LMFDB - Embedded newform 7569.2.a.bg.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:	$1$
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	$-0.737512\pi$
$5$	−3.03582	−1.35766	−0.678829	+	0.734296i	$0.737512\pi$
$6$	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	$-0.636224\pi$
$19$	−3.61803	−0.830034	−0.415017	+	0.909814i	$0.636224\pi$
$20$	−10.1176	−2.26237
$21$	0	0
$22$	−13.1178	−2.79671
$23$	−2.69640	−0.562238	−0.281119	−	0.959673i	$-0.590705\pi$
$23$	−2.69640	−0.562238	−0.281119	+	0.959673i	$0.590705\pi$
$24$	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	$-0.523567\pi$
$31$	−0.823684	−0.147938	−0.0739690	+	0.997261i	$0.523567\pi$
$32$	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	$-0.652434\pi$
$37$	−5.60575	−0.921579	−0.460789	+	0.887509i	$0.652434\pi$
$38$	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	$-0.920993\pi$
$43$	−12.7130	−1.93871	−0.969354	+	0.245667i	$0.920993\pi$
$44$	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	$-0.343197\pi$
$53$	6.88596	0.945859	0.472929	+	0.881100i	$0.343197\pi$
$54$	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	$-0.515443\pi$
$59$	−0.745036	−0.0969954	−0.0484977	+	0.998823i	$0.515443\pi$
$60$	0	0
$61$	−7.17632	−0.918833	−0.459417	−	0.888221i	$-0.651941\pi$
$61$	−7.17632	−0.918833	−0.459417	+	0.888221i	$0.651941\pi$
$62$	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	$-0.770657\pi$
$71$	−12.6641	−1.50295	−0.751475	+	0.659761i	$0.770657\pi$
$72$	0	0
$73$	8.81434	1.03164	0.515820	−	0.856697i	$-0.327487\pi$
$73$	8.81434	1.03164	0.515820	+	0.856697i	$0.327487\pi$
$74$	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	$-0.628706\pi$
$79$	−6.99349	−0.786829	−0.393414	+	0.919361i	$0.628706\pi$
$80$	−1.34098	−0.149926
$81$	0	0
$82$	1.29004	0.142461
$83$	8.90268	0.977196	0.488598	−	0.872509i	$-0.337508\pi$
$83$	8.90268	0.977196	0.488598	+	0.872509i	$0.337508\pi$
$84$	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	$-0.229926\pi$
$97$	14.7785	1.50053	0.750264	+	0.661138i	$0.229926\pi$
$98$	−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	$-0.157692\pi$
$107$	18.2010	1.75955	0.879777	+	0.475386i	$0.157692\pi$
$108$	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	$-0.584313\pi$
$127$	−5.90048	−0.523583	−0.261792	+	0.965124i	$0.584313\pi$
$128$	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	$-0.399876\pi$
$149$	7.55310	0.618774	0.309387	+	0.950936i	$0.399876\pi$
$150$	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	$-0.241511\pi$
$157$	18.1863	1.45142	0.725712	+	0.687999i	$0.241511\pi$
$158$	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	$-0.356223\pi$
$163$	11.1453	0.872970	0.436485	+	0.899712i	$0.356223\pi$
$164$	−1.86179	−0.145382
$165$	0	0
$166$	−20.5587	−1.59567
$167$	10.7469	0.831623	0.415811	−	0.909451i	$-0.363498\pi$
$167$	10.7469	0.831623	0.415811	+	0.909451i	$0.363498\pi$
$168$	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	$-0.803124\pi$
$173$	−21.4326	−1.62949	−0.814746	+	0.579818i	$0.803124\pi$
$174$	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	$-0.712148\pi$
$179$	−16.5426	−1.23645	−0.618226	+	0.786000i	$0.712148\pi$
$180$	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	$-0.520167\pi$
$193$	−1.75917	−0.126628	−0.0633140	+	0.997994i	$0.520167\pi$
$194$	−34.1276	−2.45022
$195$	0	0
$196$	40.9730	2.92665
$197$	23.9966	1.70969	0.854845	−	0.518884i	$-0.173652\pi$
$197$	23.9966	1.70969	0.854845	+	0.518884i	$0.173652\pi$
$198$	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	$-0.649273\pi$
$211$	−13.1300	−0.903909	−0.451955	+	0.892041i	$0.649273\pi$
$212$	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	$-0.540542\pi$
$227$	−3.82760	−0.254047	−0.127023	+	0.991900i	$0.540542\pi$
$228$	0	0
$229$	2.27191	0.150132	0.0750661	−	0.997179i	$-0.476083\pi$
$229$	2.27191	0.150132	0.0750661	+	0.997179i	$0.476083\pi$
$230$	−18.9032	−1.24644
$231$	0	0
$232$	0	0
