LMFDB - Embedded newform 8001.2.a.ba.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-2.26712 q^{2} +3.13982 q^{4} -1.21847 q^{5} +1.00000 q^{7} -2.58412 q^{8} +O(q^{10})$	\(q-2.26712 q^{2} +3.13982 q^{4} -1.21847 q^{5} +1.00000 q^{7} -2.58412 q^{8} +2.76242 q^{10} -0.403543 q^{11} +6.29644 q^{13} -2.26712 q^{14} -0.421149 q^{16} -3.86152 q^{17} +4.73933 q^{19} -3.82579 q^{20} +0.914879 q^{22} +5.88252 q^{23} -3.51532 q^{25} -14.2748 q^{26} +3.13982 q^{28} +3.62685 q^{29} -1.66099 q^{31} +6.12303 q^{32} +8.75452 q^{34} -1.21847 q^{35} +9.01788 q^{37} -10.7446 q^{38} +3.14868 q^{40} -7.34207 q^{41} +3.22018 q^{43} -1.26705 q^{44} -13.3364 q^{46} +4.46555 q^{47} +1.00000 q^{49} +7.96966 q^{50} +19.7697 q^{52} +6.05072 q^{53} +0.491706 q^{55} -2.58412 q^{56} -8.22249 q^{58} -9.42554 q^{59} -4.73025 q^{61} +3.76566 q^{62} -13.0393 q^{64} -7.67204 q^{65} +7.97909 q^{67} -12.1245 q^{68} +2.76242 q^{70} -2.86285 q^{71} +12.9236 q^{73} -20.4446 q^{74} +14.8807 q^{76} -0.403543 q^{77} +12.0981 q^{79} +0.513159 q^{80} +16.6453 q^{82} -4.81525 q^{83} +4.70516 q^{85} -7.30053 q^{86} +1.04280 q^{88} +14.8745 q^{89} +6.29644 q^{91} +18.4701 q^{92} -10.1239 q^{94} -5.77475 q^{95} -9.51735 q^{97} -2.26712 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) $ 40 * q + 54 * q^4 + 40 * q^7	$ 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) $ 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.26712	−1.60309	−0.801547	−	0.597931i	$-0.795990\pi$
$2$	−2.26712	−1.60309	−0.801547	+	0.597931i	$0.795990\pi$
$3$	0	0
$3$	0	0
$4$	3.13982	1.56991
$5$	−1.21847	−0.544917	−0.272459	−	0.962167i	$-0.587837\pi$
$5$	−1.21847	−0.544917	−0.272459	+	0.962167i	$0.587837\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.58412	−0.913624
$9$	0	0
$10$	2.76242	0.873554
$11$	−0.403543	−0.121673	−0.0608364	−	0.998148i	$-0.519377\pi$
$11$	−0.403543	−0.121673	−0.0608364	+	0.998148i	$0.519377\pi$
$12$	0	0
$13$	6.29644	1.74632	0.873159	−	0.487436i	$-0.162068\pi$
$13$	6.29644	1.74632	0.873159	+	0.487436i	$0.162068\pi$
$14$	−2.26712	−0.605913
$15$	0	0
$16$	−0.421149	−0.105287
$17$	−3.86152	−0.936556	−0.468278	−	0.883581i	$-0.655125\pi$
$17$	−3.86152	−0.936556	−0.468278	+	0.883581i	$0.655125\pi$
$18$	0	0
$19$	4.73933	1.08728	0.543639	−	0.839319i	$-0.317046\pi$
$19$	4.73933	1.08728	0.543639	+	0.839319i	$0.317046\pi$
$20$	−3.82579	−0.855473
$21$	0	0
$22$	0.914879	0.195053
$23$	5.88252	1.22659	0.613295	−	0.789854i	$-0.289844\pi$
$23$	5.88252	1.22659	0.613295	+	0.789854i	$0.289844\pi$
$24$	0	0
$25$	−3.51532	−0.703065
$26$	−14.2748	−2.79951
$27$	0	0
$28$	3.13982	0.593371
$29$	3.62685	0.673489	0.336744	−	0.941596i	$-0.390674\pi$
$29$	3.62685	0.673489	0.336744	+	0.941596i	$0.390674\pi$
$30$	0	0
$31$	−1.66099	−0.298323	−0.149161	−	0.988813i	$-0.547657\pi$
$31$	−1.66099	−0.298323	−0.149161	+	0.988813i	$0.547657\pi$
$32$	6.12303	1.08241
$33$	0	0
$34$	8.75452	1.50139
$35$	−1.21847	−0.205959
$36$	0	0
$37$	9.01788	1.48253	0.741265	−	0.671212i	$-0.234226\pi$
$37$	9.01788	1.48253	0.741265	+	0.671212i	$0.234226\pi$
$38$	−10.7446	−1.74301
$39$	0	0
$40$	3.14868	0.497849
$41$	−7.34207	−1.14664	−0.573320	−	0.819332i	$-0.694345\pi$
$41$	−7.34207	−1.14664	−0.573320	+	0.819332i	$0.694345\pi$
$42$	0	0
$43$	3.22018	0.491073	0.245537	−	0.969387i	$-0.421036\pi$
$43$	3.22018	0.491073	0.245537	+	0.969387i	$0.421036\pi$
$44$	−1.26705	−0.191016
$45$	0	0
$46$	−13.3364	−1.96634
$47$	4.46555	0.651368	0.325684	−	0.945479i	$-0.394406\pi$
$47$	4.46555	0.651368	0.325684	+	0.945479i	$0.394406\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	7.96966	1.12708
$51$	0	0
$52$	19.7697	2.74157
$53$	6.05072	0.831131	0.415565	−	0.909563i	$-0.363584\pi$
$53$	6.05072	0.831131	0.415565	+	0.909563i	$0.363584\pi$
$54$	0	0
$55$	0.491706	0.0663016
$56$	−2.58412	−0.345317
$57$	0	0
$58$	−8.22249	−1.07967
$59$	−9.42554	−1.22710	−0.613551	−	0.789655i	$-0.710259\pi$
$59$	−9.42554	−1.22710	−0.613551	+	0.789655i	$0.710259\pi$
$60$	0	0
$61$	−4.73025	−0.605647	−0.302823	−	0.953047i	$-0.597929\pi$
$61$	−4.73025	−0.605647	−0.302823	+	0.953047i	$0.597929\pi$
$62$	3.76566	0.478240
$63$	0	0
$64$	−13.0393	−1.62992
$65$	−7.67204	−0.951599
$66$	0	0
$67$	7.97909	0.974802	0.487401	−	0.873178i	$-0.337945\pi$
$67$	7.97909	0.974802	0.487401	+	0.873178i	$0.337945\pi$
$68$	−12.1245	−1.47031
$69$	0	0
$70$	2.76242	0.330172
$71$	−2.86285	−0.339758	−0.169879	−	0.985465i	$-0.554338\pi$
$71$	−2.86285	−0.339758	−0.169879	+	0.985465i	$0.554338\pi$
$72$	0	0
$73$	12.9236	1.51259	0.756294	−	0.654231i	$-0.227008\pi$
$73$	12.9236	1.51259	0.756294	+	0.654231i	$0.227008\pi$
$74$	−20.4446	−2.37664
$75$	0	0
$76$	14.8807	1.70693
$77$	−0.403543	−0.0459880
$78$	0	0
$79$	12.0981	1.36114	0.680569	−	0.732684i	$-0.261733\pi$
$79$	12.0981	1.36114	0.680569	+	0.732684i	$0.261733\pi$
$80$	0.513159	0.0573729
$81$	0	0
$82$	16.6453	1.83817
$83$	−4.81525	−0.528543	−0.264271	−	0.964448i	$-0.585131\pi$
