LMFDB - Embedded newform 8001.2.a.z.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.66842 q^{2} +0.783626 q^{4} -3.76494 q^{5} -1.00000 q^{7} +2.02942 q^{8} +O(q^{10})$	\(q-1.66842 q^{2} +0.783626 q^{4} -3.76494 q^{5} -1.00000 q^{7} +2.02942 q^{8} +6.28150 q^{10} +3.16270 q^{11} -3.02168 q^{13} +1.66842 q^{14} -4.95318 q^{16} -1.05689 q^{17} +3.14601 q^{19} -2.95030 q^{20} -5.27672 q^{22} +7.12860 q^{23} +9.17476 q^{25} +5.04143 q^{26} -0.783626 q^{28} -2.67282 q^{29} -8.88393 q^{31} +4.20514 q^{32} +1.76334 q^{34} +3.76494 q^{35} +0.943469 q^{37} -5.24887 q^{38} -7.64065 q^{40} -9.90868 q^{41} -10.8524 q^{43} +2.47838 q^{44} -11.8935 q^{46} -4.25536 q^{47} +1.00000 q^{49} -15.3074 q^{50} -2.36787 q^{52} +5.14637 q^{53} -11.9074 q^{55} -2.02942 q^{56} +4.45938 q^{58} +4.15250 q^{59} +1.23055 q^{61} +14.8221 q^{62} +2.89042 q^{64} +11.3764 q^{65} -10.7930 q^{67} -0.828207 q^{68} -6.28150 q^{70} +15.1564 q^{71} +7.29623 q^{73} -1.57410 q^{74} +2.46530 q^{76} -3.16270 q^{77} +5.09038 q^{79} +18.6484 q^{80} +16.5318 q^{82} +8.93203 q^{83} +3.97913 q^{85} +18.1064 q^{86} +6.41846 q^{88} -5.97445 q^{89} +3.02168 q^{91} +5.58616 q^{92} +7.09974 q^{94} -11.8445 q^{95} -9.75834 q^{97} -1.66842 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) $ 32 * q + 30 * q^4 - 32 * q^7	$ 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) $ 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.66842	−1.17975	−0.589876	−	0.807494i	$-0.700823\pi$
$2$	−1.66842	−1.17975	−0.589876	+	0.807494i	$0.700823\pi$
$3$	0	0
$3$	0	0
$4$	0.783626	0.391813
$5$	−3.76494	−1.68373	−0.841866	−	0.539687i	$-0.818543\pi$
$5$	−3.76494	−1.68373	−0.841866	+	0.539687i	$0.818543\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.02942	0.717509
$9$	0	0
$10$	6.28150	1.98638
$11$	3.16270	0.953591	0.476795	−	0.879014i	$-0.341798\pi$
$11$	3.16270	0.953591	0.476795	+	0.879014i	$0.341798\pi$
$12$	0	0
$13$	−3.02168	−0.838063	−0.419031	−	0.907972i	$-0.637630\pi$
$13$	−3.02168	−0.838063	−0.419031	+	0.907972i	$0.637630\pi$
$14$	1.66842	0.445904
$15$	0	0
$16$	−4.95318	−1.23830
$17$	−1.05689	−0.256333	−0.128167	−	0.991753i	$-0.540909\pi$
$17$	−1.05689	−0.256333	−0.128167	+	0.991753i	$0.540909\pi$
$18$	0	0
$19$	3.14601	0.721744	0.360872	−	0.932615i	$-0.382479\pi$
$19$	3.14601	0.721744	0.360872	+	0.932615i	$0.382479\pi$
$20$	−2.95030	−0.659708
$21$	0	0
$22$	−5.27672	−1.12500
$23$	7.12860	1.48642	0.743208	−	0.669060i	$-0.233303\pi$
$23$	7.12860	1.48642	0.743208	+	0.669060i	$0.233303\pi$
$24$	0	0
$25$	9.17476	1.83495
$26$	5.04143	0.988705
$27$	0	0
$28$	−0.783626	−0.148091
$29$	−2.67282	−0.496329	−0.248165	−	0.968718i	$-0.579827\pi$
$29$	−2.67282	−0.496329	−0.248165	+	0.968718i	$0.579827\pi$
$30$	0	0
$31$	−8.88393	−1.59560	−0.797801	−	0.602921i	$-0.794003\pi$
$31$	−8.88393	−1.59560	−0.797801	+	0.602921i	$0.794003\pi$
$32$	4.20514	0.743372
$33$	0	0
$34$	1.76334	0.302410
$35$	3.76494	0.636391
$36$	0	0
$37$	0.943469	0.155105	0.0775527	−	0.996988i	$-0.475289\pi$
$37$	0.943469	0.155105	0.0775527	+	0.996988i	$0.475289\pi$
$38$	−5.24887	−0.851479
$39$	0	0
$40$	−7.64065	−1.20809
$41$	−9.90868	−1.54748	−0.773738	−	0.633506i	$-0.781615\pi$
$41$	−9.90868	−1.54748	−0.773738	+	0.633506i	$0.781615\pi$
$42$	0	0
$43$	−10.8524	−1.65498	−0.827489	−	0.561482i	$-0.810232\pi$
$43$	−10.8524	−1.65498	−0.827489	+	0.561482i	$0.810232\pi$
$44$	2.47838	0.373629
$45$	0	0
$46$	−11.8935	−1.75360
$47$	−4.25536	−0.620709	−0.310354	−	0.950621i	$-0.600448\pi$
$47$	−4.25536	−0.620709	−0.310354	+	0.950621i	$0.600448\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−15.3074	−2.16479
$51$	0	0
$52$	−2.36787	−0.328364
$53$	5.14637	0.706908	0.353454	−	0.935452i	$-0.385007\pi$
$53$	5.14637	0.706908	0.353454	+	0.935452i	$0.385007\pi$
$54$	0	0
$55$	−11.9074	−1.60559
$56$	−2.02942	−0.271193
$57$	0	0
$58$	4.45938	0.585545
$59$	4.15250	0.540609	0.270305	−	0.962775i	$-0.412876\pi$
$59$	4.15250	0.540609	0.270305	+	0.962775i	$0.412876\pi$
$60$	0	0
$61$	1.23055	0.157556	0.0787778	−	0.996892i	$-0.474898\pi$
$61$	1.23055	0.157556	0.0787778	+	0.996892i	$0.474898\pi$
$62$	14.8221	1.88241
$63$	0	0
$64$	2.89042	0.361302
$65$	11.3764	1.41107
$66$	0	0
$67$	−10.7930	−1.31857	−0.659285	−	0.751893i	$-0.729141\pi$
$67$	−10.7930	−1.31857	−0.659285	+	0.751893i	$0.729141\pi$
$68$	−0.828207	−0.100435
$69$	0	0
$70$	−6.28150	−0.750783
$71$	15.1564	1.79874	0.899369	−	0.437191i	$-0.144027\pi$
$71$	15.1564	1.79874	0.899369	+	0.437191i	$0.144027\pi$
$72$	0	0
$73$	7.29623	0.853959	0.426980	−	0.904261i	$-0.359578\pi$
$73$	7.29623	0.853959	0.426980	+	0.904261i	$0.359578\pi$
$74$	−1.57410	−0.182986
$75$	0	0
$76$	2.46530	0.282789
$77$	−3.16270	−0.360423
$78$	0	0
$79$	5.09038	0.572713	0.286357	−	0.958123i	$-0.407556\pi$
$79$	5.09038	0.572713	0.286357	+	0.958123i	$0.407556\pi$
