LMFDB - Embedded newform 2400.4.a.bt.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2400,4,Mod(1,2400)] 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("2400.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,9,0,0,0,-3,0,27,0,44,0,-13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$141.604584014$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.115636.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 69x - 81 $ x^3 - x^2 - 69*x - 81
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$9.32800$ of defining polynomial
Character	$\chi$	$=$	2400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +2.13303 q^{7} +9.00000 q^{9} -24.4450 q^{11} -34.0459 q^{13} +123.069 q^{17} -9.17896 q^{19} +6.39909 q^{21} -142.225 q^{23} +27.0000 q^{27} +140.491 q^{29} +158.711 q^{31} -73.3351 q^{33} -58.5783 q^{37} -102.138 q^{39} -108.895 q^{41} +246.565 q^{43} -466.056 q^{47} -338.450 q^{49} +369.207 q^{51} +312.757 q^{53} -27.5369 q^{57} +410.973 q^{59} -44.5685 q^{61} +19.1973 q^{63} +368.079 q^{67} -426.675 q^{69} -108.097 q^{71} +627.753 q^{73} -52.1419 q^{77} +196.404 q^{79} +81.0000 q^{81} +107.596 q^{83} +421.473 q^{87} +685.708 q^{89} -72.6210 q^{91} +476.134 q^{93} -73.7529 q^{97} -220.005 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} - 3 q^{7} + 27 q^{9} + 44 q^{11} - 13 q^{13} + 36 q^{17} + 71 q^{19} - 9 q^{21} + 160 q^{23} + 81 q^{27} - 184 q^{29} - 59 q^{31} + 132 q^{33} + 350 q^{37} - 39 q^{39} + 166 q^{41} + 341 q^{43}+ \cdots + 396 q^{99}+O(q^{100}) $ 3 * q + 9 * q^3 - 3 * q^7 + 27 * q^9 + 44 * q^11 - 13 * q^13 + 36 * q^17 + 71 * q^19 - 9 * q^21 + 160 * q^23 + 81 * q^27 - 184 * q^29 - 59 * q^31 + 132 * q^33 + 350 * q^37 - 39 * q^39 + 166 * q^41 + 341 * q^43 - 394 * q^47 + 158 * q^49 + 108 * q^51 + 314 * q^53 + 213 * q^57 + 538 * q^59 - 523 * q^61 - 27 * q^63 - 543 * q^67 + 480 * q^69 + 910 * q^71 + 870 * q^73 - 828 * q^77 - 96 * q^79 + 243 * q^81 + 210 * q^83 - 552 * q^87 - 64 * q^89 + 2781 * q^91 - 177 * q^93 + 943 * q^97 + 396 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.13303	0.115173	0.0575864	−	0.998341i	$-0.481660\pi$
$7$	2.13303	0.115173	0.0575864	+	0.998341i	$0.481660\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−24.4450	−0.670041	−0.335020	−	0.942211i	$-0.608743\pi$
$11$	−24.4450	−0.670041	−0.335020	+	0.942211i	$0.608743\pi$
$12$	0	0
$13$	−34.0459	−0.726357	−0.363179	−	0.931720i	$-0.618308\pi$
$13$	−34.0459	−0.726357	−0.363179	+	0.931720i	$0.618308\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	123.069	1.75580	0.877900	−	0.478843i	$-0.158944\pi$
$17$	123.069	1.75580	0.877900	+	0.478843i	$0.158944\pi$
$18$	0	0
$19$	−9.17896	−0.110831	−0.0554157	−	0.998463i	$-0.517648\pi$
$19$	−9.17896	−0.110831	−0.0554157	+	0.998463i	$0.517648\pi$
$20$	0	0
$21$	6.39909	0.0664950
$22$	0	0
$23$	−142.225	−1.28939	−0.644695	−	0.764440i	$-0.723016\pi$
$23$	−142.225	−1.28939	−0.644695	+	0.764440i	$0.723016\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	140.491	0.899605	0.449803	−	0.893128i	$-0.351494\pi$
$29$	140.491	0.899605	0.449803	+	0.893128i	$0.351494\pi$
$30$	0	0
$31$	158.711	0.919529	0.459764	−	0.888041i	$-0.347934\pi$
$31$	158.711	0.919529	0.459764	+	0.888041i	$0.347934\pi$
$32$	0	0
$33$	−73.3351	−0.386848
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−58.5783	−0.260276	−0.130138	−	0.991496i	$-0.541542\pi$
$37$	−58.5783	−0.260276	−0.130138	+	0.991496i	$0.541542\pi$
$38$	0	0
$39$	−102.138	−0.419362
$40$	0	0
$41$	−108.895	−0.414793	−0.207396	−	0.978257i	$-0.566499\pi$
$41$	−108.895	−0.414793	−0.207396	+	0.978257i	$0.566499\pi$
$42$	0	0
$43$	246.565	0.874436	0.437218	−	0.899356i	$-0.355964\pi$
$43$	246.565	0.874436	0.437218	+	0.899356i	$0.355964\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−466.056	−1.44641	−0.723204	−	0.690634i	$-0.757332\pi$
$47$	−466.056	−1.44641	−0.723204	+	0.690634i	$0.757332\pi$
$48$	0	0
$49$	−338.450	−0.986735
$50$	0	0
$51$	369.207	1.01371
$52$	0	0
$53$	312.757	0.810576	0.405288	−	0.914189i	$-0.367171\pi$
$53$	312.757	0.810576	0.405288	+	0.914189i	$0.367171\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−27.5369	−0.0639886
$58$	0	0
$59$	410.973	0.906849	0.453425	−	0.891295i	$-0.350202\pi$
$59$	410.973	0.906849	0.453425	+	0.891295i	$0.350202\pi$
$60$	0	0
$61$	−44.5685	−0.0935478	−0.0467739	−	0.998906i	$-0.514894\pi$
$61$	−44.5685	−0.0935478	−0.0467739	+	0.998906i	$0.514894\pi$
$62$	0	0
$63$	19.1973	0.0383909
$64$	0	0
$65$	0	0
$66$	0	0
$67$	368.079	0.671164	0.335582	−	0.942011i	$-0.391067\pi$
$67$	368.079	0.671164	0.335582	+	0.942011i	$0.391067\pi$
$68$	0	0
