LMFDB - Embedded newform 8001.2.a.y.1.9

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:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-1.34710 q^{2} -0.185322 q^{4} +0.124848 q^{5} -1.00000 q^{7} +2.94385 q^{8} +O(q^{10})$	\(q-1.34710 q^{2} -0.185322 q^{4} +0.124848 q^{5} -1.00000 q^{7} +2.94385 q^{8} -0.168183 q^{10} +0.0232879 q^{11} -1.81631 q^{13} +1.34710 q^{14} -3.59501 q^{16} +5.85992 q^{17} -6.33861 q^{19} -0.0231371 q^{20} -0.0313711 q^{22} +5.01952 q^{23} -4.98441 q^{25} +2.44676 q^{26} +0.185322 q^{28} -5.62396 q^{29} +7.17849 q^{31} -1.04485 q^{32} -7.89390 q^{34} -0.124848 q^{35} +3.76017 q^{37} +8.53874 q^{38} +0.367534 q^{40} -1.39770 q^{41} +1.00299 q^{43} -0.00431575 q^{44} -6.76180 q^{46} -1.58115 q^{47} +1.00000 q^{49} +6.71450 q^{50} +0.336602 q^{52} -9.43367 q^{53} +0.00290745 q^{55} -2.94385 q^{56} +7.57604 q^{58} -8.78661 q^{59} +10.4912 q^{61} -9.67015 q^{62} +8.59755 q^{64} -0.226764 q^{65} +5.56958 q^{67} -1.08597 q^{68} +0.168183 q^{70} +9.21819 q^{71} +8.24252 q^{73} -5.06532 q^{74} +1.17468 q^{76} -0.0232879 q^{77} -4.90911 q^{79} -0.448831 q^{80} +1.88284 q^{82} -1.62186 q^{83} +0.731601 q^{85} -1.35112 q^{86} +0.0685559 q^{88} -9.82593 q^{89} +1.81631 q^{91} -0.930227 q^{92} +2.12996 q^{94} -0.791364 q^{95} -3.84997 q^{97} -1.34710 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) $ 28 * q + 30 * q^4 - 28 * q^7	$ 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) $ 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * 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 $

$2$	−1.34710	−0.952544	−0.476272	−	0.879298i	$-0.658012\pi$
$2$	−1.34710	−0.952544	−0.476272	+	0.879298i	$0.658012\pi$
$3$	0	0
$3$	0	0
$4$	−0.185322	−0.0926609
$5$	0.124848	0.0558338	0.0279169	−	0.999610i	$-0.491113\pi$
$5$	0.124848	0.0558338	0.0279169	+	0.999610i	$0.491113\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.94385	1.04081
$9$	0	0
$10$	−0.168183	−0.0531841
$11$	0.0232879	0.00702156	0.00351078	−	0.999994i	$-0.498882\pi$
$11$	0.0232879	0.00702156	0.00351078	+	0.999994i	$0.498882\pi$
$12$	0	0
$13$	−1.81631	−0.503755	−0.251877	−	0.967759i	$-0.581048\pi$
$13$	−1.81631	−0.503755	−0.251877	+	0.967759i	$0.581048\pi$
$14$	1.34710	0.360028
$15$	0	0
$16$	−3.59501	−0.898753
$17$	5.85992	1.42124	0.710620	−	0.703576i	$-0.248415\pi$
$17$	5.85992	1.42124	0.710620	+	0.703576i	$0.248415\pi$
$18$	0	0
$19$	−6.33861	−1.45418	−0.727088	−	0.686544i	$-0.759127\pi$
$19$	−6.33861	−1.45418	−0.727088	+	0.686544i	$0.759127\pi$
$20$	−0.0231371	−0.00517361
$21$	0	0
$22$	−0.0313711	−0.00668834
$23$	5.01952	1.04664	0.523322	−	0.852135i	$-0.324693\pi$
$23$	5.01952	1.04664	0.523322	+	0.852135i	$0.324693\pi$
$24$	0	0
$25$	−4.98441	−0.996883
$26$	2.44676	0.479849
$27$	0	0
$28$	0.185322	0.0350225
$29$	−5.62396	−1.04434	−0.522172	−	0.852840i	$-0.674878\pi$
$29$	−5.62396	−1.04434	−0.522172	+	0.852840i	$0.674878\pi$
$30$	0	0
$31$	7.17849	1.28930	0.644648	−	0.764480i	$-0.277004\pi$
$31$	7.17849	1.28930	0.644648	+	0.764480i	$0.277004\pi$
$32$	−1.04485	−0.184706
$33$	0	0
$34$	−7.89390	−1.35379
$35$	−0.124848	−0.0211032
$36$	0	0
$37$	3.76017	0.618167	0.309084	−	0.951035i	$-0.399978\pi$
$37$	3.76017	0.618167	0.309084	+	0.951035i	$0.399978\pi$
$38$	8.53874	1.38517
$39$	0	0
$40$	0.367534	0.0581122
$41$	−1.39770	−0.218284	−0.109142	−	0.994026i	$-0.534810\pi$
$41$	−1.39770	−0.218284	−0.109142	+	0.994026i	$0.534810\pi$
$42$	0	0
$43$	1.00299	0.152954	0.0764771	−	0.997071i	$-0.475633\pi$
$43$	1.00299	0.152954	0.0764771	+	0.997071i	$0.475633\pi$
$44$	−0.00431575	−0.000650624 0
$45$	0	0
$46$	−6.76180	−0.996973
$47$	−1.58115	−0.230634	−0.115317	−	0.993329i	$-0.536788\pi$
$47$	−1.58115	−0.230634	−0.115317	+	0.993329i	$0.536788\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	6.71450	0.949574
$51$	0	0
$52$	0.336602	0.0466784
$53$	−9.43367	−1.29581	−0.647907	−	0.761719i	$-0.724356\pi$
$53$	−9.43367	−1.29581	−0.647907	+	0.761719i	$0.724356\pi$
$54$	0	0
$55$	0.00290745	0.000392040 0
$56$	−2.94385	−0.393388
$57$	0	0
$58$	7.57604	0.994782
$59$	−8.78661	−1.14392	−0.571960	−	0.820282i	$-0.693817\pi$
$59$	−8.78661	−1.14392	−0.571960	+	0.820282i	$0.693817\pi$
$60$	0	0
$61$	10.4912	1.34327	0.671633	−	0.740884i	$-0.265593\pi$
$61$	10.4912	1.34327	0.671633	+	0.740884i	$0.265593\pi$
$62$	−9.67015	−1.22811
$63$	0	0
$64$	8.59755	1.07469
$65$	−0.226764	−0.0281266
$66$	0	0
$67$	5.56958	0.680432	0.340216	−	0.940347i	$-0.389500\pi$
$67$	5.56958	0.680432	0.340216	+	0.940347i	$0.389500\pi$
$68$	−1.08597	−0.131693
$69$	0	0
$70$	0.168183	0.0201017
$71$	9.21819	1.09400	0.546999	−	0.837133i	$-0.315770\pi$
$71$	9.21819	1.09400	0.546999	+	0.837133i	$0.315770\pi$
$72$	0	0
$73$	8.24252	0.964714	0.482357	−	0.875975i	$-0.339781\pi$
$73$	8.24252	0.964714	0.482357	+	0.875975i	$0.339781\pi$
$74$	−5.06532	−0.588831
$75$	0	0
$76$	1.17468	0.134745
$77$	−0.0232879	−0.00265390
$78$	0	0
