LMFDB - Embedded newform 1134.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1134,2,Mod(1,1134)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1134, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1134.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1134 = 2 \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1134.a (trivial)

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:	yes
Analytic conductor:	$9.05503558921$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	1134.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.44949 q^{5} -1.00000 q^{7} -1.00000 q^{8} +3.44949 q^{10} -2.00000 q^{11} -4.89898 q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +7.44949 q^{19} -3.44949 q^{20} +2.00000 q^{22} +1.00000 q^{23} +6.89898 q^{25} +4.89898 q^{26} -1.00000 q^{28} -2.89898 q^{29} +6.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} +3.44949 q^{35} -7.79796 q^{37} -7.44949 q^{38} +3.44949 q^{40} +9.79796 q^{41} -2.89898 q^{43} -2.00000 q^{44} -1.00000 q^{46} +9.79796 q^{47} +1.00000 q^{49} -6.89898 q^{50} -4.89898 q^{52} +1.10102 q^{53} +6.89898 q^{55} +1.00000 q^{56} +2.89898 q^{58} +2.00000 q^{59} +11.4495 q^{61} -6.00000 q^{62} +1.00000 q^{64} +16.8990 q^{65} -3.10102 q^{67} -2.00000 q^{68} -3.44949 q^{70} -9.89898 q^{71} +2.89898 q^{73} +7.79796 q^{74} +7.44949 q^{76} +2.00000 q^{77} +7.89898 q^{79} -3.44949 q^{80} -9.79796 q^{82} -2.00000 q^{83} +6.89898 q^{85} +2.89898 q^{86} +2.00000 q^{88} +7.10102 q^{89} +4.89898 q^{91} +1.00000 q^{92} -9.79796 q^{94} -25.6969 q^{95} -6.89898 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{14} + 2 q^{16} - 4 q^{17} + 10 q^{19} - 2 q^{20} + 4 q^{22} + 2 q^{23} + 4 q^{25} - 2 q^{28} + 4 q^{29} + 12 q^{31} - 2 q^{32}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 2 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^14 + 2 * q^16 - 4 * q^17 + 10 * q^19 - 2 * q^20 + 4 * q^22 + 2 * q^23 + 4 * q^25 - 2 * q^28 + 4 * q^29 + 12 * q^31 - 2 * q^32 + 4 * q^34 + 2 * q^35 + 4 * q^37 - 10 * q^38 + 2 * q^40 + 4 * q^43 - 4 * q^44 - 2 * q^46 + 2 * q^49 - 4 * q^50 + 12 * q^53 + 4 * q^55 + 2 * q^56 - 4 * q^58 + 4 * q^59 + 18 * q^61 - 12 * q^62 + 2 * q^64 + 24 * q^65 - 16 * q^67 - 4 * q^68 - 2 * q^70 - 10 * q^71 - 4 * q^73 - 4 * q^74 + 10 * q^76 + 4 * q^77 + 6 * q^79 - 2 * q^80 - 4 * q^83 + 4 * q^85 - 4 * q^86 + 4 * q^88 + 24 * q^89 + 2 * q^92 - 22 * q^95 - 4 * q^97 - 2 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.44949	−1.54266	−0.771329	−	0.636436i	$-0.780408\pi$
$5$	−3.44949	−1.54266	−0.771329	+	0.636436i	$0.780408\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.44949	1.09082
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−4.89898	−1.35873	−0.679366	−	0.733799i	$-0.737745\pi$
$13$	−4.89898	−1.35873	−0.679366	+	0.733799i	$0.737745\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	7.44949	1.70903	0.854515	−	0.519427i	$-0.173854\pi$
$19$	7.44949	1.70903	0.854515	+	0.519427i	$0.173854\pi$
$20$	−3.44949	−0.771329
$21$	0	0
$22$	2.00000	0.426401
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	6.89898	1.37980
$26$	4.89898	0.960769
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−2.89898	−0.538327	−0.269163	−	0.963095i	$-0.586747\pi$
$29$	−2.89898	−0.538327	−0.269163	+	0.963095i	$0.586747\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	3.44949	0.583070
$36$	0	0
$37$	−7.79796	−1.28198	−0.640988	−	0.767551i	$-0.721475\pi$
$37$	−7.79796	−1.28198	−0.640988	+	0.767551i	$0.721475\pi$
$38$	−7.44949	−1.20847
$39$	0	0
$40$	3.44949	0.545412
$41$	9.79796	1.53018	0.765092	−	0.643921i	$-0.222693\pi$
$41$	9.79796	1.53018	0.765092	+	0.643921i	$0.222693\pi$
$42$	0	0
$43$	−2.89898	−0.442090	−0.221045	−	0.975264i	$-0.570947\pi$
$43$	−2.89898	−0.442090	−0.221045	+	0.975264i	$0.570947\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−1.00000	−0.147442
$47$	9.79796	1.42918	0.714590	−	0.699544i	$-0.246613\pi$
$47$	9.79796	1.42918	0.714590	+	0.699544i	$0.246613\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−6.89898	−0.975663
$51$	0	0
$52$	−4.89898	−0.679366
$53$	1.10102	0.151237	0.0756184	−	0.997137i	$-0.475907\pi$
$53$	1.10102	0.151237	0.0756184	+	0.997137i	$0.475907\pi$
$54$	0	0
$55$	6.89898	0.930258
$56$	1.00000	0.133631
$57$	0	0
$58$	2.89898	0.380655
$59$	2.00000	0.260378	0.130189	−	0.991489i	$-0.458442\pi$
$59$	2.00000	0.260378	0.130189	+	0.991489i	$0.458442\pi$
$60$	0	0
$61$	11.4495	1.46596	0.732978	−	0.680252i	$-0.238130\pi$
$61$	11.4495	1.46596	0.732978	+	0.680252i	$0.238130\pi$
$62$	−6.00000	−0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	16.8990	2.09606
$66$	0	0
$67$	−3.10102	−0.378850	−0.189425	−	0.981895i	$-0.560662\pi$
$67$	−3.10102	−0.378850	−0.189425	+	0.981895i	$0.560662\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	−3.44949	−0.412293
$71$	−9.89898	−1.17479	−0.587396	−	0.809299i	$-0.699847\pi$
$71$	−9.89898	−1.17479	−0.587396	+	0.809299i	$0.699847\pi$
$72$	0	0
$73$	2.89898	0.339300	0.169650	−	0.985504i	$-0.445736\pi$
$73$	2.89898	0.339300	0.169650	+	0.985504i	$0.445736\pi$
$74$	7.79796	0.906494
$75$	0	0
$76$	7.44949	0.854515
$77$	2.00000	0.227921
$78$	0	0
