LMFDB - Embedded newform 1089.2.a.p.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1089,2,Mod(1,1089)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1089.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1089, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1089.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,6,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.69570878012$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 363)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} +3.00000 q^{4} -2.00000 q^{5} +4.47214 q^{7} +2.23607 q^{8} -4.47214 q^{10} +10.0000 q^{14} -1.00000 q^{16} +4.47214 q^{17} +4.47214 q^{19} -6.00000 q^{20} +4.00000 q^{23} -1.00000 q^{25} +13.4164 q^{28} +4.47214 q^{29} -6.70820 q^{32} +10.0000 q^{34} -8.94427 q^{35} +2.00000 q^{37} +10.0000 q^{38} -4.47214 q^{40} -4.47214 q^{41} -4.47214 q^{43} +8.94427 q^{46} -8.00000 q^{47} +13.0000 q^{49} -2.23607 q^{50} -6.00000 q^{53} +10.0000 q^{56} +10.0000 q^{58} -8.94427 q^{61} -13.0000 q^{64} -12.0000 q^{67} +13.4164 q^{68} -20.0000 q^{70} +8.00000 q^{71} +8.94427 q^{73} +4.47214 q^{74} +13.4164 q^{76} -13.4164 q^{79} +2.00000 q^{80} -10.0000 q^{82} -8.94427 q^{83} -8.94427 q^{85} -10.0000 q^{86} +14.0000 q^{89} +12.0000 q^{92} -17.8885 q^{94} -8.94427 q^{95} +2.00000 q^{97} +29.0689 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} - 4 q^{5} + 20 q^{14} - 2 q^{16} - 12 q^{20} + 8 q^{23} - 2 q^{25} + 20 q^{34} + 4 q^{37} + 20 q^{38} - 16 q^{47} + 26 q^{49} - 12 q^{53} + 20 q^{56} + 20 q^{58} - 26 q^{64} - 24 q^{67} - 40 q^{70}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q + 6 * q^4 - 4 * q^5 + 20 * q^14 - 2 * q^16 - 12 * q^20 + 8 * q^23 - 2 * q^25 + 20 * q^34 + 4 * q^37 + 20 * q^38 - 16 * q^47 + 26 * q^49 - 12 * q^53 + 20 * q^56 + 20 * q^58 - 26 * q^64 - 24 * q^67 - 40 * q^70 + 16 * q^71 + 4 * q^80 - 20 * q^82 - 20 * q^86 + 28 * q^89 + 24 * q^92 + 4 * q^97

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 $

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	0	0
$3$	0	0
$4$	3.00000	1.50000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	4.47214	1.69031	0.845154	−	0.534522i	$-0.179509\pi$
$7$	4.47214	1.69031	0.845154	+	0.534522i	$0.179509\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	−4.47214	−1.41421
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	10.0000	2.67261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	−6.00000	−1.34164
$21$	0	0
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	13.4164	2.53546
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−6.70820	−1.18585
$33$	0	0
$34$	10.0000	1.71499
$35$	−8.94427	−1.51186
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	10.0000	1.62221
$39$	0	0
$40$	−4.47214	−0.707107
$41$	−4.47214	−0.698430	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214	−0.698430	−0.349215	+	0.937043i	$0.613552\pi$
$42$	0	0
$43$	−4.47214	−0.681994	−0.340997	−	0.940064i	$-0.610765\pi$
$43$	−4.47214	−0.681994	−0.340997	+	0.940064i	$0.610765\pi$
$44$	0	0
$45$	0	0
$46$	8.94427	1.31876
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	13.0000	1.85714
$50$	−2.23607	−0.316228
$51$	0	0
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	10.0000	1.33631
$57$	0	0
$58$	10.0000	1.31306
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−8.94427	−1.14520	−0.572598	−	0.819836i	$-0.694065\pi$
$61$	−8.94427	−1.14520	−0.572598	+	0.819836i	$0.694065\pi$
$62$	0	0
$63$	0	0
$64$	−13.0000	−1.62500
$65$	0	0
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	13.4164	1.62698
$69$	0	0
$70$	−20.0000	−2.39046
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	8.94427	1.04685	0.523424	−	0.852072i	$-0.324654\pi$
$73$	8.94427	1.04685	0.523424	+	0.852072i	$0.324654\pi$
$74$	4.47214	0.519875
$75$	0	0
$76$	13.4164	1.53897
$77$	0	0
$78$	0	0
$79$	−13.4164	−1.50946	−0.754732	−	0.656033i	$-0.772233\pi$
