LMFDB - Embedded newform 2888.2.a.u.1.4

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 = [6,0,3,0,2,0,-2,0,9,0,-5,0,10,0,0] 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:	$0$
Dimension:	$6$
Coefficient field:	6.6.3022625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 7x^{3} + 17x^{2} + 4x - 1 $ x^6 - 2x^5 - 7x^4 + 7x^3 + 17x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.496915$ 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)

$f(q)$	$=$	$q+1.45214 q^{3} +3.79176 q^{5} -3.95766 q^{7} -0.891299 q^{9} -2.06636 q^{11} +3.18371 q^{13} +5.50615 q^{15} +6.58249 q^{17} -5.74706 q^{21} +7.10283 q^{23} +9.37743 q^{25} -5.65070 q^{27} -3.19207 q^{29} +3.24708 q^{31} -3.00063 q^{33} -15.0065 q^{35} +0.364154 q^{37} +4.62318 q^{39} +11.3651 q^{41} -4.39395 q^{43} -3.37959 q^{45} -4.46998 q^{47} +8.66304 q^{49} +9.55868 q^{51} +4.70439 q^{53} -7.83513 q^{55} +10.3237 q^{59} -6.96631 q^{61} +3.52745 q^{63} +12.0719 q^{65} -1.11636 q^{67} +10.3143 q^{69} +10.9421 q^{71} +1.44215 q^{73} +13.6173 q^{75} +8.17793 q^{77} -4.72706 q^{79} -5.53169 q^{81} -4.04937 q^{83} +24.9592 q^{85} -4.63532 q^{87} +2.77258 q^{89} -12.6000 q^{91} +4.71520 q^{93} +14.3211 q^{97} +1.84174 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3} + 2 q^{5} - 2 q^{7} + 9 q^{9} - 5 q^{11} + 10 q^{13} + 4 q^{17} + 23 q^{21} - 15 q^{23} + 12 q^{25} + 18 q^{27} - 7 q^{29} + 5 q^{31} - q^{33} - 38 q^{35} + 17 q^{37} + 37 q^{39} + 4 q^{41}+ \cdots - 47 q^{99}+O(q^{100}) $ 6 * q + 3 * q^3 + 2 * q^5 - 2 * q^7 + 9 * q^9 - 5 * q^11 + 10 * q^13 + 4 * q^17 + 23 * q^21 - 15 * q^23 + 12 * q^25 + 18 * q^27 - 7 * q^29 + 5 * q^31 - q^33 - 38 * q^35 + 17 * q^37 + 37 * q^39 + 4 * q^41 - 11 * q^43 + 6 * q^45 + 18 * q^47 + 20 * q^49 + 48 * q^51 - 7 * q^53 - 42 * q^55 + q^59 - 4 * q^61 + 19 * q^63 + 6 * q^65 + 20 * q^67 + 6 * q^69 - 6 * q^71 - 2 * q^73 + 19 * q^75 + 41 * q^77 + 8 * q^79 - 6 * q^81 - 22 * q^83 + 32 * q^85 + 29 * q^87 + 14 * q^89 + 26 * q^91 - 5 * q^93 + 41 * q^97 - 47 * 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
$3$	1.45214	0.838392	0.419196	−	0.907896i	$-0.362312\pi$
$3$	1.45214	0.838392	0.419196	+	0.907896i	$0.362312\pi$
$4$	0	0
$5$	3.79176	1.69573	0.847863	−	0.530215i	$-0.177889\pi$
$5$	3.79176	1.69573	0.847863	+	0.530215i	$0.177889\pi$
$6$	0	0
$7$	−3.95766	−1.49585	−0.747927	−	0.663781i	$-0.768951\pi$
$7$	−3.95766	−1.49585	−0.747927	+	0.663781i	$0.768951\pi$
$8$	0	0
$9$	−0.891299	−0.297100
$10$	0	0
$11$	−2.06636	−0.623030	−0.311515	−	0.950241i	$-0.600836\pi$
$11$	−2.06636	−0.623030	−0.311515	+	0.950241i	$0.600836\pi$
$12$	0	0
$13$	3.18371	0.883002	0.441501	−	0.897261i	$-0.354446\pi$
$13$	3.18371	0.883002	0.441501	+	0.897261i	$0.354446\pi$
$14$	0	0
$15$	5.50615	1.42168
$16$	0	0
$17$	6.58249	1.59649	0.798245	−	0.602333i	$-0.205762\pi$
$17$	6.58249	1.59649	0.798245	+	0.602333i	$0.205762\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−5.74706	−1.25411
$22$	0	0
$23$	7.10283	1.48104	0.740522	−	0.672032i	$-0.234578\pi$
$23$	7.10283	1.48104	0.740522	+	0.672032i	$0.234578\pi$
$24$	0	0
$25$	9.37743	1.87549
$26$	0	0
$27$	−5.65070	−1.08748
$28$	0	0
$29$	−3.19207	−0.592752	−0.296376	−	0.955071i	$-0.595778\pi$
$29$	−3.19207	−0.592752	−0.296376	+	0.955071i	$0.595778\pi$
$30$	0	0
$31$	3.24708	0.583192	0.291596	−	0.956542i	$-0.405814\pi$
$31$	3.24708	0.583192	0.291596	+	0.956542i	$0.405814\pi$
$32$	0	0
$33$	−3.00063	−0.522343
$34$	0	0
$35$	−15.0065	−2.53656
$36$	0	0
$37$	0.364154	0.0598665	0.0299333	−	0.999552i	$-0.490471\pi$
$37$	0.364154	0.0598665	0.0299333	+	0.999552i	$0.490471\pi$
$38$	0	0
$39$	4.62318	0.740301
$40$	0	0
$41$	11.3651	1.77493	0.887465	−	0.460876i	$-0.152465\pi$
$41$	11.3651	1.77493	0.887465	+	0.460876i	$0.152465\pi$
$42$	0	0
$43$	−4.39395	−0.670071	−0.335035	−	0.942206i	$-0.608748\pi$
$43$	−4.39395	−0.670071	−0.335035	+	0.942206i	$0.608748\pi$
$44$	0	0
$45$	−3.37959	−0.503800
$46$	0	0
$47$	−4.46998	−0.652013	−0.326007	−	0.945367i	$-0.605703\pi$
$47$	−4.46998	−0.652013	−0.326007	+	0.945367i	$0.605703\pi$
$48$	0	0
$49$	8.66304	1.23758
$50$	0	0
$51$	9.55868	1.33848
$52$	0	0
$53$	4.70439	0.646198	0.323099	−	0.946365i	$-0.395275\pi$
$53$	4.70439	0.646198	0.323099	+	0.946365i	$0.395275\pi$
$54$	0	0
$55$	−7.83513	−1.05649
$56$	0	0
$57$	0	0
$58$	0	0
$59$	10.3237	1.34404	0.672018	−	0.740535i	$-0.265428\pi$
$59$	10.3237	1.34404	0.672018	+	0.740535i	$0.265428\pi$
$60$	0	0
$61$	−6.96631	−0.891944	−0.445972	−	0.895047i	$-0.647142\pi$
$61$	−6.96631	−0.891944	−0.445972	+	0.895047i	$0.647142\pi$
$62$	0	0
$63$	3.52745	0.444418
$64$	0	0
$65$	12.0719	1.49733
$66$	0	0
$67$	−1.11636	−0.136385	−0.0681924	−	0.997672i	$-0.521723\pi$
