LMFDB - Embedded newform 648.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,4,Mod(1,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("648.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.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:	$38.2332376837$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 19x^{3} + 4x^{2} + 81x + 12 $ x^5 - x^4 - 19x^3 + 4x^2 + 81*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 72)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.150045$ of defining polynomial
Character	$\chi$	$=$	648.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-20.2222 q^{5} -24.7425 q^{7} +O(q^{10})$	$q-20.2222 q^{5} -24.7425 q^{7} -18.8052 q^{11} -48.4309 q^{13} -40.4983 q^{17} +7.82714 q^{19} -157.763 q^{23} +283.937 q^{25} -219.845 q^{29} -139.373 q^{31} +500.347 q^{35} +270.503 q^{37} -30.7644 q^{41} +57.2502 q^{43} +143.357 q^{47} +269.190 q^{49} -180.047 q^{53} +380.282 q^{55} +317.084 q^{59} +759.323 q^{61} +979.379 q^{65} +428.267 q^{67} -29.0901 q^{71} -327.554 q^{73} +465.287 q^{77} -1023.43 q^{79} -454.472 q^{83} +818.965 q^{85} -677.749 q^{89} +1198.30 q^{91} -158.282 q^{95} -397.800 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} + 3 q^{7}+O(q^{10}) $ 5 * q - 5 * q^5 + 3 * q^7	$ 5 q - 5 q^{5} + 3 q^{7} + 25 q^{11} + 29 q^{13} - 28 q^{17} + 64 q^{19} - 89 q^{23} + 322 q^{25} - 129 q^{29} + 241 q^{31} + 243 q^{35} + 366 q^{37} - 171 q^{41} + 803 q^{43} + 477 q^{47} + 1072 q^{49} - 374 q^{53} + 1469 q^{55} + 607 q^{59} + 1349 q^{61} - 527 q^{65} + 1549 q^{67} + 812 q^{71} + 1928 q^{73} - 903 q^{77} + 1727 q^{79} + 1025 q^{83} + 2902 q^{85} - 2310 q^{89} + 2985 q^{91} + 2012 q^{95} + 2875 q^{97}+O(q^{100}) $ 5 * q - 5 * q^5 + 3 * q^7 + 25 * q^11 + 29 * q^13 - 28 * q^17 + 64 * q^19 - 89 * q^23 + 322 * q^25 - 129 * q^29 + 241 * q^31 + 243 * q^35 + 366 * q^37 - 171 * q^41 + 803 * q^43 + 477 * q^47 + 1072 * q^49 - 374 * q^53 + 1469 * q^55 + 607 * q^59 + 1349 * q^61 - 527 * q^65 + 1549 * q^67 + 812 * q^71 + 1928 * q^73 - 903 * q^77 + 1727 * q^79 + 1025 * q^83 + 2902 * q^85 - 2310 * q^89 + 2985 * q^91 + 2012 * q^95 + 2875 * 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
$5$	−20.2222	−1.80873	−0.904363	−	0.426763i	$-0.859654\pi$
$5$	−20.2222	−1.80873	−0.904363	+	0.426763i	$0.859654\pi$
$6$	0	0
$7$	−24.7425	−1.33597	−0.667984	−	0.744176i	$-0.732843\pi$
$7$	−24.7425	−1.33597	−0.667984	+	0.744176i	$0.732843\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−18.8052	−0.515453	−0.257726	−	0.966218i	$-0.582973\pi$
$11$	−18.8052	−0.515453	−0.257726	+	0.966218i	$0.582973\pi$
$12$	0	0
$13$	−48.4309	−1.03326	−0.516628	−	0.856210i	$-0.672813\pi$
$13$	−48.4309	−1.03326	−0.516628	+	0.856210i	$0.672813\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−40.4983	−0.577782	−0.288891	−	0.957362i	$-0.593286\pi$
$17$	−40.4983	−0.577782	−0.288891	+	0.957362i	$0.593286\pi$
$18$	0	0
$19$	7.82714	0.0945089	0.0472545	−	0.998883i	$-0.484953\pi$
$19$	7.82714	0.0945089	0.0472545	+	0.998883i	$0.484953\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−157.763	−1.43026	−0.715128	−	0.698994i	$-0.753632\pi$
$23$	−157.763	−1.43026	−0.715128	+	0.698994i	$0.753632\pi$
$24$	0	0
$25$	283.937	2.27149
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−219.845	−1.40773	−0.703866	−	0.710333i	$-0.748544\pi$
$29$	−219.845	−1.40773	−0.703866	+	0.710333i	$0.748544\pi$
$30$	0	0
$31$	−139.373	−0.807491	−0.403745	−	0.914871i	$-0.632292\pi$
$31$	−139.373	−0.807491	−0.403745	+	0.914871i	$0.632292\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	500.347	2.41640
$36$	0	0
$37$	270.503	1.20190	0.600952	−	0.799285i	$-0.294788\pi$
$37$	270.503	1.20190	0.600952	+	0.799285i	$0.294788\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−30.7644	−0.117185	−0.0585926	−	0.998282i	$-0.518661\pi$
$41$	−30.7644	−0.117185	−0.0585926	+	0.998282i	$0.518661\pi$
$42$	0	0
$43$	57.2502	0.203037	0.101518	−	0.994834i	$-0.467630\pi$
$43$	57.2502	0.203037	0.101518	+	0.994834i	$0.467630\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	143.357	0.444910	0.222455	−	0.974943i	$-0.428593\pi$
$47$	143.357	0.444910	0.222455	+	0.974943i	$0.428593\pi$
$48$	0	0
$49$	269.190	0.784810
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−180.047	−0.466630	−0.233315	−	0.972401i	$-0.574957\pi$
$53$	−180.047	−0.466630	−0.233315	+	0.972401i	$0.574957\pi$
$54$	0	0
$55$	380.282	0.932313
$56$	0	0
$57$	0	0
$58$	0	0
$59$	317.084	0.699675	0.349838	−	0.936810i	$-0.386237\pi$
$59$	317.084	0.699675	0.349838	+	0.936810i	$0.386237\pi$
$60$	0	0
$61$	759.323	1.59379	0.796896	−	0.604117i	$-0.206474\pi$
$61$	759.323	1.59379	0.796896	+	0.604117i	$0.206474\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	979.379	1.86888
