LMFDB - Embedded newform 162.3.b.b.161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,3,Mod(161,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("162.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	162.b (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:	$4.41418028264$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.3
Root			$1.93185i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.3.b.b.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -5.79555i q^{5} -8.39230 q^{7} -2.82843i q^{8} +O(q^{10})$	\(q+1.41421i q^{2} -2.00000 q^{4} -5.79555i q^{5} -8.39230 q^{7} -2.82843i q^{8} +8.19615 q^{10} -14.6969i q^{11} -21.1962 q^{13} -11.8685i q^{14} +4.00000 q^{16} -7.76457i q^{17} +24.3923 q^{19} +11.5911i q^{20} +20.7846 q^{22} +14.6969i q^{23} -8.58846 q^{25} -29.9759i q^{26} +16.7846 q^{28} -35.4940i q^{29} +8.00000 q^{31} +5.65685i q^{32} +10.9808 q^{34} +48.6381i q^{35} -60.5692 q^{37} +34.4959i q^{38} -16.3923 q^{40} +33.6365i q^{41} +9.17691 q^{43} +29.3939i q^{44} -20.7846 q^{46} +16.9706i q^{47} +21.4308 q^{49} -12.1459i q^{50} +42.3923 q^{52} -25.7605i q^{53} -85.1769 q^{55} +23.7370i q^{56} +50.1962 q^{58} -61.6706i q^{59} -13.0000 q^{61} +11.3137i q^{62} -8.00000 q^{64} +122.843i q^{65} +21.1769 q^{67} +15.5291i q^{68} -68.7846 q^{70} +101.214i q^{71} +40.4115 q^{73} -85.6578i q^{74} -48.7846 q^{76} +123.341i q^{77} +98.7461 q^{79} -23.1822i q^{80} -47.5692 q^{82} -103.488i q^{83} -45.0000 q^{85} +12.9781i q^{86} -41.5692 q^{88} -134.130i q^{89} +177.885 q^{91} -29.3939i q^{92} -24.0000 q^{94} -141.367i q^{95} -75.1384 q^{97} +30.3077i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + 8 q^{7}+O(q^{10}) $ 4 * q - 8 * q^4 + 8 * q^7	$ 4 q - 8 q^{4} + 8 q^{7} + 12 q^{10} - 64 q^{13} + 16 q^{16} + 56 q^{19} + 28 q^{25} - 16 q^{28} + 32 q^{31} - 60 q^{34} - 76 q^{37} - 24 q^{40} - 88 q^{43} + 252 q^{49} + 128 q^{52} - 216 q^{55} + 180 q^{58} - 52 q^{61} - 32 q^{64} - 40 q^{67} - 192 q^{70} + 224 q^{73} - 112 q^{76} + 104 q^{79} - 24 q^{82} - 180 q^{85} + 88 q^{91} - 96 q^{94} + 32 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 + 8 * q^7 + 12 * q^10 - 64 * q^13 + 16 * q^16 + 56 * q^19 + 28 * q^25 - 16 * q^28 + 32 * q^31 - 60 * q^34 - 76 * q^37 - 24 * q^40 - 88 * q^43 + 252 * q^49 + 128 * q^52 - 216 * q^55 + 180 * q^58 - 52 * q^61 - 32 * q^64 - 40 * q^67 - 192 * q^70 + 224 * q^73 - 112 * q^76 + 104 * q^79 - 24 * q^82 - 180 * q^85 + 88 * q^91 - 96 * q^94 + 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$-1$

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 $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	− 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	$-0.704614\pi$
$7$	−8.39230	−1.19890	−0.599450	+	0.800412i	$0.704614\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	8.19615	0.819615
$11$	− 14.6969i	− 1.33609i	−0.744123	−	0.668043i	$-0.767132\pi$
$11$	− 14.6969i	− 1.33609i	0.744123	−	0.668043i	$-0.232868\pi$
$12$	0	0
$13$	−21.1962	−1.63047	−0.815237	−	0.579128i	$-0.803393\pi$
$13$	−21.1962	−1.63047	−0.815237	+	0.579128i	$0.803393\pi$
$14$	− 11.8685i	− 0.847751i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 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	$-0.278145\pi$
$19$	24.3923	1.28381	0.641903	+	0.766786i	$0.278145\pi$
$20$	11.5911i	0.579555i
$21$	0	0
$22$	20.7846	0.944755
$23$	14.6969i	0.638997i	0.947587	+	0.319499i	$0.103514\pi$
$23$	14.6969i	0.638997i	−0.947587	+	0.319499i	$0.896486\pi$
$24$	0	0
$25$	−8.58846	−0.343538
$26$	− 29.9759i	− 1.15292i
$27$	0	0
$28$	16.7846	0.599450
$29$	− 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	$-0.458813\pi$
$31$	8.00000	0.258065	0.129032	+	0.991640i	$0.458813\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	10.9808	0.322964
$35$	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	$0.805196\pi$
$38$	34.4959i	0.907788i
$39$	0	0
$40$	−16.3923	−0.409808
$41$	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	$-0.465969\pi$
$43$	9.17691	0.213417	0.106708	+	0.994290i	$0.465969\pi$
$44$	29.3939i	0.668043i
$45$	0	0
$46$	−20.7846	−0.451839
$47$	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
$49$	21.4308	0.437363
$50$	− 12.1459i	− 0.242918i
$51$	0	0
$52$	42.3923	0.815237
$53$	− 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
$56$	23.7370i	0.423875i
$57$	0	0
$58$	50.1962	0.865451
$59$	− 61.6706i	− 1.04526i	−0.852558	−	0.522632i	$-0.824950\pi$
