LMFDB - Embedded newform 2888.2.a.m.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2888,2,Mod(1,2888)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2888.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2888 = 2^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2888.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,-3,0,-6,0,12,0,-6,0,0,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.0607961037$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	2888.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41147 q^{3} -1.34730 q^{5} -0.467911 q^{7} +2.81521 q^{9} -0.120615 q^{11} -4.10607 q^{13} -3.24897 q^{15} -7.63816 q^{17} -1.12836 q^{21} +6.45336 q^{23} -3.18479 q^{25} -0.445622 q^{27} +4.65270 q^{29} -5.59627 q^{31} -0.290859 q^{33} +0.630415 q^{35} +0.588526 q^{37} -9.90167 q^{39} -7.66044 q^{41} +2.49020 q^{43} -3.79292 q^{45} +4.24897 q^{47} -6.78106 q^{49} -18.4192 q^{51} -9.55438 q^{53} +0.162504 q^{55} -8.14796 q^{59} +2.51754 q^{61} -1.31727 q^{63} +5.53209 q^{65} +8.53714 q^{67} +15.5621 q^{69} -16.0993 q^{71} -13.6459 q^{73} -7.68004 q^{75} +0.0564370 q^{77} +16.0496 q^{79} -9.52023 q^{81} -3.65270 q^{83} +10.2909 q^{85} +11.2199 q^{87} +2.42602 q^{89} +1.92127 q^{91} -13.4953 q^{93} +9.36959 q^{97} -0.339556 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} - 6 q^{7} + 12 q^{9} - 6 q^{11} + 3 q^{15} - 6 q^{17} + 15 q^{21} + 6 q^{23} - 6 q^{25} - 12 q^{27} + 15 q^{29} - 3 q^{31} + 15 q^{33} + 9 q^{35} + 12 q^{37} - 18 q^{39} + 6 q^{43}+ \cdots - 24 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 - 6 * q^7 + 12 * q^9 - 6 * q^11 + 3 * q^15 - 6 * q^17 + 15 * q^21 + 6 * q^23 - 6 * q^25 - 12 * q^27 + 15 * q^29 - 3 * q^31 + 15 * q^33 + 9 * q^35 + 12 * q^37 - 18 * q^39 + 6 * q^43 - 21 * q^45 - 3 * q^49 - 21 * q^51 - 18 * q^53 + 3 * q^55 - 9 * q^59 - 15 * q^61 - 33 * q^63 + 12 * q^65 + 6 * q^67 + 12 * q^69 + 6 * q^71 - 3 * q^75 + 15 * q^77 + 21 * q^79 + 3 * q^81 - 12 * q^83 + 15 * q^85 - 15 * q^87 + 15 * q^89 - 3 * q^91 - 24 * q^93 + 21 * q^97 - 24 * 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
$3$	2.41147	1.39227	0.696133	−	0.717913i	$-0.254903\pi$
$3$	2.41147	1.39227	0.696133	+	0.717913i	$0.254903\pi$
$4$	0	0
$5$	−1.34730	−0.602529	−0.301265	−	0.953541i	$-0.597409\pi$
$5$	−1.34730	−0.602529	−0.301265	+	0.953541i	$0.597409\pi$
$6$	0	0
$7$	−0.467911	−0.176854	−0.0884269	−	0.996083i	$-0.528184\pi$
$7$	−0.467911	−0.176854	−0.0884269	+	0.996083i	$0.528184\pi$
$8$	0	0
$9$	2.81521	0.938402
$10$	0	0
$11$	−0.120615	−0.0363667	−0.0181834	−	0.999835i	$-0.505788\pi$
$11$	−0.120615	−0.0363667	−0.0181834	+	0.999835i	$0.505788\pi$
$12$	0	0
$13$	−4.10607	−1.13882	−0.569409	−	0.822054i	$-0.692828\pi$
$13$	−4.10607	−1.13882	−0.569409	+	0.822054i	$0.692828\pi$
$14$	0	0
$15$	−3.24897	−0.838881
$16$	0	0
$17$	−7.63816	−1.85252	−0.926262	−	0.376879i	$-0.876997\pi$
$17$	−7.63816	−1.85252	−0.926262	+	0.376879i	$0.876997\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−1.12836	−0.246227
$22$	0	0
$23$	6.45336	1.34562	0.672810	−	0.739816i	$-0.265087\pi$
$23$	6.45336	1.34562	0.672810	+	0.739816i	$0.265087\pi$
$24$	0	0
$25$	−3.18479	−0.636959
$26$	0	0
$27$	−0.445622	−0.0857601
$28$	0	0
$29$	4.65270	0.863985	0.431993	−	0.901877i	$-0.357811\pi$
$29$	4.65270	0.863985	0.431993	+	0.901877i	$0.357811\pi$
$30$	0	0
$31$	−5.59627	−1.00512	−0.502560	−	0.864543i	$-0.667608\pi$
$31$	−5.59627	−1.00512	−0.502560	+	0.864543i	$0.667608\pi$
$32$	0	0
$33$	−0.290859	−0.0506321
$34$	0	0
$35$	0.630415	0.106560
$36$	0	0
$37$	0.588526	0.0967531	0.0483765	−	0.998829i	$-0.484595\pi$
$37$	0.588526	0.0967531	0.0483765	+	0.998829i	$0.484595\pi$
$38$	0	0
$39$	−9.90167	−1.58554
$40$	0	0
$41$	−7.66044	−1.19636	−0.598180	−	0.801362i	$-0.704109\pi$
$41$	−7.66044	−1.19636	−0.598180	+	0.801362i	$0.704109\pi$
$42$	0	0
$43$	2.49020	0.379752	0.189876	−	0.981808i	$-0.439191\pi$
$43$	2.49020	0.379752	0.189876	+	0.981808i	$0.439191\pi$
$44$	0	0
$45$	−3.79292	−0.565415
$46$	0	0
$47$	4.24897	0.619776	0.309888	−	0.950773i	$-0.399708\pi$
$47$	4.24897	0.619776	0.309888	+	0.950773i	$0.399708\pi$
$48$	0	0
$49$	−6.78106	−0.968723
$50$	0	0
$51$	−18.4192	−2.57921
$52$	0	0
$53$	−9.55438	−1.31239	−0.656197	−	0.754589i	$-0.727836\pi$
$53$	−9.55438	−1.31239	−0.656197	+	0.754589i	$0.727836\pi$
$54$	0	0
$55$	0.162504	0.0219120
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.14796	−1.06077	−0.530387	−	0.847756i	$-0.677953\pi$
$59$	−8.14796	−1.06077	−0.530387	+	0.847756i	$0.677953\pi$
$60$	0	0
$61$	2.51754	0.322338	0.161169	−	0.986927i	$-0.448474\pi$
$61$	2.51754	0.322338	0.161169	+	0.986927i	$0.448474\pi$
$62$	0	0
$63$	−1.31727	−0.165960
$64$	0	0
$65$	5.53209	0.686171
$66$	0	0