$79$	−4.90911	−0.552318	−0.276159	−	0.961112i	$-0.589062\pi$
$79$	−4.90911	−0.552318	−0.276159	+	0.961112i	$0.589062\pi$
$80$	−0.448831	−0.0501808
$81$	0	0
$82$	1.88284	0.207925
$83$	−1.62186	−0.178022	−0.0890112	−	0.996031i	$-0.528371\pi$
$83$	−1.62186	−0.178022	−0.0890112	+	0.996031i	$0.528371\pi$
$84$	0	0
$85$	0.731601	0.0793532
$86$	−1.35112	−0.145695
$87$	0	0
$88$	0.0685559	0.00730809
$89$	−9.82593	−1.04155	−0.520773	−	0.853695i	$-0.674356\pi$
$89$	−9.82593	−1.04155	−0.520773	+	0.853695i	$0.674356\pi$
$90$	0	0
$91$	1.81631	0.190401
$92$	−0.930227	−0.0969829
$93$	0	0
$94$	2.12996	0.219689
$95$	−0.791364	−0.0811923
$96$	0	0
$97$	−3.84997	−0.390905	−0.195453	−	0.980713i	$-0.562618\pi$
$97$	−3.84997	−0.390905	−0.195453	+	0.980713i	$0.562618\pi$
$98$	−1.34710	−0.136078
$99$	0	0
$100$	0.923720	0.0923720
$101$	0.486490	0.0484076	0.0242038	−	0.999707i	$-0.492295\pi$
$101$	0.486490	0.0484076	0.0242038	+	0.999707i	$0.492295\pi$
$102$	0	0
$103$	−1.22701	−0.120901	−0.0604507	−	0.998171i	$-0.519254\pi$
$103$	−1.22701	−0.120901	−0.0604507	+	0.998171i	$0.519254\pi$
$104$	−5.34695	−0.524312
$105$	0	0
$106$	12.7081	1.23432
$107$	−12.9702	−1.25388	−0.626941	−	0.779067i	$-0.715693\pi$
$107$	−12.9702	−1.25388	−0.626941	+	0.779067i	$0.715693\pi$
$108$	0	0
$109$	−1.10456	−0.105797	−0.0528987	−	0.998600i	$-0.516846\pi$
$109$	−1.10456	−0.105797	−0.0528987	+	0.998600i	$0.516846\pi$
$110$	−0.00391662	−0.000373436 0
$111$	0	0
$112$	3.59501	0.339697
$113$	19.9356	1.87538	0.937691	−	0.347472i	$-0.112960\pi$
$113$	19.9356	1.87538	0.937691	+	0.347472i	$0.112960\pi$
$114$	0	0
$115$	0.626679	0.0584381
$116$	1.04224	0.0967697
$117$	0	0
$118$	11.8364	1.08963
$119$	−5.85992	−0.537178
$120$	0	0
$121$	−10.9995	−0.999951
$122$	−14.1328	−1.27952
$123$	0	0
$124$	−1.33033	−0.119467
$125$	−1.24654	−0.111494
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	−9.49205	−0.838986
$129$	0	0
$130$	0.305473	0.0267918
$131$	−2.17649	−0.190161	−0.0950805	−	0.995470i	$-0.530311\pi$
$131$	−2.17649	−0.190161	−0.0950805	+	0.995470i	$0.530311\pi$
$132$	0	0
$133$	6.33861	0.549627
$134$	−7.50278	−0.648141
$135$	0	0
$136$	17.2507	1.47924
$137$	14.1303	1.20724	0.603618	−	0.797274i	$-0.293725\pi$
$137$	14.1303	1.20724	0.603618	+	0.797274i	$0.293725\pi$
$138$	0	0
$139$	11.7203	0.994105	0.497052	−	0.867721i	$-0.334416\pi$
$139$	11.7203	0.994105	0.497052	+	0.867721i	$0.334416\pi$
$140$	0.0231371	0.00195544
$141$	0	0
$142$	−12.4178	−1.04208
$143$	−0.0422981	−0.00353714
$144$	0	0
$145$	−0.702141	−0.0583097
$146$	−11.1035	−0.918932
$147$	0	0
$148$	−0.696840	−0.0572799
$149$	16.5132	1.35281	0.676407	−	0.736528i	$-0.263536\pi$
$149$	16.5132	1.35281	0.676407	+	0.736528i	$0.263536\pi$
$150$	0	0
$151$	−9.59928	−0.781178	−0.390589	−	0.920565i	$-0.627729\pi$
$151$	−9.59928	−0.781178	−0.390589	+	0.920565i	$0.627729\pi$
$152$	−18.6599	−1.51352
$153$	0	0
$154$	0.0313711	0.00252795
$155$	0.896222	0.0719863
$156$	0	0
$157$	−9.16452	−0.731408	−0.365704	−	0.930731i	$-0.619172\pi$
$157$	−9.16452	−0.731408	−0.365704	+	0.930731i	$0.619172\pi$
$158$	6.61306	0.526107
$159$	0	0
$160$	−0.130448	−0.0103128
$161$	−5.01952	−0.395594
$162$	0	0
$163$	17.0298	1.33388	0.666939	−	0.745112i	$-0.267604\pi$
$163$	17.0298	1.33388	0.666939	+	0.745112i	$0.267604\pi$
$164$	0.259024	0.0202264
$165$	0	0
$166$	2.18481	0.169574
$167$	10.5430	0.815845	0.407923	−	0.913016i	$-0.366253\pi$
$167$	10.5430	0.815845	0.407923	+	0.913016i	$0.366253\pi$
$168$	0	0
$169$	−9.70100	−0.746231
$170$	−0.985539	−0.0755874
$171$	0	0
$172$	−0.185875	−0.0141729
$173$	−3.46156	−0.263178	−0.131589	−	0.991304i	$-0.542008\pi$
$173$	−3.46156	−0.263178	−0.131589	+	0.991304i	$0.542008\pi$
$174$	0	0
$175$	4.98441	0.376786
$176$	−0.0837202	−0.00631065
$177$	0	0
$178$	13.2365	0.992118
$179$	−5.32899	−0.398308	−0.199154	−	0.979968i	$-0.563819\pi$
$179$	−5.32899	−0.398308	−0.199154	+	0.979968i	$0.563819\pi$
$180$	0	0
$181$	−19.2408	−1.43016	−0.715078	−	0.699045i	$-0.753609\pi$
$181$	−19.2408	−1.43016	−0.715078	+	0.699045i	$0.753609\pi$
$182$	−2.44676	−0.181366
$183$	0	0
$184$	14.7767	1.08935
$185$	0.469450	0.0345146
$186$	0	0
$187$	0.136465	0.00997932
$188$	0.293021	0.0213708
$189$	0	0
$190$	1.06605	0.0773392
$191$	−20.7182	−1.49911	−0.749557	−	0.661939i	$-0.769734\pi$
$191$	−20.7182	−1.49911	−0.749557	+	0.661939i	$0.769734\pi$
$192$	0	0
$193$	25.9852	1.87045	0.935227	−	0.354050i	$-0.115196\pi$
$193$	25.9852	1.87045	0.935227	+	0.354050i	$0.115196\pi$
$194$	5.18629	0.372354
$195$	0	0
$196$	−0.185322	−0.0132373
$197$	−24.2925	−1.73077	−0.865386	−	0.501105i	$-0.832927\pi$
$197$	−24.2925	−1.73077	−0.865386	+	0.501105i	$0.832927\pi$
$198$	0	0
$199$	21.9908	1.55889	0.779443	−	0.626473i	$-0.215502\pi$
$199$	21.9908	1.55889	0.779443	+	0.626473i	$0.215502\pi$
$200$	−14.6733	−1.03756
$201$	0	0
$202$	−0.655351	−0.0461104
$203$	5.62396	0.394725
$204$	0	0
$205$	−0.174501	−0.0121876
