LMFDB - Embedded newform 1872.4.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1872,4,Mod(1,1872)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ 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)

$f(q)$	$=$	$q+3.56155 q^{5} +27.1771 q^{7} +O(q^{10})$	$q+3.56155 q^{5} +27.1771 q^{7} +15.2614 q^{11} -13.0000 q^{13} -44.5464 q^{17} -23.9697 q^{19} +122.739 q^{23} -112.315 q^{25} +219.909 q^{29} -27.0928 q^{31} +96.7926 q^{35} +94.1922 q^{37} +160.354 q^{41} +151.302 q^{43} +466.948 q^{47} +395.594 q^{49} +120.847 q^{53} +54.3542 q^{55} -439.633 q^{59} -137.305 q^{61} -46.3002 q^{65} -512.280 q^{67} +410.719 q^{71} -308.004 q^{73} +414.759 q^{77} +586.462 q^{79} +1354.20 q^{83} -158.654 q^{85} -439.882 q^{89} -353.302 q^{91} -85.3693 q^{95} -1511.27 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{5} + 9 q^{7}+O(q^{10}) $ 2 * q + 3 * q^5 + 9 * q^7	$ 2 q + 3 q^{5} + 9 q^{7} + 80 q^{11} - 26 q^{13} - 19 q^{17} + 84 q^{19} + 196 q^{23} - 237 q^{25} + 44 q^{29} + 86 q^{31} + 107 q^{35} + 209 q^{37} + 230 q^{41} - 287 q^{43} + 435 q^{47} + 383 q^{49} + 118 q^{53} + 18 q^{55} - 368 q^{59} - 1058 q^{61} - 39 q^{65} - 68 q^{67} - 131 q^{71} + 456 q^{73} - 762 q^{77} + 1008 q^{79} + 1958 q^{83} - 173 q^{85} + 720 q^{89} - 117 q^{91} - 146 q^{95} - 928 q^{97}+O(q^{100}) $ 2 * q + 3 * q^5 + 9 * q^7 + 80 * q^11 - 26 * q^13 - 19 * q^17 + 84 * q^19 + 196 * q^23 - 237 * q^25 + 44 * q^29 + 86 * q^31 + 107 * q^35 + 209 * q^37 + 230 * q^41 - 287 * q^43 + 435 * q^47 + 383 * q^49 + 118 * q^53 + 18 * q^55 - 368 * q^59 - 1058 * q^61 - 39 * q^65 - 68 * q^67 - 131 * q^71 + 456 * q^73 - 762 * q^77 + 1008 * q^79 + 1958 * q^83 - 173 * q^85 + 720 * q^89 - 117 * q^91 - 146 * q^95 - 928 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.56155	0.318555	0.159277	−	0.987234i	$-0.449084\pi$
$5$	3.56155	0.318555	0.159277	+	0.987234i	$0.449084\pi$
$6$	0	0
$7$	27.1771	1.46742	0.733712	−	0.679460i	$-0.237786\pi$
$7$	27.1771	1.46742	0.733712	+	0.679460i	$0.237786\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	15.2614	0.418316	0.209158	−	0.977882i	$-0.432928\pi$
$11$	15.2614	0.418316	0.209158	+	0.977882i	$0.432928\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−44.5464	−0.635535	−0.317767	−	0.948169i	$-0.602933\pi$
$17$	−44.5464	−0.635535	−0.317767	+	0.948169i	$0.602933\pi$
$18$	0	0
$19$	−23.9697	−0.289422	−0.144711	−	0.989474i	$-0.546225\pi$
$19$	−23.9697	−0.289422	−0.144711	+	0.989474i	$0.546225\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	122.739	1.11273	0.556365	−	0.830938i	$-0.312196\pi$
$23$	122.739	1.11273	0.556365	+	0.830938i	$0.312196\pi$
$24$	0	0
$25$	−112.315	−0.898523
$26$	0	0
$27$	0	0
$28$	0	0
$29$	219.909	1.40814	0.704071	−	0.710130i	$-0.251364\pi$
$29$	219.909	1.40814	0.704071	+	0.710130i	$0.251364\pi$
$30$	0	0
$31$	−27.0928	−0.156968	−0.0784840	−	0.996915i	$-0.525008\pi$
$31$	−27.0928	−0.156968	−0.0784840	+	0.996915i	$0.525008\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	96.7926	0.467455
$36$	0	0
$37$	94.1922	0.418516	0.209258	−	0.977860i	$-0.432895\pi$
$37$	94.1922	0.418516	0.209258	+	0.977860i	$0.432895\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	160.354	0.610808	0.305404	−	0.952223i	$-0.401209\pi$
$41$	160.354	0.610808	0.305404	+	0.952223i	$0.401209\pi$
$42$	0	0
$43$	151.302	0.536589	0.268295	−	0.963337i	$-0.413540\pi$
$43$	151.302	0.536589	0.268295	+	0.963337i	$0.413540\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	466.948	1.44918	0.724589	−	0.689181i	$-0.242030\pi$
$47$	466.948	1.44918	0.724589	+	0.689181i	$0.242030\pi$
$48$	0	0
$49$	395.594	1.15333
$50$	0	0
$51$	0	0
$52$	0	0
$53$	120.847	0.313199	0.156600	−	0.987662i	$-0.449947\pi$
$53$	120.847	0.313199	0.156600	+	0.987662i	$0.449947\pi$
$54$	0	0
$55$	54.3542	0.133257
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−439.633	−0.970090	−0.485045	−	0.874489i	$-0.661197\pi$
$59$	−439.633	−0.970090	−0.485045	+	0.874489i	$0.661197\pi$
$60$	0	0
$61$	−137.305	−0.288198	−0.144099	−	0.989563i	$-0.546028\pi$
$61$	−137.305	−0.288198	−0.144099	+	0.989563i	$0.546028\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−46.3002	−0.0883513
$66$	0	0
$67$	−512.280	−0.934104	−0.467052	−	0.884230i	$-0.654684\pi$
$67$	−512.280	−0.934104	−0.467052	+	0.884230i	$0.654684\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	410.719	0.686526	0.343263	−	0.939239i	$-0.388468\pi$
$71$	410.719	0.686526	0.343263	+	0.939239i	$0.388468\pi$
$72$	0	0
$73$	−308.004	−0.493823	−0.246912	−	0.969038i	$-0.579416\pi$
$73$	−308.004	−0.493823	−0.246912	+	0.969038i	$0.579416\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	414.759	0.613847
$78$	0	0
$79$	586.462	0.835217	0.417608	−	0.908627i	$-0.362868\pi$
