LMFDB - Embedded newform 1872.4.a.ba.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1872,4,Mod(1,1872)] 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("1872.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1872, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1872.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,44] 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:	$110.451575531$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.64575$ of defining polynomial
Character	$\chi$	$=$	1872.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+10.5830 q^{5} +22.0000 q^{7} -5.29150 q^{11} +13.0000 q^{13} -116.413 q^{17} +126.000 q^{19} +31.7490 q^{23} -13.0000 q^{25} -52.9150 q^{29} +182.000 q^{31} +232.826 q^{35} -86.0000 q^{37} +444.486 q^{41} -96.0000 q^{43} +365.114 q^{47} +141.000 q^{49} +190.494 q^{53} -56.0000 q^{55} -587.357 q^{59} +574.000 q^{61} +137.579 q^{65} +530.000 q^{67} +809.600 q^{71} -154.000 q^{73} -116.413 q^{77} +460.000 q^{79} -322.782 q^{83} -1232.00 q^{85} -1439.29 q^{89} +286.000 q^{91} +1333.46 q^{95} +70.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 44 q^{7} + 26 q^{13} + 252 q^{19} - 26 q^{25} + 364 q^{31} - 172 q^{37} - 192 q^{43} + 282 q^{49} - 112 q^{55} + 1148 q^{61} + 1060 q^{67} - 308 q^{73} + 920 q^{79} - 2464 q^{85} + 572 q^{91} + 140 q^{97}+O(q^{100}) $ 2 * q + 44 * q^7 + 26 * q^13 + 252 * q^19 - 26 * q^25 + 364 * q^31 - 172 * q^37 - 192 * q^43 + 282 * q^49 - 112 * q^55 + 1148 * q^61 + 1060 * q^67 - 308 * q^73 + 920 * q^79 - 2464 * q^85 + 572 * q^91 + 140 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	10.5830	0.946573	0.473286	−	0.880909i	$-0.343068\pi$
$5$	10.5830	0.946573	0.473286	+	0.880909i	$0.343068\pi$
$6$	0	0
$7$	22.0000	1.18789	0.593944	−	0.804506i	$-0.297570\pi$
$7$	22.0000	1.18789	0.593944	+	0.804506i	$0.297570\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.29150	−0.145041	−0.0725204	−	0.997367i	$-0.523104\pi$
$11$	−5.29150	−0.145041	−0.0725204	+	0.997367i	$0.523104\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−116.413	−1.66084	−0.830421	−	0.557136i	$-0.811900\pi$
$17$	−116.413	−1.66084	−0.830421	+	0.557136i	$0.811900\pi$
$18$	0	0
$19$	126.000	1.52139	0.760694	−	0.649110i	$-0.224859\pi$
$19$	126.000	1.52139	0.760694	+	0.649110i	$0.224859\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	31.7490	0.287832	0.143916	−	0.989590i	$-0.454031\pi$
$23$	31.7490	0.287832	0.143916	+	0.989590i	$0.454031\pi$
$24$	0	0
$25$	−13.0000	−0.104000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−52.9150	−0.338830	−0.169415	−	0.985545i	$-0.554188\pi$
$29$	−52.9150	−0.338830	−0.169415	+	0.985545i	$0.554188\pi$
$30$	0	0
$31$	182.000	1.05446	0.527228	−	0.849724i	$-0.323231\pi$
$31$	182.000	1.05446	0.527228	+	0.849724i	$0.323231\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	232.826	1.12442
$36$	0	0
$37$	−86.0000	−0.382117	−0.191058	−	0.981579i	$-0.561192\pi$
$37$	−86.0000	−0.382117	−0.191058	+	0.981579i	$0.561192\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	444.486	1.69310	0.846550	−	0.532310i	$-0.178676\pi$
$41$	444.486	1.69310	0.846550	+	0.532310i	$0.178676\pi$
$42$	0	0
$43$	−96.0000	−0.340462	−0.170231	−	0.985404i	$-0.554451\pi$
$43$	−96.0000	−0.340462	−0.170231	+	0.985404i	$0.554451\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	365.114	1.13313	0.566567	−	0.824016i	$-0.308271\pi$
$47$	365.114	1.13313	0.566567	+	0.824016i	$0.308271\pi$
$48$	0	0
$49$	141.000	0.411079
$50$	0	0
$51$	0	0
$52$	0	0
$53$	190.494	0.493705	0.246853	−	0.969053i	$-0.420604\pi$
$53$	190.494	0.493705	0.246853	+	0.969053i	$0.420604\pi$
$54$	0	0
$55$	−56.0000	−0.137292
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−587.357	−1.29606	−0.648028	−	0.761616i	$-0.724406\pi$
$59$	−587.357	−1.29606	−0.648028	+	0.761616i	$0.724406\pi$
$60$	0	0
$61$	574.000	1.20481	0.602403	−	0.798192i	$-0.294210\pi$
$61$	574.000	1.20481	0.602403	+	0.798192i	$0.294210\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	137.579	0.262532
$66$	0	0
$67$	530.000	0.966415	0.483208	−	0.875506i	$-0.339472\pi$
$67$	530.000	0.966415	0.483208	+	0.875506i	$0.339472\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	809.600	1.35327	0.676633	−	0.736321i	$-0.263439\pi$
$71$	809.600	1.35327	0.676633	+	0.736321i	$0.263439\pi$
$72$	0	0
$73$	−154.000	−0.246909	−0.123454	−	0.992350i	$-0.539397\pi$
$73$	−154.000	−0.246909	−0.123454	+	0.992350i	$0.539397\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−116.413	−0.172292
