LMFDB - Embedded newform 1296.3.e.d.161.1

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			161.1
Root			$-1.93185i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.3.e.d.161.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.79555i q^{5} +8.39230 q^{7} +O(q^{10})$	$q-5.79555i q^{5} +8.39230 q^{7} +14.6969i q^{11} -21.1962 q^{13} -7.76457i q^{17} -24.3923 q^{19} -14.6969i q^{23} -8.58846 q^{25} -35.4940i q^{29} -8.00000 q^{31} -48.6381i q^{35} -60.5692 q^{37} +33.6365i q^{41} -9.17691 q^{43} -16.9706i q^{47} +21.4308 q^{49} -25.7605i q^{53} +85.1769 q^{55} +61.6706i q^{59} -13.0000 q^{61} +122.843i q^{65} -21.1769 q^{67} -101.214i q^{71} +40.4115 q^{73} +123.341i q^{77} -98.7461 q^{79} +103.488i q^{83} -45.0000 q^{85} -134.130i q^{89} -177.885 q^{91} +141.367i q^{95} -75.1384 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 5.79555i	− 1.15911i	−0.814933	−	0.579555i	$-0.803226\pi$
$5$	− 5.79555i	− 1.15911i	0.814933	−	0.579555i	$-0.196774\pi$
$6$	0	0
$7$	8.39230	1.19890	0.599450	−	0.800412i	$-0.295386\pi$
$7$	8.39230	1.19890	0.599450	+	0.800412i	$0.295386\pi$
$8$	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
$13$	−21.1962	−1.63047	−0.815237	−	0.579128i	$-0.803393\pi$
$13$	−21.1962	−1.63047	−0.815237	+	0.579128i	$0.803393\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.76457i	− 0.456739i	−0.973574	−	0.228370i	$-0.926660\pi$
$17$	− 7.76457i	− 0.456739i	0.973574	−	0.228370i	$-0.0733395\pi$
$18$	0	0
$19$	−24.3923	−1.28381	−0.641903	−	0.766786i	$-0.721855\pi$
$19$	−24.3923	−1.28381	−0.641903	+	0.766786i	$0.721855\pi$
$20$	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
$25$	−8.58846	−0.343538
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 35.4940i	− 1.22393i	−0.790884	−	0.611966i	$-0.790379\pi$
$29$	− 35.4940i	− 1.22393i	0.790884	−	0.611966i	$-0.209621\pi$
$30$	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
$35$	− 48.6381i	− 1.38966i
$36$	0	0
$37$	−60.5692	−1.63701	−0.818503	−	0.574502i	$-0.805196\pi$
$37$	−60.5692	−1.63701	−0.818503	+	0.574502i	$0.805196\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	33.6365i	0.820403i	0.911995	+	0.410201i	$0.134542\pi$
$41$	33.6365i	0.820403i	−0.911995	+	0.410201i	$0.865458\pi$
$42$	0	0
$43$	−9.17691	−0.213417	−0.106708	−	0.994290i	$-0.534031\pi$
$43$	−9.17691	−0.213417	−0.106708	+	0.994290i	$0.534031\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 16.9706i	− 0.361076i	−0.983568	−	0.180538i	$-0.942216\pi$
$47$	− 16.9706i	− 0.361076i	0.983568	−	0.180538i	$-0.0577838\pi$
$48$	0	0
$49$	21.4308	0.437363
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 25.7605i	− 0.486046i	−0.970020	−	0.243023i	$-0.921861\pi$
$53$	− 25.7605i	− 0.486046i	0.970020	−	0.243023i	$-0.0781391\pi$
$54$	0	0
$55$	85.1769	1.54867
$56$	0	0
$57$	0	0
$58$	0	0
$59$	61.6706i	1.04526i	0.852558	+	0.522632i	$0.175050\pi$
$59$	61.6706i	1.04526i	−0.852558	+	0.522632i	$0.824950\pi$
$60$	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
$65$	122.843i	1.88990i
$66$	0	0
$67$	−21.1769	−0.316073	−0.158037	−	0.987433i	$-0.550516\pi$
$67$	−21.1769	−0.316073	−0.158037	+	0.987433i	$0.550516\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 101.214i	− 1.42555i	−0.701392	−	0.712776i	$-0.747438\pi$
$71$	− 101.214i	− 1.42555i	0.701392	−	0.712776i	$-0.252562\pi$
$72$	0	0
$73$	40.4115	0.553583	0.276791	−	0.960930i	$-0.410729\pi$
$73$	40.4115	0.553583	0.276791	+	0.960930i	$0.410729\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	123.341i	1.60183i
$78$	0	0
$79$	−98.7461	−1.24995	−0.624976	−	0.780644i	$-0.714891\pi$
$79$	−98.7461	−1.24995	−0.624976	+	0.780644i	$0.714891\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	103.488i	1.24684i	0.781887	+	0.623420i	$0.214257\pi$
$83$	103.488i	1.24684i	−0.781887	+	0.623420i	$0.785743\pi$
$84$	0	0
$85$	−45.0000	−0.529412
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 134.130i	− 1.50708i	−0.657403	−	0.753539i	$-0.728345\pi$
$89$	− 134.130i	− 1.50708i	0.657403	−	0.753539i	$-0.271655\pi$
$90$	0	0
$91$	−177.885	−1.95478
$92$	0	0
$93$	0	0
$94$	0	0
$95$	141.367i	1.48807i
$96$	0	0
$97$	−75.1384	−0.774623	−0.387312	−	0.921949i	$-0.626596\pi$
$97$	−75.1384	−0.774623	−0.387312	+	0.921949i	$0.626596\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	29.0893i	0.288013i	0.989577	+	0.144006i	$0.0459986\pi$
$101$	29.0893i	0.288013i	−0.989577	+	0.144006i	$0.954001\pi$
