LMFDB - Embedded newform 7569.2.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7569,2,Mod(1,7569)] 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("7569.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7569, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,1,0,-1,-1,0,0,0,0,7,5,0,4,5,0,-3,11,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$60.4387692899$
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:	$ 1 $
Twist minimal:	no (minimal twist has level 841)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.618034 q^{2} -1.61803 q^{4} -3.85410 q^{5} -2.23607 q^{7} +2.23607 q^{8} +2.38197 q^{10} +1.38197 q^{11} -0.236068 q^{13} +1.38197 q^{14} +1.85410 q^{16} +4.38197 q^{17} -4.85410 q^{19} +6.23607 q^{20} -0.854102 q^{22} +1.23607 q^{23} +9.85410 q^{25} +0.145898 q^{26} +3.61803 q^{28} -10.0902 q^{31} -5.61803 q^{32} -2.70820 q^{34} +8.61803 q^{35} +4.70820 q^{37} +3.00000 q^{38} -8.61803 q^{40} -3.85410 q^{41} -7.23607 q^{43} -2.23607 q^{44} -0.763932 q^{46} +7.00000 q^{47} -2.00000 q^{49} -6.09017 q^{50} +0.381966 q^{52} +2.00000 q^{53} -5.32624 q^{55} -5.00000 q^{56} -6.09017 q^{59} -0.618034 q^{61} +6.23607 q^{62} -0.236068 q^{64} +0.909830 q^{65} -1.52786 q^{67} -7.09017 q^{68} -5.32624 q^{70} -10.4721 q^{71} -13.7082 q^{73} -2.90983 q^{74} +7.85410 q^{76} -3.09017 q^{77} -6.09017 q^{79} -7.14590 q^{80} +2.38197 q^{82} +9.94427 q^{83} -16.8885 q^{85} +4.47214 q^{86} +3.09017 q^{88} -4.70820 q^{89} +0.527864 q^{91} -2.00000 q^{92} -4.32624 q^{94} +18.7082 q^{95} +3.56231 q^{97} +1.23607 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - q^{5} + 7 q^{10} + 5 q^{11} + 4 q^{13} + 5 q^{14} - 3 q^{16} + 11 q^{17} - 3 q^{19} + 8 q^{20} + 5 q^{22} - 2 q^{23} + 13 q^{25} + 7 q^{26} + 5 q^{28} - 9 q^{31} - 9 q^{32} + 8 q^{34}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - q^5 + 7 * q^10 + 5 * q^11 + 4 * q^13 + 5 * q^14 - 3 * q^16 + 11 * q^17 - 3 * q^19 + 8 * q^20 + 5 * q^22 - 2 * q^23 + 13 * q^25 + 7 * q^26 + 5 * q^28 - 9 * q^31 - 9 * q^32 + 8 * q^34 + 15 * q^35 - 4 * q^37 + 6 * q^38 - 15 * q^40 - q^41 - 10 * q^43 - 6 * q^46 + 14 * q^47 - 4 * q^49 - q^50 + 3 * q^52 + 4 * q^53 + 5 * q^55 - 10 * q^56 - q^59 + q^61 + 8 * q^62 + 4 * q^64 + 13 * q^65 - 12 * q^67 - 3 * q^68 + 5 * q^70 - 12 * q^71 - 14 * q^73 - 17 * q^74 + 9 * q^76 + 5 * q^77 - q^79 - 21 * q^80 + 7 * q^82 + 2 * q^83 + 2 * q^85 - 5 * q^88 + 4 * q^89 + 10 * q^91 - 4 * q^92 + 7 * q^94 + 24 * q^95 - 13 * 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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	−3.85410	−1.72361	−0.861803	−	0.507242i	$-0.830665\pi$
$5$	−3.85410	−1.72361	−0.861803	+	0.507242i	$0.830665\pi$
$6$	0	0
$7$	−2.23607	−0.845154	−0.422577	−	0.906327i	$-0.638874\pi$
$7$	−2.23607	−0.845154	−0.422577	+	0.906327i	$0.638874\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	2.38197	0.753244
$11$	1.38197	0.416678	0.208339	−	0.978057i	$-0.433194\pi$
$11$	1.38197	0.416678	0.208339	+	0.978057i	$0.433194\pi$
$12$	0	0
$13$	−0.236068	−0.0654735	−0.0327367	−	0.999464i	$-0.510422\pi$
$13$	−0.236068	−0.0654735	−0.0327367	+	0.999464i	$0.510422\pi$
$14$	1.38197	0.369346
$15$	0	0
$16$	1.85410	0.463525
$17$	4.38197	1.06278	0.531391	−	0.847126i	$-0.321669\pi$
$17$	4.38197	1.06278	0.531391	+	0.847126i	$0.321669\pi$
$18$	0	0
$19$	−4.85410	−1.11361	−0.556804	−	0.830644i	$-0.687972\pi$
$19$	−4.85410	−1.11361	−0.556804	+	0.830644i	$0.687972\pi$
$20$	6.23607	1.39443
$21$	0	0
$22$	−0.854102	−0.182095
$23$	1.23607	0.257738	0.128869	−	0.991662i	$-0.458865\pi$
$23$	1.23607	0.257738	0.128869	+	0.991662i	$0.458865\pi$
$24$	0	0
$25$	9.85410	1.97082
$26$	0.145898	0.0286130
$27$	0	0
$28$	3.61803	0.683744
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−10.0902	−1.81225	−0.906124	−	0.423012i	$-0.860973\pi$
$31$	−10.0902	−1.81225	−0.906124	+	0.423012i	$0.860973\pi$
$32$	−5.61803	−0.993137
$33$	0	0
$34$	−2.70820	−0.464453
$35$	8.61803	1.45671
$36$	0	0
$37$	4.70820	0.774024	0.387012	−	0.922075i	$-0.373507\pi$
$37$	4.70820	0.774024	0.387012	+	0.922075i	$0.373507\pi$
$38$	3.00000	0.486664
$39$	0	0
$40$	−8.61803	−1.36263
$41$	−3.85410	−0.601910	−0.300955	−	0.953638i	$-0.597305\pi$
$41$	−3.85410	−0.601910	−0.300955	+	0.953638i	$0.597305\pi$
$42$	0	0
$43$	−7.23607	−1.10349	−0.551745	−	0.834013i	$-0.686038\pi$
$43$	−7.23607	−1.10349	−0.551745	+	0.834013i	$0.686038\pi$
$44$	−2.23607	−0.337100
$45$	0	0
$46$	−0.763932	−0.112636
$47$	7.00000	1.02105	0.510527	−	0.859861i	$-0.329450\pi$
$47$	7.00000	1.02105	0.510527	+	0.859861i	$0.329450\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	−6.09017	−0.861280
$51$	0	0
$52$	0.381966	0.0529692
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	−5.32624	−0.718190
$56$	−5.00000	−0.668153
$57$	0	0
$58$	0	0
$59$	−6.09017	−0.792873	−0.396436	−	0.918062i	$-0.629753\pi$
$59$	−6.09017	−0.792873	−0.396436	+	0.918062i	$0.629753\pi$
$60$	0	0
$61$	−0.618034	−0.0791311	−0.0395656	−	0.999217i	$-0.512597\pi$
$61$	−0.618034	−0.0791311	−0.0395656	+	0.999217i	$0.512597\pi$
$62$	6.23607	0.791981
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	0.909830	0.112851
$66$	0	0
$67$	−1.52786	−0.186658	−0.0933292	−	0.995635i	$-0.529751\pi$
