LMFDB - Embedded newform 324.4.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,4,Mod(1,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("324.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	324.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:	$19.1166188419$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 4 $ x^3 - x^2 - 7*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 36)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.551929$ of defining polynomial
Character	$\chi$	$=$	324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+12.7419 q^{5} -14.0535 q^{7} -42.5491 q^{11} -72.4632 q^{13} -59.6114 q^{17} +105.570 q^{19} -0.225180 q^{23} +37.3563 q^{25} -225.710 q^{29} +201.194 q^{31} -179.068 q^{35} -152.926 q^{37} -489.648 q^{41} -7.58743 q^{43} +373.391 q^{47} -145.500 q^{49} -43.6780 q^{53} -542.157 q^{55} -671.899 q^{59} +74.0354 q^{61} -923.320 q^{65} +420.871 q^{67} -730.840 q^{71} +473.927 q^{73} +597.963 q^{77} -529.622 q^{79} +26.1534 q^{83} -759.563 q^{85} +415.949 q^{89} +1018.36 q^{91} +1345.17 q^{95} +927.485 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{5} + 6 q^{7} - 51 q^{11} - 12 q^{13} - 111 q^{17} + 15 q^{19} - 210 q^{23} + 3 q^{25} - 456 q^{29} - 48 q^{31} - 552 q^{35} - 48 q^{37} - 897 q^{41} - 129 q^{43} - 522 q^{47} + 225 q^{49} - 1104 q^{53}+ \cdots - 93 q^{97}+O(q^{100}) $ 3 * q - 6 * q^5 + 6 * q^7 - 51 * q^11 - 12 * q^13 - 111 * q^17 + 15 * q^19 - 210 * q^23 + 3 * q^25 - 456 * q^29 - 48 * q^31 - 552 * q^35 - 48 * q^37 - 897 * q^41 - 129 * q^43 - 522 * q^47 + 225 * q^49 - 1104 * q^53 - 108 * q^55 - 453 * q^59 + 402 * q^61 - 1110 * q^65 + 213 * q^67 + 60 * q^71 + 375 * q^73 - 1128 * q^77 - 552 * q^79 + 612 * q^83 - 1188 * q^85 - 462 * q^89 - 132 * q^91 + 2184 * q^95 - 93 * 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
$5$	12.7419	1.13967	0.569836	−	0.821759i	$-0.307007\pi$
$5$	12.7419	1.13967	0.569836	+	0.821759i	$0.307007\pi$
$6$	0	0
$7$	−14.0535	−0.758817	−0.379408	−	0.925229i	$-0.623872\pi$
$7$	−14.0535	−0.758817	−0.379408	+	0.925229i	$0.623872\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−42.5491	−1.16628	−0.583138	−	0.812373i	$-0.698175\pi$
$11$	−42.5491	−1.16628	−0.583138	+	0.812373i	$0.698175\pi$
$12$	0	0
$13$	−72.4632	−1.54598	−0.772988	−	0.634421i	$-0.781239\pi$
$13$	−72.4632	−1.54598	−0.772988	+	0.634421i	$0.781239\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−59.6114	−0.850464	−0.425232	−	0.905084i	$-0.639807\pi$
$17$	−59.6114	−0.850464	−0.425232	+	0.905084i	$0.639807\pi$
$18$	0	0
$19$	105.570	1.27471	0.637354	−	0.770571i	$-0.280029\pi$
$19$	105.570	1.27471	0.637354	+	0.770571i	$0.280029\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.225180	−0.00204144	−0.00102072	−	0.999999i	$-0.500325\pi$
$23$	−0.225180	−0.00204144	−0.00102072	+	0.999999i	$0.500325\pi$
$24$	0	0
$25$	37.3563	0.298850
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−225.710	−1.44528	−0.722642	−	0.691223i	$-0.757072\pi$
$29$	−225.710	−1.44528	−0.722642	+	0.691223i	$0.757072\pi$
$30$	0	0
$31$	201.194	1.16566	0.582831	−	0.812594i	$-0.301945\pi$
$31$	201.194	1.16566	0.582831	+	0.812594i	$0.301945\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−179.068	−0.864802
$36$	0	0
$37$	−152.926	−0.679485	−0.339743	−	0.940518i	$-0.610340\pi$
$37$	−152.926	−0.679485	−0.339743	+	0.940518i	$0.610340\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−489.648	−1.86513	−0.932563	−	0.361008i	$-0.882433\pi$
$41$	−489.648	−1.86513	−0.932563	+	0.361008i	$0.882433\pi$
$42$	0	0
$43$	−7.58743	−0.0269087	−0.0134543	−	0.999909i	$-0.504283\pi$
$43$	−7.58743	−0.0269087	−0.0134543	+	0.999909i	$0.504283\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	373.391	1.15882	0.579412	−	0.815035i	$-0.303282\pi$
$47$	373.391	1.15882	0.579412	+	0.815035i	$0.303282\pi$
$48$	0	0
$49$	−145.500	−0.424197
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−43.6780	−0.113201	−0.0566003	−	0.998397i	$-0.518026\pi$
$53$	−43.6780	−0.113201	−0.0566003	+	0.998397i	$0.518026\pi$
$54$	0	0
$55$	−542.157	−1.32917
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−671.899	−1.48261	−0.741303	−	0.671171i	$-0.765792\pi$
$59$	−671.899	−1.48261	−0.741303	+	0.671171i	$0.765792\pi$
$60$	0	0
$61$	74.0354	0.155398	0.0776988	−	0.996977i	$-0.475243\pi$
$61$	74.0354	0.155398	0.0776988	+	0.996977i	$0.475243\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−923.320	−1.76190
$66$	0	0
$67$	420.871	0.767427	0.383713	−	0.923452i	$-0.374645\pi$
$67$	420.871	0.767427	0.383713	+	0.923452i	$0.374645\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−730.840	−1.22162	−0.610808	−	0.791778i	$-0.709155\pi$
