LMFDB - Embedded newform 1296.3.e.d.161.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,3,Mod(161,1296)]

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, 1]))

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

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

chi := DirichletCharacter("1296.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.e (of order $2$, degree $1$, not 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:	$35.3134422611$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 162)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.2
Root			$-0.517638i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.3.e.d.161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.55291i q^{5} -12.3923 q^{7} +O(q^{10})$	$q-1.55291i q^{5} -12.3923 q^{7} +14.6969i q^{11} -10.8038 q^{13} -28.9778i q^{17} -3.60770 q^{19} -14.6969i q^{23} +22.5885 q^{25} +28.1456i q^{29} -8.00000 q^{31} +19.2442i q^{35} +22.5692 q^{37} +25.1512i q^{41} +53.1769 q^{43} +16.9706i q^{47} +104.569 q^{49} +84.5482i q^{53} +22.8231 q^{55} -91.0645i q^{59} -13.0000 q^{61} +16.7774i q^{65} +41.1769 q^{67} -16.3613i q^{71} +71.5885 q^{73} -182.129i q^{77} +46.7461 q^{79} -15.3062i q^{83} -45.0000 q^{85} -78.9756i q^{89} +133.885 q^{91} +5.60244i q^{95} +91.1384 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7}+O(q^{10}) $ 4 * q - 8 * q^7	$ 4 q - 8 q^{7} - 64 q^{13} - 56 q^{19} + 28 q^{25} - 32 q^{31} - 76 q^{37} + 88 q^{43} + 252 q^{49} + 216 q^{55} - 52 q^{61} + 40 q^{67} + 224 q^{73} - 104 q^{79} - 180 q^{85} - 88 q^{91} + 32 q^{97}+O(q^{100}) $ 4 * q - 8 * q^7 - 64 * q^13 - 56 * q^19 + 28 * q^25 - 32 * q^31 - 76 * q^37 + 88 * q^43 + 252 * q^49 + 216 * q^55 - 52 * q^61 + 40 * q^67 + 224 * q^73 - 104 * q^79 - 180 * q^85 - 88 * q^91 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	− 1.55291i	− 0.310583i	−0.987869	−	0.155291i	$-0.950368\pi$
$5$	− 1.55291i	− 0.310583i	0.987869	−	0.155291i	$-0.0496317\pi$
$6$	0	0
$7$	−12.3923	−1.77033	−0.885165	−	0.465278i	$-0.845954\pi$
$7$	−12.3923	−1.77033	−0.885165	+	0.465278i	$0.845954\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	14.6969i	1.33609i	0.744123	+	0.668043i	$0.232868\pi$
$11$	14.6969i	1.33609i	−0.744123	+	0.668043i	$0.767132\pi$
$12$	0	0
$13$	−10.8038	−0.831065	−0.415533	−	0.909578i	$-0.636405\pi$
$13$	−10.8038	−0.831065	−0.415533	+	0.909578i	$0.636405\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 28.9778i	− 1.70457i	−0.523074	−	0.852287i	$-0.675215\pi$
$17$	− 28.9778i	− 1.70457i	0.523074	−	0.852287i	$-0.324785\pi$
$18$	0	0
$19$	−3.60770	−0.189879	−0.0949393	−	0.995483i	$-0.530266\pi$
$19$	−3.60770	−0.189879	−0.0949393	+	0.995483i	$0.530266\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 14.6969i	− 0.638997i	−0.947587	−	0.319499i	$-0.896486\pi$
$23$	− 14.6969i	− 0.638997i	0.947587	−	0.319499i	$-0.103514\pi$
$24$	0	0
$25$	22.5885	0.903538
$26$	0	0
$27$	0	0
$28$	0	0
$29$	28.1456i	0.970537i	0.874365	+	0.485268i	$0.161278\pi$
$29$	28.1456i	0.970537i	−0.874365	+	0.485268i	$0.838722\pi$
$30$	0	0
$31$	−8.00000	−0.258065	−0.129032	−	0.991640i	$-0.541187\pi$
$31$	−8.00000	−0.258065	−0.129032	+	0.991640i	$0.541187\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	19.2442i	0.549834i
$36$	0	0
$37$	22.5692	0.609979	0.304989	−	0.952356i	$-0.401347\pi$
$37$	22.5692	0.609979	0.304989	+	0.952356i	$0.401347\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	25.1512i	0.613445i	0.951799	+	0.306722i	$0.0992323\pi$
$41$	25.1512i	0.613445i	−0.951799	+	0.306722i	$0.900768\pi$
$42$	0	0
$43$	53.1769	1.23667	0.618336	−	0.785914i	$-0.287807\pi$
$43$	53.1769	1.23667	0.618336	+	0.785914i	$0.287807\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	16.9706i	0.361076i	0.983568	+	0.180538i	$0.0577838\pi$
$47$	16.9706i	0.361076i	−0.983568	+	0.180538i	$0.942216\pi$
$48$	0	0
$49$	104.569	2.13407
$50$	0	0
$51$	0	0
$52$	0	0
$53$	84.5482i	1.59525i	0.603154	+	0.797625i	$0.293910\pi$
$53$	84.5482i	1.59525i	−0.603154	+	0.797625i	$0.706090\pi$
$54$	0	0
$55$	22.8231	0.414965
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 91.0645i	− 1.54347i	−0.635947	−	0.771733i	$-0.719390\pi$
$59$	− 91.0645i	− 1.54347i	0.635947	−	0.771733i	$-0.280610\pi$
$60$	0	0
$61$	−13.0000	−0.213115	−0.106557	−	0.994307i	$-0.533983\pi$
$61$	−13.0000	−0.213115	−0.106557	+	0.994307i	$0.533983\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16.7774i	0.258115i
$66$	0	0
$67$	41.1769	0.614581	0.307290	−	0.951616i	$-0.400578\pi$
$67$	41.1769	0.614581	0.307290	+	0.951616i	$0.400578\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 16.3613i	− 0.230442i	−0.993340	−	0.115221i	$-0.963242\pi$