$206$	1.65291	0.115164
$207$	0	0
$208$	6.52967	0.452751
$209$	−0.147613	−0.0102106
$210$	0	0
$211$	8.17029	0.562466	0.281233	−	0.959640i	$-0.409257\pi$
$211$	8.17029	0.562466	0.281233	+	0.959640i	$0.409257\pi$
$212$	1.74826	0.120071
$213$	0	0
$214$	17.4722	1.19438
$215$	0.125221	0.00854001
$216$	0	0
$217$	−7.17849	−0.487308
$218$	1.48795	0.100777
$219$	0	0
$220$	−0.000538813 0	−3.63268e−5 0
$221$	−10.6435	−0.715957
$222$	0	0
$223$	−18.4912	−1.23826	−0.619131	−	0.785288i	$-0.712515\pi$
$223$	−18.4912	−1.23826	−0.619131	+	0.785288i	$0.712515\pi$
$224$	1.04485	0.0698121
$225$	0	0
$226$	−26.8552	−1.78638
$227$	−5.25708	−0.348925	−0.174462	−	0.984664i	$-0.555819\pi$
$227$	−5.25708	−0.348925	−0.174462	+	0.984664i	$0.555819\pi$
$228$	0	0
$229$	5.22204	0.345082	0.172541	−	0.985002i	$-0.444802\pi$
$229$	5.22204	0.345082	0.172541	+	0.985002i	$0.444802\pi$
$230$	−0.844199	−0.0556648
$231$	0	0
$232$	−16.5561	−1.08696
$233$	8.58212	0.562234	0.281117	−	0.959674i	$-0.409295\pi$
$233$	8.58212	0.562234	0.281117	+	0.959674i	$0.409295\pi$
$234$	0	0
$235$	−0.197404	−0.0128772
$236$	1.62835	0.105997
$237$	0	0
$238$	7.89390	0.511686
$239$	−8.10547	−0.524300	−0.262150	−	0.965027i	$-0.584431\pi$
$239$	−8.10547	−0.524300	−0.262150	+	0.965027i	$0.584431\pi$
$240$	0	0
$241$	−25.0876	−1.61603	−0.808017	−	0.589160i	$-0.799459\pi$
$241$	−25.0876	−1.61603	−0.808017	+	0.589160i	$0.799459\pi$
$242$	14.8174	0.952497
$243$	0	0
$244$	−1.94426	−0.124468
$245$	0.124848	0.00797626
$246$	0	0
$247$	11.5129	0.732549
$248$	21.1324	1.34191
$249$	0	0
$250$	1.67921	0.106202
$251$	−0.464425	−0.0293143	−0.0146571	−	0.999893i	$-0.504666\pi$
$251$	−0.464425	−0.0293143	−0.0146571	+	0.999893i	$0.504666\pi$
$252$	0	0
$253$	0.116894	0.00734907
$254$	−1.34710	−0.0845246
$255$	0	0
$256$	−4.40835	−0.275522
$257$	−6.25568	−0.390219	−0.195109	−	0.980781i	$-0.562506\pi$
$257$	−6.25568	−0.390219	−0.195109	+	0.980781i	$0.562506\pi$
$258$	0	0
$259$	−3.76017	−0.233645
$260$	0.0420242	0.00260623
$261$	0	0
$262$	2.93195	0.181137
$263$	29.6611	1.82898	0.914492	−	0.404604i	$-0.132591\pi$
$263$	29.6611	1.82898	0.914492	+	0.404604i	$0.132591\pi$
$264$	0	0
$265$	−1.17778	−0.0723503
$266$	−8.53874	−0.523544
$267$	0	0
$268$	−1.03216	−0.0630494
$269$	15.2324	0.928733	0.464366	−	0.885643i	$-0.346282\pi$
$269$	15.2324	0.928733	0.464366	+	0.885643i	$0.346282\pi$
$270$	0	0
$271$	−20.3142	−1.23400	−0.617000	−	0.786963i	$-0.711652\pi$
$271$	−20.3142	−1.23400	−0.617000	+	0.786963i	$0.711652\pi$
$272$	−21.0665	−1.27734
$273$	0	0
$274$	−19.0350	−1.14995
$275$	−0.116076	−0.00699967
$276$	0	0
$277$	25.0157	1.50305	0.751523	−	0.659707i	$-0.229319\pi$
$277$	25.0157	1.50305	0.751523	+	0.659707i	$0.229319\pi$
$278$	−15.7884	−0.946928
$279$	0	0
$280$	−0.367534	−0.0219644
$281$	−20.2609	−1.20867	−0.604333	−	0.796732i	$-0.706560\pi$
$281$	−20.2609	−1.20867	−0.604333	+	0.796732i	$0.706560\pi$
$282$	0	0
$283$	4.30742	0.256050	0.128025	−	0.991771i	$-0.459136\pi$
$283$	4.30742	0.256050	0.128025	+	0.991771i	$0.459136\pi$
$284$	−1.70833	−0.101371
$285$	0	0
$286$	0.0569798	0.00336928
$287$	1.39770	0.0825037
$288$	0	0
$289$	17.3387	1.01992
$290$	0.945855	0.0555425
$291$	0	0
$292$	−1.52752	−0.0893912
$293$	4.11421	0.240355	0.120177	−	0.992752i	$-0.461654\pi$
$293$	4.11421	0.240355	0.120177	+	0.992752i	$0.461654\pi$
$294$	0	0
$295$	−1.09699	−0.0638694
$296$	11.0694	0.643393
$297$	0	0
$298$	−22.2449	−1.28861
$299$	−9.11703	−0.527252
$300$	0	0
$301$	−1.00299	−0.0578112
$302$	12.9312	0.744106
$303$	0	0
$304$	22.7874	1.30695
$305$	1.30981	0.0749997
$306$	0	0
$307$	14.6296	0.834956	0.417478	−	0.908687i	$-0.362914\pi$
$307$	14.6296	0.834956	0.417478	+	0.908687i	$0.362914\pi$
$308$	0.00431575	0.000245913 0
$309$	0	0
$310$	−1.20730	−0.0685701
$311$	14.6918	0.833096	0.416548	−	0.909114i	$-0.363240\pi$
$311$	14.6918	0.833096	0.416548	+	0.909114i	$0.363240\pi$
$312$	0	0
$313$	31.7208	1.79297	0.896483	−	0.443079i	$-0.146114\pi$
$313$	31.7208	1.79297	0.896483	+	0.443079i	$0.146114\pi$
$314$	12.3455	0.696698
$315$	0	0
$316$	0.909764	0.0511782
$317$	10.6694	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$317$	10.6694	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$318$	0	0
$319$	−0.130970	−0.00733292
$320$	1.07339	0.0600042
$321$	0	0
$322$	6.76180	0.376820
$323$	−37.1438	−2.06673
$324$	0	0
$325$	9.05326	0.502185
$326$	−22.9409	−1.27058
$327$	0	0
$328$	−4.11462	−0.227192
$329$	1.58115	0.0871715
$330$	0	0
$331$	31.9750	1.75750	0.878752	−	0.477278i	$-0.158377\pi$
$331$	31.9750	1.75750	0.878752	+	0.477278i	$0.158377\pi$
$332$	0.300566	0.0164957
$333$	0	0
$334$	−14.2025	−0.777128
$335$	0.695352	0.0379911
$336$	0	0
$337$	−20.7774	−1.13182	−0.565908	−	0.824468i	$-0.691474\pi$
$337$	−20.7774	−1.13182	−0.565908	+	0.824468i	$0.691474\pi$
$338$	13.0682	0.710817
$339$	0	0
$340$	−0.135581	−0.00735294
$341$	0.167172	0.00905286
