LMFDB - Embedded newform 2888.2.a.m.1.1

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			1.1
Root			$1.87939$ 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)

$f(q)$	$=$	$q-3.22668 q^{3} -2.53209 q^{5} -3.87939 q^{7} +7.41147 q^{9} -2.34730 q^{11} -0.837496 q^{13} +8.17024 q^{15} -0.958111 q^{17} +12.5175 q^{21} +4.36959 q^{23} +1.41147 q^{25} -14.2344 q^{27} +3.46791 q^{29} +4.63816 q^{31} +7.57398 q^{33} +9.82295 q^{35} +6.22668 q^{37} +2.70233 q^{39} +9.39693 q^{41} -4.47565 q^{43} -18.7665 q^{45} -7.17024 q^{47} +8.04963 q^{49} +3.09152 q^{51} +4.23442 q^{53} +5.94356 q^{55} -8.43376 q^{59} -6.38919 q^{61} -28.7520 q^{63} +2.12061 q^{65} +13.5621 q^{67} -14.0993 q^{69} +8.53714 q^{71} +8.90673 q^{73} -4.55438 q^{75} +9.10607 q^{77} +3.73143 q^{79} +23.6955 q^{81} -2.46791 q^{83} +2.42602 q^{85} -11.1898 q^{87} +2.28312 q^{89} +3.24897 q^{91} -14.9659 q^{93} +0.177052 q^{97} -17.3969 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.22668	−1.86293	−0.931463	−	0.363837i	$-0.881467\pi$
$3$	−3.22668	−1.86293	−0.931463	+	0.363837i	$0.881467\pi$
$4$	0	0
$5$	−2.53209	−1.13238	−0.566192	−	0.824273i	$-0.691584\pi$
$5$	−2.53209	−1.13238	−0.566192	+	0.824273i	$0.691584\pi$
$6$	0	0
$7$	−3.87939	−1.46627	−0.733135	−	0.680083i	$-0.761944\pi$
$7$	−3.87939	−1.46627	−0.733135	+	0.680083i	$0.761944\pi$
$8$	0	0
$9$	7.41147	2.47049
$10$	0	0
$11$	−2.34730	−0.707736	−0.353868	−	0.935295i	$-0.615134\pi$
$11$	−2.34730	−0.707736	−0.353868	+	0.935295i	$0.615134\pi$
$12$	0	0
$13$	−0.837496	−0.232280	−0.116140	−	0.993233i	$-0.537052\pi$
$13$	−0.837496	−0.232280	−0.116140	+	0.993233i	$0.537052\pi$
$14$	0	0
$15$	8.17024	2.10955
$16$	0	0
$17$	−0.958111	−0.232376	−0.116188	−	0.993227i	$-0.537068\pi$
$17$	−0.958111	−0.232376	−0.116188	+	0.993227i	$0.537068\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	12.5175	2.73155
$22$	0	0
$23$	4.36959	0.911121	0.455561	−	0.890205i	$-0.349439\pi$
$23$	4.36959	0.911121	0.455561	+	0.890205i	$0.349439\pi$
$24$	0	0
$25$	1.41147	0.282295
$26$	0	0
$27$	−14.2344	−2.73942
$28$	0	0
$29$	3.46791	0.643975	0.321987	−	0.946744i	$-0.395649\pi$
$29$	3.46791	0.643975	0.321987	+	0.946744i	$0.395649\pi$
$30$	0	0
$31$	4.63816	0.833037	0.416519	−	0.909127i	$-0.363250\pi$
$31$	4.63816	0.833037	0.416519	+	0.909127i	$0.363250\pi$
$32$	0	0
$33$	7.57398	1.31846
$34$	0	0
$35$	9.82295	1.66038
$36$	0	0
$37$	6.22668	1.02366	0.511830	−	0.859087i	$-0.328968\pi$
$37$	6.22668	1.02366	0.511830	+	0.859087i	$0.328968\pi$
$38$	0	0
$39$	2.70233	0.432720
$40$	0	0
$41$	9.39693	1.46755	0.733777	−	0.679391i	$-0.237756\pi$
$41$	9.39693	1.46755	0.733777	+	0.679391i	$0.237756\pi$
$42$	0	0
$43$	−4.47565	−0.682531	−0.341265	−	0.939967i	$-0.610855\pi$
$43$	−4.47565	−0.682531	−0.341265	+	0.939967i	$0.610855\pi$
$44$	0	0
$45$	−18.7665	−2.79755
$46$	0	0
$47$	−7.17024	−1.04589	−0.522944	−	0.852367i	$-0.675166\pi$
$47$	−7.17024	−1.04589	−0.522944	+	0.852367i	$0.675166\pi$
$48$	0	0
$49$	8.04963	1.14995
$50$	0	0
$51$	3.09152	0.432899
$52$	0	0
$53$	4.23442	0.581643	0.290821	−	0.956777i	$-0.406071\pi$
$53$	4.23442	0.581643	0.290821	+	0.956777i	$0.406071\pi$
$54$	0	0
$55$	5.94356	0.801430
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.43376	−1.09798	−0.548991	−	0.835828i	$-0.684988\pi$
$59$	−8.43376	−1.09798	−0.548991	+	0.835828i	$0.684988\pi$
$60$	0	0
$61$	−6.38919	−0.818051	−0.409026	−	0.912523i	$-0.634131\pi$
$61$	−6.38919	−0.818051	−0.409026	+	0.912523i	$0.634131\pi$
$62$	0	0
$63$	−28.7520	−3.62241
$64$	0	0
$65$	2.12061	0.263030
$66$	0	0
$67$	13.5621	1.65688	0.828438	−	0.560080i	$-0.189230\pi$
$67$	13.5621	1.65688	0.828438	+	0.560080i	$0.189230\pi$
$68$	0	0
$69$	−14.0993	−1.69735
$70$	0	0
$71$	8.53714	1.01317	0.506586	−	0.862189i	$-0.330907\pi$
$71$	8.53714	1.01317	0.506586	+	0.862189i	$0.330907\pi$
$72$	0	0
$73$	8.90673	1.04245	0.521227	−	0.853418i	$-0.325475\pi$
$73$	8.90673	1.04245	0.521227	+	0.853418i	$0.325475\pi$
$74$	0	0
$75$	−4.55438	−0.525894
$76$	0	0
$77$	9.10607	1.03773
$78$	0	0
$79$	3.73143	0.419819	0.209909	−	0.977721i	$-0.432683\pi$
$79$	3.73143	0.419819	0.209909	+	0.977721i	$0.432683\pi$
$80$	0	0
$81$	23.6955	2.63284
$82$	0	0
$83$	−2.46791	−0.270888	−0.135444	−	0.990785i	$-0.543246\pi$
$83$	−2.46791	−0.270888	−0.135444	+	0.990785i	$0.543246\pi$
$84$	0	0
$85$	2.42602	0.263139
$86$	0	0
$87$	−11.1898	−1.19968
$88$	0	0
$89$	2.28312	0.242010	0.121005	−	0.992652i	$-0.461388\pi$
$89$	2.28312	0.242010	0.121005	+	0.992652i	$0.461388\pi$