$79$	7.89898	0.888705	0.444352	−	0.895852i	$-0.353434\pi$
$79$	7.89898	0.888705	0.444352	+	0.895852i	$0.353434\pi$
$80$	−3.44949	−0.385665
$81$	0	0
$82$	−9.79796	−1.08200
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	0	0
$85$	6.89898	0.748299
$86$	2.89898	0.312605
$87$	0	0
$88$	2.00000	0.213201
$89$	7.10102	0.752707	0.376353	−	0.926476i	$-0.377178\pi$
$89$	7.10102	0.752707	0.376353	+	0.926476i	$0.377178\pi$
$90$	0	0
$91$	4.89898	0.513553
$92$	1.00000	0.104257
$93$	0	0
$94$	−9.79796	−1.01058
$95$	−25.6969	−2.63645
$96$	0	0
$97$	−6.89898	−0.700485	−0.350243	−	0.936659i	$-0.613901\pi$
$97$	−6.89898	−0.700485	−0.350243	+	0.936659i	$0.613901\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	6.89898	0.689898
$101$	−7.24745	−0.721148	−0.360574	−	0.932731i	$-0.617419\pi$
$101$	−7.24745	−0.721148	−0.360574	+	0.932731i	$0.617419\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	4.89898	0.480384
$105$	0	0
$106$	−1.10102	−0.106941
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−16.6969	−1.59928	−0.799638	−	0.600482i	$-0.794975\pi$
$109$	−16.6969	−1.59928	−0.799638	+	0.600482i	$0.794975\pi$
$110$	−6.89898	−0.657792
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	15.8990	1.49565	0.747825	−	0.663896i	$-0.231098\pi$
$113$	15.8990	1.49565	0.747825	+	0.663896i	$0.231098\pi$
$114$	0	0
$115$	−3.44949	−0.321667
$116$	−2.89898	−0.269163
$117$	0	0
$118$	−2.00000	−0.184115
$119$	2.00000	0.183340
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−11.4495	−1.03659
$123$	0	0
$124$	6.00000	0.538816
$125$	−6.55051	−0.585895
$126$	0	0
$127$	−3.00000	−0.266207	−0.133103	−	0.991102i	$-0.542494\pi$
$127$	−3.00000	−0.266207	−0.133103	+	0.991102i	$0.542494\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−16.8990	−1.48214
$131$	13.4495	1.17509	0.587544	−	0.809192i	$-0.300095\pi$
$131$	13.4495	1.17509	0.587544	+	0.809192i	$0.300095\pi$
$132$	0	0
$133$	−7.44949	−0.645953
$134$	3.10102	0.267887
$135$	0	0
$136$	2.00000	0.171499
$137$	11.7980	1.00797	0.503984	−	0.863713i	$-0.331867\pi$
$137$	11.7980	1.00797	0.503984	+	0.863713i	$0.331867\pi$
$138$	0	0
$139$	9.44949	0.801495	0.400748	−	0.916188i	$-0.368750\pi$
$139$	9.44949	0.801495	0.400748	+	0.916188i	$0.368750\pi$
$140$	3.44949	0.291535
$141$	0	0
$142$	9.89898	0.830704
$143$	9.79796	0.819346
$144$	0	0
$145$	10.0000	0.830455
$146$	−2.89898	−0.239921
$147$	0	0
$148$	−7.79796	−0.640988
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	−7.44949	−0.604233
$153$	0	0
$154$	−2.00000	−0.161165
$155$	−20.6969	−1.66242
$156$	0	0
$157$	6.34847	0.506663	0.253332	−	0.967380i	$-0.418474\pi$
$157$	6.34847	0.506663	0.253332	+	0.967380i	$0.418474\pi$
$158$	−7.89898	−0.628409
$159$	0	0
$160$	3.44949	0.272706
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−0.202041	−0.0158251	−0.00791254	−	0.999969i	$-0.502519\pi$
$163$	−0.202041	−0.0158251	−0.00791254	+	0.999969i	$0.502519\pi$
$164$	9.79796	0.765092
$165$	0	0
$166$	2.00000	0.155230
$167$	−18.6969	−1.44681	−0.723406	−	0.690423i	$-0.757425\pi$
$167$	−18.6969	−1.44681	−0.723406	+	0.690423i	$0.757425\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	−6.89898	−0.529128
$171$	0	0
$172$	−2.89898	−0.221045
$173$	12.8990	0.980691	0.490346	−	0.871528i	$-0.336871\pi$
$173$	12.8990	0.980691	0.490346	+	0.871528i	$0.336871\pi$
$174$	0	0
$175$	−6.89898	−0.521514
$176$	−2.00000	−0.150756
$177$	0	0
$178$	−7.10102	−0.532244
$179$	8.69694	0.650040	0.325020	−	0.945707i	$-0.394629\pi$
$179$	8.69694	0.650040	0.325020	+	0.945707i	$0.394629\pi$
$180$	0	0
$181$	4.34847	0.323219	0.161610	−	0.986855i	$-0.448331\pi$
$181$	4.34847	0.323219	0.161610	+	0.986855i	$0.448331\pi$
$182$	−4.89898	−0.363137
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	26.8990	1.97765
$186$	0	0
$187$	4.00000	0.292509
$188$	9.79796	0.714590
$189$	0	0
$190$	25.6969	1.86425
$191$	−13.8990	−1.00569	−0.502847	−	0.864375i	$-0.667714\pi$
$191$	−13.8990	−1.00569	−0.502847	+	0.864375i	$0.667714\pi$
$192$	0	0
$193$	−8.10102	−0.583124	−0.291562	−	0.956552i	$-0.594175\pi$
$193$	−8.10102	−0.583124	−0.291562	+	0.956552i	$0.594175\pi$
$194$	6.89898	0.495318
$195$	0	0
$196$	1.00000	0.0714286
$197$	12.6969	0.904619	0.452310	−	0.891861i	$-0.350600\pi$
$197$	12.6969	0.904619	0.452310	+	0.891861i	$0.350600\pi$
$198$	0	0
$199$	6.89898	0.489056	0.244528	−	0.969642i	$-0.421367\pi$
$199$	6.89898	0.489056	0.244528	+	0.969642i	$0.421367\pi$
$200$	−6.89898	−0.487832
$201$	0	0
$202$	7.24745	0.509929
$203$	2.89898	0.203468
$204$	0	0
$205$	−33.7980	−2.36055
$206$	−14.0000	−0.975426
$207$	0	0
$208$	−4.89898	−0.339683
$209$	−14.8990	−1.03058
$210$	0	0
$211$	3.10102	0.213483	0.106742	−	0.994287i	$-0.465958\pi$
$211$	3.10102	0.213483	0.106742	+	0.994287i	$0.465958\pi$
$212$	1.10102	0.0756184
$213$	0	0
$214$	−12.0000	−0.820303
$215$	10.0000	0.681994
$216$	0	0
$217$	−6.00000	−0.407307
$218$	16.6969	1.13086
$219$	0	0
$220$	6.89898	0.465129
$221$	9.79796	0.659082
$222$	0	0
$223$	−20.8990	−1.39950	−0.699750	−	0.714388i	$-0.746705\pi$
$223$	−20.8990	−1.39950	−0.699750	+	0.714388i	$0.746705\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−15.8990	−1.05758
$227$	0.550510	0.0365386	0.0182693	−	0.999833i	$-0.494184\pi$