$67$	−1.52786	−0.186658	−0.0933292	+	0.995635i	$0.529751\pi$
$68$	−7.09017	−0.859809
$69$	0	0
$70$	−5.32624	−0.636607
$71$	−10.4721	−1.24281	−0.621407	−	0.783488i	$-0.713439\pi$
$71$	−10.4721	−1.24281	−0.621407	+	0.783488i	$0.713439\pi$
$72$	0	0
$73$	−13.7082	−1.60442	−0.802212	−	0.597039i	$-0.796344\pi$
$73$	−13.7082	−1.60442	−0.802212	+	0.597039i	$0.796344\pi$
$74$	−2.90983	−0.338261
$75$	0	0
$76$	7.85410	0.900927
$77$	−3.09017	−0.352158
$78$	0	0
$79$	−6.09017	−0.685198	−0.342599	−	0.939482i	$-0.611307\pi$
$79$	−6.09017	−0.685198	−0.342599	+	0.939482i	$0.611307\pi$
$80$	−7.14590	−0.798936
$81$	0	0
$82$	2.38197	0.263044
$83$	9.94427	1.09153	0.545763	−	0.837940i	$-0.316240\pi$
$83$	9.94427	1.09153	0.545763	+	0.837940i	$0.316240\pi$
$84$	0	0
$85$	−16.8885	−1.83182
$86$	4.47214	0.482243
$87$	0	0
$88$	3.09017	0.329413
$89$	−4.70820	−0.499069	−0.249534	−	0.968366i	$-0.580278\pi$
$89$	−4.70820	−0.499069	−0.249534	+	0.968366i	$0.580278\pi$
$90$	0	0
$91$	0.527864	0.0553352
$92$	−2.00000	−0.208514
$93$	0	0
$94$	−4.32624	−0.446217
$95$	18.7082	1.91942
$96$	0	0
$97$	3.56231	0.361697	0.180849	−	0.983511i	$-0.442116\pi$
$97$	3.56231	0.361697	0.180849	+	0.983511i	$0.442116\pi$
$98$	1.23607	0.124862
$99$	0	0
$100$	−15.9443	−1.59443
$101$	0.618034	0.0614967	0.0307483	−	0.999527i	$-0.490211\pi$
$101$	0.618034	0.0614967	0.0307483	+	0.999527i	$0.490211\pi$
$102$	0	0
$103$	9.18034	0.904566	0.452283	−	0.891875i	$-0.350610\pi$
$103$	9.18034	0.904566	0.452283	+	0.891875i	$0.350610\pi$
$104$	−0.527864	−0.0517613
$105$	0	0
$106$	−1.23607	−0.120058
$107$	−6.76393	−0.653894	−0.326947	−	0.945043i	$-0.606020\pi$
$107$	−6.76393	−0.653894	−0.326947	+	0.945043i	$0.606020\pi$
$108$	0	0
$109$	−14.3820	−1.37754	−0.688771	−	0.724979i	$-0.741850\pi$
$109$	−14.3820	−1.37754	−0.688771	+	0.724979i	$0.741850\pi$
$110$	3.29180	0.313860
$111$	0	0
$112$	−4.14590	−0.391751
$113$	−7.94427	−0.747334	−0.373667	−	0.927563i	$-0.621900\pi$
$113$	−7.94427	−0.747334	−0.373667	+	0.927563i	$0.621900\pi$
$114$	0	0
$115$	−4.76393	−0.444239
$116$	0	0
$117$	0	0
$118$	3.76393	0.346498
$119$	−9.79837	−0.898215
$120$	0	0
$121$	−9.09017	−0.826379
$122$	0.381966	0.0345816
$123$	0	0
$124$	16.3262	1.46614
$125$	−18.7082	−1.67331
$126$	0	0
$127$	15.9443	1.41483	0.707413	−	0.706801i	$-0.249862\pi$
$127$	15.9443	1.41483	0.707413	+	0.706801i	$0.249862\pi$
$128$	11.3820	1.00603
$129$	0	0
$130$	−0.562306	−0.0493175
$131$	−14.3262	−1.25169	−0.625845	−	0.779948i	$-0.715246\pi$
$131$	−14.3262	−1.25169	−0.625845	+	0.779948i	$0.715246\pi$
$132$	0	0
$133$	10.8541	0.941170
$134$	0.944272	0.0815727
$135$	0	0
$136$	9.79837	0.840204
$137$	−7.14590	−0.610515	−0.305258	−	0.952270i	$-0.598743\pi$
$137$	−7.14590	−0.610515	−0.305258	+	0.952270i	$0.598743\pi$
$138$	0	0
$139$	1.29180	0.109569	0.0547844	−	0.998498i	$-0.482553\pi$
$139$	1.29180	0.109569	0.0547844	+	0.998498i	$0.482553\pi$
$140$	−13.9443	−1.17851
$141$	0	0
$142$	6.47214	0.543130
$143$	−0.326238	−0.0272814
$144$	0	0
$145$	0	0
$146$	8.47214	0.701159
$147$	0	0
$148$	−7.61803	−0.626199
$149$	−9.61803	−0.787940	−0.393970	−	0.919123i	$-0.628899\pi$
$149$	−9.61803	−0.787940	−0.393970	+	0.919123i	$0.628899\pi$
$150$	0	0
$151$	2.67376	0.217588	0.108794	−	0.994064i	$-0.465301\pi$
$151$	2.67376	0.217588	0.108794	+	0.994064i	$0.465301\pi$
$152$	−10.8541	−0.880384
$153$	0	0
$154$	1.90983	0.153898
$155$	38.8885	3.12360
$156$	0	0
$157$	14.5623	1.16220	0.581099	−	0.813833i	$-0.302623\pi$
$157$	14.5623	1.16220	0.581099	+	0.813833i	$0.302623\pi$
$158$	3.76393	0.299442
$159$	0	0
$160$	21.6525	1.71178
$161$	−2.76393	−0.217828
$162$	0	0
$163$	−6.03444	−0.472654	−0.236327	−	0.971674i	$-0.575944\pi$
$163$	−6.03444	−0.472654	−0.236327	+	0.971674i	$0.575944\pi$
$164$	6.23607	0.486955
$165$	0	0
$166$	−6.14590	−0.477014
$167$	−10.5279	−0.814671	−0.407335	−	0.913279i	$-0.633542\pi$
$167$	−10.5279	−0.814671	−0.407335	+	0.913279i	$0.633542\pi$
$168$	0	0
$169$	−12.9443	−0.995713
$170$	10.4377	0.800535
$171$	0	0
$172$	11.7082	0.892742
$173$	−4.09017	−0.310970	−0.155485	−	0.987838i	$-0.549694\pi$
$173$	−4.09017	−0.310970	−0.155485	+	0.987838i	$0.549694\pi$
$174$	0	0
$175$	−22.0344	−1.66565
$176$	2.56231	0.193141
$177$	0	0
$178$	2.90983	0.218101
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	5.94427	0.441834	0.220917	−	0.975293i	$-0.429095\pi$
$181$	5.94427	0.441834	0.220917	+	0.975293i	$0.429095\pi$
$182$	−0.326238	−0.0241824
$183$	0	0
$184$	2.76393	0.203760
$185$	−18.1459	−1.33411
$186$	0	0
$187$	6.05573	0.442839
$188$	−11.3262	−0.826051
$189$	0	0
$190$	−11.5623	−0.838818
$191$	−17.0344	−1.23257	−0.616284	−	0.787524i	$-0.711363\pi$
$191$	−17.0344	−1.23257	−0.616284	+	0.787524i	$0.711363\pi$
$192$	0	0
$193$	12.4721	0.897764	0.448882	−	0.893591i	$-0.351822\pi$
$193$	12.4721	0.897764	0.448882	+	0.893591i	$0.351822\pi$
$194$	−2.20163	−0.158068
$195$	0	0
$196$	3.23607	0.231148
$197$	6.29180	0.448272	0.224136	−	0.974558i	$-0.428044\pi$
$197$	6.29180	0.448272	0.224136	+	0.974558i	$0.428044\pi$
$198$	0	0
$199$	−5.85410	−0.414986	−0.207493	−	0.978236i	$-0.566530\pi$
$199$	−5.85410	−0.414986	−0.207493	+	0.978236i	$0.566530\pi$
$200$	22.0344	1.55807
$201$	0	0
