LMFDB - Embedded newform 1368.4.a.e.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1368,4,Mod(1,1368)] 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("1368.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1368.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-2,0,-35,0,0,0,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$80.7146128879$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.3221.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 2 $ x^3 - x^2 - 9*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.218090$ of defining polynomial
Character	$\chi$	$=$	1368.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+18.0754 q^{5} +0.213413 q^{7} +3.39329 q^{11} -90.7156 q^{13} +2.59392 q^{17} +19.0000 q^{19} -26.6112 q^{23} +201.720 q^{25} -60.1034 q^{29} -176.070 q^{31} +3.85753 q^{35} -154.115 q^{37} -434.137 q^{41} -365.511 q^{43} -204.021 q^{47} -342.954 q^{49} +135.726 q^{53} +61.3351 q^{55} -759.895 q^{59} +284.941 q^{61} -1639.72 q^{65} +590.922 q^{67} +972.291 q^{71} +368.462 q^{73} +0.724174 q^{77} +204.854 q^{79} +782.229 q^{83} +46.8861 q^{85} -213.620 q^{89} -19.3599 q^{91} +343.433 q^{95} -1219.54 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{5} - 35 q^{7} + 28 q^{11} - 109 q^{13} + 123 q^{17} + 57 q^{19} + 193 q^{23} + 187 q^{25} + 297 q^{29} - 140 q^{31} + 246 q^{35} + 38 q^{37} - 736 q^{41} - 514 q^{43} - 134 q^{47} - 42 q^{49}+ \cdots - 2178 q^{97}+O(q^{100}) $ 3 * q - 2 * q^5 - 35 * q^7 + 28 * q^11 - 109 * q^13 + 123 * q^17 + 57 * q^19 + 193 * q^23 + 187 * q^25 + 297 * q^29 - 140 * q^31 + 246 * q^35 + 38 * q^37 - 736 * q^41 - 514 * q^43 - 134 * q^47 - 42 * q^49 - 311 * q^53 - 130 * q^55 - 199 * q^59 + 56 * q^61 - 1156 * q^65 - 509 * q^67 + 874 * q^71 + 203 * q^73 - 616 * q^77 + 242 * q^79 + 62 * q^83 - 1430 * q^85 - 1764 * q^89 - 687 * q^91 - 38 * q^95 - 2178 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	18.0754	1.61671	0.808356	−	0.588693i	$-0.200357\pi$
$5$	18.0754	1.61671	0.808356	+	0.588693i	$0.200357\pi$
$6$	0	0
$7$	0.213413	0.0115232	0.00576162	−	0.999983i	$-0.498166\pi$
$7$	0.213413	0.0115232	0.00576162	+	0.999983i	$0.498166\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.39329	0.0930106	0.0465053	−	0.998918i	$-0.485192\pi$
$11$	3.39329	0.0930106	0.0465053	+	0.998918i	$0.485192\pi$
$12$	0	0
$13$	−90.7156	−1.93538	−0.967692	−	0.252135i	$-0.918867\pi$
$13$	−90.7156	−1.93538	−0.967692	+	0.252135i	$0.918867\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.59392	0.0370069	0.0185035	−	0.999829i	$-0.494110\pi$
$17$	2.59392	0.0370069	0.0185035	+	0.999829i	$0.494110\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−26.6112	−0.241253	−0.120626	−	0.992698i	$-0.538490\pi$
$23$	−26.6112	−0.241253	−0.120626	+	0.992698i	$0.538490\pi$
$24$	0	0
$25$	201.720	1.61376
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−60.1034	−0.384859	−0.192430	−	0.981311i	$-0.561637\pi$
$29$	−60.1034	−0.384859	−0.192430	+	0.981311i	$0.561637\pi$
$30$	0	0
$31$	−176.070	−1.02010	−0.510050	−	0.860145i	$-0.670373\pi$
$31$	−176.070	−1.02010	−0.510050	+	0.860145i	$0.670373\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.85753	0.0186298
$36$	0	0
$37$	−154.115	−0.684768	−0.342384	−	0.939560i	$-0.611234\pi$
$37$	−154.115	−0.684768	−0.342384	+	0.939560i	$0.611234\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−434.137	−1.65368	−0.826839	−	0.562439i	$-0.809863\pi$
$41$	−434.137	−1.65368	−0.826839	+	0.562439i	$0.809863\pi$
$42$	0	0
$43$	−365.511	−1.29628	−0.648138	−	0.761523i	$-0.724452\pi$
$43$	−365.511	−1.29628	−0.648138	+	0.761523i	$0.724452\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−204.021	−0.633180	−0.316590	−	0.948562i	$-0.602538\pi$
$47$	−204.021	−0.633180	−0.316590	+	0.948562i	$0.602538\pi$
$48$	0	0
$49$	−342.954	−0.999867
$50$	0	0
$51$	0	0
$52$	0	0
$53$	135.726	0.351763	0.175881	−	0.984411i	$-0.443722\pi$
$53$	135.726	0.351763	0.175881	+	0.984411i	$0.443722\pi$
$54$	0	0
$55$	61.3351	0.150371
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−759.895	−1.67678	−0.838389	−	0.545072i	$-0.816502\pi$
$59$	−759.895	−1.67678	−0.838389	+	0.545072i	$0.816502\pi$
$60$	0	0
$61$	284.941	0.598082	0.299041	−	0.954240i	$-0.403333\pi$
$61$	284.941	0.598082	0.299041	+	0.954240i	$0.403333\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1639.72	−3.12896
$66$	0	0
$67$	590.922	1.07750	0.538750	−	0.842465i	$-0.318897\pi$
$67$	590.922	1.07750	0.538750	+	0.842465i	$0.318897\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	972.291	1.62521	0.812604	−	0.582817i	$-0.198050\pi$
