LMFDB - Embedded newform 8019.2.a.l.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8019,2,Mod(1,8019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8019, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8019 = 3^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8019.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:	$64.0320373809$
Analytic rank:	$0$
Dimension:	$51$
Twist minimal:	no (minimal twist has level 297)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8019.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.65065 q^{2} +5.02592 q^{4} -0.556977 q^{5} +3.79141 q^{7} -8.02065 q^{8} +O(q^{10})$	\(q-2.65065 q^{2} +5.02592 q^{4} -0.556977 q^{5} +3.79141 q^{7} -8.02065 q^{8} +1.47635 q^{10} +1.00000 q^{11} +2.27225 q^{13} -10.0497 q^{14} +11.2080 q^{16} +6.78209 q^{17} -1.19748 q^{19} -2.79932 q^{20} -2.65065 q^{22} -6.56677 q^{23} -4.68978 q^{25} -6.02294 q^{26} +19.0554 q^{28} -5.80966 q^{29} +6.58724 q^{31} -13.6673 q^{32} -17.9769 q^{34} -2.11173 q^{35} -10.0977 q^{37} +3.17408 q^{38} +4.46732 q^{40} +0.148388 q^{41} +2.78134 q^{43} +5.02592 q^{44} +17.4062 q^{46} -5.16941 q^{47} +7.37482 q^{49} +12.4309 q^{50} +11.4202 q^{52} +0.0350266 q^{53} -0.556977 q^{55} -30.4096 q^{56} +15.3994 q^{58} +6.28511 q^{59} +8.74946 q^{61} -17.4604 q^{62} +13.8110 q^{64} -1.26559 q^{65} +7.58278 q^{67} +34.0863 q^{68} +5.59745 q^{70} -7.52724 q^{71} +0.629010 q^{73} +26.7655 q^{74} -6.01842 q^{76} +3.79141 q^{77} +1.16111 q^{79} -6.24263 q^{80} -0.393323 q^{82} +6.23807 q^{83} -3.77747 q^{85} -7.37234 q^{86} -8.02065 q^{88} +4.77319 q^{89} +8.61506 q^{91} -33.0041 q^{92} +13.7023 q^{94} +0.666967 q^{95} -9.12923 q^{97} -19.5480 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7}+O(q^{10}) $ 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7	$ 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7} + 18 q^{10} + 51 q^{11} + 30 q^{13} + 12 q^{14} + 78 q^{16} + 30 q^{19} + 18 q^{20} + 3 q^{23} + 75 q^{25} + 9 q^{26} + 36 q^{28} + 42 q^{31} - 15 q^{32} + 42 q^{34} - 9 q^{35} + 48 q^{37} - 3 q^{38} + 54 q^{40} + 18 q^{43} + 60 q^{44} + 42 q^{46} + 30 q^{47} + 99 q^{49} - 30 q^{50} + 60 q^{52} + 18 q^{53} + 6 q^{55} + 21 q^{56} + 30 q^{58} + 24 q^{59} + 99 q^{61} + 114 q^{64} - 15 q^{65} + 39 q^{67} - 39 q^{68} + 48 q^{70} + 30 q^{71} + 69 q^{73} + 90 q^{76} + 12 q^{77} + 48 q^{79} + 42 q^{80} + 42 q^{82} - 21 q^{83} + 84 q^{85} + 24 q^{86} + 15 q^{89} + 69 q^{91} - 66 q^{92} + 66 q^{94} - 12 q^{95} + 72 q^{97} - 57 q^{98}+O(q^{100}) $ 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7 + 18 * q^10 + 51 * q^11 + 30 * q^13 + 12 * q^14 + 78 * q^16 + 30 * q^19 + 18 * q^20 + 3 * q^23 + 75 * q^25 + 9 * q^26 + 36 * q^28 + 42 * q^31 - 15 * q^32 + 42 * q^34 - 9 * q^35 + 48 * q^37 - 3 * q^38 + 54 * q^40 + 18 * q^43 + 60 * q^44 + 42 * q^46 + 30 * q^47 + 99 * q^49 - 30 * q^50 + 60 * q^52 + 18 * q^53 + 6 * q^55 + 21 * q^56 + 30 * q^58 + 24 * q^59 + 99 * q^61 + 114 * q^64 - 15 * q^65 + 39 * q^67 - 39 * q^68 + 48 * q^70 + 30 * q^71 + 69 * q^73 + 90 * q^76 + 12 * q^77 + 48 * q^79 + 42 * q^80 + 42 * q^82 - 21 * q^83 + 84 * q^85 + 24 * q^86 + 15 * q^89 + 69 * q^91 - 66 * q^92 + 66 * q^94 - 12 * q^95 + 72 * q^97 - 57 * q^98

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 $

$2$	−2.65065	−1.87429	−0.937145	−	0.348941i	$-0.886541\pi$
$2$	−2.65065	−1.87429	−0.937145	+	0.348941i	$0.886541\pi$
$3$	0	0
$3$	0	0
$4$	5.02592	2.51296
$5$	−0.556977	−0.249088	−0.124544	−	0.992214i	$-0.539747\pi$
$5$	−0.556977	−0.249088	−0.124544	+	0.992214i	$0.539747\pi$
$6$	0	0
$7$	3.79141	1.43302	0.716510	−	0.697577i	$-0.245738\pi$
$7$	3.79141	1.43302	0.716510	+	0.697577i	$0.245738\pi$
$8$	−8.02065	−2.83573
$9$	0	0
$10$	1.47635	0.466863
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.27225	0.630210	0.315105	−	0.949057i	$-0.397960\pi$
$13$	2.27225	0.630210	0.315105	+	0.949057i	$0.397960\pi$
$14$	−10.0497	−2.68589
$15$	0	0
$16$	11.2080	2.80201
$17$	6.78209	1.64490	0.822449	−	0.568838i	$-0.192607\pi$
$17$	6.78209	1.64490	0.822449	+	0.568838i	$0.192607\pi$
$18$	0	0
$19$	−1.19748	−0.274720	−0.137360	−	0.990521i	$-0.543862\pi$
$19$	−1.19748	−0.274720	−0.137360	+	0.990521i	$0.543862\pi$
$20$	−2.79932	−0.625948
$21$	0	0
$22$	−2.65065	−0.565120
$23$	−6.56677	−1.36927	−0.684634	−	0.728887i	$-0.740038\pi$
$23$	−6.56677	−1.36927	−0.684634	+	0.728887i	$0.740038\pi$
$24$	0	0
$25$	−4.68978	−0.937955
$26$	−6.02294	−1.18120
$27$	0	0
$28$	19.0554	3.60112
$29$	−5.80966	−1.07883	−0.539414	−	0.842041i	$-0.681354\pi$
$29$	−5.80966	−1.07883	−0.539414	+	0.842041i	$0.681354\pi$
$30$	0	0
$31$	6.58724	1.18310	0.591552	−	0.806267i	$-0.298516\pi$
$31$	6.58724	1.18310	0.591552	+	0.806267i	$0.298516\pi$
$32$	−13.6673	−2.41605
$33$	0	0
$34$	−17.9769	−3.08302
$35$	−2.11173	−0.356948
$36$	0	0
$37$	−10.0977	−1.66006	−0.830028	−	0.557721i	$-0.811676\pi$
$37$	−10.0977	−1.66006	−0.830028	+	0.557721i	$0.811676\pi$
$38$	3.17408	0.514904
$39$	0	0
$40$	4.46732	0.706345
$41$	0.148388	0.0231743	0.0115871	−	0.999933i	$-0.496312\pi$
$41$	0.148388	0.0231743	0.0115871	+	0.999933i	$0.496312\pi$
$42$	0	0
$43$	2.78134	0.424150	0.212075	−	0.977253i	$-0.431978\pi$
$43$	2.78134	0.424150	0.212075	+	0.977253i	$0.431978\pi$
$44$	5.02592	0.757686
$45$	0	0
$46$	17.4062	2.56640
