LMFDB - Embedded newform 8001.2.a.x.1.5

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:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
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)

$f(q)$	$=$	$q-1.69586 q^{2} +0.875937 q^{4} -2.42277 q^{5} +1.00000 q^{7} +1.90625 q^{8} +O(q^{10})$	\(q-1.69586 q^{2} +0.875937 q^{4} -2.42277 q^{5} +1.00000 q^{7} +1.90625 q^{8} +4.10867 q^{10} +1.73392 q^{11} -5.13465 q^{13} -1.69586 q^{14} -4.98461 q^{16} +0.851260 q^{17} +2.28603 q^{19} -2.12219 q^{20} -2.94048 q^{22} -0.763693 q^{23} +0.869801 q^{25} +8.70764 q^{26} +0.875937 q^{28} +1.78230 q^{29} -0.617901 q^{31} +4.64069 q^{32} -1.44362 q^{34} -2.42277 q^{35} -0.416163 q^{37} -3.87679 q^{38} -4.61840 q^{40} +10.0613 q^{41} +1.13021 q^{43} +1.51880 q^{44} +1.29512 q^{46} -7.29504 q^{47} +1.00000 q^{49} -1.47506 q^{50} -4.49763 q^{52} +7.96281 q^{53} -4.20088 q^{55} +1.90625 q^{56} -3.02252 q^{58} -4.85174 q^{59} -11.0868 q^{61} +1.04787 q^{62} +2.09926 q^{64} +12.4401 q^{65} -2.64984 q^{67} +0.745650 q^{68} +4.10867 q^{70} -6.20885 q^{71} +5.81806 q^{73} +0.705753 q^{74} +2.00242 q^{76} +1.73392 q^{77} -9.99008 q^{79} +12.0765 q^{80} -17.0625 q^{82} +13.8723 q^{83} -2.06241 q^{85} -1.91668 q^{86} +3.30528 q^{88} +9.74339 q^{89} -5.13465 q^{91} -0.668947 q^{92} +12.3714 q^{94} -5.53852 q^{95} +8.66426 q^{97} -1.69586 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) $ 22 * q + 10 * q^4 + 22 * q^7	$ 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52} - 28 q^{55} - 26 q^{58} - 22 q^{61} - 76 q^{64} - 48 q^{67} - 4 q^{70} - 48 q^{73} - 20 q^{76} - 94 q^{79} - 8 q^{82} - 40 q^{85} - 26 q^{88} - 10 q^{91} - 26 q^{94} - 56 q^{97}+O(q^{100}) $ 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * 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 $

$2$	−1.69586	−1.19915	−0.599577	−	0.800317i	$-0.704664\pi$
$2$	−1.69586	−1.19915	−0.599577	+	0.800317i	$0.704664\pi$
$3$	0	0
$3$	0	0
$4$	0.875937	0.437969
$5$	−2.42277	−1.08349	−0.541747	−	0.840541i	$-0.682237\pi$
$5$	−2.42277	−1.08349	−0.541747	+	0.840541i	$0.682237\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.90625	0.673962
$9$	0	0
$10$	4.10867	1.29928
$11$	1.73392	0.522795	0.261398	−	0.965231i	$-0.415817\pi$
$11$	1.73392	0.522795	0.261398	+	0.965231i	$0.415817\pi$
$12$	0	0
$13$	−5.13465	−1.42410	−0.712048	−	0.702131i	$-0.752232\pi$
$13$	−5.13465	−1.42410	−0.712048	+	0.702131i	$0.752232\pi$
$14$	−1.69586	−0.453237
$15$	0	0
$16$	−4.98461	−1.24615
$17$	0.851260	0.206461	0.103230	−	0.994657i	$-0.467082\pi$
$17$	0.851260	0.206461	0.103230	+	0.994657i	$0.467082\pi$
$18$	0	0
$19$	2.28603	0.524452	0.262226	−	0.965007i	$-0.415543\pi$
$19$	2.28603	0.524452	0.262226	+	0.965007i	$0.415543\pi$
$20$	−2.12219	−0.474537
$21$	0	0
$22$	−2.94048	−0.626912
$23$	−0.763693	−0.159241	−0.0796205	−	0.996825i	$-0.525371\pi$
$23$	−0.763693	−0.159241	−0.0796205	+	0.996825i	$0.525371\pi$
$24$	0	0
$25$	0.869801	0.173960
$26$	8.70764	1.70771
$27$	0	0
$28$	0.875937	0.165537
$29$	1.78230	0.330964	0.165482	−	0.986213i	$-0.447082\pi$
$29$	1.78230	0.330964	0.165482	+	0.986213i	$0.447082\pi$
$30$	0	0
$31$	−0.617901	−0.110978	−0.0554891	−	0.998459i	$-0.517672\pi$
$31$	−0.617901	−0.110978	−0.0554891	+	0.998459i	$0.517672\pi$
$32$	4.64069	0.820366
$33$	0	0
$34$	−1.44362	−0.247578
$35$	−2.42277	−0.409522
$36$	0	0
$37$	−0.416163	−0.0684167	−0.0342083	−	0.999415i	$-0.510891\pi$
$37$	−0.416163	−0.0684167	−0.0342083	+	0.999415i	$0.510891\pi$
$38$	−3.87679	−0.628898
$39$	0	0
$40$	−4.61840	−0.730234
$41$	10.0613	1.57130	0.785652	−	0.618669i	$-0.212328\pi$
$41$	10.0613	1.57130	0.785652	+	0.618669i	$0.212328\pi$
$42$	0	0
$43$	1.13021	0.172356	0.0861781	−	0.996280i	$-0.472535\pi$
$43$	1.13021	0.172356	0.0861781	+	0.996280i	$0.472535\pi$
$44$	1.51880	0.228968
$45$	0	0
$46$	1.29512	0.190954
$47$	−7.29504	−1.06409	−0.532046	−	0.846716i	$-0.678577\pi$
$47$	−7.29504	−1.06409	−0.532046	+	0.846716i	$0.678577\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.47506	−0.208605
$51$	0	0
$52$	−4.49763	−0.623709
$53$	7.96281	1.09378	0.546888	−	0.837206i	$-0.315812\pi$
$53$	7.96281	1.09378	0.546888	+	0.837206i	$0.315812\pi$
$54$	0	0
$55$	−4.20088	−0.566446
$56$	1.90625	0.254734
$57$	0	0
$58$	−3.02252	−0.396877
$59$	−4.85174	−0.631643	−0.315821	−	0.948819i	$-0.602280\pi$
$59$	−4.85174	−0.631643	−0.315821	+	0.948819i	$0.602280\pi$
$60$	0	0
$61$	−11.0868	−1.41953	−0.709763	−	0.704441i	$-0.751198\pi$
$61$	−11.0868	−1.41953	−0.709763	+	0.704441i	$0.751198\pi$
$62$	1.04787	0.133080
$63$	0	0
$64$	2.09926	0.262408
$65$	12.4401	1.54300
$66$	0	0
$67$	−2.64984	−0.323730	−0.161865	−	0.986813i	$-0.551751\pi$
$67$	−2.64984	−0.323730	−0.161865	+	0.986813i	$0.551751\pi$
$68$	0.745650	0.0904234
$69$	0	0
$70$	4.10867	0.491080
$71$	−6.20885	−0.736854	−0.368427	−	0.929657i	$-0.620104\pi$
$71$	−6.20885	−0.736854	−0.368427	+	0.929657i	$0.620104\pi$
$72$	0	0
$73$	5.81806	0.680952	0.340476	−	0.940253i	$-0.389412\pi$
