LMFDB - Embedded newform 1296.4.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,4,Mod(1,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, 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("1296.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1296.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:	$76.4664753674$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{57}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 14 $ x^2 - x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 81)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.27492$ of defining polynomial
Character	$\chi$	$=$	1296.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13.5498 q^{5} -27.6495 q^{7} +O(q^{10})$	$q-13.5498 q^{5} -27.6495 q^{7} +13.4502 q^{11} +54.6495 q^{13} +99.5980 q^{17} +2.94851 q^{19} -140.048 q^{23} +58.5980 q^{25} -21.8522 q^{29} +185.196 q^{31} +374.646 q^{35} +24.0515 q^{37} +441.698 q^{41} -207.650 q^{43} +153.100 q^{47} +421.495 q^{49} +162.000 q^{53} -182.248 q^{55} +245.292 q^{59} -416.939 q^{61} -740.492 q^{65} -695.444 q^{67} -493.341 q^{71} +220.691 q^{73} -371.890 q^{77} -711.650 q^{79} -204.508 q^{83} -1349.54 q^{85} -428.980 q^{89} -1511.03 q^{91} -39.9518 q^{95} -209.505 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 12 q^{5} - 10 q^{7}+O(q^{10}) $ 2 * q - 12 * q^5 - 10 * q^7	$ 2 q - 12 q^{5} - 10 q^{7} + 42 q^{11} + 64 q^{13} + 18 q^{17} - 130 q^{19} - 114 q^{23} - 64 q^{25} - 240 q^{29} + 8 q^{31} + 402 q^{35} + 184 q^{37} + 672 q^{41} - 370 q^{43} + 276 q^{47} + 390 q^{49} + 324 q^{53} - 138 q^{55} - 204 q^{59} + 208 q^{61} - 726 q^{65} - 802 q^{67} - 126 q^{71} - 374 q^{73} + 132 q^{77} - 1378 q^{79} - 1164 q^{83} - 1476 q^{85} + 954 q^{89} - 1346 q^{91} - 246 q^{95} - 872 q^{97}+O(q^{100}) $ 2 * q - 12 * q^5 - 10 * q^7 + 42 * q^11 + 64 * q^13 + 18 * q^17 - 130 * q^19 - 114 * q^23 - 64 * q^25 - 240 * q^29 + 8 * q^31 + 402 * q^35 + 184 * q^37 + 672 * q^41 - 370 * q^43 + 276 * q^47 + 390 * q^49 + 324 * q^53 - 138 * q^55 - 204 * q^59 + 208 * q^61 - 726 * q^65 - 802 * q^67 - 126 * q^71 - 374 * q^73 + 132 * q^77 - 1378 * q^79 - 1164 * q^83 - 1476 * q^85 + 954 * q^89 - 1346 * q^91 - 246 * q^95 - 872 * 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
$5$	−13.5498	−1.21193	−0.605967	−	0.795490i	$-0.707214\pi$
$5$	−13.5498	−1.21193	−0.605967	+	0.795490i	$0.707214\pi$
$6$	0	0
$7$	−27.6495	−1.49293	−0.746466	−	0.665423i	$-0.768251\pi$
$7$	−27.6495	−1.49293	−0.746466	+	0.665423i	$0.768251\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	13.4502	0.368671	0.184335	−	0.982863i	$-0.440987\pi$
$11$	13.4502	0.368671	0.184335	+	0.982863i	$0.440987\pi$
$12$	0	0
$13$	54.6495	1.16593	0.582963	−	0.812499i	$-0.301893\pi$
$13$	54.6495	1.16593	0.582963	+	0.812499i	$0.301893\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	99.5980	1.42095	0.710473	−	0.703725i	$-0.248481\pi$
$17$	99.5980	1.42095	0.710473	+	0.703725i	$0.248481\pi$
$18$	0	0
$19$	2.94851	0.0356018	0.0178009	−	0.999842i	$-0.494333\pi$
$19$	2.94851	0.0356018	0.0178009	+	0.999842i	$0.494333\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−140.048	−1.26965	−0.634827	−	0.772654i	$-0.718929\pi$
$23$	−140.048	−1.26965	−0.634827	+	0.772654i	$0.718929\pi$
$24$	0	0
$25$	58.5980	0.468784
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−21.8522	−0.139926	−0.0699628	−	0.997550i	$-0.522288\pi$
$29$	−21.8522	−0.139926	−0.0699628	+	0.997550i	$0.522288\pi$
$30$	0	0
$31$	185.196	1.07297	0.536487	−	0.843909i	$-0.319751\pi$
$31$	185.196	1.07297	0.536487	+	0.843909i	$0.319751\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	374.646	1.80934
$36$	0	0
$37$	24.0515	0.106866	0.0534330	−	0.998571i	$-0.482984\pi$
$37$	24.0515	0.106866	0.0534330	+	0.998571i	$0.482984\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	441.698	1.68248	0.841239	−	0.540664i	$-0.181827\pi$
$41$	441.698	1.68248	0.841239	+	0.540664i	$0.181827\pi$
$42$	0	0
$43$	−207.650	−0.736424	−0.368212	−	0.929742i	$-0.620030\pi$
$43$	−207.650	−0.736424	−0.368212	+	0.929742i	$0.620030\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	153.100	0.475146	0.237573	−	0.971370i	$-0.423648\pi$
$47$	153.100	0.475146	0.237573	+	0.971370i	$0.423648\pi$
$48$	0	0
$49$	421.495	1.22885
$50$	0	0
$51$	0	0
$52$	0	0
$53$	162.000	0.419857	0.209928	−	0.977717i	$-0.432677\pi$
$53$	162.000	0.419857	0.209928	+	0.977717i	$0.432677\pi$
$54$	0	0
$55$	−182.248	−0.446805
$56$	0	0
$57$	0	0
$58$	0	0
$59$	245.292	0.541260	0.270630	−	0.962683i	$-0.412768\pi$
$59$	245.292	0.541260	0.270630	+	0.962683i	$0.412768\pi$
$60$	0	0
$61$	−416.939	−0.875140	−0.437570	−	0.899184i	$-0.644161\pi$