$202$	−0.381966	−0.0268750
$203$	0	0
$204$	0	0
$205$	14.8541	1.03746
$206$	−5.67376	−0.395310
$207$	0	0
$208$	−0.437694	−0.0303486
$209$	−6.70820	−0.464016
$210$	0	0
$211$	−11.6525	−0.802190	−0.401095	−	0.916037i	$-0.631370\pi$
$211$	−11.6525	−0.802190	−0.401095	+	0.916037i	$0.631370\pi$
$212$	−3.23607	−0.222254
$213$	0	0
$214$	4.18034	0.285762
$215$	27.8885	1.90198
$216$	0	0
$217$	22.5623	1.53163
$218$	8.88854	0.602008
$219$	0	0
$220$	8.61803	0.581028
$221$	−1.03444	−0.0695841
$222$	0	0
$223$	2.67376	0.179048	0.0895242	−	0.995985i	$-0.471465\pi$
$223$	2.67376	0.179048	0.0895242	+	0.995985i	$0.471465\pi$
$224$	12.5623	0.839354
$225$	0	0
$226$	4.90983	0.326597
$227$	20.8885	1.38642	0.693211	−	0.720735i	$-0.256196\pi$
$227$	20.8885	1.38642	0.693211	+	0.720735i	$0.256196\pi$
$228$	0	0
$229$	2.29180	0.151446	0.0757231	−	0.997129i	$-0.475874\pi$
$229$	2.29180	0.151446	0.0757231	+	0.997129i	$0.475874\pi$
$230$	2.94427	0.194140
$231$	0	0
$232$	0	0
$233$	−15.2361	−0.998148	−0.499074	−	0.866559i	$-0.666326\pi$
$233$	−15.2361	−0.998148	−0.499074	+	0.866559i	$0.666326\pi$
$234$	0	0
$235$	−26.9787	−1.75990
$236$	9.85410	0.641447
$237$	0	0
$238$	6.05573	0.392535
$239$	27.7426	1.79452	0.897261	−	0.441500i	$-0.145553\pi$
$239$	27.7426	1.79452	0.897261	+	0.441500i	$0.145553\pi$
$240$	0	0
$241$	−4.65248	−0.299692	−0.149846	−	0.988709i	$-0.547878\pi$
$241$	−4.65248	−0.299692	−0.149846	+	0.988709i	$0.547878\pi$
$242$	5.61803	0.361141
$243$	0	0
$244$	1.00000	0.0640184
$245$	7.70820	0.492459
$246$	0	0
$247$	1.14590	0.0729117
$248$	−22.5623	−1.43271
$249$	0	0
$250$	11.5623	0.731264
$251$	19.6525	1.24045	0.620227	−	0.784423i	$-0.287041\pi$
$251$	19.6525	1.24045	0.620227	+	0.784423i	$0.287041\pi$
$252$	0	0
$253$	1.70820	0.107394
$254$	−9.85410	−0.618301
$255$	0	0
$256$	−6.56231	−0.410144
$257$	23.1803	1.44595	0.722975	−	0.690874i	$-0.242774\pi$
$257$	23.1803	1.44595	0.722975	+	0.690874i	$0.242774\pi$
$258$	0	0
$259$	−10.5279	−0.654170
$260$	−1.47214	−0.0912980
$261$	0	0
$262$	8.85410	0.547008
$263$	16.7082	1.03027	0.515136	−	0.857108i	$-0.327741\pi$
$263$	16.7082	1.03027	0.515136	+	0.857108i	$0.327741\pi$
$264$	0	0
$265$	−7.70820	−0.473511
$266$	−6.70820	−0.411306
$267$	0	0
$268$	2.47214	0.151010
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	10.1803	0.618412	0.309206	−	0.950995i	$-0.399937\pi$
$271$	10.1803	0.618412	0.309206	+	0.950995i	$0.399937\pi$
$272$	8.12461	0.492627
$273$	0	0
$274$	4.41641	0.266805
$275$	13.6180	0.821198
$276$	0	0
$277$	−21.3820	−1.28472	−0.642359	−	0.766404i	$-0.722044\pi$
$277$	−21.3820	−1.28472	−0.642359	+	0.766404i	$0.722044\pi$
$278$	−0.798374	−0.0478833
$279$	0	0
$280$	19.2705	1.15163
$281$	−23.1246	−1.37950	−0.689749	−	0.724048i	$-0.742279\pi$
$281$	−23.1246	−1.37950	−0.689749	+	0.724048i	$0.742279\pi$
$282$	0	0
$283$	−5.23607	−0.311252	−0.155626	−	0.987816i	$-0.549739\pi$
$283$	−5.23607	−0.311252	−0.155626	+	0.987816i	$0.549739\pi$
$284$	16.9443	1.00546
$285$	0	0
$286$	0.201626	0.0119224
$287$	8.61803	0.508706
$288$	0	0
$289$	2.20163	0.129507
$290$	0	0
$291$	0	0
$292$	22.1803	1.29801
$293$	−8.52786	−0.498203	−0.249102	−	0.968477i	$-0.580135\pi$
$293$	−8.52786	−0.498203	−0.249102	+	0.968477i	$0.580135\pi$
$294$	0	0
$295$	23.4721	1.36660
$296$	10.5279	0.611920
$297$	0	0
$298$	5.94427	0.344342
$299$	−0.291796	−0.0168750
$300$	0	0
$301$	16.1803	0.932619
$302$	−1.65248	−0.0950893
$303$	0	0
$304$	−9.00000	−0.516185
$305$	2.38197	0.136391
$306$	0	0
$307$	−19.1803	−1.09468	−0.547340	−	0.836910i	$-0.684360\pi$
$307$	−19.1803	−1.09468	−0.547340	+	0.836910i	$0.684360\pi$
$308$	5.00000	0.284901
$309$	0	0
$310$	−24.0344	−1.36506
$311$	−2.09017	−0.118523	−0.0592613	−	0.998243i	$-0.518875\pi$
$311$	−2.09017	−0.118523	−0.0592613	+	0.998243i	$0.518875\pi$
$312$	0	0
$313$	−12.9098	−0.729707	−0.364853	−	0.931065i	$-0.618881\pi$
$313$	−12.9098	−0.729707	−0.364853	+	0.931065i	$0.618881\pi$
$314$	−9.00000	−0.507899
$315$	0	0
$316$	9.85410	0.554337
$317$	−27.7082	−1.55625	−0.778124	−	0.628111i	$-0.783828\pi$
$317$	−27.7082	−1.55625	−0.778124	+	0.628111i	$0.783828\pi$
$318$	0	0
$319$	0	0
$320$	0.909830	0.0508610
$321$	0	0
$322$	1.70820	0.0951945
$323$	−21.2705	−1.18352
$324$	0	0
$325$	−2.32624	−0.129036
$326$	3.72949	0.206557
$327$	0	0
$328$	−8.61803	−0.475851
$329$	−15.6525	−0.862949
$330$	0	0
$331$	21.1803	1.16418	0.582088	−	0.813126i	$-0.302236\pi$
$331$	21.1803	1.16418	0.582088	+	0.813126i	$0.302236\pi$
$332$	−16.0902	−0.883063
$333$	0	0
$334$	6.50658	0.356024
$335$	5.88854	0.321726
$336$	0	0
$337$	−34.0689	−1.85585	−0.927925	−	0.372767i	$-0.878409\pi$
$337$	−34.0689	−1.85585	−0.927925	+	0.372767i	$0.878409\pi$
$338$	8.00000	0.435143
$339$	0	0
$340$	27.3262	1.48197
$341$	−13.9443	−0.755125
$342$	0	0
$343$	20.1246	1.08663
$344$	−16.1803	−0.872385
$345$	0	0
$346$	2.52786	0.135899
$347$	32.1246	1.72454	0.862270	−	0.506449i	$-0.169042\pi$
$347$	32.1246	1.72454	0.862270	+	0.506449i	$0.169042\pi$
$348$	0	0
$349$	4.52786	0.242371	0.121186	−	0.992630i	$-0.461330\pi$
$349$	4.52786	0.242371	0.121186	+	0.992630i	$0.461330\pi$
$350$	13.6180	0.727915
$351$	0	0
$352$	−7.76393	−0.413819
$353$	19.1246	1.01790	0.508950	−	0.860796i	$-0.330034\pi$