$80$	18.6484	2.08496
$81$	0	0
$82$	16.5318	1.82564
$83$	8.93203	0.980417	0.490209	−	0.871605i	$-0.336921\pi$
$83$	8.93203	0.980417	0.490209	+	0.871605i	$0.336921\pi$
$84$	0	0
$85$	3.97913	0.431597
$86$	18.1064	1.95246
$87$	0	0
$88$	6.41846	0.684210
$89$	−5.97445	−0.633291	−0.316645	−	0.948544i	$-0.602557\pi$
$89$	−5.97445	−0.633291	−0.316645	+	0.948544i	$0.602557\pi$
$90$	0	0
$91$	3.02168	0.316758
$92$	5.58616	0.582397
$93$	0	0
$94$	7.09974	0.732282
$95$	−11.8445	−1.21522
$96$	0	0
$97$	−9.75834	−0.990809	−0.495405	−	0.868662i	$-0.664980\pi$
$97$	−9.75834	−0.990809	−0.495405	+	0.868662i	$0.664980\pi$
$98$	−1.66842	−0.168536
$99$	0	0
$100$	7.18958	0.718958
$101$	11.6815	1.16235	0.581174	−	0.813779i	$-0.302593\pi$
$101$	11.6815	1.16235	0.581174	+	0.813779i	$0.302593\pi$
$102$	0	0
$103$	13.9511	1.37465	0.687323	−	0.726352i	$-0.258786\pi$
$103$	13.9511	1.37465	0.687323	+	0.726352i	$0.258786\pi$
$104$	−6.13226	−0.601318
$105$	0	0
$106$	−8.58631	−0.833976
$107$	16.0629	1.55286	0.776432	−	0.630202i	$-0.217028\pi$
$107$	16.0629	1.55286	0.776432	+	0.630202i	$0.217028\pi$
$108$	0	0
$109$	−1.53849	−0.147360	−0.0736802	−	0.997282i	$-0.523474\pi$
$109$	−1.53849	−0.147360	−0.0736802	+	0.997282i	$0.523474\pi$
$110$	19.8665	1.89420
$111$	0	0
$112$	4.95318	0.468032
$113$	15.3117	1.44040	0.720202	−	0.693765i	$-0.244049\pi$
$113$	15.3117	1.44040	0.720202	+	0.693765i	$0.244049\pi$
$114$	0	0
$115$	−26.8387	−2.50273
$116$	−2.09449	−0.194468
$117$	0	0
$118$	−6.92811	−0.637784
$119$	1.05689	0.0968849
$120$	0	0
$121$	−0.997312	−0.0906647
$122$	−2.05307	−0.185876
$123$	0	0
$124$	−6.96168	−0.625178
$125$	−15.7177	−1.40583
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−13.2327	−1.16962
$129$	0	0
$130$	−18.9807	−1.66471
$131$	3.78335	0.330553	0.165277	−	0.986247i	$-0.447148\pi$
$131$	3.78335	0.330553	0.165277	+	0.986247i	$0.447148\pi$
$132$	0	0
$133$	−3.14601	−0.272794
$134$	18.0072	1.55559
$135$	0	0
$136$	−2.14488	−0.183922
$137$	21.8842	1.86969	0.934847	−	0.355050i	$-0.115536\pi$
$137$	21.8842	1.86969	0.934847	+	0.355050i	$0.115536\pi$
$138$	0	0
$139$	9.85829	0.836169	0.418085	−	0.908408i	$-0.362702\pi$
$139$	9.85829	0.836169	0.418085	+	0.908408i	$0.362702\pi$
$140$	2.95030	0.249346
$141$	0	0
$142$	−25.2873	−2.12206
$143$	−9.55667	−0.799169
$144$	0	0
$145$	10.0630	0.835685
$146$	−12.1732	−1.00746
$147$	0	0
$148$	0.739327	0.0607723
$149$	−1.98780	−0.162847	−0.0814233	−	0.996680i	$-0.525947\pi$
$149$	−1.98780	−0.162847	−0.0814233	+	0.996680i	$0.525947\pi$
$150$	0	0
$151$	−8.05456	−0.655471	−0.327735	−	0.944770i	$-0.606285\pi$
$151$	−8.05456	−0.655471	−0.327735	+	0.944770i	$0.606285\pi$
$152$	6.38459	0.517858
$153$	0	0
$154$	5.27672	0.425210
$155$	33.4475	2.68656
$156$	0	0
$157$	−9.72055	−0.775784	−0.387892	−	0.921705i	$-0.626797\pi$
$157$	−9.72055	−0.775784	−0.387892	+	0.921705i	$0.626797\pi$
$158$	−8.49290	−0.675659
$159$	0	0
$160$	−15.8321	−1.25164
$161$	−7.12860	−0.561812
$162$	0	0
$163$	4.77862	0.374290	0.187145	−	0.982332i	$-0.440077\pi$
$163$	4.77862	0.374290	0.187145	+	0.982332i	$0.440077\pi$
$164$	−7.76470	−0.606321
$165$	0	0
$166$	−14.9024	−1.15665
$167$	−20.9276	−1.61943	−0.809713	−	0.586827i	$-0.800377\pi$
$167$	−20.9276	−1.61943	−0.809713	+	0.586827i	$0.800377\pi$
$168$	0	0
$169$	−3.86947	−0.297651
$170$	−6.63885	−0.509177
$171$	0	0
$172$	−8.50424	−0.648442
$173$	−13.7591	−1.04609	−0.523044	−	0.852306i	$-0.675204\pi$
$173$	−13.7591	−1.04609	−0.523044	+	0.852306i	$0.675204\pi$
$174$	0	0
$175$	−9.17476	−0.693547
$176$	−15.6654	−1.18083
$177$	0	0
$178$	9.96790	0.747126
$179$	9.24850	0.691265	0.345633	−	0.938370i	$-0.387664\pi$
$179$	9.24850	0.691265	0.345633	+	0.938370i	$0.387664\pi$
$180$	0	0
$181$	14.1657	1.05293	0.526465	−	0.850197i	$-0.323517\pi$
$181$	14.1657	1.05293	0.526465	+	0.850197i	$0.323517\pi$
$182$	−5.04143	−0.373696
$183$	0	0
$184$	14.4669	1.06652
$185$	−3.55210	−0.261156
$186$	0	0
$187$	−3.34263	−0.244437
$188$	−3.33461	−0.243202
$189$	0	0
$190$	19.7617	1.43366
$191$	3.34589	0.242100	0.121050	−	0.992646i	$-0.461374\pi$
$191$	3.34589	0.242100	0.121050	+	0.992646i	$0.461374\pi$
$192$	0	0
$193$	6.91295	0.497605	0.248802	−	0.968554i	$-0.419963\pi$
$193$	6.91295	0.497605	0.248802	+	0.968554i	$0.419963\pi$
$194$	16.2810	1.16891
$195$	0	0
$196$	0.783626	0.0559733
$197$	−17.8080	−1.26877	−0.634385	−	0.773017i	$-0.718747\pi$
$197$	−17.8080	−1.26877	−0.634385	+	0.773017i	$0.718747\pi$
$198$	0	0
$199$	−16.6199	−1.17815	−0.589077	−	0.808077i	$-0.700509\pi$
$199$	−16.6199	−1.17815	−0.589077	+	0.808077i	$0.700509\pi$
$200$	18.6195	1.31659
$201$	0	0
$202$	−19.4896	−1.37128
$203$	2.67282	0.187595
$204$	0	0
$205$	37.3056	2.60553
$206$	−23.2764	−1.62174
$207$	0	0
$208$	14.9669	1.03777
$209$	9.94990	0.688249
$210$	0	0
$211$	−14.6764	−1.01037	−0.505184	−	0.863012i	$-0.668575\pi$