$67$	8.53714	1.04298	0.521489	−	0.853258i	$-0.325377\pi$
$67$	8.53714	1.04298	0.521489	+	0.853258i	$0.325377\pi$
$68$	0	0
$69$	15.5621	1.87346
$70$	0	0
$71$	−16.0993	−1.91063	−0.955315	−	0.295589i	$-0.904484\pi$
$71$	−16.0993	−1.91063	−0.955315	+	0.295589i	$0.904484\pi$
$72$	0	0
$73$	−13.6459	−1.59713	−0.798566	−	0.601908i	$-0.794408\pi$
$73$	−13.6459	−1.59713	−0.798566	+	0.601908i	$0.794408\pi$
$74$	0	0
$75$	−7.68004	−0.886815
$76$	0	0
$77$	0.0564370	0.00643159
$78$	0	0
$79$	16.0496	1.80572	0.902862	−	0.429930i	$-0.141462\pi$
$79$	16.0496	1.80572	0.902862	+	0.429930i	$0.141462\pi$
$80$	0	0
$81$	−9.52023	−1.05780
$82$	0	0
$83$	−3.65270	−0.400936	−0.200468	−	0.979700i	$-0.564246\pi$
$83$	−3.65270	−0.400936	−0.200468	+	0.979700i	$0.564246\pi$
$84$	0	0
$85$	10.2909	1.11620
$86$	0	0
$87$	11.2199	1.20290
$88$	0	0
$89$	2.42602	0.257158	0.128579	−	0.991699i	$-0.458958\pi$
$89$	2.42602	0.257158	0.128579	+	0.991699i	$0.458958\pi$
$90$	0	0
$91$	1.92127	0.201404
$92$	0	0
$93$	−13.4953	−1.39939
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.36959	0.951337	0.475669	−	0.879625i	$-0.342206\pi$
$97$	9.36959	0.951337	0.475669	+	0.879625i	$0.342206\pi$
$98$	0	0
$99$	−0.339556	−0.0341266
$100$	0	0
$101$	6.17024	0.613962	0.306981	−	0.951716i	$-0.400681\pi$
$101$	6.17024	0.613962	0.306981	+	0.951716i	$0.400681\pi$
$102$	0	0
$103$	7.53983	0.742921	0.371461	−	0.928449i	$-0.378857\pi$
$103$	7.53983	0.742921	0.371461	+	0.928449i	$0.378857\pi$
$104$	0	0
$105$	1.52023	0.148359
$106$	0	0
$107$	5.41921	0.523895	0.261948	−	0.965082i	$-0.415635\pi$
$107$	5.41921	0.523895	0.261948	+	0.965082i	$0.415635\pi$
$108$	0	0
$109$	−8.54664	−0.818619	−0.409310	−	0.912396i	$-0.634230\pi$
$109$	−8.54664	−0.818619	−0.409310	+	0.912396i	$0.634230\pi$
$110$	0	0
$111$	1.41921	0.134706
$112$	0	0
$113$	6.37464	0.599675	0.299838	−	0.953990i	$-0.403067\pi$
$113$	6.37464	0.599675	0.299838	+	0.953990i	$0.403067\pi$
$114$	0	0
$115$	−8.69459	−0.810775
$116$	0	0
$117$	−11.5594	−1.06867
$118$	0	0
$119$	3.57398	0.327626
$120$	0	0
$121$	−10.9855	−0.998677
$122$	0	0
$123$	−18.4730	−1.66565
$124$	0	0
$125$	11.0273	0.986315
$126$	0	0
$127$	0.162504	0.0144199	0.00720994	−	0.999974i	$-0.497705\pi$
$127$	0.162504	0.0144199	0.00720994	+	0.999974i	$0.497705\pi$
$128$	0	0
$129$	6.00505	0.528715
$130$	0	0
$131$	−0.985452	−0.0860993	−0.0430497	−	0.999073i	$-0.513707\pi$
$131$	−0.985452	−0.0860993	−0.0430497	+	0.999073i	$0.513707\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.600385	0.0516730
$136$	0	0
$137$	−1.96316	−0.167724	−0.0838622	−	0.996477i	$-0.526726\pi$
$137$	−1.96316	−0.167724	−0.0838622	+	0.996477i	$0.526726\pi$
$138$	0	0
$139$	−13.7888	−1.16955	−0.584775	−	0.811195i	$-0.698817\pi$
$139$	−13.7888	−1.16955	−0.584775	+	0.811195i	$0.698817\pi$
$140$	0	0
$141$	10.2463	0.862893
$142$	0	0
$143$	0.495252	0.0414151
$144$	0	0
$145$	−6.26857	−0.520576
$146$	0	0
$147$	−16.3523	−1.34872
$148$	0	0
$149$	−8.14290	−0.667093	−0.333546	−	0.942734i	$-0.608245\pi$
$149$	−8.14290	−0.667093	−0.333546	+	0.942734i	$0.608245\pi$
$150$	0	0
$151$	17.6236	1.43419	0.717094	−	0.696976i	$-0.245472\pi$
$151$	17.6236	1.43419	0.717094	+	0.696976i	$0.245472\pi$
$152$	0	0
$153$	−21.5030	−1.73841
$154$	0	0
$155$	7.53983	0.605614
$156$	0	0
$157$	−20.0419	−1.59952	−0.799758	−	0.600322i	$-0.795039\pi$
$157$	−20.0419	−1.59952	−0.799758	+	0.600322i	$0.795039\pi$
$158$	0	0
$159$	−23.0401	−1.82720
$160$	0	0
$161$	−3.01960	−0.237978
$162$	0	0
$163$	9.28405	0.727183	0.363592	−	0.931558i	$-0.381550\pi$
$163$	9.28405	0.727183	0.363592	+	0.931558i	$0.381550\pi$
$164$	0	0
$165$	0.391874	0.0305073
$166$	0	0
$167$	11.8229	0.914887	0.457444	−	0.889239i	$-0.348765\pi$
$167$	11.8229	0.914887	0.457444	+	0.889239i	$0.348765\pi$
$168$	0	0
$169$	3.85978	0.296907
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.4688	1.70827	0.854137	−	0.520048i	$-0.174086\pi$
$173$	22.4688	1.70827	0.854137	+	0.520048i	$0.174086\pi$
$174$	0	0
$175$	1.49020	0.112649
$176$	0	0
$177$	−19.6486	−1.47688
$178$	0	0
$179$	−3.03684	−0.226984	−0.113492	−	0.993539i	$-0.536204\pi$
$179$	−3.03684	−0.226984	−0.113492	+	0.993539i	$0.536204\pi$
$180$	0	0
$181$	−11.4338	−0.849865	−0.424932	−	0.905225i	$-0.639702\pi$
$181$	−11.4338	−0.849865	−0.424932	+	0.905225i	$0.639702\pi$
$182$	0	0
$183$	6.07098	0.448780
$184$	0	0
$185$	−0.792919	−0.0582965
$186$	0	0
$187$	0.921274	0.0673702
$188$	0	0
$189$	0.208512	0.0151670
$190$	0	0
$191$	4.02734	0.291408	0.145704	−	0.989328i	$-0.453455\pi$
$191$	4.02734	0.291408	0.145704	+	0.989328i	$0.453455\pi$
$192$	0	0
$193$	−16.5790	−1.19338	−0.596692	−	0.802470i	$-0.703519\pi$
$193$	−16.5790	−1.19338	−0.596692	+	0.802470i	$0.703519\pi$
$194$	0	0
