LMFDB - Embedded newform 2912.2.a.s.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2912,2,Mod(1,2912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2912, 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("2912.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2912 = 2^{5} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2912.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,3,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.2524370686$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.6329476.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 13x^{3} - 2x^{2} + 40x + 18 $ x^5 - 13x^3 - 2x^2 + 40*x + 18
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.86011$ of defining polynomial
Character	$\chi$	$=$	2912.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.86011 q^{3} +2.19539 q^{5} +1.00000 q^{7} +5.18025 q^{9} -0.804606 q^{11} +1.00000 q^{13} +6.27907 q^{15} +5.37564 q^{17} -6.55589 q^{19} +2.86011 q^{21} -0.238713 q^{23} -0.180247 q^{25} +6.23575 q^{27} -4.08368 q^{29} +9.70508 q^{31} -2.30127 q^{33} +2.19539 q^{35} +7.76355 q^{37} +2.86011 q^{39} +0.155033 q^{41} -3.18025 q^{43} +11.3727 q^{45} -1.55885 q^{47} +1.00000 q^{49} +15.3749 q^{51} +4.53998 q^{53} -1.76643 q^{55} -18.7506 q^{57} -7.29422 q^{59} -6.60625 q^{61} +5.18025 q^{63} +2.19539 q^{65} +0.636546 q^{67} -0.682747 q^{69} +7.80165 q^{71} -9.97485 q^{73} -0.515528 q^{75} -0.804606 q^{77} -11.4160 q^{79} +2.29422 q^{81} +9.85011 q^{83} +11.8016 q^{85} -11.6798 q^{87} +5.26167 q^{89} +1.00000 q^{91} +27.7576 q^{93} -14.3928 q^{95} +1.02591 q^{97} -4.16806 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{5} + 5 q^{7} + 11 q^{9} - 12 q^{11} + 5 q^{13} + 2 q^{15} + 4 q^{17} + 5 q^{19} - q^{23} + 14 q^{25} - 6 q^{27} + q^{29} + 13 q^{31} + 2 q^{33} + 3 q^{35} + 18 q^{37} + 22 q^{41} - q^{43}+ \cdots - 36 q^{99}+O(q^{100}) $ 5 * q + 3 * q^5 + 5 * q^7 + 11 * q^9 - 12 * q^11 + 5 * q^13 + 2 * q^15 + 4 * q^17 + 5 * q^19 - q^23 + 14 * q^25 - 6 * q^27 + q^29 + 13 * q^31 + 2 * q^33 + 3 * q^35 + 18 * q^37 + 22 * q^41 - q^43 - 3 * q^45 - 7 * q^47 + 5 * q^49 - 4 * q^51 + 9 * q^53 + 30 * q^55 + 10 * q^57 - 14 * q^59 - 12 * q^61 + 11 * q^63 + 3 * q^65 - 4 * q^67 + 14 * q^69 + 10 * q^71 + 3 * q^73 + 6 * q^75 - 12 * q^77 - 5 * q^79 - 11 * q^81 - 11 * q^83 + 30 * q^85 - 4 * q^87 + 11 * q^89 + 5 * q^91 + 44 * q^93 - 15 * q^95 + 27 * q^97 - 36 * 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
$3$	2.86011	1.65129	0.825644	−	0.564192i	$-0.190812\pi$
$3$	2.86011	1.65129	0.825644	+	0.564192i	$0.190812\pi$
$4$	0	0
$5$	2.19539	0.981810	0.490905	−	0.871213i	$-0.336666\pi$
$5$	2.19539	0.981810	0.490905	+	0.871213i	$0.336666\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	5.18025	1.72675
$10$	0	0
$11$	−0.804606	−0.242598	−0.121299	−	0.992616i	$-0.538706\pi$
$11$	−0.804606	−0.242598	−0.121299	+	0.992616i	$0.538706\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	6.27907	1.62125
$16$	0	0
$17$	5.37564	1.30378	0.651892	−	0.758312i	$-0.273975\pi$
$17$	5.37564	1.30378	0.651892	+	0.758312i	$0.273975\pi$
$18$	0	0
$19$	−6.55589	−1.50402	−0.752012	−	0.659149i	$-0.770916\pi$
$19$	−6.55589	−1.50402	−0.752012	+	0.659149i	$0.770916\pi$
$20$	0	0
$21$	2.86011	0.624128
$22$	0	0
$23$	−0.238713	−0.0497752	−0.0248876	−	0.999690i	$-0.507923\pi$
$23$	−0.238713	−0.0497752	−0.0248876	+	0.999690i	$0.507923\pi$
$24$	0	0
$25$	−0.180247	−0.0360495
$26$	0	0
$27$	6.23575	1.20007
$28$	0	0
$29$	−4.08368	−0.758320	−0.379160	−	0.925331i	$-0.623787\pi$
$29$	−4.08368	−0.758320	−0.379160	+	0.925331i	$0.623787\pi$
$30$	0	0
$31$	9.70508	1.74308	0.871542	−	0.490321i	$-0.163120\pi$
$31$	9.70508	1.74308	0.871542	+	0.490321i	$0.163120\pi$
$32$	0	0
$33$	−2.30127	−0.400599
$34$	0	0
$35$	2.19539	0.371089
$36$	0	0
$37$	7.76355	1.27632	0.638160	−	0.769904i	$-0.279696\pi$
$37$	7.76355	1.27632	0.638160	+	0.769904i	$0.279696\pi$
$38$	0	0
$39$	2.86011	0.457985
$40$	0	0
$41$	0.155033	0.0242121	0.0121060	−	0.999927i	$-0.496146\pi$
$41$	0.155033	0.0242121	0.0121060	+	0.999927i	$0.496146\pi$
$42$	0	0
$43$	−3.18025	−0.484983	−0.242492	−	0.970154i	$-0.577965\pi$
$43$	−3.18025	−0.484983	−0.242492	+	0.970154i	$0.577965\pi$
$44$	0	0
$45$	11.3727	1.69534
$46$	0	0
$47$	−1.55885	−0.227381	−0.113691	−	0.993516i	$-0.536267\pi$
$47$	−1.55885	−0.227381	−0.113691	+	0.993516i	$0.536267\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	15.3749	2.15292
$52$	0	0
$53$	4.53998	0.623614	0.311807	−	0.950145i	$-0.399066\pi$
$53$	4.53998	0.623614	0.311807	+	0.950145i	$0.399066\pi$
$54$	0	0
$55$	−1.76643	−0.238185
$56$	0	0
$57$	−18.7506	−2.48358
$58$	0	0
$59$	−7.29422	−0.949627	−0.474813	−	0.880087i	$-0.657484\pi$