$90$	0	0
$91$	3.24897	0.340585
$92$	0	0
$93$	−14.9659	−1.55189
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.177052	0.0179769	0.00898844	−	0.999960i	$-0.497139\pi$
$97$	0.177052	0.0179769	0.00898844	+	0.999960i	$0.497139\pi$
$98$	0	0
$99$	−17.3969	−1.74846
$100$	0	0
$101$	−3.92127	−0.390181	−0.195091	−	0.980785i	$-0.562500\pi$
$101$	−3.92127	−0.390181	−0.195091	+	0.980785i	$0.562500\pi$
$102$	0	0
$103$	−11.7442	−1.15719	−0.578596	−	0.815614i	$-0.696399\pi$
$103$	−11.7442	−1.15719	−0.578596	+	0.815614i	$0.696399\pi$
$104$	0	0
$105$	−31.6955	−3.09317
$106$	0	0
$107$	−16.0915	−1.55563	−0.777813	−	0.628496i	$-0.783671\pi$
$107$	−16.0915	−1.55563	−0.777813	+	0.628496i	$0.783671\pi$
$108$	0	0
$109$	−10.6304	−1.01821	−0.509105	−	0.860705i	$-0.670023\pi$
$109$	−10.6304	−1.01821	−0.509105	+	0.860705i	$0.670023\pi$
$110$	0	0
$111$	−20.0915	−1.90700
$112$	0	0
$113$	5.61856	0.528549	0.264275	−	0.964447i	$-0.414867\pi$
$113$	5.61856	0.528549	0.264275	+	0.964447i	$0.414867\pi$
$114$	0	0
$115$	−11.0642	−1.03174
$116$	0	0
$117$	−6.20708	−0.573845
$118$	0	0
$119$	3.71688	0.340726
$120$	0	0
$121$	−5.49020	−0.499109
$122$	0	0
$123$	−30.3209	−2.73394
$124$	0	0
$125$	9.08647	0.812718
$126$	0	0
$127$	5.94356	0.527406	0.263703	−	0.964604i	$-0.415056\pi$
$127$	5.94356	0.527406	0.263703	+	0.964604i	$0.415056\pi$
$128$	0	0
$129$	14.4415	1.27150
$130$	0	0
$131$	4.50980	0.394023	0.197012	−	0.980401i	$-0.436876\pi$
$131$	4.50980	0.394023	0.197012	+	0.980401i	$0.436876\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	36.0428	3.10207
$136$	0	0
$137$	−6.84524	−0.584828	−0.292414	−	0.956292i	$-0.594459\pi$
$137$	−6.84524	−0.584828	−0.292414	+	0.956292i	$0.594459\pi$
$138$	0	0
$139$	16.9145	1.43467	0.717333	−	0.696730i	$-0.245363\pi$
$139$	16.9145	1.43467	0.717333	+	0.696730i	$0.245363\pi$
$140$	0	0
$141$	23.1361	1.94841
$142$	0	0
$143$	1.96585	0.164393
$144$	0	0
$145$	−8.78106	−0.729227
$146$	0	0
$147$	−25.9736	−2.14227
$148$	0	0
$149$	0.00774079	0.000634150 0	0.000317075	−	1.00000i	$-0.499899\pi$
$149$	0.00774079	0.000634150 0	0.000317075		1.00000i	$0.499899\pi$
$150$	0	0
$151$	5.44831	0.443377	0.221689	−	0.975118i	$-0.428843\pi$
$151$	5.44831	0.443377	0.221689	+	0.975118i	$0.428843\pi$
$152$	0	0
$153$	−7.10101	−0.574083
$154$	0	0
$155$	−11.7442	−0.943319
$156$	0	0
$157$	−23.5963	−1.88319	−0.941594	−	0.336752i	$-0.890672\pi$
$157$	−23.5963	−1.88319	−0.941594	+	0.336752i	$0.890672\pi$
$158$	0	0
$159$	−13.6631	−1.08356
$160$	0	0
$161$	−16.9513	−1.33595
$162$	0	0
$163$	−19.9486	−1.56250	−0.781248	−	0.624221i	$-0.785417\pi$
$163$	−19.9486	−1.56250	−0.781248	+	0.624221i	$0.785417\pi$
$164$	0	0
$165$	−19.1780	−1.49300
$166$	0	0
$167$	0.546637	0.0423000	0.0211500	−	0.999776i	$-0.493267\pi$
$167$	0.546637	0.0423000	0.0211500	+	0.999776i	$0.493267\pi$
$168$	0	0
$169$	−12.2986	−0.946046
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.3601	−0.863692	−0.431846	−	0.901947i	$-0.642138\pi$
$173$	−11.3601	−0.863692	−0.431846	+	0.901947i	$0.642138\pi$
$174$	0	0
$175$	−5.47565	−0.413920
$176$	0	0
$177$	27.2131	2.04546
$178$	0	0
$179$	1.84524	0.137919	0.0689597	−	0.997619i	$-0.478032\pi$
$179$	1.84524	0.137919	0.0689597	+	0.997619i	$0.478032\pi$
$180$	0	0
$181$	4.58172	0.340556	0.170278	−	0.985396i	$-0.445533\pi$
$181$	4.58172	0.340556	0.170278	+	0.985396i	$0.445533\pi$
$182$	0	0
$183$	20.6159	1.52397
$184$	0	0
$185$	−15.7665	−1.15918
$186$	0	0
$187$	2.24897	0.164461
$188$	0	0
$189$	55.2208	4.01672
$190$	0	0
$191$	2.08647	0.150971	0.0754857	−	0.997147i	$-0.475949\pi$
$191$	2.08647	0.150971	0.0754857	+	0.997147i	$0.475949\pi$
$192$	0	0
$193$	−25.1584	−1.81094	−0.905470	−	0.424410i	$-0.860482\pi$
$193$	−25.1584	−1.81094	−0.905470	+	0.424410i	$0.860482\pi$
$194$	0	0
$195$	−6.84255	−0.490005
$196$	0	0
$197$	24.8307	1.76911	0.884557	−	0.466433i	$-0.154461\pi$
$197$	24.8307	1.76911	0.884557	+	0.466433i	$0.154461\pi$
$198$	0	0
$199$	12.4243	0.880733	0.440366	−	0.897818i	$-0.354849\pi$
$199$	12.4243	0.880733	0.440366	+	0.897818i	$0.354849\pi$
$200$	0	0
$201$	−43.7606	−3.08664
$202$	0	0
$203$	−13.4534	−0.944241
$204$	0	0
$205$	−23.7939	−1.66183
$206$	0	0
$207$	32.3851	2.25092
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.84524	0.127031	0.0635157	−	0.997981i	$-0.479769\pi$
$211$	1.84524	0.127031	0.0635157	+	0.997981i	$0.479769\pi$
$212$	0	0
$213$	−27.5466	−1.88746
$214$	0	0
$215$	11.3327	0.772887
$216$	0	0
$217$	−17.9932	−1.22146
$218$	0	0
$219$	−28.7392	−1.94201