$79$	586.462	0.835217	0.417608	+	0.908627i	$0.362868\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1354.20	1.79088	0.895440	−	0.445182i	$-0.146861\pi$
$83$	1354.20	1.79088	0.895440	+	0.445182i	$0.146861\pi$
$84$	0	0
$85$	−158.654	−0.202453
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−439.882	−0.523904	−0.261952	−	0.965081i	$-0.584366\pi$
$89$	−439.882	−0.523904	−0.261952	+	0.965081i	$0.584366\pi$
$90$	0	0
$91$	−353.302	−0.406990
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−85.3693	−0.0921969
$96$	0	0
$97$	−1511.27	−1.58192	−0.790959	−	0.611869i	$-0.790418\pi$
$97$	−1511.27	−1.58192	−0.790959	+	0.611869i	$0.790418\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−336.260	−0.331278	−0.165639	−	0.986186i	$-0.552969\pi$
$101$	−336.260	−0.331278	−0.165639	+	0.986186i	$0.552969\pi$
$102$	0	0
$103$	−322.712	−0.308716	−0.154358	−	0.988015i	$-0.549331\pi$
$103$	−322.712	−0.308716	−0.154358	+	0.988015i	$0.549331\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1434.62	1.29617	0.648083	−	0.761570i	$-0.275571\pi$
$107$	1434.62	1.29617	0.648083	+	0.761570i	$0.275571\pi$
$108$	0	0
$109$	849.147	0.746179	0.373089	−	0.927795i	$-0.378298\pi$
$109$	849.147	0.746179	0.373089	+	0.927795i	$0.378298\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1614.53	−1.34409	−0.672044	−	0.740511i	$-0.734583\pi$
$113$	−1614.53	−1.34409	−0.672044	+	0.740511i	$0.734583\pi$
$114$	0	0
$115$	437.140	0.354465
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1210.64	−0.932599
$120$	0	0
$121$	−1098.09	−0.825012
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−845.211	−0.604784
$126$	0	0
$127$	−865.174	−0.604502	−0.302251	−	0.953228i	$-0.597738\pi$
$127$	−865.174	−0.604502	−0.302251	+	0.953228i	$0.597738\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−281.400	−0.187680	−0.0938400	−	0.995587i	$-0.529914\pi$
$131$	−281.400	−0.187680	−0.0938400	+	0.995587i	$0.529914\pi$
$132$	0	0
$133$	−651.426	−0.424705
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2641.43	1.64725	0.823624	−	0.567137i	$-0.191949\pi$
$137$	2641.43	1.64725	0.823624	+	0.567137i	$0.191949\pi$
$138$	0	0
$139$	1998.64	1.21958	0.609791	−	0.792562i	$-0.291253\pi$
$139$	1998.64	1.21958	0.609791	+	0.792562i	$0.291253\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−198.398	−0.116020
$144$	0	0
$145$	783.218	0.448570
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1752.98	0.963824	0.481912	−	0.876220i	$-0.339942\pi$
$149$	1752.98	0.963824	0.481912	+	0.876220i	$0.339942\pi$
$150$	0	0
$151$	2794.64	1.50613	0.753063	−	0.657949i	$-0.228576\pi$
$151$	2794.64	1.50613	0.753063	+	0.657949i	$0.228576\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−96.4924	−0.0500030
$156$	0	0
$157$	3244.87	1.64949	0.824743	−	0.565508i	$-0.191320\pi$
$157$	3244.87	1.64949	0.824743	+	0.565508i	$0.191320\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3335.68	1.63285
$162$	0	0
$163$	−3281.47	−1.57684	−0.788418	−	0.615139i	$-0.789100\pi$
$163$	−3281.47	−1.57684	−0.788418	+	0.615139i	$0.789100\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3126.52	−1.44873	−0.724364	−	0.689418i	$-0.757866\pi$
$167$	−3126.52	−1.44873	−0.724364	+	0.689418i	$0.757866\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−97.5698	−0.0428792	−0.0214396	−	0.999770i	$-0.506825\pi$
$173$	−97.5698	−0.0428792	−0.0214396	+	0.999770i	$0.506825\pi$
$174$	0	0
$175$	−3052.40	−1.31851
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−34.7150	−0.0144956	−0.00724782	−	0.999974i	$-0.502307\pi$
$179$	−34.7150	−0.0144956	−0.00724782	+	0.999974i	$0.502307\pi$
$180$	0	0
$181$	−1229.35	−0.504843	−0.252422	−	0.967617i	$-0.581227\pi$
$181$	−1229.35	−0.504843	−0.252422	+	0.967617i	$0.581227\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	335.471	0.133320
$186$	0	0
$187$	−679.839	−0.265854
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4280.80	1.62172	0.810858	−	0.585243i	$-0.199001\pi$
$191$	4280.80	1.62172	0.810858	+	0.585243i	$0.199001\pi$
$192$	0	0
$193$	472.320	0.176157	0.0880786	−	0.996114i	$-0.471927\pi$
$193$	472.320	0.176157	0.0880786	+	0.996114i	$0.471927\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4484.37	1.62182	0.810908	−	0.585173i	$-0.198974\pi$
$197$	4484.37	1.62182	0.810908	+	0.585173i	$0.198974\pi$
$198$	0	0
$199$	366.240	0.130463	0.0652314	−	0.997870i	$-0.479221\pi$
$199$	366.240	0.130463	0.0652314	+	0.997870i	$0.479221\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5976.49	2.06634
$204$	0	0
$205$	571.110	0.194576
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−365.810	−0.121070
$210$	0	0
$211$	−2122.55	−0.692524	−0.346262	−	0.938138i	$-0.612549\pi$
$211$	−2122.55	−0.692524	−0.346262	+	0.938138i	$0.612549\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	538.870	0.170933
