LMFDB - Embedded newform 1368.4.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,4,Mod(1,1368)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$80.7146128879$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 429x^{3} + 1657x^{2} + 46980x - 289104 $ x^5 - x^4 - 429x^3 + 1657x^2 + 46980*x - 289104
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 456)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-14.4806$ 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)

$f(q)$	$=$	$q+13.4806 q^{5} +33.1709 q^{7} +O(q^{10})$	$q+13.4806 q^{5} +33.1709 q^{7} +11.2817 q^{11} -57.6774 q^{13} -22.2836 q^{17} -19.0000 q^{19} +37.1511 q^{23} +56.7253 q^{25} +185.717 q^{29} +304.292 q^{31} +447.162 q^{35} +301.915 q^{37} -138.414 q^{41} +250.601 q^{43} -64.8079 q^{47} +757.308 q^{49} -707.886 q^{53} +152.083 q^{55} -394.168 q^{59} +294.532 q^{61} -777.524 q^{65} -296.343 q^{67} +528.201 q^{71} +285.570 q^{73} +374.223 q^{77} -581.579 q^{79} +848.567 q^{83} -300.395 q^{85} -1410.25 q^{89} -1913.21 q^{91} -256.131 q^{95} +1561.78 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 6 q^{5} + 10 q^{7}+O(q^{10}) $ 5 * q - 6 * q^5 + 10 * q^7	$ 5 q - 6 q^{5} + 10 q^{7} + 32 q^{11} + 36 q^{13} + 8 q^{17} - 95 q^{19} + 152 q^{23} + 241 q^{25} - 248 q^{29} + 118 q^{31} - 162 q^{35} + 472 q^{37} - 944 q^{41} + 150 q^{43} - 38 q^{47} + 1719 q^{49} - 788 q^{53} + 1270 q^{55} - 396 q^{59} + 1724 q^{61} - 2200 q^{65} + 204 q^{67} - 480 q^{71} + 2608 q^{73} - 2458 q^{77} + 786 q^{79} + 1118 q^{83} + 1798 q^{85} - 792 q^{89} + 1040 q^{91} + 114 q^{95} + 4638 q^{97}+O(q^{100}) $ 5 * q - 6 * q^5 + 10 * q^7 + 32 * q^11 + 36 * q^13 + 8 * q^17 - 95 * q^19 + 152 * q^23 + 241 * q^25 - 248 * q^29 + 118 * q^31 - 162 * q^35 + 472 * q^37 - 944 * q^41 + 150 * q^43 - 38 * q^47 + 1719 * q^49 - 788 * q^53 + 1270 * q^55 - 396 * q^59 + 1724 * q^61 - 2200 * q^65 + 204 * q^67 - 480 * q^71 + 2608 * q^73 - 2458 * q^77 + 786 * q^79 + 1118 * q^83 + 1798 * q^85 - 792 * q^89 + 1040 * q^91 + 114 * q^95 + 4638 * q^97

Display 100 coefficients 10 coefficients

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
$5$	13.4806	1.20574	0.602869	−	0.797840i	$-0.294024\pi$
$5$	13.4806	1.20574	0.602869	+	0.797840i	$0.294024\pi$
$6$	0	0
$7$	33.1709	1.79106	0.895530	−	0.445001i	$-0.146797\pi$
$7$	33.1709	1.79106	0.895530	+	0.445001i	$0.146797\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	11.2817	0.309232	0.154616	−	0.987975i	$-0.450586\pi$
$11$	11.2817	0.309232	0.154616	+	0.987975i	$0.450586\pi$
$12$	0	0
$13$	−57.6774	−1.23053	−0.615263	−	0.788322i	$-0.710950\pi$
$13$	−57.6774	−1.23053	−0.615263	+	0.788322i	$0.710950\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−22.2836	−0.317915	−0.158958	−	0.987285i	$-0.550813\pi$
$17$	−22.2836	−0.317915	−0.158958	+	0.987285i	$0.550813\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	37.1511	0.336806	0.168403	−	0.985718i	$-0.446139\pi$
$23$	37.1511	0.336806	0.168403	+	0.985718i	$0.446139\pi$
$24$	0	0
$25$	56.7253	0.453803
$26$	0	0
$27$	0	0
$28$	0	0
$29$	185.717	1.18920	0.594601	−	0.804021i	$-0.297310\pi$
$29$	185.717	1.18920	0.594601	+	0.804021i	$0.297310\pi$
$30$	0	0
$31$	304.292	1.76298	0.881492	−	0.472200i	$-0.156540\pi$
$31$	304.292	1.76298	0.881492	+	0.472200i	$0.156540\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	447.162	2.15955
$36$	0	0
$37$	301.915	1.34147	0.670736	−	0.741696i	$-0.265979\pi$
$37$	301.915	1.34147	0.670736	+	0.741696i	$0.265979\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−138.414	−0.527233	−0.263617	−	0.964628i	$-0.584915\pi$
$41$	−138.414	−0.527233	−0.263617	+	0.964628i	$0.584915\pi$
$42$	0	0
$43$	250.601	0.888752	0.444376	−	0.895840i	$-0.353425\pi$
$43$	250.601	0.888752	0.444376	+	0.895840i	$0.353425\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−64.8079	−0.201132	−0.100566	−	0.994930i	$-0.532065\pi$
$47$	−64.8079	−0.201132	−0.100566	+	0.994930i	$0.532065\pi$
$48$	0	0
$49$	757.308	2.20790
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−707.886	−1.83463	−0.917317	−	0.398158i	$-0.869650\pi$
$53$	−707.886	−1.83463	−0.917317	+	0.398158i	$0.869650\pi$
$54$	0	0
$55$	152.083	0.372853
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−394.168	−0.869769	−0.434884	−	0.900486i	$-0.643211\pi$
$59$	−394.168	−0.869769	−0.434884	+	0.900486i	$0.643211\pi$
$60$	0	0
$61$	294.532	0.618213	0.309106	−	0.951028i	$-0.399970\pi$
$61$	294.532	0.618213	0.309106	+	0.951028i	$0.399970\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−777.524	−1.48369
$66$	0	0
$67$	−296.343	−0.540358	−0.270179	−	0.962810i	$-0.587083\pi$