$353$	19.1246	1.01790	0.508950	+	0.860796i	$0.330034\pi$
$354$	0	0
$355$	40.3607	2.14212
$356$	7.61803	0.403755
$357$	0	0
$358$	−9.88854	−0.522626
$359$	23.7639	1.25421	0.627106	−	0.778934i	$-0.284239\pi$
$359$	23.7639	1.25421	0.627106	+	0.778934i	$0.284239\pi$
$360$	0	0
$361$	4.56231	0.240121
$362$	−3.67376	−0.193089
$363$	0	0
$364$	−0.854102	−0.0447671
$365$	52.8328	2.76540
$366$	0	0
$367$	27.2705	1.42351	0.711755	−	0.702428i	$-0.247901\pi$
$367$	27.2705	1.42351	0.711755	+	0.702428i	$0.247901\pi$
$368$	2.29180	0.119468
$369$	0	0
$370$	11.2148	0.583029
$371$	−4.47214	−0.232182
$372$	0	0
$373$	20.6180	1.06756	0.533781	−	0.845623i	$-0.320771\pi$
$373$	20.6180	1.06756	0.533781	+	0.845623i	$0.320771\pi$
$374$	−3.74265	−0.193528
$375$	0	0
$376$	15.6525	0.807215
$377$	0	0
$378$	0	0
$379$	−24.2918	−1.24779	−0.623893	−	0.781510i	$-0.714450\pi$
$379$	−24.2918	−1.24779	−0.623893	+	0.781510i	$0.714450\pi$
$380$	−30.2705	−1.55284
$381$	0	0
$382$	10.5279	0.538652
$383$	28.8541	1.47438	0.737188	−	0.675688i	$-0.236153\pi$
$383$	28.8541	1.47438	0.737188	+	0.675688i	$0.236153\pi$
$384$	0	0
$385$	11.9098	0.606981
$386$	−7.70820	−0.392337
$387$	0	0
$388$	−5.76393	−0.292619
$389$	19.1246	0.969656	0.484828	−	0.874609i	$-0.338882\pi$
$389$	19.1246	0.969656	0.484828	+	0.874609i	$0.338882\pi$
$390$	0	0
$391$	5.41641	0.273920
$392$	−4.47214	−0.225877
$393$	0	0
$394$	−3.88854	−0.195902
$395$	23.4721	1.18101
$396$	0	0
$397$	−14.0557	−0.705437	−0.352718	−	0.935730i	$-0.614743\pi$
$397$	−14.0557	−0.705437	−0.352718	+	0.935730i	$0.614743\pi$
$398$	3.61803	0.181356
$399$	0	0
$400$	18.2705	0.913525
$401$	−25.0689	−1.25188	−0.625940	−	0.779871i	$-0.715285\pi$
$401$	−25.0689	−1.25188	−0.625940	+	0.779871i	$0.715285\pi$
$402$	0	0
$403$	2.38197	0.118654
$404$	−1.00000	−0.0497519
$405$	0	0
$406$	0	0
$407$	6.50658	0.322519
$408$	0	0
$409$	27.4164	1.35565	0.677827	−	0.735221i	$-0.262922\pi$
$409$	27.4164	1.35565	0.677827	+	0.735221i	$0.262922\pi$
$410$	−9.18034	−0.453385
$411$	0	0
$412$	−14.8541	−0.731809
$413$	13.6180	0.670100
$414$	0	0
$415$	−38.3262	−1.88136
$416$	1.32624	0.0650242
$417$	0	0
$418$	4.14590	0.202783
$419$	17.5623	0.857975	0.428987	−	0.903310i	$-0.358870\pi$
$419$	17.5623	0.857975	0.428987	+	0.903310i	$0.358870\pi$
$420$	0	0
$421$	−31.0344	−1.51253	−0.756263	−	0.654268i	$-0.772977\pi$
$421$	−31.0344	−1.51253	−0.756263	+	0.654268i	$0.772977\pi$
$422$	7.20163	0.350570
$423$	0	0
$424$	4.47214	0.217186
$425$	43.1803	2.09455
$426$	0	0
$427$	1.38197	0.0668780
$428$	10.9443	0.529011
$429$	0	0
$430$	−17.2361	−0.831197
$431$	−14.5967	−0.703101	−0.351550	−	0.936169i	$-0.614345\pi$
$431$	−14.5967	−0.703101	−0.351550	+	0.936169i	$0.614345\pi$
$432$	0	0
$433$	10.3820	0.498925	0.249463	−	0.968384i	$-0.419746\pi$
$433$	10.3820	0.498925	0.249463	+	0.968384i	$0.419746\pi$
$434$	−13.9443	−0.669346
$435$	0	0
$436$	23.2705	1.11446
$437$	−6.00000	−0.287019
$438$	0	0
$439$	−20.9443	−0.999616	−0.499808	−	0.866136i	$-0.666596\pi$
$439$	−20.9443	−0.999616	−0.499808	+	0.866136i	$0.666596\pi$
$440$	−11.9098	−0.567779
$441$	0	0
$442$	0.639320	0.0304094
$443$	1.90983	0.0907388	0.0453694	−	0.998970i	$-0.485554\pi$
$443$	1.90983	0.0907388	0.0453694	+	0.998970i	$0.485554\pi$
$444$	0	0
$445$	18.1459	0.860198
$446$	−1.65248	−0.0782470
$447$	0	0
$448$	0.527864	0.0249392
$449$	26.1246	1.23290	0.616448	−	0.787395i	$-0.288571\pi$
$449$	26.1246	1.23290	0.616448	+	0.787395i	$0.288571\pi$
$450$	0	0
$451$	−5.32624	−0.250803
$452$	12.8541	0.604606
$453$	0	0
$454$	−12.9098	−0.605888
$455$	−2.03444	−0.0953761
$456$	0	0
$457$	18.7082	0.875133	0.437566	−	0.899186i	$-0.355840\pi$
$457$	18.7082	0.875133	0.437566	+	0.899186i	$0.355840\pi$
$458$	−1.41641	−0.0661844
$459$	0	0
$460$	7.70820	0.359397
$461$	38.9787	1.81542	0.907710	−	0.419598i	$-0.137829\pi$
$461$	38.9787	1.81542	0.907710	+	0.419598i	$0.137829\pi$
$462$	0	0
$463$	10.7082	0.497652	0.248826	−	0.968548i	$-0.419955\pi$
$463$	10.7082	0.497652	0.248826	+	0.968548i	$0.419955\pi$
$464$	0	0
$465$	0	0
$466$	9.41641	0.436207
$467$	17.9443	0.830362	0.415181	−	0.909739i	$-0.363718\pi$
$467$	17.9443	0.830362	0.415181	+	0.909739i	$0.363718\pi$
$468$	0	0
$469$	3.41641	0.157755
$470$	16.6738	0.769103
$471$	0	0
$472$	−13.6180	−0.626821
$473$	−10.0000	−0.459800
$474$	0	0
$475$	−47.8328	−2.19472
$476$	15.8541	0.726672
$477$	0	0
$478$	−17.1459	−0.784235
$479$	11.1803	0.510843	0.255421	−	0.966830i	$-0.417786\pi$
$479$	11.1803	0.510843	0.255421	+	0.966830i	$0.417786\pi$
$480$	0	0
$481$	−1.11146	−0.0506780
$482$	2.87539	0.130970
$483$	0	0
$484$	14.7082	0.668555
$485$	−13.7295	−0.623424
$486$	0	0
$487$	42.5623	1.92868	0.964341	−	0.264663i	$-0.0852606\pi$
$487$	42.5623	1.92868	0.964341	+	0.264663i	$0.0852606\pi$
$488$	−1.38197	−0.0625587
$489$	0	0
$490$	−4.76393	−0.215213
$491$	15.1246	0.682564	0.341282	−	0.939961i	$-0.389139\pi$
$491$	15.1246	0.682564	0.341282	+	0.939961i	$0.389139\pi$
$492$	0	0
$493$	0	0
$494$	−0.708204	−0.0318636
$495$	0	0
$496$	−18.7082	−0.840023
$497$	23.4164	1.05037
$498$	0	0
$499$	24.6869	1.10514	0.552569	−	0.833467i	$-0.313648\pi$
$499$	24.6869	1.10514	0.552569	+	0.833467i	$0.313648\pi$
$500$	30.2705	1.35374
$501$	0	0
