LMFDB - Embedded newform 1452.4.a.r.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1452,4,Mod(1,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1452 = 2^{2} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1452.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-18,0,-12,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$85.6707733283$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 74x^{4} - 18x^{3} + 1206x^{2} + 2088x - 99 $ x^6 - 2x^5 - 74x^4 - 18x^3 + 1206x^2 + 2088*x - 99
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-5.89143$ of defining polynomial
Character	$\chi$	$=$	1452.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -2.54018 q^{5} -26.1494 q^{7} +9.00000 q^{9} -38.2737 q^{13} +7.62055 q^{15} +106.022 q^{17} +56.8376 q^{19} +78.4481 q^{21} +75.4162 q^{23} -118.547 q^{25} -27.0000 q^{27} +78.1574 q^{29} -141.481 q^{31} +66.4241 q^{35} +258.738 q^{37} +114.821 q^{39} +410.838 q^{41} -64.8313 q^{43} -22.8616 q^{45} +446.234 q^{47} +340.789 q^{49} -318.065 q^{51} -320.890 q^{53} -170.513 q^{57} -50.4518 q^{59} -63.2866 q^{61} -235.344 q^{63} +97.2222 q^{65} -297.845 q^{67} -226.249 q^{69} -315.811 q^{71} -1111.77 q^{73} +355.642 q^{75} -216.539 q^{79} +81.0000 q^{81} -434.074 q^{83} -269.314 q^{85} -234.472 q^{87} -1363.23 q^{89} +1000.83 q^{91} +424.444 q^{93} -144.378 q^{95} +757.176 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 18 q^{3} - 12 q^{5} + 54 q^{9} + 36 q^{15} - 492 q^{23} + 534 q^{25} - 162 q^{27} - 270 q^{31} + 582 q^{37} - 108 q^{45} + 756 q^{47} + 780 q^{49} - 72 q^{53} + 564 q^{59} - 1590 q^{67} + 1476 q^{69}+ \cdots - 4242 q^{97}+O(q^{100}) $ 6 * q - 18 * q^3 - 12 * q^5 + 54 * q^9 + 36 * q^15 - 492 * q^23 + 534 * q^25 - 162 * q^27 - 270 * q^31 + 582 * q^37 - 108 * q^45 + 756 * q^47 + 780 * q^49 - 72 * q^53 + 564 * q^59 - 1590 * q^67 + 1476 * q^69 + 984 * q^71 - 1602 * q^75 + 486 * q^81 - 3324 * q^89 + 2544 * q^91 + 810 * q^93 - 4242 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−2.54018	−0.227201	−0.113600	−	0.993527i	$-0.536238\pi$
$5$	−2.54018	−0.227201	−0.113600	+	0.993527i	$0.536238\pi$
$6$	0	0
$7$	−26.1494	−1.41193	−0.705966	−	0.708246i	$-0.749487\pi$
$7$	−26.1494	−1.41193	−0.705966	+	0.708246i	$0.749487\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−38.2737	−0.816555	−0.408278	−	0.912858i	$-0.633870\pi$
$13$	−38.2737	−0.816555	−0.408278	+	0.912858i	$0.633870\pi$
$14$	0	0
$15$	7.62055	0.131174
$16$	0	0
$17$	106.022	1.51259	0.756295	−	0.654231i	$-0.227007\pi$
$17$	106.022	1.51259	0.756295	+	0.654231i	$0.227007\pi$
$18$	0	0
$19$	56.8376	0.686287	0.343143	−	0.939283i	$-0.388508\pi$
$19$	56.8376	0.686287	0.343143	+	0.939283i	$0.388508\pi$
$20$	0	0
$21$	78.4481	0.815179
$22$	0	0
$23$	75.4162	0.683711	0.341856	−	0.939752i	$-0.388945\pi$
$23$	75.4162	0.683711	0.341856	+	0.939752i	$0.388945\pi$
$24$	0	0
$25$	−118.547	−0.948380
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	78.1574	0.500464	0.250232	−	0.968186i	$-0.419493\pi$
$29$	78.1574	0.500464	0.250232	+	0.968186i	$0.419493\pi$
$30$	0	0
$31$	−141.481	−0.819703	−0.409851	−	0.912152i	$-0.634419\pi$
$31$	−141.481	−0.819703	−0.409851	+	0.912152i	$0.634419\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	66.4241	0.320792
$36$	0	0
$37$	258.738	1.14963	0.574814	−	0.818284i	$-0.305075\pi$
$37$	258.738	1.14963	0.574814	+	0.818284i	$0.305075\pi$
$38$	0	0
$39$	114.821	0.471438
$40$	0	0
$41$	410.838	1.56493	0.782465	−	0.622695i	$-0.213962\pi$
$41$	410.838	1.56493	0.782465	+	0.622695i	$0.213962\pi$
$42$	0	0
$43$	−64.8313	−0.229923	−0.114961	−	0.993370i	$-0.536674\pi$
$43$	−64.8313	−0.229923	−0.114961	+	0.993370i	$0.536674\pi$
$44$	0	0
$45$	−22.8616	−0.0757336
$46$	0	0
$47$	446.234	1.38489	0.692446	−	0.721469i	$-0.256533\pi$
$47$	446.234	1.38489	0.692446	+	0.721469i	$0.256533\pi$
$48$	0	0
$49$	340.789	0.993553
$50$	0	0
$51$	−318.065	−0.873294
$52$	0	0
$53$	−320.890	−0.831654	−0.415827	−	0.909444i	$-0.636508\pi$
$53$	−320.890	−0.831654	−0.415827	+	0.909444i	$0.636508\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−170.513	−0.396228
$58$	0	0
$59$	−50.4518	−0.111327	−0.0556633	−	0.998450i	$-0.517727\pi$
$59$	−50.4518	−0.111327	−0.0556633	+	0.998450i	$0.517727\pi$
$60$	0	0
$61$	−63.2866	−0.132836	−0.0664181	−	0.997792i	$-0.521157\pi$
$61$	−63.2866	−0.132836	−0.0664181	+	0.997792i	$0.521157\pi$
$62$	0	0
$63$	−235.344	−0.470644
$64$	0	0
$65$	97.2222	0.185522
$66$	0	0
$67$	−297.845	−0.543099	−0.271549	−	0.962425i	$-0.587536\pi$
$67$	−297.845	−0.543099	−0.271549	+	0.962425i	$0.587536\pi$
$68$	0	0
$69$	−226.249	−0.394741
