LMFDB - Embedded newform 8001.2.a.ba.1.25

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:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.25
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)

$f(q)$	$=$	$q+0.997785 q^{2} -1.00442 q^{4} +3.25439 q^{5} +1.00000 q^{7} -2.99777 q^{8} +O(q^{10})$	\(q+0.997785 q^{2} -1.00442 q^{4} +3.25439 q^{5} +1.00000 q^{7} -2.99777 q^{8} +3.24718 q^{10} +1.13258 q^{11} -2.48306 q^{13} +0.997785 q^{14} -0.982283 q^{16} +2.73411 q^{17} -4.35505 q^{19} -3.26879 q^{20} +1.13008 q^{22} -1.12705 q^{23} +5.59105 q^{25} -2.47756 q^{26} -1.00442 q^{28} +4.75284 q^{29} -10.7720 q^{31} +5.01543 q^{32} +2.72805 q^{34} +3.25439 q^{35} +7.74896 q^{37} -4.34541 q^{38} -9.75591 q^{40} -3.17067 q^{41} +3.00279 q^{43} -1.13760 q^{44} -1.12455 q^{46} +11.1790 q^{47} +1.00000 q^{49} +5.57867 q^{50} +2.49404 q^{52} +7.30168 q^{53} +3.68587 q^{55} -2.99777 q^{56} +4.74231 q^{58} -3.14018 q^{59} +6.04867 q^{61} -10.7482 q^{62} +6.96889 q^{64} -8.08084 q^{65} +13.4063 q^{67} -2.74621 q^{68} +3.24718 q^{70} +13.8654 q^{71} -12.0216 q^{73} +7.73180 q^{74} +4.37432 q^{76} +1.13258 q^{77} +14.4326 q^{79} -3.19673 q^{80} -3.16365 q^{82} +7.53954 q^{83} +8.89786 q^{85} +2.99614 q^{86} -3.39523 q^{88} +9.91053 q^{89} -2.48306 q^{91} +1.13203 q^{92} +11.1543 q^{94} -14.1730 q^{95} -11.6643 q^{97} +0.997785 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) $ 40 * q + 54 * q^4 + 40 * q^7	$ 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) $ 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * 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 $

$2$	0.997785	0.705541	0.352770	−	0.935710i	$-0.385240\pi$
$2$	0.997785	0.705541	0.352770	+	0.935710i	$0.385240\pi$
$3$	0	0
$3$	0	0
$4$	−1.00442	−0.502212
$5$	3.25439	1.45541	0.727704	−	0.685892i	$-0.240588\pi$
$5$	3.25439	1.45541	0.727704	+	0.685892i	$0.240588\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.99777	−1.05987
$9$	0	0
$10$	3.24718	1.02685
$11$	1.13258	0.341487	0.170744	−	0.985316i	$-0.445383\pi$
$11$	1.13258	0.341487	0.170744	+	0.985316i	$0.445383\pi$
$12$	0	0
$13$	−2.48306	−0.688676	−0.344338	−	0.938846i	$-0.611897\pi$
$13$	−2.48306	−0.688676	−0.344338	+	0.938846i	$0.611897\pi$
$14$	0.997785	0.266669
$15$	0	0
$16$	−0.982283	−0.245571
$17$	2.73411	0.663119	0.331559	−	0.943434i	$-0.392425\pi$
$17$	2.73411	0.663119	0.331559	+	0.943434i	$0.392425\pi$
$18$	0	0
$19$	−4.35505	−0.999117	−0.499559	−	0.866280i	$-0.666504\pi$
$19$	−4.35505	−0.999117	−0.499559	+	0.866280i	$0.666504\pi$
$20$	−3.26879	−0.730923
$21$	0	0
$22$	1.13008	0.240933
$23$	−1.12705	−0.235005	−0.117503	−	0.993073i	$-0.537489\pi$
$23$	−1.12705	−0.235005	−0.117503	+	0.993073i	$0.537489\pi$
$24$	0	0
$25$	5.59105	1.11821
$26$	−2.47756	−0.485889
$27$	0	0
$28$	−1.00442	−0.189818
$29$	4.75284	0.882580	0.441290	−	0.897365i	$-0.354521\pi$
$29$	4.75284	0.882580	0.441290	+	0.897365i	$0.354521\pi$
$30$	0	0
$31$	−10.7720	−1.93471	−0.967357	−	0.253417i	$-0.918446\pi$
$31$	−10.7720	−1.93471	−0.967357	+	0.253417i	$0.918446\pi$
$32$	5.01543	0.886612
$33$	0	0
$34$	2.72805	0.467857
$35$	3.25439	0.550092
$36$	0	0
$37$	7.74896	1.27392	0.636961	−	0.770896i	$-0.280191\pi$
$37$	7.74896	1.27392	0.636961	+	0.770896i	$0.280191\pi$
$38$	−4.34541	−0.704918
$39$	0	0
$40$	−9.75591	−1.54255
$41$	−3.17067	−0.495176	−0.247588	−	0.968865i	$-0.579638\pi$
$41$	−3.17067	−0.495176	−0.247588	+	0.968865i	$0.579638\pi$
$42$	0	0
$43$	3.00279	0.457921	0.228961	−	0.973436i	$-0.426467\pi$
$43$	3.00279	0.457921	0.228961	+	0.973436i	$0.426467\pi$
$44$	−1.13760	−0.171499
$45$	0	0
$46$	−1.12455	−0.165806
$47$	11.1790	1.63063	0.815315	−	0.579018i	$-0.196564\pi$
$47$	11.1790	1.63063	0.815315	+	0.579018i	$0.196564\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.57867	0.788943
$51$	0	0
$52$	2.49404	0.345862
$53$	7.30168	1.00296	0.501481	−	0.865169i	$-0.332789\pi$
$53$	7.30168	1.00296	0.501481	+	0.865169i	$0.332789\pi$
$54$	0	0
$55$	3.68587	0.497003
$56$	−2.99777	−0.400594
$57$	0	0
$58$	4.74231	0.622696
$59$	−3.14018	−0.408816	−0.204408	−	0.978886i	$-0.565527\pi$
$59$	−3.14018	−0.408816	−0.204408	+	0.978886i	$0.565527\pi$
$60$	0	0
$61$	6.04867	0.774453	0.387226	−	0.921985i	$-0.373433\pi$
$61$	6.04867	0.774453	0.387226	+	0.921985i	$0.373433\pi$
$62$	−10.7482	−1.36502
$63$	0	0
$64$	6.96889	0.871112
$65$	−8.08084	−1.00230
$66$	0	0
$67$	13.4063	1.63784	0.818919	−	0.573909i	$-0.194574\pi$
$67$	13.4063	1.63784	0.818919	+	0.573909i	$0.194574\pi$
$68$	−2.74621	−0.333026
$69$	0	0
$70$	3.24718	0.388113
$71$	13.8654	1.64552	0.822762	−	0.568387i	$-0.192432\pi$
$71$	13.8654	1.64552	0.822762	+	0.568387i	$0.192432\pi$
$72$	0	0
$73$	−12.0216	−1.40702	−0.703511	−	0.710684i	$-0.748386\pi$
$73$	−12.0216	−1.40702	−0.703511	+	0.710684i	$0.748386\pi$
$74$	7.73180	0.898803
$75$	0	0
$76$	4.37432	0.501769
