LMFDB - Embedded newform 324.3.c.b.161.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,3,Mod(161,324)]

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, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("324.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	324.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$8.82836056527$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.2
Root			$1.68614 + 0.396143i$ of defining polynomial
Character	$\chi$	$=$	324.161
Dual form			324.3.c.b.161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37686i q^{5} -8.11684 q^{7} +O(q^{10})$	$q-2.37686i q^{5} -8.11684 q^{7} +20.3422i q^{11} +6.11684 q^{13} +17.9653i q^{17} +9.11684 q^{19} +33.5538i q^{23} +19.3505 q^{25} -16.6380i q^{29} -22.3505 q^{31} +19.2926i q^{35} -50.4674 q^{37} +34.6033i q^{41} +23.0000 q^{43} +38.3075i q^{47} +16.8832 q^{49} +19.0149i q^{53} +48.3505 q^{55} -3.42643i q^{59} -46.3505 q^{61} -14.5389i q^{65} -6.29894 q^{67} -35.9306i q^{71} +47.3505 q^{73} -165.114i q^{77} -84.5842 q^{79} +38.3075i q^{83} +42.7011 q^{85} -143.723i q^{89} -49.6495 q^{91} -21.6695i q^{95} +80.7663 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{7}+O(q^{10}) $ 4 * q + 2 * q^7	$ 4 q + 2 q^{7} - 10 q^{13} + 2 q^{19} - 26 q^{25} + 14 q^{31} - 64 q^{37} + 92 q^{43} + 102 q^{49} + 90 q^{55} - 82 q^{61} - 232 q^{67} + 86 q^{73} - 166 q^{79} - 36 q^{85} - 302 q^{91} + 392 q^{97}+O(q^{100}) $ 4 * q + 2 * q^7 - 10 * q^13 + 2 * q^19 - 26 * q^25 + 14 * q^31 - 64 * q^37 + 92 * q^43 + 102 * q^49 + 90 * q^55 - 82 * q^61 - 232 * q^67 + 86 * q^73 - 166 * q^79 - 36 * q^85 - 302 * q^91 + 392 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$-1$

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
$5$	− 2.37686i	− 0.475372i	−0.971342	−	0.237686i	$-0.923611\pi$
$5$	− 2.37686i	− 0.475372i	0.971342	−	0.237686i	$-0.0763890\pi$
$6$	0	0
$7$	−8.11684	−1.15955	−0.579775	−	0.814777i	$-0.696859\pi$
$7$	−8.11684	−1.15955	−0.579775	+	0.814777i	$0.696859\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	20.3422i	1.84929i	0.380831	+	0.924645i	$0.375638\pi$
$11$	20.3422i	1.84929i	−0.380831	+	0.924645i	$0.624362\pi$
$12$	0	0
$13$	6.11684	0.470526	0.235263	−	0.971932i	$-0.424405\pi$
$13$	6.11684	0.470526	0.235263	+	0.971932i	$0.424405\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	17.9653i	1.05678i	0.849001	+	0.528392i	$0.177205\pi$
$17$	17.9653i	1.05678i	−0.849001	+	0.528392i	$0.822795\pi$
$18$	0	0
$19$	9.11684	0.479834	0.239917	−	0.970793i	$-0.422880\pi$
$19$	9.11684	0.479834	0.239917	+	0.970793i	$0.422880\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	33.5538i	1.45886i	0.684056	+	0.729430i	$0.260215\pi$
$23$	33.5538i	1.45886i	−0.684056	+	0.729430i	$0.739785\pi$
$24$	0	0
$25$	19.3505	0.774021
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 16.6380i	− 0.573725i	−0.957972	−	0.286863i	$-0.907388\pi$
$29$	− 16.6380i	− 0.573725i	0.957972	−	0.286863i	$-0.0926123\pi$
$30$	0	0
$31$	−22.3505	−0.720985	−0.360492	−	0.932762i	$-0.617391\pi$
$31$	−22.3505	−0.720985	−0.360492	+	0.932762i	$0.617391\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	19.2926i	0.551217i
$36$	0	0
$37$	−50.4674	−1.36398	−0.681992	−	0.731360i	$-0.738886\pi$
$37$	−50.4674	−1.36398	−0.681992	+	0.731360i	$0.738886\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	34.6033i	0.843984i	0.906600	+	0.421992i	$0.138669\pi$
$41$	34.6033i	0.843984i	−0.906600	+	0.421992i	$0.861331\pi$
$42$	0	0
$43$	23.0000	0.534884	0.267442	−	0.963574i	$-0.413822\pi$
$43$	23.0000	0.534884	0.267442	+	0.963574i	$0.413822\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	38.3075i	0.815053i	0.913193	+	0.407527i	$0.133609\pi$
$47$	38.3075i	0.815053i	−0.913193	+	0.407527i	$0.866391\pi$
$48$	0	0
$49$	16.8832	0.344554
$50$	0	0
$51$	0	0
$52$	0	0
$53$	19.0149i	0.358771i	0.983779	+	0.179386i	$0.0574110\pi$
$53$	19.0149i	0.358771i	−0.983779	+	0.179386i	$0.942589\pi$
$54$	0	0
$55$	48.3505	0.879101
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 3.42643i	− 0.0580751i	−0.999578	−	0.0290375i	$-0.990756\pi$
$59$	− 3.42643i	− 0.0580751i	0.999578	−	0.0290375i	$-0.00924424\pi$
$60$	0	0
$61$	−46.3505	−0.759845	−0.379922	−	0.925018i	$-0.624049\pi$
$61$	−46.3505	−0.759845	−0.379922	+	0.925018i	$0.624049\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 14.5389i	− 0.223675i
$66$	0	0
$67$	−6.29894	−0.0940140	−0.0470070	−	0.998895i	$-0.514968\pi$
$67$	−6.29894	−0.0940140	−0.0470070	+	0.998895i	$0.514968\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 35.9306i	− 0.506065i	−0.967458	−	0.253033i	$-0.918572\pi$
$71$	− 35.9306i	− 0.506065i	0.967458	−	0.253033i	$-0.0814280\pi$
$72$	0	0
$73$	47.3505	0.648637	0.324319	−	0.945948i	$-0.394865\pi$
$73$	47.3505	0.648637	0.324319	+	0.945948i	$0.394865\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 165.114i	− 2.14434i