$102$	0	0
$103$	−96.7077	−0.938909	−0.469455	−	0.882957i	$-0.655549\pi$
$103$	−96.7077	−0.938909	−0.469455	+	0.882957i	$0.655549\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	177.582i	1.65964i	0.558030	+	0.829821i	$0.311558\pi$
$107$	177.582i	1.65964i	−0.558030	+	0.829821i	$0.688442\pi$
$108$	0	0
$109$	61.9423	0.568278	0.284139	−	0.958783i	$-0.408292\pi$
$109$	61.9423	0.568278	0.284139	+	0.958783i	$0.408292\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 109.811i	− 0.971778i	−0.874020	−	0.485889i	$-0.838496\pi$
$113$	− 109.811i	− 0.971778i	0.874020	−	0.485889i	$-0.161504\pi$
$114$	0	0
$115$	−85.1769	−0.740669
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 65.1626i	− 0.547585i
$120$	0	0
$121$	−95.0000	−0.785124
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 95.1140i	− 0.760912i
$126$	0	0
$127$	−141.177	−1.11163	−0.555815	−	0.831306i	$-0.687594\pi$
$127$	−141.177	−1.11163	−0.555815	+	0.831306i	$0.687594\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 53.1853i	− 0.405995i	−0.979179	−	0.202997i	$-0.934932\pi$
$131$	− 53.1853i	− 0.405995i	0.979179	−	0.202997i	$-0.0650683\pi$
$132$	0	0
$133$	−204.708	−1.53916
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.63453i	− 0.0119309i	−0.999982	−	0.00596545i	$-0.998101\pi$
$137$	− 1.63453i	− 0.0119309i	0.999982	−	0.00596545i	$-0.00189887\pi$
$138$	0	0
$139$	19.2154	0.138240	0.0691201	−	0.997608i	$-0.477981\pi$
$139$	19.2154	0.138240	0.0691201	+	0.997608i	$0.477981\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 311.519i	− 2.17845i
$144$	0	0
$145$	−205.708	−1.41867
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 94.2818i	− 0.632764i	−0.948632	−	0.316382i	$-0.897532\pi$
$149$	− 94.2818i	− 0.632764i	0.948632	−	0.316382i	$-0.102468\pi$
$150$	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
$155$	46.3644i	0.299125i
$156$	0	0
$157$	0.292342	0.00186205	0.000931025	−	1.00000i	$-0.499704\pi$
$157$	0.292342	0.00186205	0.000931025		1.00000i	$0.499704\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 123.341i	− 0.766094i
$162$	0	0
$163$	−28.7846	−0.176593	−0.0882963	−	0.996094i	$-0.528142\pi$
$163$	−28.7846	−0.176593	−0.0882963	+	0.996094i	$0.528142\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	56.0682i	0.335737i	0.985809	+	0.167869i	$0.0536885\pi$
$167$	56.0682i	0.335737i	−0.985809	+	0.167869i	$0.946312\pi$
$168$	0	0
$169$	280.277	1.65844
$170$	0	0
$171$	0	0
$172$	0	0
$173$	220.952i	1.27718i	0.769548	+	0.638589i	$0.220482\pi$
$173$	220.952i	1.27718i	−0.769548	+	0.638589i	$0.779518\pi$
$174$	0	0
$175$	−72.0770	−0.411868
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 248.347i	− 1.38741i	−0.720258	−	0.693706i	$-0.755977\pi$
$179$	− 248.347i	− 1.38741i	0.720258	−	0.693706i	$-0.244023\pi$
$180$	0	0
$181$	−158.277	−0.874458	−0.437229	−	0.899350i	$-0.644040\pi$
$181$	−158.277	−0.874458	−0.437229	+	0.899350i	$0.644040\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	351.032i	1.89747i
$186$	0	0
$187$	114.115	0.610243
$188$	0	0
$189$	0	0
$190$	0	0
$191$	34.5503i	0.180892i	0.995901	+	0.0904459i	$0.0288292\pi$
$191$	34.5503i	0.180892i	−0.995901	+	0.0904459i	$0.971171\pi$
$192$	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
$197$	− 35.7170i	− 0.181305i	−0.995883	−	0.0906524i	$-0.971105\pi$
$197$	− 35.7170i	− 0.181305i	0.995883	−	0.0906524i	$-0.0288952\pi$
$198$	0	0
$199$	375.138	1.88512	0.942559	−	0.334040i	$-0.108412\pi$
$199$	375.138	1.88512	0.942559	+	0.334040i	$0.108412\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 297.877i	− 1.46737i
$204$	0	0
$205$	194.942	0.950938
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 358.492i	− 1.71527i
$210$	0	0
$211$	−307.454	−1.45713	−0.728563	−	0.684978i	$-0.759812\pi$
$211$	−307.454	−1.45713	−0.728563	+	0.684978i	$0.759812\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	53.1853i	0.247374i
$216$	0	0
$217$	−67.1384	−0.309394
$218$	0	0
$219$	0	0
$220$	0	0
$221$	164.579i	0.744702i
$222$	0	0
$223$	338.592	1.51835	0.759175	−	0.650886i	$-0.225602\pi$
$223$	338.592	1.51835	0.759175	+	0.650886i	$0.225602\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	123.950i	0.546037i	0.962009	+	0.273019i	$0.0880220\pi$
$227$	123.950i	0.546037i	−0.962009	+	0.273019i	$0.911978\pi$
$228$	0	0
$229$	74.0577	0.323396	0.161698	−	0.986840i	$-0.448303\pi$