$67$	−296.343	−0.540358	−0.270179	+	0.962810i	$0.587083\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	528.201	0.882901	0.441451	−	0.897286i	$-0.354464\pi$
$71$	528.201	0.882901	0.441451	+	0.897286i	$0.354464\pi$
$72$	0	0
$73$	285.570	0.457855	0.228927	−	0.973444i	$-0.426478\pi$
$73$	285.570	0.457855	0.228927	+	0.973444i	$0.426478\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	374.223	0.553853
$78$	0	0
$79$	−581.579	−0.828263	−0.414132	−	0.910217i	$-0.635915\pi$
$79$	−581.579	−0.828263	−0.414132	+	0.910217i	$0.635915\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	848.567	1.12220	0.561098	−	0.827749i	$-0.310379\pi$
$83$	848.567	1.12220	0.561098	+	0.827749i	$0.310379\pi$
$84$	0	0
$85$	−300.395	−0.383322
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1410.25	−1.67962	−0.839811	−	0.542879i	$-0.817334\pi$
$89$	−1410.25	−1.67962	−0.839811	+	0.542879i	$0.817334\pi$
$90$	0	0
$91$	−1913.21	−2.20395
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−256.131	−0.276615
$96$	0	0
$97$	1561.78	1.63479	0.817395	−	0.576078i	$-0.195417\pi$
$97$	1561.78	1.63479	0.817395	+	0.576078i	$0.195417\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1022.66	1.00751	0.503757	−	0.863845i	$-0.331951\pi$
$101$	1022.66	1.00751	0.503757	+	0.863845i	$0.331951\pi$
$102$	0	0
$103$	−152.882	−0.146252	−0.0731259	−	0.997323i	$-0.523298\pi$
$103$	−152.882	−0.146252	−0.0731259	+	0.997323i	$0.523298\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2018.27	−1.82349	−0.911747	−	0.410753i	$-0.865266\pi$
$107$	−2018.27	−1.82349	−0.911747	+	0.410753i	$0.865266\pi$
$108$	0	0
$109$	1687.40	1.48278	0.741392	−	0.671072i	$-0.234166\pi$
$109$	1687.40	1.48278	0.741392	+	0.671072i	$0.234166\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1136.53	0.946154	0.473077	−	0.881021i	$-0.343143\pi$
$113$	1136.53	0.946154	0.473077	+	0.881021i	$0.343143\pi$
$114$	0	0
$115$	500.817	0.406100
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−739.166	−0.569405
$120$	0	0
$121$	−1203.72	−0.904376
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−920.380	−0.658571
$126$	0	0
$127$	2021.04	1.41212	0.706058	−	0.708154i	$-0.250472\pi$
$127$	2021.04	1.41212	0.706058	+	0.708154i	$0.250472\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1533.12	−1.02252	−0.511258	−	0.859427i	$-0.670820\pi$
$131$	−1533.12	−1.02252	−0.511258	+	0.859427i	$0.670820\pi$
$132$	0	0
$133$	−630.247	−0.410897
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1790.08	1.11633	0.558165	−	0.829730i	$-0.311506\pi$
$137$	1790.08	1.11633	0.558165	+	0.829730i	$0.311506\pi$
$138$	0	0
$139$	−1144.57	−0.698425	−0.349212	−	0.937044i	$-0.613551\pi$
$139$	−1144.57	−0.698425	−0.349212	+	0.937044i	$0.613551\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−650.698	−0.380518
$144$	0	0
$145$	2503.57	1.43386
$146$	0	0
$147$	0	0
$148$	0	0
$149$	815.511	0.448384	0.224192	−	0.974545i	$-0.428026\pi$
$149$	815.511	0.448384	0.224192	+	0.974545i	$0.428026\pi$
$150$	0	0
$151$	1797.12	0.968525	0.484263	−	0.874923i	$-0.339088\pi$
$151$	1797.12	0.968525	0.484263	+	0.874923i	$0.339088\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4102.03	2.12569
$156$	0	0
$157$	1864.10	0.947590	0.473795	−	0.880635i	$-0.342884\pi$
$157$	1864.10	0.947590	0.473795	+	0.880635i	$0.342884\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1232.33	0.603240
$162$	0	0
$163$	−8.40262	−0.00403769	−0.00201885	−	0.999998i	$-0.500643\pi$
$163$	−8.40262	−0.00403769	−0.00201885	+	0.999998i	$0.500643\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1408.67	0.652730	0.326365	−	0.945244i	$-0.394176\pi$
$167$	1408.67	0.652730	0.326365	+	0.945244i	$0.394176\pi$
$168$	0	0
$169$	1129.69	0.514195
$170$	0	0
$171$	0	0
$172$	0	0
$173$	431.506	0.189634	0.0948172	−	0.995495i	$-0.469773\pi$
$173$	431.506	0.189634	0.0948172	+	0.995495i	$0.469773\pi$
$174$	0	0
$175$	1881.63	0.812788
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2581.91	−1.07810	−0.539052	−	0.842272i	$-0.681217\pi$
$179$	−2581.91	−1.07810	−0.539052	+	0.842272i	$0.681217\pi$
$180$	0	0
$181$	745.223	0.306033	0.153017	−	0.988224i	$-0.451101\pi$
$181$	745.223	0.306033	0.153017	+	0.988224i	$0.451101\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4069.97	1.61746
$186$	0	0
$187$	−251.396	−0.0983095
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3574.01	1.35396	0.676979	−	0.736002i	$-0.263289\pi$
$191$	3574.01	1.35396	0.676979	+	0.736002i	$0.263289\pi$
$192$	0	0
$193$	−3184.67	−1.18776	−0.593879	−	0.804554i	$-0.702404\pi$