$47$	−5.16941	−0.754036	−0.377018	−	0.926206i	$-0.623051\pi$
$47$	−5.16941	−0.754036	−0.377018	+	0.926206i	$0.623051\pi$
$48$	0	0
$49$	7.37482	1.05355
$50$	12.4309	1.75800
$51$	0	0
$52$	11.4202	1.58369
$53$	0.0350266	0.00481127	0.00240564	−	0.999997i	$-0.499234\pi$
$53$	0.0350266	0.00481127	0.00240564	+	0.999997i	$0.499234\pi$
$54$	0	0
$55$	−0.556977	−0.0751028
$56$	−30.4096	−4.06365
$57$	0	0
$58$	15.3994	2.02203
$59$	6.28511	0.818251	0.409126	−	0.912478i	$-0.365834\pi$
$59$	6.28511	0.818251	0.409126	+	0.912478i	$0.365834\pi$
$60$	0	0
$61$	8.74946	1.12025	0.560127	−	0.828407i	$-0.310753\pi$
$61$	8.74946	1.12025	0.560127	+	0.828407i	$0.310753\pi$
$62$	−17.4604	−2.21748
$63$	0	0
$64$	13.8110	1.72637
$65$	−1.26559	−0.156978
$66$	0	0
$67$	7.58278	0.926384	0.463192	−	0.886258i	$-0.346704\pi$
$67$	7.58278	0.926384	0.463192	+	0.886258i	$0.346704\pi$
$68$	34.0863	4.13357
$69$	0	0
$70$	5.59745	0.669024
$71$	−7.52724	−0.893319	−0.446660	−	0.894704i	$-0.647387\pi$
$71$	−7.52724	−0.893319	−0.446660	+	0.894704i	$0.647387\pi$
$72$	0	0
$73$	0.629010	0.0736201	0.0368101	−	0.999322i	$-0.488280\pi$
$73$	0.629010	0.0736201	0.0368101	+	0.999322i	$0.488280\pi$
$74$	26.7655	3.11143
$75$	0	0
$76$	−6.01842	−0.690360
$77$	3.79141	0.432072
$78$	0	0
$79$	1.16111	0.130635	0.0653175	−	0.997865i	$-0.479194\pi$
$79$	1.16111	0.130635	0.0653175	+	0.997865i	$0.479194\pi$
$80$	−6.24263	−0.697947
$81$	0	0
$82$	−0.393323	−0.0434353
$83$	6.23807	0.684717	0.342359	−	0.939569i	$-0.388774\pi$
$83$	6.23807	0.684717	0.342359	+	0.939569i	$0.388774\pi$
$84$	0	0
$85$	−3.77747	−0.409724
$86$	−7.37234	−0.794980
$87$	0	0
$88$	−8.02065	−0.855004
$89$	4.77319	0.505957	0.252978	−	0.967472i	$-0.418590\pi$
$89$	4.77319	0.505957	0.252978	+	0.967472i	$0.418590\pi$
$90$	0	0
$91$	8.61506	0.903103
$92$	−33.0041	−3.44091
$93$	0	0
$94$	13.7023	1.41328
$95$	0.666967	0.0684293
$96$	0	0
$97$	−9.12923	−0.926933	−0.463466	−	0.886114i	$-0.653395\pi$
$97$	−9.12923	−0.926933	−0.463466	+	0.886114i	$0.653395\pi$
$98$	−19.5480	−1.97465
$99$	0	0
$100$	−23.5704	−2.35704
$101$	−11.4645	−1.14076	−0.570379	−	0.821381i	$-0.693204\pi$
$101$	−11.4645	−1.14076	−0.570379	+	0.821381i	$0.693204\pi$
$102$	0	0
$103$	18.0615	1.77966	0.889828	−	0.456297i	$-0.150825\pi$
$103$	18.0615	1.77966	0.889828	+	0.456297i	$0.150825\pi$
$104$	−18.2249	−1.78710
$105$	0	0
$106$	−0.0928431	−0.00901772
$107$	5.35682	0.517863	0.258932	−	0.965896i	$-0.416630\pi$
$107$	5.35682	0.517863	0.258932	+	0.965896i	$0.416630\pi$
$108$	0	0
$109$	−11.1968	−1.07246	−0.536232	−	0.844071i	$-0.680153\pi$
$109$	−11.1968	−1.07246	−0.536232	+	0.844071i	$0.680153\pi$
$110$	1.47635	0.140764
$111$	0	0
$112$	42.4943	4.01534
$113$	20.9568	1.97145	0.985727	−	0.168352i	$-0.0538444\pi$
$113$	20.9568	1.97145	0.985727	+	0.168352i	$0.0538444\pi$
$114$	0	0
$115$	3.65754	0.341068
$116$	−29.1989	−2.71105
$117$	0	0
$118$	−16.6596	−1.53364
$119$	25.7137	2.35717
$120$	0	0
$121$	1.00000	0.0909091
$122$	−23.1917	−2.09968
$123$	0	0
$124$	33.1069	2.97309
$125$	5.39699	0.482721
$126$	0	0
$127$	−7.73798	−0.686635	−0.343318	−	0.939219i	$-0.611551\pi$
$127$	−7.73798	−0.686635	−0.343318	+	0.939219i	$0.611551\pi$
$128$	−9.27349	−0.819669
$129$	0	0
$130$	3.35464	0.294221
$131$	−0.550816	−0.0481250	−0.0240625	−	0.999710i	$-0.507660\pi$
$131$	−0.550816	−0.0481250	−0.0240625	+	0.999710i	$0.507660\pi$
$132$	0	0
$133$	−4.54013	−0.393679
$134$	−20.0993	−1.73631
$135$	0	0
$136$	−54.3968	−4.66448
$137$	20.7988	1.77696	0.888482	−	0.458911i	$-0.151760\pi$
$137$	20.7988	1.77696	0.888482	+	0.458911i	$0.151760\pi$
$138$	0	0
$139$	−2.64061	−0.223973	−0.111987	−	0.993710i	$-0.535721\pi$
$139$	−2.64061	−0.223973	−0.111987	+	0.993710i	$0.535721\pi$
$140$	−10.6134	−0.896996
$141$	0	0
$142$	19.9521	1.67434
$143$	2.27225	0.190015
$144$	0	0
$145$	3.23585	0.268723
$146$	−1.66728	−0.137985
$147$	0	0
$148$	−50.7504	−4.17166
$149$	16.0121	1.31176	0.655880	−	0.754865i	$-0.272298\pi$
$149$	16.0121	1.31176	0.655880	+	0.754865i	$0.272298\pi$
$150$	0	0
$151$	9.09796	0.740381	0.370191	−	0.928956i	$-0.379292\pi$
$151$	9.09796	0.740381	0.370191	+	0.928956i	$0.379292\pi$
$152$	9.60453	0.779030
$153$	0	0
$154$	−10.0497	−0.809828
$155$	−3.66894	−0.294697
$156$	0	0
$157$	−7.44633	−0.594282	−0.297141	−	0.954834i	$-0.596033\pi$
$157$	−7.44633	−0.594282	−0.297141	+	0.954834i	$0.596033\pi$
$158$	−3.07769	−0.244848
$159$	0	0
$160$	7.61236	0.601810
$161$	−24.8974	−1.96219
$162$	0	0
$163$	3.55705	0.278610	0.139305	−	0.990250i	$-0.455513\pi$
$163$	3.55705	0.278610	0.139305	+	0.990250i	$0.455513\pi$
$164$	0.745785	0.0582360
$165$	0	0
$166$	−16.5349	−1.28336
$167$	−1.61497	−0.124970	−0.0624850	−	0.998046i	$-0.519903\pi$
$167$	−1.61497	−0.124970	−0.0624850	+	0.998046i	$0.519903\pi$
$168$	0	0
$169$	−7.83686	−0.602836
$170$	10.0127	0.767942
$171$	0	0
$172$	13.9788	1.06587
$173$	13.8641	1.05407	0.527034	−	0.849844i	$-0.323304\pi$
$173$	13.8641	1.05407	0.527034	+	0.849844i	$0.323304\pi$
$174$	0	0
$175$	−17.7809	−1.34411
$176$	11.2080	0.844838
$177$	0	0
$178$	−12.6520	−0.948309
$179$	9.24661	0.691124	0.345562	−	0.938396i	$-0.387688\pi$
$179$	9.24661	0.691124	0.345562	+	0.938396i	$0.387688\pi$
$180$	0	0
$181$	23.0650	1.71441	0.857203	−	0.514978i	$-0.172200\pi$
$181$	23.0650	1.71441	0.857203	+	0.514978i	$0.172200\pi$
