LMFDB - Embedded newform 162.3.b.b.161.1

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			161.1
Root			$0.517638i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.3.b.b.161.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} -1.55291i q^{5} +12.3923 q^{7} +2.82843i q^{8} +O(q^{10})$	\(q-1.41421i q^{2} -2.00000 q^{4} -1.55291i q^{5} +12.3923 q^{7} +2.82843i q^{8} -2.19615 q^{10} -14.6969i q^{11} -10.8038 q^{13} -17.5254i q^{14} +4.00000 q^{16} -28.9778i q^{17} +3.60770 q^{19} +3.10583i q^{20} -20.7846 q^{22} +14.6969i q^{23} +22.5885 q^{25} +15.2789i q^{26} -24.7846 q^{28} +28.1456i q^{29} +8.00000 q^{31} -5.65685i q^{32} -40.9808 q^{34} -19.2442i q^{35} +22.5692 q^{37} -5.10205i q^{38} +4.39230 q^{40} +25.1512i q^{41} -53.1769 q^{43} +29.3939i q^{44} +20.7846 q^{46} -16.9706i q^{47} +104.569 q^{49} -31.9449i q^{50} +21.6077 q^{52} +84.5482i q^{53} -22.8231 q^{55} +35.0507i q^{56} +39.8038 q^{58} +91.0645i q^{59} -13.0000 q^{61} -11.3137i q^{62} -8.00000 q^{64} +16.7774i q^{65} -41.1769 q^{67} +57.9555i q^{68} -27.2154 q^{70} +16.3613i q^{71} +71.5885 q^{73} -31.9177i q^{74} -7.21539 q^{76} -182.129i q^{77} -46.7461 q^{79} -6.21166i q^{80} +35.5692 q^{82} +15.3062i q^{83} -45.0000 q^{85} +75.2035i q^{86} +41.5692 q^{88} -78.9756i q^{89} -133.885 q^{91} -29.3939i q^{92} -24.0000 q^{94} -5.60244i q^{95} +91.1384 q^{97} -147.883i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 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	$-0.154046\pi$
$7$	12.3923	1.77033	0.885165	+	0.465278i	$0.154046\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−2.19615	−0.219615
$11$	− 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
$13$	−10.8038	−0.831065	−0.415533	−	0.909578i	$-0.636405\pi$
$13$	−10.8038	−0.831065	−0.415533	+	0.909578i	$0.636405\pi$
$14$	− 17.5254i	− 1.25181i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 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	$-0.469734\pi$
$19$	3.60770	0.189879	0.0949393	+	0.995483i	$0.469734\pi$
$20$	3.10583i	0.155291i
$21$	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
$25$	22.5885	0.903538
$26$	15.2789i	0.587652i
$27$	0	0
$28$	−24.7846	−0.885165
$29$	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	$-0.458813\pi$
$31$	8.00000	0.258065	0.129032	+	0.991640i	$0.458813\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−40.9808	−1.20532
$35$	− 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	$0.401347\pi$
$38$	− 5.10205i	− 0.134265i
$39$	0	0
$40$	4.39230	0.109808
$41$	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	$-0.712193\pi$
$43$	−53.1769	−1.23667	−0.618336	+	0.785914i	$0.712193\pi$
$44$	29.3939i	0.668043i
$45$	0	0
$46$	20.7846	0.451839
$47$	− 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
$49$	104.569	2.13407
$50$	− 31.9449i	− 0.638898i
$51$	0	0
$52$	21.6077	0.415533
$53$	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
$56$	35.0507i	0.625906i
$57$	0	0
$58$	39.8038	0.686273
$59$	91.0645i	1.54347i	0.635947	+	0.771733i	$0.280610\pi$
$59$	91.0645i	1.54347i	−0.635947	+	0.771733i	$0.719390\pi$
$60$	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
$65$	16.7774i	0.258115i
$66$	0	0
$67$	−41.1769	−0.614581	−0.307290	−	0.951616i	$-0.599422\pi$
$67$	−41.1769	−0.614581	−0.307290	+	0.951616i	$0.599422\pi$
$68$	57.9555i	0.852287i
$69$	0	0
$70$	−27.2154	−0.388791
$71$	16.3613i	0.230442i	0.993340	+	0.115221i	$0.0367575\pi$
$71$	16.3613i	0.230442i	−0.993340	+	0.115221i	$0.963242\pi$
$72$	0	0
$73$	71.5885	0.980664	0.490332	−	0.871536i	$-0.336876\pi$
$73$	71.5885	0.980664	0.490332	+	0.871536i	$0.336876\pi$
$74$	− 31.9177i	− 0.431320i
$75$	0	0
$76$	−7.21539	−0.0949393
$77$	− 182.129i	− 2.36531i
$78$	0	0
$79$	−46.7461	−0.591723	−0.295862	−	0.955231i	$-0.595607\pi$
$79$	−46.7461	−0.591723	−0.295862	+	0.955231i	$0.595607\pi$
$80$	− 6.21166i	− 0.0776457i
$81$	0	0
$82$	35.5692	0.433771
$83$	15.3062i	0.184411i	0.995740	+	0.0922057i	$0.0293917\pi$
$83$	15.3062i	0.184411i	−0.995740	+	0.0922057i	$0.970608\pi$
$84$	0	0
$85$	−45.0000	−0.529412
$86$	75.2035i	0.874459i
