LMFDB - Embedded newform 7616.2.a.bb.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7616,2,Mod(1,7616)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7616, 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("7616.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,5,0,-3,0,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$60.8140661794$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 952)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.860806 q^{3} +2.46260 q^{5} -1.00000 q^{7} -2.25901 q^{9} -3.72161 q^{11} -4.64681 q^{13} +2.11982 q^{15} -1.00000 q^{17} +0.925197 q^{19} -0.860806 q^{21} +6.64681 q^{23} +1.06439 q^{25} -4.52699 q^{27} +4.92520 q^{29} +5.90582 q^{31} -3.20359 q^{33} -2.46260 q^{35} +8.92520 q^{37} -4.00000 q^{39} -3.93561 q^{41} +9.50761 q^{43} -5.56304 q^{45} -3.44322 q^{47} +1.00000 q^{49} -0.860806 q^{51} +4.33382 q^{53} -9.16484 q^{55} +0.796415 q^{57} -13.2936 q^{59} +10.8608 q^{61} +2.25901 q^{63} -11.4432 q^{65} +7.38780 q^{67} +5.72161 q^{69} -15.8116 q^{71} +7.37883 q^{73} +0.916234 q^{75} +3.72161 q^{77} -4.00000 q^{79} +2.88018 q^{81} -3.07480 q^{83} -2.46260 q^{85} +4.23964 q^{87} +6.49720 q^{89} +4.64681 q^{91} +5.08377 q^{93} +2.27839 q^{95} -10.5526 q^{97} +8.40717 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 5 q^{5} - 3 q^{7} + 2 q^{9} + 2 q^{13} - 8 q^{15} - 3 q^{17} - 2 q^{19} + 3 q^{21} + 4 q^{23} + 4 q^{25} - 12 q^{27} + 10 q^{29} - 7 q^{31} - 16 q^{33} - 5 q^{35} + 22 q^{37} - 12 q^{39}+ \cdots + 38 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 5 * q^5 - 3 * q^7 + 2 * q^9 + 2 * q^13 - 8 * q^15 - 3 * q^17 - 2 * q^19 + 3 * q^21 + 4 * q^23 + 4 * q^25 - 12 * q^27 + 10 * q^29 - 7 * q^31 - 16 * q^33 - 5 * q^35 + 22 * q^37 - 12 * q^39 - 11 * q^41 + 7 * q^43 + 20 * q^45 + 12 * q^47 + 3 * q^49 + 3 * q^51 + 9 * q^53 + 6 * q^55 - 4 * q^57 - 8 * q^59 + 27 * q^61 - 2 * q^63 - 12 * q^65 + 15 * q^67 + 6 * q^69 + 2 * q^71 - q^73 - 18 * q^75 - 12 * q^79 + 23 * q^81 - 14 * q^83 - 5 * q^85 - 16 * q^87 - 6 * q^89 - 2 * q^91 + 36 * q^93 + 18 * q^95 + 9 * q^97 + 38 * 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
$3$	0.860806	0.496986	0.248493	−	0.968634i	$-0.420065\pi$
$3$	0.860806	0.496986	0.248493	+	0.968634i	$0.420065\pi$
$4$	0	0
$5$	2.46260	1.10131	0.550654	−	0.834734i	$-0.314378\pi$
$5$	2.46260	1.10131	0.550654	+	0.834734i	$0.314378\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.25901	−0.753004
$10$	0	0
$11$	−3.72161	−1.12211	−0.561054	−	0.827779i	$-0.689604\pi$
$11$	−3.72161	−1.12211	−0.561054	+	0.827779i	$0.689604\pi$
$12$	0	0
$13$	−4.64681	−1.28879	−0.644396	−	0.764692i	$-0.722891\pi$
$13$	−4.64681	−1.28879	−0.644396	+	0.764692i	$0.722891\pi$
$14$	0	0
$15$	2.11982	0.547335
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	0.925197	0.212255	0.106127	−	0.994353i	$-0.466155\pi$
$19$	0.925197	0.212255	0.106127	+	0.994353i	$0.466155\pi$
$20$	0	0
$21$	−0.860806	−0.187843
$22$	0	0
$23$	6.64681	1.38596	0.692978	−	0.720959i	$-0.256298\pi$
$23$	6.64681	1.38596	0.692978	+	0.720959i	$0.256298\pi$
$24$	0	0
$25$	1.06439	0.212878
$26$	0	0
$27$	−4.52699	−0.871220
$28$	0	0
$29$	4.92520	0.914586	0.457293	−	0.889316i	$-0.348819\pi$
$29$	4.92520	0.914586	0.457293	+	0.889316i	$0.348819\pi$
$30$	0	0
$31$	5.90582	1.06072	0.530358	−	0.847773i	$-0.322057\pi$
$31$	5.90582	1.06072	0.530358	+	0.847773i	$0.322057\pi$
$32$	0	0
$33$	−3.20359	−0.557673
$34$	0	0
$35$	−2.46260	−0.416255
$36$	0	0
$37$	8.92520	1.46729	0.733647	−	0.679531i	$-0.237817\pi$
$37$	8.92520	1.46729	0.733647	+	0.679531i	$0.237817\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−3.93561	−0.614639	−0.307319	−	0.951606i	$-0.599432\pi$
$41$	−3.93561	−0.614639	−0.307319	+	0.951606i	$0.599432\pi$
$42$	0	0
$43$	9.50761	1.44990	0.724949	−	0.688803i	$-0.241863\pi$
$43$	9.50761	1.44990	0.724949	+	0.688803i	$0.241863\pi$
$44$	0	0
$45$	−5.56304	−0.829289
$46$	0	0
$47$	−3.44322	−0.502246	−0.251123	−	0.967955i	$-0.580800\pi$
$47$	−3.44322	−0.502246	−0.251123	+	0.967955i	$0.580800\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.860806	−0.120537
$52$	0	0
$53$	4.33382	0.595295	0.297648	−	0.954676i	$-0.403798\pi$
$53$	4.33382	0.595295	0.297648	+	0.954676i	$0.403798\pi$
$54$	0	0
$55$	−9.16484	−1.23579
$56$	0	0
$57$	0.796415	0.105488
$58$	0	0
$59$	−13.2936	−1.73068	−0.865341	−	0.501184i	$-0.832898\pi$
$59$	−13.2936	−1.73068	−0.865341	+	0.501184i	$0.832898\pi$
$60$	0	0
$61$	10.8608	1.39058	0.695292	−	0.718728i	$-0.255275\pi$
$61$	10.8608	1.39058	0.695292	+	0.718728i	$0.255275\pi$
$62$	0	0
$63$	2.25901	0.284609
$64$	0	0
$65$	−11.4432	−1.41936
$66$	0	0
$67$	7.38780	0.902563	0.451281	−	0.892382i	$-0.350967\pi$