$193$	−3184.67	−1.18776	−0.593879	+	0.804554i	$0.702404\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1380.30	−0.499198	−0.249599	−	0.968349i	$-0.580299\pi$
$197$	−1380.30	−0.499198	−0.249599	+	0.968349i	$0.580299\pi$
$198$	0	0
$199$	−3358.71	−1.19645	−0.598223	−	0.801330i	$-0.704126\pi$
$199$	−3358.71	−1.19645	−0.598223	+	0.801330i	$0.704126\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6160.41	2.12993
$204$	0	0
$205$	−1865.89	−0.635705
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−214.352	−0.0709427
$210$	0	0
$211$	−3488.21	−1.13810	−0.569048	−	0.822304i	$-0.692688\pi$
$211$	−3488.21	−1.13810	−0.569048	+	0.822304i	$0.692688\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3378.24	1.07160
$216$	0	0
$217$	10093.6	3.15761
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1285.26	0.391203
$222$	0	0
$223$	−456.148	−0.136977	−0.0684886	−	0.997652i	$-0.521818\pi$
$223$	−456.148	−0.136977	−0.0684886	+	0.997652i	$0.521818\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2144.22	−0.626947	−0.313473	−	0.949597i	$-0.601493\pi$
$227$	−2144.22	−0.626947	−0.313473	+	0.949597i	$0.601493\pi$
$228$	0	0
$229$	4332.76	1.25029	0.625146	−	0.780508i	$-0.285039\pi$
$229$	4332.76	1.25029	0.625146	+	0.780508i	$0.285039\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1587.60	−0.446383	−0.223192	−	0.974775i	$-0.571648\pi$
$233$	−1587.60	−0.446383	−0.223192	+	0.974775i	$0.571648\pi$
$234$	0	0
$235$	−873.647	−0.242513
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4942.23	−1.33760	−0.668800	−	0.743442i	$-0.733192\pi$
$239$	−4942.23	−1.33760	−0.668800	+	0.743442i	$0.733192\pi$
$240$	0	0
$241$	−2726.78	−0.728827	−0.364413	−	0.931237i	$-0.618730\pi$
$241$	−2726.78	−0.728827	−0.364413	+	0.931237i	$0.618730\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	10208.9	2.66214
$246$	0	0
$247$	1095.87	0.282302
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2418.97	−0.608304	−0.304152	−	0.952624i	$-0.598373\pi$
$251$	−2418.97	−0.608304	−0.304152	+	0.952624i	$0.598373\pi$
$252$	0	0
$253$	419.126	0.104151
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1516.06	−0.367974	−0.183987	−	0.982929i	$-0.558900\pi$
$257$	−1516.06	−0.367974	−0.183987	+	0.982929i	$0.558900\pi$
$258$	0	0
$259$	10014.8	2.40266
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1794.92	0.420835	0.210418	−	0.977612i	$-0.432518\pi$
$263$	1794.92	0.420835	0.210418	+	0.977612i	$0.432518\pi$
$264$	0	0
$265$	−9542.69	−2.21209
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1153.84	0.261527	0.130764	−	0.991414i	$-0.458257\pi$
$269$	1153.84	0.261527	0.130764	+	0.991414i	$0.458257\pi$
$270$	0	0
$271$	6069.12	1.36042	0.680208	−	0.733019i	$-0.261889\pi$
$271$	6069.12	1.36042	0.680208	+	0.733019i	$0.261889\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	639.957	0.140330
$276$	0	0
$277$	3930.37	0.852538	0.426269	−	0.904597i	$-0.359828\pi$
$277$	3930.37	0.852538	0.426269	+	0.904597i	$0.359828\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2396.28	0.508719	0.254359	−	0.967110i	$-0.418135\pi$
$281$	2396.28	0.508719	0.254359	+	0.967110i	$0.418135\pi$
$282$	0	0
$283$	1307.62	0.274663	0.137332	−	0.990525i	$-0.456147\pi$
$283$	1307.62	0.274663	0.137332	+	0.990525i	$0.456147\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4591.30	−0.944306
$288$	0	0
$289$	−4416.44	−0.898930
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8904.31	−1.77541	−0.887706	−	0.460411i	$-0.847702\pi$
$293$	−8904.31	−1.77541	−0.887706	+	0.460411i	$0.847702\pi$
$294$	0	0
$295$	−5313.61	−1.04871
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2142.78	−0.414449
$300$	0	0
$301$	8312.67	1.59181
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3970.46	0.745402
$306$	0	0
$307$	−8886.06	−1.65197	−0.825984	−	0.563694i	$-0.809380\pi$
$307$	−8886.06	−1.65197	−0.825984	+	0.563694i	$0.809380\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8124.81	1.48140	0.740700	−	0.671836i	$-0.234494\pi$
$311$	8124.81	1.48140	0.740700	+	0.671836i	$0.234494\pi$
$312$	0	0
$313$	−6047.61	−1.09211	−0.546056	−	0.837748i	$-0.683872\pi$
$313$	−6047.61	−1.09211	−0.546056	+	0.837748i	$0.683872\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7177.30	1.27166	0.635831	−	0.771828i	$-0.280657\pi$
$317$	7177.30	1.27166	0.635831	+	0.771828i	$0.280657\pi$
$318$	0	0
$319$	2095.20	0.367739
$320$	0	0
$321$	0	0
$322$	0	0
$323$	423.388	0.0729347
$324$	0	0
$325$	−3271.77	−0.558416
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−2149.74	−0.360240
$330$	0	0
$331$	−9108.19	−1.51248	−0.756240	−	0.654294i	$-0.772966\pi$
$331$	−9108.19	−1.51248	−0.756240	+	0.654294i	$0.772966\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3994.86	−0.651530