$211$	−14.6764	−1.01037	−0.505184	+	0.863012i	$0.668575\pi$
$212$	4.03283	0.276976
$213$	0	0
$214$	−26.7997	−1.83199
$215$	40.8587	2.78654
$216$	0	0
$217$	8.88393	0.603081
$218$	2.56685	0.173849
$219$	0	0
$220$	−9.33094	−0.629092
$221$	3.19358	0.214823
$222$	0	0
$223$	17.2818	1.15728	0.578639	−	0.815584i	$-0.303584\pi$
$223$	17.2818	1.15728	0.578639	+	0.815584i	$0.303584\pi$
$224$	−4.20514	−0.280968
$225$	0	0
$226$	−25.5464	−1.69932
$227$	−7.13775	−0.473749	−0.236875	−	0.971540i	$-0.576123\pi$
$227$	−7.13775	−0.473749	−0.236875	+	0.971540i	$0.576123\pi$
$228$	0	0
$229$	8.65153	0.571709	0.285855	−	0.958273i	$-0.407723\pi$
$229$	8.65153	0.571709	0.285855	+	0.958273i	$0.407723\pi$
$230$	44.7783	2.95259
$231$	0	0
$232$	−5.42427	−0.356121
$233$	−2.76388	−0.181068	−0.0905340	−	0.995893i	$-0.528857\pi$
$233$	−2.76388	−0.181068	−0.0905340	+	0.995893i	$0.528857\pi$
$234$	0	0
$235$	16.0212	1.04511
$236$	3.25401	0.211818
$237$	0	0
$238$	−1.76334	−0.114300
$239$	23.8172	1.54061	0.770304	−	0.637677i	$-0.220104\pi$
$239$	23.8172	1.54061	0.770304	+	0.637677i	$0.220104\pi$
$240$	0	0
$241$	−18.6840	−1.20354	−0.601770	−	0.798670i	$-0.705537\pi$
$241$	−18.6840	−1.20354	−0.601770	+	0.798670i	$0.705537\pi$
$242$	1.66394	0.106962
$243$	0	0
$244$	0.964290	0.0617324
$245$	−3.76494	−0.240533
$246$	0	0
$247$	−9.50623	−0.604867
$248$	−18.0293	−1.14486
$249$	0	0
$250$	26.2237	1.65854
$251$	19.4770	1.22938	0.614690	−	0.788769i	$-0.289281\pi$
$251$	19.4770	1.22938	0.614690	+	0.788769i	$0.289281\pi$
$252$	0	0
$253$	22.5456	1.41743
$254$	1.66842	0.104686
$255$	0	0
$256$	16.2969	1.01856
$257$	−9.85146	−0.614517	−0.307259	−	0.951626i	$-0.599412\pi$
$257$	−9.85146	−0.614517	−0.307259	+	0.951626i	$0.599412\pi$
$258$	0	0
$259$	−0.943469	−0.0586243
$260$	8.91487	0.552877
$261$	0	0
$262$	−6.31222	−0.389971
$263$	−13.8741	−0.855514	−0.427757	−	0.903894i	$-0.640696\pi$
$263$	−13.8741	−0.855514	−0.427757	+	0.903894i	$0.640696\pi$
$264$	0	0
$265$	−19.3758	−1.19024
$266$	5.24887	0.321829
$267$	0	0
$268$	−8.45765	−0.516633
$269$	−26.2278	−1.59914	−0.799568	−	0.600575i	$-0.794938\pi$
$269$	−26.2278	−1.59914	−0.799568	+	0.600575i	$0.794938\pi$
$270$	0	0
$271$	19.6970	1.19651	0.598255	−	0.801306i	$-0.295861\pi$
$271$	19.6970	1.19651	0.598255	+	0.801306i	$0.295861\pi$
$272$	5.23497	0.317417
$273$	0	0
$274$	−36.5121	−2.20577
$275$	29.0170	1.74979
$276$	0	0
$277$	−1.14232	−0.0686351	−0.0343176	−	0.999411i	$-0.510926\pi$
$277$	−1.14232	−0.0686351	−0.0343176	+	0.999411i	$0.510926\pi$
$278$	−16.4478	−0.986472
$279$	0	0
$280$	7.64065	0.456616
$281$	16.7929	1.00178	0.500891	−	0.865511i	$-0.333006\pi$
$281$	16.7929	1.00178	0.500891	+	0.865511i	$0.333006\pi$
$282$	0	0
$283$	13.1223	0.780042	0.390021	−	0.920806i	$-0.372468\pi$
$283$	13.1223	0.780042	0.390021	+	0.920806i	$0.372468\pi$
$284$	11.8770	0.704769
$285$	0	0
$286$	15.9445	0.942820
$287$	9.90868	0.584891
$288$	0	0
$289$	−15.8830	−0.934293
$290$	−16.7893	−0.985901
$291$	0	0
$292$	5.71752	0.334593
$293$	−8.84515	−0.516739	−0.258370	−	0.966046i	$-0.583185\pi$
$293$	−8.84515	−0.516739	−0.258370	+	0.966046i	$0.583185\pi$
$294$	0	0
$295$	−15.6339	−0.910241
$296$	1.91470	0.111290
$297$	0	0
$298$	3.31648	0.192118
$299$	−21.5403	−1.24571
$300$	0	0
$301$	10.8524	0.625523
$302$	13.4384	0.773292
$303$	0	0
$304$	−15.5828	−0.893733
$305$	−4.63294	−0.265281
$306$	0	0
$307$	8.38864	0.478765	0.239382	−	0.970925i	$-0.423055\pi$
$307$	8.38864	0.478765	0.239382	+	0.970925i	$0.423055\pi$
$308$	−2.47838	−0.141219
$309$	0	0
$310$	−55.8044	−3.16948
$311$	−3.66921	−0.208062	−0.104031	−	0.994574i	$-0.533174\pi$
$311$	−3.66921	−0.208062	−0.104031	+	0.994574i	$0.533174\pi$
$312$	0	0
$313$	27.7568	1.56891	0.784453	−	0.620188i	$-0.212944\pi$
$313$	27.7568	1.56891	0.784453	+	0.620188i	$0.212944\pi$
$314$	16.2180	0.915232
$315$	0	0
$316$	3.98896	0.224397
$317$	−8.76792	−0.492455	−0.246228	−	0.969212i	$-0.579191\pi$
$317$	−8.76792	−0.492455	−0.246228	+	0.969212i	$0.579191\pi$
$318$	0	0
$319$	−8.45332	−0.473295
$320$	−10.8822	−0.608335
$321$	0	0
$322$	11.8935	0.662799
$323$	−3.32499	−0.185007
$324$	0	0
$325$	−27.7232	−1.53780
$326$	−7.97274	−0.441569
$327$	0	0
$328$	−20.1089	−1.11033
$329$	4.25536	0.234606
$330$	0	0
$331$	26.3095	1.44610	0.723051	−	0.690795i	$-0.242739\pi$
$331$	26.3095	1.44610	0.723051	+	0.690795i	$0.242739\pi$
$332$	6.99937	0.384140
$333$	0	0
$334$	34.9160	1.91052
$335$	40.6348	2.22012
$336$	0	0
$337$	0.194439	0.0105917	0.00529587	−	0.999986i	$-0.498314\pi$
$337$	0.194439	0.0105917	0.00529587	+	0.999986i	$0.498314\pi$
$338$	6.45589	0.351154
$339$	0	0
$340$	3.11815	0.169105
$341$	−28.0972	−1.52155
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−22.0241	−1.18746
$345$	0	0
$346$	22.9560	1.23412
$347$	−20.8704	−1.12038	−0.560191	−	0.828363i	$-0.689272\pi$
