LMFDB - Embedded newform 456.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [456,2,Mod(1,456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(456, 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("456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 456 = 2^{3} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	456.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:	$3.64117833217$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.70156 q^{5} -4.70156 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.70156 q^{5} -4.70156 q^{7} +1.00000 q^{9} +4.70156 q^{11} +6.00000 q^{13} -2.70156 q^{15} -2.70156 q^{17} +1.00000 q^{19} +4.70156 q^{21} +4.00000 q^{23} +2.29844 q^{25} -1.00000 q^{27} +2.00000 q^{29} +9.40312 q^{31} -4.70156 q^{33} -12.7016 q^{35} -3.40312 q^{37} -6.00000 q^{39} -3.40312 q^{41} +10.1047 q^{43} +2.70156 q^{45} -0.701562 q^{47} +15.1047 q^{49} +2.70156 q^{51} -6.00000 q^{53} +12.7016 q^{55} -1.00000 q^{57} -4.00000 q^{59} +1.29844 q^{61} -4.70156 q^{63} +16.2094 q^{65} -12.0000 q^{67} -4.00000 q^{69} +6.70156 q^{73} -2.29844 q^{75} -22.1047 q^{77} -10.8062 q^{79} +1.00000 q^{81} -10.8062 q^{83} -7.29844 q^{85} -2.00000 q^{87} -12.8062 q^{89} -28.2094 q^{91} -9.40312 q^{93} +2.70156 q^{95} -6.00000 q^{97} +4.70156 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9} + 3 q^{11} + 12 q^{13} + q^{15} + q^{17} + 2 q^{19} + 3 q^{21} + 8 q^{23} + 11 q^{25} - 2 q^{27} + 4 q^{29} + 6 q^{31} - 3 q^{33} - 19 q^{35} + 6 q^{37} - 12 q^{39} + 6 q^{41} + q^{43} - q^{45} + 5 q^{47} + 11 q^{49} - q^{51} - 12 q^{53} + 19 q^{55} - 2 q^{57} - 8 q^{59} + 9 q^{61} - 3 q^{63} - 6 q^{65} - 24 q^{67} - 8 q^{69} + 7 q^{73} - 11 q^{75} - 25 q^{77} + 4 q^{79} + 2 q^{81} + 4 q^{83} - 21 q^{85} - 4 q^{87} - 18 q^{91} - 6 q^{93} - q^{95} - 12 q^{97} + 3 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9 + 3 * q^11 + 12 * q^13 + q^15 + q^17 + 2 * q^19 + 3 * q^21 + 8 * q^23 + 11 * q^25 - 2 * q^27 + 4 * q^29 + 6 * q^31 - 3 * q^33 - 19 * q^35 + 6 * q^37 - 12 * q^39 + 6 * q^41 + q^43 - q^45 + 5 * q^47 + 11 * q^49 - q^51 - 12 * q^53 + 19 * q^55 - 2 * q^57 - 8 * q^59 + 9 * q^61 - 3 * q^63 - 6 * q^65 - 24 * q^67 - 8 * q^69 + 7 * q^73 - 11 * q^75 - 25 * q^77 + 4 * q^79 + 2 * q^81 + 4 * q^83 - 21 * q^85 - 4 * q^87 - 18 * q^91 - 6 * q^93 - q^95 - 12 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.70156	1.20818	0.604088	−	0.796918i	$-0.293538\pi$
$5$	2.70156	1.20818	0.604088	+	0.796918i	$0.293538\pi$
$6$	0	0
$7$	−4.70156	−1.77702	−0.888512	−	0.458854i	$-0.848260\pi$
$7$	−4.70156	−1.77702	−0.888512	+	0.458854i	$0.848260\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.70156	1.41757	0.708787	−	0.705422i	$-0.249243\pi$
$11$	4.70156	1.41757	0.708787	+	0.705422i	$0.249243\pi$
$12$	0	0
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	−2.70156	−0.697540
$16$	0	0
$17$	−2.70156	−0.655225	−0.327613	−	0.944812i	$-0.606244\pi$
$17$	−2.70156	−0.655225	−0.327613	+	0.944812i	$0.606244\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	4.70156	1.02596
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	2.29844	0.459688
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	9.40312	1.68885	0.844425	−	0.535673i	$-0.179942\pi$
$31$	9.40312	1.68885	0.844425	+	0.535673i	$0.179942\pi$
$32$	0	0
$33$	−4.70156	−0.818437
$34$	0	0
$35$	−12.7016	−2.14696
$36$	0	0
$37$	−3.40312	−0.559470	−0.279735	−	0.960077i	$-0.590247\pi$
$37$	−3.40312	−0.559470	−0.279735	+	0.960077i	$0.590247\pi$
$38$	0	0
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−3.40312	−0.531479	−0.265739	−	0.964045i	$-0.585616\pi$
$41$	−3.40312	−0.531479	−0.265739	+	0.964045i	$0.585616\pi$
$42$	0	0
$43$	10.1047	1.54095	0.770475	−	0.637470i	$-0.220019\pi$
$43$	10.1047	1.54095	0.770475	+	0.637470i	$0.220019\pi$
$44$	0	0
$45$	2.70156	0.402725
$46$	0	0
$47$	−0.701562	−0.102333	−0.0511667	−	0.998690i	$-0.516294\pi$
$47$	−0.701562	−0.102333	−0.0511667	+	0.998690i	$0.516294\pi$
$48$	0	0
$49$	15.1047	2.15781
$50$	0	0
$51$	2.70156	0.378294
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	12.7016	1.71268
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	1.29844	0.166248	0.0831240	−	0.996539i	$-0.473510\pi$
$61$	1.29844	0.166248	0.0831240	+	0.996539i	$0.473510\pi$
$62$	0	0
$63$	−4.70156	−0.592341
$64$	0	0
$65$	16.2094	2.01053
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	−4.00000	−0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	6.70156	0.784359	0.392179	−	0.919889i	$-0.371721\pi$