$66$	0	0
$67$	428.267	0.780913	0.390456	−	0.920621i	$-0.372317\pi$
$67$	428.267	0.780913	0.390456	+	0.920621i	$0.372317\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−29.0901	−0.0486247	−0.0243124	−	0.999704i	$-0.507740\pi$
$71$	−29.0901	−0.0486247	−0.0243124	+	0.999704i	$0.507740\pi$
$72$	0	0
$73$	−327.554	−0.525169	−0.262585	−	0.964909i	$-0.584575\pi$
$73$	−327.554	−0.525169	−0.262585	+	0.964909i	$0.584575\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	465.287	0.688628
$78$	0	0
$79$	−1023.43	−1.45753	−0.728764	−	0.684765i	$-0.759905\pi$
$79$	−1023.43	−1.45753	−0.728764	+	0.684765i	$0.759905\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−454.472	−0.601021	−0.300511	−	0.953778i	$-0.597157\pi$
$83$	−454.472	−0.601021	−0.300511	+	0.953778i	$0.597157\pi$
$84$	0	0
$85$	818.965	1.04505
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−677.749	−0.807205	−0.403602	−	0.914935i	$-0.632242\pi$
$89$	−677.749	−0.807205	−0.403602	+	0.914935i	$0.632242\pi$
$90$	0	0
$91$	1198.30	1.38040
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−158.282	−0.170941
$96$	0	0
$97$	−397.800	−0.416397	−0.208198	−	0.978087i	$-0.566760\pi$
$97$	−397.800	−0.416397	−0.208198	+	0.978087i	$0.566760\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1422.19	−1.40112	−0.700560	−	0.713594i	$-0.747066\pi$
$101$	−1422.19	−1.40112	−0.700560	+	0.713594i	$0.747066\pi$
$102$	0	0
$103$	1900.65	1.81822	0.909108	−	0.416562i	$-0.136765\pi$
$103$	1900.65	1.81822	0.909108	+	0.416562i	$0.136765\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−351.195	−0.317302	−0.158651	−	0.987335i	$-0.550714\pi$
$107$	−351.195	−0.317302	−0.158651	+	0.987335i	$0.550714\pi$
$108$	0	0
$109$	−816.359	−0.717367	−0.358683	−	0.933459i	$-0.616774\pi$
$109$	−816.359	−0.717367	−0.358683	+	0.933459i	$0.616774\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	69.1664	0.0575808	0.0287904	−	0.999585i	$-0.490834\pi$
$113$	69.1664	0.0575808	0.0287904	+	0.999585i	$0.490834\pi$
$114$	0	0
$115$	3190.31	2.58694
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1002.03	0.771898
$120$	0	0
$121$	−977.365	−0.734308
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3214.04	−2.29978
$126$	0	0
$127$	−427.879	−0.298962	−0.149481	−	0.988765i	$-0.547760\pi$
$127$	−427.879	−0.298962	−0.149481	+	0.988765i	$0.547760\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	45.4153	0.0302898	0.0151449	−	0.999885i	$-0.495179\pi$
$131$	45.4153	0.0302898	0.0151449	+	0.999885i	$0.495179\pi$
$132$	0	0
$133$	−193.663	−0.126261
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1499.52	−0.935127	−0.467563	−	0.883960i	$-0.654868\pi$
$137$	−1499.52	−0.935127	−0.467563	+	0.883960i	$0.654868\pi$
$138$	0	0
$139$	2620.62	1.59912	0.799562	−	0.600584i	$-0.205065\pi$
$139$	2620.62	1.59912	0.799562	+	0.600584i	$0.205065\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	910.754	0.532595
$144$	0	0
$145$	4445.75	2.54620
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2055.90	−1.13037	−0.565186	−	0.824963i	$-0.691196\pi$
$149$	−2055.90	−1.13037	−0.565186	+	0.824963i	$0.691196\pi$
$150$	0	0
$151$	−943.391	−0.508424	−0.254212	−	0.967148i	$-0.581816\pi$
$151$	−943.391	−0.508424	−0.254212	+	0.967148i	$0.581816\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2818.44	1.46053
$156$	0	0
$157$	1974.49	1.00370	0.501851	−	0.864954i	$-0.332652\pi$
$157$	1974.49	1.00370	0.501851	+	0.864954i	$0.332652\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3903.45	1.91078
$162$	0	0
$163$	1739.38	0.835820	0.417910	−	0.908489i	$-0.362763\pi$
$163$	1739.38	0.835820	0.417910	+	0.908489i	$0.362763\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3065.23	1.42033	0.710163	−	0.704037i	$-0.248621\pi$
$167$	3065.23	1.42033	0.710163	+	0.704037i	$0.248621\pi$
$168$	0	0
$169$	148.557	0.0676180
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−405.063	−0.178014	−0.0890068	−	0.996031i	$-0.528369\pi$
$173$	−405.063	−0.178014	−0.0890068	+	0.996031i	$0.528369\pi$
$174$	0	0
$175$	−7025.29	−3.03464
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3463.47	1.44621	0.723107	−	0.690736i	$-0.242714\pi$
$179$	3463.47	1.44621	0.723107	+	0.690736i	$0.242714\pi$
$180$	0	0
$181$	−4227.72	−1.73615	−0.868076	−	0.496431i	$-0.834644\pi$
$181$	−4227.72	−1.73615	−0.868076	+	0.496431i	$0.834644\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5470.16	−2.17392
$186$	0	0
$187$	761.579	0.297819
$188$	0	0
$189$	0	0
$190$	0	0
$191$	60.5902	0.0229537	0.0114768	−	0.999934i	$-0.496347\pi$
$191$	60.5902	0.0229537	0.0114768	+	0.999934i	$0.496347\pi$
$192$	0	0
$193$	−3002.50	−1.11982	−0.559909	−	0.828554i	$-0.689164\pi$