$347$	−20.8704	−1.12038	−0.560191	+	0.828363i	$0.689272\pi$
$348$	0	0
$349$	−6.06183	−0.324483	−0.162241	−	0.986751i	$-0.551872\pi$
$349$	−6.06183	−0.324483	−0.162241	+	0.986751i	$0.551872\pi$
$350$	15.3074	0.818212
$351$	0	0
$352$	13.2996	0.708872
$353$	−18.0447	−0.960424	−0.480212	−	0.877152i	$-0.659440\pi$
$353$	−18.0447	−0.960424	−0.480212	+	0.877152i	$0.659440\pi$
$354$	0	0
$355$	−57.0630	−3.02859
$356$	−4.68174	−0.248132
$357$	0	0
$358$	−15.4304	−0.815521
$359$	−34.6535	−1.82894	−0.914472	−	0.404650i	$-0.867394\pi$
$359$	−34.6535	−1.82894	−0.914472	+	0.404650i	$0.867394\pi$
$360$	0	0
$361$	−9.10261	−0.479085
$362$	−23.6344	−1.24220
$363$	0	0
$364$	2.36787	0.124110
$365$	−27.4699	−1.43784
$366$	0	0
$367$	−11.3162	−0.590699	−0.295349	−	0.955389i	$-0.595436\pi$
$367$	−11.3162	−0.590699	−0.295349	+	0.955389i	$0.595436\pi$
$368$	−35.3093	−1.84062
$369$	0	0
$370$	5.92640	0.308099
$371$	−5.14637	−0.267186
$372$	0	0
$373$	−3.90440	−0.202162	−0.101081	−	0.994878i	$-0.532230\pi$
$373$	−3.90440	−0.202162	−0.101081	+	0.994878i	$0.532230\pi$
$374$	5.57691	0.288375
$375$	0	0
$376$	−8.63593	−0.445364
$377$	8.07639	0.415955
$378$	0	0
$379$	23.6160	1.21307	0.606537	−	0.795055i	$-0.292558\pi$
$379$	23.6160	1.21307	0.606537	+	0.795055i	$0.292558\pi$
$380$	−9.28169	−0.476141
$381$	0	0
$382$	−5.58235	−0.285618
$383$	25.6133	1.30878	0.654388	−	0.756159i	$-0.272926\pi$
$383$	25.6133	1.30878	0.654388	+	0.756159i	$0.272926\pi$
$384$	0	0
$385$	11.9074	0.606856
$386$	−11.5337	−0.587050
$387$	0	0
$388$	−7.64689	−0.388212
$389$	−15.4262	−0.782140	−0.391070	−	0.920361i	$-0.627895\pi$
$389$	−15.4262	−0.782140	−0.391070	+	0.920361i	$0.627895\pi$
$390$	0	0
$391$	−7.53415	−0.381018
$392$	2.02942	0.102501
$393$	0	0
$394$	29.7113	1.49683
$395$	−19.1650	−0.964295
$396$	0	0
$397$	−5.68868	−0.285507	−0.142753	−	0.989758i	$-0.545596\pi$
$397$	−5.68868	−0.285507	−0.142753	+	0.989758i	$0.545596\pi$
$398$	27.7290	1.38993
$399$	0	0
$400$	−45.4443	−2.27221
$401$	−22.9927	−1.14820	−0.574101	−	0.818785i	$-0.694648\pi$
$401$	−22.9927	−1.14820	−0.574101	+	0.818785i	$0.694648\pi$
$402$	0	0
$403$	26.8444	1.33721
$404$	9.15390	0.455423
$405$	0	0
$406$	−4.45938	−0.221315
$407$	2.98391	0.147907
$408$	0	0
$409$	−18.0347	−0.891761	−0.445880	−	0.895093i	$-0.647109\pi$
$409$	−18.0347	−0.891761	−0.445880	+	0.895093i	$0.647109\pi$
$410$	−62.2414	−3.07388
$411$	0	0
$412$	10.9325	0.538604
$413$	−4.15250	−0.204331
$414$	0	0
$415$	−33.6285	−1.65076
$416$	−12.7066	−0.622992
$417$	0	0
$418$	−16.6006	−0.811962
$419$	−13.0874	−0.639360	−0.319680	−	0.947525i	$-0.603575\pi$
$419$	−13.0874	−0.639360	−0.319680	+	0.947525i	$0.603575\pi$
$420$	0	0
$421$	−40.1201	−1.95533	−0.977667	−	0.210160i	$-0.932602\pi$
$421$	−40.1201	−1.95533	−0.977667	+	0.210160i	$0.932602\pi$
$422$	24.4865	1.19198
$423$	0	0
$424$	10.4442	0.507213
$425$	−9.69671	−0.470360
$426$	0	0
$427$	−1.23055	−0.0595504
$428$	12.5873	0.608432
$429$	0	0
$430$	−68.1695	−3.28742
$431$	−16.9188	−0.814951	−0.407476	−	0.913216i	$-0.633591\pi$
$431$	−16.9188	−0.814951	−0.407476	+	0.913216i	$0.633591\pi$
$432$	0	0
$433$	−8.66554	−0.416439	−0.208220	−	0.978082i	$-0.566767\pi$
$433$	−8.66554	−0.416439	−0.208220	+	0.978082i	$0.566767\pi$
$434$	−14.8221	−0.711485
$435$	0	0
$436$	−1.20560	−0.0577378
$437$	22.4267	1.07281
$438$	0	0
$439$	−25.4905	−1.21660	−0.608298	−	0.793709i	$-0.708147\pi$
$439$	−25.4905	−1.21660	−0.608298	+	0.793709i	$0.708147\pi$
$440$	−24.1651	−1.15203
$441$	0	0
$442$	−5.32823	−0.253438
$443$	34.9265	1.65941	0.829703	−	0.558205i	$-0.188510\pi$
$443$	34.9265	1.65941	0.829703	+	0.558205i	$0.188510\pi$
$444$	0	0
$445$	22.4934	1.06629
$446$	−28.8334	−1.36530
$447$	0	0
$448$	−2.89042	−0.136559
$449$	29.0615	1.37150	0.685749	−	0.727838i	$-0.259475\pi$
$449$	29.0615	1.37150	0.685749	+	0.727838i	$0.259475\pi$
$450$	0	0
$451$	−31.3382	−1.47566
$452$	11.9987	0.564369
$453$	0	0
$454$	11.9088	0.558906
$455$	−11.3764	−0.533335
$456$	0	0
$457$	16.6346	0.778133	0.389066	−	0.921210i	$-0.372798\pi$
$457$	16.6346	0.778133	0.389066	+	0.921210i	$0.372798\pi$
$458$	−14.4344	−0.674475
$459$	0	0
$460$	−21.0315	−0.980601
$461$	−1.69635	−0.0790067	−0.0395034	−	0.999219i	$-0.512578\pi$
$461$	−1.69635	−0.0790067	−0.0395034	+	0.999219i	$0.512578\pi$
$462$	0	0
$463$	−33.3579	−1.55027	−0.775136	−	0.631795i	$-0.782318\pi$
$463$	−33.3579	−1.55027	−0.775136	+	0.631795i	$0.782318\pi$
$464$	13.2389	0.614602
$465$	0	0
$466$	4.61132	0.213615
$467$	22.2821	1.03109	0.515546	−	0.856862i	$-0.327589\pi$
$467$	22.2821	1.03109	0.515546	+	0.856862i	$0.327589\pi$
$468$	0	0
$469$	10.7930	0.498373
$470$	−26.7301	−1.23297
$471$	0	0
$472$	8.42717	0.387892
$473$	−34.3230	−1.57817
$474$	0	0
$475$	28.8639	1.32437
$476$	0.828207	0.0379608
$477$	0	0
$478$	−39.7371	−1.81753
