LMFDB - Embedded newform 7569.2.a.bf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7569,2,Mod(1,7569)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.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:	$60.4387692899$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.14884000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.30927$ 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)

$f(q)$	$=$	$q-2.30927 q^{2} +3.33275 q^{4} +3.03582 q^{5} -4.39250 q^{7} -3.07768 q^{8} -7.01054 q^{10} +5.68046 q^{11} +3.95078 q^{13} +10.1435 q^{14} +0.441718 q^{16} +3.21721 q^{17} +3.61803 q^{19} +10.1176 q^{20} -13.1178 q^{22} +2.69640 q^{23} +4.21619 q^{25} -9.12344 q^{26} -14.6391 q^{28} +0.823684 q^{31} +5.13532 q^{32} -7.42943 q^{34} -13.3348 q^{35} +5.60575 q^{37} -8.35503 q^{38} -9.34328 q^{40} -0.558636 q^{41} +12.7130 q^{43} +18.9316 q^{44} -6.22672 q^{46} -0.129647 q^{47} +12.2941 q^{49} -9.73633 q^{50} +13.1670 q^{52} -6.88596 q^{53} +17.2449 q^{55} +13.5187 q^{56} +0.745036 q^{59} +7.17632 q^{61} -1.90211 q^{62} -12.7423 q^{64} +11.9939 q^{65} +7.06735 q^{67} +10.7222 q^{68} +30.7938 q^{70} +12.6641 q^{71} -8.81434 q^{73} -12.9452 q^{74} +12.0580 q^{76} -24.9514 q^{77} +6.99349 q^{79} +1.34098 q^{80} +1.29004 q^{82} -8.90268 q^{83} +9.76687 q^{85} -29.3577 q^{86} -17.4827 q^{88} -1.51344 q^{89} -17.3538 q^{91} +8.98641 q^{92} +0.299391 q^{94} +10.9837 q^{95} -14.7785 q^{97} -28.3904 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4} - 12 q^{10} + 2 q^{13} - 2 q^{16} + 20 q^{19} - 14 q^{22} + 2 q^{25} - 20 q^{28} + 10 q^{31} - 36 q^{34} + 18 q^{37} - 10 q^{40} + 28 q^{43} + 26 q^{46} + 4 q^{49} + 44 q^{52} + 14 q^{55}+ \cdots - 6 q^{97}+O(q^{100}) $ 8 * q + 6 * q^4 - 12 * q^10 + 2 * q^13 - 2 * q^16 + 20 * q^19 - 14 * q^22 + 2 * q^25 - 20 * q^28 + 10 * q^31 - 36 * q^34 + 18 * q^37 - 10 * q^40 + 28 * q^43 + 26 * q^46 + 4 * q^49 + 44 * q^52 + 14 * q^55 + 54 * q^61 - 44 * q^64 + 38 * q^67 + 70 * q^70 - 16 * q^73 + 20 * q^76 + 14 * q^79 + 2 * q^82 + 56 * q^85 - 60 * q^88 - 30 * q^91 + 52 * q^94 - 6 * 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 $

$2$	−2.30927	−1.63290	−0.816452	−	0.577414i	$-0.804062\pi$
$2$	−2.30927	−1.63290	−0.816452	+	0.577414i	$0.804062\pi$
$3$	0	0
$3$	0	0
$4$	3.33275	1.66637
$5$	3.03582	1.35766	0.678829	−	0.734296i	$-0.262488\pi$
$5$	3.03582	1.35766	0.678829	+	0.734296i	$0.262488\pi$
$6$	0	0
$7$	−4.39250	−1.66021	−0.830105	−	0.557608i	$-0.811719\pi$
$7$	−4.39250	−1.66021	−0.830105	+	0.557608i	$0.811719\pi$
$8$	−3.07768	−1.08813
$9$	0	0
$10$	−7.01054	−2.21693
$11$	5.68046	1.71272	0.856362	−	0.516376i	$-0.172719\pi$
$11$	5.68046	1.71272	0.856362	+	0.516376i	$0.172719\pi$
$12$	0	0
$13$	3.95078	1.09575	0.547875	−	0.836560i	$-0.315437\pi$
$13$	3.95078	1.09575	0.547875	+	0.836560i	$0.315437\pi$
$14$	10.1435	2.71096
$15$	0	0
$16$	0.441718	0.110430
$17$	3.21721	0.780289	0.390144	−	0.920754i	$-0.372425\pi$
$17$	3.21721	0.780289	0.390144	+	0.920754i	$0.372425\pi$
$18$	0	0
$19$	3.61803	0.830034	0.415017	−	0.909814i	$-0.363776\pi$
$19$	3.61803	0.830034	0.415017	+	0.909814i	$0.363776\pi$
$20$	10.1176	2.26237
$21$	0	0
$22$	−13.1178	−2.79671
$23$	2.69640	0.562238	0.281119	−	0.959673i	$-0.409295\pi$
$23$	2.69640	0.562238	0.281119	+	0.959673i	$0.409295\pi$
$24$	0	0
$25$	4.21619	0.843237
$26$	−9.12344	−1.78925
$27$	0	0
$28$	−14.6391	−2.76653
$29$	0	0
$29$	0	0
$30$	0	0
$31$	0.823684	0.147938	0.0739690	−	0.997261i	$-0.476433\pi$
$31$	0.823684	0.147938	0.0739690	+	0.997261i	$0.476433\pi$
$32$	5.13532	0.907805
$33$	0	0
$34$	−7.42943	−1.27414
$35$	−13.3348	−2.25400
$36$	0	0
$37$	5.60575	0.921579	0.460789	−	0.887509i	$-0.347566\pi$
$37$	5.60575	0.921579	0.460789	+	0.887509i	$0.347566\pi$
$38$	−8.35503	−1.35537
$39$	0	0
$40$	−9.34328	−1.47730
$41$	−0.558636	−0.0872442	−0.0436221	−	0.999048i	$-0.513890\pi$
$41$	−0.558636	−0.0872442	−0.0436221	+	0.999048i	$0.513890\pi$
$42$	0	0
$43$	12.7130	1.93871	0.969354	−	0.245667i	$-0.0790070\pi$
$43$	12.7130	1.93871	0.969354	+	0.245667i	$0.0790070\pi$
$44$	18.9316	2.85404
$45$	0	0
$46$	−6.22672	−0.918080
$47$	−0.129647	−0.0189110	−0.00945548	−	0.999955i	$-0.503010\pi$
$47$	−0.129647	−0.0189110	−0.00945548	+	0.999955i	$0.503010\pi$
$48$	0	0
$49$	12.2941	1.75630
$50$	−9.73633	−1.37692
$51$	0	0
$52$	13.1670	1.82593
$53$	−6.88596	−0.945859	−0.472929	−	0.881100i	$-0.656803\pi$
$53$	−6.88596	−0.945859	−0.472929	+	0.881100i	$0.656803\pi$
$54$	0	0
$55$	17.2449	2.32530
$56$	13.5187	1.80652
$57$	0	0
$58$	0	0
$59$	0.745036	0.0969954	0.0484977	−	0.998823i	$-0.484557\pi$
$59$	0.745036	0.0969954	0.0484977	+	0.998823i	$0.484557\pi$
$60$	0	0
$61$	7.17632	0.918833	0.459417	−	0.888221i	$-0.348059\pi$
$61$	7.17632	0.918833	0.459417	+	0.888221i	$0.348059\pi$
$62$	−1.90211	−0.241569
$63$	0	0
$64$	−12.7423	−1.59279
$65$	11.9939	1.48765
$66$	0	0
$67$	7.06735	0.863414	0.431707	−	0.902014i	$-0.357911\pi$