$182$	−22.8355	−1.69268
$183$	0	0
$184$	52.6698	3.88287
$185$	5.62421	0.413500
$186$	0	0
$187$	6.78209	0.495956
$188$	−25.9811	−1.89486
$189$	0	0
$190$	−1.76789	−0.128256
$191$	3.25804	0.235744	0.117872	−	0.993029i	$-0.462393\pi$
$191$	3.25804	0.235744	0.117872	+	0.993029i	$0.462393\pi$
$192$	0	0
$193$	−6.17662	−0.444603	−0.222302	−	0.974978i	$-0.571357\pi$
$193$	−6.17662	−0.444603	−0.222302	+	0.974978i	$0.571357\pi$
$194$	24.1984	1.73734
$195$	0	0
$196$	37.0653	2.64752
$197$	14.4985	1.03298	0.516489	−	0.856294i	$-0.327239\pi$
$197$	14.4985	1.03298	0.516489	+	0.856294i	$0.327239\pi$
$198$	0	0
$199$	16.7844	1.18981	0.594907	−	0.803794i	$-0.297189\pi$
$199$	16.7844	1.18981	0.594907	+	0.803794i	$0.297189\pi$
$200$	37.6150	2.65978
$201$	0	0
$202$	30.3883	2.13811
$203$	−22.0268	−1.54598
$204$	0	0
$205$	−0.0826486	−0.00577243
$206$	−47.8747	−3.33559
$207$	0	0
$208$	25.4675	1.76586
$209$	−1.19748	−0.0828311
$210$	0	0
$211$	21.8562	1.50464	0.752322	−	0.658796i	$-0.228934\pi$
$211$	21.8562	1.50464	0.752322	+	0.658796i	$0.228934\pi$
$212$	0.176041	0.0120905
$213$	0	0
$214$	−14.1990	−0.970625
$215$	−1.54914	−0.105651
$216$	0	0
$217$	24.9750	1.69541
$218$	29.6789	2.01011
$219$	0	0
$220$	−2.79932	−0.188730
$221$	15.4106	1.03663
$222$	0	0
$223$	−1.44052	−0.0964645	−0.0482322	−	0.998836i	$-0.515359\pi$
$223$	−1.44052	−0.0964645	−0.0482322	+	0.998836i	$0.515359\pi$
$224$	−51.8183	−3.46225
$225$	0	0
$226$	−55.5492	−3.69508
$227$	7.04073	0.467310	0.233655	−	0.972320i	$-0.424931\pi$
$227$	7.04073	0.467310	0.233655	+	0.972320i	$0.424931\pi$
$228$	0	0
$229$	22.3312	1.47569	0.737843	−	0.674972i	$-0.235844\pi$
$229$	22.3312	1.47569	0.737843	+	0.674972i	$0.235844\pi$
$230$	−9.69485	−0.639260
$231$	0	0
$232$	46.5972	3.05926
$233$	−19.4917	−1.27695	−0.638473	−	0.769644i	$-0.720434\pi$
$233$	−19.4917	−1.27695	−0.638473	+	0.769644i	$0.720434\pi$
$234$	0	0
$235$	2.87925	0.187821
$236$	31.5885	2.05623
$237$	0	0
$238$	−68.1580	−4.41802
$239$	21.2286	1.37317	0.686583	−	0.727052i	$-0.259110\pi$
$239$	21.2286	1.37317	0.686583	+	0.727052i	$0.259110\pi$
$240$	0	0
$241$	−8.45695	−0.544760	−0.272380	−	0.962190i	$-0.587811\pi$
$241$	−8.45695	−0.544760	−0.272380	+	0.962190i	$0.587811\pi$
$242$	−2.65065	−0.170390
$243$	0	0
$244$	43.9741	2.81515
$245$	−4.10761	−0.262426
$246$	0	0
$247$	−2.72097	−0.173131
$248$	−52.8339	−3.35496
$249$	0	0
$250$	−14.3055	−0.904759
$251$	−24.7174	−1.56015	−0.780075	−	0.625686i	$-0.784819\pi$
$251$	−24.7174	−1.56015	−0.780075	+	0.625686i	$0.784819\pi$
$252$	0	0
$253$	−6.56677	−0.412850
$254$	20.5107	1.28695
$255$	0	0
$256$	−3.04123	−0.190077
$257$	3.34928	0.208923	0.104461	−	0.994529i	$-0.466688\pi$
$257$	3.34928	0.208923	0.104461	+	0.994529i	$0.466688\pi$
$258$	0	0
$259$	−38.2847	−2.37889
$260$	−6.36078	−0.394479
$261$	0	0
$262$	1.46002	0.0902002
$263$	0.875411	0.0539802	0.0269901	−	0.999636i	$-0.491408\pi$
$263$	0.875411	0.0539802	0.0269901	+	0.999636i	$0.491408\pi$
$264$	0	0
$265$	−0.0195090	−0.00119843
$266$	12.0343	0.737868
$267$	0	0
$268$	38.1104	2.32797
$269$	4.42989	0.270095	0.135048	−	0.990839i	$-0.456881\pi$
$269$	4.42989	0.270095	0.135048	+	0.990839i	$0.456881\pi$
$270$	0	0
$271$	−11.4996	−0.698552	−0.349276	−	0.937020i	$-0.613572\pi$
$271$	−11.4996	−0.698552	−0.349276	+	0.937020i	$0.613572\pi$
$272$	76.0140	4.60903
$273$	0	0
$274$	−55.1303	−3.33055
$275$	−4.68978	−0.282804
$276$	0	0
$277$	−8.57871	−0.515445	−0.257722	−	0.966219i	$-0.582972\pi$
$277$	−8.57871	−0.515445	−0.257722	+	0.966219i	$0.582972\pi$
$278$	6.99931	0.419791
$279$	0	0
$280$	16.9375	1.01221
$281$	−11.3574	−0.677526	−0.338763	−	0.940872i	$-0.610009\pi$
$281$	−11.3574	−0.677526	−0.338763	+	0.940872i	$0.610009\pi$
$282$	0	0
$283$	−15.0774	−0.896257	−0.448129	−	0.893969i	$-0.647909\pi$
$283$	−15.0774	−0.896257	−0.448129	+	0.893969i	$0.647909\pi$
$284$	−37.8313	−2.24488
$285$	0	0
$286$	−6.02294	−0.356144
$287$	0.562599	0.0332092
$288$	0	0
$289$	28.9968	1.70569
$290$	−8.57709	−0.503664
$291$	0	0
$292$	3.16136	0.185004
$293$	11.6304	0.679454	0.339727	−	0.940524i	$-0.389665\pi$
$293$	11.6304	0.679454	0.339727	+	0.940524i	$0.389665\pi$
$294$	0	0
$295$	−3.50066	−0.203816
$296$	80.9903	4.70747
$297$	0	0
$298$	−42.4423	−2.45862
$299$	−14.9214	−0.862926
$300$	0	0
$301$	10.5452	0.607816
$302$	−24.1155	−1.38769
$303$	0	0
$304$	−13.4214	−0.769768
$305$	−4.87325	−0.279041
$306$	0	0
$307$	−2.46141	−0.140480	−0.0702399	−	0.997530i	$-0.522376\pi$
$307$	−2.46141	−0.140480	−0.0702399	+	0.997530i	$0.522376\pi$
$308$	19.0554	1.08578
$309$	0	0
$310$	9.72507	0.552347
$311$	19.7156	1.11797	0.558985	−	0.829178i	$-0.311191\pi$
$311$	19.7156	1.11797	0.558985	+	0.829178i	$0.311191\pi$
$312$	0	0
$313$	−31.7853	−1.79661	−0.898305	−	0.439372i	$-0.855201\pi$
$313$	−31.7853	−1.79661	−0.898305	+	0.439372i	$0.855201\pi$
$314$	19.7376	1.11386
$315$	0	0
$316$	5.83564	0.328281
$317$	22.5280	1.26530	0.632649	−	0.774439i	$-0.281968\pi$
$317$	22.5280	1.26530	0.632649	+	0.774439i	$0.281968\pi$
$318$	0	0
$319$	−5.80966	−0.325279
$320$	−7.69240	−0.430018
$321$	0	0
$322$	65.9941	3.67771
$323$	−8.12139	−0.451886
$324$	0	0
$325$	−10.6564	−0.591109
$326$	−9.42848	−0.522195
$327$	0	0
$328$	−1.19017	−0.0657159
$329$	−19.5994	−1.08055
$330$	0	0
$331$	25.6797	1.41149	0.705743	−	0.708468i	$-0.250613\pi$