$479$	−10.4324	−0.476667	−0.238333	−	0.971183i	$-0.576601\pi$
$479$	−10.4324	−0.476667	−0.238333	+	0.971183i	$0.576601\pi$
$480$	0	0
$481$	−2.85086	−0.129988
$482$	31.1727	1.41988
$483$	0	0
$484$	−0.781520	−0.0355236
$485$	36.7395	1.66826
$486$	0	0
$487$	4.27431	0.193687	0.0968437	−	0.995300i	$-0.469125\pi$
$487$	4.27431	0.193687	0.0968437	+	0.995300i	$0.469125\pi$
$488$	2.49730	0.113048
$489$	0	0
$490$	6.28150	0.283769
$491$	12.2377	0.552281	0.276141	−	0.961117i	$-0.410944\pi$
$491$	12.2377	0.552281	0.276141	+	0.961117i	$0.410944\pi$
$492$	0	0
$493$	2.82487	0.127226
$494$	15.8604	0.713593
$495$	0	0
$496$	44.0037	1.97583
$497$	−15.1564	−0.679859
$498$	0	0
$499$	18.0798	0.809361	0.404681	−	0.914458i	$-0.367383\pi$
$499$	18.0798	0.809361	0.404681	+	0.914458i	$0.367383\pi$
$500$	−12.3168	−0.550824
$501$	0	0
$502$	−32.4959	−1.45036
$503$	15.4918	0.690747	0.345373	−	0.938465i	$-0.387752\pi$
$503$	15.4918	0.690747	0.345373	+	0.938465i	$0.387752\pi$
$504$	0	0
$505$	−43.9800	−1.95708
$506$	−37.6156	−1.67222
$507$	0	0
$508$	−0.783626	−0.0347678
$509$	28.3379	1.25606	0.628029	−	0.778190i	$-0.283862\pi$
$509$	28.3379	1.25606	0.628029	+	0.778190i	$0.283862\pi$
$510$	0	0
$511$	−7.29623	−0.322766
$512$	−0.724649	−0.0320253
$513$	0	0
$514$	16.4364	0.724978
$515$	−52.5251	−2.31453
$516$	0	0
$517$	−13.4584	−0.591902
$518$	1.57410	0.0691621
$519$	0	0
$520$	23.0876	1.01246
$521$	−24.5238	−1.07441	−0.537205	−	0.843452i	$-0.680520\pi$
$521$	−24.5238	−1.07441	−0.537205	+	0.843452i	$0.680520\pi$
$522$	0	0
$523$	36.5848	1.59974	0.799870	−	0.600173i	$-0.204902\pi$
$523$	36.5848	1.59974	0.799870	+	0.600173i	$0.204902\pi$
$524$	2.96474	0.129515
$525$	0	0
$526$	23.1478	1.00929
$527$	9.38934	0.409006
$528$	0	0
$529$	27.8169	1.20943
$530$	32.3269	1.40419
$531$	0	0
$532$	−2.46530	−0.106884
$533$	29.9408	1.29688
$534$	0	0
$535$	−60.4760	−2.61460
$536$	−21.9035	−0.946087
$537$	0	0
$538$	43.7590	1.88658
$539$	3.16270	0.136227
$540$	0	0
$541$	−26.7095	−1.14833	−0.574166	−	0.818739i	$-0.694674\pi$
$541$	−26.7095	−1.14833	−0.574166	+	0.818739i	$0.694674\pi$
$542$	−32.8629	−1.41158
$543$	0	0
$544$	−4.44438	−0.190551
$545$	5.79231	0.248115
$546$	0	0
$547$	44.3462	1.89611	0.948054	−	0.318109i	$-0.103048\pi$
$547$	44.3462	1.89611	0.948054	+	0.318109i	$0.103048\pi$
$548$	17.1490	0.732571
$549$	0	0
$550$	−48.4126	−2.06432
$551$	−8.40871	−0.358223
$552$	0	0
$553$	−5.09038	−0.216465
$554$	1.90586	0.0809724
$555$	0	0
$556$	7.72521	0.327622
$557$	−25.7466	−1.09092	−0.545460	−	0.838137i	$-0.683645\pi$
$557$	−25.7466	−1.09092	−0.545460	+	0.838137i	$0.683645\pi$
$558$	0	0
$559$	32.7925	1.38698
$560$	−18.6484	−0.788040
$561$	0	0
$562$	−28.0176	−1.18185
$563$	−28.0882	−1.18378	−0.591888	−	0.806021i	$-0.701617\pi$
$563$	−28.0882	−1.18378	−0.591888	+	0.806021i	$0.701617\pi$
$564$	0	0
$565$	−57.6476	−2.42525
$566$	−21.8936	−0.920256
$567$	0	0
$568$	30.7588	1.29061
$569$	−19.8532	−0.832289	−0.416144	−	0.909299i	$-0.636619\pi$
$569$	−19.8532	−0.832289	−0.416144	+	0.909299i	$0.636619\pi$
$570$	0	0
$571$	−20.5202	−0.858745	−0.429372	−	0.903128i	$-0.641265\pi$
$571$	−20.5202	−0.858745	−0.429372	+	0.903128i	$0.641265\pi$
$572$	−7.48886	−0.313125
$573$	0	0
$574$	−16.5318	−0.690026
$575$	65.4032	2.72750
$576$	0	0
$577$	−15.0348	−0.625908	−0.312954	−	0.949768i	$-0.601319\pi$
$577$	−15.0348	−0.625908	−0.312954	+	0.949768i	$0.601319\pi$
$578$	26.4995	1.10223
$579$	0	0
$580$	7.88562	0.327432
$581$	−8.93203	−0.370563
$582$	0	0
$583$	16.2764	0.674101
$584$	14.8071	0.612724
$585$	0	0
$586$	14.7574	0.609624
$587$	−39.4649	−1.62889	−0.814444	−	0.580242i	$-0.802958\pi$
$587$	−39.4649	−1.62889	−0.814444	+	0.580242i	$0.802958\pi$
$588$	0	0
$589$	−27.9489	−1.15162
$590$	26.0839	1.07386
$591$	0	0
$592$	−4.67317	−0.192066
$593$	−7.71037	−0.316627	−0.158314	−	0.987389i	$-0.550606\pi$
$593$	−7.71037	−0.316627	−0.158314	+	0.987389i	$0.550606\pi$
$594$	0	0
$595$	−3.97913	−0.163128
$596$	−1.55769	−0.0638054
$597$	0	0
$598$	35.9383	1.46963
$599$	45.7827	1.87063	0.935315	−	0.353817i	$-0.115116\pi$
$599$	45.7827	1.87063	0.935315	+	0.353817i	$0.115116\pi$
$600$	0	0
$601$	−46.7608	−1.90741	−0.953706	−	0.300739i	$-0.902767\pi$
$601$	−46.7608	−1.90741	−0.953706	+	0.300739i	$0.902767\pi$
$602$	−18.1064	−0.737962
$603$	0	0
$604$	−6.31176	−0.256822
$605$	3.75482	0.152655
$606$	0	0
$607$	43.1360	1.75084	0.875419	−	0.483366i	$-0.160586\pi$
$607$	43.1360	1.75084	0.875419	+	0.483366i	$0.160586\pi$
$608$	13.2294	0.536524
$609$	0	0
$610$	7.72969	0.312966
$611$	12.8583	0.520193
$612$	0	0
$613$	19.6862	0.795120	0.397560	−	0.917576i	$-0.369857\pi$
$613$	19.6862	0.795120	0.397560	+	0.917576i	$0.369857\pi$
$614$	−13.9958	−0.564824
$615$	0	0
$616$	−6.41846	−0.258607
$617$	5.42454	0.218384	0.109192	−	0.994021i	$-0.465174\pi$
$617$	5.42454	0.218384	0.109192	+	0.994021i	$0.465174\pi$