$195$	13.3405	0.955332
$196$	0	0
$197$	7.48751	0.533463	0.266732	−	0.963771i	$-0.414056\pi$
$197$	7.48751	0.533463	0.266732	+	0.963771i	$0.414056\pi$
$198$	0	0
$199$	−23.7743	−1.68531	−0.842656	−	0.538452i	$-0.819009\pi$
$199$	−23.7743	−1.68531	−0.842656	+	0.538452i	$0.819009\pi$
$200$	0	0
$201$	20.5871	1.45210
$202$	0	0
$203$	−2.17705	−0.152799
$204$	0	0
$205$	10.3209	0.720842
$206$	0	0
$207$	18.1676	1.26273
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−3.03684	−0.209064	−0.104532	−	0.994521i	$-0.533335\pi$
$211$	−3.03684	−0.209064	−0.104532	+	0.994521i	$0.533335\pi$
$212$	0	0
$213$	−38.8229	−2.66010
$214$	0	0
$215$	−3.35504	−0.228812
$216$	0	0
$217$	2.61856	0.177759
$218$	0	0
$219$	−32.9067	−2.22363
$220$	0	0
$221$	31.3628	2.10969
$222$	0	0
$223$	−6.28312	−0.420749	−0.210374	−	0.977621i	$-0.567468\pi$
$223$	−6.28312	−0.420749	−0.210374	+	0.977621i	$0.567468\pi$
$224$	0	0
$225$	−8.96585	−0.597723
$226$	0	0
$227$	−4.09833	−0.272015	−0.136008	−	0.990708i	$-0.543427\pi$
$227$	−4.09833	−0.272015	−0.136008	+	0.990708i	$0.543427\pi$
$228$	0	0
$229$	−1.21213	−0.0801000	−0.0400500	−	0.999198i	$-0.512752\pi$
$229$	−1.21213	−0.0801000	−0.0400500	+	0.999198i	$0.512752\pi$
$230$	0	0
$231$	0.136096	0.00895448
$232$	0	0
$233$	6.22399	0.407747	0.203874	−	0.978997i	$-0.434647\pi$
$233$	6.22399	0.407747	0.203874	+	0.978997i	$0.434647\pi$
$234$	0	0
$235$	−5.72462	−0.373433
$236$	0	0
$237$	38.7033	2.51405
$238$	0	0
$239$	12.7365	0.823855	0.411927	−	0.911217i	$-0.364856\pi$
$239$	12.7365	0.823855	0.411927	+	0.911217i	$0.364856\pi$
$240$	0	0
$241$	20.7442	1.33625	0.668126	−	0.744048i	$-0.267097\pi$
$241$	20.7442	1.33625	0.668126	+	0.744048i	$0.267097\pi$
$242$	0	0
$243$	−21.6209	−1.38698
$244$	0	0
$245$	9.13610	0.583684
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−8.80840	−0.558210
$250$	0	0
$251$	7.04189	0.444480	0.222240	−	0.974992i	$-0.428663\pi$
$251$	7.04189	0.444480	0.222240	+	0.974992i	$0.428663\pi$
$252$	0	0
$253$	−0.778371	−0.0489358
$254$	0	0
$255$	24.8161	1.55405
$256$	0	0
$257$	16.8357	1.05018	0.525092	−	0.851045i	$-0.324031\pi$
$257$	16.8357	1.05018	0.525092	+	0.851045i	$0.324031\pi$
$258$	0	0
$259$	−0.275378	−0.0171111
$260$	0	0
$261$	13.0983	0.810766
$262$	0	0
$263$	−7.85978	−0.484655	−0.242328	−	0.970194i	$-0.577911\pi$
$263$	−7.85978	−0.484655	−0.242328	+	0.970194i	$0.577911\pi$
$264$	0	0
$265$	12.8726	0.790756
$266$	0	0
$267$	5.85029	0.358032
$268$	0	0
$269$	32.4611	1.97919	0.989594	−	0.143886i	$-0.0459600\pi$
$269$	32.4611	1.97919	0.989594	+	0.143886i	$0.0459600\pi$
$270$	0	0
$271$	1.44562	0.0878153	0.0439077	−	0.999036i	$-0.486019\pi$
$271$	1.44562	0.0878153	0.0439077	+	0.999036i	$0.486019\pi$
$272$	0	0
$273$	4.63310	0.280408
$274$	0	0
$275$	0.384133	0.0231641
$276$	0	0
$277$	17.0770	1.02606	0.513028	−	0.858372i	$-0.328524\pi$
$277$	17.0770	1.02606	0.513028	+	0.858372i	$0.328524\pi$
$278$	0	0
$279$	−15.7547	−0.943206
$280$	0	0
$281$	17.7442	1.05853	0.529266	−	0.848456i	$-0.322467\pi$
$281$	17.7442	1.05853	0.529266	+	0.848456i	$0.322467\pi$
$282$	0	0
$283$	−26.1334	−1.55347	−0.776735	−	0.629828i	$-0.783126\pi$
$283$	−26.1334	−1.55347	−0.776735	+	0.629828i	$0.783126\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.58441	0.211581
$288$	0	0
$289$	41.3414	2.43185
$290$	0	0
$291$	22.5945	1.32451
$292$	0	0
$293$	−6.67736	−0.390095	−0.195048	−	0.980794i	$-0.562486\pi$
$293$	−6.67736	−0.390095	−0.195048	+	0.980794i	$0.562486\pi$
$294$	0	0
$295$	10.9777	0.639147
$296$	0	0
$297$	0.0537486	0.00311881
$298$	0	0
$299$	−26.4979	−1.53242
$300$	0	0
$301$	−1.16519	−0.0671606
$302$	0	0
$303$	14.8794	0.854798
$304$	0	0
$305$	−3.39187	−0.194218
$306$	0	0
$307$	−20.6382	−1.17788	−0.588941	−	0.808176i	$-0.700455\pi$
$307$	−20.6382	−1.17788	−0.588941	+	0.808176i	$0.700455\pi$
$308$	0	0
$309$	18.1821	1.03434
$310$	0	0
$311$	−0.667252	−0.0378364	−0.0189182	−	0.999821i	$-0.506022\pi$
$311$	−0.667252	−0.0378364	−0.0189182	+	0.999821i	$0.506022\pi$
$312$	0	0
$313$	−22.1506	−1.25203	−0.626014	−	0.779812i	$-0.715315\pi$
$313$	−22.1506	−1.25203	−0.626014	+	0.779812i	$0.715315\pi$
$314$	0	0
$315$	1.77475	0.0999958
$316$	0	0
$317$	−8.50568	−0.477727	−0.238863	−	0.971053i	$-0.576775\pi$
$317$	−8.50568	−0.477727	−0.238863	+	0.971053i	$0.576775\pi$
$318$	0	0
$319$	−0.561185	−0.0314203
$320$	0	0
$321$	13.0683	0.729401
$322$	0	0
$323$	0	0
$324$	0	0
$325$	13.0770	0.725380
$326$	0	0
$327$	−20.6100	−1.13974
$328$	0	0
$329$	−1.98814	−0.109610
$330$	0	0
$331$	−13.9067	−0.764383	−0.382191	−	0.924083i	$-0.624830\pi$
$331$	−13.9067	−0.764383	−0.382191	+	0.924083i	$0.624830\pi$
$332$	0	0
$333$	1.65682	0.0907933
$334$	0	0
$335$	−11.5021	−0.628425
$336$	0	0
$337$	18.9486	1.03220	0.516098	−	0.856529i	$-0.327384\pi$
