LMFDB - Embedded newform 8001.2.a.z.1.14

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			1.14
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	$-0.603549\pi$
$2$	−0.903971	−0.639204	−0.319602	+	0.947552i	$0.603549\pi$
$3$	0	0
$3$	0	0
$4$	−1.18284	−0.591418
$5$	2.97865	1.33209	0.666046	−	0.745910i	$-0.267985\pi$
$5$	2.97865	1.33209	0.666046	+	0.745910i	$0.267985\pi$
$6$	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	$-0.457814\pi$
$11$	0.876539	0.264286	0.132143	+	0.991231i	$0.457814\pi$
$12$	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	$-0.522402\pi$
$17$	−0.579876	−0.140641	−0.0703203	+	0.997524i	$0.522402\pi$
$18$	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	$-0.619546\pi$
$23$	−3.51820	−0.733595	−0.366798	+	0.930301i	$0.619546\pi$
$24$	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	$-0.448045\pi$
$29$	1.75016	0.324997	0.162498	+	0.986709i	$0.448045\pi$
$30$	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	$-0.646461\pi$
$41$	−5.68670	−0.888114	−0.444057	+	0.895999i	$0.646461\pi$
$42$	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	$-0.808770\pi$
$47$	−11.3105	−1.64980	−0.824902	+	0.565275i	$0.808770\pi$
$48$	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	$-0.668761\pi$
$53$	−7.36292	−1.01138	−0.505688	+	0.862717i	$0.668761\pi$
$54$	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	$-0.684755\pi$
$59$	−8.42438	−1.09676	−0.548380	+	0.836229i	$0.684755\pi$
$60$	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	$-0.292056\pi$
$71$	10.2426	1.21558	0.607789	+	0.794098i	$0.292056\pi$
$72$	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	$-0.108333\pi$
$83$	17.1757	1.88528	0.942641	+	0.333807i	$0.108333\pi$
$84$	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	$-0.488146\pi$
$89$	0.702467	0.0744613	0.0372307	+	0.999307i	$0.488146\pi$
$90$	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	$-0.444114\pi$
$101$	3.51085	0.349343	0.174671	+	0.984627i	$0.444114\pi$
$102$	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	$-0.733674\pi$
$107$	−13.8595	−1.33985	−0.669925	+	0.742428i	$0.733674\pi$
$108$	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	$-0.600881\pi$
$113$	−6.62574	−0.623297	−0.311648	+	0.950198i	$0.600881\pi$
$114$	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	$-0.547083\pi$
$131$	−3.37363	−0.294755	−0.147378	+	0.989080i	$0.547083\pi$
$132$	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	$-0.256536\pi$
$137$	16.2096	1.38488	0.692439	+	0.721476i	$0.256536\pi$
$138$	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	$-0.693710\pi$
$149$	−13.9566	−1.14337	−0.571685	+	0.820473i	$0.693710\pi$
$150$	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	$-0.822660\pi$
$167$	−21.9372	−1.69755	−0.848776	+	0.528753i	$0.822660\pi$
$168$	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	$-0.397993\pi$
$173$	8.28652	0.630013	0.315006	+	0.949090i	$0.397993\pi$
$174$	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	$-0.468760\pi$
$179$	2.62195	0.195974	0.0979870	+	0.995188i	$0.468760\pi$
$180$	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	$-0.603422\pi$
$191$	−8.82348	−0.638445	−0.319222	+	0.947680i	$0.603422\pi$
$192$	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	$-0.185992\pi$
$197$	23.4141	1.66818	0.834092	+	0.551626i	$0.185992\pi$
$198$	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	$-0.257136\pi$
$227$	20.8243	1.38216	0.691078	+	0.722780i	$0.257136\pi$
$228$	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	$-0.373132\pi$
$233$	11.8482	0.776200	0.388100	+	0.921617i	$0.373132\pi$
$234$	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	$-0.591641\pi$
$239$	−8.77917	−0.567877	−0.283938	+	0.958842i	$0.591641\pi$
$240$	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	$-0.765413\pi$
$251$	−23.4635	−1.48101	−0.740503	+	0.672053i	$0.765413\pi$
$252$	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	$-0.230741\pi$
$257$	24.0009	1.49714	0.748569	+	0.663057i	$0.230741\pi$