$70$	0	0
$71$	−315.811	−0.527886	−0.263943	−	0.964538i	$-0.585023\pi$
$71$	−315.811	−0.527886	−0.263943	+	0.964538i	$0.585023\pi$
$72$	0	0
$73$	−1111.77	−1.78251	−0.891253	−	0.453506i	$-0.850173\pi$
$73$	−1111.77	−1.78251	−0.891253	+	0.453506i	$0.850173\pi$
$74$	0	0
$75$	355.642	0.547547
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−216.539	−0.308387	−0.154193	−	0.988041i	$-0.549278\pi$
$79$	−216.539	−0.308387	−0.154193	+	0.988041i	$0.549278\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−434.074	−0.574046	−0.287023	−	0.957924i	$-0.592666\pi$
$83$	−434.074	−0.574046	−0.287023	+	0.957924i	$0.592666\pi$
$84$	0	0
$85$	−269.314	−0.343662
$86$	0	0
$87$	−234.472	−0.288943
$88$	0	0
$89$	−1363.23	−1.62362	−0.811809	−	0.583924i	$-0.801517\pi$
$89$	−1363.23	−1.62362	−0.811809	+	0.583924i	$0.801517\pi$
$90$	0	0
$91$	1000.83	1.15292
$92$	0	0
$93$	424.444	0.473256
$94$	0	0
$95$	−144.378	−0.155925
$96$	0	0
$97$	757.176	0.792573	0.396287	−	0.918127i	$-0.370299\pi$
$97$	757.176	0.792573	0.396287	+	0.918127i	$0.370299\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1765.66	1.73950	0.869750	−	0.493493i	$-0.164280\pi$
$101$	1765.66	1.73950	0.869750	+	0.493493i	$0.164280\pi$
$102$	0	0
$103$	−1350.20	−1.29164	−0.645821	−	0.763489i	$-0.723485\pi$
$103$	−1350.20	−1.29164	−0.645821	+	0.763489i	$0.723485\pi$
$104$	0	0
$105$	−199.272	−0.185209
$106$	0	0
$107$	616.304	0.556825	0.278413	−	0.960462i	$-0.410192\pi$
$107$	616.304	0.556825	0.278413	+	0.960462i	$0.410192\pi$
$108$	0	0
$109$	1340.39	1.17785	0.588927	−	0.808186i	$-0.299550\pi$
$109$	1340.39	1.17785	0.588927	+	0.808186i	$0.299550\pi$
$110$	0	0
$111$	−776.213	−0.663738
$112$	0	0
$113$	1631.82	1.35849	0.679243	−	0.733913i	$-0.262308\pi$
$113$	1631.82	1.35849	0.679243	+	0.733913i	$0.262308\pi$
$114$	0	0
$115$	−191.571	−0.155340
$116$	0	0
$117$	−344.463	−0.272185
$118$	0	0
$119$	−2772.40	−2.13567
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−1232.51	−0.903513
$124$	0	0
$125$	618.655	0.442674
$126$	0	0
$127$	298.202	0.208356	0.104178	−	0.994559i	$-0.466779\pi$
$127$	298.202	0.208356	0.104178	+	0.994559i	$0.466779\pi$
$128$	0	0
$129$	194.494	0.132746
$130$	0	0
$131$	201.576	0.134441	0.0672206	−	0.997738i	$-0.478587\pi$
$131$	201.576	0.134441	0.0672206	+	0.997738i	$0.478587\pi$
$132$	0	0
$133$	−1486.27	−0.968990
$134$	0	0
$135$	68.5849	0.0437248
$136$	0	0
$137$	−1390.51	−0.867150	−0.433575	−	0.901117i	$-0.642748\pi$
$137$	−1390.51	−0.867150	−0.433575	+	0.901117i	$0.642748\pi$
$138$	0	0
$139$	−1636.71	−0.998734	−0.499367	−	0.866390i	$-0.666434\pi$
$139$	−1636.71	−0.998734	−0.499367	+	0.866390i	$0.666434\pi$
$140$	0	0
$141$	−1338.70	−0.799568
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−198.534	−0.113706
$146$	0	0
$147$	−1022.37	−0.573628
$148$	0	0
$149$	−1112.20	−0.611510	−0.305755	−	0.952110i	$-0.598909\pi$
$149$	−1112.20	−0.611510	−0.305755	+	0.952110i	$0.598909\pi$
$150$	0	0
$151$	1696.08	0.914075	0.457038	−	0.889447i	$-0.348910\pi$
$151$	1696.08	0.914075	0.457038	+	0.889447i	$0.348910\pi$
$152$	0	0
$153$	954.195	0.504197
$154$	0	0
$155$	359.388	0.186237
$156$	0	0
$157$	−1888.14	−0.959810	−0.479905	−	0.877320i	$-0.659329\pi$
$157$	−1888.14	−0.959810	−0.479905	+	0.877320i	$0.659329\pi$
$158$	0	0
$159$	962.671	0.480156
$160$	0	0
$161$	−1972.08	−0.965354
$162$	0	0
$163$	−3346.75	−1.60821	−0.804104	−	0.594488i	$-0.797355\pi$
$163$	−3346.75	−1.60821	−0.804104	+	0.594488i	$0.797355\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2966.19	−1.37443	−0.687217	−	0.726453i	$-0.741168\pi$
$167$	−2966.19	−1.37443	−0.687217	+	0.726453i	$0.741168\pi$
$168$	0	0
$169$	−732.123	−0.333238
$170$	0	0
$171$	511.539	0.228762
$172$	0	0
$173$	1957.75	0.860375	0.430188	−	0.902740i	$-0.358447\pi$
$173$	1957.75	0.860375	0.430188	+	0.902740i	$0.358447\pi$
$174$	0	0
$175$	3099.94	1.33905
$176$	0	0
$177$	151.355	0.0642744
$178$	0	0
$179$	−4415.78	−1.84386	−0.921930	−	0.387357i	$-0.873388\pi$
$179$	−4415.78	−1.84386	−0.921930	+	0.387357i	$0.873388\pi$
$180$	0	0
$181$	2412.52	0.990723	0.495362	−	0.868687i	$-0.335036\pi$
$181$	2412.52	0.990723	0.495362	+	0.868687i	$0.335036\pi$
$182$	0	0
$183$	189.860	0.0766931
$184$	0	0
$185$	−657.241	−0.261196
$186$	0	0
$187$	0	0
$188$	0	0
$189$	706.032	0.271726
$190$	0	0
$191$	−2243.91	−0.850072	−0.425036	−	0.905177i	$-0.639738\pi$
$191$	−2243.91	−0.850072	−0.425036	+	0.905177i	$0.639738\pi$
$192$	0	0
$193$	−1838.33	−0.685628	−0.342814	−	0.939403i	$-0.611380\pi$
$193$	−1838.33	−0.685628	−0.342814	+	0.939403i	$0.611380\pi$
$194$	0	0
$195$	−291.667	−0.107111