$337$	18.9486	1.03220	0.516098	+	0.856529i	$0.327384\pi$
$338$	0	0
$339$	15.3723	0.834907
$340$	0	0
$341$	0.674992	0.0365529
$342$	0	0
$343$	6.44831	0.348176
$344$	0	0
$345$	−20.9668	−1.12881
$346$	0	0
$347$	−24.6313	−1.32228	−0.661140	−	0.750263i	$-0.729927\pi$
$347$	−24.6313	−1.32228	−0.661140	+	0.750263i	$0.729927\pi$
$348$	0	0
$349$	23.1043	1.23675	0.618373	−	0.785885i	$-0.287792\pi$
$349$	23.1043	1.23675	0.618373	+	0.785885i	$0.287792\pi$
$350$	0	0
$351$	1.82976	0.0976651
$352$	0	0
$353$	−28.9118	−1.53882	−0.769409	−	0.638756i	$-0.779449\pi$
$353$	−28.9118	−1.53882	−0.769409	+	0.638756i	$0.779449\pi$
$354$	0	0
$355$	21.6905	1.15121
$356$	0	0
$357$	8.61856	0.456142
$358$	0	0
$359$	15.8256	0.835245	0.417623	−	0.908621i	$-0.362863\pi$
$359$	15.8256	0.835245	0.417623	+	0.908621i	$0.362863\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−26.4911	−1.39042
$364$	0	0
$365$	18.3851	0.962318
$366$	0	0
$367$	−19.8331	−1.03528	−0.517638	−	0.855599i	$-0.673189\pi$
$367$	−19.8331	−1.03528	−0.517638	+	0.855599i	$0.673189\pi$
$368$	0	0
$369$	−21.5657	−1.12267
$370$	0	0
$371$	4.47060	0.232102
$372$	0	0
$373$	−17.9786	−0.930899	−0.465449	−	0.885074i	$-0.654107\pi$
$373$	−17.9786	−0.930899	−0.465449	+	0.885074i	$0.654107\pi$
$374$	0	0
$375$	26.5921	1.37321
$376$	0	0
$377$	−19.1043	−0.983922
$378$	0	0
$379$	−3.35504	−0.172337	−0.0861683	−	0.996281i	$-0.527462\pi$
$379$	−3.35504	−0.172337	−0.0861683	+	0.996281i	$0.527462\pi$
$380$	0	0
$381$	0.391874	0.0200763
$382$	0	0
$383$	11.0665	0.565474	0.282737	−	0.959197i	$-0.408758\pi$
$383$	11.0665	0.565474	0.282737	+	0.959197i	$0.408758\pi$
$384$	0	0
$385$	−0.0760373	−0.00387522
$386$	0	0
$387$	7.01043	0.356360
$388$	0	0
$389$	6.04694	0.306592	0.153296	−	0.988180i	$-0.451011\pi$
$389$	6.04694	0.306592	0.153296	+	0.988180i	$0.451011\pi$
$390$	0	0
$391$	−49.2918	−2.49279
$392$	0	0
$393$	−2.37639	−0.119873
$394$	0	0
$395$	−21.6236	−1.08800
$396$	0	0
$397$	−14.9949	−0.752575	−0.376287	−	0.926503i	$-0.622799\pi$
$397$	−14.9949	−0.752575	−0.376287	+	0.926503i	$0.622799\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.6212	0.830025	0.415013	−	0.909816i	$-0.363777\pi$
$401$	16.6212	0.830025	0.415013	+	0.909816i	$0.363777\pi$
$402$	0	0
$403$	22.9786	1.14465
$404$	0	0
$405$	12.8266	0.637357
$406$	0	0
$407$	−0.0709849	−0.00351859
$408$	0	0
$409$	12.7196	0.628942	0.314471	−	0.949267i	$-0.398173\pi$
$409$	12.7196	0.628942	0.314471	+	0.949267i	$0.398173\pi$
$410$	0	0
$411$	−4.73412	−0.233517
$412$	0	0
$413$	3.81252	0.187602
$414$	0	0
$415$	4.92127	0.241576
$416$	0	0
$417$	−33.2513	−1.62832
$418$	0	0
$419$	26.5990	1.29944	0.649722	−	0.760172i	$-0.274885\pi$
$419$	26.5990	1.29944	0.649722	+	0.760172i	$0.274885\pi$
$420$	0	0
$421$	10.6946	0.521223	0.260611	−	0.965444i	$-0.416076\pi$
$421$	10.6946	0.521223	0.260611	+	0.965444i	$0.416076\pi$
$422$	0	0
$423$	11.9617	0.581599
$424$	0	0
$425$	24.3259	1.17998
$426$	0	0
$427$	−1.17799	−0.0570067
$428$	0	0
$429$	1.19429	0.0576608
$430$	0	0
$431$	−14.0273	−0.675673	−0.337837	−	0.941205i	$-0.609695\pi$
$431$	−14.0273	−0.675673	−0.337837	+	0.941205i	$0.609695\pi$
$432$	0	0
$433$	12.8571	0.617873	0.308936	−	0.951083i	$-0.400027\pi$
$433$	12.8571	0.617873	0.308936	+	0.951083i	$0.400027\pi$
$434$	0	0
$435$	−15.1165	−0.724781
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−16.1138	−0.769070	−0.384535	−	0.923110i	$-0.625638\pi$
$439$	−16.1138	−0.769070	−0.384535	+	0.923110i	$0.625638\pi$
$440$	0	0
$441$	−19.0901	−0.909052
$442$	0	0
$443$	3.53714	0.168055	0.0840273	−	0.996463i	$-0.473222\pi$
$443$	3.53714	0.168055	0.0840273	+	0.996463i	$0.473222\pi$
$444$	0	0
$445$	−3.26857	−0.154945
$446$	0	0
$447$	−19.6364	−0.928770
$448$	0	0
$449$	−23.3705	−1.10292	−0.551461	−	0.834200i	$-0.685930\pi$
$449$	−23.3705	−1.10292	−0.551461	+	0.834200i	$0.685930\pi$
$450$	0	0
$451$	0.923963	0.0435077
$452$	0	0
$453$	42.4989	1.99677
$454$	0	0
$455$	−2.58853	−0.121352
$456$	0	0
$457$	−7.53890	−0.352655	−0.176327	−	0.984332i	$-0.556422\pi$
$457$	−7.53890	−0.352655	−0.176327	+	0.984332i	$0.556422\pi$
$458$	0	0
$459$	3.40373	0.158873
$460$	0	0
$461$	−9.26588	−0.431555	−0.215778	−	0.976443i	$-0.569229\pi$
$461$	−9.26588	−0.431555	−0.215778	+	0.976443i	$0.569229\pi$
$462$	0	0
$463$	−40.6222	−1.88787	−0.943936	−	0.330128i	$-0.892908\pi$
$463$	−40.6222	−1.88787	−0.943936	+	0.330128i	$0.892908\pi$
$464$	0	0
$465$	18.1821	0.843175
$466$	0	0
$467$	−27.7469	−1.28397	−0.641987	−	0.766716i	$-0.721890\pi$
$467$	−27.7469	−1.28397	−0.641987	+	0.766716i	$0.721890\pi$
$468$	0	0
$469$	−3.99462	−0.184455
$470$	0	0
$471$	−48.3305	−2.22695
$472$	0	0
$473$	−0.300355	−0.0138103
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−26.8976	−1.23155