$229$	74.0577	0.323396	0.161698	+	0.986840i	$0.448303\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	273.223i	1.17263i	0.810082	+	0.586316i	$0.199422\pi$
$233$	273.223i	1.17263i	−0.810082	+	0.586316i	$0.800578\pi$
$234$	0	0
$235$	−98.3538	−0.418527
$236$	0	0
$237$	0	0
$238$	0	0
$239$	452.885i	1.89492i	0.319877	+	0.947459i	$0.396359\pi$
$239$	452.885i	1.89492i	−0.319877	+	0.947459i	$0.603641\pi$
$240$	0	0
$241$	−382.688	−1.58792	−0.793959	−	0.607971i	$-0.791984\pi$
$241$	−382.688	−1.58792	−0.793959	+	0.607971i	$0.791984\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 124.203i	− 0.506952i
$246$	0	0
$247$	517.023	2.09321
$248$	0	0
$249$	0	0
$250$	0	0
$251$	73.9307i	0.294544i	0.989096	+	0.147272i	$0.0470493\pi$
$251$	73.9307i	0.294544i	−0.989096	+	0.147272i	$0.952951\pi$
$252$	0	0
$253$	216.000	0.853755
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 159.809i	− 0.621824i	−0.950439	−	0.310912i	$-0.899366\pi$
$257$	− 159.809i	− 0.621824i	0.950439	−	0.310912i	$-0.100634\pi$
$258$	0	0
$259$	−508.315	−1.96261
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 259.106i	− 0.985193i	−0.870258	−	0.492596i	$-0.836048\pi$
$263$	− 259.106i	− 0.985193i	0.870258	−	0.492596i	$-0.163952\pi$
$264$	0	0
$265$	−149.296	−0.563382
$266$	0	0
$267$	0	0
$268$	0	0
$269$	24.8168i	0.0922556i	0.998936	+	0.0461278i	$0.0146881\pi$
$269$	24.8168i	0.0922556i	−0.998936	+	0.0461278i	$0.985312\pi$
$270$	0	0
$271$	−98.1154	−0.362050	−0.181025	−	0.983479i	$-0.557941\pi$
$271$	−98.1154	−0.362050	−0.181025	+	0.983479i	$0.557941\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 126.224i	− 0.458996i
$276$	0	0
$277$	−1.41532	−0.00510945	−0.00255472	−	0.999997i	$-0.500813\pi$
$277$	−1.41532	−0.00510945	−0.00255472	+	0.999997i	$0.500813\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 100.716i	− 0.358421i	−0.983811	−	0.179211i	$-0.942646\pi$
$281$	− 100.716i	− 0.358421i	0.983811	−	0.179211i	$-0.0573544\pi$
$282$	0	0
$283$	47.2923	0.167111	0.0835554	−	0.996503i	$-0.473372\pi$
$283$	47.2923	0.167111	0.0835554	+	0.996503i	$0.473372\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	282.288i	0.983582i
$288$	0	0
$289$	228.711	0.791389
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 333.616i	− 1.13862i	−0.822123	−	0.569310i	$-0.807210\pi$
$293$	− 333.616i	− 1.13862i	0.822123	−	0.569310i	$-0.192790\pi$
$294$	0	0
$295$	357.415	1.21158
$296$	0	0
$297$	0	0
$298$	0	0
$299$	311.519i	1.04187i
$300$	0	0
$301$	−77.0155	−0.255865
$302$	0	0
$303$	0	0
$304$	0	0
$305$	75.3422i	0.247024i
$306$	0	0
$307$	−10.3538	−0.0337258	−0.0168629	−	0.999858i	$-0.505368\pi$
$307$	−10.3538	−0.0337258	−0.0168629	+	0.999858i	$0.505368\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.54047i	0.0306768i	0.999882	+	0.0153384i	$0.00488255\pi$
$311$	9.54047i	0.0306768i	−0.999882	+	0.0153384i	$0.995117\pi$
$312$	0	0
$313$	−479.277	−1.53124	−0.765618	−	0.643295i	$-0.777567\pi$
$313$	−479.277	−1.53124	−0.765618	+	0.643295i	$0.777567\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	179.662i	0.566758i	0.959008	+	0.283379i	$0.0914554\pi$
$317$	179.662i	0.566758i	−0.959008	+	0.283379i	$0.908545\pi$
$318$	0	0
$319$	521.654	1.63528
$320$	0	0
$321$	0	0
$322$	0	0
$323$	189.396i	0.586365i
$324$	0	0
$325$	182.042	0.560130
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 142.422i	− 0.432894i
$330$	0	0
$331$	−295.454	−0.892610	−0.446305	−	0.894881i	$-0.647260\pi$
$331$	−295.454	−0.892610	−0.446305	+	0.894881i	$0.647260\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	122.732i	0.366364i
$336$	0	0
$337$	489.261	1.45181	0.725907	−	0.687793i	$-0.241420\pi$
$337$	489.261	1.45181	0.725907	+	0.687793i	$0.241420\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 117.576i	− 0.344796i
$342$	0	0
$343$	−231.369	−0.674546
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 666.682i	− 1.92127i	−0.277807	−	0.960637i	$-0.589608\pi$
$347$	− 666.682i	− 1.92127i	0.277807	−	0.960637i	$-0.410392\pi$
$348$	0	0
$349$	511.969	1.46696	0.733480	−	0.679711i	$-0.237895\pi$
$349$	511.969	1.46696	0.733480	+	0.679711i	$0.237895\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	586.681i	1.66199i	0.556283	+	0.830993i	$0.312227\pi$
$353$	586.681i	1.66199i	−0.556283	+	0.830993i	$0.687773\pi$
$354$	0	0
$355$	−586.592	−1.65237
$356$	0	0
$357$	0	0
$358$	0	0
$359$	534.573i	1.48906i	0.667589	+	0.744530i	$0.267326\pi$
$359$	534.573i	1.48906i	−0.667589	+	0.744530i	$0.732674\pi$
$360$	0	0
$361$	233.985	0.648157
