LMFDB - Embedded newform 2888.2.a.q.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 = [3,0,0,0,0,0,-3,0,-3,0,3,0,0,0,3] 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:	$\Z[a_1, a_2, a_3]$
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			$-1.53209$ 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)

$f(q)$	$=$	$q-1.53209 q^{3} -1.87939 q^{5} +2.06418 q^{7} -0.652704 q^{9} -2.06418 q^{11} +1.53209 q^{13} +2.87939 q^{15} +0.347296 q^{17} -3.16250 q^{21} +2.94356 q^{23} -1.46791 q^{25} +5.59627 q^{27} +10.4757 q^{29} -5.45336 q^{31} +3.16250 q^{33} -3.87939 q^{35} +5.51754 q^{37} -2.34730 q^{39} -9.98545 q^{41} -6.94356 q^{43} +1.22668 q^{45} +5.17024 q^{47} -2.73917 q^{49} -0.532089 q^{51} +7.76651 q^{53} +3.87939 q^{55} -0.263518 q^{59} -10.6186 q^{61} -1.34730 q^{63} -2.87939 q^{65} +7.04189 q^{67} -4.50980 q^{69} -6.46110 q^{71} -15.3746 q^{73} +2.24897 q^{75} -4.26083 q^{77} +12.5963 q^{79} -6.61587 q^{81} +13.0993 q^{83} -0.652704 q^{85} -16.0496 q^{87} -4.59627 q^{89} +3.16250 q^{91} +8.35504 q^{93} +1.73648 q^{97} +1.34730 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{7} - 3 q^{9} + 3 q^{11} + 3 q^{15} - 12 q^{21} - 6 q^{23} - 9 q^{25} + 3 q^{27} + 12 q^{29} - 3 q^{31} + 12 q^{33} - 6 q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 6 q^{43} - 3 q^{45} - 6 q^{47}+ \cdots + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^7 - 3 * q^9 + 3 * q^11 + 3 * q^15 - 12 * q^21 - 6 * q^23 - 9 * q^25 + 3 * q^27 + 12 * q^29 - 3 * q^31 + 12 * q^33 - 6 * q^35 - 6 * q^37 - 6 * q^39 - 12 * q^41 - 6 * q^43 - 3 * q^45 - 6 * q^47 + 6 * q^49 + 3 * q^51 - 12 * q^53 + 6 * q^55 - 6 * q^59 - 12 * q^61 - 3 * q^63 - 3 * q^65 + 18 * q^67 - 15 * q^69 + 18 * q^71 - 24 * q^73 - 6 * q^75 - 27 * q^77 + 24 * q^79 - 9 * q^81 - 15 * q^83 - 3 * q^85 - 21 * q^87 + 12 * q^91 + 3 * 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
$3$	−1.53209	−0.884552	−0.442276	−	0.896879i	$-0.645829\pi$
$3$	−1.53209	−0.884552	−0.442276	+	0.896879i	$0.645829\pi$
$4$	0	0
$5$	−1.87939	−0.840487	−0.420243	−	0.907411i	$-0.638055\pi$
$5$	−1.87939	−0.840487	−0.420243	+	0.907411i	$0.638055\pi$
$6$	0	0
$7$	2.06418	0.780186	0.390093	−	0.920775i	$-0.372443\pi$
$7$	2.06418	0.780186	0.390093	+	0.920775i	$0.372443\pi$
$8$	0	0
$9$	−0.652704	−0.217568
$10$	0	0
$11$	−2.06418	−0.622373	−0.311187	−	0.950349i	$-0.600726\pi$
$11$	−2.06418	−0.622373	−0.311187	+	0.950349i	$0.600726\pi$
$12$	0	0
$13$	1.53209	0.424925	0.212463	−	0.977169i	$-0.431852\pi$
$13$	1.53209	0.424925	0.212463	+	0.977169i	$0.431852\pi$
$14$	0	0
$15$	2.87939	0.743454
$16$	0	0
$17$	0.347296	0.0842317	0.0421159	−	0.999113i	$-0.486590\pi$
$17$	0.347296	0.0842317	0.0421159	+	0.999113i	$0.486590\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−3.16250	−0.690115
$22$	0	0
$23$	2.94356	0.613775	0.306888	−	0.951746i	$-0.400712\pi$
$23$	2.94356	0.613775	0.306888	+	0.951746i	$0.400712\pi$
$24$	0	0
$25$	−1.46791	−0.293582
$26$	0	0
$27$	5.59627	1.07700
$28$	0	0
$29$	10.4757	1.94528	0.972640	−	0.232318i	$-0.0746310\pi$
$29$	10.4757	1.94528	0.972640	+	0.232318i	$0.0746310\pi$
$30$	0	0
$31$	−5.45336	−0.979453	−0.489726	−	0.871876i	$-0.662903\pi$
$31$	−5.45336	−0.979453	−0.489726	+	0.871876i	$0.662903\pi$
$32$	0	0
$33$	3.16250	0.550521
$34$	0	0
$35$	−3.87939	−0.655736
$36$	0	0
$37$	5.51754	0.907078	0.453539	−	0.891236i	$-0.350161\pi$
$37$	5.51754	0.907078	0.453539	+	0.891236i	$0.350161\pi$
$38$	0	0
$39$	−2.34730	−0.375868
$40$	0	0
$41$	−9.98545	−1.55947	−0.779733	−	0.626112i	$-0.784645\pi$
$41$	−9.98545	−1.55947	−0.779733	+	0.626112i	$0.784645\pi$
$42$	0	0
$43$	−6.94356	−1.05888	−0.529442	−	0.848346i	$-0.677599\pi$
$43$	−6.94356	−1.05888	−0.529442	+	0.848346i	$0.677599\pi$
$44$	0	0
$45$	1.22668	0.182863
$46$	0	0
$47$	5.17024	0.754158	0.377079	−	0.926181i	$-0.376929\pi$
$47$	5.17024	0.754158	0.377079	+	0.926181i	$0.376929\pi$
$48$	0	0
$49$	−2.73917	−0.391310
$50$	0	0
$51$	−0.532089	−0.0745073
$52$	0	0
$53$	7.76651	1.06681	0.533406	−	0.845859i	$-0.320912\pi$
$53$	7.76651	1.06681	0.533406	+	0.845859i	$0.320912\pi$
$54$	0	0
$55$	3.87939	0.523096
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.263518	−0.0343072	−0.0171536	−	0.999853i	$-0.505460\pi$
$59$	−0.263518	−0.0343072	−0.0171536	+	0.999853i	$0.505460\pi$
$60$	0	0
$61$	−10.6186	−1.35957	−0.679783	−	0.733413i	$-0.737926\pi$
$61$	−10.6186	−1.35957	−0.679783	+	0.733413i	$0.737926\pi$
$62$	0	0
$63$	−1.34730	−0.169743
$64$	0	0
$65$	−2.87939	−0.357144
$66$	0	0
$67$	7.04189	0.860304	0.430152	−	0.902757i	$-0.358460\pi$
$67$	7.04189	0.860304	0.430152	+	0.902757i	$0.358460\pi$
$68$	0	0
$69$	−4.50980	−0.542916
$70$	0	0