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	2.95264	0.159196
$345$	0	0
$346$	4.66307	0.250688
$347$	1.89823	0.101903	0.0509513	−	0.998701i	$-0.483775\pi$
$347$	1.89823	0.101903	0.0509513	+	0.998701i	$0.483775\pi$
$348$	0	0
$349$	30.6543	1.64089	0.820443	−	0.571729i	$-0.193727\pi$
$349$	30.6543	1.64089	0.820443	+	0.571729i	$0.193727\pi$
$350$	−6.71450	−0.358905
$351$	0	0
$352$	−0.0243324	−0.00129692
$353$	26.7737	1.42502	0.712510	−	0.701662i	$-0.247558\pi$
$353$	26.7737	1.42502	0.712510	+	0.701662i	$0.247558\pi$
$354$	0	0
$355$	1.15087	0.0610821
$356$	1.82096	0.0965106
$357$	0	0
$358$	7.17869	0.379405
$359$	−20.6079	−1.08764	−0.543821	−	0.839201i	$-0.683023\pi$
$359$	−20.6079	−1.08764	−0.543821	+	0.839201i	$0.683023\pi$
$360$	0	0
$361$	21.1780	1.11463
$362$	25.9193	1.36229
$363$	0	0
$364$	−0.336602	−0.0176428
$365$	1.02906	0.0538637
$366$	0	0
$367$	6.09130	0.317963	0.158982	−	0.987282i	$-0.449179\pi$
$367$	6.09130	0.317963	0.158982	+	0.987282i	$0.449179\pi$
$368$	−18.0453	−0.940674
$369$	0	0
$370$	−0.632396	−0.0328767
$371$	9.43367	0.489772
$372$	0	0
$373$	−5.81287	−0.300979	−0.150490	−	0.988612i	$-0.548085\pi$
$373$	−5.81287	−0.300979	−0.150490	+	0.988612i	$0.548085\pi$
$374$	−0.183832	−0.00950573
$375$	0	0
$376$	−4.65466	−0.240046
$377$	10.2149	0.526093
$378$	0	0
$379$	−31.8302	−1.63501	−0.817504	−	0.575922i	$-0.804643\pi$
$379$	−31.8302	−1.63501	−0.817504	+	0.575922i	$0.804643\pi$
$380$	0.146657	0.00752334
$381$	0	0
$382$	27.9095	1.42797
$383$	−29.5476	−1.50981	−0.754906	−	0.655833i	$-0.772317\pi$
$383$	−29.5476	−1.50981	−0.754906	+	0.655833i	$0.772317\pi$
$384$	0	0
$385$	−0.00290745	−0.000148177 0
$386$	−35.0046	−1.78169
$387$	0	0
$388$	0.713483	0.0362216
$389$	−25.8440	−1.31034	−0.655171	−	0.755480i	$-0.727404\pi$
$389$	−25.8440	−1.31034	−0.655171	+	0.755480i	$0.727404\pi$
$390$	0	0
$391$	29.4140	1.48753
$392$	2.94385	0.148687
$393$	0	0
$394$	32.7245	1.64864
$395$	−0.612893	−0.0308380
$396$	0	0
$397$	19.4251	0.974920	0.487460	−	0.873145i	$-0.337924\pi$
$397$	19.4251	0.974920	0.487460	+	0.873145i	$0.337924\pi$
$398$	−29.6238	−1.48491
$399$	0	0
$400$	17.9190	0.895951
$401$	39.6369	1.97937	0.989686	−	0.143254i	$-0.0457566\pi$
$401$	39.6369	1.97937	0.989686	+	0.143254i	$0.0457566\pi$
$402$	0	0
$403$	−13.0384	−0.649489
$404$	−0.0901572	−0.00448549
$405$	0	0
$406$	−7.57604	−0.375992
$407$	0.0875663	0.00434050
$408$	0	0
$409$	35.7310	1.76678	0.883392	−	0.468635i	$-0.155254\pi$
$409$	35.7310	1.76678	0.883392	+	0.468635i	$0.155254\pi$
$410$	0.235070	0.0116093
$411$	0	0
$412$	0.227392	0.0112028
$413$	8.78661	0.432361
$414$	0	0
$415$	−0.202487	−0.00993967
$416$	1.89778	0.0930463
$417$	0	0
$418$	0.198849	0.00972603
$419$	−36.2048	−1.76872	−0.884360	−	0.466805i	$-0.845405\pi$
$419$	−36.2048	−1.76872	−0.884360	+	0.466805i	$0.845405\pi$
$420$	0	0
$421$	−1.08095	−0.0526824	−0.0263412	−	0.999653i	$-0.508386\pi$
$421$	−1.08095	−0.0526824	−0.0263412	+	0.999653i	$0.508386\pi$
$422$	−11.0062	−0.535773
$423$	0	0
$424$	−27.7713	−1.34869
$425$	−29.2083	−1.41681
$426$	0	0
$427$	−10.4912	−0.507707
$428$	2.40367	0.116186
$429$	0	0
$430$	−0.168685	−0.00813473
$431$	29.0605	1.39980	0.699898	−	0.714243i	$-0.253229\pi$
$431$	29.0605	1.39980	0.699898	+	0.714243i	$0.253229\pi$
$432$	0	0
$433$	23.0990	1.11007	0.555033	−	0.831828i	$-0.312706\pi$
$433$	23.0990	1.11007	0.555033	+	0.831828i	$0.312706\pi$
$434$	9.67015	0.464182
$435$	0	0
$436$	0.204698	0.00980327
$437$	−31.8168	−1.52200
$438$	0	0
$439$	5.62374	0.268407	0.134203	−	0.990954i	$-0.457152\pi$
$439$	5.62374	0.268407	0.134203	+	0.990954i	$0.457152\pi$
$440$	0.00855909	0.000408038 0
$441$	0	0
$442$	14.3378	0.681980
$443$	19.9387	0.947315	0.473658	−	0.880709i	$-0.342933\pi$
$443$	19.9387	0.947315	0.473658	+	0.880709i	$0.342933\pi$
$444$	0	0
$445$	−1.22675	−0.0581535
$446$	24.9095	1.17950
$447$	0	0
$448$	−8.59755	−0.406196
$449$	−33.6204	−1.58664	−0.793322	−	0.608802i	$-0.791650\pi$
$449$	−33.6204	−1.58664	−0.793322	+	0.608802i	$0.791650\pi$
$450$	0	0
$451$	−0.0325495	−0.00153270
$452$	−3.69449	−0.173774
$453$	0	0
$454$	7.08181	0.332366
$455$	0.226764	0.0106308
$456$	0	0
$457$	3.43769	0.160808	0.0804042	−	0.996762i	$-0.474379\pi$
$457$	3.43769	0.160808	0.0804042	+	0.996762i	$0.474379\pi$
$458$	−7.03461	−0.328706
$459$	0	0
$460$	−0.116137	−0.00541492
$461$	−34.0720	−1.58689	−0.793445	−	0.608642i	$-0.791715\pi$
$461$	−34.0720	−1.58689	−0.793445	+	0.608642i	$0.791715\pi$
$462$	0	0
$463$	34.1909	1.58898	0.794492	−	0.607274i	$-0.207737\pi$
$463$	34.1909	1.58898	0.794492	+	0.607274i	$0.207737\pi$
$464$	20.2182	0.938607
$465$	0	0
$466$	−11.5610	−0.535552
$467$	−4.91008	−0.227211	−0.113606	−	0.993526i	$-0.536240\pi$
$467$	−4.91008	−0.227211	−0.113606	+	0.993526i	$0.536240\pi$
$468$	0	0
$469$	−5.56958	−0.257179
$470$	0.265922	0.0122661
$471$	0	0
$472$	−25.8664	−1.19060
$473$	0.0233574	0.00107398
$474$	0	0
$475$	31.5943	1.44964
$476$	1.08597	0.0497754