$258$	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	$-0.450796\pi$
$263$	4.99373	0.307926	0.153963	+	0.988077i	$0.450796\pi$
$264$	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	$-0.371048\pi$
$269$	12.9283	0.788250	0.394125	+	0.919057i	$0.371048\pi$
$270$	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	$-0.408182\pi$
$281$	9.53726	0.568945	0.284473	+	0.958684i	$0.408182\pi$
$282$	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	$-0.159934\pi$
$293$	30.0033	1.75281	0.876407	+	0.481572i	$0.159934\pi$
$294$	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	$-0.542808\pi$
$311$	−4.72911	−0.268163	−0.134082	+	0.990970i	$0.542808\pi$
$312$	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	$-0.562003\pi$
$317$	−6.89241	−0.387117	−0.193558	+	0.981089i	$0.562003\pi$
$318$	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	$-0.552690\pi$
$347$	−6.13890	−0.329553	−0.164777	+	0.986331i	$0.552690\pi$
$348$	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	$-0.235408\pi$
$353$	27.7604	1.47754	0.738768	+	0.673959i	$0.235408\pi$
$354$	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	$-0.534555\pi$
$359$	−4.10568	−0.216689	−0.108345	+	0.994113i	$0.534555\pi$
$360$	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	$-0.767082\pi$
$383$	−29.1214	−1.48804	−0.744018	+	0.668159i	$0.767082\pi$
$384$	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	$-0.503891\pi$
$389$	−0.482130	−0.0244449	−0.0122225	+	0.999925i	$0.503891\pi$
$390$	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	$-0.628238\pi$
$401$	−15.7020	−0.784123	−0.392061	+	0.919939i	$0.628238\pi$
$402$	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	$-0.781295\pi$
$419$	−31.6499	−1.54620	−0.773100	+	0.634284i	$0.781295\pi$
$420$	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	$-0.724237\pi$
$431$	−26.8900	−1.29525	−0.647623	+	0.761961i	$0.724237\pi$
$432$	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	$-0.367245\pi$
$443$	17.0518	0.810154	0.405077	+	0.914283i	$0.367245\pi$
$444$	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	$-0.389951\pi$
$449$	14.3616	0.677766	0.338883	+	0.940829i	$0.389951\pi$
$450$	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	$-0.433750\pi$
$461$	8.87317	0.413265	0.206632	+	0.978419i	$0.433750\pi$
$462$	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	$-0.448281\pi$
$467$	6.99157	0.323531	0.161766	+	0.986829i	$0.448281\pi$
$468$	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	$-0.648547\pi$
$479$	−19.6939	−0.899837	−0.449919	+	0.893070i	$0.648547\pi$
$480$	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	$-0.868674\pi$
$491$	−40.5985	−1.83218	−0.916092	+	0.400967i	$0.868674\pi$
$492$	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	$-0.402751\pi$
$503$	13.4918	0.601571	0.300786	+	0.953692i	$0.402751\pi$
$504$	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	$-0.461640\pi$
$509$	5.42460	0.240441	0.120221	+	0.992747i	$0.461640\pi$
$510$	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	$-0.252861\pi$
$521$	31.9886	1.40145	0.700723	+	0.713433i	$0.252861\pi$
$522$	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	$-0.445764\pi$
$557$	8.00375	0.339130	0.169565	+	0.985519i	$0.445764\pi$
$558$	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	$-0.621444\pi$
$563$	−17.6695	−0.744680	−0.372340	+	0.928096i	$0.621444\pi$
$564$	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	$-0.706150\pi$
$569$	−28.7821	−1.20661	−0.603305	+	0.797510i	$0.706150\pi$
$570$	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	$-0.423932\pi$
$587$	11.4699	0.473414	0.236707	+	0.971581i	$0.423932\pi$
$588$	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	$-0.583568\pi$
$593$	−12.6400	−0.519061	−0.259530	+	0.965735i	$0.583568\pi$
$594$	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	$-0.539082\pi$
$599$	−5.99492	−0.244946	−0.122473	+	0.992472i	$0.539082\pi$
$600$	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	$-0.673475\pi$
$617$	−25.7540	−1.03682	−0.518408	+	0.855134i	$0.673475\pi$