$73$	5.81806	0.680952	0.340476	+	0.940253i	$0.389412\pi$
$74$	0.705753	0.0820421
$75$	0	0
$76$	2.00242	0.229693
$77$	1.73392	0.197598
$78$	0	0
$79$	−9.99008	−1.12397	−0.561986	−	0.827147i	$-0.689962\pi$
$79$	−9.99008	−1.12397	−0.561986	+	0.827147i	$0.689962\pi$
$80$	12.0765	1.35020
$81$	0	0
$82$	−17.0625	−1.88423
$83$	13.8723	1.52268	0.761340	−	0.648353i	$-0.224542\pi$
$83$	13.8723	1.52268	0.761340	+	0.648353i	$0.224542\pi$
$84$	0	0
$85$	−2.06241	−0.223699
$86$	−1.91668	−0.206681
$87$	0	0
$88$	3.30528	0.352344
$89$	9.74339	1.03280	0.516398	−	0.856348i	$-0.327272\pi$
$89$	9.74339	1.03280	0.516398	+	0.856348i	$0.327272\pi$
$90$	0	0
$91$	−5.13465	−0.538258
$92$	−0.668947	−0.0697425
$93$	0	0
$94$	12.3714	1.27601
$95$	−5.53852	−0.568240
$96$	0	0
$97$	8.66426	0.879722	0.439861	−	0.898066i	$-0.355028\pi$
$97$	8.66426	0.879722	0.439861	+	0.898066i	$0.355028\pi$
$98$	−1.69586	−0.171308
$99$	0	0
$100$	0.761891	0.0761891
$101$	14.0069	1.39374	0.696870	−	0.717197i	$-0.254575\pi$
$101$	14.0069	1.39374	0.696870	+	0.717197i	$0.254575\pi$
$102$	0	0
$103$	−13.6920	−1.34912	−0.674559	−	0.738221i	$-0.735666\pi$
$103$	−13.6920	−1.34912	−0.674559	+	0.738221i	$0.735666\pi$
$104$	−9.78794	−0.959786
$105$	0	0
$106$	−13.5038	−1.31160
$107$	8.20220	0.792937	0.396468	−	0.918048i	$-0.370236\pi$
$107$	8.20220	0.792937	0.396468	+	0.918048i	$0.370236\pi$
$108$	0	0
$109$	−13.3855	−1.28210	−0.641049	−	0.767500i	$-0.721500\pi$
$109$	−13.3855	−1.28210	−0.641049	+	0.767500i	$0.721500\pi$
$110$	7.12409	0.679255
$111$	0	0
$112$	−4.98461	−0.471001
$113$	−15.8349	−1.48962	−0.744811	−	0.667275i	$-0.767460\pi$
$113$	−15.8349	−1.48962	−0.744811	+	0.667275i	$0.767460\pi$
$114$	0	0
$115$	1.85025	0.172537
$116$	1.56118	0.144952
$117$	0	0
$118$	8.22787	0.757437
$119$	0.851260	0.0780349
$120$	0	0
$121$	−7.99353	−0.726685
$122$	18.8017	1.70223
$123$	0	0
$124$	−0.541242	−0.0486050
$125$	10.0065	0.895010
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−12.8414	−1.13503
$129$	0	0
$130$	−21.0966	−1.85029
$131$	−7.38950	−0.645624	−0.322812	−	0.946463i	$-0.604628\pi$
$131$	−7.38950	−0.645624	−0.322812	+	0.946463i	$0.604628\pi$
$132$	0	0
$133$	2.28603	0.198224
$134$	4.49376	0.388202
$135$	0	0
$136$	1.62272	0.139147
$137$	−3.60485	−0.307983	−0.153991	−	0.988072i	$-0.549213\pi$
$137$	−3.60485	−0.307983	−0.153991	+	0.988072i	$0.549213\pi$
$138$	0	0
$139$	14.5161	1.23124	0.615619	−	0.788044i	$-0.288906\pi$
$139$	14.5161	1.23124	0.615619	+	0.788044i	$0.288906\pi$
$140$	−2.12219	−0.179358
$141$	0	0
$142$	10.5293	0.883601
$143$	−8.90306	−0.744511
$144$	0	0
$145$	−4.31809	−0.358598
$146$	−9.86661	−0.816566
$147$	0	0
$148$	−0.364532	−0.0299644
$149$	3.71268	0.304155	0.152077	−	0.988369i	$-0.451404\pi$
$149$	3.71268	0.304155	0.152077	+	0.988369i	$0.451404\pi$
$150$	0	0
$151$	−15.7946	−1.28535	−0.642673	−	0.766141i	$-0.722175\pi$
$151$	−15.7946	−1.28535	−0.642673	+	0.766141i	$0.722175\pi$
$152$	4.35775	0.353460
$153$	0	0
$154$	−2.94048	−0.236950
$155$	1.49703	0.120244
$156$	0	0
$157$	−19.8049	−1.58060	−0.790301	−	0.612719i	$-0.790076\pi$
$157$	−19.8049	−1.58060	−0.790301	+	0.612719i	$0.790076\pi$
$158$	16.9418	1.34781
$159$	0	0
$160$	−11.2433	−0.888862
$161$	−0.763693	−0.0601874
$162$	0	0
$163$	25.0668	1.96338	0.981692	−	0.190477i	$-0.0610034\pi$
$163$	25.0668	1.96338	0.981692	+	0.190477i	$0.0610034\pi$
$164$	8.81303	0.688182
$165$	0	0
$166$	−23.5254	−1.82593
$167$	21.9625	1.69951	0.849753	−	0.527181i	$-0.176751\pi$
$167$	21.9625	1.69951	0.849753	+	0.527181i	$0.176751\pi$
$168$	0	0
$169$	13.3647	1.02805
$170$	3.49755	0.268250
$171$	0	0
$172$	0.989997	0.0754866
$173$	−8.96659	−0.681717	−0.340859	−	0.940115i	$-0.610718\pi$
$173$	−8.96659	−0.681717	−0.340859	+	0.940115i	$0.610718\pi$
$174$	0	0
$175$	0.869801	0.0657508
$176$	−8.64289	−0.651483
$177$	0	0
$178$	−16.5234	−1.23848
$179$	−5.80927	−0.434205	−0.217103	−	0.976149i	$-0.569661\pi$
$179$	−5.80927	−0.434205	−0.217103	+	0.976149i	$0.569661\pi$
$180$	0	0
$181$	17.4833	1.29953	0.649763	−	0.760137i	$-0.274868\pi$
$181$	17.4833	1.29953	0.649763	+	0.760137i	$0.274868\pi$
$182$	8.70764	0.645454
$183$	0	0
$184$	−1.45579	−0.107322
$185$	1.00826	0.0741291
$186$	0	0
$187$	1.47601	0.107937
$188$	−6.39000	−0.466039
$189$	0	0
$190$	9.39255	0.681407
$191$	2.83289	0.204981	0.102490	−	0.994734i	$-0.467319\pi$
$191$	2.83289	0.204981	0.102490	+	0.994734i	$0.467319\pi$
$192$	0	0
$193$	8.87295	0.638689	0.319344	−	0.947639i	$-0.396537\pi$
$193$	8.87295	0.638689	0.319344	+	0.947639i	$0.396537\pi$
$194$	−14.6934	−1.05492
$195$	0	0
$196$	0.875937	0.0625669
$197$	4.35954	0.310605	0.155302	−	0.987867i	$-0.450365\pi$
$197$	4.35954	0.310605	0.155302	+	0.987867i	$0.450365\pi$
$198$	0	0
$199$	18.7456	1.32884	0.664422	−	0.747358i	$-0.268678\pi$
$199$	18.7456	1.32884	0.664422	+	0.747358i	$0.268678\pi$
$200$	1.65806	0.117243
$201$	0	0
$202$	−23.7538	−1.67131
$203$	1.78230	0.125093
$204$	0	0
$205$	−24.3761	−1.70250
$206$	23.2198	1.61780
$207$	0	0
$208$	25.5942	1.77464
$209$	3.96379	0.274181