$71$	−730.840	−1.22162	−0.610808	+	0.791778i	$0.709155\pi$
$72$	0	0
$73$	473.927	0.759849	0.379925	−	0.925017i	$-0.375950\pi$
$73$	473.927	0.759849	0.379925	+	0.925017i	$0.375950\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	597.963	0.884990
$78$	0	0
$79$	−529.622	−0.754267	−0.377134	−	0.926159i	$-0.623090\pi$
$79$	−529.622	−0.754267	−0.377134	+	0.926159i	$0.623090\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	26.1534	0.0345868	0.0172934	−	0.999850i	$-0.494495\pi$
$83$	26.1534	0.0345868	0.0172934	+	0.999850i	$0.494495\pi$
$84$	0	0
$85$	−759.563	−0.969249
$86$	0	0
$87$	0	0
$88$	0	0
$89$	415.949	0.495399	0.247700	−	0.968837i	$-0.420325\pi$
$89$	415.949	0.495399	0.247700	+	0.968837i	$0.420325\pi$
$90$	0	0
$91$	1018.36	1.17311
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1345.17	1.45275
$96$	0	0
$97$	927.485	0.970844	0.485422	−	0.874280i	$-0.338666\pi$
$97$	927.485	0.970844	0.485422	+	0.874280i	$0.338666\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−90.1180	−0.0887829	−0.0443915	−	0.999014i	$-0.514135\pi$
$101$	−90.1180	−0.0887829	−0.0443915	+	0.999014i	$0.514135\pi$
$102$	0	0
$103$	−325.652	−0.311529	−0.155765	−	0.987794i	$-0.549784\pi$
$103$	−325.652	−0.311529	−0.155765	+	0.987794i	$0.549784\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1073.49	0.969888	0.484944	−	0.874545i	$-0.338840\pi$
$107$	1073.49	0.969888	0.484944	+	0.874545i	$0.338840\pi$
$108$	0	0
$109$	601.488	0.528552	0.264276	−	0.964447i	$-0.414867\pi$
$109$	601.488	0.528552	0.264276	+	0.964447i	$0.414867\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−420.039	−0.349681	−0.174840	−	0.984597i	$-0.555941\pi$
$113$	−420.039	−0.349681	−0.174840	+	0.984597i	$0.555941\pi$
$114$	0	0
$115$	−2.86922	−0.00232658
$116$	0	0
$117$	0	0
$118$	0	0
$119$	837.748	0.645346
$120$	0	0
$121$	479.425	0.360199
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1116.75	−0.799080
$126$	0	0
$127$	−980.264	−0.684916	−0.342458	−	0.939533i	$-0.611259\pi$
$127$	−980.264	−0.684916	−0.342458	+	0.939533i	$0.611259\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1355.39	0.903976	0.451988	−	0.892024i	$-0.350715\pi$
$131$	1355.39	0.903976	0.451988	+	0.892024i	$0.350715\pi$
$132$	0	0
$133$	−1483.63	−0.967271
$134$	0	0
$135$	0	0
$136$	0	0
$137$	906.522	0.565324	0.282662	−	0.959220i	$-0.408783\pi$
$137$	906.522	0.565324	0.282662	+	0.959220i	$0.408783\pi$
$138$	0	0
$139$	−818.418	−0.499405	−0.249703	−	0.968323i	$-0.580333\pi$
$139$	−818.418	−0.499405	−0.249703	+	0.968323i	$0.580333\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3083.25	1.80303
$144$	0	0
$145$	−2875.97	−1.64715
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1159.50	−0.637514	−0.318757	−	0.947836i	$-0.603265\pi$
$149$	−1159.50	−0.637514	−0.318757	+	0.947836i	$0.603265\pi$
$150$	0	0
$151$	−637.973	−0.343825	−0.171912	−	0.985112i	$-0.554995\pi$
$151$	−637.973	−0.343825	−0.171912	+	0.985112i	$0.554995\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2563.59	1.32847
$156$	0	0
$157$	3291.72	1.67330	0.836649	−	0.547739i	$-0.184511\pi$
$157$	3291.72	1.67330	0.836649	+	0.547739i	$0.184511\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.16456	0.00154908
$162$	0	0
$163$	−1197.14	−0.575258	−0.287629	−	0.957742i	$-0.592867\pi$
$163$	−1197.14	−0.575258	−0.287629	+	0.957742i	$0.592867\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	838.838	0.388690	0.194345	−	0.980933i	$-0.437742\pi$
$167$	838.838	0.388690	0.194345	+	0.980933i	$0.437742\pi$
$168$	0	0
$169$	3053.92	1.39004
$170$	0	0
$171$	0	0
$172$	0	0
$173$	2307.18	1.01394	0.506970	−	0.861964i	$-0.330765\pi$
$173$	2307.18	1.01394	0.506970	+	0.861964i	$0.330765\pi$
$174$	0	0
$175$	−524.986	−0.226773
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3114.47	1.30048	0.650241	−	0.759728i	$-0.274668\pi$
$179$	3114.47	1.30048	0.650241	+	0.759728i	$0.274668\pi$
$180$	0	0
$181$	3902.75	1.60270	0.801350	−	0.598195i	$-0.204115\pi$
$181$	3902.75	1.60270	0.801350	+	0.598195i	$0.204115\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1948.58	−0.774390
$186$	0	0
$187$	2536.41	0.991875
$188$	0	0
$189$	0	0
$190$	0	0
$191$	105.739	0.0400578	0.0200289	−	0.999799i	$-0.493624\pi$
$191$	105.739	0.0400578	0.0200289	+	0.999799i	$0.493624\pi$
$192$	0	0
$193$	−1585.37	−0.591282	−0.295641	−	0.955299i	$-0.595533\pi$
$193$	−1585.37	−0.591282	−0.295641	+	0.955299i	$0.595533\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3905.37	−1.41242	−0.706208	−	0.708005i	$-0.749595\pi$
$197$	−3905.37	−1.41242	−0.706208	+	0.708005i	$0.749595\pi$
$198$	0	0