$67$	7.38780	0.902563	0.451281	+	0.892382i	$0.350967\pi$
$68$	0	0
$69$	5.72161	0.688801
$70$	0	0
$71$	−15.8116	−1.87650	−0.938248	−	0.345962i	$-0.887553\pi$
$71$	−15.8116	−1.87650	−0.938248	+	0.345962i	$0.887553\pi$
$72$	0	0
$73$	7.37883	0.863627	0.431814	−	0.901963i	$-0.357874\pi$
$73$	7.37883	0.863627	0.431814	+	0.901963i	$0.357874\pi$
$74$	0	0
$75$	0.916234	0.105798
$76$	0	0
$77$	3.72161	0.424117
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	2.88018	0.320020
$82$	0	0
$83$	−3.07480	−0.337503	−0.168752	−	0.985659i	$-0.553974\pi$
$83$	−3.07480	−0.337503	−0.168752	+	0.985659i	$0.553974\pi$
$84$	0	0
$85$	−2.46260	−0.267106
$86$	0	0
$87$	4.23964	0.454537
$88$	0	0
$89$	6.49720	0.688702	0.344351	−	0.938841i	$-0.388099\pi$
$89$	6.49720	0.688702	0.344351	+	0.938841i	$0.388099\pi$
$90$	0	0
$91$	4.64681	0.487118
$92$	0	0
$93$	5.08377	0.527162
$94$	0	0
$95$	2.27839	0.233758
$96$	0	0
$97$	−10.5526	−1.07146	−0.535729	−	0.844390i	$-0.679963\pi$
$97$	−10.5526	−1.07146	−0.535729	+	0.844390i	$0.679963\pi$
$98$	0	0
$99$	8.40717	0.844952
$100$	0	0
$101$	17.0152	1.69308	0.846539	−	0.532326i	$-0.178682\pi$
$101$	17.0152	1.69308	0.846539	+	0.532326i	$0.178682\pi$
$102$	0	0
$103$	−4.92520	−0.485294	−0.242647	−	0.970115i	$-0.578016\pi$
$103$	−4.92520	−0.485294	−0.242647	+	0.970115i	$0.578016\pi$
$104$	0	0
$105$	−2.11982	−0.206873
$106$	0	0
$107$	20.0900	1.94218	0.971088	−	0.238720i	$-0.0767279\pi$
$107$	20.0900	1.94218	0.971088	+	0.238720i	$0.0767279\pi$
$108$	0	0
$109$	12.4972	1.19701	0.598507	−	0.801117i	$-0.295761\pi$
$109$	12.4972	1.19701	0.598507	+	0.801117i	$0.295761\pi$
$110$	0	0
$111$	7.68286	0.729225
$112$	0	0
$113$	19.5333	1.83753	0.918767	−	0.394800i	$-0.129186\pi$
$113$	19.5333	1.83753	0.918767	+	0.394800i	$0.129186\pi$
$114$	0	0
$115$	16.3684	1.52636
$116$	0	0
$117$	10.4972	0.970467
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	2.85039	0.259127
$122$	0	0
$123$	−3.38780	−0.305467
$124$	0	0
$125$	−9.69182	−0.866863
$126$	0	0
$127$	2.83102	0.251212	0.125606	−	0.992080i	$-0.459912\pi$
$127$	2.83102	0.251212	0.125606	+	0.992080i	$0.459912\pi$
$128$	0	0
$129$	8.18421	0.720580
$130$	0	0
$131$	−10.7964	−0.943287	−0.471644	−	0.881789i	$-0.656339\pi$
$131$	−10.7964	−0.943287	−0.471644	+	0.881789i	$0.656339\pi$
$132$	0	0
$133$	−0.925197	−0.0802247
$134$	0	0
$135$	−11.1482	−0.959481
$136$	0	0
$137$	13.6572	1.16682	0.583408	−	0.812180i	$-0.301719\pi$
$137$	13.6572	1.16682	0.583408	+	0.812180i	$0.301719\pi$
$138$	0	0
$139$	−5.51658	−0.467910	−0.233955	−	0.972247i	$-0.575167\pi$
$139$	−5.51658	−0.467910	−0.233955	+	0.972247i	$0.575167\pi$
$140$	0	0
$141$	−2.96395	−0.249609
$142$	0	0
$143$	17.2936	1.44616
$144$	0	0
$145$	12.1288	1.00724
$146$	0	0
$147$	0.860806	0.0709981
$148$	0	0
$149$	2.34278	0.191928	0.0959640	−	0.995385i	$-0.469407\pi$
$149$	2.34278	0.191928	0.0959640	+	0.995385i	$0.469407\pi$
$150$	0	0
$151$	5.63640	0.458683	0.229342	−	0.973346i	$-0.426343\pi$
$151$	5.63640	0.458683	0.229342	+	0.973346i	$0.426343\pi$
$152$	0	0
$153$	2.25901	0.182630
$154$	0	0
$155$	14.5437	1.16818
$156$	0	0
$157$	0.518027	0.0413430	0.0206715	−	0.999786i	$-0.493420\pi$
$157$	0.518027	0.0413430	0.0206715	+	0.999786i	$0.493420\pi$
$158$	0	0
$159$	3.73057	0.295854
$160$	0	0
$161$	−6.64681	−0.523842
$162$	0	0
$163$	−0.149606	−0.0117181	−0.00585904	−	0.999983i	$-0.501865\pi$
$163$	−0.149606	−0.0117181	−0.00585904	+	0.999983i	$0.501865\pi$
$164$	0	0
$165$	−7.88914	−0.614169
$166$	0	0
$167$	−12.9508	−1.00217	−0.501083	−	0.865399i	$-0.667065\pi$
$167$	−12.9508	−1.00217	−0.501083	+	0.865399i	$0.667065\pi$
$168$	0	0
$169$	8.59283	0.660987
$170$	0	0
$171$	−2.09003	−0.159829
$172$	0	0
$173$	3.41758	0.259834	0.129917	−	0.991525i	$-0.458529\pi$
$173$	3.41758	0.259834	0.129917	+	0.991525i	$0.458529\pi$
$174$	0	0
$175$	−1.06439	−0.0804604
$176$	0	0
$177$	−11.4432	−0.860125
$178$	0	0
$179$	−0.471561	−0.0352461	−0.0176231	−	0.999845i	$-0.505610\pi$
$179$	−0.471561	−0.0352461	−0.0176231	+	0.999845i	$0.505610\pi$
$180$	0	0
$181$	16.2396	1.20708	0.603541	−	0.797332i	$-0.293756\pi$
$181$	16.2396	1.20708	0.603541	+	0.797332i	$0.293756\pi$
$182$	0	0
$183$	9.34905	0.691101
$184$	0	0
$185$	21.9792	1.61594
$186$	0	0
$187$	3.72161	0.272151
$188$	0	0
$189$	4.52699	0.329290
$190$	0	0
$191$	14.1634	1.02483	0.512413	−	0.858739i	$-0.328752\pi$
$191$	14.1634	1.02483	0.512413	+	0.858739i	$0.328752\pi$
$192$	0	0
$193$	4.94602	0.356022	0.178011	−	0.984028i	$-0.443034\pi$
$193$	4.94602	0.356022	0.178011	+	0.984028i	$0.443034\pi$
$194$	0	0
$195$	−9.85039	−0.705401
$196$	0	0
$197$	21.5333	1.53418	0.767090	−	0.641539i	$-0.221704\pi$
$197$	21.5333	1.53418	0.767090	+	0.641539i	$0.221704\pi$