$67$	7.06735	0.863414	0.431707	+	0.902014i	$0.357911\pi$
$68$	10.7222	1.30025
$69$	0	0
$70$	30.7938	3.68056
$71$	12.6641	1.50295	0.751475	−	0.659761i	$-0.229343\pi$
$71$	12.6641	1.50295	0.751475	+	0.659761i	$0.229343\pi$
$72$	0	0
$73$	−8.81434	−1.03164	−0.515820	−	0.856697i	$-0.672513\pi$
$73$	−8.81434	−1.03164	−0.515820	+	0.856697i	$0.672513\pi$
$74$	−12.9452	−1.50485
$75$	0	0
$76$	12.0580	1.38315
$77$	−24.9514	−2.84348
$78$	0	0
$79$	6.99349	0.786829	0.393414	−	0.919361i	$-0.371294\pi$
$79$	6.99349	0.786829	0.393414	+	0.919361i	$0.371294\pi$
$80$	1.34098	0.149926
$81$	0	0
$82$	1.29004	0.142461
$83$	−8.90268	−0.977196	−0.488598	−	0.872509i	$-0.662492\pi$
$83$	−8.90268	−0.977196	−0.488598	+	0.872509i	$0.662492\pi$
$84$	0	0
$85$	9.76687	1.05937
$86$	−29.3577	−3.16572
$87$	0	0
$88$	−17.4827	−1.86366
$89$	−1.51344	−0.160425	−0.0802124	−	0.996778i	$-0.525560\pi$
$89$	−1.51344	−0.160425	−0.0802124	+	0.996778i	$0.525560\pi$
$90$	0	0
$91$	−17.3538	−1.81917
$92$	8.98641	0.936898
$93$	0	0
$94$	0.299391	0.0308798
$95$	10.9837	1.12690
$96$	0	0
$97$	−14.7785	−1.50053	−0.750264	−	0.661138i	$-0.770074\pi$
$97$	−14.7785	−1.50053	−0.750264	+	0.661138i	$0.770074\pi$
$98$	−28.3904	−2.86786
$99$	0	0
$100$	14.0515	1.40515
$101$	7.59901	0.756130	0.378065	−	0.925779i	$-0.376590\pi$
$101$	7.59901	0.756130	0.378065	+	0.925779i	$0.376590\pi$
$102$	0	0
$103$	4.37262	0.430847	0.215423	−	0.976521i	$-0.430887\pi$
$103$	4.37262	0.430847	0.215423	+	0.976521i	$0.430887\pi$
$104$	−12.1593	−1.19231
$105$	0	0
$106$	15.9016	1.54450
$107$	−18.2010	−1.75955	−0.879777	−	0.475386i	$-0.842308\pi$
$107$	−18.2010	−1.75955	−0.879777	+	0.475386i	$0.842308\pi$
$108$	0	0
$109$	−6.72525	−0.644162	−0.322081	−	0.946712i	$-0.604382\pi$
$109$	−6.72525	−0.644162	−0.322081	+	0.946712i	$0.604382\pi$
$110$	−39.8231	−3.79698
$111$	0	0
$112$	−1.94025	−0.183336
$113$	−4.84586	−0.455860	−0.227930	−	0.973678i	$-0.573196\pi$
$113$	−4.84586	−0.455860	−0.227930	+	0.973678i	$0.573196\pi$
$114$	0	0
$115$	8.18577	0.763327
$116$	0	0
$117$	0	0
$118$	−1.72049	−0.158384
$119$	−14.1316	−1.29544
$120$	0	0
$121$	21.2677	1.93342
$122$	−16.5721	−1.50037
$123$	0	0
$124$	2.74513	0.246520
$125$	−2.37952	−0.212831
$126$	0	0
$127$	5.90048	0.523583	0.261792	−	0.965124i	$-0.415687\pi$
$127$	5.90048	0.523583	0.261792	+	0.965124i	$0.415687\pi$
$128$	19.1548	1.69306
$129$	0	0
$130$	−27.6971	−2.42920
$131$	0.678840	0.0593106	0.0296553	−	0.999560i	$-0.490559\pi$
$131$	0.678840	0.0593106	0.0296553	+	0.999560i	$0.490559\pi$
$132$	0	0
$133$	−15.8922	−1.37803
$134$	−16.3204	−1.40987
$135$	0	0
$136$	−9.90157	−0.849052
$137$	16.9527	1.44836	0.724182	−	0.689609i	$-0.242218\pi$
$137$	16.9527	1.44836	0.724182	+	0.689609i	$0.242218\pi$
$138$	0	0
$139$	−5.46461	−0.463502	−0.231751	−	0.972775i	$-0.574445\pi$
$139$	−5.46461	−0.463502	−0.231751	+	0.972775i	$0.574445\pi$
$140$	−44.4416	−3.75600
$141$	0	0
$142$	−29.2449	−2.45417
$143$	22.4423	1.87672
$144$	0	0
$145$	0	0
$146$	20.3547	1.68457
$147$	0	0
$148$	18.6825	1.53570
$149$	−7.55310	−0.618774	−0.309387	−	0.950936i	$-0.600124\pi$
$149$	−7.55310	−0.618774	−0.309387	+	0.950936i	$0.600124\pi$
$150$	0	0
$151$	−1.69659	−0.138066	−0.0690331	−	0.997614i	$-0.521991\pi$
$151$	−1.69659	−0.138066	−0.0690331	+	0.997614i	$0.521991\pi$
$152$	−11.1352	−0.903181
$153$	0	0
$154$	57.6197	4.64313
$155$	2.50055	0.200849
$156$	0	0
$157$	−18.1863	−1.45142	−0.725712	−	0.687999i	$-0.758489\pi$
$157$	−18.1863	−1.45142	−0.725712	+	0.687999i	$0.758489\pi$
$158$	−16.1499	−1.28482
$159$	0	0
$160$	15.5899	1.23249
$161$	−11.8439	−0.933432
$162$	0	0
$163$	−11.1453	−0.872970	−0.436485	−	0.899712i	$-0.643777\pi$
$163$	−11.1453	−0.872970	−0.436485	+	0.899712i	$0.643777\pi$
$164$	−1.86179	−0.145382
$165$	0	0
$166$	20.5587	1.59567
$167$	−10.7469	−0.831623	−0.415811	−	0.909451i	$-0.636502\pi$
$167$	−10.7469	−0.831623	−0.415811	+	0.909451i	$0.636502\pi$
$168$	0	0
$169$	2.60869	0.200668
$170$	−22.5544	−1.72984
$171$	0	0
$172$	42.3691	3.23061
$173$	21.4326	1.62949	0.814746	−	0.579818i	$-0.196876\pi$
$173$	21.4326	1.62949	0.814746	+	0.579818i	$0.196876\pi$
$174$	0	0
$175$	−18.5196	−1.39995
$176$	2.50916	0.189135
$177$	0	0
$178$	3.49496	0.261958
$179$	16.5426	1.23645	0.618226	−	0.786000i	$-0.287852\pi$
$179$	16.5426	1.23645	0.618226	+	0.786000i	$0.287852\pi$
$180$	0	0
$181$	14.5336	1.08028	0.540139	−	0.841576i	$-0.318372\pi$
$181$	14.5336	1.08028	0.540139	+	0.841576i	$0.318372\pi$
$182$	40.0747	2.97054
$183$	0	0
$184$	−8.29866	−0.611785
$185$	17.0180	1.25119
$186$	0	0
$187$	18.2753	1.33642
$188$	−0.432081	−0.0315127
$189$	0	0
$190$	−25.3644	−1.84012
$191$	5.79567	0.419360	0.209680	−	0.977770i	$-0.432758\pi$
$191$	5.79567	0.419360	0.209680	+	0.977770i	$0.432758\pi$
$192$	0	0
$193$	1.75917	0.126628	0.0633140	−	0.997994i	$-0.479833\pi$
$193$	1.75917	0.126628	0.0633140	+	0.997994i	$0.479833\pi$
$194$	34.1276	2.45022
$195$	0	0
$196$	40.9730	2.92665