$87$	0	0
$88$	41.5692	0.472377
$89$	− 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
$92$	− 29.3939i	− 0.319499i
$93$	0	0
$94$	−24.0000	−0.255319
$95$	− 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	$0.344331\pi$
$98$	− 147.883i	− 1.50901i
$99$	0	0
$100$	−45.1769	−0.451769
$101$	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	$-0.765790\pi$
$103$	−152.708	−1.48260	−0.741299	+	0.671175i	$0.765790\pi$
$104$	− 30.5579i	− 0.293826i
$105$	0	0
$106$	119.569	1.12801
$107$	60.0062i	0.560805i	0.959882	+	0.280403i	$0.0904680\pi$
$107$	60.0062i	0.560805i	−0.959882	+	0.280403i	$0.909532\pi$
$108$	0	0
$109$	−93.9423	−0.861856	−0.430928	−	0.902386i	$-0.641814\pi$
$109$	−93.9423	−0.861856	−0.430928	+	0.902386i	$0.641814\pi$
$110$	32.2767i	0.293425i
$111$	0	0
$112$	49.5692	0.442582
$113$	− 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
$116$	− 56.2911i	− 0.485268i
$117$	0	0
$118$	128.785	1.09139
$119$	− 359.101i	− 3.01766i
$120$	0	0
$121$	−95.0000	−0.785124
$122$	18.3848i	0.150695i
$123$	0	0
$124$	−16.0000	−0.129032
$125$	− 73.9008i	− 0.591206i
$126$	0	0
$127$	78.8231	0.620654	0.310327	−	0.950630i	$-0.399561\pi$
$127$	78.8231	0.620654	0.310327	+	0.950630i	$0.399561\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	23.7269	0.182515
$131$	− 82.5792i	− 0.630375i	−0.949029	−	0.315188i	$-0.897933\pi$
$131$	− 82.5792i	− 0.630375i	0.949029	−	0.315188i	$-0.102067\pi$
$132$	0	0
$133$	44.7077	0.336148
$134$	58.2330i	0.434574i
$135$	0	0
$136$	81.9615	0.602658
$137$	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	$-0.570165\pi$
$139$	−60.7846	−0.437299	−0.218650	+	0.975803i	$0.570165\pi$
$140$	38.4884i	0.274917i
$141$	0	0
$142$	23.1384	0.162947
$143$	158.783i	1.11037i
$144$	0	0
$145$	43.7077	0.301432
$146$	− 101.241i	− 0.693434i
$147$	0	0
$148$	−45.1384	−0.304989
$149$	− 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	$-0.466208\pi$
$151$	32.0000	0.211921	0.105960	+	0.994370i	$0.466208\pi$
$152$	10.2041i	0.0671323i
$153$	0	0
$154$	−257.569	−1.67253
$155$	− 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	$0.207341\pi$
$158$	66.1090i	0.418411i
$159$	0	0
$160$	−8.78461	−0.0549038
$161$	182.129i	1.13124i
$162$	0	0
$163$	−12.7846	−0.0784332	−0.0392166	−	0.999231i	$-0.512486\pi$
$163$	−12.7846	−0.0784332	−0.0392166	+	0.999231i	$0.512486\pi$
$164$	− 50.3025i	− 0.306722i
$165$	0	0
$166$	21.6462	0.130399
$167$	232.431i	1.39180i	0.718136	+	0.695902i	$0.244995\pi$
$167$	232.431i	1.39180i	−0.718136	+	0.695902i	$0.755005\pi$
$168$	0	0
$169$	−52.2769	−0.309331
$170$	63.6396i	0.374351i
$171$	0	0
$172$	106.354	0.618336
$173$	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
$176$	− 58.7878i	− 0.334021i
$177$	0	0
$178$	−111.688	−0.627463
$179$	− 277.741i	− 1.55162i	−0.630964	−	0.775812i	$-0.717341\pi$
$179$	− 277.741i	− 1.55162i	0.630964	−	0.775812i	$-0.282659\pi$
$180$	0	0
$181$	174.277	0.962856	0.481428	−	0.876486i	$-0.340118\pi$
$181$	174.277	0.962856	0.481428	+	0.876486i	$0.340118\pi$
$182$	189.341i	1.04034i
$183$	0	0
$184$	−41.5692	−0.225920
$185$	− 35.0481i	− 0.189449i
$186$	0	0
$187$	−425.885	−2.27746
$188$	33.9411i	0.180538i
$189$	0	0
$190$	−7.92305	−0.0417003
$191$	152.126i	0.796470i	0.917283	+	0.398235i	$0.130377\pi$
$191$	152.126i	0.796470i	−0.917283	+	0.398235i	$0.869623\pi$
$192$	0	0
$193$	−55.0000	−0.284974	−0.142487	−	0.989797i	$-0.545510\pi$
$193$	−55.0000	−0.284974	−0.142487	+	0.989797i	$0.545510\pi$
$194$	− 128.889i	− 0.664377i
$195$	0	0
$196$	−209.138	−1.06703
$197$	− 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	$-0.675852\pi$
$199$	−208.862	−1.04956	−0.524778	+	0.851239i	$0.675852\pi$
$200$	63.8898i	0.319449i
$201$	0	0
$202$	125.138	0.619497
$203$	348.788i	1.71817i
$204$	0	0
$205$	39.0577	0.190525
$206$	215.961i	1.04836i
$207$	0	0
$208$	−43.2154	−0.207766
$209$	− 53.0221i	− 0.253694i
$210$	0	0
$211$	−87.4538	−0.414473	−0.207236	−	0.978291i	$-0.566447\pi$
$211$	−87.4538	−0.414473	−0.207236	+	0.978291i	$0.566447\pi$
$212$	− 169.096i	− 0.797625i
$213$	0	0
$214$	84.8616	0.396549
$215$	82.5792i	0.384089i
$216$	0	0