$193$	−3002.50	−1.11982	−0.559909	+	0.828554i	$0.689164\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2196.69	0.794456	0.397228	−	0.917720i	$-0.369972\pi$
$197$	2196.69	0.794456	0.397228	+	0.917720i	$0.369972\pi$
$198$	0	0
$199$	−4514.63	−1.60821	−0.804104	−	0.594488i	$-0.797355\pi$
$199$	−4514.63	−1.60821	−0.804104	+	0.594488i	$0.797355\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5439.51	1.88068
$204$	0	0
$205$	622.123	0.211956
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−147.191	−0.0487149
$210$	0	0
$211$	−2652.05	−0.865283	−0.432641	−	0.901566i	$-0.642418\pi$
$211$	−2652.05	−0.865283	−0.432641	+	0.901566i	$0.642418\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1157.72	−0.367238
$216$	0	0
$217$	3448.44	1.07878
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1961.37	0.596997
$222$	0	0
$223$	4501.25	1.35169	0.675843	−	0.737045i	$-0.263780\pi$
$223$	4501.25	1.35169	0.675843	+	0.737045i	$0.263780\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−483.739	−0.141440	−0.0707200	−	0.997496i	$-0.522530\pi$
$227$	−483.739	−0.141440	−0.0707200	+	0.997496i	$0.522530\pi$
$228$	0	0
$229$	768.462	0.221753	0.110876	−	0.993834i	$-0.464634\pi$
$229$	768.462	0.221753	0.110876	+	0.993834i	$0.464634\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−225.688	−0.0634563	−0.0317282	−	0.999497i	$-0.510101\pi$
$233$	−225.688	−0.0634563	−0.0317282	+	0.999497i	$0.510101\pi$
$234$	0	0
$235$	−2898.99	−0.804721
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4151.17	−1.12350	−0.561751	−	0.827307i	$-0.689872\pi$
$239$	−4151.17	−1.12350	−0.561751	+	0.827307i	$0.689872\pi$
$240$	0	0
$241$	−4576.05	−1.22311	−0.611555	−	0.791202i	$-0.709456\pi$
$241$	−4576.05	−1.22311	−0.611555	+	0.791202i	$0.709456\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5443.60	−1.41951
$246$	0	0
$247$	−379.076	−0.0976519
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−3740.17	−0.940548	−0.470274	−	0.882520i	$-0.655845\pi$
$251$	−3740.17	−0.940548	−0.470274	+	0.882520i	$0.655845\pi$
$252$	0	0
$253$	2966.77	0.737229
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5049.23	−1.22553	−0.612767	−	0.790264i	$-0.709943\pi$
$257$	−5049.23	−1.22553	−0.612767	+	0.790264i	$0.709943\pi$
$258$	0	0
$259$	−6692.91	−1.60570
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−241.506	−0.0566231	−0.0283115	−	0.999599i	$-0.509013\pi$
$263$	−241.506	−0.0566231	−0.0283115	+	0.999599i	$0.509013\pi$
$264$	0	0
$265$	3640.95	0.844006
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1976.08	−0.447894	−0.223947	−	0.974601i	$-0.571894\pi$
$269$	−1976.08	−0.447894	−0.223947	+	0.974601i	$0.571894\pi$
$270$	0	0
$271$	−1562.95	−0.350342	−0.175171	−	0.984538i	$-0.556048\pi$
$271$	−1562.95	−0.350342	−0.175171	+	0.984538i	$0.556048\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5339.48	−1.17085
$276$	0	0
$277$	−3043.54	−0.660175	−0.330087	−	0.943950i	$-0.607078\pi$
$277$	−3043.54	−0.660175	−0.330087	+	0.943950i	$0.607078\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5583.07	1.18526	0.592630	−	0.805475i	$-0.298090\pi$
$281$	5583.07	1.18526	0.592630	+	0.805475i	$0.298090\pi$
$282$	0	0
$283$	1779.53	0.373788	0.186894	−	0.982380i	$-0.440158\pi$
$283$	1779.53	0.373788	0.186894	+	0.982380i	$0.440158\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	761.188	0.156556
$288$	0	0
$289$	−3272.88	−0.666168
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9527.73	1.89971	0.949856	−	0.312687i	$-0.101229\pi$
$293$	9527.73	1.89971	0.949856	+	0.312687i	$0.101229\pi$
$294$	0	0
$295$	−6412.13	−1.26552
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7640.62	1.47782
$300$	0	0
$301$	−1416.51	−0.271250
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−15355.2	−2.88273
$306$	0	0
$307$	−549.492	−0.102154	−0.0510768	−	0.998695i	$-0.516265\pi$
$307$	−549.492	−0.102154	−0.0510768	+	0.998695i	$0.516265\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8603.41	1.56866	0.784332	−	0.620341i	$-0.213006\pi$
$311$	8603.41	1.56866	0.784332	+	0.620341i	$0.213006\pi$
$312$	0	0
$313$	3037.74	0.548573	0.274287	−	0.961648i	$-0.411558\pi$
$313$	3037.74	0.548573	0.274287	+	0.961648i	$0.411558\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−6673.25	−1.18236	−0.591178	−	0.806541i	$-0.701337\pi$
$317$	−6673.25	−1.18236	−0.591178	+	0.806541i	$0.701337\pi$
$318$	0	0
$319$	4134.23	0.725619
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−316.986	−0.0546055
$324$	0	0
$325$	−13751.3	−2.34703
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−3547.01	−0.594386
$330$	0	0
$331$	−6246.69	−1.03731	−0.518655	−	0.854984i	$-0.673567\pi$
$331$	−6246.69	−1.03731	−0.518655	+	0.854984i	$0.673567\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−8660.49	−1.41246
