LMFDB - Embedded newform 8001.2.a.z.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+0.903971 q^{2} -1.18284 q^{4} -2.97865 q^{5} -1.00000 q^{7} -2.87719 q^{8} +O(q^{10})$	\(q+0.903971 q^{2} -1.18284 q^{4} -2.97865 q^{5} -1.00000 q^{7} -2.87719 q^{8} -2.69261 q^{10} -0.876539 q^{11} -1.42672 q^{13} -0.903971 q^{14} -0.235226 q^{16} +0.579876 q^{17} +7.77454 q^{19} +3.52326 q^{20} -0.792366 q^{22} +3.51820 q^{23} +3.87236 q^{25} -1.28972 q^{26} +1.18284 q^{28} -1.75016 q^{29} -5.47829 q^{31} +5.54175 q^{32} +0.524191 q^{34} +2.97865 q^{35} -8.19490 q^{37} +7.02796 q^{38} +8.57015 q^{40} +5.68670 q^{41} +7.37898 q^{43} +1.03680 q^{44} +3.18035 q^{46} +11.3105 q^{47} +1.00000 q^{49} +3.50050 q^{50} +1.68758 q^{52} +7.36292 q^{53} +2.61090 q^{55} +2.87719 q^{56} -1.58209 q^{58} +8.42438 q^{59} -14.0554 q^{61} -4.95221 q^{62} +5.48003 q^{64} +4.24971 q^{65} +1.17313 q^{67} -0.685899 q^{68} +2.69261 q^{70} -10.2426 q^{71} +11.1051 q^{73} -7.40795 q^{74} -9.19601 q^{76} +0.876539 q^{77} -1.45557 q^{79} +0.700655 q^{80} +5.14061 q^{82} -17.1757 q^{83} -1.72725 q^{85} +6.67039 q^{86} +2.52197 q^{88} -0.702467 q^{89} +1.42672 q^{91} -4.16145 q^{92} +10.2244 q^{94} -23.1576 q^{95} -17.4791 q^{97} +0.903971 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) $ 32 * q + 30 * q^4 - 32 * q^7	$ 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) $ 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.903971	0.639204	0.319602	−	0.947552i	$-0.396451\pi$
$2$	0.903971	0.639204	0.319602	+	0.947552i	$0.396451\pi$
$3$	0	0
$3$	0	0
$4$	−1.18284	−0.591418
$5$	−2.97865	−1.33209	−0.666046	−	0.745910i	$-0.732015\pi$
$5$	−2.97865	−1.33209	−0.666046	+	0.745910i	$0.732015\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.87719	−1.01724
$9$	0	0
$10$	−2.69261	−0.851479
$11$	−0.876539	−0.264286	−0.132143	−	0.991231i	$-0.542186\pi$
$11$	−0.876539	−0.264286	−0.132143	+	0.991231i	$0.542186\pi$
$12$	0	0
$13$	−1.42672	−0.395701	−0.197851	−	0.980232i	$-0.563396\pi$
$13$	−1.42672	−0.395701	−0.197851	+	0.980232i	$0.563396\pi$
$14$	−0.903971	−0.241596
$15$	0	0
$16$	−0.235226	−0.0588064
$17$	0.579876	0.140641	0.0703203	−	0.997524i	$-0.477598\pi$
$17$	0.579876	0.140641	0.0703203	+	0.997524i	$0.477598\pi$
$18$	0	0
$19$	7.77454	1.78360	0.891801	−	0.452427i	$-0.149442\pi$
$19$	7.77454	1.78360	0.891801	+	0.452427i	$0.149442\pi$
$20$	3.52326	0.787824
$21$	0	0
$22$	−0.792366	−0.168933
$23$	3.51820	0.733595	0.366798	−	0.930301i	$-0.380454\pi$
$23$	3.51820	0.733595	0.366798	+	0.930301i	$0.380454\pi$
$24$	0	0
$25$	3.87236	0.774471
$26$	−1.28972	−0.252934
$27$	0	0
$28$	1.18284	0.223535
$29$	−1.75016	−0.324997	−0.162498	−	0.986709i	$-0.551955\pi$
$29$	−1.75016	−0.324997	−0.162498	+	0.986709i	$0.551955\pi$
$30$	0	0
$31$	−5.47829	−0.983929	−0.491965	−	0.870615i	$-0.663721\pi$
$31$	−5.47829	−0.983929	−0.491965	+	0.870615i	$0.663721\pi$
$32$	5.54175	0.979652
$33$	0	0
$34$	0.524191	0.0898981
$35$	2.97865	0.503484
$36$	0	0
$37$	−8.19490	−1.34723	−0.673617	−	0.739081i	$-0.735260\pi$
$37$	−8.19490	−1.34723	−0.673617	+	0.739081i	$0.735260\pi$
$38$	7.02796	1.14009
$39$	0	0
$40$	8.57015	1.35506
$41$	5.68670	0.888114	0.444057	−	0.895999i	$-0.353539\pi$
$41$	5.68670	0.888114	0.444057	+	0.895999i	$0.353539\pi$
$42$	0	0
$43$	7.37898	1.12528	0.562642	−	0.826701i	$-0.309785\pi$
$43$	7.37898	1.12528	0.562642	+	0.826701i	$0.309785\pi$
$44$	1.03680	0.156304
$45$	0	0
$46$	3.18035	0.468917
$47$	11.3105	1.64980	0.824902	−	0.565275i	$-0.191230\pi$
$47$	11.3105	1.64980	0.824902	+	0.565275i	$0.191230\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.50050	0.495045
$51$	0	0
$52$	1.68758	0.234025
$53$	7.36292	1.01138	0.505688	−	0.862717i	$-0.331239\pi$
$53$	7.36292	1.01138	0.505688	+	0.862717i	$0.331239\pi$
$54$	0	0
$55$	2.61090	0.352054
$56$	2.87719	0.384481
$57$	0	0
$58$	−1.58209	−0.207739
$59$	8.42438	1.09676	0.548380	−	0.836229i	$-0.315245\pi$
$59$	8.42438	1.09676	0.548380	+	0.836229i	$0.315245\pi$
$60$	0	0
$61$	−14.0554	−1.79961	−0.899803	−	0.436296i	$-0.856290\pi$
$61$	−14.0554	−1.79961	−0.899803	+	0.436296i	$0.856290\pi$
$62$	−4.95221	−0.628932
$63$	0	0
$64$	5.48003	0.685004
$65$	4.24971	0.527111
$66$	0	0
$67$	1.17313	0.143320	0.0716602	−	0.997429i	$-0.477170\pi$
$67$	1.17313	0.143320	0.0716602	+	0.997429i	$0.477170\pi$
$68$	−0.685899	−0.0831775
$69$	0	0
$70$	2.69261	0.321829
$71$	−10.2426	−1.21558	−0.607789	−	0.794098i	$-0.707944\pi$
$71$	−10.2426	−1.21558	−0.607789	+	0.794098i	$0.707944\pi$
$72$	0	0
$73$	11.1051	1.29975	0.649874	−	0.760042i	$-0.274821\pi$
$73$	11.1051	1.29975	0.649874	+	0.760042i	$0.274821\pi$
$74$	−7.40795	−0.861157
$75$	0	0
$76$	−9.19601	−1.05485
$77$	0.876539	0.0998909
$78$	0	0
$79$	−1.45557	−0.163764	−0.0818822	−	0.996642i	$-0.526093\pi$
$79$	−1.45557	−0.163764	−0.0818822	+	0.996642i	$0.526093\pi$
$80$	0.700655	0.0783356
$81$	0	0
$82$	5.14061	0.567686
$83$	−17.1757	−1.88528	−0.942641	−	0.333807i	$-0.891667\pi$