$227$	0.550510	0.0365386	0.0182693	+	0.999833i	$0.494184\pi$
$228$	0	0
$229$	−23.2474	−1.53623	−0.768117	−	0.640309i	$-0.778806\pi$
$229$	−23.2474	−1.53623	−0.768117	+	0.640309i	$0.778806\pi$
$230$	3.44949	0.227453
$231$	0	0
$232$	2.89898	0.190327
$233$	7.00000	0.458585	0.229293	−	0.973358i	$-0.426359\pi$
$233$	7.00000	0.458585	0.229293	+	0.973358i	$0.426359\pi$
$234$	0	0
$235$	−33.7980	−2.20474
$236$	2.00000	0.130189
$237$	0	0
$238$	−2.00000	−0.129641
$239$	12.7980	0.827831	0.413916	−	0.910315i	$-0.364161\pi$
$239$	12.7980	0.827831	0.413916	+	0.910315i	$0.364161\pi$
$240$	0	0
$241$	−8.89898	−0.573234	−0.286617	−	0.958045i	$-0.592531\pi$
$241$	−8.89898	−0.573234	−0.286617	+	0.958045i	$0.592531\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	11.4495	0.732978
$245$	−3.44949	−0.220380
$246$	0	0
$247$	−36.4949	−2.32211
$248$	−6.00000	−0.381000
$249$	0	0
$250$	6.55051	0.414291
$251$	−12.5505	−0.792181	−0.396091	−	0.918211i	$-0.629633\pi$
$251$	−12.5505	−0.792181	−0.396091	+	0.918211i	$0.629633\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	3.00000	0.188237
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.7980	−1.73399	−0.866995	−	0.498318i	$-0.833951\pi$
$257$	−27.7980	−1.73399	−0.866995	+	0.498318i	$0.833951\pi$
$258$	0	0
$259$	7.79796	0.484542
$260$	16.8990	1.04803
$261$	0	0
$262$	−13.4495	−0.830912
$263$	−16.1010	−0.992831	−0.496416	−	0.868085i	$-0.665351\pi$
$263$	−16.1010	−0.992831	−0.496416	+	0.868085i	$0.665351\pi$
$264$	0	0
$265$	−3.79796	−0.233307
$266$	7.44949	0.456758
$267$	0	0
$268$	−3.10102	−0.189425
$269$	3.65153	0.222638	0.111319	−	0.993785i	$-0.464493\pi$
$269$	3.65153	0.222638	0.111319	+	0.993785i	$0.464493\pi$
$270$	0	0
$271$	16.8990	1.02654	0.513270	−	0.858227i	$-0.328434\pi$
$271$	16.8990	1.02654	0.513270	+	0.858227i	$0.328434\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	−11.7980	−0.712741
$275$	−13.7980	−0.832048
$276$	0	0
$277$	10.6969	0.642717	0.321358	−	0.946958i	$-0.395861\pi$
$277$	10.6969	0.642717	0.321358	+	0.946958i	$0.395861\pi$
$278$	−9.44949	−0.566743
$279$	0	0
$280$	−3.44949	−0.206146
$281$	19.0000	1.13344	0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000	1.13344	0.566722	+	0.823909i	$0.308211\pi$
$282$	0	0
$283$	−20.5505	−1.22160	−0.610801	−	0.791785i	$-0.709152\pi$
$283$	−20.5505	−1.22160	−0.610801	+	0.791785i	$0.709152\pi$
$284$	−9.89898	−0.587396
$285$	0	0
$286$	−9.79796	−0.579365
$287$	−9.79796	−0.578355
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−10.0000	−0.587220
$291$	0	0
$292$	2.89898	0.169650
$293$	−27.2474	−1.59181	−0.795906	−	0.605420i	$-0.793005\pi$
$293$	−27.2474	−1.59181	−0.795906	+	0.605420i	$0.793005\pi$
$294$	0	0
$295$	−6.89898	−0.401674
$296$	7.79796	0.453247
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−4.89898	−0.283315
$300$	0	0
$301$	2.89898	0.167094
$302$	5.00000	0.287718
$303$	0	0
$304$	7.44949	0.427258
$305$	−39.4949	−2.26147
$306$	0	0
$307$	0.752551	0.0429504	0.0214752	−	0.999769i	$-0.493164\pi$
$307$	0.752551	0.0429504	0.0214752	+	0.999769i	$0.493164\pi$
$308$	2.00000	0.113961
$309$	0	0
$310$	20.6969	1.17551
$311$	−1.30306	−0.0738898	−0.0369449	−	0.999317i	$-0.511763\pi$
$311$	−1.30306	−0.0738898	−0.0369449	+	0.999317i	$0.511763\pi$
$312$	0	0
$313$	24.6969	1.39595	0.697977	−	0.716120i	$-0.254084\pi$
$313$	24.6969	1.39595	0.697977	+	0.716120i	$0.254084\pi$
$314$	−6.34847	−0.358265
$315$	0	0
$316$	7.89898	0.444352
$317$	8.69694	0.488469	0.244234	−	0.969716i	$-0.421463\pi$
$317$	8.69694	0.488469	0.244234	+	0.969716i	$0.421463\pi$
$318$	0	0
$319$	5.79796	0.324623
$320$	−3.44949	−0.192832
$321$	0	0
$322$	1.00000	0.0557278
$323$	−14.8990	−0.829001
$324$	0	0
$325$	−33.7980	−1.87477
$326$	0.202041	0.0111900
$327$	0	0
$328$	−9.79796	−0.541002
$329$	−9.79796	−0.540179
$330$	0	0
$331$	−24.6969	−1.35747	−0.678733	−	0.734385i	$-0.737471\pi$
$331$	−24.6969	−1.35747	−0.678733	+	0.734385i	$0.737471\pi$
$332$	−2.00000	−0.109764
$333$	0	0
$334$	18.6969	1.02305
$335$	10.6969	0.584436
$336$	0	0
$337$	35.3939	1.92803	0.964014	−	0.265853i	$-0.0856535\pi$
$337$	35.3939	1.92803	0.964014	+	0.265853i	$0.0856535\pi$
$338$	−11.0000	−0.598321
$339$	0	0
$340$	6.89898	0.374150
$341$	−12.0000	−0.649836
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	2.89898	0.156302
$345$	0	0
$346$	−12.8990	−0.693453
$347$	19.5959	1.05196	0.525982	−	0.850496i	$-0.323698\pi$
$347$	19.5959	1.05196	0.525982	+	0.850496i	$0.323698\pi$
$348$	0	0
$349$	20.8990	1.11870	0.559348	−	0.828933i	$-0.311051\pi$
$349$	20.8990	1.11870	0.559348	+	0.828933i	$0.311051\pi$
$350$	6.89898	0.368766
$351$	0	0
$352$	2.00000	0.106600
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	34.1464	1.81230
$356$	7.10102	0.376353
$357$	0	0
$358$	−8.69694	−0.459647
$359$	−10.7980	−0.569894	−0.284947	−	0.958543i	$-0.591976\pi$
$359$	−10.7980	−0.569894	−0.284947	+	0.958543i	$0.591976\pi$
$360$	0	0
$361$	36.4949	1.92078
$362$	−4.34847	−0.228550
$363$	0	0
$364$	4.89898	0.256776
$365$	−10.0000	−0.523424
$366$	0	0
$367$	5.79796	0.302651	0.151325	−	0.988484i	$-0.451646\pi$
$367$	5.79796	0.302651	0.151325	+	0.988484i	$0.451646\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−26.8990	−1.39841