$210$	0	0
$211$	2.40710	0.165711	0.0828557	−	0.996562i	$-0.473596\pi$
$211$	2.40710	0.165711	0.0828557	+	0.996562i	$0.473596\pi$
$212$	6.97492	0.479039
$213$	0	0
$214$	−13.9098	−0.950852
$215$	−2.73825	−0.186747
$216$	0	0
$217$	−0.617901	−0.0419459
$218$	22.6999	1.53743
$219$	0	0
$220$	−3.67970	−0.248086
$221$	−4.37092	−0.294020
$222$	0	0
$223$	−15.3987	−1.03118	−0.515588	−	0.856837i	$-0.672426\pi$
$223$	−15.3987	−1.03118	−0.515588	+	0.856837i	$0.672426\pi$
$224$	4.64069	0.310069
$225$	0	0
$226$	26.8538	1.78629
$227$	18.0030	1.19490	0.597450	−	0.801906i	$-0.296181\pi$
$227$	18.0030	1.19490	0.597450	+	0.801906i	$0.296181\pi$
$228$	0	0
$229$	5.95469	0.393497	0.196749	−	0.980454i	$-0.436962\pi$
$229$	5.95469	0.393497	0.196749	+	0.980454i	$0.436962\pi$
$230$	−3.13776	−0.206898
$231$	0	0
$232$	3.39751	0.223057
$233$	23.3672	1.53083	0.765417	−	0.643535i	$-0.222533\pi$
$233$	23.3672	1.53083	0.765417	+	0.643535i	$0.222533\pi$
$234$	0	0
$235$	17.6742	1.15294
$236$	−4.24982	−0.276640
$237$	0	0
$238$	−1.44362	−0.0935758
$239$	−18.5556	−1.20026	−0.600131	−	0.799902i	$-0.704885\pi$
$239$	−18.5556	−1.20026	−0.600131	+	0.799902i	$0.704885\pi$
$240$	0	0
$241$	19.0571	1.22757	0.613787	−	0.789472i	$-0.289645\pi$
$241$	19.0571	1.22757	0.613787	+	0.789472i	$0.289645\pi$
$242$	13.5559	0.871407
$243$	0	0
$244$	−9.71138	−0.621707
$245$	−2.42277	−0.154785
$246$	0	0
$247$	−11.7380	−0.746870
$248$	−1.17787	−0.0747951
$249$	0	0
$250$	−16.9696	−1.07325
$251$	−30.6144	−1.93236	−0.966182	−	0.257859i	$-0.916983\pi$
$251$	−30.6144	−1.93236	−0.966182	+	0.257859i	$0.916983\pi$
$252$	0	0
$253$	−1.32418	−0.0832504
$254$	−1.69586	−0.106408
$255$	0	0
$256$	17.5787	1.09867
$257$	12.0807	0.753571	0.376786	−	0.926300i	$-0.377029\pi$
$257$	12.0807	0.753571	0.376786	+	0.926300i	$0.377029\pi$
$258$	0	0
$259$	−0.416163	−0.0258591
$260$	10.8967	0.675786
$261$	0	0
$262$	12.5315	0.774202
$263$	15.2700	0.941591	0.470796	−	0.882242i	$-0.343967\pi$
$263$	15.2700	0.941591	0.470796	+	0.882242i	$0.343967\pi$
$264$	0	0
$265$	−19.2920	−1.18510
$266$	−3.87679	−0.237701
$267$	0	0
$268$	−2.32110	−0.141784
$269$	−3.93075	−0.239662	−0.119831	−	0.992794i	$-0.538235\pi$
$269$	−3.93075	−0.239662	−0.119831	+	0.992794i	$0.538235\pi$
$270$	0	0
$271$	−11.6607	−0.708340	−0.354170	−	0.935181i	$-0.615237\pi$
$271$	−11.6607	−0.708340	−0.354170	+	0.935181i	$0.615237\pi$
$272$	−4.24320	−0.257282
$273$	0	0
$274$	6.11331	0.369319
$275$	1.50816	0.0909456
$276$	0	0
$277$	13.1023	0.787239	0.393619	−	0.919274i	$-0.371223\pi$
$277$	13.1023	0.787239	0.393619	+	0.919274i	$0.371223\pi$
$278$	−24.6172	−1.47644
$279$	0	0
$280$	−4.61840	−0.276002
$281$	−1.00292	−0.0598293	−0.0299146	−	0.999552i	$-0.509524\pi$
$281$	−1.00292	−0.0598293	−0.0299146	+	0.999552i	$0.509524\pi$
$282$	0	0
$283$	−4.54951	−0.270441	−0.135220	−	0.990816i	$-0.543174\pi$
$283$	−4.54951	−0.270441	−0.135220	+	0.990816i	$0.543174\pi$
$284$	−5.43856	−0.322719
$285$	0	0
$286$	15.0983	0.892783
$287$	10.0613	0.593897
$288$	0	0
$289$	−16.2754	−0.957374
$290$	7.32287	0.430014
$291$	0	0
$292$	5.09626	0.298236
$293$	−26.1899	−1.53003	−0.765015	−	0.644012i	$-0.777269\pi$
$293$	−26.1899	−1.53003	−0.765015	+	0.644012i	$0.777269\pi$
$294$	0	0
$295$	11.7546	0.684382
$296$	−0.793311	−0.0461102
$297$	0	0
$298$	−6.29619	−0.364728
$299$	3.92130	0.226774
$300$	0	0
$301$	1.13021	0.0651445
$302$	26.7854	1.54133
$303$	0	0
$304$	−11.3950	−0.653547
$305$	26.8608	1.53805
$306$	0	0
$307$	3.82385	0.218238	0.109119	−	0.994029i	$-0.465197\pi$
$307$	3.82385	0.218238	0.109119	+	0.994029i	$0.465197\pi$
$308$	1.51880	0.0865418
$309$	0	0
$310$	−2.53875	−0.144191
$311$	12.8378	0.727963	0.363981	−	0.931406i	$-0.381417\pi$
$311$	12.8378	0.727963	0.363981	+	0.931406i	$0.381417\pi$
$312$	0	0
$313$	−21.1630	−1.19620	−0.598101	−	0.801421i	$-0.704078\pi$
$313$	−21.1630	−1.19620	−0.598101	+	0.801421i	$0.704078\pi$
$314$	33.5863	1.89538
$315$	0	0
$316$	−8.75068	−0.492264
$317$	−17.1220	−0.961666	−0.480833	−	0.876812i	$-0.659666\pi$
$317$	−17.1220	−0.961666	−0.480833	+	0.876812i	$0.659666\pi$
$318$	0	0
$319$	3.09035	0.173026
$320$	−5.08603	−0.284318
$321$	0	0
$322$	1.29512	0.0721740
$323$	1.94601	0.108279
$324$	0	0
$325$	−4.46613	−0.247736
$326$	−42.5098	−2.35440
$327$	0	0
$328$	19.1793	1.05900
$329$	−7.29504	−0.402189
$330$	0	0
$331$	3.15269	0.173287	0.0866437	−	0.996239i	$-0.472386\pi$
$331$	3.15269	0.173287	0.0866437	+	0.996239i	$0.472386\pi$
$332$	12.1512	0.666886
$333$	0	0
$334$	−37.2452	−2.03797
$335$	6.41996	0.350760
$336$	0	0
$337$	10.6795	0.581748	0.290874	−	0.956761i	$-0.406054\pi$
$337$	10.6795	0.581748	0.290874	+	0.956761i	$0.406054\pi$
$338$	−22.6646	−1.23279
$339$	0	0
$340$	−1.80654	−0.0979732
$341$	−1.07139	−0.0580189
$342$	0	0
$343$	1.00000	0.0539949
$344$	2.15447	0.116161
$345$	0	0
$346$	15.2061	0.817483
$347$	−10.0725	−0.540718	−0.270359	−	0.962760i	$-0.587142\pi$
$347$	−10.0725	−0.540718	−0.270359	+	0.962760i	$0.587142\pi$
$348$	0	0