$199$	1538.77	0.548143	0.274072	−	0.961709i	$-0.411629\pi$
$199$	1538.77	0.548143	0.274072	+	0.961709i	$0.411629\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3172.01	1.09671
$204$	0	0
$205$	−6239.05	−2.12563
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4491.92	−1.48666
$210$	0	0
$211$	941.467	0.307172	0.153586	−	0.988135i	$-0.450918\pi$
$211$	941.467	0.307172	0.153586	+	0.988135i	$0.450918\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−96.6784	−0.0306670
$216$	0	0
$217$	−2827.48	−0.884523
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4319.63	1.31480
$222$	0	0
$223$	3321.57	0.997438	0.498719	−	0.866764i	$-0.333804\pi$
$223$	3321.57	0.997438	0.498719	+	0.866764i	$0.333804\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4211.84	−1.23150	−0.615749	−	0.787943i	$-0.711146\pi$
$227$	−4211.84	−1.23150	−0.615749	+	0.787943i	$0.711146\pi$
$228$	0	0
$229$	−354.145	−0.102195	−0.0510973	−	0.998694i	$-0.516272\pi$
$229$	−354.145	−0.102195	−0.0510973	+	0.998694i	$0.516272\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	868.789	0.244276	0.122138	−	0.992513i	$-0.461025\pi$
$233$	868.789	0.244276	0.122138	+	0.992513i	$0.461025\pi$
$234$	0	0
$235$	4757.72	1.32068
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1204.35	−0.325955	−0.162977	−	0.986630i	$-0.552110\pi$
$239$	−1204.35	−0.325955	−0.162977	+	0.986630i	$0.552110\pi$
$240$	0	0
$241$	−5087.40	−1.35979	−0.679893	−	0.733311i	$-0.737974\pi$
$241$	−5087.40	−1.35979	−0.679893	+	0.733311i	$0.737974\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1853.94	−0.483445
$246$	0	0
$247$	−7649.96	−1.97067
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−5463.91	−1.37402	−0.687010	−	0.726648i	$-0.741077\pi$
$251$	−5463.91	−1.37402	−0.687010	+	0.726648i	$0.741077\pi$
$252$	0	0
$253$	9.58119	0.00238089
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7402.12	−1.79662	−0.898311	−	0.439361i	$-0.855205\pi$
$257$	−7402.12	−1.79662	−0.898311	+	0.439361i	$0.855205\pi$
$258$	0	0
$259$	2149.15	0.515605
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−200.031	−0.0468989	−0.0234495	−	0.999725i	$-0.507465\pi$
$263$	−200.031	−0.0468989	−0.0234495	+	0.999725i	$0.507465\pi$
$264$	0	0
$265$	−556.541	−0.129011
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5493.50	−1.24515	−0.622574	−	0.782561i	$-0.713913\pi$
$269$	−5493.50	−1.24515	−0.622574	+	0.782561i	$0.713913\pi$
$270$	0	0
$271$	−1861.89	−0.417350	−0.208675	−	0.977985i	$-0.566915\pi$
$271$	−1861.89	−0.417350	−0.208675	+	0.977985i	$0.566915\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1589.48	−0.348542
$276$	0	0
$277$	5639.03	1.22316	0.611582	−	0.791181i	$-0.290533\pi$
$277$	5639.03	1.22316	0.611582	+	0.791181i	$0.290533\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2558.04	0.543059	0.271530	−	0.962430i	$-0.412470\pi$
$281$	2558.04	0.543059	0.271530	+	0.962430i	$0.412470\pi$
$282$	0	0
$283$	7467.95	1.56863	0.784317	−	0.620360i	$-0.213014\pi$
$283$	7467.95	1.56863	0.784317	+	0.620360i	$0.213014\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6881.26	1.41529
$288$	0	0
$289$	−1359.48	−0.276712
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3561.32	−0.710083	−0.355041	−	0.934851i	$-0.615533\pi$
$293$	−3561.32	−0.710083	−0.355041	+	0.934851i	$0.615533\pi$
$294$	0	0
$295$	−8561.27	−1.68968
$296$	0	0
$297$	0	0
$298$	0	0
$299$	16.3173	0.00315602
$300$	0	0
$301$	106.630	0.0204188
$302$	0	0
$303$	0	0
$304$	0	0
$305$	943.352	0.177102
$306$	0	0
$307$	−6101.93	−1.13438	−0.567192	−	0.823586i	$-0.691970\pi$
$307$	−6101.93	−1.13438	−0.567192	+	0.823586i	$0.691970\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	371.235	0.0676875	0.0338438	−	0.999427i	$-0.489225\pi$
$311$	371.235	0.0676875	0.0338438	+	0.999427i	$0.489225\pi$
$312$	0	0
$313$	9321.10	1.68326	0.841629	−	0.540056i	$-0.181597\pi$
$313$	9321.10	1.68326	0.841629	+	0.540056i	$0.181597\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6383.57	−1.13103	−0.565516	−	0.824737i	$-0.691323\pi$
$317$	−6383.57	−1.13103	−0.565516	+	0.824737i	$0.691323\pi$
$318$	0	0
$319$	9603.74	1.68560
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6293.19	−1.08409
$324$	0	0
$325$	−2706.96	−0.462015
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5247.45	−0.879335
$330$	0	0
$331$	−2529.42	−0.420028	−0.210014	−	0.977698i	$-0.567351\pi$
$331$	−2529.42	−0.420028	−0.210014	+	0.977698i	$0.567351\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5362.70	0.874614
$336$	0	0
$337$	3599.81	0.581882	0.290941	−	0.956741i	$-0.406032\pi$
$337$	3599.81	0.581882	0.290941	+	0.956741i	$0.406032\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−8560.62	−1.35948