$198$	0	0
$199$	−15.3580	−1.08870	−0.544350	−	0.838858i	$-0.683224\pi$
$199$	−15.3580	−1.08870	−0.544350	+	0.838858i	$0.683224\pi$
$200$	0	0
$201$	6.35946	0.448562
$202$	0	0
$203$	−4.92520	−0.345681
$204$	0	0
$205$	−9.69182	−0.676906
$206$	0	0
$207$	−15.0152	−1.04363
$208$	0	0
$209$	−3.44322	−0.238173
$210$	0	0
$211$	16.5180	1.13715	0.568574	−	0.822632i	$-0.307495\pi$
$211$	16.5180	1.13715	0.568574	+	0.822632i	$0.307495\pi$
$212$	0	0
$213$	−13.6108	−0.932594
$214$	0	0
$215$	23.4134	1.59678
$216$	0	0
$217$	−5.90582	−0.400913
$218$	0	0
$219$	6.35174	0.429211
$220$	0	0
$221$	4.64681	0.312578
$222$	0	0
$223$	7.63158	0.511048	0.255524	−	0.966803i	$-0.417752\pi$
$223$	7.63158	0.511048	0.255524	+	0.966803i	$0.417752\pi$
$224$	0	0
$225$	−2.40447	−0.160298
$226$	0	0
$227$	17.8850	1.18707	0.593534	−	0.804809i	$-0.297732\pi$
$227$	17.8850	1.18707	0.593534	+	0.804809i	$0.297732\pi$
$228$	0	0
$229$	4.27839	0.282724	0.141362	−	0.989958i	$-0.454852\pi$
$229$	4.27839	0.282724	0.141362	+	0.989958i	$0.454852\pi$
$230$	0	0
$231$	3.20359	0.210780
$232$	0	0
$233$	−18.0692	−1.18375	−0.591877	−	0.806029i	$-0.701613\pi$
$233$	−18.0692	−1.18375	−0.591877	+	0.806029i	$0.701613\pi$
$234$	0	0
$235$	−8.47928	−0.553127
$236$	0	0
$237$	−3.44322	−0.223661
$238$	0	0
$239$	−0.770774	−0.0498572	−0.0249286	−	0.999689i	$-0.507936\pi$
$239$	−0.770774	−0.0498572	−0.0249286	+	0.999689i	$0.507936\pi$
$240$	0	0
$241$	−13.9959	−0.901552	−0.450776	−	0.892637i	$-0.648853\pi$
$241$	−13.9959	−0.901552	−0.450776	+	0.892637i	$0.648853\pi$
$242$	0	0
$243$	16.0602	1.03027
$244$	0	0
$245$	2.46260	0.157330
$246$	0	0
$247$	−4.29921	−0.273552
$248$	0	0
$249$	−2.64681	−0.167735
$250$	0	0
$251$	4.92520	0.310876	0.155438	−	0.987846i	$-0.450321\pi$
$251$	4.92520	0.310876	0.155438	+	0.987846i	$0.450321\pi$
$252$	0	0
$253$	−24.7368	−1.55519
$254$	0	0
$255$	−2.11982	−0.132748
$256$	0	0
$257$	26.9765	1.68275	0.841373	−	0.540454i	$-0.181748\pi$
$257$	26.9765	1.68275	0.841373	+	0.540454i	$0.181748\pi$
$258$	0	0
$259$	−8.92520	−0.554585
$260$	0	0
$261$	−11.1261	−0.688687
$262$	0	0
$263$	−10.5872	−0.652837	−0.326418	−	0.945225i	$-0.605842\pi$
$263$	−10.5872	−0.652837	−0.326418	+	0.945225i	$0.605842\pi$
$264$	0	0
$265$	10.6724	0.655603
$266$	0	0
$267$	5.59283	0.342276
$268$	0	0
$269$	−23.9404	−1.45967	−0.729837	−	0.683621i	$-0.760404\pi$
$269$	−23.9404	−1.45967	−0.729837	+	0.683621i	$0.760404\pi$
$270$	0	0
$271$	−1.90997	−0.116022	−0.0580111	−	0.998316i	$-0.518476\pi$
$271$	−1.90997	−0.116022	−0.0580111	+	0.998316i	$0.518476\pi$
$272$	0	0
$273$	4.00000	0.242091
$274$	0	0
$275$	−3.96125	−0.238872
$276$	0	0
$277$	−14.3476	−0.862063	−0.431032	−	0.902337i	$-0.641850\pi$
$277$	−14.3476	−0.862063	−0.431032	+	0.902337i	$0.641850\pi$
$278$	0	0
$279$	−13.3413	−0.798725
$280$	0	0
$281$	−2.55263	−0.152277	−0.0761386	−	0.997097i	$-0.524259\pi$
$281$	−2.55263	−0.152277	−0.0761386	+	0.997097i	$0.524259\pi$
$282$	0	0
$283$	20.1455	1.19752	0.598762	−	0.800927i	$-0.295660\pi$
$283$	20.1455	1.19752	0.598762	+	0.800927i	$0.295660\pi$
$284$	0	0
$285$	1.96125	0.116174
$286$	0	0
$287$	3.93561	0.232312
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−9.08377	−0.532500
$292$	0	0
$293$	10.7964	0.630733	0.315367	−	0.948970i	$-0.397873\pi$
$293$	10.7964	0.630733	0.315367	+	0.948970i	$0.397873\pi$
$294$	0	0
$295$	−32.7368	−1.90601
$296$	0	0
$297$	16.8477	0.977603
$298$	0	0
$299$	−30.8864	−1.78621
$300$	0	0
$301$	−9.50761	−0.548010
$302$	0	0
$303$	14.6468	0.841437
$304$	0	0
$305$	26.7458	1.53146
$306$	0	0
$307$	−11.8296	−0.675149	−0.337575	−	0.941299i	$-0.609607\pi$
$307$	−11.8296	−0.675149	−0.337575	+	0.941299i	$0.609607\pi$
$308$	0	0
$309$	−4.23964	−0.241185
$310$	0	0
$311$	9.23819	0.523850	0.261925	−	0.965088i	$-0.415643\pi$
$311$	9.23819	0.523850	0.261925	+	0.965088i	$0.415643\pi$
$312$	0	0
$313$	−13.5166	−0.764002	−0.382001	−	0.924162i	$-0.624765\pi$
$313$	−13.5166	−0.764002	−0.382001	+	0.924162i	$0.624765\pi$
$314$	0	0
$315$	5.56304	0.313442
$316$	0	0
$317$	6.70638	0.376668	0.188334	−	0.982105i	$-0.439691\pi$
$317$	6.70638	0.376668	0.188334	+	0.982105i	$0.439691\pi$
$318$	0	0
$319$	−18.3297	−1.02626
$320$	0	0
$321$	17.2936	0.965236
$322$	0	0
$323$	−0.925197	−0.0514793
$324$	0	0
$325$	−4.94602	−0.274356
$326$	0	0
$327$	10.7577	0.594900
$328$	0	0
$329$	3.44322	0.189831
$330$	0	0
$331$	9.60661	0.528027	0.264013	−	0.964519i	$-0.414954\pi$
$331$	9.60661	0.528027	0.264013	+	0.964519i	$0.414954\pi$
$332$	0	0
$333$	−20.1621	−1.10488
$334$	0	0
$335$	18.1932	0.993999
$336$	0	0
$337$	−35.2340	−1.91932	−0.959660	−	0.281162i	$-0.909280\pi$
$337$	−35.2340	−1.91932	−0.959660	+	0.281162i	$0.909280\pi$
$338$	0	0
