LMFDB - Embedded newform 1089.3.c.g.604.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$29.6731007888$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 363)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.3
Root			$0.517638i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.g.604.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.517638i q^{2} +3.73205 q^{4} +7.19615 q^{5} +8.86422i q^{7} +4.00240i q^{8} +O(q^{10})$	$q+0.517638i q^{2} +3.73205 q^{4} +7.19615 q^{5} +8.86422i q^{7} +4.00240i q^{8} +3.72500i q^{10} -16.0740i q^{13} -4.58846 q^{14} +12.8564 q^{16} +27.0088i q^{17} -13.3843i q^{19} +26.8564 q^{20} +28.0526 q^{23} +26.7846 q^{25} +8.32051 q^{26} +33.0817i q^{28} +3.79933i q^{29} -37.3205 q^{31} +22.6646i q^{32} -13.9808 q^{34} +63.7883i q^{35} +26.1244 q^{37} +6.92820 q^{38} +28.8019i q^{40} +35.3182i q^{41} -22.5259i q^{43} +14.5211i q^{46} +54.8897 q^{47} -29.5744 q^{49} +13.8647i q^{50} -59.9889i q^{52} -55.7846 q^{53} -35.4782 q^{56} -1.96668 q^{58} -38.0910 q^{59} -12.0444i q^{61} -19.3185i q^{62} +39.6936 q^{64} -115.671i q^{65} -81.9090 q^{67} +100.798i q^{68} -33.0192 q^{70} +52.6936 q^{71} -129.526i q^{73} +13.5230i q^{74} -49.9507i q^{76} +59.7487i q^{79} +92.5167 q^{80} -18.2820 q^{82} +85.4076i q^{83} +194.359i q^{85} +11.6603 q^{86} -123.837 q^{89} +142.483 q^{91} +104.694 q^{92} +28.4130i q^{94} -96.3152i q^{95} -22.2295 q^{97} -15.3088i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} + 8 q^{5}+O(q^{10}) $ 4 * q + 8 * q^4 + 8 * q^5	$ 4 q + 8 q^{4} + 8 q^{5} + 44 q^{14} - 4 q^{16} + 52 q^{20} + 36 q^{23} + 24 q^{25} - 36 q^{26} - 80 q^{31} + 48 q^{34} + 56 q^{37} - 16 q^{47} - 340 q^{49} - 140 q^{53} + 156 q^{56} - 188 q^{58} - 284 q^{59} - 56 q^{64} - 196 q^{67} - 236 q^{70} - 4 q^{71} + 280 q^{80} + 204 q^{82} + 12 q^{86} - 336 q^{89} + 660 q^{91} + 204 q^{92} + 112 q^{97}+O(q^{100}) $ 4 * q + 8 * q^4 + 8 * q^5 + 44 * q^14 - 4 * q^16 + 52 * q^20 + 36 * q^23 + 24 * q^25 - 36 * q^26 - 80 * q^31 + 48 * q^34 + 56 * q^37 - 16 * q^47 - 340 * q^49 - 140 * q^53 + 156 * q^56 - 188 * q^58 - 284 * q^59 - 56 * q^64 - 196 * q^67 - 236 * q^70 - 4 * q^71 + 280 * q^80 + 204 * q^82 + 12 * q^86 - 336 * q^89 + 660 * q^91 + 204 * q^92 + 112 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.517638i	0.258819i	0.991591	+	0.129410i	$0.0413082\pi$
$2$	0.517638i	0.258819i	−0.991591	+	0.129410i	$0.958692\pi$
$3$	0	0
$3$	0	0
$4$	3.73205	0.933013
$5$	7.19615	1.43923	0.719615	−	0.694373i	$-0.244318\pi$
$5$	7.19615	1.43923	0.719615	+	0.694373i	$0.244318\pi$
$6$	0	0
$7$	8.86422i	1.26632i	0.774022	+	0.633158i	$0.218242\pi$
$7$	8.86422i	1.26632i	−0.774022	+	0.633158i	$0.781758\pi$
$8$	4.00240i	0.500301i
$9$	0	0
$10$	3.72500i	0.372500i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 16.0740i	− 1.23646i	−0.785997	−	0.618230i	$-0.787850\pi$
$13$	− 16.0740i	− 1.23646i	0.785997	−	0.618230i	$-0.212150\pi$
$14$	−4.58846	−0.327747
$15$	0	0
$16$	12.8564	0.803525
$17$	27.0088i	1.58875i	0.607427	+	0.794375i	$0.292202\pi$
$17$	27.0088i	1.58875i	−0.607427	+	0.794375i	$0.707798\pi$
$18$	0	0
$19$	− 13.3843i	− 0.704435i	−0.935918	−	0.352217i	$-0.885428\pi$
$19$	− 13.3843i	− 0.704435i	0.935918	−	0.352217i	$-0.114572\pi$
$20$	26.8564	1.34282
$21$	0	0
$22$	0	0
$23$	28.0526	1.21968	0.609838	−	0.792526i	$-0.291234\pi$
$23$	28.0526	1.21968	0.609838	+	0.792526i	$0.291234\pi$
$24$	0	0
$25$	26.7846	1.07138
$26$	8.32051	0.320020
$27$	0	0
$28$	33.0817i	1.18149i
$29$	3.79933i	0.131011i	0.997852	+	0.0655057i	$0.0208661\pi$
$29$	3.79933i	0.131011i	−0.997852	+	0.0655057i	$0.979134\pi$
$30$	0	0
$31$	−37.3205	−1.20389	−0.601944	−	0.798539i	$-0.705607\pi$
$31$	−37.3205	−1.20389	−0.601944	+	0.798539i	$0.705607\pi$
$32$	22.6646i	0.708268i
$33$	0	0
$34$	−13.9808	−0.411199
$35$	63.7883i	1.82252i
$36$	0	0
$37$	26.1244	0.706064	0.353032	−	0.935611i	$-0.385151\pi$
$37$	26.1244	0.706064	0.353032	+	0.935611i	$0.385151\pi$
$38$	6.92820	0.182321
$39$	0	0
$40$	28.8019i	0.720048i
$41$	35.3182i	0.861419i	0.902491	+	0.430709i	$0.141737\pi$
$41$	35.3182i	0.861419i	−0.902491	+	0.430709i	$0.858263\pi$
$42$	0	0
$43$	− 22.5259i	− 0.523858i	−0.965087	−	0.261929i	$-0.915641\pi$
$43$	− 22.5259i	− 0.523858i	0.965087	−	0.261929i	$-0.0843586\pi$
$44$	0	0
$45$	0	0
$46$	14.5211i	0.315676i
$47$	54.8897	1.16787	0.583933	−	0.811802i	$-0.301513\pi$
$47$	54.8897	1.16787	0.583933	+	0.811802i	$0.301513\pi$
$48$	0	0
$49$	−29.5744	−0.603559
$50$	13.8647i	0.277295i
$51$	0	0
$52$	− 59.9889i	− 1.15363i
$53$	−55.7846	−1.05254	−0.526270	−	0.850318i	$-0.676410\pi$
$53$	−55.7846	−1.05254	−0.526270	+	0.850318i	$0.676410\pi$
$54$	0	0
$55$	0	0
$56$	−35.4782	−0.633539
$57$	0	0
$58$	−1.96668	−0.0339083
$59$	−38.0910	−0.645611	−0.322805	−	0.946465i	$-0.604626\pi$
$59$	−38.0910	−0.645611	−0.322805	+	0.946465i	$0.604626\pi$
$60$	0	0
$61$	− 12.0444i	− 0.197449i	−0.995115	−	0.0987244i	$-0.968524\pi$
$61$	− 12.0444i	− 0.197449i	0.995115	−	0.0987244i	$-0.0314762\pi$
$62$	− 19.3185i	− 0.311589i
$63$	0	0
$64$	39.6936	0.620212
$65$	− 115.671i	− 1.77955i
$66$	0	0
$67$	−81.9090	−1.22252	−0.611261	−	0.791429i	$-0.709337\pi$