$59$	−7.29422	−0.949627	−0.474813	+	0.880087i	$0.657484\pi$
$60$	0	0
$61$	−6.60625	−0.845844	−0.422922	−	0.906166i	$-0.638996\pi$
$61$	−6.60625	−0.845844	−0.422922	+	0.906166i	$0.638996\pi$
$62$	0	0
$63$	5.18025	0.652650
$64$	0	0
$65$	2.19539	0.272305
$66$	0	0
$67$	0.636546	0.0777665	0.0388832	−	0.999244i	$-0.487620\pi$
$67$	0.636546	0.0777665	0.0388832	+	0.999244i	$0.487620\pi$
$68$	0	0
$69$	−0.682747	−0.0821931
$70$	0	0
$71$	7.80165	0.925885	0.462943	−	0.886388i	$-0.346794\pi$
$71$	7.80165	0.925885	0.462943	+	0.886388i	$0.346794\pi$
$72$	0	0
$73$	−9.97485	−1.16747	−0.583734	−	0.811945i	$-0.698409\pi$
$73$	−9.97485	−1.16747	−0.583734	+	0.811945i	$0.698409\pi$
$74$	0	0
$75$	−0.515528	−0.0595280
$76$	0	0
$77$	−0.804606	−0.0916934
$78$	0	0
$79$	−11.4160	−1.28440	−0.642200	−	0.766537i	$-0.721978\pi$
$79$	−11.4160	−1.28440	−0.642200	+	0.766537i	$0.721978\pi$
$80$	0	0
$81$	2.29422	0.254913
$82$	0	0
$83$	9.85011	1.08119	0.540595	−	0.841283i	$-0.318199\pi$
$83$	9.85011	1.08119	0.540595	+	0.841283i	$0.318199\pi$
$84$	0	0
$85$	11.8016	1.28007
$86$	0	0
$87$	−11.6798	−1.25220
$88$	0	0
$89$	5.26167	0.557736	0.278868	−	0.960329i	$-0.410041\pi$
$89$	5.26167	0.557736	0.278868	+	0.960329i	$0.410041\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	27.7576	2.87833
$94$	0	0
$95$	−14.3928	−1.47667
$96$	0	0
$97$	1.02591	0.104166	0.0520829	−	0.998643i	$-0.483414\pi$
$97$	1.02591	0.104166	0.0520829	+	0.998643i	$0.483414\pi$
$98$	0	0
$99$	−4.16806	−0.418906
$100$	0	0
$101$	4.09883	0.407848	0.203924	−	0.978987i	$-0.434630\pi$
$101$	4.09883	0.407848	0.203924	+	0.978987i	$0.434630\pi$
$102$	0	0
$103$	10.5581	1.04033	0.520163	−	0.854067i	$-0.325871\pi$
$103$	10.5581	1.04033	0.520163	+	0.854067i	$0.325871\pi$
$104$	0	0
$105$	6.27907	0.612775
$106$	0	0
$107$	−13.5681	−1.31168	−0.655838	−	0.754902i	$-0.727684\pi$
$107$	−13.5681	−1.31168	−0.655838	+	0.754902i	$0.727684\pi$
$108$	0	0
$109$	7.88829	0.755561	0.377780	−	0.925895i	$-0.376687\pi$
$109$	7.88829	0.755561	0.377780	+	0.925895i	$0.376687\pi$
$110$	0	0
$111$	22.2046	2.10757
$112$	0	0
$113$	−9.14623	−0.860405	−0.430203	−	0.902732i	$-0.641558\pi$
$113$	−9.14623	−0.860405	−0.430203	+	0.902732i	$0.641558\pi$
$114$	0	0
$115$	−0.524070	−0.0488698
$116$	0	0
$117$	5.18025	0.478914
$118$	0	0
$119$	5.37564	0.492784
$120$	0	0
$121$	−10.3526	−0.941146
$122$	0	0
$123$	0.443412	0.0399811
$124$	0	0
$125$	−11.3727	−1.01720
$126$	0	0
$127$	−10.4686	−0.928940	−0.464470	−	0.885589i	$-0.653755\pi$
$127$	−10.4686	−0.928940	−0.464470	+	0.885589i	$0.653755\pi$
$128$	0	0
$129$	−9.09587	−0.800846
$130$	0	0
$131$	−13.0800	−1.14280	−0.571401	−	0.820671i	$-0.693600\pi$
$131$	−13.0800	−1.14280	−0.571401	+	0.820671i	$0.693600\pi$
$132$	0	0
$133$	−6.55589	−0.568468
$134$	0	0
$135$	13.6899	1.17824
$136$	0	0
$137$	16.5529	1.41421	0.707106	−	0.707107i	$-0.250000\pi$
$137$	16.5529	1.41421	0.707106	+	0.707107i	$0.250000\pi$
$138$	0	0
$139$	6.04620	0.512832	0.256416	−	0.966567i	$-0.417458\pi$
$139$	6.04620	0.512832	0.256416	+	0.966567i	$0.417458\pi$
$140$	0	0
$141$	−4.45848	−0.375472
$142$	0	0
$143$	−0.804606	−0.0672846
$144$	0	0
$145$	−8.96529	−0.744526
$146$	0	0
$147$	2.86011	0.235898
$148$	0	0
$149$	7.76355	0.636014	0.318007	−	0.948088i	$-0.396986\pi$
$149$	7.76355	0.636014	0.318007	+	0.948088i	$0.396986\pi$
$150$	0	0
$151$	−9.88829	−0.804697	−0.402349	−	0.915487i	$-0.631806\pi$
$151$	−9.88829	−0.804697	−0.402349	+	0.915487i	$0.631806\pi$
$152$	0	0
$153$	27.8471	2.25131
$154$	0	0
$155$	21.3065	1.71138
$156$	0	0
$157$	−0.0988266	−0.00788722	−0.00394361	−	0.999992i	$-0.501255\pi$
$157$	−0.0988266	−0.00788722	−0.00394361	+	0.999992i	$0.501255\pi$
$158$	0	0
$159$	12.9849	1.02977
$160$	0	0
$161$	−0.238713	−0.0188133
$162$	0	0
$163$	−15.8312	−1.24000	−0.619999	−	0.784602i	$-0.712867\pi$
$163$	−15.8312	−1.24000	−0.619999	+	0.784602i	$0.712867\pi$
$164$	0	0
$165$	−5.05218	−0.393312
$166$	0	0
$167$	−19.7166	−1.52571	−0.762857	−	0.646567i	$-0.776204\pi$
$167$	−19.7166	−1.52571	−0.762857	+	0.646567i	$0.776204\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−33.9611	−2.59707
$172$	0	0
$173$	−13.6843	−1.04040	−0.520199	−	0.854045i	$-0.674142\pi$
$173$	−13.6843	−1.04040	−0.520199	+	0.854045i	$0.674142\pi$
$174$	0	0
$175$	−0.180247	−0.0136254
$176$	0	0
$177$	−20.8623	−1.56811
$178$	0	0
$179$	−7.37790	−0.551450	−0.275725	−	0.961237i	$-0.588918\pi$
$179$	−7.37790	−0.551450	−0.275725	+	0.961237i	$0.588918\pi$
$180$	0	0
$181$	22.4534	1.66895	0.834474	−	0.551047i	$-0.185772\pi$
$181$	22.4534	1.66895	0.834474	+	0.551047i	$0.185772\pi$
$182$	0	0
$183$	−18.8946	−1.39673
$184$	0	0
$185$	17.0440	1.25310
$186$	0	0
$187$	−4.32528	−0.316295
$188$	0	0
$189$	6.23575	0.453584
$190$	0	0
$191$	7.48735	0.541766	0.270883	−	0.962612i	$-0.412684\pi$