$477$	0	0
$478$	10.9189	0.499418
$479$	34.0644	1.55644	0.778222	−	0.627989i	$-0.216122\pi$
$479$	34.0644	1.55644	0.778222	+	0.627989i	$0.216122\pi$
$480$	0	0
$481$	−6.82964	−0.311405
$482$	33.7955	1.53934
$483$	0	0
$484$	2.03844	0.0926563
$485$	−0.480662	−0.0218257
$486$	0	0
$487$	−18.0362	−0.817298	−0.408649	−	0.912692i	$-0.634000\pi$
$487$	−18.0362	−0.817298	−0.408649	+	0.912692i	$0.634000\pi$
$488$	30.8846	1.39808
$489$	0	0
$490$	−0.168183	−0.00759773
$491$	16.9027	0.762807	0.381403	−	0.924409i	$-0.375441\pi$
$491$	16.9027	0.762807	0.381403	+	0.924409i	$0.375441\pi$
$492$	0	0
$493$	−32.9560	−1.48426
$494$	−15.5090	−0.697785
$495$	0	0
$496$	−25.8068	−1.15876
$497$	−9.21819	−0.413492
$498$	0	0
$499$	−30.6444	−1.37183	−0.685916	−	0.727681i	$-0.740598\pi$
$499$	−30.6444	−1.37183	−0.685916	+	0.727681i	$0.740598\pi$
$500$	0.231010	0.0103311
$501$	0	0
$502$	0.625627	0.0279231
$503$	6.83573	0.304790	0.152395	−	0.988320i	$-0.451301\pi$
$503$	6.83573	0.304790	0.152395	+	0.988320i	$0.451301\pi$
$504$	0	0
$505$	0.0607375	0.00270278
$506$	−0.157468	−0.00700031
$507$	0	0
$508$	−0.185322	−0.00822232
$509$	−29.0466	−1.28747	−0.643735	−	0.765249i	$-0.722616\pi$
$509$	−29.0466	−1.28747	−0.643735	+	0.765249i	$0.722616\pi$
$510$	0	0
$511$	−8.24252	−0.364628
$512$	24.9226	1.10143
$513$	0	0
$514$	8.42703	0.371700
$515$	−0.153191	−0.00675038
$516$	0	0
$517$	−0.0368216	−0.00161941
$518$	5.06532	0.222557
$519$	0	0
$520$	−0.667557	−0.0292743
$521$	−0.707723	−0.0310059	−0.0155030	−	0.999880i	$-0.504935\pi$
$521$	−0.707723	−0.0310059	−0.0155030	+	0.999880i	$0.504935\pi$
$522$	0	0
$523$	35.0484	1.53256	0.766280	−	0.642507i	$-0.222106\pi$
$523$	35.0484	1.53256	0.766280	+	0.642507i	$0.222106\pi$
$524$	0.403351	0.0176205
$525$	0	0
$526$	−39.9565	−1.74219
$527$	42.0654	1.83240
$528$	0	0
$529$	2.19563	0.0954622
$530$	1.58658	0.0689168
$531$	0	0
$532$	−1.17468	−0.0509289
$533$	2.53867	0.109962
$534$	0	0
$535$	−1.61931	−0.0700090
$536$	16.3960	0.708199
$537$	0	0
$538$	−20.5195	−0.884658
$539$	0.0232879	0.00100308
$540$	0	0
$541$	20.6213	0.886577	0.443289	−	0.896379i	$-0.353812\pi$
$541$	20.6213	0.886577	0.443289	+	0.896379i	$0.353812\pi$
$542$	27.3653	1.17544
$543$	0	0
$544$	−6.12275	−0.262511
$545$	−0.137902	−0.00590707
$546$	0	0
$547$	26.7235	1.14261	0.571306	−	0.820737i	$-0.306437\pi$
$547$	26.7235	1.14261	0.571306	+	0.820737i	$0.306437\pi$
$548$	−2.61866	−0.111864
$549$	0	0
$550$	0.156366	0.00666749
$551$	35.6481	1.51866
$552$	0	0
$553$	4.90911	0.208756
$554$	−33.6986	−1.43172
$555$	0	0
$556$	−2.17203	−0.0921146
$557$	30.1554	1.27773	0.638863	−	0.769321i	$-0.279405\pi$
$557$	30.1554	1.27773	0.638863	+	0.769321i	$0.279405\pi$
$558$	0	0
$559$	−1.82174	−0.0770514
$560$	0.448831	0.0189666
$561$	0	0
$562$	27.2935	1.15131
$563$	38.3423	1.61593	0.807967	−	0.589227i	$-0.200568\pi$
$563$	38.3423	1.61593	0.807967	+	0.589227i	$0.200568\pi$
$564$	0	0
$565$	2.48892	0.104710
$566$	−5.80253	−0.243898
$567$	0	0
$568$	27.1369	1.13864
$569$	−12.3771	−0.518875	−0.259437	−	0.965760i	$-0.583537\pi$
$569$	−12.3771	−0.518875	−0.259437	+	0.965760i	$0.583537\pi$
$570$	0	0
$571$	31.8770	1.33401	0.667006	−	0.745052i	$-0.267576\pi$
$571$	31.8770	1.33401	0.667006	+	0.745052i	$0.267576\pi$
$572$	0.00783876	0.000327755 0
$573$	0	0
$574$	−1.88284	−0.0785884
$575$	−25.0194	−1.04338
$576$	0	0
$577$	7.83358	0.326116	0.163058	−	0.986616i	$-0.447864\pi$
$577$	7.83358	0.326116	0.163058	+	0.986616i	$0.447864\pi$
$578$	−23.3569	−0.971520
$579$	0	0
$580$	0.130122	0.00540302
$581$	1.62186	0.0672862
$582$	0	0
$583$	−0.219690	−0.00909864
$584$	24.2647	1.00408
$585$	0	0
$586$	−5.54225	−0.228948
$587$	42.9733	1.77370	0.886849	−	0.462060i	$-0.152889\pi$
$587$	42.9733	1.77370	0.886849	+	0.462060i	$0.152889\pi$
$588$	0	0
$589$	−45.5017	−1.87486
$590$	1.47776	0.0608384
$591$	0	0
$592$	−13.5178	−0.555580
$593$	20.2941	0.833379	0.416689	−	0.909049i	$-0.363190\pi$
$593$	20.2941	0.833379	0.416689	+	0.909049i	$0.363190\pi$
$594$	0	0
$595$	−0.731601	−0.0299927
$596$	−3.06025	−0.125353
$597$	0	0
$598$	12.2816	0.502230
$599$	35.2549	1.44048	0.720239	−	0.693727i	$-0.244032\pi$
$599$	35.2549	1.44048	0.720239	+	0.693727i	$0.244032\pi$
$600$	0	0
$601$	2.68415	0.109489	0.0547443	−	0.998500i	$-0.482566\pi$
$601$	2.68415	0.109489	0.0547443	+	0.998500i	$0.482566\pi$
$602$	1.35112	0.0550677
$603$	0	0
$604$	1.77895	0.0723846
$605$	−1.37326	−0.0558311
$606$	0	0
$607$	43.9417	1.78354	0.891768	−	0.452492i	$-0.149465\pi$
$607$	43.9417	1.78354	0.891768	+	0.452492i	$0.149465\pi$
$608$	6.62291	0.268595
$609$	0	0
$610$	−1.76445	−0.0714405
$611$	2.87186	0.116183
$612$	0	0
$613$	−46.4561	−1.87634	−0.938171	−	0.346171i	$-0.887482\pi$
$613$	−46.4561	−1.87634	−0.938171	+	0.346171i	$0.887482\pi$
$614$	−19.7075	−0.795332
$615$	0	0
$616$	−0.0685559	−0.00276220
$617$	4.73332	0.190556	0.0952781	−	0.995451i	$-0.469626\pi$