$197$	−23.9966	−1.70969	−0.854845	−	0.518884i	$-0.826348\pi$
$197$	−23.9966	−1.70969	−0.854845	+	0.518884i	$0.826348\pi$
$198$	0	0
$199$	−2.99349	−0.212203	−0.106101	−	0.994355i	$-0.533837\pi$
$199$	−2.99349	−0.212203	−0.106101	+	0.994355i	$0.533837\pi$
$200$	−12.9761	−0.917548
$201$	0	0
$202$	−17.5482	−1.23469
$203$	0	0
$204$	0	0
$205$	−1.69592	−0.118448
$206$	−10.0976	−0.703532
$207$	0	0
$208$	1.74513	0.121003
$209$	20.5521	1.42162
$210$	0	0
$211$	13.1300	0.903909	0.451955	−	0.892041i	$-0.350727\pi$
$211$	13.1300	0.903909	0.451955	+	0.892041i	$0.350727\pi$
$212$	−22.9492	−1.57616
$213$	0	0
$214$	42.0310	2.87318
$215$	38.5942	2.63210
$216$	0	0
$217$	−3.61803	−0.245608
$218$	15.5304	1.05185
$219$	0	0
$220$	57.4728	3.87481
$221$	12.7105	0.855002
$222$	0	0
$223$	−3.06343	−0.205142	−0.102571	−	0.994726i	$-0.532707\pi$
$223$	−3.06343	−0.205142	−0.102571	+	0.994726i	$0.532707\pi$
$224$	−22.5569	−1.50715
$225$	0	0
$226$	11.1904	0.744376
$227$	3.82760	0.254047	0.127023	−	0.991900i	$-0.459458\pi$
$227$	3.82760	0.254047	0.127023	+	0.991900i	$0.459458\pi$
$228$	0	0
$229$	−2.27191	−0.150132	−0.0750661	−	0.997179i	$-0.523917\pi$
$229$	−2.27191	−0.150132	−0.0750661	+	0.997179i	$0.523917\pi$
$230$	−18.9032	−1.24644
$231$	0	0
$232$	0	0
$233$	5.59312	0.366418	0.183209	−	0.983074i	$-0.441352\pi$
$233$	5.59312	0.366418	0.183209	+	0.983074i	$0.441352\pi$
$234$	0	0
$235$	−0.393585	−0.0256746
$236$	2.48302	0.161631
$237$	0	0
$238$	32.6338	2.11533
$239$	−2.70501	−0.174972	−0.0874862	−	0.996166i	$-0.527883\pi$
$239$	−2.70501	−0.174972	−0.0874862	+	0.996166i	$0.527883\pi$
$240$	0	0
$241$	−24.2707	−1.56341	−0.781706	−	0.623647i	$-0.785650\pi$
$241$	−24.2707	−1.56341	−0.781706	+	0.623647i	$0.785650\pi$
$242$	−49.1129	−3.15710
$243$	0	0
$244$	23.9169	1.53112
$245$	37.3225	2.38445
$246$	0	0
$247$	14.2941	0.909510
$248$	−2.53504	−0.160975
$249$	0	0
$250$	5.49496	0.347532
$251$	−23.3913	−1.47644	−0.738221	−	0.674558i	$-0.764334\pi$
$251$	−23.3913	−1.47644	−0.738221	+	0.674558i	$0.764334\pi$
$252$	0	0
$253$	15.3168	0.962958
$254$	−13.6258	−0.854961
$255$	0	0
$256$	−18.7492	−1.17182
$257$	−23.5267	−1.46756	−0.733779	−	0.679388i	$-0.762245\pi$
$257$	−23.5267	−1.46756	−0.733779	+	0.679388i	$0.762245\pi$
$258$	0	0
$259$	−24.6232	−1.53001
$260$	39.9725	2.47899
$261$	0	0
$262$	−1.56763	−0.0968484
$263$	−12.2963	−0.758222	−0.379111	−	0.925351i	$-0.623770\pi$
$263$	−12.2963	−0.758222	−0.379111	+	0.925351i	$0.623770\pi$
$264$	0	0
$265$	−20.9045	−1.28415
$266$	36.6995	2.25019
$267$	0	0
$268$	23.5537	1.43877
$269$	−23.1823	−1.41345	−0.706725	−	0.707488i	$-0.749828\pi$
$269$	−23.1823	−1.41345	−0.706725	+	0.707488i	$0.749828\pi$
$270$	0	0
$271$	14.0675	0.854537	0.427268	−	0.904125i	$-0.359476\pi$
$271$	14.0675	0.854537	0.427268	+	0.904125i	$0.359476\pi$
$272$	1.42110	0.0861670
$273$	0	0
$274$	−39.1483	−2.36504
$275$	23.9499	1.44423
$276$	0	0
$277$	2.23029	0.134005	0.0670026	−	0.997753i	$-0.478656\pi$
$277$	2.23029	0.134005	0.0670026	+	0.997753i	$0.478656\pi$
$278$	12.6193	0.756854
$279$	0	0
$280$	41.0404	2.45263
$281$	9.75566	0.581974	0.290987	−	0.956727i	$-0.406016\pi$
$281$	9.75566	0.581974	0.290987	+	0.956727i	$0.406016\pi$
$282$	0	0
$283$	27.4962	1.63448	0.817240	−	0.576297i	$-0.195503\pi$
$283$	27.4962	1.63448	0.817240	+	0.576297i	$0.195503\pi$
$284$	42.2062	2.50448
$285$	0	0
$286$	−51.8254	−3.06450
$287$	2.45381	0.144844
$288$	0	0
$289$	−6.64954	−0.391149
$290$	0	0
$291$	0	0
$292$	−29.3760	−1.71910
$293$	−13.8751	−0.810592	−0.405296	−	0.914186i	$-0.632832\pi$
$293$	−13.8751	−0.810592	−0.405296	+	0.914186i	$0.632832\pi$
$294$	0	0
$295$	2.26179	0.131687
$296$	−17.2527	−1.00279
$297$	0	0
$298$	17.4422	1.01040
$299$	10.6529	0.616072
$300$	0	0
$301$	−55.8417	−3.21866
$302$	3.91788	0.225449
$303$	0	0
$304$	1.59815	0.0916603
$305$	21.7860	1.24746
$306$	0	0
$307$	9.49020	0.541634	0.270817	−	0.962631i	$-0.412706\pi$
$307$	9.49020	0.541634	0.270817	+	0.962631i	$0.412706\pi$
$308$	−83.1569	−4.73830
$309$	0	0
$310$	−5.77447	−0.327968
$311$	−18.7276	−1.06195	−0.530973	−	0.847389i	$-0.678173\pi$
$311$	−18.7276	−1.06195	−0.530973	+	0.847389i	$0.678173\pi$
$312$	0	0
$313$	22.9794	1.29888	0.649438	−	0.760415i	$-0.275004\pi$
$313$	22.9794	1.29888	0.649438	+	0.760415i	$0.275004\pi$
$314$	41.9971	2.37004
$315$	0	0
$316$	23.3075	1.31115
$317$	−28.7899	−1.61700	−0.808501	−	0.588495i	$-0.799720\pi$
$317$	−28.7899	−1.61700	−0.808501	+	0.588495i	$0.799720\pi$
$318$	0	0
$319$	0	0
$320$	−38.6833	−2.16246
$321$	0	0
$322$	27.3509	1.52420
$323$	11.6400	0.647666
$324$	0	0
$325$	16.6572	0.923977
$326$	25.7376	1.42548
$327$	0	0
$328$	1.71930	0.0949327
$329$	0.569475	0.0313961
$330$	0	0
$331$	−2.33575	−0.128385	−0.0641923	−	0.997938i	$-0.520447\pi$
$331$	−2.33575	−0.128385	−0.0641923	+	0.997938i	$0.520447\pi$
$332$	−29.6704	−1.62837
$333$	0	0
$334$	24.8176	1.35796
$335$	21.4552	1.17222
$336$	0	0
$337$	2.36492	0.128825	0.0644127	−	0.997923i	$-0.479483\pi$