$331$	25.6797	1.41149	0.705743	+	0.708468i	$0.250613\pi$
$332$	31.3521	1.72067
$333$	0	0
$334$	4.28071	0.234230
$335$	−4.22343	−0.230751
$336$	0	0
$337$	−21.7915	−1.18706	−0.593530	−	0.804812i	$-0.702266\pi$
$337$	−21.7915	−1.18706	−0.593530	+	0.804812i	$0.702266\pi$
$338$	20.7727	1.12989
$339$	0	0
$340$	−18.9853	−1.02962
$341$	6.58724	0.356719
$342$	0	0
$343$	1.42111	0.0767327
$344$	−22.3081	−1.20277
$345$	0	0
$346$	−36.7488	−1.97563
$347$	0.102787	0.00551789	0.00275894	−	0.999996i	$-0.499122\pi$
$347$	0.102787	0.00551789	0.00275894	+	0.999996i	$0.499122\pi$
$348$	0	0
$349$	11.8822	0.636041	0.318020	−	0.948084i	$-0.396982\pi$
$349$	11.8822	0.636041	0.318020	+	0.948084i	$0.396982\pi$
$350$	47.1308	2.51925
$351$	0	0
$352$	−13.6673	−0.728468
$353$	−24.0732	−1.28129	−0.640644	−	0.767838i	$-0.721333\pi$
$353$	−24.0732	−1.28129	−0.640644	+	0.767838i	$0.721333\pi$
$354$	0	0
$355$	4.19250	0.222515
$356$	23.9897	1.27145
$357$	0	0
$358$	−24.5095	−1.29537
$359$	−6.86403	−0.362270	−0.181135	−	0.983458i	$-0.557977\pi$
$359$	−6.86403	−0.362270	−0.181135	+	0.983458i	$0.557977\pi$
$360$	0	0
$361$	−17.5661	−0.924529
$362$	−61.1371	−3.21329
$363$	0	0
$364$	43.2986	2.26946
$365$	−0.350345	−0.0183379
$366$	0	0
$367$	25.1060	1.31052	0.655261	−	0.755402i	$-0.272559\pi$
$367$	25.1060	1.31052	0.655261	+	0.755402i	$0.272559\pi$
$368$	−73.6007	−3.83670
$369$	0	0
$370$	−14.9078	−0.775019
$371$	0.132800	0.00689465
$372$	0	0
$373$	35.7392	1.85051	0.925253	−	0.379352i	$-0.123853\pi$
$373$	35.7392	1.85051	0.925253	+	0.379352i	$0.123853\pi$
$374$	−17.9769	−0.929565
$375$	0	0
$376$	41.4620	2.13824
$377$	−13.2010	−0.679888
$378$	0	0
$379$	−11.3435	−0.582678	−0.291339	−	0.956620i	$-0.594101\pi$
$379$	−11.3435	−0.582678	−0.291339	+	0.956620i	$0.594101\pi$
$380$	3.35212	0.171960
$381$	0	0
$382$	−8.63592	−0.441852
$383$	8.17101	0.417519	0.208760	−	0.977967i	$-0.433057\pi$
$383$	8.17101	0.417519	0.208760	+	0.977967i	$0.433057\pi$
$384$	0	0
$385$	−2.11173	−0.107624
$386$	16.3720	0.833315
$387$	0	0
$388$	−45.8828	−2.32935
$389$	21.4931	1.08974	0.544872	−	0.838519i	$-0.316578\pi$
$389$	21.4931	1.08974	0.544872	+	0.838519i	$0.316578\pi$
$390$	0	0
$391$	−44.5365	−2.25231
$392$	−59.1508	−2.98757
$393$	0	0
$394$	−38.4305	−1.93610
$395$	−0.646712	−0.0325396
$396$	0	0
$397$	−24.0226	−1.20566	−0.602831	−	0.797869i	$-0.705961\pi$
$397$	−24.0226	−1.20566	−0.602831	+	0.797869i	$0.705961\pi$
$398$	−44.4895	−2.23006
$399$	0	0
$400$	−52.5632	−2.62816
$401$	30.4497	1.52058	0.760292	−	0.649581i	$-0.225056\pi$
$401$	30.4497	1.52058	0.760292	+	0.649581i	$0.225056\pi$
$402$	0	0
$403$	14.9679	0.745603
$404$	−57.6196	−2.86668
$405$	0	0
$406$	58.3853	2.89762
$407$	−10.0977	−0.500526
$408$	0	0
$409$	20.0217	0.990011	0.495005	−	0.868890i	$-0.335166\pi$
$409$	20.0217	0.990011	0.495005	+	0.868890i	$0.335166\pi$
$410$	0.219072	0.0108192
$411$	0	0
$412$	90.7758	4.47220
$413$	23.8294	1.17257
$414$	0	0
$415$	−3.47447	−0.170555
$416$	−31.0555	−1.52262
$417$	0	0
$418$	3.17408	0.155249
$419$	−5.41629	−0.264603	−0.132302	−	0.991210i	$-0.542237\pi$
$419$	−5.41629	−0.264603	−0.132302	+	0.991210i	$0.542237\pi$
$420$	0	0
$421$	0.566278	0.0275987	0.0137993	−	0.999905i	$-0.495607\pi$
$421$	0.566278	0.0275987	0.0137993	+	0.999905i	$0.495607\pi$
$422$	−57.9331	−2.82014
$423$	0	0
$424$	−0.280936	−0.0136435
$425$	−31.8065	−1.54284
$426$	0	0
$427$	33.1728	1.60534
$428$	26.9229	1.30137
$429$	0	0
$430$	4.10623	0.198020
$431$	−27.6803	−1.33331	−0.666657	−	0.745365i	$-0.732275\pi$
$431$	−27.6803	−1.33331	−0.666657	+	0.745365i	$0.732275\pi$
$432$	0	0
$433$	3.14072	0.150933	0.0754666	−	0.997148i	$-0.475955\pi$
$433$	3.14072	0.150933	0.0754666	+	0.997148i	$0.475955\pi$
$434$	−66.1997	−3.17769
$435$	0	0
$436$	−56.2745	−2.69506
$437$	7.86355	0.376165
$438$	0	0
$439$	14.4133	0.687910	0.343955	−	0.938986i	$-0.388233\pi$
$439$	14.4133	0.687910	0.343955	+	0.938986i	$0.388233\pi$
$440$	4.46732	0.212971
$441$	0	0
$442$	−40.8481	−1.94295
$443$	−3.57460	−0.169834	−0.0849171	−	0.996388i	$-0.527063\pi$
$443$	−3.57460	−0.169834	−0.0849171	+	0.996388i	$0.527063\pi$
$444$	0	0
$445$	−2.65856	−0.126028
$446$	3.81831	0.180802
$447$	0	0
$448$	52.3632	2.47393
$449$	4.26373	0.201218	0.100609	−	0.994926i	$-0.467921\pi$
$449$	4.26373	0.201218	0.100609	+	0.994926i	$0.467921\pi$
$450$	0	0
$451$	0.148388	0.00698730
$452$	105.327	4.95419
$453$	0	0
$454$	−18.6625	−0.875874
$455$	−4.79839	−0.224952
$456$	0	0
$457$	−31.7090	−1.48328	−0.741642	−	0.670796i	$-0.765953\pi$
$457$	−31.7090	−1.48328	−0.741642	+	0.670796i	$0.765953\pi$
$458$	−59.1921	−2.76586
$459$	0	0
$460$	18.3825	0.857090
$461$	−17.3145	−0.806417	−0.403208	−	0.915108i	$-0.632105\pi$
$461$	−17.3145	−0.806417	−0.403208	+	0.915108i	$0.632105\pi$
$462$	0	0
$463$	−7.86921	−0.365713	−0.182857	−	0.983140i	$-0.558534\pi$
$463$	−7.86921	−0.365713	−0.182857	+	0.983140i	$0.558534\pi$
$464$	−65.1150	−3.02289
$465$	0	0
$466$	51.6657	2.39337
$467$	−23.7683	−1.09987	−0.549934	−	0.835208i	$-0.685347\pi$
$467$	−23.7683	−1.09987	−0.549934	+	0.835208i	$0.685347\pi$
$468$	0	0
$469$	28.7494	1.32753
$470$	−7.63186	−0.352031
$471$	0	0
$472$	−50.4106	−2.32034
$473$	2.78134	0.127886
$474$	0	0
$475$	5.61589	0.257675
$476$	129.235	5.92348
$477$	0	0
$478$	−56.2695	−2.57371