$71$	−6.46110	−0.766792	−0.383396	−	0.923584i	$-0.625246\pi$
$71$	−6.46110	−0.766792	−0.383396	+	0.923584i	$0.625246\pi$
$72$	0	0
$73$	−15.3746	−1.79947	−0.899733	−	0.436442i	$-0.856239\pi$
$73$	−15.3746	−1.79947	−0.899733	+	0.436442i	$0.856239\pi$
$74$	0	0
$75$	2.24897	0.259689
$76$	0	0
$77$	−4.26083	−0.485567
$78$	0	0
$79$	12.5963	1.41719	0.708595	−	0.705615i	$-0.249329\pi$
$79$	12.5963	1.41719	0.708595	+	0.705615i	$0.249329\pi$
$80$	0	0
$81$	−6.61587	−0.735096
$82$	0	0
$83$	13.0993	1.43783	0.718915	−	0.695098i	$-0.244639\pi$
$83$	13.0993	1.43783	0.718915	+	0.695098i	$0.244639\pi$
$84$	0	0
$85$	−0.652704	−0.0707957
$86$	0	0
$87$	−16.0496	−1.72070
$88$	0	0
$89$	−4.59627	−0.487203	−0.243602	−	0.969875i	$-0.578329\pi$
$89$	−4.59627	−0.487203	−0.243602	+	0.969875i	$0.578329\pi$
$90$	0	0
$91$	3.16250	0.331520
$92$	0	0
$93$	8.35504	0.866377
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.73648	0.176313	0.0881565	−	0.996107i	$-0.471902\pi$
$97$	1.73648	0.176313	0.0881565	+	0.996107i	$0.471902\pi$
$98$	0	0
$99$	1.34730	0.135408
$100$	0	0
$101$	−9.98545	−0.993590	−0.496795	−	0.867868i	$-0.665490\pi$
$101$	−9.98545	−0.993590	−0.496795	+	0.867868i	$0.665490\pi$
$102$	0	0
$103$	1.28581	0.126694	0.0633472	−	0.997992i	$-0.479822\pi$
$103$	1.28581	0.126694	0.0633472	+	0.997992i	$0.479822\pi$
$104$	0	0
$105$	5.94356	0.580032
$106$	0	0
$107$	0.674992	0.0652540	0.0326270	−	0.999468i	$-0.489613\pi$
$107$	0.674992	0.0652540	0.0326270	+	0.999468i	$0.489613\pi$
$108$	0	0
$109$	−9.87939	−0.946273	−0.473137	−	0.880989i	$-0.656878\pi$
$109$	−9.87939	−0.946273	−0.473137	+	0.880989i	$0.656878\pi$
$110$	0	0
$111$	−8.45336	−0.802358
$112$	0	0
$113$	−4.61081	−0.433749	−0.216874	−	0.976199i	$-0.569586\pi$
$113$	−4.61081	−0.433749	−0.216874	+	0.976199i	$0.569586\pi$
$114$	0	0
$115$	−5.53209	−0.515870
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	0.716881	0.0657164
$120$	0	0
$121$	−6.73917	−0.612652
$122$	0	0
$123$	15.2986	1.37943
$124$	0	0
$125$	12.1557	1.08724
$126$	0	0
$127$	−9.33544	−0.828386	−0.414193	−	0.910189i	$-0.635936\pi$
$127$	−9.33544	−0.828386	−0.414193	+	0.910189i	$0.635936\pi$
$128$	0	0
$129$	10.6382	0.936637
$130$	0	0
$131$	13.2148	1.15458	0.577292	−	0.816538i	$-0.304109\pi$
$131$	13.2148	1.15458	0.577292	+	0.816538i	$0.304109\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−10.5175	−0.905206
$136$	0	0
$137$	−21.8794	−1.86928	−0.934641	−	0.355593i	$-0.884279\pi$
$137$	−21.8794	−1.86928	−0.934641	+	0.355593i	$0.884279\pi$
$138$	0	0
$139$	−20.5963	−1.74695	−0.873476	−	0.486867i	$-0.838140\pi$
$139$	−20.5963	−1.74695	−0.873476	+	0.486867i	$0.838140\pi$
$140$	0	0
$141$	−7.92127	−0.667092
$142$	0	0
$143$	−3.16250	−0.264462
$144$	0	0
$145$	−19.6878	−1.63498
$146$	0	0
$147$	4.19665	0.346134
$148$	0	0
$149$	8.14290	0.667093	0.333546	−	0.942734i	$-0.391755\pi$
$149$	8.14290	0.667093	0.333546	+	0.942734i	$0.391755\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	−0.226682	−0.0183261
$154$	0	0
$155$	10.2490	0.823217
$156$	0	0
$157$	−5.39693	−0.430722	−0.215361	−	0.976535i	$-0.569093\pi$
$157$	−5.39693	−0.430722	−0.215361	+	0.976535i	$0.569093\pi$
$158$	0	0
$159$	−11.8990	−0.943651
$160$	0	0
$161$	6.07604	0.478859
$162$	0	0
$163$	−22.4884	−1.76143	−0.880715	−	0.473646i	$-0.842938\pi$
$163$	−22.4884	−1.76143	−0.880715	+	0.473646i	$0.842938\pi$
$164$	0	0
$165$	−5.94356	−0.462706
$166$	0	0
$167$	17.7219	1.37136	0.685682	−	0.727901i	$-0.259504\pi$
$167$	17.7219	1.37136	0.685682	+	0.727901i	$0.259504\pi$
$168$	0	0
$169$	−10.6527	−0.819439
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−19.1702	−1.45749	−0.728743	−	0.684787i	$-0.759895\pi$
$173$	−19.1702	−1.45749	−0.728743	+	0.684787i	$0.759895\pi$
$174$	0	0
$175$	−3.03003	−0.229049
$176$	0	0
$177$	0.403733	0.0303465
$178$	0	0
$179$	4.54664	0.339832	0.169916	−	0.985459i	$-0.445650\pi$
$179$	4.54664	0.339832	0.169916	+	0.985459i	$0.445650\pi$
$180$	0	0
$181$	−15.1702	−1.12760	−0.563798	−	0.825913i	$-0.690660\pi$
$181$	−15.1702	−1.12760	−0.563798	+	0.825913i	$0.690660\pi$
$182$	0	0
$183$	16.2686	1.20261
$184$	0	0
$185$	−10.3696	−0.762387
$186$	0	0
$187$	−0.716881	−0.0524236
$188$	0	0
$189$	11.5517	0.840262
$190$	0	0
$191$	−13.6459	−0.987382	−0.493691	−	0.869637i	$-0.664353\pi$
$191$	−13.6459	−0.987382	−0.493691	+	0.869637i	$0.664353\pi$
$192$	0	0
$193$	−9.33544	−0.671979	−0.335990	−	0.941866i	$-0.609071\pi$
$193$	−9.33544	−0.671979	−0.335990	+	0.941866i	$0.609071\pi$
$194$	0	0
$195$	4.41147	0.315912
$196$	0	0
$197$	4.64590	0.331006	0.165503	−	0.986209i	$-0.447075\pi$
$197$	4.64590	0.331006	0.165503	+	0.986209i	$0.447075\pi$
$198$	0	0