$78$	0	0
$79$	460.000	0.655114	0.327557	−	0.944831i	$-0.393775\pi$
$79$	460.000	0.655114	0.327557	+	0.944831i	$0.393775\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−322.782	−0.426866	−0.213433	−	0.976958i	$-0.568465\pi$
$83$	−322.782	−0.426866	−0.213433	+	0.976958i	$0.568465\pi$
$84$	0	0
$85$	−1232.00	−1.57211
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1439.29	−1.71421	−0.857103	−	0.515145i	$-0.827738\pi$
$89$	−1439.29	−1.71421	−0.857103	+	0.515145i	$0.827738\pi$
$90$	0	0
$91$	286.000	0.329461
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1333.46	1.44010
$96$	0	0
$97$	70.0000	0.0732724	0.0366362	−	0.999329i	$-0.488336\pi$
$97$	70.0000	0.0732724	0.0366362	+	0.999329i	$0.488336\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1460.45	1.43882	0.719409	−	0.694586i	$-0.244413\pi$
$101$	1460.45	1.43882	0.719409	+	0.694586i	$0.244413\pi$
$102$	0	0
$103$	1428.00	1.36607	0.683034	−	0.730387i	$-0.260660\pi$
$103$	1428.00	1.36607	0.683034	+	0.730387i	$0.260660\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1619.20	−1.46293	−0.731467	−	0.681877i	$-0.761164\pi$
$107$	−1619.20	−1.46293	−0.731467	+	0.681877i	$0.761164\pi$
$108$	0	0
$109$	−338.000	−0.297014	−0.148507	−	0.988911i	$-0.547447\pi$
$109$	−338.000	−0.297014	−0.148507	+	0.988911i	$0.547447\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1682.70	1.40084	0.700420	−	0.713731i	$-0.252996\pi$
$113$	1682.70	1.40084	0.700420	+	0.713731i	$0.252996\pi$
$114$	0	0
$115$	336.000	0.272454
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2561.09	−1.97289
$120$	0	0
$121$	−1303.00	−0.978963
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1460.45	−1.04502
$126$	0	0
$127$	376.000	0.262713	0.131357	−	0.991335i	$-0.458067\pi$
$127$	376.000	0.262713	0.131357	+	0.991335i	$0.458067\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−687.895	−0.458792	−0.229396	−	0.973333i	$-0.573675\pi$
$131$	−687.895	−0.458792	−0.229396	+	0.973333i	$0.573675\pi$
$132$	0	0
$133$	2772.00	1.80724
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1396.96	−0.871168	−0.435584	−	0.900148i	$-0.643458\pi$
$137$	−1396.96	−0.871168	−0.435584	+	0.900148i	$0.643458\pi$
$138$	0	0
$139$	−2100.00	−1.28144	−0.640718	−	0.767776i	$-0.721363\pi$
$139$	−2100.00	−1.28144	−0.640718	+	0.767776i	$0.721363\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−68.7895	−0.0402271
$144$	0	0
$145$	−560.000	−0.320727
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2000.19	1.09974	0.549872	−	0.835249i	$-0.314677\pi$
$149$	2000.19	1.09974	0.549872	+	0.835249i	$0.314677\pi$
$150$	0	0
$151$	−3526.00	−1.90028	−0.950138	−	0.311828i	$-0.899059\pi$
$151$	−3526.00	−1.90028	−0.950138	+	0.311828i	$0.899059\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1926.11	0.998120
$156$	0	0
$157$	3066.00	1.55856	0.779278	−	0.626678i	$-0.215586\pi$
$157$	3066.00	1.55856	0.779278	+	0.626678i	$0.215586\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	698.478	0.341912
$162$	0	0
$163$	3442.00	1.65398	0.826988	−	0.562219i	$-0.190052\pi$
$163$	3442.00	1.65398	0.826988	+	0.562219i	$0.190052\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2693.37	1.24802	0.624011	−	0.781416i	$-0.285502\pi$
$167$	2693.37	1.24802	0.624011	+	0.781416i	$0.285502\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3492.39	−1.53481	−0.767404	−	0.641164i	$-0.778452\pi$
$173$	−3492.39	−1.53481	−0.767404	+	0.641164i	$0.778452\pi$
$174$	0	0
$175$	−286.000	−0.123540
$176$	0	0
$177$	0	0
$178$	0	0
$179$	169.328	0.0707049	0.0353524	−	0.999375i	$-0.488745\pi$
$179$	169.328	0.0707049	0.0353524	+	0.999375i	$0.488745\pi$
$180$	0	0
$181$	3374.00	1.38557	0.692783	−	0.721146i	$-0.256384\pi$
$181$	3374.00	1.38557	0.692783	+	0.721146i	$0.256384\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−910.138	−0.361701
$186$	0	0
$187$	616.000	0.240890
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1185.30	−0.449032	−0.224516	−	0.974470i	$-0.572080\pi$
$191$	−1185.30	−0.449032	−0.224516	+	0.974470i	$0.572080\pi$
$192$	0	0
$193$	−1542.00	−0.575107	−0.287553	−	0.957765i	$-0.592842\pi$
$193$	−1542.00	−0.575107	−0.287553	+	0.957765i	$0.592842\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2127.18	−0.769318	−0.384659	−	0.923059i	$-0.625681\pi$
$197$	−2127.18	−0.769318	−0.384659	+	0.923059i	$0.625681\pi$
$198$	0	0
$199$	952.000	0.339123	0.169562	−	0.985520i	$-0.445765\pi$
$199$	952.000	0.339123	0.169562	+	0.985520i	$0.445765\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1164.13	−0.402492