$479$	−28.6414	−1.30866	−0.654330	−	0.756209i	$-0.727049\pi$
$479$	−28.6414	−1.30866	−0.654330	+	0.756209i	$0.727049\pi$
$480$	0	0
$481$	−22.9446	−1.04618
$482$	22.4164	1.02104
$483$	0	0
$484$	5.02592	0.228451
$485$	5.08477	0.230888
$486$	0	0
$487$	−5.46658	−0.247715	−0.123857	−	0.992300i	$-0.539527\pi$
$487$	−5.46658	−0.247715	−0.123857	+	0.992300i	$0.539527\pi$
$488$	−70.1763	−3.17673
$489$	0	0
$490$	10.8878	0.491861
$491$	6.73166	0.303796	0.151898	−	0.988396i	$-0.451462\pi$
$491$	6.73166	0.303796	0.151898	+	0.988396i	$0.451462\pi$
$492$	0	0
$493$	−39.4017	−1.77456
$494$	7.21232	0.324498
$495$	0	0
$496$	73.8301	3.31507
$497$	−28.5389	−1.28014
$498$	0	0
$499$	3.93986	0.176372	0.0881861	−	0.996104i	$-0.471893\pi$
$499$	3.93986	0.176372	0.0881861	+	0.996104i	$0.471893\pi$
$500$	27.1248	1.21306
$501$	0	0
$502$	65.5171	2.92417
$503$	9.62240	0.429041	0.214521	−	0.976719i	$-0.431181\pi$
$503$	9.62240	0.429041	0.214521	+	0.976719i	$0.431181\pi$
$504$	0	0
$505$	6.38546	0.284149
$506$	17.4062	0.773800
$507$	0	0
$508$	−38.8905	−1.72549
$509$	12.5594	0.556687	0.278344	−	0.960482i	$-0.410215\pi$
$509$	12.5594	0.556687	0.278344	+	0.960482i	$0.410215\pi$
$510$	0	0
$511$	2.38484	0.105499
$512$	26.6082	1.17593
$513$	0	0
$514$	−8.87776	−0.391581
$515$	−10.0599	−0.443290
$516$	0	0
$517$	−5.16941	−0.227351
$518$	101.479	4.45874
$519$	0	0
$520$	10.1509	0.445146
$521$	25.1427	1.10152	0.550760	−	0.834664i	$-0.314338\pi$
$521$	25.1427	1.10152	0.550760	+	0.834664i	$0.314338\pi$
$522$	0	0
$523$	6.58890	0.288113	0.144056	−	0.989569i	$-0.453985\pi$
$523$	6.58890	0.288113	0.144056	+	0.989569i	$0.453985\pi$
$524$	−2.76836	−0.120936
$525$	0	0
$526$	−2.32041	−0.101174
$527$	44.6753	1.94608
$528$	0	0
$529$	20.1225	0.874893
$530$	0.0517115	0.00224620
$531$	0	0
$532$	−22.8183	−0.989299
$533$	0.337175	0.0146046
$534$	0	0
$535$	−2.98363	−0.128993
$536$	−60.8188	−2.62697
$537$	0	0
$538$	−11.7421	−0.506237
$539$	7.37482	0.317656
$540$	0	0
$541$	19.0193	0.817705	0.408853	−	0.912600i	$-0.365929\pi$
$541$	19.0193	0.817705	0.408853	+	0.912600i	$0.365929\pi$
$542$	30.4814	1.30929
$543$	0	0
$544$	−92.6927	−3.97416
$545$	6.23639	0.267138
$546$	0	0
$547$	17.4953	0.748046	0.374023	−	0.927419i	$-0.377978\pi$
$547$	17.4953	0.748046	0.374023	+	0.927419i	$0.377978\pi$
$548$	104.533	4.46544
$549$	0	0
$550$	12.4309	0.530057
$551$	6.95693	0.296375
$552$	0	0
$553$	4.40225	0.187203
$554$	22.7391	0.966093
$555$	0	0
$556$	−13.2715	−0.562836
$557$	−1.99780	−0.0846495	−0.0423248	−	0.999104i	$-0.513476\pi$
$557$	−1.99780	−0.0846495	−0.0423248	+	0.999104i	$0.513476\pi$
$558$	0	0
$559$	6.31991	0.267304
$560$	−23.6684	−1.00017
$561$	0	0
$562$	30.1045	1.26988
$563$	−19.5439	−0.823678	−0.411839	−	0.911257i	$-0.635113\pi$
$563$	−19.5439	−0.823678	−0.411839	+	0.911257i	$0.635113\pi$
$564$	0	0
$565$	−11.6725	−0.491065
$566$	39.9648	1.67985
$567$	0	0
$568$	60.3733	2.53321
$569$	−17.7537	−0.744272	−0.372136	−	0.928178i	$-0.621375\pi$
$569$	−17.7537	−0.744272	−0.372136	+	0.928178i	$0.621375\pi$
$570$	0	0
$571$	−40.6686	−1.70193	−0.850965	−	0.525223i	$-0.823982\pi$
$571$	−40.6686	−1.70193	−0.850965	+	0.525223i	$0.823982\pi$
$572$	11.4202	0.477501
$573$	0	0
$574$	−1.49125	−0.0622436
$575$	30.7967	1.28431
$576$	0	0
$577$	−4.03476	−0.167969	−0.0839846	−	0.996467i	$-0.526765\pi$
$577$	−4.03476	−0.167969	−0.0839846	+	0.996467i	$0.526765\pi$
$578$	−76.8602	−3.19696
$579$	0	0
$580$	16.2631	0.675290
$581$	23.6511	0.981214
$582$	0	0
$583$	0.0350266	0.00145065
$584$	−5.04507	−0.208767
$585$	0	0
$586$	−30.8280	−1.27349
$587$	−18.0832	−0.746374	−0.373187	−	0.927756i	$-0.621735\pi$
$587$	−18.0832	−0.746374	−0.373187	+	0.927756i	$0.621735\pi$
$588$	0	0
$589$	−7.88806	−0.325022
$590$	9.27902	0.382011
$591$	0	0
$592$	−113.176	−4.65150
$593$	−5.33660	−0.219148	−0.109574	−	0.993979i	$-0.534949\pi$
$593$	−5.33660	−0.219148	−0.109574	+	0.993979i	$0.534949\pi$
$594$	0	0
$595$	−14.3220	−0.587143
$596$	80.4755	3.29640
$597$	0	0
$598$	39.5513	1.61737
$599$	23.1182	0.944582	0.472291	−	0.881443i	$-0.343427\pi$
$599$	23.1182	0.944582	0.472291	+	0.881443i	$0.343427\pi$
$600$	0	0
$601$	−28.5927	−1.16632	−0.583160	−	0.812357i	$-0.698184\pi$
$601$	−28.5927	−1.16632	−0.583160	+	0.812357i	$0.698184\pi$
$602$	−27.9516	−1.13922
$603$	0	0
$604$	45.7256	1.86055
$605$	−0.556977	−0.0226443
$606$	0	0
$607$	36.3998	1.47742	0.738710	−	0.674023i	$-0.235435\pi$
$607$	36.3998	1.47742	0.738710	+	0.674023i	$0.235435\pi$
$608$	16.3662	0.663738
$609$	0	0
$610$	12.9173	0.523004
$611$	−11.7462	−0.475201
$612$	0	0
$613$	−9.93192	−0.401146	−0.200573	−	0.979679i	$-0.564280\pi$
$613$	−9.93192	−0.401146	−0.200573	+	0.979679i	$0.564280\pi$
$614$	6.52431	0.263300
$615$	0	0
$616$	−30.4096	−1.22524
$617$	−4.86148	−0.195716	−0.0978578	−	0.995200i	$-0.531199\pi$
$617$	−4.86148	−0.195716	−0.0978578	+	0.995200i	$0.531199\pi$
$618$	0	0
$619$	−10.9002	−0.438115	−0.219058	−	0.975712i	$-0.570298\pi$
$619$	−10.9002	−0.438115	−0.219058	+	0.975712i	$0.570298\pi$
$620$	−18.4398	−0.740561
$621$	0	0
$622$	−52.2591	−2.09540
$623$	18.0971	0.725046
$624$	0	0
$625$	20.4429	0.817715
$626$	84.2515	3.36737
$627$	0	0
$628$	−37.4247	−1.49341
$629$	−68.4837	−2.73063
$630$	0	0
$631$	25.1650	1.00180	0.500901	−	0.865504i	$-0.333002\pi$
$631$	25.1650	1.00180	0.500901	+	0.865504i	$0.333002\pi$