$71$	972.291	1.62521	0.812604	+	0.582817i	$0.198050\pi$
$72$	0	0
$73$	368.462	0.590756	0.295378	−	0.955380i	$-0.404554\pi$
$73$	368.462	0.590756	0.295378	+	0.955380i	$0.404554\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.724174	0.00107178
$78$	0	0
$79$	204.854	0.291745	0.145873	−	0.989303i	$-0.453401\pi$
$79$	204.854	0.291745	0.145873	+	0.989303i	$0.453401\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	782.229	1.03447	0.517234	−	0.855844i	$-0.326962\pi$
$83$	782.229	1.03447	0.517234	+	0.855844i	$0.326962\pi$
$84$	0	0
$85$	46.8861	0.0598296
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−213.620	−0.254423	−0.127211	−	0.991876i	$-0.540603\pi$
$89$	−213.620	−0.254423	−0.127211	+	0.991876i	$0.540603\pi$
$90$	0	0
$91$	−19.3599	−0.0223019
$92$	0	0
$93$	0	0
$94$	0	0
$95$	343.433	0.370899
$96$	0	0
$97$	−1219.54	−1.27655	−0.638274	−	0.769809i	$-0.720351\pi$
$97$	−1219.54	−1.27655	−0.638274	+	0.769809i	$0.720351\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	53.9766	0.0531770	0.0265885	−	0.999646i	$-0.491536\pi$
$101$	53.9766	0.0531770	0.0265885	+	0.999646i	$0.491536\pi$
$102$	0	0
$103$	1987.50	1.90130	0.950652	−	0.310260i	$-0.100416\pi$
$103$	1987.50	1.90130	0.950652	+	0.310260i	$0.100416\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1076.92	0.972991	0.486496	−	0.873683i	$-0.338275\pi$
$107$	1076.92	0.972991	0.486496	+	0.873683i	$0.338275\pi$
$108$	0	0
$109$	−1200.57	−1.05499	−0.527494	−	0.849559i	$-0.676868\pi$
$109$	−1200.57	−1.05499	−0.527494	+	0.849559i	$0.676868\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1402.72	−1.16776	−0.583880	−	0.811840i	$-0.698466\pi$
$113$	−1402.72	−1.16776	−0.583880	+	0.811840i	$0.698466\pi$
$114$	0	0
$115$	−481.007	−0.390036
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.553577	0.000426440 0
$120$	0	0
$121$	−1319.49	−0.991349
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1386.75	0.992275
$126$	0	0
$127$	−1055.22	−0.737289	−0.368644	−	0.929570i	$-0.620178\pi$
$127$	−1055.22	−0.737289	−0.368644	+	0.929570i	$0.620178\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1026.98	0.684944	0.342472	−	0.939528i	$-0.388736\pi$
$131$	1026.98	0.684944	0.342472	+	0.939528i	$0.388736\pi$
$132$	0	0
$133$	4.05485	0.00264361
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1728.02	1.07763	0.538813	−	0.842425i	$-0.318873\pi$
$137$	1728.02	1.07763	0.538813	+	0.842425i	$0.318873\pi$
$138$	0	0
$139$	624.188	0.380885	0.190442	−	0.981698i	$-0.439008\pi$
$139$	624.188	0.380885	0.190442	+	0.981698i	$0.439008\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−307.825	−0.180011
$144$	0	0
$145$	−1086.39	−0.622207
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−57.5590	−0.0316471	−0.0158235	−	0.999875i	$-0.505037\pi$
$149$	−57.5590	−0.0316471	−0.0158235	+	0.999875i	$0.505037\pi$
$150$	0	0
$151$	−2567.35	−1.38363	−0.691814	−	0.722076i	$-0.743188\pi$
$151$	−2567.35	−1.38363	−0.691814	+	0.722076i	$0.743188\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−3182.53	−1.64921
$156$	0	0
$157$	−3015.67	−1.53297	−0.766486	−	0.642261i	$-0.777997\pi$
$157$	−3015.67	−1.53297	−0.766486	+	0.642261i	$0.777997\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.67918	−0.00278001
$162$	0	0
$163$	−3331.49	−1.60088	−0.800438	−	0.599416i	$-0.795400\pi$
$163$	−3331.49	−1.60088	−0.800438	+	0.599416i	$0.795400\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2465.56	−1.14246	−0.571229	−	0.820791i	$-0.693533\pi$
$167$	−2465.56	−1.14246	−0.571229	+	0.820791i	$0.693533\pi$
$168$	0	0
$169$	6032.33	2.74571
$170$	0	0
$171$	0	0
$172$	0	0
$173$	549.111	0.241319	0.120659	−	0.992694i	$-0.461499\pi$
$173$	549.111	0.241319	0.120659	+	0.992694i	$0.461499\pi$
$174$	0	0
$175$	43.0498	0.0185958
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4186.74	1.74822	0.874110	−	0.485728i	$-0.161445\pi$
$179$	4186.74	1.74822	0.874110	+	0.485728i	$0.161445\pi$
$180$	0	0
$181$	−3954.07	−1.62378	−0.811890	−	0.583811i	$-0.801561\pi$
$181$	−3954.07	−1.62378	−0.811890	+	0.583811i	$0.801561\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2785.70	−1.10707
$186$	0	0
$187$	8.80193	0.00344204
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1623.62	0.615084	0.307542	−	0.951535i	$-0.400494\pi$
$191$	1623.62	0.615084	0.307542	+	0.951535i	$0.400494\pi$
$192$	0	0
$193$	1817.44	0.677833	0.338917	−	0.940816i	$-0.389940\pi$
$193$	1817.44	0.677833	0.338917	+	0.940816i	$0.389940\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−151.771	−0.0548894	−0.0274447	−	0.999623i	$-0.508737\pi$
