LMFDB - Embedded newform 9464.2.a.br.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9464,2,Mod(1,9464)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9464 = 2^{3} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,0,-3,0,-4,0,-15,0,16,0,-15,0,0,0,-8,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$75.5704204729$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 26 x^{13} + 78 x^{12} + 253 x^{11} - 782 x^{10} - 1087 x^{9} + 3776 x^{8} + \cdots - 344 $ x^15 - 3x^14 - 26x^13 + 78x^12 + 253x^11 - 782x^10 - 1087x^9 + 3776x^8 + 1576x^7 - 8849x^6 + 1791x^5 + 8496x^4 - 5353x^3 - 1252x^2 + 1724x - 344
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-0.619176$ of defining polynomial
Character	$\chi$	$=$	9464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.619176 q^{3} +2.60169 q^{5} -1.00000 q^{7} -2.61662 q^{9} +4.07771 q^{11} +1.61090 q^{15} -7.64859 q^{17} +3.87020 q^{19} -0.619176 q^{21} -7.02679 q^{23} +1.76880 q^{25} -3.47768 q^{27} -4.09415 q^{29} -3.74320 q^{31} +2.52482 q^{33} -2.60169 q^{35} +8.83955 q^{37} -1.62764 q^{41} +3.25603 q^{43} -6.80764 q^{45} +8.01809 q^{47} +1.00000 q^{49} -4.73582 q^{51} +9.25086 q^{53} +10.6089 q^{55} +2.39633 q^{57} -6.25887 q^{59} -10.5394 q^{61} +2.61662 q^{63} -9.24894 q^{67} -4.35082 q^{69} -12.8346 q^{71} -2.59859 q^{73} +1.09520 q^{75} -4.07771 q^{77} -0.869671 q^{79} +5.69657 q^{81} +6.34814 q^{83} -19.8993 q^{85} -2.53500 q^{87} -0.774962 q^{89} -2.31770 q^{93} +10.0691 q^{95} -14.1122 q^{97} -10.6698 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - 3 q^{3} - 4 q^{5} - 15 q^{7} + 16 q^{9} - 15 q^{11} - 8 q^{15} + 2 q^{17} - 13 q^{19} + 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} - 19 q^{31} + 24 q^{33} + 4 q^{35} + 2 q^{37} - 30 q^{41}+ \cdots - 70 q^{99}+O(q^{100}) $ 15 * q - 3 * q^3 - 4 * q^5 - 15 * q^7 + 16 * q^9 - 15 * q^11 - 8 * q^15 + 2 * q^17 - 13 * q^19 + 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 - 19 * q^31 + 24 * q^33 + 4 * q^35 + 2 * q^37 - 30 * q^41 + 15 * q^43 + 2 * q^45 - 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 + 2 * q^57 - 22 * q^59 - q^61 - 16 * q^63 - 29 * q^67 + 32 * q^69 - 20 * q^71 - 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 - 23 * q^83 - 8 * q^85 - 22 * q^87 - 9 * q^89 - 10 * q^93 + 28 * q^95 - 11 * q^97 - 70 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0.619176	0.357481	0.178741	−	0.983896i	$-0.442798\pi$
$3$	0.619176	0.357481	0.178741	+	0.983896i	$0.442798\pi$
$4$	0	0
$5$	2.60169	1.16351	0.581756	−	0.813363i	$-0.302366\pi$
$5$	2.60169	1.16351	0.581756	+	0.813363i	$0.302366\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.61662	−0.872207
$10$	0	0
$11$	4.07771	1.22948	0.614738	−	0.788732i	$-0.289262\pi$
$11$	4.07771	1.22948	0.614738	+	0.788732i	$0.289262\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.61090	0.415934
$16$	0	0
$17$	−7.64859	−1.85506	−0.927528	−	0.373755i	$-0.878070\pi$
$17$	−7.64859	−1.85506	−0.927528	+	0.373755i	$0.878070\pi$
$18$	0	0
$19$	3.87020	0.887884	0.443942	−	0.896056i	$-0.353580\pi$
$19$	3.87020	0.887884	0.443942	+	0.896056i	$0.353580\pi$
$20$	0	0
$21$	−0.619176	−0.135115
$22$	0	0
$23$	−7.02679	−1.46519	−0.732593	−	0.680667i	$-0.761690\pi$
$23$	−7.02679	−1.46519	−0.732593	+	0.680667i	$0.761690\pi$
$24$	0	0
$25$	1.76880	0.353761
$26$	0	0
$27$	−3.47768	−0.669279
$28$	0	0
$29$	−4.09415	−0.760264	−0.380132	−	0.924932i	$-0.624121\pi$
$29$	−4.09415	−0.760264	−0.380132	+	0.924932i	$0.624121\pi$
$30$	0	0
$31$	−3.74320	−0.672298	−0.336149	−	0.941809i	$-0.609125\pi$
$31$	−3.74320	−0.672298	−0.336149	+	0.941809i	$0.609125\pi$
$32$	0	0
$33$	2.52482	0.439514
$34$	0	0
$35$	−2.60169	−0.439766
$36$	0	0
$37$	8.83955	1.45321	0.726606	−	0.687054i	$-0.241096\pi$
$37$	8.83955	1.45321	0.726606	+	0.687054i	$0.241096\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.62764	−0.254194	−0.127097	−	0.991890i	$-0.540566\pi$
$41$	−1.62764	−0.254194	−0.127097	+	0.991890i	$0.540566\pi$
$42$	0	0
$43$	3.25603	0.496540	0.248270	−	0.968691i	$-0.420138\pi$
$43$	3.25603	0.496540	0.248270	+	0.968691i	$0.420138\pi$
$44$	0	0
$45$	−6.80764	−1.01482
$46$	0	0
$47$	8.01809	1.16956	0.584780	−	0.811192i	$-0.301181\pi$
$47$	8.01809	1.16956	0.584780	+	0.811192i	$0.301181\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.73582	−0.663148
$52$	0	0
$53$	9.25086	1.27070	0.635352	−	0.772223i	$-0.280855\pi$
$53$	9.25086	1.27070	0.635352	+	0.772223i	$0.280855\pi$
$54$	0	0
$55$	10.6089	1.43051
$56$	0	0
$57$	2.39633	0.317402
$58$	0	0
$59$	−6.25887	−0.814835	−0.407418	−	0.913242i	$-0.633571\pi$
$59$	−6.25887	−0.814835	−0.407418	+	0.913242i	$0.633571\pi$
$60$	0	0
$61$	−10.5394	−1.34943	−0.674717	−	0.738076i	$-0.735735\pi$