$632$	−9.31285	−0.370445
$633$	0	0
$634$	−59.7137	−2.37153
$635$	4.30988	0.171032
$636$	0	0
$637$	16.7575	0.663955
$638$	15.3994	0.609666
$639$	0	0
$640$	5.16512	0.204169
$641$	−30.3119	−1.19725	−0.598623	−	0.801031i	$-0.704285\pi$
$641$	−30.3119	−1.19725	−0.598623	+	0.801031i	$0.704285\pi$
$642$	0	0
$643$	−16.5682	−0.653384	−0.326692	−	0.945131i	$-0.605934\pi$
$643$	−16.5682	−0.653384	−0.326692	+	0.945131i	$0.605934\pi$
$644$	−125.132	−4.93090
$645$	0	0
$646$	21.5269	0.846965
$647$	−3.48914	−0.137172	−0.0685861	−	0.997645i	$-0.521849\pi$
$647$	−3.48914	−0.137172	−0.0685861	+	0.997645i	$0.521849\pi$
$648$	0	0
$649$	6.28511	0.246712
$650$	28.2462	1.10791
$651$	0	0
$652$	17.8775	0.700135
$653$	−22.7463	−0.890130	−0.445065	−	0.895498i	$-0.646819\pi$
$653$	−22.7463	−0.890130	−0.445065	+	0.895498i	$0.646819\pi$
$654$	0	0
$655$	0.306792	0.0119873
$656$	1.66314	0.0649346
$657$	0	0
$658$	51.9510	2.02526
$659$	2.07746	0.0809264	0.0404632	−	0.999181i	$-0.487117\pi$
$659$	2.07746	0.0809264	0.0404632	+	0.999181i	$0.487117\pi$
$660$	0	0
$661$	−5.14194	−0.199998	−0.0999992	−	0.994988i	$-0.531884\pi$
$661$	−5.14194	−0.199998	−0.0999992	+	0.994988i	$0.531884\pi$
$662$	−68.0679	−2.64553
$663$	0	0
$664$	−50.0334	−1.94167
$665$	2.52875	0.0980606
$666$	0	0
$667$	38.1507	1.47720
$668$	−8.11670	−0.314045
$669$	0	0
$670$	11.1948	0.432494
$671$	8.74946	0.337769
$672$	0	0
$673$	−7.61005	−0.293346	−0.146673	−	0.989185i	$-0.546856\pi$
$673$	−7.61005	−0.293346	−0.146673	+	0.989185i	$0.546856\pi$
$674$	57.7616	2.22489
$675$	0	0
$676$	−39.3875	−1.51490
$677$	16.2837	0.625833	0.312917	−	0.949781i	$-0.398694\pi$
$677$	16.2837	0.625833	0.312917	+	0.949781i	$0.398694\pi$
$678$	0	0
$679$	−34.6127	−1.32831
$680$	30.2978	1.16187
$681$	0	0
$682$	−17.4604	−0.668595
$683$	−2.78266	−0.106476	−0.0532378	−	0.998582i	$-0.516954\pi$
$683$	−2.78266	−0.106476	−0.0532378	+	0.998582i	$0.516954\pi$
$684$	0	0
$685$	−11.5845	−0.442620
$686$	−3.76686	−0.143819
$687$	0	0
$688$	31.1734	1.18847
$689$	0.0795893	0.00303211
$690$	0	0
$691$	47.8794	1.82142	0.910709	−	0.413049i	$-0.135536\pi$
$691$	47.8794	1.82142	0.910709	+	0.413049i	$0.135536\pi$
$692$	69.6799	2.64883
$693$	0	0
$694$	−0.272452	−0.0103421
$695$	1.47076	0.0557891
$696$	0	0
$697$	1.00638	0.0381193
$698$	−31.4956	−1.19212
$699$	0	0
$700$	−89.3653	−3.37769
$701$	19.9183	0.752304	0.376152	−	0.926558i	$-0.377247\pi$
$701$	19.9183	0.752304	0.376152	+	0.926558i	$0.377247\pi$
$702$	0	0
$703$	12.0918	0.456050
$704$	13.8110	0.520521
$705$	0	0
$706$	63.8096	2.40151
$707$	−43.4666	−1.63473
$708$	0	0
$709$	15.2356	0.572185	0.286092	−	0.958202i	$-0.407644\pi$
$709$	15.2356	0.572185	0.286092	+	0.958202i	$0.407644\pi$
$710$	−11.1128	−0.417057
$711$	0	0
$712$	−38.2840	−1.43475
$713$	−43.2569	−1.61998
$714$	0	0
$715$	−1.26559	−0.0473305
$716$	46.4727	1.73677
$717$	0	0
$718$	18.1941	0.678998
$719$	32.8742	1.22600	0.613001	−	0.790082i	$-0.289962\pi$
$719$	32.8742	1.22600	0.613001	+	0.790082i	$0.289962\pi$
$720$	0	0
$721$	68.4787	2.55028
$722$	46.5614	1.73284
$723$	0	0
$724$	115.923	4.30824
$725$	27.2460	1.01189
$726$	0	0
$727$	−10.6496	−0.394971	−0.197486	−	0.980306i	$-0.563278\pi$
$727$	−10.6496	−0.394971	−0.197486	+	0.980306i	$0.563278\pi$
$728$	−69.0983	−2.56095
$729$	0	0
$730$	0.928639	0.0343705
$731$	18.8633	0.697684
$732$	0	0
$733$	−34.3944	−1.27039	−0.635193	−	0.772354i	$-0.719079\pi$
$733$	−34.3944	−1.27039	−0.635193	+	0.772354i	$0.719079\pi$
$734$	−66.5471	−2.45630
$735$	0	0
$736$	89.7499	3.30822
$737$	7.58278	0.279315
$738$	0	0
$739$	50.2109	1.84704	0.923519	−	0.383552i	$-0.125299\pi$
$739$	50.2109	1.84704	0.923519	+	0.383552i	$0.125299\pi$
$740$	28.2668	1.03911
$741$	0	0
$742$	−0.352007	−0.0129226
$743$	−13.1792	−0.483499	−0.241750	−	0.970339i	$-0.577721\pi$
$743$	−13.1792	−0.483499	−0.241750	+	0.970339i	$0.577721\pi$
$744$	0	0
$745$	−8.91837	−0.326744
$746$	−94.7319	−3.46838
$747$	0	0
$748$	34.0863	1.24632
$749$	20.3099	0.742108
$750$	0	0
$751$	−33.6825	−1.22909	−0.614546	−	0.788881i	$-0.710661\pi$
$751$	−33.6825	−1.22909	−0.614546	+	0.788881i	$0.710661\pi$
$752$	−57.9390	−2.11282
$753$	0	0
$754$	34.9912	1.27431
$755$	−5.06736	−0.184420
$756$	0	0
$757$	7.26781	0.264153	0.132076	−	0.991240i	$-0.457836\pi$
$757$	7.26781	0.264153	0.132076	+	0.991240i	$0.457836\pi$
$758$	30.0677	1.09211
$759$	0	0
$760$	−5.34950	−0.194047
$761$	10.4590	0.379137	0.189569	−	0.981867i	$-0.439291\pi$
$761$	10.4590	0.379137	0.189569	+	0.981867i	$0.439291\pi$
$762$	0	0
$763$	−42.4519	−1.53686
$764$	16.3747	0.592415
$765$	0	0
$766$	−21.6585	−0.782552
$767$	14.2814	0.515670
$768$	0	0
$769$	25.1597	0.907281	0.453640	−	0.891185i	$-0.350125\pi$
$769$	25.1597	0.907281	0.453640	+	0.891185i	$0.350125\pi$
$770$	5.59745	0.201718
$771$	0	0
$772$	−31.0432	−1.11727
$773$	41.0503	1.47648	0.738239	−	0.674539i	$-0.235658\pi$
$773$	41.0503	1.47648	0.738239	+	0.674539i	$0.235658\pi$
$774$	0	0
$775$	−30.8927	−1.10970
$776$	73.2223	2.62853
$777$	0	0
$778$	−56.9706	−2.04250
$779$	−0.177691	−0.00636643
$780$	0	0
$781$	−7.52724	−0.269346
$782$	118.050	4.22147
$783$	0	0
$784$	82.6574	2.95205
$785$	4.14744	0.148028
$786$	0	0
$787$	−9.13416	−0.325598	−0.162799	−	0.986659i	$-0.552052\pi$
$787$	−9.13416	−0.325598	−0.162799	+	0.986659i	$0.552052\pi$