$197$	−151.771	−0.0548894	−0.0274447	+	0.999623i	$0.508737\pi$
$198$	0	0
$199$	1229.91	0.438119	0.219060	−	0.975711i	$-0.429701\pi$
$199$	1229.91	0.438119	0.219060	+	0.975711i	$0.429701\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−12.8269	−0.00443483
$204$	0	0
$205$	−7847.20	−2.67352
$206$	0	0
$207$	0	0
$208$	0	0
$209$	64.4726	0.0213381
$210$	0	0
$211$	−2998.12	−0.978194	−0.489097	−	0.872229i	$-0.662674\pi$
$211$	−2998.12	−0.978194	−0.489097	+	0.872229i	$0.662674\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6606.75	−2.09571
$216$	0	0
$217$	−37.5757	−0.0117549
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−235.309	−0.0716226
$222$	0	0
$223$	−781.297	−0.234617	−0.117308	−	0.993096i	$-0.537427\pi$
$223$	−781.297	−0.234617	−0.117308	+	0.993096i	$0.537427\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2695.16	0.788035	0.394017	−	0.919103i	$-0.371085\pi$
$227$	2695.16	0.788035	0.394017	+	0.919103i	$0.371085\pi$
$228$	0	0
$229$	3952.95	1.14069	0.570346	−	0.821405i	$-0.306809\pi$
$229$	3952.95	1.14069	0.570346	+	0.821405i	$0.306809\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4315.28	−1.21332	−0.606659	−	0.794962i	$-0.707491\pi$
$233$	−4315.28	−1.21332	−0.606659	+	0.794962i	$0.707491\pi$
$234$	0	0
$235$	−3687.75	−1.02367
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3808.38	−1.03073	−0.515363	−	0.856972i	$-0.672343\pi$
$239$	−3808.38	−1.03073	−0.515363	+	0.856972i	$0.672343\pi$
$240$	0	0
$241$	4274.22	1.14243	0.571217	−	0.820799i	$-0.306471\pi$
$241$	4274.22	1.14243	0.571217	+	0.820799i	$0.306471\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6199.04	−1.61650
$246$	0	0
$247$	−1723.60	−0.444008
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2552.20	−0.641806	−0.320903	−	0.947112i	$-0.603986\pi$
$251$	−2552.20	−0.641806	−0.320903	+	0.947112i	$0.603986\pi$
$252$	0	0
$253$	−90.2995	−0.0224390
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2099.29	−0.509533	−0.254766	−	0.967003i	$-0.581999\pi$
$257$	−2099.29	−0.509533	−0.254766	+	0.967003i	$0.581999\pi$
$258$	0	0
$259$	−32.8903	−0.00789075
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7691.71	1.80339	0.901695	−	0.432374i	$-0.142324\pi$
$263$	7691.71	1.80339	0.901695	+	0.432374i	$0.142324\pi$
$264$	0	0
$265$	2453.31	0.568700
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1460.77	−0.331096	−0.165548	−	0.986202i	$-0.552939\pi$
$269$	−1460.77	−0.331096	−0.165548	+	0.986202i	$0.552939\pi$
$270$	0	0
$271$	3180.10	0.712832	0.356416	−	0.934327i	$-0.383999\pi$
$271$	3180.10	0.712832	0.356416	+	0.934327i	$0.383999\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	684.495	0.150097
$276$	0	0
$277$	−1227.66	−0.266293	−0.133146	−	0.991096i	$-0.542508\pi$
$277$	−1227.66	−0.266293	−0.133146	+	0.991096i	$0.542508\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6497.13	−1.37931	−0.689655	−	0.724138i	$-0.742238\pi$
$281$	−6497.13	−1.37931	−0.689655	+	0.724138i	$0.742238\pi$
$282$	0	0
$283$	−48.1723	−0.0101185	−0.00505927	−	0.999987i	$-0.501610\pi$
$283$	−48.1723	−0.0101185	−0.00505927	+	0.999987i	$0.501610\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−92.6506	−0.0190557
$288$	0	0
$289$	−4906.27	−0.998630
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1140.74	−0.227449	−0.113724	−	0.993512i	$-0.536278\pi$
$293$	−1140.74	−0.227449	−0.113724	+	0.993512i	$0.536278\pi$
$294$	0	0
$295$	−13735.4	−2.71087
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2414.05	0.466916
$300$	0	0
$301$	−78.0049	−0.0149373
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5150.43	0.966927
$306$	0	0
$307$	−6728.53	−1.25087	−0.625435	−	0.780276i	$-0.715079\pi$
$307$	−6728.53	−1.25087	−0.625435	+	0.780276i	$0.715079\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9758.17	1.77921	0.889606	−	0.456728i	$-0.150979\pi$
$311$	9758.17	1.77921	0.889606	+	0.456728i	$0.150979\pi$
$312$	0	0
$313$	−1660.82	−0.299920	−0.149960	−	0.988692i	$-0.547915\pi$
$313$	−1660.82	−0.299920	−0.149960	+	0.988692i	$0.547915\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6578.84	−1.16563	−0.582814	−	0.812605i	$-0.698049\pi$
$317$	−6578.84	−1.16563	−0.582814	+	0.812605i	$0.698049\pi$
$318$	0	0
$319$	−203.948	−0.0357960
$320$	0	0
$321$	0	0
$322$	0	0
$323$	49.2845	0.00848997
$324$	0	0
$325$	−18299.2	−3.12325
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−43.5407	−0.00729629
$330$	0	0
$331$	−5522.33	−0.917024	−0.458512	−	0.888688i	$-0.651617\pi$
$331$	−5522.33	−0.917024	−0.458512	+	0.888688i	$0.651617\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10681.1	1.74201