$73$	6.70156	0.784359	0.392179	+	0.919889i	$0.371721\pi$
$74$	0	0
$75$	−2.29844	−0.265401
$76$	0	0
$77$	−22.1047	−2.51906
$78$	0	0
$79$	−10.8062	−1.21580	−0.607899	−	0.794014i	$-0.707987\pi$
$79$	−10.8062	−1.21580	−0.607899	+	0.794014i	$0.707987\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.8062	−1.18614	−0.593070	−	0.805151i	$-0.702084\pi$
$83$	−10.8062	−1.18614	−0.593070	+	0.805151i	$0.702084\pi$
$84$	0	0
$85$	−7.29844	−0.791627
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−12.8062	−1.35746	−0.678730	−	0.734388i	$-0.737469\pi$
$89$	−12.8062	−1.35746	−0.678730	+	0.734388i	$0.737469\pi$
$90$	0	0
$91$	−28.2094	−2.95715
$92$	0	0
$93$	−9.40312	−0.975059
$94$	0	0
$95$	2.70156	0.277174
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	4.70156	0.472525
$100$	0	0
$101$	16.8062	1.67228	0.836142	−	0.548513i	$-0.184806\pi$
$101$	16.8062	1.67228	0.836142	+	0.548513i	$0.184806\pi$
$102$	0	0
$103$	−9.40312	−0.926517	−0.463259	−	0.886223i	$-0.653320\pi$
$103$	−9.40312	−0.926517	−0.463259	+	0.886223i	$0.653320\pi$
$104$	0	0
$105$	12.7016	1.23955
$106$	0	0
$107$	5.40312	0.522340	0.261170	−	0.965293i	$-0.415892\pi$
$107$	5.40312	0.522340	0.261170	+	0.965293i	$0.415892\pi$
$108$	0	0
$109$	−3.40312	−0.325960	−0.162980	−	0.986629i	$-0.552111\pi$
$109$	−3.40312	−0.325960	−0.162980	+	0.986629i	$0.552111\pi$
$110$	0	0
$111$	3.40312	0.323010
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	10.8062	1.00769
$116$	0	0
$117$	6.00000	0.554700
$118$	0	0
$119$	12.7016	1.16435
$120$	0	0
$121$	11.1047	1.00952
$122$	0	0
$123$	3.40312	0.306849
$124$	0	0
$125$	−7.29844	−0.652792
$126$	0	0
$127$	9.40312	0.834392	0.417196	−	0.908816i	$-0.363013\pi$
$127$	9.40312	0.834392	0.417196	+	0.908816i	$0.363013\pi$
$128$	0	0
$129$	−10.1047	−0.889668
$130$	0	0
$131$	−12.7016	−1.10974	−0.554870	−	0.831937i	$-0.687232\pi$
$131$	−12.7016	−1.10974	−0.554870	+	0.831937i	$0.687232\pi$
$132$	0	0
$133$	−4.70156	−0.407677
$134$	0	0
$135$	−2.70156	−0.232513
$136$	0	0
$137$	−10.7016	−0.914296	−0.457148	−	0.889391i	$-0.651129\pi$
$137$	−10.7016	−0.914296	−0.457148	+	0.889391i	$0.651129\pi$
$138$	0	0
$139$	15.2984	1.29760	0.648798	−	0.760960i	$-0.275272\pi$
$139$	15.2984	1.29760	0.648798	+	0.760960i	$0.275272\pi$
$140$	0	0
$141$	0.701562	0.0590822
$142$	0	0
$143$	28.2094	2.35899
$144$	0	0
$145$	5.40312	0.448705
$146$	0	0
$147$	−15.1047	−1.24581
$148$	0	0
$149$	20.1047	1.64704	0.823520	−	0.567287i	$-0.192007\pi$
$149$	20.1047	1.64704	0.823520	+	0.567287i	$0.192007\pi$
$150$	0	0
$151$	2.80625	0.228369	0.114185	−	0.993460i	$-0.463574\pi$
$151$	2.80625	0.228369	0.114185	+	0.993460i	$0.463574\pi$
$152$	0	0
$153$	−2.70156	−0.218408
$154$	0	0
$155$	25.4031	2.04043
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	−18.8062	−1.48214
$162$	0	0
$163$	−14.8062	−1.15971	−0.579857	−	0.814718i	$-0.696892\pi$
$163$	−14.8062	−1.15971	−0.579857	+	0.814718i	$0.696892\pi$
$164$	0	0
$165$	−12.7016	−0.988815
$166$	0	0
$167$	20.2094	1.56385	0.781924	−	0.623374i	$-0.214238\pi$
$167$	20.2094	1.56385	0.781924	+	0.623374i	$0.214238\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−8.80625	−0.669527	−0.334763	−	0.942302i	$-0.608656\pi$
$173$	−8.80625	−0.669527	−0.334763	+	0.942302i	$0.608656\pi$
$174$	0	0
$175$	−10.8062	−0.816876
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	−5.40312	−0.403848	−0.201924	−	0.979401i	$-0.564719\pi$
$179$	−5.40312	−0.403848	−0.201924	+	0.979401i	$0.564719\pi$
$180$	0	0
$181$	−4.80625	−0.357246	−0.178623	−	0.983918i	$-0.557164\pi$
$181$	−4.80625	−0.357246	−0.178623	+	0.983918i	$0.557164\pi$
$182$	0	0
$183$	−1.29844	−0.0959833
$184$	0	0
$185$	−9.19375	−0.675938
$186$	0	0
$187$	−12.7016	−0.928830
$188$	0	0
$189$	4.70156	0.341988
$190$	0	0
$191$	−20.9109	−1.51306	−0.756531	−	0.653958i	$-0.773107\pi$
$191$	−20.9109	−1.51306	−0.756531	+	0.653958i	$0.773107\pi$
$192$	0	0
$193$	11.4031	0.820815	0.410407	−	0.911902i	$-0.365386\pi$
$193$	11.4031	0.820815	0.410407	+	0.911902i	$0.365386\pi$
$194$	0	0
$195$	−16.2094	−1.16078
$196$	0	0
$197$	22.0000	1.56744	0.783718	−	0.621117i	$-0.213321\pi$
$197$	22.0000	1.56744	0.783718	+	0.621117i	$0.213321\pi$
$198$	0	0
$199$	3.29844	0.233820	0.116910	−	0.993143i	$-0.462701\pi$
$199$	3.29844	0.233820	0.116910	+	0.993143i	$0.462701\pi$
$200$	0	0
$201$	12.0000	0.846415
$202$	0	0
$203$	−9.40312	−0.659970
$204$	0	0
$205$	−9.19375	−0.642119
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	4.70156	0.325214
$210$	0	0
$211$	6.80625	0.468561	0.234281	−	0.972169i	$-0.424727\pi$