$79$	−13.4164	−1.50946	−0.754732	+	0.656033i	$0.772233\pi$
$80$	2.00000	0.223607
$81$	0	0
$82$	−10.0000	−1.10432
$83$	−8.94427	−0.981761	−0.490881	−	0.871227i	$-0.663325\pi$
$83$	−8.94427	−0.981761	−0.490881	+	0.871227i	$0.663325\pi$
$84$	0	0
$85$	−8.94427	−0.970143
$86$	−10.0000	−1.07833
$87$	0	0
$88$	0	0
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	0	0
$92$	12.0000	1.25109
$93$	0	0
$94$	−17.8885	−1.84506
$95$	−8.94427	−0.917663
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	29.0689	2.93640
$99$	0	0
$100$	−3.00000	−0.300000
$101$	4.47214	0.444994	0.222497	−	0.974933i	$-0.428579\pi$
$101$	4.47214	0.444994	0.222497	+	0.974933i	$0.428579\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	0	0
$106$	−13.4164	−1.30312
$107$	8.94427	0.864675	0.432338	−	0.901712i	$-0.357689\pi$
$107$	8.94427	0.864675	0.432338	+	0.901712i	$0.357689\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	−4.47214	−0.422577
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−8.00000	−0.746004
$116$	13.4164	1.24568
$117$	0	0
$118$	0	0
$119$	20.0000	1.83340
$120$	0	0
$121$	0	0
$122$	−20.0000	−1.81071
$123$	0	0
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−13.4164	−1.19051	−0.595257	−	0.803535i	$-0.702950\pi$
$127$	−13.4164	−1.19051	−0.595257	+	0.803535i	$0.702950\pi$
$128$	−15.6525	−1.38350
$129$	0	0
$130$	0	0
$131$	−17.8885	−1.56293	−0.781465	−	0.623949i	$-0.785527\pi$
$131$	−17.8885	−1.56293	−0.781465	+	0.623949i	$0.785527\pi$
$132$	0	0
$133$	20.0000	1.73422
$134$	−26.8328	−2.31800
$135$	0	0
$136$	10.0000	0.857493
$137$	−22.0000	−1.87959	−0.939793	−	0.341743i	$-0.888983\pi$
$137$	−22.0000	−1.87959	−0.939793	+	0.341743i	$0.888983\pi$
$138$	0	0
$139$	−13.4164	−1.13796	−0.568982	−	0.822350i	$-0.692663\pi$
$139$	−13.4164	−1.13796	−0.568982	+	0.822350i	$0.692663\pi$
$140$	−26.8328	−2.26779
$141$	0	0
$142$	17.8885	1.50117
$143$	0	0
$144$	0	0
$145$	−8.94427	−0.742781
$146$	20.0000	1.65521
$147$	0	0
$148$	6.00000	0.493197
$149$	−22.3607	−1.83186	−0.915929	−	0.401340i	$-0.868545\pi$
$149$	−22.3607	−1.83186	−0.915929	+	0.401340i	$0.868545\pi$
$150$	0	0
$151$	−13.4164	−1.09181	−0.545906	−	0.837846i	$-0.683814\pi$
$151$	−13.4164	−1.09181	−0.545906	+	0.837846i	$0.683814\pi$
$152$	10.0000	0.811107
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	−30.0000	−2.38667
$159$	0	0
$160$	13.4164	1.06066
$161$	17.8885	1.40981
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−13.4164	−1.04765
$165$	0	0
$166$	−20.0000	−1.55230
$167$	8.94427	0.692129	0.346064	−	0.938211i	$-0.387518\pi$
$167$	8.94427	0.692129	0.346064	+	0.938211i	$0.387518\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−20.0000	−1.53393
$171$	0	0
$172$	−13.4164	−1.02299
$173$	−13.4164	−1.02003	−0.510015	−	0.860165i	$-0.670360\pi$
$173$	−13.4164	−1.02003	−0.510015	+	0.860165i	$0.670360\pi$
$174$	0	0
$175$	−4.47214	−0.338062
$176$	0	0
$177$	0	0
$178$	31.3050	2.34641
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	0	0
$184$	8.94427	0.659380
$185$	−4.00000	−0.294086
$186$	0	0
$187$	0	0
$188$	−24.0000	−1.75038
$189$	0	0
$190$	−20.0000	−1.45095
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	17.8885	1.28765	0.643823	−	0.765175i	$-0.277347\pi$
$193$	17.8885	1.28765	0.643823	+	0.765175i	$0.277347\pi$
$194$	4.47214	0.321081
$195$	0	0
$196$	39.0000	2.78571
$197$	22.3607	1.59313	0.796566	−	0.604551i	$-0.206648\pi$
$197$	22.3607	1.59313	0.796566	+	0.604551i	$0.206648\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	−2.23607	−0.158114
$201$	0	0
$202$	10.0000	0.703598
$203$	20.0000	1.40372
$204$	0	0
$205$	8.94427	0.624695
$206$	35.7771	2.49271
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.47214	0.307875	0.153937	−	0.988081i	$-0.450805\pi$
$211$	4.47214	0.307875	0.153937	+	0.988081i	$0.450805\pi$
$212$	−18.0000	−1.23625
$213$	0	0
$214$	20.0000	1.36717
$215$	8.94427	0.609994