$77$	1.13258	0.129070
$78$	0	0
$79$	14.4326	1.62379	0.811895	−	0.583803i	$-0.198436\pi$
$79$	14.4326	1.62379	0.811895	+	0.583803i	$0.198436\pi$
$80$	−3.19673	−0.357406
$81$	0	0
$82$	−3.16365	−0.349367
$83$	7.53954	0.827572	0.413786	−	0.910374i	$-0.364206\pi$
$83$	7.53954	0.827572	0.413786	+	0.910374i	$0.364206\pi$
$84$	0	0
$85$	8.89786	0.965108
$86$	2.99614	0.323082
$87$	0	0
$88$	−3.39523	−0.361933
$89$	9.91053	1.05051	0.525257	−	0.850943i	$-0.323969\pi$
$89$	9.91053	1.05051	0.525257	+	0.850943i	$0.323969\pi$
$90$	0	0
$91$	−2.48306	−0.260295
$92$	1.13203	0.118022
$93$	0	0
$94$	11.1543	1.15048
$95$	−14.1730	−1.45412
$96$	0	0
$97$	−11.6643	−1.18433	−0.592167	−	0.805815i	$-0.701727\pi$
$97$	−11.6643	−1.18433	−0.592167	+	0.805815i	$0.701727\pi$
$98$	0.997785	0.100792
$99$	0	0
$100$	−5.61579	−0.561579
$101$	9.18558	0.913999	0.457000	−	0.889467i	$-0.348924\pi$
$101$	9.18558	0.913999	0.457000	+	0.889467i	$0.348924\pi$
$102$	0	0
$103$	−11.3010	−1.11352	−0.556758	−	0.830675i	$-0.687955\pi$
$103$	−11.3010	−1.11352	−0.556758	+	0.830675i	$0.687955\pi$
$104$	7.44364	0.729909
$105$	0	0
$106$	7.28551	0.707631
$107$	15.5832	1.50648	0.753242	−	0.657743i	$-0.228489\pi$
$107$	15.5832	1.50648	0.753242	+	0.657743i	$0.228489\pi$
$108$	0	0
$109$	−16.6472	−1.59451	−0.797256	−	0.603642i	$-0.793716\pi$
$109$	−16.6472	−1.59451	−0.797256	+	0.603642i	$0.793716\pi$
$110$	3.67771	0.350656
$111$	0	0
$112$	−0.982283	−0.0928170
$113$	−15.5085	−1.45891	−0.729457	−	0.684027i	$-0.760227\pi$
$113$	−15.5085	−1.45891	−0.729457	+	0.684027i	$0.760227\pi$
$114$	0	0
$115$	−3.66785	−0.342028
$116$	−4.77387	−0.443242
$117$	0	0
$118$	−3.13322	−0.288437
$119$	2.73411	0.250635
$120$	0	0
$121$	−9.71725	−0.883387
$122$	6.03527	0.546408
$123$	0	0
$124$	10.8197	0.971637
$125$	1.92352	0.172045
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−3.07741	−0.272007
$129$	0	0
$130$	−8.06294	−0.707167
$131$	5.90447	0.515876	0.257938	−	0.966161i	$-0.416957\pi$
$131$	5.90447	0.515876	0.257938	+	0.966161i	$0.416957\pi$
$132$	0	0
$133$	−4.35505	−0.377631
$134$	13.3766	1.15556
$135$	0	0
$136$	−8.19623	−0.702821
$137$	12.3486	1.05502	0.527508	−	0.849550i	$-0.323127\pi$
$137$	12.3486	1.05502	0.527508	+	0.849550i	$0.323127\pi$
$138$	0	0
$139$	4.10843	0.348473	0.174236	−	0.984704i	$-0.444254\pi$
$139$	4.10843	0.348473	0.174236	+	0.984704i	$0.444254\pi$
$140$	−3.26879	−0.276263
$141$	0	0
$142$	13.8347	1.16098
$143$	−2.81227	−0.235174
$144$	0	0
$145$	15.4676	1.28451
$146$	−11.9950	−0.992712
$147$	0	0
$148$	−7.78324	−0.639779
$149$	9.92770	0.813309	0.406654	−	0.913582i	$-0.366695\pi$
$149$	9.92770	0.813309	0.406654	+	0.913582i	$0.366695\pi$
$150$	0	0
$151$	15.1257	1.23091	0.615456	−	0.788171i	$-0.288972\pi$
$151$	15.1257	1.23091	0.615456	+	0.788171i	$0.288972\pi$
$152$	13.0554	1.05894
$153$	0	0
$154$	1.13008	0.0910641
$155$	−35.0564	−2.81580
$156$	0	0
$157$	13.2624	1.05845	0.529225	−	0.848481i	$-0.322483\pi$
$157$	13.2624	1.05845	0.529225	+	0.848481i	$0.322483\pi$
$158$	14.4006	1.14565
$159$	0	0
$160$	16.3222	1.29038
$161$	−1.12705	−0.0888236
$162$	0	0
$163$	12.8980	1.01025	0.505124	−	0.863047i	$-0.331447\pi$
$163$	12.8980	1.01025	0.505124	+	0.863047i	$0.331447\pi$
$164$	3.18470	0.248683
$165$	0	0
$166$	7.52284	0.583886
$167$	−5.48974	−0.424809	−0.212404	−	0.977182i	$-0.568129\pi$
$167$	−5.48974	−0.424809	−0.212404	+	0.977182i	$0.568129\pi$
$168$	0	0
$169$	−6.83443	−0.525725
$170$	8.87815	0.680923
$171$	0	0
$172$	−3.01607	−0.229974
$173$	−4.62814	−0.351871	−0.175936	−	0.984402i	$-0.556295\pi$
$173$	−4.62814	−0.351871	−0.175936	+	0.984402i	$0.556295\pi$
$174$	0	0
$175$	5.59105	0.422644
$176$	−1.11252	−0.0838592
$177$	0	0
$178$	9.88859	0.741181
$179$	10.2441	0.765677	0.382838	−	0.923815i	$-0.374947\pi$
$179$	10.2441	0.765677	0.382838	+	0.923815i	$0.374947\pi$
$180$	0	0
$181$	−12.3060	−0.914696	−0.457348	−	0.889288i	$-0.651201\pi$
$181$	−12.3060	−0.914696	−0.457348	+	0.889288i	$0.651201\pi$
$182$	−2.47756	−0.183649
$183$	0	0
$184$	3.37862	0.249075
$185$	25.2181	1.85407
$186$	0	0
$187$	3.09661	0.226446
$188$	−11.2285	−0.818922
$189$	0	0
$190$	−14.1416	−1.02594
$191$	0.652954	0.0472461	0.0236231	−	0.999721i	$-0.492480\pi$
$191$	0.652954	0.0472461	0.0236231	+	0.999721i	$0.492480\pi$
$192$	0	0
$193$	−8.97285	−0.645880	−0.322940	−	0.946419i	$-0.604671\pi$
$193$	−8.97285	−0.645880	−0.322940	+	0.946419i	$0.604671\pi$
$194$	−11.6385	−0.835596
$195$	0	0
$196$	−1.00442	−0.0717446
$197$	−2.85259	−0.203239	−0.101619	−	0.994823i	$-0.532402\pi$
$197$	−2.85259	−0.203239	−0.101619	+	0.994823i	$0.532402\pi$
$198$	0	0
$199$	−20.9884	−1.48783	−0.743913	−	0.668277i	$-0.767032\pi$
$199$	−20.9884	−1.48783	−0.743913	+	0.668277i	$0.767032\pi$
$200$	−16.7607	−1.18516
$201$	0	0
$202$	9.16524	0.644864
$203$	4.75284	0.333584
$204$	0	0
$205$	−10.3186	−0.720683