$211$	6.80625	0.468561	0.234281	+	0.972169i	$0.424727\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	27.2984	1.86174
$216$	0	0
$217$	−44.2094	−3.00113
$218$	0	0
$219$	−6.70156	−0.452850
$220$	0	0
$221$	−16.2094	−1.09036
$222$	0	0
$223$	26.8062	1.79508	0.897540	−	0.440934i	$-0.145353\pi$
$223$	26.8062	1.79508	0.897540	+	0.440934i	$0.145353\pi$
$224$	0	0
$225$	2.29844	0.153229
$226$	0	0
$227$	−10.5969	−0.703339	−0.351670	−	0.936124i	$-0.614386\pi$
$227$	−10.5969	−0.703339	−0.351670	+	0.936124i	$0.614386\pi$
$228$	0	0
$229$	5.50781	0.363966	0.181983	−	0.983302i	$-0.441748\pi$
$229$	5.50781	0.363966	0.181983	+	0.983302i	$0.441748\pi$
$230$	0	0
$231$	22.1047	1.45438
$232$	0	0
$233$	−12.1047	−0.793004	−0.396502	−	0.918034i	$-0.629776\pi$
$233$	−12.1047	−0.793004	−0.396502	+	0.918034i	$0.629776\pi$
$234$	0	0
$235$	−1.89531	−0.123637
$236$	0	0
$237$	10.8062	0.701941
$238$	0	0
$239$	−2.10469	−0.136141	−0.0680704	−	0.997681i	$-0.521684\pi$
$239$	−2.10469	−0.136141	−0.0680704	+	0.997681i	$0.521684\pi$
$240$	0	0
$241$	−15.4031	−0.992202	−0.496101	−	0.868265i	$-0.665236\pi$
$241$	−15.4031	−0.992202	−0.496101	+	0.868265i	$0.665236\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	40.8062	2.60702
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	10.8062	0.684818
$250$	0	0
$251$	19.2984	1.21811	0.609053	−	0.793129i	$-0.291550\pi$
$251$	19.2984	1.21811	0.609053	+	0.793129i	$0.291550\pi$
$252$	0	0
$253$	18.8062	1.18234
$254$	0	0
$255$	7.29844	0.457046
$256$	0	0
$257$	−19.4031	−1.21033	−0.605167	−	0.796099i	$-0.706894\pi$
$257$	−19.4031	−1.21033	−0.605167	+	0.796099i	$0.706894\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	−0.701562	−0.0432602	−0.0216301	−	0.999766i	$-0.506886\pi$
$263$	−0.701562	−0.0432602	−0.0216301	+	0.999766i	$0.506886\pi$
$264$	0	0
$265$	−16.2094	−0.995734
$266$	0	0
$267$	12.8062	0.783730
$268$	0	0
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	10.8062	0.656433	0.328216	−	0.944603i	$-0.393552\pi$
$271$	10.8062	0.656433	0.328216	+	0.944603i	$0.393552\pi$
$272$	0	0
$273$	28.2094	1.70731
$274$	0	0
$275$	10.8062	0.651641
$276$	0	0
$277$	2.70156	0.162321	0.0811606	−	0.996701i	$-0.474137\pi$
$277$	2.70156	0.162321	0.0811606	+	0.996701i	$0.474137\pi$
$278$	0	0
$279$	9.40312	0.562950
$280$	0	0
$281$	0.806248	0.0480968	0.0240484	−	0.999711i	$-0.492344\pi$
$281$	0.806248	0.0480968	0.0240484	+	0.999711i	$0.492344\pi$
$282$	0	0
$283$	−8.70156	−0.517254	−0.258627	−	0.965977i	$-0.583270\pi$
$283$	−8.70156	−0.517254	−0.258627	+	0.965977i	$0.583270\pi$
$284$	0	0
$285$	−2.70156	−0.160027
$286$	0	0
$287$	16.0000	0.944450
$288$	0	0
$289$	−9.70156	−0.570680
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	−23.4031	−1.36723	−0.683613	−	0.729845i	$-0.739592\pi$
$293$	−23.4031	−1.36723	−0.683613	+	0.729845i	$0.739592\pi$
$294$	0	0
$295$	−10.8062	−0.629164
$296$	0	0
$297$	−4.70156	−0.272812
$298$	0	0
$299$	24.0000	1.38796
$300$	0	0
$301$	−47.5078	−2.73830
$302$	0	0
$303$	−16.8062	−0.965494
$304$	0	0
$305$	3.50781	0.200857
$306$	0	0
$307$	−1.19375	−0.0681310	−0.0340655	−	0.999420i	$-0.510845\pi$
$307$	−1.19375	−0.0681310	−0.0340655	+	0.999420i	$0.510845\pi$
$308$	0	0
$309$	9.40312	0.534925
$310$	0	0
$311$	−27.5078	−1.55982	−0.779912	−	0.625889i	$-0.784736\pi$
$311$	−27.5078	−1.55982	−0.779912	+	0.625889i	$0.784736\pi$
$312$	0	0
$313$	28.8062	1.62823	0.814113	−	0.580707i	$-0.197224\pi$
$313$	28.8062	1.62823	0.814113	+	0.580707i	$0.197224\pi$
$314$	0	0
$315$	−12.7016	−0.715652
$316$	0	0
$317$	3.40312	0.191138	0.0955692	−	0.995423i	$-0.469533\pi$
$317$	3.40312	0.191138	0.0955692	+	0.995423i	$0.469533\pi$
$318$	0	0
$319$	9.40312	0.526474
$320$	0	0
$321$	−5.40312	−0.301573
$322$	0	0
$323$	−2.70156	−0.150319
$324$	0	0
$325$	13.7906	0.764966
$326$	0	0
$327$	3.40312	0.188193
$328$	0	0
$329$	3.29844	0.181849
$330$	0	0
$331$	−30.8062	−1.69326	−0.846632	−	0.532178i	$-0.821374\pi$
$331$	−30.8062	−1.69326	−0.846632	+	0.532178i	$0.821374\pi$
$332$	0	0
$333$	−3.40312	−0.186490
$334$	0	0
$335$	−32.4187	−1.77123
$336$	0	0
$337$	12.8062	0.697601	0.348800	−	0.937197i	$-0.386589\pi$
$337$	12.8062	0.697601	0.348800	+	0.937197i	$0.386589\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	44.2094	2.39407
$342$	0	0
$343$	−38.1047	−2.05746
$344$	0	0
$345$	−10.8062	−0.581789
$346$	0	0
$347$	−20.7016	−1.11132	−0.555659	−	0.831410i	$-0.687534\pi$
$347$	−20.7016	−1.11132	−0.555659	+	0.831410i	$0.687534\pi$
$348$	0	0
$349$	−24.1047	−1.29029	−0.645147	−	0.764058i	$-0.723204\pi$