$339$	16.8143	0.913230
$340$	0	0
$341$	−21.9792	−1.19024
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	14.0900	0.758582
$346$	0	0
$347$	−33.6233	−1.80499	−0.902496	−	0.430698i	$-0.858268\pi$
$347$	−33.6233	−1.80499	−0.902496	+	0.430698i	$0.858268\pi$
$348$	0	0
$349$	12.7756	0.683862	0.341931	−	0.939725i	$-0.388919\pi$
$349$	12.7756	0.683862	0.341931	+	0.939725i	$0.388919\pi$
$350$	0	0
$351$	21.0361	1.12282
$352$	0	0
$353$	28.2009	1.50098	0.750491	−	0.660881i	$-0.229817\pi$
$353$	28.2009	1.50098	0.750491	+	0.660881i	$0.229817\pi$
$354$	0	0
$355$	−38.9377	−2.06660
$356$	0	0
$357$	0.860806	0.0455587
$358$	0	0
$359$	−17.6066	−0.929241	−0.464621	−	0.885510i	$-0.653809\pi$
$359$	−17.6066	−0.929241	−0.464621	+	0.885510i	$0.653809\pi$
$360$	0	0
$361$	−18.1440	−0.954948
$362$	0	0
$363$	2.45364	0.128782
$364$	0	0
$365$	18.1711	0.951119
$366$	0	0
$367$	25.3788	1.32476	0.662382	−	0.749166i	$-0.269546\pi$
$367$	25.3788	1.32476	0.662382	+	0.749166i	$0.269546\pi$
$368$	0	0
$369$	8.89059	0.462826
$370$	0	0
$371$	−4.33382	−0.225001
$372$	0	0
$373$	30.1365	1.56041	0.780204	−	0.625525i	$-0.215115\pi$
$373$	30.1365	1.56041	0.780204	+	0.625525i	$0.215115\pi$
$374$	0	0
$375$	−8.34278	−0.430819
$376$	0	0
$377$	−22.8864	−1.17871
$378$	0	0
$379$	−2.86562	−0.147197	−0.0735986	−	0.997288i	$-0.523448\pi$
$379$	−2.86562	−0.147197	−0.0735986	+	0.997288i	$0.523448\pi$
$380$	0	0
$381$	2.43696	0.124849
$382$	0	0
$383$	33.9612	1.73534	0.867669	−	0.497142i	$-0.165617\pi$
$383$	33.9612	1.73534	0.867669	+	0.497142i	$0.165617\pi$
$384$	0	0
$385$	9.16484	0.467083
$386$	0	0
$387$	−21.4778	−1.09178
$388$	0	0
$389$	15.7354	0.797816	0.398908	−	0.916991i	$-0.369389\pi$
$389$	15.7354	0.797816	0.398908	+	0.916991i	$0.369389\pi$
$390$	0	0
$391$	−6.64681	−0.336144
$392$	0	0
$393$	−9.29362	−0.468801
$394$	0	0
$395$	−9.85039	−0.495627
$396$	0	0
$397$	−11.3878	−0.571537	−0.285769	−	0.958299i	$-0.592249\pi$
$397$	−11.3878	−0.571537	−0.285769	+	0.958299i	$0.592249\pi$
$398$	0	0
$399$	−0.796415	−0.0398706
$400$	0	0
$401$	−25.2549	−1.26117	−0.630584	−	0.776121i	$-0.717185\pi$
$401$	−25.2549	−1.26117	−0.630584	+	0.776121i	$0.717185\pi$
$402$	0	0
$403$	−27.4432	−1.36704
$404$	0	0
$405$	7.09273	0.352441
$406$	0	0
$407$	−33.2161	−1.64646
$408$	0	0
$409$	18.6676	0.923055	0.461528	−	0.887126i	$-0.347302\pi$
$409$	18.6676	0.923055	0.461528	+	0.887126i	$0.347302\pi$
$410$	0	0
$411$	11.7562	0.579891
$412$	0	0
$413$	13.2936	0.654136
$414$	0	0
$415$	−7.57201	−0.371695
$416$	0	0
$417$	−4.74870	−0.232545
$418$	0	0
$419$	−8.07625	−0.394551	−0.197275	−	0.980348i	$-0.563209\pi$
$419$	−8.07625	−0.394551	−0.197275	+	0.980348i	$0.563209\pi$
$420$	0	0
$421$	−31.2597	−1.52350	−0.761752	−	0.647869i	$-0.775660\pi$
$421$	−31.2597	−1.52350	−0.761752	+	0.647869i	$0.775660\pi$
$422$	0	0
$423$	7.77829	0.378193
$424$	0	0
$425$	−1.06439	−0.0516305
$426$	0	0
$427$	−10.8608	−0.525591
$428$	0	0
$429$	14.8864	0.718724
$430$	0	0
$431$	−16.8656	−0.812388	−0.406194	−	0.913787i	$-0.633144\pi$
$431$	−16.8656	−0.812388	−0.406194	+	0.913787i	$0.633144\pi$
$432$	0	0
$433$	−32.3297	−1.55366	−0.776832	−	0.629707i	$-0.783175\pi$
$433$	−32.3297	−1.55366	−0.776832	+	0.629707i	$0.783175\pi$
$434$	0	0
$435$	10.4405	0.500585
$436$	0	0
$437$	6.14961	0.294176
$438$	0	0
$439$	−24.2653	−1.15812	−0.579059	−	0.815285i	$-0.696580\pi$
$439$	−24.2653	−1.15812	−0.579059	+	0.815285i	$0.696580\pi$
$440$	0	0
$441$	−2.25901	−0.107572
$442$	0	0
$443$	−13.2936	−0.631599	−0.315799	−	0.948826i	$-0.602273\pi$
$443$	−13.2936	−0.631599	−0.315799	+	0.948826i	$0.602273\pi$
$444$	0	0
$445$	16.0000	0.758473
$446$	0	0
$447$	2.01668	0.0953856
$448$	0	0
$449$	34.4376	1.62521	0.812606	−	0.582813i	$-0.198048\pi$
$449$	34.4376	1.62521	0.812606	+	0.582813i	$0.198048\pi$
$450$	0	0
$451$	14.6468	0.689691
$452$	0	0
$453$	4.85184	0.227959
$454$	0	0
$455$	11.4432	0.536467
$456$	0	0
$457$	28.7714	1.34587	0.672936	−	0.739701i	$-0.265033\pi$
$457$	28.7714	1.34587	0.672936	+	0.739701i	$0.265033\pi$
$458$	0	0
$459$	4.52699	0.211302
$460$	0	0
$461$	34.4376	1.60392	0.801960	−	0.597378i	$-0.203791\pi$
$461$	34.4376	1.60392	0.801960	+	0.597378i	$0.203791\pi$
$462$	0	0
$463$	−6.80123	−0.316080	−0.158040	−	0.987433i	$-0.550518\pi$
$463$	−6.80123	−0.316080	−0.158040	+	0.987433i	$0.550518\pi$
$464$	0	0
$465$	12.5193	0.580567
$466$	0	0
$467$	−30.3892	−1.40625	−0.703123	−	0.711068i	$-0.748212\pi$
$467$	−30.3892	−1.40625	−0.703123	+	0.711068i	$0.748212\pi$
$468$	0	0
$469$	−7.38780	−0.341137
$470$	0	0
$471$	0.445920	0.0205469
$472$	0	0
$473$	−35.3836	−1.62694
$474$	0	0
$475$	0.984771	0.0451844
$476$	0	0
$477$	−9.79015	−0.448260
$478$	0	0
$479$	6.36360	0.290760	0.145380	−	0.989376i	$-0.453559\pi$