$191$	7.48735	0.541766	0.270883	+	0.962612i	$0.412684\pi$
$192$	0	0
$193$	5.24280	0.377385	0.188692	−	0.982036i	$-0.439575\pi$
$193$	5.24280	0.377385	0.188692	+	0.982036i	$0.439575\pi$
$194$	0	0
$195$	6.27907	0.449654
$196$	0	0
$197$	25.4645	1.81427	0.907134	−	0.420842i	$-0.138265\pi$
$197$	25.4645	1.81427	0.907134	+	0.420842i	$0.138265\pi$
$198$	0	0
$199$	12.8679	0.912183	0.456091	−	0.889933i	$-0.349249\pi$
$199$	12.8679	0.912183	0.456091	+	0.889933i	$0.349249\pi$
$200$	0	0
$201$	1.82059	0.128415
$202$	0	0
$203$	−4.08368	−0.286618
$204$	0	0
$205$	0.340359	0.0237717
$206$	0	0
$207$	−1.23659	−0.0859493
$208$	0	0
$209$	5.27491	0.364873
$210$	0	0
$211$	−12.0534	−0.829789	−0.414895	−	0.909869i	$-0.636182\pi$
$211$	−12.0534	−0.829789	−0.414895	+	0.909869i	$0.636182\pi$
$212$	0	0
$213$	22.3136	1.52890
$214$	0	0
$215$	−6.98189	−0.476161
$216$	0	0
$217$	9.70508	0.658824
$218$	0	0
$219$	−28.5292	−1.92782
$220$	0	0
$221$	5.37564	0.361605
$222$	0	0
$223$	−19.8968	−1.33239	−0.666195	−	0.745778i	$-0.732078\pi$
$223$	−19.8968	−1.33239	−0.666195	+	0.745778i	$0.732078\pi$
$224$	0	0
$225$	−0.933726	−0.0622484
$226$	0	0
$227$	−24.8926	−1.65218	−0.826090	−	0.563539i	$-0.809439\pi$
$227$	−24.8926	−1.65218	−0.826090	+	0.563539i	$0.809439\pi$
$228$	0	0
$229$	−26.7044	−1.76468	−0.882338	−	0.470617i	$-0.844031\pi$
$229$	−26.7044	−1.76468	−0.882338	+	0.470617i	$0.844031\pi$
$230$	0	0
$231$	−2.30127	−0.151412
$232$	0	0
$233$	25.8809	1.69551	0.847757	−	0.530384i	$-0.177952\pi$
$233$	25.8809	1.69551	0.847757	+	0.530384i	$0.177952\pi$
$234$	0	0
$235$	−3.42228	−0.223245
$236$	0	0
$237$	−32.6511	−2.12091
$238$	0	0
$239$	−1.60329	−0.103708	−0.0518542	−	0.998655i	$-0.516513\pi$
$239$	−1.60329	−0.103708	−0.0518542	+	0.998655i	$0.516513\pi$
$240$	0	0
$241$	10.0023	0.644302	0.322151	−	0.946688i	$-0.395594\pi$
$241$	10.0023	0.644302	0.322151	+	0.946688i	$0.395594\pi$
$242$	0	0
$243$	−12.1455	−0.779136
$244$	0	0
$245$	2.19539	0.140259
$246$	0	0
$247$	−6.55589	−0.417141
$248$	0	0
$249$	28.1724	1.78535
$250$	0	0
$251$	−6.31225	−0.398426	−0.199213	−	0.979956i	$-0.563838\pi$
$251$	−6.31225	−0.398426	−0.199213	+	0.979956i	$0.563838\pi$
$252$	0	0
$253$	0.192070	0.0120754
$254$	0	0
$255$	33.7540	2.11376
$256$	0	0
$257$	−30.0952	−1.87729	−0.938644	−	0.344886i	$-0.887917\pi$
$257$	−30.0952	−1.87729	−0.938644	+	0.344886i	$0.887917\pi$
$258$	0	0
$259$	7.76355	0.482403
$260$	0	0
$261$	−21.1545	−1.30943
$262$	0	0
$263$	15.2121	0.938016	0.469008	−	0.883194i	$-0.344612\pi$
$263$	15.2121	0.938016	0.469008	+	0.883194i	$0.344612\pi$
$264$	0	0
$265$	9.96704	0.612270
$266$	0	0
$267$	15.0490	0.920982
$268$	0	0
$269$	8.83420	0.538631	0.269315	−	0.963052i	$-0.413203\pi$
$269$	8.83420	0.538631	0.269315	+	0.963052i	$0.413203\pi$
$270$	0	0
$271$	14.3945	0.874404	0.437202	−	0.899363i	$-0.355969\pi$
$271$	14.3945	0.874404	0.437202	+	0.899363i	$0.355969\pi$
$272$	0	0
$273$	2.86011	0.173102
$274$	0	0
$275$	0.145028	0.00874553
$276$	0	0
$277$	22.6359	1.36006	0.680030	−	0.733184i	$-0.261967\pi$
$277$	22.6359	1.36006	0.680030	+	0.733184i	$0.261967\pi$
$278$	0	0
$279$	50.2747	3.00987
$280$	0	0
$281$	2.39009	0.142581	0.0712903	−	0.997456i	$-0.477288\pi$
$281$	2.39009	0.142581	0.0712903	+	0.997456i	$0.477288\pi$
$282$	0	0
$283$	−33.2795	−1.97826	−0.989130	−	0.147040i	$-0.953025\pi$
$283$	−33.2795	−1.97826	−0.989130	+	0.147040i	$0.953025\pi$
$284$	0	0
$285$	−41.1649	−2.43840
$286$	0	0
$287$	0.155033	0.00915131
$288$	0	0
$289$	11.8975	0.699854
$290$	0	0
$291$	2.93423	0.172008
$292$	0	0
$293$	−21.6570	−1.26521	−0.632607	−	0.774473i	$-0.718015\pi$
$293$	−21.6570	−1.26521	−0.632607	+	0.774473i	$0.718015\pi$
$294$	0	0
$295$	−16.0137	−0.932353
$296$	0	0
$297$	−5.01733	−0.291135
$298$	0	0
$299$	−0.238713	−0.0138052
$300$	0	0
$301$	−3.18025	−0.183306
$302$	0	0
$303$	11.7231	0.673475
$304$	0	0
$305$	−14.5033	−0.830458
$306$	0	0
$307$	30.0721	1.71631	0.858154	−	0.513393i	$-0.171612\pi$
$307$	30.0721	1.71631	0.858154	+	0.513393i	$0.171612\pi$
$308$	0	0
$309$	30.1975	1.71788
$310$	0	0
$311$	−9.71753	−0.551030	−0.275515	−	0.961297i	$-0.588848\pi$
$311$	−9.71753	−0.551030	−0.275515	+	0.961297i	$0.588848\pi$
$312$	0	0
$313$	−5.18251	−0.292933	−0.146466	−	0.989216i	$-0.546790\pi$
$313$	−5.18251	−0.292933	−0.146466	+	0.989216i	$0.546790\pi$
$314$	0	0
$315$	11.3727	0.640778
$316$	0	0
$317$	20.7376	1.16474	0.582369	−	0.812925i	$-0.302126\pi$
$317$	20.7376	1.16474	0.582369	+	0.812925i	$0.302126\pi$
$318$	0	0
$319$	3.28576	0.183967
$320$	0	0
$321$	−38.8062	−2.16595
$322$	0	0
$323$	−35.2421	−1.96092
$324$	0	0
$325$	−0.180247	−0.00999832
$326$	0	0
$327$	22.5614	1.24765
$328$	0	0
$329$	−1.55885	−0.0859420
$330$	0	0