$69$	−426.675	−0.744430
$70$	0	0
$71$	−108.097	−0.180687	−0.0903434	−	0.995911i	$-0.528796\pi$
$71$	−108.097	−0.180687	−0.0903434	+	0.995911i	$0.528796\pi$
$72$	0	0
$73$	627.753	1.00648	0.503239	−	0.864147i	$-0.332141\pi$
$73$	627.753	1.00648	0.503239	+	0.864147i	$0.332141\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−52.1419	−0.0771705
$78$	0	0
$79$	196.404	0.279711	0.139856	−	0.990172i	$-0.455336\pi$
$79$	196.404	0.279711	0.139856	+	0.990172i	$0.455336\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	107.596	0.142292	0.0711460	−	0.997466i	$-0.477334\pi$
$83$	107.596	0.142292	0.0711460	+	0.997466i	$0.477334\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	421.473	0.519387
$88$	0	0
$89$	685.708	0.816684	0.408342	−	0.912829i	$-0.366107\pi$
$89$	685.708	0.816684	0.408342	+	0.912829i	$0.366107\pi$
$90$	0	0
$91$	−72.6210	−0.0836566
$92$	0	0
$93$	476.134	0.530890
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−73.7529	−0.0772007	−0.0386003	−	0.999255i	$-0.512290\pi$
$97$	−73.7529	−0.0772007	−0.0386003	+	0.999255i	$0.512290\pi$
$98$	0	0
$99$	−220.005	−0.223347
$100$	0	0
$101$	120.869	0.119078	0.0595389	−	0.998226i	$-0.481037\pi$
$101$	120.869	0.119078	0.0595389	+	0.998226i	$0.481037\pi$
$102$	0	0
$103$	1136.78	1.08748	0.543740	−	0.839254i	$-0.317008\pi$
$103$	1136.78	1.08748	0.543740	+	0.839254i	$0.317008\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1043.08	−0.942414	−0.471207	−	0.882023i	$-0.656181\pi$
$107$	−1043.08	−0.942414	−0.471207	+	0.882023i	$0.656181\pi$
$108$	0	0
$109$	1903.91	1.67304	0.836521	−	0.547935i	$-0.184586\pi$
$109$	1903.91	1.67304	0.836521	+	0.547935i	$0.184586\pi$
$110$	0	0
$111$	−175.735	−0.150270
$112$	0	0
$113$	−553.984	−0.461190	−0.230595	−	0.973050i	$-0.574067\pi$
$113$	−553.984	−0.461190	−0.230595	+	0.973050i	$0.574067\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−306.413	−0.242119
$118$	0	0
$119$	262.510	0.202220
$120$	0	0
$121$	−733.441	−0.551045
$122$	0	0
$123$	−326.684	−0.239481
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−887.643	−0.620201	−0.310101	−	0.950704i	$-0.600363\pi$
$127$	−887.643	−0.620201	−0.310101	+	0.950704i	$0.600363\pi$
$128$	0	0
$129$	739.694	0.504856
$130$	0	0
$131$	2320.16	1.54743	0.773716	−	0.633533i	$-0.218396\pi$
$131$	2320.16	1.54743	0.773716	+	0.633533i	$0.218396\pi$
$132$	0	0
$133$	−19.5790	−0.0127648
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1841.31	1.14828	0.574139	−	0.818758i	$-0.305337\pi$
$137$	1841.31	1.14828	0.574139	+	0.818758i	$0.305337\pi$
$138$	0	0
$139$	686.396	0.418844	0.209422	−	0.977825i	$-0.432842\pi$
$139$	686.396	0.418844	0.209422	+	0.977825i	$0.432842\pi$
$140$	0	0
$141$	−1398.17	−0.835084
$142$	0	0
$143$	832.253	0.486689
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1015.35	−0.569692
$148$	0	0
$149$	680.173	0.373973	0.186986	−	0.982363i	$-0.440128\pi$
$149$	680.173	0.373973	0.186986	+	0.982363i	$0.440128\pi$
$150$	0	0
$151$	750.511	0.404475	0.202238	−	0.979336i	$-0.435179\pi$
$151$	750.511	0.404475	0.202238	+	0.979336i	$0.435179\pi$
$152$	0	0
$153$	1107.62	0.585267
$154$	0	0
$155$	0	0
$156$	0	0
$157$	941.269	0.478480	0.239240	−	0.970960i	$-0.423102\pi$
$157$	941.269	0.478480	0.239240	+	0.970960i	$0.423102\pi$
$158$	0	0
$159$	938.272	0.467986
$160$	0	0
$161$	−303.370	−0.148503
$162$	0	0
$163$	745.142	0.358062	0.179031	−	0.983843i	$-0.442704\pi$
$163$	745.142	0.358062	0.179031	+	0.983843i	$0.442704\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2744.97	1.27193	0.635965	−	0.771718i	$-0.280602\pi$
$167$	2744.97	1.27193	0.635965	+	0.771718i	$0.280602\pi$
$168$	0	0
$169$	−1037.87	−0.472405
$170$	0	0
$171$	−82.6106	−0.0369438
$172$	0	0
$173$	1891.42	0.831224	0.415612	−	0.909542i	$-0.363567\pi$
$173$	1891.42	0.831224	0.415612	+	0.909542i	$0.363567\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1232.92	0.523570
$178$	0	0
$179$	−1127.81	−0.470930	−0.235465	−	0.971883i	$-0.575661\pi$
$179$	−1127.81	−0.470930	−0.235465	+	0.971883i	$0.575661\pi$
$180$	0	0
$181$	−3421.68	−1.40515	−0.702574	−	0.711611i	$-0.747966\pi$
$181$	−3421.68	−1.40515	−0.702574	+	0.711611i	$0.747966\pi$
$182$	0	0
$183$	−133.706	−0.0540098
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3008.42	−1.17646
$188$	0	0
$189$	57.5918	0.0221650
$190$	0	0
$191$	51.0506	0.0193398	0.00966988	−	0.999953i	$-0.496922\pi$
$191$	51.0506	0.0193398	0.00966988	+	0.999953i	$0.496922\pi$
$192$	0	0
$193$	5176.59	1.93067	0.965334	−	0.261016i	$-0.0840576\pi$
$193$	5176.59	1.93067	0.965334	+	0.261016i	$0.0840576\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1867.19	−0.675287	−0.337644	−	0.941274i	$-0.609630\pi$
