0.999919i	$0.504046\pi$
$810$	0	0
$811$	42.5073	1.49263	0.746317	−	0.665590i	$-0.231820\pi$
$811$	42.5073	1.49263	0.746317	+	0.665590i	$0.231820\pi$
$812$	0	0
$813$	−2.37316	−0.0832305
$814$	0	0
$815$	−10.4403	−0.365706
$816$	0	0
$817$	−7.71713	−0.269988
$818$	0	0
$819$	−13.5422	−0.473201
$820$	0	0
$821$	−4.90371	−0.171141	−0.0855704	−	0.996332i	$-0.527271\pi$
$821$	−4.90371	−0.171141	−0.0855704	+	0.996332i	$0.527271\pi$
$822$	0	0
$823$	1.37915	0.0480743	0.0240371	−	0.999711i	$-0.492348\pi$
$823$	1.37915	0.0480743	0.0240371	+	0.999711i	$0.492348\pi$
$824$	0	0
$825$	1.45186	0.0505472
$826$	0	0
$827$	16.9764	0.590328	0.295164	−	0.955447i	$-0.404626\pi$
$827$	16.9764	0.590328	0.295164	+	0.955447i	$0.404626\pi$
$828$	0	0
$829$	−13.4343	−0.466591	−0.233296	−	0.972406i	$-0.574951\pi$
$829$	−13.4343	−0.466591	−0.233296	+	0.972406i	$0.574951\pi$
$830$	0	0
$831$	−16.3380	−0.566758
$832$	0	0
$833$	101.303	3.50995
$834$	0	0
$835$	13.8921	0.480756
$836$	0	0
$837$	−8.26527	−0.285690
$838$	0	0
$839$	43.2185	1.49207	0.746034	−	0.665908i	$-0.231956\pi$
$839$	43.2185	1.49207	0.746034	+	0.665908i	$0.231956\pi$
$840$	0	0
$841$	−28.6995	−0.989639
$842$	0	0
$843$	5.64443	0.194404
$844$	0	0
$845$	−5.08468	−0.174918
$846$	0	0
$847$	−42.8014	−1.47067
$848$	0	0
$849$	21.3440	0.732523
$850$	0	0
$851$	−42.6879	−1.46332
$852$	0	0
$853$	−19.0963	−0.653844	−0.326922	−	0.945051i	$-0.606012\pi$
$853$	−19.0963	−0.653844	−0.326922	+	0.945051i	$0.606012\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	−25.5422	−0.872503	−0.436252	−	0.899825i	$-0.643694\pi$
$857$	−25.5422	−0.872503	−0.436252	+	0.899825i	$0.643694\pi$
$858$	0	0
$859$	−51.3991	−1.75371	−0.876857	−	0.480751i	$-0.840364\pi$
$859$	−51.3991	−1.75371	−0.876857	+	0.480751i	$0.840364\pi$
$860$	0	0
$861$	21.8921	0.746081
$862$	0	0
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	0	0
$865$	−2.63844	−0.0897096
$866$	0	0
$867$	22.2537	0.755774
$868$	0	0
$869$	−15.5684	−0.528123
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−6.81342	−0.230599
$874$	0	0
$875$	4.81342	0.162723
$876$	0	0
$877$	−49.6819	−1.67764	−0.838820	−	0.544409i	$-0.816754\pi$
$877$	−49.6819	−1.67764	−0.838820	+	0.544409i	$0.816754\pi$
$878$	0	0
$879$	10.6384	0.358826
$880$	0	0
$881$	−21.2889	−0.717240	−0.358620	−	0.933484i	$-0.616753\pi$
$881$	−21.2889	−0.717240	−0.358620	+	0.933484i	$0.616753\pi$
$882$	0	0
$883$	30.6208	1.03047	0.515237	−	0.857048i	$-0.327704\pi$
$883$	30.6208	1.03047	0.515237	+	0.857048i	$0.327704\pi$
$884$	0	0
$885$	−2.90371	−0.0976073
$886$	0	0
$887$	33.0611	1.11008	0.555042	−	0.831823i	$-0.312702\pi$
$887$	33.0611	1.11008	0.555042	+	0.831823i	$0.312702\pi$
$888$	0	0
$889$	−19.2537	−0.645747
$890$	0	0
$891$	1.45186	0.0486391
$892$	0	0
$893$	−10.2653	−0.343514
$894$	0	0
$895$	2.37316	0.0793261
$896$	0	0
$897$	17.6268	0.588543