$478$	0	0
$479$	−34.5827	−1.58012	−0.790061	−	0.613028i	$-0.789951\pi$
$479$	−34.5827	−1.58012	−0.790061	+	0.613028i	$0.789951\pi$
$480$	0	0
$481$	−2.41653	−0.110184
$482$	0	0
$483$	−7.28169	−0.331328
$484$	0	0
$485$	−12.6236	−0.573209
$486$	0	0
$487$	−17.2882	−0.783402	−0.391701	−	0.920093i	$-0.628113\pi$
$487$	−17.2882	−0.783402	−0.391701	+	0.920093i	$0.628113\pi$
$488$	0	0
$489$	22.3883	1.01243
$490$	0	0
$491$	24.4483	1.10334	0.551668	−	0.834064i	$-0.313991\pi$
$491$	24.4483	1.10334	0.551668	+	0.834064i	$0.313991\pi$
$492$	0	0
$493$	−35.5381	−1.60055
$494$	0	0
$495$	0.457482	0.0205623
$496$	0	0
$497$	7.53302	0.337902
$498$	0	0
$499$	21.2175	0.949826	0.474913	−	0.880033i	$-0.342479\pi$
$499$	21.2175	0.949826	0.474913	+	0.880033i	$0.342479\pi$
$500$	0	0
$501$	28.5107	1.27377
$502$	0	0
$503$	32.4816	1.44828	0.724142	−	0.689651i	$-0.242236\pi$
$503$	32.4816	1.44828	0.724142	+	0.689651i	$0.242236\pi$
$504$	0	0
$505$	−8.31315	−0.369930
$506$	0	0
$507$	9.30777	0.413373
$508$	0	0
$509$	−26.2772	−1.16472	−0.582359	−	0.812932i	$-0.697870\pi$
$509$	−26.2772	−1.16472	−0.582359	+	0.812932i	$0.697870\pi$
$510$	0	0
$511$	6.38507	0.282459
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−10.1584	−0.447632
$516$	0	0
$517$	−0.512489	−0.0225392
$518$	0	0
$519$	54.1830	2.37837
$520$	0	0
$521$	0.610815	0.0267603	0.0133801	−	0.999910i	$-0.495741\pi$
$521$	0.610815	0.0267603	0.0133801	+	0.999910i	$0.495741\pi$
$522$	0	0
$523$	9.01186	0.394061	0.197031	−	0.980397i	$-0.436870\pi$
$523$	9.01186	0.394061	0.197031	+	0.980397i	$0.436870\pi$
$524$	0	0
$525$	3.59358	0.156837
$526$	0	0
$527$	42.7452	1.86201
$528$	0	0
$529$	18.6459	0.810691
$530$	0	0
$531$	−22.9382	−0.995432
$532$	0	0
$533$	31.4543	1.36244
$534$	0	0
$535$	−7.30129	−0.315662
$536$	0	0
$537$	−7.32325	−0.316022
$538$	0	0
$539$	0.817896	0.0352293
$540$	0	0
$541$	−16.3814	−0.704293	−0.352147	−	0.935945i	$-0.614548\pi$
$541$	−16.3814	−0.704293	−0.352147	+	0.935945i	$0.614548\pi$
$542$	0	0
$543$	−27.5722	−1.18324
$544$	0	0
$545$	11.5149	0.493242
$546$	0	0
$547$	39.4338	1.68607	0.843033	−	0.537862i	$-0.180768\pi$
$547$	39.4338	1.68607	0.843033	+	0.537862i	$0.180768\pi$
$548$	0	0
$549$	7.08740	0.302483
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−7.50980	−0.319349
$554$	0	0
$555$	−1.91210	−0.0811643
$556$	0	0
$557$	40.9249	1.73404	0.867022	−	0.498270i	$-0.166031\pi$
$557$	40.9249	1.73404	0.867022	+	0.498270i	$0.166031\pi$
$558$	0	0
$559$	−10.2249	−0.432468
$560$	0	0
$561$	2.22163	0.0937973
$562$	0	0
$563$	7.15301	0.301463	0.150732	−	0.988575i	$-0.451837\pi$
$563$	7.15301	0.301463	0.150732	+	0.988575i	$0.451837\pi$
$564$	0	0
$565$	−8.58853	−0.361322
$566$	0	0
$567$	4.45462	0.187077
$568$	0	0
$569$	−4.13516	−0.173355	−0.0866775	−	0.996236i	$-0.527625\pi$
$569$	−4.13516	−0.173355	−0.0866775	+	0.996236i	$0.527625\pi$
$570$	0	0
$571$	36.9513	1.54636	0.773182	−	0.634184i	$-0.218664\pi$
$571$	36.9513	1.54636	0.773182	+	0.634184i	$0.218664\pi$
$572$	0	0
$573$	9.71183	0.405717
$574$	0	0
$575$	−20.5526	−0.857104
$576$	0	0
$577$	29.6195	1.23308	0.616538	−	0.787325i	$-0.288535\pi$
$577$	29.6195	1.23308	0.616538	+	0.787325i	$0.288535\pi$
$578$	0	0
$579$	−39.9799	−1.66151
$580$	0	0
$581$	1.70914	0.0709071
$582$	0	0
$583$	1.15240	0.0477275
$584$	0	0
$585$	15.5740	0.643905
$586$	0	0
$587$	−44.2285	−1.82551	−0.912754	−	0.408510i	$-0.866048\pi$
$587$	−44.2285	−1.82551	−0.912754	+	0.408510i	$0.866048\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	18.0559	0.742722
$592$	0	0
$593$	18.4142	0.756179	0.378090	−	0.925769i	$-0.376581\pi$
$593$	18.4142	0.756179	0.378090	+	0.925769i	$0.376581\pi$
$594$	0	0
$595$	−4.81521	−0.197404
$596$	0	0
$597$	−57.3310	−2.34640
$598$	0	0
$599$	42.3063	1.72859	0.864295	−	0.502985i	$-0.167765\pi$
$599$	42.3063	1.72859	0.864295	+	0.502985i	$0.167765\pi$
$600$	0	0
$601$	−43.0009	−1.75404	−0.877022	−	0.480450i	$-0.840473\pi$
$601$	−43.0009	−1.75404	−0.877022	+	0.480450i	$0.840473\pi$
$602$	0	0
$603$	24.0338	0.978733
$604$	0	0
$605$	14.8007	0.601732
$606$	0	0
$607$	13.0787	0.530849	0.265425	−	0.964132i	$-0.414488\pi$
$607$	13.0787	0.530849	0.265425	+	0.964132i	$0.414488\pi$
$608$	0	0
$609$	−5.24990	−0.212737
$610$	0	0
$611$	−17.4466	−0.705812
$612$	0	0
$613$	−13.4439	−0.542993	−0.271496	−	0.962439i	$-0.587518\pi$
$613$	−13.4439	−0.542993	−0.271496	+	0.962439i	$0.587518\pi$
$614$	0	0
$615$	24.8886	1.00360
$616$	0	0
$617$	12.1165	0.487792	0.243896	−	0.969801i	$-0.421574\pi$
$617$	12.1165	0.487792	0.243896	+	0.969801i	$0.421574\pi$
$618$	0	0
$619$	−27.1343	−1.09062	−0.545311	−	0.838234i	$-0.683588\pi$
$619$	−27.1343	−1.09062	−0.545311	+	0.838234i	$0.683588\pi$
$620$	0	0