$83$	−4.81525	−0.528543	−0.264271	+	0.964448i	$0.585131\pi$
$84$	0	0
$85$	4.70516	0.510346
$86$	−7.30053	−0.787237
$87$	0	0
$88$	1.04280	0.111163
$89$	14.8745	1.57670	0.788348	−	0.615229i	$-0.210937\pi$
$89$	14.8745	1.57670	0.788348	+	0.615229i	$0.210937\pi$
$90$	0	0
$91$	6.29644	0.660046
$92$	18.4701	1.92564
$93$	0	0
$94$	−10.1239	−1.04420
$95$	−5.77475	−0.592477
$96$	0	0
$97$	−9.51735	−0.966341	−0.483170	−	0.875526i	$-0.660515\pi$
$97$	−9.51735	−0.966341	−0.483170	+	0.875526i	$0.660515\pi$
$98$	−2.26712	−0.229014
$99$	0	0
$100$	−11.0375	−1.10375
$101$	4.14790	0.412731	0.206366	−	0.978475i	$-0.433836\pi$
$101$	4.14790	0.412731	0.206366	+	0.978475i	$0.433836\pi$
$102$	0	0
$103$	−13.9983	−1.37929	−0.689647	−	0.724145i	$-0.742234\pi$
$103$	−13.9983	−1.37929	−0.689647	+	0.724145i	$0.742234\pi$
$104$	−16.2707	−1.59548
$105$	0	0
$106$	−13.7177	−1.33238
$107$	19.6665	1.90123	0.950617	−	0.310366i	$-0.100452\pi$
$107$	19.6665	1.90123	0.950617	+	0.310366i	$0.100452\pi$
$108$	0	0
$109$	−6.41208	−0.614166	−0.307083	−	0.951683i	$-0.599353\pi$
$109$	−6.41208	−0.614166	−0.307083	+	0.951683i	$0.599353\pi$
$110$	−1.11476	−0.106288
$111$	0	0
$112$	−0.421149	−0.0397949
$113$	3.70618	0.348648	0.174324	−	0.984688i	$-0.444226\pi$
$113$	3.70618	0.348648	0.174324	+	0.984688i	$0.444226\pi$
$114$	0	0
$115$	−7.16768	−0.668390
$116$	11.3877	1.05732
$117$	0	0
$118$	21.3688	1.96716
$119$	−3.86152	−0.353985
$120$	0	0
$121$	−10.8372	−0.985196
$122$	10.7240	0.970909
$123$	0	0
$124$	−5.21522	−0.468341
$125$	10.3757	0.928030
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	17.3157	1.53050
$129$	0	0
$130$	17.3934	1.52550
$131$	−18.1245	−1.58354	−0.791771	−	0.610818i	$-0.790841\pi$
$131$	−18.1245	−1.58354	−0.791771	+	0.610818i	$0.790841\pi$
$132$	0	0
$133$	4.73933	0.410952
$134$	−18.0896	−1.56270
$135$	0	0
$136$	9.97862	0.855660
$137$	−17.6316	−1.50637	−0.753184	−	0.657810i	$-0.771483\pi$
$137$	−17.6316	−1.50637	−0.753184	+	0.657810i	$0.771483\pi$
$138$	0	0
$139$	1.80333	0.152956	0.0764781	−	0.997071i	$-0.475632\pi$
$139$	1.80333	0.152956	0.0764781	+	0.997071i	$0.475632\pi$
$140$	−3.82579	−0.323338
$141$	0	0
$142$	6.49042	0.544664
$143$	−2.54088	−0.212479
$144$	0	0
$145$	−4.41922	−0.366996
$146$	−29.2993	−2.42482
$147$	0	0
$148$	28.3146	2.32744
$149$	13.8231	1.13244	0.566218	−	0.824256i	$-0.308406\pi$
$149$	13.8231	1.13244	0.566218	+	0.824256i	$0.308406\pi$
$150$	0	0
$151$	13.9476	1.13504	0.567521	−	0.823359i	$-0.307903\pi$
$151$	13.9476	1.13504	0.567521	+	0.823359i	$0.307903\pi$
$152$	−12.2470	−0.993363
$153$	0	0
$154$	0.914879	0.0737231
$155$	2.02387	0.162561
$156$	0	0
$157$	9.25592	0.738703	0.369352	−	0.929290i	$-0.379580\pi$
$157$	9.25592	0.738703	0.369352	+	0.929290i	$0.379580\pi$
$158$	−27.4277	−2.18203
$159$	0	0
$160$	−7.46075	−0.589824
$161$	5.88252	0.463607
$162$	0	0
$163$	−2.02749	−0.158805	−0.0794026	−	0.996843i	$-0.525301\pi$
$163$	−2.02749	−0.158805	−0.0794026	+	0.996843i	$0.525301\pi$
$164$	−23.0528	−1.80012
$165$	0	0
$166$	10.9168	0.847304
$167$	0.373895	0.0289328	0.0144664	−	0.999895i	$-0.495395\pi$
$167$	0.373895	0.0289328	0.0144664	+	0.999895i	$0.495395\pi$
$168$	0	0
$169$	26.6451	2.04963
$170$	−10.6671	−0.818133
$171$	0	0
$172$	10.1108	0.770942
$173$	−3.75274	−0.285316	−0.142658	−	0.989772i	$-0.545565\pi$
$173$	−3.75274	−0.285316	−0.142658	+	0.989772i	$0.545565\pi$
$174$	0	0
$175$	−3.51532	−0.265734
$176$	0.169952	0.0128106
$177$	0	0
$178$	−33.7223	−2.52759
$179$	23.3292	1.74371	0.871853	−	0.489768i	$-0.162919\pi$
$179$	23.3292	1.74371	0.871853	+	0.489768i	$0.162919\pi$
$180$	0	0
$181$	−7.41835	−0.551402	−0.275701	−	0.961244i	$-0.588910\pi$
$181$	−7.41835	−0.551402	−0.275701	+	0.961244i	$0.588910\pi$
$182$	−14.2748	−1.05812
$183$	0	0
$184$	−15.2011	−1.12064
$185$	−10.9880	−0.807857
$186$	0	0
$187$	1.55829	0.113953
$188$	14.0210	1.02259
$189$	0	0
$190$	13.0920	0.949796
$191$	−18.4923	−1.33805	−0.669027	−	0.743238i	$-0.733289\pi$
$191$	−18.4923	−1.33805	−0.669027	+	0.743238i	$0.733289\pi$
$192$	0	0
$193$	−16.0686	−1.15664	−0.578320	−	0.815810i	$-0.696292\pi$
$193$	−16.0686	−1.15664	−0.578320	+	0.815810i	$0.696292\pi$
$194$	21.5770	1.54914
$195$	0	0
$196$	3.13982	0.224273
$197$	−15.2326	−1.08527	−0.542637	−	0.839967i	$-0.682574\pi$
$197$	−15.2326	−1.08527	−0.542637	+	0.839967i	$0.682574\pi$
$198$	0	0
$199$	−11.1933	−0.793472	−0.396736	−	0.917933i	$-0.629857\pi$
$199$	−11.1933	−0.793472	−0.396736	+	0.917933i	$0.629857\pi$
$200$	9.08401	0.642337
$201$	0	0
$202$	−9.40377	−0.661647
$203$	3.62685	0.254555
$204$	0	0
$205$	8.94612	0.624824
$206$	31.7358	2.21114
$207$	0	0
$208$	−2.65174	−0.183865
$209$	−1.91252	−0.132292
$210$	0	0
$211$	13.3925	0.921980	0.460990	−	0.887405i	$-0.347494\pi$
$211$	13.3925	0.921980	0.460990	+	0.887405i	$0.347494\pi$
$212$	18.9982	1.30480
$213$	0	0
$214$	−44.5863	−3.04786
$215$	−3.92370	−0.267594
$216$	0	0