$197$	−1867.19	−0.675287	−0.337644	+	0.941274i	$0.609630\pi$
$198$	0	0
$199$	−2891.08	−1.02987	−0.514933	−	0.857230i	$-0.672183\pi$
$199$	−2891.08	−1.02987	−0.514933	+	0.857230i	$0.672183\pi$
$200$	0	0
$201$	1104.24	0.387497
$202$	0	0
$203$	299.672	0.103610
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1280.03	−0.429797
$208$	0	0
$209$	224.380	0.0742616
$210$	0	0
$211$	122.931	0.0401085	0.0200543	−	0.999799i	$-0.493616\pi$
$211$	122.931	0.0401085	0.0200543	+	0.999799i	$0.493616\pi$
$212$	0	0
$213$	−324.291	−0.104320
$214$	0	0
$215$	0	0
$216$	0	0
$217$	338.536	0.105905
$218$	0	0
$219$	1883.26	0.581091
$220$	0	0
$221$	−4190.00	−1.27534
$222$	0	0
$223$	−6307.07	−1.89396	−0.946979	−	0.321295i	$-0.895882\pi$
$223$	−6307.07	−1.89396	−0.946979	+	0.321295i	$0.895882\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2795.36	0.817332	0.408666	−	0.912684i	$-0.365994\pi$
$227$	2795.36	0.817332	0.408666	+	0.912684i	$0.365994\pi$
$228$	0	0
$229$	1328.44	0.383343	0.191672	−	0.981459i	$-0.438609\pi$
$229$	1328.44	0.383343	0.191672	+	0.981459i	$0.438609\pi$
$230$	0	0
$231$	−156.426	−0.0445544
$232$	0	0
$233$	6117.63	1.72008	0.860041	−	0.510226i	$-0.170438\pi$
$233$	6117.63	1.72008	0.860041	+	0.510226i	$0.170438\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	589.212	0.161491
$238$	0	0
$239$	−1222.76	−0.330936	−0.165468	−	0.986215i	$-0.552913\pi$
$239$	−1222.76	−0.330936	−0.165468	+	0.986215i	$0.552913\pi$
$240$	0	0
$241$	194.893	0.0520919	0.0260460	−	0.999661i	$-0.491708\pi$
$241$	194.893	0.0520919	0.0260460	+	0.999661i	$0.491708\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	312.506	0.0805032
$248$	0	0
$249$	322.789	0.0821523
$250$	0	0
$251$	−348.453	−0.0876262	−0.0438131	−	0.999040i	$-0.513951\pi$
$251$	−348.453	−0.0876262	−0.0438131	+	0.999040i	$0.513951\pi$
$252$	0	0
$253$	3476.70	0.863944
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3969.83	0.963544	0.481772	−	0.876296i	$-0.339993\pi$
$257$	3969.83	0.963544	0.481772	+	0.876296i	$0.339993\pi$
$258$	0	0
$259$	−124.949	−0.0299767
$260$	0	0
$261$	1264.42	0.299868
$262$	0	0
$263$	−756.907	−0.177463	−0.0887317	−	0.996056i	$-0.528281\pi$
$263$	−756.907	−0.177463	−0.0887317	+	0.996056i	$0.528281\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2057.12	0.471513
$268$	0	0
$269$	6105.26	1.38381	0.691904	−	0.721989i	$-0.256772\pi$
$269$	6105.26	1.38381	0.691904	+	0.721989i	$0.256772\pi$
$270$	0	0
$271$	−3824.23	−0.857216	−0.428608	−	0.903491i	$-0.640996\pi$
$271$	−3824.23	−0.857216	−0.428608	+	0.903491i	$0.640996\pi$
$272$	0	0
$273$	−217.863	−0.0482991
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1930.18	0.418676	0.209338	−	0.977843i	$-0.432869\pi$
$277$	1930.18	0.418676	0.209338	+	0.977843i	$0.432869\pi$
$278$	0	0
$279$	1428.40	0.306510
$280$	0	0
$281$	−2942.33	−0.624643	−0.312322	−	0.949976i	$-0.601107\pi$
$281$	−2942.33	−0.624643	−0.312322	+	0.949976i	$0.601107\pi$
$282$	0	0
$283$	8483.92	1.78204	0.891019	−	0.453967i	$-0.149992\pi$
$283$	8483.92	1.78204	0.891019	+	0.453967i	$0.149992\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−232.276	−0.0477728
$288$	0	0
$289$	10233.0	2.08284
$290$	0	0
$291$	−221.259	−0.0445718
$292$	0	0
$293$	8526.45	1.70007	0.850035	−	0.526726i	$-0.176581\pi$
$293$	8526.45	1.70007	0.850035	+	0.526726i	$0.176581\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−660.016	−0.128949
$298$	0	0
$299$	4842.19	0.936558
$300$	0	0
$301$	525.929	0.100711
$302$	0	0
$303$	362.606	0.0687497
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1977.34	−0.367599	−0.183800	−	0.982964i	$-0.558840\pi$
$307$	−1977.34	−0.367599	−0.183800	+	0.982964i	$0.558840\pi$
$308$	0	0
$309$	3410.34	0.627856
$310$	0	0
$311$	−5243.29	−0.956012	−0.478006	−	0.878356i	$-0.658640\pi$
$311$	−5243.29	−0.956012	−0.478006	+	0.878356i	$0.658640\pi$
$312$	0	0
$313$	−755.186	−0.136376	−0.0681879	−	0.997672i	$-0.521722\pi$
$313$	−755.186	−0.136376	−0.0681879	+	0.997672i	$0.521722\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4018.11	0.711922	0.355961	−	0.934501i	$-0.384154\pi$
$317$	4018.11	0.711922	0.355961	+	0.934501i	$0.384154\pi$
$318$	0	0
$319$	−3434.31	−0.602772
$320$	0	0
$321$	−3129.24	−0.544103
$322$	0	0
$323$	−1129.65	−0.194598
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5711.73	0.965931
$328$	0	0
$329$	−994.110	−0.166587
$330$	0	0
$331$	5297.98	0.879769	0.439884	−	0.898054i	$-0.355019\pi$
$331$	5297.98	0.879769	0.439884	+	0.898054i	$0.355019\pi$
$332$	0	0
$333$	−527.204	−0.0867586
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9911.99	1.60220	0.801099	−	0.598532i	$-0.204249\pi$