$67$	−1.11636	−0.136385	−0.0681924	+	0.997672i	$0.521723\pi$
$68$	0	0
$69$	10.3143	1.24169
$70$	0	0
$71$	10.9421	1.29859	0.649295	−	0.760537i	$-0.275064\pi$
$71$	10.9421	1.29859	0.649295	+	0.760537i	$0.275064\pi$
$72$	0	0
$73$	1.44215	0.168791	0.0843956	−	0.996432i	$-0.473104\pi$
$73$	1.44215	0.168791	0.0843956	+	0.996432i	$0.473104\pi$
$74$	0	0
$75$	13.6173	1.57239
$76$	0	0
$77$	8.17793	0.931962
$78$	0	0
$79$	−4.72706	−0.531836	−0.265918	−	0.963996i	$-0.585675\pi$
$79$	−4.72706	−0.531836	−0.265918	+	0.963996i	$0.585675\pi$
$80$	0	0
$81$	−5.53169	−0.614632
$82$	0	0
$83$	−4.04937	−0.444476	−0.222238	−	0.974992i	$-0.571336\pi$
$83$	−4.04937	−0.444476	−0.222238	+	0.974992i	$0.571336\pi$
$84$	0	0
$85$	24.9592	2.70721
$86$	0	0
$87$	−4.63532	−0.496958
$88$	0	0
$89$	2.77258	0.293893	0.146947	−	0.989144i	$-0.453055\pi$
$89$	2.77258	0.293893	0.146947	+	0.989144i	$0.453055\pi$
$90$	0	0
$91$	−12.6000	−1.32084
$92$	0	0
$93$	4.71520	0.488943
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.3211	1.45409	0.727044	−	0.686591i	$-0.240894\pi$
$97$	14.3211	1.45409	0.727044	+	0.686591i	$0.240894\pi$
$98$	0	0
$99$	1.84174	0.185102
$100$	0	0
$101$	0.0483452	0.00481053	0.00240526	−	0.999997i	$-0.499234\pi$
$101$	0.0483452	0.00481053	0.00240526	+	0.999997i	$0.499234\pi$
$102$	0	0
$103$	19.5843	1.92970	0.964851	−	0.262797i	$-0.0846448\pi$
$103$	19.5843	1.92970	0.964851	+	0.262797i	$0.0846448\pi$
$104$	0	0
$105$	−21.7915	−2.12663
$106$	0	0
$107$	0.554667	0.0536217	0.0268108	−	0.999641i	$-0.491465\pi$
$107$	0.554667	0.0536217	0.0268108	+	0.999641i	$0.491465\pi$
$108$	0	0
$109$	−11.2401	−1.07660	−0.538301	−	0.842752i	$-0.680934\pi$
$109$	−11.2401	−1.07660	−0.538301	+	0.842752i	$0.680934\pi$
$110$	0	0
$111$	0.528801	0.0501916
$112$	0	0
$113$	1.20277	0.113147	0.0565736	−	0.998398i	$-0.481982\pi$
$113$	1.20277	0.113147	0.0565736	+	0.998398i	$0.481982\pi$
$114$	0	0
$115$	26.9322	2.51144
$116$	0	0
$117$	−2.83764	−0.262340
$118$	0	0
$119$	−26.0512	−2.38811
$120$	0	0
$121$	−6.73017	−0.611834
$122$	0	0
$123$	16.5037	1.48809
$124$	0	0
$125$	16.5982	1.48459
$126$	0	0
$127$	−14.8301	−1.31596	−0.657978	−	0.753037i	$-0.728588\pi$
$127$	−14.8301	−1.31596	−0.657978	+	0.753037i	$0.728588\pi$
$128$	0	0
$129$	−6.38061	−0.561781
$130$	0	0
$131$	−1.36177	−0.118978	−0.0594891	−	0.998229i	$-0.518947\pi$
$131$	−1.36177	−0.118978	−0.0594891	+	0.998229i	$0.518947\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−21.4261	−1.84406
$136$	0	0
$137$	−13.9423	−1.19117	−0.595585	−	0.803292i	$-0.703080\pi$
$137$	−13.9423	−1.19117	−0.595585	+	0.803292i	$0.703080\pi$
$138$	0	0
$139$	2.72025	0.230729	0.115364	−	0.993323i	$-0.463196\pi$
$139$	2.72025	0.230729	0.115364	+	0.993323i	$0.463196\pi$
$140$	0	0
$141$	−6.49102	−0.546642
$142$	0	0
$143$	−6.57868	−0.550137
$144$	0	0
$145$	−12.1035	−1.00515
$146$	0	0
$147$	12.5799	1.03757
$148$	0	0
$149$	6.55050	0.536637	0.268319	−	0.963330i	$-0.413532\pi$
$149$	6.55050	0.536637	0.268319	+	0.963330i	$0.413532\pi$
$150$	0	0
$151$	11.3481	0.923494	0.461747	−	0.887012i	$-0.347223\pi$
$151$	11.3481	0.923494	0.461747	+	0.887012i	$0.347223\pi$
$152$	0	0
$153$	−5.86697	−0.474316
$154$	0	0
$155$	12.3121	0.988934
$156$	0	0
$157$	−17.1978	−1.37254	−0.686269	−	0.727348i	$-0.740753\pi$
$157$	−17.1978	−1.37254	−0.686269	+	0.727348i	$0.740753\pi$
$158$	0	0
$159$	6.83142	0.541767
$160$	0	0
$161$	−28.1106	−2.21542
$162$	0	0
$163$	−17.0946	−1.33896	−0.669478	−	0.742832i	$-0.733482\pi$
$163$	−17.0946	−1.33896	−0.669478	+	0.742832i	$0.733482\pi$
$164$	0	0
$165$	−11.3777	−0.885751
$166$	0	0
$167$	17.6013	1.36203	0.681016	−	0.732269i	$-0.261538\pi$
$167$	17.6013	1.36203	0.681016	+	0.732269i	$0.261538\pi$
$168$	0	0
$169$	−2.86400	−0.220307
$170$	0	0
$171$	0	0
$172$	0	0
$173$	19.9418	1.51615	0.758073	−	0.652170i	$-0.226141\pi$
$173$	19.9418	1.51615	0.758073	+	0.652170i	$0.226141\pi$
$174$	0	0
$175$	−37.1127	−2.80545
$176$	0	0
$177$	14.9915	1.12683
$178$	0	0
$179$	2.85313	0.213253	0.106626	−	0.994299i	$-0.465995\pi$
$179$	2.85313	0.213253	0.106626	+	0.994299i	$0.465995\pi$
$180$	0	0
$181$	−18.0757	−1.34355	−0.671777	−	0.740753i	$-0.734469\pi$
$181$	−18.0757	−1.34355	−0.671777	+	0.740753i	$0.734469\pi$
$182$	0	0
$183$	−10.1160	−0.747798
$184$	0	0
$185$	1.38078	0.101517
$186$	0	0
$187$	−13.6018	−0.994661
$188$	0	0
$189$	22.3635	1.62671
$190$	0	0
$191$	15.8231	1.14492	0.572460	−	0.819933i	$-0.305989\pi$
$191$	15.8231	1.14492	0.572460	+	0.819933i	$0.305989\pi$
$192$	0	0
$193$	−16.2193	−1.16749	−0.583745	−	0.811937i	$-0.698413\pi$
$193$	−16.2193	−1.16749	−0.583745	+	0.811937i	$0.698413\pi$
$194$	0	0
$195$	17.5300	1.25535
$196$	0	0