$217$	−1.66099	−0.112755
$218$	14.5370	0.984567
$219$	0	0
$220$	1.54387	0.104088
$221$	−24.3138	−1.63552
$222$	0	0
$223$	−22.9827	−1.53903	−0.769517	−	0.638627i	$-0.779503\pi$
$223$	−22.9827	−1.53903	−0.769517	+	0.638627i	$0.779503\pi$
$224$	6.12303	0.409112
$225$	0	0
$226$	−8.40236	−0.558916
$227$	−11.4302	−0.758650	−0.379325	−	0.925263i	$-0.623844\pi$
$227$	−11.4302	−0.758650	−0.379325	+	0.925263i	$0.623844\pi$
$228$	0	0
$229$	23.3287	1.54160	0.770802	−	0.637075i	$-0.219856\pi$
$229$	23.3287	1.54160	0.770802	+	0.637075i	$0.219856\pi$
$230$	16.2500	1.07149
$231$	0	0
$232$	−9.37220	−0.615315
$233$	10.9115	0.714837	0.357418	−	0.933944i	$-0.383657\pi$
$233$	10.9115	0.714837	0.357418	+	0.933944i	$0.383657\pi$
$234$	0	0
$235$	−5.44115	−0.354942
$236$	−29.5946	−1.92644
$237$	0	0
$238$	8.75452	0.567471
$239$	−23.5940	−1.52617	−0.763085	−	0.646299i	$-0.776316\pi$
$239$	−23.5940	−1.52617	−0.763085	+	0.646299i	$0.776316\pi$
$240$	0	0
$241$	−10.3516	−0.666802	−0.333401	−	0.942785i	$-0.608196\pi$
$241$	−10.3516	−0.666802	−0.333401	+	0.942785i	$0.608196\pi$
$242$	24.5691	1.57936
$243$	0	0
$244$	−14.8522	−0.950812
$245$	−1.21847	−0.0778454
$246$	0	0
$247$	29.8409	1.89873
$248$	4.29220	0.272555
$249$	0	0
$250$	−23.5229	−1.48772
$251$	3.45334	0.217973	0.108986	−	0.994043i	$-0.465240\pi$
$251$	3.45334	0.217973	0.108986	+	0.994043i	$0.465240\pi$
$252$	0	0
$253$	−2.37385	−0.149243
$254$	2.26712	0.142252
$255$	0	0
$256$	−13.1780	−0.823623
$257$	25.0017	1.55956	0.779782	−	0.626051i	$-0.215330\pi$
$257$	25.0017	1.55956	0.779782	+	0.626051i	$0.215330\pi$
$258$	0	0
$259$	9.01788	0.560344
$260$	−24.0889	−1.49393
$261$	0	0
$262$	41.0903	2.53857
$263$	−24.7045	−1.52335	−0.761673	−	0.647962i	$-0.775622\pi$
$263$	−24.7045	−1.52335	−0.761673	+	0.647962i	$0.775622\pi$
$264$	0	0
$265$	−7.37264	−0.452898
$266$	−10.7446	−0.658796
$267$	0	0
$268$	25.0530	1.53035
$269$	−2.45095	−0.149437	−0.0747185	−	0.997205i	$-0.523806\pi$
$269$	−2.45095	−0.149437	−0.0747185	+	0.997205i	$0.523806\pi$
$270$	0	0
$271$	−9.46087	−0.574707	−0.287354	−	0.957825i	$-0.592775\pi$
$271$	−9.46087	−0.574707	−0.287354	+	0.957825i	$0.592775\pi$
$272$	1.62628	0.0986075
$273$	0	0
$274$	39.9729	2.41485
$275$	1.41858	0.0855438
$276$	0	0
$277$	19.6811	1.18253	0.591263	−	0.806479i	$-0.298630\pi$
$277$	19.6811	1.18253	0.591263	+	0.806479i	$0.298630\pi$
$278$	−4.08835	−0.245203
$279$	0	0
$280$	3.14868	0.188169
$281$	15.2211	0.908016	0.454008	−	0.890998i	$-0.349994\pi$
$281$	15.2211	0.908016	0.454008	+	0.890998i	$0.349994\pi$
$282$	0	0
$283$	20.0366	1.19105	0.595526	−	0.803336i	$-0.296944\pi$
$283$	20.0366	1.19105	0.595526	+	0.803336i	$0.296944\pi$
$284$	−8.98885	−0.533390
$285$	0	0
$286$	5.76048	0.340624
$287$	−7.34207	−0.433389
$288$	0	0
$289$	−2.08867	−0.122863
$290$	10.0189	0.588329
$291$	0	0
$292$	40.5777	2.37463
$293$	0.0602401	0.00351927	0.00175963	−	0.999998i	$-0.499440\pi$
$293$	0.0602401	0.00351927	0.00175963	+	0.999998i	$0.499440\pi$
$294$	0	0
$295$	11.4848	0.668669
$296$	−23.3033	−1.35447
$297$	0	0
$298$	−31.3387	−1.81540
$299$	37.0389	2.14201
$300$	0	0
$301$	3.22018	0.185608
$302$	−31.6209	−1.81958
$303$	0	0
$304$	−1.99597	−0.114477
$305$	5.76368	0.330027
$306$	0	0
$307$	27.0672	1.54480	0.772402	−	0.635134i	$-0.219055\pi$
$307$	27.0672	1.54480	0.772402	+	0.635134i	$0.219055\pi$
$308$	−1.26705	−0.0721971
$309$	0	0
$310$	−4.58836	−0.260601
$311$	−0.554210	−0.0314264	−0.0157132	−	0.999877i	$-0.505002\pi$
$311$	−0.554210	−0.0314264	−0.0157132	+	0.999877i	$0.505002\pi$
$312$	0	0
$313$	30.0815	1.70031	0.850153	−	0.526536i	$-0.176510\pi$
$313$	30.0815	1.70031	0.850153	+	0.526536i	$0.176510\pi$
$314$	−20.9843	−1.18421
$315$	0	0
$316$	37.9858	2.13687
$317$	21.2962	1.19611	0.598056	−	0.801454i	$-0.295940\pi$
$317$	21.2962	1.19611	0.598056	+	0.801454i	$0.295940\pi$
$318$	0	0
$319$	−1.46359	−0.0819452
$320$	15.8881	0.888170
$321$	0	0
$322$	−13.3364	−0.743206
$323$	−18.3010	−1.01830
$324$	0	0
$325$	−22.1340	−1.22777
$326$	4.59656	0.254580
$327$	0	0
$328$	18.9728	1.04760
$329$	4.46555	0.246194
$330$	0	0
$331$	25.6783	1.41141	0.705705	−	0.708506i	$-0.250630\pi$
$331$	25.6783	1.41141	0.705705	+	0.708506i	$0.250630\pi$
$332$	−15.1191	−0.829766
$333$	0	0
$334$	−0.847663	−0.0463821
$335$	−9.72231	−0.531186
$336$	0	0
$337$	−35.1488	−1.91468	−0.957339	−	0.288966i	$-0.906689\pi$
$337$	−35.1488	−1.91468	−0.957339	+	0.288966i	$0.906689\pi$
$338$	−60.4077	−3.28574
$339$	0	0
$340$	14.7734	0.801198
$341$	0.670281	0.0362978
$342$	0	0
$343$	1.00000	0.0539949
$344$	−8.32133	−0.448656
$345$	0	0
$346$	8.50792	0.457388
$347$	−20.7165	−1.11212	−0.556061	−	0.831141i	$-0.687688\pi$
$347$	−20.7165	−1.11212	−0.556061	+	0.831141i	$0.687688\pi$
$348$	0	0
$349$	−30.0600	−1.60908	−0.804538	−	0.593902i	$-0.797587\pi$
$349$	−30.0600	−1.60908	−0.804538	+	0.593902i	$0.797587\pi$
$350$	7.96966	0.425996
$351$	0	0
$352$	−2.47091	−0.131700