$217$	99.1384	0.456859
$218$	132.854i	0.609424i
$219$	0	0
$220$	45.6462	0.207483
$221$	313.071i	1.41661i
$222$	0	0
$223$	222.592	0.998171	0.499086	−	0.866553i	$-0.333669\pi$
$223$	222.592	0.998171	0.499086	+	0.866553i	$0.333669\pi$
$224$	− 70.1015i	− 0.312953i
$225$	0	0
$226$	−125.296	−0.554408
$227$	300.314i	1.32297i	0.749959	+	0.661484i	$0.230073\pi$
$227$	300.314i	1.32297i	−0.749959	+	0.661484i	$0.769927\pi$
$228$	0	0
$229$	229.942	1.00411	0.502057	−	0.864834i	$-0.332577\pi$
$229$	229.942	1.00411	0.502057	+	0.864834i	$0.332577\pi$
$230$	− 32.2767i	− 0.140334i
$231$	0	0
$232$	−79.6077	−0.343137
$233$	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
$236$	− 182.129i	− 0.771733i
$237$	0	0
$238$	−507.846	−2.13381
$239$	− 164.386i	− 0.687807i	−0.939005	−	0.343904i	$-0.888251\pi$
$239$	− 164.386i	− 0.687807i	0.939005	−	0.343904i	$-0.111749\pi$
$240$	0	0
$241$	−81.3116	−0.337392	−0.168696	−	0.985668i	$-0.553956\pi$
$241$	−81.3116	−0.337392	−0.168696	+	0.985668i	$0.553956\pi$
$242$	134.350i	0.555166i
$243$	0	0
$244$	26.0000	0.106557
$245$	− 162.387i	− 0.662804i
$246$	0	0
$247$	−38.9770	−0.157802
$248$	22.6274i	0.0912396i
$249$	0	0
$250$	−104.512	−0.418046
$251$	− 396.371i	− 1.57917i	−0.613642	−	0.789584i	$-0.710296\pi$
$251$	− 396.371i	− 1.57917i	0.613642	−	0.789584i	$-0.289704\pi$
$252$	0	0
$253$	216.000	0.853755
$254$	− 111.473i	− 0.438869i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 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
$260$	− 33.5549i	− 0.129057i
$261$	0	0
$262$	−116.785	−0.445743
$263$	− 317.893i	− 1.20872i	−0.796711	−	0.604360i	$-0.793429\pi$
$263$	− 317.893i	− 1.20872i	0.796711	−	0.604360i	$-0.206571\pi$
$264$	0	0
$265$	131.296	0.495457
$266$	− 63.2262i	− 0.237692i
$267$	0	0
$268$	82.3538	0.307290
$269$	− 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	$-0.227032\pi$
$271$	409.885	1.51249	0.756245	+	0.654289i	$0.227032\pi$
$272$	− 115.911i	− 0.426144i
$273$	0	0
$274$	303.688	1.10835
$275$	− 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	$0.145120\pi$
$278$	85.9624i	0.309217i
$279$	0	0
$280$	54.4308	0.194396
$281$	− 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	$-0.675642\pi$
$283$	−296.708	−1.04844	−0.524218	+	0.851584i	$0.675642\pi$
$284$	− 32.7227i	− 0.115221i
$285$	0	0
$286$	224.554	0.785153
$287$	311.682i	1.08600i
$288$	0	0
$289$	−550.711	−1.90558
$290$	− 61.8120i	− 0.213145i
$291$	0	0
$292$	−143.177	−0.490332
$293$	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
$296$	63.8354i	0.215660i
$297$	0	0
$298$	−43.3346	−0.145418
$299$	− 158.783i	− 0.531048i
$300$	0	0
$301$	−658.985	−2.18932
$302$	− 45.2548i	− 0.149850i
$303$	0	0
$304$	14.4308	0.0474697
$305$	20.1879i	0.0661898i
$306$	0	0
$307$	−114.354	−0.372488	−0.186244	−	0.982504i	$-0.559632\pi$
$307$	−114.354	−0.372488	−0.186244	+	0.982504i	$0.559632\pi$
$308$	364.258i	1.18266i
$309$	0	0
$310$	−17.5692	−0.0566749
$311$	− 196.217i	− 0.630922i	−0.948939	−	0.315461i	$-0.897841\pi$
$311$	− 196.217i	− 0.630922i	0.948939	−	0.315461i	$-0.102159\pi$
$312$	0	0
$313$	−146.723	−0.468764	−0.234382	−	0.972145i	$-0.575307\pi$
$313$	−146.723	−0.468764	−0.234382	+	0.972145i	$0.575307\pi$
$314$	− 353.140i	− 1.12465i
$315$	0	0
$316$	93.4923	0.295862
$317$	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
$320$	12.4233i	0.0388229i
$321$	0	0
$322$	257.569	0.799904
$323$	− 104.543i	− 0.323662i
$324$	0	0
$325$	−244.042	−0.750899
$326$	18.0802i	0.0554606i
$327$	0	0
$328$	−71.1384	−0.216885
$329$	− 210.304i	− 0.639223i
$330$	0	0
$331$	−99.4538	−0.300465	−0.150232	−	0.988651i	$-0.548002\pi$
$331$	−99.4538	−0.300465	−0.150232	+	0.988651i	$0.548002\pi$
$332$	− 30.6123i	− 0.0922057i
$333$	0	0
$334$	328.708	0.984155
$335$	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	$0.717335\pi$
$338$	73.9307i	0.218730i
$339$	0	0
$340$	90.0000	0.264706
$341$	− 117.576i	− 0.344796i
$342$	0	0
$343$	688.631	2.00767
$344$	− 150.407i	− 0.437230i
$345$	0	0
$346$	30.4730	0.0880723
$347$	38.7711i	0.111732i	0.998438	+	0.0558662i	$0.0177920\pi$
$347$	38.7711i	0.111732i	−0.998438	+	0.0558662i	$0.982208\pi$
