LMFDB - Embedded newform 4761.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4761,2,Mod(1,4761)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4761, 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("4761.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.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:	$38.0167764023$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{2} +3.00000 q^{4} -3.23607 q^{5} +1.23607 q^{7} -2.23607 q^{8} +7.23607 q^{10} +4.00000 q^{11} +4.47214 q^{13} -2.76393 q^{14} -1.00000 q^{16} -7.23607 q^{17} -2.76393 q^{19} -9.70820 q^{20} -8.94427 q^{22} +5.47214 q^{25} -10.0000 q^{26} +3.70820 q^{28} +4.47214 q^{29} +2.47214 q^{31} +6.70820 q^{32} +16.1803 q^{34} -4.00000 q^{35} +4.47214 q^{37} +6.18034 q^{38} +7.23607 q^{40} -6.94427 q^{41} -7.70820 q^{43} +12.0000 q^{44} +4.00000 q^{47} -5.47214 q^{49} -12.2361 q^{50} +13.4164 q^{52} -0.763932 q^{53} -12.9443 q^{55} -2.76393 q^{56} -10.0000 q^{58} -12.9443 q^{59} +4.47214 q^{61} -5.52786 q^{62} -13.0000 q^{64} -14.4721 q^{65} -5.23607 q^{67} -21.7082 q^{68} +8.94427 q^{70} +8.00000 q^{71} -10.9443 q^{73} -10.0000 q^{74} -8.29180 q^{76} +4.94427 q^{77} +3.70820 q^{79} +3.23607 q^{80} +15.5279 q^{82} +4.00000 q^{83} +23.4164 q^{85} +17.2361 q^{86} -8.94427 q^{88} +3.23607 q^{89} +5.52786 q^{91} -8.94427 q^{94} +8.94427 q^{95} +0.472136 q^{97} +12.2361 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{4} - 2 q^{5} - 2 q^{7} + 10 q^{10} + 8 q^{11} - 10 q^{14} - 2 q^{16} - 10 q^{17} - 10 q^{19} - 6 q^{20} + 2 q^{25} - 20 q^{26} - 6 q^{28} - 4 q^{31} + 10 q^{34} - 8 q^{35} - 10 q^{38} + 10 q^{40}+ \cdots + 20 q^{98}+O(q^{100}) $ 2 * q + 6 * q^4 - 2 * q^5 - 2 * q^7 + 10 * q^10 + 8 * q^11 - 10 * q^14 - 2 * q^16 - 10 * q^17 - 10 * q^19 - 6 * q^20 + 2 * q^25 - 20 * q^26 - 6 * q^28 - 4 * q^31 + 10 * q^34 - 8 * q^35 - 10 * q^38 + 10 * q^40 + 4 * q^41 - 2 * q^43 + 24 * q^44 + 8 * q^47 - 2 * q^49 - 20 * q^50 - 6 * q^53 - 8 * q^55 - 10 * q^56 - 20 * q^58 - 8 * q^59 - 20 * q^62 - 26 * q^64 - 20 * q^65 - 6 * q^67 - 30 * q^68 + 16 * q^71 - 4 * q^73 - 20 * q^74 - 30 * q^76 - 8 * q^77 - 6 * q^79 + 2 * q^80 + 40 * q^82 + 8 * q^83 + 20 * q^85 + 30 * q^86 + 2 * q^89 + 20 * q^91 - 8 * q^97 + 20 * 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 $

$2$	−2.23607	−1.58114	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607	−1.58114	−0.790569	+	0.612372i	$0.790215\pi$
$3$	0	0
$3$	0	0
$4$	3.00000	1.50000
$5$	−3.23607	−1.44721	−0.723607	−	0.690212i	$-0.757517\pi$
$5$	−3.23607	−1.44721	−0.723607	+	0.690212i	$0.757517\pi$
$6$	0	0
$7$	1.23607	0.467190	0.233595	−	0.972334i	$-0.424951\pi$
$7$	1.23607	0.467190	0.233595	+	0.972334i	$0.424951\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	7.23607	2.28825
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	−2.76393	−0.738692
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−7.23607	−1.75500	−0.877502	−	0.479573i	$-0.840792\pi$
$17$	−7.23607	−1.75500	−0.877502	+	0.479573i	$0.840792\pi$
$18$	0	0
$19$	−2.76393	−0.634089	−0.317045	−	0.948411i	$-0.602691\pi$
$19$	−2.76393	−0.634089	−0.317045	+	0.948411i	$0.602691\pi$
$20$	−9.70820	−2.17082
$21$	0	0
$22$	−8.94427	−1.90693
$23$	0	0
$23$	0	0
$24$	0	0
$25$	5.47214	1.09443
$26$	−10.0000	−1.96116
$27$	0	0
$28$	3.70820	0.700785
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	6.70820	1.18585
$33$	0	0
$34$	16.1803	2.77491
$35$	−4.00000	−0.676123
$36$	0	0
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	6.18034	1.00258
$39$	0	0
$40$	7.23607	1.14412
$41$	−6.94427	−1.08451	−0.542257	−	0.840213i	$-0.682430\pi$
$41$	−6.94427	−1.08451	−0.542257	+	0.840213i	$0.682430\pi$
$42$	0	0
$43$	−7.70820	−1.17549	−0.587745	−	0.809046i	$-0.699984\pi$
$43$	−7.70820	−1.17549	−0.587745	+	0.809046i	$0.699984\pi$
$44$	12.0000	1.80907
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	−12.2361	−1.73044
$51$	0	0
$52$	13.4164	1.86052
$53$	−0.763932	−0.104934	−0.0524671	−	0.998623i	$-0.516708\pi$
$53$	−0.763932	−0.104934	−0.0524671	+	0.998623i	$0.516708\pi$
$54$	0	0
$55$	−12.9443	−1.74541
$56$	−2.76393	−0.369346
$57$	0	0
$58$	−10.0000	−1.31306
$59$	−12.9443	−1.68520	−0.842600	−	0.538539i	$-0.818976\pi$
$59$	−12.9443	−1.68520	−0.842600	+	0.538539i	$0.818976\pi$
$60$	0	0
$61$	4.47214	0.572598	0.286299	−	0.958140i	$-0.407575\pi$
$61$	4.47214	0.572598	0.286299	+	0.958140i	$0.407575\pi$
$62$	−5.52786	−0.702039
$63$	0	0
$64$	−13.0000	−1.62500
$65$	−14.4721	−1.79505
$66$	0	0
$67$	−5.23607	−0.639688	−0.319844	−	0.947470i	$-0.603630\pi$
$67$	−5.23607	−0.639688	−0.319844	+	0.947470i	$0.603630\pi$
$68$	−21.7082	−2.63251
$69$	0	0
$70$	8.94427	1.06904
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−10.9443	−1.28093	−0.640465	−	0.767987i	$-0.721258\pi$
$73$	−10.9443	−1.28093	−0.640465	+	0.767987i	$0.721258\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−8.29180	−0.951134