$353$	18.0660	0.961556	0.480778	−	0.876842i	$-0.340354\pi$
$353$	18.0660	0.961556	0.480778	+	0.876842i	$0.340354\pi$
$354$	0	0
$355$	3.48830	0.185140
$356$	46.7034	2.47528
$357$	0	0
$358$	−52.8900	−2.79532
$359$	−6.69367	−0.353279	−0.176639	−	0.984276i	$-0.556523\pi$
$359$	−6.69367	−0.353279	−0.176639	+	0.984276i	$0.556523\pi$
$360$	0	0
$361$	3.46129	0.182173
$362$	16.8183	0.883949
$363$	0	0
$364$	19.7697	1.03621
$365$	−15.7470	−0.824236
$366$	0	0
$367$	−21.6728	−1.13131	−0.565655	−	0.824642i	$-0.691377\pi$
$367$	−21.6728	−1.13131	−0.565655	+	0.824642i	$0.691377\pi$
$368$	−2.47742	−0.129144
$369$	0	0
$370$	24.9112	1.29507
$371$	6.05072	0.314138
$372$	0	0
$373$	−17.1149	−0.886177	−0.443089	−	0.896478i	$-0.646117\pi$
$373$	−17.1149	−0.886177	−0.443089	+	0.896478i	$0.646117\pi$
$374$	−3.53282	−0.182678
$375$	0	0
$376$	−11.5395	−0.595105
$377$	22.8362	1.17613
$378$	0	0
$379$	−13.3781	−0.687189	−0.343594	−	0.939118i	$-0.611645\pi$
$379$	−13.3781	−0.687189	−0.343594	+	0.939118i	$0.611645\pi$
$380$	−18.1317	−0.930137
$381$	0	0
$382$	41.9242	2.14503
$383$	−19.5773	−1.00035	−0.500176	−	0.865924i	$-0.666731\pi$
$383$	−19.5773	−1.00035	−0.500176	+	0.865924i	$0.666731\pi$
$384$	0	0
$385$	0.491706	0.0250597
$386$	36.4293	1.85420
$387$	0	0
$388$	−29.8828	−1.51707
$389$	−25.2025	−1.27782	−0.638908	−	0.769283i	$-0.720613\pi$
$389$	−25.2025	−1.27782	−0.638908	+	0.769283i	$0.720613\pi$
$390$	0	0
$391$	−22.7155	−1.14877
$392$	−2.58412	−0.130518
$393$	0	0
$394$	34.5340	1.73980
$395$	−14.7411	−0.741707
$396$	0	0
$397$	8.11535	0.407298	0.203649	−	0.979044i	$-0.434720\pi$
$397$	8.11535	0.407298	0.203649	+	0.979044i	$0.434720\pi$
$398$	25.3765	1.27201
$399$	0	0
$400$	1.48048	0.0740238
$401$	0.700314	0.0349720	0.0174860	−	0.999847i	$-0.494434\pi$
$401$	0.700314	0.0349720	0.0174860	+	0.999847i	$0.494434\pi$
$402$	0	0
$403$	−10.4583	−0.520966
$404$	13.0237	0.647952
$405$	0	0
$406$	−8.22249	−0.408076
$407$	−3.63910	−0.180384
$408$	0	0
$409$	−10.9186	−0.539888	−0.269944	−	0.962876i	$-0.587005\pi$
$409$	−10.9186	−0.539888	−0.269944	+	0.962876i	$0.587005\pi$
$410$	−20.2819	−1.00165
$411$	0	0
$412$	−43.9523	−2.16537
$413$	−9.42554	−0.463801
$414$	0	0
$415$	5.86726	0.288012
$416$	38.5533	1.89023
$417$	0	0
$418$	4.33592	0.212077
$419$	4.82983	0.235952	0.117976	−	0.993016i	$-0.462359\pi$
$419$	4.82983	0.235952	0.117976	+	0.993016i	$0.462359\pi$
$420$	0	0
$421$	1.86078	0.0906888	0.0453444	−	0.998971i	$-0.485561\pi$
$421$	1.86078	0.0906888	0.0453444	+	0.998971i	$0.485561\pi$
$422$	−30.3625	−1.47802
$423$	0	0
$424$	−15.6358	−0.759341
$425$	13.5745	0.658460
$426$	0	0
$427$	−4.73025	−0.228913
$428$	61.7494	2.98477
$429$	0	0
$430$	8.89550	0.428979
$431$	−8.27255	−0.398475	−0.199237	−	0.979951i	$-0.563846\pi$
$431$	−8.27255	−0.398475	−0.199237	+	0.979951i	$0.563846\pi$
$432$	0	0
$433$	−31.0032	−1.48992	−0.744959	−	0.667110i	$-0.767531\pi$
$433$	−31.0032	−1.48992	−0.744959	+	0.667110i	$0.767531\pi$
$434$	3.76566	0.180758
$435$	0	0
$436$	−20.1328	−0.964187
$437$	27.8792	1.33364
$438$	0	0
$439$	16.5451	0.789656	0.394828	−	0.918755i	$-0.370804\pi$
$439$	16.5451	0.789656	0.394828	+	0.918755i	$0.370804\pi$
$440$	−1.27063	−0.0605747
$441$	0	0
$442$	55.1223	2.62190
$443$	−16.0560	−0.762844	−0.381422	−	0.924401i	$-0.624565\pi$
$443$	−16.0560	−0.762844	−0.381422	+	0.924401i	$0.624565\pi$
$444$	0	0
$445$	−18.1242	−0.859170
$446$	52.1044	2.46722
$447$	0	0
$448$	−13.0393	−0.616051
$449$	13.0752	0.617057	0.308529	−	0.951215i	$-0.400163\pi$
$449$	13.0752	0.617057	0.308529	+	0.951215i	$0.400163\pi$
$450$	0	0
$451$	2.96284	0.139515
$452$	11.6368	0.547347
$453$	0	0
$454$	25.9137	1.21619
$455$	−7.67204	−0.359671
$456$	0	0
$457$	−4.39981	−0.205815	−0.102907	−	0.994691i	$-0.532814\pi$
$457$	−4.39981	−0.205815	−0.102907	+	0.994691i	$0.532814\pi$
$458$	−52.8889	−2.47134
$459$	0	0
$460$	−22.5053	−1.04931
$461$	−5.12901	−0.238882	−0.119441	−	0.992841i	$-0.538110\pi$
$461$	−5.12901	−0.238882	−0.119441	+	0.992841i	$0.538110\pi$
$462$	0	0
$463$	14.9297	0.693840	0.346920	−	0.937895i	$-0.387227\pi$
$463$	14.9297	0.693840	0.346920	+	0.937895i	$0.387227\pi$
$464$	−1.52744	−0.0709098
$465$	0	0
$466$	−24.7377	−1.14595
$467$	27.3857	1.26726	0.633630	−	0.773637i	$-0.281564\pi$
$467$	27.3857	1.26726	0.633630	+	0.773637i	$0.281564\pi$
$468$	0	0
$469$	7.97909	0.368440
$470$	12.3357	0.569005
$471$	0	0
$472$	24.3567	1.12111
$473$	−1.29948	−0.0597502
$474$	0	0
$475$	−16.6603	−0.764427
$476$	−12.1245	−0.555725
$477$	0	0
$478$	53.4904	2.44659
$479$	−15.8235	−0.722992	−0.361496	−	0.932374i	$-0.617734\pi$
$479$	−15.8235	−0.722992	−0.361496	+	0.932374i	$0.617734\pi$
$480$	0	0
$481$	56.7805	2.58897
$482$	23.4682	1.06895
$483$	0	0
$484$	−34.0268	−1.54667
$485$	11.5966	0.526576
$486$	0	0
$487$	4.42400	0.200471	0.100235	−	0.994964i	$-0.468040\pi$
$487$	4.42400	0.200471	0.100235	+	0.994964i	$0.468040\pi$
$488$	12.2235	0.553333
$489$	0	0
$490$	2.76242	0.124793