$336$	0	0
$337$	−97.0153	−0.0156818	−0.00784089	−	0.999969i	$-0.502496\pi$
$337$	−97.0153	−0.0156818	−0.00784089	+	0.999969i	$0.502496\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3432.92	0.545171
$342$	0	0
$343$	13743.0	2.16341
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4211.96	0.651614	0.325807	−	0.945436i	$-0.394364\pi$
$347$	4211.96	0.651614	0.325807	+	0.945436i	$0.394364\pi$
$348$	0	0
$349$	6397.37	0.981213	0.490606	−	0.871381i	$-0.336775\pi$
$349$	6397.37	0.981213	0.490606	+	0.871381i	$0.336775\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3189.40	−0.480891	−0.240445	−	0.970663i	$-0.577294\pi$
$353$	−3189.40	−0.480891	−0.240445	+	0.970663i	$0.577294\pi$
$354$	0	0
$355$	7120.45	1.06455
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3423.06	0.503237	0.251618	−	0.967827i	$-0.419037\pi$
$359$	3423.06	0.503237	0.251618	+	0.967827i	$0.419037\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3849.64	0.552052
$366$	0	0
$367$	−3255.44	−0.463031	−0.231516	−	0.972831i	$-0.574368\pi$
$367$	−3255.44	−0.463031	−0.231516	+	0.972831i	$0.574368\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−23481.2	−3.28594
$372$	0	0
$373$	−3992.57	−0.554229	−0.277114	−	0.960837i	$-0.589378\pi$
$373$	−3992.57	−0.554229	−0.277114	+	0.960837i	$0.589378\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10711.7	−1.46334
$378$	0	0
$379$	8455.38	1.14597	0.572987	−	0.819565i	$-0.305785\pi$
$379$	8455.38	1.14597	0.572987	+	0.819565i	$0.305785\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7443.81	−0.993109	−0.496554	−	0.868006i	$-0.665402\pi$
$383$	−7443.81	−0.993109	−0.496554	+	0.868006i	$0.665402\pi$
$384$	0	0
$385$	5044.74	0.667801
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9477.11	1.23524	0.617620	−	0.786476i	$-0.288097\pi$
$389$	9477.11	1.23524	0.617620	+	0.786476i	$0.288097\pi$
$390$	0	0
$391$	−827.859	−0.107076
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−7840.01	−0.998668
$396$	0	0
$397$	5253.16	0.664103	0.332051	−	0.943261i	$-0.392259\pi$
$397$	5253.16	0.664103	0.332051	+	0.943261i	$0.392259\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2429.55	0.302558	0.151279	−	0.988491i	$-0.451661\pi$
$401$	2429.55	0.302558	0.151279	+	0.988491i	$0.451661\pi$
$402$	0	0
$403$	−17550.8	−2.16940
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3406.10	0.414826
$408$	0	0
$409$	−3968.24	−0.479747	−0.239874	−	0.970804i	$-0.577106\pi$
$409$	−3968.24	−0.479747	−0.239874	+	0.970804i	$0.577106\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13074.9	−1.55781
$414$	0	0
$415$	11439.2	1.35307
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10907.8	−1.27180	−0.635899	−	0.771773i	$-0.719370\pi$
$419$	−10907.8	−1.27180	−0.635899	+	0.771773i	$0.719370\pi$
$420$	0	0
$421$	7789.33	0.901731	0.450865	−	0.892592i	$-0.351115\pi$
$421$	7789.33	0.901731	0.450865	+	0.892592i	$0.351115\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1264.04	−0.144271
$426$	0	0
$427$	9769.89	1.10726
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7051.60	−0.788083	−0.394041	−	0.919093i	$-0.628923\pi$
$431$	−7051.60	−0.788083	−0.394041	+	0.919093i	$0.628923\pi$
$432$	0	0
$433$	10411.1	1.15549	0.577744	−	0.816218i	$-0.303933\pi$
$433$	10411.1	1.15549	0.577744	+	0.816218i	$0.303933\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−705.871	−0.0772686
$438$	0	0
$439$	9759.35	1.06102	0.530510	−	0.847678i	$-0.322000\pi$
$439$	9759.35	1.06102	0.530510	+	0.847678i	$0.322000\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−482.951	−0.0517961	−0.0258980	−	0.999665i	$-0.508245\pi$
$443$	−482.951	−0.0517961	−0.0258980	+	0.999665i	$0.508245\pi$
$444$	0	0
$445$	−19011.0	−2.02518
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14650.7	1.53989	0.769946	−	0.638109i	$-0.220283\pi$
$449$	14650.7	1.53989	0.769946	+	0.638109i	$0.220283\pi$
$450$	0	0
$451$	−1561.54	−0.163037
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−25791.2	−2.65738
$456$	0	0
$457$	19350.6	1.98070	0.990352	−	0.138576i	$-0.0442525\pi$
$457$	19350.6	1.98070	0.990352	+	0.138576i	$0.0442525\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	151.593	0.0153154	0.00765768	−	0.999971i	$-0.497562\pi$
$461$	151.593	0.0153154	0.00765768	+	0.999971i	$0.497562\pi$
$462$	0	0
$463$	455.326	0.0457036	0.0228518	−	0.999739i	$-0.492725\pi$
$463$	455.326	0.0457036	0.0228518	+	0.999739i	$0.492725\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9597.43	0.950998	0.475499	−	0.879716i	$-0.342267\pi$
$467$	9597.43	0.950998	0.475499	+	0.879716i	$0.342267\pi$