$617$	4.73332	0.190556	0.0952781	+	0.995451i	$0.469626\pi$
$618$	0	0
$619$	−16.0979	−0.647030	−0.323515	−	0.946223i	$-0.604865\pi$
$619$	−16.0979	−0.647030	−0.323515	+	0.946223i	$0.604865\pi$
$620$	−0.166089	−0.00667031
$621$	0	0
$622$	−19.7913	−0.793561
$623$	9.82593	0.393668
$624$	0	0
$625$	24.7664	0.990657
$626$	−42.7311	−1.70788
$627$	0	0
$628$	1.69838	0.0677729
$629$	22.0343	0.878564
$630$	0	0
$631$	2.43050	0.0967567	0.0483783	−	0.998829i	$-0.484595\pi$
$631$	2.43050	0.0967567	0.0483783	+	0.998829i	$0.484595\pi$
$632$	−14.4517	−0.574856
$633$	0	0
$634$	−14.3727	−0.570813
$635$	0.124848	0.00495445
$636$	0	0
$637$	−1.81631	−0.0719650
$638$	0.176430	0.00698492
$639$	0	0
$640$	−1.18507	−0.0468438
$641$	11.4361	0.451699	0.225849	−	0.974162i	$-0.427484\pi$
$641$	11.4361	0.451699	0.225849	+	0.974162i	$0.427484\pi$
$642$	0	0
$643$	−10.3693	−0.408927	−0.204464	−	0.978874i	$-0.565545\pi$
$643$	−10.3693	−0.408927	−0.204464	+	0.978874i	$0.565545\pi$
$644$	0.930227	0.0366561
$645$	0	0
$646$	50.0364	1.96865
$647$	45.9780	1.80758	0.903791	−	0.427975i	$-0.140773\pi$
$647$	45.9780	1.80758	0.903791	+	0.427975i	$0.140773\pi$
$648$	0	0
$649$	−0.204621	−0.00803209
$650$	−12.1956	−0.478353
$651$	0	0
$652$	−3.15600	−0.123598
$653$	−1.83975	−0.0719950	−0.0359975	−	0.999352i	$-0.511461\pi$
$653$	−1.83975	−0.0719950	−0.0359975	+	0.999352i	$0.511461\pi$
$654$	0	0
$655$	−0.271731	−0.0106174
$656$	5.02475	0.196184
$657$	0	0
$658$	−2.12996	−0.0830347
$659$	−29.7649	−1.15948	−0.579738	−	0.814803i	$-0.696845\pi$
$659$	−29.7649	−1.15948	−0.579738	+	0.814803i	$0.696845\pi$
$660$	0	0
$661$	18.0666	0.702710	0.351355	−	0.936242i	$-0.385721\pi$
$661$	18.0666	0.702710	0.351355	+	0.936242i	$0.385721\pi$
$662$	−43.0735	−1.67410
$663$	0	0
$664$	−4.77451	−0.185287
$665$	0.791364	0.0306878
$666$	0	0
$667$	−28.2296	−1.09305
$668$	−1.95386	−0.0755969
$669$	0	0
$670$	−0.936708	−0.0361882
$671$	0.244319	0.00943183
$672$	0	0
$673$	15.6827	0.604525	0.302262	−	0.953225i	$-0.402258\pi$
$673$	15.6827	0.604525	0.302262	+	0.953225i	$0.402258\pi$
$674$	27.9892	1.07810
$675$	0	0
$676$	1.79781	0.0691464
$677$	−28.5923	−1.09889	−0.549445	−	0.835530i	$-0.685161\pi$
$677$	−28.5923	−1.09889	−0.549445	+	0.835530i	$0.685161\pi$
$678$	0	0
$679$	3.84997	0.147748
$680$	2.15372	0.0825914
$681$	0	0
$682$	−0.225197	−0.00862325
$683$	3.63560	0.139112	0.0695561	−	0.997578i	$-0.477842\pi$
$683$	3.63560	0.139112	0.0695561	+	0.997578i	$0.477842\pi$
$684$	0	0
$685$	1.76415	0.0674046
$686$	1.34710	0.0514325
$687$	0	0
$688$	−3.60575	−0.137468
$689$	17.1345	0.652773
$690$	0	0
$691$	47.1042	1.79193	0.895965	−	0.444125i	$-0.146485\pi$
$691$	47.1042	1.79193	0.895965	+	0.444125i	$0.146485\pi$
$692$	0.641503	0.0243863
$693$	0	0
$694$	−2.55711	−0.0970666
$695$	1.46326	0.0555047
$696$	0	0
$697$	−8.19042	−0.310234
$698$	−41.2944	−1.56301
$699$	0	0
$700$	−0.923720	−0.0349133
$701$	31.6487	1.19535	0.597677	−	0.801737i	$-0.296091\pi$
$701$	31.6487	1.19535	0.597677	+	0.801737i	$0.296091\pi$
$702$	0	0
$703$	−23.8342	−0.898925
$704$	0.200219	0.00754602
$705$	0	0
$706$	−36.0668	−1.35739
$707$	−0.486490	−0.0182964
$708$	0	0
$709$	18.4600	0.693280	0.346640	−	0.937998i	$-0.387323\pi$
$709$	18.4600	0.693280	0.346640	+	0.937998i	$0.387323\pi$
$710$	−1.55034	−0.0581834
$711$	0	0
$712$	−28.9260	−1.08405
$713$	36.0326	1.34943
$714$	0	0
$715$	−0.00528084	−0.000197492 0
$716$	0.987578	0.0369075
$717$	0	0
$718$	27.7609	1.03603
$719$	−18.6752	−0.696469	−0.348235	−	0.937407i	$-0.613219\pi$
$719$	−18.6752	−0.696469	−0.348235	+	0.937407i	$0.613219\pi$
$720$	0	0
$721$	1.22701	0.0456964
$722$	−28.5289	−1.06173
$723$	0	0
$724$	3.56573	0.132519
$725$	28.0321	1.04109
$726$	0	0
$727$	26.1626	0.970315	0.485158	−	0.874427i	$-0.338762\pi$
$727$	26.1626	0.970315	0.485158	+	0.874427i	$0.338762\pi$
$728$	5.34695	0.198171
$729$	0	0
$730$	−1.38625	−0.0513075
$731$	5.87743	0.217384
$732$	0	0
$733$	−31.5201	−1.16422	−0.582111	−	0.813109i	$-0.697773\pi$
$733$	−31.5201	−1.16422	−0.582111	+	0.813109i	$0.697773\pi$
$734$	−8.20559	−0.302874
$735$	0	0
$736$	−5.24466	−0.193321
$737$	0.129704	0.00477769
$738$	0	0
$739$	9.78068	0.359788	0.179894	−	0.983686i	$-0.442425\pi$
$739$	9.78068	0.359788	0.179894	+	0.983686i	$0.442425\pi$
$740$	−0.0869993	−0.00319816
$741$	0	0
$742$	−12.7081	−0.466529
$743$	−10.6755	−0.391645	−0.195823	−	0.980639i	$-0.562738\pi$
$743$	−10.6755	−0.391645	−0.195823	+	0.980639i	$0.562738\pi$
$744$	0	0
$745$	2.06164	0.0755328
$746$	7.83052	0.286696
$747$	0	0
$748$	−0.0252899	−0.000924692 0
$749$	12.9702	0.473923
$750$	0	0
$751$	−10.4434	−0.381085	−0.190543	−	0.981679i	$-0.561025\pi$
$751$	−10.4434	−0.381085	−0.190543	+	0.981679i	$0.561025\pi$
$752$	5.68425	0.207283
$753$	0	0
$754$	−13.7605	−0.501127
$755$	−1.19845	−0.0436161
$756$	0	0
$757$	−41.1218	−1.49460	−0.747298	−	0.664489i	$-0.768649\pi$
$757$	−41.1218	−1.49460	−0.747298	+	0.664489i	$0.768649\pi$
$758$	42.8785	1.55742