$336$	0	0
$337$	1350.63	0.218319	0.109159	−	0.994024i	$-0.465184\pi$
$337$	1350.63	0.218319	0.109159	+	0.994024i	$0.465184\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−597.457	−0.0948800
$342$	0	0
$343$	−146.392	−0.0230450
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7266.15	−1.12411	−0.562057	−	0.827099i	$-0.689990\pi$
$347$	−7266.15	−1.12411	−0.562057	+	0.827099i	$0.689990\pi$
$348$	0	0
$349$	10360.4	1.58905	0.794524	−	0.607233i	$-0.207721\pi$
$349$	10360.4	1.58905	0.794524	+	0.607233i	$0.207721\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4014.20	0.605253	0.302626	−	0.953109i	$-0.402137\pi$
$353$	4014.20	0.605253	0.302626	+	0.953109i	$0.402137\pi$
$354$	0	0
$355$	17574.5	2.62749
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7120.01	−1.04674	−0.523370	−	0.852106i	$-0.675325\pi$
$359$	−7120.01	−1.04674	−0.523370	+	0.852106i	$0.675325\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6660.09	0.955083
$366$	0	0
$367$	12447.9	1.77050	0.885251	−	0.465114i	$-0.153987\pi$
$367$	12447.9	1.77050	0.885251	+	0.465114i	$0.153987\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	28.9658	0.00405345
$372$	0	0
$373$	−2379.11	−0.330257	−0.165128	−	0.986272i	$-0.552804\pi$
$373$	−2379.11	−0.330257	−0.165128	+	0.986272i	$0.552804\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5452.32	0.744850
$378$	0	0
$379$	10559.0	1.43108	0.715541	−	0.698571i	$-0.246180\pi$
$379$	10559.0	1.43108	0.715541	+	0.698571i	$0.246180\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3074.48	−0.410180	−0.205090	−	0.978743i	$-0.565749\pi$
$383$	−3074.48	−0.410180	−0.205090	+	0.978743i	$0.565749\pi$
$384$	0	0
$385$	13.0897	0.00173277
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7437.27	−0.969369	−0.484684	−	0.874689i	$-0.661066\pi$
$389$	−7437.27	−0.969369	−0.484684	+	0.874689i	$0.661066\pi$
$390$	0	0
$391$	−69.0272	−0.00892802
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3702.82	0.471668
$396$	0	0
$397$	−10690.7	−1.35151	−0.675756	−	0.737125i	$-0.736183\pi$
$397$	−10690.7	−1.35151	−0.675756	+	0.737125i	$0.736183\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8536.44	1.06307	0.531533	−	0.847038i	$-0.321616\pi$
$401$	8536.44	1.06307	0.531533	+	0.847038i	$0.321616\pi$
$402$	0	0
$403$	15972.3	1.97428
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−522.959	−0.0636907
$408$	0	0
$409$	5072.21	0.613214	0.306607	−	0.951836i	$-0.400806\pi$
$409$	5072.21	0.613214	0.306607	+	0.951836i	$0.400806\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−162.172	−0.0193219
$414$	0	0
$415$	14139.1	1.67244
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13968.1	1.62861	0.814306	−	0.580436i	$-0.197118\pi$
$419$	13968.1	1.62861	0.814306	+	0.580436i	$0.197118\pi$
$420$	0	0
$421$	14983.2	1.73453	0.867264	−	0.497849i	$-0.165877\pi$
$421$	14983.2	1.73453	0.867264	+	0.497849i	$0.165877\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	523.246	0.0597203
$426$	0	0
$427$	60.8103	0.00689184
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9101.07	1.01713	0.508565	−	0.861023i	$-0.330176\pi$
$431$	9101.07	1.01713	0.508565	+	0.861023i	$0.330176\pi$
$432$	0	0
$433$	−8406.65	−0.933020	−0.466510	−	0.884516i	$-0.654489\pi$
$433$	−8406.65	−0.933020	−0.466510	+	0.884516i	$0.654489\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−505.612	−0.0553472
$438$	0	0
$439$	4687.92	0.509664	0.254832	−	0.966985i	$-0.417980\pi$
$439$	4687.92	0.509664	0.254832	+	0.966985i	$0.417980\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9503.30	−1.01922	−0.509611	−	0.860405i	$-0.670211\pi$
$443$	−9503.30	−1.01922	−0.509611	+	0.860405i	$0.670211\pi$
$444$	0	0
$445$	−3861.26	−0.411329
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2589.35	0.272159	0.136079	−	0.990698i	$-0.456550\pi$
$449$	2589.35	0.272159	0.136079	+	0.990698i	$0.456550\pi$
$450$	0	0
$451$	−1473.15	−0.153810
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−349.939	−0.0360558
$456$	0	0
$457$	−11145.8	−1.14087	−0.570436	−	0.821342i	$-0.693226\pi$
$457$	−11145.8	−1.14087	−0.570436	+	0.821342i	$0.693226\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4634.89	−0.468261	−0.234131	−	0.972205i	$-0.575224\pi$
$461$	−4634.89	−0.468261	−0.234131	+	0.972205i	$0.575224\pi$
$462$	0	0
$463$	8329.82	0.836112	0.418056	−	0.908421i	$-0.362712\pi$
$463$	8329.82	0.836112	0.418056	+	0.908421i	$0.362712\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13878.1	−1.37516	−0.687582	−	0.726107i	$-0.741328\pi$