$479$	6.36360	0.290760	0.145380	+	0.989376i	$0.453559\pi$
$480$	0	0
$481$	−41.4737	−1.89104
$482$	0	0
$483$	−5.72161	−0.260342
$484$	0	0
$485$	−25.9869	−1.18000
$486$	0	0
$487$	35.4529	1.60652	0.803261	−	0.595627i	$-0.203096\pi$
$487$	35.4529	1.60652	0.803261	+	0.595627i	$0.203096\pi$
$488$	0	0
$489$	−0.128782	−0.00582372
$490$	0	0
$491$	−9.90582	−0.447043	−0.223522	−	0.974699i	$-0.571755\pi$
$491$	−9.90582	−0.447043	−0.223522	+	0.974699i	$0.571755\pi$
$492$	0	0
$493$	−4.92520	−0.221820
$494$	0	0
$495$	20.7035	0.930552
$496$	0	0
$497$	15.8116	0.709249
$498$	0	0
$499$	7.78119	0.348334	0.174167	−	0.984716i	$-0.444277\pi$
$499$	7.78119	0.348334	0.174167	+	0.984716i	$0.444277\pi$
$500$	0	0
$501$	−11.1482	−0.498063
$502$	0	0
$503$	23.1994	1.03441	0.517206	−	0.855861i	$-0.326972\pi$
$503$	23.1994	1.03441	0.517206	+	0.855861i	$0.326972\pi$
$504$	0	0
$505$	41.9017	1.86460
$506$	0	0
$507$	7.39676	0.328502
$508$	0	0
$509$	3.85039	0.170666	0.0853328	−	0.996353i	$-0.472805\pi$
$509$	3.85039	0.170666	0.0853328	+	0.996353i	$0.472805\pi$
$510$	0	0
$511$	−7.37883	−0.326420
$512$	0	0
$513$	−4.18836	−0.184920
$514$	0	0
$515$	−12.1288	−0.534458
$516$	0	0
$517$	12.8143	0.563574
$518$	0	0
$519$	2.94187	0.129134
$520$	0	0
$521$	−32.5437	−1.42576	−0.712882	−	0.701284i	$-0.752610\pi$
$521$	−32.5437	−1.42576	−0.712882	+	0.701284i	$0.752610\pi$
$522$	0	0
$523$	−19.1261	−0.836325	−0.418163	−	0.908372i	$-0.637326\pi$
$523$	−19.1261	−0.836325	−0.418163	+	0.908372i	$0.637326\pi$
$524$	0	0
$525$	−0.916234	−0.0399877
$526$	0	0
$527$	−5.90582	−0.257262
$528$	0	0
$529$	21.1801	0.920872
$530$	0	0
$531$	30.0305	1.30321
$532$	0	0
$533$	18.2880	0.792142
$534$	0	0
$535$	49.4737	2.13893
$536$	0	0
$537$	−0.405923	−0.0175169
$538$	0	0
$539$	−3.72161	−0.160301
$540$	0	0
$541$	32.2396	1.38609	0.693045	−	0.720894i	$-0.256269\pi$
$541$	32.2396	1.38609	0.693045	+	0.720894i	$0.256269\pi$
$542$	0	0
$543$	13.9792	0.599904
$544$	0	0
$545$	30.7756	1.31828
$546$	0	0
$547$	−14.5485	−0.622048	−0.311024	−	0.950402i	$-0.600672\pi$
$547$	−14.5485	−0.622048	−0.311024	+	0.950402i	$0.600672\pi$
$548$	0	0
$549$	−24.5347	−1.04712
$550$	0	0
$551$	4.55678	0.194125
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	18.9198	0.803101
$556$	0	0
$557$	−17.6233	−0.746723	−0.373361	−	0.927686i	$-0.621795\pi$
$557$	−17.6233	−0.746723	−0.373361	+	0.927686i	$0.621795\pi$
$558$	0	0
$559$	−44.1801	−1.86862
$560$	0	0
$561$	3.20359	0.135255
$562$	0	0
$563$	−29.4045	−1.23925	−0.619625	−	0.784898i	$-0.712715\pi$
$563$	−29.4045	−1.23925	−0.619625	+	0.784898i	$0.712715\pi$
$564$	0	0
$565$	48.1026	2.02369
$566$	0	0
$567$	−2.88018	−0.120956
$568$	0	0
$569$	−9.52844	−0.399453	−0.199726	−	0.979852i	$-0.564005\pi$
$569$	−9.52844	−0.399453	−0.199726	+	0.979852i	$0.564005\pi$
$570$	0	0
$571$	32.1592	1.34582	0.672911	−	0.739723i	$-0.265044\pi$
$571$	32.1592	1.34582	0.672911	+	0.739723i	$0.265044\pi$
$572$	0	0
$573$	12.1919	0.509325
$574$	0	0
$575$	7.07480	0.295040
$576$	0	0
$577$	15.5928	0.649138	0.324569	−	0.945862i	$-0.394781\pi$
$577$	15.5928	0.649138	0.324569	+	0.945862i	$0.394781\pi$
$578$	0	0
$579$	4.25756	0.176938
$580$	0	0
$581$	3.07480	0.127564
$582$	0	0
$583$	−16.1288	−0.667986
$584$	0	0
$585$	25.8504	1.06878
$586$	0	0
$587$	−2.94602	−0.121595	−0.0607977	−	0.998150i	$-0.519364\pi$
$587$	−2.94602	−0.121595	−0.0607977	+	0.998150i	$0.519364\pi$
$588$	0	0
$589$	5.46405	0.225142
$590$	0	0
$591$	18.5360	0.762467
$592$	0	0
$593$	−15.4224	−0.633322	−0.316661	−	0.948539i	$-0.602562\pi$
$593$	−15.4224	−0.633322	−0.316661	+	0.948539i	$0.602562\pi$
$594$	0	0
$595$	2.46260	0.100957
$596$	0	0
$597$	−13.2203	−0.541069
$598$	0	0
$599$	−18.6724	−0.762936	−0.381468	−	0.924382i	$-0.624581\pi$
$599$	−18.6724	−0.762936	−0.381468	+	0.924382i	$0.624581\pi$
$600$	0	0
$601$	−9.44322	−0.385197	−0.192599	−	0.981278i	$-0.561692\pi$
$601$	−9.44322	−0.385197	−0.192599	+	0.981278i	$0.561692\pi$
$602$	0	0
$603$	−16.6891	−0.679634
$604$	0	0
$605$	7.01938	0.285378
$606$	0	0
$607$	−30.6427	−1.24375	−0.621874	−	0.783118i	$-0.713628\pi$
$607$	−30.6427	−1.24375	−0.621874	+	0.783118i	$0.713628\pi$
$608$	0	0
$609$	−4.23964	−0.171799
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	−39.2895	−1.58689	−0.793443	−	0.608644i	$-0.791714\pi$
$613$	−39.2895	−1.58689	−0.793443	+	0.608644i	$0.791714\pi$
$614$	0	0
$615$	−8.34278	−0.336413
$616$	0	0
$617$	8.71602	0.350894	0.175447	−	0.984489i	$-0.443863\pi$
$617$	8.71602	0.350894	0.175447	+	0.984489i	$0.443863\pi$
$618$	0	0
$619$	2.49720	0.100371	0.0501855	−	0.998740i	$-0.484019\pi$
$619$	2.49720	0.100371	0.0501855	+	0.998740i	$0.484019\pi$
$620$	0	0
$621$	−30.0900	−1.20747