$59$	− 61.6706i	− 1.04526i	0.852558	−	0.522632i	$-0.175050\pi$
$60$	0	0
$61$	−13.0000	−0.213115	−0.106557	−	0.994307i	$-0.533983\pi$
$61$	−13.0000	−0.213115	−0.106557	+	0.994307i	$0.533983\pi$
$62$	11.3137i	0.182479i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	122.843i	1.88990i
$66$	0	0
$67$	21.1769	0.316073	0.158037	−	0.987433i	$-0.449484\pi$
$67$	21.1769	0.316073	0.158037	+	0.987433i	$0.449484\pi$
$68$	15.5291i	0.228370i
$69$	0	0
$70$	−68.7846	−0.982637
$71$	101.214i	1.42555i	0.701392	+	0.712776i	$0.252562\pi$
$71$	101.214i	1.42555i	−0.701392	+	0.712776i	$0.747438\pi$
$72$	0	0
$73$	40.4115	0.553583	0.276791	−	0.960930i	$-0.410729\pi$
$73$	40.4115	0.553583	0.276791	+	0.960930i	$0.410729\pi$
$74$	− 85.6578i	− 1.15754i
$75$	0	0
$76$	−48.7846	−0.641903
$77$	123.341i	1.60183i
$78$	0	0
$79$	98.7461	1.24995	0.624976	−	0.780644i	$-0.285109\pi$
$79$	98.7461	1.24995	0.624976	+	0.780644i	$0.285109\pi$
$80$	− 23.1822i	− 0.289778i
$81$	0	0
$82$	−47.5692	−0.580112
$83$	− 103.488i	− 1.24684i	−0.781887	−	0.623420i	$-0.785743\pi$
$83$	− 103.488i	− 1.24684i	0.781887	−	0.623420i	$-0.214257\pi$
$84$	0	0
$85$	−45.0000	−0.529412
$86$	12.9781i	0.150908i
$87$	0	0
$88$	−41.5692	−0.472377
$89$	− 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
$92$	− 29.3939i	− 0.319499i
$93$	0	0
$94$	−24.0000	−0.255319
$95$	− 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	$0.626596\pi$
$98$	30.3077i	0.309262i
$99$	0	0
$100$	17.1769	0.171769
$101$	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	$-0.344451\pi$
$103$	96.7077	0.938909	0.469455	+	0.882957i	$0.344451\pi$
$104$	59.9518i	0.576459i
$105$	0	0
$106$	36.4308	0.343687
$107$	− 177.582i	− 1.65964i	−0.558030	−	0.829821i	$-0.688442\pi$
$107$	− 177.582i	− 1.65964i	0.558030	−	0.829821i	$-0.311558\pi$
$108$	0	0
$109$	61.9423	0.568278	0.284139	−	0.958783i	$-0.408292\pi$
$109$	61.9423	0.568278	0.284139	+	0.958783i	$0.408292\pi$
$110$	− 120.458i	− 1.09508i
$111$	0	0
$112$	−33.5692	−0.299725
$113$	− 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
$116$	70.9881i	0.611966i
$117$	0	0
$118$	87.2154	0.739113
$119$	65.1626i	0.547585i
$120$	0	0
$121$	−95.0000	−0.785124
$122$	− 18.3848i	− 0.150695i
$123$	0	0
$124$	−16.0000	−0.129032
$125$	− 95.1140i	− 0.760912i
$126$	0	0
$127$	141.177	1.11163	0.555815	−	0.831306i	$-0.312406\pi$
$127$	141.177	1.11163	0.555815	+	0.831306i	$0.312406\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	−173.727	−1.33636
$131$	53.1853i	0.405995i	0.979179	+	0.202997i	$0.0650683\pi$
$131$	53.1853i	0.405995i	−0.979179	+	0.202997i	$0.934932\pi$
$132$	0	0
$133$	−204.708	−1.53916
$134$	29.9487i	0.223498i
$135$	0	0
$136$	−21.9615	−0.161482
$137$	− 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	$-0.522019\pi$
$139$	−19.2154	−0.138240	−0.0691201	+	0.997608i	$0.522019\pi$
$140$	− 97.2761i	− 0.694829i
$141$	0	0
$142$	−143.138	−1.00802
$143$	311.519i	2.17845i
$144$	0	0
$145$	−205.708	−1.41867
$146$	57.1506i	0.391442i
$147$	0	0
$148$	121.138	0.818503
$149$	− 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	$-0.466208\pi$
$151$	32.0000	0.211921	0.105960	+	0.994370i	$0.466208\pi$
$152$	− 68.9919i	− 0.453894i
$153$	0	0
$154$	−174.431	−1.13267
$155$	− 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	$0.499704\pi$
$158$	139.648i	0.883849i
$159$	0	0
$160$	32.7846	0.204904
$161$	− 123.341i	− 0.766094i
$162$	0	0
$163$	28.7846	0.176593	0.0882963	−	0.996094i	$-0.471858\pi$
$163$	28.7846	0.176593	0.0882963	+	0.996094i	$0.471858\pi$
$164$	− 67.2730i	− 0.410201i
$165$	0	0
$166$	146.354	0.881650
$167$	− 56.0682i	− 0.335737i	−0.985809	−	0.167869i	$-0.946312\pi$
$167$	− 56.0682i	− 0.335737i	0.985809	−	0.167869i	$-0.0536885\pi$
$168$	0	0
$169$	280.277	1.65844
$170$	− 63.6396i	− 0.374351i
$171$	0	0
$172$	−18.3538	−0.106708
$173$	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
$176$	− 58.7878i	− 0.334021i
$177$	0	0
$178$	189.688	1.06567
$179$	248.347i	1.38741i	0.720258	+	0.693706i	$0.244023\pi$
$179$	248.347i	1.38741i	−0.720258	+	0.693706i	$0.755977\pi$
$180$	0	0
$181$	−158.277	−0.874458	−0.437229	−	0.899350i	$-0.644040\pi$
$181$	−158.277	−0.874458	−0.437229	+	0.899350i	$0.644040\pi$
$182$	251.567i	1.38224i
$183$	0	0
$184$	41.5692	0.225920
$185$	351.032i	1.89747i
$186$	0	0
$187$	−114.115	−0.610243
$188$	− 33.9411i	− 0.180538i
$189$	0	0
$190$	199.923	1.05223
$191$	− 34.5503i	− 0.180892i	−0.995901	−	0.0904459i	$-0.971171\pi$