$621$	−2.87576	−0.115400
$622$	0	0
$623$	−1.13516	−0.0454793
$624$	0	0
$625$	1.06687	0.0426746
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4.49525	−0.179237
$630$	0	0
$631$	26.9513	1.07291	0.536457	−	0.843928i	$-0.319762\pi$
$631$	26.9513	1.07291	0.536457	+	0.843928i	$0.319762\pi$
$632$	0	0
$633$	−7.32325	−0.291073
$634$	0	0
$635$	−0.218941	−0.00868840
$636$	0	0
$637$	27.8435	1.10320
$638$	0	0
$639$	−45.3228	−1.79294
$640$	0	0
$641$	0.467911	0.0184814	0.00924069	−	0.999957i	$-0.497059\pi$
$641$	0.467911	0.0184814	0.00924069	+	0.999957i	$0.497059\pi$
$642$	0	0
$643$	−31.9290	−1.25916	−0.629579	−	0.776937i	$-0.716772\pi$
$643$	−31.9290	−1.25916	−0.629579	+	0.776937i	$0.716772\pi$
$644$	0	0
$645$	−8.09059	−0.318566
$646$	0	0
$647$	−24.2594	−0.953735	−0.476868	−	0.878975i	$-0.658228\pi$
$647$	−24.2594	−0.953735	−0.476868	+	0.878975i	$0.658228\pi$
$648$	0	0
$649$	0.982764	0.0385769
$650$	0	0
$651$	6.31458	0.247488
$652$	0	0
$653$	−25.5134	−0.998417	−0.499209	−	0.866482i	$-0.666376\pi$
$653$	−25.5134	−0.998417	−0.499209	+	0.866482i	$0.666376\pi$
$654$	0	0
$655$	1.32770	0.0518774
$656$	0	0
$657$	−38.4160	−1.49875
$658$	0	0
$659$	4.20027	0.163619	0.0818097	−	0.996648i	$-0.473930\pi$
$659$	4.20027	0.163619	0.0818097	+	0.996648i	$0.473930\pi$
$660$	0	0
$661$	−17.1548	−0.667243	−0.333621	−	0.942707i	$-0.608271\pi$
$661$	−17.1548	−0.667243	−0.333621	+	0.942707i	$0.608271\pi$
$662$	0	0
$663$	75.6305	2.93725
$664$	0	0
$665$	0	0
$666$	0	0
$667$	30.0256	1.16260
$668$	0	0
$669$	−15.1516	−0.585794
$670$	0	0
$671$	−0.303653	−0.0117224
$672$	0	0
$673$	0.248970	0.00959710	0.00479855	−	0.999988i	$-0.498473\pi$
$673$	0.248970	0.00959710	0.00479855	+	0.999988i	$0.498473\pi$
$674$	0	0
$675$	1.41921	0.0546256
$676$	0	0
$677$	34.2300	1.31556	0.657782	−	0.753208i	$-0.271495\pi$
$677$	34.2300	1.31556	0.657782	+	0.753208i	$0.271495\pi$
$678$	0	0
$679$	−4.38413	−0.168248
$680$	0	0
$681$	−9.88301	−0.378718
$682$	0	0
$683$	1.97771	0.0756750	0.0378375	−	0.999284i	$-0.487953\pi$
$683$	1.97771	0.0756750	0.0378375	+	0.999284i	$0.487953\pi$
$684$	0	0
$685$	2.64496	0.101059
$686$	0	0
$687$	−2.92303	−0.111521
$688$	0	0
$689$	39.2309	1.49458
$690$	0	0
$691$	36.8485	1.40178	0.700892	−	0.713267i	$-0.252785\pi$
$691$	36.8485	1.40178	0.700892	+	0.713267i	$0.252785\pi$
$692$	0	0
$693$	0.158882	0.00603542
$694$	0	0
$695$	18.5776	0.704689
$696$	0	0
$697$	58.5117	2.21629
$698$	0	0
$699$	15.0090	0.567692
$700$	0	0
$701$	6.29767	0.237860	0.118930	−	0.992903i	$-0.462054\pi$
$701$	6.29767	0.237860	0.118930	+	0.992903i	$0.462054\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−13.8048	−0.519918
$706$	0	0
$707$	−2.88713	−0.108582
$708$	0	0
$709$	−18.7888	−0.705628	−0.352814	−	0.935693i	$-0.614775\pi$
$709$	−18.7888	−0.705628	−0.352814	+	0.935693i	$0.614775\pi$
$710$	0	0
$711$	45.1830	1.69450
$712$	0	0
$713$	−36.1147	−1.35251
$714$	0	0
$715$	−0.667252	−0.0249538
$716$	0	0
$717$	30.7137	1.14702
$718$	0	0
$719$	30.9923	1.15582	0.577908	−	0.816102i	$-0.303869\pi$
$719$	30.9923	1.15582	0.577908	+	0.816102i	$0.303869\pi$
$720$	0	0
$721$	−3.52797	−0.131388
$722$	0	0
$723$	50.0242	1.86042
$724$	0	0
$725$	−14.8179	−0.550323
$726$	0	0
$727$	13.1361	0.487191	0.243595	−	0.969877i	$-0.421673\pi$
$727$	13.1361	0.487191	0.243595	+	0.969877i	$0.421673\pi$
$728$	0	0
$729$	−23.5776	−0.873244
$730$	0	0
$731$	−19.0205	−0.703500
$732$	0	0
$733$	−29.9668	−1.10685	−0.553424	−	0.832900i	$-0.686679\pi$
$733$	−29.9668	−1.10685	−0.553424	+	0.832900i	$0.686679\pi$
$734$	0	0
$735$	22.0315	0.812643
$736$	0	0
$737$	−1.02971	−0.0379297
$738$	0	0
$739$	26.0283	0.957466	0.478733	−	0.877961i	$-0.341096\pi$
$739$	26.0283	0.957466	0.478733	+	0.877961i	$0.341096\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−21.3405	−0.782907	−0.391453	−	0.920198i	$-0.628028\pi$
$743$	−21.3405	−0.782907	−0.391453	+	0.920198i	$0.628028\pi$
$744$	0	0
$745$	10.9709	0.401943
$746$	0	0
$747$	−10.2831	−0.376240
$748$	0	0
$749$	−2.53571	−0.0926529
$750$	0	0
$751$	−22.2158	−0.810664	−0.405332	−	0.914169i	$-0.632844\pi$
$751$	−22.2158	−0.810664	−0.405332	+	0.914169i	$0.632844\pi$
$752$	0	0
$753$	16.9813	0.618834
$754$	0	0
$755$	−23.7442	−0.864141
$756$	0	0
$757$	−40.6064	−1.47586	−0.737932	−	0.674875i	$-0.764197\pi$
$757$	−40.6064	−1.47586	−0.737932	+	0.674875i	$0.764197\pi$
$758$	0	0
$759$	−1.87702	−0.0681316
$760$	0	0
$761$	39.0455	1.41540	0.707699	−	0.706514i	$-0.249733\pi$
$761$	39.0455	1.41540	0.707699	+	0.706514i	$0.249733\pi$
$762$	0	0
$763$	3.99907	0.144776
$764$	0	0
$765$	28.9709	1.04745
$766$	0	0
$767$	33.4561	1.20803
$768$	0	0
$769$	−29.5294	−1.06486	−0.532429	−	0.846475i	$-0.678721\pi$
$769$	−29.5294	−1.06486	−0.532429	+	0.846475i	$0.678721\pi$