$220$	0	0
$221$	0.802414	0.0539762
$222$	0	0
$223$	−14.2909	−0.956987	−0.478493	−	0.878091i	$-0.658817\pi$
$223$	−14.2909	−0.956987	−0.478493	+	0.878091i	$0.658817\pi$
$224$	0	0
$225$	10.4611	0.697407
$226$	0	0
$227$	−16.7023	−1.10857	−0.554286	−	0.832326i	$-0.687009\pi$
$227$	−16.7023	−1.10857	−0.554286	+	0.832326i	$0.687009\pi$
$228$	0	0
$229$	5.32501	0.351886	0.175943	−	0.984400i	$-0.443702\pi$
$229$	5.32501	0.351886	0.175943	+	0.984400i	$0.443702\pi$
$230$	0	0
$231$	−29.3824	−1.93322
$232$	0	0
$233$	29.4911	1.93203	0.966014	−	0.258489i	$-0.0832246\pi$
$233$	29.4911	1.93203	0.966014	+	0.258489i	$0.0832246\pi$
$234$	0	0
$235$	18.1557	1.18435
$236$	0	0
$237$	−12.0401	−0.782091
$238$	0	0
$239$	18.6604	1.20704	0.603522	−	0.797346i	$-0.293764\pi$
$239$	18.6604	1.20704	0.603522	+	0.797346i	$0.293764\pi$
$240$	0	0
$241$	10.7956	0.695406	0.347703	−	0.937605i	$-0.386962\pi$
$241$	10.7956	0.695406	0.347703	+	0.937605i	$0.386962\pi$
$242$	0	0
$243$	−33.7547	−2.16536
$244$	0	0
$245$	−20.3824	−1.30218
$246$	0	0
$247$	0	0
$248$	0	0
$249$	7.96316	0.504645
$250$	0	0
$251$	10.5963	0.668830	0.334415	−	0.942426i	$-0.391461\pi$
$251$	10.5963	0.668830	0.334415	+	0.942426i	$0.391461\pi$
$252$	0	0
$253$	−10.2567	−0.644834
$254$	0	0
$255$	−7.82800	−0.490208
$256$	0	0
$257$	−1.87670	−0.117065	−0.0585326	−	0.998285i	$-0.518642\pi$
$257$	−1.87670	−0.117065	−0.0585326	+	0.998285i	$0.518642\pi$
$258$	0	0
$259$	−24.1557	−1.50096
$260$	0	0
$261$	25.7023	1.59093
$262$	0	0
$263$	8.29860	0.511714	0.255857	−	0.966715i	$-0.417642\pi$
$263$	8.29860	0.511714	0.255857	+	0.966715i	$0.417642\pi$
$264$	0	0
$265$	−10.7219	−0.658643
$266$	0	0
$267$	−7.36690	−0.450847
$268$	0	0
$269$	14.5047	0.884370	0.442185	−	0.896924i	$-0.354203\pi$
$269$	14.5047	0.884370	0.442185	+	0.896924i	$0.354203\pi$
$270$	0	0
$271$	15.2344	0.925425	0.462713	−	0.886508i	$-0.346876\pi$
$271$	15.2344	0.925425	0.462713	+	0.886508i	$0.346876\pi$
$272$	0	0
$273$	−10.4834	−0.634484
$274$	0	0
$275$	−3.31315	−0.199790
$276$	0	0
$277$	2.81790	0.169311	0.0846555	−	0.996410i	$-0.473021\pi$
$277$	2.81790	0.169311	0.0846555	+	0.996410i	$0.473021\pi$
$278$	0	0
$279$	34.3756	2.05801
$280$	0	0
$281$	7.79561	0.465047	0.232523	−	0.972591i	$-0.425302\pi$
$281$	7.79561	0.465047	0.232523	+	0.972591i	$0.425302\pi$
$282$	0	0
$283$	−20.9240	−1.24380	−0.621900	−	0.783096i	$-0.713639\pi$
$283$	−20.9240	−1.24380	−0.621900	+	0.783096i	$0.713639\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−36.4543	−2.15183
$288$	0	0
$289$	−16.0820	−0.946001
$290$	0	0
$291$	−0.571290	−0.0334896
$292$	0	0
$293$	−27.8607	−1.62764	−0.813820	−	0.581117i	$-0.802616\pi$
$293$	−27.8607	−1.62764	−0.813820	+	0.581117i	$0.802616\pi$
$294$	0	0
$295$	21.3550	1.24334
$296$	0	0
$297$	33.4124	1.93878
$298$	0	0
$299$	−3.65951	−0.211635
$300$	0	0
$301$	17.3628	1.00077
$302$	0	0
$303$	12.6527	0.726879
$304$	0	0
$305$	16.1780	0.926349
$306$	0	0
$307$	−13.9581	−0.796631	−0.398316	−	0.917248i	$-0.630405\pi$
$307$	−13.9581	−0.796631	−0.398316	+	0.917248i	$0.630405\pi$
$308$	0	0
$309$	37.8949	2.15576
$310$	0	0
$311$	−4.97771	−0.282260	−0.141130	−	0.989991i	$-0.545074\pi$
$311$	−4.97771	−0.282260	−0.141130	+	0.989991i	$0.545074\pi$
$312$	0	0
$313$	1.87258	0.105844	0.0529222	−	0.998599i	$-0.483146\pi$
$313$	1.87258	0.105844	0.0529222	+	0.998599i	$0.483146\pi$
$314$	0	0
$315$	72.8025	4.10196
$316$	0	0
$317$	30.2053	1.69650	0.848250	−	0.529596i	$-0.177656\pi$
$317$	30.2053	1.69650	0.848250	+	0.529596i	$0.177656\pi$
$318$	0	0
$319$	−8.14022	−0.455765
$320$	0	0
$321$	51.9222	2.89802
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.18210	−0.0655713
$326$	0	0
$327$	34.3010	1.89685
$328$	0	0
$329$	27.8161	1.53355
$330$	0	0
$331$	−9.73917	−0.535313	−0.267657	−	0.963514i	$-0.586249\pi$
$331$	−9.73917	−0.535313	−0.267657	+	0.963514i	$0.586249\pi$
$332$	0	0
$333$	46.1489	2.52894
$334$	0	0
$335$	−34.3405	−1.87622
$336$	0	0
$337$	18.3354	0.998795	0.499397	−	0.866373i	$-0.333555\pi$
$337$	18.3354	0.998795	0.499397	+	0.866373i	$0.333555\pi$
$338$	0	0
$339$	−18.1293	−0.984648
$340$	0	0
$341$	−10.8871	−0.589571
$342$	0	0
$343$	−4.07192	−0.219863
$344$	0	0
$345$	35.7006	1.92205
$346$	0	0
$347$	3.41653	0.183409	0.0917044	−	0.995786i	$-0.470769\pi$
$347$	3.41653	0.183409	0.0917044	+	0.995786i	$0.470769\pi$
$348$	0	0
$349$	6.90436	0.369582	0.184791	−	0.982778i	$-0.440839\pi$
$349$	6.90436	0.369582	0.184791	+	0.982778i	$0.440839\pi$
$350$	0	0
$351$	11.9213	0.636311