$61$	−416.939	−0.875140	−0.437570	+	0.899184i	$0.644161\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−740.492	−1.41303
$66$	0	0
$67$	−695.444	−1.26809	−0.634044	−	0.773297i	$-0.718606\pi$
$67$	−695.444	−1.26809	−0.634044	+	0.773297i	$0.718606\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−493.341	−0.824630	−0.412315	−	0.911041i	$-0.635280\pi$
$71$	−493.341	−0.824630	−0.412315	+	0.911041i	$0.635280\pi$
$72$	0	0
$73$	220.691	0.353835	0.176917	−	0.984226i	$-0.443387\pi$
$73$	220.691	0.353835	0.176917	+	0.984226i	$0.443387\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−371.890	−0.550401
$78$	0	0
$79$	−711.650	−1.01350	−0.506752	−	0.862092i	$-0.669154\pi$
$79$	−711.650	−1.01350	−0.506752	+	0.862092i	$0.669154\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−204.508	−0.270454	−0.135227	−	0.990815i	$-0.543176\pi$
$83$	−204.508	−0.270454	−0.135227	+	0.990815i	$0.543176\pi$
$84$	0	0
$85$	−1349.54	−1.72209
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−428.980	−0.510919	−0.255460	−	0.966820i	$-0.582227\pi$
$89$	−428.980	−0.510919	−0.255460	+	0.966820i	$0.582227\pi$
$90$	0	0
$91$	−1511.03	−1.74065
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−39.9518	−0.0431471
$96$	0	0
$97$	−209.505	−0.219299	−0.109650	−	0.993970i	$-0.534973\pi$
$97$	−209.505	−0.219299	−0.109650	+	0.993970i	$0.534973\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	176.282	0.173671	0.0868354	−	0.996223i	$-0.472325\pi$
$101$	176.282	0.173671	0.0868354	+	0.996223i	$0.472325\pi$
$102$	0	0
$103$	−908.289	−0.868897	−0.434448	−	0.900697i	$-0.643057\pi$
$103$	−908.289	−0.868897	−0.434448	+	0.900697i	$0.643057\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	972.000	0.878194	0.439097	−	0.898440i	$-0.355298\pi$
$107$	972.000	0.878194	0.439097	+	0.898440i	$0.355298\pi$
$108$	0	0
$109$	−2087.33	−1.83422	−0.917110	−	0.398634i	$-0.869484\pi$
$109$	−2087.33	−1.83422	−0.917110	+	0.398634i	$0.869484\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1008.37	0.839463	0.419732	−	0.907648i	$-0.362124\pi$
$113$	1008.37	0.839463	0.419732	+	0.907648i	$0.362124\pi$
$114$	0	0
$115$	1897.63	1.53874
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2753.84	−2.12138
$120$	0	0
$121$	−1150.09	−0.864082
$122$	0	0
$123$	0	0
$124$	0	0
$125$	899.736	0.643799
$126$	0	0
$127$	2311.40	1.61499	0.807496	−	0.589873i	$-0.200822\pi$
$127$	2311.40	1.61499	0.807496	+	0.589873i	$0.200822\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	340.570	0.227143	0.113571	−	0.993530i	$-0.463771\pi$
$131$	340.570	0.227143	0.113571	+	0.993530i	$0.463771\pi$
$132$	0	0
$133$	−81.5248	−0.0531511
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2663.77	−1.66118	−0.830589	−	0.556886i	$-0.811996\pi$
$137$	−2663.77	−1.66118	−0.830589	+	0.556886i	$0.811996\pi$
$138$	0	0
$139$	538.887	0.328833	0.164417	−	0.986391i	$-0.447426\pi$
$139$	538.887	0.328833	0.164417	+	0.986391i	$0.447426\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	735.045	0.429843
$144$	0	0
$145$	296.093	0.169581
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3286.50	−1.80698	−0.903492	−	0.428605i	$-0.859005\pi$
$149$	−3286.50	−1.80698	−0.903492	+	0.428605i	$0.859005\pi$
$150$	0	0
$151$	−2791.05	−1.50419	−0.752095	−	0.659055i	$-0.770957\pi$
$151$	−2791.05	−1.50419	−0.752095	+	0.659055i	$0.770957\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2509.38	−1.30037
$156$	0	0
$157$	−851.659	−0.432929	−0.216464	−	0.976291i	$-0.569453\pi$
$157$	−851.659	−0.432929	−0.216464	+	0.976291i	$0.569453\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3872.26	1.89551
$162$	0	0
$163$	−389.897	−0.187356	−0.0936782	−	0.995603i	$-0.529862\pi$
$163$	−389.897	−0.187356	−0.0936782	+	0.995603i	$0.529862\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	787.580	0.364939	0.182469	−	0.983212i	$-0.441591\pi$
$167$	787.580	0.364939	0.182469	+	0.983212i	$0.441591\pi$
$168$	0	0
$169$	789.568	0.359385
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−144.862	−0.0636628	−0.0318314	−	0.999493i	$-0.510134\pi$
$173$	−144.862	−0.0636628	−0.0318314	+	0.999493i	$0.510134\pi$
$174$	0	0
$175$	−1620.21	−0.699863
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2193.61	0.915966	0.457983	−	0.888961i	$-0.348572\pi$
$179$	2193.61	0.915966	0.457983	+	0.888961i	$0.348572\pi$
$180$	0	0
$181$	856.970	0.351923	0.175962	−	0.984397i	$-0.443697\pi$
$181$	856.970	0.351923	0.175962	+	0.984397i	$0.443697\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−325.894	−0.129514
$186$	0	0
$187$	1339.61	0.523861
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2701.56	−1.02345	−0.511723	−	0.859151i	$-0.670992\pi$
$191$	−2701.56	−1.02345	−0.511723	+	0.859151i	$0.670992\pi$
$192$	0	0