$348$	0	0
$349$	−651.969	−1.86811	−0.934053	−	0.357134i	$-0.883754\pi$
$349$	−651.969	−1.86811	−0.934053	+	0.357134i	$0.883754\pi$
$350$	− 395.871i	− 1.13106i
$351$	0	0
$352$	−83.1384	−0.236189
$353$	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
$356$	157.951i	0.443683i
$357$	0	0
$358$	−392.785	−1.09716
$359$	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
$361$	−347.985	−0.963946
$362$	− 246.465i	− 0.680842i
$363$	0	0
$364$	267.769	0.735630
$365$	− 111.171i	− 0.304577i
$366$	0	0
$367$	264.708	0.721274	0.360637	−	0.932706i	$-0.382559\pi$
$367$	264.708	0.721274	0.360637	+	0.932706i	$0.382559\pi$
$368$	58.7878i	0.159749i
$369$	0	0
$370$	−49.5654	−0.133961
$371$	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	$0.469653\pi$
$374$	602.292i	1.61041i
$375$	0	0
$376$	48.0000	0.127660
$377$	− 304.080i	− 0.806579i
$378$	0	0
$379$	−696.785	−1.83848	−0.919241	−	0.393696i	$-0.871196\pi$
$379$	−696.785	−1.83848	−0.919241	+	0.393696i	$0.871196\pi$
$380$	11.2049i	0.0294865i
$381$	0	0
$382$	215.138	0.563190
$383$	− 435.306i	− 1.13657i	−0.822832	−	0.568284i	$-0.807607\pi$
$383$	− 435.306i	− 1.13657i	0.822832	−	0.568284i	$-0.192393\pi$
$384$	0	0
$385$	−282.831	−0.734625
$386$	77.7817i	0.201507i
$387$	0	0
$388$	−182.277	−0.469786
$389$	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
$392$	295.766i	0.754506i
$393$	0	0
$394$	−188.512	−0.478456
$395$	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	$0.474432\pi$
$398$	295.375i	0.742148i
$399$	0	0
$400$	90.3538	0.225885
$401$	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
$404$	− 176.972i	− 0.438051i
$405$	0	0
$406$	493.261	1.21493
$407$	− 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	$0.272931\pi$
$410$	− 55.2359i	− 0.134722i
$411$	0	0
$412$	305.415	0.741299
$413$	1128.50i	2.73244i
$414$	0	0
$415$	23.7691	0.0572750
$416$	61.1158i	0.146913i
$417$	0	0
$418$	−74.9845	−0.179389
$419$	− 385.330i	− 0.919641i	−0.888012	−	0.459821i	$-0.847914\pi$
$419$	− 385.330i	− 0.919641i	0.888012	−	0.459821i	$-0.152086\pi$
$420$	0	0
$421$	−755.319	−1.79411	−0.897054	−	0.441922i	$-0.854297\pi$
$421$	−755.319	−1.79411	−0.897054	+	0.441922i	$0.854297\pi$
$422$	123.678i	0.293077i
$423$	0	0
$424$	−239.138	−0.564006
$425$	− 654.563i	− 1.54015i
$426$	0	0
$427$	−161.100	−0.377283
$428$	− 120.012i	− 0.280403i
$429$	0	0
$430$	116.785	0.271592
$431$	107.709i	0.249904i	0.992163	+	0.124952i	$0.0398777\pi$
$431$	107.709i	0.249904i	−0.992163	+	0.124952i	$0.960122\pi$
$432$	0	0
$433$	655.123	1.51299	0.756493	−	0.654002i	$-0.226911\pi$
$433$	655.123	1.51299	0.756493	+	0.654002i	$0.226911\pi$
$434$	− 140.203i	− 0.323048i
$435$	0	0
$436$	187.885	0.430928
$437$	53.0221i	0.121332i
$438$	0	0
$439$	−472.000	−1.07517	−0.537585	−	0.843209i	$-0.680663\pi$
$439$	−472.000	−1.07517	−0.537585	+	0.843209i	$0.680663\pi$
$440$	− 64.5534i	− 0.146712i
$441$	0	0
$442$	442.750	1.00170
$443$	841.218i	1.89891i	0.313902	+	0.949455i	$0.398364\pi$
$443$	841.218i	1.89891i	−0.313902	+	0.949455i	$0.601636\pi$
$444$	0	0
$445$	−122.642	−0.275601
$446$	− 314.793i	− 0.705814i
$447$	0	0
$448$	−99.1384	−0.221291
$449$	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
$452$	177.195i	0.392025i
$453$	0	0
$454$	424.708	0.935479
$455$	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	$0.384766\pi$
$458$	− 325.187i	− 0.710016i
$459$	0	0
$460$	−45.6462	−0.0992308
$461$	− 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	$-0.455341\pi$
$463$	129.492	0.279681	0.139840	+	0.990174i	$0.455341\pi$
$464$	112.582i	0.242634i
$465$	0	0
$466$	164.396	0.352781
$467$	− 672.730i	− 1.44054i	−0.693696	−	0.720268i	$-0.744019\pi$
$467$	− 672.730i	− 1.44054i	0.693696	−	0.720268i	$-0.255981\pi$
$468$	0	0
$469$	−510.277	−1.08801
$470$	37.2699i	0.0792977i
$471$	0	0
$472$	−257.569	−0.545697
$473$	781.538i	1.65230i
$474$	0	0
$475$	81.4923	0.171563
$476$	718.203i	1.50883i
$477$	0	0
$478$	−232.477	−0.486353
$479$	− 125.615i	− 0.262244i	−0.991366	−	0.131122i	$-0.958142\pi$
$479$	− 125.615i	− 0.262244i	0.991366	−	0.131122i	$-0.0418579\pi$