$349$	−18.6921	−1.00056	−0.500282	−	0.865862i	$-0.666770\pi$
$349$	−18.6921	−1.00056	−0.500282	+	0.865862i	$0.666770\pi$
$350$	−1.47506	−0.0788453
$351$	0	0
$352$	8.04656	0.428883
$353$	−5.87312	−0.312594	−0.156297	−	0.987710i	$-0.549956\pi$
$353$	−5.87312	−0.312594	−0.156297	+	0.987710i	$0.549956\pi$
$354$	0	0
$355$	15.0426	0.798378
$356$	8.53459	0.452333
$357$	0	0
$358$	9.85170	0.520679
$359$	−7.58043	−0.400080	−0.200040	−	0.979788i	$-0.564107\pi$
$359$	−7.58043	−0.400080	−0.200040	+	0.979788i	$0.564107\pi$
$360$	0	0
$361$	−13.7741	−0.724950
$362$	−29.6493	−1.55833
$363$	0	0
$364$	−4.49763	−0.235740
$365$	−14.0958	−0.737808
$366$	0	0
$367$	33.6521	1.75662	0.878311	−	0.478089i	$-0.158670\pi$
$367$	33.6521	1.75662	0.878311	+	0.478089i	$0.158670\pi$
$368$	3.80671	0.198438
$369$	0	0
$370$	−1.70987	−0.0888922
$371$	7.96281	0.413408
$372$	0	0
$373$	−6.89540	−0.357030	−0.178515	−	0.983937i	$-0.557129\pi$
$373$	−6.89540	−0.357030	−0.178515	+	0.983937i	$0.557129\pi$
$374$	−2.50311	−0.129433
$375$	0	0
$376$	−13.9062	−0.717157
$377$	−9.15147	−0.471325
$378$	0	0
$379$	−9.10862	−0.467879	−0.233939	−	0.972251i	$-0.575162\pi$
$379$	−9.10862	−0.467879	−0.233939	+	0.972251i	$0.575162\pi$
$380$	−4.85140	−0.248871
$381$	0	0
$382$	−4.80418	−0.245803
$383$	9.52730	0.486822	0.243411	−	0.969923i	$-0.421734\pi$
$383$	9.52730	0.486822	0.243411	+	0.969923i	$0.421734\pi$
$384$	0	0
$385$	−4.20088	−0.214096
$386$	−15.0473	−0.765886
$387$	0	0
$388$	7.58935	0.385291
$389$	−37.6745	−1.91017	−0.955087	−	0.296324i	$-0.904239\pi$
$389$	−37.6745	−1.91017	−0.955087	+	0.296324i	$0.904239\pi$
$390$	0	0
$391$	−0.650101	−0.0328770
$392$	1.90625	0.0962803
$393$	0	0
$394$	−7.39317	−0.372462
$395$	24.2036	1.21782
$396$	0	0
$397$	−22.0009	−1.10419	−0.552096	−	0.833780i	$-0.686172\pi$
$397$	−22.0009	−1.10419	−0.552096	+	0.833780i	$0.686172\pi$
$398$	−31.7900	−1.59349
$399$	0	0
$400$	−4.33562	−0.216781
$401$	0.904318	0.0451595	0.0225797	−	0.999745i	$-0.492812\pi$
$401$	0.904318	0.0451595	0.0225797	+	0.999745i	$0.492812\pi$
$402$	0	0
$403$	3.17271	0.158044
$404$	12.2692	0.610415
$405$	0	0
$406$	−3.02252	−0.150005
$407$	−0.721591	−0.0357679
$408$	0	0
$409$	25.7199	1.27177	0.635885	−	0.771784i	$-0.280635\pi$
$409$	25.7199	1.27177	0.635885	+	0.771784i	$0.280635\pi$
$410$	41.3384	2.04156
$411$	0	0
$412$	−11.9934	−0.590871
$413$	−4.85174	−0.238739
$414$	0	0
$415$	−33.6093	−1.64982
$416$	−23.8283	−1.16828
$417$	0	0
$418$	−6.72202	−0.328785
$419$	−10.6580	−0.520677	−0.260338	−	0.965517i	$-0.583834\pi$
$419$	−10.6580	−0.520677	−0.260338	+	0.965517i	$0.583834\pi$
$420$	0	0
$421$	−19.3285	−0.942013	−0.471007	−	0.882130i	$-0.656109\pi$
$421$	−19.3285	−0.942013	−0.471007	+	0.882130i	$0.656109\pi$
$422$	−4.08210	−0.198713
$423$	0	0
$424$	15.1791	0.737163
$425$	0.740427	0.0359160
$426$	0	0
$427$	−11.0868	−0.536530
$428$	7.18461	0.347281
$429$	0	0
$430$	4.64368	0.223938
$431$	29.3932	1.41582	0.707910	−	0.706302i	$-0.249638\pi$
$431$	29.3932	1.41582	0.707910	+	0.706302i	$0.249638\pi$
$432$	0	0
$433$	17.9349	0.861896	0.430948	−	0.902377i	$-0.358179\pi$
$433$	17.9349	0.861896	0.430948	+	0.902377i	$0.358179\pi$
$434$	1.04787	0.0502995
$435$	0	0
$436$	−11.7248	−0.561518
$437$	−1.74583	−0.0835142
$438$	0	0
$439$	−3.92761	−0.187455	−0.0937274	−	0.995598i	$-0.529878\pi$
$439$	−3.92761	−0.187455	−0.0937274	+	0.995598i	$0.529878\pi$
$440$	−8.00793	−0.381763
$441$	0	0
$442$	7.41247	0.352575
$443$	−37.0565	−1.76061	−0.880303	−	0.474412i	$-0.842661\pi$
$443$	−37.0565	−1.76061	−0.880303	+	0.474412i	$0.842661\pi$
$444$	0	0
$445$	−23.6060	−1.11903
$446$	26.1141	1.23654
$447$	0	0
$448$	2.09926	0.0991809
$449$	−21.4769	−1.01356	−0.506780	−	0.862075i	$-0.669164\pi$
$449$	−21.4769	−1.01356	−0.506780	+	0.862075i	$0.669164\pi$
$450$	0	0
$451$	17.4454	0.821471
$452$	−13.8704	−0.652408
$453$	0	0
$454$	−30.5305	−1.43287
$455$	12.4401	0.583199
$456$	0	0
$457$	−22.0414	−1.03106	−0.515528	−	0.856873i	$-0.672404\pi$
$457$	−22.0414	−1.03106	−0.515528	+	0.856873i	$0.672404\pi$
$458$	−10.0983	−0.471863
$459$	0	0
$460$	1.62070	0.0755657
$461$	−21.7986	−1.01526	−0.507632	−	0.861574i	$-0.669479\pi$
$461$	−21.7986	−1.01526	−0.507632	+	0.861574i	$0.669479\pi$
$462$	0	0
$463$	−34.0075	−1.58046	−0.790231	−	0.612809i	$-0.790039\pi$
$463$	−34.0075	−1.58046	−0.790231	+	0.612809i	$0.790039\pi$
$464$	−8.88405	−0.412432
$465$	0	0
$466$	−39.6274	−1.83570
$467$	24.2455	1.12195	0.560975	−	0.827833i	$-0.310426\pi$
$467$	24.2455	1.12195	0.560975	+	0.827833i	$0.310426\pi$
$468$	0	0
$469$	−2.64984	−0.122358
$470$	−29.9729	−1.38255
$471$	0	0
$472$	−9.24864	−0.425703
$473$	1.95970	0.0901070
$474$	0	0
$475$	1.98839	0.0912337
$476$	0.745650	0.0341768
$477$	0	0
$478$	31.4677	1.43930
$479$	28.6302	1.30815	0.654074	−	0.756430i	$-0.273058\pi$
$479$	28.6302	1.30815	0.654074	+	0.756430i	$0.273058\pi$
$480$	0	0
$481$	2.13685	0.0974320
$482$	−32.3181	−1.47205
$483$	0	0
$484$	−7.00183	−0.318265
$485$	−20.9915	−0.953174
$486$	0	0
$487$	−7.66945	−0.347536	−0.173768	−	0.984787i	$-0.555594\pi$