$467$	−13878.1	−1.37516	−0.687582	+	0.726107i	$0.741328\pi$
$468$	0	0
$469$	126.111	0.0124163
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1240.29	−0.120567
$474$	0	0
$475$	3832.68	0.370222
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15896.9	−1.51639	−0.758194	−	0.652029i	$-0.773918\pi$
$479$	−15896.9	−1.51639	−0.758194	+	0.652029i	$0.773918\pi$
$480$	0	0
$481$	13980.7	1.32529
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−22043.6	−2.06381
$486$	0	0
$487$	−10987.2	−1.02233	−0.511166	−	0.859482i	$-0.670786\pi$
$487$	−10987.2	−1.02233	−0.511166	+	0.859482i	$0.670786\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4422.00	−0.406440	−0.203220	−	0.979133i	$-0.565141\pi$
$491$	−4422.00	−0.406440	−0.203220	+	0.979133i	$0.565141\pi$
$492$	0	0
$493$	−155.903	−0.0142425
$494$	0	0
$495$	0	0
$496$	0	0
$497$	207.500	0.0187277
$498$	0	0
$499$	9203.68	0.825678	0.412839	−	0.910804i	$-0.364537\pi$
$499$	9203.68	0.825678	0.412839	+	0.910804i	$0.364537\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5394.92	0.478226	0.239113	−	0.970992i	$-0.423143\pi$
$503$	5394.92	0.478226	0.239113	+	0.970992i	$0.423143\pi$
$504$	0	0
$505$	975.649	0.0859719
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14409.8	−1.25482	−0.627410	−	0.778689i	$-0.715885\pi$
$509$	−14409.8	−1.25482	−0.627410	+	0.778689i	$0.715885\pi$
$510$	0	0
$511$	78.6347	0.00680742
$512$	0	0
$513$	0	0
$514$	0	0
$515$	35924.9	3.07386
$516$	0	0
$517$	−692.302	−0.0588924
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2249.98	0.189201	0.0946003	−	0.995515i	$-0.469843\pi$
$521$	2249.98	0.189201	0.0946003	+	0.995515i	$0.469843\pi$
$522$	0	0
$523$	−19298.1	−1.61348	−0.806738	−	0.590909i	$-0.798769\pi$
$523$	−19298.1	−1.61348	−0.806738	+	0.590909i	$0.798769\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−456.711	−0.0377507
$528$	0	0
$529$	−11458.8	−0.941797
$530$	0	0
$531$	0	0
$532$	0	0
$533$	39383.0	3.20050
$534$	0	0
$535$	19465.8	1.57305
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1163.75	−0.0929982
$540$	0	0
$541$	−307.192	−0.0244126	−0.0122063	−	0.999926i	$-0.503885\pi$
$541$	−307.192	−0.0244126	−0.0122063	+	0.999926i	$0.503885\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−21700.8	−1.70561
$546$	0	0
$547$	−13468.0	−1.05274	−0.526371	−	0.850255i	$-0.676448\pi$
$547$	−13468.0	−1.05274	−0.526371	+	0.850255i	$0.676448\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1141.96	−0.0882928
$552$	0	0
$553$	43.7186	0.00336185
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3808.21	−0.289693	−0.144846	−	0.989454i	$-0.546269\pi$
$557$	−3808.21	−0.289693	−0.144846	+	0.989454i	$0.546269\pi$
$558$	0	0
$559$	33157.5	2.50879
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22451.9	1.68070	0.840351	−	0.542042i	$-0.182349\pi$
$563$	22451.9	1.68070	0.840351	+	0.542042i	$0.182349\pi$
$564$	0	0
$565$	−25354.7	−1.88793
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21111.7	−1.55545	−0.777723	−	0.628607i	$-0.783625\pi$
$569$	−21111.7	−1.55545	−0.777723	+	0.628607i	$0.783625\pi$
$570$	0	0
$571$	−7839.51	−0.574559	−0.287279	−	0.957847i	$-0.592751\pi$
$571$	−7839.51	−0.574559	−0.287279	+	0.957847i	$0.592751\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5368.01	−0.389324
$576$	0	0
$577$	−2092.09	−0.150944	−0.0754720	−	0.997148i	$-0.524046\pi$
$577$	−2092.09	−0.150944	−0.0754720	+	0.997148i	$0.524046\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	166.938	0.0119204
$582$	0	0
$583$	460.559	0.0327177
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19301.8	1.35719	0.678595	−	0.734513i	$-0.262589\pi$
$587$	19301.8	1.35719	0.678595	+	0.734513i	$0.262589\pi$
$588$	0	0
$589$	−3345.33	−0.234027
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22422.7	1.55276	0.776381	−	0.630264i	$-0.217053\pi$
$593$	22422.7	1.55276	0.776381	+	0.630264i	$0.217053\pi$
$594$	0	0
$595$	10.0061	0.000689431 0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6779.52	0.462443	0.231222	−	0.972901i	$-0.425728\pi$
$599$	6779.52	0.462443	0.231222	+	0.972901i	$0.425728\pi$
$600$	0	0
$601$	13065.1	0.886746	0.443373	−	0.896337i	$-0.353782\pi$
$601$	13065.1	0.886746	0.443373	+	0.896337i	$0.353782\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−23850.2	−1.60273
$606$	0	0
$607$	−17590.3	−1.17622	−0.588112	−	0.808780i	$-0.700128\pi$
$607$	−17590.3	−1.17622	−0.588112	+	0.808780i	$0.700128\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	18507.9	1.22545
$612$	0	0
$613$	10723.1	0.706530	0.353265	−	0.935523i	$-0.385071\pi$