$618$	0	0
$619$	−23.7411	−0.954235	−0.477118	−	0.878839i	$-0.658318\pi$
$619$	−23.7411	−0.954235	−0.477118	+	0.878839i	$0.658318\pi$
$620$	26.2103	1.05263
$621$	0	0
$622$	6.12178	0.245461
$623$	5.97445	0.239361
$624$	0	0
$625$	13.3024	0.532096
$626$	−46.3100	−1.85092
$627$	0	0
$628$	−7.61728	−0.303962
$629$	−0.997143	−0.0397587
$630$	0	0
$631$	29.7531	1.18445	0.592226	−	0.805772i	$-0.298249\pi$
$631$	29.7531	1.18445	0.592226	+	0.805772i	$0.298249\pi$
$632$	10.3305	0.410927
$633$	0	0
$634$	14.6286	0.580975
$635$	3.76494	0.149407
$636$	0	0
$637$	−3.02168	−0.119723
$638$	14.1037	0.558370
$639$	0	0
$640$	49.8204	1.96932
$641$	6.06350	0.239494	0.119747	−	0.992804i	$-0.461792\pi$
$641$	6.06350	0.239494	0.119747	+	0.992804i	$0.461792\pi$
$642$	0	0
$643$	15.0189	0.592287	0.296144	−	0.955143i	$-0.404299\pi$
$643$	15.0189	0.592287	0.296144	+	0.955143i	$0.404299\pi$
$644$	−5.58616	−0.220125
$645$	0	0
$646$	5.54748	0.218263
$647$	−8.33725	−0.327771	−0.163886	−	0.986479i	$-0.552403\pi$
$647$	−8.33725	−0.327771	−0.163886	+	0.986479i	$0.552403\pi$
$648$	0	0
$649$	13.1331	0.515520
$650$	46.2539	1.81423
$651$	0	0
$652$	3.74465	0.146652
$653$	2.97846	0.116556	0.0582781	−	0.998300i	$-0.481439\pi$
$653$	2.97846	0.116556	0.0582781	+	0.998300i	$0.481439\pi$
$654$	0	0
$655$	−14.2441	−0.556563
$656$	49.0795	1.91623
$657$	0	0
$658$	−7.09974	−0.276776
$659$	−0.632856	−0.0246526	−0.0123263	−	0.999924i	$-0.503924\pi$
$659$	−0.632856	−0.0246526	−0.0123263	+	0.999924i	$0.503924\pi$
$660$	0	0
$661$	−32.4745	−1.26311	−0.631557	−	0.775330i	$-0.717584\pi$
$661$	−32.4745	−1.26311	−0.631557	+	0.775330i	$0.717584\pi$
$662$	−43.8953	−1.70604
$663$	0	0
$664$	18.1269	0.703459
$665$	11.8445	0.459311
$666$	0	0
$667$	−19.0534	−0.737752
$668$	−16.3994	−0.634512
$669$	0	0
$670$	−67.7960	−2.61919
$671$	3.89186	0.150244
$672$	0	0
$673$	−9.45807	−0.364582	−0.182291	−	0.983245i	$-0.558351\pi$
$673$	−9.45807	−0.364582	−0.182291	+	0.983245i	$0.558351\pi$
$674$	−0.324405	−0.0124956
$675$	0	0
$676$	−3.03221	−0.116624
$677$	10.2613	0.394372	0.197186	−	0.980366i	$-0.436820\pi$
$677$	10.2613	0.394372	0.197186	+	0.980366i	$0.436820\pi$
$678$	0	0
$679$	9.75834	0.374491
$680$	8.07533	0.309675
$681$	0	0
$682$	46.8780	1.79505
$683$	−1.93744	−0.0741339	−0.0370670	−	0.999313i	$-0.511801\pi$
$683$	−1.93744	−0.0741339	−0.0370670	+	0.999313i	$0.511801\pi$
$684$	0	0
$685$	−82.3927	−3.14806
$686$	1.66842	0.0637006
$687$	0	0
$688$	53.7540	2.04935
$689$	−15.5507	−0.592433
$690$	0	0
$691$	27.6381	1.05140	0.525702	−	0.850669i	$-0.323803\pi$
$691$	27.6381	1.05140	0.525702	+	0.850669i	$0.323803\pi$
$692$	−10.7820	−0.409871
$693$	0	0
$694$	34.8206	1.32177
$695$	−37.1158	−1.40788
$696$	0	0
$697$	10.4724	0.396670
$698$	10.1137	0.382809
$699$	0	0
$700$	−7.18958	−0.271741
$701$	18.7994	0.710043	0.355022	−	0.934858i	$-0.384474\pi$
$701$	18.7994	0.710043	0.355022	+	0.934858i	$0.384474\pi$
$702$	0	0
$703$	2.96816	0.111946
$704$	9.14153	0.344534
$705$	0	0
$706$	30.1062	1.13306
$707$	−11.6815	−0.439327
$708$	0	0
$709$	−36.4454	−1.36874	−0.684369	−	0.729136i	$-0.739922\pi$
$709$	−36.4454	−1.36874	−0.684369	+	0.729136i	$0.739922\pi$
$710$	95.2051	3.57298
$711$	0	0
$712$	−12.1247	−0.454392
$713$	−63.3300	−2.37173
$714$	0	0
$715$	35.9803	1.34559
$716$	7.24737	0.270847
$717$	0	0
$718$	57.8167	2.15770
$719$	25.7559	0.960533	0.480267	−	0.877123i	$-0.340540\pi$
$719$	25.7559	0.960533	0.480267	+	0.877123i	$0.340540\pi$
$720$	0	0
$721$	−13.9511	−0.519567
$722$	15.1870	0.565201
$723$	0	0
$724$	11.1006	0.412552
$725$	−24.5224	−0.910740
$726$	0	0
$727$	0.347207	0.0128772	0.00643859	−	0.999979i	$-0.497951\pi$
$727$	0.347207	0.0128772	0.00643859	+	0.999979i	$0.497951\pi$
$728$	6.13226	0.227277
$729$	0	0
$730$	45.8313	1.69629
$731$	11.4698	0.424226
$732$	0	0
$733$	−35.7358	−1.31993	−0.659966	−	0.751295i	$-0.729429\pi$
$733$	−35.7358	−1.31993	−0.659966	+	0.751295i	$0.729429\pi$
$734$	18.8801	0.696878
$735$	0	0
$736$	29.9768	1.10496
$737$	−34.1349	−1.25738
$738$	0	0
$739$	−45.6054	−1.67762	−0.838811	−	0.544422i	$-0.816749\pi$
$739$	−45.6054	−1.67762	−0.838811	+	0.544422i	$0.816749\pi$
$740$	−2.78352	−0.102324
$741$	0	0
$742$	8.58631	0.315213
$743$	−32.1925	−1.18103	−0.590513	−	0.807028i	$-0.701075\pi$
$743$	−32.1925	−1.18103	−0.590513	+	0.807028i	$0.701075\pi$
$744$	0	0
$745$	7.48393	0.274190
$746$	6.51418	0.238501
$747$	0	0
$748$	−2.61937	−0.0957737
$749$	−16.0629	−0.586927
$750$	0	0
$751$	−38.9426	−1.42104	−0.710518	−	0.703679i	$-0.751540\pi$
$751$	−38.9426	−1.42104	−0.710518	+	0.703679i	$0.751540\pi$
$752$	21.0776	0.768621
$753$	0	0
$754$	−13.4748	−0.490723
$755$	30.3249	1.10364
$756$	0	0
$757$	17.7030	0.643427	0.321714	−	0.946837i	$-0.395741\pi$
$757$	17.7030	0.643427	0.321714	+	0.946837i	$0.395741\pi$
$758$	−39.4015	−1.43113