$759$	0	0
$760$	−2.32965	−0.0845055
$761$	22.5152	0.816176	0.408088	−	0.912943i	$-0.366196\pi$
$761$	22.5152	0.816176	0.408088	+	0.912943i	$0.366196\pi$
$762$	0	0
$763$	1.10456	0.0399876
$764$	3.83953	0.138909
$765$	0	0
$766$	39.8036	1.43816
$767$	15.9592	0.576255
$768$	0	0
$769$	16.2486	0.585940	0.292970	−	0.956122i	$-0.405356\pi$
$769$	16.2486	0.585940	0.292970	+	0.956122i	$0.405356\pi$
$770$	0.00391662	0.000141145 0
$771$	0	0
$772$	−4.81562	−0.173318
$773$	−43.5565	−1.56662	−0.783308	−	0.621633i	$-0.786469\pi$
$773$	−43.5565	−1.56662	−0.783308	+	0.621633i	$0.786469\pi$
$774$	0	0
$775$	−35.7806	−1.28528
$776$	−11.3337	−0.406857
$777$	0	0
$778$	34.8144	1.24816
$779$	8.85949	0.317424
$780$	0	0
$781$	0.214672	0.00768157
$782$	−39.6236	−1.41694
$783$	0	0
$784$	−3.59501	−0.128393
$785$	−1.14417	−0.0408373
$786$	0	0
$787$	14.6810	0.523323	0.261661	−	0.965160i	$-0.415730\pi$
$787$	14.6810	0.523323	0.261661	+	0.965160i	$0.415730\pi$
$788$	4.50194	0.160375
$789$	0	0
$790$	0.825628	0.0293745
$791$	−19.9356	−0.708827
$792$	0	0
$793$	−19.0554	−0.676677
$794$	−26.1676	−0.928654
$795$	0	0
$796$	−4.07537	−0.144448
$797$	−13.6198	−0.482437	−0.241219	−	0.970471i	$-0.577547\pi$
$797$	−13.6198	−0.482437	−0.241219	+	0.970471i	$0.577547\pi$
$798$	0	0
$799$	−9.26541	−0.327786
$800$	5.20798	0.184130
$801$	0	0
$802$	−53.3949	−1.88544
$803$	0.191951	0.00677380
$804$	0	0
$805$	−0.626679	−0.0220875
$806$	17.5640	0.618667
$807$	0	0
$808$	1.43215	0.0503830
$809$	52.6461	1.85094	0.925469	−	0.378824i	$-0.123671\pi$
$809$	52.6461	1.85094	0.925469	+	0.378824i	$0.123671\pi$
$810$	0	0
$811$	31.7302	1.11420	0.557099	−	0.830446i	$-0.311914\pi$
$811$	31.7302	1.11420	0.557099	+	0.830446i	$0.311914\pi$
$812$	−1.04224	−0.0365755
$813$	0	0
$814$	−0.117961	−0.00413451
$815$	2.12614	0.0744756
$816$	0	0
$817$	−6.35754	−0.222422
$818$	−48.1332	−1.68294
$819$	0	0
$820$	0.0323387	0.00112932
$821$	33.7481	1.17782	0.588908	−	0.808200i	$-0.299558\pi$
$821$	33.7481	1.17782	0.588908	+	0.808200i	$0.299558\pi$
$822$	0	0
$823$	−12.4138	−0.432717	−0.216359	−	0.976314i	$-0.569418\pi$
$823$	−12.4138	−0.432717	−0.216359	+	0.976314i	$0.569418\pi$
$824$	−3.61214	−0.125835
$825$	0	0
$826$	−11.8364	−0.411842
$827$	13.4423	0.467435	0.233718	−	0.972304i	$-0.424911\pi$
$827$	13.4423	0.467435	0.233718	+	0.972304i	$0.424911\pi$
$828$	0	0
$829$	12.1533	0.422101	0.211051	−	0.977475i	$-0.432312\pi$
$829$	12.1533	0.422101	0.211051	+	0.977475i	$0.432312\pi$
$830$	0.272770	0.00946797
$831$	0	0
$832$	−15.6158	−0.541382
$833$	5.85992	0.203034
$834$	0	0
$835$	1.31628	0.0455518
$836$	0.0273558	0.000946122 0
$837$	0	0
$838$	48.7715	1.68478
$839$	−33.4257	−1.15398	−0.576991	−	0.816750i	$-0.695773\pi$
$839$	−33.4257	−1.15398	−0.576991	+	0.816750i	$0.695773\pi$
$840$	0	0
$841$	2.62894	0.0906529
$842$	1.45615	0.0501823
$843$	0	0
$844$	−1.51413	−0.0521186
$845$	−1.21115	−0.0416649
$846$	0	0
$847$	10.9995	0.377946
$848$	33.9142	1.16462
$849$	0	0
$850$	39.3465	1.34957
$851$	18.8742	0.647001
$852$	0	0
$853$	−43.0658	−1.47454	−0.737272	−	0.675596i	$-0.763886\pi$
$853$	−43.0658	−1.47454	−0.737272	+	0.675596i	$0.763886\pi$
$854$	14.1328	0.483613
$855$	0	0
$856$	−38.1824	−1.30505
$857$	15.2431	0.520694	0.260347	−	0.965515i	$-0.416163\pi$
$857$	15.2431	0.520694	0.260347	+	0.965515i	$0.416163\pi$
$858$	0	0
$859$	43.5085	1.48449	0.742246	−	0.670127i	$-0.233761\pi$
$859$	43.5085	1.48449	0.742246	+	0.670127i	$0.233761\pi$
$860$	−0.0232062	−0.000791325 0
$861$	0	0
$862$	−39.1474	−1.33337
$863$	37.0693	1.26185	0.630927	−	0.775842i	$-0.282675\pi$
$863$	37.0693	1.26185	0.630927	+	0.775842i	$0.282675\pi$
$864$	0	0
$865$	−0.432170	−0.0146942
$866$	−31.1166	−1.05739
$867$	0	0
$868$	1.33033	0.0451544
$869$	−0.114323	−0.00387813
$870$	0	0
$871$	−10.1161	−0.342771
$872$	−3.25164	−0.110115
$873$	0	0
$874$	42.8604	1.44978
$875$	1.24654	0.0421406
$876$	0	0
$877$	3.61667	0.122126	0.0610631	−	0.998134i	$-0.480551\pi$
$877$	3.61667	0.122126	0.0610631	+	0.998134i	$0.480551\pi$
$878$	−7.57575	−0.255669
$879$	0	0
$880$	−0.0104523	−0.000352348 0
$881$	12.9935	0.437764	0.218882	−	0.975751i	$-0.429759\pi$
$881$	12.9935	0.437764	0.218882	+	0.975751i	$0.429759\pi$
$882$	0	0
$883$	43.0994	1.45041	0.725205	−	0.688533i	$-0.241745\pi$
$883$	43.0994	1.45041	0.725205	+	0.688533i	$0.241745\pi$
$884$	1.97246	0.0663411
$885$	0	0
$886$	−26.8594	−0.902359
$887$	22.1654	0.744241	0.372121	−	0.928184i	$-0.378631\pi$
$887$	22.1654	0.744241	0.372121	+	0.928184i	$0.378631\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	1.65255	0.0553938
$891$	0	0
$892$	3.42682	0.114738
$893$	10.0223	0.335383
$894$	0	0
$895$	−0.665315	−0.0222390
$896$	9.49205	0.317107
$897$	0	0
$898$	45.2900	1.51135
$899$	−40.3716	−1.34647
$900$	0	0
$901$	−55.2806	−1.84166
$902$	0.0438474	0.00145996
$903$	0	0
$904$	58.6873	1.95191
$905$	−2.40218	−0.0798511
$906$	0	0