$193$	442.053	0.164869	0.0824344	−	0.996596i	$-0.473730\pi$
$193$	442.053	0.164869	0.0824344	+	0.996596i	$0.473730\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3665.56	1.32569	0.662843	−	0.748759i	$-0.269350\pi$
$197$	3665.56	1.32569	0.662843	+	0.748759i	$0.269350\pi$
$198$	0	0
$199$	−3502.97	−1.24783	−0.623917	−	0.781491i	$-0.714460\pi$
$199$	−3502.97	−1.24783	−0.623917	+	0.781491i	$0.714460\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	604.201	0.208900
$204$	0	0
$205$	−5984.93	−2.03905
$206$	0	0
$207$	0	0
$208$	0	0
$209$	39.6579	0.0131253
$210$	0	0
$211$	1507.73	0.491927	0.245963	−	0.969279i	$-0.420896\pi$
$211$	1507.73	0.491927	0.245963	+	0.969279i	$0.420896\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2813.62	0.892498
$216$	0	0
$217$	−5120.58	−1.60188
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5442.98	1.65672
$222$	0	0
$223$	−4828.27	−1.44989	−0.724943	−	0.688809i	$-0.758134\pi$
$223$	−4828.27	−1.44989	−0.724943	+	0.688809i	$0.758134\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5151.43	1.50622	0.753112	−	0.657893i	$-0.228552\pi$
$227$	5151.43	1.50622	0.753112	+	0.657893i	$0.228552\pi$
$228$	0	0
$229$	151.885	0.0438291	0.0219145	−	0.999760i	$-0.493024\pi$
$229$	151.885	0.0438291	0.0219145	+	0.999760i	$0.493024\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3017.06	−0.848302	−0.424151	−	0.905591i	$-0.639427\pi$
$233$	−3017.06	−0.848302	−0.424151	+	0.905591i	$0.639427\pi$
$234$	0	0
$235$	−2074.48	−0.575846
$236$	0	0
$237$	0	0
$238$	0	0
$239$	502.076	0.135885	0.0679427	−	0.997689i	$-0.478356\pi$
$239$	502.076	0.135885	0.0679427	+	0.997689i	$0.478356\pi$
$240$	0	0
$241$	−3455.72	−0.923663	−0.461832	−	0.886968i	$-0.652808\pi$
$241$	−3455.72	−0.923663	−0.461832	+	0.886968i	$0.652808\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5711.19	−1.48928
$246$	0	0
$247$	161.135	0.0415091
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7318.87	1.84049	0.920245	−	0.391343i	$-0.127989\pi$
$251$	7318.87	1.84049	0.920245	+	0.391343i	$0.127989\pi$
$252$	0	0
$253$	−1883.67	−0.468085
$254$	0	0
$255$	0	0
$256$	0	0
$257$	886.345	0.215131	0.107566	−	0.994198i	$-0.465694\pi$
$257$	886.345	0.215131	0.107566	+	0.994198i	$0.465694\pi$
$258$	0	0
$259$	−665.012	−0.159544
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2460.43	−0.576869	−0.288434	−	0.957500i	$-0.593135\pi$
$263$	−2460.43	−0.576869	−0.288434	+	0.957500i	$0.593135\pi$
$264$	0	0
$265$	−2195.07	−0.508839
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2951.48	−0.668977	−0.334488	−	0.942400i	$-0.608563\pi$
$269$	−2951.48	−0.668977	−0.334488	+	0.942400i	$0.608563\pi$
$270$	0	0
$271$	−3915.61	−0.877699	−0.438849	−	0.898561i	$-0.644614\pi$
$271$	−3915.61	−0.877699	−0.438849	+	0.898561i	$0.644614\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	788.153	0.172827
$276$	0	0
$277$	4343.57	0.942165	0.471083	−	0.882089i	$-0.343863\pi$
$277$	4343.57	0.942165	0.471083	+	0.882089i	$0.343863\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1122.74	−0.238352	−0.119176	−	0.992873i	$-0.538025\pi$
$281$	−1122.74	−0.238352	−0.119176	+	0.992873i	$0.538025\pi$
$282$	0	0
$283$	−8091.10	−1.69953	−0.849763	−	0.527165i	$-0.823255\pi$
$283$	−8091.10	−1.69953	−0.849763	+	0.527165i	$0.823255\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12212.7	−2.51183
$288$	0	0
$289$	5006.76	1.01908
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6945.75	1.38490	0.692449	−	0.721467i	$-0.256532\pi$
$293$	6945.75	1.38490	0.692449	+	0.721467i	$0.256532\pi$
$294$	0	0
$295$	−3323.67	−0.655972
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7653.56	−1.48032
$300$	0	0
$301$	5741.41	1.09943
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5649.45	1.06061
$306$	0	0
$307$	2017.53	0.375070	0.187535	−	0.982258i	$-0.439950\pi$
$307$	2017.53	0.375070	0.187535	+	0.982258i	$0.439950\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2557.99	0.466400	0.233200	−	0.972429i	$-0.425080\pi$
$311$	2557.99	0.466400	0.233200	+	0.972429i	$0.425080\pi$
$312$	0	0
$313$	−4105.58	−0.741409	−0.370704	−	0.928751i	$-0.620884\pi$
$313$	−4105.58	−0.741409	−0.370704	+	0.928751i	$0.620884\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5842.43	−1.03515	−0.517577	−	0.855637i	$-0.673166\pi$
$317$	−5842.43	−1.03515	−0.517577	+	0.855637i	$0.673166\pi$
$318$	0	0
$319$	−293.915	−0.0515865
$320$	0	0
$321$	0	0
$322$	0	0
$323$	293.666	0.0505882
$324$	0	0
$325$	3202.35	0.546568
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−4233.13	−0.709362
$330$	0	0
$331$	−6372.56	−1.05821	−0.529105	−	0.848556i	$-0.677472\pi$
$331$	−6372.56	−1.05821	−0.529105	+	0.848556i	$0.677472\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	9423.14	1.53684