$337$	9911.99	1.60220	0.801099	+	0.598532i	$0.204249\pi$
$338$	0	0
$339$	−1661.95	−0.266268
$340$	0	0
$341$	−3879.70	−0.616122
$342$	0	0
$343$	−1453.55	−0.228818
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4218.61	−0.652643	−0.326321	−	0.945259i	$-0.605809\pi$
$347$	−4218.61	−0.652643	−0.326321	+	0.945259i	$0.605809\pi$
$348$	0	0
$349$	2128.88	0.326522	0.163261	−	0.986583i	$-0.447799\pi$
$349$	2128.88	0.326522	0.163261	+	0.986583i	$0.447799\pi$
$350$	0	0
$351$	−919.240	−0.139787
$352$	0	0
$353$	8492.19	1.28044	0.640218	−	0.768194i	$-0.278844\pi$
$353$	8492.19	1.28044	0.640218	+	0.768194i	$0.278844\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	787.529	0.116752
$358$	0	0
$359$	6606.71	0.971277	0.485639	−	0.874160i	$-0.338587\pi$
$359$	6606.71	0.971277	0.485639	+	0.874160i	$0.338587\pi$
$360$	0	0
$361$	−6774.75	−0.987716
$362$	0	0
$363$	−2200.32	−0.318146
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7180.57	1.02132	0.510658	−	0.859784i	$-0.329402\pi$
$367$	7180.57	1.02132	0.510658	+	0.859784i	$0.329402\pi$
$368$	0	0
$369$	−980.053	−0.138264
$370$	0	0
$371$	667.120	0.0933562
$372$	0	0
$373$	7436.71	1.03233	0.516164	−	0.856490i	$-0.327359\pi$
$373$	7436.71	1.03233	0.516164	+	0.856490i	$0.327359\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4783.15	−0.653435
$378$	0	0
$379$	−11591.3	−1.57099	−0.785496	−	0.618867i	$-0.787592\pi$
$379$	−11591.3	−1.57099	−0.785496	+	0.618867i	$0.787592\pi$
$380$	0	0
$381$	−2662.93	−0.358073
$382$	0	0
$383$	−7706.49	−1.02815	−0.514077	−	0.857744i	$-0.671866\pi$
$383$	−7706.49	−1.02815	−0.514077	+	0.857744i	$0.671866\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2219.08	0.291479
$388$	0	0
$389$	2067.75	0.269509	0.134755	−	0.990879i	$-0.456975\pi$
$389$	2067.75	0.269509	0.134755	+	0.990879i	$0.456975\pi$
$390$	0	0
$391$	−17503.5	−2.26391
$392$	0	0
$393$	6960.49	0.893410
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−11330.1	−1.43235	−0.716175	−	0.697921i	$-0.754109\pi$
$397$	−11330.1	−1.43235	−0.716175	+	0.697921i	$0.754109\pi$
$398$	0	0
$399$	−58.7370	−0.00736974
$400$	0	0
$401$	−8551.98	−1.06500	−0.532500	−	0.846430i	$-0.678747\pi$
$401$	−8551.98	−1.06500	−0.532500	+	0.846430i	$0.678747\pi$
$402$	0	0
$403$	−5403.47	−0.667906
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1431.95	0.174395
$408$	0	0
$409$	448.744	0.0542517	0.0271259	−	0.999632i	$-0.491365\pi$
$409$	448.744	0.0542517	0.0271259	+	0.999632i	$0.491365\pi$
$410$	0	0
$411$	5523.94	0.662958
$412$	0	0
$413$	876.617	0.104444
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2059.19	0.241820
$418$	0	0
$419$	5153.24	0.600841	0.300420	−	0.953807i	$-0.402873\pi$
$419$	5153.24	0.600841	0.300420	+	0.953807i	$0.402873\pi$
$420$	0	0
$421$	−12633.7	−1.46254	−0.731269	−	0.682089i	$-0.761072\pi$
$421$	−12633.7	−1.46254	−0.731269	+	0.682089i	$0.761072\pi$
$422$	0	0
$423$	−4194.50	−0.482136
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−95.0660	−0.0107742
$428$	0	0
$429$	2496.76	0.280990
$430$	0	0
$431$	−9746.12	−1.08922	−0.544610	−	0.838689i	$-0.683322\pi$
$431$	−9746.12	−1.08922	−0.544610	+	0.838689i	$0.683322\pi$
$432$	0	0
$433$	12953.8	1.43768	0.718842	−	0.695173i	$-0.244672\pi$
$433$	12953.8	1.43768	0.718842	+	0.695173i	$0.244672\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1305.48	0.142905
$438$	0	0
$439$	−11877.5	−1.29130	−0.645649	−	0.763634i	$-0.723413\pi$
$439$	−11877.5	−1.29130	−0.645649	+	0.763634i	$0.723413\pi$
$440$	0	0
$441$	−3046.05	−0.328912
$442$	0	0
$443$	−1165.96	−0.125049	−0.0625243	−	0.998043i	$-0.519915\pi$
$443$	−1165.96	−0.125049	−0.0625243	+	0.998043i	$0.519915\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2040.52	0.215913
$448$	0	0
$449$	1415.70	0.148799	0.0743996	−	0.997229i	$-0.476296\pi$
$449$	1415.70	0.148799	0.0743996	+	0.997229i	$0.476296\pi$
$450$	0	0
$451$	2661.94	0.277928
$452$	0	0
$453$	2251.53	0.233524
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12279.3	1.25690	0.628449	−	0.777851i	$-0.283690\pi$
$457$	12279.3	1.25690	0.628449	+	0.777851i	$0.283690\pi$
$458$	0	0
$459$	3322.86	0.337904
$460$	0	0
$461$	−5591.54	−0.564911	−0.282455	−	0.959280i	$-0.591149\pi$
$461$	−5591.54	−0.564911	−0.282455	+	0.959280i	$0.591149\pi$
$462$	0	0
$463$	18810.7	1.88813	0.944067	−	0.329755i	$-0.106966\pi$
$463$	18810.7	1.88813	0.944067	+	0.329755i	$0.106966\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6580.13	−0.652017	−0.326008	−	0.945367i	$-0.605704\pi$
$467$	−6580.13	−0.652017	−0.326008	+	0.945367i	$0.605704\pi$
$468$	0	0
$469$	785.123	0.0772998