$349$	−24.1047	−1.29029	−0.645147	+	0.764058i	$0.723204\pi$
$350$	0	0
$351$	−6.00000	−0.320256
$352$	0	0
$353$	7.19375	0.382885	0.191442	−	0.981504i	$-0.438683\pi$
$353$	7.19375	0.382885	0.191442	+	0.981504i	$0.438683\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−12.7016	−0.672238
$358$	0	0
$359$	−4.49219	−0.237089	−0.118544	−	0.992949i	$-0.537823\pi$
$359$	−4.49219	−0.237089	−0.118544	+	0.992949i	$0.537823\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−11.1047	−0.582845
$364$	0	0
$365$	18.1047	0.947643
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	0	0
$369$	−3.40312	−0.177160
$370$	0	0
$371$	28.2094	1.46456
$372$	0	0
$373$	−3.40312	−0.176207	−0.0881035	−	0.996111i	$-0.528081\pi$
$373$	−3.40312	−0.176207	−0.0881035	+	0.996111i	$0.528081\pi$
$374$	0	0
$375$	7.29844	0.376890
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	−21.4031	−1.09940	−0.549702	−	0.835361i	$-0.685259\pi$
$379$	−21.4031	−1.09940	−0.549702	+	0.835361i	$0.685259\pi$
$380$	0	0
$381$	−9.40312	−0.481737
$382$	0	0
$383$	20.2094	1.03265	0.516325	−	0.856393i	$-0.327300\pi$
$383$	20.2094	1.03265	0.516325	+	0.856393i	$0.327300\pi$
$384$	0	0
$385$	−59.7172	−3.04347
$386$	0	0
$387$	10.1047	0.513650
$388$	0	0
$389$	−34.9109	−1.77005	−0.885027	−	0.465539i	$-0.845860\pi$
$389$	−34.9109	−1.77005	−0.885027	+	0.465539i	$0.845860\pi$
$390$	0	0
$391$	−10.8062	−0.546495
$392$	0	0
$393$	12.7016	0.640709
$394$	0	0
$395$	−29.1938	−1.46890
$396$	0	0
$397$	29.5078	1.48095	0.740477	−	0.672081i	$-0.234600\pi$
$397$	29.5078	1.48095	0.740477	+	0.672081i	$0.234600\pi$
$398$	0	0
$399$	4.70156	0.235373
$400$	0	0
$401$	10.2094	0.509832	0.254916	−	0.966963i	$-0.417952\pi$
$401$	10.2094	0.509832	0.254916	+	0.966963i	$0.417952\pi$
$402$	0	0
$403$	56.4187	2.81042
$404$	0	0
$405$	2.70156	0.134242
$406$	0	0
$407$	−16.0000	−0.793091
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	10.7016	0.527869
$412$	0	0
$413$	18.8062	0.925395
$414$	0	0
$415$	−29.1938	−1.43306
$416$	0	0
$417$	−15.2984	−0.749168
$418$	0	0
$419$	18.8062	0.918745	0.459373	−	0.888244i	$-0.348074\pi$
$419$	18.8062	0.918745	0.459373	+	0.888244i	$0.348074\pi$
$420$	0	0
$421$	0.806248	0.0392941	0.0196471	−	0.999807i	$-0.493746\pi$
$421$	0.806248	0.0392941	0.0196471	+	0.999807i	$0.493746\pi$
$422$	0	0
$423$	−0.701562	−0.0341111
$424$	0	0
$425$	−6.20937	−0.301199
$426$	0	0
$427$	−6.10469	−0.295426
$428$	0	0
$429$	−28.2094	−1.36196
$430$	0	0
$431$	−14.5969	−0.703107	−0.351553	−	0.936168i	$-0.614346\pi$
$431$	−14.5969	−0.703107	−0.351553	+	0.936168i	$0.614346\pi$
$432$	0	0
$433$	23.6125	1.13474	0.567372	−	0.823462i	$-0.307960\pi$
$433$	23.6125	1.13474	0.567372	+	0.823462i	$0.307960\pi$
$434$	0	0
$435$	−5.40312	−0.259060
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	6.59688	0.314852	0.157426	−	0.987531i	$-0.449680\pi$
$439$	6.59688	0.314852	0.157426	+	0.987531i	$0.449680\pi$
$440$	0	0
$441$	15.1047	0.719271
$442$	0	0
$443$	−7.50781	−0.356707	−0.178353	−	0.983966i	$-0.557077\pi$
$443$	−7.50781	−0.356707	−0.178353	+	0.983966i	$0.557077\pi$
$444$	0	0
$445$	−34.5969	−1.64005
$446$	0	0
$447$	−20.1047	−0.950919
$448$	0	0
$449$	−0.596876	−0.0281683	−0.0140842	−	0.999901i	$-0.504483\pi$
$449$	−0.596876	−0.0281683	−0.0140842	+	0.999901i	$0.504483\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	0	0
$453$	−2.80625	−0.131849
$454$	0	0
$455$	−76.2094	−3.57275
$456$	0	0
$457$	−15.8953	−0.743551	−0.371776	−	0.928323i	$-0.621251\pi$
$457$	−15.8953	−0.743551	−0.371776	+	0.928323i	$0.621251\pi$
$458$	0	0
$459$	2.70156	0.126098
$460$	0	0
$461$	28.1047	1.30897	0.654483	−	0.756077i	$-0.272886\pi$
$461$	28.1047	1.30897	0.654483	+	0.756077i	$0.272886\pi$
$462$	0	0
$463$	7.50781	0.348918	0.174459	−	0.984664i	$-0.444182\pi$
$463$	7.50781	0.348918	0.174459	+	0.984664i	$0.444182\pi$
$464$	0	0
$465$	−25.4031	−1.17804
$466$	0	0
$467$	−17.8953	−0.828096	−0.414048	−	0.910255i	$-0.635886\pi$
$467$	−17.8953	−0.828096	−0.414048	+	0.910255i	$0.635886\pi$
$468$	0	0
$469$	56.4187	2.60518
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	47.5078	2.18441
$474$	0	0
$475$	2.29844	0.105460
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	22.8062	1.04204	0.521022	−	0.853543i	$-0.325551\pi$
$479$	22.8062	1.04204	0.521022	+	0.853543i	$0.325551\pi$
$480$	0	0
$481$	−20.4187	−0.931015
$482$	0	0
$483$	18.8062	0.855714
$484$	0	0
$485$	−16.2094	−0.736030
$486$	0	0
$487$	5.19375	0.235351	0.117676	−	0.993052i	$-0.462456\pi$