$199$	0.431074	0.0305581	0.0152790	−	0.999883i	$-0.495136\pi$
$199$	0.431074	0.0305581	0.0152790	+	0.999883i	$0.495136\pi$
$200$	0	0
$201$	−10.7888	−0.760983
$202$	0	0
$203$	21.6236	1.51768
$204$	0	0
$205$	18.7665	1.31071
$206$	0	0
$207$	−1.92127	−0.133538
$208$	0	0
$209$	0	0
$210$	0	0
$211$	17.3824	1.19665	0.598326	−	0.801253i	$-0.295833\pi$
$211$	17.3824	1.19665	0.598326	+	0.801253i	$0.295833\pi$
$212$	0	0
$213$	9.89899	0.678267
$214$	0	0
$215$	13.0496	0.889977
$216$	0	0
$217$	−11.2567	−0.764155
$218$	0	0
$219$	23.5553	1.59172
$220$	0	0
$221$	0.532089	0.0357922
$222$	0	0
$223$	−13.2395	−0.886581	−0.443290	−	0.896378i	$-0.646189\pi$
$223$	−13.2395	−0.886581	−0.443290	+	0.896378i	$0.646189\pi$
$224$	0	0
$225$	0.958111	0.0638741
$226$	0	0
$227$	−7.77425	−0.515995	−0.257998	−	0.966146i	$-0.583063\pi$
$227$	−7.77425	−0.515995	−0.257998	+	0.966146i	$0.583063\pi$
$228$	0	0
$229$	2.73917	0.181009	0.0905047	−	0.995896i	$-0.471152\pi$
$229$	2.73917	0.181009	0.0905047	+	0.995896i	$0.471152\pi$
$230$	0	0
$231$	6.52797	0.429509
$232$	0	0
$233$	−1.87939	−0.123123	−0.0615613	−	0.998103i	$-0.519608\pi$
$233$	−1.87939	−0.123123	−0.0615613	+	0.998103i	$0.519608\pi$
$234$	0	0
$235$	−9.71688	−0.633859
$236$	0	0
$237$	−19.2986	−1.25358
$238$	0	0
$239$	−18.8033	−1.21629	−0.608144	−	0.793827i	$-0.708086\pi$
$239$	−18.8033	−1.21629	−0.608144	+	0.793827i	$0.708086\pi$
$240$	0	0
$241$	−11.6313	−0.749241	−0.374621	−	0.927178i	$-0.622227\pi$
$241$	−11.6313	−0.749241	−0.374621	+	0.927178i	$0.622227\pi$
$242$	0	0
$243$	−6.65270	−0.426771
$244$	0	0
$245$	5.14796	0.328891
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−20.0692	−1.27184
$250$	0	0
$251$	−8.09152	−0.510732	−0.255366	−	0.966844i	$-0.582196\pi$
$251$	−8.09152	−0.510732	−0.255366	+	0.966844i	$0.582196\pi$
$252$	0	0
$253$	−6.07604	−0.381997
$254$	0	0
$255$	1.00000	0.0626224
$256$	0	0
$257$	−9.94862	−0.620578	−0.310289	−	0.950642i	$-0.600426\pi$
$257$	−9.94862	−0.620578	−0.310289	+	0.950642i	$0.600426\pi$
$258$	0	0
$259$	11.3892	0.707690
$260$	0	0
$261$	−6.83750	−0.423230
$262$	0	0
$263$	16.2814	1.00395	0.501976	−	0.864882i	$-0.332607\pi$
$263$	16.2814	1.00395	0.501976	+	0.864882i	$0.332607\pi$
$264$	0	0
$265$	−14.5963	−0.896642
$266$	0	0
$267$	7.04189	0.430957
$268$	0	0
$269$	−14.2422	−0.868360	−0.434180	−	0.900826i	$-0.642962\pi$
$269$	−14.2422	−0.868360	−0.434180	+	0.900826i	$0.642962\pi$
$270$	0	0
$271$	25.4807	1.54784	0.773921	−	0.633282i	$-0.218293\pi$
$271$	25.4807	1.54784	0.773921	+	0.633282i	$0.218293\pi$
$272$	0	0
$273$	−4.84524	−0.293247
$274$	0	0
$275$	3.03003	0.182718
$276$	0	0
$277$	11.2567	0.676350	0.338175	−	0.941083i	$-0.390190\pi$
$277$	11.2567	0.676350	0.338175	+	0.941083i	$0.390190\pi$
$278$	0	0
$279$	3.55943	0.213098
$280$	0	0
$281$	2.28817	0.136501	0.0682504	−	0.997668i	$-0.478258\pi$
$281$	2.28817	0.136501	0.0682504	+	0.997668i	$0.478258\pi$
$282$	0	0
$283$	−7.78106	−0.462536	−0.231268	−	0.972890i	$-0.574287\pi$
$283$	−7.78106	−0.462536	−0.231268	+	0.972890i	$0.574287\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−20.6117	−1.21667
$288$	0	0
$289$	−16.8794	−0.992905
$290$	0	0
$291$	−2.66044	−0.155958
$292$	0	0
$293$	−4.09327	−0.239132	−0.119566	−	0.992826i	$-0.538150\pi$
$293$	−4.09327	−0.239132	−0.119566	+	0.992826i	$0.538150\pi$
$294$	0	0
$295$	0.495252	0.0288347
$296$	0	0
$297$	−11.5517	−0.670297
$298$	0	0
$299$	4.50980	0.260808
$300$	0	0
$301$	−14.3327	−0.826126
$302$	0	0
$303$	15.2986	0.878882
$304$	0	0
$305$	19.9564	1.14270
$306$	0	0
$307$	−17.0496	−0.973074	−0.486537	−	0.873660i	$-0.661740\pi$
$307$	−17.0496	−0.973074	−0.486537	+	0.873660i	$0.661740\pi$
$308$	0	0
$309$	−1.96997	−0.112068
$310$	0	0
$311$	18.0642	1.02433	0.512163	−	0.858888i	$-0.328844\pi$
$311$	18.0642	1.02433	0.512163	+	0.858888i	$0.328844\pi$
$312$	0	0
$313$	−3.17024	−0.179193	−0.0895964	−	0.995978i	$-0.528558\pi$
$313$	−3.17024	−0.179193	−0.0895964	+	0.995978i	$0.528558\pi$
$314$	0	0
$315$	2.53209	0.142667
$316$	0	0
$317$	−15.3746	−0.863526	−0.431763	−	0.901987i	$-0.642108\pi$
$317$	−15.3746	−0.863526	−0.431763	+	0.901987i	$0.642108\pi$
$318$	0	0
$319$	−21.6236	−1.21069
$320$	0	0
$321$	−1.03415	−0.0577205
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−2.24897	−0.124750
$326$	0	0
$327$	15.1361	0.837028
$328$	0	0
$329$	10.6723	0.588383
$330$	0	0
$331$	25.7101	1.41315	0.706577	−	0.707636i	$-0.250239\pi$
$331$	25.7101	1.41315	0.706577	+	0.707636i	$0.250239\pi$
$332$	0	0
$333$	−3.60132	−0.197351
$334$	0	0
$335$	−13.2344	−0.723074
$336$	0	0
$337$	34.8016	1.89576	0.947882	−	0.318622i	$-0.103220\pi$
$337$	34.8016	1.89576	0.947882	+	0.318622i	$0.103220\pi$
$338$	0	0
$339$	7.06418	0.383673