$337$	2.36492	0.128825	0.0644127	+	0.997923i	$0.479483\pi$
$338$	−6.02417	−0.327672
$339$	0	0
$340$	32.5505	1.76530
$341$	4.67891	0.253377
$342$	0	0
$343$	−23.2542	−1.25561
$344$	−39.1265	−2.10956
$345$	0	0
$346$	−49.4938	−2.66080
$347$	−34.2637	−1.83937	−0.919685	−	0.392657i	$-0.871556\pi$
$347$	−34.2637	−1.83937	−0.919685	+	0.392657i	$0.871556\pi$
$348$	0	0
$349$	25.0903	1.34306	0.671528	−	0.740980i	$-0.265638\pi$
$349$	25.0903	1.34306	0.671528	+	0.740980i	$0.265638\pi$
$350$	42.7668	2.28598
$351$	0	0
$352$	29.1710	1.55482
$353$	9.88447	0.526097	0.263049	−	0.964783i	$-0.415272\pi$
$353$	9.88447	0.526097	0.263049	+	0.964783i	$0.415272\pi$
$354$	0	0
$355$	38.4458	2.04049
$356$	−5.04393	−0.267328
$357$	0	0
$358$	−38.2014	−2.01901
$359$	−0.912501	−0.0481600	−0.0240800	−	0.999710i	$-0.507666\pi$
$359$	−0.912501	−0.0481600	−0.0240800	+	0.999710i	$0.507666\pi$
$360$	0	0
$361$	−5.90983	−0.311044
$362$	−33.5622	−1.76399
$363$	0	0
$364$	−57.8359	−3.03143
$365$	−26.7587	−1.40062
$366$	0	0
$367$	21.9402	1.14527	0.572636	−	0.819810i	$-0.305921\pi$
$367$	21.9402	1.14527	0.572636	+	0.819810i	$0.305921\pi$
$368$	1.19105	0.0620876
$369$	0	0
$370$	−39.2993	−2.04307
$371$	30.2466	1.57032
$372$	0	0
$373$	−19.1235	−0.990179	−0.495089	−	0.868842i	$-0.664865\pi$
$373$	−19.1235	−0.990179	−0.495089	+	0.868842i	$0.664865\pi$
$374$	−42.2026	−2.18225
$375$	0	0
$376$	0.399012	0.0205775
$377$	0	0
$378$	0	0
$379$	10.6333	0.546197	0.273099	−	0.961986i	$-0.411951\pi$
$379$	10.6333	0.546197	0.273099	+	0.961986i	$0.411951\pi$
$380$	36.6059	1.87784
$381$	0	0
$382$	−13.3838	−0.684774
$383$	24.6435	1.25922	0.629612	−	0.776910i	$-0.283214\pi$
$383$	24.6435	1.25922	0.629612	+	0.776910i	$0.283214\pi$
$384$	0	0
$385$	−75.7480	−3.86048
$386$	−4.06242	−0.206771
$387$	0	0
$388$	−49.2530	−2.50044
$389$	−13.5454	−0.686781	−0.343390	−	0.939193i	$-0.611575\pi$
$389$	−13.5454	−0.686781	−0.343390	+	0.939193i	$0.611575\pi$
$390$	0	0
$391$	8.67489	0.438708
$392$	−37.8372	−1.91107
$393$	0	0
$394$	55.4148	2.79176
$395$	21.2310	1.06825
$396$	0	0
$397$	−31.1851	−1.56514	−0.782568	−	0.622566i	$-0.786090\pi$
$397$	−31.1851	−1.56514	−0.782568	+	0.622566i	$0.786090\pi$
$398$	6.91279	0.346507
$399$	0	0
$400$	1.86237	0.0931183
$401$	−20.3905	−1.01825	−0.509126	−	0.860692i	$-0.670031\pi$
$401$	−20.3905	−1.01825	−0.509126	+	0.860692i	$0.670031\pi$
$402$	0	0
$403$	3.25420	0.162103
$404$	25.3256	1.26000
$405$	0	0
$406$	0	0
$407$	31.8432	1.57841
$408$	0	0
$409$	−3.54900	−0.175487	−0.0877434	−	0.996143i	$-0.527966\pi$
$409$	−3.54900	−0.175487	−0.0877434	+	0.996143i	$0.527966\pi$
$410$	3.91634	0.193414
$411$	0	0
$412$	14.5728	0.717952
$413$	−3.27257	−0.161033
$414$	0	0
$415$	−27.0269	−1.32670
$416$	20.2885	0.994727
$417$	0	0
$418$	−47.4605	−2.32137
$419$	−10.5213	−0.514000	−0.257000	−	0.966411i	$-0.582734\pi$
$419$	−10.5213	−0.514000	−0.257000	+	0.966411i	$0.582734\pi$
$420$	0	0
$421$	−8.27884	−0.403486	−0.201743	−	0.979439i	$-0.564661\pi$
$421$	−8.27884	−0.403486	−0.201743	+	0.979439i	$0.564661\pi$
$422$	−30.3209	−1.47600
$423$	0	0
$424$	21.1928	1.02921
$425$	13.5644	0.657969
$426$	0	0
$427$	−31.5220	−1.52546
$428$	−60.6593	−2.93208
$429$	0	0
$430$	−89.1247	−4.29797
$431$	−0.546934	−0.0263449	−0.0131724	−	0.999913i	$-0.504193\pi$
$431$	−0.546934	−0.0263449	−0.0131724	+	0.999913i	$0.504193\pi$
$432$	0	0
$433$	−2.61555	−0.125695	−0.0628476	−	0.998023i	$-0.520018\pi$
$433$	−2.61555	−0.125695	−0.0628476	+	0.998023i	$0.520018\pi$
$434$	8.35503	0.401054
$435$	0	0
$436$	−22.4136	−1.07342
$437$	9.75566	0.466676
$438$	0	0
$439$	33.4898	1.59838	0.799191	−	0.601078i	$-0.205262\pi$
$439$	33.4898	1.59838	0.799191	+	0.601078i	$0.205262\pi$
$440$	−53.0742	−2.53021
$441$	0	0
$442$	−29.3521	−1.39614
$443$	−19.1594	−0.910290	−0.455145	−	0.890417i	$-0.650413\pi$
$443$	−19.1594	−0.910290	−0.455145	+	0.890417i	$0.650413\pi$
$444$	0	0
$445$	−4.59454	−0.217802
$446$	7.07429	0.334978
$447$	0	0
$448$	55.9706	2.64436
$449$	26.1010	1.23178	0.615892	−	0.787831i	$-0.288796\pi$
$449$	26.1010	1.23178	0.615892	+	0.787831i	$0.288796\pi$
$450$	0	0
$451$	−3.17331	−0.149425
$452$	−16.1500	−0.759634
$453$	0	0
$454$	−8.83898	−0.414834
$455$	−52.6830	−2.46982
$456$	0	0
$457$	10.3432	0.483835	0.241918	−	0.970297i	$-0.422224\pi$
$457$	10.3432	0.483835	0.241918	+	0.970297i	$0.422224\pi$
$458$	5.24647	0.245152
$459$	0	0
$460$	27.2811	1.27199
$461$	33.4724	1.55897	0.779483	−	0.626424i	$-0.215482\pi$
$461$	33.4724	1.55897	0.779483	+	0.626424i	$0.215482\pi$
$462$	0	0
$463$	−11.2612	−0.523351	−0.261675	−	0.965156i	$-0.584275\pi$
$463$	−11.2612	−0.523351	−0.261675	+	0.965156i	$0.584275\pi$
$464$	0	0
$465$	0	0
$466$	−12.9161	−0.598325
$467$	−30.6211	−1.41697	−0.708487	−	0.705724i	$-0.750622\pi$
$467$	−30.6211	−1.41697	−0.708487	+	0.705724i	$0.750622\pi$
$468$	0	0
$469$	−31.0433	−1.43345
$470$	0.908895	0.0419242
$471$	0	0
$472$	−2.29298	−0.105543
$473$	72.2155	3.32047
$474$	0	0