$233$	−5.59312	−0.366418	−0.183209	−	0.983074i	$-0.558648\pi$
$233$	−5.59312	−0.366418	−0.183209	+	0.983074i	$0.558648\pi$
$234$	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	$-0.472117\pi$
$239$	2.70501	0.174972	0.0874862	+	0.996166i	$0.472117\pi$
$240$	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	$-0.237755\pi$
$257$	23.5267	1.46756	0.733779	+	0.679388i	$0.237755\pi$
$258$	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	$-0.640524\pi$
$271$	−14.0675	−0.854537	−0.427268	+	0.904125i	$0.640524\pi$
$272$	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	$-0.593984\pi$
$281$	−9.75566	−0.581974	−0.290987	+	0.956727i	$0.593984\pi$
$282$	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	$-0.587294\pi$
$307$	−9.49020	−0.541634	−0.270817	+	0.962631i	$0.587294\pi$
$308$	−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	$-0.479553\pi$
$331$	2.33575	0.128385	0.0641923	+	0.997938i	$0.479553\pi$
$332$	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	$-0.520517\pi$
$337$	−2.36492	−0.128825	−0.0644127	+	0.997923i	$0.520517\pi$
$338$	−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	$-0.128444\pi$
$347$	34.2637	1.83937	0.919685	+	0.392657i	$0.128444\pi$
$348$	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	$-0.584728\pi$
$353$	−9.88447	−0.526097	−0.263049	+	0.964783i	$0.584728\pi$
$354$	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	$-0.694079\pi$
$367$	−21.9402	−1.14527	−0.572636	+	0.819810i	$0.694079\pi$
$368$	−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	$-0.588049\pi$
$379$	−10.6333	−0.546197	−0.273099	+	0.961986i	$0.588049\pi$
$380$	36.6059	1.87784
$381$	0	0
$382$	−13.3838	−0.684774
$383$	−24.6435	−1.25922	−0.629612	−	0.776910i	$-0.716786\pi$
$383$	−24.6435	−1.25922	−0.629612	+	0.776910i	$0.716786\pi$
$384$	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	$-0.329969\pi$
$401$	20.3905	1.01825	0.509126	+	0.860692i	$0.329969\pi$
$402$	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	$-0.472034\pi$
$409$	3.54900	0.175487	0.0877434	+	0.996143i	$0.472034\pi$
$410$	−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	$-0.417266\pi$
$419$	10.5213	0.514000	0.257000	+	0.966411i	$0.417266\pi$
$420$	0	0
$421$	8.27884	0.403486	0.201743	−	0.979439i	$-0.435339\pi$
$421$	8.27884	0.403486	0.201743	+	0.979439i	$0.435339\pi$
$422$	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	$-0.495807\pi$
$431$	0.546934	0.0263449	0.0131724	+	0.999913i	$0.495807\pi$
$432$	0	0
$433$	2.61555	0.125695	0.0628476	−	0.998023i	$-0.479982\pi$
$433$	2.61555	0.125695	0.0628476	+	0.998023i	$0.479982\pi$
$434$	−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	$-0.497898\pi$
$509$	0.297945	0.0132062	0.00660309	+	0.999978i	$0.497898\pi$
$510$	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	$-0.343096\pi$
$521$	21.6023	0.946415	0.473208	+	0.880951i	$0.343096\pi$
$522$	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	$-0.259048\pi$
$541$	31.9456	1.37345	0.686725	+	0.726917i	$0.259048\pi$
$542$	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	$-0.548272\pi$
$557$	−7.13081	−0.302142	−0.151071	+	0.988523i	$0.548272\pi$
$558$	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	$-0.428412\pi$
$577$	10.7137	0.446017	0.223009	+	0.974816i	$0.428412\pi$
$578$	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	$-0.164732\pi$
$587$	42.1108	1.73810	0.869049	+	0.494726i	$0.164732\pi$
$588$	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	$-0.832806\pi$
$593$	−42.1378	−1.73039	−0.865196	+	0.501433i	$0.832806\pi$
$594$	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	$-0.589940\pi$
$601$	−13.6703	−0.557621	−0.278811	+	0.960346i	$0.589940\pi$
$602$	−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	$-0.394556\pi$
$607$	16.0259	0.650471	0.325235	+	0.945633i	$0.394556\pi$
$608$	−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	$-0.843109\pi$