$206$	−11.2759	−0.785631
$207$	0	0
$208$	2.43907	0.169119
$209$	−4.93246	−0.341186
$210$	0	0
$211$	−4.90341	−0.337565	−0.168782	−	0.985653i	$-0.553983\pi$
$211$	−4.90341	−0.337565	−0.168782	+	0.985653i	$0.553983\pi$
$212$	−7.33398	−0.503700
$213$	0	0
$214$	15.5487	1.06289
$215$	9.77225	0.666462
$216$	0	0
$217$	−10.7720	−0.731253
$218$	−16.6103	−1.12499
$219$	0	0
$220$	−3.70218	−0.249601
$221$	−6.78895	−0.456674
$222$	0	0
$223$	24.2821	1.62605	0.813023	−	0.582231i	$-0.197820\pi$
$223$	24.2821	1.62605	0.813023	+	0.582231i	$0.197820\pi$
$224$	5.01543	0.335108
$225$	0	0
$226$	−15.4741	−1.02932
$227$	26.6809	1.77087	0.885437	−	0.464760i	$-0.153859\pi$
$227$	26.6809	1.77087	0.885437	+	0.464760i	$0.153859\pi$
$228$	0	0
$229$	0.645784	0.0426746	0.0213373	−	0.999772i	$-0.493208\pi$
$229$	0.645784	0.0426746	0.0213373	+	0.999772i	$0.493208\pi$
$230$	−3.65972	−0.241315
$231$	0	0
$232$	−14.2479	−0.935422
$233$	16.3172	1.06897	0.534487	−	0.845177i	$-0.320505\pi$
$233$	16.3172	1.06897	0.534487	+	0.845177i	$0.320505\pi$
$234$	0	0
$235$	36.3809	2.37323
$236$	3.15407	0.205312
$237$	0	0
$238$	2.72805	0.176833
$239$	−5.56324	−0.359856	−0.179928	−	0.983680i	$-0.557587\pi$
$239$	−5.56324	−0.359856	−0.179928	+	0.983680i	$0.557587\pi$
$240$	0	0
$241$	−4.58194	−0.295149	−0.147574	−	0.989051i	$-0.547147\pi$
$241$	−4.58194	−0.295149	−0.147574	+	0.989051i	$0.547147\pi$
$242$	−9.69573	−0.623265
$243$	0	0
$244$	−6.07543	−0.388940
$245$	3.25439	0.207915
$246$	0	0
$247$	10.8138	0.688068
$248$	32.2921	2.05055
$249$	0	0
$250$	1.91926	0.121384
$251$	21.9990	1.38856	0.694282	−	0.719703i	$-0.255722\pi$
$251$	21.9990	1.38856	0.694282	+	0.719703i	$0.255722\pi$
$252$	0	0
$253$	−1.27647	−0.0802512
$254$	−0.997785	−0.0626066
$255$	0	0
$256$	−17.0084	−1.06302
$257$	−18.4804	−1.15278	−0.576388	−	0.817176i	$-0.695538\pi$
$257$	−18.4804	−1.15278	−0.576388	+	0.817176i	$0.695538\pi$
$258$	0	0
$259$	7.74896	0.481497
$260$	8.11659	0.503369
$261$	0	0
$262$	5.89139	0.363971
$263$	−9.46225	−0.583467	−0.291734	−	0.956500i	$-0.594232\pi$
$263$	−9.46225	−0.583467	−0.291734	+	0.956500i	$0.594232\pi$
$264$	0	0
$265$	23.7625	1.45972
$266$	−4.34541	−0.266434
$267$	0	0
$268$	−13.4656	−0.822542
$269$	15.8033	0.963546	0.481773	−	0.876296i	$-0.339993\pi$
$269$	15.8033	0.963546	0.481773	+	0.876296i	$0.339993\pi$
$270$	0	0
$271$	15.9591	0.969443	0.484722	−	0.874668i	$-0.338921\pi$
$271$	15.9591	0.969443	0.484722	+	0.874668i	$0.338921\pi$
$272$	−2.68567	−0.162843
$273$	0	0
$274$	12.3213	0.744357
$275$	6.33234	0.381854
$276$	0	0
$277$	12.2037	0.733251	0.366626	−	0.930369i	$-0.380513\pi$
$277$	12.2037	0.733251	0.366626	+	0.930369i	$0.380513\pi$
$278$	4.09933	0.245862
$279$	0	0
$280$	−9.75591	−0.583027
$281$	−19.6950	−1.17491	−0.587454	−	0.809258i	$-0.699870\pi$
$281$	−19.6950	−1.17491	−0.587454	+	0.809258i	$0.699870\pi$
$282$	0	0
$283$	−32.8138	−1.95058	−0.975289	−	0.220931i	$-0.929090\pi$
$283$	−32.8138	−1.95058	−0.975289	+	0.220931i	$0.929090\pi$
$284$	−13.9268	−0.826402
$285$	0	0
$286$	−2.80604	−0.165925
$287$	−3.17067	−0.187159
$288$	0	0
$289$	−9.52465	−0.560274
$290$	15.4333	0.906276
$291$	0	0
$292$	12.0748	0.706624
$293$	−5.45112	−0.318458	−0.159229	−	0.987242i	$-0.550901\pi$
$293$	−5.45112	−0.318458	−0.159229	+	0.987242i	$0.550901\pi$
$294$	0	0
$295$	−10.2194	−0.594994
$296$	−23.2296	−1.35019
$297$	0	0
$298$	9.90571	0.573823
$299$	2.79852	0.161843
$300$	0	0
$301$	3.00279	0.173078
$302$	15.0922	0.868459
$303$	0	0
$304$	4.27789	0.245354
$305$	19.6847	1.12714
$306$	0	0
$307$	10.5920	0.604517	0.302258	−	0.953226i	$-0.402259\pi$
$307$	10.5920	0.604517	0.302258	+	0.953226i	$0.402259\pi$
$308$	−1.13760	−0.0648205
$309$	0	0
$310$	−34.9788	−1.98666
$311$	−23.1085	−1.31036	−0.655180	−	0.755473i	$-0.727407\pi$
$311$	−23.1085	−1.31036	−0.655180	+	0.755473i	$0.727407\pi$
$312$	0	0
$313$	15.0856	0.852690	0.426345	−	0.904561i	$-0.359801\pi$
$313$	15.0856	0.852690	0.426345	+	0.904561i	$0.359801\pi$
$314$	13.2330	0.746780
$315$	0	0
$316$	−14.4964	−0.815487
$317$	−26.6722	−1.49806	−0.749030	−	0.662536i	$-0.769480\pi$
$317$	−26.6722	−1.49806	−0.749030	+	0.662536i	$0.769480\pi$
$318$	0	0
$319$	5.38299	0.301390
$320$	22.6795	1.26782
$321$	0	0
$322$	−1.12455	−0.0626687
$323$	−11.9072	−0.662533
$324$	0	0
$325$	−13.8829	−0.770085
$326$	12.8694	0.712771
$327$	0	0
$328$	9.50495	0.524823
$329$	11.1790	0.616320
$330$	0	0
$331$	31.6366	1.73891	0.869453	−	0.494016i	$-0.164472\pi$
$331$	31.6366	1.73891	0.869453	+	0.494016i	$0.164472\pi$
$332$	−7.57289	−0.415617
$333$	0	0
$334$	−5.47758	−0.299720
$335$	43.6293	2.38372
$336$	0	0
$337$	18.3807	1.00126	0.500630	−	0.865662i	$-0.333102\pi$
$337$	18.3807	1.00126	0.500630	+	0.865662i	$0.333102\pi$
$338$	−6.81929	−0.370921
$339$	0	0
$340$	−8.93722	−0.484689
$341$	−12.2002	−0.660680
$342$	0	0
$343$	1.00000	0.0539949
$344$	−9.00167	−0.485338