$216$	0	0
$217$	−736.303	−0.230339
$218$	0	0
$219$	0	0
$220$	0	0
$221$	579.103	0.176266
$222$	0	0
$223$	5926.42	1.77965	0.889826	−	0.456301i	$-0.150826\pi$
$223$	5926.42	1.77965	0.889826	+	0.456301i	$0.150826\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−895.661	−0.261881	−0.130941	−	0.991390i	$-0.541800\pi$
$227$	−895.661	−0.261881	−0.130941	+	0.991390i	$0.541800\pi$
$228$	0	0
$229$	627.717	0.181138	0.0905692	−	0.995890i	$-0.471131\pi$
$229$	627.717	0.181138	0.0905692	+	0.995890i	$0.471131\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2303.72	−0.647734	−0.323867	−	0.946103i	$-0.604983\pi$
$233$	−2303.72	−0.647734	−0.323867	+	0.946103i	$0.604983\pi$
$234$	0	0
$235$	1663.06	0.461643
$236$	0	0
$237$	0	0
$238$	0	0
$239$	544.622	0.147400	0.0737001	−	0.997280i	$-0.476519\pi$
$239$	544.622	0.147400	0.0737001	+	0.997280i	$0.476519\pi$
$240$	0	0
$241$	5426.10	1.45031	0.725157	−	0.688584i	$-0.241767\pi$
$241$	5426.10	1.45031	0.725157	+	0.688584i	$0.241767\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1408.93	0.367400
$246$	0	0
$247$	311.606	0.0802713
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5221.22	−1.31299	−0.656494	−	0.754331i	$-0.727961\pi$
$251$	−5221.22	−1.31299	−0.656494	+	0.754331i	$0.727961\pi$
$252$	0	0
$253$	1873.16	0.465472
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−658.206	−0.159758	−0.0798789	−	0.996805i	$-0.525453\pi$
$257$	−658.206	−0.159758	−0.0798789	+	0.996805i	$0.525453\pi$
$258$	0	0
$259$	2559.87	0.614141
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3246.45	0.761160	0.380580	−	0.924748i	$-0.375724\pi$
$263$	3246.45	0.761160	0.380580	+	0.924748i	$0.375724\pi$
$264$	0	0
$265$	430.401	0.0997711
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2585.80	0.586093	0.293047	−	0.956098i	$-0.405331\pi$
$269$	2585.80	0.586093	0.293047	+	0.956098i	$0.405331\pi$
$270$	0	0
$271$	−988.933	−0.221673	−0.110836	−	0.993839i	$-0.535353\pi$
$271$	−988.933	−0.221673	−0.110836	+	0.993839i	$0.535353\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1714.09	−0.375866
$276$	0	0
$277$	8142.40	1.76617	0.883086	−	0.469211i	$-0.155462\pi$
$277$	8142.40	1.76617	0.883086	+	0.469211i	$0.155462\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1534.21	−0.325705	−0.162853	−	0.986650i	$-0.552070\pi$
$281$	−1534.21	−0.325705	−0.162853	+	0.986650i	$0.552070\pi$
$282$	0	0
$283$	6965.00	1.46299	0.731495	−	0.681847i	$-0.238823\pi$
$283$	6965.00	1.46299	0.731495	+	0.681847i	$0.238823\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4357.96	0.896314
$288$	0	0
$289$	−2928.62	−0.596096
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−640.029	−0.127614	−0.0638070	−	0.997962i	$-0.520324\pi$
$293$	−640.029	−0.127614	−0.0638070	+	0.997962i	$0.520324\pi$
$294$	0	0
$295$	−1565.77	−0.309027
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1595.60	−0.308616
$300$	0	0
$301$	4111.95	0.787404
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−489.019	−0.0918070
$306$	0	0
$307$	100.406	0.0186660	0.00933299	−	0.999956i	$-0.497029\pi$
$307$	100.406	0.0186660	0.00933299	+	0.999956i	$0.497029\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3878.92	−0.707245	−0.353623	−	0.935388i	$-0.615050\pi$
$311$	−3878.92	−0.707245	−0.353623	+	0.935388i	$0.615050\pi$
$312$	0	0
$313$	−3789.39	−0.684311	−0.342155	−	0.939643i	$-0.611157\pi$
$313$	−3789.39	−0.684311	−0.342155	+	0.939643i	$0.611157\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4406.81	−0.780791	−0.390396	−	0.920647i	$-0.627662\pi$
$317$	−4406.81	−0.780791	−0.390396	+	0.920647i	$0.627662\pi$
$318$	0	0
$319$	3356.11	0.589048
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1067.76	0.183938
$324$	0	0
$325$	1460.10	0.249205
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12690.3	2.12656
$330$	0	0
$331$	4131.49	0.686064	0.343032	−	0.939324i	$-0.388546\pi$
$331$	4131.49	0.686064	0.343032	+	0.939324i	$0.388546\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1824.51	−0.297564
$336$	0	0
$337$	−4560.82	−0.737221	−0.368611	−	0.929584i	$-0.620166\pi$
$337$	−4560.82	−0.737221	−0.368611	+	0.929584i	$0.620166\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−413.473	−0.0656622
$342$	0	0
$343$	1429.34	0.225007
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10069.4	1.55779	0.778896	−	0.627153i	$-0.215780\pi$
$347$	10069.4	1.55779	0.778896	+	0.627153i	$0.215780\pi$
$348$	0	0
$349$	5879.32	0.901757	0.450878	−	0.892585i	$-0.351111\pi$
$349$	5879.32	0.901757	0.450878	+	0.892585i	$0.351111\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9142.56	1.37850	0.689249	−	0.724525i	$-0.257941\pi$
$353$	9142.56	1.37850	0.689249	+	0.724525i	$0.257941\pi$
$354$	0	0