$470$	0	0
$471$	2823.81	0.276251
$472$	0	0
$473$	−6027.27	−0.585908
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2814.81	0.270192
$478$	0	0
$479$	−12089.0	−1.15316	−0.576578	−	0.817042i	$-0.695612\pi$
$479$	−12089.0	−1.15316	−0.576578	+	0.817042i	$0.695612\pi$
$480$	0	0
$481$	1994.35	0.189053
$482$	0	0
$483$	−910.111	−0.0857381
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7259.45	−0.675477	−0.337738	−	0.941240i	$-0.609662\pi$
$487$	−7259.45	−0.675477	−0.337738	+	0.941240i	$0.609662\pi$
$488$	0	0
$489$	2235.43	0.206727
$490$	0	0
$491$	16499.5	1.51652	0.758260	−	0.651953i	$-0.226050\pi$
$491$	16499.5	1.51652	0.758260	+	0.651953i	$0.226050\pi$
$492$	0	0
$493$	17290.1	1.57953
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−230.574	−0.0208102
$498$	0	0
$499$	10733.6	0.962932	0.481466	−	0.876465i	$-0.340105\pi$
$499$	10733.6	0.962932	0.481466	+	0.876465i	$0.340105\pi$
$500$	0	0
$501$	8234.92	0.734349
$502$	0	0
$503$	−17777.5	−1.57586	−0.787930	−	0.615764i	$-0.788847\pi$
$503$	−17777.5	−1.57586	−0.787930	+	0.615764i	$0.788847\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3113.62	−0.272743
$508$	0	0
$509$	11995.3	1.04456	0.522279	−	0.852775i	$-0.325082\pi$
$509$	11995.3	1.04456	0.522279	+	0.852775i	$0.325082\pi$
$510$	0	0
$511$	1339.02	0.115919
$512$	0	0
$513$	−247.832	−0.0213295
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11392.7	0.969153
$518$	0	0
$519$	5674.25	0.479907
$520$	0	0
$521$	18050.7	1.51788	0.758938	−	0.651162i	$-0.225718\pi$
$521$	18050.7	1.51788	0.758938	+	0.651162i	$0.225718\pi$
$522$	0	0
$523$	12834.3	1.07305	0.536523	−	0.843886i	$-0.319738\pi$
$523$	12834.3	1.07305	0.536523	+	0.843886i	$0.319738\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19532.4	1.61451
$528$	0	0
$529$	8060.98	0.662528
$530$	0	0
$531$	3698.76	0.302283
$532$	0	0
$533$	3707.42	0.301288
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3383.43	−0.271892
$538$	0	0
$539$	8273.42	0.661153
$540$	0	0
$541$	6839.45	0.543532	0.271766	−	0.962363i	$-0.412392\pi$
$541$	6839.45	0.543532	0.271766	+	0.962363i	$0.412392\pi$
$542$	0	0
$543$	−10265.0	−0.811262
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−24487.2	−1.91407	−0.957034	−	0.289976i	$-0.906353\pi$
$547$	−24487.2	−1.91407	−0.957034	+	0.289976i	$0.906353\pi$
$548$	0	0
$549$	−401.117	−0.0311826
$550$	0	0
$551$	−1289.56	−0.0997046
$552$	0	0
$553$	418.936	0.0322151
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21516.5	1.63677	0.818386	−	0.574669i	$-0.194869\pi$
$557$	21516.5	1.63677	0.818386	+	0.574669i	$0.194869\pi$
$558$	0	0
$559$	−8394.52	−0.635153
$560$	0	0
$561$	−9025.27	−0.679229
$562$	0	0
$563$	14612.4	1.09385	0.546926	−	0.837181i	$-0.315798\pi$
$563$	14612.4	1.09385	0.546926	+	0.837181i	$0.315798\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	172.775	0.0127970
$568$	0	0
$569$	17738.6	1.30693	0.653464	−	0.756958i	$-0.273315\pi$
$569$	17738.6	1.30693	0.653464	+	0.756958i	$0.273315\pi$
$570$	0	0
$571$	−8033.63	−0.588786	−0.294393	−	0.955684i	$-0.595117\pi$
$571$	−8033.63	−0.588786	−0.294393	+	0.955684i	$0.595117\pi$
$572$	0	0
$573$	153.152	0.0111658
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−20620.8	−1.48779	−0.743894	−	0.668297i	$-0.767023\pi$
$577$	−20620.8	−1.48779	−0.743894	+	0.668297i	$0.767023\pi$
$578$	0	0
$579$	15529.8	1.11467
$580$	0	0
$581$	229.506	0.0163882
$582$	0	0
$583$	−7645.36	−0.543119
$584$	0	0
$585$	0	0
$586$	0	0
$587$	10364.5	0.728772	0.364386	−	0.931248i	$-0.381279\pi$
$587$	10364.5	0.728772	0.364386	+	0.931248i	$0.381279\pi$
$588$	0	0
$589$	−1456.80	−0.101913
$590$	0	0
$591$	−5601.56	−0.389877
$592$	0	0
$593$	9074.45	0.628403	0.314202	−	0.949356i	$-0.398263\pi$
$593$	9074.45	0.628403	0.314202	+	0.949356i	$0.398263\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8673.25	−0.594594
$598$	0	0
$599$	−6917.00	−0.471821	−0.235911	−	0.971775i	$-0.575807\pi$
$599$	−6917.00	−0.471821	−0.235911	+	0.971775i	$0.575807\pi$
$600$	0	0
$601$	−16258.3	−1.10348	−0.551738	−	0.834017i	$-0.686035\pi$
$601$	−16258.3	−1.10348	−0.551738	+	0.834017i	$0.686035\pi$
$602$	0	0
$603$	3312.71	0.223721
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1406.90	−0.0940764	−0.0470382	−	0.998893i	$-0.514978\pi$
$607$	−1406.90	−0.0940764	−0.0470382	+	0.998893i	$0.514978\pi$
$608$	0	0
$609$	899.015	0.0598193
$610$	0	0
$611$	15867.3	1.05061
$612$	0	0
$613$	−21539.2	−1.41918	−0.709592	−	0.704613i	$-0.751121\pi$
$613$	−21539.2	−1.41918	−0.709592	+	0.704613i	$0.751121\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	798.965	0.0521314	0.0260657	−	0.999660i	$-0.491702\pi$