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 234.207i	− 0.641664i
$366$	0	0
$367$	−15.2923	−0.0416685	−0.0208343	−	0.999783i	$-0.506632\pi$
$367$	−15.2923	−0.0416685	−0.0208343	+	0.999783i	$0.506632\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 216.190i	− 0.582721i
$372$	0	0
$373$	652.985	1.75063	0.875314	−	0.483554i	$-0.160654\pi$
$373$	652.985	1.75063	0.875314	+	0.483554i	$0.160654\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	752.337i	1.99559i
$378$	0	0
$379$	655.215	1.72880	0.864400	−	0.502804i	$-0.167698\pi$
$379$	655.215	1.72880	0.864400	+	0.502804i	$0.167698\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	299.541i	0.782092i	0.920371	+	0.391046i	$0.127887\pi$
$383$	299.541i	0.782092i	−0.920371	+	0.391046i	$0.872113\pi$
$384$	0	0
$385$	714.831	1.85670
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 464.558i	− 1.19424i	−0.802153	−	0.597119i	$-0.796312\pi$
$389$	− 464.558i	− 1.19424i	0.802153	−	0.597119i	$-0.203688\pi$
$390$	0	0
$391$	−114.115	−0.291855
$392$	0	0
$393$	0	0
$394$	0	0
$395$	572.289i	1.44883i
$396$	0	0
$397$	−185.708	−0.467777	−0.233889	−	0.972263i	$-0.575145\pi$
$397$	−185.708	−0.467777	−0.233889	+	0.972263i	$0.575145\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 62.8592i	− 0.156756i	−0.996924	−	0.0783780i	$-0.975026\pi$
$401$	− 62.8592i	− 0.156756i	0.996924	−	0.0783780i	$-0.0249741\pi$
$402$	0	0
$403$	169.569	0.420767
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 890.182i	− 2.18718i
$408$	0	0
$409$	−327.281	−0.800197	−0.400099	−	0.916472i	$-0.631024\pi$
$409$	−327.281	−0.800197	−0.400099	+	0.916472i	$0.631024\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	517.558i	1.25317i
$414$	0	0
$415$	599.769	1.44523
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 649.875i	− 1.55101i	−0.631339	−	0.775507i	$-0.717494\pi$
$419$	− 649.875i	− 1.55101i	0.631339	−	0.775507i	$-0.282506\pi$
$420$	0	0
$421$	3.31913	0.00788391	0.00394196	−	0.999992i	$-0.498745\pi$
$421$	3.31913	0.00788391	0.00394196	+	0.999992i	$0.498745\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	66.6857i	0.156908i
$426$	0	0
$427$	−109.100	−0.255503
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 803.502i	− 1.86427i	−0.362107	−	0.932136i	$-0.617943\pi$
$431$	− 803.502i	− 1.86427i	0.362107	−	0.932136i	$-0.382057\pi$
$432$	0	0
$433$	−93.1230	−0.215065	−0.107532	−	0.994202i	$-0.534295\pi$
$433$	−93.1230	−0.215065	−0.107532	+	0.994202i	$0.534295\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	358.492i	0.820348i
$438$	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
$443$	− 569.689i	− 1.28598i	−0.765875	−	0.642989i	$-0.777694\pi$
$443$	− 569.689i	− 1.28598i	0.765875	−	0.642989i	$-0.222306\pi$
$444$	0	0
$445$	−777.358	−1.74687
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 559.115i	− 1.24524i	−0.782523	−	0.622622i	$-0.786067\pi$
$449$	− 559.115i	− 1.24524i	0.782523	−	0.622622i	$-0.213933\pi$
$450$	0	0
$451$	−494.354	−1.09613
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1030.94i	2.26580i
$456$	0	0
$457$	604.296	1.32231	0.661155	−	0.750249i	$-0.270066\pi$
$457$	604.296	1.32231	0.661155	+	0.750249i	$0.270066\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 5.13459i	− 0.0111379i	−0.999984	−	0.00556897i	$-0.998227\pi$
$461$	− 5.13459i	− 0.0111379i	0.999984	−	0.00556897i	$-0.00177267\pi$
$462$	0	0
$463$	161.492	0.348795	0.174398	−	0.984675i	$-0.444202\pi$
$463$	161.492	0.348795	0.174398	+	0.984675i	$0.444202\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	503.025i	1.07714i	0.842581	+	0.538570i	$0.181035\pi$
$467$	503.025i	1.07714i	−0.842581	+	0.538570i	$0.818965\pi$
$468$	0	0
$469$	−177.723	−0.378941
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 134.873i	− 0.285143i
$474$	0	0
$475$	209.492	0.441036
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 213.796i	− 0.446339i	−0.974780	−	0.223170i	$-0.928360\pi$
$479$	− 213.796i	− 0.446339i	0.974780	−	0.223170i	$-0.0716404\pi$
$480$	0	0
$481$	1283.83	2.66909
$482$	0	0
$483$	0	0
$484$	0	0
$485$	435.469i	0.897874i
$486$	0	0
$487$	−8.63071	−0.0177222	−0.00886110	−	0.999961i	$-0.502821\pi$
$487$	−8.63071	−0.0177222	−0.00886110	+	0.999961i	$0.502821\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 922.459i	− 1.87873i	−0.342912	−	0.939367i	$-0.611413\pi$
$491$	− 922.459i	− 1.87873i	0.342912	−	0.939367i	$-0.388587\pi$
$492$	0	0
$493$	−275.596	−0.559018