$61$	−10.5394	−1.34943	−0.674717	+	0.738076i	$0.735735\pi$
$62$	0	0
$63$	2.61662	0.329663
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.24894	−1.12994	−0.564969	−	0.825112i	$-0.691112\pi$
$67$	−9.24894	−1.12994	−0.564969	+	0.825112i	$0.691112\pi$
$68$	0	0
$69$	−4.35082	−0.523777
$70$	0	0
$71$	−12.8346	−1.52319	−0.761595	−	0.648053i	$-0.775583\pi$
$71$	−12.8346	−1.52319	−0.761595	+	0.648053i	$0.775583\pi$
$72$	0	0
$73$	−2.59859	−0.304142	−0.152071	−	0.988370i	$-0.548594\pi$
$73$	−2.59859	−0.304142	−0.152071	+	0.988370i	$0.548594\pi$
$74$	0	0
$75$	1.09520	0.126463
$76$	0	0
$77$	−4.07771	−0.464698
$78$	0	0
$79$	−0.869671	−0.0978456	−0.0489228	−	0.998803i	$-0.515579\pi$
$79$	−0.869671	−0.0978456	−0.0489228	+	0.998803i	$0.515579\pi$
$80$	0	0
$81$	5.69657	0.632952
$82$	0	0
$83$	6.34814	0.696799	0.348400	−	0.937346i	$-0.386725\pi$
$83$	6.34814	0.696799	0.348400	+	0.937346i	$0.386725\pi$
$84$	0	0
$85$	−19.8993	−2.15838
$86$	0	0
$87$	−2.53500	−0.271780
$88$	0	0
$89$	−0.774962	−0.0821458	−0.0410729	−	0.999156i	$-0.513078\pi$
$89$	−0.774962	−0.0821458	−0.0410729	+	0.999156i	$0.513078\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.31770	−0.240334
$94$	0	0
$95$	10.0691	1.03306
$96$	0	0
$97$	−14.1122	−1.43288	−0.716439	−	0.697650i	$-0.754229\pi$
$97$	−14.1122	−1.43288	−0.716439	+	0.697650i	$0.754229\pi$
$98$	0	0
$99$	−10.6698	−1.07236
$100$	0	0
$101$	−3.00119	−0.298629	−0.149315	−	0.988790i	$-0.547707\pi$
$101$	−3.00119	−0.298629	−0.149315	+	0.988790i	$0.547707\pi$
$102$	0	0
$103$	−19.7122	−1.94230	−0.971150	−	0.238469i	$-0.923355\pi$
$103$	−19.7122	−1.94230	−0.971150	+	0.238469i	$0.923355\pi$
$104$	0	0
$105$	−1.61090	−0.157208
$106$	0	0
$107$	1.80259	0.174263	0.0871316	−	0.996197i	$-0.472230\pi$
$107$	1.80259	0.174263	0.0871316	+	0.996197i	$0.472230\pi$
$108$	0	0
$109$	−8.49662	−0.813829	−0.406914	−	0.913466i	$-0.633395\pi$
$109$	−8.49662	−0.813829	−0.406914	+	0.913466i	$0.633395\pi$
$110$	0	0
$111$	5.47323	0.519496
$112$	0	0
$113$	−19.2502	−1.81090	−0.905452	−	0.424449i	$-0.860468\pi$
$113$	−19.2502	−1.81090	−0.905452	+	0.424449i	$0.860468\pi$
$114$	0	0
$115$	−18.2815	−1.70476
$116$	0	0
$117$	0	0
$118$	0	0
$119$	7.64859	0.701145
$120$	0	0
$121$	5.62770	0.511609
$122$	0	0
$123$	−1.00779	−0.0908696
$124$	0	0
$125$	−8.40658	−0.751907
$126$	0	0
$127$	4.46686	0.396370	0.198185	−	0.980165i	$-0.436495\pi$
$127$	4.46686	0.396370	0.198185	+	0.980165i	$0.436495\pi$
$128$	0	0
$129$	2.01606	0.177504
$130$	0	0
$131$	17.5622	1.53442	0.767209	−	0.641397i	$-0.221645\pi$
$131$	17.5622	1.53442	0.767209	+	0.641397i	$0.221645\pi$
$132$	0	0
$133$	−3.87020	−0.335588
$134$	0	0
$135$	−9.04784	−0.778714
$136$	0	0
$137$	10.0427	0.858004	0.429002	−	0.903304i	$-0.358865\pi$
$137$	10.0427	0.858004	0.429002	+	0.903304i	$0.358865\pi$
$138$	0	0
$139$	7.23741	0.613869	0.306935	−	0.951731i	$-0.400697\pi$
$139$	7.23741	0.613869	0.306935	+	0.951731i	$0.400697\pi$
$140$	0	0
$141$	4.96461	0.418096
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−10.6517	−0.884576
$146$	0	0
$147$	0.619176	0.0510688
$148$	0	0
$149$	11.1546	0.913822	0.456911	−	0.889512i	$-0.348956\pi$
$149$	11.1546	0.913822	0.456911	+	0.889512i	$0.348956\pi$
$150$	0	0
$151$	−0.207386	−0.0168768	−0.00843842	−	0.999964i	$-0.502686\pi$
$151$	−0.207386	−0.0168768	−0.00843842	+	0.999964i	$0.502686\pi$
$152$	0	0
$153$	20.0135	1.61799
$154$	0	0
$155$	−9.73865	−0.782227
$156$	0	0
$157$	−13.6061	−1.08588	−0.542941	−	0.839771i	$-0.682689\pi$
$157$	−13.6061	−1.08588	−0.542941	+	0.839771i	$0.682689\pi$
$158$	0	0
$159$	5.72791	0.454253
$160$	0	0
$161$	7.02679	0.553789
$162$	0	0
$163$	−6.84998	−0.536532	−0.268266	−	0.963345i	$-0.586451\pi$
$163$	−6.84998	−0.536532	−0.268266	+	0.963345i	$0.586451\pi$
$164$	0	0
$165$	6.56880	0.511380
$166$	0	0
$167$	−6.64278	−0.514034	−0.257017	−	0.966407i	$-0.582740\pi$
$167$	−6.64278	−0.514034	−0.257017	+	0.966407i	$0.582740\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−10.1268	−0.774418
$172$	0	0
$173$	−0.267894	−0.0203676	−0.0101838	−	0.999948i	$-0.503242\pi$
$173$	−0.267894	−0.0203676	−0.0101838	+	0.999948i	$0.503242\pi$
$174$	0	0
$175$	−1.76880	−0.133709
$176$	0	0
$177$	−3.87534	−0.291288
$178$	0	0
$179$	−17.9819	−1.34403	−0.672015	−	0.740537i	$-0.734571\pi$
$179$	−17.9819	−1.34403	−0.672015	+	0.740537i	$0.734571\pi$
$180$	0	0
$181$	−1.62862	−0.121054	−0.0605272	−	0.998167i	$-0.519278\pi$
$181$	−1.62862	−0.121054	−0.0605272	+	0.998167i	$0.519278\pi$
$182$	0	0
$183$	−6.52576	−0.482398
$184$	0	0
$185$	22.9978	1.69083
$186$	0	0
$187$	−31.1887	−2.28074
$188$	0	0
$189$	3.47768	0.252964
$190$	0	0
$191$	12.3448	0.893241	0.446620	−	0.894724i	$-0.352627\pi$