$71$	− 16.3613i	− 0.230442i	0.993340	−	0.115221i	$-0.0367575\pi$
$72$	0	0
$73$	71.5885	0.980664	0.490332	−	0.871536i	$-0.336876\pi$
$73$	71.5885	0.980664	0.490332	+	0.871536i	$0.336876\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 182.129i	− 2.36531i
$78$	0	0
$79$	46.7461	0.591723	0.295862	−	0.955231i	$-0.404393\pi$
$79$	46.7461	0.591723	0.295862	+	0.955231i	$0.404393\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 15.3062i	− 0.184411i	−0.995740	−	0.0922057i	$-0.970608\pi$
$83$	− 15.3062i	− 0.184411i	0.995740	−	0.0922057i	$-0.0293917\pi$
$84$	0	0
$85$	−45.0000	−0.529412
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 78.9756i	− 0.887367i	−0.896184	−	0.443683i	$-0.853671\pi$
$89$	− 78.9756i	− 0.887367i	0.896184	−	0.443683i	$-0.146329\pi$
$90$	0	0
$91$	133.885	1.47126
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.60244i	0.0589731i
$96$	0	0
$97$	91.1384	0.939572	0.469786	−	0.882780i	$-0.344331\pi$
$97$	91.1384	0.939572	0.469786	+	0.882780i	$0.344331\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	88.4862i	0.876101i	0.898950	+	0.438051i	$0.144331\pi$
$101$	88.4862i	0.876101i	−0.898950	+	0.438051i	$0.855669\pi$
$102$	0	0
$103$	152.708	1.48260	0.741299	−	0.671175i	$-0.234210\pi$
$103$	152.708	1.48260	0.741299	+	0.671175i	$0.234210\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 60.0062i	− 0.560805i	−0.959882	−	0.280403i	$-0.909532\pi$
$107$	− 60.0062i	− 0.560805i	0.959882	−	0.280403i	$-0.0904680\pi$
$108$	0	0
$109$	−93.9423	−0.861856	−0.430928	−	0.902386i	$-0.641814\pi$
$109$	−93.9423	−0.861856	−0.430928	+	0.902386i	$0.641814\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 88.5977i	− 0.784051i	−0.919954	−	0.392025i	$-0.871775\pi$
$113$	− 88.5977i	− 0.784051i	0.919954	−	0.392025i	$-0.128225\pi$
$114$	0	0
$115$	−22.8231	−0.198462
$116$	0	0
$117$	0	0
$118$	0	0
$119$	359.101i	3.01766i
$120$	0	0
$121$	−95.0000	−0.785124
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 73.9008i	− 0.591206i
$126$	0	0
$127$	−78.8231	−0.620654	−0.310327	−	0.950630i	$-0.600439\pi$
$127$	−78.8231	−0.620654	−0.310327	+	0.950630i	$0.600439\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	82.5792i	0.630375i	0.949029	+	0.315188i	$0.102067\pi$
$131$	82.5792i	0.630375i	−0.949029	+	0.315188i	$0.897933\pi$
$132$	0	0
$133$	44.7077	0.336148
$134$	0	0
$135$	0	0
$136$	0	0
$137$	214.740i	1.56745i	0.621110	+	0.783723i	$0.286682\pi$
$137$	214.740i	1.56745i	−0.621110	+	0.783723i	$0.713318\pi$
$138$	0	0
$139$	60.7846	0.437299	0.218650	−	0.975803i	$-0.429835\pi$
$139$	60.7846	0.437299	0.218650	+	0.975803i	$0.429835\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 158.783i	− 1.11037i
$144$	0	0
$145$	43.7077	0.301432
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 30.6422i	− 0.205652i	−0.994699	−	0.102826i	$-0.967211\pi$
$149$	− 30.6422i	− 0.205652i	0.994699	−	0.102826i	$-0.0327885\pi$
$150$	0	0
$151$	−32.0000	−0.211921	−0.105960	−	0.994370i	$-0.533792\pi$
$151$	−32.0000	−0.211921	−0.105960	+	0.994370i	$0.533792\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	12.4233i	0.0801504i
$156$	0	0
$157$	249.708	1.59049	0.795247	−	0.606285i	$-0.207341\pi$
$157$	249.708	1.59049	0.795247	+	0.606285i	$0.207341\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	182.129i	1.13124i
$162$	0	0
$163$	12.7846	0.0784332	0.0392166	−	0.999231i	$-0.487514\pi$
$163$	12.7846	0.0784332	0.0392166	+	0.999231i	$0.487514\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 232.431i	− 1.39180i	−0.718136	−	0.695902i	$-0.755005\pi$
$167$	− 232.431i	− 1.39180i	0.718136	−	0.695902i	$-0.244995\pi$
$168$	0	0
$169$	−52.2769	−0.309331
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.5477i	0.124553i	0.998059	+	0.0622765i	$0.0198361\pi$
$173$	21.5477i	0.124553i	−0.998059	+	0.0622765i	$0.980164\pi$
$174$	0	0
$175$	−279.923	−1.59956
$176$	0	0
$177$	0	0
$178$	0	0
$179$	277.741i	1.55162i	0.630964	+	0.775812i	$0.282659\pi$
$179$	277.741i	1.55162i	−0.630964	+	0.775812i	$0.717341\pi$
$180$	0	0
$181$	174.277	0.962856	0.481428	−	0.876486i	$-0.340118\pi$
$181$	174.277	0.962856	0.481428	+	0.876486i	$0.340118\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 35.0481i	− 0.189449i
$186$	0	0
$187$	425.885	2.27746
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 152.126i	− 0.796470i	−0.917283	−	0.398235i	$-0.869623\pi$
$191$	− 152.126i	− 0.796470i	0.917283	−	0.398235i	$-0.130377\pi$
$192$	0	0
$193$	−55.0000	−0.284974	−0.142487	−	0.989797i	$-0.545510\pi$
$193$	−55.0000	−0.284974	−0.142487	+	0.989797i	$0.545510\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 133.298i	− 0.676638i	−0.941031	−	0.338319i	$-0.890142\pi$