$371$	−1.10102	−0.0571621
$372$	0	0
$373$	2.89898	0.150103	0.0750517	−	0.997180i	$-0.476088\pi$
$373$	2.89898	0.150103	0.0750517	+	0.997180i	$0.476088\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	−9.79796	−0.505291
$377$	14.2020	0.731442
$378$	0	0
$379$	−26.4949	−1.36095	−0.680476	−	0.732771i	$-0.738227\pi$
$379$	−26.4949	−1.36095	−0.680476	+	0.732771i	$0.738227\pi$
$380$	−25.6969	−1.31823
$381$	0	0
$382$	13.8990	0.711134
$383$	−6.89898	−0.352521	−0.176261	−	0.984344i	$-0.556400\pi$
$383$	−6.89898	−0.352521	−0.176261	+	0.984344i	$0.556400\pi$
$384$	0	0
$385$	−6.89898	−0.351605
$386$	8.10102	0.412331
$387$	0	0
$388$	−6.89898	−0.350243
$389$	15.1010	0.765652	0.382826	−	0.923820i	$-0.374951\pi$
$389$	15.1010	0.765652	0.382826	+	0.923820i	$0.374951\pi$
$390$	0	0
$391$	−2.00000	−0.101144
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−12.6969	−0.639663
$395$	−27.2474	−1.37097
$396$	0	0
$397$	9.30306	0.466907	0.233454	−	0.972368i	$-0.424997\pi$
$397$	9.30306	0.466907	0.233454	+	0.972368i	$0.424997\pi$
$398$	−6.89898	−0.345815
$399$	0	0
$400$	6.89898	0.344949
$401$	10.1010	0.504421	0.252210	−	0.967672i	$-0.418842\pi$
$401$	10.1010	0.504421	0.252210	+	0.967672i	$0.418842\pi$
$402$	0	0
$403$	−29.3939	−1.46421
$404$	−7.24745	−0.360574
$405$	0	0
$406$	−2.89898	−0.143874
$407$	15.5959	0.773061
$408$	0	0
$409$	5.79796	0.286691	0.143345	−	0.989673i	$-0.454214\pi$
$409$	5.79796	0.286691	0.143345	+	0.989673i	$0.454214\pi$
$410$	33.7980	1.66916
$411$	0	0
$412$	14.0000	0.689730
$413$	−2.00000	−0.0984136
$414$	0	0
$415$	6.89898	0.338658
$416$	4.89898	0.240192
$417$	0	0
$418$	14.8990	0.728733
$419$	−24.5505	−1.19937	−0.599685	−	0.800236i	$-0.704708\pi$
$419$	−24.5505	−1.19937	−0.599685	+	0.800236i	$0.704708\pi$
$420$	0	0
$421$	13.1010	0.638505	0.319252	−	0.947670i	$-0.396568\pi$
$421$	13.1010	0.638505	0.319252	+	0.947670i	$0.396568\pi$
$422$	−3.10102	−0.150955
$423$	0	0
$424$	−1.10102	−0.0534703
$425$	−13.7980	−0.669299
$426$	0	0
$427$	−11.4495	−0.554080
$428$	12.0000	0.580042
$429$	0	0
$430$	−10.0000	−0.482243
$431$	7.59592	0.365882	0.182941	−	0.983124i	$-0.441438\pi$
$431$	7.59592	0.365882	0.182941	+	0.983124i	$0.441438\pi$
$432$	0	0
$433$	11.7980	0.566974	0.283487	−	0.958976i	$-0.408509\pi$
$433$	11.7980	0.566974	0.283487	+	0.958976i	$0.408509\pi$
$434$	6.00000	0.288009
$435$	0	0
$436$	−16.6969	−0.799638
$437$	7.44949	0.356357
$438$	0	0
$439$	21.7980	1.04036	0.520180	−	0.854057i	$-0.325865\pi$
$439$	21.7980	1.04036	0.520180	+	0.854057i	$0.325865\pi$
$440$	−6.89898	−0.328896
$441$	0	0
$442$	−9.79796	−0.466041
$443$	5.10102	0.242357	0.121178	−	0.992631i	$-0.461333\pi$
$443$	5.10102	0.242357	0.121178	+	0.992631i	$0.461333\pi$
$444$	0	0
$445$	−24.4949	−1.16117
$446$	20.8990	0.989595
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	18.5959	0.877596	0.438798	−	0.898586i	$-0.355404\pi$
$449$	18.5959	0.877596	0.438798	+	0.898586i	$0.355404\pi$
$450$	0	0
$451$	−19.5959	−0.922736
$452$	15.8990	0.747825
$453$	0	0
$454$	−0.550510	−0.0258367
$455$	−16.8990	−0.792236
$456$	0	0
$457$	31.4949	1.47327	0.736635	−	0.676291i	$-0.236414\pi$
$457$	31.4949	1.47327	0.736635	+	0.676291i	$0.236414\pi$
$458$	23.2474	1.08628
$459$	0	0
$460$	−3.44949	−0.160833
$461$	−20.3485	−0.947723	−0.473861	−	0.880599i	$-0.657140\pi$
$461$	−20.3485	−0.947723	−0.473861	+	0.880599i	$0.657140\pi$
$462$	0	0
$463$	−25.6969	−1.19424	−0.597119	−	0.802153i	$-0.703688\pi$
$463$	−25.6969	−1.19424	−0.597119	+	0.802153i	$0.703688\pi$
$464$	−2.89898	−0.134582
$465$	0	0
$466$	−7.00000	−0.324269
$467$	−10.0000	−0.462745	−0.231372	−	0.972865i	$-0.574322\pi$
$467$	−10.0000	−0.462745	−0.231372	+	0.972865i	$0.574322\pi$
$468$	0	0
$469$	3.10102	0.143192
$470$	33.7980	1.55898
$471$	0	0
$472$	−2.00000	−0.0920575
$473$	5.79796	0.266590
$474$	0	0
$475$	51.3939	2.35811
$476$	2.00000	0.0916698
$477$	0	0
$478$	−12.7980	−0.585365
$479$	29.5959	1.35227	0.676136	−	0.736777i	$-0.263653\pi$
$479$	29.5959	1.35227	0.676136	+	0.736777i	$0.263653\pi$
$480$	0	0
$481$	38.2020	1.74186
$482$	8.89898	0.405337
$483$	0	0
$484$	−7.00000	−0.318182
$485$	23.7980	1.08061
$486$	0	0
$487$	22.3939	1.01476	0.507382	−	0.861721i	$-0.330613\pi$
$487$	22.3939	1.01476	0.507382	+	0.861721i	$0.330613\pi$
$488$	−11.4495	−0.518294
$489$	0	0
$490$	3.44949	0.155832
$491$	3.79796	0.171399	0.0856997	−	0.996321i	$-0.472687\pi$
$491$	3.79796	0.171399	0.0856997	+	0.996321i	$0.472687\pi$
$492$	0	0
$493$	5.79796	0.261127
$494$	36.4949	1.64198
$495$	0	0
$496$	6.00000	0.269408
$497$	9.89898	0.444030
$498$	0	0
$499$	33.3939	1.49492	0.747458	−	0.664309i	$-0.231274\pi$
$499$	33.3939	1.49492	0.747458	+	0.664309i	$0.231274\pi$
$500$	−6.55051	−0.292948
$501$	0	0
$502$	12.5505	0.560157
$503$	−24.4949	−1.09217	−0.546087	−	0.837729i	$-0.683883\pi$
$503$	−24.4949	−1.09217	−0.546087	+	0.837729i	$0.683883\pi$
$504$	0	0
$505$	25.0000	1.11249
$506$	2.00000	0.0889108
$507$	0	0
$508$	−3.00000	−0.133103
$509$	16.8990	0.749034	0.374517	−	0.927220i	$-0.377809\pi$
$509$	16.8990	0.749034	0.374517	+	0.927220i	$0.377809\pi$
$510$	0	0
$511$	−2.89898	−0.128243
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	27.7980	1.22612