$336$	0	0
$337$	−10811.7	−1.74763	−0.873816	−	0.486258i	$-0.838362\pi$
$337$	−10811.7	−1.74763	−0.873816	+	0.486258i	$0.838362\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2620.95	0.416223
$342$	0	0
$343$	1826.25	0.287487
$344$	0	0
$345$	0	0
$346$	0	0
$347$	3711.90	0.574251	0.287126	−	0.957893i	$-0.407300\pi$
$347$	3711.90	0.574251	0.287126	+	0.957893i	$0.407300\pi$
$348$	0	0
$349$	1654.03	0.253691	0.126846	−	0.991922i	$-0.459515\pi$
$349$	1654.03	0.253691	0.126846	+	0.991922i	$0.459515\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4080.06	0.615184	0.307592	−	0.951518i	$-0.400477\pi$
$353$	4080.06	0.615184	0.307592	+	0.951518i	$0.400477\pi$
$354$	0	0
$355$	588.264	0.0879488
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2626.25	0.386096	0.193048	−	0.981189i	$-0.438163\pi$
$359$	2626.25	0.386096	0.193048	+	0.981189i	$0.438163\pi$
$360$	0	0
$361$	−6797.74	−0.991068
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6623.86	0.949887
$366$	0	0
$367$	−356.222	−0.0506667	−0.0253333	−	0.999679i	$-0.508065\pi$
$367$	−356.222	−0.0506667	−0.0253333	+	0.999679i	$0.508065\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4454.81	0.623402
$372$	0	0
$373$	−3860.84	−0.535943	−0.267972	−	0.963427i	$-0.586353\pi$
$373$	−3860.84	−0.535943	−0.267972	+	0.963427i	$0.586353\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10647.3	1.45455
$378$	0	0
$379$	1767.05	0.239492	0.119746	−	0.992805i	$-0.461792\pi$
$379$	1767.05	0.239492	0.119746	+	0.992805i	$0.461792\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−12356.6	−1.64855	−0.824275	−	0.566189i	$-0.808417\pi$
$383$	−12356.6	−1.64855	−0.824275	+	0.566189i	$0.808417\pi$
$384$	0	0
$385$	−9409.12	−1.24554
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14304.1	1.86438	0.932191	−	0.361968i	$-0.117895\pi$
$389$	14304.1	1.86438	0.932191	+	0.361968i	$0.117895\pi$
$390$	0	0
$391$	6389.14	0.826376
$392$	0	0
$393$	0	0
$394$	0	0
$395$	20696.0	2.63627
$396$	0	0
$397$	7003.81	0.885419	0.442709	−	0.896665i	$-0.354017\pi$
$397$	7003.81	0.885419	0.442709	+	0.896665i	$0.354017\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	653.469	0.0813782	0.0406891	−	0.999172i	$-0.487045\pi$
$401$	653.469	0.0813782	0.0406891	+	0.999172i	$0.487045\pi$
$402$	0	0
$403$	6749.99	0.834345
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5086.86	−0.619525
$408$	0	0
$409$	498.335	0.0602471	0.0301236	−	0.999546i	$-0.490410\pi$
$409$	498.335	0.0602471	0.0301236	+	0.999546i	$0.490410\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7845.45	−0.934744
$414$	0	0
$415$	9190.41	1.08708
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12477.8	1.45484	0.727421	−	0.686191i	$-0.240719\pi$
$419$	12477.8	1.45484	0.727421	+	0.686191i	$0.240719\pi$
$420$	0	0
$421$	4377.22	0.506728	0.253364	−	0.967371i	$-0.418463\pi$
$421$	4377.22	0.506728	0.253364	+	0.967371i	$0.418463\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−11499.0	−1.31243
$426$	0	0
$427$	−18787.5	−2.12925
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12308.9	−1.37564	−0.687820	−	0.725881i	$-0.741432\pi$
$431$	−12308.9	−1.37564	−0.687820	+	0.725881i	$0.741432\pi$
$432$	0	0
$433$	2479.42	0.275181	0.137591	−	0.990489i	$-0.456064\pi$
$433$	2479.42	0.275181	0.137591	+	0.990489i	$0.456064\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1234.83	−0.135172
$438$	0	0
$439$	5922.44	0.643878	0.321939	−	0.946760i	$-0.395665\pi$
$439$	5922.44	0.643878	0.321939	+	0.946760i	$0.395665\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4835.63	0.518618	0.259309	−	0.965794i	$-0.416505\pi$
$443$	4835.63	0.518618	0.259309	+	0.965794i	$0.416505\pi$
$444$	0	0
$445$	13705.6	1.46001
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−174.505	−0.0183417	−0.00917083	−	0.999958i	$-0.502919\pi$
$449$	−174.505	−0.0183417	−0.00917083	+	0.999958i	$0.502919\pi$
$450$	0	0
$451$	578.531	0.0604034
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−24232.3	−2.49676
$456$	0	0
$457$	11211.1	1.14755	0.573777	−	0.819011i	$-0.305478\pi$
$457$	11211.1	1.14755	0.573777	+	0.819011i	$0.305478\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	715.604	0.0722972	0.0361486	−	0.999346i	$-0.488491\pi$
$461$	715.604	0.0722972	0.0361486	+	0.999346i	$0.488491\pi$
$462$	0	0
$463$	−395.385	−0.0396870	−0.0198435	−	0.999803i	$-0.506317\pi$
$463$	−395.385	−0.0396870	−0.0198435	+	0.999803i	$0.506317\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−157.166	−0.0155734	−0.00778672	−	0.999970i	$-0.502479\pi$
$467$	−157.166	−0.0155734	−0.00778672	+	0.999970i	$0.502479\pi$
$468$	0	0
$469$	−10596.4	−1.04327
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1076.60	−0.104656
$474$	0	0