$788$	72.8685	2.59583
$789$	0	0
$790$	1.71420	0.0609886
$791$	79.4561	2.82513
$792$	0	0
$793$	19.8810	0.705994
$794$	63.6755	2.25976
$795$	0	0
$796$	84.3571	2.98996
$797$	22.3554	0.791870	0.395935	−	0.918279i	$-0.370421\pi$
$797$	22.3554	0.791870	0.395935	+	0.918279i	$0.370421\pi$
$798$	0	0
$799$	−35.0594	−1.24031
$800$	64.0964	2.26615
$801$	0	0
$802$	−80.7113	−2.85001
$803$	0.629010	0.0221973
$804$	0	0
$805$	13.8673	0.488757
$806$	−39.6745	−1.39748
$807$	0	0
$808$	91.9526	3.23488
$809$	22.3117	0.784437	0.392219	−	0.919872i	$-0.371708\pi$
$809$	22.3117	0.784437	0.392219	+	0.919872i	$0.371708\pi$
$810$	0	0
$811$	32.4795	1.14051	0.570254	−	0.821468i	$-0.306845\pi$
$811$	32.4795	1.14051	0.570254	+	0.821468i	$0.306845\pi$
$812$	−110.705	−3.88499
$813$	0	0
$814$	26.7655	0.938131
$815$	−1.98120	−0.0693983
$816$	0	0
$817$	−3.33058	−0.116522
$818$	−53.0705	−1.85557
$819$	0	0
$820$	−0.415385	−0.0145059
$821$	−26.1876	−0.913954	−0.456977	−	0.889479i	$-0.651068\pi$
$821$	−26.1876	−0.913954	−0.456977	+	0.889479i	$0.651068\pi$
$822$	0	0
$823$	0.681134	0.0237428	0.0118714	−	0.999930i	$-0.496221\pi$
$823$	0.681134	0.0237428	0.0118714	+	0.999930i	$0.496221\pi$
$824$	−144.865	−5.04662
$825$	0	0
$826$	−63.1634	−2.19774
$827$	26.7695	0.930867	0.465434	−	0.885083i	$-0.345898\pi$
$827$	26.7695	0.930867	0.465434	+	0.885083i	$0.345898\pi$
$828$	0	0
$829$	−2.66365	−0.0925125	−0.0462562	−	0.998930i	$-0.514729\pi$
$829$	−2.66365	−0.0925125	−0.0462562	+	0.998930i	$0.514729\pi$
$830$	9.20958	0.319669
$831$	0	0
$832$	31.3821	1.08798
$833$	50.0167	1.73298
$834$	0	0
$835$	0.899500	0.0311285
$836$	−6.01842	−0.208151
$837$	0	0
$838$	14.3567	0.495943
$839$	12.7892	0.441532	0.220766	−	0.975327i	$-0.429144\pi$
$839$	12.7892	0.441532	0.220766	+	0.975327i	$0.429144\pi$
$840$	0	0
$841$	4.75218	0.163868
$842$	−1.50100	−0.0517279
$843$	0	0
$844$	109.848	3.78111
$845$	4.36495	0.150159
$846$	0	0
$847$	3.79141	0.130275
$848$	0.392580	0.0134812
$849$	0	0
$850$	84.3077	2.89173
$851$	66.3095	2.27306
$852$	0	0
$853$	34.3220	1.17516	0.587581	−	0.809165i	$-0.300080\pi$
$853$	34.3220	1.17516	0.587581	+	0.809165i	$0.300080\pi$
$854$	−87.9294	−3.00888
$855$	0	0
$856$	−42.9651	−1.46852
$857$	51.0286	1.74310	0.871551	−	0.490304i	$-0.163114\pi$
$857$	51.0286	1.74310	0.871551	+	0.490304i	$0.163114\pi$
$858$	0	0
$859$	−29.1275	−0.993818	−0.496909	−	0.867803i	$-0.665532\pi$
$859$	−29.1275	−0.993818	−0.496909	+	0.867803i	$0.665532\pi$
$860$	−7.78587	−0.265496
$861$	0	0
$862$	73.3707	2.49902
$863$	8.31789	0.283144	0.141572	−	0.989928i	$-0.454784\pi$
$863$	8.31789	0.283144	0.141572	+	0.989928i	$0.454784\pi$
$864$	0	0
$865$	−7.72199	−0.262555
$866$	−8.32493	−0.282893
$867$	0	0
$868$	125.522	4.26050
$869$	1.16111	0.0393879
$870$	0	0
$871$	17.2300	0.583816
$872$	89.8060	3.04121
$873$	0	0
$874$	−20.8435	−0.705042
$875$	20.4622	0.691749
$876$	0	0
$877$	12.0137	0.405675	0.202837	−	0.979212i	$-0.434984\pi$
$877$	12.0137	0.405675	0.202837	+	0.979212i	$0.434984\pi$
$878$	−38.2046	−1.28934
$879$	0	0
$880$	−6.24263	−0.210439
$881$	6.56937	0.221328	0.110664	−	0.993858i	$-0.464702\pi$
$881$	6.56937	0.221328	0.110664	+	0.993858i	$0.464702\pi$
$882$	0	0
$883$	−15.3838	−0.517707	−0.258853	−	0.965917i	$-0.583345\pi$
$883$	−15.3838	−0.517707	−0.258853	+	0.965917i	$0.583345\pi$
$884$	77.4526	2.60501
$885$	0	0
$886$	9.47499	0.318319
$887$	−20.7023	−0.695117	−0.347558	−	0.937658i	$-0.612989\pi$
$887$	−20.7023	−0.695117	−0.347558	+	0.937658i	$0.612989\pi$
$888$	0	0
$889$	−29.3379	−0.983962
$890$	7.04689	0.236212
$891$	0	0
$892$	−7.23995	−0.242411
$893$	6.19025	0.207149
$894$	0	0
$895$	−5.15015	−0.172151
$896$	−35.1596	−1.17460
$897$	0	0
$898$	−11.3016	−0.377141
$899$	−38.2696	−1.27636
$900$	0	0
$901$	0.237554	0.00791406
$902$	−0.393323	−0.0130962
$903$	0	0
$904$	−168.087	−5.59050
$905$	−12.8467	−0.427038
$906$	0	0
$907$	36.0556	1.19721	0.598603	−	0.801046i	$-0.295723\pi$
$907$	36.0556	1.19721	0.598603	+	0.801046i	$0.295723\pi$
$908$	35.3862	1.17433
$909$	0	0
$910$	12.7188	0.421625
$911$	−22.2540	−0.737308	−0.368654	−	0.929567i	$-0.620181\pi$
$911$	−22.2540	−0.737308	−0.368654	+	0.929567i	$0.620181\pi$
$912$	0	0
$913$	6.23807	0.206450
$914$	84.0493	2.78010
$915$	0	0
$916$	112.235	3.70834
$917$	−2.08837	−0.0689641
$918$	0	0
$919$	13.7446	0.453393	0.226696	−	0.973965i	$-0.427207\pi$
$919$	13.7446	0.453393	0.226696	+	0.973965i	$0.427207\pi$
$920$	−29.3359	−0.967175
$921$	0	0
$922$	45.8946	1.51146
$923$	−17.1038	−0.562979
$924$	0	0
$925$	47.3561	1.55706
$926$	20.8585	0.685453
$927$	0	0
$928$	79.4022	2.60650
$929$	−6.87723	−0.225635	−0.112817	−	0.993616i	$-0.535987\pi$
$929$	−6.87723	−0.225635	−0.112817	+	0.993616i	$0.535987\pi$
$930$	0	0
$931$	−8.83117	−0.289430
$932$	−97.9639	−3.20892
$933$	0	0
$934$	63.0015	2.06147
$935$	−3.77747	−0.123537
$936$	0	0
$937$	20.5781	0.672258	0.336129	−	0.941816i	$-0.390882\pi$
$937$	20.5781	0.672258	0.336129	+	0.941816i	$0.390882\pi$
$938$	−76.2046	−2.48817
$939$	0	0
$940$	14.4709	0.471988
$941$	19.2556	0.627713	0.313857	−	0.949470i	$-0.398379\pi$
$941$	19.2556	0.627713	0.313857	+	0.949470i	$0.398379\pi$
$942$	0	0
$943$	−0.974429	−0.0317318
$944$	70.4438	2.29275
$945$	0	0
$946$	−7.37234	−0.239696
$947$	28.1258	0.913964	0.456982	−	0.889476i	$-0.348930\pi$