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	−30.0000	−2.00446
$225$	0	0
$226$	−13.4164	−0.892446
$227$	8.94427	0.593652	0.296826	−	0.954932i	$-0.404072\pi$
$227$	8.94427	0.593652	0.296826	+	0.954932i	$0.404072\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	−17.8885	−1.17954
$231$	0	0
$232$	10.0000	0.656532
$233$	−4.47214	−0.292979	−0.146490	−	0.989212i	$-0.546798\pi$
$233$	−4.47214	−0.292979	−0.146490	+	0.989212i	$0.546798\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	0	0
$237$	0	0
$238$	44.7214	2.89886
$239$	8.94427	0.578557	0.289278	−	0.957245i	$-0.406585\pi$
$239$	8.94427	0.578557	0.289278	+	0.957245i	$0.406585\pi$
$240$	0	0
$241$	−8.94427	−0.576151	−0.288076	−	0.957608i	$-0.593015\pi$
$241$	−8.94427	−0.576151	−0.288076	+	0.957608i	$0.593015\pi$
$242$	0	0
$243$	0	0
$244$	−26.8328	−1.71780
$245$	−26.0000	−1.66108
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	26.8328	1.69706
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	−30.0000	−1.88237
$255$	0	0
$256$	−9.00000	−0.562500
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	8.94427	0.555770
$260$	0	0
$261$	0	0
$262$	−40.0000	−2.47121
$263$	8.94427	0.551527	0.275764	−	0.961225i	$-0.411069\pi$
$263$	8.94427	0.551527	0.275764	+	0.961225i	$0.411069\pi$
$264$	0	0
$265$	12.0000	0.737154
$266$	44.7214	2.74204
$267$	0	0
$268$	−36.0000	−2.19905
$269$	10.0000	0.609711	0.304855	−	0.952399i	$-0.401392\pi$
$269$	10.0000	0.609711	0.304855	+	0.952399i	$0.401392\pi$
$270$	0	0
$271$	13.4164	0.814989	0.407494	−	0.913208i	$-0.366403\pi$
$271$	13.4164	0.814989	0.407494	+	0.913208i	$0.366403\pi$
$272$	−4.47214	−0.271163
$273$	0	0
$274$	−49.1935	−2.97189
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	−30.0000	−1.79928
$279$	0	0
$280$	−20.0000	−1.19523
$281$	31.3050	1.86750	0.933748	−	0.357930i	$-0.116517\pi$
$281$	31.3050	1.86750	0.933748	+	0.357930i	$0.116517\pi$
$282$	0	0
$283$	13.4164	0.797523	0.398761	−	0.917055i	$-0.369440\pi$
$283$	13.4164	0.797523	0.398761	+	0.917055i	$0.369440\pi$
$284$	24.0000	1.42414
$285$	0	0
$286$	0	0
$287$	−20.0000	−1.18056
$288$	0	0
$289$	3.00000	0.176471
$290$	−20.0000	−1.17444
$291$	0	0
$292$	26.8328	1.57027
$293$	−22.3607	−1.30632	−0.653162	−	0.757218i	$-0.726558\pi$
$293$	−22.3607	−1.30632	−0.653162	+	0.757218i	$0.726558\pi$
$294$	0	0
$295$	0	0
$296$	4.47214	0.259938
$297$	0	0
$298$	−50.0000	−2.89642
$299$	0	0
$300$	0	0
$301$	−20.0000	−1.15278
$302$	−30.0000	−1.72631
$303$	0	0
$304$	−4.47214	−0.256495
$305$	17.8885	1.02430
$306$	0	0
$307$	4.47214	0.255238	0.127619	−	0.991823i	$-0.459266\pi$
$307$	4.47214	0.255238	0.127619	+	0.991823i	$0.459266\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	4.47214	0.252377
$315$	0	0
$316$	−40.2492	−2.26420
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	0	0
$320$	26.0000	1.45344
$321$	0	0
$322$	40.0000	2.22911
$323$	20.0000	1.11283
$324$	0	0
$325$	0	0
$326$	8.94427	0.495377
$327$	0	0
$328$	−10.0000	−0.552158
$329$	−35.7771	−1.97245
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−26.8328	−1.47264
$333$	0	0
$334$	20.0000	1.09435
$335$	24.0000	1.31126
$336$	0	0
$337$	−8.94427	−0.487226	−0.243613	−	0.969873i	$-0.578333\pi$
$337$	−8.94427	−0.487226	−0.243613	+	0.969873i	$0.578333\pi$
$338$	−29.0689	−1.58114
$339$	0	0
$340$	−26.8328	−1.45521
$341$	0	0
$342$	0	0
$343$	26.8328	1.44884
$344$	−10.0000	−0.539164
$345$	0	0
$346$	−30.0000	−1.61281
$347$	8.94427	0.480154	0.240077	−	0.970754i	$-0.422827\pi$
$347$	8.94427	0.480154	0.240077	+	0.970754i	$0.422827\pi$
$348$	0	0
$349$	26.8328	1.43633	0.718164	−	0.695874i	$-0.244983\pi$
$349$	26.8328	1.43633	0.718164	+	0.695874i	$0.244983\pi$
$350$	−10.0000	−0.534522
$351$	0	0
$352$	0	0
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	0	0
$355$	−16.0000	−0.849192
$356$	42.0000	2.22600
$357$	0	0
$358$	8.94427	0.472719
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−22.3607	−1.17525