$342$	0	0
$343$	6865.12	1.08070
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11373.2	−1.75950	−0.879748	−	0.475440i	$-0.842289\pi$
$347$	−11373.2	−1.75950	−0.879748	+	0.475440i	$0.842289\pi$
$348$	0	0
$349$	−1788.95	−0.274385	−0.137193	−	0.990544i	$-0.543808\pi$
$349$	−1788.95	−0.274385	−0.137193	+	0.990544i	$0.543808\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1110.37	−0.167419	−0.0837096	−	0.996490i	$-0.526677\pi$
$353$	−1110.37	−0.167419	−0.0837096	+	0.996490i	$0.526677\pi$
$354$	0	0
$355$	−9312.30	−1.39224
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2250.71	0.330886	0.165443	−	0.986219i	$-0.447095\pi$
$359$	2250.71	0.330886	0.165443	+	0.986219i	$0.447095\pi$
$360$	0	0
$361$	4286.07	0.624883
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6038.74	0.865978
$366$	0	0
$367$	2507.38	0.356632	0.178316	−	0.983973i	$-0.442935\pi$
$367$	2507.38	0.356632	0.178316	+	0.983973i	$0.442935\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	613.828	0.0858985
$372$	0	0
$373$	−4965.41	−0.689274	−0.344637	−	0.938736i	$-0.611998\pi$
$373$	−4965.41	−0.689274	−0.344637	+	0.938736i	$0.611998\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16355.6	2.23437
$378$	0	0
$379$	−13541.7	−1.83534	−0.917668	−	0.397349i	$-0.869930\pi$
$379$	−13541.7	−1.83534	−0.917668	+	0.397349i	$0.869930\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	8350.08	1.11402	0.557009	−	0.830506i	$-0.311949\pi$
$383$	8350.08	1.11402	0.557009	+	0.830506i	$0.311949\pi$
$384$	0	0
$385$	7619.19	1.00860
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2324.62	−0.302989	−0.151495	−	0.988458i	$-0.548409\pi$
$389$	−2324.62	−0.302989	−0.151495	+	0.988458i	$0.548409\pi$
$390$	0	0
$391$	13.4233	0.00173617
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6748.39	−0.859617
$396$	0	0
$397$	13253.5	1.67550	0.837749	−	0.546056i	$-0.183871\pi$
$397$	13253.5	1.67550	0.837749	+	0.546056i	$0.183871\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15580.6	1.94029	0.970144	−	0.242528i	$-0.0779767\pi$
$401$	15580.6	1.94029	0.970144	+	0.242528i	$0.0779767\pi$
$402$	0	0
$403$	−14579.2	−1.80208
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6506.88	0.792467
$408$	0	0
$409$	−12239.2	−1.47968	−0.739840	−	0.672783i	$-0.765099\pi$
$409$	−12239.2	−1.47968	−0.739840	+	0.672783i	$0.765099\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9442.52	1.12503
$414$	0	0
$415$	333.244	0.0394176
$416$	0	0
$417$	0	0
$418$	0	0
$419$	6393.00	0.745390	0.372695	−	0.927954i	$-0.378434\pi$
$419$	6393.00	0.745390	0.372695	+	0.927954i	$0.378434\pi$
$420$	0	0
$421$	−15832.6	−1.83286	−0.916431	−	0.400193i	$-0.868943\pi$
$421$	−15832.6	−1.83286	−0.916431	+	0.400193i	$0.868943\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2226.86	−0.254161
$426$	0	0
$427$	−1040.45	−0.117918
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−9339.01	−1.04372	−0.521861	−	0.853030i	$-0.674762\pi$
$431$	−9339.01	−1.04372	−0.521861	+	0.853030i	$0.674762\pi$
$432$	0	0
$433$	−3379.19	−0.375043	−0.187522	−	0.982260i	$-0.560045\pi$
$433$	−3379.19	−0.375043	−0.187522	+	0.982260i	$0.560045\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−23.7723	−0.00260225
$438$	0	0
$439$	14547.0	1.58153	0.790766	−	0.612119i	$-0.209683\pi$
$439$	14547.0	1.58153	0.790766	+	0.612119i	$0.209683\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5902.00	−0.632985	−0.316492	−	0.948595i	$-0.602505\pi$
$443$	−5902.00	−0.632985	−0.316492	+	0.948595i	$0.602505\pi$
$444$	0	0
$445$	5299.99	0.564592
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2448.48	0.257352	0.128676	−	0.991687i	$-0.458927\pi$
$449$	2448.48	0.257352	0.128676	+	0.991687i	$0.458927\pi$
$450$	0	0
$451$	20834.1	2.17525
$452$	0	0
$453$	0	0
$454$	0	0
$455$	12975.9	1.33696
$456$	0	0
$457$	4473.74	0.457928	0.228964	−	0.973435i	$-0.426466\pi$
$457$	4473.74	0.457928	0.228964	+	0.973435i	$0.426466\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	502.213	0.0507384	0.0253692	−	0.999678i	$-0.491924\pi$
$461$	502.213	0.0507384	0.0253692	+	0.999678i	$0.491924\pi$
$462$	0	0
$463$	7086.74	0.711337	0.355668	−	0.934612i	$-0.384253\pi$
$463$	7086.74	0.711337	0.355668	+	0.934612i	$0.384253\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8057.18	0.798376	0.399188	−	0.916869i	$-0.369292\pi$
$467$	8057.18	0.798376	0.399188	+	0.916869i	$0.369292\pi$
$468$	0	0
$469$	−5914.71	−0.582336
$470$	0	0
$471$	0	0
$472$	0	0
$473$	322.838	0.0313829
$474$	0	0
$475$	3943.71	0.380947
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17160.9	1.63695	0.818477	−	0.574539i	$-0.194819\pi$