$67$	−81.9090	−1.22252	−0.611261	+	0.791429i	$0.709337\pi$
$68$	100.798i	1.48232i
$69$	0	0
$70$	−33.0192	−0.471703
$71$	52.6936	0.742163	0.371082	−	0.928600i	$-0.378987\pi$
$71$	52.6936	0.742163	0.371082	+	0.928600i	$0.378987\pi$
$72$	0	0
$73$	− 129.526i	− 1.77432i	−0.461458	−	0.887162i	$-0.652674\pi$
$73$	− 129.526i	− 1.77432i	0.461458	−	0.887162i	$-0.347326\pi$
$74$	13.5230i	0.182743i
$75$	0	0
$76$	− 49.9507i	− 0.657247i
$77$	0	0
$78$	0	0
$79$	59.7487i	0.756313i	0.925742	+	0.378156i	$0.123442\pi$
$79$	59.7487i	0.756313i	−0.925742	+	0.378156i	$0.876558\pi$
$80$	92.5167	1.15646
$81$	0	0
$82$	−18.2820	−0.222952
$83$	85.4076i	1.02901i	0.857488	+	0.514504i	$0.172024\pi$
$83$	85.4076i	1.02901i	−0.857488	+	0.514504i	$0.827976\pi$
$84$	0	0
$85$	194.359i	2.28658i
$86$	11.6603	0.135584
$87$	0	0
$88$	0	0
$89$	−123.837	−1.39143	−0.695714	−	0.718318i	$-0.744912\pi$
$89$	−123.837	−1.39143	−0.695714	+	0.718318i	$0.744912\pi$
$90$	0	0
$91$	142.483	1.56575
$92$	104.694	1.13797
$93$	0	0
$94$	28.4130i	0.302266i
$95$	− 96.3152i	− 1.01384i
$96$	0	0
$97$	−22.2295	−0.229170	−0.114585	−	0.993413i	$-0.536554\pi$
$97$	−22.2295	−0.229170	−0.114585	+	0.993413i	$0.536554\pi$
$98$	− 15.3088i	− 0.156212i
$99$	0	0
$100$	99.9615	0.999615
$101$	− 150.786i	− 1.49293i	−0.665424	−	0.746465i	$-0.731749\pi$
$101$	− 150.786i	− 1.49293i	0.665424	−	0.746465i	$-0.268251\pi$
$102$	0	0
$103$	60.7987	0.590279	0.295139	−	0.955454i	$-0.404634\pi$
$103$	60.7987	0.590279	0.295139	+	0.955454i	$0.404634\pi$
$104$	64.3346	0.618602
$105$	0	0
$106$	− 28.8762i	− 0.272417i
$107$	− 73.2345i	− 0.684435i	−0.939621	−	0.342217i	$-0.888822\pi$
$107$	− 73.2345i	− 0.684435i	0.939621	−	0.342217i	$-0.111178\pi$
$108$	0	0
$109$	7.06111i	0.0647808i	0.999475	+	0.0323904i	$0.0103120\pi$
$109$	7.06111i	0.0647808i	−0.999475	+	0.0323904i	$0.989688\pi$
$110$	0	0
$111$	0	0
$112$	113.962i	1.01752i
$113$	86.2154	0.762968	0.381484	−	0.924375i	$-0.375413\pi$
$113$	86.2154	0.762968	0.381484	+	0.924375i	$0.375413\pi$
$114$	0	0
$115$	201.870	1.75540
$116$	14.1793i	0.122235i
$117$	0	0
$118$	− 19.7174i	− 0.167096i
$119$	−239.412	−2.01186
$120$	0	0
$121$	0	0
$122$	6.23463	0.0511035
$123$	0	0
$124$	−139.282	−1.12324
$125$	12.8423	0.102739
$126$	0	0
$127$	57.9011i	0.455914i	0.973671	+	0.227957i	$0.0732046\pi$
$127$	57.9011i	0.455914i	−0.973671	+	0.227957i	$0.926795\pi$
$128$	111.205i	0.868791i
$129$	0	0
$130$	59.8756	0.460582
$131$	− 88.4862i	− 0.675467i	−0.941242	−	0.337734i	$-0.890340\pi$
$131$	− 88.4862i	− 0.675467i	0.941242	−	0.337734i	$-0.109660\pi$
$132$	0	0
$133$	118.641	0.892038
$134$	− 42.3992i	− 0.316412i
$135$	0	0
$136$	−108.100	−0.794853
$137$	128.067	0.934793	0.467397	−	0.884048i	$-0.345192\pi$
$137$	128.067	0.934793	0.467397	+	0.884048i	$0.345192\pi$
$138$	0	0
$139$	203.511i	1.46411i	0.681248	+	0.732053i	$0.261438\pi$
$139$	203.511i	1.46411i	−0.681248	+	0.732053i	$0.738562\pi$
$140$	238.061i	1.70044i
$141$	0	0
$142$	27.2762i	0.192086i
$143$	0	0
$144$	0	0
$145$	27.3406i	0.188556i
$146$	67.0474	0.459229
$147$	0	0
$148$	97.4974	0.658766
$149$	173.052i	1.16142i	0.814110	+	0.580710i	$0.197225\pi$
$149$	173.052i	1.16142i	−0.814110	+	0.580710i	$0.802775\pi$
$150$	0	0
$151$	10.7589i	0.0712510i	0.999365	+	0.0356255i	$0.0113424\pi$
$151$	10.7589i	0.0712510i	−0.999365	+	0.0356255i	$0.988658\pi$
$152$	53.5692	0.352429
$153$	0	0
$154$	0	0
$155$	−268.564	−1.73267
$156$	0	0
$157$	−5.96152	−0.0379715	−0.0189857	−	0.999820i	$-0.506044\pi$
$157$	−5.96152	−0.0379715	−0.0189857	+	0.999820i	$0.506044\pi$
$158$	−30.9282	−0.195748
$159$	0	0
$160$	163.098i	1.01936i
$161$	248.664i	1.54450i
$162$	0	0
$163$	−76.5218	−0.469459	−0.234729	−	0.972061i	$-0.575420\pi$
$163$	−76.5218	−0.469459	−0.234729	+	0.972061i	$0.575420\pi$
$164$	131.809i	0.803715i
$165$	0	0
$166$	−44.2102	−0.266327
$167$	311.120i	1.86299i	0.363751	+	0.931496i	$0.381496\pi$
$167$	311.120i	1.86299i	−0.363751	+	0.931496i	$0.618504\pi$
$168$	0	0
$169$	−89.3731	−0.528835
$170$	−100.608	−0.591810
$171$	0	0
$172$	− 84.0677i	− 0.488766i
$173$	− 335.709i	− 1.94051i	−0.242078	−	0.970257i	$-0.577829\pi$
$173$	− 335.709i	− 1.94051i	0.242078	−	0.970257i	$-0.422171\pi$
$174$	0	0
$175$	237.425i	1.35671i
$176$	0	0
$177$	0	0
$178$	− 64.1028i	− 0.360128i
$179$	25.3205	0.141455	0.0707277	−	0.997496i	$-0.477468\pi$
$179$	25.3205	0.141455	0.0707277	+	0.997496i	$0.477468\pi$
$180$	0	0
$181$	−167.186	−0.923679	−0.461839	−	0.886964i	$-0.652810\pi$
$181$	−167.186	−0.923679	−0.461839	+	0.886964i	$0.652810\pi$
$182$	73.7548i	0.405246i
$183$	0	0
$184$	112.278i	0.610205i
$185$	187.995	1.01619
$186$	0	0
$187$	0	0
$188$	204.851	1.08963
$189$	0	0
$190$	49.8564	0.262402
$191$	116.133	0.608028	0.304014	−	0.952668i	$-0.401673\pi$
$191$	116.133	0.608028	0.304014	+	0.952668i	$0.401673\pi$
$192$	0	0
$193$	− 261.105i	− 1.35287i	−0.736501	−	0.676437i	$-0.763523\pi$
$193$	− 261.105i	− 1.35287i	0.736501	−	0.676437i	$-0.236477\pi$
$194$	− 11.5068i	− 0.0593135i
$195$	0	0
$196$	−110.373	−0.563128
$197$	− 78.1235i	− 0.396566i	−0.980145	−	0.198283i	$-0.936463\pi$
$197$	− 78.1235i	− 0.396566i	0.980145	−	0.198283i	$-0.0635365\pi$
$198$	0	0
$199$	−177.478	−0.891850	−0.445925	−	0.895070i	$-0.647125\pi$