$197$	−7.46439	−0.531816	−0.265908	−	0.963998i	$-0.585672\pi$
$197$	−7.46439	−0.531816	−0.265908	+	0.963998i	$0.585672\pi$
$198$	0	0
$199$	−0.979896	−0.0694630	−0.0347315	−	0.999397i	$-0.511058\pi$
$199$	−0.979896	−0.0694630	−0.0347315	+	0.999397i	$0.511058\pi$
$200$	0	0
$201$	−1.62111	−0.114344
$202$	0	0
$203$	12.6331	0.886670
$204$	0	0
$205$	43.0937	3.00979
$206$	0	0
$207$	−6.33075	−0.440017
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−11.6647	−0.803032	−0.401516	−	0.915852i	$-0.631517\pi$
$211$	−11.6647	−0.803032	−0.401516	+	0.915852i	$0.631517\pi$
$212$	0	0
$213$	15.8894	1.08873
$214$	0	0
$215$	−16.6608	−1.13626
$216$	0	0
$217$	−12.8508	−0.872370
$218$	0	0
$219$	2.09420	0.141513
$220$	0	0
$221$	20.9567	1.40970
$222$	0	0
$223$	−26.4274	−1.76971	−0.884856	−	0.465865i	$-0.845743\pi$
$223$	−26.4274	−1.76971	−0.884856	+	0.465865i	$0.845743\pi$
$224$	0	0
$225$	−8.35810	−0.557206
$226$	0	0
$227$	−23.1268	−1.53498	−0.767491	−	0.641060i	$-0.778495\pi$
$227$	−23.1268	−1.53498	−0.767491	+	0.641060i	$0.778495\pi$
$228$	0	0
$229$	−5.99359	−0.396068	−0.198034	−	0.980195i	$-0.563456\pi$
$229$	−5.99359	−0.396068	−0.198034	+	0.980195i	$0.563456\pi$
$230$	0	0
$231$	11.8755	0.781349
$232$	0	0
$233$	−12.3459	−0.808805	−0.404403	−	0.914581i	$-0.632521\pi$
$233$	−12.3459	−0.808805	−0.404403	+	0.914581i	$0.632521\pi$
$234$	0	0
$235$	−16.9491	−1.10564
$236$	0	0
$237$	−6.86433	−0.445886
$238$	0	0
$239$	−0.705772	−0.0456526	−0.0228263	−	0.999739i	$-0.507266\pi$
$239$	−0.705772	−0.0456526	−0.0228263	+	0.999739i	$0.507266\pi$
$240$	0	0
$241$	−25.3762	−1.63463	−0.817313	−	0.576195i	$-0.804537\pi$
$241$	−25.3762	−1.63463	−0.817313	+	0.576195i	$0.804537\pi$
$242$	0	0
$243$	8.91933	0.572175
$244$	0	0
$245$	32.8482	2.09859
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−5.88024	−0.372645
$250$	0	0
$251$	29.6381	1.87074	0.935370	−	0.353671i	$-0.115067\pi$
$251$	29.6381	1.87074	0.935370	+	0.353671i	$0.115067\pi$
$252$	0	0
$253$	−14.6770	−0.922735
$254$	0	0
$255$	36.2442	2.26970
$256$	0	0
$257$	−0.953128	−0.0594545	−0.0297272	−	0.999558i	$-0.509464\pi$
$257$	−0.953128	−0.0594545	−0.0297272	+	0.999558i	$0.509464\pi$
$258$	0	0
$259$	−1.44120	−0.0895516
$260$	0	0
$261$	2.84509	0.176106
$262$	0	0
$263$	9.65822	0.595551	0.297776	−	0.954636i	$-0.403755\pi$
$263$	9.65822	0.595551	0.297776	+	0.954636i	$0.403755\pi$
$264$	0	0
$265$	17.8379	1.09577
$266$	0	0
$267$	4.02617	0.246398
$268$	0	0
$269$	6.78406	0.413632	0.206816	−	0.978380i	$-0.433690\pi$
$269$	6.78406	0.413632	0.206816	+	0.978380i	$0.433690\pi$
$270$	0	0
$271$	9.00012	0.546719	0.273359	−	0.961912i	$-0.411865\pi$
$271$	9.00012	0.546719	0.273359	+	0.961912i	$0.411865\pi$
$272$	0	0
$273$	−18.2970	−1.10738
$274$	0	0
$275$	−19.3771	−1.16848
$276$	0	0
$277$	−19.4125	−1.16639	−0.583193	−	0.812333i	$-0.698197\pi$
$277$	−19.4125	−1.16639	−0.583193	+	0.812333i	$0.698197\pi$
$278$	0	0
$279$	−2.89412	−0.173266
$280$	0	0
$281$	−0.218245	−0.0130194	−0.00650969	−	0.999979i	$-0.502072\pi$
$281$	−0.218245	−0.0130194	−0.00650969	+	0.999979i	$0.502072\pi$
$282$	0	0
$283$	−7.68891	−0.457058	−0.228529	−	0.973537i	$-0.573392\pi$
$283$	−7.68891	−0.457058	−0.228529	+	0.973537i	$0.573392\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−44.9791	−2.65503
$288$	0	0
$289$	26.3292	1.54878
$290$	0	0
$291$	20.7962	1.21910
$292$	0	0
$293$	10.3008	0.601777	0.300889	−	0.953659i	$-0.402717\pi$
$293$	10.3008	0.601777	0.300889	+	0.953659i	$0.402717\pi$
$294$	0	0
$295$	39.1451	2.27912
$296$	0	0
$297$	11.6764	0.677531
$298$	0	0
$299$	22.6134	1.30776
$300$	0	0
$301$	17.3897	1.00233
$302$	0	0
$303$	0.0702038	0.00403310
$304$	0	0
$305$	−26.4146	−1.51249
$306$	0	0
$307$	−3.95460	−0.225701	−0.112850	−	0.993612i	$-0.535998\pi$
$307$	−3.95460	−0.225701	−0.112850	+	0.993612i	$0.535998\pi$
$308$	0	0
$309$	28.4391	1.61785
$310$	0	0
$311$	9.45362	0.536066	0.268033	−	0.963410i	$-0.413626\pi$
$311$	9.45362	0.536066	0.268033	+	0.963410i	$0.413626\pi$
$312$	0	0
$313$	15.3475	0.867492	0.433746	−	0.901035i	$-0.357192\pi$
$313$	15.3475	0.867492	0.433746	+	0.901035i	$0.357192\pi$
$314$	0	0
$315$	13.3753	0.753610
$316$	0	0
$317$	18.4012	1.03351	0.516757	−	0.856132i	$-0.327139\pi$
$317$	18.4012	1.03351	0.516757	+	0.856132i	$0.327139\pi$
$318$	0	0
$319$	6.59595	0.369302
$320$	0	0
$321$	0.805452	0.0449560
$322$	0	0
$323$	0	0
$324$	0	0
$325$	29.8550	1.65606
$326$	0	0
$327$	−16.3221	−0.902615
$328$	0	0
$329$	17.6906	0.975316
$330$	0	0
$331$	−24.1178	−1.32563	−0.662816	−	0.748782i	$-0.730639\pi$
$331$	−24.1178	−1.32563	−0.662816	+	0.748782i	$0.730639\pi$
$332$	0	0
$333$	−0.324570	−0.0177863
$334$	0	0
$335$	−4.23296	−0.231271
$336$	0	0