$515$	−48.2929	−2.12804
$516$	0	0
$517$	−19.5959	−0.861827
$518$	−7.79796	−0.342623
$519$	0	0
$520$	−16.8990	−0.741069
$521$	38.6969	1.69534	0.847672	−	0.530521i	$-0.178004\pi$
$521$	38.6969	1.69534	0.847672	+	0.530521i	$0.178004\pi$
$522$	0	0
$523$	0.348469	0.0152375	0.00761875	−	0.999971i	$-0.497575\pi$
$523$	0.348469	0.0152375	0.00761875	+	0.999971i	$0.497575\pi$
$524$	13.4495	0.587544
$525$	0	0
$526$	16.1010	0.702038
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−22.0000	−0.956522
$530$	3.79796	0.164973
$531$	0	0
$532$	−7.44949	−0.322976
$533$	−48.0000	−2.07911
$534$	0	0
$535$	−41.3939	−1.78961
$536$	3.10102	0.133944
$537$	0	0
$538$	−3.65153	−0.157429
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	30.4949	1.31108	0.655539	−	0.755161i	$-0.272441\pi$
$541$	30.4949	1.31108	0.655539	+	0.755161i	$0.272441\pi$
$542$	−16.8990	−0.725873
$543$	0	0
$544$	2.00000	0.0857493
$545$	57.5959	2.46714
$546$	0	0
$547$	31.5959	1.35094	0.675472	−	0.737386i	$-0.263940\pi$
$547$	31.5959	1.35094	0.675472	+	0.737386i	$0.263940\pi$
$548$	11.7980	0.503984
$549$	0	0
$550$	13.7980	0.588347
$551$	−21.5959	−0.920017
$552$	0	0
$553$	−7.89898	−0.335899
$554$	−10.6969	−0.454469
$555$	0	0
$556$	9.44949	0.400748
$557$	3.10102	0.131394	0.0656972	−	0.997840i	$-0.479073\pi$
$557$	3.10102	0.131394	0.0656972	+	0.997840i	$0.479073\pi$
$558$	0	0
$559$	14.2020	0.600682
$560$	3.44949	0.145768
$561$	0	0
$562$	−19.0000	−0.801467
$563$	−13.9444	−0.587686	−0.293843	−	0.955854i	$-0.594934\pi$
$563$	−13.9444	−0.587686	−0.293843	+	0.955854i	$0.594934\pi$
$564$	0	0
$565$	−54.8434	−2.30728
$566$	20.5505	0.863802
$567$	0	0
$568$	9.89898	0.415352
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	14.2020	0.594337	0.297168	−	0.954825i	$-0.403958\pi$
$571$	14.2020	0.594337	0.297168	+	0.954825i	$0.403958\pi$
$572$	9.79796	0.409673
$573$	0	0
$574$	9.79796	0.408959
$575$	6.89898	0.287707
$576$	0	0
$577$	23.5959	0.982311	0.491155	−	0.871072i	$-0.336575\pi$
$577$	23.5959	0.982311	0.491155	+	0.871072i	$0.336575\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	10.0000	0.415227
$581$	2.00000	0.0829740
$582$	0	0
$583$	−2.20204	−0.0911992
$584$	−2.89898	−0.119961
$585$	0	0
$586$	27.2474	1.12558
$587$	18.1464	0.748983	0.374492	−	0.927230i	$-0.377817\pi$
$587$	18.1464	0.748983	0.374492	+	0.927230i	$0.377817\pi$
$588$	0	0
$589$	44.6969	1.84171
$590$	6.89898	0.284026
$591$	0	0
$592$	−7.79796	−0.320494
$593$	14.6969	0.603531	0.301765	−	0.953382i	$-0.402424\pi$
$593$	14.6969	0.603531	0.301765	+	0.953382i	$0.402424\pi$
$594$	0	0
$595$	−6.89898	−0.282831
$596$	6.00000	0.245770
$597$	0	0
$598$	4.89898	0.200334
$599$	14.2020	0.580280	0.290140	−	0.956984i	$-0.406298\pi$
$599$	14.2020	0.580280	0.290140	+	0.956984i	$0.406298\pi$
$600$	0	0
$601$	−12.6969	−0.517919	−0.258959	−	0.965888i	$-0.583380\pi$
$601$	−12.6969	−0.517919	−0.258959	+	0.965888i	$0.583380\pi$
$602$	−2.89898	−0.118154
$603$	0	0
$604$	−5.00000	−0.203447
$605$	24.1464	0.981692
$606$	0	0
$607$	−8.69694	−0.352998	−0.176499	−	0.984301i	$-0.556477\pi$
$607$	−8.69694	−0.352998	−0.176499	+	0.984301i	$0.556477\pi$
$608$	−7.44949	−0.302117
$609$	0	0
$610$	39.4949	1.59910
$611$	−48.0000	−1.94187
$612$	0	0
$613$	14.6969	0.593604	0.296802	−	0.954939i	$-0.404080\pi$
$613$	14.6969	0.593604	0.296802	+	0.954939i	$0.404080\pi$
$614$	−0.752551	−0.0303705
$615$	0	0
$616$	−2.00000	−0.0805823
$617$	−43.3939	−1.74697	−0.873486	−	0.486850i	$-0.838146\pi$
$617$	−43.3939	−1.74697	−0.873486	+	0.486850i	$0.838146\pi$
$618$	0	0
$619$	−4.14643	−0.166659	−0.0833295	−	0.996522i	$-0.526555\pi$
$619$	−4.14643	−0.166659	−0.0833295	+	0.996522i	$0.526555\pi$
$620$	−20.6969	−0.831209
$621$	0	0
$622$	1.30306	0.0522480
$623$	−7.10102	−0.284496
$624$	0	0
$625$	−11.8990	−0.475959
$626$	−24.6969	−0.987088
$627$	0	0
$628$	6.34847	0.253332
$629$	15.5959	0.621850
$630$	0	0
$631$	18.1010	0.720590	0.360295	−	0.932838i	$-0.382676\pi$
$631$	18.1010	0.720590	0.360295	+	0.932838i	$0.382676\pi$
$632$	−7.89898	−0.314205
$633$	0	0
$634$	−8.69694	−0.345400
$635$	10.3485	0.410666
$636$	0	0
$637$	−4.89898	−0.194105
$638$	−5.79796	−0.229543
$639$	0	0
$640$	3.44949	0.136353
$641$	−41.4949	−1.63895	−0.819475	−	0.573115i	$-0.805735\pi$
$641$	−41.4949	−1.63895	−0.819475	+	0.573115i	$0.805735\pi$
$642$	0	0
$643$	19.3939	0.764820	0.382410	−	0.923993i	$-0.375094\pi$
$643$	19.3939	0.764820	0.382410	+	0.923993i	$0.375094\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	14.8990	0.586193
$647$	21.3031	0.837510	0.418755	−	0.908099i	$-0.362467\pi$
$647$	21.3031	0.837510	0.418755	+	0.908099i	$0.362467\pi$
$648$	0	0
$649$	−4.00000	−0.157014
$650$	33.7980	1.32567
$651$	0	0
$652$	−0.202041	−0.00791254
$653$	9.79796	0.383424	0.191712	−	0.981451i	$-0.438596\pi$
$653$	9.79796	0.383424	0.191712	+	0.981451i	$0.438596\pi$
$654$	0	0
$655$	−46.3939	−1.81276
$656$	9.79796	0.382546
$657$	0	0
$658$	9.79796	0.381964
$659$	−4.69694	−0.182967	−0.0914834	−	0.995807i	$-0.529161\pi$
$659$	−4.69694	−0.182967	−0.0914834	+	0.995807i	$0.529161\pi$
$660$	0	0
$661$	9.44949	0.367543	0.183771	−	0.982969i	$-0.441169\pi$
$661$	9.44949	0.367543	0.183771	+	0.982969i	$0.441169\pi$