$491$	42.3151	1.90965	0.954827	−	0.297161i	$-0.0960400\pi$
$491$	42.3151	1.90965	0.954827	+	0.297161i	$0.0960400\pi$
$492$	0	0
$493$	−14.0051	−0.630760
$494$	−67.6529	−3.04385
$495$	0	0
$496$	0.699525	0.0314096
$497$	−2.86285	−0.128416
$498$	0	0
$499$	−19.5942	−0.877158	−0.438579	−	0.898693i	$-0.644518\pi$
$499$	−19.5942	−0.877158	−0.438579	+	0.898693i	$0.644518\pi$
$500$	32.5778	1.45693
$501$	0	0
$502$	−7.82913	−0.349431
$503$	29.4333	1.31237	0.656184	−	0.754601i	$-0.272170\pi$
$503$	29.4333	1.31237	0.656184	+	0.754601i	$0.272170\pi$
$504$	0	0
$505$	−5.05410	−0.224904
$506$	5.38179	0.239250
$507$	0	0
$508$	−3.13982	−0.139307
$509$	36.1437	1.60204	0.801021	−	0.598637i	$-0.204291\pi$
$509$	36.1437	1.60204	0.801021	+	0.598637i	$0.204291\pi$
$510$	0	0
$511$	12.9236	0.571705
$512$	−4.75531	−0.210157
$513$	0	0
$514$	−56.6819	−2.50013
$515$	17.0566	0.751602
$516$	0	0
$517$	−1.80204	−0.0792537
$518$	−20.4446	−0.898284
$519$	0	0
$520$	19.8254	0.869403
$521$	22.4187	0.982182	0.491091	−	0.871108i	$-0.336598\pi$
$521$	22.4187	0.982182	0.491091	+	0.871108i	$0.336598\pi$
$522$	0	0
$523$	29.0989	1.27241	0.636203	−	0.771522i	$-0.280504\pi$
$523$	29.0989	1.27241	0.636203	+	0.771522i	$0.280504\pi$
$524$	−56.9077	−2.48602
$525$	0	0
$526$	56.0081	2.44207
$527$	6.41395	0.279396
$528$	0	0
$529$	11.6040	0.504522
$530$	16.7146	0.726038
$531$	0	0
$532$	14.8807	0.645159
$533$	−46.2289	−2.00240
$534$	0	0
$535$	−23.9631	−1.03602
$536$	−20.6189	−0.890602
$537$	0	0
$538$	5.55659	0.239562
$539$	−0.403543	−0.0173818
$540$	0	0
$541$	−4.99218	−0.214631	−0.107315	−	0.994225i	$-0.534225\pi$
$541$	−4.99218	−0.214631	−0.107315	+	0.994225i	$0.534225\pi$
$542$	21.4489	0.921310
$543$	0	0
$544$	−23.6442	−1.01374
$545$	7.81295	0.334670
$546$	0	0
$547$	−31.3250	−1.33936	−0.669680	−	0.742650i	$-0.733569\pi$
$547$	−31.3250	−1.33936	−0.669680	+	0.742650i	$0.733569\pi$
$548$	−55.3601	−2.36487
$549$	0	0
$550$	−3.21610	−0.137135
$551$	17.1888	0.732270
$552$	0	0
$553$	12.0981	0.514462
$554$	−44.6195	−1.89570
$555$	0	0
$556$	5.66213	0.240128
$557$	−10.9950	−0.465872	−0.232936	−	0.972492i	$-0.574833\pi$
$557$	−10.9950	−0.465872	−0.232936	+	0.972492i	$0.574833\pi$
$558$	0	0
$559$	20.2757	0.857570
$560$	0.513159	0.0216849
$561$	0	0
$562$	−34.5081	−1.45563
$563$	36.7412	1.54846	0.774229	−	0.632906i	$-0.218138\pi$
$563$	36.7412	1.54846	0.774229	+	0.632906i	$0.218138\pi$
$564$	0	0
$565$	−4.51588	−0.189985
$566$	−45.4253	−1.90937
$567$	0	0
$568$	7.39794	0.310411
$569$	−1.18547	−0.0496977	−0.0248488	−	0.999691i	$-0.507910\pi$
$569$	−1.18547	−0.0496977	−0.0248488	+	0.999691i	$0.507910\pi$
$570$	0	0
$571$	2.39349	0.100164	0.0500822	−	0.998745i	$-0.484052\pi$
$571$	2.39349	0.100164	0.0500822	+	0.998745i	$0.484052\pi$
$572$	−7.97793	−0.333574
$573$	0	0
$574$	16.6453	0.694763
$575$	−20.6790	−0.862372
$576$	0	0
$577$	−38.3835	−1.59793	−0.798964	−	0.601379i	$-0.794618\pi$
$577$	−38.3835	−1.59793	−0.798964	+	0.601379i	$0.794618\pi$
$578$	4.73526	0.196961
$579$	0	0
$580$	−13.8756	−0.576151
$581$	−4.81525	−0.199770
$582$	0	0
$583$	−2.44173	−0.101126
$584$	−33.3960	−1.38194
$585$	0	0
$586$	−0.136572	−0.00564172
$587$	−8.44224	−0.348448	−0.174224	−	0.984706i	$-0.555742\pi$
$587$	−8.44224	−0.348448	−0.174224	+	0.984706i	$0.555742\pi$
$588$	0	0
$589$	−7.87199	−0.324360
$590$	−26.0373	−1.07194
$591$	0	0
$592$	−3.79787	−0.156092
$593$	−8.53231	−0.350380	−0.175190	−	0.984535i	$-0.556054\pi$
$593$	−8.53231	−0.350380	−0.175190	+	0.984535i	$0.556054\pi$
$594$	0	0
$595$	4.70516	0.192893
$596$	43.4022	1.77782
$597$	0	0
$598$	−83.9716	−3.43385
$599$	44.8675	1.83324	0.916618	−	0.399765i	$-0.130908\pi$
$599$	44.8675	1.83324	0.916618	+	0.399765i	$0.130908\pi$
$600$	0	0
$601$	12.8649	0.524770	0.262385	−	0.964963i	$-0.415491\pi$
$601$	12.8649	0.524770	0.262385	+	0.964963i	$0.415491\pi$
$602$	−7.30053	−0.297547
$603$	0	0
$604$	43.7931	1.78192
$605$	13.2048	0.536850
$606$	0	0
$607$	−37.6254	−1.52717	−0.763585	−	0.645708i	$-0.776563\pi$
$607$	−37.6254	−1.52717	−0.763585	+	0.645708i	$0.776563\pi$
$608$	29.0191	1.17688
$609$	0	0
$610$	−13.0669	−0.529065
$611$	28.1171	1.13749
$612$	0	0
$613$	−26.7751	−1.08144	−0.540718	−	0.841204i	$-0.681848\pi$
$613$	−26.7751	−1.08144	−0.540718	+	0.841204i	$0.681848\pi$
$614$	−61.3645	−2.47647
$615$	0	0
$616$	1.04280	0.0420157
$617$	45.1369	1.81714	0.908571	−	0.417730i	$-0.137174\pi$
$617$	45.1369	1.81714	0.908571	+	0.417730i	$0.137174\pi$
$618$	0	0
$619$	0.494593	0.0198794	0.00993969	−	0.999951i	$-0.496836\pi$
$619$	0.494593	0.0198794	0.00993969	+	0.999951i	$0.496836\pi$
$620$	6.35460	0.255207
$621$	0	0
$622$	1.25646	0.0503795
$623$	14.8745	0.595935
$624$	0	0
$625$	4.93413	0.197365
$626$	−68.1983	−2.72575
$627$	0	0
$628$	29.0620	1.15970
$629$	−34.8227	−1.38847
$630$	0	0
$631$	−24.9411	−0.992890	−0.496445	−	0.868068i	$-0.665362\pi$
$631$	−24.9411	−0.992890	−0.496445	+	0.868068i	$0.665362\pi$