$759$	0	0
$760$	−24.0376	−0.871934
$761$	13.0843	0.474305	0.237153	−	0.971472i	$-0.423786\pi$
$761$	13.0843	0.474305	0.237153	+	0.971472i	$0.423786\pi$
$762$	0	0
$763$	1.53849	0.0556970
$764$	2.62193	0.0948580
$765$	0	0
$766$	−42.7337	−1.54403
$767$	−12.5475	−0.453064
$768$	0	0
$769$	19.4773	0.702371	0.351185	−	0.936306i	$-0.385779\pi$
$769$	19.4773	0.702371	0.351185	+	0.936306i	$0.385779\pi$
$770$	−19.8665	−0.715939
$771$	0	0
$772$	5.41717	0.194968
$773$	−13.8332	−0.497546	−0.248773	−	0.968562i	$-0.580027\pi$
$773$	−13.8332	−0.497546	−0.248773	+	0.968562i	$0.580027\pi$
$774$	0	0
$775$	−81.5079	−2.92785
$776$	−19.8038	−0.710915
$777$	0	0
$778$	25.7374	0.922731
$779$	−31.1728	−1.11688
$780$	0	0
$781$	47.9353	1.71526
$782$	12.5701	0.449507
$783$	0	0
$784$	−4.95318	−0.176899
$785$	36.5973	1.30621
$786$	0	0
$787$	−7.07202	−0.252090	−0.126045	−	0.992025i	$-0.540228\pi$
$787$	−7.07202	−0.252090	−0.126045	+	0.992025i	$0.540228\pi$
$788$	−13.9548	−0.497121
$789$	0	0
$790$	31.9752	1.13763
$791$	−15.3117	−0.544421
$792$	0	0
$793$	−3.71832	−0.132041
$794$	9.49112	0.336827
$795$	0	0
$796$	−13.0238	−0.461617
$797$	14.5337	0.514809	0.257405	−	0.966304i	$-0.417133\pi$
$797$	14.5337	0.514809	0.257405	+	0.966304i	$0.417133\pi$
$798$	0	0
$799$	4.49745	0.159108
$800$	38.5812	1.36405
$801$	0	0
$802$	38.3615	1.35459
$803$	23.0758	0.814328
$804$	0	0
$805$	26.8387	0.945941
$806$	−44.7877	−1.57758
$807$	0	0
$808$	23.7066	0.833996
$809$	−31.3659	−1.10277	−0.551383	−	0.834252i	$-0.685900\pi$
$809$	−31.3659	−1.10277	−0.551383	+	0.834252i	$0.685900\pi$
$810$	0	0
$811$	−18.3232	−0.643413	−0.321707	−	0.946839i	$-0.604257\pi$
$811$	−18.3232	−0.643413	−0.321707	+	0.946839i	$0.604257\pi$
$812$	2.09449	0.0735021
$813$	0	0
$814$	−4.97842	−0.174494
$815$	−17.9912	−0.630204
$816$	0	0
$817$	−34.1418	−1.19447
$818$	30.0895	1.05206
$819$	0	0
$820$	29.2336	1.02088
$821$	−9.46589	−0.330362	−0.165181	−	0.986263i	$-0.552821\pi$
$821$	−9.46589	−0.330362	−0.165181	+	0.986263i	$0.552821\pi$
$822$	0	0
$823$	11.5049	0.401036	0.200518	−	0.979690i	$-0.435738\pi$
$823$	11.5049	0.401036	0.200518	+	0.979690i	$0.435738\pi$
$824$	28.3127	0.986321
$825$	0	0
$826$	6.92811	0.241060
$827$	−15.4743	−0.538094	−0.269047	−	0.963127i	$-0.586709\pi$
$827$	−15.4743	−0.538094	−0.269047	+	0.963127i	$0.586709\pi$
$828$	0	0
$829$	14.4745	0.502720	0.251360	−	0.967894i	$-0.419122\pi$
$829$	14.4745	0.502720	0.251360	+	0.967894i	$0.419122\pi$
$830$	56.1065	1.94749
$831$	0	0
$832$	−8.73390	−0.302794
$833$	−1.05689	−0.0366191
$834$	0	0
$835$	78.7911	2.72668
$836$	7.79700	0.269665
$837$	0	0
$838$	21.8353	0.754286
$839$	−24.9719	−0.862127	−0.431063	−	0.902322i	$-0.641861\pi$
$839$	−24.9719	−0.862127	−0.431063	+	0.902322i	$0.641861\pi$
$840$	0	0
$841$	−21.8561	−0.753657
$842$	66.9372	2.30681
$843$	0	0
$844$	−11.5008	−0.395875
$845$	14.5683	0.501165
$846$	0	0
$847$	0.997312	0.0342680
$848$	−25.4909	−0.875361
$849$	0	0
$850$	16.1782	0.554907
$851$	6.72561	0.230551
$852$	0	0
$853$	0.627948	0.0215005	0.0107503	−	0.999942i	$-0.496578\pi$
$853$	0.627948	0.0215005	0.0107503	+	0.999942i	$0.496578\pi$
$854$	2.05307	0.0702547
$855$	0	0
$856$	32.5985	1.11419
$857$	37.4706	1.27997	0.639986	−	0.768387i	$-0.278940\pi$
$857$	37.4706	1.27997	0.639986	+	0.768387i	$0.278940\pi$
$858$	0	0
$859$	0.464206	0.0158385	0.00791926	−	0.999969i	$-0.497479\pi$
$859$	0.464206	0.0158385	0.00791926	+	0.999969i	$0.497479\pi$
$860$	32.0179	1.09180
$861$	0	0
$862$	28.2277	0.961440
$863$	13.3762	0.455333	0.227666	−	0.973739i	$-0.426890\pi$
$863$	13.3762	0.455333	0.227666	+	0.973739i	$0.426890\pi$
$864$	0	0
$865$	51.8023	1.76133
$866$	14.4578	0.491295
$867$	0	0
$868$	6.96168	0.236295
$869$	16.0994	0.546134
$870$	0	0
$871$	32.6129	1.10504
$872$	−3.12224	−0.105733
$873$	0	0
$874$	−37.4171	−1.26565
$875$	15.7177	0.531356
$876$	0	0
$877$	−4.07242	−0.137516	−0.0687580	−	0.997633i	$-0.521904\pi$
$877$	−4.07242	−0.137516	−0.0687580	+	0.997633i	$0.521904\pi$
$878$	42.5289	1.43528
$879$	0	0
$880$	58.9794	1.98820
$881$	−4.54811	−0.153230	−0.0766149	−	0.997061i	$-0.524411\pi$
$881$	−4.54811	−0.153230	−0.0766149	+	0.997061i	$0.524411\pi$
$882$	0	0
$883$	−22.3985	−0.753771	−0.376885	−	0.926260i	$-0.623005\pi$
$883$	−22.3985	−0.753771	−0.376885	+	0.926260i	$0.623005\pi$
$884$	2.50257	0.0841707
$885$	0	0
$886$	−58.2720	−1.95769
$887$	4.37232	0.146808	0.0734040	−	0.997302i	$-0.476614\pi$
$887$	4.37232	0.146808	0.0734040	+	0.997302i	$0.476614\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−37.5285	−1.25796
$891$	0	0
$892$	13.5425	0.453436
$893$	−13.3874	−0.447993
$894$	0	0
$895$	−34.8200	−1.16391
$896$	13.2327	0.442074
$897$	0	0
$898$	−48.4868	−1.61803
$899$	23.7451	0.791944
$900$	0	0
$901$	−5.43915	−0.181204
$902$	52.2853	1.74091
$903$	0	0
$904$	31.0739	1.03350