$199$	−177.478	−0.891850	−0.445925	+	0.895070i	$0.647125\pi$
$200$	107.203i	0.536014i
$201$	0	0
$202$	78.0526	0.386399
$203$	−33.6781	−0.165902
$204$	0	0
$205$	254.155i	1.23978i
$206$	31.4717i	0.152775i
$207$	0	0
$208$	− 206.654i	− 0.993527i
$209$	0	0
$210$	0	0
$211$	20.2323i	0.0958879i	0.998850	+	0.0479439i	$0.0152669\pi$
$211$	20.2323i	0.0958879i	−0.998850	+	0.0479439i	$0.984733\pi$
$212$	−208.191	−0.982033
$213$	0	0
$214$	37.9090	0.177145
$215$	− 162.100i	− 0.753952i
$216$	0	0
$217$	− 330.817i	− 1.52450i
$218$	−3.65510	−0.0167665
$219$	0	0
$220$	0	0
$221$	434.138	1.96443
$222$	0	0
$223$	26.6025	0.119294	0.0596470	−	0.998220i	$-0.481003\pi$
$223$	26.6025	0.119294	0.0596470	+	0.998220i	$0.481003\pi$
$224$	−200.904	−0.896892
$225$	0	0
$226$	44.6284i	0.197471i
$227$	− 102.703i	− 0.452435i	−0.974077	−	0.226217i	$-0.927364\pi$
$227$	− 102.703i	− 0.452435i	0.974077	−	0.226217i	$-0.0726360\pi$
$228$	0	0
$229$	228.755	0.998930	0.499465	−	0.866334i	$-0.333530\pi$
$229$	228.755	0.998930	0.499465	+	0.866334i	$0.333530\pi$
$230$	104.496i	0.454330i
$231$	0	0
$232$	−15.2065	−0.0655451
$233$	− 182.424i	− 0.782934i	−0.920192	−	0.391467i	$-0.871968\pi$
$233$	− 182.424i	− 0.782934i	0.920192	−	0.391467i	$-0.128032\pi$
$234$	0	0
$235$	394.995	1.68083
$236$	−142.158	−0.602363
$237$	0	0
$238$	− 123.929i	− 0.520708i
$239$	153.959i	0.644179i	0.946709	+	0.322090i	$0.104385\pi$
$239$	153.959i	0.644179i	−0.946709	+	0.322090i	$0.895615\pi$
$240$	0	0
$241$	− 314.929i	− 1.30676i	−0.757030	−	0.653380i	$-0.773351\pi$
$241$	− 314.929i	− 1.30676i	0.757030	−	0.653380i	$-0.226649\pi$
$242$	0	0
$243$	0	0
$244$	− 44.9502i	− 0.184222i
$245$	−212.822	−0.868660
$246$	0	0
$247$	−215.138	−0.871006
$248$	− 149.372i	− 0.602305i
$249$	0	0
$250$	6.64768i	0.0265907i
$251$	203.594	0.811130	0.405565	−	0.914066i	$-0.367075\pi$
$251$	203.594	0.811130	0.405565	+	0.914066i	$0.367075\pi$
$252$	0	0
$253$	0	0
$254$	−29.9718	−0.117999
$255$	0	0
$256$	101.210	0.395352
$257$	−87.5500	−0.340661	−0.170331	−	0.985387i	$-0.554484\pi$
$257$	−87.5500	−0.340661	−0.170331	+	0.985387i	$0.554484\pi$
$258$	0	0
$259$	231.572i	0.894100i
$260$	− 431.690i	− 1.66034i
$261$	0	0
$262$	45.8038	0.174824
$263$	− 97.7566i	− 0.371698i	−0.982578	−	0.185849i	$-0.940496\pi$
$263$	− 97.7566i	− 0.371698i	0.982578	−	0.185849i	$-0.0595035\pi$
$264$	0	0
$265$	−401.435	−1.51485
$266$	61.4131i	0.230876i
$267$	0	0
$268$	−305.688	−1.14063
$269$	111.172	0.413278	0.206639	−	0.978417i	$-0.433747\pi$
$269$	111.172	0.413278	0.206639	+	0.978417i	$0.433747\pi$
$270$	0	0
$271$	− 61.3315i	− 0.226315i	−0.993577	−	0.113158i	$-0.963903\pi$
$271$	− 61.3315i	− 0.226315i	0.993577	−	0.113158i	$-0.0360965\pi$
$272$	347.236i	1.27660i
$273$	0	0
$274$	66.2922i	0.241942i
$275$	0	0
$276$	0	0
$277$	− 501.506i	− 1.81049i	−0.424888	−	0.905246i	$-0.639686\pi$
$277$	− 501.506i	− 1.81049i	0.424888	−	0.905246i	$-0.360314\pi$
$278$	−105.345	−0.378938
$279$	0	0
$280$	−255.306	−0.911809
$281$	− 315.023i	− 1.12108i	−0.828128	−	0.560540i	$-0.810594\pi$
$281$	− 315.023i	− 1.12108i	0.828128	−	0.560540i	$-0.189406\pi$
$282$	0	0
$283$	− 104.605i	− 0.369628i	−0.982774	−	0.184814i	$-0.940832\pi$
$283$	− 104.605i	− 0.369628i	0.982774	−	0.184814i	$-0.0591682\pi$
$284$	196.655	0.692448
$285$	0	0
$286$	0	0
$287$	−313.068	−1.09083
$288$	0	0
$289$	−440.473	−1.52413
$290$	−14.1525	−0.0488018
$291$	0	0
$292$	− 483.396i	− 1.65547i
$293$	− 11.5068i	− 0.0392724i	−0.999807	−	0.0196362i	$-0.993749\pi$
$293$	− 11.5068i	− 0.0392724i	0.999807	−	0.0196362i	$-0.00625080\pi$
$294$	0	0
$295$	−274.109	−0.929183
$296$	104.560i	0.353244i
$297$	0	0
$298$	−89.5781	−0.300598
$299$	− 450.916i	− 1.50808i
$300$	0	0
$301$	199.674	0.663370
$302$	−5.56922	−0.0184411
$303$	0	0
$304$	− 172.073i	− 0.566031i
$305$	− 86.6732i	− 0.284174i
$306$	0	0
$307$	− 239.814i	− 0.781154i	−0.920570	−	0.390577i	$-0.872275\pi$
$307$	− 239.814i	− 0.781154i	0.920570	−	0.390577i	$-0.127725\pi$
$308$	0	0
$309$	0	0
$310$	− 139.019i	− 0.448448i
$311$	−271.951	−0.874441	−0.437221	−	0.899354i	$-0.644037\pi$
$311$	−271.951	−0.874441	−0.437221	+	0.899354i	$0.644037\pi$
$312$	0	0
$313$	37.6410	0.120259	0.0601294	−	0.998191i	$-0.480849\pi$
$313$	37.6410	0.120259	0.0601294	+	0.998191i	$0.480849\pi$
$314$	− 3.08591i	− 0.00982775i
$315$	0	0
$316$	222.985i	0.705649i
$317$	−79.9897	−0.252333	−0.126167	−	0.992009i	$-0.540267\pi$
$317$	−79.9897	−0.252333	−0.126167	+	0.992009i	$0.540267\pi$
$318$	0	0
$319$	0	0
$320$	285.641	0.892628
$321$	0	0
$322$	−128.718	−0.399745
$323$	361.492	1.11917
$324$	0	0
$325$	− 430.535i	− 1.32472i
$326$	− 39.6106i	− 0.121505i
$327$	0	0
$328$	−141.358	−0.430968
$329$	486.555i	1.47889i
$330$	0	0
$331$	616.515	1.86258	0.931292	−	0.364274i	$-0.118683\pi$
$331$	616.515	1.86258	0.931292	+	0.364274i	$0.118683\pi$
$332$	318.746i	0.960077i
$333$	0	0
$334$	−161.047	−0.482178
$335$	−589.429	−1.75949
$336$	0	0
$337$	− 225.256i	− 0.668416i	−0.942499	−	0.334208i	$-0.891531\pi$
$337$	− 225.256i	− 0.668416i	0.942499	−	0.334208i	$-0.108469\pi$
$338$	− 46.2629i	− 0.136872i
$339$	0	0
$340$	725.358i	2.13341i
$341$	0	0
$342$	0	0
$343$	172.193i	0.502020i