$770$	0	0
$771$	40.5990	1.46214
$772$	0	0
$773$	24.9135	0.896078	0.448039	−	0.894014i	$-0.352123\pi$
$773$	24.9135	0.896078	0.448039	+	0.894014i	$0.352123\pi$
$774$	0	0
$775$	17.8229	0.640219
$776$	0	0
$777$	−0.664066	−0.0238233
$778$	0	0
$779$	0	0
$780$	0	0
$781$	1.94181	0.0694834
$782$	0	0
$783$	−2.07335	−0.0740954
$784$	0	0
$785$	27.0024	0.963756
$786$	0	0
$787$	−43.4861	−1.55011	−0.775056	−	0.631893i	$-0.782278\pi$
$787$	−43.4861	−1.55011	−0.775056	+	0.631893i	$0.782278\pi$
$788$	0	0
$789$	−18.9537	−0.674768
$790$	0	0
$791$	−2.98276	−0.106055
$792$	0	0
$793$	−10.3372	−0.367084
$794$	0	0
$795$	31.0419	1.10094
$796$	0	0
$797$	36.2158	1.28283	0.641414	−	0.767195i	$-0.278348\pi$
$797$	36.2158	1.28283	0.641414	+	0.767195i	$0.278348\pi$
$798$	0	0
$799$	−32.4543	−1.14815
$800$	0	0
$801$	6.82976	0.241318
$802$	0	0
$803$	1.64590	0.0580824
$804$	0	0
$805$	4.06830	0.143389
$806$	0	0
$807$	78.2791	2.75556
$808$	0	0
$809$	28.5030	1.00211	0.501056	−	0.865415i	$-0.332945\pi$
$809$	28.5030	1.00211	0.501056	+	0.865415i	$0.332945\pi$
$810$	0	0
$811$	9.47390	0.332674	0.166337	−	0.986069i	$-0.446806\pi$
$811$	9.47390	0.332674	0.166337	+	0.986069i	$0.446806\pi$
$812$	0	0
$813$	3.48608	0.122262
$814$	0	0
$815$	−12.5084	−0.438149
$816$	0	0
$817$	0	0
$818$	0	0
$819$	5.40879	0.188998
$820$	0	0
$821$	−9.12660	−0.318521	−0.159260	−	0.987237i	$-0.550911\pi$
$821$	−9.12660	−0.318521	−0.159260	+	0.987237i	$0.550911\pi$
$822$	0	0
$823$	−1.86390	−0.0649716	−0.0324858	−	0.999472i	$-0.510342\pi$
$823$	−1.86390	−0.0649716	−0.0324858	+	0.999472i	$0.510342\pi$
$824$	0	0
$825$	0.926327	0.0322506
$826$	0	0
$827$	−40.6459	−1.41340	−0.706698	−	0.707515i	$-0.749816\pi$
$827$	−40.6459	−1.41340	−0.706698	+	0.707515i	$0.749816\pi$
$828$	0	0
$829$	−28.5354	−0.991075	−0.495537	−	0.868587i	$-0.665029\pi$
$829$	−28.5354	−0.991075	−0.495537	+	0.868587i	$0.665029\pi$
$830$	0	0
$831$	41.1807	1.42854
$832$	0	0
$833$	51.7948	1.79458
$834$	0	0
$835$	−15.9290	−0.551246
$836$	0	0
$837$	2.49382	0.0861991
$838$	0	0
$839$	27.2621	0.941192	0.470596	−	0.882349i	$-0.344039\pi$
$839$	27.2621	0.941192	0.470596	+	0.882349i	$0.344039\pi$
$840$	0	0
$841$	−7.35235	−0.253529
$842$	0	0
$843$	42.7897	1.47376
$844$	0	0
$845$	−5.20027	−0.178895
$846$	0	0
$847$	5.14022	0.176620
$848$	0	0
$849$	−63.0200	−2.16284
$850$	0	0
$851$	3.79797	0.130193
$852$	0	0
$853$	45.7333	1.56588	0.782939	−	0.622098i	$-0.213720\pi$
$853$	45.7333	1.56588	0.782939	+	0.622098i	$0.213720\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.8280	0.540674	0.270337	−	0.962766i	$-0.412865\pi$
$857$	15.8280	0.540674	0.270337	+	0.962766i	$0.412865\pi$
$858$	0	0
$859$	−37.3618	−1.27477	−0.637385	−	0.770546i	$-0.719984\pi$
$859$	−37.3618	−1.27477	−0.637385	+	0.770546i	$0.719984\pi$
$860$	0	0
$861$	8.64370	0.294577
$862$	0	0
$863$	−15.2249	−0.518263	−0.259131	−	0.965842i	$-0.583436\pi$
$863$	−15.2249	−0.518263	−0.259131	+	0.965842i	$0.583436\pi$
$864$	0	0
$865$	−30.2722	−1.02929
$866$	0	0
$867$	99.6938	3.38578
$868$	0	0
$869$	−1.93582	−0.0656683
$870$	0	0
$871$	−35.0541	−1.18776
$872$	0	0
$873$	26.3773	0.892737
$874$	0	0
$875$	−5.15982	−0.174434
$876$	0	0
$877$	−50.5844	−1.70811	−0.854057	−	0.520179i	$-0.825865\pi$
$877$	−50.5844	−1.70811	−0.854057	+	0.520179i	$0.825865\pi$
$878$	0	0
$879$	−16.1023	−0.543116
$880$	0	0
$881$	−21.5449	−0.725866	−0.362933	−	0.931815i	$-0.618225\pi$
$881$	−21.5449	−0.725866	−0.362933	+	0.931815i	$0.618225\pi$
$882$	0	0
$883$	−18.4037	−0.619335	−0.309667	−	0.950845i	$-0.600218\pi$
$883$	−18.4037	−0.619335	−0.309667	+	0.950845i	$0.600218\pi$
$884$	0	0
$885$	26.4725	0.889862
$886$	0	0
$887$	27.4935	0.923141	0.461571	−	0.887103i	$-0.347286\pi$
$887$	27.4935	0.923141	0.461571	+	0.887103i	$0.347286\pi$
$888$	0	0
$889$	−0.0760373	−0.00255021
$890$	0	0
$891$	1.14828	0.0384688
$892$	0	0
$893$	0	0
$894$	0	0
$895$	4.09152	0.136764
$896$	0	0
$897$	−63.8991	−2.13353
$898$	0	0
$899$	−26.0378	−0.868408
$900$	0	0
$901$	72.9778	2.43124
$902$	0	0
$903$	−2.80983	−0.0935053
$904$	0	0
$905$	15.4047	0.512068
$906$	0	0
$907$	43.5313	1.44543	0.722716	−	0.691145i	$-0.242893\pi$
$907$	43.5313	1.44543	0.722716	+	0.691145i	$0.242893\pi$
$908$	0	0
$909$	17.3705	0.576144
$910$	0	0
$911$	−0.662199	−0.0219396	−0.0109698	−	0.999940i	$-0.503492\pi$
$911$	−0.662199	−0.0219396	−0.0109698	+	0.999940i	$0.503492\pi$
$912$	0	0
$913$	0.440570	0.0145807
$914$	0	0
$915$	−8.17942	−0.270403
$916$	0	0
$917$	0.461104	0.0152270
$918$	0	0
$919$	−48.9172	−1.61363	−0.806814	−	0.590805i	$-0.798810\pi$
$919$	−48.9172	−1.61363	−0.806814	+	0.590805i	$0.798810\pi$
$920$	0	0
$921$	−49.7684	−1.63992
$922$	0	0
$923$	66.1046	2.17586
$924$	0	0
$925$	−1.87433	−0.0616277
$926$	0	0