$83$	−17.1757	−1.88528	−0.942641	+	0.333807i	$0.891667\pi$
$84$	0	0
$85$	−1.72725	−0.187346
$86$	6.67039	0.719286
$87$	0	0
$88$	2.52197	0.268843
$89$	−0.702467	−0.0744613	−0.0372307	−	0.999307i	$-0.511854\pi$
$89$	−0.702467	−0.0744613	−0.0372307	+	0.999307i	$0.511854\pi$
$90$	0	0
$91$	1.42672	0.149561
$92$	−4.16145	−0.433862
$93$	0	0
$94$	10.2244	1.05456
$95$	−23.1576	−2.37592
$96$	0	0
$97$	−17.4791	−1.77474	−0.887368	−	0.461062i	$-0.847469\pi$
$97$	−17.4791	−1.77474	−0.887368	+	0.461062i	$0.847469\pi$
$98$	0.903971	0.0913149
$99$	0	0
$100$	−4.58036	−0.458036
$101$	−3.51085	−0.349343	−0.174671	−	0.984627i	$-0.555886\pi$
$101$	−3.51085	−0.349343	−0.174671	+	0.984627i	$0.555886\pi$
$102$	0	0
$103$	−5.78567	−0.570079	−0.285039	−	0.958516i	$-0.592007\pi$
$103$	−5.78567	−0.570079	−0.285039	+	0.958516i	$0.592007\pi$
$104$	4.10495	0.402524
$105$	0	0
$106$	6.65587	0.646475
$107$	13.8595	1.33985	0.669925	−	0.742428i	$-0.266326\pi$
$107$	13.8595	1.33985	0.669925	+	0.742428i	$0.266326\pi$
$108$	0	0
$109$	−2.29482	−0.219804	−0.109902	−	0.993942i	$-0.535054\pi$
$109$	−2.29482	−0.219804	−0.109902	+	0.993942i	$0.535054\pi$
$110$	2.36018	0.225034
$111$	0	0
$112$	0.235226	0.0222267
$113$	6.62574	0.623297	0.311648	−	0.950198i	$-0.399119\pi$
$113$	6.62574	0.623297	0.311648	+	0.950198i	$0.399119\pi$
$114$	0	0
$115$	−10.4795	−0.977217
$116$	2.07015	0.192209
$117$	0	0
$118$	7.61539	0.701054
$119$	−0.579876	−0.0531572
$120$	0	0
$121$	−10.2317	−0.930153
$122$	−12.7057	−1.15032
$123$	0	0
$124$	6.47992	0.581914
$125$	3.35885	0.300425
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−6.12970	−0.541794
$129$	0	0
$130$	3.84161	0.336932
$131$	3.37363	0.294755	0.147378	−	0.989080i	$-0.452917\pi$
$131$	3.37363	0.294755	0.147378	+	0.989080i	$0.452917\pi$
$132$	0	0
$133$	−7.77454	−0.674138
$134$	1.06047	0.0916110
$135$	0	0
$136$	−1.66842	−0.143065
$137$	−16.2096	−1.38488	−0.692439	−	0.721476i	$-0.743464\pi$
$137$	−16.2096	−1.38488	−0.692439	+	0.721476i	$0.743464\pi$
$138$	0	0
$139$	12.8614	1.09089	0.545445	−	0.838146i	$-0.316361\pi$
$139$	12.8614	1.09089	0.545445	+	0.838146i	$0.316361\pi$
$140$	−3.52326	−0.297769
$141$	0	0
$142$	−9.25906	−0.777003
$143$	1.25058	0.104579
$144$	0	0
$145$	5.21312	0.432926
$146$	10.0387	0.830805
$147$	0	0
$148$	9.69323	0.796778
$149$	13.9566	1.14337	0.571685	−	0.820473i	$-0.306290\pi$
$149$	13.9566	1.14337	0.571685	+	0.820473i	$0.306290\pi$
$150$	0	0
$151$	−1.84657	−0.150272	−0.0751360	−	0.997173i	$-0.523939\pi$
$151$	−1.84657	−0.150272	−0.0751360	+	0.997173i	$0.523939\pi$
$152$	−22.3689	−1.81435
$153$	0	0
$154$	0.792366	0.0638507
$155$	16.3179	1.31069
$156$	0	0
$157$	8.49539	0.678006	0.339003	−	0.940785i	$-0.389910\pi$
$157$	8.49539	0.678006	0.339003	+	0.940785i	$0.389910\pi$
$158$	−1.31579	−0.104679
$159$	0	0
$160$	−16.5069	−1.30499
$161$	−3.51820	−0.277273
$162$	0	0
$163$	−14.0518	−1.10062	−0.550311	−	0.834960i	$-0.685491\pi$
$163$	−14.0518	−1.10062	−0.550311	+	0.834960i	$0.685491\pi$
$164$	−6.72644	−0.525247
$165$	0	0
$166$	−15.5264	−1.20508
$167$	21.9372	1.69755	0.848776	−	0.528753i	$-0.177340\pi$
$167$	21.9372	1.69755	0.848776	+	0.528753i	$0.177340\pi$
$168$	0	0
$169$	−10.9645	−0.843420
$170$	−1.56138	−0.119753
$171$	0	0
$172$	−8.72813	−0.665513
$173$	−8.28652	−0.630013	−0.315006	−	0.949090i	$-0.602007\pi$
$173$	−8.28652	−0.630013	−0.315006	+	0.949090i	$0.602007\pi$
$174$	0	0
$175$	−3.87236	−0.292723
$176$	0.206184	0.0155417
$177$	0	0
$178$	−0.635009	−0.0475960
$179$	−2.62195	−0.195974	−0.0979870	−	0.995188i	$-0.531240\pi$
$179$	−2.62195	−0.195974	−0.0979870	+	0.995188i	$0.531240\pi$
$180$	0	0
$181$	−18.9051	−1.40520	−0.702601	−	0.711584i	$-0.747978\pi$
$181$	−18.9051	−1.40520	−0.702601	+	0.711584i	$0.747978\pi$
$182$	1.28972	0.0956001
$183$	0	0
$184$	−10.1225	−0.746243
$185$	24.4097	1.79464
$186$	0	0
$187$	−0.508284	−0.0371694
$188$	−13.3785	−0.975724
$189$	0	0
$190$	−20.9338	−1.51870
$191$	8.82348	0.638445	0.319222	−	0.947680i	$-0.396578\pi$
$191$	8.82348	0.638445	0.319222	+	0.947680i	$0.396578\pi$
$192$	0	0
$193$	16.4173	1.18174	0.590872	−	0.806765i	$-0.298784\pi$
$193$	16.4173	1.18174	0.590872	+	0.806765i	$0.298784\pi$
$194$	−15.8006	−1.13442
$195$	0	0
$196$	−1.18284	−0.0844883
$197$	−23.4141	−1.66818	−0.834092	−	0.551626i	$-0.814008\pi$
$197$	−23.4141	−1.66818	−0.834092	+	0.551626i	$0.814008\pi$
$198$	0	0
$199$	6.84358	0.485128	0.242564	−	0.970135i	$-0.422011\pi$
$199$	6.84358	0.485128	0.242564	+	0.970135i	$0.422011\pi$
$200$	−11.1415	−0.787824
$201$	0	0
$202$	−3.17371	−0.223301
$203$	1.75016	0.122837
$204$	0	0
$205$	−16.9387	−1.18305
$206$	−5.23008	−0.364397
$207$	0	0
$208$	0.335602	0.0232698
$209$	−6.81469	−0.471382
$210$	0	0
$211$	15.1368	1.04206	0.521029	−	0.853539i	$-0.325548\pi$
$211$	15.1368	1.04206	0.521029	+	0.853539i	$0.325548\pi$
$212$	−8.70913	−0.598146
$213$	0	0
$214$	12.5286	0.856438
$215$	−21.9794	−1.49898
$216$	0	0
$217$	5.47829	0.371890