$77$	4.94427	0.563452
$78$	0	0
$79$	3.70820	0.417206	0.208603	−	0.978000i	$-0.433108\pi$
$79$	3.70820	0.417206	0.208603	+	0.978000i	$0.433108\pi$
$80$	3.23607	0.361803
$81$	0	0
$82$	15.5279	1.71477
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	23.4164	2.53987
$86$	17.2361	1.85861
$87$	0	0
$88$	−8.94427	−0.953463
$89$	3.23607	0.343023	0.171511	−	0.985182i	$-0.445135\pi$
$89$	3.23607	0.343023	0.171511	+	0.985182i	$0.445135\pi$
$90$	0	0
$91$	5.52786	0.579478
$92$	0	0
$93$	0	0
$94$	−8.94427	−0.922531
$95$	8.94427	0.917663
$96$	0	0
$97$	0.472136	0.0479381	0.0239691	−	0.999713i	$-0.492370\pi$
$97$	0.472136	0.0479381	0.0239691	+	0.999713i	$0.492370\pi$
$98$	12.2361	1.23603
$99$	0	0
$100$	16.4164	1.64164
$101$	−10.9443	−1.08900	−0.544498	−	0.838762i	$-0.683280\pi$
$101$	−10.9443	−1.08900	−0.544498	+	0.838762i	$0.683280\pi$
$102$	0	0
$103$	6.76393	0.666470	0.333235	−	0.942844i	$-0.391860\pi$
$103$	6.76393	0.666470	0.333235	+	0.942844i	$0.391860\pi$
$104$	−10.0000	−0.980581
$105$	0	0
$106$	1.70820	0.165915
$107$	−0.944272	−0.0912862	−0.0456431	−	0.998958i	$-0.514534\pi$
$107$	−0.944272	−0.0912862	−0.0456431	+	0.998958i	$0.514534\pi$
$108$	0	0
$109$	−2.94427	−0.282010	−0.141005	−	0.990009i	$-0.545033\pi$
$109$	−2.94427	−0.282010	−0.141005	+	0.990009i	$0.545033\pi$
$110$	28.9443	2.75973
$111$	0	0
$112$	−1.23607	−0.116797
$113$	16.1803	1.52212	0.761059	−	0.648682i	$-0.224680\pi$
$113$	16.1803	1.52212	0.761059	+	0.648682i	$0.224680\pi$
$114$	0	0
$115$	0	0
$116$	13.4164	1.24568
$117$	0	0
$118$	28.9443	2.66454
$119$	−8.94427	−0.819920
$120$	0	0
$121$	5.00000	0.454545
$122$	−10.0000	−0.905357
$123$	0	0
$124$	7.41641	0.666013
$125$	−1.52786	−0.136656
$126$	0	0
$127$	10.4721	0.929252	0.464626	−	0.885507i	$-0.346189\pi$
$127$	10.4721	0.929252	0.464626	+	0.885507i	$0.346189\pi$
$128$	15.6525	1.38350
$129$	0	0
$130$	32.3607	2.83822
$131$	0.944272	0.0825014	0.0412507	−	0.999149i	$-0.486866\pi$
$131$	0.944272	0.0825014	0.0412507	+	0.999149i	$0.486866\pi$
$132$	0	0
$133$	−3.41641	−0.296240
$134$	11.7082	1.01143
$135$	0	0
$136$	16.1803	1.38745
$137$	3.23607	0.276476	0.138238	−	0.990399i	$-0.455856\pi$
$137$	3.23607	0.276476	0.138238	+	0.990399i	$0.455856\pi$
$138$	0	0
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	−12.0000	−1.01419
$141$	0	0
$142$	−17.8885	−1.50117
$143$	17.8885	1.49592
$144$	0	0
$145$	−14.4721	−1.20185
$146$	24.4721	2.02533
$147$	0	0
$148$	13.4164	1.10282
$149$	1.70820	0.139942	0.0699708	−	0.997549i	$-0.477709\pi$
$149$	1.70820	0.139942	0.0699708	+	0.997549i	$0.477709\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	6.18034	0.501292
$153$	0	0
$154$	−11.0557	−0.890896
$155$	−8.00000	−0.642575
$156$	0	0
$157$	12.4721	0.995385	0.497692	−	0.867354i	$-0.334181\pi$
$157$	12.4721	0.995385	0.497692	+	0.867354i	$0.334181\pi$
$158$	−8.29180	−0.659660
$159$	0	0
$160$	−21.7082	−1.71618
$161$	0	0
$162$	0	0
$163$	−19.4164	−1.52081	−0.760405	−	0.649449i	$-0.775000\pi$
$163$	−19.4164	−1.52081	−0.760405	+	0.649449i	$0.775000\pi$
$164$	−20.8328	−1.62677
$165$	0	0
$166$	−8.94427	−0.694210
$167$	4.94427	0.382599	0.191300	−	0.981532i	$-0.438730\pi$
$167$	4.94427	0.382599	0.191300	+	0.981532i	$0.438730\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	−52.3607	−4.01588
$171$	0	0
$172$	−23.1246	−1.76324
$173$	4.47214	0.340010	0.170005	−	0.985443i	$-0.445622\pi$
$173$	4.47214	0.340010	0.170005	+	0.985443i	$0.445622\pi$
$174$	0	0
$175$	6.76393	0.511305
$176$	−4.00000	−0.301511
$177$	0	0
$178$	−7.23607	−0.542366
$179$	−3.05573	−0.228396	−0.114198	−	0.993458i	$-0.536430\pi$
$179$	−3.05573	−0.228396	−0.114198	+	0.993458i	$0.536430\pi$
$180$	0	0
$181$	−23.8885	−1.77562	−0.887811	−	0.460209i	$-0.847775\pi$
$181$	−23.8885	−1.77562	−0.887811	+	0.460209i	$0.847775\pi$
$182$	−12.3607	−0.916235
$183$	0	0
$184$	0	0
$185$	−14.4721	−1.06401
$186$	0	0
$187$	−28.9443	−2.11661
$188$	12.0000	0.875190
$189$	0	0
$190$	−20.0000	−1.45095
$191$	10.4721	0.757737	0.378869	−	0.925450i	$-0.376313\pi$
$191$	10.4721	0.757737	0.378869	+	0.925450i	$0.376313\pi$
$192$	0	0
$193$	−9.41641	−0.677808	−0.338904	−	0.940821i	$-0.610056\pi$
$193$	−9.41641	−0.677808	−0.338904	+	0.940821i	$0.610056\pi$
$194$	−1.05573	−0.0757969
$195$	0	0
$196$	−16.4164	−1.17260
$197$	9.41641	0.670891	0.335446	−	0.942060i	$-0.391113\pi$
$197$	9.41641	0.670891	0.335446	+	0.942060i	$0.391113\pi$
$198$	0	0
$199$	3.70820	0.262868	0.131434	−	0.991325i	$-0.458042\pi$
$199$	3.70820	0.262868	0.131434	+	0.991325i	$0.458042\pi$
$200$	−12.2361	−0.865221
$201$	0	0
$202$	24.4721	1.72185
$203$	5.52786	0.387980
$204$	0	0
$205$	22.4721	1.56952
$206$	−15.1246	−1.05378
$207$	0	0
$208$	−4.47214	−0.310087
$209$	−11.0557	−0.764741
$210$	0	0
$211$	22.4721	1.54705	0.773523	−	0.633768i	$-0.218493\pi$