$204$	0	0
$205$	4704.00	1.60264
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−666.729	−0.220663
$210$	0	0
$211$	1640.00	0.535082	0.267541	−	0.963547i	$-0.413789\pi$
$211$	1640.00	0.535082	0.267541	+	0.963547i	$0.413789\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1015.97	−0.322272
$216$	0	0
$217$	4004.00	1.25258
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1513.37	−0.460635
$222$	0	0
$223$	4886.00	1.46722	0.733612	−	0.679569i	$-0.237833\pi$
$223$	4886.00	1.46722	0.733612	+	0.679569i	$0.237833\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1867.90	0.546154	0.273077	−	0.961992i	$-0.411959\pi$
$227$	1867.90	0.546154	0.273077	+	0.961992i	$0.411959\pi$
$228$	0	0
$229$	5558.00	1.60386	0.801928	−	0.597421i	$-0.203808\pi$
$229$	5558.00	1.60386	0.801928	+	0.597421i	$0.203808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3577.06	1.00575	0.502877	−	0.864358i	$-0.332275\pi$
$233$	3577.06	1.00575	0.502877	+	0.864358i	$0.332275\pi$
$234$	0	0
$235$	3864.00	1.07259
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2936.78	−0.794832	−0.397416	−	0.917639i	$-0.630093\pi$
$239$	−2936.78	−0.794832	−0.397416	+	0.917639i	$0.630093\pi$
$240$	0	0
$241$	−602.000	−0.160906	−0.0804528	−	0.996758i	$-0.525637\pi$
$241$	−602.000	−0.160906	−0.0804528	+	0.996758i	$0.525637\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1492.20	0.389116
$246$	0	0
$247$	1638.00	0.421957
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3524.14	−0.886222	−0.443111	−	0.896467i	$-0.646125\pi$
$251$	−3524.14	−0.886222	−0.443111	+	0.896467i	$0.646125\pi$
$252$	0	0
$253$	−168.000	−0.0417473
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2942.08	−0.714092	−0.357046	−	0.934087i	$-0.616216\pi$
$257$	−2942.08	−0.714092	−0.357046	+	0.934087i	$0.616216\pi$
$258$	0	0
$259$	−1892.00	−0.453912
$260$	0	0
$261$	0	0
$262$	0	0
$263$	857.223	0.200984	0.100492	−	0.994938i	$-0.467958\pi$
$263$	857.223	0.200984	0.100492	+	0.994938i	$0.467958\pi$
$264$	0	0
$265$	2016.00	0.467328
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−328.073	−0.0743605	−0.0371802	−	0.999309i	$-0.511838\pi$
$269$	−328.073	−0.0743605	−0.0371802	+	0.999309i	$0.511838\pi$
$270$	0	0
$271$	2814.00	0.630769	0.315384	−	0.948964i	$-0.397867\pi$
$271$	2814.00	0.630769	0.315384	+	0.948964i	$0.397867\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	68.7895	0.0150842
$276$	0	0
$277$	−3190.00	−0.691944	−0.345972	−	0.938245i	$-0.612451\pi$
$277$	−3190.00	−0.691944	−0.345972	+	0.938245i	$0.612451\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6116.98	−1.29861	−0.649303	−	0.760530i	$-0.724939\pi$
$281$	−6116.98	−1.29861	−0.649303	+	0.760530i	$0.724939\pi$
$282$	0	0
$283$	4788.00	1.00571	0.502857	−	0.864370i	$-0.332282\pi$
$283$	4788.00	1.00571	0.502857	+	0.864370i	$0.332282\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9778.70	2.01121
$288$	0	0
$289$	8639.00	1.75840
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6699.04	1.33571	0.667854	−	0.744293i	$-0.267213\pi$
$293$	6699.04	1.33571	0.667854	+	0.744293i	$0.267213\pi$
$294$	0	0
$295$	−6216.00	−1.22681
$296$	0	0
$297$	0	0
$298$	0	0
$299$	412.737	0.0798301
$300$	0	0
$301$	−2112.00	−0.404431
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6074.65	1.14044
$306$	0	0
$307$	406.000	0.0754777	0.0377388	−	0.999288i	$-0.487985\pi$
$307$	406.000	0.0754777	0.0377388	+	0.999288i	$0.487985\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8286.49	1.51088	0.755440	−	0.655217i	$-0.227423\pi$
$311$	8286.49	1.51088	0.755440	+	0.655217i	$0.227423\pi$
$312$	0	0
$313$	−5586.00	−1.00875	−0.504376	−	0.863484i	$-0.668277\pi$
$313$	−5586.00	−1.00875	−0.504376	+	0.863484i	$0.668277\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8392.32	1.48694	0.743470	−	0.668770i	$-0.233179\pi$
$317$	8392.32	1.48694	0.743470	+	0.668770i	$0.233179\pi$
$318$	0	0
$319$	280.000	0.0491442
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−14668.0	−2.52679
$324$	0	0
$325$	−169.000	−0.0288444
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8032.50	1.34604
$330$	0	0
$331$	4426.00	0.734970	0.367485	−	0.930030i	$-0.380219\pi$
$331$	4426.00	0.734970	0.367485	+	0.930030i	$0.380219\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5608.99	0.914782
$336$	0	0
$337$	8370.00	1.35295	0.676473	−	0.736467i	$-0.263507\pi$
$337$	8370.00	1.35295	0.676473	+	0.736467i	$0.263507\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−963.053	−0.152939
$342$	0	0