$344$	90.1577	0.262086
$345$	0	0
$346$	173.776	0.502242
$347$	− 233.189i	− 0.672015i	−0.941859	−	0.336008i	$-0.890923\pi$
$347$	− 233.189i	− 0.672015i	0.941859	−	0.336008i	$-0.109077\pi$
$348$	0	0
$349$	− 524.418i	− 1.50263i	−0.659943	−	0.751316i	$-0.729420\pi$
$349$	− 524.418i	− 1.50263i	0.659943	−	0.751316i	$-0.270580\pi$
$350$	−122.900	−0.351143
$351$	0	0
$352$	0	0
$353$	318.804	0.903127	0.451564	−	0.892239i	$-0.350866\pi$
$353$	318.804	0.903127	0.451564	+	0.892239i	$0.350866\pi$
$354$	0	0
$355$	379.191	1.06814
$356$	−462.167	−1.29822
$357$	0	0
$358$	13.1069i	0.0366113i
$359$	131.527i	0.366371i	0.983078	+	0.183186i	$0.0586409\pi$
$359$	131.527i	0.366371i	−0.983078	+	0.183186i	$0.941359\pi$
$360$	0	0
$361$	181.862	0.503772
$362$	− 86.5418i	− 0.239066i
$363$	0	0
$364$	531.755	1.46087
$365$	− 932.086i	− 2.55366i
$366$	0	0
$367$	−385.191	−1.04957	−0.524783	−	0.851236i	$-0.675854\pi$
$367$	−385.191	−1.04957	−0.524783	+	0.851236i	$0.675854\pi$
$368$	360.655	0.980041
$369$	0	0
$370$	97.3133i	0.263009i
$371$	− 494.487i	− 1.33285i
$372$	0	0
$373$	165.651i	0.444106i	0.975035	+	0.222053i	$0.0712758\pi$
$373$	165.651i	0.444106i	−0.975035	+	0.222053i	$0.928724\pi$
$374$	0	0
$375$	0	0
$376$	219.691i	0.584284i
$377$	61.0704	0.161990
$378$	0	0
$379$	155.611	0.410584	0.205292	−	0.978701i	$-0.434186\pi$
$379$	155.611	0.410584	0.205292	+	0.978701i	$0.434186\pi$
$380$	− 359.453i	− 0.945929i
$381$	0	0
$382$	60.1150i	0.157369i
$383$	−552.070	−1.44144	−0.720719	−	0.693228i	$-0.756188\pi$
$383$	−552.070	−1.44144	−0.720719	+	0.693228i	$0.756188\pi$
$384$	0	0
$385$	0	0
$386$	135.158	0.350149
$387$	0	0
$388$	−82.9615	−0.213818
$389$	−559.242	−1.43764	−0.718820	−	0.695196i	$-0.755318\pi$
$389$	−559.242	−1.43764	−0.718820	+	0.695196i	$0.755318\pi$
$390$	0	0
$391$	757.665i	1.93776i
$392$	− 118.369i	− 0.301961i
$393$	0	0
$394$	40.4397	0.102639
$395$	429.961i	1.08851i
$396$	0	0
$397$	17.9282	0.0451592	0.0225796	−	0.999745i	$-0.492812\pi$
$397$	17.9282	0.0451592	0.0225796	+	0.999745i	$0.492812\pi$
$398$	− 91.8695i	− 0.230828i
$399$	0	0
$400$	344.354	0.860885
$401$	−600.138	−1.49660	−0.748302	−	0.663358i	$-0.769131\pi$
$401$	−600.138	−1.49660	−0.748302	+	0.663358i	$0.769131\pi$
$402$	0	0
$403$	599.889i	1.48856i
$404$	− 562.741i	− 1.39292i
$405$	0	0
$406$	− 17.4331i	− 0.0429386i
$407$	0	0
$408$	0	0
$409$	− 117.816i	− 0.288058i	−0.989573	−	0.144029i	$-0.953994\pi$
$409$	− 117.816i	− 0.288058i	0.989573	−	0.144029i	$-0.0460059\pi$
$410$	−131.560	−0.320879
$411$	0	0
$412$	226.904	0.550737
$413$	− 337.647i	− 0.817548i
$414$	0	0
$415$	614.606i	1.48098i
$416$	364.310	0.875746
$417$	0	0
$418$	0	0
$419$	−637.324	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$419$	−637.324	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$420$	0	0
$421$	−86.8282	−0.206243	−0.103121	−	0.994669i	$-0.532883\pi$
$421$	−86.8282	−0.206243	−0.103121	+	0.994669i	$0.532883\pi$
$422$	−10.4730	−0.0248176
$423$	0	0
$424$	− 223.273i	− 0.526586i
$425$	723.419i	1.70216i
$426$	0	0
$427$	106.764	0.250033
$428$	− 273.315i	− 0.638586i
$429$	0	0
$430$	83.9090	0.195137
$431$	− 444.239i	− 1.03072i	−0.856975	−	0.515358i	$-0.827659\pi$
$431$	− 444.239i	− 1.03072i	0.856975	−	0.515358i	$-0.172341\pi$
$432$	0	0
$433$	−458.842	−1.05968	−0.529841	−	0.848097i	$-0.677748\pi$
$433$	−458.842	−1.05968	−0.529841	+	0.848097i	$0.677748\pi$
$434$	171.244	0.394570
$435$	0	0
$436$	26.3524i	0.0604413i
$437$	− 375.463i	− 0.859183i
$438$	0	0
$439$	853.249i	1.94362i	0.235765	+	0.971810i	$0.424240\pi$
$439$	853.249i	1.94362i	−0.235765	+	0.971810i	$0.575760\pi$
$440$	0	0
$441$	0	0
$442$	224.727i	0.508431i
$443$	100.788	0.227513	0.113757	−	0.993509i	$-0.463712\pi$
$443$	100.788	0.227513	0.113757	+	0.993509i	$0.463712\pi$
$444$	0	0
$445$	−891.151	−2.00259
$446$	13.7705i	0.0308755i
$447$	0	0
$448$	351.853i	0.785385i
$449$	−119.722	−0.266641	−0.133320	−	0.991073i	$-0.542564\pi$
$449$	−119.722	−0.266641	−0.133320	+	0.991073i	$0.542564\pi$
$450$	0	0
$451$	0	0
$452$	321.760	0.711859
$453$	0	0
$454$	53.1628	0.117099
$455$	1025.33	2.25348
$456$	0	0
$457$	− 212.019i	− 0.463936i	−0.972723	−	0.231968i	$-0.925484\pi$
$457$	− 212.019i	− 0.463936i	0.972723	−	0.231968i	$-0.0745164\pi$
$458$	118.412i	0.258542i
$459$	0	0
$460$	753.391	1.63781
$461$	− 35.0063i	− 0.0759355i	−0.999279	−	0.0379678i	$-0.987912\pi$
$461$	− 35.0063i	− 0.0759355i	0.999279	−	0.0379678i	$-0.0120884\pi$
$462$	0	0
$463$	377.381	0.815077	0.407538	−	0.913188i	$-0.366387\pi$
$463$	377.381	0.815077	0.407538	+	0.913188i	$0.366387\pi$
$464$	48.8458i	0.105271i
$465$	0	0
$466$	94.4294	0.202638
$467$	−496.081	−1.06227	−0.531136	−	0.847287i	$-0.678235\pi$
$467$	−496.081	−1.06227	−0.531136	+	0.847287i	$0.678235\pi$
$468$	0	0
$469$	− 726.059i	− 1.54810i
$470$	204.464i	0.435031i
$471$	0	0
$472$	− 152.456i	− 0.322999i
$473$	0	0
$474$	0	0
$475$	− 358.492i	− 0.754720i
$476$	−893.496	−1.87709
$477$	0	0
$478$	−79.6950	−0.166726
$479$	502.878i	1.04985i	0.851149	+	0.524925i	$0.175907\pi$
$479$	502.878i	1.04985i	−0.851149	+	0.524925i	$0.824093\pi$
$480$	0	0
$481$	− 419.923i	− 0.873020i
$482$	163.019	0.338214
$483$	0	0
$484$	0	0