$211$	22.4721	1.54705	0.773523	+	0.633768i	$0.218493\pi$
$212$	−2.29180	−0.157401
$213$	0	0
$214$	2.11146	0.144336
$215$	24.9443	1.70119
$216$	0	0
$217$	3.05573	0.207436
$218$	6.58359	0.445897
$219$	0	0
$220$	−38.8328	−2.61811
$221$	−32.3607	−2.17681
$222$	0	0
$223$	9.88854	0.662186	0.331093	−	0.943598i	$-0.392583\pi$
$223$	9.88854	0.662186	0.331093	+	0.943598i	$0.392583\pi$
$224$	8.29180	0.554019
$225$	0	0
$226$	−36.1803	−2.40668
$227$	−22.4721	−1.49153	−0.745764	−	0.666210i	$-0.767915\pi$
$227$	−22.4721	−1.49153	−0.745764	+	0.666210i	$0.767915\pi$
$228$	0	0
$229$	14.9443	0.987545	0.493773	−	0.869591i	$-0.335618\pi$
$229$	14.9443	0.987545	0.493773	+	0.869591i	$0.335618\pi$
$230$	0	0
$231$	0	0
$232$	−10.0000	−0.656532
$233$	14.0000	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	−12.9443	−0.844391
$236$	−38.8328	−2.52780
$237$	0	0
$238$	20.0000	1.29641
$239$	12.9443	0.837295	0.418648	−	0.908149i	$-0.362504\pi$
$239$	12.9443	0.837295	0.418648	+	0.908149i	$0.362504\pi$
$240$	0	0
$241$	−28.4721	−1.83405	−0.917026	−	0.398828i	$-0.869417\pi$
$241$	−28.4721	−1.83405	−0.917026	+	0.398828i	$0.869417\pi$
$242$	−11.1803	−0.718699
$243$	0	0
$244$	13.4164	0.858898
$245$	17.7082	1.13134
$246$	0	0
$247$	−12.3607	−0.786491
$248$	−5.52786	−0.351020
$249$	0	0
$250$	3.41641	0.216073
$251$	−1.52786	−0.0964379	−0.0482190	−	0.998837i	$-0.515355\pi$
$251$	−1.52786	−0.0964379	−0.0482190	+	0.998837i	$0.515355\pi$
$252$	0	0
$253$	0	0
$254$	−23.4164	−1.46928
$255$	0	0
$256$	−9.00000	−0.562500
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	0	0
$259$	5.52786	0.343485
$260$	−43.4164	−2.69257
$261$	0	0
$262$	−2.11146	−0.130446
$263$	−2.47214	−0.152438	−0.0762192	−	0.997091i	$-0.524285\pi$
$263$	−2.47214	−0.152438	−0.0762192	+	0.997091i	$0.524285\pi$
$264$	0	0
$265$	2.47214	0.151862
$266$	7.63932	0.468397
$267$	0	0
$268$	−15.7082	−0.959531
$269$	−8.47214	−0.516555	−0.258278	−	0.966071i	$-0.583155\pi$
$269$	−8.47214	−0.516555	−0.258278	+	0.966071i	$0.583155\pi$
$270$	0	0
$271$	−26.4721	−1.60807	−0.804034	−	0.594583i	$-0.797317\pi$
$271$	−26.4721	−1.60807	−0.804034	+	0.594583i	$0.797317\pi$
$272$	7.23607	0.438751
$273$	0	0
$274$	−7.23607	−0.437147
$275$	21.8885	1.31993
$276$	0	0
$277$	−19.8885	−1.19499	−0.597493	−	0.801874i	$-0.703837\pi$
$277$	−19.8885	−1.19499	−0.597493	+	0.801874i	$0.703837\pi$
$278$	−20.0000	−1.19952
$279$	0	0
$280$	8.94427	0.534522
$281$	−6.65248	−0.396853	−0.198427	−	0.980116i	$-0.563583\pi$
$281$	−6.65248	−0.396853	−0.198427	+	0.980116i	$0.563583\pi$
$282$	0	0
$283$	7.70820	0.458205	0.229103	−	0.973402i	$-0.426421\pi$
$283$	7.70820	0.458205	0.229103	+	0.973402i	$0.426421\pi$
$284$	24.0000	1.42414
$285$	0	0
$286$	−40.0000	−2.36525
$287$	−8.58359	−0.506673
$288$	0	0
$289$	35.3607	2.08004
$290$	32.3607	1.90028
$291$	0	0
$292$	−32.8328	−1.92140
$293$	−5.70820	−0.333477	−0.166738	−	0.986001i	$-0.553324\pi$
$293$	−5.70820	−0.333477	−0.166738	+	0.986001i	$0.553324\pi$
$294$	0	0
$295$	41.8885	2.43885
$296$	−10.0000	−0.581238
$297$	0	0
$298$	−3.81966	−0.221267
$299$	0	0
$300$	0	0
$301$	−9.52786	−0.549177
$302$	35.7771	2.05874
$303$	0	0
$304$	2.76393	0.158522
$305$	−14.4721	−0.828672
$306$	0	0
$307$	−6.47214	−0.369384	−0.184692	−	0.982796i	$-0.559129\pi$
$307$	−6.47214	−0.369384	−0.184692	+	0.982796i	$0.559129\pi$
$308$	14.8328	0.845178
$309$	0	0
$310$	17.8885	1.01600
$311$	−24.9443	−1.41446	−0.707230	−	0.706984i	$-0.750055\pi$
$311$	−24.9443	−1.41446	−0.707230	+	0.706984i	$0.750055\pi$
$312$	0	0
$313$	−22.9443	−1.29689	−0.648443	−	0.761263i	$-0.724580\pi$
$313$	−22.9443	−1.29689	−0.648443	+	0.761263i	$0.724580\pi$
$314$	−27.8885	−1.57384
$315$	0	0
$316$	11.1246	0.625808
$317$	−29.4164	−1.65219	−0.826095	−	0.563531i	$-0.809443\pi$
$317$	−29.4164	−1.65219	−0.826095	+	0.563531i	$0.809443\pi$
$318$	0	0
$319$	17.8885	1.00157
$320$	42.0689	2.35172
$321$	0	0
$322$	0	0
$323$	20.0000	1.11283
$324$	0	0
$325$	24.4721	1.35747
$326$	43.4164	2.40461
$327$	0	0
$328$	15.5279	0.857383
$329$	4.94427	0.272587
$330$	0	0
$331$	−3.41641	−0.187783	−0.0938914	−	0.995582i	$-0.529931\pi$
$331$	−3.41641	−0.187783	−0.0938914	+	0.995582i	$0.529931\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−11.0557	−0.604943
$335$	16.9443	0.925764
$336$	0	0
$337$	−30.3607	−1.65385	−0.826926	−	0.562311i	$-0.809912\pi$
$337$	−30.3607	−1.65385	−0.826926	+	0.562311i	$0.809912\pi$
$338$	−15.6525	−0.851382
$339$	0	0
$340$	70.2492	3.80980
$341$	9.88854	0.535495
$342$	0	0
$343$	−15.4164	−0.832408
$344$	17.2361	0.929307
$345$	0	0
$346$	−10.0000	−0.537603
$347$	−25.8885	−1.38977	−0.694885	−	0.719121i	$-0.744545\pi$
$347$	−25.8885	−1.38977	−0.694885	+	0.719121i	$0.744545\pi$
$348$	0	0
$349$	−32.4721	−1.73819	−0.869097	−	0.494642i	$-0.835299\pi$
$349$	−32.4721	−1.73819	−0.869097	+	0.494642i	$0.835299\pi$
$350$	−15.1246	−0.808445