$343$	−4444.00	−0.699573
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6095.81	0.943056	0.471528	−	0.881851i	$-0.343703\pi$
$347$	6095.81	0.943056	0.471528	+	0.881851i	$0.343703\pi$
$348$	0	0
$349$	4354.00	0.667806	0.333903	−	0.942607i	$-0.391634\pi$
$349$	4354.00	0.667806	0.333903	+	0.942607i	$0.391634\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3407.73	−0.513810	−0.256905	−	0.966437i	$-0.582703\pi$
$353$	−3407.73	−0.513810	−0.256905	+	0.966437i	$0.582703\pi$
$354$	0	0
$355$	8568.00	1.28096
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7762.63	−1.14121	−0.570607	−	0.821223i	$-0.693292\pi$
$359$	−7762.63	−1.14121	−0.570607	+	0.821223i	$0.693292\pi$
$360$	0	0
$361$	9017.00	1.31462
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1629.78	−0.233717
$366$	0	0
$367$	7784.00	1.10714	0.553572	−	0.832802i	$-0.313265\pi$
$367$	7784.00	1.10714	0.553572	+	0.832802i	$0.313265\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4190.87	0.586467
$372$	0	0
$373$	−8510.00	−1.18132	−0.590658	−	0.806922i	$-0.701132\pi$
$373$	−8510.00	−1.18132	−0.590658	+	0.806922i	$0.701132\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−687.895	−0.0939746
$378$	0	0
$379$	−1650.00	−0.223627	−0.111814	−	0.993729i	$-0.535666\pi$
$379$	−1650.00	−0.223627	−0.111814	+	0.993729i	$0.535666\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8662.19	−1.15566	−0.577829	−	0.816158i	$-0.696100\pi$
$383$	−8662.19	−1.15566	−0.577829	+	0.816158i	$0.696100\pi$
$384$	0	0
$385$	−1232.00	−0.163087
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2423.51	−0.315879	−0.157939	−	0.987449i	$-0.550485\pi$
$389$	−2423.51	−0.315879	−0.157939	+	0.987449i	$0.550485\pi$
$390$	0	0
$391$	−3696.00	−0.478043
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4868.18	0.620114
$396$	0	0
$397$	−1414.00	−0.178757	−0.0893786	−	0.995998i	$-0.528488\pi$
$397$	−1414.00	−0.178757	−0.0893786	+	0.995998i	$0.528488\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5228.00	0.651058	0.325529	−	0.945532i	$-0.394458\pi$
$401$	5228.00	0.651058	0.325529	+	0.945532i	$0.394458\pi$
$402$	0	0
$403$	2366.00	0.292454
$404$	0	0
$405$	0	0
$406$	0	0
$407$	455.069	0.0554225
$408$	0	0
$409$	5782.00	0.699026	0.349513	−	0.936932i	$-0.386347\pi$
$409$	5782.00	0.699026	0.349513	+	0.936932i	$0.386347\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−12921.8	−1.53957
$414$	0	0
$415$	−3416.00	−0.404060
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11482.6	−1.33881	−0.669403	−	0.742899i	$-0.733450\pi$
$419$	−11482.6	−1.33881	−0.669403	+	0.742899i	$0.733450\pi$
$420$	0	0
$421$	−14194.0	−1.64317	−0.821583	−	0.570088i	$-0.806909\pi$
$421$	−14194.0	−1.64317	−0.821583	+	0.570088i	$0.806909\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1513.37	0.172728
$426$	0	0
$427$	12628.0	1.43118
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5222.71	0.583687	0.291844	−	0.956466i	$-0.405731\pi$
$431$	5222.71	0.583687	0.291844	+	0.956466i	$0.405731\pi$
$432$	0	0
$433$	−686.000	−0.0761364	−0.0380682	−	0.999275i	$-0.512120\pi$
$433$	−686.000	−0.0761364	−0.0380682	+	0.999275i	$0.512120\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4000.38	0.437904
$438$	0	0
$439$	1372.00	0.149162	0.0745809	−	0.997215i	$-0.476238\pi$
$439$	1372.00	0.149162	0.0745809	+	0.997215i	$0.476238\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2338.84	0.250839	0.125420	−	0.992104i	$-0.459972\pi$
$443$	2338.84	0.250839	0.125420	+	0.992104i	$0.459972\pi$
$444$	0	0
$445$	−15232.0	−1.62262
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17250.3	−1.81312	−0.906561	−	0.422074i	$-0.861302\pi$
$449$	−17250.3	−1.81312	−0.906561	+	0.422074i	$0.861302\pi$
$450$	0	0
$451$	−2352.00	−0.245568
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3026.74	0.311859
$456$	0	0
$457$	−17866.0	−1.82874	−0.914372	−	0.404875i	$-0.867315\pi$
$457$	−17866.0	−1.82874	−0.914372	+	0.404875i	$0.867315\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2106.02	−0.212770	−0.106385	−	0.994325i	$-0.533928\pi$
$461$	−2106.02	−0.212770	−0.106385	+	0.994325i	$0.533928\pi$
$462$	0	0
$463$	13718.0	1.37695	0.688477	−	0.725258i	$-0.258280\pi$
$463$	13718.0	1.37695	0.688477	+	0.725258i	$0.258280\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4095.62	0.405830	0.202915	−	0.979196i	$-0.434958\pi$
$467$	4095.62	0.405830	0.202915	+	0.979196i	$0.434958\pi$
$468$	0	0
$469$	11660.0	1.14799
$470$	0	0
$471$	0	0
$472$	0	0
$473$	507.984	0.0493808
$474$	0	0
$475$	−1638.00	−0.158224