$191$	− 34.5503i	− 0.180892i	0.995901	−	0.0904459i	$-0.0288292\pi$
$192$	0	0
$193$	−55.0000	−0.284974	−0.142487	−	0.989797i	$-0.545510\pi$
$193$	−55.0000	−0.284974	−0.142487	+	0.989797i	$0.545510\pi$
$194$	− 106.262i	− 0.547741i
$195$	0	0
$196$	−42.8616	−0.218681
$197$	− 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	$-0.891588\pi$
$199$	−375.138	−1.88512	−0.942559	+	0.334040i	$0.891588\pi$
$200$	24.2918i	0.121459i
$201$	0	0
$202$	−41.1384	−0.203656
$203$	297.877i	1.46737i
$204$	0	0
$205$	194.942	0.950938
$206$	136.765i	0.663909i
$207$	0	0
$208$	−84.7846	−0.407618
$209$	− 358.492i	− 1.71527i
$210$	0	0
$211$	307.454	1.45713	0.728563	−	0.684978i	$-0.240188\pi$
$211$	307.454	1.45713	0.728563	+	0.684978i	$0.240188\pi$
$212$	51.5209i	0.243023i
$213$	0	0
$214$	251.138	1.17354
$215$	− 53.1853i	− 0.247374i
$216$	0	0
$217$	−67.1384	−0.309394
$218$	87.5996i	0.401833i
$219$	0	0
$220$	170.354	0.774336
$221$	164.579i	0.744702i
$222$	0	0
$223$	−338.592	−1.51835	−0.759175	−	0.650886i	$-0.774398\pi$
$223$	−338.592	−1.51835	−0.759175	+	0.650886i	$0.774398\pi$
$224$	− 47.4740i	− 0.211938i
$225$	0	0
$226$	155.296	0.687151
$227$	− 123.950i	− 0.546037i	−0.962009	−	0.273019i	$-0.911978\pi$
$227$	− 123.950i	− 0.546037i	0.962009	−	0.273019i	$-0.0880220\pi$
$228$	0	0
$229$	74.0577	0.323396	0.161698	−	0.986840i	$-0.448303\pi$
$229$	74.0577	0.323396	0.161698	+	0.986840i	$0.448303\pi$
$230$	120.458i	0.523732i
$231$	0	0
$232$	−100.392	−0.432725
$233$	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
$236$	123.341i	0.522632i
$237$	0	0
$238$	−92.1539	−0.387201
$239$	− 452.885i	− 1.89492i	−0.319877	−	0.947459i	$-0.603641\pi$
$239$	− 452.885i	− 1.89492i	0.319877	−	0.947459i	$-0.396359\pi$
$240$	0	0
$241$	−382.688	−1.58792	−0.793959	−	0.607971i	$-0.791984\pi$
$241$	−382.688	−1.58792	−0.793959	+	0.607971i	$0.791984\pi$
$242$	− 134.350i	− 0.555166i
$243$	0	0
$244$	26.0000	0.106557
$245$	− 124.203i	− 0.506952i
$246$	0	0
$247$	−517.023	−2.09321
$248$	− 22.6274i	− 0.0912396i
$249$	0	0
$250$	134.512	0.538046
$251$	− 73.9307i	− 0.294544i	−0.989096	−	0.147272i	$-0.952951\pi$
$251$	− 73.9307i	− 0.294544i	0.989096	−	0.147272i	$-0.0470493\pi$
$252$	0	0
$253$	216.000	0.853755
$254$	199.654i	0.786041i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 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
$260$	− 245.687i	− 0.944950i
$261$	0	0
$262$	−75.2154	−0.287082
$263$	259.106i	0.985193i	0.870258	+	0.492596i	$0.163952\pi$
$263$	259.106i	0.985193i	−0.870258	+	0.492596i	$0.836048\pi$
$264$	0	0
$265$	−149.296	−0.563382
$266$	− 289.500i	− 1.08835i
$267$	0	0
$268$	−42.3538	−0.158037
$269$	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	$-0.442059\pi$
$271$	98.1154	0.362050	0.181025	+	0.983479i	$0.442059\pi$
$272$	− 31.0583i	− 0.114185i
$273$	0	0
$274$	2.31158	0.00843642
$275$	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	$0.500813\pi$
$278$	− 27.1747i	− 0.0977506i
$279$	0	0
$280$	137.569	0.491319
$281$	− 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	$-0.526628\pi$
$283$	−47.2923	−0.167111	−0.0835554	+	0.996503i	$0.526628\pi$
$284$	− 202.428i	− 0.712776i
$285$	0	0
$286$	−440.554	−1.54040
$287$	− 282.288i	− 0.983582i
$288$	0	0
$289$	228.711	0.791389
$290$	− 290.915i	− 1.00315i
$291$	0	0
$292$	−80.8231	−0.276791
$293$	− 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
$296$	171.316i	0.578769i
$297$	0	0
$298$	133.335	0.447432
$299$	− 311.519i	− 1.04187i
$300$	0	0
$301$	−77.0155	−0.255865
$302$	45.2548i	0.149850i
$303$	0	0
$304$	97.5692	0.320951
$305$	75.3422i	0.247024i
$306$	0	0
$307$	10.3538	0.0337258	0.0168629	−	0.999858i	$-0.494632\pi$
$307$	10.3538	0.0337258	0.0168629	+	0.999858i	$0.494632\pi$
$308$	− 246.682i	− 0.800917i
$309$	0	0
$310$	65.5692	0.211514
$311$	− 9.54047i	− 0.0306768i	−0.999882	−	0.0153384i	$-0.995117\pi$
$311$	− 9.54047i	− 0.0306768i	0.999882	−	0.0153384i	$-0.00488255\pi$
$312$	0	0
$313$	−479.277	−1.53124	−0.765618	−	0.643295i	$-0.777567\pi$
$313$	−479.277	−1.53124	−0.765618	+	0.643295i	$0.777567\pi$
$314$	0.413434i	0.00131667i
$315$	0	0
$316$	−197.492	−0.624976
$317$	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
$320$	46.3644i	0.144889i
$321$	0	0
$322$	174.431	0.541710
$323$	− 189.396i	− 0.586365i
$324$	0	0
$325$	182.042	0.560130
$326$	40.7076i	0.124870i
$327$	0	0
$328$	95.1384	0.290056
$329$	− 142.422i	− 0.432894i
$330$	0	0
$331$	295.454	0.892610	0.446305	−	0.894881i	$-0.352740\pi$