$502$	−12.1459	−0.542098
$503$	−14.2705	−0.636291	−0.318145	−	0.948042i	$-0.603060\pi$
$503$	−14.2705	−0.636291	−0.318145	+	0.948042i	$0.603060\pi$
$504$	0	0
$505$	−2.38197	−0.105996
$506$	−1.05573	−0.0469328
$507$	0	0
$508$	−25.7984	−1.14462
$509$	−31.5623	−1.39897	−0.699487	−	0.714645i	$-0.746588\pi$
$509$	−31.5623	−1.39897	−0.699487	+	0.714645i	$0.746588\pi$
$510$	0	0
$511$	30.6525	1.35599
$512$	−18.7082	−0.826794
$513$	0	0
$514$	−14.3262	−0.631903
$515$	−35.3820	−1.55912
$516$	0	0
$517$	9.67376	0.425452
$518$	6.50658	0.285883
$519$	0	0
$520$	2.03444	0.0892162
$521$	−4.09017	−0.179194	−0.0895968	−	0.995978i	$-0.528558\pi$
$521$	−4.09017	−0.179194	−0.0895968	+	0.995978i	$0.528558\pi$
$522$	0	0
$523$	−20.3820	−0.891241	−0.445621	−	0.895222i	$-0.647017\pi$
$523$	−20.3820	−0.891241	−0.445621	+	0.895222i	$0.647017\pi$
$524$	23.1803	1.01264
$525$	0	0
$526$	−10.3262	−0.450245
$527$	−44.2148	−1.92603
$528$	0	0
$529$	−21.4721	−0.933571
$530$	4.76393	0.206932
$531$	0	0
$532$	−17.5623	−0.761423
$533$	0.909830	0.0394091
$534$	0	0
$535$	26.0689	1.12706
$536$	−3.41641	−0.147566
$537$	0	0
$538$	−3.70820	−0.159872
$539$	−2.76393	−0.119051
$540$	0	0
$541$	−14.5967	−0.627563	−0.313782	−	0.949495i	$-0.601596\pi$
$541$	−14.5967	−0.627563	−0.313782	+	0.949495i	$0.601596\pi$
$542$	−6.29180	−0.270256
$543$	0	0
$544$	−24.6180	−1.05549
$545$	55.4296	2.37434
$546$	0	0
$547$	7.38197	0.315630	0.157815	−	0.987469i	$-0.449555\pi$
$547$	7.38197	0.315630	0.157815	+	0.987469i	$0.449555\pi$
$548$	11.5623	0.493917
$549$	0	0
$550$	−8.41641	−0.358877
$551$	0	0
$552$	0	0
$553$	13.6180	0.579098
$554$	13.2148	0.561442
$555$	0	0
$556$	−2.09017	−0.0886430
$557$	5.50658	0.233321	0.116661	−	0.993172i	$-0.462781\pi$
$557$	5.50658	0.233321	0.116661	+	0.993172i	$0.462781\pi$
$558$	0	0
$559$	1.70820	0.0722493
$560$	15.9787	0.675224
$561$	0	0
$562$	14.2918	0.602863
$563$	−28.3951	−1.19671	−0.598356	−	0.801230i	$-0.704179\pi$
$563$	−28.3951	−1.19671	−0.598356	+	0.801230i	$0.704179\pi$
$564$	0	0
$565$	30.6180	1.28811
$566$	3.23607	0.136022
$567$	0	0
$568$	−23.4164	−0.982531
$569$	1.94427	0.0815081	0.0407541	−	0.999169i	$-0.487024\pi$
$569$	1.94427	0.0815081	0.0407541	+	0.999169i	$0.487024\pi$
$570$	0	0
$571$	−34.5066	−1.44406	−0.722028	−	0.691864i	$-0.756790\pi$
$571$	−34.5066	−1.44406	−0.722028	+	0.691864i	$0.756790\pi$
$572$	0.527864	0.0220711
$573$	0	0
$574$	−5.32624	−0.222313
$575$	12.1803	0.507955
$576$	0	0
$577$	−2.76393	−0.115064	−0.0575320	−	0.998344i	$-0.518323\pi$
$577$	−2.76393	−0.115064	−0.0575320	+	0.998344i	$0.518323\pi$
$578$	−1.36068	−0.0565968
$579$	0	0
$580$	0	0
$581$	−22.2361	−0.922508
$582$	0	0
$583$	2.76393	0.114470
$584$	−30.6525	−1.26841
$585$	0	0
$586$	5.27051	0.217723
$587$	−46.6180	−1.92413	−0.962066	−	0.272816i	$-0.912045\pi$
$587$	−46.6180	−1.92413	−0.962066	+	0.272816i	$0.912045\pi$
$588$	0	0
$589$	48.9787	2.01813
$590$	−14.5066	−0.597226
$591$	0	0
$592$	8.72949	0.358780
$593$	14.4377	0.592885	0.296443	−	0.955051i	$-0.404200\pi$
$593$	14.4377	0.592885	0.296443	+	0.955051i	$0.404200\pi$
$594$	0	0
$595$	37.7639	1.54817
$596$	15.5623	0.637457
$597$	0	0
$598$	0.180340	0.00737465
$599$	−13.0689	−0.533980	−0.266990	−	0.963699i	$-0.586029\pi$
$599$	−13.0689	−0.533980	−0.266990	+	0.963699i	$0.586029\pi$
$600$	0	0
$601$	29.1591	1.18942	0.594711	−	0.803939i	$-0.297266\pi$
$601$	29.1591	1.18942	0.594711	+	0.803939i	$0.297266\pi$
$602$	−10.0000	−0.407570
$603$	0	0
$604$	−4.32624	−0.176032
$605$	35.0344	1.42435
$606$	0	0
$607$	10.9787	0.445612	0.222806	−	0.974863i	$-0.428478\pi$
$607$	10.9787	0.445612	0.222806	+	0.974863i	$0.428478\pi$
$608$	27.2705	1.10597
$609$	0	0
$610$	−1.47214	−0.0596050
$611$	−1.65248	−0.0668520
$612$	0	0
$613$	27.5410	1.11237	0.556186	−	0.831058i	$-0.312264\pi$
$613$	27.5410	1.11237	0.556186	+	0.831058i	$0.312264\pi$
$614$	11.8541	0.478393
$615$	0	0
$616$	−6.90983	−0.278405
$617$	14.1803	0.570879	0.285439	−	0.958397i	$-0.407860\pi$
$617$	14.1803	0.570879	0.285439	+	0.958397i	$0.407860\pi$
$618$	0	0
$619$	7.05573	0.283594	0.141797	−	0.989896i	$-0.454712\pi$
$619$	7.05573	0.283594	0.141797	+	0.989896i	$0.454712\pi$
$620$	−62.9230	−2.52705
$621$	0	0
$622$	1.29180	0.0517963
$623$	10.5279	0.421790
$624$	0	0
$625$	22.8328	0.913313
$626$	7.97871	0.318894
$627$	0	0
$628$	−23.5623	−0.940238
$629$	20.6312	0.822619
$630$	0	0
$631$	−28.2148	−1.12321	−0.561606	−	0.827405i	$-0.689816\pi$
$631$	−28.2148	−1.12321	−0.561606	+	0.827405i	$0.689816\pi$
$632$	−13.6180	−0.541696
$633$	0	0
$634$	17.1246	0.680105
$635$	−61.4508	−2.43860
$636$	0	0
$637$	0.472136	0.0187067
$638$	0	0
$639$	0	0
$640$	−43.8673	−1.73401
$641$	−11.0557	−0.436675	−0.218338	−	0.975873i	$-0.570063\pi$
$641$	−11.0557	−0.436675	−0.218338	+	0.975873i	$0.570063\pi$
$642$	0	0
$643$	−37.4164	−1.47556	−0.737780	−	0.675042i	$-0.764126\pi$
$643$	−37.4164	−1.47556	−0.737780	+	0.675042i	$0.764126\pi$
$644$	4.47214	0.176227
$645$	0	0
$646$	13.1459	0.517218
$647$	30.5279	1.20017	0.600087	−	0.799935i	$-0.295133\pi$
$647$	30.5279	1.20017	0.600087	+	0.799935i	$0.295133\pi$
$648$	0	0
$649$	−8.41641	−0.330373
$650$	1.43769	0.0563910
$651$	0	0
$652$	9.76393	0.382385