$617$	798.965	0.0521314	0.0260657	+	0.999660i	$0.491702\pi$
$618$	0	0
$619$	−23307.9	−1.51345	−0.756724	−	0.653735i	$-0.773201\pi$
$619$	−23307.9	−1.51345	−0.756724	+	0.653735i	$0.773201\pi$
$620$	0	0
$621$	−3840.08	−0.248143
$622$	0	0
$623$	1462.63	0.0940597
$624$	0	0
$625$	0	0
$626$	0	0
$627$	673.140	0.0428750
$628$	0	0
$629$	−7209.17	−0.456993
$630$	0	0
$631$	−15480.0	−0.976625	−0.488312	−	0.872669i	$-0.662387\pi$
$631$	−15480.0	−0.976625	−0.488312	+	0.872669i	$0.662387\pi$
$632$	0	0
$633$	368.792	0.0231567
$634$	0	0
$635$	0	0
$636$	0	0
$637$	11522.9	0.716722
$638$	0	0
$639$	−972.873	−0.0602289
$640$	0	0
$641$	−8650.11	−0.533009	−0.266505	−	0.963834i	$-0.585869\pi$
$641$	−8650.11	−0.533009	−0.266505	+	0.963834i	$0.585869\pi$
$642$	0	0
$643$	27769.2	1.70313	0.851563	−	0.524252i	$-0.175655\pi$
$643$	27769.2	1.70313	0.851563	+	0.524252i	$0.175655\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15716.3	0.954977	0.477489	−	0.878638i	$-0.341547\pi$
$647$	15716.3	0.954977	0.477489	+	0.878638i	$0.341547\pi$
$648$	0	0
$649$	−10046.2	−0.607626
$650$	0	0
$651$	1015.61	0.0611441
$652$	0	0
$653$	22537.5	1.35063	0.675313	−	0.737531i	$-0.264009\pi$
$653$	22537.5	1.35063	0.675313	+	0.737531i	$0.264009\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5649.78	0.335493
$658$	0	0
$659$	18711.0	1.10603	0.553017	−	0.833170i	$-0.313477\pi$
$659$	18711.0	1.10603	0.553017	+	0.833170i	$0.313477\pi$
$660$	0	0
$661$	14966.5	0.880678	0.440339	−	0.897832i	$-0.354858\pi$
$661$	14966.5	0.880678	0.440339	+	0.897832i	$0.354858\pi$
$662$	0	0
$663$	−12570.0	−0.736317
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−19981.4	−1.15994
$668$	0	0
$669$	−18921.2	−1.09348
$670$	0	0
$671$	1089.48	0.0626808
$672$	0	0
$673$	1398.61	0.0801075	0.0400538	−	0.999198i	$-0.487247\pi$
$673$	1398.61	0.0801075	0.0400538	+	0.999198i	$0.487247\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6643.40	−0.377144	−0.188572	−	0.982059i	$-0.560386\pi$
$677$	−6643.40	−0.377144	−0.188572	+	0.982059i	$0.560386\pi$
$678$	0	0
$679$	−157.317	−0.00889142
$680$	0	0
$681$	8386.07	0.471887
$682$	0	0
$683$	11293.0	0.632673	0.316337	−	0.948647i	$-0.397547\pi$
$683$	11293.0	0.632673	0.316337	+	0.948647i	$0.397547\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3985.31	0.221323
$688$	0	0
$689$	−10648.1	−0.588767
$690$	0	0
$691$	17827.1	0.981438	0.490719	−	0.871318i	$-0.336734\pi$
$691$	17827.1	0.981438	0.490719	+	0.871318i	$0.336734\pi$
$692$	0	0
$693$	−469.277	−0.0257235
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−13401.6	−0.728294
$698$	0	0
$699$	18352.9	0.993089
$700$	0	0
$701$	−27124.4	−1.46145	−0.730725	−	0.682672i	$-0.760818\pi$
$701$	−27124.4	−1.46145	−0.730725	+	0.682672i	$0.760818\pi$
$702$	0	0
$703$	537.687	0.0288467
$704$	0	0
$705$	0	0
$706$	0	0
$707$	257.816	0.0137145
$708$	0	0
$709$	−11294.6	−0.598278	−0.299139	−	0.954210i	$-0.596699\pi$
$709$	−11294.6	−0.598278	−0.299139	+	0.954210i	$0.596699\pi$
$710$	0	0
$711$	1767.64	0.0932371
$712$	0	0
$713$	−22572.7	−1.18563
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3668.28	−0.191066
$718$	0	0
$719$	31113.8	1.61384	0.806920	−	0.590661i	$-0.201133\pi$
$719$	31113.8	1.61384	0.806920	+	0.590661i	$0.201133\pi$
$720$	0	0
$721$	2424.79	0.125248
$722$	0	0
$723$	584.679	0.0300753
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−8773.62	−0.447587	−0.223793	−	0.974637i	$-0.571844\pi$
$727$	−8773.62	−0.447587	−0.223793	+	0.974637i	$0.571844\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	30344.4	1.53533
$732$	0	0
$733$	−3745.09	−0.188715	−0.0943576	−	0.995538i	$-0.530080\pi$
$733$	−3745.09	−0.188715	−0.0943576	+	0.995538i	$0.530080\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8997.70	−0.449707
$738$	0	0
$739$	−14923.3	−0.742844	−0.371422	−	0.928464i	$-0.621130\pi$
$739$	−14923.3	−0.742844	−0.371422	+	0.928464i	$0.621130\pi$
$740$	0	0
$741$	937.519	0.0464786
$742$	0	0
$743$	888.743	0.0438827	0.0219413	−	0.999759i	$-0.493015\pi$
$743$	888.743	0.0438827	0.0219413	+	0.999759i	$0.493015\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	968.367	0.0474307
$748$	0	0
$749$	−2224.92	−0.108540
$750$	0	0
$751$	−29588.6	−1.43769	−0.718843	−	0.695172i	$-0.755328\pi$
$751$	−29588.6	−1.43769	−0.718843	+	0.695172i	$0.755328\pi$
$752$	0	0
$753$	−1045.36	−0.0505910
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8485.87	0.407429	0.203715	−	0.979030i	$-0.434698\pi$
$757$	8485.87	0.407429	0.203715	+	0.979030i	$0.434698\pi$
$758$	0	0
$759$	10430.1	0.498799
$760$	0	0
$761$	9803.23	0.466974	0.233487	−	0.972360i	$-0.424986\pi$