$468$	0	0
$469$	−9829.95	−0.967814
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2827.20	0.274830
$474$	0	0
$475$	−1077.78	−0.104109
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10730.3	−1.02355	−0.511775	−	0.859120i	$-0.671012\pi$
$479$	−10730.3	−1.02355	−0.511775	+	0.859120i	$0.671012\pi$
$480$	0	0
$481$	−17413.7	−1.65072
$482$	0	0
$483$	0	0
$484$	0	0
$485$	21053.6	1.97113
$486$	0	0
$487$	553.932	0.0515422	0.0257711	−	0.999668i	$-0.491796\pi$
$487$	553.932	0.0515422	0.0257711	+	0.999668i	$0.491796\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10136.1	0.931643	0.465822	−	0.884879i	$-0.345759\pi$
$491$	10136.1	0.931643	0.465822	+	0.884879i	$0.345759\pi$
$492$	0	0
$493$	−4138.44	−0.378065
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17520.9	1.58133
$498$	0	0
$499$	−6028.65	−0.540841	−0.270420	−	0.962742i	$-0.587163\pi$
$499$	−6028.65	−0.540841	−0.270420	+	0.962742i	$0.587163\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−13203.4	−1.17040	−0.585199	−	0.810889i	$-0.698984\pi$
$503$	−13203.4	−1.17040	−0.585199	+	0.810889i	$0.698984\pi$
$504$	0	0
$505$	13786.1	1.21480
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20939.0	−1.82339	−0.911694	−	0.410870i	$-0.865225\pi$
$509$	−20939.0	−1.82339	−0.911694	+	0.410870i	$0.865225\pi$
$510$	0	0
$511$	9472.60	0.820045
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2060.94	−0.176341
$516$	0	0
$517$	−731.142	−0.0621965
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−11440.3	−0.962009	−0.481004	−	0.876718i	$-0.659728\pi$
$521$	−11440.3	−0.962009	−0.481004	+	0.876718i	$0.659728\pi$
$522$	0	0
$523$	−18885.9	−1.57901	−0.789505	−	0.613745i	$-0.789662\pi$
$523$	−18885.9	−1.57901	−0.789505	+	0.613745i	$0.789662\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6780.71	−0.560479
$528$	0	0
$529$	−10786.8	−0.886562
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7983.34	0.648774
$534$	0	0
$535$	−27207.4	−2.19865
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8543.70	0.682752
$540$	0	0
$541$	13125.1	1.04306	0.521529	−	0.853234i	$-0.325362\pi$
$541$	13125.1	1.04306	0.521529	+	0.853234i	$0.325362\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22747.1	1.78785
$546$	0	0
$547$	−20504.9	−1.60279	−0.801394	−	0.598137i	$-0.795908\pi$
$547$	−20504.9	−1.60279	−0.801394	+	0.598137i	$0.795908\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3528.63	−0.272821
$552$	0	0
$553$	−19291.5	−1.48347
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1392.70	0.105944	0.0529718	−	0.998596i	$-0.483131\pi$
$557$	1392.70	0.105944	0.0529718	+	0.998596i	$0.483131\pi$
$558$	0	0
$559$	−14454.0	−1.09363
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15945.2	1.19362	0.596810	−	0.802382i	$-0.296434\pi$
$563$	15945.2	1.19362	0.596810	+	0.802382i	$0.296434\pi$
$564$	0	0
$565$	15321.0	1.14081
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2793.50	−0.205816	−0.102908	−	0.994691i	$-0.532815\pi$
$569$	−2793.50	−0.205816	−0.102908	+	0.994691i	$0.532815\pi$
$570$	0	0
$571$	10635.8	0.779499	0.389750	−	0.920921i	$-0.372562\pi$
$571$	10635.8	0.779499	0.389750	+	0.920921i	$0.372562\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2107.41	0.152843
$576$	0	0
$577$	2014.27	0.145330	0.0726649	−	0.997356i	$-0.476850\pi$
$577$	2014.27	0.145330	0.0726649	+	0.997356i	$0.476850\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	28147.7	2.00992
$582$	0	0
$583$	−7986.14	−0.567327
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5206.37	0.366082	0.183041	−	0.983105i	$-0.441406\pi$
$587$	5206.37	0.366082	0.183041	+	0.983105i	$0.441406\pi$
$588$	0	0
$589$	−5781.55	−0.404456
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10304.8	0.713607	0.356803	−	0.934179i	$-0.383867\pi$
$593$	10304.8	0.713607	0.356803	+	0.934179i	$0.383867\pi$
$594$	0	0
$595$	−9964.36	−0.686553
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24850.2	−1.69508	−0.847540	−	0.530732i	$-0.821917\pi$
$599$	−24850.2	−1.69508	−0.847540	+	0.530732i	$0.821917\pi$
$600$	0	0
$601$	12850.6	0.872194	0.436097	−	0.899900i	$-0.356360\pi$
$601$	12850.6	0.872194	0.436097	+	0.899900i	$0.356360\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−16226.9	−1.09044
$606$	0	0
$607$	−4115.74	−0.275210	−0.137605	−	0.990487i	$-0.543941\pi$
$607$	−4115.74	−0.275210	−0.137605	+	0.990487i	$0.543941\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3737.96	0.247498
$612$	0	0
$613$	4639.73	0.305705	0.152852	−	0.988249i	$-0.451154\pi$
$613$	4639.73	0.305705	0.152852	+	0.988249i	$0.451154\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10175.0	−0.663903	−0.331952	−	0.943296i	$-0.607707\pi$