$191$	12.3448	0.893241	0.446620	+	0.894724i	$0.352627\pi$
$192$	0	0
$193$	23.1367	1.66542	0.832708	−	0.553712i	$-0.186789\pi$
$193$	23.1367	1.66542	0.832708	+	0.553712i	$0.186789\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.9839	1.28130	0.640650	−	0.767833i	$-0.278665\pi$
$197$	17.9839	1.28130	0.640650	+	0.767833i	$0.278665\pi$
$198$	0	0
$199$	−2.44650	−0.173428	−0.0867138	−	0.996233i	$-0.527637\pi$
$199$	−2.44650	−0.173428	−0.0867138	+	0.996233i	$0.527637\pi$
$200$	0	0
$201$	−5.72672	−0.403932
$202$	0	0
$203$	4.09415	0.287353
$204$	0	0
$205$	−4.23461	−0.295758
$206$	0	0
$207$	18.3864	1.27795
$208$	0	0
$209$	15.7815	1.09163
$210$	0	0
$211$	−8.62463	−0.593744	−0.296872	−	0.954917i	$-0.595944\pi$
$211$	−8.62463	−0.593744	−0.296872	+	0.954917i	$0.595944\pi$
$212$	0	0
$213$	−7.94689	−0.544512
$214$	0	0
$215$	8.47119	0.577731
$216$	0	0
$217$	3.74320	0.254105
$218$	0	0
$219$	−1.60898	−0.108725
$220$	0	0
$221$	0	0
$222$	0	0
$223$	18.2880	1.22466	0.612328	−	0.790604i	$-0.290233\pi$
$223$	18.2880	1.22466	0.612328	+	0.790604i	$0.290233\pi$
$224$	0	0
$225$	−4.62829	−0.308552
$226$	0	0
$227$	−7.30170	−0.484631	−0.242316	−	0.970197i	$-0.577907\pi$
$227$	−7.30170	−0.484631	−0.242316	+	0.970197i	$0.577907\pi$
$228$	0	0
$229$	−24.0876	−1.59175	−0.795877	−	0.605459i	$-0.792990\pi$
$229$	−24.0876	−1.59175	−0.795877	+	0.605459i	$0.792990\pi$
$230$	0	0
$231$	−2.52482	−0.166121
$232$	0	0
$233$	−29.1103	−1.90708	−0.953540	−	0.301267i	$-0.902590\pi$
$233$	−29.1103	−1.90708	−0.953540	+	0.301267i	$0.902590\pi$
$234$	0	0
$235$	20.8606	1.36080
$236$	0	0
$237$	−0.538479	−0.0349780
$238$	0	0
$239$	−28.2255	−1.82575	−0.912876	−	0.408236i	$-0.866144\pi$
$239$	−28.2255	−1.82575	−0.912876	+	0.408236i	$0.866144\pi$
$240$	0	0
$241$	−2.46887	−0.159034	−0.0795170	−	0.996834i	$-0.525338\pi$
$241$	−2.46887	−0.159034	−0.0795170	+	0.996834i	$0.525338\pi$
$242$	0	0
$243$	13.9602	0.895548
$244$	0	0
$245$	2.60169	0.166216
$246$	0	0
$247$	0	0
$248$	0	0
$249$	3.93062	0.249093
$250$	0	0
$251$	−26.2601	−1.65753	−0.828763	−	0.559600i	$-0.810955\pi$
$251$	−26.2601	−1.65753	−0.828763	+	0.559600i	$0.810955\pi$
$252$	0	0
$253$	−28.6532	−1.80141
$254$	0	0
$255$	−12.3211	−0.771580
$256$	0	0
$257$	−10.6526	−0.664492	−0.332246	−	0.943193i	$-0.607806\pi$
$257$	−10.6526	−0.664492	−0.332246	+	0.943193i	$0.607806\pi$
$258$	0	0
$259$	−8.83955	−0.549263
$260$	0	0
$261$	10.7128	0.663108
$262$	0	0
$263$	−8.95248	−0.552033	−0.276017	−	0.961153i	$-0.589015\pi$
$263$	−8.95248	−0.552033	−0.276017	+	0.961153i	$0.589015\pi$
$264$	0	0
$265$	24.0679	1.47848
$266$	0	0
$267$	−0.479838	−0.0293656
$268$	0	0
$269$	22.0773	1.34608	0.673039	−	0.739607i	$-0.264989\pi$
$269$	22.0773	1.34608	0.673039	+	0.739607i	$0.264989\pi$
$270$	0	0
$271$	5.88890	0.357725	0.178863	−	0.983874i	$-0.442758\pi$
$271$	5.88890	0.357725	0.178863	+	0.983874i	$0.442758\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.21266	0.434940
$276$	0	0
$277$	5.66739	0.340520	0.170260	−	0.985399i	$-0.445539\pi$
$277$	5.66739	0.340520	0.170260	+	0.985399i	$0.445539\pi$
$278$	0	0
$279$	9.79453	0.586383
$280$	0	0
$281$	31.6914	1.89055	0.945274	−	0.326278i	$-0.105795\pi$
$281$	31.6914	1.89055	0.945274	+	0.326278i	$0.105795\pi$
$282$	0	0
$283$	−13.6414	−0.810896	−0.405448	−	0.914118i	$-0.632884\pi$
$283$	−13.6414	−0.810896	−0.405448	+	0.914118i	$0.632884\pi$
$284$	0	0
$285$	6.23452	0.369301
$286$	0	0
$287$	1.62764	0.0960763
$288$	0	0
$289$	41.5009	2.44123
$290$	0	0
$291$	−8.73794	−0.512227
$292$	0	0
$293$	17.7561	1.03733	0.518663	−	0.854979i	$-0.326430\pi$
$293$	17.7561	1.03733	0.518663	+	0.854979i	$0.326430\pi$
$294$	0	0
$295$	−16.2837	−0.948071
$296$	0	0
$297$	−14.1809	−0.822862
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−3.25603	−0.187675
$302$	0	0
$303$	−1.85826	−0.106754
$304$	0	0
$305$	−27.4203	−1.57008
$306$	0	0
$307$	−27.3147	−1.55893	−0.779465	−	0.626446i	$-0.784509\pi$
$307$	−27.3147	−1.55893	−0.779465	+	0.626446i	$0.784509\pi$
$308$	0	0
$309$	−12.2053	−0.694336
$310$	0	0
$311$	−10.3671	−0.587867	−0.293934	−	0.955826i	$-0.594964\pi$
$311$	−10.3671	−0.587867	−0.293934	+	0.955826i	$0.594964\pi$
$312$	0	0
$313$	17.9287	1.01339	0.506694	−	0.862126i	$-0.330867\pi$
$313$	17.9287	1.01339	0.506694	+	0.862126i	$0.330867\pi$
$314$	0	0
$315$	6.80764	0.383567
$316$	0	0
$317$	−10.1856	−0.572081	−0.286041	−	0.958217i	$-0.592339\pi$
$317$	−10.1856	−0.572081	−0.286041	+	0.958217i	$0.592339\pi$
$318$	0	0
$319$	−16.6947	−0.934726
$320$	0	0
$321$	1.11612	0.0622959
$322$	0	0
$323$	−29.6015	−1.64707
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.26090	−0.290929
$328$	0	0
$329$	−8.01809	−0.442052
$330$	0	0