$622$	0	0
$623$	−6.49720	−0.260305
$624$	0	0
$625$	−29.1890	−1.16756
$626$	0	0
$627$	−2.96395	−0.118369
$628$	0	0
$629$	−8.92520	−0.355871
$630$	0	0
$631$	31.4987	1.25394	0.626971	−	0.779043i	$-0.284295\pi$
$631$	31.4987	1.25394	0.626971	+	0.779043i	$0.284295\pi$
$632$	0	0
$633$	14.2188	0.565147
$634$	0	0
$635$	6.97166	0.276662
$636$	0	0
$637$	−4.64681	−0.184113
$638$	0	0
$639$	35.7187	1.41301
$640$	0	0
$641$	9.07480	0.358433	0.179217	−	0.983810i	$-0.442644\pi$
$641$	9.07480	0.358433	0.179217	+	0.983810i	$0.442644\pi$
$642$	0	0
$643$	−33.4176	−1.31786	−0.658930	−	0.752204i	$-0.728991\pi$
$643$	−33.4176	−1.31786	−0.658930	+	0.752204i	$0.728991\pi$
$644$	0	0
$645$	20.1544	0.793580
$646$	0	0
$647$	0.110856	0.00435821	0.00217911	−	0.999998i	$-0.499306\pi$
$647$	0.110856	0.00435821	0.00217911	+	0.999998i	$0.499306\pi$
$648$	0	0
$649$	49.4737	1.94201
$650$	0	0
$651$	−5.08377	−0.199248
$652$	0	0
$653$	−26.5277	−1.03811	−0.519054	−	0.854742i	$-0.673716\pi$
$653$	−26.5277	−1.03811	−0.519054	+	0.854742i	$0.673716\pi$
$654$	0	0
$655$	−26.5872	−1.03885
$656$	0	0
$657$	−16.6689	−0.650315
$658$	0	0
$659$	−25.0498	−0.975803	−0.487901	−	0.872899i	$-0.662237\pi$
$659$	−25.0498	−0.975803	−0.487901	+	0.872899i	$0.662237\pi$
$660$	0	0
$661$	49.6925	1.93281	0.966407	−	0.257016i	$-0.0827395\pi$
$661$	49.6925	1.93281	0.966407	+	0.257016i	$0.0827395\pi$
$662$	0	0
$663$	4.00000	0.155347
$664$	0	0
$665$	−2.27839	−0.0883521
$666$	0	0
$667$	32.7368	1.26758
$668$	0	0
$669$	6.56931	0.253984
$670$	0	0
$671$	−40.4197	−1.56039
$672$	0	0
$673$	30.9944	1.19475	0.597373	−	0.801963i	$-0.296211\pi$
$673$	30.9944	1.19475	0.597373	+	0.801963i	$0.296211\pi$
$674$	0	0
$675$	−4.81849	−0.185464
$676$	0	0
$677$	34.8269	1.33851	0.669253	−	0.743035i	$-0.266614\pi$
$677$	34.8269	1.33851	0.669253	+	0.743035i	$0.266614\pi$
$678$	0	0
$679$	10.5526	0.404973
$680$	0	0
$681$	15.3955	0.589957
$682$	0	0
$683$	20.5872	0.787749	0.393874	−	0.919164i	$-0.371135\pi$
$683$	20.5872	0.787749	0.393874	+	0.919164i	$0.371135\pi$
$684$	0	0
$685$	33.6323	1.28502
$686$	0	0
$687$	3.68286	0.140510
$688$	0	0
$689$	−20.1384	−0.767213
$690$	0	0
$691$	−2.83998	−0.108038	−0.0540190	−	0.998540i	$-0.517203\pi$
$691$	−2.83998	−0.108038	−0.0540190	+	0.998540i	$0.517203\pi$
$692$	0	0
$693$	−8.40717	−0.319362
$694$	0	0
$695$	−13.5851	−0.515313
$696$	0	0
$697$	3.93561	0.149072
$698$	0	0
$699$	−15.5541	−0.588309
$700$	0	0
$701$	8.96395	0.338564	0.169282	−	0.985568i	$-0.445855\pi$
$701$	8.96395	0.338564	0.169282	+	0.985568i	$0.445855\pi$
$702$	0	0
$703$	8.25756	0.311440
$704$	0	0
$705$	−7.29901	−0.274897
$706$	0	0
$707$	−17.0152	−0.639924
$708$	0	0
$709$	24.4793	0.919339	0.459669	−	0.888090i	$-0.347968\pi$
$709$	24.4793	0.919339	0.459669	+	0.888090i	$0.347968\pi$
$710$	0	0
$711$	9.03605	0.338878
$712$	0	0
$713$	39.2549	1.47011
$714$	0	0
$715$	42.5872	1.59267
$716$	0	0
$717$	−0.663487	−0.0247784
$718$	0	0
$719$	−23.1004	−0.861501	−0.430751	−	0.902471i	$-0.641751\pi$
$719$	−23.1004	−0.861501	−0.430751	+	0.902471i	$0.641751\pi$
$720$	0	0
$721$	4.92520	0.183424
$722$	0	0
$723$	−12.0477	−0.448059
$724$	0	0
$725$	5.24234	0.194695
$726$	0	0
$727$	5.97918	0.221755	0.110878	−	0.993834i	$-0.464634\pi$
$727$	5.97918	0.221755	0.110878	+	0.993834i	$0.464634\pi$
$728$	0	0
$729$	5.18421	0.192008
$730$	0	0
$731$	−9.50761	−0.351652
$732$	0	0
$733$	2.55678	0.0944367	0.0472183	−	0.998885i	$-0.484964\pi$
$733$	2.55678	0.0944367	0.0472183	+	0.998885i	$0.484964\pi$
$734$	0	0
$735$	2.11982	0.0781907
$736$	0	0
$737$	−27.4945	−1.01277
$738$	0	0
$739$	36.5645	1.34505	0.672523	−	0.740076i	$-0.265210\pi$
$739$	36.5645	1.34505	0.672523	+	0.740076i	$0.265210\pi$
$740$	0	0
$741$	−3.70079	−0.135952
$742$	0	0
$743$	11.0748	0.406295	0.203148	−	0.979148i	$-0.434883\pi$
$743$	11.0748	0.406295	0.203148	+	0.979148i	$0.434883\pi$
$744$	0	0
$745$	5.76932	0.211372
$746$	0	0
$747$	6.94602	0.254142
$748$	0	0
$749$	−20.0900	−0.734074
$750$	0	0
$751$	−21.4820	−0.783888	−0.391944	−	0.919989i	$-0.628197\pi$
$751$	−21.4820	−0.783888	−0.391944	+	0.919989i	$0.628197\pi$
$752$	0	0
$753$	4.23964	0.154501
$754$	0	0
$755$	13.8802	0.505152
$756$	0	0
$757$	8.80123	0.319886	0.159943	−	0.987126i	$-0.448869\pi$
$757$	8.80123	0.319886	0.159943	+	0.987126i	$0.448869\pi$
$758$	0	0
$759$	−21.2936	−0.772909
$760$	0	0
$761$	25.0152	0.906801	0.453401	−	0.891307i	$-0.350211\pi$
$761$	25.0152	0.906801	0.453401	+	0.891307i	$0.350211\pi$
$762$	0	0
$763$	−12.4972	−0.452429
$764$	0	0
$765$	5.56304	0.201132
$766$	0	0
$767$	61.7729	2.23049
$768$	0	0
$769$	−7.78119	−0.280597	−0.140298	−	0.990109i	$-0.544806\pi$
$769$	−7.78119	−0.280597	−0.140298	+	0.990109i	$0.544806\pi$
$770$	0	0