$617$	−10175.0	−0.663903	−0.331952	+	0.943296i	$0.607707\pi$
$618$	0	0
$619$	−28253.1	−1.83455	−0.917275	−	0.398255i	$-0.869616\pi$
$619$	−28253.1	−1.83455	−0.917275	+	0.398255i	$0.869616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−46779.3	−3.00830
$624$	0	0
$625$	−19497.9	−1.24787
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−6727.73	−0.426474
$630$	0	0
$631$	−21515.0	−1.35737	−0.678685	−	0.734429i	$-0.737450\pi$
$631$	−21515.0	−1.35737	−0.678685	+	0.734429i	$0.737450\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	27244.8	1.70264
$636$	0	0
$637$	−43679.6	−2.71687
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−7263.54	−0.447570	−0.223785	−	0.974639i	$-0.571841\pi$
$641$	−7263.54	−0.447570	−0.223785	+	0.974639i	$0.571841\pi$
$642$	0	0
$643$	−12913.5	−0.792004	−0.396002	−	0.918250i	$-0.629603\pi$
$643$	−12913.5	−0.792004	−0.396002	+	0.918250i	$0.629603\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30247.7	1.83796	0.918979	−	0.394308i	$-0.129016\pi$
$647$	30247.7	1.83796	0.918979	+	0.394308i	$0.129016\pi$
$648$	0	0
$649$	−4446.88	−0.268960
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10095.7	0.605016	0.302508	−	0.953147i	$-0.402176\pi$
$653$	10095.7	0.605016	0.302508	+	0.953147i	$0.402176\pi$
$654$	0	0
$655$	−20667.4	−1.23289
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−4439.77	−0.262441	−0.131221	−	0.991353i	$-0.541890\pi$
$659$	−4439.77	−0.262441	−0.131221	+	0.991353i	$0.541890\pi$
$660$	0	0
$661$	23931.4	1.40820	0.704102	−	0.710098i	$-0.251350\pi$
$661$	23931.4	1.40820	0.704102	+	0.710098i	$0.251350\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8496.08	−0.495434
$666$	0	0
$667$	6899.60	0.400530
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3322.81	0.191171
$672$	0	0
$673$	−20204.4	−1.15724	−0.578621	−	0.815597i	$-0.696409\pi$
$673$	−20204.4	−1.15724	−0.578621	+	0.815597i	$0.696409\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1902.19	0.107987	0.0539935	−	0.998541i	$-0.482805\pi$
$677$	1902.19	0.107987	0.0539935	+	0.998541i	$0.482805\pi$
$678$	0	0
$679$	51805.6	2.92801
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5569.51	−0.312022	−0.156011	−	0.987755i	$-0.549864\pi$
$683$	−5569.51	−0.312022	−0.156011	+	0.987755i	$0.549864\pi$
$684$	0	0
$685$	24131.3	1.34600
$686$	0	0
$687$	0	0
$688$	0	0
$689$	40829.0	2.25757
$690$	0	0
$691$	−19800.3	−1.09007	−0.545034	−	0.838414i	$-0.683483\pi$
$691$	−19800.3	−1.09007	−0.545034	+	0.838414i	$0.683483\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15429.4	−0.842117
$696$	0	0
$697$	3084.35	0.167615
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−23490.2	−1.26564	−0.632821	−	0.774298i	$-0.718103\pi$
$701$	−23490.2	−1.26564	−0.632821	+	0.774298i	$0.718103\pi$
$702$	0	0
$703$	−5736.38	−0.307755
$704$	0	0
$705$	0	0
$706$	0	0
$707$	33922.7	1.80452
$708$	0	0
$709$	1526.39	0.0808530	0.0404265	−	0.999183i	$-0.487128\pi$
$709$	1526.39	0.0808530	0.0404265	+	0.999183i	$0.487128\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	11304.8	0.593783
$714$	0	0
$715$	−8771.77	−0.458805
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−22362.0	−1.15989	−0.579945	−	0.814655i	$-0.696926\pi$
$719$	−22362.0	−1.15989	−0.579945	+	0.814655i	$0.696926\pi$
$720$	0	0
$721$	−5071.24	−0.261946
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10534.9	0.539663
$726$	0	0
$727$	73.0829	0.00372833	0.00186417	−	0.999998i	$-0.499407\pi$
$727$	73.0829	0.00372833	0.00186417	+	0.999998i	$0.499407\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5584.29	−0.282548
$732$	0	0
$733$	−25446.8	−1.28226	−0.641131	−	0.767432i	$-0.721534\pi$
$733$	−25446.8	−1.28226	−0.641131	+	0.767432i	$0.721534\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3343.24	−0.167096
$738$	0	0
$739$	−13743.2	−0.684102	−0.342051	−	0.939681i	$-0.611121\pi$
$739$	−13743.2	−0.684102	−0.342051	+	0.939681i	$0.611121\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17827.4	0.880250	0.440125	−	0.897937i	$-0.354934\pi$
$743$	17827.4	0.880250	0.440125	+	0.897937i	$0.354934\pi$
$744$	0	0
$745$	10993.5	0.540633
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−66947.9	−3.26599
$750$	0	0
$751$	31881.4	1.54909	0.774546	−	0.632518i	$-0.217979\pi$
$751$	31881.4	1.54909	0.774546	+	0.632518i	$0.217979\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	24226.1	1.16779
$756$	0	0
$757$	−27943.8	−1.34166	−0.670829	−	0.741612i	$-0.734061\pi$
$757$	−27943.8	−1.34166	−0.670829	+	0.741612i	$0.734061\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−17778.8	−0.846886	−0.423443	−	0.905923i	$-0.639179\pi$