$475$	15.2543	0.699915
$476$	−47.0971	−2.15869
$477$	0	0
$478$	6.24660	0.285713
$479$	1.29283	0.0590710	0.0295355	−	0.999564i	$-0.490597\pi$
$479$	1.29283	0.0590710	0.0295355	+	0.999564i	$0.490597\pi$
$480$	0	0
$481$	22.1471	1.00982
$482$	56.0477	2.55290
$483$	0	0
$484$	70.8798	3.22181
$485$	−44.8648	−2.03721
$486$	0	0
$487$	−0.933841	−0.0423164	−0.0211582	−	0.999776i	$-0.506735\pi$
$487$	−0.933841	−0.0423164	−0.0211582	+	0.999776i	$0.506735\pi$
$488$	−22.0864	−0.999806
$489$	0	0
$490$	−86.1880	−3.89358
$491$	−7.91557	−0.357225	−0.178612	−	0.983920i	$-0.557161\pi$
$491$	−7.91557	−0.357225	−0.178612	+	0.983920i	$0.557161\pi$
$492$	0	0
$493$	0	0
$494$	−33.0089	−1.48514
$495$	0	0
$496$	0.363836	0.0163367
$497$	−55.6270	−2.49521
$498$	0	0
$499$	3.89505	0.174367	0.0871833	−	0.996192i	$-0.472213\pi$
$499$	3.89505	0.174367	0.0871833	+	0.996192i	$0.472213\pi$
$500$	−7.93033	−0.354655
$501$	0	0
$502$	54.0168	2.41089
$503$	18.2986	0.815896	0.407948	−	0.913005i	$-0.366244\pi$
$503$	18.2986	0.815896	0.407948	+	0.913005i	$0.366244\pi$
$504$	0	0
$505$	23.0692	1.02657
$506$	−35.3707	−1.57242
$507$	0	0
$508$	19.6648	0.872486
$509$	−0.297945	−0.0132062	−0.00660309	−	0.999978i	$-0.502102\pi$
$509$	−0.297945	−0.0132062	−0.00660309	+	0.999978i	$0.502102\pi$
$510$	0	0
$511$	38.7170	1.71274
$512$	4.98730	0.220410
$513$	0	0
$514$	54.3297	2.39638
$515$	13.2745	0.584943
$516$	0	0
$517$	−0.736455	−0.0323893
$518$	56.8618	2.49837
$519$	0	0
$520$	−36.9133	−1.61875
$521$	−21.6023	−0.946415	−0.473208	−	0.880951i	$-0.656904\pi$
$521$	−21.6023	−0.946415	−0.473208	+	0.880951i	$0.656904\pi$
$522$	0	0
$523$	−28.4828	−1.24546	−0.622732	−	0.782435i	$-0.713977\pi$
$523$	−28.4828	−1.24546	−0.622732	+	0.782435i	$0.713977\pi$
$524$	2.26240	0.0988336
$525$	0	0
$526$	28.3955	1.23810
$527$	2.64997	0.115434
$528$	0	0
$529$	−15.7294	−0.683889
$530$	48.2742	2.09690
$531$	0	0
$532$	−52.9648	−2.29631
$533$	−2.20705	−0.0955979
$534$	0	0
$535$	−55.2548	−2.38887
$536$	−21.7511	−0.939503
$537$	0	0
$538$	53.5343	2.30803
$539$	69.8360	3.00805
$540$	0	0
$541$	−31.9456	−1.37345	−0.686725	−	0.726917i	$-0.740952\pi$
$541$	−31.9456	−1.37345	−0.686725	+	0.726917i	$0.740952\pi$
$542$	−32.4856	−1.39538
$543$	0	0
$544$	16.5214	0.708350
$545$	−20.4166	−0.874552
$546$	0	0
$547$	−29.4737	−1.26020	−0.630102	−	0.776513i	$-0.716987\pi$
$547$	−29.4737	−1.26020	−0.630102	+	0.776513i	$0.716987\pi$
$548$	56.4990	2.41352
$549$	0	0
$550$	−55.3069	−2.35829
$551$	0	0
$552$	0	0
$553$	−30.7189	−1.30630
$554$	−5.15035	−0.218818
$555$	0	0
$556$	−18.2122	−0.772368
$557$	7.13081	0.302142	0.151071	−	0.988523i	$-0.451728\pi$
$557$	7.13081	0.302142	0.151071	+	0.988523i	$0.451728\pi$
$558$	0	0
$559$	50.2262	2.12434
$560$	−5.89024	−0.248908
$561$	0	0
$562$	−22.5285	−0.950307
$563$	−17.2395	−0.726558	−0.363279	−	0.931680i	$-0.618343\pi$
$563$	−17.2395	−0.726558	−0.363279	+	0.931680i	$0.618343\pi$
$564$	0	0
$565$	−14.7111	−0.618903
$566$	−63.4963	−2.66895
$567$	0	0
$568$	−38.9760	−1.63540
$569$	25.0950	1.05204	0.526019	−	0.850473i	$-0.323684\pi$
$569$	25.0950	1.05204	0.526019	+	0.850473i	$0.323684\pi$
$570$	0	0
$571$	15.8448	0.663082	0.331541	−	0.943441i	$-0.392431\pi$
$571$	15.8448	0.663082	0.331541	+	0.943441i	$0.392431\pi$
$572$	74.7945	3.12731
$573$	0	0
$574$	−5.66652	−0.236516
$575$	11.3685	0.474100
$576$	0	0
$577$	−10.7137	−0.446017	−0.223009	−	0.974816i	$-0.571588\pi$
$577$	−10.7137	−0.446017	−0.223009	+	0.974816i	$0.571588\pi$
$578$	15.3556	0.638709
$579$	0	0
$580$	0	0
$581$	39.1050	1.62235
$582$	0	0
$583$	−39.1154	−1.62000
$584$	27.1277	1.12255
$585$	0	0
$586$	32.0414	1.32362
$587$	−42.1108	−1.73810	−0.869049	−	0.494726i	$-0.835268\pi$
$587$	−42.1108	−1.73810	−0.869049	+	0.494726i	$0.835268\pi$
$588$	0	0
$589$	2.98012	0.122794
$590$	−5.22310	−0.215032
$591$	0	0
$592$	2.47616	0.101770
$593$	42.1378	1.73039	0.865196	−	0.501433i	$-0.167194\pi$
$593$	42.1378	1.73039	0.865196	+	0.501433i	$0.167194\pi$
$594$	0	0
$595$	−42.9010	−1.75877
$596$	−25.1726	−1.03111
$597$	0	0
$598$	−24.6004	−1.00599
$599$	28.2376	1.15376	0.576879	−	0.816830i	$-0.304270\pi$
$599$	28.2376	1.15376	0.576879	+	0.816830i	$0.304270\pi$
$600$	0	0
$601$	13.6703	0.557621	0.278811	−	0.960346i	$-0.410060\pi$
$601$	13.6703	0.557621	0.278811	+	0.960346i	$0.410060\pi$
$602$	128.954	5.25577
$603$	0	0
$604$	−5.65429	−0.230070
$605$	64.5648	2.62493
$606$	0	0
$607$	−16.0259	−0.650471	−0.325235	−	0.945633i	$-0.605444\pi$
$607$	−16.0259	−0.650471	−0.325235	+	0.945633i	$0.605444\pi$
$608$	18.5798	0.753509
$609$	0	0
$610$	−50.3098	−2.03698
$611$	−0.512207	−0.0207217
$612$	0	0
$613$	−19.2847	−0.778903	−0.389451	−	0.921047i	$-0.627335\pi$
$613$	−19.2847	−0.778903	−0.389451	+	0.921047i	$0.627335\pi$
$614$	−21.9155	−0.884437
$615$	0	0
$616$	76.7927	3.09406
$617$	−18.8384	−0.758405	−0.379202	−	0.925314i	$-0.623802\pi$
$617$	−18.8384	−0.758405	−0.379202	+	0.925314i	$0.623802\pi$
$618$	0	0