$336$	0	0
$337$	−6156.56	−0.995161	−0.497581	−	0.867418i	$-0.665778\pi$
$337$	−6156.56	−0.995161	−0.497581	+	0.867418i	$0.665778\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2490.92	0.395574
$342$	0	0
$343$	−2170.35	−0.341655
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11601.4	−1.79480	−0.897401	−	0.441215i	$-0.854548\pi$
$347$	−11601.4	−1.79480	−0.897401	+	0.441215i	$0.854548\pi$
$348$	0	0
$349$	12465.8	1.91198	0.955990	−	0.293398i	$-0.0947860\pi$
$349$	12465.8	1.91198	0.955990	+	0.293398i	$0.0947860\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3058.35	0.461131	0.230566	−	0.973057i	$-0.425942\pi$
$353$	3058.35	0.461131	0.230566	+	0.973057i	$0.425942\pi$
$354$	0	0
$355$	6684.68	0.999398
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5731.16	0.842560	0.421280	−	0.906931i	$-0.361581\pi$
$359$	5731.16	0.842560	0.421280	+	0.906931i	$0.361581\pi$
$360$	0	0
$361$	−6850.31	−0.998733
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2990.33	−0.428824
$366$	0	0
$367$	13924.5	1.98052	0.990260	−	0.139231i	$-0.0444629\pi$
$367$	13924.5	1.98052	0.990260	+	0.139231i	$0.0444629\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−4479.22	−0.626818
$372$	0	0
$373$	−7451.81	−1.03442	−0.517212	−	0.855857i	$-0.673030\pi$
$373$	−7451.81	−1.03442	−0.517212	+	0.855857i	$0.673030\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1194.21	−0.163143
$378$	0	0
$379$	9791.79	1.32710	0.663550	−	0.748132i	$-0.269049\pi$
$379$	9791.79	1.32710	0.663550	+	0.748132i	$0.269049\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1335.74	0.178207	0.0891034	−	0.996022i	$-0.471600\pi$
$383$	1335.74	0.178207	0.0891034	+	0.996022i	$0.471600\pi$
$384$	0	0
$385$	5039.05	0.667049
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6042.22	0.787539	0.393770	−	0.919209i	$-0.371171\pi$
$389$	6042.22	0.787539	0.393770	+	0.919209i	$0.371171\pi$
$390$	0	0
$391$	−13948.5	−1.80411
$392$	0	0
$393$	0	0
$394$	0	0
$395$	9642.73	1.22830
$396$	0	0
$397$	64.5068	0.00815492	0.00407746	−	0.999992i	$-0.498702\pi$
$397$	64.5068	0.00815492	0.00407746	+	0.999992i	$0.498702\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10922.2	−1.36017	−0.680085	−	0.733133i	$-0.738057\pi$
$401$	−10922.2	−1.36017	−0.680085	+	0.733133i	$0.738057\pi$
$402$	0	0
$403$	10120.9	1.25101
$404$	0	0
$405$	0	0
$406$	0	0
$407$	323.497	0.0393983
$408$	0	0
$409$	4515.50	0.545909	0.272955	−	0.962027i	$-0.411999\pi$
$409$	4515.50	0.545909	0.272955	+	0.962027i	$0.411999\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6782.21	−0.808065
$414$	0	0
$415$	2771.05	0.327773
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14489.7	−1.68942	−0.844710	−	0.535224i	$-0.820227\pi$
$419$	−14489.7	−1.68942	−0.844710	+	0.535224i	$0.820227\pi$
$420$	0	0
$421$	−6798.72	−0.787053	−0.393526	−	0.919313i	$-0.628745\pi$
$421$	−6798.72	−0.787053	−0.393526	+	0.919313i	$0.628745\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5836.25	0.666116
$426$	0	0
$427$	11528.1	1.30652
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10100.8	−1.12886	−0.564429	−	0.825482i	$-0.690903\pi$
$431$	−10100.8	−1.12886	−0.564429	+	0.825482i	$0.690903\pi$
$432$	0	0
$433$	754.263	0.0837126	0.0418563	−	0.999124i	$-0.486673\pi$
$433$	754.263	0.0837126	0.0418563	+	0.999124i	$0.486673\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−412.933	−0.0452020
$438$	0	0
$439$	2.47517	0.000269096 0	0.000134548	−	1.00000i	$-0.499957\pi$
$439$	2.47517	0.000269096 0	0.000134548		1.00000i	$0.499957\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9677.17	−1.03787	−0.518935	−	0.854814i	$-0.673671\pi$
$443$	−9677.17	−1.03787	−0.518935	+	0.854814i	$0.673671\pi$
$444$	0	0
$445$	5812.61	0.619200
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4744.97	−0.498729	−0.249364	−	0.968410i	$-0.580222\pi$
$449$	−4744.97	−0.498729	−0.249364	+	0.968410i	$0.580222\pi$
$450$	0	0
$451$	5940.91	0.620280
$452$	0	0
$453$	0	0
$454$	0	0
$455$	20474.2	2.10955
$456$	0	0
$457$	−4252.18	−0.435249	−0.217624	−	0.976033i	$-0.569831\pi$
$457$	−4252.18	−0.435249	−0.217624	+	0.976033i	$0.569831\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6893.62	−0.696460	−0.348230	−	0.937409i	$-0.613217\pi$
$461$	−6893.62	−0.696460	−0.348230	+	0.937409i	$0.613217\pi$
$462$	0	0
$463$	10248.1	1.02866	0.514331	−	0.857592i	$-0.328040\pi$
$463$	10248.1	1.02866	0.514331	+	0.857592i	$0.328040\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14346.0	−1.42153	−0.710766	−	0.703429i	$-0.751651\pi$
$467$	−14346.0	−1.42153	−0.710766	+	0.703429i	$0.751651\pi$
$468$	0	0
$469$	19228.7	1.89317
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2792.92	−0.271498
$474$	0	0