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 849.420i	− 1.70909i
$498$	0	0
$499$	434.592	0.870926	0.435463	−	0.900207i	$-0.356585\pi$
$499$	434.592	0.870926	0.435463	+	0.900207i	$0.356585\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 144.087i	− 0.286454i	−0.989690	−	0.143227i	$-0.954252\pi$
$503$	− 144.087i	− 0.286454i	0.989690	−	0.143227i	$-0.0457480\pi$
$504$	0	0
$505$	168.588	0.333839
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 697.272i	− 1.36989i	−0.728596	−	0.684943i	$-0.759827\pi$
$509$	− 697.272i	− 1.36989i	0.728596	−	0.684943i	$-0.240173\pi$
$510$	0	0
$511$	339.146	0.663691
$512$	0	0
$513$	0	0
$514$	0	0
$515$	560.475i	1.08830i
$516$	0	0
$517$	249.415	0.482428
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 426.962i	− 0.819504i	−0.912197	−	0.409752i	$-0.865615\pi$
$521$	− 426.962i	− 0.819504i	0.912197	−	0.409752i	$-0.134385\pi$
$522$	0	0
$523$	−179.762	−0.343712	−0.171856	−	0.985122i	$-0.554976\pi$
$523$	−179.762	−0.343712	−0.171856	+	0.985122i	$0.554976\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	62.1166i	0.117868i
$528$	0	0
$529$	313.000	0.591682
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 712.965i	− 1.33764i
$534$	0	0
$535$	1029.18	1.92371
$536$	0	0
$537$	0	0
$538$	0	0
$539$	314.967i	0.584354i
$540$	0	0
$541$	708.734	1.31005	0.655023	−	0.755609i	$-0.272659\pi$
$541$	708.734	1.31005	0.655023	+	0.755609i	$0.272659\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 358.990i	− 0.658697i
$546$	0	0
$547$	196.231	0.358740	0.179370	−	0.983782i	$-0.442594\pi$
$547$	196.231	0.358740	0.179370	+	0.983782i	$0.442594\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	865.781i	1.57129i
$552$	0	0
$553$	−828.708	−1.49857
$554$	0	0
$555$	0	0
$556$	0	0
$557$	353.610i	0.634848i	0.948284	+	0.317424i	$0.102818\pi$
$557$	353.610i	0.634848i	−0.948284	+	0.317424i	$0.897182\pi$
$558$	0	0
$559$	194.515	0.347970
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 373.026i	− 0.662568i	−0.943531	−	0.331284i	$-0.892518\pi$
$563$	− 373.026i	− 0.662568i	0.943531	−	0.331284i	$-0.107482\pi$
$564$	0	0
$565$	−636.415	−1.12640
$566$	0	0
$567$	0	0
$568$	0	0
$569$	60.1177i	0.105655i	0.998604	+	0.0528275i	$0.0168233\pi$
$569$	60.1177i	0.105655i	−0.998604	+	0.0528275i	$0.983177\pi$
$570$	0	0
$571$	−86.1999	−0.150963	−0.0754815	−	0.997147i	$-0.524049\pi$
$571$	−86.1999	−0.150963	−0.0754815	+	0.997147i	$0.524049\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	126.224i	0.219520i
$576$	0	0
$577$	709.123	1.22898	0.614491	−	0.788924i	$-0.289361\pi$
$577$	709.123	1.22898	0.614491	+	0.788924i	$0.289361\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	868.501i	1.49484i
$582$	0	0
$583$	378.600	0.649399
$584$	0	0
$585$	0	0
$586$	0	0
$587$	962.285i	1.63933i	0.572845	+	0.819664i	$0.305840\pi$
$587$	962.285i	1.63933i	−0.572845	+	0.819664i	$0.694160\pi$
$588$	0	0
$589$	195.138	0.331305
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 104.350i	− 0.175969i	−0.996122	−	0.0879847i	$-0.971957\pi$
$593$	− 104.350i	− 0.175969i	0.996122	−	0.0879847i	$-0.0280427\pi$
$594$	0	0
$595$	−377.654	−0.634712
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 286.063i	− 0.477567i	−0.971073	−	0.238784i	$-0.923251\pi$
$599$	− 286.063i	− 0.477567i	0.971073	−	0.238784i	$-0.0767486\pi$
$600$	0	0
$601$	−280.415	−0.466581	−0.233291	−	0.972407i	$-0.574949\pi$
$601$	−280.415	−0.466581	−0.233291	+	0.972407i	$0.574949\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	550.578i	0.910046i
$606$	0	0
$607$	−737.731	−1.21537	−0.607686	−	0.794177i	$-0.707902\pi$
$607$	−737.731	−1.21537	−0.607686	+	0.794177i	$0.707902\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	359.711i	0.588724i
$612$	0	0
$613$	−679.415	−1.10834	−0.554172	−	0.832402i	$-0.686965\pi$
$613$	−679.415	−1.10834	−0.554172	+	0.832402i	$0.686965\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 472.649i	− 0.766044i	−0.923739	−	0.383022i	$-0.874883\pi$
$617$	− 472.649i	− 0.766044i	0.923739	−	0.383022i	$-0.125117\pi$
$618$	0	0
$619$	−886.354	−1.43191	−0.715956	−	0.698145i	$-0.754009\pi$
$619$	−886.354	−1.43191	−0.715956	+	0.698145i	$0.754009\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1125.66i	− 1.80684i
$624$	0	0
$625$	−765.950	−1.22552
$626$	0	0
$627$	0	0
$628$	0	0
$629$	470.294i	0.747685i
$630$	0	0
$631$	729.108	1.15548	0.577740	−	0.816221i	$-0.303935\pi$