$479$	17160.9	1.63695	0.818477	+	0.574539i	$0.194819\pi$
$480$	0	0
$481$	11081.5	1.05047
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11817.9	1.10644
$486$	0	0
$487$	−9909.84	−0.922090	−0.461045	−	0.887377i	$-0.652525\pi$
$487$	−9909.84	−0.922090	−0.461045	+	0.887377i	$0.652525\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−19488.5	−1.79125	−0.895627	−	0.444806i	$-0.853273\pi$
$491$	−19488.5	−1.79125	−0.895627	+	0.444806i	$0.853273\pi$
$492$	0	0
$493$	13454.9	1.22916
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10270.9	0.926984
$498$	0	0
$499$	−925.456	−0.0830242	−0.0415121	−	0.999138i	$-0.513218\pi$
$499$	−925.456	−0.0830242	−0.0415121	+	0.999138i	$0.513218\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8723.45	0.773279	0.386640	−	0.922231i	$-0.373636\pi$
$503$	8723.45	0.773279	0.386640	+	0.922231i	$0.373636\pi$
$504$	0	0
$505$	−1148.28	−0.101183
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3048.70	0.265484	0.132742	−	0.991151i	$-0.457622\pi$
$509$	3048.70	0.265484	0.132742	+	0.991151i	$0.457622\pi$
$510$	0	0
$511$	−6660.33	−0.576587
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4149.43	−0.355041
$516$	0	0
$517$	−15887.5	−1.35151
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−15593.0	−1.31121	−0.655607	−	0.755103i	$-0.727587\pi$
$521$	−15593.0	−1.31121	−0.655607	+	0.755103i	$0.727587\pi$
$522$	0	0
$523$	−9052.67	−0.756875	−0.378438	−	0.925627i	$-0.623539\pi$
$523$	−9052.67	−0.756875	−0.378438	+	0.925627i	$0.623539\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11993.4	−0.991352
$528$	0	0
$529$	−12166.9	−0.999996
$530$	0	0
$531$	0	0
$532$	0	0
$533$	35481.5	2.88344
$534$	0	0
$535$	13678.3	1.10535
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6190.87	0.494730
$540$	0	0
$541$	−11742.8	−0.933200	−0.466600	−	0.884469i	$-0.654521\pi$
$541$	−11742.8	−0.933200	−0.466600	+	0.884469i	$0.654521\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	7664.11	0.602375
$546$	0	0
$547$	3946.91	0.308515	0.154257	−	0.988031i	$-0.450701\pi$
$547$	3946.91	0.308515	0.154257	+	0.988031i	$0.450701\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−23828.2	−1.84232
$552$	0	0
$553$	7443.03	0.572351
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3129.69	0.238078	0.119039	−	0.992890i	$-0.462019\pi$
$557$	3129.69	0.238078	0.119039	+	0.992890i	$0.462019\pi$
$558$	0	0
$559$	549.810	0.0416001
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−192.036	−0.0143754	−0.00718769	−	0.999974i	$-0.502288\pi$
$563$	−192.036	−0.0143754	−0.00718769	+	0.999974i	$0.502288\pi$
$564$	0	0
$565$	−5352.09	−0.398521
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3261.01	0.240261	0.120131	−	0.992758i	$-0.461669\pi$
$569$	3261.01	0.240261	0.120131	+	0.992758i	$0.461669\pi$
$570$	0	0
$571$	−9414.44	−0.689986	−0.344993	−	0.938605i	$-0.612119\pi$
$571$	−9414.44	−0.689986	−0.344993	+	0.938605i	$0.612119\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.41187	−0.000610086 0
$576$	0	0
$577$	−9739.86	−0.702731	−0.351366	−	0.936238i	$-0.614283\pi$
$577$	−9739.86	−0.702731	−0.351366	+	0.936238i	$0.614283\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−367.546	−0.0262451
$582$	0	0
$583$	1858.46	0.132023
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25405.7	−1.78638	−0.893190	−	0.449680i	$-0.851538\pi$
$587$	−25405.7	−1.78638	−0.893190	+	0.449680i	$0.851538\pi$
$588$	0	0
$589$	21240.1	1.48588
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2751.26	0.190524	0.0952620	−	0.995452i	$-0.469631\pi$
$593$	2751.26	0.190524	0.0952620	+	0.995452i	$0.469631\pi$
$594$	0	0
$595$	10674.5	0.735482
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3316.22	0.226206	0.113103	−	0.993583i	$-0.463921\pi$
$599$	3316.22	0.226206	0.113103	+	0.993583i	$0.463921\pi$
$600$	0	0
$601$	−7919.69	−0.537522	−0.268761	−	0.963207i	$-0.586614\pi$
$601$	−7919.69	−0.537522	−0.268761	+	0.963207i	$0.586614\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6108.79	0.410509
$606$	0	0
$607$	−9474.47	−0.633537	−0.316768	−	0.948503i	$-0.602598\pi$
$607$	−9474.47	−0.633537	−0.316768	+	0.948503i	$0.602598\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−27057.2	−1.79151
$612$	0	0
$613$	−3165.90	−0.208596	−0.104298	−	0.994546i	$-0.533260\pi$
$613$	−3165.90	−0.208596	−0.104298	+	0.994546i	$0.533260\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−630.345	−0.0411292	−0.0205646	−	0.999789i	$-0.506546\pi$
$617$	−630.345	−0.0411292	−0.0205646	+	0.999789i	$0.506546\pi$
$618$	0	0
$619$	1172.54	0.0761365	0.0380683	−	0.999275i	$-0.487880\pi$