$78$	0	0
$79$	−84.5842	−1.07069	−0.535343	−	0.844635i	$-0.679818\pi$
$79$	−84.5842	−1.07069	−0.535343	+	0.844635i	$0.679818\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	38.3075i	0.461536i	0.973009	+	0.230768i	$0.0741239\pi$
$83$	38.3075i	0.461536i	−0.973009	+	0.230768i	$0.925876\pi$
$84$	0	0
$85$	42.7011	0.502365
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 143.723i	− 1.61486i	−0.589963	−	0.807430i	$-0.700858\pi$
$89$	− 143.723i	− 1.61486i	0.589963	−	0.807430i	$-0.299142\pi$
$90$	0	0
$91$	−49.6495	−0.545599
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 21.6695i	− 0.228100i
$96$	0	0
$97$	80.7663	0.832642	0.416321	−	0.909218i	$-0.363319\pi$
$97$	80.7663	0.832642	0.416321	+	0.909218i	$0.363319\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 122.331i	− 1.21120i	−0.795771	−	0.605598i	$-0.792934\pi$
$101$	− 122.331i	− 1.21120i	0.795771	−	0.605598i	$-0.207066\pi$
$102$	0	0
$103$	73.6495	0.715043	0.357522	−	0.933905i	$-0.383622\pi$
$103$	73.6495	0.715043	0.357522	+	0.933905i	$0.383622\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	72.9108i	0.681410i	0.940170	+	0.340705i	$0.110666\pi$
$107$	72.9108i	0.681410i	−0.940170	+	0.340705i	$0.889334\pi$
$108$	0	0
$109$	31.2989	0.287146	0.143573	−	0.989640i	$-0.454141\pi$
$109$	31.2989	0.287146	0.143573	+	0.989640i	$0.454141\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.7372i	0.165816i	0.996557	+	0.0829078i	$0.0264207\pi$
$113$	18.7372i	0.165816i	−0.996557	+	0.0829078i	$0.973579\pi$
$114$	0	0
$115$	79.7527	0.693501
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 145.822i	− 1.22539i
$120$	0	0
$121$	−292.804	−2.41987
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 105.415i	− 0.843320i
$126$	0	0
$127$	−126.103	−0.992939	−0.496469	−	0.868054i	$-0.665370\pi$
$127$	−126.103	−0.992939	−0.496469	+	0.868054i	$0.665370\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	162.460i	1.24015i	0.784542	+	0.620075i	$0.212898\pi$
$131$	162.460i	1.24015i	−0.784542	+	0.620075i	$0.787102\pi$
$132$	0	0
$133$	−74.0000	−0.556391
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 104.365i	− 0.761792i	−0.924618	−	0.380896i	$-0.875616\pi$
$137$	− 104.365i	− 0.761792i	0.924618	−	0.380896i	$-0.124384\pi$
$138$	0	0
$139$	61.2337	0.440530	0.220265	−	0.975440i	$-0.429308\pi$
$139$	61.2337	0.440530	0.220265	+	0.975440i	$0.429308\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	124.430i	0.870139i
$144$	0	0
$145$	−39.5463	−0.272733
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 148.199i	− 0.994621i	−0.867573	−	0.497311i	$-0.834321\pi$
$149$	− 148.199i	− 0.994621i	0.867573	−	0.497311i	$-0.165679\pi$
$150$	0	0
$151$	255.052	1.68908	0.844542	−	0.535490i	$-0.179873\pi$
$151$	255.052	1.68908	0.844542	+	0.535490i	$0.179873\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	53.1241i	0.342736i
$156$	0	0
$157$	292.454	1.86276	0.931381	−	0.364045i	$-0.118605\pi$
$157$	292.454	1.86276	0.931381	+	0.364045i	$0.118605\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 272.351i	− 1.69162i
$162$	0	0
$163$	93.5326	0.573820	0.286910	−	0.957958i	$-0.407372\pi$
$163$	93.5326	0.573820	0.286910	+	0.957958i	$0.407372\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 112.268i	− 0.672263i	−0.941815	−	0.336131i	$-0.890881\pi$
$167$	− 112.268i	− 0.672263i	0.941815	−	0.336131i	$-0.109119\pi$
$168$	0	0
$169$	−131.584	−0.778605
$170$	0	0
$171$	0	0
$172$	0	0
$173$	236.976i	1.36980i	0.728637	+	0.684900i	$0.240154\pi$
$173$	236.976i	1.36980i	−0.728637	+	0.684900i	$0.759846\pi$
$174$	0	0
$175$	−157.065	−0.897516
$176$	0	0
$177$	0	0
$178$	0	0
$179$	234.599i	1.31061i	0.755366	+	0.655304i	$0.227459\pi$
$179$	234.599i	1.31061i	−0.755366	+	0.655304i	$0.772541\pi$
$180$	0	0
$181$	221.636	1.22451	0.612254	−	0.790661i	$-0.290263\pi$
$181$	221.636	1.22451	0.612254	+	0.790661i	$0.290263\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	119.954i	0.648400i
$186$	0	0
$187$	−365.454	−1.95430
$188$	0	0
$189$	0	0
$190$	0	0
$191$	150.298i	0.786899i	0.919346	+	0.393449i	$0.128718\pi$
$191$	150.298i	0.786899i	−0.919346	+	0.393449i	$0.871282\pi$
$192$	0	0
$193$	−49.0000	−0.253886	−0.126943	−	0.991910i	$-0.540517\pi$
$193$	−49.0000	−0.253886	−0.126943	+	0.991910i	$0.540517\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	276.827i	1.40521i	0.711579	+	0.702606i	$0.247980\pi$
$197$	276.827i	1.40521i	−0.711579	+	0.702606i	$0.752020\pi$
$198$	0	0
$199$	−198.935	−0.999672	−0.499836	−	0.866120i	$-0.666606\pi$
$199$	−198.935	−0.999672	−0.499836	+	0.866120i	$0.666606\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	135.048i	0.665262i
$204$	0	0
$205$	82.2473	0.401207
$206$	0	0
$207$	0	0
$208$	0	0
$209$	185.456i	0.887352i
$210$	0	0
$211$	94.0137	0.445562	0.222781	−	0.974868i	$-0.428486\pi$