$340$	0	0
$341$	11.2567	0.609585
$342$	0	0
$343$	−20.1034	−1.08548
$344$	0	0
$345$	8.47565	0.456314
$346$	0	0
$347$	−0.647651	−0.0347677	−0.0173839	−	0.999849i	$-0.505534\pi$
$347$	−0.647651	−0.0347677	−0.0173839	+	0.999849i	$0.505534\pi$
$348$	0	0
$349$	11.0000	0.588817	0.294408	−	0.955680i	$-0.404877\pi$
$349$	11.0000	0.588817	0.294408	+	0.955680i	$0.404877\pi$
$350$	0	0
$351$	8.57398	0.457645
$352$	0	0
$353$	17.3851	0.925314	0.462657	−	0.886537i	$-0.346896\pi$
$353$	17.3851	0.925314	0.462657	+	0.886537i	$0.346896\pi$
$354$	0	0
$355$	12.1429	0.644478
$356$	0	0
$357$	−1.09833	−0.0581296
$358$	0	0
$359$	−18.3337	−0.967615	−0.483807	−	0.875175i	$-0.660746\pi$
$359$	−18.3337	−0.967615	−0.483807	+	0.875175i	$0.660746\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	10.3250	0.541922
$364$	0	0
$365$	28.8949	1.51243
$366$	0	0
$367$	11.8280	0.617417	0.308708	−	0.951157i	$-0.400103\pi$
$367$	11.8280	0.617417	0.308708	+	0.951157i	$0.400103\pi$
$368$	0	0
$369$	6.51754	0.339290
$370$	0	0
$371$	16.0315	0.832312
$372$	0	0
$373$	2.26083	0.117061	0.0585307	−	0.998286i	$-0.481358\pi$
$373$	2.26083	0.117061	0.0585307	+	0.998286i	$0.481358\pi$
$374$	0	0
$375$	−18.6236	−0.961719
$376$	0	0
$377$	16.0496	0.826598
$378$	0	0
$379$	−12.4243	−0.638192	−0.319096	−	0.947722i	$-0.603379\pi$
$379$	−12.4243	−0.638192	−0.319096	+	0.947722i	$0.603379\pi$
$380$	0	0
$381$	14.3027	0.732750
$382$	0	0
$383$	19.3746	0.989998	0.494999	−	0.868894i	$-0.335168\pi$
$383$	19.3746	0.989998	0.494999	+	0.868894i	$0.335168\pi$
$384$	0	0
$385$	8.00774	0.408112
$386$	0	0
$387$	4.53209	0.230379
$388$	0	0
$389$	21.9932	1.11510	0.557550	−	0.830144i	$-0.311742\pi$
$389$	21.9932	1.11510	0.557550	+	0.830144i	$0.311742\pi$
$390$	0	0
$391$	1.02229	0.0516994
$392$	0	0
$393$	−20.2463	−1.02129
$394$	0	0
$395$	−23.6732	−1.19113
$396$	0	0
$397$	−0.135163	−0.00678362	−0.00339181	−	0.999994i	$-0.501080\pi$
$397$	−0.135163	−0.00678362	−0.00339181	+	0.999994i	$0.501080\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	38.2499	1.91011	0.955055	−	0.296430i	$-0.0957962\pi$
$401$	38.2499	1.91011	0.955055	+	0.296430i	$0.0957962\pi$
$402$	0	0
$403$	−8.35504	−0.416194
$404$	0	0
$405$	12.4338	0.617839
$406$	0	0
$407$	−11.3892	−0.564541
$408$	0	0
$409$	−18.5202	−0.915766	−0.457883	−	0.889012i	$-0.651392\pi$
$409$	−18.5202	−0.915766	−0.457883	+	0.889012i	$0.651392\pi$
$410$	0	0
$411$	33.5212	1.65348
$412$	0	0
$413$	−0.543948	−0.0267660
$414$	0	0
$415$	−24.6186	−1.20848
$416$	0	0
$417$	31.5553	1.54527
$418$	0	0
$419$	−20.2567	−0.989605	−0.494803	−	0.869005i	$-0.664760\pi$
$419$	−20.2567	−0.989605	−0.494803	+	0.869005i	$0.664760\pi$
$420$	0	0
$421$	−2.55707	−0.124624	−0.0623119	−	0.998057i	$-0.519847\pi$
$421$	−2.55707	−0.124624	−0.0623119	+	0.998057i	$0.519847\pi$
$422$	0	0
$423$	−3.37464	−0.164080
$424$	0	0
$425$	−0.509800	−0.0247289
$426$	0	0
$427$	−21.9186	−1.06071
$428$	0	0
$429$	4.84524	0.233930
$430$	0	0
$431$	9.69965	0.467215	0.233608	−	0.972331i	$-0.424947\pi$
$431$	9.69965	0.467215	0.233608	+	0.972331i	$0.424947\pi$
$432$	0	0
$433$	−3.43613	−0.165130	−0.0825649	−	0.996586i	$-0.526311\pi$
$433$	−3.43613	−0.165130	−0.0825649	+	0.996586i	$0.526311\pi$
$434$	0	0
$435$	30.1634	1.44623
$436$	0	0
$437$	0	0
$438$	0	0
$439$	38.1215	1.81944	0.909721	−	0.415220i	$-0.136295\pi$
$439$	38.1215	1.81944	0.909721	+	0.415220i	$0.136295\pi$
$440$	0	0
$441$	1.78787	0.0851365
$442$	0	0
$443$	−4.76382	−0.226336	−0.113168	−	0.993576i	$-0.536100\pi$
$443$	−4.76382	−0.226336	−0.113168	+	0.993576i	$0.536100\pi$
$444$	0	0
$445$	8.63816	0.409488
$446$	0	0
$447$	−12.4757	−0.590078
$448$	0	0
$449$	6.29179	0.296928	0.148464	−	0.988918i	$-0.452567\pi$
$449$	6.29179	0.296928	0.148464	+	0.988918i	$0.452567\pi$
$450$	0	0
$451$	20.6117	0.970569
$452$	0	0
$453$	24.5134	1.15174
$454$	0	0
$455$	−5.94356	−0.278639
$456$	0	0
$457$	35.6459	1.66744	0.833722	−	0.552184i	$-0.186205\pi$
$457$	35.6459	1.66744	0.833722	+	0.552184i	$0.186205\pi$
$458$	0	0
$459$	1.94356	0.0907178
$460$	0	0
$461$	−31.2104	−1.45361	−0.726806	−	0.686843i	$-0.758996\pi$
$461$	−31.2104	−1.45361	−0.726806	+	0.686843i	$0.758996\pi$
$462$	0	0
$463$	5.89662	0.274039	0.137020	−	0.990568i	$-0.456248\pi$
$463$	5.89662	0.274039	0.137020	+	0.990568i	$0.456248\pi$
$464$	0	0
$465$	−15.7023	−0.728178
$466$	0	0
$467$	33.4534	1.54804	0.774019	−	0.633163i	$-0.218244\pi$
$467$	33.4534	1.54804	0.774019	+	0.633163i	$0.218244\pi$
$468$	0	0
$469$	14.5357	0.671197
$470$	0	0
$471$	8.26857	0.380996
$472$	0	0
$473$	14.3327	0.659020
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.06923	−0.232104
$478$	0	0