$197$	− 133.298i	− 0.676638i	0.941031	−	0.338319i	$-0.109858\pi$
$198$	0	0
$199$	208.862	1.04956	0.524778	−	0.851239i	$-0.324148\pi$
$199$	208.862	1.04956	0.524778	+	0.851239i	$0.324148\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 348.788i	− 1.71817i
$204$	0	0
$205$	39.0577	0.190525
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 53.0221i	− 0.253694i
$210$	0	0
$211$	87.4538	0.414473	0.207236	−	0.978291i	$-0.433553\pi$
$211$	87.4538	0.414473	0.207236	+	0.978291i	$0.433553\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 82.5792i	− 0.384089i
$216$	0	0
$217$	99.1384	0.456859
$218$	0	0
$219$	0	0
$220$	0	0
$221$	313.071i	1.41661i
$222$	0	0
$223$	−222.592	−0.998171	−0.499086	−	0.866553i	$-0.666331\pi$
$223$	−222.592	−0.998171	−0.499086	+	0.866553i	$0.666331\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 300.314i	− 1.32297i	−0.749959	−	0.661484i	$-0.769927\pi$
$227$	− 300.314i	− 1.32297i	0.749959	−	0.661484i	$-0.230073\pi$
$228$	0	0
$229$	229.942	1.00411	0.502057	−	0.864834i	$-0.332577\pi$
$229$	229.942	1.00411	0.502057	+	0.864834i	$0.332577\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	116.246i	0.498908i	0.968387	+	0.249454i	$0.0802512\pi$
$233$	116.246i	0.498908i	−0.968387	+	0.249454i	$0.919749\pi$
$234$	0	0
$235$	26.3538	0.112144
$236$	0	0
$237$	0	0
$238$	0	0
$239$	164.386i	0.687807i	0.939005	+	0.343904i	$0.111749\pi$
$239$	164.386i	0.687807i	−0.939005	+	0.343904i	$0.888251\pi$
$240$	0	0
$241$	−81.3116	−0.337392	−0.168696	−	0.985668i	$-0.553956\pi$
$241$	−81.3116	−0.337392	−0.168696	+	0.985668i	$0.553956\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 162.387i	− 0.662804i
$246$	0	0
$247$	38.9770	0.157802
$248$	0	0
$249$	0	0
$250$	0	0
$251$	396.371i	1.57917i	0.613642	+	0.789584i	$0.289704\pi$
$251$	396.371i	1.57917i	−0.613642	+	0.789584i	$0.710296\pi$
$252$	0	0
$253$	216.000	0.853755
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 214.963i	− 0.836432i	−0.908348	−	0.418216i	$-0.862655\pi$
$257$	− 214.963i	− 0.836432i	0.908348	−	0.418216i	$-0.137345\pi$
$258$	0	0
$259$	−279.685	−1.07986
$260$	0	0
$261$	0	0
$262$	0	0
$263$	317.893i	1.20872i	0.796711	+	0.604360i	$0.206571\pi$
$263$	317.893i	1.20872i	−0.796711	+	0.604360i	$0.793429\pi$
$264$	0	0
$265$	131.296	0.495457
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 208.528i	− 0.775199i	−0.921828	−	0.387599i	$-0.873304\pi$
$269$	− 208.528i	− 0.775199i	0.921828	−	0.387599i	$-0.126696\pi$
$270$	0	0
$271$	−409.885	−1.51249	−0.756245	−	0.654289i	$-0.772968\pi$
$271$	−409.885	−1.51249	−0.756245	+	0.654289i	$0.772968\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	331.981i	1.20720i
$276$	0	0
$277$	497.415	1.79572	0.897862	−	0.440278i	$-0.145120\pi$
$277$	497.415	1.79572	0.897862	+	0.440278i	$0.145120\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 215.268i	− 0.766077i	−0.923732	−	0.383039i	$-0.874878\pi$
$281$	− 215.268i	− 0.766077i	0.923732	−	0.383039i	$-0.125122\pi$
$282$	0	0
$283$	296.708	1.04844	0.524218	−	0.851584i	$-0.324358\pi$
$283$	296.708	1.04844	0.524218	+	0.851584i	$0.324358\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 311.682i	− 1.08600i
$288$	0	0
$289$	−550.711	−1.90558
$290$	0	0
$291$	0	0
$292$	0	0
$293$	340.964i	1.16370i	0.813296	+	0.581850i	$0.197671\pi$
$293$	340.964i	1.16370i	−0.813296	+	0.581850i	$0.802329\pi$
$294$	0	0
$295$	−141.415	−0.479374
$296$	0	0
$297$	0	0
$298$	0	0
$299$	158.783i	0.531048i
$300$	0	0
$301$	−658.985	−2.18932
$302$	0	0
$303$	0	0
$304$	0	0
$305$	20.1879i	0.0661898i
$306$	0	0
$307$	114.354	0.372488	0.186244	−	0.982504i	$-0.440368\pi$
$307$	114.354	0.372488	0.186244	+	0.982504i	$0.440368\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	196.217i	0.630922i	0.948939	+	0.315461i	$0.102159\pi$
$311$	196.217i	0.630922i	−0.948939	+	0.315461i	$0.897841\pi$
$312$	0	0
$313$	−146.723	−0.468764	−0.234382	−	0.972145i	$-0.575307\pi$
$313$	−146.723	−0.468764	−0.234382	+	0.972145i	$0.575307\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	48.1403i	0.151862i	0.997113	+	0.0759311i	$0.0241929\pi$
$317$	48.1403i	0.151862i	−0.997113	+	0.0759311i	$0.975807\pi$
$318$	0	0
$319$	−413.654	−1.29672
$320$	0	0
$321$	0	0
$322$	0	0
$323$	104.543i	0.323662i
$324$	0	0
$325$	−244.042	−0.750899
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 210.304i	− 0.639223i
$330$	0	0
$331$	99.4538	0.300465	0.150232	−	0.988651i	$-0.451998\pi$
$331$	99.4538	0.300465	0.150232	+	0.988651i	$0.451998\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 63.9442i	− 0.190878i
$336$	0	0
$337$	−425.261	−1.26190	−0.630952	−	0.775822i	$-0.717335\pi$
$337$	−425.261	−1.26190	−0.630952	+	0.775822i	$0.717335\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 117.576i	− 0.344796i