$355$	1462.80	0.218696
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2754.32	−0.404924	−0.202462	−	0.979290i	$-0.564894\pi$
$359$	−2754.32	−0.404924	−0.202462	+	0.979290i	$0.564894\pi$
$360$	0	0
$361$	−6284.45	−0.916235
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1096.97	−0.157310
$366$	0	0
$367$	−3040.19	−0.432416	−0.216208	−	0.976347i	$-0.569369\pi$
$367$	−3040.19	−0.432416	−0.216208	+	0.976347i	$0.569369\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3284.26	0.459596
$372$	0	0
$373$	−5384.72	−0.747481	−0.373740	−	0.927533i	$-0.621925\pi$
$373$	−5384.72	−0.747481	−0.373740	+	0.927533i	$0.621925\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2858.82	−0.390548
$378$	0	0
$379$	3424.27	0.464097	0.232049	−	0.972704i	$-0.425457\pi$
$379$	3424.27	0.464097	0.232049	+	0.972704i	$0.425457\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−382.985	−0.0510956	−0.0255478	−	0.999674i	$-0.508133\pi$
$383$	−382.985	−0.0510956	−0.0255478	+	0.999674i	$0.508133\pi$
$384$	0	0
$385$	1477.19	0.195544
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−8588.34	−1.11940	−0.559699	−	0.828696i	$-0.689083\pi$
$389$	−8588.34	−1.11940	−0.559699	+	0.828696i	$0.689083\pi$
$390$	0	0
$391$	−5467.56	−0.707178
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2088.72	0.266063
$396$	0	0
$397$	−7239.16	−0.915171	−0.457586	−	0.889166i	$-0.651286\pi$
$397$	−7239.16	−0.915171	−0.457586	+	0.889166i	$0.651286\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4269.62	−0.531708	−0.265854	−	0.964013i	$-0.585654\pi$
$401$	−4269.62	−0.531708	−0.265854	+	0.964013i	$0.585654\pi$
$402$	0	0
$403$	352.206	0.0435351
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1437.50	0.175072
$408$	0	0
$409$	13562.5	1.63967	0.819834	−	0.572602i	$-0.194066\pi$
$409$	13562.5	1.63967	0.819834	+	0.572602i	$0.194066\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−11947.9	−1.42353
$414$	0	0
$415$	4823.06	0.570494
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14576.9	−1.69959	−0.849794	−	0.527114i	$-0.823274\pi$
$419$	−14576.9	−1.69959	−0.849794	+	0.527114i	$0.823274\pi$
$420$	0	0
$421$	15848.4	1.83469	0.917343	−	0.398099i	$-0.130330\pi$
$421$	15848.4	1.83469	0.917343	+	0.398099i	$0.130330\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5003.24	0.571042
$426$	0	0
$427$	−3731.55	−0.422909
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10694.7	1.19524	0.597618	−	0.801781i	$-0.296114\pi$
$431$	10694.7	1.19524	0.597618	+	0.801781i	$0.296114\pi$
$432$	0	0
$433$	−16079.0	−1.78454	−0.892272	−	0.451498i	$-0.850890\pi$
$433$	−16079.0	−1.78454	−0.892272	+	0.451498i	$0.850890\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2942.01	−0.322049
$438$	0	0
$439$	−6035.80	−0.656203	−0.328101	−	0.944643i	$-0.606409\pi$
$439$	−6035.80	−0.656203	−0.328101	+	0.944643i	$0.606409\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10201.3	1.09409	0.547043	−	0.837105i	$-0.315753\pi$
$443$	10201.3	1.09409	0.547043	+	0.837105i	$0.315753\pi$
$444$	0	0
$445$	−1566.66	−0.166892
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5822.54	0.611988	0.305994	−	0.952033i	$-0.401011\pi$
$449$	5822.54	0.611988	0.305994	+	0.952033i	$0.401011\pi$
$450$	0	0
$451$	2447.22	0.255511
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1258.30	−0.129649
$456$	0	0
$457$	4621.60	0.473062	0.236531	−	0.971624i	$-0.423990\pi$
$457$	4621.60	0.473062	0.236531	+	0.971624i	$0.423990\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5127.77	−0.518056	−0.259028	−	0.965870i	$-0.583402\pi$
$461$	−5127.77	−0.518056	−0.259028	+	0.965870i	$0.583402\pi$
$462$	0	0
$463$	−6486.27	−0.651064	−0.325532	−	0.945531i	$-0.605543\pi$
$463$	−6486.27	−0.651064	−0.325532	+	0.945531i	$0.605543\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12978.0	1.28598	0.642990	−	0.765875i	$-0.277694\pi$
$467$	12978.0	1.28598	0.642990	+	0.765875i	$0.277694\pi$
$468$	0	0
$469$	−13922.3	−1.37073
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2309.08	0.224464
$474$	0	0
$475$	2692.16	0.260053
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5808.96	−0.554109	−0.277055	−	0.960854i	$-0.589358\pi$
$479$	−5808.96	−0.554109	−0.277055	+	0.960854i	$0.589358\pi$
$480$	0	0
$481$	−1224.50	−0.116076
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5382.46	−0.503928
$486$	0	0
$487$	5387.14	0.501262	0.250631	−	0.968083i	$-0.419362\pi$
$487$	5387.14	0.501262	0.250631	+	0.968083i	$0.419362\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15259.1	1.40251	0.701255	−	0.712911i	$-0.252624\pi$
$491$	15259.1	1.40251	0.701255	+	0.712911i	$0.252624\pi$
$492$	0	0
$493$	−9796.16	−0.894922
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11162.1	1.00742
$498$	0	0