$479$	25.8111	1.17934	0.589669	−	0.807645i	$-0.299258\pi$
$479$	25.8111	1.17934	0.589669	+	0.807645i	$0.299258\pi$
$480$	0	0
$481$	8.45336	0.385440
$482$	0	0
$483$	−9.30903	−0.423575
$484$	0	0
$485$	−3.26352	−0.148189
$486$	0	0
$487$	−33.7992	−1.53159	−0.765795	−	0.643085i	$-0.777654\pi$
$487$	−33.7992	−1.53159	−0.765795	+	0.643085i	$0.777654\pi$
$488$	0	0
$489$	34.4543	1.55808
$490$	0	0
$491$	−6.69553	−0.302165	−0.151082	−	0.988521i	$-0.548276\pi$
$491$	−6.69553	−0.302165	−0.151082	+	0.988521i	$0.548276\pi$
$492$	0	0
$493$	3.63816	0.163854
$494$	0	0
$495$	−2.53209	−0.113809
$496$	0	0
$497$	−13.3369	−0.598240
$498$	0	0
$499$	3.35235	0.150072	0.0750359	−	0.997181i	$-0.476093\pi$
$499$	3.35235	0.150072	0.0750359	+	0.997181i	$0.476093\pi$
$500$	0	0
$501$	−27.1516	−1.21304
$502$	0	0
$503$	−22.7716	−1.01533	−0.507667	−	0.861553i	$-0.669492\pi$
$503$	−22.7716	−1.01533	−0.507667	+	0.861553i	$0.669492\pi$
$504$	0	0
$505$	18.7665	0.835099
$506$	0	0
$507$	16.3209	0.724836
$508$	0	0
$509$	−40.4320	−1.79212	−0.896059	−	0.443936i	$-0.853582\pi$
$509$	−40.4320	−1.79212	−0.896059	+	0.443936i	$0.853582\pi$
$510$	0	0
$511$	−31.7360	−1.40392
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.41653	−0.106485
$516$	0	0
$517$	−10.6723	−0.469367
$518$	0	0
$519$	29.3705	1.28922
$520$	0	0
$521$	−11.6108	−0.508679	−0.254340	−	0.967115i	$-0.581858\pi$
$521$	−11.6108	−0.508679	−0.254340	+	0.967115i	$0.581858\pi$
$522$	0	0
$523$	11.2986	0.494053	0.247027	−	0.969009i	$-0.420546\pi$
$523$	11.2986	0.494053	0.247027	+	0.969009i	$0.420546\pi$
$524$	0	0
$525$	4.64227	0.202605
$526$	0	0
$527$	−1.89393	−0.0825010
$528$	0	0
$529$	−14.3354	−0.623280
$530$	0	0
$531$	0.171999	0.00746413
$532$	0	0
$533$	−15.2986	−0.662656
$534$	0	0
$535$	−1.26857	−0.0548451
$536$	0	0
$537$	−6.96585	−0.300599
$538$	0	0
$539$	5.65413	0.243541
$540$	0	0
$541$	14.3081	0.615153	0.307577	−	0.951523i	$-0.400482\pi$
$541$	14.3081	0.615153	0.307577	+	0.951523i	$0.400482\pi$
$542$	0	0
$543$	23.2422	0.997417
$544$	0	0
$545$	18.5672	0.795330
$546$	0	0
$547$	−35.2942	−1.50907	−0.754535	−	0.656260i	$-0.772137\pi$
$547$	−35.2942	−1.50907	−0.754535	+	0.656260i	$0.772137\pi$
$548$	0	0
$549$	6.93077	0.295798
$550$	0	0
$551$	0	0
$552$	0	0
$553$	26.0009	1.10567
$554$	0	0
$555$	15.8871	0.674371
$556$	0	0
$557$	−29.2772	−1.24052	−0.620258	−	0.784398i	$-0.712972\pi$
$557$	−29.2772	−1.24052	−0.620258	+	0.784398i	$0.712972\pi$
$558$	0	0
$559$	−10.6382	−0.449946
$560$	0	0
$561$	1.09833	0.0463714
$562$	0	0
$563$	11.9358	0.503035	0.251517	−	0.967853i	$-0.419070\pi$
$563$	11.9358	0.503035	0.251517	+	0.967853i	$0.419070\pi$
$564$	0	0
$565$	8.66550	0.364560
$566$	0	0
$567$	−13.6563	−0.573512
$568$	0	0
$569$	−20.5526	−0.861611	−0.430805	−	0.902445i	$-0.641770\pi$
$569$	−20.5526	−0.861611	−0.430805	+	0.902445i	$0.641770\pi$
$570$	0	0
$571$	34.7202	1.45299	0.726497	−	0.687169i	$-0.241147\pi$
$571$	34.7202	1.45299	0.726497	+	0.687169i	$0.241147\pi$
$572$	0	0
$573$	20.9067	0.873391
$574$	0	0
$575$	−4.32089	−0.180194
$576$	0	0
$577$	−9.33511	−0.388626	−0.194313	−	0.980940i	$-0.562248\pi$
$577$	−9.33511	−0.388626	−0.194313	+	0.980940i	$0.562248\pi$
$578$	0	0
$579$	14.3027	0.594401
$580$	0	0
$581$	27.0392	1.12178
$582$	0	0
$583$	−16.0315	−0.663955
$584$	0	0
$585$	1.87939	0.0777030
$586$	0	0
$587$	−32.2053	−1.32926	−0.664628	−	0.747174i	$-0.731410\pi$
$587$	−32.2053	−1.32926	−0.664628	+	0.747174i	$0.731410\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−7.11793	−0.292792
$592$	0	0
$593$	27.7665	1.14023	0.570117	−	0.821564i	$-0.306898\pi$
$593$	27.7665	1.14023	0.570117	+	0.821564i	$0.306898\pi$
$594$	0	0
$595$	−1.34730	−0.0552338
$596$	0	0
$597$	−0.660444	−0.0270302
$598$	0	0
$599$	−24.8999	−1.01738	−0.508692	−	0.860949i	$-0.669871\pi$
$599$	−24.8999	−1.01738	−0.508692	+	0.860949i	$0.669871\pi$
$600$	0	0
$601$	15.7392	0.642014	0.321007	−	0.947077i	$-0.395979\pi$
$601$	15.7392	0.642014	0.321007	+	0.947077i	$0.395979\pi$
$602$	0	0
$603$	−4.59627	−0.187174
$604$	0	0
$605$	12.6655	0.514926
$606$	0	0
$607$	4.16756	0.169156	0.0845779	−	0.996417i	$-0.473046\pi$
$607$	4.16756	0.169156	0.0845779	+	0.996417i	$0.473046\pi$
$608$	0	0
$609$	−33.1293	−1.34247
$610$	0	0
$611$	7.92127	0.320460
$612$	0	0
$613$	14.7124	0.594230	0.297115	−	0.954842i	$-0.403976\pi$
$613$	14.7124	0.594230	0.297115	+	0.954842i	$0.403976\pi$
$614$	0	0
$615$	−28.7520	−1.15939
$616$	0	0
$617$	−31.8512	−1.28228	−0.641141	−	0.767423i	$-0.721539\pi$
$617$	−31.8512	−1.28228	−0.641141	+	0.767423i	$0.721539\pi$
$618$	0	0
$619$	−21.2668	−0.854786	−0.427393	−	0.904066i	$-0.640568\pi$
$619$	−21.2668	−0.854786	−0.427393	+	0.904066i	$0.640568\pi$
$620$	0	0