$345$	0	0
$346$	−4.61789	−0.248259
$347$	16.6620	0.894463	0.447232	−	0.894418i	$-0.352410\pi$
$347$	16.6620	0.894463	0.447232	+	0.894418i	$0.352410\pi$
$348$	0	0
$349$	−11.1402	−0.596322	−0.298161	−	0.954516i	$-0.596373\pi$
$349$	−11.1402	−0.596322	−0.298161	+	0.954516i	$0.596373\pi$
$350$	5.57867	0.298193
$351$	0	0
$352$	5.68040	0.302766
$353$	−0.764639	−0.0406976	−0.0203488	−	0.999793i	$-0.506478\pi$
$353$	−0.764639	−0.0406976	−0.0203488	+	0.999793i	$0.506478\pi$
$354$	0	0
$355$	45.1235	2.39491
$356$	−9.95438	−0.527581
$357$	0	0
$358$	10.2214	0.540216
$359$	−17.7953	−0.939202	−0.469601	−	0.882879i	$-0.655602\pi$
$359$	−17.7953	−0.939202	−0.469601	+	0.882879i	$0.655602\pi$
$360$	0	0
$361$	−0.0335229	−0.00176436
$362$	−12.2787	−0.645355
$363$	0	0
$364$	2.49404	0.130723
$365$	−39.1230	−2.04779
$366$	0	0
$367$	−5.95917	−0.311066	−0.155533	−	0.987831i	$-0.549710\pi$
$367$	−5.95917	−0.311066	−0.155533	+	0.987831i	$0.549710\pi$
$368$	1.10708	0.0577104
$369$	0	0
$370$	25.1623	1.30813
$371$	7.30168	0.379084
$372$	0	0
$373$	14.0069	0.725249	0.362624	−	0.931935i	$-0.381881\pi$
$373$	14.0069	0.725249	0.362624	+	0.931935i	$0.381881\pi$
$374$	3.08975	0.159767
$375$	0	0
$376$	−33.5122	−1.72826
$377$	−11.8016	−0.607812
$378$	0	0
$379$	9.04010	0.464359	0.232179	−	0.972673i	$-0.425414\pi$
$379$	9.04010	0.464359	0.232179	+	0.972673i	$0.425414\pi$
$380$	14.2357	0.730278
$381$	0	0
$382$	0.651508	0.0333341
$383$	−33.2595	−1.69948	−0.849741	−	0.527201i	$-0.823242\pi$
$383$	−33.2595	−1.69948	−0.849741	+	0.527201i	$0.823242\pi$
$384$	0	0
$385$	3.68587	0.187849
$386$	−8.95298	−0.455695
$387$	0	0
$388$	11.7159	0.594787
$389$	−27.8525	−1.41218	−0.706089	−	0.708123i	$-0.749542\pi$
$389$	−27.8525	−1.41218	−0.706089	+	0.708123i	$0.749542\pi$
$390$	0	0
$391$	−3.08146	−0.155836
$392$	−2.99777	−0.151410
$393$	0	0
$394$	−2.84627	−0.143393
$395$	46.9692	2.36328
$396$	0	0
$397$	21.0932	1.05864	0.529318	−	0.848424i	$-0.322448\pi$
$397$	21.0932	1.05864	0.529318	+	0.848424i	$0.322448\pi$
$398$	−20.9419	−1.04972
$399$	0	0
$400$	−5.49200	−0.274600
$401$	−8.10329	−0.404659	−0.202329	−	0.979318i	$-0.564851\pi$
$401$	−8.10329	−0.404659	−0.202329	+	0.979318i	$0.564851\pi$
$402$	0	0
$403$	26.7476	1.33239
$404$	−9.22622	−0.459022
$405$	0	0
$406$	4.74231	0.235357
$407$	8.77635	0.435028
$408$	0	0
$409$	11.5475	0.570985	0.285492	−	0.958381i	$-0.407843\pi$
$409$	11.5475	0.570985	0.285492	+	0.958381i	$0.407843\pi$
$410$	−10.2958	−0.508471
$411$	0	0
$412$	11.3510	0.559221
$413$	−3.14018	−0.154518
$414$	0	0
$415$	24.5366	1.20445
$416$	−12.4536	−0.610588
$417$	0	0
$418$	−4.92154	−0.240720
$419$	−15.6759	−0.765820	−0.382910	−	0.923786i	$-0.625078\pi$
$419$	−15.6759	−0.765820	−0.382910	+	0.923786i	$0.625078\pi$
$420$	0	0
$421$	−10.0125	−0.487980	−0.243990	−	0.969778i	$-0.578456\pi$
$421$	−10.0125	−0.487980	−0.243990	+	0.969778i	$0.578456\pi$
$422$	−4.89255	−0.238166
$423$	0	0
$424$	−21.8888	−1.06301
$425$	15.2865	0.741506
$426$	0	0
$427$	6.04867	0.292716
$428$	−15.6521	−0.756575
$429$	0	0
$430$	9.75061	0.470216
$431$	20.8613	1.00485	0.502426	−	0.864620i	$-0.332441\pi$
$431$	20.8613	1.00485	0.502426	+	0.864620i	$0.332441\pi$
$432$	0	0
$433$	13.1483	0.631866	0.315933	−	0.948781i	$-0.397683\pi$
$433$	13.1483	0.631866	0.315933	+	0.948781i	$0.397683\pi$
$434$	−10.7482	−0.515929
$435$	0	0
$436$	16.7208	0.800783
$437$	4.90834	0.234798
$438$	0	0
$439$	−10.4408	−0.498312	−0.249156	−	0.968463i	$-0.580153\pi$
$439$	−10.4408	−0.498312	−0.249156	+	0.968463i	$0.580153\pi$
$440$	−11.0494	−0.526759
$441$	0	0
$442$	−6.77391	−0.322202
$443$	38.0686	1.80869	0.904346	−	0.426800i	$-0.140359\pi$
$443$	38.0686	1.80869	0.904346	+	0.426800i	$0.140359\pi$
$444$	0	0
$445$	32.2527	1.52893
$446$	24.2283	1.14724
$447$	0	0
$448$	6.96889	0.329249
$449$	−12.2833	−0.579684	−0.289842	−	0.957075i	$-0.593603\pi$
$449$	−12.2833	−0.579684	−0.289842	+	0.957075i	$0.593603\pi$
$450$	0	0
$451$	−3.59106	−0.169096
$452$	15.5771	0.732684
$453$	0	0
$454$	26.6218	1.24942
$455$	−8.08084	−0.378835
$456$	0	0
$457$	−30.3632	−1.42033	−0.710166	−	0.704034i	$-0.751380\pi$
$457$	−30.3632	−1.42033	−0.710166	+	0.704034i	$0.751380\pi$
$458$	0.644354	0.0301087
$459$	0	0
$460$	3.68407	0.171771
$461$	−25.5425	−1.18963	−0.594816	−	0.803862i	$-0.702775\pi$
$461$	−25.5425	−1.18963	−0.594816	+	0.803862i	$0.702775\pi$
$462$	0	0
$463$	−9.64367	−0.448179	−0.224090	−	0.974569i	$-0.571941\pi$
$463$	−9.64367	−0.448179	−0.224090	+	0.974569i	$0.571941\pi$
$464$	−4.66863	−0.216736
$465$	0	0
$466$	16.2810	0.754204
$467$	31.0894	1.43865	0.719323	−	0.694675i	$-0.244452\pi$
$467$	31.0894	1.43865	0.719323	+	0.694675i	$0.244452\pi$
$468$	0	0
$469$	13.4063	0.619045
$470$	36.3004	1.67441
$471$	0	0
$472$	9.41353	0.433293
$473$	3.40091	0.156374
$474$	0	0
$475$	−24.3493	−1.11722
$476$	−2.74621	−0.125872