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8715.10	0.831322	0.415661	−	0.909520i	$-0.363550\pi$
$479$	8715.10	0.831322	0.415661	+	0.909520i	$0.363550\pi$
$480$	0	0
$481$	−1118.00	−0.105980
$482$	0	0
$483$	0	0
$484$	0	0
$485$	740.810	0.0693577
$486$	0	0
$487$	−4506.00	−0.419274	−0.209637	−	0.977779i	$-0.567228\pi$
$487$	−4506.00	−0.419274	−0.209637	+	0.977779i	$0.567228\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−21621.1	−1.98726	−0.993631	−	0.112683i	$-0.964056\pi$
$491$	−21621.1	−1.98726	−0.993631	+	0.112683i	$0.964056\pi$
$492$	0	0
$493$	6160.00	0.562743
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17811.2	1.60753
$498$	0	0
$499$	786.000	0.0705134	0.0352567	−	0.999378i	$-0.488775\pi$
$499$	786.000	0.0705134	0.0352567	+	0.999378i	$0.488775\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	2106.02	0.186685	0.0933426	−	0.995634i	$-0.470245\pi$
$503$	2106.02	0.186685	0.0933426	+	0.995634i	$0.470245\pi$
$504$	0	0
$505$	15456.0	1.36195
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−8392.32	−0.730812	−0.365406	−	0.930848i	$-0.619070\pi$
$509$	−8392.32	−0.730812	−0.365406	+	0.930848i	$0.619070\pi$
$510$	0	0
$511$	−3388.00	−0.293300
$512$	0	0
$513$	0	0
$514$	0	0
$515$	15112.5	1.29308
$516$	0	0
$517$	−1932.00	−0.164351
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−3905.13	−0.328382	−0.164191	−	0.986429i	$-0.552501\pi$
$521$	−3905.13	−0.328382	−0.164191	+	0.986429i	$0.552501\pi$
$522$	0	0
$523$	17668.0	1.47718	0.738592	−	0.674152i	$-0.235491\pi$
$523$	17668.0	1.47718	0.738592	+	0.674152i	$0.235491\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21187.2	−1.75129
$528$	0	0
$529$	−11159.0	−0.917153
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5778.32	0.469581
$534$	0	0
$535$	−17136.0	−1.38477
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−746.102	−0.0596232
$540$	0	0
$541$	−1650.00	−0.131126	−0.0655629	−	0.997848i	$-0.520884\pi$
$541$	−1650.00	−0.131126	−0.0655629	+	0.997848i	$0.520884\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3577.06	−0.281145
$546$	0	0
$547$	−3796.00	−0.296719	−0.148359	−	0.988934i	$-0.547399\pi$
$547$	−3796.00	−0.296719	−0.148359	+	0.988934i	$0.547399\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6667.29	−0.515492
$552$	0	0
$553$	10120.0	0.778203
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2063.69	0.156986	0.0784930	−	0.996915i	$-0.474989\pi$
$557$	2063.69	0.156986	0.0784930	+	0.996915i	$0.474989\pi$
$558$	0	0
$559$	−1248.00	−0.0944271
$560$	0	0
$561$	0	0
$562$	0	0
$563$	26034.2	1.94886	0.974432	−	0.224683i	$-0.0721346\pi$
$563$	26034.2	1.94886	0.974432	+	0.224683i	$0.0721346\pi$
$564$	0	0
$565$	17808.0	1.32600
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3640.55	0.268225	0.134112	−	0.990966i	$-0.457182\pi$
$569$	3640.55	0.268225	0.134112	+	0.990966i	$0.457182\pi$
$570$	0	0
$571$	19612.0	1.43737	0.718684	−	0.695337i	$-0.244745\pi$
$571$	19612.0	1.43737	0.718684	+	0.695337i	$0.244745\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−412.737	−0.0299345
$576$	0	0
$577$	15722.0	1.13434	0.567171	−	0.823600i	$-0.308038\pi$
$577$	15722.0	1.13434	0.567171	+	0.823600i	$0.308038\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7101.20	−0.507069
$582$	0	0
$583$	−1008.00	−0.0716074
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2725.12	0.191615	0.0958074	−	0.995400i	$-0.469457\pi$
$587$	2725.12	0.191615	0.0958074	+	0.995400i	$0.469457\pi$
$588$	0	0
$589$	22932.0	1.60424
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−18012.3	−1.24734	−0.623672	−	0.781686i	$-0.714360\pi$
$593$	−18012.3	−1.24734	−0.623672	+	0.781686i	$0.714360\pi$
$594$	0	0
$595$	−27104.0	−1.86749
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12181.0	0.830891	0.415446	−	0.909618i	$-0.363626\pi$
$599$	12181.0	0.830891	0.415446	+	0.909618i	$0.363626\pi$
$600$	0	0
$601$	5950.00	0.403836	0.201918	−	0.979402i	$-0.435283\pi$
$601$	5950.00	0.403836	0.201918	+	0.979402i	$0.435283\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−13789.7	−0.926660
$606$	0	0
$607$	−14168.0	−0.947383	−0.473691	−	0.880691i	$-0.657079\pi$
$607$	−14168.0	−0.947383	−0.473691	+	0.880691i	$0.657079\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4746.48	0.314275
$612$	0	0
$613$	−6326.00	−0.416810	−0.208405	−	0.978043i	$-0.566827\pi$
$613$	−6326.00	−0.416810	−0.208405	+	0.978043i	$0.566827\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−8805.06	−0.574519	−0.287260	−	0.957853i	$-0.592744\pi$