$662$	24.6969	0.959874
$663$	0	0
$664$	2.00000	0.0776151
$665$	25.6969	0.996485
$666$	0	0
$667$	−2.89898	−0.112249
$668$	−18.6969	−0.723406
$669$	0	0
$670$	−10.6969	−0.413259
$671$	−22.8990	−0.884005
$672$	0	0
$673$	30.5959	1.17939	0.589693	−	0.807628i	$-0.299249\pi$
$673$	30.5959	1.17939	0.589693	+	0.807628i	$0.299249\pi$
$674$	−35.3939	−1.36332
$675$	0	0
$676$	11.0000	0.423077
$677$	14.6969	0.564849	0.282425	−	0.959289i	$-0.408861\pi$
$677$	14.6969	0.564849	0.282425	+	0.959289i	$0.408861\pi$
$678$	0	0
$679$	6.89898	0.264759
$680$	−6.89898	−0.264564
$681$	0	0
$682$	12.0000	0.459504
$683$	−32.2020	−1.23218	−0.616088	−	0.787677i	$-0.711284\pi$
$683$	−32.2020	−1.23218	−0.616088	+	0.787677i	$0.711284\pi$
$684$	0	0
$685$	−40.6969	−1.55495
$686$	1.00000	0.0381802
$687$	0	0
$688$	−2.89898	−0.110523
$689$	−5.39388	−0.205490
$690$	0	0
$691$	6.95459	0.264565	0.132283	−	0.991212i	$-0.457769\pi$
$691$	6.95459	0.264565	0.132283	+	0.991212i	$0.457769\pi$
$692$	12.8990	0.490346
$693$	0	0
$694$	−19.5959	−0.743851
$695$	−32.5959	−1.23643
$696$	0	0
$697$	−19.5959	−0.742248
$698$	−20.8990	−0.791038
$699$	0	0
$700$	−6.89898	−0.260757
$701$	−51.3939	−1.94112	−0.970560	−	0.240860i	$-0.922571\pi$
$701$	−51.3939	−1.94112	−0.970560	+	0.240860i	$0.922571\pi$
$702$	0	0
$703$	−58.0908	−2.19094
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−6.00000	−0.225813
$707$	7.24745	0.272568
$708$	0	0
$709$	−11.5959	−0.435494	−0.217747	−	0.976005i	$-0.569871\pi$
$709$	−11.5959	−0.435494	−0.217747	+	0.976005i	$0.569871\pi$
$710$	−34.1464	−1.28149
$711$	0	0
$712$	−7.10102	−0.266122
$713$	6.00000	0.224702
$714$	0	0
$715$	−33.7980	−1.26397
$716$	8.69694	0.325020
$717$	0	0
$718$	10.7980	0.402976
$719$	9.79796	0.365402	0.182701	−	0.983169i	$-0.441516\pi$
$719$	9.79796	0.365402	0.182701	+	0.983169i	$0.441516\pi$
$720$	0	0
$721$	−14.0000	−0.521387
$722$	−36.4949	−1.35820
$723$	0	0
$724$	4.34847	0.161610
$725$	−20.0000	−0.742781
$726$	0	0
$727$	40.4949	1.50187	0.750936	−	0.660375i	$-0.229603\pi$
$727$	40.4949	1.50187	0.750936	+	0.660375i	$0.229603\pi$
$728$	−4.89898	−0.181568
$729$	0	0
$730$	10.0000	0.370117
$731$	5.79796	0.214445
$732$	0	0
$733$	−12.5505	−0.463564	−0.231782	−	0.972768i	$-0.574456\pi$
$733$	−12.5505	−0.463564	−0.231782	+	0.972768i	$0.574456\pi$
$734$	−5.79796	−0.214007
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	6.20204	0.228455
$738$	0	0
$739$	−25.5959	−0.941561	−0.470781	−	0.882250i	$-0.656028\pi$
$739$	−25.5959	−0.941561	−0.470781	+	0.882250i	$0.656028\pi$
$740$	26.8990	0.988826
$741$	0	0
$742$	1.10102	0.0404197
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−20.6969	−0.758277
$746$	−2.89898	−0.106139
$747$	0	0
$748$	4.00000	0.146254
$749$	−12.0000	−0.438470
$750$	0	0
$751$	40.5959	1.48137	0.740683	−	0.671855i	$-0.234502\pi$
$751$	40.5959	1.48137	0.740683	+	0.671855i	$0.234502\pi$
$752$	9.79796	0.357295
$753$	0	0
$754$	−14.2020	−0.517208
$755$	17.2474	0.627699
$756$	0	0
$757$	23.3939	0.850265	0.425132	−	0.905131i	$-0.360228\pi$
$757$	23.3939	0.850265	0.425132	+	0.905131i	$0.360228\pi$
$758$	26.4949	0.962338
$759$	0	0
$760$	25.6969	0.932126
$761$	−2.00000	−0.0724999	−0.0362500	−	0.999343i	$-0.511541\pi$
$761$	−2.00000	−0.0724999	−0.0362500	+	0.999343i	$0.511541\pi$
$762$	0	0
$763$	16.6969	0.604470
$764$	−13.8990	−0.502847
$765$	0	0
$766$	6.89898	0.249270
$767$	−9.79796	−0.353784
$768$	0	0
$769$	54.0908	1.95056	0.975282	−	0.220962i	$-0.0709198\pi$
$769$	54.0908	1.95056	0.975282	+	0.220962i	$0.0709198\pi$
$770$	6.89898	0.248622
$771$	0	0
$772$	−8.10102	−0.291562
$773$	19.9444	0.717350	0.358675	−	0.933463i	$-0.383229\pi$
$773$	19.9444	0.717350	0.358675	+	0.933463i	$0.383229\pi$
$774$	0	0
$775$	41.3939	1.48691
$776$	6.89898	0.247659
$777$	0	0
$778$	−15.1010	−0.541398
$779$	72.9898	2.61513
$780$	0	0
$781$	19.7980	0.708427
$782$	2.00000	0.0715199
$783$	0	0
$784$	1.00000	0.0357143
$785$	−21.8990	−0.781608
$786$	0	0
$787$	47.3939	1.68941	0.844705	−	0.535233i	$-0.179776\pi$
$787$	47.3939	1.68941	0.844705	+	0.535233i	$0.179776\pi$
$788$	12.6969	0.452310
$789$	0	0
$790$	27.2474	0.969421
$791$	−15.8990	−0.565303
$792$	0	0
$793$	−56.0908	−1.99184
$794$	−9.30306	−0.330153
$795$	0	0
$796$	6.89898	0.244528
$797$	−35.9444	−1.27322	−0.636608	−	0.771188i	$-0.719663\pi$
$797$	−35.9444	−1.27322	−0.636608	+	0.771188i	$0.719663\pi$
$798$	0	0
$799$	−19.5959	−0.693254
$800$	−6.89898	−0.243916
$801$	0	0
$802$	−10.1010	−0.356679
$803$	−5.79796	−0.204606
$804$	0	0
$805$	3.44949	0.121579
$806$	29.3939	1.03536
$807$	0	0
$808$	7.24745	0.254964
$809$	35.7980	1.25859	0.629295	−	0.777167i	$-0.283344\pi$
$809$	35.7980	1.25859	0.629295	+	0.777167i	$0.283344\pi$
$810$	0	0
$811$	−2.00000	−0.0702295	−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000	−0.0702295	−0.0351147	+	0.999383i	$0.511180\pi$
$812$	2.89898	0.101734
$813$	0	0
$814$	−15.5959	−0.546637
$815$	0.696938	0.0244127
$816$	0	0
$817$	−21.5959	−0.755546
$818$	−5.79796	−0.202721
$819$	0	0
$820$	−33.7980	−1.18028
$821$	−39.5959	−1.38191	−0.690954	−	0.722899i	$-0.742809\pi$
$821$	−39.5959	−1.38191	−0.690954	+	0.722899i	$0.742809\pi$
$822$	0	0
$823$	45.3939	1.58233	0.791166	−	0.611602i	$-0.209475\pi$