$342$	0	0
$343$	−688.631	−2.00767
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 38.7711i	− 0.111732i	−0.998438	−	0.0558662i	$-0.982208\pi$
$347$	− 38.7711i	− 0.111732i	0.998438	−	0.0558662i	$-0.0177920\pi$
$348$	0	0
$349$	−651.969	−1.86811	−0.934053	−	0.357134i	$-0.883754\pi$
$349$	−651.969	−1.86811	−0.934053	+	0.357134i	$0.883754\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.19656i	0.00338969i	0.999999	+	0.00169485i	$0.000539487\pi$
$353$	1.19656i	0.00338969i	−0.999999	+	0.00169485i	$0.999461\pi$
$354$	0	0
$355$	−25.4078	−0.0715712
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 534.573i	− 1.48906i	−0.667589	−	0.744530i	$-0.732674\pi$
$359$	− 534.573i	− 1.48906i	0.667589	−	0.744530i	$-0.267326\pi$
$360$	0	0
$361$	−347.985	−0.963946
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 111.171i	− 0.304577i
$366$	0	0
$367$	−264.708	−0.721274	−0.360637	−	0.932706i	$-0.617441\pi$
$367$	−264.708	−0.721274	−0.360637	+	0.932706i	$0.617441\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1047.75i	− 2.82412i
$372$	0	0
$373$	71.0155	0.190390	0.0951950	−	0.995459i	$-0.469653\pi$
$373$	71.0155	0.190390	0.0951950	+	0.995459i	$0.469653\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 304.080i	− 0.806579i
$378$	0	0
$379$	696.785	1.83848	0.919241	−	0.393696i	$-0.128804\pi$
$379$	696.785	1.83848	0.919241	+	0.393696i	$0.128804\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	435.306i	1.13657i	0.822832	+	0.568284i	$0.192393\pi$
$383$	435.306i	1.13657i	−0.822832	+	0.568284i	$0.807607\pi$
$384$	0	0
$385$	−282.831	−0.734625
$386$	0	0
$387$	0	0
$388$	0	0
$389$	53.0439i	0.136360i	0.997673	+	0.0681799i	$0.0217192\pi$
$389$	53.0439i	0.136360i	−0.997673	+	0.0681799i	$0.978281\pi$
$390$	0	0
$391$	−425.885	−1.08922
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 72.5927i	− 0.183779i
$396$	0	0
$397$	63.7077	0.160473	0.0802363	−	0.996776i	$-0.474432\pi$
$397$	63.7077	0.160473	0.0802363	+	0.996776i	$0.474432\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	628.691i	1.56781i	0.620882	+	0.783904i	$0.286775\pi$
$401$	628.691i	1.56781i	−0.620882	+	0.783904i	$0.713225\pi$
$402$	0	0
$403$	86.4308	0.214468
$404$	0	0
$405$	0	0
$406$	0	0
$407$	331.698i	0.814984i
$408$	0	0
$409$	535.281	1.30875	0.654377	−	0.756168i	$-0.272931\pi$
$409$	535.281	1.30875	0.654377	+	0.756168i	$0.272931\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	1128.50i	2.73244i
$414$	0	0
$415$	−23.7691	−0.0572750
$416$	0	0
$417$	0	0
$418$	0	0
$419$	385.330i	0.919641i	0.888012	+	0.459821i	$0.152086\pi$
$419$	385.330i	0.919641i	−0.888012	+	0.459821i	$0.847914\pi$
$420$	0	0
$421$	−755.319	−1.79411	−0.897054	−	0.441922i	$-0.854297\pi$
$421$	−755.319	−1.79411	−0.897054	+	0.441922i	$0.854297\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 654.563i	− 1.54015i
$426$	0	0
$427$	161.100	0.377283
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 107.709i	− 0.249904i	−0.992163	−	0.124952i	$-0.960122\pi$
$431$	− 107.709i	− 0.249904i	0.992163	−	0.124952i	$-0.0398777\pi$
$432$	0	0
$433$	655.123	1.51299	0.756493	−	0.654002i	$-0.226911\pi$
$433$	655.123	1.51299	0.756493	+	0.654002i	$0.226911\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	53.0221i	0.121332i
$438$	0	0
$439$	472.000	1.07517	0.537585	−	0.843209i	$-0.319337\pi$
$439$	472.000	1.07517	0.537585	+	0.843209i	$0.319337\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 841.218i	− 1.89891i	−0.313902	−	0.949455i	$-0.601636\pi$
$443$	− 841.218i	− 1.89891i	0.313902	−	0.949455i	$-0.398364\pi$
$444$	0	0
$445$	−122.642	−0.275601
$446$	0	0
$447$	0	0
$448$	0	0
$449$	382.751i	0.852453i	0.904616	+	0.426227i	$0.140157\pi$
$449$	382.751i	0.852453i	−0.904616	+	0.426227i	$0.859843\pi$
$450$	0	0
$451$	−369.646	−0.819615
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 207.911i	− 0.456948i
$456$	0	0
$457$	323.704	0.708324	0.354162	−	0.935184i	$-0.384766\pi$
$457$	323.704	0.708324	0.354162	+	0.935184i	$0.384766\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 641.531i	− 1.39161i	−0.718232	−	0.695803i	$-0.755049\pi$
$461$	− 641.531i	− 1.39161i	0.718232	−	0.695803i	$-0.244951\pi$
$462$	0	0
$463$	−129.492	−0.279681	−0.139840	−	0.990174i	$-0.544659\pi$
$463$	−129.492	−0.279681	−0.139840	+	0.990174i	$0.544659\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	672.730i	1.44054i	0.693696	+	0.720268i	$0.255981\pi$
$467$	672.730i	1.44054i	−0.693696	+	0.720268i	$0.744019\pi$
$468$	0	0
$469$	−510.277	−1.08801
$470$	0	0
$471$	0	0
$472$	0	0
$473$	781.538i	1.65230i
$474$	0	0
$475$	−81.4923	−0.171563
$476$	0	0
$477$	0	0
$478$	0	0