$351$	0	0
$352$	26.8328	1.43019
$353$	−9.41641	−0.501185	−0.250592	−	0.968093i	$-0.580625\pi$
$353$	−9.41641	−0.501185	−0.250592	+	0.968093i	$0.580625\pi$
$354$	0	0
$355$	−25.8885	−1.37402
$356$	9.70820	0.514534
$357$	0	0
$358$	6.83282	0.361126
$359$	−2.47214	−0.130474	−0.0652372	−	0.997870i	$-0.520780\pi$
$359$	−2.47214	−0.130474	−0.0652372	+	0.997870i	$0.520780\pi$
$360$	0	0
$361$	−11.3607	−0.597931
$362$	53.4164	2.80750
$363$	0	0
$364$	16.5836	0.869216
$365$	35.4164	1.85378
$366$	0	0
$367$	−12.2918	−0.641627	−0.320813	−	0.947142i	$-0.603956\pi$
$367$	−12.2918	−0.641627	−0.320813	+	0.947142i	$0.603956\pi$
$368$	0	0
$369$	0	0
$370$	32.3607	1.68235
$371$	−0.944272	−0.0490242
$372$	0	0
$373$	−11.5279	−0.596890	−0.298445	−	0.954427i	$-0.596468\pi$
$373$	−11.5279	−0.596890	−0.298445	+	0.954427i	$0.596468\pi$
$374$	64.7214	3.34666
$375$	0	0
$376$	−8.94427	−0.461266
$377$	20.0000	1.03005
$378$	0	0
$379$	28.0689	1.44180	0.720901	−	0.693038i	$-0.243728\pi$
$379$	28.0689	1.44180	0.720901	+	0.693038i	$0.243728\pi$
$380$	26.8328	1.37649
$381$	0	0
$382$	−23.4164	−1.19809
$383$	34.4721	1.76144	0.880722	−	0.473634i	$-0.157058\pi$
$383$	34.4721	1.76144	0.880722	+	0.473634i	$0.157058\pi$
$384$	0	0
$385$	−16.0000	−0.815436
$386$	21.0557	1.07171
$387$	0	0
$388$	1.41641	0.0719072
$389$	−10.6525	−0.540102	−0.270051	−	0.962846i	$-0.587041\pi$
$389$	−10.6525	−0.540102	−0.270051	+	0.962846i	$0.587041\pi$
$390$	0	0
$391$	0	0
$392$	12.2361	0.618015
$393$	0	0
$394$	−21.0557	−1.06077
$395$	−12.0000	−0.603786
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−8.29180	−0.415630
$399$	0	0
$400$	−5.47214	−0.273607
$401$	−38.0689	−1.90107	−0.950535	−	0.310618i	$-0.899464\pi$
$401$	−38.0689	−1.90107	−0.950535	+	0.310618i	$0.899464\pi$
$402$	0	0
$403$	11.0557	0.550725
$404$	−32.8328	−1.63349
$405$	0	0
$406$	−12.3607	−0.613450
$407$	17.8885	0.886702
$408$	0	0
$409$	−9.41641	−0.465611	−0.232806	−	0.972523i	$-0.574791\pi$
$409$	−9.41641	−0.465611	−0.232806	+	0.972523i	$0.574791\pi$
$410$	−50.2492	−2.48163
$411$	0	0
$412$	20.2918	0.999705
$413$	−16.0000	−0.787309
$414$	0	0
$415$	−12.9443	−0.635409
$416$	30.0000	1.47087
$417$	0	0
$418$	24.7214	1.20916
$419$	−21.8885	−1.06933	−0.534663	−	0.845066i	$-0.679561\pi$
$419$	−21.8885	−1.06933	−0.534663	+	0.845066i	$0.679561\pi$
$420$	0	0
$421$	13.0557	0.636297	0.318149	−	0.948041i	$-0.396939\pi$
$421$	13.0557	0.636297	0.318149	+	0.948041i	$0.396939\pi$
$422$	−50.2492	−2.44609
$423$	0	0
$424$	1.70820	0.0829577
$425$	−39.5967	−1.92072
$426$	0	0
$427$	5.52786	0.267512
$428$	−2.83282	−0.136929
$429$	0	0
$430$	−55.7771	−2.68981
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	0	0
$433$	7.88854	0.379099	0.189550	−	0.981871i	$-0.439297\pi$
$433$	7.88854	0.379099	0.189550	+	0.981871i	$0.439297\pi$
$434$	−6.83282	−0.327986
$435$	0	0
$436$	−8.83282	−0.423015
$437$	0	0
$438$	0	0
$439$	25.8885	1.23559	0.617796	−	0.786338i	$-0.288026\pi$
$439$	25.8885	1.23559	0.617796	+	0.786338i	$0.288026\pi$
$440$	28.9443	1.37986
$441$	0	0
$442$	72.3607	3.44185
$443$	−18.8328	−0.894774	−0.447387	−	0.894340i	$-0.647645\pi$
$443$	−18.8328	−0.894774	−0.447387	+	0.894340i	$0.647645\pi$
$444$	0	0
$445$	−10.4721	−0.496427
$446$	−22.1115	−1.04701
$447$	0	0
$448$	−16.0689	−0.759183
$449$	2.94427	0.138949	0.0694744	−	0.997584i	$-0.477868\pi$
$449$	2.94427	0.138949	0.0694744	+	0.997584i	$0.477868\pi$
$450$	0	0
$451$	−27.7771	−1.30797
$452$	48.5410	2.28318
$453$	0	0
$454$	50.2492	2.35831
$455$	−17.8885	−0.838628
$456$	0	0
$457$	0.472136	0.0220856	0.0110428	−	0.999939i	$-0.496485\pi$
$457$	0.472136	0.0220856	0.0110428	+	0.999939i	$0.496485\pi$
$458$	−33.4164	−1.56145
$459$	0	0
$460$	0	0
$461$	−21.4164	−0.997462	−0.498731	−	0.866757i	$-0.666200\pi$
$461$	−21.4164	−0.997462	−0.498731	+	0.866757i	$0.666200\pi$
$462$	0	0
$463$	4.94427	0.229780	0.114890	−	0.993378i	$-0.463348\pi$
$463$	4.94427	0.229780	0.114890	+	0.993378i	$0.463348\pi$
$464$	−4.47214	−0.207614
$465$	0	0
$466$	−31.3050	−1.45017
$467$	−24.3607	−1.12728	−0.563639	−	0.826021i	$-0.690599\pi$
$467$	−24.3607	−1.12728	−0.563639	+	0.826021i	$0.690599\pi$
$468$	0	0
$469$	−6.47214	−0.298855
$470$	28.9443	1.33510
$471$	0	0
$472$	28.9443	1.33227
$473$	−30.8328	−1.41769
$474$	0	0
$475$	−15.1246	−0.693965
$476$	−26.8328	−1.22988
$477$	0	0
$478$	−28.9443	−1.32388
$479$	1.88854	0.0862898	0.0431449	−	0.999069i	$-0.486262\pi$
$479$	1.88854	0.0862898	0.0431449	+	0.999069i	$0.486262\pi$
$480$	0	0
$481$	20.0000	0.911922
$482$	63.6656	2.89989
$483$	0	0
$484$	15.0000	0.681818
$485$	−1.52786	−0.0693767
$486$	0	0
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	−10.0000	−0.452679
$489$	0	0
$490$	−39.5967	−1.78880
$491$	−40.0000	−1.80517	−0.902587	−	0.430507i	$-0.858335\pi$
$491$	−40.0000	−1.80517	−0.902587	+	0.430507i	$0.858335\pi$
$492$	0	0