$480$	0	0
$481$	−243.834	−0.506932
$482$	114.992i	0.238572i
$483$	0	0
$484$	190.000	0.392562
$485$	− 141.530i	− 0.291815i
$486$	0	0
$487$	−448.631	−0.921213	−0.460606	−	0.887604i	$-0.652368\pi$
$487$	−448.631	−0.921213	−0.460606	+	0.887604i	$0.652368\pi$
$488$	− 36.7696i	− 0.0753474i
$489$	0	0
$490$	−229.650	−0.468673
$491$	− 452.157i	− 0.920890i	−0.887688	−	0.460445i	$-0.847690\pi$
$491$	− 452.157i	− 0.920890i	0.887688	−	0.460445i	$-0.152310\pi$
$492$	0	0
$493$	815.596	1.65435
$494$	55.1218i	0.111583i
$495$	0	0
$496$	32.0000	0.0645161
$497$	202.755i	0.407957i
$498$	0	0
$499$	126.592	0.253692	0.126846	−	0.991922i	$-0.459515\pi$
$499$	126.592	0.253692	0.126846	+	0.991922i	$0.459515\pi$
$500$	147.802i	0.295603i
$501$	0	0
$502$	−560.554	−1.11664
$503$	296.822i	0.590103i	0.955481	+	0.295051i	$0.0953367\pi$
$503$	296.822i	0.590103i	−0.955481	+	0.295051i	$0.904663\pi$
$504$	0	0
$505$	137.412	0.272102
$506$	− 305.470i	− 0.603696i
$507$	0	0
$508$	−157.646	−0.310327
$509$	− 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
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−304.004	−0.591447
$515$	237.142i	0.460470i
$516$	0	0
$517$	−249.415	−0.482428
$518$	− 395.534i	− 0.763579i
$519$	0	0
$520$	−47.4538	−0.0912573
$521$	− 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	$-0.299467\pi$
$523$	616.238	1.17828	0.589138	+	0.808032i	$0.299467\pi$
$524$	165.158i	0.315188i
$525$	0	0
$526$	−449.569	−0.854694
$527$	− 231.822i	− 0.439890i
$528$	0	0
$529$	313.000	0.591682
$530$	− 185.681i	− 0.350341i
$531$	0	0
$532$	−89.4153	−0.168074
$533$	− 271.730i	− 0.509813i
$534$	0	0
$535$	93.1845	0.174177
$536$	− 116.466i	− 0.217287i
$537$	0	0
$538$	−294.904	−0.548148
$539$	− 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	$0.669300\pi$
$542$	− 579.664i	− 1.06949i
$543$	0	0
$544$	−163.923	−0.301329
$545$	145.884i	0.267678i
$546$	0	0
$547$	−819.769	−1.49866	−0.749332	−	0.662195i	$-0.769625\pi$
$547$	−819.769	−1.49866	−0.749332	+	0.662195i	$0.769625\pi$
$548$	− 429.480i	− 0.783723i
$549$	0	0
$550$	−469.492	−0.853622
$551$	101.541i	0.184284i
$552$	0	0
$553$	−579.292	−1.04754
$554$	− 703.451i	− 1.26977i
$555$	0	0
$556$	121.569	0.218650
$557$	− 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
$560$	− 76.9767i	− 0.137458i
$561$	0	0
$562$	−304.435	−0.541698
$563$	508.790i	0.903713i	0.892091	+	0.451856i	$0.149238\pi$
$563$	508.790i	0.903713i	−0.892091	+	0.451856i	$0.850762\pi$
$564$	0	0
$565$	−137.585	−0.243513
$566$	419.608i	0.741357i
$567$	0	0
$568$	−46.2769	−0.0814734
$569$	− 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	$-0.630200\pi$
$571$	−454.200	−0.795446	−0.397723	+	0.917505i	$0.630200\pi$
$572$	− 317.567i	− 0.555187i
$573$	0	0
$574$	440.785	0.767917
$575$	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	$0.510793\pi$
$578$	778.824i	1.34745i
$579$	0	0
$580$	−87.4153	−0.150716
$581$	189.679i	0.326469i
$582$	0	0
$583$	1242.60	2.13139
$584$	202.483i	0.346717i
$585$	0	0
$586$	482.196	0.822860
$587$	− 419.227i	− 0.714186i	−0.934069	−	0.357093i	$-0.883768\pi$
$587$	− 419.227i	− 0.714186i	0.934069	−	0.357093i	$-0.116232\pi$
$588$	0	0
$589$	28.8616	0.0490010
$590$	− 199.991i	− 0.338969i
$591$	0	0
$592$	90.2769	0.152495
$593$	− 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
$596$	61.2844i	0.102826i
$597$	0	0
$598$	−224.554	−0.375508
$599$	184.239i	0.307578i	0.988104	+	0.153789i	$0.0491476\pi$
$599$	184.239i	0.307578i	−0.988104	+	0.153789i	$0.950852\pi$
$600$	0	0
$601$	218.415	0.363420	0.181710	−	0.983352i	$-0.441837\pi$
$601$	218.415	0.363420	0.181710	+	0.983352i	$0.441837\pi$
$602$	931.945i	1.54808i
$603$	0	0
$604$	−64.0000	−0.105960
$605$	147.527i	0.243846i
$606$	0	0
$607$	10.2693	0.0169182	0.00845909	−	0.999964i	$-0.497307\pi$
$607$	10.2693	0.0169182	0.00845909	+	0.999964i	$0.497307\pi$
$608$	− 20.4082i	− 0.0335661i
$609$	0	0
$610$	28.5500	0.0468032
$611$	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	$0.547057\pi$
$614$	161.721i	0.263389i
$615$	0	0
$616$	515.138	0.836264
$617$	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	$-0.289066\pi$