$337$	19.0198	1.03608	0.518038	−	0.855358i	$-0.326663\pi$
$337$	19.0198	1.03608	0.518038	+	0.855358i	$0.326663\pi$
$338$	0	0
$339$	1.74659	0.0948616
$340$	0	0
$341$	−6.70962	−0.363346
$342$	0	0
$343$	−6.58174	−0.355381
$344$	0	0
$345$	39.1093	2.10557
$346$	0	0
$347$	−27.7287	−1.48856	−0.744278	−	0.667870i	$-0.767206\pi$
$347$	−27.7287	−1.48856	−0.744278	+	0.667870i	$0.767206\pi$
$348$	0	0
$349$	3.40561	0.182298	0.0911490	−	0.995837i	$-0.470946\pi$
$349$	3.40561	0.182298	0.0911490	+	0.995837i	$0.470946\pi$
$350$	0	0
$351$	−17.9902	−0.960245
$352$	0	0
$353$	13.9944	0.744849	0.372425	−	0.928062i	$-0.378527\pi$
$353$	13.9944	0.744849	0.372425	+	0.928062i	$0.378527\pi$
$354$	0	0
$355$	41.4899	2.20205
$356$	0	0
$357$	−37.8300	−2.00217
$358$	0	0
$359$	7.39821	0.390462	0.195231	−	0.980757i	$-0.437454\pi$
$359$	7.39821	0.390462	0.195231	+	0.980757i	$0.437454\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−9.77313	−0.512956
$364$	0	0
$365$	5.46829	0.286223
$366$	0	0
$367$	−16.6057	−0.866809	−0.433404	−	0.901200i	$-0.642688\pi$
$367$	−16.6057	−0.866809	−0.433404	+	0.901200i	$0.642688\pi$
$368$	0	0
$369$	−10.1297	−0.527331
$370$	0	0
$371$	−18.6184	−0.966617
$372$	0	0
$373$	14.8149	0.767084	0.383542	−	0.923523i	$-0.374704\pi$
$373$	14.8149	0.767084	0.383542	+	0.923523i	$0.374704\pi$
$374$	0	0
$375$	24.1028	1.24466
$376$	0	0
$377$	−10.1626	−0.523401
$378$	0	0
$379$	−0.782253	−0.0401817	−0.0200908	−	0.999798i	$-0.506396\pi$
$379$	−0.782253	−0.0401817	−0.0200908	+	0.999798i	$0.506396\pi$
$380$	0	0
$381$	−21.5353	−1.10329
$382$	0	0
$383$	−15.4724	−0.790605	−0.395303	−	0.918551i	$-0.629360\pi$
$383$	−15.4724	−0.790605	−0.395303	+	0.918551i	$0.629360\pi$
$384$	0	0
$385$	31.0087	1.58035
$386$	0	0
$387$	3.91632	0.199078
$388$	0	0
$389$	−5.32893	−0.270187	−0.135094	−	0.990833i	$-0.543134\pi$
$389$	−5.32893	−0.270187	−0.135094	+	0.990833i	$0.543134\pi$
$390$	0	0
$391$	46.7544	2.36447
$392$	0	0
$393$	−1.97747	−0.0997503
$394$	0	0
$395$	−17.9239	−0.901847
$396$	0	0
$397$	−31.6126	−1.58659	−0.793295	−	0.608838i	$-0.791636\pi$
$397$	−31.6126	−1.58659	−0.793295	+	0.608838i	$0.791636\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.49084	−0.473950	−0.236975	−	0.971516i	$-0.576156\pi$
$401$	−9.49084	−0.473950	−0.236975	+	0.971516i	$0.576156\pi$
$402$	0	0
$403$	10.3377	0.514960
$404$	0	0
$405$	−20.9748	−1.04225
$406$	0	0
$407$	−0.752472	−0.0372987
$408$	0	0
$409$	−8.22153	−0.406529	−0.203264	−	0.979124i	$-0.565155\pi$
$409$	−8.22153	−0.406529	−0.203264	+	0.979124i	$0.565155\pi$
$410$	0	0
$411$	−20.2461	−0.998667
$412$	0	0
$413$	−40.8578	−2.01048
$414$	0	0
$415$	−15.3542	−0.753709
$416$	0	0
$417$	3.95018	0.193441
$418$	0	0
$419$	−21.6079	−1.05561	−0.527807	−	0.849365i	$-0.676985\pi$
$419$	−21.6079	−1.05561	−0.527807	+	0.849365i	$0.676985\pi$
$420$	0	0
$421$	−1.24974	−0.0609085	−0.0304543	−	0.999536i	$-0.509695\pi$
$421$	−1.24974	−0.0609085	−0.0304543	+	0.999536i	$0.509695\pi$
$422$	0	0
$423$	3.98409	0.193713
$424$	0	0
$425$	61.7269	2.99419
$426$	0	0
$427$	27.5702	1.33422
$428$	0	0
$429$	−9.55314	−0.461230
$430$	0	0
$431$	5.07027	0.244226	0.122113	−	0.992516i	$-0.461033\pi$
$431$	5.07027	0.244226	0.122113	+	0.992516i	$0.461033\pi$
$432$	0	0
$433$	−7.96323	−0.382688	−0.191344	−	0.981523i	$-0.561285\pi$
$433$	−7.96323	−0.382688	−0.191344	+	0.981523i	$0.561285\pi$
$434$	0	0
$435$	−17.5760	−0.842705
$436$	0	0
$437$	0	0
$438$	0	0
$439$	18.2883	0.872851	0.436425	−	0.899740i	$-0.356244\pi$
$439$	18.2883	0.872851	0.436425	+	0.899740i	$0.356244\pi$
$440$	0	0
$441$	−7.72136	−0.367684
$442$	0	0
$443$	−14.1054	−0.670167	−0.335083	−	0.942188i	$-0.608765\pi$
$443$	−14.1054	−0.670167	−0.335083	+	0.942188i	$0.608765\pi$
$444$	0	0
$445$	10.5130	0.498362
$446$	0	0
$447$	9.51221	0.449912
$448$	0	0
$449$	15.3169	0.722850	0.361425	−	0.932401i	$-0.382290\pi$
$449$	15.3169	0.722850	0.361425	+	0.932401i	$0.382290\pi$
$450$	0	0
$451$	−23.4843	−1.10583
$452$	0	0
$453$	16.4790	0.774249
$454$	0	0
$455$	−47.7763	−2.23979
$456$	0	0
$457$	−3.72346	−0.174176	−0.0870880	−	0.996201i	$-0.527756\pi$
$457$	−3.72346	−0.174176	−0.0870880	+	0.996201i	$0.527756\pi$
$458$	0	0
$459$	−37.1957	−1.73615
$460$	0	0
$461$	−7.80423	−0.363479	−0.181740	−	0.983347i	$-0.558173\pi$
$461$	−7.80423	−0.363479	−0.181740	+	0.983347i	$0.558173\pi$
$462$	0	0
$463$	−31.8423	−1.47984	−0.739919	−	0.672696i	$-0.765136\pi$
$463$	−31.8423	−1.47984	−0.739919	+	0.672696i	$0.765136\pi$
$464$	0	0
$465$	17.8789	0.829114
$466$	0	0
$467$	−36.1118	−1.67105	−0.835526	−	0.549450i	$-0.814837\pi$
$467$	−36.1118	−1.67105	−0.835526	+	0.549450i	$0.814837\pi$
$468$	0	0
$469$	4.41816	0.204012
$470$	0	0
$471$	−24.9736	−1.15072