$211$	94.0137	0.445562	0.222781	+	0.974868i	$0.428486\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 54.6678i	− 0.254269i
$216$	0	0
$217$	181.416	0.836017
$218$	0	0
$219$	0	0
$220$	0	0
$221$	109.891i	0.497245i
$222$	0	0
$223$	−155.753	−0.698442	−0.349221	−	0.937040i	$-0.613554\pi$
$223$	−155.753	−0.698442	−0.349221	+	0.937040i	$0.613554\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	159.866i	0.704257i	0.935952	+	0.352129i	$0.114542\pi$
$227$	159.866i	0.704257i	−0.935952	+	0.352129i	$0.885458\pi$
$228$	0	0
$229$	−38.2473	−0.167019	−0.0835095	−	0.996507i	$-0.526613\pi$
$229$	−38.2473	−0.167019	−0.0835095	+	0.996507i	$0.526613\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	157.490i	0.675921i	0.941160	+	0.337960i	$0.109737\pi$
$233$	157.490i	0.675921i	−0.941160	+	0.337960i	$0.890263\pi$
$234$	0	0
$235$	91.0516	0.387454
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 72.1390i	− 0.301837i	−0.988546	−	0.150918i	$-0.951777\pi$
$239$	− 72.1390i	− 0.301837i	0.988546	−	0.150918i	$-0.0482231\pi$
$240$	0	0
$241$	226.739	0.940826	0.470413	−	0.882446i	$-0.344105\pi$
$241$	226.739	0.940826	0.470413	+	0.882446i	$0.344105\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 40.1289i	− 0.163791i
$246$	0	0
$247$	55.7663	0.225775
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 222.931i	− 0.888171i	−0.895985	−	0.444085i	$-0.853529\pi$
$251$	− 222.931i	− 0.888171i	0.895985	−	0.444085i	$-0.146471\pi$
$252$	0	0
$253$	−682.557	−2.69785
$254$	0	0
$255$	0	0
$256$	0	0
$257$	106.465i	0.414259i	0.978314	+	0.207130i	$0.0664122\pi$
$257$	106.465i	0.414259i	−0.978314	+	0.207130i	$0.933588\pi$
$258$	0	0
$259$	409.636	1.58161
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 179.375i	− 0.682036i	−0.940057	−	0.341018i	$-0.889228\pi$
$263$	− 179.375i	− 0.682036i	0.940057	−	0.341018i	$-0.110772\pi$
$264$	0	0
$265$	45.1957	0.170550
$266$	0	0
$267$	0	0
$268$	0	0
$269$	416.351i	1.54777i	0.633324	+	0.773887i	$0.281690\pi$
$269$	416.351i	1.54777i	−0.633324	+	0.773887i	$0.718310\pi$
$270$	0	0
$271$	396.907	1.46460	0.732302	−	0.680980i	$-0.238446\pi$
$271$	396.907	1.46460	0.732302	+	0.680980i	$0.238446\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	393.632i	1.43139i
$276$	0	0
$277$	−115.546	−0.417135	−0.208567	−	0.978008i	$-0.566880\pi$
$277$	−115.546	−0.417135	−0.208567	+	0.978008i	$0.566880\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 487.935i	− 1.73642i	−0.496195	−	0.868211i	$-0.665270\pi$
$281$	− 487.935i	− 1.73642i	0.496195	−	0.868211i	$-0.334730\pi$
$282$	0	0
$283$	−339.649	−1.20017	−0.600087	−	0.799934i	$-0.704868\pi$
$283$	−339.649	−1.20017	−0.600087	+	0.799934i	$0.704868\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 280.870i	− 0.978641i
$288$	0	0
$289$	−33.7527	−0.116791
$290$	0	0
$291$	0	0
$292$	0	0
$293$	141.346i	0.482408i	0.970474	+	0.241204i	$0.0775424\pi$
$293$	141.346i	0.482408i	−0.970474	+	0.241204i	$0.922458\pi$
$294$	0	0
$295$	−8.14415	−0.0276073
$296$	0	0
$297$	0	0
$298$	0	0
$299$	205.243i	0.686432i
$300$	0	0
$301$	−186.687	−0.620224
$302$	0	0
$303$	0	0
$304$	0	0
$305$	110.169i	0.361209i
$306$	0	0
$307$	−120.649	−0.392995	−0.196498	−	0.980504i	$-0.562957\pi$
$307$	−120.649	−0.392995	−0.196498	+	0.980504i	$0.562957\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 137.703i	− 0.442774i	−0.975186	−	0.221387i	$-0.928942\pi$
$311$	− 137.703i	− 0.442774i	0.975186	−	0.221387i	$-0.0710585\pi$
$312$	0	0
$313$	258.533	0.825983	0.412991	−	0.910735i	$-0.364484\pi$
$313$	258.533	0.825983	0.412991	+	0.910735i	$0.364484\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.2926i	0.0608600i	0.999537	+	0.0304300i	$0.00968766\pi$
$317$	19.2926i	0.0608600i	−0.999537	+	0.0304300i	$0.990312\pi$
$318$	0	0
$319$	338.454	1.06098
$320$	0	0
$321$	0	0
$322$	0	0
$323$	163.787i	0.507081i
$324$	0	0
$325$	118.364	0.364197
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 310.936i	− 0.945094i
$330$	0	0
$331$	−196.791	−0.594534	−0.297267	−	0.954794i	$-0.596075\pi$
$331$	−196.791	−0.594534	−0.297267	+	0.954794i	$0.596075\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	14.9717i	0.0446916i
$336$	0	0
$337$	317.440	0.941959	0.470979	−	0.882144i	$-0.343901\pi$
$337$	317.440	0.941959	0.470979	+	0.882144i	$0.343901\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 454.659i	− 1.33331i
$342$	0	0
$343$	260.687	0.760022
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 620.545i	− 1.78831i	−0.447754	−	0.894157i	$-0.647776\pi$
$347$	− 620.545i	− 1.78831i	0.447754	−	0.894157i	$-0.352224\pi$
$348$	0	0
$349$	−379.024	−1.08603	−0.543015	−	0.839723i	$-0.682717\pi$
$349$	−379.024	−1.08603	−0.543015	+	0.839723i	$0.682717\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	246.544i	0.698426i	0.937043	+	0.349213i	$0.113551\pi$