$905$	−53.3331	−1.77285
$906$	0	0
$907$	22.6987	0.753697	0.376849	−	0.926275i	$-0.377008\pi$
$907$	22.6987	0.753697	0.376849	+	0.926275i	$0.377008\pi$
$908$	−5.59333	−0.185621
$909$	0	0
$910$	18.9807	0.629203
$911$	−26.5960	−0.881165	−0.440582	−	0.897712i	$-0.645228\pi$
$911$	−26.5960	−0.881165	−0.440582	+	0.897712i	$0.645228\pi$
$912$	0	0
$913$	28.2493	0.934917
$914$	−27.7535	−0.918003
$915$	0	0
$916$	6.77957	0.224003
$917$	−3.78335	−0.124937
$918$	0	0
$919$	−12.3633	−0.407828	−0.203914	−	0.978989i	$-0.565366\pi$
$919$	−12.3633	−0.407828	−0.203914	+	0.978989i	$0.565366\pi$
$920$	−54.4671	−1.79573
$921$	0	0
$922$	2.83022	0.0932083
$923$	−45.7979	−1.50745
$924$	0	0
$925$	8.65610	0.284611
$926$	55.6549	1.82893
$927$	0	0
$928$	−11.2396	−0.368957
$929$	−40.0029	−1.31245	−0.656227	−	0.754563i	$-0.727849\pi$
$929$	−40.0029	−1.31245	−0.656227	+	0.754563i	$0.727849\pi$
$930$	0	0
$931$	3.14601	0.103106
$932$	−2.16585	−0.0709448
$933$	0	0
$934$	−37.1759	−1.21643
$935$	12.5848	0.411567
$936$	0	0
$937$	−51.5727	−1.68481	−0.842403	−	0.538848i	$-0.818860\pi$
$937$	−51.5727	−1.68481	−0.842403	+	0.538848i	$0.818860\pi$
$938$	−18.0072	−0.587956
$939$	0	0
$940$	12.5546	0.409487
$941$	−43.1695	−1.40729	−0.703644	−	0.710553i	$-0.748445\pi$
$941$	−43.1695	−1.40729	−0.703644	+	0.710553i	$0.748445\pi$
$942$	0	0
$943$	−70.6350	−2.30019
$944$	−20.5681	−0.669434
$945$	0	0
$946$	57.2652	1.86185
$947$	32.9398	1.07040	0.535200	−	0.844726i	$-0.320236\pi$
$947$	32.9398	1.07040	0.535200	+	0.844726i	$0.320236\pi$
$948$	0	0
$949$	−22.0469	−0.715671
$950$	−48.1571	−1.56242
$951$	0	0
$952$	2.14488	0.0695158
$953$	−23.7506	−0.769357	−0.384679	−	0.923051i	$-0.625688\pi$
$953$	−23.7506	−0.769357	−0.384679	+	0.923051i	$0.625688\pi$
$954$	0	0
$955$	−12.5971	−0.407632
$956$	18.6638	0.603631
$957$	0	0
$958$	17.4056	0.562348
$959$	−21.8842	−0.706678
$960$	0	0
$961$	47.9242	1.54594
$962$	4.75643	0.153353
$963$	0	0
$964$	−14.6412	−0.471563
$965$	−26.0268	−0.837833
$966$	0	0
$967$	1.31829	0.0423934	0.0211967	−	0.999775i	$-0.493252\pi$
$967$	1.31829	0.0423934	0.0211967	+	0.999775i	$0.493252\pi$
$968$	−2.02397	−0.0650528
$969$	0	0
$970$	−61.2970	−1.96813
$971$	−46.5792	−1.49480	−0.747398	−	0.664376i	$-0.768697\pi$
$971$	−46.5792	−1.49480	−0.747398	+	0.664376i	$0.768697\pi$
$972$	0	0
$973$	−9.85829	−0.316042
$974$	−7.13134	−0.228503
$975$	0	0
$976$	−6.09513	−0.195100
$977$	46.4171	1.48501	0.742507	−	0.669839i	$-0.233637\pi$
$977$	46.4171	1.48501	0.742507	+	0.669839i	$0.233637\pi$
$978$	0	0
$979$	−18.8954	−0.603900
$980$	−2.95030	−0.0942440
$981$	0	0
$982$	−20.4177	−0.651554
$983$	−56.9936	−1.81781	−0.908907	−	0.416998i	$-0.863082\pi$
$983$	−56.9936	−1.81781	−0.908907	+	0.416998i	$0.863082\pi$
$984$	0	0
$985$	67.0462	2.13627
$986$	−4.71307	−0.150095
$987$	0	0
$988$	−7.44933	−0.236995
$989$	−77.3626	−2.45999
$990$	0	0
$991$	6.27979	0.199484	0.0997421	−	0.995013i	$-0.468198\pi$
$991$	6.27979	0.199484	0.0997421	+	0.995013i	$0.468198\pi$
$992$	−37.3582	−1.18612
$993$	0	0
$994$	25.2873	0.802064
$995$	62.5730	1.98370
$996$	0	0
$997$	16.8353	0.533179	0.266589	−	0.963810i	$-0.414103\pi$
$997$	16.8353	0.533179	0.266589	+	0.963810i	$0.414103\pi$
$998$	−30.1646	−0.954845
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.7	✓	32
3.2	odd	2	inner	8001.2.a.z.1.26	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.7	✓	32	1.1	even	1	trivial
8001.2.a.z.1.26	yes	32	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-1.66842 q^{2} +0.783626 q^{4} -3.76494 q^{5} -1.00000 q^{7} +2.02942 q^{8} +O(q^{10})\)	\(q-1.66842 q^{2} +0.783626 q^{4} -3.76494 q^{5} -1.00000 q^{7} +2.02942 q^{8} +6.28150 q^{10} +3.16270 q^{11} -3.02168 q^{13} +1.66842 q^{14} -4.95318 q^{16} -1.05689 q^{17} +3.14601 q^{19} -2.95030 q^{20} -5.27672 q^{22} +7.12860 q^{23} +9.17476 q^{25} +5.04143 q^{26} -0.783626 q^{28} -2.67282 q^{29} -8.88393 q^{31} +4.20514 q^{32} +1.76334 q^{34} +3.76494 q^{35} +0.943469 q^{37} -5.24887 q^{38} -7.64065 q^{40} -9.90868 q^{41} -10.8524 q^{43} +2.47838 q^{44} -11.8935 q^{46} -4.25536 q^{47} +1.00000 q^{49} -15.3074 q^{50} -2.36787 q^{52} +5.14637 q^{53} -11.9074 q^{55} -2.02942 q^{56} +4.45938 q^{58} +4.15250 q^{59} +1.23055 q^{61} +14.8221 q^{62} +2.89042 q^{64} +11.3764 q^{65} -10.7930 q^{67} -0.828207 q^{68} -6.28150 q^{70} +15.1564 q^{71} +7.29623 q^{73} -1.57410 q^{74} +2.46530 q^{76} -3.16270 q^{77} +5.09038 q^{79} +18.6484 q^{80} +16.5318 q^{82} +8.93203 q^{83} +3.97913 q^{85} +18.1064 q^{86} +6.41846 q^{88} -5.97445 q^{89} +3.02168 q^{91} +5.58616 q^{92} +7.09974 q^{94} -11.8445 q^{95} -9.75834 q^{97} -1.66842 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 32 * q + 30 * q^4 - 32 * q^7	\( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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