$619$	−43.8366	−1.76194	−0.880970	+	0.473172i	$0.843109\pi$
$620$	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	$-0.371959\pi$
$647$	19.9161	0.782982	0.391491	+	0.920182i	$0.371959\pi$
$648$	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	$-0.528383\pi$
$683$	−4.65446	−0.178098	−0.0890489	+	0.996027i	$0.528383\pi$
$684$	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	$-0.389980\pi$
$701$	17.9403	0.677597	0.338798	+	0.940859i	$0.389980\pi$
$702$	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	$-0.821802\pi$
$719$	−45.4419	−1.69470	−0.847348	+	0.531039i	$0.821802\pi$
$720$	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	$-0.757593\pi$
$727$	−39.0300	−1.44754	−0.723771	+	0.690041i	$0.757593\pi$
$728$	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	$-0.625580\pi$
$733$	−20.8126	−0.768730	−0.384365	+	0.923181i	$0.625580\pi$
$734$	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	$-0.759478\pi$
$739$	−39.5723	−1.45569	−0.727845	+	0.685742i	$0.759478\pi$
$740$	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	$-0.558992\pi$
$751$	−10.0995	−0.368537	−0.184268	+	0.982876i	$0.558992\pi$
$752$	−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	$-0.665565\pi$
$757$	−27.3486	−0.994000	−0.497000	+	0.867750i	$0.665565\pi$
$758$	24.5553	0.891888
$759$	0	0
$760$	−33.8043	−1.22621
$761$	−20.1609	−0.730834	−0.365417	−	0.930844i	$-0.619074\pi$
$761$	−20.1609	−0.730834	−0.365417	+	0.930844i	$0.619074\pi$
$762$	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	$-0.524444\pi$
$769$	−4.25483	−0.153433	−0.0767165	+	0.997053i	$0.524444\pi$
$770$	−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	$-0.535600\pi$
$821$	−6.39578	−0.223214	−0.111607	+	0.993752i	$0.535600\pi$
$822$	0	0
$823$	19.8845	0.693131	0.346565	−	0.938026i	$-0.387348\pi$
$823$	19.8845	0.693131	0.346565	+	0.938026i	$0.387348\pi$
$824$	−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	$-0.525429\pi$
$829$	−4.59538	−0.159604	−0.0798021	+	0.996811i	$0.525429\pi$
$830$	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	$-0.745746\pi$
$853$	−40.7481	−1.39519	−0.697595	+	0.716492i	$0.745746\pi$
$854$	−72.7929	−2.49092
$855$	0	0
$856$	−56.0168	−1.91462
$857$	−44.6261	−1.52440	−0.762200	−	0.647342i	$-0.775881\pi$
$857$	−44.6261	−1.52440	−0.762200	+	0.647342i	$0.775881\pi$
$858$	0	0
$859$	−18.5044	−0.631361	−0.315680	−	0.948866i	$-0.602233\pi$
$859$	−18.5044	−0.631361	−0.315680	+	0.948866i	$0.602233\pi$
$860$	128.625	4.38607
$861$	0	0
$862$	−1.26302	−0.0430186
$863$	9.37702	0.319197	0.159599	−	0.987182i	$-0.448980\pi$
$863$	9.37702	0.319197	0.159599	+	0.987182i	$0.448980\pi$
$864$	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	$-0.404574\pi$
$907$	17.7879	0.590637	0.295318	+	0.955399i	$0.404574\pi$
$908$	−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	$-0.267611\pi$
$929$	40.6550	1.33385	0.666923	+	0.745127i	$0.267611\pi$
$930$	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	$-0.704368\pi$
$941$	−36.7391	−1.19766	−0.598830	+	0.800876i	$0.704368\pi$
$942$	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	$-0.634659\pi$
$953$	−25.3471	−0.821073	−0.410537	+	0.911844i	$0.634659\pi$
$954$	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	$-0.416430\pi$
$967$	16.1414	0.519072	0.259536	+	0.965733i	$0.416430\pi$
$968$	−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	$-0.495301\pi$
$977$	0.922826	0.0295238	0.0147619	+	0.999891i	$0.495301\pi$
$978$	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	$-0.378297\pi$
$997$	23.5611	0.746188	0.373094	+	0.927794i	$0.378297\pi$
$998$	−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.bg.1.1	yes	8
3.2	odd	2	inner	7569.2.a.bg.1.8	yes	8
29.28	even	2		7569.2.a.bf.1.8	yes	8
87.86	odd	2		7569.2.a.bf.1.1	✓	8

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

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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