$761$	9803.23	0.466974	0.233487	+	0.972360i	$0.424986\pi$
$762$	0	0
$763$	4061.10	0.192689
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13992.0	−0.658696
$768$	0	0
$769$	−1920.45	−0.0900561	−0.0450281	−	0.998986i	$-0.514338\pi$
$769$	−1920.45	−0.0900561	−0.0450281	+	0.998986i	$0.514338\pi$
$770$	0	0
$771$	11909.5	0.556303
$772$	0	0
$773$	−9679.15	−0.450368	−0.225184	−	0.974316i	$-0.572298\pi$
$773$	−9679.15	−0.450368	−0.225184	+	0.974316i	$0.572298\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−374.847	−0.0173070
$778$	0	0
$779$	999.541	0.0459721
$780$	0	0
$781$	2642.43	0.121067
$782$	0	0
$783$	3793.26	0.173129
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16275.2	−0.737165	−0.368582	−	0.929595i	$-0.620157\pi$
$787$	−16275.2	−0.737165	−0.368582	+	0.929595i	$0.620157\pi$
$788$	0	0
$789$	−2270.72	−0.102459
$790$	0	0
$791$	−1181.67	−0.0531165
$792$	0	0
$793$	1517.38	0.0679491
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−9219.64	−0.409757	−0.204879	−	0.978787i	$-0.565680\pi$
$797$	−9219.64	−0.409757	−0.204879	+	0.978787i	$0.565680\pi$
$798$	0	0
$799$	−57357.0	−2.53961
$800$	0	0
$801$	6171.37	0.272228
$802$	0	0
$803$	−15345.4	−0.674382
$804$	0	0
$805$	0	0
$806$	0	0
$807$	18315.8	0.798942
$808$	0	0
$809$	−5045.18	−0.219257	−0.109629	−	0.993973i	$-0.534966\pi$
$809$	−5045.18	−0.219257	−0.109629	+	0.993973i	$0.534966\pi$
$810$	0	0
$811$	45242.3	1.95890	0.979452	−	0.201677i	$-0.0646393\pi$
$811$	45242.3	1.95890	0.979452	+	0.201677i	$0.0646393\pi$
$812$	0	0
$813$	−11472.7	−0.494914
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−2263.21	−0.0969150
$818$	0	0
$819$	−653.589	−0.0278855
$820$	0	0
$821$	25431.0	1.08106	0.540528	−	0.841326i	$-0.318225\pi$
$821$	25431.0	1.08106	0.540528	+	0.841326i	$0.318225\pi$
$822$	0	0
$823$	6907.29	0.292555	0.146278	−	0.989244i	$-0.453271\pi$
$823$	6907.29	0.292555	0.146278	+	0.989244i	$0.453271\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−38201.5	−1.60628	−0.803141	−	0.595789i	$-0.796840\pi$
$827$	−38201.5	−1.60628	−0.803141	+	0.595789i	$0.796840\pi$
$828$	0	0
$829$	20573.4	0.861933	0.430966	−	0.902368i	$-0.358173\pi$
$829$	20573.4	0.861933	0.430966	+	0.902368i	$0.358173\pi$
$830$	0	0
$831$	5790.53	0.241722
$832$	0	0
$833$	−41652.7	−1.73251
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4285.20	0.176963
$838$	0	0
$839$	16604.9	0.683272	0.341636	−	0.939832i	$-0.389019\pi$
$839$	16604.9	0.683272	0.341636	+	0.939832i	$0.389019\pi$
$840$	0	0
$841$	−4651.24	−0.190710
$842$	0	0
$843$	−8826.99	−0.360638
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1564.45	−0.0634654
$848$	0	0
$849$	25451.7	1.02886
$850$	0	0
$851$	8331.30	0.335597
$852$	0	0
$853$	−39897.4	−1.60148	−0.800739	−	0.599013i	$-0.795560\pi$
$853$	−39897.4	−1.60148	−0.800739	+	0.599013i	$0.795560\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−14165.4	−0.564623	−0.282311	−	0.959323i	$-0.591101\pi$
$857$	−14165.4	−0.564623	−0.282311	+	0.959323i	$0.591101\pi$
$858$	0	0
$859$	−18842.9	−0.748441	−0.374220	−	0.927340i	$-0.622090\pi$
$859$	−18842.9	−0.748441	−0.374220	+	0.927340i	$0.622090\pi$
$860$	0	0
$861$	−696.827	−0.0275817
$862$	0	0
$863$	−35799.2	−1.41207	−0.706037	−	0.708175i	$-0.749519\pi$
$863$	−35799.2	−1.41207	−0.706037	+	0.708175i	$0.749519\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	30698.9	1.20253
$868$	0	0
$869$	−4801.10	−0.187418
$870$	0	0
$871$	−12531.6	−0.487505
$872$	0	0
$873$	−663.776	−0.0257336
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−17739.1	−0.683019	−0.341510	−	0.939878i	$-0.610938\pi$
$877$	−17739.1	−0.683019	−0.341510	+	0.939878i	$0.610938\pi$
$878$	0	0
$879$	25579.3	0.981536
$880$	0	0
$881$	19788.8	0.756757	0.378379	−	0.925651i	$-0.376482\pi$
$881$	19788.8	0.756757	0.378379	+	0.925651i	$0.376482\pi$
$882$	0	0
$883$	15831.4	0.603363	0.301682	−	0.953409i	$-0.402452\pi$
$883$	15831.4	0.603363	0.301682	+	0.953409i	$0.402452\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30374.8	1.14981	0.574907	−	0.818219i	$-0.305038\pi$
$887$	30374.8	1.14981	0.574907	+	0.818219i	$0.305038\pi$
$888$	0	0
$889$	−1893.37	−0.0714303
$890$	0	0
$891$	−1980.05	−0.0744490
$892$	0	0
$893$	4277.91	0.160308
$894$	0	0
$895$	0	0
$896$	0	0
$897$	14526.6	0.540722
$898$	0	0
$899$	22297.5	0.827213
$900$	0	0
$901$	38490.7	1.42321
$902$	0	0
$903$	1577.79	0.0581456
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−35591.3	−1.30297	−0.651483	−	0.758664i	$-0.725853\pi$
$907$	−35591.3	−1.30297	−0.651483	+	0.758664i	$0.725853\pi$
$908$	0	0
$909$	1087.82	0.0396926
$910$	0	0
$911$	−53121.1	−1.93192	−0.965961	−	0.258689i	$-0.916710\pi$