$331$	−7.06960	−0.388580	−0.194290	−	0.980944i	$-0.562240\pi$
$331$	−7.06960	−0.388580	−0.194290	+	0.980944i	$0.562240\pi$
$332$	0	0
$333$	−23.1297	−1.26750
$334$	0	0
$335$	−24.0629	−1.31470
$336$	0	0
$337$	24.9764	1.36055	0.680277	−	0.732955i	$-0.261860\pi$
$337$	24.9764	1.36055	0.680277	+	0.732955i	$0.261860\pi$
$338$	0	0
$339$	−11.9192	−0.647364
$340$	0	0
$341$	−15.2637	−0.826574
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−11.3195	−0.609421
$346$	0	0
$347$	−28.6475	−1.53788	−0.768939	−	0.639323i	$-0.779215\pi$
$347$	−28.6475	−1.53788	−0.768939	+	0.639323i	$0.779215\pi$
$348$	0	0
$349$	−4.67144	−0.250056	−0.125028	−	0.992153i	$-0.539902\pi$
$349$	−4.67144	−0.250056	−0.125028	+	0.992153i	$0.539902\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.0372	1.59872	0.799359	−	0.600854i	$-0.205173\pi$
$353$	30.0372	1.59872	0.799359	+	0.600854i	$0.205173\pi$
$354$	0	0
$355$	−33.3917	−1.77225
$356$	0	0
$357$	4.73582	0.250646
$358$	0	0
$359$	−16.6008	−0.876157	−0.438079	−	0.898937i	$-0.644341\pi$
$359$	−16.6008	−0.876157	−0.438079	+	0.898937i	$0.644341\pi$
$360$	0	0
$361$	−4.02159	−0.211663
$362$	0	0
$363$	3.48454	0.182891
$364$	0	0
$365$	−6.76073	−0.353873
$366$	0	0
$367$	6.69549	0.349502	0.174751	−	0.984613i	$-0.444088\pi$
$367$	6.69549	0.349502	0.174751	+	0.984613i	$0.444088\pi$
$368$	0	0
$369$	4.25891	0.221710
$370$	0	0
$371$	−9.25086	−0.480281
$372$	0	0
$373$	−9.21141	−0.476948	−0.238474	−	0.971149i	$-0.576647\pi$
$373$	−9.21141	−0.476948	−0.238474	+	0.971149i	$0.576647\pi$
$374$	0	0
$375$	−5.20515	−0.268793
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−3.75851	−0.193061	−0.0965307	−	0.995330i	$-0.530775\pi$
$379$	−3.75851	−0.193061	−0.0965307	+	0.995330i	$0.530775\pi$
$380$	0	0
$381$	2.76577	0.141695
$382$	0	0
$383$	23.1457	1.18269	0.591345	−	0.806418i	$-0.298597\pi$
$383$	23.1457	1.18269	0.591345	+	0.806418i	$0.298597\pi$
$384$	0	0
$385$	−10.6089	−0.540682
$386$	0	0
$387$	−8.51980	−0.433086
$388$	0	0
$389$	25.4849	1.29214	0.646068	−	0.763280i	$-0.276412\pi$
$389$	25.4849	1.29214	0.646068	+	0.763280i	$0.276412\pi$
$390$	0	0
$391$	53.7450	2.71800
$392$	0	0
$393$	10.8741	0.548526
$394$	0	0
$395$	−2.26262	−0.113845
$396$	0	0
$397$	29.3041	1.47073	0.735365	−	0.677671i	$-0.237011\pi$
$397$	29.3041	1.47073	0.735365	+	0.677671i	$0.237011\pi$
$398$	0	0
$399$	−2.39633	−0.119967
$400$	0	0
$401$	−17.6977	−0.883782	−0.441891	−	0.897069i	$-0.645692\pi$
$401$	−17.6977	−0.883782	−0.441891	+	0.897069i	$0.645692\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	14.8207	0.736448
$406$	0	0
$407$	36.0451	1.78669
$408$	0	0
$409$	−1.77124	−0.0875823	−0.0437911	−	0.999041i	$-0.513944\pi$
$409$	−1.77124	−0.0875823	−0.0437911	+	0.999041i	$0.513944\pi$
$410$	0	0
$411$	6.21818	0.306720
$412$	0	0
$413$	6.25887	0.307979
$414$	0	0
$415$	16.5159	0.810734
$416$	0	0
$417$	4.48123	0.219447
$418$	0	0
$419$	−25.1001	−1.22622	−0.613110	−	0.789997i	$-0.710082\pi$
$419$	−25.1001	−1.22622	−0.613110	+	0.789997i	$0.710082\pi$
$420$	0	0
$421$	22.1032	1.07724	0.538622	−	0.842548i	$-0.318945\pi$
$421$	22.1032	1.07724	0.538622	+	0.842548i	$0.318945\pi$
$422$	0	0
$423$	−20.9803	−1.02010
$424$	0	0
$425$	−13.5288	−0.656245
$426$	0	0
$427$	10.5394	0.510038
$428$	0	0
$429$	0	0
$430$	0	0
$431$	35.6726	1.71829	0.859144	−	0.511735i	$-0.170997\pi$
$431$	35.6726	1.71829	0.859144	+	0.511735i	$0.170997\pi$
$432$	0	0
$433$	16.0480	0.771219	0.385609	−	0.922662i	$-0.373991\pi$
$433$	16.0480	0.771219	0.385609	+	0.922662i	$0.373991\pi$
$434$	0	0
$435$	−6.59528	−0.316220
$436$	0	0
$437$	−27.1950	−1.30092
$438$	0	0
$439$	−9.68485	−0.462233	−0.231116	−	0.972926i	$-0.574238\pi$
$439$	−9.68485	−0.462233	−0.231116	+	0.972926i	$0.574238\pi$
$440$	0	0
$441$	−2.61662	−0.124601
$442$	0	0
$443$	7.30236	0.346946	0.173473	−	0.984839i	$-0.444501\pi$
$443$	7.30236	0.346946	0.173473	+	0.984839i	$0.444501\pi$
$444$	0	0
$445$	−2.01621	−0.0955776
$446$	0	0
$447$	6.90667	0.326674
$448$	0	0
$449$	17.2179	0.812562	0.406281	−	0.913748i	$-0.366825\pi$
$449$	17.2179	0.812562	0.406281	+	0.913748i	$0.366825\pi$
$450$	0	0
$451$	−6.63702	−0.312525
$452$	0	0
$453$	−0.128408	−0.00603315
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−28.1146	−1.31515	−0.657573	−	0.753390i	$-0.728417\pi$
$457$	−28.1146	−1.31515	−0.657573	+	0.753390i	$0.728417\pi$
$458$	0	0
$459$	26.5993	1.24155
$460$	0	0
$461$	−0.452615	−0.0210804	−0.0105402	−	0.999944i	$-0.503355\pi$
$461$	−0.452615	−0.0210804	−0.0105402	+	0.999944i	$0.503355\pi$
$462$	0	0
$463$	−22.7792	−1.05864	−0.529319	−	0.848423i	$-0.677553\pi$
$463$	−22.7792	−1.05864	−0.529319	+	0.848423i	$0.677553\pi$
$464$	0	0
$465$	−6.02993	−0.279632
$466$	0	0