$487$	−7.66945	−0.347536	−0.173768	+	0.984787i	$0.555594\pi$
$488$	−21.1343	−0.956706
$489$	0	0
$490$	4.10867	0.185611
$491$	−20.5630	−0.927994	−0.463997	−	0.885837i	$-0.653585\pi$
$491$	−20.5630	−0.927994	−0.463997	+	0.885837i	$0.653585\pi$
$492$	0	0
$493$	1.51720	0.0683311
$494$	19.9060	0.895611
$495$	0	0
$496$	3.07999	0.138296
$497$	−6.20885	−0.278505
$498$	0	0
$499$	−17.8949	−0.801084	−0.400542	−	0.916278i	$-0.631178\pi$
$499$	−17.8949	−0.801084	−0.400542	+	0.916278i	$0.631178\pi$
$500$	8.76507	0.391986
$501$	0	0
$502$	51.9177	2.31720
$503$	−33.2084	−1.48069	−0.740345	−	0.672227i	$-0.765338\pi$
$503$	−33.2084	−1.48069	−0.740345	+	0.672227i	$0.765338\pi$
$504$	0	0
$505$	−33.9355	−1.51011
$506$	2.24562	0.0998300
$507$	0	0
$508$	0.875937	0.0388634
$509$	−14.0191	−0.621386	−0.310693	−	0.950510i	$-0.600561\pi$
$509$	−14.0191	−0.621386	−0.310693	+	0.950510i	$0.600561\pi$
$510$	0	0
$511$	5.81806	0.257376
$512$	−4.12818	−0.182441
$513$	0	0
$514$	−20.4871	−0.903648
$515$	33.1726	1.46176
$516$	0	0
$517$	−12.6490	−0.556302
$518$	0.705753	0.0310090
$519$	0	0
$520$	23.7139	1.03992
$521$	−41.5954	−1.82233	−0.911163	−	0.412046i	$-0.864814\pi$
$521$	−41.5954	−1.82233	−0.911163	+	0.412046i	$0.864814\pi$
$522$	0	0
$523$	23.9941	1.04919	0.524594	−	0.851353i	$-0.324217\pi$
$523$	23.9941	1.04919	0.524594	+	0.851353i	$0.324217\pi$
$524$	−6.47274	−0.282763
$525$	0	0
$526$	−25.8958	−1.12911
$527$	−0.525994	−0.0229127
$528$	0	0
$529$	−22.4168	−0.974642
$530$	32.7166	1.42112
$531$	0	0
$532$	2.00242	0.0868159
$533$	−51.6610	−2.23769
$534$	0	0
$535$	−19.8720	−0.859142
$536$	−5.05127	−0.218182
$537$	0	0
$538$	6.66600	0.287391
$539$	1.73392	0.0746851
$540$	0	0
$541$	−8.14683	−0.350259	−0.175130	−	0.984545i	$-0.556034\pi$
$541$	−8.14683	−0.350259	−0.175130	+	0.984545i	$0.556034\pi$
$542$	19.7750	0.849408
$543$	0	0
$544$	3.95043	0.169373
$545$	32.4299	1.38915
$546$	0	0
$547$	−1.28630	−0.0549982	−0.0274991	−	0.999622i	$-0.508754\pi$
$547$	−1.28630	−0.0549982	−0.0274991	+	0.999622i	$0.508754\pi$
$548$	−3.15762	−0.134887
$549$	0	0
$550$	−2.55763	−0.109058
$551$	4.07439	0.173575
$552$	0	0
$553$	−9.99008	−0.424821
$554$	−22.2196	−0.944020
$555$	0	0
$556$	12.7152	0.539243
$557$	10.7918	0.457264	0.228632	−	0.973513i	$-0.426575\pi$
$557$	10.7918	0.457264	0.228632	+	0.973513i	$0.426575\pi$
$558$	0	0
$559$	−5.80326	−0.245452
$560$	12.0765	0.510327
$561$	0	0
$562$	1.70081	0.0717445
$563$	9.52664	0.401500	0.200750	−	0.979643i	$-0.435662\pi$
$563$	9.52664	0.401500	0.200750	+	0.979643i	$0.435662\pi$
$564$	0	0
$565$	38.3643	1.61400
$566$	7.71533	0.324300
$567$	0	0
$568$	−11.8356	−0.496612
$569$	−12.1297	−0.508506	−0.254253	−	0.967138i	$-0.581830\pi$
$569$	−12.1297	−0.508506	−0.254253	+	0.967138i	$0.581830\pi$
$570$	0	0
$571$	−24.7691	−1.03655	−0.518277	−	0.855213i	$-0.673426\pi$
$571$	−24.7691	−1.03655	−0.518277	+	0.855213i	$0.673426\pi$
$572$	−7.79852	−0.326072
$573$	0	0
$574$	−17.0625	−0.712174
$575$	−0.664261	−0.0277016
$576$	0	0
$577$	−14.2754	−0.594293	−0.297146	−	0.954832i	$-0.596035\pi$
$577$	−14.2754	−0.594293	−0.297146	+	0.954832i	$0.596035\pi$
$578$	27.6007	1.14804
$579$	0	0
$580$	−3.78237	−0.157055
$581$	13.8723	0.575519
$582$	0	0
$583$	13.8068	0.571821
$584$	11.0907	0.458936
$585$	0	0
$586$	44.4144	1.83474
$587$	32.3310	1.33444	0.667221	−	0.744860i	$-0.267484\pi$
$587$	32.3310	1.33444	0.667221	+	0.744860i	$0.267484\pi$
$588$	0	0
$589$	−1.41254	−0.0582028
$590$	−19.9342	−0.820678
$591$	0	0
$592$	2.07441	0.0852576
$593$	3.51543	0.144362	0.0721808	−	0.997392i	$-0.477004\pi$
$593$	3.51543	0.144362	0.0721808	+	0.997392i	$0.477004\pi$
$594$	0	0
$595$	−2.06241	−0.0845504
$596$	3.25208	0.133210
$597$	0	0
$598$	−6.64997	−0.271937
$599$	−17.4721	−0.713889	−0.356945	−	0.934125i	$-0.616182\pi$
$599$	−17.4721	−0.713889	−0.356945	+	0.934125i	$0.616182\pi$
$600$	0	0
$601$	−31.8825	−1.30052	−0.650258	−	0.759714i	$-0.725339\pi$
$601$	−31.8825	−1.30052	−0.650258	+	0.759714i	$0.725339\pi$
$602$	−1.91668	−0.0781182
$603$	0	0
$604$	−13.8351	−0.562941
$605$	19.3665	0.787359
$606$	0	0
$607$	−31.8753	−1.29378	−0.646889	−	0.762585i	$-0.723930\pi$
$607$	−31.8753	−1.29378	−0.646889	+	0.762585i	$0.723930\pi$
$608$	10.6088	0.430242
$609$	0	0
$610$	−45.5522	−1.84435
$611$	37.4575	1.51537
$612$	0	0
$613$	−29.4259	−1.18850	−0.594249	−	0.804281i	$-0.702551\pi$
$613$	−29.4259	−1.18850	−0.594249	+	0.804281i	$0.702551\pi$
$614$	−6.48470	−0.261701
$615$	0	0
$616$	3.30528	0.133174
$617$	−36.1364	−1.45480	−0.727398	−	0.686216i	$-0.759271\pi$
$617$	−36.1364	−1.45480	−0.727398	+	0.686216i	$0.759271\pi$
$618$	0	0
$619$	−3.55043	−0.142704	−0.0713518	−	0.997451i	$-0.522731\pi$
$619$	−3.55043	−0.142704	−0.0713518	+	0.997451i	$0.522731\pi$
$620$	1.31130	0.0526633
$621$	0	0
$622$	−21.7710	−0.872939
$623$	9.74339	0.390361
$624$	0	0
$625$	−28.5925	−1.14370
$626$	35.8894	1.43443
$627$	0	0
$628$	−17.3478	−0.692254
$629$	−0.354263	−0.0141254
$630$	0	0