$472$	0	0
$473$	9.07946	0.417474
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−4.19302	−0.191985
$478$	0	0
$479$	−18.9928	−0.867802	−0.433901	−	0.900960i	$-0.642863\pi$
$479$	−18.9928	−0.867802	−0.433901	+	0.900960i	$0.642863\pi$
$480$	0	0
$481$	1.15936	0.0528623
$482$	0	0
$483$	−40.8204	−1.85739
$484$	0	0
$485$	54.3022	2.46574
$486$	0	0
$487$	−5.40887	−0.245099	−0.122550	−	0.992462i	$-0.539107\pi$
$487$	−5.40887	−0.245099	−0.122550	+	0.992462i	$0.539107\pi$
$488$	0	0
$489$	−24.8238	−1.12257
$490$	0	0
$491$	−14.5310	−0.655773	−0.327886	−	0.944717i	$-0.606336\pi$
$491$	−14.5310	−0.655773	−0.327886	+	0.944717i	$0.606336\pi$
$492$	0	0
$493$	−21.0118	−0.946322
$494$	0	0
$495$	6.98344	0.313882
$496$	0	0
$497$	−43.3051	−1.94250
$498$	0	0
$499$	−11.8269	−0.529446	−0.264723	−	0.964324i	$-0.585281\pi$
$499$	−11.8269	−0.529446	−0.264723	+	0.964324i	$0.585281\pi$
$500$	0	0
$501$	25.5595	1.14192
$502$	0	0
$503$	26.0673	1.16228	0.581141	−	0.813803i	$-0.302607\pi$
$503$	26.0673	1.16228	0.581141	+	0.813803i	$0.302607\pi$
$504$	0	0
$505$	0.183313	0.00815733
$506$	0	0
$507$	−4.15891	−0.184704
$508$	0	0
$509$	−15.3073	−0.678485	−0.339242	−	0.940699i	$-0.610171\pi$
$509$	−15.3073	−0.678485	−0.339242	+	0.940699i	$0.610171\pi$
$510$	0	0
$511$	−5.70754	−0.252487
$512$	0	0
$513$	0	0
$514$	0	0
$515$	74.2591	3.27225
$516$	0	0
$517$	9.23657	0.406224
$518$	0	0
$519$	28.9582	1.27112
$520$	0	0
$521$	−24.2492	−1.06238	−0.531188	−	0.847254i	$-0.678254\pi$
$521$	−24.2492	−1.06238	−0.531188	+	0.847254i	$0.678254\pi$
$522$	0	0
$523$	38.6783	1.69128	0.845642	−	0.533750i	$-0.179218\pi$
$523$	38.6783	1.69128	0.845642	+	0.533750i	$0.179218\pi$
$524$	0	0
$525$	−53.8926	−2.35207
$526$	0	0
$527$	21.3739	0.931060
$528$	0	0
$529$	27.4503	1.19349
$530$	0	0
$531$	−9.20154	−0.399313
$532$	0	0
$533$	36.1831	1.56727
$534$	0	0
$535$	2.10316	0.0909277
$536$	0	0
$537$	4.14313	0.178789
$538$	0	0
$539$	−17.9009	−0.771048
$540$	0	0
$541$	12.3886	0.532629	0.266315	−	0.963886i	$-0.414194\pi$
$541$	12.3886	0.532629	0.266315	+	0.963886i	$0.414194\pi$
$542$	0	0
$543$	−26.2484	−1.12643
$544$	0	0
$545$	−42.6196	−1.82562
$546$	0	0
$547$	−25.7067	−1.09914	−0.549570	−	0.835448i	$-0.685208\pi$
$547$	−25.7067	−1.09914	−0.549570	+	0.835448i	$0.685208\pi$
$548$	0	0
$549$	6.20906	0.264996
$550$	0	0
$551$	0	0
$552$	0	0
$553$	18.7081	0.795548
$554$	0	0
$555$	2.00509	0.0851112
$556$	0	0
$557$	43.9504	1.86224	0.931119	−	0.364715i	$-0.118834\pi$
$557$	43.9504	1.86224	0.931119	+	0.364715i	$0.118834\pi$
$558$	0	0
$559$	−13.9890	−0.591674
$560$	0	0
$561$	−19.7516	−0.833915
$562$	0	0
$563$	−3.39837	−0.143224	−0.0716121	−	0.997433i	$-0.522814\pi$
$563$	−3.39837	−0.143224	−0.0716121	+	0.997433i	$0.522814\pi$
$564$	0	0
$565$	4.56062	0.191867
$566$	0	0
$567$	21.8925	0.919400
$568$	0	0
$569$	−12.3585	−0.518097	−0.259049	−	0.965864i	$-0.583409\pi$
$569$	−12.3585	−0.518097	−0.259049	+	0.965864i	$0.583409\pi$
$570$	0	0
$571$	−2.77755	−0.116237	−0.0581184	−	0.998310i	$-0.518510\pi$
$571$	−2.77755	−0.116237	−0.0581184	+	0.998310i	$0.518510\pi$
$572$	0	0
$573$	22.9773	0.959891
$574$	0	0
$575$	66.6064	2.77768
$576$	0	0
$577$	18.0762	0.752520	0.376260	−	0.926514i	$-0.377210\pi$
$577$	18.0762	0.752520	0.376260	+	0.926514i	$0.377210\pi$
$578$	0	0
$579$	−23.5526	−0.978813
$580$	0	0
$581$	16.0260	0.664871
$582$	0	0
$583$	−9.72095	−0.402601
$584$	0	0
$585$	−10.7596	−0.444856
$586$	0	0
$587$	−8.56572	−0.353545	−0.176773	−	0.984252i	$-0.556566\pi$
$587$	−8.56572	−0.353545	−0.176773	+	0.984252i	$0.556566\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−10.8393	−0.445870
$592$	0	0
$593$	−38.2809	−1.57201	−0.786004	−	0.618221i	$-0.787854\pi$
$593$	−38.2809	−1.57201	−0.786004	+	0.618221i	$0.787854\pi$
$594$	0	0
$595$	−98.7800	−4.04959
$596$	0	0
$597$	−1.42294	−0.0582372
$598$	0	0
$599$	−6.37376	−0.260425	−0.130212	−	0.991486i	$-0.541566\pi$
$599$	−6.37376	−0.260425	−0.130212	+	0.991486i	$0.541566\pi$
$600$	0	0
$601$	12.2960	0.501564	0.250782	−	0.968044i	$-0.419312\pi$
$601$	12.2960	0.501564	0.250782	+	0.968044i	$0.419312\pi$
$602$	0	0
$603$	0.995009	0.0405199
$604$	0	0
$605$	−25.5192	−1.03750
$606$	0	0
$607$	−13.7106	−0.556497	−0.278248	−	0.960509i	$-0.589754\pi$
$607$	−13.7106	−0.556497	−0.278248	+	0.960509i	$0.589754\pi$
$608$	0	0
$609$	18.3450	0.743377
$610$	0	0
$611$	−14.2311	−0.575729
$612$	0	0
$613$	42.4337	1.71388	0.856941	−	0.515414i	$-0.172362\pi$
$613$	42.4337	1.71388	0.856941	+	0.515414i	$0.172362\pi$
$614$	0	0
$615$	62.5779	2.52339
$616$	0	0
$617$	−27.0083	−1.08731	−0.543656	−	0.839308i	$-0.682960\pi$
$617$	−27.0083	−1.08731	−0.543656	+	0.839308i	$0.682960\pi$
$618$	0	0