$475$	172.777	0.0166896
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17013.8	1.62292	0.811461	−	0.584406i	$-0.198672\pi$
$479$	17013.8	1.62292	0.811461	+	0.584406i	$0.198672\pi$
$480$	0	0
$481$	1314.40	0.124598
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2838.76	0.265776
$486$	0	0
$487$	−10890.0	−1.01329	−0.506647	−	0.862154i	$-0.669115\pi$
$487$	−10890.0	−1.01329	−0.506647	+	0.862154i	$0.669115\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11722.3	1.07743	0.538717	−	0.842487i	$-0.318909\pi$
$491$	11722.3	1.07743	0.538717	+	0.842487i	$0.318909\pi$
$492$	0	0
$493$	−2176.43	−0.198827
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13640.6	1.23112
$498$	0	0
$499$	14794.2	1.32721	0.663607	−	0.748081i	$-0.269025\pi$
$499$	14794.2	1.32721	0.663607	+	0.748081i	$0.269025\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11738.4	−1.04053	−0.520267	−	0.854003i	$-0.674168\pi$
$503$	−11738.4	−1.04053	−0.520267	+	0.854003i	$0.674168\pi$
$504$	0	0
$505$	−2388.60	−0.210478
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11310.1	−0.984891	−0.492446	−	0.870343i	$-0.663897\pi$
$509$	−11310.1	−0.984891	−0.492446	+	0.870343i	$0.663897\pi$
$510$	0	0
$511$	−6102.00	−0.528251
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12307.2	1.05305
$516$	0	0
$517$	2059.22	0.175173
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2080.55	−0.174953	−0.0874767	−	0.996167i	$-0.527880\pi$
$521$	−2080.55	−0.174953	−0.0874767	+	0.996167i	$0.527880\pi$
$522$	0	0
$523$	7667.48	0.641062	0.320531	−	0.947238i	$-0.396139\pi$
$523$	7667.48	0.641062	0.320531	+	0.947238i	$0.396139\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	18445.2	1.52464
$528$	0	0
$529$	7446.49	0.612024
$530$	0	0
$531$	0	0
$532$	0	0
$533$	24138.6	1.96165
$534$	0	0
$535$	−13170.4	−1.06431
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5669.18	0.453040
$540$	0	0
$541$	−23452.0	−1.86374	−0.931869	−	0.362795i	$-0.881823\pi$
$541$	−23452.0	−1.86374	−0.931869	+	0.362795i	$0.881823\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	28283.0	2.22295
$546$	0	0
$547$	6736.33	0.526553	0.263277	−	0.964720i	$-0.415197\pi$
$547$	6736.33	0.526553	0.263277	+	0.964720i	$0.415197\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−64.4313	−0.00498161
$552$	0	0
$553$	19676.8	1.51309
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18147.2	−1.38047	−0.690233	−	0.723587i	$-0.742492\pi$
$557$	−18147.2	−1.38047	−0.690233	+	0.723587i	$0.742492\pi$
$558$	0	0
$559$	−11347.9	−0.858617
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−21126.6	−1.58149	−0.790747	−	0.612143i	$-0.790308\pi$
$563$	−21126.6	−1.58149	−0.790747	+	0.612143i	$0.790308\pi$
$564$	0	0
$565$	−13663.2	−1.01737
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4810.74	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$569$	4810.74	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$570$	0	0
$571$	7553.68	0.553611	0.276805	−	0.960926i	$-0.410724\pi$
$571$	7553.68	0.553611	0.276805	+	0.960926i	$0.410724\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8206.55	−0.595194
$576$	0	0
$577$	22144.7	1.59774	0.798870	−	0.601504i	$-0.205432\pi$
$577$	22144.7	1.59774	0.798870	+	0.601504i	$0.205432\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5654.55	0.403770
$582$	0	0
$583$	2178.93	0.154789
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18267.7	1.28448	0.642240	−	0.766504i	$-0.278005\pi$
$587$	18267.7	1.28448	0.642240	+	0.766504i	$0.278005\pi$
$588$	0	0
$589$	546.052	0.0381998
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−14734.9	−1.02039	−0.510194	−	0.860059i	$-0.670426\pi$
$593$	−14734.9	−1.02039	−0.510194	+	0.860059i	$0.670426\pi$
$594$	0	0
$595$	37314.0	2.57097
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1473.75	0.100527	0.0502635	−	0.998736i	$-0.483994\pi$
$599$	1473.75	0.100527	0.0502635	+	0.998736i	$0.483994\pi$
$600$	0	0
$601$	8596.11	0.583432	0.291716	−	0.956505i	$-0.405774\pi$
$601$	8596.11	0.583432	0.291716	+	0.956505i	$0.405774\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	15583.6	1.04721
$606$	0	0
$607$	2966.30	0.198350	0.0991750	−	0.995070i	$-0.468380\pi$
$607$	2966.30	0.198350	0.0991750	+	0.995070i	$0.468380\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8366.82	0.553986
$612$	0	0
$613$	−14156.4	−0.932744	−0.466372	−	0.884589i	$-0.654439\pi$
$613$	−14156.4	−0.932744	−0.466372	+	0.884589i	$0.654439\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−980.990	−0.0640084	−0.0320042	−	0.999488i	$-0.510189\pi$
$617$	−980.990	−0.0640084	−0.0320042	+	0.999488i	$0.510189\pi$
$618$	0	0
$619$	−11629.6	−0.755144	−0.377572	−	0.925980i	$-0.623241\pi$