$485$	−159.967	−0.329828
$486$	0	0
$487$	−773.734	−1.58878	−0.794389	−	0.607410i	$-0.792209\pi$
$487$	−773.734	−1.58878	−0.794389	+	0.607410i	$0.792209\pi$
$488$	48.2065	0.0987837
$489$	0	0
$490$	− 110.165i	− 0.224826i
$491$	406.922i	0.828761i	0.910104	+	0.414381i	$0.136002\pi$
$491$	406.922i	0.828761i	−0.910104	+	0.414381i	$0.863998\pi$
$492$	0	0
$493$	−102.615	−0.208145
$494$	− 111.364i	− 0.225433i
$495$	0	0
$496$	−479.808	−0.967354
$497$	467.087i	0.939814i
$498$	0	0
$499$	−357.486	−0.716404	−0.358202	−	0.933644i	$-0.616610\pi$
$499$	−357.486	−0.716404	−0.358202	+	0.933644i	$0.616610\pi$
$500$	47.9282	0.0958564
$501$	0	0
$502$	105.388i	0.209936i
$503$	518.908i	1.03163i	0.856701	+	0.515813i	$0.172510\pi$
$503$	518.908i	1.03163i	−0.856701	+	0.515813i	$0.827490\pi$
$504$	0	0
$505$	− 1085.08i	− 2.14867i
$506$	0	0
$507$	0	0
$508$	216.090i	0.425374i
$509$	270.480	0.531394	0.265697	−	0.964057i	$-0.414398\pi$
$509$	270.480	0.531394	0.265697	+	0.964057i	$0.414398\pi$
$510$	0	0
$511$	1148.14	2.24686
$512$	497.211i	0.971116i
$513$	0	0
$514$	− 45.3192i	− 0.0881697i
$515$	437.517	0.849547
$516$	0	0
$517$	0	0
$518$	−119.870	−0.231410
$519$	0	0
$520$	462.962	0.890311
$521$	336.841	0.646528	0.323264	−	0.946309i	$-0.395220\pi$
$521$	336.841	0.646528	0.323264	+	0.946309i	$0.395220\pi$
$522$	0	0
$523$	− 490.507i	− 0.937872i	−0.883232	−	0.468936i	$-0.844637\pi$
$523$	− 490.507i	− 0.937872i	0.883232	−	0.468936i	$-0.155363\pi$
$524$	− 330.235i	− 0.630220i
$525$	0	0
$526$	50.6025	0.0962025
$527$	− 1007.98i	− 1.91268i
$528$	0	0
$529$	257.946	0.487611
$530$	− 207.798i	− 0.392071i
$531$	0	0
$532$	442.774	0.832283
$533$	567.704	1.06511
$534$	0	0
$535$	− 527.007i	− 0.985059i
$536$	− 327.833i	− 0.611628i
$537$	0	0
$538$	57.5467i	0.106964i
$539$	0	0
$540$	0	0
$541$	468.516i	0.866019i	0.901389	+	0.433009i	$0.142548\pi$
$541$	468.516i	0.866019i	−0.901389	+	0.433009i	$0.857452\pi$
$542$	31.7475	0.0585748
$543$	0	0
$544$	−612.142	−1.12526
$545$	50.8128i	0.0932345i
$546$	0	0
$547$	− 295.717i	− 0.540617i	−0.962774	−	0.270308i	$-0.912874\pi$
$547$	− 295.717i	− 0.540617i	0.962774	−	0.270308i	$-0.0871256\pi$
$548$	477.951	0.872174
$549$	0	0
$550$	0	0
$551$	50.8513	0.0922890
$552$	0	0
$553$	−529.626	−0.957732
$554$	259.599	0.468590
$555$	0	0
$556$	759.512i	1.36603i
$557$	− 590.668i	− 1.06045i	−0.847858	−	0.530223i	$-0.822108\pi$
$557$	− 590.668i	− 1.06045i	0.847858	−	0.530223i	$-0.177892\pi$
$558$	0	0
$559$	−362.081	−0.647729
$560$	820.088i	1.46444i
$561$	0	0
$562$	163.068	0.290157
$563$	− 112.337i	− 0.199534i	−0.995011	−	0.0997668i	$-0.968190\pi$
$563$	− 112.337i	− 0.199534i	0.995011	−	0.0997668i	$-0.0318097\pi$
$564$	0	0
$565$	620.419	1.09809
$566$	54.1474	0.0956667
$567$	0	0
$568$	210.901i	0.371305i
$569$	355.007i	0.623915i	0.950096	+	0.311957i	$0.100985\pi$
$569$	355.007i	0.623915i	−0.950096	+	0.311957i	$0.899015\pi$
$570$	0	0
$571$	− 646.598i	− 1.13240i	−0.824269	−	0.566198i	$-0.808414\pi$
$571$	− 646.598i	− 1.13240i	0.824269	−	0.566198i	$-0.191586\pi$
$572$	0	0
$573$	0	0
$574$	− 162.056i	− 0.282327i
$575$	751.377	1.30674
$576$	0	0
$577$	−840.037	−1.45587	−0.727935	−	0.685646i	$-0.759520\pi$
$577$	−840.037	−1.45587	−0.727935	+	0.685646i	$0.759520\pi$
$578$	− 228.006i	− 0.394473i
$579$	0	0
$580$	102.036i	0.175925i
$581$	−757.072	−1.30305
$582$	0	0
$583$	0	0
$584$	518.414	0.887695
$585$	0	0
$586$	5.95637	0.0101645
$587$	11.6743	0.0198881	0.00994407	−	0.999951i	$-0.496835\pi$
$587$	11.6743	0.0198881	0.00994407	+	0.999951i	$0.496835\pi$
$588$	0	0
$589$	499.507i	0.848060i
$590$	− 141.889i	− 0.240490i
$591$	0	0
$592$	335.865	0.567340
$593$	326.775i	0.551054i	0.961293	+	0.275527i	$0.0888523\pi$
$593$	326.775i	0.551054i	−0.961293	+	0.275527i	$0.911148\pi$
$594$	0	0
$595$	−1722.84	−2.89553
$596$	645.838i	1.08362i
$597$	0	0
$598$	233.412	0.390320
$599$	6.79492	0.0113438	0.00567189	−	0.999984i	$-0.498195\pi$
$599$	6.79492	0.0113438	0.00567189	+	0.999984i	$0.498195\pi$
$600$	0	0
$601$	557.279i	0.927253i	0.886031	+	0.463627i	$0.153452\pi$
$601$	557.279i	0.927253i	−0.886031	+	0.463627i	$0.846548\pi$
$602$	103.359i	0.171693i
$603$	0	0
$604$	40.1528i	0.0664781i
$605$	0	0
$606$	0	0
$607$	− 529.476i	− 0.872283i	−0.899878	−	0.436142i	$-0.856345\pi$
$607$	− 529.476i	− 0.872283i	0.899878	−	0.436142i	$-0.143655\pi$
$608$	303.349	0.498929
$609$	0	0
$610$	44.8653	0.0735497
$611$	− 882.297i	− 1.44402i
$612$	0	0
$613$	− 137.811i	− 0.224813i	−0.993662	−	0.112407i	$-0.964144\pi$
$613$	− 137.811i	− 0.224813i	0.993662	−	0.112407i	$-0.0358559\pi$
$614$	124.137	0.202178
$615$	0	0
$616$	0	0
$617$	−882.556	−1.43040	−0.715199	−	0.698920i	$-0.753664\pi$
$617$	−882.556	−1.43040	−0.715199	+	0.698920i	$0.753664\pi$
$618$	0	0
$619$	1075.54	1.73755	0.868774	−	0.495209i	$-0.164909\pi$
$619$	1075.54	1.73755	0.868774	+	0.495209i	$0.164909\pi$
$620$	−1002.29	−1.61660
$621$	0	0
$622$	− 140.772i	− 0.226322i
$623$	− 1097.72i	− 1.76199i
$624$	0	0
$625$	−577.200	−0.923520
$626$	19.4844i	0.0311253i
$627$	0	0
$628$	−22.2487	−0.0354279
$629$	705.586i	1.12176i
$630$	0	0
$631$	218.830	0.346798	0.173399	−	0.984852i	$-0.444525\pi$
$631$	218.830	0.346798	0.173399	+	0.984852i	$0.444525\pi$