$653$	−48.0132	−1.87890	−0.939450	−	0.342686i	$-0.888663\pi$
$653$	−48.0132	−1.87890	−0.939450	+	0.342686i	$0.888663\pi$
$654$	0	0
$655$	55.2148	2.15742
$656$	−7.14590	−0.279000
$657$	0	0
$658$	9.67376	0.377123
$659$	7.05573	0.274852	0.137426	−	0.990512i	$-0.456117\pi$
$659$	7.05573	0.274852	0.137426	+	0.990512i	$0.456117\pi$
$660$	0	0
$661$	−37.4508	−1.45667	−0.728335	−	0.685222i	$-0.759705\pi$
$661$	−37.4508	−1.45667	−0.728335	+	0.685222i	$0.759705\pi$
$662$	−13.0902	−0.508764
$663$	0	0
$664$	22.2361	0.862927
$665$	−41.8328	−1.62221
$666$	0	0
$667$	0	0
$668$	17.0344	0.659082
$669$	0	0
$670$	−3.63932	−0.140599
$671$	−0.854102	−0.0329722
$672$	0	0
$673$	−6.47214	−0.249483	−0.124741	−	0.992189i	$-0.539810\pi$
$673$	−6.47214	−0.249483	−0.124741	+	0.992189i	$0.539810\pi$
$674$	21.0557	0.811036
$675$	0	0
$676$	20.9443	0.805549
$677$	40.8328	1.56933	0.784666	−	0.619918i	$-0.212834\pi$
$677$	40.8328	1.56933	0.784666	+	0.619918i	$0.212834\pi$
$678$	0	0
$679$	−7.96556	−0.305690
$680$	−37.7639	−1.44818
$681$	0	0
$682$	8.61803	0.330002
$683$	−20.8541	−0.797960	−0.398980	−	0.916960i	$-0.630636\pi$
$683$	−20.8541	−0.797960	−0.398980	+	0.916960i	$0.630636\pi$
$684$	0	0
$685$	27.5410	1.05229
$686$	−12.4377	−0.474873
$687$	0	0
$688$	−13.4164	−0.511496
$689$	−0.472136	−0.0179869
$690$	0	0
$691$	11.8328	0.450142	0.225071	−	0.974342i	$-0.427739\pi$
$691$	11.8328	0.450142	0.225071	+	0.974342i	$0.427739\pi$
$692$	6.61803	0.251580
$693$	0	0
$694$	−19.8541	−0.753651
$695$	−4.97871	−0.188853
$696$	0	0
$697$	−16.8885	−0.639699
$698$	−2.79837	−0.105920
$699$	0	0
$700$	35.6525	1.34754
$701$	21.0557	0.795264	0.397632	−	0.917545i	$-0.369832\pi$
$701$	21.0557	0.795264	0.397632	+	0.917545i	$0.369832\pi$
$702$	0	0
$703$	−22.8541	−0.861959
$704$	−0.326238	−0.0122956
$705$	0	0
$706$	−11.8197	−0.444839
$707$	−1.38197	−0.0519742
$708$	0	0
$709$	−41.5066	−1.55881	−0.779406	−	0.626519i	$-0.784479\pi$
$709$	−41.5066	−1.55881	−0.779406	+	0.626519i	$0.784479\pi$
$710$	−24.9443	−0.936142
$711$	0	0
$712$	−10.5279	−0.394548
$713$	−12.4721	−0.467085
$714$	0	0
$715$	1.25735	0.0470224
$716$	−25.8885	−0.967500
$717$	0	0
$718$	−14.6869	−0.548111
$719$	−8.50658	−0.317242	−0.158621	−	0.987340i	$-0.550705\pi$
$719$	−8.50658	−0.317242	−0.158621	+	0.987340i	$0.550705\pi$
$720$	0	0
$721$	−20.5279	−0.764498
$722$	−2.81966	−0.104937
$723$	0	0
$724$	−9.61803	−0.357451
$725$	0	0
$726$	0	0
$727$	28.0557	1.04053	0.520265	−	0.854005i	$-0.325833\pi$
$727$	28.0557	1.04053	0.520265	+	0.854005i	$0.325833\pi$
$728$	1.18034	0.0437463
$729$	0	0
$730$	−32.6525	−1.20852
$731$	−31.7082	−1.17277
$732$	0	0
$733$	14.8197	0.547377	0.273688	−	0.961818i	$-0.411756\pi$
$733$	14.8197	0.547377	0.273688	+	0.961818i	$0.411756\pi$
$734$	−16.8541	−0.622096
$735$	0	0
$736$	−6.94427	−0.255969
$737$	−2.11146	−0.0777765
$738$	0	0
$739$	−50.0689	−1.84181	−0.920907	−	0.389783i	$-0.872550\pi$
$739$	−50.0689	−1.84181	−0.920907	+	0.389783i	$0.872550\pi$
$740$	29.3607	1.07932
$741$	0	0
$742$	2.76393	0.101467
$743$	−35.2361	−1.29269	−0.646343	−	0.763047i	$-0.723702\pi$
$743$	−35.2361	−1.29269	−0.646343	+	0.763047i	$0.723702\pi$
$744$	0	0
$745$	37.0689	1.35810
$746$	−12.7426	−0.466541
$747$	0	0
$748$	−9.79837	−0.358264
$749$	15.1246	0.552641
$750$	0	0
$751$	−18.5279	−0.676091	−0.338046	−	0.941130i	$-0.609766\pi$
$751$	−18.5279	−0.676091	−0.338046	+	0.941130i	$0.609766\pi$
$752$	12.9787	0.473285
$753$	0	0
$754$	0	0
$755$	−10.3050	−0.375036
$756$	0	0
$757$	−0.0212862	−0.000773661 0	−0.000386831	−	1.00000i	$-0.500123\pi$
$757$	−0.0212862	−0.000773661 0	−0.000386831		1.00000i	$0.500123\pi$
$758$	15.0132	0.545302
$759$	0	0
$760$	41.8328	1.51744
$761$	−25.2016	−0.913558	−0.456779	−	0.889580i	$-0.650997\pi$
$761$	−25.2016	−0.913558	−0.456779	+	0.889580i	$0.650997\pi$
$762$	0	0
$763$	32.1591	1.16424
$764$	27.5623	0.997169
$765$	0	0
$766$	−17.8328	−0.644326
$767$	1.43769	0.0519121
$768$	0	0
$769$	−19.3607	−0.698164	−0.349082	−	0.937092i	$-0.613507\pi$
$769$	−19.3607	−0.698164	−0.349082	+	0.937092i	$0.613507\pi$
$770$	−7.36068	−0.265260
$771$	0	0
$772$	−20.1803	−0.726306
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	−99.4296	−3.57162
$776$	7.96556	0.285947
$777$	0	0
$778$	−11.8197	−0.423755
$779$	18.7082	0.670291
$780$	0	0
$781$	−14.4721	−0.517854
$782$	−3.34752	−0.119707
$783$	0	0
$784$	−3.70820	−0.132436
$785$	−56.1246	−2.00317
$786$	0	0
$787$	19.3607	0.690134	0.345067	−	0.938578i	$-0.387856\pi$
$787$	19.3607	0.690134	0.345067	+	0.938578i	$0.387856\pi$
$788$	−10.1803	−0.362660
$789$	0	0
$790$	−14.5066	−0.516121
$791$	17.7639	0.631613
$792$	0	0
$793$	0.145898	0.00518099
$794$	8.68692	0.308287
$795$	0	0
$796$	9.47214	0.335731
$797$	−43.1803	−1.52953	−0.764763	−	0.644312i	$-0.777144\pi$
$797$	−43.1803	−1.52953	−0.764763	+	0.644312i	$0.777144\pi$
$798$	0	0
$799$	30.6738	1.08516
$800$	−55.3607	−1.95730
$801$	0	0
$802$	15.4934	0.547092
$803$	−18.9443	−0.668529
$804$	0	0
$805$	10.6525	0.375450
$806$	−1.47214	−0.0518538
$807$	0	0
$808$	1.38197	0.0486174
$809$	43.0689	1.51422	0.757111	−	0.653287i	$-0.226610\pi$
$809$	43.0689	1.51422	0.757111	+	0.653287i	$0.226610\pi$
$810$	0	0
$811$	5.65248	0.198485	0.0992426	−	0.995063i	$-0.468358\pi$