$477$	0	0
$478$	−5.55092	−0.253893
$479$	−24.8092	−1.13356	−0.566782	−	0.823868i	$-0.691812\pi$
$479$	−24.8092	−1.13356	−0.566782	+	0.823868i	$0.691812\pi$
$480$	0	0
$481$	−19.2411	−0.877319
$482$	−4.57179	−0.208239
$483$	0	0
$484$	9.76024	0.443647
$485$	−37.9603	−1.72369
$486$	0	0
$487$	31.7284	1.43775	0.718875	−	0.695140i	$-0.244658\pi$
$487$	31.7284	1.43775	0.718875	+	0.695140i	$0.244658\pi$
$488$	−18.1325	−0.820821
$489$	0	0
$490$	3.24718	0.146693
$491$	32.4173	1.46297	0.731486	−	0.681856i	$-0.238827\pi$
$491$	32.4173	1.46297	0.731486	+	0.681856i	$0.238827\pi$
$492$	0	0
$493$	12.9948	0.585255
$494$	10.7899	0.485460
$495$	0	0
$496$	10.5812	0.475109
$497$	13.8654	0.621949
$498$	0	0
$499$	−15.0664	−0.674465	−0.337232	−	0.941421i	$-0.609491\pi$
$499$	−15.0664	−0.674465	−0.337232	+	0.941421i	$0.609491\pi$
$500$	−1.93203	−0.0864029
$501$	0	0
$502$	21.9503	0.979688
$503$	−21.8757	−0.975389	−0.487695	−	0.873014i	$-0.662162\pi$
$503$	−21.8757	−0.975389	−0.487695	+	0.873014i	$0.662162\pi$
$504$	0	0
$505$	29.8935	1.33024
$506$	−1.27365	−0.0566205
$507$	0	0
$508$	1.00442	0.0445641
$509$	1.37865	0.0611076	0.0305538	−	0.999533i	$-0.490273\pi$
$509$	1.37865	0.0611076	0.0305538	+	0.999533i	$0.490273\pi$
$510$	0	0
$511$	−12.0216	−0.531804
$512$	−10.8159	−0.478000
$513$	0	0
$514$	−18.4395	−0.813331
$515$	−36.7777	−1.62062
$516$	0	0
$517$	12.6612	0.556839
$518$	7.73180	0.339716
$519$	0	0
$520$	24.2245	1.06231
$521$	14.0946	0.617495	0.308748	−	0.951144i	$-0.400090\pi$
$521$	14.0946	0.617495	0.308748	+	0.951144i	$0.400090\pi$
$522$	0	0
$523$	39.5023	1.72731	0.863657	−	0.504081i	$-0.168169\pi$
$523$	39.5023	1.72731	0.863657	+	0.504081i	$0.168169\pi$
$524$	−5.93059	−0.259079
$525$	0	0
$526$	−9.44129	−0.411660
$527$	−29.4519	−1.28295
$528$	0	0
$529$	−21.7298	−0.944773
$530$	23.7099	1.02989
$531$	0	0
$532$	4.37432	0.189651
$533$	7.87297	0.341016
$534$	0	0
$535$	50.7138	2.19255
$536$	−40.1890	−1.73590
$537$	0	0
$538$	15.7683	0.679821
$539$	1.13258	0.0487839
$540$	0	0
$541$	−11.0639	−0.475673	−0.237837	−	0.971305i	$-0.576438\pi$
$541$	−11.0639	−0.475673	−0.237837	+	0.971305i	$0.576438\pi$
$542$	15.9237	0.683982
$543$	0	0
$544$	13.7127	0.587929
$545$	−54.1764	−2.32066
$546$	0	0
$547$	−23.1956	−0.991773	−0.495887	−	0.868387i	$-0.665157\pi$
$547$	−23.1956	−0.991773	−0.495887	+	0.868387i	$0.665157\pi$
$548$	−12.4033	−0.529842
$549$	0	0
$550$	6.31832	0.269414
$551$	−20.6989	−0.881801
$552$	0	0
$553$	14.4326	0.613735
$554$	12.1767	0.517339
$555$	0	0
$556$	−4.12661	−0.175007
$557$	41.1183	1.74224	0.871120	−	0.491071i	$-0.163394\pi$
$557$	41.1183	1.74224	0.871120	+	0.491071i	$0.163394\pi$
$558$	0	0
$559$	−7.45610	−0.315359
$560$	−3.19673	−0.135087
$561$	0	0
$562$	−19.6514	−0.828946
$563$	−7.98256	−0.336425	−0.168212	−	0.985751i	$-0.553799\pi$
$563$	−7.98256	−0.336425	−0.168212	+	0.985751i	$0.553799\pi$
$564$	0	0
$565$	−50.4706	−2.12331
$566$	−32.7411	−1.37621
$567$	0	0
$568$	−41.5654	−1.74404
$569$	4.71194	0.197535	0.0987675	−	0.995111i	$-0.468510\pi$
$569$	4.71194	0.197535	0.0987675	+	0.995111i	$0.468510\pi$
$570$	0	0
$571$	7.97860	0.333894	0.166947	−	0.985966i	$-0.446609\pi$
$571$	7.97860	0.333894	0.166947	+	0.985966i	$0.446609\pi$
$572$	2.82471	0.118107
$573$	0	0
$574$	−3.16365	−0.132048
$575$	−6.30137	−0.262785
$576$	0	0
$577$	17.5919	0.732360	0.366180	−	0.930544i	$-0.380665\pi$
$577$	17.5919	0.732360	0.366180	+	0.930544i	$0.380665\pi$
$578$	−9.50356	−0.395296
$579$	0	0
$580$	−15.5360	−0.645098
$581$	7.53954	0.312793
$582$	0	0
$583$	8.26976	0.342499
$584$	36.0380	1.49126
$585$	0	0
$586$	−5.43905	−0.224685
$587$	−16.3578	−0.675160	−0.337580	−	0.941297i	$-0.609608\pi$
$587$	−16.3578	−0.675160	−0.337580	+	0.941297i	$0.609608\pi$
$588$	0	0
$589$	46.9128	1.93301
$590$	−10.1967	−0.419793
$591$	0	0
$592$	−7.61167	−0.312838
$593$	−35.5872	−1.46139	−0.730696	−	0.682703i	$-0.760804\pi$
$593$	−35.5872	−1.46139	−0.730696	+	0.682703i	$0.760804\pi$
$594$	0	0
$595$	8.89786	0.364777
$596$	−9.97162	−0.408454
$597$	0	0
$598$	2.79232	0.114186
$599$	29.1437	1.19078	0.595389	−	0.803437i	$-0.296998\pi$
$599$	29.1437	1.19078	0.595389	+	0.803437i	$0.296998\pi$
$600$	0	0
$601$	40.6898	1.65977	0.829886	−	0.557933i	$-0.188405\pi$
$601$	40.6898	1.65977	0.829886	+	0.557933i	$0.188405\pi$
$602$	2.99614	0.122114
$603$	0	0
$604$	−15.1926	−0.618179
$605$	−31.6237	−1.28569
$606$	0	0
$607$	3.14547	0.127671	0.0638353	−	0.997960i	$-0.479667\pi$
$607$	3.14547	0.127671	0.0638353	+	0.997960i	$0.479667\pi$
$608$	−21.8425	−0.885829
$609$	0	0
$610$	19.6411	0.795246
$611$	−27.7582	−1.12298
$612$	0	0
$613$	28.3596	1.14543	0.572716	−	0.819754i	$-0.305890\pi$
$613$	28.3596	1.14543	0.572716	+	0.819754i	$0.305890\pi$
$614$	10.5685	0.426511
$615$	0	0
$616$	−3.39523	−0.136798
$617$	−27.8358	−1.12063	−0.560314	−	0.828280i	$-0.689320\pi$