$467$	−14.8115	−0.685395	−0.342697	−	0.939446i	$-0.611341\pi$
$467$	−14.8115	−0.685395	−0.342697	+	0.939446i	$0.611341\pi$
$468$	0	0
$469$	9.24894	0.427076
$470$	0	0
$471$	−8.42455	−0.388183
$472$	0	0
$473$	13.2771	0.610484
$474$	0	0
$475$	6.84561	0.314098
$476$	0	0
$477$	−24.2060	−1.10832
$478$	0	0
$479$	−15.2597	−0.697235	−0.348617	−	0.937265i	$-0.613349\pi$
$479$	−15.2597	−0.697235	−0.348617	+	0.937265i	$0.613349\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	4.35082	0.197969
$484$	0	0
$485$	−36.7156	−1.66717
$486$	0	0
$487$	7.67715	0.347885	0.173942	−	0.984756i	$-0.444349\pi$
$487$	7.67715	0.347885	0.173942	+	0.984756i	$0.444349\pi$
$488$	0	0
$489$	−4.24134	−0.191800
$490$	0	0
$491$	−1.63021	−0.0735703	−0.0367851	−	0.999323i	$-0.511712\pi$
$491$	−1.63021	−0.0735703	−0.0367851	+	0.999323i	$0.511712\pi$
$492$	0	0
$493$	31.3144	1.41033
$494$	0	0
$495$	−27.7596	−1.24770
$496$	0	0
$497$	12.8346	0.575712
$498$	0	0
$499$	−11.6590	−0.521929	−0.260965	−	0.965348i	$-0.584041\pi$
$499$	−11.6590	−0.521929	−0.260965	+	0.965348i	$0.584041\pi$
$500$	0	0
$501$	−4.11305	−0.183758
$502$	0	0
$503$	−35.6042	−1.58751	−0.793757	−	0.608235i	$-0.791878\pi$
$503$	−35.6042	−1.58751	−0.793757	+	0.608235i	$0.791878\pi$
$504$	0	0
$505$	−7.80816	−0.347459
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13.1638	−0.583475	−0.291738	−	0.956498i	$-0.594233\pi$
$509$	−13.1638	−0.583475	−0.291738	+	0.956498i	$0.594233\pi$
$510$	0	0
$511$	2.59859	0.114955
$512$	0	0
$513$	−13.4593	−0.594242
$514$	0	0
$515$	−51.2851	−2.25989
$516$	0	0
$517$	32.6954	1.43794
$518$	0	0
$519$	−0.165874	−0.00728104
$520$	0	0
$521$	37.9650	1.66328	0.831639	−	0.555317i	$-0.187403\pi$
$521$	37.9650	1.66328	0.831639	+	0.555317i	$0.187403\pi$
$522$	0	0
$523$	16.6628	0.728612	0.364306	−	0.931279i	$-0.381306\pi$
$523$	16.6628	0.728612	0.364306	+	0.931279i	$0.381306\pi$
$524$	0	0
$525$	−1.09520	−0.0477984
$526$	0	0
$527$	28.6302	1.24715
$528$	0	0
$529$	26.3758	1.14677
$530$	0	0
$531$	16.3771	0.710705
$532$	0	0
$533$	0	0
$534$	0	0
$535$	4.68979	0.202757
$536$	0	0
$537$	−11.1340	−0.480466
$538$	0	0
$539$	4.07771	0.175639
$540$	0	0
$541$	−16.6562	−0.716105	−0.358052	−	0.933702i	$-0.616559\pi$
$541$	−16.6562	−0.716105	−0.358052	+	0.933702i	$0.616559\pi$
$542$	0	0
$543$	−1.00840	−0.0432747
$544$	0	0
$545$	−22.1056	−0.946900
$546$	0	0
$547$	31.4415	1.34434	0.672170	−	0.740397i	$-0.265362\pi$
$547$	31.4415	1.34434	0.672170	+	0.740397i	$0.265362\pi$
$548$	0	0
$549$	27.5777	1.17699
$550$	0	0
$551$	−15.8451	−0.675026
$552$	0	0
$553$	0.869671	0.0369822
$554$	0	0
$555$	14.2397	0.604440
$556$	0	0
$557$	−27.0229	−1.14500	−0.572500	−	0.819905i	$-0.694026\pi$
$557$	−27.0229	−1.14500	−0.572500	+	0.819905i	$0.694026\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−19.3113	−0.815323
$562$	0	0
$563$	22.6244	0.953504	0.476752	−	0.879038i	$-0.341814\pi$
$563$	22.6244	0.953504	0.476752	+	0.879038i	$0.341814\pi$
$564$	0	0
$565$	−50.0830	−2.10701
$566$	0	0
$567$	−5.69657	−0.239233
$568$	0	0
$569$	18.8419	0.789894	0.394947	−	0.918704i	$-0.370763\pi$
$569$	18.8419	0.789894	0.394947	+	0.918704i	$0.370763\pi$
$570$	0	0
$571$	−24.7338	−1.03508	−0.517539	−	0.855660i	$-0.673152\pi$
$571$	−24.7338	−1.03508	−0.517539	+	0.855660i	$0.673152\pi$
$572$	0	0
$573$	7.64362	0.319317
$574$	0	0
$575$	−12.4290	−0.518325
$576$	0	0
$577$	−0.123930	−0.00515926	−0.00257963	−	0.999997i	$-0.500821\pi$
$577$	−0.123930	−0.00515926	−0.00257963	+	0.999997i	$0.500821\pi$
$578$	0	0
$579$	14.3257	0.595355
$580$	0	0
$581$	−6.34814	−0.263365
$582$	0	0
$583$	37.7223	1.56230
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.2209	0.628234	0.314117	−	0.949384i	$-0.398292\pi$
$587$	15.2209	0.628234	0.314117	+	0.949384i	$0.398292\pi$
$588$	0	0
$589$	−14.4869	−0.596922
$590$	0	0
$591$	11.1352	0.458041
$592$	0	0
$593$	6.30812	0.259043	0.129522	−	0.991577i	$-0.458656\pi$
$593$	6.30812	0.259043	0.129522	+	0.991577i	$0.458656\pi$
$594$	0	0
$595$	19.8993	0.815791
$596$	0	0
$597$	−1.51481	−0.0619971
$598$	0	0
$599$	−36.0588	−1.47332	−0.736660	−	0.676263i	$-0.763598\pi$
$599$	−36.0588	−1.47332	−0.736660	+	0.676263i	$0.763598\pi$
$600$	0	0
$601$	−12.7739	−0.521057	−0.260529	−	0.965466i	$-0.583897\pi$
$601$	−12.7739	−0.521057	−0.260529	+	0.965466i	$0.583897\pi$
$602$	0	0
$603$	24.2010	0.985540
$604$	0	0
$605$	14.6415	0.595263
$606$	0	0
$607$	−17.7815	−0.721729	−0.360865	−	0.932618i	$-0.617518\pi$
$607$	−17.7815	−0.721729	−0.360865	+	0.932618i	$0.617518\pi$
$608$	0	0
$609$	2.53500	0.102723
$610$	0	0
$611$	0	0
$612$	0	0
$613$	3.28173	0.132548	0.0662739	−	0.997801i	$-0.478889\pi$
$613$	3.28173	0.132548	0.0662739	+	0.997801i	$0.478889\pi$
$614$	0	0
$615$	−2.62197	−0.105728