$619$	43.8366	1.76194	0.880970	−	0.473172i	$-0.156891\pi$
$619$	43.8366	1.76194	0.880970	+	0.473172i	$0.156891\pi$
$620$	8.33372	0.334690
$621$	0	0
$622$	43.2472	1.73406
$623$	6.64781	0.266339
$624$	0	0
$625$	−28.3047	−1.13219
$626$	−53.0659	−2.12094
$627$	0	0
$628$	−60.6103	−2.41862
$629$	18.0349	0.719098
$630$	0	0
$631$	4.64369	0.184863	0.0924313	−	0.995719i	$-0.470536\pi$
$631$	4.64369	0.184863	0.0924313	+	0.995719i	$0.470536\pi$
$632$	−21.5237	−0.856169
$633$	0	0
$634$	66.4837	2.64041
$635$	17.9128	0.710847
$636$	0	0
$637$	48.5712	1.92446
$638$	0	0
$639$	0	0
$640$	58.1505	2.29860
$641$	45.8894	1.81252	0.906261	−	0.422718i	$-0.138924\pi$
$641$	45.8894	1.81252	0.906261	+	0.422718i	$0.138924\pi$
$642$	0	0
$643$	19.8523	0.782898	0.391449	−	0.920200i	$-0.371974\pi$
$643$	19.8523	0.782898	0.391449	+	0.920200i	$0.371974\pi$
$644$	−39.4728	−1.55545
$645$	0	0
$646$	−26.8799	−1.05758
$647$	−19.9161	−0.782982	−0.391491	−	0.920182i	$-0.628041\pi$
$647$	−19.9161	−0.782982	−0.391491	+	0.920182i	$0.628041\pi$
$648$	0	0
$649$	4.23215	0.166126
$650$	−38.4661	−1.50877
$651$	0	0
$652$	−37.1446	−1.45469
$653$	−22.3060	−0.872900	−0.436450	−	0.899729i	$-0.643764\pi$
$653$	−22.3060	−0.872900	−0.436450	+	0.899729i	$0.643764\pi$
$654$	0	0
$655$	2.06084	0.0805235
$656$	−0.246760	−0.00963434
$657$	0	0
$658$	−1.31507	−0.0512669
$659$	6.11978	0.238393	0.119196	−	0.992871i	$-0.461968\pi$
$659$	6.11978	0.238393	0.119196	+	0.992871i	$0.461968\pi$
$660$	0	0
$661$	30.5599	1.18864	0.594322	−	0.804227i	$-0.297421\pi$
$661$	30.5599	1.18864	0.594322	+	0.804227i	$0.297421\pi$
$662$	5.39390	0.209640
$663$	0	0
$664$	27.3996	1.06331
$665$	−48.2459	−1.87089
$666$	0	0
$667$	0	0
$668$	−35.8168	−1.38580
$669$	0	0
$670$	−49.5459	−1.91412
$671$	40.7648	1.57371
$672$	0	0
$673$	−38.5751	−1.48696	−0.743481	−	0.668757i	$-0.766826\pi$
$673$	−38.5751	−1.48696	−0.743481	+	0.668757i	$0.766826\pi$
$674$	−5.46125	−0.210359
$675$	0	0
$676$	8.69410	0.334388
$677$	−17.6382	−0.677890	−0.338945	−	0.940806i	$-0.610070\pi$
$677$	−17.6382	−0.677890	−0.338945	+	0.940806i	$0.610070\pi$
$678$	0	0
$679$	64.9145	2.49119
$680$	−30.0593	−1.15272
$681$	0	0
$682$	−10.8049	−0.413740
$683$	4.65446	0.178098	0.0890489	−	0.996027i	$-0.471617\pi$
$683$	4.65446	0.178098	0.0890489	+	0.996027i	$0.471617\pi$
$684$	0	0
$685$	51.4652	1.96638
$686$	53.7003	2.05029
$687$	0	0
$688$	5.61555	0.214091
$689$	−27.2049	−1.03643
$690$	0	0
$691$	9.86464	0.375268	0.187634	−	0.982239i	$-0.439918\pi$
$691$	9.86464	0.375268	0.187634	+	0.982239i	$0.439918\pi$
$692$	71.4296	2.71535
$693$	0	0
$694$	79.1242	3.00351
$695$	−16.5895	−0.629277
$696$	0	0
$697$	−1.79725	−0.0680757
$698$	−57.9405	−2.19308
$699$	0	0
$700$	−61.7212	−2.33284
$701$	−17.9403	−0.677597	−0.338798	−	0.940859i	$-0.610020\pi$
$701$	−17.9403	−0.677597	−0.338798	+	0.940859i	$0.610020\pi$
$702$	0	0
$703$	20.2818	0.764942
$704$	−72.3822	−2.72801
$705$	0	0
$706$	−22.8260	−0.859066
$707$	−33.3787	−1.25533
$708$	0	0
$709$	−12.6352	−0.474524	−0.237262	−	0.971446i	$-0.576250\pi$
$709$	−12.6352	−0.474524	−0.237262	+	0.971446i	$0.576250\pi$
$710$	−88.7820	−3.33193
$711$	0	0
$712$	4.65790	0.174562
$713$	2.22098	0.0831763
$714$	0	0
$715$	68.1307	2.54794
$716$	55.1323	2.06039
$717$	0	0
$718$	2.10722	0.0786406
$719$	45.4419	1.69470	0.847348	−	0.531039i	$-0.178198\pi$
$719$	45.4419	1.69470	0.847348	+	0.531039i	$0.178198\pi$
$720$	0	0
$721$	−19.2067	−0.715296
$722$	13.6474	0.507904
$723$	0	0
$724$	48.4370	1.80015
$725$	0	0
$726$	0	0
$727$	39.0300	1.44754	0.723771	−	0.690041i	$-0.242407\pi$
$727$	39.0300	1.44754	0.723771	+	0.690041i	$0.242407\pi$
$728$	53.4096	1.97949
$729$	0	0
$730$	61.7932	2.28707
$731$	40.9003	1.51275
$732$	0	0
$733$	20.8126	0.768730	0.384365	−	0.923181i	$-0.374420\pi$
$733$	20.8126	0.768730	0.384365	+	0.923181i	$0.374420\pi$
$734$	−50.6661	−1.87012
$735$	0	0
$736$	13.8469	0.510402
$737$	40.1458	1.47879
$738$	0	0
$739$	39.5723	1.45569	0.727845	−	0.685742i	$-0.240522\pi$
$739$	39.5723	1.45569	0.727845	+	0.685742i	$0.240522\pi$
$740$	56.7168	2.08495
$741$	0	0
$742$	−69.8476	−2.56419
$743$	36.0498	1.32254	0.661270	−	0.750148i	$-0.270018\pi$
$743$	36.0498	1.32254	0.661270	+	0.750148i	$0.270018\pi$
$744$	0	0
$745$	−22.9298	−0.840084
$746$	44.1615	1.61687
$747$	0	0
$748$	60.9069	2.22698
$749$	79.9478	2.92123
$750$	0	0
$751$	10.0995	0.368537	0.184268	−	0.982876i	$-0.441008\pi$
$751$	10.0995	0.368537	0.184268	+	0.982876i	$0.441008\pi$
$752$	−0.0572674	−0.00208833
$753$	0	0
$754$	0	0
$755$	−5.15052	−0.187447
$756$	0	0
$757$	27.3486	0.994000	0.497000	−	0.867750i	$-0.334435\pi$
$757$	27.3486	0.994000	0.497000	+	0.867750i	$0.334435\pi$
$758$	−24.5553	−0.891888
$759$	0	0
$760$	−33.8043	−1.22621
$761$	20.1609	0.730834	0.365417	−	0.930844i	$-0.380926\pi$
$761$	20.1609	0.730834	0.365417	+	0.930844i	$0.380926\pi$
$762$	0	0
$763$	29.5407	1.06944
$764$	19.3155	0.698810
$765$	0	0
$766$	−56.9086	−2.05619
$767$	2.94347	0.106283