$363$	0	0
$364$	0	0
$365$	−17.8885	−0.936329
$366$	0	0
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	−8.94427	−0.464991
$371$	−26.8328	−1.39309
$372$	0	0
$373$	26.8328	1.38935	0.694675	−	0.719323i	$-0.255548\pi$
$373$	26.8328	1.38935	0.694675	+	0.719323i	$0.255548\pi$
$374$	0	0
$375$	0	0
$376$	−17.8885	−0.922531
$377$	0	0
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−26.8328	−1.37649
$381$	0	0
$382$	0	0
$383$	36.0000	1.83951	0.919757	−	0.392488i	$-0.128386\pi$
$383$	36.0000	1.83951	0.919757	+	0.392488i	$0.128386\pi$
$384$	0	0
$385$	0	0
$386$	40.0000	2.03595
$387$	0	0
$388$	6.00000	0.304604
$389$	−10.0000	−0.507020	−0.253510	−	0.967333i	$-0.581585\pi$
$389$	−10.0000	−0.507020	−0.253510	+	0.967333i	$0.581585\pi$
$390$	0	0
$391$	17.8885	0.904663
$392$	29.0689	1.46820
$393$	0	0
$394$	50.0000	2.51896
$395$	26.8328	1.35011
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	0	0
$400$	1.00000	0.0500000
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	0	0
$404$	13.4164	0.667491
$405$	0	0
$406$	44.7214	2.21948
$407$	0	0
$408$	0	0
$409$	26.8328	1.32680	0.663399	−	0.748266i	$-0.269113\pi$
$409$	26.8328	1.32680	0.663399	+	0.748266i	$0.269113\pi$
$410$	20.0000	0.987730
$411$	0	0
$412$	48.0000	2.36479
$413$	0	0
$414$	0	0
$415$	17.8885	0.878114
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.00000	−0.195413	−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000	−0.195413	−0.0977064	+	0.995215i	$0.531151\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	10.0000	0.486792
$423$	0	0
$424$	−13.4164	−0.651558
$425$	−4.47214	−0.216930
$426$	0	0
$427$	−40.0000	−1.93574
$428$	26.8328	1.29701
$429$	0	0
$430$	20.0000	0.964486
$431$	−8.94427	−0.430830	−0.215415	−	0.976523i	$-0.569110\pi$
$431$	−8.94427	−0.430830	−0.215415	+	0.976523i	$0.569110\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.8885	0.855725
$438$	0	0
$439$	13.4164	0.640330	0.320165	−	0.947362i	$-0.396262\pi$
$439$	13.4164	0.640330	0.320165	+	0.947362i	$0.396262\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	−28.0000	−1.32733
$446$	−35.7771	−1.69409
$447$	0	0
$448$	−58.1378	−2.74675
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	0	0
$452$	−18.0000	−0.846649
$453$	0	0
$454$	20.0000	0.938647
$455$	0	0
$456$	0	0
$457$	−26.8328	−1.25519	−0.627593	−	0.778542i	$-0.715960\pi$
$457$	−26.8328	−1.25519	−0.627593	+	0.778542i	$0.715960\pi$
$458$	22.3607	1.04485
$459$	0	0
$460$	−24.0000	−1.11901
$461$	13.4164	0.624864	0.312432	−	0.949940i	$-0.398856\pi$
$461$	13.4164	0.624864	0.312432	+	0.949940i	$0.398856\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	−4.47214	−0.207614
$465$	0	0
$466$	−10.0000	−0.463241
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	0	0
$469$	−53.6656	−2.47805
$470$	35.7771	1.65027
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.47214	−0.205196
$476$	60.0000	2.75010
$477$	0	0
$478$	20.0000	0.914779
$479$	−8.94427	−0.408674	−0.204337	−	0.978901i	$-0.565504\pi$
$479$	−8.94427	−0.408674	−0.204337	+	0.978901i	$0.565504\pi$
$480$	0	0
$481$	0	0
$482$	−20.0000	−0.910975
$483$	0	0
$484$	0	0
$485$	−4.00000	−0.181631
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−20.0000	−0.905357
$489$	0	0
$490$	−58.1378	−2.62640
$491$	−26.8328	−1.21095	−0.605474	−	0.795865i	$-0.707016\pi$
$491$	−26.8328	−1.21095	−0.605474	+	0.795865i	$0.707016\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	35.7771	1.60482
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	36.0000	1.60997
$501$	0	0
$502$	26.8328	1.19761
$503$	−26.8328	−1.19642	−0.598208	−	0.801341i	$-0.704120\pi$
$503$	−26.8328	−1.19642	−0.598208	+	0.801341i	$0.704120\pi$
$504$	0	0
$505$	−8.94427	−0.398015
$506$	0	0
$507$	0	0
$508$	−40.2492	−1.78577
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	11.1803	0.494106
$513$	0	0
$514$	49.1935	2.16983
$515$	−32.0000	−1.41009