$613$	10723.1	0.706530	0.353265	+	0.935523i	$0.385071\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2406.12	0.156997	0.0784983	−	0.996914i	$-0.474987\pi$
$617$	2406.12	0.156997	0.0784983	+	0.996914i	$0.474987\pi$
$618$	0	0
$619$	5375.50	0.349046	0.174523	−	0.984653i	$-0.444162\pi$
$619$	5375.50	0.349046	0.174523	+	0.984653i	$0.444162\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−45.5893	−0.00293178
$624$	0	0
$625$	−149.015	−0.00953698
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−399.763	−0.0253412
$630$	0	0
$631$	−16557.2	−1.04458	−0.522291	−	0.852767i	$-0.674923\pi$
$631$	−16557.2	−1.04458	−0.522291	+	0.852767i	$0.674923\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19073.5	−1.19198
$636$	0	0
$637$	31111.3	1.93513
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1184.92	0.0730131	0.0365066	−	0.999333i	$-0.488377\pi$
$641$	1184.92	0.0730131	0.0365066	+	0.999333i	$0.488377\pi$
$642$	0	0
$643$	−8126.38	−0.498403	−0.249202	−	0.968452i	$-0.580168\pi$
$643$	−8126.38	−0.498403	−0.249202	+	0.968452i	$0.580168\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16438.6	−0.998868	−0.499434	−	0.866352i	$-0.666459\pi$
$647$	−16438.6	−0.998868	−0.499434	+	0.866352i	$0.666459\pi$
$648$	0	0
$649$	−2578.55	−0.155958
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10561.8	0.632947	0.316473	−	0.948601i	$-0.397501\pi$
$653$	10561.8	0.632947	0.316473	+	0.948601i	$0.397501\pi$
$654$	0	0
$655$	18563.1	1.10736
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28057.4	1.65851	0.829257	−	0.558868i	$-0.188764\pi$
$659$	28057.4	1.65851	0.829257	+	0.558868i	$0.188764\pi$
$660$	0	0
$661$	16123.6	0.948769	0.474385	−	0.880318i	$-0.342671\pi$
$661$	16123.6	0.948769	0.474385	+	0.880318i	$0.342671\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	73.2931	0.00427396
$666$	0	0
$667$	1599.42	0.0928483
$668$	0	0
$669$	0	0
$670$	0	0
$671$	966.889	0.0556279
$672$	0	0
$673$	−31833.0	−1.82329	−0.911643	−	0.410982i	$-0.865186\pi$
$673$	−31833.0	−1.82329	−0.911643	+	0.410982i	$0.865186\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18505.4	1.05055	0.525273	−	0.850934i	$-0.323963\pi$
$677$	18505.4	1.05055	0.525273	+	0.850934i	$0.323963\pi$
$678$	0	0
$679$	−260.266	−0.0147100
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−2660.37	−0.149043	−0.0745213	−	0.997219i	$-0.523743\pi$
$683$	−2660.37	−0.149043	−0.0745213	+	0.997219i	$0.523743\pi$
$684$	0	0
$685$	31234.7	1.74221
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12312.5	−0.680796
$690$	0	0
$691$	23883.5	1.31487	0.657433	−	0.753513i	$-0.271642\pi$
$691$	23883.5	1.31487	0.657433	+	0.753513i	$0.271642\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11282.5	0.615781
$696$	0	0
$697$	−1126.12	−0.0611975
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−25949.8	−1.39816	−0.699079	−	0.715044i	$-0.746406\pi$
$701$	−25949.8	−1.39816	−0.699079	+	0.715044i	$0.746406\pi$
$702$	0	0
$703$	−2928.19	−0.157097
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.5193	0.000612771 0
$708$	0	0
$709$	12320.8	0.652633	0.326316	−	0.945261i	$-0.394193\pi$
$709$	12320.8	0.652633	0.326316	+	0.945261i	$0.394193\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4685.42	0.246102
$714$	0	0
$715$	−5564.06	−0.291026
$716$	0	0
$717$	0	0
$718$	0	0
$719$	27252.6	1.41356	0.706781	−	0.707432i	$-0.250146\pi$
$719$	27252.6	1.41356	0.706781	+	0.707432i	$0.250146\pi$
$720$	0	0
$721$	424.159	0.0219092
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12124.1	−0.621071
$726$	0	0
$727$	24073.8	1.22813	0.614063	−	0.789257i	$-0.289534\pi$
$727$	24073.8	1.22813	0.614063	+	0.789257i	$0.289534\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−948.105	−0.0479712
$732$	0	0
$733$	10232.2	0.515600	0.257800	−	0.966198i	$-0.417002\pi$
$733$	10232.2	0.515600	0.257800	+	0.966198i	$0.417002\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2005.17	0.100219
$738$	0	0
$739$	5327.77	0.265203	0.132602	−	0.991169i	$-0.457667\pi$
$739$	5327.77	0.265203	0.132602	+	0.991169i	$0.457667\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−20416.6	−1.00809	−0.504046	−	0.863677i	$-0.668156\pi$
$743$	−20416.6	−1.00809	−0.504046	+	0.863677i	$0.668156\pi$
$744$	0	0
$745$	−1040.40	−0.0511642
$746$	0	0
$747$	0	0
$748$	0	0
$749$	229.830	0.0112120
$750$	0	0
$751$	11444.8	0.556092	0.278046	−	0.960568i	$-0.410313\pi$
$751$	11444.8	0.556092	0.278046	+	0.960568i	$0.410313\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−46405.8	−2.23693
$756$	0	0
$757$	31158.8	1.49602	0.748009	−	0.663689i	$-0.231010\pi$