$617$	−27.8358	−1.12063	−0.560314	+	0.828280i	$0.689320\pi$
$618$	0	0
$619$	1.35292	0.0543784	0.0271892	−	0.999630i	$-0.491344\pi$
$619$	1.35292	0.0543784	0.0271892	+	0.999630i	$0.491344\pi$
$620$	35.2115	1.41413
$621$	0	0
$622$	−23.0573	−0.924513
$623$	9.91053	0.397057
$624$	0	0
$625$	−21.6954	−0.867816
$626$	15.0522	0.601608
$627$	0	0
$628$	−13.3210	−0.531567
$629$	21.1865	0.844761
$630$	0	0
$631$	−4.37058	−0.173990	−0.0869951	−	0.996209i	$-0.527726\pi$
$631$	−4.37058	−0.173990	−0.0869951	+	0.996209i	$0.527726\pi$
$632$	−43.2655	−1.72101
$633$	0	0
$634$	−26.6131	−1.05694
$635$	−3.25439	−0.129147
$636$	0	0
$637$	−2.48306	−0.0983823
$638$	5.37107	0.212643
$639$	0	0
$640$	−10.0151	−0.395881
$641$	−44.0690	−1.74062	−0.870310	−	0.492505i	$-0.836081\pi$
$641$	−44.0690	−1.74062	−0.870310	+	0.492505i	$0.836081\pi$
$642$	0	0
$643$	37.4155	1.47552	0.737762	−	0.675061i	$-0.235883\pi$
$643$	37.4155	1.47552	0.737762	+	0.675061i	$0.235883\pi$
$644$	1.13203	0.0446083
$645$	0	0
$646$	−11.8808	−0.467444
$647$	19.2258	0.755845	0.377922	−	0.925837i	$-0.376639\pi$
$647$	19.2258	0.755845	0.377922	+	0.925837i	$0.376639\pi$
$648$	0	0
$649$	−3.55652	−0.139605
$650$	−13.8522	−0.543326
$651$	0	0
$652$	−12.9550	−0.507359
$653$	12.9241	0.505759	0.252879	−	0.967498i	$-0.418622\pi$
$653$	12.9241	0.505759	0.252879	+	0.967498i	$0.418622\pi$
$654$	0	0
$655$	19.2154	0.750809
$656$	3.11450	0.121601
$657$	0	0
$658$	11.1543	0.434839
$659$	−8.64135	−0.336619	−0.168310	−	0.985734i	$-0.553831\pi$
$659$	−8.64135	−0.336619	−0.168310	+	0.985734i	$0.553831\pi$
$660$	0	0
$661$	−16.0193	−0.623079	−0.311539	−	0.950233i	$-0.600845\pi$
$661$	−16.0193	−0.623079	−0.311539	+	0.950233i	$0.600845\pi$
$662$	31.5666	1.22687
$663$	0	0
$664$	−22.6018	−0.877120
$665$	−14.1730	−0.549607
$666$	0	0
$667$	−5.35666	−0.207411
$668$	5.51403	0.213344
$669$	0	0
$670$	43.5326	1.68181
$671$	6.85063	0.264466
$672$	0	0
$673$	44.4269	1.71253	0.856266	−	0.516536i	$-0.172779\pi$
$673$	44.4269	1.71253	0.856266	+	0.516536i	$0.172779\pi$
$674$	18.3400	0.706429
$675$	0	0
$676$	6.86466	0.264026
$677$	−20.0606	−0.770990	−0.385495	−	0.922710i	$-0.625969\pi$
$677$	−20.0606	−0.770990	−0.385495	+	0.922710i	$0.625969\pi$
$678$	0	0
$679$	−11.6643	−0.447636
$680$	−26.6737	−1.02289
$681$	0	0
$682$	−12.1732	−0.466137
$683$	−26.6204	−1.01860	−0.509301	−	0.860589i	$-0.670096\pi$
$683$	−26.6204	−1.01860	−0.509301	+	0.860589i	$0.670096\pi$
$684$	0	0
$685$	40.1873	1.53548
$686$	0.997785	0.0380956
$687$	0	0
$688$	−2.94959	−0.112452
$689$	−18.1305	−0.690716
$690$	0	0
$691$	−32.0927	−1.22087	−0.610433	−	0.792068i	$-0.709004\pi$
$691$	−32.0927	−1.22087	−0.610433	+	0.792068i	$0.709004\pi$
$692$	4.64862	0.176714
$693$	0	0
$694$	16.6251	0.631081
$695$	13.3704	0.507170
$696$	0	0
$697$	−8.66897	−0.328361
$698$	−11.1155	−0.420729
$699$	0	0
$700$	−5.61579	−0.212257
$701$	−50.8147	−1.91924	−0.959622	−	0.281291i	$-0.909237\pi$
$701$	−50.8147	−1.91924	−0.959622	+	0.281291i	$0.909237\pi$
$702$	0	0
$703$	−33.7471	−1.27280
$704$	7.89286	0.297473
$705$	0	0
$706$	−0.762946	−0.0287138
$707$	9.18558	0.345459
$708$	0	0
$709$	48.1422	1.80802	0.904009	−	0.427513i	$-0.140610\pi$
$709$	48.1422	1.80802	0.904009	+	0.427513i	$0.140610\pi$
$710$	45.0236	1.68970
$711$	0	0
$712$	−29.7095	−1.11341
$713$	12.1406	0.454668
$714$	0	0
$715$	−9.15223	−0.342274
$716$	−10.2894	−0.384532
$717$	0	0
$718$	−17.7559	−0.662645
$719$	35.3715	1.31913	0.659567	−	0.751645i	$-0.270739\pi$
$719$	35.3715	1.31913	0.659567	+	0.751645i	$0.270739\pi$
$720$	0	0
$721$	−11.3010	−0.420870
$722$	−0.0334487	−0.00124483
$723$	0	0
$724$	12.3604	0.459371
$725$	26.5734	0.986910
$726$	0	0
$727$	27.3924	1.01593	0.507964	−	0.861378i	$-0.330398\pi$
$727$	27.3924	1.01593	0.507964	+	0.861378i	$0.330398\pi$
$728$	7.44364	0.275880
$729$	0	0
$730$	−39.0363	−1.44480
$731$	8.20995	0.303656
$732$	0	0
$733$	−10.0207	−0.370121	−0.185061	−	0.982727i	$-0.559248\pi$
$733$	−10.0207	−0.370121	−0.185061	+	0.982727i	$0.559248\pi$
$734$	−5.94597	−0.219470
$735$	0	0
$736$	−5.65262	−0.208358
$737$	15.1837	0.559300
$738$	0	0
$739$	−29.9761	−1.10269	−0.551344	−	0.834278i	$-0.685885\pi$
$739$	−29.9761	−1.10269	−0.551344	+	0.834278i	$0.685885\pi$
$740$	−25.3297	−0.931139
$741$	0	0
$742$	7.28551	0.267459
$743$	18.4703	0.677611	0.338806	−	0.940856i	$-0.389977\pi$
$743$	18.4703	0.677611	0.338806	+	0.940856i	$0.389977\pi$
$744$	0	0
$745$	32.3086	1.18370
$746$	13.9759	0.511693
$747$	0	0
$748$	−3.11031	−0.113724
$749$	15.5832	0.569398
$750$	0	0
$751$	−29.5002	−1.07648	−0.538239	−	0.842792i	$-0.680910\pi$
$751$	−29.5002	−1.07648	−0.538239	+	0.842792i	$0.680910\pi$
$752$	−10.9810	−0.400435
$753$	0	0
$754$	−11.7754	−0.428836
$755$	49.2249	1.79148
$756$	0	0
$757$	−35.2983	−1.28294	−0.641469	−	0.767149i	$-0.721675\pi$
$757$	−35.2983	−1.28294	−0.641469	+	0.767149i	$0.721675\pi$