$352$	0	0
$353$	−33.1807	−1.76603	−0.883015	−	0.469346i	$-0.844490\pi$
$353$	−33.1807	−1.76603	−0.883015	+	0.469346i	$0.844490\pi$
$354$	0	0
$355$	−21.6168	−1.14730
$356$	0	0
$357$	−11.9932	−0.634747
$358$	0	0
$359$	−19.7597	−1.04288	−0.521439	−	0.853289i	$-0.674604\pi$
$359$	−19.7597	−1.04288	−0.521439	+	0.853289i	$0.674604\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	17.7151	0.929803
$364$	0	0
$365$	−22.5526	−1.18046
$366$	0	0
$367$	−25.4296	−1.32742	−0.663708	−	0.747991i	$-0.731018\pi$
$367$	−25.4296	−1.32742	−0.663708	+	0.747991i	$0.731018\pi$
$368$	0	0
$369$	69.6451	3.62558
$370$	0	0
$371$	−16.4270	−0.852845
$372$	0	0
$373$	8.88444	0.460019	0.230009	−	0.973188i	$-0.426124\pi$
$373$	8.88444	0.460019	0.230009	+	0.973188i	$0.426124\pi$
$374$	0	0
$375$	−29.3191	−1.51403
$376$	0	0
$377$	−2.90436	−0.149582
$378$	0	0
$379$	11.3327	0.582124	0.291062	−	0.956704i	$-0.405991\pi$
$379$	11.3327	0.582124	0.291062	+	0.956704i	$0.405991\pi$
$380$	0	0
$381$	−19.1780	−0.982518
$382$	0	0
$383$	36.9891	1.89005	0.945027	−	0.326993i	$-0.106035\pi$
$383$	36.9891	1.89005	0.945027	+	0.326993i	$0.106035\pi$
$384$	0	0
$385$	−23.0574	−1.17511
$386$	0	0
$387$	−33.1712	−1.68619
$388$	0	0
$389$	18.0378	0.914551	0.457276	−	0.889325i	$-0.348825\pi$
$389$	18.0378	0.914551	0.457276	+	0.889325i	$0.348825\pi$
$390$	0	0
$391$	−4.18655	−0.211723
$392$	0	0
$393$	−14.5517	−0.734036
$394$	0	0
$395$	−9.44831	−0.475396
$396$	0	0
$397$	−6.55850	−0.329161	−0.164581	−	0.986364i	$-0.552627\pi$
$397$	−6.55850	−0.329161	−0.164581	+	0.986364i	$0.552627\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−28.2995	−1.41321	−0.706606	−	0.707608i	$-0.749774\pi$
$401$	−28.2995	−1.41321	−0.706606	+	0.707608i	$0.749774\pi$
$402$	0	0
$403$	−3.88444	−0.193498
$404$	0	0
$405$	−59.9992	−2.98138
$406$	0	0
$407$	−14.6159	−0.724482
$408$	0	0
$409$	−19.5972	−0.969019	−0.484510	−	0.874786i	$-0.661002\pi$
$409$	−19.5972	−0.969019	−0.484510	+	0.874786i	$0.661002\pi$
$410$	0	0
$411$	22.0874	1.08949
$412$	0	0
$413$	32.7178	1.60994
$414$	0	0
$415$	6.24897	0.306750
$416$	0	0
$417$	−54.5776	−2.67268
$418$	0	0
$419$	−7.94450	−0.388114	−0.194057	−	0.980990i	$-0.562165\pi$
$419$	−7.94450	−0.388114	−0.194057	+	0.980990i	$0.562165\pi$
$420$	0	0
$421$	13.0642	0.636709	0.318355	−	0.947972i	$-0.396870\pi$
$421$	13.0642	0.636709	0.318355	+	0.947972i	$0.396870\pi$
$422$	0	0
$423$	−53.1421	−2.58386
$424$	0	0
$425$	−1.35235	−0.0655986
$426$	0	0
$427$	24.7861	1.19948
$428$	0	0
$429$	−6.34318	−0.306251
$430$	0	0
$431$	−12.0865	−0.582185	−0.291092	−	0.956695i	$-0.594019\pi$
$431$	−12.0865	−0.582185	−0.291092	+	0.956695i	$0.594019\pi$
$432$	0	0
$433$	21.0077	1.00957	0.504784	−	0.863246i	$-0.331572\pi$
$433$	21.0077	1.00957	0.504784	+	0.863246i	$0.331572\pi$
$434$	0	0
$435$	28.3337	1.35850
$436$	0	0
$437$	0	0
$438$	0	0
$439$	3.02734	0.144487	0.0722436	−	0.997387i	$-0.476984\pi$
$439$	3.02734	0.144487	0.0722436	+	0.997387i	$0.476984\pi$
$440$	0	0
$441$	59.6596	2.84093
$442$	0	0
$443$	8.56212	0.406799	0.203399	−	0.979096i	$-0.434801\pi$
$443$	8.56212	0.406799	0.203399	+	0.979096i	$0.434801\pi$
$444$	0	0
$445$	−5.78106	−0.274048
$446$	0	0
$447$	−0.0249771	−0.00118137
$448$	0	0
$449$	23.0624	1.08838	0.544192	−	0.838961i	$-0.316836\pi$
$449$	23.0624	1.08838	0.544192	+	0.838961i	$0.316836\pi$
$450$	0	0
$451$	−22.0574	−1.03864
$452$	0	0
$453$	−17.5800	−0.825979
$454$	0	0
$455$	−8.22668	−0.385673
$456$	0	0
$457$	−25.4953	−1.19262	−0.596309	−	0.802755i	$-0.703367\pi$
$457$	−25.4953	−1.19262	−0.596309	+	0.802755i	$0.703367\pi$
$458$	0	0
$459$	13.6382	0.636575
$460$	0	0
$461$	−36.0874	−1.68076	−0.840379	−	0.541999i	$-0.817668\pi$
$461$	−36.0874	−1.68076	−0.840379	+	0.541999i	$0.817668\pi$
$462$	0	0
$463$	41.5390	1.93048	0.965241	−	0.261362i	$-0.0841718\pi$
$463$	41.5390	1.93048	0.965241	+	0.261362i	$0.0841718\pi$
$464$	0	0
$465$	37.8949	1.75733
$466$	0	0
$467$	6.51073	0.301281	0.150640	−	0.988589i	$-0.451866\pi$
$467$	6.51073	0.301281	0.150640	+	0.988589i	$0.451866\pi$
$468$	0	0
$469$	−52.6127	−2.42943
$470$	0	0
$471$	76.1376	3.50824
$472$	0	0
$473$	10.5057	0.483052
$474$	0	0
$475$	0	0
$476$	0	0
$477$	31.3833	1.43694
$478$	0	0
$479$	18.3874	0.840143	0.420072	−	0.907491i	$-0.362005\pi$
$479$	18.3874	0.840143	0.420072	+	0.907491i	$0.362005\pi$
$480$	0	0
$481$	−5.21482	−0.237775
$482$	0	0
$483$	54.6965	2.48878
$484$	0	0
$485$	−0.448311	−0.0203567
$486$	0	0
$487$	−33.7324	−1.52856	−0.764280	−	0.644885i	$-0.776905\pi$