$331$	295.454	0.892610	0.446305	+	0.894881i	$0.352740\pi$
$332$	206.976i	0.623420i
$333$	0	0
$334$	79.2923	0.237402
$335$	− 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	$0.241420\pi$
$338$	396.371i	1.17270i
$339$	0	0
$340$	90.0000	0.264706
$341$	− 117.576i	− 0.344796i
$342$	0	0
$343$	231.369	0.674546
$344$	− 25.9562i	− 0.0754542i
$345$	0	0
$346$	−312.473	−0.903101
$347$	666.682i	1.92127i	0.277807	+	0.960637i	$0.410392\pi$
$347$	666.682i	1.92127i	−0.277807	+	0.960637i	$0.589608\pi$
$348$	0	0
$349$	511.969	1.46696	0.733480	−	0.679711i	$-0.237895\pi$
$349$	511.969	1.46696	0.733480	+	0.679711i	$0.237895\pi$
$350$	101.932i	0.291235i
$351$	0	0
$352$	83.1384	0.236189
$353$	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
$356$	268.260i	0.753539i
$357$	0	0
$358$	−351.215	−0.981049
$359$	− 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
$361$	233.985	0.648157
$362$	− 223.837i	− 0.618335i
$363$	0	0
$364$	−355.769	−0.977388
$365$	− 234.207i	− 0.641664i
$366$	0	0
$367$	15.2923	0.0416685	0.0208343	−	0.999783i	$-0.493368\pi$
$367$	15.2923	0.0416685	0.0208343	+	0.999783i	$0.493368\pi$
$368$	58.7878i	0.159749i
$369$	0	0
$370$	−496.435	−1.34172
$371$	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	$0.160654\pi$
$374$	− 161.384i	− 0.431507i
$375$	0	0
$376$	48.0000	0.127660
$377$	752.337i	1.99559i
$378$	0	0
$379$	−655.215	−1.72880	−0.864400	−	0.502804i	$-0.832302\pi$
$379$	−655.215	−1.72880	−0.864400	+	0.502804i	$0.832302\pi$
$380$	282.734i	0.744037i
$381$	0	0
$382$	48.8616	0.127910
$383$	− 299.541i	− 0.782092i	−0.920371	−	0.391046i	$-0.872113\pi$
$383$	− 299.541i	− 0.782092i	0.920371	−	0.391046i	$-0.127887\pi$
$384$	0	0
$385$	714.831	1.85670
$386$	− 77.7817i	− 0.201507i
$387$	0	0
$388$	150.277	0.387312
$389$	− 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
$392$	− 60.6154i	− 0.154631i
$393$	0	0
$394$	50.5115	0.128202
$395$	− 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	$0.575145\pi$
$398$	− 530.526i	− 1.33298i
$399$	0	0
$400$	−34.3538	−0.0858846
$401$	− 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
$404$	− 58.1785i	− 0.144006i
$405$	0	0
$406$	−421.261	−1.03759
$407$	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	$0.631024\pi$
$410$	275.690i	0.672415i
$411$	0	0
$412$	−193.415	−0.469455
$413$	517.558i	1.25317i
$414$	0	0
$415$	−599.769	−1.44523
$416$	− 119.904i	− 0.288230i
$417$	0	0
$418$	506.985	1.21288
$419$	649.875i	1.55101i	0.631339	+	0.775507i	$0.282506\pi$
$419$	649.875i	1.55101i	−0.631339	+	0.775507i	$0.717494\pi$
$420$	0	0
$421$	3.31913	0.00788391	0.00394196	−	0.999992i	$-0.498745\pi$
$421$	3.31913	0.00788391	0.00394196	+	0.999992i	$0.498745\pi$
$422$	434.805i	1.03034i
$423$	0	0
$424$	−72.8616	−0.171843
$425$	66.6857i	0.156908i
$426$	0	0
$427$	109.100	0.255503
$428$	355.163i	0.829821i
$429$	0	0
$430$	75.2154	0.174920
$431$	803.502i	1.86427i	0.362107	+	0.932136i	$0.382057\pi$
$431$	803.502i	1.86427i	−0.362107	+	0.932136i	$0.617943\pi$
$432$	0	0
$433$	−93.1230	−0.215065	−0.107532	−	0.994202i	$-0.534295\pi$
$433$	−93.1230	−0.215065	−0.107532	+	0.994202i	$0.534295\pi$
$434$	− 94.9481i	− 0.218774i
$435$	0	0
$436$	−123.885	−0.284139
$437$	358.492i	0.820348i
$438$	0	0
$439$	−472.000	−1.07517	−0.537585	−	0.843209i	$-0.680663\pi$
$439$	−472.000	−1.07517	−0.537585	+	0.843209i	$0.680663\pi$
$440$	240.917i	0.547538i
$441$	0	0
$442$	−232.750	−0.526584
$443$	569.689i	1.28598i	0.765875	+	0.642989i	$0.222306\pi$
$443$	569.689i	1.28598i	−0.765875	+	0.642989i	$0.777694\pi$
$444$	0	0
$445$	−777.358	−1.74687
$446$	− 478.842i	− 1.07364i
$447$	0	0
$448$	67.1384	0.149863
$449$	− 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
$452$	219.622i	0.485889i
$453$	0	0
$454$	175.292	0.386106
$455$	− 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	$0.270066\pi$
$458$	104.733i	0.228676i
$459$	0	0
$460$	−170.354	−0.370334
$461$	− 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	$-0.555798\pi$
$463$	−161.492	−0.348795	−0.174398	+	0.984675i	$0.555798\pi$
$464$	− 141.976i	− 0.305983i
$465$	0	0
$466$	−386.396	−0.829176
$467$	− 503.025i	− 1.07714i	−0.842581	−	0.538570i	$-0.818965\pi$
$467$	− 503.025i	− 1.07714i	0.842581	−	0.538570i	$-0.181035\pi$
$468$	0	0
$469$	−177.723	−0.378941
$470$	139.093i	0.295943i
$471$	0	0
$472$	−174.431	−0.369557
$473$	− 134.873i	− 0.285143i
$474$	0	0
$475$	−209.492	−0.441036
$476$	− 130.325i	− 0.273793i
$477$	0	0
$478$	640.477	1.33991