$616$	0	0
$617$	−32.8337	−1.32184	−0.660918	−	0.750458i	$-0.729833\pi$
$617$	−32.8337	−1.32184	−0.660918	+	0.750458i	$0.729833\pi$
$618$	0	0
$619$	27.2348	1.09466	0.547329	−	0.836917i	$-0.315645\pi$
$619$	27.2348	1.09466	0.547329	+	0.836917i	$0.315645\pi$
$620$	0	0
$621$	24.4369	0.980619
$622$	0	0
$623$	0.774962	0.0310482
$624$	0	0
$625$	−30.7154	−1.22861
$626$	0	0
$627$	9.77154	0.390238
$628$	0	0
$629$	−67.6101	−2.69579
$630$	0	0
$631$	−16.7593	−0.667177	−0.333588	−	0.942719i	$-0.608260\pi$
$631$	−16.7593	−0.667177	−0.333588	+	0.942719i	$0.608260\pi$
$632$	0	0
$633$	−5.34016	−0.212252
$634$	0	0
$635$	11.6214	0.461181
$636$	0	0
$637$	0	0
$638$	0	0
$639$	33.5834	1.32854
$640$	0	0
$641$	2.13766	0.0844326	0.0422163	−	0.999108i	$-0.486558\pi$
$641$	2.13766	0.0844326	0.0422163	+	0.999108i	$0.486558\pi$
$642$	0	0
$643$	36.7877	1.45077	0.725384	−	0.688345i	$-0.241662\pi$
$643$	36.7877	1.45077	0.725384	+	0.688345i	$0.241662\pi$
$644$	0	0
$645$	5.24516	0.206528
$646$	0	0
$647$	11.2476	0.442188	0.221094	−	0.975253i	$-0.429037\pi$
$647$	11.2476	0.442188	0.221094	+	0.975253i	$0.429037\pi$
$648$	0	0
$649$	−25.5218	−1.00182
$650$	0	0
$651$	2.31770	0.0908377
$652$	0	0
$653$	28.5431	1.11698	0.558489	−	0.829512i	$-0.311381\pi$
$653$	28.5431	1.11698	0.558489	+	0.829512i	$0.311381\pi$
$654$	0	0
$655$	45.6915	1.78531
$656$	0	0
$657$	6.79953	0.265275
$658$	0	0
$659$	−0.348020	−0.0135569	−0.00677846	−	0.999977i	$-0.502158\pi$
$659$	−0.348020	−0.0135569	−0.00677846	+	0.999977i	$0.502158\pi$
$660$	0	0
$661$	−7.58414	−0.294989	−0.147494	−	0.989063i	$-0.547121\pi$
$661$	−7.58414	−0.294989	−0.147494	+	0.989063i	$0.547121\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−10.0691	−0.390461
$666$	0	0
$667$	28.7687	1.11393
$668$	0	0
$669$	11.3235	0.437792
$670$	0	0
$671$	−42.9767	−1.65910
$672$	0	0
$673$	−48.0175	−1.85094	−0.925470	−	0.378820i	$-0.876330\pi$
$673$	−48.0175	−1.85094	−0.925470	+	0.378820i	$0.876330\pi$
$674$	0	0
$675$	−6.15132	−0.236765
$676$	0	0
$677$	30.8431	1.18540	0.592699	−	0.805424i	$-0.298062\pi$
$677$	30.8431	1.18540	0.592699	+	0.805424i	$0.298062\pi$
$678$	0	0
$679$	14.1122	0.541577
$680$	0	0
$681$	−4.52104	−0.173247
$682$	0	0
$683$	−8.75533	−0.335013	−0.167507	−	0.985871i	$-0.553572\pi$
$683$	−8.75533	−0.335013	−0.167507	+	0.985871i	$0.553572\pi$
$684$	0	0
$685$	26.1279	0.998298
$686$	0	0
$687$	−14.9145	−0.569022
$688$	0	0
$689$	0	0
$690$	0	0
$691$	28.9115	1.09984	0.549922	−	0.835216i	$-0.314657\pi$
$691$	28.9115	1.09984	0.549922	+	0.835216i	$0.314657\pi$
$692$	0	0
$693$	10.6698	0.405313
$694$	0	0
$695$	18.8295	0.714244
$696$	0	0
$697$	12.4491	0.471544
$698$	0	0
$699$	−18.0244	−0.681745
$700$	0	0
$701$	36.4816	1.37789	0.688945	−	0.724814i	$-0.258074\pi$
$701$	36.4816	1.37789	0.688945	+	0.724814i	$0.258074\pi$
$702$	0	0
$703$	34.2108	1.29028
$704$	0	0
$705$	12.9164	0.486459
$706$	0	0
$707$	3.00119	0.112871
$708$	0	0
$709$	−8.58828	−0.322540	−0.161270	−	0.986910i	$-0.551559\pi$
$709$	−8.58828	−0.322540	−0.161270	+	0.986910i	$0.551559\pi$
$710$	0	0
$711$	2.27560	0.0853417
$712$	0	0
$713$	26.3027	0.985042
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−17.4765	−0.652673
$718$	0	0
$719$	−46.0390	−1.71697	−0.858483	−	0.512842i	$-0.828593\pi$
$719$	−46.0390	−1.71697	−0.858483	+	0.512842i	$0.828593\pi$
$720$	0	0
$721$	19.7122	0.734121
$722$	0	0
$723$	−1.52867	−0.0568517
$724$	0	0
$725$	−7.24174	−0.268951
$726$	0	0
$727$	−1.42797	−0.0529605	−0.0264802	−	0.999649i	$-0.508430\pi$
$727$	−1.42797	−0.0529605	−0.0264802	+	0.999649i	$0.508430\pi$
$728$	0	0
$729$	−8.44589	−0.312811
$730$	0	0
$731$	−24.9040	−0.921109
$732$	0	0
$733$	−1.35203	−0.0499385	−0.0249693	−	0.999688i	$-0.507949\pi$
$733$	−1.35203	−0.0499385	−0.0249693	+	0.999688i	$0.507949\pi$
$734$	0	0
$735$	1.61090	0.0594191
$736$	0	0
$737$	−37.7145	−1.38923
$738$	0	0
$739$	12.2069	0.449037	0.224518	−	0.974470i	$-0.427919\pi$
$739$	12.2069	0.449037	0.224518	+	0.974470i	$0.427919\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−7.50395	−0.275293	−0.137647	−	0.990481i	$-0.543954\pi$
$743$	−7.50395	−0.275293	−0.137647	+	0.990481i	$0.543954\pi$
$744$	0	0
$745$	29.0209	1.06324
$746$	0	0
$747$	−16.6107	−0.607753
$748$	0	0
$749$	−1.80259	−0.0658653
$750$	0	0
$751$	9.54217	0.348199	0.174099	−	0.984728i	$-0.444299\pi$
$751$	9.54217	0.348199	0.174099	+	0.984728i	$0.444299\pi$
$752$	0	0
$753$	−16.2596	−0.592535
$754$	0	0
$755$	−0.539555	−0.0196364
$756$	0	0
$757$	25.8225	0.938534	0.469267	−	0.883056i	$-0.344518\pi$
$757$	25.8225	0.938534	0.469267	+	0.883056i	$0.344518\pi$
$758$	0	0
$759$	−17.7414	−0.643971
$760$	0	0
$761$	24.7826	0.898368	0.449184	−	0.893439i	$-0.351715\pi$