$479$	125.615i	0.262244i	0.991366	+	0.131122i	$0.0418579\pi$
$479$	125.615i	0.262244i	−0.991366	+	0.131122i	$0.958142\pi$
$480$	0	0
$481$	−243.834	−0.506932
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 141.530i	− 0.291815i
$486$	0	0
$487$	448.631	0.921213	0.460606	−	0.887604i	$-0.347632\pi$
$487$	448.631	0.921213	0.460606	+	0.887604i	$0.347632\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	452.157i	0.920890i	0.887688	+	0.460445i	$0.152310\pi$
$491$	452.157i	0.920890i	−0.887688	+	0.460445i	$0.847690\pi$
$492$	0	0
$493$	815.596	1.65435
$494$	0	0
$495$	0	0
$496$	0	0
$497$	202.755i	0.407957i
$498$	0	0
$499$	−126.592	−0.253692	−0.126846	−	0.991922i	$-0.540485\pi$
$499$	−126.592	−0.253692	−0.126846	+	0.991922i	$0.540485\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 296.822i	− 0.590103i	−0.955481	−	0.295051i	$-0.904663\pi$
$503$	− 296.822i	− 0.590103i	0.955481	−	0.295051i	$-0.0953367\pi$
$504$	0	0
$505$	137.412	0.272102
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 654.846i	− 1.28653i	−0.765642	−	0.643267i	$-0.777578\pi$
$509$	− 654.846i	− 1.28653i	0.765642	−	0.643267i	$-0.222422\pi$
$510$	0	0
$511$	−887.146	−1.73610
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 237.142i	− 0.460470i
$516$	0	0
$517$	−249.415	−0.482428
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 690.006i	− 1.32439i	−0.749333	−	0.662193i	$-0.769626\pi$
$521$	− 690.006i	− 1.32439i	0.749333	−	0.662193i	$-0.230374\pi$
$522$	0	0
$523$	−616.238	−1.17828	−0.589138	−	0.808032i	$-0.700533\pi$
$523$	−616.238	−1.17828	−0.589138	+	0.808032i	$0.700533\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	231.822i	0.439890i
$528$	0	0
$529$	313.000	0.591682
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 271.730i	− 0.509813i
$534$	0	0
$535$	−93.1845	−0.174177
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1536.85i	2.85129i
$540$	0	0
$541$	−548.734	−1.01430	−0.507148	−	0.861859i	$-0.669300\pi$
$541$	−548.734	−1.01430	−0.507148	+	0.861859i	$0.669300\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	145.884i	0.267678i
$546$	0	0
$547$	819.769	1.49866	0.749332	−	0.662195i	$-0.230375\pi$
$547$	819.769	1.49866	0.749332	+	0.662195i	$0.230375\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 101.541i	− 0.184284i
$552$	0	0
$553$	−579.292	−1.04754
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 125.808i	− 0.225867i	−0.993603	−	0.112934i	$-0.963975\pi$
$557$	− 125.808i	− 0.225867i	0.993603	−	0.112934i	$-0.0360247\pi$
$558$	0	0
$559$	−574.515	−1.02776
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 508.790i	− 0.903713i	−0.892091	−	0.451856i	$-0.850762\pi$
$563$	− 508.790i	− 0.903713i	0.892091	−	0.451856i	$-0.149238\pi$
$564$	0	0
$565$	−137.585	−0.243513
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 96.8600i	− 0.170229i	−0.996371	−	0.0851143i	$-0.972874\pi$
$569$	− 96.8600i	− 0.170229i	0.996371	−	0.0851143i	$-0.0271255\pi$
$570$	0	0
$571$	454.200	0.795446	0.397723	−	0.917505i	$-0.369800\pi$
$571$	454.200	0.795446	0.397723	+	0.917505i	$0.369800\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 331.981i	− 0.577359i
$576$	0	0
$577$	−39.1230	−0.0678041	−0.0339021	−	0.999425i	$-0.510793\pi$
$577$	−39.1230	−0.0678041	−0.0339021	+	0.999425i	$0.510793\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	189.679i	0.326469i
$582$	0	0
$583$	−1242.60	−2.13139
$584$	0	0
$585$	0	0
$586$	0	0
$587$	419.227i	0.714186i	0.934069	+	0.357093i	$0.116232\pi$
$587$	419.227i	0.714186i	−0.934069	+	0.357093i	$0.883768\pi$
$588$	0	0
$589$	28.8616	0.0490010
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 329.210i	− 0.555160i	−0.960703	−	0.277580i	$-0.910468\pi$
$593$	− 329.210i	− 0.555160i	0.960703	−	0.277580i	$-0.0895323\pi$
$594$	0	0
$595$	557.654	0.937233
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 184.239i	− 0.307578i	−0.988104	−	0.153789i	$-0.950852\pi$
$599$	− 184.239i	− 0.307578i	0.988104	−	0.153789i	$-0.0491476\pi$
$600$	0	0
$601$	218.415	0.363420	0.181710	−	0.983352i	$-0.441837\pi$
$601$	218.415	0.363420	0.181710	+	0.983352i	$0.441837\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	147.527i	0.243846i
$606$	0	0
$607$	−10.2693	−0.0169182	−0.00845909	−	0.999964i	$-0.502693\pi$
$607$	−10.2693	−0.0169182	−0.00845909	+	0.999964i	$0.502693\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 183.347i	− 0.300078i
$612$	0	0
$613$	−180.585	−0.294592	−0.147296	−	0.989092i	$-0.547057\pi$
$613$	−180.585	−0.294592	−0.147296	+	0.989092i	$0.547057\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	906.209i	1.46873i	0.678753	+	0.734367i	$0.262521\pi$
$617$	906.209i	1.46873i	−0.678753	+	0.734367i	$0.737479\pi$
$618$	0	0