$631$	729.108	1.15548	0.577740	+	0.816221i	$0.303935\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	818.199i	1.28850i
$636$	0	0
$637$	−454.250	−0.713108
$638$	0	0
$639$	0	0
$640$	0	0
$641$	870.195i	1.35756i	0.734342	+	0.678780i	$0.237491\pi$
$641$	870.195i	1.35756i	−0.734342	+	0.678780i	$0.762509\pi$
$642$	0	0
$643$	618.123	0.961311	0.480656	−	0.876910i	$-0.340399\pi$
$643$	618.123	0.961311	0.480656	+	0.876910i	$0.340399\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 425.439i	− 0.657556i	−0.944407	−	0.328778i	$-0.893363\pi$
$647$	− 425.439i	− 0.657556i	0.944407	−	0.328778i	$-0.106637\pi$
$648$	0	0
$649$	−906.369	−1.39656
$650$	0	0
$651$	0	0
$652$	0	0
$653$	188.808i	0.289140i	0.989495	+	0.144570i	$0.0461799\pi$
$653$	188.808i	0.289140i	−0.989495	+	0.144570i	$0.953820\pi$
$654$	0	0
$655$	−308.238	−0.470593
$656$	0	0
$657$	0	0
$658$	0	0
$659$	660.470i	1.00223i	0.865380	+	0.501116i	$0.167077\pi$
$659$	660.470i	1.00223i	−0.865380	+	0.501116i	$0.832923\pi$
$660$	0	0
$661$	−589.831	−0.892331	−0.446165	−	0.894951i	$-0.647211\pi$
$661$	−589.831	−0.892331	−0.446165	+	0.894951i	$0.647211\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1186.39i	1.78405i
$666$	0	0
$667$	−521.654	−0.782090
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 191.060i	− 0.284739i
$672$	0	0
$673$	−391.431	−0.581621	−0.290810	−	0.956781i	$-0.593925\pi$
$673$	−391.431	−0.581621	−0.290810	+	0.956781i	$0.593925\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 693.661i	− 1.02461i	−0.858804	−	0.512305i	$-0.828792\pi$
$677$	− 693.661i	− 1.02461i	0.858804	−	0.512305i	$-0.171208\pi$
$678$	0	0
$679$	−630.585	−0.928696
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 678.170i	− 0.992928i	−0.868057	−	0.496464i	$-0.834632\pi$
$683$	− 678.170i	− 0.992928i	0.868057	−	0.496464i	$-0.165368\pi$
$684$	0	0
$685$	−9.47303	−0.0138292
$686$	0	0
$687$	0	0
$688$	0	0
$689$	546.022i	0.792485i
$690$	0	0
$691$	−675.777	−0.977969	−0.488985	−	0.872292i	$-0.662633\pi$
$691$	−675.777	−0.977969	−0.488985	+	0.872292i	$0.662633\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 111.364i	− 0.160236i
$696$	0	0
$697$	261.173	0.374710
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 797.260i	− 1.13732i	−0.822573	−	0.568659i	$-0.807462\pi$
$701$	− 797.260i	− 1.13732i	0.822573	−	0.568659i	$-0.192538\pi$
$702$	0	0
$703$	1477.42	2.10160
$704$	0	0
$705$	0	0
$706$	0	0
$707$	244.126i	0.345298i
$708$	0	0
$709$	173.504	0.244716	0.122358	−	0.992486i	$-0.460954\pi$
$709$	173.504	0.244716	0.122358	+	0.992486i	$0.460954\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	117.576i	0.164903i
$714$	0	0
$715$	−1805.42	−2.52507
$716$	0	0
$717$	0	0
$718$	0	0
$719$	188.177i	0.261721i	0.991401	+	0.130860i	$0.0417740\pi$
$719$	188.177i	0.261721i	−0.991401	+	0.130860i	$0.958226\pi$
$720$	0	0
$721$	−811.600	−1.12566
$722$	0	0
$723$	0	0
$724$	0	0
$725$	304.839i	0.420468i
$726$	0	0
$727$	−711.777	−0.979060	−0.489530	−	0.871986i	$-0.662832\pi$
$727$	−711.777	−0.979060	−0.489530	+	0.871986i	$0.662832\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	71.2548i	0.0974758i
$732$	0	0
$733$	1226.83	1.67371	0.836856	−	0.547423i	$-0.184391\pi$
$733$	1226.83	1.67371	0.836856	+	0.547423i	$0.184391\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 311.236i	− 0.422301i
$738$	0	0
$739$	−741.892	−1.00391	−0.501957	−	0.864893i	$-0.667386\pi$
$739$	−741.892	−1.00391	−0.501957	+	0.864893i	$0.667386\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	781.984i	1.05247i	0.850340	+	0.526234i	$0.176396\pi$
$743$	781.984i	1.05247i	−0.850340	+	0.526234i	$0.823604\pi$
$744$	0	0
$745$	−546.415	−0.733443
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1490.32i	1.98975i
$750$	0	0
$751$	1033.33	1.37594	0.687970	−	0.725739i	$-0.258502\pi$
$751$	1033.33	1.37594	0.687970	+	0.725739i	$0.258502\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	185.458i	0.245639i
$756$	0	0
$757$	473.877	0.625993	0.312997	−	0.949754i	$-0.398667\pi$
$757$	473.877	0.625993	0.312997	+	0.949754i	$0.398667\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	940.455i	1.23581i	0.786251	+	0.617907i	$0.212019\pi$
$761$	940.455i	1.23581i	−0.786251	+	0.617907i	$0.787981\pi$
$762$	0	0
$763$	519.839	0.681309
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1307.18i	− 1.70428i
$768$	0	0
$769$	980.815	1.27544	0.637721	−	0.770267i	$-0.279877\pi$