$761$	24.7826	0.898368	0.449184	+	0.893439i	$0.351715\pi$
$762$	0	0
$763$	8.49662	0.307598
$764$	0	0
$765$	52.0689	1.88255
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−2.75790	−0.0994525	−0.0497262	−	0.998763i	$-0.515835\pi$
$769$	−2.75790	−0.0994525	−0.0497262	+	0.998763i	$0.515835\pi$
$770$	0	0
$771$	−6.59585	−0.237544
$772$	0	0
$773$	−41.6893	−1.49946	−0.749730	−	0.661744i	$-0.769817\pi$
$773$	−41.6893	−1.49946	−0.749730	+	0.661744i	$0.769817\pi$
$774$	0	0
$775$	−6.62098	−0.237832
$776$	0	0
$777$	−5.47323	−0.196351
$778$	0	0
$779$	−6.29927	−0.225695
$780$	0	0
$781$	−52.3359	−1.87272
$782$	0	0
$783$	14.2381	0.508829
$784$	0	0
$785$	−35.3988	−1.26344
$786$	0	0
$787$	−22.5927	−0.805341	−0.402671	−	0.915345i	$-0.631918\pi$
$787$	−22.5927	−0.805341	−0.402671	+	0.915345i	$0.631918\pi$
$788$	0	0
$789$	−5.54316	−0.197342
$790$	0	0
$791$	19.2502	0.684457
$792$	0	0
$793$	0	0
$794$	0	0
$795$	14.9023	0.528529
$796$	0	0
$797$	−27.4859	−0.973601	−0.486801	−	0.873513i	$-0.661836\pi$
$797$	−27.4859	−0.973601	−0.486801	+	0.873513i	$0.661836\pi$
$798$	0	0
$799$	−61.3271	−2.16960
$800$	0	0
$801$	2.02778	0.0716482
$802$	0	0
$803$	−10.5963	−0.373935
$804$	0	0
$805$	18.2815	0.644340
$806$	0	0
$807$	13.6697	0.481198
$808$	0	0
$809$	28.3701	0.997438	0.498719	−	0.866764i	$-0.333804\pi$
$809$	28.3701	0.997438	0.498719	+	0.866764i	$0.333804\pi$
$810$	0	0
$811$	−53.9253	−1.89357	−0.946787	−	0.321860i	$-0.895692\pi$
$811$	−53.9253	−1.89357	−0.946787	+	0.321860i	$0.895692\pi$
$812$	0	0
$813$	3.64627	0.127880
$814$	0	0
$815$	−17.8215	−0.624262
$816$	0	0
$817$	12.6015	0.440870
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.3523	−0.989502	−0.494751	−	0.869035i	$-0.664741\pi$
$821$	−28.3523	−0.989502	−0.494751	+	0.869035i	$0.664741\pi$
$822$	0	0
$823$	12.4445	0.433787	0.216893	−	0.976195i	$-0.430408\pi$
$823$	12.4445	0.433787	0.216893	+	0.976195i	$0.430408\pi$
$824$	0	0
$825$	4.46590	0.155483
$826$	0	0
$827$	54.2614	1.88685	0.943427	−	0.331581i	$-0.107582\pi$
$827$	54.2614	1.88685	0.943427	+	0.331581i	$0.107582\pi$
$828$	0	0
$829$	−18.1806	−0.631437	−0.315719	−	0.948853i	$-0.602246\pi$
$829$	−18.1806	−0.631437	−0.315719	+	0.948853i	$0.602246\pi$
$830$	0	0
$831$	3.50911	0.121730
$832$	0	0
$833$	−7.64859	−0.265008
$834$	0	0
$835$	−17.2825	−0.598085
$836$	0	0
$837$	13.0176	0.449955
$838$	0	0
$839$	−2.25075	−0.0777045	−0.0388522	−	0.999245i	$-0.512370\pi$
$839$	−2.25075	−0.0777045	−0.0388522	+	0.999245i	$0.512370\pi$
$840$	0	0
$841$	−12.2380	−0.421999
$842$	0	0
$843$	19.6225	0.675836
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.62770	−0.193370
$848$	0	0
$849$	−8.44641	−0.289880
$850$	0	0
$851$	−62.1136	−2.12923
$852$	0	0
$853$	49.4253	1.69229	0.846146	−	0.532952i	$-0.178917\pi$
$853$	49.4253	1.69229	0.846146	+	0.532952i	$0.178917\pi$
$854$	0	0
$855$	−26.3469	−0.901045
$856$	0	0
$857$	6.30457	0.215360	0.107680	−	0.994186i	$-0.465658\pi$
$857$	6.30457	0.215360	0.107680	+	0.994186i	$0.465658\pi$
$858$	0	0
$859$	−2.87727	−0.0981713	−0.0490856	−	0.998795i	$-0.515631\pi$
$859$	−2.87727	−0.0981713	−0.0490856	+	0.998795i	$0.515631\pi$
$860$	0	0
$861$	1.00779	0.0343455
$862$	0	0
$863$	43.4449	1.47888	0.739441	−	0.673221i	$-0.235090\pi$
$863$	43.4449	1.47888	0.739441	+	0.673221i	$0.235090\pi$
$864$	0	0
$865$	−0.696978	−0.0236980
$866$	0	0
$867$	25.6964	0.872694
$868$	0	0
$869$	−3.54626	−0.120299
$870$	0	0
$871$	0	0
$872$	0	0
$873$	36.9263	1.24977
$874$	0	0
$875$	8.40658	0.284194
$876$	0	0
$877$	−24.7065	−0.834278	−0.417139	−	0.908843i	$-0.636967\pi$
$877$	−24.7065	−0.834278	−0.417139	+	0.908843i	$0.636967\pi$
$878$	0	0
$879$	10.9942	0.370824
$880$	0	0
$881$	−4.47629	−0.150810	−0.0754050	−	0.997153i	$-0.524025\pi$
$881$	−4.47629	−0.150810	−0.0754050	+	0.997153i	$0.524025\pi$
$882$	0	0
$883$	−1.96708	−0.0661977	−0.0330988	−	0.999452i	$-0.510538\pi$
$883$	−1.96708	−0.0661977	−0.0330988	+	0.999452i	$0.510538\pi$
$884$	0	0
$885$	−10.0824	−0.338918
$886$	0	0
$887$	−59.4072	−1.99470	−0.997349	−	0.0727641i	$-0.976818\pi$
$887$	−59.4072	−1.99470	−0.997349	+	0.0727641i	$0.976818\pi$
$888$	0	0
$889$	−4.46686	−0.149814
$890$	0	0
$891$	23.2290	0.778199
$892$	0	0
$893$	31.0316	1.03843
$894$	0	0
$895$	−46.7834	−1.56380
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.3252	0.511124
$900$	0	0
$901$	−70.7561	−2.35723
$902$	0	0
$903$	−2.01606	−0.0670901
$904$	0	0
$905$	−4.23717	−0.140848
$906$	0	0
$907$	−30.9828	−1.02877	−0.514383	−	0.857561i	$-0.671979\pi$
$907$	−30.9828	−1.02877	−0.514383	+	0.857561i	$0.671979\pi$
$908$	0	0
$909$	7.85297	0.260466
$910$	0	0
$911$	10.8420	0.359211	0.179606	−	0.983739i	$-0.442518\pi$
$911$	10.8420	0.359211	0.179606	+	0.983739i	$0.442518\pi$
$912$	0	0