$757$	31158.8	1.49602	0.748009	+	0.663689i	$0.231010\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	8933.92	0.425564	0.212782	−	0.977100i	$-0.431748\pi$
$761$	8933.92	0.425564	0.212782	+	0.977100i	$0.431748\pi$
$762$	0	0
$763$	−256.217	−0.0121569
$764$	0	0
$765$	0	0
$766$	0	0
$767$	68934.4	3.24521
$768$	0	0
$769$	29688.6	1.39220	0.696099	−	0.717946i	$-0.254918\pi$
$769$	29688.6	1.39220	0.696099	+	0.717946i	$0.254918\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	35704.1	1.66130	0.830651	−	0.556793i	$-0.187968\pi$
$773$	35704.1	1.66130	0.830651	+	0.556793i	$0.187968\pi$
$774$	0	0
$775$	−35516.8	−1.64620
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−8248.60	−0.379380
$780$	0	0
$781$	3299.27	0.151161
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−54509.4	−2.47838
$786$	0	0
$787$	−26828.3	−1.21515	−0.607577	−	0.794261i	$-0.707858\pi$
$787$	−26828.3	−1.21515	−0.607577	+	0.794261i	$0.707858\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−299.360	−0.0134564
$792$	0	0
$793$	−25848.6	−1.15752
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42689.8	1.89730	0.948650	−	0.316327i	$-0.102450\pi$
$797$	42689.8	1.89730	0.948650	+	0.316327i	$0.102450\pi$
$798$	0	0
$799$	−529.213	−0.0234320
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1250.30	0.0549465
$804$	0	0
$805$	−102.653	−0.00449448
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21486.1	0.933759	0.466880	−	0.884321i	$-0.345378\pi$
$809$	21486.1	0.933759	0.466880	+	0.884321i	$0.345378\pi$
$810$	0	0
$811$	3074.32	0.133112	0.0665560	−	0.997783i	$-0.478799\pi$
$811$	3074.32	0.133112	0.0665560	+	0.997783i	$0.478799\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−60218.1	−2.58816
$816$	0	0
$817$	−6944.70	−0.297386
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32460.2	1.37987	0.689933	−	0.723873i	$-0.257640\pi$
$821$	32460.2	1.37987	0.689933	+	0.723873i	$0.257640\pi$
$822$	0	0
$823$	−25382.1	−1.07505	−0.537523	−	0.843249i	$-0.680640\pi$
$823$	−25382.1	−1.07505	−0.537523	+	0.843249i	$0.680640\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9830.51	0.413350	0.206675	−	0.978410i	$-0.433736\pi$
$827$	9830.51	0.413350	0.206675	+	0.978410i	$0.433736\pi$
$828$	0	0
$829$	47474.0	1.98895	0.994476	−	0.104968i	$-0.0334739\pi$
$829$	47474.0	1.98895	0.994476	+	0.104968i	$0.0334739\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−889.596	−0.0370020
$834$	0	0
$835$	−44565.9	−1.84703
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−29050.3	−1.19539	−0.597693	−	0.801725i	$-0.703916\pi$
$839$	−29050.3	−1.19539	−0.597693	+	0.801725i	$0.703916\pi$
$840$	0	0
$841$	−20776.6	−0.851883
$842$	0	0
$843$	0	0
$844$	0	0
$845$	109037.	4.43903
$846$	0	0
$847$	−281.596	−0.0114236
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4101.19	0.165202
$852$	0	0
$853$	21804.2	0.875217	0.437609	−	0.899166i	$-0.355826\pi$
$853$	21804.2	0.875217	0.437609	+	0.899166i	$0.355826\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−30879.5	−1.23083	−0.615416	−	0.788203i	$-0.711012\pi$
$857$	−30879.5	−1.23083	−0.615416	+	0.788203i	$0.711012\pi$
$858$	0	0
$859$	6065.94	0.240940	0.120470	−	0.992717i	$-0.461560\pi$
$859$	6065.94	0.240940	0.120470	+	0.992717i	$0.461560\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1527.39	0.0602467	0.0301233	−	0.999546i	$-0.490410\pi$
$863$	1527.39	0.0602467	0.0301233	+	0.999546i	$0.490410\pi$
$864$	0	0
$865$	9925.40	0.390143
$866$	0	0
$867$	0	0
$868$	0	0
$869$	695.129	0.0271354
$870$	0	0
$871$	−53605.8	−2.08538
$872$	0	0
$873$	0	0
$874$	0	0
$875$	295.950	0.0114342
$876$	0	0
$877$	−565.469	−0.0217725	−0.0108863	−	0.999941i	$-0.503465\pi$
$877$	−565.469	−0.0217725	−0.0108863	+	0.999941i	$0.503465\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	28897.8	1.10510	0.552549	−	0.833481i	$-0.313655\pi$
$881$	28897.8	1.10510	0.552549	+	0.833481i	$0.313655\pi$
$882$	0	0
$883$	26073.0	0.993687	0.496844	−	0.867840i	$-0.334492\pi$
$883$	26073.0	0.993687	0.496844	+	0.867840i	$0.334492\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	859.344	0.0325298	0.0162649	−	0.999868i	$-0.494822\pi$
$887$	859.344	0.0325298	0.0162649	+	0.999868i	$0.494822\pi$
$888$	0	0
$889$	−225.198	−0.00849596
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3876.39	−0.145261
$894$	0	0
$895$	75676.9	2.82637
$896$	0	0
$897$	0	0
$898$	0	0
$899$	10582.4	0.392595
$900$	0	0
$901$	352.063	0.0130177
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−71471.5	−2.62519
$906$	0	0
$907$	36272.4	1.32790	0.663951	−	0.747776i	$-0.268878\pi$