$196$	0	0
$197$	3351.47	1.21209	0.606047	−	0.795429i	$-0.292754\pi$
$197$	3351.47	1.21209	0.606047	+	0.795429i	$0.292754\pi$
$198$	0	0
$199$	−1032.38	−0.367758	−0.183879	−	0.982949i	$-0.558865\pi$
$199$	−1032.38	−0.367758	−0.183879	+	0.982949i	$0.558865\pi$
$200$	0	0
$201$	893.536	0.313558
$202$	0	0
$203$	−2043.77	−0.706622
$204$	0	0
$205$	−1043.60	−0.355553
$206$	0	0
$207$	678.746	0.227904
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1360.30	−0.443824	−0.221912	−	0.975067i	$-0.571230\pi$
$211$	−1360.30	−0.443824	−0.221912	+	0.975067i	$0.571230\pi$
$212$	0	0
$213$	947.433	0.304775
$214$	0	0
$215$	164.683	0.0522386
$216$	0	0
$217$	3699.64	1.15736
$218$	0	0
$219$	3335.31	1.02913
$220$	0	0
$221$	−4057.84	−1.23511
$222$	0	0
$223$	2909.37	0.873658	0.436829	−	0.899544i	$-0.356101\pi$
$223$	2909.37	0.873658	0.436829	+	0.899544i	$0.356101\pi$
$224$	0	0
$225$	−1066.93	−0.316127
$226$	0	0
$227$	−5121.69	−1.49753	−0.748763	−	0.662837i	$-0.769352\pi$
$227$	−5121.69	−1.49753	−0.748763	+	0.662837i	$0.769352\pi$
$228$	0	0
$229$	−5803.70	−1.67476	−0.837378	−	0.546624i	$-0.815913\pi$
$229$	−5803.70	−1.67476	−0.837378	+	0.546624i	$0.815913\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1483.89	0.417224	0.208612	−	0.977999i	$-0.433105\pi$
$233$	1483.89	0.417224	0.208612	+	0.977999i	$0.433105\pi$
$234$	0	0
$235$	−1133.52	−0.314649
$236$	0	0
$237$	649.617	0.178047
$238$	0	0
$239$	5090.19	1.37764	0.688822	−	0.724931i	$-0.258128\pi$
$239$	5090.19	1.37764	0.688822	+	0.724931i	$0.258128\pi$
$240$	0	0
$241$	2272.07	0.607289	0.303645	−	0.952785i	$-0.401796\pi$
$241$	2272.07	0.607289	0.303645	+	0.952785i	$0.401796\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−865.665	−0.225736
$246$	0	0
$247$	−2175.39	−0.560391
$248$	0	0
$249$	1302.22	0.331426
$250$	0	0
$251$	504.097	0.126766	0.0633831	−	0.997989i	$-0.479811\pi$
$251$	504.097	0.126766	0.0633831	+	0.997989i	$0.479811\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	807.943	0.198413
$256$	0	0
$257$	−8044.64	−1.95257	−0.976286	−	0.216485i	$-0.930541\pi$
$257$	−8044.64	−1.95257	−0.976286	+	0.216485i	$0.930541\pi$
$258$	0	0
$259$	−6765.82	−1.62320
$260$	0	0
$261$	703.417	0.166821
$262$	0	0
$263$	1070.47	0.250980	0.125490	−	0.992095i	$-0.459950\pi$
$263$	1070.47	0.250980	0.125490	+	0.992095i	$0.459950\pi$
$264$	0	0
$265$	815.120	0.188953
$266$	0	0
$267$	4089.69	0.937396
$268$	0	0
$269$	475.225	0.107714	0.0538568	−	0.998549i	$-0.482849\pi$
$269$	475.225	0.107714	0.0538568	+	0.998549i	$0.482849\pi$
$270$	0	0
$271$	3752.90	0.841228	0.420614	−	0.907240i	$-0.361815\pi$
$271$	3752.90	0.841228	0.420614	+	0.907240i	$0.361815\pi$
$272$	0	0
$273$	−3002.50	−0.665639
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1294.84	−0.280865	−0.140433	−	0.990090i	$-0.544849\pi$
$277$	−1294.84	−0.280865	−0.140433	+	0.990090i	$0.544849\pi$
$278$	0	0
$279$	−1273.33	−0.273234
$280$	0	0
$281$	−5245.82	−1.11366	−0.556832	−	0.830625i	$-0.687983\pi$
$281$	−5245.82	−1.11366	−0.556832	+	0.830625i	$0.687983\pi$
$282$	0	0
$283$	−7793.87	−1.63709	−0.818547	−	0.574439i	$-0.805220\pi$
$283$	−7793.87	−1.63709	−0.818547	+	0.574439i	$0.805220\pi$
$284$	0	0
$285$	433.134	0.0900233
$286$	0	0
$287$	−10743.1	−2.20958
$288$	0	0
$289$	6327.59	1.28793
$290$	0	0
$291$	−2271.53	−0.457592
$292$	0	0
$293$	−9089.55	−1.81235	−0.906173	−	0.422908i	$-0.861009\pi$
$293$	−9089.55	−1.81235	−0.906173	+	0.422908i	$0.861009\pi$
$294$	0	0
$295$	128.157	0.0252935
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2886.46	−0.558288
$300$	0	0
$301$	1695.30	0.324635
$302$	0	0
$303$	−5296.97	−1.00430
$304$	0	0
$305$	160.759	0.0301805
$306$	0	0
$307$	6200.52	1.15271	0.576356	−	0.817199i	$-0.304474\pi$
$307$	6200.52	1.15271	0.576356	+	0.817199i	$0.304474\pi$
$308$	0	0
$309$	4050.60	0.745730
$310$	0	0
$311$	−7229.44	−1.31815	−0.659073	−	0.752079i	$-0.729051\pi$
$311$	−7229.44	−1.31815	−0.659073	+	0.752079i	$0.729051\pi$
$312$	0	0
$313$	1588.15	0.286798	0.143399	−	0.989665i	$-0.454197\pi$
$313$	1588.15	0.286798	0.143399	+	0.989665i	$0.454197\pi$
$314$	0	0
$315$	597.817	0.106931
$316$	0	0
$317$	6915.07	1.22520	0.612601	−	0.790392i	$-0.290123\pi$
$317$	6915.07	1.22520	0.612601	+	0.790392i	$0.290123\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1848.91	−0.321483
$322$	0	0
$323$	6026.02	1.03807
$324$	0	0
$325$	4537.25	0.774404
$326$	0	0
$327$	−4021.17	−0.680035
$328$	0	0
$329$	−11668.7	−1.95537
$330$	0	0
$331$	−9801.17	−1.62756	−0.813778	−	0.581175i	$-0.802593\pi$
$331$	−9801.17	−1.62756	−0.813778	+	0.581175i	$0.802593\pi$
$332$	0	0
$333$	2328.64	0.383209