$619$	−39.5183	−1.58838	−0.794188	−	0.607672i	$-0.792104\pi$
$619$	−39.5183	−1.58838	−0.794188	+	0.607672i	$0.792104\pi$
$620$	0	0
$621$	−40.1360	−1.61060
$622$	0	0
$623$	−10.9729	−0.439621
$624$	0	0
$625$	16.0491	0.641963
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.39704	0.0955763
$630$	0	0
$631$	37.6515	1.49888	0.749441	−	0.662071i	$-0.230322\pi$
$631$	37.6515	1.49888	0.749441	+	0.662071i	$0.230322\pi$
$632$	0	0
$633$	−16.9388	−0.673255
$634$	0	0
$635$	−56.2321	−2.23150
$636$	0	0
$637$	27.5806	1.09278
$638$	0	0
$639$	−9.75270	−0.385811
$640$	0	0
$641$	21.6697	0.855900	0.427950	−	0.903802i	$-0.359236\pi$
$641$	21.6697	0.855900	0.427950	+	0.903802i	$0.359236\pi$
$642$	0	0
$643$	35.9395	1.41731	0.708657	−	0.705553i	$-0.249301\pi$
$643$	35.9395	1.41731	0.708657	+	0.705553i	$0.249301\pi$
$644$	0	0
$645$	−24.1937	−0.952627
$646$	0	0
$647$	0.915638	0.0359974	0.0179987	−	0.999838i	$-0.494271\pi$
$647$	0.915638	0.0359974	0.0179987	+	0.999838i	$0.494271\pi$
$648$	0	0
$649$	−21.3325	−0.837375
$650$	0	0
$651$	−18.6611	−0.731388
$652$	0	0
$653$	13.9357	0.545346	0.272673	−	0.962107i	$-0.412092\pi$
$653$	13.9357	0.545346	0.272673	+	0.962107i	$0.412092\pi$
$654$	0	0
$655$	−5.16349	−0.201754
$656$	0	0
$657$	−1.28539	−0.0501478
$658$	0	0
$659$	−43.2164	−1.68347	−0.841736	−	0.539890i	$-0.818466\pi$
$659$	−43.2164	−1.68347	−0.841736	+	0.539890i	$0.818466\pi$
$660$	0	0
$661$	9.81904	0.381916	0.190958	−	0.981598i	$-0.438841\pi$
$661$	9.81904	0.381916	0.190958	+	0.981598i	$0.438841\pi$
$662$	0	0
$663$	30.4321	1.18188
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−22.6727	−0.877892
$668$	0	0
$669$	−38.3762	−1.48371
$670$	0	0
$671$	14.3949	0.555708
$672$	0	0
$673$	11.0877	0.427399	0.213699	−	0.976899i	$-0.431449\pi$
$673$	11.0877	0.427399	0.213699	+	0.976899i	$0.431449\pi$
$674$	0	0
$675$	−52.9890	−2.03955
$676$	0	0
$677$	−36.1996	−1.39126	−0.695632	−	0.718398i	$-0.744876\pi$
$677$	−36.1996	−1.39126	−0.695632	+	0.718398i	$0.744876\pi$
$678$	0	0
$679$	−56.6780	−2.17510
$680$	0	0
$681$	−33.5833	−1.28692
$682$	0	0
$683$	45.4612	1.73953	0.869763	−	0.493470i	$-0.164272\pi$
$683$	45.4612	1.73953	0.869763	+	0.493470i	$0.164272\pi$
$684$	0	0
$685$	−52.8658	−2.01990
$686$	0	0
$687$	−8.70351	−0.332060
$688$	0	0
$689$	14.9774	0.570594
$690$	0	0
$691$	−11.0639	−0.420890	−0.210445	−	0.977606i	$-0.567491\pi$
$691$	−11.0639	−0.420890	−0.210445	+	0.977606i	$0.567491\pi$
$692$	0	0
$693$	−7.28898	−0.276885
$694$	0	0
$695$	10.3145	0.391253
$696$	0	0
$697$	74.8106	2.83366
$698$	0	0
$699$	−17.9279	−0.678095
$700$	0	0
$701$	−26.8397	−1.01372	−0.506861	−	0.862028i	$-0.669194\pi$
$701$	−26.8397	−1.01372	−0.506861	+	0.862028i	$0.669194\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−24.6124	−0.926956
$706$	0	0
$707$	−0.191334	−0.00719584
$708$	0	0
$709$	27.2769	1.02440	0.512202	−	0.858865i	$-0.328830\pi$
$709$	27.2769	1.02440	0.512202	+	0.858865i	$0.328830\pi$
$710$	0	0
$711$	4.21322	0.158008
$712$	0	0
$713$	23.0634	0.863733
$714$	0	0
$715$	−24.9448	−0.932881
$716$	0	0
$717$	−1.02488	−0.0382747
$718$	0	0
$719$	4.39717	0.163987	0.0819933	−	0.996633i	$-0.473871\pi$
$719$	4.39717	0.163987	0.0819933	+	0.996633i	$0.473871\pi$
$720$	0	0
$721$	−77.5081	−2.88655
$722$	0	0
$723$	−36.8497	−1.37046
$724$	0	0
$725$	−29.9334	−1.11170
$726$	0	0
$727$	0.801638	0.0297311	0.0148656	−	0.999890i	$-0.495268\pi$
$727$	0.801638	0.0297311	0.0148656	+	0.999890i	$0.495268\pi$
$728$	0	0
$729$	29.5471	1.09434
$730$	0	0
$731$	−28.9231	−1.06976
$732$	0	0
$733$	−39.0667	−1.44296	−0.721480	−	0.692435i	$-0.756538\pi$
$733$	−39.0667	−1.44296	−0.721480	+	0.692435i	$0.756538\pi$
$734$	0	0
$735$	47.7000	1.75944
$736$	0	0
$737$	2.30680	0.0849719
$738$	0	0
$739$	33.0869	1.21712	0.608562	−	0.793507i	$-0.291747\pi$
$739$	33.0869	1.21712	0.608562	+	0.793507i	$0.291747\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.8634	−0.655345	−0.327672	−	0.944791i	$-0.606264\pi$
$743$	−17.8634	−0.655345	−0.327672	+	0.944791i	$0.606264\pi$
$744$	0	0
$745$	24.8379	0.909990
$746$	0	0
$747$	3.60920	0.132054
$748$	0	0
$749$	−2.19518	−0.0802102
$750$	0	0
$751$	29.7141	1.08428	0.542141	−	0.840287i	$-0.317614\pi$
$751$	29.7141	1.08428	0.542141	+	0.840287i	$0.317614\pi$
$752$	0	0
$753$	43.0386	1.56841
$754$	0	0
$755$	43.0292	1.56599
$756$	0	0
$757$	21.5838	0.784476	0.392238	−	0.919864i	$-0.371701\pi$
$757$	21.5838	0.784476	0.392238	+	0.919864i	$0.371701\pi$
$758$	0	0
$759$	−21.3130	−0.773613
$760$	0	0
$761$	−21.2021	−0.768574	−0.384287	−	0.923214i	$-0.625553\pi$
$761$	−21.2021	−0.768574	−0.384287	+	0.923214i	$0.625553\pi$
$762$	0	0
$763$	44.4843	1.61044
$764$	0	0
$765$	−22.2461	−0.804311
$766$	0	0