$811$	5.65248	0.198485	0.0992426	+	0.995063i	$0.468358\pi$
$812$	0	0
$813$	0	0
$814$	−4.02129	−0.140946
$815$	23.2574	0.814670
$816$	0	0
$817$	35.1246	1.22885
$818$	−16.9443	−0.592443
$819$	0	0
$820$	−24.0344	−0.839319
$821$	−8.58359	−0.299569	−0.149785	−	0.988719i	$-0.547858\pi$
$821$	−8.58359	−0.299569	−0.149785	+	0.988719i	$0.547858\pi$
$822$	0	0
$823$	−5.47214	−0.190747	−0.0953733	−	0.995442i	$-0.530404\pi$
$823$	−5.47214	−0.190747	−0.0953733	+	0.995442i	$0.530404\pi$
$824$	20.5279	0.715122
$825$	0	0
$826$	−8.41641	−0.292844
$827$	−26.9656	−0.937684	−0.468842	−	0.883282i	$-0.655329\pi$
$827$	−26.9656	−0.937684	−0.468842	+	0.883282i	$0.655329\pi$
$828$	0	0
$829$	32.7984	1.13913	0.569567	−	0.821945i	$-0.307111\pi$
$829$	32.7984	1.13913	0.569567	+	0.821945i	$0.307111\pi$
$830$	23.6869	0.822185
$831$	0	0
$832$	0.0557281	0.00193202
$833$	−8.76393	−0.303652
$834$	0	0
$835$	40.5755	1.40417
$836$	10.8541	0.375397
$837$	0	0
$838$	−10.8541	−0.374949
$839$	48.2148	1.66456	0.832280	−	0.554356i	$-0.187035\pi$
$839$	48.2148	1.66456	0.832280	+	0.554356i	$0.187035\pi$
$840$	0	0
$841$	0	0
$842$	19.1803	0.660998
$843$	0	0
$844$	18.8541	0.648985
$845$	49.8885	1.71622
$846$	0	0
$847$	20.3262	0.698418
$848$	3.70820	0.127340
$849$	0	0
$850$	−26.6869	−0.915354
$851$	5.81966	0.199495
$852$	0	0
$853$	−45.0000	−1.54077	−0.770385	−	0.637579i	$-0.779936\pi$
$853$	−45.0000	−1.54077	−0.770385	+	0.637579i	$0.779936\pi$
$854$	−0.854102	−0.0292268
$855$	0	0
$856$	−15.1246	−0.516949
$857$	45.3262	1.54831	0.774157	−	0.632993i	$-0.218174\pi$
$857$	45.3262	1.54831	0.774157	+	0.632993i	$0.218174\pi$
$858$	0	0
$859$	32.7082	1.11599	0.557995	−	0.829844i	$-0.311571\pi$
$859$	32.7082	1.11599	0.557995	+	0.829844i	$0.311571\pi$
$860$	−45.1246	−1.53874
$861$	0	0
$862$	9.02129	0.307266
$863$	−17.7426	−0.603967	−0.301983	−	0.953313i	$-0.597649\pi$
$863$	−17.7426	−0.603967	−0.301983	+	0.953313i	$0.597649\pi$
$864$	0	0
$865$	15.7639	0.535990
$866$	−6.41641	−0.218038
$867$	0	0
$868$	−36.5066	−1.23911
$869$	−8.41641	−0.285507
$870$	0	0
$871$	0.360680	0.0122212
$872$	−32.1591	−1.08904
$873$	0	0
$874$	3.70820	0.125432
$875$	41.8328	1.41421
$876$	0	0
$877$	4.38197	0.147968	0.0739842	−	0.997259i	$-0.476429\pi$
$877$	4.38197	0.147968	0.0739842	+	0.997259i	$0.476429\pi$
$878$	12.9443	0.436848
$879$	0	0
$880$	−9.87539	−0.332899
$881$	−4.27051	−0.143877	−0.0719386	−	0.997409i	$-0.522919\pi$
$881$	−4.27051	−0.143877	−0.0719386	+	0.997409i	$0.522919\pi$
$882$	0	0
$883$	−20.6869	−0.696170	−0.348085	−	0.937463i	$-0.613168\pi$
$883$	−20.6869	−0.696170	−0.348085	+	0.937463i	$0.613168\pi$
$884$	1.67376	0.0562947
$885$	0	0
$886$	−1.18034	−0.0396543
$887$	38.0689	1.27823	0.639114	−	0.769112i	$-0.279301\pi$
$887$	38.0689	1.27823	0.639114	+	0.769112i	$0.279301\pi$
$888$	0	0
$889$	−35.6525	−1.19575
$890$	−11.2148	−0.375920
$891$	0	0
$892$	−4.32624	−0.144853
$893$	−33.9787	−1.13705
$894$	0	0
$895$	−61.6656	−2.06125
$896$	−25.4508	−0.850253
$897$	0	0
$898$	−16.1459	−0.538796
$899$	0	0
$900$	0	0
$901$	8.76393	0.291969
$902$	3.29180	0.109605
$903$	0	0
$904$	−17.7639	−0.590820
$905$	−22.9098	−0.761549
$906$	0	0
$907$	9.76393	0.324206	0.162103	−	0.986774i	$-0.448172\pi$
$907$	9.76393	0.324206	0.162103	+	0.986774i	$0.448172\pi$
$908$	−33.7984	−1.12164
$909$	0	0
$910$	1.25735	0.0416809
$911$	10.9443	0.362600	0.181300	−	0.983428i	$-0.441970\pi$
$911$	10.9443	0.362600	0.181300	+	0.983428i	$0.441970\pi$
$912$	0	0
$913$	13.7426	0.454815
$914$	−11.5623	−0.382447
$915$	0	0
$916$	−3.70820	−0.122523
$917$	32.0344	1.05787
$918$	0	0
$919$	31.3050	1.03266	0.516328	−	0.856391i	$-0.327299\pi$
$919$	31.3050	1.03266	0.516328	+	0.856391i	$0.327299\pi$
$920$	−10.6525	−0.351202
$921$	0	0
$922$	−24.0902	−0.793367
$923$	2.47214	0.0813713
$924$	0	0
$925$	46.3951	1.52546
$926$	−6.61803	−0.217482
$927$	0	0
$928$	0	0
$929$	3.65248	0.119834	0.0599169	−	0.998203i	$-0.480916\pi$
$929$	3.65248	0.119834	0.0599169	+	0.998203i	$0.480916\pi$
$930$	0	0
$931$	9.70820	0.318174
$932$	24.6525	0.807519
$933$	0	0
$934$	−11.0902	−0.362881
$935$	−23.3394	−0.763280
$936$	0	0
$937$	6.65248	0.217327	0.108663	−	0.994079i	$-0.465343\pi$
$937$	6.65248	0.217327	0.108663	+	0.994079i	$0.465343\pi$
$938$	−2.11146	−0.0689415
$939$	0	0
$940$	43.6525	1.42379
$941$	34.8885	1.13733	0.568667	−	0.822568i	$-0.307459\pi$
$941$	34.8885	1.13733	0.568667	+	0.822568i	$0.307459\pi$
$942$	0	0
$943$	−4.76393	−0.155135
$944$	−11.2918	−0.367517
$945$	0	0
$946$	6.18034	0.200940
$947$	−43.0344	−1.39843	−0.699216	−	0.714911i	$-0.746467\pi$
$947$	−43.0344	−1.39843	−0.699216	+	0.714911i	$0.746467\pi$
$948$	0	0
$949$	3.23607	0.105047
$950$	29.5623	0.959128
$951$	0	0
$952$	−21.9098	−0.710102
$953$	−42.6312	−1.38096	−0.690480	−	0.723352i	$-0.742601\pi$
$953$	−42.6312	−1.38096	−0.690480	+	0.723352i	$0.742601\pi$
$954$	0	0
$955$	65.6525	2.12446
$956$	−44.8885	−1.45180
$957$	0	0
$958$	−6.90983	−0.223246
$959$	15.9787	0.515980
$960$	0	0
$961$	70.8115	2.28424
$962$	0.686918	0.0221471
$963$	0	0
$964$	7.52786	0.242456
$965$	−48.0689	−1.54739
$966$	0	0
$967$	3.56231	0.114556	0.0572780	−	0.998358i	$-0.481758\pi$
$967$	3.56231	0.114556	0.0572780	+	0.998358i	$0.481758\pi$
$968$	−20.3262	−0.653310