$331$	25.7424	1.41493	0.707465	−	0.706748i	$-0.249839\pi$
$331$	25.7424	1.41493	0.707465	+	0.706748i	$0.249839\pi$
$332$	0	0
$333$	40.2171	2.20388
$334$	0	0
$335$	1.39747	0.0763519
$336$	0	0
$337$	−1.23864	−0.0674728	−0.0337364	−	0.999431i	$-0.510741\pi$
$337$	−1.23864	−0.0674728	−0.0337364	+	0.999431i	$0.510741\pi$
$338$	0	0
$339$	−26.1593	−1.42078
$340$	0	0
$341$	−7.80877	−0.422869
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−1.49890	−0.0806980
$346$	0	0
$347$	−15.4405	−0.828887	−0.414443	−	0.910075i	$-0.636024\pi$
$347$	−15.4405	−0.828887	−0.414443	+	0.910075i	$0.636024\pi$
$348$	0	0
$349$	−19.6813	−1.05352	−0.526759	−	0.850015i	$-0.676593\pi$
$349$	−19.6813	−1.05352	−0.526759	+	0.850015i	$0.676593\pi$
$350$	0	0
$351$	6.23575	0.332840
$352$	0	0
$353$	−15.2763	−0.813073	−0.406537	−	0.913635i	$-0.633264\pi$
$353$	−15.2763	−0.813073	−0.406537	+	0.913635i	$0.633264\pi$
$354$	0	0
$355$	17.1277	0.909043
$356$	0	0
$357$	15.3749	0.813728
$358$	0	0
$359$	12.6641	0.668386	0.334193	−	0.942505i	$-0.391536\pi$
$359$	12.6641	0.668386	0.334193	+	0.942505i	$0.391536\pi$
$360$	0	0
$361$	23.9797	1.26209
$362$	0	0
$363$	−29.6096	−1.55410
$364$	0	0
$365$	−21.8987	−1.14623
$366$	0	0
$367$	8.14553	0.425193	0.212597	−	0.977140i	$-0.431808\pi$
$367$	8.14553	0.425193	0.212597	+	0.977140i	$0.431808\pi$
$368$	0	0
$369$	0.803110	0.0418082
$370$	0	0
$371$	4.53998	0.235704
$372$	0	0
$373$	4.07003	0.210738	0.105369	−	0.994433i	$-0.466398\pi$
$373$	4.07003	0.210738	0.105369	+	0.994433i	$0.466398\pi$
$374$	0	0
$375$	−32.5272	−1.67970
$376$	0	0
$377$	−4.08368	−0.210320
$378$	0	0
$379$	−0.421779	−0.0216653	−0.0108327	−	0.999941i	$-0.503448\pi$
$379$	−0.421779	−0.0216653	−0.0108327	+	0.999941i	$0.503448\pi$
$380$	0	0
$381$	−29.9415	−1.53395
$382$	0	0
$383$	8.55583	0.437182	0.218591	−	0.975817i	$-0.429854\pi$
$383$	8.55583	0.437182	0.218591	+	0.975817i	$0.429854\pi$
$384$	0	0
$385$	−1.76643	−0.0900255
$386$	0	0
$387$	−16.4745	−0.837444
$388$	0	0
$389$	−29.5622	−1.49886	−0.749430	−	0.662083i	$-0.769673\pi$
$389$	−29.5622	−1.49886	−0.749430	+	0.662083i	$0.769673\pi$
$390$	0	0
$391$	−1.28324	−0.0648961
$392$	0	0
$393$	−37.4102	−1.88709
$394$	0	0
$395$	−25.0626	−1.26104
$396$	0	0
$397$	−2.76271	−0.138656	−0.0693281	−	0.997594i	$-0.522086\pi$
$397$	−2.76271	−0.138656	−0.0693281	+	0.997594i	$0.522086\pi$
$398$	0	0
$399$	−18.7506	−0.938703
$400$	0	0
$401$	21.2414	1.06074	0.530372	−	0.847765i	$-0.322052\pi$
$401$	21.2414	1.06074	0.530372	+	0.847765i	$0.322052\pi$
$402$	0	0
$403$	9.70508	0.483444
$404$	0	0
$405$	5.03672	0.250276
$406$	0	0
$407$	−6.24660	−0.309632
$408$	0	0
$409$	31.7459	1.56973	0.784866	−	0.619665i	$-0.212732\pi$
$409$	31.7459	1.56973	0.784866	+	0.619665i	$0.212732\pi$
$410$	0	0
$411$	47.3432	2.33527
$412$	0	0
$413$	−7.29422	−0.358925
$414$	0	0
$415$	21.6249	1.06152
$416$	0	0
$417$	17.2928	0.846833
$418$	0	0
$419$	3.94007	0.192485	0.0962425	−	0.995358i	$-0.469318\pi$
$419$	3.94007	0.192485	0.0962425	+	0.995358i	$0.469318\pi$
$420$	0	0
$421$	−15.0730	−0.734612	−0.367306	−	0.930100i	$-0.619720\pi$
$421$	−15.0730	−0.734612	−0.367306	+	0.930100i	$0.619720\pi$
$422$	0	0
$423$	−8.07522	−0.392630
$424$	0	0
$425$	−0.968945	−0.0470007
$426$	0	0
$427$	−6.60625	−0.319699
$428$	0	0
$429$	−2.30127	−0.111106
$430$	0	0
$431$	32.4715	1.56410	0.782049	−	0.623217i	$-0.214175\pi$
$431$	32.4715	1.56410	0.782049	+	0.623217i	$0.214175\pi$
$432$	0	0
$433$	36.0001	1.73006	0.865028	−	0.501724i	$-0.167301\pi$
$433$	36.0001	1.73006	0.865028	+	0.501724i	$0.167301\pi$
$434$	0	0
$435$	−25.6417	−1.22943
$436$	0	0
$437$	1.56498	0.0748631
$438$	0	0
$439$	0.491518	0.0234589	0.0117294	−	0.999931i	$-0.496266\pi$
$439$	0.491518	0.0234589	0.0117294	+	0.999931i	$0.496266\pi$
$440$	0	0
$441$	5.18025	0.246678
$442$	0	0
$443$	−34.1067	−1.62046	−0.810228	−	0.586115i	$-0.800657\pi$
$443$	−34.1067	−1.62046	−0.810228	+	0.586115i	$0.800657\pi$
$444$	0	0
$445$	11.5514	0.547590
$446$	0	0
$447$	22.2046	1.05024
$448$	0	0
$449$	−33.7749	−1.59394	−0.796968	−	0.604022i	$-0.793564\pi$
$449$	−33.7749	−1.59394	−0.796968	+	0.604022i	$0.793564\pi$
$450$	0	0
$451$	−0.124741	−0.00587381
$452$	0	0
$453$	−28.2816	−1.32879
$454$	0	0
$455$	2.19539	0.102922
$456$	0	0
$457$	25.5019	1.19293	0.596463	−	0.802640i	$-0.296572\pi$
$457$	25.5019	1.19293	0.596463	+	0.802640i	$0.296572\pi$
$458$	0	0
$459$	33.5212	1.56463
$460$	0	0
$461$	−42.1963	−1.96528	−0.982638	−	0.185535i	$-0.940598\pi$
$461$	−42.1963	−1.96528	−0.982638	+	0.185535i	$0.940598\pi$
$462$	0	0
$463$	−28.4722	−1.32322	−0.661608	−	0.749850i	$-0.730126\pi$
$463$	−28.4722	−1.32322	−0.661608	+	0.749850i	$0.730126\pi$
$464$	0	0
$465$	60.9389	2.82597
$466$	0	0
$467$	19.9782	0.924480	0.462240	−	0.886755i	$-0.347046\pi$