$761$	−17778.8	−0.846886	−0.423443	+	0.905923i	$0.639179\pi$
$762$	0	0
$763$	55972.5	2.65575
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22734.6	1.07027
$768$	0	0
$769$	27392.0	1.28450	0.642251	−	0.766494i	$-0.278001\pi$
$769$	27392.0	1.28450	0.642251	+	0.766494i	$0.278001\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−42710.2	−1.98730	−0.993648	−	0.112534i	$-0.964103\pi$
$773$	−42710.2	−1.98730	−0.993648	+	0.112534i	$0.964103\pi$
$774$	0	0
$775$	17261.1	0.800047
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2629.86	0.120956
$780$	0	0
$781$	5959.00	0.273021
$782$	0	0
$783$	0	0
$784$	0	0
$785$	25129.2	1.14255
$786$	0	0
$787$	1304.06	0.0590658	0.0295329	−	0.999564i	$-0.490598\pi$
$787$	1304.06	0.0590658	0.0295329	+	0.999564i	$0.490598\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	37699.6	1.69462
$792$	0	0
$793$	−16987.9	−0.760727
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32495.9	−1.44425	−0.722123	−	0.691765i	$-0.756833\pi$
$797$	−32495.9	−1.44425	−0.722123	+	0.691765i	$0.756833\pi$
$798$	0	0
$799$	1444.15	0.0639429
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3221.70	0.141583
$804$	0	0
$805$	16612.6	0.727349
$806$	0	0
$807$	0	0
$808$	0	0
$809$	12386.2	0.538287	0.269144	−	0.963100i	$-0.413259\pi$
$809$	12386.2	0.538287	0.269144	+	0.963100i	$0.413259\pi$
$810$	0	0
$811$	14133.4	0.611949	0.305975	−	0.952040i	$-0.401018\pi$
$811$	14133.4	0.611949	0.305975	+	0.952040i	$0.401018\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−113.272	−0.00486840
$816$	0	0
$817$	−4761.42	−0.203894
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−39636.1	−1.68491	−0.842453	−	0.538770i	$-0.818889\pi$
$821$	−39636.1	−1.68491	−0.842453	+	0.538770i	$0.818889\pi$
$822$	0	0
$823$	−37263.1	−1.57826	−0.789132	−	0.614224i	$-0.789469\pi$
$823$	−37263.1	−1.57826	−0.789132	+	0.614224i	$0.789469\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13613.3	0.572408	0.286204	−	0.958169i	$-0.407606\pi$
$827$	13613.3	0.572408	0.286204	+	0.958169i	$0.407606\pi$
$828$	0	0
$829$	−36264.2	−1.51931	−0.759655	−	0.650327i	$-0.774632\pi$
$829$	−36264.2	−1.51931	−0.759655	+	0.650327i	$0.774632\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−16875.5	−0.701923
$834$	0	0
$835$	18989.6	0.787021
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−40472.8	−1.66540	−0.832702	−	0.553721i	$-0.813207\pi$
$839$	−40472.8	−1.66540	−0.832702	+	0.553721i	$0.813207\pi$
$840$	0	0
$841$	10101.9	0.414199
$842$	0	0
$843$	0	0
$844$	0	0
$845$	15228.8	0.619984
$846$	0	0
$847$	−39928.6	−1.61979
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11216.5	0.451816
$852$	0	0
$853$	−43209.2	−1.73441	−0.867207	−	0.497948i	$-0.834087\pi$
$853$	−43209.2	−1.73441	−0.867207	+	0.497948i	$0.834087\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3388.98	−0.135082	−0.0675411	−	0.997716i	$-0.521515\pi$
$857$	−3388.98	−0.135082	−0.0675411	+	0.997716i	$0.521515\pi$
$858$	0	0
$859$	20387.2	0.809783	0.404891	−	0.914365i	$-0.367309\pi$
$859$	20387.2	0.809783	0.404891	+	0.914365i	$0.367309\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31127.9	−1.22782	−0.613908	−	0.789377i	$-0.710403\pi$
$863$	−31127.9	−1.22782	−0.613908	+	0.789377i	$0.710403\pi$
$864$	0	0
$865$	5816.93	0.228649
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−6561.19	−0.256125
$870$	0	0
$871$	17092.3	0.664925
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−30529.8	−1.17954
$876$	0	0
$877$	−23125.7	−0.890419	−0.445210	−	0.895426i	$-0.646871\pi$
$877$	−23125.7	−0.890419	−0.445210	+	0.895426i	$0.646871\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	36896.9	1.41100	0.705498	−	0.708712i	$-0.250724\pi$
$881$	36896.9	1.41100	0.705498	+	0.708712i	$0.250724\pi$
$882$	0	0
$883$	−28830.8	−1.09879	−0.549397	−	0.835561i	$-0.685143\pi$
$883$	−28830.8	−1.09879	−0.549397	+	0.835561i	$0.685143\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38307.6	1.45011	0.725053	−	0.688693i	$-0.241815\pi$
$887$	38307.6	1.45011	0.725053	+	0.688693i	$0.241815\pi$
$888$	0	0
$889$	67039.8	2.52918
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1231.35	0.0461429
$894$	0	0
$895$	−34805.5	−1.29991
$896$	0	0
$897$	0	0
$898$	0	0
$899$	56512.3	2.09654
$900$	0	0
$901$	15774.2	0.583258
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10046.0	0.368995
$906$	0	0
$907$	−45680.3	−1.67232	−0.836158	−	0.548489i	$-0.815203\pi$
$907$	−45680.3	−1.67232	−0.836158	+	0.548489i	$0.815203\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−42906.7	−1.56044	−0.780220	−	0.625505i	$-0.784893\pi$