$499$	−1856.04	−0.166509	−0.0832544	−	0.996528i	$-0.526531\pi$
$499$	−1856.04	−0.166509	−0.0832544	+	0.996528i	$0.526531\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1049.46	0.0930283	0.0465142	−	0.998918i	$-0.485189\pi$
$503$	1049.46	0.0930283	0.0465142	+	0.998918i	$0.485189\pi$
$504$	0	0
$505$	−1197.61	−0.105530
$506$	0	0
$507$	0	0
$508$	0	0
$509$	551.106	0.0479909	0.0239954	−	0.999712i	$-0.492361\pi$
$509$	551.106	0.0479909	0.0239954	+	0.999712i	$0.492361\pi$
$510$	0	0
$511$	−8370.64	−0.724649
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1149.36	−0.0983431
$516$	0	0
$517$	7126.26	0.606214
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8995.30	0.756413	0.378206	−	0.925721i	$-0.376541\pi$
$521$	8995.30	0.756413	0.378206	+	0.925721i	$0.376541\pi$
$522$	0	0
$523$	−2663.91	−0.222724	−0.111362	−	0.993780i	$-0.535521\pi$
$523$	−2663.91	−0.222724	−0.111362	+	0.993780i	$0.535521\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1206.89	0.0997586
$528$	0	0
$529$	2897.77	0.238167
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2084.60	−0.169408
$534$	0	0
$535$	5109.47	0.412900
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6037.30	0.482458
$540$	0	0
$541$	−6169.23	−0.490270	−0.245135	−	0.969489i	$-0.578832\pi$
$541$	−6169.23	−0.490270	−0.245135	+	0.969489i	$0.578832\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3024.28	0.237699
$546$	0	0
$547$	−5140.42	−0.401807	−0.200904	−	0.979611i	$-0.564388\pi$
$547$	−5140.42	−0.401807	−0.200904	+	0.979611i	$0.564388\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5271.15	−0.407547
$552$	0	0
$553$	15938.3	1.22562
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2778.56	−0.211367	−0.105683	−	0.994400i	$-0.533703\pi$
$557$	−2778.56	−0.211367	−0.105683	+	0.994400i	$0.533703\pi$
$558$	0	0
$559$	−1966.93	−0.148823
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4906.14	0.367263	0.183632	−	0.982995i	$-0.441215\pi$
$563$	4906.14	0.367263	0.183632	+	0.982995i	$0.441215\pi$
$564$	0	0
$565$	−5750.22	−0.428166
$566$	0	0
$567$	0	0
$568$	0	0
$569$	9363.15	0.689849	0.344924	−	0.938631i	$-0.387905\pi$
$569$	9363.15	0.689849	0.344924	+	0.938631i	$0.387905\pi$
$570$	0	0
$571$	−7199.32	−0.527640	−0.263820	−	0.964572i	$-0.584982\pi$
$571$	−7199.32	−0.527640	−0.263820	+	0.964572i	$0.584982\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−13785.4	−0.999813
$576$	0	0
$577$	−11449.6	−0.826086	−0.413043	−	0.910711i	$-0.635534\pi$
$577$	−11449.6	−0.826086	−0.413043	+	0.910711i	$0.635534\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	36803.3	2.62798
$582$	0	0
$583$	1844.28	0.131016
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5439.39	−0.382466	−0.191233	−	0.981545i	$-0.561249\pi$
$587$	−5439.39	−0.382466	−0.191233	+	0.981545i	$0.561249\pi$
$588$	0	0
$589$	649.406	0.0454301
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28405.8	1.96709	0.983547	−	0.180651i	$-0.0578204\pi$
$593$	28405.8	1.96709	0.983547	+	0.180651i	$0.0578204\pi$
$594$	0	0
$595$	−4311.76	−0.297084
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10482.3	−0.715020	−0.357510	−	0.933909i	$-0.616374\pi$
$599$	−10482.3	−0.715020	−0.357510	+	0.933909i	$0.616374\pi$
$600$	0	0
$601$	3199.54	0.217158	0.108579	−	0.994088i	$-0.465370\pi$
$601$	3199.54	0.217158	0.108579	+	0.994088i	$0.465370\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3910.91	−0.262812
$606$	0	0
$607$	−11342.8	−0.758468	−0.379234	−	0.925301i	$-0.623812\pi$
$607$	−11342.8	−0.758468	−0.379234	+	0.925301i	$0.623812\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6070.32	−0.401930
$612$	0	0
$613$	14385.4	0.947831	0.473916	−	0.880570i	$-0.342840\pi$
$613$	14385.4	0.947831	0.473916	+	0.880570i	$0.342840\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22056.8	−1.43918	−0.719588	−	0.694401i	$-0.755669\pi$
$617$	−22056.8	−1.43918	−0.719588	+	0.694401i	$0.755669\pi$
$618$	0	0
$619$	−13621.4	−0.884477	−0.442238	−	0.896898i	$-0.645815\pi$
$619$	−13621.4	−0.884477	−0.442238	+	0.896898i	$0.645815\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11954.7	−0.768789
$624$	0	0
$625$	11029.2	0.705866
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−4195.92	−0.265982
$630$	0	0
$631$	18737.5	1.18214	0.591068	−	0.806622i	$-0.298707\pi$
$631$	18737.5	1.18214	0.591068	+	0.806622i	$0.298707\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3081.36	−0.192567
$636$	0	0
$637$	−5142.72	−0.319877
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−29798.7	−1.83616	−0.918081	−	0.396394i	$-0.870261\pi$
$641$	−29798.7	−1.83616	−0.918081	+	0.396394i	$0.870261\pi$
$642$	0	0
$643$	22983.5	1.40961	0.704807	−	0.709399i	$-0.251034\pi$