$927$	21.2262	0.697159
$928$	0	0
$929$	−9.82201	−0.322250	−0.161125	−	0.986934i	$-0.551512\pi$
$929$	−9.82201	−0.322250	−0.161125	+	0.986934i	$0.551512\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.60906	−0.0526783
$934$	0	0
$935$	−1.24123	−0.0405925
$936$	0	0
$937$	−19.7382	−0.644820	−0.322410	−	0.946600i	$-0.604493\pi$
$937$	−19.7382	−0.644820	−0.322410	+	0.946600i	$0.604493\pi$
$938$	0	0
$939$	−53.4157	−1.74316
$940$	0	0
$941$	−18.9162	−0.616651	−0.308326	−	0.951281i	$-0.599769\pi$
$941$	−18.9162	−0.616651	−0.308326	+	0.951281i	$0.599769\pi$
$942$	0	0
$943$	−49.4356	−1.60985
$944$	0	0
$945$	−0.280927	−0.00913856
$946$	0	0
$947$	−36.1652	−1.17521	−0.587605	−	0.809148i	$-0.699929\pi$
$947$	−36.1652	−1.17521	−0.587605	+	0.809148i	$0.699929\pi$
$948$	0	0
$949$	56.0310	1.81884
$950$	0	0
$951$	−20.5112	−0.665122
$952$	0	0
$953$	22.6932	0.735104	0.367552	−	0.930003i	$-0.380196\pi$
$953$	22.6932	0.735104	0.367552	+	0.930003i	$0.380196\pi$
$954$	0	0
$955$	−5.42602	−0.175582
$956$	0	0
$957$	−1.35328	−0.0437454
$958$	0	0
$959$	0.918586	0.0296627
$960$	0	0
$961$	0.318201	0.0102645
$962$	0	0
$963$	15.2562	0.491625
$964$	0	0
$965$	22.3369	0.719049
$966$	0	0
$967$	17.0743	0.549072	0.274536	−	0.961577i	$-0.411476\pi$
$967$	17.0743	0.549072	0.274536	+	0.961577i	$0.411476\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	30.7701	0.987461	0.493730	−	0.869615i	$-0.335633\pi$
$971$	30.7701	0.987461	0.493730	+	0.869615i	$0.335633\pi$
$972$	0	0
$973$	6.45193	0.206839
$974$	0	0
$975$	31.5348	1.00992
$976$	0	0
$977$	9.53714	0.305120	0.152560	−	0.988294i	$-0.451248\pi$
$977$	9.53714	0.305120	0.152560	+	0.988294i	$0.451248\pi$
$978$	0	0
$979$	−0.292614	−0.00935199
$980$	0	0
$981$	−24.0606	−0.768194
$982$	0	0
$983$	−1.53951	−0.0491026	−0.0245513	−	0.999699i	$-0.507816\pi$
$983$	−1.53951	−0.0491026	−0.0245513	+	0.999699i	$0.507816\pi$
$984$	0	0
$985$	−10.0879	−0.321427
$986$	0	0
$987$	−4.79435	−0.152606
$988$	0	0
$989$	16.0702	0.511001
$990$	0	0
$991$	19.5452	0.620874	0.310437	−	0.950594i	$-0.399525\pi$
$991$	19.5452	0.620874	0.310437	+	0.950594i	$0.399525\pi$
$992$	0	0
$993$	−33.5357	−1.06422
$994$	0	0
$995$	32.0310	1.01545
$996$	0	0
$997$	−26.8060	−0.848956	−0.424478	−	0.905438i	$-0.639542\pi$
$997$	−26.8060	−0.848956	−0.424478	+	0.905438i	$0.639542\pi$
$998$	0	0
$999$	−0.262260	−0.00829755

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.m.1.3		3
4.3	odd	2		5776.2.a.bs.1.1		3
19.14	odd	18		152.2.q.b.25.1	✓	6
19.15	odd	18		152.2.q.b.73.1	yes	6
19.18	odd	2		2888.2.a.s.1.1		3
76.15	even	18		304.2.u.a.225.1		6
76.71	even	18		304.2.u.a.177.1		6
76.75	even	2		5776.2.a.bj.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.q.b.25.1	✓	6	19.14	odd	18
152.2.q.b.73.1	yes	6	19.15	odd	18
304.2.u.a.177.1		6	76.71	even	18
304.2.u.a.225.1		6	76.15	even	18
2888.2.a.m.1.3		3	1.1	even	1	trivial
2888.2.a.s.1.1		3	19.18	odd	2
5776.2.a.bj.1.3		3	76.75	even	2
5776.2.a.bs.1.1		3	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2888.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41147 q^{3} -1.34730 q^{5} -0.467911 q^{7} +2.81521 q^{9} -0.120615 q^{11} -4.10607 q^{13} -3.24897 q^{15} -7.63816 q^{17} -1.12836 q^{21} +6.45336 q^{23} -3.18479 q^{25} -0.445622 q^{27} +4.65270 q^{29} -5.59627 q^{31} -0.290859 q^{33} +0.630415 q^{35} +0.588526 q^{37} -9.90167 q^{39} -7.66044 q^{41} +2.49020 q^{43} -3.79292 q^{45} +4.24897 q^{47} -6.78106 q^{49} -18.4192 q^{51} -9.55438 q^{53} +0.162504 q^{55} -8.14796 q^{59} +2.51754 q^{61} -1.31727 q^{63} +5.53209 q^{65} +8.53714 q^{67} +15.5621 q^{69} -16.0993 q^{71} -13.6459 q^{73} -7.68004 q^{75} +0.0564370 q^{77} +16.0496 q^{79} -9.52023 q^{81} -3.65270 q^{83} +10.2909 q^{85} +11.2199 q^{87} +2.42602 q^{89} +1.92127 q^{91} -13.4953 q^{93} +9.36959 q^{97} -0.339556 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 6 q^{7} + 12 q^{9} - 6 q^{11} + 3 q^{15} - 6 q^{17} + 15 q^{21} + 6 q^{23} - 6 q^{25} - 12 q^{27} + 15 q^{29} - 3 q^{31} + 15 q^{33} + 9 q^{35} + 12 q^{37} - 18 q^{39} + 6 q^{43}+ \cdots - 24 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 - 6 * q^7 + 12 * q^9 - 6 * q^11 + 3 * q^15 - 6 * q^17 + 15 * q^21 + 6 * q^23 - 6 * q^25 - 12 * q^27 + 15 * q^29 - 3 * q^31 + 15 * q^33 + 9 * q^35 + 12 * q^37 - 18 * q^39 + 6 * q^43 - 21 * q^45 - 3 * q^49 - 21 * q^51 - 18 * q^53 + 3 * q^55 - 9 * q^59 - 15 * q^61 - 33 * q^63 + 12 * q^65 + 6 * q^67 + 12 * q^69 + 6 * q^71 - 3 * q^75 + 15 * q^77 + 21 * q^79 + 3 * q^81 - 12 * q^83 + 15 * q^85 - 15 * q^87 + 15 * q^89 - 3 * q^91 - 24 * q^93 + 21 * q^97 - 24 * q^99

Introduction

L-functions

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Database

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