$619$	−761.646	−1.23045	−0.615223	−	0.788353i	$-0.710934\pi$
$619$	−761.646	−1.23045	−0.615223	+	0.788353i	$0.710934\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	978.690i	1.57093i
$624$	0	0
$625$	449.950	0.719920
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 654.006i	− 1.03975i
$630$	0	0
$631$	−601.108	−0.952627	−0.476313	−	0.879276i	$-0.658027\pi$
$631$	−601.108	−0.952627	−0.476313	+	0.879276i	$0.658027\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	122.405i	0.192765i
$636$	0	0
$637$	−1129.75	−1.77355
$638$	0	0
$639$	0	0
$640$	0	0
$641$	577.453i	0.900863i	0.892811	+	0.450431i	$0.148730\pi$
$641$	577.453i	0.900863i	−0.892811	+	0.450431i	$0.851270\pi$
$642$	0	0
$643$	−130.123	−0.202369	−0.101184	−	0.994868i	$-0.532263\pi$
$643$	−130.123	−0.202369	−0.101184	+	0.994868i	$0.532263\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 985.467i	− 1.52313i	−0.648087	−	0.761567i	$-0.724430\pi$
$647$	− 985.467i	− 1.52313i	0.648087	−	0.761567i	$-0.275570\pi$
$648$	0	0
$649$	1338.37	2.06220
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 600.323i	− 0.919330i	−0.888092	−	0.459665i	$-0.847969\pi$
$653$	− 600.323i	− 0.919330i	0.888092	−	0.459665i	$-0.152031\pi$
$654$	0	0
$655$	128.238	0.195784
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.5889i	0.0236554i	0.999930	+	0.0118277i	$0.00376496\pi$
$659$	15.5889i	0.0236554i	−0.999930	+	0.0118277i	$0.996235\pi$
$660$	0	0
$661$	407.831	0.616990	0.308495	−	0.951226i	$-0.400175\pi$
$661$	407.831	0.616990	0.308495	+	0.951226i	$0.400175\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 69.4272i	− 0.104402i
$666$	0	0
$667$	413.654	0.620170
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 191.060i	− 0.284739i
$672$	0	0
$673$	−474.569	−0.705155	−0.352577	−	0.935783i	$-0.614695\pi$
$673$	−474.569	−0.705155	−0.352577	+	0.935783i	$0.614695\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	282.147i	0.416760i	0.978048	+	0.208380i	$0.0668191\pi$
$677$	282.147i	0.416760i	−0.978048	+	0.208380i	$0.933181\pi$
$678$	0	0
$679$	−1129.42	−1.66335
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1085.46i	− 1.58926i	−0.607095	−	0.794629i	$-0.707665\pi$
$683$	− 1085.46i	− 1.58926i	0.607095	−	0.794629i	$-0.292335\pi$
$684$	0	0
$685$	333.473	0.486822
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 913.446i	− 1.32576i
$690$	0	0
$691$	1007.78	1.45843	0.729216	−	0.684283i	$-0.239885\pi$
$691$	1007.78	1.45843	0.729216	+	0.684283i	$0.239885\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 94.3933i	− 0.135818i
$696$	0	0
$697$	728.827	1.04566
$698$	0	0
$699$	0	0
$700$	0	0
$701$	216.731i	0.309174i	0.987979	+	0.154587i	$0.0494047\pi$
$701$	216.731i	0.309174i	−0.987979	+	0.154587i	$0.950595\pi$
$702$	0	0
$703$	−81.4229	−0.115822
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1096.55i	− 1.55099i
$708$	0	0
$709$	994.496	1.40267	0.701337	−	0.712830i	$-0.252587\pi$
$709$	994.496	1.40267	0.701337	+	0.712830i	$0.252587\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	117.576i	0.164903i
$714$	0	0
$715$	−246.577	−0.344863
$716$	0	0
$717$	0	0
$718$	0	0
$719$	340.912i	0.474148i	0.971492	+	0.237074i	$0.0761884\pi$
$719$	340.912i	0.474148i	−0.971492	+	0.237074i	$0.923812\pi$
$720$	0	0
$721$	−1892.40	−2.62469
$722$	0	0
$723$	0	0
$724$	0	0
$725$	635.765i	0.876917i
$726$	0	0
$727$	971.777	1.33669	0.668347	−	0.743850i	$-0.267002\pi$
$727$	971.777	1.33669	0.668347	+	0.743850i	$0.267002\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1540.95i	− 2.10800i
$732$	0	0
$733$	229.169	0.312646	0.156323	−	0.987706i	$-0.450036\pi$
$733$	229.169	0.312646	0.156323	+	0.987706i	$0.450036\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	605.175i	0.821132i
$738$	0	0
$739$	629.892	0.852357	0.426179	−	0.904639i	$-0.359859\pi$
$739$	629.892	0.852357	0.426179	+	0.904639i	$0.359859\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	188.014i	0.253047i	0.991964	+	0.126524i	$0.0403820\pi$
$743$	188.014i	0.253047i	−0.991964	+	0.126524i	$0.959618\pi$
$744$	0	0
$745$	−47.5847	−0.0638721
$746$	0	0
$747$	0	0
$748$	0	0
$749$	743.615i	0.992810i
$750$	0	0
$751$	1386.67	1.84643	0.923215	−	0.384283i	$-0.125551\pi$
$751$	1386.67	1.84643	0.923215	+	0.384283i	$0.125551\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	49.6933i	0.0658189i
$756$	0	0
$757$	1222.12	1.61443	0.807215	−	0.590258i	$-0.200974\pi$
$757$	1222.12	1.61443	0.807215	+	0.590258i	$0.200974\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 565.683i	− 0.743341i	−0.928365	−	0.371671i	$-0.878785\pi$
$761$	− 565.683i	− 0.743341i	0.928365	−	0.371671i	$-0.121215\pi$
$762$	0	0
$763$	1164.16	1.52577
$764$	0	0
$765$	0	0