$487$	5.19375	0.235351	0.117676	+	0.993052i	$0.462456\pi$
$488$	0	0
$489$	14.8062	0.669562
$490$	0	0
$491$	−8.00000	−0.361035	−0.180517	−	0.983572i	$-0.557777\pi$
$491$	−8.00000	−0.361035	−0.180517	+	0.983572i	$0.557777\pi$
$492$	0	0
$493$	−5.40312	−0.243344
$494$	0	0
$495$	12.7016	0.570893
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−20.9109	−0.936102	−0.468051	−	0.883701i	$-0.655044\pi$
$499$	−20.9109	−0.936102	−0.468051	+	0.883701i	$0.655044\pi$
$500$	0	0
$501$	−20.2094	−0.902888
$502$	0	0
$503$	−38.8062	−1.73029	−0.865143	−	0.501526i	$-0.832772\pi$
$503$	−38.8062	−1.73029	−0.865143	+	0.501526i	$0.832772\pi$
$504$	0	0
$505$	45.4031	2.02041
$506$	0	0
$507$	−23.0000	−1.02147
$508$	0	0
$509$	−43.6125	−1.93309	−0.966545	−	0.256497i	$-0.917432\pi$
$509$	−43.6125	−1.93309	−0.966545	+	0.256497i	$0.917432\pi$
$510$	0	0
$511$	−31.5078	−1.39382
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−25.4031	−1.11940
$516$	0	0
$517$	−3.29844	−0.145065
$518$	0	0
$519$	8.80625	0.386551
$520$	0	0
$521$	12.5969	0.551879	0.275940	−	0.961175i	$-0.411011\pi$
$521$	12.5969	0.551879	0.275940	+	0.961175i	$0.411011\pi$
$522$	0	0
$523$	−16.2094	−0.708786	−0.354393	−	0.935097i	$-0.615313\pi$
$523$	−16.2094	−0.708786	−0.354393	+	0.935097i	$0.615313\pi$
$524$	0	0
$525$	10.8062	0.471623
$526$	0	0
$527$	−25.4031	−1.10658
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−20.4187	−0.884434
$534$	0	0
$535$	14.5969	0.631078
$536$	0	0
$537$	5.40312	0.233162
$538$	0	0
$539$	71.0156	3.05886
$540$	0	0
$541$	25.7172	1.10567	0.552834	−	0.833291i	$-0.313546\pi$
$541$	25.7172	1.10567	0.552834	+	0.833291i	$0.313546\pi$
$542$	0	0
$543$	4.80625	0.206256
$544$	0	0
$545$	−9.19375	−0.393817
$546$	0	0
$547$	−19.0156	−0.813049	−0.406525	−	0.913640i	$-0.633259\pi$
$547$	−19.0156	−0.813049	−0.406525	+	0.913640i	$0.633259\pi$
$548$	0	0
$549$	1.29844	0.0554160
$550$	0	0
$551$	2.00000	0.0852029
$552$	0	0
$553$	50.8062	2.16050
$554$	0	0
$555$	9.19375	0.390253
$556$	0	0
$557$	−38.7016	−1.63984	−0.819919	−	0.572480i	$-0.805982\pi$
$557$	−38.7016	−1.63984	−0.819919	+	0.572480i	$0.805982\pi$
$558$	0	0
$559$	60.6281	2.56430
$560$	0	0
$561$	12.7016	0.536260
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	−48.6281	−2.04580
$566$	0	0
$567$	−4.70156	−0.197447
$568$	0	0
$569$	0.806248	0.0337997	0.0168998	−	0.999857i	$-0.494620\pi$
$569$	0.806248	0.0337997	0.0168998	+	0.999857i	$0.494620\pi$
$570$	0	0
$571$	1.19375	0.0499569	0.0249785	−	0.999688i	$-0.492048\pi$
$571$	1.19375	0.0499569	0.0249785	+	0.999688i	$0.492048\pi$
$572$	0	0
$573$	20.9109	0.873567
$574$	0	0
$575$	9.19375	0.383406
$576$	0	0
$577$	2.49219	0.103751	0.0518756	−	0.998654i	$-0.483480\pi$
$577$	2.49219	0.103751	0.0518756	+	0.998654i	$0.483480\pi$
$578$	0	0
$579$	−11.4031	−0.473898
$580$	0	0
$581$	50.8062	2.10780
$582$	0	0
$583$	−28.2094	−1.16831
$584$	0	0
$585$	16.2094	0.670175
$586$	0	0
$587$	3.29844	0.136141	0.0680706	−	0.997681i	$-0.478316\pi$
$587$	3.29844	0.136141	0.0680706	+	0.997681i	$0.478316\pi$
$588$	0	0
$589$	9.40312	0.387449
$590$	0	0
$591$	−22.0000	−0.904959
$592$	0	0
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	0	0
$595$	34.3141	1.40674
$596$	0	0
$597$	−3.29844	−0.134996
$598$	0	0
$599$	−10.8062	−0.441531	−0.220766	−	0.975327i	$-0.570856\pi$
$599$	−10.8062	−0.441531	−0.220766	+	0.975327i	$0.570856\pi$
$600$	0	0
$601$	22.2094	0.905939	0.452970	−	0.891526i	$-0.350365\pi$
$601$	22.2094	0.905939	0.452970	+	0.891526i	$0.350365\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	30.0000	1.21967
$606$	0	0
$607$	28.2094	1.14498	0.572492	−	0.819911i	$-0.305977\pi$
$607$	28.2094	1.14498	0.572492	+	0.819911i	$0.305977\pi$
$608$	0	0
$609$	9.40312	0.381034
$610$	0	0
$611$	−4.20937	−0.170293
$612$	0	0
$613$	−10.4922	−0.423776	−0.211888	−	0.977294i	$-0.567961\pi$
$613$	−10.4922	−0.423776	−0.211888	+	0.977294i	$0.567961\pi$
$614$	0	0
$615$	9.19375	0.370728
$616$	0	0
$617$	−29.5078	−1.18794	−0.593970	−	0.804487i	$-0.702440\pi$
$617$	−29.5078	−1.18794	−0.593970	+	0.804487i	$0.702440\pi$
$618$	0	0
$619$	12.0000	0.482321	0.241160	−	0.970485i	$-0.422472\pi$
$619$	12.0000	0.482321	0.241160	+	0.970485i	$0.422472\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	0	0
$623$	60.2094	2.41224
$624$	0	0
$625$	−31.2094	−1.24837
$626$	0	0
$627$	−4.70156	−0.187762
$628$	0	0
$629$	9.19375	0.366579
$630$	0	0
$631$	8.49219	0.338069	0.169034	−	0.985610i	$-0.445935\pi$
$631$	8.49219	0.338069	0.169034	+	0.985610i	$0.445935\pi$
$632$	0	0
$633$	−6.80625	−0.270524
$634$	0	0