$619$	−11629.6	−0.755144	−0.377572	+	0.925980i	$0.623241\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11861.1	0.762768
$624$	0	0
$625$	−19516.0	−1.24903
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2395.48	0.151851
$630$	0	0
$631$	−17594.2	−1.11001	−0.555003	−	0.831848i	$-0.687283\pi$
$631$	−17594.2	−1.11001	−0.555003	+	0.831848i	$0.687283\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−31319.1	−1.95726
$636$	0	0
$637$	23034.5	1.43275
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−20949.0	−1.29085	−0.645426	−	0.763823i	$-0.723320\pi$
$641$	−20949.0	−1.29085	−0.645426	+	0.763823i	$0.723320\pi$
$642$	0	0
$643$	12089.6	0.741471	0.370736	−	0.928738i	$-0.379106\pi$
$643$	12089.6	0.741471	0.370736	+	0.928738i	$0.379106\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18149.9	−1.10285	−0.551426	−	0.834224i	$-0.685916\pi$
$647$	−18149.9	−1.10285	−0.551426	+	0.834224i	$0.685916\pi$
$648$	0	0
$649$	3299.22	0.199547
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16512.3	−0.989550	−0.494775	−	0.869021i	$-0.664750\pi$
$653$	−16512.3	−0.989550	−0.494775	+	0.869021i	$0.664750\pi$
$654$	0	0
$655$	−4614.66	−0.275282
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15597.7	0.922006	0.461003	−	0.887399i	$-0.347490\pi$
$659$	15597.7	0.922006	0.461003	+	0.887399i	$0.347490\pi$
$660$	0	0
$661$	4826.44	0.284004	0.142002	−	0.989866i	$-0.454646\pi$
$661$	4826.44	0.284004	0.142002	+	0.989866i	$0.454646\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1104.65	0.0644157
$666$	0	0
$667$	3060.35	0.177657
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5607.89	−0.322638
$672$	0	0
$673$	−2081.42	−0.119216	−0.0596082	−	0.998222i	$-0.518985\pi$
$673$	−2081.42	−0.119216	−0.0596082	+	0.998222i	$0.518985\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26839.5	1.52367	0.761836	−	0.647770i	$-0.224298\pi$
$677$	26839.5	1.52367	0.761836	+	0.647770i	$0.224298\pi$
$678$	0	0
$679$	5792.71	0.327399
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6892.12	0.386119	0.193060	−	0.981187i	$-0.438159\pi$
$683$	6892.12	0.386119	0.193060	+	0.981187i	$0.438159\pi$
$684$	0	0
$685$	36093.7	2.01324
$686$	0	0
$687$	0	0
$688$	0	0
$689$	8853.22	0.489522
$690$	0	0
$691$	24434.9	1.34522	0.672611	−	0.739996i	$-0.265173\pi$
$691$	24434.9	1.34522	0.672611	+	0.739996i	$0.265173\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7301.83	−0.398524
$696$	0	0
$697$	43992.2	2.39071
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20137.7	1.08501	0.542503	−	0.840054i	$-0.317477\pi$
$701$	20137.7	1.08501	0.542503	+	0.840054i	$0.317477\pi$
$702$	0	0
$703$	70.9161	0.00380462
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4874.12	−0.259279
$708$	0	0
$709$	9050.94	0.479429	0.239715	−	0.970843i	$-0.422946\pi$
$709$	9050.94	0.479429	0.239715	+	0.970843i	$0.422946\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−25936.4	−1.36231
$714$	0	0
$715$	−9959.74	−0.520941
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−33217.3	−1.72295	−0.861473	−	0.507804i	$-0.830457\pi$
$719$	−33217.3	−1.72295	−0.861473	+	0.507804i	$0.830457\pi$
$720$	0	0
$721$	25113.7	1.29720
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1280.49	−0.0655949
$726$	0	0
$727$	−22399.0	−1.14269	−0.571343	−	0.820711i	$-0.693577\pi$
$727$	−22399.0	−1.14269	−0.571343	+	0.820711i	$0.693577\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−20681.5	−1.04642
$732$	0	0
$733$	12191.6	0.614333	0.307166	−	0.951656i	$-0.400619\pi$
$733$	12191.6	0.614333	0.307166	+	0.951656i	$0.400619\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9353.83	−0.467507
$738$	0	0
$739$	9986.27	0.497092	0.248546	−	0.968620i	$-0.420047\pi$
$739$	9986.27	0.497092	0.248546	+	0.968620i	$0.420047\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2031.83	−0.100324	−0.0501619	−	0.998741i	$-0.515974\pi$
$743$	−2031.83	−0.100324	−0.0501619	+	0.998741i	$0.515974\pi$
$744$	0	0
$745$	44531.5	2.18995
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−26875.3	−1.31109
$750$	0	0
$751$	−28259.5	−1.37311	−0.686553	−	0.727080i	$-0.740877\pi$
$751$	−28259.5	−1.37311	−0.686553	+	0.727080i	$0.740877\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	37818.3	1.82298
$756$	0	0
$757$	2906.88	0.139567	0.0697835	−	0.997562i	$-0.477769\pi$
$757$	2906.88	0.139567	0.0697835	+	0.997562i	$0.477769\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−31188.0	−1.48563	−0.742815	−	0.669497i	$-0.766510\pi$
$761$	−31188.0	−1.48563	−0.742815	+	0.669497i	$0.766510\pi$
$762$	0	0
$763$	57713.7	2.73837
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13405.1	0.631070
$768$	0	0
$769$	−12793.1	−0.599910	−0.299955	−	0.953953i	$-0.596972\pi$