$334$	0	0
$335$	756.582	0.123392
$336$	0	0
$337$	−9503.73	−1.53621	−0.768103	−	0.640327i	$-0.778799\pi$
$337$	−9503.73	−1.53621	−0.768103	+	0.640327i	$0.778799\pi$
$338$	0	0
$339$	−4895.47	−0.784323
$340$	0	0
$341$	0	0
$342$	0	0
$343$	57.8275	0.00910318
$344$	0	0
$345$	574.713	0.0896855
$346$	0	0
$347$	2040.66	0.315701	0.157851	−	0.987463i	$-0.449544\pi$
$347$	2040.66	0.315701	0.157851	+	0.987463i	$0.449544\pi$
$348$	0	0
$349$	4067.71	0.623896	0.311948	−	0.950099i	$-0.399019\pi$
$349$	4067.71	0.623896	0.311948	+	0.950099i	$0.399019\pi$
$350$	0	0
$351$	1033.39	0.157146
$352$	0	0
$353$	−5470.21	−0.824787	−0.412394	−	0.911006i	$-0.635307\pi$
$353$	−5470.21	−0.824787	−0.412394	+	0.911006i	$0.635307\pi$
$354$	0	0
$355$	802.218	0.119936
$356$	0	0
$357$	8317.19	1.23303
$358$	0	0
$359$	6413.83	0.942922	0.471461	−	0.881887i	$-0.343727\pi$
$359$	6413.83	0.942922	0.471461	+	0.881887i	$0.343727\pi$
$360$	0	0
$361$	−3628.48	−0.529011
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2824.10	0.404987
$366$	0	0
$367$	−7863.74	−1.11848	−0.559242	−	0.829004i	$-0.688908\pi$
$367$	−7863.74	−1.11848	−0.559242	+	0.829004i	$0.688908\pi$
$368$	0	0
$369$	3697.54	0.521643
$370$	0	0
$371$	8391.08	1.17424
$372$	0	0
$373$	2794.19	0.387875	0.193938	−	0.981014i	$-0.437874\pi$
$373$	2794.19	0.387875	0.193938	+	0.981014i	$0.437874\pi$
$374$	0	0
$375$	−1855.97	−0.255578
$376$	0	0
$377$	−2991.37	−0.408657
$378$	0	0
$379$	−3318.55	−0.449768	−0.224884	−	0.974385i	$-0.572200\pi$
$379$	−3318.55	−0.449768	−0.224884	+	0.974385i	$0.572200\pi$
$380$	0	0
$381$	−894.606	−0.120294
$382$	0	0
$383$	−4808.32	−0.641498	−0.320749	−	0.947164i	$-0.603935\pi$
$383$	−4808.32	−0.641498	−0.320749	+	0.947164i	$0.603935\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−583.482	−0.0766409
$388$	0	0
$389$	2474.39	0.322511	0.161255	−	0.986913i	$-0.448446\pi$
$389$	2474.39	0.322511	0.161255	+	0.986913i	$0.448446\pi$
$390$	0	0
$391$	7995.75	1.03418
$392$	0	0
$393$	−604.729	−0.0776197
$394$	0	0
$395$	550.049	0.0700657
$396$	0	0
$397$	11862.7	1.49967	0.749836	−	0.661624i	$-0.230132\pi$
$397$	11862.7	1.49967	0.749836	+	0.661624i	$0.230132\pi$
$398$	0	0
$399$	4458.80	0.559447
$400$	0	0
$401$	−5696.44	−0.709393	−0.354697	−	0.934981i	$-0.615416\pi$
$401$	−5696.44	−0.709393	−0.354697	+	0.934981i	$0.615416\pi$
$402$	0	0
$403$	5415.01	0.669332
$404$	0	0
$405$	−205.755	−0.0252445
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−40.5788	−0.00490585	−0.00245292	−	0.999997i	$-0.500781\pi$
$409$	−40.5788	−0.00490585	−0.00245292	+	0.999997i	$0.500781\pi$
$410$	0	0
$411$	4171.54	0.500649
$412$	0	0
$413$	1319.28	0.157186
$414$	0	0
$415$	1102.63	0.130424
$416$	0	0
$417$	4910.14	0.576620
$418$	0	0
$419$	14629.9	1.70577	0.852887	−	0.522096i	$-0.174850\pi$
$419$	14629.9	1.70577	0.852887	+	0.522096i	$0.174850\pi$
$420$	0	0
$421$	11909.0	1.37865	0.689325	−	0.724452i	$-0.257907\pi$
$421$	11909.0	1.37865	0.689325	+	0.724452i	$0.257907\pi$
$422$	0	0
$423$	4016.11	0.461631
$424$	0	0
$425$	−12568.6	−1.43451
$426$	0	0
$427$	1654.90	0.187556
$428$	0	0
$429$	0	0
$430$	0	0
$431$	15478.7	1.72989	0.864947	−	0.501864i	$-0.167352\pi$
$431$	15478.7	1.72989	0.864947	+	0.501864i	$0.167352\pi$
$432$	0	0
$433$	7213.51	0.800599	0.400299	−	0.916384i	$-0.368906\pi$
$433$	7213.51	0.800599	0.400299	+	0.916384i	$0.368906\pi$
$434$	0	0
$435$	595.602	0.0656481
$436$	0	0
$437$	4286.48	0.469222
$438$	0	0
$439$	−546.724	−0.0594390	−0.0297195	−	0.999558i	$-0.509461\pi$
$439$	−546.724	−0.0594390	−0.0297195	+	0.999558i	$0.509461\pi$
$440$	0	0
$441$	3067.10	0.331184
$442$	0	0
$443$	−12830.7	−1.37608	−0.688040	−	0.725672i	$-0.741529\pi$
$443$	−12830.7	−1.37608	−0.688040	+	0.725672i	$0.741529\pi$
$444$	0	0
$445$	3462.85	0.368887
$446$	0	0
$447$	3336.60	0.353056
$448$	0	0
$449$	−2643.74	−0.277875	−0.138938	−	0.990301i	$-0.544369\pi$
$449$	−2643.74	−0.277875	−0.138938	+	0.990301i	$0.544369\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−5088.25	−0.527742
$454$	0	0
$455$	−2542.30	−0.261945
$456$	0	0
$457$	13267.2	1.35801	0.679007	−	0.734132i	$-0.262411\pi$
$457$	13267.2	1.35801	0.679007	+	0.734132i	$0.262411\pi$
$458$	0	0
$459$	−2862.59	−0.291098
$460$	0	0
$461$	−15562.1	−1.57223	−0.786115	−	0.618080i	$-0.787911\pi$
$461$	−15562.1	−1.57223	−0.786115	+	0.618080i	$0.787911\pi$
$462$	0	0
$463$	7233.58	0.726075	0.363038	−	0.931774i	$-0.381740\pi$
$463$	7233.58	0.726075	0.363038	+	0.931774i	$0.381740\pi$
$464$	0	0
$465$	−1078.16	−0.107524
$466$	0	0
$467$	−16335.6	−1.61867	−0.809335	−	0.587347i	$-0.800173\pi$
$467$	−16335.6	−1.61867	−0.809335	+	0.587347i	$0.800173\pi$