$479$	213.796i	0.446339i	0.974780	+	0.223170i	$0.0716404\pi$
$479$	213.796i	0.446339i	−0.974780	+	0.223170i	$0.928360\pi$
$480$	0	0
$481$	1283.83	2.66909
$482$	− 541.203i	− 1.12283i
$483$	0	0
$484$	190.000	0.392562
$485$	435.469i	0.897874i
$486$	0	0
$487$	8.63071	0.0177222	0.00886110	−	0.999961i	$-0.497179\pi$
$487$	8.63071	0.0177222	0.00886110	+	0.999961i	$0.497179\pi$
$488$	36.7696i	0.0753474i
$489$	0	0
$490$	175.650	0.358469
$491$	922.459i	1.87873i	0.342912	+	0.939367i	$0.388587\pi$
$491$	922.459i	1.87873i	−0.342912	+	0.939367i	$0.611413\pi$
$492$	0	0
$493$	−275.596	−0.559018
$494$	− 731.181i	− 1.48012i
$495$	0	0
$496$	32.0000	0.0645161
$497$	− 849.420i	− 1.70909i
$498$	0	0
$499$	−434.592	−0.870926	−0.435463	−	0.900207i	$-0.643415\pi$
$499$	−434.592	−0.870926	−0.435463	+	0.900207i	$0.643415\pi$
$500$	190.228i	0.380456i
$501$	0	0
$502$	104.554	0.208274
$503$	144.087i	0.286454i	0.989690	+	0.143227i	$0.0457480\pi$
$503$	144.087i	0.286454i	−0.989690	+	0.143227i	$0.954252\pi$
$504$	0	0
$505$	168.588	0.333839
$506$	305.470i	0.603696i
$507$	0	0
$508$	−282.354	−0.555815
$509$	− 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
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	226.004	0.439696
$515$	− 560.475i	− 1.08830i
$516$	0	0
$517$	249.415	0.482428
$518$	718.866i	1.38777i
$519$	0	0
$520$	347.454	0.668180
$521$	− 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	$-0.445024\pi$
$523$	179.762	0.343712	0.171856	+	0.985122i	$0.445024\pi$
$524$	− 106.371i	− 0.202997i
$525$	0	0
$526$	−366.431	−0.696636
$527$	− 62.1166i	− 0.117868i
$528$	0	0
$529$	313.000	0.591682
$530$	− 211.137i	− 0.398371i
$531$	0	0
$532$	409.415	0.769578
$533$	− 712.965i	− 1.33764i
$534$	0	0
$535$	−1029.18	−1.92371
$536$	− 59.8974i	− 0.111749i
$537$	0	0
$538$	−35.0962	−0.0652346
$539$	− 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	$0.272659\pi$
$542$	138.756i	0.256008i
$543$	0	0
$544$	43.9230	0.0807409
$545$	− 358.990i	− 0.658697i
$546$	0	0
$547$	−196.231	−0.358740	−0.179370	−	0.983782i	$-0.557406\pi$
$547$	−196.231	−0.358740	−0.179370	+	0.983782i	$0.557406\pi$
$548$	3.26907i	0.00596545i
$549$	0	0
$550$	−178.508	−0.324560
$551$	− 865.781i	− 1.57129i
$552$	0	0
$553$	−828.708	−1.49857
$554$	− 2.00156i	− 0.00361292i
$555$	0	0
$556$	38.4308	0.0691201
$557$	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
$560$	194.552i	0.347415i
$561$	0	0
$562$	142.435	0.253442
$563$	373.026i	0.662568i	0.943531	+	0.331284i	$0.107482\pi$
$563$	373.026i	0.662568i	−0.943531	+	0.331284i	$0.892518\pi$
$564$	0	0
$565$	−636.415	−1.12640
$566$	− 66.8815i	− 0.118165i
$567$	0	0
$568$	286.277	0.504009
$569$	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	$-0.475951\pi$
$571$	86.1999	0.150963	0.0754815	+	0.997147i	$0.475951\pi$
$572$	− 623.037i	− 1.08923i
$573$	0	0
$574$	399.215	0.695497
$575$	− 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	$0.289361\pi$
$578$	323.447i	0.559597i
$579$	0	0
$580$	411.415	0.709337
$581$	868.501i	1.49484i
$582$	0	0
$583$	−378.600	−0.649399
$584$	− 114.301i	− 0.195721i
$585$	0	0
$586$	471.804	0.805126
$587$	− 962.285i	− 1.63933i	−0.572845	−	0.819664i	$-0.694160\pi$
$587$	− 962.285i	− 1.63933i	0.572845	−	0.819664i	$-0.305840\pi$
$588$	0	0
$589$	195.138	0.331305
$590$	− 505.462i	− 0.856715i
$591$	0	0
$592$	−242.277	−0.409251
$593$	− 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
$596$	188.564i	0.316382i
$597$	0	0
$598$	440.554	0.736712
$599$	286.063i	0.477567i	0.971073	+	0.238784i	$0.0767486\pi$
$599$	286.063i	0.477567i	−0.971073	+	0.238784i	$0.923251\pi$
$600$	0	0
$601$	−280.415	−0.466581	−0.233291	−	0.972407i	$-0.574949\pi$
$601$	−280.415	−0.466581	−0.233291	+	0.972407i	$0.574949\pi$
$602$	− 108.916i	− 0.180924i
$603$	0	0
$604$	−64.0000	−0.105960
$605$	550.578i	0.910046i
$606$	0	0
$607$	737.731	1.21537	0.607686	−	0.794177i	$-0.292098\pi$
$607$	737.731	1.21537	0.607686	+	0.794177i	$0.292098\pi$
$608$	137.984i	0.226947i
$609$	0	0
$610$	−106.550	−0.174672
$611$	− 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	$0.686965\pi$
$614$	14.6425i	0.0238478i
$615$	0	0
$616$	348.862	0.566334
$617$	− 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	$-0.245991\pi$
$619$	886.354	1.43191	0.715956	+	0.698145i	$0.245991\pi$
$620$	92.7289i	0.149563i
$621$	0	0
$622$	13.4923	0.0216917
$623$	1125.66i	1.80684i
$624$	0	0
$625$	−765.950	−1.22552
$626$	− 677.800i	− 1.08275i
$627$	0	0