$218$	−2.07445	−0.140499
$219$	0	0
$220$	−3.08827	−0.208211
$221$	−0.827322	−0.0556517
$222$	0	0
$223$	−13.0515	−0.873993	−0.436997	−	0.899463i	$-0.643958\pi$
$223$	−13.0515	−0.873993	−0.436997	+	0.899463i	$0.643958\pi$
$224$	−5.54175	−0.370274
$225$	0	0
$226$	5.98947	0.398414
$227$	−20.8243	−1.38216	−0.691078	−	0.722780i	$-0.742864\pi$
$227$	−20.8243	−1.38216	−0.691078	+	0.722780i	$0.742864\pi$
$228$	0	0
$229$	−20.9369	−1.38355	−0.691776	−	0.722113i	$-0.743171\pi$
$229$	−20.9369	−1.38355	−0.691776	+	0.722113i	$0.743171\pi$
$230$	−9.47315	−0.624641
$231$	0	0
$232$	5.03555	0.330600
$233$	−11.8482	−0.776200	−0.388100	−	0.921617i	$-0.626868\pi$
$233$	−11.8482	−0.776200	−0.388100	+	0.921617i	$0.626868\pi$
$234$	0	0
$235$	−33.6900	−2.19769
$236$	−9.96466	−0.648644
$237$	0	0
$238$	−0.524191	−0.0339783
$239$	8.77917	0.567877	0.283938	−	0.958842i	$-0.408359\pi$
$239$	8.77917	0.567877	0.283938	+	0.958842i	$0.408359\pi$
$240$	0	0
$241$	−4.46764	−0.287786	−0.143893	−	0.989593i	$-0.545962\pi$
$241$	−4.46764	−0.287786	−0.143893	+	0.989593i	$0.545962\pi$
$242$	−9.24914	−0.594557
$243$	0	0
$244$	16.6252	1.06432
$245$	−2.97865	−0.190299
$246$	0	0
$247$	−11.0921	−0.705774
$248$	15.7621	1.00089
$249$	0	0
$250$	3.03631	0.192033
$251$	23.4635	1.48101	0.740503	−	0.672053i	$-0.234587\pi$
$251$	23.4635	1.48101	0.740503	+	0.672053i	$0.234587\pi$
$252$	0	0
$253$	−3.08384	−0.193879
$254$	−0.903971	−0.0567202
$255$	0	0
$256$	−16.5011	−1.03132
$257$	−24.0009	−1.49714	−0.748569	−	0.663057i	$-0.769259\pi$
$257$	−24.0009	−1.49714	−0.748569	+	0.663057i	$0.769259\pi$
$258$	0	0
$259$	8.19490	0.509206
$260$	−5.02671	−0.311743
$261$	0	0
$262$	3.04966	0.188409
$263$	−4.99373	−0.307926	−0.153963	−	0.988077i	$-0.549204\pi$
$263$	−4.99373	−0.307926	−0.153963	+	0.988077i	$0.549204\pi$
$264$	0	0
$265$	−21.9316	−1.34725
$266$	−7.02796	−0.430912
$267$	0	0
$268$	−1.38762	−0.0847623
$269$	−12.9283	−0.788250	−0.394125	−	0.919057i	$-0.628952\pi$
$269$	−12.9283	−0.788250	−0.394125	+	0.919057i	$0.628952\pi$
$270$	0	0
$271$	−23.0369	−1.39939	−0.699695	−	0.714441i	$-0.746681\pi$
$271$	−23.0369	−1.39939	−0.699695	+	0.714441i	$0.746681\pi$
$272$	−0.136402	−0.00827057
$273$	0	0
$274$	−14.6530	−0.885220
$275$	−3.39427	−0.204682
$276$	0	0
$277$	21.4193	1.28696	0.643481	−	0.765462i	$-0.277490\pi$
$277$	21.4193	1.28696	0.643481	+	0.765462i	$0.277490\pi$
$278$	11.6263	0.697302
$279$	0	0
$280$	−8.57015	−0.512164
$281$	−9.53726	−0.568945	−0.284473	−	0.958684i	$-0.591818\pi$
$281$	−9.53726	−0.568945	−0.284473	+	0.958684i	$0.591818\pi$
$282$	0	0
$283$	−15.7458	−0.935992	−0.467996	−	0.883731i	$-0.655024\pi$
$283$	−15.7458	−0.935992	−0.467996	+	0.883731i	$0.655024\pi$
$284$	12.1154	0.718915
$285$	0	0
$286$	1.13049	0.0668470
$287$	−5.68670	−0.335675
$288$	0	0
$289$	−16.6637	−0.980220
$290$	4.71251	0.276728
$291$	0	0
$292$	−13.1355	−0.768695
$293$	−30.0033	−1.75281	−0.876407	−	0.481572i	$-0.840066\pi$
$293$	−30.0033	−1.75281	−0.876407	+	0.481572i	$0.840066\pi$
$294$	0	0
$295$	−25.0933	−1.46099
$296$	23.5783	1.37046
$297$	0	0
$298$	12.6164	0.730847
$299$	−5.01949	−0.290285
$300$	0	0
$301$	−7.37898	−0.425317
$302$	−1.66925	−0.0960545
$303$	0	0
$304$	−1.82877	−0.104887
$305$	41.8660	2.39724
$306$	0	0
$307$	22.2703	1.27103	0.635515	−	0.772088i	$-0.280788\pi$
$307$	22.2703	1.27103	0.635515	+	0.772088i	$0.280788\pi$
$308$	−1.03680	−0.0590773
$309$	0	0
$310$	14.7509	0.837795
$311$	4.72911	0.268163	0.134082	−	0.990970i	$-0.457192\pi$
$311$	4.72911	0.268163	0.134082	+	0.990970i	$0.457192\pi$
$312$	0	0
$313$	−4.61634	−0.260931	−0.130465	−	0.991453i	$-0.541647\pi$
$313$	−4.61634	−0.260931	−0.130465	+	0.991453i	$0.541647\pi$
$314$	7.67959	0.433384
$315$	0	0
$316$	1.72170	0.0968533
$317$	6.89241	0.387117	0.193558	−	0.981089i	$-0.437997\pi$
$317$	6.89241	0.387117	0.193558	+	0.981089i	$0.437997\pi$
$318$	0	0
$319$	1.53408	0.0858922
$320$	−16.3231	−0.912489
$321$	0	0
$322$	−3.18035	−0.177234
$323$	4.50827	0.250847
$324$	0	0
$325$	−5.52478	−0.306459
$326$	−12.7024	−0.703522
$327$	0	0
$328$	−16.3617	−0.903426
$329$	−11.3105	−0.623567
$330$	0	0
$331$	12.1593	0.668338	0.334169	−	0.942513i	$-0.391544\pi$
$331$	12.1593	0.668338	0.334169	+	0.942513i	$0.391544\pi$
$332$	20.3161	1.11499
$333$	0	0
$334$	19.8306	1.08508
$335$	−3.49434	−0.190916
$336$	0	0
$337$	0.169189	0.00921633	0.00460817	−	0.999989i	$-0.498533\pi$
$337$	0.169189	0.00921633	0.00460817	+	0.999989i	$0.498533\pi$
$338$	−9.91156	−0.539118
$339$	0	0
$340$	2.04305	0.110800
$341$	4.80193	0.260039
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−21.2307	−1.14469
$345$	0	0
$346$	−7.49078	−0.402707
$347$	6.13890	0.329553	0.164777	−	0.986331i	$-0.447310\pi$
$347$	6.13890	0.329553	0.164777	+	0.986331i	$0.447310\pi$
$348$	0	0
$349$	27.6545	1.48031	0.740157	−	0.672434i	$-0.234751\pi$
$349$	27.6545	1.48031	0.740157	+	0.672434i	$0.234751\pi$
$350$	−3.50050	−0.187110
$351$	0	0
$352$	−4.85756	−0.258909