$493$	−32.3607	−1.45745
$494$	27.6393	1.24355
$495$	0	0
$496$	−2.47214	−0.111002
$497$	9.88854	0.443562
$498$	0	0
$499$	32.3607	1.44866	0.724331	−	0.689452i	$-0.242149\pi$
$499$	32.3607	1.44866	0.724331	+	0.689452i	$0.242149\pi$
$500$	−4.58359	−0.204984
$501$	0	0
$502$	3.41641	0.152482
$503$	−36.9443	−1.64726	−0.823632	−	0.567125i	$-0.808056\pi$
$503$	−36.9443	−1.64726	−0.823632	+	0.567125i	$0.808056\pi$
$504$	0	0
$505$	35.4164	1.57601
$506$	0	0
$507$	0	0
$508$	31.4164	1.39388
$509$	7.52786	0.333667	0.166833	−	0.985985i	$-0.446646\pi$
$509$	7.52786	0.333667	0.166833	+	0.985985i	$0.446646\pi$
$510$	0	0
$511$	−13.5279	−0.598437
$512$	−11.1803	−0.494106
$513$	0	0
$514$	−49.1935	−2.16983
$515$	−21.8885	−0.964524
$516$	0	0
$517$	16.0000	0.703679
$518$	−12.3607	−0.543097
$519$	0	0
$520$	32.3607	1.41911
$521$	−28.7639	−1.26017	−0.630085	−	0.776526i	$-0.716980\pi$
$521$	−28.7639	−1.26017	−0.630085	+	0.776526i	$0.716980\pi$
$522$	0	0
$523$	33.5967	1.46908	0.734542	−	0.678564i	$-0.237397\pi$
$523$	33.5967	1.46908	0.734542	+	0.678564i	$0.237397\pi$
$524$	2.83282	0.123752
$525$	0	0
$526$	5.52786	0.241026
$527$	−17.8885	−0.779237
$528$	0	0
$529$	0	0
$530$	−5.52786	−0.240115
$531$	0	0
$532$	−10.2492	−0.444360
$533$	−31.0557	−1.34517
$534$	0	0
$535$	3.05573	0.132111
$536$	11.7082	0.505717
$537$	0	0
$538$	18.9443	0.816746
$539$	−21.8885	−0.942806
$540$	0	0
$541$	−24.4721	−1.05214	−0.526070	−	0.850441i	$-0.676335\pi$
$541$	−24.4721	−1.05214	−0.526070	+	0.850441i	$0.676335\pi$
$542$	59.1935	2.54258
$543$	0	0
$544$	−48.5410	−2.08118
$545$	9.52786	0.408129
$546$	0	0
$547$	22.4721	0.960839	0.480420	−	0.877039i	$-0.340484\pi$
$547$	22.4721	0.960839	0.480420	+	0.877039i	$0.340484\pi$
$548$	9.70820	0.414714
$549$	0	0
$550$	−48.9443	−2.08699
$551$	−12.3607	−0.526583
$552$	0	0
$553$	4.58359	0.194914
$554$	44.4721	1.88944
$555$	0	0
$556$	26.8328	1.13796
$557$	−19.8197	−0.839786	−0.419893	−	0.907574i	$-0.637932\pi$
$557$	−19.8197	−0.839786	−0.419893	+	0.907574i	$0.637932\pi$
$558$	0	0
$559$	−34.4721	−1.45802
$560$	4.00000	0.169031
$561$	0	0
$562$	14.8754	0.627480
$563$	19.4164	0.818304	0.409152	−	0.912466i	$-0.365825\pi$
$563$	19.4164	0.818304	0.409152	+	0.912466i	$0.365825\pi$
$564$	0	0
$565$	−52.3607	−2.20283
$566$	−17.2361	−0.724486
$567$	0	0
$568$	−17.8885	−0.750587
$569$	−15.2361	−0.638729	−0.319365	−	0.947632i	$-0.603469\pi$
$569$	−15.2361	−0.638729	−0.319365	+	0.947632i	$0.603469\pi$
$570$	0	0
$571$	−16.2918	−0.681790	−0.340895	−	0.940101i	$-0.610730\pi$
$571$	−16.2918	−0.681790	−0.340895	+	0.940101i	$0.610730\pi$
$572$	53.6656	2.24387
$573$	0	0
$574$	19.1935	0.801121
$575$	0	0
$576$	0	0
$577$	16.4721	0.685744	0.342872	−	0.939382i	$-0.388600\pi$
$577$	16.4721	0.685744	0.342872	+	0.939382i	$0.388600\pi$
$578$	−79.0689	−3.28883
$579$	0	0
$580$	−43.4164	−1.80277
$581$	4.94427	0.205123
$582$	0	0
$583$	−3.05573	−0.126555
$584$	24.4721	1.01266
$585$	0	0
$586$	12.7639	0.527273
$587$	−16.9443	−0.699365	−0.349682	−	0.936868i	$-0.613711\pi$
$587$	−16.9443	−0.699365	−0.349682	+	0.936868i	$0.613711\pi$
$588$	0	0
$589$	−6.83282	−0.281541
$590$	−93.6656	−3.85615
$591$	0	0
$592$	−4.47214	−0.183804
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	0	0
$595$	28.9443	1.18660
$596$	5.12461	0.209912
$597$	0	0
$598$	0	0
$599$	20.9443	0.855760	0.427880	−	0.903836i	$-0.359261\pi$
$599$	20.9443	0.855760	0.427880	+	0.903836i	$0.359261\pi$
$600$	0	0
$601$	2.36068	0.0962941	0.0481471	−	0.998840i	$-0.484668\pi$
$601$	2.36068	0.0962941	0.0481471	+	0.998840i	$0.484668\pi$
$602$	21.3050	0.868325
$603$	0	0
$604$	−48.0000	−1.95309
$605$	−16.1803	−0.657824
$606$	0	0
$607$	38.8328	1.57618	0.788088	−	0.615563i	$-0.211071\pi$
$607$	38.8328	1.57618	0.788088	+	0.615563i	$0.211071\pi$
$608$	−18.5410	−0.751938
$609$	0	0
$610$	32.3607	1.31025
$611$	17.8885	0.723693
$612$	0	0
$613$	31.5279	1.27340	0.636699	−	0.771112i	$-0.280299\pi$
$613$	31.5279	1.27340	0.636699	+	0.771112i	$0.280299\pi$
$614$	14.4721	0.584048
$615$	0	0
$616$	−11.0557	−0.445448
$617$	−33.7082	−1.35704	−0.678521	−	0.734581i	$-0.737379\pi$
$617$	−33.7082	−1.35704	−0.678521	+	0.734581i	$0.737379\pi$
$618$	0	0
$619$	−31.1246	−1.25100	−0.625502	−	0.780223i	$-0.715106\pi$
$619$	−31.1246	−1.25100	−0.625502	+	0.780223i	$0.715106\pi$
$620$	−24.0000	−0.963863
$621$	0	0
$622$	55.7771	2.23646
$623$	4.00000	0.160257
$624$	0	0
$625$	−22.4164	−0.896656
$626$	51.3050	2.05056
$627$	0	0
$628$	37.4164	1.49308
$629$	−32.3607	−1.29030
$630$	0	0
$631$	−6.18034	−0.246035	−0.123018	−	0.992404i	$-0.539257\pi$
$631$	−6.18034	−0.246035	−0.123018	+	0.992404i	$0.539257\pi$
$632$	−8.29180	−0.329830
$633$	0	0
$634$	65.7771	2.61234
$635$	−33.8885	−1.34483
$636$	0	0
$637$	−24.4721	−0.969621
$638$	−40.0000	−1.58362
$639$	0	0
$640$	−50.6525	−2.00221
$641$	−27.0132	−1.06696	−0.533478	−	0.845814i	$-0.679115\pi$