$628$	−0.584684	−0.000931025 0
$629$	470.294i	0.747685i
$630$	0	0
$631$	−729.108	−1.15548	−0.577740	−	0.816221i	$-0.696065\pi$
$631$	−729.108	−1.15548	−0.577740	+	0.816221i	$0.696065\pi$
$632$	− 279.296i	− 0.441924i
$633$	0	0
$634$	−254.081	−0.400758
$635$	− 818.199i	− 1.28850i
$636$	0	0
$637$	−454.250	−0.713108
$638$	− 737.730i	− 1.15632i
$639$	0	0
$640$	−65.5692	−0.102452
$641$	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	$-0.659601\pi$
$643$	−618.123	−0.961311	−0.480656	+	0.876910i	$0.659601\pi$
$644$	246.682i	0.383047i
$645$	0	0
$646$	267.846	0.414622
$647$	425.439i	0.657556i	0.944407	+	0.328778i	$0.106637\pi$
$647$	425.439i	0.657556i	−0.944407	+	0.328778i	$0.893363\pi$
$648$	0	0
$649$	−906.369	−1.39656
$650$	257.447i	0.396072i
$651$	0	0
$652$	−57.5692	−0.0882963
$653$	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
$656$	134.546i	0.205101i
$657$	0	0
$658$	201.415	0.306102
$659$	− 660.470i	− 1.00223i	−0.865380	−	0.501116i	$-0.832923\pi$
$659$	− 660.470i	− 1.00223i	0.865380	−	0.501116i	$-0.167077\pi$
$660$	0	0
$661$	−589.831	−0.892331	−0.446165	−	0.894951i	$-0.647211\pi$
$661$	−589.831	−0.892331	−0.446165	+	0.894951i	$0.647211\pi$
$662$	417.835i	0.631170i
$663$	0	0
$664$	−292.708	−0.440825
$665$	1186.39i	1.78405i
$666$	0	0
$667$	521.654	0.782090
$668$	112.136i	0.167869i
$669$	0	0
$670$	173.569	0.259059
$671$	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	$0.593925\pi$
$674$	691.920i	1.02659i
$675$	0	0
$676$	−560.554	−0.829222
$677$	− 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
$680$	127.279i	0.187175i
$681$	0	0
$682$	166.277	0.243808
$683$	678.170i	0.992928i	0.868057	+	0.496464i	$0.165368\pi$
$683$	678.170i	0.992928i	−0.868057	+	0.496464i	$0.834632\pi$
$684$	0	0
$685$	−9.47303	−0.0138292
$686$	327.206i	0.476976i
$687$	0	0
$688$	36.7077	0.0533542
$689$	546.022i	0.792485i
$690$	0	0
$691$	675.777	0.977969	0.488985	−	0.872292i	$-0.337367\pi$
$691$	675.777	0.977969	0.488985	+	0.872292i	$0.337367\pi$
$692$	− 441.904i	− 0.638589i
$693$	0	0
$694$	−942.831	−1.35855
$695$	111.364i	0.160236i
$696$	0	0
$697$	261.173	0.374710
$698$	724.034i	1.03730i
$699$	0	0
$700$	−144.154	−0.205934
$701$	− 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
$704$	117.576i	0.167011i
$705$	0	0
$706$	−829.692	−1.17520
$707$	− 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	$0.460954\pi$
$710$	829.567i	1.16840i
$711$	0	0
$712$	−379.377	−0.532833
$713$	117.576i	0.164903i
$714$	0	0
$715$	1805.42	2.52507
$716$	− 496.694i	− 0.693706i
$717$	0	0
$718$	756.000	1.05292
$719$	− 188.177i	− 0.261721i	−0.991401	−	0.130860i	$-0.958226\pi$
$719$	− 188.177i	− 0.261721i	0.991401	−	0.130860i	$-0.0417740\pi$
$720$	0	0
$721$	−811.600	−1.12566
$722$	330.904i	0.458316i
$723$	0	0
$724$	316.554	0.437229
$725$	304.839i	0.420468i
$726$	0	0
$727$	711.777	0.979060	0.489530	−	0.871986i	$-0.337168\pi$
$727$	711.777	0.979060	0.489530	+	0.871986i	$0.337168\pi$
$728$	− 503.134i	− 0.691118i
$729$	0	0
$730$	331.219	0.453725
$731$	− 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	$0.184391\pi$
$734$	21.6266i	0.0294641i
$735$	0	0
$736$	−83.1384	−0.112960
$737$	− 311.236i	− 0.422301i
$738$	0	0
$739$	741.892	1.00391	0.501957	−	0.864893i	$-0.332614\pi$
$739$	741.892	1.00391	0.501957	+	0.864893i	$0.332614\pi$
$740$	− 702.064i	− 0.948736i
$741$	0	0
$742$	−305.738	−0.412046
$743$	− 781.984i	− 1.05247i	−0.850340	−	0.526234i	$-0.823604\pi$
$743$	− 781.984i	− 1.05247i	0.850340	−	0.526234i	$-0.176396\pi$
$744$	0	0
$745$	−546.415	−0.733443
$746$	923.460i	1.23788i
$747$	0	0
$748$	228.231	0.305121
$749$	1490.32i	1.98975i
$750$	0	0
$751$	−1033.33	−1.37594	−0.687970	−	0.725739i	$-0.741498\pi$
$751$	−1033.33	−1.37594	−0.687970	+	0.725739i	$0.741498\pi$
$752$	67.8823i	0.0902690i
$753$	0	0
$754$	−1063.97	−1.41109
$755$	− 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	$0.398667\pi$
$758$	− 926.614i	− 1.22245i
$759$	0	0
$760$	−399.846	−0.526113
$761$	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
$764$	69.1007i	0.0904459i
$765$	0	0
$766$	423.615	0.553023
$767$	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	$0.279877\pi$
$770$	1010.92i	1.31289i
$771$	0	0
$772$	110.000	0.142487
$773$	− 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
$776$	212.524i	0.273871i
$777$	0	0
$778$	656.985	0.844453
$779$	820.472i	1.05324i
$780$	0	0
$781$	1487.54	1.90466
$782$	161.384i	0.206373i
$783$	0	0
$784$	85.7231	0.109341