$467$	19.9782	0.924480	0.462240	+	0.886755i	$0.347046\pi$
$468$	0	0
$469$	0.636546	0.0293930
$470$	0	0
$471$	−0.282655	−0.0130241
$472$	0	0
$473$	2.55885	0.117656
$474$	0	0
$475$	1.18168	0.0542192
$476$	0	0
$477$	23.5182	1.07683
$478$	0	0
$479$	20.2624	0.925814	0.462907	−	0.886407i	$-0.346806\pi$
$479$	20.2624	0.925814	0.462907	+	0.886407i	$0.346806\pi$
$480$	0	0
$481$	7.76355	0.353987
$482$	0	0
$483$	−0.682747	−0.0310661
$484$	0	0
$485$	2.25228	0.102271
$486$	0	0
$487$	4.80763	0.217854	0.108927	−	0.994050i	$-0.465258\pi$
$487$	4.80763	0.217854	0.108927	+	0.994050i	$0.465258\pi$
$488$	0	0
$489$	−45.2791	−2.04759
$490$	0	0
$491$	34.6844	1.56528	0.782642	−	0.622472i	$-0.213872\pi$
$491$	34.6844	1.56528	0.782642	+	0.622472i	$0.213872\pi$
$492$	0	0
$493$	−21.9524	−0.988686
$494$	0	0
$495$	−9.15053	−0.411286
$496$	0	0
$497$	7.80165	0.349952
$498$	0	0
$499$	15.3871	0.688822	0.344411	−	0.938819i	$-0.388079\pi$
$499$	15.3871	0.688822	0.344411	+	0.938819i	$0.388079\pi$
$500$	0	0
$501$	−56.3916	−2.51939
$502$	0	0
$503$	−21.1910	−0.944859	−0.472429	−	0.881369i	$-0.656623\pi$
$503$	−21.1910	−0.944859	−0.472429	+	0.881369i	$0.656623\pi$
$504$	0	0
$505$	8.99854	0.400430
$506$	0	0
$507$	2.86011	0.127022
$508$	0	0
$509$	19.0626	0.844936	0.422468	−	0.906378i	$-0.361164\pi$
$509$	19.0626	0.844936	0.422468	+	0.906378i	$0.361164\pi$
$510$	0	0
$511$	−9.97485	−0.441261
$512$	0	0
$513$	−40.8809	−1.80494
$514$	0	0
$515$	23.1793	1.02140
$516$	0	0
$517$	1.25426	0.0551622
$518$	0	0
$519$	−39.1387	−1.71800
$520$	0	0
$521$	18.5663	0.813405	0.406703	−	0.913561i	$-0.366679\pi$
$521$	18.5663	0.813405	0.406703	+	0.913561i	$0.366679\pi$
$522$	0	0
$523$	8.96690	0.392095	0.196048	−	0.980594i	$-0.437189\pi$
$523$	8.96690	0.392095	0.196048	+	0.980594i	$0.437189\pi$
$524$	0	0
$525$	−0.515528	−0.0224995
$526$	0	0
$527$	52.1710	2.27261
$528$	0	0
$529$	−22.9430	−0.997522
$530$	0	0
$531$	−37.7859	−1.63977
$532$	0	0
$533$	0.155033	0.00671523
$534$	0	0
$535$	−29.7873	−1.28782
$536$	0	0
$537$	−21.1016	−0.910603
$538$	0	0
$539$	−0.804606	−0.0346569
$540$	0	0
$541$	5.36043	0.230463	0.115231	−	0.993339i	$-0.463239\pi$
$541$	5.36043	0.230463	0.115231	+	0.993339i	$0.463239\pi$
$542$	0	0
$543$	64.2193	2.75591
$544$	0	0
$545$	17.3179	0.741817
$546$	0	0
$547$	−2.62169	−0.112095	−0.0560477	−	0.998428i	$-0.517850\pi$
$547$	−2.62169	−0.112095	−0.0560477	+	0.998428i	$0.517850\pi$
$548$	0	0
$549$	−34.2220	−1.46056
$550$	0	0
$551$	26.7722	1.14053
$552$	0	0
$553$	−11.4160	−0.485458
$554$	0	0
$555$	48.7479	2.06923
$556$	0	0
$557$	−22.2949	−0.944667	−0.472333	−	0.881420i	$-0.656588\pi$
$557$	−22.2949	−0.944667	−0.472333	+	0.881420i	$0.656588\pi$
$558$	0	0
$559$	−3.18025	−0.134510
$560$	0	0
$561$	−12.3708	−0.522295
$562$	0	0
$563$	−20.3957	−0.859575	−0.429788	−	0.902930i	$-0.641412\pi$
$563$	−20.3957	−0.859575	−0.429788	+	0.902930i	$0.641412\pi$
$564$	0	0
$565$	−20.0796	−0.844754
$566$	0	0
$567$	2.29422	0.0963482
$568$	0	0
$569$	−15.4832	−0.649089	−0.324545	−	0.945870i	$-0.605211\pi$
$569$	−15.4832	−0.649089	−0.324545	+	0.945870i	$0.605211\pi$
$570$	0	0
$571$	42.8075	1.79144	0.895719	−	0.444620i	$-0.146661\pi$
$571$	42.8075	1.79144	0.895719	+	0.444620i	$0.146661\pi$
$572$	0	0
$573$	21.4147	0.894611
$574$	0	0
$575$	0.0430275	0.00179437
$576$	0	0
$577$	19.2287	0.800502	0.400251	−	0.916406i	$-0.368923\pi$
$577$	19.2287	0.800502	0.400251	+	0.916406i	$0.368923\pi$
$578$	0	0
$579$	14.9950	0.623171
$580$	0	0
$581$	9.85011	0.408651
$582$	0	0
$583$	−3.65290	−0.151288
$584$	0	0
$585$	11.3727	0.470203
$586$	0	0
$587$	40.1383	1.65669	0.828343	−	0.560221i	$-0.189284\pi$
$587$	40.1383	1.65669	0.828343	+	0.560221i	$0.189284\pi$
$588$	0	0
$589$	−63.6254	−2.62164
$590$	0	0
$591$	72.8312	2.99588
$592$	0	0
$593$	18.2014	0.747443	0.373722	−	0.927541i	$-0.378081\pi$
$593$	18.2014	0.747443	0.373722	+	0.927541i	$0.378081\pi$
$594$	0	0
$595$	11.8016	0.483820
$596$	0	0
$597$	36.8037	1.50628
$598$	0	0
$599$	−28.2568	−1.15454	−0.577270	−	0.816553i	$-0.695882\pi$
$599$	−28.2568	−1.15454	−0.577270	+	0.816553i	$0.695882\pi$
$600$	0	0
$601$	16.1399	0.658361	0.329180	−	0.944267i	$-0.393228\pi$
$601$	16.1399	0.658361	0.329180	+	0.944267i	$0.393228\pi$
$602$	0	0
$603$	3.29747	0.134283
$604$	0	0
$605$	−22.7280	−0.924027
$606$	0	0
$607$	−17.2690	−0.700927	−0.350463	−	0.936576i	$-0.613976\pi$
$607$	−17.2690	−0.700927	−0.350463	+	0.936576i	$0.613976\pi$
$608$	0	0
$609$	−11.6798	−0.473289
$610$	0	0
$611$	−1.55885	−0.0630642
$612$	0	0
$613$	5.04762	0.203871	0.101936	−	0.994791i	$-0.467496\pi$
$613$	5.04762	0.203871	0.101936	+	0.994791i	$0.467496\pi$
$614$	0	0
$615$	0.973464	0.0392539
$616$	0	0
$617$	15.8009	0.636122	0.318061	−	0.948070i	$-0.396968\pi$