$641$	−27.0132	−1.06696	−0.533478	+	0.845814i	$0.679115\pi$
$642$	0	0
$643$	20.0689	0.791440	0.395720	−	0.918371i	$-0.370495\pi$
$643$	20.0689	0.791440	0.395720	+	0.918371i	$0.370495\pi$
$644$	0	0
$645$	0	0
$646$	−44.7214	−1.75954
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	0	0
$649$	−51.7771	−2.03243
$650$	−54.7214	−2.14635
$651$	0	0
$652$	−58.2492	−2.28122
$653$	5.05573	0.197846	0.0989230	−	0.995095i	$-0.468460\pi$
$653$	5.05573	0.197846	0.0989230	+	0.995095i	$0.468460\pi$
$654$	0	0
$655$	−3.05573	−0.119397
$656$	6.94427	0.271128
$657$	0	0
$658$	−11.0557	−0.430997
$659$	−8.36068	−0.325686	−0.162843	−	0.986652i	$-0.552066\pi$
$659$	−8.36068	−0.325686	−0.162843	+	0.986652i	$0.552066\pi$
$660$	0	0
$661$	20.4721	0.796274	0.398137	−	0.917326i	$-0.369657\pi$
$661$	20.4721	0.796274	0.398137	+	0.917326i	$0.369657\pi$
$662$	7.63932	0.296911
$663$	0	0
$664$	−8.94427	−0.347105
$665$	11.0557	0.428723
$666$	0	0
$667$	0	0
$668$	14.8328	0.573899
$669$	0	0
$670$	−37.8885	−1.46376
$671$	17.8885	0.690580
$672$	0	0
$673$	29.4164	1.13392	0.566960	−	0.823746i	$-0.308120\pi$
$673$	29.4164	1.13392	0.566960	+	0.823746i	$0.308120\pi$
$674$	67.8885	2.61497
$675$	0	0
$676$	21.0000	0.807692
$677$	37.4853	1.44068	0.720338	−	0.693623i	$-0.243987\pi$
$677$	37.4853	1.44068	0.720338	+	0.693623i	$0.243987\pi$
$678$	0	0
$679$	0.583592	0.0223962
$680$	−52.3607	−2.00794
$681$	0	0
$682$	−22.1115	−0.846691
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	0	0
$685$	−10.4721	−0.400120
$686$	34.4721	1.31615
$687$	0	0
$688$	7.70820	0.293873
$689$	−3.41641	−0.130155
$690$	0	0
$691$	−40.3607	−1.53539	−0.767696	−	0.640814i	$-0.778597\pi$
$691$	−40.3607	−1.53539	−0.767696	+	0.640814i	$0.778597\pi$
$692$	13.4164	0.510015
$693$	0	0
$694$	57.8885	2.19742
$695$	−28.9443	−1.09792
$696$	0	0
$697$	50.2492	1.90333
$698$	72.6099	2.74833
$699$	0	0
$700$	20.2918	0.766958
$701$	20.7639	0.784243	0.392121	−	0.919913i	$-0.371741\pi$
$701$	20.7639	0.784243	0.392121	+	0.919913i	$0.371741\pi$
$702$	0	0
$703$	−12.3607	−0.466192
$704$	−52.0000	−1.95982
$705$	0	0
$706$	21.0557	0.792443
$707$	−13.5279	−0.508768
$708$	0	0
$709$	−20.8328	−0.782393	−0.391196	−	0.920307i	$-0.627939\pi$
$709$	−20.8328	−0.782393	−0.391196	+	0.920307i	$0.627939\pi$
$710$	57.8885	2.17252
$711$	0	0
$712$	−7.23607	−0.271183
$713$	0	0
$714$	0	0
$715$	−57.8885	−2.16491
$716$	−9.16718	−0.342594
$717$	0	0
$718$	5.52786	0.206298
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	0	0
$721$	8.36068	0.311368
$722$	25.4033	0.945411
$723$	0	0
$724$	−71.6656	−2.66343
$725$	24.4721	0.908872
$726$	0	0
$727$	−32.6525	−1.21101	−0.605507	−	0.795840i	$-0.707029\pi$
$727$	−32.6525	−1.21101	−0.605507	+	0.795840i	$0.707029\pi$
$728$	−12.3607	−0.458117
$729$	0	0
$730$	−79.1935	−2.93108
$731$	55.7771	2.06299
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	27.4853	1.01450
$735$	0	0
$736$	0	0
$737$	−20.9443	−0.771492
$738$	0	0
$739$	26.8328	0.987061	0.493531	−	0.869728i	$-0.335706\pi$
$739$	26.8328	0.987061	0.493531	+	0.869728i	$0.335706\pi$
$740$	−43.4164	−1.59602
$741$	0	0
$742$	2.11146	0.0775140
$743$	20.3607	0.746961	0.373480	−	0.927638i	$-0.378164\pi$
$743$	20.3607	0.746961	0.373480	+	0.927638i	$0.378164\pi$
$744$	0	0
$745$	−5.52786	−0.202525
$746$	25.7771	0.943766
$747$	0	0
$748$	−86.8328	−3.17492
$749$	−1.16718	−0.0426480
$750$	0	0
$751$	8.06888	0.294438	0.147219	−	0.989104i	$-0.452968\pi$
$751$	8.06888	0.294438	0.147219	+	0.989104i	$0.452968\pi$
$752$	−4.00000	−0.145865
$753$	0	0
$754$	−44.7214	−1.62866
$755$	51.7771	1.88436
$756$	0	0
$757$	−4.11146	−0.149433	−0.0747167	−	0.997205i	$-0.523805\pi$
$757$	−4.11146	−0.149433	−0.0747167	+	0.997205i	$0.523805\pi$
$758$	−62.7639	−2.27969
$759$	0	0
$760$	−20.0000	−0.725476
$761$	2.36068	0.0855746	0.0427873	−	0.999084i	$-0.486376\pi$
$761$	2.36068	0.0855746	0.0427873	+	0.999084i	$0.486376\pi$
$762$	0	0
$763$	−3.63932	−0.131752
$764$	31.4164	1.13661
$765$	0	0
$766$	−77.0820	−2.78509
$767$	−57.8885	−2.09023
$768$	0	0
$769$	36.8328	1.32823	0.664113	−	0.747633i	$-0.268810\pi$
$769$	36.8328	1.32823	0.664113	+	0.747633i	$0.268810\pi$
$770$	35.7771	1.28932
$771$	0	0
$772$	−28.2492	−1.01671
$773$	49.7082	1.78788	0.893940	−	0.448187i	$-0.147930\pi$
$773$	49.7082	1.78788	0.893940	+	0.448187i	$0.147930\pi$
$774$	0	0
$775$	13.5279	0.485935
$776$	−1.05573	−0.0378984
$777$	0	0
$778$	23.8197	0.853976
$779$	19.1935	0.687678
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	0	0
$784$	5.47214	0.195433
$785$	−40.3607	−1.44053
$786$	0	0
$787$	0.291796	0.0104014	0.00520070	−	0.999986i	$-0.498345\pi$
$787$	0.291796	0.0104014	0.00520070	+	0.999986i	$0.498345\pi$
$788$	28.2492	1.00634
$789$	0	0
$790$	26.8328	0.954669
$791$	20.0000	0.711118
$792$	0	0
$793$	20.0000	0.710221
$794$	4.47214	0.158710
$795$	0	0