$475$	2222.41	0.214676
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1099.56	0.104885	0.0524425	−	0.998624i	$-0.483299\pi$
$479$	1099.56	0.104885	0.0524425	+	0.998624i	$0.483299\pi$
$480$	0	0
$481$	−13100.7	−1.24187
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8044.39	0.753148
$486$	0	0
$487$	15410.3	1.43390	0.716948	−	0.697127i	$-0.245539\pi$
$487$	15410.3	1.43390	0.716948	+	0.697127i	$0.245539\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−7328.91	−0.673624	−0.336812	−	0.941572i	$-0.609349\pi$
$491$	−7328.91	−0.673624	−0.336812	+	0.941572i	$0.609349\pi$
$492$	0	0
$493$	8903.37	0.813362
$494$	0	0
$495$	0	0
$496$	0	0
$497$	719.760	0.0649611
$498$	0	0
$499$	14808.4	1.32849	0.664243	−	0.747517i	$-0.268754\pi$
$499$	14808.4	1.32849	0.664243	+	0.747517i	$0.268754\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−675.499	−0.0598788	−0.0299394	−	0.999552i	$-0.509531\pi$
$503$	−675.499	−0.0598788	−0.0299394	+	0.999552i	$0.509531\pi$
$504$	0	0
$505$	28759.7	2.53424
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14037.5	−1.22240	−0.611201	−	0.791476i	$-0.709313\pi$
$509$	−14037.5	−1.22240	−0.611201	+	0.791476i	$0.709313\pi$
$510$	0	0
$511$	8104.51	0.701609
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−38435.2	−3.28865
$516$	0	0
$517$	−2695.86	−0.229330
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12189.0	1.02497	0.512487	−	0.858695i	$-0.328724\pi$
$521$	12189.0	1.02497	0.512487	+	0.858695i	$0.328724\pi$
$522$	0	0
$523$	−18385.2	−1.53715	−0.768575	−	0.639760i	$-0.779034\pi$
$523$	−18385.2	−1.53715	−0.768575	+	0.639760i	$0.779034\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5644.40	0.466554
$528$	0	0
$529$	12722.2	1.04563
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1489.95	0.121082
$534$	0	0
$535$	7101.93	0.573913
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5062.17	−0.404532
$540$	0	0
$541$	7621.88	0.605712	0.302856	−	0.953036i	$-0.402060\pi$
$541$	7621.88	0.605712	0.302856	+	0.953036i	$0.402060\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	16508.6	1.29752
$546$	0	0
$547$	−20212.6	−1.57994	−0.789969	−	0.613146i	$-0.789904\pi$
$547$	−20212.6	−1.57994	−0.789969	+	0.613146i	$0.789904\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1720.76	−0.133043
$552$	0	0
$553$	25322.2	1.94721
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8673.96	0.659834	0.329917	−	0.944010i	$-0.392979\pi$
$557$	8673.96	0.659834	0.329917	+	0.944010i	$0.392979\pi$
$558$	0	0
$559$	−2772.68	−0.209789
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3800.36	−0.284487	−0.142244	−	0.989832i	$-0.545432\pi$
$563$	−3800.36	−0.284487	−0.142244	+	0.989832i	$0.545432\pi$
$564$	0	0
$565$	−1398.70	−0.104148
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7024.57	0.517549	0.258774	−	0.965938i	$-0.416681\pi$
$569$	7024.57	0.517549	0.258774	+	0.965938i	$0.416681\pi$
$570$	0	0
$571$	−20813.7	−1.52544	−0.762720	−	0.646729i	$-0.776136\pi$
$571$	−20813.7	−1.52544	−0.762720	+	0.646729i	$0.776136\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−44794.7	−3.24881
$576$	0	0
$577$	−6708.36	−0.484008	−0.242004	−	0.970275i	$-0.577805\pi$
$577$	−6708.36	−0.484008	−0.242004	+	0.970275i	$0.577805\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	11244.8	0.802945
$582$	0	0
$583$	3385.82	0.240526
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16969.7	1.19321	0.596606	−	0.802534i	$-0.296516\pi$
$587$	16969.7	1.19321	0.596606	+	0.802534i	$0.296516\pi$
$588$	0	0
$589$	−1090.90	−0.0763151
$590$	0	0
$591$	0	0
$592$	0	0
$593$	10223.4	0.707968	0.353984	−	0.935251i	$-0.384827\pi$
$593$	10223.4	0.707968	0.353984	+	0.935251i	$0.384827\pi$
$594$	0	0
$595$	−20263.2	−1.39615
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6316.22	0.430841	0.215420	−	0.976521i	$-0.430888\pi$
$599$	6316.22	0.430841	0.215420	+	0.976521i	$0.430888\pi$
$600$	0	0
$601$	−13291.5	−0.902119	−0.451060	−	0.892494i	$-0.648954\pi$
$601$	−13291.5	−0.902119	−0.451060	+	0.892494i	$0.648954\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	19764.4	1.32816
$606$	0	0
$607$	7355.04	0.491815	0.245908	−	0.969293i	$-0.420914\pi$
$607$	7355.04	0.491815	0.245908	+	0.969293i	$0.420914\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6942.92	−0.459706
$612$	0	0
$613$	22969.6	1.51343	0.756716	−	0.653744i	$-0.226803\pi$
$613$	22969.6	1.51343	0.756716	+	0.653744i	$0.226803\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22387.9	1.46078	0.730391	−	0.683029i	$-0.239338\pi$
$617$	22387.9	1.46078	0.730391	+	0.683029i	$0.239338\pi$
$618$	0	0
$619$	27633.4	1.79431	0.897157	−	0.441712i	$-0.145629\pi$
$619$	27633.4	1.79431	0.897157	+	0.441712i	$0.145629\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	16769.2	1.07840