$617$	15.8009	0.636122	0.318061	+	0.948070i	$0.396968\pi$
$618$	0	0
$619$	45.1221	1.81361	0.906804	−	0.421552i	$-0.138514\pi$
$619$	45.1221	1.81361	0.906804	+	0.421552i	$0.138514\pi$
$620$	0	0
$621$	−1.48856	−0.0597338
$622$	0	0
$623$	5.26167	0.210804
$624$	0	0
$625$	−24.0663	−0.962651
$626$	0	0
$627$	15.0868	0.602510
$628$	0	0
$629$	41.7340	1.66404
$630$	0	0
$631$	−28.2436	−1.12436	−0.562179	−	0.827015i	$-0.690037\pi$
$631$	−28.2436	−1.12436	−0.562179	+	0.827015i	$0.690037\pi$
$632$	0	0
$633$	−34.4741	−1.37022
$634$	0	0
$635$	−22.9828	−0.912043
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	40.4145	1.59877
$640$	0	0
$641$	−24.3049	−0.959986	−0.479993	−	0.877272i	$-0.659361\pi$
$641$	−24.3049	−0.959986	−0.479993	+	0.877272i	$0.659361\pi$
$642$	0	0
$643$	−36.8320	−1.45251	−0.726256	−	0.687424i	$-0.758741\pi$
$643$	−36.8320	−1.45251	−0.726256	+	0.687424i	$0.758741\pi$
$644$	0	0
$645$	−19.9690	−0.786279
$646$	0	0
$647$	−24.7569	−0.973294	−0.486647	−	0.873599i	$-0.661780\pi$
$647$	−24.7569	−0.973294	−0.486647	+	0.873599i	$0.661780\pi$
$648$	0	0
$649$	5.86898	0.230377
$650$	0	0
$651$	27.7576	1.08791
$652$	0	0
$653$	−25.4648	−0.996516	−0.498258	−	0.867029i	$-0.666027\pi$
$653$	−25.4648	−0.996516	−0.498258	+	0.867029i	$0.666027\pi$
$654$	0	0
$655$	−28.7157	−1.12201
$656$	0	0
$657$	−51.6722	−2.01592
$658$	0	0
$659$	−28.2207	−1.09932	−0.549662	−	0.835387i	$-0.685244\pi$
$659$	−28.2207	−1.09932	−0.549662	+	0.835387i	$0.685244\pi$
$660$	0	0
$661$	−12.1133	−0.471152	−0.235576	−	0.971856i	$-0.575698\pi$
$661$	−12.1133	−0.471152	−0.235576	+	0.971856i	$0.575698\pi$
$662$	0	0
$663$	15.3749	0.597113
$664$	0	0
$665$	−14.3928	−0.558127
$666$	0	0
$667$	0.974829	0.0377455
$668$	0	0
$669$	−56.9071	−2.20016
$670$	0	0
$671$	5.31543	0.205200
$672$	0	0
$673$	6.25941	0.241282	0.120641	−	0.992696i	$-0.461505\pi$
$673$	6.25941	0.241282	0.120641	+	0.992696i	$0.461505\pi$
$674$	0	0
$675$	−1.12398	−0.0432619
$676$	0	0
$677$	6.45340	0.248024	0.124012	−	0.992281i	$-0.460424\pi$
$677$	6.45340	0.248024	0.124012	+	0.992281i	$0.460424\pi$
$678$	0	0
$679$	1.02591	0.0393710
$680$	0	0
$681$	−71.1956	−2.72822
$682$	0	0
$683$	−26.0066	−0.995113	−0.497557	−	0.867431i	$-0.665769\pi$
$683$	−26.0066	−0.995113	−0.497557	+	0.867431i	$0.665769\pi$
$684$	0	0
$685$	36.3402	1.38849
$686$	0	0
$687$	−76.3775	−2.91399
$688$	0	0
$689$	4.53998	0.172959
$690$	0	0
$691$	−31.7625	−1.20830	−0.604151	−	0.796870i	$-0.706488\pi$
$691$	−31.7625	−1.20830	−0.604151	+	0.796870i	$0.706488\pi$
$692$	0	0
$693$	−4.16806	−0.158332
$694$	0	0
$695$	13.2738	0.503504
$696$	0	0
$697$	0.833402	0.0315674
$698$	0	0
$699$	74.0223	2.79978
$700$	0	0
$701$	−18.8595	−0.712313	−0.356157	−	0.934426i	$-0.615913\pi$
$701$	−18.8595	−0.712313	−0.356157	+	0.934426i	$0.615913\pi$
$702$	0	0
$703$	−50.8969	−1.91961
$704$	0	0
$705$	−9.78812	−0.368642
$706$	0	0
$707$	4.09883	0.154152
$708$	0	0
$709$	33.0211	1.24014	0.620068	−	0.784548i	$-0.287105\pi$
$709$	33.0211	1.24014	0.620068	+	0.784548i	$0.287105\pi$
$710$	0	0
$711$	−59.1377	−2.21784
$712$	0	0
$713$	−2.31673	−0.0867623
$714$	0	0
$715$	−1.76643	−0.0660607
$716$	0	0
$717$	−4.58560	−0.171252
$718$	0	0
$719$	−8.84215	−0.329756	−0.164878	−	0.986314i	$-0.552723\pi$
$719$	−8.84215	−0.329756	−0.164878	+	0.986314i	$0.552723\pi$
$720$	0	0
$721$	10.5581	0.393206
$722$	0	0
$723$	28.6076	1.06393
$724$	0	0
$725$	0.736072	0.0273370
$726$	0	0
$727$	19.6885	0.730207	0.365104	−	0.930967i	$-0.381034\pi$
$727$	19.6885	0.730207	0.365104	+	0.930967i	$0.381034\pi$
$728$	0	0
$729$	−41.6203	−1.54149
$730$	0	0
$731$	−17.0959	−0.632313
$732$	0	0
$733$	4.57590	0.169015	0.0845073	−	0.996423i	$-0.473068\pi$
$733$	4.57590	0.169015	0.0845073	+	0.996423i	$0.473068\pi$
$734$	0	0
$735$	6.27907	0.231607
$736$	0	0
$737$	−0.512169	−0.0188660
$738$	0	0
$739$	−13.2783	−0.488451	−0.244225	−	0.969719i	$-0.578534\pi$
$739$	−13.2783	−0.488451	−0.244225	+	0.969719i	$0.578534\pi$
$740$	0	0
$741$	−18.7506	−0.688820
$742$	0	0
$743$	−12.6373	−0.463619	−0.231810	−	0.972761i	$-0.574465\pi$
$743$	−12.6373	−0.463619	−0.231810	+	0.972761i	$0.574465\pi$
$744$	0	0
$745$	17.0440	0.624445
$746$	0	0
$747$	51.0260	1.86694
$748$	0	0
$749$	−13.5681	−0.495767
$750$	0	0
$751$	−34.2435	−1.24956	−0.624781	−	0.780800i	$-0.714812\pi$
$751$	−34.2435	−1.24956	−0.624781	+	0.780800i	$0.714812\pi$
$752$	0	0
$753$	−18.0537	−0.657915
$754$	0	0
$755$	−21.7087	−0.790060
$756$	0	0
$757$	−24.6011	−0.894142	−0.447071	−	0.894498i	$-0.647533\pi$
$757$	−24.6011	−0.894142	−0.447071	+	0.894498i	$0.647533\pi$
$758$	0	0
$759$	0.549343	0.0199399
$760$	0	0
$761$	−38.8019	−1.40657	−0.703284	−	0.710909i	$-0.748284\pi$
$761$	−38.8019	−1.40657	−0.703284	+	0.710909i	$0.748284\pi$
$762$	0	0
$763$	7.88829	0.285575