$635$	25.4031	1.00809
$636$	0	0
$637$	90.6281	3.59082
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	−19.5078	−0.769313	−0.384656	−	0.923060i	$-0.625680\pi$
$643$	−19.5078	−0.769313	−0.384656	+	0.923060i	$0.625680\pi$
$644$	0	0
$645$	−27.2984	−1.07487
$646$	0	0
$647$	−31.2984	−1.23047	−0.615234	−	0.788344i	$-0.710939\pi$
$647$	−31.2984	−1.23047	−0.615234	+	0.788344i	$0.710939\pi$
$648$	0	0
$649$	−18.8062	−0.738210
$650$	0	0
$651$	44.2094	1.73270
$652$	0	0
$653$	28.1047	1.09982	0.549911	−	0.835223i	$-0.314662\pi$
$653$	28.1047	1.09982	0.549911	+	0.835223i	$0.314662\pi$
$654$	0	0
$655$	−34.3141	−1.34076
$656$	0	0
$657$	6.70156	0.261453
$658$	0	0
$659$	18.5969	0.724431	0.362216	−	0.932094i	$-0.382020\pi$
$659$	18.5969	0.724431	0.362216	+	0.932094i	$0.382020\pi$
$660$	0	0
$661$	−38.2094	−1.48617	−0.743086	−	0.669196i	$-0.766639\pi$
$661$	−38.2094	−1.48617	−0.743086	+	0.669196i	$0.766639\pi$
$662$	0	0
$663$	16.2094	0.629520
$664$	0	0
$665$	−12.7016	−0.492545
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	−26.8062	−1.03639
$670$	0	0
$671$	6.10469	0.235669
$672$	0	0
$673$	−7.40312	−0.285369	−0.142685	−	0.989768i	$-0.545574\pi$
$673$	−7.40312	−0.285369	−0.142685	+	0.989768i	$0.545574\pi$
$674$	0	0
$675$	−2.29844	−0.0884669
$676$	0	0
$677$	12.8062	0.492184	0.246092	−	0.969246i	$-0.420853\pi$
$677$	12.8062	0.492184	0.246092	+	0.969246i	$0.420853\pi$
$678$	0	0
$679$	28.2094	1.08258
$680$	0	0
$681$	10.5969	0.406073
$682$	0	0
$683$	−48.2094	−1.84468	−0.922340	−	0.386379i	$-0.873726\pi$
$683$	−48.2094	−1.84468	−0.922340	+	0.386379i	$0.873726\pi$
$684$	0	0
$685$	−28.9109	−1.10463
$686$	0	0
$687$	−5.50781	−0.210136
$688$	0	0
$689$	−36.0000	−1.37149
$690$	0	0
$691$	11.5078	0.437778	0.218889	−	0.975750i	$-0.429757\pi$
$691$	11.5078	0.437778	0.218889	+	0.975750i	$0.429757\pi$
$692$	0	0
$693$	−22.1047	−0.839688
$694$	0	0
$695$	41.3297	1.56772
$696$	0	0
$697$	9.19375	0.348238
$698$	0	0
$699$	12.1047	0.457841
$700$	0	0
$701$	3.19375	0.120626	0.0603132	−	0.998180i	$-0.480790\pi$
$701$	3.19375	0.120626	0.0603132	+	0.998180i	$0.480790\pi$
$702$	0	0
$703$	−3.40312	−0.128351
$704$	0	0
$705$	1.89531	0.0713816
$706$	0	0
$707$	−79.0156	−2.97169
$708$	0	0
$709$	27.6125	1.03701	0.518505	−	0.855075i	$-0.326489\pi$
$709$	27.6125	1.03701	0.518505	+	0.855075i	$0.326489\pi$
$710$	0	0
$711$	−10.8062	−0.405266
$712$	0	0
$713$	37.6125	1.40860
$714$	0	0
$715$	76.2094	2.85007
$716$	0	0
$717$	2.10469	0.0786010
$718$	0	0
$719$	24.7016	0.921213	0.460606	−	0.887604i	$-0.347632\pi$
$719$	24.7016	0.921213	0.460606	+	0.887604i	$0.347632\pi$
$720$	0	0
$721$	44.2094	1.64644
$722$	0	0
$723$	15.4031	0.572848
$724$	0	0
$725$	4.59688	0.170724
$726$	0	0
$727$	15.5078	0.575153	0.287576	−	0.957758i	$-0.407150\pi$
$727$	15.5078	0.575153	0.287576	+	0.957758i	$0.407150\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−27.2984	−1.00967
$732$	0	0
$733$	−20.8062	−0.768496	−0.384248	−	0.923230i	$-0.625539\pi$
$733$	−20.8062	−0.768496	−0.384248	+	0.923230i	$0.625539\pi$
$734$	0	0
$735$	−40.8062	−1.50516
$736$	0	0
$737$	−56.4187	−2.07821
$738$	0	0
$739$	23.2984	0.857047	0.428523	−	0.903531i	$-0.359034\pi$
$739$	23.2984	0.857047	0.428523	+	0.903531i	$0.359034\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	2.80625	0.102951	0.0514756	−	0.998674i	$-0.483608\pi$
$743$	2.80625	0.102951	0.0514756	+	0.998674i	$0.483608\pi$
$744$	0	0
$745$	54.3141	1.98991
$746$	0	0
$747$	−10.8062	−0.395380
$748$	0	0
$749$	−25.4031	−0.928210
$750$	0	0
$751$	29.6125	1.08058	0.540288	−	0.841480i	$-0.318315\pi$
$751$	29.6125	1.08058	0.540288	+	0.841480i	$0.318315\pi$
$752$	0	0
$753$	−19.2984	−0.703274
$754$	0	0
$755$	7.58125	0.275910
$756$	0	0
$757$	25.2984	0.919487	0.459744	−	0.888052i	$-0.347941\pi$
$757$	25.2984	0.919487	0.459744	+	0.888052i	$0.347941\pi$
$758$	0	0
$759$	−18.8062	−0.682624
$760$	0	0
$761$	26.9109	0.975521	0.487760	−	0.872978i	$-0.337814\pi$
$761$	26.9109	0.975521	0.487760	+	0.872978i	$0.337814\pi$
$762$	0	0
$763$	16.0000	0.579239
$764$	0	0
$765$	−7.29844	−0.263876
$766$	0	0
$767$	−24.0000	−0.866590
$768$	0	0
$769$	−35.1203	−1.26647	−0.633235	−	0.773959i	$-0.718273\pi$
$769$	−35.1203	−1.26647	−0.633235	+	0.773959i	$0.718273\pi$
$770$	0	0
$771$	19.4031	0.698786
$772$	0	0
$773$	33.0156	1.18749	0.593745	−	0.804654i	$-0.297649\pi$
$773$	33.0156	1.18749	0.593745	+	0.804654i	$0.297649\pi$
$774$	0	0
$775$	21.6125	0.776344
$776$	0	0
$777$	−16.0000	−0.573997
$778$	0	0
$779$	−3.40312	−0.121930