$624$	0	0
$625$	29502.9	1.88818
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−10954.9	−0.694438
$630$	0	0
$631$	−1076.09	−0.0678899	−0.0339449	−	0.999424i	$-0.510807\pi$
$631$	−1076.09	−0.0678899	−0.0339449	+	0.999424i	$0.510807\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	8652.64	0.540740
$636$	0	0
$637$	−13037.1	−0.810910
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7159.58	0.441164	0.220582	−	0.975368i	$-0.429204\pi$
$641$	7159.58	0.441164	0.220582	+	0.975368i	$0.429204\pi$
$642$	0	0
$643$	30459.5	1.86813	0.934064	−	0.357105i	$-0.116236\pi$
$643$	30459.5	1.86813	0.934064	+	0.357105i	$0.116236\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	24555.0	1.49205	0.746024	−	0.665919i	$-0.231960\pi$
$647$	24555.0	1.49205	0.746024	+	0.665919i	$0.231960\pi$
$648$	0	0
$649$	−5962.83	−0.360650
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−727.746	−0.0436124	−0.0218062	−	0.999762i	$-0.506942\pi$
$653$	−727.746	−0.0436124	−0.0218062	+	0.999762i	$0.506942\pi$
$654$	0	0
$655$	−918.397	−0.0547859
$656$	0	0
$657$	0	0
$658$	0	0
$659$	33042.7	1.95321	0.976603	−	0.215051i	$-0.0689918\pi$
$659$	33042.7	1.95321	0.976603	+	0.215051i	$0.0689918\pi$
$660$	0	0
$661$	9000.51	0.529621	0.264810	−	0.964300i	$-0.414691\pi$
$661$	9000.51	0.529621	0.264810	+	0.964300i	$0.414691\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3916.28	0.228371
$666$	0	0
$667$	34683.5	2.01342
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−14279.2	−0.821524
$672$	0	0
$673$	−20547.3	−1.17688	−0.588441	−	0.808540i	$-0.700258\pi$
$673$	−20547.3	−1.17688	−0.588441	+	0.808540i	$0.700258\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−29292.6	−1.66293	−0.831466	−	0.555576i	$-0.812498\pi$
$677$	−29292.6	−1.66293	−0.831466	+	0.555576i	$0.812498\pi$
$678$	0	0
$679$	9842.56	0.556293
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−13236.0	−0.741523	−0.370761	−	0.928728i	$-0.620903\pi$
$683$	−13236.0	−0.741523	−0.370761	+	0.928728i	$0.620903\pi$
$684$	0	0
$685$	30323.5	1.69139
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8719.85	0.482148
$690$	0	0
$691$	8325.22	0.458331	0.229165	−	0.973388i	$-0.426400\pi$
$691$	8325.22	0.458331	0.229165	+	0.973388i	$0.426400\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−52994.7	−2.89238
$696$	0	0
$697$	1245.91	0.0677075
$698$	0	0
$699$	0	0
$700$	0	0
$701$	19324.2	1.04118	0.520589	−	0.853807i	$-0.325712\pi$
$701$	19324.2	1.04118	0.520589	+	0.853807i	$0.325712\pi$
$702$	0	0
$703$	2117.27	0.113591
$704$	0	0
$705$	0	0
$706$	0	0
$707$	35188.4	1.87185
$708$	0	0
$709$	−12965.6	−0.686790	−0.343395	−	0.939191i	$-0.611577\pi$
$709$	−12965.6	−0.686790	−0.343395	+	0.939191i	$0.611577\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	21988.0	1.15492
$714$	0	0
$715$	−18417.4	−0.963318
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27536.2	−1.42827	−0.714137	−	0.700006i	$-0.753180\pi$
$719$	−27536.2	−1.42827	−0.714137	+	0.700006i	$0.753180\pi$
$720$	0	0
$721$	−47026.7	−2.42908
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−62422.1	−3.19765
$726$	0	0
$727$	9461.60	0.482684	0.241342	−	0.970440i	$-0.422412\pi$
$727$	9461.60	0.482684	0.241342	+	0.970440i	$0.422412\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2318.54	−0.117311
$732$	0	0
$733$	−6366.10	−0.320787	−0.160394	−	0.987053i	$-0.551276\pi$
$733$	−6366.10	−0.320787	−0.160394	+	0.987053i	$0.551276\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8053.65	−0.402524
$738$	0	0
$739$	6382.72	0.317716	0.158858	−	0.987301i	$-0.449219\pi$
$739$	6382.72	0.317716	0.158858	+	0.987301i	$0.449219\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2052.97	0.101368	0.0506838	−	0.998715i	$-0.483860\pi$
$743$	2052.97	0.101368	0.0506838	+	0.998715i	$0.483860\pi$
$744$	0	0
$745$	41574.7	2.04453
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8689.43	0.423905
$750$	0	0
$751$	−4996.54	−0.242778	−0.121389	−	0.992605i	$-0.538735\pi$
$751$	−4996.54	−0.242778	−0.121389	+	0.992605i	$0.538735\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	19077.4	0.919600
$756$	0	0
$757$	−1840.15	−0.0883505	−0.0441752	−	0.999024i	$-0.514066\pi$
$757$	−1840.15	−0.0883505	−0.0441752	+	0.999024i	$0.514066\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1895.85	−0.0903079	−0.0451540	−	0.998980i	$-0.514378\pi$
$761$	−1895.85	−0.0903079	−0.0451540	+	0.998980i	$0.514378\pi$
$762$	0	0
$763$	20198.7	0.958379
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−15356.7	−0.722944
$768$	0	0
$769$	−28583.8	−1.34039	−0.670193	−	0.742187i	$-0.733789\pi$
$769$	−28583.8	−1.34039	−0.670193	+	0.742187i	$0.733789\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10336.6	0.480960	0.240480	−	0.970654i	$-0.422695\pi$