$516$	0	0
$517$	0	0
$518$	20.0000	0.878750
$519$	0	0
$520$	0	0
$521$	−30.0000	−1.31432	−0.657162	−	0.753749i	$-0.728243\pi$
$521$	−30.0000	−1.31432	−0.657162	+	0.753749i	$0.728243\pi$
$522$	0	0
$523$	−4.47214	−0.195553	−0.0977764	−	0.995208i	$-0.531173\pi$
$523$	−4.47214	−0.195553	−0.0977764	+	0.995208i	$0.531173\pi$
$524$	−53.6656	−2.34439
$525$	0	0
$526$	20.0000	0.872041
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	26.8328	1.16554
$531$	0	0
$532$	60.0000	2.60133
$533$	0	0
$534$	0	0
$535$	−17.8885	−0.773389
$536$	−26.8328	−1.15900
$537$	0	0
$538$	22.3607	0.964037
$539$	0	0
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	30.0000	1.28861
$543$	0	0
$544$	−30.0000	−1.28624
$545$	0	0
$546$	0	0
$547$	−40.2492	−1.72093	−0.860466	−	0.509507i	$-0.829828\pi$
$547$	−40.2492	−1.72093	−0.860466	+	0.509507i	$0.829828\pi$
$548$	−66.0000	−2.81938
$549$	0	0
$550$	0	0
$551$	20.0000	0.852029
$552$	0	0
$553$	−60.0000	−2.55146
$554$	0	0
$555$	0	0
$556$	−40.2492	−1.70695
$557$	−40.2492	−1.70541	−0.852707	−	0.522389i	$-0.825041\pi$
$557$	−40.2492	−1.70541	−0.852707	+	0.522389i	$0.825041\pi$
$558$	0	0
$559$	0	0
$560$	8.94427	0.377964
$561$	0	0
$562$	70.0000	2.95277
$563$	17.8885	0.753912	0.376956	−	0.926231i	$-0.376971\pi$
$563$	17.8885	0.753912	0.376956	+	0.926231i	$0.376971\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	30.0000	1.26099
$567$	0	0
$568$	17.8885	0.750587
$569$	4.47214	0.187482	0.0937408	−	0.995597i	$-0.470117\pi$
$569$	4.47214	0.187482	0.0937408	+	0.995597i	$0.470117\pi$
$570$	0	0
$571$	13.4164	0.561459	0.280730	−	0.959787i	$-0.409424\pi$
$571$	13.4164	0.561459	0.280730	+	0.959787i	$0.409424\pi$
$572$	0	0
$573$	0	0
$574$	−44.7214	−1.86663
$575$	−4.00000	−0.166812
$576$	0	0
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	6.70820	0.279024
$579$	0	0
$580$	−26.8328	−1.11417
$581$	−40.0000	−1.65948
$582$	0	0
$583$	0	0
$584$	20.0000	0.827606
$585$	0	0
$586$	−50.0000	−2.06548
$587$	8.00000	0.330195	0.165098	−	0.986277i	$-0.447206\pi$
$587$	8.00000	0.330195	0.165098	+	0.986277i	$0.447206\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−2.00000	−0.0821995
$593$	−22.3607	−0.918243	−0.459122	−	0.888373i	$-0.651836\pi$
$593$	−22.3607	−0.918243	−0.459122	+	0.888373i	$0.651836\pi$
$594$	0	0
$595$	−40.0000	−1.63984
$596$	−67.0820	−2.74779
$597$	0	0
$598$	0	0
$599$	36.0000	1.47092	0.735460	−	0.677568i	$-0.236966\pi$
$599$	36.0000	1.47092	0.735460	+	0.677568i	$0.236966\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	−44.7214	−1.82271
$603$	0	0
$604$	−40.2492	−1.63772
$605$	0	0
$606$	0	0
$607$	4.47214	0.181518	0.0907592	−	0.995873i	$-0.471071\pi$
$607$	4.47214	0.181518	0.0907592	+	0.995873i	$0.471071\pi$
$608$	−30.0000	−1.21666
$609$	0	0
$610$	40.0000	1.61955
$611$	0	0
$612$	0	0
$613$	−44.7214	−1.80628	−0.903139	−	0.429348i	$-0.858744\pi$
$613$	−44.7214	−1.80628	−0.903139	+	0.429348i	$0.858744\pi$
$614$	10.0000	0.403567
$615$	0	0
$616$	0	0
$617$	2.00000	0.0805170	0.0402585	−	0.999189i	$-0.487182\pi$
$617$	2.00000	0.0805170	0.0402585	+	0.999189i	$0.487182\pi$
$618$	0	0
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	0	0
$621$	0	0
$622$	26.8328	1.07590
$623$	62.6099	2.50841
$624$	0	0
$625$	−19.0000	−0.760000
$626$	31.3050	1.25120
$627$	0	0
$628$	6.00000	0.239426
$629$	8.94427	0.356631
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	−30.0000	−1.19334
$633$	0	0
$634$	−40.2492	−1.59850
$635$	26.8328	1.06483
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	31.3050	1.23744
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	53.6656	2.11472
$645$	0	0
$646$	44.7214	1.75954
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	12.0000	0.469956
$653$	−46.0000	−1.80012	−0.900060	−	0.435767i	$-0.856477\pi$
$653$	−46.0000	−1.80012	−0.900060	+	0.435767i	$0.856477\pi$
$654$	0	0
$655$	35.7771	1.39793
$656$	4.47214	0.174608
$657$	0	0