$619$	1172.54	0.0761365	0.0380683	+	0.999275i	$0.487880\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5845.54	−0.375918
$624$	0	0
$625$	−18899.0	−1.20954
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9116.16	0.577878
$630$	0	0
$631$	17925.9	1.13093	0.565465	−	0.824772i	$-0.308697\pi$
$631$	17925.9	1.13093	0.565465	+	0.824772i	$0.308697\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12490.4	−0.780579
$636$	0	0
$637$	10543.4	0.655798
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6696.85	−0.412652	−0.206326	−	0.978483i	$-0.566151\pi$
$641$	−6696.85	−0.412652	−0.206326	+	0.978483i	$0.566151\pi$
$642$	0	0
$643$	−29691.0	−1.82100	−0.910498	−	0.413513i	$-0.864301\pi$
$643$	−29691.0	−1.82100	−0.910498	+	0.413513i	$0.864301\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12607.6	−0.766084	−0.383042	−	0.923731i	$-0.625124\pi$
$647$	−12607.6	−0.766084	−0.383042	+	0.923731i	$0.625124\pi$
$648$	0	0
$649$	28588.7	1.72913
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−939.460	−0.0563000	−0.0281500	−	0.999604i	$-0.508962\pi$
$653$	−939.460	−0.0563000	−0.0281500	+	0.999604i	$0.508962\pi$
$654$	0	0
$655$	17270.2	1.03024
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13883.2	−0.820656	−0.410328	−	0.911938i	$-0.634586\pi$
$659$	−13883.2	−0.820656	−0.410328	+	0.911938i	$0.634586\pi$
$660$	0	0
$661$	8144.59	0.479256	0.239628	−	0.970865i	$-0.422975\pi$
$661$	8144.59	0.479256	0.239628	+	0.970865i	$0.422975\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−18904.3	−1.10237
$666$	0	0
$667$	50.8252	0.00295047
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3150.14	−0.181236
$672$	0	0
$673$	−28922.4	−1.65658	−0.828290	−	0.560300i	$-0.810686\pi$
$673$	−28922.4	−1.65658	−0.828290	+	0.560300i	$0.810686\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27180.8	−1.54305	−0.771524	−	0.636200i	$-0.780505\pi$
$677$	−27180.8	−1.54305	−0.771524	+	0.636200i	$0.780505\pi$
$678$	0	0
$679$	−13034.4	−0.736693
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7985.87	−0.447395	−0.223697	−	0.974659i	$-0.571813\pi$
$683$	−7985.87	−0.447395	−0.223697	+	0.974659i	$0.571813\pi$
$684$	0	0
$685$	11550.8	0.644283
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3165.05	0.175005
$690$	0	0
$691$	9250.05	0.509246	0.254623	−	0.967040i	$-0.418049\pi$
$691$	9250.05	0.509246	0.254623	+	0.967040i	$0.418049\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10428.2	−0.569158
$696$	0	0
$697$	29188.6	1.58622
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−10267.4	−0.553201	−0.276601	−	0.960985i	$-0.589208\pi$
$701$	−10267.4	−0.553201	−0.276601	+	0.960985i	$0.589208\pi$
$702$	0	0
$703$	−16144.5	−0.866146
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1266.47	0.0673700
$708$	0	0
$709$	13958.7	0.739392	0.369696	−	0.929153i	$-0.379462\pi$
$709$	13958.7	0.739392	0.369696	+	0.929153i	$0.379462\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−45.3048	−0.00237963
$714$	0	0
$715$	39286.4	2.05487
$716$	0	0
$717$	0	0
$718$	0	0
$719$	26373.7	1.36798	0.683988	−	0.729493i	$-0.260244\pi$
$719$	26373.7	1.36798	0.683988	+	0.729493i	$0.260244\pi$
$720$	0	0
$721$	4576.55	0.236394
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−8431.67	−0.431923
$726$	0	0
$727$	−8793.84	−0.448618	−0.224309	−	0.974518i	$-0.572013\pi$
$727$	−8793.84	−0.448618	−0.224309	+	0.974518i	$0.572013\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	452.297	0.0228848
$732$	0	0
$733$	−10674.0	−0.537862	−0.268931	−	0.963159i	$-0.586670\pi$
$733$	−10674.0	−0.537862	−0.268931	+	0.963159i	$0.586670\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−17907.7	−0.895031
$738$	0	0
$739$	−12678.1	−0.631084	−0.315542	−	0.948912i	$-0.602186\pi$
$739$	−12678.1	−0.631084	−0.315542	+	0.948912i	$0.602186\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	22111.3	1.09177	0.545884	−	0.837861i	$-0.316194\pi$
$743$	22111.3	1.09177	0.545884	+	0.837861i	$0.316194\pi$
$744$	0	0
$745$	−14774.2	−0.726556
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−15086.2	−0.735967
$750$	0	0
$751$	−22151.0	−1.07630	−0.538151	−	0.842848i	$-0.680877\pi$
$751$	−22151.0	−1.07630	−0.538151	+	0.842848i	$0.680877\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8129.00	−0.391847
$756$	0	0
$757$	−25282.2	−1.21387	−0.606933	−	0.794753i	$-0.707601\pi$
$757$	−25282.2	−1.21387	−0.606933	+	0.794753i	$0.707601\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−10128.9	−0.482488	−0.241244	−	0.970464i	$-0.577555\pi$
$761$	−10128.9	−0.482488	−0.241244	+	0.970464i	$0.577555\pi$
$762$	0	0
$763$	−8453.01	−0.401074
$764$	0	0
$765$	0	0
$766$	0	0