$773$	10336.6	0.480960	0.240480	+	0.970654i	$0.422695\pi$
$774$	0	0
$775$	−39573.2	−1.83421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−240.797	−0.0110750
$780$	0	0
$781$	547.044	0.0250637
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−39928.5	−1.81542
$786$	0	0
$787$	−3257.97	−0.147566	−0.0737828	−	0.997274i	$-0.523507\pi$
$787$	−3257.97	−0.147566	−0.0737828	+	0.997274i	$0.523507\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1711.35	−0.0769261
$792$	0	0
$793$	−36774.7	−1.64680
$794$	0	0
$795$	0	0
$796$	0	0
$797$	12739.9	0.566211	0.283105	−	0.959089i	$-0.408635\pi$
$797$	12739.9	0.566211	0.283105	+	0.959089i	$0.408635\pi$
$798$	0	0
$799$	−5805.72	−0.257061
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6159.73	0.270700
$804$	0	0
$805$	−78936.2	−3.45607
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−31914.5	−1.38696	−0.693482	−	0.720473i	$-0.743925\pi$
$809$	−31914.5	−1.38696	−0.693482	+	0.720473i	$0.743925\pi$
$810$	0	0
$811$	37156.0	1.60878	0.804392	−	0.594099i	$-0.202491\pi$
$811$	37156.0	1.60878	0.804392	+	0.594099i	$0.202491\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−35174.0	−1.51177
$816$	0	0
$817$	448.105	0.0191888
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12946.9	0.550364	0.275182	−	0.961392i	$-0.411262\pi$
$821$	12946.9	0.550364	0.275182	+	0.961392i	$0.411262\pi$
$822$	0	0
$823$	−28733.9	−1.21701	−0.608506	−	0.793549i	$-0.708231\pi$
$823$	−28733.9	−1.21701	−0.608506	+	0.793549i	$0.708231\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34976.0	1.47066	0.735329	−	0.677710i	$-0.237028\pi$
$827$	34976.0	1.47066	0.735329	+	0.677710i	$0.237028\pi$
$828$	0	0
$829$	1001.73	0.0419680	0.0209840	−	0.999780i	$-0.493320\pi$
$829$	1001.73	0.0419680	0.0209840	+	0.999780i	$0.493320\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−10901.7	−0.453449
$834$	0	0
$835$	−61985.6	−2.56898
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−37120.6	−1.52747	−0.763733	−	0.645532i	$-0.776636\pi$
$839$	−37120.6	−1.52747	−0.763733	+	0.645532i	$0.776636\pi$
$840$	0	0
$841$	23942.9	0.981709
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3004.14	−0.122303
$846$	0	0
$847$	24182.4	0.981012
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−42675.4	−1.71903
$852$	0	0
$853$	−45401.6	−1.82242	−0.911208	−	0.411946i	$-0.864849\pi$
$853$	−45401.6	−1.82242	−0.911208	+	0.411946i	$0.864849\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6003.97	−0.239314	−0.119657	−	0.992815i	$-0.538179\pi$
$857$	−6003.97	−0.239314	−0.119657	+	0.992815i	$0.538179\pi$
$858$	0	0
$859$	45828.3	1.82031	0.910153	−	0.414272i	$-0.135964\pi$
$859$	45828.3	1.82031	0.910153	+	0.414272i	$0.135964\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19545.3	0.770949	0.385475	−	0.922718i	$-0.374038\pi$
$863$	19545.3	0.770949	0.385475	+	0.922718i	$0.374038\pi$
$864$	0	0
$865$	8191.25	0.321978
$866$	0	0
$867$	0	0
$868$	0	0
$869$	19245.8	0.751287
$870$	0	0
$871$	−20741.4	−0.806883
$872$	0	0
$873$	0	0
$874$	0	0
$875$	79523.4	3.07243
$876$	0	0
$877$	−8775.65	−0.337894	−0.168947	−	0.985625i	$-0.554037\pi$
$877$	−8775.65	−0.337894	−0.168947	+	0.985625i	$0.554037\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	23809.8	0.910527	0.455263	−	0.890357i	$-0.349545\pi$
$881$	23809.8	0.910527	0.455263	+	0.890357i	$0.349545\pi$
$882$	0	0
$883$	−30409.1	−1.15894	−0.579471	−	0.814993i	$-0.696741\pi$
$883$	−30409.1	−1.15894	−0.579471	+	0.814993i	$0.696741\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−49636.6	−1.87896	−0.939478	−	0.342609i	$-0.888690\pi$
$887$	−49636.6	−1.87896	−0.939478	+	0.342609i	$0.888690\pi$
$888$	0	0
$889$	10586.8	0.399403
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1122.08	0.0420480
$894$	0	0
$895$	−70039.0	−2.61580
$896$	0	0
$897$	0	0
$898$	0	0
$899$	30640.6	1.13673
$900$	0	0
$901$	7291.61	0.269610
$902$	0	0
$903$	0	0
$904$	0	0
$905$	85493.6	3.14023
$906$	0	0
$907$	3936.55	0.144114	0.0720568	−	0.997401i	$-0.477044\pi$
$907$	3936.55	0.144114	0.0720568	+	0.997401i	$0.477044\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6508.02	−0.236685	−0.118343	−	0.992973i	$-0.537758\pi$
$911$	−6508.02	−0.236685	−0.118343	+	0.992973i	$0.537758\pi$
$912$	0	0
$913$	8546.43	0.309798
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1123.69	−0.0404661
$918$	0	0
$919$	−28312.0	−1.01624	−0.508121	−	0.861286i	$-0.669660\pi$
$919$	−28312.0	−1.01624	−0.508121	+	0.861286i	$0.669660\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1408.86	0.0502418
$924$	0	0
$925$	76805.7	2.73011
$926$	0	0
$927$	0	0
$928$	0	0
$929$	2226.80	0.0786427	0.0393213	−	0.999227i	$-0.487480\pi$