$969$	0	0
$970$	8.48529	0.272446
$971$	19.5066	0.625996	0.312998	−	0.949754i	$-0.398667\pi$
$971$	19.5066	0.625996	0.312998	+	0.949754i	$0.398667\pi$
$972$	0	0
$973$	−2.88854	−0.0926025
$974$	−26.3050	−0.842865
$975$	0	0
$976$	−1.14590	−0.0366793
$977$	45.2148	1.44655	0.723275	−	0.690561i	$-0.242636\pi$
$977$	45.2148	1.44655	0.723275	+	0.690561i	$0.242636\pi$
$978$	0	0
$979$	−6.50658	−0.207951
$980$	−12.4721	−0.398408
$981$	0	0
$982$	−9.34752	−0.298291
$983$	10.9443	0.349068	0.174534	−	0.984651i	$-0.444158\pi$
$983$	10.9443	0.349068	0.174534	+	0.984651i	$0.444158\pi$
$984$	0	0
$985$	−24.2492	−0.772645
$986$	0	0
$987$	0	0
$988$	−1.85410	−0.0589868
$989$	−8.94427	−0.284411
$990$	0	0
$991$	7.34752	0.233402	0.116701	−	0.993167i	$-0.462768\pi$
$991$	7.34752	0.233402	0.116701	+	0.993167i	$0.462768\pi$
$992$	56.6869	1.79981
$993$	0	0
$994$	−14.4721	−0.459028
$995$	22.5623	0.715273
$996$	0	0
$997$	16.9098	0.535540	0.267770	−	0.963483i	$-0.413713\pi$
$997$	16.9098	0.535540	0.267770	+	0.963483i	$0.413713\pi$
$998$	−15.2574	−0.482963
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.l.1.1		2
3.2	odd	2		841.2.a.a.1.2	✓	2
29.28	even	2		7569.2.a.d.1.2		2
87.2	even	28		841.2.e.j.236.3		24
87.5	odd	14		841.2.d.g.605.2		12
87.8	even	28		841.2.e.j.267.3		24
87.11	even	28		841.2.e.j.63.3		24
87.14	even	28		841.2.e.j.196.2		24
87.17	even	4		841.2.b.b.840.2		4
87.20	odd	14		841.2.d.i.574.2		12
87.23	odd	14		841.2.d.i.645.1		12
87.26	even	28		841.2.e.j.270.2		24
87.32	even	28		841.2.e.j.270.3		24
87.35	odd	14		841.2.d.g.645.2		12
87.38	odd	14		841.2.d.g.574.1		12
87.41	even	4		841.2.b.b.840.3		4
87.44	even	28		841.2.e.j.196.3		24
87.47	even	28		841.2.e.j.63.2		24
87.50	even	28		841.2.e.j.267.2		24
87.53	odd	14		841.2.d.i.605.1		12
87.56	even	28		841.2.e.j.236.2		24
87.62	odd	14		841.2.d.g.190.2		12
87.65	odd	14		841.2.d.i.571.1		12
87.68	even	28		841.2.e.j.651.3		24
87.71	odd	14		841.2.d.g.778.1		12
87.74	odd	14		841.2.d.i.778.2		12
87.77	even	28		841.2.e.j.651.2		24
87.80	odd	14		841.2.d.g.571.2		12
87.83	odd	14		841.2.d.i.190.1		12
87.86	odd	2		841.2.a.c.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
841.2.a.a.1.2	✓	2	3.2	odd	2
841.2.a.c.1.1	yes	2	87.86	odd	2
841.2.b.b.840.2		4	87.17	even	4
841.2.b.b.840.3		4	87.41	even	4
841.2.d.g.190.2		12	87.62	odd	14
841.2.d.g.571.2		12	87.80	odd	14
841.2.d.g.574.1		12	87.38	odd	14
841.2.d.g.605.2		12	87.5	odd	14
841.2.d.g.645.2		12	87.35	odd	14
841.2.d.g.778.1		12	87.71	odd	14
841.2.d.i.190.1		12	87.83	odd	14
841.2.d.i.571.1		12	87.65	odd	14
841.2.d.i.574.2		12	87.20	odd	14
841.2.d.i.605.1		12	87.53	odd	14
841.2.d.i.645.1		12	87.23	odd	14
841.2.d.i.778.2		12	87.74	odd	14
841.2.e.j.63.2		24	87.47	even	28
841.2.e.j.63.3		24	87.11	even	28
841.2.e.j.196.2		24	87.14	even	28
841.2.e.j.196.3		24	87.44	even	28
841.2.e.j.236.2		24	87.56	even	28
841.2.e.j.236.3		24	87.2	even	28
841.2.e.j.267.2		24	87.50	even	28
841.2.e.j.267.3		24	87.8	even	28
841.2.e.j.270.2		24	87.26	even	28
841.2.e.j.270.3		24	87.32	even	28
841.2.e.j.651.2		24	87.77	even	28
841.2.e.j.651.3		24	87.68	even	28
7569.2.a.d.1.2		2	29.28	even	2
7569.2.a.l.1.1		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.61803 q^{4} -3.85410 q^{5} -2.23607 q^{7} +2.23607 q^{8} +2.38197 q^{10} +1.38197 q^{11} -0.236068 q^{13} +1.38197 q^{14} +1.85410 q^{16} +4.38197 q^{17} -4.85410 q^{19} +6.23607 q^{20} -0.854102 q^{22} +1.23607 q^{23} +9.85410 q^{25} +0.145898 q^{26} +3.61803 q^{28} -10.0902 q^{31} -5.61803 q^{32} -2.70820 q^{34} +8.61803 q^{35} +4.70820 q^{37} +3.00000 q^{38} -8.61803 q^{40} -3.85410 q^{41} -7.23607 q^{43} -2.23607 q^{44} -0.763932 q^{46} +7.00000 q^{47} -2.00000 q^{49} -6.09017 q^{50} +0.381966 q^{52} +2.00000 q^{53} -5.32624 q^{55} -5.00000 q^{56} -6.09017 q^{59} -0.618034 q^{61} +6.23607 q^{62} -0.236068 q^{64} +0.909830 q^{65} -1.52786 q^{67} -7.09017 q^{68} -5.32624 q^{70} -10.4721 q^{71} -13.7082 q^{73} -2.90983 q^{74} +7.85410 q^{76} -3.09017 q^{77} -6.09017 q^{79} -7.14590 q^{80} +2.38197 q^{82} +9.94427 q^{83} -16.8885 q^{85} +4.47214 q^{86} +3.09017 q^{88} -4.70820 q^{89} +0.527864 q^{91} -2.00000 q^{92} -4.32624 q^{94} +18.7082 q^{95} +3.56231 q^{97} +1.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - q^{5} + 7 q^{10} + 5 q^{11} + 4 q^{13} + 5 q^{14} - 3 q^{16} + 11 q^{17} - 3 q^{19} + 8 q^{20} + 5 q^{22} - 2 q^{23} + 13 q^{25} + 7 q^{26} + 5 q^{28} - 9 q^{31} - 9 q^{32} + 8 q^{34}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - q^5 + 7 * q^10 + 5 * q^11 + 4 * q^13 + 5 * q^14 - 3 * q^16 + 11 * q^17 - 3 * q^19 + 8 * q^20 + 5 * q^22 - 2 * q^23 + 13 * q^25 + 7 * q^26 + 5 * q^28 - 9 * q^31 - 9 * q^32 + 8 * q^34 + 15 * q^35 - 4 * q^37 + 6 * q^38 - 15 * q^40 - q^41 - 10 * q^43 - 6 * q^46 + 14 * q^47 - 4 * q^49 - q^50 + 3 * q^52 + 4 * q^53 + 5 * q^55 - 10 * q^56 - q^59 + q^61 + 8 * q^62 + 4 * q^64 + 13 * q^65 - 12 * q^67 - 3 * q^68 + 5 * q^70 - 12 * q^71 - 14 * q^73 - 17 * q^74 + 9 * q^76 + 5 * q^77 - q^79 - 21 * q^80 + 7 * q^82 + 2 * q^83 + 2 * q^85 - 5 * q^88 + 4 * q^89 + 10 * q^91 - 4 * q^92 + 7 * q^94 + 24 * q^95 - 13 * q^97 - 2 * q^98

Introduction

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