$796$	11.1246	0.394301
$797$	−18.6525	−0.660705	−0.330352	−	0.943858i	$-0.607168\pi$
$797$	−18.6525	−0.660705	−0.330352	+	0.943858i	$0.607168\pi$
$798$	0	0
$799$	−28.9443	−1.02397
$800$	36.7082	1.29783
$801$	0	0
$802$	85.1246	3.00585
$803$	−43.7771	−1.54486
$804$	0	0
$805$	0	0
$806$	−24.7214	−0.870773
$807$	0	0
$808$	24.4721	0.860927
$809$	−19.3050	−0.678726	−0.339363	−	0.940655i	$-0.610211\pi$
$809$	−19.3050	−0.678726	−0.339363	+	0.940655i	$0.610211\pi$
$810$	0	0
$811$	3.41641	0.119966	0.0599832	−	0.998199i	$-0.480895\pi$
$811$	3.41641	0.119966	0.0599832	+	0.998199i	$0.480895\pi$
$812$	16.5836	0.581970
$813$	0	0
$814$	−40.0000	−1.40200
$815$	62.8328	2.20094
$816$	0	0
$817$	21.3050	0.745366
$818$	21.0557	0.736196
$819$	0	0
$820$	67.4164	2.35428
$821$	−47.8885	−1.67132	−0.835661	−	0.549246i	$-0.814915\pi$
$821$	−47.8885	−1.67132	−0.835661	+	0.549246i	$0.814915\pi$
$822$	0	0
$823$	−34.4721	−1.20162	−0.600812	−	0.799391i	$-0.705156\pi$
$823$	−34.4721	−1.20162	−0.600812	+	0.799391i	$0.705156\pi$
$824$	−15.1246	−0.526891
$825$	0	0
$826$	35.7771	1.24484
$827$	35.4164	1.23155	0.615775	−	0.787922i	$-0.288843\pi$
$827$	35.4164	1.23155	0.615775	+	0.787922i	$0.288843\pi$
$828$	0	0
$829$	−34.3607	−1.19340	−0.596698	−	0.802466i	$-0.703521\pi$
$829$	−34.3607	−1.19340	−0.596698	+	0.802466i	$0.703521\pi$
$830$	28.9443	1.00467
$831$	0	0
$832$	−58.1378	−2.01556
$833$	39.5967	1.37195
$834$	0	0
$835$	−16.0000	−0.553703
$836$	−33.1672	−1.14711
$837$	0	0
$838$	48.9443	1.69075
$839$	−39.4164	−1.36081	−0.680403	−	0.732838i	$-0.738195\pi$
$839$	−39.4164	−1.36081	−0.680403	+	0.732838i	$0.738195\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	−29.1935	−1.00607
$843$	0	0
$844$	67.4164	2.32057
$845$	−22.6525	−0.779269
$846$	0	0
$847$	6.18034	0.212359
$848$	0.763932	0.0262335
$849$	0	0
$850$	88.5410	3.03693
$851$	0	0
$852$	0	0
$853$	22.0000	0.753266	0.376633	−	0.926363i	$-0.377082\pi$
$853$	22.0000	0.753266	0.376633	+	0.926363i	$0.377082\pi$
$854$	−12.3607	−0.422974
$855$	0	0
$856$	2.11146	0.0721681
$857$	25.0557	0.855887	0.427944	−	0.903805i	$-0.359238\pi$
$857$	25.0557	0.855887	0.427944	+	0.903805i	$0.359238\pi$
$858$	0	0
$859$	−24.9443	−0.851088	−0.425544	−	0.904938i	$-0.639917\pi$
$859$	−24.9443	−0.851088	−0.425544	+	0.904938i	$0.639917\pi$
$860$	74.8328	2.55178
$861$	0	0
$862$	−17.8885	−0.609286
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	−14.4721	−0.492067
$866$	−17.6393	−0.599409
$867$	0	0
$868$	9.16718	0.311155
$869$	14.8328	0.503169
$870$	0	0
$871$	−23.4164	−0.793435
$872$	6.58359	0.222949
$873$	0	0
$874$	0	0
$875$	−1.88854	−0.0638444
$876$	0	0
$877$	50.9443	1.72027	0.860133	−	0.510070i	$-0.170381\pi$
$877$	50.9443	1.72027	0.860133	+	0.510070i	$0.170381\pi$
$878$	−57.8885	−1.95364
$879$	0	0
$880$	12.9443	0.436351
$881$	−40.5410	−1.36586	−0.682931	−	0.730483i	$-0.739295\pi$
$881$	−40.5410	−1.36586	−0.682931	+	0.730483i	$0.739295\pi$
$882$	0	0
$883$	19.4164	0.653414	0.326707	−	0.945126i	$-0.394061\pi$
$883$	19.4164	0.653414	0.326707	+	0.945126i	$0.394061\pi$
$884$	−97.0820	−3.26522
$885$	0	0
$886$	42.1115	1.41476
$887$	−48.9443	−1.64339	−0.821694	−	0.569929i	$-0.806971\pi$
$887$	−48.9443	−1.64339	−0.821694	+	0.569929i	$0.806971\pi$
$888$	0	0
$889$	12.9443	0.434137
$890$	23.4164	0.784920
$891$	0	0
$892$	29.6656	0.993279
$893$	−11.0557	−0.369966
$894$	0	0
$895$	9.88854	0.330538
$896$	19.3475	0.646355
$897$	0	0
$898$	−6.58359	−0.219697
$899$	11.0557	0.368729
$900$	0	0
$901$	5.52786	0.184160
$902$	62.1115	2.06809
$903$	0	0
$904$	−36.1803	−1.20334
$905$	77.3050	2.56970
$906$	0	0
$907$	−39.1246	−1.29911	−0.649556	−	0.760314i	$-0.725045\pi$
$907$	−39.1246	−1.29911	−0.649556	+	0.760314i	$0.725045\pi$
$908$	−67.4164	−2.23729
$909$	0	0
$910$	40.0000	1.32599
$911$	48.7214	1.61421	0.807105	−	0.590407i	$-0.201033\pi$
$911$	48.7214	1.61421	0.807105	+	0.590407i	$0.201033\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	−1.05573	−0.0349204
$915$	0	0
$916$	44.8328	1.48132
$917$	1.16718	0.0385438
$918$	0	0
$919$	−53.0132	−1.74874	−0.874371	−	0.485257i	$-0.838726\pi$
$919$	−53.0132	−1.74874	−0.874371	+	0.485257i	$0.838726\pi$
$920$	0	0
$921$	0	0
$922$	47.8885	1.57713
$923$	35.7771	1.17762
$924$	0	0
$925$	24.4721	0.804639
$926$	−11.0557	−0.363314
$927$	0	0
$928$	30.0000	0.984798
$929$	−24.8328	−0.814738	−0.407369	−	0.913264i	$-0.633554\pi$
$929$	−24.8328	−0.814738	−0.407369	+	0.913264i	$0.633554\pi$
$930$	0	0
$931$	15.1246	0.495689
$932$	42.0000	1.37576
$933$	0	0
$934$	54.4721	1.78238
$935$	93.6656	3.06319
$936$	0	0
$937$	−40.8328	−1.33395	−0.666975	−	0.745080i	$-0.732411\pi$
$937$	−40.8328	−1.33395	−0.666975	+	0.745080i	$0.732411\pi$
$938$	14.4721	0.472532
$939$	0	0
$940$	−38.8328	−1.26659
$941$	24.5410	0.800014	0.400007	−	0.916512i	$-0.369008\pi$
$941$	24.5410	0.800014	0.400007	+	0.916512i	$0.369008\pi$