$785$	− 1.69428i	− 0.00215832i
$786$	0	0
$787$	−144.700	−0.183863	−0.0919315	−	0.995765i	$-0.529304\pi$
$787$	−144.700	−0.183863	−0.0919315	+	0.995765i	$0.529304\pi$
$788$	71.4341i	0.0906524i
$789$	0	0
$790$	809.338	1.02448
$791$	921.567i	1.16507i
$792$	0	0
$793$	275.550	0.347478
$794$	− 262.630i	− 0.330769i
$795$	0	0
$796$	750.277	0.942559
$797$	− 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
$800$	− 48.5837i	− 0.0607296i
$801$	0	0
$802$	88.8963	0.110843
$803$	− 593.926i	− 0.739634i
$804$	0	0
$805$	−714.831	−0.887988
$806$	− 239.807i	− 0.297527i
$807$	0	0
$808$	82.2769	0.101828
$809$	− 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	$-0.221983\pi$
$811$	1243.31	1.53305	0.766527	+	0.642212i	$0.221983\pi$
$812$	− 595.754i	− 0.733687i
$813$	0	0
$814$	−1258.91	−1.54657
$815$	− 166.823i	− 0.204691i
$816$	0	0
$817$	223.846	0.273985
$818$	− 462.845i	− 0.565825i
$819$	0	0
$820$	−389.885	−0.475469
$821$	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	$-0.541541\pi$
$823$	−214.200	−0.260267	−0.130134	+	0.991496i	$0.541541\pi$
$824$	− 273.531i	− 0.331955i
$825$	0	0
$826$	−731.938	−0.886124
$827$	1221.27i	1.47675i	0.674391	+	0.738374i	$0.264406\pi$
$827$	1221.27i	1.47675i	−0.674391	+	0.738374i	$0.735594\pi$
$828$	0	0
$829$	−358.431	−0.432365	−0.216183	−	0.976353i	$-0.569361\pi$
$829$	−358.431	−0.432365	−0.216183	+	0.976353i	$0.569361\pi$
$830$	− 848.202i	− 1.02193i
$831$	0	0
$832$	169.569	0.203809
$833$	− 166.401i	− 0.199761i
$834$	0	0
$835$	−324.946	−0.389157
$836$	716.984i	0.857637i
$837$	0	0
$838$	−919.061	−1.09673
$839$	− 327.597i	− 0.390461i	−0.980757	−	0.195231i	$-0.937454\pi$
$839$	− 327.597i	− 0.390461i	0.980757	−	0.195231i	$-0.0625456\pi$
$840$	0	0
$841$	−418.827	−0.498011
$842$	4.69395i	0.00557477i
$843$	0	0
$844$	−614.908	−0.728563
$845$	− 1624.36i	− 1.92232i
$846$	0	0
$847$	797.269	0.941286
$848$	− 103.042i	− 0.121512i
$849$	0	0
$850$	−94.3078	−0.110950
$851$	− 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	$0.509859\pi$
$854$	154.291i	0.180668i
$855$	0	0
$856$	−502.277	−0.586772
$857$	− 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	$-0.583734\pi$
$859$	−446.739	−0.520068	−0.260034	+	0.965599i	$0.583734\pi$
$860$	106.371i	0.123687i
$861$	0	0
$862$	−1136.32	−1.31824
$863$	635.297i	0.736150i	0.929796	+	0.368075i	$0.119983\pi$
$863$	635.297i	0.736150i	−0.929796	+	0.368075i	$0.880017\pi$
$864$	0	0
$865$	1280.54	1.48039
$866$	− 131.696i	− 0.152074i
$867$	0	0
$868$	134.277	0.154697
$869$	− 1451.27i	− 1.67004i
$870$	0	0
$871$	−448.869	−0.515349
$872$	− 175.199i	− 0.200917i
$873$	0	0
$874$	−506.985	−0.580074
$875$	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	$0.648453\pi$
$878$	− 667.509i	− 0.760261i
$879$	0	0
$880$	−340.708	−0.387168
$881$	− 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	$-0.443222\pi$
$883$	313.338	0.354857	0.177428	+	0.984134i	$0.443222\pi$
$884$	− 329.158i	− 0.372351i
$885$	0	0
$886$	−805.661	−0.909324
$887$	− 597.701i	− 0.673845i	−0.941532	−	0.336923i	$-0.890614\pi$
$887$	− 597.701i	− 0.673845i	0.941532	−	0.336923i	$-0.109386\pi$
$888$	0	0
$889$	−1184.80	−1.33273
$890$	− 1099.35i	− 1.23522i
$891$	0	0
$892$	677.184	0.759175
$893$	413.951i	0.463551i
$894$	0	0
$895$	1439.31	1.60816
$896$	94.9481i	0.105969i
$897$	0	0
$898$	790.708	0.880521
$899$	− 283.952i	− 0.315854i
$900$	0	0
$901$	−200.019	−0.221997
$902$	699.122i	0.775080i
$903$	0	0
$904$	−310.592	−0.343575
$905$	917.302i	1.01359i
$906$	0	0
$907$	1086.51	1.19791	0.598957	−	0.800781i	$-0.295582\pi$
$907$	1086.51	1.19791	0.598957	+	0.800781i	$0.295582\pi$
$908$	247.901i	0.273019i
$909$	0	0
$910$	1457.97	1.60216
$911$	− 710.283i	− 0.779674i	−0.920884	−	0.389837i	$-0.872531\pi$
$911$	− 710.283i	− 0.779674i	0.920884	−	0.389837i	$-0.127469\pi$
$912$	0	0
$913$	−1520.95	−1.66589
$914$	854.604i	0.935015i
$915$	0	0
$916$	−148.115	−0.161698
$917$	− 446.347i	− 0.486747i
$918$	0	0
$919$	1540.01	1.67574	0.837871	−	0.545868i	$-0.183800\pi$
$919$	1540.01	1.67574	0.837871	+	0.545868i	$0.183800\pi$
$920$	− 240.917i	− 0.261866i
$921$	0	0
$922$	7.26141	0.00787572
$923$	− 2145.35i	− 2.32432i
$924$	0	0
$925$	520.196	0.562374
$926$	− 228.385i	− 0.246636i
$927$	0	0
$928$	200.785	0.216363
$929$	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
$932$	− 546.447i	− 0.586316i
$933$	0	0
$934$	711.384	0.761654
$935$	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	$0.882328\pi$
$938$	− 251.338i	− 0.267951i