$643$	22983.5	1.40961	0.704807	+	0.709399i	$0.251034\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−24905.4	−1.51334	−0.756672	−	0.653794i	$-0.773176\pi$
$647$	−24905.4	−1.51334	−0.756672	+	0.653794i	$0.773176\pi$
$648$	0	0
$649$	−6709.39	−0.405804
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10077.8	−0.603946	−0.301973	−	0.953316i	$-0.597645\pi$
$653$	−10077.8	−0.603946	−0.301973	+	0.953316i	$0.597645\pi$
$654$	0	0
$655$	−1002.22	−0.0597864
$656$	0	0
$657$	0	0
$658$	0	0
$659$	12334.6	0.729116	0.364558	−	0.931181i	$-0.381220\pi$
$659$	12334.6	0.729116	0.364558	+	0.931181i	$0.381220\pi$
$660$	0	0
$661$	−12749.1	−0.750202	−0.375101	−	0.926984i	$-0.622392\pi$
$661$	−12749.1	−0.750202	−0.375101	+	0.926984i	$0.622392\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2320.09	−0.135292
$666$	0	0
$667$	26991.3	1.56688
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2095.46	−0.120558
$672$	0	0
$673$	−13618.2	−0.780007	−0.390004	−	0.920813i	$-0.627526\pi$
$673$	−13618.2	−0.780007	−0.390004	+	0.920813i	$0.627526\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9655.67	−0.548150	−0.274075	−	0.961708i	$-0.588372\pi$
$677$	−9655.67	−0.548150	−0.274075	+	0.961708i	$0.588372\pi$
$678$	0	0
$679$	−41071.9	−2.32135
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16316.8	0.914119	0.457060	−	0.889436i	$-0.348903\pi$
$683$	16316.8	0.914119	0.457060	+	0.889436i	$0.348903\pi$
$684$	0	0
$685$	9407.61	0.524739
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1571.01	−0.0868658
$690$	0	0
$691$	−2350.84	−0.129421	−0.0647106	−	0.997904i	$-0.520612\pi$
$691$	−2350.84	−0.129421	−0.0647106	+	0.997904i	$0.520612\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7118.24	0.388504
$696$	0	0
$697$	−7143.20	−0.388189
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8076.90	0.435179	0.217589	−	0.976040i	$-0.430181\pi$
$701$	8076.90	0.435179	0.217589	+	0.976040i	$0.430181\pi$
$702$	0	0
$703$	−2257.76	−0.121128
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9138.55	−0.486125
$708$	0	0
$709$	−13624.9	−0.721712	−0.360856	−	0.932622i	$-0.617515\pi$
$709$	−13624.9	−0.721712	−0.360856	+	0.932622i	$0.617515\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3325.33	−0.174663
$714$	0	0
$715$	−706.604	−0.0369587
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16235.8	0.842131	0.421066	−	0.907030i	$-0.361656\pi$
$719$	16235.8	0.842131	0.421066	+	0.907030i	$0.361656\pi$
$720$	0	0
$721$	−8770.37	−0.453018
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−24699.2	−1.26525
$726$	0	0
$727$	−24181.2	−1.23361	−0.616803	−	0.787118i	$-0.711572\pi$
$727$	−24181.2	−1.23361	−0.616803	+	0.787118i	$0.711572\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−6739.96	−0.341021
$732$	0	0
$733$	3053.70	0.153876	0.0769379	−	0.997036i	$-0.475486\pi$
$733$	3053.70	0.153876	0.0769379	+	0.997036i	$0.475486\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7818.10	−0.390751
$738$	0	0
$739$	8033.62	0.399894	0.199947	−	0.979807i	$-0.435923\pi$
$739$	8033.62	0.399894	0.199947	+	0.979807i	$0.435923\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16139.6	−0.796912	−0.398456	−	0.917187i	$-0.630454\pi$
$743$	−16139.6	−0.796912	−0.398456	+	0.917187i	$0.630454\pi$
$744$	0	0
$745$	6243.33	0.307031
$746$	0	0
$747$	0	0
$748$	0	0
$749$	38988.7	1.90202
$750$	0	0
$751$	18491.1	0.898469	0.449235	−	0.893414i	$-0.351697\pi$
$751$	18491.1	0.898469	0.449235	+	0.893414i	$0.351697\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	9953.28	0.479784
$756$	0	0
$757$	160.630	0.00771227	0.00385613	−	0.999993i	$-0.498773\pi$
$757$	160.630	0.00771227	0.00385613	+	0.999993i	$0.498773\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−26799.1	−1.27656	−0.638282	−	0.769803i	$-0.720355\pi$
$761$	−26799.1	−1.27656	−0.638282	+	0.769803i	$0.720355\pi$
$762$	0	0
$763$	23077.3	1.09496
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5715.22	0.269054
$768$	0	0
$769$	−5145.82	−0.241304	−0.120652	−	0.992695i	$-0.538499\pi$
$769$	−5145.82	−0.241304	−0.120652	+	0.992695i	$0.538499\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−12810.6	−0.596072	−0.298036	−	0.954555i	$-0.596332\pi$
$773$	−12810.6	−0.596072	−0.298036	+	0.954555i	$0.596332\pi$
$774$	0	0
$775$	3042.94	0.141039
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3843.64	−0.176781
$780$	0	0
$781$	6268.13	0.287185
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11556.8	0.525452
$786$	0	0
$787$	−28073.0	−1.27153	−0.635764	−	0.771883i	$-0.719315\pi$
$787$	−28073.0	−1.27153	−0.635764	+	0.771883i	$0.719315\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−43878.1	−1.97235
$792$	0	0
$793$	1784.96	0.0799318
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30093.1	−1.33746	−0.668729	−	0.743507i	$-0.733161\pi$