$621$	16.4730	0.661037
$622$	0	0
$623$	−9.48751	−0.380109
$624$	0	0
$625$	−15.5057	−0.620227
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.91622	0.0764048
$630$	0	0
$631$	−0.741534	−0.0295200	−0.0147600	−	0.999891i	$-0.504698\pi$
$631$	−0.741534	−0.0295200	−0.0147600	+	0.999891i	$0.504698\pi$
$632$	0	0
$633$	−26.6313	−1.05850
$634$	0	0
$635$	17.5449	0.696247
$636$	0	0
$637$	−4.19665	−0.166277
$638$	0	0
$639$	4.21719	0.166829
$640$	0	0
$641$	−24.8253	−0.980541	−0.490270	−	0.871570i	$-0.663102\pi$
$641$	−24.8253	−0.980541	−0.490270	+	0.871570i	$0.663102\pi$
$642$	0	0
$643$	46.4989	1.83374	0.916868	−	0.399190i	$-0.130709\pi$
$643$	46.4989	1.83374	0.916868	+	0.399190i	$0.130709\pi$
$644$	0	0
$645$	−19.9932	−0.787231
$646$	0	0
$647$	−11.2918	−0.443926	−0.221963	−	0.975055i	$-0.571246\pi$
$647$	−11.2918	−0.443926	−0.221963	+	0.975055i	$0.571246\pi$
$648$	0	0
$649$	0.543948	0.0213518
$650$	0	0
$651$	17.2463	0.675935
$652$	0	0
$653$	16.2216	0.634801	0.317401	−	0.948292i	$-0.397190\pi$
$653$	16.2216	0.634801	0.317401	+	0.948292i	$0.397190\pi$
$654$	0	0
$655$	−24.8357	−0.970413
$656$	0	0
$657$	10.0351	0.391506
$658$	0	0
$659$	22.2713	0.867565	0.433783	−	0.901018i	$-0.357179\pi$
$659$	22.2713	0.867565	0.433783	+	0.901018i	$0.357179\pi$
$660$	0	0
$661$	−12.0077	−0.467047	−0.233523	−	0.972351i	$-0.575026\pi$
$661$	−12.0077	−0.467047	−0.233523	+	0.972351i	$0.575026\pi$
$662$	0	0
$663$	−0.815207	−0.0316600
$664$	0	0
$665$	0	0
$666$	0	0
$667$	30.8357	1.19396
$668$	0	0
$669$	20.2841	0.784227
$670$	0	0
$671$	21.9186	0.846158
$672$	0	0
$673$	22.0351	0.849390	0.424695	−	0.905336i	$-0.360381\pi$
$673$	22.0351	0.849390	0.424695	+	0.905336i	$0.360381\pi$
$674$	0	0
$675$	−8.21482	−0.316189
$676$	0	0
$677$	8.07428	0.310320	0.155160	−	0.987889i	$-0.450411\pi$
$677$	8.07428	0.310320	0.155160	+	0.987889i	$0.450411\pi$
$678$	0	0
$679$	3.58441	0.137557
$680$	0	0
$681$	11.9108	0.456425
$682$	0	0
$683$	23.0933	0.883640	0.441820	−	0.897104i	$-0.354333\pi$
$683$	23.0933	0.883640	0.441820	+	0.897104i	$0.354333\pi$
$684$	0	0
$685$	41.1198	1.57111
$686$	0	0
$687$	−4.19665	−0.160112
$688$	0	0
$689$	11.8990	0.453315
$690$	0	0
$691$	−27.7493	−1.05563	−0.527816	−	0.849359i	$-0.676989\pi$
$691$	−27.7493	−1.05563	−0.527816	+	0.849359i	$0.676989\pi$
$692$	0	0
$693$	2.78106	0.105644
$694$	0	0
$695$	38.7083	1.46829
$696$	0	0
$697$	−3.46791	−0.131357
$698$	0	0
$699$	2.87939	0.108908
$700$	0	0
$701$	−3.90941	−0.147657	−0.0738283	−	0.997271i	$-0.523522\pi$
$701$	−3.90941	−0.147657	−0.0738283	+	0.997271i	$0.523522\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	14.8871	0.560682
$706$	0	0
$707$	−20.6117	−0.775185
$708$	0	0
$709$	41.7306	1.56723	0.783613	−	0.621249i	$-0.213375\pi$
$709$	41.7306	1.56723	0.783613	+	0.621249i	$0.213375\pi$
$710$	0	0
$711$	−8.22163	−0.308335
$712$	0	0
$713$	−16.0523	−0.601164
$714$	0	0
$715$	5.94356	0.222277
$716$	0	0
$717$	28.8084	1.07587
$718$	0	0
$719$	−9.75784	−0.363906	−0.181953	−	0.983307i	$-0.558242\pi$
$719$	−9.75784	−0.363906	−0.181953	+	0.983307i	$0.558242\pi$
$720$	0	0
$721$	2.65413	0.0988451
$722$	0	0
$723$	17.8203	0.662743
$724$	0	0
$725$	−15.3773	−0.571100
$726$	0	0
$727$	−30.2044	−1.12022	−0.560109	−	0.828419i	$-0.689241\pi$
$727$	−30.2044	−1.12022	−0.560109	+	0.828419i	$0.689241\pi$
$728$	0	0
$729$	30.0401	1.11260
$730$	0	0
$731$	−2.41147	−0.0891916
$732$	0	0
$733$	−7.52166	−0.277819	−0.138909	−	0.990305i	$-0.544360\pi$
$733$	−7.52166	−0.277819	−0.138909	+	0.990305i	$0.544360\pi$
$734$	0	0
$735$	−7.88713	−0.290921
$736$	0	0
$737$	−14.5357	−0.535430
$738$	0	0
$739$	3.76744	0.138588	0.0692939	−	0.997596i	$-0.477925\pi$
$739$	3.76744	0.138588	0.0692939	+	0.997596i	$0.477925\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26.0770	−0.956671	−0.478336	−	0.878177i	$-0.658760\pi$
$743$	−26.0770	−0.956671	−0.478336	+	0.878177i	$0.658760\pi$
$744$	0	0
$745$	−15.3037	−0.560682
$746$	0	0
$747$	−8.54993	−0.312826
$748$	0	0
$749$	1.39330	0.0509102
$750$	0	0
$751$	−31.7365	−1.15808	−0.579040	−	0.815299i	$-0.696573\pi$
$751$	−31.7365	−1.15808	−0.579040	+	0.815299i	$0.696573\pi$
$752$	0	0
$753$	12.3969	0.451769
$754$	0	0
$755$	30.0702	1.09437
$756$	0	0
$757$	19.3455	0.703126	0.351563	−	0.936164i	$-0.385650\pi$
$757$	19.3455	0.703126	0.351563	+	0.936164i	$0.385650\pi$
$758$	0	0
$759$	9.30903	0.337896
$760$	0	0
$761$	12.1284	0.439653	0.219826	−	0.975539i	$-0.429451\pi$
$761$	12.1284	0.439653	0.219826	+	0.975539i	$0.429451\pi$
$762$	0	0
$763$	−20.3928	−0.738269
$764$	0	0
$765$	0.426022	0.0154029
$766$	0	0
$767$	−0.403733	−0.0145780
$768$	0	0
$769$	−27.1702	−0.979784	−0.489892	−	0.871783i	$-0.662964\pi$