$768$	0	0
$769$	4.25483	0.153433	0.0767165	−	0.997053i	$-0.475556\pi$
$769$	4.25483	0.153433	0.0767165	+	0.997053i	$0.475556\pi$
$770$	174.923	6.30379
$771$	0	0
$772$	5.86288	0.211010
$773$	−40.4349	−1.45434	−0.727171	−	0.686457i	$-0.759165\pi$
$773$	−40.4349	−1.45434	−0.727171	+	0.686457i	$0.759165\pi$
$774$	0	0
$775$	3.47281	0.124747
$776$	45.4835	1.63276
$777$	0	0
$778$	31.2801	1.12145
$779$	−2.02116	−0.0724157
$780$	0	0
$781$	71.9379	2.57414
$782$	−20.0327	−0.716368
$783$	0	0
$784$	5.43051	0.193947
$785$	−55.2102	−1.97054
$786$	0	0
$787$	24.7692	0.882927	0.441463	−	0.897279i	$-0.354459\pi$
$787$	24.7692	0.882927	0.441463	+	0.897279i	$0.354459\pi$
$788$	−79.9748	−2.84898
$789$	0	0
$790$	−49.0281	−1.74434
$791$	21.2854	0.756823
$792$	0	0
$793$	28.3521	1.00681
$794$	72.0150	2.55572
$795$	0	0
$796$	−9.97655	−0.353609
$797$	−1.60802	−0.0569589	−0.0284794	−	0.999594i	$-0.509067\pi$
$797$	−1.60802	−0.0569589	−0.0284794	+	0.999594i	$0.509067\pi$
$798$	0	0
$799$	−0.417102	−0.0147560
$800$	21.6515	0.765495
$801$	0	0
$802$	47.0872	1.66271
$803$	−50.0695	−1.76692
$804$	0	0
$805$	−35.9560	−1.26728
$806$	−7.51484	−0.264699
$807$	0	0
$808$	−23.3873	−0.822764
$809$	2.09411	0.0736249	0.0368124	−	0.999322i	$-0.488280\pi$
$809$	2.09411	0.0736249	0.0368124	+	0.999322i	$0.488280\pi$
$810$	0	0
$811$	26.6256	0.934952	0.467476	−	0.884006i	$-0.345163\pi$
$811$	26.6256	0.934952	0.467476	+	0.884006i	$0.345163\pi$
$812$	0	0
$813$	0	0
$814$	−73.5348	−2.57739
$815$	−33.8352	−1.18520
$816$	0	0
$817$	45.9959	1.60919
$818$	8.19561	0.286553
$819$	0	0
$820$	−5.65206	−0.197379
$821$	6.39578	0.223214	0.111607	−	0.993752i	$-0.464400\pi$
$821$	6.39578	0.223214	0.111607	+	0.993752i	$0.464400\pi$
$822$	0	0
$823$	−19.8845	−0.693131	−0.346565	−	0.938026i	$-0.612652\pi$
$823$	−19.8845	−0.693131	−0.346565	+	0.938026i	$0.612652\pi$
$824$	−13.4575	−0.468816
$825$	0	0
$826$	7.55726	0.262951
$827$	40.8266	1.41968	0.709839	−	0.704364i	$-0.248768\pi$
$827$	40.8266	1.41968	0.709839	+	0.704364i	$0.248768\pi$
$828$	0	0
$829$	4.59538	0.159604	0.0798021	−	0.996811i	$-0.474571\pi$
$829$	4.59538	0.159604	0.0798021	+	0.996811i	$0.474571\pi$
$830$	62.4126	2.16637
$831$	0	0
$832$	−50.3421	−1.74530
$833$	39.5526	1.37042
$834$	0	0
$835$	−32.6257	−1.12906
$836$	68.4950	2.36895
$837$	0	0
$838$	24.2966	0.839313
$839$	−38.9935	−1.34620	−0.673102	−	0.739549i	$-0.735039\pi$
$839$	−38.9935	−1.34620	−0.673102	+	0.739549i	$0.735039\pi$
$840$	0	0
$841$	0	0
$842$	19.1181	0.658854
$843$	0	0
$844$	43.7591	1.50625
$845$	7.91950	0.272439
$846$	0	0
$847$	−93.4183	−3.20989
$848$	−3.04165	−0.104451
$849$	0	0
$850$	−31.3239	−1.07440
$851$	15.1153	0.518146
$852$	0	0
$853$	40.7481	1.39519	0.697595	−	0.716492i	$-0.254254\pi$
$853$	40.7481	1.39519	0.697595	+	0.716492i	$0.254254\pi$
$854$	72.7929	2.49092
$855$	0	0
$856$	56.0168	1.91462
$857$	44.6261	1.52440	0.762200	−	0.647342i	$-0.224119\pi$
$857$	44.6261	1.52440	0.762200	+	0.647342i	$0.224119\pi$
$858$	0	0
$859$	18.5044	0.631361	0.315680	−	0.948866i	$-0.397767\pi$
$859$	18.5044	0.631361	0.315680	+	0.948866i	$0.397767\pi$
$860$	128.625	4.38607
$861$	0	0
$862$	1.26302	0.0430186
$863$	−9.37702	−0.319197	−0.159599	−	0.987182i	$-0.551020\pi$
$863$	−9.37702	−0.319197	−0.159599	+	0.987182i	$0.551020\pi$
$864$	0	0
$865$	65.0655	2.21229
$866$	6.04002	0.205248
$867$	0	0
$868$	−12.0580	−0.409275
$869$	39.7263	1.34762
$870$	0	0
$871$	27.9216	0.946086
$872$	20.6982	0.700929
$873$	0	0
$874$	−22.5285	−0.762037
$875$	10.4520	0.353343
$876$	0	0
$877$	−5.93854	−0.200530	−0.100265	−	0.994961i	$-0.531969\pi$
$877$	−5.93854	−0.200530	−0.100265	+	0.994961i	$0.531969\pi$
$878$	−77.3372	−2.61000
$879$	0	0
$880$	7.61736	0.256781
$881$	−1.02910	−0.0346712	−0.0173356	−	0.999850i	$-0.505518\pi$
$881$	−1.02910	−0.0346712	−0.0173356	+	0.999850i	$0.505518\pi$
$882$	0	0
$883$	−28.3201	−0.953046	−0.476523	−	0.879162i	$-0.658103\pi$
$883$	−28.3201	−0.953046	−0.476523	+	0.879162i	$0.658103\pi$
$884$	42.3610	1.42475
$885$	0	0
$886$	44.2443	1.48642
$887$	11.9079	0.399827	0.199913	−	0.979814i	$-0.435934\pi$
$887$	11.9079	0.399827	0.199913	+	0.979814i	$0.435934\pi$
$888$	0	0
$889$	−25.9179	−0.869258
$890$	10.6101	0.355650
$891$	0	0
$892$	−10.2096	−0.341844
$893$	−0.469067	−0.0156967
$894$	0	0
$895$	50.2203	1.67868
$896$	−84.1376	−2.81084
$897$	0	0
$898$	−60.2744	−2.01138
$899$	0	0
$900$	0	0
$901$	−22.1536	−0.738043
$902$	7.32804	0.243997
$903$	0	0
$904$	14.9140	0.496033
$905$	44.1215	1.46665
$906$	0	0
$907$	−17.7879	−0.590637	−0.295318	−	0.955399i	$-0.595426\pi$
$907$	−17.7879	−0.590637	−0.295318	+	0.955399i	$0.595426\pi$
$908$	12.7564	0.423337
$909$	0	0
$910$	121.660	4.03298
$911$	−41.3338	−1.36945	−0.684725	−	0.728802i	$-0.740078\pi$
$911$	−41.3338	−1.36945	−0.684725	+	0.728802i	$0.740078\pi$
$912$	0	0
$913$	−50.5714	−1.67367
$914$	−23.8853	−0.790057
$915$	0	0
$916$	−7.57172	−0.250177
$917$	−2.98181	−0.0984680
$918$	0	0