$769$	−12793.1	−0.599910	−0.299955	+	0.953953i	$0.596972\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−15770.5	−0.733799	−0.366899	−	0.930261i	$-0.619581\pi$
$773$	−15770.5	−0.733799	−0.366899	+	0.930261i	$0.619581\pi$
$774$	0	0
$775$	10852.1	0.502993
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1302.35	0.0598993
$780$	0	0
$781$	−6635.51	−0.304017
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11539.8	0.524681
$786$	0	0
$787$	355.971	0.0161232	0.00806162	−	0.999968i	$-0.497434\pi$
$787$	355.971	0.0161232	0.00806162	+	0.999968i	$0.497434\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−27880.9	−1.25326
$792$	0	0
$793$	−22785.5	−1.02035
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2850.91	−0.126706	−0.0633529	−	0.997991i	$-0.520179\pi$
$797$	−2850.91	−0.126706	−0.0633529	+	0.997991i	$0.520179\pi$
$798$	0	0
$799$	15248.4	0.675157
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2968.33	0.130448
$804$	0	0
$805$	−52468.5	−2.29723
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−20748.9	−0.901722	−0.450861	−	0.892594i	$-0.648883\pi$
$809$	−20748.9	−0.901722	−0.450861	+	0.892594i	$0.648883\pi$
$810$	0	0
$811$	−38381.1	−1.66183	−0.830913	−	0.556402i	$-0.812181\pi$
$811$	−38381.1	−1.66183	−0.830913	+	0.556402i	$0.812181\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5283.04	0.227064
$816$	0	0
$817$	−612.257	−0.0262181
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−8240.92	−0.350317	−0.175158	−	0.984540i	$-0.556044\pi$
$821$	−8240.92	−0.350317	−0.175158	+	0.984540i	$0.556044\pi$
$822$	0	0
$823$	4436.90	0.187923	0.0939616	−	0.995576i	$-0.470047\pi$
$823$	4436.90	0.187923	0.0939616	+	0.995576i	$0.470047\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43845.7	1.84361	0.921804	−	0.387656i	$-0.126715\pi$
$827$	43845.7	1.84361	0.921804	+	0.387656i	$0.126715\pi$
$828$	0	0
$829$	−32363.3	−1.35588	−0.677939	−	0.735118i	$-0.737127\pi$
$829$	−32363.3	−1.35588	−0.677939	+	0.735118i	$0.737127\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	41980.1	1.74613
$834$	0	0
$835$	−10671.6	−0.442281
$836$	0	0
$837$	0	0
$838$	0	0
$839$	25582.3	1.05268	0.526341	−	0.850274i	$-0.323564\pi$
$839$	25582.3	1.05268	0.526341	+	0.850274i	$0.323564\pi$
$840$	0	0
$841$	−23911.5	−0.980421
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10698.5	−0.435551
$846$	0	0
$847$	31799.5	1.29002
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−3368.37	−0.135683
$852$	0	0
$853$	−36107.3	−1.44935	−0.724673	−	0.689093i	$-0.758009\pi$
$853$	−36107.3	−1.44935	−0.724673	+	0.689093i	$0.758009\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	35375.0	1.41002	0.705011	−	0.709197i	$-0.250942\pi$
$857$	35375.0	1.41002	0.705011	+	0.709197i	$0.250942\pi$
$858$	0	0
$859$	23378.5	0.928596	0.464298	−	0.885679i	$-0.346307\pi$
$859$	23378.5	0.928596	0.464298	+	0.885679i	$0.346307\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19474.2	−0.768146	−0.384073	−	0.923303i	$-0.625479\pi$
$863$	−19474.2	−0.768146	−0.384073	+	0.923303i	$0.625479\pi$
$864$	0	0
$865$	1962.86	0.0771551
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9571.80	−0.373649
$870$	0	0
$871$	−38005.6	−1.47850
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−24877.3	−0.961148
$876$	0	0
$877$	29529.8	1.13700	0.568500	−	0.822683i	$-0.307524\pi$
$877$	29529.8	1.13700	0.568500	+	0.822683i	$0.307524\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	31785.8	1.21554	0.607769	−	0.794114i	$-0.292065\pi$
$881$	31785.8	1.21554	0.607769	+	0.794114i	$0.292065\pi$
$882$	0	0
$883$	−21881.2	−0.833930	−0.416965	−	0.908922i	$-0.636906\pi$
$883$	−21881.2	−0.833930	−0.416965	+	0.908922i	$0.636906\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26384.9	0.998780	0.499390	−	0.866377i	$-0.333557\pi$
$887$	26384.9	0.998780	0.499390	+	0.866377i	$0.333557\pi$
$888$	0	0
$889$	−63909.2	−2.41107
$890$	0	0
$891$	0	0
$892$	0	0
$893$	451.416	0.0169161
$894$	0	0
$895$	−29723.0	−1.11009
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4046.93	−0.150137
$900$	0	0
$901$	16134.9	0.596593
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11611.8	−0.426508
$906$	0	0
$907$	9381.52	0.343449	0.171725	−	0.985145i	$-0.445066\pi$
$907$	9381.52	0.343449	0.171725	+	0.985145i	$0.445066\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−33920.7	−1.23364	−0.616818	−	0.787106i	$-0.711579\pi$
$911$	−33920.7	−1.23364	−0.616818	+	0.787106i	$0.711579\pi$
$912$	0	0
$913$	−2750.67	−0.0997085
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9416.58	−0.339109
$918$	0	0
$919$	−12858.4	−0.461546	−0.230773	−	0.973008i	$-0.574125\pi$
$919$	−12858.4	−0.461546	−0.230773	+	0.973008i	$0.574125\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−26960.8	−0.961458