$619$	761.646	1.23045	0.615223	+	0.788353i	$0.289066\pi$
$620$	24.8466i	0.0400752i
$621$	0	0
$622$	−277.492	−0.446129
$623$	− 978.690i	− 1.57093i
$624$	0	0
$625$	449.950	0.719920
$626$	207.498i	0.331466i
$627$	0	0
$628$	−499.415	−0.795247
$629$	− 654.006i	− 1.03975i
$630$	0	0
$631$	601.108	0.952627	0.476313	−	0.879276i	$-0.341973\pi$
$631$	601.108	0.952627	0.476313	+	0.879276i	$0.341973\pi$
$632$	− 132.218i	− 0.209206i
$633$	0	0
$634$	68.0807	0.107383
$635$	− 122.405i	− 0.192765i
$636$	0	0
$637$	−1129.75	−1.77355
$638$	− 584.995i	− 0.916920i
$639$	0	0
$640$	17.5692	0.0274519
$641$	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	$-0.467737\pi$
$643$	130.123	0.202369	0.101184	+	0.994868i	$0.467737\pi$
$644$	− 364.258i	− 0.565618i
$645$	0	0
$646$	−147.846	−0.228864
$647$	985.467i	1.52313i	0.648087	+	0.761567i	$0.275570\pi$
$647$	985.467i	1.52313i	−0.648087	+	0.761567i	$0.724430\pi$
$648$	0	0
$649$	1338.37	2.06220
$650$	345.128i	0.530966i
$651$	0	0
$652$	25.5692	0.0392166
$653$	− 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
$656$	100.605i	0.153361i
$657$	0	0
$658$	−297.415	−0.451999
$659$	− 15.5889i	− 0.0236554i	−0.999930	−	0.0118277i	$-0.996235\pi$
$659$	− 15.5889i	− 0.0236554i	0.999930	−	0.0118277i	$-0.00376496\pi$
$660$	0	0
$661$	407.831	0.616990	0.308495	−	0.951226i	$-0.400175\pi$
$661$	407.831	0.616990	0.308495	+	0.951226i	$0.400175\pi$
$662$	140.649i	0.212461i
$663$	0	0
$664$	−43.2923	−0.0651993
$665$	− 69.4272i	− 0.104402i
$666$	0	0
$667$	−413.654	−0.620170
$668$	− 464.863i	− 0.695902i
$669$	0	0
$670$	90.4308	0.134971
$671$	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	$0.614695\pi$
$674$	601.410i	0.892300i
$675$	0	0
$676$	104.554	0.154665
$677$	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
$680$	− 127.279i	− 0.187175i
$681$	0	0
$682$	−166.277	−0.243808
$683$	1085.46i	1.58926i	0.607095	+	0.794629i	$0.292335\pi$
$683$	1085.46i	1.58926i	−0.607095	+	0.794629i	$0.707665\pi$
$684$	0	0
$685$	333.473	0.486822
$686$	− 973.871i	− 1.41964i
$687$	0	0
$688$	−212.708	−0.309168
$689$	− 913.446i	− 1.32576i
$690$	0	0
$691$	−1007.78	−1.45843	−0.729216	−	0.684283i	$-0.760115\pi$
$691$	−1007.78	−1.45843	−0.729216	+	0.684283i	$0.760115\pi$
$692$	− 43.0954i	− 0.0622765i
$693$	0	0
$694$	54.8306	0.0790067
$695$	94.3933i	0.135818i
$696$	0	0
$697$	728.827	1.04566
$698$	922.024i	1.32095i
$699$	0	0
$700$	−559.846	−0.799780
$701$	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
$704$	117.576i	0.167011i
$705$	0	0
$706$	1.69219	0.00239688
$707$	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	$0.252587\pi$
$710$	− 35.9320i	− 0.0506085i
$711$	0	0
$712$	223.377	0.313732
$713$	117.576i	0.164903i
$714$	0	0
$715$	246.577	0.344863
$716$	555.481i	0.775812i
$717$	0	0
$718$	756.000	1.05292
$719$	− 340.912i	− 0.474148i	−0.971492	−	0.237074i	$-0.923812\pi$
$719$	− 340.912i	− 0.474148i	0.971492	−	0.237074i	$-0.0761884\pi$
$720$	0	0
$721$	−1892.40	−2.62469
$722$	492.124i	0.681613i
$723$	0	0
$724$	−348.554	−0.481428
$725$	635.765i	0.876917i
$726$	0	0
$727$	−971.777	−1.33669	−0.668347	−	0.743850i	$-0.732998\pi$
$727$	−971.777	−1.33669	−0.668347	+	0.743850i	$0.732998\pi$
$728$	− 378.683i	− 0.520169i
$729$	0	0
$730$	−157.219	−0.215369
$731$	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	$0.450036\pi$
$734$	− 374.353i	− 0.510018i
$735$	0	0
$736$	83.1384	0.112960
$737$	605.175i	0.821132i
$738$	0	0
$739$	−629.892	−0.852357	−0.426179	−	0.904639i	$-0.640141\pi$
$739$	−629.892	−0.852357	−0.426179	+	0.904639i	$0.640141\pi$
$740$	70.0961i	0.0947245i
$741$	0	0
$742$	1481.74	1.99695
$743$	− 188.014i	− 0.253047i	−0.991964	−	0.126524i	$-0.959618\pi$
$743$	− 188.014i	− 0.253047i	0.991964	−	0.126524i	$-0.0403820\pi$
$744$	0	0
$745$	−47.5847	−0.0638721
$746$	− 100.431i	− 0.134626i
$747$	0	0
$748$	851.769	1.13873
$749$	743.615i	0.992810i
$750$	0	0
$751$	−1386.67	−1.84643	−0.923215	−	0.384283i	$-0.874449\pi$
$751$	−1386.67	−1.84643	−0.923215	+	0.384283i	$0.874449\pi$
$752$	− 67.8823i	− 0.0902690i
$753$	0	0
$754$	−430.035	−0.570338