$631$	−8.37167	−0.333271	−0.166635	−	0.986019i	$-0.553290\pi$
$631$	−8.37167	−0.333271	−0.166635	+	0.986019i	$0.553290\pi$
$632$	−19.0436	−0.757514
$633$	0	0
$634$	29.0365	1.15319
$635$	−2.42277	−0.0961446
$636$	0	0
$637$	−5.13465	−0.203442
$638$	−5.24080	−0.207485
$639$	0	0
$640$	31.1118	1.22980
$641$	−0.727653	−0.0287406	−0.0143703	−	0.999897i	$-0.504574\pi$
$641$	−0.727653	−0.0287406	−0.0143703	+	0.999897i	$0.504574\pi$
$642$	0	0
$643$	23.5424	0.928422	0.464211	−	0.885725i	$-0.346338\pi$
$643$	23.5424	0.928422	0.464211	+	0.885725i	$0.346338\pi$
$644$	−0.668947	−0.0263602
$645$	0	0
$646$	−3.30015	−0.129843
$647$	−1.24356	−0.0488894	−0.0244447	−	0.999701i	$-0.507782\pi$
$647$	−1.24356	−0.0488894	−0.0244447	+	0.999701i	$0.507782\pi$
$648$	0	0
$649$	−8.41251	−0.330220
$650$	7.57392	0.297074
$651$	0	0
$652$	21.9569	0.859900
$653$	28.5121	1.11576	0.557882	−	0.829920i	$-0.311614\pi$
$653$	28.5121	1.11576	0.557882	+	0.829920i	$0.311614\pi$
$654$	0	0
$655$	17.9030	0.699530
$656$	−50.1514	−1.95808
$657$	0	0
$658$	12.3714	0.482286
$659$	0.186042	0.00724718	0.00362359	−	0.999993i	$-0.498847\pi$
$659$	0.186042	0.00724718	0.00362359	+	0.999993i	$0.498847\pi$
$660$	0	0
$661$	21.2916	0.828149	0.414074	−	0.910243i	$-0.364105\pi$
$661$	21.2916	0.828149	0.414074	+	0.910243i	$0.364105\pi$
$662$	−5.34651	−0.207798
$663$	0	0
$664$	26.4441	1.02623
$665$	−5.53852	−0.214775
$666$	0	0
$667$	−1.36113	−0.0527030
$668$	19.2377	0.744330
$669$	0	0
$670$	−10.8873	−0.420615
$671$	−19.2237	−0.742121
$672$	0	0
$673$	13.4570	0.518730	0.259365	−	0.965779i	$-0.416487\pi$
$673$	13.4570	0.518730	0.259365	+	0.965779i	$0.416487\pi$
$674$	−18.1109	−0.697605
$675$	0	0
$676$	11.7066	0.450254
$677$	−0.0830843	−0.00319319	−0.00159660	−	0.999999i	$-0.500508\pi$
$677$	−0.0830843	−0.00319319	−0.00159660	+	0.999999i	$0.500508\pi$
$678$	0	0
$679$	8.66426	0.332504
$680$	−3.93146	−0.150765
$681$	0	0
$682$	1.81692	0.0695736
$683$	17.5526	0.671633	0.335816	−	0.941927i	$-0.390988\pi$
$683$	17.5526	0.671633	0.335816	+	0.941927i	$0.390988\pi$
$684$	0	0
$685$	8.73370	0.333698
$686$	−1.69586	−0.0647482
$687$	0	0
$688$	−5.63368	−0.214782
$689$	−40.8862	−1.55764
$690$	0	0
$691$	18.3750	0.699017	0.349509	−	0.936933i	$-0.386349\pi$
$691$	18.3750	0.699017	0.349509	+	0.936933i	$0.386349\pi$
$692$	−7.85417	−0.298571
$693$	0	0
$694$	17.0815	0.648403
$695$	−35.1691	−1.33404
$696$	0	0
$697$	8.56475	0.324413
$698$	31.6992	1.19983
$699$	0	0
$700$	0.761891	0.0287968
$701$	2.29396	0.0866418	0.0433209	−	0.999061i	$-0.486206\pi$
$701$	2.29396	0.0866418	0.0433209	+	0.999061i	$0.486206\pi$
$702$	0	0
$703$	−0.951361	−0.0358813
$704$	3.63995	0.137186
$705$	0	0
$706$	9.95998	0.374849
$707$	14.0069	0.526785
$708$	0	0
$709$	45.3496	1.70314	0.851569	−	0.524242i	$-0.175651\pi$
$709$	45.3496	1.70314	0.851569	+	0.524242i	$0.175651\pi$
$710$	−25.5101	−0.957377
$711$	0	0
$712$	18.5733	0.696066
$713$	0.471887	0.0176723
$714$	0	0
$715$	21.5700	0.806673
$716$	−5.08856	−0.190168
$717$	0	0
$718$	12.8553	0.479757
$719$	5.80935	0.216652	0.108326	−	0.994115i	$-0.465451\pi$
$719$	5.80935	0.216652	0.108326	+	0.994115i	$0.465451\pi$
$720$	0	0
$721$	−13.6920	−0.509918
$722$	23.3589	0.869327
$723$	0	0
$724$	15.3143	0.569152
$725$	1.55024	0.0575746
$726$	0	0
$727$	−33.9757	−1.26009	−0.630045	−	0.776559i	$-0.716963\pi$
$727$	−33.9757	−1.26009	−0.630045	+	0.776559i	$0.716963\pi$
$728$	−9.78794	−0.362765
$729$	0	0
$730$	23.9045	0.884745
$731$	0.962107	0.0355848
$732$	0	0
$733$	3.90157	0.144108	0.0720539	−	0.997401i	$-0.477045\pi$
$733$	3.90157	0.144108	0.0720539	+	0.997401i	$0.477045\pi$
$734$	−57.0691	−2.10646
$735$	0	0
$736$	−3.54406	−0.130636
$737$	−4.59461	−0.169245
$738$	0	0
$739$	−13.0397	−0.479672	−0.239836	−	0.970813i	$-0.577094\pi$
$739$	−13.0397	−0.479672	−0.239836	+	0.970813i	$0.577094\pi$
$740$	0.883177	0.0324662
$741$	0	0
$742$	−13.5038	−0.495740
$743$	−19.9142	−0.730580	−0.365290	−	0.930894i	$-0.619030\pi$
$743$	−19.9142	−0.730580	−0.365290	+	0.930894i	$0.619030\pi$
$744$	0	0
$745$	−8.99497	−0.329550
$746$	11.6936	0.428134
$747$	0	0
$748$	1.29290	0.0472729
$749$	8.20220	0.299702
$750$	0	0
$751$	−14.7889	−0.539654	−0.269827	−	0.962909i	$-0.586967\pi$
$751$	−14.7889	−0.539654	−0.269827	+	0.962909i	$0.586967\pi$
$752$	36.3629	1.32602
$753$	0	0
$754$	15.5196	0.565191
$755$	38.2666	1.39267
$756$	0	0
$757$	−13.3397	−0.484839	−0.242420	−	0.970171i	$-0.577941\pi$
$757$	−13.3397	−0.484839	−0.242420	+	0.970171i	$0.577941\pi$
$758$	15.4469	0.561058
$759$	0	0
$760$	−10.5578	−0.382972
$761$	−24.7854	−0.898470	−0.449235	−	0.893414i	$-0.648303\pi$
$761$	−24.7854	−0.898470	−0.449235	+	0.893414i	$0.648303\pi$
$762$	0	0
$763$	−13.3855	−0.484587
$764$	2.48144	0.0897752
$765$	0	0
$766$	−16.1570	−0.583775
$767$	24.9120	0.899520
$768$	0	0
$769$	53.3224	1.92285	0.961427	−	0.275061i	$-0.0886980\pi$
$769$	53.3224	1.92285	0.961427	+	0.275061i	$0.0886980\pi$
$770$	7.12409	0.256734
$771$	0	0
$772$	7.77215	0.279726
$773$	5.47408	0.196889	0.0984445	−	0.995143i	$-0.468613\pi$