$766$	0	0
$767$	983.847i	1.28272i
$768$	0	0
$769$	−598.815	−0.778693	−0.389347	−	0.921091i	$-0.627299\pi$
$769$	−598.815	−0.778693	−0.389347	+	0.921091i	$0.627299\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	446.970i	0.578228i	0.957295	+	0.289114i	$0.0933607\pi$
$773$	446.970i	0.578228i	−0.957295	+	0.289114i	$0.906639\pi$
$774$	0	0
$775$	−180.708	−0.233171
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 90.7380i	− 0.116480i
$780$	0	0
$781$	240.462	0.307890
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 387.775i	− 0.493980i
$786$	0	0
$787$	955.300	1.21385	0.606925	−	0.794759i	$-0.292403\pi$
$787$	955.300	1.21385	0.606925	+	0.794759i	$0.292403\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1097.93i	1.38803i
$792$	0	0
$793$	140.450	0.177112
$794$	0	0
$795$	0	0
$796$	0	0
$797$	221.866i	0.278376i	0.990266	+	0.139188i	$0.0444492\pi$
$797$	221.866i	0.278376i	−0.990266	+	0.139188i	$0.955551\pi$
$798$	0	0
$799$	491.769	0.615481
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1052.13i	1.31025i
$804$	0	0
$805$	282.831	0.351342
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1180.17i	1.45881i	0.684084	+	0.729403i	$0.260202\pi$
$809$	1180.17i	1.45881i	−0.684084	+	0.729403i	$0.739798\pi$
$810$	0	0
$811$	627.307	0.773499	0.386749	−	0.922185i	$-0.373598\pi$
$811$	627.307	0.773499	0.386749	+	0.922185i	$0.373598\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 19.8534i	− 0.0243600i
$816$	0	0
$817$	−191.846	−0.234818
$818$	0	0
$819$	0	0
$820$	0	0
$821$	390.880i	0.476103i	0.971253	+	0.238051i	$0.0765087\pi$
$821$	390.880i	0.476103i	−0.971253	+	0.238051i	$0.923491\pi$
$822$	0	0
$823$	−326.200	−0.396355	−0.198177	−	0.980166i	$-0.563502\pi$
$823$	−326.200	−0.396355	−0.198177	+	0.980166i	$0.563502\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1103.70i	1.33458i	0.744799	+	0.667289i	$0.232545\pi$
$827$	1103.70i	1.33458i	−0.744799	+	0.667289i	$0.767455\pi$
$828$	0	0
$829$	−441.569	−0.532653	−0.266326	−	0.963883i	$-0.585810\pi$
$829$	−441.569	−0.532653	−0.266326	+	0.963883i	$0.585810\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3030.18i	− 3.63768i
$834$	0	0
$835$	−360.946	−0.432271
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 503.960i	− 0.600668i	−0.953834	−	0.300334i	$-0.902902\pi$
$839$	− 503.960i	− 0.600668i	0.953834	−	0.300334i	$-0.0970981\pi$
$840$	0	0
$841$	48.8269	0.0580581
$842$	0	0
$843$	0	0
$844$	0	0
$845$	81.1815i	0.0960728i
$846$	0	0
$847$	1177.27	1.38993
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 331.698i	− 0.389775i
$852$	0	0
$853$	944.831	1.10766	0.553828	−	0.832631i	$-0.313166\pi$
$853$	944.831	1.10766	0.553828	+	0.832631i	$0.313166\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	741.377i	0.865084i	0.901614	+	0.432542i	$0.142383\pi$
$857$	741.377i	0.865084i	−0.901614	+	0.432542i	$0.857617\pi$
$858$	0	0
$859$	1361.26	1.58470	0.792352	−	0.610064i	$-0.208856\pi$
$859$	1361.26	1.58470	0.792352	+	0.610064i	$0.208856\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 805.003i	− 0.932796i	−0.884575	−	0.466398i	$-0.845551\pi$
$863$	− 805.003i	− 0.932796i	0.884575	−	0.466398i	$-0.154449\pi$
$864$	0	0
$865$	33.4617	0.0386841
$866$	0	0
$867$	0	0
$868$	0	0
$869$	687.025i	0.790593i
$870$	0	0
$871$	−444.869	−0.510757
$872$	0	0
$873$	0	0
$874$	0	0
$875$	915.801i	1.04663i
$876$	0	0
$877$	42.6922	0.0486798	0.0243399	−	0.999704i	$-0.492252\pi$
$877$	42.6922	0.0486798	0.0243399	+	0.999704i	$0.492252\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	264.567i	0.300303i	0.988663	+	0.150151i	$0.0479761\pi$
$881$	264.567i	0.300303i	−0.988663	+	0.150151i	$0.952024\pi$
$882$	0	0
$883$	393.338	0.445457	0.222728	−	0.974881i	$-0.428504\pi$
$883$	393.338	0.445457	0.222728	+	0.974881i	$0.428504\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1548.05i	1.74527i	0.488375	+	0.872634i	$0.337590\pi$
$887$	1548.05i	1.74527i	−0.488375	+	0.872634i	$0.662410\pi$
$888$	0	0
$889$	976.800	1.09876
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 61.2246i	− 0.0685606i
$894$	0	0
$895$	431.307	0.481908
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 225.165i	− 0.250461i
$900$	0	0
$901$	2450.02	2.71922
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 270.637i	− 0.299046i
$906$	0	0
$907$	−1377.49	−1.51873	−0.759367	−	0.650662i	$-0.774491\pi$
$907$	−1377.49	−1.51873	−0.759367	+	0.650662i	$0.774491\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1406.08i	1.54344i	0.635961	+	0.771721i	$0.280604\pi$
$911$	1406.08i	1.54344i	−0.635961	+	0.771721i	$0.719396\pi$
$912$	0	0
$913$	224.954	0.246389
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1023.35i	− 1.11597i