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	−5.40312	−0.192846
$786$	0	0
$787$	38.8062	1.38329	0.691647	−	0.722236i	$-0.256886\pi$
$787$	38.8062	1.38329	0.691647	+	0.722236i	$0.256886\pi$
$788$	0	0
$789$	0.701562	0.0249763
$790$	0	0
$791$	84.6281	3.00903
$792$	0	0
$793$	7.79063	0.276653
$794$	0	0
$795$	16.2094	0.574887
$796$	0	0
$797$	−12.5969	−0.446204	−0.223102	−	0.974795i	$-0.571618\pi$
$797$	−12.5969	−0.446204	−0.223102	+	0.974795i	$0.571618\pi$
$798$	0	0
$799$	1.89531	0.0670514
$800$	0	0
$801$	−12.8062	−0.452487
$802$	0	0
$803$	31.5078	1.11189
$804$	0	0
$805$	−50.8062	−1.79068
$806$	0	0
$807$	−10.0000	−0.352017
$808$	0	0
$809$	−49.7172	−1.74796	−0.873982	−	0.485959i	$-0.838470\pi$
$809$	−49.7172	−1.74796	−0.873982	+	0.485959i	$0.838470\pi$
$810$	0	0
$811$	−2.59688	−0.0911886	−0.0455943	−	0.998960i	$-0.514518\pi$
$811$	−2.59688	−0.0911886	−0.0455943	+	0.998960i	$0.514518\pi$
$812$	0	0
$813$	−10.8062	−0.378992
$814$	0	0
$815$	−40.0000	−1.40114
$816$	0	0
$817$	10.1047	0.353518
$818$	0	0
$819$	−28.2094	−0.985715
$820$	0	0
$821$	14.4922	0.505781	0.252890	−	0.967495i	$-0.418619\pi$
$821$	14.4922	0.505781	0.252890	+	0.967495i	$0.418619\pi$
$822$	0	0
$823$	−10.3141	−0.359525	−0.179763	−	0.983710i	$-0.557533\pi$
$823$	−10.3141	−0.359525	−0.179763	+	0.983710i	$0.557533\pi$
$824$	0	0
$825$	−10.8062	−0.376225
$826$	0	0
$827$	−44.4187	−1.54459	−0.772296	−	0.635263i	$-0.780892\pi$
$827$	−44.4187	−1.54459	−0.772296	+	0.635263i	$0.780892\pi$
$828$	0	0
$829$	−22.2094	−0.771363	−0.385682	−	0.922632i	$-0.626034\pi$
$829$	−22.2094	−0.771363	−0.385682	+	0.922632i	$0.626034\pi$
$830$	0	0
$831$	−2.70156	−0.0937162
$832$	0	0
$833$	−40.8062	−1.41385
$834$	0	0
$835$	54.5969	1.88940
$836$	0	0
$837$	−9.40312	−0.325020
$838$	0	0
$839$	−31.0156	−1.07078	−0.535389	−	0.844606i	$-0.679835\pi$
$839$	−31.0156	−1.07078	−0.535389	+	0.844606i	$0.679835\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−0.806248	−0.0277687
$844$	0	0
$845$	62.1359	2.13754
$846$	0	0
$847$	−52.2094	−1.79394
$848$	0	0
$849$	8.70156	0.298637
$850$	0	0
$851$	−13.6125	−0.466630
$852$	0	0
$853$	3.19375	0.109352	0.0546760	−	0.998504i	$-0.482587\pi$
$853$	3.19375	0.109352	0.0546760	+	0.998504i	$0.482587\pi$
$854$	0	0
$855$	2.70156	0.0923915
$856$	0	0
$857$	53.0156	1.81098	0.905490	−	0.424369i	$-0.139504\pi$
$857$	53.0156	1.81098	0.905490	+	0.424369i	$0.139504\pi$
$858$	0	0
$859$	31.2984	1.06789	0.533944	−	0.845520i	$-0.320709\pi$
$859$	31.2984	1.06789	0.533944	+	0.845520i	$0.320709\pi$
$860$	0	0
$861$	−16.0000	−0.545279
$862$	0	0
$863$	−26.8062	−0.912495	−0.456248	−	0.889853i	$-0.650807\pi$
$863$	−26.8062	−0.912495	−0.456248	+	0.889853i	$0.650807\pi$
$864$	0	0
$865$	−23.7906	−0.808906
$866$	0	0
$867$	9.70156	0.329482
$868$	0	0
$869$	−50.8062	−1.72348
$870$	0	0
$871$	−72.0000	−2.43963
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	34.3141	1.16003
$876$	0	0
$877$	−52.8062	−1.78314	−0.891570	−	0.452883i	$-0.850396\pi$
$877$	−52.8062	−1.78314	−0.891570	+	0.452883i	$0.850396\pi$
$878$	0	0
$879$	23.4031	0.789368
$880$	0	0
$881$	32.1047	1.08163	0.540817	−	0.841140i	$-0.318115\pi$
$881$	32.1047	1.08163	0.540817	+	0.841140i	$0.318115\pi$
$882$	0	0
$883$	44.9109	1.51137	0.755687	−	0.654933i	$-0.227303\pi$
$883$	44.9109	1.51137	0.755687	+	0.654933i	$0.227303\pi$
$884$	0	0
$885$	10.8062	0.363248
$886$	0	0
$887$	−29.6125	−0.994290	−0.497145	−	0.867667i	$-0.665618\pi$
$887$	−29.6125	−0.994290	−0.497145	+	0.867667i	$0.665618\pi$
$888$	0	0
$889$	−44.2094	−1.48273
$890$	0	0
$891$	4.70156	0.157508
$892$	0	0
$893$	−0.701562	−0.0234769
$894$	0	0
$895$	−14.5969	−0.487920
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	18.8062	0.627224
$900$	0	0
$901$	16.2094	0.540013
$902$	0	0
$903$	47.5078	1.58096
$904$	0	0
$905$	−12.9844	−0.431615
$906$	0	0
$907$	27.0156	0.897039	0.448519	−	0.893773i	$-0.351951\pi$
$907$	27.0156	0.897039	0.448519	+	0.893773i	$0.351951\pi$
$908$	0	0
$909$	16.8062	0.557428
$910$	0	0
$911$	−28.2094	−0.934618	−0.467309	−	0.884094i	$-0.654776\pi$
$911$	−28.2094	−0.934618	−0.467309	+	0.884094i	$0.654776\pi$
$912$	0	0
$913$	−50.8062	−1.68144
$914$	0	0
$915$	−3.50781	−0.115965
$916$	0	0
$917$	59.7172	1.97204
$918$	0	0
$919$	−29.6125	−0.976826	−0.488413	−	0.872613i	$-0.662424\pi$
$919$	−29.6125	−0.976826	−0.488413	+	0.872613i	$0.662424\pi$
$920$	0	0
$921$	1.19375	0.0393355
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−7.82187	−0.257181
$926$	0	0
$927$	−9.40312	−0.308839
$928$	0	0
$929$	36.8062	1.20757	0.603787	−	0.797146i	$-0.293658\pi$