$755$	− 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	$0.200974\pi$
$758$	985.402i	1.30000i
$759$	0	0
$760$	15.8461	0.0208501
$761$	− 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
$764$	− 304.252i	− 0.398235i
$765$	0	0
$766$	−615.615	−0.803675
$767$	− 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	$0.627299\pi$
$770$	399.983i	0.519458i
$771$	0	0
$772$	110.000	0.142487
$773$	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
$776$	257.778i	0.332189i
$777$	0	0
$778$	75.0155	0.0964209
$779$	90.7380i	0.116480i
$780$	0	0
$781$	240.462	0.307890
$782$	− 602.292i	− 0.770194i
$783$	0	0
$784$	418.277	0.533516
$785$	− 387.775i	− 0.493980i
$786$	0	0
$787$	−955.300	−1.21385	−0.606925	−	0.794759i	$-0.707597\pi$
$787$	−955.300	−1.21385	−0.606925	+	0.794759i	$0.707597\pi$
$788$	266.596i	0.338319i
$789$	0	0
$790$	102.662	0.129951
$791$	− 1097.93i	− 1.38803i
$792$	0	0
$793$	140.450	0.177112
$794$	− 90.0962i	− 0.113471i
$795$	0	0
$796$	417.723	0.524778
$797$	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
$800$	− 127.780i	− 0.159725i
$801$	0	0
$802$	889.104	1.10861
$803$	− 1052.13i	− 1.31025i
$804$	0	0
$805$	282.831	0.351342
$806$	122.232i	0.151652i
$807$	0	0
$808$	−250.277	−0.309749
$809$	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	$-0.626402\pi$
$811$	−627.307	−0.773499	−0.386749	+	0.922185i	$0.626402\pi$
$812$	− 697.577i	− 0.859085i
$813$	0	0
$814$	−469.092	−0.576281
$815$	19.8534i	0.0243600i
$816$	0	0
$817$	−191.846	−0.234818
$818$	− 757.001i	− 0.925429i
$819$	0	0
$820$	−78.1154	−0.0952627
$821$	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	$-0.436498\pi$
$823$	326.200	0.396355	0.198177	+	0.980166i	$0.436498\pi$
$824$	− 431.922i	− 0.524178i
$825$	0	0
$826$	1595.94	1.93213
$827$	− 1103.70i	− 1.33458i	−0.744799	−	0.667289i	$-0.767455\pi$
$827$	− 1103.70i	− 1.33458i	0.744799	−	0.667289i	$-0.232545\pi$
$828$	0	0
$829$	−441.569	−0.532653	−0.266326	−	0.963883i	$-0.585810\pi$
$829$	−441.569	−0.532653	−0.266326	+	0.963883i	$0.585810\pi$
$830$	− 33.6146i	− 0.0404996i
$831$	0	0
$832$	86.4308	0.103883
$833$	− 3030.18i	− 3.63768i
$834$	0	0
$835$	360.946	0.432271
$836$	106.044i	0.126847i
$837$	0	0
$838$	−544.939	−0.650285
$839$	503.960i	0.600668i	0.953834	+	0.300334i	$0.0970981\pi$
$839$	503.960i	0.600668i	−0.953834	+	0.300334i	$0.902902\pi$
$840$	0	0
$841$	48.8269	0.0580581
$842$	1068.18i	1.26863i
$843$	0	0
$844$	174.908	0.207236
$845$	81.1815i	0.0960728i
$846$	0	0
$847$	−1177.27	−1.38993
$848$	338.193i	0.398812i
$849$	0	0
$850$	−925.692	−1.08905
$851$	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	$0.313166\pi$
$854$	227.830i	0.266780i
$855$	0	0
$856$	−169.723	−0.198275
$857$	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	$-0.791144\pi$
$859$	−1361.26	−1.58470	−0.792352	+	0.610064i	$0.791144\pi$
$860$	− 165.158i	− 0.192045i
$861$	0	0
$862$	152.323	0.176709
$863$	805.003i	0.932796i	0.884575	+	0.466398i	$0.154449\pi$
$863$	805.003i	0.932796i	−0.884575	+	0.466398i	$0.845551\pi$
$864$	0	0
$865$	33.4617	0.0386841
$866$	− 926.484i	− 1.06984i
$867$	0	0
$868$	−198.277	−0.228430
$869$	687.025i	0.790593i
$870$	0	0
$871$	444.869	0.510757
$872$	− 265.709i	− 0.304712i
$873$	0	0
$874$	74.9845	0.0857947
$875$	− 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	$0.492252\pi$
$878$	667.509i	0.760261i
$879$	0	0
$880$	−91.2923	−0.103741
$881$	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	$-0.571496\pi$
$883$	−393.338	−0.445457	−0.222728	+	0.974881i	$0.571496\pi$
$884$	− 626.143i	− 0.708306i
$885$	0	0
$886$	1189.66	1.34273
$887$	− 1548.05i	− 1.74527i	−0.488375	−	0.872634i	$-0.662410\pi$
$887$	− 1548.05i	− 1.74527i	0.488375	−	0.872634i	$-0.337590\pi$
$888$	0	0
$889$	976.800	1.09876
$890$	173.443i	0.194879i
$891$	0	0
$892$	−445.184	−0.499086
$893$	− 61.2246i	− 0.0685606i
$894$	0	0
$895$	−431.307	−0.481908
$896$	140.203i	0.156476i
$897$	0	0
$898$	541.292	0.602775
$899$	225.165i	0.250461i
$900$	0	0
$901$	2450.02	2.71922
$902$	− 522.759i	− 0.579555i
$903$	0	0
$904$	250.592	0.277204
$905$	− 270.637i	− 0.299046i
$906$	0	0