$918$	0	0
$919$	−479.992	−0.522299	−0.261149	−	0.965298i	$-0.584101\pi$
$919$	−479.992	−0.522299	−0.261149	+	0.965298i	$0.584101\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	176.765i	0.191512i
$924$	0	0
$925$	509.804	0.551139
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 678.733i	− 0.730606i	−0.930889	−	0.365303i	$-0.880965\pi$
$929$	− 678.733i	− 0.730606i	0.930889	−	0.365303i	$-0.119035\pi$
$930$	0	0
$931$	−377.254	−0.405214
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 661.362i	− 0.707339i
$936$	0	0
$937$	−666.600	−0.711420	−0.355710	−	0.934596i	$-0.615761\pi$
$937$	−666.600	−0.711420	−0.355710	+	0.934596i	$0.615761\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1197.96i	− 1.27307i	−0.771249	−	0.636533i	$-0.780368\pi$
$941$	− 1197.96i	− 1.27307i	0.771249	−	0.636533i	$-0.219632\pi$
$942$	0	0
$943$	369.646	0.391990
$944$	0	0
$945$	0	0
$946$	0	0
$947$	454.713i	0.480162i	0.970753	+	0.240081i	$0.0771740\pi$
$947$	454.713i	0.480162i	−0.970753	+	0.240081i	$0.922826\pi$
$948$	0	0
$949$	−773.431	−0.814996
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1060.16i	− 1.11245i	−0.831033	−	0.556224i	$-0.812250\pi$
$953$	− 1060.16i	− 1.11245i	0.831033	−	0.556224i	$-0.187750\pi$
$954$	0	0
$955$	−236.238	−0.247370
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 2661.13i	− 2.77490i
$960$	0	0
$961$	−897.000	−0.933403
$962$	0	0
$963$	0	0
$964$	0	0
$965$	85.4103i	0.0885081i
$966$	0	0
$967$	1655.31	1.71180	0.855902	−	0.517138i	$-0.173002\pi$
$967$	1655.31	1.71180	0.855902	+	0.517138i	$0.173002\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 797.780i	− 0.821606i	−0.911724	−	0.410803i	$-0.865248\pi$
$971$	− 797.780i	− 0.821606i	0.911724	−	0.410803i	$-0.134752\pi$
$972$	0	0
$973$	−753.261	−0.774164
$974$	0	0
$975$	0	0
$976$	0	0
$977$	312.432i	0.319787i	0.987134	+	0.159894i	$0.0511152\pi$
$977$	312.432i	0.319787i	−0.987134	+	0.159894i	$0.948885\pi$
$978$	0	0
$979$	1160.70	1.18560
$980$	0	0
$981$	0	0
$982$	0	0
$983$	370.143i	0.376544i	0.982117	+	0.188272i	$0.0602887\pi$
$983$	370.143i	0.376544i	−0.982117	+	0.188272i	$0.939711\pi$
$984$	0	0
$985$	−207.000	−0.210152
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 781.538i	− 0.790230i
$990$	0	0
$991$	1324.48	1.33651	0.668256	−	0.743931i	$-0.267041\pi$
$991$	1324.48	1.33651	0.668256	+	0.743931i	$0.267041\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 324.344i	− 0.325974i
$996$	0	0
$997$	879.246	0.881892	0.440946	−	0.897534i	$-0.354643\pi$
$997$	879.246	0.881892	0.440946	+	0.897534i	$0.354643\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.e.d.161.2		4
3.2	odd	2	inner	1296.3.e.d.161.3		4
4.3	odd	2		162.3.b.b.161.1	✓	4
9.2	odd	6		1296.3.q.o.1025.2		8
9.4	even	3		1296.3.q.o.593.2		8
9.5	odd	6		1296.3.q.o.593.3		8
9.7	even	3		1296.3.q.o.1025.3		8
12.11	even	2		162.3.b.b.161.4	yes	4
36.7	odd	6		162.3.d.c.53.2		8
36.11	even	6		162.3.d.c.53.3		8
36.23	even	6		162.3.d.c.107.2		8
36.31	odd	6		162.3.d.c.107.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.b.b.161.1	✓	4	4.3	odd	2
162.3.b.b.161.4	yes	4	12.11	even	2
162.3.d.c.53.2		8	36.7	odd	6
162.3.d.c.53.3		8	36.11	even	6
162.3.d.c.107.2		8	36.23	even	6
162.3.d.c.107.3		8	36.31	odd	6
1296.3.e.d.161.2		4	1.1	even	1	trivial
1296.3.e.d.161.3		4	3.2	odd	2	inner
1296.3.q.o.593.2		8	9.4	even	3
1296.3.q.o.593.3		8	9.5	odd	6
1296.3.q.o.1025.2		8	9.2	odd	6
1296.3.q.o.1025.3		8	9.7	even	3

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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.55291i q^{5} -12.3923 q^{7} +O(q^{10})\)	\(q-1.55291i q^{5} -12.3923 q^{7} +14.6969i q^{11} -10.8038 q^{13} -28.9778i q^{17} -3.60770 q^{19} -14.6969i q^{23} +22.5885 q^{25} +28.1456i q^{29} -8.00000 q^{31} +19.2442i q^{35} +22.5692 q^{37} +25.1512i q^{41} +53.1769 q^{43} +16.9706i q^{47} +104.569 q^{49} +84.5482i q^{53} +22.8231 q^{55} -91.0645i q^{59} -13.0000 q^{61} +16.7774i q^{65} +41.1769 q^{67} -16.3613i q^{71} +71.5885 q^{73} -182.129i q^{77} +46.7461 q^{79} -15.3062i q^{83} -45.0000 q^{85} -78.9756i q^{89} +133.885 q^{91} +5.60244i q^{95} +91.1384 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7}+O(q^{10}) \) 4 * q - 8 * q^7	\( 4 q - 8 q^{7} - 64 q^{13} - 56 q^{19} + 28 q^{25} - 32 q^{31} - 76 q^{37} + 88 q^{43} + 252 q^{49} + 216 q^{55} - 52 q^{61} + 40 q^{67} + 224 q^{73} - 104 q^{79} - 180 q^{85} - 88 q^{91} + 32 q^{97}+O(q^{100}) \) 4 * q - 8 * q^7 - 64 * q^13 - 56 * q^19 + 28 * q^25 - 32 * q^31 - 76 * q^37 + 88 * q^43 + 252 * q^49 + 216 * q^55 - 52 * q^61 + 40 * q^67 + 224 * q^73 - 104 * q^79 - 180 * q^85 - 88 * q^91 + 32 * q^97

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