$767$	48688.0	2.29207
$768$	0	0
$769$	−25234.5	−1.18333	−0.591663	−	0.806185i	$-0.701528\pi$
$769$	−25234.5	−1.18333	−0.591663	+	0.806185i	$0.701528\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	30384.6	1.41379	0.706895	−	0.707319i	$-0.250096\pi$
$773$	30384.6	1.41379	0.706895	+	0.707319i	$0.250096\pi$
$774$	0	0
$775$	7515.85	0.348358
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−51692.2	−2.37749
$780$	0	0
$781$	31096.6	1.42474
$782$	0	0
$783$	0	0
$784$	0	0
$785$	41942.8	1.90701
$786$	0	0
$787$	30144.7	1.36536	0.682682	−	0.730716i	$-0.260814\pi$
$787$	30144.7	1.36536	0.682682	+	0.730716i	$0.260814\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5903.01	0.265344
$792$	0	0
$793$	−5364.84	−0.240241
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−16128.0	−0.716793	−0.358397	−	0.933569i	$-0.616676\pi$
$797$	−16128.0	−0.716793	−0.358397	+	0.933569i	$0.616676\pi$
$798$	0	0
$799$	−22258.4	−0.985538
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20165.2	−0.886194
$804$	0	0
$805$	40.3225	0.00176544
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23446.4	−1.01895	−0.509475	−	0.860485i	$-0.670160\pi$
$809$	−23446.4	−1.01895	−0.509475	+	0.860485i	$0.670160\pi$
$810$	0	0
$811$	−37762.1	−1.63503	−0.817513	−	0.575910i	$-0.804648\pi$
$811$	−37762.1	−1.63503	−0.817513	+	0.575910i	$0.804648\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−15253.8	−0.655605
$816$	0	0
$817$	−801.007	−0.0343007
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−11884.7	−0.505213	−0.252606	−	0.967569i	$-0.581288\pi$
$821$	−11884.7	−0.505213	−0.252606	+	0.967569i	$0.581288\pi$
$822$	0	0
$823$	−9064.05	−0.383904	−0.191952	−	0.981404i	$-0.561482\pi$
$823$	−9064.05	−0.383904	−0.191952	+	0.981404i	$0.561482\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36451.4	−1.53270	−0.766348	−	0.642426i	$-0.777928\pi$
$827$	−36451.4	−1.53270	−0.766348	+	0.642426i	$0.777928\pi$
$828$	0	0
$829$	−15293.4	−0.640725	−0.320362	−	0.947295i	$-0.603805\pi$
$829$	−15293.4	−0.640725	−0.320362	+	0.947295i	$0.603805\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8673.43	0.360764
$834$	0	0
$835$	10688.4	0.442979
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−353.539	−0.0145477	−0.00727385	−	0.999974i	$-0.502315\pi$
$839$	−353.539	−0.0145477	−0.00727385	+	0.999974i	$0.502315\pi$
$840$	0	0
$841$	26555.8	1.08884
$842$	0	0
$843$	0	0
$844$	0	0
$845$	38912.8	1.58419
$846$	0	0
$847$	−6737.59	−0.273325
$848$	0	0
$849$	0	0
$850$	0	0
$851$	34.4359	0.00138713
$852$	0	0
$853$	25692.7	1.03130	0.515651	−	0.856799i	$-0.327550\pi$
$853$	25692.7	1.03130	0.515651	+	0.856799i	$0.327550\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13018.0	−0.518886	−0.259443	−	0.965758i	$-0.583539\pi$
$857$	−13018.0	−0.518886	−0.259443	+	0.965758i	$0.583539\pi$
$858$	0	0
$859$	−2132.08	−0.0846862	−0.0423431	−	0.999103i	$-0.513482\pi$
$859$	−2132.08	−0.0846862	−0.0423431	+	0.999103i	$0.513482\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	39300.5	1.55018	0.775090	−	0.631851i	$-0.217705\pi$
$863$	39300.5	1.55018	0.775090	+	0.631851i	$0.217705\pi$
$864$	0	0
$865$	29397.9	1.15556
$866$	0	0
$867$	0	0
$868$	0	0
$869$	22534.9	0.879684
$870$	0	0
$871$	−30497.7	−1.18642
$872$	0	0
$873$	0	0
$874$	0	0
$875$	15694.2	0.606356
$876$	0	0
$877$	−10608.5	−0.408463	−0.204232	−	0.978923i	$-0.565470\pi$
$877$	−10608.5	−0.408463	−0.204232	+	0.978923i	$0.565470\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41310.4	1.57978	0.789888	−	0.613251i	$-0.210139\pi$
$881$	41310.4	1.57978	0.789888	+	0.613251i	$0.210139\pi$
$882$	0	0
$883$	47180.5	1.79813	0.899066	−	0.437814i	$-0.144247\pi$
$883$	47180.5	1.79813	0.899066	+	0.437814i	$0.144247\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49777.7	1.88430	0.942148	−	0.335197i	$-0.108803\pi$
$887$	49777.7	1.88430	0.942148	+	0.335197i	$0.108803\pi$
$888$	0	0
$889$	13776.1	0.519726
$890$	0	0
$891$	0	0
$892$	0	0
$893$	39419.0	1.47716
$894$	0	0
$895$	39684.2	1.48212
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−45411.4	−1.68471
$900$	0	0
$901$	2603.70	0.0962730
$902$	0	0
$903$	0	0
$904$	0	0
$905$	49728.4	1.82655
$906$	0	0
$907$	1036.51	0.0379456	0.0189728	−	0.999820i	$-0.493960\pi$
$907$	1036.51	0.0379456	0.0189728	+	0.999820i	$0.493960\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24780.9	0.901239	0.450620	−	0.892716i	$-0.351203\pi$
$911$	24780.9	0.901239	0.450620	+	0.892716i	$0.351203\pi$
$912$	0	0
$913$	−1112.80	−0.0403378
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−19047.9	−0.685953
$918$	0	0
$919$	25210.4	0.904912	0.452456	−	0.891787i	$-0.350548\pi$