$487$	−33.7324	−1.52856	−0.764280	+	0.644885i	$0.776905\pi$
$488$	0	0
$489$	64.3678	2.91081
$490$	0	0
$491$	13.9281	0.628566	0.314283	−	0.949329i	$-0.398236\pi$
$491$	13.9281	0.628566	0.314283	+	0.949329i	$0.398236\pi$
$492$	0	0
$493$	−3.32264	−0.149644
$494$	0	0
$495$	44.0506	1.97993
$496$	0	0
$497$	−33.1189	−1.48558
$498$	0	0
$499$	−33.9377	−1.51926	−0.759630	−	0.650356i	$-0.774620\pi$
$499$	−33.9377	−1.51926	−0.759630	+	0.650356i	$0.774620\pi$
$500$	0	0
$501$	−1.76382	−0.0788018
$502$	0	0
$503$	−8.78342	−0.391633	−0.195817	−	0.980641i	$-0.562736\pi$
$503$	−8.78342	−0.391633	−0.195817	+	0.980641i	$0.562736\pi$
$504$	0	0
$505$	9.92902	0.441835
$506$	0	0
$507$	39.6837	1.76241
$508$	0	0
$509$	24.3233	1.07811	0.539055	−	0.842271i	$-0.318782\pi$
$509$	24.3233	1.07811	0.539055	+	0.842271i	$0.318782\pi$
$510$	0	0
$511$	−34.5526	−1.52852
$512$	0	0
$513$	0	0
$514$	0	0
$515$	29.7374	1.31039
$516$	0	0
$517$	16.8307	0.740213
$518$	0	0
$519$	36.6554	1.60899
$520$	0	0
$521$	−4.12836	−0.180867	−0.0904333	−	0.995903i	$-0.528825\pi$
$521$	−4.12836	−0.180867	−0.0904333	+	0.995903i	$0.528825\pi$
$522$	0	0
$523$	38.8161	1.69731	0.848656	−	0.528946i	$-0.177412\pi$
$523$	38.8161	1.69731	0.848656	+	0.528946i	$0.177412\pi$
$524$	0	0
$525$	17.6682	0.771103
$526$	0	0
$527$	−4.44387	−0.193578
$528$	0	0
$529$	−3.90673	−0.169858
$530$	0	0
$531$	−62.5066	−2.71256
$532$	0	0
$533$	−7.86989	−0.340883
$534$	0	0
$535$	40.7452	1.76157
$536$	0	0
$537$	−5.95399	−0.256934
$538$	0	0
$539$	−18.8949	−0.813860
$540$	0	0
$541$	−36.9932	−1.59046	−0.795231	−	0.606307i	$-0.792650\pi$
$541$	−36.9932	−1.59046	−0.795231	+	0.606307i	$0.792650\pi$
$542$	0	0
$543$	−14.7837	−0.634431
$544$	0	0
$545$	26.9172	1.15300
$546$	0	0
$547$	23.4183	1.00129	0.500647	−	0.865652i	$-0.333096\pi$
$547$	23.4183	1.00129	0.500647	+	0.865652i	$0.333096\pi$
$548$	0	0
$549$	−47.3533	−2.02099
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−14.4757	−0.615567
$554$	0	0
$555$	50.8735	2.15946
$556$	0	0
$557$	−19.2968	−0.817634	−0.408817	−	0.912616i	$-0.634058\pi$
$557$	−19.2968	−0.817634	−0.408817	+	0.912616i	$0.634058\pi$
$558$	0	0
$559$	3.74834	0.158538
$560$	0	0
$561$	−7.25671	−0.306379
$562$	0	0
$563$	15.8753	0.669063	0.334531	−	0.942385i	$-0.391422\pi$
$563$	15.8753	0.669063	0.334531	+	0.942385i	$0.391422\pi$
$564$	0	0
$565$	−14.2267	−0.598521
$566$	0	0
$567$	−91.9241	−3.86045
$568$	0	0
$569$	−11.8571	−0.497075	−0.248538	−	0.968622i	$-0.579950\pi$
$569$	−11.8571	−0.497075	−0.248538	+	0.968622i	$0.579950\pi$
$570$	0	0
$571$	12.0291	0.503402	0.251701	−	0.967805i	$-0.419010\pi$
$571$	12.0291	0.503402	0.251701	+	0.967805i	$0.419010\pi$
$572$	0	0
$573$	−6.73236	−0.281248
$574$	0	0
$575$	6.16756	0.257205
$576$	0	0
$577$	−28.2327	−1.17534	−0.587671	−	0.809100i	$-0.699955\pi$
$577$	−28.2327	−1.17534	−0.587671	+	0.809100i	$0.699955\pi$
$578$	0	0
$579$	81.1781	3.37365
$580$	0	0
$581$	9.57398	0.397196
$582$	0	0
$583$	−9.93944	−0.411650
$584$	0	0
$585$	15.7169	0.649813
$586$	0	0
$587$	31.2942	1.29165	0.645824	−	0.763486i	$-0.276514\pi$
$587$	31.2942	1.29165	0.645824	+	0.763486i	$0.276514\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−80.1207	−3.29573
$592$	0	0
$593$	−11.5330	−0.473604	−0.236802	−	0.971558i	$-0.576099\pi$
$593$	−11.5330	−0.473604	−0.236802	+	0.971558i	$0.576099\pi$
$594$	0	0
$595$	−9.41147	−0.385833
$596$	0	0
$597$	−40.0892	−1.64074
$598$	0	0
$599$	2.69635	0.110170	0.0550849	−	0.998482i	$-0.482457\pi$
$599$	2.69635	0.110170	0.0550849	+	0.998482i	$0.482457\pi$
$600$	0	0
$601$	−5.76053	−0.234977	−0.117488	−	0.993074i	$-0.537484\pi$
$601$	−5.76053	−0.234977	−0.117488	+	0.993074i	$0.537484\pi$
$602$	0	0
$603$	100.515	4.09330
$604$	0	0
$605$	13.9017	0.565183
$606$	0	0
$607$	11.7510	0.476960	0.238480	−	0.971147i	$-0.423351\pi$
$607$	11.7510	0.476960	0.238480	+	0.971147i	$0.423351\pi$
$608$	0	0
$609$	43.4097	1.75905
$610$	0	0
$611$	6.00505	0.242938
$612$	0	0
$613$	−14.3013	−0.577624	−0.288812	−	0.957386i	$-0.593260\pi$
$613$	−14.3013	−0.577624	−0.288812	+	0.957386i	$0.593260\pi$
$614$	0	0
$615$	76.7752	3.09587
$616$	0	0
$617$	−31.3337	−1.26145	−0.630723	−	0.776008i	$-0.717242\pi$
$617$	−31.3337	−1.26145	−0.630723	+	0.776008i	$0.717242\pi$
$618$	0	0
$619$	15.3155	0.615582	0.307791	−	0.951454i	$-0.400410\pi$
$619$	15.3155	0.615582	0.307791	+	0.951454i	$0.400410\pi$
$620$	0	0
$621$	−62.1985	−2.49594
$622$	0	0
$623$	−8.85710	−0.354852
$624$	0	0
$625$	−30.0651	−1.20260