$758$	9.02008	0.327624
$759$	0	0
$760$	42.4875	1.54118
$761$	−9.75795	−0.353725	−0.176863	−	0.984236i	$-0.556595\pi$
$761$	−9.75795	−0.353725	−0.176863	+	0.984236i	$0.556595\pi$
$762$	0	0
$763$	−16.6472	−0.602669
$764$	−0.655843	−0.0237276
$765$	0	0
$766$	−33.1859	−1.19905
$767$	7.79724	0.281542
$768$	0	0
$769$	−21.8327	−0.787307	−0.393653	−	0.919259i	$-0.628789\pi$
$769$	−21.8327	−0.787307	−0.393653	+	0.919259i	$0.628789\pi$
$770$	3.67771	0.132535
$771$	0	0
$772$	9.01255	0.324369
$773$	53.8588	1.93717	0.968584	−	0.248688i	$-0.0799995\pi$
$773$	53.8588	1.93717	0.968584	+	0.248688i	$0.0799995\pi$
$774$	0	0
$775$	−60.2270	−2.16342
$776$	34.9670	1.25524
$777$	0	0
$778$	−27.7908	−0.996349
$779$	13.8085	0.494739
$780$	0	0
$781$	15.7038	0.561925
$782$	−3.07464	−0.109949
$783$	0	0
$784$	−0.982283	−0.0350815
$785$	43.1609	1.54048
$786$	0	0
$787$	4.77951	0.170371	0.0851856	−	0.996365i	$-0.472852\pi$
$787$	4.77951	0.170371	0.0851856	+	0.996365i	$0.472852\pi$
$788$	2.86521	0.102069
$789$	0	0
$790$	46.8652	1.66739
$791$	−15.5085	−0.551418
$792$	0	0
$793$	−15.0192	−0.533347
$794$	21.0465	0.746911
$795$	0	0
$796$	21.0812	0.747204
$797$	51.6219	1.82854	0.914271	−	0.405104i	$-0.132765\pi$
$797$	51.6219	1.82854	0.914271	+	0.405104i	$0.132765\pi$
$798$	0	0
$799$	30.5647	1.08130
$800$	28.0416	0.991419
$801$	0	0
$802$	−8.08534	−0.285503
$803$	−13.6155	−0.480480
$804$	0	0
$805$	−3.66785	−0.129275
$806$	26.6883	0.940057
$807$	0	0
$808$	−27.5363	−0.968722
$809$	21.5876	0.758981	0.379490	−	0.925196i	$-0.376099\pi$
$809$	21.5876	0.758981	0.379490	+	0.925196i	$0.376099\pi$
$810$	0	0
$811$	−37.7627	−1.32603	−0.663013	−	0.748608i	$-0.730723\pi$
$811$	−37.7627	−1.32603	−0.663013	+	0.748608i	$0.730723\pi$
$812$	−4.77387	−0.167530
$813$	0	0
$814$	8.75691	0.306930
$815$	41.9751	1.47032
$816$	0	0
$817$	−13.0773	−0.457517
$818$	11.5219	0.402853
$819$	0	0
$820$	10.3643	0.361936
$821$	−21.0481	−0.734585	−0.367292	−	0.930106i	$-0.619715\pi$
$821$	−21.0481	−0.734585	−0.367292	+	0.930106i	$0.619715\pi$
$822$	0	0
$823$	7.83366	0.273064	0.136532	−	0.990636i	$-0.456404\pi$
$823$	7.83366	0.273064	0.136532	+	0.990636i	$0.456404\pi$
$824$	33.8777	1.18018
$825$	0	0
$826$	−3.13322	−0.109019
$827$	−6.19814	−0.215530	−0.107765	−	0.994176i	$-0.534369\pi$
$827$	−6.19814	−0.215530	−0.107765	+	0.994176i	$0.534369\pi$
$828$	0	0
$829$	−41.2494	−1.43265	−0.716325	−	0.697767i	$-0.754177\pi$
$829$	−41.2494	−1.43265	−0.716325	+	0.697767i	$0.754177\pi$
$830$	24.4823	0.849791
$831$	0	0
$832$	−17.3042	−0.599914
$833$	2.73411	0.0947312
$834$	0	0
$835$	−17.8657	−0.618270
$836$	4.95429	0.171348
$837$	0	0
$838$	−15.6412	−0.540317
$839$	−23.4121	−0.808274	−0.404137	−	0.914698i	$-0.632428\pi$
$839$	−23.4121	−0.808274	−0.404137	+	0.914698i	$0.632428\pi$
$840$	0	0
$841$	−6.41053	−0.221053
$842$	−9.99034	−0.344290
$843$	0	0
$844$	4.92510	0.169529
$845$	−22.2419	−0.765144
$846$	0	0
$847$	−9.71725	−0.333889
$848$	−7.17231	−0.246298
$849$	0	0
$850$	15.2527	0.523163
$851$	−8.73343	−0.299378
$852$	0	0
$853$	23.2571	0.796307	0.398153	−	0.917319i	$-0.369651\pi$
$853$	23.2571	0.796307	0.398153	+	0.917319i	$0.369651\pi$
$854$	6.03527	0.206523
$855$	0	0
$856$	−46.7149	−1.59668
$857$	44.7658	1.52917	0.764585	−	0.644523i	$-0.222944\pi$
$857$	44.7658	1.52917	0.764585	+	0.644523i	$0.222944\pi$
$858$	0	0
$859$	−36.4491	−1.24363	−0.621814	−	0.783165i	$-0.713604\pi$
$859$	−36.4491	−1.24363	−0.621814	+	0.783165i	$0.713604\pi$
$860$	−9.81548	−0.334705
$861$	0	0
$862$	20.8151	0.708965
$863$	−50.1696	−1.70779	−0.853896	−	0.520444i	$-0.825767\pi$
$863$	−50.1696	−1.70779	−0.853896	+	0.520444i	$0.825767\pi$
$864$	0	0
$865$	−15.0618	−0.512116
$866$	13.1192	0.445807
$867$	0	0
$868$	10.8197	0.367244
$869$	16.3461	0.554503
$870$	0	0
$871$	−33.2886	−1.12794
$872$	49.9045	1.68998
$873$	0	0
$874$	4.89747	0.165659
$875$	1.92352	0.0650267
$876$	0	0
$877$	6.75208	0.228002	0.114001	−	0.993481i	$-0.463633\pi$
$877$	6.75208	0.228002	0.114001	+	0.993481i	$0.463633\pi$
$878$	−10.4177	−0.351579
$879$	0	0
$880$	−3.62057	−0.122049
$881$	−0.796637	−0.0268394	−0.0134197	−	0.999910i	$-0.504272\pi$
$881$	−0.796637	−0.0268394	−0.0134197	+	0.999910i	$0.504272\pi$
$882$	0	0
$883$	16.4569	0.553818	0.276909	−	0.960896i	$-0.410690\pi$
$883$	16.4569	0.553818	0.276909	+	0.960896i	$0.410690\pi$
$884$	6.81898	0.229347
$885$	0	0
$886$	37.9843	1.27611
$887$	−28.6628	−0.962403	−0.481201	−	0.876610i	$-0.659799\pi$
$887$	−28.6628	−0.962403	−0.481201	+	0.876610i	$0.659799\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	32.1813	1.07872
$891$	0	0
$892$	−24.3895	−0.816620
$893$	−48.6853	−1.62919
$894$	0	0
$895$	33.3382	1.11437
$896$	−3.07741	−0.102809
$897$	0	0
$898$	−12.2561	−0.408991
$899$	−51.1977	−1.70754
$900$	0	0
$901$	19.9636	0.665083
$902$	−3.58310	−0.119304
$903$	0	0
$904$	46.4908	1.54626
$905$	−40.0484	−1.33125