$919$	25210.4	0.904912	0.452456	+	0.891787i	$0.350548\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	52959.1	1.88859
$924$	0	0
$925$	−5712.76	−0.203064
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−27878.6	−0.984570	−0.492285	−	0.870434i	$-0.663838\pi$
$929$	−27878.6	−0.984570	−0.492285	+	0.870434i	$0.663838\pi$
$930$	0	0
$931$	−15360.4	−0.540727
$932$	0	0
$933$	0	0
$934$	0	0
$935$	32318.7	1.13041
$936$	0	0
$937$	5091.76	0.177525	0.0887623	−	0.996053i	$-0.471709\pi$
$937$	5091.76	0.177525	0.0887623	+	0.996053i	$0.471709\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−43099.4	−1.49309	−0.746547	−	0.665333i	$-0.768290\pi$
$941$	−43099.4	−1.49309	−0.746547	+	0.665333i	$0.768290\pi$
$942$	0	0
$943$	110.259	0.00380755
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−24685.7	−0.847072	−0.423536	−	0.905879i	$-0.639211\pi$
$947$	−24685.7	−0.847072	−0.423536	+	0.905879i	$0.639211\pi$
$948$	0	0
$949$	−34342.3	−1.17471
$950$	0	0
$951$	0	0
$952$	0	0
$953$	9714.42	0.330200	0.165100	−	0.986277i	$-0.447205\pi$
$953$	9714.42	0.330200	0.165100	+	0.986277i	$0.447205\pi$
$954$	0	0
$955$	1347.32	0.0456527
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−12739.8	−0.428977
$960$	0	0
$961$	10688.0	0.358766
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−20200.6	−0.673867
$966$	0	0
$967$	−34368.2	−1.14292	−0.571461	−	0.820629i	$-0.693623\pi$
$967$	−34368.2	−1.14292	−0.571461	+	0.820629i	$0.693623\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31313.7	1.03492	0.517458	−	0.855708i	$-0.326878\pi$
$971$	31313.7	1.03492	0.517458	+	0.855708i	$0.326878\pi$
$972$	0	0
$973$	11501.6	0.378957
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25398.2	0.831688	0.415844	−	0.909436i	$-0.363486\pi$
$977$	25398.2	0.831688	0.415844	+	0.909436i	$0.363486\pi$
$978$	0	0
$979$	−17698.3	−0.577772
$980$	0	0
$981$	0	0
$982$	0	0
$983$	48894.2	1.58645	0.793227	−	0.608926i	$-0.208399\pi$
$983$	48894.2	1.58645	0.793227	+	0.608926i	$0.208399\pi$
$984$	0	0
$985$	−49761.8	−1.60969
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.70854	5.49325e−5 0
$990$	0	0
$991$	−46870.9	−1.50242	−0.751212	−	0.660061i	$-0.770530\pi$
$991$	−46870.9	−1.50242	−0.751212	+	0.660061i	$0.770530\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	19606.9	0.624703
$996$	0	0
$997$	−31408.2	−0.997702	−0.498851	−	0.866688i	$-0.666245\pi$
$997$	−31408.2	−0.997702	−0.498851	+	0.866688i	$0.666245\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.4.a.c.1.3		3
3.2	odd	2		324.4.a.d.1.1		3
4.3	odd	2		1296.4.a.v.1.3		3
9.2	odd	6		108.4.e.a.37.3		6
9.4	even	3		36.4.e.a.25.2	yes	6
9.5	odd	6		108.4.e.a.73.3		6
9.7	even	3		36.4.e.a.13.2	✓	6
12.11	even	2		1296.4.a.w.1.1		3
36.7	odd	6		144.4.i.d.49.2		6
36.11	even	6		432.4.i.d.145.3		6
36.23	even	6		432.4.i.d.289.3		6
36.31	odd	6		144.4.i.d.97.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.4.e.a.13.2	✓	6	9.7	even	3
36.4.e.a.25.2	yes	6	9.4	even	3
108.4.e.a.37.3		6	9.2	odd	6
108.4.e.a.73.3		6	9.5	odd	6
144.4.i.d.49.2		6	36.7	odd	6
144.4.i.d.97.2		6	36.31	odd	6
324.4.a.c.1.3		3	1.1	even	1	trivial
324.4.a.d.1.1		3	3.2	odd	2
432.4.i.d.145.3		6	36.11	even	6
432.4.i.d.289.3		6	36.23	even	6
1296.4.a.v.1.3		3	4.3	odd	2
1296.4.a.w.1.1		3	12.11	even	2

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	324.a (trivial)

\(f(q)\)	\(=\)	\(q+12.7419 q^{5} -14.0535 q^{7} -42.5491 q^{11} -72.4632 q^{13} -59.6114 q^{17} +105.570 q^{19} -0.225180 q^{23} +37.3563 q^{25} -225.710 q^{29} +201.194 q^{31} -179.068 q^{35} -152.926 q^{37} -489.648 q^{41} -7.58743 q^{43} +373.391 q^{47} -145.500 q^{49} -43.6780 q^{53} -542.157 q^{55} -671.899 q^{59} +74.0354 q^{61} -923.320 q^{65} +420.871 q^{67} -730.840 q^{71} +473.927 q^{73} +597.963 q^{77} -529.622 q^{79} +26.1534 q^{83} -759.563 q^{85} +415.949 q^{89} +1018.36 q^{91} +1345.17 q^{95} +927.485 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{5} + 6 q^{7} - 51 q^{11} - 12 q^{13} - 111 q^{17} + 15 q^{19} - 210 q^{23} + 3 q^{25} - 456 q^{29} - 48 q^{31} - 552 q^{35} - 48 q^{37} - 897 q^{41} - 129 q^{43} - 522 q^{47} + 225 q^{49} - 1104 q^{53}+ \cdots - 93 q^{97}+O(q^{100}) \) 3 * q - 6 * q^5 + 6 * q^7 - 51 * q^11 - 12 * q^13 - 111 * q^17 + 15 * q^19 - 210 * q^23 + 3 * q^25 - 456 * q^29 - 48 * q^31 - 552 * q^35 - 48 * q^37 - 897 * q^41 - 129 * q^43 - 522 * q^47 + 225 * q^49 - 1104 * q^53 - 108 * q^55 - 453 * q^59 + 402 * q^61 - 1110 * q^65 + 213 * q^67 + 60 * q^71 + 375 * q^73 - 1128 * q^77 - 552 * q^79 + 612 * q^83 - 1188 * q^85 - 462 * q^89 - 132 * q^91 + 2184 * q^95 - 93 * q^97

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