$823$	45.3939	1.58233	0.791166	+	0.611602i	$0.209475\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	2.00000	0.0695889
$827$	−12.4949	−0.434490	−0.217245	−	0.976117i	$-0.569707\pi$
$827$	−12.4949	−0.434490	−0.217245	+	0.976117i	$0.569707\pi$
$828$	0	0
$829$	30.6969	1.06615	0.533074	−	0.846068i	$-0.321037\pi$
$829$	30.6969	1.06615	0.533074	+	0.846068i	$0.321037\pi$
$830$	−6.89898	−0.239467
$831$	0	0
$832$	−4.89898	−0.169842
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	64.4949	2.23194
$836$	−14.8990	−0.515292
$837$	0	0
$838$	24.5505	0.848083
$839$	44.8990	1.55008	0.775042	−	0.631909i	$-0.217728\pi$
$839$	44.8990	1.55008	0.775042	+	0.631909i	$0.217728\pi$
$840$	0	0
$841$	−20.5959	−0.710204
$842$	−13.1010	−0.451491
$843$	0	0
$844$	3.10102	0.106742
$845$	−37.9444	−1.30533
$846$	0	0
$847$	7.00000	0.240523
$848$	1.10102	0.0378092
$849$	0	0
$850$	13.7980	0.473266
$851$	−7.79796	−0.267311
$852$	0	0
$853$	−38.8434	−1.32997	−0.664986	−	0.746856i	$-0.731562\pi$
$853$	−38.8434	−1.32997	−0.664986	+	0.746856i	$0.731562\pi$
$854$	11.4495	0.391793
$855$	0	0
$856$	−12.0000	−0.410152
$857$	25.1010	0.857435	0.428717	−	0.903439i	$-0.358966\pi$
$857$	25.1010	0.857435	0.428717	+	0.903439i	$0.358966\pi$
$858$	0	0
$859$	−10.0000	−0.341196	−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000	−0.341196	−0.170598	+	0.985341i	$0.554570\pi$
$860$	10.0000	0.340997
$861$	0	0
$862$	−7.59592	−0.258718
$863$	−2.10102	−0.0715196	−0.0357598	−	0.999360i	$-0.511385\pi$
$863$	−2.10102	−0.0715196	−0.0357598	+	0.999360i	$0.511385\pi$
$864$	0	0
$865$	−44.4949	−1.51287
$866$	−11.7980	−0.400911
$867$	0	0
$868$	−6.00000	−0.203653
$869$	−15.7980	−0.535909
$870$	0	0
$871$	15.1918	0.514756
$872$	16.6969	0.565430
$873$	0	0
$874$	−7.44949	−0.251983
$875$	6.55051	0.221448
$876$	0	0
$877$	−26.4949	−0.894669	−0.447335	−	0.894367i	$-0.647627\pi$
$877$	−26.4949	−0.894669	−0.447335	+	0.894367i	$0.647627\pi$
$878$	−21.7980	−0.735645
$879$	0	0
$880$	6.89898	0.232565
$881$	19.5959	0.660203	0.330102	−	0.943945i	$-0.392917\pi$
$881$	19.5959	0.660203	0.330102	+	0.943945i	$0.392917\pi$
$882$	0	0
$883$	−0.202041	−0.00679922	−0.00339961	−	0.999994i	$-0.501082\pi$
$883$	−0.202041	−0.00679922	−0.00339961	+	0.999994i	$0.501082\pi$
$884$	9.79796	0.329541
$885$	0	0
$886$	−5.10102	−0.171372
$887$	33.7980	1.13482	0.567412	−	0.823434i	$-0.307945\pi$
$887$	33.7980	1.13482	0.567412	+	0.823434i	$0.307945\pi$
$888$	0	0
$889$	3.00000	0.100617
$890$	24.4949	0.821071
$891$	0	0
$892$	−20.8990	−0.699750
$893$	72.9898	2.44251
$894$	0	0
$895$	−30.0000	−1.00279
$896$	1.00000	0.0334077
$897$	0	0
$898$	−18.5959	−0.620554
$899$	−17.3939	−0.580118
$900$	0	0
$901$	−2.20204	−0.0733606
$902$	19.5959	0.652473
$903$	0	0
$904$	−15.8990	−0.528792
$905$	−15.0000	−0.498617
$906$	0	0
$907$	−26.6969	−0.886457	−0.443229	−	0.896409i	$-0.646167\pi$
$907$	−26.6969	−0.886457	−0.443229	+	0.896409i	$0.646167\pi$
$908$	0.550510	0.0182693
$909$	0	0
$910$	16.8990	0.560196
$911$	45.9898	1.52371	0.761855	−	0.647748i	$-0.224289\pi$
$911$	45.9898	1.52371	0.761855	+	0.647748i	$0.224289\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	−31.4949	−1.04176
$915$	0	0
$916$	−23.2474	−0.768117
$917$	−13.4495	−0.444141
$918$	0	0
$919$	3.69694	0.121951	0.0609754	−	0.998139i	$-0.480579\pi$
$919$	3.69694	0.121951	0.0609754	+	0.998139i	$0.480579\pi$
$920$	3.44949	0.113726
$921$	0	0
$922$	20.3485	0.670141
$923$	48.4949	1.59623
$924$	0	0
$925$	−53.7980	−1.76887
$926$	25.6969	0.844454
$927$	0	0
$928$	2.89898	0.0951637
$929$	34.2929	1.12511	0.562556	−	0.826759i	$-0.309818\pi$
$929$	34.2929	1.12511	0.562556	+	0.826759i	$0.309818\pi$
$930$	0	0
$931$	7.44949	0.244147
$932$	7.00000	0.229293
$933$	0	0
$934$	10.0000	0.327210
$935$	−13.7980	−0.451242
$936$	0	0
$937$	6.40408	0.209212	0.104606	−	0.994514i	$-0.466642\pi$
$937$	6.40408	0.209212	0.104606	+	0.994514i	$0.466642\pi$
$938$	−3.10102	−0.101252
$939$	0	0
$940$	−33.7980	−1.10237
$941$	−3.44949	−0.112450	−0.0562251	−	0.998418i	$-0.517906\pi$
$941$	−3.44949	−0.112450	−0.0562251	+	0.998418i	$0.517906\pi$
$942$	0	0
$943$	9.79796	0.319065
$944$	2.00000	0.0650945
$945$	0	0
$946$	−5.79796	−0.188508
$947$	−3.50510	−0.113901	−0.0569503	−	0.998377i	$-0.518138\pi$
$947$	−3.50510	−0.113901	−0.0569503	+	0.998377i	$0.518138\pi$
$948$	0	0
$949$	−14.2020	−0.461018
$950$	−51.3939	−1.66744
$951$	0	0
$952$	−2.00000	−0.0648204
$953$	−55.3939	−1.79438	−0.897192	−	0.441641i	$-0.854396\pi$
$953$	−55.3939	−1.79438	−0.897192	+	0.441641i	$0.854396\pi$
$954$	0	0
$955$	47.9444	1.55144
$956$	12.7980	0.413916
$957$	0	0
$958$	−29.5959	−0.956201
$959$	−11.7980	−0.380976
$960$	0	0
$961$	5.00000	0.161290
$962$	−38.2020	−1.23168
$963$	0	0
$964$	−8.89898	−0.286617
$965$	27.9444	0.899562
$966$	0	0
$967$	−14.5959	−0.469373	−0.234687	−	0.972071i	$-0.575406\pi$
$967$	−14.5959	−0.469373	−0.234687	+	0.972071i	$0.575406\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	−23.7980	−0.764106
$971$	53.9444	1.73116	0.865579	−	0.500773i	$-0.166951\pi$
$971$	53.9444	1.73116	0.865579	+	0.500773i	$0.166951\pi$
$972$	0	0
$973$	−9.44949	−0.302937
$974$	−22.3939	−0.717546
$975$	0	0
$976$	11.4495	0.366489