$632$	−31.2628	−1.24357
$633$	0	0
$634$	−48.2809	−1.91748
$635$	1.21847	0.0483536
$636$	0	0
$637$	6.29644	0.249474
$638$	3.31813	0.131366
$639$	0	0
$640$	−21.0986	−0.833997
$641$	9.50497	0.375424	0.187712	−	0.982224i	$-0.439893\pi$
$641$	9.50497	0.375424	0.187712	+	0.982224i	$0.439893\pi$
$642$	0	0
$643$	−10.2435	−0.403964	−0.201982	−	0.979389i	$-0.564738\pi$
$643$	−10.2435	−0.403964	−0.201982	+	0.979389i	$0.564738\pi$
$644$	18.4701	0.727823
$645$	0	0
$646$	41.4906	1.63243
$647$	−30.1171	−1.18403	−0.592014	−	0.805928i	$-0.701667\pi$
$647$	−30.1171	−1.18403	−0.592014	+	0.805928i	$0.701667\pi$
$648$	0	0
$649$	3.80361	0.149305
$650$	50.1804	1.96824
$651$	0	0
$652$	−6.36596	−0.249310
$653$	−0.123624	−0.00483779	−0.00241890	−	0.999997i	$-0.500770\pi$
$653$	−0.123624	−0.00483779	−0.00241890	+	0.999997i	$0.500770\pi$
$654$	0	0
$655$	22.0842	0.862900
$656$	3.09211	0.120727
$657$	0	0
$658$	−10.1239	−0.394672
$659$	25.1499	0.979701	0.489850	−	0.871806i	$-0.337051\pi$
$659$	25.1499	0.979701	0.489850	+	0.871806i	$0.337051\pi$
$660$	0	0
$661$	26.1903	1.01869	0.509343	−	0.860563i	$-0.329888\pi$
$661$	26.1903	1.01869	0.509343	+	0.860563i	$0.329888\pi$
$662$	−58.2158	−2.26262
$663$	0	0
$664$	12.4432	0.482889
$665$	−5.77475	−0.223935
$666$	0	0
$667$	21.3350	0.826094
$668$	1.17396	0.0454220
$669$	0	0
$670$	22.0416	0.851542
$671$	1.90886	0.0736907
$672$	0	0
$673$	−36.2724	−1.39820	−0.699100	−	0.715024i	$-0.746416\pi$
$673$	−36.2724	−1.39820	−0.699100	+	0.715024i	$0.746416\pi$
$674$	79.6866	3.06941
$675$	0	0
$676$	83.6611	3.21773
$677$	12.8070	0.492213	0.246106	−	0.969243i	$-0.420849\pi$
$677$	12.8070	0.492213	0.246106	+	0.969243i	$0.420849\pi$
$678$	0	0
$679$	−9.51735	−0.365242
$680$	−12.1587	−0.466264
$681$	0	0
$682$	−1.51961	−0.0581887
$683$	−0.284197	−0.0108745	−0.00543725	−	0.999985i	$-0.501731\pi$
$683$	−0.284197	−0.0108745	−0.00543725	+	0.999985i	$0.501731\pi$
$684$	0	0
$685$	21.4836	0.820846
$686$	−2.26712	−0.0865590
$687$	0	0
$688$	−1.35618	−0.0517038
$689$	38.0980	1.45142
$690$	0	0
$691$	−11.6706	−0.443969	−0.221984	−	0.975050i	$-0.571253\pi$
$691$	−11.6706	−0.443969	−0.221984	+	0.975050i	$0.571253\pi$
$692$	−11.7830	−0.447921
$693$	0	0
$694$	46.9668	1.78284
$695$	−2.19730	−0.0833485
$696$	0	0
$697$	28.3516	1.07389
$698$	68.1496	2.57950
$699$	0	0
$700$	−11.0375	−0.417178
$701$	41.0436	1.55020	0.775099	−	0.631840i	$-0.217700\pi$
$701$	41.0436	1.55020	0.775099	+	0.631840i	$0.217700\pi$
$702$	0	0
$703$	42.7388	1.61192
$704$	5.26193	0.198316
$705$	0	0
$706$	−40.9577	−1.54146
$707$	4.14790	0.155998
$708$	0	0
$709$	30.8323	1.15793	0.578966	−	0.815351i	$-0.303456\pi$
$709$	30.8323	1.15793	0.578966	+	0.815351i	$0.303456\pi$
$710$	−7.90840	−0.296797
$711$	0	0
$712$	−38.4375	−1.44051
$713$	−9.77081	−0.365920
$714$	0	0
$715$	3.09600	0.115784
$716$	73.2496	2.73746
$717$	0	0
$718$	15.1754	0.566339
$719$	42.9266	1.60089	0.800446	−	0.599405i	$-0.204596\pi$
$719$	42.9266	1.60089	0.800446	+	0.599405i	$0.204596\pi$
$720$	0	0
$721$	−13.9983	−0.521324
$722$	−7.84716	−0.292041
$723$	0	0
$724$	−23.2923	−0.865652
$725$	−12.7495	−0.473506
$726$	0	0
$727$	9.93051	0.368302	0.184151	−	0.982898i	$-0.441046\pi$
$727$	9.93051	0.368302	0.184151	+	0.982898i	$0.441046\pi$
$728$	−16.2707	−0.603034
$729$	0	0
$730$	35.7003	1.32133
$731$	−12.4348	−0.459917
$732$	0	0
$733$	46.0680	1.70156	0.850780	−	0.525522i	$-0.176130\pi$
$733$	46.0680	1.70156	0.850780	+	0.525522i	$0.176130\pi$
$734$	49.1348	1.81360
$735$	0	0
$736$	36.0188	1.32767
$737$	−3.21991	−0.118607
$738$	0	0
$739$	−21.5802	−0.793841	−0.396920	−	0.917853i	$-0.629921\pi$
$739$	−21.5802	−0.793841	−0.396920	+	0.917853i	$0.629921\pi$
$740$	−34.5005	−1.26826
$741$	0	0
$742$	−13.7177	−0.503593
$743$	30.0182	1.10126	0.550630	−	0.834749i	$-0.314387\pi$
$743$	30.0182	1.10126	0.550630	+	0.834749i	$0.314387\pi$
$744$	0	0
$745$	−16.8431	−0.617084
$746$	38.8016	1.42063
$747$	0	0
$748$	4.89275	0.178897
$749$	19.6665	0.718599
$750$	0	0
$751$	15.8616	0.578796	0.289398	−	0.957209i	$-0.406545\pi$
$751$	15.8616	0.578796	0.289398	+	0.957209i	$0.406545\pi$
$752$	−1.88066	−0.0685807
$753$	0	0
$754$	−51.7724	−1.88544
$755$	−16.9948	−0.618504
$756$	0	0
$757$	37.6885	1.36981	0.684905	−	0.728632i	$-0.259844\pi$
$757$	37.6885	1.36981	0.684905	+	0.728632i	$0.259844\pi$
$758$	30.3298	1.10163
$759$	0	0
$760$	14.9226	0.541301
$761$	6.65557	0.241264	0.120632	−	0.992697i	$-0.461508\pi$
$761$	6.65557	0.241264	0.120632	+	0.992697i	$0.461508\pi$
$762$	0	0
$763$	−6.41208	−0.232133
$764$	−58.0625	−2.10063
$765$	0	0
$766$	44.3840	1.60366
$767$	−59.3473	−2.14291
$768$	0	0
$769$	−12.3964	−0.447027	−0.223513	−	0.974701i	$-0.571753\pi$
$769$	−12.3964	−0.447027	−0.223513	+	0.974701i	$0.571753\pi$
$770$	−1.11476	−0.0401730
$771$	0	0
$772$	−50.4525	−1.81582
$773$	17.9252	0.644725	0.322362	−	0.946616i	$-0.395523\pi$
$773$	17.9252	0.644725	0.322362	+	0.946616i	$0.395523\pi$