$617$	−8805.06	−0.574519	−0.287260	+	0.957853i	$0.592744\pi$
$618$	0	0
$619$	24486.0	1.58994	0.794972	−	0.606646i	$-0.207485\pi$
$619$	24486.0	1.58994	0.794972	+	0.606646i	$0.207485\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−31664.4	−2.03628
$624$	0	0
$625$	−13831.0	−0.885184
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10011.5	0.634635
$630$	0	0
$631$	−22430.0	−1.41509	−0.707547	−	0.706666i	$-0.750198\pi$
$631$	−22430.0	−1.41509	−0.707547	+	0.706666i	$0.750198\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3979.21	0.248677
$636$	0	0
$637$	1833.00	0.114013
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−27484.1	−1.69353	−0.846767	−	0.531964i	$-0.821454\pi$
$641$	−27484.1	−1.69353	−0.846767	+	0.531964i	$0.821454\pi$
$642$	0	0
$643$	−16478.0	−1.01062	−0.505310	−	0.862938i	$-0.668622\pi$
$643$	−16478.0	−1.01062	−0.505310	+	0.862938i	$0.668622\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26563.3	−1.61408	−0.807042	−	0.590494i	$-0.798933\pi$
$647$	−26563.3	−1.61408	−0.807042	+	0.590494i	$0.798933\pi$
$648$	0	0
$649$	3108.00	0.187981
$650$	0	0
$651$	0	0
$652$	0	0
$653$	20287.6	1.21580	0.607899	−	0.794014i	$-0.292013\pi$
$653$	20287.6	1.21580	0.607899	+	0.794014i	$0.292013\pi$
$654$	0	0
$655$	−7280.00	−0.434280
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−656.146	−0.0387858	−0.0193929	−	0.999812i	$-0.506173\pi$
$659$	−656.146	−0.0387858	−0.0193929	+	0.999812i	$0.506173\pi$
$660$	0	0
$661$	−14238.0	−0.837812	−0.418906	−	0.908030i	$-0.637586\pi$
$661$	−14238.0	−0.837812	−0.418906	+	0.908030i	$0.637586\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	29336.1	1.71068
$666$	0	0
$667$	−1680.00	−0.0975260
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3037.32	−0.174746
$672$	0	0
$673$	−4874.00	−0.279166	−0.139583	−	0.990210i	$-0.544576\pi$
$673$	−4874.00	−0.279166	−0.139583	+	0.990210i	$0.544576\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21801.0	1.23764	0.618818	−	0.785534i	$-0.287612\pi$
$677$	21801.0	1.23764	0.618818	+	0.785534i	$0.287612\pi$
$678$	0	0
$679$	1540.00	0.0870394
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8746.85	−0.490028	−0.245014	−	0.969520i	$-0.578793\pi$
$683$	−8746.85	−0.490028	−0.245014	+	0.969520i	$0.578793\pi$
$684$	0	0
$685$	−14784.0	−0.824624
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2476.42	0.136929
$690$	0	0
$691$	−294.000	−0.0161857	−0.00809283	−	0.999967i	$-0.502576\pi$
$691$	−294.000	−0.0161857	−0.00809283	+	0.999967i	$0.502576\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−22224.3	−1.21297
$696$	0	0
$697$	−51744.0	−2.81197
$698$	0	0
$699$	0	0
$700$	0	0
$701$	15758.1	0.849037	0.424519	−	0.905419i	$-0.360443\pi$
$701$	15758.1	0.849037	0.424519	+	0.905419i	$0.360443\pi$
$702$	0	0
$703$	−10836.0	−0.581348
$704$	0	0
$705$	0	0
$706$	0	0
$707$	32130.0	1.70916
$708$	0	0
$709$	−6722.00	−0.356065	−0.178032	−	0.984025i	$-0.556973\pi$
$709$	−6722.00	−0.356065	−0.178032	+	0.984025i	$0.556973\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5778.32	0.303506
$714$	0	0
$715$	−728.000	−0.0380778
$716$	0	0
$717$	0	0
$718$	0	0
$719$	31.7490	0.00164679	0.000823393	−	1.00000i	$-0.499738\pi$
$719$	31.7490	0.00164679	0.000823393		1.00000i	$0.499738\pi$
$720$	0	0
$721$	31416.0	1.62274
$722$	0	0
$723$	0	0
$724$	0	0
$725$	687.895	0.0352383
$726$	0	0
$727$	−12824.0	−0.654217	−0.327109	−	0.944987i	$-0.606074\pi$
$727$	−12824.0	−0.654217	−0.327109	+	0.944987i	$0.606074\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	11175.7	0.565453
$732$	0	0
$733$	−29610.0	−1.49205	−0.746023	−	0.665920i	$-0.768039\pi$
$733$	−29610.0	−1.49205	−0.746023	+	0.665920i	$0.768039\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2804.50	−0.140170
$738$	0	0
$739$	15622.0	0.777625	0.388812	−	0.921317i	$-0.372885\pi$
$739$	15622.0	0.777625	0.388812	+	0.921317i	$0.372885\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8588.11	−0.424047	−0.212024	−	0.977265i	$-0.568005\pi$
$743$	−8588.11	−0.424047	−0.212024	+	0.977265i	$0.568005\pi$
$744$	0	0
$745$	21168.0	1.04099
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−35622.4	−1.73780
$750$	0	0
$751$	−29468.0	−1.43183	−0.715914	−	0.698189i	$-0.753990\pi$
$751$	−29468.0	−1.43183	−0.715914	+	0.698189i	$0.753990\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−37315.7	−1.79875
$756$	0	0
$757$	−35030.0	−1.68189	−0.840943	−	0.541124i	$-0.817999\pi$
$757$	−35030.0	−1.68189	−0.840943	+	0.541124i	$0.817999\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22330.1	1.06369	0.531844	−	0.846842i	$-0.321499\pi$