$769$	−27.1702	−0.979784	−0.489892	+	0.871783i	$0.662964\pi$
$770$	0	0
$771$	15.2422	0.548933
$772$	0	0
$773$	13.0496	0.469363	0.234681	−	0.972072i	$-0.424595\pi$
$773$	13.0496	0.469363	0.234681	+	0.972072i	$0.424595\pi$
$774$	0	0
$775$	8.00505	0.287550
$776$	0	0
$777$	−17.4492	−0.625988
$778$	0	0
$779$	0	0
$780$	0	0
$781$	13.3369	0.477231
$782$	0	0
$783$	58.6245	2.09507
$784$	0	0
$785$	10.1429	0.362016
$786$	0	0
$787$	−41.1884	−1.46821	−0.734104	−	0.679037i	$-0.762398\pi$
$787$	−41.1884	−1.46821	−0.734104	+	0.679037i	$0.762398\pi$
$788$	0	0
$789$	−24.9445	−0.888048
$790$	0	0
$791$	−9.51754	−0.338405
$792$	0	0
$793$	−16.2686	−0.577714
$794$	0	0
$795$	22.3628	0.793126
$796$	0	0
$797$	−4.20676	−0.149011	−0.0745055	−	0.997221i	$-0.523738\pi$
$797$	−4.20676	−0.149011	−0.0745055	+	0.997221i	$0.523738\pi$
$798$	0	0
$799$	1.79561	0.0635240
$800$	0	0
$801$	3.00000	0.106000
$802$	0	0
$803$	31.7360	1.11994
$804$	0	0
$805$	−11.4192	−0.402474
$806$	0	0
$807$	21.8203	0.768110
$808$	0	0
$809$	−32.8527	−1.15504	−0.577519	−	0.816377i	$-0.695979\pi$
$809$	−32.8527	−1.15504	−0.577519	+	0.816377i	$0.695979\pi$
$810$	0	0
$811$	−33.8381	−1.18822	−0.594108	−	0.804385i	$-0.702495\pi$
$811$	−33.8381	−1.18822	−0.594108	+	0.804385i	$0.702495\pi$
$812$	0	0
$813$	−39.0387	−1.36915
$814$	0	0
$815$	42.2645	1.48046
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−2.06418	−0.0721282
$820$	0	0
$821$	1.30304	0.0454765	0.0227383	−	0.999741i	$-0.492762\pi$
$821$	1.30304	0.0454765	0.0227383	+	0.999741i	$0.492762\pi$
$822$	0	0
$823$	53.4157	1.86195	0.930977	−	0.365077i	$-0.118957\pi$
$823$	53.4157	1.86195	0.930977	+	0.365077i	$0.118957\pi$
$824$	0	0
$825$	−4.64227	−0.161623
$826$	0	0
$827$	−44.5850	−1.55037	−0.775186	−	0.631732i	$-0.782344\pi$
$827$	−44.5850	−1.55037	−0.775186	+	0.631732i	$0.782344\pi$
$828$	0	0
$829$	1.35410	0.0470300	0.0235150	−	0.999723i	$-0.492514\pi$
$829$	1.35410	0.0470300	0.0235150	+	0.999723i	$0.492514\pi$
$830$	0	0
$831$	−17.2463	−0.598267
$832$	0	0
$833$	−0.951304	−0.0329607
$834$	0	0
$835$	−33.3063	−1.15261
$836$	0	0
$837$	−30.5185	−1.05487
$838$	0	0
$839$	43.0907	1.48766	0.743828	−	0.668371i	$-0.233008\pi$
$839$	43.0907	1.48766	0.743828	+	0.668371i	$0.233008\pi$
$840$	0	0
$841$	80.7393	2.78411
$842$	0	0
$843$	−3.50568	−0.120742
$844$	0	0
$845$	20.0205	0.688727
$846$	0	0
$847$	−13.9108	−0.477982
$848$	0	0
$849$	11.9213	0.409137
$850$	0	0
$851$	16.2412	0.556742
$852$	0	0
$853$	−40.3337	−1.38100	−0.690499	−	0.723333i	$-0.742609\pi$
$853$	−40.3337	−1.38100	−0.690499	+	0.723333i	$0.742609\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.2012	−0.929176	−0.464588	−	0.885527i	$-0.653797\pi$
$857$	−27.2012	−0.929176	−0.464588	+	0.885527i	$0.653797\pi$
$858$	0	0
$859$	−17.1456	−0.585000	−0.292500	−	0.956266i	$-0.594487\pi$
$859$	−17.1456	−0.585000	−0.292500	+	0.956266i	$0.594487\pi$
$860$	0	0
$861$	31.5790	1.07621
$862$	0	0
$863$	38.2235	1.30114	0.650571	−	0.759445i	$-0.274529\pi$
$863$	38.2235	1.30114	0.650571	+	0.759445i	$0.274529\pi$
$864$	0	0
$865$	36.0283	1.22500
$866$	0	0
$867$	25.8607	0.878276
$868$	0	0
$869$	−26.0009	−0.882021
$870$	0	0
$871$	10.7888	0.365565
$872$	0	0
$873$	−1.13341	−0.0383600
$874$	0	0
$875$	25.0915	0.848248
$876$	0	0
$877$	7.01043	0.236725	0.118363	−	0.992970i	$-0.462235\pi$
$877$	7.01043	0.236725	0.118363	+	0.992970i	$0.462235\pi$
$878$	0	0
$879$	6.27126	0.211524
$880$	0	0
$881$	−28.5485	−0.961824	−0.480912	−	0.876769i	$-0.659694\pi$
$881$	−28.5485	−0.961824	−0.480912	+	0.876769i	$0.659694\pi$
$882$	0	0
$883$	15.3878	0.517839	0.258920	−	0.965899i	$-0.416634\pi$
$883$	15.3878	0.517839	0.258920	+	0.965899i	$0.416634\pi$
$884$	0	0
$885$	−0.758770	−0.0255058
$886$	0	0
$887$	−44.9421	−1.50901	−0.754505	−	0.656295i	$-0.772123\pi$
$887$	−44.9421	−1.50901	−0.754505	+	0.656295i	$0.772123\pi$
$888$	0	0
$889$	−19.2700	−0.646295
$890$	0	0
$891$	13.6563	0.457504
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−8.54488	−0.285624
$896$	0	0
$897$	−6.90941	−0.230699
$898$	0	0
$899$	−57.1275	−1.90531
$900$	0	0
$901$	2.69728	0.0898595
$902$	0	0
$903$	21.9590	0.730751
$904$	0	0
$905$	28.5107	0.947729
$906$	0	0
$907$	45.1658	1.49971	0.749853	−	0.661605i	$-0.230124\pi$
$907$	45.1658	1.49971	0.749853	+	0.661605i	$0.230124\pi$
$908$	0	0
$909$	6.51754	0.216173
$910$	0	0
$911$	47.2336	1.56492	0.782460	−	0.622701i	$-0.213965\pi$
$911$	47.2336	1.56492	0.782460	+	0.622701i	$0.213965\pi$
$912$	0	0
$913$	−27.0392	−0.894867
$914$	0	0
$915$	−30.5749	−1.01078
$916$	0	0
$917$	27.2777	0.900790
$918$	0	0
$919$	36.6750	1.20980	0.604898	−	0.796303i	$-0.293214\pi$
$919$	36.6750	1.20980	0.604898	+	0.796303i	$0.293214\pi$