$924$	0	0
$925$	1409.37	0.0500971
$926$	0	0
$927$	0	0
$928$	0	0
$929$	11351.7	0.400902	0.200451	−	0.979704i	$-0.435759\pi$
$929$	11351.7	0.400902	0.200451	+	0.979704i	$0.435759\pi$
$930$	0	0
$931$	1242.78	0.0437492
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18151.5	−0.634885
$936$	0	0
$937$	−14940.6	−0.520906	−0.260453	−	0.965486i	$-0.583872\pi$
$937$	−14940.6	−0.520906	−0.260453	+	0.965486i	$0.583872\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	13912.2	0.481961	0.240980	−	0.970530i	$-0.422531\pi$
$941$	13912.2	0.481961	0.240980	+	0.970530i	$0.422531\pi$
$942$	0	0
$943$	−61859.0	−2.13617
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28187.0	−0.967215	−0.483608	−	0.875285i	$-0.660674\pi$
$947$	−28187.0	−0.967215	−0.483608	+	0.875285i	$0.660674\pi$
$948$	0	0
$949$	12060.7	0.412545
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22113.0	0.751636	0.375818	−	0.926693i	$-0.377362\pi$
$953$	22113.0	0.751636	0.375818	+	0.926693i	$0.377362\pi$
$954$	0	0
$955$	36605.7	1.24035
$956$	0	0
$957$	0	0
$958$	0	0
$959$	73651.9	2.48003
$960$	0	0
$961$	4506.57	0.151273
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5989.75	−0.199810
$966$	0	0
$967$	8105.27	0.269543	0.134771	−	0.990877i	$-0.456970\pi$
$967$	8105.27	0.269543	0.134771	+	0.990877i	$0.456970\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−8668.66	−0.286499	−0.143250	−	0.989687i	$-0.545755\pi$
$971$	−8668.66	−0.286499	−0.143250	+	0.989687i	$0.545755\pi$
$972$	0	0
$973$	−14900.0	−0.490926
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28801.8	−0.943143	−0.471572	−	0.881828i	$-0.656313\pi$
$977$	−28801.8	−0.943143	−0.471572	+	0.881828i	$0.656313\pi$
$978$	0	0
$979$	−5769.85	−0.188361
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−16252.2	−0.527328	−0.263664	−	0.964615i	$-0.584931\pi$
$983$	−16252.2	−0.527328	−0.263664	+	0.964615i	$0.584931\pi$
$984$	0	0
$985$	−49667.7	−1.60664
$986$	0	0
$987$	0	0
$988$	0	0
$989$	29080.9	0.935005
$990$	0	0
$991$	−31687.9	−1.01574	−0.507871	−	0.861433i	$-0.669567\pi$
$991$	−31687.9	−1.01574	−0.507871	+	0.861433i	$0.669567\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	47464.7	1.51229
$996$	0	0
$997$	29704.3	0.943576	0.471788	−	0.881712i	$-0.343609\pi$
$997$	29704.3	0.943576	0.471788	+	0.881712i	$0.343609\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.4.a.k.1.1		2
3.2	odd	2		1296.4.a.t.1.2		2
4.3	odd	2		81.4.a.e.1.1	yes	2
12.11	even	2		81.4.a.b.1.2	✓	2
20.19	odd	2		2025.4.a.h.1.2		2
36.7	odd	6		81.4.c.d.28.2		4
36.11	even	6		81.4.c.g.28.1		4
36.23	even	6		81.4.c.g.55.1		4
36.31	odd	6		81.4.c.d.55.2		4
60.59	even	2		2025.4.a.o.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.4.a.b.1.2	✓	2	12.11	even	2
81.4.a.e.1.1	yes	2	4.3	odd	2
81.4.c.d.28.2		4	36.7	odd	6
81.4.c.d.55.2		4	36.31	odd	6
81.4.c.g.28.1		4	36.11	even	6
81.4.c.g.55.1		4	36.23	even	6
1296.4.a.k.1.1		2	1.1	even	1	trivial
1296.4.a.t.1.2		2	3.2	odd	2
2025.4.a.h.1.2		2	20.19	odd	2
2025.4.a.o.1.1		2	60.59	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1296.a (trivial)

\(f(q)\)	\(=\)	\(q-13.5498 q^{5} -27.6495 q^{7} +O(q^{10})\)	\(q-13.5498 q^{5} -27.6495 q^{7} +13.4502 q^{11} +54.6495 q^{13} +99.5980 q^{17} +2.94851 q^{19} -140.048 q^{23} +58.5980 q^{25} -21.8522 q^{29} +185.196 q^{31} +374.646 q^{35} +24.0515 q^{37} +441.698 q^{41} -207.650 q^{43} +153.100 q^{47} +421.495 q^{49} +162.000 q^{53} -182.248 q^{55} +245.292 q^{59} -416.939 q^{61} -740.492 q^{65} -695.444 q^{67} -493.341 q^{71} +220.691 q^{73} -371.890 q^{77} -711.650 q^{79} -204.508 q^{83} -1349.54 q^{85} -428.980 q^{89} -1511.03 q^{91} -39.9518 q^{95} -209.505 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 12 q^{5} - 10 q^{7}+O(q^{10}) \) 2 * q - 12 * q^5 - 10 * q^7	\( 2 q - 12 q^{5} - 10 q^{7} + 42 q^{11} + 64 q^{13} + 18 q^{17} - 130 q^{19} - 114 q^{23} - 64 q^{25} - 240 q^{29} + 8 q^{31} + 402 q^{35} + 184 q^{37} + 672 q^{41} - 370 q^{43} + 276 q^{47} + 390 q^{49} + 324 q^{53} - 138 q^{55} - 204 q^{59} + 208 q^{61} - 726 q^{65} - 802 q^{67} - 126 q^{71} - 374 q^{73} + 132 q^{77} - 1378 q^{79} - 1164 q^{83} - 1476 q^{85} + 954 q^{89} - 1346 q^{91} - 246 q^{95} - 872 q^{97}+O(q^{100}) \) 2 * q - 12 * q^5 - 10 * q^7 + 42 * q^11 + 64 * q^13 + 18 * q^17 - 130 * q^19 - 114 * q^23 - 64 * q^25 - 240 * q^29 + 8 * q^31 + 402 * q^35 + 184 * q^37 + 672 * q^41 - 370 * q^43 + 276 * q^47 + 390 * q^49 + 324 * q^53 - 138 * q^55 - 204 * q^59 + 208 * q^61 - 726 * q^65 - 802 * q^67 - 126 * q^71 - 374 * q^73 + 132 * q^77 - 1378 * q^79 - 1164 * q^83 - 1476 * q^85 + 954 * q^89 - 1346 * q^91 - 246 * q^95 - 872 * q^97

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