$353$	−27.7604	−1.47754	−0.738768	−	0.673959i	$-0.764592\pi$
$353$	−27.7604	−1.47754	−0.738768	+	0.673959i	$0.764592\pi$
$354$	0	0
$355$	30.5093	1.61926
$356$	0.830903	0.0440378
$357$	0	0
$358$	−2.37017	−0.125267
$359$	4.10568	0.216689	0.108345	−	0.994113i	$-0.465445\pi$
$359$	4.10568	0.216689	0.108345	+	0.994113i	$0.465445\pi$
$360$	0	0
$361$	41.4435	2.18124
$362$	−17.0896	−0.898211
$363$	0	0
$364$	−1.68758	−0.0884531
$365$	−33.0781	−1.73139
$366$	0	0
$367$	−25.1833	−1.31456	−0.657278	−	0.753648i	$-0.728292\pi$
$367$	−25.1833	−1.31456	−0.657278	+	0.753648i	$0.728292\pi$
$368$	−0.827571	−0.0431401
$369$	0	0
$370$	22.0657	1.14714
$371$	−7.36292	−0.382264
$372$	0	0
$373$	25.8322	1.33754	0.668770	−	0.743469i	$-0.266821\pi$
$373$	25.8322	1.33754	0.668770	+	0.743469i	$0.266821\pi$
$374$	−0.459474	−0.0237589
$375$	0	0
$376$	−32.5424	−1.67825
$377$	2.49699	0.128602
$378$	0	0
$379$	−3.13475	−0.161021	−0.0805107	−	0.996754i	$-0.525655\pi$
$379$	−3.13475	−0.161021	−0.0805107	+	0.996754i	$0.525655\pi$
$380$	27.3917	1.40516
$381$	0	0
$382$	7.97618	0.408097
$383$	29.1214	1.48804	0.744018	−	0.668159i	$-0.232918\pi$
$383$	29.1214	1.48804	0.744018	+	0.668159i	$0.232918\pi$
$384$	0	0
$385$	−2.61090	−0.133064
$386$	14.8408	0.755376
$387$	0	0
$388$	20.6749	1.04961
$389$	0.482130	0.0244449	0.0122225	−	0.999925i	$-0.496109\pi$
$389$	0.482130	0.0244449	0.0122225	+	0.999925i	$0.496109\pi$
$390$	0	0
$391$	2.04012	0.103173
$392$	−2.87719	−0.145320
$393$	0	0
$394$	−21.1656	−1.06631
$395$	4.33564	0.218150
$396$	0	0
$397$	−34.2111	−1.71701	−0.858503	−	0.512808i	$-0.828605\pi$
$397$	−34.2111	−1.71701	−0.858503	+	0.512808i	$0.828605\pi$
$398$	6.18640	0.310096
$399$	0	0
$400$	−0.910878	−0.0455439
$401$	15.7020	0.784123	0.392061	−	0.919939i	$-0.371762\pi$
$401$	15.7020	0.784123	0.392061	+	0.919939i	$0.371762\pi$
$402$	0	0
$403$	7.81599	0.389342
$404$	4.15276	0.206608
$405$	0	0
$406$	1.58209	0.0785180
$407$	7.18315	0.356056
$408$	0	0
$409$	−18.6655	−0.922948	−0.461474	−	0.887154i	$-0.652679\pi$
$409$	−18.6655	−0.922948	−0.461474	+	0.887154i	$0.652679\pi$
$410$	−15.3121	−0.756210
$411$	0	0
$412$	6.84350	0.337155
$413$	−8.42438	−0.414537
$414$	0	0
$415$	51.1605	2.51137
$416$	−7.90653	−0.387650
$417$	0	0
$418$	−6.16028	−0.301309
$419$	31.6499	1.54620	0.773100	−	0.634284i	$-0.218705\pi$
$419$	31.6499	1.54620	0.773100	+	0.634284i	$0.218705\pi$
$420$	0	0
$421$	−5.12267	−0.249664	−0.124832	−	0.992178i	$-0.539839\pi$
$421$	−5.12267	−0.249664	−0.124832	+	0.992178i	$0.539839\pi$
$422$	13.6832	0.666088
$423$	0	0
$424$	−21.1845	−1.02881
$425$	2.24549	0.108922
$426$	0	0
$427$	14.0554	0.680187
$428$	−16.3936	−0.792412
$429$	0	0
$430$	−19.8687	−0.958156
$431$	26.8900	1.29525	0.647623	−	0.761961i	$-0.275763\pi$
$431$	26.8900	1.29525	0.647623	+	0.761961i	$0.275763\pi$
$432$	0	0
$433$	16.9919	0.816577	0.408289	−	0.912853i	$-0.366126\pi$
$433$	16.9919	0.816577	0.408289	+	0.912853i	$0.366126\pi$
$434$	4.95221	0.237714
$435$	0	0
$436$	2.71439	0.129996
$437$	27.3524	1.30844
$438$	0	0
$439$	−12.5338	−0.598207	−0.299104	−	0.954221i	$-0.596688\pi$
$439$	−12.5338	−0.598207	−0.299104	+	0.954221i	$0.596688\pi$
$440$	−7.51207	−0.358124
$441$	0	0
$442$	−0.747875	−0.0355728
$443$	−17.0518	−0.810154	−0.405077	−	0.914283i	$-0.632755\pi$
$443$	−17.0518	−0.810154	−0.405077	+	0.914283i	$0.632755\pi$
$444$	0	0
$445$	2.09240	0.0991894
$446$	−11.7982	−0.558660
$447$	0	0
$448$	−5.48003	−0.258907
$449$	−14.3616	−0.677766	−0.338883	−	0.940829i	$-0.610049\pi$
$449$	−14.3616	−0.677766	−0.338883	+	0.940829i	$0.610049\pi$
$450$	0	0
$451$	−4.98462	−0.234716
$452$	−7.83716	−0.368629
$453$	0	0
$454$	−18.8246	−0.883480
$455$	−4.24971	−0.199229
$456$	0	0
$457$	2.47963	0.115992	0.0579960	−	0.998317i	$-0.481529\pi$
$457$	2.47963	0.115992	0.0579960	+	0.998317i	$0.481529\pi$
$458$	−18.9264	−0.884372
$459$	0	0
$460$	12.3955	0.577944
$461$	−8.87317	−0.413265	−0.206632	−	0.978419i	$-0.566250\pi$
$461$	−8.87317	−0.413265	−0.206632	+	0.978419i	$0.566250\pi$
$462$	0	0
$463$	−27.1704	−1.26271	−0.631357	−	0.775492i	$-0.717502\pi$
$463$	−27.1704	−1.26271	−0.631357	+	0.775492i	$0.717502\pi$
$464$	0.411683	0.0191119
$465$	0	0
$466$	−10.7104	−0.496150
$467$	−6.99157	−0.323531	−0.161766	−	0.986829i	$-0.551719\pi$
$467$	−6.99157	−0.323531	−0.161766	+	0.986829i	$0.551719\pi$
$468$	0	0
$469$	−1.17313	−0.0541700
$470$	−30.4548	−1.40477
$471$	0	0
$472$	−24.2386	−1.11567
$473$	−6.46797	−0.297397
$474$	0	0
$475$	30.1058	1.38135
$476$	0.685899	0.0314381
$477$	0	0
$478$	7.93611	0.362989
$479$	19.6939	0.899837	0.449919	−	0.893070i	$-0.351453\pi$
$479$	19.6939	0.899837	0.449919	+	0.893070i	$0.351453\pi$
$480$	0	0
$481$	11.6918	0.533102
$482$	−4.03862	−0.183954
$483$	0	0
$484$	12.1024	0.550109
$485$	52.0642	2.36411
$486$	0	0
$487$	−22.9375	−1.03940	−0.519700	−	0.854349i	$-0.673956\pi$
$487$	−22.9375	−1.03940	−0.519700	+	0.854349i	$0.673956\pi$
$488$	40.4400	1.83063
$489$	0	0