$468$	0	0
$469$	7788.46	0.766818
$470$	0	0
$471$	5664.43	0.554147
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−6737.96	−0.650860
$476$	0	0
$477$	−2888.01	−0.277218
$478$	0	0
$479$	2714.83	0.258964	0.129482	−	0.991582i	$-0.458668\pi$
$479$	2714.83	0.258964	0.129482	+	0.991582i	$0.458668\pi$
$480$	0	0
$481$	−9902.85	−0.938734
$482$	0	0
$483$	5916.25	0.557348
$484$	0	0
$485$	−1923.37	−0.180073
$486$	0	0
$487$	−17888.5	−1.66448	−0.832242	−	0.554413i	$-0.812943\pi$
$487$	−17888.5	−1.66448	−0.832242	+	0.554413i	$0.812943\pi$
$488$	0	0
$489$	10040.3	0.928500
$490$	0	0
$491$	4786.96	0.439985	0.219992	−	0.975502i	$-0.429397\pi$
$491$	4786.96	0.439985	0.219992	+	0.975502i	$0.429397\pi$
$492$	0	0
$493$	8286.38	0.756997
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8258.26	0.745339
$498$	0	0
$499$	16093.0	1.44373	0.721863	−	0.692036i	$-0.243286\pi$
$499$	16093.0	1.44373	0.721863	+	0.692036i	$0.243286\pi$
$500$	0	0
$501$	8898.56	0.793529
$502$	0	0
$503$	11735.7	1.04030	0.520149	−	0.854076i	$-0.325876\pi$
$503$	11735.7	1.04030	0.520149	+	0.854076i	$0.325876\pi$
$504$	0	0
$505$	−4485.09	−0.395216
$506$	0	0
$507$	2196.37	0.192395
$508$	0	0
$509$	11803.6	1.02787	0.513934	−	0.857830i	$-0.328188\pi$
$509$	11803.6	1.02787	0.513934	+	0.857830i	$0.328188\pi$
$510$	0	0
$511$	29072.1	2.51678
$512$	0	0
$513$	−1534.62	−0.132076
$514$	0	0
$515$	3429.76	0.293462
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−5873.25	−0.496738
$520$	0	0
$521$	13862.4	1.16569	0.582844	−	0.812584i	$-0.301940\pi$
$521$	13862.4	1.16569	0.582844	+	0.812584i	$0.301940\pi$
$522$	0	0
$523$	−10358.4	−0.866041	−0.433021	−	0.901384i	$-0.642552\pi$
$523$	−10358.4	−0.866041	−0.433021	+	0.901384i	$0.642552\pi$
$524$	0	0
$525$	−9299.82	−0.773100
$526$	0	0
$527$	−15000.1	−1.23987
$528$	0	0
$529$	−6479.40	−0.532539
$530$	0	0
$531$	−454.066	−0.0371088
$532$	0	0
$533$	−15724.3	−1.27785
$534$	0	0
$535$	−1565.52	−0.126511
$536$	0	0
$537$	13247.3	1.06455
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−4978.41	−0.395635	−0.197818	−	0.980239i	$-0.563385\pi$
$541$	−4978.41	−0.395635	−0.197818	+	0.980239i	$0.563385\pi$
$542$	0	0
$543$	−7237.55	−0.571994
$544$	0	0
$545$	−3404.84	−0.267610
$546$	0	0
$547$	−23106.8	−1.80617	−0.903085	−	0.429461i	$-0.858704\pi$
$547$	−23106.8	−1.80617	−0.903085	+	0.429461i	$0.858704\pi$
$548$	0	0
$549$	−569.579	−0.0442788
$550$	0	0
$551$	4442.28	0.343462
$552$	0	0
$553$	5662.35	0.435421
$554$	0	0
$555$	1971.72	0.150802
$556$	0	0
$557$	5777.91	0.439530	0.219765	−	0.975553i	$-0.429471\pi$
$557$	5777.91	0.439530	0.219765	+	0.975553i	$0.429471\pi$
$558$	0	0
$559$	2481.33	0.187745
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11244.8	0.841764	0.420882	−	0.907115i	$-0.361721\pi$
$563$	11244.8	0.841764	0.420882	+	0.907115i	$0.361721\pi$
$564$	0	0
$565$	−4145.13	−0.308649
$566$	0	0
$567$	−2118.10	−0.156881
$568$	0	0
$569$	6793.91	0.500555	0.250277	−	0.968174i	$-0.419478\pi$
$569$	6793.91	0.500555	0.250277	+	0.968174i	$0.419478\pi$
$570$	0	0
$571$	−2275.66	−0.166784	−0.0833919	−	0.996517i	$-0.526575\pi$
$571$	−2275.66	−0.166784	−0.0833919	+	0.996517i	$0.526575\pi$
$572$	0	0
$573$	6731.73	0.490789
$574$	0	0
$575$	−8940.40	−0.648418
$576$	0	0
$577$	14473.4	1.04425	0.522127	−	0.852868i	$-0.325139\pi$
$577$	14473.4	1.04425	0.522127	+	0.852868i	$0.325139\pi$
$578$	0	0
$579$	5515.00	0.395847
$580$	0	0
$581$	11350.8	0.810514
$582$	0	0
$583$	0	0
$584$	0	0
$585$	875.000	0.0618407
$586$	0	0
$587$	−6820.80	−0.479599	−0.239800	−	0.970822i	$-0.577082\pi$
$587$	−6820.80	−0.479599	−0.239800	+	0.970822i	$0.577082\pi$
$588$	0	0
$589$	−8041.46	−0.562551
$590$	0	0
$591$	−10054.4	−0.699803
$592$	0	0
$593$	−25294.8	−1.75166	−0.875830	−	0.482620i	$-0.839685\pi$
$593$	−25294.8	−1.75166	−0.875830	+	0.482620i	$0.839685\pi$
$594$	0	0
$595$	7042.40	0.485227
$596$	0	0
$597$	3097.15	0.212325
$598$	0	0
$599$	−29190.2	−1.99111	−0.995557	−	0.0941563i	$-0.969985\pi$
$599$	−29190.2	−1.99111	−0.995557	+	0.0941563i	$0.969985\pi$
$600$	0	0
$601$	25784.6	1.75005	0.875023	−	0.484081i	$-0.160846\pi$
$601$	25784.6	1.75005	0.875023	+	0.484081i	$0.160846\pi$
$602$	0	0
$603$	−2680.61	−0.181033
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−19498.5	−1.30382	−0.651910	−	0.758296i	$-0.726032\pi$
$607$	−19498.5	−1.30382	−0.651910	+	0.758296i	$0.726032\pi$
$608$	0	0
$609$	6131.30	0.407968
$610$	0	0
$611$	−17079.0	−1.13084
$612$	0	0
$613$	−9010.88	−0.593713	−0.296856	−	0.954922i	$-0.595938\pi$
$613$	−9010.88	−0.593713	−0.296856	+	0.954922i	$0.595938\pi$
$614$	0	0