$911$	−42906.7	−1.56044	−0.780220	+	0.625505i	$0.784893\pi$
$912$	0	0
$913$	9573.26	0.347019
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−50855.1	−1.83139
$918$	0	0
$919$	34909.8	1.25307	0.626534	−	0.779394i	$-0.284473\pi$
$919$	34909.8	1.25307	0.626534	+	0.779394i	$0.284473\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−30465.3	−1.08643
$924$	0	0
$925$	17126.2	0.608763
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−47469.1	−1.67644	−0.838218	−	0.545335i	$-0.816403\pi$
$929$	−47469.1	−1.67644	−0.838218	+	0.545335i	$0.816403\pi$
$930$	0	0
$931$	−14388.9	−0.506526
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3388.95	−0.118535
$936$	0	0
$937$	−22636.2	−0.789213	−0.394606	−	0.918850i	$-0.629119\pi$
$937$	−22636.2	−0.789213	−0.394606	+	0.918850i	$0.629119\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2091.76	0.0724649	0.0362324	−	0.999343i	$-0.488464\pi$
$941$	2091.76	0.0724649	0.0362324	+	0.999343i	$0.488464\pi$
$942$	0	0
$943$	−5142.21	−0.177575
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38021.9	1.30469	0.652347	−	0.757921i	$-0.273785\pi$
$947$	38021.9	1.30469	0.652347	+	0.757921i	$0.273785\pi$
$948$	0	0
$949$	−16470.9	−0.563402
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−9281.67	−0.315491	−0.157745	−	0.987480i	$-0.550423\pi$
$953$	−9281.67	−0.315491	−0.157745	+	0.987480i	$0.550423\pi$
$954$	0	0
$955$	48179.6	1.63252
$956$	0	0
$957$	0	0
$958$	0	0
$959$	59378.7	1.99941
$960$	0	0
$961$	62802.7	2.10811
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−42931.1	−1.43212
$966$	0	0
$967$	49765.7	1.65497	0.827486	−	0.561487i	$-0.189770\pi$
$967$	49765.7	1.65497	0.827486	+	0.561487i	$0.189770\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11784.0	−0.389460	−0.194730	−	0.980857i	$-0.562383\pi$
$971$	−11784.0	−0.389460	−0.194730	+	0.980857i	$0.562383\pi$
$972$	0	0
$973$	−37966.4	−1.25092
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8888.07	0.291049	0.145524	−	0.989355i	$-0.453513\pi$
$977$	8888.07	0.291049	0.145524	+	0.989355i	$0.453513\pi$
$978$	0	0
$979$	−15910.0	−0.519393
$980$	0	0
$981$	0	0
$982$	0	0
$983$	27703.9	0.898898	0.449449	−	0.893306i	$-0.351620\pi$
$983$	27703.9	0.898898	0.449449	+	0.893306i	$0.351620\pi$
$984$	0	0
$985$	−18607.1	−0.601902
$986$	0	0
$987$	0	0
$988$	0	0
$989$	9310.11	0.299337
$990$	0	0
$991$	−53022.0	−1.69960	−0.849798	−	0.527108i	$-0.823276\pi$
$991$	−53022.0	−1.69960	−0.849798	+	0.527108i	$0.823276\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−45277.3	−1.44260
$996$	0	0
$997$	40990.8	1.30210	0.651048	−	0.759036i	$-0.274330\pi$
$997$	40990.8	1.30210	0.651048	+	0.759036i	$0.274330\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.4.a.l.1.4		5
3.2	odd	2		456.4.a.h.1.2	✓	5
12.11	even	2		912.4.a.w.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
456.4.a.h.1.2	✓	5	3.2	odd	2
912.4.a.w.1.2		5	12.11	even	2
1368.4.a.l.1.4		5	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+13.4806 q^{5} +33.1709 q^{7} +O(q^{10})\)	\(q+13.4806 q^{5} +33.1709 q^{7} +11.2817 q^{11} -57.6774 q^{13} -22.2836 q^{17} -19.0000 q^{19} +37.1511 q^{23} +56.7253 q^{25} +185.717 q^{29} +304.292 q^{31} +447.162 q^{35} +301.915 q^{37} -138.414 q^{41} +250.601 q^{43} -64.8079 q^{47} +757.308 q^{49} -707.886 q^{53} +152.083 q^{55} -394.168 q^{59} +294.532 q^{61} -777.524 q^{65} -296.343 q^{67} +528.201 q^{71} +285.570 q^{73} +374.223 q^{77} -581.579 q^{79} +848.567 q^{83} -300.395 q^{85} -1410.25 q^{89} -1913.21 q^{91} -256.131 q^{95} +1561.78 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 6 q^{5} + 10 q^{7}+O(q^{10}) \) 5 * q - 6 * q^5 + 10 * q^7	\( 5 q - 6 q^{5} + 10 q^{7} + 32 q^{11} + 36 q^{13} + 8 q^{17} - 95 q^{19} + 152 q^{23} + 241 q^{25} - 248 q^{29} + 118 q^{31} - 162 q^{35} + 472 q^{37} - 944 q^{41} + 150 q^{43} - 38 q^{47} + 1719 q^{49} - 788 q^{53} + 1270 q^{55} - 396 q^{59} + 1724 q^{61} - 2200 q^{65} + 204 q^{67} - 480 q^{71} + 2608 q^{73} - 2458 q^{77} + 786 q^{79} + 1118 q^{83} + 1798 q^{85} - 792 q^{89} + 1040 q^{91} + 114 q^{95} + 4638 q^{97}+O(q^{100}) \) 5 * q - 6 * q^5 + 10 * q^7 + 32 * q^11 + 36 * q^13 + 8 * q^17 - 95 * q^19 + 152 * q^23 + 241 * q^25 - 248 * q^29 + 118 * q^31 - 162 * q^35 + 472 * q^37 - 944 * q^41 + 150 * q^43 - 38 * q^47 + 1719 * q^49 - 788 * q^53 + 1270 * q^55 - 396 * q^59 + 1724 * q^61 - 2200 * q^65 + 204 * q^67 - 480 * q^71 + 2608 * q^73 - 2458 * q^77 + 786 * q^79 + 1118 * q^83 + 1798 * q^85 - 792 * q^89 + 1040 * q^91 + 114 * q^95 + 4638 * q^97

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