$939$	0	0
$940$	−196.708	−0.209263
$941$	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
$944$	− 246.682i	− 0.261316i
$945$	0	0
$946$	190.739	0.201626
$947$	190.168i	0.200811i	0.994947	+	0.100406i	$0.0320140\pi$
$947$	190.168i	0.200811i	−0.994947	+	0.100406i	$0.967986\pi$
$948$	0	0
$949$	−856.569	−0.902602
$950$	− 296.267i	− 0.311860i
$951$	0	0
$952$	184.308	0.193601
$953$	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
$956$	905.771i	0.947459i
$957$	0	0
$958$	−302.354	−0.315609
$959$	13.7175i	0.0143040i
$960$	0	0
$961$	−897.000	−0.933403
$962$	1815.62i	1.88733i
$963$	0	0
$964$	765.377	0.793959
$965$	318.756i	0.330317i
$966$	0	0
$967$	1275.31	1.31884	0.659418	−	0.751776i	$-0.270803\pi$
$967$	1275.31	1.31884	0.659418	+	0.751776i	$0.270803\pi$
$968$	268.701i	0.277583i
$969$	0	0
$970$	−615.846	−0.634893
$971$	− 1238.69i	− 1.27568i	−0.770168	−	0.637841i	$-0.779828\pi$
$971$	− 1238.69i	− 1.27568i	0.770168	−	0.637841i	$-0.220172\pi$
$972$	0	0
$973$	161.261	0.165736
$974$	12.2057i	0.0125315i
$975$	0	0
$976$	−52.0000	−0.0532787
$977$	− 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
$980$	248.407i	0.253476i
$981$	0	0
$982$	−1304.55	−1.32847
$983$	− 658.643i	− 0.670033i	−0.942212	−	0.335017i	$-0.891258\pi$
$983$	− 658.643i	− 0.670033i	0.942212	−	0.335017i	$-0.108742\pi$
$984$	0	0
$985$	−207.000	−0.210152
$986$	− 389.752i	− 0.395286i
$987$	0	0
$988$	1034.05	1.04661
$989$	134.873i	0.136373i
$990$	0	0
$991$	608.484	0.614010	0.307005	−	0.951708i	$-0.400673\pi$
$991$	608.484	0.614010	0.307005	+	0.951708i	$0.400673\pi$
$992$	45.2548i	0.0456198i
$993$	0	0
$994$	1201.26	1.20851
$995$	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	$0.600179\pi$
$998$	− 614.606i	− 0.615838i
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.b.b.161.2	✓	4	3.2	odd	2	inner
162.3.b.b.161.3	yes	4	1.1	even	1	trivial
162.3.d.c.53.1		8	9.2	odd	6
162.3.d.c.53.4		8	9.7	even	3
162.3.d.c.107.1		8	9.4	even	3
162.3.d.c.107.4		8	9.5	odd	6
1296.3.e.d.161.1		4	4.3	odd	2
1296.3.e.d.161.4		4	12.11	even	2
1296.3.q.o.593.1		8	36.31	odd	6
1296.3.q.o.593.4		8	36.23	even	6
1296.3.q.o.1025.1		8	36.11	even	6
1296.3.q.o.1025.4		8	36.7	odd	6

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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	162.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -5.79555i q^{5} -8.39230 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} -5.79555i q^{5} -8.39230 q^{7} -2.82843i q^{8} +8.19615 q^{10} -14.6969i q^{11} -21.1962 q^{13} -11.8685i q^{14} +4.00000 q^{16} -7.76457i q^{17} +24.3923 q^{19} +11.5911i q^{20} +20.7846 q^{22} +14.6969i q^{23} -8.58846 q^{25} -29.9759i q^{26} +16.7846 q^{28} -35.4940i q^{29} +8.00000 q^{31} +5.65685i q^{32} +10.9808 q^{34} +48.6381i q^{35} -60.5692 q^{37} +34.4959i q^{38} -16.3923 q^{40} +33.6365i q^{41} +9.17691 q^{43} +29.3939i q^{44} -20.7846 q^{46} +16.9706i q^{47} +21.4308 q^{49} -12.1459i q^{50} +42.3923 q^{52} -25.7605i q^{53} -85.1769 q^{55} +23.7370i q^{56} +50.1962 q^{58} -61.6706i q^{59} -13.0000 q^{61} +11.3137i q^{62} -8.00000 q^{64} +122.843i q^{65} +21.1769 q^{67} +15.5291i q^{68} -68.7846 q^{70} +101.214i q^{71} +40.4115 q^{73} -85.6578i q^{74} -48.7846 q^{76} +123.341i q^{77} +98.7461 q^{79} -23.1822i q^{80} -47.5692 q^{82} -103.488i q^{83} -45.0000 q^{85} +12.9781i q^{86} -41.5692 q^{88} -134.130i q^{89} +177.885 q^{91} -29.3939i q^{92} -24.0000 q^{94} -141.367i q^{95} -75.1384 q^{97} +30.3077i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 8 q^{7}+O(q^{10}) \) 4 * q - 8 * q^4 + 8 * q^7	\( 4 q - 8 q^{4} + 8 q^{7} + 12 q^{10} - 64 q^{13} + 16 q^{16} + 56 q^{19} + 28 q^{25} - 16 q^{28} + 32 q^{31} - 60 q^{34} - 76 q^{37} - 24 q^{40} - 88 q^{43} + 252 q^{49} + 128 q^{52} - 216 q^{55} + 180 q^{58} - 52 q^{61} - 32 q^{64} - 40 q^{67} - 192 q^{70} + 224 q^{73} - 112 q^{76} + 104 q^{79} - 24 q^{82} - 180 q^{85} + 88 q^{91} - 96 q^{94} + 32 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 + 8 * q^7 + 12 * q^10 - 64 * q^13 + 16 * q^16 + 56 * q^19 + 28 * q^25 - 16 * q^28 + 32 * q^31 - 60 * q^34 - 76 * q^37 - 24 * q^40 - 88 * q^43 + 252 * q^49 + 128 * q^52 - 216 * q^55 + 180 * q^58 - 52 * q^61 - 32 * q^64 - 40 * q^67 - 192 * q^70 + 224 * q^73 - 112 * q^76 + 104 * q^79 - 24 * q^82 - 180 * q^85 + 88 * q^91 - 96 * q^94 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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