$490$	−2.69261	−0.121640
$491$	40.5985	1.83218	0.916092	−	0.400967i	$-0.131326\pi$
$491$	40.5985	1.83218	0.916092	+	0.400967i	$0.131326\pi$
$492$	0	0
$493$	−1.01488	−0.0457078
$494$	−10.0269	−0.451134
$495$	0	0
$496$	1.28863	0.0578614
$497$	10.2426	0.459446
$498$	0	0
$499$	−15.2512	−0.682739	−0.341369	−	0.939929i	$-0.610891\pi$
$499$	−15.2512	−0.682739	−0.341369	+	0.939929i	$0.610891\pi$
$500$	−3.97297	−0.177677
$501$	0	0
$502$	21.2104	0.946665
$503$	−13.4918	−0.601571	−0.300786	−	0.953692i	$-0.597249\pi$
$503$	−13.4918	−0.601571	−0.300786	+	0.953692i	$0.597249\pi$
$504$	0	0
$505$	10.4576	0.465357
$506$	−2.78770	−0.123928
$507$	0	0
$508$	1.18284	0.0524799
$509$	−5.42460	−0.240441	−0.120221	−	0.992747i	$-0.538360\pi$
$509$	−5.42460	−0.240441	−0.120221	+	0.992747i	$0.538360\pi$
$510$	0	0
$511$	−11.1051	−0.491259
$512$	−2.65714	−0.117430
$513$	0	0
$514$	−21.6962	−0.956976
$515$	17.2335	0.759398
$516$	0	0
$517$	−9.91409	−0.436021
$518$	7.40795	0.325487
$519$	0	0
$520$	−12.2272	−0.536199
$521$	−31.9886	−1.40145	−0.700723	−	0.713433i	$-0.747139\pi$
$521$	−31.9886	−1.40145	−0.700723	+	0.713433i	$0.747139\pi$
$522$	0	0
$523$	−42.1807	−1.84443	−0.922217	−	0.386672i	$-0.873625\pi$
$523$	−42.1807	−1.84443	−0.922217	+	0.386672i	$0.873625\pi$
$524$	−3.99045	−0.174324
$525$	0	0
$526$	−4.51418	−0.196828
$527$	−3.17673	−0.138380
$528$	0	0
$529$	−10.6223	−0.461838
$530$	−19.8255	−0.861165
$531$	0	0
$532$	9.19601	0.398698
$533$	−8.11334	−0.351428
$534$	0	0
$535$	−41.2827	−1.78481
$536$	−3.37531	−0.145791
$537$	0	0
$538$	−11.6868	−0.503852
$539$	−0.876539	−0.0377552
$540$	0	0
$541$	−39.4783	−1.69731	−0.848653	−	0.528950i	$-0.822586\pi$
$541$	−39.4783	−1.69731	−0.848653	+	0.528950i	$0.822586\pi$
$542$	−20.8247	−0.894496
$543$	0	0
$544$	3.21353	0.137779
$545$	6.83546	0.292799
$546$	0	0
$547$	−6.56204	−0.280573	−0.140286	−	0.990111i	$-0.544802\pi$
$547$	−6.56204	−0.280573	−0.140286	+	0.990111i	$0.544802\pi$
$548$	19.1733	0.819042
$549$	0	0
$550$	−3.06832	−0.130834
$551$	−13.6067	−0.579665
$552$	0	0
$553$	1.45557	0.0618972
$554$	19.3624	0.822631
$555$	0	0
$556$	−15.2129	−0.645173
$557$	−8.00375	−0.339130	−0.169565	−	0.985519i	$-0.554236\pi$
$557$	−8.00375	−0.339130	−0.169565	+	0.985519i	$0.554236\pi$
$558$	0	0
$559$	−10.5278	−0.445277
$560$	−0.700655	−0.0296081
$561$	0	0
$562$	−8.62141	−0.363672
$563$	17.6695	0.744680	0.372340	−	0.928096i	$-0.378556\pi$
$563$	17.6695	0.744680	0.372340	+	0.928096i	$0.378556\pi$
$564$	0	0
$565$	−19.7357	−0.830289
$566$	−14.2338	−0.598290
$567$	0	0
$568$	29.4701	1.23654
$569$	28.7821	1.20661	0.603305	−	0.797510i	$-0.293850\pi$
$569$	28.7821	1.20661	0.603305	+	0.797510i	$0.293850\pi$
$570$	0	0
$571$	12.4448	0.520798	0.260399	−	0.965501i	$-0.416146\pi$
$571$	12.4448	0.520798	0.260399	+	0.965501i	$0.416146\pi$
$572$	−1.47923	−0.0618496
$573$	0	0
$574$	−5.14061	−0.214565
$575$	13.6237	0.568148
$576$	0	0
$577$	46.9052	1.95269	0.976344	−	0.216222i	$-0.0693736\pi$
$577$	46.9052	1.95269	0.976344	+	0.216222i	$0.0693736\pi$
$578$	−15.0635	−0.626561
$579$	0	0
$580$	−6.16626	−0.256040
$581$	17.1757	0.712570
$582$	0	0
$583$	−6.45389	−0.267293
$584$	−31.9514	−1.32216
$585$	0	0
$586$	−27.1221	−1.12041
$587$	−11.4699	−0.473414	−0.236707	−	0.971581i	$-0.576068\pi$
$587$	−11.4699	−0.473414	−0.236707	+	0.971581i	$0.576068\pi$
$588$	0	0
$589$	−42.5912	−1.75494
$590$	−22.6836	−0.933869
$591$	0	0
$592$	1.92765	0.0792260
$593$	12.6400	0.519061	0.259530	−	0.965735i	$-0.416432\pi$
$593$	12.6400	0.519061	0.259530	+	0.965735i	$0.416432\pi$
$594$	0	0
$595$	1.72725	0.0708103
$596$	−16.5084	−0.676210
$597$	0	0
$598$	−4.53748	−0.185551
$599$	5.99492	0.244946	0.122473	−	0.992472i	$-0.460918\pi$
$599$	5.99492	0.244946	0.122473	+	0.992472i	$0.460918\pi$
$600$	0	0
$601$	40.2347	1.64121	0.820604	−	0.571498i	$-0.193637\pi$
$601$	40.2347	1.64121	0.820604	+	0.571498i	$0.193637\pi$
$602$	−6.67039	−0.271865
$603$	0	0
$604$	2.18419	0.0888736
$605$	30.4766	1.23905
$606$	0	0
$607$	−22.8886	−0.929020	−0.464510	−	0.885568i	$-0.653770\pi$
$607$	−22.8886	−0.929020	−0.464510	+	0.885568i	$0.653770\pi$
$608$	43.0846	1.74731
$609$	0	0
$610$	37.8457	1.53233
$611$	−16.1369	−0.652830
$612$	0	0
$613$	−44.0392	−1.77872	−0.889362	−	0.457203i	$-0.848851\pi$
$613$	−44.0392	−1.77872	−0.889362	+	0.457203i	$0.848851\pi$
$614$	20.1317	0.812448
$615$	0	0
$616$	−2.52197	−0.101613
$617$	25.7540	1.03682	0.518408	−	0.855134i	$-0.326525\pi$
$617$	25.7540	1.03682	0.518408	+	0.855134i	$0.326525\pi$
$618$	0	0
$619$	−1.87469	−0.0753501	−0.0376750	−	0.999290i	$-0.511995\pi$
$619$	−1.87469	−0.0753501	−0.0376750	+	0.999290i	$0.511995\pi$
$620$	−19.3014	−0.775163
$621$	0	0
$622$	4.27498	0.171411
$623$	0.702467	0.0281437
$624$	0	0
$625$	−29.3666	−1.17467
$626$	−4.17304	−0.166788
$627$	0	0
$628$	−10.0487	−0.400985
$629$	−4.75203	−0.189476
$630$	0	0
$631$	13.6693	0.544167	0.272083	−	0.962274i	$-0.412287\pi$
$631$	13.6693	0.544167	0.272083	+	0.962274i	$0.412287\pi$