$615$	3130.81	0.205279
$616$	0	0
$617$	−10980.3	−0.716453	−0.358227	−	0.933635i	$-0.616618\pi$
$617$	−10980.3	−0.716453	−0.358227	+	0.933635i	$0.616618\pi$
$618$	0	0
$619$	12017.4	0.780322	0.390161	−	0.920747i	$-0.372419\pi$
$619$	12017.4	0.780322	0.390161	+	0.920747i	$0.372419\pi$
$620$	0	0
$621$	−2036.24	−0.131580
$622$	0	0
$623$	35647.5	2.29244
$624$	0	0
$625$	13246.9	0.847804
$626$	0	0
$627$	0	0
$628$	0	0
$629$	27431.8	1.73891
$630$	0	0
$631$	−23630.7	−1.49084	−0.745422	−	0.666593i	$-0.767752\pi$
$631$	−23630.7	−1.49084	−0.745422	+	0.666593i	$0.767752\pi$
$632$	0	0
$633$	4080.90	0.256242
$634$	0	0
$635$	−757.488	−0.0473386
$636$	0	0
$637$	−13043.2	−0.811291
$638$	0	0
$639$	−2842.30	−0.175962
$640$	0	0
$641$	−3972.80	−0.244799	−0.122400	−	0.992481i	$-0.539059\pi$
$641$	−3972.80	−0.244799	−0.122400	+	0.992481i	$0.539059\pi$
$642$	0	0
$643$	−32135.8	−1.97094	−0.985469	−	0.169855i	$-0.945670\pi$
$643$	−32135.8	−1.97094	−0.985469	+	0.169855i	$0.945670\pi$
$644$	0	0
$645$	−494.050	−0.0301600
$646$	0	0
$647$	30595.4	1.85909	0.929544	−	0.368711i	$-0.120201\pi$
$647$	30595.4	1.85909	0.929544	+	0.368711i	$0.120201\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−11098.9	−0.668205
$652$	0	0
$653$	18867.1	1.13067	0.565333	−	0.824863i	$-0.308748\pi$
$653$	18867.1	1.13067	0.565333	+	0.824863i	$0.308748\pi$
$654$	0	0
$655$	−512.041	−0.0305452
$656$	0	0
$657$	−10005.9	−0.594169
$658$	0	0
$659$	−14577.4	−0.861693	−0.430847	−	0.902425i	$-0.641785\pi$
$659$	−14577.4	−0.861693	−0.430847	+	0.902425i	$0.641785\pi$
$660$	0	0
$661$	27302.3	1.60656	0.803280	−	0.595601i	$-0.203086\pi$
$661$	27302.3	1.60656	0.803280	+	0.595601i	$0.203086\pi$
$662$	0	0
$663$	12173.5	0.713093
$664$	0	0
$665$	3775.39	0.220155
$666$	0	0
$667$	5894.33	0.342173
$668$	0	0
$669$	−8728.11	−0.504407
$670$	0	0
$671$	0	0
$672$	0	0
$673$	9449.41	0.541230	0.270615	−	0.962688i	$-0.412773\pi$
$673$	9449.41	0.541230	0.270615	+	0.962688i	$0.412773\pi$
$674$	0	0
$675$	3200.78	0.182516
$676$	0	0
$677$	12013.6	0.682008	0.341004	−	0.940062i	$-0.389233\pi$
$677$	12013.6	0.682008	0.341004	+	0.940062i	$0.389233\pi$
$678$	0	0
$679$	−19799.7	−1.11906
$680$	0	0
$681$	15365.1	0.864597
$682$	0	0
$683$	17624.3	0.987374	0.493687	−	0.869640i	$-0.335649\pi$
$683$	17624.3	0.987374	0.493687	+	0.869640i	$0.335649\pi$
$684$	0	0
$685$	3532.16	0.197017
$686$	0	0
$687$	17411.1	0.966921
$688$	0	0
$689$	12281.7	0.679092
$690$	0	0
$691$	19020.2	1.04713	0.523563	−	0.851987i	$-0.324603\pi$
$691$	19020.2	1.04713	0.523563	+	0.851987i	$0.324603\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4157.55	0.226913
$696$	0	0
$697$	43557.7	2.36710
$698$	0	0
$699$	−4451.68	−0.240884
$700$	0	0
$701$	4820.48	0.259725	0.129862	−	0.991532i	$-0.458546\pi$
$701$	4820.48	0.259725	0.129862	+	0.991532i	$0.458546\pi$
$702$	0	0
$703$	14706.0	0.788974
$704$	0	0
$705$	3400.55	0.181663
$706$	0	0
$707$	−46170.8	−2.45606
$708$	0	0
$709$	9785.59	0.518343	0.259172	−	0.965831i	$-0.416550\pi$
$709$	9785.59	0.518343	0.259172	+	0.965831i	$0.416550\pi$
$710$	0	0
$711$	−1948.85	−0.102796
$712$	0	0
$713$	−10670.0	−0.560440
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−15270.6	−0.795383
$718$	0	0
$719$	−27008.2	−1.40088	−0.700442	−	0.713709i	$-0.747014\pi$
$719$	−27008.2	−1.40088	−0.700442	+	0.713709i	$0.747014\pi$
$720$	0	0
$721$	35306.9	1.82371
$722$	0	0
$723$	−6816.20	−0.350619
$724$	0	0
$725$	−9265.36	−0.474630
$726$	0	0
$727$	22367.8	1.14109	0.570547	−	0.821265i	$-0.306731\pi$
$727$	22367.8	1.14109	0.570547	+	0.821265i	$0.306731\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−6873.52	−0.347779
$732$	0	0
$733$	−12665.6	−0.638218	−0.319109	−	0.947718i	$-0.603384\pi$
$733$	−12665.6	−0.638218	−0.319109	+	0.947718i	$0.603384\pi$
$734$	0	0
$735$	2597.00	0.130329
$736$	0	0
$737$	0	0
$738$	0	0
$739$	10981.0	0.546606	0.273303	−	0.961928i	$-0.411884\pi$
$739$	10981.0	0.546606	0.273303	+	0.961928i	$0.411884\pi$
$740$	0	0
$741$	6526.16	0.323542
$742$	0	0
$743$	−7465.40	−0.368613	−0.184306	−	0.982869i	$-0.559004\pi$
$743$	−7465.40	−0.368613	−0.184306	+	0.982869i	$0.559004\pi$
$744$	0	0
$745$	2825.19	0.138936
$746$	0	0
$747$	−3906.67	−0.191349
$748$	0	0
$749$	−16115.9	−0.786200
$750$	0	0
$751$	−12937.3	−0.628616	−0.314308	−	0.949321i	$-0.601772\pi$
$751$	−12937.3	−0.628616	−0.314308	+	0.949321i	$0.601772\pi$
$752$	0	0
$753$	−1512.29	−0.0731885
$754$	0	0
$755$	−4308.36	−0.207679
$756$	0	0
$757$	17792.8	0.854282	0.427141	−	0.904185i	$-0.359521\pi$
$757$	17792.8	0.854282	0.427141	+	0.904185i	$0.359521\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8522.14	−0.405949	−0.202975	−	0.979184i	$-0.565061\pi$