$353$	246.544i	0.698426i	−0.937043	+	0.349213i	$0.886449\pi$
$354$	0	0
$355$	−85.4021	−0.240569
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 572.791i	− 1.59552i	−0.602976	−	0.797759i	$-0.706019\pi$
$359$	− 572.791i	− 1.59552i	0.602976	−	0.797759i	$-0.293981\pi$
$360$	0	0
$361$	−277.883	−0.769759
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 112.546i	− 0.308344i
$366$	0	0
$367$	−187.856	−0.511869	−0.255934	−	0.966694i	$-0.582383\pi$
$367$	−187.856	−0.511869	−0.255934	+	0.966694i	$0.582383\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 154.341i	− 0.416013i
$372$	0	0
$373$	150.117	0.402458	0.201229	−	0.979544i	$-0.435506\pi$
$373$	150.117	0.402458	0.201229	+	0.979544i	$0.435506\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 101.772i	− 0.269953i
$378$	0	0
$379$	−26.6222	−0.0702432	−0.0351216	−	0.999383i	$-0.511182\pi$
$379$	−26.6222	−0.0702432	−0.0351216	+	0.999383i	$0.511182\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 513.802i	− 1.34152i	−0.741674	−	0.670760i	$-0.765968\pi$
$383$	− 513.802i	− 1.34152i	0.741674	−	0.670760i	$-0.234032\pi$
$384$	0	0
$385$	−392.454	−1.01936
$386$	0	0
$387$	0	0
$388$	0	0
$389$	25.5900i	0.0657841i	0.999459	+	0.0328921i	$0.0104718\pi$
$389$	25.5900i	0.0657841i	−0.999459	+	0.0328921i	$0.989528\pi$
$390$	0	0
$391$	−602.804	−1.54170
$392$	0	0
$393$	0	0
$394$	0	0
$395$	201.045i	0.508975i
$396$	0	0
$397$	388.804	0.979356	0.489678	−	0.871903i	$-0.337114\pi$
$397$	388.804	0.979356	0.489678	+	0.871903i	$0.337114\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 39.3571i	− 0.0981473i	−0.998795	−	0.0490736i	$-0.984373\pi$
$401$	− 39.3571i	− 0.0981473i	0.998795	−	0.0490736i	$-0.0156269\pi$
$402$	0	0
$403$	−136.715	−0.339242
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1026.62i	− 2.52240i
$408$	0	0
$409$	173.440	0.424059	0.212029	−	0.977263i	$-0.431993\pi$
$409$	173.440	0.424059	0.212029	+	0.977263i	$0.431993\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	27.8118i	0.0673409i
$414$	0	0
$415$	91.0516	0.219401
$416$	0	0
$417$	0	0
$418$	0	0
$419$	132.826i	0.317008i	0.987358	+	0.158504i	$0.0506671\pi$
$419$	132.826i	0.317008i	−0.987358	+	0.158504i	$0.949333\pi$
$420$	0	0
$421$	634.894	1.50806	0.754031	−	0.656839i	$-0.228107\pi$
$421$	634.894	1.50806	0.754031	+	0.656839i	$0.228107\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	347.638i	0.817973i
$426$	0	0
$427$	376.220	0.881077
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 602.424i	− 1.39774i	−0.715251	−	0.698868i	$-0.753687\pi$
$431$	− 602.424i	− 1.39774i	0.715251	−	0.698868i	$-0.246313\pi$
$432$	0	0
$433$	266.155	0.614676	0.307338	−	0.951600i	$-0.400562\pi$
$433$	266.155	0.614676	0.307338	+	0.951600i	$0.400562\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	305.905i	0.700010i
$438$	0	0
$439$	500.660	1.14046	0.570228	−	0.821487i	$-0.306855\pi$
$439$	500.660	1.14046	0.570228	+	0.821487i	$0.306855\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	301.490i	0.680564i	0.940323	+	0.340282i	$0.110523\pi$
$443$	301.490i	0.680564i	−0.940323	+	0.340282i	$0.889477\pi$
$444$	0	0
$445$	−341.609	−0.767660
$446$	0	0
$447$	0	0
$448$	0	0
$449$	565.321i	1.25907i	0.776973	+	0.629534i	$0.216754\pi$
$449$	565.321i	1.25907i	−0.776973	+	0.629534i	$0.783246\pi$
$450$	0	0
$451$	−703.907	−1.56077
$452$	0	0
$453$	0	0
$454$	0	0
$455$	118.010i	0.259362i
$456$	0	0
$457$	52.2989	0.114440	0.0572198	−	0.998362i	$-0.481776\pi$
$457$	52.2989	0.114440	0.0572198	+	0.998362i	$0.481776\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 192.093i	− 0.416687i	−0.978056	−	0.208344i	$-0.933193\pi$
$461$	− 192.093i	− 0.416687i	0.978056	−	0.208344i	$-0.0668073\pi$
$462$	0	0
$463$	566.220	1.22294	0.611469	−	0.791269i	$-0.290579\pi$
$463$	566.220	1.22294	0.611469	+	0.791269i	$0.290579\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	174.405i	0.373459i	0.982411	+	0.186729i	$0.0597888\pi$
$467$	174.405i	0.373459i	−0.982411	+	0.186729i	$0.940211\pi$
$468$	0	0
$469$	51.1275	0.109014
$470$	0	0
$471$	0	0
$472$	0	0
$473$	467.870i	0.989155i
$474$	0	0
$475$	176.416	0.371402
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 547.078i	− 1.14213i	−0.820906	−	0.571063i	$-0.806531\pi$
$479$	− 547.078i	− 1.14213i	0.820906	−	0.571063i	$-0.193469\pi$
$480$	0	0
$481$	−308.701	−0.641790
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 191.970i	− 0.395815i
$486$	0	0
$487$	−769.945	−1.58100	−0.790498	−	0.612464i	$-0.790178\pi$
$487$	−769.945	−1.58100	−0.790498	+	0.612464i	$0.790178\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 178.881i	− 0.364320i	−0.983269	−	0.182160i	$-0.941691\pi$
$491$	− 178.881i	− 0.364320i	0.983269	−	0.182160i	$-0.0583090\pi$
$492$	0	0
$493$	298.907	0.606303
$494$	0	0
$495$	0	0
$496$	0	0
$497$	291.643i	0.586808i
$498$	0	0