$773$	5.47408	0.196889	0.0984445	+	0.995143i	$0.468613\pi$
$774$	0	0
$775$	−0.537451	−0.0193058
$776$	16.5163	0.592899
$777$	0	0
$778$	63.8907	2.29059
$779$	23.0004	0.824073
$780$	0	0
$781$	−10.7656	−0.385224
$782$	1.10248	0.0394246
$783$	0	0
$784$	−4.98461	−0.178022
$785$	47.9826	1.71257
$786$	0	0
$787$	−33.8863	−1.20791	−0.603957	−	0.797017i	$-0.706410\pi$
$787$	−33.8863	−1.20791	−0.603957	+	0.797017i	$0.706410\pi$
$788$	3.81868	0.136035
$789$	0	0
$790$	−41.0459	−1.46035
$791$	−15.8349	−0.563024
$792$	0	0
$793$	56.9271	2.02154
$794$	37.3104	1.32410
$795$	0	0
$796$	16.4200	0.581992
$797$	−46.7694	−1.65666	−0.828329	−	0.560243i	$-0.810708\pi$
$797$	−46.7694	−1.65666	−0.828329	+	0.560243i	$0.810708\pi$
$798$	0	0
$799$	−6.20998	−0.219693
$800$	4.03648	0.142711
$801$	0	0
$802$	−1.53360	−0.0541531
$803$	10.0880	0.355999
$804$	0	0
$805$	1.85025	0.0652127
$806$	−5.38046	−0.189519
$807$	0	0
$808$	26.7007	0.939328
$809$	−6.32417	−0.222346	−0.111173	−	0.993801i	$-0.535461\pi$
$809$	−6.32417	−0.222346	−0.111173	+	0.993801i	$0.535461\pi$
$810$	0	0
$811$	1.49724	0.0525751	0.0262875	−	0.999654i	$-0.491631\pi$
$811$	1.49724	0.0525751	0.0262875	+	0.999654i	$0.491631\pi$
$812$	1.56118	0.0547867
$813$	0	0
$814$	1.22372	0.0428912
$815$	−60.7310	−2.12731
$816$	0	0
$817$	2.58371	0.0903925
$818$	−43.6174	−1.52505
$819$	0	0
$820$	−21.3519	−0.745641
$821$	46.0452	1.60699	0.803495	−	0.595312i	$-0.202971\pi$
$821$	46.0452	1.60699	0.803495	+	0.595312i	$0.202971\pi$
$822$	0	0
$823$	−30.4575	−1.06168	−0.530840	−	0.847472i	$-0.678124\pi$
$823$	−30.4575	−1.06168	−0.530840	+	0.847472i	$0.678124\pi$
$824$	−26.1005	−0.909254
$825$	0	0
$826$	8.22787	0.286284
$827$	−0.319612	−0.0111140	−0.00555699	−	0.999985i	$-0.501769\pi$
$827$	−0.319612	−0.0111140	−0.00555699	+	0.999985i	$0.501769\pi$
$828$	0	0
$829$	6.51140	0.226150	0.113075	−	0.993586i	$-0.463930\pi$
$829$	6.51140	0.226150	0.113075	+	0.993586i	$0.463930\pi$
$830$	56.9966	1.97838
$831$	0	0
$832$	−10.7790	−0.373694
$833$	0.851260	0.0294944
$834$	0	0
$835$	−53.2099	−1.84141
$836$	3.47203	0.120083
$837$	0	0
$838$	18.0744	0.624371
$839$	−9.43335	−0.325675	−0.162838	−	0.986653i	$-0.552065\pi$
$839$	−9.43335	−0.325675	−0.162838	+	0.986653i	$0.552065\pi$
$840$	0	0
$841$	−25.8234	−0.890463
$842$	32.7784	1.12962
$843$	0	0
$844$	2.10847	0.0725764
$845$	−32.3794	−1.11389
$846$	0	0
$847$	−7.99353	−0.274661
$848$	−39.6915	−1.36301
$849$	0	0
$850$	−1.25566	−0.0430688
$851$	0.317820	0.0108947
$852$	0	0
$853$	32.1801	1.10182	0.550912	−	0.834563i	$-0.314280\pi$
$853$	32.1801	1.10182	0.550912	+	0.834563i	$0.314280\pi$
$854$	18.8017	0.643382
$855$	0	0
$856$	15.6355	0.534409
$857$	6.54678	0.223634	0.111817	−	0.993729i	$-0.464333\pi$
$857$	6.54678	0.223634	0.111817	+	0.993729i	$0.464333\pi$
$858$	0	0
$859$	−25.8503	−0.882002	−0.441001	−	0.897507i	$-0.645376\pi$
$859$	−25.8503	−0.882002	−0.441001	+	0.897507i	$0.645376\pi$
$860$	−2.39853	−0.0817893
$861$	0	0
$862$	−49.8467	−1.69779
$863$	4.20981	0.143304	0.0716518	−	0.997430i	$-0.477173\pi$
$863$	4.20981	0.143304	0.0716518	+	0.997430i	$0.477173\pi$
$864$	0	0
$865$	21.7240	0.738637
$866$	−30.4150	−1.03355
$867$	0	0
$868$	−0.541242	−0.0183710
$869$	−17.3220	−0.587607
$870$	0	0
$871$	13.6060	0.461023
$872$	−25.5161	−0.864085
$873$	0	0
$874$	2.96067	0.100146
$875$	10.0065	0.338282
$876$	0	0
$877$	−26.3849	−0.890954	−0.445477	−	0.895293i	$-0.646966\pi$
$877$	−26.3849	−0.890954	−0.445477	+	0.895293i	$0.646966\pi$
$878$	6.66068	0.224787
$879$	0	0
$880$	20.9397	0.705878
$881$	−37.1621	−1.25202	−0.626011	−	0.779814i	$-0.715314\pi$
$881$	−37.1621	−1.25202	−0.626011	+	0.779814i	$0.715314\pi$
$882$	0	0
$883$	−26.0075	−0.875223	−0.437611	−	0.899164i	$-0.644175\pi$
$883$	−26.0075	−0.875223	−0.437611	+	0.899164i	$0.644175\pi$
$884$	−3.82866	−0.128772
$885$	0	0
$886$	62.8426	2.11124
$887$	−35.6770	−1.19792	−0.598959	−	0.800780i	$-0.704419\pi$
$887$	−35.6770	−1.19792	−0.598959	+	0.800780i	$0.704419\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	40.0324	1.34189
$891$	0	0
$892$	−13.4883	−0.451623
$893$	−16.6767	−0.558064
$894$	0	0
$895$	14.0745	0.470459
$896$	−12.8414	−0.429002
$897$	0	0
$898$	36.4219	1.21541
$899$	−1.10128	−0.0367298
$900$	0	0
$901$	6.77842	0.225822
$902$	−29.5849	−0.985069
$903$	0	0
$904$	−30.1853	−1.00395
$905$	−42.3581	−1.40803
$906$	0	0
$907$	35.9045	1.19219	0.596094	−	0.802915i	$-0.296718\pi$
$907$	35.9045	1.19219	0.596094	+	0.802915i	$0.296718\pi$
$908$	15.7695	0.523329
$909$	0	0
$910$	−21.0966	−0.699345
$911$	−24.9579	−0.826892	−0.413446	−	0.910529i	$-0.635675\pi$
$911$	−24.9579	−0.826892	−0.413446	+	0.910529i	$0.635675\pi$
$912$	0	0
$913$	24.0534	0.796050
$914$	37.3792	1.23639
$915$	0	0
$916$	5.21594	0.172339
$917$	−7.38950	−0.244023
$918$	0	0
$919$	−32.8965	−1.08515	−0.542577	−	0.840006i	$-0.682551\pi$
$919$	−32.8965	−1.08515	−0.542577	+	0.840006i	$0.682551\pi$
$920$	3.52704	0.116283
$921$	0	0
$922$	36.9674	1.21746
$923$	31.8803	1.04935
$924$	0	0