$797$	−30093.1	−1.33746	−0.668729	+	0.743507i	$0.733161\pi$
$798$	0	0
$799$	−20800.8	−0.921003
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−4700.56	−0.206574
$804$	0	0
$805$	11880.2	0.520151
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24337.1	1.05766	0.528831	−	0.848727i	$-0.322631\pi$
$809$	24337.1	1.05766	0.528831	+	0.848727i	$0.322631\pi$
$810$	0	0
$811$	−19078.7	−0.826071	−0.413035	−	0.910715i	$-0.635531\pi$
$811$	−19078.7	−0.826071	−0.413035	+	0.910715i	$0.635531\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11687.1	−0.502309
$816$	0	0
$817$	−3626.66	−0.155301
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2013.92	−0.0856104	−0.0428052	−	0.999083i	$-0.513629\pi$
$821$	−2013.92	−0.0856104	−0.0428052	+	0.999083i	$0.513629\pi$
$822$	0	0
$823$	7692.10	0.325795	0.162898	−	0.986643i	$-0.447916\pi$
$823$	7692.10	0.325795	0.162898	+	0.986643i	$0.447916\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4762.76	−0.200263	−0.100131	−	0.994974i	$-0.531926\pi$
$827$	−4762.76	−0.200263	−0.100131	+	0.994974i	$0.531926\pi$
$828$	0	0
$829$	19977.7	0.836976	0.418488	−	0.908222i	$-0.362560\pi$
$829$	19977.7	0.836976	0.418488	+	0.908222i	$0.362560\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−17622.3	−0.732984
$834$	0	0
$835$	−11135.3	−0.461499
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−30615.8	−1.25980	−0.629901	−	0.776676i	$-0.716905\pi$
$839$	−30615.8	−1.25980	−0.629901	+	0.776676i	$0.716905\pi$
$840$	0	0
$841$	23971.0	0.982861
$842$	0	0
$843$	0	0
$844$	0	0
$845$	601.902	0.0245042
$846$	0	0
$847$	−29842.9	−1.21064
$848$	0	0
$849$	0	0
$850$	0	0
$851$	11561.0	0.465696
$852$	0	0
$853$	5660.88	0.227227	0.113614	−	0.993525i	$-0.463757\pi$
$853$	5660.88	0.227227	0.113614	+	0.993525i	$0.463757\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41346.1	−1.64802	−0.824012	−	0.566572i	$-0.808269\pi$
$857$	−41346.1	−1.64802	−0.824012	+	0.566572i	$0.808269\pi$
$858$	0	0
$859$	34810.5	1.38268	0.691339	−	0.722530i	$-0.257021\pi$
$859$	34810.5	1.38268	0.691339	+	0.722530i	$0.257021\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−8360.51	−0.329774	−0.164887	−	0.986312i	$-0.552726\pi$
$863$	−8360.51	−0.329774	−0.164887	+	0.986312i	$0.552726\pi$
$864$	0	0
$865$	−347.500	−0.0136594
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8950.21	0.349385
$870$	0	0
$871$	6659.64	0.259074
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−22970.4	−0.887475
$876$	0	0
$877$	−40579.3	−1.56245	−0.781223	−	0.624251i	$-0.785404\pi$
$877$	−40579.3	−1.56245	−0.781223	+	0.624251i	$0.785404\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−10445.2	−0.399442	−0.199721	−	0.979853i	$-0.564004\pi$
$881$	−10445.2	−0.399442	−0.199721	+	0.979853i	$0.564004\pi$
$882$	0	0
$883$	−18227.6	−0.694685	−0.347343	−	0.937738i	$-0.612916\pi$
$883$	−18227.6	−0.694685	−0.347343	+	0.937738i	$0.612916\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−23517.7	−0.890245	−0.445122	−	0.895470i	$-0.646840\pi$
$887$	−23517.7	−0.890245	−0.445122	+	0.895470i	$0.646840\pi$
$888$	0	0
$889$	−23512.9	−0.887061
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11192.6	−0.419424
$894$	0	0
$895$	−123.639	−0.00461766
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−5957.95	−0.221033
$900$	0	0
$901$	−5383.28	−0.199049
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4378.38	−0.160820
$906$	0	0
$907$	30564.6	1.11894	0.559471	−	0.828850i	$-0.311004\pi$
$907$	30564.6	1.11894	0.559471	+	0.828850i	$0.311004\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−32766.5	−1.19166	−0.595831	−	0.803110i	$-0.703177\pi$
$911$	−32766.5	−1.19166	−0.595831	+	0.803110i	$0.703177\pi$
$912$	0	0
$913$	20667.0	0.749154
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7647.64	−0.275406
$918$	0	0
$919$	−20686.7	−0.742538	−0.371269	−	0.928525i	$-0.621077\pi$
$919$	−20686.7	−0.742538	−0.371269	+	0.928525i	$0.621077\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5339.34	−0.190408
$924$	0	0
$925$	−10579.2	−0.376047
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−45632.2	−1.61156	−0.805782	−	0.592212i	$-0.798255\pi$
$929$	−45632.2	−1.61156	−0.805782	+	0.592212i	$0.798255\pi$
$930$	0	0
$931$	−9482.26	−0.333801
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2421.28	−0.0846892
$936$	0	0
$937$	−17761.4	−0.619253	−0.309626	−	0.950858i	$-0.600204\pi$
$937$	−17761.4	−0.619253	−0.309626	+	0.950858i	$0.600204\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	44888.3	1.55507	0.777534	−	0.628841i	$-0.216470\pi$
$941$	44888.3	1.55507	0.777534	+	0.628841i	$0.216470\pi$
$942$	0	0
$943$	19681.7	0.679664
$944$	0	0
$945$	0	0
$946$	0	0
$947$	16069.6	0.551415	0.275708	−	0.961242i	$-0.411088\pi$
$947$	16069.6	0.551415	0.275708	+	0.961242i	$0.411088\pi$