$774$	0	0
$775$	5.83892	0.209740
$776$	24.5940	0.882872
$777$	0	0
$778$	57.1369	2.04846
$779$	−34.7965	−1.24672
$780$	0	0
$781$	1.15528	0.0413393
$782$	51.4986	1.84159
$783$	0	0
$784$	−0.421149	−0.0150410
$785$	−11.2781	−0.402532
$786$	0	0
$787$	35.2116	1.25516	0.627578	−	0.778553i	$-0.284046\pi$
$787$	35.2116	1.25516	0.627578	+	0.778553i	$0.284046\pi$
$788$	−47.8276	−1.70379
$789$	0	0
$790$	33.4199	1.18903
$791$	3.70618	0.131777
$792$	0	0
$793$	−29.7837	−1.05765
$794$	−18.3985	−0.652937
$795$	0	0
$796$	−35.1450	−1.24568
$797$	−50.5215	−1.78956	−0.894782	−	0.446504i	$-0.852669\pi$
$797$	−50.5215	−1.78956	−0.894782	+	0.446504i	$0.852669\pi$
$798$	0	0
$799$	−17.2438	−0.610042
$800$	−21.5244	−0.761004
$801$	0	0
$802$	−1.58769	−0.0560634
$803$	−5.21521	−0.184041
$804$	0	0
$805$	−7.16768	−0.252628
$806$	23.7103	0.835158
$807$	0	0
$808$	−10.7187	−0.377081
$809$	41.9172	1.47373	0.736864	−	0.676041i	$-0.236306\pi$
$809$	41.9172	1.47373	0.736864	+	0.676041i	$0.236306\pi$
$810$	0	0
$811$	49.5079	1.73846	0.869228	−	0.494411i	$-0.164616\pi$
$811$	49.5079	1.73846	0.869228	+	0.494411i	$0.164616\pi$
$812$	11.3877	0.399629
$813$	0	0
$814$	8.25027	0.289172
$815$	2.47044	0.0865357
$816$	0	0
$817$	15.2615	0.533933
$818$	24.7537	0.865491
$819$	0	0
$820$	28.0892	0.980919
$821$	7.26516	0.253556	0.126778	−	0.991931i	$-0.459536\pi$
$821$	7.26516	0.253556	0.126778	+	0.991931i	$0.459536\pi$
$822$	0	0
$823$	−38.5363	−1.34329	−0.671645	−	0.740873i	$-0.734412\pi$
$823$	−38.5363	−1.34329	−0.671645	+	0.740873i	$0.734412\pi$
$824$	36.1733	1.26016
$825$	0	0
$826$	21.3688	0.743516
$827$	30.6705	1.06652	0.533260	−	0.845952i	$-0.320967\pi$
$827$	30.6705	1.06652	0.533260	+	0.845952i	$0.320967\pi$
$828$	0	0
$829$	32.2744	1.12094	0.560468	−	0.828176i	$-0.310621\pi$
$829$	32.2744	1.12094	0.560468	+	0.828176i	$0.310621\pi$
$830$	−13.3018	−0.461711
$831$	0	0
$832$	−82.1014	−2.84635
$833$	−3.86152	−0.133794
$834$	0	0
$835$	−0.455580	−0.0157660
$836$	−6.00499	−0.207687
$837$	0	0
$838$	−10.9498	−0.378254
$839$	27.3044	0.942654	0.471327	−	0.881958i	$-0.343775\pi$
$839$	27.3044	0.942654	0.471327	+	0.881958i	$0.343775\pi$
$840$	0	0
$841$	−15.8460	−0.546413
$842$	−4.21861	−0.145383
$843$	0	0
$844$	42.0502	1.44743
$845$	−32.4664	−1.11688
$846$	0	0
$847$	−10.8372	−0.372369
$848$	−2.54826	−0.0875075
$849$	0	0
$850$	−30.7750	−1.05557
$851$	53.0478	1.81846
$852$	0	0
$853$	−43.5379	−1.49071	−0.745355	−	0.666668i	$-0.767720\pi$
$853$	−43.5379	−1.49071	−0.745355	+	0.666668i	$0.767720\pi$
$854$	10.7240	0.366969
$855$	0	0
$856$	−50.8206	−1.73701
$857$	−13.9649	−0.477032	−0.238516	−	0.971139i	$-0.576661\pi$
$857$	−13.9649	−0.477032	−0.238516	+	0.971139i	$0.576661\pi$
$858$	0	0
$859$	−19.7875	−0.675141	−0.337571	−	0.941300i	$-0.609605\pi$
$859$	−19.7875	−0.675141	−0.337571	+	0.941300i	$0.609605\pi$
$860$	−12.3197	−0.420100
$861$	0	0
$862$	18.7548	0.638793
$863$	5.63431	0.191794	0.0958971	−	0.995391i	$-0.469428\pi$
$863$	5.63431	0.191794	0.0958971	+	0.995391i	$0.469428\pi$
$864$	0	0
$865$	4.57262	0.155474
$866$	70.2879	2.38848
$867$	0	0
$868$	−5.21522	−0.177016
$869$	−4.88208	−0.165613
$870$	0	0
$871$	50.2399	1.70231
$872$	16.5696	0.561117
$873$	0	0
$874$	−63.2055	−2.13796
$875$	10.3757	0.350762
$876$	0	0
$877$	−19.5022	−0.658543	−0.329271	−	0.944235i	$-0.606803\pi$
$877$	−19.5022	−0.658543	−0.329271	+	0.944235i	$0.606803\pi$
$878$	−37.5098	−1.26589
$879$	0	0
$880$	−0.207082	−0.00698072
$881$	−49.5689	−1.67002	−0.835010	−	0.550235i	$-0.814538\pi$
$881$	−49.5689	−1.67002	−0.835010	+	0.550235i	$0.814538\pi$
$882$	0	0
$883$	43.7367	1.47186	0.735928	−	0.677060i	$-0.236746\pi$
$883$	43.7367	1.47186	0.735928	+	0.677060i	$0.236746\pi$
$884$	−76.3411	−2.56763
$885$	0	0
$886$	36.4009	1.22291
$887$	40.1921	1.34952	0.674760	−	0.738037i	$-0.264247\pi$
$887$	40.1921	1.34952	0.674760	+	0.738037i	$0.264247\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	41.0897	1.37733
$891$	0	0
$892$	−72.1615	−2.41615
$893$	21.1637	0.708218
$894$	0	0
$895$	−28.4260	−0.950175
$896$	17.3157	0.578475
$897$	0	0
$898$	−29.6430	−0.989201
$899$	−6.02416	−0.200917
$900$	0	0
$901$	−23.3650	−0.778400
$902$	−6.71711	−0.223655
$903$	0	0
$904$	−9.57721	−0.318533
$905$	9.03905	0.300468
$906$	0	0
$907$	43.4324	1.44215	0.721074	−	0.692858i	$-0.243649\pi$
$907$	43.4324	1.44215	0.721074	+	0.692858i	$0.243649\pi$
$908$	−35.8889	−1.19101
$909$	0	0
$910$	17.3934	0.576586
$911$	56.8041	1.88201	0.941003	−	0.338400i	$-0.109886\pi$
$911$	56.8041	1.88201	0.941003	+	0.338400i	$0.109886\pi$
$912$	0	0
$913$	1.94316	0.0643093
$914$	9.97489	0.329940
$915$	0	0
$916$	73.2481	2.42018
$917$	−18.1245	−0.598523
$918$	0	0
$919$	3.03048	0.0999662	0.0499831	−	0.998750i	$-0.484083\pi$
$919$	3.03048	0.0999662	0.0499831	+	0.998750i	$0.484083\pi$
$920$	18.5221	0.610657
$921$	0	0
$922$	11.6281	0.382950
$923$	−18.0258	−0.593325
$924$	0	0
$925$	−31.7008	−1.04232
$926$	−33.8473	−1.11229