$898$	0	0
$899$	−4.53055	−0.151102
$900$	0	0
$901$	−1.66202	−0.0553699
$902$	0	0
$903$	37.1458	1.23613
$904$	0	0
$905$	0.373165	0.0124044
$906$	0	0
$907$	−51.7490	−1.71830	−0.859149	−	0.511725i	$-0.829007\pi$
$907$	−51.7490	−1.71830	−0.859149	+	0.511725i	$0.829007\pi$
$908$	0	0
$909$	17.4343	0.578258
$910$	0	0
$911$	−3.78422	−0.125377	−0.0626884	−	0.998033i	$-0.519967\pi$
$911$	−3.78422	−0.125377	−0.0626884	+	0.998033i	$0.519967\pi$
$912$	0	0
$913$	−13.3121	−0.440565
$914$	0	0
$915$	11.3616	0.375602
$916$	0	0
$917$	20.9652	0.692331
$918$	0	0
$919$	2.72312	0.0898275	0.0449137	−	0.998991i	$-0.485699\pi$
$919$	2.72312	0.0898275	0.0449137	+	0.998991i	$0.485699\pi$
$920$	0	0
$921$	−22.1574	−0.730111
$922$	0	0
$923$	19.4230	0.639317
$924$	0	0
$925$	6.81342	0.224024
$926$	0	0
$927$	−16.5305	−0.542934
$928$	0	0
$929$	19.8426	0.651015	0.325508	−	0.945539i	$-0.394465\pi$
$929$	19.8426	0.651015	0.325508	+	0.945539i	$0.394465\pi$
$930$	0	0
$931$	−16.1690	−0.529917
$932$	0	0
$933$	−4.17498	−0.136683
$934$	0	0
$935$	9.09629	0.297480
$936$	0	0
$937$	−1.28886	−0.0421051	−0.0210525	−	0.999778i	$-0.506702\pi$
$937$	−1.28886	−0.0421051	−0.0210525	+	0.999778i	$0.506702\pi$
$938$	0	0
$939$	−12.3732	−0.403783
$940$	0	0
$941$	−21.7898	−0.710328	−0.355164	−	0.934804i	$-0.615575\pi$
$941$	−21.7898	−0.710328	−0.355164	+	0.934804i	$0.615575\pi$
$942$	0	0
$943$	−28.4954	−0.927937
$944$	0	0
$945$	4.81342	0.156580
$946$	0	0
$947$	−12.6384	−0.410694	−0.205347	−	0.978689i	$-0.565832\pi$
$947$	−12.6384	−0.410694	−0.205347	+	0.978689i	$0.565832\pi$
$948$	0	0
$949$	18.3732	0.596418
$950$	0	0
$951$	−3.73473	−0.121107
$952$	0	0
$953$	29.6763	0.961311	0.480655	−	0.876910i	$-0.340399\pi$
$953$	29.6763	0.961311	0.480655	+	0.876910i	$0.340399\pi$
$954$	0	0
$955$	−4.17498	−0.135099
$956$	0	0
$957$	0.795825	0.0257254
$958$	0	0
$959$	−28.8805	−0.932600
$960$	0	0
$961$	37.3148	1.20370
$962$	0	0
$963$	16.0000	0.515593
$964$	0	0
$965$	15.3440	0.493940
$966$	0	0
$967$	−10.2597	−0.329928	−0.164964	−	0.986300i	$-0.552751\pi$
$967$	−10.2597	−0.329928	−0.164964	+	0.986300i	$0.552751\pi$
$968$	0	0
$969$	−6.26527	−0.201270
$970$	0	0
$971$	49.5916	1.59147	0.795736	−	0.605644i	$-0.207084\pi$
$971$	49.5916	1.59147	0.795736	+	0.605644i	$0.207084\pi$
$972$	0	0
$973$	13.9768	0.448075
$974$	0	0
$975$	−2.81342	−0.0901015
$976$	0	0
$977$	−9.71152	−0.310699	−0.155349	−	0.987860i	$-0.549650\pi$
$977$	−9.71152	−0.310699	−0.155349	+	0.987860i	$0.549650\pi$
$978$	0	0
$979$	−27.1578	−0.867966
$980$	0	0
$981$	3.09629	0.0988568
$982$	0	0
$983$	−4.83101	−0.154085	−0.0770427	−	0.997028i	$-0.524548\pi$
$983$	−4.83101	−0.154085	−0.0770427	+	0.997028i	$0.524548\pi$
$984$	0	0
$985$	−22.2653	−0.709431
$986$	0	0
$987$	49.4111	1.57277
$988$	0	0
$989$	−48.3500	−1.53744
$990$	0	0
$991$	10.1926	0.323778	0.161889	−	0.986809i	$-0.448241\pi$