$767$	32.8678	1.18679
$768$	0	0
$769$	−3.91027	−0.141008	−0.0705040	−	0.997511i	$-0.522461\pi$
$769$	−3.91027	−0.141008	−0.0705040	+	0.997511i	$0.522461\pi$
$770$	0	0
$771$	−1.38407	−0.0498461
$772$	0	0
$773$	5.45337	0.196144	0.0980720	−	0.995179i	$-0.468732\pi$
$773$	5.45337	0.196144	0.0980720	+	0.995179i	$0.468732\pi$
$774$	0	0
$775$	30.4492	1.09377
$776$	0	0
$777$	−2.09281	−0.0750793
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−22.6103	−0.809061
$782$	0	0
$783$	18.0374	0.644604
$784$	0	0
$785$	−65.2101	−2.32745
$786$	0	0
$787$	39.6644	1.41388	0.706941	−	0.707272i	$-0.250074\pi$
$787$	39.6644	1.41388	0.706941	+	0.707272i	$0.250074\pi$
$788$	0	0
$789$	14.0250	0.499305
$790$	0	0
$791$	−4.76015	−0.169252
$792$	0	0
$793$	−22.1787	−0.787589
$794$	0	0
$795$	25.9031	0.918688
$796$	0	0
$797$	30.4396	1.07823	0.539113	−	0.842233i	$-0.318760\pi$
$797$	30.4396	1.07823	0.539113	+	0.842233i	$0.318760\pi$
$798$	0	0
$799$	−29.4236	−1.04093
$800$	0	0
$801$	−2.47120	−0.0873156
$802$	0	0
$803$	−2.98000	−0.105162
$804$	0	0
$805$	−106.589	−3.75675
$806$	0	0
$807$	9.85139	0.346785
$808$	0	0
$809$	31.9492	1.12327	0.561637	−	0.827384i	$-0.310172\pi$
$809$	31.9492	1.12327	0.561637	+	0.827384i	$0.310172\pi$
$810$	0	0
$811$	−31.2468	−1.09722	−0.548612	−	0.836077i	$-0.684843\pi$
$811$	−31.2468	−1.09722	−0.548612	+	0.836077i	$0.684843\pi$
$812$	0	0
$813$	13.0694	0.458364
$814$	0	0
$815$	−64.8188	−2.27050
$816$	0	0
$817$	0	0
$818$	0	0
$819$	11.2304	0.392422
$820$	0	0
$821$	24.5387	0.856405	0.428203	−	0.903683i	$-0.359147\pi$
$821$	24.5387	0.856405	0.428203	+	0.903683i	$0.359147\pi$
$822$	0	0
$823$	35.6092	1.24126	0.620630	−	0.784104i	$-0.286877\pi$
$823$	35.6092	1.24126	0.620630	+	0.784104i	$0.286877\pi$
$824$	0	0
$825$	−28.1382	−0.979648
$826$	0	0
$827$	4.91861	0.171037	0.0855184	−	0.996337i	$-0.472745\pi$
$827$	4.91861	0.171037	0.0855184	+	0.996337i	$0.472745\pi$
$828$	0	0
$829$	21.5808	0.749532	0.374766	−	0.927119i	$-0.377723\pi$
$829$	21.5808	0.749532	0.374766	+	0.927119i	$0.377723\pi$
$830$	0	0
$831$	−28.1897	−0.977889
$832$	0	0
$833$	57.0244	1.97578
$834$	0	0
$835$	66.7400	2.30963
$836$	0	0
$837$	−18.3482	−0.634208
$838$	0	0
$839$	51.0772	1.76338	0.881691	−	0.471827i	$-0.156405\pi$
$839$	51.0772	1.76338	0.881691	+	0.471827i	$0.156405\pi$
$840$	0	0
$841$	−18.8107	−0.648645
$842$	0	0
$843$	−0.316921	−0.0109153
$844$	0	0
$845$	−10.8596	−0.373581
$846$	0	0
$847$	26.6357	0.915213
$848$	0	0
$849$	−11.1653	−0.383194
$850$	0	0
$851$	2.58653	0.0886650
$852$	0	0
$853$	−30.2837	−1.03689	−0.518447	−	0.855110i	$-0.673490\pi$
$853$	−30.2837	−1.03689	−0.518447	+	0.855110i	$0.673490\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−39.2037	−1.33917	−0.669586	−	0.742735i	$-0.733528\pi$
$857$	−39.2037	−1.33917	−0.669586	+	0.742735i	$0.733528\pi$
$858$	0	0
$859$	34.6728	1.18302	0.591510	−	0.806298i	$-0.298532\pi$
$859$	34.6728	1.18302	0.591510	+	0.806298i	$0.298532\pi$
$860$	0	0
$861$	−65.3158	−2.22596
$862$	0	0
$863$	1.08878	0.0370624	0.0185312	−	0.999828i	$-0.494101\pi$
$863$	1.08878	0.0370624	0.0185312	+	0.999828i	$0.494101\pi$
$864$	0	0
$865$	75.6144	2.57097
$866$	0	0
$867$	38.2336	1.29848
$868$	0	0
$869$	9.76779	0.331350
$870$	0	0
$871$	−3.55416	−0.120428
$872$	0	0
$873$	−12.7644	−0.432009
$874$	0	0
$875$	−65.6898	−2.22072
$876$	0	0
$877$	13.4413	0.453882	0.226941	−	0.973909i	$-0.427128\pi$
$877$	13.4413	0.453882	0.226941	+	0.973909i	$0.427128\pi$
$878$	0	0
$879$	14.9581	0.504525
$880$	0	0
$881$	−40.0708	−1.35002	−0.675010	−	0.737809i	$-0.735861\pi$
$881$	−40.0708	−1.35002	−0.675010	+	0.737809i	$0.735861\pi$
$882$	0	0
$883$	−37.4146	−1.25910	−0.629551	−	0.776959i	$-0.716761\pi$
$883$	−37.4146	−1.25910	−0.629551	+	0.776959i	$0.716761\pi$
$884$	0	0
$885$	56.8441	1.91079
$886$	0	0
$887$	−2.70328	−0.0907674	−0.0453837	−	0.998970i	$-0.514451\pi$
$887$	−2.70328	−0.0907674	−0.0453837	+	0.998970i	$0.514451\pi$
$888$	0	0
$889$	58.6923	1.96848
$890$	0	0
$891$	11.4304	0.382934
$892$	0	0
$893$	0	0
$894$	0	0
$895$	10.8184	0.361619
$896$	0	0
$897$	32.8377	1.09642
$898$	0	0
$899$	−10.3649	−0.345688
$900$	0	0
$901$	30.9666	1.03165
$902$	0	0
$903$	25.2523	0.840343
$904$	0	0
$905$	−68.5386	−2.27830
$906$	0	0
$907$	35.8898	1.19170	0.595850	−	0.803096i	$-0.296815\pi$
$907$	35.8898	1.19170	0.595850	+	0.803096i	$0.296815\pi$
$908$	0	0
$909$	−0.0430900	−0.00142921
$910$	0	0
$911$	−27.3039	−0.904617	−0.452309	−	0.891861i	$-0.649399\pi$
$911$	−27.3039	−0.904617	−0.452309	+	0.891861i	$0.649399\pi$
$912$	0	0
$913$	8.36744	0.276922
$914$	0	0
$915$	−38.3575	−1.26806
$916$	0	0
$917$	5.38941	0.177974
$918$	0	0
$919$	29.0584	0.958547	0.479273	−	0.877666i	$-0.340900\pi$