$907$	1377.49	1.51873	0.759367	−	0.650662i	$-0.225509\pi$
$907$	1377.49	1.51873	0.759367	+	0.650662i	$0.225509\pi$
$908$	− 600.627i	− 0.661484i
$909$	0	0
$910$	294.031	0.323111
$911$	− 1406.08i	− 1.54344i	−0.635961	−	0.771721i	$-0.719396\pi$
$911$	− 1406.08i	− 1.54344i	0.635961	−	0.771721i	$-0.280604\pi$
$912$	0	0
$913$	224.954	0.246389
$914$	− 457.786i	− 0.500860i
$915$	0	0
$916$	−459.885	−0.502057
$917$	− 1023.35i	− 1.11597i
$918$	0	0
$919$	479.992	0.522299	0.261149	−	0.965298i	$-0.415899\pi$
$919$	479.992	0.522299	0.261149	+	0.965298i	$0.415899\pi$
$920$	64.5534i	0.0701668i
$921$	0	0
$922$	−907.261	−0.984015
$923$	− 176.765i	− 0.191512i
$924$	0	0
$925$	509.804	0.551139
$926$	− 183.130i	− 0.197764i
$927$	0	0
$928$	159.215	0.171568
$929$	− 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
$932$	− 232.491i	− 0.249454i
$933$	0	0
$934$	−951.384	−1.01861
$935$	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	$0.615761\pi$
$938$	721.640i	0.769340i
$939$	0	0
$940$	52.7077	0.0560720
$941$	− 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
$944$	364.258i	0.385866i
$945$	0	0
$946$	1105.26	1.16835
$947$	− 454.713i	− 0.480162i	−0.970753	−	0.240081i	$-0.922826\pi$
$947$	− 454.713i	− 0.480162i	0.970753	−	0.240081i	$-0.0771740\pi$
$948$	0	0
$949$	−773.431	−0.814996
$950$	− 115.247i	− 0.121313i
$951$	0	0
$952$	1015.69	1.06690
$953$	− 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
$956$	328.772i	0.343904i
$957$	0	0
$958$	−177.646	−0.185434
$959$	2661.13i	2.77490i
$960$	0	0
$961$	−897.000	−0.933403
$962$	344.834i	0.358455i
$963$	0	0
$964$	162.623	0.168696
$965$	85.4103i	0.0885081i
$966$	0	0
$967$	−1655.31	−1.71180	−0.855902	−	0.517138i	$-0.826998\pi$
$967$	−1655.31	−1.71180	−0.855902	+	0.517138i	$0.826998\pi$
$968$	− 268.701i	− 0.277583i
$969$	0	0
$970$	−200.154	−0.206344
$971$	797.780i	0.821606i	0.911724	+	0.410803i	$0.134752\pi$
$971$	797.780i	0.821606i	−0.911724	+	0.410803i	$0.865248\pi$
$972$	0	0
$973$	−753.261	−0.774164
$974$	634.460i	0.651396i
$975$	0	0
$976$	−52.0000	−0.0532787
$977$	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
$980$	324.774i	0.331402i
$981$	0	0
$982$	−639.446	−0.651167
$983$	− 370.143i	− 0.376544i	−0.982117	−	0.188272i	$-0.939711\pi$
$983$	− 370.143i	− 0.376544i	0.982117	−	0.188272i	$-0.0602887\pi$
$984$	0	0
$985$	−207.000	−0.210152
$986$	− 1153.43i	− 1.16980i
$987$	0	0
$988$	77.9540	0.0789008
$989$	− 781.538i	− 0.790230i
$990$	0	0
$991$	−1324.48	−1.33651	−0.668256	−	0.743931i	$-0.732959\pi$
$991$	−1324.48	−1.33651	−0.668256	+	0.743931i	$0.732959\pi$
$992$	− 45.2548i	− 0.0456198i
$993$	0	0
$994$	286.739	0.288469
$995$	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	$0.354643\pi$
$998$	− 179.028i	− 0.179387i
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -1.55291i q^{5} +12.3923 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} -1.55291i q^{5} +12.3923 q^{7} +2.82843i q^{8} -2.19615 q^{10} -14.6969i q^{11} -10.8038 q^{13} -17.5254i q^{14} +4.00000 q^{16} -28.9778i q^{17} +3.60770 q^{19} +3.10583i q^{20} -20.7846 q^{22} +14.6969i q^{23} +22.5885 q^{25} +15.2789i q^{26} -24.7846 q^{28} +28.1456i q^{29} +8.00000 q^{31} -5.65685i q^{32} -40.9808 q^{34} -19.2442i q^{35} +22.5692 q^{37} -5.10205i q^{38} +4.39230 q^{40} +25.1512i q^{41} -53.1769 q^{43} +29.3939i q^{44} +20.7846 q^{46} -16.9706i q^{47} +104.569 q^{49} -31.9449i q^{50} +21.6077 q^{52} +84.5482i q^{53} -22.8231 q^{55} +35.0507i q^{56} +39.8038 q^{58} +91.0645i q^{59} -13.0000 q^{61} -11.3137i q^{62} -8.00000 q^{64} +16.7774i q^{65} -41.1769 q^{67} +57.9555i q^{68} -27.2154 q^{70} +16.3613i q^{71} +71.5885 q^{73} -31.9177i q^{74} -7.21539 q^{76} -182.129i q^{77} -46.7461 q^{79} -6.21166i q^{80} +35.5692 q^{82} +15.3062i q^{83} -45.0000 q^{85} +75.2035i q^{86} +41.5692 q^{88} -78.9756i q^{89} -133.885 q^{91} -29.3939i q^{92} -24.0000 q^{94} -5.60244i q^{95} +91.1384 q^{97} -147.883i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 8 q^{7}+O(q^{10}) \) 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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