$919$	26.3691	0.869837	0.434918	−	0.900470i	$-0.356777\pi$
$919$	26.3691	0.869837	0.434918	+	0.900470i	$0.356777\pi$
$920$	−25.1932	−0.830595
$921$	0	0
$922$	−77.2970	−2.54564
$923$	50.0331	1.64686
$924$	0	0
$925$	23.6349	0.777110
$926$	26.0051	0.854581
$927$	0	0
$928$	0	0
$929$	−40.6550	−1.33385	−0.666923	−	0.745127i	$-0.732389\pi$
$929$	−40.6550	−1.33385	−0.666923	+	0.745127i	$0.732389\pi$
$930$	0	0
$931$	44.4804	1.45778
$932$	18.6405	0.610589
$933$	0	0
$934$	70.7125	2.31378
$935$	55.4804	1.81440
$936$	0	0
$937$	−14.8322	−0.484548	−0.242274	−	0.970208i	$-0.577893\pi$
$937$	−14.8322	−0.484548	−0.242274	+	0.970208i	$0.577893\pi$
$938$	71.6876	2.34068
$939$	0	0
$940$	−1.31172	−0.0427835
$941$	36.7391	1.19766	0.598830	−	0.800876i	$-0.295632\pi$
$941$	36.7391	1.19766	0.598830	+	0.800876i	$0.295632\pi$
$942$	0	0
$943$	−1.50630	−0.0490520
$944$	0.329096	0.0107112
$945$	0	0
$946$	−166.765	−5.42201
$947$	24.7731	0.805019	0.402509	−	0.915416i	$-0.368138\pi$
$947$	24.7731	0.805019	0.402509	+	0.915416i	$0.368138\pi$
$948$	0	0
$949$	−34.8235	−1.13042
$950$	−35.2264	−1.14289
$951$	0	0
$952$	43.4926	1.40960
$953$	25.3471	0.821073	0.410537	−	0.911844i	$-0.365341\pi$
$953$	25.3471	0.821073	0.410537	+	0.911844i	$0.365341\pi$
$954$	0	0
$955$	17.5946	0.569347
$956$	−9.01511	−0.291569
$957$	0	0
$958$	−2.98551	−0.0964573
$959$	−74.4646	−2.40459
$960$	0	0
$961$	−30.3215	−0.978114
$962$	−51.1437	−1.64894
$963$	0	0
$964$	−80.8881	−2.60523
$965$	5.34053	0.171918
$966$	0	0
$967$	−16.1414	−0.519072	−0.259536	−	0.965733i	$-0.583570\pi$
$967$	−16.1414	−0.519072	−0.259536	+	0.965733i	$0.583570\pi$
$968$	−65.4552	−2.10381
$969$	0	0
$970$	103.605	3.32656
$971$	−39.1025	−1.25486	−0.627430	−	0.778673i	$-0.715893\pi$
$971$	−39.1025	−1.25486	−0.627430	+	0.778673i	$0.715893\pi$
$972$	0	0
$973$	24.0033	0.769510
$974$	2.15650	0.0690986
$975$	0	0
$976$	3.16991	0.101466
$977$	−0.922826	−0.0295238	−0.0147619	−	0.999891i	$-0.504699\pi$
$977$	−0.922826	−0.0295238	−0.0147619	+	0.999891i	$0.504699\pi$
$978$	0	0
$979$	−8.59707	−0.274763
$980$	124.387	3.97339
$981$	0	0
$982$	18.2792	0.583314
$983$	31.5174	1.00525	0.502625	−	0.864505i	$-0.332368\pi$
$983$	31.5174	1.00525	0.502625	+	0.864505i	$0.332368\pi$
$984$	0	0
$985$	−72.8494	−2.32117
$986$	0	0
$987$	0	0
$988$	47.6385	1.51558
$989$	34.2792	1.09001
$990$	0	0
$991$	−35.3884	−1.12415	−0.562075	−	0.827086i	$-0.689997\pi$
$991$	−35.3884	−1.12415	−0.562075	+	0.827086i	$0.689997\pi$
$992$	4.22988	0.134299
$993$	0	0
$994$	128.458	4.07444
$995$	−9.08768	−0.288099
$996$	0	0
$997$	−23.5611	−0.746188	−0.373094	−	0.927794i	$-0.621703\pi$
$997$	−23.5611	−0.746188	−0.373094	+	0.927794i	$0.621703\pi$
$998$	−8.99475	−0.284724
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bf.1.1	✓	8
3.2	odd	2	inner	7569.2.a.bf.1.8	yes	8
29.28	even	2		7569.2.a.bg.1.8	yes	8
87.86	odd	2		7569.2.a.bg.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7569.2.a.bf.1.1	✓	8	1.1	even	1	trivial
7569.2.a.bf.1.8	yes	8	3.2	odd	2	inner
7569.2.a.bg.1.1	yes	8	87.86	odd	2
7569.2.a.bg.1.8	yes	8	29.28	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:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30927 q^{2} +3.33275 q^{4} +3.03582 q^{5} -4.39250 q^{7} -3.07768 q^{8} -7.01054 q^{10} +5.68046 q^{11} +3.95078 q^{13} +10.1435 q^{14} +0.441718 q^{16} +3.21721 q^{17} +3.61803 q^{19} +10.1176 q^{20} -13.1178 q^{22} +2.69640 q^{23} +4.21619 q^{25} -9.12344 q^{26} -14.6391 q^{28} +0.823684 q^{31} +5.13532 q^{32} -7.42943 q^{34} -13.3348 q^{35} +5.60575 q^{37} -8.35503 q^{38} -9.34328 q^{40} -0.558636 q^{41} +12.7130 q^{43} +18.9316 q^{44} -6.22672 q^{46} -0.129647 q^{47} +12.2941 q^{49} -9.73633 q^{50} +13.1670 q^{52} -6.88596 q^{53} +17.2449 q^{55} +13.5187 q^{56} +0.745036 q^{59} +7.17632 q^{61} -1.90211 q^{62} -12.7423 q^{64} +11.9939 q^{65} +7.06735 q^{67} +10.7222 q^{68} +30.7938 q^{70} +12.6641 q^{71} -8.81434 q^{73} -12.9452 q^{74} +12.0580 q^{76} -24.9514 q^{77} +6.99349 q^{79} +1.34098 q^{80} +1.29004 q^{82} -8.90268 q^{83} +9.76687 q^{85} -29.3577 q^{86} -17.4827 q^{88} -1.51344 q^{89} -17.3538 q^{91} +8.98641 q^{92} +0.299391 q^{94} +10.9837 q^{95} -14.7785 q^{97} -28.3904 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4} - 12 q^{10} + 2 q^{13} - 2 q^{16} + 20 q^{19} - 14 q^{22} + 2 q^{25} - 20 q^{28} + 10 q^{31} - 36 q^{34} + 18 q^{37} - 10 q^{40} + 28 q^{43} + 26 q^{46} + 4 q^{49} + 44 q^{52} + 14 q^{55}+ \cdots - 6 q^{97}+O(q^{100}) \) 8 * q + 6 * q^4 - 12 * q^10 + 2 * q^13 - 2 * q^16 + 20 * q^19 - 14 * q^22 + 2 * q^25 - 20 * q^28 + 10 * q^31 - 36 * q^34 + 18 * q^37 - 10 * q^40 + 28 * q^43 + 26 * q^46 + 4 * q^49 + 44 * q^52 + 14 * q^55 + 54 * q^61 - 44 * q^64 + 38 * q^67 + 70 * q^70 - 16 * q^73 + 20 * q^76 + 14 * q^79 + 2 * q^82 + 56 * q^85 - 60 * q^88 - 30 * q^91 + 52 * q^94 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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