$632$	4.18796	0.166588
$633$	0	0
$634$	6.23054	0.247446
$635$	2.97865	0.118204
$636$	0	0
$637$	−1.42672	−0.0565288
$638$	1.38677	0.0549027
$639$	0	0
$640$	18.2582	0.721721
$641$	−3.35578	−0.132545	−0.0662726	−	0.997802i	$-0.521111\pi$
$641$	−3.35578	−0.132545	−0.0662726	+	0.997802i	$0.521111\pi$
$642$	0	0
$643$	37.4676	1.47758	0.738788	−	0.673938i	$-0.235398\pi$
$643$	37.4676	1.47758	0.738788	+	0.673938i	$0.235398\pi$
$644$	4.16145	0.163984
$645$	0	0
$646$	4.07535	0.160342
$647$	22.3128	0.877208	0.438604	−	0.898680i	$-0.355473\pi$
$647$	22.3128	0.877208	0.438604	+	0.898680i	$0.355473\pi$
$648$	0	0
$649$	−7.38430	−0.289859
$650$	−4.99424	−0.195890
$651$	0	0
$652$	16.6210	0.650927
$653$	5.24950	0.205429	0.102714	−	0.994711i	$-0.467247\pi$
$653$	5.24950	0.205429	0.102714	+	0.994711i	$0.467247\pi$
$654$	0	0
$655$	−10.0489	−0.392641
$656$	−1.33766	−0.0522268
$657$	0	0
$658$	−10.2244	−0.398587
$659$	23.8972	0.930904	0.465452	−	0.885073i	$-0.345892\pi$
$659$	23.8972	0.930904	0.465452	+	0.885073i	$0.345892\pi$
$660$	0	0
$661$	−31.8842	−1.24015	−0.620076	−	0.784541i	$-0.712898\pi$
$661$	−31.8842	−1.24015	−0.620076	+	0.784541i	$0.712898\pi$
$662$	10.9917	0.427204
$663$	0	0
$664$	49.4179	1.91779
$665$	23.1576	0.898015
$666$	0	0
$667$	−6.15741	−0.238416
$668$	−25.9481	−1.00396
$669$	0	0
$670$	−3.15878	−0.122034
$671$	12.3201	0.475612
$672$	0	0
$673$	29.9344	1.15389	0.576943	−	0.816785i	$-0.304246\pi$
$673$	29.9344	1.15389	0.576943	+	0.816785i	$0.304246\pi$
$674$	0.152942	0.00589112
$675$	0	0
$676$	12.9692	0.498814
$677$	−10.5137	−0.404076	−0.202038	−	0.979378i	$-0.564756\pi$
$677$	−10.5137	−0.404076	−0.202038	+	0.979378i	$0.564756\pi$
$678$	0	0
$679$	17.4791	0.670787
$680$	4.96963	0.190576
$681$	0	0
$682$	4.34081	0.166218
$683$	3.60843	0.138073	0.0690363	−	0.997614i	$-0.478008\pi$
$683$	3.60843	0.138073	0.0690363	+	0.997614i	$0.478008\pi$
$684$	0	0
$685$	48.2827	1.84479
$686$	−0.903971	−0.0345138
$687$	0	0
$688$	−1.73573	−0.0661739
$689$	−10.5048	−0.400203
$690$	0	0
$691$	−29.5114	−1.12267	−0.561334	−	0.827589i	$-0.689712\pi$
$691$	−29.5114	−1.12267	−0.561334	+	0.827589i	$0.689712\pi$
$692$	9.80160	0.372601
$693$	0	0
$694$	5.54939	0.210652
$695$	−38.3096	−1.45317
$696$	0	0
$697$	3.29758	0.124905
$698$	24.9989	0.946222
$699$	0	0
$700$	4.58036	0.173121
$701$	28.7494	1.08585	0.542924	−	0.839782i	$-0.317317\pi$
$701$	28.7494	1.08585	0.542924	+	0.839782i	$0.317317\pi$
$702$	0	0
$703$	−63.7116	−2.40293
$704$	−4.80346	−0.181037
$705$	0	0
$706$	−25.0946	−0.944448
$707$	3.51085	0.132039
$708$	0	0
$709$	−14.2070	−0.533555	−0.266777	−	0.963758i	$-0.585959\pi$
$709$	−14.2070	−0.533555	−0.266777	+	0.963758i	$0.585959\pi$
$710$	27.5795	1.03504
$711$	0	0
$712$	2.02113	0.0757451
$713$	−19.2737	−0.721806
$714$	0	0
$715$	−3.72503	−0.139308
$716$	3.10134	0.115903
$717$	0	0
$718$	3.71142	0.138509
$719$	−49.3434	−1.84020	−0.920098	−	0.391688i	$-0.871891\pi$
$719$	−49.3434	−1.84020	−0.920098	+	0.391688i	$0.871891\pi$
$720$	0	0
$721$	5.78567	0.215470
$722$	37.4638	1.39426
$723$	0	0
$724$	22.3616	0.831062
$725$	−6.77725	−0.251701
$726$	0	0
$727$	−16.1812	−0.600128	−0.300064	−	0.953919i	$-0.597008\pi$
$727$	−16.1812	−0.600128	−0.300064	+	0.953919i	$0.597008\pi$
$728$	−4.10495	−0.152140
$729$	0	0
$730$	−29.9016	−1.10671
$731$	4.27890	0.158261
$732$	0	0
$733$	38.5103	1.42241	0.711206	−	0.702984i	$-0.248149\pi$
$733$	38.5103	1.42241	0.711206	+	0.702984i	$0.248149\pi$
$734$	−22.7649	−0.840270
$735$	0	0
$736$	19.4970	0.718668
$737$	−1.02829	−0.0378776
$738$	0	0
$739$	−34.7101	−1.27683	−0.638416	−	0.769692i	$-0.720410\pi$
$739$	−34.7101	−1.27683	−0.638416	+	0.769692i	$0.720410\pi$
$740$	−28.8727	−1.06138
$741$	0	0
$742$	−6.65587	−0.244345
$743$	−31.5544	−1.15762	−0.578810	−	0.815462i	$-0.696483\pi$
$743$	−31.5544	−1.15762	−0.578810	+	0.815462i	$0.696483\pi$
$744$	0	0
$745$	−41.5719	−1.52308
$746$	23.3516	0.854962
$747$	0	0
$748$	0.601217	0.0219827
$749$	−13.8595	−0.506416
$750$	0	0
$751$	19.5122	0.712011	0.356005	−	0.934484i	$-0.384139\pi$
$751$	19.5122	0.712011	0.356005	+	0.934484i	$0.384139\pi$
$752$	−2.66052	−0.0970191
$753$	0	0
$754$	2.25721	0.0822027
$755$	5.50030	0.200176
$756$	0	0
$757$	17.6410	0.641174	0.320587	−	0.947219i	$-0.396120\pi$
$757$	17.6410	0.641174	0.320587	+	0.947219i	$0.396120\pi$
$758$	−2.83372	−0.102925
$759$	0	0
$760$	66.6290	2.41689
$761$	14.6906	0.532534	0.266267	−	0.963899i	$-0.414210\pi$
$761$	14.6906	0.532534	0.266267	+	0.963899i	$0.414210\pi$
$762$	0	0
$763$	2.29482	0.0830780
$764$	−10.4367	−0.377588
$765$	0	0
$766$	26.3249	0.951159
$767$	−12.0192	−0.433990
$768$	0	0
$769$	−28.0912	−1.01299	−0.506497	−	0.862242i	$-0.669060\pi$
$769$	−28.0912	−1.01299	−0.506497	+	0.862242i	$0.669060\pi$
$770$	−2.36018	−0.0850550
$771$	0	0
$772$	−19.4190	−0.698905
$773$	35.2304	1.26715	0.633574	−	0.773682i	$-0.281587\pi$
$773$	35.2304	1.26715	0.633574	+	0.773682i	$0.281587\pi$
$774$	0	0