$769$	980.815	1.27544	0.637721	+	0.770267i	$0.279877\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1042.20i	− 1.34825i	−0.738618	−	0.674124i	$-0.764521\pi$
$773$	− 1042.20i	− 1.34825i	0.738618	−	0.674124i	$-0.235479\pi$
$774$	0	0
$775$	68.7077	0.0886550
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 820.472i	− 1.05324i
$780$	0	0
$781$	1487.54	1.90466
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1.69428i	− 0.00215832i
$786$	0	0
$787$	144.700	0.183863	0.0919315	−	0.995765i	$-0.470696\pi$
$787$	144.700	0.183863	0.0919315	+	0.995765i	$0.470696\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 921.567i	− 1.16507i
$792$	0	0
$793$	275.550	0.347478
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 155.729i	− 0.195394i	−0.995216	−	0.0976972i	$-0.968852\pi$
$797$	− 155.729i	− 0.195394i	0.995216	−	0.0976972i	$-0.0311477\pi$
$798$	0	0
$799$	−131.769	−0.164918
$800$	0	0
$801$	0	0
$802$	0	0
$803$	593.926i	0.739634i
$804$	0	0
$805$	−714.831	−0.887988
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 1216.92i	− 1.50422i	−0.659035	−	0.752112i	$-0.729035\pi$
$809$	− 1216.92i	− 1.50422i	0.659035	−	0.752112i	$-0.270965\pi$
$810$	0	0
$811$	−1243.31	−1.53305	−0.766527	−	0.642212i	$-0.778017\pi$
$811$	−1243.31	−1.53305	−0.766527	+	0.642212i	$0.778017\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	166.823i	0.204691i
$816$	0	0
$817$	223.846	0.273985
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.2854i	0.0161820i	0.999967	+	0.00809098i	$0.00257547\pi$
$821$	13.2854i	0.0161820i	−0.999967	+	0.00809098i	$0.997425\pi$
$822$	0	0
$823$	214.200	0.260267	0.130134	−	0.991496i	$-0.458459\pi$
$823$	214.200	0.260267	0.130134	+	0.991496i	$0.458459\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1221.27i	− 1.47675i	−0.674391	−	0.738374i	$-0.735594\pi$
$827$	− 1221.27i	− 1.47675i	0.674391	−	0.738374i	$-0.264406\pi$
$828$	0	0
$829$	−358.431	−0.432365	−0.216183	−	0.976353i	$-0.569361\pi$
$829$	−358.431	−0.432365	−0.216183	+	0.976353i	$0.569361\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 166.401i	− 0.199761i
$834$	0	0
$835$	324.946	0.389157
$836$	0	0
$837$	0	0
$838$	0	0
$839$	327.597i	0.390461i	0.980757	+	0.195231i	$0.0625456\pi$
$839$	327.597i	0.390461i	−0.980757	+	0.195231i	$0.937454\pi$
$840$	0	0
$841$	−418.827	−0.498011
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1624.36i	− 1.92232i
$846$	0	0
$847$	−797.269	−0.941286
$848$	0	0
$849$	0	0
$850$	0	0
$851$	890.182i	1.04604i
$852$	0	0
$853$	−52.8306	−0.0619351	−0.0309675	−	0.999520i	$-0.509859\pi$
$853$	−52.8306	−0.0619351	−0.0309675	+	0.999520i	$0.509859\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1248.42i	− 1.45673i	−0.685187	−	0.728367i	$-0.740280\pi$
$857$	− 1248.42i	− 1.45673i	0.685187	−	0.728367i	$-0.259720\pi$
$858$	0	0
$859$	446.739	0.520068	0.260034	−	0.965599i	$-0.416266\pi$
$859$	446.739	0.520068	0.260034	+	0.965599i	$0.416266\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 635.297i	− 0.736150i	−0.929796	−	0.368075i	$-0.880017\pi$
$863$	− 635.297i	− 0.736150i	0.929796	−	0.368075i	$-0.119983\pi$
$864$	0	0
$865$	1280.54	1.48039
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 1451.27i	− 1.67004i
$870$	0	0
$871$	448.869	0.515349
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 798.226i	− 0.912258i
$876$	0	0
$877$	−788.692	−0.899307	−0.449653	−	0.893203i	$-0.648453\pi$
$877$	−788.692	−0.899307	−0.449653	+	0.893203i	$0.648453\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 558.506i	− 0.633945i	−0.948435	−	0.316972i	$-0.897334\pi$
$881$	− 558.506i	− 0.633945i	0.948435	−	0.316972i	$-0.102666\pi$
$882$	0	0
$883$	−313.338	−0.354857	−0.177428	−	0.984134i	$-0.556778\pi$
$883$	−313.338	−0.354857	−0.177428	+	0.984134i	$0.556778\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	597.701i	0.673845i	0.941532	+	0.336923i	$0.109386\pi$
$887$	597.701i	0.673845i	−0.941532	+	0.336923i	$0.890614\pi$
$888$	0	0
$889$	−1184.80	−1.33273
$890$	0	0
$891$	0	0
$892$	0	0
$893$	413.951i	0.463551i
$894$	0	0
$895$	−1439.31	−1.60816
$896$	0	0
$897$	0	0
$898$	0	0
$899$	283.952i	0.315854i
$900$	0	0
$901$	−200.019	−0.221997
$902$	0	0
$903$	0	0
$904$	0	0
$905$	917.302i	1.01359i
$906$	0	0
$907$	−1086.51	−1.19791	−0.598957	−	0.800781i	$-0.704418\pi$
$907$	−1086.51	−1.19791	−0.598957	+	0.800781i	$0.704418\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	710.283i	0.779674i	0.920884	+	0.389837i	$0.127469\pi$