$929$	36.8062	1.20757	0.603787	+	0.797146i	$0.293658\pi$
$930$	0	0
$931$	15.1047	0.495036
$932$	0	0
$933$	27.5078	0.900565
$934$	0	0
$935$	−34.3141	−1.12219
$936$	0	0
$937$	6.70156	0.218930	0.109465	−	0.993991i	$-0.465086\pi$
$937$	6.70156	0.218930	0.109465	+	0.993991i	$0.465086\pi$
$938$	0	0
$939$	−28.8062	−0.940056
$940$	0	0
$941$	10.0000	0.325991	0.162995	−	0.986627i	$-0.447884\pi$
$941$	10.0000	0.325991	0.162995	+	0.986627i	$0.447884\pi$
$942$	0	0
$943$	−13.6125	−0.443284
$944$	0	0
$945$	12.7016	0.413182
$946$	0	0
$947$	18.8062	0.611121	0.305560	−	0.952173i	$-0.401156\pi$
$947$	18.8062	0.611121	0.305560	+	0.952173i	$0.401156\pi$
$948$	0	0
$949$	40.2094	1.30525
$950$	0	0
$951$	−3.40312	−0.110354
$952$	0	0
$953$	7.40312	0.239811	0.119905	−	0.992785i	$-0.461741\pi$
$953$	7.40312	0.239811	0.119905	+	0.992785i	$0.461741\pi$
$954$	0	0
$955$	−56.4922	−1.82804
$956$	0	0
$957$	−9.40312	−0.303960
$958$	0	0
$959$	50.3141	1.62473
$960$	0	0
$961$	57.4187	1.85222
$962$	0	0
$963$	5.40312	0.174113
$964$	0	0
$965$	30.8062	0.991688
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	0	0
$969$	2.70156	0.0867867
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	0	0
$973$	−71.9266	−2.30586
$974$	0	0
$975$	−13.7906	−0.441654
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	−60.2094	−1.92430
$980$	0	0
$981$	−3.40312	−0.108653
$982$	0	0
$983$	32.4187	1.03400	0.516999	−	0.855986i	$-0.327049\pi$
$983$	32.4187	1.03400	0.516999	+	0.855986i	$0.327049\pi$
$984$	0	0
$985$	59.4344	1.89374
$986$	0	0
$987$	−3.29844	−0.104990
$988$	0	0
$989$	40.4187	1.28524
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	0	0
$993$	30.8062	0.977607
$994$	0	0
$995$	8.91093	0.282496
$996$	0	0
$997$	15.8953	0.503410	0.251705	−	0.967804i	$-0.419009\pi$
$997$	15.8953	0.503410	0.251705	+	0.967804i	$0.419009\pi$
$998$	0	0
$999$	3.40312	0.107670

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	456.2.a.e.1.2	✓	2
3.2	odd	2		1368.2.a.l.1.1		2
4.3	odd	2		912.2.a.o.1.2		2
8.3	odd	2		3648.2.a.bn.1.1		2
8.5	even	2		3648.2.a.bs.1.1		2
12.11	even	2		2736.2.a.bb.1.1		2
19.18	odd	2		8664.2.a.v.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.2.a.e.1.2	✓	2	1.1	even	1	trivial
912.2.a.o.1.2		2	4.3	odd	2
1368.2.a.l.1.1		2	3.2	odd	2
2736.2.a.bb.1.1		2	12.11	even	2
3648.2.a.bn.1.1		2	8.3	odd	2
3648.2.a.bs.1.1		2	8.5	even	2
8664.2.a.v.1.2		2	19.18	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	456.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.70156 q^{5} -4.70156 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.70156 q^{5} -4.70156 q^{7} +1.00000 q^{9} +4.70156 q^{11} +6.00000 q^{13} -2.70156 q^{15} -2.70156 q^{17} +1.00000 q^{19} +4.70156 q^{21} +4.00000 q^{23} +2.29844 q^{25} -1.00000 q^{27} +2.00000 q^{29} +9.40312 q^{31} -4.70156 q^{33} -12.7016 q^{35} -3.40312 q^{37} -6.00000 q^{39} -3.40312 q^{41} +10.1047 q^{43} +2.70156 q^{45} -0.701562 q^{47} +15.1047 q^{49} +2.70156 q^{51} -6.00000 q^{53} +12.7016 q^{55} -1.00000 q^{57} -4.00000 q^{59} +1.29844 q^{61} -4.70156 q^{63} +16.2094 q^{65} -12.0000 q^{67} -4.00000 q^{69} +6.70156 q^{73} -2.29844 q^{75} -22.1047 q^{77} -10.8062 q^{79} +1.00000 q^{81} -10.8062 q^{83} -7.29844 q^{85} -2.00000 q^{87} -12.8062 q^{89} -28.2094 q^{91} -9.40312 q^{93} +2.70156 q^{95} -6.00000 q^{97} +4.70156 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - q^{5} - 3 q^{7} + 2 q^{9} + 3 q^{11} + 12 q^{13} + q^{15} + q^{17} + 2 q^{19} + 3 q^{21} + 8 q^{23} + 11 q^{25} - 2 q^{27} + 4 q^{29} + 6 q^{31} - 3 q^{33} - 19 q^{35} + 6 q^{37} - 12 q^{39} + 6 q^{41} + q^{43} - q^{45} + 5 q^{47} + 11 q^{49} - q^{51} - 12 q^{53} + 19 q^{55} - 2 q^{57} - 8 q^{59} + 9 q^{61} - 3 q^{63} - 6 q^{65} - 24 q^{67} - 8 q^{69} + 7 q^{73} - 11 q^{75} - 25 q^{77} + 4 q^{79} + 2 q^{81} + 4 q^{83} - 21 q^{85} - 4 q^{87} - 18 q^{91} - 6 q^{93} - q^{95} - 12 q^{97} + 3 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - q^5 - 3 * q^7 + 2 * q^9 + 3 * q^11 + 12 * q^13 + q^15 + q^17 + 2 * q^19 + 3 * q^21 + 8 * q^23 + 11 * q^25 - 2 * q^27 + 4 * q^29 + 6 * q^31 - 3 * q^33 - 19 * q^35 + 6 * q^37 - 12 * q^39 + 6 * q^41 + q^43 - q^45 + 5 * q^47 + 11 * q^49 - q^51 - 12 * q^53 + 19 * q^55 - 2 * q^57 - 8 * q^59 + 9 * q^61 - 3 * q^63 - 6 * q^65 - 24 * q^67 - 8 * q^69 + 7 * q^73 - 11 * q^75 - 25 * q^77 + 4 * q^79 + 2 * q^81 + 4 * q^83 - 21 * q^85 - 4 * q^87 - 18 * q^91 - 6 * q^93 - q^95 - 12 * q^97 + 3 * q^99

Introduction

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