$771$	23.2215	0.836302
$772$	0	0
$773$	27.2757	0.981038	0.490519	−	0.871430i	$-0.336807\pi$
$773$	27.2757	0.981038	0.490519	+	0.871430i	$0.336807\pi$
$774$	0	0
$775$	6.28610	0.225804
$776$	0	0
$777$	−7.68286	−0.275621
$778$	0	0
$779$	−3.64121	−0.130460
$780$	0	0
$781$	58.8448	2.10563
$782$	0	0
$783$	−22.2963	−0.796805
$784$	0	0
$785$	1.27569	0.0455314
$786$	0	0
$787$	3.35319	0.119528	0.0597642	−	0.998213i	$-0.480965\pi$
$787$	3.35319	0.119528	0.0597642	+	0.998213i	$0.480965\pi$
$788$	0	0
$789$	−9.11355	−0.324451
$790$	0	0
$791$	−19.5333	−0.694523
$792$	0	0
$793$	−50.4681	−1.79217
$794$	0	0
$795$	9.18691	0.325826
$796$	0	0
$797$	−27.6800	−0.980475	−0.490237	−	0.871589i	$-0.663090\pi$
$797$	−27.6800	−0.980475	−0.490237	+	0.871589i	$0.663090\pi$
$798$	0	0
$799$	3.44322	0.121812
$800$	0	0
$801$	−14.6773	−0.518596
$802$	0	0
$803$	−27.4611	−0.969083
$804$	0	0
$805$	−16.3684	−0.576911
$806$	0	0
$807$	−20.6081	−0.725438
$808$	0	0
$809$	−5.70079	−0.200429	−0.100215	−	0.994966i	$-0.531953\pi$
$809$	−5.70079	−0.200429	−0.100215	+	0.994966i	$0.531953\pi$
$810$	0	0
$811$	3.39676	0.119276	0.0596382	−	0.998220i	$-0.481005\pi$
$811$	3.39676	0.119276	0.0596382	+	0.998220i	$0.481005\pi$
$812$	0	0
$813$	−1.64411	−0.0576615
$814$	0	0
$815$	−0.368420	−0.0129052
$816$	0	0
$817$	8.79641	0.307748
$818$	0	0
$819$	−10.4972	−0.366802
$820$	0	0
$821$	−16.6676	−0.581704	−0.290852	−	0.956768i	$-0.593939\pi$
$821$	−16.6676	−0.581704	−0.290852	+	0.956768i	$0.593939\pi$
$822$	0	0
$823$	−25.9196	−0.903501	−0.451750	−	0.892144i	$-0.649200\pi$
$823$	−25.9196	−0.903501	−0.451750	+	0.892144i	$0.649200\pi$
$824$	0	0
$825$	−3.40987	−0.118716
$826$	0	0
$827$	30.9252	1.07537	0.537687	−	0.843144i	$-0.319298\pi$
$827$	30.9252	1.07537	0.537687	+	0.843144i	$0.319298\pi$
$828$	0	0
$829$	−49.3836	−1.71517	−0.857583	−	0.514346i	$-0.828035\pi$
$829$	−49.3836	−1.71517	−0.857583	+	0.514346i	$0.828035\pi$
$830$	0	0
$831$	−12.3505	−0.428434
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−31.8927	−1.10369
$836$	0	0
$837$	−26.7356	−0.924117
$838$	0	0
$839$	−27.1857	−0.938553	−0.469277	−	0.883051i	$-0.655485\pi$
$839$	−27.1857	−0.938553	−0.469277	+	0.883051i	$0.655485\pi$
$840$	0	0
$841$	−4.74244	−0.163532
$842$	0	0
$843$	−2.19732	−0.0756797
$844$	0	0
$845$	21.1607	0.727950
$846$	0	0
$847$	−2.85039	−0.0979407
$848$	0	0
$849$	17.3413	0.595153
$850$	0	0
$851$	59.3241	2.03360
$852$	0	0
$853$	4.23964	0.145162	0.0725812	−	0.997363i	$-0.476876\pi$
$853$	4.23964	0.145162	0.0725812	+	0.997363i	$0.476876\pi$
$854$	0	0
$855$	−5.14691	−0.176021
$856$	0	0
$857$	−52.2563	−1.78504	−0.892521	−	0.451006i	$-0.851065\pi$
$857$	−52.2563	−1.78504	−0.892521	+	0.451006i	$0.851065\pi$
$858$	0	0
$859$	37.7729	1.28879	0.644397	−	0.764691i	$-0.277108\pi$
$859$	37.7729	1.28879	0.644397	+	0.764691i	$0.277108\pi$
$860$	0	0
$861$	3.38780	0.115456
$862$	0	0
$863$	−24.8698	−0.846577	−0.423288	−	0.905995i	$-0.639124\pi$
$863$	−24.8698	−0.846577	−0.423288	+	0.905995i	$0.639124\pi$
$864$	0	0
$865$	8.41613	0.286157
$866$	0	0
$867$	0.860806	0.0292345
$868$	0	0
$869$	14.8864	0.504988
$870$	0	0
$871$	−34.3297	−1.16322
$872$	0	0
$873$	23.8385	0.806812
$874$	0	0
$875$	9.69182	0.327643
$876$	0	0
$877$	33.0540	1.11615	0.558077	−	0.829789i	$-0.311539\pi$
$877$	33.0540	1.11615	0.558077	+	0.829789i	$0.311539\pi$
$878$	0	0
$879$	9.29362	0.313466
$880$	0	0
$881$	−10.7410	−0.361873	−0.180937	−	0.983495i	$-0.557913\pi$
$881$	−10.7410	−0.361873	−0.180937	+	0.983495i	$0.557913\pi$
$882$	0	0
$883$	33.3490	1.12228	0.561142	−	0.827719i	$-0.310362\pi$
$883$	33.3490	1.12228	0.561142	+	0.827719i	$0.310362\pi$
$884$	0	0
$885$	−28.1801	−0.947262
$886$	0	0
$887$	35.6371	1.19658	0.598288	−	0.801281i	$-0.295848\pi$
$887$	35.6371	1.19658	0.598288	+	0.801281i	$0.295848\pi$
$888$	0	0
$889$	−2.83102	−0.0949493
$890$	0	0
$891$	−10.7189	−0.359097
$892$	0	0
$893$	−3.18566	−0.106604
$894$	0	0
$895$	−1.16127	−0.0388168
$896$	0	0
$897$	−26.5872	−0.887722
$898$	0	0
$899$	29.0873	0.970117
$900$	0	0
$901$	−4.33382	−0.144380
$902$	0	0
$903$	−8.18421	−0.272353
$904$	0	0
$905$	39.9917	1.32937
$906$	0	0
$907$	3.96954	0.131806	0.0659032	−	0.997826i	$-0.479007\pi$
$907$	3.96954	0.131806	0.0659032	+	0.997826i	$0.479007\pi$
$908$	0	0
$909$	−38.4376	−1.27490
$910$	0	0
$911$	29.4640	0.976187	0.488094	−	0.872791i	$-0.337692\pi$
$911$	29.4640	0.976187	0.488094	+	0.872791i	$0.337692\pi$
$912$	0	0
$913$	11.4432	0.378715
$914$	0	0
$915$	23.0229	0.761115
$916$	0	0
$917$	10.7964	0.356529
$918$	0	0
$919$	13.6877	0.451515	0.225757	−	0.974184i	$-0.427514\pi$
$919$	13.6877	0.451515	0.225757	+	0.974184i	$0.427514\pi$
$920$	0	0
$921$	−10.1830	−0.335540
$922$	0	0