$942$	0	0
$943$	0	0
$944$	12.9443	0.421300
$945$	0	0
$946$	68.9443	2.24157
$947$	54.8328	1.78183	0.890914	−	0.454173i	$-0.150065\pi$
$947$	54.8328	1.78183	0.890914	+	0.454173i	$0.150065\pi$
$948$	0	0
$949$	−48.9443	−1.58880
$950$	33.8197	1.09725
$951$	0	0
$952$	20.0000	0.648204
$953$	−46.0689	−1.49232	−0.746159	−	0.665768i	$-0.768104\pi$
$953$	−46.0689	−1.49232	−0.746159	+	0.665768i	$0.768104\pi$
$954$	0	0
$955$	−33.8885	−1.09661
$956$	38.8328	1.25594
$957$	0	0
$958$	−4.22291	−0.136436
$959$	4.00000	0.129167
$960$	0	0
$961$	−24.8885	−0.802856
$962$	−44.7214	−1.44187
$963$	0	0
$964$	−85.4164	−2.75108
$965$	30.4721	0.980933
$966$	0	0
$967$	−52.3607	−1.68381	−0.841903	−	0.539629i	$-0.818565\pi$
$967$	−52.3607	−1.68381	−0.841903	+	0.539629i	$0.818565\pi$
$968$	−11.1803	−0.359350
$969$	0	0
$970$	3.41641	0.109694
$971$	−45.3050	−1.45391	−0.726953	−	0.686688i	$-0.759064\pi$
$971$	−45.3050	−1.45391	−0.726953	+	0.686688i	$0.759064\pi$
$972$	0	0
$973$	11.0557	0.354430
$974$	−53.6656	−1.71956
$975$	0	0
$976$	−4.47214	−0.143150
$977$	26.0689	0.834017	0.417009	−	0.908902i	$-0.363078\pi$
$977$	26.0689	0.834017	0.417009	+	0.908902i	$0.363078\pi$
$978$	0	0
$979$	12.9443	0.413701
$980$	53.1246	1.69700
$981$	0	0
$982$	89.4427	2.85423
$983$	−30.8328	−0.983414	−0.491707	−	0.870761i	$-0.663627\pi$
$983$	−30.8328	−0.983414	−0.491707	+	0.870761i	$0.663627\pi$
$984$	0	0
$985$	−30.4721	−0.970923
$986$	72.3607	2.30443
$987$	0	0
$988$	−37.0820	−1.17974
$989$	0	0
$990$	0	0
$991$	−10.4721	−0.332658	−0.166329	−	0.986070i	$-0.553191\pi$
$991$	−10.4721	−0.332658	−0.166329	+	0.986070i	$0.553191\pi$
$992$	16.5836	0.526530
$993$	0	0
$994$	−22.1115	−0.701333
$995$	−12.0000	−0.380426
$996$	0	0
$997$	−2.36068	−0.0747635	−0.0373817	−	0.999301i	$-0.511902\pi$
$997$	−2.36068	−0.0747635	−0.0373817	+	0.999301i	$0.511902\pi$
$998$	−72.3607	−2.29054
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.v.1.1		2
3.2	odd	2		1587.2.a.i.1.2		2
23.22	odd	2		207.2.a.c.1.1		2
69.68	even	2		69.2.a.b.1.2	✓	2
92.91	even	2		3312.2.a.bb.1.2		2
115.114	odd	2		5175.2.a.bk.1.2		2
276.275	odd	2		1104.2.a.m.1.1		2
345.68	odd	4		1725.2.b.o.1174.1		4
345.137	odd	4		1725.2.b.o.1174.4		4
345.344	even	2		1725.2.a.ba.1.1		2
483.482	odd	2		3381.2.a.t.1.2		2
552.275	odd	2		4416.2.a.bg.1.2		2
552.413	even	2		4416.2.a.bm.1.2		2
759.758	odd	2		8349.2.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.2.a.b.1.2	✓	2	69.68	even	2
207.2.a.c.1.1		2	23.22	odd	2
1104.2.a.m.1.1		2	276.275	odd	2
1587.2.a.i.1.2		2	3.2	odd	2
1725.2.a.ba.1.1		2	345.344	even	2
1725.2.b.o.1174.1		4	345.68	odd	4
1725.2.b.o.1174.4		4	345.137	odd	4
3312.2.a.bb.1.2		2	92.91	even	2
3381.2.a.t.1.2		2	483.482	odd	2
4416.2.a.bg.1.2		2	552.275	odd	2
4416.2.a.bm.1.2		2	552.413	even	2
4761.2.a.v.1.1		2	1.1	even	1	trivial
5175.2.a.bk.1.2		2	115.114	odd	2
8349.2.a.i.1.1		2	759.758	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} +3.00000 q^{4} -3.23607 q^{5} +1.23607 q^{7} -2.23607 q^{8} +7.23607 q^{10} +4.00000 q^{11} +4.47214 q^{13} -2.76393 q^{14} -1.00000 q^{16} -7.23607 q^{17} -2.76393 q^{19} -9.70820 q^{20} -8.94427 q^{22} +5.47214 q^{25} -10.0000 q^{26} +3.70820 q^{28} +4.47214 q^{29} +2.47214 q^{31} +6.70820 q^{32} +16.1803 q^{34} -4.00000 q^{35} +4.47214 q^{37} +6.18034 q^{38} +7.23607 q^{40} -6.94427 q^{41} -7.70820 q^{43} +12.0000 q^{44} +4.00000 q^{47} -5.47214 q^{49} -12.2361 q^{50} +13.4164 q^{52} -0.763932 q^{53} -12.9443 q^{55} -2.76393 q^{56} -10.0000 q^{58} -12.9443 q^{59} +4.47214 q^{61} -5.52786 q^{62} -13.0000 q^{64} -14.4721 q^{65} -5.23607 q^{67} -21.7082 q^{68} +8.94427 q^{70} +8.00000 q^{71} -10.9443 q^{73} -10.0000 q^{74} -8.29180 q^{76} +4.94427 q^{77} +3.70820 q^{79} +3.23607 q^{80} +15.5279 q^{82} +4.00000 q^{83} +23.4164 q^{85} +17.2361 q^{86} -8.94427 q^{88} +3.23607 q^{89} +5.52786 q^{91} -8.94427 q^{94} +8.94427 q^{95} +0.472136 q^{97} +12.2361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{4} - 2 q^{5} - 2 q^{7} + 10 q^{10} + 8 q^{11} - 10 q^{14} - 2 q^{16} - 10 q^{17} - 10 q^{19} - 6 q^{20} + 2 q^{25} - 20 q^{26} - 6 q^{28} - 4 q^{31} + 10 q^{34} - 8 q^{35} - 10 q^{38} + 10 q^{40}+ \cdots + 20 q^{98}+O(q^{100}) \) 2 * q + 6 * q^4 - 2 * q^5 - 2 * q^7 + 10 * q^10 + 8 * q^11 - 10 * q^14 - 2 * q^16 - 10 * q^17 - 10 * q^19 - 6 * q^20 + 2 * q^25 - 20 * q^26 - 6 * q^28 - 4 * q^31 + 10 * q^34 - 8 * q^35 - 10 * q^38 + 10 * q^40 + 4 * q^41 - 2 * q^43 + 24 * q^44 + 8 * q^47 - 2 * q^49 - 20 * q^50 - 6 * q^53 - 8 * q^55 - 10 * q^56 - 20 * q^58 - 8 * q^59 - 20 * q^62 - 26 * q^64 - 20 * q^65 - 6 * q^67 - 30 * q^68 + 16 * q^71 - 4 * q^73 - 20 * q^74 - 30 * q^76 - 8 * q^77 - 6 * q^79 + 2 * q^80 + 40 * q^82 + 8 * q^83 + 20 * q^85 + 30 * q^86 + 2 * q^89 + 20 * q^91 - 8 * q^97 + 20 * q^98

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