$907$	36272.4	1.32790	0.663951	+	0.747776i	$0.268878\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20484.7	0.744993	0.372497	−	0.928034i	$-0.378502\pi$
$911$	20484.7	0.744993	0.372497	+	0.928034i	$0.378502\pi$
$912$	0	0
$913$	2654.33	0.0962165
$914$	0	0
$915$	0	0
$916$	0	0
$917$	219.171	0.00789277
$918$	0	0
$919$	6354.25	0.228082	0.114041	−	0.993476i	$-0.463620\pi$
$919$	6354.25	0.228082	0.114041	+	0.993476i	$0.463620\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−88202.0	−3.14540
$924$	0	0
$925$	−31088.2	−1.10505
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−33316.3	−1.17661	−0.588307	−	0.808638i	$-0.700205\pi$
$929$	−33316.3	−1.17661	−0.588307	+	0.808638i	$0.700205\pi$
$930$	0	0
$931$	−6516.13	−0.229385
$932$	0	0
$933$	0	0
$934$	0	0
$935$	159.098	0.00556478
$936$	0	0
$937$	−6181.13	−0.215506	−0.107753	−	0.994178i	$-0.534365\pi$
$937$	−6181.13	−0.215506	−0.107753	+	0.994178i	$0.534365\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−609.315	−0.0211085	−0.0105543	−	0.999944i	$-0.503360\pi$
$941$	−609.315	−0.0211085	−0.0105543	+	0.999944i	$0.503360\pi$
$942$	0	0
$943$	11552.9	0.398954
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9348.71	0.320794	0.160397	−	0.987053i	$-0.448722\pi$
$947$	9348.71	0.320794	0.160397	+	0.987053i	$0.448722\pi$
$948$	0	0
$949$	−33425.2	−1.14334
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36990.5	1.25734	0.628668	−	0.777674i	$-0.283601\pi$
$953$	36990.5	1.25734	0.628668	+	0.777674i	$0.283601\pi$
$954$	0	0
$955$	29347.6	0.994414
$956$	0	0
$957$	0	0
$958$	0	0
$959$	368.783	0.0124177
$960$	0	0
$961$	1209.59	0.0406027
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32850.9	1.09586
$966$	0	0
$967$	−48806.2	−1.62306	−0.811532	−	0.584308i	$-0.801366\pi$
$967$	−48806.2	−1.62306	−0.811532	+	0.584308i	$0.801366\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−401.490	−0.0132692	−0.00663462	−	0.999978i	$-0.502112\pi$
$971$	−401.490	−0.0132692	−0.00663462	+	0.999978i	$0.502112\pi$
$972$	0	0
$973$	133.210	0.00438903
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6786.27	−0.222223	−0.111112	−	0.993808i	$-0.535441\pi$
$977$	−6786.27	−0.222223	−0.111112	+	0.993808i	$0.535441\pi$
$978$	0	0
$979$	−724.874	−0.0236640
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17163.1	0.556884	0.278442	−	0.960453i	$-0.410182\pi$
$983$	17163.1	0.556884	0.278442	+	0.960453i	$0.410182\pi$
$984$	0	0
$985$	−2743.31	−0.0887404
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9726.67	0.312730
$990$	0	0
$991$	9323.61	0.298864	0.149432	−	0.988772i	$-0.452255\pi$
$991$	9323.61	0.298864	0.149432	+	0.988772i	$0.452255\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	22231.0	0.708313
$996$	0	0
$997$	14764.7	0.469011	0.234506	−	0.972115i	$-0.424653\pi$
$997$	14764.7	0.469011	0.234506	+	0.972115i	$0.424653\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.4.a.e.1.3		3
3.2	odd	2		152.4.a.b.1.3	✓	3
12.11	even	2		304.4.a.j.1.1		3
24.5	odd	2		1216.4.a.x.1.1		3
24.11	even	2		1216.4.a.q.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.b.1.3	✓	3	3.2	odd	2
304.4.a.j.1.1		3	12.11	even	2
1216.4.a.q.1.3		3	24.11	even	2
1216.4.a.x.1.1		3	24.5	odd	2
1368.4.a.e.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1368.a (trivial)

\(f(q)\)	\(=\)	\(q+18.0754 q^{5} +0.213413 q^{7} +3.39329 q^{11} -90.7156 q^{13} +2.59392 q^{17} +19.0000 q^{19} -26.6112 q^{23} +201.720 q^{25} -60.1034 q^{29} -176.070 q^{31} +3.85753 q^{35} -154.115 q^{37} -434.137 q^{41} -365.511 q^{43} -204.021 q^{47} -342.954 q^{49} +135.726 q^{53} +61.3351 q^{55} -759.895 q^{59} +284.941 q^{61} -1639.72 q^{65} +590.922 q^{67} +972.291 q^{71} +368.462 q^{73} +0.724174 q^{77} +204.854 q^{79} +782.229 q^{83} +46.8861 q^{85} -213.620 q^{89} -19.3599 q^{91} +343.433 q^{95} -1219.54 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{5} - 35 q^{7} + 28 q^{11} - 109 q^{13} + 123 q^{17} + 57 q^{19} + 193 q^{23} + 187 q^{25} + 297 q^{29} - 140 q^{31} + 246 q^{35} + 38 q^{37} - 736 q^{41} - 514 q^{43} - 134 q^{47} - 42 q^{49}+ \cdots - 2178 q^{97}+O(q^{100}) \) 3 * q - 2 * q^5 - 35 * q^7 + 28 * q^11 - 109 * q^13 + 123 * q^17 + 57 * q^19 + 193 * q^23 + 187 * q^25 + 297 * q^29 - 140 * q^31 + 246 * q^35 + 38 * q^37 - 736 * q^41 - 514 * q^43 - 134 * q^47 - 42 * q^49 - 311 * q^53 - 130 * q^55 - 199 * q^59 + 56 * q^61 - 1156 * q^65 - 509 * q^67 + 874 * q^71 + 203 * q^73 - 616 * q^77 + 242 * q^79 + 62 * q^83 - 1430 * q^85 - 1764 * q^89 - 687 * q^91 - 38 * q^95 - 2178 * q^97

Introduction

L-functions

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