$618$	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	$-0.478889\pi$
$641$	3.35578	0.132545	0.0662726	+	0.997802i	$0.478889\pi$
$642$	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	$-0.644527\pi$
$647$	−22.3128	−0.877208	−0.438604	+	0.898680i	$0.644527\pi$
$648$	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	$-0.532753\pi$
$653$	−5.24950	−0.205429	−0.102714	+	0.994711i	$0.532753\pi$
$654$	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	$-0.654108\pi$
$659$	−23.8972	−0.930904	−0.465452	+	0.885073i	$0.654108\pi$
$660$	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	$-0.435244\pi$
$677$	10.5137	0.404076	0.202038	+	0.979378i	$0.435244\pi$
$678$	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	$-0.521992\pi$
$683$	−3.60843	−0.138073	−0.0690363	+	0.997614i	$0.521992\pi$
$684$	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	$-0.682683\pi$
$701$	−28.7494	−1.08585	−0.542924	+	0.839782i	$0.682683\pi$
$702$	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	$-0.128109\pi$
$719$	49.3434	1.84020	0.920098	+	0.391688i	$0.128109\pi$
$720$	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	$-0.303517\pi$
$743$	31.5544	1.15762	0.578810	+	0.815462i	$0.303517\pi$
$744$	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	$-0.585790\pi$
$761$	−14.6906	−0.532534	−0.266267	+	0.963899i	$0.585790\pi$
$762$	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	$-0.718413\pi$
$773$	−35.2304	−1.26715	−0.633574	+	0.773682i	$0.718413\pi$
$774$	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	$-0.625011\pi$
$797$	−21.6089	−0.765428	−0.382714	+	0.923867i	$0.625011\pi$
$798$	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	$-0.425259\pi$
$809$	13.2348	0.465310	0.232655	+	0.972559i	$0.425259\pi$
$810$	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	$-0.704906\pi$
$821$	−34.3943	−1.20037	−0.600184	+	0.799862i	$0.704906\pi$
$822$	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	$-0.635269\pi$
$827$	−23.7127	−0.824571	−0.412285	+	0.911055i	$0.635269\pi$
$828$	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	$-0.109937\pi$
$839$	54.5101	1.88190	0.940948	+	0.338550i	$0.109937\pi$
$840$	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	$-0.260470\pi$
$857$	40.0166	1.36694	0.683471	+	0.729978i	$0.260470\pi$
$858$	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	$-0.669769\pi$
$863$	−29.8714	−1.01683	−0.508417	+	0.861111i	$0.669769\pi$
$864$	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	$-0.594368\pi$
$881$	−17.3426	−0.584286	−0.292143	+	0.956375i	$0.594368\pi$
$882$	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	$-0.351023\pi$
$887$	26.8712	0.902247	0.451124	+	0.892461i	$0.351023\pi$
$888$	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	$-0.526726\pi$
$911$	−5.06252	−0.167729	−0.0838644	+	0.996477i	$0.526726\pi$
$912$	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	$-0.266415\pi$
$929$	40.8254	1.33944	0.669719	+	0.742615i	$0.266415\pi$
$930$	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	$-0.352754\pi$
$941$	27.3790	0.892528	0.446264	+	0.894901i	$0.352754\pi$
$942$	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	$-0.553722\pi$
$947$	−10.3382	−0.335948	−0.167974	+	0.985791i	$0.553722\pi$
$948$	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	$-0.112863\pi$
$953$	57.9009	1.87559	0.937796	+	0.347186i	$0.112863\pi$
$954$	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	$-0.405893\pi$
$971$	18.1580	0.582717	0.291358	+	0.956614i	$0.405893\pi$
$972$	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	$-0.176786\pi$
$977$	53.1178	1.69939	0.849695	+	0.527274i	$0.176786\pi$
$978$	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	$-0.662965\pi$
$983$	−30.7192	−0.979790	−0.489895	+	0.871781i	$0.662965\pi$
$984$	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.1.14	✓	32
3.2	odd	2	inner	8001.2.a.z.1.19	yes	32

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

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