$925$	−0.361979	−0.0119018
$926$	57.6719	1.89522
$927$	0	0
$928$	8.27108	0.271512
$929$	13.5342	0.444041	0.222021	−	0.975042i	$-0.428735\pi$
$929$	13.5342	0.444041	0.222021	+	0.975042i	$0.428735\pi$
$930$	0	0
$931$	2.28603	0.0749217
$932$	20.4682	0.670457
$933$	0	0
$934$	−41.1170	−1.34539
$935$	−3.57604	−0.116949
$936$	0	0
$937$	−47.8950	−1.56466	−0.782330	−	0.622864i	$-0.785969\pi$
$937$	−47.8950	−1.56466	−0.782330	+	0.622864i	$0.785969\pi$
$938$	4.49376	0.146727
$939$	0	0
$940$	15.4815	0.504950
$941$	9.32200	0.303889	0.151944	−	0.988389i	$-0.451447\pi$
$941$	9.32200	0.303889	0.151944	+	0.988389i	$0.451447\pi$
$942$	0	0
$943$	−7.68371	−0.250216
$944$	24.1840	0.787123
$945$	0	0
$946$	−3.32337	−0.108052
$947$	−13.9821	−0.454356	−0.227178	−	0.973853i	$-0.572950\pi$
$947$	−13.9821	−0.454356	−0.227178	+	0.973853i	$0.572950\pi$
$948$	0	0
$949$	−29.8737	−0.969742
$950$	−3.37203	−0.109403
$951$	0	0
$952$	1.62272	0.0525925
$953$	−12.4685	−0.403893	−0.201947	−	0.979397i	$-0.564727\pi$
$953$	−12.4685	−0.403893	−0.201947	+	0.979397i	$0.564727\pi$
$954$	0	0
$955$	−6.86344	−0.222096
$956$	−16.2535	−0.525677
$957$	0	0
$958$	−48.5528	−1.56867
$959$	−3.60485	−0.116407
$960$	0	0
$961$	−30.6182	−0.987684
$962$	−3.62380	−0.116836
$963$	0	0
$964$	16.6928	0.537639
$965$	−21.4971	−0.692016
$966$	0	0
$967$	27.8149	0.894468	0.447234	−	0.894417i	$-0.352409\pi$
$967$	27.8149	0.894468	0.447234	+	0.894417i	$0.352409\pi$
$968$	−15.2377	−0.489758
$969$	0	0
$970$	35.5986	1.14300
$971$	−44.9360	−1.44206	−0.721032	−	0.692901i	$-0.756332\pi$
$971$	−44.9360	−1.44206	−0.721032	+	0.692901i	$0.756332\pi$
$972$	0	0
$973$	14.5161	0.465364
$974$	13.0063	0.416749
$975$	0	0
$976$	55.2636	1.76894
$977$	−1.13333	−0.0362583	−0.0181292	−	0.999836i	$-0.505771\pi$
$977$	−1.13333	−0.0362583	−0.0181292	+	0.999836i	$0.505771\pi$
$978$	0	0
$979$	16.8942	0.539941
$980$	−2.12219	−0.0677909
$981$	0	0
$982$	34.8719	1.11281
$983$	−27.0551	−0.862924	−0.431462	−	0.902131i	$-0.642002\pi$
$983$	−27.0551	−0.862924	−0.431462	+	0.902131i	$0.642002\pi$
$984$	0	0
$985$	−10.5622	−0.336538
$986$	−2.57295	−0.0819395
$987$	0	0
$988$	−10.2817	−0.327105
$989$	−0.863137	−0.0274462
$990$	0	0
$991$	21.9508	0.697292	0.348646	−	0.937255i	$-0.386642\pi$
$991$	21.9508	0.697292	0.348646	+	0.937255i	$0.386642\pi$
$992$	−2.86749	−0.0910428
$993$	0	0
$994$	10.5293	0.333970
$995$	−45.4163	−1.43979
$996$	0	0
$997$	32.6653	1.03452	0.517261	−	0.855828i	$-0.326952\pi$
$997$	32.6653	1.03452	0.517261	+	0.855828i	$0.326952\pi$
$998$	30.3472	0.960623
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.x.1.5	✓	22
3.2	odd	2	inner	8001.2.a.x.1.18	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.x.1.5	✓	22	1.1	even	1	trivial
8001.2.a.x.1.18	yes	22	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)

\(f(q)\)	\(=\)	\(q-1.69586 q^{2} +0.875937 q^{4} -2.42277 q^{5} +1.00000 q^{7} +1.90625 q^{8} +O(q^{10})\)	\(q-1.69586 q^{2} +0.875937 q^{4} -2.42277 q^{5} +1.00000 q^{7} +1.90625 q^{8} +4.10867 q^{10} +1.73392 q^{11} -5.13465 q^{13} -1.69586 q^{14} -4.98461 q^{16} +0.851260 q^{17} +2.28603 q^{19} -2.12219 q^{20} -2.94048 q^{22} -0.763693 q^{23} +0.869801 q^{25} +8.70764 q^{26} +0.875937 q^{28} +1.78230 q^{29} -0.617901 q^{31} +4.64069 q^{32} -1.44362 q^{34} -2.42277 q^{35} -0.416163 q^{37} -3.87679 q^{38} -4.61840 q^{40} +10.0613 q^{41} +1.13021 q^{43} +1.51880 q^{44} +1.29512 q^{46} -7.29504 q^{47} +1.00000 q^{49} -1.47506 q^{50} -4.49763 q^{52} +7.96281 q^{53} -4.20088 q^{55} +1.90625 q^{56} -3.02252 q^{58} -4.85174 q^{59} -11.0868 q^{61} +1.04787 q^{62} +2.09926 q^{64} +12.4401 q^{65} -2.64984 q^{67} +0.745650 q^{68} +4.10867 q^{70} -6.20885 q^{71} +5.81806 q^{73} +0.705753 q^{74} +2.00242 q^{76} +1.73392 q^{77} -9.99008 q^{79} +12.0765 q^{80} -17.0625 q^{82} +13.8723 q^{83} -2.06241 q^{85} -1.91668 q^{86} +3.30528 q^{88} +9.74339 q^{89} -5.13465 q^{91} -0.668947 q^{92} +12.3714 q^{94} -5.53852 q^{95} +8.66426 q^{97} -1.69586 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 10 q^{4} + 22 q^{7}+O(q^{10}) \) 22 * q + 10 * q^4 + 22 * q^7	\( 22 q + 10 q^{4} + 22 q^{7} - 4 q^{10} - 10 q^{13} - 6 q^{16} - 18 q^{19} - 34 q^{22} - 26 q^{25} + 10 q^{28} - 42 q^{31} - 16 q^{34} - 36 q^{37} - 46 q^{40} - 54 q^{43} - 12 q^{46} + 22 q^{49} - 22 q^{52} - 28 q^{55} - 26 q^{58} - 22 q^{61} - 76 q^{64} - 48 q^{67} - 4 q^{70} - 48 q^{73} - 20 q^{76} - 94 q^{79} - 8 q^{82} - 40 q^{85} - 26 q^{88} - 10 q^{91} - 26 q^{94} - 56 q^{97}+O(q^{100}) \) 22 * q + 10 * q^4 + 22 * q^7 - 4 * q^10 - 10 * q^13 - 6 * q^16 - 18 * q^19 - 34 * q^22 - 26 * q^25 + 10 * q^28 - 42 * q^31 - 16 * q^34 - 36 * q^37 - 46 * q^40 - 54 * q^43 - 12 * q^46 + 22 * q^49 - 22 * q^52 - 28 * q^55 - 26 * q^58 - 22 * q^61 - 76 * q^64 - 48 * q^67 - 4 * q^70 - 48 * q^73 - 20 * q^76 - 94 * q^79 - 8 * q^82 - 40 * q^85 - 26 * q^88 - 10 * q^91 - 26 * q^94 - 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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