$764$	0	0
$765$	61.1354	2.21036
$766$	0	0
$767$	−7.29422	−0.263379
$768$	0	0
$769$	−17.5918	−0.634377	−0.317189	−	0.948362i	$-0.602739\pi$
$769$	−17.5918	−0.634377	−0.317189	+	0.948362i	$0.602739\pi$
$770$	0	0
$771$	−86.0758	−3.09994
$772$	0	0
$773$	−34.8320	−1.25282	−0.626410	−	0.779494i	$-0.715476\pi$
$773$	−34.8320	−1.25282	−0.626410	+	0.779494i	$0.715476\pi$
$774$	0	0
$775$	−1.74931	−0.0628372
$776$	0	0
$777$	22.2046	0.796586
$778$	0	0
$779$	−1.01638	−0.0364156
$780$	0	0
$781$	−6.27726	−0.224618
$782$	0	0
$783$	−25.4648	−0.910039
$784$	0	0
$785$	−0.216963	−0.00774375
$786$	0	0
$787$	10.9115	0.388951	0.194476	−	0.980907i	$-0.437699\pi$
$787$	10.9115	0.388951	0.194476	+	0.980907i	$0.437699\pi$
$788$	0	0
$789$	43.5082	1.54893
$790$	0	0
$791$	−9.14623	−0.325203
$792$	0	0
$793$	−6.60625	−0.234595
$794$	0	0
$795$	28.5069	1.01103
$796$	0	0
$797$	−22.0119	−0.779701	−0.389851	−	0.920878i	$-0.627473\pi$
$797$	−22.0119	−0.779701	−0.389851	+	0.920878i	$0.627473\pi$
$798$	0	0
$799$	−8.37980	−0.296456
$800$	0	0
$801$	27.2567	0.963070
$802$	0	0
$803$	8.02583	0.283225
$804$	0	0
$805$	−0.524070	−0.0184710
$806$	0	0
$807$	25.2668	0.889434
$808$	0	0
$809$	40.8981	1.43790	0.718950	−	0.695062i	$-0.244623\pi$
$809$	40.8981	1.43790	0.718950	+	0.695062i	$0.244623\pi$
$810$	0	0
$811$	−9.90912	−0.347956	−0.173978	−	0.984750i	$-0.555662\pi$
$811$	−9.90912	−0.347956	−0.173978	+	0.984750i	$0.555662\pi$
$812$	0	0
$813$	41.1699	1.44389
$814$	0	0
$815$	−34.7558	−1.21744
$816$	0	0
$817$	20.8493	0.729426
$818$	0	0
$819$	5.18025	0.181012
$820$	0	0
$821$	4.38502	0.153038	0.0765192	−	0.997068i	$-0.475619\pi$
$821$	4.38502	0.153038	0.0765192	+	0.997068i	$0.475619\pi$
$822$	0	0
$823$	−20.3459	−0.709215	−0.354607	−	0.935015i	$-0.615385\pi$
$823$	−20.3459	−0.709215	−0.354607	+	0.935015i	$0.615385\pi$
$824$	0	0
$825$	0.414797	0.0144414
$826$	0	0
$827$	−0.168760	−0.00586837	−0.00293419	−	0.999996i	$-0.500934\pi$
$827$	−0.168760	−0.00586837	−0.00293419	+	0.999996i	$0.500934\pi$
$828$	0	0
$829$	39.3179	1.36557	0.682783	−	0.730621i	$-0.260769\pi$
$829$	39.3179	1.36557	0.682783	+	0.730621i	$0.260769\pi$
$830$	0	0
$831$	64.7413	2.24585
$832$	0	0
$833$	5.37564	0.186255
$834$	0	0
$835$	−43.2856	−1.49796
$836$	0	0
$837$	60.5185	2.09183
$838$	0	0
$839$	−15.1889	−0.524380	−0.262190	−	0.965016i	$-0.584445\pi$
$839$	−15.1889	−0.524380	−0.262190	+	0.965016i	$0.584445\pi$
$840$	0	0
$841$	−12.3236	−0.424950
$842$	0	0
$843$	6.83592	0.235442
$844$	0	0
$845$	2.19539	0.0755238
$846$	0	0
$847$	−10.3526	−0.355720
$848$	0	0
$849$	−95.1831	−3.26668
$850$	0	0
$851$	−1.85326	−0.0635290
$852$	0	0
$853$	43.7180	1.49687	0.748437	−	0.663206i	$-0.230804\pi$
$853$	43.7180	1.49687	0.748437	+	0.663206i	$0.230804\pi$
$854$	0	0
$855$	−74.5580	−2.54983
$856$	0	0
$857$	−13.6884	−0.467587	−0.233794	−	0.972286i	$-0.575114\pi$
$857$	−13.6884	−0.467587	−0.233794	+	0.972286i	$0.575114\pi$
$858$	0	0
$859$	−19.5697	−0.667710	−0.333855	−	0.942624i	$-0.608350\pi$
$859$	−19.5697	−0.667710	−0.333855	+	0.942624i	$0.608350\pi$
$860$	0	0
$861$	0.443412	0.0151114
$862$	0	0
$863$	11.8126	0.402107	0.201053	−	0.979580i	$-0.435564\pi$
$863$	11.8126	0.402107	0.201053	+	0.979580i	$0.435564\pi$
$864$	0	0
$865$	−30.0424	−1.02147
$866$	0	0
$867$	34.0282	1.15566
$868$	0	0
$869$	9.18539	0.311593
$870$	0	0
$871$	0.636546	0.0215685
$872$	0	0
$873$	5.31449	0.179868
$874$	0	0
$875$	−11.3727	−0.384467
$876$	0	0
$877$	21.7449	0.734274	0.367137	−	0.930167i	$-0.380338\pi$
$877$	21.7449	0.734274	0.367137	+	0.930167i	$0.380338\pi$
$878$	0	0
$879$	−61.9414	−2.08923
$880$	0	0
$881$	20.4760	0.689855	0.344927	−	0.938629i	$-0.387904\pi$
$881$	20.4760	0.689855	0.344927	+	0.938629i	$0.387904\pi$
$882$	0	0
$883$	8.03098	0.270264	0.135132	−	0.990828i	$-0.456854\pi$
$883$	8.03098	0.270264	0.135132	+	0.990828i	$0.456854\pi$
$884$	0	0
$885$	−45.8009	−1.53958
$886$	0	0
$887$	5.03981	0.169220	0.0846102	−	0.996414i	$-0.473036\pi$
$887$	5.03981	0.169220	0.0846102	+	0.996414i	$0.473036\pi$
$888$	0	0
$889$	−10.4686	−0.351106
$890$	0	0
$891$	−1.84594	−0.0618415
$892$	0	0
$893$	10.2196	0.341987
$894$	0	0
$895$	−16.1974	−0.541419
$896$	0	0
$897$	−0.682747	−0.0227963
$898$	0	0
$899$	−39.6324	−1.32182
$900$	0	0
$901$	24.4053	0.813058
$902$	0	0
$903$	−9.09587	−0.302692
$904$	0	0
$905$	49.2941	1.63859
$906$	0	0
$907$	−42.2437	−1.40268	−0.701340	−	0.712827i	$-0.747414\pi$
$907$	−42.2437	−1.40268	−0.701340	+	0.712827i	$0.747414\pi$
$908$	0	0
$909$	21.2329	0.704252
$910$	0	0
$911$	50.3587	1.66846	0.834228	−	0.551419i	$-0.185913\pi$
$911$	50.3587	1.66846	0.834228	+	0.551419i	$0.185913\pi$
$912$	0	0
$913$	−7.92546	−0.262294
$914$	0	0
$915$	−41.4812	−1.37132
$916$	0	0
$917$	−13.0800	−0.431938
$918$	0	0