$911$	710.283i	0.779674i	−0.920884	+	0.389837i	$0.872531\pi$
$912$	0	0
$913$	−1520.95	−1.66589
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 446.347i	− 0.486747i
$918$	0	0
$919$	−1540.01	−1.67574	−0.837871	−	0.545868i	$-0.816200\pi$
$919$	−1540.01	−1.67574	−0.837871	+	0.545868i	$0.816200\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2145.35i	2.32432i
$924$	0	0
$925$	520.196	0.562374
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1582.59i	1.70355i	0.523911	+	0.851773i	$0.324473\pi$
$929$	1582.59i	1.70355i	−0.523911	+	0.851773i	$0.675527\pi$
$930$	0	0
$931$	−522.746	−0.561489
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 661.362i	− 0.707339i
$936$	0	0
$937$	−1747.40	−1.86489	−0.932444	−	0.361315i	$-0.882328\pi$
$937$	−1747.40	−1.86489	−0.932444	+	0.361315i	$0.882328\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	367.579i	0.390626i	0.980741	+	0.195313i	$0.0625722\pi$
$941$	367.579i	0.390626i	−0.980741	+	0.195313i	$0.937428\pi$
$942$	0	0
$943$	494.354	0.524235
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 190.168i	− 0.200811i	−0.994947	−	0.100406i	$-0.967986\pi$
$947$	− 190.168i	− 0.200811i	0.994947	−	0.100406i	$-0.0320140\pi$
$948$	0	0
$949$	−856.569	−0.902602
$950$	0	0
$951$	0	0
$952$	0	0
$953$	861.754i	0.904254i	0.891954	+	0.452127i	$0.149335\pi$
$953$	861.754i	0.904254i	−0.891954	+	0.452127i	$0.850665\pi$
$954$	0	0
$955$	200.238	0.209674
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 13.7175i	− 0.0143040i
$960$	0	0
$961$	−897.000	−0.933403
$962$	0	0
$963$	0	0
$964$	0	0
$965$	318.756i	0.330317i
$966$	0	0
$967$	−1275.31	−1.31884	−0.659418	−	0.751776i	$-0.729197\pi$
$967$	−1275.31	−1.31884	−0.659418	+	0.751776i	$0.729197\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1238.69i	1.27568i	0.770168	+	0.637841i	$0.220172\pi$
$971$	1238.69i	1.27568i	−0.770168	+	0.637841i	$0.779828\pi$
$972$	0	0
$973$	161.261	0.165736
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 18.4936i	− 0.0189290i	−0.999955	−	0.00946448i	$-0.996987\pi$
$977$	− 18.4936i	− 0.0189290i	0.999955	−	0.00946448i	$-0.00301268\pi$
$978$	0	0
$979$	1971.30	2.01359
$980$	0	0
$981$	0	0
$982$	0	0
$983$	658.643i	0.670033i	0.942212	+	0.335017i	$0.108742\pi$
$983$	658.643i	0.670033i	−0.942212	+	0.335017i	$0.891258\pi$
$984$	0	0
$985$	−207.000	−0.210152
$986$	0	0
$987$	0	0
$988$	0	0
$989$	134.873i	0.136373i
$990$	0	0
$991$	−608.484	−0.614010	−0.307005	−	0.951708i	$-0.599327\pi$
$991$	−608.484	−0.614010	−0.307005	+	0.951708i	$0.599327\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 2174.14i	− 2.18506i
$996$	0	0
$997$	−617.246	−0.619103	−0.309552	−	0.950883i	$-0.600179\pi$
$997$	−617.246	−0.619103	−0.309552	+	0.950883i	$0.600179\pi$
$998$	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.161.1		4
3.2	odd	2	inner	1296.3.e.d.161.4		4
4.3	odd	2		162.3.b.b.161.3	yes	4
9.2	odd	6		1296.3.q.o.1025.1		8
9.4	even	3		1296.3.q.o.593.1		8
9.5	odd	6		1296.3.q.o.593.4		8
9.7	even	3		1296.3.q.o.1025.4		8
12.11	even	2		162.3.b.b.161.2	✓	4
36.7	odd	6		162.3.d.c.53.4		8
36.11	even	6		162.3.d.c.53.1		8
36.23	even	6		162.3.d.c.107.4		8
36.31	odd	6		162.3.d.c.107.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.b.b.161.2	✓	4	12.11	even	2
162.3.b.b.161.3	yes	4	4.3	odd	2
162.3.d.c.53.1		8	36.11	even	6
162.3.d.c.53.4		8	36.7	odd	6
162.3.d.c.107.1		8	36.31	odd	6
162.3.d.c.107.4		8	36.23	even	6
1296.3.e.d.161.1		4	1.1	even	1	trivial
1296.3.e.d.161.4		4	3.2	odd	2	inner
1296.3.q.o.593.1		8	9.4	even	3
1296.3.q.o.593.4		8	9.5	odd	6
1296.3.q.o.1025.1		8	9.2	odd	6
1296.3.q.o.1025.4		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)

\(f(q)\)	\(=\)	\(q-5.79555i q^{5} +8.39230 q^{7} +O(q^{10})\)	\(q-5.79555i q^{5} +8.39230 q^{7} +14.6969i q^{11} -21.1962 q^{13} -7.76457i q^{17} -24.3923 q^{19} -14.6969i q^{23} -8.58846 q^{25} -35.4940i q^{29} -8.00000 q^{31} -48.6381i q^{35} -60.5692 q^{37} +33.6365i q^{41} -9.17691 q^{43} -16.9706i q^{47} +21.4308 q^{49} -25.7605i q^{53} +85.1769 q^{55} +61.6706i q^{59} -13.0000 q^{61} +122.843i q^{65} -21.1769 q^{67} -101.214i q^{71} +40.4115 q^{73} +123.341i q^{77} -98.7461 q^{79} +103.488i q^{83} -45.0000 q^{85} -134.130i q^{89} -177.885 q^{91} +141.367i q^{95} -75.1384 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7}+O(q^{10}) \) 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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