$761$	−8522.14	−0.405949	−0.202975	+	0.979184i	$0.565061\pi$
$762$	0	0
$763$	−35050.4	−1.66305
$764$	0	0
$765$	−2423.83	−0.114554
$766$	0	0
$767$	1930.98	0.0909043
$768$	0	0
$769$	23114.2	1.08390	0.541950	−	0.840411i	$-0.317686\pi$
$769$	23114.2	1.08390	0.541950	+	0.840411i	$0.317686\pi$
$770$	0	0
$771$	24133.9	1.12732
$772$	0	0
$773$	−33994.2	−1.58174	−0.790871	−	0.611982i	$-0.790372\pi$
$773$	−33994.2	−1.58174	−0.790871	+	0.611982i	$0.790372\pi$
$774$	0	0
$775$	16772.2	0.777390
$776$	0	0
$777$	20297.5	0.937153
$778$	0	0
$779$	23351.1	1.07399
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−2110.25	−0.0963144
$784$	0	0
$785$	4796.23	0.218070
$786$	0	0
$787$	−28390.5	−1.28591	−0.642956	−	0.765903i	$-0.722292\pi$
$787$	−28390.5	−1.28591	−0.642956	+	0.765903i	$0.722292\pi$
$788$	0	0
$789$	−3211.40	−0.144904
$790$	0	0
$791$	−42671.1	−1.91809
$792$	0	0
$793$	2422.21	0.108468
$794$	0	0
$795$	−2445.36	−0.109092
$796$	0	0
$797$	17874.1	0.794394	0.397197	−	0.917733i	$-0.369983\pi$
$797$	17874.1	0.794394	0.397197	+	0.917733i	$0.369983\pi$
$798$	0	0
$799$	47310.5	2.09477
$800$	0	0
$801$	−12269.1	−0.541206
$802$	0	0
$803$	0	0
$804$	0	0
$805$	5009.46	0.219329
$806$	0	0
$807$	−1425.67	−0.0621885
$808$	0	0
$809$	28314.9	1.23053	0.615264	−	0.788321i	$-0.289049\pi$
$809$	28314.9	1.23053	0.615264	+	0.788321i	$0.289049\pi$
$810$	0	0
$811$	17968.1	0.777983	0.388991	−	0.921241i	$-0.372824\pi$
$811$	17968.1	0.777983	0.388991	+	0.921241i	$0.372824\pi$
$812$	0	0
$813$	−11258.7	−0.485683
$814$	0	0
$815$	8501.37	0.365386
$816$	0	0
$817$	−3684.86	−0.157793
$818$	0	0
$819$	9007.49	0.384307
$820$	0	0
$821$	−24621.7	−1.04666	−0.523328	−	0.852131i	$-0.675310\pi$
$821$	−24621.7	−1.04666	−0.523328	+	0.852131i	$0.675310\pi$
$822$	0	0
$823$	−39965.3	−1.69271	−0.846356	−	0.532618i	$-0.821208\pi$
$823$	−39965.3	−1.69271	−0.846356	+	0.532618i	$0.821208\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34404.2	−1.44662	−0.723308	−	0.690526i	$-0.757379\pi$
$827$	−34404.2	−1.44662	−0.723308	+	0.690526i	$0.757379\pi$
$828$	0	0
$829$	18352.3	0.768878	0.384439	−	0.923150i	$-0.374395\pi$
$829$	18352.3	0.768878	0.384439	+	0.923150i	$0.374395\pi$
$830$	0	0
$831$	3884.53	0.162158
$832$	0	0
$833$	36131.0	1.50284
$834$	0	0
$835$	7534.65	0.312272
$836$	0	0
$837$	3819.99	0.157752
$838$	0	0
$839$	−11399.6	−0.469081	−0.234541	−	0.972106i	$-0.575359\pi$
$839$	−11399.6	−0.469081	−0.234541	+	0.972106i	$0.575359\pi$
$840$	0	0
$841$	−18280.4	−0.749536
$842$	0	0
$843$	15737.5	0.642974
$844$	0	0
$845$	1859.73	0.0757119
$846$	0	0
$847$	0	0
$848$	0	0
$849$	23381.6	0.945177
$850$	0	0
$851$	19513.0	0.786013
$852$	0	0
$853$	27862.9	1.11842	0.559208	−	0.829027i	$-0.311105\pi$
$853$	27862.9	1.11842	0.559208	+	0.829027i	$0.311105\pi$
$854$	0	0
$855$	−1299.40	−0.0519750
$856$	0	0
$857$	−13579.1	−0.541253	−0.270626	−	0.962684i	$-0.587231\pi$
$857$	−13579.1	−0.541253	−0.270626	+	0.962684i	$0.587231\pi$
$858$	0	0
$859$	40540.1	1.61026	0.805128	−	0.593102i	$-0.202097\pi$
$859$	40540.1	1.61026	0.805128	+	0.593102i	$0.202097\pi$
$860$	0	0
$861$	32229.4	1.27570
$862$	0	0
$863$	−15045.5	−0.593457	−0.296729	−	0.954962i	$-0.595896\pi$
$863$	−15045.5	−0.593457	−0.296729	+	0.954962i	$0.595896\pi$
$864$	0	0
$865$	−4973.04	−0.195478
$866$	0	0
$867$	−18982.8	−0.743586
$868$	0	0
$869$	0	0
$870$	0	0
$871$	11399.6	0.443470
$872$	0	0
$873$	6814.59	0.264191
$874$	0	0
$875$	−16177.4	−0.625025
$876$	0	0
$877$	−47716.1	−1.83724	−0.918620	−	0.395142i	$-0.870695\pi$
$877$	−47716.1	−1.83724	−0.918620	+	0.395142i	$0.870695\pi$
$878$	0	0
$879$	27268.6	1.04636
$880$	0	0
$881$	7586.63	0.290125	0.145062	−	0.989423i	$-0.453662\pi$
$881$	7586.63	0.290125	0.145062	+	0.989423i	$0.453662\pi$
$882$	0	0
$883$	23306.0	0.888232	0.444116	−	0.895969i	$-0.353518\pi$
$883$	23306.0	0.888232	0.444116	+	0.895969i	$0.353518\pi$
$884$	0	0
$885$	−384.470	−0.0146032
$886$	0	0
$887$	−23091.1	−0.874094	−0.437047	−	0.899439i	$-0.643976\pi$
$887$	−23091.1	−0.874094	−0.437047	+	0.899439i	$0.643976\pi$
$888$	0	0
$889$	−7797.79	−0.294184
$890$	0	0
$891$	0	0
$892$	0	0
$893$	25362.9	0.950433
$894$	0	0
$895$	11216.9	0.418926
$896$	0	0
$897$	8659.37	0.322328
$898$	0	0
$899$	−11057.8	−0.410232
$900$	0	0
$901$	−34021.3	−1.25795
$902$	0	0
$903$	−5085.89	−0.187428
$904$	0	0
$905$	−6128.23	−0.225093
$906$	0	0
$907$	29259.3	1.07116	0.535578	−	0.844486i	$-0.320094\pi$
$907$	29259.3	1.07116	0.535578	+	0.844486i	$0.320094\pi$
$908$	0	0
$909$	15890.9	0.579833
$910$	0	0
$911$	9754.80	0.354765	0.177383	−	0.984142i	$-0.443237\pi$