$632$	−239.138	−0.378384
$633$	0	0
$634$	− 41.4057i	− 0.0653087i
$635$	416.665i	0.656166i
$636$	0	0
$637$	475.378i	0.746276i
$638$	0	0
$639$	0	0
$640$	800.250i	1.25039i
$641$	−1079.33	−1.68382	−0.841908	−	0.539620i	$-0.818568\pi$
$641$	−1079.33	−1.68382	−0.841908	+	0.539620i	$0.818568\pi$
$642$	0	0
$643$	1005.91	1.56441	0.782203	−	0.623024i	$-0.214096\pi$
$643$	1005.91	1.56441	0.782203	+	0.623024i	$0.214096\pi$
$644$	928.027i	1.44104i
$645$	0	0
$646$	187.122i	0.289663i
$647$	663.885	1.02610	0.513048	−	0.858360i	$-0.328516\pi$
$647$	663.885	1.02610	0.513048	+	0.858360i	$0.328516\pi$
$648$	0	0
$649$	0	0
$650$	222.862	0.342864
$651$	0	0
$652$	−285.583	−0.438011
$653$	−975.426	−1.49376	−0.746880	−	0.664959i	$-0.768449\pi$
$653$	−975.426	−1.49376	−0.746880	+	0.664959i	$0.768449\pi$
$654$	0	0
$655$	− 636.760i	− 0.972153i
$656$	454.065i	0.692172i
$657$	0	0
$658$	−251.859	−0.382765
$659$	− 83.8666i	− 0.127263i	−0.997973	−	0.0636317i	$-0.979732\pi$
$659$	− 83.8666i	− 0.127263i	0.997973	−	0.0636317i	$-0.0202683\pi$
$660$	0	0
$661$	−848.349	−1.28343	−0.641716	−	0.766942i	$-0.721777\pi$
$661$	−848.349	−1.28343	−0.641716	+	0.766942i	$0.721777\pi$
$662$	319.132i	0.482072i
$663$	0	0
$664$	−341.836	−0.514813
$665$	853.759	1.28385
$666$	0	0
$667$	106.581i	0.159792i
$668$	1161.11i	1.73820i
$669$	0	0
$670$	− 305.111i	− 0.455390i
$671$	0	0
$672$	0	0
$673$	− 518.155i	− 0.769918i	−0.922934	−	0.384959i	$-0.874216\pi$
$673$	− 518.155i	− 0.769918i	0.922934	−	0.384959i	$-0.125784\pi$
$674$	116.601	0.172999
$675$	0	0
$676$	−333.545	−0.493410
$677$	917.541i	1.35530i	0.735383	+	0.677652i	$0.237002\pi$
$677$	917.541i	1.35530i	−0.735383	+	0.677652i	$0.762998\pi$
$678$	0	0
$679$	− 197.047i	− 0.290202i
$680$	−777.904	−1.14398
$681$	0	0
$682$	0	0
$683$	−14.7217	−0.0215545	−0.0107773	−	0.999942i	$-0.503431\pi$
$683$	−14.7217	−0.0215545	−0.0107773	+	0.999942i	$0.503431\pi$
$684$	0	0
$685$	921.587	1.34538
$686$	−89.1337	−0.129932
$687$	0	0
$688$	− 289.602i	− 0.420933i
$689$	896.681i	1.30142i
$690$	0	0
$691$	−1274.40	−1.84429	−0.922144	−	0.386846i	$-0.873565\pi$
$691$	−1274.40	−1.84429	−0.922144	+	0.386846i	$0.873565\pi$
$692$	− 1252.88i	− 1.81052i
$693$	0	0
$694$	120.708	0.173930
$695$	1464.49i	2.10719i
$696$	0	0
$697$	−953.900	−1.36858
$698$	271.459	0.388910
$699$	0	0
$700$	886.081i	1.26583i
$701$	− 286.714i	− 0.409008i	−0.978866	−	0.204504i	$-0.934442\pi$
$701$	− 286.714i	− 0.409008i	0.978866	−	0.204504i	$-0.0655581\pi$
$702$	0	0
$703$	− 349.655i	− 0.497376i
$704$	0	0
$705$	0	0
$706$	165.025i	0.233746i
$707$	1336.60	1.89052
$708$	0	0
$709$	1261.12	1.77873	0.889364	−	0.457200i	$-0.151148\pi$
$709$	1261.12	1.77873	0.889364	+	0.457200i	$0.151148\pi$
$710$	196.284i	0.276456i
$711$	0	0
$712$	− 495.646i	− 0.696133i
$713$	−1046.94	−1.46835
$714$	0	0
$715$	0	0
$716$	94.4974	0.131980
$717$	0	0
$718$	−68.0835	−0.0948238
$719$	−22.7884	−0.0316946	−0.0158473	−	0.999874i	$-0.505045\pi$
$719$	−22.7884	−0.0316946	−0.0158473	+	0.999874i	$0.505045\pi$
$720$	0	0
$721$	538.933i	0.747480i
$722$	94.1385i	0.130386i
$723$	0	0
$724$	−623.946	−0.861804
$725$	101.764i	0.140364i
$726$	0	0
$727$	984.301	1.35392	0.676961	−	0.736019i	$-0.263297\pi$
$727$	984.301	1.35392	0.676961	+	0.736019i	$0.263297\pi$
$728$	570.276i	0.783346i
$729$	0	0
$730$	482.483	0.660936
$731$	608.396	0.832279
$732$	0	0
$733$	− 97.5708i	− 0.133112i	−0.997783	−	0.0665558i	$-0.978799\pi$
$733$	− 97.5708i	− 0.133112i	0.997783	−	0.0665558i	$-0.0212010\pi$
$734$	− 199.390i	− 0.271648i
$735$	0	0
$736$	635.800i	0.863858i
$737$	0	0
$738$	0	0
$739$	− 746.447i	− 1.01008i	−0.863097	−	0.505039i	$-0.831478\pi$
$739$	− 746.447i	− 1.01008i	0.863097	−	0.505039i	$-0.168522\pi$
$740$	701.606	0.948117
$741$	0	0
$742$	255.965	0.344967
$743$	− 246.646i	− 0.331960i	−0.986129	−	0.165980i	$-0.946921\pi$
$743$	− 246.646i	− 0.331960i	0.986129	−	0.165980i	$-0.0530786\pi$
$744$	0	0
$745$	1245.31i	1.67155i
$746$	−85.7475	−0.114943
$747$	0	0
$748$	0	0
$749$	649.167	0.866711
$750$	0	0
$751$	616.676	0.821139	0.410570	−	0.911829i	$-0.365330\pi$
$751$	616.676	0.821139	0.410570	+	0.911829i	$0.365330\pi$
$752$	705.685	0.938410
$753$	0	0
$754$	31.6124i	0.0419262i
$755$	77.4227i	0.102547i
$756$	0	0
$757$	750.661	0.991627	0.495813	−	0.868429i	$-0.334870\pi$
$757$	750.661	0.991627	0.495813	+	0.868429i	$0.334870\pi$
$758$	80.5504i	0.106267i
$759$	0	0
$760$	385.492	0.507227
$761$	− 1081.21i	− 1.42077i	−0.703813	−	0.710386i	$-0.748521\pi$
$761$	− 1081.21i	− 1.42077i	0.703813	−	0.710386i	$-0.251479\pi$
$762$	0	0
$763$	−62.5912	−0.0820331
$764$	433.415	0.567298
$765$	0	0
$766$	− 285.773i	− 0.373071i
$767$	612.275i	0.798272i
$768$	0	0
$769$	− 311.835i	− 0.405507i	−0.979230	−	0.202754i	$-0.935011\pi$
$769$	− 311.835i	− 0.405507i	0.979230	−	0.202754i	$-0.0649891\pi$
$770$	0	0
$771$	0	0
$772$	− 974.456i	− 1.26225i
$773$	1308.18	1.69234	0.846172	−	0.532910i	$-0.178901\pi$
$773$	1308.18	1.69234	0.846172	+	0.532910i	$0.178901\pi$
$774$	0	0
$775$	−999.615	−1.28983
$776$	− 88.9713i	− 0.114654i
$777$	0	0
$778$	− 289.485i	− 0.372089i
$779$	472.708	0.606813
$780$	0	0
$781$	0	0
$782$	−392.196	−0.501530