$913$	25.8859	0.856697
$914$	0	0
$915$	−16.9780	−0.561276
$916$	0	0
$917$	−17.5622	−0.579955
$918$	0	0
$919$	34.3176	1.13203	0.566016	−	0.824394i	$-0.308484\pi$
$919$	34.3176	1.13203	0.566016	+	0.824394i	$0.308484\pi$
$920$	0	0
$921$	−16.9126	−0.557288
$922$	0	0
$923$	0	0
$924$	0	0
$925$	15.6354	0.514089
$926$	0	0
$927$	51.5794	1.69409
$928$	0	0
$929$	31.9184	1.04721	0.523604	−	0.851962i	$-0.324587\pi$
$929$	31.9184	1.04721	0.523604	+	0.851962i	$0.324587\pi$
$930$	0	0
$931$	3.87020	0.126841
$932$	0	0
$933$	−6.41909	−0.210152
$934$	0	0
$935$	−81.1434	−2.65367
$936$	0	0
$937$	−11.9524	−0.390468	−0.195234	−	0.980757i	$-0.562547\pi$
$937$	−11.9524	−0.390468	−0.195234	+	0.980757i	$0.562547\pi$
$938$	0	0
$939$	11.1010	0.362267
$940$	0	0
$941$	−22.7232	−0.740754	−0.370377	−	0.928882i	$-0.620771\pi$
$941$	−22.7232	−0.740754	−0.370377	+	0.928882i	$0.620771\pi$
$942$	0	0
$943$	11.4371	0.372442
$944$	0	0
$945$	9.04784	0.294326
$946$	0	0
$947$	−28.7874	−0.935463	−0.467732	−	0.883871i	$-0.654929\pi$
$947$	−28.7874	−0.935463	−0.467732	+	0.883871i	$0.654929\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−6.30669	−0.204508
$952$	0	0
$953$	34.3797	1.11367	0.556834	−	0.830624i	$-0.312016\pi$
$953$	34.3797	1.11367	0.556834	+	0.830624i	$0.312016\pi$
$954$	0	0
$955$	32.1175	1.03930
$956$	0	0
$957$	−10.3370	−0.334147
$958$	0	0
$959$	−10.0427	−0.324295
$960$	0	0
$961$	−16.9885	−0.548015
$962$	0	0
$963$	−4.71670	−0.151994
$964$	0	0
$965$	60.1946	1.93773
$966$	0	0
$967$	−42.2318	−1.35808	−0.679042	−	0.734099i	$-0.737605\pi$
$967$	−42.2318	−1.35808	−0.679042	+	0.734099i	$0.737605\pi$
$968$	0	0
$969$	−18.3286	−0.588798
$970$	0	0
$971$	−49.1872	−1.57849	−0.789246	−	0.614077i	$-0.789528\pi$
$971$	−49.1872	−1.57849	−0.789246	+	0.614077i	$0.789528\pi$
$972$	0	0
$973$	−7.23741	−0.232021
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.72093	−0.119043	−0.0595216	−	0.998227i	$-0.518958\pi$
$977$	−3.72093	−0.119043	−0.0595216	+	0.998227i	$0.518958\pi$
$978$	0	0
$979$	−3.16007	−0.100996
$980$	0	0
$981$	22.2324	0.709827
$982$	0	0
$983$	36.3136	1.15822	0.579111	−	0.815248i	$-0.303400\pi$
$983$	36.3136	1.15822	0.579111	+	0.815248i	$0.303400\pi$
$984$	0	0
$985$	46.7886	1.49081
$986$	0	0
$987$	−4.96461	−0.158025
$988$	0	0
$989$	−22.8794	−0.727524
$990$	0	0
$991$	11.8610	0.376778	0.188389	−	0.982095i	$-0.439674\pi$
$991$	11.8610	0.376778	0.188389	+	0.982095i	$0.439674\pi$
$992$	0	0
$993$	−4.37732	−0.138910
$994$	0	0
$995$	−6.36503	−0.201785
$996$	0	0
$997$	9.62529	0.304836	0.152418	−	0.988316i	$-0.451294\pi$
$997$	9.62529	0.304836	0.152418	+	0.988316i	$0.451294\pi$
$998$	0	0
$999$	−30.7411	−0.972605

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9464.2.a.br.1.10	✓	15
13.12	even	2		9464.2.a.bs.1.10	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9464.2.a.br.1.10	✓	15	1.1	even	1	trivial
9464.2.a.bs.1.10	yes	15	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9464 = 2^{3} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9464.a (trivial)

\(f(q)\)	\(=\)	\(q+0.619176 q^{3} +2.60169 q^{5} -1.00000 q^{7} -2.61662 q^{9} +4.07771 q^{11} +1.61090 q^{15} -7.64859 q^{17} +3.87020 q^{19} -0.619176 q^{21} -7.02679 q^{23} +1.76880 q^{25} -3.47768 q^{27} -4.09415 q^{29} -3.74320 q^{31} +2.52482 q^{33} -2.60169 q^{35} +8.83955 q^{37} -1.62764 q^{41} +3.25603 q^{43} -6.80764 q^{45} +8.01809 q^{47} +1.00000 q^{49} -4.73582 q^{51} +9.25086 q^{53} +10.6089 q^{55} +2.39633 q^{57} -6.25887 q^{59} -10.5394 q^{61} +2.61662 q^{63} -9.24894 q^{67} -4.35082 q^{69} -12.8346 q^{71} -2.59859 q^{73} +1.09520 q^{75} -4.07771 q^{77} -0.869671 q^{79} +5.69657 q^{81} +6.34814 q^{83} -19.8993 q^{85} -2.53500 q^{87} -0.774962 q^{89} -2.31770 q^{93} +10.0691 q^{95} -14.1122 q^{97} -10.6698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - 3 q^{3} - 4 q^{5} - 15 q^{7} + 16 q^{9} - 15 q^{11} - 8 q^{15} + 2 q^{17} - 13 q^{19} + 3 q^{21} - 10 q^{23} + 23 q^{25} - 9 q^{27} + 25 q^{29} - 19 q^{31} + 24 q^{33} + 4 q^{35} + 2 q^{37} - 30 q^{41}+ \cdots - 70 q^{99}+O(q^{100}) \) 15 * q - 3 * q^3 - 4 * q^5 - 15 * q^7 + 16 * q^9 - 15 * q^11 - 8 * q^15 + 2 * q^17 - 13 * q^19 + 3 * q^21 - 10 * q^23 + 23 * q^25 - 9 * q^27 + 25 * q^29 - 19 * q^31 + 24 * q^33 + 4 * q^35 + 2 * q^37 - 30 * q^41 + 15 * q^43 + 2 * q^45 - 13 * q^47 + 15 * q^49 + q^51 + 62 * q^53 - 3 * q^55 + 2 * q^57 - 22 * q^59 - q^61 - 16 * q^63 - 29 * q^67 + 32 * q^69 - 20 * q^71 - 21 * q^73 - 46 * q^75 + 15 * q^77 + 29 * q^79 + 7 * q^81 - 23 * q^83 - 8 * q^85 - 22 * q^87 - 9 * q^89 - 10 * q^93 + 28 * q^95 - 11 * q^97 - 70 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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