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.96585	−0.237874
$630$	0	0
$631$	2.02910	0.0807770	0.0403885	−	0.999184i	$-0.487140\pi$
$631$	2.02910	0.0807770	0.0403885	+	0.999184i	$0.487140\pi$
$632$	0	0
$633$	−5.95399	−0.236650
$634$	0	0
$635$	−15.0496	−0.597226
$636$	0	0
$637$	−6.74153	−0.267109
$638$	0	0
$639$	63.2728	2.50303
$640$	0	0
$641$	3.87939	0.153227	0.0766133	−	0.997061i	$-0.475589\pi$
$641$	3.87939	0.153227	0.0766133	+	0.997061i	$0.475589\pi$
$642$	0	0
$643$	−17.3841	−0.685563	−0.342782	−	0.939415i	$-0.611369\pi$
$643$	−17.3841	−0.685563	−0.342782	+	0.939415i	$0.611369\pi$
$644$	0	0
$645$	−36.5672	−1.43983
$646$	0	0
$647$	27.3414	1.07490	0.537451	−	0.843295i	$-0.319387\pi$
$647$	27.3414	1.07490	0.537451	+	0.843295i	$0.319387\pi$
$648$	0	0
$649$	19.7965	0.777082
$650$	0	0
$651$	58.0583	2.27548
$652$	0	0
$653$	29.0702	1.13760	0.568802	−	0.822475i	$-0.307407\pi$
$653$	29.0702	1.13760	0.568802	+	0.822475i	$0.307407\pi$
$654$	0	0
$655$	−11.4192	−0.446186
$656$	0	0
$657$	66.0120	2.57537
$658$	0	0
$659$	−32.1411	−1.25204	−0.626021	−	0.779806i	$-0.715317\pi$
$659$	−32.1411	−1.25204	−0.626021	+	0.779806i	$0.715317\pi$
$660$	0	0
$661$	−38.8084	−1.50947	−0.754736	−	0.656029i	$-0.772235\pi$
$661$	−38.8084	−1.50947	−0.754736	+	0.656029i	$0.772235\pi$
$662$	0	0
$663$	−2.58914	−0.100554
$664$	0	0
$665$	0	0
$666$	0	0
$667$	15.1533	0.586739
$668$	0	0
$669$	46.1121	1.78279
$670$	0	0
$671$	14.9973	0.578965
$672$	0	0
$673$	−11.1702	−0.430581	−0.215291	−	0.976550i	$-0.569070\pi$
$673$	−11.1702	−0.430581	−0.215291	+	0.976550i	$0.569070\pi$
$674$	0	0
$675$	−20.0915	−0.773323
$676$	0	0
$677$	28.6932	1.10277	0.551384	−	0.834252i	$-0.314100\pi$
$677$	28.6932	1.10277	0.551384	+	0.834252i	$0.314100\pi$
$678$	0	0
$679$	−0.686852	−0.0263590
$680$	0	0
$681$	53.8931	2.06519
$682$	0	0
$683$	12.3550	0.472752	0.236376	−	0.971662i	$-0.424040\pi$
$683$	12.3550	0.472752	0.236376	+	0.971662i	$0.424040\pi$
$684$	0	0
$685$	17.3327	0.662250
$686$	0	0
$687$	−17.1821	−0.655538
$688$	0	0
$689$	−3.54631	−0.135104
$690$	0	0
$691$	10.7000	0.407046	0.203523	−	0.979070i	$-0.434761\pi$
$691$	10.7000	0.407046	0.203523	+	0.979070i	$0.434761\pi$
$692$	0	0
$693$	67.4894	2.56371
$694$	0	0
$695$	−42.8289	−1.62459
$696$	0	0
$697$	−9.00330	−0.341024
$698$	0	0
$699$	−95.1585	−3.59922
$700$	0	0
$701$	19.8007	0.747861	0.373930	−	0.927457i	$-0.378010\pi$
$701$	19.8007	0.747861	0.373930	+	0.927457i	$0.378010\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−58.5827	−2.20635
$706$	0	0
$707$	15.2121	0.572111
$708$	0	0
$709$	11.9145	0.447457	0.223729	−	0.974651i	$-0.428177\pi$
$709$	11.9145	0.447457	0.223729	+	0.974651i	$0.428177\pi$
$710$	0	0
$711$	27.6554	1.03716
$712$	0	0
$713$	20.2668	0.758998
$714$	0	0
$715$	−4.97771	−0.186156
$716$	0	0
$717$	−60.2113	−2.24863
$718$	0	0
$719$	46.8648	1.74776	0.873882	−	0.486139i	$-0.161595\pi$
$719$	46.8648	1.74776	0.873882	+	0.486139i	$0.161595\pi$
$720$	0	0
$721$	45.5604	1.69676
$722$	0	0
$723$	−34.8340	−1.29549
$724$	0	0
$725$	4.89487	0.181791
$726$	0	0
$727$	−16.3824	−0.607589	−0.303794	−	0.952738i	$-0.598254\pi$
$727$	−16.3824	−0.607589	−0.303794	+	0.952738i	$0.598254\pi$
$728$	0	0
$729$	37.8289	1.40107
$730$	0	0
$731$	4.28817	0.158604
$732$	0	0
$733$	26.7006	0.986208	0.493104	−	0.869970i	$-0.335862\pi$
$733$	26.7006	0.986208	0.493104	+	0.869970i	$0.335862\pi$
$734$	0	0
$735$	65.7674	2.42587
$736$	0	0
$737$	−31.8343	−1.17263
$738$	0	0
$739$	−13.1530	−0.483841	−0.241921	−	0.970296i	$-0.577777\pi$
$739$	−13.1530	−0.483841	−0.241921	+	0.970296i	$0.577777\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.15745	−0.0424628	−0.0212314	−	0.999775i	$-0.506759\pi$
$743$	−1.15745	−0.0424628	−0.0212314	+	0.999775i	$0.506759\pi$
$744$	0	0
$745$	−0.0196004	−0.000718102 0
$746$	0	0
$747$	−18.2909	−0.669228
$748$	0	0
$749$	62.4252	2.28097
$750$	0	0
$751$	45.8708	1.67385	0.836925	−	0.547317i	$-0.184351\pi$
$751$	45.8708	1.67385	0.836925	+	0.547317i	$0.184351\pi$
$752$	0	0
$753$	−34.1908	−1.24598
$754$	0	0
$755$	−13.7956	−0.502074
$756$	0	0
$757$	−47.2449	−1.71714	−0.858572	−	0.512694i	$-0.828648\pi$
$757$	−47.2449	−1.71714	−0.858572	+	0.512694i	$0.828648\pi$
$758$	0	0
$759$	33.0951	1.20128
$760$	0	0
$761$	−18.9495	−0.686921	−0.343460	−	0.939167i	$-0.611599\pi$
$761$	−18.9495	−0.686921	−0.343460	+	0.939167i	$0.611599\pi$
$762$	0	0
$763$	41.2395	1.49297
$764$	0	0
$765$	17.9804	0.650083
$766$	0	0
$767$	7.06324	0.255039
$768$	0	0
$769$	−50.4270	−1.81844	−0.909221	−	0.416313i	$-0.863322\pi$