$499$	−385.310	−0.772163	−0.386082	−	0.922465i	$-0.626172\pi$
$499$	−385.310	−0.772163	−0.386082	+	0.922465i	$0.626172\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	67.6630i	0.134519i	0.997736	+	0.0672594i	$0.0214255\pi$
$503$	67.6630i	0.134519i	−0.997736	+	0.0672594i	$0.978574\pi$
$504$	0	0
$505$	−290.763	−0.575769
$506$	0	0
$507$	0	0
$508$	0	0
$509$	604.246i	1.18712i	0.804789	+	0.593562i	$0.202279\pi$
$509$	604.246i	1.18712i	−0.804789	+	0.593562i	$0.797721\pi$
$510$	0	0
$511$	−384.337	−0.752127
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 175.055i	− 0.339912i
$516$	0	0
$517$	−779.258	−1.50727
$518$	0	0
$519$	0	0
$520$	0	0
$521$	273.678i	0.525294i	0.964892	+	0.262647i	$0.0845954\pi$
$521$	273.678i	0.525294i	−0.964892	+	0.262647i	$0.915405\pi$
$522$	0	0
$523$	687.402	1.31434	0.657172	−	0.753740i	$-0.271752\pi$
$523$	687.402	1.31434	0.657172	+	0.753740i	$0.271752\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 401.534i	− 0.761925i
$528$	0	0
$529$	−596.856	−1.12827
$530$	0	0
$531$	0	0
$532$	0	0
$533$	211.663i	0.397117i
$534$	0	0
$535$	173.299	0.323923
$536$	0	0
$537$	0	0
$538$	0	0
$539$	343.440i	0.637180i
$540$	0	0
$541$	664.543	1.22836	0.614180	−	0.789166i	$-0.289487\pi$
$541$	664.543	1.22836	0.614180	+	0.789166i	$0.289487\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 74.3932i	− 0.136501i
$546$	0	0
$547$	519.206	0.949189	0.474594	−	0.880205i	$-0.342595\pi$
$547$	519.206	0.949189	0.474594	+	0.880205i	$0.342595\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 151.686i	− 0.275293i
$552$	0	0
$553$	686.557	1.24151
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 422.648i	− 0.758794i	−0.925234	−	0.379397i	$-0.876131\pi$
$557$	− 422.648i	− 0.758794i	0.925234	−	0.379397i	$-0.123869\pi$
$558$	0	0
$559$	140.687	0.251677
$560$	0	0
$561$	0	0
$562$	0	0
$563$	922.374i	1.63832i	0.573566	+	0.819159i	$0.305560\pi$
$563$	922.374i	1.63832i	−0.573566	+	0.819159i	$0.694440\pi$
$564$	0	0
$565$	44.5356	0.0788241
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1055.91i	1.85573i	0.372916	+	0.927865i	$0.378358\pi$
$569$	1055.91i	1.85573i	−0.372916	+	0.927865i	$0.621642\pi$
$570$	0	0
$571$	803.049	1.40639	0.703195	−	0.710997i	$-0.251756\pi$
$571$	803.049	1.40639	0.703195	+	0.710997i	$0.251756\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	649.283i	1.12919i
$576$	0	0
$577$	−96.6495	−0.167503	−0.0837517	−	0.996487i	$-0.526690\pi$
$577$	−96.6495	−0.167503	−0.0837517	+	0.996487i	$0.526690\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 310.936i	− 0.535174i
$582$	0	0
$583$	−386.804	−0.663472
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1005.16i	1.71237i	0.516666	+	0.856187i	$0.327173\pi$
$587$	1005.16i	1.71237i	−0.516666	+	0.856187i	$0.672827\pi$
$588$	0	0
$589$	−203.766	−0.345953
$590$	0	0
$591$	0	0
$592$	0	0
$593$	752.444i	1.26888i	0.772973	+	0.634439i	$0.218769\pi$
$593$	752.444i	1.26888i	−0.772973	+	0.634439i	$0.781231\pi$
$594$	0	0
$595$	−346.598	−0.582517
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 27.8118i	− 0.0464304i	−0.999730	−	0.0232152i	$-0.992610\pi$
$599$	− 27.8118i	− 0.0464304i	0.999730	−	0.0232152i	$-0.00739029\pi$
$600$	0	0
$601$	−950.712	−1.58188	−0.790942	−	0.611892i	$-0.790409\pi$
$601$	−950.712	−1.58188	−0.790942	+	0.611892i	$0.790409\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	695.955i	1.15034i
$606$	0	0
$607$	−322.612	−0.531485	−0.265743	−	0.964044i	$-0.585617\pi$
$607$	−322.612	−0.531485	−0.265743	+	0.964044i	$0.585617\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	234.321i	0.383504i
$612$	0	0
$613$	138.206	0.225459	0.112730	−	0.993626i	$-0.464041\pi$
$613$	138.206	0.225459	0.112730	+	0.993626i	$0.464041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 787.603i	− 1.27650i	−0.769827	−	0.638252i	$-0.779658\pi$
$617$	− 787.603i	− 1.27650i	0.769827	−	0.638252i	$-0.220342\pi$
$618$	0	0
$619$	−243.495	−0.393368	−0.196684	−	0.980467i	$-0.563017\pi$
$619$	−243.495	−0.393368	−0.196684	+	0.980467i	$0.563017\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1166.57i	1.87251i
$624$	0	0
$625$	233.206	0.373130
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 906.662i	− 1.44143i
$630$	0	0
$631$	111.924	0.177376	0.0886879	−	0.996059i	$-0.471733\pi$
$631$	111.924	0.177376	0.0886879	+	0.996059i	$0.471733\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	299.730i	0.472015i
$636$	0	0
$637$	103.272	0.162122
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 729.880i	− 1.13866i	−0.822110	−	0.569329i	$-0.807203\pi$
$641$	− 729.880i	− 1.13866i	0.822110	−	0.569329i	$-0.192797\pi$
$642$	0	0
$643$	−577.000	−0.897356	−0.448678	−	0.893693i	$-0.648105\pi$
$643$	−577.000	−0.897356	−0.448678	+	0.893693i	$0.648105\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	129.029i	0.199426i	0.995016	+	0.0997130i	$0.0317925\pi$