$919$	29.0584	0.958547	0.479273	+	0.877666i	$0.340900\pi$
$920$	0	0
$921$	−5.74261	−0.189226
$922$	0	0
$923$	34.8365	1.14666
$924$	0	0
$925$	3.41483	0.112279
$926$	0	0
$927$	−17.4555	−0.573314
$928$	0	0
$929$	8.78904	0.288359	0.144179	−	0.989552i	$-0.453946\pi$
$929$	8.78904	0.288359	0.144179	+	0.989552i	$0.453946\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	13.7280	0.449433
$934$	0	0
$935$	−51.5747	−1.68667
$936$	0	0
$937$	−28.9746	−0.946560	−0.473280	−	0.880912i	$-0.656930\pi$
$937$	−28.9746	−0.946560	−0.473280	+	0.880912i	$0.656930\pi$
$938$	0	0
$939$	22.2867	0.727298
$940$	0	0
$941$	−59.8921	−1.95243	−0.976213	−	0.216814i	$-0.930433\pi$
$941$	−59.8921	−1.95243	−0.976213	+	0.216814i	$0.930433\pi$
$942$	0	0
$943$	80.7244	2.62875
$944$	0	0
$945$	84.7971	2.75845
$946$	0	0
$947$	−9.88241	−0.321135	−0.160568	−	0.987025i	$-0.551332\pi$
$947$	−9.88241	−0.321135	−0.160568	+	0.987025i	$0.551332\pi$
$948$	0	0
$949$	4.59139	0.149043
$950$	0	0
$951$	26.7210	0.866489
$952$	0	0
$953$	−34.9048	−1.13068	−0.565339	−	0.824859i	$-0.691255\pi$
$953$	−34.9048	−1.13068	−0.565339	+	0.824859i	$0.691255\pi$
$954$	0	0
$955$	59.9974	1.94147
$956$	0	0
$957$	9.57822	0.309620
$958$	0	0
$959$	55.1788	1.78182
$960$	0	0
$961$	−20.4565	−0.659887
$962$	0	0
$963$	−0.494374	−0.0159310
$964$	0	0
$965$	−61.4996	−1.97974
$966$	0	0
$967$	48.5434	1.56105	0.780525	−	0.625125i	$-0.214952\pi$
$967$	48.5434	1.56105	0.780525	+	0.625125i	$0.214952\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−38.2627	−1.22791	−0.613955	−	0.789341i	$-0.710422\pi$
$971$	−38.2627	−1.22791	−0.613955	+	0.789341i	$0.710422\pi$
$972$	0	0
$973$	−10.7658	−0.345137
$974$	0	0
$975$	43.3536	1.38843
$976$	0	0
$977$	28.1346	0.900105	0.450052	−	0.893002i	$-0.351405\pi$
$977$	28.1346	0.900105	0.450052	+	0.893002i	$0.351405\pi$
$978$	0	0
$979$	−5.72915	−0.183104
$980$	0	0
$981$	10.0183	0.319858
$982$	0	0
$983$	−15.3684	−0.490175	−0.245087	−	0.969501i	$-0.578817\pi$
$983$	−15.3684	−0.490175	−0.245087	+	0.969501i	$0.578817\pi$
$984$	0	0
$985$	−28.3032	−0.901814
$986$	0	0
$987$	25.6892	0.817697
$988$	0	0
$989$	−31.2095	−0.992404
$990$	0	0
$991$	−5.33357	−0.169426	−0.0847132	−	0.996405i	$-0.526997\pi$
$991$	−5.33357	−0.169426	−0.0847132	+	0.996405i	$0.526997\pi$
$992$	0	0
$993$	−35.0223	−1.11140
$994$	0	0
$995$	−3.71553	−0.117790
$996$	0	0
$997$	−25.7143	−0.814379	−0.407190	−	0.913344i	$-0.633491\pi$
$997$	−25.7143	−0.814379	−0.407190	+	0.913344i	$0.633491\pi$
$998$	0	0
$999$	−2.05772	−0.0651035

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.u.1.4	yes	6
4.3	odd	2		5776.2.a.bx.1.3		6
19.18	odd	2		2888.2.a.t.1.3	✓	6
76.75	even	2		5776.2.a.bz.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2888.2.a.t.1.3	✓	6	19.18	odd	2
2888.2.a.u.1.4	yes	6	1.1	even	1	trivial
5776.2.a.bx.1.3		6	4.3	odd	2
5776.2.a.bz.1.4		6	76.75	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:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2888.a (trivial)

\(f(q)\)	\(=\)	\(q+1.45214 q^{3} +3.79176 q^{5} -3.95766 q^{7} -0.891299 q^{9} -2.06636 q^{11} +3.18371 q^{13} +5.50615 q^{15} +6.58249 q^{17} -5.74706 q^{21} +7.10283 q^{23} +9.37743 q^{25} -5.65070 q^{27} -3.19207 q^{29} +3.24708 q^{31} -3.00063 q^{33} -15.0065 q^{35} +0.364154 q^{37} +4.62318 q^{39} +11.3651 q^{41} -4.39395 q^{43} -3.37959 q^{45} -4.46998 q^{47} +8.66304 q^{49} +9.55868 q^{51} +4.70439 q^{53} -7.83513 q^{55} +10.3237 q^{59} -6.96631 q^{61} +3.52745 q^{63} +12.0719 q^{65} -1.11636 q^{67} +10.3143 q^{69} +10.9421 q^{71} +1.44215 q^{73} +13.6173 q^{75} +8.17793 q^{77} -4.72706 q^{79} -5.53169 q^{81} -4.04937 q^{83} +24.9592 q^{85} -4.63532 q^{87} +2.77258 q^{89} -12.6000 q^{91} +4.71520 q^{93} +14.3211 q^{97} +1.84174 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} + 2 q^{5} - 2 q^{7} + 9 q^{9} - 5 q^{11} + 10 q^{13} + 4 q^{17} + 23 q^{21} - 15 q^{23} + 12 q^{25} + 18 q^{27} - 7 q^{29} + 5 q^{31} - q^{33} - 38 q^{35} + 17 q^{37} + 37 q^{39} + 4 q^{41}+ \cdots - 47 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 + 2 * q^5 - 2 * q^7 + 9 * q^9 - 5 * q^11 + 10 * q^13 + 4 * q^17 + 23 * q^21 - 15 * q^23 + 12 * q^25 + 18 * q^27 - 7 * q^29 + 5 * q^31 - q^33 - 38 * q^35 + 17 * q^37 + 37 * q^39 + 4 * q^41 - 11 * q^43 + 6 * q^45 + 18 * q^47 + 20 * q^49 + 48 * q^51 - 7 * q^53 - 42 * q^55 + q^59 - 4 * q^61 + 19 * q^63 + 6 * q^65 + 20 * q^67 + 6 * q^69 - 6 * q^71 - 2 * q^73 + 19 * q^75 + 41 * q^77 + 8 * q^79 - 6 * q^81 - 22 * q^83 + 32 * q^85 + 29 * q^87 + 14 * q^89 + 26 * q^91 - 5 * q^93 + 41 * q^97 - 47 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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