$761$	22330.1	1.06369	0.531844	+	0.846842i	$0.321499\pi$
$762$	0	0
$763$	−7436.00	−0.352819
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7635.64	−0.359461
$768$	0	0
$769$	−27342.0	−1.28216	−0.641078	−	0.767476i	$-0.721512\pi$
$769$	−27342.0	−1.28216	−0.641078	+	0.767476i	$0.721512\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−11884.7	−0.552993	−0.276496	−	0.961015i	$-0.589173\pi$
$773$	−11884.7	−0.552993	−0.276496	+	0.961015i	$0.589173\pi$
$774$	0	0
$775$	−2366.00	−0.109664
$776$	0	0
$777$	0	0
$778$	0	0
$779$	56005.3	2.57586
$780$	0	0
$781$	−4284.00	−0.196279
$782$	0	0
$783$	0	0
$784$	0	0
$785$	32447.5	1.47529
$786$	0	0
$787$	−22666.0	−1.02663	−0.513314	−	0.858201i	$-0.671582\pi$
$787$	−22666.0	−1.02663	−0.513314	+	0.858201i	$0.671582\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	37019.4	1.66404
$792$	0	0
$793$	7462.00	0.334153
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−582.065	−0.0258693	−0.0129346	−	0.999916i	$-0.504117\pi$
$797$	−582.065	−0.0258693	−0.0129346	+	0.999916i	$0.504117\pi$
$798$	0	0
$799$	−42504.0	−1.88196
$800$	0	0
$801$	0	0
$802$	0	0
$803$	814.891	0.0358118
$804$	0	0
$805$	7392.00	0.323644
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−793.725	−0.0344943	−0.0172472	−	0.999851i	$-0.505490\pi$
$809$	−793.725	−0.0344943	−0.0172472	+	0.999851i	$0.505490\pi$
$810$	0	0
$811$	9478.00	0.410379	0.205190	−	0.978722i	$-0.434219\pi$
$811$	9478.00	0.410379	0.205190	+	0.978722i	$0.434219\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	36426.7	1.56561
$816$	0	0
$817$	−12096.0	−0.517975
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3227.82	−0.137213	−0.0686063	−	0.997644i	$-0.521855\pi$
$821$	−3227.82	−0.137213	−0.0686063	+	0.997644i	$0.521855\pi$
$822$	0	0
$823$	−40476.0	−1.71434	−0.857172	−	0.515031i	$-0.827781\pi$
$823$	−40476.0	−1.71434	−0.857172	+	0.515031i	$0.827781\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7169.99	0.301481	0.150741	−	0.988573i	$-0.451834\pi$
$827$	7169.99	0.301481	0.150741	+	0.988573i	$0.451834\pi$
$828$	0	0
$829$	27482.0	1.15137	0.575687	−	0.817670i	$-0.304735\pi$
$829$	27482.0	1.15137	0.575687	+	0.817670i	$0.304735\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−16414.2	−0.682737
$834$	0	0
$835$	28504.0	1.18134
$836$	0	0
$837$	0	0
$838$	0	0
$839$	19128.8	0.787126	0.393563	−	0.919298i	$-0.371242\pi$
$839$	19128.8	0.787126	0.393563	+	0.919298i	$0.371242\pi$
$840$	0	0
$841$	−21589.0	−0.885194
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1788.53	0.0728133
$846$	0	0
$847$	−28666.0	−1.16290
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2730.42	−0.109985
$852$	0	0
$853$	31962.0	1.28295	0.641476	−	0.767143i	$-0.278322\pi$
$853$	31962.0	1.28295	0.641476	+	0.767143i	$0.278322\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4931.68	0.196573	0.0982865	−	0.995158i	$-0.468664\pi$
$857$	4931.68	0.196573	0.0982865	+	0.995158i	$0.468664\pi$
$858$	0	0
$859$	−11704.0	−0.464884	−0.232442	−	0.972610i	$-0.574672\pi$
$859$	−11704.0	−0.464884	−0.232442	+	0.972610i	$0.574672\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2280.64	0.0899581	0.0449790	−	0.998988i	$-0.485678\pi$
$863$	2280.64	0.0899581	0.0449790	+	0.998988i	$0.485678\pi$
$864$	0	0
$865$	−36960.0	−1.45281
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2434.09	−0.0950183
$870$	0	0
$871$	6890.00	0.268035
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−32130.0	−1.24136
$876$	0	0
$877$	−1006.00	−0.0387346	−0.0193673	−	0.999812i	$-0.506165\pi$
$877$	−1006.00	−0.0387346	−0.0193673	+	0.999812i	$0.506165\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−40681.1	−1.55571	−0.777855	−	0.628444i	$-0.783692\pi$
$881$	−40681.1	−1.55571	−0.777855	+	0.628444i	$0.783692\pi$
$882$	0	0
$883$	−124.000	−0.00472586	−0.00236293	−	0.999997i	$-0.500752\pi$
$883$	−124.000	−0.00472586	−0.00236293	+	0.999997i	$0.500752\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16573.0	−0.627358	−0.313679	−	0.949529i	$-0.601562\pi$
$887$	−16573.0	−0.627358	−0.313679	+	0.949529i	$0.601562\pi$
$888$	0	0
$889$	8272.00	0.312074
$890$	0	0
$891$	0	0
$892$	0	0
$893$	46004.3	1.72394
$894$	0	0
$895$	1792.00	0.0669273
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9630.53	−0.357282
$900$	0	0
$901$	−22176.0	−0.819966
$902$	0	0
$903$	0	0
$904$	0	0
$905$	35707.1	1.31154
$906$	0	0
$907$	42996.0	1.57404	0.787022	−	0.616924i	$-0.211622\pi$