$658$	−80.0000	−3.11872
$659$	17.8885	0.696839	0.348419	−	0.937339i	$-0.386719\pi$
$659$	17.8885	0.696839	0.348419	+	0.937339i	$0.386719\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	−44.7214	−1.73814
$663$	0	0
$664$	−20.0000	−0.776151
$665$	−40.0000	−1.55113
$666$	0	0
$667$	17.8885	0.692647
$668$	26.8328	1.03819
$669$	0	0
$670$	53.6656	2.07328
$671$	0	0
$672$	0	0
$673$	17.8885	0.689553	0.344776	−	0.938685i	$-0.387955\pi$
$673$	17.8885	0.689553	0.344776	+	0.938685i	$0.387955\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	−39.0000	−1.50000
$677$	−31.3050	−1.20315	−0.601574	−	0.798817i	$-0.705459\pi$
$677$	−31.3050	−1.20315	−0.601574	+	0.798817i	$0.705459\pi$
$678$	0	0
$679$	8.94427	0.343250
$680$	−20.0000	−0.766965
$681$	0	0
$682$	0	0
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	0	0
$685$	44.0000	1.68115
$686$	60.0000	2.29081
$687$	0	0
$688$	4.47214	0.170499
$689$	0	0
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	−40.2492	−1.53005
$693$	0	0
$694$	20.0000	0.759190
$695$	26.8328	1.01783
$696$	0	0
$697$	−20.0000	−0.757554
$698$	60.0000	2.27103
$699$	0	0
$700$	−13.4164	−0.507093
$701$	−22.3607	−0.844551	−0.422276	−	0.906467i	$-0.638769\pi$
$701$	−22.3607	−0.844551	−0.422276	+	0.906467i	$0.638769\pi$
$702$	0	0
$703$	8.94427	0.337340
$704$	0	0
$705$	0	0
$706$	31.3050	1.17818
$707$	20.0000	0.752177
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	−35.7771	−1.34269
$711$	0	0
$712$	31.3050	1.17320
$713$	0	0
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	71.5542	2.66482
$722$	2.23607	0.0832178
$723$	0	0
$724$	−30.0000	−1.11494
$725$	−4.47214	−0.166091
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	0	0
$730$	−40.0000	−1.48047
$731$	−20.0000	−0.739727
$732$	0	0
$733$	17.8885	0.660728	0.330364	−	0.943854i	$-0.392828\pi$
$733$	17.8885	0.660728	0.330364	+	0.943854i	$0.392828\pi$
$734$	17.8885	0.660278
$735$	0	0
$736$	−26.8328	−0.989071
$737$	0	0
$738$	0	0
$739$	−4.47214	−0.164510	−0.0822551	−	0.996611i	$-0.526212\pi$
$739$	−4.47214	−0.164510	−0.0822551	+	0.996611i	$0.526212\pi$
$740$	−12.0000	−0.441129
$741$	0	0
$742$	−60.0000	−2.20267
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	44.7214	1.63846
$746$	60.0000	2.19676
$747$	0	0
$748$	0	0
$749$	40.0000	1.46157
$750$	0	0
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	8.00000	0.291730
$753$	0	0
$754$	0	0
$755$	26.8328	0.976546
$756$	0	0
$757$	42.0000	1.52652	0.763258	−	0.646094i	$-0.223599\pi$
$757$	42.0000	1.52652	0.763258	+	0.646094i	$0.223599\pi$
$758$	−44.7214	−1.62435
$759$	0	0
$760$	−20.0000	−0.725476
$761$	−31.3050	−1.13480	−0.567402	−	0.823441i	$-0.692051\pi$
$761$	−31.3050	−1.13480	−0.567402	+	0.823441i	$0.692051\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	80.4984	2.90853
$767$	0	0
$768$	0	0
$769$	−35.7771	−1.29015	−0.645077	−	0.764117i	$-0.723175\pi$
$769$	−35.7771	−1.29015	−0.645077	+	0.764117i	$0.723175\pi$
$770$	0	0
$771$	0	0
$772$	53.6656	1.93147
$773$	−14.0000	−0.503545	−0.251773	−	0.967786i	$-0.581013\pi$
$773$	−14.0000	−0.503545	−0.251773	+	0.967786i	$0.581013\pi$
$774$	0	0
$775$	0	0
$776$	4.47214	0.160540
$777$	0	0
$778$	−22.3607	−0.801669
$779$	−20.0000	−0.716574
$780$	0	0
$781$	0	0
$782$	40.0000	1.43040
$783$	0	0
$784$	−13.0000	−0.464286
$785$	−4.00000	−0.142766
$786$	0	0
$787$	−40.2492	−1.43473	−0.717365	−	0.696698i	$-0.754652\pi$
$787$	−40.2492	−1.43473	−0.717365	+	0.696698i	$0.754652\pi$
$788$	67.0820	2.38970
$789$	0	0
$790$	60.0000	2.13470
$791$	−26.8328	−0.954065
$792$	0	0
$793$	0	0
$794$	49.1935	1.74581
$795$	0	0
$796$	0	0
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	−35.7771	−1.26570
$800$	6.70820	0.237171
$801$	0	0
$802$	−67.0820	−2.36875
$803$	0	0
$804$	0	0
$805$	−35.7771	−1.26098
$806$	0	0
$807$	0	0
$808$	10.0000	0.351799
$809$	−13.4164	−0.471696	−0.235848	−	0.971790i	$-0.575787\pi$