$647$	129.029i	0.199426i	−0.995016	+	0.0997130i	$0.968208\pi$
$648$	0	0
$649$	69.7011	0.107398
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1185.43i	1.81536i	0.419658	+	0.907682i	$0.362150\pi$
$653$	1185.43i	1.81536i	−0.419658	+	0.907682i	$0.637850\pi$
$654$	0	0
$655$	386.144	0.589533
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1094.43i	− 1.66075i	−0.557205	−	0.830375i	$-0.688126\pi$
$659$	− 1094.43i	− 1.66075i	0.557205	−	0.830375i	$-0.311874\pi$
$660$	0	0
$661$	1209.75	1.83019	0.915093	−	0.403243i	$-0.132117\pi$
$661$	1209.75	1.83019	0.915093	+	0.403243i	$0.132117\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	175.888i	0.264493i
$666$	0	0
$667$	558.269	0.836984
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 942.871i	− 1.40517i
$672$	0	0
$673$	1017.23	1.51149	0.755743	−	0.654868i	$-0.227276\pi$
$673$	1017.23	1.51149	0.755743	+	0.654868i	$0.227276\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 796.616i	− 1.17669i	−0.808612	−	0.588343i	$-0.799781\pi$
$677$	− 796.616i	− 1.17669i	0.808612	−	0.588343i	$-0.200219\pi$
$678$	0	0
$679$	−655.568	−0.965490
$680$	0	0
$681$	0	0
$682$	0	0
$683$	400.485i	0.586361i	0.956057	+	0.293181i	$0.0947138\pi$
$683$	400.485i	0.586361i	−0.956057	+	0.293181i	$0.905286\pi$
$684$	0	0
$685$	−248.062	−0.362135
$686$	0	0
$687$	0	0
$688$	0	0
$689$	116.311i	0.168811i
$690$	0	0
$691$	−432.845	−0.626404	−0.313202	−	0.949687i	$-0.601402\pi$
$691$	−432.845	−0.626404	−0.313202	+	0.949687i	$0.601402\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 145.544i	− 0.209416i
$696$	0	0
$697$	−621.660	−0.891908
$698$	0	0
$699$	0	0
$700$	0	0
$701$	65.4412i	0.0933541i	0.998910	+	0.0466770i	$0.0148632\pi$
$701$	65.4412i	0.0933541i	−0.998910	+	0.0466770i	$0.985137\pi$
$702$	0	0
$703$	−460.103	−0.654485
$704$	0	0
$705$	0	0
$706$	0	0
$707$	992.940i	1.40444i
$708$	0	0
$709$	200.921	0.283387	0.141693	−	0.989911i	$-0.454745\pi$
$709$	200.921	0.283387	0.141693	+	0.989911i	$0.454745\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 749.945i	− 1.05182i
$714$	0	0
$715$	295.753	0.413640
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1062.98i	− 1.47841i	−0.673478	−	0.739207i	$-0.735200\pi$
$719$	− 1062.98i	− 1.47841i	0.673478	−	0.739207i	$-0.264800\pi$
$720$	0	0
$721$	−597.801	−0.829128
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 321.955i	− 0.444075i
$726$	0	0
$727$	991.258	1.36349	0.681746	−	0.731589i	$-0.261221\pi$
$727$	991.258	1.36349	0.681746	+	0.731589i	$0.261221\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	413.202i	0.565256i
$732$	0	0
$733$	−1180.27	−1.61019	−0.805095	−	0.593146i	$-0.797885\pi$
$733$	−1180.27	−1.61019	−0.805095	+	0.593146i	$0.797885\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 128.134i	− 0.173859i
$738$	0	0
$739$	599.351	0.811029	0.405515	−	0.914089i	$-0.367092\pi$
$739$	599.351	0.811029	0.405515	+	0.914089i	$0.367092\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	331.495i	0.446157i	0.974800	+	0.223078i	$0.0716106\pi$
$743$	331.495i	0.446157i	−0.974800	+	0.223078i	$0.928389\pi$
$744$	0	0
$745$	−352.247	−0.472815
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 591.806i	− 0.790128i
$750$	0	0
$751$	152.090	0.202516	0.101258	−	0.994860i	$-0.467713\pi$
$751$	152.090	0.202516	0.101258	+	0.994860i	$0.467713\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 606.222i	− 0.802943i
$756$	0	0
$757$	−1179.61	−1.55827	−0.779134	−	0.626858i	$-0.784341\pi$
$757$	−1179.61	−1.55827	−0.779134	+	0.626858i	$0.784341\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1342.43i	− 1.76403i	−0.471219	−	0.882016i	$-0.656186\pi$
$761$	− 1342.43i	− 1.76403i	0.471219	−	0.882016i	$-0.343814\pi$
$762$	0	0
$763$	−254.049	−0.332960
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 20.9589i	− 0.0273259i
$768$	0	0
$769$	1097.02	1.42656	0.713280	−	0.700879i	$-0.247209\pi$
$769$	1097.02	1.42656	0.713280	+	0.700879i	$0.247209\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1181.39i	1.52832i	0.645028	+	0.764159i	$0.276846\pi$
$773$	1181.39i	1.52832i	−0.645028	+	0.764159i	$0.723154\pi$
$774$	0	0
$775$	−432.495	−0.558058
$776$	0	0
$777$	0	0
$778$	0	0
$779$	315.473i	0.404972i
$780$	0	0
$781$	730.907	0.935861
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 695.122i	− 0.885506i
$786$	0	0
$787$	−36.0622	−0.0458224	−0.0229112	−	0.999738i	$-0.507294\pi$
$787$	−36.0622	−0.0458224	−0.0229112	+	0.999738i	$0.507294\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 152.087i	− 0.192271i
$792$	0	0
$793$	−283.519	−0.357527
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 133.505i	− 0.167509i	−0.996486	−	0.0837544i	$-0.973309\pi$
$797$	− 133.505i	− 0.167509i	0.996486	−	0.0837544i	$-0.0266911\pi$
$798$	0	0
$799$	−688.206	−0.861335
$800$	0	0
$801$	0	0
$802$	0	0
$803$	963.213i	1.19952i