$775$	−21.2139	−0.762025
$776$	50.2908	1.80533
$777$	0	0
$778$	0.435831	0.0156253
$779$	44.2115	1.58404
$780$	0	0
$781$	8.97808	0.321261
$782$	1.84421	0.0659488
$783$	0	0
$784$	−0.235226	−0.00840092
$785$	−25.3048	−0.903167
$786$	0	0
$787$	31.1134	1.10907	0.554537	−	0.832159i	$-0.312896\pi$
$787$	31.1134	1.10907	0.554537	+	0.832159i	$0.312896\pi$
$788$	27.6950	0.986594
$789$	0	0
$790$	3.91929	0.139442
$791$	−6.62574	−0.235584
$792$	0	0
$793$	20.0531	0.712107
$794$	−30.9259	−1.09752
$795$	0	0
$796$	−8.09483	−0.286914
$797$	21.6089	0.765428	0.382714	−	0.923867i	$-0.374989\pi$
$797$	21.6089	0.765428	0.382714	+	0.923867i	$0.374989\pi$
$798$	0	0
$799$	6.55869	0.232030
$800$	21.4596	0.758712
$801$	0	0
$802$	14.1942	0.501214
$803$	−9.73402	−0.343506
$804$	0	0
$805$	10.4795	0.369353
$806$	7.06543	0.248869
$807$	0	0
$808$	10.1014	0.355366
$809$	−13.2348	−0.465310	−0.232655	−	0.972559i	$-0.574741\pi$
$809$	−13.2348	−0.465310	−0.232655	+	0.972559i	$0.574741\pi$
$810$	0	0
$811$	−6.57091	−0.230736	−0.115368	−	0.993323i	$-0.536805\pi$
$811$	−6.57091	−0.230736	−0.115368	+	0.993323i	$0.536805\pi$
$812$	−2.07015	−0.0726481
$813$	0	0
$814$	6.49336	0.227592
$815$	41.8554	1.46613
$816$	0	0
$817$	57.3682	2.00706
$818$	−16.8730	−0.589952
$819$	0	0
$820$	20.0357	0.699677
$821$	34.3943	1.20037	0.600184	−	0.799862i	$-0.295094\pi$
$821$	34.3943	1.20037	0.600184	+	0.799862i	$0.295094\pi$
$822$	0	0
$823$	−8.93365	−0.311407	−0.155704	−	0.987804i	$-0.549764\pi$
$823$	−8.93365	−0.311407	−0.155704	+	0.987804i	$0.549764\pi$
$824$	16.6465	0.579908
$825$	0	0
$826$	−7.61539	−0.264973
$827$	23.7127	0.824571	0.412285	−	0.911055i	$-0.364731\pi$
$827$	23.7127	0.824571	0.412285	+	0.911055i	$0.364731\pi$
$828$	0	0
$829$	−47.6433	−1.65472	−0.827360	−	0.561672i	$-0.810158\pi$
$829$	−47.6433	−1.65472	−0.827360	+	0.561672i	$0.810158\pi$
$830$	46.2476	1.60528
$831$	0	0
$832$	−7.81848	−0.271057
$833$	0.579876	0.0200915
$834$	0	0
$835$	−65.3432	−2.26130
$836$	8.06066	0.278784
$837$	0	0
$838$	28.6106	0.988337
$839$	−54.5101	−1.88190	−0.940948	−	0.338550i	$-0.890063\pi$
$839$	−54.5101	−1.88190	−0.940948	+	0.338550i	$0.890063\pi$
$840$	0	0
$841$	−25.9369	−0.894377
$842$	−4.63074	−0.159586
$843$	0	0
$844$	−17.9043	−0.616293
$845$	32.6593	1.12351
$846$	0	0
$847$	10.2317	0.351565
$848$	−1.73195	−0.0594754
$849$	0	0
$850$	2.02986	0.0696235
$851$	−28.8313	−0.988324
$852$	0	0
$853$	−38.9289	−1.33290	−0.666450	−	0.745550i	$-0.732187\pi$
$853$	−38.9289	−1.33290	−0.666450	+	0.745550i	$0.732187\pi$
$854$	12.7057	0.434778
$855$	0	0
$856$	−39.8765	−1.36295
$857$	−40.0166	−1.36694	−0.683471	−	0.729978i	$-0.739530\pi$
$857$	−40.0166	−1.36694	−0.683471	+	0.729978i	$0.739530\pi$
$858$	0	0
$859$	−22.7166	−0.775079	−0.387540	−	0.921853i	$-0.626675\pi$
$859$	−22.7166	−0.775079	−0.387540	+	0.921853i	$0.626675\pi$
$860$	25.9980	0.886526
$861$	0	0
$862$	24.3078	0.827927
$863$	29.8714	1.01683	0.508417	−	0.861111i	$-0.330231\pi$
$863$	29.8714	1.01683	0.508417	+	0.861111i	$0.330231\pi$
$864$	0	0
$865$	24.6827	0.839236
$866$	15.3602	0.521959
$867$	0	0
$868$	−6.47992	−0.219943
$869$	1.27586	0.0432807
$870$	0	0
$871$	−1.67373	−0.0567121
$872$	6.60263	0.223593
$873$	0	0
$874$	24.7258	0.836362
$875$	−3.35885	−0.113550
$876$	0	0
$877$	16.3145	0.550901	0.275450	−	0.961315i	$-0.411173\pi$
$877$	16.3145	0.550901	0.275450	+	0.961315i	$0.411173\pi$
$878$	−11.3302	−0.382377
$879$	0	0
$880$	−0.614151	−0.0207030
$881$	17.3426	0.584286	0.292143	−	0.956375i	$-0.405632\pi$
$881$	17.3426	0.584286	0.292143	+	0.956375i	$0.405632\pi$
$882$	0	0
$883$	14.5659	0.490182	0.245091	−	0.969500i	$-0.421182\pi$
$883$	14.5659	0.490182	0.245091	+	0.969500i	$0.421182\pi$
$884$	0.978587	0.0329134
$885$	0	0
$886$	−15.4143	−0.517854
$887$	−26.8712	−0.902247	−0.451124	−	0.892461i	$-0.648977\pi$
$887$	−26.8712	−0.902247	−0.451124	+	0.892461i	$0.648977\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	1.89147	0.0634022
$891$	0	0
$892$	15.4378	0.516895
$893$	87.9339	2.94260
$894$	0	0
$895$	7.80988	0.261056
$896$	6.12970	0.204779
$897$	0	0
$898$	−12.9825	−0.433231
$899$	9.58788	0.319774
$900$	0	0
$901$	4.26958	0.142240
$902$	−4.50595	−0.150032
$903$	0	0
$904$	−19.0635	−0.634043
$905$	56.3116	1.87186
$906$	0	0
$907$	43.7507	1.45272	0.726358	−	0.687316i	$-0.241211\pi$
$907$	43.7507	1.45272	0.726358	+	0.687316i	$0.241211\pi$
$908$	24.6317	0.817433
$909$	0	0
$910$	−3.84161	−0.127348
$911$	5.06252	0.167729	0.0838644	−	0.996477i	$-0.473274\pi$
$911$	5.06252	0.167729	0.0838644	+	0.996477i	$0.473274\pi$
$912$	0	0
$913$	15.0552	0.498255
$914$	2.24151	0.0741426
$915$	0	0
$916$	24.7650	0.818257
$917$	−3.37363	−0.111407
$918$	0	0
$919$	26.4799	0.873492	0.436746	−	0.899585i	$-0.356131\pi$
$919$	26.4799	0.873492	0.436746	+	0.899585i	$0.356131\pi$
$920$	30.1515	0.994065
$921$	0	0
$922$	−8.02109	−0.264160
$923$	14.6134	0.481006
$924$	0	0
$925$	−31.7336	−1.04339
$926$	−24.5612	−0.807132
$927$	0	0