$809$	−13.4164	−0.471696	−0.235848	+	0.971790i	$0.575787\pi$
$810$	0	0
$811$	4.47214	0.157038	0.0785190	−	0.996913i	$-0.474981\pi$
$811$	4.47214	0.157038	0.0785190	+	0.996913i	$0.474981\pi$
$812$	60.0000	2.10559
$813$	0	0
$814$	0	0
$815$	−8.00000	−0.280228
$816$	0	0
$817$	−20.0000	−0.699711
$818$	60.0000	2.09785
$819$	0	0
$820$	26.8328	0.937043
$821$	22.3607	0.780393	0.390197	−	0.920732i	$-0.372407\pi$
$821$	22.3607	0.780393	0.390197	+	0.920732i	$0.372407\pi$
$822$	0	0
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	35.7771	1.24635
$825$	0	0
$826$	0	0
$827$	44.7214	1.55511	0.777557	−	0.628812i	$-0.216459\pi$
$827$	44.7214	1.55511	0.777557	+	0.628812i	$0.216459\pi$
$828$	0	0
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	40.0000	1.38842
$831$	0	0
$832$	0	0
$833$	58.1378	2.01435
$834$	0	0
$835$	−17.8885	−0.619059
$836$	0	0
$837$	0	0
$838$	−8.94427	−0.308975
$839$	20.0000	0.690477	0.345238	−	0.938515i	$-0.387798\pi$
$839$	20.0000	0.690477	0.345238	+	0.938515i	$0.387798\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	22.3607	0.770600
$843$	0	0
$844$	13.4164	0.461812
$845$	26.0000	0.894427
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	−10.0000	−0.342997
$851$	8.00000	0.274236
$852$	0	0
$853$	−8.94427	−0.306246	−0.153123	−	0.988207i	$-0.548933\pi$
$853$	−8.94427	−0.306246	−0.153123	+	0.988207i	$0.548933\pi$
$854$	−89.4427	−3.06067
$855$	0	0
$856$	20.0000	0.683586
$857$	40.2492	1.37489	0.687444	−	0.726238i	$-0.258733\pi$
$857$	40.2492	1.37489	0.687444	+	0.726238i	$0.258733\pi$
$858$	0	0
$859$	4.00000	0.136478	0.0682391	−	0.997669i	$-0.478262\pi$
$859$	4.00000	0.136478	0.0682391	+	0.997669i	$0.478262\pi$
$860$	26.8328	0.914991
$861$	0	0
$862$	−20.0000	−0.681203
$863$	−4.00000	−0.136162	−0.0680808	−	0.997680i	$-0.521688\pi$
$863$	−4.00000	−0.136162	−0.0680808	+	0.997680i	$0.521688\pi$
$864$	0	0
$865$	26.8328	0.912343
$866$	−13.4164	−0.455908
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	40.0000	1.35302
$875$	53.6656	1.81423
$876$	0	0
$877$	8.94427	0.302027	0.151013	−	0.988532i	$-0.451746\pi$
$877$	8.94427	0.302027	0.151013	+	0.988532i	$0.451746\pi$
$878$	30.0000	1.01245
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	0	0
$886$	53.6656	1.80293
$887$	−53.6656	−1.80192	−0.900958	−	0.433907i	$-0.857135\pi$
$887$	−53.6656	−1.80192	−0.900958	+	0.433907i	$0.857135\pi$
$888$	0	0
$889$	−60.0000	−2.01234
$890$	−62.6099	−2.09869
$891$	0	0
$892$	−48.0000	−1.60716
$893$	−35.7771	−1.19723
$894$	0	0
$895$	−8.00000	−0.267411
$896$	−70.0000	−2.33854
$897$	0	0
$898$	13.4164	0.447711
$899$	0	0
$900$	0	0
$901$	−26.8328	−0.893931
$902$	0	0
$903$	0	0
$904$	−13.4164	−0.446223
$905$	20.0000	0.664822
$906$	0	0
$907$	−12.0000	−0.398453	−0.199227	−	0.979953i	$-0.563843\pi$
$907$	−12.0000	−0.398453	−0.199227	+	0.979953i	$0.563843\pi$
$908$	26.8328	0.890478
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	−60.0000	−1.98462
$915$	0	0
$916$	30.0000	0.991228
$917$	−80.0000	−2.64183
$918$	0	0
$919$	−22.3607	−0.737611	−0.368805	−	0.929507i	$-0.620233\pi$
$919$	−22.3607	−0.737611	−0.368805	+	0.929507i	$0.620233\pi$
$920$	−17.8885	−0.589768
$921$	0	0
$922$	30.0000	0.987997
$923$	0	0
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	53.6656	1.76356
$927$	0	0
$928$	−30.0000	−0.984798
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	58.1378	1.90539
$932$	−13.4164	−0.439469
$933$	0	0
$934$	17.8885	0.585331
$935$	0	0
$936$	0	0
$937$	−53.6656	−1.75318	−0.876590	−	0.481238i	$-0.840187\pi$
$937$	−53.6656	−1.75318	−0.876590	+	0.481238i	$0.840187\pi$
$938$	−120.000	−3.91814
$939$	0	0
$940$	48.0000	1.56559
$941$	22.3607	0.728937	0.364469	−	0.931216i	$-0.381251\pi$
$941$	22.3607	0.728937	0.364469	+	0.931216i	$0.381251\pi$
$942$	0	0
$943$	−17.8885	−0.582531
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	0	0
$949$	0	0
$950$	−10.0000	−0.324443
$951$	0	0
$952$	44.7214	1.44943