$947$	28.1258	0.913964	0.456982	+	0.889476i	$0.348930\pi$
$948$	0	0
$949$	1.42927	0.0463961
$950$	−14.8857	−0.482957
$951$	0	0
$952$	−206.241	−6.68430
$953$	−22.4325	−0.726660	−0.363330	−	0.931661i	$-0.618360\pi$
$953$	−22.4325	−0.726660	−0.363330	+	0.931661i	$0.618360\pi$
$954$	0	0
$955$	−1.81466	−0.0587209
$956$	106.693	3.45071
$957$	0	0
$958$	75.9183	2.45281
$959$	78.8570	2.54643
$960$	0	0
$961$	12.3917	0.399733
$962$	60.8180	1.96085
$963$	0	0
$964$	−42.5040	−1.36896
$965$	3.44024	0.110745
$966$	0	0
$967$	−34.6182	−1.11325	−0.556623	−	0.830765i	$-0.687903\pi$
$967$	−34.6182	−1.11325	−0.556623	+	0.830765i	$0.687903\pi$
$968$	−8.02065	−0.257793
$969$	0	0
$970$	−13.4779	−0.432750
$971$	6.62970	0.212757	0.106379	−	0.994326i	$-0.466074\pi$
$971$	6.62970	0.212757	0.106379	+	0.994326i	$0.466074\pi$
$972$	0	0
$973$	−10.0116	−0.320958
$974$	14.4900	0.464289
$975$	0	0
$976$	98.0643	3.13896
$977$	37.0721	1.18604	0.593021	−	0.805187i	$-0.297935\pi$
$977$	37.0721	1.18604	0.593021	+	0.805187i	$0.297935\pi$
$978$	0	0
$979$	4.77319	0.152552
$980$	−20.6445	−0.659465
$981$	0	0
$982$	−17.8432	−0.569401
$983$	−13.2073	−0.421249	−0.210624	−	0.977567i	$-0.567550\pi$
$983$	−13.2073	−0.421249	−0.210624	+	0.977567i	$0.567550\pi$
$984$	0	0
$985$	−8.07536	−0.257302
$986$	104.440	3.32604
$987$	0	0
$988$	−13.6754	−0.435072
$989$	−18.2644	−0.580775
$990$	0	0
$991$	−12.8106	−0.406943	−0.203472	−	0.979081i	$-0.565222\pi$
$991$	−12.8106	−0.406943	−0.203472	+	0.979081i	$0.565222\pi$
$992$	−90.0295	−2.85844
$993$	0	0
$994$	75.6465	2.39936
$995$	−9.34853	−0.296368
$996$	0	0
$997$	−13.2443	−0.419450	−0.209725	−	0.977760i	$-0.567257\pi$
$997$	−13.2443	−0.419450	−0.209725	+	0.977760i	$0.567257\pi$
$998$	−10.4432	−0.330573
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.l.1.4		51
3.2	odd	2		8019.2.a.k.1.48		51
27.5	odd	18		891.2.j.c.397.17		102
27.11	odd	18		891.2.j.c.496.17		102
27.16	even	9		297.2.j.c.67.1	✓	102
27.22	even	9		297.2.j.c.133.1	yes	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.67.1	✓	102	27.16	even	9
297.2.j.c.133.1	yes	102	27.22	even	9
891.2.j.c.397.17		102	27.5	odd	18
891.2.j.c.496.17		102	27.11	odd	18
8019.2.a.k.1.48		51	3.2	odd	2
8019.2.a.l.1.4		51	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:	\( N \)	\(=\)	\( 8019 = 3^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8019.a (trivial)

\(f(q)\)	\(=\)	\(q-2.65065 q^{2} +5.02592 q^{4} -0.556977 q^{5} +3.79141 q^{7} -8.02065 q^{8} +O(q^{10})\)	\(q-2.65065 q^{2} +5.02592 q^{4} -0.556977 q^{5} +3.79141 q^{7} -8.02065 q^{8} +1.47635 q^{10} +1.00000 q^{11} +2.27225 q^{13} -10.0497 q^{14} +11.2080 q^{16} +6.78209 q^{17} -1.19748 q^{19} -2.79932 q^{20} -2.65065 q^{22} -6.56677 q^{23} -4.68978 q^{25} -6.02294 q^{26} +19.0554 q^{28} -5.80966 q^{29} +6.58724 q^{31} -13.6673 q^{32} -17.9769 q^{34} -2.11173 q^{35} -10.0977 q^{37} +3.17408 q^{38} +4.46732 q^{40} +0.148388 q^{41} +2.78134 q^{43} +5.02592 q^{44} +17.4062 q^{46} -5.16941 q^{47} +7.37482 q^{49} +12.4309 q^{50} +11.4202 q^{52} +0.0350266 q^{53} -0.556977 q^{55} -30.4096 q^{56} +15.3994 q^{58} +6.28511 q^{59} +8.74946 q^{61} -17.4604 q^{62} +13.8110 q^{64} -1.26559 q^{65} +7.58278 q^{67} +34.0863 q^{68} +5.59745 q^{70} -7.52724 q^{71} +0.629010 q^{73} +26.7655 q^{74} -6.01842 q^{76} +3.79141 q^{77} +1.16111 q^{79} -6.24263 q^{80} -0.393323 q^{82} +6.23807 q^{83} -3.77747 q^{85} -7.37234 q^{86} -8.02065 q^{88} +4.77319 q^{89} +8.61506 q^{91} -33.0041 q^{92} +13.7023 q^{94} +0.666967 q^{95} -9.12923 q^{97} -19.5480 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7}+O(q^{10}) \) 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7	\( 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7} + 18 q^{10} + 51 q^{11} + 30 q^{13} + 12 q^{14} + 78 q^{16} + 30 q^{19} + 18 q^{20} + 3 q^{23} + 75 q^{25} + 9 q^{26} + 36 q^{28} + 42 q^{31} - 15 q^{32} + 42 q^{34} - 9 q^{35} + 48 q^{37} - 3 q^{38} + 54 q^{40} + 18 q^{43} + 60 q^{44} + 42 q^{46} + 30 q^{47} + 99 q^{49} - 30 q^{50} + 60 q^{52} + 18 q^{53} + 6 q^{55} + 21 q^{56} + 30 q^{58} + 24 q^{59} + 99 q^{61} + 114 q^{64} - 15 q^{65} + 39 q^{67} - 39 q^{68} + 48 q^{70} + 30 q^{71} + 69 q^{73} + 90 q^{76} + 12 q^{77} + 48 q^{79} + 42 q^{80} + 42 q^{82} - 21 q^{83} + 84 q^{85} + 24 q^{86} + 15 q^{89} + 69 q^{91} - 66 q^{92} + 66 q^{94} - 12 q^{95} + 72 q^{97} - 57 q^{98}+O(q^{100}) \) 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7 + 18 * q^10 + 51 * q^11 + 30 * q^13 + 12 * q^14 + 78 * q^16 + 30 * q^19 + 18 * q^20 + 3 * q^23 + 75 * q^25 + 9 * q^26 + 36 * q^28 + 42 * q^31 - 15 * q^32 + 42 * q^34 - 9 * q^35 + 48 * q^37 - 3 * q^38 + 54 * q^40 + 18 * q^43 + 60 * q^44 + 42 * q^46 + 30 * q^47 + 99 * q^49 - 30 * q^50 + 60 * q^52 + 18 * q^53 + 6 * q^55 + 21 * q^56 + 30 * q^58 + 24 * q^59 + 99 * q^61 + 114 * q^64 - 15 * q^65 + 39 * q^67 - 39 * q^68 + 48 * q^70 + 30 * q^71 + 69 * q^73 + 90 * q^76 + 12 * q^77 + 48 * q^79 + 42 * q^80 + 42 * q^82 - 21 * q^83 + 84 * q^85 + 24 * q^86 + 15 * q^89 + 69 * q^91 - 66 * q^92 + 66 * q^94 - 12 * q^95 + 72 * q^97 - 57 * q^98

Introduction

L-functions

Modular forms

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Representations

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Database

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