$906$	0	0
$907$	−36.4674	−1.21088	−0.605440	−	0.795891i	$-0.707003\pi$
$907$	−36.4674	−1.21088	−0.605440	+	0.795891i	$0.707003\pi$
$908$	−26.7989	−0.889354
$909$	0	0
$910$	−8.06294	−0.267284
$911$	30.9163	1.02430	0.512151	−	0.858896i	$-0.328849\pi$
$911$	30.9163	1.02430	0.512151	+	0.858896i	$0.328849\pi$
$912$	0	0
$913$	8.53916	0.282605
$914$	−30.2960	−1.00210
$915$	0	0
$916$	−0.648641	−0.0214317
$917$	5.90447	0.194983
$918$	0	0
$919$	−1.41810	−0.0467787	−0.0233893	−	0.999726i	$-0.507446\pi$
$919$	−1.41810	−0.0467787	−0.0233893	+	0.999726i	$0.507446\pi$
$920$	10.9954	0.362506
$921$	0	0
$922$	−25.4859	−0.839334
$923$	−34.4286	−1.13323
$924$	0	0
$925$	43.3248	1.42451
$926$	−9.62231	−0.316209
$927$	0	0
$928$	23.8375	0.782506
$929$	7.77224	0.254999	0.127500	−	0.991839i	$-0.459305\pi$
$929$	7.77224	0.254999	0.127500	+	0.991839i	$0.459305\pi$
$930$	0	0
$931$	−4.35505	−0.142731
$932$	−16.3894	−0.536851
$933$	0	0
$934$	31.0206	1.01502
$935$	10.0776	0.329572
$936$	0	0
$937$	−45.7151	−1.49345	−0.746723	−	0.665135i	$-0.768374\pi$
$937$	−45.7151	−1.49345	−0.746723	+	0.665135i	$0.768374\pi$
$938$	13.3766	0.436761
$939$	0	0
$940$	−36.5419	−1.19187
$941$	6.60714	0.215387	0.107693	−	0.994184i	$-0.465654\pi$
$941$	6.60714	0.215387	0.107693	+	0.994184i	$0.465654\pi$
$942$	0	0
$943$	3.57349	0.116369
$944$	3.08454	0.100393
$945$	0	0
$946$	3.39338	0.110328
$947$	28.7943	0.935688	0.467844	−	0.883811i	$-0.345031\pi$
$947$	28.7943	0.935688	0.467844	+	0.883811i	$0.345031\pi$
$948$	0	0
$949$	29.8503	0.968983
$950$	−24.2954	−0.788247
$951$	0	0
$952$	−8.19623	−0.265641
$953$	−43.0676	−1.39510	−0.697549	−	0.716537i	$-0.745726\pi$
$953$	−43.0676	−1.39510	−0.697549	+	0.716537i	$0.745726\pi$
$954$	0	0
$955$	2.12497	0.0687623
$956$	5.58786	0.180724
$957$	0	0
$958$	−24.7543	−0.799775
$959$	12.3486	0.398759
$960$	0	0
$961$	85.0367	2.74312
$962$	−19.1985	−0.618985
$963$	0	0
$964$	4.60221	0.148227
$965$	−29.2012	−0.940018
$966$	0	0
$967$	−27.2985	−0.877859	−0.438930	−	0.898521i	$-0.644642\pi$
$967$	−27.2985	−0.877859	−0.438930	+	0.898521i	$0.644642\pi$
$968$	29.1301	0.936277
$969$	0	0
$970$	−37.8762	−1.21613
$971$	13.6106	0.436785	0.218392	−	0.975861i	$-0.429919\pi$
$971$	13.6106	0.436785	0.218392	+	0.975861i	$0.429919\pi$
$972$	0	0
$973$	4.10843	0.131710
$974$	31.6581	1.01439
$975$	0	0
$976$	−5.94151	−0.190183
$977$	−12.5918	−0.402849	−0.201424	−	0.979504i	$-0.564557\pi$
$977$	−12.5918	−0.402849	−0.201424	+	0.979504i	$0.564557\pi$
$978$	0	0
$979$	11.2245	0.358737
$980$	−3.26879	−0.104418
$981$	0	0
$982$	32.3455	1.03219
$983$	−61.9971	−1.97740	−0.988701	−	0.149901i	$-0.952104\pi$
$983$	−61.9971	−1.97740	−0.988701	+	0.149901i	$0.952104\pi$
$984$	0	0
$985$	−9.28344	−0.295795
$986$	12.9660	0.412921
$987$	0	0
$988$	−10.8617	−0.345556
$989$	−3.38428	−0.107614
$990$	0	0
$991$	10.4117	0.330738	0.165369	−	0.986232i	$-0.447119\pi$
$991$	10.4117	0.330738	0.165369	+	0.986232i	$0.447119\pi$
$992$	−54.0264	−1.71534
$993$	0	0
$994$	13.8347	0.438811
$995$	−68.3043	−2.16539
$996$	0	0
$997$	4.54947	0.144083	0.0720416	−	0.997402i	$-0.477049\pi$
$997$	4.54947	0.144083	0.0720416	+	0.997402i	$0.477049\pi$
$998$	−15.0330	−0.475863
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.25	yes	40
3.2	odd	2	inner	8001.2.a.ba.1.16	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.16	✓	40	3.2	odd	2	inner
8001.2.a.ba.1.25	yes	40	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+0.997785 q^{2} -1.00442 q^{4} +3.25439 q^{5} +1.00000 q^{7} -2.99777 q^{8} +O(q^{10})\)	\(q+0.997785 q^{2} -1.00442 q^{4} +3.25439 q^{5} +1.00000 q^{7} -2.99777 q^{8} +3.24718 q^{10} +1.13258 q^{11} -2.48306 q^{13} +0.997785 q^{14} -0.982283 q^{16} +2.73411 q^{17} -4.35505 q^{19} -3.26879 q^{20} +1.13008 q^{22} -1.12705 q^{23} +5.59105 q^{25} -2.47756 q^{26} -1.00442 q^{28} +4.75284 q^{29} -10.7720 q^{31} +5.01543 q^{32} +2.72805 q^{34} +3.25439 q^{35} +7.74896 q^{37} -4.34541 q^{38} -9.75591 q^{40} -3.17067 q^{41} +3.00279 q^{43} -1.13760 q^{44} -1.12455 q^{46} +11.1790 q^{47} +1.00000 q^{49} +5.57867 q^{50} +2.49404 q^{52} +7.30168 q^{53} +3.68587 q^{55} -2.99777 q^{56} +4.74231 q^{58} -3.14018 q^{59} +6.04867 q^{61} -10.7482 q^{62} +6.96889 q^{64} -8.08084 q^{65} +13.4063 q^{67} -2.74621 q^{68} +3.24718 q^{70} +13.8654 q^{71} -12.0216 q^{73} +7.73180 q^{74} +4.37432 q^{76} +1.13258 q^{77} +14.4326 q^{79} -3.19673 q^{80} -3.16365 q^{82} +7.53954 q^{83} +8.89786 q^{85} +2.99614 q^{86} -3.39523 q^{88} +9.91053 q^{89} -2.48306 q^{91} +1.13203 q^{92} +11.1543 q^{94} -14.1730 q^{95} -11.6643 q^{97} +0.997785 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 40 * q + 54 * q^4 + 40 * q^7	\( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) \) 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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