$920$	0	0
$921$	26.1215	0.860734
$922$	0	0
$923$	−9.89899	−0.325829
$924$	0	0
$925$	−8.09926	−0.266302
$926$	0	0
$927$	−0.839251	−0.0275646
$928$	0	0
$929$	−8.95273	−0.293730	−0.146865	−	0.989157i	$-0.546918\pi$
$929$	−8.95273	−0.293730	−0.146865	+	0.989157i	$0.546918\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−27.6759	−0.906069
$934$	0	0
$935$	1.34730	0.0440613
$936$	0	0
$937$	3.60225	0.117680	0.0588402	−	0.998267i	$-0.481260\pi$
$937$	3.60225	0.117680	0.0588402	+	0.998267i	$0.481260\pi$
$938$	0	0
$939$	4.85710	0.158505
$940$	0	0
$941$	26.6432	0.868544	0.434272	−	0.900782i	$-0.357006\pi$
$941$	26.6432	0.868544	0.434272	+	0.900782i	$0.357006\pi$
$942$	0	0
$943$	−29.3928	−0.957162
$944$	0	0
$945$	−21.7101	−0.706229
$946$	0	0
$947$	50.8922	1.65377	0.826887	−	0.562368i	$-0.190110\pi$
$947$	50.8922	1.65377	0.826887	+	0.562368i	$0.190110\pi$
$948$	0	0
$949$	−23.5553	−0.764638
$950$	0	0
$951$	23.5553	0.763833
$952$	0	0
$953$	41.5631	1.34636	0.673180	−	0.739479i	$-0.264928\pi$
$953$	41.5631	1.34636	0.673180	+	0.739479i	$0.264928\pi$
$954$	0	0
$955$	25.6459	0.829882
$956$	0	0
$957$	33.1293	1.07092
$958$	0	0
$959$	−45.1629	−1.45839
$960$	0	0
$961$	−1.26083	−0.0406719
$962$	0	0
$963$	−0.440570	−0.0141972
$964$	0	0
$965$	17.5449	0.564790
$966$	0	0
$967$	−50.6486	−1.62875	−0.814374	−	0.580340i	$-0.802920\pi$
$967$	−50.6486	−1.62875	−0.814374	+	0.580340i	$0.802920\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−14.5594	−0.467234	−0.233617	−	0.972329i	$-0.575056\pi$
$971$	−14.5594	−0.467234	−0.233617	+	0.972329i	$0.575056\pi$
$972$	0	0
$973$	−42.5144	−1.36295
$974$	0	0
$975$	3.44562	0.110348
$976$	0	0
$977$	−13.0310	−0.416897	−0.208449	−	0.978033i	$-0.566841\pi$
$977$	−13.0310	−0.416897	−0.208449	+	0.978033i	$0.566841\pi$
$978$	0	0
$979$	9.48751	0.303222
$980$	0	0
$981$	6.44831	0.205879
$982$	0	0
$983$	9.99413	0.318763	0.159382	−	0.987217i	$-0.449050\pi$
$983$	9.99413	0.318763	0.159382	+	0.987217i	$0.449050\pi$
$984$	0	0
$985$	−8.73143	−0.278206
$986$	0	0
$987$	−16.3509	−0.520455
$988$	0	0
$989$	−20.4388	−0.649917
$990$	0	0
$991$	−52.9130	−1.68084	−0.840419	−	0.541937i	$-0.817691\pi$
$991$	−52.9130	−1.68084	−0.840419	+	0.541937i	$0.817691\pi$
$992$	0	0
$993$	−39.3901	−1.25001
$994$	0	0
$995$	−0.810155	−0.0256836
$996$	0	0
$997$	−41.9486	−1.32853	−0.664263	−	0.747499i	$-0.731254\pi$
$997$	−41.9486	−1.32853	−0.664263	+	0.747499i	$0.731254\pi$
$998$	0	0
$999$	30.8776	0.976925

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.q.1.1		3
4.3	odd	2		5776.2.a.bl.1.3		3
19.6	even	9		152.2.q.a.17.1	yes	6
19.16	even	9		152.2.q.a.9.1	✓	6
19.18	odd	2		2888.2.a.p.1.3		3
76.35	odd	18		304.2.u.d.161.1		6
76.63	odd	18		304.2.u.d.17.1		6
76.75	even	2		5776.2.a.bm.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.q.a.9.1	✓	6	19.16	even	9
152.2.q.a.17.1	yes	6	19.6	even	9
304.2.u.d.17.1		6	76.63	odd	18
304.2.u.d.161.1		6	76.35	odd	18
2888.2.a.p.1.3		3	19.18	odd	2
2888.2.a.q.1.1		3	1.1	even	1	trivial
5776.2.a.bl.1.3		3	4.3	odd	2
5776.2.a.bm.1.1		3	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)

\(f(q)\)	\(=\)	\(q-1.53209 q^{3} -1.87939 q^{5} +2.06418 q^{7} -0.652704 q^{9} -2.06418 q^{11} +1.53209 q^{13} +2.87939 q^{15} +0.347296 q^{17} -3.16250 q^{21} +2.94356 q^{23} -1.46791 q^{25} +5.59627 q^{27} +10.4757 q^{29} -5.45336 q^{31} +3.16250 q^{33} -3.87939 q^{35} +5.51754 q^{37} -2.34730 q^{39} -9.98545 q^{41} -6.94356 q^{43} +1.22668 q^{45} +5.17024 q^{47} -2.73917 q^{49} -0.532089 q^{51} +7.76651 q^{53} +3.87939 q^{55} -0.263518 q^{59} -10.6186 q^{61} -1.34730 q^{63} -2.87939 q^{65} +7.04189 q^{67} -4.50980 q^{69} -6.46110 q^{71} -15.3746 q^{73} +2.24897 q^{75} -4.26083 q^{77} +12.5963 q^{79} -6.61587 q^{81} +13.0993 q^{83} -0.652704 q^{85} -16.0496 q^{87} -4.59627 q^{89} +3.16250 q^{91} +8.35504 q^{93} +1.73648 q^{97} +1.34730 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{7} - 3 q^{9} + 3 q^{11} + 3 q^{15} - 12 q^{21} - 6 q^{23} - 9 q^{25} + 3 q^{27} + 12 q^{29} - 3 q^{31} + 12 q^{33} - 6 q^{35} - 6 q^{37} - 6 q^{39} - 12 q^{41} - 6 q^{43} - 3 q^{45} - 6 q^{47}+ \cdots + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^7 - 3 * q^9 + 3 * q^11 + 3 * q^15 - 12 * q^21 - 6 * q^23 - 9 * q^25 + 3 * q^27 + 12 * q^29 - 3 * q^31 + 12 * q^33 - 6 * q^35 - 6 * q^37 - 6 * q^39 - 12 * q^41 - 6 * q^43 - 3 * q^45 - 6 * q^47 + 6 * q^49 + 3 * q^51 - 12 * q^53 + 6 * q^55 - 6 * q^59 - 12 * q^61 - 3 * q^63 - 3 * q^65 + 18 * q^67 - 15 * q^69 + 18 * q^71 - 24 * q^73 - 6 * q^75 - 27 * q^77 + 24 * q^79 - 9 * q^81 - 15 * q^83 - 3 * q^85 - 21 * q^87 + 12 * q^91 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more