$907$	−50.0794	−1.66286	−0.831430	−	0.555629i	$-0.812477\pi$
$907$	−50.0794	−1.66286	−0.831430	+	0.555629i	$0.812477\pi$
$908$	0.974251	0.0323317
$909$	0	0
$910$	−0.305473	−0.0101263
$911$	−6.00788	−0.199050	−0.0995249	−	0.995035i	$-0.531732\pi$
$911$	−6.00788	−0.199050	−0.0995249	+	0.995035i	$0.531732\pi$
$912$	0	0
$913$	−0.0377697	−0.00125000
$914$	−4.63091	−0.153177
$915$	0	0
$916$	−0.967757	−0.0319756
$917$	2.17649	0.0718741
$918$	0	0
$919$	17.3917	0.573701	0.286850	−	0.957975i	$-0.407392\pi$
$919$	17.3917	0.573701	0.286850	+	0.957975i	$0.407392\pi$
$920$	1.84485	0.0608228
$921$	0	0
$922$	45.8984	1.51158
$923$	−16.7431	−0.551107
$924$	0	0
$925$	−18.7422	−0.616240
$926$	−46.0585	−1.51358
$927$	0	0
$928$	5.87621	0.192896
$929$	−3.52532	−0.115662	−0.0578311	−	0.998326i	$-0.518418\pi$
$929$	−3.52532	−0.115662	−0.0578311	+	0.998326i	$0.518418\pi$
$930$	0	0
$931$	−6.33861	−0.207740
$932$	−1.59045	−0.0520970
$933$	0	0
$934$	6.61436	0.216429
$935$	0.0170374	0.000557183 0
$936$	0	0
$937$	−53.1948	−1.73780	−0.868899	−	0.494989i	$-0.835172\pi$
$937$	−53.1948	−1.73780	−0.868899	+	0.494989i	$0.835172\pi$
$938$	7.50278	0.244974
$939$	0	0
$940$	0.0365832	0.00119321
$941$	−17.0900	−0.557117	−0.278559	−	0.960419i	$-0.589857\pi$
$941$	−17.0900	−0.557117	−0.278559	+	0.960419i	$0.589857\pi$
$942$	0	0
$943$	−7.01580	−0.228466
$944$	31.5880	1.02810
$945$	0	0
$946$	−0.0314648	−0.00102301
$947$	−32.8715	−1.06818	−0.534090	−	0.845428i	$-0.679345\pi$
$947$	−32.8715	−1.06818	−0.534090	+	0.845428i	$0.679345\pi$
$948$	0	0
$949$	−14.9710	−0.485980
$950$	−42.5606	−1.38085
$951$	0	0
$952$	−17.2507	−0.559099
$953$	−48.2051	−1.56152	−0.780758	−	0.624833i	$-0.785167\pi$
$953$	−48.2051	−1.56152	−0.780758	+	0.624833i	$0.785167\pi$
$954$	0	0
$955$	−2.58663	−0.0837013
$956$	1.50212	0.0485820
$957$	0	0
$958$	−45.8882	−1.48258
$959$	−14.1303	−0.456292
$960$	0	0
$961$	20.5308	0.662283
$962$	9.20021	0.296627
$963$	0	0
$964$	4.64927	0.149743
$965$	3.24420	0.104435
$966$	0	0
$967$	52.7810	1.69732	0.848661	−	0.528937i	$-0.177409\pi$
$967$	52.7810	1.69732	0.848661	+	0.528937i	$0.177409\pi$
$968$	−32.3807	−1.04076
$969$	0	0
$970$	0.647499	0.0207900
$971$	15.9585	0.512133	0.256066	−	0.966659i	$-0.417573\pi$
$971$	15.9585	0.512133	0.256066	+	0.966659i	$0.417573\pi$
$972$	0	0
$973$	−11.7203	−0.375736
$974$	24.2965	0.778511
$975$	0	0
$976$	−37.7162	−1.20727
$977$	−3.79640	−0.121457	−0.0607287	−	0.998154i	$-0.519342\pi$
$977$	−3.79640	−0.121457	−0.0607287	+	0.998154i	$0.519342\pi$
$978$	0	0
$979$	−0.228825	−0.00731328
$980$	−0.0231371	−0.000739087 0
$981$	0	0
$982$	−22.7696	−0.726607
$983$	9.99776	0.318879	0.159440	−	0.987208i	$-0.449031\pi$
$983$	9.99776	0.318879	0.159440	+	0.987208i	$0.449031\pi$
$984$	0	0
$985$	−3.03288	−0.0966356
$986$	44.3950	1.41382
$987$	0	0
$988$	−2.13359	−0.0678786
$989$	5.03452	0.160088
$990$	0	0
$991$	19.7568	0.627595	0.313798	−	0.949490i	$-0.398399\pi$
$991$	19.7568	0.627595	0.313798	+	0.949490i	$0.398399\pi$
$992$	−7.50047	−0.238140
$993$	0	0
$994$	12.4178	0.393870
$995$	2.74551	0.0870386
$996$	0	0
$997$	34.9981	1.10840	0.554201	−	0.832383i	$-0.313024\pi$
$997$	34.9981	1.10840	0.554201	+	0.832383i	$0.313024\pi$
$998$	41.2811	1.30673
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.y.1.9	✓	28
3.2	odd	2	inner	8001.2.a.y.1.20	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.y.1.9	✓	28	1.1	even	1	trivial
8001.2.a.y.1.20	yes	28	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)

\(f(q)\)	\(=\)	\(q-1.34710 q^{2} -0.185322 q^{4} +0.124848 q^{5} -1.00000 q^{7} +2.94385 q^{8} +O(q^{10})\)	\(q-1.34710 q^{2} -0.185322 q^{4} +0.124848 q^{5} -1.00000 q^{7} +2.94385 q^{8} -0.168183 q^{10} +0.0232879 q^{11} -1.81631 q^{13} +1.34710 q^{14} -3.59501 q^{16} +5.85992 q^{17} -6.33861 q^{19} -0.0231371 q^{20} -0.0313711 q^{22} +5.01952 q^{23} -4.98441 q^{25} +2.44676 q^{26} +0.185322 q^{28} -5.62396 q^{29} +7.17849 q^{31} -1.04485 q^{32} -7.89390 q^{34} -0.124848 q^{35} +3.76017 q^{37} +8.53874 q^{38} +0.367534 q^{40} -1.39770 q^{41} +1.00299 q^{43} -0.00431575 q^{44} -6.76180 q^{46} -1.58115 q^{47} +1.00000 q^{49} +6.71450 q^{50} +0.336602 q^{52} -9.43367 q^{53} +0.00290745 q^{55} -2.94385 q^{56} +7.57604 q^{58} -8.78661 q^{59} +10.4912 q^{61} -9.67015 q^{62} +8.59755 q^{64} -0.226764 q^{65} +5.56958 q^{67} -1.08597 q^{68} +0.168183 q^{70} +9.21819 q^{71} +8.24252 q^{73} -5.06532 q^{74} +1.17468 q^{76} -0.0232879 q^{77} -4.90911 q^{79} -0.448831 q^{80} +1.88284 q^{82} -1.62186 q^{83} +0.731601 q^{85} -1.35112 q^{86} +0.0685559 q^{88} -9.82593 q^{89} +1.81631 q^{91} -0.930227 q^{92} +2.12996 q^{94} -0.791364 q^{95} -3.84997 q^{97} -1.34710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) \) 28 * q + 30 * q^4 - 28 * q^7	\( 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) \) 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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