$804$	0	0
$805$	−647.340	−0.804149
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1053.66i	− 1.30242i	−0.758898	−	0.651209i	$-0.774262\pi$
$809$	− 1053.66i	− 1.30242i	0.758898	−	0.651209i	$-0.225738\pi$
$810$	0	0
$811$	434.464	0.535714	0.267857	−	0.963459i	$-0.413684\pi$
$811$	434.464	0.535714	0.267857	+	0.963459i	$0.413684\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 222.314i	− 0.272778i
$816$	0	0
$817$	209.687	0.256655
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 291.488i	− 0.355041i	−0.984117	−	0.177520i	$-0.943192\pi$
$821$	− 291.488i	− 0.355041i	0.984117	−	0.177520i	$-0.0568075\pi$
$822$	0	0
$823$	336.818	0.409256	0.204628	−	0.978840i	$-0.434401\pi$
$823$	336.818	0.409256	0.204628	+	0.978840i	$0.434401\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1029.27i	− 1.24458i	−0.782785	−	0.622292i	$-0.786202\pi$
$827$	− 1029.27i	− 1.24458i	0.782785	−	0.622292i	$-0.213798\pi$
$828$	0	0
$829$	790.674	0.953768	0.476884	−	0.878966i	$-0.341766\pi$
$829$	790.674	0.953768	0.476884	+	0.878966i	$0.341766\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	303.311i	0.364119i
$834$	0	0
$835$	−266.845	−0.319575
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 756.365i	− 0.901507i	−0.892648	−	0.450754i	$-0.851155\pi$
$839$	− 756.365i	− 0.901507i	0.892648	−	0.450754i	$-0.148845\pi$
$840$	0	0
$841$	564.176	0.670840
$842$	0	0
$843$	0	0
$844$	0	0
$845$	312.757i	0.370127i
$846$	0	0
$847$	2376.65	2.80596
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1693.37i	− 1.98986i
$852$	0	0
$853$	−1198.35	−1.40487	−0.702433	−	0.711750i	$-0.747903\pi$
$853$	−1198.35	−1.40487	−0.702433	+	0.711750i	$0.747903\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 699.320i	− 0.816009i	−0.912980	−	0.408005i	$-0.866225\pi$
$857$	− 699.320i	− 0.816009i	0.912980	−	0.408005i	$-0.133775\pi$
$858$	0	0
$859$	−557.467	−0.648972	−0.324486	−	0.945890i	$-0.605191\pi$
$859$	−557.467	−0.648972	−0.324486	+	0.945890i	$0.605191\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	99.3954i	0.115174i	0.998340	+	0.0575871i	$0.0183407\pi$
$863$	99.3954i	0.115174i	−0.998340	+	0.0575871i	$0.981659\pi$
$864$	0	0
$865$	563.258	0.651165
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1720.63i	− 1.98001i
$870$	0	0
$871$	−38.5296	−0.0442361
$872$	0	0
$873$	0	0
$874$	0	0
$875$	855.638i	0.977871i
$876$	0	0
$877$	802.584	0.915147	0.457574	−	0.889172i	$-0.348719\pi$
$877$	802.584	0.915147	0.457574	+	0.889172i	$0.348719\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 524.266i	− 0.595080i	−0.954709	−	0.297540i	$-0.903834\pi$
$881$	− 524.266i	− 0.595080i	0.954709	−	0.297540i	$-0.0961662\pi$
$882$	0	0
$883$	−993.894	−1.12559	−0.562794	−	0.826597i	$-0.690273\pi$
$883$	−993.894	−1.12559	−0.562794	+	0.826597i	$0.690273\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	712.038i	0.802748i	0.915914	+	0.401374i	$0.131467\pi$
$887$	712.038i	0.802748i	−0.915914	+	0.401374i	$0.868533\pi$
$888$	0	0
$889$	1023.56	1.15136
$890$	0	0
$891$	0	0
$892$	0	0
$893$	349.243i	0.391090i
$894$	0	0
$895$	557.609	0.623026
$896$	0	0
$897$	0	0
$898$	0	0
$899$	371.869i	0.413647i
$900$	0	0
$901$	−341.609	−0.379144
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 526.798i	− 0.582097i
$906$	0	0
$907$	−748.945	−0.825739	−0.412870	−	0.910790i	$-0.635473\pi$
$907$	−748.945	−0.825739	−0.412870	+	0.910790i	$0.635473\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.47600i	0.00491328i	0.999997	+	0.00245664i	$0.000781974\pi$
$911$	4.47600i	0.00491328i	−0.999997	+	0.00245664i	$0.999218\pi$
$912$	0	0
$913$	−779.258	−0.853514
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1318.66i	− 1.43802i
$918$	0	0
$919$	−1592.91	−1.73331	−0.866653	−	0.498912i	$-0.833733\pi$
$919$	−1592.91	−1.73331	−0.866653	+	0.498912i	$0.833733\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 219.782i	− 0.238117i
$924$	0	0
$925$	−976.571	−1.05575
$926$	0	0
$927$	0	0
$928$	0	0
$929$	890.024i	0.958046i	0.877802	+	0.479023i	$0.159009\pi$
$929$	890.024i	0.958046i	−0.877802	+	0.479023i	$0.840991\pi$
$930$	0	0
$931$	153.921	0.165329
$932$	0	0
$933$	0	0
$934$	0	0
$935$	868.633i	0.929019i
$936$	0	0
$937$	443.554	0.473377	0.236688	−	0.971586i	$-0.423938\pi$
$937$	443.554	0.473377	0.236688	+	0.971586i	$0.423938\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 80.5356i	− 0.0855851i	−0.999084	−	0.0427925i	$-0.986375\pi$
$941$	− 80.5356i	− 0.0855851i	0.999084	−	0.0427925i	$-0.0136254\pi$
$942$	0	0
$943$	−1161.07	−1.23125
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 283.896i	− 0.299785i	−0.988702	−	0.149892i	$-0.952107\pi$
$947$	− 283.896i	− 0.299785i	0.988702	−	0.149892i	$-0.0478927\pi$
$948$	0	0
$949$	289.636	0.305201
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1123.17i	− 1.17857i	−0.807927	−	0.589283i	$-0.799410\pi$