$919$	−18.2660	−0.602541	−0.301271	−	0.953539i	$-0.597411\pi$
$919$	−18.2660	−0.602541	−0.301271	+	0.953539i	$0.597411\pi$
$920$	0	0
$921$	86.0097	2.83412
$922$	0	0
$923$	7.80165	0.256794
$924$	0	0
$925$	−1.39936	−0.0460106
$926$	0	0
$927$	54.6938	1.79638
$928$	0	0
$929$	20.7281	0.680068	0.340034	−	0.940413i	$-0.389561\pi$
$929$	20.7281	0.680068	0.340034	+	0.940413i	$0.389561\pi$
$930$	0	0
$931$	−6.55589	−0.214861
$932$	0	0
$933$	−27.7932	−0.909909
$934$	0	0
$935$	−9.49568	−0.310542
$936$	0	0
$937$	−59.1752	−1.93317	−0.966585	−	0.256348i	$-0.917481\pi$
$937$	−59.1752	−1.93317	−0.966585	+	0.256348i	$0.917481\pi$
$938$	0	0
$939$	−14.8226	−0.483716
$940$	0	0
$941$	−50.7678	−1.65498	−0.827491	−	0.561479i	$-0.810232\pi$
$941$	−50.7678	−1.65498	−0.827491	+	0.561479i	$0.810232\pi$
$942$	0	0
$943$	−0.0370085	−0.00120516
$944$	0	0
$945$	13.6899	0.445334
$946$	0	0
$947$	51.3353	1.66817	0.834087	−	0.551633i	$-0.185995\pi$
$947$	51.3353	1.66817	0.834087	+	0.551633i	$0.185995\pi$
$948$	0	0
$949$	−9.97485	−0.323797
$950$	0	0
$951$	59.3118	1.92332
$952$	0	0
$953$	−6.71227	−0.217432	−0.108716	−	0.994073i	$-0.534674\pi$
$953$	−6.71227	−0.217432	−0.108716	+	0.994073i	$0.534674\pi$
$954$	0	0
$955$	16.4377	0.531911
$956$	0	0
$957$	9.39763	0.303782
$958$	0	0
$959$	16.5529	0.534522
$960$	0	0
$961$	63.1886	2.03834
$962$	0	0
$963$	−70.2860	−2.26493
$964$	0	0
$965$	11.5100	0.370520
$966$	0	0
$967$	41.2207	1.32557	0.662784	−	0.748811i	$-0.269375\pi$
$967$	41.2207	1.32557	0.662784	+	0.748811i	$0.269375\pi$
$968$	0	0
$969$	−100.796	−3.23805
$970$	0	0
$971$	3.70790	0.118992	0.0594961	−	0.998229i	$-0.481051\pi$
$971$	3.70790	0.118992	0.0594961	+	0.998229i	$0.481051\pi$
$972$	0	0
$973$	6.04620	0.193832
$974$	0	0
$975$	−0.515528	−0.0165101
$976$	0	0
$977$	−16.3769	−0.523943	−0.261971	−	0.965076i	$-0.584373\pi$
$977$	−16.3769	−0.523943	−0.261971	+	0.965076i	$0.584373\pi$
$978$	0	0
$979$	−4.23357	−0.135306
$980$	0	0
$981$	40.8633	1.30466
$982$	0	0
$983$	60.0468	1.91519	0.957597	−	0.288110i	$-0.0930270\pi$
$983$	60.0468	1.91519	0.957597	+	0.288110i	$0.0930270\pi$
$984$	0	0
$985$	55.9045	1.78127
$986$	0	0
$987$	−4.45848	−0.141915
$988$	0	0
$989$	0.759168	0.0241401
$990$	0	0
$991$	−10.7296	−0.340838	−0.170419	−	0.985372i	$-0.554512\pi$
$991$	−10.7296	−0.340838	−0.170419	+	0.985372i	$0.554512\pi$
$992$	0	0
$993$	73.6262	2.33646
$994$	0	0
$995$	28.2502	0.895590
$996$	0	0
$997$	55.4812	1.75711	0.878554	−	0.477643i	$-0.158509\pi$
$997$	55.4812	1.75711	0.878554	+	0.477643i	$0.158509\pi$
$998$	0	0
$999$	48.4116	1.53167

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2912.2.a.s.1.5	yes	5
4.3	odd	2		2912.2.a.r.1.1	✓	5
8.3	odd	2		5824.2.a.cj.1.5		5
8.5	even	2		5824.2.a.ck.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2912.2.a.r.1.1	✓	5	4.3	odd	2
2912.2.a.s.1.5	yes	5	1.1	even	1	trivial
5824.2.a.cj.1.5		5	8.3	odd	2
5824.2.a.ck.1.1		5	8.5	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:	\( N \)	\(=\)	\( 2912 = 2^{5} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2912.a (trivial)

\(f(q)\)	\(=\)	\(q+2.86011 q^{3} +2.19539 q^{5} +1.00000 q^{7} +5.18025 q^{9} -0.804606 q^{11} +1.00000 q^{13} +6.27907 q^{15} +5.37564 q^{17} -6.55589 q^{19} +2.86011 q^{21} -0.238713 q^{23} -0.180247 q^{25} +6.23575 q^{27} -4.08368 q^{29} +9.70508 q^{31} -2.30127 q^{33} +2.19539 q^{35} +7.76355 q^{37} +2.86011 q^{39} +0.155033 q^{41} -3.18025 q^{43} +11.3727 q^{45} -1.55885 q^{47} +1.00000 q^{49} +15.3749 q^{51} +4.53998 q^{53} -1.76643 q^{55} -18.7506 q^{57} -7.29422 q^{59} -6.60625 q^{61} +5.18025 q^{63} +2.19539 q^{65} +0.636546 q^{67} -0.682747 q^{69} +7.80165 q^{71} -9.97485 q^{73} -0.515528 q^{75} -0.804606 q^{77} -11.4160 q^{79} +2.29422 q^{81} +9.85011 q^{83} +11.8016 q^{85} -11.6798 q^{87} +5.26167 q^{89} +1.00000 q^{91} +27.7576 q^{93} -14.3928 q^{95} +1.02591 q^{97} -4.16806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{5} + 5 q^{7} + 11 q^{9} - 12 q^{11} + 5 q^{13} + 2 q^{15} + 4 q^{17} + 5 q^{19} - q^{23} + 14 q^{25} - 6 q^{27} + q^{29} + 13 q^{31} + 2 q^{33} + 3 q^{35} + 18 q^{37} + 22 q^{41} - q^{43}+ \cdots - 36 q^{99}+O(q^{100}) \) 5 * q + 3 * q^5 + 5 * q^7 + 11 * q^9 - 12 * q^11 + 5 * q^13 + 2 * q^15 + 4 * q^17 + 5 * q^19 - q^23 + 14 * q^25 - 6 * q^27 + q^29 + 13 * q^31 + 2 * q^33 + 3 * q^35 + 18 * q^37 + 22 * q^41 - q^43 - 3 * q^45 - 7 * q^47 + 5 * q^49 - 4 * q^51 + 9 * q^53 + 30 * q^55 + 10 * q^57 - 14 * q^59 - 12 * q^61 + 11 * q^63 + 3 * q^65 - 4 * q^67 + 14 * q^69 + 10 * q^71 + 3 * q^73 + 6 * q^75 - 12 * q^77 - 5 * q^79 - 11 * q^81 - 11 * q^83 + 30 * q^85 - 4 * q^87 + 11 * q^89 + 5 * q^91 + 44 * q^93 - 15 * q^95 + 27 * q^97 - 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more