$928$	−9.69895	−0.318384
$929$	−40.8254	−1.33944	−0.669719	−	0.742615i	$-0.733585\pi$
$929$	−40.8254	−1.33944	−0.669719	+	0.742615i	$0.733585\pi$
$930$	0	0
$931$	7.77454	0.254800
$932$	14.0144	0.459059
$933$	0	0
$934$	−6.32018	−0.206802
$935$	1.51400	0.0495131
$936$	0	0
$937$	−33.3098	−1.08818	−0.544092	−	0.839025i	$-0.683126\pi$
$937$	−33.3098	−1.08818	−0.544092	+	0.839025i	$0.683126\pi$
$938$	−1.06047	−0.0346257
$939$	0	0
$940$	39.8497	1.29976
$941$	−27.3790	−0.892528	−0.446264	−	0.894901i	$-0.647246\pi$
$941$	−27.3790	−0.892528	−0.446264	+	0.894901i	$0.647246\pi$
$942$	0	0
$943$	20.0070	0.651516
$944$	−1.98163	−0.0644966
$945$	0	0
$946$	−5.84685	−0.190098
$947$	10.3382	0.335948	0.167974	−	0.985791i	$-0.446278\pi$
$947$	10.3382	0.335948	0.167974	+	0.985791i	$0.446278\pi$
$948$	0	0
$949$	−15.8438	−0.514313
$950$	27.2148	0.882964
$951$	0	0
$952$	1.66842	0.0540737
$953$	−57.9009	−1.87559	−0.937796	−	0.347186i	$-0.887137\pi$
$953$	−57.9009	−1.87559	−0.937796	+	0.347186i	$0.887137\pi$
$954$	0	0
$955$	−26.2821	−0.850468
$956$	−10.3843	−0.335853
$957$	0	0
$958$	17.8027	0.575179
$959$	16.2096	0.523435
$960$	0	0
$961$	−0.988371	−0.0318829
$962$	10.5691	0.340761
$963$	0	0
$964$	5.28448	0.170202
$965$	−48.9015	−1.57419
$966$	0	0
$967$	−47.6104	−1.53105	−0.765523	−	0.643408i	$-0.777520\pi$
$967$	−47.6104	−1.53105	−0.765523	+	0.643408i	$0.777520\pi$
$968$	29.4385	0.946189
$969$	0	0
$970$	47.0645	1.51115
$971$	−18.1580	−0.582717	−0.291358	−	0.956614i	$-0.594107\pi$
$971$	−18.1580	−0.582717	−0.291358	+	0.956614i	$0.594107\pi$
$972$	0	0
$973$	−12.8614	−0.412318
$974$	−20.7349	−0.664388
$975$	0	0
$976$	3.30618	0.105828
$977$	−53.1178	−1.69939	−0.849695	−	0.527274i	$-0.823214\pi$
$977$	−53.1178	−1.69939	−0.849695	+	0.527274i	$0.823214\pi$
$978$	0	0
$979$	0.615739	0.0196791
$980$	3.52326	0.112546
$981$	0	0
$982$	36.6999	1.17114
$983$	30.7192	0.979790	0.489895	−	0.871781i	$-0.337035\pi$
$983$	30.7192	0.979790	0.489895	+	0.871781i	$0.337035\pi$
$984$	0	0
$985$	69.7423	2.22217
$986$	−0.917419	−0.0292166
$987$	0	0
$988$	13.1202	0.417408
$989$	25.9607	0.825503
$990$	0	0
$991$	−36.4388	−1.15752	−0.578759	−	0.815499i	$-0.696463\pi$
$991$	−36.4388	−1.15752	−0.578759	+	0.815499i	$0.696463\pi$
$992$	−30.3593	−0.963908
$993$	0	0
$994$	9.25906	0.293679
$995$	−20.3846	−0.646236
$996$	0	0
$997$	−37.1698	−1.17718	−0.588589	−	0.808432i	$-0.700316\pi$
$997$	−37.1698	−1.17718	−0.588589	+	0.808432i	$0.700316\pi$
$998$	−13.7867	−0.436409
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.z.1.19	yes	32
3.2	odd	2	inner	8001.2.a.z.1.14	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.z.1.14	✓	32	3.2	odd	2	inner
8001.2.a.z.1.19	yes	32	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)

\(f(q)\)	\(=\)	\(q+0.903971 q^{2} -1.18284 q^{4} -2.97865 q^{5} -1.00000 q^{7} -2.87719 q^{8} +O(q^{10})\)	\(q+0.903971 q^{2} -1.18284 q^{4} -2.97865 q^{5} -1.00000 q^{7} -2.87719 q^{8} -2.69261 q^{10} -0.876539 q^{11} -1.42672 q^{13} -0.903971 q^{14} -0.235226 q^{16} +0.579876 q^{17} +7.77454 q^{19} +3.52326 q^{20} -0.792366 q^{22} +3.51820 q^{23} +3.87236 q^{25} -1.28972 q^{26} +1.18284 q^{28} -1.75016 q^{29} -5.47829 q^{31} +5.54175 q^{32} +0.524191 q^{34} +2.97865 q^{35} -8.19490 q^{37} +7.02796 q^{38} +8.57015 q^{40} +5.68670 q^{41} +7.37898 q^{43} +1.03680 q^{44} +3.18035 q^{46} +11.3105 q^{47} +1.00000 q^{49} +3.50050 q^{50} +1.68758 q^{52} +7.36292 q^{53} +2.61090 q^{55} +2.87719 q^{56} -1.58209 q^{58} +8.42438 q^{59} -14.0554 q^{61} -4.95221 q^{62} +5.48003 q^{64} +4.24971 q^{65} +1.17313 q^{67} -0.685899 q^{68} +2.69261 q^{70} -10.2426 q^{71} +11.1051 q^{73} -7.40795 q^{74} -9.19601 q^{76} +0.876539 q^{77} -1.45557 q^{79} +0.700655 q^{80} +5.14061 q^{82} -17.1757 q^{83} -1.72725 q^{85} +6.67039 q^{86} +2.52197 q^{88} -0.702467 q^{89} +1.42672 q^{91} -4.16145 q^{92} +10.2244 q^{94} -23.1576 q^{95} -17.4791 q^{97} +0.903971 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 30 q^{4} - 32 q^{7}+O(q^{10}) \) 32 * q + 30 * q^4 - 32 * q^7	\( 32 q + 30 q^{4} - 32 q^{7} - 16 q^{10} - 14 q^{13} + 18 q^{16} - 30 q^{19} - 10 q^{22} + 36 q^{25} - 30 q^{28} - 58 q^{31} - 34 q^{34} + 8 q^{37} - 34 q^{40} + 6 q^{43} - 36 q^{46} + 32 q^{49} - 56 q^{52} - 88 q^{55} - 22 q^{58} - 46 q^{61} + 20 q^{64} - 8 q^{67} + 16 q^{70} - 60 q^{73} - 128 q^{76} - 74 q^{79} - 52 q^{82} - 16 q^{85} - 64 q^{88} + 14 q^{91} - 58 q^{94} - 44 q^{97}+O(q^{100}) \) 32 * q + 30 * q^4 - 32 * q^7 - 16 * q^10 - 14 * q^13 + 18 * q^16 - 30 * q^19 - 10 * q^22 + 36 * q^25 - 30 * q^28 - 58 * q^31 - 34 * q^34 + 8 * q^37 - 34 * q^40 + 6 * q^43 - 36 * q^46 + 32 * q^49 - 56 * q^52 - 88 * q^55 - 22 * q^58 - 46 * q^61 + 20 * q^64 - 8 * q^67 + 16 * q^70 - 60 * q^73 - 128 * q^76 - 74 * q^79 - 52 * q^82 - 16 * q^85 - 64 * q^88 + 14 * q^91 - 58 * q^94 - 44 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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