$783$	0	0
$784$	−380.220	−0.484975
$785$	−42.9000	−0.0546497
$786$	0	0
$787$	388.436i	0.493565i	0.969071	+	0.246782i	$0.0793733\pi$
$787$	388.436i	0.493565i	−0.969071	+	0.246782i	$0.920627\pi$
$788$	− 291.561i	− 0.370001i
$789$	0	0
$790$	−222.564	−0.281727
$791$	764.232i	0.966159i
$792$	0	0
$793$	−193.601	−0.244138
$794$	9.28032i	0.0116881i
$795$	0	0
$796$	−662.358	−0.832108
$797$	948.410	1.18998	0.594988	−	0.803735i	$-0.297157\pi$
$797$	948.410	1.18998	0.594988	+	0.803735i	$0.297157\pi$
$798$	0	0
$799$	1482.50i	1.85545i
$800$	607.062i	0.758827i
$801$	0	0
$802$	− 310.655i	− 0.387350i
$803$	0	0
$804$	0	0
$805$	1789.42i	2.22289i
$806$	−310.526	−0.385267
$807$	0	0
$808$	603.506	0.746914
$809$	1380.27i	1.70614i	0.521793	+	0.853072i	$0.325263\pi$
$809$	1380.27i	1.70614i	−0.521793	+	0.853072i	$0.674737\pi$
$810$	0	0
$811$	73.6678i	0.0908358i	0.998968	+	0.0454179i	$0.0144619\pi$
$811$	73.6678i	0.0908358i	−0.998968	+	0.0454179i	$0.985538\pi$
$812$	−125.688	−0.154789
$813$	0	0
$814$	0	0
$815$	−550.663	−0.675660
$816$	0	0
$817$	−301.492	−0.369024
$818$	60.9859	0.0745549
$819$	0	0
$820$	948.519i	1.15673i
$821$	− 994.669i	− 1.21153i	−0.795642	−	0.605767i	$-0.792867\pi$
$821$	− 994.669i	− 1.21153i	0.795642	−	0.605767i	$-0.207133\pi$
$822$	0	0
$823$	1615.57	1.96303	0.981513	−	0.191396i	$-0.0613015\pi$
$823$	1615.57	1.96303	0.981513	+	0.191396i	$0.0613015\pi$
$824$	243.341i	0.295317i
$825$	0	0
$826$	174.779	0.211597
$827$	190.636i	0.230515i	0.993336	+	0.115258i	$0.0367694\pi$
$827$	190.636i	0.230515i	−0.993336	+	0.115258i	$0.963231\pi$
$828$	0	0
$829$	607.049	0.732266	0.366133	−	0.930562i	$-0.380681\pi$
$829$	607.049	0.732266	0.366133	+	0.930562i	$0.380681\pi$
$830$	−318.144	−0.383306
$831$	0	0
$832$	− 638.034i	− 0.766868i
$833$	− 798.767i	− 0.958904i
$834$	0	0
$835$	2238.86i	2.68128i
$836$	0	0
$837$	0	0
$838$	− 329.903i	− 0.393679i
$839$	16.9742	0.0202315	0.0101157	−	0.999949i	$-0.496780\pi$
$839$	16.9742	0.0202315	0.0101157	+	0.999949i	$0.496780\pi$
$840$	0	0
$841$	826.565	0.982836
$842$	− 44.9456i	− 0.0533796i
$843$	0	0
$844$	75.5081i	0.0894646i
$845$	−643.142	−0.761115
$846$	0	0
$847$	0	0
$848$	−717.190	−0.845742
$849$	0	0
$850$	−374.469	−0.440552
$851$	732.855	0.861169
$852$	0	0
$853$	1546.20i	1.81266i	0.422567	+	0.906332i	$0.361129\pi$
$853$	1546.20i	1.81266i	−0.422567	+	0.906332i	$0.638871\pi$
$854$	55.2651i	0.0647132i
$855$	0	0
$856$	293.114	0.342423
$857$	− 1385.15i	− 1.61628i	−0.588992	−	0.808139i	$-0.700475\pi$
$857$	− 1385.15i	− 1.61628i	0.588992	−	0.808139i	$-0.299525\pi$
$858$	0	0
$859$	−642.852	−0.748373	−0.374186	−	0.927354i	$-0.622078\pi$
$859$	−642.852	−0.748373	−0.374186	+	0.927354i	$0.622078\pi$
$860$	− 604.964i	− 0.703447i
$861$	0	0
$862$	229.955	0.266769
$863$	−668.827	−0.775002	−0.387501	−	0.921869i	$-0.626662\pi$
$863$	−668.827	−0.775002	−0.387501	+	0.921869i	$0.626662\pi$
$864$	0	0
$865$	− 2415.81i	− 2.79285i
$866$	− 237.514i	− 0.274266i
$867$	0	0
$868$	− 1234.63i	− 1.42238i
$869$	0	0
$870$	0	0
$871$	1316.60i	1.51160i
$872$	−28.2614	−0.0324099
$873$	0	0
$874$	194.354	0.222373
$875$	113.837i	0.130100i
$876$	0	0
$877$	− 336.858i	− 0.384102i	−0.981385	−	0.192051i	$-0.938486\pi$
$877$	− 336.858i	− 0.384102i	0.981385	−	0.192051i	$-0.0615139\pi$
$878$	−441.674	−0.503046
$879$	0	0
$880$	0	0
$881$	−449.841	−0.510603	−0.255301	−	0.966862i	$-0.582175\pi$
$881$	−449.841	−0.510603	−0.255301	+	0.966862i	$0.582175\pi$
$882$	0	0
$883$	901.864	1.02136	0.510682	−	0.859770i	$-0.329393\pi$
$883$	901.864	1.02136	0.510682	+	0.859770i	$0.329393\pi$
$884$	1620.23	1.83284
$885$	0	0
$886$	52.1719i	0.0588848i
$887$	1051.52i	1.18548i	0.805395	+	0.592739i	$0.201953\pi$
$887$	1051.52i	1.18548i	−0.805395	+	0.592739i	$0.798047\pi$
$888$	0	0
$889$	−513.248	−0.577332
$890$	− 461.294i	− 0.518308i
$891$	0	0
$892$	99.2820	0.111303
$893$	− 734.658i	− 0.822686i
$894$	0	0
$895$	182.210	0.203587
$896$	−985.748	−1.10016
$897$	0	0
$898$	− 61.9725i	− 0.0690117i
$899$	− 141.793i	− 0.157723i
$900$	0	0
$901$	− 1506.67i	− 1.67222i
$902$	0	0
$903$	0	0
$904$	345.069i	0.381713i
$905$	−1203.09	−1.32939
$906$	0	0
$907$	−1311.01	−1.44543	−0.722717	−	0.691144i	$-0.757107\pi$
$907$	−1311.01	−1.44543	−0.722717	+	0.691144i	$0.757107\pi$
$908$	− 383.292i	− 0.422127i
$909$	0	0
$910$	530.751i	0.583243i
$911$	1716.62	1.88433	0.942163	−	0.335154i	$-0.108788\pi$
$911$	1716.62	1.88433	0.942163	+	0.335154i	$0.108788\pi$
$912$	0	0
$913$	0	0
$914$	109.749	0.120075
$915$	0	0
$916$	853.726	0.932015
$917$	784.361	0.855356
$918$	0	0
$919$	− 1535.42i	− 1.67075i	−0.549681	−	0.835375i	$-0.685251\pi$
$919$	− 1535.42i	− 1.67075i	0.549681	−	0.835375i	$-0.314749\pi$
$920$	807.967i	0.878225i
$921$	0	0
$922$	18.1206	0.0196536
$923$	− 846.996i	− 0.917655i
$924$	0	0
$925$	699.731	0.756466
$926$	195.347i	0.210957i
$927$	0	0
$928$	−86.1103	−0.0927912
$929$	1194.93	1.28625	0.643126	−	0.765760i	$-0.277637\pi$
$929$	1194.93	1.28625	0.643126	+	0.765760i	$0.277637\pi$
$930$	0	0
$931$	395.831i	0.425168i
$932$	− 680.814i	− 0.730487i
$933$	0	0
$934$	− 256.790i	− 0.274936i
$935$	0	0
$936$	0	0
$937$	1572.33i	1.67804i	0.544098	+	0.839022i	$0.316872\pi$