$953$	− 1123.17i	− 1.17857i	0.807927	−	0.589283i	$-0.200590\pi$
$954$	0	0
$955$	357.237	0.374070
$956$	0	0
$957$	0	0
$958$	0	0
$959$	847.118i	0.883335i
$960$	0	0
$961$	−461.454	−0.480181
$962$	0	0
$963$	0	0
$964$	0	0
$965$	116.466i	0.120690i
$966$	0	0
$967$	−1399.07	−1.44682	−0.723409	−	0.690420i	$-0.757426\pi$
$967$	−1399.07	−1.44682	−0.723409	+	0.690420i	$0.757426\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1705.41i	1.75634i	0.478345	+	0.878172i	$0.341237\pi$
$971$	1705.41i	1.75634i	−0.478345	+	0.878172i	$0.658763\pi$
$972$	0	0
$973$	−497.024	−0.510816
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1161.05i	1.18838i	0.804324	+	0.594190i	$0.202527\pi$
$977$	1161.05i	1.18838i	−0.804324	+	0.594190i	$0.797473\pi$
$978$	0	0
$979$	2923.63	2.98634
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1181.79i	1.20223i	0.799163	+	0.601114i	$0.205276\pi$
$983$	1181.79i	1.20223i	−0.799163	+	0.601114i	$0.794724\pi$
$984$	0	0
$985$	657.979	0.667999
$986$	0	0
$987$	0	0
$988$	0	0
$989$	771.737i	0.780320i
$990$	0	0
$991$	−969.527	−0.978332	−0.489166	−	0.872191i	$-0.662699\pi$
$991$	−969.527	−0.978332	−0.489166	+	0.872191i	$0.662699\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	472.840i	0.475216i
$996$	0	0
$997$	780.247	0.782595	0.391298	−	0.920264i	$-0.372026\pi$
$997$	780.247	0.782595	0.391298	+	0.920264i	$0.372026\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.3.c.b.161.2		4
3.2	odd	2	inner	324.3.c.b.161.3		4
4.3	odd	2		1296.3.e.e.161.2		4
9.2	odd	6		36.3.g.a.5.2	✓	4
9.4	even	3		36.3.g.a.29.2	yes	4
9.5	odd	6		108.3.g.a.89.2		4
9.7	even	3		108.3.g.a.17.2		4
12.11	even	2		1296.3.e.e.161.3		4
36.7	odd	6		432.3.q.b.17.2		4
36.11	even	6		144.3.q.b.113.1		4
36.23	even	6		432.3.q.b.305.2		4
36.31	odd	6		144.3.q.b.65.1		4
45.2	even	12		900.3.u.a.149.2		8
45.4	even	6		900.3.p.a.101.1		4
45.7	odd	12		2700.3.u.b.449.2		8
45.13	odd	12		900.3.u.a.749.2		8
45.14	odd	6		2700.3.p.b.1601.1		4
45.22	odd	12		900.3.u.a.749.3		8
45.23	even	12		2700.3.u.b.2249.2		8
45.29	odd	6		900.3.p.a.401.1		4
45.32	even	12		2700.3.u.b.2249.3		8
45.34	even	6		2700.3.p.b.2501.1		4
45.38	even	12		900.3.u.a.149.3		8
45.43	odd	12		2700.3.u.b.449.3		8
72.5	odd	6		1728.3.q.g.1601.1		4
72.11	even	6		576.3.q.g.257.2		4
72.13	even	6		576.3.q.d.65.1		4
72.29	odd	6		576.3.q.d.257.1		4
72.43	odd	6		1728.3.q.h.449.1		4
72.59	even	6		1728.3.q.h.1601.1		4
72.61	even	6		1728.3.q.g.449.1		4
72.67	odd	6		576.3.q.g.65.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.3.g.a.5.2	✓	4	9.2	odd	6
36.3.g.a.29.2	yes	4	9.4	even	3
108.3.g.a.17.2		4	9.7	even	3
108.3.g.a.89.2		4	9.5	odd	6
144.3.q.b.65.1		4	36.31	odd	6
144.3.q.b.113.1		4	36.11	even	6
324.3.c.b.161.2		4	1.1	even	1	trivial
324.3.c.b.161.3		4	3.2	odd	2	inner
432.3.q.b.17.2		4	36.7	odd	6
432.3.q.b.305.2		4	36.23	even	6
576.3.q.d.65.1		4	72.13	even	6
576.3.q.d.257.1		4	72.29	odd	6
576.3.q.g.65.2		4	72.67	odd	6
576.3.q.g.257.2		4	72.11	even	6
900.3.p.a.101.1		4	45.4	even	6
900.3.p.a.401.1		4	45.29	odd	6
900.3.u.a.149.2		8	45.2	even	12
900.3.u.a.149.3		8	45.38	even	12
900.3.u.a.749.2		8	45.13	odd	12
900.3.u.a.749.3		8	45.22	odd	12
1296.3.e.e.161.2		4	4.3	odd	2
1296.3.e.e.161.3		4	12.11	even	2
1728.3.q.g.449.1		4	72.61	even	6
1728.3.q.g.1601.1		4	72.5	odd	6
1728.3.q.h.449.1		4	72.43	odd	6
1728.3.q.h.1601.1		4	72.59	even	6
2700.3.p.b.1601.1		4	45.14	odd	6
2700.3.p.b.2501.1		4	45.34	even	6
2700.3.u.b.449.2		8	45.7	odd	12
2700.3.u.b.449.3		8	45.43	odd	12
2700.3.u.b.2249.2		8	45.23	even	12
2700.3.u.b.2249.3		8	45.32	even	12

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	324.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.37686i q^{5} -8.11684 q^{7} +O(q^{10})\)	\(q-2.37686i q^{5} -8.11684 q^{7} +20.3422i q^{11} +6.11684 q^{13} +17.9653i q^{17} +9.11684 q^{19} +33.5538i q^{23} +19.3505 q^{25} -16.6380i q^{29} -22.3505 q^{31} +19.2926i q^{35} -50.4674 q^{37} +34.6033i q^{41} +23.0000 q^{43} +38.3075i q^{47} +16.8832 q^{49} +19.0149i q^{53} +48.3505 q^{55} -3.42643i q^{59} -46.3505 q^{61} -14.5389i q^{65} -6.29894 q^{67} -35.9306i q^{71} +47.3505 q^{73} -165.114i q^{77} -84.5842 q^{79} +38.3075i q^{83} +42.7011 q^{85} -143.723i q^{89} -49.6495 q^{91} -21.6695i q^{95} +80.7663 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{7}+O(q^{10}) \) 4 * q + 2 * q^7	\( 4 q + 2 q^{7} - 10 q^{13} + 2 q^{19} - 26 q^{25} + 14 q^{31} - 64 q^{37} + 92 q^{43} + 102 q^{49} + 90 q^{55} - 82 q^{61} - 232 q^{67} + 86 q^{73} - 166 q^{79} - 36 q^{85} - 302 q^{91} + 392 q^{97}+O(q^{100}) \) 4 * q + 2 * q^7 - 10 * q^13 + 2 * q^19 - 26 * q^25 + 14 * q^31 - 64 * q^37 + 92 * q^43 + 102 * q^49 + 90 * q^55 - 82 * q^61 - 232 * q^67 + 86 * q^73 - 166 * q^79 - 36 * q^85 - 302 * q^91 + 392 * q^97

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