$948$	0	0
$949$	4004.05	0.136962
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3512.03	−0.119377	−0.0596883	−	0.998217i	$-0.519011\pi$
$953$	−3512.03	−0.119377	−0.0596883	+	0.998217i	$0.519011\pi$
$954$	0	0
$955$	15246.3	0.516605
$956$	0	0
$957$	0	0
$958$	0	0
$959$	71786.5	2.41721
$960$	0	0
$961$	−29057.0	−0.975361
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1682.19	0.0561158
$966$	0	0
$967$	37011.9	1.23084	0.615421	−	0.788199i	$-0.288986\pi$
$967$	37011.9	1.23084	0.615421	+	0.788199i	$0.288986\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19532.3	0.645542	0.322771	−	0.946477i	$-0.395386\pi$
$971$	19532.3	0.645542	0.322771	+	0.946477i	$0.395386\pi$
$972$	0	0
$973$	54317.1	1.78965
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30201.2	−0.988970	−0.494485	−	0.869186i	$-0.664643\pi$
$977$	−30201.2	−0.988970	−0.494485	+	0.869186i	$0.664643\pi$
$978$	0	0
$979$	−6713.21	−0.219157
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−38774.9	−1.25812	−0.629058	−	0.777359i	$-0.716559\pi$
$983$	−38774.9	−1.25812	−0.629058	+	0.777359i	$0.716559\pi$
$984$	0	0
$985$	15971.3	0.516638
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18570.6	0.597079
$990$	0	0
$991$	27728.9	0.888838	0.444419	−	0.895819i	$-0.353410\pi$
$991$	27728.9	0.888838	0.444419	+	0.895819i	$0.353410\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1304.38	0.0415596
$996$	0	0
$997$	−48918.2	−1.55392	−0.776958	−	0.629552i	$-0.783239\pi$
$997$	−48918.2	−1.55392	−0.776958	+	0.629552i	$0.783239\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.4.a.bb.1.2		2
3.2	odd	2		208.4.a.h.1.1		2
4.3	odd	2		117.4.a.d.1.2		2
12.11	even	2		13.4.a.b.1.1	✓	2
24.5	odd	2		832.4.a.z.1.2		2
24.11	even	2		832.4.a.s.1.1		2
52.51	odd	2		1521.4.a.r.1.1		2
60.23	odd	4		325.4.b.e.274.3		4
60.47	odd	4		325.4.b.e.274.2		4
60.59	even	2		325.4.a.f.1.2		2
84.83	odd	2		637.4.a.b.1.1		2
132.131	odd	2		1573.4.a.b.1.2		2
156.11	odd	12		169.4.e.f.147.3		8
156.23	even	6		169.4.c.j.22.1		4
156.35	even	6		169.4.c.g.146.2		4
156.47	odd	4		169.4.b.f.168.2		4
156.59	odd	12		169.4.e.f.23.2		8
156.71	odd	12		169.4.e.f.23.3		8
156.83	odd	4		169.4.b.f.168.3		4
156.95	even	6		169.4.c.j.146.1		4
156.107	even	6		169.4.c.g.22.2		4
156.119	odd	12		169.4.e.f.147.2		8
156.155	even	2		169.4.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.b.1.1	✓	2	12.11	even	2
117.4.a.d.1.2		2	4.3	odd	2
169.4.a.g.1.2		2	156.155	even	2
169.4.b.f.168.2		4	156.47	odd	4
169.4.b.f.168.3		4	156.83	odd	4
169.4.c.g.22.2		4	156.107	even	6
169.4.c.g.146.2		4	156.35	even	6
169.4.c.j.22.1		4	156.23	even	6
169.4.c.j.146.1		4	156.95	even	6
169.4.e.f.23.2		8	156.59	odd	12
169.4.e.f.23.3		8	156.71	odd	12
169.4.e.f.147.2		8	156.119	odd	12
169.4.e.f.147.3		8	156.11	odd	12
208.4.a.h.1.1		2	3.2	odd	2
325.4.a.f.1.2		2	60.59	even	2
325.4.b.e.274.2		4	60.47	odd	4
325.4.b.e.274.3		4	60.23	odd	4
637.4.a.b.1.1		2	84.83	odd	2
832.4.a.s.1.1		2	24.11	even	2
832.4.a.z.1.2		2	24.5	odd	2
1521.4.a.r.1.1		2	52.51	odd	2
1573.4.a.b.1.2		2	132.131	odd	2
1872.4.a.bb.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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+3.56155 q^{5} +27.1771 q^{7} +O(q^{10})\)	\(q+3.56155 q^{5} +27.1771 q^{7} +15.2614 q^{11} -13.0000 q^{13} -44.5464 q^{17} -23.9697 q^{19} +122.739 q^{23} -112.315 q^{25} +219.909 q^{29} -27.0928 q^{31} +96.7926 q^{35} +94.1922 q^{37} +160.354 q^{41} +151.302 q^{43} +466.948 q^{47} +395.594 q^{49} +120.847 q^{53} +54.3542 q^{55} -439.633 q^{59} -137.305 q^{61} -46.3002 q^{65} -512.280 q^{67} +410.719 q^{71} -308.004 q^{73} +414.759 q^{77} +586.462 q^{79} +1354.20 q^{83} -158.654 q^{85} -439.882 q^{89} -353.302 q^{91} -85.3693 q^{95} -1511.27 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{5} + 9 q^{7}+O(q^{10}) \) 2 * q + 3 * q^5 + 9 * q^7	\( 2 q + 3 q^{5} + 9 q^{7} + 80 q^{11} - 26 q^{13} - 19 q^{17} + 84 q^{19} + 196 q^{23} - 237 q^{25} + 44 q^{29} + 86 q^{31} + 107 q^{35} + 209 q^{37} + 230 q^{41} - 287 q^{43} + 435 q^{47} + 383 q^{49} + 118 q^{53} + 18 q^{55} - 368 q^{59} - 1058 q^{61} - 39 q^{65} - 68 q^{67} - 131 q^{71} + 456 q^{73} - 762 q^{77} + 1008 q^{79} + 1958 q^{83} - 173 q^{85} + 720 q^{89} - 117 q^{91} - 146 q^{95} - 928 q^{97}+O(q^{100}) \) 2 * q + 3 * q^5 + 9 * q^7 + 80 * q^11 - 26 * q^13 - 19 * q^17 + 84 * q^19 + 196 * q^23 - 237 * q^25 + 44 * q^29 + 86 * q^31 + 107 * q^35 + 209 * q^37 + 230 * q^41 - 287 * q^43 + 435 * q^47 + 383 * q^49 + 118 * q^53 + 18 * q^55 - 368 * q^59 - 1058 * q^61 - 39 * q^65 - 68 * q^67 - 131 * q^71 + 456 * q^73 - 762 * q^77 + 1008 * q^79 + 1958 * q^83 - 173 * q^85 + 720 * q^89 - 117 * q^91 - 146 * q^95 - 928 * q^97

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