$911$	−53121.1	−1.93192	−0.965961	+	0.258689i	$0.916710\pi$
$912$	0	0
$913$	−2630.19	−0.0953415
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4948.98	0.178222
$918$	0	0
$919$	−23275.5	−0.835461	−0.417731	−	0.908571i	$-0.637174\pi$
$919$	−23275.5	−0.835461	−0.417731	+	0.908571i	$0.637174\pi$
$920$	0	0
$921$	−5932.03	−0.212234
$922$	0	0
$923$	3680.26	0.131243
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10231.0	0.362493
$928$	0	0
$929$	−37535.8	−1.32563	−0.662814	−	0.748784i	$-0.730638\pi$
$929$	−37535.8	−1.32563	−0.662814	+	0.748784i	$0.730638\pi$
$930$	0	0
$931$	3106.62	0.109361
$932$	0	0
$933$	−15729.9	−0.551954
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−5622.57	−0.196031	−0.0980156	−	0.995185i	$-0.531250\pi$
$937$	−5622.57	−0.196031	−0.0980156	+	0.995185i	$0.531250\pi$
$938$	0	0
$939$	−2265.56	−0.0787367
$940$	0	0
$941$	53452.3	1.85175	0.925875	−	0.377831i	$-0.123330\pi$
$941$	53452.3	1.85175	0.925875	+	0.377831i	$0.123330\pi$
$942$	0	0
$943$	15487.6	0.534830
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20301.2	0.696622	0.348311	−	0.937379i	$-0.386755\pi$
$947$	20301.2	0.696622	0.348311	+	0.937379i	$0.386755\pi$
$948$	0	0
$949$	−21372.4	−0.731063
$950$	0	0
$951$	12054.3	0.411028
$952$	0	0
$953$	−46002.1	−1.56365	−0.781823	−	0.623500i	$-0.785710\pi$
$953$	−46002.1	−1.56365	−0.781823	+	0.623500i	$0.785710\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−10302.9	−0.348011
$958$	0	0
$959$	3927.58	0.132250
$960$	0	0
$961$	−4601.73	−0.154467
$962$	0	0
$963$	−9387.71	−0.314138
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12199.2	0.405688	0.202844	−	0.979211i	$-0.434982\pi$
$967$	12199.2	0.405688	0.202844	+	0.979211i	$0.434982\pi$
$968$	0	0
$969$	−3388.94	−0.112351
$970$	0	0
$971$	−37476.7	−1.23860	−0.619302	−	0.785153i	$-0.712584\pi$
$971$	−37476.7	−1.23860	−0.619302	+	0.785153i	$0.712584\pi$
$972$	0	0
$973$	1464.10	0.0482394
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−14592.7	−0.477851	−0.238926	−	0.971038i	$-0.576795\pi$
$977$	−14592.7	−0.477851	−0.238926	+	0.971038i	$0.576795\pi$
$978$	0	0
$979$	−16762.1	−0.547212
$980$	0	0
$981$	17135.2	0.557680
$982$	0	0
$983$	−28500.9	−0.924758	−0.462379	−	0.886682i	$-0.653004\pi$
$983$	−28500.9	−0.924758	−0.462379	+	0.886682i	$0.653004\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2982.33	−0.0961790
$988$	0	0
$989$	−35067.7	−1.12749
$990$	0	0
$991$	−38654.0	−1.23904	−0.619518	−	0.784982i	$-0.712672\pi$
$991$	−38654.0	−1.23904	−0.619518	+	0.784982i	$0.712672\pi$
$992$	0	0
$993$	15893.9	0.507935
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−25637.7	−0.814398	−0.407199	−	0.913339i	$-0.633494\pi$
$997$	−25637.7	−0.814398	−0.407199	+	0.913339i	$0.633494\pi$
$998$	0	0
$999$	−1581.61	−0.0500901

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bt.1.2	yes	3
4.3	odd	2		2400.4.a.bi.1.2	✓	3
5.4	even	2		2400.4.a.bj.1.2	yes	3
20.19	odd	2		2400.4.a.bs.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.4.a.bi.1.2	✓	3	4.3	odd	2
2400.4.a.bj.1.2	yes	3	5.4	even	2
2400.4.a.bs.1.2	yes	3	20.19	odd	2
2400.4.a.bt.1.2	yes	3	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +2.13303 q^{7} +9.00000 q^{9} -24.4450 q^{11} -34.0459 q^{13} +123.069 q^{17} -9.17896 q^{19} +6.39909 q^{21} -142.225 q^{23} +27.0000 q^{27} +140.491 q^{29} +158.711 q^{31} -73.3351 q^{33} -58.5783 q^{37} -102.138 q^{39} -108.895 q^{41} +246.565 q^{43} -466.056 q^{47} -338.450 q^{49} +369.207 q^{51} +312.757 q^{53} -27.5369 q^{57} +410.973 q^{59} -44.5685 q^{61} +19.1973 q^{63} +368.079 q^{67} -426.675 q^{69} -108.097 q^{71} +627.753 q^{73} -52.1419 q^{77} +196.404 q^{79} +81.0000 q^{81} +107.596 q^{83} +421.473 q^{87} +685.708 q^{89} -72.6210 q^{91} +476.134 q^{93} -73.7529 q^{97} -220.005 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} - 3 q^{7} + 27 q^{9} + 44 q^{11} - 13 q^{13} + 36 q^{17} + 71 q^{19} - 9 q^{21} + 160 q^{23} + 81 q^{27} - 184 q^{29} - 59 q^{31} + 132 q^{33} + 350 q^{37} - 39 q^{39} + 166 q^{41} + 341 q^{43}+ \cdots + 396 q^{99}+O(q^{100}) \) 3 * q + 9 * q^3 - 3 * q^7 + 27 * q^9 + 44 * q^11 - 13 * q^13 + 36 * q^17 + 71 * q^19 - 9 * q^21 + 160 * q^23 + 81 * q^27 - 184 * q^29 - 59 * q^31 + 132 * q^33 + 350 * q^37 - 39 * q^39 + 166 * q^41 + 341 * q^43 - 394 * q^47 + 158 * q^49 + 108 * q^51 + 314 * q^53 + 213 * q^57 + 538 * q^59 - 523 * q^61 - 27 * q^63 - 543 * q^67 + 480 * q^69 + 910 * q^71 + 870 * q^73 - 828 * q^77 - 96 * q^79 + 243 * q^81 + 210 * q^83 - 552 * q^87 - 64 * q^89 + 2781 * q^91 - 177 * q^93 + 943 * q^97 + 396 * q^99

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