$953$	22.3607	0.724333	0.362167	−	0.932113i	$-0.382037\pi$
$953$	22.3607	0.724333	0.362167	+	0.932113i	$0.382037\pi$
$954$	0	0
$955$	0	0
$956$	26.8328	0.867835
$957$	0	0
$958$	−20.0000	−0.646171
$959$	−98.3870	−3.17708
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	−26.8328	−0.864227
$965$	−35.7771	−1.15171
$966$	0	0
$967$	−13.4164	−0.431443	−0.215721	−	0.976455i	$-0.569210\pi$
$967$	−13.4164	−0.431443	−0.215721	+	0.976455i	$0.569210\pi$
$968$	0	0
$969$	0	0
$970$	−8.94427	−0.287183
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−60.0000	−1.92351
$974$	17.8885	0.573186
$975$	0	0
$976$	8.94427	0.286299
$977$	42.0000	1.34370	0.671850	−	0.740688i	$-0.265500\pi$
$977$	42.0000	1.34370	0.671850	+	0.740688i	$0.265500\pi$
$978$	0	0
$979$	0	0
$980$	−78.0000	−2.49162
$981$	0	0
$982$	−60.0000	−1.91468
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	−44.7214	−1.42494
$986$	44.7214	1.42422
$987$	0	0
$988$	0	0
$989$	−17.8885	−0.568823
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	0	0
$993$	0	0
$994$	80.0000	2.53745
$995$	0	0
$996$	0	0
$997$	26.8328	0.849804	0.424902	−	0.905239i	$-0.360309\pi$
$997$	26.8328	0.849804	0.424902	+	0.905239i	$0.360309\pi$
$998$	−44.7214	−1.41563
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.a.p.1.2		2
3.2	odd	2		363.2.a.g.1.1	✓	2
11.10	odd	2	inner	1089.2.a.p.1.1		2
12.11	even	2		5808.2.a.bx.1.1		2
15.14	odd	2		9075.2.a.bi.1.2		2
33.2	even	10		363.2.e.a.202.1		4
33.5	odd	10		363.2.e.l.124.1		4
33.8	even	10		363.2.e.l.130.1		4
33.14	odd	10		363.2.e.a.130.1		4
33.17	even	10		363.2.e.a.124.1		4
33.20	odd	10		363.2.e.l.202.1		4
33.26	odd	10		363.2.e.a.148.1		4
33.29	even	10		363.2.e.l.148.1		4
33.32	even	2		363.2.a.g.1.2	yes	2
132.131	odd	2		5808.2.a.bx.1.2		2
165.164	even	2		9075.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.g.1.1	✓	2	3.2	odd	2
363.2.a.g.1.2	yes	2	33.32	even	2
363.2.e.a.124.1		4	33.17	even	10
363.2.e.a.130.1		4	33.14	odd	10
363.2.e.a.148.1		4	33.26	odd	10
363.2.e.a.202.1		4	33.2	even	10
363.2.e.l.124.1		4	33.5	odd	10
363.2.e.l.130.1		4	33.8	even	10
363.2.e.l.148.1		4	33.29	even	10
363.2.e.l.202.1		4	33.20	odd	10
1089.2.a.p.1.1		2	11.10	odd	2	inner
1089.2.a.p.1.2		2	1.1	even	1	trivial
5808.2.a.bx.1.1		2	12.11	even	2
5808.2.a.bx.1.2		2	132.131	odd	2
9075.2.a.bi.1.1		2	165.164	even	2
9075.2.a.bi.1.2		2	15.14	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} +3.00000 q^{4} -2.00000 q^{5} +4.47214 q^{7} +2.23607 q^{8} -4.47214 q^{10} +10.0000 q^{14} -1.00000 q^{16} +4.47214 q^{17} +4.47214 q^{19} -6.00000 q^{20} +4.00000 q^{23} -1.00000 q^{25} +13.4164 q^{28} +4.47214 q^{29} -6.70820 q^{32} +10.0000 q^{34} -8.94427 q^{35} +2.00000 q^{37} +10.0000 q^{38} -4.47214 q^{40} -4.47214 q^{41} -4.47214 q^{43} +8.94427 q^{46} -8.00000 q^{47} +13.0000 q^{49} -2.23607 q^{50} -6.00000 q^{53} +10.0000 q^{56} +10.0000 q^{58} -8.94427 q^{61} -13.0000 q^{64} -12.0000 q^{67} +13.4164 q^{68} -20.0000 q^{70} +8.00000 q^{71} +8.94427 q^{73} +4.47214 q^{74} +13.4164 q^{76} -13.4164 q^{79} +2.00000 q^{80} -10.0000 q^{82} -8.94427 q^{83} -8.94427 q^{85} -10.0000 q^{86} +14.0000 q^{89} +12.0000 q^{92} -17.8885 q^{94} -8.94427 q^{95} +2.00000 q^{97} +29.0689 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} - 4 q^{5} + 20 q^{14} - 2 q^{16} - 12 q^{20} + 8 q^{23} - 2 q^{25} + 20 q^{34} + 4 q^{37} + 20 q^{38} - 16 q^{47} + 26 q^{49} - 12 q^{53} + 20 q^{56} + 20 q^{58} - 26 q^{64} - 24 q^{67} - 40 q^{70}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q + 6 * q^4 - 4 * q^5 + 20 * q^14 - 2 * q^16 - 12 * q^20 + 8 * q^23 - 2 * q^25 + 20 * q^34 + 4 * q^37 + 20 * q^38 - 16 * q^47 + 26 * q^49 - 12 * q^53 + 20 * q^56 + 20 * q^58 - 26 * q^64 - 24 * q^67 - 40 * q^70 + 16 * q^71 + 4 * q^80 - 20 * q^82 - 20 * q^86 + 28 * q^89 + 24 * q^92 + 4 * q^97

Introduction

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