$929$	2226.80	0.0786427	0.0393213	+	0.999227i	$0.487480\pi$
$930$	0	0
$931$	2106.99	0.0741715
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−15400.8	−0.538674
$936$	0	0
$937$	36687.9	1.27913	0.639563	−	0.768738i	$-0.279115\pi$
$937$	36687.9	1.27913	0.639563	+	0.768738i	$0.279115\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−10746.7	−0.372297	−0.186148	−	0.982522i	$-0.559600\pi$
$941$	−10746.7	−0.372297	−0.186148	+	0.982522i	$0.559600\pi$
$942$	0	0
$943$	4853.49	0.167605
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−51985.2	−1.78384	−0.891918	−	0.452197i	$-0.850640\pi$
$947$	−51985.2	−1.78384	−0.891918	+	0.452197i	$0.850640\pi$
$948$	0	0
$949$	15863.8	0.542634
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−32244.5	−1.09601	−0.548007	−	0.836474i	$-0.684613\pi$
$953$	−32244.5	−1.09601	−0.548007	+	0.836474i	$0.684613\pi$
$954$	0	0
$955$	−1225.27	−0.0415170
$956$	0	0
$957$	0	0
$958$	0	0
$959$	37101.8	1.24930
$960$	0	0
$961$	−10366.0	−0.347958
$962$	0	0
$963$	0	0
$964$	0	0
$965$	60717.1	2.02544
$966$	0	0
$967$	30347.2	1.00920	0.504601	−	0.863352i	$-0.331639\pi$
$967$	30347.2	1.00920	0.504601	+	0.863352i	$0.331639\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−5343.51	−0.176603	−0.0883014	−	0.996094i	$-0.528144\pi$
$971$	−5343.51	−0.176603	−0.0883014	+	0.996094i	$0.528144\pi$
$972$	0	0
$973$	−64840.6	−2.13638
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−44796.2	−1.46689	−0.733447	−	0.679746i	$-0.762090\pi$
$977$	−44796.2	−1.46689	−0.733447	+	0.679746i	$0.762090\pi$
$978$	0	0
$979$	12745.2	0.416076
$980$	0	0
$981$	0	0
$982$	0	0
$983$	34414.2	1.11662	0.558312	−	0.829631i	$-0.311449\pi$
$983$	34414.2	1.11662	0.558312	+	0.829631i	$0.311449\pi$
$984$	0	0
$985$	−44421.9	−1.43695
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9031.97	−0.290394
$990$	0	0
$991$	−54446.8	−1.74527	−0.872633	−	0.488377i	$-0.837589\pi$
$991$	−54446.8	−1.74527	−0.872633	+	0.488377i	$0.837589\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	91295.6	2.90881
$996$	0	0
$997$	−8410.68	−0.267171	−0.133585	−	0.991037i	$-0.542649\pi$
$997$	−8410.68	−0.267171	−0.133585	+	0.991037i	$0.542649\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.a.k.1.1		5
3.2	odd	2		648.4.a.l.1.5		5
4.3	odd	2		1296.4.a.bc.1.1		5
9.2	odd	6		72.4.i.b.49.5	yes	10
9.4	even	3		216.4.i.b.73.5		10
9.5	odd	6		72.4.i.b.25.5	✓	10
9.7	even	3		216.4.i.b.145.5		10
12.11	even	2		1296.4.a.bd.1.5		5
36.7	odd	6		432.4.i.f.145.5		10
36.11	even	6		144.4.i.f.49.1		10
36.23	even	6		144.4.i.f.97.1		10
36.31	odd	6		432.4.i.f.289.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.4.i.b.25.5	✓	10	9.5	odd	6
72.4.i.b.49.5	yes	10	9.2	odd	6
144.4.i.f.49.1		10	36.11	even	6
144.4.i.f.97.1		10	36.23	even	6
216.4.i.b.73.5		10	9.4	even	3
216.4.i.b.145.5		10	9.7	even	3
432.4.i.f.145.5		10	36.7	odd	6
432.4.i.f.289.5		10	36.31	odd	6
648.4.a.k.1.1		5	1.1	even	1	trivial
648.4.a.l.1.5		5	3.2	odd	2
1296.4.a.bc.1.1		5	4.3	odd	2
1296.4.a.bd.1.5		5	12.11	even	2

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Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.a (trivial)

\(f(q)\)	\(=\)	\(q-20.2222 q^{5} -24.7425 q^{7} +O(q^{10})\)	\(q-20.2222 q^{5} -24.7425 q^{7} -18.8052 q^{11} -48.4309 q^{13} -40.4983 q^{17} +7.82714 q^{19} -157.763 q^{23} +283.937 q^{25} -219.845 q^{29} -139.373 q^{31} +500.347 q^{35} +270.503 q^{37} -30.7644 q^{41} +57.2502 q^{43} +143.357 q^{47} +269.190 q^{49} -180.047 q^{53} +380.282 q^{55} +317.084 q^{59} +759.323 q^{61} +979.379 q^{65} +428.267 q^{67} -29.0901 q^{71} -327.554 q^{73} +465.287 q^{77} -1023.43 q^{79} -454.472 q^{83} +818.965 q^{85} -677.749 q^{89} +1198.30 q^{91} -158.282 q^{95} -397.800 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} + 3 q^{7}+O(q^{10}) \) 5 * q - 5 * q^5 + 3 * q^7	\( 5 q - 5 q^{5} + 3 q^{7} + 25 q^{11} + 29 q^{13} - 28 q^{17} + 64 q^{19} - 89 q^{23} + 322 q^{25} - 129 q^{29} + 241 q^{31} + 243 q^{35} + 366 q^{37} - 171 q^{41} + 803 q^{43} + 477 q^{47} + 1072 q^{49} - 374 q^{53} + 1469 q^{55} + 607 q^{59} + 1349 q^{61} - 527 q^{65} + 1549 q^{67} + 812 q^{71} + 1928 q^{73} - 903 q^{77} + 1727 q^{79} + 1025 q^{83} + 2902 q^{85} - 2310 q^{89} + 2985 q^{91} + 2012 q^{95} + 2875 q^{97}+O(q^{100}) \) 5 * q - 5 * q^5 + 3 * q^7 + 25 * q^11 + 29 * q^13 - 28 * q^17 + 64 * q^19 - 89 * q^23 + 322 * q^25 - 129 * q^29 + 241 * q^31 + 243 * q^35 + 366 * q^37 - 171 * q^41 + 803 * q^43 + 477 * q^47 + 1072 * q^49 - 374 * q^53 + 1469 * q^55 + 607 * q^59 + 1349 * q^61 - 527 * q^65 + 1549 * q^67 + 812 * q^71 + 1928 * q^73 - 903 * q^77 + 1727 * q^79 + 1025 * q^83 + 2902 * q^85 - 2310 * q^89 + 2985 * q^91 + 2012 * q^95 + 2875 * q^97

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