$907$	42996.0	1.57404	0.787022	+	0.616924i	$0.211622\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	53169.0	1.93366	0.966832	−	0.255413i	$-0.0822113\pi$
$911$	53169.0	1.93366	0.966832	+	0.255413i	$0.0822113\pi$
$912$	0	0
$913$	1708.00	0.0619130
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15133.7	−0.544993
$918$	0	0
$919$	−8244.00	−0.295913	−0.147957	−	0.988994i	$-0.547270\pi$
$919$	−8244.00	−0.295913	−0.147957	+	0.988994i	$0.547270\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	10524.8	0.375328
$924$	0	0
$925$	1118.00	0.0397401
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−16192.0	−0.571843	−0.285922	−	0.958253i	$-0.592300\pi$
$929$	−16192.0	−0.571843	−0.285922	+	0.958253i	$0.592300\pi$
$930$	0	0
$931$	17766.0	0.625410
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6519.13	0.228020
$936$	0	0
$937$	−18214.0	−0.635032	−0.317516	−	0.948253i	$-0.602849\pi$
$937$	−18214.0	−0.635032	−0.317516	+	0.948253i	$0.602849\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−13916.7	−0.482115	−0.241057	−	0.970511i	$-0.577494\pi$
$941$	−13916.7	−0.482115	−0.241057	+	0.970511i	$0.577494\pi$
$942$	0	0
$943$	14112.0	0.487328
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40400.6	−1.38632	−0.693159	−	0.720784i	$-0.743782\pi$
$947$	−40400.6	−1.38632	−0.693159	+	0.720784i	$0.743782\pi$
$948$	0	0
$949$	−2002.00	−0.0684802
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21271.8	0.723046	0.361523	−	0.932363i	$-0.382257\pi$
$953$	21271.8	0.723046	0.361523	+	0.932363i	$0.382257\pi$
$954$	0	0
$955$	−12544.0	−0.425041
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−30733.0	−1.03485
$960$	0	0
$961$	3333.00	0.111879
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−16319.0	−0.544380
$966$	0	0
$967$	−9578.00	−0.318519	−0.159259	−	0.987237i	$-0.550911\pi$
$967$	−9578.00	−0.318519	−0.159259	+	0.987237i	$0.550911\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−44956.6	−1.48581	−0.742907	−	0.669394i	$-0.766554\pi$
$971$	−44956.6	−1.48581	−0.742907	+	0.669394i	$0.766554\pi$
$972$	0	0
$973$	−46200.0	−1.52220
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32765.0	−1.07292	−0.536461	−	0.843925i	$-0.680239\pi$
$977$	−32765.0	−1.07292	−0.536461	+	0.843925i	$0.680239\pi$
$978$	0	0
$979$	7616.00	0.248630
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−47438.3	−1.53921	−0.769607	−	0.638518i	$-0.779548\pi$
$983$	−47438.3	−1.53921	−0.769607	+	0.638518i	$0.779548\pi$
$984$	0	0
$985$	−22512.0	−0.728215
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3047.91	−0.0979957
$990$	0	0
$991$	6848.00	0.219509	0.109755	−	0.993959i	$-0.464993\pi$
$991$	6848.00	0.219509	0.109755	+	0.993959i	$0.464993\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	10075.0	0.321005
$996$	0	0
$997$	−5810.00	−0.184558	−0.0922791	−	0.995733i	$-0.529415\pi$
$997$	−5810.00	−0.184558	−0.0922791	+	0.995733i	$0.529415\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.4.a.ba.1.2		2
3.2	odd	2	inner	1872.4.a.ba.1.1		2
4.3	odd	2		117.4.a.e.1.1	✓	2
12.11	even	2		117.4.a.e.1.2	yes	2
52.51	odd	2		1521.4.a.p.1.2		2
156.155	even	2		1521.4.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.a.e.1.1	✓	2	4.3	odd	2
117.4.a.e.1.2	yes	2	12.11	even	2
1521.4.a.p.1.1		2	156.155	even	2
1521.4.a.p.1.2		2	52.51	odd	2
1872.4.a.ba.1.1		2	3.2	odd	2	inner
1872.4.a.ba.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1872.a (trivial)

\(f(q)\)	\(=\)	\(q+10.5830 q^{5} +22.0000 q^{7} -5.29150 q^{11} +13.0000 q^{13} -116.413 q^{17} +126.000 q^{19} +31.7490 q^{23} -13.0000 q^{25} -52.9150 q^{29} +182.000 q^{31} +232.826 q^{35} -86.0000 q^{37} +444.486 q^{41} -96.0000 q^{43} +365.114 q^{47} +141.000 q^{49} +190.494 q^{53} -56.0000 q^{55} -587.357 q^{59} +574.000 q^{61} +137.579 q^{65} +530.000 q^{67} +809.600 q^{71} -154.000 q^{73} -116.413 q^{77} +460.000 q^{79} -322.782 q^{83} -1232.00 q^{85} -1439.29 q^{89} +286.000 q^{91} +1333.46 q^{95} +70.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 44 q^{7} + 26 q^{13} + 252 q^{19} - 26 q^{25} + 364 q^{31} - 172 q^{37} - 192 q^{43} + 282 q^{49} - 112 q^{55} + 1148 q^{61} + 1060 q^{67} - 308 q^{73} + 920 q^{79} - 2464 q^{85} + 572 q^{91} + 140 q^{97}+O(q^{100}) \) 2 * q + 44 * q^7 + 26 * q^13 + 252 * q^19 - 26 * q^25 + 364 * q^31 - 172 * q^37 - 192 * q^43 + 282 * q^49 - 112 * q^55 + 1148 * q^61 + 1060 * q^67 - 308 * q^73 + 920 * q^79 - 2464 * q^85 + 572 * q^91 + 140 * q^97

Introduction

L-functions

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