$977$	−1.59592	−0.0510579	−0.0255290	−	0.999674i	$-0.508127\pi$
$977$	−1.59592	−0.0510579	−0.0255290	+	0.999674i	$0.508127\pi$
$978$	0	0
$979$	−14.2020	−0.453899
$980$	−3.44949	−0.110190
$981$	0	0
$982$	−3.79796	−0.121198
$983$	−45.1918	−1.44140	−0.720698	−	0.693249i	$-0.756178\pi$
$983$	−45.1918	−1.44140	−0.720698	+	0.693249i	$0.756178\pi$
$984$	0	0
$985$	−43.7980	−1.39552
$986$	−5.79796	−0.184645
$987$	0	0
$988$	−36.4949	−1.16106
$989$	−2.89898	−0.0921822
$990$	0	0
$991$	17.7980	0.565371	0.282685	−	0.959213i	$-0.408775\pi$
$991$	17.7980	0.565371	0.282685	+	0.959213i	$0.408775\pi$
$992$	−6.00000	−0.190500
$993$	0	0
$994$	−9.89898	−0.313977
$995$	−23.7980	−0.754446
$996$	0	0
$997$	17.8536	0.565428	0.282714	−	0.959204i	$-0.408765\pi$
$997$	17.8536	0.565428	0.282714	+	0.959204i	$0.408765\pi$
$998$	−33.3939	−1.05706
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.2.a.i.1.1		2
3.2	odd	2		1134.2.a.p.1.2		2
4.3	odd	2		9072.2.a.bd.1.1		2
7.6	odd	2		7938.2.a.bm.1.2		2
9.2	odd	6		126.2.f.c.85.2	yes	4
9.4	even	3		378.2.f.d.127.2		4
9.5	odd	6		126.2.f.c.43.1	✓	4
9.7	even	3		378.2.f.d.253.2		4
12.11	even	2		9072.2.a.bk.1.2		2
21.20	even	2		7938.2.a.bn.1.1		2
36.7	odd	6		3024.2.r.e.1009.2		4
36.11	even	6		1008.2.r.e.337.1		4
36.23	even	6		1008.2.r.e.673.2		4
36.31	odd	6		3024.2.r.e.2017.2		4
63.2	odd	6		882.2.h.k.67.1		4
63.4	even	3		2646.2.h.m.667.1		4
63.5	even	6		882.2.e.n.655.1		4
63.11	odd	6		882.2.e.m.373.2		4
63.13	odd	6		2646.2.f.k.883.1		4
63.16	even	3		2646.2.h.m.361.1		4
63.20	even	6		882.2.f.j.589.1		4
63.23	odd	6		882.2.e.m.655.2		4
63.25	even	3		2646.2.e.l.1549.2		4
63.31	odd	6		2646.2.h.n.667.2		4
63.32	odd	6		882.2.h.k.79.1		4
63.34	odd	6		2646.2.f.k.1765.1		4
63.38	even	6		882.2.e.n.373.1		4
63.40	odd	6		2646.2.e.k.2125.1		4
63.41	even	6		882.2.f.j.295.2		4
63.47	even	6		882.2.h.l.67.2		4
63.52	odd	6		2646.2.e.k.1549.1		4
63.58	even	3		2646.2.e.l.2125.2		4
63.59	even	6		882.2.h.l.79.2		4
63.61	odd	6		2646.2.h.n.361.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.f.c.43.1	✓	4	9.5	odd	6
126.2.f.c.85.2	yes	4	9.2	odd	6
378.2.f.d.127.2		4	9.4	even	3
378.2.f.d.253.2		4	9.7	even	3
882.2.e.m.373.2		4	63.11	odd	6
882.2.e.m.655.2		4	63.23	odd	6
882.2.e.n.373.1		4	63.38	even	6
882.2.e.n.655.1		4	63.5	even	6
882.2.f.j.295.2		4	63.41	even	6
882.2.f.j.589.1		4	63.20	even	6
882.2.h.k.67.1		4	63.2	odd	6
882.2.h.k.79.1		4	63.32	odd	6
882.2.h.l.67.2		4	63.47	even	6
882.2.h.l.79.2		4	63.59	even	6
1008.2.r.e.337.1		4	36.11	even	6
1008.2.r.e.673.2		4	36.23	even	6
1134.2.a.i.1.1		2	1.1	even	1	trivial
1134.2.a.p.1.2		2	3.2	odd	2
2646.2.e.k.1549.1		4	63.52	odd	6
2646.2.e.k.2125.1		4	63.40	odd	6
2646.2.e.l.1549.2		4	63.25	even	3
2646.2.e.l.2125.2		4	63.58	even	3
2646.2.f.k.883.1		4	63.13	odd	6
2646.2.f.k.1765.1		4	63.34	odd	6
2646.2.h.m.361.1		4	63.16	even	3
2646.2.h.m.667.1		4	63.4	even	3
2646.2.h.n.361.2		4	63.61	odd	6
2646.2.h.n.667.2		4	63.31	odd	6
3024.2.r.e.1009.2		4	36.7	odd	6
3024.2.r.e.2017.2		4	36.31	odd	6
7938.2.a.bm.1.2		2	7.6	odd	2
7938.2.a.bn.1.1		2	21.20	even	2
9072.2.a.bd.1.1		2	4.3	odd	2
9072.2.a.bk.1.2		2	12.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1134.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.44949 q^{5} -1.00000 q^{7} -1.00000 q^{8} +3.44949 q^{10} -2.00000 q^{11} -4.89898 q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +7.44949 q^{19} -3.44949 q^{20} +2.00000 q^{22} +1.00000 q^{23} +6.89898 q^{25} +4.89898 q^{26} -1.00000 q^{28} -2.89898 q^{29} +6.00000 q^{31} -1.00000 q^{32} +2.00000 q^{34} +3.44949 q^{35} -7.79796 q^{37} -7.44949 q^{38} +3.44949 q^{40} +9.79796 q^{41} -2.89898 q^{43} -2.00000 q^{44} -1.00000 q^{46} +9.79796 q^{47} +1.00000 q^{49} -6.89898 q^{50} -4.89898 q^{52} +1.10102 q^{53} +6.89898 q^{55} +1.00000 q^{56} +2.89898 q^{58} +2.00000 q^{59} +11.4495 q^{61} -6.00000 q^{62} +1.00000 q^{64} +16.8990 q^{65} -3.10102 q^{67} -2.00000 q^{68} -3.44949 q^{70} -9.89898 q^{71} +2.89898 q^{73} +7.79796 q^{74} +7.44949 q^{76} +2.00000 q^{77} +7.89898 q^{79} -3.44949 q^{80} -9.79796 q^{82} -2.00000 q^{83} +6.89898 q^{85} +2.89898 q^{86} +2.00000 q^{88} +7.10102 q^{89} +4.89898 q^{91} +1.00000 q^{92} -9.79796 q^{94} -25.6969 q^{95} -6.89898 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} - 2 q^{7} - 2 q^{8} + 2 q^{10} - 4 q^{11} + 2 q^{14} + 2 q^{16} - 4 q^{17} + 10 q^{19} - 2 q^{20} + 4 q^{22} + 2 q^{23} + 4 q^{25} - 2 q^{28} + 4 q^{29} + 12 q^{31} - 2 q^{32}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 - 2 * q^7 - 2 * q^8 + 2 * q^10 - 4 * q^11 + 2 * q^14 + 2 * q^16 - 4 * q^17 + 10 * q^19 - 2 * q^20 + 4 * q^22 + 2 * q^23 + 4 * q^25 - 2 * q^28 + 4 * q^29 + 12 * q^31 - 2 * q^32 + 4 * q^34 + 2 * q^35 + 4 * q^37 - 10 * q^38 + 2 * q^40 + 4 * q^43 - 4 * q^44 - 2 * q^46 + 2 * q^49 - 4 * q^50 + 12 * q^53 + 4 * q^55 + 2 * q^56 - 4 * q^58 + 4 * q^59 + 18 * q^61 - 12 * q^62 + 2 * q^64 + 24 * q^65 - 16 * q^67 - 4 * q^68 - 2 * q^70 - 10 * q^71 - 4 * q^73 - 4 * q^74 + 10 * q^76 + 4 * q^77 + 6 * q^79 - 2 * q^80 - 4 * q^83 + 4 * q^85 - 4 * q^86 + 4 * q^88 + 24 * q^89 + 2 * q^92 - 22 * q^95 - 4 * q^97 - 2 * q^98

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