$769$	−50.4270	−1.81844	−0.909221	+	0.416313i	$0.863322\pi$
$770$	0	0
$771$	6.05550	0.218084
$772$	0	0
$773$	42.1138	1.51473	0.757364	−	0.652993i	$-0.226487\pi$
$773$	42.1138	1.51473	0.757364	+	0.652993i	$0.226487\pi$
$774$	0	0
$775$	6.54664	0.235162
$776$	0	0
$777$	77.9427	2.79618
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−20.0392	−0.717059
$782$	0	0
$783$	−49.3637	−1.76412
$784$	0	0
$785$	59.7478	2.13249
$786$	0	0
$787$	9.15663	0.326399	0.163199	−	0.986593i	$-0.447819\pi$
$787$	9.15663	0.326399	0.163199	+	0.986593i	$0.447819\pi$
$788$	0	0
$789$	−26.7769	−0.953284
$790$	0	0
$791$	−21.7965	−0.774996
$792$	0	0
$793$	5.35092	0.190017
$794$	0	0
$795$	34.5963	1.22700
$796$	0	0
$797$	−31.8708	−1.12892	−0.564461	−	0.825460i	$-0.690916\pi$
$797$	−31.8708	−1.12892	−0.564461	+	0.825460i	$0.690916\pi$
$798$	0	0
$799$	6.86989	0.243039
$800$	0	0
$801$	16.9213	0.597884
$802$	0	0
$803$	−20.9067	−0.737782
$804$	0	0
$805$	42.9222	1.51281
$806$	0	0
$807$	−46.8022	−1.64752
$808$	0	0
$809$	14.1010	0.495765	0.247883	−	0.968790i	$-0.420265\pi$
$809$	14.1010	0.495765	0.247883	+	0.968790i	$0.420265\pi$
$810$	0	0
$811$	−15.9186	−0.558977	−0.279489	−	0.960149i	$-0.590165\pi$
$811$	−15.9186	−0.558977	−0.279489	+	0.960149i	$0.590165\pi$
$812$	0	0
$813$	−49.1566	−1.72400
$814$	0	0
$815$	50.5117	1.76935
$816$	0	0
$817$	0	0
$818$	0	0
$819$	24.0797	0.841411
$820$	0	0
$821$	17.4507	0.609033	0.304516	−	0.952507i	$-0.401505\pi$
$821$	17.4507	0.609033	0.304516	+	0.952507i	$0.401505\pi$
$822$	0	0
$823$	−31.3824	−1.09392	−0.546960	−	0.837158i	$-0.684215\pi$
$823$	−31.3824	−1.09392	−0.546960	+	0.837158i	$0.684215\pi$
$824$	0	0
$825$	10.6905	0.372195
$826$	0	0
$827$	−18.0933	−0.629165	−0.314582	−	0.949230i	$-0.601864\pi$
$827$	−18.0933	−0.629165	−0.314582	+	0.949230i	$0.601864\pi$
$828$	0	0
$829$	−20.6290	−0.716474	−0.358237	−	0.933631i	$-0.616622\pi$
$829$	−20.6290	−0.716474	−0.358237	+	0.933631i	$0.616622\pi$
$830$	0	0
$831$	−9.09245	−0.315414
$832$	0	0
$833$	−7.71244	−0.267220
$834$	0	0
$835$	−1.38413	−0.0478999
$836$	0	0
$837$	−66.0215	−2.28204
$838$	0	0
$839$	−48.6478	−1.67951	−0.839754	−	0.542968i	$-0.817301\pi$
$839$	−48.6478	−1.67951	−0.839754	+	0.542968i	$0.817301\pi$
$840$	0	0
$841$	−16.9736	−0.585296
$842$	0	0
$843$	−25.1539	−0.866348
$844$	0	0
$845$	31.1411	1.07129
$846$	0	0
$847$	21.2986	0.731829
$848$	0	0
$849$	67.5150	2.31711
$850$	0	0
$851$	27.2080	0.932679
$852$	0	0
$853$	−31.2600	−1.07032	−0.535161	−	0.844750i	$-0.679749\pi$
$853$	−31.2600	−1.07032	−0.535161	+	0.844750i	$0.679749\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.9881	0.443666	0.221833	−	0.975085i	$-0.428796\pi$
$857$	12.9881	0.443666	0.221833	+	0.975085i	$0.428796\pi$
$858$	0	0
$859$	−44.0419	−1.50269	−0.751345	−	0.659910i	$-0.770595\pi$
$859$	−44.0419	−1.50269	−0.751345	+	0.659910i	$0.770595\pi$
$860$	0	0
$861$	117.626	4.00870
$862$	0	0
$863$	−1.25166	−0.0426070	−0.0213035	−	0.999773i	$-0.506782\pi$
$863$	−1.25166	−0.0426070	−0.0213035	+	0.999773i	$0.506782\pi$
$864$	0	0
$865$	28.7648	0.978031
$866$	0	0
$867$	51.8916	1.76233
$868$	0	0
$869$	−8.75877	−0.297121
$870$	0	0
$871$	−11.3582	−0.384859
$872$	0	0
$873$	1.31221	0.0444117
$874$	0	0
$875$	−35.2499	−1.19166
$876$	0	0
$877$	−10.5457	−0.356103	−0.178052	−	0.984021i	$-0.556979\pi$
$877$	−10.5457	−0.356103	−0.178052	+	0.984021i	$0.556979\pi$
$878$	0	0
$879$	89.8977	3.03217
$880$	0	0
$881$	−10.6973	−0.360401	−0.180200	−	0.983630i	$-0.557675\pi$
$881$	−10.6973	−0.360401	−0.180200	+	0.983630i	$0.557675\pi$
$882$	0	0
$883$	−28.6382	−0.963751	−0.481875	−	0.876240i	$-0.660044\pi$
$883$	−28.6382	−0.963751	−0.481875	+	0.876240i	$0.660044\pi$
$884$	0	0
$885$	−68.9059	−2.31625
$886$	0	0
$887$	16.0327	0.538326	0.269163	−	0.963095i	$-0.413253\pi$
$887$	16.0327	0.538326	0.269163	+	0.963095i	$0.413253\pi$
$888$	0	0
$889$	−23.0574	−0.773319
$890$	0	0
$891$	−55.6204	−1.86335
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−4.67230	−0.156178
$896$	0	0
$897$	11.8081	0.394260
$898$	0	0
$899$	16.0847	0.536455
$900$	0	0
$901$	−4.05705	−0.135160
$902$	0	0
$903$	−56.0242	−1.86437
$904$	0	0
$905$	−11.6013	−0.385641
$906$	0	0
$907$	−10.0520	−0.333771	−0.166885	−	0.985976i	$-0.553371\pi$
$907$	−10.0520	−0.333771	−0.166885	+	0.985976i	$0.553371\pi$
$908$	0	0
$909$	−29.0624	−0.963940
$910$	0	0
$911$	3.46379	0.114761	0.0573803	−	0.998352i	$-0.481725\pi$
$911$	3.46379	0.114761	0.0573803	+	0.998352i	$0.481725\pi$