$927$	0	0
$928$	22.2073	0.728991
$929$	48.0527	1.57656	0.788279	−	0.615317i	$-0.210972\pi$
$929$	48.0527	1.57656	0.788279	+	0.615317i	$0.210972\pi$
$930$	0	0
$931$	4.73933	0.155325
$932$	34.2602	1.12223
$933$	0	0
$934$	−62.0866	−2.03154
$935$	−1.89873	−0.0620952
$936$	0	0
$937$	−22.3785	−0.731075	−0.365538	−	0.930797i	$-0.619115\pi$
$937$	−22.3785	−0.731075	−0.365538	+	0.930797i	$0.619115\pi$
$938$	−18.0896	−0.590645
$939$	0	0
$940$	−17.0843	−0.557227
$941$	14.8322	0.483515	0.241758	−	0.970337i	$-0.422276\pi$
$941$	14.8322	0.483515	0.241758	+	0.970337i	$0.422276\pi$
$942$	0	0
$943$	−43.1899	−1.40646
$944$	3.96956	0.129198
$945$	0	0
$946$	2.94608	0.0957853
$947$	3.48245	0.113165	0.0565823	−	0.998398i	$-0.481980\pi$
$947$	3.48245	0.113165	0.0565823	+	0.998398i	$0.481980\pi$
$948$	0	0
$949$	81.3724	2.64146
$950$	37.7709	1.22545
$951$	0	0
$952$	9.97862	0.323409
$953$	19.4366	0.629612	0.314806	−	0.949156i	$-0.398060\pi$
$953$	19.4366	0.629612	0.314806	+	0.949156i	$0.398060\pi$
$954$	0	0
$955$	22.5323	0.729129
$956$	−74.0810	−2.39595
$957$	0	0
$958$	35.8736	1.15902
$959$	−17.6316	−0.569354
$960$	0	0
$961$	−28.2411	−0.911004
$962$	−128.728	−4.15036
$963$	0	0
$964$	−32.5021	−1.04682
$965$	19.5791	0.630274
$966$	0	0
$967$	28.6414	0.921046	0.460523	−	0.887648i	$-0.347662\pi$
$967$	28.6414	0.921046	0.460523	+	0.887648i	$0.347662\pi$
$968$	28.0045	0.900098
$969$	0	0
$970$	−26.2909	−0.844151
$971$	−1.53315	−0.0492012	−0.0246006	−	0.999697i	$-0.507831\pi$
$971$	−1.53315	−0.0492012	−0.0246006	+	0.999697i	$0.507831\pi$
$972$	0	0
$973$	1.80333	0.0578120
$974$	−10.0297	−0.321374
$975$	0	0
$976$	1.99214	0.0637669
$977$	27.6378	0.884212	0.442106	−	0.896963i	$-0.354232\pi$
$977$	27.6378	0.884212	0.442106	+	0.896963i	$0.354232\pi$
$978$	0	0
$979$	−6.00251	−0.191841
$980$	−3.82579	−0.122210
$981$	0	0
$982$	−95.9334	−3.06136
$983$	−9.86003	−0.314486	−0.157243	−	0.987560i	$-0.550261\pi$
$983$	−9.86003	−0.314486	−0.157243	+	0.987560i	$0.550261\pi$
$984$	0	0
$985$	18.5605	0.591385
$986$	31.7513	1.01117
$987$	0	0
$988$	93.6953	2.98084
$989$	18.9428	0.602345
$990$	0	0
$991$	34.1339	1.08430	0.542150	−	0.840282i	$-0.317611\pi$
$991$	34.1339	1.08430	0.542150	+	0.840282i	$0.317611\pi$
$992$	−10.1703	−0.322907
$993$	0	0
$994$	6.49042	0.205864
$995$	13.6387	0.432377
$996$	0	0
$997$	−1.91194	−0.0605517	−0.0302759	−	0.999542i	$-0.509639\pi$
$997$	−1.91194	−0.0605517	−0.0302759	+	0.999542i	$0.509639\pi$
$998$	44.4224	1.40617
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.ba.1.7	✓	40
3.2	odd	2	inner	8001.2.a.ba.1.34	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.ba.1.7	✓	40	1.1	even	1	trivial
8001.2.a.ba.1.34	yes	40	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)

\(f(q)\)	\(=\)	\(q-2.26712 q^{2} +3.13982 q^{4} -1.21847 q^{5} +1.00000 q^{7} -2.58412 q^{8} +O(q^{10})\)	\(q-2.26712 q^{2} +3.13982 q^{4} -1.21847 q^{5} +1.00000 q^{7} -2.58412 q^{8} +2.76242 q^{10} -0.403543 q^{11} +6.29644 q^{13} -2.26712 q^{14} -0.421149 q^{16} -3.86152 q^{17} +4.73933 q^{19} -3.82579 q^{20} +0.914879 q^{22} +5.88252 q^{23} -3.51532 q^{25} -14.2748 q^{26} +3.13982 q^{28} +3.62685 q^{29} -1.66099 q^{31} +6.12303 q^{32} +8.75452 q^{34} -1.21847 q^{35} +9.01788 q^{37} -10.7446 q^{38} +3.14868 q^{40} -7.34207 q^{41} +3.22018 q^{43} -1.26705 q^{44} -13.3364 q^{46} +4.46555 q^{47} +1.00000 q^{49} +7.96966 q^{50} +19.7697 q^{52} +6.05072 q^{53} +0.491706 q^{55} -2.58412 q^{56} -8.22249 q^{58} -9.42554 q^{59} -4.73025 q^{61} +3.76566 q^{62} -13.0393 q^{64} -7.67204 q^{65} +7.97909 q^{67} -12.1245 q^{68} +2.76242 q^{70} -2.86285 q^{71} +12.9236 q^{73} -20.4446 q^{74} +14.8807 q^{76} -0.403543 q^{77} +12.0981 q^{79} +0.513159 q^{80} +16.6453 q^{82} -4.81525 q^{83} +4.70516 q^{85} -7.30053 q^{86} +1.04280 q^{88} +14.8745 q^{89} +6.29644 q^{91} +18.4701 q^{92} -10.1239 q^{94} -5.77475 q^{95} -9.51735 q^{97} -2.26712 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 54 q^{4} + 40 q^{7}+O(q^{10}) \) 40 * q + 54 * q^4 + 40 * q^7	\( 40 q + 54 q^{4} + 40 q^{7} + 20 q^{10} + 10 q^{13} + 90 q^{16} + 38 q^{19} + 14 q^{22} + 84 q^{25} + 54 q^{28} + 66 q^{31} + 22 q^{34} + 40 q^{37} + 26 q^{40} + 38 q^{43} + 28 q^{46} + 40 q^{49} + 28 q^{52} + 60 q^{55} + 42 q^{58} + 54 q^{61} + 124 q^{64} + 48 q^{67} + 20 q^{70} + 16 q^{76} + 102 q^{79} + 48 q^{82} + 104 q^{85} + 48 q^{88} + 10 q^{91} - 10 q^{94} + 8 q^{97}+O(q^{100}) \) 40 * q + 54 * q^4 + 40 * q^7 + 20 * q^10 + 10 * q^13 + 90 * q^16 + 38 * q^19 + 14 * q^22 + 84 * q^25 + 54 * q^28 + 66 * q^31 + 22 * q^34 + 40 * q^37 + 26 * q^40 + 38 * q^43 + 28 * q^46 + 40 * q^49 + 28 * q^52 + 60 * q^55 + 42 * q^58 + 54 * q^61 + 124 * q^64 + 48 * q^67 + 20 * q^70 + 16 * q^76 + 102 * q^79 + 48 * q^82 + 104 * q^85 + 48 * q^88 + 10 * q^91 - 10 * q^94 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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