$991$	10.1926	0.323778	0.161889	+	0.986809i	$0.448241\pi$
$992$	0	0
$993$	−14.9884	−0.475642
$994$	0	0
$995$	−6.15739	−0.195202
$996$	0	0
$997$	−17.9648	−0.568951	−0.284476	−	0.958683i	$-0.591820\pi$
$997$	−17.9648	−0.568951	−0.284476	+	0.958683i	$0.591820\pi$
$998$	0	0
$999$	6.81342	0.215567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.bu.1.3		3
4.3	odd	2		1140.2.a.f.1.1	✓	3
12.11	even	2		3420.2.a.k.1.1		3
20.3	even	4		5700.2.f.p.3649.3		6
20.7	even	4		5700.2.f.p.3649.4		6
20.19	odd	2		5700.2.a.z.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.f.1.1	✓	3	4.3	odd	2
3420.2.a.k.1.1		3	12.11	even	2
4560.2.a.bu.1.3		3	1.1	even	1	trivial
5700.2.a.z.1.3		3	20.19	odd	2
5700.2.f.p.3649.3		6	20.3	even	4
5700.2.f.p.3649.4		6	20.7	even	4

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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +4.81342 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +4.81342 q^{7} +1.00000 q^{9} +1.45186 q^{11} -2.81342 q^{13} +1.00000 q^{15} +6.26527 q^{17} -1.00000 q^{19} +4.81342 q^{21} -6.26527 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.548143 q^{29} -8.26527 q^{31} +1.45186 q^{33} +4.81342 q^{35} +6.81342 q^{37} -2.81342 q^{39} +4.54814 q^{41} +7.71713 q^{43} +1.00000 q^{45} +10.2653 q^{47} +16.1690 q^{49} +6.26527 q^{51} -0.265275 q^{53} +1.45186 q^{55} -1.00000 q^{57} -2.90371 q^{59} +11.3616 q^{61} +4.81342 q^{63} -2.81342 q^{65} -6.26527 q^{69} -6.90371 q^{71} -6.53055 q^{73} +1.00000 q^{75} +6.98840 q^{77} -10.7231 q^{79} +1.00000 q^{81} -9.16899 q^{83} +6.26527 q^{85} +0.548143 q^{87} -18.7055 q^{89} -13.5422 q^{91} -8.26527 q^{93} -1.00000 q^{95} -6.81342 q^{97} +1.45186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + 2 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} - 3 q^{19} - 2 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 8 q^{31} + 2 q^{33} + 6 q^{37} + 6 q^{39} + 16 q^{41} + 4 q^{43} + 3 q^{45} + 14 q^{47} + 27 q^{49} + 2 q^{51} + 16 q^{53} + 2 q^{55} - 3 q^{57} - 4 q^{59} + 22 q^{61} + 6 q^{65} - 2 q^{69} - 16 q^{71} + 14 q^{73} + 3 q^{75} - 20 q^{77} - 8 q^{79} + 3 q^{81} - 6 q^{83} + 2 q^{85} + 4 q^{87} + 4 q^{89} - 48 q^{91} - 8 q^{93} - 3 q^{95} - 6 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9 + 2 * q^11 + 6 * q^13 + 3 * q^15 + 2 * q^17 - 3 * q^19 - 2 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 8 * q^31 + 2 * q^33 + 6 * q^37 + 6 * q^39 + 16 * q^41 + 4 * q^43 + 3 * q^45 + 14 * q^47 + 27 * q^49 + 2 * q^51 + 16 * q^53 + 2 * q^55 - 3 * q^57 - 4 * q^59 + 22 * q^61 + 6 * q^65 - 2 * q^69 - 16 * q^71 + 14 * q^73 + 3 * q^75 - 20 * q^77 - 8 * q^79 + 3 * q^81 - 6 * q^83 + 2 * q^85 + 4 * q^87 + 4 * q^89 - 48 * q^91 - 8 * q^93 - 3 * q^95 - 6 * q^97 + 2 * q^99

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