$911$	9754.80	0.354765	0.177383	+	0.984142i	$0.443237\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−482.278	−0.0174247
$916$	0	0
$917$	−5271.09	−0.189822
$918$	0	0
$919$	−12161.0	−0.436512	−0.218256	−	0.975892i	$-0.570037\pi$
$919$	−12161.0	−0.436512	−0.218256	+	0.975892i	$0.570037\pi$
$920$	0	0
$921$	−18601.6	−0.665518
$922$	0	0
$923$	12087.3	0.431048
$924$	0	0
$925$	−30672.7	−1.09028
$926$	0	0
$927$	−12151.8	−0.430547
$928$	0	0
$929$	−42434.8	−1.49865	−0.749323	−	0.662205i	$-0.769621\pi$
$929$	−42434.8	−1.49865	−0.749323	+	0.662205i	$0.769621\pi$
$930$	0	0
$931$	19369.6	0.681862
$932$	0	0
$933$	21688.3	0.761032
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−20602.8	−0.718319	−0.359159	−	0.933276i	$-0.616937\pi$
$937$	−20602.8	−0.718319	−0.359159	+	0.933276i	$0.616937\pi$
$938$	0	0
$939$	−4764.45	−0.165583
$940$	0	0
$941$	11310.8	0.391839	0.195920	−	0.980620i	$-0.437231\pi$
$941$	11310.8	0.391839	0.195920	+	0.980620i	$0.437231\pi$
$942$	0	0
$943$	30983.8	1.06996
$944$	0	0
$945$	−1793.45	−0.0617365
$946$	0	0
$947$	−31990.9	−1.09774	−0.548872	−	0.835907i	$-0.684942\pi$
$947$	−31990.9	−1.09774	−0.548872	+	0.835907i	$0.684942\pi$
$948$	0	0
$949$	42551.6	1.45551
$950$	0	0
$951$	−20745.2	−0.707371
$952$	0	0
$953$	−55966.0	−1.90233	−0.951164	−	0.308687i	$-0.900111\pi$
$953$	−55966.0	−1.90233	−0.951164	+	0.308687i	$0.900111\pi$
$954$	0	0
$955$	5699.94	0.193137
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36361.0	1.22436
$960$	0	0
$961$	−9774.05	−0.328087
$962$	0	0
$963$	5546.73	0.185608
$964$	0	0
$965$	4669.70	0.155775
$966$	0	0
$967$	14964.4	0.497644	0.248822	−	0.968549i	$-0.419957\pi$
$967$	14964.4	0.497644	0.248822	+	0.968549i	$0.419957\pi$
$968$	0	0
$969$	−18078.1	−0.599330
$970$	0	0
$971$	−1529.46	−0.0505487	−0.0252743	−	0.999681i	$-0.508046\pi$
$971$	−1529.46	−0.0505487	−0.0252743	+	0.999681i	$0.508046\pi$
$972$	0	0
$973$	42799.0	1.41015
$974$	0	0
$975$	−13611.8	−0.447103
$976$	0	0
$977$	18436.2	0.603712	0.301856	−	0.953353i	$-0.402394\pi$
$977$	18436.2	0.603712	0.301856	+	0.953353i	$0.402394\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	12063.5	0.392618
$982$	0	0
$983$	−7901.56	−0.256379	−0.128189	−	0.991750i	$-0.540917\pi$
$983$	−7901.56	−0.256379	−0.128189	+	0.991750i	$0.540917\pi$
$984$	0	0
$985$	−8513.36	−0.275389
$986$	0	0
$987$	35006.2	1.12894
$988$	0	0
$989$	−4889.33	−0.157201
$990$	0	0
$991$	−20000.2	−0.641097	−0.320548	−	0.947232i	$-0.603867\pi$
$991$	−20000.2	−0.641097	−0.320548	+	0.947232i	$0.603867\pi$
$992$	0	0
$993$	29403.5	0.939670
$994$	0	0
$995$	2622.44	0.0835548
$996$	0	0
$997$	−23991.9	−0.762117	−0.381058	−	0.924551i	$-0.624440\pi$
$997$	−23991.9	−0.762117	−0.381058	+	0.924551i	$0.624440\pi$
$998$	0	0
$999$	−6985.92	−0.221246

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.4.a.r.1.3	✓	6
11.10	odd	2	inner	1452.4.a.r.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1452.4.a.r.1.3	✓	6	1.1	even	1	trivial
1452.4.a.r.1.4	yes	6	11.10	odd	2	inner

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:	\( N \)	\(=\)	\( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1452.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -2.54018 q^{5} -26.1494 q^{7} +9.00000 q^{9} -38.2737 q^{13} +7.62055 q^{15} +106.022 q^{17} +56.8376 q^{19} +78.4481 q^{21} +75.4162 q^{23} -118.547 q^{25} -27.0000 q^{27} +78.1574 q^{29} -141.481 q^{31} +66.4241 q^{35} +258.738 q^{37} +114.821 q^{39} +410.838 q^{41} -64.8313 q^{43} -22.8616 q^{45} +446.234 q^{47} +340.789 q^{49} -318.065 q^{51} -320.890 q^{53} -170.513 q^{57} -50.4518 q^{59} -63.2866 q^{61} -235.344 q^{63} +97.2222 q^{65} -297.845 q^{67} -226.249 q^{69} -315.811 q^{71} -1111.77 q^{73} +355.642 q^{75} -216.539 q^{79} +81.0000 q^{81} -434.074 q^{83} -269.314 q^{85} -234.472 q^{87} -1363.23 q^{89} +1000.83 q^{91} +424.444 q^{93} -144.378 q^{95} +757.176 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 18 q^{3} - 12 q^{5} + 54 q^{9} + 36 q^{15} - 492 q^{23} + 534 q^{25} - 162 q^{27} - 270 q^{31} + 582 q^{37} - 108 q^{45} + 756 q^{47} + 780 q^{49} - 72 q^{53} + 564 q^{59} - 1590 q^{67} + 1476 q^{69}+ \cdots - 4242 q^{97}+O(q^{100}) \) 6 * q - 18 * q^3 - 12 * q^5 + 54 * q^9 + 36 * q^15 - 492 * q^23 + 534 * q^25 - 162 * q^27 - 270 * q^31 + 582 * q^37 - 108 * q^45 + 756 * q^47 + 780 * q^49 - 72 * q^53 + 564 * q^59 - 1590 * q^67 + 1476 * q^69 + 984 * q^71 - 1602 * q^75 + 486 * q^81 - 3324 * q^89 + 2544 * q^91 + 810 * q^93 - 4242 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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