$923$	73.4737	2.41842
$924$	0	0
$925$	9.49990	0.312355
$926$	0	0
$927$	11.1261	0.365429
$928$	0	0
$929$	14.8102	0.485907	0.242953	−	0.970038i	$-0.421884\pi$
$929$	14.8102	0.485907	0.242953	+	0.970038i	$0.421884\pi$
$930$	0	0
$931$	0.925197	0.0303221
$932$	0	0
$933$	7.95229	0.260346
$934$	0	0
$935$	9.16484	0.299722
$936$	0	0
$937$	29.0844	0.950147	0.475074	−	0.879946i	$-0.342421\pi$
$937$	29.0844	0.950147	0.475074	+	0.879946i	$0.342421\pi$
$938$	0	0
$939$	−11.6351	−0.379699
$940$	0	0
$941$	8.41199	0.274223	0.137111	−	0.990556i	$-0.456218\pi$
$941$	8.41199	0.274223	0.137111	+	0.990556i	$0.456218\pi$
$942$	0	0
$943$	−26.1592	−0.851862
$944$	0	0
$945$	11.1482	0.362650
$946$	0	0
$947$	−22.6081	−0.734663	−0.367332	−	0.930090i	$-0.619729\pi$
$947$	−22.6081	−0.734663	−0.367332	+	0.930090i	$0.619729\pi$
$948$	0	0
$949$	−34.2880	−1.11304
$950$	0	0
$951$	5.77289	0.187199
$952$	0	0
$953$	9.81579	0.317965	0.158982	−	0.987281i	$-0.449179\pi$
$953$	9.81579	0.317965	0.158982	+	0.987281i	$0.449179\pi$
$954$	0	0
$955$	34.8787	1.12865
$956$	0	0
$957$	−15.7783	−0.510040
$958$	0	0
$959$	−13.6572	−0.441015
$960$	0	0
$961$	3.87873	0.125120
$962$	0	0
$963$	−45.3836	−1.46247
$964$	0	0
$965$	12.1801	0.392090
$966$	0	0
$967$	−10.0227	−0.322310	−0.161155	−	0.986929i	$-0.551522\pi$
$967$	−10.0227	−0.322310	−0.161155	+	0.986929i	$0.551522\pi$
$968$	0	0
$969$	−0.796415	−0.0255845
$970$	0	0
$971$	−43.6925	−1.40216	−0.701079	−	0.713083i	$-0.747298\pi$
$971$	−43.6925	−1.40216	−0.701079	+	0.713083i	$0.747298\pi$
$972$	0	0
$973$	5.51658	0.176853
$974$	0	0
$975$	−4.25756	−0.136351
$976$	0	0
$977$	−44.9598	−1.43839	−0.719196	−	0.694808i	$-0.755490\pi$
$977$	−44.9598	−1.43839	−0.719196	+	0.694808i	$0.755490\pi$
$978$	0	0
$979$	−24.1801	−0.772798
$980$	0	0
$981$	−28.2313	−0.901358
$982$	0	0
$983$	−23.3103	−0.743483	−0.371741	−	0.928336i	$-0.621239\pi$
$983$	−23.3103	−0.743483	−0.371741	+	0.928336i	$0.621239\pi$
$984$	0	0
$985$	53.0278	1.68960
$986$	0	0
$987$	2.96395	0.0943435
$988$	0	0
$989$	63.1953	2.00949
$990$	0	0
$991$	23.6412	0.750988	0.375494	−	0.926825i	$-0.377473\pi$
$991$	23.6412	0.750988	0.375494	+	0.926825i	$0.377473\pi$
$992$	0	0
$993$	8.26943	0.262422
$994$	0	0
$995$	−37.8206	−1.19899
$996$	0	0
$997$	−20.5408	−0.650533	−0.325266	−	0.945622i	$-0.605454\pi$
$997$	−20.5408	−0.650533	−0.325266	+	0.945622i	$0.605454\pi$
$998$	0	0
$999$	−40.4043	−1.27833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bb.1.3		3
4.3	odd	2		7616.2.a.bh.1.1		3
8.3	odd	2		952.2.a.c.1.3	✓	3
8.5	even	2		1904.2.a.o.1.1		3
24.11	even	2		8568.2.a.be.1.2		3
56.27	even	2		6664.2.a.m.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
952.2.a.c.1.3	✓	3	8.3	odd	2
1904.2.a.o.1.1		3	8.5	even	2
6664.2.a.m.1.1		3	56.27	even	2
7616.2.a.bb.1.3		3	1.1	even	1	trivial
7616.2.a.bh.1.1		3	4.3	odd	2
8568.2.a.be.1.2		3	24.11	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:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q+0.860806 q^{3} +2.46260 q^{5} -1.00000 q^{7} -2.25901 q^{9} -3.72161 q^{11} -4.64681 q^{13} +2.11982 q^{15} -1.00000 q^{17} +0.925197 q^{19} -0.860806 q^{21} +6.64681 q^{23} +1.06439 q^{25} -4.52699 q^{27} +4.92520 q^{29} +5.90582 q^{31} -3.20359 q^{33} -2.46260 q^{35} +8.92520 q^{37} -4.00000 q^{39} -3.93561 q^{41} +9.50761 q^{43} -5.56304 q^{45} -3.44322 q^{47} +1.00000 q^{49} -0.860806 q^{51} +4.33382 q^{53} -9.16484 q^{55} +0.796415 q^{57} -13.2936 q^{59} +10.8608 q^{61} +2.25901 q^{63} -11.4432 q^{65} +7.38780 q^{67} +5.72161 q^{69} -15.8116 q^{71} +7.37883 q^{73} +0.916234 q^{75} +3.72161 q^{77} -4.00000 q^{79} +2.88018 q^{81} -3.07480 q^{83} -2.46260 q^{85} +4.23964 q^{87} +6.49720 q^{89} +4.64681 q^{91} +5.08377 q^{93} +2.27839 q^{95} -10.5526 q^{97} +8.40717 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 5 q^{5} - 3 q^{7} + 2 q^{9} + 2 q^{13} - 8 q^{15} - 3 q^{17} - 2 q^{19} + 3 q^{21} + 4 q^{23} + 4 q^{25} - 12 q^{27} + 10 q^{29} - 7 q^{31} - 16 q^{33} - 5 q^{35} + 22 q^{37} - 12 q^{39}+ \cdots + 38 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 5 * q^5 - 3 * q^7 + 2 * q^9 + 2 * q^13 - 8 * q^15 - 3 * q^17 - 2 * q^19 + 3 * q^21 + 4 * q^23 + 4 * q^25 - 12 * q^27 + 10 * q^29 - 7 * q^31 - 16 * q^33 - 5 * q^35 + 22 * q^37 - 12 * q^39 - 11 * q^41 + 7 * q^43 + 20 * q^45 + 12 * q^47 + 3 * q^49 + 3 * q^51 + 9 * q^53 + 6 * q^55 - 4 * q^57 - 8 * q^59 + 27 * q^61 - 2 * q^63 - 12 * q^65 + 15 * q^67 + 6 * q^69 + 2 * q^71 - q^73 - 18 * q^75 - 12 * q^79 + 23 * q^81 - 14 * q^83 - 5 * q^85 - 16 * q^87 - 6 * q^89 - 2 * q^91 + 36 * q^93 + 18 * q^95 + 9 * q^97 + 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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