$937$	1572.33i	1.67804i	−0.544098	+	0.839022i	$0.683128\pi$
$938$	375.836	0.400678
$939$	0	0
$940$	1474.14	1.56823
$941$	1478.64i	1.57135i	0.618642	+	0.785673i	$0.287683\pi$
$941$	1478.64i	1.57135i	−0.618642	+	0.785673i	$0.712317\pi$
$942$	0	0
$943$	990.765i	1.05065i
$944$	−489.714	−0.518765
$945$	0	0
$946$	0	0
$947$	−745.460	−0.787181	−0.393590	−	0.919286i	$-0.628767\pi$
$947$	−745.460	−0.787181	−0.393590	+	0.919286i	$0.628767\pi$
$948$	0	0
$949$	−2081.99	−2.19388
$950$	185.569	0.195336
$951$	0	0
$952$	− 958.222i	− 1.00654i
$953$	106.841i	0.112110i	0.998428	+	0.0560552i	$0.0178523\pi$
$953$	106.841i	0.112110i	−0.998428	+	0.0560552i	$0.982148\pi$
$954$	0	0
$955$	835.713	0.875092
$956$	574.582i	0.601027i
$957$	0	0
$958$	−260.309	−0.271721
$959$	1135.21i	1.18374i
$960$	0	0
$961$	431.820	0.449345
$962$	217.368	0.225954
$963$	0	0
$964$	− 1175.33i	− 1.21922i
$965$	− 1878.95i	− 1.94710i
$966$	0	0
$967$	269.645i	0.278847i	0.990233	+	0.139423i	$0.0445249\pi$
$967$	269.645i	0.278847i	−0.990233	+	0.139423i	$0.955475\pi$
$968$	0	0
$969$	0	0
$970$	− 82.8048i	− 0.0853658i
$971$	−437.163	−0.450219	−0.225110	−	0.974333i	$-0.572274\pi$
$971$	−437.163	−0.450219	−0.225110	+	0.974333i	$0.572274\pi$
$972$	0	0
$973$	−1803.96	−1.85402
$974$	− 400.514i	− 0.411206i
$975$	0	0
$976$	− 154.847i	− 0.158655i
$977$	268.061	0.274372	0.137186	−	0.990545i	$-0.456194\pi$
$977$	268.061	0.274372	0.137186	+	0.990545i	$0.456194\pi$
$978$	0	0
$979$	0	0
$980$	−794.261	−0.810471
$981$	0	0
$982$	−210.638	−0.214499
$983$	−1581.88	−1.60924	−0.804618	−	0.593793i	$-0.797630\pi$
$983$	−1581.88	−1.60924	−0.804618	+	0.593793i	$0.797630\pi$
$984$	0	0
$985$	− 562.189i	− 0.570750i
$986$	− 53.1176i	− 0.0538718i
$987$	0	0
$988$	−802.908	−0.812659
$989$	− 631.909i	− 0.638937i
$990$	0	0
$991$	1058.64	1.06825	0.534126	−	0.845405i	$-0.320641\pi$
$991$	1058.64	1.06825	0.534126	+	0.845405i	$0.320641\pi$
$992$	− 845.854i	− 0.852675i
$993$	0	0
$994$	−241.782	−0.243242
$995$	−1277.16	−1.28358
$996$	0	0
$997$	993.442i	0.996432i	0.867053	+	0.498216i	$0.166011\pi$
$997$	993.442i	0.996432i	−0.867053	+	0.498216i	$0.833989\pi$
$998$	− 185.048i	− 0.185419i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.g.604.3		4
3.2	odd	2		363.3.c.c.241.2	✓	4
11.10	odd	2	inner	1089.3.c.g.604.2		4
33.2	even	10		363.3.g.b.40.3		16
33.5	odd	10		363.3.g.b.118.3		16
33.8	even	10		363.3.g.b.112.3		16
33.14	odd	10		363.3.g.b.112.2		16
33.17	even	10		363.3.g.b.118.2		16
33.20	odd	10		363.3.g.b.40.2		16
33.26	odd	10		363.3.g.b.94.3		16
33.29	even	10		363.3.g.b.94.2		16
33.32	even	2		363.3.c.c.241.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.3.c.c.241.2	✓	4	3.2	odd	2
363.3.c.c.241.3	yes	4	33.32	even	2
363.3.g.b.40.2		16	33.20	odd	10
363.3.g.b.40.3		16	33.2	even	10
363.3.g.b.94.2		16	33.29	even	10
363.3.g.b.94.3		16	33.26	odd	10
363.3.g.b.112.2		16	33.14	odd	10
363.3.g.b.112.3		16	33.8	even	10
363.3.g.b.118.2		16	33.17	even	10
363.3.g.b.118.3		16	33.5	odd	10
1089.3.c.g.604.2		4	11.10	odd	2	inner
1089.3.c.g.604.3		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.517638i q^{2} +3.73205 q^{4} +7.19615 q^{5} +8.86422i q^{7} +4.00240i q^{8} +O(q^{10})\)	\(q+0.517638i q^{2} +3.73205 q^{4} +7.19615 q^{5} +8.86422i q^{7} +4.00240i q^{8} +3.72500i q^{10} -16.0740i q^{13} -4.58846 q^{14} +12.8564 q^{16} +27.0088i q^{17} -13.3843i q^{19} +26.8564 q^{20} +28.0526 q^{23} +26.7846 q^{25} +8.32051 q^{26} +33.0817i q^{28} +3.79933i q^{29} -37.3205 q^{31} +22.6646i q^{32} -13.9808 q^{34} +63.7883i q^{35} +26.1244 q^{37} +6.92820 q^{38} +28.8019i q^{40} +35.3182i q^{41} -22.5259i q^{43} +14.5211i q^{46} +54.8897 q^{47} -29.5744 q^{49} +13.8647i q^{50} -59.9889i q^{52} -55.7846 q^{53} -35.4782 q^{56} -1.96668 q^{58} -38.0910 q^{59} -12.0444i q^{61} -19.3185i q^{62} +39.6936 q^{64} -115.671i q^{65} -81.9090 q^{67} +100.798i q^{68} -33.0192 q^{70} +52.6936 q^{71} -129.526i q^{73} +13.5230i q^{74} -49.9507i q^{76} +59.7487i q^{79} +92.5167 q^{80} -18.2820 q^{82} +85.4076i q^{83} +194.359i q^{85} +11.6603 q^{86} -123.837 q^{89} +142.483 q^{91} +104.694 q^{92} +28.4130i q^{94} -96.3152i q^{95} -22.2295 q^{97} -15.3088i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} + 8 q^{5}+O(q^{10}) \) 4 * q + 8 * q^4 + 8 * q^5	\( 4 q + 8 q^{4} + 8 q^{5} + 44 q^{14} - 4 q^{16} + 52 q^{20} + 36 q^{23} + 24 q^{25} - 36 q^{26} - 80 q^{31} + 48 q^{34} + 56 q^{37} - 16 q^{47} - 340 q^{49} - 140 q^{53} + 156 q^{56} - 188 q^{58} - 284 q^{59} - 56 q^{64} - 196 q^{67} - 236 q^{70} - 4 q^{71} + 280 q^{80} + 204 q^{82} + 12 q^{86} - 336 q^{89} + 660 q^{91} + 204 q^{92} + 112 q^{97}+O(q^{100}) \) 4 * q + 8 * q^4 + 8 * q^5 + 44 * q^14 - 4 * q^16 + 52 * q^20 + 36 * q^23 + 24 * q^25 - 36 * q^26 - 80 * q^31 + 48 * q^34 + 56 * q^37 - 16 * q^47 - 340 * q^49 - 140 * q^53 + 156 * q^56 - 188 * q^58 - 284 * q^59 - 56 * q^64 - 196 * q^67 - 236 * q^70 - 4 * q^71 + 280 * q^80 + 204 * q^82 + 12 * q^86 - 336 * q^89 + 660 * q^91 + 204 * q^92 + 112 * q^97

Introduction

L-functions

Modular forms

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Database

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