$912$	0	0
$913$	5.79292	0.191718
$914$	0	0
$915$	−52.2012	−1.72572
$916$	0	0
$917$	−17.4953	−0.577744
$918$	0	0
$919$	−4.56799	−0.150684	−0.0753421	−	0.997158i	$-0.524005\pi$
$919$	−4.56799	−0.150684	−0.0753421	+	0.997158i	$0.524005\pi$
$920$	0	0
$921$	45.0384	1.48407
$922$	0	0
$923$	−7.14982	−0.235339
$924$	0	0
$925$	8.78880	0.288974
$926$	0	0
$927$	−87.0420	−2.85883
$928$	0	0
$929$	−35.7861	−1.17410	−0.587052	−	0.809549i	$-0.699712\pi$
$929$	−35.7861	−1.17410	−0.587052	+	0.809549i	$0.699712\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	16.0615	0.525830
$934$	0	0
$935$	−5.69459	−0.186233
$936$	0	0
$937$	−38.5936	−1.26080	−0.630399	−	0.776272i	$-0.717109\pi$
$937$	−38.5936	−1.26080	−0.630399	+	0.776272i	$0.717109\pi$
$938$	0	0
$939$	−6.04221	−0.197180
$940$	0	0
$941$	−11.8075	−0.384912	−0.192456	−	0.981306i	$-0.561645\pi$
$941$	−11.8075	−0.384912	−0.192456	+	0.981306i	$0.561645\pi$
$942$	0	0
$943$	41.0607	1.33712
$944$	0	0
$945$	−139.824	−4.54847
$946$	0	0
$947$	−17.6372	−0.573133	−0.286566	−	0.958060i	$-0.592514\pi$
$947$	−17.6372	−0.573133	−0.286566	+	0.958060i	$0.592514\pi$
$948$	0	0
$949$	−7.45935	−0.242141
$950$	0	0
$951$	−97.4630	−3.16045
$952$	0	0
$953$	−44.9231	−1.45520	−0.727602	−	0.686000i	$-0.759365\pi$
$953$	−44.9231	−1.45520	−0.727602	+	0.686000i	$0.759365\pi$
$954$	0	0
$955$	−5.28312	−0.170958
$956$	0	0
$957$	26.2659	0.849055
$958$	0	0
$959$	26.5553	0.857516
$960$	0	0
$961$	−9.48751	−0.306049
$962$	0	0
$963$	−119.262	−3.84316
$964$	0	0
$965$	63.7033	2.05068
$966$	0	0
$967$	27.1242	0.872257	0.436128	−	0.899884i	$-0.356349\pi$
$967$	27.1242	0.872257	0.436128	+	0.899884i	$0.356349\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−51.1052	−1.64005	−0.820023	−	0.572331i	$-0.806039\pi$
$971$	−51.1052	−1.64005	−0.820023	+	0.572331i	$0.806039\pi$
$972$	0	0
$973$	−65.6177	−2.10361
$974$	0	0
$975$	3.81427	0.122155
$976$	0	0
$977$	14.5621	0.465883	0.232942	−	0.972491i	$-0.425165\pi$
$977$	14.5621	0.465883	0.232942	+	0.972491i	$0.425165\pi$
$978$	0	0
$979$	−5.35916	−0.171279
$980$	0	0
$981$	−78.7870	−2.51548
$982$	0	0
$983$	−39.3100	−1.25379	−0.626897	−	0.779103i	$-0.715675\pi$
$983$	−39.3100	−1.25379	−0.626897	+	0.779103i	$0.715675\pi$
$984$	0	0
$985$	−62.8735	−2.00332
$986$	0	0
$987$	−89.7538	−2.85690
$988$	0	0
$989$	−19.5567	−0.621868
$990$	0	0
$991$	−48.3569	−1.53611	−0.768054	−	0.640385i	$-0.778775\pi$
$991$	−48.3569	−1.53611	−0.768054	+	0.640385i	$0.778775\pi$
$992$	0	0
$993$	31.4252	0.997249
$994$	0	0
$995$	−31.4593	−0.997328
$996$	0	0
$997$	22.7110	0.719265	0.359632	−	0.933094i	$-0.382902\pi$
$997$	22.7110	0.719265	0.359632	+	0.933094i	$0.382902\pi$
$998$	0	0
$999$	−88.6332	−2.80423

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.m.1.1		3
4.3	odd	2		5776.2.a.bs.1.3		3
19.2	odd	18		152.2.q.b.137.1	yes	6
19.10	odd	18		152.2.q.b.81.1	✓	6
19.18	odd	2		2888.2.a.s.1.3		3
76.59	even	18		304.2.u.a.289.1		6
76.67	even	18		304.2.u.a.81.1		6
76.75	even	2		5776.2.a.bj.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.q.b.81.1	✓	6	19.10	odd	18
152.2.q.b.137.1	yes	6	19.2	odd	18
304.2.u.a.81.1		6	76.67	even	18
304.2.u.a.289.1		6	76.59	even	18
2888.2.a.m.1.1		3	1.1	even	1	trivial
2888.2.a.s.1.3		3	19.18	odd	2
5776.2.a.bj.1.1		3	76.75	even	2
5776.2.a.bs.1.3		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)

\(f(q)\)	\(=\)	\(q-3.22668 q^{3} -2.53209 q^{5} -3.87939 q^{7} +7.41147 q^{9} -2.34730 q^{11} -0.837496 q^{13} +8.17024 q^{15} -0.958111 q^{17} +12.5175 q^{21} +4.36959 q^{23} +1.41147 q^{25} -14.2344 q^{27} +3.46791 q^{29} +4.63816 q^{31} +7.57398 q^{33} +9.82295 q^{35} +6.22668 q^{37} +2.70233 q^{39} +9.39693 q^{41} -4.47565 q^{43} -18.7665 q^{45} -7.17024 q^{47} +8.04963 q^{49} +3.09152 q^{51} +4.23442 q^{53} +5.94356 q^{55} -8.43376 q^{59} -6.38919 q^{61} -28.7520 q^{63} +2.12061 q^{65} +13.5621 q^{67} -14.0993 q^{69} +8.53714 q^{71} +8.90673 q^{73} -4.55438 q^{75} +9.10607 q^{77} +3.73143 q^{79} +23.6955 q^{81} -2.46791 q^{83} +2.42602 q^{85} -11.1898 q^{87} +2.28312 q^{89} +3.24897 q^{91} -14.9659 q^{93} +0.177052 q^{97} -17.3969 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 6 q^{7} + 12 q^{9} - 6 q^{11} + 3 q^{15} - 6 q^{17} + 15 q^{21} + 6 q^{23} - 6 q^{25} - 12 q^{27} + 15 q^{29} - 3 q^{31} + 15 q^{33} + 9 q^{35} + 12 q^{37} - 18 q^{39} + 6 q^{43}+ \cdots - 24 q^{99}+O(q^{100}) \) 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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